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This is PART 14: Centers X(26001) - X(28000)

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

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Collineation mappings involving Gemini triangle 38: X(26001)-X(26026)

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Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 38, as in centers X(26001)-X(26026). Then

m(X) = 2 b c (a - b + c) (a + b - c) x + (a - b - c) (a^2 + b^2 - c^2) y + (a - b - c) (a^2 - b^2 + c^2) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 29, 2018)

X(26001) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26001) lies on these lines: {1, 2}, {4, 24590}, {7, 20262}, {11, 26002}, {56, 25931}, {57, 23058}, {63, 6554}, {75, 25019}, {142, 26540}, {241, 1146}, {269, 5942}, {281, 1445}, {515, 11349}, {594, 25067}, {673, 1861}, {908, 26005}, {1449, 24553}, {2262, 21239}, {2270, 21279}, {2321, 26669}, {3218, 5199}, {3666, 21049}, {3739, 25964}, {4000, 24005}, {4025, 4391}, {4357, 20905}, {4359, 25002}, {4416, 26651}, {4431, 25243}, {4858, 22464}, {4967, 25001}, {5179, 20367}, {5249, 13567}, {5257, 24554}, {5435, 20205}, {5787, 11347}, {6245, 24604}, {7291, 8074}, {7384, 27000}, {8756, 16560}, {16608, 21617}, {17275, 25878}, {20888, 26592}, {21495, 25954}, {24789, 26958}, {25023, 26538}, {26004, 26007}, {26010, 26019}

X(26002) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26002) lies on these lines: {2, 3}, {11, 26001}, {77, 15849}, {241, 6506}, {1329, 25930}, {7681, 24590}

X(26003) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26003) lies on these lines: {2, 3}, {9, 273}, {34, 25930}, {53, 17337}, {63, 1847}, {92, 3305}, {142, 7282}, {144, 1119}, {239, 1897}, {264, 2322}, {275, 17758}, {278, 18228}, {281, 18230}, {317, 17234}, {318, 4384}, {333, 18736}, {340, 17297}, {342, 1445}, {673, 1861}, {908, 4564}, {1021, 1577}, {1235, 26592}, {1753, 24590}, {1785, 3008}, {1839, 25993}, {1841, 25067}, {3087, 4648}, {3912, 5081}, {5174, 25935}, {5222, 7952}, {5226, 17917}, {6748, 17245}, {6749, 17392}, {9308, 17349}, {17300, 27377}, {17352, 17907}

X(26003) = orthocentroidal-circle-inverse of X(37448)
X(26003) = {X(2),X(4)}-harmonic conjugate of X(37448)

X(26004) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26004) lies on these lines: {2, 3}, {14838, 26017}, {26001, 26007}

X(26005) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26005) lies on these lines: {2, 6}, {11, 26010}, {238, 25968}, {440, 21363}, {594, 26591}, {899, 25882}, {908, 26001}, {1736, 2968}, {1788, 20306}, {1834, 24983}, {2887, 25973}, {3452, 26942}, {3687, 25091}, {3911, 26932}, {4364, 26635}, {4415, 17862}, {5219, 16608}, {5723, 17923}, {6247, 6848}, {6847, 15873}, {6949, 26879}, {6959, 12359}, {14557, 21621}, {17810, 26118}, {20905, 26580}, {25019, 25939}, {26014, 26016}

X(26006) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26006) lies on these lines: {1, 2}, {6, 25019}, {9, 26668}, {40, 24580}, {48, 18589}, {63, 348}, {77, 27509}, {86, 2327}, {110, 2741}, {125, 20754}, {142, 2289}, {205, 21062}, {219, 307}, {223, 27540}, {226, 9310}, {241, 17044}, {278, 27413}, {347, 27382}, {379, 946}, {394, 4001}, {441, 525}, {515, 857}, {516, 14953}, {517, 1375}, {534, 2173}, {610, 4329}, {908, 4564}, {962, 24604}, {968, 26649}, {1100, 25964}, {1214, 22070}, {1813, 6518}, {1819, 16054}, {1944, 22464}, {2187, 24605}, {2328, 26647}, {2360, 24606}, {3007, 14543}, {3430, 26252}, {3576, 14021}, {3589, 25067}, {3663, 26651}, {3686, 25000}, {3879, 26540}, {3946, 20905}, {4466, 9028}, {4657, 25878}, {4855, 25932}, {5227, 25915}, {5250, 24609}, {5294, 23292}, {5717, 25017}, {5750, 25001}, {5930, 27410}, {6510, 26932}, {6684, 24581}, {8804, 17134}, {10436, 24553}, {13161, 26678}, {15988, 25023}, {17086, 27420}, {17353, 26669}, {17355, 25243}, {17859, 26165}, {18594, 20061}, {24179, 24779}, {24203, 24781}, {25082, 25087}

X(26006) = isogonal conjugate of polar conjugate of X(35517)
X(26006) = isotomic conjugate of polar conjugate of X(516)
X(26006) = complement of polar conjugate of X(917)
X(26006) = crossdifference of every pair of points on line X(25)X(649)
X(26006) = X(19)-isoconjugate of X(103)

X(26007) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 38

Barycentrics    2 a^4 - 2 a^3 b + a^2 b^2 - 2 a b^3 + b^4 - 2 a^3 c + 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(26007) lies on these lines: {2, 11}, {12, 17682}, {41, 21258}, {101, 4904}, {142, 24685}, {169, 3665}, {241, 514}, {479, 658}, {664, 4534}, {1086, 9318}, {1146, 9317}, {1194, 3752}, {1358, 3732}, {1438, 17060}, {1479, 17675}, {1565, 5540}, {2098, 26658}, {2170, 17044}, {2246, 5845}, {2348, 9436}, {3666, 25070}, {3689, 3912}, {4000, 26273}, {4209, 7354}, {4258, 26101}, {4422, 14439}, {5305, 24790}, {6284, 17671}, {6547, 8649}, {6710, 17761}, {7819, 25992}, {8256, 26653}, {10950, 26531}, {11349, 20989}, {17056, 21341}, {17683, 25466}, {17728, 24600}, {26001, 26004}

X(26008) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26008) lies on these lines: {2, 3}

X(26009) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26009) lies on these lines: {2, 3}

X(26010) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26010) lies on these lines: {2, 31}, {11, 26005}, {124, 3911}, {343, 3840}, {726, 26611}, {899, 23541}, {978, 17555}, {1193, 24983}, {3816, 13567}, {5087, 26011}, {5741, 25941}, {24984, 27627}, {26001, 26019}

X(26011) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26011) lies on these lines: {2, 37}, {11, 1861}, {92, 1427}, {226, 6708}, {518, 26013}, {525, 3239}, {908, 26001}, {1104, 11109}, {1150, 26651}, {1465, 4858}, {1738, 25882}, {3011, 25968}, {3706, 25941}, {3713, 25934}, {4054, 25019}, {5087, 26010}, {7270, 25983}, {9371, 26095}, {11679, 17811}, {12618, 14022}, {15852, 26027}, {17102, 20320}, {25000, 26580}

X(26012) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26012) lies on these lines: {1, 7380}, {2, 41}, {5, 226}, {6, 20305}, {11, 20358}, {37, 24317}, {44, 8287}, {57, 4911}, {69, 21244}, {116, 3008}, {150, 1429}, {325, 3912}, {524, 21237}, {604, 21270}, {672, 857}, {908, 26019}, {1211, 3831}, {1400, 5740}, {1737, 16609}, {1825, 1848}, {1837, 24268}, {2347, 25000}, {3589, 21236}, {3666, 24211}, {3782, 24172}, {4357, 25371}, {5249, 17048}, {5712, 10588}, {5750, 17052}, {7146, 17181}, {7291, 24712}, {8609, 21091}, {16888, 17861}, {17023, 17062}, {17303, 25363}, {21069, 25078}, {21232, 25007}, {24318, 25083}, {26013, 26020}, {26176, 26963}

X(26013) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26013) lies on these lines: {1, 2}, {11, 26005}, {38, 17862}, {46, 14058}, {243, 1861}, {291, 16082}, {343, 2887}, {515, 851}, {516, 14956}, {518, 26011}, {774, 23528}, {850, 4025}, {946, 1985}, {1468, 24537}, {1725, 23580}, {1736, 24026}, {1776, 24410}, {1818, 26031}, {2886, 13567}, {3580, 21241}, {3696, 25939}, {3706, 25091}, {3925, 25970}, {4191, 6796}, {5173, 6708}, {5247, 11109}, {5278, 25885}, {10601, 25496}, {11433, 26098}, {11499, 16059}, {17871, 24218}, {25024, 26587}, {26012, 26020}

X(26014) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26014) lies on these lines: {2, 37}, {239, 26025}, {6063, 20310}, {26005, 26016}

X(26015) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26015) lies on these lines: {1, 2}, {5, 3555}, {6, 17721}, {7, 24389}, {11, 518}, {36, 12750}, {38, 24210}, {56, 1004}, {57, 3434}, {63, 497}, {65, 3813}, {72, 496}, {84, 10431}, {100, 2078}, {142, 11025}, {149, 516}, {165, 20075}, {210, 3816}, {226, 3873}, {238, 1331}, {244, 1738}, {283, 1067}, {329, 5274}, {354, 2886}, {377, 3333}, {390, 5744}, {442, 5045}, {515, 13279}, {517, 6075}, {522, 693}, {527, 1156}, {528, 1155}, {553, 20292}, {740, 12080}, {912, 1519}, {942, 20612}, {946, 3868}, {950, 1005}, {952, 1512}, {956, 5722}, {962, 6245}, {982, 3914}, {984, 24217}, {999, 3419}, {1058, 5250}, {1086, 3999}, {1100, 17726}, {1150, 3883}, {1280, 2006}, {1320, 10265}, {1376, 4863}, {1420, 12625}, {1445, 6601}, {1476, 6598}, {1538, 13257}, {1621, 5745}, {1699, 5905}, {1836, 11235}, {1837, 12513}, {1861, 1897}, {1864, 15845}, {1936, 2342}, {1996, 6604}, {2321, 8568}, {2323, 4700}, {2475, 4298}, {2476, 3889}, {2550, 3306}, {2784, 5990}, {3035, 3689}, {3058, 4640}, {3120, 17449}, {3175, 4884}, {3189, 4855}, {3242, 17720}, {3243, 5219}, {3303, 26066}, {3304, 5794}, {3305, 26105}, {3361, 4190}, {3436, 6762}, {3452, 3681}, {3485, 11520}, {3574, 5777}, {3600, 5175}, {3649, 10957}, {3660, 10427}, {3663, 4392}, {3674, 20247}, {3677, 19785}, {3685, 3977}, {3693, 3943}, {3697, 17527}, {3712, 4702}, {3717, 4358}, {3742, 3925}, {3748, 6690}, {3755, 4850}, {3756, 16610}, {3772, 17597}, {3782, 21342}, {3817, 4430}, {3822, 3892}, {3826, 17051}, {3829, 17605}, {3834, 20541}, {3869, 12053}, {3871, 6684}, {3874, 12047}, {3875, 24388}, {3880, 13996}, {3881, 13407}, {3885, 11362}, {3886, 17740}, {3890, 5837}, {3893, 8256}, {3894, 18393}, {3895, 5657}, {3913, 24914}, {3916, 15171}, {3928, 9580}, {3936, 4684}, {3937, 15310}, {3947, 5141}, {3952, 4899}, {3953, 23537}, {3962, 26475}, {3976, 23536}, {3994, 4712}, {4001, 4388}, {4018, 8727}, {4054, 24349}, {4104, 25960}, {4189, 4314}, {4193, 21075}, {4253, 21073}, {4294, 4652}, {4349, 14996}, {4434, 17765}, {4514, 14829}, {4649, 17722}, {4656, 7226}, {4661, 21060}, {4679, 5220}, {4706, 8758}, {4848, 14923}, {4857, 6763}, {4860, 5880}, {4864, 17724}, {4867, 16173}, {4875, 21049}, {4883, 17056}, {4956, 17132}, {4996, 17010}, {5046, 12527}, {5048, 5855}, {5086, 10106}, {5126, 10609}, {5177, 11037}, {5178, 5253}, {5208, 17167}, {5290, 6871}, {5316, 24393}, {5435, 17784}, {5440, 15325}, {5442, 14798}, {5534, 6834}, {5537, 11219}, {5563, 17647}, {5572, 6067}, {5586, 10941}, {5691, 20076}, {5709, 6361}, {5730, 11373}, {5735, 9812}, {5815, 6919}, {5839, 24005}, {5850, 17484}, {5854, 20118}, {5904, 21616}, {5927, 7956}, {6769, 6890}, {7290, 24597}, {7330, 10531}, {7411, 11012}, {7580, 11249}, {7681, 14872}, {7741, 21077}, {7982, 12616}, {8666, 10572}, {9284, 17448}, {9335, 24175}, {9614, 11415}, {10025, 17036}, {10395, 11523}, {10589, 25568}, {10680, 18525}, {10950, 11260}, {11113, 18527}, {11238, 17781}, {11376, 12635}, {11522, 12617}, {12512, 20066}, {12607, 17606}, {12609, 18398}, {12619, 25416}, {12675, 15908}, {13138, 15499}, {13226, 17613}, {14956, 18206}, {15185, 21617}, {16418, 18530}, {17474, 21029}, {17491, 23821}, {17609, 25466}, {17774, 18134}, {18201, 24715}, {18239, 18243}, {18492, 26332}, {18653, 19642}, {19925, 20060}, {20835, 26357}, {21096, 25082}, {21242, 24325}, {21255, 25959}, {21296, 24213}

X(26015) = complement of X(3935)
X(26015) = anticomplement of X(6745)
X(26015) = inverse-in-inellipse-centered-at-X(10) of X(2)

X(26016) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26016) lies on these lines: {1, 2}, {7291, 21382}, {20911, 25002}, {26005, 26014}

X(26017) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26017) lies on these lines: {2, 661}, {9, 4077}, {514, 24562}, {649, 25009}, {657, 693}, {812, 26546}, {850, 4529}, {1021, 1577}, {2522, 14837}, {4379, 26695}, {4885, 14298}, {8062, 24718}, {14838, 26004}, {17072, 18344}, {17811, 18199}, {21146, 25926}

X(26018) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26018) lies on these lines: {2, 3}

X(26019) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26019) lies on these lines: {2, 3}, {11, 239}, {12, 16826}, {325, 3948}, {496, 4393}, {908, 26012}, {1329, 3661}, {1778, 24895}, {1959, 21044}, {2893, 25679}, {3580, 17174}, {3662, 21239}, {3814, 3912}, {3816, 17397}, {3825, 17023}, {3847, 17367}, {4384, 7741}, {5254, 24598}, {5949, 6707}, {6542, 17757}, {7173, 16815}, {7951, 16831}, {9722, 18747}, {10593, 16816}, {11681, 17316}, {12607, 17389}, {17167, 25977}, {19719, 19754}, {19791, 19839}, {20486, 20531}, {21926, 27483}, {24603, 25639}, {26001, 26010}

X(26020) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26020) lies on these lines: {2, 3}, {11, 1861}, {33, 3816}, {34, 1329}, {120, 13999}, {123, 1465}, {908, 1876}, {1376, 11393}, {1395, 25938}, {1398, 3436}, {1753, 7681}, {1785, 5121}, {1829, 24982}, {1870, 17757}, {1892, 3306}, {1897, 5211}, {5081, 5205}, {5090, 19861}, {5554, 11396}, {10200, 11399}, {11392, 25524}, {11398, 26364}, {16082, 17987}, {16997, 27377}, {17721, 23050}, {26012, 26013}

X(26021) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26021) lies on these lines: {2, 3}

X(26022) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26022) lies on these lines:

X(26023) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26023) lies on these lines: {2, 3}, {239, 17923}, {273, 27483}, {286, 1213}, {1838, 24603}, {5081, 27399}, {5174, 16826}, {17917, 26626}, {17924, 27486}

X(26024) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 38

Barycentrics    a^8 - a^7 b + a^6 b^2 + a^5 b^3 - 7 a^4 b^4 + a^3 b^5 + 7 a^2 b^6 - a b^7 - 2 b^8 - a^7 c - 5 a^6 b c - 3 a^5 b^2 c + 9 a^4 b^3 c + 9 a^3 b^4 c - 3 a^2 b^5 c - 5 a b^6 c - b^7 c + a^6 c^2 - 3 a^5 b c^2 + 4 a^4 b^2 c^2 - 10 a^3 b^3 c^2 + a^2 b^4 c^2 - 3 a b^5 c^2 + 10 b^6 c^2 + a^5 c^3 + 9 a^4 b c^3 - 10 a^3 b^2 c^3 - 10 a^2 b^3 c^3 + 9 a b^4 c^3 + b^5 c^3 - 7 a^4 c^4 + 9 a^3 b c^4 + a^2 b^2 c^4 + 9 a b^3 c^4 - 16 b^4 c^4 + a^3 c^5 - 3 a^2 b c^5 - 3 a b^2 c^5 + b^3 c^5 + 7 a^2 c^6 - 5 a b c^6 + 10 b^2 c^6 - a c^7 - b c^7 - 2 c^8 : :

X(26024) lies on these lines: {2, 3}

X(26025) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26025) lies on these lines: {2, 3}, {239, 26014}

X(26026) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 38

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

X(26026) lies on these lines: {2, 3}

leftri

Collineation mappings involving Gemini triangle 39: X(26027)-X(26084)

rightri

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 39, as in centers X(26027)-X(26084). Then

m(X) = 2 b c (a - b - c) x - a c(a + b + c) y - a b (a + b + c) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 29, 2018)

X(26027) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26027) lies on these lines: {2, 3}, {8, 73}, {10, 1745}, {318, 1214}, {966, 3330}, {1788, 19366}, {2183, 5749}, {2551, 26031}, {2635, 9780}, {2654, 3616}, {4645, 5552}, {5342, 6708}, {6349, 7952}, {6734, 27339}, {7080, 26942}, {9612, 27287}, {15852, 26011}, {17080, 23661}, {26041, 26043}

X(26028) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26028) lies on these lines: {2, 3}, {8, 2594}, {4417, 5552}, {4645, 27529}, {9780, 26031}, {17095, 18738}, {22300, 26115}, {26034, 26364}

X(26029) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26029) lies on these lines: {1, 2}, {46, 17350}, {100, 17697}, {341, 3752}, {346, 21796}, {377, 26073}, {442, 26772}, {740, 27291}, {986, 27538}, {1058, 26139}, {1089, 1278}, {1220, 4413}, {1329, 4429}, {1376, 4195}, {1575, 25610}, {2345, 21892}, {2551, 4201}, {3210, 3701}, {3303, 25531}, {3662, 21075}, {3672, 18140}, {3697, 27311}, {3760, 4452}, {3820, 16062}, {4188, 15654}, {4352, 6376}, {4385, 17490}, {4454, 4721}, {4642, 19582}, {4646, 18743}, {4657, 25109}, {4673, 21896}, {4695, 25591}, {4737, 17480}, {4968, 24620}, {5260, 19278}, {5687, 13741}, {6210, 26685}, {9709, 13740}, {11415, 26791}, {17303, 25629}, {17691, 26687}, {17756, 27523}, {17869, 26612}, {20498, 26132}, {24174, 24349}, {25242, 25994}, {26040, 26051}, {26041, 26042}, {26050, 26062}, {26077, 26083}, {27102, 27334}

X(26030) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 b + 2 a^2 b^2 + a b^3 + a^3 c + 2 a^2 b c + b^3 c + 2 a^2 c^2 + 2 b^2 c^2 + a c^3 + b c^3 : :

X(26030) lies on these lines: {1, 2}, {5, 4972}, {12, 4202}, {35, 11319}, {46, 26223}, {55, 5192}, {100, 13740}, {256, 27033}, {404, 1220}, {740, 27261}, {964, 1376}, {1089, 17147}, {1215, 24443}, {1329, 5051}, {1469, 17077}, {1575, 25629}, {1621, 13741}, {1706, 14554}, {1909, 27162}, {2183, 5749}, {2228, 26042}, {2276, 27040}, {2277, 14624}, {2347, 5750}, {2476, 4429}, {3264, 17321}, {3454, 27041}, {3666, 3701}, {3670, 17165}, {3697, 4981}, {3702, 4646}, {3752, 4968}, {3761, 18600}, {3820, 13728}, {3826, 27042}, {3923, 27078}, {3931, 4358}, {4201, 5080}, {4385, 4850}, {4413, 16454}, {4424, 25253}, {4645, 26067}, {4649, 27145}, {4698, 24751}, {4754, 25350}, {5010, 17539}, {5218, 17526}, {5251, 16347}, {5252, 26126}, {5260, 19270}, {5294, 6684}, {5432, 8240}, {5482, 11231}, {5687, 24552}, {6376, 16705}, {6381, 25599}, {6690, 25992}, {8728, 24988}, {9596, 26085}, {11115, 25440}, {11681, 16062}, {15888, 25914}, {17140, 24046}, {17184, 21077}, {17674, 25466}, {20140, 27169}, {24325, 27311}, {25017, 25882}, {25499, 27076}, {25611, 27032}, {26051, 26060}, {26057, 26065}

X(26031) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26031) lies on these lines: {2, 11}, {10, 73}, {474, 26126}, {1362, 27339}, {1698, 5400}, {1788, 10822}, {1818, 26013}, {2254, 26078}, {2551, 26027}, {2887, 21912}, {3120, 21914}, {3698, 22313}, {4425, 21913}, {5229, 26050}, {9780, 26028}, {16578, 24026}, {18134, 27517}, {18141, 27518}

X(26032) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26032) lies on these lines: {2, 3}, {144, 17007}, {1853, 26579}, {3219, 26034}, {4123, 16580}, {4463, 17481}, {4645, 5905}, {5800, 17778}, {12588, 25308}

X(26033) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26033) lies on these lines: {2, 3}, {659, 25299}, {3952, 4645}

X(26034) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    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(26034) lies on these lines: {2, 31}, {8, 38}, {9, 15487}, {10, 46}, {42, 69}, {43, 5739}, {55, 141}, {58, 19784}, {200, 17272}, {210, 4643}, {306, 17594}, {312, 24723}, {321, 24248}, {329, 4683}, {333, 4429}, {345, 4414}, {474, 27657}, {498, 3454}, {595, 19836}, {612, 4357}, {614, 3883}, {672, 966}, {756, 4019}, {846, 17776}, {851, 1211}, {896, 9780}, {899, 14555}, {902, 3619}, {940, 4026}, {958, 1473}, {968, 3912}, {984, 10327}, {993, 7293}, {1036, 19527}, {1150, 4972}, {1215, 4655}, {1403, 12588}, {1654, 2227}, {1698, 1707}, {1709, 12618}, {1738, 5271}, {1755, 26063}, {1761, 2345}, {1824, 18252}, {1962, 17316}, {2177, 3620}, {2187, 14826}, {2221, 5711}, {2223, 7800}, {2225, 26036}, {2232, 26043}, {2236, 26042}, {2308, 3618}, {2478, 3831}, {2550, 6817}, {2895, 3240}, {3011, 25527}, {3052, 3763}, {3219, 26032}, {3242, 4030}, {3416, 3666}, {3434, 3741}, {3616, 17469}, {3662, 3757}, {3683, 17279}, {3705, 24627}, {3715, 17332}, {3720, 18141}, {3745, 4657}, {3747, 27248}, {3751, 4001}, {3752, 3966}, {3755, 17156}, {3769, 19786}, {3821, 4362}, {3826, 19732}, {3844, 4640}, {3914, 11679}, {3925, 5737}, {3974, 4419}, {4003, 4914}, {4046, 4445}, {4259, 22275}, {4363, 11246}, {4384, 23682}, {4413, 5743}, {4450, 24552}, {4512, 17284}, {4646, 10371}, {4849, 17344}, {4865, 6682}, {5230, 16062}, {5256, 5847}, {5269, 17306}, {5311, 17321}, {5314, 25440}, {5552, 26057}, {5774, 11359}, {5793, 7354}, {5846, 17599}, {6057, 17262}, {6999, 9778}, {7081, 27184}, {9598, 21024}, {11031, 18391}, {11269, 14829}, {12586, 15621}, {16570, 19875}, {16825, 24169}, {17184, 26227}, {17596, 17740}, {17598, 19993}, {17676, 17751}, {17792, 26893}, {20368, 26118}, {21000, 21358}, {21240, 26101}, {24349, 26840}, {24597, 25453}, {24693, 27798}, {24695, 26223}, {26028, 26364}, {26038, 26073}, {26128, 26228}

X(26035) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    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(26035) lies on these lines: {1, 21070}, {2, 39}, {6, 8}, {10, 672}, {32, 11115}, {37, 4968}, {75, 17489}, {105, 405}, {141, 4754}, {281, 4185}, {377, 966}, {379, 5273}, {391, 4274}, {404, 26244}, {573, 15971}, {894, 17137}, {1010, 5276}, {1150, 5021}, {1213, 4202}, {1334, 17355}, {1475, 3741}, {1575, 25629}, {1851, 17920}, {1909, 17289}, {2276, 26115}, {2549, 17676}, {3053, 16393}, {3496, 4418}, {3720, 21071}, {3735, 17164}, {3739, 20880}, {3954, 17165}, {3975, 19808}, {4253, 10479}, {4359, 16583}, {5051, 5254}, {5206, 16397}, {5257, 23536}, {5275, 16454}, {5277, 19284}, {5278, 19281}, {5300, 17275}, {5308, 19701}, {6376, 27026}, {6542, 19717}, {9780, 20331}, {10472, 16713}, {11319, 24275}, {13728, 15048}, {16502, 24552}, {16604, 26094}, {16818, 20888}, {16998, 17688}, {17033, 17368}, {17135, 20963}, {17277, 17686}, {17303, 19874}, {17316, 19684}, {17359, 24656}, {17750, 17751}, {19743, 20055}, {21024, 24512}, {21808, 24325}, {24989, 27376}, {25000, 26550}, {26058, 26072}, {26059, 26961}, {27071, 27251}

X(26036) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26036) lies on these lines: {2, 41}, {4, 9}, {6, 25466}, {8, 3930}, {101, 26363}, {198, 5742}, {213, 26098}, {218, 442}, {220, 2886}, {226, 4384}, {377, 672}, {388, 21384}, {391, 1405}, {405, 8299}, {443, 17754}, {607, 25985}, {910, 26066}, {978, 7736}, {1212, 5794}, {1334, 3434}, {1478, 16552}, {1479, 3294}, {1714, 5280}, {1738, 9593}, {2082, 24987}, {2225, 26034}, {2246, 9780}, {2329, 19843}, {2893, 26045}, {3008, 25525}, {3085, 3684}, {3207, 4999}, {3208, 5082}, {3419, 16601}, {3436, 3691}, {3487, 16825}, {3679, 7323}, {4251, 10198}, {4258, 6690}, {4520, 12701}, {4662, 17275}, {4875, 5252}, {5230, 5276}, {5273, 6999}, {5436, 19868}, {5749, 26051}, {5783, 15973}, {7384, 18228}, {7774, 16827}, {9310, 10527}, {11236, 17330}, {12649, 21808}, {13161, 16517}, {16788, 19854}, {17170, 24694}, {22127, 24512}, {24318, 25583}, {26037, 26052}

X(26037) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a b^2 + 3 a b c + b^2 c + a c^2 + b c^2 : :

X(26037) lies on these lines: {1, 2}, {9, 4418}, {31, 17277}, {38, 19804}, {55, 17259}, {75, 756}, {141, 25961}, {171, 5278}, {210, 3739}, {291, 24988}, {310, 6376}, {312, 21020}, {333, 750}, {649, 25627}, {672, 966}, {748, 5263}, {749, 16709}, {851, 26066}, {958, 4191}, {982, 4981}, {984, 4359}, {993, 4210}, {1011, 1376}, {1150, 17122}, {1211, 3826}, {1213, 2276}, {1268, 2296}, {1329, 3136}, {1574, 21838}, {1575, 25624}, {1861, 4207}, {1962, 4687}, {2238, 17303}, {2239, 19808}, {2308, 17349}, {2345, 25623}, {2350, 21384}, {2550, 6818}, {2551, 6817}, {2886, 5241}, {3210, 3989}, {3219, 3980}, {3681, 24325}, {3715, 4363}, {3745, 17348}, {3761, 16748}, {3791, 9347}, {3923, 27065}, {3925, 5743}, {3971, 9330}, {4023, 17056}, {4046, 17243}, {4104, 5249}, {4147, 4379}, {4184, 25440}, {4192, 26446}, {4199, 5955}, {4413, 5737}, {4441, 4967}, {4665, 6057}, {4683, 5880}, {4703, 20292}, {4751, 21805}, {4893, 17072}, {5235, 13588}, {5247, 16454}, {5791, 16056}, {7226, 24165}, {7308, 13576}, {9568, 12435}, {9708, 16059}, {9709, 16058}, {10440, 10478}, {11246, 17332}, {14829, 17124}, {17123, 24552}, {17248, 17759}, {17251, 24690}, {17275, 24512}, {17289, 25611}, {17750, 21753}, {17889, 26580}, {18154, 21727}, {20347, 25590}, {21223, 27318}, {23791, 26777}, {24342, 26223}, {25385, 27131}, {26036, 26052}, {26044, 26073}, {26060, 26064}, {26128, 26724}

X(26038) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26038) lies on these lines: {1, 2}, {37, 4734}, {38, 24620}, {69, 25144}, {75, 3740}, {100, 16058}, {171, 17349}, {210, 19804}, {321, 4903}, {333, 4413}, {391, 17754}, {756, 3210}, {956, 16409}, {966, 1575}, {984, 17490}, {1150, 9342}, {1215, 4699}, {1278, 3971}, {1376, 4203}, {1654, 26135}, {2238, 5749}, {2239, 26065}, {2276, 5296}, {2550, 25135}, {2975, 16059}, {3061, 22173}, {3434, 6822}, {3436, 6821}, {3662, 4104}, {3681, 24589}, {3685, 7308}, {3696, 18743}, {3759, 4682}, {3769, 17348}, {3826, 4417}, {3921, 4737}, {3925, 5233}, {3980, 17350}, {3996, 4423}, {4023, 18134}, {4210, 15654}, {4429, 5743}, {4640, 17335}, {4645, 14555}, {4704, 4970}, {4748, 25349}, {5080, 6817}, {5278, 11358}, {5328, 20545}, {5657, 19540}, {5744, 16056}, {6210, 9778}, {6384, 25280}, {7155, 27439}, {7229, 24514}, {9330, 17147}, {10440, 10446}, {16604, 24528}, {17236, 24169}, {17251, 25350}, {17260, 17594}, {17275, 25311}, {17280, 25623}, {17358, 25611}, {17592, 27268}, {17756, 21838}, {19808, 26083}, {21060, 24199}, {21264, 25116}, {24749, 27345}, {26034, 26073}

X(26039) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    5 a^2 + 2 a b + 3 b^2 + 2 a c + 6 b c + 3 c^2 : :

X(26039) lies on these lines: {1, 2321}, {2, 45}, {6, 3617}, {7, 17385}, {8, 16666}, {9, 3634}, {10, 16670}, {37, 5550}, {44, 966}, {144, 17327}, {346, 16672}, {551, 4873}, {594, 3621}, {599, 4747}, {1100, 20050}, {1449, 3625}, {1698, 24695}, {2246, 26040}, {2325, 3624}, {3247, 15808}, {3616, 17281}, {3618, 16816}, {3622, 3943}, {3626, 5839}, {3672, 7227}, {3707, 19875}, {3945, 17293}, {4029, 25055}, {4461, 17045}, {4644, 17308}, {4648, 17241}, {4657, 7229}, {4665, 17014}, {4677, 4982}, {4678, 4969}, {4708, 6172}, {4727, 20057}, {4798, 5308}, {4887, 7222}, {5257, 19872}, {5746, 15650}, {5936, 17348}, {16676, 17355}, {16815, 17368}, {17012, 19822}, {17067, 25590}, {17160, 17381}, {17572, 19297}

X(26040) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 - a^2 b + a b^2 - b^3 - a^2 c + 6 a b c + b^2 c + a c^2 + b c^2 - c^3 : :

X(26040) lies on these lines: {1, 12521}, {2, 11}, {3, 19855}, {4, 165}, {5, 6244}, {7, 210}, {8, 354}, {9, 3474}, {10, 57}, {12, 4208}, {33, 25993}, {35, 16845}, {40, 6864}, {42, 4648}, {43, 5712}, {56, 17580}, {65, 11024}, {75, 3974}, {142, 200}, {144, 3715}, {226, 8580}, {329, 3740}, {355, 11227}, {376, 5251}, {377, 1155}, {442, 10588}, {474, 1617}, {496, 16863}, {515, 10857}, {516, 7308}, {518, 9776}, {553, 5223}, {612, 4000}, {631, 6796}, {672, 966}, {756, 4419}, {899, 10460}, {910, 17303}, {936, 3485}, {946, 7994}, {958, 6904}, {962, 25917}, {1002, 4651}, {1010, 5324}, {1056, 3679}, {1058, 3624}, {1125, 5082}, {1329, 5177}, {1478, 19875}, {1479, 17559}, {1699, 5316}, {1709, 5817}, {1722, 5716}, {1738, 5268}, {1836, 18228}, {1864, 15587}, {2078, 6681}, {2246, 26039}, {2272, 26063}, {2348, 5749}, {2478, 19877}, {3008, 5269}, {3052, 17337}, {3085, 8728}, {3086, 16408}, {3090, 5537}, {3091, 7965}, {3158, 6601}, {3256, 3841}, {3305, 5698}, {3306, 24477}, {3340, 12447}, {3476, 9623}, {3486, 10383}, {3523, 24953}, {3525, 12116}, {3579, 6849}, {3583, 19876}, {3616, 3748}, {3617, 4860}, {3632, 17706}, {3634, 5084}, {3646, 10624}, {3660, 4002}, {3663, 7322}, {3677, 24175}, {3683, 9778}, {3689, 10578}, {3744, 16020}, {3745, 5222}, {3753, 5173}, {3755, 17022}, {3782, 7613}, {3817, 20196}, {3820, 10590}, {3838, 5748}, {3844, 5800}, {3983, 5815}, {4061, 17296}, {4082, 4659}, {4190, 5260}, {4197, 5552}, {4293, 9708}, {4294, 11108}, {4295, 5044}, {4307, 4383}, {4309, 25542}, {4355, 4866}, {4356, 25430}, {4359, 10327}, {4433, 27253}, {4461, 6057}, {4470, 24315}, {4512, 6666}, {4645, 14555}, {4654, 21060}, {4675, 4849}, {4679, 9812}, {4699, 16990}, {4731, 5252}, {4847, 5437}, {4863, 10580}, {5067, 10531}, {5128, 18249}, {5129, 6284}, {5217, 17558}, {5219, 20103}, {5220, 9965}, {5231, 6692}, {5248, 17552}, {5249, 25568}, {5261, 21031}, {5297, 19785}, {5328, 17605}, {5536, 10532}, {5587, 6916}, {5657, 6854}, {5686, 21454}, {5687, 17529}, {5739, 20290}, {5794, 17603}, {5818, 6897}, {5836, 17642}, {5853, 10582}, {6361, 6896}, {6743, 11518}, {6745, 25525}, {6764, 17609}, {6826, 26446}, {6827, 11231}, {6835, 7964}, {6846, 10310}, {6850, 9956}, {6857, 25440}, {6887, 11248}, {6951, 17057}, {6964, 15908}, {6989, 11499}, {7069, 24341}, {7074, 25878}, {7080, 25466}, {7174, 24177}, {7392, 11677}, {7967, 7993}, {8165, 10895}, {8171, 15325}, {8583, 10388}, {9535, 10824}, {9579, 18250}, {9589, 11379}, {9710, 25524}, {10172, 26333}, {10178, 10430}, {10527, 17531}, {10591, 17527}, {10855, 17625}, {11018, 18391}, {11106, 15338}, {11221, 18698}, {11269, 17124}, {15171, 16853}, {16043, 16819}, {16569, 26098}, {16862, 24390}, {17570, 20066}, {21010, 27304}, {21912, 26939}, {23207, 25932}, {26029, 26051}, {26228, 26724}

X(26041) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    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 - 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(26041) lies on these lines: {2, 6}, {10, 1716}, {75, 16605}, {264, 25021}, {345, 21857}, {1108, 25895}, {2183, 26685}, {2551, 4429}, {3718, 16583}, {3975, 4000}, {4352, 25470}, {4357, 27299}, {17270, 27248}, {20336, 21216}, {21035, 27549}, {26027, 26043}, {26029, 26042}, {26056, 26072}, {27047, 27280}

X(26042) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    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(26042) lies on these lines: {2, 37}, {8, 1964}, {10, 1740}, {39, 3596}, {45, 27111}, {69, 26752}, {194, 313}, {322, 25918}, {894, 24315}, {966, 2235}, {984, 25120}, {1441, 26134}, {1698, 16571}, {1755, 26053}, {2227, 26061}, {2228, 26030}, {2234, 9780}, {2236, 26034}, {2237, 26085}, {2245, 17350}, {3097, 21080}, {3247, 25510}, {3616, 17445}, {3764, 7155}, {3778, 24351}, {3875, 26959}, {3963, 24598}, {4357, 27091}, {4393, 5153}, {4446, 24327}, {4741, 26756}, {4967, 17030}, {5294, 19591}, {5749, 26076}, {7187, 20930}, {9596, 26058}, {10436, 27020}, {17178, 17373}, {17230, 27145}, {17232, 27017}, {17233, 26979}, {17236, 27095}, {17238, 27044}, {17323, 25534}, {21238, 21299}, {25504, 27272}, {25538, 25590}, {25635, 26069}, {26029, 26041}, {26063, 26081}

X(26043) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26043) lies on these lines: {2, 39}, {377, 26072}, {672, 27091}, {966, 2231}, {2228, 26030}, {2230, 9780}, {2232, 26034}, {2233, 26058}, {17486, 27801}, {26027, 26041}

X(26044) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 - 2 a^2 b - 4 a b^2 - b^3 - 2 a^2 c - 7 a b c - 4 b^2 c - 4 a c^2 - 4 b c^2 - c^3 : :

X(26044) lies on these lines: {2, 6}, {8, 1962}, {10, 846}, {896, 9780}, {1330, 1698}, {1655, 3210}, {1761, 3219}, {1999, 5257}, {2183, 27065}, {2475, 2551}, {3151, 26063}, {3617, 3704}, {3739, 26840}, {3770, 19804}, {3882, 7308}, {3975, 4359}, {4708, 19786}, {5249, 17252}, {5271, 17248}, {9791, 21020}, {14005, 20077}, {16589, 25058}, {17250, 24789}, {17326, 26723}, {17499, 24603}, {19877, 26131}, {20929, 27705}, {24697, 27798}, {26037, 26073}, {26053, 26059}, {26070, 26081}

X(26045) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 b^2 + a^2 b^3 + a^3 b c + 5 a^2 b^2 c + 3 a b^3 c + a^3 c^2 + 5 a^2 b c^2 + 5 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 3 a b c^3 + b^2 c^3 : :

X(26045) lies on these lines: {2, 6}, {8, 2667}, {10, 1045}, {21, 22369}, {75, 1655}, {261, 5277}, {314, 16589}, {1444, 16917}, {2183, 17260}, {2234, 9780}, {2550, 26117}, {2551, 26051}, {2893, 26036}, {3739, 3770}, {4357, 16819}, {4645, 19874}, {5257, 25427}, {5839, 25426}, {10436, 17499}, {16696, 25457}, {16705, 25470}, {17250, 26149}, {17270, 27255}, {17303, 26076}, {17321, 18904}, {17322, 26801}, {17762, 27565}, {19877, 26135}, {26055, 26068}

X(26046) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 b + 2 a^2 b^2 + a b^3 + a^3 c - a^2 b c - 9 a b^2 c + b^3 c + 2 a^2 c^2 - 9 a b c^2 + 2 b^2 c^2 + a c^3 + b c^3 : :

X(26046) lies on these lines: {1, 2}, {341, 24620}, {1574, 27523}, {1575, 25612}, {2551, 26073}, {9709, 17697}, {25631, 26077}

X(26047) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26047) lies on these lines: {1, 2}, {461, 5101}, {2348, 5749}, {3677, 10005}, {3914, 8055}, {3974, 4402}, {4000, 5423}, {4082, 4452}, {4429, 18228}, {5772, 19804}, {9776, 24988}, {9778, 26685}, {26065, 26073}

X(26048) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26048) lies on these lines: {1, 2}, {44, 26076}, {594, 27111}, {649, 22224}, {966, 2235}, {1268, 27042}, {1575, 3975}, {1654, 27102}, {2238, 18278}, {3210, 18135}, {3752, 25107}, {3948, 17759}, {3965, 25975}, {4395, 25534}, {4473, 27036}, {4699, 26149}, {5687, 11353}, {5749, 26077}, {6645, 25946}, {9263, 25298}, {17787, 21892}, {21226, 24598}, {21858, 25660}, {24478, 25120}, {26756, 26806}

X(26049) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c + a^3 b c - 2 a^2 b^2 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(26049) lies on these lines: {2, 650}, {75, 25271}, {513, 25636}, {649, 27527}, {652, 26652}, {659, 23301}, {661, 27345}, {798, 20295}, {812, 27293}, {850, 21225}, {1491, 6133}, {3210, 25098}, {3835, 4063}, {4147, 23655}, {4416, 23725}, {6586, 25258}, {16751, 17496}, {21127, 25008}, {21727, 26115}, {23806, 27184}, {27013, 27114}

X(26050) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26050) lies on these lines: {2, 3}, {8, 1042}, {10, 1044}, {145, 1066}, {1448, 7360}, {3000, 9780}, {3701, 25242}, {4645, 5906}, {5229, 26031}, {26029, 26062}

X(26051) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26051) lies on these lines: {2, 3}, {8, 2650}, {10, 894}, {58, 25446}, {86, 1834}, {148, 5988}, {239, 5717}, {333, 20077}, {341, 3770}, {387, 17379}, {896, 9780}, {942, 26806}, {1043, 17056}, {1220, 3925}, {1478, 19853}, {1655, 25242}, {1706, 3882}, {2550, 26110}, {2551, 26045}, {2893, 10436}, {2895, 3617}, {3583, 25512}, {3585, 16828}, {3616, 24161}, {3624, 26139}, {3786, 10381}, {4418, 21674}, {5080, 19874}, {5263, 25466}, {5295, 6542}, {5712, 20018}, {5716, 19851}, {5749, 26036}, {9791, 24851}, {10449, 17300}, {12572, 17260}, {13161, 16830}, {17248, 19859}, {17302, 23537}, {20533, 27255}, {24440, 24693}, {26029, 26040}, {26030, 26060}

X(26052) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26052) lies on these lines: {2, 3}, {8, 17441}, {9, 15487}, {10, 1763}, {33, 18589}, {55, 11677}, {69, 18138}, {72, 10327}, {184, 26668}, {197, 23305}, {226, 612}, {329, 4645}, {388, 1427}, {497, 1279}, {614, 950}, {910, 17303}, {1211, 1853}, {1441, 7102}, {1824, 4329}, {1861, 10319}, {1890, 9816}, {1899, 5739}, {1901, 5275}, {2000, 18651}, {2550, 3198}, {3434, 3757}, {3487, 3920}, {3488, 7191}, {3586, 5272}, {3917, 10477}, {5268, 9612}, {5276, 5746}, {5297, 5714}, {5712, 5800}, {5744, 26929}, {7172, 20344}, {14547, 26130}, {17810, 25964}, {21015, 27540}, {26036, 26037}

X(26053) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26053) lies on these lines: {2, 3}, {92, 18666}, {1214, 1947}, {1755, 26042}, {2893, 27339}, {26044, 26059}

X(26054) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26054) lies on these lines: {2, 3}, {7, 26131}, {8, 18673}, {10, 2939}, {63, 1330}, {71, 1761}, {846, 1770}, {2173, 9780}, {2292, 4295}, {2893, 6734}, {2947, 12520}, {3868, 17778}, {5262, 14547}, {5273, 26064}

X(26055) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26055) lies on these lines: {2, 3}, {8, 2658}, {10, 1047}, {318, 18667}, {26045, 26068}

X(26056) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26056) lies on these lines: {2, 3}, {26041, 26072}

X(26057) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26057) lies on these lines: {2, 3}, {46, 894}, {1210, 27305}, {1714, 5145}, {3085, 4645}, {3550, 10198}, {5552, 26034}, {9612, 27254}, {26029, 26041}, {26030, 26065}

X(26058) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26058) lies on these lines: {2, 3}, {148, 27262}, {2233, 26043}, {2896, 27312}, {4645, 26752}, {9596, 26042}, {26035, 26072}

X(26059) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26059) lies on these lines: {2, 7}, {8, 2293}, {10, 1742}, {75, 1212}, {86, 220}, {192, 27317}, {219, 17379}, {239, 25601}, {314, 346}, {333, 3713}, {391, 27514}, {958, 4195}, {1441, 3177}, {1757, 3085}, {2324, 16826}, {2551, 26045}, {3000, 9780}, {3730, 10446}, {3923, 19843}, {3945, 27253}, {4772, 4858}, {5234, 19853}, {6603, 17394}, {7379, 26939}, {10456, 17355}, {10460, 10578}, {15817, 19308}, {16050, 16738}, {17238, 26932}, {19855, 24342}, {20072, 27267}, {24456, 24744}, {24547, 26690}, {24635, 25001}, {26029, 26041}, {26035, 26961}, {26044, 26053}

X(26060) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26060) lies on these lines: {2, 35}, {4, 11231}, {5, 9342}, {8, 443}, {9, 3648}, {10, 3218}, {11, 17535}, {21, 3826}, {43, 26131}, {79, 26792}, {100, 8728}, {149, 3624}, {165, 6894}, {377, 1155}, {404, 3925}, {442, 27529}, {750, 24883}, {962, 6854}, {1329, 6175}, {1376, 4197}, {1621, 17529}, {1698, 2475}, {1770, 27065}, {2077, 6884}, {2476, 4413}, {2550, 3616}, {2886, 17531}, {3434, 5550}, {3524, 18517}, {3579, 6900}, {3585, 3828}, {3634, 5046}, {3678, 17483}, {3811, 27186}, {3868, 9782}, {3876, 5880}, {4002, 5176}, {4188, 19854}, {4190, 19855}, {4201, 19874}, {4208, 5552}, {4302, 16859}, {4420, 5249}, {4429, 16454}, {4857, 19878}, {4872, 25585}, {5010, 15674}, {5015, 24589}, {5044, 20292}, {5067, 10525}, {5178, 5439}, {5260, 11112}, {5263, 17674}, {5266, 26724}, {5297, 23537}, {5300, 19804}, {5303, 17563}, {5791, 9352}, {5904, 26842}, {6224, 19860}, {6284, 17536}, {6684, 6839}, {6835, 9778}, {6864, 9812}, {6895, 10164}, {6901, 26446}, {6951, 9956}, {6991, 10310}, {7486, 26333}, {9668, 16854}, {9669, 16864}, {10527, 17580}, {11680, 16408}, {11681, 17528}, {12436, 25006}, {13587, 24953}, {13740, 24988}, {15338, 16858}, {15586, 17303}, {17572, 26363}, {17680, 27026}, {26030, 26051}, {26037, 26064}\

X(26060) = anticomplement of X(25542)

X(26061) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 + a^2 b + b^3 + a^2 c + b^2 c + b c^2 + c^3 : :

X(26061) lies on these lines: {2, 38}, {6, 15523}, {8, 16478}, {10, 31}, {45, 6536}, {63, 1698}, {69, 4722}, {321, 25453}, {354, 17357}, {498, 11031}, {518, 24943}, {672, 17303}, {748, 17353}, {894, 16991}, {896, 9780}, {976, 17698}, {993, 19867}, {1089, 20083}, {1213, 5282}, {1473, 4413}, {1707, 19875}, {1962, 17776}, {2225, 25616}, {2227, 26042}, {2239, 19808}, {2292, 19784}, {2308, 3416}, {2312, 26063}, {2345, 21020}, {2887, 24725}, {3006, 25496}, {3187, 3773}, {3589, 3703}, {3706, 17359}, {3720, 17279}, {3844, 4641}, {3869, 19879}, {3914, 17355}, {3923, 4972}, {3925, 17369}, {3932, 5311}, {3989, 4657}, {4042, 17293}, {4365, 17281}, {4418, 4429}, {4672, 6327}, {4683, 17350}, {4854, 17340}, {5251, 5314}, {5749, 21026}, {6679, 26227}, {7085, 21671}, {10453, 17358}, {12526, 19880}, {16706, 17155}, {17156, 17286}, {17275, 21764}, {24295, 24552}, {25760, 27064}

X(26062) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26062) lies on these lines: {2, 40}, {4, 17613}, {7, 5552}, {8, 56}, {10, 4293}, {20, 10270}, {46, 329}, {57, 7080}, {65, 27383}, {100, 938}, {165, 452}, {377, 1155}, {443, 26446}, {474, 5657}, {516, 6919}, {517, 17567}, {631, 3753}, {944, 16371}, {952, 17573}, {1004, 9799}, {1167, 1771}, {1210, 17784}, {1329, 3474}, {1482, 17564}, {1697, 6692}, {1698, 1770}, {1706, 3911}, {1737, 5175}, {2093, 6700}, {2094, 3336}, {2183, 5749}, {2476, 3826}, {2478, 9778}, {2550, 24914}, {3035, 3485}, {3057, 3616}, {3085, 9776}, {3218, 5815}, {3241, 20323}, {3339, 6745}, {3359, 6848}, {3361, 6736}, {3434, 5704}, {3436, 9352}, {3452, 5128}, {3523, 19860}, {3579, 5084}, {3600, 6735}, {3623, 17706}, {3871, 10580}, {3872, 5265}, {4187, 6361}, {4188, 5554}, {4190, 12616}, {4193, 9812}, {4295, 5748}, {4679, 6933}, {4848, 5438}, {5129, 25011}, {5183, 24954}, {5221, 25568}, {5226, 27529}, {5328, 11415}, {5550, 6690}, {5603, 13747}, {5690, 16417}, {5768, 11499}, {5790, 17563}, {5804, 11248}, {5818, 11112}, {5825, 17668}, {5828, 20060}, {5836, 7288}, {5880, 10588}, {6856, 11231}, {6931, 9779}, {7982, 24558}, {9800, 19541}, {9965, 21075}, {10303, 24541}, {10528, 11037}, {11240, 12541}, {12245, 17614}, {12526, 20103}, {13996, 20057}, {17580, 24987}, {18391, 25440}, {21454, 27525}, {25019, 27530}, {26029, 26050}

X(26063) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26063) lies on these lines: {2, 48}, {4, 9}, {5, 219}, {6, 12}, {8, 1953}, {11, 2256}, {37, 1837}, {80, 2335}, {119, 5778}, {150, 25521}, {197, 1011}, {198, 1213}, {210, 2262}, {284, 498}, {318, 6520}, {329, 3958}, {377, 2252}, {388, 2260}, {442, 19350}, {579, 1478}, {610, 1698}, {612, 14547}, {631, 22054}, {908, 5271}, {958, 5742}, {965, 1329}, {1100, 17718}, {1108, 5252}, {1436, 4413}, {1441, 24316}, {1630, 19854}, {1656, 20818}, {1714, 5747}, {1723, 10827}, {1751, 2259}, {1755, 26034}, {1765, 6256}, {1781, 18395}, {1802, 6846}, {1836, 21866}, {1853, 3197}, {1857, 7069}, {1901, 10895}, {2173, 9780}, {2182, 17303}, {2238, 2911}, {2257, 9578}, {2261, 5750}, {2265, 5749}, {2272, 26040}, {2273, 3767}, {2287, 11681}, {2289, 6824}, {2294, 18391}, {2300, 2548}, {2302, 10198}, {2312, 26061}, {3085, 5802}, {3090, 22356}, {3151, 26044}, {3211, 6881}, {3419, 3694}, {3525, 22357}, {3616, 17438}, {3628, 23073}, {3686, 21075}, {3826, 5781}, {3975, 20927}, {4329, 21231}, {4362, 5839}, {5055, 22147}, {5086, 27396}, {5220, 5829}, {5227, 6734}, {5251, 13726}, {5282, 21014}, {5433, 37519}, {5746, 10590}, {5755, 10526}, {5776, 18242}, {5792, 17327}, {5798, 10894}, {7522, 26942}, {9599, 21769}, {9958, 18491}, {10327, 21278}, {16713, 21286}, {16788, 19784}, {17582, 22088}, {18594, 19875}, {21239, 25878}, {26042, 26081}

X(26064) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^4 - 2 a^2 b^2 - 2 a b^3 - b^4 - 3 a^2 b c - 5 a b^2 c - 2 b^3 c - 2 a^2 c^2 - 5 a b c^2 - 2 b^2 c^2 - 2 a c^3 - 2 b c^3 - c^4 : :

X(26064) lies on these lines: {1, 2895}, {2, 58}, {8, 192}, {10, 191}, {21, 1211}, {81, 4205}, {141, 5047}, {238, 27270}, {333, 5051}, {377, 966}, {404, 5743}, {442, 5235}, {452, 2893}, {846, 20653}, {896, 9780}, {964, 5224}, {1046, 27714}, {1213, 1778}, {2476, 5737}, {3616, 16478}, {3770, 4968}, {3868, 4643}, {3882, 5250}, {3936, 11110}, {4104, 4420}, {4197, 19732}, {4202, 17277}, {4357, 5262}, {4417, 16342}, {4425, 27368}, {4645, 19874}, {4658, 20086}, {4683, 14450}, {4748, 5716}, {4981, 5015}, {5046, 10479}, {5241, 17531}, {5273, 26054}, {5277, 6537}, {5278, 16062}, {5292, 5361}, {5333, 17514}, {5550, 26109}, {5739, 13725}, {5741, 19270}, {5810, 19262}, {6327, 19853}, {7270, 17256}, {9534, 17676}, {11114, 17251}, {11115, 27081}, {12579, 21085}, {14020, 17271}, {15674, 25645}, {15676, 24946}, {16817, 17184}, {17056, 17557}, {17238, 17697}, {17588, 25650}, {18228, 26120}, {19854, 25958}, {21020, 24851}, {26037, 26060}

X(26064) = anticomplement of X(25526)

X(26065) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    3 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(26065) lies on these lines: {2, 7}, {6, 345}, {8, 31}, {10, 1707}, {21, 7085}, {38, 3616}, {44, 14555}, {45, 6703}, {69, 4641}, {81, 7123}, {92, 26665}, {189, 17743}, {191, 19784}, {193, 306}, {321, 24597}, {333, 1778}, {344, 940}, {346, 1999}, {387, 7283}, {404, 1473}, {497, 4676}, {612, 27549}, {896, 9780}, {938, 17697}, {942, 13742}, {966, 19808}, {1009, 20760}, {1264, 2273}, {1698, 16570}, {1730, 26961}, {1743, 3687}, {1755, 26042}, {1812, 2911}, {2221, 17740}, {2239, 26038}, {2247, 26081}, {2887, 24695}, {2899, 8258}, {3210, 5222}, {3241, 17469}, {3474, 4429}, {3488, 13735}, {3618, 3666}, {3620, 4001}, {3661, 14552}, {3710, 20009}, {3717, 5269}, {3730, 17185}, {3758, 5712}, {3769, 3974}, {3772, 17351}, {3868, 17526}, {3870, 10460}, {3914, 24280}, {3927, 17698}, {3977, 5256}, {4188, 7293}, {4189, 5314}, {4332, 19860}, {4419, 19786}, {4438, 4672}, {4472, 19744}, {4644, 18134}, {4656, 25728}, {4712, 10578}, {5221, 25992}, {5253, 25879}, {5278, 19281}, {5320, 17977}, {5703, 11031}, {5737, 17369}, {5743, 16885}, {6350, 15988}, {6763, 19836}, {7102, 14006}, {10327, 17126}, {11319, 12649}, {11342, 19716}, {11679, 17355}, {11681, 25984}, {13461, 19066}, {14001, 25083}, {14829, 17354}, {16061, 23151}, {16298, 22458}, {17022, 25101}, {17121, 20043}, {17141, 26626}, {17165, 26228}, {17256, 19827}, {17258, 19812}, {17279, 18141}, {18206, 27248}, {18651, 27127}, {20073, 25734}, {21526, 23089}, {24248, 25453}, {25091, 26658}, {26029, 26050}, {26030, 26057}, {26047, 26073}

X(26066) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26066) lies on these lines: {1, 4999}, {2, 65}, {3, 10}, {4, 4640}, {5, 12514}, {6, 5530}, {8, 2320}, {9, 46}, {11, 5250}, {12, 63}, {19, 5742}, {21, 1837}, {35, 3419}, {40, 2886}, {55, 6734}, {56, 24987}, {57, 25466}, {58, 5725}, {72, 498}, {78, 5432}, {140, 997}, {210, 5552}, {281, 14018}, {283, 5348}, {329, 10588}, {345, 3714}, {377, 1155}, {388, 5744}, {392, 499}, {405, 1737}, {517, 6862}, {518, 3085}, {527, 3947}, {529, 9578}, {551, 17706}, {573, 5831}, {750, 21674}, {758, 11374}, {851, 26037}, {910, 26036}, {912, 26487}, {936, 3035}, {942, 10198}, {956, 10039}, {962, 6860}, {965, 19350}, {966, 2182}, {986, 3772}, {1001, 1210}, {1125, 5289}, {1150, 10371}, {1158, 6907}, {1159, 19862}, {1191, 24239}, {1212, 1575}, {1452, 25985}, {1478, 3916}, {1656, 21616}, {1697, 3813}, {1706, 9588}, {1714, 4261}, {1770, 17532}, {1834, 17594}, {1836, 2476}, {1940, 17555}, {2099, 24541}, {2268, 21014}, {2278, 17275}, {2292, 17720}, {2478, 3683}, {2550, 6836}, {2975, 5252}, {3036, 9897}, {3057, 10527}, {3090, 5087}, {3091, 5698}, {3178, 4851}, {3185, 13731}, {3189, 5281}, {3218, 10404}, {3219, 11681}, {3295, 10916}, {3303, 26015}, {3305, 25973}, {3339, 25525}, {3416, 5135}, {3452, 3634}, {3474, 5177}, {3555, 10056}, {3556, 19544}, {3579, 18407}, {3584, 5904}, {3585, 17057}, {3612, 3679}, {3616, 17728}, {3624, 15829}, {3666, 5230}, {3701, 19807}, {3704, 11679}, {3740, 6889}, {3753, 19854}, {3781, 24655}, {3828, 5325}, {3829, 9614}, {3831, 17279}, {3838, 4295}, {3868, 17718}, {3871, 4863}, {3874, 10197}, {3876, 27529}, {3877, 11376}, {3878, 5886}, {3884, 11373}, {3899, 5443}, {3911, 25524}, {3913, 4847}, {3915, 17721}, {3927, 21077}, {3928, 5290}, {3931, 5292}, {4047, 5747}, {4185, 5155}, {4189, 5086}, {4193, 4679}, {4414, 21935}, {4512, 9581}, {4642, 24892}, {4643, 24315}, {4645, 25613}, {4652, 7354}, {4657, 17048}, {4662, 7080}, {5044, 5694}, {5057, 5141}, {5084, 15254}, {5090, 20832}, {5119, 24390}, {5219, 6668}, {5220, 21075}, {5221, 5249}, {5234, 11112}, {5235, 16049}, {5248, 5722}, {5251, 18395}, {5260, 25005}, {5433, 19861}, {5657, 5836}, {5686, 27525}, {5703, 5775}, {5704, 26105}, {5709, 7680}, {5719, 12559}, {5743, 20306}, {5770, 12675}, {5784, 12669}, {5818, 6934}, {5887, 6863}, {5905, 10585}, {5919, 10529}, {6001, 6825}, {6667, 25522}, {6691, 8583}, {6735, 22768}, {6824, 7686}, {6838, 12688}, {6857, 18391}, {6908, 9943}, {6917, 9956}, {6932, 12679}, {6933, 11415}, {6980, 18233}, {6991, 24329}, {7082, 10958}, {7330, 18242}, {8167, 9843}, {8256, 9623}, {9564, 10974}, {10106, 11194}, {10175, 12572}, {10179, 14986}, {10395, 13615}, {10441, 22276}, {10479, 16455}, {10572, 16370}, {10587, 17609}, {10624, 11235}, {10786, 14872}, {10826, 11113}, {11236, 12527}, {11281, 11529}, {11344, 11502}, {11509, 24982}, {11680, 12701}, {11682, 15950}, {11827, 21165}, {12575, 24386}, {12617, 19541}, {12635, 13411}, {12699, 25639}, {13405, 24391}, {15843, 17700}, {15865, 18389}, {16968, 21965}, {17278, 24174}, {17595, 23536}, {17597, 28027}, {18228, 19877}, {19860, 24953}, {21231, 25104}, {24443, 24789}, {24583, 26621}, {26029, 26041}

X(26067) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^5 - a^4 b - a^3 b^2 - 3 a^2 b^3 - a b^4 - b^5 - a^4 c - 3 a^2 b^2 c - b^4 c - a^3 c^2 - 3 a^2 b c^2 - a b^2 c^2 - 3 b^3 c^2 - 3 a^2 c^3 - 3 b^2 c^3 - a c^4 - b c^4 - c^5 : :

X(26067) lies on these lines: {2, 82}, {8, 17457}, {10, 16556}, {2236, 26034}, {2244, 9780}, {2896, 18082}, {4645, 26030}

X(26068) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26068) lies on these lines: {2, 85}, {9, 27020}, {76, 16588}, {349, 21218}, {958, 19312}, {7770, 15288}, {16819, 23058}, {24505, 27326}, {26029, 26041}, {26045, 26055}, {26110, 27382}

X(26069) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    a^3 b^2 - a^2 b^3 - 2 a^3 b c + 3 a^2 b^2 c + 2 a b^3 c + a^3 c^2 + 3 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(26069) lies on these lines: {2, 87}, {8, 192}, {9, 20667}, {37, 25311}, {69, 26105}, {966, 1575}, {2551, 4645}, {3226, 17321}, {3248, 25535}, {4704, 25292}, {6376, 7155}, {9780, 25624}, {10453, 17343}, {16706, 24753}, {17275, 24717}, {17300, 26103}, {17792, 27538}, {18194, 26143}, {25635, 26042}

X(26070) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26070) lies on these lines: {1, 8258}, {2, 45}, {8, 678}, {10, 9324}, {89, 17300}, {100, 958}, {244, 5550}, {966, 26071}, {1054, 3634}, {2246, 5273}, {3246, 5211}, {3722, 20050}, {4201, 9780}, {4438, 17601}, {16816, 17740}, {20072, 27757}, {24620, 25242}, {26044, 26081}

X(26071) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    8 a^3 - 5 a^2 b - 13 a b^2 - 5 a^2 c - 17 a b c - 9 b^2 c - 13 a c^2 - 9 b c^2 : :

X(26071) lies on these lines: {2, 44}, {333, 16672}, {966, 26070}, {3617, 23937}, {5302, 18231}

X(26072) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26072) lies on these lines: {2, 99}, {8, 4128}, {10, 5539}, {377, 26043}, {668, 21220}, {6625, 26752}, {7257, 16592}, {20349, 27044}, {26035, 26058}, {26041, 26056}, {26074, 26076}

X(26073) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26073) lies on these lines: {2, 11}, {8, 244}, {10, 1054}, {43, 17778}, {88, 1219}, {145, 3315}, {210, 26840}, {377, 26029}, {404, 20999}, {659, 26076}, {678, 5550}, {899, 4645}, {966, 20331}, {1086, 3699}, {1283, 25440}, {1635, 26074}, {1654, 21220}, {1698, 26117}, {1699, 27130}, {1738, 5205}, {1739, 16086}, {1836, 26791}, {1837, 25979}, {2246, 5749}, {2551, 26046}, {3120, 9458}, {3240, 17300}, {3616, 3722}, {3634, 9324}, {3820, 17678}, {3952, 4440}, {4188, 23843}, {4201, 9780}, {4383, 20101}, {4388, 16569}, {4427, 4473}, {4514, 16602}, {4689, 17263}, {4847, 27002}, {5082, 26093}, {5211, 16610}, {5296, 14439}, {5297, 17302}, {6702, 10774}, {8580, 27184}, {9350, 25957}, {9508, 26075}, {10327, 17490}, {17531, 23858}, {17719, 25351}, {17724, 27191}, {17777, 24003}, {17780, 24188}, {18141, 20012}, {19278, 19855}, {23833, 24193}, {26030, 26051}, {26034, 26038}, {26037, 26044}, {26047, 26065}

X(26073) = anticomplement of X(25531)

X(26074) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26074) lies on these lines: {2, 101}, {8, 2170}, {9, 11604}, {10, 5540}, {11, 644}, {41, 27529}, {80, 24036}, {149, 1018}, {169, 25005}, {218, 11681}, {220, 4193}, {355, 26690}, {391, 2316}, {672, 5080}, {1213, 3196}, {1635, 26073}, {1837, 25082}, {2161, 2345}, {2246, 9780}, {2265, 5749}, {2348, 5123}, {2475, 16549}, {3036, 4534}, {3207, 17566}, {3616, 17439}, {3730, 5046}, {3814, 5526}, {4253, 20060}, {5030, 20067}, {5086, 25066}, {5701, 21859}, {5750, 16554}, {5816, 12034}, {7291, 25007}, {9317, 24318}, {9956, 27068}, {15680, 24047}, {17181, 26653}, {17750, 26131}, {21053, 26075}, {21232, 24712}, {26072, 26076}

X(26074) = anticomplement of X(25532)

X(26075) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26075) lies on these lines: {2, 98}, {8, 2611}, {10, 21381}, {100, 21221}, {643, 8287}, {966, 2503}, {1158, 2475}, {1654, 3909}, {1793, 4189}, {9508, 26073}, {21053, 26074}

X(26075) = anticomplement of X(25533)

X(26076) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26076) lies on these lines: {2, 45}, {8, 3248}, {10, 9359}, {44, 26048}, {292, 2345}, {646, 1015}, {659, 26073}, {966, 26077}, {1654, 20355}, {2325, 25510}, {3271, 24485}, {3758, 26752}, {4033, 9263}, {5749, 26042}, {5750, 24578}, {6542, 26975}, {7240, 25140}, {17264, 26113}, {17300, 27136}, {17303, 26045}, {19951, 23774}, {20072, 27044}, {24487, 25048}, {26072, 26074}

X(26077) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26077) lies on these lines: {2, 37}, {10, 87}, {966, 26076}, {3617, 25293}, {3963, 27318}, {4110, 16604}, {5749, 26048}, {9780, 25624}, {17238, 20343}, {25631, 26046}, {26029, 26083}

X(26078) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26078) lies on these lines: {2, 900}, {522, 3582}, {659, 26073}, {665, 2345}, {1769, 25380}, {2254, 26031}, {2517, 23880}, {2815, 5657}, {3960, 4768}, {5749, 22108}, {13266, 24988}, {14315, 27342}

X(26079) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26079) lies on these lines: {2, 187}, {6, 17679}, {39, 17690}, {377, 966}, {649, 17072}, {754, 25468}, {1055, 21241}, {2475, 27040}, {3230, 21282}, {3285, 4202}, {5276, 17678}, {5300, 17299}, {6175, 26244}, {6781, 24956}, {7267, 25383}, {7745, 17674}, {7779, 16711}, {7784, 17683}, {16910, 26100}, {17300, 17680}, {17307, 17686}, {17345, 20880}, {17491, 21839}

X(26080) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26080) lies on these lines: {2, 647}, {8, 21719}, {10, 1021}, {649, 17072}, {650, 2517}, {652, 20316}, {661, 4581}, {966, 9404}, {2345, 3700}, {2522, 4391}, {2523, 17496}, {3239, 21186}, {4086, 16612}, {4397, 6591}, {4467, 19822}, {7252, 21721}, {8062, 8611}, {17924, 25009}, {18155, 19808}, {20293, 22383}

X(26081) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26081) lies on these lines: {2, 662}, {8, 2643}, {10, 2640}, {75, 1654}, {115, 645}, {148, 190}, {238, 20558}, {897, 3617}, {1213, 9509}, {2247, 26065}, {2652, 5794}, {3616, 17467}, {3758, 6625}, {3772, 17778}, {5207, 15994}, {9508, 26073}, {20072, 20349}, {21254, 24711}, {21277, 27321}, {26042, 26063}, {26044, 26070}, {26072, 26074}

X(26081) = anticomplement of X(25536)

X(26082) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26082) lies on these lines: {2, 7}, {8, 2309}, {192, 26801}, {966, 2235}, {1107, 17787}, {1654, 26752}, {3729, 17030}, {3758, 26110}, {3963, 21226}, {3986, 25510}, {4416, 27020}, {4431, 16829}, {4473, 27261}, {9780, 25624}, {16738, 17280}, {17249, 26142}, {17300, 27032}, {17303, 26045}, {17369, 27164}, {26113, 27268}, {26769, 26812}

X(26082) = anticomplement of X(25538)

X(26083) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    2 a^3 + 3 a^2 b + a b^2 + 2 b^3 + 3 a^2 c + a b c + 3 b^2 c + a c^2 + 3 b c^2 + 2 c^3 : :

X(26083) lies on these lines: {1, 17268}, {2, 38}, {7, 10588}, {8, 1386}, {10, 16468}, {44, 966}, {518, 17371}, {726, 17383}, {894, 1698}, {1757, 17238}, {2228, 26030}, {3616, 5772}, {3634, 3662}, {3740, 19827}, {3751, 17292}, {3758, 3844}, {3773, 4393}, {3790, 17023}, {3932, 17381}, {3967, 19812}, {4026, 17354}, {4078, 17397}, {4429, 17369}, {4649, 17230}, {4663, 17228}, {5220, 17307}, {5550, 17263}, {15569, 17342}, {19808, 26038}, {26029, 26077}

X(26083) = anticomplement of X(25539)

X(26084) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26084) lies on these lines: {2, 3}

X(26085) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 39

Barycentrics    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(26085) lies on these lines: {2, 32}, {4, 27040}, {6, 4202}, {8, 3721}, {10, 1759}, {37, 5300}, {41, 2887}, {69, 26978}, {76, 16910}, {213, 6327}, {377, 966}, {379, 1211}, {384, 16991}, {385, 16906}, {964, 1213}, {1334, 4660}, {1654, 17680}, {2225, 26034}, {2233, 26043}, {2237, 26042}, {2243, 9780}, {2271, 3936}, {2345, 5341}, {2476, 26244}, {2549, 26770}, {3686, 23536}, {3972, 16909}, {4201, 22380}, {4262, 25645}, {4372, 25345}, {4450, 14974}, {4643, 20880}, {4680, 16600}, {4805, 21240}, {4968, 17275}, {5015, 26242}, {5016, 16583}, {5051, 5275}, {5192, 7745}, {5224, 17686}, {5276, 16062}, {5283, 17676}, {5816, 15971}, {7737, 11319}, {7758, 18600}, {7774, 27162}, {7791, 27109}, {9596, 26030}, {9599, 26094}, {16589, 22430}, {16908, 17003}, {16998, 17673}, {17259, 17672}, {17330, 17679}, {20553, 27248}, {22426, 26117}, {24586, 24995}, {26961, 27039}

X(26085) = anticomplement of X(25497)

X(26086) =  X(1)X(3)∩X(5)X(24042)

Barycentrics    a^2 (2 a^5-2 a^4 (b+c)-4 a^3 (b^2-b c+c^2)-(b-c)^2 (2 b^3+3 b^2 c+3 b c^2+2 c^3)+a^2 (4 b^3+b^2 c+b c^2+4 c^3)+2 a (b^4-2 b^3 c+b^2 c^2-2 b c^3+c^4)) : :
X(26086) = 3*X[376]+X[10526], X[548]-X[5841], 5*X[631]-X[10525]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28545.

X(26086) lies on these lines: {1,3}, {5,24042}, {21,11231}, {140,3825}, {186,1872}, {355,6950}, {376,10526}, {404,11230}, {548,5841}, {631,10525}, {4188,5886}, {4302,6958}, {4996,10914}, {5267,5690}, {5428,10164}, {5440,5694}, {5657,17548}, {5881,18515}, {5887,17100}, {6684,7508}, {6713,15171}, {6833,18407}, {6842,24466}, {6882,15338}, {6905,22793}, {6906,18480}, {6914,9956}, {6924,9955}, {6935,18517}, {6942,12699}, {10572,12619}, {10993,24390}, {12515,21740}

X(26086) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,23961}, {3,35,1385}, {3,1482,7280}, {3,2077,3579}, {3,10679,5204}, {3,10902,17502}, {3,11849,36}, {35,14792,3057}, {36,11849,10222}, {6914,25440,9956}

X(26087) =  X(1)X(3)∩X(10)X(19907)

Barycentrics    a (2 a^6-4 a^5 b-2 a^4 b^2+8 a^3 b^3-2 a^2 b^4-4 a b^5+2 b^6-4 a^5 c+12 a^4 b c-9 a^3 b^2 c-8 a^2 b^3 c+13 a b^4 c-4 b^5 c-2 a^4 c^2-9 a^3 b c^2+22 a^2 b^2 c^2-9 a b^3 c^2-2 b^4 c^2+8 a^3 c^3-8 a^2 b c^3-9 a b^2 c^3+8 b^3 c^3-2 a^2 c^4+13 a b c^4-2 b^2 c^4-4 a c^5-4 b c^5+2 c^6) : :
X(26087) = X[952]-X[24387], 3*X[7967]+X[10525], X[10526]-5*X[10595]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28548.

X(26087) lies on these lines: {1,3}, {10,19907}, {952,24387}, {1317,6842}, {3825,5901}, {4861,6265}, {5154,5886}, {5882,21630}, {7967,10525}, {10526,10595}, {10914,22935}, {11230,17619}, {12737,21740}

X(26087) = reflection of X(11567) in X(1)
X(26087) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1385,10222,3057}, {13145,15178,1385}, {21842,25413,23961}

X(26088) =  MIDPOINT OF X(6583) AND X(12672)

Barycentrics    -a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-4 a^4 b c+9 a^3 b^2 c+6 a^2 b^3 c-10 a b^4 c-2 b^5 c-a^4 c^2+9 a^3 b c^2-18 a^2 b^2 c^2+9 a b^3 c^2+b^4 c^2-2 a^3 c^3+6 a^2 b c^3+9 a b^2 c^3+4 b^3 c^3+2 a^2 c^4-10 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6) : :
X(26088) = 7*X[1385]-3*X[5918], 3*X[1699]+X[10284], X[2771]-X[3881], 9*X[3656]-X[3868], 5*X[3890]+3*X[12699], 5*X[5439]-3*X[13145], 3*X[5603]-X[5885], X[6583]+X[12672]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28549.

X(26088) lies on these lines: {1,22461}, {5,10}, {149,18480}, {1385,5918}, {1621,13624}, {1699,10284}, {2771,3881}, {3585,9957}, {3656,3868}, {3890,12699}, {5180,12600}, {5439,13145}, { 5603,5885}, {6264,10222}, {6583, 12672}, {15178,18444}, {16160, 21630}

X(26088) = midpoint X(6583) and X(12672)

X(26089) =  MIDPOINT OF X(944) AND X(5885)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+12 a^4 b c-7 a^3 b^2 c-10 a^2 b^3 c+6 a b^4 c-2 b^5 c-a^4 c^2-7 a^3 b c^2+14 a^2 b^2 c^2-7 a b^3 c^2+b^4 c^2-2 a^3 c^3-10 a^2 b c^3-7 a b^2 c^3+4 b^3 c^3+2 a^2 c^4+6 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6) : :
X(26089) = X[517]-X[550], X[944]+X[5885], X[2771]-X[3884], 9*X[3655]-X[3869], 5*X[3889]+3*X[18481], 2*X[4540]-3*X[11812]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28549.

X(26089) lies on these lines: {1,22461}, {517,550}, {944,5885}, {1385,5251}, {2771,3884}, { 2975,4420}, {3655,3869}, {3889,18481}, {4540,11812}, {4857, 5049}, {5045,5434}, {6224,10914}, {6912,15178}

X(26089) = midpoint X(944) and X(5885)

X(26090) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 39

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

X(26090) lies on these lines: {2, 3}

leftri

Collineation mappings involving Gemini triangle 40: X(26091)-X(26152)

rightri

Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 40, as in centers X(26091)-X(26152). Then

m(X) = b c (a + b + c) x + a c (a - b + c) y + a b (a + b - c) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 29, 2018)

X(26091) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26091) lies on these lines: {1, 14058}, {2, 3}, {31, 3075}, {92, 17102}, {388, 26095}, {1457, 3616}, {1465, 5342}, {1936, 10527}, {3085, 10448}, {4512, 19863}, {4652, 27339}, {5433, 20992}, {13411, 27287}, {14986, 15501}, {20256, 23085}, {26094, 26129}, {26105, 26126}, {26106, 26108}

X(26092) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26092) lies on these lines: {2, 3}, {499, 595}, {3193, 14829}, {3616, 26095}, {3897, 26115}

X(26093) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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(26093) lies on these lines: {1, 2}, {40, 27002}, {56, 17697}, {346, 17053}, {740, 27343}, {958, 25531}, {982, 19582}, {999, 13741}, {1001, 19278}, {1284, 7288}, {1463, 11375}, {2275, 27523}, {2478, 26139}, {3333, 27064}, {3672, 27162}, {3701, 17480}, {3702, 17490}, {3976, 25079}, {4195, 5253}, {4657, 24668}, {4673, 16610}, {5082, 26073}, {5084, 5484}, {9335, 17164}, {9669, 17678}, {11319, 19769}, {15717, 26997}, {16342, 27145}, {16738, 17557}, {17279, 24739}, {20530, 24652}, {21075, 27130}, {22220, 24349}, {26105, 26117}, {26116, 26129}, {26143, 26150}

X(26094) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26094) lies on these lines: {1, 2}, {11, 4202}, {36, 11319}, {38, 25079}, {56, 5192}, {238, 27145}, {350, 27162}, {404, 23383}, {474, 24552}, {496, 4972}, {740, 27311}, {964, 25524}, {982, 25591}, {1284, 5433}, {2275, 27040}, {2345, 8610}, {2886, 17674}, {2975, 13741}, {3338, 26223}, {3670, 25253}, {3702, 3752}, {3760, 18600}, {3777, 4874}, {3816, 5051}, {3923, 27017}, {3953, 17165}, {4054, 24171}, {4423, 16342}, {4645, 26133}, {4646, 4742}, {5047, 25531}, {5253, 13740}, {5259, 16347}, {5263, 17531}, {5284, 19270}, {5300, 17721}, {5303, 13735}, {5482, 11230}, {7280, 17539}, {7288, 17526}, {7483, 24542}, {9599, 26085}, {10483, 17537}, {11375, 26126}, {16468, 17178}, {16604, 26035}, {17164, 24046}, {17184, 21616}, {20244, 24170}, {20530, 24668}, {22220, 24325}, {23541, 25877}, {26091, 26129}, {26107, 26108}, {26117, 26127}, {26123, 26132}

X(26095) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26095) lies on these lines: {2, 11}, {4, 15666}, {56, 27506}, {86, 6649}, {388, 26091}, {406, 10321}, {1066, 14058}, {1361, 3485}, {1769, 3716}, {1846, 4194}, {2551, 25513}, {3041, 25568}, {3085, 25490}, {3086, 16483}, {3616, 26092}, {3952, 24433}, {4858, 24025}, {5136, 8069}, {6712, 20266}, {9371, 26011}, {10523, 11105}

X(26096) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26096) lies on these lines: {2, 3}, {192, 497}, {614, 3944}, {1352, 3794}, {1479, 3705}, {1853, 26530}, {3421, 20056}, {4388, 7155}, {7295, 27512}

X(26096) = anticomplement of X(37099)

X(26097) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26097) lies on these lines: {2, 3}, {3120, 7292}, {3837, 26148}, {4459, 5057}, {5211, 5992}, {24436, 27548}

X(26098) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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(26098) lies on these lines: {1, 4}, {2, 31}, {3, 21321}, {5, 5711}, {6, 2886}, {7, 256}, {8, 1215}, {10, 14555}, {11, 940}, {12, 5710}, {36, 19262}, {37, 24703}, {38, 5905}, {40, 5530}, {42, 3434}, {43, 2550}, {55, 4192}, {56, 9840}, {57, 6210}, {58, 26363}, {63, 24695}, {69, 3741}, {81, 11269}, {142, 1716}, {149, 17018}, {193, 21242}, {213, 26036}, {221, 15844}, {329, 984}, {330, 6625}, {344, 4011}, {345, 3923}, {354, 1463}, {377, 1193}, {390, 3750}, {442, 16466}, {443, 978}, {498, 5264}, {499, 26126}, {511, 10473}, {516, 17594}, {517, 5725}, {553, 18193}, {595, 10198}, {601, 6833}, {602, 6889}, {612, 908}, {614, 5249}, {846, 5698}, {870, 7018}, {894, 3705}, {975, 21616}, {986, 4295}, {988, 4292}, {1001, 4199}, {1008, 18134}, {1036, 4185}, {1125, 4138}, {1191, 25466}, {1203, 1714}, {1245, 12609}, {1386, 3772}, {1460, 19544}, {1468, 10527}, {1582, 4212}, {1707, 5745}, {1724, 19854}, {1738, 2999}, {1836, 3666}, {1909, 21590}, {1985, 19734}, {2099, 5724}, {2177, 20075}, {2292, 11415}, {2295, 9596}, {2308, 24597}, {2345, 4071}, {2476, 5230}, {2548, 17750}, {2650, 12649}, {3006, 26223}, {3052, 6690}, {3072, 6825}, {3073, 6824}, {3085, 5255}, {3086, 4340}, {3120, 17017}, {3195, 25985}, {3333, 28039}, {3421, 20498}, {3436, 10459}, {3452, 5268}, {3474, 17596}, {3550, 5218}, {3616, 4892}, {3618, 21241}, {3664, 11019}, {3672, 17600}, {3677, 4654}, {3742, 4675}, {3744, 17718}, {3745, 17605}, {3749, 13405}, {3751, 4847}, {3752, 5880}, {3782, 17599}, {3817, 4349}, {3840, 18141}, {3886, 4028}, {3914, 5256}, {3925, 4383}, {3931, 12699}, {3936, 24552}, {3945, 4038}, {3961, 25568}, {4000, 17889}, {4220, 5329}, {4224, 7295}, {4252, 4999}, {4300, 6836}, {4310, 17598}, {4331, 17080}, {4339, 5703}, {4344, 5226}, {4362, 25385}, {4392, 17483}, {4417, 5263}, {4418, 17740}, {4423, 16850}, {4425, 17321}, {4438, 4672}, {4644, 24333}, {4648, 20335}, {4650, 5744}, {4655, 6682}, {4660, 6685}, {4667, 24386}, {4682, 5087}, {4703, 17257}, {4716, 20043}, {4850, 20292}, {4854, 20182}, {4888, 10980}, {5018, 7365}, {5121, 5437}, {5219, 5269}, {5247, 19843}, {5266, 11374}, {5273, 7262}, {5292, 25639}, {5297, 27131}, {5573, 6173}, {5706, 15908}, {5847, 11679}, {6284, 19765}, {6871, 21935}, {6872, 10448}, {7083, 25514}, {7226, 17484}, {7290, 25525}, {7292, 27186}, {7736, 17754}, {8167, 17245}, {8731, 20992}, {9599, 24512}, {9776, 17063}, {9778, 17601}, {9812, 17592}, {10453, 17778}, {10458, 14956}, {10480, 15488}, {10578, 17715}, {11246, 17595}, {11263, 24159}, {11433, 26013}, {11512, 12436}, {16475, 17064}, {16478, 24161}, {16569, 26040}, {17300, 21299}, {17314, 21101}, {17469, 26228}, {17724, 17775}, {17732, 25092}, {18067, 18135}, {18201, 21454}, {19725, 21015}, {20011, 21283}, {20430, 21333}, {20964, 27254}, {26099, 26101}, {26103, 26139}, {26107, 26133}

X(26099) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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(26099) lies on these lines: {1, 5074}, {2, 32}, {4, 26978}, {8, 4950}, {31, 17046}, {69, 27040}, {86, 17550}, {116, 5264}, {141, 5192}, {213, 21285}, {316, 16910}, {857, 940}, {1572, 26526}, {2478, 4648}, {2549, 18600}, {3701, 4851}, {3915, 17062}, {4056, 16600}, {4202, 7784}, {4766, 24549}, {4911, 26242}, {5051, 15668}, {5276, 17671}, {6327, 21240}, {7758, 26770}, {7768, 17007}, {7774, 27109}, {7791, 27162}, {7795, 11319}, {7832, 16909}, {7885, 16906}, {7901, 17003}, {7931, 16905}, {7939, 16991}, {17234, 17541}, {17300, 18135}, {18635, 27378}, {20553, 27299}, {26098, 26101}, {26108, 26124}

X(26100) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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(26100) lies on these lines: {2, 39}, {37, 20435}, {38, 17048}, {75, 27026}, {86, 17541}, {141, 5051}, {1015, 27146}, {1500, 27096}, {2478, 4648}, {3214, 3946}, {3616, 24654}, {3634, 24790}, {3701, 3739}, {3734, 11115}, {3954, 20247}, {4000, 9780}, {4657, 26115}, {5192, 15668}, {5254, 17672}, {5276, 17681}, {6376, 26965}, {6381, 16818}, {7800, 17676}, {16020, 16846}, {16910, 26079}, {16975, 26964}, {17234, 17550}, {17302, 26752}, {17382, 25107}, {17750, 20347}, {18139, 26601}, {20530, 24668}, {24254, 25253}, {26124, 26138}, {27008, 27302}

X(26101) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^4 - 2 a^2 b^2 + 2 a b^3 - b^4 - 2 a^2 b c - 2 a b^2 c + 2 b^3 c - 2 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(26101) lies on these lines: {2, 41}, {7, 21808}, {55, 21258}, {57, 14021}, {69, 3691}, {116, 498}, {142, 377}, {226, 4350}, {277, 3488}, {388, 1458}, {405, 3423}, {1086, 9598}, {1334, 6604}, {1475, 14548}, {1478, 17758}, {1479, 2140}, {1837, 6706}, {2478, 20335}, {3295, 4904}, {3434, 17050}, {3486, 9317}, {3616, 26140}, {3720, 7386}, {3785, 24602}, {4059, 4675}, {4258, 26007}, {4302, 14377}, {5554, 21232}, {5722, 24774}, {5738, 27626}, {7247, 27475}, {10200, 25532}, {17170, 17451}, {18639, 25907}, {20269, 24929}, {21240, 26034}, {26098, 26099}, {26102, 26118}

X(26102) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26102) lies on these lines: {1, 2}, {6, 4038}, {7, 25421}, {9, 24512}, {31, 5284}, {33, 4212}, {34, 4213}, {35, 4191}, {36, 1011}, {37, 982}, {38, 22220}, {55, 16059}, {56, 16058}, {57, 846}, {81, 748}, {86, 87}, {100, 17124}, {142, 4335}, {171, 1001}, {192, 24165}, {210, 4883}, {226, 4334}, {238, 940}, {244, 2108}, {291, 3677}, {310, 3760}, {312, 24325}, {320, 4703}, {350, 10436}, {354, 984}, {388, 6822}, {405, 19715}, {497, 6821}, {672, 3731}, {740, 19804}, {750, 1621}, {756, 3873}, {851, 3612}, {894, 4011}, {968, 3306}, {986, 5439}, {988, 1009}, {991, 3817}, {1010, 19803}, {1044, 12047}, {1054, 5437}, {1197, 21001}, {1203, 16355}, {1215, 18743}, {1279, 4682}, {1376, 3750}, {1385, 19540}, {1449, 2238}, {1458, 5226}, {1464, 11375}, {1468, 5047}, {1478, 6818}, {1479, 6817}, {1575, 16777}, {1613, 23660}, {1695, 10441}, {1699, 1742}, {1716, 17306}, {1721, 10857}, {1724, 19714}, {1740, 15668}, {1745, 1985}, {1757, 3305}, {1962, 4850}, {2162, 23417}, {2275, 21838}, {2276, 3247}, {2293, 5274}, {2308, 14996}, {2309, 25528}, {2356, 8889}, {2667, 4751}, {2886, 17245}, {2887, 17234}, {2979, 20961}, {3094, 22200}, {3120, 27186}, {3136, 7741}, {3210, 3993}, {3295, 16409}, {3303, 16421}, {3510, 20530}, {3576, 4192}, {3601, 16056}, {3662, 4425}, {3666, 17063}, {3670, 27785}, {3683, 4650}, {3685, 3980}, {3696, 4891}, {3736, 25507}, {3743, 24046}, {3751, 7308}, {3752, 3848}, {3761, 18152}, {3795, 20182}, {3816, 17056}, {3819, 21746}, {3835, 24666}, {3846, 18134}, {3919, 17461}, {3931, 24174}, {3936, 25960}, {3944, 5249}, {3945, 25572}, {3971, 24349}, {3989, 4392}, {3995, 17155}, {4040, 4379}, {4104, 4684}, {4184, 7280}, {4199, 5436}, {4203, 5253}, {4204, 5429}, {4210, 5010}, {4322, 5261}, {4356, 24175}, {4364, 24691}, {4383, 4649}, {4389, 25422}, {4414, 27003}, {4415, 25557}, {4418, 26627}, {4430, 9330}, {4441, 25590}, {4465, 9359}, {4648, 20335}, {4653, 13588}, {4656, 24231}, {4670, 4713}, {4675, 24703}, {4676, 4697}, {4888, 20347}, {4966, 5743}, {4970, 17490}, {4972, 25961}, {5247, 11108}, {5259, 16343}, {5275, 16503}, {5276, 16779}, {5333, 10458}, {5563, 16373}, {6688, 23638}, {7226, 17449}, {7262, 15254}, {7322, 16496}, {8025, 18192}, {8543, 9316}, {9347, 17469}, {9776, 24248}, {10013, 17259}, {10439, 21363}, {10476, 13731}, {10589, 14547}, {11358, 25524}, {11451, 20962}, {15950, 24806}, {16478, 16846}, {16589, 21384}, {16678, 19341}, {16884, 21904}, {17149, 18140}, {17182, 17194}, {17263, 24736}, {17321, 25420}, {17394, 18194}, {17445, 24766}, {17793, 25531}, {18139, 25760}, {18173, 20984}, {18197, 25537}, {20284, 21827}, {20923, 25124}, {24406, 24495}, {26101, 26118}, {26109, 26139}, {26127, 26131}

X(26102) = {X(1),X(2)}-harmonic conjugate of X(43)

X(26103) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^2 b - a b^2 + a^2 c + 7 a b c - b^2 c - a c^2 - b c^2 : :

X(26103) lies on these lines: {1, 2}, {7, 24495}, {75, 3848}, {100, 16409}, {192, 17063}, {354, 27538}, {940, 25531}, {968, 27002}, {982, 22220}, {1100, 24753}, {1284, 5435}, {3161, 17754}, {3685, 5437}, {3742, 3967}, {3816, 17234}, {3846, 17232}, {3995, 9335}, {4645, 26105}, {4648, 20530}, {4704, 17591}, {4734, 16610}, {5080, 6822}, {5731, 19540}, {6384, 18135}, {6682, 27268}, {8167, 14829}, {12014, 17777}, {17261, 18193}, {17300, 26069}, {17317, 25311}, {26098, 26139}

X(26104) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26104) lies on these lines: {2, 45}, {7, 17384}, {8, 17382}, {10, 4000}, {69, 17383}, {141, 145}, {142, 3624}, {344, 17324}, {346, 17323}, {599, 17014}, {966, 16706}, {1266, 2345}, {1279, 3616}, {1633, 4423}, {1698, 17067}, {3008, 4748}, {3617, 4395}, {3618, 17236}, {3619, 17230}, {3620, 17380}, {3622, 17313}, {3632, 3946}, {3635, 17296}, {3636, 21255}, {3672, 3763}, {3739, 19877}, {4029, 17284}, {4361, 4678}, {4371, 4668}, {4373, 7227}, {4393, 21356}, {4402, 17239}, {4445, 20052}, {4452, 17293}, {4700, 17272}, {4747, 7238}, {4851, 20057}, {4852, 20053}, {4869, 17045}, {5084, 15434}, {5222, 17237}, {5232, 17366}, {5296, 17356}, {5550, 24723}, {5749, 17235}, {6361, 12610}, {15668, 16347}, {17227, 26626}, {17244, 17291}, {17249, 26685}, {17251, 24599}, {17257, 17370}, {17316, 17399}, {17318, 20582}, {17395, 21358}, {24248, 25539}

X(26105) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 - a^2 b + a b^2 - b^3 - a^2 c - 6 a b c + b^2 c + a c^2 + b c^2 - c^3 : :

X(26105) lies on these lines: {1, 2551}, {2, 11}, {3, 7956}, {4, 1125}, {7, 3742}, {8, 3740}, {9, 11019}, {10, 1058}, {12, 6919}, {20, 25524}, {21, 1470}, {35, 17567}, {36, 11111}, {37, 7736}, {40, 9843}, {56, 452}, {57, 5698}, {69, 26069}, {85, 2898}, {104, 6976}, {142, 1699}, {144, 17051}, {165, 6692}, {200, 5316}, {226, 4321}, {329, 354}, {344, 3705}, {377, 5225}, {388, 1319}, {392, 18391}, {405, 3086}, {442, 10591}, {443, 1479}, {474, 4294}, {496, 11108}, {499, 5259}, {515, 6939}, {516, 5437}, {518, 10580}, {527, 10980}, {551, 1056}, {631, 2077}, {748, 11269}, {908, 3475}, {936, 3189}, {938, 960}, {944, 6898}, {946, 6865}, {950, 8583}, {958, 5129}, {962, 3812}, {966, 3741}, {982, 4419}, {997, 2900}, {1000, 3898}, {1385, 6893}, {1478, 25055}, {1486, 19649}, {1697, 8582}, {1698, 5082}, {1706, 12575}, {1750, 10863}, {1788, 5250}, {1836, 9776}, {1997, 7081}, {2267, 25496}, {2975, 10586}, {3085, 4187}, {3090, 3825}, {3091, 7958}, {3158, 20103}, {3219, 15297}, {3243, 21060}, {3295, 17527}, {3303, 7080}, {3305, 26015}, {3306, 3474}, {3333, 12572}, {3436, 3622}, {3486, 19861}, {3487, 21616}, {3523, 6691}, {3545, 3822}, {3582, 17561}, {3663, 5573}, {3677, 4656}, {3683, 5744}, {3711, 20015}, {3720, 5712}, {3755, 23511}, {3772, 16020}, {3789, 10453}, {3814, 8164}, {3820, 6767}, {3838, 9779}, {3847, 5056}, {3848, 5880}, {3884, 12245}, {3890, 5554}, {3911, 4512}, {3967, 8055}, {3974, 4358}, {4000, 5272}, {4193, 10588}, {4293, 11113}, {4295, 5439}, {4305, 17614}, {4310, 4415}, {4314, 5438}, {4344, 4682}, {4388, 18141}, {4425, 4466}, {4640, 5435}, {4645, 26103}, {4648, 20335}, {4657, 26118}, {4662, 6764}, {4847, 7308}, {4860, 9965}, {4999, 17558}, {5046, 5229}, {5047, 10527}, {5121, 17594}, {5154, 10585}, {5177, 10896}, {5204, 17576}, {5249, 8544}, {5251, 10072}, {5253, 6872}, {5260, 10529}, {5265, 11106}, {5273, 15254}, {5328, 10578}, {5333, 14956}, {5603, 6947}, {5657, 10596}, {5687, 17575}, {5703, 25681}, {5704, 26066}, {5731, 6957}, {5748, 17718}, {5758, 13374}, {5804, 14110}, {5809, 17604}, {5811, 12675}, {5818, 10806}, {5836, 9785}, {5853, 8580}, {5886, 6827}, {6284, 6904}, {6326, 7967}, {6601, 6666}, {6668, 7486}, {6738, 15829}, {6744, 11523}, {6745, 10389}, {6762, 18250}, {6821, 25501}, {6826, 11230}, {6851, 9955}, {6856, 7741}, {6887, 26470}, {6892, 26492}, {6899, 12609}, {6902, 10532}, {6908, 7681}, {6916, 10165}, {6920, 10785}, {6926, 11496}, {6927, 10902}, {6929, 22799}, {6930, 10269}, {6937, 10598}, {6944, 10267}, {6964, 11500}, {6965, 12115}, {6975, 10786}, {6981, 26487}, {6983, 11491}, {6987, 22753}, {7179, 17321}, {7226, 24433}, {7292, 19785}, {7738, 16604}, {8165, 12607}, {8728, 9669}, {9709, 15172}, {9957, 17648}, {10177, 11018}, {10587, 11681}, {10590, 17556}, {10855, 17668}, {11934, 26695}, {12447, 12625}, {13411, 25522}, {15171, 16408}, {15296, 27065}, {15325, 16418}, {16842, 19855}, {16845, 26363}, {17063, 24248}, {17123, 24217}, {17183, 18165}, {17552, 19854}, {17582, 19862}, {17768, 21454}, {24954, 27383}, {26091, 26126}, {26093, 26117}

X(26106) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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 - 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(26106) lies on these lines: {2, 6}, {37, 24652}, {75, 24663}, {304, 20227}, {322, 25975}, {941, 27162}, {1449, 27299}, {1463, 11375}, {2345, 24654}, {5749, 27097}, {5750, 27248}, {17303, 24656}, {17754, 27264}, {20255, 21785}, {21281, 21769}, {24549, 27332}, {25521, 26959}, {26091, 26108}, {26122, 26138}, {27343, 27487}

X(26107) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    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(26107) lies on these lines: {1, 21257}, {2, 37}, {6, 27262}, {8, 21238}, {9, 26959}, {76, 17053}, {330, 3770}, {384, 2178}, {583, 17350}, {672, 27158}, {966, 16525}, {1001, 19312}, {1108, 25994}, {1125, 5145}, {1213, 16515}, {1269, 24621}, {1463, 11375}, {1740, 24717}, {1964, 21299}, {2092, 20170}, {2260, 24514}, {2321, 27091}, {3247, 27020}, {3616, 26110}, {3763, 25534}, {4272, 4393}, {4277, 20168}, {4357, 25369}, {4361, 27111}, {4389, 26979}, {4648, 26113}, {4741, 17178}, {5257, 17030}, {5301, 7793}, {5749, 27019}, {10436, 25510}, {12263, 17065}, {16831, 25538}, {17144, 21857}, {17230, 27095}, {17236, 27145}, {17314, 26752}, {17373, 26756}, {17379, 27166}, {17719, 24653}, {20271, 23481}, {24520, 25688}, {24667, 25504}, {24672, 26135}, {25079, 27680}, {26094, 26108}, {26098, 26133}, {26119, 26132}, {26130, 26147}

X(26108) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26108) lies on these lines: {2, 39}, {7, 26986}, {2478, 26138}, {21071, 27105}, {21384, 26974}, {26091, 26106}, {26094, 26107}, {26099, 26124}

X(26109) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26109) lies on these lines: {1, 26051}, {2, 6}, {8, 27798}, {148, 15903}, {226, 6625}, {329, 27268}, {497, 2475}, {846, 23812}, {1125, 1330}, {1655, 5308}, {2893, 25525}, {2999, 27147}, {3151, 17134}, {3616, 4892}, {3666, 26806}, {3770, 18743}, {3772, 17394}, {3882, 5437}, {4208, 19783}, {4473, 26223}, {4654, 17247}, {4658, 25446}, {4798, 19827}, {5249, 17302}, {5253, 21321}, {5550, 26064}, {6542, 24656}, {6999, 10478}, {9791, 10180}, {11110, 20077}, {11679, 17391}, {16736, 24530}, {17032, 20533}, {17396, 23681}, {17397, 25527}, {19786, 24663}, {25526, 25650}, {25660, 27792}, {26102, 26139}, {26119, 26125}, {26136, 26147}

X(26110) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 b^2 + a^2 b^3 + 3 a^3 b c + 3 a^2 b^2 c + a b^3 c + a^3 c^2 + 3 a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 : :

X(26110) lies on these lines: {2, 6}, {9, 17499}, {10, 2663}, {37, 1655}, {71, 894}, {256, 25124}, {257, 2294}, {274, 2092}, {388, 1284}, {870, 17321}, {941, 1218}, {1030, 17693}, {1100, 26801}, {1449, 17030}, {1966, 2345}, {2305, 17103}, {2550, 26051}, {3616, 26107}, {3686, 16819}, {3758, 26082}, {3882, 10436}, {4254, 11321}, {4441, 20170}, {4645, 26115}, {4657, 26142}, {5484, 16684}, {5750, 27020}, {13588, 22369}, {16709, 24530}, {16752, 25470}, {17023, 25538}, {17303, 26752}, {17322, 24663}, {19581, 25054}, {24325, 24478}, {26068, 27382}, {26121, 26134}

X(26110) = anticomplement of X(27164)

X(26111) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 b + 2 a^2 b^2 + a b^3 + a^3 c - 11 a^2 b c - 3 a b^2 c + b^3 c + 2 a^2 c^2 - 3 a b c^2 + 2 b^2 c^2 + a c^3 + b c^3 : :

X(26111) lies on these lines: {1, 2}, {346, 16604}, {388, 26139}, {1284, 5265}, {3304, 25531}, {3333, 17350}, {3976, 22220}, {4461, 27318}, {4719, 27343}, {11110, 17178}, {17480, 18743}, {20530, 24654}, {24669, 26143}

X(26112) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    3 a^3 - a^2 b + 5 a b^2 + b^3 - a^2 c - 14 a b c + 3 b^2 c + 5 a c^2 + 3 b c^2 + c^3 : :

X(26112) lies on these lines: {1, 2}, {346, 5573}, {461, 1878}, {982, 3161}, {3742, 5749}, {3967, 15590}, {4011, 4488}, {4310, 8055}, {5274, 17282}, {5296, 8167}, {5423, 17597}, {11037, 13741}, {18228, 25531}, {26132, 26139}

X(26113) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26113) lies on these lines: {1, 2}, {335, 22220}, {2275, 18743}, {3619, 25535}, {3834, 26142}, {3975, 9263}, {4366, 11349}, {4473, 26975}, {4648, 26107}, {5749, 27291}, {17264, 26076}, {17390, 27111}, {26082, 27268}

X(26114) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    (b - c) (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(26114) lies on these lines: {2, 650}, {37, 21611}, {192, 21438}, {513, 24674}, {514, 27527}, {647, 21225}, {649, 17204}, {661, 27265}, {812, 27345}, {3261, 6589}, {3310, 24622}, {3716, 4017}, {3766, 7180}, {3837, 24533}, {4147, 25637}, {4449, 25128}, {7234, 21301}, {8640, 23818}, {16754, 17496}, {17379, 22383}, {17383, 25603}, {20293, 24718}, {20295, 26983}, {27013, 27167}

X(26115) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26115) lies on these lines: {1, 2}, {12, 1284}, {21, 1220}, {35, 11115}, {37, 3701}, {55, 964}, {65, 22325}, {71, 5749}, {100, 1010}, {227, 1441}, {313, 17321}, {321, 3931}, {406, 7102}, {442, 4972}, {495, 13728}, {941, 2345}, {956, 19273}, {958, 16342}, {993, 16347}, {1001, 5192}, {1089, 3743}, {1215, 2292}, {1319, 26126}, {1376, 16454}, {1478, 17676}, {1621, 13740}, {1788, 17077}, {1826, 4194}, {1869, 4200}, {1909, 16705}, {2049, 5687}, {2269, 5750}, {2276, 26035}, {2901, 27804}, {2975, 19270}, {3295, 24552}, {3436, 13725}, {3666, 4968}, {3670, 17140}, {3698, 22313}, {3728, 3842}, {3868, 22275}, {3871, 5263}, {3877, 22299}, {3896, 5295}, {3897, 26092}, {3915, 25496}, {4160, 27114}, {4197, 4429}, {4202, 25466}, {4205, 17757}, {4358, 6051}, {4424, 17164}, {4645, 26110}, {4647, 4868}, {4649, 16738}, {4657, 26100}, {4658, 27163}, {4754, 25349}, {4761, 26983}, {5016, 5725}, {5080, 5143}, {5125, 17913}, {5217, 16393}, {5247, 10457}, {5248, 11319}, {5251, 17588}, {5260, 11110}, {5284, 13741}, {5686, 22312}, {5711, 19684}, {5793, 19765}, {7148, 27033}, {9709, 16458}, {9711, 15571}, {9782, 26806}, {12514, 26223}, {13407, 17184}, {16346, 27410}, {17175, 24170}, {17303, 21858}, {17529, 24988}, {17551, 25508}, {18600, 25599}, {19284, 25440}, {20005, 27918}, {20133, 27169}, {21077, 26580}, {21727, 26049}, {22279, 22281}, {22300, 26028}, {24325, 24443}, {25092, 26770}, {25107, 25498}

X(26115) = anticomplement of X(19863)

X(26116) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26116) lies on these lines: {2, 3}, {41, 27508}, {1458, 3616}, {1468, 14986}, {4512, 19853}, {11415, 17950}, {26093, 26129}

X(26117) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26117) lies on these lines: {1, 1330}, {2, 3}, {8, 192}, {10, 846}, {34, 17086}, {37, 7270}, {65, 17950}, {81, 20077}, {145, 2895}, {148, 1281}, {149, 12746}, {153, 13265}, {333, 1834}, {355, 9959}, {388, 1284}, {497, 8240}, {500, 18465}, {515, 8235}, {519, 11533}, {540, 4658}, {938, 9852}, {942, 26840}, {950, 2893}, {958, 27319}, {966, 27523}, {1043, 1211}, {1056, 11043}, {1104, 19786}, {1210, 24627}, {1220, 4026}, {1245, 17016}, {1283, 5248}, {1503, 25898}, {1697, 3882}, {1698, 26073}, {1837, 17611}, {2345, 9598}, {2550, 26045}, {2551, 18235}, {2650, 4683}, {2652, 5794}, {3421, 13097}, {3436, 11688}, {3454, 4653}, {3583, 19863}, {3616, 4892}, {3710, 17261}, {3757, 13161}, {3868, 6646}, {3890, 3909}, {3897, 26141}, {3914, 16824}, {3951, 17333}, {4255, 5233}, {4417, 19765}, {4418, 27714}, {4972, 5260}, {4981, 5178}, {5080, 5143}, {5208, 10381}, {5250, 6210}, {5262, 17302}, {5263, 6284}, {5296, 21811}, {5436, 25527}, {5691, 8245}, {5711, 20101}, {5716, 17321}, {5739, 20018}, {6625, 18757}, {9579, 10436}, {9780, 17601}, {9843, 27002}, {10025, 12527}, {10448, 25760}, {11518, 17274}, {12247, 12770}, {12567, 19853}, {12572, 27064}, {16817, 23537}, {16823, 23536}, {19785, 19851}, {22426, 26085}, {25531, 25914}, {26093, 26105}, {26094, 26127}, {27410, 27547}

X(26118) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26118) lies on these lines: {1, 8900}, {2, 3}, {7, 26929}, {69, 24523}, {81, 6776}, {355, 10327}, {388, 1455}, {497, 3666}, {511, 5739}, {515, 612}, {516, 3980}, {614, 946}, {940, 1503}, {944, 3920}, {952, 20020}, {980, 8721}, {1029, 7612}, {1038, 1891}, {1040, 1848}, {1211, 1350}, {1333, 7735}, {1479, 24239}, {1482, 19993}, {1486, 23304}, {1699, 1721}, {1714, 7683}, {2807, 17617}, {2886, 11677}, {3011, 26332}, {3421, 7172}, {3434, 3705}, {3436, 7081}, {4261, 7736}, {4383, 5480}, {4425, 24728}, {4657, 26105}, {5268, 5691}, {5273, 26939}, {5322, 5450}, {5603, 7191}, {5928, 10391}, {7179, 21279}, {10532, 26228}, {10595, 17024}, {12588, 20359}, {17810, 26005}, {20368, 26034}, {23291, 26540}, {24320, 27540}, {26101, 26102}

X(26119) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26119) lies on these lines: {2, 3}, {92, 18667}, {286, 18592}, {1214, 18666}, {26107, 26132}, {26109, 26125}

X(26120) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26120) lies on these lines: {2, 3}, {73, 1442}, {78, 2893}, {908, 1330}, {975, 1745}, {1654, 3781}, {2303, 3330}, {2654, 5262}, {3616, 26130}, {5226, 26131}, {18228, 26064}

X(26121) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26121) lies on these lines: {2, 3}, {17102, 18667}, {26110, 26134}

X(26122) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26122) lies on these lines: {2, 3}, {391, 644}, {3217, 5802}, {4512, 19870}, {26106, 26138}

X(26123) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26123) lies on these lines: {2, 3}, {238, 10527}, {1463, 11375}, {1728, 27064}, {4652, 27305}, {10529, 16466}, {21616, 27184}, {26094, 26132}

X(26124) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26124) lies on these lines: {2, 3}, {148, 27312}, {2896, 27262}, {26099, 26108}, {26100, 26138}

X(26125) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26125) lies on these lines: {2, 7}, {6, 27142}, {12, 4429}, {37, 85}, {75, 3965}, {76, 346}, {77, 16826}, {86, 6180}, {150, 5816}, {192, 1441}, {198, 4209}, {239, 7190}, {241, 4687}, {269, 16831}, {284, 26802}, {347, 18666}, {388, 1284}, {391, 27304}, {573, 17753}, {604, 20146}, {651, 17379}, {664, 16777}, {941, 2481}, {948, 17086}, {954, 13727}, {966, 6604}, {1125, 4334}, {1418, 4698}, {1434, 25508}, {1446, 27250}, {1458, 3616}, {1463, 11375}, {1469, 3485}, {1901, 27021}, {2171, 3212}, {2263, 16830}, {2270, 27000}, {2345, 10030}, {3085, 24248}, {3247, 9312}, {3600, 13736}, {3622, 10571}, {3671, 19853}, {3674, 27248}, {3729, 27544}, {3986, 10481}, {4327, 16823}, {4328, 4384}, {4331, 9791}, {4335, 13405}, {4343, 10578}, {4355, 25512}, {4393, 7269}, {4454, 27514}, {4552, 4704}, {4747, 27161}, {5228, 17277}, {5723, 17380}, {6817, 21319}, {7011, 25908}, {7201, 16609}, {7274, 16832}, {7384, 21279}, {17247, 22464}, {20072, 27317}, {20262, 26531}, {21068, 27129}, {25242, 27396}, {26109, 26119}, {26976, 27252}

X(26126) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26126) lies on these lines: {2, 12}, {10, 1450}, {201, 6682}, {226, 19864}, {474, 26031}, {498, 24222}, {499, 26098}, {603, 25496}, {964, 1470}, {1001, 27506}, {1125, 1457}, {1319, 26115}, {2122, 25490}, {3086, 5711}, {3616, 26092}, {3911, 19863}, {4202, 26481}, {4551, 20108}, {4647, 26740}, {4972, 10957}, {5252, 26030}, {7098, 24627}, {11375, 26094}, {11509, 24552}, {26091, 26105}

X(26127) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26127) lies on these lines: {2, 35}, {4, 5550}, {5, 5284}, {8, 392}, {11, 5047}, {21, 3816}, {57, 3648}, {100, 10386}, {149, 1698}, {377, 19832}, {388, 1319}, {404, 15338}, {452, 5265}, {496, 5260}, {497, 9780}, {499, 16865}, {551, 20060}, {632, 10738}, {748, 24883}, {962, 6947}, {1001, 4193}, {1125, 3585}, {1385, 6965}, {1621, 4187}, {1836, 9782}, {2475, 3624}, {2476, 4423}, {2551, 3241}, {2886, 17536}, {3421, 20057}, {3434, 17559}, {3523, 26333}, {3525, 10525}, {3576, 13729}, {3583, 19862}, {3634, 4857}, {3817, 6895}, {3826, 17546}, {3829, 17547}, {3847, 7504}, {3868, 4679}, {3874, 26792}, {3925, 17534}, {4189, 10200}, {4197, 8167}, {4202, 25531}, {4302, 17572}, {4999, 16858}, {5057, 5439}, {5071, 18517}, {5129, 10527}, {5154, 10198}, {5178, 18527}, {5253, 11113}, {5270, 15808}, {5731, 6893}, {5886, 6902}, {6224, 19861}, {6284, 17531}, {6691, 17549}, {6836, 9779}, {6840, 8227}, {6857, 10584}, {6865, 9812}, {6894, 7988}, {6903, 9955}, {6975, 10267}, {6979, 10902}, {6986, 7681}, {6989, 10598}, {7280, 15677}, {8165, 11239}, {9668, 16862}, {9669, 16842}, {10624, 25011}, {10916, 27065}, {11108, 11680}, {11114, 25524}, {11604, 15674}, {14450, 24703}, {15171, 17575}, {16859, 26363}, {16861, 24953}, {17484, 18398}, {17570, 19854}, {17676, 25492}, {17717, 24936}, {24955, 25463}, {26094, 26117}, {26102, 26131}

X(26128) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 + a b^2 + b^3 + a c^2 + c^3 : :

X(26128) lies on these lines: {1, 977}, {2, 38}, {7, 4697}, {10, 24789}, {31, 4655}, {43, 16706}, {55, 3821}, {56, 226}, {63, 6679}, {69, 3791}, {86, 17203}, {141, 4362}, {171, 3662}, {238, 4703}, {306, 19834}, {321, 24943}, {518, 25453}, {551, 4138}, {612, 3836}, {614, 3846}, {740, 19785}, {748, 26580}, {846, 4389}, {976, 4202}, {1001, 1626}, {1086, 3980}, {1211, 16825}, {1330, 16478}, {1376, 17290}, {1707, 17274}, {1909, 18067}, {1961, 7194}, {3008, 4104}, {3120, 24552}, {3616, 4892}, {3666, 3771}, {3705, 17598}, {3740, 17356}, {3741, 3772}, {3744, 4660}, {3769, 17227}, {3775, 5271}, {3782, 3923}, {3834, 4682}, {3840, 17720}, {3870, 4085}, {3874, 20083}, {3891, 15523}, {3920, 25957}, {3936, 17017}, {3938, 4972}, {3946, 4028}, {3961, 4429}, {3967, 17357}, {3971, 17279}, {4011, 4415}, {4071, 16777}, {4353, 20106}, {4357, 16992}, {4361, 21085}, {4364, 24333}, {4640, 17235}, {4645, 17716}, {4650, 26840}, {4657, 20335}, {4672, 5905}, {4683, 17127}, {4970, 17301}, {4974, 5739}, {5117, 7009}, {5249, 5329}, {5263, 17889}, {5268, 17282}, {5297, 25961}, {5311, 18139}, {6327, 17469}, {6646, 7262}, {6685, 17718}, {6703, 25557}, {7081, 16986}, {7191, 25760}, {7292, 25960}, {8616, 24723}, {10180, 17321}, {13161, 19768}, {16887, 25598}, {17024, 25958}, {17064, 21242}, {17302, 17592}, {17303, 21101}, {17304, 17594}, {18398, 25441}, {24694, 25345}, {26034, 26228}, {26037, 26724}, {26181, 26188}

X(26129) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26129) lies on these lines: {1, 5748}, {2, 40}, {4, 17614}, {8, 11}, {21, 4423}, {78, 5274}, {191, 499}, {226, 7091}, {329, 1728}, {377, 9779}, {388, 1319}, {390, 27385}, {392, 3090}, {404, 9812}, {443, 9955}, {452, 1125}, {497, 25681}, {515, 24558}, {908, 14986}, {944, 10711}, {960, 10589}, {997, 5175}, {1519, 6926}, {1699, 6904}, {1770, 3624}, {2094, 10199}, {2136, 7080}, {2476, 7958}, {2550, 24954}, {2551, 11376}, {3091, 19861}, {3189, 11238}, {3421, 11373}, {3474, 6691}, {3485, 3816}, {3487, 14022}, {3701, 6557}, {3817, 5177}, {3825, 18391}, {3869, 5704}, {3872, 8165}, {3877, 4731}, {3895, 27525}, {4187, 5603}, {4295, 10200}, {4310, 28018}, {4512, 19862}, {5046, 5731}, {5056, 24987}, {5082, 7743}, {5084, 5886}, {5129, 24541}, {5284, 11344}, {5433, 5698}, {5435, 7098}, {5552, 9785}, {5554, 5734}, {5811, 10785}, {5815, 10529}, {5818, 17533}, {5828, 12648}, {5880, 6910}, {6361, 13747}, {6700, 9614}, {6857, 11230}, {6921, 9778}, {7288, 24703}, {8582, 11522}, {9776, 12047}, {10165, 17576}, {10248, 17579}, {10527, 18228}, {10586, 11037}, {12245, 17619}, {12699, 17567}, {17527, 18493}, {18135, 20449}, {19843, 23708}, {25492, 27506}, {26091, 26094}, {26093, 26116}

X(26130) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26130) lies on these lines: {1, 5800}, {2, 48}, {3, 16608}, {4, 15669}, {7, 2294}, {8, 21231}, {9, 9028}, {19, 18650}, {56, 18635}, {71, 14021}, {77, 5236}, {141, 958}, {142, 515}, {198, 25964}, {226, 4341}, {278, 25361}, {281, 24315}, {388, 1458}, {464, 24310}, {518, 3781}, {529, 17313}, {1001, 1503}, {1385, 17073}, {1953, 4329}, {2260, 5738}, {2293, 11677}, {2317, 26668}, {2345, 21091}, {3475, 5311}, {3486, 3924}, {3576, 18634}, {3616, 26120}, {3739, 5794}, {3912, 5227}, {5249, 5307}, {5786, 15668}, {10246, 17043}, {14547, 26052}, {16713, 21285}, {17052, 26363}, {17170, 17442}, {17306, 19869}, {18162, 27509}, {21280, 23407}, {21483, 26942}, {22054, 24580}, {24220, 26332}, {26107, 26147}, {26639, 27180}

X(26131) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26131) lies on these lines: {1, 149}, {2, 58}, {4, 500}, {6, 4197}, {7, 26054}, {8, 2650}, {10, 2895}, {12, 651}, {20, 5713}, {21, 17056}, {43, 26060}, {79, 3743}, {81, 442}, {86, 5051}, {162, 451}, {225, 1442}, {226, 4296}, {229, 2915}, {377, 5712}, {388, 1464}, {404, 5718}, {445, 8747}, {498, 6149}, {581, 6839}, {750, 27529}, {846, 3648}, {940, 2476}, {964, 18134}, {977, 5716}, {991, 6895}, {1010, 3936}, {1046, 21674}, {1211, 14005}, {1213, 17551}, {1654, 9780}, {1655, 6625}, {1834, 6175}, {1962, 24851}, {2292, 14450}, {2478, 4648}, {2893, 3945}, {3152, 5703}, {3178, 4418}, {3194, 25987}, {3448, 6126}, {3616, 4892}, {3651, 13408}, {3664, 6734}, {3670, 26842}, {3701, 3770}, {3836, 27320}, {3909, 5725}, {3920, 13407}, {3931, 20292}, {4205, 5333}, {4417, 16454}, {4645, 26110}, {4653, 15680}, {5057, 6051}, {5125, 5736}, {5192, 17234}, {5226, 26120}, {5249, 5262}, {5277, 5546}, {5287, 9612}, {5292, 14996}, {5297, 21077}, {5396, 6901}, {5492, 16116}, {5707, 6937}, {6675, 16948}, {9782, 24443}, {10198, 17126}, {11115, 25650}, {11374, 26738}, {12609, 17016}, {13740, 18139}, {15844, 17074}, {15988, 25984}, {16062, 19684}, {16704, 25446}, {17011, 23537}, {17245, 17536}, {17392, 17577}, {17550, 20131}, {17579, 19765}, {17750, 26074}, {18666, 25255}, {19784, 25959}, {19877, 26044}, {20653, 24342}, {24968, 24971}, {26102, 26127}

X(26132) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26132) lies on these lines: {1, 4138}, {2, 7}, {8, 2887}, {56, 25906}, {69, 3772}, {278, 297}, {344, 4415}, {345, 3782}, {948, 26561}, {1125, 13736}, {1215, 9780}, {1458, 24551}, {1763, 26998}, {3241, 4865}, {3454, 24159}, {3487, 16062}, {3488, 17677}, {3616, 4892}, {3620, 11679}, {3687, 23681}, {3705, 4310}, {3729, 20106}, {3771, 24248}, {3875, 4035}, {3936, 19785}, {4000, 4417}, {4201, 5703}, {4429, 25568}, {4470, 19827}, {4517, 25137}, {5550, 25496}, {5712, 19786}, {5714, 13740}, {5719, 11359}, {6327, 26228}, {6679, 24695}, {8165, 25965}, {9308, 18678}, {10327, 25959}, {14555, 24789}, {15934, 16052}, {17011, 19823}, {17056, 17321}, {17103, 25507}, {17170, 17211}, {17182, 18648}, {17316, 18134}, {17720, 18141}, {18135, 21590}, {20498, 26029}, {21062, 27127}, {21609, 26563}, {25681, 25912}, {25990, 27410}, {26093, 26116}, {26094, 26123}, {26107, 26119}, {26112, 26139}

X(26133) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26133) lies on these lines: {2, 82}, {75, 5211}, {83, 17055}, {4645, 26094}, {26098, 26107}

X(26134) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    (a + b - c) (a - b + c) (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(26134) lies on these lines: {2, 85}, {7, 27019}, {39, 6063}, {194, 349}, {226, 1424}, {269, 25538}, {1441, 26042}, {1463, 11375}, {4554, 5283}, {6516, 16915}, {6604, 26801}, {9312, 27020}, {9436, 17030}, {26110, 26121}

X(26135) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26135) lies on these lines: {2, 87}, {7, 8}, {1278, 25284}, {1654, 26038}, {2345, 20532}, {3616, 24661}, {4648, 20530}, {4772, 25292}, {4851, 24717}, {5550, 25535}, {7155, 20917}, {9780, 25121}, {10453, 17375}, {17278, 24753}, {17300, 21299}, {17786, 24451}, {19877, 26045}, {24672, 26107}, {25570, 26752}

X(26136) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26136) lies on these lines: {2, 45}, {11, 145}, {908, 20072}, {3616, 17719}, {3624, 11814}, {3699, 4678}, {4648, 26137}, {4928, 21222}, {5219, 9312}, {16732, 18743}, {19877, 24003}, {26109, 26147}

X(26137) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    13 a^2 b + 5 a b^2 - 8 b^3 + 13 a^2 c + 7 a b c + 9 b^2 c + 5 a c^2 + 9 b c^2 - 8 c^3 : :

X(26137) lies on these lines: {2, 44}, {3486, 10129}, {4080, 4704}, {4648, 26136}, {17379, 25529}

X(26138) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26138) lies on these lines: {2, 99}, {799, 16613}, {1015, 21220}, {2170, 24505}, {2478, 26108}, {20349, 27166}, {26100, 26124}, {26106, 26122}, {26140, 26142}

X(26139) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 + 2 a b^2 - b^3 - 7 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 - c^3 : :

X(26139) lies on these lines: {1, 11814}, {2, 11}, {8, 17460}, {145, 3699}, {190, 3756}, {214, 10774}, {244, 4440}, {388, 26111}, {1054, 25377}, {1058, 26029}, {1357, 4499}, {1647, 4473}, {2478, 26093}, {2899, 17480}, {3600, 8686}, {3616, 17719}, {3622, 4997}, {3624, 26051}, {3685, 5121}, {3837, 26142}, {3870, 27130}, {3873, 26791}, {4076, 5516}, {4152, 20014}, {4201, 25492}, {4358, 5211}, {4645, 4871}, {4679, 6646}, {4928, 26140}, {5231, 17338}, {6999, 25510}, {12053, 25965}, {14923, 25979}, {18149, 20345}, {26094, 26117}, {26098, 26103}, {26102, 26109}, {26112, 26132}, {26141, 26147}

X(26140) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26140) lies on these lines: {1, 20344}, {2, 101}, {8, 21232}, {100, 4904}, {142, 6224}, {149, 17761}, {404, 21258}, {644, 16593}, {1385, 27006}, {1477, 3600}, {2140, 2475}, {3616, 26101}, {4107, 26141}, {4675, 7200}, {4928, 26139}, {5080, 20335}, {5086, 24774}, {5519, 6065}, {8299, 18343}, {9263, 17300}, {17234, 18047}, {26138, 26142}

X(26141) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26141) lies on these lines: {1, 149}, {2, 98}, {11, 21221}, {662, 8286}, {1330, 8666}, {1469, 3873}, {2895, 3705}, {3897, 26117}, {4107, 26140}, {4188, 25650}, {4645, 5143}, {5347, 18134}, {17300, 24523}, {26139, 26147}

X(26142) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26142) lies on these lines: {2, 45}, {334, 17321}, {1654, 27011}, {3662, 4466}, {3834, 26113}, {3837, 26139}, {4000, 20333}, {4499, 24485}, {4648, 26143}, {4657, 26110}, {6386, 18135}, {6542, 27106}, {17237, 26801}, {17249, 26082}, {17300, 20355}, {17301, 26752}, {17314, 20532}, {20072, 26982}, {26138, 26140}

X(26143) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c - 3 a^2 b^2 c - 2 a b^3 c + a^3 c^2 - 3 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(26143) lies on these lines: {1, 25311}, {2, 37}, {7, 24509}, {8, 25121}, {1001, 20676}, {1125, 25528}, {3616, 24661}, {4021, 27091}, {4648, 26142}, {4941, 24451}, {7155, 24456}, {16709, 26852}, {16777, 20532}, {17236, 27166}, {17304, 25510}, {17343, 26821}, {17379, 20332}, {17397, 20146}, {18133, 21219}, {18194, 26069}, {24669, 26111}, {26093, 26150}

X(26144) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26144) lies on these lines: {2, 900}, {522, 14429}, {966, 4435}, {1769, 3716}, {2345, 4526}, {2815, 5603}, {3738, 16173}, {3766, 17321}, {3837, 26139}, {5296, 22108}, {6615, 8062}, {7650, 23882}, {13266, 24542}, {17320, 21606}

X(26145) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26145) lies on these lines: {2, 187}, {148, 16711}, {663, 3835}, {754, 25683}, {1654, 24958}, {2478, 4648}, {3701, 17372}, {5046, 26978}, {5051, 6707}, {5192, 17327}, {6781, 24918}, {7778, 11346}, {7842, 17690}, {16705, 17685}, {17283, 17541}, {17375, 18135}, {17381, 17550}

X(26146) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26146) lies on these lines: {2, 647}, {278, 17094}, {650, 7212}, {663, 3835}, {693, 905}, {2517, 4885}, {2522, 17896}, {4000, 17069}, {4017, 4369}, {4077, 16612}, {4379, 20521}, {4467, 19785}, {6590, 14837}, {7658, 21186}, {8642, 26249}, {14296, 27527}, {18155, 19786}, {21173, 23803}

X(26147) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26147) lies on these lines: {2, 662}, {8, 21254}, {86, 24957}, {99, 17058}, {145, 16597}, {148, 1086}, {1654, 17228}, {3836, 20558}, {3942, 24504}, {4675, 6625}, {4851, 20529}, {17300, 18133}, {17374, 20536}, {17387, 17778}, {21277, 27272}, {26107, 26130}, {26109, 26136}, {26138, 26140}, {26139, 26141}

X(26148) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 40

Barycentrics    (b - c) (-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(26148) lies on these lines: {2, 669}, {320, 350}, {661, 3907}, {663, 3835}, {667, 27345}, {3005, 25258}, {3741, 18197}, {3837, 26097}, {4455, 27527}, {20979, 25128}, {20983, 25301}, {21191, 24666}, {24663, 24674}

X(26148) = anticomplement of X(24533)

X(26149) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26149) lies on these lines: {2, 7}, {69, 26801}, {75, 21021}, {1125, 7184}, {3616, 24661}, {3663, 27020}, {3664, 26959}, {4648, 26107}, {4657, 26110}, {4675, 25505}, {4699, 26048}, {16924, 21279}, {17030, 17272}, {17250, 26045}, {17280, 26976}, {17300, 26971}, {17305, 27042}, {17398, 25534}, {25590, 27091}, {26756, 26812}

X(26150) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26150) lies on these lines: {1, 17232}, {2, 38}, {7, 5550}, {8, 3619}, {238, 17236}, {518, 17370}, {726, 17358}, {894, 3624}, {1001, 17305}, {1125, 3662}, {1279, 3616}, {1386, 17227}, {3210, 24943}, {3685, 17304}, {3742, 19812}, {3775, 16816}, {3790, 4353}, {4676, 17235}, {4741, 16468}, {4966, 17380}, {4974, 17343}, {5263, 17290}, {5749, 16814}, {7155, 15315}, {9780, 17278}, {15569, 17399}, {16475, 17288}, {16823, 17306}, {16825, 17238}, {16830, 17282}, {17368, 19862}, {17381, 25557}, {17480, 19879}, {19853, 27154}, {26093, 26143}, {26094, 26107}

X(26151) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26151) lies on these lines: {2, 3}

X(26152) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 40

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

X(26152) lies on these lines: {2, 3}

leftri

Collineation mappings involving Gemini triangle 41: X(26153)-X(6180)

rightri

Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 41, Gemini triangle 42, Gemini triangle 43, etc. See the preamble just before X(24537) for the definitions of Gemini triangles 1-40. (Clark Kimberling, October 30, 2018)

Gemini 41      b^2 + c^2 : a^2 : a^2
Gemini 42      a^2 + b^2 + c^2 : a^2 : a^2

Gemini 43      a^2 : b^2 + c^2 : b^2 + c^2
Gemini 44      - a^2 : b^2 + c^2 : b^2 + c^2 (circum-medial triangle, TCCT 6.19

Gemini 45      (b - c)^2 : a^2 : a^2
Gemini 46      (b + c)^2 : a^2 : a^2

Gemini 47      a^2 : (b + c)^2 : (b + c)^2
Gemini 48      a^2 : (b - c)^2 : (b - c)^2

Gemini 49      (b + c)^2 : (b - c)^2 : (b - c)^2
Gemini 50      (b - c)^2 : (b + c)^2 : (b + c)^2

Gemini 51      (b - c)^2 : b^2 + c^2 : b^2 + c^2
Gemini 52      (b + c)^2 : b^2 + c^2 : b^2 + c^2

Gemini 53      b^2 + c^2 : (b - c)^2 : (b - c)^2
Gemini 54      b^2 + c^2 : (b + c)^2 : (b + c)^2

Gemini 55      a^2 : 2 b c : 2 b c
Gemini 56      - a^2 : 2 b c : 2 b c

Gemini 57      b^2 + c^2 : b c : b c
Gemini 58      b^2 + c^2 : - b c : - b c

Gemini 59      - b c + c a + a b : b c + c a + a b : b c + c a + a b
Gemini 60      b c + c a + a b : - b c + c a + a b : - b c + c a + a b

If T is a central triangle A'B'C' with A' of the form f(a,b,c) : g(a,b,c) : g(a,b,c), then the (A,B,C,X(2); A',B',C',X(2)) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 30-60.

Let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 41, as in centers X(26153)-X(26180). Then

m(X) = (a^2 - b^2 + c^2) (a^2 + b^2 - c^2) (b^2 + c^2 ) x + (b^2 (b^2 + c^2 - a^2) ( axxx : : ,

and m(X) is on the Euler line if and only if X is on the Euler line.

X(26153) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26153) lies on these lines: {1, 2}, {141, 1231}, {379, 5090}, {857, 1829}, {1861, 26961}, {5081, 26678}, {17184, 26161}, {18636, 20235}, {20911, 26165}, {23661, 26550}, {26156, 26163}, {26178, 26179}

X(26154) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26154) lies on these lines: {2, 3}, {141, 22416}, {185, 15595}, {287, 14516}, {1105, 6330}, {9289, 26156}, {16890, 26224}

X(26155) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26155) lies on these lines: {2, 3}, {1970, 3589}, {9729, 15595}, {23115, 27377}

X(26156) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26156) lies on these lines: {2, 6}, {22, 15812}, {74, 18358}, {110, 26926}, {125, 14913}, {468, 19121}, {858, 1843}, {1352, 17928}, {1368, 12220}, {1503, 22467}, {1568, 21851}, {3564, 26879}, {5133, 9822}, {5866, 7789}, {5895, 10516}, {5972, 21637}, {6403, 11585}, {6656, 26162}, {6816, 10519}, {7391, 7716}, {7399, 11459}, {7762, 26212}, {8263, 12272}, {9289, 26154}, {10018, 19131}, {11188, 23300}, {13160, 24206}, {15059, 15128}, {16238, 19128}, {18639, 27180}, {18642, 21511}, {18911, 19459}, {19588, 26869}, {26153, 26163}, {26166, 26177}, {26175, 26179}

X(26157) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26157) lies on these lines: {1, 2}, {141, 26165}, {318, 26528}, {321, 26171}, {1375, 12135}, {5090, 24584}, {7270, 26219}, {16607, 18669}, {17184, 26170}, {17233, 26215}, {17492, 18596}, {18657, 21063}, {23661, 26527}

X(26158) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26158) lies on these lines: {1, 2}, {318, 26556}, {1441, 18639}, {1826, 18659}, {5090, 24605}, {7718, 24580}, {17184, 26174}, {17670, 26213}, {18671, 20305}, {26165, 26166}, {26168, 26177}

X(26159) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26159) lies on these lines: {2, 3}

X(26160) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26160) lies on these lines: {2, 3}

X(26161) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26161) lies on these lines: {2, 31}, {17184, 26153}

X(26162) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26162) lies on these lines: {2, 32}, {141, 26214}, {6656, 26156}, {7879, 26206}, {26166, 26175}

X(26163) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26163) lies on these lines: {2, 37}, {226, 21406}, {3912, 18692}, {26153, 26156}, {26164, 26169}

X(26164) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26164) lies on these lines: {2, 39}, {4, 11382}, {6, 26212}, {83, 1236}, {339, 7819}, {1235, 7770}, {3260, 7745}, {6656, 26156}, {7754, 26206}, {12203, 22467}, {26163, 26169}, {26175, 26177}

X(26165) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26165) lies on these lines: {2, 37}, {29, 3100}, {92, 4329}, {141, 26157}, {142, 23581}, {318, 2478}, {390, 23528}, {968, 23556}, {1040, 27386}, {3262, 4150}, {4319, 17860}, {17858, 25935}, {17859, 26006}, {18589, 20883}, {20911, 26153}, {23978, 26601}, {23983, 26543}, {26158, 26166}

X(26166) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26166) lies on these lines: {2, 39}, {3, 1235}, {20, 264}, {69, 5889}, {97, 276}, {99, 14118}, {140, 339}, {141, 22416}, {183, 17928}, {237, 12143}, {308, 26224}, {311, 1975}, {317, 7544}, {325, 13160}, {1078, 1236}, {1232, 1238}, {3096, 26170}, {3260, 7750}, {3933, 7399}, {26156, 26177}, {26158, 26165}, {26162, 26175}

X(26167) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26167) lies on these lines: {2, 6}, {21, 18642}, {286, 26605}, {858, 17171}, {3868, 16608}, {20911, 26153}, {26168, 26169}, {26171, 26563}

X(26168) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26168) lies on these lines: {2, 31}, {26153, 26156}, {26158, 26177}, {26167, 26169}

X(26169) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26169) lies on these lines: {1, 2}, {26163, 26164}, {26167, 26168}

X(26170) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26170) lies on these lines: {2, 3}, {3096, 26166}, {4045, 26216}, {8743, 13219}, {12111, 15595}, {17184, 26157}

X(26171) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26171) lies on these lines: {2, 3}, {321, 26157}, {17184, 26153}, {26167, 26563}

X(26172) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26172) lies on these lines: {2, 3}

X(26173) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26173) lies on these lines: {2, 3}

X(26174) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26174) lies on these lines: {2, 3}, {141, 26157}, {17184, 26158}

X(26175) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26175) lies on these lines: {2, 3}, {26156, 26179}, {26162, 26166}, {26164, 26177}

X(26176) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 41

Barycentrics    a^3 b^2 + b^5 + a^3 c^2 + c^5 : :

X(26176) lies on these lines: {2, 48}, {6, 26589}, {31, 21275}, {80, 17291}, {141, 313}, {1964, 21235}, {2887, 21278}, {3662, 17861}, {6679, 21298}, {17046, 27145}, {21236, 26979}, {21244, 27095}, {26012, 26963}

X(26177) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26177) lies on these lines: {2, 32}, {6815, 15062}, {26156, 26166}, {26158, 26168}, {26164, 26175}

X(26178) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26178) lies on these lines: {2, 37}, {16580, 20884}, {17481, 21582}, {26153, 26179}

X(26179) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26179) lies on these lines: {2, 39}, {32, 1236}, {98, 22467}, {112, 384}, {264, 14035}, {311, 17128}, {339, 7807}, {1352, 12111}, {3260, 7823}, {6655, 17984}, {7791, 15075}, {7929, 8920}, {26153, 26178}, {26156, 26175}

X(26180) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 41

Barycentrics    a^14 b^2 - a^12 b^4 - 3 a^10 b^6 + 3 a^8 b^8 + 3 a^6 b^10 - 3 a^4 b^12 - a^2 b^14 + b^16 + a^14 c^2 - 4 a^12 b^2 c^2 - 3 a^10 b^4 c^2 + 6 a^8 b^6 c^2 + 3 a^6 b^8 c^2 - a^2 b^12 c^2 - 2 b^14 c^2 - a^12 c^4 - 3 a^10 b^2 c^4 + 10 a^8 b^4 c^4 - 10 a^6 b^6 c^4 + 7 a^4 b^8 c^4 - 3 a^2 b^10 c^4 - 3 a^10 c^6 + 6 a^8 b^2 c^6 - 10 a^6 b^4 c^6 - 8 a^4 b^6 c^6 + 5 a^2 b^8 c^6 + 2 b^10 c^6 + 3 a^8 c^8 + 3 a^6 b^2 c^8 + 7 a^4 b^4 c^8 + 5 a^2 b^6 c^8 - 2 b^8 c^8 + 3 a^6 c^10 - 3 a^2 b^4 c^10 + 2 b^6 c^10 - 3 a^4 c^12 - a^2 b^2 c^12 - a^2 c^14 - 2 b^2 c^14 + c^16 : :

X(26180) lies on these lines: {2, 3}

leftri

Collineation mappings involving Gemini triangle 42: X(26181)-X(26199)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 42, as in centers X(26181)-X(26199). Then

m(X) = (a^2 + b^2) (a^2 + c^2) (a^2 + b^2 + c^2)x + b^2 (a^2 + b^2) (b^2 + c^2) y + c^2 (a^2 + c^2) (b^2 + c^2) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 30, 2018)

X(26181) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26181) lies on these lines: {1, 2}, {26128, 26188}

X(26182) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26182) lies on these lines: {2, 3}, {827, 3096}, {7834, 26185}, {26192, 26197}

X(26183) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26183) lies on these lines: {2, 3}, {26189, 26198}, {26190, 26192}

X(26184) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26184) lies on these lines: {2, 3}, {7834, 26198}

X(26185) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26185) lies on these lines: {2, 6}, {6680, 23322}, {7834, 26182}, {26195, 26199}

X(26186) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26186) lies on these lines: {2, 3}

X(26187) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26187) lies on these lines: {2, 3}

X(26188) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 41

Barycentrics    a^9 + 2 a^7 b^2 + a^5 b^4 + a^2 b^7 + b^9 + 2 a^7 c^2 + 3 a^5 b^2 c^2 + a^3 b^4 c^2 + a^2 b^5 c^2 + b^7 c^2 + a^5 c^4 + a^3 b^2 c^4 + a^2 b^2 c^5 + a^2 c^7 + b^2 c^7 + c^9 : :

X(26188) lies on these lines: {2, 31}, {26128, 26181}

X(26189) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26189) lies on these lines: {2, 32}, {7834, 26182}, {26183, 26198}, {26192, 26195}, {26197, 26199}

X(26190) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26190) lies on these lines: {2, 6}, {1078, 6697}, {3313, 11056}, {26183, 26192}, {26191, 26196}, {26194, 26197}

X(26191) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26191) lies on these lines: {2, 37}, {3112, 21249}, {26190, 26196}

X(26192) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26192) lies on these lines: {2, 39}, {83, 10339}, {308, 6292}, {3096, 14970}, {26182, 26197}, {26183, 26190}, {26189, 26195}

X(26193) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26193) lies on these lines: {2, 3}

X(26194) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26194) lies on these lines: {2, 3}, {26190, 26197}

X(26195) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26195) lies on these lines: {2, 3}, {7834, 26197}, {26185, 26199}, {26189, 26192}

X(26196) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26196) lies on these lines: {2, 3}, {26190, 26191}

X(26197) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26197) lies on these lines: {2, 99}, {7834, 26195}, {26182, 26192}, {26189, 26199}, {26190, 26194}

X(26198) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26198) lies on these lines: {2, 99}, {141, 14990}, {7834, 26184}, {26183, 26189}

X(26199) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 41

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

X(26199) lies on these lines: {2, 39}, {827, 7787}, {5103, 16285}, {26185, 26195}, {26189, 26197}

X(26200) =  MIDPOINT OF X(4) AND X(10284)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-4 a^4 b c+7 a^3 b^2 c+4 a^2 b^3 c-8 a b^4 c-a^4 c^2+7 a^3 b c^2-14 a^2 b^2 c^2+7 a b^3 c^2+b^4 c^2-2 a^3 c^3+4 a^2 b c^3+7 a b^2 c^3+2 a^2 c^4-8 a b c^4+b^2 c^4+a c^5-c^6) : :
X(26200) = X[4]+X[10284], X[546]-X[2802], X[550]-3*X[3898], 3*X[1482]+X[5693], X[2771]-X[7984], X[2800]-X[6583], X[5694]+X[7982], X[5885]-2*X[13464], X[5887]+X[11278], 2*X[5901]-X[13145], 5*X[11522]-X[25413]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28554.

X(26200) lies on these lines: {4,10284}, {5,10}, {65,16173}, {392,17531}, {546,2802}, {550,3898}, {962,6951}, {1385,6909}, {1482,5693}, {2771,7984}, {2800,6583}, {3057,3585}, {3579,6940}, {5441,5919}, {5603,6972}, {5694,7982}, {5697,10895}, {5885,13464}, {5887,11278}, {5901,13145}, {6284,9957}, {10058,24928}, {10738,12751}, {11009,17638}, {11522,25413}, {15558,18990}, {18393,25414}

X(26200) = midpoint of X(i) and X(j) for these {i,j}: {4,10284}, {3057,22793}, {5887,11278}, {10222,12672}, {18480,23340}
X(26200) = reflection of X(i) in X(j) for these {i,j}: {5885,13464}, {13145,5901}

X(26201) =  MIDPOINT OF X(550) AND X(3874)

Barycentrics    -a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+4 a^4 b c-a^3 b^2 c-4 a^2 b^3 c-a^4 c^2-a^3 b c^2+2 a^2 b^2 c^2-a b^3 c^2+b^4 c^2-2 a^3 c^3-4 a^2 b c^3-a b^2 c^3+2 a^2 c^4+b^2 c^4+a c^5-c^6) : :
X(26201) = 5*X[3]-X[5904], X[30]-X[6583], X[140]-X[2801], X[382]-5*X[18398], X[389]-2*X[15229], X[515]-X[5885], X[517]-X[550], X[518]-X[14810], 5*X[632]-3*X[15064], X[912]-X[12038], X[952]-X[13145], X[971]-X[9955], 2*X[3530]-X[3678], 3*X[3576]-X[5694], X[3579]-3*X[10167], 3*X[3656]+X[9961], X[6001]-X[15178], X[6102]+X[23156], 3*X[7967]-X[10284], 3*X[10202]+X[12680], 3*X[10246]+X[15071], 3*X[11220]+X[12699], 3*X[11231]-X[14872], 5*X[15016]-X[18525]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28554.

X(26201) lies on these lines: {3,5904}, {21,104}, {30,6583}, {35,17660}, {65,4325}, {72,5303}, {79,354}, {140,2801}, {382,18398}, {389,15229}, {515,5885}, {517,550}, {518,14810}, {632,15064}, {912,12038}, {942,7354}, {946,12267}, {952,13145}, {971,9955}, {1858,5126}, {3057,11571}, {3530,3678}, {3576,5694}, {3579,10167}, {3583,13751}, {3656,9961}, {5045,10391}, {5083,15171}, {5536,16117}, {5563,17637}, {6001,15178}, {6102,23156}, {6940,12738}, {7967,10284}, {8582,8728}, {10202,12680}, {10225,11491}, {10246,15071}, {10268,24645}, {11220,12699}, {11231,14872}, {15016,18525}, {15931,22937}, {16132,22765}

X(26201) = midpoint of X(i) and X(j) for these {i,j}: {550,3874}, {6102,23156}, {12675,13369}, {12680,18480}
X(26201) = reflection of X(i) in X(j) for these {i,j}: {389,15229}, {3678,3530}, {6583,12005}, {9955,13373}, {9956,9940}
X(26201) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {10202,12680,18480}

X(26202) =  MIDPOINT OF X(21) AND X(16138)

Barycentrics    a (2 a^6-a^5 b-5 a^4 b^2+2 a^3 b^3+4 a^2 b^4-a b^5-b^6-a^5 c+6 a^4 b c+a^3 b^2 c-2 a^2 b^3 c-4 b^5 c-5 a^4 c^2+a^3 b c^2+2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2+2 a^3 c^3-2 a^2 b c^3+a b^2 c^3+8 b^3 c^3+4 a^2 c^4+b^2 c^4-a c^5-4 b c^5-c^6) : :
X(26202) = X[10]-X[30], 3*X[191]-X[12702], 3*X[381]-X[16118], X[517]-X[3652], X[758]-X[11278], 3*X[1699]-X[16150], 4*X[3634]-3*X[5499], X[3648]+X[12699], 3*X[5886]-X[16116], X[8148]+3*X[13465], 6*X[10021]-5*X[19862], 4*X[12104]-3*X[17502], 3*X[15677]-X[18481], 3*X[16159]-X[20084], 3*X[16160]-2*X[18483]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28556.

X(26202) lies on these lines: {1,399}, {3,5506}, {10,30}, {21,4881}, {79,3582}, {191,12702}, {355,13199}, {381,16118}, {517,3652}, {758,11278}, {942,16141}, {1012,5694}, {1699,16150}, {2475,7705}, {3634,5499}, {3648,12699}, {5428,15254}, {5659,16113}, {5885,6912}, {5886,16116}, {6841,7173}, {6888,16128}, {6906,22935}, {7743,16153}, {8148,13465}, {9780,18516}, {9957,16140}, {10021,19862}, {10225,19925}, {11230,12608}, {11263,12611}, {12104,17502}, {15677,18481}, {16117,18540}, {16159,20084}, {16160,18483}

X(26202) = midpoint of X(i) and X(j) for these {i,j}: {21,16138}, {7701,13743}
X(26202) = reflection of X(i) in X(j) for these {i,j}: {79,9955}, {3579,3647}, {18480,22798}, {22937,22936}
X(26202) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3579,3647,22937}, {3579,22936,3647}

leftri

Collineation mappings involving Gemini triangle 43: X(26203)-X(26226)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 43, as in centers X(26203)-X(26226). Then

m(X) = a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) x + (a^2 + c^2) (b^2 + c^2 - a^2) (a^2 + b^2 - c^2) y + (a^2 + b^2) (b^2 + c^2 - a^2) (a^2 - b^2 + c^2) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 31, 2018)

X(26203) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26203) lies on these lines: {1, 2}, {6, 1231}, {33, 27022}, {34, 26961}, {318, 26678}, {379, 1829}, {607, 1441}, {857, 5090}, {1038, 27143}, {1040, 27093}, {1973, 26260}, {20811, 26206}, {23620, 24252}, {26211, 26219}

X(26204) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26204) lies on these lines: {2, 3}, {1968, 6389}, {3618, 26216}, {15595, 19467}

X(26205) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26205) lies on these lines: {2, 3}, {141, 1970}, {8721, 20792}, {10316, 27377}

X(26206) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26206) lies on these lines: {2, 6}, {3, 19118}, {22, 1974}, {24, 9967}, {25, 12220}, {74, 12017}, {110, 19459}, {155, 6804}, {182, 185}, {206, 6800}, {511, 17928}, {607, 27059}, {608, 26998}, {1176, 19153}, {1350, 22467}, {1351, 3567}, {1599, 11513}, {1600, 11514}, {1843, 1995}, {2211, 7791}, {2916, 27085}, {3098, 15078}, {3313, 19136}, {3564, 18912}, {3796, 19132}, {3867, 7394}, {5012, 19122}, {5013, 5866}, {5020, 11416}, {5050, 7395}, {5063, 9723}, {5085, 8567}, {5093, 13363}, {5622, 12825}, {5651, 14913}, {5921, 17814}, {6090, 15531}, {6225, 19149}, {6403, 6642}, {6467, 9306}, {6644, 18438}, {6656, 8743}, {6776, 6816}, {6815, 14853}, {7399, 13142}, {7485, 19126}, {7509, 19131}, {7514, 19129}, {7716, 13595}, {7754, 26164}, {7770, 14965}, {7819, 22120}, {7879, 26162}, {8541, 9822}, {8745, 17907}, {9605, 22241}, {10602, 12272}, {11413, 12294}, {11442, 13562}, {11487, 19458}, {12215, 26221}, {13160, 14561}, {14001, 23115}, {15056, 19460}, {16072, 18440}, {17710, 20987}, {17847, 25321}, {18911, 26926}, {20811, 26203}, {26216, 26224}

X(26207) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 43

Barycentrics    (a^16 - a^14 b^2 - 3 a^12 b^4 + 3 a^10 b^6 + 3 a^8 b^8 - 3 a^6 b^10 - a^4 b^12 + a^2 b^14 - a^14 c^2 - 4 a^12 b^2 c^2 + 3 a^10 b^4 c^2 + 2 a^8 b^6 c^2 + 5 a^6 b^8 c^2 - 7 a^2 b^12 c^2 + 2 b^14 c^2 - 3 a^12 c^4 + 3 a^10 b^2 c^4 + 2 a^8 b^4 c^4 - 6 a^6 b^6 c^4 + 5 a^4 b^8 c^4 + 3 a^2 b^10 c^4 - 4 b^12 c^4 + 3 a^10 c^6 + 2 a^8 b^2 c^6 - 6 a^6 b^4 c^6 - 8 a^4 b^6 c^6 + 3 a^2 b^8 c^6 - 2 b^10 c^6 + 3 a^8 c^8 + 5 a^6 b^2 c^8 + 5 a^4 b^4 c^8 + 3 a^2 b^6 c^8 + 8 b^8 c^8 - 3 a^6 c^10 + 3 a^2 b^4 c^10 - 2 b^6 c^10 - a^4 c^12 - 7 a^2 b^2 c^12 - 4 b^4 c^12 + a^2 c^14 + 2 b^2 c^14) : :

X(26207) lies on these lines: {2, 3}

X(26208) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26208) lies on these lines: {1, 2}, {26215, 26216}

X(26209) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26209) lies on these lines: {2, 3}, {8743, 18018}

X(26210) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26210) lies on these lines: {2, 3}, {4580, 26225}

X(26211) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26211) lies on these lines: {2, 31}, {1395, 26990}, {2212, 27051}, {26203, 26219}

X(26212) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26212) lies on these lines: {2, 32}, {6, 26164}, {7762, 26156}, {7770, 14965}, {26216, 26221}

X(26213) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26213) lies on these lines: {2, 37}, {1441, 27059}, {5745, 21406}, {17023, 18692}, {17670, 26158}, {20811, 26203}

X(26214) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26214) lies on these lines: {2, 39}, {4, 9967}, {141, 26162}, {311, 5254}, {324, 27376}, {339, 8362}, {384, 10313}, {1235, 5523}, {1236, 3096}, {3260, 7784}, {7467, 12143}, {7770, 14965}, {12203, 14118}, {26221, 26224}

X(26215) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26215) lies on these lines: {2, 37}, {3, 3100}, {1060, 4227}, {1214, 4329}, {6356, 17080}, {12610, 22464}, {17233, 26157}, {26208, 26216}

X(26216) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26216) lies on these lines: {2, 39}, {3, 5481}, {4, 22240}, {5, 5523}, {6, 5889}, {20, 216}, {32, 14118}, {54, 23128}, {112, 7526}, {217, 12111}, {232, 3091}, {570, 6815}, {574, 22467}, {631, 14961}, {1625, 15058}, {1658, 10986}, {1968, 5158}, {2079, 5013}, {3172, 15851}, {3199, 3832}, {3269, 10574}, {3289, 11444}, {3331, 11439}, {3523, 22401}, {3618, 26204}, {4045, 26170}, {5133, 27376}, {5169, 27371}, {5254, 13160}, {6509, 11348}, {6816, 7736}, {7395, 9605}, {7399, 15048}, {7488, 10311}, {7509, 23115}, {7514, 22120}, {8743, 9818}, {9607, 13351}, {11174, 26226}, {11325, 23635}, {15078, 15815}, {26206, 26224}, {26208, 26215}, {26212, 26221}

X(26217) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26217) lies on these lines: {2, 650}, {2485, 16757}

X(26218) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 43

Barycentrics    5 a^10 - 3 a^8 b^2 - 6 a^6 b^4 + 2 a^4 b^6 + a^2 b^8 + b^10 - 3 a^8 c^2 + 8 a^6 b^2 c^2 - 2 a^4 b^4 c^2 - 8 a^2 b^6 c^2 + 5 b^8 c^2 - 6 a^6 c^4 - 2 a^4 b^2 c^4 + 14 a^2 b^4 c^4 - 6 b^6 c^4 + 2 a^4 c^6 - 8 a^2 b^2 c^6 - 6 b^4 c^6 + a^2 c^8 + 5 b^2 c^8 + c^10 : :

X(26218) lies on these lines: {2, 3}

X(26219) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26219) lies on these lines: {2, 3}, {7270, 26157}, {26203, 26211}

X(26220) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 43

Barycentrics    7 a^10 - 3 a^8 b^2 - 10 a^6 b^4 + 2 a^4 b^6 + 3 a^2 b^8 + b^10 - 3 a^8 c^2 + 12 a^6 b^2 c^2 - 6 a^4 b^4 c^2 - 12 a^2 b^6 c^2 + 9 b^8 c^2 - 10 a^6 c^4 - 6 a^4 b^2 c^4 + 18 a^2 b^4 c^4 - 10 b^6 c^4 + 2 a^4 c^6 - 12 a^2 b^2 c^6 - 10 b^4 c^6 + 3 a^2 c^8 + 9 b^2 c^8 + c^10 : :

X(26220) lies on these lines: {2, 3}

X(26221) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26221) lies on these lines: {2, 3}, {5063, 17128}, {12215, 26206}, {26212, 26216}, {26214, 26224}

X(26222) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 43

Barycentrics    a^5 + a^2 b^3 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(26222) lies on these lines: {2, 48}, {6, 313}, {8, 238}, {31, 21278}, {41, 26772}, {71, 11320}, {80, 5150}, {81, 19806}, {141, 26634}, {312, 3187}, {560, 21238}, {604, 26963}, {894, 17861}, {1837, 2330}, {1914, 5278}, {1958, 27102}, {2273, 3948}, {2887, 21275}, {3778, 4112}, {7770, 20747}, {10791, 20964}, {18042, 25505}, {21221, 27320}, {25940, 27095}

X(26223) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 43

Barycentrics    a^3 + a^2 b + a^2 c + b^2 c + b c^2 : :

X(26223) lies on these lines: {1, 3159}, {2, 7}, {6, 321}, {10, 6327}, {31, 1215}, {33, 14954}, {37, 19684}, {38, 25496}, {42, 3923}, {43, 4418}, {44, 5278}, {45, 19701}, {46, 26030}, {72, 964}, {78, 11115}, {81, 312}, {192, 17011}, {218, 19281}, {228, 11322}, {274, 27643}, {284, 17587}, {306, 17355}, {318, 3194}, {394, 26591}, {404, 23206}, {474, 23169}, {518, 24552}, {593, 27958}, {612, 3952}, {614, 17140}, {726, 17017}, {748, 24325}, {750, 4697}, {899, 3980}, {936, 19284}, {940, 4358}, {942, 5192}, {1009, 21319}, {1010, 3876}, {1100, 3175}, {1150, 4641}, {1185, 2235}, {1211, 17369}, {1220, 3869}, {1255, 17394}, {1386, 3891}, {1449, 19743}, {1621, 4676}, {1698, 17491}, {1743, 5271}, {1836, 4972}, {1877, 5554}, {1999, 4671}, {2049, 15650}, {2051, 21375}, {2295, 3765}, {2308, 4362}, {2321, 20017}, {2345, 5739}, {2887, 24725}, {2895, 3661}, {2999, 17495}, {3006, 26098}, {3120, 25453}, {3210, 17012}, {3247, 19741}, {3338, 26094}, {3487, 17526}, {3488, 4217}, {3586, 17537}, {3589, 3782}, {3601, 17539}, {3618, 19785}, {3666, 17351}, {3671, 25904}, {3677, 17154}, {3679, 6539}, {3681, 5263}, {3685, 17018}, {3701, 5711}, {3706, 4663}, {3710, 5717}, {3720, 4011}, {3729, 5256}, {3731, 19740}, {3745, 3967}, {3751, 17135}, {3757, 17127}, {3868, 13740}, {3886, 20011}, {3896, 5695}, {3940, 16394}, {3947, 25982}, {3948, 17750}, {3969, 17281}, {3971, 5311}, {4009, 4682}, {4082, 4349}, {4307, 10327}, {4344, 20020}, {4359, 4363}, {4361, 4980}, {4402, 19826}, {4414, 6685}, {4427, 17594}, {4429, 20292}, {4461, 20043}, {4473, 26109}, {4687, 5333}, {4696, 5710}, {4884, 17726}, {4968, 16466}, {4981, 5220}, {5044, 16454}, {5222, 19789}, {5287, 8025}, {5297, 27538}, {5440, 16393}, {5712, 17776}, {5928, 27052}, {6358, 21741}, {6651, 17032}, {7081, 17126}, {7191, 24349}, {7229, 19825}, {7283, 19767}, {10391, 27394}, {10601, 17862}, {10791, 24255}, {11263, 19846}, {11342, 16601}, {11679, 16704}, {12514, 26115}, {14555, 19822}, {14997, 17116}, {16050, 25082}, {16405, 20760}, {16475, 17150}, {16549, 21361}, {16666, 22034}, {16672, 19747}, {16674, 19745}, {16677, 19746}, {16777, 19722}, {16788, 22001}, {16884, 19739}, {16885, 19732}, {17019, 17379}, {17020, 17490}, {17124, 24003}, {17262, 20182}, {17279, 18139}, {17280, 17778}, {17352, 26724}, {17354, 18134}, {17479, 25245}, {17825, 20905}, {18206, 27163}, {18607, 25099}, {20444, 20896}, {21327, 23543}, {21362, 27070}, {24295, 24943}, {24342, 26037}, {24695, 26034}, {24892, 25385}, {26203, 26211}, {27318, 27646}

X(26224) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26224) lies on these lines: {2, 32}, {64, 1176}, {112, 8362}, {308, 26166}, {6815, 17500}, {7395, 10547}, {7544, 10550}, {7770, 10313}, {10316, 16045}, {11380, 14096}, {16890, 26154}, {26206, 26216}, {26214, 26221}

X(26225) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26225) lies on these lines: {2, 669}, {2501, 7770}, {4580, 26210}

X(26226) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 43

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

X(26226) lies on these lines: {2, 3}, {287, 11441}, {11174, 26216}

leftri

Collineation mappings involving Gemini triangle 44: X(26227)-X(26284)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 44, as in centers X(26227)-X(26284). Then

m(X) = a^2 x - (a^2 + c^2) y - (a^2 + b^2) z : :

A point X lies on the Euler line if and only if m(X) also lies on the Euler line. Also, X lies on the circumcircle if and only if m(X) lies on the circumcircle; specifically, the line XX(2) meets the circumcircle in X and m(X). Moreover, m(m(X)) = X for every point X. (Clark Kimberling, October 31, 2018)

X(26227) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    a^3 - a^2 b - a^2 c - b^2 c - b c^2 : :

X(26227) lies on these lines: {1, 2}, {3, 4968}, {9, 3952}, {12, 5016}, {21, 4385}, {22, 23843}, {31, 1215}, {40, 17164}, {55, 321}, {57, 17140}, {63, 17165}, {75, 100}, {81, 3769}, {86, 9347}, {92, 7466}, {98, 9070}, {105, 9059}, {110, 27958}, {141, 17724}, {183, 3262}, {210, 5278}, {226, 6327}, {251, 18099}, {312, 1621}, {333, 3681}, {341, 5260}, {355, 8229}, {385, 24345}, {405, 3701}, {442, 5300}, {516, 4054}, {518, 1150}, {536, 4689}, {726, 4414}, {740, 2177}, {748, 26688}, {750, 4434}, {752, 24725}, {850, 4477}, {894, 17002}, {902, 3923}, {908, 3883}, {958, 4696}, {964, 5266}, {968, 3995}, {993, 4692}, {1001, 4358}, {1004, 20880}, {1089, 5248}, {1230, 4199}, {1311, 9058}, {1376, 4359}, {1759, 22011}, {1792, 19799}, {1836, 4450}, {1842, 6995}, {1995, 26241}, {2223, 11322}, {2476, 5015}, {2886, 4030}, {3120, 4660}, {3158, 17163}, {3218, 24349}, {3243, 17145}, {3247, 27811}, {3263, 16992}, {3295, 3702}, {3416, 3936}, {3550, 4418}, {3666, 3891}, {3683, 3967}, {3685, 4671}, {3689, 3696}, {3699, 17277}, {3703, 6690}, {3729, 4427}, {3744, 24552}, {3745, 19684}, {3751, 16704}, {3772, 4972}, {3822, 4680}, {3873, 14829}, {3933, 25581}, {3966, 5741}, {3974, 17776}, {4009, 15254}, {4026, 17602}, {4220, 11491}, {4232, 8756}, {4239, 26232}, {4387, 4428}, {4392, 24627}, {4396, 24357}, {4413, 24589}, {4421, 4980}, {4426, 21021}, {4430, 5372}, {4514, 11680}, {4519, 4702}, {4613, 6187}, {4647, 8715}, {4659, 4781}, {4661, 5361}, {4723, 9708}, {4742, 6767}, {4756, 17336}, {4767, 17335}, {4860, 24593}, {4894, 25639}, {4981, 5737}, {5218, 17740}, {5250, 25253}, {5269, 5764}, {5282, 21101}, {5284, 18743}, {5336, 14624}, {5718, 5846}, {5739, 25568}, {5853, 21283}, {6679, 26061}, {7426, 16305}, {7495, 26231}, {8707, 9077}, {9056, 26703}, {9071, 9075}, {9083, 9104}, {9330, 17260}, {9335, 27002}, {13161, 17676}, {16998, 18900}, {17125, 24003}, {17127, 27064}, {17147, 17594}, {17155, 17596}, {17184, 26034}, {17278, 24988}, {17279, 24542}, {17469, 25496}, {17719, 25760}, {17765, 21242}, {17766, 25385}, {26242, 26244}, {26253, 26260}, {26271, 26274}, {27065, 27538}

X(26228) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    3 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(26228) lies on these lines: {1, 2}, {6, 17724}, {7, 109}, {20, 1072}, {23, 11809}, {25, 1068}, {31, 5905}, {55, 7465}, {81, 3475}, {100, 4000}, {105, 1995}, {225, 6995}, {238, 17725}, {278, 7466}, {329, 17127}, {344, 24542}, {345, 3891}, {377, 5266}, {468, 13869}, {518, 24597}, {595, 11415}, {902, 24248}, {908, 7290}, {944, 8229}, {1001, 17602}, {1070, 7398}, {1104, 3436}, {1279, 17720}, {1311, 9088}, {1386, 17718}, {1707, 20078}, {2475, 4339}, {3052, 3782}, {3218, 4310}, {3246, 4679}, {3434, 3744}, {3598, 22464}, {3699, 17352}, {3701, 13742}, {3749, 3914}, {3952, 26685}, {4190, 23536}, {4220, 10267}, {4232, 23710}, {4239, 26241}, {4383, 12595}, {4385, 17526}, {4392, 5744}, {4428, 4854}, {4648, 9347}, {4689, 17301}, {4850, 5218}, {4906, 17728}, {5249, 5269}, {5264, 24159}, {5273, 7226}, {5304, 8557}, {5310, 14798}, {6327, 26132}, {6690, 17599}, {6872, 13161}, {7426, 16272}, {7485, 26357}, {7493, 8758}, {7735, 8609}, {8193, 19850}, {9330, 18230}, {9465, 26278}, {10532, 26118}, {10680, 16434}, {11249, 19649}, {16202, 19544}, {17002, 17257}, {17165, 26065}, {17469, 26098}, {26034, 26128}, {26040, 26724}, {26274, 26281}

X(26229) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    a^4 - a^2 b^2 + 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(26229) lies on these lines: {1, 21208}, {2, 7}, {41, 17048}, {56, 26563}, {75, 26263}, {78, 20247}, {85, 934}, {105, 9086}, {140, 25581}, {183, 3262}, {239, 17001}, {404, 3673}, {474, 20880}, {675, 9058}, {901, 9073}, {976, 24172}, {1055, 24249}, {1210, 21285}, {1329, 7198}, {2082, 26964}, {2280, 24685}, {3007, 7493}, {3665, 6691}, {3814, 7272}, {3825, 4056}, {4193, 4911}, {4239, 26236}, {4376, 20530}, {4386, 27918}, {5433, 27187}, {5804, 7390}, {5826, 17023}, {6745, 10520}, {7247, 11681}, {7264, 25440}, {7289, 27161}, {9310, 26653}, {16609, 26621}, {16862, 25585}, {17683, 24774}, {20930, 26232}, {24471, 24540}, {26241, 26246}, {26242, 26273}, {26247, 26274}

X(26230) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    2 a^3 + a b^2 + b^3 + a c^2 + c^3 : :

X(26230) lies on these lines: {1, 2}, {22, 23383}, {31, 4655}, {37, 24542}, {38, 6679}, {69, 16798}, {81, 16791}, {86, 110}, {100, 16706}, {105, 4239}, {238, 26580}, {321, 17061}, {385, 24348}, {902, 3821}, {940, 16790}, {1385, 8229}, {1386, 3936}, {1441, 15253}, {1621, 7465}, {2887, 17469}, {3007, 7493}, {3589, 17724}, {3618, 16799}, {3662, 17126}, {3663, 4427}, {3722, 4085}, {3744, 4972}, {3745, 18139}, {3772, 24552}, {3952, 17353}, {3953, 6693}, {3977, 4353}, {4202, 5266}, {4358, 17602}, {4689, 17382}, {4968, 17698}, {5294, 17165}, {6327, 25527}, {7466, 17923}, {8610, 9465}, {9059, 9109}, {9330, 17338}, {9347, 17234}, {10130, 26250}, {11115, 23536}, {11319, 13161}, {16793, 17379}, {16795, 24512}, {17002, 17248}, {17127, 27184}, {17356, 24988}, {17716, 25957}, {17770, 21747}, {19284, 24178}, {20905, 25968}, {26256, 26267}, {26259, 26268}

X(26231) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26231) lies on these lines: {2, 11}, {23, 5520}, {119, 7427}, {140, 16823}, {468, 5205}, {498, 19310}, {1329, 17522}, {1478, 19326}, {2862, 4998}, {2968, 6676}, {3011, 16586}, {3756, 7191}, {3757, 7499}, {3912, 11712}, {4223, 27529}, {4242, 20621}, {4579, 26932}, {7426, 26262}, {7493, 9058}, {7495, 26227}, {16020, 17566}, {16048, 26364}, {17004, 26274}

X(26232) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26232) lies on these lines: {2, 31}, {22, 23380}, {48, 21278}, {82, 25505}, {100, 312}, {105, 26238}, {183, 18613}, {251, 18093}, {313, 1631}, {560, 21238}, {561, 789}, {675, 9067}, {1078, 23407}, {1150, 3966}, {2177, 27804}, {3416, 19561}, {3570, 3681}, {3757, 26281}, {3765, 17798}, {3891, 4396}, {3920, 16997}, {4112, 8626}, {4239, 26227}, {7081, 17860}, {8709, 9073}, {9059, 9093}, {10327, 26258}, {17001, 17018}, {20305, 21275}, {20544, 24587}, {20930, 26229}, {26233, 26236}, {26242, 26270}

X(26233) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26233) lies on these lines: {2, 32}, {3, 3266}, {22, 1975}, {23, 76}, {25, 1235}, {39, 15822}, {69, 110}, {98, 9066}, {99, 5987}, {111, 2998}, {183, 1995}, {305, 6636}, {311, 26284}, {316, 5169}, {325, 7495}, {385, 9465}, {468, 7767}, {599, 9516}, {689, 1502}, {733, 9102}, {858, 7750}, {1180, 7839}, {1194, 7805}, {1495, 14994}, {2770, 9150}, {3098, 4576}, {3124, 8177}, {3291, 7780}, {4048, 8627}, {4232, 15589}, {5189, 7802}, {5354, 6179}, {5971, 7496}, {5986, 20023}, {6655, 19577}, {7426, 16335}, {7467, 14880}, {7519, 11185}, {7824, 15302}, {7840, 9829}, {7845, 10163}, {8667, 19221}, {8891, 16932}, {10989, 11057}, {12215, 15080}, {14907, 16063}, {15107, 18906}, {15574, 26283}, {26232, 26236}

X(26234) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26234) lies on these lines: {1, 20911}, {2, 37}, {7, 4388}, {10, 4986}, {21, 99}, {22, 1602}, {38, 3778}, {65, 17152}, {69, 3873}, {72, 17141}, {76, 4968}, {85, 3598}, {86, 7191}, {141, 3726}, {142, 4071}, {183, 3262}, {239, 5276}, {304, 3616}, {322, 15589}, {354, 16739}, {551, 14210}, {612, 3875}, {614, 10436}, {672, 24631}, {675, 9070}, {742, 24512}, {870, 16998}, {942, 17137}, {1125, 1930}, {1228, 4205}, {1269, 8024}, {1290, 2862}, {1402, 1441}, {1909, 5484}, {1962, 18697}, {3230, 24254}, {3264, 26235}, {3622, 18156}, {3663, 4425}, {3670, 24166}, {3673, 13725}, {3701, 18140}, {3877, 24282}, {3896, 3920}, {3953, 16887}, {4021, 4970}, {4223, 16817}, {4361, 5275}, {4385, 18135}, {4514, 20553}, {4692, 6381}, {4696, 6376}, {4981, 5224}, {5268, 17151}, {5272, 25590}, {5297, 17160}, {7081, 20895}, {7264, 20888}, {7763, 25581}, {8682, 16971}, {9310, 16822}, {9318, 27916}, {16583, 26965}, {16600, 16818}, {16601, 27109}, {16604, 16720}, {16830, 17143}, {17007, 17275}, {17023, 21840}, {17024, 17394}, {17140, 20347}, {17754, 24629}, {20271, 26562}, {20955, 25303}, {21443, 23689}, {22232, 27846}, {25261, 26770}, {25263, 27148}, {26244, 26273}

X(26235) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26235) lies on these lines: {2, 39}, {23, 1078}, {69, 5640}, {75, 24988}, {83, 5354}, {98, 9069}, {99, 7496}, {111, 308}, {141, 3124}, {183, 1995}, {264, 4232}, {311, 7495}, {316, 7533}, {338, 11168}, {350, 5297}, {373, 14994}, {468, 1235}, {524, 13410}, {850, 8371}, {1236, 9176}, {1239, 8770}, {1627, 16950}, {1799, 13595}, {1909, 7292}, {3231, 24256}, {3264, 26234}, {4576, 5650}, {5092, 10330}, {5741, 18052}, {7191, 25303}, {7492, 7771}, {7519, 14907}, {7998, 18906}, {9185, 14295}, {11185, 16063}, {15246, 16276}, {18067, 25960}, {21590, 27186}

X(26236) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26236) lies on these lines: {2, 41}, {22, 16681}, {75, 100}, {183, 26264}, {1233, 1626}, {3598, 26245}, {4228, 26238}, {4239, 26229}, {17002, 27624}, {20045, 20247}, {24596, 24789}, {26232, 26233}

X(26237) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26237) lies on these lines: {1, 2}, {22, 16681}, {71, 17142}, {76, 23407}, {99, 310}, {105, 26243}, {183, 18613}, {313, 16684}, {321, 8299}, {350, 1621}, {672, 17165}, {902, 24259}, {1009, 4968}, {1269, 8053}, {2223, 20913}, {2276, 3891}, {3219, 17794}, {3744, 21264}, {3747, 12263}, {3789, 5278}, {4115, 22013}, {4797, 24330}, {7453, 26261}, {7465, 19787}, {17002, 17127}, {26277, 27855}

X(26238) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26238) lies on these lines: {1, 2}, {105, 26232}, {183, 18043}, {675, 932}, {748, 17793}, {902, 24260}, {1447, 7243}, {2108, 17155}, {3941, 18143}, {4228, 26236}, {6327, 20335}, {7465, 19803}, {16684, 18044}, {17140, 17754}, {21264, 24552}, {26241, 26250}

X(26239) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26239) lies on these lines: {2, 44}, {105, 9089}, {183, 3262}, {659, 693}, {1447, 3263}, {4766, 25342}, {9093, 20568}, {26247, 26273}

X(26240) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26240) lies on these lines: {2, 45}, {56, 85}, {75, 4413}, {183, 3262}, {320, 4860}, {350, 5695}, {2726, 20569}, {3304, 20955}, {4361, 16997}, {5211, 17378}, {8649, 24262}, {9318, 24629}, {17274, 18201}, {20172, 27918}

X(26241) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26241) lies on these lines: {1, 19310}, {2, 11}, {3, 16823}, {8, 4223}, {10, 16048}, {22, 1602}, {25, 92}, {35, 19314}, {36, 19326}, {75, 1486}, {111, 9096}, {171, 614}, {183, 18613}, {274, 16876}, {333, 4228}, {379, 20556}, {385, 26274}, {404, 16020}, {612, 3750}, {675, 9086}, {894, 7083}, {927, 2862}, {940, 7191}, {958, 17522}, {999, 19322}, {1281, 20834}, {1311, 9057}, {1447, 1617}, {1958, 2293}, {1995, 26227}, {2175, 17049}, {2223, 11329}, {3290, 3744}, {3295, 16830}, {3303, 19318}, {3550, 5272}, {3684, 3870}, {3705, 25514}, {3746, 19316}, {3920, 5275}, {3996, 10327}, {4224, 5744}, {4239, 26228}, {4336, 17868}, {4339, 17518}, {4363, 16686}, {4436, 23855}, {4438, 25494}, {4459, 26659}, {5010, 19325}, {5015, 7535}, {5020, 7081}, {5205, 11284}, {5248, 19845}, {5276, 17018}, {6998, 10267}, {7295, 24325}, {7379, 11496}, {7385, 11500}, {7427, 22758}, {7453, 26243}, {7493, 26259}, {8193, 16817}, {8298, 17715}, {9059, 9095}, {9746, 15931}, {11248, 21554}, {12329, 17277}, {12410, 16824}, {16608, 21280}, {17000, 20992}, {17792, 26657}, {23865, 26277}, {24199, 24309}, {24320, 24349}, {25279, 25878}, {26229, 26246}, {26238, 26250}

X(26242) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26242) lies on these lines: {1, 41}, {2, 37}, {6, 3726}, {8, 16583}, {9, 38}, {22, 2178}, {31, 3509}, {39, 16614}, {43, 3930}, {45, 7292}, {58, 17736}, {63, 16970}, {81, 16972}, {111, 9070}, {172, 16974}, {183, 26247}, {213, 3868}, {238, 5282}, {241, 3598}, {244, 17754}, {304, 17489}, {319, 17007}, {335, 24514}, {386, 3970}, {451, 17916}, {595, 1759}, {612, 1962}, {644, 9620}, {672, 982}, {675, 9072}, {743, 9068}, {910, 3744}, {941, 6601}, {966, 4981}, {986, 1334}, {1100, 17024}, {1108, 5304}, {1194, 17053}, {1196, 21827}, {1201, 3061}, {1475, 3976}, {1627, 5301}, {1766, 19649}, {1841, 6995}, {2176, 3721}, {2238, 3681}, {2243, 21793}, {2275, 26690}, {2295, 20271}, {2298, 4224}, {2303, 4228}, {2329, 3924}, {2975, 16968}, {3116, 24513}, {3208, 4642}, {3230, 3735}, {3496, 3915}, {3501, 24443}, {3617, 16605}, {3670, 3730}, {3673, 26978}, {3679, 16611}, {3684, 3938}, {3705, 21796}, {3727, 3890}, {3731, 5272}, {3782, 17747}, {3876, 3954}, {3889, 20963}, {3896, 10327}, {3920, 5275}, {3950, 4970}, {3953, 4253}, {3959, 14923}, {3997, 5902}, {4071, 25957}, {4385, 27040}, {4868, 9331}, {4911, 26099}, {5015, 26085}, {5089, 6353}, {5262, 16048}, {5266, 17562}, {5268, 16673}, {5279, 25494}, {5283, 16823}, {5297, 16672}, {5749, 20227}, {5839, 19993}, {6998, 25090}, {7426, 16307}, {7735, 8609}, {8607, 22240}, {8610, 9465}, {9347, 20998}, {11115, 16716}, {14482, 16020}, {16549, 24046}, {17355, 24165}, {17750, 21802}, {18600, 25237}, {20875, 20990}, {20911, 27248}, {21073, 23537}, {21281, 26562}, {21813, 27184}, {22021, 22196}, {26227, 26244}, {26229, 26273}, {26232, 26270}, {26252, 26260}

X(26242) = complement of X(31130)
X(26242) = anticomplement of X(30748)

X(26243) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    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(26243) lies on these lines: {2, 6}, {8, 6998}, {21, 3948}, {35, 4044}, {76, 21511}, {92, 4231}, {98, 100}, {105, 26237}, {111, 9067}, {187, 16046}, {226, 4987}, {274, 25946}, {329, 7413}, {980, 7751}, {1078, 21495}, {1230, 27174}, {1444, 3770}, {1447, 4359}, {1959, 17739}, {1975, 21508}, {2857, 9090}, {2975, 3765}, {3666, 4396}, {4239, 26227}, {4683, 5988}, {5249, 24602}, {5277, 26643}, {5337, 7780}, {7438, 26268}, {7449, 26264}, {7453, 26241}, {9070, 9093}, {11349, 20913}, {16050, 27040}, {16609, 25998}, {19649, 22712}, {26252, 26258}

X(26244) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26244) lies on these lines: {2, 6}, {8, 21965}, {9, 1755}, {10, 98}, {21, 27040}, {32, 13740}, {37, 893}, {76, 16060}, {99, 21937}, {111, 9059}, {115, 17677}, {172, 1220}, {187, 4234}, {198, 1376}, {232, 2322}, {257, 27954}, {281, 4231}, {339, 22366}, {404, 26035}, {612, 3725}, {673, 21264}, {846, 3985}, {904, 27880}, {958, 20471}, {1010, 5277}, {1043, 18755}, {1078, 16061}, {1215, 3509}, {1222, 17962}, {1384, 11354}, {1434, 4754}, {1447, 3739}, {1975, 22267}, {2247, 25607}, {2271, 10449}, {2476, 26085}, {2759, 9136}, {3053, 4195}, {3207, 5793}, {3290, 3757}, {3684, 3741}, {3686, 24239}, {3705, 17275}, {3767, 16062}, {3769, 16972}, {3840, 16503}, {3934, 17681}, {4201, 5254}, {4239, 26258}, {4386, 5263}, {4643, 7179}, {4972, 17737}, {5283, 19270}, {5299, 19864}, {5750, 17122}, {5976, 6626}, {5980, 21898}, {5981, 21869}, {5988, 24697}, {5989, 9509}, {6175, 26079}, {7172, 17314}, {7380, 9753}, {7453, 15621}, {7793, 17688}, {10311, 11109}, {11110, 16589}, {11683, 27697}, {16605, 16824}, {16823, 17448}, {17206, 17499}, {17388, 20056}, {17763, 21840}, {19278, 27523}, {21554, 22712}, {26227, 26242}, {26234, 26273}, {26250, 26251}

X(26245) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26245) lies on these lines: {1, 2}, {69, 17724}, {105, 9104}, {144, 17002}, {675, 1293}, {902, 24280}, {3210, 5281}, {3475, 3769}, {3598, 26236}, {7426, 16304}, {7465, 19789}, {7474, 16704}, {10565, 20222}

X(26246) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26246) lies on these lines: {1, 2}, {105, 9057}, {675, 934}, {902, 24283}, {3693, 3891}, {4184, 16750}, {7465, 19790}, {10025, 17127}, {26229, 26241}

X(26247) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26247) lies on these lines: {1, 2}, {9, 17002}, {37, 4396}, {183, 26242}, {335, 675}, {902, 17738}, {1447, 4552}, {4434, 24602}, {4766, 17719}, {4968, 16061}, {5266, 17686}, {6590, 11068}, {7465, 19791}, {7754, 25082}, {9347, 20131}, {26229, 26274}, {26239, 26273}

X(26248) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26248) lies on these lines: {2, 661}, {22, 23864}, {523, 21205}, {649, 4486}, {650, 16757}, {659, 693}, {675, 2752}, {798, 8060}, {850, 14296}, {1311, 2856}, {1447, 4077}, {3716, 20295}, {3733, 18160}, {4122, 4467}, {4761, 16830}, {4897, 18004}, {4913, 17161}, {5224, 9013}, {6133, 20906}, {8062, 17217}, {18155, 18158}, {27193, 27294}

X(26249) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    a (b - c) (a^4 + 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(26249) lies on these lines: {2, 667}, {23, 5990}, {25, 17924}, {81, 9010}, {105, 9081}, {513, 5040}, {612, 4063}, {649, 24462}, {650, 18108}, {669, 804}, {675, 9073}, {693, 21005}, {901, 1633}, {2517, 4057}, {3309, 4220}, {3835, 8635}, {3920, 4083}, {4782, 5297}, {5996, 8639}, {8642, 26146}, {8646, 20295}, {8654, 25537}, {9082, 9111}

X(26250) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26250) lies on these lines: {2, 31}, {100, 350}, {183, 3262}, {321, 1376}, {334, 9075}, {901, 1311}, {1155, 20716}, {1281, 5205}, {1429, 20352}, {1631, 18044}, {2517, 4057}, {3006, 5137}, {3240, 17001}, {4495, 4613}, {5297, 16999}, {7081, 20237}, {8626, 24294}, {9059, 9081}, {10130, 26230}, {26238, 26241}, {26244, 26251}, {26264, 26271}

X(26251) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    2 a^2 b + a b^2 + b^3 + 2 a^2 c + 2 b^2 c + a c^2 + 2 b c^2 + c^3 : :

X(26251) lies on these lines: {1, 2}, {100, 9077}, {675, 1268}, {902, 24295}, {1213, 3124}, {1215, 17184}, {1995, 23854}, {2243, 17369}, {3264, 26234}, {3699, 17307}, {3701, 13728}, {3739, 24988}, {3775, 21805}, {3844, 3936}, {3952, 4357}, {4026, 4358}, {4239, 26262}, {4427, 17355}, {4689, 17359}, {4756, 17258}, {4970, 6535}, {7465, 19808}, {7485, 23361}, {8229, 9956}, {9330, 17248}, {9347, 17381}, {17126, 17368}, {17357, 24542}, {24695, 26034}, {25001, 25882}, {26244, 26250}

X(26252) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26252) lies on these lines: {2, 3}, {101, 306}, {111, 1305}, {1297, 9057}, {3430, 26006}, {26242, 26260}, {26243, 26258}

X(26253) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26253) lies on these lines: {2, 3}, {100, 2373}, {111, 13397}, {1297, 9058}, {3101, 5297}, {9070, 26703}, {26227, 26260}, {26265, 26266}

X(26254) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26254) lies on these lines: {2, 3}, {109, 307}, {1297, 9056}

X(26255) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    7 a^6 - a^4 b^2 - 7 a^2 b^4 + b^6 - a^4 c^2 + 18 a^2 b^2 c^2 - b^4 c^2 - 7 a^2 c^4 - b^2 c^4 + c^6 : :

X(26255) lies on these lines: {2, 3}, {6, 20192}, {69, 10546}, {110, 1992}, {111, 1302}, {476, 10102}, {597, 3066}, {1007, 7664}, {1285, 1383}, {1384, 16317}, {1495, 11179}, {2373, 9064}, {2393, 5640}, {2770, 9060}, {3580, 11180}, {3618, 10545}, {5642, 20423}, {7665, 7774}, {7737, 10418}, {8263, 11160}, {8585, 21843}, {8644, 21732}, {9058, 9061}, {9143, 20772}, {11002, 14984}, {11693, 13352}, {16279, 16319}, {18928, 26881}

X(26255) = anticomplement of X(32216)

X(26256) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26256) lies on these lines: {2, 3}, {7735, 8609}, {26230, 26267}

X(26257) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26257) lies on these lines: {2, 3}, {111, 308}, {115, 11056}, {141, 7665}, {305, 7781}, {385, 9465}, {574, 11059}, {1078, 3291}, {1194, 7760}, {1196, 1799}, {2373, 9229}, {3266, 7783}, {3329, 26276}, {5254, 19577}, {7664, 7931}, {7831, 10418}, {7842, 15820}, {7898, 9745}, {10163, 14061}, {24726, 25344}

X(26258) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26258) lies on these lines: {2, 7}, {8, 101}, {41, 12649}, {169, 10527}, {388, 27068}, {631, 25082}, {644, 5657}, {910, 3434}, {1055, 24247}, {1759, 11415}, {2082, 10529}, {2329, 5554}, {2975, 6554}, {3554, 5304}, {3872, 8074}, {4232, 8756}, {4239, 26244}, {4302, 21090}, {4936, 9588}, {5227, 27522}, {5552, 17742}, {5819, 11680}, {6910, 16601}, {6921, 25066}, {7195, 27006}, {7288, 26690}, {7735, 8609}, {10327, 26232}, {17001, 17316}, {17744, 26364}, {26243, 26252}

X(26259) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26259) lies on these lines: {2, 12}, {140, 5205}, {468, 16823}, {993, 16067}, {3757, 6676}, {7081, 7499}, {7426, 26261}, {7493, 26241}, {7495, 26227}, {26230, 26268}

X(26260) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26260) lies on these lines: {2, 19}, {22, 1602}, {25, 1441}, {105, 1305}, {183, 26268}, {304, 1310}, {347, 1447}, {1231, 3556}, {1370, 20291}, {1973, 26203}, {2373, 9070}, {3007, 7493}, {6360, 26274}, {7520, 16823}, {8193, 20235}, {9086, 26703}, {26227, 26253}, {26242, 26252}

X(26261) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26261) lies on these lines: {2, 35}, {23, 16823}, {100, 17263}, {105, 4239}, {678, 5297}, {931, 9094}, {1302, 1311}, {1995, 26227}, {3006, 4223}, {3757, 13595}, {4359, 20988}, {5205, 16042}, {7295, 26627}, {7426, 26259}, {7453, 26237}, {20872, 24589}

X(26262) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26262) lies on these lines: {2, 36}, {23, 5205}, {100, 17264}, {1995, 26227}, {2517, 4057}, {2726, 9059}, {2752, 9070}, {4239, 26251}, {4358, 20989}, {5329, 26688}, {7081, 13595}, {7426, 26231}, {7449, 26266}, {16042, 16823}, {20875, 23386}

X(26263) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26263) lies on these lines: {2, 38}, {75, 26229}, {518, 5741}, {1311, 26711}, {3112, 9073}, {4239, 26227}, {5258, 16823}, {7081, 20237}

X(26264) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26264) lies on these lines: {2, 12}, {3, 5205}, {22, 1603}, {25, 318}, {45, 2243}, {63, 9364}, {100, 198}, {105, 9104}, {183, 26236}, {197, 312}, {612, 5250}, {1089, 19845}, {1311, 9059}, {1460, 27064}, {1698, 19844}, {1995, 26227}, {2223, 11345}, {3011, 16048}, {3596, 8707}, {3699, 12329}, {3701, 11337}, {3757, 5020}, {3890, 3920}, {4220, 26935}, {4239, 26244}, {4434, 7295}, {5121, 8666}, {5211, 12513}, {7085, 27538}, {7449, 26243}, {7493, 9058}, {11284, 16823}, {26250, 26271}

X(26265) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26265) lies on these lines: {2, 7}, {77, 20248}, {100, 312}, {198, 1229}, {220, 24633}, {1055, 24266}, {1696, 24547}, {1995, 26227}, {2324, 21273}, {5227, 27108}, {6078, 9073}, {9057, 26703}, {9095, 9104}, {9310, 26621}, {11683, 26669}, {16609, 26653}, {17134, 20927}, {20244, 24590}, {20262, 21286}, {26253, 26266}

X(26266) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 44

Barycentrics    a^6 + a^5 b - 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^3 b 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(26266) lies on these lines: {2, 58}, {98, 9059}, {100, 4043}, {183, 1995}, {199, 1230}, {313, 835}, {1311, 9070}, {2373, 9057}, {3006, 6998}, {4239, 26227}, {7449, 26262}, {7453, 26237}, {26244, 26250}, {26253, 26265}

X(26267) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26267) lies on these lines: {2, 7}, {92, 108}, {100, 20173}, {105, 9057}, {198, 17863}, {321, 1376}, {612, 9746}, {614, 8054}, {919, 6654}, {1055, 24268}, {1696, 25001}, {1999, 17001}, {2178, 17134}, {3086, 5813}, {3187, 3684}, {3550, 24428}, {3673, 11349}, {3742, 19684}, {4232, 23710}, {4239, 26227}, {4414, 4656}, {5227, 27039}, {5739, 24477}, {7191, 20277}, {8557, 14543}, {9310, 16609}, {14557, 17626}, {16412, 20880}, {21270, 24005}, {26230, 26256}, {26242, 26252}

X(26268) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26268) lies on these lines: {2, 65}, {100, 1229}, {183, 26260}, {314, 931}, {1302, 26703}, {1311, 9070}, {1995, 26227}, {3757, 4223}, {4385, 19256}, {4968, 19245}, {7081, 23528}, {7438, 26243}, {7735, 8609}, {11688, 17862}, {26230, 26259}

X(26269) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26269) lies on these lines: {2, 66}, {98, 7505}, {232, 800}, {315, 827}, {1995, 7792}, {3090, 7852}, {3518, 9753}, {7556, 12253}

X(26270) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26270) lies on these lines: {2, 82}, {251, 18082}, {321, 16277}, {831, 1930}, {1402, 1441}, {9070, 9076}, {10130, 26230}, {26232, 26242}

X(26271) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26271) lies on these lines: {2, 87}, {183, 18043}, {932, 6376}, {9059, 9082}, {26227, 26274}, {26250, 26264}

X(26272) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26272) lies on these lines: {2, 45}, {100, 2726}, {105, 9059}, {5260, 9369}, {8649, 24277}

X(26273) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26273) lies on these lines: {1, 24685}, {2, 45}, {19, 28023}, {100, 24403}, {101, 21208}, {105, 659}, {106, 514}, {111, 675}, {183, 26274}, {241, 292}, {244, 9318}, {335, 27912}, {385, 3226}, {524, 5211}, {527, 5121}, {536, 5205}, {544, 6788}, {614, 3248}, {664, 9259}, {673, 27918}, {743, 9073}, {1015, 3732}, {1054, 24398}, {1647, 24712}, {3125, 24203}, {3699, 9055}, {3756, 5845}, {4000, 26007}, {4360, 16997}, {4644, 4860}, {5272, 9359}, {5275, 16518}, {5304, 23972}, {8649, 24281}, {9083, 9109}, {9094, 9110}, {9095, 9097}, {17063, 24333}, {17321, 26629}, {17719, 25342}, {24358, 25531}, {24841, 27921}, {26229, 26242}, {26234, 26244}, {26239, 26247}

X(26274) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26274) lies on these lines: {1, 21216}, {2, 37}, {38, 17257}, {69, 3726}, {105, 330}, {183, 26273}, {193, 3873}, {194, 16823}, {385, 26241}, {612, 17319}, {614, 894}, {3230, 24282}, {3241, 17497}, {3616, 17489}, {3729, 4011}, {3730, 24166}, {4223, 19851}, {4360, 5275}, {4393, 5276}, {4970, 5268}, {5211, 7774}, {5550, 25263}, {6360, 26260}, {7191, 17379}, {16020, 25242}, {17001, 20045}, {17004, 26231}, {17480, 21226}, {17760, 21214}, {20271, 21281}, {21840, 26626}, {24349, 24514}, {26227, 26271}, {26228, 26281}, {26229, 26247}

X(26275) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26275) lies on these lines: {2, 900}, {105, 659}, {351, 523}, {513, 1638}, {522, 4763}, {551, 23888}, {665, 3290}, {918, 4448}, {1960, 10015}, {2786, 3716}, {2804, 11124}, {2826, 14419}, {3004, 26277}, {3776, 8689}, {4435, 5275}, {4555, 9089}, {6050, 21185}, {6366, 25569}, {6550, 14422}, {8638, 20875}, {11712, 24685}

X(26276) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26276) lies on these lines: {2, 187}, {23, 99}, {25, 8024}, {32, 16055}, {69, 10546}, {76, 14002}, {98, 9080}, {111, 385}, {126, 6781}, {183, 1995}, {325, 3233}, {328, 476}, {340, 4232}, {511, 5468}, {524, 2502}, {669, 804}, {754, 10418}, {1007, 7493}, {1078, 16042}, {1236, 13595}, {1302, 2857}, {2374, 2858}, {3329, 26257}, {3793, 16317}, {5104, 5108}, {5914, 22329}, {6082, 9084}, {6325, 18023}, {7492, 11059}, {7533, 11056}, {7665, 7779}, {7766, 9465}, {9146, 15107}, {9775, 11676}, {9855, 10717}, {10754, 13192}

X(26276) = isotomic conjugate of the isogonal conjugate of X(32217)

X(26277) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26277) lies on these lines: {2, 649}, {23, 5991}, {86, 9002}, {105, 9073}, {514, 5029}, {659, 693}, {661, 4817}, {667, 3766}, {669, 804}, {675, 2726}, {927, 9057}, {1311, 2862}, {1443, 1447}, {1978, 8709}, {3004, 26275}, {3261, 4057}, {4025, 13246}, {4106, 4782}, {4406, 4491}, {4885, 24623}, {6586, 10566}, {6590, 11068}, {9059, 9089}, {17072, 21303}, {20316, 21304}, {23865, 26241}, {26237, 27855}

X(26278) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26278) lies on these lines: {2, 668}, {98, 9079}, {105, 111}, {106, 14438}, {385, 17961}, {513, 739}, {675, 743}, {1180, 13006}, {1415, 1627}, {5304, 23980}, {9082, 9111}, {9465, 26228}

X(26279) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26279) lies on these lines: {1, 17001}, {2, 7}, {105, 9096}, {183, 26242}, {257, 1311}, {385, 7191}, {614, 17002}, {1055, 24291}, {1201, 17739}, {2975, 25994}, {3705, 17007}, {3920, 16997}, {5297, 16999}, {7292, 16998}, {11285, 25082}, {26227, 26271}, {26234, 26244}, {26561, 27068}, {26959, 27010}, {26971, 26977}

X(26280) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26280) lies on these lines: {2, 896}, {659, 693}, {1281, 5205}, {1290, 1311}, {3248, 7292}, {4239, 26227}, {5563, 16823}

X(26281) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26281) lies on these lines: {2, 38}, {105, 9068}, {183, 3262}, {675, 9071}, {870, 9073}, {3757, 26232}, {3873, 4417}, {4359, 4413}, {8610, 9465}, {8666, 16823}, {26228, 26274}

X(26282) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26282) lies on these lines: {2, 6}, {31, 908}, {105, 1995}, {187, 24296}, {609, 24630}, {985, 17719}, {1447, 4850}, {1914, 17720}, {2298, 27254}, {3972, 11352}, {5988, 24725}, {6998, 19767}, {8229, 9753}, {16020, 19316}, {16412, 16752}

X(26283) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26283) lies on these lines: {2, 3}, {74, 19376}, {110, 159}, {111, 13398}, {161, 394}, {925, 2373}, {1351, 15135}, {1993, 15073}, {2697, 16167}, {3100, 10833}, {4296, 18954}, {5640, 19121}, {9464, 22241}, {9465, 10313}, {9914, 12279}, {9919, 11820}, {9937, 11412}, {10316, 14580}, {11064, 15577}, {11416, 11422}, {11750, 19908}, {12289, 12301}, {12310, 15106}, {15574, 26233}, {19377, 19381}

X(26284) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 44

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

X(26284) lies on these lines: {2, 3}, {110, 20987}, {161, 3060}, {311, 26233}, {1176, 5640}, {1288, 2373}, {19153, 27085}

X(26285) =  COMPLEMENT OF X(10525)

Barycentrics    a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c+a^2 b^2 c-2 a b^3 c-2 a^3 c^2+a^2 b c^2+b^3 c^2+2 a^2 c^3-2 a b c^3+b^2 c^3+a c^4-c^5) : :
X(26285) = 3*X[2]-X[10525], X[20]+X[10526], X[30]-X[6796], X[511]-X[5495], X[528]-X[10943], X[550]-X[5841], X[952]-X[5450], X[1158]-X[2771], X[2829]-X[10942], X[3189]+3*X[5770], 3*X[4421]+X[12114], X[5844]-X[8666], X[14988]-X[22836]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28557.

X(26285) lies on these lines: {1,3}, {2,10525}, {5,3035}, {8,6950}, {10,6914}, {12,24466}, {20,10526}, {21,25005}, {24,1872}, {30,6796}, {78,5694}, {100,355}, {104,3871}, {140,3816}, {404,5886}, {405,11231}, {474,11230}, {496,6713}, {497,6961}, {498,6923}, {511,5495}, {528,10943}, {549,10199}, {550,5841}, {601,5396}, {603,5399}, {946,6924}, {952,5450}, {962,6942}, {993,5690}, {1012,11499}, {1030,1766}, {1158,2771}, {1376,3560}, {1479,6958}, {1483,25439}, {1490,12660}, {1538,3149}, {1621,6940}, {1698,7489}, {1837,10058}, {2550,6892}, {2829,10942}, {2932,4855}, {3085,6948}, {3189,5770}, {3434,6977}, {3474,5761}, {3526,5259}, {3583,6971}, {3651,5812}, {3654,17549}, {3656,13587}, {3811,12341}, {3885,12737}, {4188,5603}, {4189,5657}, {4276,15952}, {4294,6891}, {4302,6928}, {4421,12114}, {4640,14454}, {4848,17010}, {4996,14923}, {5218,6850}, {5225,6978}, {5250,19524}, {5267,11362}, {5310,16434}, {5432,6842}, {5440,5887}, {5552,6938}, {5587,13743}, {5687,22758}, {5691,18524}, {5777,11517}, {5844,8666}, {5881,12331}, {6265,17100}, {6284,6882}, {6831,18407}, {6847,18517}, {6876,9778}, {6905,12699}, {6909,11491}, {6911,9955}, {6921,10531}, {6952,13199}, {6966,12116}, {6972,20066}, {7491,15338}, {7701,13146}, {7741,10738}, {8553,21853}, {9817,13222}, {10090,11376}, {10785,20075}, {11929,12943}, {12528,12738}, {12611,12775}, {12645,18515}, {14988,22836}, {15171,15845}, {17662,18976}, {19525,19860}

Let A'B'C' be the medial triangle. Let LA be the reflection of line B'C' in the internal angle bisector of A, and define LB and LC cyclically. Let A" = LB∩LC, B" = LC∩LA, C" = LA∩LB. A"B"C" is the mid-triangle of the intangents and tangential triangles. A"B"C" is homothetic to the intangents, extangents, and tangential triangles at X(55), and to the Kosnita triangle at X(26285). (Randy Hutson, June 7, 2019)

X(26285) = midpoint of X(i) and X(j) for these {i,j}: {3,11248}, {3811,24467}, {5450,8715}
X(26285 ) = reflection of X(1482) in X(11567)
X(26285) = complement of X(10525)
X(26285) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,40,25413}, {3,55,1385}, {3,56,23961}, {3,1482,36}, {3,3295,10269}, {3,10267,13624}, {3,10306,11249}, {3,10310,3579}, {3,10679,56}, {3,10680,5204}, {3,11508,18857}, {3,11849,1}, {3,12702,11012}, {3,22765,7280}, {35,2077,3}, {40,5010,3}, {55,8071,9957}, {56,10679,10222}, {100,6906,355}, {1012,11499,18480}, {1376,3560,9956}, {1385,10222,25405}, {1385,10284,1}, {1470,11508,24928}, {3295,10269,15178}, {5217,10310,3}, {5432,11826,6842}, {5537,11012,12702}, {6909,11491,18481}, {6911,11496,9955}, {7280,7982,22765}, {8069,11509,942}, {10222,23961,56}, {11248,11249,10306}

X(26285) = X(10224)-of-excentral-triangle

X(26286) =  COMPLEMENT OF X(10526)

Barycentrics    a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c-a^2 b^2 c-2 a b^3 c+2 b^4 c-2 a^3 c^2-a^2 b c^2+4 a b^2 c^2-b^3 c^2+2 a^2 c^3-2 a b c^3-b^2 c^3+a c^4+2 b c^4-c^5) : :
X(26286) = 3*X[2]-X[10526], X[20]+X[10525], X[30]-X[3829], X[529]-X[10942], X[952]-X[6796], X[2771]-X[6261], X[2818]-X[10282], X[5842]-X[10943], X[5844]-X[8715], X[6985]+X[12114], 3*X[11194]+X[11500]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28557.

X(26286) lies on these lines: {1,3}, {2,10526}, {5,993}, {8,6942}, {10,6924}, {11,7491}, {20,10525}, {21,5886}, {30,3829}, {48,5755}, {63,5694}, {78,22935}, {84,6597}, {104,411}, {140,25466}, {355,2975}, {378,1872}, {382,18515}, {388,6954}, {405,11230}, {474,11231}, {495,15865}, {499,6928}, {529,10942}, {548,12511}, {549,10197}, {550,1484}, {573,7113}, {946,5267}, {952,6796}, {956,11499}, {958,6911}, {962,6950}, {1006,5253}, {1012,22793}, {1193,5398}, {1437,3417}, {1468,5396}, {1478,6863}, {1656,5251}, {1699,13743}, {1766,5124}, {2551,6970}, {2771,6261}, {2818,10282}, {2915,8279}, {3086,6868}, {3149,18480}, {3218,21740}, {3436,6880}, {3560,9955}, {3583,15446}, {3585,6980}, {3616,6875}, {3632,12331}, {3653,21161}, {3654,13587}, {3656,17549}, {3869,4996}, {3916,5887}, {4188,5657}, {4189,5603}, {4278,15952}, {4293,6825}, {4299,6923}, {4973,5884}, {5080,6949}, {5248,5901}, {5250,19525}, {5258,5790}, {5260,6946}, {5265,6987}, {5288,12645}, {5303,6906}, {5322,19544}, {5428,11281}, {5433,6882}, {5690,25440}, {5731,6876}, {5842,10943}, {5844,8715}, {5881,18524}, {6326,6763}, {6713,6922}, {6827,7288}, {6842,7354}, {6848,18516}, {6881,24953}, {6883,25524}, {6885,19843}, {6910,10532}, {6934,10527}, {6960,20067}, {6962,12115}, {6985,12114}, {7489,8227}, {10058,12701}, {10090,12619}, {10786,20076}, {10913,18763}, {11194,11500}, {11483,11512}, {11928,12953}, {12053,17010}, {12515,18861}, {12556,12913}, {15326,15908}, {15844,18990}, {15888,21155}, {17734,19550}, {18761,19541}, {19524,19860}, {19861,21165}

X(26286) = midpoint of X(i) and X(j) for these {i,j}: {3,11249}, {20,10525}, {6261,24467}, {6796,8666}, {6985,12114}, {11248,22770}
X(26286) = complement of X(10526)
X(26286) = X(13406)-of-excentral-triangle
X(26286) = 2nd-isogonal-triangle-of-X(1)-to-ABC similarity image of X(3)
X(26286) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,56,1385}, {3,999,10267}, {3,1482,35}, {3,3428,3579}, {3,5204,23961}, {3,10246,10902}, {3,10269,13624}, {3,10679,5217}, {3,10680,55}, {3,11849,5010}, {3,12702,2077}, {3,22765,1}, {3,22770,11248}, {36,11012,3}, {40,7280,3}, {55,10680,10222}, {56,5204,7742}, {56,7742,5126}, {56,8071,942}, {104,411,18481}, {484,11014,25413}, {946,5267,6914}, {958,6911,9956}, {999,10267,15178}, {2975,6905,355}, {3149,22758,18480}, {3428,5204,3}, {3560,22753,9955}, {3579,23961,3}, {5010,7982,11849}, {5433,11827,6882}, {5563,10902,10246}, {5901,7508,5248}, {8069,10966,9957}, {11248,11249,22770}, {13373,13624,1385}

X(26287) =  X(1)X(3)∩X(5)X(214)

Barycentrics    -a (2 a^6-3 a^5 b-3 a^4 b^2+6 a^3 b^3-3 a b^5+b^6-3 a^5 c+8 a^4 b c-3 a^3 b^2 c-6 a^2 b^3 c+6 a b^4 c-2 b^5 c-3 a^4 c^2-3 a^3 b c^2+10 a^2 b^2 c^2-3 a b^3 c^2-b^4 c^2+6 a^3 c^3-6 a^2 b c^3-3 a b^2 c^3+4 b^3 c^3+6 a b c^4-b^2 c^4-3 a c^5-2 b c^5+c^6) : :
X(26287) = X[2771]-X[5450], X[5840]-X[5901]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28557.

X(26287) lies on these lines: {1,3}, {5,214}, {355,6224}, {631,2320}, {944,6972}, {1389,13587}, {1483,11715}, {2475,5886}, {2476,11230}, {2771,5450}, {3616,6951}, {3871,12737}, {4511,5694}, {5443,12119}, {5693,18515}, {5731,6903}, {5840,5901}, {6261,12524}, {6265,6906}, {6830,18480}, {6840,18481}, {10950,12619}, {11231,25005}, {18357,20400}

X(26287) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,35,25414}, {1,11849,10284}, {3,1482,484}, {3,5903,10225}, {1385,10222,1319}, {1385,24929,15178}, {6224,6952,355}, {10222,10225,5903}, {13624,15178,9940}

X(26288) =  REFLECTION OF X(4) IN X(591)

Barycentrics    5 a^4-4 a^2 b^2-b^4-4 a^2 c^2+2 b^2 c^2-c^4+4 (2 a^2-b^2-c^2) S : :
X(26288) =2 X[1991] - 3 X[3524], 7 X[488] - 4 X[6311], X[5871] - 4 X[9733].

See Tran Quang Hung, Antreas Hatzipolakis and Peter Moses, Hyacinthos 28564.

X(26288) lies on these lines: {2,372}, {3,5861}, {4,591}, {30,1160}, {148,22601}, {193,9541}, {194,13678}, {371,13712}, {376,524}, {490,1588}, {492,23249}, {754,8982}, {1270,6560}, {1271,6396}, {1991,3524}, {3593,6564}, {3594,7375}, {5591,6398}, {5871,9733}, {6231,9880}, {9770,13674}, {10783,12305}, {23269,23311}

X(26288) = reflection of X(i) in X(j) for these {i,j}: {4, 591}, {5861, 3}

X(26289) =  REFLECTION OF X(4) IN X(1991)

Barycentrics    5 a^4-4 a^2 b^2-b^4-4 a^2 c^2+2 b^2 c^2-c^4-4 (2 a^2-b^2-c^2) S : :
X(26289) =2 X[591] - 3 X[3524], 7 X[487] - 4 X[6315], X[5870] - 4 X[9732].

See Tran Quang Hung, Antreas Hatzipolakis and Peter Moses, Hyacinthos 28564.

X(26289) lies on these lines: {2,371}, {3,5860}, {4,1991}, {30,1161}, {69,9541}, {148,22630}, {194,13798}, {372,13835}, {376,524}, {489,1587}, {591,3524}, {1270,6200}, {1271,6561}, {3592,7376}, {3595,6565}, {5590,6221}, {5870,9732}, {6230,9880}, {9770,13794}, {10784,12306}, {23275,23312}

X(26289) = reflection of X(i) in X(j) for these {i,j}: {4, 1991}, {5860, 3}

leftri

Endo-homothetic centers: X(26290)-X(26525)

rightri

This preamble and centers X(26290)-X(26525) were contributed by César Eliud Lozada, October 31, 2018.

This section comprises the endo-homothetic centers of the family of triangles homothetic with the reference triangle ABC. This family is composed by the following 40 triangles:

ABC, ABC-X3 reflections, anti-Aquila, anti-Ara, 5th anti-Brocard, 2nd anti-circumperp-tangential, anti-Euler, anti-inner-Grebe, anti-outer-Grebe, anti-Mandart-incircle, anticomplementary, Aquila, Ara, 1st Auriga, 2nd Auriga, 5th Brocard, 2nd circumperp tangential, Ehrmann-mid, Euler, outer-Garcia, Gossard, inner-Grebe, outer-Grebe, Johnson, inner-Johnson, outer-Johnson, 1st Johnson-Yff, 2nd Johnson-Yff, Lucas homothetic, Lucas(-1) homothetic, Mandart-incircle, medial, 5th mixtilinear, 3rd tri-squares-central, 4th tri-squares-central, X3-ABC reflections, inner-Yff, outer-Yff, inner-Yff tangents, outer-Yff tangents.

For definitions and coordinates of these triangles, see the index of triangles referenced in ETC.

X(26290) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND 1st AURIGA

Barycentrics    a*(4*S^2*D+a*(a+b+c)*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)*(b-c)^3)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26290) = (-D+4*R*S)*X(1)-4*(r+2*R)*S*X(3)

X(26290) lies on these lines: {1,3}, {2,26326}, {4,26359}, {20,26394}, {30,26383}, {182,26379}, {371,26385}, {372,26384}, {376,26381}, {515,26382}, {1593,26371}, {1657,18496}, {3098,26310}, {6284,26387}, {7354,26388}, {11414,26302}, {11825,26344}, {11826,26390}, {11827,26389}, {15908,26413}, {26292,26391}, {26293,26392}, {26294,26396}, {26295,26397}

X(26290) = reflection of X(11822) in X(3)
X(26290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3428, 26291), (999, 14110, 26291)

X(26291) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND 2nd AURIGA

Barycentrics    a*(-4*S^2*D+a*(a+b+c)*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)*(b-c)^3)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26291) = (D+4*R*S)*X(1)-4*(r+2*R)*S*X(3)

X(26291) lies on these lines: {1,3}, {2,26327}, {4,26360}, {20,26418}, {30,26407}, {182,26403}, {371,26409}, {372,26408}, {376,26405}, {515,26406}, {1593,26372}, {1657,18498}, {3098,26311}, {6284,26411}, {7354,26412}, {11414,26303}, {11824,26335}, {11825,26345}, {11826,26414}, {11827,26413}, {15908,26389}, {26292,26415}, {26293,26416}, {26294,26420}, {26295,26421}

X(26291) = reflection of X(11823) in X(3)
X(26291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3428, 26290), (999, 14110, 26290)

X(26292) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND LUCAS HOMOTHETIC

Barycentrics    (SB+SC)*(-(2*S^2-(4*R^2-SA+2*SW)*SA)*S+(SA-SW)*S^2+SA*SW^2) : :

X(26292) lies on these lines: {2,26328}, {3,493}, {4,488}, {6,13011}, {20,26494}, {30,26447}, {55,26433}, {56,26353}, {165,26298}, {182,26427}, {371,26460}, {372,26454}, {376,26439}, {515,26442}, {517,26495}, {1160,12164}, {1306,11412}, {1593,26373}, {1657,18521}, {2077,26500}, {3098,26312}, {3428,26322}, {3576,26367}, {6284,26471}, {6464,26293}, {7354,26477}, {8950,13023}, {10310,26493}, {11012,26499}, {11249,26501}, {11414,26304}, {11824,26337}, {11825,26347}, {11826,26488}, {11827,26483}, {12305,13021}, {19443,19497}, {26290,26391}, {26291,26415}, {26294,26496}, {26295,26497}

X(26292) = reflection of X(11828) in X(3)
X(26292) = {X(3), X(11949)}-harmonic conjugate of X(11198)

X(26293) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND LUCAS(-1) HOMOTHETIC

Barycentrics    (SB+SC)*((2*S^2-(4*R^2-SA+2*SW)*SA)*S+(SA-SW)*S^2+SA*SW^2) : :

X(26293) lies on these lines: {2,26329}, {3,494}, {4,487}, {6,13012}, {20,26503}, {30,26448}, {55,26434}, {56,26354}, {165,26299}, {182,26428}, {371,26461}, {372,26455}, {376,26440}, {515,26443}, {517,26504}, {1161,12164}, {1307,11412}, {1593,26374}, {1657,18523}, {2077,26509}, {3098,26313}, {3428,26323}, {3576,26368}, {6284,26472}, {6464,26292}, {7354,26478}, {10310,26502}, {11012,26508}, {11248,26511}, {11249,26510}, {11414,26305}, {11825,26338}, {11826,26489}, {11827,26484}, {12306,13022}, {19442,19496}, {26290,26392}, {26291,26416}, {26294,26505}, {26295,26506}

X(26293) = reflection of X(11829) in X(3)

X(26294) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND 3rd TRI-SQUARES-CENTRAL

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

X(26294) lies on these lines: {2,26330}, {3,1587}, {4,641}, {20,492}, {30,26449}, {55,26435}, {56,26355}, {165,26300}, {182,26429}, {193,1350}, {230,6410}, {371,26462}, {372,26456}, {376,5860}, {490,6337}, {515,26444}, {517,26514}, {1152,7738}, {1160,9541}, {1593,26375}, {1657,18539}, {2077,26518}, {3069,9739}, {3098,26314}, {3127,13019}, {3428,26324}, {3528,11824}, {3576,26369}, {3593,14233}, {6284,26473}, {6459,9733}, {7354,26479}, {10304,12306}, {10310,26512}, {11012,26517}, {11248,26520}, {11249,26519}, {11414,26306}, {11826,26490}, {11827,26485}, {12314,19053}, {13666,15682}, {26290,26396}, {26291,26420}, {26292,26496}, {26293,26505}

X(26294) = reflection of X(9540) in X(3)
X(26294) = {X(1350), X(3522)}-harmonic conjugate of X(26295)

X(26295) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND 4th TRI-SQUARES-CENTRAL

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

X(26295) lies on these lines: {2,26331}, {3,1588}, {4,642}, {20,491}, {30,26450}, {55,26436}, {56,26356}, {165,26301}, {182,26430}, {193,1350}, {230,6409}, {371,26463}, {372,26457}, {376,5861}, {489,6337}, {515,26445}, {517,26515}, {1151,7738}, {1593,26376}, {1657,26438}, {2077,26523}, {3068,9738}, {3098,26315}, {3128,13020}, {3428,26325}, {3528,11825}, {3576,26370}, {3595,14230}, {6284,26474}, {6460,9732}, {7354,26480}, {10304,12305}, {10310,26513}, {11012,26522}, {11248,26525}, {11249,26524}, {11414,26307}, {11826,26491}, {11827,26486}, {12313,19054}, {13786,15682}, {26290,26397}, {26291,26421}, {26292,26497}, {26293,26506}

X(26295) = reflection of X(13935) in X(3)
X(26295) = {X(1350), X(3522)}-harmonic conjugate of X(26294)

X(26296) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 1st AURIGA

Barycentrics    a*(2*D+a^3+(b+c)*a^2-(b^2+4*b*c+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26296) = (-D+2*(R-r)*S)*X(1)+4*S*r*X(3)

X(26296) lies on these lines: {1,3}, {10,26394}, {515,26381}, {1698,26359}, {1699,26326}, {3099,26310}, {3679,26382}, {5587,26386}, {5588,26344}, {7713,26371}, {8185,26302}, {9578,26388}, {9581,26387}, {10789,26379}, {10826,26390}, {10827,26389}, {11852,26383}, {18480,18496}, {19003,26384}, {19004,26385}, {26298,26391}, {26299,26392}, {26300,26396}, {26301,26397}

X(26296) = reflection of X(1) in X(11366)
X(26296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3338, 5902, 26297), (5597, 26395, 26365), (26365, 26395, 1)

X(26297) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 2nd AURIGA

Barycentrics    a*(-2*D+a^3+(b+c)*a^2-(b^2+4*b*c+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26297) = (D+2*(R-r)*S)*X(1)+4*S*r*X(3)

X(26297) lies on these lines: {1,3}, {10,26418}, {515,26405}, {1698,26360}, {1699,26327}, {3099,26311}, {3679,26406}, {5587,26410}, {5588,26345}, {5589,26335}, {7713,26372}, {8185,26303}, {9578,26412}, {9581,26411}, {10789,26403}, {10826,26414}, {10827,26413}, {11852,26407}, {18480,18498}, {19003,26408}, {19004,26409}, {26298,26415}, {26299,26416}, {26300,26420}, {26301,26421}

X(26297) = reflection of X(1) in X(11367)
X(26297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5119, 26296), (3338, 5902, 26296), (26366, 26419, 1)

X(26298) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND LUCAS HOMOTHETIC

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

X(26298) lies on these lines: {1,493}, {10,26494}, {35,26493}, {36,26322}, {165,26292}, {515,26439}, {1698,5490}, {1699,26328}, {3099,26312}, {3576,26498}, {3679,26442}, {5587,26466}, {5588,26347}, {5589,26337}, {6464,26299}, {7713,26373}, {8185,26304}, {9578,26477}, {9581,26471}, {10789,26427}, {10826,26488}, {10827,26483}, {11852,26447}, {18480,18521}, {19003,26454}, {19004,26460}, {26296,26391}, {26297,26415}, {26300,26496}, {26301,26497}

X(26298) = reflection of X(1) in X(11377)
X(26298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (493, 26495, 26367), (26367, 26495, 1)

X(26299) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND LUCAS(-1) HOMOTHETIC

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

X(26299) lies on these lines: {1,494}, {10,26503}, {35,26502}, {36,26323}, {57,26434}, {165,26293}, {515,26440}, {1697,26354}, {1698,5491}, {1699,26329}, {3099,26313}, {3576,26507}, {3679,26443}, {5587,26467}, {5588,26338}, {6464,26298}, {7713,26374}, {8185,26305}, {9578,26478}, {9581,26472}, {10789,26428}, {10826,26489}, {10827,26484}, {11852,26448}, {18480,18523}, {19003,26455}, {19004,26461}, {26296,26392}, {26297,26416}, {26300,26505}, {26301,26506}

X(26299) = reflection of X(1) in X(11378)
X(26299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (494, 26504, 26368), (26368, 26504, 1)

X(26300) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    2*(-a+b+c)*S+a*(a^2+2*(b+c)*a-b^2-c^2) : :
X(26300) = 3*X(3679)-2*X(5688)

X(26300) lies on these lines: {1,1336}, {4,9907}, {8,193}, {10,492}, {35,26512}, {36,26324}, {57,26435}, {165,26294}, {230,7968}, {515,26441}, {1697,26355}, {1698,26361}, {1699,26330}, {3099,26314}, {3576,26516}, {3632,5589}, {3679,5588}, {5587,26468}, {7713,26375}, {8185,26306}, {8960,12269}, {9578,26479}, {9581,26473}, {10789,26429}, {10826,26490}, {10827,26485}, {11852,26449}, {13386,24210}, {13679,15682}, {18480,18539}, {19004,26462}, {26296,26396}, {26297,26420}, {26298,26496}, {26299,26505}

X(26300) = reflection of X(1) in X(13883)
X(26300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3068, 26514, 26369), (26369, 26514, 1)

X(26301) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 4th TRI-SQUARES-CENTRAL

Barycentrics    -2*(-a+b+c)*S+a*(a^2+2*(b+c)*a-b^2-c^2) : :
X(26301) = 3*X(3679)-2*X(5689)

X(26301) lies on these lines: {1,1123}, {4,9906}, {8,193}, {10,491}, {35,26513}, {36,26325}, {57,26436}, {165,26295}, {230,7969}, {515,8982}, {1697,26356}, {1698,26362}, {1699,26331}, {3099,26315}, {3576,26521}, {3632,5588}, {3679,5589}, {4028,13461}, {5587,26469}, {7713,26376}, {8185,26307}, {9578,26480}, {9581,26474}, {10789,26430}, {10826,26491}, {10827,26486}, {11852,26450}, {13387,24210}, {13799,15682}, {18480,26438}, {19003,26457}, {19004,26463}, {26296,26397}, {26297,26421}, {26298,26497}, {26299,26506}

X(26301) = reflection of X(1) in X(13936)
X(26301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3069, 26515, 26370), (26370, 26515, 1)

X(26302) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 1st AURIGA

Barycentrics    a^2*(-4*a*D*b^2*c^2+(a+b+c)*(a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^2+c^2)^2*a^3+(b+c)*(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26302) lies on these lines: {1,26303}, {3,18496}, {22,26394}, {24,26381}, {25,5597}, {197,26393}, {1598,26326}, {5594,26344}, {6642,26398}, {8185,26296}, {8192,26395}, {8193,26382}, {10790,26379}, {10828,26310}, {10829,26390}, {10830,26389}, {10831,26388}, {10832,26387}, {10833,26351}, {10834,26402}, {10835,26401}, {11365,26365}, {11414,26290}, {11853,26383}, {18954,26380}, {19005,26384}, {19006,26385}, {22654,26319}, {26304,26391}, {26305,26392}, {26306,26396}, {26307,26397}, {26308,26399}, {26309,26400}

X(26303) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 2nd AURIGA

Barycentrics    a^2*(4*a*D*b^2*c^2+(a+b+c)*(a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^2+c^2)^2*a^3+(b+c)*(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26303) lies on these lines: {1,26302}, {3,18498}, {22,26418}, {24,26405}, {25,5598}, {197,26417}, {1598,26327}, {5594,26345}, {5595,26335}, {6642,26422}, {8185,26297}, {8192,26419}, {8193,26406}, {10790,26403}, {10828,26311}, {10829,26414}, {10830,26413}, {10831,26412}, {10832,26411}, {10833,26352}, {10834,26426}, {10835,26425}, {11365,26366}, {11414,26291}, {11853,26407}, {18954,26404}, {19005,26408}, {19006,26409}, {22654,26320}, {26304,26415}, {26305,26416}, {26306,26420}, {26307,26421}, {26308,26423}, {26309,26424}

X(26304) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND LUCAS HOMOTHETIC

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

X(26304) lies on these lines: {3,5490}, {22,26494}, {24,26439}, {25,371}, {197,26493}, {1598,26328}, {5594,26347}, {5595,26337}, {6289,19446}, {6464,26305}, {6642,26498}, {8185,26298}, {8192,26495}, {8193,26442}, {10790,26427}, {10828,26312}, {10829,26488}, {10830,26483}, {10831,26477}, {10832,26471}, {10833,26353}, {10835,26501}, {11365,26367}, {11414,26292}, {11853,26447}, {18954,26433}, {19005,26454}, {19006,26460}, {22654,26322}, {26302,26391}, {26303,26415}, {26306,26496}, {26307,26497}, {26308,26499}, {26309,26500}

X(26305) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND LUCAS(-1) HOMOTHETIC

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

X(26305) lies on these lines: {3,5491}, {22,26503}, {24,26440}, {25,372}, {197,26502}, {1598,26329}, {5594,26338}, {6290,19447}, {6464,26304}, {6642,26507}, {8185,26299}, {8192,26504}, {8193,26443}, {10790,26428}, {10828,26313}, {10829,26489}, {10830,26484}, {10831,26478}, {10832,26472}, {10833,26354}, {10834,26511}, {10835,26510}, {11365,26368}, {11414,26293}, {11853,26448}, {18954,26434}, {19005,26455}, {19006,26461}, {22654,26323}, {26302,26392}, {26303,26416}, {26306,26505}, {26307,26506}, {26308,26508}, {26309,26509}

X(26306) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 3rd TRI-SQUARES-CENTRAL

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

X(26306) lies on these lines: {3,18539}, {4,9922}, {22,492}, {23,159}, {24,26441}, {25,3068}, {197,26512}, {1598,26330}, {5594,5860}, {5595,20850}, {6642,26516}, {8185,26300}, {8192,26514}, {8193,26444}, {10790,26429}, {10828,26314}, {10829,26490}, {10830,26485}, {10831,26479}, {10832,26473}, {10833,26355}, {10834,26520}, {10835,26519}, {11365,26369}, {11414,26294}, {11853,26449}, {13680,15682}, {18954,26435}, {19005,26456}, {19006,26462}, {22654,26324}, {26302,26396}, {26303,26420}, {26304,26496}, {26305,26505}, {26308,26517}, {26309,26518}

X(26306) = {X(23), X(159)}-harmonic conjugate of X(26307)

X(26307) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 4th TRI-SQUARES-CENTRAL

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

X(26307) lies on these lines: {3,26362}, {4,9921}, {22,491}, {23,159}, {24,8982}, {25,3069}, {197,26513}, {1598,26331}, {5594,20850}, {5595,5861}, {6642,26521}, {8185,26301}, {8192,26515}, {8193,26445}, {10790,26430}, {10828,26315}, {10829,26491}, {10830,26486}, {10831,26480}, {10832,26474}, {10833,26356}, {10834,26525}, {10835,26524}, {11365,26370}, {11414,26295}, {11853,26450}, {13800,15682}, {18954,26436}, {19005,26457}, {19006,26463}, {22654,26325}, {26302,26397}, {26303,26421}, {26304,26497}, {26305,26506}, {26308,26522}, {26309,26523}

X(26307) = {X(23), X(159)}-harmonic conjugate of X(26306)

X(26308) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND INNER-YFF

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

X(26308) lies on these lines: {1,25}, {3,2886}, {5,10830}, {22,10527}, {23,10529}, {24,12116}, {26,10829}, {197,6642}, {497,14017}, {1598,26332}, {1602,3651}, {2070,18543}, {3220,12704}, {3518,10806}, {3556,24474}, {5020,10198}, {5594,26349}, {5595,26342}, {5709,9911}, {6585,23843}, {6734,8193}, {7387,11249}, {7506,16202}, {7517,10680}, {9658,18967}, {9673,10966}, {10532,10594}, {10587,13595}, {10790,26431}, {10828,26317}, {10831,26481}, {10832,26475}, {10833,13730}, {11012,11414}, {11510,20989}, {11853,26452}, {12001,18378}, {12595,20987}, {18954,26437}, {19005,26458}, {19006,26464}, {26302,26399}, {26303,26423}, {26304,26499}, {26305,26508}, {26306,26517}, {26307,26522}

X(26308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 8192, 10037), (25, 9798, 26309), (25, 10835, 1)

X(26309) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND OUTER-YFF

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

X(26309) lies on these lines: {1,25}, {3,119}, {5,10829}, {22,5552}, {23,10528}, {24,12115}, {26,10830}, {197,7387}, {1324,13730}, {1470,15654}, {1598,26333}, {1603,6906}, {2070,18545}, {2077,11414}, {3435,22758}, {3518,10805}, {5020,10200}, {5594,26350}, {5595,26343}, {6642,10269}, {6735,8193}, {7506,16203}, {7517,10679}, {9658,11509}, {9673,10965}, {9912,12751}, {10531,10594}, {10586,13595}, {10790,26432}, {10828,26318}, {10831,26482}, {10832,26476}, {10833,26358}, {11853,26453}, {12000,18378}, {12594,20987}, {13222,25438}, {19005,26459}, {19006,26465}, {26302,26400}, {26303,26424}, {26304,26500}, {26305,26509}, {26306,26518}, {26307,26523}

X(26309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 8192, 10046), (25, 9798, 26308), (25, 10834, 1)

X(26310) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 1st AURIGA

Barycentrics    a*((a^4+(b^2+c^2)*a^2+(b^2+c^2)^2-b^2*c^2)*D+a*(a+b+c)*(2*(b^2+c^2)*a^3-2*(b^3+c^3)*a^2+(b^4+b^2*c^2+c^4)*a-(b^3+c^3)*(b^2+b*c+c^2))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26310) lies on these lines: {1,26311}, {32,5597}, {2896,26394}, {3096,26359}, {3098,26290}, {3099,26296}, {9857,26382}, {9862,26381}, {9993,26326}, {9995,26344}, {9996,26386}, {9997,26395}, {10828,26302}, {10871,26390}, {10872,26389}, {10873,26388}, {10874,26387}, {10877,26351}, {10878,26402}, {10879,26401}, {11368,26365}, {11386,26371}, {11494,26393}, {11885,26383}, {18496,18503}, {18957,26380}, {19011,26384}, {19012,26385}, {22744,26319}, {26312,26391}, {26313,26392}, {26314,26396}, {26315,26397}, {26316,26398}, {26317,26399}, {26318,26400}

X(26311) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 2nd AURIGA

Barycentrics    a*(-(a^4+(b^2+c^2)*a^2+(b^2+c^2)^2-b^2*c^2)*D+a*(a+b+c)*(2*(b^2+c^2)*a^3-2*(b^3+c^3)*a^2+(b^4+b^2*c^2+c^4)*a-(b^3+c^3)*(b^2+b*c+c^2))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26311) lies on these lines: {1,26310}, {32,5598}, {2896,26418}, {3096,26360}, {3098,26291}, {3099,26297}, {9857,26406}, {9862,26405}, {9993,26327}, {9994,26335}, {9995,26345}, {9996,26410}, {9997,26419}, {10828,26303}, {10871,26414}, {10872,26413}, {10873,26412}, {10874,26411}, {10877,26352}, {10878,26426}, {11368,26366}, {11386,26372}, {11494,26417}, {11885,26407}, {18498,18503}, {18957,26404}, {19011,26408}, {19012,26409}, {22744,26320}, {26312,26415}, {26313,26416}, {26314,26420}, {26315,26421}, {26316,26422}, {26317,26423}, {26318,26424}

X(26312) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND LUCAS HOMOTHETIC

Barycentrics    (SB+SC)*(((SA-3*SW)*S^2+2*(2*SA+SW)*SW^2)*S+(4*(SA-SW)*R^2+SA^2-2*SB*SC+4*SW^2)*S^2+3*SA^2*SW^2) : :

X(26312) lies on these lines: {32,493}, {2896,26494}, {3096,5490}, {3098,26292}, {3099,26298}, {6464,26313}, {9857,26442}, {9862,26439}, {9993,26328}, {9994,26337}, {9995,26347}, {9996,26466}, {9997,26495}, {10828,26304}, {10871,26488}, {10872,26483}, {10873,26477}, {10874,26471}, {10877,26353}, {10879,26501}, {11368,26367}, {11386,26373}, {11494,26493}, {11885,26447}, {18503,18521}, {18957,26433}, {19011,26454}, {19012,26460}, {22744,26322}, {26310,26391}, {26311,26415}, {26314,26496}, {26315,26497}, {26316,26498}, {26317,26499}, {26318,26500}

X(26313) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND LUCAS(-1) HOMOTHETIC

Barycentrics    (SB+SC)*(-((SA-3*SW)*S^2+2*(2*SA+SW)*SW^2)*S+(4*(SA-SW)*R^2+SA^2-2*SB*SC+4*SW^2)*S^2+3*SA^2*SW^2) : :

X(26313) lies on these lines: {32,494}, {2896,26503}, {3096,5491}, {3098,26293}, {3099,26299}

X(26314) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 3rd TRI-SQUARES-CENTRAL

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

X(26314) lies on these lines: {4,9987}, {32,638}, {193,3094}, {492,2896}, {3096,26361}, {3098,26294}, {3099,26300}, {5860,7811}, {9857,26444}, {9862,26441}, {9993,26330}, {9994,26339}, {9996,26468}, {9997,26514}, {10828,26306}, {10871,26490}, {10872,26485}, {10873,26479}, {10874,26473}, {10877,26355}, {10878,26520}, {10879,26519}, {11368,26369}, {11386,26375}, {11494,26512}, {11885,26449}, {13685,15682}, {18503,18539}, {18957,26435}, {19011,26456}, {19012,26462}, {22744,26324}, {26310,26396}, {26311,26420}, {26312,26496}, {26313,26505}, {26316,26516}, {26317,26517}, {26318,26518}

X(26315) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 4th TRI-SQUARES-CENTRAL

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

X(26315) lies on these lines: {4,9986}, {32,637}, {193,3094}, {491,2896}, {3096,26362}, {3098,26295}, {3099,26301}, {5861,7811}, {8982,9862}, {9857,26445}, {9993,26331}, {9995,26340}, {9996,26469}, {9997,26515}, {10828,26307}, {10871,26491}, {10872,26486}, {10873,26480}, {10874,26474}, {10877,26356}, {10878,26525}, {10879,26524}, {11368,26370}, {11386,26376}, {11494,26513}, {11885,26450}, {13805,15682}, {18503,26438}, {18957,26436}, {19011,26457}, {19012,26463}, {22744,26325}, {26310,26397}, {26311,26421}, {26312,26497}, {26313,26506}, {26316,26521}, {26317,26522}, {26318,26523}

X(26316) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND X3-ABC REFLECTIONS

Barycentrics    a^2*(2*a^6-2*(b^2+c^2)*a^4+(b^4-3*b^2*c^2+c^4)*a^2-(b^4+b^2*c^2+c^4)*(b^2+c^2)) : :
X(26316) = 3*(S^2-SW^2)*X(3)-2*SW^2*X(6)

X(26316) lies on these lines: {2,5191}, {3,6}, {4,7932}, {5,7846}, {24,11386}, {30,3972}, {35,10877}, {36,18957}, {55,10047}, {56,10038}, {98,7697}, {125,12501}, {140,3096}, {237,15080}, {262,10348}, {325,549}, {353,9486}, {376,16989}, {381,7804}, {384,14880}, {498,10873}, {499,10874}, {517,11368}, {542,7820}, {631,2896}, {1385,9941}, {1495,11328}, {1511,13210}, {1656,10356}, {2782,8289}, {3099,3576}, {3357,12502}, {3523,10357}, {3524,7774}, {3526,7914}, {3579,12497}, {3734,12188}, {4550,19576}, {5026,8724}, {5054,7778}, {5690,12495}, {5939,11185}, {5999,10796}, {6642,10828}, {6771,9982}, {6774,9981}, {7583,13892}, {7584,13946}, {7622,9766}, {7709,8782}, {7787,14881}, {7819,18358}, {7824,10349}, {7919,10722}, {8546,9145}, {8570,8627}, {8703,19661}, {9155,11003}, {9744,15561}, {9857,26446}, {9923,12359}, {9984,12041}, {9985,10610}, {9997,10246}, {10267,11494}, {10269,22744}, {10346,10359}, {10871,26492}, {10872,26487}, {10878,16203}, {10879,16202}, {10991,24206}, {11885,26451}, {12176,14931}, {12498,12619}, {14355,14601}, {14650,15921}, {26310,26398}, {26311,26422}, {26312,26498}, {26313,26507}, {26314,26516}, {26315,26521}

X(26316) = midpoint of X(3) and X(11842)
X(26316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9862, 9996), (8722, 17508, 3), (9821, 11171, 3094)

X(26317) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND INNER-YFF

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

X(26317) lies on these lines: {1,32}, {5,10872}, {2076,12595}, {2896,10527}, {3096,26363}, {3098,11012}, {5709,12497}, {6734,9857}, {7846,10198}, {9301,10680}, {9821,11249}, {9862,12116}, {9993,26332}, {9994,26342}, {9995,26349}, {9996,26470}, {10267,11494}, {10828,26308}, {10871,10943}, {10873,26481}, {10874,26475}, {10877,26357}, {11386,26377}, {11885,26452}, {18503,18544}, {18957,26437}, {19011,26458}, {19012,26464}, {26310,26399}, {26311,26423}, {26312,26499}, {26313,26508}, {26314,26517}, {26315,26522}

X(26317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 9941, 26318), (32, 9997, 10038), (32, 10879, 1)

X(26318) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND OUTER-YFF

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

X(26318) lies on these lines: {1,32}, {5,10871}, {119,9996}, {1470,18957}, {2076,12594}, {2077,3098}, {2896,5552}, {3096,26364}, {6256,9873}, {6735,9857}, {7846,10200}, {9301,10679}, {9821,11248}, {9862,12115}, {9993,26333}, {9994,26343}, {9995,26350}, {10269,22744}, {10828,26309}, {10872,10942}, {10873,26482}, {10874,26476}, {10877,26358}, {11386,26378}, {11885,26453}, {12498,12751}, {13235,25438}, {18503,18542}, {19011,26459}, {19012,26465}, {26310,26400}, {26311,26424}, {26312,26500}, {26313,26509}, {26314,26518}, {26315,26523}

X(26318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 9941, 26317), (32, 9997, 10047), (32, 10878, 1)

X(26319) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 1st AURIGA

Barycentrics    a^2*(-2*D*b*c+a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)*(b-c)^3) : : , where D=4*S*sqrt(R*(4*R+r))
X(26319) = R*(D+4*S*r)*X(1)-4*S*r*(r+2*R)*X(3)

X(26319) lies on these lines: {1,3}, {104,26381}, {956,26382}, {958,26359}, {2975,26394}, {12114,26390}, {18496,26321}, {19013,26384}, {19014,26385}, {22479,26371}, {22520,26379}, {22654,26302}, {22744,26310}, {22753,26326}, {22755,26383}, {22757,26344}, {22758,26386}, {22759,26388}, {22760,26387}, {26322,26391}, {26323,26392}, {26324,26396}, {26325,26397}

X(26319) = {X(1), X(3428)}-harmonic conjugate of X(26320)

X(26320) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 2nd AURIGA

Barycentrics    a^2*(2*D*b*c+a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)*(b-c)^3) : : , where D=4*S*sqrt(R*(4*R+r))
X(26320) = R*(D+4*S*r)*X(1)-4*S*r*(r+2*R)*X(3)

X(26320) lies on these lines: {1,3}, {104,26405}, {956,26406}, {958,26360}, {2975,26418}, {12114,26414}, {18498,26321}, {19013,26408}, {19014,26409}, {22479,26372}, {22520,26403}, {22654,26303}, {22744,26311}, {22753,26327}, {22755,26407}, {22756,26335}, {22757,26345}, {22758,26410}, {22759,26412}, {22760,26411}, {26322,26415}, {26323,26416}, {26324,26420}, {26325,26421}

X(26320) = {X(1), X(3428)}-harmonic conjugate of X(26319)

X(26321) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND EHRMANN-MID

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

X(26321) lies on these lines: {1,399}, {3,10}, {4,20067}, {5,104}, {30,2975}, {36,17606}, {55,18526}, {56,381}, {153,6952}, {382,11249}, {404,18357}, {474,7705}, {499,18516}, {517,5288}, {549,5260}, {944,6914}, {952,3871}, {956,12702}, {999,10404}, {1012,1482}, {1158,25413}, {1385,5259}, {1420,18540}, {1455,18447}, {1656,10269}, {1657,3428}, {2475,12248}, {2829,26470}, {3295,22759}, {3560,10246}, {3579,5258}, {3652,3878}, {3655,5248}, {3830,11194}, {3843,22753}, {3869,13465}, {3897,12919}, {4299,18517}, {4325,18406}, {5055,25524}, {5172,15446}, {5204,18491}, {5251,13624}, {5563,9955}, {5584,15696}, {5690,6909}, {5881,12331}, {5901,6912}, {6256,6980}, {6264,10284}, {6841,18990}, {6862,10585}, {6863,12667}, {6891,8165}, {6913,16203}, {6929,10785}, {6974,10805}, {7330,15829}, {7702,18541}, {7987,18528}, {8148,12513}, {8666,12699}, {9654,22766}, {9668,10966}, {9669,22767}, {10058,10944}, {10247,11496}, {10738,10943}, {11248,12645}, {12164,22659}, {12749,17662}, {12902,22586}, {18440,22769}, {18494,22479}, {18496,26319}, {18498,26320}, {18501,22520}, {18503,22744}, {18508,22755}, {18510,19013}, {18512,19014}, {18521,26322}, {18523,26323}, {18539,26324}, {18545,22768}, {21669,22791}, {22756,26336}, {22757,26346}, {26325,26438}

X(26321) = reflection of X(11849) in X(6906)
X(26321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18519, 18525), (993, 18481, 3), (12773, 13743, 1)

X(26322) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND LUCAS HOMOTHETIC

Barycentrics
a^2*((a^6-(7*b^2-2*b*c+7*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^2+8*b*c*(b+c)*(b^2+c^2)*a-((b^2-c^2)^2-4*b^2*c^2)*(b-c)^2)*S+(a-b-c)*(a^7+(b+c)*a^6-(b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4-(b^4+10*b^2*c^2+c^4)*a^3-(b+c)*(b^4+c^4-2*b*c*(2*b-c)*(b-2*c))*a^2+(b^6+c^6-(2*b^4+2*c^4-b*c*(7*b^2-20*b*c+7*c^2))*b*c)*a+(b^4-c^4)*(b-c)^3)) : :

X(26322) lies on these lines: {3,26493}, {36,26298}, {56,493}, {104,26439}, {956,26442}, {958,5490}, {999,26367}, {2975,26494}, {3428,26292}, {6464,26323}, {10269,26498}, {11249,26499}, {12114,26488}, {18521,26321}, {19013,26454}, {19014,26460}, {22479,26373}, {22520,26427}, {22654,26304}, {22744,26312}, {22753,26328}, {22755,26447}, {22756,26337}, {22757,26347}, {22758,26466}, {22759,26477}, {22760,26471}, {26319,26391}, {26320,26415}, {26324,26496}, {26325,26497}

X(26323) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND LUCAS(-1) HOMOTHETIC

Barycentrics
a^2*((a^6-(7*b^2-2*b*c+7*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^2+8*b*c*(b+c)*(b^2+c^2)*a-((b^2-c^2)^2-4*b^2*c^2)*(b-c)^2)*S-(a-b-c)*(a^7+(b+c)*a^6-(b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4-(b^4+10*b^2*c^2+c^4)*a^3-(b+c)*(b^4+c^4-2*b*c*(2*b-c)*(b-2*c))*a^2+(b^6+c^6-(2*b^4+2*c^4-b*c*(7*b^2-20*b*c+7*c^2))*b*c)*a+(b^4-c^4)*(b-c)^3)) : :

X(26323) lies on these lines: {3,26502}, {36,26299}, {55,26504}, {56,494}, {104,26440}, {956,26443}, {958,5491}, {999,26368}, {2975,26503}, {3428,26293}, {6464,26322}, {10269,26507}, {10966,26354}, {11249,26508}, {12114,26489}, {18523,26321}, {19013,26455}, {19014,26461}, {22479,26374}, {22520,26428}, {22654,26305}, {22744,26313}, {22753,26329}, {22755,26448}, {22757,26338}, {22758,26467}, {22759,26478}, {22760,26472}, {22768,26511}, {26319,26392}, {26320,26416}, {26324,26505}, {26325,26506}

X(26324) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 3rd TRI-SQUARES-CENTRAL

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

X(26324) lies on these lines: {3,26512}, {4,22624}, {36,26300}, {55,26514}, {56,3068}, {104,26441}, {193,22769}, {492,2975}, {956,26444}, {958,26361}, {999,26369}, {3428,26294}, {5860,11194}, {10269,26516}, {10966,26355}, {11249,26517}, {12114,26490}, {15682,22783}, {18539,26321}, {19013,26456}, {19014,26462}, {22479,26375}, {22520,26429}, {22654,26306}, {22744,26314}, {22753,26330}, {22755,26449}, {22756,26339}, {22758,26468}, {22759,26479}, {22760,26473}, {22768,26520}, {26319,26396}, {26320,26420}, {26322,26496}, {26323,26505}

X(26325) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 4th TRI-SQUARES-CENTRAL

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

X(26325) lies on these lines: {3,26513}, {4,22595}, {36,26301}, {55,26515}, {56,3069}, {104,8982}, {193,22769}, {491,2975}, {956,26445}, {958,26362}, {999,26370}, {3428,26295}, {5861,11194}, {10269,26521}, {10966,26356}, {11249,26522}, {12114,26491}, {15682,22784}, {19013,26457}, {19014,26463}, {22479,26376}, {22520,26430}, {22654,26307}, {22744,26315}, {22753,26331}, {22755,26450}, {22757,26340}, {22758,26469}, {22759,26480}, {22760,26474}, {22768,26525}, {26319,26397}, {26320,26421}, {26321,26438}, {26322,26497}, {26323,26506}

X(26326) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND 1st AURIGA

Barycentrics    4*S^2*a*D+(a+b+c)*((b^2+c^2)*a^5-(b+c)^3*a^4-2*(b^2-c^2)^2*a^3+2*(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26326) lies on these lines: {1,6831}, {2,26290}, {4,5597}, {5,26359}, {11,26380}, {12,26351}, {30,26398}, {55,26412}, {98,26379}, {235,26371}, {381,26386}, {515,26365}, {517,26360}, {1587,26384}, {1588,26385}, {1598,26302}, {1699,26296}, {3091,26394}, {3843,18496}, {5587,26382}, {5603,26395}, {5842,8186}, {6201,26344}, {6833,26425}, {9993,26310}, {10531,26402}, {10532,26401}, {10679,26410}, {10893,26390}, {10894,26389}, {10895,26388}, {10896,26387}, {11496,26393}, {11897,26383}, {22753,26319}, {26328,26391}, {26329,26392}, {26330,26396}, {26331,26397}, {26332,26399}, {26333,26400}

X(26326) = midpoint of X(4) and X(11843)
X(26326) = {X(1), X(7680)}-harmonic conjugate of X(26327)

X(26327) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND 2nd AURIGA

Barycentrics    -4*S^2*a*D+(a+b+c)*((b^2+c^2)*a^5-(b+c)^3*a^4-2*(b^2-c^2)^2*a^3+2*(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26327) lies on these lines: {1,6831}, {2,26291}, {4,5598}, {5,26360}, {11,26404}, {12,26352}, {30,26422}, {55,26388}, {98,26403}, {235,26372}, {381,26410}, {515,26366}, {517,26359}, {1587,26408}, {1588,26409}, {1598,26303}, {1699,26297}, {3091,26418}, {3843,18498}, {5587,26406}, {5603,26419}, {5842,8187}, {6201,26345}, {6202,26335}, {6833,26401}, {9993,26311}, {10531,26426}, {10532,26425}, {10679,26386}, {10893,26414}, {10894,26413}, {10895,26412}, {10896,26411}, {11496,26389}, {11897,26407}, {22753,26320}, {26328,26415}, {26329,26416}, {26330,26420}, {26331,26421}, {26332,26423}, {26333,26424}

X(26327) = midpoint of X(4) and X(11844)
X(26327) = {X(1), X(7680)}-harmonic conjugate of X(26326)

X(26328) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND LUCAS HOMOTHETIC

Barycentrics    ((4*R^2-SA+SW)*S^2+2*(2*R^2+SW)*SB*SC)*S+(-SA^2+2*SB*SC+SW^2)*S^2+SB*SC*SW^2 : :

X(26328) lies on these lines: {2,26292}, {4,493}, {5,5490}, {11,26433}, {12,26353}, {30,26498}, {98,26427}, {235,26373}, {381,26466}, {515,26367}, {1093,24244}, {1587,26454}, {1588,26460}, {1598,26304}, {1699,26298}, {3089,8948}, {3091,26494}, {3843,18521}, {5587,26442}, {5603,26495}, {6201,26347}, {6202,26337}, {6464,26329}, {9993,26312}, {10532,26501}, {10893,26488}, {10894,26483}, {10895,26477}, {10896,26471}, {11496,26493}, {11897,26447}, {22753,26322}, {26326,26391}, {26327,26415}, {26330,26496}, {26331,26497}, {26332,26499}, {26333,26500}

X(26328) = midpoint of X(4) and X(11846)

X(26329) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND LUCAS(-1) HOMOTHETIC

Barycentrics    ((4*R^2-SA+SW)*S^2+2*(2*R^2+SW)*SB*SC)*S+(SA^2-2*SB*SC-SW^2)*S^2-SB*SC*SW^2 : :

X(26329) lies on these lines: {2,26293}, {4,494}, {5,5491}, {11,26434}, {12,26354}, {30,26507}, {98,26428}, {235,26374}, {381,26467}, {515,26368}, {1093,24243}, {1587,26455}, {1588,26461}, {1598,26305}, {1699,26299}, {3089,8946}, {3091,26503}, {3843,18523}, {5587,26443}, {5603,26504}, {6201,26338}, {6464,26328}, {9993,26313}, {10531,26511}, {10532,26510}, {10893,26489}, {10894,26484}, {10895,26478}, {10896,26472}, {11496,26502}, {11897,26448}, {22753,26323}, {26326,26392}, {26327,26416}, {26330,26505}, {26331,26506}, {26332,26508}, {26333,26509}

X(26329) = midpoint of X(4) and X(11847)

X(26330) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND 3rd TRI-SQUARES-CENTRAL

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

X(26330) lies on these lines: {2,26294}, {4,371}, {5,26361}, {11,26435}, {12,26355}, {30,26516}, {98,26429}, {193,3832}, {230,7374}, {235,26375}, {381,5860}, {492,3091}, {515,26369}, {546,5875}, {1131,1503}, {1587,26456}, {1588,26462}, {1598,26306}, {1699,26300}, {3843,18539}, {5200,13019}, {5587,26444}, {5603,26514}, {5870,13665}, {6251,22484}, {6526,24244}, {7585,14233}, {9993,26314}, {10531,26520}, {10532,26519}, {10893,26490}, {10894,26485}, {10895,26479}, {10896,26473}, {11496,26512}, {11897,26449}, {13687,15682}, {22753,26324}, {26326,26396}, {26327,26420}, {26328,26496}, {26329,26505}, {26332,26517}, {26333,26518}

X(26330) = midpoint of X(4) and X(13886)
X(26330) = {X(3832), X(5480)}-harmonic conjugate of X(26331)

X(26331) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND 4th TRI-SQUARES-CENTRAL

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

X(26331) lies on these lines: {2,26295}, {4,372}, {5,26362}, {11,26436}, {12,26356}, {30,26521}, {98,26430}, {193,3832}, {230,7000}, {235,26376}, {381,5861}, {491,3091}, {515,26370}, {546,5874}, {1132,1503}, {1587,26457}, {1588,26463}, {1598,26307}, {1699,26301}, {3843,26438}, {5587,26445}, {5603,26515}, {5871,13785}, {6250,22485}, {6526,24243}, {7586,14230}, {9993,26315}, {10531,26525}, {10532,26524}, {10893,26491}, {10894,26486}, {10895,26480}, {10896,26474}, {11496,26513}, {11897,26450}, {13807,15682}, {22753,26325}, {26326,26397}, {26327,26421}, {26328,26497}, {26329,26506}, {26332,26522}, {26333,26523}

X(26331) = midpoint of X(4) and X(13939)
X(26331) = {X(3832), X(5480)}-harmonic conjugate of X(26330)

X(26332) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND INNER-YFF

Barycentrics    a^7-(b+c)*a^6-(b-c)^2*a^5+(b+c)*(b^2-4*b*c+c^2)*a^4-(b^2+4*b*c+c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
X(26332) = X(944)-3*X(3475) = 3*X(1699)+X(9613) = X(3486)-3*X(5603) = R*X(1)+(R+r)*X(4)

X(26332) lies on these lines: {1,4}, {2,11012}, {3,6690}, {5,958}, {7,5884}, {8,2894}, {10,5709}, {11,26437}, {12,3149}, {20,10902}, {30,4428}, {35,6934}, {36,6833}, {40,377}, {56,6831}, {57,12616}, {98,26431}, {104,4317}, {117,5230}, {119,11929}, {149,5734}, {165,6897}, {219,5798}, {235,26377}, {329,20117}, {355,518}, {381,529}, {382,16202}, {405,11827}, {442,3428}, {443,6684}, {495,11500}, {498,6905}, {499,6830}, {516,6850}, {517,5794}, {535,12558}, {546,7956}, {908,10522}, {952,18517}, {960,5812}, {962,2475}, {993,6824}, {1012,7354}, {1125,6827}, {1158,4292}, {1329,6918}, {1482,13463}, {1503,13408}, {1512,10827}, {1532,10895}, {1537,13273}, {1587,26458}, {1588,26464}, {1598,26308}, {1698,6854}, {1836,12672}, {1837,18962}, {1853,13095}, {2077,4190}, {2478,8227}, {2550,5735}, {2551,5705}, {2800,4295}, {2829,9655}, {2886,22770}, {2975,6828}, {3070,19049}, {3071,19050}, {3085,6796}, {3086,6844}, {3091,5080}, {3146,10587}, {3295,5842}, {3333,12687}, {3434,7982}, {3436,5587}, {3576,6836}, {3577,5881}, {3616,6840}, {3624,6947}, {3814,6944}, {3817,6893}, {3822,6825}, {3832,10529}, {3839,11240}, {3843,10742}, {3855,8166}, {4293,5450}, {4297,6851}, {4298,6245}, {4299,6906}, {4308,11715}, {4430,6894}, {5198,10835}, {5248,6868}, {5251,6832}, {5253,6943}, {5259,6936}, {5260,6991}, {5267,6892}, {5536,5818}, {5563,10785}, {5657,6901}, {5693,5905}, {5707,5786}, {5720,21077}, {5722,13374}, {5731,6895}, {5761,22836}, {5768,12005}, {5787,12675}, {5806,18480}, {5886,6928}, {6201,26349}, {6202,26342}, {6253,15888}, {6361,6951}, {6459,13907}, {6460,13965}, {6829,19854}, {6834,7951}, {6841,22758}, {6843,19843}, {6848,10590}, {6849,7682}, {6865,10165}, {6866,8666}, {6867,25639}, {6870,20076}, {6882,10200}, {6885,25440}, {6896,7989}, {6898,7988}, {6899,7987}, {6911,26364}, {6915,11681}, {6916,10268}, {6922,25524}, {6923,12699}, {6927,10588}, {6929,9955}, {6938,10483}, {6942,14794}, {6956,7288}, {6962,10585}, {6977,7280}, {7330,12617}, {7497,9798}, {7548,11680}, {8727,12114}, {9579,12705}, {9654,18242}, {9779,13729}, {9993,26317}, {10056,11491}, {10310,11112}, {10525,22791}, {10896,18967}, {10942,18491}, {10953,11375}, {11510,12943}, {11897,26452}, {12000,18499}, {12019,12762}, {12190,14639}, {12382,14644}, {12433,20330}, {12688,16127}, {12702,15346}, {14054,14872}, {14647,15932}, {15908,17532}, {18481,24299}, {26326,26399}, {26327,26423}, {26328,26499}, {26329,26508}, {26330,26517}, {26331,26522}

X(26332) = midpoint of X(i) and X(j) for these {i,j}: {4, 388}, {9579, 12705}
X(26332) = reflection of X(i) in X(j) for these (i,j): (3, 25466), (958, 5), (6868, 5248), (7330, 12617)
X(26332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3583, 11522, 10531), (10532, 12116, 10597), (10597, 12116, 1)

X(26333) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND OUTER-YFF

Barycentrics    a^7-(b+c)*a^6-(b-c)^2*a^5+(b+c)*(b^2+c^2)*a^4-(b^2+c^2)*(b-c)^2*a^3+(b^4-c^4)*(b-c)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(26333) = R*X(1)+(R-r)*X(4)

X(26333) lies on these lines: {1,4}, {2,2077}, {3,3816}, {5,1376}, {7,15528}, {8,13729}, {10,6893}, {11,1012}, {12,26358}, {30,7956}, {35,6834}, {36,6938}, {40,2478}, {55,1532}, {79,5553}, {84,5555}, {98,26432}, {100,6945}, {104,10072}, {119,381}, {153,3241}, {165,6947}, {235,26378}, {355,3880}, {376,8166}, {377,8227}, {382,16203}, {405,15908}, {474,11826}, {496,12114}, {498,6941}, {499,6906}, {516,3359}, {517,6929}, {546,10894}, {913,5190}, {938,5884}, {952,18516}, {962,5046}, {971,18527}, {993,6930}, {999,2829}, {1001,6907}, {1125,6850}, {1158,1210}, {1329,10306}, {1352,9025}, {1387,12761}, {1512,5119}, {1537,2099}, {1538,24929}, {1587,26459}, {1588,26465}, {1598,26309}, {1621,6932}, {1698,6898}, {1836,18838}, {1837,12672}, {1853,13094}, {2095,17768}, {2550,6939}, {2551,11362}, {2800,18391}, {2886,6913}, {2950,10265}, {3070,19047}, {3071,19048}, {3073,5292}, {3086,5450}, {3091,5552}, {3146,10586}, {3149,6284}, {3256,6844}, {3295,18242}, {3428,11113}, {3434,5587}, {3436,7982}, {3560,26363}, {3576,6925}, {3624,6897}, {3746,10786}, {3814,6973}, {3817,6826}, {3822,6982}, {3825,6891}, {3832,10528}, {3838,5886}, {3839,11239}, {3841,6887}, {3843,12000}, {3899,12245}, {4187,10310}, {4293,5193}, {4294,6796}, {4295,5804}, {4302,6905}, {4309,11491}, {4512,5084}, {4863,18908}, {5010,6880}, {5045,22792}, {5198,10834}, {5218,6969}, {5248,6825}, {5251,6976}, {5252,10947}, {5259,6889}, {5440,17618}, {5657,6965}, {5693,12649}, {5722,6001}, {5734,20060}, {5805,10202}, {5806,22793}, {5836,12700}, {5840,6911}, {5842,9668}, {5924,10398}, {6201,26350}, {6202,26343}, {6259,12675}, {6361,6902}, {6459,13906}, {6460,13964}, {6824,25639}, {6830,12775}, {6831,10896}, {6833,7741}, {6836,10860}, {6838,10902}, {6839,9779}, {6840,9812}, {6842,10198}, {6847,10591}, {6849,12571}, {6854,7988}, {6865,10270}, {6866,12558}, {6872,11012}, {6895,15016}, {6899,16209}, {6912,11680}, {6916,10165}, {6917,9955}, {6920,19854}, {6928,7686}, {6934,14803}, {6935,10589}, {6944,25440}, {6966,10584}, {6968,7951}, {6992,7688}, {7330,10916}, {9581,12616}, {9993,26318}, {10247,10742}, {10248,10430}, {10526,22791}, {10724,17579}, {10895,10965}, {10915,19925}, {10943,18761}, {10953,12701}, {11372,12686}, {11376,18961}, {11500,15171}, {11729,22938}, {11897,26453}, {11928,26470}, {12189,14639}, {12381,14644}, {12676,18238}, {12953,22768}, {15254,26446}, {16371,24466}, {18481,24927}, {26326,26400}, {26327,26424}, {26328,26500}, {26329,26509}, {26330,26518}, {26331,26523}

X(26333) = midpoint of X(i) and X(j) for these {i,j}: {4, 497}, {9668, 19541}
X(26333) = reflection of X(i) in X(j) for these (i,j): (3, 3816), (1376, 5), (22753, 7956)
X(26333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12116, 5691), (1058, 12667, 5882), (1699, 3583, 4)

X(26334) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 1st AURIGA

Barycentrics    a*((a^2+b^2+c^2-S)*D+(a+b+c)*a*((-a+b+c)*S+2*(b^2+c^2)*a-2*b^3-2*c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26334) lies on these lines: {1,26335}, {6,5597}, {1271,26394}, {5589,26296}, {5591,26359}, {5595,26302}, {5605,26395}, {5689,26382}, {5861,26397}, {6202,26326}, {6215,26386}, {9994,26310}, {10783,26381}, {10792,26379}, {10919,26390}, {10921,26389}, {10923,26388}, {10925,26387}, {10927,26351}, {10929,26402}, {10931,26401}, {11370,26365}, {11388,26371}, {11497,26393}, {11824,26290}, {11901,26383}, {18496,26336}, {18959,26380}, {22756,26319}, {26337,26391}, {26339,26396}, {26341,26398}, {26342,26399}, {26343,26400}

X(26335) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd AURIGA

Barycentrics    a*((a^2+b^2+c^2-S)*D+(a+b+c)*a*(-(-a+b+c)*S-2*(b^2+c^2)*a+2*b^3+2*c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26335) lies on these lines: {1,26334}, {6,5598}, {1271,26418}, {5589,26297}, {5591,26360}, {5595,26303}, {5605,26419}, {5689,26406}, {5861,26421}, {6202,26327}, {6215,26410}, {9994,26311}, {10783,26405}, {10792,26403}, {10919,26414}, {10921,26413}, {10923,26412}, {10925,26411}, {10929,26426}, {10931,26425}, {11370,26366}, {11388,26372}, {11497,26417}, {11824,26291}, {11901,26407}, {18498,26336}, {18959,26404}, {22756,26320}, {26337,26415}, {26339,26420}, {26341,26422}, {26342,26423}, {26343,26424}

X(26336) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND EHRMANN-MID

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

X(26336) lies on these lines: {3,5591}, {4,5875}, {5,10783}, {6,13}, {30,1271}, {382,1161}, {550,10517}, {999,10925}, {1656,10514}, {1657,11824}, {3295,10923}, {3534,13810}, {3641,18525}, {3830,5861}, {3843,6202}, {5589,18480}, {5605,18526}, {5689,12702}, {7732,12902}, {8148,12627}, {9654,10040}, {9655,18959}, {9668,10927}, {9669,10048}, {9929,12164}, {9994,18503}, {10792,18501}, {10919,18519}, {10921,18518}, {10929,18545}, {10931,18543}, {11370,18493}, {11388,18494}, {11497,18524}, {11901,18508}, {13782,22807}, {14269,18539}, {18498,26335}, {18521,26337}, {18542,26343}, {18544,26342}, {22756,26321}

X(26336) = reflection of X(13782) in X(22807)
X(26336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5875, 11916), (5871, 6215, 3)

X(26337) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND LUCAS HOMOTHETIC

Barycentrics    (SB+SC)*((2*S^2-SA^2+2*SB*SC-SW^2)*S+(4*R^2+SA-SW)*S^2-2*SA^2*SW) : :

X(26337) lies on these lines: {6,493}, {1271,26494}, {5490,5591}, {5589,26298}, {5595,26304}, {5605,26495}, {5689,26442}, {5861,26497}, {6202,26328}, {6215,26466}, {9994,26312}, {10783,26439}, {10792,26427}, {10919,26488}, {10921,26483}, {10923,26477}, {10925,26471}, {10927,26353}, {10931,26501}, {11370,26367}, {11388,26373}, {11497,26493}, {11824,26292}, {11901,26447}, {18521,26336}, {18959,26433}, {22756,26322}, {26335,26415}, {26339,26496}, {26341,26498}, {26342,26499}, {26343,26500}

X(26338) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND LUCAS(-1) HOMOTHETIC

Barycentrics    (SB+SC)*(-(2*S^2-SA^2+2*SB*SC-SW^2)*S+(4*R^2+SA-SW)*S^2-2*SA^2*SW) : :

X(26338) lies on these lines: {6,494}, {1270,26503}, {5491,5590}, {5588,26299}, {5594,26305}, {5604,26504}, {5688,26443}, {5860,26505}, {6201,26329}, {6214,26467}, {6464,26347}, {10784,26440}, {10793,26428}, {10920,26489}, {10922,26484}, {10924,26478}, {10926,26472}, {10928,26354}, {10930,26511}, {10932,26510}, {11371,26368}, {11389,26374}, {11498,26502}, {11825,26293}, {11902,26448}, {18523,26346}, {18960,26434}, {22757,26323}, {26340,26506}, {26344,26392}, {26345,26416}, {26348,26507}, {26349,26508}, {26350,26509}

X(26339) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    -b^2-c^2+S+4*a^2 : :
X(26339) = 3*(S-2*SW)*X(2)+10*SW*X(6)

X(26339) lies on these lines: {2,6}, {4,6279}, {382,5871}, {546,5875}, {550,1161}, {3244,3641}, {3528,11824}, {3529,10783}, {3530,26341}, {3544,10514}, {3632,5589}, {3636,11370}, {3851,6215}, {3855,6281}, {5102,7000}, {5595,20850}, {5605,20057}, {5689,26444}, {6154,13269}, {7732,24981}, {9994,26314}, {10301,11388}, {10792,26429}, {10919,26490}, {10921,26485}, {10923,26479}, {10925,26473}, {10927,26355}, {10929,26520}, {10931,26519}, {11497,26512}, {11901,26449}, {13690,15682}, {14269,18539}, {18959,26435}, {22756,26324}, {26335,26420}, {26337,26496}, {26342,26517}, {26343,26518}

X(26339) = reflection of X(5590) in X(7585)
X(26339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3629, 26340), (193, 3068, 5860), (3068, 5860, 26361)

X(26340) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 4th TRI-SQUARES-CENTRAL

Barycentrics    -b^2-c^2-S+4*a^2 : :
X(26340) = 3*(S+2*SW)*X(2)-10*SW*X(6)

X(26340) lies on these lines: {2,6}, {4,6280}, {382,5870}, {546,5874}, {550,1160}, {3244,3640}, {3528,11825}, {3529,8982}, {3530,26348}, {3544,10515}, {3632,5588}, {3636,11371}, {3851,6214}, {3855,6278}, {5102,7374}, {5594,20850}, {5604,20057}, {5688,26445}, {6154,13270}, {7733,24981}, {9995,26315}, {10301,11389}, {10793,26430}, {10920,26491}, {10922,26486}, {10924,26480}, {10926,26474}, {10928,26356}, {10930,26525}, {10932,26524}, {11498,26513}, {11902,26450}, {13811,15682}, {14269,26346}, {18960,26436}, {22757,26325}, {26338,26506}, {26344,26397}, {26345,26421}, {26347,26497}, {26349,26522}, {26350,26523}

X(26340) = reflection of X(5591) in X(7586)
X(26340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3629, 26339), (6, 5860, 5590), (591, 7585, 26361)

X(26341) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND X3-ABC REFLECTIONS

Barycentrics    a^2*((-a^2+b^2+c^2)*S+2*a^4-2*(b^2+c^2)*a^2-4*b^2*c^2) : :
X(26341) = (S-2*SW)*X(3)-2*SW*X(6)

X(26341) lies on these lines: {2,6215}, {3,6}, {5,5871}, {24,11388}, {30,6202}, {35,10927}, {36,18959}, {55,10048}, {56,10040}, {125,12803}, {140,5591}, {498,10923}, {499,10925}, {517,11370}, {549,5861}, {631,1271}, {642,15834}, {1385,3641}, {1511,7732}, {1584,5012}, {1656,10514}, {3357,6267}, {3523,10517}, {3526,6281}, {3530,26339}, {3576,5589}, {3579,12697}, {5054,6279}, {5595,6642}, {5605,10246}, {5689,26446}, {5690,12627}, {6214,7375}, {6227,12042}, {6263,12619}, {6270,6771}, {6271,6774}, {6277,10610}, {7583,8974}, {7584,13949}, {7725,12041}, {8903,15805}, {9929,12359}, {10267,11497}, {10269,22756}, {10919,26492}, {10921,26487}, {10929,16203}, {10931,16202}, {11901,26451}, {19351,19360}, {26335,26422}, {26337,26498}

X(26341) = midpoint of X(3) and X(6418)
X(26341) = inverse of X(1161) in the Brocard circle
X(26341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10783, 6215), (6, 11824, 11916), (5092, 9733, 3)

X(26342) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND INNER-YFF

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

X(26342) lies on these lines: {1,6}, {5,10921}, {1161,11249}, {1271,10527}, {5591,26363}, {5595,26308}, {5689,6734}, {5709,12697}, {5861,26522}, {5875,10919}, {6202,26332}, {6215,26470}, {9994,26317}, {10267,11497}, {10680,11916}, {10783,12116}, {10792,26431}, {10923,26481}, {10925,26475}, {10927,26357}, {11012,11824}, {11388,26377}, {11901,26452}, {18544,26336}, {18959,26437}, {26335,26423}, {26337,26499}, {26339,26517}

X(26342) = reflection of X(26350) in X(3299)
X(26342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5605, 10040), (6, 10931, 1), (12595, 19050, 1)

X(26343) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND OUTER-YFF

Barycentrics    a^2*(2*S*b*c+a^4-2*(b^2+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-2*b^3*c-2*b^2*c^2-2*b*c^3) : :
X(26343) = R*(S-2*SW)*X(1)+2*SW*(R-r)*X(6)

X(26343) lies on these lines: {1,6}, {5,10919}, {119,6215}, {1161,11248}, {1271,5552}, {1470,18959}, {2077,11824}, {5591,26364}, {5595,26309}, {5689,6735}, {5861,26523}, {5871,6256}, {5875,10921}, {6202,26333}, {6263,12751}, {9994,26318}, {10269,22756}, {10679,11916}, {10783,12115}, {10792,26432}, {10923,26482}, {10925,26476}, {10927,26358}, {11388,26378}, {11901,26453}, {13269,25438}, {18542,26336}, {26335,26424}, {26337,26500}, {26339,26518}

X(26343) = reflection of X(26349) in X(3301)
X(26343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5605, 10048), (6, 10929, 1), (12594, 19048, 1)

X(26344) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 1st AURIGA

Barycentrics    a*((a^2+b^2+c^2+S)*D+(a+b+c)*a*((a-b-c)*S+2*(b^2+c^2)*a-2*b^3-2*c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26344) lies on these lines: {1,26345}, {6,5597}, {1270,26394}, {5588,26296}, {5590,26359}, {5594,26302}, {5604,26395}, {5688,26382}, {5860,26396}, {6201,26326}, {6214,26386}, {9995,26310}, {10784,26381}, {10793,26379}, {10920,26390}, {10922,26389}, {10924,26388}, {10926,26387}, {10928,26351}, {10930,26402}, {10932,26401}, {11371,26365}, {11389,26371}, {11498,26393}, {11825,26290}, {11902,26383}, {18496,26346}, {18960,26380}, {22757,26319}, {26338,26392}, {26340,26397}, {26347,26391}, {26348,26398}, {26349,26399}, {26350,26400}

X(26345) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd AURIGA

Barycentrics    a*(-(a^2+b^2+c^2+S)*D+(a+b+c)*a*((a-b-c)*S+2*(b^2+c^2)*a-2*b^3-2*c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26345) lies on these lines: {1,26344}, {6,5598}, {1270,26418}, {5588,26297}, {5590,26360}, {5594,26303}, {5604,26419}, {5688,26406}, {5860,26420}, {6201,26327}, {6214,26410}, {9995,26311}, {10784,26405}, {10793,26403}, {10920,26414}, {10922,26413}, {10924,26412}, {10926,26411}, {10928,26352}, {10932,26425}, {11389,26372}, {11498,26417}, {11825,26291}, {11902,26407}, {18498,26346}, {18960,26404}, {22757,26320}, {26338,26416}, {26340,26421}, {26347,26415}, {26348,26422}, {26349,26423}, {26350,26424}

X(26346) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND EHRMANN-MID

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

X(26346) lies on these lines: {3,5590}, {4,5874}, {5,10784}, {6,13}, {30,1270}, {382,1160}, {550,10518}, {999,10926}, {1656,10515}, {1657,11825}, {3295,10924}, {3534,13691}, {3640,18525}, {3830,5860}, {3843,6201}, {5588,18480}, {5604,18526}, {5688,12702}, {7733,12902}, {8148,12628}, {9654,10041}, {9655,18960}, {9668,10928}, {9669,10049}, {9930,12164}, {9995,18503}, {10793,18501}, {10920,18519}, {10922,18518}, {10930,18545}, {10932,18543}, {11371,18493}, {11389,18494}, {11498,18524}, {11902,18508}, {13662,22806}, {14269,26340}, {18496,26344}, {18498,26345}, {18521,26347}, {18523,26338}, {18542,26350}, {18544,26349}, {22757,26321}

X(26346) = reflection of X(13662) in X(22806)
X(26346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5874, 11917), (5870, 6214, 3)

X(26347) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND LUCAS HOMOTHETIC

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

X(26347) lies on these lines: {6,493}, {76,5490}, {755,1306}, {1270,26494}, {2353,6457}, {5588,26298}, {5594,26304}, {5604,26495}, {5688,26442}, {5860,26496}, {6201,26328}, {6214,26466}, {6464,26338}, {9995,26312}, {10784,26439}, {10793,26427}, {10920,26488}, {10922,26483}, {10924,26477}, {10926,26471}, {10928,26353}, {11371,26367}, {11389,26373}, {11498,26493}, {11825,26292}, {11902,26447}, {18521,26346}, {18960,26433}, {22757,26322}, {26340,26497}, {26344,26391}, {26345,26415}, {26348,26498}, {26349,26499}, {26350,26500}

X(26348) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND X3-ABC REFLECTIONS

Barycentrics    a^2*((a^2-b^2-c^2)*S+2*a^4-2*(b^2+c^2)*a^2-4*b^2*c^2) : :
X(26348) = (S+2*SW)*X(3)+2*SW*X(6)

X(26348) lies on these lines: {2,6214}, {3,6}, {5,5870}, {24,11389}, {30,6201}, {35,10928}, {36,18960}, {55,10049}, {56,10041}, {125,12804}, {140,5590}, {498,10924}, {499,10926}, {517,11371}, {549,5860}, {631,1270}, {641,15835}, {1385,3640}, {1511,7733}, {1583,5012}, {1656,10515}, {3357,6266}, {3523,10518}, {3526,6278}, {3530,26340}, {3576,5588}, {3579,12698}, {5054,6280}, {5594,6642}, {5604,10246}, {5688,26446}, {5690,12628}, {6215,7376}, {6226,12042}, {6262,12619}, {6268,6771}, {6269,6774}, {6276,10610}, {7583,8975}, {7584,13950}, {7726,12041}, {8904,15805}, {9930,12359}, {10267,11498}, {10269,22757}, {10920,26492}, {10922,26487}, {10930,16203}, {10932,16202}, {11902,26451}, {19352,19360}, {26338,26507}, {26344,26398}, {26345,26422}, {26347,26498}

X(26348) = midpoint of X(3) and X(6417)
X(26348) = inverse of X(1160) in the Brocard circle
X(26348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10784, 6214), (574, 8406, 8400), (1151, 8406, 574)

X(26349) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND INNER-YFF

Barycentrics    a^2*(2*S*b*c+a^4-2*(b^2+c^2)*a^2-2*b*c*(b+c)*a+b^4+c^4+2*b*c*(b^2-b*c+c^2)) : :
X(26349) = R*(2*SW+S)*X(1)-2*SW*(R+r)*X(6)

X(26349) lies on these lines: {1,6}, {5,10922}, {1160,11249}, {1270,10527}, {5590,26363}, {5594,26308}, {5688,6734}, {5709,12698}, {5860,26517}, {5874,10920}, {6201,26332}, {6214,26470}, {9995,26317}, {10267,11498}, {10680,11917}, {10784,12116}, {10793,26431}, {10924,26481}, {10926,26475}, {10928,26357}, {11012,11825}, {11389,26377}, {11902,26452}, {18544,26346}, {18960,26437}, {26338,26508}, {26340,26522}, {26344,26399}, {26345,26423}, {26347,26499}

X(26349) = reflection of X(26343) in X(3301)
X(26349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3640, 26350), (6, 10932, 1), (12595, 19049, 1)

X(26350) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND OUTER-YFF

Barycentrics    a^2*(-2*S*b*c+a^4-2*(b^2+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-2*b*c*(b^2+b*c+c^2)) : :
X(26350) = R*(2*SW+S)*X(1)-2*SW*(R-r)*X(6)

X(26350) lies on these lines: {1,6}, {5,10920}, {119,6214}, {1160,11248}, {1270,5552}, {1470,18960}, {2077,11825}, {5590,26364}, {5594,26309}, {5688,6735}, {5860,26518}, {5870,6256}, {5874,10922}, {6201,26333}, {6262,12751}, {9995,26318}, {10269,22757}, {10679,11917}, {10784,12115}, {10793,26432}, {10924,26482}, {10926,26476}, {10928,26358}, {11389,26378}, {11902,26453}, {13270,25438}, {18542,26346}, {26338,26509}, {26340,26523}, {26344,26400}, {26345,26424}, {26347,26500}

X(26350) = reflection of X(26342) in X(3299)
X(26350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3640, 26349), (6, 5604, 10049), (12594, 19047, 1)

X(26351) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 1st AURIGA

Barycentrics    a*(-a+b+c)*((a+b-c)*(a-b+c)*D+2*a^2*b*c*(a+b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26351) = (8*R^2*s+D)*X(1)+4*R*S*X(3)

X(26351) lies on these lines: {1,3}, {4,26388}, {11,26359}, {12,26326}, {33,26371}, {78,8197}, {497,26387}, {997,5599}, {1479,26386}, {1837,26382}, {3434,26411}, {3811,12454}, {4294,26381}, {4511,5601}, {4861,5602}, {6261,9834}, {6264,12461}, {6326,12460}, {9668,18496}, {10799,26379}, {10833,26302}, {10877,26310}, {10927,26334}, {10928,26344}, {10947,26390}, {10953,26389}, {11843,21740}, {11909,26383}, {16121,16132}, {19037,26384}, {19038,26385}, {26353,26391}, {26354,26392}, {26355,26396}, {26356,26397}

X(26351) = reflection of X(26404) in X(1)
X(26351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 55, 26352), (5597, 5598, 26393), (11881, 11884, 26417)

X(26352) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 2nd AURIGA

Barycentrics    a*(-a+b+c)*(-(a+b-c)*(a-b+c)*D+2*a^2*b*c*(a+b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26352) = (8*R^2*s-D)*X(1)+4*R*S*X(3)

X(26352) lies on these lines: {1,3}, {4,26412}, {11,26360}, {12,26327}, {33,26372}, {78,8204}, {497,26411}, {997,5600}, {1479,26410}, {1837,26406}, {3434,26387}, {3811,12455}, {4294,26405}, {4511,5602}, {4861,5601}, {6261,9835}, {6264,12460}, {6326,12461}, {9668,18498}, {10799,26403}, {10833,26303}, {10877,26311}, {10928,26345}, {10947,26414}, {10953,26413}, {11844,21740}, {11909,26407}, {16122,16132}, {19037,26408}, {19038,26409}, {26354,26416}, {26355,26420}, {26356,26421}

X(26352) = reflection of X(26380) in X(1)
X(26352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 55, 26351), (5597, 5598, 26417), (11882, 11883, 26393)

X(26353) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND LUCAS HOMOTHETIC

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

X(26353) lies on these lines: {1,26433}, {4,26477}, {11,5490}, {12,26328}, {33,26373}, {35,26498}, {55,493}, {56,26292}, {497,26471}, {1479,26466}, {1837,26442}, {2098,26495}, {2646,26367}, {4294,26439}, {6464,26354}, {9668,18521}, {10799,26427}, {10833,26304}, {10877,26312}, {10927,26337}, {10928,26347}, {10947,26488}, {10953,26483}, {11909,26447}, {19037,26454}, {19038,26460}, {26351,26391}, {26352,26415}, {26355,26496}, {26356,26497}, {26357,26499}, {26358,26500}

X(26354) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND LUCAS(-1) HOMOTHETIC

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

X(26354) lies on these lines: {1,26434}, {4,26478}, {11,5491}, {12,26329}, {33,26374}, {35,26507}, {55,494}, {56,26293}, {497,26472}, {1479,26467}, {1697,26299}, {2098,26504}, {2646,26368}, {4294,26440}, {6464,26353}, {9668,18523}, {10799,26428}, {10833,26305}, {10877,26313}, {10928,26338}, {10947,26489}, {10953,26484}, {10965,26511}, {10966,26323}, {11909,26448}, {19037,26455}, {19038,26461}, {26351,26392}, {26352,26416}, {26355,26505}, {26356,26506}, {26357,26508}, {26358,26509}

X(26355) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 3rd TRI-SQUARES-CENTRAL

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

X(26355) lies on these lines: {1,26435}, {4,12949}, {11,26361}, {12,26330}, {20,6283}, {33,26375}, {35,26516}, {55,3068}, {56,26294}, {144,145}, {492,497}, {1007,26474}, {1479,26468}, {1837,26444}, {2098,26514}, {2646,26369}, {3058,5860}, {4294,26441}, {9668,18539}, {10799,26429}, {10833,26306}, {10947,26490}, {10953,26485}, {10965,26520}, {10966,26324}, {11909,26449}, {13699,15682}, {19037,26456}, {19038,26462}, {26351,26396}, {26352,26420}, {26353,26496}, {26354,26505}, {26357,26517}, {26358,26518}

X(26355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390, 3056, 26356), (492, 497, 26473)

X(26356) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 4th TRI-SQUARES-CENTRAL

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

X(26356) lies on these lines: {1,26436}, {4,12948}, {12,26331}, {20,6405}, {33,26376}, {35,26521}, {55,3069}, {56,26295}, {144,145}, {491,497}, {1007,26473}, {1479,26469}, {1697,26301}, {1837,26445}, {2098,26515}, {2646,26370}, {3058,5861}, {4294,8982}, {9668,26438}, {10799,26430}, {10833,26307}, {10877,26315}, {10928,26340}, {10947,26491}, {10953,26486}, {10965,26525}, {10966,26325}, {11909,26450}, {13819,15682}, {19037,26457}, {19038,26463}, {26351,26397}, {26352,26421}, {26353,26497}, {26354,26506}, {26357,26522}, {26358,26523}

X(26356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390, 3056, 26355), (491, 497, 26474)

X(26357) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND INNER-YFF

Barycentrics    a^2*(-a+b+c)*(a^4-2*(b^2+c^2)*a^2-2*b*c*(b+c)*a+(b^2+c^2)*(b-c)^2) : :
X(26357) = R^2*X(1)+r*(R+r)*X(3)

X(26357) lies on these lines: {1,3}, {4,26481}, {5,10953}, {6,22070}, {10,11502}, {11,405}, {12,3149}, {21,497}, {25,23361}, {31,22361}, {33,26377}, {48,836}, {63,1858}, {73,1496}, {104,4305}, {212,1193}, {221,13095}, {225,1593}, {255,1064}, {283,1036}, {378,1068}, {388,411}, {390,4189}, {404,5218}, {474,5432}, {498,6911}, {499,6883}, {515,22759}, {603,4300}, {859,11365}, {950,993}, {956,10950}, {958,1837}, {960,1259}, {997,11517}, {1001,5832}, {1006,3086}, {1011,11269}, {1012,6284}, {1056,6876}, {1058,6875}, {1069,3422}, {1070,21312}, {1072,7395}, {1106,22053}, {1253,22072}, {1376,24987}, {1455,15852}, {1468,14547}, {1478,6985}, {1479,3560}, {1486,16872}, {1682,6056}, {1898,7330}, {2066,19050}, {2071,16272}, {2260,2268}, {2323,4254}, {2360,4276}, {2361,16466}, {2478,26476}, {2654,10448}, {2975,3486}, {3011,7484}, {3058,10959}, {3085,6905}, {3145,8240}, {3516,23710}, {3556,22345}, {3651,4293}, {3895,8668}, {3916,12711}, {3925,19520}, {4188,5281}, {4255,7074}, {4265,10387}, {4294,6906}, {4304,5450}, {4309,10058}, {4314,5267}, {4995,16371}, {4996,9785}, {5047,10589}, {5132,16295}, {5225,6912}, {5231,13615}, {5248,12053}, {5251,9581}, {5252,11500}, {5258,5727}, {5274,16865}, {5292,16287}, {5326,16862}, {5414,19049}, {5687,19524}, {5705,16293}, {5713,7420}, {5715,17605}, {6796,11501}, {6825,10629}, {6863,10523}, {6872,10530}, {6907,18961}, {6913,10896}, {6914,10943}, {6915,10588}, {6920,10591}, {6942,10597}, {6950,10806}, {6954,10321}, {6986,7288}, {7071,11401}, {7354,7580}, {7489,9669}, {7508,15172}, {8053,10934}, {8192,23843}, {8614,23072}, {9668,13743}, {9673,20831}, {9798,11334}, {10039,11499}, {10385,11240}, {10393,14054}, {10572,22758}, {10786,26482}, {10799,26431}, {10827,18491}, {10833,13730}, {10877,26317}, {10895,19541}, {10927,26342}, {10928,26349}, {11238,16418}, {11375,22753}, {11496,12701}, {11809,18859}, {11909,26452}, {12739,22775}, {13738,21321}, {16344,19858}, {19037,26458}, {19038,26464}, {26353,26499}, {26354,26508}, {26355,26517}, {26356,26522}

X(26357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 37579), (55, 56, 2646), (3, 10680, 6585), (3295, 10680, 1), (11012, 12704, 11249)

X(26358) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND OUTER-YFF

Barycentrics    a^2*(-a+b+c)*(a^4-2*(b^2+c^2)*a^2+2*b*c*(b+c)*a+(b^2+4*b*c+c^2)*(b-c)^2) : :
X(26358) = R*(R-2*r)*X(1)+r*(R-r)*X(3)

X(26358) lies on these lines: {1,3}, {4,26482}, {5,10947}, {8,4571}, {11,5687}, {12,26333}, {33,26378}, {78,8668}, {119,1479}, {221,13094}, {390,5046}, {497,3871}, {519,22760}, {944,12775}, {946,11501}, {950,10915}, {1001,24982}, {1012,10944}, {1259,3880}, {1260,12625}, {1376,11376}, {1519,11500}, {1621,5554}, {1837,3913}, {1858,3870}, {1898,5534}, {2057,3689}, {2066,19048}, {2346,5555}, {2348,7368}, {2950,17660}, {3058,10958}, {3085,6941}, {3486,12648}, {3560,12647}, {3583,18518}, {4294,12115}, {5218,17566}, {5252,11496}, {5281,10586}, {5414,19047}, {5432,10200}, {5440,17622}, {6256,6284}, {6913,17662}, {6949,10596}, {6958,10948}, {7071,11400}, {8068,11928}, {8192,23844}, {8715,11502}, {9668,18542}, {9669,12331}, {10385,11114}, {10387,12594}, {10624,12608}, {10799,26432}, {10833,26309}, {10877,26318}, {10927,26343}, {10928,26350}, {10942,10953}, {11909,26453}, {12332,20586}, {12740,13205}, {12743,12751}, {19037,26459}, {19038,26465}, {26353,26500}, {26354,26509}, {26355,26518}, {26356,26523}

X(26538) = complement of X(25245)

X(26359) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND 1st AURIGA

Barycentrics    a*r*D+S*((b^2+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26359) lies on these lines: {1,442}, {2,5597}, {3,18496}, {4,26290}, {5,26326}, {8,26395}, {11,26351}, {12,26380}, {55,26387}, {56,26388}, {83,26379}, {140,26398}, {377,26425}, {427,26371}, {517,26327}, {528,8187}, {631,26381}, {958,26319}, {1004,11492}, {1125,26365}, {1376,26390}, {1650,26383}, {1698,26296}, {3068,26385}, {3069,26384}, {3096,26310}, {3434,5598}, {5490,26391}, {5491,26392}, {5552,26402}, {5590,26344}, {5591,26334}, {6690,8186}, {10527,26401}, {26361,26396}, {26362,26397}, {26363,26399}, {26364,26400}

X(26359) = complement of X(5601)
X(26359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 26360), (3813, 24392, 26360)

X(26360) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND 2nd AURIGA

Barycentrics    -a*r*D+S*((b^2+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26360) lies on these lines: {1,442}, {2,5598}, {3,18498}, {4,26291}, {5,26327}, {8,26419}, {11,26352}, {12,26404}, {55,26411}, {56,26412}, {83,26403}, {140,26422}, {377,26401}, {427,26372}, {517,26326}, {528,8186}, {631,26405}, {958,26320}, {1004,11493}, {1125,26366}, {1376,26414}, {1650,26407}, {1698,26297}, {3068,26409}, {3069,26408}, {3096,26311}, {3434,5597}, {5490,26415}, {5491,26416}, {5552,26426}, {5590,26345}, {5591,26335}, {6690,8187}, {10527,26425}, {26361,26420}, {26362,26421}, {26363,26423}, {26364,26424}

X(26360) = complement of X(5602)
X(26360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 26359), (3813, 24392, 26359), (25466, 25525, 26359)

X(26361) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    3*S+b^2+c^2 : :
X(26361) = 3*(3*S+2*SW)*X(2)-2*SW*X(6)

X(26361) lies on these lines: {1,26444}, {2,6}, {3,18539}, {4,641}, {5,26330}, {8,26514}, {11,26355}, {12,26435}, {20,23311}, {55,26473}, {56,26479}, {83,26429}, {140,26516}, {427,26375}, {625,6460}, {631,639}, {640,5067}, {642,3533}, {958,26324}, {1125,26369}, {1376,26490}, {1586,24244}, {1588,11315}, {1650,26449}, {3096,26314}, {5420,7375}, {5490,7763}, {5491,26505}, {5552,26520}, {6118,13886}, {7376,10577}, {7486,23312}, {10194,18840}, {10527,26519}, {13701,15682}, {18819,21463}, {26359,26396}, {26360,26420}, {26363,26517}, {26364,26518}

X(26361) = complement of X(8972)
X(26361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1271, 8253), (591, 7585, 26340), (3068, 5860, 26339)

X(26362) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND 4th TRI-SQUARES-CENTRAL

Barycentrics    3*S-b^2-c^2 : :
X(26362) = 3*(3*S-2*SW)*X(2)+2*SW*X(6)

X(26362) lies on these lines: {1,26445}, {2,6}, {3,26307}, {4,642}, {5,26331}, {8,26515}, {12,26436}, {20,23312}, {55,26474}, {56,26480}, {83,26430}, {140,26521}, {427,26376}, {625,6459}, {631,640}, {639,5067}, {641,3533}, {958,26325}, {1125,26370}, {1376,26491}, {1585,24243}, {1587,11316}, {1650,26450}, {1698,26301}, {3096,26315}, {5418,7376}, {5490,26497}, {5491,7763}, {5552,26525}, {6119,13939}, {7375,10576}, {7486,23311}, {9540,11314}, {10195,18840}, {10527,26524}, {13821,15682}, {18820,21464}, {26359,26397}, {26360,26421}, {26363,26522}, {26364,26523}

X(26362) = complement of X(13941)
X(26362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 491, 3069), (2, 1271, 615), (491, 3069, 5861)

X(26363) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND INNER-YFF

Barycentrics    a^4-2*(b^2+c^2)*a^2-2*b*c*(b+c)*a+(b^2-c^2)^2 : :
X(26363) = R*X(1)-3*(R+r)*X(2)

X(26363) lies on these lines: {1,2}, {3,2886}, {4,993}, {5,958}, {7,11263}, {9,6832}, {11,405}, {12,956}, {20,5267}, {21,1479}, {35,3434}, {36,377}, {37,5831}, {40,6833}, {55,7483}, {56,442}, {57,12609}, {63,12047}, {72,11375}, {75,7763}, {83,26431}, {104,6937}, {119,15843}, {140,1376}, {149,4309}, {165,6890}, {191,11415}, {197,19547}, {219,5742}, {225,475}, {238,25519}, {281,7537}, {283,17188}, {354,14054}, {355,6863}, {388,3822}, {392,11376}, {427,26377}, {443,3841}, {452,10591}, {474,3925}, {495,12513}, {496,1001}, {497,5248}, {515,6825}, {516,6847}, {517,6862}, {518,11374}, {529,9654}, {535,5229}, {590,1378}, {615,1377}, {631,2550}, {758,3485}, {944,6853}, {946,5709}, {952,26487}, {954,6067}, {960,5791}, {962,6888}, {966,2323}, {982,24159}, {988,17064}, {999,25466}, {1012,15908}, {1068,17917}, {1107,3767}, {1203,24597}, {1329,1656}, {1385,5794}, {1478,2476}, {1573,7746}, {1650,26452}, {1699,6837}, {1706,6967}, {1770,4652}, {1788,3754}, {1836,3916}, {1861,3541}, {1936,25490}, {2006,15065}, {2049,19720}, {2077,6977}, {2078,6681}, {2345,25078}, {2475,4299}, {2478,5251}, {2548,4426}, {2551,3090}, {2646,3419}, {2949,5758}, {3035,3526}, {3068,26464}, {3071,9678}, {3096,26317}, {3295,3813}, {3333,25525}, {3338,5249}, {3421,10588}, {3428,6831}, {3436,5258}, {3452,6887}, {3475,3881}, {3487,3874}, {3488,6598}, {3525,10806}, {3555,17718}, {3560,26333}, {3576,6889}, {3583,6872}, {3585,6871}, {3628,3820}, {3647,5698}, {3739,6389}, {3753,24914}, {3816,11108}, {3817,5715}, {3825,5084}, {3826,6691}, {3829,9669}, {3847,5713}, {3878,5603}, {3897,5086}, {3926,20888}, {3962,4870}, {4187,10966}, {4189,4302}, {4190,7280}, {4193,5260}, {4197,5253}, {4208,5265}, {4293,5177}, {4295,5744}, {4297,6908}, {4305,5175}, {4323,5775}, {4331,17077}, {4357,24179}, {4359,17869}, {4413,11510}, {4428,15172}, {4512,9614}, {4640,12699}, {4647,17740}, {5044,11230}, {5054,18543}, {5067,10597}, {5070,9711}, {5080,5141}, {5082,5218}, {5087,5302}, {5094,11401}, {5204,11112}, {5219,21077}, {5225,11111}, {5234,6886}, {5247,17717}, {5259,15175}, {5270,20076}, {5273,5536}, {5274,17558}, {5288,10585}, {5289,5901}, {5291,9596}, {5303,17579}, {5432,5687}, {5435,15932}, {5439,17728}, {5443,5692}, {5450,6850}, {5490,26499}, {5491,26508}, {5587,6834}, {5657,6952}, {5691,6838}, {5730,15950}, {5770,5884}, {5795,6944}, {5811,21635}, {5818,6949}, {5836,6958}, {5837,13464}, {5850,8232}, {5881,10786}, {5905,6763}, {6245,12520}, {6256,6842}, {6284,16370}, {6585,6911}, {6668,12607}, {6684,6891}, {6796,6954}, {6848,19925}, {6899,7688}, {6907,12114}, {6913,7681}, {6914,10525}, {6921,14798}, {6926,10164}, {6953,7989}, {6959,9956}, {6976,10598}, {6989,10165}, {7173,17556}, {7294,10949}, {7308,25522}, {7330,12608}, {7354,17532}, {7484,10835}, {7486,8165}, {7504,11681}, {7506,9713}, {7680,22770}, {7786,13110}, {7795,21264}, {7800,20541}, {7807,20172}, {7808,10804}, {7914,10879}, {8609,17303}, {8728,15325}, {9624,15829}, {9668,17571}, {9785,21630}, {9798,19544}, {9840,15654}, {9940,18251}, {9943,17646}, {9955,24703}, {10171,18250}, {10473,10974}, {10592,11236}, {10895,17530}, {10896,11113}, {11194,18990}, {11235,15171}, {11238,15670}, {11281,15934}, {11365,25514}, {11915,15184}, {12559,24391}, {13190,14061}, {13218,15059}, {15338,19535}, {16062,19794}, {16252,20306}, {16342,23518}, {16415,20470}, {17321,25598}, {17757,18967}, {18253,18493}, {18839,24954}, {19548,23850}, {19763,21321}, {19888,19941}, {19894,19930}, {21530,23304}, {22464,25590}, {26359,26399}, {26360,26423}

X(26363) = midpoint of X(4305) and X(5175)
X(26363) = reflection of X(10894) in X(5)
X(26363) = complement of X(3085)
X(26363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 499, 10200), (499, 19854, 2), (3616, 12649, 1)

X(26364) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND OUTER-YFF

Barycentrics    a^4-2*(b^2+c^2)*a^2+2*b*c*(b+c)*a+(b^2-c^2)^2 : :
X(26364) = R*X(1)-3*(R-r)*X(2)

X(26364) lies on these lines: {1,2}, {3,119}, {4,2077}, {5,1376}, {9,2252}, {11,5687}, {12,474}, {35,2478}, {36,3436}, {40,1519}, {46,908}, {55,4187}, {56,13747}, {57,21077}, {72,18838}, {83,26432}, {100,1479}, {140,958}, {165,6838}, {191,10940}, {214,944}, {281,25078}, {345,20320}, {355,5123}, {377,7951}, {388,17567}, {392,24954}, {404,1478}, {405,5432}, {427,26378}, {442,4413}, {443,3822}, {475,1877}, {484,11415}, {495,25524}, {496,3913}, {497,3825}, {515,6891}, {516,6848}, {517,6959}, {528,9669}, {590,1377}, {615,1378}, {631,993}, {758,1788}, {946,6944}, {952,26492}, {956,5433}, {960,6863}, {962,6979}, {999,6691}, {1001,17527}, {1058,25439}, {1145,2098}, {1213,5783}, {1259,10523}, {1324,13732}, {1387,10912}, {1482,8256}, {1532,10310}, {1574,7746}, {1575,3767}, {1650,26453}, {1656,2886}, {1697,25522}, {1699,6953}, {1706,6983}, {1837,5440}, {1861,3542}, {2049,19721}, {2548,4386}, {2550,3090}, {2950,21635}, {2975,17566}, {3036,12645}, {3068,26465}, {3069,26459}, {3071,9679}, {3096,26318}, {3256,3841}, {3295,3816}, {3306,13407}, {3336,5905}, {3359,3452}, {3419,17606}, {3421,5193}, {3434,6931}, {3485,3754}, {3487,5883}, {3523,5267}, {3525,10805}, {3526,4999}, {3555,17728}, {3576,6967}, {3579,24703}, {3583,5187}, {3585,4190}, {3614,17532}, {3740,5791}, {3753,11375}, {3763,12594}, {3812,11374}, {3813,6667}, {3817,6964}, {3826,6668}, {3836,23693}, {3874,25568}, {3878,5657}, {3880,11373}, {3911,21075}, {3922,4870}, {3926,6381}, {3947,12436}, {4188,4299}, {4197,9342}, {4294,6919}, {4295,5748}, {4297,6926}, {4302,5046}, {4308,5828}, {4310,24167}, {4317,20060}, {4358,17869}, {4421,15171}, {4423,17575}, {4855,10572}, {4857,20075}, {5010,6872}, {5044,5694}, {5054,18545}, {5067,10596}, {5070,9710}, {5082,10589}, {5084,5218}, {5086,7705}, {5087,12699}, {5094,11400}, {5217,11113}, {5219,12609}, {5226,11263}, {5251,6910}, {5252,17614}, {5277,9596}, {5289,5690}, {5326,10955}, {5328,6960}, {5438,5587}, {5439,17718}, {5445,5692}, {5450,6961}, {5490,26500}, {5491,26509}, {5590,26350}, {5591,26343}, {5660,15071}, {5691,6890}, {5693,18254}, {5720,12616}, {5745,6989}, {5770,15528}, {5794,6862}, {5795,10165}, {5818,6952}, {5836,5886}, {5850,8732}, {5881,10785}, {6174,6284}, {6376,7763}, {6554,24036}, {6690,11108}, {6692,21620}, {6796,6827}, {6824,10175}, {6837,7989}, {6847,19925}, {6853,10176}, {6880,11012}, {6882,11499}, {6887,10172}, {6904,10590}, {6908,10164}, {6911,26332}, {6918,7680}, {6922,11500}, {6924,10526}, {6941,12775}, {6947,10902}, {6963,11491}, {7354,16371}, {7483,22768}, {7484,10834}, {7506,9712}, {7629,8062}, {7681,10306}, {7786,13109}, {7808,10803}, {7914,10878}, {7952,24025}, {9654,16417}, {9655,17573}, {9656,17583}, {9798,16434}, {10591,17784}, {10593,11235}, {10895,11112}, {10914,11376}, {10965,24390}, {11236,17564}, {11358,19754}, {11502,11517}, {11849,15813}, {11914,15184}, {11928,23513}, {12513,15325}, {12679,17613}, {12700,22835}, {12749,21842}, {13189,14061}, {13217,15059}, {13465,18253}, {14561,17792}, {15326,19537}, {15654,19514}, {15842,26470}, {15844,16410}, {16062,19795}, {16252,20307}, {16408,25466}, {16593,17675}, {17719,24159}, {18250,21164}, {19550,23361}, {26359,26400}, {26360,26424}, {26361,26518}, {26362,26523}

X(26364) = midpoint of X(3086) and X(7080)
X(26364) = reflection of X(10893) in X(5)
X(26364) = complement of X(3086)
X(26364) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1125, 10915, 1), (3244, 10199, 14986), (3616, 12648, 1)

X(26365) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 1st AURIGA

Barycentrics    a*(D+2*a^3-(b+c)*a^2-2*(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26365) = (-D+4*(r+2*R)*S)*X(1)+4*S*r*X(3)

X(26365) lies on these lines: {1,3}, {2,26382}, {515,26326}, {1125,26359}, {3616,26394}, {5603,26381}, {5886,26386}, {11363,26371}, {11364,26379}, {11365,26302}, {11368,26310}, {11370,26334}, {11371,26344}, {11373,26390}, {11374,26389}, {11375,26388}, {11376,26387}, {11831,26383}, {18493,18496}, {18991,26384}, {18992,26385}, {26367,26391}, {26368,26392}, {26369,26396}, {26370,26397}

X(26365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26296, 26395), (999, 2646, 26366), (5597, 26395, 26296)

X(26366) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 2nd AURIGA

Barycentrics    a*(D-2*a^3+(b+c)*a^2+2*(b^2+b*c+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26366) = (D+4*(r+2*R)*S)*X(1)+4*S*r*X(3)

X(26366) lies on these lines: {1,3}, {2,26406}, {515,26327}, {1125,26360}, {3616,26418}, {5603,26405}, {5886,26410}, {11363,26372}, {11364,26403}, {11365,26303}, {11368,26311}, {11370,26335}, {11373,26414}, {11374,26413}, {11375,26412}, {11376,26411}, {11831,26407}, {18493,18498}, {18991,26408}, {18992,26409}, {26367,26415}, {26368,26416}, {26369,26420}, {26370,26421}

X(26366) = midpoint of X(1) and X(8187)
X(26366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24929, 26365), (1, 26297, 26419), (999, 2646, 26365)

X(26367) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND LUCAS HOMOTHETIC

Barycentrics
a*((a^4-10*(b^2+c^2)*a^2-4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2-4*b^2*c^2)*S+2*a^6+(b+c)*a^5-3*(b^2+c^2)*a^4-2*(b^2+c^2)*(b+c)*a^3-16*b^2*c^2*a^2+((b^2-c^2)^2-4*b^2*c^2)*(b+c)*a+(b^4-c^4)*(b^2-c^2)) : :

X(26367) lies on these lines: {1,493}, {2,26442}, {515,26328}, {517,26498}, {999,26322}, {1125,5490}, {1319,26433}, {2646,26353}, {3295,26493}, {3576,26292}, {3616,26494}, {5603,26439}, {5886,26466}, {6464,26368}, {11363,26373}, {11364,26427}, {11365,26304}, {11368,26312}, {11370,26337}, {11371,26347}, {11373,26488}, {11374,26483}, {11375,26477}, {11376,26471}, {11831,26447}, {18493,18521}, {18991,26454}, {18992,26460}, {26365,26391}, {26366,26415}, {26369,26496}, {26370,26497}

X(26367) = midpoint of X(1) and X(8188)
X(26367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26298, 26495), (493, 26495, 26298)

X(26368) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND LUCAS(-1) HOMOTHETIC

Barycentrics
a*(-(a^4-10*(b^2+c^2)*a^2-4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2-4*b^2*c^2)*S+2*a^6+(b+c)*a^5-3*(b^2+c^2)*a^4-2*(b^2+c^2)*(b+c)*a^3-16*b^2*c^2*a^2+((b^2-c^2)^2-4*b^2*c^2)*(b+c)*a+(b^4-c^4)*(b^2-c^2)) : :

X(26368) lies on these lines: {1,494}, {2,26443}, {515,26329}, {517,26507}, {999,26323}, {1125,5491}, {1319,26434}, {2646,26354}, {3295,26502}, {3576,26293}, {3616,26503}, {5603,26440}, {5886,26467}, {6464,26367}, {11363,26374}, {11364,26428}, {11365,26305}, {11368,26313}, {11371,26338}, {11373,26489}, {11374,26484}, {11375,26478}, {11376,26472}, {11831,26448}, {18493,18523}, {18991,26455}, {18992,26461}, {26365,26392}, {26366,26416}, {26369,26505}, {26370,26506}

X(26368) = midpoint of X(1) and X(8189)
X(26368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26299, 26504), (494, 26504, 26299)

X(26369) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 3rd TRI-SQUARES-CENTRAL

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

X(26369) lies on these lines: {1,1336}, {2,26444}, {4,12269}, {193,1386}, {492,3616}, {515,26330}, {517,26516}, {999,26324}, {1125,26361}, {1319,26435}, {2646,26355}, {3295,26512}, {3576,26294}, {3636,11370}, {5603,26441}, {5886,26468}, {7981,8960}, {11363,26375}, {11364,26429}, {11365,26306}, {11368,26314}, {11373,26490}, {11374,26485}, {11375,26479}, {11376,26473}, {11831,26449}, {13667,15682}, {18493,18539}, {18991,26456}, {18992,26462}, {26365,26396}, {26366,26420}, {26367,26496}, {26368,26505}

X(26369) = midpoint of X(1) and X(13888)
X(26369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26300, 26514), (3068, 26514, 26300)

X(26370) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 4th TRI-SQUARES-CENTRAL

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

X(26370) lies on these lines: {1,1123}, {2,26445}, {4,12268}, {193,1386}, {491,3616}, {515,26331}, {517,26521}, {551,5861}, {999,26325}, {1125,26362}, {1319,26436}, {2646,26356}, {3295,26513}, {3576,26295}, {3636,11371}, {5603,8982}, {5886,26469}, {11363,26376}, {11364,26430}, {11365,26307}, {11368,26315}, {11373,26491}, {11374,26486}, {11375,26480}, {11376,26474}, {11831,26450}, {13787,15682}, {18493,26438}, {18991,26457}, {18992,26463}, {26365,26397}, {26366,26421}, {26367,26497}, {26368,26506}

X(26370) = midpoint of X(1) and X(13942)
X(26370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26301, 26515), (3069, 26515, 26301)

X(26371) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 1st AURIGA

Barycentrics    a*(-(-a^2+b^2+c^2)*D+2*a*b*c*(b+c)*(a+b+c))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : : , where D=4*S*sqrt(R*(4*R+r))

X(26371) lies on these lines: {1,1824}, {4,26386}, {24,26398}, {25,5597}, {33,26351}, {34,26380}, {235,26326}, {427,26359}, {1593,26290}, {5090,26382}, {5410,26385}, {5411,26384}, {7487,26381}, {7713,26296}, {11363,26365}, {11380,26379}, {11383,26393}, {11386,26310}, {11388,26334}, {11389,26344}, {11390,26390}, {11391,26389}, {11392,26388}, {11393,26387}, {11396,26395}, {11400,26402}, {11401,26401}, {11832,26383}, {18494,18496}, {22479,26319}, {26373,26391}, {26374,26392}, {26375,26396}, {26376,26397}, {26377,26399}, {26378,26400}

X(26371) = {X(1), X(1824)}-harmonic conjugate of X(26372)

X(26372) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2nd AURIGA

Barycentrics    a*((-a^2+b^2+c^2)*D+2*a*b*c*(b+c)*(a+b+c))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : : , where D=4*S*sqrt(R*(4*R+r))

X(26372) lies on these lines: {1,1824}, {4,26410}, {24,26422}, {25,5598}, {33,26352}, {34,26404}, {235,26327}, {427,26360}, {1593,26291}, {5090,26406}, {5410,26409}, {5411,26408}, {7487,26405}, {7713,26297}, {11363,26366}, {11380,26403}, {11383,26417}, {11386,26311}, {11388,26335}, {11389,26345}, {11390,26414}, {11391,26413}, {11392,26412}, {11393,26411}, {11396,26419}, {11400,26426}, {11401,26425}, {11832,26407}, {18494,18498}, {22479,26320}, {26373,26415}, {26374,26416}, {26375,26420}, {26376,26421}, {26377,26423}, {26378,26424}

X(26372) = {X(1), X(1824)}-harmonic conjugate of X(26371)

X(26373) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND LUCAS HOMOTHETIC

Barycentrics    a^2*((a^4-2*(b^2+c^2)*a^2-14*b^2*c^2-3*c^4-3*b^4)*S+2*(b^2+c^2)*(a^4-2*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(26373) lies on these lines: {4,26466}, {24,26498}, {25,371}, {33,26353}, {34,26433}, {69,24244}, {235,26328}, {427,5490}, {1593,26292}, {5090,26442}, {5410,26460}, {5411,26454}, {6464,26374}, {7487,26439}, {7713,26298}, {11363,26367}, {11380,26427}, {11383,26493}, {11386,26312}, {11388,26337}, {11389,26347}, {11390,26488}, {11391,26483}, {11392,26477}, {11393,26471}, {11396,26495}, {11401,26501}, {11832,26447}, {18494,18521}, {22479,26322}, {26371,26391}, {26372,26415}, {26375,26496}, {26376,26497}, {26377,26499}, {26378,26500}

X(26373) = {X(493), X(8948)}-harmonic conjugate of X(25)

X(26374) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(-(a^4-2*(b^2+c^2)*a^2-14*b^2*c^2-3*c^4-3*b^4)*S+2*(b^2+c^2)*(a^4-2*(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(26374) lies on these lines: {4,26467}, {24,26507}, {25,372}, {33,26354}, {34,26434}, {69,24243}, {235,26329}, {427,5491}, {1593,26293}, {5090,26443}, {5410,26461}, {5411,26455}, {6464,26373}, {7487,26440}, {7713,26299}, {11363,26368}, {11380,26428}, {11383,26502}, {11386,26313}, {11389,26338}, {11390,26489}, {11391,26484}, {11392,26478}, {11393,26472}, {11396,26504}, {11400,26511}, {11401,26510}, {11832,26448}, {18494,18523}, {22479,26323}, {26371,26392}, {26372,26416}, {26375,26505}, {26376,26506}, {26377,26508}, {26378,26509}

X(26374) = {X(494), X(8946)}-harmonic conjugate of X(25)

X(26375) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 3rd TRI-SQUARES-CENTRAL

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

X(26375) lies on these lines: {4,488}, {20,6291}, {24,26516}, {25,3068}, {33,26355}, {34,26435}, {193,1843}, {235,26330}, {393,5200}, {427,26361}, {428,5860}, {1593,26294}, {5090,26444}, {5410,26462}, {5411,26456}, {7487,26441}, {7713,26300}, {8408,11473}, {10301,11388}, {11363,26369}, {11380,26429}, {11383,26512}, {11386,26314}, {11390,26490}, {11391,26485}, {11392,26479}, {11393,26473}, {11396,26514}, {11400,26520}, {11401,26519}, {11832,26449}, {13668,15682}, {18494,18539}, {22479,26324}, {26371,26396}, {26372,26420}, {26373,26496}, {26374,26505}, {26377,26517}, {26378,26518}

X(26375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1843, 6995, 26376), (8948, 12148, 4)

X(26376) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 4th TRI-SQUARES-CENTRAL

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

X(26376) lies on these lines: {4,487}, {20,6406}, {24,26521}, {25,3069}, {33,26356}, {34,26436}, {193,1843}, {235,26331}, {393,5412}, {427,26362}, {428,5861}, {1163,5200}, {1593,26295}, {5090,26445}, {5410,26463}, {5411,26457}, {7487,8982}, {7713,26301}, {8420,11474}, {10301,11389}, {11363,26370}, {11380,26430}, {11383,26513}, {11386,26315}, {11390,26491}, {11391,26486}, {11392,26480}, {11393,26474}, {11396,26515}, {11400,26525}, {11401,26524}, {11832,26450}, {13788,15682}, {18494,26438}, {22479,26325}, {26371,26397}, {26372,26421}, {26373,26497}, {26374,26506}, {26377,26522}, {26378,26523}

X(26376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1843, 6995, 26375), (8946, 12147, 4)

X(26377) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND INNER-YFF

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

X(26377) lies on these lines: {1,25}, {3,1824}, {4,2975}, {5,11391}, {8,4231}, {19,1609}, {24,10267}, {28,1068}, {33,26357}, {34,26437}, {55,20832}, {56,225}, {232,607}, {235,26332}, {283,24320}, {427,26363}, {429,958}, {431,1478}, {444,5230}, {468,10198}, {956,5130}, {1593,1900}, {1598,1828}, {1825,11509}, {1871,6585}, {1878,5198}, {1902,5709}, {2333,9310}, {2905,11107}, {3089,10532}, {3515,10902}, {3517,16202}, {4186,10966}, {4232,10587}, {5090,6734}, {5410,26464}, {5411,26458}, {5412,19050}, {5413,19049}, {6198,14017}, {6756,10943}, {6995,10529}, {7466,7718}, {7487,12116}, {7714,11240}, {7716,12595}, {8946,26510}, {8948,26501}, {9645,13730}, {11380,26431}, {11386,26317}, {11388,26342}, {11389,26349}, {11392,26481}, {11393,26475}, {11832,26452}, {13095,15811}, {14018,19850}, {17523,23710}, {18494,18544}, {26371,26399}, {26372,26423}, {26373,26499}, {26374,26508}, {26375,26517}, {26376,26522}

X(26377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 1829, 26378), (25, 11396, 11398), (25, 11401, 1)

X(26378) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND OUTER-YFF

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

X(26378) lies on these lines: {1,25}, {3,1828}, {4,100}, {5,11390}, {24,10269}, {33,26358}, {34,1470}, {55,1842}, {56,1866}, {235,26333}, {427,26364}, {607,10311}, {1376,1883}, {1452,18838}, {1593,1878}, {1598,1824}, {1831,10965}, {1851,7412}, {1862,25438}, {1877,4185}, {1900,5198}, {3089,10531}, {3517,16203}, {3575,6256}, {4232,10586}, {5090,6735}, {5101,5687}, {5151,13205}, {5410,26465}, {5411,26459}, {5412,19048}, {5413,19047}, {6756,10942}, {6995,10528}, {7487,12115}, {7714,11239}, {7716,12594}, {7718,12648}, {8946,26511}, {11380,26432}, {11386,26318}, {11388,26343}, {11389,26350}, {11392,26482}, {11393,26476}, {11832,26453}, {12137,12751}, {13094,15811}, {18494,18542}, {20619,23404}, {20832,22768}, {26371,26400}, {26372,26424}, {26373,26500}, {26374,26509}, {26375,26518}, {26376,26523}

X(26378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 1829, 26377), (25, 11396, 11399), (25, 11400, 1)

X(26379) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 5th BROCARD

Barycentrics    a*(((b^2+c^2)*a^2+b^2*c^2)*D+a*(a+b+c)*(a^5-(b+c)*a^4-2*b*c*(b+c)*a^2+b^2*c^2*a-b^2*c^2*(b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26379) lies on these lines: {1,26403}, {32,5597}, {83,26359}, {98,26326}, {182,26290}, {2080,26398}, {7787,26394}, {10788,26381}, {10789,26296}, {10790,26302}, {10791,26382}, {10792,26334}, {10793,26344}, {10794,26390}, {10795,26389}, {10796,26386}, {10797,26388}, {10798,26387}, {10799,26351}, {10800,26395}, {10803,26402}, {10804,26401}, {11364,26365}, {11380,26371}, {11490,26393}, {11839,26383}, {12835,26380}, {18496,18501}, {18994,26385}, {22520,26319}, {26391,26427}, {26392,26428}, {26396,26429}, {26397,26430}, {26399,26431}, {26400,26432}

X(26380) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 2nd CIRCUMPERP TANGENTIAL

Barycentrics    a*(a+b-c)*(a-b+c)*((-a+b+c)*D+2*a*b*c*(a-2*b-2*c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26380) = (8*R*(R*s+S)-D)*X(1)-4*R*S*X(3)

X(26380) lies on these lines: {1,3}, {4,26387}, {11,26326}, {12,26359}, {34,26371}, {388,26388}, {1478,26386}, {3434,26412}, {4293,26381}, {5252,26382}, {9655,18496}, {12835,26379}, {18954,26302}, {18957,26310}, {18958,26383}, {18959,26334}, {18960,26344}, {18961,26390}, {18962,26389}, {18995,26384}, {18996,26385}, {26391,26433}, {26392,26434}, {26396,26435}, {26397,26436}

X(26380) = reflection of X(26352) in X(1)
X(26380) = inverse of X(5903) in the Moses-Longuet-Higgins circle
X(26380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2099, 26404), (65, 1319, 5598), (26402, 26425, 26393)

X(26381) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND EULER

Barycentrics
-2*a*(-a+b+c)*(a+b-c)*(a-b+c)*D+3*a^7-3*(b+c)*a^6-5*(b^2+c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b^2-c^2)^2*a^3-(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : : , where D=4*S*sqrt(R*(4*R+r))

X(26381) lies on these lines: {1,6934}, {2,26386}, {3,26394}, {4,5597}, {5,18496}, {24,26302}, {104,26319}, {376,26290}, {515,26296}, {631,26359}, {3085,26388}, {3086,26387}, {4293,26380}, {4294,26351}, {5603,26365}, {5657,26382}, {5842,11366}, {7487,26371}, {7581,26385}, {7582,26384}, {7967,26395}, {8982,26397}, {9862,26310}, {10783,26334}, {10784,26344}, {10785,26390}, {10786,26389}, {10788,26379}, {10805,26402}, {10806,26401}, {11491,26393}, {11845,26383}, {12115,26400}, {12116,26399}, {26391,26439}, {26392,26440}, {26396,26441}

X(26381) = reflection of X(4) in X(8196)

X(26382) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-GARCIA

Barycentrics    a*D+a^4-(b+c)*a^3+(b^2+c^2)*(b+c)*a-(b^2-c^2)^2 : : , where D=4*S*sqrt(R*(4*R+r))

X(26382) lies on these lines: {1,442}, {2,26365}, {8,26394}, {10,5597}, {65,26388}, {72,26389}, {515,26290}, {517,26386}, {519,26395}, {956,26319}, {1837,26351}, {3057,26387}, {3679,26296}, {5090,26371}, {5252,26380}, {5587,26326}, {5657,26381}, {5687,26393}, {5688,26344}, {5689,26334}, {6734,26399}, {6735,26400}, {8193,26302}, {9857,26310}, {10791,26379}, {10914,26390}, {10915,26402}, {10916,26401}, {11900,26383}, {12702,18496}, {13883,26385}, {13936,26384}, {17647,26425}, {26391,26442}, {26392,26443}, {26396,26444}, {26397,26445}, {26398,26446}

X(26382) = reflection of X(8197) in X(10)
X(26382) = {X(1), X(3419)}-harmonic conjugate of X(26406)

X(26383) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND GOSSARD

Barycentrics
(a*(a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))*(a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*D+(a+b+c)*((b^2+c^2)*a^9-(b^2-c^2)*(b-c)*a^8-2*(b^4+c^4)*a^7+2*(b^3-c^3)*(b^2-c^2)*a^6+b^2*c^2*(b^2+c^2)*a^5-(b^2-c^2)*(b-c)*b*c*(4*b^2+9*b*c+4*c^2)*a^4+2*(b^4-c^4)^2*a^3-2*(b^4-c^4)*(b^2-c^2)*(b+c)*(b^2-3*b*c+c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4)*a+(b^2-c^2)^3*(b-c)*(b^4+3*b^2*c^2+c^4)))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))

X(26383) lies on these lines: {1,26407}, {30,26290}, {402,5597}, {1650,26359}, {4240,26394}, {11831,26365}, {11832,26371}, {11839,26379}, {11845,26381}, {11848,26393}, {11852,26296}, {11853,26302}, {11885,26310}, {11897,26326}, {11900,26382}, {11901,26334}, {11902,26344}, {11903,26390}, {11904,26389}, {11905,26388}, {11906,26387}, {11909,26351}, {11910,26395}, {11914,26402}, {11915,26401}, {18496,18508}, {18958,26380}, {19017,26384}, {19018,26385}, {22755,26319}, {26396,26449}, {26398,26451}, {26399,26452}, {26400,26453}

X(26384) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND INNER-GREBE

Barycentrics    a*((a+b-c)*(-a+b+c)*(a-b+c)*D-a*((4*a^3-4*(b+c)*a^2-4*(b^2+c^2)*a+4*(b^2-c^2)*(b-c))*S+4*S^2*(-a+b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26384) lies on these lines: {1,26408}, {6,5597}, {55,26409}, {372,26290}, {1587,26326}, {3069,26359}, {3311,26398}, {5411,26371}, {7582,26381}, {7584,26386}, {7586,26394}, {7968,26395}, {13936,26382}, {18496,18510}, {18991,26365}, {18995,26380}, {18999,26393}, {19003,26296}, {19005,26302}, {19011,26310}, {19013,26319}, {19017,26383}, {19023,26390}, {19025,26389}, {19027,26388}, {19029,26387}, {19037,26351}, {19047,26402}, {19049,26401}, {26391,26454}, {26392,26455}, {26396,26456}, {26397,26457}, {26399,26458}, {26400,26459}

X(26384) = {X(6), X(5597)}-harmonic conjugate of X(26385)

X(26385) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-GREBE

Barycentrics    a*((a+b-c)*(-a+b+c)*(a-b+c)*D-a*(-(4*a^3-4*(b+c)*a^2-4*(b^2+c^2)*a+4*(b^2-c^2)*(b-c))*S+4*S^2*(-a+b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26385) lies on these lines: {1,26409}, {6,5597}, {55,26408}, {371,26290}, {1588,26326}, {3068,26359}, {3312,26398}, {5410,26371}, {7581,26381}, {7583,26386}, {7585,26394}, {7969,26395}, {13883,26382}, {18496,18512}, {18992,26365}, {18996,26380}, {19004,26296}, {19006,26302}, {19012,26310}, {19014,26319}, {19018,26383}, {19026,26389}, {19028,26388}, {19030,26387}, {19038,26351}, {19048,26402}, {19050,26401}, {26391,26460}, {26392,26461}, {26396,26462}, {26399,26464}, {26400,26465}

X(26385) = {X(6), X(5597)}-harmonic conjugate of X(26384)

X(26386) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND JOHNSON

Barycentrics    -a*(-a+b+c)*(a-b+c)*(a+b-c)*D+a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2+c^2)*(b+c)*a^4-(b^2+c^2)^2*a^3+(b^4-c^4)*(b-c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : : , where D=4*S*sqrt(R*(4*R+r))

X(26386) lies on these lines: {1,6917}, {2,26381}, {3,18496}, {4,26371}, {5,5597}, {30,26290}, {119,26400}, {355,26389}, {381,26326}, {517,26382}, {952,26395}, {1478,26380}, {1479,26351}, {5587,26296}, {5886,26365}, {6214,26344}, {6215,26334}, {7583,26385}, {7584,26384}, {9996,26310}, {10679,26327}, {10796,26379}, {10942,26402}, {10943,26401}, {11499,26393}, {22758,26319}, {26391,26466}, {26392,26467}, {26396,26468}, {26397,26469}, {26399,26470}

X(26386) = reflection of X(8200) in X(5)
X(26386) = {X(26387), X(26388)}-harmonic conjugate of X(1)

X(26387) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND INNER-JOHNSON

Barycentrics    (-a+b+c)*(a*(a+b-c)*(a-b+c)*D+(a+b+c)*(a^5-(b+c)*a^4-(b^2+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26387) lies on these lines: {1,6917}, {4,26380}, {11,5597}, {55,26359}, {497,26351}, {499,26398}, {999,18496}, {3057,26382}, {3086,26381}, {3434,26352}, {6284,26290}, {9581,26296}, {10798,26379}, {10832,26302}, {10874,26310}, {10896,26326}, {10925,26334}, {10926,26344}, {10950,26389}, {10958,26402}, {10959,26401}, {11376,26365}, {11393,26371}, {11502,26393}, {11906,26383}, {19029,26384}, {19030,26385}, {22760,26319}, {26396,26473}, {26397,26474}, {26399,26475}, {26400,26476}

X(26387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26386, 26388), (497, 26394, 26351)

X(26388) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-JOHNSON

Barycentrics    (a+b-c)*(a-b+c)*(-a*(-a+b+c)*D+a^5-(b+c)*a^4-(b^2+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26388) lies on these lines: {1,6917}, {4,26351}, {12,5597}, {55,26327}, {56,26359}, {65,26382}, {388,26380}, {498,26398}, {3085,26381}, {3295,18496}, {3434,26404}, {7354,26290}, {9578,26296}, {10797,26379}, {10831,26302}, {10873,26310}, {10895,26326}, {10923,26334}, {10924,26344}, {10944,26390}, {10956,26402}, {10957,26401}, {11375,26365}, {11392,26371}, {11501,26393}, {11905,26383}, {19027,26384}, {19028,26385}, {22759,26319}, {26396,26479}, {26397,26480}, {26399,26481}, {26400,26482}

X(26388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26386, 26387), (388, 26394, 26380)

X(26389) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 1st JOHNSON-YFF

Barycentrics    D*a+(a^2+b^2-c^2)*(a^2-b^2+c^2) : : , where D=4*S*sqrt(R*(4*R+r))
X(26389) = D*X(1)+4*S*r*X(4)

X(26389) lies on these lines: {1,4}, {5,26399}, {11,26401}, {12,5597}, {30,26423}, {72,26382}, {355,26386}, {958,26319}, {3085,8186}, {4294,8187}, {5598,6284}, {5842,11878}, {7354,26425}, {7680,11877}, {10525,26414}, {10786,26381}, {10795,26379}, {10827,26296}, {10830,26302}, {10872,26310}, {10894,26326}, {10895,11366}, {10921,26334}, {10922,26344}, {10942,26400}, {10950,26387}, {10953,26351}, {10955,26402}, {11367,12953}, {11374,26365}, {11391,26371}, {11496,26327}, {11500,26393}, {11827,26290}, {11879,18242}, {11904,26383}, {15908,26291}, {18496,18518}, {18962,26380}, {19025,26384}, {19026,26385}, {26391,26483}, {26392,26484}, {26396,26485}, {26397,26486}, {26398,26487}

X(26389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4, 26413), (388, 5290, 26413), (1478, 21620, 26413)

X(26390) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 2nd JOHNSON-YFF

Barycentrics    a*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*D-4*S^2*(a^3-(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26390) lies on these lines: {1,224}, {5,26400}, {11,5597}, {12,26402}, {355,26386}, {528,11880}, {1376,26359}, {2886,11879}, {10525,26413}, {10785,26381}, {10794,26379}, {10826,26296}, {10829,26302}, {10871,26310}, {10893,26326}, {10914,26382}, {10919,26334}, {10920,26344}, {10943,26399}, {10944,26388}, {10947,26351}, {10949,26401}, {11373,26365}, {11390,26371}, {11826,26290}, {11903,26383}, {12114,26319}, {18496,18519}, {18961,26380}, {19023,26384}, {19024,26385}, {26391,26488}, {26392,26489}, {26396,26490}, {26397,26491}, {26398,26492}

X(26391) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND LUCAS HOMOTHETIC

Barycentrics
a*(((4*a^6-4*(b^2+c^2)*a^4-4*(b^4+10*b^2*c^2+c^4)*a^2+4*(b^4-c^4)*(b^2-c^2))*S+4*S^2*(a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2-4*b^2*c^2))*D+(a+b+c)*a*((8*(b^2+c^2)*a^5-8*(b^3+c^3)*a^4-16*(b^2+c^2)^2*a^3+16*(b^3+c^3)*(b^2+c^2)*a^2+8*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a+(b^2-c^2)*(b-c)*(-8*b^4-8*c^4-8*b*c*(b-c)^2))*S+4*S^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^4+14*b^2*c^2+3*c^4)*a+(b+c)*(3*b^2-2*b*c+c^2)*(b^2-2*b*c+3*c^2)))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26391) lies on these lines: {493,5597}, {5490,26359}, {18496,18521}, {26290,26292}, {26296,26298}, {26302,26304}, {26310,26312}, {26319,26322}, {26326,26328}, {26334,26337}, {26344,26347}, {26351,26353}, {26365,26367}, {26371,26373}, {26379,26427}, {26380,26433}, {26381,26439}, {26382,26442}, {26384,26454}, {26385,26460}, {26386,26466}, {26389,26483}, {26390,26488}, {26393,26493}, {26394,26494}, {26395,26495}, {26396,26496}, {26397,26497}, {26398,26498}, {26399,26499}

X(26392) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND LUCAS(-1) HOMOTHETIC

Barycentrics
a*((-(4*a^6-4*(b^2+c^2)*a^4-4*(b^4+10*b^2*c^2+c^4)*a^2+4*(b^4-c^4)*(b^2-c^2))*S+4*S^2*(a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2-4*b^2*c^2))*D+(a+b+c)*a*(-(8*(b^2+c^2)*a^5-8*(b^3+c^3)*a^4-16*(b^2+c^2)^2*a^3+16*(b^3+c^3)*(b^2+c^2)*a^2+8*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a+(b^2-c^2)*(b-c)*(-8*b^4-8*c^4-8*b*c*(b-c)^2))*S+4*S^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^4+14*b^2*c^2+3*c^4)*a+(b+c)*(3*b^2-2*b*c+c^2)*(b^2-2*b*c+3*c^2)))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26392) lies on these lines: {494,5597}, {5491,26359}, {18496,18523}, {26290,26293}, {26296,26299}, {26302,26305}, {26310,26313}, {26319,26323}, {26326,26329}, {26338,26344}, {26351,26354}, {26365,26368}, {26371,26374}, {26379,26428}, {26380,26434}, {26381,26440}, {26382,26443}, {26384,26455}, {26385,26461}, {26386,26467}, {26389,26484}, {26393,26502}, {26394,26503}, {26395,26504}, {26396,26505}, {26397,26506}, {26398,26507}, {26401,26510}

X(26393) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND MANDART-INCIRCLE

Barycentrics    a^2*(a+b+c)*(2*D*b*c+4*S^2*(-a+b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26393) = (4*R*r*S+D*R)*X(1)+4*S*r^2*X(3)

X(26393) lies on these lines: {1,3}, {100,26394}, {197,26302}, {355,11869}, {1376,26359}, {1737,5599}, {1837,8200}, {1905,11384}, {3476,11844}, {3486,11843}, {5252,8207}, {5600,10039}, {5601,18391}, {5687,26382}, {5722,11871}, {8196,12047}, {8197,10573}, {8204,12647}, {9834,10572}, {11383,26371}, {11490,26379}, {11491,26381}, {11494,26310}, {11496,26326}, {11497,26334}, {11498,26344}, {11499,26386}, {11500,26389}, {11501,26388}, {11502,26387}, {11570,12462}, {11848,26383}, {12456,15071}, {12463,12758}, {18496,18524}, {18999,26384}, {19000,26385}, {26391,26493}, {26392,26502}, {26396,26512}, {26397,26513}

X(26393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5597, 5598, 26351), (11882, 11883, 26352), (26402, 26425, 26380)

X(26394) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND MEDIAL

Barycentrics    -2*a*r*(a+b+c)*D+a*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S-r*(a+b+c)*(2*a^4-(b+c)*a^3-(b+c)^2*a^2+(b+c)*(b^2+c^2)*a-(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))
Barycentrics    (a+b+c) (a^3-a^2 b+a b^2-b^3-a^2 c+b^2 c+a c^2+b c^2-c^3)+8 a Sqrt[R (r+4 R)] S : :      (Peter Moses, November 1 2018)

X(26394) lies on these lines: {1,224}, {2,5597}, {3,26381}, {4,26371}, {8,26382}, {10,26296}, {20,26290}, {22,26302}, {30,18496}, {100,26393}, {145,26395}, {388,26380}, {491,26397}, {492,26396}, {497,26351}, {528,11367}, {631,26398}, {1270,26344}, {1271,26334}, {2896,26310}, {2975,26319}, {3091,26326}, {4190,26425}, {4240,26383}, {5598,20075}, {7585,26385}, {7586,26384}, {7787,26379}, {10527,26399}, {10528,26402}, {10529,26401}, {26391,26494}, {26392,26503}

X(26394) = anticomplement of X(5599)
X(26394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3434, 26418), (5597, 26359, 2)

X(26395) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 5th MIXTILINEAR

Barycentrics    a*(-r*(a+b+c)*(D+(b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))+(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S) : : , where D=4*S*sqrt(R*(4*R+r))
Barycentrics    a ((a-2 b-2 c) (a+b-c) (a-b+c)-4 Sqrt[R (r+4 R)] S) : :      (Peter Moses, November 1 2018)
X(26395) = (4*S*(R+2*r)+D)*X(1)-4*S*r*X(3)

X(26395) lies on these lines: {1,3}, {8,26359}, {145,26394}, {519,26382}, {952,26386}, {5603,26326}, {5604,26344}, {5605,26334}, {7967,26381}, {7968,26384}, {7969,26385}, {8192,26302}, {9997,26310}, {10800,26379}, {10944,26388}, {10950,26387}, {11396,26371}, {11910,26383}, {18496,18526}, {26391,26495}, {26392,26504}, {26396,26514}, {26397,26515}

X(26395) = reflection of X(5598) in X(1)
X(26395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5903, 26423), (1482, 5919, 26419), (8162, 11009, 26419)

X(26396) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    a*(a^2+b^2+c^2+4*S)*D+(a+b+c)*((3*a^3-3*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c))*S+2*a^2*((b^2+c^2)*a-b^3-c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26396) lies on these lines: {1,26420}, {193,26397}, {492,26394}, {3068,5597}, {5860,26344}, {18496,18539}, {26290,26294}, {26296,26300}, {26302,26306}, {26310,26314}, {26319,26324}, {26326,26330}, {26334,26339}, {26351,26355}, {26359,26361}, {26365,26369}, {26371,26375}, {26379,26429}, {26380,26435}, {26381,26441}, {26382,26444}, {26383,26449}, {26384,26456}, {26385,26462}, {26386,26468}, {26387,26473}, {26388,26479}, {26389,26485}, {26390,26490}, {26391,26496}, {26392,26505}, {26393,26512}, {26395,26514}, {26398,26516}, {26399,26517}, {26400,26518}, {26401,26519}, {26402,26520}

X(26397) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 4th TRI-SQUARES-CENTRAL

Barycentrics    a*(a^2+b^2+c^2-4*S)*D+(a+b+c)*(-(3*a^3-3*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c))*S+2*a^2*((b^2+c^2)*a-b^3-c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26397) lies on these lines: {1,26421}, {193,26396}, {491,26394}, {3069,5597}, {5861,26334}, {8982,26381}, {18496,26438}, {26290,26295}, {26296,26301}, {26302,26307}, {26310,26315}, {26319,26325}, {26326,26331}, {26340,26344}, {26351,26356}, {26359,26362}, {26365,26370}, {26371,26376}, {26379,26430}, {26380,26436}, {26382,26445}, {26383,26450}, {26384,26457}, {26385,26463}, {26386,26469}, {26387,26474}, {26388,26480}, {26389,26486}, {26390,26491}, {26391,26497}, {26392,26506}, {26393,26513}, {26395,26515}, {26398,26521}, {26399,26522}, {26400,26523}, {26401,26524}, {26402,26525}

X(26398) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND X3-ABC REFLECTIONS

Barycentrics    a*(-(a-b+c)*(-a+b+c)*(a+b-c)*D+2*a*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+(b+c)*(2*b^2-b*c+2*c^2)*a^2+(b^4+c^4)*a-(b^2-c^2)*(b^3-c^3))) : : , where D=4*S*sqrt(R*(4*R+r))
X(26398) = (4*R*S-D)*X(1)+4*S*(R+2*r)*X(3)

X(26398) lies on these lines: {1,3}, {2,26381}, {24,26371}, {30,26326}, {140,26359}, {498,26388}, {499,26387}, {631,26394}, {1656,18496}, {2080,26379}, {3311,26384}, {3312,26385}, {6642,26302}, {26310,26316}, {26334,26341}, {26344,26348}, {26382,26446}, {26383,26451}, {26389,26487}, {26390,26492}, {26391,26498}, {26392,26507}, {26396,26516}, {26397,26521}

X(26398) = midpoint of X(3) and X(11875)

X(26399) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND INNER-YFF

Barycentrics    a*(-D+a*(-a^2+b^2+c^2)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26399) = D*X(1)-4*S*r*X(3)

X(26399) lies on these lines: {1,3}, {5,26389}, {30,26413}, {5601,16845}, {6734,26382}, {6846,8200}, {10527,26394}, {10943,26390}, {12116,26381}, {18496,18544}, {26302,26308}, {26310,26317}, {26326,26332}, {26334,26342}, {26344,26349}, {26359,26363}, {26371,26377}, {26379,26431}, {26383,26452}, {26384,26458}, {26385,26464}, {26386,26470}, {26387,26475}, {26388,26481}, {26396,26517}, {26397,26522}

X(26399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 26425), (1, 5903, 26419), (1, 11248, 26424)

X(26400) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-YFF

Barycentrics
a*((a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*D+a*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+((b^2-c^2)^2-4*b^2*c^2)*a-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2))) : : , where D=4*S*sqrt(R*(4*R+r))
X(26400) = (8*R*r*S+(R-r)*D)*X(1)-4*S*r*(R-r)*X(3)

X(26400) lies on these lines: {1,3}, {5,26390}, {119,26386}, {6735,26382}, {10942,26389}, {12115,26381}, {18496,18542}, {26302,26309}, {26310,26318}, {26326,26333}, {26334,26343}, {26344,26350}, {26359,26364}, {26371,26378}, {26379,26432}, {26383,26453}, {26384,26459}, {26385,26465}, {26387,26476}, {26388,26482}, {26396,26518}, {26397,26523}

X(26400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10679, 26424), (1, 11248, 26423), (5597, 26402, 1)

X(26401) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND INNER-YFF TANGENTS

Barycentrics    a*(D+a*(a+b-c)*(a-b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26401) = (4*R*S+D)*X(1)-4*S*r*X(3)

X(26401) lies on these lines: {1,3}, {11,26389}, {377,26360}, {4190,26418}, {6833,26327}, {7354,26413}, {10527,26359}, {10529,26394}, {10532,26326}, {10804,26379}, {10806,26381}, {10835,26302}, {10879,26310}, {10916,26382}, {10931,26334}, {10932,26344}, {10943,26386}, {10949,26390}, {10957,26388}, {10959,26387}, {11401,26371}, {11915,26383}, {17647,26406}, {18496,18543}, {19049,26384}, {19050,26385}, {26396,26519}, {26397,26524}

X(26401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 26423), (1, 56, 26425), (2223, 19765, 26425)

X(26402) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-YFF TANGENTS

Barycentrics    a*((a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*D+a*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+4*b*c+c^2)*a+(b^2+6*b*c+c^2)*(b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26402) lies on these lines: {1,3}, {12,26390}, {5552,26359}, {10528,26394}, {10531,26326}, {10803,26379}, {10805,26381}, {10834,26302}, {10878,26310}, {10915,26382}, {10929,26334}, {10930,26344}, {10942,26386}, {10955,26389}, {10956,26388}, {10958,26387}, {11400,26371}, {11914,26383}, {18496,18545}, {19047,26384}, {19048,26385}, {26396,26520}, {26397,26525}

X(26402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10679, 5598), (5597, 26395, 26401), (26380, 26393, 26425)

X(26403) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 5th BROCARD

Barycentrics    a*(-((b^2+c^2)*a^2+b^2*c^2)*D+a*(a+b+c)*(a^5-(b+c)*a^4-2*b*c*(b+c)*a^2+b^2*c^2*a-b^2*c^2*(b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26403) lies on these lines: {1,26379}, {32,5598}, {83,26360}, {98,26327}, {182,26291}, {2080,26422}, {7787,26418}, {10788,26405}, {10789,26297}, {10790,26303}, {10791,26406}, {10792,26335}, {10793,26345}, {10794,26414}, {10795,26413}, {10796,26410}, {10797,26412}, {10798,26411}, {10799,26352}, {10800,26419}, {10803,26426}, {10804,26425}, {11364,26366}, {11380,26372}, {11839,26407}, {12835,26404}, {18498,18501}, {18993,26408}, {18994,26409}, {22520,26320}, {26415,26427}, {26416,26428}, {26420,26429}, {26421,26430}, {26423,26431}, {26424,26432}

X(26404) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 2nd CIRCUMPERP TANGENTIAL

Barycentrics    a*(a+b-c)*(a-b+c)*(-(-a+b+c)*D+2*a*b*c*(a-2*b-2*c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26404) = (8*R*s*(R+2*r)+D)*X(1)-4*R*S*X(3)

X(26404) lies on these lines: {1,3}, {4,26411}, {11,26327}, {12,26360}, {34,26372}, {388,26412}, {1478,26410}, {3434,26388}, {4293,26405}, {5252,26406}, {9655,18498}, {12835,26403}, {18954,26303}, {18957,26311}, {18958,26407}, {18959,26335}, {18960,26345}, {18961,26414}, {18962,26413}, {18995,26408}, {18996,26409}, {26420,26435}, {26421,26436}

X(26404) = reflection of X(26351) in X(1)
X(26404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2099, 26380), (65, 1319, 5597), (26401, 26426, 26417)

X(26405) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND EULER

Barycentrics
2*a*(-a+b+c)*(a+b-c)*(a-b+c)*D+3*a^7-3*(b+c)*a^6-5*(b^2+c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b^2-c^2)^2*a^3-(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : : , where D=4*S*sqrt(R*(4*R+r))

X(26405) lies on these lines: {1,6934}, {2,26410}, {3,26418}, {4,5598}, {5,18498}, {24,26303}, {104,26320}, {376,26291}, {515,26297}, {631,26360}, {3085,26412}, {3086,26411}, {4293,26404}, {4294,26352}, {5603,26366}, {5657,26406}, {5842,11367}, {7487,26372}, {7581,26409}, {7582,26408}, {7967,26419}, {8982,26421}, {9862,26311}, {10783,26335}, {10784,26345}, {10785,26414}, {10786,26413}, {10788,26403}, {10805,26426}, {10806,26425}, {11491,26417}, {11845,26407}, {12116,26423}, {26416,26440}, {26420,26441}

X(26406) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-GARCIA

Barycentrics    -a*D+a^4-(b+c)*a^3+(b^2+c^2)*(b+c)*a-(b^2-c^2)^2 : : , where D=4*S*sqrt(R*(4*R+r))

X(26406) lies on these lines: {1,442}, {2,26366}, {8,26418}, {10,5598}, {65,26412}, {72,26413}, {515,26291}, {517,26410}, {519,26419}, {956,26320}, {1837,26352}, {3057,26411}, {3679,26297}, {5090,26372}, {5252,26404}, {5587,26327}, {5657,26405}, {5687,26417}, {5688,26345}, {5689,26335}, {6734,26423}, {6735,26424}, {8193,26303}, {9857,26311}, {10791,26403}, {10914,26414}, {10915,26426}, {10916,26425}, {11900,26407}, {12702,18498}, {13883,26409}, {13936,26408}, {17647,26401}, {26415,26442}, {26416,26443}, {26420,26444}, {26421,26445}, {26422,26446}

X(26406) = reflection of X(8204) in X(10)
X(26406) = {X(1), X(3419)}-harmonic conjugate of X(26382)

X(26407) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND GOSSARD

Barycentrics
(-a*(a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))*(a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*D+(a+b+c)*((b^2+c^2)*a^9-(b^2-c^2)*(b-c)*a^8-2*(b^4+c^4)*a^7+2*(b^3-c^3)*(b^2-c^2)*a^6+b^2*c^2*(b^2+c^2)*a^5-(b^2-c^2)*(b-c)*b*c*(4*b^2+9*b*c+4*c^2)*a^4+2*(b^4-c^4)^2*a^3-2*(b^4-c^4)*(b^2-c^2)*(b+c)*(b^2-3*b*c+c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4)*a+(b^2-c^2)^3*(b-c)*(b^4+3*b^2*c^2+c^4)))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))

X(26407) lies on these lines: {1,26383}, {30,26291}, {402,5598}, {1650,26360}, {4240,26418}, {11831,26366}, {11832,26372}, {11839,26403}, {11845,26405}, {11848,26417}, {11852,26297}, {11853,26303}, {11885,26311}, {11897,26327}, {11900,26406}, {11901,26335}, {11902,26345}, {11903,26414}, {11904,26413}, {11905,26412}, {11906,26411}, {11909,26352}, {11910,26419}, {11914,26426}, {11915,26425}, {18498,18508}, {18958,26404}, {19017,26408}, {19018,26409}, {22755,26320}, {26420,26449}, {26421,26450}, {26422,26451}, {26423,26452}, {26424,26453}

X(26408) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND INNER-GREBE

Barycentrics    a*(-(a+b-c)*(-a+b+c)*(a-b+c)*D-a*((4*a^3-4*(b+c)*a^2-4*(b^2+c^2)*a+4*(b^2-c^2)*(b-c))*S+4*S^2*(-a+b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26408) lies on these lines: {1,26384}, {6,5598}, {55,26385}, {372,26291}, {1587,26327}, {3069,26360}, {3311,26422}, {5411,26372}, {7582,26405}, {7584,26410}, {7586,26418}, {7968,26419}, {13936,26406}, {18498,18510}, {18991,26366}, {18993,26403}, {18995,26404}, {18999,26417}, {19003,26297}, {19005,26303}, {19011,26311}, {19013,26320}, {19017,26407}, {19025,26413}, {19027,26412}, {19029,26411}, {19037,26352}, {19047,26426}, {19049,26425}, {26415,26454}, {26416,26455}, {26420,26456}, {26421,26457}, {26423,26458}, {26424,26459}

X(26408) = {X(6), X(5598)}-harmonic conjugate of X(26409)

X(26409) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-GREBE

Barycentrics    a*(-(a+b-c)*(-a+b+c)*(a-b+c)*D-a*(-(4*a^3-4*(b+c)*a^2-4*(b^2+c^2)*a+4*(b^2-c^2)*(b-c))*S+4*S^2*(-a+b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26409) lies on these lines: {1,26385}, {6,5598}, {55,26384}, {371,26291}, {1588,26327}, {3068,26360}, {3312,26422}, {5410,26372}, {7581,26405}, {7583,26410}, {7585,26418}, {7969,26419}, {13883,26406}, {18498,18512}, {18992,26366}, {18994,26403}, {18996,26404}, {19000,26417}, {19004,26297}, {19006,26303}, {19012,26311}, {19014,26320}, {19018,26407}, {19024,26414}, {19026,26413}, {19028,26412}, {19030,26411}, {19038,26352}, {19048,26426}, {19050,26425}, {26415,26460}, {26416,26461}, {26420,26462}, {26421,26463}, {26423,26464}, {26424,26465}

X(26409) = {X(6), X(5598)}-harmonic conjugate of X(26408)

X(26410) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND JOHNSON

Barycentrics    a*(-a+b+c)*(a-b+c)*(a+b-c)*D+a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2+c^2)*(b+c)*a^4-(b^2+c^2)^2*a^3+(b^4-c^4)*(b-c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : : , where D=4*S*sqrt(R*(4*R+r))

X(26410) lies on these lines: {1,6917}, {2,26405}, {3,18498}, {4,26372}, {5,5598}, {30,26291}, {119,26424}, {355,26413}, {381,26327}, {517,26406}, {952,26419}, {1478,26404}, {1479,26352}, {5587,26297}, {5886,26366}, {6214,26345}, {6215,26335}, {7583,26409}, {7584,26408}, {9996,26311}, {10679,26326}, {10796,26403}, {10942,26426}, {10943,26425}, {11499,26417}, {22758,26320}, {26415,26466}, {26416,26467}, {26420,26468}, {26421,26469}, {26423,26470}

X(26410) = reflection of X(8207) in X(5)
X(26410) = {X(26411), X(26412)}-harmonic conjugate of X(1)

X(26411) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND INNER-JOHNSON

Barycentrics    (-a+b+c)*(-a*(a+b-c)*(a-b+c)*D+(a+b+c)*(a^5-(b+c)*a^4-(b^2+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26411) lies on these lines: {1,6917}, {4,26404}, {11,5598}, {55,26360}, {497,26352}, {499,26422}, {3057,26406}, {3086,26405}, {3434,26351}, {6284,26291}, {9581,26297}, {10798,26403}, {10832,26303}, {10874,26311}, {10896,26327}, {10926,26345}, {10950,26413}, {10958,26426}, {10959,26425}, {11376,26366}, {11393,26372}, {11502,26417}, {11906,26407}, {19029,26408}, {19030,26409}, {22760,26320}, {26420,26473}, {26421,26474}, {26423,26475}, {26424,26476}

X(26411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26410, 26412), (497, 26418, 26352)

X(26412) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-JOHNSON

Barycentrics    (a+b-c)*(a-b+c)*(a*(-a+b+c)*D+a^5-(b+c)*a^4-(b^2+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26412) lies on these lines: {1,6917}, {4,26352}, {12,5598}, {55,26326}, {56,26360}, {65,26406}, {388,26404}, {498,26422}, {3085,26405}, {3295,18498}, {3434,26380}, {7354,26291}, {9578,26297}, {10797,26403}, {10831,26303}, {10895,26327}, {10923,26335}, {10924,26345}, {10944,26414}, {10956,26426}, {10957,26425}, {11375,26366}, {11392,26372}, {11501,26417}, {11905,26407}, {19027,26408}, {19028,26409}, {22759,26320}, {26420,26479}, {26421,26480}, {26423,26481}, {26424,26482}

X(26412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26410, 26411), (388, 26418, 26404)

X(26413) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 1st JOHNSON-YFF

Barycentrics    -D*a+(a^2+b^2-c^2)*(a^2-b^2+c^2) : : , where D=4*S*sqrt(R*(4*R+r))
X(26413) = D*X(1)-4*S*r*X(4)

X(26413) lies on these lines: {1,4}, {5,26423}, {11,26425}, {12,5598}, {30,26399}, {72,26406}, {355,26410}, {958,26320}, {3085,8187}, {3436,26418}, {5597,6284}, {5842,11877}, {7354,26401}, {7680,11878}, {10786,26405}, {10795,26403}, {10827,26297}, {10830,26303}, {10872,26311}, {10894,26327}, {10895,11367}, {10921,26335}, {10922,26345}, {10942,26424}, {10950,26411}, {10953,26352}, {11366,12953}, {11374,26366}, {11391,26372}, {11496,26326}, {11500,26417}, {11827,26291}, {11880,18242}, {11904,26407}, {18498,18518}, {18962,26404}, {19025,26408}, {26415,26483}, {26416,26484}, {26420,26485}, {26421,26486}, {26422,26487}

X(26413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4, 26389), (388, 5290, 26389), (1478, 21620, 26389)

X(26414) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 2nd JOHNSON-YFF

Barycentrics    -a*(a^3-(b+c)*a^2-(b^2+c^2-4*b*c)*a+(b^2-c^2)*(b-c))*D-4*S^2*(a^3-(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26414) lies on these lines: {1,224}, {5,26424}, {11,5598}, {12,26426}, {355,26410}, {528,11879}, {1376,26360}, {2886,11880}, {10525,26389}, {10785,26405}, {10794,26403}, {10826,26297}, {10829,26303}, {10871,26311}, {10893,26327}, {10914,26406}, {10919,26335}, {10920,26345}, {10943,26423}, {10944,26412}, {10947,26352}, {10949,26425}, {11373,26366}, {11390,26372}, {11826,26291}, {11903,26407}, {12114,26320}, {18498,18519}, {18961,26404}, {19024,26409}, {26415,26488}, {26420,26490}, {26421,26491}, {26422,26492}

X(26415) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND LUCAS HOMOTHETIC

Barycentrics
a*(-((4*a^6-4*(b^2+c^2)*a^4-4*(b^4+10*b^2*c^2+c^4)*a^2+4*(b^4-c^4)*(b^2-c^2))*S+4*S^2*(a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2-4*b^2*c^2))*D+(a+b+c)*a*((8*(b^2+c^2)*a^5-8*(b^3+c^3)*a^4-16*(b^2+c^2)^2*a^3+16*(b^3+c^3)*(b^2+c^2)*a^2+8*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a+(b^2-c^2)*(b-c)*(-8*b^4-8*c^4-8*b*c*(b-c)^2))*S+4*S^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^4+14*b^2*c^2+3*c^4)*a+(b+c)*(3*b^2-2*b*c+c^2)*(b^2-2*b*c+3*c^2)))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26415) lies on these lines: {493,5598}, {5490,26360}, {26291,26292}, {26297,26298}, {26303,26304}, {26311,26312}, {26320,26322}, {26327,26328}, {26335,26337}, {26345,26347}, {26352,26353}, {26366,26367}, {26372,26373}, {26403,26427}, {26404,26433}, {26405,26439}, {26406,26442}, {26408,26454}, {26409,26460}, {26413,26483}, {26414,26488}, {26417,26493}, {26418,26494}, {26419,26495}, {26420,26496}, {26421,26497}, {26422,26498}, {26425,26501}

X(26416) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND LUCAS(-1) HOMOTHETIC

Barycentrics
a*(-(-(4*a^6-4*(b^2+c^2)*a^4-4*(b^4+10*b^2*c^2+c^4)*a^2+4*(b^4-c^4)*(b^2-c^2))*S+4*S^2*(a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2-4*b^2*c^2))*D+(a+b+c)*a*(-(8*(b^2+c^2)*a^5-8*(b^3+c^3)*a^4-16*(b^2+c^2)^2*a^3+16*(b^3+c^3)*(b^2+c^2)*a^2+8*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a+(b^2-c^2)*(b-c)*(-8*b^4-8*c^4-8*b*c*(b-c)^2))*S+4*S^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^4+14*b^2*c^2+3*c^4)*a+(b+c)*(3*b^2-2*b*c+c^2)*(b^2-2*b*c+3*c^2)))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26416) lies on these lines: {494,5598}, {5491,26360}, {18498,18523}, {26291,26293}, {26297,26299}, {26303,26305}, {26311,26313}, {26320,26323}, {26327,26329}, {26338,26345}, {26352,26354}, {26366,26368}, {26372,26374}, {26403,26428}, {26404,26434}, {26405,26440}, {26406,26443}, {26408,26455}, {26409,26461}, {26410,26467}, {26413,26484}, {26414,26489}, {26417,26502}, {26418,26503}, {26419,26504}, {26420,26505}, {26421,26506}, {26422,26507}

X(26417) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND MANDART-INCIRCLE

Barycentrics    a^2*(a+b+c)*(-2*D*b*c+4*S^2*(-a+b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26417) = R*(D-4*r*S)*X(1)-4*S*r^2*X(3)

X(26417) lies on these lines: {1,3}, {100,26418}, {197,26303}, {355,11870}, {1376,26360}, {1737,5600}, {1837,8207}, {1905,11385}, {3476,11843}, {3486,11844}, {5252,8200}, {5599,10039}, {5602,18391}, {5687,26406}, {5722,11872}, {8197,12647}, {8203,12047}, {8204,10573}, {9835,10572}, {11383,26372}, {11491,26405}, {11494,26311}, {11496,26327}, {11497,26335}, {11498,26345}, {11499,26410}, {11500,26413}, {11501,26412}, {11502,26411}, {11570,12463}, {11848,26407}, {12457,15071}, {12462,12758}, {18498,18524}, {18999,26408}, {19000,26409}, {26415,26493}, {26416,26502}, {26420,26512}, {26421,26513}

X(26417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5597, 5598, 26352), (11881, 11884, 26351), (26401, 26426, 26404)

X(26418) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND MEDIAL

Barycentrics    2*a*r*(a+b+c)*D+a*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S-r*(a+b+c)*(2*a^4-(b+c)*a^3-(b+c)^2*a^2+(b+c)*(b^2+c^2)*a-(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))
Barycentrics    (a+b+c) (a^3-a^2 b+a b^2-b^3-a^2 c+b^2 c+a c^2+b c^2-c^3)-8 a Sqrt[R (r+4 R)] S : :      (Peter Moses, November 1 2018)

X(26418) lies on these lines: {1,224}, {2,5598}, {3,26405}, {4,26372}, {8,26406}, {10,26297}, {20,26291}, {22,26303}, {30,18498}, {100,26417}, {145,26419}, {388,26404}, {491,26421}, {492,26420}, {497,26352}, {528,11366}, {631,26422}, {1270,26345}, {1271,26335}, {2886,11367}, {2896,26311}, {2975,26320}, {3091,26327}, {3436,26413}, {3616,26366}, {4190,26401}, {4240,26407}, {5552,26424}, {5597,20075}, {7585,26409}, {7586,26408}, {7787,26403}, {10527,26423}, {10528,26426}, {10529,26425}, {26415,26494}, {26416,26503}

X(26418) = anticomplement of X(5600)
X(26418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3434, 26394), (5598, 26360, 2)

X(26419) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 5th MIXTILINEAR

Barycentrics    a*(-r*(a+b+c)*(-D+(b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))+(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S) : : , where D=4*S*sqrt(R*(4*R+r))
Barycentrics    (a+b+c) (a^3-a^2 b+a b^2-b^3-a^2 c+b^2 c+a c^2+b c^2-c^3)-8 a Sqrt[R (r+4 R)] S : :      (Peter Moses, November 1 2018)
Barycentrics    a ((a-2 b-2 c) (a+b-c) (a-b+c)+4 Sqrt[R (r+4 R)] S) : :      (Peter Moses, November 1 2018)
X(26419) = (4*S*(R+2*r)-D)*X(1)-4*S*r*X(3)

X(26419) lies on these lines: {1,3}, {8,26360}, {145,26418}, {519,26406}, {952,26410}, {5603,26327}, {5604,26345}, {5605,26335}, {7967,26405}, {7968,26408}, {7969,26409}, {8192,26303}, {9997,26311}, {10800,26403}, {10944,26412}, {10950,26411}, {11396,26372}, {11910,26407}, {18498,18526}, {26415,26495}, {26416,26504}, {26420,26514}, {26421,26515}

X(26419) = reflection of X(5597) in X(1)
X(26419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5903, 26399), (1482, 5919, 26395), (8162, 11009, 26395)

X(26420) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    -a*(a^2+b^2+c^2+4*S)*D+(a+b+c)*((3*a^3-3*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c))*S+2*a^2*((b^2+c^2)*a-b^3-c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26420) lies on these lines: {1,26396}, {193,26421}, {492,26418}, {3068,5598}, {5860,26345}, {18498,18539}, {26291,26294}, {26297,26300}, {26303,26306}, {26311,26314}, {26320,26324}, {26327,26330}, {26335,26339}, {26352,26355}, {26360,26361}, {26366,26369}, {26372,26375}, {26403,26429}, {26404,26435}, {26405,26441}, {26406,26444}, {26407,26449}, {26408,26456}, {26409,26462}, {26410,26468}, {26411,26473}, {26412,26479}, {26413,26485}, {26414,26490}, {26415,26496}, {26416,26505}, {26417,26512}, {26419,26514}, {26422,26516}, {26423,26517}, {26424,26518}, {26425,26519}, {26426,26520}

X(26421) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 4th TRI-SQUARES-CENTRAL

Barycentrics    -a*(a^2+b^2+c^2-4*S)*D+(a+b+c)*(-(3*a^3-3*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c))*S+2*a^2*((b^2+c^2)*a-b^3-c^3)) : : , where D=4*S*sqrt(R*(4*R+r))

X(26421) lies on these lines: {1,26397}, {193,26420}, {491,26418}, {3069,5598}, {5861,26335}, {8982,26405}, {18498,26438}, {26291,26295}, {26297,26301}, {26303,26307}, {26311,26315}, {26320,26325}, {26327,26331}, {26340,26345}, {26352,26356}, {26360,26362}, {26366,26370}, {26372,26376}, {26403,26430}, {26404,26436}, {26406,26445}, {26407,26450}, {26408,26457}, {26409,26463}, {26410,26469}, {26411,26474}, {26412,26480}, {26413,26486}, {26414,26491}, {26415,26497}, {26416,26506}, {26417,26513}, {26419,26515}, {26422,26521}, {26423,26522}, {26424,26523}, {26425,26524}, {26426,26525}

X(26422) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND X3-ABC REFLECTIONS

Barycentrics    a*((a-b+c)*(-a+b+c)*(a+b-c)*D+2*a*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+(b+c)*(2*b^2-b*c+2*c^2)*a^2+(b^4+c^4)*a-(b^2-c^2)*(b^3-c^3))) : : , where D=4*S*sqrt(R*(4*R+r))
X(26422) = (4*R*S+D)*X(1)+4*S*(R+2*r)*X(3)

X(26422) lies on these lines: {1,3}, {2,26405}, {24,26372}, {30,26327}, {140,26360}, {498,26412}, {499,26411}, {631,26418}, {1656,18498}, {2080,26403}, {3311,26408}, {3312,26409}, {6642,26303}, {26311,26316}, {26335,26341}, {26345,26348}, {26406,26446}, {26407,26451}, {26413,26487}, {26414,26492}, {26415,26498}, {26416,26507}, {26420,26516}, {26421,26521}

X(26422) = midpoint of X(3) and X(11876)

X(26423) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND INNER-YFF

Barycentrics    a*(D+a*(-a^2+b^2+c^2)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26423) = D*X(1)+4*S*r*X(3)

X(26423) lies on these lines: {1,3}, {5,26413}, {30,26389}, {5602,16845}, {6734,26406}, {6846,8207}, {10527,26418}, {10943,26414}, {12116,26405}, {18498,18544}, {26303,26308}, {26311,26317}, {26327,26332}, {26335,26342}, {26345,26349}, {26360,26363}, {26372,26377}, {26403,26431}, {26407,26452}, {26408,26458}, {26409,26464}, {26410,26470}, {26411,26475}, {26412,26481}, {26415,26499}, {26416,26508}, {26420,26517}, {26421,26522}

X(26423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 26401), (1, 5903, 26395), (1, 11248, 26400)

X(26424) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-YFF

Barycentrics
a*(-(a^3-(b+c)*a^2-(-4*b*c+b^2+c^2)*a+(b^2-c^2)*(b-c))*D+a*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+((b^2-c^2)^2-4*b^2*c^2)*a-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26424) lies on these lines: {1,3}, {5,26414}, {119,26410}, {5552,26418}, {6735,26406}, {10942,26413}, {12115,26405}, {18498,18542}, {26303,26309}, {26311,26318}, {26327,26333}, {26335,26343}, {26345,26350}, {26360,26364}, {26372,26378}, {26403,26432}, {26407,26453}, {26408,26459}, {26409,26465}, {26411,26476}, {26412,26482}, {26420,26518}, {26421,26523}

X(26424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10679, 26400), (1, 11248, 26399), (5598, 26426, 1)

X(26425) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND INNER-YFF TANGENTS

Barycentrics    a*(-D+a*(a+b-c)*(a-b+c)) : : , where D=4*S*sqrt(R*(4*R+r))
X(26425) = (4*R*S-D)*X(1)-4*S*r*X(3)

X(26425) lies on these lines: {1,3}, {11,26413}, {377,26359}, {4190,26394}, {6833,26326}, {7354,26389}, {10527,26360}, {10529,26418}, {10532,26327}, {10804,26403}, {10806,26405}, {10835,26303}, {10879,26311}, {10916,26406}, {10931,26335}, {10932,26345}, {10943,26410}, {10949,26414}, {10957,26412}, {10959,26411}, {11401,26372}, {11915,26407}, {17647,26382}, {18498,18543}, {19049,26408}, {19050,26409}, {26415,26501}, {26416,26510}, {26420,26519}, {26421,26524}

X(26425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 26399), (2223, 19765, 26401), (26380, 26393, 26402)

X(26426) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-YFF TANGENTS

Barycentrics    a*(-(a^3-(b+c)*a^2-(-4*b*c+b^2+c^2)*a+(b^2-c^2)*(b-c))*D+a*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+4*b*c+c^2)*a+(6*b*c+c^2+b^2)*(b+c))) : : , where D=4*S*sqrt(R*(4*R+r))

X(26426) lies on these lines: {1,3}, {12,26414}, {5552,26360}, {10528,26418}, {10531,26327}, {10803,26403}, {10805,26405}, {10834,26303}, {10878,26311}, {10915,26406}, {10929,26335}, {10930,26345}, {10942,26410}, {10955,26413}, {10956,26412}, {10958,26411}, {11400,26372}, {11914,26407}, {18498,18545}, {19047,26408}, {19048,26409}, {26416,26511}, {26420,26520}, {26421,26525}

X(26426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10679, 5597), (1, 11509, 26401), (26404, 26417, 26401)

X(26427) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND LUCAS HOMOTHETIC

Barycentrics
a^2*((a^6-6*(b^2+c^2)*a^4-(3*b^4+14*b^2*c^2+3*c^4)*a^2-4*b^2*c^2*(b^2+c^2))*S+a^8-(3*b^4+13*b^2*c^2+3*c^4)*a^4+2*(b^2+c^2)*(b^4-5*b^2*c^2+c^4)*a^2+((b^2-c^2)^2-4*b^2*c^2)*b^2*c^2) : :

X(26427) lies on these lines: {32,493}, {83,3069}, {98,26328}, {182,26292}, {2080,26498}, {6464,26428}, {7787,26494}, {10788,26439}, {10789,26298}, {10790,26304}, {10791,26442}, {10792,26337}, {10793,26347}, {10794,26488}, {10795,26483}, {10796,26466}, {10797,26477}, {10798,26471}, {10799,26353}, {10800,26495}, {10804,26501}, {11364,26367}, {11380,26373}, {11490,26493}, {11839,26447}, {12835,26433}, {18501,18521}, {18994,26460}, {22520,26322}, {26379,26391}, {26403,26415}, {26429,26496}, {26430,26497}, {26431,26499}, {26432,26500}

X(26428) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND LUCAS(-1) HOMOTHETIC

Barycentrics
a^2*(-(a^6-6*(b^2+c^2)*a^4-(3*b^4+14*b^2*c^2+3*c^4)*a^2-4*b^2*c^2*(b^2+c^2))*S+a^8-(3*b^4+13*b^2*c^2+3*c^4)*a^4+2*(b^2+c^2)*(b^4-5*b^2*c^2+c^4)*a^2+((b^2-c^2)^2-4*b^2*c^2)*b^2*c^2) : :

X(26428) lies on these lines: {32,494}, {83,3068}, {98,26329}, {182,26293}, {2080,26507}, {6464,26427}, {7787,26503}, {10788,26440}, {10789,26299}, {10790,26305}, {10791,26443}, {10793,26338}, {10794,26489}, {10795,26484}, {10796,26467}, {10797,26478}, {10798,26472}, {10799,26354}, {10800,26504}, {10803,26511}, {10804,26510}, {11364,26368}, {11380,26374}, {11490,26502}, {11839,26448}, {12835,26434}, {18501,18523}, {18993,26455}, {22520,26323}, {26379,26392}, {26403,26416}, {26429,26505}, {26430,26506}, {26431,26508}, {26432,26509}

X(26429) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 3rd TRI-SQUARES-CENTRAL

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

X(26429) lies on these lines: {4,12211}, {32,638}, {83,26361}, {98,26330}, {182,26294}, {193,12212}, {492,7787}, {2080,26516}, {5860,10793}, {10788,26441}, {10789,26300}, {10790,26306}, {10791,26444}, {10792,26339}, {10794,26490}, {10795,26485}, {10796,26468}, {10797,26479}, {10798,26473}, {10799,26355}, {10800,26514}, {10803,26520}, {10804,26519}, {11364,26369}, {11380,26375}, {11490,26512}, {11839,26449}, {12835,26435}, {13672,15682}, {18501,18539}, {18993,26456}, {18994,26462}, {22520,26324}, {26379,26396}, {26403,26420}, {26427,26496}, {26428,26505}, {26431,26517}, {26432,26518}

X(26430) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 4th TRI-SQUARES-CENTRAL

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

X(26430) lies on these lines: {4,12210}, {32,637}, {83,26362}, {98,26331}, {182,26295}, {193,12212}, {491,7787}, {2080,26521}, {5861,10792}, {8982,10788}, {10789,26301}, {10790,26307}, {10791,26445}, {10793,26340}, {10794,26491}, {10795,26486}, {10796,26469}, {10797,26480}, {10798,26474}, {10799,26356}, {10800,26515}, {10803,26525}, {10804,26524}, {11364,26370}, {11380,26376}, {11490,26513}, {11839,26450}, {12835,26436}, {13792,15682}, {18501,26438}, {18993,26457}, {18994,26463}, {22520,26325}, {26379,26397}, {26403,26421}, {26427,26497}, {26428,26506}, {26431,26522}, {26432,26523}

X(26431) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND INNER-YFF

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

X(26431) lies on these lines: {1,32}, {5,10795}, {83,26363}, {98,26332}, {182,11012}, {1078,10198}, {2080,10267}, {3398,11249}, {3972,13110}, {5171,10902}, {5709,12197}, {6734,10791}, {7787,10527}, {10680,11842}, {10788,12116}, {10790,26308}, {10792,26342}, {10793,26349}, {10794,10943}, {10796,26470}, {10797,26481}, {10798,26475}, {10799,26357}, {11380,26377}, {11839,26452}, {12835,26437}, {18501,18544}, {18993,26458}, {18994,26464}, {26379,26399}, {26403,26423}, {26427,26499}, {26428,26508}, {26429,26517}, {26430,26522}

X(26431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 10800, 10801), (32, 10804, 1), (32, 12194, 26432)

X(26432) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND OUTER-YFF

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

X(26432) lies on these lines: {1,32}, {5,10794}, {83,26364}, {98,26333}, {119,10796}, {182,2077}, {1078,10200}, {1470,12835}, {2080,10269}, {3398,11248}, {3972,13109}, {5552,7787}, {6256,12110}, {6735,10791}, {10679,11842}, {10788,12115}, {10790,26309}, {10793,26350}, {10795,10942}, {10797,26482}, {10798,26476}, {10799,26358}, {11380,26378}, {11839,26453}, {12198,12751}, {13194,25438}, {18501,18542}, {18993,26459}, {18994,26465}, {26379,26400}, {26403,26424}, {26427,26500}, {26428,26509}, {26429,26518}, {26430,26523}

X(26432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 10800, 10802), (32, 10803, 1), (32, 12194, 26431)

X(26433) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND LUCAS HOMOTHETIC

Barycentrics
a^2*((8*a^6+8*(b+c)*a^5+16*(b+c)^2*a^4-16*(b+c)*(b^2+c^2)*a^3-8*(11*b^4+11*c^4+2*b*c*(8*b^2+15*b*c+8*c^2))*a^2-8*(b+c)*(3*b^2+2*b*c+c^2)*(b^2+2*b*c+3*c^2)*a-16*(4*b^4+4*c^4+b*c*(17*b^2+24*b*c+17*c^2))*b*c)*S-3*a^8+4*(7*b^2+4*b*c+7*c^2)*a^6+16*(b+c)*(b^2+b*c+c^2)*a^5-2*(11*b^4+11*c^4-4*b*c*(3*b^2+b*c+3*c^2))*a^4-32*(b+c)*(b^2+c^2)*(b^2+b*c+c^2)*a^3-4*(7*b^6+7*c^6+(24*b^4+24*c^4+b*c*(71*b^2+80*b*c+71*c^2))*b*c)*a^2+16*(b^4+c^4-b*c*(b+c)^2)*(b+c)^3*a+(25*b^6+25*c^6+(6*b^4+6*c^4-b*c*(29*b^2+100*b*c+29*c^2))*b*c)*(b+c)^2)*(a+b-c)*(a-b+c) : :

X(26433) lies on these lines: {1,26353}, {4,26471}, {11,26328}, {12,5490}, {34,26373}, {36,26498}, {55,26292}, {56,493}, {388,26477}, {1319,26367}, {1470,26500}, {1478,26466}, {4293,26439}, {5252,26442}, {6464,26434}, {9655,18521}, {12835,26427}, {18954,26304}, {18957,26312}, {18958,26447}, {18959,26337}, {18960,26347}, {18961,26488}, {18962,26483}, {18967,26501}, {18995,26454}, {18996,26460}, {26380,26391}, {26404,26415}, {26435,26496}, {26436,26497}, {26437,26499}

X(26434) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND LUCAS(-1) HOMOTHETIC

Barycentrics
a^2*(-(8*a^6+8*(b+c)*a^5+16*(b+c)^2*a^4-16*(b+c)*(b^2+c^2)*a^3-8*(11*b^4+11*c^4+2*b*c*(8*b^2+15*b*c+8*c^2))*a^2-8*(b+c)*(3*b^2+2*b*c+c^2)*(b^2+2*b*c+3*c^2)*a-16*(4*b^4+4*c^4+b*c*(17*b^2+24*b*c+17*c^2))*b*c)*S-3*a^8+4*(7*b^2+4*b*c+7*c^2)*a^6+16*(b+c)*(b^2+b*c+c^2)*a^5-2*(11*b^4+11*c^4-4*b*c*(3*b^2+b*c+3*c^2))*a^4-32*(b+c)*(b^2+c^2)*(b^2+b*c+c^2)*a^3-4*(7*b^6+7*c^6+(24*b^4+24*c^4+b*c*(71*b^2+80*b*c+71*c^2))*b*c)*a^2+16*(b^4+c^4-b*c*(b+c)^2)*(b+c)^3*a+(25*b^6+25*c^6+(6*b^4+6*c^4-b*c*(29*b^2+100*b*c+29*c^2))*b*c)*(b+c)^2)*(a+b-c)*(a-b+c) : :

X(26434) lies on these lines: {1,26354}, {4,26472}, {11,26329}, {12,5491}, {34,26374}, {36,26507}, {55,26293}, {56,494}, {57,26299}, {388,26478}, {1319,26368}, {1470,26509}, {1478,26467}, {2099,26504}, {4293,26440}, {5252,26443}, {6464,26433}, {9655,18523}, {11509,26502}, {12835,26428}, {18954,26305}, {18957,26313}, {18958,26448}, {18960,26338}, {18961,26489}, {18962,26484}, {18967,26510}, {18995,26455}, {18996,26461}, {26380,26392}, {26404,26416}, {26435,26505}, {26436,26506}, {26437,26508}

X(26435) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 3rd TRI-SQUARES-CENTRAL

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

X(26435) lies on these lines: {1,26355}, {4,12959}, {11,26330}, {12,26361}, {20,7362}, {36,26516}, {55,26294}, {56,3068}, {57,26300}, {193,330}, {388,492}, {1007,26480}, {1319,26369}, {1470,26518}, {1478,26468}, {2099,26514}, {4293,26441}, {5434,5860}, {9655,18539}, {11509,26512}, {12835,26429}, {15682,18986}, {18954,26306}, {18957,26314}, {18959,26339}, {18961,26490}, {18962,26485}, {18967,26519}, {18995,26456}, {18996,26462}, {26380,26396}, {26404,26420}, {26433,26496}, {26434,26505}, {26437,26517}

X(26435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 492, 26479), (1469, 3600, 26436)

X(26436) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 4th TRI-SQUARES-CENTRAL

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

X(26436) lies on these lines: {1,26356}, {4,12958}, {11,26331}, {12,26362}, {20,7353}, {34,26376}, {36,26521}, {55,26295}, {56,3069}, {57,26301}, {193,330}, {388,491}, {1007,26479}, {1319,26370}, {1470,26523}, {1478,26469}, {2099,26515}, {4293,8982}, {5252,26445}, {5434,5861}, {9655,26438}, {11509,26513}, {12835,26430}, {15682,18987}, {18954,26307}, {18957,26315}, {18958,26450}, {18960,26340}, {18961,26491}, {18962,26486}, {18967,26524}, {18995,26457}, {18996,26463}, {26380,26397}, {26404,26421}, {26433,26497}, {26434,26506}, {26437,26522}

X(26436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 491, 26480), (1469, 3600, 26435)

X(26437) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND INNER-YFF

Barycentrics    a^2*(a^3-(b+c)*a^2-(b-c)^2*a+(b+c)*(b^2-4*b*c+c^2))*(a+b-c)*(a-b+c) : :
X(26437) = R*(R+2*r)*X(1)-r*(R+r)*X(3)

X(26437) lies on these lines: {1,3}, {4,26475}, {5,18962}, {11,26332}, {12,956}, {25,1866}, {34,26377}, {104,4295}, {225,1398}, {226,8666}, {388,2476}, {405,15950}, {519,11501}, {908,958}, {946,22760}, {953,3567}, {959,2990}, {1056,6853}, {1201,1451}, {1405,22356}, {1457,1468}, {1478,26470}, {1593,1830}, {1616,15306}, {1788,5253}, {1836,12114}, {1837,22753}, {1875,11399}, {1898,12687}, {2067,19050}, {2192,13095}, {2285,8609}, {2475,3600}, {2975,3485}, {3086,6830}, {3149,10950}, {3476,12649}, {3585,18519}, {3877,7098}, {4293,12116}, {4308,6224}, {4317,10074}, {4559,5021}, {5219,5258}, {5252,6734}, {5265,10587}, {5288,9578}, {5433,10198}, {5434,10957}, {6502,19049}, {6840,14986}, {6863,10954}, {6911,10573}, {6952,10597}, {7354,10959}, {8068,11929}, {9655,12773}, {10106,10916}, {10943,18961}, {12047,22758}, {12247,12776}, {12739,22560}, {12835,26431}, {18954,26308}, {18957,26317}, {18958,26452}, {18959,26342}, {18960,26349}, {18995,26458}, {18996,26464}, {24914,24987}, {26433,26499}, {26434,26508}, {26435,26517}, {26436,26522}

X(26437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (36, 3340, 11509), (999, 10680, 1), (1482, 8069, 26358)

X(26438) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-MID AND 4th TRI-SQUARES-CENTRAL

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

X(26438) lies on these lines: {3,26307}, {4,193}, {5,8982}, {30,491}, {141,14230}, {230,6564}, {372,22596}, {381,3069}, {999,26474}, {1656,26521}, {1657,26295}, {3070,19130}, {3071,22820}, {3295,26480}, {3830,5861}, {3843,26331}, {9655,26436}, {9668,26356}, {12702,26445}, {13665,18907}, {14269,26340}, {18480,26301}, {18493,26370}, {18494,26376}, {18496,26397}, {18498,26421}, {18501,26430}, {18503,26315}, {18508,26450}, {18510,26457}, {18512,26463}, {18518,26486}, {18519,26491}, {18521,26497}, {18523,26506}, {18524,26513}, {18526,26515}, {18542,26523}, {18543,26524}, {18544,26522}, {18545,26525}, {26321,26325}

X(26438) = {X(4), X(18440)}-harmonic conjugate of X(18539)

X(26439) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND LUCAS HOMOTHETIC

Barycentrics    2*((4*R^2-SA+SW)*S^2-(2*R^2+SW)*SB*SC)*S-2*(SA^2+SB*SC-SW^2)*S^2-SB*SC*SW^2 : :

X(26439) lies on these lines: {2,26466}, {3,26494}, {4,493}, {5,18521}, {24,26304}, {104,26322}, {376,26292}, {515,26298}, {631,5490}, {3085,26477}, {3086,26471}, {4293,26433}, {4294,26353}, {5603,26367}, {5657,26442}, {6464,26440}, {7487,26373}, {7581,26460}, {7582,26454}, {7967,26495}, {8982,26497}, {9862,26312}, {10783,26337}, {10784,26347}, {10785,26488}, {10786,26483}, {10788,26427}, {10806,26501}, {11491,26493}, {11845,26447}, {12116,26499}, {26381,26391}, {26405,26415}, {26441,26496}

X(26439) = reflection of X(4) in X(8212)
X(26439) = {X(26466), X(26498)}-harmonic conjugate of X(2)

X(26440) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND LUCAS(-1) HOMOTHETIC

Barycentrics    -2*((4*R^2-SA+SW)*S^2-(2*R^2+SW)*SB*SC)*S-2*(SA^2+SB*SC-SW^2)*S^2-SB*SC*SW^2 : :

X(26440) lies on these lines: {2,26467}, {3,26503}, {4,494}, {5,18523}, {24,26305}, {104,26323}, {376,26293}, {515,26299}, {631,5491}, {3085,26478}, {3086,26472}, {4293,26434}, {4294,26354}, {5603,26368}, {5657,26443}, {6464,26439}, {7487,26374}, {7581,26461}, {7582,26455}, {7967,26504}, {8982,26506}, {10784,26338}, {10785,26489}, {10786,26484}, {10788,26428}, {10805,26511}, {10806,26510}, {11491,26502}, {11845,26448}, {12115,26509}, {12116,26508}, {26381,26392}, {26405,26416}, {26441,26505}

X(26440) = reflection of X(4) in X(8213)
X(26440) = {X(26467), X(26507)}-harmonic conjugate of X(2)

X(26441) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND 3rd TRI-SQUARES-CENTRAL

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

X(26441) lies on these lines: {2,14234}, {3,489}, {4,371}, {5,18539}, {20,185}, {24,26306}, {32,1588}, {99,488}, {104,26324}, {182,11293}, {230,3071}, {315,487}, {376,5860}, {490,3564}, {515,26300}, {590,14233}, {631,639}, {671,12296}, {1131,14240}, {1132,7607}, {1151,6811}, {1352,11294}, {1504,1587}, {1585,10132}, {2351,13428}, {2794,5871}, {3070,12962}, {3085,26479}, {3086,26473}, {3524,13794}, {3529,10783}, {4293,26435}, {4294,26355}, {5603,26369}, {5657,26444}, {5870,8721}, {6460,14912}, {7000,9753}, {7487,26375}, {7581,26462}, {7582,26456}, {7967,26514}, {8884,24244}, {9675,23259}, {9738,9744}, {9766,12306}, {9862,26314}, {10785,26490}, {10786,26485}, {10788,26429}, {10805,26520}, {10806,26519}, {10845,12601}, {11491,26512}, {11845,26449}, {12115,26518}, {12116,26517}, {13674,15682}, {26381,26396}, {26405,26420}, {26439,26496}, {26440,26505}

X(26441) = reflection of X(i) in X(j) for these (i,j): (4, 371), (637, 3)
X(26441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 6776, 8982), (26468, 26516, 2)

X(26442) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND LUCAS HOMOTHETIC

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

X(26442) lies on these lines: {1,5490}, {2,26367}, {8,26494}, {10,493}, {65,26477}, {72,26483}, {515,26292}, {517,26466}, {519,26495}, {956,26322}, {1837,26353}, {3057,26471}, {3679,26298}, {5090,26373}, {5252,26433}, {5587,26328}, {5657,26439}, {5687,26493}, {5688,26347}, {5689,26337}, {6464,26443}, {6734,26499}, {6735,26500}, {8193,26304}, {9857,26312}, {10791,26427}, {10914,26488}, {10916,26501}, {11900,26447}, {12702,18521}, {13883,26460}, {13936,26454}, {26382,26391}, {26406,26415}, {26444,26496}, {26445,26497}, {26446,26498}

X(26442) = reflection of X(8214) in X(10)

X(26443) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND LUCAS(-1) HOMOTHETIC

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

X(26443) lies on these lines: {1,5491}, {2,26368}, {8,26503}, {10,494}, {65,26478}, {72,26484}, {515,26293}, {517,26467}, {519,26504}, {956,26323}, {1837,26354}, {3057,26472}, {3679,26299}, {5090,26374}, {5252,26434}, {5587,26329}, {5657,26440}, {5687,26502}, {5688,26338}, {6464,26442}, {6734,26508}, {6735,26509}, {8193,26305}, {9857,26313}, {10791,26428}, {10914,26489}, {10915,26511}, {10916,26510}, {11900,26448}, {12702,18523}, {13883,26461}, {13936,26455}, {26382,26392}, {26406,26416}, {26444,26505}, {26445,26506}, {26446,26507}

X(26443) = reflection of X(8215) in X(10)

X(26444) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND 3rd TRI-SQUARES-CENTRAL

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

X(26444) lies on these lines: {1,26361}, {2,26369}, {4,12788}, {8,492}, {10,3068}, {65,26479}, {72,26485}, {193,3416}, {515,26294}, {517,26468}, {519,26514}, {956,26324}, {1837,26355}, {3057,26473}, {3679,5588}, {5090,26375}, {5252,26435}, {5587,26330}, {5657,26441}, {5687,26512}, {5689,26339}, {6735,26518}, {8193,26306}, {9857,26314}, {10791,26429}, {10914,26490}, {10915,26520}, {10916,26519}, {11900,26449}, {12702,18539}, {13688,15682}, {13883,26462}, {13936,26456}, {26382,26396}, {26406,26420}, {26442,26496}, {26443,26505}, {26446,26516}

X(26444) = reflection of X(13893) in X(10)
X(26444) = {X(3416), X(3617)}-harmonic conjugate of X(26445)

X(26445) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND 4th TRI-SQUARES-CENTRAL

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

X(26445) lies on these lines: {1,26362}, {2,26370}, {4,12787}, {8,491}, {10,3069}, {65,26480}, {72,26486}, {193,3416}, {515,26295}, {517,26469}, {519,26515}, {956,26325}, {1837,26356}, {3057,26474}, {3679,5589}, {5090,26376}, {5252,26436}, {5587,26331}, {5657,8982}, {5687,26513}, {5688,26340}, {6734,26522}, {6735,26523}, {8193,26307}, {9857,26315}, {10791,26430}, {10914,26491}, {10915,26525}, {10916,26524}, {11900,26450}, {12702,26438}, {13808,15682}, {13883,26463}, {13936,26457}, {26382,26397}, {26406,26421}, {26442,26497}, {26443,26506}, {26446,26521}

X(26445) = reflection of X(13947) in X(10)
X(26445) = {X(3416), X(3617)}-harmonic conjugate of X(26444)

X(26446) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND X3-ABC REFLECTIONS

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

In the plane of a triangle ABC, let
D = point on line AC with angle BAD = π - angle BAC, such that |AD| = |AB|
E = point on line AB with angle CAE = π - angle BAC, such that |AE|=|AC|
F = point on line BC with angle FCA = π - angle ACB, such that |CF| = |AC|
G = point on line AC with angle GCB = π - angle ACB, such that |CG| = |CB|
H = point on line AB with angle HBC = π- angle ABC, such that |HB| = |BC|
J = point on line BC with angle JBA = π - angle ABC, such that |JB| = |AB|
The centroids of ABC, AFJ, BDG, and CEH are concyclic about X(26446). (Benjamin Warren, November 13, 2024)

X(26446) lies on these lines: {1,140}, {2,392}, {3,10}, {4,2355}, {5,40}, {7,8164}, {8,631}, {9,119}, {11,5119}, {12,46}, {20,5818}, {21,25005}, {24,5090}, {30,165}, {35,1837}, {36,5252}, {43,5396}, {48,21012}, {55,1737}, {56,10039}, {57,495}, {63,17757}, {65,498}, {72,5552}, {80,5010}, {100,1006}, {125,12778}, {142,2095}, {145,10303}, {171,5398}, {182,3416}, {191,5499}, {200,18443}, {210,912}, {214,19914}, {230,9620}, {354,10056}, {371,13973}, {372,13911}, {377,10526}, {381,516}, {382,19925}, {390,18527}, {405,11248}, {406,1872}, {442,5812}, {474,11249}, {484,1836}, {496,1697}, {499,3057}, {500,6048}, {518,10202}, {519,3653}, {546,7989}, {547,7988}, {548,16192}, {549,952}, {550,5691}, {551,10247}, {572,17275}, {573,17303}, {582,3072}, {632,3624}, {730,11171}, {899,1064}, {942,1788}, {944,3523}, {946,1656}, {956,6735}, {960,6863}, {962,3090}, {971,14647}, {997,3035}, {999,3911}, {1001,10679}, {1012,1512}, {1056,5435}, {1058,5704}, {1125,1482}, {1155,1478}, {1158,3652}, {1210,3295}, {1213,1766}, {1319,12647}, {1329,6842}, {1352,3844}, {1387,7962}, {1479,17606}, {1483,3632}, {1484,5541}, {1511,13211}, {1532,3305}, {1538,6969}, {1571,5254}, {1572,3815}, {1595,7713}, {1657,12512}, {1702,7584}, {1703,7583}, {1706,5705}, {1739,24789}, {1768,11698}, {1770,10895}, {1829,3541}, {1902,3542}, {2077,5251}, {2080,10791}, {2093,5219}, {2362,9646}, {2475,22937}, {2478,10525}, {2550,6827}, {2551,6850}, {2646,10573}, {2783,17281}, {2800,10176}, {2801,3956}, {2807,5891}, {2886,6882}, {2948,10264}, {2951,18529}, {2975,6940}, {3086,9957}, {3091,6361}, {3147,11363}, {3241,15702}, {3245,18393}, {3309,4448}, {3311,13912}, {3312,13883}, {3336,10404}, {3338,15888}, {3339,6147}, {3357,12779}, {3421,5744}, {3428,4413}, {3434,6947}, {3436,3916}, {3474,10590}, {3476,5126}, {3488,5281}, {3524,5731}, {3525,3616}, {3530,5881}, {3533,5550}, {3545,9812}, {3555,10528}, {3560,10310}, {3573,6998}, {3584,5902}, {3586,10993}, {3587,8727}, {3612,10950}, {3625,13607}, {3626,5882}, {3627,18492}, {3628,7991}, {3678,5884}, {3683,6929}, {3687,5774}, {3698,6862}, {3740,6001}, {3772,17734}, {3773,24257}, {3812,10198}, {3814,6980}, {3817,5055}, {3822,5880}, {3826,5805}, {3842,20430}, {3851,5493}, {3868,5885}, {3869,6853}, {3876,5694}, {3878,25413}, {3890,10284}, {3898,10199}, {3913,10916}, {3921,10167}, {3927,21075}, {3940,6745}, {3983,13369}, {4002,6833}, {4187,5250}, {4221,5235}, {4292,9654}, {4293,5122}, {4295,10588}, {4301,5070}, {4390,21013}, {4424,17720}, {4640,5123}, {4643,24324}, {4646,5292}, {4662,12675}, {4668,12108}, {4669,15701}, {4677,11812}, {4691,18526}, {4695,24892}, {4745,15693}, {4769,13335}, {4848,13411}, {4857,15079}, {4866,24645}, {4999,8256}, {5044,5887}, {5046,7705}, {5050,5847}, {5071,9779}, {5072,12571}, {5080,6951}, {5086,6875}, {5128,9612}, {5142,6197}, {5174,7531}, {5176,23961}, {5183,17605}, {5217,10572}, {5218,18391}, {5221,13407}, {5234,10270}, {5248,11849}, {5260,6906}, {5273,6916}, {5290,24470}, {5302,6256}, {5305,9593}, {5326,15950}, {5418,7969}, {5420,7968}, {5441,12104}, {5530,5711}, {5534,8726}, {5554,6910}, {5584,6985}, {5658,5777}, {5686,21151}, {5687,6734}, {5688,26348}, {5689,26341}, {5692,14988}, {5697,11376}, {5698,6982}, {5708,21620}, {5709,8728}, {5719,11529}, {5720,8580}, {5727,11545}, {5732,18528}, {5747,21866}, {5754,9568}, {5758,11024}, {5759,6843}, {5770,11227}, {5804,17552}, {5806,6887}, {5836,6958}, {5837,6700}, {5840,11113}, {5883,10197}, {5903,11375}, {5904,15016}, {5919,10072}, {6244,6913}, {6284,10826}, {6347,16433}, {6348,16432}, {6642,8193}, {6644,15177}, {6666,7682}, {6685,9567}, {6702,10738}, {6767,11019}, {6771,12781}, {6774,12780}, {6824,19855}, {6834,12672}, {6836,18517}, {6848,9856}, {6861,7686}, {6891,19843}, {6921,17614}, {6924,11012}, {6925,17613}, {6937,11681}, {6939,18230}, {6946,9342}, {6963,11680}, {6967,10527}, {6971,25639}, {6986,11491}, {7026,11752}, {7043,11789}, {7080,9940}, {7288,24928}, {7354,10827}, {7483,19860}, {7502,9590}, {7525,9626}, {7529,9911}, {7580,18491}, {7741,11010}, {7742,11501}, {7743,10589}, {8148,13464}, {8158,16863}, {8251,21530}, {8582,10306}, {8981,18991}, {9458,14026}, {9540,19065}, {9548,15973}, {9574,15048}, {9578,15803}, {9581,15171}, {9614,10593}, {9624,11531}, {9625,12106}, {9669,10624}, {9857,26316}, {9864,12042}, {9905,21230}, {9928,12359}, {10087,20118}, {10104,12197}, {10124,11224}, {10156,24477}, {10200,23340}, {10265,12331}, {10283,11539}, {10610,12785}, {10680,25524}, {10744,14664}, {10747,14690}, {10860,18540}, {10915,12513}, {10942,21031}, {10954,17700}, {11260,24927}, {11277,16132}, {11343,25007}, {11471,15763}, {11522,19872}, {11900,26451}, {12041,12368}, {12247,22935}, {12261,15059}, {12610,17327}, {12738,18446}, {13405,15934}, {13634,24808}, {13747,19861}, {13935,19066}, {13966,18992}, {14839,15819}, {15228,18513}, {15254,26333}, {15310,24482}, {15489,19858}, {15556,15865}, {15644,23841}, {16266,16473}, {16408,22770}, {16832,19512}, {16842,25011}, {16862,24564}, {17073,21231}, {20195,20330}, {24833,25351}, {26382,26398}, {26406,26422}, {26442,26498}, {26443,26507}, {26444,26516}, {26445,26521}

X(26446) = midpoint of X(i) and X(j) for these {i,j}: {2, 5657}, {3, 5790}, {4, 9778}, {8, 7967}, {10, 10164}, {40, 1699}, {165, 5587}, {3576, 3679}, {3654, 5886}, {5686, 21151}, {10167, 18908}
X(26446) = reflection of X(i) in X(j) for these (i,j): (2, 11231), (3, 10164), (355, 5790), (381, 10175), (946, 10171), (1699, 5), (3576, 549), (3653, 5054), (3654, 5657), (3655, 3576), (3656, 5886), (3817, 10172), (5603, 11230), (5731, 17502), (5790, 10), (5886, 2), (7967, 1385), (9778, 3579), (10164, 6684), (10171, 3634), (10175, 3828), (10246, 10165), (10247, 551), (12699, 1699), (16200, 10283), (25055, 11539)
X(26446) = anticomplement of X(11230)
X(26446) = complement of X(5603)
X(26446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5445, 24914), (140, 5690, 1)

X(26447) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND LUCAS HOMOTHETIC

Barycentrics
(S^2-3*SB*SC)*((-2*(10*R^2+SA-3*SW)*S^2+(8*R^2-2*SW)*(2*(SW+3*SA)*R^2-6*SA^2+SB*SC+SW^2))*S+4*S^4+(144*R^4-8*(3*SA+8*SW)*R^2+4*SA^2-2*SB*SC+7*SW^2)*S^2-(-4*(SW^2+3*SW*SA-9*SA^2)*R^2-SW*(4*SA+SW)*(2*SA-SW))*SW) : :

X(26447) lies on these lines: {30,26292}, {402,493}, {1650,5490}, {4240,26494}, {6464,26448}, {11831,26367}, {11832,26373}, {11839,26427}, {11845,26439}, {11848,26493}, {11852,26298}, {11853,26304}, {11885,26312}, {11897,26328}, {11900,26442}, {11901,26337}, {11902,26347}, {11903,26488}, {11904,26483}, {11905,26477}, {11906,26471}, {11909,26353}, {11910,26495}, {11915,26501}, {18508,18521}, {18958,26433}, {19017,26454}, {19018,26460}, {22755,26322}, {26449,26496}, {26450,26497}, {26451,26498}, {26452,26499}, {26453,26500}

X(26448) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND LUCAS(-1) HOMOTHETIC

Barycentrics
(S^2-3*SB*SC)*(-(-2*(10*R^2+SA-3*SW)*S^2+(8*R^2-2*SW)*(2*(SW+3*SA)*R^2-6*SA^2+SB*SC+SW^2))*S+4*S^4+(144*R^4-8*(3*SA+8*SW)*R^2+4*SA^2-2*SB*SC+7*SW^2)*S^2-(-4*(SW^2+3*SW*SA-9*SA^2)*R^2-SW*(4*SA+SW)*(2*SA-SW))*SW) : :

X(26448) lies on these lines: {30,26293}, {402,494}, {1650,5491}, {4240,26503}, {6464,26447}, {11831,26368}, {11832,26374}, {11839,26428}, {11845,26440}, {11848,26502}, {11852,26299}, {11853,26305}, {11885,26313}, {11897,26329}, {11900,26443}, {11902,26338}, {11903,26489}, {11904,26484}, {11905,26478}, {11906,26472}, {11909,26354}, {11910,26504}, {11914,26511}, {11915,26510}, {18508,18523}, {18958,26434}, {19017,26455}, {19018,26461}, {22755,26323}, {26449,26505}, {26450,26506}, {26451,26507}, {26452,26508}, {26453,26509}

X(26449) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    (S^2-3*SB*SC)*((-13*S^2+4*(15*SA-4*SW)*R^2-15*SA^2+10*SB*SC+4*SW^2)*S+2*(6*R^2+SA-2*SW)*S^2+4*(3*SA+2*SW)*(3*SA-SW)*R^2-2*(5*SA^2+SA*SW-SW^2)*SW) : :

X(26449) lies on these lines: {4,12800}, {30,26294}, {193,12583}, {402,3068}, {492,4240}, {1650,26361}, {1651,5860}, {11831,26369}, {11832,26375}, {11839,26429}, {11845,26441}, {11848,26512}, {11852,26300}, {11853,26306}, {11885,26314}, {11897,26330}, {11900,26444}, {11901,26339}, {11903,26490}, {11905,26479}, {11906,26473}, {11909,26355}, {11910,26514}, {11914,26520}, {11915,26519}, {13689,15682}, {18508,18539}, {19017,26456}, {19018,26462}, {22755,26324}, {26383,26396}, {26407,26420}, {26447,26496}, {26448,26505}, {26451,26516}, {26452,26517}, {26453,26518}

X(26449) = reflection of X(13894) in X(402)

X(26450) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 4th TRI-SQUARES-CENTRAL

Barycentrics    (S^2-3*SB*SC)*(-(-13*S^2+4*(15*SA-4*SW)*R^2-15*SA^2+10*SB*SC+4*SW^2)*S+2*(6*R^2+SA-2*SW)*S^2+4*(3*SA+2*SW)*(3*SA-SW)*R^2-2*(5*SA^2+SA*SW-SW^2)*SW) : :

X(26450) lies on these lines: {4,12799}, {30,26295}, {193,12583}, {402,3069}, {491,4240}, {1650,26362}, {1651,5861}, {8982,11845}, {11831,26370}, {11832,26376}, {11839,26430}, {11848,26513}, {11852,26301}, {11853,26307}, {11885,26315}, {11897,26331}, {11900,26445}, {11902,26340}, {11903,26491}, {11904,26486}, {11905,26480}, {11906,26474}, {11909,26356}, {11910,26515}, {11914,26525}, {11915,26524}, {13809,15682}, {18508,26438}, {18958,26436}, {19017,26457}, {19018,26463}, {22755,26325}, {26383,26397}, {26407,26421}, {26447,26497}, {26448,26506}, {26451,26521}, {26452,26522}, {26453,26523}

X(26450) = reflection of X(13948) in X(402)

X(26451) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND X3-ABC REFLECTIONS

Barycentrics    (S^2-3*SB*SC)*(7*S^2-2*R^2*(36*R^2+18*SA-17*SW)+9*SA^2-6*SB*SC-4*SW^2) : :

X(26451) lies on these lines: {2,3}, {35,11909}, {36,18958}, {55,11913}, {56,11912}, {125,12790}, {182,12583}, {498,11905}, {499,11906}, {517,11831}, {952,16210}, {1385,12438}, {2080,11839}, {3311,19017}, {3312,19018}, {3357,12791}, {3576,11852}, {3579,12696}, {5657,16212}, {5690,12626}, {5844,16211}, {6771,12793}, {6774,12792}, {7583,13894}, {7584,13948}, {10246,11910}, {10267,11848}, {10269,22755}, {10610,12797}, {11885,26316}, {11900,26446}, {11901,26341}, {11902,26348}, {11903,26492}, {11904,26487}, {11914,16203}, {11915,16202}, {12041,12369}, {12042,12181}, {12359,12418}, {12619,12729}, {14643,23239}, {26383,26398}, {26407,26422}, {26447,26498}, {26448,26507}, {26449,26516}, {26450,26521}

X(26451) = midpoint of X(i) and X(j) for these {i,j}: {2, 11845}, {3, 11911}, {3576, 11852}, {5657, 16212}, {11897, 16190}
X(26451) = reflection of X(i) in X(j) for these (i,j): (11251, 11911), (11911, 402)
X(26451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 402, 11251), (5, 12113, 18507), (12113, 15183, 5)

X(26452) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND INNER-YFF

Barycentrics
(a^12-3*(b^2+c^2)*a^10-2*(b+c)*b*c*a^9+(b^4+7*b^2*c^2+c^4)*a^8+2*(b+c)*(b^2+c^2)*b*c*a^7+2*(3*b^6+3*c^6-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*b*c)*a^6+2*(b+c)*(b^2-2*c^2)*(2*b^2-c^2)*b*c*a^5-(b^2-c^2)^2*(9*b^4+9*c^4-2*b*c*(2*b-c)*(b-2*c))*a^4-6*(b^4-c^4)*(b^2-c^2)*b*c*(b+c)*a^3+(b^4-c^4)*(b^2-c^2)*(5*b^4+5*c^4-2*b*c*(b^2+c^2))*a^2+2*(b^2-c^2)^2*(b+c)*b*c*(b^4+3*b^2*c^2+c^4)*a-(b^2-c^2)^4*(b^4+3*b^2*c^2+c^4))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(26452) lies on these lines: {1,402}, {5,11904}, {30,11012}, {1650,26363}, {4240,10527}, {5709,12696}, {6734,11900}, {10198,15183}, {10267,11848}, {10680,11911}, {10943,11903}, {11249,11251}, {11832,26377}, {11839,26431}, {11845,12116}, {11853,26308}, {11885,26317}, {11897,26332}, {11901,26342}, {11902,26349}, {11905,26481}, {11906,26475}, {11909,26357}, {12649,16212}, {18508,18544}, {18958,26437}, {19017,26458}, {19018,26464}, {26383,26399}, {26407,26423}, {26447,26499}, {26448,26508}, {26449,26517}, {26450,26522}

X(26452) = reflection of X(11912) in X(402)
X(26452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (402, 11915, 1), (402, 12438, 26453)

X(26453) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND OUTER-YFF

Barycentrics
(a^12-3*(b^2+c^2)*a^10+2*(b+c)*b*c*a^9+(b^4+7*b^2*c^2+c^4)*a^8-2*(b+c)*(b^2+c^2)*b*c*a^7+2*(3*b^6+3*c^6+(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*b*c)*a^6-2*(b+c)*(b^2-2*c^2)*(2*b^2-c^2)*b*c*a^5-(b^2-c^2)^2*(9*b^4+9*c^4+2*b*c*(b+2*c)*(2*b+c))*a^4+6*(b^4-c^4)*(b^2-c^2)*b*c*(b+c)*a^3+(b^4-c^4)*(b^2-c^2)*(5*b^4+5*c^4+2*b*c*(b^2+c^2))*a^2-2*(b^2-c^2)^2*(b+c)*b*c*(b^4+3*b^2*c^2+c^4)*a-(b^2-c^2)^4*(b^4+3*b^2*c^2+c^4))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(26453) lies on these lines: {1,402}, {5,11903}, {30,119}, {1470,18958}, {1650,26364}, {4240,5552}, {6256,12113}, {6735,11900}, {10200,15183}, {10269,22755}, {10679,11911}, {10942,11904}, {11248,11251}, {11832,26378}, {11839,26432}, {11845,12115}, {11853,26309}, {11885,26318}, {11897,26333}, {11901,26343}, {11902,26350}, {11905,26482}, {11906,26476}, {11909,26358}, {12648,16212}, {12729,12751}, {13268,25438}, {18508,18542}, {19017,26459}, {19018,26465}, {26383,26400}, {26407,26424}, {26447,26500}, {26448,26509}, {26449,26518}, {26450,26523}

X(26453) = reflection of X(11913) in X(402)
X(26453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (402, 11914, 1), (402, 12438, 26452)

X(26454) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND LUCAS HOMOTHETIC

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

X(26454) lies on these lines: {6,493}, {32,8911}, {83,3069}, {213,606}, {372,26292}, {729,1306}, {1587,26328}, {2207,5413}, {3051,10318}, {3311,26498}, {5062,6414}, {5411,26373}, {6464,26455}, {7582,26439}, {7584,26466}, {7586,26494}, {7968,26495}, {13936,26442}, {18510,18521}, {18991,26367}, {18995,26433}, {18999,26493}, {19003,26298}, {19005,26304}, {19011,26312}, {19013,26322}, {19017,26447}, {19023,26488}, {19025,26483}, {19027,26477}, {19029,26471}, {19037,26353}, {19049,26501}, {26384,26391}, {26408,26415}, {26456,26496}, {26457,26497}, {26458,26499}, {26459,26500}

X(26454) = isogonal conjugate of the isotomic conjugate of X(493)
X(26454) = isogonal conjugate of the polar conjugate of X(8948)
X(26454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 493, 26460), (6, 8939, 19032)

X(26455) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND LUCAS(-1) HOMOTHETIC

Barycentrics
a^2*((6*a^6-12*(b^2+c^2)*a^4-6*(3*b^4+14*b^2*c^2+3*c^4)*a^2+8*(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*S-a^8-8*(b^2+c^2)*a^6+4*(4*b^4+9*b^2*c^2+4*c^4)*a^4-4*(b^2+c^2)*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*a^2-(3*b^4+14*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :

X(26455) lies on these lines: {6,494}, {372,26293}, {1505,6414}, {1587,26329}, {3069,5491}, {3311,26507}, {5411,26374}, {5413,8946}, {6464,26454}, {7582,26440}, {7584,26467}, {7586,26503}, {7968,26504}, {8576,19359}, {10318,26460}, {13936,26443}, {18510,18523}, {18991,26368}, {18993,26428}, {18995,26434}, {18999,26502}, {19003,26299}, {19005,26305}, {19011,26313}, {19013,26323}, {19017,26448}, {19023,26489}, {19025,26484}, {19027,26478}, {19029,26472}, {19037,26354}, {19047,26511}, {19049,26510}, {26384,26392}, {26408,26416}, {26456,26505}, {26457,26506}, {26458,26508}, {26459,26509}

X(26455) = {X(6), X(494)}-harmonic conjugate of X(26461)

X(26456) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 3rd TRI-SQUARES-CENTRAL

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

X(26456) lies on these lines: {2,6}, {4,19102}, {372,26294}, {1249,8037}, {1504,21843}, {1587,26330}, {1588,18907}, {2549,5062}, {3311,26516}, {5411,26375}, {6423,6459}, {7582,26441}, {7584,26468}, {13886,19103}, {14241,22541}, {15682,19099}, {18510,18539}, {18993,26429}, {18995,26435}, {18999,26512}, {19005,26306}, {19011,26314}, {19013,26324}, {19017,26449}, {19023,26490}, {19025,26485}, {19027,26479}, {19029,26473}, {19037,26355}, {19047,26520}, {19049,26519}, {26384,26396}, {26408,26420}, {26454,26496}, {26455,26505}, {26458,26517}, {26459,26518}

X(26456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3068, 26462), (3589, 15835, 2), (3618, 7586, 3069)

X(26457) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 4th TRI-SQUARES-CENTRAL

Barycentrics    (23*a^4+26*(b^2+c^2)*a^2-(b^2-c^2)^2)*S+2*a^6-19*(b^2+c^2)*a^4-24*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2) : :
X(26457) = 3*S^2*X(2)-2*SW*(5*S-2*SW)*X(6)

X(26457) lies on these lines: {2,6}, {4,19104}, {372,26295}, {1505,6459}, {1587,26331}, {3311,26521}, {5411,26376}, {7582,8982}, {7584,26469}, {7968,26515}, {13936,26445}, {13939,19105}, {14226,19100}, {15682,19101}, {18510,26438}, {18991,26370}, {18993,26430}, {18995,26436}, {18999,26513}, {19003,26301}, {19005,26307}, {19011,26315}, {19013,26325}, {19017,26450}, {19023,26491}, {19025,26486}, {19027,26480}, {19029,26474}, {19037,26356}, {19047,26525}, {19049,26524}, {26384,26397}, {26408,26421}, {26454,26497}, {26455,26506}, {26458,26522}, {26459,26523}

X(26457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3069, 26463), (491, 7586, 3069)

X(26458) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND INNER-YFF

Barycentrics    a^2*(2*b*c*S+(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))) : :
X(26458) = R*S*X(1)-2*SW*(R+r)*X(6)

X(26458) lies on these lines: {1,6}, {5,19025}, {371,5416}, {372,11012}, {495,19026}, {1377,5705}, {1587,26332}, {3068,10198}, {3311,10267}, {3312,11249}, {5411,26377}, {6417,16202}, {6418,10680}, {6501,12001}, {7581,10532}, {7582,12116}, {7584,26470}, {9616,10268}, {10943,19023}, {18510,18544}, {18993,26431}, {18995,26437}, {19005,26308}, {19011,26317}, {19017,26452}, {19027,26481}, {19029,26475}, {19037,26357}, {26384,26399}, {26408,26423}, {26454,26499}, {26455,26508}

X(26458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 26464), (6, 1335, 18991), (6, 19048, 19004)

X(26459) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND OUTER-YFF

Barycentrics    a^2*(2*b*c*S-(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))) : :
X(26459) = R*S*X(1)-2*SW*(R-r)*X(6)

X(26459) lies on these lines: {1,6}, {5,19023}, {119,7584}, {372,2077}, {496,19024}, {1378,13947}, {1470,18995}, {1587,26333}, {1588,6256}, {1702,3359}, {2067,5193}, {3068,10200}, {3311,10269}, {3312,11248}, {5411,26378}, {5416,6420}, {6417,16203}, {6418,10679}, {6501,12000}, {7581,10531}, {7582,12115}, {9616,10270}, {10942,19025}, {12751,19077}, {18510,18542}, {18993,26432}, {19005,26309}, {19011,26318}, {19017,26453}, {19027,26482}, {19029,26476}, {19037,26358}, {19112,25438}, {26384,26400}, {26408,26424}, {26454,26500}, {26455,26509}

X(26459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 26465), (6, 1124, 18991), (6, 3299, 26464)

X(26460) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND LUCAS HOMOTHETIC

Barycentrics
a^2*(2*(3*a^6-6*(b^2+c^2)*a^4-3*(3*b^4+14*b^2*c^2+3*c^4)*a^2+4*(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*S+a^8+8*(b^2+c^2)*a^6-4*(4*b^4+9*b^2*c^2+4*c^4)*a^4+4*(b^2+c^2)*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*a^2+(3*b^4+14*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :

X(26460) lies on these lines: {6,493}, {371,26292}, {1504,6413}, {1588,26328}, {3068,5490}, {3312,26498}, {5410,26373}, {5412,8948}, {6464,26461}, {7581,26439}, {7583,26466}, {7585,26494}, {8577,19358}, {10318,26455}, {13883,26442}, {18512,18521}, {18992,26367}, {18996,26433}, {19000,26493}, {19004,26298}, {19006,26304}, {19012,26312}, {19014,26322}, {19018,26447}, {19026,26483}, {19028,26477}, {19030,26471}, {19038,26353}, {19050,26501}, {26385,26391}, {26409,26415}, {26462,26496}, {26463,26497}, {26464,26499}, {26465,26500}

X(26460) = {X(6), X(493)}-harmonic conjugate of X(26454)

X(26461) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND LUCAS(-1) HOMOTHETIC

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

X(26461) lies on these lines: {6,494}, {83,3068}, {213,605}, {371,26293}, {729,1307}, {1588,26329}, {2207,5412}, {3051,10318}, {3312,26507}, {5058,6413}, {5410,26374}, {6464,26460}, {6531,24243}, {7581,26440}, {7583,26467}, {7585,26503}, {7969,26504}, {13883,26443}, {18512,18523}, {18992,26368}, {18996,26434}, {19000,26502}, {19004,26299}, {19006,26305}, {19012,26313}, {19014,26323}, {19018,26448}, {19024,26489}, {19026,26484}, {19028,26478}, {19030,26472}, {19038,26354}, {19048,26511}, {19050,26510}, {26385,26392}, {26409,26416}, {26462,26505}, {26463,26506}, {26464,26508}, {26465,26509}

X(26461) = isogonal conjugate of the isotomic conjugate of X(494)
X(26461) = isogonal conjugate of the polar conjugate of X(8946)
X(26461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 494, 26455), (6, 8943, 19033)

X(26462) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    (23*a^4+26*(b^2+c^2)*a^2-(b^2-c^2)^2)*S-2*a^6+19*(b^2+c^2)*a^4+24*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2) : :
X(26462) = 3*S^2*X(2)+2*SW*(5*S+2*SW)*X(6)

X(26462) lies on these lines: {2,6}, {4,19103}, {371,26294}, {1504,6460}, {1588,26330}, {3312,26516}, {5410,26375}, {7581,26441}, {7583,26468}, {7969,26514}, {13883,26444}, {13886,19102}, {14241,19099}, {15682,22541}, {18512,18539}, {18994,26429}, {18996,26435}, {19000,26512}, {19006,26306}, {19012,26314}, {19014,26324}, {19018,26449}, {19024,26490}, {19026,26485}, {19028,26479}, {19030,26473}, {19038,26355}, {19048,26520}, {19050,26519}, {26385,26396}, {26409,26420}, {26460,26496}, {26461,26505}, {26464,26517}, {26465,26518}

X(26462) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3068, 26456), (492, 7585, 3068)

X(26463) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 4th TRI-SQUARES-CENTRAL

Barycentrics    (7*a^4+10*(b^2+c^2)*a^2-(b^2-c^2)^2)*S-8*a^6-(b^2+c^2)*a^4-2*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(26463) = 3*S^2*X(2)+2*SW*(3*S-2*SW)*X(6)

X(26463) lies on these lines: {2,6}, {4,19105}, {371,26295}, {1505,21843}, {1587,18907}, {1588,26331}, {2549,5058}, {3312,26521}, {5410,26376}, {6424,6460}, {7581,8982}, {7583,26469}, {7969,26515}, {13883,26445}, {13939,19104}, {14226,19101}, {15682,19100}, {18512,26438}, {18992,26370}, {18994,26430}, {18996,26436}, {19000,26513}, {19004,26301}, {19006,26307}, {19014,26325}, {19018,26450}, {19024,26491}, {19026,26486}, {19028,26480}, {19030,26474}, {19038,26356}, {19048,26525}, {19050,26524}, {26385,26397}, {26409,26421}, {26460,26497}, {26461,26506}, {26464,26522}, {26465,26523}

X(26463) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3068, 3069, 26362), (3589, 15834, 2), (3618, 7585, 3068)

X(26464) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND INNER-YFF

Barycentrics    a^2*(2*b*c*S-(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))) : :
X(26464) = R*S*X(1)+2*SW*(R+r)*X(6)

X(26464) lies on these lines: {1,6}, {5,19026}, {371,11012}, {372,5415}, {495,19025}, {1378,5705}, {1588,26332}, {1702,5709}, {3311,11249}, {3312,10267}, {26461,26508}

X(26464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 26458), (6, 1124, 18992), (6, 3299, 26459)

X(26465) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND OUTER-YFF

Barycentrics    a^2*(2*b*c*S+(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))) : :
X(26465) = R*S*X(1)+2*SW*(R-r)*X(6)

X(26465) lies on these lines: {1,6}, {5,19024}, {119,7583}, {371,2077}, {496,19023}, {1377,13893}, {1470,18996}, {1587,6256}, {1588,26333}, {1703,3359}, {3311,11248}, {3312,10269}, {5193,6502}, {5410,26378}, {5415,6419}, {6417,10679}, {6418,16203}, {6500,12000}, {7581,12115}, {7582,10531}, {10942,19026}, {12751,19078}, {18512,18542}, {18994,26432}, {19006,26309}, {19012,26318}, {19018,26453}, {19028,26482}, {19030,26476}, {19038,26358}, {19113,25438}, {26385,26400}, {26409,26424}, {26460,26500}, {26461,26509}

X(26465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 26459), (6, 1335, 18992), (6, 19048, 1)

X(26466) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND LUCAS HOMOTHETIC

Barycentrics    2*((SA-2*R^2)*S^2+(2*R^2+SW)*SB*SC)*S+4*S^4+(4*SA^2-2*SA*SW-SW^2)*S^2+SB*SC*SW^2 : :

X(26466) lies on these lines: {1,26471}, {2,26439}, {3,5490}, {4,26373}, {5,493}, {30,26292}, {119,26500}, {355,26483}, {381,26328}, {517,26442}, {952,26495}, {1478,26433}, {1479,26353}, {5587,26298}, {5886,26367}, {6193,24244}, {6214,26347}, {6215,26337}, {6464,26467}, {6756,8948}, {7583,26460}, {7584,26454}, {9996,26312}, {10796,26427}, {10943,26501}, {11499,26493}, {22758,26322}, {26386,26391}, {26468,26496}, {26469,26497}, {26470,26499}

X(26466) = reflection of X(8220) in X(5)
X(26466) = {X(2), X(26439)}-harmonic conjugate of X(26498)

X(26467) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND LUCAS(-1) HOMOTHETIC

Barycentrics    -2*((SA-2*R^2)*S^2+(2*R^2+SW)*SB*SC)*S+4*S^4+(4*SA^2-2*SA*SW-SW^2)*S^2+SB*SC*SW^2 : :

X(26467) lies on these lines: {1,26472}, {2,26440}, {3,5491}, {4,26374}, {5,494}, {30,26293}, {119,26509}, {355,26484}, {381,26329}, {517,26443}, {952,26504}, {1478,26434}, {1479,26354}, {5587,26299}, {5886,26368}, {6193,24243}, {6214,26338}, {6464,26466}, {6756,8946}, {7583,26461}, {7584,26455}, {9996,26313}, {10796,26428}, {10942,26511}, {10943,26510}, {11499,26502}, {22758,26323}, {26386,26392}, {26410,26416}, {26468,26505}, {26469,26506}, {26470,26508}

X(26467) = reflection of X(8221) in X(5)
X(26467) = {X(2), X(26440)}-harmonic conjugate of X(26507)

X(26468) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND 3rd TRI-SQUARES-CENTRAL

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

X(26468) lies on these lines: {1,26473}, {2,14234}, {3,18539}, {4,488}, {5,1588}, {20,7690}, {30,26294}, {119,26518}, {193,576}, {355,26485}, {381,5860}, {517,26444}, {952,26514}, {1007,6811}, {1478,26435}, {1479,26355}, {3545,6290}, {3593,9739}, {3851,6215}, {5587,26300}, {5874,13665}, {5886,26369}, {6251,7620}, {6278,6564}, {6565,10515}, {7583,26462}, {7584,26456}, {9996,26314}, {10796,26429}, {10942,26520}, {10943,26519}, {11293,26521}, {11499,26512}, {13692,15682}, {13748,23311}, {18762,21309}, {22758,26324}, {26386,26396}, {26410,26420}, {26466,26496}, {26467,26505}, {26470,26517}

X(26468) = reflection of X(8976) in X(5)
X(26468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26441, 26516), (26473, 26479, 1)

X(26469) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND 4th TRI-SQUARES-CENTRAL

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

X(26469) lies on these lines: {1,26474}, {2,8982}, {3,26307}, {4,487}, {5,1587}, {20,7692}, {30,26295}, {119,26523}, {193,576}, {355,26486}, {381,5861}, {517,26445}, {640,21737}, {952,26515}, {1007,6813}, {1478,26436}, {1479,26356}, {3545,6289}, {3595,9738}, {3851,6214}, {5587,26301}, {5875,13785}, {5886,26370}, {6250,7620}, {6281,6565}, {6564,10514}, {7583,26463}, {7584,26457}, {9996,26315}, {10796,26430}, {10942,26525}, {10943,26524}, {11294,26516}, {11499,26513}, {13749,23312}, {13812,15682}, {18538,21309}, {22758,26325}, {26386,26397}, {26410,26421}, {26466,26497}, {26467,26506}, {26470,26522}

X(26469) = reflection of X(13951) in X(5)
X(26469) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 8982, 26521), (26474, 26480, 1)

X(26470) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND INNER-YFF

Barycentrics    (b-c)^2*a^5-(b^2-c^2)*(b-c)*a^4-2*(b^2+c^2)*(b^2-b*c+c^2)*a^3+2*(b^4-c^4)*(b-c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
X(26470) = R*X(1)-2*(R+r)*X(5)

X(26470) lies on these lines: {1,5}, {2,10267}, {3,2886}, {4,2975}, {8,6830}, {10,6882}, {30,11012}, {55,6862}, {56,6917}, {100,6952}, {104,2475}, {140,3925}, {149,6888}, {262,13110}, {377,10269}, {381,529}, {388,6867}, {404,6713}, {442,1385}, {474,26492}, {485,19050}, {486,19049}, {497,6824}, {499,6911}, {515,6842}, {517,6734}, {528,11849}, {602,24892}, {758,946}, {912,12047}, {944,2476}, {956,10526}, {958,6928}, {962,6845}, {993,7491}, {1001,6861}, {1012,10525}, {1058,6855}, {1125,6881}, {1329,5790}, {1352,5849}, {1376,6958}, {1478,26437}, {1479,3560}, {1482,3813}, {1532,18480}, {1621,6852}, {1656,3816}, {1699,6763}, {1706,5705}, {1836,24467}, {2550,6891}, {2829,26321}, {3085,6859}, {3086,6826}, {3090,10806}, {3091,10529}, {3149,6585}, {3193,14008}, {3434,6833}, {3526,3826}, {3545,10597}, {3574,5777}, {3616,6829}, {3649,24475}, {3652,5536}, {3754,10265}, {3822,5882}, {3825,10175}, {3838,12675}, {3841,10165}, {3851,12001}, {4187,9956}, {4193,5818}, {4294,6892}, {4295,5770}, {4857,16617}, {4996,5840}, {5056,10587}, {5082,6956}, {5225,6930}, {5231,5709}, {5249,13373}, {5253,6901}, {5260,6902}, {5274,6846}, {5433,6924}, {5552,6879}, {5603,6828}, {5654,12431}, {5657,6943}, {5693,18393}, {5707,11269}, {5715,7956}, {5731,6937}, {5762,6067}, {5771,16139}, {5779,5852}, {5805,5857}, {5811,9779}, {5817,7678}, {6214,26349}, {6215,26342}, {6256,18519}, {6284,6914}, {6597,16159}, {6827,19843}, {6834,18491}, {6837,10530}, {6843,14986}, {6863,11500}, {6871,12115}, {6873,10595}, {6874,7967}, {6883,19854}, {6885,7288}, {6893,10591}, {6907,18481}, {6913,9669}, {6923,12114}, {6929,10896}, {6933,10786}, {6944,10589}, {6957,10598}, {6959,11510}, {6963,9780}, {6980,18242}, {6982,12667}, {6983,10584}, {6993,10586}, {7395,10835}, {7403,17111}, {7507,11401}, {7583,26464}, {7584,26458}, {9996,26317}, {10202,12609}, {10246,25466}, {10320,11501}, {10356,10879}, {10358,10804}, {10514,10931}, {10515,10932}, {10516,12595}, {10738,13743}, {10796,26431}, {10894,12513}, {10895,18967}, {11235,11496}, {11263,12005}, {11585,23304}, {11813,20117}, {11928,26333}, {12357,23234}, {12607,12645}, {12906,14643}, {13190,14639}, {13218,14644}, {13243,16116}, {13279,13729}, {14794,15338}, {14872,17605}, {15842,26364}, {26386,26399}, {26410,26423}, {26466,26499}, {26467,26508}, {26468,26517}, {26469,26522}

X(26470) = midpoint of X(i) and X(j) for these {i,j}: {4, 2975}, {6831, 24390}
X(26470) = reflection of X(i) in X(j) for these (i,j): (3, 4999), (12, 5), (6842, 25639)
X(26470) = complement of X(11491)
X(26470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 7741, 23513), (5, 10942, 7951), (5587, 7741, 5)

X(26471) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND LUCAS HOMOTHETIC

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

X(26471) lies on these lines: {1,26466}, {4,26433}, {11,493}, {55,5490}, {497,26353}, {499,26498}, {999,18521}, {3057,26442}, {3086,26439}, {6284,26292}, {6464,26472}, {9581,26298}, {10798,26427}, {10832,26304}, {10874,26312}, {10896,26328}, {10926,26347}, {10950,26483}, {10959,26501}, {11376,26367}, {11393,26373}, {11502,26493}, {11906,26447}, {19029,26454}, {19030,26460}, {22760,26322}, {26473,26496}, {26474,26497}, {26475,26499}, {26476,26500}

X(26472) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND LUCAS(-1) HOMOTHETIC

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

X(26472) lies on these lines: {1,26467}, {4,26434}, {11,494}, {55,5491}, {497,26354}, {499,26507}, {999,18523}, {3057,26443}, {3086,26440}, {6284,26293}, {6464,26471}, {9581,26299}, {10798,26428}, {10832,26305}, {10874,26313}, {10896,26329}, {10926,26338}, {10950,26484}, {10958,26511}, {10959,26510}, {11376,26368}, {11393,26374}, {11502,26502}, {11906,26448}, {19029,26455}, {19030,26461}, {22760,26323}, {26473,26505}, {26474,26506}, {26475,26508}, {26476,26509}

X(26473) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 3rd TRI-SQUARES-CENTRAL

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

X(26473) lies on these lines: {1,26468}, {4,12959}, {55,26361}, {193,5274}, {492,497}, {499,26516}, {999,18539}, {1007,26356}, {3057,26444}, {3086,26441}, {5860,10926}, {6284,26294}, {9581,26300}, {10798,26429}, {10832,26306}, {10874,26314}, {10896,26330}, {10925,26339}, {10950,26485}, {10958,26520}, {10959,26519}, {11376,26369}, {11393,26375}, {11502,26512}, {11906,26449}, {13696,15682}, {19029,26456}, {19030,26462}, {22760,26324}, {26387,26396}, {26411,26420}, {26471,26496}, {26472,26505}, {26475,26517}, {26476,26518}

X(26473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26468, 26479), (5274, 12589, 26474)

X(26474) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 4th TRI-SQUARES-CENTRAL

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

X(26474) lies on these lines: {1,26469}, {4,12958}, {11,3069}, {55,26362}, {193,5274}, {491,497}, {499,26521}, {999,26438}, {1007,26355}, {3057,26445}, {3086,8982}, {5861,10925}, {6284,26295}, {9581,26301}, {10798,26430}, {10832,26307}, {10874,26315}, {10896,26331}, {10926,26340}, {10950,26486}, {10958,26525}, {10959,26524}, {11376,26370}, {11393,26376}, {11502,26513}, {11906,26450}, {13816,15682}, {19029,26457}, {19030,26463}, {22760,26325}, {26387,26397}, {26411,26421}, {26471,26497}, {26472,26506}, {26475,26522}, {26476,26523}

X(26474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26469, 26480), (5274, 12589, 26473)

X(26475) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND INNER-YFF

Barycentrics    (-a+b+c)*((b^2-4*b*c+c^2)*a^4-2*(b^2+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*b*c*a+(b^2-c^2)^2*(b-c)^2) : :
X(26475) = R*(R+2*r)*X(1)-2*r*(R+r)*X(5)

X(26475) lies on these lines: {1,5}, {4,26437}, {21,497}, {55,7483}, {84,1836}, {388,7548}, {499,10267}, {946,1858}, {950,24387}, {956,10953}, {999,18544}, {1058,6852}, {1389,18391}, {1470,10785}, {1479,7491}, {1519,1898}, {1749,16155}, {2099,6831}, {2646,2886}, {3057,3813}, {3086,6905}, {3486,11680}, {3582,14798}, {3816,17606}, {3878,10916}, {3925,5438}, {5046,5274}, {5254,11998}, {5433,10902}, {5709,12701}, {6284,11012}, {6839,14986}, {6882,10573}, {6949,10806}, {7504,10589}, {7508,15171}, {7680,11011}, {8256,17636}, {9614,12704}, {9669,10680}, {10532,10591}, {10798,26431}, {10832,26308}, {10874,26317}, {10896,18967}, {10925,26342}, {10926,26349}, {10947,19525}, {10966,11113}, {11393,26377}, {11813,14054}, {11906,26452}, {13463,25414}, {15842,24982}, {19029,26458}, {19030,26464}, {26387,26399}, {26411,26423}, {26471,26499}, {26472,26508}, {26473,26517}, {26474,26522}

X(26475) = reflection of X(26482) in X(10523)
X(26475) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 1837, 26476), (496, 1484, 10948), (5727, 7741, 10958)

X(26476) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND OUTER-YFF

Barycentrics    (-a+b+c)*((b^2+c^2)*a^4-2*(b^2+c^2)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*b*c*a+(b^2-c^2)^2*(b-c)^2) : :
X(26476) = R^2*X(1)-2*r*(R-r)*X(5)

X(26476) lies on these lines: {1,5}, {4,1470}, {55,4187}, {56,1532}, {65,1519}, {235,1877}, {388,6945}, {442,10200}, {497,3871}, {499,6842}, {950,3825}, {999,18542}, {1210,1858}, {1319,18242}, {1329,3057}, {1479,6882}, {1836,12686}, {2077,6284}, {2082,6506}, {2098,17757}, {2476,10589}, {2478,26357}, {2646,3816}, {2886,3698}, {3085,6975}, {3086,6941}, {3359,15908}, {3814,10915}, {4294,6963}, {5048,12607}, {5141,10586}, {5154,5274}, {5187,10530}, {5225,6943}, {5259,5432}, {5433,6907}, {5554,11680}, {5687,10947}, {6830,10531}, {6831,10896}, {6929,8071}, {6932,7288}, {6959,8069}, {6971,9669}, {6973,10629}, {6980,16203}, {6981,10321}, {8256,25414}, {9614,12703}, {10798,26432}, {10832,26309}, {10874,26318}, {10925,26343}, {10926,26350}, {10953,17556}, {10965,11238}, {11393,26378}, {11681,12648}, {11906,26453}, {12709,17618}, {13274,25438}, {13463,17636}, {15829,21031}, {15844,17605}, {18839,21077}, {19029,26459}, {19030,26465}, {26387,26400}, {26411,26424}, {26471,26500}, {26472,26509}, {26473,26518}, {26474,26523}

X(26476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (496, 10942, 1), (7741, 8070, 5), (7741, 9581, 11)

X(26477) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND LUCAS HOMOTHETIC

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

X(26477) lies on these lines: {1,26466}, {4,26353}, {12,493}, {56,5490}, {65,26442}, {388,26433}, {498,26498}, {3085,26439}, {3295,18521}, {6464,26478}, {7354,26292}, {9578,26298}, {10797,26427}, {10831,26304}, {10873,26312}, {10895,26328}, {10923,26337}, {10924,26347}, {10957,26501}, {11375,26367}, {11392,26373}, {11501,26493}, {11905,26447}, {19027,26454}, {19028,26460}, {22759,26322}, {26479,26496}, {26480,26497}, {26481,26499}, {26482,26500}

X(26478) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND LUCAS(-1) HOMOTHETIC

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

X(26478) lies on these lines: {1,26467}, {4,26354}, {12,494}, {56,5491}, {65,26443}, {388,26434}, {498,26507}, {3085,26440}, {3295,18523}, {6464,26477}, {7354,26293}, {9578,26299}, {10797,26428}, {10831,26305}, {10873,26313}, {10895,26329}, {10924,26338}, {10944,26489}, {10956,26511}, {10957,26510}, {11375,26368}, {11392,26374}, {11501,26502}, {11905,26448}, {19027,26455}, {19028,26461}, {22759,26323}, {26479,26505}, {26480,26506}, {26481,26508}, {26482,26509}

X(26479) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 3rd TRI-SQUARES-CENTRAL

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

X(26479) lies on these lines: {1,26468}, {4,12949}, {12,3068}, {56,26361}, {65,26444}, {193,5261}, {388,492}, {498,26516}, {1007,26436}, {3085,26441}, {3295,18539}, {5860,10924}, {7354,26294}, {9578,26300}, {10797,26429}, {10831,26306}, {10873,26314}, {10895,26330}, {10923,26339}, {10944,26490}, {10956,26520}, {11392,26375}, {11501,26512}, {11905,26449}, {13695,15682}, {19027,26456}, {19028,26462}, {22759,26324}, {26388,26396}, {26412,26420}, {26477,26496}, {26478,26505}, {26481,26517}, {26482,26518}

X(26479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26468, 26473), (5261, 12588, 26480)

X(26480) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 4th TRI-SQUARES-CENTRAL

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

X(26480) lies on these lines: {1,26469}, {4,12948}, {12,3069}, {56,26362}, {65,26445}, {193,5261}, {388,491}, {498,26521}, {1007,26435}, {3085,8982}, {3295,26438}, {5861,10923}, {7354,26295}, {9578,26301}, {10797,26430}, {10831,26307}, {10873,26315}, {10895,26331}, {10924,26340}, {10944,26491}, {10956,26525}, {10957,26524}, {11375,26370}, {11392,26376}, {11501,26513}, {11905,26450}, {13815,15682}, {19027,26457}, {19028,26463}, {22759,26325}, {26388,26397}, {26412,26421}, {26477,26497}, {26478,26506}, {26481,26522}, {26482,26523}

X(26480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26469, 26474), (5261, 12588, 26479)

X(26481) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND INNER-YFF

Barycentrics    ((b^2+c^2)*a^3-(b^2+c^2)*(b+c)*a^2-(b^2+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c))*(a+b-c)*(a-b+c) : :
X(26481) = R^2*X(1)+2*r*(R+r)*X(5)

X(26481) lies on these lines: {1,5}, {4,26357}, {55,6831}, {56,442}, {65,2886}, {225,427}, {377,1470}, {388,2476}, {497,6828}, {498,6882}, {499,6881}, {550,14794}, {956,18962}, {1056,6874}, {1058,6873}, {1068,1594}, {1070,8758}, {1319,24541}, {1329,24987}, {1451,24892}, {1478,6842}, {1479,6841}, {1532,10895}, {1836,5709}, {1894,23361}, {2072,16272}, {2078,17527}, {2099,24390}, {3057,7680}, {3085,6830}, {3086,6829}, {3136,10372}, {3142,23304}, {3295,18544}, {3485,11680}, {3660,3824}, {3813,11011}, {3822,10106}, {3829,4870}, {3841,3911}, {3925,5705}, {4187,4423}, {4193,10588}, {4197,7288}, {4293,6937}, {4294,6845}, {4331,23305}, {5141,5261}, {5154,10587}, {5172,7483}, {5218,6943}, {5225,10883}, {5229,6932}, {5231,10404}, {5432,6922}, {5433,8728}, {6284,8727}, {6585,6863}, {6859,10321}, {6862,8069}, {6867,10629}, {6871,10530}, {6907,7354}, {6917,8071}, {6941,10532}, {6971,16202}, {6980,9654}, {6990,10591}, {6991,10589}, {7681,17605}, {8164,10806}, {8226,10896}, {9612,12704}, {10797,26431}, {10831,26308}, {10873,26317}, {10923,26342}, {11237,17530}, {11392,26377}, {11905,26452}, {12047,24474}, {12609,18838}, {13751,25557}, {17532,18961}, {19027,26458}, {19028,26464}, {26388,26399}, {26412,26423}, {26477,26499}, {26478,26508}, {26479,26517}, {26480,26522}

X(26481) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 12, 11375), (11, 3614, 7958), (10959, 15888, 1)

X(26482) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND OUTER-YFF

Barycentrics    ((b^2+4*b*c+c^2)*a^3-(b^2+4*b*c+c^2)*(b+c)*a^2-(b^2-4*b*c+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c))*(a+b-c)*(a-b+c) : :
X(26482) = R*(R-2*r)*X(1)+2*r*(R-r)*X(5)

X(26482) lies on these lines: {1,5}, {4,26358}, {10,18838}, {55,6256}, {65,6735}, {226,10915}, {388,404}, {498,10269}, {1319,1329}, {1388,4187}, {1478,11248}, {1519,3057}, {1532,2098}, {2077,7354}, {2475,5261}, {3085,6906}, {3295,18542}, {3476,11681}, {3485,12648}, {3584,14803}, {3820,5193}, {5048,7681}, {5254,21859}, {5434,17564}, {5687,18961}, {6842,12647}, {6952,8164}, {9612,12703}, {9654,10679}, {10531,10590}, {10786,26357}, {10797,26432}, {10831,26309}, {10873,26318}, {10895,10965}, {10923,26343}, {10924,26350}, {11112,11237}, {11239,17577}, {11392,26378}, {12047,23340}, {12832,25005}, {13273,25438}, {13743,18545}, {15843,24987}, {17625,17665}, {19027,26459}, {19028,26465}, {21031,24914}, {24982,25466}, {26388,26400}, {26412,26424}, {26477,26500}, {26478,26509}, {26479,26518}, {26480,26523}

X(26482) = reflection of X(26475) in X(10523)
X(26482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 119, 26476), (495, 10942, 1), (10958, 15888, 1)

X(26483) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND LUCAS HOMOTHETIC

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

X(26483) lies on these lines: {5,26499}, {11,26501}, {12,493}, {72,26442}, {355,26466}, {958,5490}, {3436,26494}, {6464,26484}, {10786,26439}, {10795,26427}, {10827,26298}, {10830,26304}, {10872,26312}, {10894,26328}, {10921,26337}, {10922,26347}, {10942,26500}, {10950,26471}, {10953,26353}, {11374,26367}, {11391,26373}, {11500,26493}, {11827,26292}, {11904,26447}, {18518,18521}, {18962,26433}, {19025,26454}, {19026,26460}, {26389,26391}, {26413,26415}, {26485,26496}, {26486,26497}, {26487,26498}

X(26484) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND LUCAS(-1) HOMOTHETIC

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

X(26484) lies on these lines: {5,26508}, {11,26510}, {12,494}, {72,26443}, {355,26467}, {958,5491}, {3436,26503}, {6464,26483}, {10786,26440}, {10795,26428}, {10827,26299}, {10830,26305}, {10872,26313}, {10894,26329}, {10922,26338}, {10942,26509}, {10950,26472}, {10953,26354}, {10955,26511}, {11374,26368}, {11391,26374}, {11500,26502}, {11827,26293}, {11904,26448}, {18518,18523}, {18962,26434}, {19025,26455}, {19026,26461}, {26389,26392}, {26485,26505}, {26486,26506}, {26487,26507}

X(26485) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND 3rd TRI-SQUARES-CENTRAL

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

X(26485) lies on these lines: {4,12939}, {5,26517}, {11,26519}, {12,3068}, {72,26444}, {193,12587}, {355,26468}, {492,3436}, {958,26324}, {5860,10922}, {10786,26441}, {10795,26429}, {10827,26300}, {10830,26306}, {10872,26314}, {10894,26330}, {10921,26339}, {10942,26518}, {10950,26473}, {10953,26355}, {10955,26520}, {11374,26369}, {11391,26375}, {11500,26512}, {11827,26294}, {11904,26449}, {13694,15682}, {18518,18539}, {18962,26435}, {19025,26456}, {19026,26462}, {26389,26396}, {26413,26420}, {26483,26496}, {26484,26505}, {26487,26516}

X(26486) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND 4th TRI-SQUARES-CENTRAL

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

X(26486) lies on these lines: {4,12938}, {5,26522}, {11,26524}, {12,3069}, {72,26445}, {193,12587}, {355,26469}, {491,3436}, {958,26325}, {5861,10921}, {8982,10786}, {10795,26430}, {10827,26301}, {10830,26307}, {10872,26315}, {10894,26331}, {10922,26340}, {10942,26523}, {10950,26474}, {10953,26356}, {10955,26525}, {11374,26370}, {11391,26376}, {11500,26513}, {11827,26295}, {11904,26450}, {13814,15682}, {18518,26438}, {18962,26436}, {19025,26457}, {19026,26463}, {26389,26397}, {26413,26421}, {26483,26497}, {26484,26506}, {26487,26521}

X(26487) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND X3-ABC REFLECTIONS

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

X(26487) lies on these lines: {1,6863}, {2,355}, {3,12}, {4,10585}, {5,1001}, {8,6853}, {11,16202}, {20,10599}, {24,11391}, {30,10894}, {35,6923}, {36,18962}, {40,3584}, {55,6842}, {56,10954}, {72,5552}, {100,6937}, {119,405}, {125,12890}, {140,958}, {182,12587}, {226,15865}, {381,6253}, {388,6954}, {442,11499}, {495,11249}, {499,10246}, {515,6862}, {517,3085}, {527,6684}, {549,11236}, {581,17734}, {631,3436}, {946,10197}, {952,26363}, {1006,11681}, {1125,6959}, {1317,15868}, {1329,6883}, {1479,6980}, {1482,10056}, {1511,13214}, {1621,6941}, {1656,18518}, {1698,17857}, {1788,5885}, {1837,24299}, {2080,10795}, {2476,11491}, {2646,10320}, {3035,15843}, {3058,11928}, {3086,15178}, {3311,19025}, {3312,19026}, {3357,12930}, {3475,6583}, {3523,20067}, {3526,6713}, {3541,5130}, {3560,6690}, {3576,6958}, {3579,5714}, {3616,6949}, {3822,6796}, {4294,6982}, {4309,10738}, {4428,10893}, {4995,11826}, {5080,6875}, {5218,6850}, {5230,5396}, {5248,6929}, {5284,6975}, {5433,10955}, {5445,15016}, {5534,5705}, {5587,6861}, {5603,6960}, {5690,12635}, {5709,15298}, {5731,6952}, {5770,5791}, {5790,19854}, {5886,6834}, {6256,6914}, {6642,10830}, {6771,12932}, {6774,12931}, {6824,18480}, {6827,10588}, {6833,18481}, {6838,12699}, {6848,9955}, {6868,10590}, {6891,13624}, {6892,12667}, {6897,10522}, {6907,11248}, {6910,12115}, {6911,25466}, {6926,17502}, {6928,7951}, {6933,12116}, {6944,11230}, {6962,10532}, {6967,18857}, {6985,7680}, {6988,8164}, {7483,22758}, {7491,10895}, {7583,13896}, {7584,13953}, {9780,9803}, {10202,24914}, {10321,24929}, {10610,12936}, {10679,15908}, {10680,15888}, {10872,26316}, {10921,26341}, {10922,26348}, {11904,26451}, {12041,12372}, {12042,12183}, {12359,12423}, {14450,16139}, {17615,18856}, {17718,24474}, {26389,26398}, {26413,26422}, {26483,26498}, {26484,26507}, {26485,26516}, {26486,26521}

X(26487) = midpoint of X(i) and X(j) for these {i,j}: {3, 9654}, {3085, 6825}
X(26487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1385, 26492), (2, 10786, 355)

X(26488) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND LUCAS HOMOTHETIC

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

X(26488) lies on these lines: {5,26500}, {11,493}, {355,26466}, {1376,5490}, {3434,26494}, {6464,26489}, {10785,26439}, {10794,26427}, {10826,26298}, {10829,26304}, {10871,26312}, {10893,26328}, {10914,26442}, {10919,26337}, {10920,26347}, {10943,26499}, {10944,26477}, {10947,26353}, {10949,26501}, {11373,26367}, {11390,26373}, {11826,26292}, {11903,26447}, {12114,26322}, {18519,18521}, {18961,26433}, {19023,26454}, {19024,26460}, {26490,26496}, {26491,26497}, {26492,26498}

X(26489) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND LUCAS(-1) HOMOTHETIC

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

X(26489) lies on these lines: {5,26509}, {11,494}, {12,26511}, {355,26467}, {1376,5491}, {3434,26503}, {6464,26488}, {10785,26440}, {10794,26428}, {10826,26299}, {10829,26305}, {10871,26313}, {10893,26329}, {10914,26443}, {10920,26338}, {10943,26508}, {10944,26478}, {10947,26354}, {10949,26510}, {11373,26368}, {11390,26374}, {11826,26293}, {11903,26448}, {12114,26323}, {18519,18523}, {18961,26434}, {19023,26455}, {19024,26461}, {26490,26505}, {26491,26506}, {26492,26507}

X(26490) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND 3rd TRI-SQUARES-CENTRAL

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

X(26490) lies on these lines: {4,12929}, {5,26518}, {11,3068}, {12,26520}, {193,12586}, {355,26468}, {492,3434}, {1376,26361}, {5860,10920}, {10785,26441}, {10794,26429}, {10826,26300}, {10829,26306}, {10871,26314}, {10893,26330}, {10914,26444}, {10919,26339}, {10943,26517}, {10944,26479}, {10947,26355}, {10949,26519}, {11373,26369}, {11390,26375}, {11826,26294}, {11903,26449}, {12114,26324}, {13693,15682}, {18519,18539}, {18961,26435}, {19023,26456}, {19024,26462}, {26390,26396}, {26414,26420}, {26488,26496}, {26489,26505}, {26492,26516}

X(26491) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND 4th TRI-SQUARES-CENTRAL

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

X(26491) lies on these lines: {4,12928}, {5,26523}, {11,3069}, {12,26525}, {193,12586}, {355,26469}, {491,3434}, {1376,26362}, {5861,10919}, {8982,10785}, {10794,26430}, {10826,26301}, {10829,26307}, {10871,26315}, {10893,26331}, {10914,26445}, {10920,26340}, {10943,26522}, {10944,26480}, {10947,26356}, {10949,26524}, {11373,26370}, {11390,26376}, {11826,26295}, {11903,26450}, {12114,26325}, {13813,15682}, {18519,26438}, {18961,26436}, {19023,26457}, {19024,26463}, {26390,26397}, {26414,26421}, {26488,26497}, {26489,26506}, {26492,26521}

X(26492) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND X3-ABC REFLECTIONS

Barycentrics    a^7-(b+c)*a^6-3*(b-c)^2*a^5+(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+(b^2+c^2)*(3*b^2-8*b*c+3*c^2)*a^3-3*(b^4-c^4)*(b-c)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :
X(26492) = (R-r)*X(3)+(R-2*r)*X(11)

X(26492) lies on these lines: {1,6958}, {2,355}, {3,11}, {4,10584}, {5,6256}, {8,12619}, {12,16203}, {20,10598}, {24,11390}, {30,10893}, {35,10947}, {36,6928}, {40,3582}, {55,10948}, {56,6882}, {104,4193}, {125,12889}, {140,1376}, {182,12586}, {388,6978}, {474,26470}, {496,11248}, {497,6961}, {498,10246}, {515,6959}, {517,1788}, {549,11235}, {631,3434}, {912,25681}, {946,10199}, {952,26364}, {1125,6862}, {1319,10320}, {1329,20418}, {1478,6971}, {1482,10072}, {1484,13205}, {1511,13213}, {1656,18519}, {1709,8227}, {2080,10794}, {2975,6963}, {3085,15178}, {3311,19023}, {3312,19024}, {3357,12920}, {3485,5885}, {3523,20066}, {3526,19854}, {3541,5101}, {3560,3816}, {3576,6863}, {3579,6926}, {3616,6952}, {3624,6861}, {3825,5450}, {4187,22758}, {4999,6883}, {5204,7491}, {5252,24927}, {5253,6830}, {5298,11827}, {5432,10949}, {5434,11929}, {5439,5886}, {5443,15016}, {5550,6852}, {5554,17665}, {5603,6972}, {5657,17652}, {5690,10912}, {5693,11219}, {5694,5770}, {5731,6949}, {5761,6583}, {5927,6832}, {6361,10225}, {6642,10829}, {6667,18242}, {6681,6796}, {6691,6911}, {6771,12922}, {6774,12921}, {6824,9940}, {6825,13624}, {6827,7288}, {6834,18481}, {6837,17618}, {6847,9955}, {6850,10589}, {6890,12699}, {6908,17502}, {6921,12116}, {6922,11249}, {6923,7741}, {6931,12115}, {6940,11680}, {6944,18480}, {6947,10522}, {6948,10591}, {6966,10531}, {6967,10527}, {6981,12667}, {7330,25522}, {7583,13895}, {7584,13952}, {10057,21842}, {10165,17647}, {10202,11375}, {10321,24928}, {10610,12926}, {10871,26316}, {10919,26341}, {10920,26348}, {11041,14986}, {11231,19843}, {11374,13373}, {11491,17566}, {11499,13747}, {11903,26451}, {12041,12371}, {12042,12182}, {12053,15866}, {12359,12422}, {17728,24474}, {21616,24467}, {26390,26398}, {26414,26422}, {26488,26498}, {26489,26507}, {26490,26516}, {26491,26521}

X(26492) = midpoint of X(i) and X(j) for these {i,j}: {3, 9669}, {3086, 6891}
X(26492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1385, 26487), (2, 10785, 355)

X(26493) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND MANDART-INCIRCLE

Barycentrics
a^2*(-a^7+(b+c)*a^6+(b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4+(b^4+c^4+2*(b+2*c)*(2*b+c)*b*c)*a^3-(b+c)*(b^4+10*b^2*c^2+c^4)*a^2-(b^6+c^6+(2*b^4+2*c^4-(b^2+12*b*c+c^2)*b*c)*b*c)*a-(a^5-(b+c)*a^4-6*(b^2+c^2)*a^3+6*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(4*b^2+3*b*c+4*c^2)*b*c)*a-(-4*b^2*c^2+(b^2-c^2)^2)*(b+c))*S+(b^4-c^4)*(b^2-c^2)*(b+c)) : :

X(26493) lies on these lines: {3,26322}, {35,26298}, {55,493}, {100,26494}, {197,26304}, {1376,5490}, {3295,26367}, {5687,26442}, {6464,26502}, {10267,26498}, {10310,26292}, {11248,26500}, {11383,26373}, {11490,26427}, {11491,26439}, {11494,26312}, {11496,26328}, {11497,26337}, {11498,26347}, {11499,26466}, {11500,26483}, {11501,26477}, {11502,26471}, {11509,26433}, {11510,26501}, {11848,26447}, {18521,18524}, {18999,26454}, {19000,26460}, {26496,26512}, {26497,26513}

X(26494) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND MEDIAL

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

X(26494) lies on these lines: {2,493}, {3,26439}, {4,26373}, {8,26442}, {10,26298}, {20,26292}, {22,26304}, {30,18521}, {100,26493}, {145,26495}, {193,13428}, {388,26433}, {491,26497}, {492,19420}, {497,26353}, {631,26498}, {1270,26347}, {1271,26337}, {2896,26312}, {2975,26322}, {2996,13439}, {3091,26328}, {3434,26488}, {3436,26483}, {3616,26367}, {4240,26447}, {5552,26500}, {5905,19218}, {6392,6464}, {6995,8948}, {7585,26460}, {7586,26454}, {7787,26427}, {10527,26499}, {10529,26501}

X(26494) = isotomic conjugate of X(26503)
X(26494) = anticomplement of X(8222)
X(26494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (493, 5490, 2), (6392, 6515, 26503)

X(26495) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND 5th MIXTILINEAR

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

X(26495) lies on these lines: {1,493}, {8,5490}, {55,26322}, {56,26493}, {145,26494}, {517,26292}, {519,26442}, {952,26466}, {1829,8948}, {2098,26353}, {2099,26433}, {5603,26328}, {5604,26347}, {5605,26337}, {6464,26504}, {7967,26439}, {7968,26454}, {7969,26460}, {8192,26304}, {9997,26312}, {10246,26498}, {10800,26427}, {10944,26477}, {10950,26471}, {11396,26373}, {11910,26447}, {18521,18526}, {26496,26514}, {26497,26515}

X(26495) = reflection of X(8210) in X(1)
X(26495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26298, 26367), (26298, 26367, 493)

X(26496) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND 3rd TRI-SQUARES-CENTRAL

Barycentrics
a^8+8*(b^2+c^2)*a^6-6*(3*b^4+8*b^2*c^2+3*c^4)*a^4+8*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2+4*S*(2*a^6-5*(b^2+c^2)*a^4-16*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)) : :

X(26496) lies on these lines: {193,26497}, {393,493}, {492,19420}, {5490,7763}, {5860,26347}, {6459,8948}, {6464,26505}, {18521,18539}, {26292,26294}, {26298,26300}, {26304,26306}, {26312,26314}, {26322,26324}, {26328,26330}, {26337,26339}, {26353,26355}, {26367,26369}, {26373,26375}, {26427,26429}, {26433,26435}, {26439,26441}, {26442,26444}, {26447,26449}, {26454,26456}, {26460,26462}, {26466,26468}, {26471,26473}, {26477,26479}, {26483,26485}, {26488,26490}, {26493,26512}, {26495,26514}, {26498,26516}, {26499,26517}, {26501,26519}

X(26496) = {X(493), X(24244)}-harmonic conjugate of X(3068)

X(26497) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND 4th TRI-SQUARES-CENTRAL

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

X(26497) lies on these lines: {193,26496}, {491,26494}, {493,3069}, {5490,26362}, {5861,26337}, {6464,26506}, {8982,26439}, {18521,26438}, {26292,26295}, {26298,26301}, {26304,26307}, {26312,26315}, {26322,26325}, {26328,26331}, {26340,26347}, {26353,26356}, {26367,26370}, {26373,26376}, {26427,26430}, {26433,26436}, {26442,26445}, {26447,26450}, {26454,26457}, {26460,26463}, {26466,26469}, {26471,26474}, {26477,26480}, {26483,26486}, {26488,26491}, {26493,26513}, {26495,26515}, {26498,26521}, {26499,26522}, {26500,26523}, {26501,26524}

X(26498) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND X3-ABC REFLECTIONS

Barycentrics
a^2*((a^6-11*(b^2+c^2)*a^4+(15*b^4+22*b^2*c^2+15*c^4)*a^2-(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4))*S+2*a^8-6*(b^2+c^2)*a^6+2*(3*b^4-4*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(b^4-12*b^2*c^2+c^4)*a^2-8*(b^2-c^2)^2*b^2*c^2) : :

X(26498) lies on these lines: {2,26439}, {3,493}, {24,26373}, {30,26328}, {35,26353}, {36,26433}, {140,5490}, {498,26477}, {499,26471}, {517,26367}, {631,26494}, {1151,12978}, {1656,18521}, {2080,26427}, {3311,26454}, {3312,26460}, {3517,8948}, {3576,26298}, {6464,26507}, {6642,26304}, {10246,26495}, {10267,26493}, {10269,26322}, {26312,26316}, {26337,26341}, {26347,26348}, {26415,26422}, {26442,26446}, {26447,26451}, {26483,26487}, {26488,26492}, {26496,26516}, {26497,26521}

X(26498) = midpoint of X(3) and X(11949)
X(26498) = {X(2), X(26439)}-harmonic conjugate of X(26466)

X(26499) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND INNER-YFF

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

X(26499) lies on these lines: {1,493}, {5,26483}, {5490,26363}, {6464,26508}, {6734,26442}, {10267,26493}, {10527,26494}, {10943,26488}, {11012,26292}, {11249,26322}, {12116,26439}, {18521,18544}, {26304,26308}, {26312,26317}, {26328,26332}, {26337,26342}, {26347,26349}, {26353,26357}, {26373,26377}, {26427,26431}, {26433,26437}, {26447,26452}, {26454,26458}, {26460,26464}, {26466,26470}, {26471,26475}, {26477,26481}, {26496,26517}, {26497,26522}

X(26500) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND OUTER-YFF

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

X(26500) lies on these lines: {1,493}, {5,26488}, {119,26466}, {1470,26433}, {2077,26292}, {5490,26364}, {5552,26494}, {6464,26509}, {6735,26442}, {10269,26322}, {10942,26483}, {11248,26493}, {12115,26439}, {18521,18542}, {26304,26309}, {26312,26318}, {26328,26333}, {26337,26343}, {26347,26350}, {26353,26358}, {26373,26378}, {26427,26432}, {26447,26453}, {26454,26459}, {26460,26465}, {26471,26476}, {26477,26482}, {26497,26523}

X(26501) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND INNER-YFF TANGENTS

Barycentrics
a^2*(4*((b^2-b*c+c^2)*a^4-2*(b^2+c^2)*(b-c)^2*a^2-2*(b^2+c^2)*(b+c)*b*c*a+c^6+b^6-(b^4+c^4+(b^2-6*b*c+c^2)*b*c)*b*c)*S+(-a+b+c)*(a^7+(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5-(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^3+(3*b^4-2*b^2*c^2+3*c^4)*(b+c)*a^2-(b^6+c^6-(13*b^2-32*b*c+13*c^2)*b^2*c^2)*a+(b^2-c^2)*(b-c)*(-b^4-c^4-2*(2*b^2-3*b*c+2*c^2)*b*c))) : :

X(26501) lies on these lines: {1,493}, {11,26483}, {5490,10527}, {6464,26510}, {8948,26377}, {10529,26494}, {10532,26328}, {10804,26427}, {10806,26439}, {10835,26304}, {10879,26312}, {10916,26442}, {10931,26337}, {10932,26347}, {10943,26466}, {10949,26488}, {10957,26477}, {10959,26471}, {11249,26292}, {11401,26373}, {11510,26493}, {11915,26447}, {16202,26498}, {18521,18543}, {18967,26433}, {19049,26454}, {19050,26460}, {24244,26517}, {26496,26519}, {26497,26524}

X(26502) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND MANDART-INCIRCLE

Barycentrics
a^2*(-a^7+(b+c)*a^6+(b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4+(b^4+c^4+2*(b+2*c)*(2*b+c)*b*c)*a^3-(b+c)*(b^4+10*b^2*c^2+c^4)*a^2-(b^6+c^6+(2*b^4+2*c^4-(b^2+12*b*c+c^2)*b*c)*b*c)*a+(a^5-(b+c)*a^4-6*(b^2+c^2)*a^3+6*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(4*b^2+3*b*c+4*c^2)*b*c)*a-(-4*b^2*c^2+(b^2-c^2)^2)*(b+c))*S+(b^4-c^4)*(b^2-c^2)*(b+c)) : :

X(26502) lies on these lines: {3,26323}, {35,26299}, {55,494}, {56,26504}, {100,26503}, {197,26305}, {1376,5491}, {3295,26368}, {5687,26443}, {6464,26493}, {10267,26507}, {10310,26293}, {11248,26509}, {11383,26374}, {11490,26428}, {11491,26440}, {11494,26313}, {11496,26329}, {11498,26338}, {11499,26467}, {11500,26484}, {11501,26478}, {11502,26472}, {11509,26434}, {11510,26510}, {11848,26448}, {18523,18524}, {18999,26455}, {19000,26461}, {26505,26512}, {26506,26513}

X(26503) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND MEDIAL

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

X(26503) lies on these lines: {2,494}, {3,26440}, {4,26374}, {8,26443}, {10,26299}, {20,26293}, {22,26305}, {30,18523}, {100,26502}, {145,26504}, {193,13439}, {388,26434}, {491,19421}, {492,26505}, {497,26354}, {631,26507}, {1270,26338}, {2896,26313}, {2975,26323}, {2996,13428}, {3091,26329}, {3434,26489}, {3436,26484}, {3616,26368}, {4240,26448}, {5552,26509}, {5905,19217}, {6392,6464}, {6995,8946}, {7585,26461}, {7586,26455}, {7787,26428}, {10527,26508}, {10528,26511}, {10529,26510}, {26392,26394}, {26416,26418}

X(26503) = isotomic conjugate of X(26494)
X(26503) = anticomplement of X(8223)
X(26503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (494, 5491, 2), (6392, 6515, 26494)

X(26504) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND 5th MIXTILINEAR

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

X(26504) lies on these lines: {1,494}, {8,5491}, {55,26323}, {56,26502}, {145,26503}, {517,26293}, {519,26443}, {952,26467}, {1829,8946}, {2098,26354}, {2099,26434}, {5603,26329}, {5604,26338}, {6464,26495}, {7967,26440}, {7968,26455}, {7969,26461}, {8192,26305}, {9997,26313}, {10246,26507}, {10800,26428}, {10944,26478}, {10950,26472}, {11396,26374}, {11910,26448}, {18523,18526}, {26392,26395}, {26416,26419}, {26505,26514}, {26506,26515}

X(26504) = reflection of X(8211) in X(1)
X(26504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26299, 26368), (26299, 26368, 494)

X(26505) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND 3rd TRI-SQUARES-CENTRAL

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

X(26505) lies on these lines: {193,26506}, {492,26503}, {494,3068}, {5491,26361}, {5860,26338}, {6464,26496}, {18523,18539}, {26293,26294}, {26299,26300}, {26305,26306}, {26313,26314}, {26323,26324}, {26329,26330}, {26354,26355}, {26368,26369}, {26374,26375}, {26428,26429}, {26434,26435}, {26440,26441}, {26443,26444}, {26448,26449}, {26455,26456}, {26461,26462}, {26467,26468}, {26472,26473}, {26478,26479}, {26484,26485}, {26489,26490}, {26502,26512}, {26504,26514}, {26507,26516}, {26508,26517}, {26509,26518}, {26510,26519}, {26511,26520}

X(26506) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND 4th TRI-SQUARES-CENTRAL

Barycentrics
a^8+8*(b^2+c^2)*a^6-6*(3*b^4+8*b^2*c^2+3*c^4)*a^4+8*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2-4*S*(2*a^6-5*(b^2+c^2)*a^4-16*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)) : :

X(26506) lies on these lines: {193,26505}, {393,494}, {491,19421}, {5491,7763}, {6460,8946}, {6464,26497}, {8982,26440}, {18523,26438}, {26293,26295}, {26299,26301}, {26305,26307}, {26313,26315}, {26323,26325}, {26329,26331}, {26338,26340}, {26354,26356}, {26368,26370}, {26374,26376}, {26428,26430}, {26434,26436}, {26443,26445}, {26448,26450}, {26455,26457}, {26461,26463}, {26467,26469}, {26472,26474}, {26478,26480}, {26484,26486}, {26489,26491}, {26502,26513}, {26504,26515}, {26507,26521}, {26508,26522}, {26509,26523}, {26510,26524}, {26511,26525}

X(26506) = {X(494), X(24243)}-harmonic conjugate of X(3069)

X(26507) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND X3-ABC REFLECTIONS

Barycentrics
a^2*(-(a^6-11*(b^2+c^2)*a^4+(22*b^2*c^2+15*c^4+15*b^4)*a^2-(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4))*S+2*a^8-6*(b^2+c^2)*a^6+2*(3*b^4-4*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(b^4-12*b^2*c^2+c^4)*a^2-8*(b^2-c^2)^2*b^2*c^2) : :

X(26507) lies on these lines: {2,26440}, {3,494}, {24,26374}, {30,26329}, {35,26354}, {36,26434}, {140,5491}, {498,26478}, {499,26472}, {517,26368}, {631,26503}, {1152,12979}, {1656,18523}, {2080,26428}, {3311,26455}, {3312,26461}, {3517,8946}, {3576,26299}, {6464,26498}, {6642,26305}, {10246,26504}, {10267,26502}, {10269,26323}, {16202,26510}, {16203,26511}, {26313,26316}, {26338,26348}, {26392,26398}, {26416,26422}, {26443,26446}, {26448,26451}, {26484,26487}, {26489,26492}, {26505,26516}, {26506,26521}

X(26507) = midpoint of X(3) and X(11950)
X(26507) = {X(2), X(26440)}-harmonic conjugate of X(26467)

X(26508) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND INNER-YFF

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

X(26508) lies on these lines: {1,494}, {5,26484}, {5491,26363}, {6464,26499}, {6734,26443}, {10267,26502}, {10527,26503}, {10943,26489}, {11012,26293}, {11249,26323}, {12116,26440}, {18523,18544}, {26305,26308}, {26313,26317}, {26329,26332}, {26338,26349}, {26354,26357}, {26374,26377}, {26428,26431}, {26434,26437}, {26448,26452}, {26455,26458}, {26461,26464}, {26467,26470}, {26472,26475}, {26478,26481}, {26505,26517}, {26506,26522}

X(26509) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND OUTER-YFF

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

X(26509) lies on these lines: {1,494}, {5,26489}, {119,26467}, {1470,26434}, {2077,26293}, {5491,26364}, {5552,26503}, {6464,26500}, {6735,26443}, {10269,26323}, {10942,26484}, {11248,26502}, {12115,26440}, {18523,18542}, {26305,26309}, {26313,26318}, {26329,26333}, {26338,26350}, {26354,26358}, {26374,26378}, {26428,26432}, {26448,26453}, {26455,26459}, {26461,26465}, {26472,26476}, {26478,26482}, {26505,26518}, {26506,26523}

X(26510) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND INNER-YFF TANGENTS

Barycentrics
a^2*(4*((b^2-b*c+c^2)*a^4-2*(b^2+c^2)*(b-c)^2*a^2-2*(b^2+c^2)*(b+c)*b*c*a+c^6+b^6-(b^4+c^4+(b^2-6*b*c+c^2)*b*c)*b*c)*S-(-a+b+c)*(a^7+(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5-(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^3+(3*b^4-2*b^2*c^2+3*c^4)*(b+c)*a^2-(b^6+c^6-(13*b^2-32*b*c+13*c^2)*b^2*c^2)*a+(b^2-c^2)*(b-c)*(-b^4-c^4-2*(2*b^2-3*b*c+2*c^2)*b*c))) : :

X(26510) lies on these lines: {1,494}, {11,26484}, {5491,10527}, {6464,26501}, {8946,26377}, {10529,26503}, {10532,26329}, {10804,26428}, {10806,26440}, {10835,26305}, {10879,26313}, {10916,26443}, {10932,26338}, {10943,26467}, {10949,26489}, {10957,26478}, {10959,26472}, {10966,26323}, {11249,26293}, {11401,26374}, {11510,26502}, {11915,26448}, {16202,26507}, {18523,18543}, {18967,26434}, {19049,26455}, {19050,26461}, {24243,26522}, {26505,26519}, {26506,26524}

X(26511) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND OUTER-YFF TANGENTS

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

X(26511) lies on these lines: {1,494}, {12,26489}, {5491,5552}, {8946,26378}, {10528,26503}, {10531,26329}, {10803,26428}, {10805,26440}, {10834,26305}, {10878,26313}, {10915,26443}, {10930,26338}, {10942,26467}, {10955,26484}, {10956,26478}, {10958,26472}, {10965,26354}, {11248,26293}, {11400,26374}, {11509,26434}, {11914,26448}, {16203,26507}, {18523,18545}, {19047,26455}, {19048,26461}, {22768,26323}, {24243,26523}, {26505,26520}, {26506,26525}

X(26512) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE AND 3rd TRI-SQUARES-CENTRAL

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

X(26512) lies on these lines: {3,26324}, {4,12344}, {35,26300}, {55,3068}, {56,26514}, {100,492}, {193,12329}, {197,26306}, {1376,26361}, {3295,26369}, {4421,5860}, {5687,26444}, {10267,26516}, {10310,26294}, {11248,26518}, {11383,26375}, {11490,26429}, {11491,26441}, {11494,26314}, {11496,26330}, {11497,26339}, {11499,26468}, {11500,26485}, {11501,26479}, {11502,26473}, {11509,26435}, {11510,26519}, {11848,26449}, {13675,15682}, {18524,18539}, {18999,26456}, {19000,26462}, {26393,26396}, {26417,26420}, {26493,26496}, {26502,26505}

X(26513) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE AND 4th TRI-SQUARES-CENTRAL

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

X(26513) lies on these lines: {3,26325}, {4,12343}, {35,26301}, {55,3069}, {56,26515}, {100,491}, {193,12329}, {197,26307}, {1376,26362}, {3295,26370}, {4421,5861}, {5687,26445}, {8982,11491}, {10267,26521}, {10310,26295}, {11248,26523}, {11383,26376}, {11490,26430}, {11494,26315}, {11496,26331}, {11498,26340}, {11499,26469}, {11500,26486}, {11501,26480}, {11502,26474}, {11509,26436}, {11510,26524}, {11848,26450}, {13795,15682}, {18524,26438}, {18999,26457}, {19000,26463}, {26393,26397}, {26417,26421}, {26493,26497}, {26502,26506}

X(26514) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    (7*a-b-c)*S+a*(a^2-(b+c)*a+2*b^2+2*c^2) : :

X(26514) lies on these lines: {1,1336}, {4,7981}, {8,26361}, {55,26324}, {56,26512}, {145,492}, {193,3242}, {517,26294}, {519,26444}, {952,26468}, {2098,26355}, {2099,26435}, {3241,5604}, {5603,26330}, {5605,20057}, {7967,26441}, {7968,26456}, {7969,26462}, {8192,26306}, {9997,26314}, {10246,26516}, {10800,26429}, {10944,26479}, {10950,26473}, {11396,26375}, {11910,26449}, {13702,15682}, {18526,18539}, {26395,26396}, {26419,26420}, {26495,26496}, {26504,26505}

X(26514) = reflection of X(13902) in X(1)
X(26514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26300, 26369), (26300, 26369, 3068), (26519, 26520, 3068)

X(26515) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 4th TRI-SQUARES-CENTRAL

Barycentrics    (7*a-b-c)*S-a*(a^2-(b+c)*a+2*b^2+2*c^2) : :

X(26515) lies on these lines: {1,1123}, {4,7980}, {8,26362}, {55,26325}, {56,26513}, {145,491}, {193,3242}, {517,26295}, {519,26445}, {952,26469}, {2098,26356}, {2099,26436}, {3241,5605}, {5603,26331}, {5604,20057}, {7967,8982}, {7968,26457}, {7969,26463}, {8192,26307}, {9997,26315}, {10246,26521}, {10800,26430}, {10944,26480}, {10950,26474}, {11396,26376}, {11910,26450}, {13822,15682}, {18526,26438}, {26395,26397}, {26419,26421}, {26495,26497}, {26504,26506}

X(26515) = reflection of X(13959) in X(1)
X(26515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26301, 26370), (26301, 26370, 3069), (26524, 26525, 3069)

X(26516) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND X3-ABC REFLECTIONS

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

X(26516) lies on these lines: {2,14234}, {3,1587}, {20,12974}, {24,26375}, {30,26330}, {35,26355}, {36,26435}, {140,26361}, {182,193}, {230,1151}, {371,19102}, {487,492}, {488,21445}, {498,26479}, {499,26473}, {517,26369}, {549,5860}, {1656,18539}, {2080,26429}, {3311,26456}, {3312,26462}, {3530,26339}, {3576,26300}, {3767,15885}, {5305,8407}, {6200,12123}, {6459,12601}, {6642,26306}, {7585,9739}, {7735,15883}, {8960,12124}, {9680,11824}, {10246,26514}, {10267,26512}, {10269,26324}, {11294,26469}, {12314,19054}, {12975,15692}, {16202,26519}, {16203,26520}, {26314,26316}, {26396,26398}, {26420,26422}, {26444,26446}, {26449,26451}, {26485,26487}, {26490,26492}, {26496,26498}, {26505,26507}

X(26516) = midpoint of X(3) and X(13903)
X(26516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26441, 26468), (182, 3523, 26521)

X(26517) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND INNER-YFF

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

X(26517) lies on these lines: {1,1336}, {5,26485}, {193,10529}, {371,13135}, {492,10527}, {5860,26349}, {6734,26444}, {10267,26512}, {10943,26490}, {11012,26294}, {11249,26324}, {12116,26441}, {18539,18544}, {24244,26501}, {26306,26308}, {26314,26317}, {26330,26332}, {26339,26342}, {26355,26357}, {26361,26363}, {26375,26377}, {26396,26399}, {26420,26423}, {26429,26431}, {26435,26437}, {26449,26452}, {26456,26458}, {26462,26464}, {26468,26470}, {26473,26475}, {26479,26481}, {26496,26499}, {26505,26508}

X(26517) = {X(3068), X(26519)}-harmonic conjugate of X(1)

X(26518) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-YFF

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

X(26518) lies on these lines: {1,1336}, {5,26490}, {119,26468}, {193,10528}, {371,13134}, {492,5552}, {1470,26435}, {2077,26294}, {5860,26350}, {6735,26444}, {10269,26324}, {10942,26485}, {11248,26512}, {12115,26441}, {18539,18542}, {26306,26309}, {26314,26318}, {26330,26333}, {26339,26343}, {26355,26358}, {26361,26364}, {26375,26378}, {26396,26400}, {26420,26424}, {26429,26432}, {26449,26453}, {26456,26459}, {26462,26465}, {26473,26476}, {26479,26482}, {26505,26509}

X(26518) = {X(3068), X(26520)}-harmonic conjugate of X(1)

X(26519) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND INNER-YFF TANGENTS

Barycentrics    (a^4-2*(b^2-7*b*c+c^2)*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^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)) : :

X(26519) lies on these lines: {1,1336}, {4,13135}, {11,26485}, {193,12595}, {492,10529}, {5860,10932}, {10527,26361}, {10532,26330}, {10804,26429}, {10806,26441}, {10835,26306}, {10879,26314}, {10916,26444}, {10931,26339}, {10943,26468}, {10949,26490}, {10957,26479}, {10959,26473}, {10966,26324}, {11249,26294}, {11401,26375}, {11510,26512}, {11915,26449}, {13717,15682}, {16202,26516}, {18539,18543}, {18967,26435}, {19049,26456}, {19050,26462}, {26396,26401}, {26420,26425}, {26496,26501}, {26505,26510}

X(26519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26517, 3068), (3068, 26514, 26520)

X(26520) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL AND OUTER-YFF TANGENTS

Barycentrics    (a^4-2*(b^2+7*b*c+c^2)*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^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)) : :

X(26520) lies on these lines: {1,1336}, {4,13134}, {12,26490}, {193,12594}, {492,10528}, {5552,26361}, {5860,10930}, {10531,26330}, {10803,26429}, {10805,26441}, {10834,26306}, {10878,26314}, {10915,26444}, {10929,26339}, {10942,26468}, {10955,26485}, {10956,26479}, {10958,26473}, {10965,26355}, {11248,26294}, {11400,26375}, {11509,26435}, {11914,26449}, {13716,15682}, {16203,26516}, {18539,18545}, {19047,26456}, {19048,26462}, {22768,26324}, {26396,26402}, {26420,26426}, {26505,26511}

X(26520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26518, 3068), (3068, 26514, 26519)

X(26521) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND X3-ABC REFLECTIONS

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

X(26521) lies on these lines: {2,8982}, {3,1588}, {20,12975}, {24,26376}, {30,26331}, {35,26356}, {36,26436}, {140,26362}, {182,193}, {230,1152}, {372,19105}, {487,21445}, {488,491}, {498,26480}, {499,26474}, {517,26370}, {549,5861}, {1656,26438}, {2080,26430}, {3311,26457}, {3312,26463}, {3530,26340}, {3576,26301}, {3767,15886}, {5305,8400}, {5420,21737}, {6396,12124}, {6460,12602}, {6642,26307}, {7586,9738}, {7735,15884}, {10246,26515}, {10267,26513}, {10269,26325}, {11293,26468}, {12313,19053}, {12974,15692}, {16202,26524}, {16203,26525}, {26315,26316}, {26397,26398}, {26421,26422}, {26445,26446}, {26450,26451}, {26486,26487}, {26491,26492}, {26497,26498}, {26506,26507}

X(26521) = midpoint of X(3) and X(13961)
X(26521) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 8982, 26469), (182, 3523, 26516)

X(26522) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-YFF

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

X(26522) lies on these lines: {1,1123}, {5,26486}, {193,10529}, {372,13133}, {491,10527}, {5861,26342}, {6734,26445}, {8982,12116}, {10267,26513}, {10943,26491}, {11012,26295}, {11249,26325}, {18544,26438}, {24243,26510}, {26307,26308}, {26315,26317}, {26331,26332}, {26340,26349}, {26356,26357}, {26362,26363}, {26376,26377}, {26397,26399}, {26421,26423}, {26430,26431}, {26436,26437}, {26450,26452}, {26457,26458}, {26463,26464}, {26469,26470}, {26474,26475}, {26480,26481}, {26497,26499}, {26506,26508}

X(26522) = {X(3069), X(26524)}-harmonic conjugate of X(1)

X(26523) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-YFF

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

X(26523) lies on these lines: {1,1123}, {5,26491}, {119,26469}, {193,10528}, {372,13132}, {491,5552}, {1470,26436}, {2077,26295}, {5861,26343}, {6735,26445}, {8982,12115}, {10269,26325}, {10942,26486}, {11248,26513}, {18542,26438}, {24243,26511}, {26307,26309}, {26315,26318}, {26331,26333}, {26340,26350}, {26356,26358}, {26362,26364}, {26376,26378}, {26397,26400}, {26421,26424}, {26430,26432}, {26450,26453}, {26457,26459}, {26463,26465}, {26474,26476}, {26480,26482}, {26497,26500}, {26506,26509}

X(26523) = {X(3069), X(26525)}-harmonic conjugate of X(1)

X(26524) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND INNER-YFF TANGENTS

Barycentrics    -(a^4-2*(b^2-7*b*c+c^2)*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^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)) : :

X(26524) lies on these lines: {1,1123}, {4,13133}, {11,26486}, {193,12595}, {491,10529}, {5861,10931}, {8982,10806}, {10527,26362}, {10532,26331}, {10804,26430}, {10835,26307}, {10879,26315}, {10916,26445}, {10932,26340}, {10943,26469}, {10949,26491}, {10957,26480}, {10959,26474}, {10966,26325}, {11249,26295}, {11401,26376}, {11510,26513}, {11915,26450}, {13840,15682}, {16202,26521}, {18543,26438}, {18967,26436}, {19049,26457}, {19050,26463}, {26397,26401}, {26421,26425}, {26497,26501}, {26506,26510}

X(26524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26522, 3069), (3069, 26515, 26525)

X(26525) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL AND OUTER-YFF TANGENTS

Barycentrics    -(a^4-2*(b^2+7*b*c+c^2)*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^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)) : :

X(26525) lies on these lines: {1,1123}, {4,13132}, {12,26491}, {193,12594}, {491,10528}, {5552,26362}, {5861,10929}, {8982,10805}, {10531,26331}, {10803,26430}, {10834,26307}, {10878,26315}, {10915,26445}, {10930,26340}, {10942,26469}, {10955,26486}, {10956,26480}, {10958,26474}, {10965,26356}, {11248,26295}, {11400,26376}, {11509,26436}, {11914,26450}, {13839,15682}, {16203,26521}, {18545,26438}, {19047,26457}, {19048,26463}, {22768,26325}, {26397,26402}, {26421,26426}, {26506,26511}

X(26525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 26523, 3069), (3069, 26515, 26524)

leftri

Collineation mappings involving Gemini triangle 45: X(26526)-X(26574)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 45, as in centers X(26526)-X(26574). Then

m(X) = (b + c - a)(b - c)^2 x + b^2 (a - b + c) y + c^2 (a + b - c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 1, 2018)

X(26526) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26526) lies on these lines: {1, 2}, {21, 24619}, {220, 27132}, {664, 27006}, {673, 5086}, {1146, 26563}, {1572, 26099}, {2082, 21285}, {2170, 17046}, {2241, 25886}, {2475, 27000}, {3662, 26549}, {3753, 17672}, {3877, 17671}, {4904, 20880}, {4967, 25966}, {5141, 27183}, {5794, 24596}, {10025, 26793}, {17062, 17451}, {17184, 26528}, {17270, 25880}, {20257, 21029}, {20905, 26543}, {21240, 24548}, {24547, 25964}, {24986, 25887}, {26527, 26561}, {26529, 26533}, {26530, 26538}, {26536, 26542}, {26567, 26569}

X(26527) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26527) lies on these lines: {2, 3}, {7761, 25886}, {23661, 26157}, {26526, 26561}, {26530, 26537}, {26531, 26565}, {26541, 26564}, {26582, 26653}

X(26528) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26528) lies on these lines: {2, 3}, {318, 26157}, {8735, 18639}, {17184, 26526}, {26533, 26561}, {26540, 26541}

X(26529) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26529) lies on these lines: {2, 3}, {26526, 26533}, {26590, 26653}

X(26530) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26530) lies on these lines: {2, 6}, {120, 25279}, {125, 16067}, {894, 16608}, {1086, 26567}, {1330, 25990}, {1352, 16048}, {1368, 3794}, {1503, 17522}, {1853, 26096}, {1899, 25494}, {3271, 17047}, {3662, 26932}, {7083, 21280}, {17236, 27288}, {20905, 26570}, {21258, 26806}, {25007, 25966}, {26526, 26538}, {26527, 26537}, {26536, 26559}, {26557, 26569}

X(26531) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26531) lies on these lines: {1, 2}, {4, 27000}, {5, 27183}, {75, 25002}, {85, 1146}, {116, 17181}, {150, 169}, {192, 25019}, {319, 25878}, {355, 17682}, {404, 25954}, {515, 4209}, {517, 17671}, {673, 1837}, {1107, 24555}, {1482, 17675}, {1699, 26839}, {3177, 9436}, {3662, 17435}, {3673, 4904}, {4534, 9311}, {5086, 24596}, {5179, 17753}, {6554, 6604}, {6999, 24590}, {7179, 17451}, {7190, 27547}, {7991, 26790}, {8256, 16593}, {10481, 20089}, {10950, 26007}, {11101, 24619}, {11109, 14621}, {13567, 26558}, {15888, 27475}, {17086, 18634}, {17121, 26668}, {17194, 26804}, {17233, 25067}, {17242, 26669}, {17247, 25238}, {17248, 24554}, {17364, 26651}, {18928, 27064}, {19786, 26958}, {20262, 26125}, {23893, 26985}, {26527, 26565}, {26533, 26611}, {26541, 26572}, {26567, 26574}

X(26532) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26532) lies on these lines: {1, 2}, {355, 17683}, {409, 24619}, {894, 26793}, {1837, 24596}, {2646, 24582}, {5046, 27000}, {5154, 27183}, {17050, 21044}, {17062, 21921}, {17184, 25977}, {17862, 26592}, {20905, 26541}, {21258, 26563}, {24993, 25964}, {26533, 26587}, {26550, 26565}

X(26533) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26533) lies on these lines: {2, 11}, {1146, 20940}, {26526, 26529}, {26528, 26561}, {26531, 26611}, {26532, 26587}

X(26534) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26534) lies on these lines: {2, 3}

X(26535) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26535) lies on these lines: {2, 3}

X(26536) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26536) lies on these lines: {2, 31}, {21912, 27149}, {26526, 26542}, {26530, 26559}, {26560, 26565}

X(26537) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26537) lies on these lines: {2, 32}, {26527, 26530}, {26541, 26557}, {26542, 26633}, {26564, 26569}

X(26538) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26538) lies on these lines: {2, 37}, {10, 20633}, {86, 26639}, {141, 3262}, {239, 15988}, {322, 3620}, {594, 26594}, {693, 24098}, {726, 25024}, {1086, 18179}, {1125, 1733}, {1441, 3662}, {1738, 4642}, {2550, 14923}, {3218, 11683}, {3616, 4008}, {3661, 20895}, {3663, 20236}, {3821, 23690}, {4357, 4858}, {4419, 20927}, {4872, 26837}, {4967, 25007}, {5294, 20879}, {10030, 26806}, {16725, 16738}, {16732, 17235}, {16817, 25906}, {17236, 26563}, {17258, 18151}, {17277, 26699}, {17304, 17861}, {17355, 20881}, {18252, 20556}, {18698, 24199}, {20172, 26621}, {20432, 25997}, {20911, 21442}, {21020, 24997}, {24342, 24563}, {25023, 26001}, {25082, 25601}, {25964, 26570}, {26526, 26530}, {26539, 26548}, {26581, 26582}

X(26358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11248, 1470), (1482, 8069, 26437), (3295, 10679, 1)

X(26539) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26539) lies on these lines: {2, 39}, {7187, 23989}, {26527, 26530}, {26538, 26548}, {26557, 26564}

X(26540) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26540) lies on these lines: {2, 6}, {7, 281}, {10, 18412}, {77, 18634}, {105, 12589}, {142, 26001}, {189, 6355}, {307, 24635}, {314, 26607}, {315, 26678}, {320, 26651}, {451, 5707}, {857, 10446}, {948, 5942}, {1352, 4223}, {1442, 17073}, {1486, 21293}, {1861, 10394}, {1899, 4224}, {3240, 25882}, {3661, 25001}, {3662, 17435}, {3823, 25005}, {3879, 26006}, {3912, 25019}, {4228, 11442}, {4357, 24554}, {4466, 18161}, {5142, 18180}, {5273, 26942}, {6740, 17579}, {6824, 12359}, {6833, 26879}, {7671, 24388}, {10449, 25017}, {15466, 17862}, {16696, 26636}, {16948, 24538}, {17074, 20266}, {17126, 25968}, {17139, 18747}, {17170, 18636}, {17231, 25067}, {17233, 25243}, {17287, 25584}, {17296, 25930}, {20262, 21617}, {21911, 24430}, {21931, 24341}, {23291, 26118}, {26528, 26541}, {26555, 26564}

X(26541) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26541) lies on these lines: {2, 39}, {75, 24999}, {85, 18359}, {86, 311}, {264, 3945}, {313, 24993}, {321, 26581}, {338, 17392}, {343, 1231}, {1226, 17863}, {1232, 5224}, {1235, 11109}, {1269, 24547}, {2995, 10401}, {3006, 20436}, {3260, 17378}, {3673, 24984}, {3760, 19861}, {3761, 19860}, {3936, 21596}, {5741, 21581}, {6376, 25005}, {6381, 24982}, {18133, 24986}, {18147, 24540}, {20888, 24987}, {20905, 26532}, {22028, 26595}, {26527, 26564}, {26528, 26540}, {26531, 26572}, {26537, 26557}

X(26542) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26542) lies on these lines: {2, 6}, {858, 3794}, {1086, 20886}, {3662, 26552}, {11442, 25494}, {16067, 23293}, {17184, 26932}, {26526, 26536}, {26537, 26633}, {26559, 26565}

X(26543) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26543) lies on these lines: {2, 6}, {7, 11683}, {21, 1503}, {85, 257}, {142, 16609}, {182, 7483}, {189, 18632}, {274, 6393}, {286, 297}, {287, 25536}, {377, 1350}, {405, 1352}, {440, 17185}, {441, 2193}, {442, 511}, {443, 10519}, {518, 8261}, {542, 15670}, {594, 26665}, {611, 10198}, {613, 26363}, {857, 17183}, {958, 12588}, {1001, 12589}, {1086, 18179}, {1386, 24541}, {1428, 4999}, {1469, 25466}, {1723, 17272}, {1762, 7289}, {1843, 25985}, {1901, 17139}, {2330, 6690}, {2476, 5480}, {2478, 10516}, {2781, 12826}, {2886, 3056}, {3002, 16887}, {3098, 11112}, {3416, 19860}, {3434, 10387}, {3564, 6675}, {3818, 11113}, {3844, 24982}, {3912, 25099}, {3925, 17792}, {3932, 25024}, {3943, 25245}, {4187, 24206}, {4188, 21167}, {4357, 15595}, {4437, 25001}, {4904, 24199}, {5085, 6910}, {5249, 24471}, {5398, 17698}, {5721, 16062}, {5723, 17291}, {5724, 7270}, {5798, 10446}, {5830, 17289}, {5831, 17308}, {5921, 17558}, {6776, 6857}, {6856, 14853}, {9015, 26641}, {10436, 16608}, {11180, 17561}, {11645, 17525}, {14927, 17576}, {15812, 25907}, {16418, 18440}, {16603, 20258}, {17045, 26639}, {17052, 17197}, {17202, 26601}, {17237, 25887}, {17239, 25007}, {17298, 17739}, {17322, 24559}, {17332, 26699}, {17530, 19130}, {18750, 27184}, {20905, 26526}, {23983, 26165}, {26547, 26548}, {26554, 26563}

X(26544) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26544) lies on these lines: {1, 2}, {76, 26572}, {5330, 17675}, {6604, 26793}, {26561, 26565}

X(26545) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26545) lies on these lines: {2, 661}, {297, 525}, {513, 25981}, {693, 26596}, {4077, 27184}, {9013, 25898}, {17420, 17494}, {18199, 26625}, {25009, 26571}

X(26546) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 45

Barycentrics    b c (b - c) (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(26546) lies on these lines: {2, 650}, {297, 525}, {377, 8760}, {812, 26017}, {1577, 25007}, {1738, 23793}, {2517, 24990}, {2788, 8642}, {3434, 11934}, {4077, 4468}, {4379, 25955}, {4382, 25924}, {23989, 26565}

X(26547) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26547) lies on these lines: {2, 31}, {24993, 25964}, {26526, 26530}, {26543, 26548}

X(26548) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26548) lies on these lines: {1, 2}, {517, 17672}, {1573, 25888}, {16912, 24809}, {17184, 26561}, {17236, 20089}, {24547, 26574}, {26538, 26539}, {26543, 26547}

X(26549) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26549) lies on these lines: {2, 3}, {3662, 26526}

X(26550) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26550) lies on these lines: {2, 3}, {85, 257}, {23536, 25935}, {23661, 26153}, {25000, 26035}, {26526, 26536}, {26532, 26565}

X(26551) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26551) lies on these lines: {2, 3}

X(26552) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26552) lies on these lines: {2, 3}, {3662, 26542}

X(26553) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26553) lies on these lines: {2, 3}

X(26554) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26554) lies on these lines: {2, 3}, {26543, 26563}

X(26555) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26555) lies on these lines: {2, 3}, {26540, 26564}

X(26556) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26556) lies on these lines: {2, 3}, {318, 26158}, {3662, 17435}, {26590, 26658}

X(26557) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26557) lies on these lines: {2, 3}, {26530, 26569}, {26537, 26541}, {26539, 26564}

X(26558) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 45

Barycentrics    a^2 b^2 + b^4 + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + c^4 : :

X(26558) lies on these lines: {2, 12}, {4, 20172}, {5, 17030}, {8, 17550}, {10, 6656}, {11, 17669}, {21, 26629}, {36, 17694}, {75, 5254}, {116, 16887}, {141, 6376}, {239, 1834}, {257, 1146}, {325, 1107}, {341, 3661}, {442, 16819}, {495, 27255}, {626, 1573}, {993, 7807}, {1211, 3975}, {1376, 7791}, {1478, 11321}, {1574, 4045}, {1698, 17670}, {2886, 5025}, {3035, 7824}, {3662, 21258}, {3691, 24995}, {3704, 3797}, {3820, 8362}, {4187, 26959}, {4357, 17062}, {4366, 17685}, {4386, 7750}, {4426, 7792}, {4766, 10459}, {5051, 26965}, {5080, 17686}, {5432, 17684}, {6292, 27076}, {6554, 17257}, {7354, 16915}, {7745, 20179}, {7866, 9708}, {7876, 9711}, {7887, 26363}, {7933, 9710}, {8356, 25440}, {9709, 11287}, {11285, 26364}, {13567, 26531}, {14064, 19843}, {15326, 17693}, {16062, 27299}, {16829, 24390}, {16910, 24596}, {17045, 23905}, {17184, 25977}, {17671, 27248}, {17672, 27026}, {17757, 27020}, {21031, 26752}, {21485, 22654}, {21935, 24592}, {26526, 26529}, {26576, 26621}, {26804, 27149}

X(26559) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26559) lies on these lines: {1, 2}, {26530, 26536}, {26542, 26565}

X(26560) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26560) lies on these lines: {1, 2}, {17046, 22173}, {26536, 26565}

X(26561) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26561) lies on these lines: {1, 6656}, {2, 12}, {3, 26629}, {5, 26959}, {8, 26582}, {10, 17670}, {11, 5025}, {34, 297}, {35, 8356}, {36, 7807}, {55, 7791}, {85, 257}, {141, 1909}, {172, 7792}, {192, 7864}, {239, 7270}, {315, 16502}, {325, 2275}, {330, 3314}, {334, 20255}, {350, 5254}, {377, 20172}, {384, 7354}, {442, 17030}, {458, 11392}, {495, 8362}, {498, 11285}, {499, 7887}, {626, 1015}, {673, 17680}, {894, 7247}, {948, 26132}, {999, 7866}, {1003, 4299}, {1201, 4766}, {1475, 24995}, {1478, 7770}, {1479, 7841}, {1500, 4045}, {1834, 17027}, {1914, 7750}, {1975, 9597}, {2241, 7761}, {2242, 7834}, {2886, 26801}, {3058, 7924}, {3085, 16043}, {3086, 14064}, {3295, 11287}, {3552, 15326}, {3585, 8370}, {3614, 16921}, {3616, 17550}, {3665, 7187}, {3734, 9651}, {3782, 17789}, {3816, 17669}, {4202, 26965}, {4293, 14001}, {4324, 8353}, {4366, 6284}, {4904, 24190}, {5080, 17541}, {5204, 16925}, {5299, 7762}, {5432, 7824}, {5563, 8363}, {5716, 26626}, {6690, 17684}, {7179, 25918}, {7773, 9599}, {7784, 16781}, {7808, 9650}, {7819, 18990}, {7825, 9665}, {7833, 15338}, {7872, 9664}, {7876, 15888}, {8352, 18514}, {8357, 15171}, {8361, 15325}, {8728, 16819}, {9596, 11174}, {9655, 11286}, {9657, 16898}, {10350, 12835}, {10352, 12184}, {10483, 19687}, {10591, 16041}, {10895, 16924}, {10896, 14063}, {12607, 26752}, {12943, 14035}, {15048, 25264}, {17144, 21956}, {17184, 26548}, {17366, 24366}, {17397, 26601}, {17448, 20541}, {17757, 27091}, {17798, 21993}, {26279, 27068}, {26526, 26527}, {26528, 26533}, {26544, 26565}, {26578, 26621}, {26802, 26977}

X(26562) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26562) lies on these lines: {1, 24602}, {2, 65}, {116, 17211}, {141, 21951}, {321, 22202}, {335, 4696}, {517, 27097}, {942, 26965}, {1837, 16910}, {3125, 20911}, {3263, 3721}, {3290, 17152}, {3662, 17435}, {3701, 24080}, {3742, 26807}, {3868, 27299}, {3912, 4642}, {3924, 24586}, {4357, 21921}, {5086, 17680}, {5836, 26759}, {5883, 16818}, {16583, 17137}, {16720, 21331}, {17184, 25977}, {18180, 27185}, {18191, 26841}, {20271, 26234}, {20347, 25994}, {20880, 24190}, {21281, 26242}, {26526, 26527}

X(26563) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26563) lies on these lines: {2, 85}, {7, 3436}, {8, 3673}, {10, 1111}, {37, 20448}, {38, 20436}, {41, 24249}, {56, 26229}, {57, 24612}, {65, 20347}, {69, 5016}, {75, 3617}, {76, 321}, {77, 24540}, {88, 274}, {141, 1229}, {142, 21921}, {145, 16284}, {220, 26653}, {244, 24215}, {257, 18031}, {304, 4358}, {307, 24986}, {322, 3672}, {333, 16749}, {343, 1231}, {349, 23989}, {350, 20955}, {354, 21967}, {404, 5088}, {498, 25581}, {518, 20247}, {519, 7264}, {529, 7198}, {551, 7278}, {693, 21132}, {908, 3674}, {984, 20435}, {986, 21422}, {1086, 21951}, {1146, 26526}, {1211, 1233}, {1329, 3665}, {1334, 21232}, {1434, 27003}, {1441, 4357}, {1447, 2975}, {1475, 17048}, {1565, 4187}, {1828, 17183}, {1837, 21285}, {1909, 5484}, {1921, 20892}, {1930, 3701}, {2329, 9318}, {2478, 17170}, {2551, 7195}, {3057, 21272}, {3212, 3869}, {3262, 4389}, {3263, 6376}, {3452, 24994}, {3619, 20927}, {3620, 20171}, {3621, 17158}, {3634, 25585}, {3662, 17435}, {3663, 4642}, {3666, 21596}, {3693, 25237}, {3702, 3760}, {3721, 21138}, {3732, 17681}, {3739, 27170}, {3752, 18600}, {3761, 4968}, {3765, 4359}, {3812, 4059}, {3953, 21208}, {4193, 17181}, {4202, 20235}, {4352, 4850}, {4391, 17192}, {4487, 25278}, {4513, 24352}, {4515, 26757}, {4721, 24254}, {4861, 24203}, {4872, 5046}, {4911, 5080}, {5253, 7176}, {5439, 17169}, {5554, 6604}, {5836, 20244}, {6646, 10030}, {6691, 7181}, {7112, 17291}, {7179, 11681}, {7200, 16604}, {7223, 25524}, {7247, 20060}, {8582, 10481}, {9312, 19861}, {9436, 24982}, {10521, 12527}, {10587, 17321}, {14552, 19788}, {16583, 26978}, {16609, 24633}, {16611, 24790}, {16732, 17237}, {17044, 26660}, {17046, 21044}, {17233, 22040}, {17236, 26538}, {17257, 25001}, {17266, 18140}, {17272, 17861}, {17320, 17791}, {17448, 27918}, {17451, 20335}, {18133, 20336}, {18743, 21605}, {20245, 24471}, {20891, 21615}, {21258, 26532}, {21609, 26132}, {24214, 24443}, {24268, 25940}, {24326, 25102}, {25244, 27025}, {26167, 26171}, {26543, 26554}, {26669, 27282}, {27813, 27814}

X(26564) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26564) lies on these lines: {2, 99}, {1086, 26566}, {26527, 26541}, {26537, 26569}, {26539, 26557}, {26540, 26555}, {26565, 26572}

X(26565) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26565) lies on these lines: {2, 11}, {1086, 26568}, {4904, 26566}, {14936, 26641}, {23989, 26546}, {26527, 26531}, {26532, 26550}, {26536, 26560}, {26542, 26559}, {26544, 26561}, {26564, 26572}

X(26566) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26566) lies on these lines: {2, 101}, {1086, 26564}, {4904, 26565}

X(26567) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26567) lies on these lines: {2, 37}, {1086, 26530}, {3262, 17232}, {3662, 4858}, {4361, 26657}, {4440, 20927}, {7336, 17047}, {17230, 20895}, {17276, 18151}, {17339, 20881}, {26526, 26569}, {26531, 26574}

X(26568) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26568) lies on these lines: {2, 900}, {1086, 26565}, {4435, 26657}

X(26569) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26569) lies on these lines: {2, 39}, {26526, 26567}, {26530, 26557}, {26537, 26564}

X(26570) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26570) lies on these lines: {1, 2}, {524, 26674}, {3762, 26571}, {20905, 26530}, {25964, 26538}

X(26571) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26571) lies on these lines: {2, 649}, {3662, 4468}, {3762, 26570}, {4521, 27184}, {25009, 26545}

X(26572) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26572) lies on these lines: {2, 668}, {6, 26693}, {76, 26544}, {693, 4534}, {1146, 23989}, {1358, 4462}, {4366, 26691}, {4391, 4904}, {26531, 26541}, {26564, 26565}

X(26573) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26573) lies on these lines: {2, 7}, {141, 26594}, {320, 15988}, {335, 26581}, {1086, 18179}, {3663, 25241}, {3834, 25099}, {3836, 25024}, {3912, 25245}, {4201, 18444}, {14621, 26628}, {17273, 26671}, {17302, 26639}, {17324, 24559}, {20905, 26530}, {24231, 24987}, {26526, 26567}

X(26574) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 45

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

X(26574) lies on these lines: {2, 38}, {20905, 26532}, {24547, 26548}, {26526, 26530}, {26531, 26567}

leftri

Collineation mappings involving Gemini triangle 46: X(26575)-X(26612)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 46, as in centers X(26575)-X(26612). Then

m(X) = (a + b - c) (a - b + c) (b + c)^2 x + b^2 (b + c - a) (a + b - c) y + c^2 (b + c - a) (a - b + c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 1, 2018)

X(26575) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26575) lies on these lines: {1, 2}, {37, 24986}, {141, 24993}, {594, 24547}, {1229, 21933}, {2171, 21244}, {2285, 21286}, {3553, 27507}, {4364, 24998}, {4437, 20905}, {4851, 24540}, {17237, 24999}, {20262, 27058}, {25000, 25099}, {25023, 25243}, {26576, 26590}, {26578, 26582}, {26579, 26580}, {26585, 26601}, {26793, 27064}

X(26576) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26576) lies on these lines: {2, 3}, {26558, 26621}, {26575, 26590}, {26579, 26586}

X(26577) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26577) lies on these lines: {2, 3}, {26580, 26581}

X(26578) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26578) lies on these lines: {2, 3}, {17184, 26581}, {26561, 26621}, {26575, 26582}

X(26579) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26579) lies on these lines: {2, 6}, {12, 25005}, {42, 24991}, {511, 16067}, {908, 21244}, {1853, 26032}, {2887, 24996}, {4671, 26610}, {26575, 26580}, {26576, 26586}, {26585, 26590}, {26591, 26594}, {26611, 26612}, {26942, 27184}

X(26580) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26580) lies on these lines: {2, 7}, {10, 3120}, {31, 4703}, {37, 3936}, {38, 3846}, {42, 4425}, {72, 5051}, {73, 5484}, {78, 17676}, {100, 24723}, {141, 4358}, {145, 4101}, {171, 4683}, {210, 4972}, {238, 26230}, {257, 18359}, {306, 3950}, {312, 17228}, {313, 321}, {349, 23989}, {612, 6327}, {651, 26637}, {748, 26128}, {750, 4655}, {756, 2887}, {899, 3821}, {982, 25960}, {984, 3006}, {1086, 5241}, {1125, 19740}, {1150, 4396}, {1999, 2895}, {2886, 4981}, {3124, 3721}, {3175, 3969}, {3187, 5739}, {3622, 16485}, {3661, 4044}, {3663, 17495}, {3666, 5741}, {3687, 17147}, {3696, 4442}, {3705, 7226}, {3755, 19998}, {3772, 5278}, {3773, 3994}, {3782, 4359}, {3842, 4892}, {3844, 4009}, {3876, 16062}, {3883, 20045}, {3891, 3966}, {3896, 4819}, {3914, 4104}, {3920, 4388}, {3971, 15523}, {3993, 4062}, {4011, 24943}, {4028, 27804}, {4052, 6539}, {4085, 21805}, {4135, 6535}, {4199, 21319}, {4202, 5044}, {4292, 19284}, {4364, 5718}, {4365, 21085}, {4389, 4850}, {4402, 19824}, {4407, 21242}, {4416, 16704}, {4419, 17740}, {4424, 21042}, {4645, 5297}, {4667, 26860}, {5057, 5263}, {5222, 19823}, {5235, 17256}, {5269, 20064}, {5719, 13745}, {8620, 9284}, {9330, 25959}, {11263, 16828}, {11319, 12572}, {11374, 16342}, {12609, 19874}, {13411, 16347}, {14554, 26844}, {14555, 19785}, {14996, 17364}, {14997, 17367}, {15254, 24542}, {16610, 17235}, {16738, 17174}, {16887, 17198}, {17012, 17302}, {17013, 17396}, {17019, 17778}, {17021, 17300}, {17135, 24210}, {17182, 27163}, {17230, 21071}, {17255, 17595}, {17719, 24697}, {17889, 26037}, {18249, 25982}, {18250, 25904}, {18541, 19290}, {20234, 21810}, {20905, 26005}, {21077, 26115}, {22020, 27041}, {24217, 25378}, {24441, 27739}, {24552, 24703}, {25000, 26011}, {26526, 26529}, {26575, 26579}, {26577, 26581}, {26589, 26601}, {26590, 26593}, {26594, 26612}, {26609, 26942}, {27493, 27495}

X(26581) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26581) lies on these lines: {1, 2}, {9, 21286}, {141, 24547}, {321, 26541}, {335, 26573}, {536, 24999}, {594, 24993}, {1319, 24583}, {1332, 17289}, {3877, 7377}, {4357, 4552}, {4437, 25001}, {5252, 24612}, {17184, 26578}, {17239, 24986}, {17303, 24540}, {26538, 26582}, {26577, 26580}, {26591, 26592}

X(26582) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26582) lies on these lines: {1, 17670}, {2, 11}, {4, 26687}, {5, 27091}, {8, 26561}, {10, 6656}, {12, 26752}, {37, 25357}, {75, 141}, {115, 27076}, {157, 11329}, {190, 2345}, {239, 5846}, {297, 1861}, {325, 1575}, {350, 21956}, {404, 26686}, {442, 27020}, {537, 3775}, {545, 17254}, {626, 1574}, {740, 1738}, {812, 21261}, {857, 27047}, {894, 5845}, {899, 4766}, {900, 19964}, {958, 7791}, {993, 8356}, {1111, 20431}, {1211, 18037}, {1213, 4422}, {1329, 5025}, {1573, 4045}, {3008, 17766}, {3136, 27035}, {3589, 20179}, {3763, 20181}, {3797, 3932}, {4000, 4360}, {4085, 17023}, {4386, 7792}, {4426, 7750}, {4440, 17238}, {4966, 6542}, {4971, 17310}, {4999, 7824}, {5051, 27026}, {5254, 6376}, {5819, 26685}, {6284, 16916}, {6645, 17565}, {7807, 25440}, {7866, 9709}, {7876, 9710}, {7887, 26364}, {7933, 9711}, {8362, 17030}, {8728, 27255}, {9708, 11287}, {9780, 17550}, {11285, 26363}, {11349, 27323}, {15338, 17692}, {16043, 19843}, {16706, 16826}, {17303, 24358}, {17308, 17738}, {17446, 21035}, {17672, 26965}, {17674, 27097}, {17684, 24953}, {18082, 18095}, {20356, 25748}, {23891, 24222}, {24390, 26959}, {26527, 26653}, {26538, 26581}, {26575, 26578}, {26605, 27059}, {26772, 27058}

X(26582) = complement of X(4366)

X(26583) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26583) lies on these lines: {2, 3}

X(26584) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26584) lies on these lines: {2, 3}

X(26585) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26585) lies on these lines: {2, 31}, {213, 24992}, {5025, 25005}, {17555, 26653}, {26575, 26601}, {26579, 26590}, {26586, 26589}

X(26586) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26586) lies on these lines: {2, 32}, {26576, 26579}, {26585, 26589}, {26592, 26608}

X(26587) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26587) lies on these lines: {2, 37}, {329, 5933}, {756, 24996}, {908, 2171}, {1220, 1411}, {2292, 24982}, {2551, 3869}, {3124, 26611}, {3262, 5718}, {3816, 21333}, {4415, 18179}, {4425, 24991}, {5311, 24545}, {5712, 20928}, {23690, 25385}, {25024, 26013}, {26532, 26533}, {26575, 26579}, {26588, 26599}, {26590, 26602}

X(26588) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26588) lies on these lines: {2, 39}, {1232, 26979}, {26576, 26579}, {26587, 26599}

X(26589) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26589) lies on these lines: {2, 41}, {6, 26176}, {872, 21235}, {1211, 21025}, {4805, 24587}, {17671, 26688}, {20305, 26772}, {21244, 26756}, {26580, 26601}, {26585, 26586}, {26595, 26602}

X(26590) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 46

Barycentrics    a^2 b^2 + b^4 + 2 a^2 b c + a^2 c^2 + c^4 : :

X(26590) lies on these lines: {1, 6656}, {2, 11}, {3, 26686}, {5, 27020}, {8, 17550}, {12, 5025}, {33, 297}, {35, 7807}, {36, 8356}, {37, 20541}, {42, 4766}, {56, 7791}, {75, 21956}, {141, 350}, {172, 7750}, {192, 3314}, {239, 4514}, {257, 312}, {321, 8024}, {325, 2276}, {330, 7864}, {335, 3782}, {384, 6284}, {442, 27255}, {458, 11393}, {496, 8362}, {498, 7887}, {499, 11285}, {626, 1500}, {894, 4872}, {999, 11287}, {1003, 4302}, {1015, 4045}, {1125, 17670}, {1329, 17669}, {1334, 24995}, {1478, 7841}, {1479, 7770}, {1834, 17033}, {1909, 5254}, {1914, 7792}, {1975, 9598}, {2241, 7834}, {2242, 7761}, {2478, 26687}, {2887, 3912}, {3085, 14064}, {3086, 16043}, {3295, 7866}, {3552, 15338}, {3583, 8370}, {3703, 3797}, {3734, 9664}, {3746, 8363}, {3813, 26801}, {3933, 25264}, {3970, 17211}, {4187, 27091}, {4202, 27097}, {4294, 14001}, {4316, 8353}, {4415, 4437}, {4660, 24586}, {4999, 17684}, {5217, 16925}, {5276, 20553}, {5280, 7762}, {5433, 7824}, {5434, 7924}, {6645, 6655}, {7173, 16921}, {7264, 17192}, {7773, 9596}, {7803, 16502}, {7808, 9665}, {7819, 15171}, {7825, 9650}, {7833, 15326}, {7872, 9651}, {7933, 15888}, {8352, 18513}, {8357, 18990}, {8359, 15325}, {8364, 15172}, {9599, 11174}, {9668, 11286}, {9670, 16898}, {10350, 10799}, {10352, 12185}, {10483, 19695}, {10590, 16041}, {10895, 14063}, {10896, 16924}, {11343, 27309}, {12953, 14035}, {13728, 27274}, {16062, 27248}, {16826, 19786}, {17030, 24390}, {17032, 17056}, {17316, 18134}, {17671, 27299}, {17694, 25440}, {17747, 24514}, {20173, 27184}, {22370, 25978}, {24424, 25364}, {26529, 26653}, {26556, 26658}, {26575, 26576}, {26579, 26585}, {26580, 26593}, {26587, 26602}, {26597, 26598}

X(26591) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26591) lies on these lines: {2, 37}, {10, 23528}, {92, 18228}, {349, 23989}, {394, 26223}, {594, 26005}, {908, 1441}, {936, 23661}, {1211, 26603}, {1215, 25941}, {1265, 4696}, {1698, 17869}, {3187, 10601}, {3262, 5233}, {3452, 6358}, {3661, 26607}, {3701, 24987}, {3702, 19860}, {3980, 25938}, {4011, 25885}, {4054, 20880}, {4363, 25934}, {4647, 8582}, {4656, 24213}, {4858, 5316}, {4968, 19861}, {8580, 17860}, {14555, 20928}, {19875, 23580}, {21438, 26695}, {26579, 26594}, {26581, 26592}

X(26592) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26592) lies on these lines: {2, 39}, {75, 23978}, {264, 391}, {311, 17277}, {313, 25001}, {321, 13567}, {338, 17330}, {1232, 17234}, {1235, 26003}, {1269, 20905}, {3260, 17346}, {3761, 25930}, {3963, 25243}, {4044, 25935}, {4385, 25017}, {17862, 26532}, {20888, 26001}, {26581, 26591}, {26586, 26608}

X(26593) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26593) lies on these lines: {1, 2}, {321, 1233}, {3219, 20533}, {3555, 17672}, {3693, 17229}, {3773, 4712}, {10025, 17287}, {14828, 17295}, {17231, 20483}, {21096, 25261}, {26580, 26590}

X(26594) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26594) lies on these lines: {1, 2}, {104, 21495}, {141, 26573}, {594, 26538}, {740, 25010}, {1577, 26596}, {3775, 25024}, {4357, 25245}, {15988, 17289}, {17239, 25099}, {17280, 26699}, {17285, 26671}, {26579, 26591}, {26580, 26612}

X(26595) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26595) lies on these lines: {1, 2}, {22028, 26541}, {26579, 26585}, {26589, 26602}

X(26596) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26596) lies on these lines: {2, 649}, {513, 26640}, {693, 26545}, {1577, 26594}, {3676, 27184}, {4106, 25981}, {4728, 25008}, {4776, 25902}

X(26597) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26597) lies on these lines: {1, 2}, {26590, 26598}

X(26598) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26598) lies on these lines: {2, 31}, {26575, 26579}, {26590, 26597}

X(26599) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26599) lies on these lines: {1, 2}, {1086, 24993}, {4665, 24547}, {4708, 24986}, {25004, 25099}, {26587, 26588}

X(26600) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26600) lies on these lines: {2, 3}

X(26601) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26601) lies on these lines: {2, 3}, {37, 4150}, {115, 5977}, {141, 18147}, {239, 1834}, {257, 312}, {321, 1228}, {894, 1901}, {1213, 4422}, {1441, 8736}, {2303, 21287}, {3454, 3912}, {3662, 18635}, {3936, 17316}, {4357, 17052}, {16826, 17056}, {17202, 26543}, {17397, 26561}, {18091, 18703}, {18096, 27067}, {18139, 26100}, {23978, 26165}, {26575, 26585}, {26580, 26589}, {27042, 27254}

X(26602) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26602) lies on these lines: {2, 3}, {26587, 26590}, {26589, 26595}

X(26603) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26603) lies on these lines: {2, 3}, {1211, 26591}, {21245, 25091}

X(26604) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 46

Barycentrics    (b + c) (a^9 b + a^8 b^2 - 2 a^5 b^5 - 2 a^4 b^6 + a b^9 + b^10 + a^9 c + 2 a^8 b c + 2 a^7 b^2 c - 2 a^6 b^3 c - 4 a^5 b^4 c - 2 a^4 b^5 c - 2 a^3 b^6 c + 2 a^2 b^7 c + 3 a b^8 c + a^8 c^2 + 2 a^7 b c^2 + 2 a^6 b^2 c^2 + 2 a^5 b^3 c^2 - 2 a^3 b^5 c^2 - 2 a^2 b^6 c^2 - 2 a b^7 c^2 - b^8 c^2 - 2 a^6 b c^3 + 2 a^5 b^2 c^3 + 12 a^4 b^3 c^3 + 8 a^3 b^4 c^3 - 2 a^2 b^5 c^3 - 2 a b^6 c^3 - 4 a^5 b c^4 + 8 a^3 b^3 c^4 + 4 a^2 b^4 c^4 - 2 a^5 c^5 - 2 a^4 b c^5 - 2 a^3 b^2 c^5 - 2 a^2 b^3 c^5 - 2 a^4 c^6 - 2 a^3 b c^6 - 2 a^2 b^2 c^6 - 2 a b^3 c^6 + 2 a^2 b c^7 - 2 a b^2 c^7 + 3 a b c^8 - b^2 c^8 + a c^9 + c^10) : :

X(26604) lies on these lines: {2, 3}

X(26605) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26605) lies on these lines: {2, 3}, {286, 26167}, {307, 17052}, {321, 349}, {948, 3936}, {1086, 17863}, {1441, 18642}, {1726, 16549}, {2287, 21287}, {2997, 16608}, {26582, 27059}

X(26606) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26606) lies on these lines: {2, 3}

X(26607) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26607) lies on these lines: {2, 3}, {314, 26540}, {1446, 27184}, {3661, 26591}, {4766, 25930}, {24210, 25935}

X(26608) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26608) lies on these lines: {2, 3}, {26586, 26592}

X(26609) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26609) lies on these lines: {2, 6}, {442, 25005}, {3060, 16067}, {3454, 24982}, {5051, 5554}, {26575, 26585}, {26580, 26942}

X(26610) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26610) lies on these lines: {1, 2}, {4664, 24998}, {4671, 26579}, {17228, 24993}, {17233, 24986}, {17295, 24540}

X(26611) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26611) lies on these lines: {2, 45}, {6, 2990}, {9, 2006}, {11, 24433}, {220, 5723}, {226, 16578}, {312, 343}, {321, 23978}, {329, 394}, {338, 1211}, {349, 23989}, {726, 26010}, {867, 24828}, {908, 1465}, {1146, 18359}, {1331, 15252}, {1407, 5905}, {3124, 26587}, {3326, 15632}, {3952, 23541}, {4671, 23970}, {26531, 26533}, {26579, 26612}

X(26612) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 46

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

X(26612) lies on these lines: {2, 37}, {149, 1837}, {3971, 24997}, {4642, 25005}, {17743, 18359}, {17869, 26029}, {25934, 26659}, {26579, 26611}, {26580, 26594}

X(26613) =  MIDPOINT OF X(187) AND X(5215)

Barycentrics    7 a^4-4 a^2 b^2+b^4-4 a^2 c^2-b^2 c^2+c^4 : :
X(26613) = X[2] + 2 X[187], 4 X[2] - X[316], 8 X[187] + X[316], 7 X[316] - 16 X[625], 7 X[2] - 4 X[625], 7 X[187] + 2 X[625], 4 X[230] - X[671], X[1992] - 4 X[2030], 2 X[549] + X[2080], X[385] + 2 X[2482], 2 X[597] + X[5104], 2 X[551] + X[5184], X[316] - 8 X[5215], 2 X[625] - 7 X[5215], 2 X[5461] + X[6781], X[691] + 2 X[7426], 4 X[2021] - X[7757], 4 X[620] - X[7840], 2 X[395] + X[8594], 2 X[396] + X[8595], 4 X[5461] - X[8597], 2 X[6781] + X[8597], 2 X[230] + X[8598], X[671] + 2 X[8598], X[843] + 2 X[9127], 2 X[115] + X[9855], 2 X[8997] + X[9893], 5 X[316] - 16 X[10150], 5 X[625] - 7 X[10150], 5 X[2] - 4 X[10150], 5 X[5215] - 2 X[10150], 5 X[187] + 2 X[10150], 2 X[99] + X[11054], 2 X[3111] + X[11673], 2 X[6055] + X[11676], 5 X[5071] - 2 X[13449], X[13677] + 2 X[13908], X[9891] + 2 X[13989], 2 X[8352] - 5 X[14061], 5 X[2482] - 2 X[14148], 5 X[385] + 4 X[14148], 2 X[13586] + X[14568], X[381] - 4 X[14693], 10 X[187] - X[14712], 5 X[2] + X[14712], 10 X[5215] + X[14712], 4 X[10150] + X[14712], 5 X[316] + 4 X[14712], 2 X[9181] + X[15360], X[9301] + 5 X[15693], X[8593] + 2 X[15993], X[842] - 4 X[18579], 5 X[15692] - 2 X[18860], 5 X[7925] - 8 X[22247], X[11054] - 4 X[22329], X[99] + 2 X[22329], 2 X[39] + X[22564]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28568.

Let AB, AC, BC, BA, CA, CB be the points on the Dao 6-point circle as defined at X(5569). Triangles BACBAC and CAABBC are perspective at X(2), and X(26613) lies on their perspectrix, with X(8704). (Randy Hutson, August 11, 2020)

The trilinear polar of X(26613) passes through X(9123).

X(26613) lies on these lines: {2,187}, {3,7827}, {30,9166}, {32,7622}, {39,22564}, {99,9136}, {115,9855}, {230,671}, {249,524}, {381,14693}, {385,2482}, {395,8594}, {396,8595}, {511,3524}, {512,15724}, {530,16267}, {531,16268}, {542,21445}, {543,5152}, {549,2080}, {551,5184}, {597,5104}, {599,7835}, {620,7840}, {691,7426}, {754,9167}, {842,18579}, {843,9127}, {1003,7610}, {1078,8369}, {1384,11163}, {1692,5032}, {1992,2030}, {2021,7618}, {3053,7769}, {3096,8366}, {3111,11673}, {3523,7878}, {3788,9939}, {5023,7841}, {5071,13449}, {5077,15655}, {5206,7828}, {5210,7790}, {5461,6781}, {6055,11676}, {7617,11361}, {7619,7753}, {7768,7870}, {7775,7907}, {7793,7801}, {7802,11318}, {7806,8588}, {7807,7883}, {7810,7832}, {7811,11288}, {7817,7847}, {7859,8359}, {7925,22247}, {7944,8365}, {8352,14061}, {8370,15597}, {8553,21395}, {8593,15993}, {8860,11159}, {8997,9893}, {9181,15360}, {9301,15693}, {9741,11055}, {9761,19781}, {9763,19780}, {9891,13989}, {11165,14614}, {11185,23055}, {13677,13908}, {14041,14971}, {15692,18860}

X(26613) = midpoint of X(i) and X(j) for these {i,j}: {187, 5215}, {8859, 13586}
X(26613) = reflection of X(i) in X(j) for these {i,j}: {2, 5215}, {5032, 1692}, {14041, 14971}, {14568, 8859}
X(26613) = X(661)-isoconjugate of X(9124)
X(26613) = crossdifference of every pair of points on line {17414, 22260}
X(26613) = centroid of X(2)X(15)X(16)
X(26613) = centroid of X(2)PU(2)
X(26613) = Dao-6-point-circle-inverse of X(2)
X(26613) = barycentric product X(99)X(9123)
X(26613) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 9124}, {9123, 523}
X(26613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 22329, 11054), (230, 8598, 671), (5461, 6781, 8597)

X(26614) =  MIDPOINT OF X(3) AND X(9166)

Barycentrics    10 a^8-21 a^6 b^2+25 a^4 b^4-18 a^2 b^6+4 b^8-21 a^6 c^2+4 a^4 b^2 c^2+7 a^2 b^4 c^2-15 b^6 c^2+25 a^4 c^4+7 a^2 b^2 c^4+22 b^4 c^4-18 a^2 c^6-15 b^2 c^6+4 c^8 : :
X(26614) =7 X[2] - X[6033], X[549] + 2 X[6036], 7 X[3526] - X[6054], 2 X[140] + X[6055], X[3845] - 4 X[6722], 2 X[5461] + X[8703], 11 X[2] + X[9862], 11 X[6033] + 7 X[9862], X[114] - 4 X[10124], 11 X[3525] + X[11177], 5 X[631] + X[11632], X[2482] - 4 X[11812], X[11161] + 5 X[12017], 2 X[9862] - 11 X[12042], 2 X[2] + X[12042], 2 X[6033] + 7 X[12042], X[115] + 2 X[12100], X[3534] + 5 X[14061], X[10723] + 5 X[14093], 5 X[9862] - 11 X[14830], 5 X[12042] - 2 X[14830], 5 X[2] + X[14830], 5 X[6033] + 7 X[14830], 2 X[11623] + 7 X[14869], 10 X[140] - X[14981], 5 X[9167] - X[14981], 5 X[6055] + X[14981], X[3830] - 4 X[15092], X[6321] + 5 X[15692], X[671] + 5 X[15693], X[98] + 5 X[15694], X[12117] - 7 X[15700], X[99] - 7 X[15701], X[8724] - 7 X[15702], X[14651] + 3 X[15708], X[15561] - 3 X[15709], 2 X[620] - 5 X[15713], X[12355] + 11 X[15718], X[148] + 11 X[15719], X[12243] + 11 X[15721], X[10991] + 8 X[16239], 5 X[15712] + 4 X[20398], 3 X[15707] - X[21166], 4 X[547] - X[22505], 4 X[5461] - X[22515], 2 X[8703] + X[22515], 4 X[6033] - 7 X[22566], 4 X[2] - X[22566], 2 X[12042] + X[22566], 4 X[14830] + 5 X[22566], 4 X[9862] + 11 X[22566], 5 X[14971] - 3 X[23514].

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28568.

X(26614) lies on these lines: {2,5191}, {3,9166}, {30,5215}, {98,15694}, {99,15701}, {114,10124}, {115,12100}, {140,6055}, {148,15719}, {542,11539}, {543,549}, {547,22505}, {620,15713}, {631,11632}, {671,15693}, {2482,11812}, {2782,5054}, {2794,15699}, {3525,11177}, {3526,6054}, {3534,14061}, {3830,15092}, {3845,6722}, {5461,8703}, {6321,15692}, {7610,13085}, {8724,15702}, {10723,14093}, {10991,16239}, {11161,12017}, {11623,14869}, {12117,15700}, {12243,15721}, {12355,15718}, {14639,15688}, {14651,15708}, {15561,15709}, {15707,21166}, {15712,20398}, {17504,23698}

X(26614) = midpoint of X(i) and X(j) for these {i,j}: {3, 9166}, {6055, 9167}, {14639, 15688}
X(26614) = reflection of X(9167) in X(140)
X(26614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12042, 22566), (5461, 8703, 22515)

X(26615) =  EULER LINE INTERCEPT OF X(1285)X(19054)

Barycentrics    (41*a^4-44*(b^2+c^2)*a^2-13*b^4+10*b^2*c^2-13*c^4)*S+18*(c^2+a^2+b^2)*(a^2-b^2-c^2)*a^2 : :
Barycentrics    6*S^2+2*SW*S-9*SB*SC : :

X(26615) = 4*(SW+3*S)*X(3)+(2*SW-3*S)*X(4)

As a point on the Euler line, X(26615) has Shinagawa coefficients (E+F+3*S, -9*S/2).

See César Lozada, ADGEOM 5001

X(26615) lies on these lines: {2, 3}, {524, 9541}, {1285, 19054}, {3068, 13662}, {3595, 6451}, {5860, 9741}, {6221, 13639}, {12158, 12256}, {13663, 23249}, {13757, 23273}

X(26615) = reflection of X(i) in X(j) for these (i,j): (13639, 6221), (23249, 13663)

X(26616) =  EULER LINE INTERCEPT OF X(1285)X(19053)

Barycentrics    -(41*a^4-44*(b^2+c^2)*a^2-13*b^4+10*b^2*c^2-13*c^4)*S+18*(c^2+a^2+b^2)*(a^2-b^2-c^2)*a^2 : :
Barycentrics    6*S^2-2*SW*S-9*SB*SC : :

X(26616) = 4*(SW-3*S)*X(3)+(2*SW+3*S)*X(4)

As a point on the Euler line, X(26616) has Shinagawa coefficients (E+F-3*S, 9*S/2).

See César Lozada, ADGEOM 5001

X(26616) lies on these lines: {2, 3}, {597, 9541}, {1285, 19053}, {3069, 13782}, {3593, 6452}, {5861, 9741}, {6398, 13759}, {12159, 12257}, {13637, 23267}, {13783, 23259}

X(26616) = reflection of X(i) in X(j) for these (i,j): (13759, 6398), (23259, 13783)

X(26617) =  EULER LINE INTERCEPT OF X(99)X(1270)

Barycentrics    -(a^2+b^2+c^2)*S+5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(26617) = 2*(SW+4*S)*X(3)+(SW-2*S)*X(4) = 4*X(1151)-X(12222) = 2*X(12313)+X(12510)

As a point on the Euler line, X(26617) has Shinagawa coefficients (E+F+4*S, -6*S).

See César Lozada, ADGEOM 5001

X(26617) lies on these lines: {2, 3}, {99, 1270}, {193, 9541}, {488, 13712}, {490, 5861}, {591, 12221}, {1151, 12222}, {1271, 14907}, {5860, 8716}, {6409, 12323}, {6462, 13678}, {6567, 13639}, {7585, 9675}, {12313, 12510}

X(26617) = reflection of X(488) in X(13712)

X(26618) =  EULER LINE INTERCEPT OF X(99)X(1271)

Barycentrics    (a^2+b^2+c^2)*S+5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(26618) = 2*(SW+4*S)*X(3)+(SW-2*S)*X(4) = 4*X(1151)-X(12222) = 2*X(12313)+X(12510)

As a point on the Euler line, X(26618) has Shinagawa coefficients (E+F-4*S, 6*S).

See César Lozada, ADGEOM 5001

X(26618) lies on these lines: {2, 3}, {99, 1271}, {487, 13835}, {489, 5860}, {1152, 12221}, {1270, 14907}, {1991, 12222}, {5861, 8716}, {6410, 12322}, {6463, 13798}, {6566, 13759}, {12314, 12509}

X(26618) = reflection of X(487) in X(13835)

X(26619) =  EULER LINE INTERCEPT OF X(488)X(7582)

Barycentrics    5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2-4*(a^2+b^2+c^2)*S : :
X(26619) = 4*(SW+S)*X(3)+(2*SW-S)*X(4)

As a point on the Euler line, X(26619) has Shinagawa coefficients (E+F+S, -3*S/2).

See César Lozada, ADGEOM 5001

X(26619) lies on these lines: {2, 3}, {141, 9541}, {371, 5861}, {372, 13712}, {488, 7582}, {490, 7581}, {492, 23273}, {591, 1588}, {1271, 6221}, {1285, 3068}, {1384, 8974}, {3593, 13785}, {5490, 14226}, {5590, 6561}, {5591, 6200}, {6202, 12305}, {7586, 14482}, {9738, 10517}, {12323, 13886}, {13789, 13794}, {13950, 15048}, {19054, 19103}, {23263, 23311}

X(26619) = {X(11292), X(11294)}-harmonic conjugate of X(4)

X(26620) =  EULER LINE INTERCEPT OF X(488)X(7581)

Barycentrics    5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2+4*(a^2+b^2+c^2)*S : :
X(26620) = 4*(SW-S)*X(3)+(2*SW+S)*X(4)

As a point on the Euler line, X(26612) has Shinagawa coefficients (E+F-S, 3*S/2).

See César Lozada, ADGEOM 5001

X(26620) lies on these lines: {2, 3}, {371, 13835}, {372, 5860}, {487, 7581}, {489, 7582}, {491, 23267}, {1270, 6398}, {1285, 3069}, {1384, 13950}, {1587, 1991}, {3589, 9541}, {3595, 13665}, {5491, 14241}, {5590, 6396}, {5591, 6560}, {6201, 12306}, {7585, 14482}, {8974, 15048}, {9739, 10518}, {12322, 13939}, {13669, 13674}, {19053, 19104}, {23253, 23312}

X(26620) = {X(11291), X(11293)}-harmonic conjugate of X(4)

leftri

Collineation mappings involving Gemini triangle 47: X(26621)-X(26652)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 47, as in centers X(26621)-X(26652). Then

m(X) = a^2 (a - b + c) (a + b - c) x + (b + c - a) (a + b - c) (a + c)^2 y + (b + c - a) (a - b + c) (a + b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 2, 2018)

X(26621) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26621) lies on these lines: {1, 2}, {6, 24547}, {56, 24633}, {77, 27170}, {269, 26836}, {604, 21233}, {960, 24612}, {1229, 2256}, {1334, 24266}, {2285, 21273}, {2324, 27058}, {3554, 16713}, {3739, 24540}, {3877, 6996}, {4361, 24993}, {5783, 20895}, {9310, 26265}, {16609, 26229}, {17251, 24998}, {17275, 24986}, {17301, 24999}, {20172, 26538}, {24583, 26066}, {25060, 26635}, {26558, 26576}, {26561, 26578}, {26624, 26629}, {26625, 26627}, {26637, 26643}

X(26622) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26622) lies on these lines: {2, 3}, {239, 1993}, {20172, 26538}, {26625, 26633}

X(26623) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26623) lies on these lines: {2, 3}, {25060, 26636}, {26627, 26628}

X(26624) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26624) lies on these lines: {2, 3}, {26621, 26629}

X(26625) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26625) lies on these lines: {2, 6}, {3, 18465}, {25, 3794}, {31, 24550}, {56, 3218}, {222, 27184}, {474, 9567}, {608, 3662}, {959, 5253}, {1010, 5707}, {1352, 16067}, {1407, 26840}, {3741, 24545}, {5651, 16048}, {5788, 14011}, {6646, 22129}, {7252, 26640}, {9306, 25494}, {13478, 17182}, {18199, 26545}, {26621, 26627}, {26622, 26633}, {26635, 26639}

X(26626) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    3 a^2 + 2 a b + b^2 + 2 a c + c^2 : :

X(26626) lies on these lines: {1, 2}, {6, 4364}, {7, 604}, {35, 21537}, {36, 21508}, {37, 3618}, {45, 597}, {55, 21495}, {56, 21511}, {57, 17081}, {63, 1475}, {69, 1100}, {75, 4470}, {81, 2221}, {86, 4000}, {141, 16884}, {144, 17120}, {192, 5749}, {193, 1449}, {226, 5395}, {278, 11341}, {304, 4359}, {319, 17400}, {320, 17399}, {344, 3589}, {345, 20182}, {346, 17319}, {348, 5228}, {377, 19834}, {379, 19719}, {391, 17121}, {458, 7952}, {469, 7718}, {524, 17325}, {673, 20131}, {894, 3672}, {940, 16781}, {942, 24609}, {944, 7377}, {946, 7406}, {966, 3759}, {980, 1015}, {999, 11343}, {1278, 7229}, {1438, 27950}, {1453, 13736}, {1580, 9791}, {1621, 16367}, {1790, 8025}, {1959, 5744}, {1992, 4643}, {2238, 16523}, {2241, 5337}, {2275, 3666}, {2280, 20769}, {2329, 18228}, {2345, 4360}, {3061, 5273}, {3161, 4704}, {3207, 5834}, {3247, 17353}, {3295, 21477}, {3303, 21540}, {3304, 21516}, {3619, 4851}, {3620, 3879}, {3629, 17253}, {3662, 3945}, {3664, 17304}, {3674, 21454}, {3723, 17279}, {3729, 4021}, {3758, 4419}, {3763, 17390}, {3765, 18135}, {3782, 11352}, {3875, 5750}, {3946, 4758}, {4007, 4464}, {4038, 24586}, {4339, 7791}, {4346, 4747}, {4361, 17398}, {4363, 17395}, {4389, 4644}, {4398, 7222}, {4402, 4699}, {4416, 16667}, {4422, 16672}, {4452, 17116}, {4472, 17119}, {4648, 16706}, {4658, 24632}, {4667, 17274}, {4670, 17301}, {4675, 17382}, {4688, 4798}, {4748, 17346}, {4852, 17303}, {4869, 17291}, {4909, 21255}, {4916, 17295}, {4969, 17251}, {5224, 5839}, {5232, 17326}, {5253, 11329}, {5263, 20162}, {5266, 16043}, {5296, 17349}, {5435, 7146}, {5603, 6996}, {5712, 19786}, {5716, 26561}, {5731, 6999}, {5905, 16783}, {6329, 16885}, {6646, 16779}, {6654, 14267}, {6703, 24384}, {6767, 21526}, {7277, 17255}, {7373, 21514}, {7397, 10595}, {7402, 7967}, {8236, 20533}, {8772, 25378}, {9345, 24602}, {9441, 10186}, {9708, 21986}, {10283, 19512}, {11008, 17344}, {11037, 17691}, {14996, 16784}, {14997, 16785}, {15668, 17366}, {16524, 24512}, {16673, 25101}, {16780, 27184}, {16786, 20072}, {16787, 17778}, {17141, 26065}, {17147, 25244}, {17227, 26104}, {17236, 20090}, {17272, 20080}, {17275, 25498}, {17289, 17314}, {17290, 17392}, {17293, 17388}, {17299, 17385}, {17300, 17383}, {17305, 17378}, {17307, 17377}, {17315, 17371}, {17317, 17370}, {17318, 17369}, {17323, 17365}, {17324, 17364}, {17327, 17362}, {17374, 21356}, {17592, 24631}, {17742, 27065}, {17776, 27109}, {17917, 26023}, {18156, 19804}, {18230, 27268}, {19281, 19684}, {20905, 24553}, {20917, 25303}, {21840, 26274}, {24554, 26668}, {24604, 27000}, {25524, 25946}, {26635, 26649}, {26818, 27170}

X(26626) = anticomplement of X(17308)

X(26627) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    a^3 + a^2 b + a^2 c + 4 a b c + b^2 c + b c^2 : :

X(26627) lies on these lines: {1, 17495}, {2, 7}, {6, 24589}, {81, 3759}, {86, 4850}, {89, 16815}, {192, 17021}, {239, 14996}, {321, 17118}, {612, 17140}, {740, 9345}, {748, 4697}, {750, 4434}, {902, 24331}, {940, 3187}, {942, 16454}, {964, 5439}, {1125, 4414}, {1150, 3739}, {1215, 17124}, {1449, 26860}, {1961, 17155}, {2226, 27922}, {2999, 19717}, {3210, 17019}, {3337, 19858}, {3720, 3980}, {3742, 24552}, {3752, 19684}, {3891, 4682}, {3936, 4675}, {3995, 17022}, {4358, 4363}, {4384, 16704}, {4392, 16830}, {4418, 26102}, {4648, 17740}, {4652, 17588}, {4670, 16610}, {4671, 17116}, {4672, 17125}, {4751, 5235}, {5241, 17365}, {5256, 8025}, {5268, 17165}, {5287, 17147}, {5297, 24349}, {5311, 24165}, {5436, 17539}, {5708, 16458}, {7174, 17154}, {7295, 26261}, {11518, 19337}, {14997, 17120}, {15668, 17595}, {15803, 16347}, {15934, 19290}, {16496, 17146}, {16823, 17126}, {17011, 17490}, {17012, 17379}, {17272, 27081}, {18141, 19822}, {19309, 26866}, {19336, 24929}, {24046, 25526}, {25001, 25934}, {26621, 26625}, {26623, 26628}, {26634, 26643}

X(26628) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26628) lies on these lines: {1, 2}, {65, 24583}, {3897, 7377}, {4670, 24999}, {4999, 24633}, {5228, 27187}, {11375, 24612}, {14621, 26573}, {14953, 17173}, {17045, 24547}, {17398, 24993}, {24986, 25498}, {26623, 26627}, {26635, 26636}

X(26629) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26629) lies on these lines: {1, 7807}, {2, 11}, {3, 26561}, {12, 384}, {21, 26558}, {35, 6656}, {56, 16925}, {140, 26959}, {192, 7806}, {230, 350}, {287, 26956}, {325, 1914}, {330, 7891}, {335, 17724}, {495, 8369}, {498, 7770}, {620, 1015}, {902, 4766}, {999, 11288}, {1003, 1478}, {1125, 17694}, {1329, 16916}, {1479, 7887}, {1500, 6680}, {1909, 7789}, {2241, 3788}, {2276, 7792}, {3085, 14001}, {3552, 7354}, {3584, 6661}, {3585, 19687}, {3614, 16044}, {3666, 5976}, {3712, 3797}, {3771, 24586}, {3912, 4434}, {4294, 14064}, {4302, 7841}, {4316, 8598}, {4324, 19695}, {4357, 24685}, {4396, 22329}, {4999, 26801}, {5010, 8356}, {5025, 6284}, {5217, 7791}, {5305, 25264}, {5433, 7907}, {5552, 26687}, {5718, 14621}, {6645, 15888}, {6655, 15338}, {6675, 16819}, {7031, 7762}, {7294, 16923}, {7483, 17030}, {7763, 16502}, {7819, 27020}, {7844, 9664}, {7851, 9598}, {7862, 9665}, {7951, 8370}, {8164, 14039}, {8361, 15171}, {9668, 11318}, {10198, 11321}, {10349, 10801}, {10352, 12835}, {10590, 14033}, {10895, 14035}, {11269, 20162}, {11681, 16920}, {12953, 14063}, {13586, 15326}, {16915, 25466}, {17321, 26273}, {17540, 27091}, {17541, 27529}, {17670, 25440}, {17719, 17738}, {26621, 26624}

X(26630) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26630) lies on these lines: {2, 3}

X(26631) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26631) lies on these lines: {2, 3}

X(26632) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26632) lies on these lines: {2, 37}, {1441, 24627}, {3218, 24633}, {5933, 9776}, {20172, 26644}, {24178, 24443}, {26621, 26625}

X(26633) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26633) lies on these lines: {2, 39}, {311, 26979}, {26526, 26527}, {26537, 26542}, {26622, 26625}

X(26634) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    a^5 + a^2 b^3 + 2 a b^3 c + b^3 c^2 + a^2 c^3 + 2 a b c^3 + b^2 c^3 : :

X(26634) lies on these lines: {2, 41}, {21, 23206}, {48, 27145}, {141, 26222}, {604, 17178}, {942, 19271}, {1468, 17751}, {1958, 27017}, {16915, 27003}, {21240, 24587}, {26627, 26643}

X(26635) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26635) lies on these lines: {2, 37}, {57, 16579}, {81, 3554}, {86, 26645}, {394, 17011}, {990, 1005}, {1040, 1621}, {1214, 9776}, {1961, 25938}, {3218, 5228}, {3219, 10601}, {3616, 17102}, {3743, 8583}, {3977, 25082}, {4364, 26005}, {5249, 17080}, {5437, 16577}, {6173, 18593}, {8025, 18603}, {16699, 16704}, {16777, 25934}, {17592, 25941}, {17811, 20182}, {18607, 21454}, {20276, 21321}, {24181, 25094}, {25009, 25098}, {25060, 26621}, {26625, 26639}, {26626, 26649}, {26628, 26636}

X(26636) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26636) lies on these lines: {2, 39}, {216, 3945}, {394, 4255}, {566, 17392}, {570, 4648}, {1993, 2271}, {3060, 17209}, {5308, 13006}, {13351, 17245}, {16696, 26540}, {25060, 26623}, {26628, 26635}

X(26637) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26637) lies on these lines: {2, 6}, {21, 104}, {58, 19861}, {63, 1412}, {274, 2990}, {404, 7998}, {405, 6090}, {511, 4239}, {651, 26580}, {960, 1408}, {1010, 3193}, {1014, 3218}, {1172, 26651}, {1396, 17184}, {1790, 17185}, {2341, 17195}, {3794, 4228}, {3869, 5323}, {3877, 4221}, {4188, 21766}, {4189, 6800}, {4234, 6580}, {4357, 22128}, {4658, 19860}, {15080, 17549}, {16370, 26864}, {16726, 25939}, {17187, 25941}, {17588, 24558}, {24987, 25526}, {26621, 26643}

X(26638) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    (a + b) (a + c) (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(26638) lies on these lines: {2, 6}, {21, 3427}, {27, 10444}, {283, 1010}, {1014, 5744}, {1412, 5745}, {1434, 3218}, {4357, 17923}, {7054, 26645}, {8822, 24547}, {10458, 25941}, {11110, 18465}, {16054, 24590}, {16696, 25939}, {25060, 26621}

X(26639) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26639) lies on these lines: {1, 2}, {6, 26699}, {40, 21537}, {48, 26998}, {86, 26538}, {193, 3554}, {238, 8772}, {297, 1870}, {323, 16784}, {394, 16781}, {401, 3100}, {441, 18455}, {458, 6198}, {517, 21495}, {740, 24563}, {894, 7269}, {1100, 15988}, {1385, 21511}, {1429, 1959}, {1442, 3662}, {1482, 21477}, {1953, 27059}, {1993, 16502}, {1994, 5299}, {2170, 20769}, {2329, 27065}, {3061, 3219}, {3576, 21508}, {3674, 26842}, {3723, 25099}, {3875, 18261}, {3877, 16367}, {4360, 26665}, {4560, 26652}, {4881, 19308}, {4904, 25593}, {7146, 27003}, {7291, 27950}, {8148, 21539}, {10222, 21540}, {10246, 11343}, {10247, 21526}, {12702, 16431}, {15018, 16785}, {15178, 21516}, {17045, 26543}, {17302, 26573}, {17319, 25245}, {17614, 25946}, {18465, 26643}, {18650, 26837}, {19512, 19907}, {20236, 24202}, {26130, 27180}, {26625, 26635}

X(26640) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26640) lies on these lines: {2, 661}, {513, 26596}, {693, 26652}, {905, 3904}, {1993, 23092}, {7252, 26625}, {26674, 26694}

X(26641) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26641) lies on these lines: {2, 650}, {21, 8760}, {647, 2799}, {905, 3904}, {1621, 11934}, {1635, 25900}, {1993, 22383}, {4705, 25901}, {4893, 25924}, {6589, 16757}, {9001, 15988}, {9015, 26543}, {14936, 26565}, {15313, 16158}

X(26642) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26642) lies on these lines: {2, 3}

X(26643) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26643) lies on these lines: {2, 3}, {10, 24632}, {58, 4384}, {75, 2303}, {81, 239}, {86, 4000}, {284, 1958}, {333, 17103}, {894, 2287}, {1014, 16738}, {1043, 17316}, {1333, 3739}, {1444, 27164}, {1468, 5271}, {1580, 24342}, {1778, 17277}, {1931, 5235}, {3666, 16716}, {4273, 4670}, {4653, 16831}, {4658, 16834}, {4720, 6542}, {5277, 26243}, {5333, 17397}, {6703, 24366}, {8025, 17014}, {8822, 17257}, {14621, 27644}, {16589, 24271}, {16756, 25060}, {16815, 16948}, {16818, 24588}, {17023, 25526}, {17189, 24199}, {18465, 26639}, {19719, 19767}, {19791, 19848}, {26621, 26637}, {26627, 26634}

X(26644) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26644) lies on these lines: {2, 3}, {20172, 26632}

X(26645) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26645) lies on these lines: {2, 3}, {86, 26635}, {333, 2988}, {7054, 26638}

X(26646) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26646) lies on these lines: {2, 3}

X(26647) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26647) lies on these lines: {2, 3}, {78, 24632}, {81, 348}, {86, 7054}, {284, 307}, {333, 24635}, {1790, 16887}, {2328, 26006}, {4288, 17171}

X(26648) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    7 a^8 - 10 a^6 b^2 + 2 a^2 b^6 + b^8 + 10 a^6 b c - 2 a^5 b^2 c - 8 a^4 b^3 c - 8 a^3 b^4 c - 2 a^2 b^5 c + 10 a b^6 c - 10 a^6 c^2 - 2 a^5 b c^2 + 4 a^4 b^2 c^2 + 4 a^3 b^3 c^2 - 2 a^2 b^4 c^2 - 2 a b^5 c^2 + 8 b^6 c^2 - 8 a^4 b c^3 + 4 a^3 b^2 c^3 + 4 a^2 b^3 c^3 - 8 a b^4 c^3 - 8 a^3 b c^4 - 2 a^2 b^2 c^4 - 8 a b^3 c^4 - 18 b^4 c^4 - 2 a^2 b c^5 - 2 a b^2 c^5 + 2 a^2 c^6 + 10 a b c^6 + 8 b^2 c^6 + c^8 : :

X(26648) lies on these lines: {2, 3}

X(26649) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 47

Barycentrics    3 a^8 - 6 a^6 b^2 + 4 a^4 b^4 - 2 a^2 b^6 + b^8 - 4 a^5 b^2 c + 4 a^4 b^3 c + 4 a^3 b^4 c - 4 a^2 b^5 c - 6 a^6 c^2 - 4 a^5 b c^2 + 8 a^4 b^2 c^2 + 16 a^3 b^3 c^2 + 6 a^2 b^4 c^2 - 4 a b^5 c^2 + 4 a^4 b c^3 + 16 a^3 b^2 c^3 + 16 a^2 b^3 c^3 + 4 a b^4 c^3 + 4 a^4 c^4 + 4 a^3 b c^4 + 6 a^2 b^2 c^4 + 4 a b^3 c^4 - 2 b^4 c^4 - 4 a^2 b c^5 - 4 a b^2 c^5 - 2 a^2 c^6 + c^8 : :

X(26649) lies on these lines: {2, 3}, {941, 26668}, {968, 26006}, {24555, 25058}, {26626, 26635}

X(26650) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26650) lies on these lines: {2, 3}

X(26651) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26651) lies on these lines: {2, 7}, {6, 20905}, {75, 1332}, {86, 16743}, {190, 26669}, {320, 26540}, {321, 17811}, {394, 3187}, {990, 11115}, {1150, 26011}, {1172, 26637}, {2257, 26818}, {2284, 26653}, {3100, 24307}, {3551, 24428}, {3663, 26006}, {3664, 25935}, {3673, 26678}, {3729, 25243}, {4000, 26668}, {4358, 25934}, {4363, 25001}, {4416, 26001}, {4643, 25000}, {5757, 16454}, {6505, 18662}, {7289, 14543}, {10444, 14953}, {10861, 13727}, {14942, 25722}, {16551, 24237}, {17321, 24553}, {17351, 25067}, {17364, 26531}, {17365, 25964}, {20172, 26538}, {26655, 26660}

X(26652) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 47

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

X(26652) lies on these lines: {2, 649}, {512, 24561}, {513, 25981}, {652, 26049}, {693, 26640}, {812, 24560}, {894, 4468}, {4380, 25902}, {4521, 27064}, {4560, 26639}, {4979, 25008}, {9002, 25898}, {17215, 26854}, {17418, 17494}

leftri

Collineation mappings involving Gemini triangle 48: X(26653)-X(26699)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 48, as in centers X(26653)-X(26699). Then

m(X) = a^2 (b + c - a) x + (a - b + c) (a - c)^2 y + (a + b - c) (a - b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 2, 2018)

X(26653) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 48

Barycentrics    {1, 2}, {9, 20248}, {41, 21232}, {220, 26563}, {644, 3673}, {664, 26690}, {673, 14923}, {1146, 27132}, {1334, 24249}, {2082, 21272}, {2284, 26651}, {3501, 9317}, {3753, 17683}, {3875, 25880}, {3877, 17681}, {3879, 25966}, {3995, 25894}, {4566, 7131}, {5046, 27129}, {5836, 24596}, {8256, 26007}, {9310, 26229}, {9593, 18600}, {9620, 26978}, {10950, 16593}, {16609, 26265}, {17181, 26074}, {17350, 20089}, {17353, 25719}, {17555, 26585}, {24540, 25971}, {24993, 25878}, {26527, 26582}, {26529, 26590}, {26654, 26667}, {26656, 26686}, {26663, 26670}, {26669, 26671}, {26677, 26685} : :

X(26653) lies on these lines:

X(26654) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26654) lies on these lines: {2, 3}, {3934, 25886}, {26653, 26667}, {26657, 26664}, {26658, 26692}

X(26655) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26655) lies on these lines: {2, 3}, {26651, 26660}, {26667, 26686}

X(26656) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26656) lies on these lines: {2, 3}, {26653, 26686}

X(26657) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26657) lies on these lines: {1, 25903}, {2, 6}, {55, 25279}, {56, 27678}, {105, 25304}, {218, 17364}, {219, 3662}, {220, 6646}, {320, 2911}, {511, 16048}, {651, 26685}, {1100, 25891}, {1278, 4513}, {1332, 4000}, {1350, 17522}, {2256, 17302}, {2284, 26651}, {2323, 17282}, {3564, 14019}, {3713, 17230}, {3888, 7083}, {3917, 25494}, {4361, 26567}, {4435, 26568}, {5782, 17358}, {5783, 17292}, {6180, 17350}, {7232, 17796}, {10449, 25990}, {17288, 23151}, {17792, 26241}, {20818, 27950}, {26654, 26664}, {26663, 26667}, {26669, 26672}

X(26658) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26658) lies on these lines: {1, 2}, {7, 9310}, {20, 27129}, {101, 17170}, {193, 25019}, {220, 348}, {277, 24203}, {279, 10025}, {347, 27420}, {664, 6554}, {672, 17081}, {883, 26668}, {944, 17671}, {952, 17675}, {962, 4209}, {2098, 26007}, {3160, 3177}, {3618, 25067}, {5603, 17682}, {5687, 25954}, {6603, 6604}, {9436, 20111}, {9778, 26790}, {17321, 25878}, {23058, 25719}, {24553, 25001}, {25091, 26065}, {25239, 25243}, {26556, 26590}, {26654, 26692}, {26667, 26678}

X(26659) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26659) lies on these lines: {2, 45}, {2284, 26651}, {4459, 26241}, {5220, 25005}, {25934, 26612}

X(26660) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26660) lies on these lines: {1, 2}, {3057, 24582}, {4188, 27129}, {5886, 17683}, {11376, 24596}, {17044, 26563}, {26651, 26655}, {26678, 26692}, {26686, 26689}

X(26661) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26661) lies on these lines: {2, 3}

X(26662) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26662) lies on these lines: {2, 3}

X(26663) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26663) lies on these lines: {2, 31}, {26653, 26670}, {26657, 26667}

X(26664) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26664) lies on these lines: {2, 32}, {26654, 26657}

X(26665) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26665) lies on these lines: {1, 24563}, {2, 37}, {6, 3262}, {8, 4008}, {10, 1733}, {19, 21368}, {38, 24997}, {86, 16740}, {92, 26065}, {141, 26573}, {190, 26671}, {193, 322}, {239, 20895}, {287, 651}, {313, 25978}, {594, 26543}, {726, 23689}, {1738, 21935}, {1958, 24334}, {1959, 20258}, {2174, 24324}, {2284, 26651}, {2550, 5086}, {3212, 20348}, {3219, 11683}, {3403, 20911}, {3663, 20881}, {3729, 17861}, {3821, 25010}, {3923, 23690}, {4357, 25007}, {4360, 26639}, {4429, 24433}, {4459, 17792}, {4644, 20930}, {4858, 17353}, {4872, 26789}, {5294, 14213}, {6646, 10030}, {7283, 25906}, {12723, 20556}, {16284, 20080}, {16732, 17351}, {17033, 21422}, {17080, 27338}, {17116, 17741}, {17132, 24208}, {17139, 21853}, {17319, 24559}, {17355, 20236}, {17752, 20436}, {17872, 24996}, {20235, 26678}, {21033, 27492}, {22019, 24224}, {25081, 25589}, {25083, 25241}, {26575, 26578}, {26666, 26676}, {26667, 26679}

X(26666) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26666) lies on these lines: {2, 39}, {26654, 26657}, {26665, 26676}, {26684, 26691}

X(26667) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26667) lies on these lines: {2, 11}, {4554, 7123}, {26653, 26654}, {26655, 26686}, {26657, 26663}, {26658, 26678}, {26665, 26679}

X(26668) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26668) lies on these lines: {2, 6}, {9, 26006}, {184, 26052}, {273, 27382}, {329, 17923}, {572, 14021}, {573, 24580}, {651, 27509}, {692, 11677}, {883, 26658}, {941, 26649}, {1439, 5744}, {1449, 25935}, {1743, 25019}, {1876, 3869}, {2182, 4329}, {2261, 18589}, {2297, 5294}, {2317, 26130}, {2398, 4012}, {3616, 5728}, {4000, 26651}, {4223, 14853}, {5222, 20905}, {5286, 26678}, {5435, 14524}, {5749, 25001}, {5751, 6857}, {5752, 7521}, {5803, 6856}, {13742, 18465}, {17121, 26531}, {17353, 25930}, {19767, 24570}, {24554, 26626}, {26682, 26691}

X(26669) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26669) lies on these lines: {2, 37}, {9, 77}, {45, 25878}, {100, 4319}, {144, 241}, {190, 26651}, {227, 8165}, {322, 27108}, {404, 990}, {527, 17092}, {883, 26658}, {908, 3668}, {1214, 18228}, {1418, 20059}, {1445, 2324}, {1465, 5328}, {1766, 11349}, {1818, 10394}, {2092, 25004}, {2310, 25722}, {2321, 26001}, {2400, 4130}, {3161, 25083}, {3218, 25934}, {3219, 17811}, {3306, 4328}, {3452, 17080}, {3661, 25000}, {3663, 25076}, {3681, 25941}, {3731, 25065}, {3755, 24982}, {3869, 21371}, {3870, 18216}, {3873, 21346}, {3912, 25019}, {4327, 5253}, {4356, 8582}, {5749, 24553}, {7174, 15839}, {7191, 25893}, {7308, 16577}, {9352, 25938}, {11683, 26265}, {17011, 17825}, {17242, 26531}, {17243, 25964}, {17353, 26006}, {18601, 24556}, {20275, 21320}, {21955, 25973}, {25078, 25097}, {25082, 25101}, {26563, 27282}, {26653, 26671}, {26657, 26672}

X(26670) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26670) lies on these lines: {2, 6}, {3909, 5324}, {26653, 26663}, {26673, 26692}, {26680, 26685}

X(26671) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26671) lies on these lines: {2, 6}, {9, 11683}, {44, 25971}, {55, 24752}, {190, 26665}, {220, 27282}, {239, 3965}, {257, 17260}, {322, 8557}, {1043, 25906}, {1100, 24559}, {1376, 25631}, {2245, 8822}, {3692, 20173}, {6180, 27334}, {6554, 26678}, {17273, 26573}, {17285, 26594}, {17289, 25007}, {17348, 25887}, {17353, 20262}, {21677, 23904}, {23693, 24982}, {24612, 27624}, {24757, 25531}, {26653, 26669}, {26675, 26676}

X(26672) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26672) lies on these lines: {1, 2}, {190, 26674}, {1442, 17338}, {9310, 27003}, {15988, 25067}, {21222, 26694}, {26657, 26669}

X(26673) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26673) lies on these lines: {1, 2}, {26657, 26663}, {26670, 26692}

X(26674) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26674) lies on these lines: {2, 44}, {190, 26672}, {524, 26570}, {2284, 26651}, {6646, 27006}, {26640, 26694}

X(26675) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26675) lies on these lines: {2, 31}, {2284, 26651}, {24547, 25878}, {26671, 26676}

X(26676) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26676) lies on these lines: {1, 2}, {1574, 25888}, {9956, 17672}, {24986, 25971}, {26665, 26666}, {26671, 26675}

X(26677) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26677) lies on these lines: {2, 3}, {26653, 26685}

X(26678) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26678) lies on these lines: {2, 3}, {169, 16564}, {294, 18299}, {315, 26540}, {318, 26203}, {894, 1446}, {3673, 26651}, {5081, 26153}, {5286, 26668}, {6554, 26671}, {7745, 25964}, {13161, 26006}, {15988, 17499}, {20235, 26665}, {26653, 26663}, {26658, 26667}, {26660, 26692}

X(26679) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26679) lies on these lines: {2, 3}, {26665, 26667}

X(26680) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26680) lies on these lines: {2, 3}, {26670, 26685}

X(26681) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26681) lies on these lines: {2, 3}

X(26682) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 48

Barycentrics    7 a^7 - 7 a^6 b - 3 a^5 b^2 + 3 a^4 b^3 - 3 a^3 b^4 + 3 a^2 b^5 - a b^6 + b^7 - 7 a^6 c + 4 a^5 b c + a^4 b^2 c + 4 a^3 b^3 c + 7 a^2 b^4 c - 8 a b^5 c - b^6 c - 3 a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - 10 a^2 b^3 c^2 + a b^4 c^2 + 9 b^5 c^2 + 3 a^4 c^3 + 4 a^3 b c^3 - 10 a^2 b^2 c^3 + 16 a b^3 c^3 - 9 b^4 c^3 - 3 a^3 c^4 + 7 a^2 b c^4 + a b^2 c^4 - 9 b^3 c^4 + 3 a^2 c^5 - 8 a b c^5 + 9 b^2 c^5 - a c^6 - b c^6 + c^7 : :

X(26682) lies on these lines: {2, 3}, {26668, 26691}

X(26683) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26683) lies on these lines: {2, 3}, {883, 26658}

X(26684) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26684) lies on these lines: {2, 3}, {26666, 26691}

X(26685) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26685) lies on these lines: {1, 4899}, {2, 7}, {6, 344}, {8, 238}, {10, 5395}, {37, 3618}, {41, 17696}, {44, 69}, {45, 3589}, {56, 25879}, {72, 13742}, {100, 7083}, {141, 16885}, {145, 3717}, {169, 27059}, {190, 4000}, {192, 3161}, {193, 1743}, {198, 21495}, {239, 346}, {281, 458}, {319, 17342}, {320, 17341}, {345, 4383}, {391, 3661}, {404, 24320}, {524, 17267}, {597, 16777}, {651, 26657}, {883, 26658}, {899, 1716}, {962, 6211}, {966, 17289}, {984, 3616}, {1104, 1265}, {1212, 25099}, {1278, 4402}, {1405, 5933}, {1453, 20009}, {1654, 17358}, {1738, 24280}, {1766, 26998}, {1992, 4851}, {2183, 26041}, {2287, 16050}, {2325, 3875}, {2345, 17277}, {2347, 22370}, {2478, 26939}, {2550, 4676}, {2899, 5230}, {3008, 3729}, {3217, 20769}, {3220, 4188}, {3271, 25304}, {3617, 3883}, {3619, 4643}, {3620, 3973}, {3621, 4901}, {3622, 7174}, {3629, 17311}, {3663, 20073}, {3672, 17261}, {3686, 17286}, {3707, 17270}, {3718, 4358}, {3730, 27299}, {3731, 17023}, {3758, 4648}, {3759, 17264}, {3763, 17332}, {3836, 24695}, {3876, 17526}, {3879, 16670}, {3888, 9309}, {3945, 17120}, {3950, 16834}, {3952, 26228}, {4078, 16475}, {4339, 7787}, {4361, 17340}, {4363, 17337}, {4370, 17262}, {4384, 17355}, {4419, 16706}, {4429, 5698}, {4431, 16833}, {4440, 4488}, {4461, 17117}, {4470, 4751}, {4480, 4862}, {4641, 18141}, {4644, 17234}, {4657, 16814}, {4660, 4759}, {4699, 7229}, {4748, 17307}, {4869, 17266}, {4969, 17309}, {5232, 17292}, {5308, 17379}, {5817, 13727}, {5819, 26582}, {5838, 20533}, {5839, 17233}, {6210, 26029}, {6329, 16884}, {6554, 26671}, {6687, 17278}, {7277, 17313}, {7406, 10445}, {9441, 9801}, {9778, 26047}, {9780, 25611}, {10327, 17127}, {11008, 17374}, {14001, 25066}, {15828, 17304}, {16020, 24349}, {16517, 16826}, {16552, 27248}, {16675, 17045}, {16831, 25072}, {16989, 27538}, {17014, 17319}, {17033, 27523}, {17232, 20072}, {17249, 26104}, {17256, 17371}, {17258, 17370}, {17259, 17369}, {17265, 17365}, {17268, 17363}, {17269, 17362}, {17275, 17359}, {17276, 17356}, {17281, 17348}, {17283, 17347}, {17285, 17346}, {17290, 17334}, {17293, 17330}, {17296, 20080}, {17344, 21356}, {20262, 25007}, {21390, 23828}, {24509, 26752}, {24890, 25659}, {26364, 27528}, {26653, 26677}, {26670, 26680}, {26772, 27021}, {27060, 27063}

X(26685) = anticomplement of X(17282)

X(26686) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26686) lies on these lines: {1, 7807}, {2, 12}, {3, 26590}, {10, 17694}, {11, 384}, {36, 6656}, {55, 16925}, {140, 27020}, {172, 325}, {192, 7891}, {230, 1909}, {287, 26955}, {330, 7806}, {350, 7789}, {404, 26582}, {496, 8369}, {499, 7770}, {594, 24384}, {609, 7762}, {620, 1500}, {754, 9341}, {894, 17095}, {1003, 1479}, {1015, 6680}, {1055, 24995}, {1478, 7887}, {2242, 3788}, {2275, 7792}, {2886, 16915}, {3035, 26752}, {3086, 14001}, {3295, 11288}, {3552, 6284}, {3582, 6661}, {3583, 19687}, {3816, 16916}, {3925, 16917}, {4293, 14064}, {4299, 7841}, {4316, 19695}, {4324, 8598}, {4400, 22329}, {5025, 7354}, {5204, 7791}, {5326, 16923}, {5432, 7907}, {6390, 25264}, {6655, 15326}, {7173, 16044}, {7181, 7187}, {7267, 16886}, {7280, 8356}, {7483, 27255}, {7741, 8370}, {7819, 15325}, {7844, 9651}, {7851, 9597}, {7862, 9650}, {8361, 18990}, {9655, 11318}, {10349, 10802}, {10352, 10799}, {10527, 20172}, {10591, 14033}, {10896, 14035}, {11321, 26363}, {11680, 16919}, {12943, 14063}, {13586, 15338}, {13747, 27091}, {26653, 26656}, {26655, 26667}, {26660, 26689}, {26755, 27027}

X(26687) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26687) lies on these lines: {2, 12}, {3, 27091}, {4, 26582}, {6, 6376}, {8, 17541}, {9, 3503}, {10, 7770}, {32, 27076}, {55, 16916}, {100, 16920}, {183, 4426}, {220, 17743}, {239, 341}, {335, 17054}, {384, 1376}, {405, 27020}, {458, 25007}, {668, 16502}, {899, 11339}, {956, 26959}, {964, 27026}, {993, 11285}, {1001, 16918}, {1003, 25440}, {1011, 27035}, {1107, 11174}, {1191, 17752}, {1478, 17670}, {1573, 7808}, {1574, 3734}, {1575, 1975}, {1616, 10027}, {1698, 11321}, {2478, 26590}, {2886, 16924}, {3035, 16925}, {3814, 7887}, {3820, 7819}, {3912, 11353}, {3913, 4366}, {3975, 4383}, {4386, 25107}, {4413, 16915}, {4462, 26697}, {5217, 17692}, {5552, 26629}, {6381, 7754}, {6554, 26671}, {7773, 20541}, {7807, 26364}, {9708, 17030}, {9709, 11286}, {9711, 16898}, {9780, 17686}, {11108, 27255}, {11319, 27025}, {11320, 27044}, {13741, 27248}, {16781, 24524}, {17540, 17757}, {17681, 27299}, {17691, 26029}, {17738, 24440}, {26653, 26654}

X(26688) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26688) lies on these lines: {2, 7}, {31, 24003}, {192, 17020}, {321, 17119}, {614, 3952}, {748, 26227}, {899, 4011}, {936, 11319}, {1215, 17125}, {1332, 18743}, {1722, 25253}, {1836, 24988}, {1997, 24597}, {1999, 14997}, {2999, 3995}, {3187, 4358}, {3550, 9458}, {3740, 24552}, {3749, 17780}, {3873, 25531}, {3876, 13741}, {3891, 4009}, {4080, 23681}, {4414, 6686}, {4672, 17124}, {4679, 4972}, {4723, 16483}, {5044, 5192}, {5205, 17127}, {5272, 17165}, {5329, 26262}, {5423, 19993}, {5438, 17539}, {5440, 11346}, {5573, 17154}, {5741, 17279}, {7191, 27538}, {12527, 25881}, {17022, 19717}, {17147, 25268}, {17495, 23511}, {17671, 26589}, {20076, 25879}, {26653, 26654}

X(26689) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26689) lies on these lines: {2, 65}, {72, 27097}, {210, 26759}, {321, 16827}, {392, 26965}, {748, 16822}, {883, 26658}, {894, 24557}, {1201, 17755}, {2176, 3263}, {3752, 25248}, {3876, 27248}, {3877, 27299}, {4358, 17033}, {4676, 16919}, {5057, 17680}, {15254, 16931}, {16910, 24703}, {25895, 27624}, {26653, 26654}, {26660, 26686}

X(26690) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26690) lies on these lines: {1, 644}, {2, 85}, {6, 27396}, {8, 25066}, {9, 604}, {37, 3622}, {39, 4850}, {75, 25244}, {78, 16572}, {100, 2082}, {145, 3693}, {169, 404}, {218, 4511}, {269, 25880}, {304, 27109}, {312, 26770}, {346, 1108}, {355, 26074}, {664, 26653}, {672, 3061}, {883, 26658}, {894, 24554}, {910, 4188}, {934, 7131}, {982, 23649}, {1018, 3885}, {1146, 25005}, {1334, 3890}, {1475, 3873}, {1743, 25078}, {1759, 5030}, {2170, 3501}, {2275, 26242}, {3039, 6691}, {3207, 4881}, {3208, 14439}, {3218, 5022}, {3241, 3991}, {3616, 16601}, {3617, 4875}, {3621, 4515}, {3668, 25966}, {3681, 21384}, {3730, 3877}, {3868, 4253}, {3876, 16552}, {3889, 3970}, {3897, 16788}, {3959, 20331}, {4073, 20978}, {4190, 5819}, {4193, 5179}, {4358, 27523}, {4534, 8256}, {4676, 5701}, {4687, 27058}, {5086, 24247}, {5120, 5279}, {5222, 25083}, {5262, 9605}, {5283, 11342}, {5540, 25440}, {7123, 16502}, {7288, 26258}, {7291, 21477}, {8568, 24982}, {8666, 17744}, {9311, 21272}, {9780, 25068}, {11115, 16699}, {16284, 27096}, {16728, 18600}, {17092, 17282}, {17141, 26065}, {17451, 17754}, {17745, 22836}, {20905, 27334}, {24540, 27420}, {24547, 26059}, {25237, 26964}, {25261, 27146}

X(26691) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26691) lies on these lines: {2, 99}, {190, 26693}, {1577, 5546}, {4366, 26572}, {4558, 15455}, {26666, 26684}, {26668, 26682}, {26692, 26698}

X(26692) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26692) lies on these lines: {2, 11}, {644, 26693}, {1332, 26696}, {26654, 26658}, {26660, 26678}, {26670, 26673}, {26691, 26698}

X(26693) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 48

Barycentrics    (a - b) (a - c) (a^5 - a^4 b + a^2 b^3 - a b^4 - a^4 c - a^2 b^2 c + a b^3 c + b^4 c - a^2 b 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(26693) lies on these lines: {2, 101}, {6, 26572}, {190, 26691}, {644, 26692}, {18047, 26698}, {21859, 24562}

X(26694) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26694) lies on these lines: {2, 649}, {652, 27139}, {3676, 27064}, {21222, 26672}, {26640, 26674}

X(26695) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26695) lies on these lines: {2, 650}, {812, 25955}, {3126, 15283}, {3835, 25900}, {4369, 25924}, {4379, 26017}, {4397, 20315}, {4811, 8062}, {4874, 25926}, {5084, 8760}, {7658, 17896}, {11934, 26105}, {20905, 23757}, {21438, 26591}, {26640, 26674}

X(26696) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26696) lies on these lines: {2, 662}, {190, 26691}, {1332, 26692}

X(26697) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26697) lies on these lines: {2, 667}, {3309, 17541}, {4462, 26687}

X(26698) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26698) lies on these lines: {2, 668}, {106, 25920}, {644, 905}, {1252, 6516}, {4767, 25925}, {8671, 14419}, {18047, 26693}, {26691, 26692}

X(26699) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 48

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

X(26699) lies on these lines: {2, 7}, {6, 26639}, {37, 1332}, {72, 25906}, {190, 26665}, {193, 8557}, {1994, 16470}, {2183, 26998}, {3729, 24209}, {3935, 4073}, {4672, 24563}, {5554, 27549}, {15492, 25887}, {16062, 26878}, {16814, 25099}, {17120, 24559}, {17261, 25245}, {17277, 26538}, {17280, 26594}, {17332, 26543}, {17355, 25007}, {20360, 25024}, {21061, 24090}, {26657, 26669}

leftri

Circumcircle-X-antipodes: X(26700)-X(26717)

rightri

Let C(P) be the circumconic with perspector P = p : q : r (barycentrics), and let U = u : v : w and F = f : g : h be distinct points, with U on C(P). Let U* be the point, other than U, that lies on C(P) and on the line FU. Then

U* = u^2 q r (h v p + f w q + f v r) (g w p + f w q + f v r) : :

If P = X(6), then C(P) is the circumcircle; in this case, the point U* is here named the circumcircle-F-antipode of U, given by

U* = b^2 c^2 u^2 (a^2 h v + b^2 f w + c^2 f v)(a^2 g w + b^2 f w + c^2 f v) : :

Note that the circumcircle-X(3)-antipode of U is the ordinary antipode of U.

Circumcircle-X(1)-antipodes:

{74, 26700}, {99, 741}, {100, 106}, {101, 105}, {102, 108}, {103, 934}, {104, 109}, {107, 26701}{110, 759}, {111, 8691}, {112, 26702}, {689, 719}, {705, 9065}, {727, 932}, {731, 789}, {753, 13396}, {761, 825}, {813, 14665}, {840, 1308}, {898, 2382}, {901, 2718}, {919, 2725}, {927, 12032}, {953, 2222}, {1292, 1477}, {1293, 8686}, {1295, 8059}, {1381, 1382}, {2291, 14074}, {2716, 2720}, {2717, 14733}, {2748, 9097}, {6079, 12029}, {7597, 13444}

Circumcircle-X(2)-antipodes:

{74, 1302}, {98, 110}, {99, 111}, {100, 105}, {101, 675}, {102, 9056}, {103, 9057}, {104, 9058}, {106, 9059}, {107, 1297}, {108, 26703}, {109, 1311}, {112, 2373}, {476, 842}, {477, 9060}, {689, 733}, {691, 2770}, {699, 3222}, {703, 9062}, {707, 9063}, {721, 9065}, {729, 9066}, {739, 9067}, {743, 789}, {753, 9068}, {755, 9069}, {759, 9070}, {761, 9071}, {767, 9072}, {813, 9073}, {815, 9074}, {825, 9075}, {827, 9076}, {831, 9077}, {833, 9078}, {839, 9079}, {843, 9080}, {898, 9081}, {901, 2726}, {919, 2862}, {925, 3563}, {930, 5966}, {932, 9082}, {1113, 1114}, {1290, 2752}, {1292, 9061}, {1293, 9083}, {1294, 9064}, {1295, 9107}, {1296, 9084}, {1304, 2697}, {1305, 9085}, {2291, 9086}, {2367, 9087}, {2370, 9088}, {2374, 3565}, {2384, 9089}, {2696, 10102}, {2715, 2857}, {2856, 9090}, {2858, 14659}, {2868, 9091}, {3067, 9092}, {4588, 9093}, {5970, 9150}, {6013, 9094}, {6014, 9095}, {6015, 9096}, {6079, 9097}, {6082, 9136}, {6135, 9098}, {6136, 9099}, {6323, 9100}, {6325, 11636}, {6572, 9101}, {6579, 9102}, {8652, 9103}, {8686, 9104}, {8694, 9105}, {8698, 9106}, {8701, 9108}, {8706, 9109}, {8708, 9110}, {8709, 9111}, {13397, 15344}

Circumcircle-X(3)-antipodes:

{74, 110}, {98, 99}, {100, 104}, {101, 103}, {102, 109}, {105, 1292}, {106, 1293}, {107, 1294}, {108, 1295}, {111, 1296}, {112, 1297}, {476, 477}, {691, 842}, {741, 6010}, {759, 6011}, {805, 2698}, {813, 12032}, {840, 2742}, {841, 9060}, {843, 2709}, {901, 953}, {915, 13397}, {917, 1305}, {925, 1300}, {927, 2724}, {929, 2723}, {930, 1141}, {932, 15323}, {933, 18401}, {934, 972}, {935, 2697}, {1113, 1114}, {1290, 2687}, {1291, 14979}, {1298, 1303}, {1299, 13398}, {1301, 5897}, {1304, 2693}, {1308, 2717}, {1309, 2734}, {1379, 1380}, {1381, 1382}, {2222, 2716}, {2374, 20187}, {2378, 9202}, {2379, 9203}, {2383, 20185}, {2688, 2690}, {2689, 2695}, {2691, 2752}, {2692, 2758}, {2694, 2766}, {2696, 2770}, {2699, 2703}, {2700, 2702}, {2701, 2708}, {2704, 2711}, {2705, 2712}, {2706, 2713}, {2707, 2714}, {2710, 2715}, {2718, 2743}, {2719, 2744}, {2720, 2745}, {2721, 2746}, {2722, 2747}, {2725, 2736}, {2726, 2737}, {2727, 2738}, {2728, 2739}, {2729, 2740}, {2730, 2751}, {2731, 2757}, {2732, 2762}, {2733, 2765}, {2735, 2768}, {3563, 3565}, {3659, 7597}, {5606, 5951}, {6082, 6093}, {6233, 6323}, {6236, 6325}, {9160, 9161}, {9831, 13241}, {10425, 23700}, {11636, 14388}, {12092, 22751}, {12507, 13238}, {13593, 13594}, {13597, 20189}, {14074, 15731}, {14719, 14720}, {16169, 16170}

Circumcircle-X(4)-antipodes:

{74, 107}, {98, 112}, {99, 3563}, {100, 915}, {101, 917}, {102, 26704}, {103, 26705}, {104, 108}, {105, 26706}, {110, 1300}, {477, 1304}, {842, 935}, {925, 1299}, {930, 2383}, {933, 1141}, {953, 1309}, {1113, 1114}, {1289, 1297}, {1292, 15344}, {1294, 1301}, {1296, 2374}, {2687, 2766}, {2693, 22239}, {2697, 10423}, {2698, 22456}, {2752, 10101}, {2770, 10098}, {18401, 20626}

Circumcircle-X(5)-antipodes:

{98, 827}, {99, 5966}, {100, 26797}, {101,26708}, {102,26709}, {103, 26710}, {104, 26711}, {105, 26712}, {106, 26713}, {107, 18401}, {110, 1141}, {476, 14979}, {477, 16166}, {842, 1287}, {925, 2383}, {1113, 1114}

Circumcircle-X(6)-antipodes:

{74, 112}, {98, 26714}, {99, 729}, {100, 739}, {101, 106}, {102, 26715}, {103, 26716}, {105, 8693}, {107, 26717}, {109, 2291}, {110, 111}, {689, 703}, {691, 843}, {699, 25424}, {717, 789}, {753, 825}, {755, 827}, {805, 5970}, {813, 2382}, {840, 919}, {842, 2715}, {901, 2384}, {1293, 17222}, {1379, 1380}, {2378, 5995}, {2379, 5994}, {2380, 16806}, {2381, 16807}, {2702, 2712}, {2709, 9136}, {3222, 6380}, {6078, 9097}, {6323, 11636}, {8694, 17223}, {8696, 8697}, {8700, 8701}, {10425, 14659}, {11651, 11652}

Circumcircle-X(7)-antipodes:

{100, 15728}, {101, 2369}, {104, 934}, {105, 6183}, {109, 675}, {840, 927}, {2720, 2861}, {2723, 24016}

Circumcircle-X(8)-antipodes: {100, 104}, {101, 1311}, {109, 2370}, {901, 2757}, {1309, 2745}
Circumcircle-X(9)-antipodes: {100, 2291}, {101, 104}, {813, 2726}, {919, 2751}, {934, 2371}
Circumcircle-X(10)-antipodes: {98, 101}, {100, 759}, {106, 8706}, {110, 2372}, {901, 2758}, {929, 2708}
Circumcircle-X(11)-antipodes: {100, 105}, {104, 108}, {110, 19628}
Circumcircle-X(12)-antipodes: {109, 2372}, {2222, 12030}
Circumcircle-X(13)-antipodes: {74, 5618}, {98, 5995}, {99, 2381}, {476, 2379}, {1141, 16806}
Circumcircle-X(14)-antipodes: {74, 5619}, {98, 5994}, {99, 2380}, {476, 2378}, {1141, 16807}
Circumcircle-X(15)-antipodes: {74, 5995}, {110, 2378}, {111, 9202}, {691, 2379}, {842, 5994}, {843, 9203}, {1379, 1380}, {2380, 10409}
Circumcircle-X(16)-antipodes: {74, 5994}, {110, 2379}, {111, 9203}, {691, 2378}, {842, 5995}, {843, 9202}, {1379, 1380}, {2381, 10410}
Circumcircle-X(17)-antipodes: {98, 16806}, {930, 2381}
Circumcircle-X(18)-antipodes: {98, 16807}, {930, 2380}
Circumcircle-X(19)-antipodes: {100, 9085}, {101, 915}, {107, 2249}, {108, 2291}, {109, 20624}, {112, 759}

Circumcircle-X(20)-antipodes: {20, {{74, 925}, {98, 3565}, {99, 1297}, {100, 1295}, {103, 1305}, {104, 13397}, {107, 5897}, {110, 1294}, {111, 20187}, {476, 2693}, {477, 10420}, {691, 2697}, {841, 16167}, {901, 2734}, {930, 18401}, {1113, 1114}, {1141, 20185}, {1290, 2694}, {1293, 2370}, {1296, 2373}, {1300, 13398}

Circumcircle-X(21)-antipodes: {99, 105}, {100, 759}, {104, 110}, {107, 1295}, {476, 2687}, {691, 2752}, {741, 932}, {915, 925}, {1113, 1114}, {1290, 12030}, {1296, 9061}, {1304, 2694}, {3565, 15344}, {8686, 8690}

Circumcircle-X(22)-antipodes: {98, 925}, {99, 2373}, {105, 13397}, {110, 1297}, {111, 3565}, {476, 2697}, {477, 16167}, {675, 1305}, {842, 10420}, {1113, 1114}, {1294, 1302}, {1295, 9058}, {2370, 9059}, {2693, 9060}, {3563, 13398}, {5897, 9064}, {5966, 20185}, {9084, 20187}

Circumcircle-X(23)-antipodes: {74, 9060}, {98, 476}, {99, 2770}, {100, 2752}, {105, 1290}, {107, 2697}, {110, 842}, {111, 691}, {477, 1302}, {675, 2690}, {935, 2373}, {1113, 1114}, {1287, 9076}, {1291, 5966}, {1296, 10102}, {1297, 1304}, {1300, 16167}, {1311, 2689}, {2687, 9058}, {2688, 9057}, {2691, 9061}, {2692, 9083}, {2693, 9064}, {2694, 9107}, {2695, 9056}, {2696, 9084}, {2758, 9059}, {3563, 10420}, {9070, 12030}, {20185, 23096}

Circumcircle-X(24)-antipodes: {74, 1301}, {98, 1289}, {107, 1300}, {108, 915}, {110, 1299}, {112, 3563}, {477, 22239}, {842, 10423}, {933, 2383}, {1113, 1114}, {1141, 20626}

Circumcircle-X(25)-antipodes: {74, 9064}, {98, 107}, {99, 2374}, {100, 15344}, {101, 9085}, {104, 9107}, {105, 108}, {106, 9088}, {110, 3563}, {111, 112}, {842, 1304}, {915, 9058}, {917, 9057}, {933, 5966}, {935, 2770}, {1113, 1114}, {1289, 2373}, {1291, 23096}, {1297, 1301}, {1300, 1302}, {1309, 2726}, {2697, 22239}, {2752, 2766}, {10098, 10102}

Circumcircle-X(26)-antipodes: {98, 1286}, {1113, 1114}

Circumcircle-X(27)-antipodes: {99, 9085}, {103, 107}, {110, 917}, {112, 675}, {1113, 1114}, {1304, 2688}

Circumcircle-X(28)-antipodes: {99, 15344}, {104, 107}, {105, 112}, {108, 759}, {110, 915}, {935, 2752}, {1113, 1114}, {1295, 1301}, {1304, 2687}, {2694, 22239}, {2766, 12030}

Circumcircle-X(29)-antipodes: {102, 107}, {112, 1311}, {1113, 1114}, {1304, 2695}

This preamble was contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas and centers), November 2, 2018.

X(26700) =  CIRCUMCIRCLE-(X(1))-ANTIPODE OF X(74)

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

X(26700) lies on these lines: {1, 74}, {35, 5951}, {36, 2687}, {56, 759}, {57, 2611}, {79, 104}, {100, 4458}, {102, 1385}, {103, 354}, {105, 5322}, {110, 9811}, {162, 1304}, {226, 14844}, {265, 12773}, {554, 11705}, {651, 8652}, {739, 16488}, {842, 18593}, {972, 8606}, {1020, 15439}, {1081, 11706}, {1108, 2160}, {1295, 11012}, {1414, 6578}, {1429, 2711}, {1464, 14158}, {2716, 22765}, {4551, 8701}, {5427, 12030}, {8707, 15455}, {20219, 23890}

X(26700) = isogonal conjugate of X(35057)
X(26700) = cevapoint of X(i) and X(j) for these {i,j}: {56, 4017}, {513, 32636}, {523, 27555}
X(26700) = crosssum of X(1) and X(9904)
X(26700) = trilinear pole of line X(6)X(1406)
X(26700) = Ψ(X(6), X(1406))
X(26700) = Λ(X(1), X(656))
X(26700) = Ψ(X(1), X(30))
X(26700) = Ψ(X(4), X(79))
X(26700) = X(14656)-of-intouch-triangle

X(26701) =  CIRCUMCIRCLE-(X(1))-ANTIPODE OF X(107)

Barycentrics    a^2*(a + b)*(a + c)*(-(a^4*b^2) + 2*a^2*b^4 - b^6 + a^5*c + 2*a^3*b^2*c - 3*a*b^4*c - 2*a^2*b^2*c^2 + 2*b^4*c^2 - 2*a^3*c^3 + 2*a*b^2*c^3 - b^2*c^4 + a*c^5)*(a^5*b - 2*a^3*b^3 + a*b^5 - a^4*c^2 + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 + 2*a^2*c^4 - 3*a*b*c^4 + 2*b^2*c^4 - c^6) : :
Trilinears    1/(b (tan A - tan B) + c (tan A - tan C)) : :
Trilinears    1/((b + c) tan A - b tan B - c tan C)) : :

X(26701) lies on these lines: {1, 107}, {36, 2719}, {48, 112}, {56, 1363}, {58, 8059}, {73, 108}, {99, 326}, {100, 1816}, {101, 3990}, {109, 2360}, {110, 255}, {336, 22456}, {933, 2169}, {1113, 2585}, {1114, 2584}, {1301, 19614}, {1309, 3465}, {2734, 3737}, {2762, 10535}

X(26701) = Ψ(X(1), X(520))
X(26701) = Λ(X(1), X(29))
X(26701) = trilinear product of circumcircle intercepts of line X(1)X(520)

X(26702) =  CIRCUMCIRCLE-(X(1))-ANTIPODE OF X(112)

Barycentrics    a*(a + b)*(a + c)*(a^4 - a^3*b - a*b^3 + b^4 + a*b*c^2 - c^4)*(a^4 - b^4 - a^3*c + a*b^2*c - a*c^3 + c^4) : :
Trilinears    1/(b (tan A - tan B) + c (tan A - tan C)) : :
Trilinears    1/((b + c) tan A - b tan B - c tan C)) : :
X(26702) lies on these lines: {1, 112}, {21, 934}, {36, 2722}, {56, 1367}, {63, 110}, {72, 101}, {92, 107}, {99, 304}, {100, 306}, {105, 4458}, {108, 226}, {109, 1214}, {293, 2715}, {859, 2728}, {919, 3509}, {933, 2167}, {1113, 2583}, {1114, 2582}, {1290, 5057}, {1301, 2184}, {1304, 2349}, {1412, 8059}, {1444, 6183}, {2751, 3733}, {6011, 7580}

X(26702) = trilinear pole of line X(6)X(656)
X(26702) = Λ(X(65), X(1439))
X(26702) = Ψ(X(1), X(525))
X(26702) = Ψ(X(6), X(656))
X(26702) = trilinear product of circumcircle intercepts of line X(1)X(525)
X(26702) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(1),X(63)}}

X(26703) =  CIRCUMCIRCLE-(X(2))-ANTIPODE OF X(108)

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

X(26703) lies on these lines: {2, 108}, {20, 1292}, {21, 112}, {22, 100}, {23, 2766}, {25, 2968}, {28, 1289}, {30, 10101}, {63, 109}, {78, 101}, {99, 16049}, {107, 4228}, {110, 1812}, {348, 934}, {858, 1290}, {915, 7427}, {917, 7445}, {919, 3100}, {927, 7112}, {929, 10538}, {935, 1325}, {1295, 2417}, {1300, 7425}, {1301, 4233}, {1302, 26268}, {1304, 7469}, {1370, 13397}, {1791, 8687}, {1995, 9107}, {2071, 2691}, {2074, 10423}, {2374, 7458}, {2731, 5205}, {2856, 6563}, {3563, 7423}, {7219, 22654}, {7493, 9058}, {9056, 26227}, {9057, 26265}, {9070, 26253}, {9086, 26260}, {13577, 22769}

X(26703) = isogonal conjugate of X(3827)
X(26703) = anticomplement of X(20621)
X(26703) = trilinear pole of line X(6)X(521)
X(26703) = Ψ(X(6), X(521))
X(26703) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(123)
X(26703) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(34188)
X(26703) = de-Longchamps-circle-inverse of X(20344)

X(26704) =  CIRCUMCIRCLE-(X(4))-ANTIPODE OF X(102)

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

X(26704) lies on these lines: {4, 102}, {25, 1311}, {74, 15232}, {99, 7463}, {100, 7461}, {103, 13478}, {104, 2217}, {106, 1068}, {109, 23987}, {110, 1897}, {186, 2695}, {242, 2717}, {925, 7450}, {1294, 7421}, {1295, 6906}, {1297, 7413}, {1305, 7460}, {1824, 19607}, {2365, 7046}, {2370, 7428}, {2373, 7449}, {2708, 17927}, {2995, 20901}, {3565, 7462}, {7451, 13397}

X(26704) = anticomplement of X(38977)
X(26704) = Ψ(X(3), X(10))
X(26704) = Ψ(X(6), X(1826))
X(26704) = trilinear pole of line X(6)X(1826)
X(26704) = inverse-in-polar-circle of X(124)
X(26704) = X(63)-isoconjugate of X(6589)
X(26704) = perspector, wrt 2nd circumperp triangle, of polar circle

X(26705) =  CIRCUMCIRCLE-(X(4))-ANTIPODE OF X(103)

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

X(26705) lies on these lines: {3, 21665}, {4, 103}, {24, 917}, {25, 675}, {74, 15320}, {99, 4249}, {100, 4250}, {102, 5603}, {110, 3732}, {186, 2688}, {242, 2725}, {925, 4243}, {1006, 1295}, {1294, 7430}, {1297, 6998}, {1783, 8693}, {1897, 8701}, {2370, 4245}, {2373, 7453}, {3565, 4237}, {6353, 9085}, {6577, 8750}, {7437, 13397}, {7479, 10420}

X(26705) = polar conjugate of X(25259)
X(26705) = trilinear pole of line X(6)X(1836)
X(26705) = Ψ(X(6), X(1836))
X(26705) = inverse-in-polar-circle of X(116)
X(26705) = reflection of X(4) in X(20622)
X(26705) = X(63)-isoconjugate of X(6586)

X(26706) =  CIRCUMCIRCLE-(X(4))-ANTIPODE OF X(105)

Barycentrics    a*(a - b)*(a - c)*(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 - b^2*c + a*c^2 + b*c^2 - c^3)*(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(26706) lies on these lines: {4, 105}, {24, 15344}, {25, 9061}, {98, 7414}, {102, 18446}, {104, 378}, {107, 4244}, {110, 4238}, {111, 4231}, {186, 2752}, {376, 1295}, {476, 7476}, {523, 10101}, {675, 4219}, {759, 4227}, {915, 18533}, {919, 1783}, {925, 4236}, {927, 18026}, {972, 11491}, {1068, 2376}, {1297, 3651}, {1302, 4246}, {1311, 7412}, {1565, 7071}, {1897, 13397}, {2373, 4220}, {2687, 10295}, {2694, 7464}, {4222, 9083}, {4242, 9058}, {4250, 9057}, {7435, 9064}, {7438, 9084}, {7461, 9056}, {7475, 10420}, {7477, 16167}

X(26707) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(100)

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

X(26707) lies on these lines: {5, 100}, {21, 930}, {28, 933}, {101, 1953}, {108, 3518}, {109, 1393}, {110, 6583}, {901, 10225}, {1290, 2070}, {1291, 1325}, {7423, 9076}, {7488, 13397}, {9058, 13595}, {16049, 20185}

X(26708) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(101)

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

X(26708) lies on these lines: {5, 101}, {27, 933}, {100, 14213}, {109, 11246}, {110, 17167}, {930, 4184}, {1291, 5196}, {1305, 7488}, {2070, 2690}, {7432, 9076}, {9057, 13595}

X(26709) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(102)

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

X(26709) lies on these lines: {5, 102}, {930, 7450}, {933, 7452}, {1311, 13595}, {2070, 2695}, {7449, 9076}

X(26710) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(103)

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

X(26710) lies on these lines: {5, 103}, {675, 13595}, {917, 3518}, {930, 4243}, {933, 4241}, {1291, 7479}, {2070, 2688}, {7453, 9076}

X(26711) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(104)

Barycentrics    a*(a - b)*(a - c)*(a^3 + a^2*b + a*b^2 + b^3 - a^2*c + a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(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(26711) lies on these lines: {5, 104}, {74, 13145}, {102, 11014}, {105, 13595}, {915, 3518}, {930, 3658}, {933, 4246}, {1291, 7477}, {1295, 7488}, {1311, 26263}, {1633, 8697}, {1897, 2766}, {2070, 2687}, {2694, 3153}, {4239, 9076}, {7435, 20626}

X(26712) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(105)

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

X(26712) lies on these lines: {5, 105}, {915, 7576}, {930, 4236}, {933, 4238}, {1291, 7475}, {2070, 2752}, {3518, 15344}, {4220, 9076}, {4244, 20626}, {9061, 13595}

X(26713) =  CIRCUMCIRCLE-(X(5))-ANTIPODE OF X(106)

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

X(26713) lies on these lines: {5, 106}, {2070, 2758}, {2370, 7488}, {9083, 13595}

X(26714) =  CIRCUMCIRCLE-(X(6))-ANTIPODE OF X(98)

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

X(26714) lies on these lines: {3, 14252}, {6, 98}, {74, 574}, {99, 1625}, {110, 14966}, {111, 263}, {163, 8685}, {187, 2698}, {323, 2857}, {327, 2367}, {352, 2770}, {353, 11593}, {648, 22456}, {689, 4563}, {729, 1384}, {733, 17970}, {741, 3402}, {759, 2186}, {842, 5104}, {1141, 11060}, {1296, 5118}, {1297, 3098}, {1576, 2715}, {2030, 5970}, {2373, 15066}, {2420, 11636}, {3288, 6037}, {5467, 6233}, {5468, 9066}, {9181, 13241}

X(26714) = Ψ(X(i), X(j)) for these (i,j): (2, 51), (4, 39), (6, 160), (76, 5)
X(26714) = trilinear pole of line X(6)X(160)
X(26714) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(523)
X(26714) = circumcircle intercept, other than X(98), of circle {{X(15),X(16),X(98)}} (or V(X(98))
X(26714) = X(182)-isoconjugate of X(1577)
X(26714) = isogonal conjugate of X(23878)
X(26714) = barycentric product X(110)*X(262)
X(26714) = barycentric quotient X(262)/X(850)
X(26714) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(523)
X(26714) = barycentric product of circumcircle intercepts of line X(2)X(51)

X(26715) =  CIRCUMCIRCLE-(X(6))-ANTIPODE OF X(102)

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

X(26715) lies on these lines: {6, 102}, {103, 4257}, {105, 16485}, {109, 2425}, {187, 2708}, {1293, 1983}, {1384, 2291}, {2750, 5526}

X(26715) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(652)
X(26715) = circumcircle intercept, other than X(102), of circle {{X(15),X(16),X(102)}} (or V(X(102))

X(26716) =  CIRCUMCIRCLE-(X(6))-ANTIPODE OF X(103)

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

X(26716) lies on these lines: {6, 103}, {101, 2426}, {102, 4262}, {105, 16487}, {106, 1384}, {163, 5545}, {187, 2700}, {906, 6575}, {1461, 24016}, {2030, 2712}

X(26716) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(657)
X(26716) = circumcircle intercept, other than X(103), of circle {{X(15),X(16),X(103)}} (or V(X(103))

X(26717) =  CIRCUMCIRCLE-(X(6))-ANTIPODE OF X(107)

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

X(26717) lies on these lines: {6, 107}, {99, 394}, {100, 3990}, {101, 4055}, {108, 1409}, {110, 577}, {112, 184}, {187, 2713}, {287, 22456}, {353, 12507}, {933, 14533}, {935, 13509}, {1294, 2430}, {1301, 14642}, {1304, 1971}, {1629, 1988}, {9064, 10311}, {15032, 23232}

X(26717) = Λ(X(2), X(216))
X(26717) = circumcircle intercept, other than X(107), of circle {{X(15),X(16),X(107)}} (or V(X(107))

leftri

Centers associated with the Gemini triangles 1-10: X(26718)-X(26751)

rightri

These centers were contributed by Randy Hutson, November 2, 2018. The Gemini triangles are introduced in the preamble just before X(24537).

X(26718) = CENTROID OF GEMINI TRIANGLE 7

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

X(26718) lies on these lines: {1, 6692}, {1125, 8834}, {1698, 6552}, {1699, 26719}

X(26718) = reflection of X(1699) in X(26719)

X(26719) = CENTROID OF MID-TRIANGLE OF GEMINI TRIANGLES 7 AND 8

Barycentrics    a^6 - 2 a^5 (b + c) + a^4 (5 b^2 - 6 b c + 5 c^2) - 16 a^3 (b - c)^2 (b + c) - a^2 (b - c)^2 (b^2 - 54 b c + c^2) + 6 a (b - 3 c) (3 b - c) (b - c)^2 (b + c) - (b^2 - c^2)^2 (5 b^2 - 14 b c + 5 c^2) : :

X(26719) lies on these lines: {5, 6552}, {1699, 26718}, {3091, 6553}

X(26719) = midpoint of X(1699) and X(26718)

X(26720) =  CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 7 AND 8

Barycentrics    2 a^4 (b^2 + c^2) - 2 a^3 (b^3 + c^3) - a^2 (2 b^4 - 17 b^3 c + 22 b^2 c^2 - 17 b c^3 + 2 c^4) + 2 a (b - c)^4 (b + c) + b c (b^2 - c^2)^2 : :

X(26720) lies on this line: {1210, 3953}

X(26721) = PERSPECTOR OF THE {GEMINI 7, GEMINI 8}-CIRCUMCONIC

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

X(26721) lies on these lines: {514, 2082}, {905, 918}, {1734, 4025}, {3309, 4897}

X(26722) = EIGENCENTER OF GEMINI TRIANGLE 8

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

X(26722) lies on these lines: {7, 101}, {314, 7259}, {5526, 9442}

X(26723) = PERSPECTOR OF GEMINI TRIANGLE 2 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 7

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

X(26723) lies on these lines: {1, 2}, {6, 5249}, {27, 162}, {31, 1738}, {44, 3782}, {57, 15474}, {63, 4000}, {75, 5294}, {81, 142}, {238, 3914}, {278, 1445}, {377, 1453}, {908, 2911}, {1086, 4641}, {1194, 16583}, {1203, 12609}, {1211, 17348}, {1386, 3925}, {1427, 5723}, {1621, 3755}, {1708, 22464}, {1724, 23537}, {1743, 5905}, {1746, 12610} et al

X(26724) = PERSPECTOR OF GEMINI TRIANGLE 7 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 7

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

X(26724) lies on these lines: {2, 37}, {44, 17483}, {63, 4859}, {81, 142}, {277, 15474}, {404, 1612}, {748, 5057}, {1086, 3219}, {1621, 1738} et al

X(26725) = PERSPECTOR OF GEMINI TRIANGLE 3 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 3 AND 7

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

X(26725) lies on these lines: {1, 442}, {2, 758}, {10, 5425}, {21, 36}, {30, 1699}, {35, 12609}, {57, 191}, {80, 3822}, {140, 5535}, {142, 10090}, {214, 5424}, {226, 5251}, {451, 1835}, {484, 6690}, {517, 11218}, {551, 6175}, {946, 3651}, {1001, 16581}, {1479, 2475}, {1698, 11374}, {1790, 2126}, {2646, 3824} et al

X(26726) = PERSPECTOR OF GEMINI TRIANGLE 6 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 6 AND 7

Barycentrics    3 a^4 - 5 a^3 (b + c) - a^2 (2 b^2 - 15 b c + 2 c^2) + a (b + c) (5 b^2 - 12 b c + 5 c^2) - (b^2 - c^2)^2 : :

X(26726) lies on these lines: {1, 1145}, {8, 6702}, {11, 3632}, {35, 13278}, {36, 25438}, {57, 1317}, {80, 519}, {100, 3244}, {104, 5537}, {119, 16200}, {145, 2802}, {149, 20050}, {214, 3241}, {952, 3627}, {1387, 3679}, {1482, 12611}, {1537, 11224} et al

X(26727) = PERSPECTOR OF GEMINI TRIANGLE 7 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 6 AND 7

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

X(26727) lies on these lines: {1, 1145}, {8, 244}, {10, 3699}, {80, 900}, {88, 2581}, {106, 519}, {109, 4848}, {141, 3679}, {291, 2401}, {644, 21950}, {905, 9260}, {952, 1054}, {986, 5554}, {1046, 14985}, {1086, 3036}, {1320, 1647}, {1421, 1722}, {1772, 10573} et al

X(26728) = PERSPECTOR OF GEMINI TRIANGLE 1 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 1 AND 8

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

X(26728) lies on these lines: {1, 224}, {31, 11551}, {86, 99}, {553, 4257}, {595, 3671}, {982, 1125}, {990, 5603}, {1086, 24929}, {1104, 6147} et al

X(26729) = PERSPECTOR OF GEMINI TRIANGLE 8 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 1 AND 8

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

X(26729) lies on these lines: {1, 11015}, {946, 3315}, {1104, 17483}, {1714, 3868}, {3487, 4850}, {3649, 7191}, {3984, 4859} et al

X(26730) = PERSPECTOR OF GEMINI TRIANGLE 4 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 4 AND 8

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

X(26730) lies on these lines: {79, 1757}, {3914, 4416}, {5223, 24851} et al

X(26731) = PERSPECTOR OF GEMINI TRIANGLE 8 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 4 AND 8

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

X(26731) lies on these lines: {69, 4683}, {79, 3751}, {193, 17491}, {518, 24851}, {1756, 7289} et al

X(26732) = X(30)X(511)∩X(3700)X(4560)

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

X(26732) is the infinite point of the perspectrix of Gemini triangles 2 and 7.

X(26732) lies on these lines: {30, 511}, {3700, 4560}, {4391, 4976} et al

X(26733) = ISOGONAL CONJUGATE OF X(26732)

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

X(26733) lies on the circumcircle and these lines: {1415, 8652}, {2291, 10460}, {4559, 8701}, {4565, 6578}

X(26734) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 2 AND 7

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

X(26734) lies on these lines: {313, 3260}, {321, 3578}

X(26734) = trilinear pole of line X(1577)X(26732)

X(26735) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 3 AND 7

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

X(26735) lies on these lines: {40, 3729}, {223, 9312}

X(26735) = trilinear pole of line X(2517)X(4885)

X(26736) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 6 AND 7

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

X(26736) lies on this line: {3729, 3732}

X(26736) = trilinear pole of line X(4000)X(4885)

X(26737) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 1 AND 8

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

X(26737) lies on these lines: (pending)

X(26738) = CENTROID OF GEMINI TRIANGLE 9

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

X(26738) lies on these lines: {1, 10031}, {2, 44}, {88, 6173}, {226, 22464}, {651, 5219}, {1086, 4850} et al

X(26739) = CENTROID OF GEMINI TRIANGLE 10

Barycentrics    16 a^3 - 13 a^2 (b + c) - a (19 b^2 - 47 b c + 19 c^2) + 10 (b - c)^2 (b + c) : :

X(26739) lies on this line: {2, 4912}

X(26740) = PERSPECTOR OF GEMINI TRIANGLE 9 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 9 AND 10

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

X(26740) lies on these lines: {1, 6940}, {2, 26741}, {42, 5083}, {57, 77}, {226, 1086}, {241, 9328}, {354, 24025}, {553, 1465}, {1319, 4868}, {1427, 4031}, {1450, 4424} et al

X(27040) = complement of X(18600)
X(26740) = {X(2),X(26742)}-harmonic conjugate of X(26741)

X(26741) = PERSPECTOR OF GEMINI TRIANGLE 10 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 9 AND 10

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

X(26741) lies on these lines: {2, 26740}, {43,5083}, {57,88}, {216,1108}, {1450,1739} et al

X(26741) = {X(2),X(26742)}-harmonic conjugate of X(26740)

X(26742) = {X(26740),X(26741)}-HARMONIC CONJUGATE OF X(2)

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

X(26742) lies on these lines: {2, 26740}, {6, 57}, {484, 1480}, {614, 3256}, {2006, 4000} et al

X(26742) = {X(26740),X(26741)}-harmonic conjugate of X(2)

X(26743) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 9 AND 10

Barycentrics    1/((a - b - c) (a^2 - b^2 - c^2 + b c) (a^3 - b^3 - c^3 + a^2 b + a^2 c - a b^2 - a c^2 - a b c + b^2 c + b c^2)) : :

X(26743) lies on these lines: {30, 80}, {2006, 6357}, {14206, 17484}

X(26743) = isogonal conjugate of X(26744)

X(26744) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 9 AND 10

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

X(26744) lies on these lines: {3, 16554}, {9, 1030}, {35, 2161}, {36, 2245}, {37, 14579}, {44, 11063}, {55, 4516}, {71, 74}, {198, 16553}, {284, 2316}, {484, 19297}, {2077, 2173} et al

X(26744) = isogonal conjugate of X(26743)

X(26745) = ISOGONAL CONJUGATE OF X(16885)

Barycentrics    a/(2 b + 2 c - 3 a) : :
Trilinears    1/(2 b + 2 c - 3 a) : :

Let A10B10C10 be the Gemini triangle 10. Let LA be the line through A10 parallel to BC, and define LB, LC cyclically. Let A'10 = LB∩C, and define B'10, C'10 cyclically. Triangle A'10B'10C'10 is homothetic to ABC at X(26745).

X(26745) lies on these lines: {1, 1392}, {2, 4912}, {88, 4383}, {89, 3752}, {105, 8697}, {1022, 4498}, {1054, 4430}, {1219, 4678}, {1224, 19877} et al

X(26745) = isogonal conjugate of X(16885)

X(26746) = PERSPECTOR OF GEMINI TRIANGLE 2 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 10

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

X(26746) lies on these lines: {2, 313}, {333, 2275}, {4850, 6703}

X(26747) = PERSPECTOR OF GEMINI TRIANGLE 10 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 10

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

X(26747) lies on these lines: {2,313}, {81,1193}, {1575,3969}, {2275,5278}, {2277,19684} et al

X(26748) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 1 AND 9

Barycentrics    1/(a^6 - 3 a^4 (b^2 - b c + c^2) + 5 a^3 b c (b + c) + a^2 (b - 3 c) (3 b - c) (b + c)^2 - a b c (b + c) (5 b^2 - 14 b c + 5 c^2) - (b^2 - c^2)^2 (b^2 - b c + c^2)) : :

X(26748) lies on these lines: (pending)

X(26749) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 2 AND 9

Barycentrics    1/(a^4 - 4 a^3 (b + c) + 9 a^2 b c + a (b + c) (4 b^2 - 7 b c + 4 c^2) - (b + c)^2 (b^2 - b c + c^2)) : :

X(26749) lies on this line: {545, 3218}

X(26749) = trilinear pole of line X(3960)X(14475)

X(26750) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 1 AND 10

Barycentrics    1/(a^6 - a^4 (3 b^2 + 5 b c + 3 c^2) - a^3 b c (b + c) + a^2 (b + c)^2 (3 b^2 - 2 b c + 3 c^2) + a b c (b + c)^3 - (b^2 - c^2)^2 (b^2 - b c + c^2)) : :

X(26750) lies on these lines: (pending)

X(26751) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 2 AND 10

Barycentrics    1/(a^4 - b^4 - c^4 + a^2 b c - a b^2 c - a b c^2 - b^3 c - b c^3) : :

The perspectrix of Gemini triangles 2 and 10 passes through X(14838).

X(26751) lies on these lines: {1211, 3219}, {4357, 5267}

X(26751) = isotomic conjugate of X(36974)

leftri

Collineation mappings involving Gemini triangle 49: X(26752)-X(26802)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 49, as in centers X(26752)-X(26802). Then

m(X) = a (b + c)^2 x + b (a - c)^2 y + c (a - b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 3, 2018)

X(26752) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    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(26752) lies on these lines: {1, 2}, {12, 26582}, {35, 17692}, {37, 25107}, {39, 668}, {41, 17743}, {55, 16916}, {69, 26042}, {75, 21021}, {76, 17759}, {100, 384}, {192, 1921}, {194, 17756}, {274, 1574}, {335, 24443}, {350, 20691}, {404, 6645}, {495, 17670}, {874, 17280}, {891, 27140}, {956, 11285}, {958, 17684}, {1018, 26765}, {1078, 5291}, {1107, 25280}, {1278, 21443}, {1329, 17669}, {1376, 16915}, {1500, 18140}, {1575, 1909}, {1621, 16918}, {1654, 26082}, {1655, 2276}, {2238, 27033}, {2275, 9263}, {2277, 17786}, {2975, 7824}, {3035, 26686}, {3249, 27013}, {3421, 16043}, {3434, 16924}, {3436, 7791}, {3501, 24514}, {3693, 25994}, {3701, 3797}, {3758, 26076}, {3871, 4366}, {3934, 17143}, {3952, 25248}, {3959, 18055}, {4429, 16906}, {4557, 18099}, {4595, 27103}, {4645, 26058}, {4967, 25538}, {4986, 24786}, {5025, 11681}, {5080, 6655}, {5263, 20148}, {5687, 7770}, {6381, 25264}, {6625, 26072}, {6646, 26756}, {6653, 16044}, {6656, 17757}, {7785, 20553}, {7786, 16975}, {9709, 11321}, {11680, 16921}, {12607, 26561}, {12782, 17794}, {16284, 25918}, {16549, 17499}, {16604, 25303}, {16720, 20955}, {17243, 27111}, {17279, 25610}, {17295, 26979}, {17299, 25505}, {17300, 20561}, {17301, 26142}, {17302, 26100}, {17303, 26110}, {17314, 26107}, {17395, 25534}, {17693, 25440}, {18040, 24530}, {18047, 21008}, {21031, 26558}, {21264, 21868}, {23632, 25286}, {24491, 27073}, {24502, 27136}, {24509, 26685}, {25570, 26135}, {26753, 26790}, {26762, 26771}, {26784, 26789}, {26797, 26799}, {27039, 27296}

X(26752) = anticomplement of X(26959)

X(26753) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26753) lies on these lines: {2, 3}, {315, 27515}, {3177, 18738}, {26752, 26790}, {26756, 26763}, {26757, 26758}, {26770, 26794}

X(26754) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26754) lies on these lines: {2, 3}, {69, 4513}, {6646, 26759}

X(26755) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26755) lies on these lines: {2, 3}, {26686, 27027}

X(26756) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    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(26756) lies on these lines: {1, 25292}, {2, 6}, {9, 27073}, {76, 1278}, {190, 26797}, {192, 4033}, {239, 27011}, {319, 26971}, {320, 27102}, {330, 27671}, {594, 26976}, {894, 27044}, {3009, 25284}, {3879, 27166}, {4361, 26850}, {4446, 17154}, {4699, 26817}, {4741, 26042}, {6646, 26752}, {7232, 27107}, {16706, 27106}, {16816, 27192}, {17121, 26982}, {17227, 27311}, {17228, 27261}, {17252, 27020}, {17256, 27032}, {17263, 27036}, {17280, 26774}, {17288, 27017}, {17292, 27078}, {17350, 21362}, {17353, 27113}, {17360, 25505}, {17364, 27091}, {17373, 26107}, {17495, 21857}, {21244, 26589}, {26048, 26806}, {26149, 26812}, {26753, 26763}, {26762, 26766}

X(26757) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26757) lies on these lines: {1, 2}, {668, 26770}, {3620, 26836}, {3991, 25261}, {4023, 27256}, {4445, 16713}, {4515, 26563}, {4595, 17152}, {17233, 27039}, {17280, 26787}, {17286, 27058}, {17373, 26818}, {26753, 26758}, {26780, 26790}, {26797, 26800}

X(26758) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26758) lies on these lines: {2, 6}, {2476, 4678}, {4033, 4671}, {4651, 21241}, {19998, 25760}, {26753, 26757}

X(26759) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    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(26759) lies on these lines: {1, 2}, {38, 25248}, {69, 20109}, {141, 17152}, {210, 26689}, {257, 21272}, {335, 17164}, {668, 27040}, {1018, 16887}, {1500, 16705}, {1621, 16931}, {3434, 16910}, {3662, 20244}, {3775, 27047}, {3871, 16060}, {4390, 24549}, {5263, 16930}, {5484, 20533}, {5836, 26562}, {6645, 11115}, {6646, 26754}, {7187, 25244}, {8682, 21802}, {12135, 15149}, {14210, 25263}, {16600, 17497}, {16920, 20139}, {16975, 27109}, {17141, 24254}, {17143, 26978}, {17280, 21226}, {17289, 25303}, {17759, 18600}, {18047, 26843}, {26781, 26795}, {26787, 26792}

X(26760) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26760) lies on these lines: {2, 3}

X(26761) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26761) lies on these lines: {2, 3}

X(26762) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26762) lies on these lines: {2, 31}, {26752, 26771}, {26756, 26766}, {26767, 26795}

X(26763) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26763) lies on these lines: {2, 32}, {26753, 26756}, {26770, 26788}

X(26764) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^2 b^2 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 : :

X(26764) lies on these lines: {2, 37}, {8, 5145}, {141, 26774}, {190, 26772}, {291, 25295}, {573, 17350}, {594, 16738}, {894, 3882}, {1100, 26975}, {1964, 20044}, {2309, 20352}, {3912, 27017}, {3943, 26979}, {3946, 26982}, {4033, 16696}, {4357, 27044}, {4360, 26821}, {4389, 27095}, {4393, 5105}, {4436, 18082}, {6542, 17178}, {6646, 26752}, {7184, 25284}, {7227, 27042}, {16814, 27036}, {17116, 27020}, {17142, 24327}, {17233, 27145}, {17234, 27107}, {17235, 27106}, {17247, 27091}, {17291, 27113}, {17300, 26816}, {17319, 27166}, {17355, 27078}, {21278, 24696}, {26765, 26779}, {26782, 26789}

X(26765) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26765) lies on these lines: {2, 39}, {1018, 26752}, {26753, 26756}, {26764, 26779}, {26788, 26794}

X(26766) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26766) lies on these lines: {1, 2}, {26756, 26762}, {26771, 26795}

X(26767) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    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(26767) lies on these lines: {1, 2}, {20284, 21224}, {26762, 26795}

X(26768) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26768) lies on these lines: {2, 44}, {141, 26799}, {144, 27136}, {190, 26774}, {391, 27192}, {524, 26821}, {527, 27044}, {1654, 26812}, {3768, 17217}, {6646, 26752}, {16819, 17252}, {17271, 26976}, {17273, 26772}, {17343, 20561}, {17344, 26971}, {17345, 27102}, {17347, 27095}

X(26769) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c + 4 a^2 b^2 c + 2 a b^3 c + a^3 c^2 + 4 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(26769) lies on these lines: {2, 45}, {141, 26797}, {192, 16696}, {194, 1278}, {3662, 27073}, {3663, 27011}, {4398, 26850}, {6646, 26752}, {7226, 24451}, {7321, 27032}, {16819, 17116}, {17236, 27136}, {17246, 26963}, {17254, 27044}, {17255, 27095}, {17258, 27102}, {17261, 27017}, {17262, 27145}, {17280, 26857}, {17320, 26975}, {17334, 26772}, {17336, 27311}, {20068, 24351}, {26082, 26812}

X(26770) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26770) lies on these lines: {2, 39}, {6, 145}, {8, 1018}, {32, 17539}, {75, 25237}, {190, 17152}, {213, 20040}, {257, 25248}, {312, 26690}, {321, 1212}, {350, 26964}, {391, 4271}, {668, 26757}, {672, 17751}, {966, 17676}, {1011, 7172}, {1089, 24036}, {1475, 21071}, {2549, 26085}, {3061, 25253}, {3263, 25244}, {3496, 4427}, {3691, 4651}, {3693, 4696}, {3701, 25066}, {3729, 20244}, {3780, 20051}, {3840, 23649}, {4095, 14439}, {4202, 15048}, {4385, 25082}, {4968, 16601}, {5192, 9605}, {5275, 19284}, {5276, 11115}, {6376, 27025}, {6554, 17740}, {7745, 17537}, {7758, 26099}, {7791, 17007}, {7798, 25497}, {7864, 16991}, {7920, 16905}, {10459, 17355}, {11320, 19742}, {16583, 17495}, {16909, 16989}, {16920, 17349}, {17002, 17696}, {17135, 21384}, {17140, 21808}, {17164, 17451}, {17264, 25303}, {17280, 21226}, {17350, 20109}, {20331, 21025}, {25092, 26115}, {25242, 26961}, {25261, 26234}, {25264, 26965}, {26753, 26794}, {26763, 26788}

X(26771) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26771) lies on these lines: {2, 6}, {3995, 4033}, {17163, 21684}, {17490, 27794}, {17495, 27792}, {20068, 20966}, {26752, 26762}, {26766, 26795}, {26774, 27040}, {27021, 27043}

X(26772) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26772) lies on these lines: {2, 6}, {7, 27107}, {10, 21803}, {37, 4033}, {41, 26222}, {42, 21257}, {75, 26976}, {190, 26764}, {192, 2092}, {239, 26971}, {320, 27017}, {321, 21857}, {386, 27262}, {442, 26029}, {661, 24130}, {869, 21278}, {872, 21238}, {874, 17280}, {894, 21362}, {1100, 27166}, {1230, 3210}, {1269, 17495}, {2245, 17350}, {2277, 3765}, {3122, 25295}, {3661, 27261}, {3662, 27311}, {3752, 27792}, {3759, 25505}, {3770, 24530}, {3952, 21035}, {3963, 21796}, {4026, 21031}, {4043, 21858}, {4272, 4393}, {4277, 18147}, {4395, 26850}, {4422, 27073}, {4429, 11681}, {4443, 25277}, {4446, 17165}, {4557, 18082}, {5051, 27282}, {16589, 27268}, {16815, 25538}, {17120, 26975}, {17121, 26959}, {17142, 24478}, {17260, 20372}, {17273, 26768}, {17285, 26774}, {17289, 27044}, {17291, 27106}, {17334, 26769}, {17340, 26797}, {17354, 27136}, {17357, 27113}, {17366, 27011}, {17368, 27091}, {20305, 26589}, {20691, 22016}, {22174, 25124}, {26582, 27058}, {26685, 27021}, {26778, 26779}, {26785, 26793}, {27030, 27034}, {27035, 27069}

X(26773) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26773) lies on these lines: {1, 2}

X(26774) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26774) lies on these lines: {1, 2}, {69, 27136}, {141, 26764}, {190, 26768}, {536, 18073}, {594, 26812}, {1654, 27073}, {4129, 21385}, {6646, 26797}, {17229, 26971}, {17231, 27102}, {17233, 27095}, {17239, 27032}, {17280, 26756}, {17285, 26772}, {17295, 26963}, {17297, 26816}, {17374, 26975}, {26771, 27040}

X(26775) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26775) lies on these lines: {2, 661}, {1019, 27013}, {3762, 4560}, {3768, 17217}, {4833, 16738}, {7199, 26985}, {7252, 16704}, {8025, 18199}, {16751, 27115}, {17494, 18155}, {18197, 20295}

X(26776) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    (b - c) (-a b - a c + b c) (-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(26776) lies on these lines: {2, 667}, {4129, 26778}

X(26777) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26777) lies on these lines: {2, 650}, {514, 27013}, {661, 26853}, {812, 26798}, {1635, 7192}, {2490, 4789}, {3522, 8760}, {3620, 9015}, {3623, 14077}, {3762, 4560}, {4024, 10196}, {4382, 27138}, {4468, 27486}, {4704, 4777}, {4765, 25259}, {4893, 20295}, {6546, 21196}, {14936, 26846}, {19998, 21727}, {21297, 25666}, {23791, 26037}

X(26778) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26778) lies on these lines: {2, 31}, {141, 17152}, {4026, 26807}, {4129, 26776}, {6646, 26752}, {16549, 17350}, {24697, 27080}, {26772, 26779}

X(26779) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26779) lies on these lines: {1, 2}, {16705, 27076}, {20148, 26825}, {26764, 26765}, {26772, 26778}

X(26780) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26780) lies on these lines: {2, 3}, {26757, 26790}

X(26781) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26781) lies on these lines: {2, 3}, {3454, 27096}, {17052, 27170}, {21245, 27514}, {26752, 26762}, {26759, 26795}, {26794, 27040}

X(26782) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26782) lies on these lines: {2, 3}, {26764, 26789}

X(26783) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26783) lies on these lines: {2, 3}, {306, 7206}, {17280, 17482}

X(26784) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26784) lies on these lines: {2, 3}, {26752, 26789}

X(26785) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26785) lies on these lines: {2, 3}, {26772, 26793}

X(26786) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    a^6 b - 5 a^5 b^2 + 4 a^4 b^3 + 4 a^3 b^4 - 5 a^2 b^5 + a b^6 + a^6 c - 14 a^5 b c - a^4 b^2 c + 4 a^3 b^3 c - a^2 b^4 c + 10 a b^5 c + b^6 c - 5 a^5 c^2 - a^4 b c^2 + 8 a^3 b^2 c^2 + 8 a^2 b^3 c^2 - a b^4 c^2 - 5 b^5 c^2 + 4 a^4 c^3 + 4 a^3 b c^3 + 8 a^2 b^2 c^3 - 20 a b^3 c^3 + 4 b^4 c^3 + 4 a^3 c^4 - a^2 b c^4 - a b^2 c^4 + 4 b^3 c^4 - 5 a^2 c^5 + 10 a b c^5 - 5 b^2 c^5 + a c^6 + b c^6 : :

X(26786) lies on these lines: {2, 3}

X(26787) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26787) lies on these lines: {2, 3}, {17280, 26757}, {26759, 26792}

X(26788) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26788) lies on these lines: {2, 3}, {26763, 26770}, {26765, 26794}

X(26789) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26789) lies on these lines: {2, 19}, {7, 17319}, {192, 17483}, {346, 17481}, {3672, 26842}, {4872, 26665}, {6646, 26754}, {11997, 20292}, {17280, 17482}, {26752, 26784}, {26764, 26782}

X(26790) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26790) lies on these lines: {2, 40}, {3730, 5195}, {3869, 20533}, {4209, 6361}, {4295, 27253}, {4872, 21872}, {6542, 25270}, {6646, 26754}, {7991, 26531}, {9778, 26658}, {12702, 17671}, {26752, 26753}, {26757, 26780}

X(26791) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26791) lies on these lines: {2, 7}, {43, 17777}, {65, 25979}, {145, 2899}, {181, 3038}, {312, 17299}, {1252, 6634}, {1572, 27546}, {1836, 26073}, {1999, 4856}, {3873, 26139}, {4096, 17722}, {5205, 20101}, {5741, 17280}, {11415, 26029}, {17387, 17778}, {26752, 26753}, {26793, 27040}

X(26792) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    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(26792) lies on these lines: {2, 7}, {8, 3583}, {72, 5046}, {78, 15680}, {79, 26060}, {149, 3681}, {165, 9809}, {190, 5741}, {191, 27529}, {200, 20095}, {210, 5057}, {312, 2895}, {321, 4886}, {346, 26837}, {497, 4661}, {960, 20060}, {962, 4678}, {997, 20067}, {1329, 11684}, {1698, 14450}, {2475, 3876}, {2476, 15650}, {3083, 17806}, {3084, 17803}, {3146, 5811}, {3487, 16859}, {3616, 17544}, {3617, 11415}, {3621, 5815}, {3648, 25440}, {3663, 17020}, {3679, 5180}, {3699, 4450}, {3703, 4756}, {3740, 20292}, {3828, 11552}, {3832, 5758}, {3873, 4679}, {3874, 26127}, {3878, 5559}, {3927, 4193}, {3935, 21060}, {3940, 11114}, {3952, 4388}, {3995, 4053}, {4005, 5178}, {4420, 20066}, {4533, 22793}, {4656, 17011}, {4671, 5739}, {4677, 9802}, {4909, 17019}, {5080, 5692}, {5211, 20068}, {5220, 11680}, {5719, 16858}, {5777, 6895}, {5812, 6894}, {6147, 17536}, {6327, 27538}, {6546, 20295}, {6960, 26921}, {7411, 13257}, {8818, 27081}, {9342, 11246}, {9785, 20014}, {11374, 15674}, {12526, 25005}, {14555, 20886}, {14997, 19785}, {15481, 17605}, {17135, 17777}, {17280, 17482}, {17535, 24470}, {17548, 27383}, {22022, 24048}, {26752, 26762}, {26757, 26780}, {26759, 26787}

X(26793) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26793) lies on these lines: {2, 85}, {8, 5526}, {9, 21066}, {12, 3039}, {149, 2082}, {169, 2475}, {894, 26532}, {1252, 11607}, {2345, 15492}, {2348, 5086}, {5046, 5179}, {5199, 24982}, {6604, 26544}, {10025, 26526}, {17280, 26757}, {23058, 25005}, {24036, 27529}, {26575, 27064}, {26772, 26785}, {26791, 27040}

X(26794) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    (a - b) (a - c) (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(26794) lies on these lines: {2, 99}, {190, 26796}, {661, 21272}, {668, 26795}, {1018, 4129}, {4781, 27045}, {23903, 26964}, {26753, 26770}, {26765, 26788}, {26781, 27040}

X(26795) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26795) lies on these lines: {2, 11}, {668, 26794}, {1018, 26796}, {26753, 26757}, {26759, 26781}, {26762, 26767}, {26766, 26771}

X(26796) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    (a - b) (a - c) (a^3 b^2 - a b^4 + 2 a^3 b c + 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(26796) lies on these lines: {2, 101}, {190, 26794}, {644, 27134}, {693, 21859}, {1018, 26795}, {3314, 27096}

X(26797) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 49

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c + 4 a^2 b^2 c - 2 a b^3 c + a^3 c^2 + 4 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(26797) lies on these lines: {2, 37}, {141, 26769}, {190, 26756}, {3663, 27113}, {3882, 17350}, {3943, 26963}, {3950, 27166}, {6646, 26774}, {17118, 26817}, {17178, 17233}, {17261, 27044}, {17262, 27095}, {17267, 27107}, {17268, 27017}, {17269, 27145}, {17315, 26975}, {17340, 26772}, {26752, 26799}, {26757, 26800}

X(26798) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26798) lies on these lines: {2, 649}, {513, 26985}, {661, 21297}, {693, 4940}, {812, 26777}, {2516, 4380}, {3620, 9002}, {3768, 17217}, {4106, 4776}, {4129, 21385}, {4671, 20952}, {4728, 7192}, {4772, 27485}, {4775, 21301}, {4928, 4979}, {4992, 21343}, {17300, 23345}

X(26799) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26799) lies on these lines: {2, 7}, {6, 26821}, {44, 26971}, {141, 26768}, {190, 26764}, {192, 4277}, {256, 3952}, {3739, 27036}, {4473, 27073}, {4643, 27261}, {16738, 17332}, {16814, 27032}, {17120, 27166}, {17178, 20072}, {17276, 27311}, {17277, 26812}, {17280, 26756}, {17347, 27145}, {17351, 27102}, {17354, 27095}, {17355, 27044}, {17357, 27106}, {17365, 26816}, {17375, 27291}, {17789, 27727}, {18082, 23343}, {22279, 24517}, {26752, 26797}

X(26800) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 49

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

X(26800) lies on these lines: {2, 38}, {2345, 3770}, {3730, 17350}, {6646, 26752}, {17280, 21226}, {26757, 26797}

leftri

Collineation mappings involving Gemini triangle 50: X(26801)-X(26862)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 50, as in centers X(26801)-X(26862). Then

m(X) = a (b - c)^2 x + b (a + c)^2 y + c (a + b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 3, 2018)

X(26801) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    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(26801) lies on these lines: {1, 2}, {7, 1424}, {11, 17669}, {21, 4366}, {36, 17693}, {39, 17143}, {55, 17684}, {56, 16915}, {69, 26149}, {75, 2275}, {76, 16975}, {83, 5291}, {100, 7824}, {141, 27155}, {172, 20179}, {192, 26082}, {194, 4441}, {238, 18756}, {257, 2170}, {274, 1015}, {350, 1107}, {384, 2975}, {668, 3934}, {891, 27015}, {894, 1475}, {941, 20168}, {956, 7770}, {958, 16916}, {966, 16525}, {993, 17692}, {999, 11321}, {1100, 26110}, {1468, 14621}, {1573, 18140}, {1654, 20561}, {1909, 9263}, {1960, 27075}, {2276, 17144}, {2345, 24737}, {2886, 26561}, {2896, 20553}, {3434, 7791}, {3436, 16924}, {3702, 3797}, {3813, 26590}, {3879, 25538}, {3954, 18061}, {4390, 17743}, {4645, 27019}, {4875, 25994}, {4999, 26629}, {5025, 11680}, {5080, 16044}, {5082, 16043}, {5253, 16917}, {5260, 16918}, {5284, 16912}, {5303, 13586}, {5687, 11285}, {6604, 26134}, {6645, 17686}, {6650, 26835}, {6656, 24390}, {7187, 20880}, {7797, 17737}, {11681, 16921}, {12263, 17794}, {15325, 17694}, {16502, 16998}, {16705, 16738}, {16781, 16992}, {16887, 17761}, {17045, 27164}, {17169, 17178}, {17209, 26802}, {17237, 26142}, {17257, 23640}, {17275, 25505}, {17277, 21788}, {17278, 24652}, {17280, 27109}, {17322, 26045}, {17688, 24552}, {18230, 27291}, {19765, 20162}, {20072, 26976}, {21024, 27033}, {21384, 24514}, {26810, 26819}, {26827, 26836}, {26831, 26837}, {26844, 26846}, {26850, 26852}

X(26801) = anticomplement of X(27020)

X(26802) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26802) lies on these lines: {2, 3}, {284, 26125}, {3177, 16699}, {4653, 27253}, {11185, 27515}, {14621, 26964}, {14964, 17753}, {17178, 26811}, {17209, 26801}, {18600, 26845}, {19591, 20244}, {26561, 26977}, {26805, 26846}

X(26803) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26803) lies on these lines: {2, 3}, {18600, 26818}, {26806, 26807}, {26811, 26849}

X(26804) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26804) lies on these lines: {2, 3}, {17167, 26839}, {17194, 26531}, {24632, 27526}, {26558, 27149}, {26813, 26849}

X(26805) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26805) lies on these lines: {1, 2}, {1015, 18600}, {1509, 26845}, {4366, 17539}, {8025, 26828}, {16713, 17045}, {17048, 21272}, {17169, 17761}, {17302, 26818}, {17474, 20347}, {26802, 26846}, {26827, 26839}, {26850, 26859}

X(26806) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26806) lies on these lines: {2, 7}, {8, 4772}, {10, 17288}, {37, 4440}, {69, 4699}, {75, 4675}, {86, 1086}, {190, 17245}, {192, 4648}, {193, 16816}, {239, 3664}, {244, 256}, {319, 4688}, {320, 1654}, {330, 27454}, {344, 7222}, {536, 17317}, {594, 17297}, {673, 20147}, {903, 17246}, {942, 26051}, {966, 4741}, {1125, 9791}, {1213, 7238}, {1266, 17319}, {1278, 17316}, {1284, 5253}, {1463, 3812}, {1909, 20892}, {2321, 17312}, {2345, 17232}, {3008, 17120}, {3589, 27191}, {3616, 24248}, {3619, 4470}, {3661, 17298}, {3663, 16826}, {3666, 26109}, {3729, 17244}, {3758, 17278}, {3834, 17289}, {3875, 17391}, {3879, 17117}, {3888, 17049}, {3912, 17116}, {3945, 4393}, {4000, 17379}, {4334, 19860}, {4335, 4666}, {4340, 19851}, {4359, 17778}, {4360, 7263}, {4361, 17378}, {4363, 17234}, {4384, 4888}, {4389, 15668}, {4398, 16777}, {4416, 4896}, {4419, 27268}, {4431, 17310}, {4454, 25269}, {4472, 17307}, {4473, 17263}, {4480, 25072}, {4643, 4751}, {4644, 17349}, {4645, 17153}, {4659, 17242}, {4665, 17295}, {4667, 17121}, {4670, 16706}, {4686, 17315}, {4687, 17276}, {4698, 17258}, {4704, 5308}, {4739, 5564}, {4740, 17314}, {4796, 16671}, {4798, 17400}, {4859, 17367}, {4862, 16831}, {4869, 17230}, {4967, 17287}, {5224, 7232}, {5263, 25557}, {5712, 17490}, {6356, 21940}, {7184, 21352}, {7227, 17285}, {7240, 22343}, {9782, 26115}, {10030, 26538}, {11110, 24470}, {16817, 20077}, {16830, 24231}, {16832, 17331}, {17118, 17233}, {17119, 17377}, {17151, 17389}, {17160, 17390}, {17169, 17178}, {17175, 17202}, {17180, 17761}, {17227, 17303}, {17235, 17322}, {17241, 17281}, {17252, 24603}, {17256, 17345}, {17259, 17347}, {17265, 17354}, {17266, 17355}, {17275, 17361}, {17277, 17365}, {17283, 17369}, {17290, 17381}, {17292, 21255}, {17301, 17394}, {17304, 17397}, {17305, 17398}, {17343, 21296}, {17695, 25500}, {17777, 25421}, {17790, 18143}, {17951, 27827}, {20295, 21211}, {20337, 27707}, {20924, 21442}, {21258, 26530}, {21330, 24463}, {26048, 26756}, {26803, 26807}, {26821, 26850}

X(26806) = anticomplement of X(17260)

X(26807) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    a^3 b + a b^3 + a^3 c + 6 a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 + a c^3 + b c^3 : :

X(26807) lies on these lines: {1, 2}, {86, 26825}, {1015, 16705}, {2975, 16931}, {3742, 26562}, {4026, 26778}, {4357, 17474}, {4366, 11115}, {4986, 25089}, {8025, 26841}, {16710, 17302}, {17152, 24512}, {17175, 17761}, {24631, 25248}, {26803, 26806}, {26828, 26846}, {26834, 26842}

X(26808) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26808) lies on these lines: {2, 3}

X(26809) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26809) lies on these lines: {2, 3}

X(26810) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26810) lies on these lines: {2, 31}, {17178, 26814}, {26801, 26819}, {26815, 26846}

X(26811) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26811) lies on these lines: {2, 32}, {17178, 26802}, {18600, 26835}, {26803, 26849}, {26845, 26852}

X(26812) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^2 b^2 c + a^3 c^2 + 2 a^2 b c^2 + 6 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(26812) lies on these lines: {2, 37}, {86, 26821}, {594, 26774}, {1086, 16738}, {1268, 25534}, {1654, 26768}, {3008, 27078}, {4395, 27042}, {4967, 27044}, {5750, 26982}, {16819, 17324}, {16829, 17288}, {17117, 25538}, {17140, 24575}, {17169, 17178}, {17202, 17761}, {17239, 27106}, {17277, 26799}, {17445, 20044}, {24199, 27017}, {26082, 26769}, {26149, 26756}, {26813, 26826}, {26829, 26837}

X(26813) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26813) lies on these lines: {2, 39}, {1909, 16742}, {7187, 16727}, {16887, 17761}, {17178, 26802}, {17205, 26959}, {26804, 26849}, {26812, 26826}, {26835, 26843}, {26964, 27011}

X(26814) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26814) lies on these lines: {1, 2}, {1015, 16748}, {17178, 26810}, {17208, 17761}, {26819, 26846}

X(26815) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    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(26815) lies on these lines: {1, 2}, {56, 16954}, {310, 1015}, {350, 23632}, {4184, 4366}, {17759, 26963}, {18152, 22199}, {21224, 21345}, {26810, 26846}

X(26816) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26816) lies on these lines: {2, 44}, {86, 26857}, {1086, 26821}, {3664, 27017}, {4869, 27136}, {17139, 26844}, {17169, 17178}, {17217, 26822}, {17297, 26774}, {17300, 26764}, {17365, 26799}, {17375, 20561}, {17376, 27102}, {17378, 27107}

X(26817) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^3 b c + 4 a^2 b^2 c - 2 a b^3 c + a^3 c^2 + 4 a^2 b c^2 + 12 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(26817) lies on these lines: {2, 45}, {86, 26850}, {4699, 26756}, {7321, 27154}, {10436, 27011}, {17116, 27073}, {17118, 26797}, {17169, 17178}, {17236, 24190}

X(26818) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (a + b) (a + c) (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(26818) lies on these lines: {2, 6}, {7, 17197}, {58, 14986}, {144, 17183}, {145, 27334}, {192, 16728}, {284, 8732}, {314, 4461}, {390, 3286}, {757, 26856}, {1014, 14953}, {1024, 17212}, {1434, 26827}, {1449, 17077}, {2257, 26651}, {3662, 26964}, {3663, 18186}, {3672, 16696}, {4000, 16726}, {4267, 5265}, {4346, 18198}, {4352, 18171}, {4772, 16740}, {5281, 18185}, {5435, 18163}, {10580, 17194}, {11019, 20978}, {17120, 27058}, {17139, 20059}, {17169, 17207}, {17175, 27304}, {17287, 27025}, {17302, 26805}, {17367, 26997}, {17373, 26757}, {18600, 26803}, {18601, 18603}, {26626, 27170}, {26833, 26845}

X(26819) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26819) lies on these lines: {2, 6}, {3736, 20011}, {4359, 16726}, {4651, 18792}, {16696, 17147}, {16705, 26821}, {17135, 17187}, {17143, 18171}, {17184, 17197}, {17495, 18601}, {26801, 26810}, {26814, 26846}, {26830, 26836}, {26844, 26856}

X(26820) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26820) lies on these lines: {1, 2}

X(26821) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26821) lies on these lines: {1, 2}, {6, 26799}, {86, 26812}, {192, 5069}, {524, 26768}, {536, 26975}, {1019, 26853}, {1086, 26816}, {1100, 26971}, {2275, 17147}, {3286, 4366}, {3723, 27032}, {3946, 27017}, {4360, 26764}, {4648, 27192}, {4852, 27102}, {16705, 26819}, {16738, 17045}, {17178, 17302}, {17300, 27011}, {17314, 27136}, {17343, 26143}, {17374, 27106}, {17377, 27095}, {17380, 27145}, {17776, 24737}, {20530, 25298}, {26806, 26850}, {26842, 26852}

X(26822) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26822) lies on these lines: {2, 661}, {1019, 17174}, {3733, 18108}, {3960, 4560}, {7199, 16751}, {7252, 8025}, {16704, 18199}, {17096, 17498}, {17217, 26816}, {18155, 26985}, {18197, 27013}, {23829, 25259}

X(26823) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (b - c) (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(26823) lies on these lines: {2, 667}, {1019, 26825}, {23470, 26846}

X(26824) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26824) lies on these lines: {2, 650}, {193, 9015}, {514, 4024}, {523, 2528}, {649, 17029}, {661, 21297}, {812, 4979}, {1278, 4777}, {3146, 8760}, {3621, 14077}, {3676, 27486}, {3960, 4560}, {4379, 27013}, {4411, 4772}, {4453, 4976}, {4467, 21104}, {4498, 27673}, {4671, 21611}, {4699, 4828}, {4765, 21183}, {4776, 23813}, {4801, 17496}, {4802, 24719}, {4810, 4977}, {4814, 21302}, {4893, 27138}, {6545, 21196}, {6548, 21212}, {6646, 23838}, {9001, 20080}, {23989, 26846}

X(26824) = anticomplement of X(17494)

X(26825) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26825) lies on these lines: {2, 31}, {86, 26807}, {1019, 26823}, {2140, 27011}, {4366, 14953}, {5263, 16930}, {16738, 26826}, {17169, 17178}, {20148, 26779}

X(26826) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    a^3 b + 4 a^2 b^2 + a b^3 + a^3 c + 2 a^2 b c + 6 a b^2 c + b^3 c + 4 a^2 c^2 + 6 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 : :

X(26826) lies on these lines: {1, 2}, {2975, 16930}, {4366, 17588}, {16738, 26825}, {17210, 17761}, {26812, 26813}

X(26827) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26827) lies on these lines: {2, 3}, {1434, 26818}, {26801, 26836}, {26805, 26839}

X(26828) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26828) lies on these lines: {2, 3}, {8025, 26805}, {16705, 26845}, {16738, 26836}, {25526, 27146}, {26801, 26810}, {26807, 26846}

X(26829) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26829) lies on these lines: {2, 3}, {26812, 26837}

X(26830) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26830) lies on these lines: {2, 3}, {284, 5905}, {1333, 19785}, {2185, 8025}, {2206, 24248}, {3189, 20017}, {3210, 16704}, {3285, 3782}, {8822, 20078}, {17173, 17190}, {17185, 18653}, {21376, 25254}, {26819, 26836}

X(26831) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (a + b) (a + c) (a^6 b^2 - a^5 b^3 - a^2 b^6 + a b^7 - 2 a^6 b c - a^5 b^2 c - a^4 b^3 c + 2 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 + 4 a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 - a^4 b c^3 + 4 a^3 b^2 c^3 - a b^4 c^3 - b^5 c^3 + 3 a^2 b^2 c^4 - a b^3 c^4 + 2 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(26831) lies on these lines: {2, 3}, {26801, 26837}, {26840, 26841}

X(26832) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26832) lies on these lines: {2, 3}

X(26833) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26833) lies on these lines: {2, 3}, {26818, 26845}

X(26834) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26834) lies on these lines: {2, 3}, {17302, 26805}, {26807, 26842}

X(26835) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26835) lies on these lines: {2, 3}, {6650, 26801}, {17178, 26852}, {18600, 26811}, {26813, 26843}

X(26836) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26836) lies on these lines: {2, 7}, {269, 26621}, {1086, 16713}, {1122, 24633}, {1418, 24547}, {3620, 26757}, {4366, 17178}, {16738, 26828}, {17183, 24237}, {17273, 27039}, {17302, 26805}, {17304, 26964}, {23830, 27043}, {26801, 26827}, {26819, 26830}

X(26837) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26837) lies on these lines: {2, 19}, {7, 17396}, {192, 17484}, {346, 26792}, {2185, 8025}, {3100, 15680}, {3672, 17481}, {4295, 19783}, {4872, 26538}, {5057, 11997}, {18650, 26639}, {26801, 26831}, {26803, 26806}, {26812, 26829}

X(26838) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26838) lies on these lines: {2, 38}, {26801, 26810}

X(26839) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26839) lies on these lines: {2, 40}, {7, 17474}, {1699, 26531}, {2140, 5195}, {4209, 5603}, {11415, 27304}, {17167, 26804}, {17209, 26801}, {17682, 22791}, {26803, 26806}, {26805, 26827}

X(26840) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26840) lies on these lines: {1, 20101}, {2, 7}, {8, 17155}, {38, 4645}, {65, 5484}, {69, 3210}, {81, 17302}, {210, 26073}, {222, 17086}, {239, 4001}, {244, 4683}, {306, 17288}, {310, 6650}, {312, 17276}, {320, 3666}, {321, 4440}, {333, 1086}, {354, 24723}, {593, 763}, {752, 17598}, {940, 4389}, {942, 26117}, {960, 24803}, {982, 4388}, {1010, 24470}, {1111, 20882}, {1211, 17273}, {1330, 3670}, {1407, 26625}, {1654, 4359}, {1757, 24169}, {1790, 27950}, {1999, 3663}, {2551, 25979}, {2895, 17495}, {2896, 6542}, {3487, 19278}, {3720, 9791}, {3739, 26044}, {3752, 17345}, {3757, 24231}, {3782, 14829}, {3794, 3937}, {3840, 17777}, {3846, 18201}, {3868, 4201}, {4030, 24841}, {4352, 4393}, {4383, 17347}, {4392, 6327}, {4416, 24177}, {4417, 17595}, {4419, 18141}, {4514, 21342}, {4641, 16706}, {4643, 19804}, {4650, 26128}, {4703, 17063}, {4741, 5739}, {4862, 11679}, {4886, 17344}, {4902, 18229}, {5256, 17364}, {5262, 20077}, {5263, 11246}, {5287, 17247}, {6147, 19270}, {7232, 18134}, {7238, 17056}, {9782, 19874}, {10453, 24248}, {14555, 24620}, {17011, 20090}, {17024, 20064}, {17182, 24237}, {17209, 26801}, {17232, 17776}, {17235, 19786}, {17237, 19808}, {17238, 19822}, {17239, 19797}, {17339, 25734}, {17378, 20182}, {17790, 18136}, {18144, 19807}, {20043, 20080}, {24349, 26034}, {26831, 26841}

X(26841) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (a + b) (a + c) (a^3 b^2 + a b^4 - 2 a^3 b c + 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(26841) lies on these lines: {2, 58}, {6645, 11115}, {8025, 26807}, {16738, 26825}, {17178, 26802}, {18191, 26562}, {26801, 26810}, {26831, 26840}

X(26842) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    {1, 20066}, {2, 7}, {8, 3894}, {10, 9782}, {21, 24470}, {81, 1086}, {145, 10044}, {149, 354}, {239, 20086}, {320, 2895}, {321, 7321}, {388, 18419}, {404, 6147}, {551, 5180}, {942, 2475}, {1071, 6894}, {1125, 14450}, {1211, 7238}, {1255, 17246}, {1373, 3084}, {1374, 3083}, {1621, 11246}, {1993, 5228}, {2185, 8025}, {2476, 5708}, {2550, 4430}, {3337, 11263}, {3487, 4188}, {3616, 5267}, {3620, 19825}, {3622, 4295}, {3623, 11037}, {3648, 5259}, {3649, 5253}, {3663, 17019}, {3664, 17011}, {3670, 26131}, {3672, 26789}, {3674, 26639}, {3742, 5057}, {3817, 9809}, {3873, 4863}, {3881, 5557}, {3916, 15674}, {3920, 24231}, {3957, 5542}, {3969, 17297}, {3995, 4440}, {4001, 24199}, {4190, 11036}, {4292, 15680}, {4307, 17024}, {4312, 4666}, {4355, 19860}, {4511, 11551}, {4641, 26724}, {4645, 17140}, {4671, 18141}, {4858, 21739}, {4860, 11680}, {4862, 5287}, {4887, 17021}, {4888, 5256}, {4896, 17012}, {4902, 17022}, {4973, 26725}, {5080, 5883}, {5154, 5714}, {5284, 17768}, {5290, 25005}, {5303, 11281}, {5422, 6180}, {5425, 6224}, {5719, 13587}, {5758, 15717}, {5805, 11220}, {5811, 15022}, {5904, 26060}, {6840, 10202}, {6884, 24467}, {6888, 26877}, {6901, 24475}, {8226, 13243}, {9352, 17718}, {9955, 16116}, {10129, 17728}, {10404, 20060}, {10940, 12874}, {11038, 20075}, {11114, 18541}, {12690, 15679}, {14996, 19785}, {15934, 17579}, {17063, 24725}, {17147, 17300}, {17167, 24237}, {17169, 17190}, {17375, 20017}, {17495, 17778}, {18653, 26860}, {20295, 21204}, {26801, 26810}, {26805, 26827}, {26807, 26834}, {26821, 26852} : :

X(26842) lies on these lines:

X(26843) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26843) lies on these lines: {2, 32}, {86, 26807}, {1019, 16887}, {2975, 3286}, {17143, 18171}, {17178, 18600}, {17200, 26965}, {18047, 26759}, {26813, 26835}

X(26844) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26844) lies on these lines: {2, 45}, {1977, 26860}, {3952, 24399}, {4033, 17147}, {14554, 26580}, {17139, 26816}, {26801, 26846}, {26819, 26856}

X(26845) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (a + b) (b - c)^2 (a + c) (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(26845) lies on these lines: {2, 99}, {1015, 26846}, {1019, 17761}, {1086, 26847}, {1111, 4560}, {1509, 26805}, {2170, 7192}, {16705, 26828}, {17103, 26964}, {18600, 26802}, {26811, 26852}, {26813, 26835}, {26818, 26833}

X(26846) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26846) lies on these lines: {2, 11}, {1015, 26845}, {1086, 26851}, {14936, 26777}, {17761, 26847}, {23470, 26823}, {23989, 26824}, {26801, 26844}, {26802, 26805}, {26807, 26828}, {26810, 26815}, {26814, 26819}, {26848, 26856}

X(26847) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26847) lies on these lines: {2, 101}, {1086, 26845}, {4904, 27009}, {11998, 17496}, {17761, 26846}

X(26848) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26848) lies on these lines: {2, 98}, {26846, 26856}

X(26849) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26849) lies on these lines: {2, 99}, {26803, 26811}, {26804, 26813}

X(26850) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26850) lies on these lines: {2, 37}, {86, 26817}, {1086, 17178}, {4361, 26756}, {4395, 26772}, {4398, 26769}, {4431, 27113}, {5564, 27106}, {7263, 26963}, {17116, 26982}, {17119, 27095}, {17154, 24575}, {17366, 26976}, {24199, 27166}, {26801, 26852}, {26805, 26859}, {26806, 26821}

X(26851) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26851) lies on these lines: {2, 900}, {1086, 26846}, {4435, 20090}

X(26852) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 50

Barycentrics    (a + b) (a + 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(26852) lies on these lines: {2, 39}, {330, 16742}, {2275, 27011}, {16709, 26143}, {16710, 16744}, {16722, 21219}, {17178, 26835}, {26801, 26850}, {26811, 26845}, {26821, 26842}

X(26853) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26853) lies on these lines: {2, 649}, {144, 4468}, {193, 9002}, {512, 14712}, {513, 4380}, {514, 14779}, {661, 26777}, {693, 4790}, {788, 20011}, {812, 4979}, {1019, 26821}, {3667, 25259}, {3676, 21454}, {4106, 26985}, {4369, 21297}, {4382, 4932}, {4394, 4776}, {4453, 23729}, {4834, 21301}, {4984, 21196}, {8663, 9147}, {9313, 20064}, {9433, 20041}, {16874, 18108}, {17217, 26816}, {17410, 24562}, {18200, 26860}, {20090, 21143}

X(26854) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26854) lies on these lines: {1, 23791}, {2, 650}, {514, 27272}, {3837, 25299}, {4382, 27345}, {4449, 25301}, {8640, 23815}, {17215, 26652}, {17217, 26816}, {21297, 26983}, {27258, 27294}

X(26855) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26855) lies on these lines: {2, 659}, {1086, 26846}, {17217, 26816}

X(26856) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26856) lies on these lines: {2, 662}, {261, 4612}, {346, 7258}, {757, 26818}, {849, 3086}, {1019, 24237}, {1086, 26845}, {2310, 7253}, {3942, 7192}, {4366, 16738}, {4560, 4858}, {14570, 14616}, {16726, 16727}, {17058, 27008}, {17197, 17219}, {26819, 26844}, {26846, 26848}

X(26857) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26857) lies on these lines: {2, 7}, {86, 26816}, {141, 26764}, {1086, 16738}, {3619, 27136}, {3834, 27032}, {4389, 27145}, {4643, 27311}, {4698, 27159}, {5224, 27107}, {7238, 27042}, {17178, 17302}, {17202, 24237}, {17235, 26971}, {17237, 27102}, {17273, 26768}, {17276, 27261}, {17280, 26769}, {17305, 26963}, {17324, 27166}, {17384, 26975}, {26801, 26850}

X(26858) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26858) lies on these lines: {2, 896}, {17217, 26816}, {26801, 26810}

X(26859) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26859) lies on these lines: {2, 38}, {16710, 17302}, {17169, 17178}, {26805, 26850}

X(26860) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 50

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

X(26860) lies on these lines: {1, 4427}, {2, 6}, {21, 7373}, {58, 3622}, {145, 4658}, {551, 21747}, {896, 5625}, {1010, 3621}, {1100, 16710}, {1412, 21454}, {1449, 26627}, {1509, 4610}, {1977, 26844}, {3187, 25590}, {3210, 25417}, {3218, 18164}, {3240, 18792}, {3617, 25526}, {3623, 11115}, {3720, 18192}, {3977, 4909}, {3995, 17351}, {4649, 19998}, {4667, 26580}, {4678, 17589}, {4697, 27804}, {4720, 20049}, {4781, 21806}, {4850, 16726}, {16666, 24589}, {16723, 27754}, {16816, 17175}, {17018, 17187}, {17019, 17261}, {17021, 17120}, {17103, 20092}, {17147, 17393}, {17162, 24342}, {17169, 17191}, {17183, 17484}, {17450, 18174}, {18163, 27003}, {18200, 26853}, {18653, 26842}, {19825, 20046}, {26802, 26805}

X(26861) =  X(4)X(11017)∩X(6)X(15720)

Barycentrics    (-a^2+b^2+c^2)*((a^2-b^2+c^2)^2-9*a^2*c^2)*((a^2+b^2-c^2)^2-9*a^2*b^2) : :
Trilinears    (4*cos(C)^2-9)*(4*cos(B)^2-9)*cos(A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28573.

X(26861) lies on the Jerabek hyperbola and these lines: {4, 11017}, {6, 15720}, {54, 15712}, {65, 5557}, {140, 1173}, {265, 5447}, {550, 16835}, {1216, 14861}, {1657, 22334}, {2889, 11592}, {3521, 3917}, {3522, 13452}, {3523, 13472}, {5562, 13623}, {7386, 14843}, {15321, 18553}, {15740, 23039}, {18296, 18531}

X(26861) = isogonal conjugate of X(26863)

X(26862) =  X(140)X(1173)∩X(3850)X(11703)

Barycentrics    (3*S^2-SA*SC)*(9*S^2+5*SB^2)*(3*S^2-SA*SB)*(9*S^2+5*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28573.

X(26862) lies these lines: {140, 1173}, {3850, 11703}

X(26863) =  EULER LINE INTERCEPT OF X(113)X(25714)

Barycentrics    a^2*((-a^2+b^2+c^2)^2-9*b^2*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Trilinears    (4*cos(A)^2-9)*cos(B)*cos(C) : :
X(26863) = 2*(4*R^2-SW)*X(3)+9*R^2*X(4)

As a point on the Euler line, X(26863) has Shinagawa coefficients (-4*F, 9*E+4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28573.

X(26863) lies on these lines: {2, 3}, {113, 25714}, {389, 12112}, {1173, 1199}, {1493, 10540}, {2914, 5609}, {3060, 15083}, {5007, 8744}, {5446, 15801}, {5943, 8718}, {6152, 16982}, {6243, 15052}, {6759, 11423}, {9781, 15032}, {12254, 16657}, {13353, 23060}, {13452, 22334}, {13474, 16835}, {14094, 16625}, {14853, 15581}, {15873, 16659}, {18296, 18532}

X(26863) = isogonal conjugate of X(26861)
X(26863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3091, 3146, 18531), (3091, 3547, 3090), (3518, 14865, 186)

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Endo-homothetic centers: X(26864)-X(26958)

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This preamble and centers X(26864)-X(26958) were contributed by César Eliud Lozada, November 3, 2018.

This section comprises the endo-homothetic centers of the family of triangles homothetic with the excentral triangle of a reference triangle ABC. This family is composed by the following 31 triangles:

Ascella, Atik, 1st circumperp, 2nd circumperp, inner-Conway, Conway, 2nd Conway, 3rd Conway, 3rd Euler, 4th Euler, excenters-reflections, excentral, 2nd extouch, hexyl, Honsberger, inner-Hutson, Hutson intouch, outer-Hutson, incircle-circles, intouch, inverse-in-incircle, 6th mixtilinear, 2nd Pamfilos-Zhou, 1st Sharygin, tangential-midarc, 2nd tangential-midarc, Ursa major, Ursa minor, Wasat, Yff central, 2nd Zaniah.

For definitions and coordinates of these triangles, see the index of triangles referenced in ETC.

X(26864) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND INNER-CONWAY

Barycentrics    a^2*(5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(26864) = 4*X(6)-3*X(11405)

The homothetic center of these triangles is X(5744)

X(26864) lies on these lines: {2,8780}, {3,74}, {4,14530}, {6,25}, {22,323}, {23,1351}, {24,15032}, {26,12160}, {49,7387}, {54,1598}, {55,23201}, {155,9715}, {182,11284}, {185,15750}, {198,23202}, {215,10833}, {235,18925}, {237,1384}, {352,15655}, {353,3148}, {378,3426}, {381,14389}, {394,3098}, {427,11206}, {428,11427}, {462,5334}, {463,5335}, {468,6776}, {511,11181}, {575,3066}, {578,5198}, {902,2187}, {1112,15073}, {1147,11414}, {1181,3515}, {1350,3292}, {1352,13394}, {1398,26888}, {1498,3516}, {1503,5094}, {1593,6759}, {1597,14157}, {1899,10192}, {1976,2502}, {1993,9909}, {1995,5050}, {2477,18954}, {3043,9919}, {3044,13175}, {3045,13222}, {3047,12310}, {3060,20850}, {3129,11485}, {3130,11486}, {3147,18914}, {3155,6221}, {3156,6398}, {3172,9408}, {3203,10790}, {3231,20885}, {3233,6795}, {3295,9638}, {3517,7592}, {3518,11432}, {3520,12315}, {3526,11457}, {3564,7493}, {3581,14070}, {3619,7499}, {3620,7494}, {3796,5092}, {3843,12289}, {4224,14996}, {4232,14912}, {4550,18451}, {5012,5020}, {5055,25739}, {5064,23292}, {5085,5651}, {5093,11422}, {5200,23267}, {5210,15504}, {5422,10545}, {5502,14685}, {5544,16042}, {5640,12283}, {5642,14982}, {5889,16195}, {6000,11410}, {6200,10132}, {6353,11245}, {6396,10133}, {6417,11463}, {6418,11462}, {6445,21097}, {6515,10154}, {6593,8547}, {6618,14569}, {7071,10535}, {7393,18350}, {7395,10539}, {7426,21970}, {7464,11820}, {7488,12164}, {7503,15052}, {7506,15037}, {7507,9833}, {7517,9704}, {7529,18874}, {7687,18396}, {8185,9587}, {8276,9677}, {8550,15448}, {8778,9412}, {8908,26953}, {9652,10831}, {9667,10832}, {9703,12083}, {9714,12161}, {9818,10540}, {10018,26944}, {10301,14853}, {10536,11406}, {10541,22112}, {10564,21312}, {10565,20080}, {10594,11426}, {10605,11202}, {10979,26898}, {11002,11482}, {11403,11425}, {12165,13289}, {13884,18924}, {13937,18923}, {14490,14528}, {15033,18535}, {15069,24981}, {15577,21284}, {15647,19504}, {16030,26887}, {16187,20190}, {16252,19467}, {17811,22352}, {18386,18400}, {19121,19588}, {19456,20773}, {22052,26865}, {26866,26884}, {26867,26885}, {26868,26886}

X(26864) = isogonal conjugate of X(36889)
X(26864) = crosssum of X(2) and X(3543)
X(26864) = crossdifference of every pair of points on line X(525)X(1637)
X(26864) = isogonal conjugate of the isotomic conjugate of X(376)
X(26864) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6800, 15066, 15080), (9707, 11456, 11464), (11456, 11464, 3)

X(26865) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND 3rd CONWAY

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

The homothetic center of these triangles is X(10856)

X(26865) lies on these lines: {2,3}, {6,26907}, {97,3167}, {184,26909}, {216,9777}, {577,11402}, {1398,26903}, {1993,26895}, {7071,26904}, {7592,26896}, {11245,26870}, {11406,26908}, {16030,26902}, {19118,26899}, {19459,23195}, {22052,26864}, {26866,26900}, {26867,26901}, {26869,26905}

X(26865) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 418, 25), (3155, 3156, 3517)

X(26866) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND INNER-HUTSON

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

The homothetic center of these triangles is X(11854)

X(26866) lies on these lines: {3,3218}, {4,26928}, {6,3937}, {22,23958}, {25,57}, {46,8192}, {55,4864}, {56,15854}, {63,7484}, {84,11403}, {182,22129}, {184,1407}, {220,5650}, {222,11402}, {418,7011}, {427,26929}, {603,1398}, {999,17126}, {1155,22769}, {1210,17516}, {1357,2175}, {1486,4860}, {1993,26910}, {2969,4000}, {3219,16419}, {3295,4392}, {3306,11284}, {3336,9798}, {3337,11365}, {3516,26927}, {3928,7085}, {4214,4292}, {4224,21454}, {4617,7053}, {5091,15635}, {5094,26933}, {5221,22654}, {5708,13730}, {5905,16434}, {6090,7193}, {7004,7071}, {7295,18201}, {7395,24467}, {7592,26914}, {9777,26892}, {9965,19649}, {11245,26871}, {11406,26934}, {16030,26931}, {19118,26923}, {26864,26884}, {26865,26900}, {26868,26930}, {26869,26932}

X(26866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57, 1473, 25), (63, 7484, 26867)

X(26867) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND OUTER-HUTSON

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

The homothetic center of these triangles is X(11855)

X(26867) lies on these lines: {3,3219}, {4,26938}, {6,3690}, {9,25}, {10,4214}, {40,11403}, {44,55}, {63,7484}, {71,11406}, {184,220}, {197,3715}, {201,1398}, {212,7071}, {219,11402}, {268,418}, {427,26939}, {756,1460}, {894,16353}, {999,7226}, {1011,1260}, {1397,7064}, {1407,5650}, {1473,3929}, {1993,26911}, {2267,2318}, {2345,7140}, {3218,16419}, {3295,17127}, {3305,11284}, {3516,26935}, {3683,12329}, {3819,22129}, {3955,6090}, {4219,21168}, {5094,21015}, {5314,24320}, {7395,26921}, {7592,26915}, {9777,26893}, {11245,26872}, {12414,18259}, {12572,17516}, {16030,26941}, {19118,26924}, {21319,21483}, {26864,26885}, {26865,26901}, {26868,26940}, {26869,26942}

X(26867) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 7085, 25), (63, 7484, 26866)

X(26868) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(10858)

X(26868) lies on these lines: {2,13960}, {3,6}, {25,8911}, {53,6561}, {154,8908}, {184,26953}, {233,8253}, {393,6459}, {427,26945}, {485,6748}, {493,8882}, {1398,26948}, {1586,3068}, {1588,6810}, {1593,6457}, {1993,26912}, {3155,19356}, {3516,26936}, {3815,18289}, {5094,26951}, {5407,8963}, {5410,6413}, {5412,10132}, {7071,26949}, {7395,26922}, {7592,26916}, {8576,19005}, {9777,26894}, {10311,15199}, {11245,26873}, {11402,26891}, {11403,26918}, {11406,26952}, {16030,26947}, {18924,21736}, {19118,26925}, {26864,26886}, {26866,26930}, {26867,26940}, {26869,26950}

X(26868) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1151, 216), (3311, 15905, 6), (5058, 5065, 6)

X(26869) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND URSA MAJOR

Barycentrics    a^6-2*(b^2+c^2)*a^4+3*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(26869) = 2*X(5)+X(18917) = X(25)+2*X(1899) = X(25)-4*X(13567) = 4*X(125)-X(15106) = 2*X(1368)+X(6515) = 5*X(1656)-2*X(15068) = X(1899)+2*X(13567) = X(10605)+2*X(18390) = 2*X(11438)+X(18396)

The homothetic center of these triangles is X(17612)

X(26869) lies on these lines: {2,3167}, {3,3580}, {4,3426}, {5,18916}, {6,67}, {23,21970}, {25,1503}, {51,1853}, {54,3526}, {184,26958}, {193,16051}, {235,5656}, {343,7484}, {373,10516}, {381,5640}, {389,7507}, {394,5965}, {427,9777}, {468,6776}, {599,5650}, {858,1351}, {1147,11232}, {1192,21659}, {1209,15805}, {1316,12079}, {1352,11284}, {1353,5159}, {1368,6515}, {1398,26955}, {1593,16657}, {1594,11432}, {1598,11457}, {1656,7592}, {1657,15107}, {1885,18913}, {1906,12324}, {1993,26913}, {1995,3448}, {2452,3154}, {2453,6070}, {2777,10605}, {3066,3818}, {3515,6146}, {3516,12241}, {3527,15559}, {3534,15360}, {3542,18914}, {3548,13292}, {3763,22112}, {5020,11442}, {5079,5643}, {5198,14216}, {5422,23293}, {5651,15069}, {6247,11403}, {6642,25738}, {7071,26956}, {7395,12359}, {7495,12017}, {7505,19347}, {7539,10601}, {7703,15019}, {8262,8547}, {8901,19166}, {9730,14852}, {9786,12173}, {10182,19357}, {10982,20299}, {10989,16981}, {11179,13394}, {11406,26957}, {11422,15059}, {11438,18396}, {11472,16003}, {11550,17810}, {11585,12160}, {11898,15066}, {12024,15750}, {12429,17928}, {12827,14643}, {13154,21230}, {13754,16072}, {13857,15534}, {13884,18923}, {13937,18924}, {14361,14569}, {16030,26954}, {16352,25977}, {18494,25739}, {19118,26926}, {19161,23049}, {19588,26156}, {26865,26905}, {26866,26932}, {26867,26942}, {26868,26950}

X(26869) = reflection of X(6090) in X(2)
X(26869) = inverse of X(12099) in the orthocentroidal circle
X(26869) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11245, 11402), (2, 18950, 11245)
X(26869) = X(25)-of-orthocentroidal-triangle

X(26870) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 3rd CONWAY

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

The homothetic center of these triangles is X(10862)

X(26870) lies on these lines: {2,26898}, {3,69}, {4,216}, {98,7494}, {418,11433}, {577,14912}, {631,6389}, {1899,26907}, {3524,12096}, {6353,26880}, {6515,26874}, {6638,18928}, {6641,11206}, {7386,9744}, {10996,11257}, {11245,26865}, {12324,26897}, {13567,26909}, {18911,26895}, {18912,26896}, {18915,26903}, {18916,26876}, {18921,26908}, {18922,26904}, {19119,26899}, {19166,26902}, {23291,26906}, {26871,26900}, {26872,26901}

X(26870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12256, 12257, 18925), (26898, 26905, 2)

X(26871) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND INNER-HUTSON

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

The homothetic center of these triangles is X(11856)

X(26871) lies on these lines: {2,222}, {4,26892}, {7,92}, {57,11433}, {63,69}, {73,25876}, {77,6349}, {81,8048}, {84,12324}, {320,18750}, {329,4358}, {343,22129}, {348,6513}, {497,1364}, {603,18915}, {631,26890}, {908,1997}, {966,14597}, {1407,13567}, {1433,14986}, {1439,9776}, {1473,6776}, {1748,7291}, {1899,3937}, {1948,6820}, {1959,18730}, {2003,11427}, {2096,10538}, {2975,19262}, {3218,6515}, {3220,11206}, {3306,18928}, {3784,7386}, {3869,18732}, {3917,26939}, {3942,6508}, {3955,7494}, {4295,20220}, {5081,5768}, {5174,9799}, {5739,5744}, {5906,6836}, {6353,26884}, {6507,20769}, {6604,20223}, {7004,18922}, {7017,18816}, {7085,10519}, {7288,7335}, {7293,25406}, {7515,23072}, {11245,26866}, {11411,24467}, {14826,24320}, {14912,26889}, {17923,18623}, {18911,26910}, {18912,26914}, {18913,26927}, {18914,26928}, {18916,26877}, {19119,26923}, {19166,26931}, {23291,26933}, {26870,26900}, {26873,26930}

X(26871) = anticomplement of X(34048)
X(26871) = isotomic conjugate of the polar conjugate of X(3086)
X(26871) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 189, 92), (222, 26932, 2)

X(26872) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND OUTER-HUTSON

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

The homothetic center of these triangles is X(11857)

X(26872) lies on these lines: {2,219}, {4,8}, {9,11433}, {40,12324}, {63,69}, {81,22132}, {144,2895}, {200,2947}, {201,18915}, {209,5800}, {212,18922}, {220,13567}, {307,6349}, {319,18750}, {348,6505}, {388,7066}, {518,11435}, {534,17781}, {631,26889}, {908,5271}, {1264,19799}, {1441,5905}, {1473,10519}, {1748,5279}, {1899,3690}, {1947,6820}, {2323,11427}, {2975,13726}, {3219,6515}, {3305,18928}, {3781,7386}, {3870,14547}, {3917,26929}, {3949,6508}, {3990,5712}, {4886,20921}, {5218,6056}, {5249,6604}, {5285,11206}, {5314,25406}, {5596,12329}, {5816,22000}, {5928,21871}, {6353,26885}, {6776,7085}, {7193,7494}, {7536,20818}, {11245,26867}, {11411,26921}, {12587,22276}, {14912,26890}, {18911,26911}, {18912,26915}, {18913,26935}, {18914,26938}, {18916,26878}, {19119,26924}, {19166,26941}, {21015,23291}, {26870,26901}, {26873,26940}

X(26872) = anticomplement of X(37543)
X(26872) = anticomplementary conjugate of the anticomplement of X(2335)
X(26872) = isotomic conjugate of the polar conjugate of X(3085)
X(26872) = anticomplement of the isogonal conjugate of X(2335)
X(26872) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 329, 92), (219, 26942, 2), (13386, 13387, 72)

X(26873) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(10867)

X(26873) lies on these lines: {2,26920}, {4,372}, {69,1589}, {159,3156}, {371,18916}, {577,1899}, {615,10133}, {1152,17845}, {6353,26886}, {6457,18909}, {6515,26875}, {6776,8911}, {8961,19061}, {11245,26868}, {11411,26922}, {11433,26919}, {12324,26918}, {13567,26953}, {14912,26891}, {18911,26912}, {18912,26916}, {18913,26936}, {18915,26948}, {18921,26952}, {18922,26949}, {19119,26925}, {19166,26947}, {23291,26951}, {26871,26930}, {26872,26940}

X(26873) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3069, 12256, 6414), (26920, 26950, 2)

X(26874) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP AND 3rd CONWAY

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

The homothetic center of these triangles is X(10434)

X(26874) lies on these lines: {2,3}, {95,1629}, {97,184}, {110,26880}, {160,11206}, {216,3060}, {394,26909}, {511,26907}, {577,5012}, {1993,26898}, {2979,26895}, {3100,26904}, {3101,26908}, {3218,26900}, {3219,26901}, {3289,3796}, {3410,18437}, {3580,26905}, {4296,26903}, {6509,7998}, {6515,26870}, {6776,23195}, {10979,15107}, {11003,23606}, {11412,26896}, {15080,22052}, {19121,26899}

X(26874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 417, 15717), (3, 418, 2), (3, 426, 15246)

X(26875) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8224)

X(26875) lies on these lines: {2,26919}, {3,6}, {4,26922}, {20,6457}, {22,8911}, {97,26947}, {110,26886}, {317,491}, {394,26953}, {858,26951}, {1370,26945}, {1993,26920}, {2979,26912}, {3060,26894}, {3069,8576}, {3100,26949}, {3101,26952}, {3146,26918}, {3155,10962}, {3218,26930}, {3219,26940}, {3580,26950}, {4296,26948}, {5012,26891}, {5889,6458}, {6290,12960}, {6413,11417}, {6515,26873}, {8855,13960}, {11412,26916}, {11413,26936}, {19121,26925}

X(26875) = {X(5409), X(5412)}-harmonic conjugate of X(10960)

X(26876) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP AND 3rd CONWAY

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

The homothetic center of these triangles is X(10882)

X(26876) lies on these lines: {2,3}, {54,577}, {97,1147}, {216,3567}, {389,26907}, {1181,26909}, {1614,26880}, {1870,26903}, {5889,26895}, {5890,26896}, {6197,26908}, {6198,26904}, {6509,7999}, {7592,26898}, {9545,19210}, {11464,22052}, {15653,18925}, {18916,26870}, {26877,26900}, {26878,26901}, {26879,26905}

X(26876) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 417, 3524), (3, 418, 4)

X(26877) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP AND INNER-HUTSON

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

The homothetic center of these triangles is X(8109)

X(26877) lies on these lines: {1,6950}, {2,24467}, {3,3218}, {4,57}, {7,6833}, {8,6955}, {9,3525}, {20,23958}, {21,10202}, {24,1473}, {25,26928}, {35,12005}, {36,5884}, {40,3244}, {46,944}, {54,26889}, {63,631}, {65,104}, {72,6940}, {79,11219}, {140,3219}, {191,10165}, {222,7592}, {226,6952}, {244,3073}, {329,6967}, {371,26930}, {376,5709}, {377,5770}, {378,26927}, {388,17700}, {389,3937}, {404,912}, {411,13369}, {474,5780}, {484,5882}, {497,17437}, {499,1776}, {515,3336}, {553,6705}, {601,982}, {602,4650}, {603,1870}, {920,7288}, {938,6938}, {942,6906}, {943,17603}, {946,1768}, {993,15016}, {1006,3916}, {1012,5708}, {1155,11491}, {1158,3338}, {1181,1407}, {1199,2003}, {1385,5303}, {1445,6927}, {1454,4293}, {1476,12776}, {1594,26933}, {1614,26884}, {1621,13373}, {1708,6880}, {1788,12115}, {2077,3874}, {2094,5758}, {2800,5563}, {3075,6198}, {3090,3306}, {3220,3518}, {3305,3533}, {3333,10595}, {3359,12245}, {3474,12116}, {3487,6977}, {3523,26921}, {3524,3928}, {3529,7171}, {3567,26892}, {3585,10265}, {3587,21735}, {3651,10167}, {3652,11230}, {3784,11412}, {3817,7701}, {3855,18540}, {3869,10269}, {3873,11248}, {3877,16203}, {3889,10679}, {3911,6949}, {3929,15702}, {3957,11849}, {4295,10785}, {4297,5535}, {4652,6875}, {4857,16767}, {4860,11496}, {4973,11012}, {5067,5437}, {5218,7162}, {5221,12114}, {5249,6852}, {5253,5887}, {5270,16763}, {5330,24927}, {5435,6834}, {5439,6920}, {5450,5902}, {5557,11218}, {5657,10805}, {5704,6968}, {5714,6879}, {5744,6889}, {5761,6966}, {5768,6934}, {5777,6946}, {5811,6983}, {5889,26910}, {5890,26914}, {5905,6891}, {6197,26934}, {6361,10806}, {6684,6763}, {6734,6951}, {6831,13226}, {6832,9776}, {6847,21454}, {6876,10884}, {6909,24474}, {6911,12528}, {6915,13243}, {6926,9965}, {6942,15803}, {6948,12649}, {6972,17483}, {6985,11220}, {7289,14912}, {7293,7512}, {7505,20266}, {8726,21165}, {9352,11499}, {9841,17538}, {10246,19535}, {10532,14647}, {10698,24928}, {11009,11715}, {11010,13607}, {11570,18861}, {12512,24468}, {12515,24680}, {12608,16116}, {14988,19525}, {16139,17502}, {17549,24299}, {18916,26871}, {19128,26923}, {20292,26470}, {26876,26900}, {26879,26932}

X(26877) = reflection of X(5330) in X(24927)
X(26877) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (36, 5884, 21740), (63, 631, 26878)

X(26878) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP AND OUTER-HUTSON

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

The homothetic center of these triangles is X(8110)

X(26878) lies on these lines: {2,26921}, {3,3219}, {4,9}, {8,6936}, {24,7085}, {25,26938}, {35,1776}, {45,5706}, {46,5714}, {54,72}, {57,3525}, {63,631}, {78,6875}, {84,3528}, {140,3218}, {191,6684}, {201,1870}, {210,11491}, {212,6198}, {219,7592}, {220,1181}, {226,3336}, {329,6889}, {371,26940}, {376,7330}, {378,26935}, {389,3690}, {405,1482}, {498,7098}, {517,5260}, {601,7262}, {602,984}, {756,3072}, {908,6853}, {912,6986}, {920,5218}, {936,6942}, {943,11428}, {954,11025}, {1158,5658}, {1199,2323}, {1490,16192}, {1594,21015}, {1614,26885}, {1708,3338}, {1728,3488}, {1782,21361}, {2077,3647}, {2095,16842}, {2949,5506}, {3090,3305}, {3295,5729}, {3306,3533}, {3452,6949}, {3467,4330}, {3518,5285}, {3523,24467}, {3524,3929}, {3529,3587}, {3567,26893}, {3579,5927}, {3634,5535}, {3651,3652}, {3678,10902}, {3681,10267}, {3715,11500}, {3781,11412}, {3817,24468}, {3868,6883}, {3916,6940}, {3928,15702}, {3951,18443}, {4187,5771}, {4294,7082}, {5044,6905}, {5047,24474}, {5067,7308}, {5227,14912}, {5250,12245}, {5273,6833}, {5302,14110}, {5314,7512}, {5690,11113}, {5692,21740}, {5720,6876}, {5744,6967}, {5745,6952}, {5758,6832}, {5791,6830}, {5812,6829}, {5889,26911}, {5890,26915}, {5905,6989}, {6734,6902}, {6763,10165}, {6834,18228}, {6937,11681}, {6984,9780}, {7171,21735}, {7701,12512}, {7987,18446}, {9841,19708}, {9956,16139}, {10176,11012}, {10323,24320}, {10806,20588}, {12710,15837}, {14872,15481}, {15492,15852}, {16845,24541}, {18916,26872}, {19128,26924}, {20104,25525}, {26876,26901}, {26879,26942}

X(26878) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5817, 6361, 4), (6191, 6192, 71)

X(26879) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP AND URSA MAJOR

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

The homothetic center of these triangles is X(17614)

X(26879) lies on these lines: {2,155}, {3,3580}, {4,64}, {5,5890}, {24,1899}, {25,11457}, {51,15559}, {52,858}, {54,140}, {68,17928}, {74,1885}, {110,16238}, {125,389}, {141,3525}, {184,10018}, {185,403}, {186,2917}, {235,6241}, {343,631}, {371,26950}, {372,26951}, {378,26937}, {427,3567}, {468,1614}, {546,7728}, {568,13371}, {632,11423}, {1181,7505}, {1192,18396}, {1199,6143}, {1204,18390}, {1209,5892}, {1368,11412}, {1503,3518}, {1511,11264}, {1595,9781}, {1596,12290}, {1870,26955}, {1906,11455}, {1993,3548}, {2072,6102}, {2935,6696}, {3060,23335}, {3147,6776}, {3448,12134}, {3520,12241}, {3526,11402}, {3541,11433}, {3542,11456}, {3546,6515}, {3564,26156}, {3575,25739}, {5012,7542}, {5094,11432}, {5133,5462}, {5449,9730}, {5576,5946}, {5640,7403}, {5889,11585}, {6197,26957}, {6198,26956}, {6240,11438}, {6640,12161}, {6642,11442}, {6644,14516}, {6833,26540}, {6949,26005}, {7399,15045}, {7405,15028}, {7495,13336}, {7576,18381}, {7577,12233}, {8901,19168}, {10024,13630}, {10095,12099}, {10114,17701}, {10257,13292}, {10295,21659}, {10545,23411}, {10574,15760}, {10594,14216}, {11424,23329}, {11441,18917}, {11462,13884}, {11463,13937}, {11799,13491}, {12006,13565}, {12079,14894}, {12118,15078}, {13399,13474}, {13403,21663}, {14157,21841}, {14788,21243}, {14940,15032}, {15061,23336}, {19128,26926}, {26876,26905}, {26877,26932}, {26878,26942}

X(26879) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18916, 7592), (3, 18912, 12022), (3, 26869, 18912)

X(26880) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND 3rd CONWAY

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

The homothetic center of these triangles is X(10882)

X(26880) lies on these lines: {2,1629}, {3,64}, {25,216}, {51,5158}, {97,9544}, {110,26874}, {122,7386}, {160,1660}, {182,6638}, {184,418}, {206,26899}, {426,22352}, {468,26905}, {1495,6641}, {1503,26906}, {1614,26876}, {2187,23207}, {3091,19169}, {3284,11402}, {3549,10600}, {5085,6617}, {6353,26870}, {6389,7494}, {10304,23608}, {10535,26904}, {10536,26908}, {15905,17809}, {18437,21243}, {22052,26864}, {26881,26895}, {26882,26896}, {26883,26897}, {26884,26900}, {26885,26901}, {26887,26902}, {26888,26903}

X(26880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 26898, 216), (154, 26909, 3)

X(26881) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND 3rd EULER

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

The homothetic center of these triangles is X(11680)

X(26881) lies on these lines: {2,1495}, {3,6030}, {4,18475}, {20,10282}, {22,110}, {23,184}, {24,10574}, {25,5012}, {26,1614}, {30,11464}, {49,17714}, {51,11003}, {54,7517}, {74,18324}, {143,11423}, {156,2937}, {159,12272}, {182,11451}, {186,15072}, {206,12220}, {305,10330}, {382,5944}, {428,14389}, {468,26913}, {511,9544}, {669,11450}, {858,10192}, {1147,12088}, {1176,20987}, {1180,1915}, {1498,11440}, {1501,9465}, {1503,23293}, {1511,3534}, {1539,18561}, {1613,8627}, {1658,6241}, {1971,22240}, {1993,9909}, {1995,3796}, {2070,5890}, {2071,11202}, {2393,11443}, {2502,21001}, {3131,14170}, {3132,14169}, {3146,13367}, {3167,23061}, {3431,15682}, {3518,15043}, {3529,12038}, {3543,11430}, {3580,10154}, {3843,10610}, {3845,14805}, {3917,7492}, {3981,14567}, {4240,15466}, {5133,13394}, {5651,15246}, {5943,14002}, {6000,10298}, {6353,18911}, {6515,15360}, {6636,7998}, {6644,20791}, {6759,7488}, {7387,9707}, {7426,13567}, {7493,11206}, {7502,10540}, {7506,15028}, {7512,10539}, {7525,7999}, {7530,15033}, {7542,16659}, {7552,18474}, {7555,23039}, {7556,13754}, {7592,9714}, {7691,9715}, {8780,15066}, {9703,13391}, {9704,10263}, {9705,16266}, {9781,18378}, {10020,23294}, {10201,25739}, {10244,12160}, {10533,11418}, {10534,11417}, {10535,11446}, {10536,11445}, {10545,10601}, {10564,11001}, {10575,21844}, {10594,13434}, {11002,13366}, {11004,21969}, {11188,19127}, {11265,11463}, {11266,11462}, {11267,11467}, {11268,11466}, {11402,20850}, {11413,17821}, {11416,19153}, {11420,11453}, {11421,11452}, {11439,26883}, {11455,18570}, {11456,14070}, {11468,15331}, {11750,16868}, {12087,13346}, {12106,15045}, {12225,16252}, {12270,13289}, {12283,19154}, {12289,15761}, {13406,18394}, {15019,17810}, {15024,15038}, {15051,20771}, {18392,18400}, {18404,18504}, {18928,26255}, {19167,26887}, {19367,26888}, {26880,26895}, {26884,26910}, {26885,26911}, {26886,26912}

X(26881) = reflection of X(11454) in X(10298)
X(26881) = gibert circumtangential conjugate of X(3357)
X(26881) = isogonal conjugate of the isotomic conjugate of X(7802)
X(26881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11550, 7703), (1495, 7712, 15080), (1495, 15080, 10546)

X(26882) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND 4th EULER

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

The homothetic center of these triangles is X(11681)

X(26882) lies on these lines: {3,6030}, {4,1495}, {23,1147}, {24,154}, {25,54}, {26,110}, {30,11449}, {49,3060}, {52,9544}, {74,1498}, {140,15080}, {143,9704}, {156,2070}, {159,12283}, {182,11465}, {184,1199}, {186,1204}, {195,12380}, {206,6403}, {217,10986}, {378,15811}, {381,5944}, {403,12289}, {468,26917}, {569,13595}, {1092,12088}, {1173,17810}, {1503,10018}, {1511,1657}, {1594,10192}, {1656,10546}, {1658,10540}, {1993,9705}, {2393,11458}, {2883,10295}, {2937,2979}, {3091,18475}, {3146,12038}, {3147,11206}, {3357,12112}, {3515,11456}, {3517,7592}, {3520,11202}, {3523,7712}, {3525,22352}, {3533,5092}, {3542,18945}, {3850,14805}, {3851,10610}, {5012,7506}, {5059,10564}, {5446,9545}, {5447,7492}, {5462,11003}, {5562,7556}, {5622,15581}, {6000,11468}, {6143,11550}, {6146,15448}, {6240,16252}, {6353,18912}, {6642,6800}, {7488,10539}, {7502,11444}, {7505,9833}, {7512,7999}, {7525,7998}, {7526,16261}, {7691,15068}, {7730,12234}, {7746,15340}, {8537,19153}, {8780,9715}, {9703,10263}, {10020,23293}, {10274,13423}, {10298,12162}, {10533,10881}, {10534,10880}, {10535,11461}, {10536,11460}, {10594,15033}, {10632,11467}, {10633,11466}, {11265,11448}, {11266,11447}, {11267,11453}, {11268,11452}, {11413,15035}, {11439,18570}, {11440,18324}, {11441,14070}, {11451,13353}, {11454,15331}, {12022,21841}, {12082,15034}, {12106,15043}, {12107,18436}, {12163,14094}, {12244,17701}, {12254,18390}, {12272,19154}, {12278,15761}, {12281,13289}, {13383,14516}, {13394,14788}, {13406,18392}, {13434,13861}, {13619,22802}, {14487,14528}, {14575,25044}, {14940,18381}, {15073,15582}, {15107,16266}, {16868,18394}, {17714,22115}, {18403,18504}, {19168,26887}, {19368,26888}, {22658,22750}, {26880,26896}, {26884,26914}, {26885,26915}, {26886,26916}

X(26882) = reflection of X(i) in X(j) for these (i,j): (11468, 21844), (18394, 16868), (23294, 10018)
X(26882) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14157, 12290), (4, 10282, 11464), (1495, 10282, 4)

X(26883) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND EXCENTERS-REFLECTIONS

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

The homothetic center of these triangles is X(11682)

X(26883) lies on these lines: {2,13347}, {3,1495}, {4,54}, {5,10984}, {6,5198}, {20,9306}, {22,5907}, {23,12111}, {24,1204}, {25,185}, {26,12162}, {30,1092}, {32,3331}, {33,26888}, {34,10535}, {40,26885}, {49,3830}, {51,1181}, {52,7530}, {64,1620}, {74,13452}, {84,26884}, {110,3146}, {113,18569}, {125,3542}, {154,1593}, {155,18534}, {156,1514}, {159,12294}, {182,3091}, {186,3357}, {206,7507}, {235,1503}, {378,10282}, {381,11572}, {382,1147}, {389,10594}, {399,6243}, {403,16659}, {427,15152}, {428,12233}, {468,6247}, {511,11441}, {546,569}, {576,15531}, {631,8718}, {1216,12083}, {1425,11399}, {1568,14790}, {1594,16658}, {1595,16654}, {1596,6146}, {1597,14530}, {1657,18350}, {1660,17845}, {1843,19149}, {1899,3089}, {1906,12241}, {1907,16656}, {1968,1971}, {1993,13598}, {1995,9729}, {2070,7689}, {2207,8779}, {2393,11470}, {2807,8185}, {2883,3575}, {2935,17701}, {2937,18435}, {2979,12087}, {3090,22112}, {3092,21640}, {3093,21641}, {3098,11444}, {3270,11398}, {3516,17821}, {3517,10605}, {3518,6241}, {3520,11202}, {3796,11479}, {3818,13160}, {3832,5012}, {3839,13434}, {3917,11414}, {4232,18913}, {5073,22115}, {5079,13339}, {5320,5706}, {5412,12970}, {5413,12964}, {5446,18445}, {5562,7387}, {5609,16105}, {5656,7487}, {5876,17714}, {5878,18533}, {5895,15139}, {5899,18436}, {6001,11363}, {6193,24981}, {6225,22750}, {6240,22802}, {6293,22972}, {6353,12324}, {6636,15056}, {6644,10575}, {6696,15448}, {6912,13323}, {7395,22352}, {7488,15305}, {7505,20299}, {7512,15058}, {7517,13754}, {7525,15060}, {7526,16194}, {7553,22660}, {7592,10110}, {7998,16661}, {8976,9687}, {9544,17578}, {9707,11430}, {9714,12163}, {9730,13861}, {9781,15032}, {9927,11799}, {9970,11663}, {9973,12175}, {10018,23329}, {10019,23324}, {10117,21650}, {10298,15062}, {10301,11745}, {10303,16187}, {10323,11793}, {10533,11473}, {10534,11474}, {10536,11471}, {10574,13595}, {10606,15750}, {10625,15068}, {10641,10676}, {10642,10675}, {10982,13366}, {10990,12250}, {11204,21844}, {11245,15873}, {11403,11425}, {11439,26881}, {11449,12086}, {11459,12088}, {11464,14865}, {12082,15644}, {12106,13491}, {12133,15647}, {12160,21969}, {12164,14531}, {12279,22467}, {12292,13289}, {12688,14529}, {13348,15066}, {13472,14487}, {13851,15125}, {14094,14448}, {15761,18474}, {15887,17810}, {16835,20421}, {16868,23325}, {17703,22261}, {19137,25406}, {21451,26913}, {26880,26897}, {26886,26918}

X(26883) = reflection of X(i) in X(j) for these (i,j): (1092, 10539), (1204, 24)
X(26883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 184, 11424), (4, 9833, 21659), (1495, 11381, 3)

X(26884) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND INNER-HUTSON

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

The homothetic center of these triangles is X(11685)

X(26884) lies on these lines: {1,5197}, {2,3955}, {7,17985}, {9,5651}, {22,3784}, {25,222}, {28,60}, {31,56}, {34,7335}, {48,8763}, {51,2003}, {57,184}, {63,9306}, {84,26883}, {105,2720}, {110,2651}, {141,26924}, {182,3306}, {199,22097}, {206,26923}, {212,4191}, {219,6090}, {243,23353}, {244,1428}, {255,13738}, {354,20986}, {394,26893}, {450,1948}, {468,26932}, {511,22128}, {614,1397}, {649,834}, {692,1155}, {750,2330}, {851,1936}, {953,4588}, {1086,5137}, {1092,5709}, {1104,1408}, {1385,1621}, {1393,19365}, {1401,5322}, {1458,20999}, {1474,14597}, {1495,3220}, {1498,26927}, {1503,26933}, {1614,26877}, {1709,15503}, {1851,18623}, {1899,20266}, {1935,13724}, {1974,7289}, {2187,9316}, {2249,2727}, {2267,16373}, {2323,3292}, {2328,22060}, {2360,22345}, {2361,20470}, {2915,11573}, {2969,6357}, {3011,5061}, {3145,4303}, {3781,15066}, {3819,5314}, {3912,17977}, {3917,5285}, {4224,17074}, {4579,5205}, {4871,5150}, {6353,26871}, {7004,10535}, {7085,17811}, {8679,20989}, {9225,16514}, {9544,23958}, {10536,26934}, {10539,24467}, {11206,26929}, {13329,23202}, {13737,23072}, {14530,26928}, {16064,22053}, {18360,23844}, {20744,20857}, {22129,24320}, {26864,26866}, {26880,26900}, {26881,26910}, {26882,26914}, {26886,26930}, {26887,26931}

X(26884) = isogonal conjugate of the isotomic conjugate of X(5088)
X(26884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3955, 26890), (25, 222, 26892)

X(26885) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND OUTER-HUTSON

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

The homothetic center of these triangles is X(11686)

X(26885) lies on these lines: {1,5320}, {9,184}, {22,3781}, {25,219}, {31,172}, {33,6056}, {37,2194}, {40,26883}, {48,1011}, {51,2323}, {55,2164}, {57,5651}, {63,9306}, {71,199}, {72,2203}, {101,228}, {110,3219}, {141,26923}, {154,205}, {182,3305}, {198,10537}, {201,26888}, {206,26924}, {209,17796}, {210,692}, {212,8761}, {222,6090}, {394,24320}, {450,1947}, {468,26942}, {517,2355}, {572,23201}, {612,2175}, {674,20988}, {748,1428}, {756,2330}, {1092,7330}, {1473,17811}, {1495,3690}, {1498,26935}, {1503,21015}, {1614,26878}, {1762,21318}, {1818,16064}, {1914,16520}, {1915,16514}, {1974,5227}, {2003,3292}, {2200,16372}, {2280,16516}, {2299,3990}, {3145,3682}, {3220,3917}, {3683,20986}, {3688,5310}, {3784,15066}, {3819,7293}, {5138,5287}, {5279,6061}, {5311,19133}, {6353,26872}, {7069,11429}, {7076,7120}, {7140,7359}, {7186,24436}, {10539,26921}, {11206,26939}, {13615,20818}, {14530,26938}, {14547,22356}, {16058,23095}, {17976,20834}, {20989,22276}, {20999,25941}, {26864,26867}, {26880,26901}, {26881,26911}, {26882,26915}, {26886,26940}, {26887,26941}

X(26885) = isogonal conjugate of the isotomic conjugate of X(7283)
X(26885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 184, 26890), (25, 219, 26893)

X(26886) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11687)

X(26886) lies on these lines: {24,6458}, {25,26894}, {110,26875}, {154,8911}, {184,26891}, {206,26925}, {371,1614}, {372,3518}, {468,26950}, {577,1495}, {1498,26936}, {1503,26951}, {3155,6413}, {5412,6414}, {6200,12112}, {6353,26873}, {6457,6759}, {10535,26949}, {10536,26952}, {10539,26922}, {10962,11417}, {11206,26945}, {26864,26868}, {26881,26912}, {26882,26916}, {26883,26918}, {26884,26930}, {26885,26940}, {26887,26947}, {26888,26948}

X(26886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 26920, 26894), (154, 26953, 8911)

X(26887) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND 1st SHARYGIN

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

The homothetic center of these triangles is X(11688)

X(26887) lies on these lines: {3,19206}, {4,54}, {25,9792}, {26,19194}, {49,13322}, {95,9306}, {97,110}, {154,19180}, {156,19210}, {159,19197}, {182,19188}, {206,19171}, {436,8795}, {468,26954}, {1495,21638}, {1498,19172}, {1503,23295}, {1971,8882}, {1988,14533}, {2393,19178}, {4993,5012}, {6000,19192}, {6353,19166}, {10282,19185}, {10533,19183}, {10534,19184}, {10535,19182}, {10536,19181}, {10539,19179}, {10540,19176}, {13289,19195}, {14530,19173}, {16030,26864}, {19167,26881}, {19168,26882}, {19175,26888}, {26880,26902}, {26884,26931}, {26885,26941}, {26886,26947}

X(26887) = barycentric product X(54)*X(3164)
X(26887) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 19170, 9792), (184, 275, 54)

X(26888) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND TANGENTIAL-MIDARC

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

The homothetic center of these triangles is X(11690)

X(26888) lies on these lines: {1,6759}, {3,7355}, {4,11429}, {11,16252}, {12,1503}, {20,12940}, {25,19349}, {26,7352}, {28,65}, {31,56}, {33,26883}, {34,184}, {35,6000}, {36,10282}, {40,6056}, {48,1950}, {55,1498}, {64,5217}, {73,3145}, {109,2360}, {110,4296}, {159,1469}, {161,9658}, {172,1971}, {182,19372}, {201,26885}, {206,1428}, {222,13730}, {227,692}, {388,11206}, {468,26955}, {498,14216}, {999,14530}, {1038,9306}, {1060,10539}, {1181,11398}, {1250,10675}, {1319,1612}, {1393,26889}, {1394,7335}, {1398,26864}, {1409,1474}, {1425,1495}, {1478,9833}, {1614,1870}, {1619,10831}, {1887,2182}, {1935,3955}, {2066,12970}, {2067,10533}, {2099,10537}, {2192,3303}, {2307,11243}, {2393,19369}, {2646,6001}, {2777,4324}, {2818,11012}, {2883,6284}, {3028,15647}, {3056,19149}, {3146,9637}, {3157,7387}, {3215,13738}, {3295,11189}, {3357,5010}, {3576,14925}, {3585,18400}, {4294,5656}, {4295,7554}, {4302,5878}, {4354,9934}, {4857,14862}, {5204,17821}, {5218,12324}, {5285,7066}, {5414,12964}, {5432,6247}, {5433,10192}, {5596,12588}, {5706,11428}, {6198,14157}, {6353,18915}, {6502,10534}, {7280,11202}, {7951,18381}, {10060,12315}, {10540,18447}, {10638,10676}, {11510,18621}, {12943,17845}, {13289,19470}, {15311,15338}, {17819,18996}, {17820,18995}, {17975,20836}, {19175,26887}, {19367,26881}, {19368,26882}, {20122,20831}, {20306,24953}, {26880,26903}, {26886,26948}

X(26888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6759, 10535), (25, 19349, 19366)
X(26888) = homothetic center of anti-tangential midarc triangle and X(3)-Ehrmann triangle

X(26889) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND INNER-HUTSON

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

The homothetic center of these triangles is X(11886)

X(26889) lies on these lines: {2,7193}, {3,26893}, {6,1473}, {31,1403}, {38,2330}, {42,20999}, {48,4191}, {51,3220}, {54,26877}, {55,12595}, {57,184}, {58,22344}, {63,182}, {84,11424}, {181,5322}, {199,22390}, {209,5096}, {219,7484}, {222,11402}, {228,13329}, {354,692}, {511,7293}, {569,24467}, {572,22060}, {577,26900}, {580,22345}, {603,19365}, {614,2175}, {631,26872}, {1155,20986}, {1393,26888}, {1407,17809}, {1471,2187}, {1851,5222}, {1993,3784}, {2003,3937}, {2194,3752}, {2317,22053}, {2323,3917}, {2999,5320}, {3218,3955}, {3306,9306}, {3666,5135}, {3741,24253}, {3781,7485}, {3914,5091}, {4652,13323}, {5085,7085}, {5092,5314}, {5138,5256}, {5157,26924}, {5221,14529}, {5285,22352}, {5398,23206}, {5437,5651}, {5709,10984}, {7004,11429}, {7308,22112}, {7499,26942}, {10601,24320}, {11003,23958}, {11245,26932}, {11422,26910}, {11423,26914}, {11425,26927}, {11426,26928}, {11427,26929}, {11428,26934}, {13336,26921}, {14547,16064}, {14912,26871}, {15299,15503}, {16059,23095}, {16560,21318}, {17017,19133}, {17188,24618}, {18162,21319}, {22394,23621}, {23292,26933}, {26891,26930}

X(26889) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7193, 26885), (6, 1473, 26892)

X(26890) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND OUTER-HUTSON

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

The homothetic center of these triangles is X(11887)

X(26890) lies on these lines: {2,3955}, {3,26892}, {6,31}, {9,184}, {38,1428}, {40,11424}, {44,2194}, {51,5285}, {54,72}, {63,182}, {78,13323}, {101,23201}, {199,2183}, {201,19365}, {210,20986}, {219,11402}, {220,17809}, {222,7484}, {228,572}, {375,20989}, {511,5314}, {569,26921}, {577,26901}, {612,1397}, {631,26871}, {692,3683}, {1211,3035}, {1437,5044}, {1473,5085}, {1743,5320}, {1829,6197}, {1993,3781}, {2003,3917}, {2203,4183}, {2317,2318}, {2323,3690}, {2328,23202}, {2352,4268}, {3219,5012}, {3220,22352}, {3271,5310}, {3305,9306}, {3687,17977}, {3741,5150}, {3757,4579}, {3784,7485}, {3796,24320}, {3819,22128}, {4415,5137}, {4641,5135}, {5092,7293}, {5130,5136}, {5157,26923}, {5197,16569}, {5437,22112}, {5651,7308}, {5749,7102}, {5752,26285}, {7069,10535}, {7330,10984}, {7499,26932}, {9957,17015}, {11245,26942}, {11422,26911}, {11423,26915}, {11425,26935}, {11426,26938}, {11427,26939}, {13329,22060}, {13336,24467}, {14153,16514}, {14912,26872}, {20683,20959}, {21015,23292}, {21319,23693}, {26891,26940}

X(26890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3955, 26884), (6, 7085, 26893), (212, 2267, 1011)

X(26891) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(10885)

X(26891) lies on these lines: {6,3156}, {54,371}, {184,26886}, {372,1199}, {569,26922}, {577,13366}, {578,6457}, {3311,19356}, {3518,5413}, {5012,26875}, {6431,17820}, {6458,7592}, {11245,26950}, {11402,26868}, {11422,26912}, {11423,26916}, {11424,26918}, {11425,26936}, {11427,26945}, {11428,26952}, {11429,26949}, {14912,26873}, {17809,26953}, {19365,26948}, {23292,26951}, {26889,26930}, {26890,26940}

X(26891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 8911, 26894), (184, 26919, 26886)

X(26892) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CONWAY AND INNER-HUTSON

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

The homothetic center of these triangles is X(9783)

X(26892) lies on these lines: {1,855}, {2,3784}, {3,26890}, {4,26871}, {6,1473}, {7,1851}, {9,3917}, {22,3955}, {25,222}, {27,2659}, {31,1469}, {33,1364}, {38,3056}, {40,16980}, {47,23850}, {51,57}, {52,24467}, {55,8679}, {63,511}, {84,185}, {182,7293}, {184,2003}, {189,7102}, {212,16064}, {216,26900}, {228,991}, {244,7248}, {255,3145}, {373,5437}, {375,4413}, {386,22344}, {394,24320}, {405,11573}, {427,26932}, {513,1836}, {517,6938}, {573,22060}, {581,22345}, {603,19366}, {614,1401}, {651,4224}, {942,1828}, {966,22412}, {970,4652}, {971,1824}, {984,7186}, {993,2392}, {1011,22097}, {1350,7085}, {1394,1425}, {1397,5322}, {1399,23843}, {1407,17810}, {1423,23440}, {1621,23155}, {1626,2361}, {1709,2807}, {1843,7289}, {1935,13733}, {1993,7193}, {2082,23630}, {2099,2390}, {2183,4191}, {2270,22440}, {2277,17187}, {2310,21328}, {2810,3870}, {2841,25415}, {2979,3219}, {3060,3218}, {3098,5314}, {3157,13730}, {3305,3819}, {3306,5943}, {3434,15310}, {3567,26877}, {3690,3929}, {3772,18191}, {3792,7262}, {3868,20077}, {3916,5752}, {3928,21969}, {4001,10477}, {4259,4641}, {4303,13738}, {4459,17871}, {4640,9037}, {4884,9024}, {5208,17364}, {5248,23156}, {5360,24635}, {5396,23206}, {5562,7330}, {5640,26910}, {5650,7308}, {6090,23140}, {7004,11436}, {7363,15508}, {8614,14529}, {9306,22128}, {9777,26866}, {9781,26914}, {9786,26927}, {9792,26931}, {10167,14557}, {10391,17441}, {10625,26921}, {11002,23958}, {11432,26928}, {11433,26929}, {11435,26934}, {13567,26933}, {14963,22420}, {15030,18540}, {18161,21318}, {20665,23636}, {20831,23070}, {20834,22161}, {20852,23131}, {22069,23619}, {26894,26930}

X(26892) = reflection of X(i) in X(j) for these (i,j): (17441, 10391), (26893, 63)
X(26892) = isogonal conjugate of the isotomic conjugate of X(17181)
X(26892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1473, 26889), (25, 222, 26884)

X(26893) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CONWAY AND OUTER-HUTSON

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

The homothetic center of these triangles is X(9787)

X(26893) lies on these lines: {1,10974}, {2,3781}, {3,26889}, {4,8}, {6,31}, {9,51}, {22,7193}, {25,219}, {34,7066}, {38,1469}, {40,185}, {48,199}, {52,26921}, {57,3917}, {63,511}, {78,970}, {181,612}, {182,5314}, {184,2323}, {201,19366}, {210,2262}, {216,26901}, {220,17810}, {228,573}, {306,10477}, {373,7308}, {375,3715}, {394,26884}, {427,26942}, {464,16465}, {518,17441}, {756,4517}, {851,24310}, {916,7580}, {941,2335}, {968,21746}, {982,3792}, {991,22060}, {1211,2886}, {1282,1763}, {1350,1473}, {1818,4191}, {1836,20718}, {1837,22299}, {1843,5227}, {1864,21871}, {1993,3955}, {2082,20683}, {2099,10459}, {2175,5310}, {2183,2318}, {2245,2352}, {2277,20966}, {2900,3169}, {2979,3218}, {3060,3219}, {3098,7293}, {3151,20243}, {3270,7070}, {3305,5943}, {3306,3819}, {3416,22275}, {3567,26878}, {3666,4259}, {3682,13738}, {3725,3764}, {3868,17778}, {3870,9052}, {3928,3937}, {3929,21969}, {3981,16514}, {4215,4269}, {4260,5256}, {4640,9047}, {4645,25308}, {4650,7186}, {4855,15489}, {5231,10439}, {5364,20684}, {5437,5650}, {5562,5709}, {5640,26911}, {5791,18180}, {6506,15508}, {6734,10441}, {6745,10440}, {7069,21801}, {7235,17871}, {9777,26867}, {9781,26915}, {9786,26935}, {9792,26941}, {10625,24467}, {11269,21334}, {11432,26938}, {11433,26939}, {13567,21015}, {13726,19767}, {17792,26034}, {20012,20075}, {20539,22321}, {20857,22126}, {26894,26940}

X(26893) = reflection of X(i) in X(j) for these (i,j): (55, 22276), (26892, 63)
X(26893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7085, 26890), (6, 12329, 26924), (3869, 25306, 4388)

X(26894) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CONWAY AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(9789)

X(26894) lies on these lines: {4,372}, {6,3156}, {25,26886}, {51,577}, {52,26922}, {185,26918}, {371,3567}, {389,6457}, {427,26950}, {571,8576}, {1589,3618}, {3060,26875}, {3312,19347}, {3594,12964}, {5408,10963}, {5640,26912}, {6420,11423}, {6423,19005}, {8908,13366}, {9777,26868}, {9781,26916}, {9786,26936}, {9792,26947}, {11242,17849}, {11433,26945}, {11435,26952}, {11436,26949}, {13567,26951}, {17810,26953}, {19366,26948}, {26892,26930}, {26893,26940}

X(26894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 372, 6458), (372, 5413, 6414)

X(26895) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 3rd EULER

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

The homothetic center of these triangles is X(10886)

X(26895) lies on these lines: {2,26907}, {3,74}, {22,26909}, {216,5640}, {418,3060}, {577,11422}, {1993,26865}, {2979,26874}, {5012,26898}, {5889,26876}, {6638,11451}, {10546,10979}, {11439,26897}, {11445,26908}, {11446,26904}, {11746,18573}, {18911,26870}, {19122,26899}, {19167,26902}, {19367,26903}, {23293,26906}, {26880,26881}, {26900,26910}, {26901,26911}, {26905,26913}

X(26896) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 4th EULER

Barycentrics
a^2*(-a^2+b^2+c^2)*(3*(b^2+c^2)*a^10-(13*b^4+15*b^2*c^2+13*c^4)*a^8+2*(b^2+c^2)*(11*b^4-6*b^2*c^2+11*c^4)*a^6-18*(b^6-c^6)*(b^2-c^2)*a^4+7*(b^4-c^4)*(b^2-c^2)^3*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^4) : :

The homothetic center of these triangles is X(10887)

X(26896) lies on these lines: {3,74}, {4,26907}, {24,26909}, {54,26898}, {216,9781}, {418,3567}, {577,11423}, {5890,26876}, {6638,11465}, {7592,26865}, {11412,26874}, {11455,26897}, {11460,26908}, {11461,26904}, {18912,26870}, {19123,26899}, {19168,26902}, {19368,26903}, {23294,26906}, {26880,26882}, {26900,26914}, {26901,26915}, {26905,26917}

X(26897) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND EXCENTERS-REFLECTIONS

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

The homothetic center of these triangles is X(11521)

X(26897) lies on these lines: {2,3}, {33,26903}, {34,26904}, {40,26901}, {54,14152}, {84,26900}, {95,1105}, {160,17845}, {185,216}, {577,11424}, {578,23606}, {1498,26898}, {2055,15033}, {2972,11793}, {5562,13409}, {6000,23719}, {6247,26905}, {11381,26907}, {11439,26895}, {11455,26896}, {11471,26908}, {12324,26870}, {15811,26909}, {19124,26899}, {19169,26902}, {19467,20775}, {21659,23195}, {26880,26883}

X(26897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 418), (3, 6905, 408), (3, 7395, 426)

X(26898) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 2nd EXTOUCH

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

The homothetic center of these triangles is X(10888)

X(26898) lies on these lines: {2,26870}, {3,49}, {6,418}, {25,216}, {54,26896}, {154,157}, {183,7494}, {219,26901}, {222,26900}, {426,5085}, {577,11402}, {852,17825}, {1073,5650}, {1350,13409}, {1498,26897}, {1899,26906}, {1993,26874}, {5012,26895}, {5158,9777}, {6389,7499}, {6509,7484}, {6638,10601}, {7503,19172}, {7592,26876}, {10979,26864}, {13366,15905}, {15004,15851}, {17809,23606}, {19125,26899}, {19170,26902}, {19349,26903}, {19350,26908}, {19354,26904}

X(26898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26870, 26905), (184, 26907, 3), (10132, 10133, 19357)

X(26899) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND HONSBERGER

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

The homothetic center of these triangles is X(10889)

X(26899) lies on these lines: {3,6}, {53,12362}, {97,193}, {206,26880}, {233,3549}, {418,1974}, {1352,10600}, {1428,26903}, {1843,6641}, {2330,26904}, {2351,6467}, {3087,7400}, {3589,26906}, {5907,17849}, {6638,19137}, {6676,10314}, {6748,6823}, {7494,10311}, {14576,15818}, {19118,26865}, {19119,26870}, {19121,26874}, {19122,26895}, {19123,26896}, {19124,26897}, {19125,26898}, {19128,26876}, {19132,26909}, {19133,26908}, {19171,26902}, {21637,26907}, {26900,26923}, {26901,26924}, {26905,26926}

X(26899) = midpoint of X(11513) and X(11514)
X(26899) = isogonal conjugate of the polar conjugate of X(7395)
X(26899) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10979, 22052, 8588), (11515, 11516, 3098), (11574, 19126, 13355)

X(26900) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND INNER-HUTSON

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

The homothetic center of these triangles is X(11892)

X(26900) lies on these lines: {3,63}, {57,418}, {84,26897}, {216,26892}, {222,26898}, {577,26889}, {603,26903}, {852,5437}, {1407,26909}, {3218,26874}, {3220,6641}, {3306,6638}, {3937,26907}, {7004,26904}, {26865,26866}, {26870,26871}, {26876,26877}, {26880,26884}, {26895,26910}, {26896,26914}, {26899,26923}, {26902,26931}, {26905,26932}, {26906,26933}, {26908,26934}

X(26900) = {X(3), X(63)}-harmonic conjugate of X(26901)

X(26901) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND OUTER-HUTSON

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

The homothetic center of these triangles is X(11893)

X(26901) lies on these lines: {3,63}, {9,418}, {40,26897}, {71,26908}, {201,26903}, {212,26904}, {216,26893}, {219,26898}, {220,26909}, {408,5438}, {577,26890}, {852,7308}, {3219,26874}, {3305,6638}, {3690,26907}, {5285,6641}, {21015,26906}, {26865,26867}, {26870,26872}, {26876,26878}, {26880,26885}, {26895,26911}, {26896,26915}, {26899,26924}, {26902,26941}, {26905,26942}

X(26901) = {X(3), X(63)}-harmonic conjugate of X(26900)

X(26902) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 1st SHARYGIN

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

The homothetic center of these triangles is X(10892)

X(26902) lies on these lines: {3,95}, {54,577}, {97,184}, {216,9792}, {275,418}, {6638,19188}, {16030,26865}, {19166,26870}, {19167,26895}, {19168,26896}, {19169,26897}, {19170,26898}, {19171,26899}, {19175,26903}, {19180,26909}, {19181,26908}, {19182,26904}, {21638,26907}, {23295,26906}, {26880,26887}, {26900,26931}, {26901,26941}, {26905,26954}

X(26903) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND TANGENTIAL-MIDARC

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

The homothetic center of these triangles is X(11894)

X(26903) lies on these lines: {1,3}, {12,26906}, {33,26897}, {34,418}, {201,26901}, {216,19366}, {221,26909}, {577,19365}, {603,26900}, {1398,26865}, {1425,26907}, {1428,26899}, {1870,26876}, {4296,26874}, {6638,19372}, {18915,26870}, {19175,26902}, {19349,26898}, {19367,26895}, {19368,26896}, {26880,26888}, {26905,26955}

X(26903) = {X(1), X(3)}-harmonic conjugate of X(26904)

X(26904) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 2nd TANGENTIAL-MIDARC

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

The homothetic center of these triangles is X(11895)

X(26904) lies on these lines: {1,3}, {11,26906}, {33,418}, {34,26897}, {97,9637}, {212,26901}, {216,11436}, {577,11429}, {2192,26909}, {2330,26899}, {3100,26874}, {3270,26907}, {6198,26876}, {6638,9817}, {7004,26900}, {7071,26865}, {10535,26880}, {11446,26895}, {11461,26896}, {18922,26870}, {19182,26902}, {19354,26898}, {26905,26956}

X(26904) = {X(1), X(3)}-harmonic conjugate of X(26903)

X(26905) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND URSA MAJOR

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

The homothetic center of these triangles is X(17617)

X(26905) lies on these lines: {2,26870}, {3,68}, {125,26906}, {216,427}, {325,7386}, {418,13567}, {468,26880}, {577,11245}, {1503,6641}, {3001,13409}, {3580,26874}, {6247,26897}, {6389,7484}, {7399,10600}, {8550,23606}, {26865,26869}, {26876,26879}, {26895,26913}, {26896,26917}, {26899,26926}, {26900,26932}, {26901,26942}, {26902,26954}, {26903,26955}, {26904,26956}, {26908,26957}, {26909,26958}

X(26905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26870, 26898), (125, 26907, 26906)

X(26906) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND URSA MINOR

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

The homothetic center of these triangles is X(10473)

X(26906) lies on these lines: {2,3}, {11,26904}, {12,26903}, {125,26905}, {141,6509}, {216,13567}, {577,23292}, {1503,26880}, {1853,26909}, {1899,26898}, {3589,26899}, {3925,26908}, {11427,15905}, {21015,26901}, {23291,26870}, {23293,26895}, {23294,26896}, {23295,26902}, {26900,26933}

X(26906) = isotomic conjugate of the polar conjugate of X(12233)
X(26906) = complement of the polar conjugate of X(13599)
X(26906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (465, 466, 549), (1589, 1590, 3523)

X(26907) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND WASAT

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

The homothetic center of these triangles is X(10478)

X(26907) lies on these lines: {2,26895}, {3,49}, {4,26896}, {5,12012}, {6,26865}, {25,26909}, {51,216}, {125,26905}, {311,7494}, {373,6638}, {389,26876}, {511,26874}, {577,13366}, {1425,26903}, {1495,6641}, {1843,3135}, {1899,26870}, {3270,26904}, {3611,26908}, {3690,26901}, {3937,26900}, {5650,6509}, {6467,23195}, {6617,22112}, {10282,23719}, {11381,26897}, {21637,26899}, {21638,26902}, {22052,23606}

X(26907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 26898, 184), (216, 418, 51)

X(26908) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND YFF CENTRAL

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

The homothetic center of these triangles is X(11896)

X(26908) lies on these lines: {1,3}, {19,418}, {71,26901}, {216,11435}, {577,11428}, {3101,26874}, {3197,26909}, {3611,26907}, {3925,26906}, {6197,26876}, {6638,9816}, {10536,26880}, {11406,26865}, {11436,18591}, {11445,26895}, {11460,26896}, {11471,26897}, {18921,26870}, {19133,26899}, {19181,26902}, {19350,26898}, {26900,26934}, {26905,26957}

X(26909) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd CONWAY AND 2nd ZANIAH

Barycentrics    a^2*(a^8+6*(b^2+c^2)*a^6-16*(b^4+c^4)*a^4+10*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)*(-a^2+b^2+c^2) : :

The homothetic center of these triangles is X(18229)

X(26909) lies on these lines: {3,64}, {6,418}, {22,26895}, {24,26896}, {25,26907}, {184,26865}, {216,17810}, {220,26901}, {221,26903}, {394,26874}, {577,17809}, {1181,26876}, {1407,26900}, {1853,26906}, {2192,26904}, {3197,26908}, {6638,17825}, {7494,15271}, {13567,26870}, {15811,26897}, {19132,26899}, {19180,26902}, {26905,26958}

X(26909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 26880, 154), (418, 26898, 6)

X(26910) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND INNER-HUTSON

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

The homothetic center of these triangles is X(8377)

X(26910) lies on these lines: {2,3937}, {3,26914}, {22,1407}, {55,840}, {57,3060}, {63,7998}, {84,11439}, {108,17074}, {110,1473}, {222,5012}, {511,23958}, {603,19367}, {1155,23155}, {1401,17126}, {1993,26866}, {2979,3218}, {3271,9335}, {3306,11451}, {4188,23154}, {5640,26892}, {5889,26877}, {7004,11446}, {7293,15080}, {7485,22129}, {8679,9352}, {11422,26889}, {11440,26927}, {11441,26928}, {11442,26929}, {11444,24467}, {11445,26934}, {17375,22413}, {18911,26871}, {19122,26923}, {19167,26931}, {26881,26884}, {26895,26900}, {26912,26930}, {26913,26932}

X(26910) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 7998, 26911), (3218, 3784, 2979)

X(26911) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND OUTER-HUTSON

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

The homothetic center of these triangles is X(8378)

X(26911) lies on these lines: {2,3690}, {3,26915}, {9,3060}, {22,220}, {40,11439}, {56,7144}, {63,7998}, {71,11445}, {110,7085}, {181,9330}, {201,19367}, {212,11446}, {219,5012}, {469,3876}, {1180,16514}, {1993,26867}, {2979,3219}, {3305,11451}, {3681,17233}, {3688,17127}, {3730,4184}, {3920,4517}, {5314,15080}, {5640,26893}, {5650,23958}, {5692,15523}, {5889,26878}, {11422,26890}, {11440,26935}, {11441,26938}, {11442,26939}, {11444,26921}, {12109,17570}, {17018,20683}, {18911,26872}, {19122,26924}, {19167,26941}, {21015,23293}, {26881,26885}, {26895,26901}, {26912,26940}, {26913,26942}

X(26911) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 7998, 26910), (3219, 3781, 2979)

X(26912) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8228)

X(26912) lies on these lines: {2,95}, {3,5410}, {6,588}, {22,26953}, {32,7585}, {50,590}, {110,8911}, {371,5889}, {372,15043}, {492,4558}, {571,3068}, {1583,15905}, {1993,26868}, {2193,16441}, {2979,26875}, {3060,26919}, {3069,5063}, {3155,11418}, {5012,26920}, {5065,7586}, {5640,26894}, {6413,11447}, {6457,12111}, {6458,10574}, {6748,15234}, {8908,9544}, {8963,22052}, {10316,11292}, {10962,11448}, {11422,26891}, {11439,26918}, {11440,26936}, {11442,26945}, {11444,26922}, {11445,26952}, {11446,26949}, {11514,12220}, {13345,19054}, {18911,26873}, {19122,26925}, {19167,26947}, {19367,26948}, {23293,26951}, {26881,26886}, {26910,26930}, {26911,26940}, {26913,26950}

X(26913) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND URSA MAJOR

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

The homothetic center of these triangles is X(17618)

X(26913) lies on these lines: {2,98}, {3,26917}, {4,13445}, {5,6241}, {22,26958}, {54,6640}, {235,12279}, {343,7998}, {403,15072}, {427,5640}, {468,26881}, {569,6143}, {631,5449}, {858,3060}, {1209,3525}, {1368,2979}, {1370,15107}, {1594,15043}, {1648,3981}, {1656,13561}, {1853,1995}, {1993,26869}, {2071,18390}, {2072,5890}, {3091,5878}, {3153,11438}, {3548,18912}, {3567,13371}, {3618,6697}, {3855,18488}, {5094,5422}, {5133,7703}, {5159,11245}, {5169,5943}, {5576,15024}, {5643,18928}, {5889,11585}, {6030,7493}, {6146,11449}, {6247,11439}, {6515,8538}, {6643,7691}, {6644,25739}, {6677,10546}, {7394,10545}, {7509,9932}, {7527,23329}, {7569,15805}, {7577,9730}, {8263,12272}, {8889,12834}, {10024,11704}, {10254,20304}, {10255,13630}, {10257,12022}, {10264,18435}, {10413,11648}, {11004,11225}, {11440,26937}, {11441,26944}, {11444,12359}, {11445,26957}, {11446,26956}, {11550,13595}, {12278,22467}, {15033,18281}, {15060,20379}, {15061,18570}, {15078,18396}, {15760,20791}, {15801,18951}, {19122,26926}, {19167,26954}, {19367,26955}, {21451,26883}, {26895,26905}, {26910,26932}, {26911,26942}, {26912,26950}

X(26913) = inverse of X(3047) in the Brocard circle
X(26913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3410, 5651), (2, 9544, 5972), (13414, 13415, 3047)

X(26914) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th EULER AND INNER-HUTSON

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

The homothetic center of these triangles is X(8380)

X(26914) lies on these lines: {3,26910}, {4,3937}, {24,1407}, {54,222}, {56,953}, {57,3567}, {63,7999}, {74,26927}, {84,11455}, {603,19368}, {1473,1614}, {3218,11412}, {3306,11465}, {5890,26877}, {6942,23154}, {7004,11461}, {7509,22129}, {7592,26866}, {9781,26892}, {11423,26889}, {11456,26928}, {11457,26929}, {11459,24467}, {11460,26934}, {18912,26871}, {19123,26923}, {19168,26931}, {23294,26933}, {26882,26884}, {26896,26900}, {26916,26930}, {26917,26932}

X(26914) = {X(63), X(7999)}-harmonic conjugate of X(26915)

X(26915) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th EULER AND OUTER-HUTSON

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

The homothetic center of these triangles is X(8381)

X(26915) lies on these lines: {3,26911}, {4,3690}, {9,3567}, {24,220}, {40,11455}, {54,219}, {55,7144}, {63,7999}, {71,11460}, {74,26935}, {201,19368}, {212,11461}, {1614,7085}, {3219,11412}, {3305,11465}, {5890,26878}, {7592,26867}, {9781,26893}, {11423,26890}, {11456,26938}, {11457,26939}, {11459,26921}, {18912,26872}, {19123,26924}, {19168,26941}, {21015,23294}, {26882,26885}, {26896,26901}, {26916,26940}, {26917,26942}

X(26915) = {X(63), X(7999)}-harmonic conjugate of X(26914)

X(26916) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th EULER AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8230)

X(26916) lies on these lines: {3,5410}, {4,577}, {24,26953}, {32,7581}, {50,3070}, {54,26920}, {74,26936}, {97,1586}, {371,5890}, {372,3567}, {571,1587}, {637,4558}, {1588,5063}, {1614,8911}, {3155,10881}, {5065,7582}, {6241,6457}, {6413,11462}, {6811,10313}, {7592,26868}, {9781,26894}, {10316,21736}, {11412,26875}, {11423,26891}, {11455,26918}, {11457,26945}, {11459,26922}, {11460,26952}, {11461,26949}, {18912,26873}, {19123,26925}, {19168,26947}, {19368,26948}, {23294,26951}, {26882,26886}, {26914,26930}, {26915,26940}, {26917,26950}

X(26916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371, 6458, 5890), (372, 26919, 3567)

X(26917) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th EULER AND URSA MAJOR

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

The homothetic center of these triangles is X(17619)

X(26917) lies on these lines: {2,54}, {3,26913}, {4,74}, {5,5890}, {24,25739}, {110,25738}, {140,12022}, {143,12099}, {184,14940}, {185,16868}, {186,12289}, {235,12290}, {265,12278}, {343,7999}, {381,13561}, {389,7577}, {403,2883}, {427,9781}, {468,26882}, {568,10224}, {578,6143}, {1594,3567}, {1614,1899}, {1656,7592}, {1853,10594}, {2072,5889}, {2929,16013}, {3060,13371}, {3090,18916}, {3448,10539}, {3518,18381}, {3520,18390}, {3542,11457}, {3549,18911}, {3580,11412}, {3628,11245}, {3839,18488}, {5012,6639}, {5067,18950}, {5070,11402}, {5448,23515}, {5576,5640}, {6102,7723}, {6146,10018}, {6240,18394}, {6247,11455}, {6640,15059}, {6697,14853}, {6723,10112}, {6794,10413}, {7547,9786}, {7552,10984}, {7569,10601}, {7699,12233}, {9544,10116}, {9703,11264}, {9927,22467}, {9938,14852}, {10024,10574}, {10113,18565}, {10182,10619}, {10254,13630}, {10264,18439}, {11202,12254}, {11250,15061}, {11456,26944}, {11459,12359}, {11460,26957}, {11461,26956}, {11465,14788}, {11468,18560}, {11695,14789}, {11799,12279}, {12106,15027}, {12118,15035}, {12161,24572}, {12293,15078}, {12824,15114}, {12897,20397}, {13160,15045}, {14516,16238}, {14865,23329}, {14912,24206}, {15072,15761}, {15559,23332}, {16659,21841}, {18350,18356}, {18383,18559}, {19123,26926}, {19168,26954}, {19368,26955}, {21659,21844}, {26896,26905}, {26914,26932}, {26915,26942}, {26916,26950}

X(26917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18912, 54), (4, 125, 23294), (4, 26937, 74)

X(26918) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11532)

X(26918) lies on these lines: {4,371}, {25,26936}, {30,26922}, {33,26948}, {34,26949}, {40,26940}, {64,1152}, {84,26930}, {185,26894}, {235,26951}, {372,6241}, {577,11381}, {1498,26920}, {1593,8911}, {3146,26875}, {3155,6409}, {6000,6458}, {6247,26950}, {8576,23261}, {11403,26868}, {11424,26891}, {11439,26912}, {11455,26916}, {11471,26952}, {11513,14927}, {12324,26873}, {15811,26953}, {19124,26925}, {19169,26947}, {26883,26886}

X(26918) = {X(4), X(6457)}-harmonic conjugate of X(26919)

X(26919) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXCENTRAL AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8231)

X(26919) lies on these lines: {2,26875}, {4,371}, {5,26922}, {6,3155}, {9,26940}, {19,26952}, {25,8911}, {33,26949}, {34,26948}, {51,577}, {57,26930}, {184,26886}, {275,26947}, {372,3567}, {389,6458}, {427,26951}, {571,8577}, {1495,8908}, {1590,3618}, {1593,26936}, {1974,26925}, {3060,26912}, {3311,19347}, {3592,12970}, {5409,10961}, {6419,11423}, {8963,9738}, {11241,17849}, {11433,26873}

X(26919) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 371, 6457), (4, 6457, 26918), (371, 5412, 6413)

X(26920) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8233)

X(26920) lies on these lines: {2,26873}, {3,6413}, {6,3155}, {24,372}, {25,26886}, {32,26461}, {48,606}, {54,26916}, {96,485}, {155,26922}, {184,418}, {185,26936}, {216,21640}, {219,26940}, {222,26930}, {371,7592}, {1152,17819}, {1181,6457}, {1300,6560}, {1498,26918}, {1590,18923}, {1600,10960}, {1899,26951}, {1993,26875}, {3068,12256}, {3070,22261}, {3156,10533}, {3284,21641}, {3365,8837}, {3390,8839}, {5012,26912}, {5408,11513}, {5409,9723}, {6423,19006}, {6776,26945}, {11402,26868}, {15905,19356}, {19125,26925}, {19170,26947}, {19349,26948}, {19350,26952}, {19354,26949}

X(26920) = isogonal conjugate of the isotomic conjugate of X(5409)
X(26920) = isogonal conjugate of the polar conjugate of X(372)
X(26920) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26873, 26950), (3, 19355, 6413)

X(26921) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HEXYL AND OUTER-HUTSON

Barycentrics    a*(a^4-2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(26921) = X(40)+3*X(3929) = X(946)-3*X(5325) = X(1482)-3*X(16418) = 2*X(3824)-3*X(11231) = 3*X(3929)-X(7330) = 3*X(5657)-X(6850) = 5*X(10595)-9*X(17561) = 3*X(11111)+X(12245)

The homothetic center of these triangles is X(8112)

X(26921) lies on these lines: {1,2361}, {2,26878}, {3,63}, {4,3219}, {5,9}, {7,6989}, {8,6868}, {10,6917}, {12,46}, {19,7534}, {26,5285}, {30,40}, {37,5707}, {38,602}, {52,26893}, {55,920}, {57,140}, {68,71}, {77,23070}, {84,550}, {90,6284}, {144,6908}, {155,219}, {165,17857}, {201,255}, {210,11499}, {212,1062}, {329,6825}, {392,10680}, {405,24474}, {484,9579}, {498,1454}, {516,18517}, {517,958}, {518,10267}, {527,6684}, {548,7171}, {549,3928}, {569,26890}, {573,5810}, {601,896}, {631,3218}, {632,5437}, {908,6863}, {936,6924}, {942,1708}, {946,5325}, {960,11249}, {971,1158}, {984,3072}, {997,26286}, {1006,3868}, {1147,3955}, {1214,3157}, {1216,3781}, {1385,11194}, {1445,5708}, {1479,7082}, {1482,5250}, {1483,6762}, {1656,3305}, {1697,5844}, {1698,5535}, {1699,24468}, {1728,5722}, {1737,10953}, {1766,5788}, {1768,12738}, {1776,4294}, {2003,16266}, {2095,11108}, {2323,12161}, {2771,12520}, {3073,7262}, {3085,7098}, {3306,3526}, {3336,4654}, {3338,5298}, {3359,10942}, {3419,7491}, {3428,5887}, {3436,5657}, {3452,6959}, {3523,26877}, {3555,16202}, {3564,5227}, {3576,6763}, {3601,7508}, {3627,18540}, {3628,7308}, {3651,12528}, {3678,6796}, {3681,11491}, {3690,5562}, {3695,3719}, {3730,8558}, {3784,5447}, {3811,22937}, {3824,11231}, {3876,6905}, {3899,11014}, {4640,11248}, {4880,15016}, {5010,16767}, {5044,6911}, {5119,10950}, {5130,7511}, {5223,5534}, {5273,5758}, {5302,7686}, {5428,11523}, {5432,17700}, {5433,17437}, {5536,8227}, {5692,11012}, {5694,6261}, {5744,6891}, {5745,6862}, {5759,6851}, {5761,6857}, {5769,21061}, {5770,6865}, {5777,6985}, {5811,6172}, {5886,12704}, {5904,10902}, {5905,6889}, {6643,26939}, {6734,6928}, {6929,12572}, {6936,12649}, {6944,18228}, {7066,7352}, {7070,8144}, {7162,17699}, {7387,24320}, {7395,26867}, {7680,18253}, {7688,15071}, {7965,12699}, {8545,11662}, {8703,9841}, {9780,10599}, {9956,10894}, {10039,18962}, {10198,15296}, {10303,23958}, {10523,24914}, {10525,18232}, {10539,26885}, {10595,17561}, {10625,26892}, {11111,12245}, {11411,26872}, {11444,26911}, {11459,26915}, {11585,21015}, {11929,17528}, {12359,26942}, {12526,14988}, {12619,13272}, {13336,26889}, {13374,15254}, {14110,22758}, {15481,18491}, {18443,24475}, {18518,18908}, {19131,26924}, {19179,26941}, {19861,22765}, {26922,26940}

X(26921) = midpoint of X(i) and X(j) for these {i,j}: {3, 3927}, {8, 6868}, {40, 7330}
X(26921) = reflection of X(i) in X(j) for these (i,j): (6147, 140), (6917, 10)
X(26921) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 63, 24467), (78, 21165, 3)
X(26921) = 2nd-extouch-to-excentral similarity image of X(5)

X(26922) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HEXYL AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8234)

X(26922) lies on these lines: {2,371}, {3,6414}, {4,26875}, {5,26919}, {6,19493}, {30,26918}, {52,26894}, {68,6413}, {97,5408}, {155,26920}, {372,5889}, {569,26891}, {577,5562}, {1060,26948}, {1062,26949}, {1151,19409}, {1217,11473}, {1297,11824}, {1322,5412}, {3071,13046}, {3092,9732}, {6458,13754}, {6643,26945}, {6776,11513}, {7395,26868}, {8251,26952}, {10539,26886}, {10880,10960}, {11411,26873}, {11444,26912}, {11459,26916}, {11585,26951}, {12313,14489}, {12359,26950}, {17814,26953}, {19131,26925}, {19179,26947}, {24467,26930}, {26921,26940}

X(26922) = isogonal conjugate of the polar conjugate of X(11091)
X(26922) = isotomic conjugate of the polar conjugate of X(6414)
X(26922) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10666, 6414), (371, 486, 8576)

X(26923) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND INNER-HUTSON

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

The homothetic center of these triangles is X(8385)

X(26923) lies on these lines: {6,1473}, {48,20731}, {57,1974}, {63,19126}, {69,7193}, {84,19124}, {141,26885}, {184,7289}, {206,26884}, {222,19125}, {603,1428}, {1176,3955}, {1407,19132}, {1843,3220}, {2330,7004}, {3218,19121}, {3306,19137}, {3589,26933}, {3618,26929}, {3784,20806}, {3937,21637}, {5050,26928}, {5085,26927}, {5157,26890}, {7293,11574}, {19118,26866}, {19119,26871}, {19122,26910}, {19123,26914}, {19128,26877}, {19131,24467}, {19133,26934}, {19171,26931}, {26899,26900}, {26925,26930}, {26926,26932}

X(26923) = {X(63), X(19126)}-harmonic conjugate of X(26924)

X(26924) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND OUTER-HUTSON

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

The homothetic center of these triangles is X(8386)

X(26924) lies on these lines: {6,31}, {9,1974}, {40,19124}, {63,19126}, {69,3955}, {72,1176}, {141,26884}, {184,5227}, {201,1428}, {206,26885}, {219,19125}, {220,19132}, {1843,5285}, {3219,19121}, {3305,19137}, {3589,21015}, {3618,26939}, {3690,21637}, {3781,20806}, {5050,26938}, {5085,26935}, {5157,26889}, {5314,11574}, {19118,26867}, {19119,26872}, {19122,26911}, {19123,26915}, {19128,26878}, {19131,26921}, {19171,26941}, {26899,26901}, {26925,26940}, {26926,26942}

X(26924) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12329, 26893), (63, 19126, 26923)

X(26925) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8237)

X(26925) lies on these lines: {6,3156}, {182,6457}, {206,26886}, {371,19128}, {577,21637}, {1176,6414}, {1428,26948}, {1974,26919}, {2330,26949}, {3589,26951}, {3618,26945}, {5085,26936}, {6467,8908}, {11514,22151}, {19118,26868}, {19119,26873}, {19121,26875}, {19122,26912}, {19123,26916}, {19124,26918}, {19125,26920}, {19131,26922}, {19132,26953}, {19133,26952}, {19171,26947}, {26923,26930}, {26924,26940}, {26926,26950}

X(26926) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND URSA MAJOR

Barycentrics    (2*a^6+(b^2+c^2)*a^4+(b^4-c^4)*(b^2-c^2))*(-a^2+b^2+c^2) : :
X(26926) = 2*X(6)-3*X(11245) = 3*X(428)-4*X(9969) = 2*X(3313)-3*X(7667)

The homothetic center of these triangles is X(17620)

X(26926) lies on these lines: {2,13562}, {3,69}, {4,20079}, {6,66}, {25,5596}, {30,10938}, {67,13198}, {110,26156}, {125,3589}, {141,184}, {159,21213}, {185,1503}, {193,1370}, {206,468}, {235,19149}, {287,14601}, {343,19126}, {394,15812}, {399,18358}, {428,9969}, {441,14575}, {511,6146}, {524,3313}, {542,974}, {1176,6676}, {1181,1352}, {1350,19467}, {1351,13292}, {1353,23335}, {1368,20806}, {1428,26955}, {1974,13567}, {2330,26956}, {2892,19504}, {3269,23642}, {3541,14912}, {3580,19121}, {3618,23291}, {3629,15826}, {3867,11550}, {5050,26944}, {5085,26937}, {5576,18583}, {5622,10264}, {5848,17660}, {5921,6815}, {5965,11577}, {6247,19124}, {8541,15583}, {9924,17818}, {10111,14984}, {10116,12421}, {10937,11188}, {11585,19139}, {12241,12294}, {12359,19131}, {12588,19349}, {12589,19354}, {13142,18945}, {16310,23333}, {18400,21851}, {18420,18440}, {18911,26206}, {18923,19022}, {18924,19021}, {19118,26869}, {19122,26913}, {19123,26917}, {19128,26879}, {19132,26958}, {19133,26957}, {19171,26954}, {26899,26905}, {26923,26932}, {26924,26942}, {26925,26950}

X(26926) = reflection of X(i) in X(j) for these (i,j): (1351, 13292), (3575, 19161), (6776, 18914), (12294, 12241)
X(26926) = anticomplement of X(13562)
X(26926) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 19119, 19125), (69, 6776, 19459), (6776, 18913, 25406)

X(26927) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND HUTSON INTOUCH

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

The homothetic center of these triangles is X(8390)

X(26927) lies on these lines: {3,63}, {4,26933}, {20,26929}, {25,84}, {31,1208}, {55,603}, {56,774}, {57,1593}, {64,1407}, {74,26914}, {185,222}, {197,12680}, {235,20266}, {378,26877}, {474,25019}, {1204,3937}, {1433,3270}, {1436,2155}, {1498,26884}, {1622,12330}, {1709,11365}, {1768,9912}, {1795,11508}, {2096,7412}, {3218,11413}, {3220,3515}, {3306,11479}, {3516,26866}, {4185,6245}, {4222,12246}, {5085,26923}, {5285,9841}, {5584,26934}, {5709,21312}, {7171,11414}, {7335,19354}, {7523,21151}, {9026,12329}, {9786,26892}, {9798,10085}, {11220,11337}, {11248,15626}, {11425,26889}, {11440,26910}, {11509,15622}, {12086,23958}, {12164,22128}, {17928,24320}, {18913,26871}, {19172,26931}, {26930,26936}, {26932,26937}

X(26927) = {X(3), X(63)}-harmonic conjugate of X(26935)

X(26928) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND INCIRCLE-CIRCLES

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

The homothetic center of these triangles is X(11039)

X(26928) lies on these lines: {3,63}, {4,26866}, {5,26929}, {25,26877}, {57,1598}, {84,1597}, {222,19347}, {603,999}, {1181,3937}, {1407,6759}, {1656,26933}, {3218,11414}, {3220,3517}, {3295,7004}, {3306,11484}, {5050,26923}, {10306,26934}, {10984,22129}, {11426,26889}, {11432,26892}, {11441,26910}, {11456,26914}, {14530,26884}, {18914,26871}, {19173,26931}, {26932,26944}

X(26928) = {X(3), X(63)}-harmonic conjugate of X(26938)

X(26929) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND INVERSE-IN-INCIRCLE

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

The homothetic center of these triangles is X(11026)

X(26929) lies on these lines: {2,1473}, {4,57}, {5,26928}, {7,26118}, {20,26927}, {63,7386}, {69,3784}, {150,8817}, {171,388}, {222,6776}, {376,5285}, {427,26866}, {464,22060}, {497,982}, {944,8270}, {990,21621}, {1056,5269}, {1058,3677}, {1364,18922}, {1370,3218}, {1407,1503}, {1460,4293}, {1479,18193}, {1899,3937}, {2003,14912}, {2550,3980}, {3220,6353}, {3306,7392}, {3487,10383}, {3618,26923}, {3917,26872}, {3955,25406}, {4425,4466}, {5225,18201}, {5744,26052}, {6643,24467}, {6804,7330}, {6821,17754}, {7009,7365}, {7182,17170}, {7248,12589}, {7289,18935}, {7293,7494}, {7391,23958}, {9364,12667}, {9436,10444}, {10519,26942}, {11206,26884}, {11427,26889}, {11433,26892}, {11442,26910}, {11457,26914}, {11677,24477}, {19174,26931}, {22344,25876}, {23291,26932}, {26930,26945}

X(26929) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 7386, 26939), (1473, 26933, 2)

X(26930) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11922)

X(26930) lies on these lines: {57,26919}, {63,26940}, {84,26918}, {222,26920}, {371,26877}, {577,3937}, {603,26948}, {1407,26953}, {1473,8911}, {3218,26875}, {7004,26949}, {24467,26922}, {26866,26868}, {26871,26873}, {26884,26886}, {26889,26891}, {26892,26894}, {26910,26912}, {26914,26916}, {26923,26925}, {26927,26936}, {26929,26945}, {26931,26947}, {26932,26950}, {26933,26951}, {26934,26952}

X(26931) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND 1st SHARYGIN

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

The homothetic center of these triangles is X(8391)

X(26931) lies on these lines: {54,26877}, {57,275}, {63,95}, {84,19169}, {97,1214}, {222,19170}, {603,19175}, {1407,19180}, {1473,19189}, {3306,19188}, {3937,21638}, {7004,19182}, {9792,26892}, {16030,26866}, {19166,26871}, {19167,26910}, {19168,26914}, {19171,26923}, {19172,26927}, {19173,26928}, {19174,26929}, {19179,24467}, {19181,26934}, {23295,26933}, {26884,26887}, {26900,26902}, {26930,26947}, {26932,26954}

X(26931) = {X(63), X(95)}-harmonic conjugate of X(26941)

X(26932) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND URSA MAJOR

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

The homothetic center of these triangles is X(17621)

X(26932) lies on these lines: {1,20306}, {2,222}, {3,23161}, {7,281}, {9,141}, {11,124}, {57,13567}, {63,343}, {69,219}, {77,17073}, {84,6247}, {85,1952}, {109,25968}, {116,5514}, {120,3041}, {123,125}, {142,1439}, {189,278}, {220,599}, {226,20205}, {255,7515}, {268,20208}, {269,282}, {297,1948}, {320,1944}, {427,26892}, {440,22097}, {468,26884}, {513,21252}, {521,3270}, {522,4081}, {523,21340}, {524,2323}, {525,20902}, {603,26955}, {608,8048}, {656,22084}, {692,5848}, {918,1086}, {960,2836}, {971,1861}, {1211,5745}, {1212,17237}, {1352,24320}, {1358,1367}, {1368,3784}, {1407,20266}, {1433,3086}, {1437,4999}, {1442,17043}, {1443,18644}, {1473,1899}, {1486,12586}, {1503,3220}, {1565,3942}, {1633,21293}, {2003,23292}, {2097,18636}, {2262,12610}, {2324,17296}, {2995,8736}, {3061,18730}, {3218,3580}, {3452,16594}, {3554,20270}, {3564,7193}, {3662,26530}, {3664,18635}, {3911,26005}, {3917,21015}, {3955,6676}, {4303,18641}, {4357,15595}, {4383,23122}, {4391,23978}, {4503,17056}, {4551,25882}, {4579,26231}, {5249,6708}, {5433,7335}, {5662,23585}, {5743,16554}, {5928,21370}, {6173,21258}, {6335,18816}, {6357,17923}, {6388,16592}, {6506,8287}, {6510,26006}, {6518,20769}, {6603,17374}, {6831,14058}, {7083,12589}, {7117,16731}, {7354,10570}, {7499,26890}, {9119,24471}, {10519,26939}, {11064,22128}, {11245,26889}, {11573,21530}, {12359,24467}, {14100,24388}, {15526,16595}, {15849,21239}, {15985,19557}, {15993,16514}, {17077,25000}, {17170,18639}, {17184,26542}, {17238,26059}, {17421,18210}, {17880,23983}, {17917,18623}, {18642,18650}, {20122,25985}, {20258,20341}, {21739,24145}, {21912,22053}, {23291,26929}, {26866,26869}, {26877,26879}, {26900,26905}, {26910,26913}, {26914,26917}, {26923,26926}, {26927,26937}, {26928,26944}, {26930,26950}, {26931,26954}, {26934,26957}

X(26932) = midpoint of X(i) and X(j) for these {i,j}: {69, 1814}, {1633, 21293}
X(26932) = anticomplement of X(36949)
X(26932) = complementary conjugate of X(4885)
X(26932) = isogonal conjugate of X(7115)
X(26932) = isotomic conjugate of the isogonal conjugate of X(7117)
X(26932) = isotomic conjugate of the polar conjugate of X(11)
X(26932) = polar conjugate of the isogonal conjugate of X(1364)
X(26932) = complement of X(651)
X(26932) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26871, 222), (3, 23161, 23198)
X(26932) = center of hyperbola {{A,B,C,X(7),X(63)}}
X(26932) = X(19)-isoconjugate of X(59)
X(26932) = trilinear pole, wrt medial triangle, of line X(5)X(10)
X(26932) = X(2)-Ceva conjugate of X(905)
X(26932) = barycentric product X(63)*X(4564)
X(26932) = barycentric product X(1)*X(17880)
X(26932) = crosssum of circumcircle-intercepts of Stevanovic circle

X(26933) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND URSA MINOR

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

The homothetic center of these triangles is X(17607)

X(26933) lies on these lines: {2,1473}, {4,26927}, {5,3306}, {11,244}, {12,603}, {25,20266}, {57,427}, {63,1368}, {84,235}, {116,5521}, {123,125}, {222,1899}, {343,3784}, {429,4292}, {440,22060}, {468,3220}, {858,3218}, {1210,1883}, {1364,26956}, {1407,1853}, {1448,5130}, {1503,26884}, {1565,2968}, {1594,26877}, {1656,26928}, {1824,21621}, {1836,23304}, {1904,9579}, {1985,21239}, {2003,11245}, {2611,17876}, {2969,4858}, {3138,6506}, {3564,22128}, {3589,26923}, {3662,16067}, {3916,21530}, {3917,26942}, {3925,26934}, {5094,26866}, {5285,7667}, {5314,10691}, {5515,5517}, {5518,5993}, {6676,7293}, {7085,7386}, {7102,7365}, {7193,11064}, {11585,24467}, {13567,26892}, {17111,17728}, {18641,22345}, {18671,21915}, {20999,25968}, {23291,26871}, {23292,26889}, {23293,26910}, {23294,26914}, {23295,26931}, {24611,24701}, {26900,26906}, {26930,26951}

X(26933) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26929, 1473), (63, 1368, 21015)

X(26934) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND YFF CENTRAL

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

The homothetic center of these triangles is X(11923)

X(26934) lies on these lines: {1,1782}, {2,1762}, {3,18673}, {6,2312}, {9,20106}, {19,57}, {31,3827}, {38,55}, {40,376}, {48,1214}, {58,14015}, {63,69}, {65,603}, {81,18161}, {84,11471}, {184,18210}, {209,8679}, {212,17441}, {222,3942}, {223,2261}, {226,1726}, {255,18732}, {527,21375}, {579,18598}, {649,23726}, {774,3556}, {940,2294}, {1040,20780}, {1150,20896}, {1155,3198}, {1210,1842}, {1407,3197}, {1427,2182}, {1451,1829}, {1708,1763}, {1869,4292}, {2083,23620}, {2173,11347}, {2187,8758}, {2264,3752}, {2385,3914}, {2504,6084}, {2550,3980}, {3101,3164}, {3188,3212}, {3306,9816}, {3611,3937}, {3925,26933}, {4376,5845}, {5584,26927}, {5745,16551}, {5905,21368}, {6197,26877}, {6211,25568}, {7066,23154}, {7193,20254}, {8251,24467}, {8680,19645}, {9536,23958}, {10306,26928}, {10536,26884}, {11406,26866}, {11428,26889}, {11435,26892}, {11445,26910}, {11460,26914}, {11683,14829}, {12587,15523}, {17889,21381}, {19133,26923}, {19181,26931}, {20256,24332}, {26900,26908}, {26930,26952}, {26932,26957}

X(26934) = isogonal conjugate of the polar conjugate of X(17861)
X(26934) = isotomic conjugate of the polar conjugate of X(3924)
X(26934) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 8897, 3719), (63, 10319, 71)

X(26935) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH AND OUTER-HUTSON

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

The homothetic center of these triangles is X(8392)

X(26935) lies on these lines: {3,63}, {4,21015}, {9,1593}, {20,24320}, {25,40}, {28,5759}, {41,4300}, {55,201}, {56,212}, {64,71}, {74,26915}, {165,16389}, {185,219}, {378,26878}, {405,25019}, {573,2983}, {958,13734}, {962,4223}, {1204,3690}, {1425,7078}, {1486,7957}, {1498,26885}, {3145,10310}, {3219,11413}, {3305,11479}, {3428,13738}, {3515,5285}, {3516,26867}, {3587,11414}, {4220,26264}, {5085,26924}, {5657,7412}, {6056,19349}, {7330,21312}, {8273,22769}, {9786,26893}, {10373,13737}, {11425,26890}, {11440,26911}, {18913,26872}, {19172,26941}, {26936,26940}, {26937,26942}

X(26935) = {X(3), X(63)}-harmonic conjugate of X(26927)

X(26936) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8239)

X(26936) lies on these lines: {3,6414}, {4,26951}, {20,26945}, {25,26918}, {55,26948}, {56,26949}, {64,1151}, {74,26916}, {185,26920}, {371,378}, {577,1204}, {1322,6561}, {1498,26886}, {1593,26919}, {2063,5409}, {3516,26868}, {5085,26925}, {5584,26952}, {6200,11456}, {6409,10132}, {6458,10605}, {9786,26894}, {9862,9987}, {11413,26875}, {11425,26891}, {11440,26912}, {18913,26873}, {19172,26947}, {26927,26930}, {26935,26940}, {26937,26950}

X(26936) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6457, 8911), (64, 1151, 26953)

X(26937) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH AND URSA MAJOR

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

The homothetic center of these triangles is X(17622)

X(26937) lies on these lines: {2,185}, {3,68}, {4,74}, {5,10605}, {20,21663}, {24,14216}, {25,6247}, {51,3088}, {55,26955}, {56,26956}, {64,235}, {69,18936}, {140,1181}, {155,10257}, {184,631}, {186,9833}, {287,16925}, {376,21659}, {378,26879}, {389,3541}, {394,16196}, {403,5878}, {417,6389}, {427,9786}, {468,1498}, {549,18914}, {550,18396}, {578,18916}, {974,5654}, {1092,11411}, {1147,6699}, {1192,1853}, {1352,17928}, {1425,3085}, {1503,3515}, {1593,6696}, {1620,17845}, {1885,10606}, {1907,17810}, {2781,15128}, {2929,5621}, {3086,3270}, {3089,11381}, {3146,13851}, {3147,6759}, {3183,6619}, {3269,3767}, {3516,12241}, {3517,16655}, {3520,18912}, {3522,18945}, {3523,3620}, {3524,18925}, {3529,18918}, {3542,6000}, {3546,5562}, {3548,13754}, {3574,8889}, {3580,11413}, {4846,10024}, {5054,19347}, {5064,11745}, {5085,26926}, {5094,12233}, {5133,9815}, {5218,18915}, {5432,19349}, {5433,19354}, {5448,20397}, {5584,26957}, {5622,15057}, {5703,10360}, {5892,14786}, {5895,10151}, {6102,16270}, {6225,6622}, {6241,7505}, {6353,12324}, {6467,10519}, {6515,13346}, {6623,12250}, {6746,23327}, {6815,21243}, {7288,18922}, {7383,16836}, {7487,11550}, {7507,13568}, {7544,15053}, {7689,18531}, {7691,16063}, {9140,12278}, {9540,21640}, {9936,22115}, {10018,11456}, {10201,13491}, {10299,10619}, {10539,16003}, {11064,12164}, {11204,13403}, {11245,11425}, {11250,19353}, {11403,15873}, {11424,11433}, {11440,26913}, {11442,22467}, {11585,12163}, {12161,23336}, {12174,16252}, {13148,15131}, {13352,18951}, {13935,21641}, {14156,15083}, {14379,15526}, {14516,15078}, {14561,15043}, {14585,21843}, {14683,17701}, {15122,16266}, {15738,18439}, {16238,18451}, {18381,18533}, {18570,18952}, {19172,26954}, {19348,19361}, {22533,22978}, {26927,26932}, {26935,26942}, {26936,26950}

X(26937) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18913, 185), (3, 26944, 6146), (9938, 12359, 68)

X(26938) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND INCIRCLE-CIRCLES

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

The homothetic center of these triangles is X(11040)

X(26938) lies on these lines: {3,63}, {4,26867}, {5,26939}, {9,1598}, {25,26878}, {40,1597}, {71,3527}, {201,999}, {212,1497}, {219,19347}, {220,6759}, {1181,3690}, {1656,21015}, {3219,11414}, {3305,11484}, {3517,5285}, {5050,26924}, {7412,21168}, {11426,26890}, {11432,26893}, {11441,26911}, {11456,26915}, {14530,26885}, {18914,26872}, {19173,26941}, {26942,26944}

X(26938) = {X(3), X(63)}-harmonic conjugate of X(26928)

X(26939) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND INVERSE-IN-INCIRCLE

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

The homothetic center of these triangles is X(11027)

X(26939) lies on these lines: {2,7085}, {4,9}, {5,26938}, {20,24320}, {37,5800}, {63,7386}, {69,72}, {201,388}, {210,5928}, {212,238}, {219,6776}, {220,1503}, {226,5268}, {228,464}, {329,4645}, {376,3220}, {377,17257}, {405,12410}, {427,26867}, {440,1260}, {443,4357}, {975,3487}, {1056,7174}, {1058,7290}, {1370,3219}, {1818,18446}, {1899,3690}, {2323,14912}, {3305,7392}, {3421,3717}, {3434,5278}, {3618,26924}, {3651,16119}, {3883,5082}, {3917,26871}, {4294,7083}, {4307,5276}, {4517,12588}, {5084,17353}, {5227,18935}, {5273,26118}, {5285,6353}, {5314,7494}, {5709,6804}, {6356,23603}, {6643,26921}, {7066,18915}, {7193,25406}, {7379,26059}, {10519,26932}, {11206,26885}, {11427,26890}, {11433,26893}, {11442,26911}, {11457,26915}, {17306,17582}, {19174,26941}, {21912,26040}, {23291,26942}, {26940,26945}

X(26939) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 7386, 26929), (7085, 21015, 2)

X(26940) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11925)

X(26940) lies on these lines: {9,26919}, {40,26918}, {63,26930}, {71,6414}, {72,6413}, {201,26948}, {212,26949}, {219,26920}, {220,26953}, {371,26878}, {577,3690}, {3219,26875}, {5415,7968}, {7085,8911}, {21015,26951}, {26867,26868}, {26872,26873}, {26885,26886}, {26890,26891}, {26893,26894}, {26911,26912}, {26915,26916}, {26921,26922}, {26924,26925}, {26935,26936}, {26939,26945}, {26941,26947}, {26942,26950}

X(26941) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND 1st SHARYGIN

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

The homothetic center of these triangles is X(11926)

X(26941) lies on these lines: {9,275}, {40,19169}, {54,72}, {63,95}, {71,8795}, {97,3219}, {201,19175}, {212,19182}, {219,19170}, {220,19180}, {3305,19188}, {3690,21638}, {7085,19189}, {9792,26893}, {16030,26867}, {19166,26872}, {19167,26911}, {19168,26915}, {19171,26924}, {19172,26935}, {19173,26938}, {19174,26939}, {19179,26921}, {21015,23295}, {26885,26887}, {26901,26902}, {26940,26947}, {26942,26954}

X(26941) = {X(63), X(95)}-harmonic conjugate of X(26931)

X(26942) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND URSA MAJOR

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

The homothetic center of these triangles is X(17623)

X(26942) lies on these lines: {2,219}, {3,23162}, {7,19822}, {8,278}, {9,13567}, {10,12}, {34,5814}, {40,6247}, {48,7536}, {57,141}, {63,343}, {66,12329}, {69,222}, {71,440}, {81,22123}, {125,3690}, {197,12587}, {200,223}, {201,3695}, {212,26956}, {220,26958}, {225,5295}, {281,329}, {297,1947}, {306,307}, {312,1952}, {319,1943}, {321,8736}, {355,5307}, {427,26893}, {468,26885}, {517,1848}, {524,2003}, {594,6354}, {599,1407}, {608,5739}, {651,2895}, {756,21717}, {908,6708}, {914,18607}, {940,22132}, {1254,20653}, {1368,3781}, {1451,17698}, {1460,11358}, {1465,3687}, {1471,24943}, {1503,5285}, {1766,5928}, {1783,18687}, {1864,12618}, {1899,7085}, {2318,21912}, {2323,23292}, {2594,3811}, {3085,5711}, {3219,3580}, {3452,26005}, {3564,3955}, {3682,18641}, {3745,13405}, {3782,17861}, {3911,20106}, {3917,26933}, {3949,6356}, {3969,4552}, {3990,17056}, {4016,4415}, {4383,22131}, {4904,24789}, {5219,5743}, {5273,26540}, {5432,6056}, {5718,22134}, {5849,20986}, {6057,7068}, {6510,18652}, {6676,7193}, {7011,20208}, {7080,26027}, {7140,21028}, {7499,26889}, {7522,26063}, {7680,10478}, {10198,19701}, {10371,21147}, {10479,15844}, {10519,26929}, {11245,26890}, {12359,26921}, {12526,20306}, {17077,18139}, {17484,24146}, {17811,20266}, {19542,24310}, {19645,21270}, {21062,21871}, {21072,22001}, {21231,25361}, {21483,26130}, {23291,26939}, {26580,26609}, {26867,26869}, {26878,26879}, {26901,26905}, {26911,26913}, {26915,26917}, {26924,26926}, {26935,26937}, {26938,26944}, {26940,26950}, {26941,26954}

X(26942) = isogonal conjugate of X(2189)
X(26942) = isotomic conjugate of the isogonal conjugate of X(2197)
X(26942) = isotomic conjugate of the polar conjugate of X(12)
X(26942) = polar conjugate of the isogonal conjugate of X(7066)
X(26942) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26872, 219), (3, 23162, 23199)

X(26943) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11042)

X(26943) lies on the line {48,26946}

X(26944) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND URSA MAJOR

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

The homothetic center of these triangles is X(17624)

X(26944) lies on these lines: {2,18914}, {3,68}, {4,3426}, {5,18909}, {6,19361}, {25,11457}, {30,18913}, {64,18390}, {125,399}, {140,3619}, {184,3526}, {185,381}, {235,12315}, {382,10605}, {389,1853}, {403,12174}, {427,11432}, {468,14530}, {495,18915}, {496,18922}, {549,18925}, {550,18931}, {858,12160}, {999,26955}, {1192,18400}, {1204,1657}, {1351,18951}, {1368,11411}, {1503,3517}, {1593,18912}, {1595,3527}, {1596,12324}, {1597,6247}, {1598,13567}, {3088,18950}, {3167,3548}, {3295,26956}, {3448,17928}, {3516,12022}, {3534,17712}, {3541,11245}, {3546,3564}, {3567,5064}, {3580,11414}, {3627,18918}, {5050,26926}, {5054,19357}, {5076,13851}, {5094,7592}, {5447,6467}, {5622,15132}, {5890,7507}, {6147,10360}, {6193,16196}, {6391,18934}, {6642,18440}, {6759,26958}, {7395,18911}, {7517,21970}, {7526,10264}, {8567,20417}, {8780,16238}, {8981,18923}, {9140,10574}, {9704,13198}, {9777,15559}, {9786,18381}, {9818,18952}, {10018,26864}, {10306,26957}, {10516,11695}, {10606,13403}, {10627,15073}, {10938,14852}, {11425,23329}, {11441,26913}, {11456,26917}, {11585,12164}, {12111,16072}, {12163,22808}, {12173,25739}, {13367,15720}, {13382,23325}, {13903,21640}, {13961,21641}, {13966,18924}, {14912,16774}, {15696,21663}, {16003,17818}, {17836,22834}, {19173,26954}, {19360,19362}, {26928,26932}, {26938,26942}

X(26944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18914, 19347), (3, 25738, 12429), (1899, 26937, 6146)

X(26945) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(11030)

X(26945) lies on these lines: {2,8911}, {4,371}, {20,26936}, {69,1590}, {159,3155}, {372,18916}, {388,26948}, {427,26868}, {497,26949}, {577,1899}, {590,10132}, {1151,17845}, {1370,26875}, {1503,26953}, {2550,26952}, {3618,26925}, {6458,18909}, {6643,26922}, {6776,26920}, {11206,26886}, {11427,26891}, {11433,26894}, {11442,26912}, {11457,26916}, {19174,26947}, {23291,26950}, {26929,26930}, {26939,26940}

X(26945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3068, 12257, 6413), (8911, 26951, 2)

X(26946) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 2nd PAMFILOS-ZHOU

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

The homothetic center of these triangles is X(8244)

X(26946) lies on the line {48,26943}

X(26947) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND 1st SHARYGIN

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

The homothetic center of these triangles is X(8246)

X(26947) lies on these lines: {54,371}, {97,26875}, {275,26919}, {577,21638}, {6413,8795}, {6457,8884}, {8911,19189}, {9792,26894}, {16030,26868}, {19166,26873}, {19167,26912}, {19168,26916}, {19169,26918}, {19170,26920}, {19171,26925}, {19172,26936}, {19174,26945}, {19175,26948}, {19179,26922}, {19180,26953}, {19181,26952}, {19182,26949}, {23295,26951}, {26886,26887}, {26930,26931}, {26940,26941}, {26950,26954}

X(26948) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND TANGENTIAL-MIDARC

Barycentrics
a^2*(2*(a^10-(3*b^2+2*b*c+3*c^2)*a^8+4*(b^4+c^4+(b^2+b*c+c^2)*b*c)*a^6-4*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(3*b^4+3*c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^4*(b-c)^2)*S+(-a^2+b^2+c^2)*(a^6-(b+c)^2*a^4+(b+c)^4*a^2-(b^2-c^2)^2*(b+c)^2)*(a+b-c)^2*(a-b+c)^2)*(-a^2+b^2+c^2) : :

The homothetic center of these triangles is X(8247)

X(26948) lies on these lines: {1,6457}, {12,26951}, {33,26918}, {34,26919}, {55,26936}, {56,8911}, {65,2067}, {73,6414}, {201,26940}, {221,26953}, {371,1870}, {388,26945}, {577,1425}, {603,26930}, {1060,26922}, {1398,26868}, {1428,26925}, {4296,26875}, {18915,26873}, {19175,26947}, {19349,26920}, {19365,26891}, {19366,26894}, {19367,26912}, {19368,26916}, {26886,26888}, {26950,26955}

X(26948) = {X(1), X(6457)}-harmonic conjugate of X(26949)

X(26949) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND 2nd TANGENTIAL-MIDARC

Barycentrics
a^2*(2*(a^10-(3*b^2-2*b*c+3*c^2)*a^8+4*(b^4+c^4-(b^2-b*c+c^2)*b*c)*a^6-4*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(3*b^4+3*c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^4*(b+c)^2)*S+(-a^2+b^2+c^2)*(a^6-(b-c)^2*a^4+(b-c)^4*a^2-(b^2-c^2)^2*(b-c)^2)*(-a+b+c)^2*(a+b+c)^2)*(-a^2+b^2+c^2) : :

The homothetic center of these triangles is X(8248)

X(26949) lies on these lines: {1,6457}, {33,26919}, {34,26918}, {55,8911}, {56,26936}, {212,26940}, {371,6198}, {497,26945}, {577,3270}, {1062,26922}, {2066,6413}, {2192,26953}, {2330,26925}, {3100,26875}, {7004,26930}, {7071,26868}, {10535,26886}, {11429,26891}, {11436,26894}, {11446,26912}, {11461,26916}, {18922,26873}, {19182,26947}, {19354,26920}, {26950,26956}

X(26949) = {X(1), X(6457)}-harmonic conjugate of X(26948)

X(26950) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND URSA MAJOR

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

The homothetic center of these triangles is X(17627)

X(26950) lies on these lines: {2,26873}, {5,6458}, {125,577}, {371,26879}, {372,1594}, {427,26894}, {468,26886}, {615,6414}, {1899,8911}, {3580,26875}, {6247,26918}, {8252,10133}, {8961,13970}, {11090,20563}, {11245,26891}, {12359,26922}, {13567,26919}, {23291,26945}, {26868,26869}, {26912,26913}, {26916,26917}, {26925,26926}, {26930,26932}, {26936,26937}, {26940,26942}, {26947,26954}, {26948,26955}, {26949,26956}, {26952,26957}, {26953,26958}

X(26950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26873, 26920), (125, 577, 26951)

X(26951) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND URSA MINOR

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

The homothetic center of these triangles is X(17610)

X(26951) lies on these lines: {2,8911}, {4,26936}, {5,6457}, {12,26948}, {125,577}, {235,26918}, {371,1594}, {372,26879}, {427,26919}, {590,6413}, {858,26875}, {1503,26886}, {1853,26953}, {1899,26920}, {3589,26925}, {3925,26952}, {5094,26868}, {8253,10132}, {11091,20563}, {11585,26922}, {13567,26894}, {21015,26940}, {23291,26873}, {23292,26891}, {23293,26912}, {23294,26916}, {23295,26947}, {26930,26933}

X(26951) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 26945, 8911), (125, 577, 26950)

X(26952) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND YFF CENTRAL

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

The homothetic center of these triangles is X(11996)

X(26952) lies on these lines: {19,26919}, {40,6457}, {55,8911}, {65,2067}, {71,6414}, {371,6197}, {577,3611}, {2550,26945}, {3101,26875}, {3197,26953}, {3925,26951}, {5584,26936}, {8251,26922}, {10536,26886}, {11406,26868}, {11428,26891}, {11435,26894}, {11445,26912}, {11460,26916}, {11471,26918}, {18921,26873}, {19133,26925}, {19181,26947}, {19350,26920}, {26930,26934}, {26950,26957}

X(26953) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND 2nd ZANIAH

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

The homothetic center of these triangles is X(18234)

X(26953) lies on these lines: {3,3093}, {5,1322}, {6,3155}, {22,26912}, {24,26916}, {25,577}, {32,19006}, {64,1151}, {97,15187}, {154,8911}, {184,26868}, {216,5410}, {220,26940}, {221,26948}, {371,1181}, {394,26875}, {1407,26930}, {1498,6457}, {1503,26945}, {1583,11513}, {1584,10961}, {1599,11417}, {1853,26951}, {2192,26949}, {3068,21736}, {3092,14152}, {3197,26952}, {3284,5411}, {5065,19005}, {5407,10960}, {5413,15905}, {6458,9786}, {8908,26864}, {10132,10533}, {13567,26873}, {15811,26918}, {17809,26891}, {17810,26894}, {17814,26922}, {19132,26925}, {19180,26947}, {26950,26958}

X(26953) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64, 1151, 26936), (1151, 17819, 6413), (26919, 26920, 6)

X(26954) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st SHARYGIN AND URSA MAJOR

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

The homothetic center of these triangles is X(17628)

X(26954) lies on these lines: {2,19166}, {54,140}, {95,343}, {97,3580}, {125,21638}, {235,19206}, {275,6749}, {427,9792}, {468,26887}, {1899,19189}, {6146,19185}, {6247,19169}, {8612,8795}, {8901,19209}, {11585,19194}, {12359,19179}, {16030,26869}, {19167,26913}, {19168,26917}, {19171,26926}, {19172,26937}, {19173,26944}, {19174,23291}, {19175,26955}, {19180,26958}, {19181,26957}, {19182,26956}, {26902,26905}, {26931,26932}, {26941,26942}, {26947,26950}

X(26954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 19166, 19170), (125, 21638, 23295)

X(26955) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC AND URSA MAJOR

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

The homothetic center of these triangles is X(17629)

X(26955) lies on these lines: {1,26956}, {2,18915}, {3,10071}, {4,10076}, {11,185}, {12,125}, {33,6247}, {34,10361}, {36,6146}, {55,26937}, {56,1899}, {65,429}, {73,18641}, {184,5433}, {201,3695}, {221,26958}, {235,7355}, {343,1038}, {388,23291}, {427,19366}, {468,26888}, {497,18913}, {499,1181}, {603,26932}, {999,26944}, {1060,12359}, {1069,18917}, {1204,6284}, {1213,1409}, {1398,26869}, {1428,26926}, {1479,10605}, {1853,11392}, {1870,26879}, {2477,13198}, {3086,18909}, {3157,3548}, {3215,7515}, {3485,10360}, {3580,4296}, {4294,18931}, {4299,18396}, {5204,19467}, {6776,7288}, {7066,21015}, {7352,11585}, {9786,11393}, {11245,19365}, {11399,14216}, {14986,18922}, {15325,18914}, {15326,21659}, {15338,21663}, {18965,21640}, {18966,21641}, {18970,25738}, {19175,26954}, {19367,26913}, {19368,26917}, {26903,26905}, {26948,26950}

X(26955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18915, 19349), (125, 1425, 12)

X(26956) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND URSA MAJOR

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

The homothetic center of these triangles is X(17630)

X(26956) lies on these lines: {1,26955}, {2,18922}, {3,10055}, {4,10060}, {11,125}, {12,185}, {33,13567}, {34,6247}, {35,6146}, {55,1899}, {56,26937}, {184,5432}, {212,26942}, {215,13198}, {235,6285}, {343,1040}, {388,18913}, {427,11436}, {468,10535}, {497,23291}, {498,1181}, {1062,12359}, {1069,3548}, {1146,8735}, {1204,7354}, {1364,26933}, {1425,15888}, {1478,10605}, {1853,11393}, {2192,26958}, {2330,26926}, {2342,25968}, {2968,7004}, {3085,18909}, {3100,3580}, {3157,18917}, {3295,26944}, {9638,10018}, {11398,14216}, {15526,17421}, {19182,26954}

X(26956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18922, 19354), (125, 3270, 11)

X(26957) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: URSA MAJOR AND YFF CENTRAL

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

The homothetic center of these triangles is X(17631)

X(26957) lies on these lines: {2,18921}, {19,5928}, {55,1899}, {65,429}, {71,440}, {125,3611}, {235,6254}, {343,10319}, {427,11435}, {468,10536}, {1409,17056}, {2550,23291}, {3101,3580}, {3197,26958}, {5584,26937}, {6146,10902}, {6197,26879}, {6237,11585}, {6247,11471}, {8251,12359}, {8896,18589}, {10306,26944}, {11245,11428}, {11406,26869}, {11445,26913}, {11460,26917}, {19133,26926}, {19181,26954}, {26905,26908}, {26932,26934}, {26950,26952}

X(26957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18921, 19350), (125, 3611, 3925)

X(26958) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: URSA MAJOR AND 2nd ZANIAH

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

The homothetic center of these triangles is X(18236)

X(26958) lies on these lines: {2,6}, {3,2929}, {4,1192}, {5,9786}, {20,1620}, {22,26913}, {24,25739}, {25,125}, {51,5094}, {55,21912}, {64,235}, {68,16238}, {140,11425}, {154,468}, {184,26869}, {186,18396}, {220,26942}, {221,26955}, {278,1146}, {281,6354}, {338,2052}, {373,7539}, {381,11438}, {389,1656}, {393,459}, {402,5877}, {403,10605}, {427,17810}, {441,3053}, {451,5706}, {465,11480}, {466,11481}, {470,5340}, {471,5339}, {542,8780}, {578,3526}, {631,12241}, {800,20208}, {1030,21482}, {1073,15526}, {1181,7505}, {1204,5895}, {1350,1368}, {1351,6723}, {1352,6677}, {1407,20266}, {1427,18634}, {1498,3542}, {1503,6353}, {1585,23251}, {1586,23261}, {1589,6409}, {1590,6410}, {1597,23329}, {1598,20299}, {1609,6617}, {1885,8567}, {1990,14361}, {1995,23293}, {2192,26956}, {2883,6622}, {3003,6509}, {3052,25968}, {3060,11746}, {3066,5133}, {3070,3535}, {3071,3536}, {3088,15873}, {3089,6247}, {3090,12233}, {3091,13568}, {3119,7147}, {3144,5786}, {3147,6146}, {3167,5972}, {3168,15274}, {3197,26957}, {3515,17845}, {3517,18381}, {3767,20207}, {3772,24005}, {3796,18911}, {3830,7687}, {4265,25907}, {5020,10516}, {5054,11430}, {5055,18388}, {5070,11432}, {5085,6676}, {5096,25947}, {5159,11477}, {5449,6642}, {5480,8889}, {5644,25555}, {5816,6678}, {5943,19161}, {6525,6619}, {6623,15311}, {6644,14852}, {6759,26944}, {6776,10192}, {7547,11704}, {7569,15024}, {7592,14940}, {7716,23300}, {8550,18950}, {9119,25525}, {9306,15069}, {9820,18951}, {10018,18912}, {10594,23294}, {11206,15448}, {11216,15118}, {11245,17809}, {11585,17834}, {12359,17814}, {12828,15131}, {13561,13861}, {14216,21841}, {15081,18559}, {15585,18935}, {15750,21659}, {15752,17578}, {16252,18909}, {18405,18533}, {18494,23325}, {19132,26926}, {19180,26954}, {19786,26531}, {24789,26001}, {26905,26909}, {26950,26953}

X(26958) = midpoint of X(i) and X(j) for these {i,j}: {6353, 23291}, {6623, 18931}
X(26958) = polar conjugate of X(18848)
X(26958) = complement of the isotomic conjugate of X(459)
X(26958) = complement of the polar conjugate of X(6526)
X(26958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (343, 17811, 599), (11433, 23292, 6), (13567, 23292, 11433)

leftri

Collineation mappings involving Gemini triangle 51: X(26959)-X(27019)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 51, as in centers X(26595)-X(27019). Then

m(X) = a (b - c)^2 x + b (a^2 + c^2) y + c (a^2 + b^2) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 4, 2018)

X(26959) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26959) lies on these lines: {1, 2}, {5, 26561}, {6, 25505}, {9, 26107}, {11, 6656}, {35, 4366}, {36, 384}, {39, 350}, {55, 11285}, {56, 7770}, {76, 2275}, {83, 172}, {86, 23660}, {140, 26629}, {183, 16502}, {192, 27351}, {194, 3760}, {238, 1923}, {274, 4602}, {291, 12263}, {314, 27633}, {315, 9599}, {330, 3761}, {335, 3953}, {385, 5299}, {458, 11399}, {474, 20172}, {496, 8362}, {497, 16043}, {609, 7787}, {667, 18102}, {668, 17448}, {894, 20372}, {946, 8924}, {956, 26687}, {993, 16916}, {1003, 5204}, {1015, 1909}, {1078, 1914}, {1107, 18140}, {1111, 7187}, {1475, 17499}, {1478, 16924}, {1479, 7791}, {1500, 6683}, {1575, 17143}, {1920, 18833}, {1966, 16706}, {2241, 7815}, {2242, 7808}, {2276, 7786}, {2886, 17670}, {2975, 17541}, {3329, 5280}, {3403, 4000}, {3405, 27004}, {3503, 3911}, {3508, 17353}, {3552, 7280}, {3583, 6655}, {3585, 16044}, {3662, 17181}, {3664, 26149}, {3673, 25918}, {3721, 18061}, {3739, 20363}, {3825, 17669}, {3873, 27285}, {3875, 26042}, {4063, 26984}, {4187, 26558}, {4253, 24514}, {4299, 14035}, {4316, 6658}, {4396, 7760}, {4649, 20148}, {5025, 7741}, {5248, 17684}, {5251, 16918}, {5253, 17686}, {5267, 17692}, {5277, 20179}, {5298, 6661}, {5322, 16950}, {5332, 6179}, {5433, 7807}, {5563, 6645}, {6284, 8356}, {6376, 16975}, {6381, 21226}, {6626, 25530}, {6691, 17694}, {7031, 7793}, {7288, 14001}, {7296, 7878}, {7354, 8370}, {7761, 9665}, {7819, 15325}, {7841, 10896}, {7951, 16921}, {8359, 15171}, {9597, 11185}, {9669, 11287}, {10069, 10352}, {10483, 11361}, {10589, 14064}, {11321, 25524}, {14210, 24786}, {15271, 16781}, {15326, 19687}, {16061, 18758}, {16552, 27262}, {16564, 26992}, {16720, 27918}, {16738, 17210}, {16887, 19579}, {16912, 25542}, {17117, 27102}, {17121, 26772}, {17178, 17288}, {17205, 26813}, {17237, 25534}, {17265, 24679}, {17277, 21760}, {17287, 27095}, {17291, 27145}, {17302, 25599}, {17348, 27111}, {17755, 25079}, {17758, 27155}, {17760, 24172}, {17761, 24170}, {18152, 23632}, {18170, 21238}, {19565, 21443}, {19792, 26746}, {21327, 21412}, {21431, 27698}, {24390, 26582}, {24945, 25660}, {25280, 27076}, {25498, 27164}, {25521, 26106}, {26279, 27010}, {26960, 26977}, {26962, 26966}, {26969, 26989}, {26988, 26997}, {27007, 27011}, {27185, 27190}

X(26959) = complement of X(26752)

X(26960) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26960) lies on these lines: {2, 3}, {1975, 27515}, {26959, 26977}, {26963, 26970}, {26964, 27009}, {26965, 27335}, {26978, 27008}

X(26961) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26961) lies on these lines: {2, 3}, {6, 6604}, {34, 26203}, {894, 20605}, {1730, 26065}, {1861, 26153}, {25242, 26770}, {26035, 26059}, {26085, 27039}

X(26962) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26962) lies on these lines: {2, 3}, {26959, 26966}, {27000, 27324}

X(26963) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26963) lies on these lines: {2, 6}, {37, 27166}, {39, 192}, {75, 27809}, {239, 27102}, {291, 17142}, {319, 27044}, {583, 17350}, {604, 26222}, {894, 20372}, {1015, 3963}, {1086, 27011}, {2275, 17148}, {2350, 24514}, {3248, 21238}, {3662, 24237}, {3758, 25505}, {3943, 26797}, {4000, 27107}, {4253, 27262}, {4360, 26764}, {4393, 5153}, {4687, 27037}, {5069, 18147}, {7032, 21278}, {7263, 26850}, {16679, 18082}, {16696, 18046}, {16706, 26982}, {16710, 20913}, {16726, 18143}, {16826, 27032}, {17169, 27155}, {17231, 27113}, {17233, 27136}, {17243, 27073}, {17246, 26769}, {17260, 25510}, {17273, 25534}, {17288, 27106}, {17295, 26774}, {17305, 26857}, {17363, 27091}, {17367, 27311}, {17368, 27261}, {17759, 26815}, {18170, 20352}, {20868, 23488}, {21257, 22343}, {24327, 25295}, {26012, 26176}, {26960, 26970}, {26969, 26973}, {26974, 27007}, {26978, 27005}

X(26964) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26964) lies on these lines: {1, 2}, {86, 27172}, {350, 26770}, {673, 5253}, {1015, 26978}, {1212, 25261}, {1475, 20347}, {1509, 27189}, {1655, 27348}, {2082, 26229}, {2140, 17169}, {2170, 17048}, {2275, 16742}, {3061, 20247}, {3618, 27058}, {3662, 26818}, {3663, 23649}, {3759, 27039}, {4000, 27161}, {4657, 16713}, {5701, 17302}, {6691, 24582}, {14621, 26802}, {16706, 26995}, {16975, 26100}, {17103, 26845}, {17141, 18061}, {17164, 24631}, {17304, 26836}, {17474, 20335}, {17672, 24390}, {17754, 20244}, {19284, 20172}, {19717, 27142}, {19743, 27181}, {23903, 26794}, {24596, 25524}, {25237, 26690}, {26813, 27011}, {26960, 27009}, {26977, 26989}, {26988, 27000}

X(26965) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b + a b^3 + a^3 c + 2 a^2 b c + b^3 c + a c^3 + b c^3 : :

X(26965) lies on these lines: {1, 2}, {6, 17137}, {75, 17489}, {81, 27185}, {86, 27169}, {100, 16061}, {141, 3780}, {213, 17152}, {350, 27040}, {392, 26689}, {607, 17913}, {673, 1220}, {894, 20605}, {942, 26562}, {964, 20172}, {1086, 4754}, {1107, 16705}, {1334, 17353}, {1468, 24586}, {1478, 16910}, {1573, 25499}, {1655, 17302}, {1829, 15149}, {1909, 16706}, {2082, 10436}, {2241, 25497}, {2275, 27162}, {2276, 27109}, {2280, 24549}, {2292, 17755}, {2295, 3589}, {2345, 20174}, {2975, 16060}, {3212, 17077}, {3263, 25263}, {3618, 21281}, {3672, 27523}, {3691, 4357}, {3721, 17141}, {3739, 17497}, {3975, 19786}, {4026, 27047}, {4202, 26561}, {4424, 25248}, {4657, 24735}, {4699, 21216}, {4972, 6656}, {5051, 26558}, {5251, 16931}, {5303, 21937}, {5826, 27300}, {6376, 26100}, {8192, 16412}, {11321, 24596}, {16583, 26234}, {16707, 16735}, {17062, 24995}, {17200, 26843}, {17356, 24656}, {17370, 24524}, {17499, 20347}, {17672, 26582}, {17694, 24582}, {17741, 27078}, {18107, 21301}, {19717, 27152}, {20255, 24512}, {20963, 21240}, {24174, 24629}, {24443, 24631}, {25264, 26770}, {26960, 27335}, {26989, 27009}, {26995, 27003}

X(26966) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26966) lies on these lines: {2, 11}, {26959, 26962}

X(26967) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26967) lies on these lines: {2, 3}

X(26968) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26968) lies on these lines: {2, 3}, {10566, 27015}

X(26969) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26969) lies on these lines: {2, 31}, {5205, 27128}, {5297, 27061}, {5329, 16949}, {16706, 27004}, {26959, 26989}, {26963, 26973}, {26974, 27009}

X(26970) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26970) lies on these lines: {2, 32}, {26960, 26963}, {26978, 26996}

X(26971) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    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(26971) lies on these lines: {1, 21278}, {2, 37}, {44, 26799}, {76, 17148}, {86, 27166}, {141, 27106}, {142, 27159}, {239, 26772}, {319, 26756}, {320, 17178}, {594, 27044}, {894, 20372}, {1086, 26979}, {1100, 26821}, {1125, 2309}, {1213, 17475}, {1654, 20561}, {1964, 24688}, {3056, 11376}, {3589, 26982}, {3616, 21299}, {3661, 27095}, {3662, 24220}, {3728, 17793}, {3778, 12263}, {3934, 3963}, {4272, 27041}, {4357, 16738}, {7155, 15315}, {16826, 25538}, {17030, 17248}, {17045, 27042}, {17053, 20913}, {17174, 17184}, {17229, 26774}, {17235, 26857}, {17277, 27036}, {17285, 27113}, {17300, 26149}, {17307, 25534}, {17319, 27020}, {17344, 26768}, {17379, 23660}, {17445, 20352}, {21035, 25347}, {21257, 21352}, {25591, 27680}, {26279, 26977}, {26972, 26987}, {27097, 27155}

X(26972) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26972) lies on these lines: {2, 39}, {17761, 24170}, {26960, 26963}, {26971, 26987}, {26996, 27005}

X(26973) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    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 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 : :

X(26973) lies on these lines: {1, 2}, {16742, 16748}, {26963, 26969}

X(26974) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26974) lies on these lines: {1, 2}, {75, 22218}, {310, 16606}, {1920, 3121}, {1978, 21345}, {21384, 26108}, {26963, 27007}, {26969, 27009}, {26977, 26986}

X(26975) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26975) lies on these lines: {2, 44}, {6, 27102}, {86, 27032}, {87, 21278}, {190, 27166}, {192, 2275}, {524, 27044}, {536, 26821}, {798, 26983}, {894, 20372}, {1086, 26982}, {1100, 26764}, {3248, 20352}, {3589, 27017}, {3618, 27311}, {4473, 26113}, {4851, 27136}, {5749, 27261}, {5750, 16738}, {6542, 26076}, {7321, 27011}, {10436, 27154}, {17120, 26772}, {17178, 17289}, {17297, 27113}, {17315, 26797}, {17317, 27073}, {17320, 26769}, {17364, 27095}, {17367, 27107}, {17368, 27145}, {17374, 26774}, {17379, 17750}, {17384, 26857}, {26979, 27078}

X(26976) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^3 b c + a^3 c^2 + 4 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(26976) lies on these lines: {2, 45}, {7, 27145}, {37, 18143}, {75, 26772}, {76, 192}, {256, 17142}, {594, 26756}, {894, 20372}, {1215, 3123}, {1654, 17152}, {2345, 27095}, {3122, 17140}, {3589, 27011}, {3662, 27261}, {4443, 17165}, {4670, 27166}, {6646, 16738}, {7321, 27017}, {16706, 27078}, {16710, 17053}, {16815, 27036}, {17030, 17333}, {17116, 27102}, {17178, 17365}, {17246, 27042}, {17257, 27142}, {17260, 27154}, {17261, 25538}, {17271, 26768}, {17277, 26799}, {17280, 26149}, {17292, 27106}, {17340, 27073}, {17352, 27192}, {17359, 27113}, {17366, 26850}, {20072, 26801}, {24325, 24399}, {26125, 27252}

X(26977) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26977) lies on these lines: {2, 11}, {894, 26981}, {26279, 26971}, {26561, 26802}, {26959, 26960}, {26963, 26969}, {26964, 26989}, {26974, 26986}

X(26978) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26978) lies on these lines: {1, 2140}, {2, 39}, {4, 26099}, {8, 141}, {10, 24790}, {37, 20880}, {69, 26085}, {86, 17686}, {142, 23536}, {213, 20347}, {244, 17048}, {277, 1390}, {315, 16910}, {335, 17141}, {350, 27097}, {377, 4648}, {379, 940}, {672, 24214}, {964, 15668}, {1015, 26964}, {1086, 2295}, {1111, 16600}, {1193, 20335}, {1334, 3663}, {1475, 24215}, {1574, 27025}, {1909, 16706}, {3008, 3691}, {3314, 16906}, {3589, 4754}, {3662, 17033}, {3666, 6706}, {3672, 27253}, {3673, 26242}, {3720, 23682}, {3734, 11319}, {3739, 4968}, {3752, 20436}, {3780, 17366}, {3924, 24249}, {3975, 18136}, {3995, 20432}, {4039, 24169}, {4260, 4310}, {4441, 27248}, {4447, 17061}, {4642, 21232}, {4851, 5300}, {5046, 26145}, {5264, 14377}, {5275, 17683}, {5276, 17682}, {5337, 14953}, {7191, 20556}, {7264, 24403}, {7816, 17539}, {7892, 17003}, {9620, 26653}, {10448, 25500}, {10459, 17050}, {15971, 24220}, {16020, 16850}, {16583, 26563}, {16927, 16994}, {17046, 21935}, {17143, 26759}, {17149, 27313}, {17164, 24254}, {17169, 24512}, {17245, 17672}, {17300, 17680}, {17313, 17679}, {17316, 18139}, {17382, 24656}, {17497, 20955}, {17751, 21240}, {18150, 24524}, {26960, 27008}, {26963, 27005}, {26970, 26996}

X(26979) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b^2 + a^2 b^3 + a b^3 c + a^3 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 : :

X(26979) lies on these lines: {2, 6}, {10, 24659}, {39, 18137}, {75, 17053}, {311, 26633}, {314, 24530}, {594, 27102}, {980, 18147}, {1086, 26971}, {1230, 18601}, {1232, 26588}, {1284, 5433}, {1966, 16706}, {3634, 25113}, {3662, 25505}, {3934, 18143}, {3943, 26764}, {3948, 16696}, {4357, 24237}, {4369, 21143}, {4389, 26107}, {4447, 18082}, {4751, 17030}, {6675, 25492}, {7263, 27107}, {10471, 24897}, {17045, 27166}, {17228, 27091}, {17233, 26042}, {17295, 26752}, {17317, 27020}, {17322, 25510}, {17366, 27311}, {17369, 27261}, {20927, 25918}, {21236, 26176}, {21330, 24327}, {26975, 27078}, {26986, 26987}, {26989, 26997}, {26993, 27006}

X(26980) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26980) lies on these lines: {1, 2}

X(26981) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26981) lies on these lines: {1, 2}, {894, 26977}, {16742, 16750}, {27003, 27009}

X(26982) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b^2 + a^2 b^3 - 4 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(26982) lies on these lines: {1, 2}, {524, 27106}, {894, 27011}, {1086, 26975}, {3589, 26971}, {3759, 27095}, {3875, 27136}, {3946, 26764}, {4063, 27013}, {5750, 26812}, {10436, 27192}, {16706, 26963}, {16738, 17384}, {17045, 27032}, {17116, 26850}, {17121, 26756}, {17178, 17291}, {17319, 27073}, {17366, 27102}, {17370, 27145}, {17398, 27154}, {18106, 18107}, {20072, 26142}

X(26983) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26983) lies on these lines: {2, 661}, {649, 23803}, {798, 26975}, {810, 21302}, {850, 7180}, {1150, 18199}, {1577, 3960}, {2978, 24674}, {4761, 26115}, {4776, 25511}, {4885, 21894}, {7199, 24900}, {7252, 19684}, {11322, 23864}, {16751, 25667}, {17494, 27345}, {20295, 26114}, {21297, 26854}, {27138, 27193}

X(26984) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    (b - c) (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(26984) lies on these lines: {2, 667}, {4063, 26959}, {23807, 27318}, {27013, 27016}

X(26985) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26985) lies on these lines: {2, 650}, {37, 4828}, {192, 4411}, {513, 26798}, {514, 17266}, {523, 7925}, {649, 21297}, {661, 4928}, {812, 24924}, {885, 15283}, {1577, 3960}, {1638, 4467}, {3004, 4789}, {3091, 8760}, {3239, 21183}, {3617, 14077}, {3618, 9015}, {3620, 9001}, {3662, 23838}, {3676, 25259}, {3700, 4453}, {3776, 6548}, {3835, 4379}, {3840, 23791}, {3995, 25271}, {4024, 14475}, {4106, 26853}, {4358, 21611}, {4369, 4728}, {4374, 4526}, {4380, 23813}, {4500, 17161}, {4554, 27134}, {4560, 4823}, {4671, 21438}, {4699, 4777}, {4791, 21222}, {4804, 25380}, {4814, 17072}, {4895, 21302}, {5274, 11934}, {7199, 26775}, {7658, 27486}, {8047, 17036}, {8142, 21734}, {14996, 22383}, {16892, 21204}, {17166, 21260}, {17291, 23810}, {18155, 26822}, {23100, 25244}, {23806, 27186}, {23893, 26531}, {27114, 27293}, {27167, 27345}

X(26985) = complement of X(26777)
X(26985) = anticomplement of X(31209)

X(26986) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26986) lies on these lines: {2, 31}, {6, 20561}, {7, 26108}, {335, 27166}, {894, 20372}, {2295, 3589}, {3662, 24491}, {12263, 24349}, {16830, 27080}, {17030, 17368}, {17232, 27341}, {17291, 27159}, {17398, 27148}, {18103, 20556}, {18111, 18705}, {20549, 20669}, {26974, 26977}, {26979, 26987}

X(26987) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26987) lies on these lines: {1, 2}, {16705, 20530}, {26971, 26972}, {26979, 26986}

X(26988) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26988) lies on these lines: {2, 3}, {26959, 26997}, {26964, 27000}

X(26989) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26989) lies on these lines: {2, 3}, {6703, 27146}, {26959, 26969}, {26964, 26977}, {26965, 27009}, {26979, 26997}

X(26990) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26990) lies on these lines: {2, 3}, {1395, 26211}, {26279, 26971}

X(26991) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26991) lies on these lines: {2, 3}, {16568, 16706}

X(26992) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26992) lies on these lines: {2, 3}, {16564, 26959}

X(26993) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26993) lies on these lines: {2, 3}, {26979, 27006}

X(26994) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26994) lies on these lines: {2, 3}

X(26995) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26995) lies on these lines: {2, 3}, {16706, 26964}, {26965, 27003}

X(26996) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26996) lies on these lines: {2, 3}, {26970, 26978}, {26972, 27005}

X(26997) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26997) lies on these lines: {2, 7}, {3619, 27025}, {4188, 20470}, {15717, 26093}, {16706, 26964}, {16713, 16726}, {17227, 27039}, {17367, 26818}, {20172, 27145}, {20946, 25237}, {21255, 24778}, {26959, 26988}, {26979, 26989}

X(26998) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26998) lies on these lines: {2, 19}, {9, 4967}, {48, 26639}, {75, 3219}, {141, 7297}, {169, 17257}, {193, 2082}, {238, 17872}, {239, 5279}, {524, 7300}, {597, 5356}, {608, 26206}, {673, 11683}, {894, 20605}, {1731, 16574}, {1760, 3218}, {1763, 26132}, {1766, 26685}, {1781, 17023}, {1861, 5046}, {1890, 2475}, {1958, 3061}, {2183, 26699}, {2262, 15988}, {2329, 17868}, {2345, 27065}, {3008, 16566}, {3100, 17522}, {3589, 5341}, {3662, 7291}, {3663, 20602}, {4357, 16547}, {4416, 5540}, {4431, 17744}, {7083, 12530}, {7191, 17446}, {16548, 17353}, {16564, 26959}, {16568, 16706}, {17787, 23978}, {18698, 24588}, {21376, 26723}, {26279, 26971}

X(26999) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(26999) lies on these lines: {2, 38}, {7292, 27030}, {17123, 27079}, {26959, 26969}

X(27000) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27000) lies on these lines: {1, 4209}, {2, 40}, {4, 26531}, {7, 2082}, {63, 27304}, {65, 673}, {169, 10025}, {239, 379}, {517, 17682}, {894, 20605}, {1697, 27253}, {1730, 16819}, {2140, 5011}, {2170, 7176}, {2270, 26125}, {2475, 26526}, {3218, 27171}, {3303, 27475}, {3339, 24600}, {3496, 17050}, {3509, 20257}, {3753, 17681}, {3869, 24596}, {3877, 17683}, {4185, 14621}, {4384, 12526}, {4872, 21258}, {4904, 4911}, {5046, 26532}, {5088, 14377}, {5819, 6604}, {6915, 25954}, {6999, 25935}, {7223, 9311}, {7384, 26001}, {7406, 9800}, {11329, 24559}, {11349, 16826}, {12699, 17671}, {17030, 24627}, {17220, 27420}, {17397, 24580}, {17541, 25965}, {17691, 19860}, {24604, 26626}, {26959, 26960}, {26962, 27324}, {26964, 26988}, {27064, 27299}

X(27001) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27001) lies on these lines: {2, 48}, {894, 27010}, {16564, 26959}

X(27002) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27002) lies on these lines: {2, 7}, {8, 11512}, {21, 22376}, {40, 26093}, {46, 25492}, {88, 321}, {244, 7081}, {333, 16602}, {968, 26103}, {982, 5205}, {1054, 3840}, {1999, 3752}, {2975, 25965}, {3336, 19847}, {3699, 21342}, {3756, 4514}, {3757, 17063}, {4388, 5121}, {4429, 17728}, {4640, 25531}, {4847, 26073}, {4871, 17596}, {5122, 13735}, {5484, 8582}, {5741, 17288}, {5795, 25979}, {8056, 11679}, {9335, 26227}, {9843, 26117}, {14829, 16610}, {15803, 17697}, {16830, 17124}, {17020, 17121}, {17595, 18743}, {19242, 23169}, {20237, 25580}, {24183, 26724}, {26959, 26960}

X(27003) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27003) lies on these lines: {1, 1392}, {2, 7}, {5, 26877}, {6, 17020}, {8, 3338}, {10, 3337}, {11, 20292}, {20, 5804}, {21, 5439}, {23, 7293}, {31, 7292}, {36, 5883}, {38, 5297}, {40, 3622}, {42, 1054}, {46, 3616}, {55, 9352}, {65, 5253}, {72, 17531}, {78, 17572}, {79, 3825}, {81, 88}, {84, 3832}, {89, 8056}, {100, 354}, {110, 26889}, {145, 3333}, {149, 11019}, {165, 4666}, {171, 244}, {191, 19862}, {200, 4430}, {210, 9342}, {214, 5425}, {222, 5422}, {320, 5741}, {333, 24589}, {388, 25005}, {404, 942}, {411, 9940}, {474, 3868}, {484, 551}, {612, 4392}, {614, 9335}, {631, 5761}, {649, 21204}, {678, 9337}, {748, 4650}, {750, 982}, {896, 17123}, {912, 6946}, {938, 4190}, {940, 4850}, {962, 10586}, {984, 17124}, {1004, 11020}, {1005, 11575}, {1019, 4049}, {1046, 27627}, {1071, 6915}, {1125, 3336}, {1150, 19804}, {1155, 1621}, {1210, 2475}, {1376, 3873}, {1385, 1389}, {1393, 4296}, {1407, 10601}, {1434, 26563}, {1454, 7288}, {1465, 17074}, {1468, 24174}, {1473, 1995}, {1706, 3621}, {1709, 9779}, {1730, 19717}, {1748, 17917}, {1749, 11263}, {1768, 3817}, {1776, 10129}, {1943, 24148}, {1962, 17593}, {1993, 23140}, {1994, 22128}, {1999, 17495}, {2096, 6957}, {2975, 3812}, {3007, 3101}, {3058, 17051}, {3060, 3784}, {3085, 17437}, {3086, 17700}, {3090, 24467}, {3091, 18540}, {3187, 17490}, {3220, 13595}, {3245, 3898}, {3262, 4359}, {3304, 14923}, {3315, 3744}, {3339, 19861}, {3361, 19860}, {3487, 6921}, {3523, 5709}, {3525, 26921}, {3526, 26878}, {3543, 7171}, {3582, 16763}, {3587, 15692}, {3600, 5554}, {3634, 6763}, {3636, 11010}, {3647, 25542}, {3649, 6691}, {3666, 3723}, {3681, 4413}, {3683, 3848}, {3720, 17596}, {3750, 17450}, {3754, 4861}, {3816, 5057}, {3833, 4973}, {3840, 4418}, {3869, 5221}, {3870, 10980}, {3871, 5045}, {3874, 4420}, {3876, 16408}, {3885, 7373}, {3889, 5687}, {3916, 5047}, {3918, 5288}, {3922, 11260}, {3927, 16862}, {3937, 5943}, {3961, 17449}, {3995, 22003}, {4000, 27059}, {4003, 4682}, {4004, 24928}, {4090, 9458}, {4187, 24470}, {4189, 15803}, {4253, 21373}, {4292, 5046}, {4298, 20060}, {4384, 5361}, {4414, 26102}, {4438, 25961}, {4440, 16561}, {4511, 5902}, {4640, 5284}, {4641, 16602}, {4652, 16865}, {4655, 25960}, {4678, 6762}, {4855, 11518}, {4880, 10176}, {4921, 17348}, {4993, 26931}, {5012, 26884}, {5020, 26866}, {5044, 17535}, {5056, 7330}, {5059, 9841}, {5122, 17549}, {5133, 26933}, {5154, 9612}, {5176, 5434}, {5183, 10179}, {5205, 17165}, {5256, 14996}, {5262, 24046}, {5268, 7226}, {5269, 17024}, {5271, 5372}, {5272, 17127}, {5278, 24594}, {5285, 15246}, {5311, 17591}, {5314, 7496}, {5432, 25557}, {5433, 7098}, {5438, 11520}, {5535, 10165}, {5536, 10164}, {5550, 12514}, {5603, 6966}, {5640, 26892}, {5704, 6871}, {5714, 6931}, {5722, 17579}, {5770, 6854}, {5826, 7291}, {5880, 11680}, {5885, 21740}, {5927, 13243}, {6147, 13747}, {6245, 6894}, {6734, 12436}, {6904, 12649}, {6905, 10202}, {6918, 12528}, {6940, 24474}, {6997, 26929}, {7081, 17140}, {7146, 26639}, {7196, 23989}, {7262, 17125}, {7411, 11227}, {7419, 22344}, {7705, 9654}, {7998, 26893}, {8025, 17168}, {8226, 13226}, {9310, 26672}, {9345, 17592}, {9347, 17599}, {9782, 12609}, {10107, 20323}, {10199, 18393}, {10273, 10698}, {10404, 11681}, {10461, 17589}, {10528, 11037}, {10566, 18087}, {10580, 20075}, {10587, 12704}, {10914, 15179}, {11015, 12433}, {11220, 19541}, {11374, 17566}, {11491, 13373}, {11552, 11813}, {11684, 25917}, {12527, 25011}, {13407, 27529}, {13587, 24929}, {14450, 21616}, {14997, 23511}, {15024, 26914}, {15650, 16863}, {15932, 24541}, {15934, 16371}, {16297, 22458}, {16421, 22149}, {16465, 17612}, {16549, 17266}, {16568, 16706}, {16586, 26740}, {16610, 16671}, {16672, 17021}, {16815, 18206}, {16826, 20367}, {16915, 26634}, {17016, 24443}, {17121, 18164}, {17556, 18541}, {17740, 18141}, {17763, 24165}, {17825, 22129}, {17862, 18359}, {18134, 27757}, {18163, 26860}, {18398, 25440}, {19241, 23169}, {19245, 23206}, {21540, 25083}, {24175, 26723}, {24586, 24629}, {24602, 24631}, {26959, 26969}, {26964, 26988}, {26965, 26995}, {26981, 27009}

X(27004) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27004) lies on these lines: {2, 82}, {2295, 3589}, {2345, 3112}, {3405, 26959}, {16706, 26969}, {16890, 18101}, {18095, 18102}

X(27005) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27005) lies on these lines: {2, 32}, {377, 17500}, {2295, 3589}, {16889, 18101}, {17686, 18092}, {18082, 19874}, {26963, 26978}, {26972, 26996}

X(27006) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27006) lies on these lines: {2, 85}, {8, 20269}, {142, 5253}, {277, 10527}, {664, 26526}, {1385, 26140}, {3661, 25593}, {4000, 10529}, {4861, 4904}, {5086, 9317}, {5337, 17397}, {6646, 26674}, {7195, 26258}, {7483, 20328}, {16706, 26964}, {17073, 25876}, {17298, 25582}, {20935, 27337}, {21255, 24780}, {24784, 27529}, {26979, 26993}

X(27007) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^4 b^4 - 2 a^4 b^3 c - a^3 b^4 c + 2 a^4 b^2 c^2 - 2 a^4 b 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(27007) lies on these lines: {2, 87}, {26959, 27011}, {26963, 26974}

X(27008) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27008) lies on these lines: {2, 99}, {1015, 27009}, {1086, 27010}, {1111, 14838}, {2170, 4369}, {7192, 20982}, {17058, 26856}, {26100, 27302}, {26960, 26978}, {26972, 26996}

X(27009) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27009) lies on these lines: {2, 11}, {767, 3767}, {1015, 27008}, {1086, 27012}, {4904, 26847}, {7192, 20974}, {14936, 17494}, {17761, 27010}, {26960, 26964}, {26965, 26989}, {26969, 26974}, {26981, 27003}, {27146, 27302}

X(27010) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27010) lies on these lines: {2, 101}, {11, 18101}, {894, 27001}, {1086, 27008}, {1311, 3086}, {4391, 11998}, {4560, 7117}, {17761, 27009}, {26279, 26959}

X(27011) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c - 2 a b^3 c + a^3 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(27011) lies on these lines: {2, 37}, {142, 27166}, {239, 26756}, {319, 27106}, {894, 26982}, {1086, 26963}, {1654, 26142}, {2140, 26825}, {2275, 26852}, {2321, 27113}, {3589, 26976}, {3662, 17178}, {3663, 26769}, {4361, 27095}, {7321, 26975}, {10436, 26817}, {16738, 17305}, {17117, 27044}, {17290, 27145}, {17300, 26821}, {17366, 26772}, {25253, 27680}, {26813, 26964}, {26959, 27007}

X(27012) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27012) lies on these lines: {2, 900}, {665, 17302}, {1086, 27009}, {4435, 17300}, {6646, 22108}, {27190, 27191}

X(27013) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27013) lies on these lines: {2, 649}, {100, 23865}, {514, 26777}, {650, 7192}, {661, 4763}, {667, 21302}, {693, 4394}, {798, 26975}, {812, 24924}, {890, 25537}, {1019, 26775}, {1635, 4369}, {2487, 4453}, {2490, 4897}, {2527, 3004}, {3239, 4786}, {3240, 23655}, {3249, 26752}, {3618, 9002}, {3676, 5435}, {3733, 20293}, {3798, 25259}, {4063, 26982}, {4359, 20952}, {4379, 26824}, {4380, 4885}, {4468, 5744}, {4521, 5273}, {4598, 8050}, {4651, 7234}, {4776, 4790}, {4789, 4976}, {4893, 4932}, {4979, 25666}, {6006, 18230}, {6586, 17159}, {6590, 17161}, {8653, 15724}, {8663, 11176}, {9463, 23575}, {16704, 18200}, {18197, 26822}, {25577, 27134}, {26049, 27114}, {26114, 27167}, {26984, 27016}

X(27014) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c - 3 a^3 b c + 2 a^2 b^2 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(27014) lies on these lines: {2, 650}, {661, 23466}, {798, 26975}, {1635, 27345}, {3210, 21438}, {3666, 21611}, {4893, 27527}, {6589, 25258}, {17215, 25955}, {25666, 27293}

X(27015) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27015) lies on these lines: {2, 659}, {798, 26975}, {891, 26801}, {1086, 27009}, {10566, 26968}, {17030, 21385}

X(27016) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27016) lies on these lines: {2, 669}, {10566, 26968}, {26984, 27013}

X(27017) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 51

Barycentrics    a^3 b^2 + a^2 b^3 + 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(27017) lies on these lines: {2, 7}, {6, 27311}, {75, 27107}, {141, 27044}, {239, 17178}, {314, 17495}, {320, 26772}, {1086, 26971}, {1958, 26634}, {3589, 26975}, {3664, 26816}, {3739, 16709}, {3912, 26764}, {3923, 26094}, {3946, 26821}, {4359, 16753}, {4363, 27261}, {4859, 27192}, {7321, 26976}, {10566, 18094}, {16706, 26963}, {16816, 27343}, {17227, 27095}, {17232, 26042}, {17245, 27032}, {17261, 26769}, {17266, 27073}, {17268, 26797}, {17273, 27111}, {17284, 27136}, {17288, 26756}, {17302, 27166}, {17324, 25510}, {17332, 27036}, {24199, 26812}, {25269, 27291}, {26959, 27007}

X(27018) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27018) lies on these lines: {2, 896}, {798, 26975}, {17122, 27061}, {26959, 26969}

X(27019) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 51

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

X(27019) lies on these lines: {2, 38}, {7, 26134}, {10, 27116}, {142, 27169}, {330, 17383}, {894, 20372}, {1909, 16706}, {2140, 3662}, {2275, 4657}, {4357, 20459}, {4645, 26801}, {5749, 26107}, {16819, 17291}, {16823, 27047}, {17278, 25610}, {17302, 19565}, {17322, 27148}, {24789, 27313}, {26813, 26964}

leftri

Collineation mappings involving Gemini triangle 52: X(27020)-X(27081)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 52, as in centers X(27020)-X(27081). Then

m(X) = a (b + c)^2 x + b (a^2 + c^2) y + c (a^2 + b^2) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 4, 2018)

X(27020) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27020) lies on these lines: {1, 2}, {5, 26590}, {9, 26068}, {12, 6656}, {35, 384}, {36, 6645}, {37, 308}, {39, 1909}, {55, 7770}, {56, 11285}, {75, 3774}, {76, 2276}, {83, 1914}, {86, 21760}, {100, 17686}, {140, 26686}, {171, 1923}, {172, 1078}, {190, 4721}, {192, 3760}, {194, 3761}, {226, 3503}, {238, 20148}, {274, 1575}, {315, 9596}, {335, 3670}, {350, 1500}, {385, 5280}, {388, 16043}, {405, 26687}, {442, 26582}, {458, 11398}, {495, 8362}, {609, 7793}, {668, 1107}, {672, 17499}, {894, 16549}, {980, 20917}, {993, 17684}, {1003, 5217}, {1015, 6683}, {1089, 3797}, {1220, 16061}, {1376, 11321}, {1478, 7791}, {1479, 16924}, {1573, 25280}, {1621, 17541}, {1655, 6381}, {1966, 17289}, {2241, 7808}, {2242, 7815}, {2275, 7786}, {2345, 3403}, {3035, 17694}, {3247, 26107}, {3329, 5299}, {3405, 27066}, {3508, 4357}, {3552, 5010}, {3583, 16044}, {3585, 6655}, {3663, 26149}, {3727, 18061}, {3730, 24514}, {3735, 18055}, {3739, 21897}, {3746, 4366}, {3814, 17669}, {4063, 27046}, {4302, 14035}, {4324, 6658}, {4400, 7760}, {4416, 26082}, {4698, 20363}, {4754, 20331}, {4995, 6661}, {5025, 7951}, {5218, 14001}, {5248, 16916}, {5259, 16918}, {5264, 14621}, {5283, 6376}, {5310, 16950}, {5332, 7878}, {5432, 7807}, {5687, 20172}, {5750, 26110}, {6179, 7296}, {6284, 8370}, {6651, 27057}, {6684, 8924}, {7031, 7787}, {7242, 14620}, {7354, 8356}, {7741, 16921}, {7761, 9650}, {7819, 26629}, {7833, 10483}, {7841, 10895}, {8359, 18990}, {8367, 15172}, {9312, 26134}, {9598, 11185}, {9654, 11287}, {10053, 10352}, {10436, 26042}, {10588, 14064}, {11174, 16502}, {11681, 17550}, {15338, 19687}, {16060, 18758}, {16564, 27053}, {16589, 27076}, {16601, 25994}, {16604, 24656}, {16720, 20924}, {16738, 17287}, {16777, 25505}, {16788, 17743}, {16915, 25440}, {16975, 24524}, {17116, 26764}, {17143, 20691}, {17239, 27164}, {17252, 26756}, {17260, 20372}, {17268, 27261}, {17277, 23660}, {17291, 27116}, {17312, 27145}, {17317, 26979}, {17319, 26971}, {17326, 27095}, {17357, 25629}, {17670, 25466}, {17755, 25073}, {17757, 26558}, {17758, 24170}, {17759, 20888}, {18040, 18148}, {19579, 27033}, {20174, 21858}, {24530, 25458}, {27021, 27038}, {27023, 27027}, {27030, 27041}, {27049, 27058}, {27069, 27073}, {27070, 27072}

X(27020) = complement of X(26801)

X(27021) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27021) lies on these lines: {2, 3}, {183, 27515}, {1901, 26125}, {17056, 27253}, {18299, 21838}, {26685, 26772}, {26771, 27043}, {27020, 27038}, {27025, 27072}, {27040, 27071}, {27097, 27256}, {27129, 27255}

X(27022) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27022) lies on these lines: {2, 3}, {33, 26203}, {346, 1228}, {17052, 27509}, {17260, 20605}, {27031, 27062}, {27039, 27040}

X(27023) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27023) lies on these lines: {2, 3}, {257, 27261}, {27020, 27027}, {27033, 27062}

X(27024) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27024) lies on these lines: {1, 2}, {1575, 16750}, {17260, 27038}, {27065, 27072}

X(27025) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27025) lies on these lines: {1, 2}, {1574, 26978}, {1575, 18600}, {3619, 26997}, {3693, 25261}, {5241, 27256}, {6376, 26770}, {6537, 27071}, {11319, 26687}, {16713, 17239}, {17287, 26818}, {17289, 27039}, {17672, 17757}, {25244, 26563}, {27021, 27072}, {27038, 27050}, {27040, 27076}, {27049, 27065}, {27073, 27080}

X(27026) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^3 b + a b^3 + a^3 c + 2 a^2 b c + 4 a b^2 c + b^3 c + 4 a b c^2 + a c^3 + b c^3 : :

X(27026) lies on these lines: {1, 2}, {35, 16931}, {75, 26100}, {964, 26687}, {1213, 27047}, {1574, 25499}, {1575, 16705}, {2345, 18135}, {3739, 21021}, {5051, 26582}, {5260, 16061}, {5263, 17541}, {6376, 26035}, {14005, 27185}, {14210, 25089}, {17260, 20605}, {17289, 27040}, {17307, 27116}, {17385, 25107}, {17672, 26558}, {17680, 26060}, {18136, 19808}, {20911, 25263}, {27050, 27072}, {27056, 27065}

X(27027) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27027) lies on these lines: {2, 11}, {26686, 26755}, {27020, 27023}, {27074, 27294}

X(27028) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27028) lies on these lines: {2, 3}

X(27029) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27029) lies on these lines: {2, 3}, {10566, 27075}

X(27030) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^4 b c + a^4 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 : :

X(27030) lies on these lines: {2, 31}, {7292, 26999}, {7295, 16949}, {16823, 27182}, {17289, 27066}, {26772, 27034}, {27020, 27041}, {27035, 27072}

X(27031) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    (b + c) (a^5 b + a^2 b^4 + a^5 c - 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(27031) lies on these lines: {2, 32}, {26685, 26772}, {27022, 27062}, {27040, 27057}

X(27032) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^3 b^2 + a^2 b^3 + 4 a^2 b^2 c + a^3 c^2 + 4 a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(27032) lies on these lines: {2, 37}, {86, 26975}, {1125, 21803}, {1213, 27044}, {3723, 26821}, {3834, 26857}, {3912, 16738}, {4357, 27106}, {4422, 27042}, {7321, 26769}, {16814, 26799}, {16826, 26963}, {17030, 17242}, {17045, 26982}, {17178, 17317}, {17239, 26774}, {17244, 27145}, {17245, 27017}, {17248, 27095}, {17256, 26756}, {17260, 20372}, {17261, 25538}, {17285, 27164}, {17300, 26082}, {17307, 27113}, {17349, 23660}, {17368, 27255}, {25611, 26030}, {27033, 27048}, {27038, 27051}, {27107, 27147}

X(27033) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27033) lies on these lines: {2, 39}, {256, 26030}, {2238, 26752}, {7148, 26115}, {9263, 23447}, {19579, 27020}, {21024, 26801}, {26685, 26772}, {27023, 27062}, {27032, 27048}, {27057, 27067}

X(27034) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    2 a^3 b^3 + 3 a^3 b^2 c + a^2 b^3 c + 3 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 : :

X(27034) lies on these lines: {1, 2}, {1575, 16748}, {4430, 27351}, {8299, 18103}, {17147, 27285}, {23632, 25102}, {26772, 27030}, {27041, 27072}

X(27035) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27035) lies on these lines: {1, 2}, {75, 27285}, {310, 1575}, {668, 23632}, {1011, 26687}, {1921, 21814}, {3136, 26582}, {16954, 25440}, {18152, 21877}, {21838, 27076}, {22199, 25286}, {26772, 27069}, {27030, 27072}, {27038, 27047}

X(27036) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27036) lies on these lines: {2, 44}, {9, 27102}, {798, 20295}, {966, 27261}, {1213, 27078}, {2245, 27070}, {3739, 26799}, {4422, 27044}, {4473, 26048}, {16814, 26764}, {16815, 26976}, {17257, 27311}, {17259, 27154}, {17260, 20372}, {17263, 26756}, {17277, 26971}, {17331, 27145}, {17332, 27017}, {17333, 27107}, {17338, 27095}, {17349, 21760}, {17368, 27116}, {20363, 27268}, {27290, 27321}

X(27037) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27037) lies on these lines: {2, 45}, {1213, 27073}, {3780, 17349}, {4687, 26963}, {4708, 27113}, {4755, 27166}, {5296, 27095}, {17260, 20372}, {17261, 27154}

X(27038) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27038) lies on these lines: {2, 11}, {1233, 25249}, {17260, 27024}, {26772, 27030}, {27020, 27021}, {27025, 27050}, {27032, 27051}, {27035, 27047}, {27096, 27283}

X(27039) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27039) lies on these lines: {2, 6}, {9, 14543}, {10, 21931}, {144, 2245}, {344, 27096}, {346, 3948}, {442, 7679}, {2092, 3672}, {3759, 26964}, {3882, 17183}, {3965, 17863}, {4199, 5281}, {4272, 17014}, {4466, 27689}, {4515, 22040}, {5051, 7080}, {5227, 26267}, {16609, 21033}, {17077, 17272}, {17227, 26997}, {17233, 26757}, {17257, 25601}, {17273, 26836}, {17289, 27025}, {18600, 24530}, {26085, 26961}, {26752, 27296}, {27022, 27040}, {27055, 27071}

X(27040) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27040) lies on these lines: {2, 39}, {4, 26085}, {6, 5192}, {8, 2176}, {10, 1018}, {21, 26244}, {32, 11319}, {37, 3701}, {42, 21071}, {45, 1213}, {69, 26099}, {115, 5992}, {145, 20970}, {187, 17539}, {213, 17751}, {281, 429}, {315, 17007}, {321, 16583}, {344, 27042}, {346, 2092}, {350, 26965}, {668, 26759}, {672, 3831}, {857, 1211}, {862, 17920}, {874, 17280}, {964, 5275}, {965, 27378}, {966, 2478}, {1089, 16600}, {1215, 21808}, {1475, 3840}, {1654, 24958}, {1909, 27097}, {2275, 26094}, {2276, 26030}, {2292, 3985}, {2295, 21025}, {2321, 3214}, {2475, 26079}, {3061, 25591}, {3125, 17164}, {3290, 4968}, {3293, 21070}, {3691, 3741}, {3735, 25253}, {3952, 3954}, {4065, 24049}, {4099, 4868}, {4109, 15523}, {4202, 5254}, {4272, 17314}, {4385, 26242}, {4441, 27299}, {4647, 16611}, {4721, 20347}, {4754, 17169}, {5025, 16991}, {5224, 17550}, {5276, 13740}, {5277, 11115}, {6155, 27804}, {6537, 6627}, {7735, 17526}, {7747, 17537}, {7751, 25497}, {7806, 16905}, {10453, 21753}, {11185, 16910}, {14953, 24271}, {15985, 17183}, {16050, 26243}, {16926, 16993}, {16997, 17688}, {17137, 24514}, {17277, 17541}, {17281, 25610}, {17289, 27026}, {17359, 25107}, {17497, 17762}, {20255, 24330}, {20911, 25994}, {20947, 25263}, {25255, 27697}, {25264, 27324}, {26771, 26774}, {26781, 26794}, {26791, 26793}, {27021, 27071}, {27022, 27039}, {27025, 27076}, {27031, 27057}

X(27041) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27041) lies on these lines: {2, 6}, {321, 21858}, {740, 21684}, {1230, 2092}, {1500, 3948}, {3454, 26030}, {3752, 27793}, {4272, 26971}, {4850, 27792}, {5051, 17757}, {16549, 21361}, {17165, 20966}, {22020, 26580}, {27020, 27030}, {27034, 27072}, {27052, 27058}

X(27042) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27042) lies on these lines: {2, 6}, {10, 872}, {12, 1284}, {37, 313}, {75, 2092}, {239, 4272}, {257, 18714}, {274, 24530}, {344, 27040}, {442, 4429}, {495, 4205}, {661, 24103}, {740, 21730}, {860, 17913}, {894, 2245}, {1084, 19581}, {1215, 21035}, {1268, 26048}, {1269, 3666}, {1500, 4043}, {1966, 17289}, {2511, 3766}, {3122, 25124}, {3759, 17030}, {3775, 19863}, {3826, 26030}, {3882, 10455}, {3934, 18046}, {4395, 26812}, {4422, 27032}, {4443, 23444}, {4446, 20966}, {4472, 27102}, {4687, 6376}, {7227, 26764}, {7238, 26857}, {16706, 25538}, {17045, 26971}, {17233, 21024}, {17243, 27261}, {17246, 26976}, {17305, 26149}, {17397, 25505}, {17719, 17954}, {26601, 27254}, {27047, 27048}, {27050, 27058}, {27054, 27068}

X(27043) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27043) lies on these lines: {1, 2}, {23830, 26836}, {26771, 27021}

X(27044) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27044) lies on these lines: {1, 2}, {9, 27136}, {75, 27095}, {141, 27017}, {319, 26963}, {335, 27116}, {524, 26975}, {527, 26768}, {594, 26971}, {894, 26756}, {1086, 27106}, {1213, 27032}, {1278, 4494}, {1574, 20913}, {2309, 25121}, {3763, 27311}, {3948, 27076}, {3995, 18140}, {4063, 4129}, {4357, 26764}, {4359, 21021}, {4422, 27036}, {4967, 26812}, {7032, 25292}, {11320, 26687}, {16738, 17239}, {17117, 27011}, {17160, 25534}, {17178, 17287}, {17227, 27107}, {17228, 27145}, {17238, 26042}, {17254, 26769}, {17260, 27073}, {17261, 26797}, {17285, 27111}, {17289, 26772}, {17293, 27261}, {17355, 26799}, {17786, 27641}, {18046, 21858}, {18091, 18093}, {19308, 21005}, {20072, 26076}, {20349, 26072}

X(27045) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27045) lies on these lines: {2, 661}, {649, 4129}, {669, 21051}, {798, 20295}, {810, 25301}, {850, 3709}, {1577, 17494}, {2978, 25636}, {4391, 27648}, {4761, 19874}, {4781, 26794}, {5278, 7252}, {18155, 24948}, {20910, 25271}, {21383, 27134}, {21960, 27588}, {24459, 27712}, {27138, 27346}

X(27046) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    (b - c) (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(27046) lies on these lines: {2, 667}, {4063, 27020}, {4129, 27047}, {16158, 18110}, {20295, 27077}, {21261, 27345}

X(27047) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27047) lies on these lines: {2, 31}, {141, 27097}, {857, 26582}, {1213, 27026}, {3230, 20549}, {3775, 26759}, {4026, 26965}, {4129, 27046}, {4429, 17550}, {16823, 27019}, {17238, 27248}, {17248, 27255}, {17260, 20372}, {17326, 27106}, {17338, 24491}, {20561, 21788}, {26041, 27280}, {27035, 27038}, {27042, 27048}

X(27048) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^3 b + 4 a^2 b^2 + a b^3 + a^3 c + 6 a^2 b c + 4 a b^2 c + b^3 c + 4 a^2 c^2 + 4 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 : :

X(27048) lies on these lines: {1, 2}, {35, 16930}, {27032, 27033}, {27042, 27047}, {27050, 27060}

X(27049) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27049) lies on these lines: {2, 3}, {27020, 27058}, {27025, 27065}

X(27050) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27050) lies on these lines: {2, 3}, {1211, 27096}, {3936, 27283}, {18635, 27170}, {27020, 27030}, {27025, 27038}, {27026, 27072}, {27042, 27058}, {27048, 27060}

X(27051) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27051) lies on these lines: {2, 3}, {2212, 26211}, {27032, 27038}

X(27052) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27052) lies on these lines: {2, 3}, {63, 17052}, {210, 4463}, {226, 21065}, {306, 1089}, {312, 1230}, {321, 4150}, {1211, 17293}, {1441, 18588}, {1901, 5905}, {5928, 26223}, {16568, 17289}, {18082, 18083}, {18139, 18147}, {18744, 19792}, {27041, 27058}

X(27053) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27053) lies on these lines: {2, 3}, {16564, 27020}

X(27054) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27054) lies on these lines: {2, 3}, {27042, 27068}

X(27055) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27055) lies on these lines: {2, 3}, {27039, 27071}

X(27056) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27056) lies on these lines: {2, 3}, {17289, 27025}, {27026, 27065}

X(27057) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27057) lies on these lines: {2, 3}, {6651, 27020}, {27031, 27040}, {27033, 27067}

X(27058) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 6 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(27058) lies on these lines: {2, 7}, {44, 16713}, {86, 23617}, {344, 18040}, {2324, 26621}, {2345, 27108}, {3618, 26964}, {4687, 26690}, {5046, 17500}, {5782, 27381}, {7146, 20248}, {17120, 26818}, {17152, 17277}, {17263, 18150}, {17286, 26757}, {17289, 27025}, {20262, 26575}, {26582, 26772}, {27020, 27049}, {27041, 27052}, {27042, 27050}

X(27059) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27059) lies on these lines: {2, 19}, {10, 16566}, {75, 1150}, {141, 5341}, {169, 26685}, {171, 17872}, {193, 2285}, {524, 5356}, {597, 7300}, {607, 26206}, {894, 7291}, {910, 25099}, {1429, 17868}, {1441, 26213}, {1738, 24883}, {1760, 2345}, {1766, 17257}, {1781, 3912}, {1861, 2475}, {1890, 5046}, {1953, 26639}, {1958, 7146}, {2171, 20769}, {2182, 15988}, {3589, 7297}, {3661, 5279}, {3920, 17446}, {4000, 27003}, {4357, 16548}, {7269, 27950}, {16547, 17353}, {16564, 27020}, {16568, 17289}, {17260, 20605}, {17355, 20602}, {26582, 26605}, {27032, 27038}

X(27060) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27060) lies on these lines: {2, 36}, {4129, 27046}, {26685, 27063}, {27020, 27021}, {27048, 27050}, {27251, 27255}, {27274, 27283}

X(27061) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^2 b^3 c + a^4 c^2 + 6 a^2 b^2 c^2 + b^4 c^2 + 2 a^2 b c^3 + a^2 c^4 + b^2 c^4 : :

X(27061) lies on these lines: {2, 38}, {5297, 26969}, {17122, 27018}, {27020, 27030}

X(27062) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27062) lies on these lines: {2, 99}, {27022, 27031}, {27023, 27033}

X(27063) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27063) lies on these lines: {2, 48}, {16564, 27020}, {26685, 27060}

X(27064) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27064) lies on these lines: {1, 979}, {2, 7}, {6, 312}, {8, 989}, {10, 4388}, {31, 7081}, {42, 3685}, {43, 3923}, {44, 333}, {55, 4676}, {72, 13740}, {75, 4383}, {78, 4195}, {81, 4358}, {83, 213}, {92, 458}, {100, 20967}, {171, 4672}, {190, 3666}, {192, 5256}, {210, 5263}, {228, 4203}, {238, 1215}, {306, 17280}, {318, 3195}, {341, 5710}, {344, 5712}, {386, 7283}, {404, 22344}, {474, 23085}, {537, 17598}, {594, 4886}, {612, 27538}, {614, 24349}, {645, 14534}, {748, 16823}, {756, 16830}, {846, 6685}, {899, 4418}, {940, 3758}, {942, 13741}, {960, 1220}, {964, 3876}, {984, 25496}, {1010, 5044}, {1046, 3831}, {1054, 6686}, {1089, 1203}, {1211, 17289}, {1255, 3227}, {1265, 5716}, {1386, 3967}, {1460, 26264}, {1468, 25591}, {1696, 25895}, {1728, 26123}, {1743, 11679}, {1757, 3741}, {1766, 9535}, {1836, 4429}, {2235, 21779}, {2258, 3240}, {2295, 3975}, {2308, 17763}, {2345, 14555}, {2895, 17287}, {2999, 3210}, {3175, 4360}, {3187, 4671}, {3333, 26093}, {3337, 19847}, {3338, 25492}, {3487, 13742}, {3589, 4415}, {3649, 25992}, {3661, 5739}, {3676, 26694}, {3681, 24552}, {3687, 17355}, {3720, 17794}, {3742, 25531}, {3745, 4009}, {3750, 4432}, {3751, 10453}, {3752, 17351}, {3765, 17752}, {3782, 16706}, {3791, 16477}, {3868, 5192}, {3886, 20012}, {3912, 17499}, {3920, 3952}, {3940, 11354}, {3944, 25453}, {3961, 4090}, {3973, 18229}, {3980, 16569}, {3995, 17011}, {3996, 4849}, {4001, 20072}, {4044, 17034}, {4054, 26723}, {4063, 23825}, {4234, 5440}, {4344, 5423}, {4359, 17116}, {4362, 16468}, {4363, 19804}, {4385, 16466}, {4395, 19820}, {4417, 17354}, {4422, 17056}, {4438, 17717}, {4521, 26652}, {4641, 14829}, {4644, 18141}, {4656, 17023}, {4664, 20182}, {4687, 19701}, {4692, 5315}, {4697, 17122}, {4698, 25507}, {4852, 22034}, {4972, 5057}, {4975, 16474}, {5271, 17349}, {5283, 11342}, {5287, 17379}, {5484, 12527}, {5506, 25512}, {5737, 16885}, {5743, 17369}, {5927, 13727}, {6537, 27068}, {6651, 19579}, {6679, 17719}, {6763, 19864}, {7123, 14621}, {7191, 17165}, {7227, 19797}, {7292, 17140}, {8025, 17021}, {10394, 27394}, {12572, 26117}, {13425, 19065}, {13458, 19066}, {13735, 24929}, {14997, 17117}, {17012, 17147}, {17016, 25253}, {17019, 19717}, {17020, 17495}, {17123, 24325}, {17127, 26227}, {17266, 18139}, {17279, 18134}, {17316, 27523}, {17335, 19732}, {17339, 17776}, {17352, 24789}, {17366, 19796}, {17367, 19785}, {17394, 19722}, {17777, 24210}, {18662, 25245}, {18928, 26531}, {23511, 24620}, {24725, 25957}, {25066, 27399}, {25760, 26061}, {25930, 27340}, {26575, 26793}, {27000, 27299}, {27020, 27021}

X(27065) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27065) lies on these lines: {1, 4134}, {2, 7}, {5, 26878}, {6, 17019}, {8, 7162}, {10, 3583}, {20, 18540}, {21, 5044}, {23, 5314}, {31, 5297}, {37, 17011}, {38, 7292}, {39, 27646}, {40, 3832}, {44, 81}, {45, 4383}, {46, 19877}, {72, 5047}, {78, 16865}, {84, 15717}, {100, 3683}, {110, 26890}, {149, 25006}, {190, 4359}, {191, 3634}, {210, 1621}, {219, 5422}, {220, 10601}, {238, 756}, {239, 3294}, {306, 25101}, {312, 5278}, {321, 17277}, {333, 4358}, {344, 5739}, {354, 15481}, {405, 3876}, {484, 3828}, {518, 5284}, {612, 9330}, {614, 7226}, {662, 17190}, {748, 984}, {750, 7262}, {846, 899}, {896, 17122}, {936, 4189}, {940, 16885}, {942, 17536}, {958, 1388}, {960, 5260}, {968, 3240}, {982, 17125}, {988, 27625}, {993, 4881}, {1001, 3681}, {1018, 6539}, {1100, 1255}, {1125, 5506}, {1150, 18743}, {1155, 9342}, {1171, 1963}, {1211, 2503}, {1212, 15889}, {1697, 4678}, {1698, 4338}, {1728, 5703}, {1743, 5287}, {1749, 14526}, {1757, 3720}, {1770, 26060}, {1776, 5432}, {1961, 2308}, {1995, 7085}, {1999, 19742}, {2183, 26044}, {2329, 26639}, {2345, 26998}, {2475, 12572}, {2895, 3912}, {2975, 5302}, {3060, 3781}, {3090, 26921}, {3100, 7069}, {3175, 17348}, {3187, 17349}, {3220, 15246}, {3245, 3968}, {3337, 19878}, {3523, 7330}, {3525, 24467}, {3526, 26877}, {3543, 3587}, {3617, 5250}, {3661, 21373}, {3666, 16814}, {3678, 5259}, {3685, 4651}, {3690, 5943}, {3691, 6542}, {3697, 3871}, {3711, 4428}, {3731, 5256}, {3746, 4015}, {3750, 21805}, {3757, 3952}, {3782, 17337}, {3812, 11684}, {3826, 20292}, {3833, 4880}, {3836, 4683}, {3868, 11108}, {3873, 4423}, {3874, 25542}, {3877, 9708}, {3916, 17531}, {3923, 26037}, {3925, 5057}, {3927, 16842}, {3938, 15485}, {3969, 4886}, {3973, 14996}, {3984, 5436}, {4038, 4722}, {4113, 4702}, {4193, 5791}, {4392, 5272}, {4414, 16569}, {4420, 5248}, {4430, 4666}, {4438, 25960}, {4473, 16561}, {4511, 5251}, {4641, 15492}, {4650, 17124}, {4652, 17572}, {4655, 25961}, {4656, 26723}, {4671, 5271}, {4679, 11680}, {4687, 19684}, {4698, 5333}, {4703, 25957}, {4745, 5541}, {4993, 26941}, {5012, 26885}, {5020, 26867}, {5056, 5709}, {5129, 12649}, {5133, 21015}, {5154, 5705}, {5218, 7082}, {5234, 19861}, {5268, 17126}, {5285, 13595}, {5311, 16468}, {5438, 17548}, {5439, 17534}, {5535, 10172}, {5536, 10171}, {5640, 26893}, {5657, 6957}, {5708, 16854}, {5729, 11020}, {5741, 27757}, {5758, 6886}, {5777, 6986}, {5779, 11220}, {5812, 6991}, {5815, 10587}, {5817, 10431}, {5927, 7411}, {6147, 17590}, {6197, 7563}, {6763, 19862}, {6871, 9780}, {6883, 18444}, {6932, 26446}, {6997, 26939}, {7171, 15692}, {7174, 17024}, {7291, 17292}, {7293, 7496}, {7322, 15601}, {7485, 24320}, {7548, 9956}, {7998, 26892}, {8025, 17120}, {9350, 17601}, {9945, 17525}, {10578, 15299}, {10580, 15298}, {10916, 26127}, {11227, 13243}, {11415, 19855}, {12527, 24564}, {13411, 15674}, {14555, 17776}, {15024, 26915}, {15064, 15931}, {15066, 23140}, {15296, 26105}, {15934, 17542}, {16296, 22458}, {16373, 20760}, {16514, 20965}, {16552, 16826}, {16568, 17289}, {16578, 16585}, {16667, 25417}, {16670, 25430}, {16675, 20182}, {16676, 17013}, {16823, 17165}, {16824, 25253}, {16858, 24929}, {17023, 17744}, {17147, 17261}, {17242, 20017}, {17263, 18139}, {17336, 19804}, {17394, 19738}, {17479, 25243}, {17742, 26626}, {17825, 24554}, {18151, 20886}, {18249, 24982}, {18250, 24987}, {18607, 25067}, {19249, 23169}, {19292, 23206}, {21511, 25066}, {21516, 25083}, {25068, 25946}, {26227, 27538}, {27020, 27030}, {27024, 27072}, {27025, 27049}, {27026, 27056}

X(27066) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27066) lies on these lines: {2, 82}, {1213, 27026}, {3112, 4000}, {3405, 27020}, {17289, 27030}, {18082, 18095}, {18092, 18101}

X(27067) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27067) lies on these lines: {2, 32}, {10, 22025}, {12, 1284}, {308, 941}, {857, 18086}, {874, 17280}, {1176, 20029}, {1213, 27026}, {1228, 2092}, {1500, 3948}, {2478, 17500}, {4129, 4375}, {8299, 18091}, {16890, 17550}, {17541, 18092}, {18096, 26601}, {27033, 27057}

X(27068) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27068) lies on these lines: {2, 85}, {9, 7679}, {10, 5526}, {21, 5179}, {41, 5086}, {169, 2476}, {388, 26258}, {498, 25082}, {644, 10039}, {894, 25000}, {910, 2475}, {1220, 1311}, {2082, 11680}, {2329, 5176}, {2345, 27522}, {3039, 6668}, {3496, 5057}, {3684, 5178}, {3746, 21090}, {3871, 21073}, {4262, 11015}, {4766, 17739}, {4850, 5286}, {5262, 5305}, {5540, 25639}, {5750, 7110}, {5819, 6871}, {6537, 27064}, {9318, 17062}, {9956, 26074}, {15492, 17303}, {16589, 23988}, {17289, 27025}, {19860, 23058}, {24547, 27547}, {25066, 27529}, {26279, 26561}, {27042, 27054}

X(27069) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^4 b^4 - a^3 b^4 c - 2 a^4 b^2 c^2 - 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(27069) lies on these lines: {2, 87}, {26772, 27035}, {27020, 27073}

X(27070) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27070) lies on these lines: {2, 45}, {2245, 27036}, {21362, 26223}, {27020, 27072}

X(27071) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27071) lies on these lines: {2, 99}, {661, 21232}, {1577, 24036}, {6537, 27025}, {16592, 27256}, {20982, 21272}, {26035, 27251}, {27021, 27040}, {27033, 27057}, {27039, 27055}, {27072, 27076}

X(27072) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27072) lies on these lines: {2, 11}, {767, 7795}, {4422, 27074}, {6184, 23989}, {17494, 23988}, {27020, 27070}, {27021, 27025}, {27024, 27065}, {27026, 27050}, {27030, 27035}, {27034, 27041}, {27071, 27076}

X(27073) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27073) lies on these lines: {2, 37}, {9, 26756}, {45, 27095}, {1213, 27037}, {1654, 26774}, {3662, 26769}, {3912, 17178}, {4357, 27113}, {4422, 26772}, {4473, 26799}, {16738, 17285}, {17116, 26817}, {17243, 26963}, {17258, 27106}, {17260, 27044}, {17265, 27107}, {17266, 27017}, {17267, 27145}, {17317, 26975}, {17319, 26982}, {17340, 26976}, {22343, 25284}, {24491, 26752}, {27020, 27069}, {27025, 27080}

X(27074) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27074) lies on these lines: {2, 900}, {190, 27134}, {3766, 17280}, {4422, 27072}, {4526, 17302}, {17281, 21606}, {27027, 27294}

X(27075) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27075) lies on these lines: {2, 659}, {798, 20295}, {1960, 26801}, {4422, 27072}, {10566, 27029}, {20979, 24356}, {21385, 27255}

X(27076) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27076) lies on these lines: {2, 668}, {10, 3934}, {32, 26687}, {39, 6376}, {75, 9466}, {76, 1574}, {115, 26582}, {116, 121}, {192, 18146}, {291, 1698}, {519, 20530}, {537, 3739}, {538, 1575}, {620, 2787}, {625, 3814}, {626, 1329}, {812, 4422}, {891, 4928}, {958, 7815}, {1018, 4465}, {1107, 6683}, {1125, 25102}, {1376, 3734}, {1500, 18140}, {1921, 21830}, {2551, 7800}, {2885, 21258}, {3008, 25125}, {3039, 20317}, {3634, 25109}, {3788, 26364}, {3948, 27044}, {4103, 9055}, {4386, 7804}, {4403, 18159}, {4426, 7780}, {4482, 9259}, {4561, 24281}, {4568, 21138}, {4986, 27918}, {6292, 26558}, {6685, 25115}, {6686, 25116}, {6702, 17239}, {7257, 25530}, {7816, 25440}, {8649, 18047}, {9317, 9458}, {9708, 15271}, {9780, 17794}, {16589, 27020}, {16705, 26779}, {17759, 18145}, {19862, 24656}, {19878, 25130}, {21838, 27035}, {24988, 25468}, {25280, 26959}, {25499, 26030}, {27025, 27040}, {27071, 27072}

X(27077) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27077) lies on these lines: {2, 669}, {10566, 27029}, {20295, 27046}

X(27078) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^3 b^2 + a^2 b^3 + 4 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(27078) lies on these lines: {2, 7}, {6, 27261}, {44, 16738}, {1213, 27036}, {3008, 26812}, {3589, 26971}, {3758, 27145}, {3923, 26030}, {4363, 27311}, {4422, 27032}, {4698, 16726}, {16706, 26976}, {17120, 17178}, {17289, 26772}, {17292, 26756}, {17337, 27154}, {17355, 26764}, {17369, 27102}, {17371, 27095}, {17741, 26965}, {20352, 21803}, {26975, 26979}, {27020, 27069}

X(27079) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27079) lies on these lines: {2, 896}, {798, 20295}, {17123, 26999}, {27020, 27030}

X(27080) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 52

Barycentrics    a^4 b^2 + a^2 b^4 + a^3 b^2 c + 3 a^2 b^3 c + a^4 c^2 + a^3 b c^2 + 8 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 + 3 a^2 b c^3 + a b^2 c^3 + a^2 c^4 + b^2 c^4 : :

X(27080) lies on these lines: {2, 38}, {4357, 27116}, {16830, 26986}, {17248, 27091}, {17260, 20372}, {17263, 27097}, {17289, 27026}, {17326, 27102}, {17338, 27255}, {24697, 26778}, {27025, 27073}

X(27081) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 52

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

X(27081) lies on these lines: {2, 6}, {10, 3120}, {145, 4205}, {306, 3986}, {321, 6539}, {1086, 27791}, {1230, 3596}, {1330, 17589}, {1834, 4678}, {2321, 3995}, {3187, 4034}, {3218, 17252}, {3454, 9780}, {3617, 5051}, {3661, 17497}, {3948, 4671}, {4026, 19998}, {4062, 25354}, {4085, 4651}, {4272, 17013}, {4357, 17495}, {4358, 17239}, {4359, 17235}, {4425, 8013}, {4427, 24697}, {4442, 4733}, {4748, 17740}, {4850, 17250}, {4938, 5625}, {6536, 21085}, {6537, 6627}, {7226, 20966}, {8818, 26792}, {11115, 26064}, {16589, 17230}, {17012, 17326}, {17021, 17287}, {17147, 17247}, {17184, 24199}, {17236, 27794}, {17237, 24589}, {17272, 26627}, {17292, 21383}, {17491, 24342}, {19804, 27793}, {27021, 27025}

X(27082) =  X(3)X(15077)∩X(4)X(5972)

Barycentrics    (a^2-b^2-c^2) (5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) : :
Barycentrics    S^2 (24 R^2-SB-SC-5 SW)+SB SC (-32 R^2+8 SW) : :
X(27082) = 4*X[3]-X[15077]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27082) lies on the cubics K041 and K934 and these lines: {3,15077}, {4,5972}, {20,154}, { 69,3522}, {159,11413}, {343, 21734}, {376,5562}, {394,16936}, {511,16879}, {631,11704}, {1092, 8718}, {2071,8907}, {3146,15748}, {3528,12254}, {3619,14118}, { 5059,11064}, {5921,8567}, {6225,16386}, {6467,25406}, {10167, 18732}, {11206,12279}, {12118, 18931}, {19467,22647}

X(27083) =  X(21)X(60)∩X(1175)X(18123)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27083) lies on the cubic K934 and these lines: {21,60}, {1175,18123}

X(27084) =  X(4)X(15462)∩X(22)X(206)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27084) lies on the cubic K934 and these lines: {4,15462}, {22,206}, {343,19127}, {1176,1899}

X(27085) =  X(3)X(15077)∩X(4)X(5972)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27085) lies on the cubic K934 and these lines: {4,83}, {23,6593}, {1177,9140}, { 2070,19381}, {3047,12367}, { 5169,19127}, {9979,13315}, {15019,19136}

X(27086) =  EULER LINE INTERCEPT OF X(35)X(3754)

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

As a point on the Euler line, X(27086) has Shinagawa coefficients {2 r^2 + 2 r R - R^2, -2 r (r + R)}.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27086) lies on these lines: {2,3}, {35,3754}, {36,214}, {100, 5172}, {191,997}, {515,17009}, { 1125,14794}, {1470,21454}, {1708,4855}, {1737,17010}, {1994, 5398}, {2206,4257}, {2646,8261}, {2771,18861}, {2975,21677}, {3002,5546}, {4256,20966}, {4861,14798}, {5010,5426}, {5204,11684}, {5253,11281}, {5303,18253}, {5445,25440}, {6796,25005}, {10090,11604}, {11263,14792}, {17653,22936}

X(27086)= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,4216,6636}, {3,4218,15246}, { 3,19525,17549}, {21,404,442}, { 21,3651,15680}, {404,1006,2}, { 442,5428,21}, {1006,6905,6882}, {1006,21161,5428}, {4188,4189,4190}, {4189,15674,21}, {6827,6921,2}, {6830,17566,2}, {11334,19245,13595}

X(27087) =  EULER LINE INTERCEPT OF X(131)X(12095)

Barycentrics    (a^2-b^2-c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (2 a^8-3 a^6 b^2+a^4 b^4-a^2 b^6+b^8-3 a^6 c^2+2 a^4 b^2 c^2+a^2 b^4 c^2-4 b^6 c^2+a^4 c^4+a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8) : :
Barycentrics    S^4 + (-20 R^4-SB SC+12 R^2 SW-2 SW^2)S^2 + SB SC (-12 R^4+4 R^2 SW) : :

As a point on the Euler line, X(27087) has Shinagawa coefficients {20 R^4 - S^2 - 12 R^2 SW + 2 SW^2, 12 R^4 + S^2 - 4 R^2 SW}.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27087) lies on these lines: {2,3}, {131,12095}, {3564,13557}

X(27087)= midpoint of X(131) and X(12095)

X(27088) =  EULER LINE INTERCEPT OF X(6)X(7618)

Barycentrics    (2 a^2-b^2-c^2) (5 a^2-b^2-c^2) : :
Barycentrics    9 S^2 - 9 SB SC - 2 SW^2 : :
X(27088) = X[115]-3*X[5215], X[625]-2*X[22247], X[6781]+3*X[9167], 3*X[8290]+X[9889], X[8591]+3*X[8859], 2*X[14148]+X[15480]

As a point on the Euler line, X(27088) has Shinagawa coefficients {2 SW^2 - 9 S^2, 9 S^2}.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27088) lies on these lines: {2,3}, {6,7618}, {32,8584}, {69,15655}, {99,9136}, {110,6093}, {115,5215}, {141,8588}, {187,524}, {230,543}, {574,597}, {598,11149}, {599,5210}, {620,3849}, {625,22247}, {671,10153}, {1384,1992}, {1499,4786}, {2021,5969}, {2080,5182}, {3053,15534}, {3054,7617}, {3055,7619}, {3564,8593}, {3589,8589}, {3734,5569}, {3815,7622}, {3933,5023}, {5008,20583}, {5032,21309}, {5104,15483}, {5206,7767}, {5305,7782}, {5475,9771}, {5476,9734}, {5585,21358}, {6781,9167}, {7610,21843}, {7737,11184}, {7750,7870}, {7789,7810}, {7820,15810}, {7891,9939}, {8030,14567}, {8290,9889}, {8591,8859}, {8860,11164}, {9486,16317}, {9489,25423}, {9741,22253}, {11151,11171}, {11161,14830}, {11162,14666}, {11163,12040}, {11645,19662}, {14148,15480}, {15993,19911}

X(27088) = midpoint of X(i) and X(j) for these {i,j}: {2,8598}, {99,22329}, {187,2482}, {376,1513}, {1551,10295}, {6661,10997}, {7426,7472}, {8352,9855}, {35303,35304}
X(27088) = reflection of X(i) in X(j) for these {i,j}: {381,10011}, {625,22247}, {6390,2482}, {8352,8355}, {22110,620}
X(27088) = complement of X(8352)
X(27088) = anticomplement of X(8355)

X(27088) = X(230)-of-anti-Artzt-triangle
X(27088) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,376,5077}, {2,8352,8355}, {2,8703,8354}, {2,9855,8352}, {2,11159,3363}, {2,11317,5}, {2,13586,8598}, {3,8369,8359}, {187,6390,3793}, {548,7807,8357}, {548,8360,7833}, {550,16925,8361}, {599,5210,8182}, {1384,11165,1992}, {3734,5569,11168}, {5077,11288,2}, {7807,7833,8360}, {7820,15810,20582}, {7833,8360,8357}, {8352,8598,9855}, {8359,8369,7819}, {8860,11164,11185}, {8860,11185,16509}, {12040,18907,11163}, {16431,16436,11350}

X(27089) =  EULER LINE INTERCEPT OF X(1503)X(11589)

Barycentrics    (a^2-b^2-c^2) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (2 a^10-a^8 b^2-8 a^6 b^4+10 a^4 b^6-2 a^2 b^8-b^10-a^8 c^2+16 a^6 b^2 c^2-10 a^4 b^4 c^2-8 a^2 b^6 c^2+3 b^8 c^2-8 a^6 c^4-10 a^4 b^2 c^4+20 a^2 b^4 c^4-2 b^6 c^4+10 a^4 c^6-8 a^2 b^2 c^6-2 b^4 c^6-2 a^2 c^8+3 b^2 c^8-c^10) : :
Barycentrics    S^4 + (160 R^4 - SB SC - 64 R^2 SW + 6 SW^2)S^2 + (-192 R^4 + 80 R^2 SW - 8 SW^2)SB SC : :

As a point on the Euler line, X(27089) has Shinagawa coefficients {160 R^4+S^2-64 R^2 SW+6 SW^2,-192 R^4-S^2+80 R^2 SW-8 SW^2}.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27089) lies on these lines: {2,3}, {1503,11589}, {3184,12096}, {5894,14379}, {8057,15427}

X(27089) = midpoint of X(i) and X(j) for these {i,j}: {20,1559}, {3184,12096}
X(27089) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {20,2060,6616}, {376,3079,20}, {550,13155,20}

X(27090) =  EULER LINE INTERCEPT OF X(6150)X(6592)

Barycentrics    (3 a^6-7 a^4 b^2+5 a^2 b^4-b^6-7 a^4 c^2-3 a^2 b^2 c^2+b^4 c^2+5 a^2 c^4+b^2 c^4-c^6) (2 a^10-7 a^8 b^2+10 a^6 b^4-8 a^4 b^6+4 a^2 b^8-b^10-7 a^8 c^2+10 a^6 b^2 c^2-a^4 b^4 c^2-5 a^2 b^6 c^2+3 b^8 c^2+10 a^6 c^4-a^4 b^2 c^4+2 a^2 b^4 c^4-2 b^6 c^4-8 a^4 c^6-5 a^2 b^2 c^6-2 b^4 c^6+4 a^2 c^8+3 b^2 c^8-c^10) : :
Barycentrics    16 S^4 + (-47 R^4-16 SB SC+44 R^2 SW-12 SW^2)S^2 + (-3 R^4-4 R^2 SW+4 SW^2)SB SC : :

As a point on the Euler line, X(27090) has Shinagawa coefficients {47 R^4-16 S^2-44 R^2 SW+12 SW^2,3 R^4+16 S^2+4 R^2 SW-4 SW^2}.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28576.

X(27090) lies on these lines: {2,3}, {930,24385}, {6150,6592}

X(27090) = midpoint of X(i) and X(j) for these {i,j}:{930,24385}, {6150,6592}
X(27090) = complement of X(24306)
X(27090) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,140,15334}, {140,5501,3628}

leftri

Collineation mappings involving Gemini triangle 53: X(27091)-X(27141)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 53, as in centers X(27091)-X(27141). Then

m(X) = a (b^2 + c^2) x + b (a - c)^2 y + c (a - b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 5, 2018)

X(27091) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27091) lies on these lines: {1, 2}, {3, 26687}, {5, 26582}, {12, 17670}, {35, 16916}, {37, 10009}, {39, 6376}, {75, 1574}, {76, 1575}, {83, 4386}, {100, 17541}, {101, 17743}, {194, 6381}, {312, 21412}, {335, 24046}, {384, 25440}, {538, 20943}, {594, 25505}, {668, 2275}, {672, 26043}, {726, 24080}, {874, 17279}, {958, 11285}, {993, 7824}, {1015, 24524}, {1018, 27103}, {1078, 4426}, {1107, 7786}, {1329, 6656}, {1376, 7770}, {1500, 30963}, {1573, 6683}, {1837, 28798}, {1921, 32453}, {1964, 31337}, {2276, 18140}, {2321, 26107}, {2550, 32968}, {2551, 16043}, {2886, 32992}, {3035, 7807}, {3125, 18055}, {3249, 31286}, {3596, 27633}, {3662, 24170}, {3760, 17759}, {3814, 5025}, {3820, 8362}, {3826, 33033}, {3841, 33045}, {3959, 18061}, {3963, 27641}, {4021, 26143}, {4075, 27481}, {4187, 26590}, {4357, 26042}, {4358, 21435}, {4359, 27285}, {4366, 8715}, {4413, 11321}, {4699, 27298}, {5010, 17692}, {5248, 16918}, {5251, 17684}, {5267, 33004}, {5280, 16997}, {7752, 20541}, {7808, 20179}, {8165, 33202}, {9709, 20172}, {12782, 17793}, {13747, 26686}, {15482, 31456}, {16549, 24514}, {16604, 25102}, {16606, 25115}, {16921, 25639}, {16975, 25280}, {17053, 17786}, {17228, 26979}, {17234, 20549}, {17242, 20501}, {17243, 20491}, {17247, 26764}, {17248, 27080}, {17301, 25534}, {17338, 24491}, {17339, 27136}, {17353, 24502}, {17363, 26963}, {17364, 26756}, {17368, 26772}, {17499, 17754}, {17540, 26629}, {17756, 18135}, {17757, 26561}, {18044, 24530}, {20335, 24190}, {20530, 20691}, {20888, 31276}, {21067, 24166}, {21385, 27140}, {22199, 25287}, {24914, 28771}, {25066, 25994}, {25092, 27269}, {25590, 26149}, {26689, 28737}, {27092, 27129}, {27100, 27110}, {27122, 27127}, {30478, 32978}, {31418, 32987}

X(27092) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27092) lies on these lines: {2, 3}, {325, 27515}, {27091, 27129}, {27095, 27101}, {27096, 27134}, {27109, 27133}

X(27093) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 4 a^5 b c - a^4 b^2 c + 4 a^3 b^3 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 + 4 a^3 b c^3 - 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(27093) lies on these lines: {2, 3}, {36, 28410}, {1040, 26203}, {3662, 27097}, {27108, 27109}

X(27094) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27094) lies on this line: {2, 3}

X(27095) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^3 b^2 + a^2 b^3 - 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(27095) lies on these lines: {2, 6}, {45, 27073}, {75, 27044}, {190, 27136}, {192, 646}, {2345, 26976}, {3009, 25140}, {3661, 26971}, {3662, 24170}, {3759, 26982}, {4360, 25534}, {4361, 27011}, {4389, 26764}, {4751, 27160}, {4851, 27166}, {5296, 27037}, {17119, 26850}, {17148, 27633}, {17227, 27017}, {17228, 25505}, {17230, 26107}, {17233, 26774}, {17236, 26042}, {17248, 27032}, {17255, 26769}, {17262, 26797}, {17279, 27113}, {17287, 26959}, {17291, 27311}, {17292, 27261}, {17302, 26100}, {17312, 25510}, {17326, 27020}, {17338, 27036}, {17347, 26768}, {17354, 26799}, {17364, 26975}, {17371, 27078}, {17377, 26821}, {17786, 28395}, {18133, 31036}, {18170, 25292}, {20352, 31337}, {20917, 27641}, {21244, 26176}, {21352, 25121}, {21858, 29764}, {25535, 29570}, {25538, 29610}, {25940, 26222}, {26076, 31300}, {26149, 28604}, {27035, 31004}, {27092, 27101}, {27100, 27104}, {27126, 27137}, {27154, 29576}

X(27096) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27096) lies on these lines: {1, 2}, {3, 31020}, {5, 31031}, {85, 25244}, {141, 27170}, {344, 27039}, {345, 18136}, {495, 17672}, {668, 27109}, {857, 31032}, {1211, 27050}, {1500, 26100}, {3061, 21272}, {3314, 26796}, {3454, 26781}, {3501, 20347}, {3693, 25237}, {3871, 17681}, {3930, 20247}, {4193, 31058}, {4851, 27161}, {5046, 20533}, {5233, 27256}, {9709, 17683}, {16284, 26690}, {16593, 21031}, {17170, 31080}, {17279, 27108}, {17756, 18600}, {20244, 20335}, {21232, 33299}, {25066, 30806}, {27021, 31037}, {27038, 27283}, {27049, 31018}, {27072, 31017}, {27092, 27134}, {27118, 27129}, {27119, 30831}, {28772, 33160}

X(27097) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27097) lies on these lines: {1, 2}, {21, 27185}, {37, 16705}, {56, 28777}, {72, 26689}, {141, 27047}, {213, 30941}, {304, 17489}, {321, 16752}, {350, 26978}, {517, 26562}, {595, 29473}, {894, 17169}, {1018, 24170}, {1475, 17353}, {1479, 16910}, {1621, 16060}, {1909, 27040}, {2176, 17137}, {2242, 25497}, {2275, 17279}, {2276, 27162}, {2345, 25504}, {3230, 17152}, {3263, 28598}, {3290, 20911}, {3294, 16887}, {3662, 27093}, {3670, 25248}, {3726, 17141}, {3915, 24586}, {4202, 26590}, {5074, 17211}, {5253, 16061}, {5255, 24602}, {5259, 16931}, {5263, 27169}, {5749, 26106}, {9310, 24549}, {11321, 24552}, {11363, 15149}, {14210, 16600}, {16583, 17497}, {16712, 32026}, {16738, 17201}, {17053, 27634}, {17081, 28739}, {17084, 27273}, {17263, 27080}, {17283, 27116}, {17674, 26582}, {17683, 20172}, {17686, 32942}, {18600, 25264}, {20244, 24190}, {25082, 25918}, {25994, 30806}, {26035, 31997}, {26041, 32099}, {26100, 30963}, {26971, 27155}, {27021, 27256}, {27119, 27134}, {27125, 27131}, {27249, 27259}

X(27098) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27098) lies on these lines: {2, 3}

X(27099) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27099) lies on these lines: {2, 3}

X(27100) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27100) lies on these lines: {2, 31}, {20965, 21250}, {27091, 27110}, {27095, 27104}, {27105, 27134}

X(27101) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27101) lies on these lines: {2, 32}, {27092, 27095}, {27109, 27126}, {27119, 27312}, {27133, 27137}

X(27102) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    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(27102) lies on these lines: {2, 37}, {6, 26975}, {9, 27036}, {10, 16738}, {38, 25120}, {141, 27017}, {190, 27111}, {239, 26963}, {319, 17178}, {320, 26756}, {594, 26979}, {894, 21362}, {1268, 27164}, {1269, 31026}, {1654, 26048}, {1740, 21278}, {1909, 16710}, {1958, 26222}, {1964, 20352}, {2234, 21238}, {2309, 20340}, {3009, 28597}, {3596, 17148}, {3661, 27145}, {3662, 24170}, {4360, 27166}, {4438, 25611}, {4472, 27042}, {4852, 26821}, {17077, 27315}, {17116, 26976}, {17117, 26959}, {17231, 26774}, {17234, 27159}, {17237, 26857}, {17258, 26769}, {17283, 27113}, {17300, 20561}, {17319, 25510}, {17326, 27080}, {17345, 26768}, {17351, 26799}, {17366, 26982}, {17369, 27078}, {17376, 26816}, {17872, 30801}, {20255, 29964}, {21858, 30939}, {22012, 24195}, {24746, 33115}, {26029, 27334}, {27103, 27117}, {27120, 27127}

X(27103) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27103) lies on these lines: {2, 39}, {1018, 27091}, {4595, 26752}, {27092, 27095}, {27102, 27117}, {27126, 27133}

X(27104) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27104) lies on these lines: {1, 2}, {27095, 27100}, {27110, 27134}

X(27105) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^3 b^3 - a^2 b^3 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(27105) lies on these lines: {1, 2}, {1978, 20284}, {2229, 6384}, {2309, 27188}, {21071, 26108}, {27100, 27134}, {27285, 30818}

X(27106) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27106) lies on these lines: {2, 44}, {141, 26971}, {190, 27113}, {319, 27011}, {524, 26982}, {536, 18073}, {1086, 27044}, {3619, 27261}, {3662, 24170}, {3768, 27114}, {4357, 27032}, {5224, 27154}, {5564, 26850}, {6542, 26142}, {16706, 26756}, {17235, 26764}, {17238, 20549}, {17239, 26812}, {17258, 27073}, {17275, 27192}, {17276, 27136}, {17288, 26963}, {17291, 26772}, {17292, 26976}, {17297, 25534}, {17326, 27047}, {17357, 26799}, {17374, 26821}, {25140, 28597}

X(27107) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a b^3 c + a^3 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(27107) lies on these lines: {2, 45}, {7, 26772}, {75, 27017}, {274, 330}, {894, 27311}, {3662, 24170}, {4000, 26963}, {4361, 17178}, {5224, 26857}, {7232, 26756}, {7263, 26979}, {17116, 27261}, {17148, 20892}, {17227, 27044}, {17234, 26764}, {17244, 27159}, {17265, 27073}, {17267, 26797}, {17283, 27136}, {17301, 27166}, {17302, 27162}, {17333, 27036}, {17349, 27343}, {17367, 26975}, {17378, 26816}, {27032, 27147}

X(27108) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27108) lies on these lines: {2, 6}, {9, 27514}, {198, 24612}, {319, 28748}, {322, 26669}, {346, 646}, {390, 2478}, {908, 17220}, {1229, 3965}, {2183, 20245}, {2269, 3452}, {2293, 6745}, {2345, 27058}, {2347, 21246}, {3262, 25243}, {3672, 27282}, {3686, 28797}, {4266, 17183}, {4384, 7190}, {4416, 17077}, {4643, 27170}, {4698, 4875}, {5227, 26265}, {6666, 28742}, {17279, 27096}, {27093, 27109}, {27124, 27133}, {27396, 30854}, {28778, 29616}

X(27109) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27109) lies on these lines: {2, 39}, {8, 4595}, {21, 5132}, {75, 25082}, {83, 11319}, {304, 26690}, {344, 3616}, {345, 5222}, {668, 27096}, {672, 17137}, {1193, 17353}, {1212, 20911}, {1280, 30701}, {1334, 30038}, {1475, 3912}, {1909, 28742}, {1930, 24036}, {2275, 17279}, {2276, 26965}, {2549, 16910}, {3263, 25066}, {3501, 30036}, {3618, 17526}, {3730, 17152}, {3735, 25248}, {3972, 17539}, {4253, 30941}, {4651, 23407}, {5030, 29473}, {5192, 11174}, {5276, 16061}, {5278, 16367}, {7772, 25497}, {7774, 26099}, {7791, 26085}, {7800, 17007}, {7864, 16906}, {7875, 16905}, {7876, 16991}, {7920, 17003}, {14021, 14555}, {16050, 32911}, {16549, 30109}, {16601, 26234}, {16818, 25092}, {16975, 26759}, {17169, 17234}, {17280, 26801}, {17303, 27156}, {17349, 17696}, {17754, 29966}, {17755, 33299}, {17756, 27299}, {17776, 26626}, {18061, 25253}, {20255, 20331}, {21070, 29742}, {21808, 24631}, {21877, 27313}, {23632, 27263}, {23649, 30821}, {27092, 27133}, {27093, 27108}, {27101, 27126}, {29590, 33168}

X(27110) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27110) lies on these lines: {2, 6}, {646, 3995}, {27091, 27100}, {27104, 27134}

X(27111) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27111) lies on these lines: {2, 6}, {37, 646}, {45, 26042}, {190, 27102}, {594, 26048}, {874, 17279}, {1100, 25510}, {2092, 25660}, {2664, 21238}, {3863, 27805}, {3948, 24530}, {4261, 30830}, {4361, 26107}, {4384, 25505}, {4698, 20363}, {16706, 25534}, {17243, 26752}, {17269, 27291}, {17273, 27017}, {17285, 27044}, {17305, 27311}, {17348, 26959}, {17390, 26113}, {17790, 21796}, {23345, 28758}, {25538, 31238}, {27073, 31333}, {27116, 27117}, {27123, 27132}

X(27112) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^3 b - 6 a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c + 6 a b^2 c + b^3 c - 6 a^2 c^2 + 6 a b c^2 - 6 b^2 c^2 + a c^3 + b c^3 : :

X(27112) lies on these lines: {1, 2}, {26687, 31020}

X(27113) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27113) lies on these lines: {1, 2}, {190, 27106}, {2321, 27011}, {3662, 27136}, {3663, 26797}, {4357, 27073}, {4431, 26850}, {4708, 27037}, {17231, 26963}, {17279, 27095}, {17283, 27102}, {17285, 26971}, {17291, 26764}, {17297, 26975}, {17307, 27032}, {17353, 26756}, {17357, 26772}, {17359, 26976}, {21385, 27138}, {27070, 31029}, {27131, 27137}

X(27114) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (b - c) (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(27114) lies on these lines: {2, 661}, {1150, 7252}, {2978, 24755}, {3762, 14838}, {3768, 27106}, {3952, 30584}, {4160, 26115}, {18155, 24900}, {18199, 19684}, {20295, 27346}, {21259, 21302}, {26049, 27013}, {26985, 27293}

X(27115) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27115) lies on these lines: {2, 650}, {149, 10006}, {514, 29607}, {661, 4763}, {812, 27138}, {1635, 20295}, {1639, 4467}, {1643, 29569}, {2516, 4380}, {3004, 14425}, {3239, 27486}, {3523, 8760}, {3619, 9015}, {3622, 14077}, {3762, 14838}, {4359, 21611}, {4394, 4776}, {4406, 27344}, {4521, 25259}, {4560, 4791}, {4777, 27268}, {4828, 31238}, {4893, 7192}, {5059, 8142}, {5281, 11934}, {6050, 31291}, {6546, 21212}, {9780, 29066}, {10196, 16892}, {16751, 26775}, {17069, 30565}, {17166, 31288}, {17260, 23808}, {17495, 25271}, {21297, 30835}, {21727, 29822}, {23806, 31053}

X(27116) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27116) lies on these lines: {2, 31}, {10, 27019}, {141, 20561}, {335, 27044}, {3662, 24170}, {4357, 27080}, {17283, 27097}, {17291, 27020}, {17307, 27026}, {17368, 27036}, {21003, 21301}, {27111, 27117}

X(27117) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27117) lies on these lines: {1, 2}, {24988, 32992}, {27102, 27103}, {27111, 27116}

X(27118) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27118) lies on these lines: {2, 3}, {346, 21579}, {4461, 21403}, {27096, 27129}

X(27119) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27119) lies on these lines: {2, 3}, {7790, 28749}, {24170, 27135}, {27025, 30832}, {27091, 27100}, {27096, 30831}, {27097, 27134}, {27101, 27312}

X(27120) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27120) lies on these lines: {2, 3}, {27102, 27127}

X(27121) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27121) lies on these lines: {2, 3}, {16581, 17279}

X(27122) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27122) lies on these lines: {2, 3}, {27091, 27127}

X(27123) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27123) lies on these lines: {2, 3}, {27111, 27132}

X(27124) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^6 b - 5 a^5 b^2 + 4 a^4 b^3 + 4 a^3 b^4 - 5 a^2 b^5 + a b^6 + a^6 c - 4 a^5 b c - a^4 b^2 c - 4 a^3 b^3 c - a^2 b^4 c + 8 a b^5 c + b^6 c - 5 a^5 c^2 - a^4 b c^2 + 8 a^3 b^2 c^2 + 8 a^2 b^3 c^2 - a b^4 c^2 - 5 b^5 c^2 + 4 a^4 c^3 - 4 a^3 b c^3 + 8 a^2 b^2 c^3 - 16 a b^3 c^3 + 4 b^4 c^3 + 4 a^3 c^4 - a^2 b c^4 - a b^2 c^4 + 4 b^3 c^4 - 5 a^2 c^5 + 8 a b c^5 - 5 b^2 c^5 + a c^6 + b c^6 : :

X(27124) lies on these lines: {2, 3}, {27108, 27133}

X(27125) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 53

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

X(27125) lies on these lines: {2, 3}, {17279, 27096}, {27097, 27131}

X(27126) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^5 b^2 + a^2 b^5 + a^4 b^2 c - 2 a^3 b^3 c + a^2 b^4 c - 2 a b^5 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 + b^5 c^2 - 2 a^3 b c^3 + a^2 b c^4 + a b^2 c^4 + a^2 c^5 - 2 a b c^5 + b^2 c^5 : :

X(27126) lies on these lines: {2, 3}, {27095, 27137}, {27101, 27109}, {27103, 27133}

X(27127) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^4 b - b^5 + a^4 c + a^3 b c - 2 a^2 b^2 c + a b^3 c + b^4 c - 2 a^2 b c^2 + a b c^3 + b c^4 - c^5 : :

X(27127) lies on these lines: {2, 19}, {35, 25582}, {37, 31019}, {344, 16580}, {3218, 28420}, {3662, 27093}, {16581, 17279}, {17073, 21495}, {17321, 27186}, {18651, 26065}, {20336, 32858}, {21062, 26132}, {27091, 27122}, {27102, 27120}

X(27128) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^4 b^2 + a^2 b^4 - 2 a^3 b^2 c + a^4 c^2 - 2 a^3 b c^2 + 6 a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - 2 a b^2 c^3 + a^2 c^4 + b^2 c^4 : :

X(27128) lies on these lines: {2, 38}, {5205, 26969}, {27091, 27100}

X(27129) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    2 a^3 b - a^2 b^2 - b^4 + 2 a^3 c - 3 a^2 b c + b^3 c - a^2 c^2 + b c^3 - c^4 : :

X(27129) lies on these lines: {2, 40}, {20, 26658}, {78, 20533}, {85, 17747}, {169, 5195}, {220, 4872}, {226, 27253}, {516, 4209}, {517, 17671}, {673, 12701}, {857, 3661}, {1334, 7179}, {3649, 27475}, {3662, 27093}, {3730, 5074}, {3868, 31038}, {3912, 19582}, {4101, 29616}, {4188, 26660}, {4329, 27420}, {5046, 26653}, {5080, 28961}, {5088, 17732}, {6999, 25930}, {10025, 17170}, {11415, 28740}, {12699, 17682}, {12702, 17675}, {14021, 16826}, {21062, 27184}, {21068, 26125}, {21872, 33298}, {27021, 27255}, {27049, 31053}, {27091, 27092}, {27096, 27118}, {31045, 32858}

X(27130) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    2 a^2 b + a b^2 - b^3 + 2 a^2 c - 7 a b c + 2 b^2 c + a c^2 + 2 b c^2 - c^3 : :

X(27130) lies on these lines: {2, 7}, {43, 11814}, {306, 30861}, {1401, 3038}, {1699, 26073}, {1997, 1999}, {3340, 25979}, {3699, 4952}, {3772, 4997}, {3782, 31233}, {3870, 26139}, {4033, 16594}, {5121, 32937}, {6557, 30699}, {6700, 17697}, {13466, 29582}, {21075, 26093}, {24003, 29641}, {27091, 27092}, {27132, 29629}, {30855, 32911}

X(27131) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    -a^2 b + b^3 - a^2 c + 3 a b c - b^2 c - b c^2 + c^3 : :

X(27131) lies on these lines: {1, 26127}, {2, 7}, {5, 3876}, {8, 5187}, {10, 3899}, {11, 3681}, {43, 33134}, {72, 4193}, {78, 3586}, {100, 24703}, {145, 21075}, {149, 200}, {165, 21635}, {210, 5087}, {238, 29665}, {244, 33101}, {312, 3969}, {321, 5233}, {333, 17174}, {344, 27180}, {474, 18541}, {497, 3935}, {517, 6945}, {612, 33107}, {614, 33153}, {748, 17719}, {750, 33096}, {756, 17717}, {899, 3944}, {912, 6963}, {936, 2475}, {946, 3617}, {960, 11681}, {984, 29680}, {993, 5444}, {997, 5080}, {1054, 33098}, {1125, 17570}, {1215, 25960}, {1329, 3869}, {1376, 5057}, {1479, 4420}, {1621, 4679}, {1656, 15650}, {1699, 33110}, {1757, 29662}, {2051, 6539}, {2476, 5044}, {2478, 3488}, {2975, 25681}, {2994, 6557}, {2999, 33155}, {3006, 27538}, {3120, 16569}, {3240, 24210}, {3266, 21590}, {3436, 3476}, {3522, 6260}, {3616, 21077}, {3621, 12053}, {3654, 12611}, {3661, 31014}, {3671, 25011}, {3678, 7741}, {3679, 11813}, {3687, 4671}, {3697, 9955}, {3699, 5014}, {3705, 3952}, {3711, 11235}, {3740, 17605}, {3752, 33151}, {3786, 14008}, {3814, 5692}, {3816, 3873}, {3817, 25006}, {3825, 5904}, {3835, 6546}, {3840, 33065}, {3846, 29667}, {3868, 4187}, {3877, 17757}, {3890, 12607}, {3916, 17566}, {3925, 10129}, {3936, 18743}, {3940, 17556}, {3947, 24564}, {3957, 25568}, {3967, 33089}, {3971, 29849}, {3984, 9581}, {3994, 32855}, {3995, 22020}, {4009, 32862}, {4011, 29846}, {4090, 33120}, {4188, 6700}, {4189, 12572}, {4292, 17572}, {4358, 4417}, {4383, 17796}, {4413, 20292}, {4415, 4850}, {4416, 5372}, {4430, 11019}, {4661, 21060}, {4677, 21630}, {4703, 32918}, {4767, 30615}, {4855, 15680}, {4863, 10707}, {4871, 33069}, {4892, 25961}, {4997, 14829}, {5047, 11374}, {5123, 31165}, {5154, 6734}, {5176, 5289}, {5178, 10896}, {5205, 6327}, {5235, 17173}, {5253, 24954}, {5260, 11375}, {5268, 33112}, {5272, 33148}, {5284, 17718}, {5297, 26098}, {5423, 31091}, {5440, 11114}, {5550, 13407}, {5554, 8165}, {5709, 6979}, {5712, 17021}, {5720, 6840}, {5737, 30824}, {5739, 28808}, {5758, 6953}, {5761, 6898}, {5777, 6943}, {5791, 7504}, {5811, 6890}, {5812, 6915}, {5815, 10529}, {5880, 9342}, {6147, 17575}, {6384, 27461}, {6536, 29825}, {6686, 33125}, {6863, 26878}, {6872, 27383}, {6919, 12649}, {6922, 12528}, {6941, 31837}, {6947, 18444}, {6949, 26921}, {6971, 31835}, {6972, 7330}, {6975, 24474}, {7226, 24239}, {7292, 33144}, {7951, 10176}, {9335, 24231}, {9350, 24715}, {9352, 17768}, {9580, 20095}, {9780, 12047}, {10157, 10883}, {10584, 24477}, {11220, 13257}, {11684, 24914}, {11814, 30957}, {12514, 27529}, {12609, 19877}, {13411, 16865}, {14923, 21031}, {15228, 25440}, {15677, 30282}, {16468, 29683}, {16581, 17279}, {16610, 33146}, {16704, 17182}, {17020, 19785}, {17063, 32856}, {17122, 24725}, {17123, 33127}, {17124, 33097}, {17125, 33130}, {17155, 21093}, {17245, 17775}, {17331, 24220}, {17339, 27141}, {17521, 27412}, {17720, 32911}, {17776, 27757}, {17777, 32929}, {18139, 30829}, {18250, 24541}, {19861, 20060}, {20052, 21627}, {21805, 33141}, {22000, 31025}, {23536, 27625}, {24003, 25957}, {25385, 26037}, {25760, 28595}, {26105, 29817}, {27091, 27100}, {27096, 27118}, {27097, 27125}, {27113, 27137}, {27489, 27493}, {29612, 31039}, {29648, 32944}, {29649, 32843}, {29666, 32775}, {30567, 32863}, {30568, 32849}, {30578, 33168}, {30818, 32782}

X(27132) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^4 - 2 a^3 b + 2 a^2 b^2 - 2 a b^3 + b^4 - 2 a^3 c + a b^2 c - b^3 c + 2 a^2 c^2 + a b c^2 - 2 a c^3 - b c^3 + c^4 : :

X(27132) lies on these lines: {2, 85}, {220, 26526}, {344, 10528}, {1146, 26653}, {2348, 21285}, {3039, 3665}, {17279, 27096}, {17353, 24982}, {21856, 27337}, {25005, 31640}, {27111, 27123}, {27130, 29629}

X(27133) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (a - b) (a - c) (a^3 b^2 - a b^4 + 2 a^2 b^2 c + 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 b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(27133) lies on these lines: {2, 99}, {190, 27135}, {668, 27134}, {27092, 27109}, {27101, 27137}, {27103, 27126}, {27108, 27124}, {28736, 28747}, {28737, 28749}

X(27134) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (a - b) (a - c) (a^2 b^2 - a b^3 + a b^2 c + b^3 c + a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(27134) lies on these lines: {2, 11}, {190, 27074}, {644, 26796}, {668, 27133}, {1018, 27135}, {4554, 26985}, {21383, 27045}, {25577, 27013}, {27092, 27096}, {27097, 27119}, {27100, 27105}, {27104, 27110}

X(27135) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (a - b) (a - c) (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(27135) lies on these lines: {2, 101}, {190, 27133}, {644, 28743}, {1018, 27134}, {4885, 21859}, {24170, 27119}, {28737, 33298}

X(27136) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c + 2 a^2 b^2 c - 2 a b^3 c + a^3 c^2 + 2 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(27136) lies on these lines: {2, 37}, {9, 27044}, {69, 26774}, {87, 25284}, {144, 26768}, {190, 27095}, {3619, 26857}, {3662, 27113}, {3875, 26982}, {4851, 26975}, {4869, 26816}, {7032, 23354}, {16738, 17293}, {17178, 17230}, {17233, 26963}, {17236, 26769}, {17242, 27166}, {17276, 27106}, {17283, 27107}, {17284, 27017}, {17285, 27145}, {17300, 26076}, {17314, 26821}, {17339, 27091}, {17350, 21362}, {17354, 26772}, {22343, 25292}, {24502, 26752}, {26082, 29591}

X(27137) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (a b + a c - b c) (a^3 b^2 + a^2 b^3 - 2 a^3 b c + a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(27137) lies on these lines: {2, 39}, {2176, 4595}, {17339, 27091}, {27095, 27126}, {27101, 27133}, {27113, 27131}

X(27138) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (b - c) (a^2 - 3 a b - 3 a c + 3 b c) : :

X(27138) lies on these lines: {2, 649}, {192, 27485}, {650, 21297}, {661, 4928}, {812, 27115}, {1960, 21301}, {2516, 4106}, {3240, 24749}, {3619, 9002}, {3676, 5226}, {3768, 27106}, {4120, 21212}, {4358, 20952}, {4379, 31290}, {4380, 31287}, {4382, 26777}, {4453, 14321}, {4468, 5748}, {4521, 5328}, {4671, 20909}, {4728, 17494}, {4775, 21260}, {4776, 4885}, {4806, 30795}, {4893, 26824}, {4940, 31250}, {5284, 23865}, {8656, 31291}, {8657, 24601}, {9404, 28834}, {9780, 29350}, {9812, 15599}, {14433, 29587}, {17234, 23345}, {20293, 31946}, {21051, 21343}, {21385, 27113}, {23655, 29814}, {23813, 31150}, {25301, 29824}, {26983, 27193}, {27045, 27346}

X(27139) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c - 3 a^3 b c + 2 a^2 b^2 c + 2 a^2 b c^2 - 5 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(27139) lies on these lines: {2, 650}, {312, 25271}, {652, 26694}, {3768, 27106}, {4379, 28758}, {4928, 27293}, {21611, 30818}, {24924, 27345}, {27527, 30835}

X(27140) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    (b - c) (a^4 b^2 - a^3 b^3 + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c + a^4 c^2 - a^3 b c^2 - 2 a b^3 c^2 - a^3 c^3 + 2 a^2 b c^3 - 2 a b^2 c^3 + b^3 c^3) : :

X(27140) lies on these lines: {2, 659}, {190, 27074}, {891, 26752}, {3768, 27106}, {21385, 27091}

X(27141) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 53

Barycentrics    2 a^3 - 3 a^2 b - a b^2 + 4 b^3 - 3 a^2 c + 6 a b c - b^2 c - a c^2 - b c^2 + 4 c^3 : :

X(27141) lies on these lines: {2, 6}, {646, 4671}, {3306, 31029}, {4080, 17740}, {5219, 31025}, {17286, 30852}, {17339, 27131}, {24589, 30823}, {27092, 27096}, {27757, 30867}

leftri

Collineation mappings involving Gemini triangle 54: X(27142)-X(27195)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 54, as in centers X(27142)-X(27195). Then

m(X) = a (b^2 + c^2) x + b (a + c)^2 y + c (a + b)^2 z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 5, 2018)

X(27142) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 - 2 a^3 b^3 c + 2 a b^5 c + a^5 c^2 - 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 - 2 a^2 b^2 c^3 - 4 a b^3 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + a^2 c^5 + 2 a b c^5 + b^2 c^5 : :

X(27142) lies on these lines: {2, 3}, {6, 26125}, {1730, 16819}, {2481, 5283}, {5278, 27304}, {10025, 16552}, {17030, 27181}, {17257, 26976}, {19717, 26964}, {19740, 27146}, {27145, 27153}, {27162, 27189}

X(27143) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c + 4 a^5 b c - a^4 b^2 c - 4 a^3 b^3 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 - 4 a^3 b c^3 - 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(27143) lies on these lines: {2, 3}, {35, 28410}, {1038, 26203}, {27147, 27148}, {27161, 27162}

X(27144) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c + 4 a^5 b c - a^4 b^2 c - 6 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + b^6 c - a^4 b c^2 - 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - a b^4 c^2 - a^4 c^3 - 6 a^3 b c^3 - 2 a^2 b^2 c^3 - 4 a b^3 c^3 - b^4 c^3 - a^3 c^4 - a^2 b c^4 - a b^2 c^4 - b^3 c^4 + 2 a b c^5 + a c^6 + b c^6 : :

X(27144) lies on these lines: {2, 3}, {17030, 27149}

X(27145) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^2 + a^2 b^3 + 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(27145) lies on these lines: {2, 6}, {7, 26976}, {48, 26634}, {75, 27017}, {192, 980}, {238, 26094}, {239, 27311}, {894, 27261}, {982, 17142}, {1001, 16347}, {1107, 29982}, {1429, 17077}, {1740, 30942}, {2274, 17751}, {2309, 3840}, {3009, 24659}, {3286, 11319}, {3661, 27102}, {3662, 24220}, {3758, 27078}, {4389, 26857}, {4649, 26030}, {4657, 27166}, {4699, 10472}, {5253, 5263}, {16342, 26093}, {16696, 18137}, {16887, 27262}, {17030, 27147}, {17046, 26176}, {17148, 20891}, {17227, 25505}, {17228, 27044}, {17230, 26042}, {17233, 26764}, {17236, 26107}, {17244, 27032}, {17262, 26769}, {17267, 27073}, {17269, 26797}, {17285, 27136}, {17290, 27011}, {17291, 26959}, {17312, 27020}, {17326, 25510}, {17331, 27036}, {17347, 26799}, {17368, 26975}, {17370, 26982}, {17380, 26821}, {18144, 31026}, {18792, 27312}, {20172, 26997}, {24437, 25277}, {25528, 29827}, {27142, 27153}, {27152, 27157}, {27158, 27188}, {27162, 30940}, {27191, 27192}

X(27146) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c + 6 a^2 b c + 2 a b^2 c + b^3 c - 2 a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 + a c^3 + b c^3 : :

X(27146) lies on these lines: {1, 2}, {496, 17672}, {1015, 26100}, {3295, 31020}, {4657, 27161}, {5333, 27172}, {6703, 26989}, {16744, 18600}, {17045, 27514}, {17164, 24629}, {17302, 24486}, {19740, 27142}, {24631, 25253}, {25261, 26690}, {25526, 26828}, {27009, 27302}, {27162, 27195}, {27171, 27183}

X(27147) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    2 a b - b^2 + 2 a c + 3 b c - c^2 : :

X(27147) lies on these lines: {2, 7}, {6, 29628}, {8, 17312}, {10, 17232}, {37, 4398}, {45, 7321}, {69, 16815}, {75, 3943}, {76, 29982}, {86, 4273}, {141, 4751}, {192, 4098}, {239, 4648}, {319, 17313}, {320, 17259}, {344, 17116}, {594, 17241}, {966, 17288}, {1086, 4687}, {1125, 17383}, {1213, 17227}, {1266, 4704}, {1278, 29599}, {1449, 29590}, {1654, 16832}, {1738, 3616}, {2321, 4772}, {2345, 17266}, {2999, 26109}, {3008, 17379}, {3596, 30044}, {3617, 4684}, {3618, 29607}, {3619, 29610}, {3622, 3755}, {3624, 3821}, {3661, 3739}, {3663, 27268}, {3664, 17349}, {3686, 17375}, {3731, 4440}, {3758, 17337}, {3759, 17392}, {3763, 28653}, {3778, 17063}, {3834, 5224}, {3875, 29569}, {3879, 16816}, {3912, 4058}, {3925, 29843}, {3945, 17121}, {3946, 29570}, {3950, 4740}, {3963, 30090}, {4000, 16826}, {4029, 4788}, {4270, 17020}, {4361, 17317}, {4363, 17263}, {4384, 17300}, {4389, 4698}, {4395, 17393}, {4399, 17386}, {4402, 29585}, {4416, 31211}, {4430, 22312}, {4431, 29600}, {4472, 17371}, {4657, 27191}, {4664, 7263}, {4665, 17240}, {4670, 17352}, {4675, 17277}, {4688, 17233}, {4798, 25357}, {4851, 29617}, {4859, 16831}, {4869, 17287}, {4888, 20072}, {4967, 17230}, {5308, 17319}, {5564, 17311}, {6707, 17400}, {7227, 17342}, {7232, 17256}, {7238, 17329}, {15668, 16706}, {16484, 24693}, {16777, 29622}, {16911, 24549}, {16917, 25500}, {16994, 24586}, {17030, 27145}, {17049, 25279}, {17067, 29595}, {17073, 21940}, {17117, 17316}, {17118, 17264}, {17119, 17315}, {17202, 25508}, {17238, 21255}, {17262, 31139}, {17265, 17289}, {17268, 29627}, {17275, 17297}, {17280, 25590}, {17283, 17303}, {17284, 28604}, {17290, 17322}, {17295, 28634}, {17299, 29618}, {17314, 29575}, {17321, 29578}, {17330, 17361}, {17334, 31285}, {17335, 17365}, {17341, 17369}, {17346, 17376}, {17348, 17378}, {17356, 17381}, {17362, 17387}, {17366, 17394}, {17370, 17398}, {17380, 28639}, {17385, 31243}, {17743, 32015}, {17889, 25501}, {20913, 20923}, {24058, 27586}, {24077, 27565}, {24325, 33165}, {24661, 27846}, {24789, 29841}, {27032, 27107}, {27143, 27148}, {27166, 27192}, {27641, 31198}, {29583, 32087}, {29603, 31312}

X(27148) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b + a b^3 + a^3 c + 6 a^2 b c + 4 a b^2 c + b^3 c + 4 a b c^2 + a c^3 + b c^3 : :

X(27148) lies on these lines: {1, 2}, {36, 16931}, {4657, 27162}, {5284, 16061}, {5333, 27185}, {16604, 16705}, {17169, 31004}, {17322, 27019}, {17398, 26986}, {24739, 25498}, {25263, 26234}, {26035, 30963}, {26100, 31997}, {27143, 27147}, {27172, 27190}, {27178, 27186}

X(27149) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^5 b - a^4 b^2 - a^2 b^4 + a b^5 + a^5 c + 2 a^4 b c - 2 a^3 b^2 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(27149) lies on these lines: {2, 11}, {17030, 27144}, {21912, 26536}, {26558, 26804}

X(27150) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^7 b^2 - a^6 b^3 - a^3 b^6 + a^2 b^7 - 2 a^5 b^3 c + 2 a b^7 c + a^7 c^2 - a^4 b^3 c^2 - a^3 b^4 c^2 + b^7 c^2 - a^6 c^3 - 2 a^5 b c^3 - a^4 b^2 c^3 - a^2 b^4 c^3 - 2 a b^5 c^3 - b^6 c^3 - a^3 b^2 c^4 - a^2 b^3 c^4 - 2 a b^3 c^5 - a^3 c^6 - b^3 c^6 + a^2 c^7 + 2 a b c^7 + b^2 c^7 : :

X(27150) lies on these lines: {2, 3}

X(27151) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^7 b^2 - a^6 b^3 - a^3 b^6 + a^2 b^7 - 2 a^5 b^3 c + 2 a b^7 c + a^7 c^2 + b^7 c^2 - a^6 c^3 - 2 a^5 b c^3 + 4 a^3 b^3 c^3 - 2 a b^5 c^3 - b^6 c^3 - 2 a b^3 c^5 - a^3 c^6 - b^3 c^6 + a^2 c^7 + 2 a b c^7 + b^2 c^7 : :

X(27151) lies on these lines: {2, 3}

X(27152) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a b^4 c + a^4 c^2 + b^4 c^2 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27152) lies on these lines: {2, 31}, {2140, 17176}, {5372, 27314}, {8267, 21415}, {16704, 27313}, {17030, 27163}, {18067, 31078}, {19717, 26965}, {20255, 20965}, {27145, 27157}, {27158, 27190}

X(27153) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^5 b^2 + a^2 b^5 + 2 a b^5 c + a^5 c^2 + b^5 c^2 + a^2 c^5 + 2 a b c^5 + b^2 c^5 : :

X(27153) lies on these lines: {2, 32}, {27142, 27145}, {27162, 27179}

X(27154) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^2 + a^2 b^3 + 4 a^2 b^2 c + a^3 c^2 + 4 a^2 b c^2 + 6 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(27154) lies on these lines: {2, 37}, {142, 16738}, {5224, 27106}, {7321, 26817}, {10436, 26975}, {16815, 25538}, {16819, 17291}, {16829, 17312}, {17030, 27145}, {17259, 27036}, {17260, 26976}, {17261, 27037}, {17337, 27078}, {17398, 26982}, {19853, 26150}, {25534, 31248}, {26110, 29590}, {26821, 28639}, {27095, 29576}, {27155, 27156}, {27164, 27191}, {27173, 27180}

X(27155) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^3 + a^3 b^4 + 2 a^3 b^3 c + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + a^4 c^3 + 2 a^3 b c^3 + 2 a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 + a^3 c^4 + b^3 c^4 : :

X(27155) lies on these lines: {2, 39}, {141, 26801}, {2275, 18143}, {17169, 26963}, {17758, 26959}, {26971, 27097}, {27142, 27145}, {27154, 27156}, {27179, 27189}

X(27156) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b + 4 a^2 b^2 + a b^3 + a^3 c + 6 a^2 b c + 8 a b^2 c + b^3 c + 4 a^2 c^2 + 8 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 : :

X(27156) lies on these lines: {1, 2}, {3739, 16705}, {3876, 31322}, {5251, 16930}, {17303, 27109}, {27154, 27155}, {27164, 27169}, {27172, 27181}

X(27157) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    2 a^3 b^3 + a^3 b^2 c + 3 a^2 b^3 c + a^3 b c^2 + 3 a b^3 c^2 + 2 a^3 c^3 + 3 a^2 b c^3 + 3 a b^2 c^3 + 2 b^3 c^3 : :

X(27157) lies on these lines: {1, 2}, {4430, 27298}, {16748, 21264}, {27145, 27152}, {27163, 27190}, {27351, 28605}

X(27158) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^3 + a^2 b^3 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(27158) lies on these lines: {1, 2}, {36, 16955}, {354, 27285}, {672, 26107}, {2275, 18152}, {2350, 24514}, {6384, 30955}, {21330, 30004}, {24512, 25505}, {27145, 27188}, {27152, 27190}

X(27159) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^2 + a^2 b^3 - 4 a^2 b^2 c + 4 a b^3 c + a^3 c^2 - 4 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 4 a b c^3 + b^2 c^3 : :

X(27159) lies on these lines: {2, 44}, {142, 26971}, {4648, 27311}, {4698, 26857}, {17030, 27145}, {17232, 20549}, {17234, 27102}, {17244, 27107}, {17245, 27017}, {17291, 26986}, {20295, 27167}, {27166, 27191}, {27272, 27342}

X(27160) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^2 + a^2 b^3 + 8 a^2 b^2 c - 2 a b^3 c + a^3 c^2 + 8 a^2 b c^2 + 12 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(27160) lies on these lines: {2, 45}, {4751, 27095}, {17030, 27145}

X(27161) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c + 4 a^3 b c + 2 a^2 b^2 c + b^4 c - a^3 c^2 + 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(27161) lies on these lines: {2, 6}, {579, 17183}, {1108, 24993}, {1475, 21246}, {2260, 20245}, {3672, 27334}, {4000, 26964}, {4657, 27146}, {4747, 26125}, {4851, 27096}, {5750, 28797}, {7289, 26229}, {8732, 24609}, {10200, 27519}, {10586, 27520}, {17023, 17077}, {27143, 27162}, {27177, 27189}

X(27162) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b + 2 a^2 b^2 + a b^3 + a^3 c + b^3 c + 2 a^2 c^2 + a c^3 + b c^3 : :

X(27162) lies on these lines: {1, 24170}, {2, 39}, {86, 404}, {99, 11319}, {304, 4850}, {325, 4202}, {348, 5435}, {350, 26094}, {386, 30941}, {536, 24668}, {574, 25497}, {941, 26106}, {982, 17141}, {995, 17152}, {1125, 25599}, {1193, 17137}, {1509, 19717}, {1909, 26030}, {1921, 27311}, {1975, 5192}, {2275, 26965}, {2276, 27097}, {2295, 25350}, {2548, 16910}, {3216, 16887}, {3616, 8299}, {3672, 26093}, {3752, 20911}, {4357, 27627}, {4398, 5550}, {4657, 27148}, {5701, 10030}, {6337, 17526}, {7758, 17007}, {7774, 26085}, {7777, 16906}, {7782, 17539}, {7791, 26099}, {7891, 16905}, {7906, 16991}, {7907, 17003}, {11329, 19684}, {11337, 19769}, {16549, 30106}, {16720, 17489}, {17205, 20108}, {17206, 32911}, {17302, 27107}, {17382, 24739}, {17756, 27248}, {19767, 30962}, {19864, 20888}, {24190, 30112}, {26115, 31997}, {27142, 27189}, {27143, 27161}, {27145, 30940}, {27146, 27195}, {27153, 27179}

X(27163) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a^2 b^2 + a b^3 + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 + a c^3 + b c^3) : :

X(27163) lies on these lines: {2, 6}, {75, 18601}, {314, 17147}, {321, 16696}, {1444, 17587}, {3286, 24552}, {3662, 17173}, {3736, 17135}, {3741, 17187}, {3995, 32026}, {4359, 16700}, {4658, 26115}, {10458, 29824}, {10471, 28605}, {16050, 31039}, {16726, 31993}, {16736, 24589}, {16753, 19804}, {16887, 16891}, {17030, 27152}, {17174, 27184}, {17182, 26580}, {18169, 30942}, {18192, 29827}, {18206, 26223}, {18600, 19789}, {18792, 31330}, {26801, 33150}, {27157, 27190}, {27170, 27174}, {28606, 30939}, {30599, 30710}

X(27164) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a b^2 + a b c + b^2 c + a c^2 + b c^2) : :

X(27164) lies on these lines: {2, 6}, {9, 10455}, {10, 3736}, {21, 5263}, {37, 314}, {58, 19858}, {142, 16887}, {238, 19863}, {261, 1333}, {274, 1107}, {286, 1841}, {334, 28653}, {958, 1010}, {1001, 4267}, {1014, 17077}, {1125, 4281}, {1220, 14005}, {1268, 27102}, {1444, 26643}, {1698, 18792}, {1740, 18169}, {1918, 32917}, {2274, 31339}, {2309, 30970}, {3616, 5331}, {3666, 20174}, {3863, 32010}, {4000, 16705}, {4269, 4357}, {4361, 33296}, {4483, 4653}, {4657, 17030}, {4687, 30939}, {4751, 16709}, {4833, 27527}, {4851, 27255}, {4852, 16829}, {5132, 19270}, {5247, 16828}, {5257, 17197}, {5283, 10471}, {5296, 17183}, {5327, 27509}, {10436, 18206}, {10458, 31330}, {15320, 24723}, {16589, 25660}, {16724, 17382}, {16726, 31238}, {17045, 26801}, {17139, 17257}, {17175, 18164}, {17185, 25515}, {17202, 17248}, {17210, 17306}, {17237, 25538}, {17239, 27020}, {17285, 27032}, {17326, 25534}, {17369, 26082}, {18192, 25528}, {18196, 21191}, {21264, 31008}, {24437, 25124}, {25498, 26959}, {25512, 28619}, {27037, 31333}, {27154, 27191}, {27156, 27169}, { 27170, 27172}, {27176, 27187}, {28639, 31996}

X(27164) = complement of X(26110)

X(27165) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b-6 a^2 b^2+a b^3+a^3 c+6 a^2 b c-2 a b^2 c+b^3 c-6 a^2 c^2-2 a b c^2-6 b^2 c^2+a c^3+b c^3 : :

X(27165) lies on these lines: {1, 2}

X(27166) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    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 + b^3 c^2 + a^2 c^3 + b^2 c^3 : :

X(27166) lies on these lines: {1, 2}, {37, 26963}, {39, 3995}, {56, 11320}, {86, 26971}, {142, 27011}, {190, 26975}, {321, 16604}, {330, 31060}, {335, 26986}, {1015, 3948}, {1019, 17174}, {1100, 26772}, {1269, 16710}, {1909, 31026}, {2260, 17350}, {2275, 31036}, {3210, 26747}, {3879, 26756}, {3950, 26797}, {3952, 20456}, {4357, 17178}, {4360, 27102}, {4366, 19308}, {4657, 27145}, {4670, 26976}, {4755, 27037}, {4851, 27095}, {12263, 17140}, {16685, 29453}, {16736, 19821}, {16738, 17322}, {17045, 26979}, {17120, 26799}, {17147, 24598}, {17148, 18147}, {17236, 26143}, {17238, 25535}, {17242, 27136}, {17297, 25534}, {17301, 27107}, {17302, 27017}, {17319, 26764}, {17324, 26857}, {17379, 26107}, {17380, 27311}, {17381, 27261}, {17394, 25505}, {19717, 27262}, {20349, 26138}, {20363, 20868}, {20530, 31061}, {24199, 26850}, {27147, 27192}, {27159, 27191}, {27318, 28605}, {28654, 29974}

X(27167) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (b - c) (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(27167) lies on these lines: {2, 661}, {850, 17066}, {4077, 17077}, {4160, 19874}, {4978, 14838}, {5278, 18199}, {7199, 24948}, {16751, 18154}, {20295, 27159}, {21191, 27673}, {21259, 25301}, {26114, 27013}, {26985, 27345}

X(27168) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (b - c) (-a^2 + a b + a c + b c) (-a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(27168) lies on these lines: {2, 667}, {194, 23807}, {1019, 27169}, {4063, 16819}, {8630, 25299}, {8632, 27293}, {9010, 20139}, {9491, 23301}, {21191, 23572}

X(27169) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^2 + a^2 b^4 - a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + a^4 c^2 - a^3 b c^2 - 4 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27169) lies on these lines: {2, 31}, {86, 26965}, {142, 27019}, {940, 27313}, {1019, 27168}, {1740, 29966}, {2309, 29968}, {3701, 20167}, {5263, 27097}, {9780, 20139}, {16738, 16819}, {17030, 27145}, {17379, 27299}, {18792, 30109}, {20133, 26115}, {20140, 26030}, {27156, 27164}

X(27170) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 2 a^2 b^2 c - 4 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 - 4 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(27170) lies on these lines: {2, 7}, {77, 26621}, {141, 27096}, {160, 4189}, {241, 24547}, {1229, 25237}, {3522, 10882}, {3663, 28797}, {3739, 26563}, {4000, 16696}, {4643, 27108}, {4657, 27146}, {7146, 21273}, {7613, 19843}, {14953, 16738}, {17030, 27171}, {17052, 26781}, {17258, 28748}, {18635, 27050}, {21255, 28742}, {24471, 24633}, {25601, 29579}, {26626, 26818}, {27163, 27174}, {27164, 27172}, {27283, 33298}

X(27171) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^6 b - 3 a^5 b^2 + 2 a^4 b^3 + 2 a^3 b^4 - 3 a^2 b^5 + a b^6 + a^6 c + 4 a^5 b c - a^4 b^2 c - a^2 b^4 c - 4 a b^5 c + b^6 c - 3 a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - a b^4 c^2 - 3 b^5 c^2 + 2 a^4 c^3 + 4 a^2 b^2 c^3 + 8 a b^3 c^3 + 2 b^4 c^3 + 2 a^3 c^4 - a^2 b c^4 - a b^2 c^4 + 2 b^3 c^4 - 3 a^2 c^5 - 4 a b c^5 - 3 b^2 c^5 + a c^6 + b c^6 : :

X(27171) lies on these lines: {2, 3}, {3218, 27000}, {17030, 27170}, {20367, 27304}, {27146, 27183}

X(27172) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a^3 b^2 - 2 a^2 b^3 + a b^4 - 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 - 2 a^2 c^3 - a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(27172) lies on these lines: {2, 3}, {86, 26964}, {3218, 16819}, {4267, 24596}, {5333, 27146}, {16704, 27304}, {17030, 27152}, {17174, 27183}, {27148, 27190}, {27156, 27181}, {27164, 27170}

X(27173) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^7 b^2 - a^6 b^3 - a^3 b^6 + a^2 b^7 - 2 a^5 b^3 c + 2 a b^7 c + a^7 c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 + b^7 c^2 - a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + 8 a^3 b^3 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 - b^6 c^3 + a^3 b^2 c^4 + a^2 b^3 c^4 - 2 a b^3 c^5 - a^3 c^6 - b^3 c^6 + a^2 c^7 + 2 a b c^7 + b^2 c^7 : :

X(27173) lies on these lines: {2, 3}, {27154, 27180}

X(27174) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a (a + b) (a + c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c - b^2 c - a c^2 - b c^2 - c^3) : :

X(27174) lies on these lines: {2, 3}, {35, 306}, {58, 4652}, {63, 284}, {81, 593}, {333, 2164}, {993, 5271}, {1014, 8025}, {1030, 1211}, {1172, 1748}, {1214, 1950}, {1230, 26243}, {1396, 17080}, {1621, 2352}, {1778, 4261}, {1790, 17185}, {1792, 33077}, {1801, 2328}, {1993, 23602}, {2194, 4640}, {2206, 4414}, {2287, 3219}, {2303, 28606}, {2360, 5250}, {2975, 3187}, {3305, 4877}, {3687, 4276}, {3871, 20017}, {4273, 4641}, {4278, 17023}, {4288, 24611}, {4384, 5358}, {4653, 5287}, {4657, 5333}, {5303, 29833}, {5905, 8822}, {11683, 25254}, {12572, 27412}, {12610, 17167}, {15817, 27540}, {16704, 20043}, {16948, 17012}, {21376, 25080}, {27163, 27170}, {27398, 31018}

X(27175) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a^6 b^2 - a^5 b^3 - a^2 b^6 + a b^7 - a^5 b^2 c - a^4 b^3 c + a b^6 c + b^7 c + a^6 c^2 - a^5 b c^2 - 2 a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 - a^4 b c^3 + 4 a^3 b^2 c^3 + 4 a^2 b^3 c^3 - a b^4 c^3 - b^5 c^3 + 3 a^2 b^2 c^4 - a b^3 c^4 - a b^2 c^5 - b^3 c^5 - a^2 c^6 + a b c^6 + a c^7 + b c^7) : :

X(27175) lies on these lines: {2, 3}, {16887, 18648}, {17030, 27180}, {27184, 27185}

X(27176) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a^6 b^2 - a^5 b^3 - a^2 b^6 + a b^7 + a^6 b c + a^5 b^2 c - a^4 b^3 c - 2 a^3 b^4 c - a^2 b^5 c + a b^6 c + b^7 c + a^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 + 2 a^2 b^3 c^3 - a b^4 c^3 - b^5 c^3 - 2 a^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 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(27176) lies on these lines: {2, 3}, {7054, 25508}, {27164, 27187}

X(27177) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^6 b - 5 a^5 b^2 + 4 a^4 b^3 + 4 a^3 b^4 - 5 a^2 b^5 + a b^6 + a^6 c + 4 a^5 b c - a^4 b^2 c + 4 a^3 b^3 c - a^2 b^4 c - 8 a b^5 c + b^6 c - 5 a^5 c^2 - a^4 b c^2 + 8 a^3 b^2 c^2 + 8 a^2 b^3 c^2 - a b^4 c^2 - 5 b^5 c^2 + 4 a^4 c^3 + 4 a^3 b c^3 + 8 a^2 b^2 c^3 + 16 a b^3 c^3 + 4 b^4 c^3 + 4 a^3 c^4 - a^2 b c^4 - a b^2 c^4 + 4 b^3 c^4 - 5 a^2 c^5 - 8 a b c^5 - 5 b^2 c^5 + a c^6 + b c^6 : :

X(27177) lies on these lines: {2, 3}, {27161, 27189}

X(27178) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c + 4 a^5 b c - 3 a^4 b^2 c - 8 a^3 b^3 c - 3 a^2 b^4 c + b^6 c - a^5 c^2 - 3 a^4 b c^2 - 12 a^3 b^2 c^2 - 12 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 8 a^3 b c^3 - 12 a^2 b^2 c^3 - 4 a b^3 c^3 - 3 a^2 b c^4 - 3 a b^2 c^4 - a^2 c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(27178) lies on these lines: {2, 3}, {4657, 27146}, {27148, 27186}

X(27179) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^5 b^2 + a^2 b^5 + a^4 b^2 c + 2 a^3 b^3 c + a^2 b^4 c + 2 a b^5 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 + b^5 c^2 + 2 a^3 b c^3 + a^2 b c^4 + a b^2 c^4 + a^2 c^5 + 2 a b c^5 + b^2 c^5 : :

X(27179) lies on these lines: {2, 3}, {27153, 27162}, {27155, 27189}

X(27180) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b - b^5 + a^4 c - a^3 b c - 2 a^2 b^2 c - a b^3 c + b^4 c - 2 a^2 b c^2 - a b c^3 + b c^4 - c^5 : :

X(27180) lies on these lines: {2, 19}, {21, 16114}, {37, 31053}, {344, 27131}, {1001, 20846}, {3219, 28420}, {3662, 17183}, {4657, 5333}, {16580, 17321}, {17030, 27175}, {17073, 21511}, {18639, 26156}, {20254, 23635}, {20336, 33077}, {26130, 26639}, {27143, 27147}, {27154, 27173}, {28022, 33146}

X(27181) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 a b^5 c + a^5 c^2 - 2 a^3 b^2 c^2 + b^5 c^2 - a^4 c^3 - 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + a^2 c^5 + 2 a b c^5 + b^2 c^5 : :

X(27181) lies on these lines: {2, 36}, {1019, 27168}, {16819, 20367}, {17030, 27142}, {19743, 26964}, {24296, 28803}, {27156, 27172}

X(27182) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^3 b^2 c + a^4 c^2 + 2 a^3 b c^2 + 6 a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 + 2 a b^2 c^3 + a^2 c^4 + b^2 c^4 : :

X(27182) lies on these lines: {2, 38}, {2140, 16891}, {16823, 27030}, {17030, 27152}

X(27183) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    -3 a^2 b^2 + 2 a b^3 + b^4 + a^2 b c - 2 a b^2 c + b^3 c - 3 a^2 c^2 - 2 a b c^2 - 4 b^2 c^2 + 2 a c^3 + b c^3 + c^4 : :

X(27183) lies on these lines: {2, 40}, {5, 26531}, {379, 17397}, {673, 11375}, {908, 27304}, {1125, 4209}, {2140, 17181}, {5141, 26526}, {5154, 26532}, {5886, 17682}, {9779, 11201}, {9955, 17671}, {11349, 29612}, {12053, 27253}, {17030, 27142}, {17174, 27172}, {17691, 24541}, {17747, 31269}, {24580, 29609}, {26964, 31019}, {27143, 27147}, {27146, 27171}

X(27184) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a b^2 + b^3 + a b c + a c^2 + c^3 : :

X(27184) lies on these lines: {1, 1330}, {2, 7}, {6, 19786}, {8, 3914}, {10, 17889}, {31, 4683}, {37, 18134}, {38, 3705}, {42, 32776}, {43, 3821}, {55, 24723}, {69, 1999}, {72, 16062}, {75, 1211}, {76, 321}, {78, 4201}, {81, 17202}, {85, 6354}, {92, 257}, {100, 32950}, {141, 312}, {171, 4655}, {190, 32777}, {192, 306}, {210, 4429}, {222, 26625}, {223, 17086}, {238, 4703}, {239, 5739}, {244, 25960}, {320, 940}, {333, 3772}, {345, 4419}, {518, 32773}, {537, 33169}, {612, 4645}, {726, 32778}, {740, 33084}, {748, 33123}, {750, 33067}, {752, 17716}, {756, 25957}, {846, 3771}, {899, 33125}, {902, 29848}, {968, 9791}, {982, 3846}, {984, 2887}, {1001, 33124}, {1086, 5743}, {1150, 33133}, {1215, 32784}, {1376, 33068}, {1426, 17555}, {1446, 26607}, {1458, 24550}, {1621, 33122}, {1654, 5271}, {1738, 4104}, {1757, 25453}, {1836, 5263}, {1931, 5333}, {2308, 29636}, {2895, 3187}, {2975, 25906}, {2999, 17304}, {3006, 7226}, {3061, 23636}, {3120, 17794}, {3175, 17233}, {3210, 3663}, {3242, 4514}, {3416, 32926}, {3419, 17677}, {3487, 13725}, {3616, 13736}, {3620, 34255}, {3666, 4389}, {3673, 21405}, {3676, 26596}, {3677, 5211}, {3681, 4972}, {3685, 33171}, {3688, 25308}, {3720, 33069}, {3721, 3981}, {3741, 3944}, {3752, 5233}, {3755, 20012}, {3757, 33144}, {3758, 19812}, {3769, 17602}, {3790, 15523}, {3794, 26892}, {3844, 3967}, {3868, 5051}, {3873, 29843}, {3876, 4202}, {3891, 33075}, {3912, 4656}, {3920, 6327}, {3923, 32783}, {3936, 17247}, {3938, 32947}, {3940, 11359}, {3952, 29679}, {3961, 4660}, {3966, 32922}, {3971, 29674}, {3980, 32857}, {3989, 29643}, {3995, 17242}, {4000, 14555}, {4001, 4741}, {4011, 29637}, {4052, 27835}, {4054, 17238}, {4077, 26545}, {4101, 20018}, {4205, 6147}, {4320, 19861}, {4358, 33172}, {4359, 33146}, {4361, 4886}, {4362, 33082}, {4363, 19808}, {4364, 17056}, {4383, 16706}, {4384, 23681}, {4414, 29846}, {4418, 33098}, {4430, 29835}, {4521, 26571}, {4641, 17347}, {4651, 33131}, {4850, 5741}, {4892, 33111}, {4981, 33108}, {5057, 24552}, {5220, 33118}, {5224, 31993}, {5256, 17302}, {5269, 20101}, {5278, 17331}, {5287, 17300}, {5311, 32949}, {5712, 17321}, {5718, 17249}, {5737, 17253}, {6376, 30631}, {6679, 7262}, {6682, 17717}, {6703, 17365}, {7018, 17149}, {7081, 26034}, {8580, 26073}, {8616, 29656}, {9284, 20284}, {9534, 23537}, {9535, 12610}, {10453, 24210}, {10468, 10478}, {11263, 19858}, {11374, 19270}, {11679, 17272}, {11814, 31242}, {12572, 17697}, {12609, 19853}, {13411, 19278}, {13567, 26531}, {14829, 17273}, {15485, 29672}, {16468, 29654}, {16569, 24169}, {16608, 25977}, {16738, 17167}, {16780, 26626}, {16817, 24159}, {16825, 33147}, {16887, 17177}, {17011, 17396}, {17017, 32843}, {17019, 17391}, {17022, 17298}, {17030, 27142}, {17073, 25908}, {17116, 19822}, {17117, 19789}, {17118, 19797}, {17119, 19820}, {17121, 19823}, {17127, 26230}, {17135, 33134}, {17147, 31037}, {17165, 29667}, {17174, 27163}, {17192, 17284}, {17227, 18743}, {17229, 22034}, {17244, 18139}, {17255, 32851}, {17256, 19732}, {17258, 30811}, {17261, 17776}, {17276, 30832}, {17277, 24789}, {17320, 20182}, {17322, 19701}, {17339, 33157}, {17349, 26723}, {17367, 32774}, {17397, 19684}, {17599, 33071}, {17719, 32916}, {17763, 33080}, {17770, 29645}, {18056, 30660}, {18541, 19276}, {18750, 26543}, {20173, 26590}, {20256, 30546}, {21062, 27129}, {21240, 30830}, {21616, 26123}, {21813, 26242}, {23806, 26049}, {24177, 24620}, {24190, 24603}, {24214, 24621}, {24248, 32932}, {24320, 25494}, {24325, 33103}, {24697, 33130}, {24703, 32942}, {24725, 32772}, {24943, 32930}, {25466, 31359}, {25496, 33096}, {25935, 27288}, {26061, 32938}, {26227, 33083}, {26579, 26942}, {26724, 29628}, {27175, 27185}, {27476, 27481}, {27479, 27495}, {28595, 33165}, {29617, 31143}, {29631, 32912}, {29635, 32913}, {29649, 33085}, {29652, 33106}, {30473, 30713}, {30831, 33113}, {30965, 31008}, {31134, 33072}, {31237, 33115}, {32771, 32856}, {32779, 32933}, {32780, 32935}, {32781, 32931}, {32852, 32928}, {32853, 33135}, {32860, 33145}, {32861, 32921}, {32864, 33128}, {32914, 33143}, {32915, 33081}, {32917, 33127}, {32920, 33076}, {32927, 33074}, {32929, 33100}, {32934, 33160}, {32936, 33156}, {32941, 33095}, {32945, 33094}

X(27184) = anticomplement of isotomic conjugate of polar conjugate of X(1891)

X(27185) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (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(27185) lies on these lines: {2, 58}, {21, 27097}, {81, 26965}, {4184, 27263}, {5333, 27148}, {11115, 27248}, {14005, 27026}, {16703, 16735}, {16704, 27299}, {16716, 20911}, {16738, 17169}, {17030, 27152}, {17187, 29960}, {18169, 29966}, {18180, 26562}, {26807, 28619}, {26959, 27190}, {26969, 30176}, {27142, 27145}, {27156, 27164}, {27175, 27184}

X(27186) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^2 b - b^3 + a^2 c + 3 a b c + b^2 c + b c^2 - c^3 : :

X(27186) lies on these lines: {2, 7}, {6, 26724}, {10, 3894}, {37, 33146}, {46, 9782}, {75, 3969}, {81, 4675}, {85, 30690}, {86, 17173}, {90, 10266}, {149, 4666}, {171, 29681}, {244, 29680}, {273, 445}, {306, 24199}, {320, 5278}, {321, 17234}, {343, 21258}, {354, 33108}, {377, 3488}, {404, 943}, {405, 18541}, {474, 25593}, {612, 33148}, {614, 33112}, {726, 29854}, {748, 33097}, {750, 29665}, {756, 33103}, {940, 33129}, {942, 4197}, {946, 3522}, {968, 33102}, {982, 29664}, {1001, 20292}, {1054, 29678}, {1071, 6991}, {1086, 28606}, {1125, 1770}, {1215, 25961}, {1230, 20923}, {1621, 5880}, {1738, 17018}, {1836, 5284}, {1961, 33143}, {1962, 33149}, {2140, 14953}, {2475, 3586}, {2476, 3824}, {2550, 3957}, {2895, 4384}, {3120, 26102}, {3187, 17300}, {3434, 29817}, {3475, 3935}, {3476, 28629}, {3550, 29689}, {3578, 17361}, {3612, 3616}, {3617, 21620}, {3624, 3648}, {3661, 6539}, {3664, 26723}, {3671, 24564}, {3681, 3826}, {3720, 17889}, {3739, 32782}, {3742, 11680}, {3782, 17245}, {3811, 26060}, {3812, 25005}, {3816, 10129}, {3833, 7951}, {3834, 31993}, {3836, 29679}, {3841, 18398}, {3848, 17605}, {3868, 8728}, {3873, 3925}, {3876, 6147}, {3889, 31419}, {3912, 28605}, {3914, 29814}, {3923, 29851}, {3936, 19804}, {3944, 30950}, {3947, 25011}, {3980, 29632}, {3995, 17244}, {4000, 17011}, {4038, 33128}, {4188, 12436}, {4208, 12649}, {4292, 16865}, {4359, 18134}, {4363, 33157}, {4393, 17050}, {4417, 24589}, {4418, 29642}, {4423, 5057}, {4430, 5542}, {4648, 17019}, {4657, 5333}, {4850, 17056}, {4859, 5256}, {4883, 21949}, {4892, 25960}, {4980, 17233}, {5068, 6260}, {5133, 25365}, {5154, 9843}, {5248, 15228}, {5253, 28628}, {5260, 10404}, {5268, 33153}, {5271, 17298}, {5272, 33107}, {5287, 23681}, {5297, 33144}, {5311, 33147}, {5436, 15680}, {5444, 26725}, {5550, 12047}, {5712, 17012}, {5722, 6175}, {5768, 6993}, {5770, 6877}, {5805, 7411}, {5886, 6909}, {6327, 16823}, {6384, 27446}, {6690, 9352}, {6701, 7741}, {6826, 18444}, {6828, 9940}, {6829, 10202}, {6839, 18443}, {6861, 26877}, {6894, 10884}, {6895, 8726}, {6990, 13369}, {7226, 24231}, {7232, 19732}, {7292, 26098}, {7321, 32933}, {7560, 20291}, {8226, 11220}, {9335, 24239}, {9345, 33135}, {9347, 17061}, {9780, 13407}, {10167, 10883}, {10389, 20095}, {10431, 21151}, {10528, 11024}, {11374, 17531}, {12572, 17570}, {13411, 17572}, {16484, 33094}, {16706, 19684}, {16708, 18045}, {16753, 26746}, {16825, 32949}, {16891, 17175}, {17013, 17067}, {17014, 24181}, {17030, 27152}, {17063, 33105}, {17117, 20017}, {17122, 33127}, {17123, 24725}, {17124, 17719}, {17125, 33096}, {17140, 29641}, {17155, 29653}, {17277, 32859}, {17278, 32911}, {17290, 19701}, {17305, 25507}, {17314, 19819}, {17315, 19820}, {17316, 19789}, {17317, 19796}, {17321, 27127}, {17364, 19742}, {17367, 19717}, {17450, 33141}, {17521, 25526}, {17591, 29682}, {17596, 29661}, {17761, 30562}, {19877, 21077}, {20269, 25946}, {20271, 20859}, {20917, 28654}, {20966, 24046}, {21020, 33087}, {21026, 33169}, {21195, 27486}, {21566, 31540}, {21567, 31541}, {21590, 26235}, {23806, 26985}, {24165, 29643}, {24325, 25957}, {24331, 32947}, {24342, 24943}, {24693, 32945}, {25385, 30957}, {25495, 25496}, {26037, 33064}, {27146, 27171}, {27148, 27178}, {29648, 33123}, {29651, 32948}, {29666, 32772}, {29820, 33104}, {29830, 32932}, {29968, 31060}, {31151, 33074}, {31178, 33162}

X(27187) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 - 2 a^2 b^2 + b^4 - a b^2 c - b^3 c - 2 a^2 c^2 - a b c^2 - b c^3 + c^4 : :

X(27187) lies on these lines: {1, 25581}, {2, 85}, {8, 32818}, {21, 17181}, {57, 24583}, {150, 3897}, {304, 33113}, {321, 3926}, {325, 5016}, {498, 30806}, {1358, 31260}, {1434, 31019}, {1565, 7483}, {1931, 5333}, {2476, 5088}, {2646, 21285}, {2975, 7179}, {3665, 4999}, {3772, 18600}, {3869, 17084}, {4056, 5267}, {4189, 4872}, {4352, 33133}, {4357, 24540}, {4657, 27146}, {5228, 26628}, {5433, 26229}, {5794, 17136}, {6910, 17170}, {7181, 25466}, {7278, 10197}, {7763, 20911}, {9310, 25353}, {9436, 24541}, {10448, 24241}, {10586, 17321}, {11375, 20347}, {16601, 28734}, {17206, 32859}, {17248, 24557}, {17257, 24553}, {17740, 32831}, {20880, 26363}, {24215, 33127}, {24627, 29614}, {27164, 27176}, {31039, 31121}

X(27188) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^4 - a^4 b^3 c - a^3 b^3 c^2 - 2 a^2 b^4 c^2 - a^4 b c^3 - a^3 b^2 c^3 + a^4 c^4 - 2 a^2 b^2 c^4 + b^4 c^4 : :

X(27188) lies on these lines: {2, 87}, {2309, 27105}, {15668, 23538}, {17030, 27192}, {27145, 27158}

X(27189) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (a + b) (a + c) (a^3 b^2 - 2 a^2 b^3 + a b^4 + a b^3 c + b^4 c + a^3 c^2 - 2 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(27189) lies on these lines: {2, 99}, {1509, 26964}, {4576, 18061}, {27142, 27162}, {27155, 27179}, {27161, 27177}, {27190, 27195}

X(27190) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^4 b^2 - 2 a^3 b^3 + a^2 b^4 - 2 a^2 b^3 c + 2 a b^4 c + a^4 c^2 + 4 a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27190) lies on these lines: {2, 11}, {19740, 27142}, {26959, 27185}, {27012, 27191}, {27148, 27172}, {27152, 27158}, {27157, 27163}, {27189, 27195}

X(27191) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^2 - a b + 2 b^2 - a c - 3 b c + 2 c^2 : :

X(27191) lies on these lines: {1, 25351}, {2, 45}, {3, 24827}, {5, 24813}, {7, 17352}, {10, 24841}, {37, 29626}, {44, 29607}, {69, 24599}, {75, 646}, {86, 142}, {140, 24833}, {141, 32025}, {192, 17265}, {238, 24692}, {239, 3834}, {314, 29756}, {319, 21255}, {320, 3008}, {333, 26724}, {335, 1268}, {344, 4398}, {524, 29590}, {528, 3616}, {536, 17266}, {537, 1698}, {590, 24819}, {599, 16816}, {615, 24818}, {631, 29243}, {668, 18150}, {726, 31252}, {812, 27195}, {894, 17356}, {900, 30795}, {918, 31640}, {1100, 32096}, {1111, 18151}, {1125, 24715}, {1222, 23675}, {1227, 25527}, {1266, 17264}, {1278, 17267}, {2224, 17682}, {2486, 30993}, {2786, 14061}, {2796, 19862}, {3090, 24828}, {3120, 24709}, {3306, 16560}, {3315, 20042}, {3525, 24817}, {3589, 26806}, {3617, 9041}, {3618, 4747}, {3619, 4437}, {3624, 4432}, {3662, 4643}, {3663, 17263}, {3699, 24988}, {3729, 17341}, {3758, 6173}, {3759, 17298}, {3763, 4699}, {3836, 17769}, {3875, 17241}, {3912, 17067}, {3946, 17317}, {4000, 4360}, {4014, 24482}, {4033, 30866}, {4357, 31211}, {4361, 17232}, {4366, 15668}, {4384, 17227}, {4393, 17313}, {4395, 6542}, {4413, 24820}, {4479, 30822}, {4499, 16482}, {4648, 17380}, {4657, 27147}, {4659, 17342}, {4665, 29587}, {4670, 29630}, {4675, 17367}, {4686, 17268}, {4687, 17304}, {4688, 17292}, {4698, 17324}, {4740, 17269}, {4751, 17306}, {4772, 17293}, {4792, 25031}, {4796, 17120}, {4852, 17312}, {4862, 17336}, {4869, 17377}, {4885, 32016}, {5043, 23958}, {5070, 24844}, {5094, 24814}, {5219, 31233}, {5222, 17378}, {5263, 24693}, {5432, 24837}, {5433, 24836}, {6547, 6631}, {6646, 17337}, {6650, 6707}, {6651, 25498}, {6653, 17045}, {6666, 17258}, {6678, 16099}, {7232, 17349}, {7238, 20072}, {7263, 17280}, {7321, 17353}, {7484, 24822}, {7808, 24815}, {7914, 24825}, {10436, 17370}, {14475, 24129}, {14829, 24789}, {15184, 24830}, {16593, 17321}, {16610, 30823}, {16672, 29599}, {16726, 24625}, {16815, 17237}, {16826, 17382}, {16831, 17399}, {16832, 17250}, {16833, 17360}, {16834, 17387}, {17116, 17357}, {17117, 17231}, {17118, 17358}, {17119, 17230}, {17121, 17376}, {17151, 17240}, {17235, 17260}, {17236, 17259}, {17244, 17301}, {17245, 17302}, {17261, 31333}, {17274, 17335}, {17276, 17338}, {17281, 29629}, {17287, 32108}, {17288, 17348}, {17289, 24199}, {17300, 17366}, {17318, 29572}, {17320, 29571}, {17323, 27268}, {17326, 31238}, {17371, 25590}, {17384, 24358}, {17395, 29569}, {17400, 17738}, {17719, 24188}, {17724, 26073}, {18044, 30090}, {18743, 23681}, {20582, 29591}, {21358, 29593}, {24177, 33116}, {24589, 30832}, {24617, 25536}, {24620, 30811}, {24627, 31205}, {24790, 33296}, {24835, 24953}, {24842, 32785}, {24843, 32786}, {24847, 26364}, {24848, 26363}, {25961, 32926}, {27012, 27190}, {27145, 27192}, {27154, 27164}, {27159, 27166}, {30598, 31312}, {30867, 31197}, {31289, 32857}, {31647, 32028}

X(27191) = isotomic conjugate of X(36954)
X(27191) = complement of X(4473)

X(27192) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c + 2 a^2 b^2 c - 2 a b^3 c + a^3 c^2 + 2 a^2 b c^2 + 6 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(27192) lies on these lines: {2, 37}, {391, 26768}, {4648, 26821}, {4859, 27017}, {10436, 26982}, {16738, 17290}, {16816, 26756}, {17030, 27188}, {17164, 27680}, {17275, 27106}, {17352, 26976}, {25277, 30982}, {25295, 31005}, {26149, 29590}, {27145, 27191}, {27147, 27166}

X(27193) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c + a^3 b c - 2 a^2 b^2 c - 2 a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(27193) lies on these lines: {2, 650}, {37, 25271}, {1019, 3835}, {3261, 25258}, {3837, 4057}, {4369, 27293}, {4379, 27527}, {4449, 25627}, {4728, 27345}, {17215, 25924}, {17217, 27673}, {19874, 21727}, {20295, 27159}, {21191, 28398}, {23301, 30795}, {23791, 25501}, {23803, 29426}, {26248, 27294}, {26983, 27138}, {28758, 30835}

X(27194) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    (b - c) (a^4 b^2 - a^3 b^3 + a^4 b c - a^3 b^2 c - 2 a^2 b^3 c + a^4 c^2 - a^3 b c^2 + 2 a b^3 c^2 - a^3 c^3 - 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3) : :

X(27194) lies on these lines: {2, 659}, {20295, 27159}, {27012, 27190}

X(27195) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 54

Barycentrics    2 a^2 b^2 - 3 a^2 b c - a b^2 c + 2 a^2 c^2 - a b c^2 + b^2 c^2 : :

X(27195) lies on these lines: {1, 4595}, {2, 668}, {39, 32026}, {83, 5253}, {86, 25532}, {106, 18047}, {244, 18061}, {274, 4602}, {291, 1125}, {537, 4687}, {812, 27191}, {1086, 24508}, {1621, 8671}, {2275, 18140}, {2787, 14061}, {2810, 3618}, {3616, 7786}, {3624, 17793}, {4389, 24497}, {5222, 31234}, {5550, 17794}, {6703, 17946}, {7200, 20568}, {7208, 18159}, {9336, 24524}, {14759, 31233}, {16723, 25498}, {16726, 25534}, {17205, 30997}, {18145, 20530}, {27146, 27162}, {27189, 27190}, {29750, 33296}, {31286, 32016}

X(27195) = isotomic conjugate of X(36957)

X(27196) =  MIDPOINT OF X(54) AND X(1141)

Barycentrics    (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (2 a^14-6 a^12 b^2+7 a^10 b^4-7 a^8 b^6+8 a^6 b^8-4 a^4 b^10-a^2 b^12+b^14-6 a^12 c^2+10 a^10 b^2 c^2-3 a^8 b^4 c^2-9 a^6 b^6 c^2+12 a^4 b^8 c^2+a^2 b^10 c^2-5 b^12 c^2+7 a^10 c^4-3 a^8 b^2 c^4+8 a^6 b^4 c^4-8 a^4 b^6 c^4+5 a^2 b^8 c^4+9 b^10 c^4-7 a^8 c^6-9 a^6 b^2 c^6-8 a^4 b^4 c^6-10 a^2 b^6 c^6-5 b^8 c^6+8 a^6 c^8+12 a^4 b^2 c^8+5 a^2 b^4 c^8-5 b^6 c^8-4 a^4 c^10+a^2 b^2 c^10+9 b^4 c^10-a^2 c^12-5 b^2 c^12+c^14) :
X(27196) = X[128]-2*X[6689]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28578.

X(27196) lies on these lines: {5,49}, {128,6689}, {137,18400}, {1154,24147}, {6592,25042}, {10610,25150}, {18370,24144}, {20424,25044}

X(27196) = midpoint of X(54) and X(1141)
X(27196) = reflection of X(128) in X(6689)

X(27197) =  MIDPOINT OF X(79) AND X(3336)

Barycentrics    a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7+6 a^5 b c+5 a^4 b^2 c-a^3 b^3 c-6 a^2 b^4 c-5 a b^5 c+b^6 c+a^5 c^2+5 a^4 b c^2+6 a^3 b^2 c^2+4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3-a^3 b c^3+4 a^2 b^2 c^3+10 a b^3 c^3-3 b^4 c^3-2 a^3 c^4-6 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5-5 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
X(27197) = X[10543]-2*X[20323]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28578.

X(27197)lies on these lines: {5,79}, {11,1354}, {12,11544}, {30,4325}, {392,11263}, {442,3828}, {517,3649}, {2475,9657}, {3647,17575}, {3654,5499}, {3813,15679}, {4309,16117}, {4317,10525}, {4338,16159}, {5221,16116}, {6175,9710}, {6701,17529}, {9711,11684}, {10543,20323}

X(27197) = midpoint of X(79) and X(3336)
X(27197) = reflection of X(10543) in X(20323)

leftri

Collineation mappings involving Gemini triangle 55: X(27198)-X(27208)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 55, as in centers X(27198)-X(27208). Then

m(X) = a^2 (b^2 - 2 a c) (c^2 - 2 a b) x + 2 a c (a^2 - 2 b c) (c^2 - 2 a b) y + 2 a b (a^2 - 2 b c) (b^2 - 2 a c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 6, 2018)

X(27198) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^3 b^3 + 4 a^4 b c - 4 a^3 b^2 c + 2 a^2 b^3 c - 4 a^3 b c^2 + a^2 b^2 c^2 + 8 a b^3 c^2 - 4 b^4 c^2 - 2 a^3 c^3 + 2 a^2 b c^3 + 8 a b^2 c^3 - 4 b^2 c^4) : :

X(27198) lies on these lines: {1, 2}

X(27199) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^6 b^3 + 2 a^4 b^5 + 4 a^7 b c - 4 a^3 b^5 c + a^5 b^2 c^2 - 9 a^3 b^4 c^2 - 2 a^2 b^5 c^2 + 8 a b^6 c^2 - 2 a^6 c^3 + 8 a^3 b^3 c^3 + 6 a^2 b^4 c^3 + 4 b^6 c^3 - 9 a^3 b^2 c^4 + 6 a^2 b^3 c^4 - 16 a b^4 c^4 - 4 b^5 c^4 + 2 a^4 c^5 - 4 a^3 b c^5 - 2 a^2 b^2 c^5 - 4 b^4 c^5 + 8 a b^2 c^6 + 4 b^3 c^6) : :

X(27199) lies on these lines:

X(27200) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-4 a^6 b^3 + 4 a^4 b^5 + 12 a^7 b c - 8 a^5 b^3 c + 4 a^4 b^4 c - 4 a^3 b^5 c - 8 a^2 b^6 c + 4 b^8 c - 15 a^5 b^2 c^2 + 16 a^3 b^4 c^2 - 4 a^2 b^5 c^2 - a b^6 c^2 - 4 a^6 c^3 - 8 a^5 b c^3 + 8 a^3 b^3 c^3 + 4 a^2 b^4 c^3 + 4 a^4 b c^4 + 16 a^3 b^2 c^4 + 4 a^2 b^3 c^4 + 2 a b^4 c^4 - 4 b^5 c^4 + 4 a^4 c^5 - 4 a^3 b c^5 - 4 a^2 b^2 c^5 - 4 b^4 c^5 - 8 a^2 b c^6 - a b^2 c^6 + 4 b c^8) : :

X(27200) lies on these lines:

X(27201) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^6 b^3 + 2 a^4 b^5 + 8 a^7 b c - 8 a^5 b^3 c + 4 a^4 b^4 c - 8 a^2 b^6 c + 4 b^8 c - 16 a^5 b^2 c^2 + 25 a^3 b^4 c^2 - 2 a^2 b^5 c^2 - 9 a b^6 c^2 - 2 a^6 c^3 - 8 a^5 b c^3 - 2 a^2 b^4 c^3 - 4 b^6 c^3 + 4 a^4 b c^4 + 25 a^3 b^2 c^4 - 2 a^2 b^3 c^4 + 18 a b^4 c^4 + 2 a^4 c^5 - 2 a^2 b^2 c^5 - 8 a^2 b c^6 - 9 a b^2 c^6 - 4 b^3 c^6 + 4 b c^8) : :

X(27201) lies on these lines:

X(27202) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^4 b^3 + 4 a^5 b c - 4 a^3 b^3 c + a^3 b^2 c^2 + 2 a^2 b^3 c^2 + 8 a b^4 c^2 - 2 a^4 c^3 - 4 a^3 b c^3 + 2 a^2 b^2 c^3 - 4 b^4 c^3 + 8 a b^2 c^4 - 4 b^3 c^4) : :

X(27202) lies on these lines: {2, 6}, {16777, 27205}

X(27203) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^8 b^3 + 2 a^4 b^7 + 4 a^9 b c + 4 a^7 b^3 c - 4 a^5 b^5 c - 4 a^3 b^7 c + a^7 b^2 c^2 - 2 a^6 b^3 c^2 - 8 a^5 b^4 c^2 - a^3 b^6 c^2 - 2 a^2 b^7 c^2 + 8 a b^8 c^2 - 2 a^8 c^3 + 4 a^7 b c^3 - 2 a^6 b^2 c^3 + 6 a^4 b^4 c^3 + 4 a^3 b^5 c^3 + 2 a^2 b^6 c^3 + 4 b^8 c^3 - 8 a^5 b^2 c^4 + 6 a^4 b^3 c^4 - 8 a b^6 c^4 - 4 b^7 c^4 - 4 a^5 b c^5 + 4 a^3 b^3 c^5 - a^3 b^2 c^6 + 2 a^2 b^3 c^6 - 8 a b^4 c^6 + 2 a^4 c^7 - 4 a^3 b c^7 - 2 a^2 b^2 c^7 - 4 b^4 c^7 + 8 a b^2 c^8 + 4 b^3 c^8) : :

X(27203) lies on these lines:

X(27204) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^8 b^3 + 2 a^4 b^7 + 4 a^9 b c + 4 a^7 b^3 c - 4 a^5 b^5 c - 4 a^3 b^7 c + a^7 b^2 c^2 - 2 a^6 b^3 c^2 - 8 a^5 b^4 c^2 - a^3 b^6 c^2 - 2 a^2 b^7 c^2 + 8 a b^8 c^2 - 2 a^8 c^3 + 4 a^7 b c^3 - 2 a^6 b^2 c^3 - 4 a^5 b^3 c^3 + 6 a^4 b^4 c^3 + 4 a^3 b^5 c^3 - 2 a^2 b^6 c^3 + 4 b^8 c^3 - 8 a^5 b^2 c^4 + 6 a^4 b^3 c^4 + 17 a^3 b^4 c^4 - 8 a b^6 c^4 - 4 b^7 c^4 - 4 a^5 b c^5 + 4 a^3 b^3 c^5 - a^3 b^2 c^6 - 2 a^2 b^3 c^6 - 8 a b^4 c^6 + 2 a^4 c^7 - 4 a^3 b c^7 - 2 a^2 b^2 c^7 - 4 b^4 c^7 + 8 a b^2 c^8 + 4 b^3 c^8) : :

X(27204) lies on these lines:

X(27205) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a^2 (-2 a^2 b^4 + 4 a^3 b^2 c + 2 a b^4 c + 4 b^5 c + 4 a^3 b c^2 - 4 a^2 b^2 c^2 - 8 a b^3 c^2 - 8 a b^2 c^3 - b^3 c^3 - 2 a^2 c^4 + 2 a b c^4 + 4 b c^5) : :

X(27205) lies on these lines: {2, 37}, {16777, 27202}

X(27206) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-8 a^6 b^3 + 8 a^4 b^5 + 20 a^7 b c - 8 a^5 b^3 c + 4 a^4 b^4 c - 12 a^3 b^5 c - 8 a^2 b^6 c + 4 b^8 c - 13 a^5 b^2 c^2 - 2 a^3 b^4 c^2 - 8 a^2 b^5 c^2 + 15 a b^6 c^2 - 8 a^6 c^3 - 8 a^5 b c^3 + 24 a^3 b^3 c^3 + 16 a^2 b^4 c^3 + 8 b^6 c^3 + 4 a^4 b c^4 - 2 a^3 b^2 c^4 + 16 a^2 b^3 c^4 - 30 a b^4 c^4 - 12 b^5 c^4 + 8 a^4 c^5 - 12 a^3 b c^5 - 8 a^2 b^2 c^5 - 12 b^4 c^5 - 8 a^2 b c^6 + 15 a b^2 c^6 + 8 b^3 c^6 + 4 b c^8) : :

X(27206) lies on these lines:

X(27207) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^6 b^3 + 2 a^4 b^5 + 4 a^7 b c - 4 a^3 b^5 c + 5 a^5 b^2 c^2 + 4 a^4 b^3 c^2 - 9 a^3 b^4 c^2 + 2 a^2 b^5 c^2 + 12 a b^6 c^2 - 2 a^6 c^3 + 4 a^4 b^2 c^3 - 9 a^3 b^3 c^3 - 11 a^2 b^4 c^3 + 4 a b^5 c^3 + 4 b^6 c^3 - 9 a^3 b^2 c^4 - 11 a^2 b^3 c^4 - 16 a b^4 c^4 - 4 b^5 c^4 + 2 a^4 c^5 - 4 a^3 b c^5 + 2 a^2 b^2 c^5 + 4 a b^3 c^5 - 4 b^4 c^5 + 12 a b^2 c^6 + 4 b^3 c^6) : :

X(27207) lies on these lines:

X(27208) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^8 b^3 + 2 a^4 b^7 + 4 a^9 b c + 4 a^7 b^3 c - 4 a^5 b^5 c - 4 a^3 b^7 c + a^7 b^2 c^2 - 2 a^6 b^3 c^2 - 8 a^5 b^4 c^2 - a^3 b^6 c^2 - 2 a^2 b^7 c^2 + 8 a b^8 c^2 - 2 a^8 c^3 + 4 a^7 b c^3 - 2 a^6 b^2 c^3 - 8 a^5 b^3 c^3 + 6 a^4 b^4 c^3 + 4 a^3 b^5 c^3 - 6 a^2 b^6 c^3 + 4 b^8 c^3 - 8 a^5 b^2 c^4 + 6 a^4 b^3 c^4 + 34 a^3 b^4 c^4 - 8 a b^6 c^4 - 4 b^7 c^4 - 4 a^5 b c^5 + 4 a^3 b^3 c^5 - a^3 b^2 c^6 - 6 a^2 b^3 c^6 - 8 a b^4 c^6 + 2 a^4 c^7 - 4 a^3 b c^7 - 2 a^2 b^2 c^7 - 4 b^4 c^7 + 8 a b^2 c^8 + 4 b^3 c^8) : :

X(27208) lies on these lines:

X(27209) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 55

Barycentrics    a (-2 a^6 b^3 + 2 a^4 b^5 + 4 a^7 b c - 4 a^5 b^3 c - 4 a^3 b^5 c - 4 a^2 b^6 c + a^5 b^2 c^2 + 8 a^3 b^4 c^2 - 2 a^2 b^5 c^2 + 8 a b^6 c^2 - 2 a^6 c^3 - 4 a^5 b c^3 + 4 a^3 b^3 c^3 + 2 a^2 b^4 c^3 + 8 a^3 b^2 c^4 + 2 a^2 b^3 c^4 + a b^4 c^4 - 4 b^5 c^4 + 2 a^4 c^5 - 4 a^3 b c^5 - 2 a^2 b^2 c^5 - 4 b^4 c^5 - 4 a^2 b c^6 + 8 a b^2 c^6) : :

X(27209) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 56: X(27210)-X(27220)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 56, as in centers X(27210)-X(27228). Then

m(X) = a^2 (b^2 + 2 a c) (c^2 + 2 a b) x - 2 a c (a^2 + 2 b c) (c^2 + 2 a b) y - 2 a b (a^2 + 2 b c) (b^2 + 2 a c) z : : ,

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 6, 2018)

X(27210) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^3 b^3 + 4 a^4 b c - 4 a^3 b^2 c - 2 a^2 b^3 c - 4 a^3 b c^2 + a^2 b^2 c^2 - 8 a b^3 c^2 - 4 b^4 c^2 + 2 a^3 c^3 - 2 a^2 b c^3 - 8 a b^2 c^3 - 4 b^2 c^4) : :

X(27210) lies on these lines: {1, 2}

X(27211) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^6 b^3 - 2 a^4 b^5 + 4 a^7 b c - 4 a^3 b^5 c + a^5 b^2 c^2 + 7 a^3 b^4 c^2 + 2 a^2 b^5 c^2 - 8 a b^6 c^2 + 2 a^6 c^3 + 8 a^3 b^3 c^3 + 2 a^2 b^4 c^3 + 4 b^6 c^3 + 7 a^3 b^2 c^4 + 2 a^2 b^3 c^4 + 16 a b^4 c^4 - 4 b^5 c^4 - 2 a^4 c^5 - 4 a^3 b c^5 + 2 a^2 b^2 c^5 - 4 b^4 c^5 - 8 a b^2 c^6 + 4 b^3 c^6) : :

X(27211) lies on these lines:

X(27212) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (4 a^6 b^3-4 a^4 b^5+12 a^7 b c-8 a^5 b^3 c+4 a^4 b^4 c-4 a^3 b^5 c-8 a^2 b^6 c+4 b^8 c+17 a^5 b^2 c^2-16 a^3 b^4 c^2+4 a^2 b^5 c^2-a b^6 c^2+4 a^6 c^3-8 a^5 b c^3+8 a^3 b^3 c^3-4 a^2 b^4 c^3+4 a^4 b c^4-16 a^3 b^2 c^4-4 a^2 b^3 c^4+2 a b^4 c^4-4 b^5 c^4-4 a^4 c^5-4 a^3 b c^5+4 a^2 b^2 c^5-4 b^4 c^5-8 a^2 b c^6-a b^2 c^6+4 b c^8) : :

X(27212) lies on these lines:

X(27213) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^6 b^3 - 2 a^4 b^5 + 8 a^7 b c - 8 a^5 b^3 c + 4 a^4 b^4 c - 8 a^2 b^6 c + 4 b^8 c + 16 a^5 b^2 c^2 - 23 a^3 b^4 c^2 + 2 a^2 b^5 c^2 + 7 a b^6 c^2 + 2 a^6 c^3 - 8 a^5 b c^3 - 6 a^2 b^4 c^3 - 4 b^6 c^3 + 4 a^4 b c^4 - 23 a^3 b^2 c^4 - 6 a^2 b^3 c^4 - 14 a b^4 c^4 - 2 a^4 c^5 + 2 a^2 b^2 c^5 - 8 a^2 b c^6 + 7 a b^2 c^6 - 4 b^3 c^6 + 4 b c^8) : :

X(27213) lies on these lines:

X(27214) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^4 b^3 + 4 a^5 b c - 4 a^3 b^3 c + a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 8 a b^4 c^2 + 2 a^4 c^3 - 4 a^3 b c^3 - 2 a^2 b^2 c^3 - 4 b^4 c^3 - 8 a b^2 c^4 - 4 b^3 c^4) : :

X(27214) lies on these lines:

X(27215) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^8 b^3 - 2 a^4 b^7 + 4 a^9 b c + 4 a^7 b^3 c - 4 a^5 b^5 c - 4 a^3 b^7 c + a^7 b^2 c^2 + 2 a^6 b^3 c^2 + 8 a^5 b^4 c^2 - a^3 b^6 c^2 + 2 a^2 b^7 c^2 - 8 a b^8 c^2 + 2 a^8 c^3 + 4 a^7 b c^3 + 2 a^6 b^2 c^3 + 2 a^4 b^4 c^3 + 4 a^3 b^5 c^3 - 2 a^2 b^6 c^3 + 4 b^8 c^3 + 8 a^5 b^2 c^4 + 2 a^4 b^3 c^4 + 8 a b^6 c^4 - 4 b^7 c^4 - 4 a^5 b c^5 + 4 a^3 b^3 c^5 - a^3 b^2 c^6 - 2 a^2 b^3 c^6 + 8 a b^4 c^6 - 2 a^4 c^7 - 4 a^3 b c^7 + 2 a^2 b^2 c^7 - 4 b^4 c^7 - 8 a b^2 c^8 + 4 b^3 c^8) : :

X(27215) lies on these lines:

X(27216) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^8 b^3 - 2 a^4 b^7 + 4 a^9 b c + 4 a^7 b^3 c - 4 a^5 b^5 c - 4 a^3 b^7 c + a^7 b^2 c^2 + 2 a^6 b^3 c^2 + 8 a^5 b^4 c^2 - a^3 b^6 c^2 + 2 a^2 b^7 c^2 - 8 a b^8 c^2 + 2 a^8 c^3 + 4 a^7 b c^3 + 2 a^6 b^2 c^3 - 4 a^5 b^3 c^3 + 2 a^4 b^4 c^3 + 4 a^3 b^5 c^3 - 6 a^2 b^6 c^3 + 4 b^8 c^3 + 8 a^5 b^2 c^4 + 2 a^4 b^3 c^4 - 15 a^3 b^4 c^4 + 8 a b^6 c^4 - 4 b^7 c^4 - 4 a^5 b c^5 + 4 a^3 b^3 c^5 - a^3 b^2 c^6 - 6 a^2 b^3 c^6 + 8 a b^4 c^6 - 2 a^4 c^7 - 4 a^3 b c^7 + 2 a^2 b^2 c^7 - 4 b^4 c^7 - 8 a b^2 c^8 + 4 b^3 c^8) : :

X(27216) lies on these lines:

X(27217) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a^2 (2 a^2 b^4 + 4 a^3 b^2 c - 2 a b^4 c + 4 b^5 c + 4 a^3 b c^2 - 4 a^2 b^2 c^2 + 8 a b^3 c^2 + 8 a b^2 c^3 - b^3 c^3 + 2 a^2 c^4 - 2 a b c^4 + 4 b c^5) : :

X(27217) lies on these lines: {2, 37}, {45, 27214}

X(27218) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (8 a^6 b^3 - 8 a^4 b^5 + 20 a^7 b c - 8 a^5 b^3 c + 4 a^4 b^4 c - 12 a^3 b^5 c - 8 a^2 b^6 c + 4 b^8 c + 19 a^5 b^2 c^2 - 2 a^3 b^4 c^2 + 8 a^2 b^5 c^2 - 17 a b^6 c^2 + 8 a^6 c^3 - 8 a^5 b c^3 + 24 a^3 b^3 c^3 + 8 b^6 c^3 + 4 a^4 b c^4 - 2 a^3 b^2 c^4 + 34 a b^4 c^4 - 12 b^5 c^4 - 8 a^4 c^5 - 12 a^3 b c^5 + 8 a^2 b^2 c^5 - 12 b^4 c^5 - 8 a^2 b c^6 - 17 a b^2 c^6 + 8 b^3 c^6 + 4 b c^8) : :

X(27218) lies on these lines:

X(27219) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^6 b^3 - 2 a^4 b^5 + 4 a^7 b c - 4 a^3 b^5 c + 5 a^5 b^2 c^2 + 4 a^4 b^3 c^2 + 7 a^3 b^4 c^2 + 6 a^2 b^5 c^2 - 4 a b^6 c^2 + 2 a^6 c^3 + 4 a^4 b^2 c^3 + 23 a^3 b^3 c^3 + 17 a^2 b^4 c^3 + 4 a b^5 c^3 + 4 b^6 c^3 + 7 a^3 b^2 c^4 + 17 a^2 b^3 c^4 + 16 a b^4 c^4 - 4 b^5 c^4 - 2 a^4 c^5 - 4 a^3 b c^5 + 6 a^2 b^2 c^5 + 4 a b^3 c^5 - 4 b^4 c^5 - 4 a b^2 c^6 + 4 b^3 c^6) : :

X(27219) lies on these lines:

X(27220) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 56

Barycentrics    a (2 a^6 b^3 - 2 a^4 b^5 + 4 a^7 b c - 4 a^5 b^3 c - 4 a^3 b^5 c - 4 a^2 b^6 c + a^5 b^2 c^2 - 8 a^3 b^4 c^2 + 2 a^2 b^5 c^2 - 8 a b^6 c^2 + 2 a^6 c^3 - 4 a^5 b c^3 + 4 a^3 b^3 c^3 - 2 a^2 b^4 c^3 - 8 a^3 b^2 c^4 - 2 a^2 b^3 c^4 + a b^4 c^4 - 4 b^5 c^4 - 2 a^4 c^5 - 4 a^3 b c^5 + 2 a^2 b^2 c^5 - 4 b^4 c^5 - 4 a^2 b c^6 - 8 a b^2 c^6) : :

X(27220) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 57: X(27221)-X(27232)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 57, as in centers X(27221)-X(27232). Then

m(X) = (b^2 +c^2) x / (b^2 - b c + c^ 2) + a c y / (c^2 - c a + a^2) + a b z / (a^2 - a b + b^2) : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 6, 2018)

X(27221) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^4 b^2 - a^3 b^3 + a^2 b^4 - a^3 b^2 c + 3 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 - a^3 c^3 + 3 a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 - 2 a b c^4 + b c^5) : :

X(27221) lies on these lines: {1, 2}, {732, 27226}, {756, 27222}, {27227, 27228}

X(27222) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^7 b^2 - a^6 b^3 + a^4 b^5 - a^3 b^6 - a^6 b^2 c + a^5 b^3 c - a^4 b^4 c + a^2 b^6 c - a b^7 c + b^8 c + a^7 c^2 - a^6 b c^2 + a^4 b^3 c^2 - 3 a^3 b^4 c^2 - b^7 c^2 - a^6 c^3 + a^5 b c^3 + a^4 b^2 c^3 + 2 a^2 b^4 c^3 + a b^5 c^3 - a^4 b c^4 - 3 a^3 b^2 c^4 + 2 a^2 b^3 c^4 - 4 a b^4 c^4 + a^4 c^5 + a b^3 c^5 - a^3 c^6 + a^2 b c^6 - a b c^7 - b^2 c^7 + b c^8) : :

X(27222) lies on these lines:

X(27223) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a^8 b^2 - 2 a^7 b^3 + a^6 b^4 + 2 a^5 b^5 - a^4 b^6 - a^2 b^8 - a^7 b^2 c + 3 a^6 b^3 c - 4 a^5 b^4 c - 2 a^4 b^5 c + 3 a^3 b^6 c - a^2 b^7 c + 2 a b^8 c + a^8 c^2 - a^7 b c^2 + a^5 b^3 c^2 + 4 a^4 b^4 c^2 - a^3 b^5 c^2 - a b^7 c^2 - b^8 c^2 - 2 a^7 c^3 + 3 a^6 b c^3 + a^5 b^2 c^3 - 4 a^4 b^3 c^3 + a^2 b^5 c^3 + a b^6 c^3 + a^6 c^4 - 4 a^5 b c^4 + 4 a^4 b^2 c^4 + 2 a^2 b^4 c^4 - 2 a b^5 c^4 + b^6 c^4 + 2 a^5 c^5 - 2 a^4 b c^5 - a^3 b^2 c^5 + a^2 b^3 c^5 - 2 a b^4 c^5 - a^4 c^6 + 3 a^3 b c^6 + a b^3 c^6 + b^4 c^6 - a^2 b c^7 - a b^2 c^7 - a^2 c^8 + 2 a b c^8 - b^2 c^8 : :

X(27223) lies on these lines:

X(27224) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a^7 b^3 - a^6 b^4 - a^5 b^5 + a^2 b^8 - 2 a^6 b^3 c + 3 a^5 b^4 c + 2 a^4 b^5 c - 2 a^3 b^6 c - a b^8 c - 7 a^4 b^4 c^2 + a^3 b^5 c^2 + b^8 c^2 + a^7 c^3 - 2 a^6 b c^3 + 4 a^4 b^3 c^3 + 2 a^3 b^4 c^3 - a b^6 c^3 - a^6 c^4 + 3 a^5 b c^4 - 7 a^4 b^2 c^4 + 2 a^3 b^3 c^4 - 6 a^2 b^4 c^4 + 2 a b^5 c^4 - b^6 c^4 - a^5 c^5 + 2 a^4 b c^5 + a^3 b^2 c^5 + 2 a b^4 c^5 - 2 a^3 b c^6 - a b^3 c^6 - b^4 c^6 + a^2 c^8 - a b c^8 + b^2 c^8 : :

X(27224) lies on these lines: {2, 3}

X(27225) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^5 b^2 - a^4 b^3 + a^3 b^4 - a^4 b^2 c + a^3 b^3 c - a b^5 c + b^6 c + a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a^2 b^3 c^2 + 2 a b^4 c^2 - b^5 c^2 - a^4 c^3 + a^3 b c^3 - a^2 b^2 c^3 - 2 a b^3 c^3 + b^4 c^3 + a^3 c^4 + 2 a b^2 c^4 + b^3 c^4 - a b c^5 - b^2 c^5 + b c^6) : :

X(27225) lies on these lines:

X(27226) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a^4 b^4 - 2 a^3 b^4 c + a^2 b^5 c + 2 a^4 b^2 c^2 - a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a b^5 c^2 - a^3 b^2 c^3 - a b^4 c^3 + b^5 c^3 + a^4 c^4 - 2 a^3 b c^4 + 3 a^2 b^2 c^4 - a b^3 c^4 + a^2 b c^5 - a b^2 c^5 + b^3 c^5 : :

X(27226) lies on these lines:

X(27227) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^6 b^2 + a^3 b^5 + 2 a^3 b^4 c + b^7 c + a^6 c^2 + a^3 b^3 c^2 + a^3 b^2 c^3 + b^5 c^3 + 2 a^3 b c^4 + 2 b^4 c^4 + a^3 c^5 + b^3 c^5 + b c^7) : :

X(27227) lies on these lines:

X(27228) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^7 b^2-a^6 b^3+a^4 b^5-a^3 b^6-a^6 b^2 c-2 a^4 b^4 c-a^3 b^5 c-a b^7 c+b^8 c+a^7 c^2-a^6 b c^2+a^4 b^3 c^2-a^3 b^4 c^2-b^7 c^2-a^6 c^3+a^4 b^2 c^3-2 a^3 b^3 c^3+a b^5 c^3-b^6 c^3-2 a^4 b c^4-a^3 b^2 c^4-2 a b^4 c^4-b^5 c^4+a^4 c^5-a^3 b c^5+a b^3 c^5-b^4 c^5-a^3 c^6-b^3 c^6-a b c^7-b^2 c^7+b c^8) : :

X(27228) lies on these lines:

X(27229) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^10 b^2 + a^7 b^5 - a^6 b^6 - a^3 b^9 - 2 a^7 b^4 c + 2 a^5 b^6 c - a^4 b^7 c + b^11 c + a^10 c^2 + a^7 b^3 c^2 + a^6 b^4 c^2 + a^3 b^7 c^2 + a^7 b^2 c^3 + 2 a^5 b^4 c^3 + a^4 b^5 c^3 - a^3 b^6 c^3 + 2 a^2 b^7 c^3 - b^9 c^3 - 2 a^7 b c^4 + a^6 b^2 c^4 + 2 a^5 b^3 c^4 - 2 a^4 b^4 c^4 + a^3 b^5 c^4 + 2 a^2 b^6 c^4 + a^7 c^5 + a^4 b^3 c^5 + a^3 b^4 c^5 - a^6 c^6 + 2 a^5 b c^6 - a^3 b^3 c^6 + 2 a^2 b^4 c^6 - a^4 b c^7 + a^3 b^2 c^7 + 2 a^2 b^3 c^7 - a^3 c^9 - b^3 c^9 + b c^11) : :

X(27229) lies on these lines:

X(27230) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a^8 b^2 - a^7 b^3 + a^6 b^4 + a^5 b^5 - a^7 b^2 c + a^6 b^3 c - 2 a^5 b^4 c - a^4 b^5 c + a^3 b^6 c - a^2 b^7 c + a b^8 c + a^8 c^2 - a^7 b c^2 + 2 a^6 b^2 c^2 + 3 a^4 b^4 c^2 - a^3 b^5 c^2 + 2 a^2 b^6 c^2 - a b^7 c^2 - a^7 c^3 + a^6 b c^3 - 2 a^4 b^3 c^3 + a^6 c^4 - 2 a^5 b c^4 + 3 a^4 b^2 c^4 + 2 a^2 b^4 c^4 - a b^5 c^4 + b^6 c^4 + a^5 c^5 - a^4 b c^5 - a^3 b^2 c^5 - a b^4 c^5 + a^3 b c^6 + 2 a^2 b^2 c^6 + b^4 c^6 - a^2 b c^7 - a b^2 c^7 + a b c^8 : :

X(27230) lies on these lines:

X(27231) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    (a^4 - a^3 b + a^2 b^2 + a b^3 - a^3 c + a^2 b c - a b^2 c - b^3 c + a^2 c^2 - a b c^2 + 2 b^2 c^2 + a c^3 - b c^3) (a^2 b^2 - a b^3 + b^3 c + a^2 c^2 - a c^3 + b c^3) : :

X(27231) lies on these lines:

X(27232) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 57

Barycentrics    a (a^7 b^2 + a^4 b^5 + a^3 b^5 c + b^8 c + a^7 c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 + a^4 b^2 c^3 + a b^5 c^3 + a^3 b^2 c^4 + b^5 c^4 + a^4 c^5 + a^3 b c^5 + a b^3 c^5 + b^4 c^5 + b c^8) : :

X(27232) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 58: X(27233)-X(27245)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 58, as in centers X(27233)-X(27245). Then

m(X) = (b^2 +c^2) x / (b^2 + b c + c^ 2) - a c y / (c^2 + c a + a^2) - a b z / (a^2 + a b + b^2) : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 6, 2018)

X(27233) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^4 b^2 + a^3 b^3 + a^2 b^4 + a^3 b^2 c - a^2 b^3 c - b^5 c + a^4 c^2 + a^3 b c^2 - a b^3 c^2 + a^3 c^3 - a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 - b c^5) : :

X(27233) lies on these lines: {1, 2}, {238, 1492}, {244, 27237}

X(27234) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^7 b^2 + a^6 b^3 - a^4 b^5 - a^3 b^6 + a^6 b^2 c + a^5 b^3 c + a^4 b^4 c - a^2 b^6 c - a b^7 c - b^8 c + a^7 c^2 + a^6 b c^2 + a^4 b^3 c^2 - a^3 b^4 c^2 - 2 a b^6 c^2 - b^7 c^2 + a^6 c^3 + a^5 b c^3 + a^4 b^2 c^3 + a b^5 c^3 + a^4 b c^4 - a^3 b^2 c^4 + 2 b^5 c^4 - a^4 c^5 + a b^3 c^5 + 2 b^4 c^5 - a^3 c^6 - a^2 b c^6 - 2 a b^2 c^6 - a b c^7 - b^2 c^7 - b c^8) : :

X(27234) lies on these lines:

X(27235) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a^8 b^2 + 2 a^7 b^3 + a^6 b^4 - 2 a^5 b^5 - a^4 b^6 - a^2 b^8 + 3 a^7 b^2 c + 3 a^6 b^3 c - 2 a^4 b^5 c - a^3 b^6 c - a^2 b^7 c - 2 a b^8 c + a^8 c^2 + 3 a^7 b c^2 + 4 a^6 b^2 c^2 + a^5 b^3 c^2 - a^3 b^5 c^2 - a b^7 c^2 - b^8 c^2 + 2 a^7 c^3 + 3 a^6 b c^3 + a^5 b^2 c^3 - 4 a^4 b^3 c^3 - 4 a^3 b^4 c^3 + a^2 b^5 c^3 + a b^6 c^3 + a^6 c^4 - 4 a^3 b^3 c^4 + 2 a^2 b^4 c^4 + 2 a b^5 c^4 + b^6 c^4 - 2 a^5 c^5 - 2 a^4 b c^5 - a^3 b^2 c^5 + a^2 b^3 c^5 + 2 a b^4 c^5 - a^4 c^6 - a^3 b c^6 + a b^3 c^6 + b^4 c^6 - a^2 b c^7 - a b^2 c^7 - a^2 c^8 - 2 a b c^8 - b^2 c^8 : :

X(27235) lies on these lines:

X(27236) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a^7 b^3+a^6 b^4-a^5 b^5-a^2 b^8+2 a^7 b^2 c+2 a^6 b^3 c-a^5 b^4 c-2 a^4 b^5 c-a b^8 c+2 a^7 b c^2+4 a^6 b^2 c^2+a^4 b^4 c^2-a^3 b^5 c^2+2 a^2 b^6 c^2-b^8 c^2+a^7 c^3+2 a^6 b c^3-4 a^4 b^3 c^3-4 a^3 b^4 c^3+a b^6 c^3+a^6 c^4-a^5 b c^4+a^4 b^2 c^4-4 a^3 b^3 c^4+2 a^2 b^4 c^4+b^6 c^4-a^5 c^5-2 a^4 b c^5-a^3 b^2 c^5+2 a^2 b^2 c^6+a b^3 c^6+b^4 c^6-a^2 c^8-a b c^8-b^2 c^8 : :

X(27236) lies on these lines:

X(27237) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^5 b^2 + a^4 b^3 + a^3 b^4 + a^4 b^2 c + a^3 b^3 c - a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 + a^4 c^3 + a^3 b c^3 - a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 + a^3 c^4 - b^3 c^4 - a b c^5 - b^2 c^5 - b c^6) : :

X(27237) lies on these lines:

X(27238) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a^4 b^4 - a^2 b^5 c + 2 a^4 b^2 c^2 + a^3 b^3 c^2 + a^2 b^4 c^2 - a b^5 c^2 + a^3 b^2 c^3 - a b^4 c^3 - b^5 c^3 + a^4 c^4 + a^2 b^2 c^4 - a b^3 c^4 - a^2 b c^5 - a b^2 c^5 - b^3 c^5 : :

X(27238) lies on these lines:

X(27239) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^7 b^2 + a^6 b^3 - a^4 b^5 - a^3 b^6 + a^6 b^2 c - a^3 b^5 c - 2 a^2 b^6 c - a b^7 c - b^8 c + a^7 c^2 + a^6 b c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 - 2 a b^6 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 + 2 a^3 b^3 c^3 + 2 a^2 b^4 c^3 + a b^5 c^3 - b^6 c^3 + a^3 b^2 c^4 + 2 a^2 b^3 c^4 + 2 a b^4 c^4 + b^5 c^4 - a^4 c^5 - a^3 b c^5 + a b^3 c^5 + b^4 c^5 - a^3 c^6 - 2 a^2 b c^6 - 2 a b^2 c^6 - b^3 c^6 - a b c^7 - b^2 c^7 - b c^8) : :

X(27239) lies on these lines:

X(27240) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a^8 b^2+a^7 b^3+a^6 b^4-a^5 b^5+a^7 b^2 c+a^6 b^3 c-a^4 b^5 c-a^3 b^6 c-a^2 b^7 c-a b^8 c+a^8 c^2+a^7 b c^2+2 a^6 b^2 c^2+a^4 b^4 c^2-a^3 b^5 c^2-a b^7 c^2+a^7 c^3+a^6 b c^3-2 a^4 b^3 c^3-2 a^3 b^4 c^3+a^6 c^4+a^4 b^2 c^4-2 a^3 b^3 c^4+2 a^2 b^4 c^4+a b^5 c^4+b^6 c^4-a^5 c^5-a^4 b c^5-a^3 b^2 c^5+a b^4 c^5-a^3 b c^6+b^4 c^6-a^2 b c^7-a b^2 c^7-a b c^8 : :

X(27240) lies on these lines:

X(27241) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a^5 b^2+a^4 b^3+a^3 b^4-a^2 b^5+2 a^4 b^2 c-2 a b^5 c+a^5 c^2+2 a^4 b c^2+2 a^3 b^2 c^2-b^5 c^2+a^4 c^3-2 a b^3 c^3-b^4 c^3+a^3 c^4-b^3 c^4-a^2 c^5-2 a b c^5-b^2 c^5 : :

X(27241) lies on these lines:

X(27242) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^5 b^2 - a^2 b^5 + a^3 b^3 c - a b^5 c - b^6 c + a^5 c^2 + 2 a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 + a^3 b c^3 + a b^2 c^4 - a^2 c^5 - a b c^5 - b^2 c^5 - b c^6) : :

X(27242) lies on these lines:

X(27243) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^6 b^2 - a^3 b^5 - b^7 c + a^6 c^2 - a^3 b^3 c^2 - a^3 b^2 c^3 + b^5 c^3 + 2 b^4 c^4 - a^3 c^5 + b^3 c^5 - b c^7) : :

X(27243) lies on these lines: {2, 11}

X(27244) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^7 b^2 - a^4 b^5 + a^3 b^5 c - b^8 c + a^7 c^2 - a^4 b^3 c^2 - a^3 b^4 c^2 - a^4 b^2 c^3 + a b^5 c^3 - a^3 b^2 c^4 + b^5 c^4 - a^4 c^5 + a^3 b c^5 + a b^3 c^5 + b^4 c^5 - b c^8) : :

X(27244) lies on these lines:

X(27245) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 58

Barycentrics    a (a^4 b^3 + a^3 b^4 + a^2 b^5 + a^4 b^2 c + a^4 b c^2 - a^2 b^3 c^2 - a b^4 c^2 + a^4 c^3 - a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 + a^3 c^4 - a b^2 c^4 - b^3 c^4 + a^2 c^5) : :

X(27245) lies on these lines:

X(27246) =  MIDPOINT OF X(195) AND X(1157)

Barycentrics    a^2 (2 a^20-15 a^18 b^2+49 a^16 b^4-92 a^14 b^6+112 a^12 b^8-98 a^10 b^10+70 a^8 b^12-44 a^6 b^14+22 a^4 b^16-7 a^2 b^18+b^20-15 a^18 c^2+76 a^16 b^2 c^2-153 a^14 b^4 c^2+149 a^12 b^6 c^2-55 a^10 b^8 c^2-35 a^8 b^10 c^2+69 a^6 b^12 c^2-57 a^4 b^14 c^2+26 a^2 b^16 c^2-5 b^18 c^2+49 a^16 c^4-153 a^14 b^2 c^4+164 a^12 b^4 c^4-72 a^10 b^6 c^4+35 a^8 b^8 c^4-60 a^6 b^10 c^4+71 a^4 b^12 c^4-47 a^2 b^14 c^4+13 b^16 c^4-92 a^14 c^6+149 a^12 b^2 c^6-72 a^10 b^4 c^6+22 a^8 b^6 c^6+8 a^6 b^8 c^6-42 a^4 b^10 c^6+55 a^2 b^12 c^6-28 b^14 c^6+112 a^12 c^8-55 a^10 b^2 c^8+35 a^8 b^4 c^8+8 a^6 b^6 c^8+12 a^4 b^8 c^8-27 a^2 b^10 c^8+50 b^12 c^8-98 a^10 c^10-35 a^8 b^2 c^10-60 a^6 b^4 c^10-42 a^4 b^6 c^10-27 a^2 b^8 c^10-62 b^10 c^10+70 a^8 c^12+69 a^6 b^2 c^12+71 a^4 b^4 c^12+55 a^2 b^6 c^12+50 b^8 c^12-44 a^6 c^14-57 a^4 b^2 c^14-47 a^2 b^4 c^14-28 b^6 c^14+22 a^4 c^16+26 a^2 b^2 c^16+13 b^4 c^16-7 a^2 c^18-5 b^2 c^18+c^20) : :
Barycentrics    (16 R^2-4 SB-4 SC-4 SW) S^4 + (-47 R^6+43 R^4 SB+43 R^4 SC-8 R^2 SB SC+37 R^4 SW-36 R^2 SB SW-36 R^2 SC SW+4 SB SC SW-8 R^2 SW^2+8 SB SW^2+8 SC SW^2) S^2 + (39 R^6 SB SC+35 R^4 SB SC SW-8 R^2 SB SC SW^2) : :
X(27246) = 2*X[8254]-X[16336]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28585.

X(27246) lies on these lines: {3,54}, {128,24147}, {1263,18400}, {3459,6288}, {8254,16336}, {10615,16768}, {16337,24385}

X(27246) = midpoint of X(195) and X(1157)
X(27246) = reflection of X(i) in X(j) for these {i,j}: {16336,8254}, {21230,10615}

X(27247) =  X(1)X(3)∩X(3814)X(20292)

Barycentrics    a (a^6-3 a^4 b^2+3 a^2 b^4-b^6+2 a^4 b c+4 a^3 b^2 c-4 a^2 b^3 c-4 a b^4 c+2 b^5 c-3 a^4 c^2+4 a^3 b c^2-3 a^2 b^2 c^2+4 a b^3 c^2+b^4 c^2-4 a^2 b c^3+4 a b^2 c^3-4 b^3 c^3+3 a^2 c^4-4 a b c^4+b^2 c^4+2 b c^5-c^6) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28585.

X(27247) lies on these lines: {1,3}, {3814,20292}, {5180,10200}

leftri

Collineation mappings involving Gemini triangle 59: X(27248)-X(27297)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 59, as in centers X(27248)-X(27297). Then

m(X) = a(a b + a c - b c) x + b (a b + a c + b c) y + c (a b + a c + b c) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 7, 2018)

X(27248) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    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(27248) lies on these lines: {1, 2}, {35, 22267}, {37, 304}, {55, 16060}, {56, 16061}, {69, 213}, {141, 2176}, {192, 1930}, {194, 17280}, {238, 20139}, {274, 2345}, {330, 17358}, {344, 5283}, {345, 980}, {346, 4352}, {388, 28777}, {894, 17742}, {966, 16782}, {993, 17696}, {1107, 17279}, {1213, 16524}, {1959, 27259}, {2223, 14001}, {2295, 30945}, {2329, 24549}, {3230, 3619}, {3294, 17170}, {3618, 20963}, {3662, 17753}, {3674, 26125}, {3721, 24282}, {3729, 24214}, {3730, 16887}, {3747, 26034}, {3763, 16969}, {3876, 26689}, {3976, 24631}, {4000, 17143}, {4201, 20533}, {4441, 26978}, {4687, 18156}, {5217, 21937}, {5255, 24586}, {5263, 11321}, {5264, 29473}, {5280, 17379}, {5291, 25497}, {5299, 17349}, {5750, 26106}, {6645, 17688}, {7176, 28739}, {7718, 15149}, {7770, 32942}, {7800, 8624}, {8193, 11329}, {11115, 27185}, {12410, 16412}, {13741, 26687}, {14210, 27268}, {16062, 26590}, {16502, 17277}, {16523, 17398}, {16552, 26685}, {16600, 21216}, {16604, 24652}, {16706, 17144}, {16738, 16788}, {16781, 17259}, {17050, 17282}, {17192, 17236}, {17233, 33296}, {17238, 27047}, {17270, 26041}, {17289, 31997}, {17303, 25504}, {17321, 25499}, {17350, 17744}, {17353, 21384}, {17355, 24215}, {17357, 17448}, {17373, 27320}, {17383, 32095}, {17385, 25130}, {17671, 26558}, {17682, 20172}, {17686, 24552}, {17750, 30962}, {17756, 27162}, {18206, 26065}, {20553, 26085}, {20911, 26242}, {21062, 27129}, {21240, 21281}, {23493, 27341}, {24790, 32104}, {25066, 25918}, {27249, 27284}, {27251, 27256}, {27252, 27254}, {27275, 27282}, {27289, 27291}, {28598, 31130}

X(27248) = anticomplement of X(30107)
X(27248) = {X(2),X(8)}-harmonic conjugate of X(27299)

X(27249) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c - a^5 b c - a^4 b^2 c - a^2 b^4 c + a b^5 c + b^6 c - a^4 b c^2 - a b^4 c^2 - a^4 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(27249) lies on these lines:

X(27250) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 3 a^5 b c - a^4 b^2 c + 2 a^3 b^3 c - a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + 2 a^3 b c^3 + 4 a^2 b^2 c^3 - 2 a b^3 c^3 - a^2 b c^4 - a b^2 c^4 - a^2 c^5 + a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(27250) lies on these lines: {2, 3}, {37, 20914}, {169, 17260}, {192, 20235}, {344, 6376}, {1446, 26125}, {2333, 4329}, {4766, 28740}, {5179, 27254}, {27267, 27269}

X(27251) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 a^5 b c - 2 a^3 b^3 c + a^5 c^2 - 4 a^3 b^2 c^2 - 4 a^2 b^3 c^2 + b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 - 4 a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + a^2 c^5 + b^2 c^5 : :

X(27251) lies on these lines: {2, 3}, {37, 21579}, {192, 21403}, {1212, 18738}, {4766, 28742}, {17338, 24491}, {26035, 27071}, {27060, 27255}, {27248, 27256}

X(27252) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b + a b^4 + a^4 c - a^3 b c + a b^3 c + b^4 c + a b c^3 + a c^4 + b c^4 : :

X(27252) lies on these lines: {2, 6}, {37, 20444}, {192, 20234}, {573, 3662}, {941, 17302}, {1740, 29846}, {1918, 6327}, {2209, 2887}, {2309, 3771}, {4270, 17367}, {4277, 16706}, {5145, 25645}, {7175, 28774}, {14963, 27249}, {17142, 33144}, {20170, 33155}, {22343, 29865}, {26125, 26976}, {27248, 27254}, {27259, 27263}, {27264, 27289}, {27268, 27272}, {27290, 27291}

X(27253) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    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(27253) lies on these lines: {1, 2}, {6, 32008}, {7, 1334}, {31, 19225}, {37, 85}, {44, 32088}, {45, 32024}, {55, 4209}, {65, 27475}, {142, 3208}, {192, 20880}, {220, 14828}, {226, 27129}, {344, 1909}, {377, 20533}, {495, 17671}, {673, 3303}, {728, 10436}, {956, 17687}, {1621, 17691}, {1697, 27000}, {2275, 24654}, {2295, 4648}, {3177, 16601}, {3212, 21808}, {3247, 25521}, {3295, 17682}, {3475, 4517}, {3672, 26978}, {3691, 18230}, {3731, 30625}, {3871, 17683}, {3945, 26059}, {3976, 25073}, {4050, 20195}, {4295, 26790}, {4433, 26040}, {4653, 26802}, {4675, 32007}, {4687, 16284}, {4704, 25241}, {4869, 17137}, {5141, 31058}, {6645, 17696}, {8232, 14189}, {9331, 24790}, {12053, 27183}, {16814, 32100}, {17056, 27021}, {17077, 32003}, {17234, 21281}, {17263, 24524}, {17279, 24656}, {17572, 31020}, {17753, 17758}, {25082, 27340}, {26839, 30305}, {27267, 27268}, {27269, 27295}, {27275, 27287}, {27291, 27298}

X(27254) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 3 a^3 b c - 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 - a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(27254) lies on these lines: {2, 7}, {37, 20927}, {190, 25601}, {192, 20236}, {193, 28797}, {239, 3553}, {344, 1234}, {498, 3923}, {1743, 27317}, {1757, 26363}, {2298, 26282}, {2663, 11269}, {2911, 17277}, {3085, 3685}, {3751, 10527}, {3886, 10528}, {3912, 27267}, {4648, 28748}, {5179, 27250}, {5692, 19853}, {9612, 26057}, {11374, 25519}, {17244, 28778}, {20964, 26098}, {24342, 26364}, {26601, 27042}, {27248, 27252}, {27268, 27290}, {27272, 27291}

X(27255) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^2 b^2 + 2 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 + b^2 c^2 : :

X(27255) lies on these lines: {1, 2}, {9, 17499}, {35, 16915}, {36, 17684}, {37, 76}, {39, 31997}, {55, 11321}, {75, 1500}, {86, 17750}, {142, 24190}, {192, 20888}, {194, 25092}, {238, 20140}, {274, 2276}, {334, 25499}, {384, 5248}, {442, 26590}, {495, 26558}, {595, 14621}, {750, 18756}, {894, 3730}, {958, 33036}, {993, 6645}, {1001, 7770}, {1107, 24656}, {1329, 33034}, {1376, 33035}, {1573, 24524}, {1621, 17686}, {1655, 3761}, {1909, 5283}, {2241, 20179}, {2296, 7109}, {2550, 33026}, {2886, 33033}, {3294, 24514}, {3295, 20172}, {3501, 10436}, {3662, 17758}, {3739, 20691}, {3814, 33046}, {3816, 32992}, {3822, 5025}, {3825, 16921}, {3934, 30963}, {3963, 29983}, {3971, 24080}, {4687, 6376}, {4698, 25102}, {4851, 27164}, {5010, 17693}, {5179, 27250}, {5259, 16916}, {5267, 33063}, {5280, 16998}, {5284, 17541}, {5432, 17694}, {5711, 20131}, {5847, 20139}, {6381, 19565}, {6656, 25466}, {6668, 33249}, {6681, 33015}, {6690, 7807}, {7483, 26686}, {7786, 16604}, {7951, 17669}, {8715, 16911}, {8728, 26582}, {9331, 32104}, {11108, 26687}, {11285, 25524}, {11375, 28771}, {12782, 24325}, {15668, 21788}, {16525, 17398}, {16549, 17175}, {16738, 17391}, {16917, 25440}, {16975, 25303}, {17234, 21240}, {17242, 21070}, {17243, 21024}, {17245, 20255}, {17248, 27047}, {17270, 26045}, {17303, 25508}, {17321, 25538}, {17338, 27080}, {17368, 27032}, {17759, 32092}, {18136, 20917}, {18140, 32009}, {18145, 32090}, {18832, 21827}, {20533, 26051}, {20913, 28606}, {21385, 27075}, {21868, 31238}, {23657, 25127}, {24068, 27481}, {24631, 25073}, {25639, 33045}, {26105, 32968}, {27021, 27129}, {27050, 31053}, {27060, 27251}, {27280, 27287}, {30478, 33044}, {31418, 33037}

X(27256) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + 2 a^4 b c - a^3 b^2 c + a^2 b^3 c + a^4 c^2 - a^3 b c^2 - 4 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + b^2 c^4 : :

X(27256) lies on these lines: {2, 11}, {37, 21580}, {192, 21404}, {4011, 28772}, {4023, 26757}, {4728, 27292}, {5233, 27096}, {5241, 27025}, {16592, 27071}, {17717, 28742}, {27021, 27097}, {27248, 27251}

X(27257) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^8 b - a^5 b^4 - a^4 b^5 + a b^8 + a^8 c - a^7 b c - a^5 b^3 c - 2 a^4 b^4 c + a^3 b^5 c + a b^7 c + b^8 c - a^5 b c^3 - a b^5 c^3 - a^5 c^4 - 2 a^4 b c^4 - 2 a b^4 c^4 - b^5 c^4 - a^4 c^5 + a^3 b c^5 - a b^3 c^5 - b^4 c^5 + a b c^7 + a c^8 + b c^8 : :

X(27257) lies on these lines: {2, 3}, {37, 21583}, {192, 21407}, {17492, 21034}

X(27258) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^8 b - a^5 b^4 - a^4 b^5 + a b^8 + a^8 c - a^7 b c - a^5 b^3 c - 2 a^4 b^4 c + a^3 b^5 c + a b^7 c + b^8 c + a^4 b^3 c^2 + a^3 b^4 c^2 - a^5 b c^3 + a^4 b^2 c^3 + a^3 b^3 c^3 + a^2 b^4 c^3 - a b^5 c^3 - a^5 c^4 - 2 a^4 b c^4 + a^3 b^2 c^4 + a^2 b^3 c^4 - 2 a b^4 c^4 - b^5 c^4 - a^4 c^5 + a^3 b c^5 - a b^3 c^5 - b^4 c^5 + a b c^7 + a c^8 + b c^8 : :

X(27258) lies on these lines: {2, 3}, {37, 21584}, {192, 21408}, {26854, 27294}

X(27259) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b + a b^5 + a^5 c - a^4 b c + a b^4 c + b^5 c + a b c^4 + a c^5 + b c^5 : :

X(27259) lies on these lines: {2, 31}, {37, 20641}, {192, 20627}, {315, 2205}, {1621, 17550}, {1959, 27248}, {8616, 30104}, {27097, 27249}, {27252, 27263}

X(27260) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b + a b^6 + a^6 c - a^5 b c + a b^5 c + b^6 c + a b c^5 + a c^6 + b c^6 : :

X(27260) lies on these lines: {2, 32}, {37, 21585}, {192, 21409}, {14963, 27249}, {27269, 27281}

X(27261) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    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(27261) lies on these lines: {2, 37}, {6, 27078}, {9, 16738}, {257, 27023}, {726, 19864}, {740, 26030}, {872, 17135}, {894, 27145}, {966, 27036}, {984, 25591}, {1215, 21330}, {2300, 17349}, {3619, 27106}, {3661, 26772}, {3662, 26976}, {3758, 17178}, {3953, 24349}, {4022, 17165}, {4363, 27017}, {4473, 26082}, {4643, 26799}, {5749, 26975}, {6375, 21327}, {16685, 17277}, {17030, 17338}, {17116, 27107}, {17228, 26756}, {17243, 27042}, {17266, 25538}, {17268, 27020}, {17276, 26857}, {17292, 27095}, {17293, 27044}, {17368, 26963}, {17369, 26979}, {17381, 27166}, {18044, 31026}, {21895, 23632}, {22220, 24325}, {25124, 31264}, {26110, 29569}, {27248, 27252}, {27262, 27274}, {27270, 27290}

X(27262) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b^3 + a^3 b^4 + a^4 b^2 c + a^2 b^4 c + a^4 b c^2 + a b^4 c^2 + a^4 c^3 + 2 a b^3 c^3 + b^4 c^3 + a^3 c^4 + a^2 b c^4 + a b^2 c^4 + b^3 c^4 : :

X(27262) lies on these lines: {2, 39}, {6, 26107}, {37, 18050}, {148, 26058}, {192, 21412}, {385, 25520}, {386, 26772}, {2896, 26124}, {3159, 19858}, {4253, 26963}, {14963, 27249}, {16552, 26959}, {16887, 27145}, {17030, 17248}, {19717, 27166}, {27261, 27274}

X(27263) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^4 b c - a^3 b^2 c + a^2 b^3 c + 2 a b^4 c + a^4 c^2 - a^3 b c^2 + a b^3 c^2 + b^4 c^2 + a^2 b c^3 + a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27263) lies on these lines: {1, 2}, {37, 18138}, {192, 21415}, {4184, 27185}, {7109, 17137}, {23632, 27109}, {27030, 31017}, {27047, 32782}, {27252, 27259}

X(27264) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^4 b c - 2 a^3 b^2 c + 2 a b^4 c + a^4 c^2 - 2 a^3 b 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(27264) lies on these lines: {1, 2}, {37, 21590}, {192, 21416}, {344, 21838}, {1575, 25504}, {16606, 17279}, {17754, 26106}, {25074, 25918}, {27252, 27289}

X(27265) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    2 a^4 b - a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 a^4 c - 4 a^3 b 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(27265) lies on these lines: {2, 44}, {37, 21591}, {192, 21417}, {26114, 27293}, {27248, 27252}, {27272, 27290}

X(27266) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + a^4 c - 5 a^3 b c - 3 a b^3 c + b^4 c - 2 a^3 c^2 - 4 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 3 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(27266) lies on these lines: {2, 45}, {37, 21592}, {192, 21418}, {27248, 27252}

X(27267) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    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 - 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(27267) lies on these lines: {2, 6}, {37, 20930}, {71, 5905}, {192, 1441}, {226, 22370}, {239, 25521}, {518, 25613}, {527, 25601}, {914, 27287}, {1234, 31060}, {2269, 30985}, {2550, 10528}, {3085, 4645}, {3475, 24752}, {3912, 27254}, {4660, 10056}, {8232, 20533}, {10198, 33082}, {17298, 27305}, {17718, 17792}, {20072, 26059}, {21299, 29839}, {26685, 28742}, {27250, 27269}, {27253, 27268}, {27518, 33111}, {28778, 29572}

X(27268) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    3 a b + 3 a c + b c : :

X(27268) lies on these lines: {1, 4991}, {2, 37}, {6, 29570}, {8, 3842}, {9, 16826}, {10, 17242}, {44, 17394}, {45, 86}, {55, 16993}, {69, 29569}, {141, 29572}, {142, 17247}, {144, 27475}, {145, 15569}, {190, 15668}, {193, 29624}, {194, 31996}, {239, 3247}, {274, 32107}, {329, 26109}, {330, 5283}, {335, 4473}, {391, 27495}, {518, 3622}, {631, 20430}, {726, 3624}, {740, 9780}, {742, 3619}, {872, 17018}, {894, 3731}, {966, 6542}, {984, 3616}, {1100, 17335}, {1107, 31999}, {1125, 17368}, {1213, 17233}, {1255, 5278}, {1449, 29580}, {1654, 5296}, {1698, 3993}, {1743, 29597}, {2321, 27480}, {2667, 3240}, {3008, 17396}, {3090, 29010}, {3091, 30273}, {3161, 31336}, {3329, 4423}, {3618, 29586}, {3620, 29621}, {3661, 5257}, {3662, 29571}, {3663, 27147}, {3664, 17333}, {3686, 17389}, {3723, 3759}, {3728, 10180}, {3758, 16814}, {3766, 27292}, {3826, 21927}, {3834, 17249}, {3875, 16815}, {3879, 17331}, {3912, 3986}, {3945, 20072}, {3946, 29628}, {3950, 24603}, {3963, 30830}, {3971, 17157}, {4022, 7226}, {4029, 4967}, {4032, 5226}, {4098, 4431}, {4357, 17232}, {4360, 16672}, {4361, 16674}, {4363, 16677}, {4364, 17234}, {4384, 16673}, {4389, 17245}, {4393, 16777}, {4416, 17391}, {4419, 26806}, {4422, 17381}, {4445, 31144}, {4643, 17317}, {4648, 6646}, {4670, 17336}, {4675, 17258}, {4678, 28581}, {4690, 17386}, {4708, 17228}, {4709, 19875}, {4741, 5308}, {4748, 29589}, {4777, 27115}, {4851, 17256}, {5224, 17230}, {5232, 29583}, {5259, 7787}, {5281, 11997}, {5435, 7201}, {5550, 24325}, {5750, 17339}, {5839, 29588}, {6381, 19565}, {6536, 29854}, {6666, 17367}, {6682, 26103}, {6685, 17038}, {6707, 17340}, {9330, 29822}, {10436, 16676}, {14206, 27277}, {14210, 27248}, {15717, 30271}, {15966, 23493}, {16578, 25521}, {16589, 26772}, {16601, 27340}, {16669, 31313}, {16832, 17117}, {16989, 29838}, {17023, 17338}, {17045, 17352}, {17145, 29814}, {17231, 17250}, {17237, 17241}, {17239, 17240}, {17251, 17295}, {17252, 17296}, {17253, 17297}, {17254, 17298}, {17265, 17305}, {17266, 17306}, {17267, 17307}, {17268, 17308}, {17270, 17310}, {17271, 17311}, {17272, 17312}, {17273, 17313}, {17275, 17315}, {17282, 17324}, {17283, 17325}, {17284, 17326}, {17285, 17327}, {17286, 29610}, {17287, 29573}, {17288, 29620}, {17309, 32025}, {17323, 27191}, {17328, 17374}, {17329, 17376}, {17330, 17377}, {17332, 17378}, {17337, 17380}, {17344, 17387}, {17346, 17390}, {17347, 17392}, {17348, 17393}, {17353, 17397}, {17354, 17398}, {17363, 29574}, {17364, 29622}, {17366, 31285}, {17592, 26038}, {18230, 26626}, {19237, 20136}, {19853, 27785}, {20140, 30667}, {20363, 27036}, {21219, 24656}, {21830, 32090}, {22343, 24661}, {25092, 27318}, {25124, 32931}, {25354, 29674}, {26082, 26113}, {27252, 27272}, {27253, 27267}, {27254, 27290}, {27299, 28594}, {27472, 31019}, {27481, 29609}, {28611, 31320}, {31276, 32453}, {31997, 32005}

X(27269) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^2 b^2 + 2 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - b^2 c^2 : :

X(27269) lies on these lines: {1, 20464}, {2, 39}, {3, 16999}, {6, 16918}, {10, 192}, {21, 7793}, {32, 16914}, {37, 2998}, {38, 22190}, {69, 33029}, {99, 33062}, {148, 377}, {183, 33047}, {193, 5129}, {315, 17685}, {325, 33046}, {330, 1125}, {344, 27296}, {384, 5275}, {385, 405}, {452, 20065}, {474, 7783}, {519, 32095}, {536, 25614}, {551, 31999}, {908, 3177}, {1007, 33053}, {1078, 33063}, {1107, 30963}, {1500, 4704}, {1654, 10449}, {1698, 25264}, {1975, 16917}, {2478, 7785}, {2549, 17565}, {2996, 4208}, {3210, 29576}, {3294, 30114}, {3501, 17261}, {3552, 5277}, {3616, 21226}, {3622, 9263}, {3634, 32107}, {3661, 21071}, {3662, 29968}, {3663, 30063}, {3686, 20168}, {3691, 17027}, {3720, 23457}, {3734, 16913}, {3760, 16819}, {3761, 31996}, {3785, 33059}, {3912, 27288}, {3933, 33034}, {3995, 29593}, {4187, 7777}, {4364, 21025}, {4389, 20255}, {4443, 21700}, {4664, 20691}, {4681, 21868}, {4687, 20943}, {4699, 20888}, {4791, 21225}, {5047, 16998}, {5051, 16991}, {5084, 7774}, {5224, 21024}, {5276, 7787}, {6337, 33054}, {6381, 19565}, {6857, 17008}, {6872, 14712}, {7483, 17004}, {7751, 16996}, {7752, 33061}, {7754, 11108}, {7823, 11113}, {7864, 17670}, {7875, 17540}, {7891, 17694}, {7912, 17669}, {7938, 17550}, {9780, 17759}, {11185, 33030}, {11321, 16993}, {15589, 33040}, {16408, 31859}, {16853, 22253}, {16859, 17002}, {16865, 17001}, {16912, 16992}, {17030, 30998}, {17034, 17349}, {17230, 31035}, {17236, 21240}, {17275, 20170}, {17302, 27299}, {17343, 33297}, {17350, 17750}, {17379, 17499}, {17383, 30107}, {17490, 24603}, {19862, 32005}, {21218, 27287}, {22011, 24080}, {24450, 24717}, {25092, 27091}, {25261, 31087}, {25918, 30829}, {27250, 27267}, {27253, 27295}, {27260, 27281}, {27340, 29571}, {29966, 31004}, {32006, 33032}, {32815, 33058}, {32817, 33043}, {32818, 33042}

X(27270) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b + a^3 b^2 + a^2 b^3 + a b^4 + a^4 c + a^3 b c + 3 a^2 b^2 c + 3 a b^3 c + b^4 c + a^3 c^2 + 3 a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 3 a b c^3 + b^2 c^3 + a c^4 + b c^4 : :

X(27270) lies on these lines: {2, 6}, {37, 20932}, {192, 18697}, {238, 26064}, {1655, 17280}, {1918, 33083}, {2209, 32784}, {2309, 32783}, {2475, 5263}, {3770, 17289}, {3882, 17306}, {3948, 24958}, {5904, 19853}, {14210, 27248}, {16818, 17121}, {17257, 17481}, {17368, 17499}, {18792, 24931}, {20174, 33150}, {27261, 27290}, {27273, 27274}, {27276, 27282}, {27278, 27288}

X(27271) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    3 a^3 b - 2 a^2 b^2 + 3 a b^3 + 3 a^3 c - 7 a^2 b c + a b^2 c + 3 b^3 c - 2 a^2 c^2 + a b c^2 - 2 b^2 c^2 + 3 a c^3 + 3 b c^3 : :

X(27271) lies on these lines: {1, 2}, {37, 21605}, {192, 21432}

X(27272) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b + a b^4 + a^4 c - a^3 b c - 3 a^2 b^2 c + a b^3 c + b^4 c - 3 a^2 b c^2 - a b^2 c^2 + a b c^3 + a c^4 + b c^4 : :

X(27272) lies on these lines: {1, 2}, {37, 17789}, {192, 20432}, {194, 17776}, {213, 17778}, {345, 24621}, {514, 26854}, {2176, 18134}, {2273, 17379}, {3685, 23682}, {3747, 4645}, {4648, 27349}, {6645, 16050}, {16752, 17759}, {17144, 24789}, {21277, 26147}, {25504, 26042}, {27159, 27342}, {27252, 27268}, {27254, 27291}, {27265, 27290}, {31997, 32777}

X(27273) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b + a b^5 + a^5 c - a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c + b^5 c - a^3 b c^2 - a^2 b^2 c^2 - a b^3 c^2 - a^2 b c^3 - a b^2 c^3 + a b c^4 + a c^5 + b c^5 : :

X(27273) lies on these lines: {2, 31}, {37, 20643}, {192, 20629}, {1001, 17550}, {17084, 27097}, {17737, 30940}, {27248, 27252}, {27270, 27274}

X(27274) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^3 b + a^2 b^2 + a b^3 + a^3 c + 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(27274) lies on these lines: {1, 2}, {39, 17289}, {213, 30966}, {319, 20970}, {350, 25499}, {894, 16887}, {993, 17688}, {2140, 17291}, {3294, 17210}, {3678, 27495}, {3841, 17673}, {4253, 17368}, {5045, 31306}, {5074, 17248}, {5259, 16927}, {5263, 16060}, {13728, 26590}, {16604, 17385}, {16705, 25264}, {16738, 17200}, {17280, 25092}, {17759, 25599}, {23657, 24665}, {27060, 27283}, {27261, 27262}, {27270, 27273}, {27276, 27284}

X(27275) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    3 a^6 b - a^5 b^2 - 2 a^4 b^3 - 2 a^3 b^4 - a^2 b^5 + 3 a b^6 + 3 a^6 c - 5 a^5 b c - 3 a^4 b^2 c + 2 a^3 b^3 c - 3 a^2 b^4 c + 3 a b^5 c + 3 b^6 c - a^5 c^2 - 3 a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 + 2 a^3 b c^3 + 4 a^2 b^2 c^3 - 6 a b^3 c^3 - 2 b^4 c^3 - 2 a^3 c^4 - 3 a^2 b c^4 - 3 a b^2 c^4 - 2 b^3 c^4 - a^2 c^5 + 3 a b c^5 - b^2 c^5 + 3 a c^6 + 3 b c^6 : :

X(27275) lies on these lines:

X(27276) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c - a^5 b c - 2 a^4 b^2 c - 2 a^3 b^3 c - 2 a^2 b^4 c + a b^5 c + b^6 c - 2 a^4 b c^2 - 3 a^3 b^2 c^2 - 3 a^2 b^3 c^2 - 2 a b^4 c^2 - a^4 c^3 - 2 a^3 b c^3 - 3 a^2 b^2 c^3 - 4 a b^3 c^3 - b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(27276) lies on these lines:

X(27277) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^8 b + a^7 b^2 - a^6 b^3 - a^5 b^4 - a^4 b^5 - a^3 b^6 + a^2 b^7 + a b^8 + a^8 c + a^7 b c - 3 a^6 b^2 c - 3 a^5 b^3 c - a^3 b^5 c + a^2 b^6 c + 3 a b^7 c + b^8 c + a^7 c^2 - 3 a^6 b c^2 - 5 a^5 b^2 c^2 + 7 a^4 b^3 c^2 + 7 a^3 b^4 c^2 - a^2 b^5 c^2 + a b^6 c^2 + b^7 c^2 - a^6 c^3 - 3 a^5 b c^3 + 7 a^4 b^2 c^3 + 14 a^3 b^3 c^3 + 3 a^2 b^4 c^3 - 3 a b^5 c^3 - b^6 c^3 - a^5 c^4 + 7 a^3 b^2 c^4 + 3 a^2 b^3 c^4 - 4 a b^4 c^4 - b^5 c^4 - a^4 c^5 - a^3 b c^5 - a^2 b^2 c^5 - 3 a b^3 c^5 - b^4 c^5 - a^3 c^6 + a^2 b c^6 + a b^2 c^6 - b^3 c^6 + a^2 c^7 + 3 a b c^7 + b^2 c^7 + a c^8 + b c^8 : :

X(27277) lies on these lines:

X(27278) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^9 b - 2 a^5 b^5 + a b^9 + a^9 c - a^8 b c + a^7 b^2 c + 3 a^6 b^3 c - 4 a^5 b^4 c - 2 a^4 b^5 c + a^3 b^6 c - a^2 b^7 c + a b^8 c + b^9 c + a^7 b c^2 + 5 a^6 b^2 c^2 + a^5 b^3 c^2 - 6 a^4 b^4 c^2 - a^3 b^5 c^2 + a^2 b^6 c^2 - a b^7 c^2 + 3 a^6 b c^3 + a^5 b^2 c^3 - 8 a^4 b^3 c^3 - 4 a^3 b^4 c^3 + a^2 b^5 c^3 - a b^6 c^3 - 4 a^5 b c^4 - 6 a^4 b^2 c^4 - 4 a^3 b^3 c^4 - 2 a^2 b^4 c^4 - 2 a^5 c^5 - 2 a^4 b c^5 - a^3 b^2 c^5 + a^2 b^3 c^5 - 2 b^5 c^5 + a^3 b c^6 + a^2 b^2 c^6 - a b^3 c^6 - a^2 b c^7 - a b^2 c^7 + a b c^8 + a c^9 + b c^9 : :

X(27278) lies on these lines:

X(27279) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    5 a^6 b - a^5 b^2 - 4 a^4 b^3 - 4 a^3 b^4 - a^2 b^5 + 5 a b^6 + 5 a^6 c - 7 a^5 b c - 5 a^4 b^2 c + 2 a^3 b^3 c - 5 a^2 b^4 c + 5 a b^5 c + 5 b^6 c - a^5 c^2 - 5 a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 5 a b^4 c^2 - b^5 c^2 - 4 a^4 c^3 + 2 a^3 b c^3 + 4 a^2 b^2 c^3 - 10 a b^3 c^3 - 4 b^4 c^3 - 4 a^3 c^4 - 5 a^2 b c^4 - 5 a b^2 c^4 - 4 b^3 c^4 - a^2 c^5 + 5 a b c^5 - b^2 c^5 + 5 a c^6 + 5 b c^6 : :

X(27279) lies on these lines:

X(27280) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c - 3 a^5 b c + a^4 b^2 c + 6 a^3 b^3 c + a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 + a^4 b c^2 + 10 a^3 b^2 c^2 + 10 a^2 b^3 c^2 + a b^4 c^2 - b^5 c^2 + 6 a^3 b c^3 + 10 a^2 b^2 c^3 + 2 a b^3 c^3 + a^2 b c^4 + a b^2 c^4 - a^2 c^5 + a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(27280) lies on these lines:

X(27281) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b + a b^6 + a^6 c - a^5 b c + a^3 b^3 c + a b^5 c + b^6 c + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + a^3 b c^3 + 2 a^2 b^2 c^3 - a b^3 c^3 + a b c^5 + a c^6 + b c^6 : :

X(27281) lies on these lines: {2, 3}, {27260, 27269}

X(27282) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 3 a^3 b c + 4 a^2 b^2 c + a b^3 c + b^4 c - a^3 c^2 + 4 a^2 b 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(27282) lies on these lines: {2, 7}, {37, 322}, {141, 28778}, {192, 20895}, {220, 26671}, {242, 4194}, {346, 3948}, {1284, 2551}, {1463, 24954}, {3672, 27108}, {3965, 20173}, {4335, 6745}, {4517, 24752}, {5051, 26772}, {5552, 9791}, {17238, 27290}, {17261, 27544}, {26563, 26669}, {27248, 27275}, {27253, 27267}, {27270, 27276}

X(27283) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 a^5 b c - 2 a^4 b^2 c - 2 a^3 b^3 c - 2 a^2 b^4 c + a^5 c^2 - 2 a^4 b c^2 - 2 a^3 b^2 c^2 - 6 a^2 b^3 c^2 - 2 a b^4 c^2 + b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 - 6 a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 + a^2 c^5 + b^2 c^5 : :

X(27283) lies on these lines: {2, 12}, {37, 21581}, {192, 21405}, {3936, 27050}, {27038, 27096}, {27060, 27274}, {27170, 33298}, {27248, 27251}, {30867, 31996}

X(27284) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c - a^5 b c + a b^5 c + b^6 c - a^3 b^2 c^2 + a^2 b^3 c^2 - a^4 c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(27284) lies on these lines: {2, 36}, {37, 21587}, {192, 21411}, {27248, 27249}, {27274, 27276}

X(27285) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^3 b^3 + a^3 b^2 c + a^3 b c^2 + a b^3 c^2 + a^3 c^3 + a b^2 c^3 + b^3 c^3 : :

X(27285) lies on these lines: {2, 38}, {37, 561}, {75, 27035}, {76, 21814}, {192, 20889}, {354, 27158}, {1959, 27248}, {3726, 25505}, {3873, 26959}, {4359, 27091}, {4493, 24327}, {4687, 17149}, {4981, 17030}, {17017, 20148}, {17147, 27034}, {17486, 28592}, {21345, 25102}, {23538, 24679}, {26752, 32860}, {27020, 28606}, {27105, 30818}

X(27286) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^4 b c + 2 a^2 b^3 c + a b^4 c + b^5 c + 2 a^2 b^2 c^2 - 2 a^3 c^3 + 2 a^2 b c^3 - 2 b^3 c^3 + a b c^4 + a c^5 + b c^5 : :

X(27286) lies on these lines: {2, 7}, {37, 20928}, {192, 20237}, {345, 3948}, {1329, 1403}, {1402, 3436}, {4364, 19721}, {5552, 17594}, {5903, 19853}, {16878, 20076}, {17332, 19720}, {17596, 26364}, {24210, 27518}, {25906, 27410}, {27248, 27249}

X(27287) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^4 b c - 2 a^3 b^2 c + a b^4 c + b^5 c - 2 a^3 b c^2 - 2 a b^3 c^2 - 2 a^3 c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + a b c^4 + a c^5 + b c^5 : :

X(27287) lies on these lines: {2, 7}, {4, 228}, {5, 20760}, {8, 3191}, {12, 17555}, {27, 198}, {37, 92}, {55, 14004}, {189, 5308}, {192, 14213}, {281, 2052}, {498, 846}, {499, 32913}, {573, 22000}, {631, 22060}, {914, 27267}, {948, 6349}, {966, 26872}, {968, 1785}, {1473, 21554}, {1656, 22149}, {1896, 7952}, {1959, 27248}, {1985, 21319}, {3177, 16585}, {3869, 19853}, {3948, 17776}, {3980, 26364}, {4415, 19721}, {4468, 23787}, {4648, 26871}, {4687, 18750}, {4699, 20879}, {5051, 11681}, {5552, 32932}, {6051, 20220}, {6350, 18592}, {6360, 16577}, {6991, 23542}, {6998, 7085}, {9612, 26027}, {10478, 21361}, {11374, 25490}, {11427, 20752}, {12526, 16828}, {13411, 26091}, {14206, 27268}, {14552, 28797}, {18141, 28748}, {19795, 28811}, {21218, 27269}, {27253, 27275}, {27255, 27280}

X(27288) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c - a^2 b c - 3 a b^2 c - b^3 c - 2 a^2 c^2 - 3 a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(27288) lies on these lines: {2, 85}, {8, 192}, {9, 3212}, {10, 25242}, {37, 16284}, {65, 144}, {145, 25261}, {169, 17691}, {194, 4384}, {330, 16823}, {344, 20955}, {405, 3732}, {728, 17261}, {894, 30625}, {960, 20535}, {3617, 25237}, {3673, 27304}, {3691, 27484}, {3912, 27269}, {3959, 4419}, {4352, 16583}, {4416, 6738}, {4699, 20880}, {5273, 16609}, {5296, 31994}, {6604, 6646}, {7146, 18228}, {7754, 19851}, {10025, 19860}, {10405, 31359}, {17090, 18230}, {17236, 26530}, {17350, 32024}, {17451, 30946}, {17480, 21226}, {17760, 27538}, {20911, 27523}, {25935, 27184}, {27253, 27267}, {27270, 27278}

X(27289) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    (a b + a c - b c) (a^4 b^2 + a^2 b^4 - 2 a^4 b c + a^4 c^2 - 3 a^2 b^2 c^2 - 2 a b^3 c^2 - b^4 c^2 - 2 a b^2 c^3 + a^2 c^4 - b^2 c^4) : :

X(27289) lies on these lines: {2, 87}, {6, 21250}, {37, 21599}, {192, 21426}, {6376, 17289}, {27248, 27291}, {27252, 27264}

X(27290) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 3 a^3 b c + 3 a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 + 3 a^2 b c^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(27290) lies on these lines: {2, 45}, {37, 18151}, {149, 4557}, {192, 4858}, {219, 17349}, {3762, 27295}, {4466, 31053}, {4699, 20881}, {6646, 28748}, {17232, 28778}, {17238, 27282}, {21320, 30993}, {27036, 27321}, {27252, 27291}, {27254, 27268}, {27261, 27270}, {27265, 27272}, {27292, 27294}

X(27291) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^3 b c - 3 a^2 b^2 c + 2 a b^3 c + a^3 c^2 - 3 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(27291) lies on these lines: {2, 37}, {87, 24672}, {145, 872}, {726, 25492}, {740, 26029}, {4022, 31302}, {4903, 21080}, {5749, 26113}, {16826, 20146}, {16969, 17277}, {17030, 25072}, {17233, 21025}, {17269, 27111}, {17349, 21769}, {17375, 26799}, {18230, 26801}, {21330, 32937}, {22019, 24190}, {22220, 24349}, {25269, 27017}, {26149, 29627}, {27078, 29570}, {27248, 27289}, {27252, 27290}, {27253, 27298}, {27254, 27272}

X(27292) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    (b - c) (3 a^3 b - 3 a^2 b^2 + 3 a^3 c - 7 a^2 b c + 2 a b^2 c - 3 a^2 c^2 + 2 a b c^2 + 3 b^2 c^2) : :

X(27292) lies on these lines: {2, 900}, {37, 21606}, {192, 21433}, {3766, 27268}, {4526, 4699}, {4728, 27256}, {4928, 14408}, {14407, 21297}, {27290, 27294}

X(27293) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c - a^3 b c - 2 a^2 b^2 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(27293) lies on these lines: {1, 25301}, {2, 649}, {37, 20952}, {192, 20909}, {513, 25511}, {514, 26854}, {647, 3766}, {650, 20954}, {659, 25686}, {661, 26114}, {665, 25594}, {669, 21301}, {24459, 27527}, {812, 26049}, {1577, 27045}, {4040, 23791}, {4083, 25636}, {4391, 28374}, {4468, 23787}, {4521, 23810}, {4728, 27346}, {4928, 27139}, {7192, 14349}, {8632, 27168}, {8640, 21260}, {8655, 31291}, {10453, 25128}, {19853, 29350}, {20908, 25271}, {21834, 25638}, {25666, 27014}, {26148, 30968}, {26985, 27114}

X(27294) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    (b - c) (a^5 b - a^2 b^4 + a^5 c - a^4 b c - 2 a^2 b^3 c + a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(27294) lies on these lines: {2, 659}, {37, 21612}, {192, 21439}, {26114, 27265}, {26248, 27193}, {26854, 27258}, {27027, 27074}, {27290, 27292}

X(27295) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^3 b - a^2 b^2 + a b^3 + a^3 c - 3 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 3 b^2 c^2 + a c^3 + b c^3 : :

X(27295) lies on these lines: {2, 668}, {37, 18159}, {148, 20533}, {150, 17300}, {192, 1111}, {335, 21232}, {1018, 4440}, {1086, 4595}, {3761, 17280}, {3762, 27290}, {4699, 4986}, {4704, 20568}, {9507, 19951}, {10027, 20335}, {17232, 24222}, {21226, 28742}, {23354, 33148}, {24190, 29697}, {27253, 27269}

X(27296) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^4 b + a b^4 + a^4 c - a^3 b c + 3 a^2 b^2 c + a b^3 c + b^4 c + 3 a^2 b c^2 + a b^2 c^2 + a b c^3 + a c^4 + b c^4 : :

X(27296) lies on these lines: {2, 7}, {37, 17788}, {192, 21442}, {344, 27269}, {966, 27321}, {2273, 17349}, {2663, 29837}, {3948, 17280}, {4388, 20964}, {4503, 17300}, {26752, 27039}, {27248, 27289}, {27252, 27268}, {27261, 27270}

X(27297) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    a^5 b - a^3 b^3 + a b^5 + a^5 c - a^4 b c - a^3 b^2 c + a b^4 c + b^5 c - a^3 b c^2 - a b^3 c^2 - a^3 c^3 - a b^2 c^3 - b^3 c^3 + a b c^4 + a c^5 + b c^5 : :

X(27297) lies on these lines: {2, 896}, {37, 20944}, {192, 20904}, {26114, 27265}, {1580, 16818}, {1959, 27248}

X(27298) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 59

Barycentrics    2 a^3 b^3 + 3 a^3 b^2 c + a^2 b^3 c + 3 a^3 b c^2 + a^2 b^2 c^2 + 3 a b^3 c^2 + 2 a^3 c^3 + a^2 b c^3 + 3 a b^2 c^3 + 2 b^3 c^3 : :

X(27298) lies on these lines: {2, 38}, {37, 18135}, {192, 3760}, {330, 31996}, {1909, 4687}, {2275, 4698}, {3774, 4441}, {4430, 27157}, {4699, 27091}, {6381, 19565}, {27034, 28605}, {27248, 27252}, {27253, 27291}

leftri

Collineation mappings involving Gemini triangle 60: X(27299)-X(27351)

rightri

Extending the preambles just before X(24537) and X(26153), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 60, as in centers X(27299)-X(27351). Then

m(X) = a(a b + a c + b c) x + b (a b - a c + b c) y + c (a c - a b + b c) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 7, 2018)

X(27299) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    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(27299) lies on these lines: {1, 2}, {6, 20255}, {7, 17499}, {35, 17696}, {69, 21240}, {75, 16583}, {76, 4000}, {169, 894}, {192, 16600}, {213, 21281}, {304, 30748}, {344, 1500}, {673, 7770}, {958, 16060}, {986, 17755}, {993, 22267}, {1100, 25504}, {1107, 24735}, {1220, 11321}, {1376, 16061}, {1449, 26106}, {1468, 24602}, {1478, 17680}, {1730, 26065}, {1930, 21216}, {2901, 27480}, {3501, 17353}, {3618, 17750}, {3673, 25994}, {3684, 24549}, {3730, 26685}, {3739, 16605}, {3761, 24790}, {3780, 30945}, {3868, 26562}, {3875, 21071}, {3877, 26689}, {3948, 19785}, {3959, 24282}, {4044, 30699}, {4253, 24170}, {4357, 26041}, {4361, 21024}, {4429, 6656}, {4441, 27040}, {4699, 16611}, {4972, 17550}, {5080, 16910}, {5247, 24586}, {5839, 33297}, {6376, 16706}, {8074, 27325}, {9798, 11329}, {10446, 20606}, {13740, 20172}, {14064, 20544}, {16062, 26558}, {16589, 17321}, {16609, 27309}, {16704, 27185}, {17062, 25521}, {17065, 23664}, {17279, 20691}, {17302, 27269}, {17356, 25102}, {17357, 21868}, {17366, 21025}, {17379, 27169}, {17384, 25614}, {17489, 31130}, {17671, 26590}, {17681, 26687}, {17686, 24596}, {17753, 24514}, {17756, 27109}, {20553, 26099}, {20963, 30962}, {24174, 24631}, {25264, 27523}, {27000, 27064}, {27268, 28594}, {27300, 27336}, {27302, 27306}, {27303, 27305}, {27341, 27343}, {31060, 33150}

X(27299) = anticomplement of X(30110)
X(27299) = {X(2),X(8)}-harmonic conjugate of X(27248)

X(27300) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c + a^5 b c - a^4 b^2 c - a^2 b^4 c - a b^5 c + b^6 c - a^4 b c^2 - a b^4 c^2 - a^4 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(27300) lies on these lines: {2, 3}, {894, 1729}, {5723, 27304}, {5826, 26965}, {17368, 27324}, {27299, 27336}, {27303, 27310}

X(27301) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c + 3 a^5 b c - a^4 b^2 c - 2 a^3 b^3 c - a^2 b^4 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 - 2 a^3 b c^3 + 4 a^2 b^2 c^3 + 2 a b^3 c^3 - a^2 b c^4 - a b^2 c^4 - a^2 c^5 - a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(27301) lies on these lines: {2, 3}, {241, 19804}, {3002, 27317}, {17030, 27305}, {20110, 22126}, {27335, 27337}

X(27302) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 - 2 a^5 b c + 2 a^3 b^3 c + a^5 c^2 - 4 a^3 b^2 c^2 - 4 a^2 b^3 c^2 + b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 - 4 a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + a^2 c^5 + b^2 c^5 : :

X(27302) lies on these lines: {2, 3}, {25242, 27397}, {26100, 27008}, {27009, 27146}, {27299, 27306}

X(27303) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b + a b^4 + a^4 c + a^3 b c - a b^3 c + b^4 c - a b c^3 + a c^4 + b c^4 : :

X(27303) lies on these lines: {2, 6}, {894, 16551}, {1468, 3836}, {2975, 4429}, {4000, 17148}, {4699, 27321}, {18805, 20274}, {21257, 29662}, {27299, 27305}, {27300, 27310}, {27309, 27313}, {27314, 27341}, {27342, 27343}

X(27304) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c + a^2 b c - 3 a b^2 c + b^3 c - 2 a^2 c^2 - 3 a b c^2 - 2 b^2 c^2 + a c^3 + b c^3 : :

X(27304) lies on these lines: {1, 2}, {7, 17050}, {9, 20257}, {37, 17158}, {63, 27000}, {72, 27484}, {75, 1212}, {85, 4875}, {144, 16552}, {192, 16601}, {218, 17349}, {220, 17277}, {241, 19804}, {274, 279}, {277, 330}, {333, 5228}, {344, 17144}, {346, 17143}, {391, 26125}, {673, 958}, {894, 16572}, {908, 27183}, {956, 17682}, {1107, 4000}, {1213, 16518}, {2975, 4209}, {3160, 17077}, {3177, 20880}, {3295, 17687}, {3662, 24181}, {3672, 5283}, {3673, 27288}, {3686, 25521}, {3691, 30946}, {3693, 31269}, {3945, 20963}, {4195, 20172}, {4307, 16476}, {4359, 24635}, {4441, 27523}, {4452, 25264}, {4461, 32104}, {4513, 32008}, {4699, 20435}, {4859, 24215}, {5082, 20533}, {5278, 27142}, {5723, 27300}, {6706, 16284}, {7176, 8732}, {9708, 17681}, {10481, 24199}, {11415, 26839}, {15853, 30854}, {16604, 24737}, {16704, 27172}, {16969, 17337}, {17175, 26818}, {17278, 17448}, {17671, 24390}, {17784, 23407}, {18206, 21454}, {19789, 31036}, {20367, 27171}, {21010, 26040}, {21264, 24735}, {27318, 27348}, {27325, 27339}, {27326, 27337}, {27343, 27351}, {27544, 32087}

X(27305) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - a^3 b c - 3 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 - 3 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(27305) lies on these lines: {2, 7}, {192, 25065}, {344, 25601}, {499, 3821}, {1210, 26057}, {1738, 10527}, {3008, 27317}, {3755, 10529}, {4384, 24778}, {4419, 28748}, {4652, 26123}, {4684, 10528}, {4699, 27342}, {17030, 27301}, {17261, 28778}, {17268, 27544}, {17298, 27267}, {20358, 24655}, {27299, 27303}, {27321, 27343}, {27514, 29579}

X(27306) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b^2 - 2 a^3 b^3 + a^2 b^4 - 2 a^4 b c + 3 a^3 b^2 c + a^2 b^3 c + a^4 c^2 + 3 a^3 b c^2 - 8 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + b^2 c^4 : :

X(27306) lies on these lines: {2, 11}, {4763, 27344}, {27299, 27302}

X(27307) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^8 b - a^5 b^4 - a^4 b^5 + a b^8 + a^8 c + a^7 b c + a^5 b^3 c - 2 a^4 b^4 c - a^3 b^5 c - a b^7 c + b^8 c + a^5 b c^3 + a b^5 c^3 - a^5 c^4 - 2 a^4 b c^4 - 2 a b^4 c^4 - b^5 c^4 - a^4 c^5 - a^3 b c^5 + a b^3 c^5 - b^4 c^5 - a b c^7 + a c^8 + b c^8 : :

X(27307) lies on these lines: {2, 3}

X(27308) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^8 b - a^5 b^4 - a^4 b^5 + a b^8 + a^8 c + a^7 b c + a^5 b^3 c - 2 a^4 b^4 c - a^3 b^5 c - a b^7 c + b^8 c + a^4 b^3 c^2 + a^3 b^4 c^2 + a^5 b c^3 + a^4 b^2 c^3 - a^3 b^3 c^3 + a^2 b^4 c^3 + a b^5 c^3 - a^5 c^4 - 2 a^4 b c^4 + a^3 b^2 c^4 + a^2 b^3 c^4 - 2 a b^4 c^4 - b^5 c^4 - a^4 c^5 - a^3 b c^5 + a b^3 c^5 - b^4 c^5 - a b c^7 + a c^8 + b c^8 : :

X(27308) lies on these lines: {2, 3}

X(27309) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b + a b^5 + a^5 c + a^4 b c - a b^4 c + b^5 c - a b c^4 + a c^5 + b c^5 : :

X(27309) lies on these lines: {2, 31}, {3925, 11321}, {11343, 26590}, {16060, 32773}, {16609, 27299}, {27303, 27313}

X(27310) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b + a b^6 + a^6 c + a^5 b c - a b^5 c + b^6 c - a b c^5 + a c^6 + b c^6 : :

X(27310) lies on these lines: {2, 32}, {27300, 27303}, {27318, 27333}

X(27311) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    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(27311) lies on these lines: {2, 37}, {6, 27017}, {38, 25106}, {239, 27145}, {740, 26094}, {894, 27107}, {1921, 27162}, {3618, 26975}, {3662, 26772}, {3697, 26029}, {3759, 17178}, {3763, 27044}, {3993, 19847}, {4022, 25277}, {4363, 27078}, {4384, 16738}, {4643, 26857}, {4648, 27159}, {16609, 17077}, {17227, 26756}, {17257, 27036}, {17276, 26799}, {17291, 27095}, {17305, 27111}, {17336, 26769}, {17366, 26979}, {17367, 26963}, {17380, 27166}, {18600, 21615}, {24046, 24349}, {24325, 26030}, {27299, 27303}, {27312, 27324}, {27320, 27342}, {27327, 27337}

X(27312) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b^3 + a^3 b^4 + a^4 b^2 c + a^2 b^4 c + a^4 b c^2 + a b^4 c^2 + a^4 c^3 - 2 a b^3 c^3 + b^4 c^3 + a^3 c^4 + a^2 b c^4 + a b^2 c^4 + b^3 c^4 : :

X(27312) lies on these lines: {2, 39}, {148, 26124}, {2896, 26058}, {3662, 24170}, {17128, 25520}, {18792, 27145}, {26963, 29455}, {27101, 27119}, {27300, 27303}, {27311, 27324}

X(27313) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^4 b c + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + a^4 c^2 + a^3 b c^2 - a b^3 c^2 + b^4 c^2 - a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27313) lies on these lines: {1, 2}, {940, 27169}, {16704, 27152}, {17149, 26978}, {21877, 27109}, {24789, 27019}, {25123, 30028}, {27047, 32773}, {27303, 27309}

X(27314) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b^2 + a^2 b^4 + 2 a^4 b c - 2 a^2 b^3 c + 2 a b^4 c + a^4 c^2 + a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(27314) lies on these lines: {1, 2}, {5372, 27152}, {27303, 27341}, {27323, 27337}

X(27315) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    2 a^4 b - a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 a^4 c - 4 a b^3 c + 2 b^4 c - a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - 4 a b c^3 - b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(27315) lies on these lines: {2, 44}, {4369, 27345}, {17077, 27102}, {27299, 27303}, {27321, 27342}

X(27316) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + a^4 c - 3 a^3 b c - 5 a b^3 c + b^4 c - 2 a^3 c^2 + 4 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 5 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(27316) lies on these lines: {2, 45}, {27299, 27303}

X(27317) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    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 - 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(27317) lies on these lines: {1, 25589}, {2, 6}, {87, 33138}, {192, 26059}, {238, 10527}, {894, 1723}, {1001, 10529}, {1212, 20171}, {1714, 5145}, {1743, 27254}, {2309, 33137}, {3002, 27301}, {3008, 27305}, {3015, 27331}, {3286, 4190}, {3434, 20992}, {3875, 25601}, {4699, 20435}, {5723, 27342}, {8053, 20075}, {10198, 28650}, {11240, 16484}, {16468, 26363}, {17364, 25521}, {20072, 26125}, {21384, 29967}, {22343, 24892}, {26685, 28797}

X(27318) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^2 b^2 - 2 a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 - b^2 c^2 : :

X(27318) lies on these lines: {2, 39}, {3, 17000}, {6, 16917}, {10, 330}, {32, 33062}, {75, 16604}, {83, 16913}, {87, 23652}, {99, 16914}, {115, 33061}, {148, 2478}, {192, 1125}, {193, 17580}, {257, 24174}, {315, 17565}, {377, 7785}, {385, 474}, {386, 17379}, {404, 7793}, {405, 7783}, {442, 7777}, {443, 7774}, {519, 31999}, {551, 32095}, {574, 33063}, {894, 978}, {1078, 16996}, {1269, 25535}, {1506, 33060}, {1509, 20145}, {1575, 24656}, {1975, 16918}, {2092, 20146}, {2271, 20147}, {2548, 33030}, {2549, 17685}, {3002, 27301}, {3008, 27340}, {3177, 29628}, {3210, 17397}, {3314, 17670}, {3329, 11321}, {3616, 17759}, {3617, 9263}, {3624, 25264}, {3634, 32005}, {3662, 29991}, {3815, 33045}, {3963, 26077}, {4190, 14712}, {4253, 6629}, {4359, 20899}, {4461, 26111}, {4699, 17030}, {5013, 33047}, {5021, 20142}, {5022, 20154}, {5024, 33036}, {5069, 25457}, {5254, 33046}, {5275, 7839}, {5277, 7766}, {6376, 25109}, {6384, 22028}, {6904, 20065}, {7735, 33054}, {7736, 33028}, {7738, 33029}, {7754, 16408}, {7760, 16995}, {7787, 16915}, {7806, 17694}, {7823, 11112}, {7824, 16992}, {9605, 33035}, {9780, 21226}, {11108, 31859}, {13747, 17004}, {15048, 33034}, {16602, 25994}, {16710, 16887}, {16827, 17754}, {16863, 22253}, {16991, 17674}, {16997, 17531}, {17002, 17572}, {17008, 17567}, {17023, 17490}, {17065, 23414}, {17148, 19858}, {17358, 30110}, {17367, 21216}, {17480, 19868}, {17499, 17749}, {19804, 25918}, {19862, 32107}, {20888, 30998}, {21223, 26037}, {22036, 27494}, {23807, 26984}, {24514, 27627}, {25092, 27268}, {26223, 27646}, {26959, 32092}, {27166, 28605}, {27304, 27348}, {27310, 27333}, {31404, 33056}, {31406, 33033}

X(27319) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b + a^4 b^2 + a^2 b^4 + a b^5 + a^5 c + 3 a^4 b c + 2 a^3 b^2 c + a b^4 c + b^5 c + a^4 c^2 + 2 a^3 b c^2 - a^2 b^2 c^2 + b^4 c^2 + a^2 c^4 + a b c^4 + b^2 c^4 + a c^5 + b c^5 : :

X(27319) lies on these lines: {2, 6}, {226, 17499}, {894, 1762}, {958, 26117}, {977, 19851}, {1107, 19786}, {1220, 3925}, {1330, 2887}, {3770, 3772}, {4281, 25650}, {16609, 27299}, {17789, 27321}, {19804, 27349}, {21384, 25527}, {23751, 27345}, {27328, 27334}

X(27320) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b + a^3 b^2 + a^2 b^3 + a b^4 + a^4 c + 3 a^3 b c + a^2 b^2 c + a b^3 c + b^4 c + a^3 c^2 + a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 + a c^4 + b c^4 : :

X(27320) lies on these lines: {2, 6}, {274, 24919}, {314, 24958}, {894, 1781}, {1269, 33150}, {1405, 28780}, {1655, 17302}, {2475, 4429}, {3662, 17499}, {3770, 16706}, {3836, 26131}, {4026, 5260}, {4446, 33170}, {4699, 16611}, {16818, 17326}, {17373, 27248}, {17391, 30110}, {21035, 33166}, {21221, 26222}, {21257, 29631}, {21857, 32779}, {26064, 32784}, {27311, 27342}, {27323, 27324}, {27326, 27334}, {27330, 27340}

X(27321) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b + a b^4 + a^4 c + a^3 b c - a^2 b^2 c - a b^3 c + b^4 c - a^2 b c^2 - 3 a b^2 c^2 - a b c^3 + a c^4 + b c^4 : :

X(27321) lies on these lines: {1, 2}, {514, 27345}, {966, 27296}, {1324, 19308}, {1500, 25059}, {2975, 24610}, {3772, 6376}, {4699, 27303}, {6645, 16054}, {14829, 20255}, {15149, 17927}, {17789, 27319}, {20691, 33116}, {20917, 24789}, {21277, 26081}, {27036, 27290}, {27305, 27343}, {27315, 27342}

X(27322) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (b - c) (a^5 b - a^2 b^4 + a^5 c + a^4 b c - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(27322) lies on these lines: {2, 667}, {514, 27347}, {21263, 21304}, {27323, 27336}

X(27323) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b + a b^5 + a^5 c + a^4 b c - a^3 b^2 c - a^2 b^3 c - a b^4 c + b^5 c - a^3 b c^2 + a^2 b^2 c^2 - a b^3 c^2 - a^2 b c^3 - a b^2 c^3 - a b c^4 + a c^5 + b c^5 : :

X(27323) lies on these lines: {2, 31}, {11349, 26582}, {27299, 27303}, {27314, 27337}, {27320, 27324}, {27322, 27336}

X(27324) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^3 b + a^2 b^2 + a b^3 + a^3 c + a^2 b c + b^3 c + a^2 c^2 + b^2 c^2 + a c^3 + b c^3 : :

X(27324) lies on these lines: {1, 2}, {894, 24170}, {1574, 17289}, {1655, 25599}, {3822, 17673}, {3934, 16706}, {4015, 27495}, {4026, 33034}, {7951, 16906}, {17368, 27300}, {17688, 25440}, {18107, 21260}, {19786, 30819}, {21904, 33297}, {23657, 24748}, {24046, 31317}, {25264, 27040}, {26962, 27000}, {27311, 27312}, {27320, 27323}, {27326, 27336}

X(27325) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    3 a^6 b - a^5 b^2 - 2 a^4 b^3 - 2 a^3 b^4 - a^2 b^5 + 3 a b^6 + 3 a^6 c + 5 a^5 b c - 3 a^4 b^2 c - 2 a^3 b^3 c - 3 a^2 b^4 c - 3 a b^5 c + 3 b^6 c - a^5 c^2 - 3 a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 - 2 a^3 b c^3 + 4 a^2 b^2 c^3 + 6 a b^3 c^3 - 2 b^4 c^3 - 2 a^3 c^4 - 3 a^2 b c^4 - 3 a b^2 c^4 - 2 b^3 c^4 - a^2 c^5 - 3 a b c^5 - b^2 c^5 + 3 a c^6 + 3 b c^6 : :

X(27325) lies on these lines:

X(27326) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c + a^5 b c - 2 a^4 b^2 c - 2 a^3 b^3 c - 2 a^2 b^4 c - a b^5 c + b^6 c - 2 a^4 b c^2 - a^3 b^2 c^2 - a^2 b^3 c^2 - 2 a b^4 c^2 - a^4 c^3 - 2 a^3 b c^3 - a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(27326) lies on these lines:

X(27327) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^8 b - a^5 b^4 - a^4 b^5 + a b^8 + a^8 c + a^7 b c + a^5 b^3 c - 2 a^4 b^4 c - a^3 b^5 c - a b^7 c + b^8 c + 2 a^4 b^3 c^2 + 2 a^3 b^4 c^2 + a^5 b c^3 + 2 a^4 b^2 c^3 - 2 a^3 b^3 c^3 + 2 a^2 b^4 c^3 + a b^5 c^3 - a^5 c^4 - 2 a^4 b c^4 + 2 a^3 b^2 c^4 + 2 a^2 b^3 c^4 - 2 a b^4 c^4 - b^5 c^4 - a^4 c^5 - a^3 b c^5 + a b^3 c^5 - b^4 c^5 - a b c^7 + a c^8 + b c^8 : :

X(27327) lies on these lines:

X(27328) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^8 b + a^7 b^2 - a^6 b^3 - a^5 b^4 - a^4 b^5 - a^3 b^6 + a^2 b^7 + a b^8 + a^8 c + 3 a^7 b c + 3 a^6 b^2 c - a^5 b^3 c - 4 a^4 b^4 c - 3 a^3 b^5 c - a^2 b^6 c + a b^7 c + b^8 c + a^7 c^2 + 3 a^6 b c^2 + a^5 b^2 c^2 + 3 a^4 b^3 c^2 + 3 a^3 b^4 c^2 - 3 a^2 b^5 c^2 - a b^6 c^2 + b^7 c^2 - a^6 c^3 - a^5 b c^3 + 3 a^4 b^2 c^3 + 10 a^3 b^3 c^3 + 7 a^2 b^4 c^3 - a b^5 c^3 - b^6 c^3 - a^5 c^4 - 4 a^4 b c^4 + 3 a^3 b^2 c^4 + 7 a^2 b^3 c^4 - b^5 c^4 - a^4 c^5 - 3 a^3 b c^5 - 3 a^2 b^2 c^5 - a b^3 c^5 - b^4 c^5 - a^3 c^6 - a^2 b c^6 - a b^2 c^6 - b^3 c^6 + a^2 c^7 + a b c^7 + b^2 c^7 + a c^8 + b c^8 : :

X(27328) lies on these lines:

X(27329) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^9 b + a^8 b^2 - a^6 b^4 - 2 a^5 b^5 - a^4 b^6 + a^2 b^8 + a b^9 + a^9 c + 3 a^8 b c + 2 a^7 b^2 c - 2 a^5 b^4 c - 4 a^4 b^5 c - 2 a^3 b^6 c + a b^8 c + b^9 c + a^8 c^2 + 2 a^7 b c^2 + 3 a^6 b^2 c^2 + 2 a^5 b^3 c^2 - a^2 b^6 c^2 + b^8 c^2 + 2 a^5 b^2 c^3 + 6 a^4 b^3 c^3 + 6 a^3 b^4 c^3 + 2 a^2 b^5 c^3 - a^6 c^4 - 2 a^5 b c^4 + 6 a^3 b^3 c^4 + 4 a^2 b^4 c^4 - 2 a b^5 c^4 - b^6 c^4 - 2 a^5 c^5 - 4 a^4 b c^5 + 2 a^2 b^3 c^5 - 2 a b^4 c^5 - 2 b^5 c^5 - a^4 c^6 - 2 a^3 b c^6 - a^2 b^2 c^6 - b^4 c^6 + a^2 c^8 + a b c^8 + b^2 c^8 + a c^9 + b c^9 : :

X(27329) lies on these lines:

X(27330) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^9 b - 2 a^5 b^5 + a b^9 + a^9 c + a^8 b c - 3 a^7 b^2 c - a^6 b^3 c + 2 a^5 b^4 c + a^3 b^6 c - a^2 b^7 c - a b^8 c + b^9 c - 3 a^7 b c^2 - a^6 b^2 c^2 + a^5 b^3 c^2 - 2 a^4 b^4 c^2 + 3 a^3 b^5 c^2 + 3 a^2 b^6 c^2 - a b^7 c^2 - a^6 b c^3 + a^5 b^2 c^3 - 4 a^4 b^3 c^3 - 8 a^3 b^4 c^3 + a^2 b^5 c^3 + 3 a b^6 c^3 + 2 a^5 b c^4 - 2 a^4 b^2 c^4 - 8 a^3 b^3 c^4 - 6 a^2 b^4 c^4 - 2 a b^5 c^4 - 2 a^5 c^5 + 3 a^3 b^2 c^5 + a^2 b^3 c^5 - 2 a b^4 c^5 - 2 b^5 c^5 + a^3 b c^6 + 3 a^2 b^2 c^6 + 3 a b^3 c^6 - a^2 b c^7 - a b^2 c^7 - a b c^8 + a c^9 + b c^9 : :

X(27330) lies on these lines:

X(27331) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    5 a^6 b - a^5 b^2 - 4 a^4 b^3 - 4 a^3 b^4 - a^2 b^5 + 5 a b^6 + 5 a^6 c + 7 a^5 b c - 5 a^4 b^2 c - 2 a^3 b^3 c - 5 a^2 b^4 c - 5 a b^5 c + 5 b^6 c - a^5 c^2 - 5 a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 5 a b^4 c^2 - b^5 c^2 - 4 a^4 c^3 - 2 a^3 b c^3 + 4 a^2 b^2 c^3 + 10 a b^3 c^3 - 4 b^4 c^3 - 4 a^3 c^4 - 5 a^2 b c^4 - 5 a b^2 c^4 - 4 b^3 c^4 - a^2 c^5 - 5 a b c^5 - b^2 c^5 + 5 a c^6 + 5 b c^6 : :

X(27331) lies on these lines:

X(27332) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b - a^5 b^2 - a^2 b^5 + a b^6 + a^6 c + 3 a^5 b c + a^4 b^2 c + 2 a^3 b^3 c + a^2 b^4 c - a b^5 c + b^6 c - a^5 c^2 + a^4 b c^2 + 6 a^3 b^2 c^2 + 6 a^2 b^3 c^2 + a b^4 c^2 - b^5 c^2 + 2 a^3 b c^3 + 6 a^2 b^2 c^3 + 6 a b^3 c^3 + a^2 b c^4 + a b^2 c^4 -a^2 c^5 - a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(27332) lies on these lines:

X(27333) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b + a b^6 + a^6 c + a^5 b c - a^3 b^3 c - a b^5 c + b^6 c + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - a^3 b c^3 + 2 a^2 b^2 c^3 + a b^3 c^3 - a b c^5 + a c^6 + b c^6 : :

X(27333) lies on these lines: {2, 3}, {27310, 27318}

X(27334) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c + 3 a^3 b c - a b^3 c + b^4 c - a^3 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(27334) lies on these lines: {2, 7}, {75, 1108}, {86, 2256}, {145, 26818}, {192, 25083}, {956, 1010}, {1418, 25971}, {1604, 11329}, {3086, 3923}, {3672, 27161}, {3685, 14986}, {3751, 7080}, {3945, 27514}, {4699, 20435}, {6180, 26671}, {7190, 24559}, {7229, 28797}, {8074, 27299}, {15149, 31917}, {16571, 33137}, {16826, 25601}, {17316, 27544}, {17379, 20742}, {19843, 24342}, {20171, 25242}, {20358, 24669}, {20905, 26690}, {23125, 27644}, {24635, 24993}, {25099, 31225}, {26029, 27102}, {26543, 33298}, {27319, 27328}, {27320, 27326}

X(27335) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b^2 - a^4 b^3 - a^3 b^4 + a^2 b^5 - 2 a^5 b c - 2 a^4 b^2 c + 2 a^3 b^3 c - 2 a^2 b^4 c + a^5 c^2 - 2 a^4 b c^2 - 6 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 2 a b^4 c^2 + b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 - 2 a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 + a^2 c^5 + b^2 c^5 : :

X(27335) lies on these lines:

X(27336) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^6 b - a^4 b^3 - a^3 b^4 + a b^6 + a^6 c + a^5 b c - a b^5 c + b^6 c + a^3 b^2 c^2 - a^2 b^3 c^2 - a^4 c^3 - a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(27336) lies on these lines:

X(27337) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b - a^4 b^2 - a^2 b^4 + a b^5 + a^5 c - a^4 b c - a^3 b^2 c + a^2 b^3 c - 3 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 - 3 a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(27337) lies on these lines:

X(27338) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b - 2 a^3 b^3 + a b^5 + a^5 c + a^4 b c + 2 a^3 b^2 c - a b^4 c + b^5 c + 2 a^3 b c^2 - 2 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - a b c^4 + a c^5 + b c^5 : :

X(27338) lies on these lines:

X(27339) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (a + b - c) (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 - 2 b^2 c^2 - a c^3 - b c^3) : :

X(27339) lies on these lines:

X(27340) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^3 b - 2 a^2 b^2 + a b^3 + a^3 c + 3 a^2 b c + a b^2 c - b^3 c - 2 a^2 c^2 + a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 : :

X(27340) lies on these lines: {1, 87}, {2, 85}, {7, 3061}, {9, 7176}, {37, 24654}, {65, 20535}, {142, 7185}, {144, 960}, {145, 25244}, {193, 20007}, {220, 6645}, {294, 16915}, {346, 18156}, {474, 3732}, {1219, 4461}, {1278, 17158}, {1323, 17368}, {1655, 5308}, {2124, 9312}, {3008, 27318}, {3160, 5749}, {3210, 17014}, {3212, 17754}, {3622, 25237}, {3662, 10481}, {4416, 12447}, {4518, 30669}, {4699, 20435}, {4741, 7960}, {5088, 17691}, {5222, 17490}, {5836, 9311}, {6743, 17363}, {8583, 30625}, {10025, 19861}, {16572, 17349}, {16601, 27268}, {17291, 21314}, {19719, 31036}, {20060, 31080}, {21384, 27484}, {23988, 24555}, {25082, 27253}, {25930, 27064}, {27269, 29571}, {27320, 27330}

X(27341) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b + a b^3 + a^3 c + a^2 b c + a b^2 c - b^3 c + a b c^2 + a c^3 - b c^3) : :

X(27341) lies on these lines:

X(27342) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - a^3 b c + a^2 b^2 c - 3 a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 5 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(27342) lies on these lines:

X(27343) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    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 - 5 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 2 a b c^3 + b^2 c^3 : :

X(27343) lies on these lines: {1, 24765}, {2, 37}, {740, 26093}, {4719, 26111}, {6686, 24182}, {9335, 25295}, {16816, 27017}, {17232, 20255}, {17349, 27107}, {22343, 24753}, {24174, 24349}, {25106, 32937}, {26106, 27487}, {27299, 27341}, {27303, 27342}, {27304, 27351}, {27305, 27321}

X(27344) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (b - c) (3 a^3 b - 3 a^2 b^2 + 3 a^3 c + a^2 b c - 2 a b^2 c - 3 a^2 c^2 - 2 a b c^2 + 3 b^2 c^2) : :

X(27344) lies on these lines:

X(27345) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (b - c) (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(27345) lies on these lines:

X(27346) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (b - c) (-a b - a c + b c) (-a^3 + a b^2 - a b c + b^2 c + a c^2 + b c^2) : :

X(27346) lies on these lines: {2, 650}, {192, 25098}, {321, 25271}, {513, 24755}, {514, 28758}, {3662, 23806}, {3835, 17217}, {4017, 25380}, {4147, 24749}, {4369, 27315}, {4449, 24675}, {4728, 27293}, {20295, 27114}, {27045, 27138}

X(27347) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    (b - c) (a^5 b - a^2 b^4 + a^5 c + a^4 b c - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(27347) lies on these lines: {2, 659}, {514, 27322}, {1638, 27342}, {4369, 27315}

X(27348) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^3 b - 3 a^2 b^2 + a b^3 + a^3 c + 3 a^2 b c - a b^2 c + b^3 c - 3 a^2 c^2 - a b c^2 - b^2 c^2 + a c^3 + b c^3 : :

X(27348) lies on these lines: {2, 668}, {101, 17349}, {192, 24036}, {1655, 26964}, {2241, 17695}, {3888, 23456}, {3960, 27342}, {4440, 17761}, {9259, 17277}, {27304, 27318}

X(27349) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^4 b + a b^4 + a^4 c + a^3 b c + a^2 b^2 c - a b^3 c + b^4 c + a^2 b c^2 + 3 a b^2 c^2 - a b c^3 + a c^4 + b c^4 : :

X(27349) lies on these lines: {2, 7}, {980, 17302}, {4000, 24621}, {4648, 27272}, {4699, 27303}, {19804, 27319}, {20358, 24678}, {27299, 27341}, {27311, 27320}

X(27350) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    a^5 b - a^3 b^3 + a b^5 + a^5 c + a^4 b c - a^2 b^3 c - a b^4 c + b^5 c - a^3 c^3 - a^2 b c^3 - b^3 c^3 - a b c^4 + a c^5 + b c^5 : :

X(27350) lies on these lines: {2, 896}, {4369, 27315}, {16609, 27299}

X(27351) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 60

Barycentrics    2 a^3 b^3 + a^3 b^2 c + 3 a^2 b^3 c + a^3 b c^2 - a^2 b^2 c^2 + a b^3 c^2 + 2 a^3 c^3 + 3 a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 : :

X(27351) lies on these lines: {2, 38}, {192, 26959}, {330, 16819}, {1909, 4751}, {2275, 3739}, {3212, 17077}, {4430, 27034}, {4699, 17030}, {10436, 20459}, {20435, 25918}, {27157, 28605}, {27299, 27303}, {27304, 27343}

X(27352) =  X(4)X(157)∩X(1899)X(2165)

Barycentrics    (a^2 (b^2+c^2)-(b^2-c^2)^2)/(a^8-2 a^6 (b^2+c^2)+2 a^4 (b^2+c^2)^2-2 a^2 (b^6+c^6)+(b^2-c^2)^2 (b^4+c^4)) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27352) lies on these lines: {4,157}, {1899, 2165}


X(27353) =  X(4)X(323)∩X(30)X(2980)

Barycentrics    ((b^2-c^2)^2-a^2 (b^2+c^2))/(a^6-3 a^4 (b^2+c^2)+a^2 (3 b^4-4 b^2 c^2+3 c^4)-(b^2-c^2)^2 (b^2+c^2)) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27353) lies on these lines: {4,323}, {30,2980}, {53,1154}, {97,1141}, {525,10412}, {2165,2549}, {8800,11591}, {13450,14918}


X(27354) =  X(4)X(325)∩X(5562)X(6751)

Barycentrics    (a^2-b^2-c^2)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^8-6 a^6 b^2+4 a^4 b^4-2 a^2 b^6+b^8-6 a^6 c^2+8 a^4 b^2 c^2+2 a^2 b^4 c^2-4 b^6 c^2+4 a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-4 b^2 c^6+c^8) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27354) lies on these lines: {4,325}, {5562,6751}


X(27355) =  MIDPOINT OF X(3855) AND X(15024)

Barycentrics    a^2 (a^4+b^4-10 b^2 c^2+c^4-2 a^2 (b^2+c^2)) (a^2 (b^2+c^2)-(b^2-c^2)^2) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27355) lies on these lines: {2,13348}, {4,373}, {5,51}, {20,6688}, {185,3091}, {381,11381}, {389,3545}, {511,5056}, {547,10625}, {632,12046}, {1216,5079}, {1495,7529}, {1656,5447}, {1995,13367}, {3060,15022}, {3066,11479}, {3090,3917}, {3146,13570}, {3543,17704}, {3544,3567}, {3819,7486}, {3832,9729}, {3839,15028}, {3843,5892}, {3850,9730}, {3851,5462}, {3854,10574}, {3855,6000}, {3857,12006}, {3858,10575}, {3859,13491}, {3861,14855}, {5020,11424}, {5055,5446}, {5066,12162}, {5067,15644}, {5068,5640}, {5071,9781}, {5072,13754}, {5198,17825}, {5642,15465}, {5651,10982}, {5876,11737}, {5946,12811}, {6101,10109}, {6467,14561}, {7398,19467}, {7528,11572}, {7544,13851}, {8227,16980}, {10219,10303}, {10263,12812}, {10545,14118}, {10594,22352}, {11414,22112}, {11444,21849}, {13474,15045}, {14269,14641}, {14831,19709}, {14892,16881}, {15004,17814}, {15056,16625}, {18369,18475}

X(27355) = midpoint of X(3855) and X(15024)

X(27356) =  X(4)X(394)∩X(53)X(5562)

Barycentrics    ((b^2-c^2)^2-a^2 (b^2+c^2))/(a^6+3 a^2 (b^2-c^2)^2-3 a^4 (b^2+c^2)-(b^2-c^2)^2 (b^2+c^2)) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27356) lies on these lines: {4,394}, {53,5562}, {343,13450}, {2165,5254}, {5891,8800}

X(27356) = X(16936)-of-orthic-triangle

X(27357) =  X(4)X(1117)∩X(137)X(24772)

Barycentrics    a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2+a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6)^2 (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6)^2 : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27357) lies on these lines: {4,1117}, {137,24772}, {1154,1263}, {1291,5899}, {13582,19552}

X(27357) = isogonal conjugate of X(32637)

X(27358) =  X(4)X(1970)∩X(5)X(53)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4 b^4-2 a^2 b^6+b^8-a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2+a^4 c^4+2 a^2 b^2 c^4-2 a^2 c^6-b^2 c^6+c^8) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27358) lies on these lines: {4,1970}, {5,53}, {52,129}

X(27358) = X(32)-of-orthic-triangle

X(27359) =  X(4)X(1987)∩X(5)X(53)

Barycentrics    (-a^2+b^2-c^2) (a^2+b^2-c^2) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^8+2 a^6 b^2-a^4 b^4+2 a^6 c^2-3 a^4 b^2 c^2+b^6 c^2-a^4 c^4-2 b^4 c^4+b^2 c^6) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27359) lies on these lines: {4,1987}, {5,53}, {6,1093}, {217,13450}, {436,1970}, {458,15466}, {6748,10110}


X(27360) =  (name pending)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^6 b^2-2 a^4 b^4+a^2 b^6-a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+2 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6)^2 (a^6 b^2-2 a^4 b^4+a^2 b^6-a^6 c^2-a^4 b^2 c^2+a^2 b^4 c^2+b^6 c^2+2 a^4 c^4-a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6)^2 : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27360) lies on this line: {4,1988}


X(27361) =  X(4)X(1994)∩X(5)X(15869)

Barycentrics    (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6+3 a^4 b^2-3 a^2 b^4+b^6+a^4 c^2-3 b^4 c^2+a^2 c^4+3 b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27361) lies on these lines: {4,1994}, {5,15869}, {26,2165}, {30,22261}, {53,143}, {343,565}, {1141,8883}, {1154,8800}, {3459,7488}, {5576,16837}, {7540,11816}, {8905,25150}, {10279,10412}, {13450,14129}


X(27362) =  X(4)X(2351)∩X(5)X(51)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^4+b^4-2 b^2 c^2+c^4) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27362) lies on these lines: {4,2351}, {5,51}, {129,134}, {132,135}, {138,139}, {230,427}, {1899,2450}, {5133,9753}, {11245,23333}

X(27362) = X(25)-of-orthic-triangle
X(27362) = X(4)-Ceva conjugate of X(6751)
X(27362) = QA-P18 (Involutary Conjugate of QA-P19) of quadrangle ABCX(4)

X(27363) =  X(4)X(2413)∩X(143)X(1510)

Barycentrics    (b-c) (b+c) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (-a^16+4 a^14 b^2-6 a^12 b^4+4 a^10 b^6-a^8 b^8+4 a^14 c^2-10 a^12 b^2 c^2+8 a^10 b^4 c^2-2 a^8 b^6 c^2+a^4 b^10 c^2-2 a^2 b^12 c^2+b^14 c^2-6 a^12 c^4+8 a^10 b^2 c^4-3 a^8 b^4 c^4+a^4 b^8 c^4+6 a^2 b^10 c^4-6 b^12 c^4+4 a^10 c^6-2 a^8 b^2 c^6-4 a^4 b^6 c^6-4 a^2 b^8 c^6+15 b^10 c^6-a^8 c^8+a^4 b^4 c^8-4 a^2 b^6 c^8-20 b^8 c^8+a^4 b^2 c^10+6 a^2 b^4 c^10+15 b^6 c^10-2 a^2 b^2 c^12-6 b^4 c^12+b^2 c^14) : :

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 28595.

X(27363) lies on these lines: {4,2413}, {143,1510}


X(27364) =  X(343)-CROSS CONJUGATE OF X(5)

Barycentrics    (a^2+b^2-3 c^2) (a^2-3 b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : :
X(27364) = (8 S^2-SW^2) X[4] + (8 R^2 SW-3 SW^2) X[193]

X(27364) lies on the conic {{A,B,C,X(4),X(5)}}, the cubic K1087, and these lines: {4,193}, {327,6374}, {1141,3565}, {2165,6676}, {6340,8797}

X(27364) = X(343)-cross conjugate of X(5)
X(27364) = X(i)-isoconjugate of X(j) for these (i,j): {54, 1707}, {193, 2148}, {2167, 3053}, {2169, 6353}, {2190, 3167}, {14587, 17876}
X(27364) = barycentric product X(i)*X(j) for these {i,j}: {5, 2996}, {53, 6340}, {311, 8770}, {324, 6391}, {3565, 18314}, {8769, 14213}
X(27364) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 193}, {51, 3053}, {53, 6353}, {216, 3167}, {343, 6337}, {1953, 1707}, {2996, 95}, {3199, 19118}, {3565, 18315}, {5562, 10607}, {6391, 97}, {8769, 2167}, {8770, 54}, {12077, 3566}, {13450, 21447}, {14213, 18156}, {14248, 8882}, {21011, 4028}, {21102, 3798}, {21807, 21874}

X(27365) =  X(2)X(15073)∩X(22)X(2393)

Barycentrics    a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2-4 a^6 b^2 c^2+4 a^4 b^4 c^2-4 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4+4 a^4 b^2 c^4-2 b^6 c^4-4 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8- c^10) : :
X(27365) = 3 (4 R^2 - SW) X[51] - (9 R^2 - 2 SW) X[110]

X(27365) lies on these lines: {2,15073}, {22,2393}, {24,12235}, {51,110}, {52,539}, {185,12278}, {193,1843}, {427,12827}, {511,7391}, {569,8907}, {2888,3153}, {2904,10539}, {2979,7396}, {3543,12111}, {3917,23293}, {5012,6467}, {5446,11441}, {5889,21651}, {6403,6515}, {6997,11188}, {7499,15074}, {8548,21213}, {12825,16194}, {15030,18392}

X(27366) =  X(4)X(15897)∩X(5)X(182)

Barycentrics    (b^2 + c^2)*(a^4 + a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 + a^2*c^2 - b^2*c^2 + c^4) : :

X(27366) lies on the cubic K1088 and these lines: {4, 15897}, {5, 182}, {32, 11550}, {211, 20021}, {217, 7753}, {315, 2387}, {427, 3203}, {2548, 11175}, {3313, 14994}, {5475, 10014}, {6145, 12202}, {6292, 14096}, {12212, 15321}

X(27366) = X(i)-isoconjugate of X(j) for these (i,j): {82, 2979}, {160, 3112}, {3202, 18833}
X(27366) = crosssum of X(i) and X(j) for these (i,j): {22, 8266}, {160, 2979}
X(27366) = barycentric product X(141)X(2980)
X(27366) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 2979}, {141, 7796}, {2980, 83}, {3051, 160}

X(27367) =  X(4)X(52)∩X(211)X(1843)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(27367) lies on the cubic K1088 and these lines: {4, 52}, {211, 1843}, {217, 2971}, {384, 925}, {1692, 1974}

X(27367) = X(i)-isoconjugate of X(j) for these (i,j): {1147, 18833}, {3112, 9723}
barycentric product X(i)*X(j) for these {i,j}: {39, 14593}, {847, 3051}, {1843, 2165}
barycentric quotient X(i)/X(j) for these {i,j}: {1843, 7763}, {3051, 9723}, {14593, 308}

X(27368) =  (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = N OBVERSE TRIANGLE OF X(1)

Barycentrics    a^4 + a^3 b + 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 - b c^3 : :

X(27368) lies on these lines:

X(27369) =  X(2)X(3)∩X(6)X(15270)

Barycentrics    a^4*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + c^2) : :
Trilinears    sin^2 A tan A sin(A + ω) : :
Trilinears    sec A sin(A + ω) sin(A - ω) : :

X(27369) lies on the cubic K1088 and these lines: {2, 3}, {6, 15270}, {32, 682}, {39, 1843}, {69, 9917}, {112, 733}, {160, 5013}, {193, 19597}, {217, 9418}, {232, 13357}, {695, 1915}, {881, 9491}, {1235, 12143}, {1973, 18265}, {2356, 18758}, {2489, 9489}, {2971, 3199}, {3051, 14820}, {3095, 6403}, {3186, 19566}, {3313, 22424}, {3398, 19128}, {3455, 3456}, {5188, 12294}, {7772, 8541}, {7800, 22062}, {9468, 14946}, {9605, 12167}, {10311, 13356}, {15257, 18374}

Let circles (O)A, (O)B, (O)C be as defined at X(8160). Let PA be the perspector of (O)A, and define PB and PC cyclically. The lines APA, BPB, CPC concur in X(32). Let LA be the polar of PA wrt (O)A and define LB and LC cyclically. Let A' = LBnLC and define B' and C' cycllically. The lines AA', BB', CC' concur in X(27369). (Randy Hutson, November 30, 2018)

The trilinear polar of X(27369) meets the line at infinity at X(688). (Randy Hutson, November 30, 2018)

X(27369) = isogonal conjugate of the complement X(10340)
X(27369) = X(i)-Ceva conjugate of X(j) for these (i,j): {25, 1843}, {1843, 3051}, {17980, 2211}
X(27369) = crosspoint of X(i) and X(j) for these (i,j): {25, 1974}, {32, 2353}
X(27369) = crossdifference of every pair of points on line {647, 3267}
X(27369) = crosssum of X(i) and X(j) for these (i,j): {2, 12220}, {69, 305}, {76, 315}, {10999, 11000}
X(27369) = circumcircle-inverse of X(37912)
X(27369) = X(i)-isoconjugate of X(j) for these (i,j): {3, 18833}, {63, 308}, {69, 3112}, {75, 1799}, {82, 305}, {83, 304}, {336, 20022}, {525, 4593}, {561, 1176}, {656, 689}, {799, 4580}, {1928, 10547}, {3267, 4599}, {4563, 18070}, {4577, 14208}
X(27369) = barycentric product X(i)*X(j) for these {i,j}: {4, 3051}, {6, 1843}, {19, 1964}, {25, 39}, {28, 21814}, {31, 17442}, {32, 427}, {38, 1973}, {92, 1923}, {112, 3005}, {141, 1974}, {162, 2084}, {217, 19174}, {393, 20775}, {560, 20883}, {607, 1401}, {608, 3688}, {648, 688}, {1096, 4020}, {1235, 1501}, {1474, 21035}, {1634, 2489}, {1918, 17171}, {2203, 3954}, {2205, 16747}, {2206, 21016}, {2207, 3917}, {2211, 20021}, {2333, 17187}, {3186, 19606}, {3199, 16030}, {3787, 14248}, {6331, 9494}, {8623, 17980}, {8750, 21123}, {13854, 23208}, {19595, 22262}
X(27369) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 18833}, {25, 308}, {32, 1799}, {39, 305}, {112, 689}, {427, 1502}, {669, 4580}, {688, 525}, {1501, 1176}, {1843, 76}, {1923, 63}, {1964, 304}, {1973, 3112}, {1974, 83}, {2084, 14208}, {2211, 20022}, {2531, 2525}, {3005, 3267}, {3051, 69}, {3118, 4121}, {9233, 10547}, {9494, 647}, {17442, 561}, {20775, 3926}, {20883, 1928}, {21814, 20336}
X(27369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 8362, 14096), (3, 11328, 14001), (3, 20960, 237), (25, 11325, 4), (39, 23208, 20775), (237, 3148, 418), (237, 11326, 3)

X(27370) =  CROSSPOINT OF X(4) AND X(10312)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(27370) = 3 X[51]-X[217]

X(27370) lies on the Kiepert hyperbola of the orthic triangle, the cubic K1088, and on these lines: {39, 1843}, {51, 217}, {52, 14978}, {53, 15897}, {211, 427}, {389, 1503}, {460, 2387}, {2909, 10311}, {6746, 12131}, {7715, 15510}

X(27370) = crosspoint of X(4) and X(10312)
X(27370) = barycentric product X(i)*X(j) for these {i,j}: {324, 3203}, {6663, 26922}
X(27370) = barycentric quotient X(3203)/X(97)

X(27371) =  X(4)X(32)∩X(5)X(53)

Barycentrics    (-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(b^2 + c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(27371) lies on the cubic K1088 and these lines: {4, 32}, {5, 53}, {6, 18381}, {24, 7749}, {25, 7746}, {39, 427}, {51, 15897}, {54, 15340}, {187, 3575}, {211, 1843}, {217, 3574}, {230, 6756}, {232, 1506}, {264, 626}, {297, 3934}, {317, 7751}, {378, 7756}, {393, 2548}, {458, 7834}, {574, 3541}, {577, 14790}, {648, 7838}, {1196, 15809}, {1235, 7794}, {1593, 7748}, {1595, 5254}, {1598, 1609}, {1970, 18400}, {1971, 13419}, {2207, 5475}, {2241, 11393}, {2242, 11392}, {3053, 18494}, {3087, 5319}, {3162, 5064}, {5007, 16318}, {5169, 26216}, {5206, 18533}, {5286, 7378}, {5305, 6748}, {5523, 7765}, {6240, 6781}, {6643, 26899}, {7528, 10314}, {7745, 14581}, {7753, 8743}, {7759, 9308}, {7808, 17907}, {10641, 22481}, {10642, 22482}, {16589, 25985}, {22401, 23335}

X(27371) = X(i)-isoconjugate of X(j) for these (i,j): {82, 97}, {83, 2169}, {1176, 2167}, {1799, 2148}, {3112, 14533}, {4599, 23286}, {15958, 18070}
X(27371) = crosspoint of X(53) and X(324)
X(27371) = crossdifference of every pair of points on line {684, 23286}
X(27371) = crosssum of X(97) and X(14533)
X(27371) = barycentric product X(i)*X(j) for these {i,j}: {5, 427}, {39, 324}, {51, 1235}, {53, 141}, {311, 1843}, {1634, 23290}, {1930, 2181}, {1953, 20883}, {3199, 8024}, {3917, 13450}, {3933, 14569}, {14213, 17442}, {16747, 21807}, {17167, 21016}, {17171, 21011}
X(27371) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 1799}, {39, 97}, {51, 1176}, {53, 83}, {324, 308}, {427, 95}, {1843, 54}, {1964, 2169}, {2181, 82}, {2525, 15414}, {3005, 23286}, {3051, 14533}, {3199, 251}, {12077, 4580}, {17442, 2167}, {20775, 19210}
X(27371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1968, 7747), (5, 53, 3199), (232, 1594, 1506), (2207, 7507, 5475), (5305, 16198, 6748)

X(27372) =  X(4)X(66)∩X(39)X(184)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 + b^4 - c^4)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - b^4 + c^4) : :
X(27372) = 3 X[51]-2 X[15897]

X(27372) lies on the cubic K1088 and these lines: {4, 66}, {39, 184}, {51, 15897}, {217, 6751}, {1216, 14376}, {1289, 1298}, {11437, 16318}

X(27372) = X(i)-isoconjugate of X(j) for these (i,j): {275, 1760}, {276, 2172}, {315, 2190}, {2167, 17907}, {8882, 20641}
X(27372) = crosssum of X(22) and X(17907)
X(27372) = barycentric product X(i)*X(j) for these {i,j}: {51, 14376}, {66, 216}, {217, 18018}, {343, 2353}, {1289, 17434}, {5562, 13854}
X(27372) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 17907}, {66, 276}, {216, 315}, {217, 22}, {418, 20806}, {2353, 275}, {13854, 8795}

X(27373) =  X(4)X(66)∩X(5)X(127)

Barycentrics    a^2*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(b^2 + c^2)*(a^4 - b^4 - c^4) : :
X(27373) = 3 X[51]-2 X[11437]

X(27373) lies on the cubic K1088 and these lines: {4, 66}, {5, 127}, {6, 22262}, {25, 32}, {51, 11437}, {83, 107}, {206, 8743}, {232, 20960}, {1843, 19595}, {3542, 9752}, {7507, 22682}, {11397, 16318}

X(27373) = X(107)-Ceva conjugate of X(2485)
X(27373) = X(326)-isoconjugate of X(16277)
X(27373) = crosspoint of X(i) and X(j) for these (i,j): {4, 8743}, {315, 19613}
X(27373) = crosssum of X(3) and X(14376)
X(27373) = polar-circle-inverse of X(34237)
X(27373) = barycentric product X(i)*X(j) for these {i,j}: {393, 3313}, {427, 8743}, {1235, 17409}, {1824, 16715}, {1843, 17907}, {2052, 23208}
X(27373) = barycentric quotient X(i)/X(j) for these {i,j}: {1843, 14376}, {2207, 16277}, {3313, 3926}, {8743, 1799}, {17409, 1176}, {23208, 394}
X(27373) = {X(25),X(3172)}-harmonic conjugate of X(20993)

X(27374) =  X(4)X(263)∩X(5)X(51)

Barycentrics    a^4*(b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(27374) lies on the cubic K1088 and these lines: {4, 263}, {5, 51}, {6, 4173}, {32, 2909}, {39, 211}, {184, 20960}, {389, 1513}, {511, 6656}, {694, 3499}, {1186, 8265}, {1285, 9292}, {1843, 19595}, {2387, 5007}, {2979, 7876}, {3051, 14820}, {3060, 5025}, {3199, 15897}, {3491, 7762}, {3852, 12212}, {3917, 8362}, {5167, 7745}, {5446, 15980}, {5640, 16921}, {5889, 13862}, {6310, 8370}, {6680, 14962}, {6784, 7755}, {6786, 7764}, {7824, 11673}, {8041, 14822}, {10547, 19558}, {11272, 11675}

X(27374) = lies on the cubic K1088 and these lines: X(i)-isoconjugate of X(j) for these (i,j): {54, 18833}, {95, 3112}, {308, 2167}, {689, 2616}, {4593, 15412}
X(27374) = crosspoint of X(1843) and X(3051)
X(27374) = crosssum of X(i) and X(j) for these (i,j): {76, 1078}, {308, 1799}
X(27374) = barycentric product X(i)*X(j) for these {i,j}: {5, 3051}, {38, 2179}, {39, 51}, {53, 20775}, {216, 1843}, {217, 427}, {688, 14570}, {1625, 3005}, {1923, 14213}, {1953, 1964}, {2084, 2617}, {2181, 4020}, {3199, 3917}, {18180, 21814}
X(27374) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 308}, {217, 1799}, {688, 15412}, {1625, 689}, {1843, 276}, {1923, 2167}, {1953, 18833}, {2179, 3112}, {3051, 95}, {9494, 2623}

X(27375) =  X(5)X(141)∩X(39)X(51)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) : :
X(27375) = X[39] - 3 X[51], X[76] + 3 X[3060], 3 X[5943] - 2 X[6683], X[6243] + 3 X[7697], X[4173] - 3 X[7753], 9 X[5640] - 5 X[7786], 3 X[262] - 7 X[9781], X[194] - 9 X[11002], 5 X[3567] - X[11257], 3 X[9971] + X[13330], X[10625] - 3 X[15819], X[5876] - 3 X[22681]

X(27375) lies on the cubics K054 and K1088, and on these lines: {5, 141}, {25, 3202}, {39, 51}, {52, 6248}, {76, 3060}, {143, 2782}, {194, 11002}, {262, 9781}, {263, 3767}, {384, 14962}, {512, 7747}, {538, 21849}, {550, 15510}, {674, 21067}, {732, 6664}, {882, 8711}, {1112, 12143}, {1843, 2211}, {2387, 7745}, {2698, 12110}, {3094, 11360}, {3491, 7843}, {3567, 11257}, {3954, 5360}, {4173, 7753}, {5462, 13334}, {5640, 7786}, {5876, 22681}, {5943, 6683}, {6243, 7697}, {7730, 18304}, {7737, 9292}, {7760, 14970}, {9466, 21969}, {9971, 13330}, {10095, 11272}, {10625, 15819}, {11426, 22655}, {11576, 22480}, {14569, 14715}, {14839, 24068}, {18322, 18502}

X(27375) = midpoint of X(i) and X(j) for these {i,j}: {52, 6248}, {1843, 5052}, {9466, 21969}
X(27375) = reflection of X(i) in X(j) for these {i,j}: {11272, 10095}, {13334, 5462}
X(27375) = isogonal conjugate of X(1078)
X(27375) = isotomic conjugate of X(33769)
X(27375) = X(115)-cross conjugate of X(512)
X(27375) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1078}, {2, 18042}, {75, 5012}, {304, 10312}, {326, 1629}, {799, 3050}, {3203, 18833}, {4564, 27010}, {7668, 24041}
X(27375) = cevapoint of X(688) and X(3124)
X(27375) = trilinear pole of line {2491, 3005}
X(27375) = crosssum of X(2) and X(8266)
X(27375) = barycentric product X(i)*X(j) for these {i,j}: {6, 3613}, {512, 11794}
X(27375) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1078}, {31, 18042}, {32, 5012}, {669, 3050}, {1974, 10312}, {2207, 1629}, {3124, 7668}, {3271, 27010}, {3613, 76}, {11794, 670}
X(27375) = {X(5403),X(5404)}-harmonic conjugate of X(3613)

X(27376) =  X(4)X(6)∩X(5)X(232)

Barycentrics    (-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(b^2 + c^2) : :

The trilinear polar of X(27376) passes through X(3005). (Randy Hutson, November 30, 2018)

X(27376) lies on the cubic K1088 and theswe lines: {4, 6}, {5, 232}, {24, 230}, {25, 2353}, {30, 1968}, {32, 3575}, {39, 427}, {76, 297}, {83, 10549}, {107, 755}, {112, 6240}, {115, 235}, {141, 1235}, {216, 7399}, {264, 6656}, {315, 9308}, {317, 7754}, {324, 26214}, {403, 16308}, {428, 5309}, {458, 7803}, {460, 2909}, {468, 7746}, {607, 11391}, {648, 7762}, {1194, 15809}, {1562, 11381}, {1593, 2549}, {1594, 3815}, {1595, 15048}, {1625, 22660}, {1799, 17037}, {1843, 19595}, {1885, 7748}, {1907, 7765}, {2501, 23105}, {2548, 7507}, {3051, 19174}, {3053, 18533}, {3054, 10018}, {3172, 7737}, {3269, 6247}, {3541, 5013}, {3542, 13881}, {5064, 7739}, {5090, 9620}, {5117, 17042}, {5133, 26216}, {5283, 25985}, {5305, 6756}, {5306, 7576}, {5412, 12147}, {5413, 12148}, {5475, 23047}, {6392, 6464}, {6528, 14970}, {6531, 8884}, {7487, 7735}, {7715, 10985}, {7747, 14581}, {7770, 17907}, {7790, 21447}, {7859, 14165}, {9607, 15559}, {9722, 14576}, {10019, 18424}, {11393, 16502}, {12085, 15075}, {12134, 23128}, {13160, 22240}, {14569, 14715}, {14790, 23115}, {14961, 23335}, {15750, 21843}, {21016, 21035}, {24989, 26035}

X(27376) = X(19174)-Ceva conjugate of X(1843)
X(27376) = X(1843)-cross conjugate of X(427)
X(27376) = X(i)-isoconjugate of X(j) for these (i,j): {48, 1799}, {63, 1176}, {82, 394}, {83, 255}, {251, 326}, {304, 10547}, {520, 4599}, {577, 3112}, {822, 4577}, {827, 24018}, {3405, 17974}, {4131, 4628}, {4575, 4580}, {14585, 18833}, {18082, 18604}
X(27376) = crosspoint of X(393) and X(2052)
X(27376) = crosssum of X(i) and X(j) for these (i,j): {3, 10316}, {32, 20993}, {394, 577}
X(27376) = polar conjugate of X(1799)
X(27376) = polar-circle-inverse of X(34137)
X(27376) = barycentric product X(i)*X(j) for these {i,j}: {4, 427}, {5, 19174}, {19, 20883}, {25, 1235}, {27, 21016}, {38, 158}, {39, 2052}, {92, 17442}, {107, 826}, {141, 393}, {264, 1843}, {823, 8061}, {1093, 3917}, {1096, 1930}, {1118, 3703}, {1824, 16747}, {1826, 17171}, {1857, 3665}, {1897, 21108}, {2207, 8024}, {2525, 6529}, {3005, 6528}, {3051, 18027}, {3867, 8801}, {3933, 6524}, {4020, 6521}, {6530, 20021}, {8747, 15523}, {13450, 16030}
X(27376) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1799}, {25, 1176}, {38, 326}, {39, 394}, {107, 4577}, {141, 3926}, {158, 3112}, {393, 83}, {427, 69}, {823, 4593}, {826, 3265}, {1096, 82}, {1235, 305}, {1401, 1804}, {1843, 3}, {1964, 255}, {1974, 10547}, {2052, 308}, {2084, 822}, {2207, 251}, {2501, 4580}, {2525, 4143}, {2530, 4131}, {3005, 520}, {3051, 577}, {3665, 7055}, {3688, 1259}, {3703, 1264}, {3787, 10607}, {3867, 3785}, {3917, 3964}, {3933, 4176}, {3954, 3998}, {4020, 6507}, {6528, 689}, {6530, 20022}, {8061, 24018}, {14569, 17500}, {17171, 17206}, {17442, 63}, {19174, 95}, {20021, 6394}, {20775, 1092}, {20883, 304}, {21016, 306}, {21035, 3682}, {21108, 4025}, {21123, 4091}, {21814, 3990}, {24019, 4599}
X(27376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 393, 2207), (4, 5523, 5254), (4, 8743, 7745), (53, 5254, 4), (115, 3199, 235), (1990, 7745, 8743), (3172, 12173, 7737), (3575, 16318, 32), (5305, 6756, 10311)

X(27377) =  X(2)X(15905)∩X(4)X(193)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2+c^4) : :
X(27377) = S^2 X[4] - (4 R^2 - SW) SW X[193]

X(27377) lies on these lines: {2,15905}, {4,193}, {5,17035}, {6,297}, {25,7774}, {30,3164}, {53,648}, {69,458}, {141,340}, {194,3575}, {235,7785}, {239,7282}, {264,524}, {273,17364}, {275,343}, {318,17363}, {325,10311}, {385,427}, {393,1992}, {428,7837}, {445,19742}, {460,3186}, {467,1994}, {468,7777}, {472,3180}, {473,3181}, {542,16264}, {576,6530}, {894,5081}, {1585,5411}, {1586,5410}, {1593,20065}, {1654,11109}, {1885,7823}, {1904,2905}, {3199,7838}, {3618,11331}, {4558,8882}, {5094,17008}, {5278,25986}, {7513,20077}, {7763,10607}, {7766,16318}, {7807,10312}, {7812,21447}, {9289,13568}, {10316,26205}, {10754,20774}, {14389,14918}, {14590,18122}, {15014,18907}, {16997,26020}, {16998,25985}, {17037,17578}, {17300,26003}, {17379,17555}, {18494,22253}, {23115,26155}

X(27377) = reflection of X(264) in X(6748)
X(27377) = anticomplement of the isotomic of X(14860)
X(27377) = X(14860)-anticomplementary conjugate of X(6327)
X(27377) = X(14860)-Ceva conjugate of X(2)
X(27377) = cevapoint of X(193) and X(17035)
X(27377) = polar conjugate of isogonal conjugate of X(34986)
X(27377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 193, 9308}, {6, 317, 297}, {53, 3629, 648}, {69, 3087, 458}

leftri

Collineation mappings involving Gemini triangles 61: X(27378)-X(27422)

rightri

Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 61, Gemini triangle 62, etc. See the preamble just before X(24537) and X(26153) for definitions of Gemini triangles 1-60. (Clark Kimberling, November 8, 2018)

Gemini 61      a (a + b + c) (a - b - c) : (b + c) (a - b + c) (a + b - c) : (b + c) (a - b + c)(a + b - c)
     Let A''B''C'' = Gemini triangle 1, defined by A' = a : b + c : b + c, and let A'B'C' = Gemini triangle 61.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 62      (b c + a c - a b ) (a b - a c + b c) (b^2 + b c + c^2) : (a b - a c - b c) (a b + a c - b c)(c^2 + a b) : (a c - a b - b c) (a c + a b - b c)(b^2 + a c)
     Let A''B''C'' = Gemini triangle 3, defined by A' = a : a + b : a + c, and let A'B'C' = Gemini triangle 62.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 63      - (b^2 + b c + c^2) : a b + 2 a c + 2 b c + c^2 : a c + 2 a b + 2 b c + b^2
     Let A''B''C'' = Gemini triangle 4, defined by A' = -a : a + b : a + c, and let A'B'C' = Gemini triangle 63.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 64      (a b - a c + b c) (a b - a c - b c) (b^2 + c^2 - b c) : (-a b + a c + b c) (a b + a c - b c)(c^2 + a b - 2 b c) : (a b - a c + b c) (a b + a c - b c) (b^2 + a c - 2 b c)
     Let A''B''C'' = Gemini triangle 5, defined by A' = -a : a - b : a - c, and let A'B'C' = Gemini triangle 64.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 65      (a - b - c) (a + b + c) : (a + b - c) (a - b + c) : (a + b - c) (a - b + c)
     Let A''B''C'' = Gemini triangle 13, defined by A' = b + c : a : a, and let A'B'C' = Gemini triangle 65.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 66      (b + c)(a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) : -a (a + b) (a + c) : -a (a + b) (a + c)
     Let A''B''C'' = Gemini triangle 21, defined by A' = a + b + c : a : a, and let A'B'C' = Gemini triangle 66.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 67      (2 a + b + c) (a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) : a (a + 2 b + c) (a + b + 2 c) : a (a + 2 b + c)(a + b + 2 c)
     Let A''B''C'' = Gemini triangle 22, defined by A' = a + b + c : -a : -a, and let A'B'C' = Gemini triangle 67.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 68      a (b^2 + c^2 + a b + c a + b c) : -b c (b + c) : -b c (b + c)
     Let A''B''C'' = Gemini triangle 23, defined by A' = a + b + c : b + c : b + c, and let A'B'C' = Gemini triangle 68.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 69      (b + c) (b^2 + c^2 - a^2 - b c) : -a (a + b) (a + c) : -a (a + b) (a + c)
     Let A''B''C'' = Gemini triangle 27, defined by A' = a - b - c : a : a, and let A'B'C' = Gemini triangle 69.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 70      (a - 2 b -2 c) (b^2 + c^2 + a b + a c - b c) : (b + c) (2 a + 2 b - c) (2 a - b + 2 c) : (b + c) (2 a + 2 b - c) (2 a - b + 2 c)
     Let A''B''C'' = Gemini triangle 28, defined by A' = a - b - c : b + c : b + c, and let A'B'C' = Gemini triangle 70.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 71      a ( a - 3 b + c) (a + b - 3 c) : (3 a - b - c) (3 c - a - b) (b - c) : (3 a - b - c) (a - 3 b + c) (b - c)
     Let A''B''C'' = Gemini triangle 30, defined by A' = a : c - b : b - c, and let A'B'C' = Gemini triangle 71.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 72      b c (a^2 - b c)^2 : a^2 (b^2 - a c) (c^2 - a b) : a^2 (b^2 - a c) (c^2 - a b)
     Let A''B''C'' = Gemini triangle 31, defined by A' = b c : a^2 : a^2, and let A'B'C' = Gemini triangle 72.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 73      b c (a^4 - b^2 c^2) : a^2 (b^2 + a c) (c^2 + a b) : a^2 (b^2 + a c) (c^2 + a b)
     Let A''B''C'' = Gemini triangle 32, defined by A' = -b c : a^2 : a^2, and let A'B'C' = Gemini triangle 73.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 74      (a^2 - b c)^2 : (b^2 - a c) (c^2 - a b) : (b^2 - a c) (c^2 - a b)
     Let A''B''C'' = Gemini triangle 33, defined by A' = a^2 : b c : b c, and let A'B'C' = Gemini triangle 74.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 75      a^4 - b^2 c^2 : -(b^2 + a c) (c^2 + a b) : -(b^2 + a c) (c^2 + a b)
     Let A''B''C'' = Gemini triangle 34, defined by A' = -a^2 : b c : b c, and let A'B'C' = Gemini triangle 75.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 76      (a + b + c) (a^2 + b^2 + c^2 - 2 b c) : 2 b c (a - b - c) : 2 b c (a - b - c)
     Let A''B''C'' = Gemini triangle 35, defined by A' = cos A : 1 : 1 = -a^2 + b^2 + c^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 76.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 77      (a - b - c)^2 (a^2 + b^2 + c^2 - 2 b c) : -2 b c (a - b + c) (a + b - c) : - 2 b c (a - b + c) (a + b - c)
     Let A''B''C'' = Gemini triangle 36, defined by A' = -cos A : 1 : 1 = a^2 - b^2 - c^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 77.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 78      (a + b + c) (a^2 + b^2 + c^2 - 2 b c) : (b + c - a) (a^2 - b^2 - c^2) : (b + c - a) (a^2 - b^2 - c^2)
     Let A''B''C'' = Gemini triangle 37, defined by A' = sec A : 1 : 1 = 2 b c : -a^2 + b^2 + c^2 : -a^2 + b^2 + c^2, and let A'B'C' = Gemini triangle 78.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 79      (a - b - c)^2 (a^2 + b^2 + c^2 - 2 b c) : (a - b + c) (a + b - c) (b^2 + c^2 - a^2) : (a - b + c) (a + b - c) (b^2 + c^2 - a^2)
     Let A''B''C'' = Gemini triangle 38, defined by A' = -sec A : 1 : 1 = -2 b c : -a^2 + b^2 + c^2 : -a^2 + b^2 + c^2, and let A'B'C' = Gemini triangle 79.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 80      a (b^2 + c^2 + a b + a c) : -b c (a + b + c) : -b c (a + b + c)
     Let A''B''C'' = Gemini triangle 39, defined by A' = -a + b + c : a + b + c : a + b + c, and let A'B'C' = Gemini triangle 80.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 81      a (b^2 + c^2 + a b + a c) : b c (a - b - c) : b c (a - b - c)
     Let A''B''C'' = Gemini triangle 40, defined by A' = a + b + c : -a + b + c : -a + b + c, and let A'B'C' = Gemini triangle 81.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 82      (a^2 - b^2 - c^2) (a^2 + b^2 + c^2 : (a^2 + b^2 - c^2) (a^2 - b^2 + c^2 : (a^2 + b^2 - c^2) (a^2 - b^2 + c^2
     Let A''B''C'' = Gemini triangle 41, defined by A' = b^2 + c^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 82.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 83      (b^2 + c^2) (a^2 + b^2 + c^2 - b c) (a^2 + b^2 + c^2 + b c) : - a^2 (a^2 + b^2) (a^2 + c^2) : - a^2 (a^2 + b^2) (a^2 + c^2)
     Let A''B''C'' = Gemini triangle 42, defined by A' = a^2 + b^2 + c^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 83.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 84      a^2 (a^2 + b^2 + c^2) (a^2 - b^2 - c^2) : (b^2 + c^2) (a^2 - b^2 + c^2) (a^2 + b^2 - c^2) : (b^2 + c^2)(a^2 - b^2 + c^2) (a^2 + b^2 - c^2)
     Let A''B''C'' = Gemini triangle 43, defined by A' = a^2 : b^2 + c^2 : b^2 + c^2 , and let A'B'C' = Gemini triangle 84.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 85      a^2 - a b - a c + 2 b c : a (a - b - c) : a (a - b - c)
     Let A''B''C'' = Gemini triangle 45, defined by A' = (b - c)^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 85.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 86      (a - b - c) (a^2 + a b + a c + 2 b c) : a (a - b + c) (a + b - c) : a (a - b + c) (a + b - c)
     Let A''B''C'' = Gemini triangle 46, defined by A' = (b + c)^2 : a^2 : a^2, and let A'B'C' = Gemini triangle 86.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 87      a (a - b - c) (a^2 + a b + a c + 2 b c) : (b + c)^2 (a - b + c) (a + b - c) : (b + c)^2 (a - b + c)(a + b - c)
     Let A''B''C'' = Gemini triangle 47, defined by A' = a^2 : (b + c)^2 : (b + c)^2 , and let A'B'C' = Gemini triangle 87.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 88      a (a^2 - a b - a c + 2 b c) : (b - c)^2 (a - b - c) : (b - c)^2 (a - b - c)
     Let A''B''C'' = Gemini triangle 48, defined by A' = a^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 88.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 89      (b + c)(a^2 + b c) : -a (b - c)^2 : -a (b - c)^2
     Let A''B''C'' = Gemini triangle 49, defined by A' = (b + c)^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 89.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 90      a^2 + b c : -a b - a c : -a b - a c
     Let A''B''C'' = Gemini triangle 50, defined by A' = (b - c)^2 : (b + c)^2 : (b + c)^2, and let A'B'C' = Gemini triangle 90.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 91      a^2 b+a^2 c - 2 a b c + b^2 c + b c^2 : -a (b^2 + c^2) : -a (b^2 + c^2)
     Let A''B''C'' = Gemini triangle 51, defined by A' = (b - c)^2 : b^2 + c^2 : b^2 + c^2, and let A'B'C' = Gemini triangle 91.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 92      a^2 b + a^2 c + 2 a b c + b^2 c + b c^2 : -a (b^2 + c^2) : -a (b^2 +c ^2)
     Let A''B''C'' = Gemini triangle 52, defined by A' = (b + c)^2 : b^2 + c^2 : b^2 + c^2, and let A'B'C' = Gemini triangle 92.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 93      a^2 b + a^2 c - 2 a b c + b^2 c + b c^2 : -a (b - c)^2 : -a (b - c)^2
     Let A''B''C'' = Gemini triangle 53, defined by A' = b^2 + c^2 : (b - c)^2 : (b - c)^2, and let A'B'C' = Gemini triangle 93.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 94      a^2 b + a^2 c + 2 a b c + b^2 c + b c^2 : -a (b + c)^2 : -a (b + c)^2
     Let A''B''C'' = Gemini triangle 54, defined by A' = b^2 + c^2 : (b + c)^2 : (b + c)^2 , and let A'B'C' = Gemini triangle 94.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 95      (a^2 - 2 b c) (4 a^2 - b c) : 2 (b^2 - 2 a c) (c^2 - 2 a b) : 2 (b^2 - 2 a c)(c^2 - 2 a b)
     Let A''B''C'' = Gemini triangle 55, defined by A' = a^2 : 2 b c : 2 b c, and let A'B'C' = Gemini triangle 95.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 96      (a^2 + 2 b c) (4 a^2 - b c) : -2 (b^2 + 2 a c) (c^2 + 2 a b) : -2 (b^2 + 2 a c) (c^2 + 2 a b)
     Let A''B''C'' = Gemini triangle 56, defined by A' = -a^2 : 2bc : 2bc , and let A'B'C' = Gemini triangle 96.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 97      (b^2 - b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 - a^2 b c + b^2 c^2) : -b c (a^2 - a b + b^2) (a^2 - a c + c^2) : -b c (a^2 - a b + b^2) (a^2 - a c + c^2)
     Let A''B''C'' = Gemini triangle 57, defined by A' = b^2 + c^2 : b c : b c, and let A'B'C' = Gemini triangle 97.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 98      (b^2 + b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 - a^2 b c + b^2 c^2) : b c (a^2 + a b + b^2) (a^2 + a c + c^2) : b c (a^2 + a b + b^2) (a^2 + a c + c^2)
     Let A''B''C'' = Gemini triangle 58, defined by A' = b^2 + c^2 : -b c : -b c, and let A'B'C' = Gemini triangle 98.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 99      a b^2 + a c^2 + b^2 c + b c^2 : -a (b c + c a + a b): -a (b c + c a + a b)
     Let A''B''C'' = Gemini triangle 59, defined by A' = -b c + c a + a b : b c + c a + a b : b c + c a + a b, and let A'B'C' = Gemini triangle 99.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

Gemini 100      ab^2+ac^2+b^2 c+bc^2 : a (b c - c a - a b): a (b c - c a - a b)
     Let A''B''C'' = Gemini triangle 60, defined by A' = -b c + c a + a b : b c + c a + a b : b c + c a + a b, and let A'B'C' = Gemini triangle 100.
     The collineation (A,B,C,X(2); A',B',C',X(2)) is the inverse of the collineation (A,B,C,X(2); A'',B'',C'',X(2)).

If T is a central triangle A'B'C' with A' of the form f(a,b,c) : g(a,b,c) : g(a,b,c), then the (A,B,C,X(2); A',B',C',X(2)) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 61, 65-70, and 72-100.

X(27378) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 + a b^3 c^2 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a c^5 - b c^5) : :

X(27378) lies on these lines: {2, 3}, {8, 7074}, {31, 1210}, {78, 3701}, {255, 14058}, {283, 1150}, {318, 3100}, {965, 27040}, {1040, 23661}, {1076, 1125}, {1479, 23518}, {1897, 9538}, {2328, 10479}, {2899, 27383}, {3883, 6734}, {5250, 24552}, {5705, 32917}, {5906, 26932}, {6253, 25882}, {10448, 13411}, {18635, 26099}, {27381, 27389}, {27397, 27415}, {27398, 28809}

X(27379) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (3 a^6 + 2 a^5 b - 3 a^4 b^2 + a^2 b^4 - 2 a b^5 - b^6 + 2 a^5 c + 2 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 3 a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 + 2 a b^2 c^3 + a^2 c^4 + b^2 c^4 - 2 a c^5 - c^6) : :

X(27379) lies on these lines: {2, 3}, {283, 5739}, {968, 13411}, {1098, 14555}, {1259, 27540}, {1792, 28807}, {3616, 17080}, {4011, 6700}, {5250, 24611}, {9371, 9375}, {27384, 27385}, {27395, 27397}

X(27380) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (2 a^6 + a^5 b - 3 a^4 b^2 + 2 a^2 b^4 - a b^5 - b^6 + a^5 c + 2 a^4 b c - a^3 b^2 c - 3 a^2 b^3 c + b^5 c - 3 a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - 3 a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + 2 a^2 c^4 + b^2 c^4 - a c^5 + b c^5 - c^6) : :

X(27380) lies on these lines: {2, 3}, {40, 6711}, {283, 5741}, {1259, 28826}, {6796, 23541}, {15252, 20222}

X(27381) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - b^3 c - a^2 c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(27381) lies on these lines: {2, 6}, {7, 16412}, {41, 21246}, {48, 24612}, {78, 1229}, {198, 20245}, {220, 27514}, {329, 11350}, {332, 28809}, {379, 2327}, {908, 1958}, {1253, 6745}, {2264, 30812}, {2268, 3452}, {3217, 20258}, {3553, 24993}, {4193, 25679}, {4254, 17183}, {4402, 24203}, {4875, 28639}, {5283, 16743}, {5782, 27058}, {11344, 27383}, {16833, 24202}, {17077, 23151}, {20769, 29965}, {21239, 21285}, {24435, 27471}, {27378, 27389}, {27388, 27391}, {27396, 27399}, {27397, 27414}

X(27382) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (3 a^4 - 2 a^2 b^2 - b^4 - 2 a^2 c^2 + 2 b^2 c^2 - c^4) : :

X(27382) lies on these lines: {2, 7}, {6, 938}, {8, 29}, {10, 3332}, {19, 962}, {20, 610}, {37, 5703}, {44, 5704}, {48, 5731}, {72, 7498}, {78, 280}, {189, 394}, {198, 411}, {220, 965}, {273, 26668}, {284, 4313}, {345, 5931}, {347, 26006}, {380, 390}, {391, 6734}, {497, 2264}, {516, 18594}, {645, 1264}, {651, 5932}, {936, 990}, {944, 20818}, {952, 22147}, {1034, 3692}, {1100, 15933}, {1103, 20224}, {1146, 5839}, {1210, 1743}, {1249, 1895}, {1259, 8805}, {1436, 6909}, {1723, 3086}, {1781, 4295}, {1783, 22132}, {1814, 6559}, {1857, 6056}, {1901, 25015}, {1903, 12528}, {1953, 5734}, {2182, 6836}, {2257, 14986}, {2277, 9367}, {2289, 7538}, {2321, 20007}, {2323, 12649}, {2551, 5800}, {3161, 27383}, {3211, 5768}, {3618, 30854}, {3682, 20226}, {3686, 23058}, {3699, 30681}, {3731, 13411}, {3868, 9119}, {4329, 14543}, {4402, 4858}, {4644, 18635}, {5175, 7518}, {5222, 17863}, {5227, 5815}, {5308, 5736}, {5776, 9799}, {5801, 18250}, {6603, 17314}, {7090, 31413}, {7102, 26885}, {8822, 24607}, {15817, 20846}, {17277, 28827}, {18249, 19859}, {18623, 18750}, {18655, 24604}, {21296, 26932}, {23603, 30625}, {24553, 24635}, {26068, 26110}

X(27383) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (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(27383) lies on these lines:

X(27384) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + a b^2 c - 2 b^3 c - a^2 c^2 + a b c^2 + 4 b^2 c^2 - a c^3 - 2 b c^3) : :

X(27384) lies on these lines:

X(27385) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (2 a^3 + a^2 b - 2 a b^2 - b^3 + a^2 c + 2 a b c + b^2 c - 2 a c^2 + b c^2 - c^3) : :

X(27385) lies on these lines:

X(27386) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (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^4 b^3 c - a^3 b^4 c - b^7 c + a^6 c^2 - 2 a^4 b^2 c^2 + a^2 b^4 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 + b^3 c^5 - a^2 c^6 - a c^7 - b c^7) : :

X(27386) lies on these lines:

X(27387) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (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^4 b^3 c - a^3 b^4 c - b^7 c + a^6 c^2 - 3 a^4 b^2 c^2 + 2 a^2 b^4 c^2 + a^5 c^3 + a^4 b c^3 - 2 a^2 b^3 c^3 + a b^4 c^3 + b^5 c^3 - a^4 c^4 - a^3 b c^4 + 2 a^2 b^2 c^4 + a b^3 c^4 - a^3 c^5 + b^3 c^5 - a^2 c^6 - a c^7 - b c^7) : :

X(27387) lies on these lines:

X(27388) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^5 + a^4 b - a^2 b^3 - a b^4 + a^4 c - b^4 c + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27388) lies on these lines:

X(27389) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - b^5 c + b^4 c^2 - a^2 c^4 + b^2 c^4 - a c^5 - b c^5) : :

X(27389) lies on these lines:

X(27390) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    b c (-a + b + c) (a^3 b - a^2 b^2 + a^3 c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + b c^3) : :

X(27390) lies on these lines:

X(27391) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    a (a - b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - a b^2 c - 2 b^3 c - a b c^2 + 2 b^2 c^2 - a c^3 - 2 b c^3) : :

X(27391) lies on these lines:

X(27392) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    a (a - b - c) (a^3 b - a b^3 + a^3 c + 3 a^2 b c - a b^2 c - 3 b^3 c - a b c^2 + 4 b^2 c^2 - a c^3 - 3 b c^3) : :

X(27392) lies on these lines:

X(27393) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 2 a b^2 c - 3 b^3 c - a^2 c^2 + 2 a b c^2 + 6 b^2 c^2 - a c^3 - 3 b c^3) : :

X(27393) lies on these lines:

X(27394) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^5 - a b^4 - 2 a^3 b c + a^2 b^2 c + 2 a b^3 c - b^4 c + a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + 2 a b c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27394) lies on these lines:

X(27395) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (3 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 - b^4 + 2 a^3 c + 2 a^2 b c - 2 a^2 c^2 + 2 b^2 c^2 - 2 a c^3 - c^4) : :

X(27395) lies on these lines:

X(27396) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    a (a - b - c) (a^2 b - b^3 + a^2 c + a b c - c^3) : :

X(27396) lies on these lines: {1, 1257}, {2, 37}, {3, 5279}, {6, 26690}, {7, 25083}, {8, 2335}, {9, 21}, {19, 100}, {22, 198}, {45, 965}, {69, 24635}, {71, 3869}, {72, 13726}, {81, 3719}, {101, 2908}, {145, 1108}, {200, 21039}, {219, 4511}, {241, 4869}, {273, 4552}, {281, 5552}, {307, 3912}, {329, 464}, {332, 16743}, {391, 1212}, {411, 1766}, {579, 2198}, {594, 5742}, {610, 4855}, {728, 3247}, {800, 23988}, {894, 5736}, {908, 8804}, {936, 3731}, {938, 3991}, {1210, 3950}, {1441, 25252}, {1612, 1723}, {1698, 25081}, {1743, 24036}, {1765, 12528}, {1781, 25440}, {1826, 11681}, {1901, 31053}, {1953, 14923}, {1959, 22370}, {2170, 3169}, {2171, 3501}, {2189, 5546}, {2257, 3870}, {2260, 3873}, {2269, 3061}, {2293, 4073}, {2321, 6734}, {2325, 27385}, {2975, 5227}, {3161, 27382}, {3208, 17452}, {3218, 21488}, {3610, 32862}, {3681, 3949}, {3730, 21078}, {3936, 18591}, {3985, 27409}, {4098, 9843}, {4165, 21798}, {4513, 16777}, {4557, 18610}, {4881, 37519}, {5086, 26063}, {5296, 13725}, {5703, 5749}, {5738, 17316}, {5740, 17242}, {5839, 20013}, {6554, 27522}, {6986, 17742}, {6991, 21073}, {7101, 17916}, {8609, 12649}, {11517, 14017}, {13411, 17355}, {15936, 17391}, {16578, 18634}, {17019, 19716}, {17073, 28757}, {17092, 17298}, {17243, 18635}, {17284, 25065}, {18721, 31047}, {20880, 25521}, {21933, 25005}, {25242, 26125}, {27108, 30854}, {27381, 27399}, {27404, 27413}, {27505, 27508}

X(27397) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    a (a - b - c) (a^3 b^2 - a b^4 + a^2 b^2 c + a b^3 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4) : :

X(27397) lies on these lines:

X(27398) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a - b - c) (a + c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(27398) lies on these lines:

X(27399) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 + 2 a^3 b - a^2 b^2 - 2 a b^3 + 2 a^3 c + a^2 b c - b^3 c - a^2 c^2 + 2 b^2 c^2 - 2 a c^3 - b c^3) : :

X(27399) lies on these lines:

X(27400) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (b - c) (a^4 + 2 a^3 b + a^2 b^2 + 2 a^3 c + 5 a^2 b c - b^3 c + a^2 c^2 - b c^3) : :

X(27400) lies on these lines:

X(27401) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^5 + a^4 b - a^2 b^3 - a b^4 + a^4 c + a^3 b c - a b^3 c - b^4 c + 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(27401) lies on these lines:

X(27402) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (5 a^6 + 4 a^5 b - 3 a^4 b^2 - a^2 b^4 - 4 a b^5 - b^6 + 4 a^5 c + 2 a^4 b c - 4 a^3 b^2 c - 2 b^5 c - 3 a^4 c^2 - 4 a^3 b c^2 + 2 a^2 b^2 c^2 + 4 a b^3 c^2 + b^4 c^2 + 4 a b^2 c^3 + 4 b^3 c^3 - a^2 c^4 + b^2 c^4 - 4 a c^5 - 2 b c^5 - c^6) : :

X(27402) lies on these lines:

X(27403) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (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^4 b^3 c - a^3 b^4 c - b^7 c + a^6 c^2 - 4 a^4 b^2 c^2 + 3 a^2 b^4 c^2 + a^5 c^3 + a^4 b c^3 - 4 a^2 b^3 c^3 + a b^4 c^3 + b^5 c^3 - a^4 c^4 - a^3 b c^4 + 3 a^2 b^2 c^4 + a b^3 c^4 - a^3 c^5 + b^3 c^5 - a^2 c^6 - a c^7 - b c^7) : :

X(27403) lies on these lines:

X(27404) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a - b - c) (a + c) (a^6 + 3 a^5 b - 2 a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + 3 a^5 c - 2 a^3 b^2 c - a b^4 c - 2 a^3 b c^2 - 2 a^2 b^2 c^2 + 2 a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 + 2 a b^2 c^3 + a^2 c^4 - a b c^4 + 2 b^2 c^4 - a c^5 - 2 c^6) : :

X(27404) lies on these lines:

X(27405) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a - b - c) (a + c) (a^7 + a^6 b + a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 - a b^6 - b^7 + a^6 c + 2 a^5 b c - a^4 b^2 c - 2 a^3 b^3 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^4 c^3 - 2 a^3 b 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^2 c^5 + b^2 c^5 - a c^6 - b c^6 - c^7) : :

X(27405) lies on these lines:

X(27406) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (7 a^6 + 6 a^5 b - 3 a^4 b^2 - 3 a^2 b^4 - 6 a b^5 - b^6 + 6 a^5 c + 2 a^4 b c - 6 a^3 b^2 c + 2 a^2 b^3 c - 4 b^5 c - 3 a^4 c^2 - 6 a^3 b c^2 + 2 a^2 b^2 c^2 + 6 a b^3 c^2 + b^4 c^2 + 2 a^2 b c^3 + 6 a b^2 c^3 + 8 b^3 c^3 - 3 a^2 c^4 + b^2 c^4 - 6 a c^5 - 4 b c^5 - c^6) : :

X(27406) lies on these lines:

X(27407) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^2 - b^2 - c^2) (3 a^4 + 2 a^3 b + 2 a b^3 + b^4 + 2 a^3 c - 2 a b^2 c - 2 a b c^2 - 2 b^2 c^2 + 2 a c^3 + c^4) : :

X(27407) lies on these lines:

X(27408) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^6 + a^5 b - a^4 b^2 - a b^5 + a^5 c - a^3 b^2 c - a^2 b^3 c - b^5 c - a^4 c^2 - a^3 b c^2 + a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - a^2 b c^3 + a b^2 c^3 + b^2 c^4 - a c^5 - b c^5) : :

X(27408) lies on these lines:

X(27409) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 b - a^2 b^3 + a^4 c - b^4 c + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - b c^4) : :

X(27409) lies on these lines:

X(27410) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^6 + a^5 b - 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 - 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^2 c^4 - a c^5 - b c^5) : :

X(27410) lies on these lines:

X(27411) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^5 + 2 a^4 b - 2 a^2 b^3 - a b^4 + 2 a^4 c - 2 a^3 b c + 2 a b^3 c - 2 b^4 c - 2 a b^2 c^2 + 2 b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - a c^4 - 2 b c^4) : :

X(27411) lies on these lines:

X(27412) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a - b - c) (a + c) (a^4 + a^3 b - a b^3 - b^4 + a^3 c - a b^2 c - a b c^2 + 2 b^2 c^2 - a c^3 - c^4) : :

X(27412) lies on these lines:

X(27413) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^5 + 2 a^4 b - 2 a^2 b^3 - a b^4 + 2 a^4 c - 2 b^4 c + 2 a b^2 c^2 + 2 b^3 c^2 - 2 a^2 c^3 + 2 b^2 c^3 - a c^4 - 2 b c^4) : :

X(27413) lies on these lines:

X(27414) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^6 + a^5 b - a^4 b^2 - a b^5 + a^5 c - 2 a^2 b^3 c - b^5 c - a^4 c^2 + a^2 b^2 c^2 + 2 b^4 c^2 - 2 a^2 b c^3 - 2 b^3 c^3 + 2 b^2 c^4 - a c^5 - b c^5) : :

X(27414) lies on these lines:

X(27415) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a + c) (a - b - c) (a^4 - a^2 b^2 + a b^3 - b^4 - a^2 b c - a^2 c^2 + 2 b^2 c^2 + a c^3 - c^4) : :

X(27415) lies on these lines:

X(27416) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (b - c) (a^4 + 2 a^3 b + a^2 b^2 + 2 a^3 c + a^2 c^2 - 2 b^2 c^2) : :

X(27416) lies on these lines:

X(27417) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    b c (b - c) (-a + b + c) (-5 a^2 + b^2 - 2 b c + c^2) : :

X(27417) lies on these lines:

X(27418) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a + b) (a - b - c) (a + c) (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^3 b c + b^4 c - a^3 c^2 + a^2 c^3 + b c^4 - c^5) : :

X(27418) lies on these lines:

X(27419) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (b - c) (a^5 + a^4 b + a^3 b^2 + a^2 b^3 + a^4 c + a^3 b c + a^3 c^2 - a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(27419) lies on these lines:

X(27420) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 - a^2 b^2 - a^2 b c - b^3 c - a^2 c^2 + 2 b^2 c^2 - b c^3) : :

X(27420) lies on these lines:

X(27421) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (2 a^5 + 3 a^4 b - 3 a^2 b^3 - 2 a b^4 + 3 a^4 c - 3 b^4 c + 2 a b^2 c^2 + 3 b^3 c^2 - 3 a^2 c^3 + 3 b^2 c^3 - 2 a c^4 - 3 b c^4) : :

X(27421) lies on these lines:

X(27422) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 61

Barycentrics    (a - b - c) (a^4 b - a^2 b^3 + a^4 c + a^3 b c - a b^3 c - b^4 c + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 - b c^4) : :

X(27422) lies on these lines:

X(27423) =  REFLECTION OF X(137) IN X(12242)

Barycentrics    (a^4+b^4-b^2 c^2-a^2 (2 b^2+c^2)) (a^4-b^2 c^2+c^4-a^2 (b^2+2 c^2)) (2 a^8+(b^2-c^2)^4-4 a^6 (b^2+c^2)-2 a^2 (b^2-c^2)^2 (b^2+c^2)+3 a^4 (b^4+c^4)) (a^6-3 a^4 (b^2+c^2)-(b^2-c^2)^2 (b^2+c^2)+a^2 (3 b^4-b^2 c^2+3 c^4)) : :
X(27423) = X[137]-2*X[12242], 3*X[195]+X[13512], X[930]+X[15801], X[6343]+X[20424]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28607.

X(27423) lies on these lines: {5,49}, {128,539}, {137,12242}, {195,13512}, {930,15801}, {1154,14071}, {1157,6592}, {1493,25150}, {5965,12060}, {6343,20424}, {14072,25044}, {14857,18400}, {15959,19468}

X(27423) = reflection of X(137) in X(12242)

leftri

Collineation mappings involving Gemini triangle 62: X(27424)-X(27470)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 62, as in centers X(27424)-X(27470). Then

m(X) = (bc+ac-ab)(ab-ac+bc)(b^2+bc+c^2)x + (c^2+ab)(ab-ac+bc)ab-ac-bc)y + (b^2+ac)(ac-ab-bc)(ac-ab-bc)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 8, 2018)

X(27424) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    b c (-a + b + c) (a b - a c + b c) (-a b + a c + b c) : :

X(27424) lies on these lines:

X(27424) = isotomic conjugate of X(1423)
X(27424) = complement of X(36858)

X(27425) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^3 b^2 + a^2 b^3 - a b^4 - b^5 + a^3 b c - a b^3 c + a^3 c^2 + b^3 c^2 + a^2 c^3 - a b c^3 + b^2 c^3 - a c^4 - c^5) : :

X(27425) lies on these lines:

X(27426) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^5 b + 2 a^4 b^2 - 2 a^2 b^4 - a b^5 + a^5 c + a^4 b c - 2 a^3 b^2 c + a b^4 c - b^5 c + 2 a^4 c^2 - 2 a^3 b c^2 + 2 b^3 c^3 - 2 a^2 c^4 + a b c^4 - a c^5 - b c^5) : :

X(27426) lies on these lines:

X(27427) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c - 2 a^3 b^2 c + a^2 b^3 c + a b^4 c - b^5 c + a^4 c^2 - 2 a^3 b c^2 - a b^3 c^2 - a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + a b c^4 - b c^5) : :

X(27427) lies on these lines:

X(27428) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - a b^3 + a^2 b c + a^2 c^2 - 2 b^2 c^2 - a c^3) : :

X(27428) lies on these lines:

X(27429) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b + 2 a^2 b^2 - 3 a b^3 + a^3 c - 3 a^2 b c + a b^2 c + b^3 c + 2 a^2 c^2 + a b c^2 - 2 b^2 c^2 - 3 a c^3 + b c^3) : :

X(27429) lies on these lines:

X(27430) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b + a b^2 + a^2 c + 3 a b c - 3 b^2 c + a c^2 - 3 b c^2) : :

X(27430) lies on these lines:

X(27431) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (2 a^2 b^2 - 2 a b^3 + a^2 b c + b^3 c + 2 a^2 c^2 - 2 b^2 c^2 - 2 a c^3 + b c^3) : :

X(27431) lies on these lines:

X(27432) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b + c) (a b - a c + b c) (-a b + a c + b c) (-a^2 - a b - a c + 2 b c) : :

X(27432) lies on these lines:

X(27433) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + 2 a b c^3 + b^2 c^3 - b c^4) : :

X(27433) lies on these lines:

X(27434) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^2 - a b^4 + a^3 b c + a^3 c^2 - b^3 c^2 - b^2 c^3 - a c^4) : :

X(27434) lies on these lines:

X(27435) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^4 b^2 - a b^5 + a^4 b c + a^4 c^2 - b^4 c^2 - b^2 c^4 - a c^5) : :

X(27435) lies on these lines:

X(27436) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - a b^3 + b^3 c + a^2 c^2 - a c^3 + b c^3) : :

X(27436) lies on these lines:

X(27437) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^3 - a^2 b^4 - a^2 b^3 c + a b^3 c^2 + b^4 c^2 + a^3 c^3 - a^2 b c^3 + a b^2 c^3 - a^2 c^4 + b^2 c^4) : :

X(27437) lies on these lines:

X(27438) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    b c (b + c) (a b - a c + b c) (-a b + a c + b c) (-2 a^2 + a b + a c + b c) : :

X(27438) lies on these lines:

X(27439) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    b c (a b - a c + b c) (-a b + a c + b c) (-3 a^2 b + a b^2 - 3 a^2 c + 3 a b c + b^2 c + a c^2 + b c^2) : :

X(27439) lies on these lines:

X(27440) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (3 a^2 b^2 - 3 a b^3 + 2 a^2 b c + b^3 c + 3 a^2 c^2 - 4 b^2 c^2 - 3 a c^3 + b c^3) : :

X(27440) lies on these lines:

X(27441) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (3 a^2 b^2 - 3 a b^3 + a^2 b c + 2 b^3 c + 3 a^2 c^2 - 2 b^2 c^2 - 3 a c^3 + 2 b c^3) : :

X(27441) lies on these lines:

X(27442) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - b^4 + a^2 b c - 2 a b^2 c + b^3 c + a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + b c^3 - c^4) : :

X(27442) lies on these lines:

X(27443) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (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 - 4 b^2 c^2 - a c^3 - b c^3) : :

X(27443) lies on these lines:

X(27444) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a b^3 + 2 a^2 b c - b^3 c - b^2 c^2 + a c^3 - b c^3) : :

X(27444) lies on these lines:

X(27445) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^4 + a^3 b^2 c + a^3 b c^2 - b^4 c^2 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(27445) lies on these lines:

X(27446) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^2 - a b^4 + a^3 b c + a^2 b^2 c - a b^3 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 - a c^4) : :

X(27446) lies on these lines:

X(27447) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b^2 + a c) (c^2 + a b) (a b - a c - b c) (a b - a c + b c) : :

X(27447) lies on these lines:

X(27447) = isotomic conjugate of X(17752)

X(27448) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b + a b^2 + a^2 c + 5 a b c - 5 b^2 c + a c^2 - 5 b c^2) : :

X(27448) lies on these lines:

X(27449) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    b c (a b - a c + b c) (-a b + a c + b c) (-a^3 + 7 a^2 b - 3 a b^2 + b^3 + 7 a^2 c - 6 a b c - b^2 c - 3 a c^2 - b c^2 + c^3) : :

X(27449) lies on these lines:

X(27450) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a^2 - b c) (a b - a c - b c) (a b - a c + b c) (b^2 + 3 b c + c^2) : :

X(27450) lies on these lines:

X(27451) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b - c) (a b - a c + b c) (-a b + a c + b c) (a^3 b + a b^3 + a^3 c - a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 + b^2 c^2 + a c^3 + b c^3) : :

X(27451) lies on these lines:

X(27452) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (b - c) (a b - a c - b c) (a b - a c + b c) (a^2 b^2 + a b^3 + a^2 b c + a b^2 c - b^3 c + a^2 c^2 + a b c^2 + a c^3 - b c^3) : :

X(27452) lies on these lines:

X(27453) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b - c) (a b - a c + b c) (-a b + a c + 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(27453) lies on these lines:

X(27454) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^2 - a b^4 + a^3 b c + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - b^2 c^3 - a c^4) : :

X(27454) lies on these lines:

X(27455) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a b + b^2 + a c + c^2) : :

X(27455) lies on these lines:

X(27456) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^3 b^2 + a^2 b^3 - a b^4 - b^5 + a^3 b c - a b^3 c + a^3 c^2 - a b^2 c^2 + 2 b^3 c^2 + a^2 c^3 - a b c^3 + 2 b^2 c^3 - a c^4 - c^5) : :

X(27456) lies on these lines:

X(27457) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^3 b^2 + a^2 b^3 - a b^4 - b^5 + a^3 b c - a b^3 c + a^3 c^2 + a b^2 c^2 + a^2 c^3 - a b c^3 - a c^4 - c^5) : :

X(27457) lies on these lines:

X(27458) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^2 - a b^4 + 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(27458) lies on these lines:

X(27459) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^3 b^2 + a^2 b^3 - a b^4 - b^5 + a^3 b c - a b^3 c + a^3 c^2 + 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 - c^5) : :

X(27459) lies on these lines:

X(27460) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (2 a^3 b^2 - 2 a b^4 + a^3 b c - a^2 b^2 c - a b^3 c + b^4 c + 2 a^3 c^2 - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a b c^3 - b^2 c^3 - 2 a c^4 + b c^4) : :

X(27460) lies on these lines:

X(27461) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - b^4 + a^2 b c - 3 a b^2 c + b^3 c + a^2 c^2 - 3 a b c^2 + 3 b^2 c^2 + b c^3 - c^4) : :

X(27461) lies on these lines:

X(27462) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^4 b^2 - a^3 b^3 + a^2 b^4 - a b^5 + a^4 b c - 2 a^3 b^2 c + a b^4 c + a^4 c^2 - 2 a^3 b c^2 + a^2 b^2 c^2 - b^4 c^2 - a^3 c^3 + 2 b^3 c^3 + a^2 c^4 + a b c^4 - b^2 c^4 - a c^5) : :

X(27462) lies on these lines:

X(27463) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^5 b + a^4 b^2 - 2 a^2 b^4 + a^5 c - 2 a^3 b^2 c + a b^4 c - b^5 c + a^4 c^2 - 2 a^3 b c^2 + b^4 c^2 + 2 b^3 c^3 - 2 a^2 c^4 + a b c^4 + b^2 c^4 - b c^5) : :

X(27463) lies on these lines:

X(27464) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (2 a^2 b^2 - 3 a b^3 - a^2 b c + 2 b^3 c + 2 a^2 c^2 - b^2 c^2 - 3 a c^3 + 2 b c^3) : :

X(27464) lies on these lines:

X(27465) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - 2 a b^3 - 2 a^2 b c + 2 b^3 c + a^2 c^2 + b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(27465) lies on these lines:

X(27466) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b - c) (a b - a c + b c) (-a b + a c + b c) (2 a^2 b^2 + 2 a^2 b c - a b^2 c + 2 a^2 c^2 - a b c^2 + b^2 c^2) : :

X(27466) lies on these lines:

X(27467) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (b - c) (a b - a c + b c) (-a b + a c + b c) (a^3 b - 2 a^2 b^2 + a b^3 + a^3 c - 3 a^2 b c + 3 a b^2 c + b^3 c - 2 a^2 c^2 + 3 a b c^2 + a c^3 + b c^3) : :

X(27467) lies on these lines:

X(27468) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (b - c) (a b - a c - b c) (a b - a c + b c) (a^2 b^2 + a b^3 + 2 a b^2 c - 3 b^3 c + a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 + a c^3 - 3 b c^3) : :

X(27468) lies on these lines:

X(27469) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    a (b - c) (a^2 b^2 + a b^3 + a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 + b c^3) : :

X(27469) lies on these lines:

X(27470) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 62

Barycentrics    (a b - a c - b c) (a^2 - 2 a b - 2 a c + b c) (a b - a c + b c) (b^2 - b c + c^2) : :

X(27470) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 63: X(27471)-X(27495)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 63, as in centers X(27471)-X(27495). Then

m(X) = -(b^2+bc+c^2)x + (ab+2ac+2bc+c^2)y + (ac+2ab+2bc+b^2)z : :

(Clark Kimberling, November 9, 2018)

X(27471) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c - a b^3 c + 2 b^4 c - a^3 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + 2 b c^4 : :

X(27471) lies on these lines: {2, 8680}, {9, 3739}, {37, 1465}, {75, 908}, {142, 5179}, {192, 27493}, {273, 1826}, {514, 18161}, {518, 5587}, {984, 7951}, {1246, 27483}, {4688, 31142}, {4699, 31018}, {5692, 20718}, {6996, 24315}, {7146, 16732}, {7384, 24682}, {7406, 24683}, {13476, 18412}, {17789, 20923}, {18697, 20236}, {20171, 27491}, {20891, 27476}, {24435, 27381}, {24773, 25651}, {26063, 27484}, {31160, 31178}

X(27472) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    3 a^4 b - a^3 b^2 - 3 a^2 b^3 + a b^4 + 3 a^4 c - a^3 b c - a b^3 c - b^4 c - a^3 c^2 + b^3 c^2 - 3 a^2 c^3 - a b c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(27472) lies on these lines: {2, 8680}, {7, 37}, {9, 5088}, {19, 31346}, {63, 27492}, {75, 5744}, {192, 3218}, {374, 31169}, {514, 573}, {518, 5731}, {984, 4293}, {1444, 27958}, {2094, 4664}, {2183, 3177}, {2245, 3212}, {3696, 5775}, {5768, 30273}, {5770, 29010}, {6999, 24316}, {7560, 21367}, {16574, 25252}, {17134, 24435}, {22001, 27339}, {25241, 27480}, {27268, 31019}

X(27473) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    3 a^4 b - 2 a^3 b^2 - 3 a^2 b^3 + 2 a b^4 + 3 a^4 c - 2 a^3 b c - 2 a b^3 c + b^4 c - 2 a^3 c^2 - b^3 c^2 - 3 a^2 c^3 - 2 a b c^3 - b^2 c^3 + 2 a c^4 + b c^4 : :

X(27473) lies on these lines: {2, 8680}, {37, 3911}, {142, 3986}, {518, 10165}, {1445, 25523}, {3306, 4687}, {4777, 15584}, {5883, 20718}, {16560, 24684}, {25456, 27474}

X(27474) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (a^2 - a b - a c - 2 b c) (b^2 + b c + c^2) : :

X(27474) lies on these lines: {2, 740}, {8, 17755}, {10, 31322}, {37, 17269}, {75, 142}, {192, 29611}, {239, 32941}, {306, 27491}, {312, 1921}, {321, 20435}, {335, 4535}, {518, 17294}, {536, 21358}, {726, 29594}, {742, 17281}, {871, 1978}, {982, 31027}, {984, 3661}, {1001, 3696}, {1043, 16822}, {1930, 20431}, {3008, 4709}, {3061, 17762}, {3687, 27489}, {3706, 17026}, {3739, 3875}, {3755, 24603}, {3943, 24357}, {3946, 4751}, {3993, 29604}, {4044, 21615}, {4085, 29576}, {4133, 29571}, {4527, 17244}, {4671, 27493}, {4673, 30038}, {4699, 5308}, {4766, 33077}, {4772, 29621}, {5695, 17738}, {6542, 31314}, {10453, 24631}, {11679, 21483}, {15569, 29603}, {16061, 17733}, {16819, 31327}, {16826, 31335}, {16834, 28581}, {17063, 31028}, {17310, 31178}, {17316, 24325}, {17389, 17769}, {17550, 20653}, {17765, 29617}, {18697, 20236}, {20131, 24342}, {20891, 20895}, {24349, 29616}, {24586, 32932}, {24629, 29824}, {25456, 27473}, {26582, 29674}, {26590, 32778}, {29593, 31323}

X(27475) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (a b - b^2 + 2 a c + b c) (2 a b + a c + b c - c^2) : :

X(27475) lies on these lines: {1, 673}, {2, 210}, {7, 37}, {9, 86}, {27, 33}, {57, 21453}, {65, 27253}, {75, 142}, {85, 21808}, {144, 27268}, {192, 4373}, {226, 1088}, {272, 2303}, {273, 1826}, {310, 312}, {335, 16593}, {341, 29968}, {390, 15569}, {480, 27399}, {675, 8693}, {726, 29600}, {740, 29573}, {742, 17313}, {857, 8818}, {871, 20917}, {903, 4664}, {984, 5542}, {1001, 14621}, {1215, 30822}, {1240, 20923}, {1268, 4751}, {1440, 1903}, {1445, 25523}, {1475, 31269}, {2250, 8545}, {2346, 11349}, {2400, 28898}, {2550, 17316}, {2886, 31038}, {3243, 4384}, {3303, 27000}, {3649, 27129}, {3661, 3826}, {3673, 17758}, {3696, 29616}, {3739, 5936}, {3797, 27494}, {3807, 30758}, {3834, 24357}, {3842, 5223}, {3957, 24596}, {3993, 29606}, {4321, 17022}, {4393, 15570}, {4678, 31352}, {4698, 18230}, {4755, 4795}, {4776, 6548}, {4871, 30869}, {5220, 29578}, {5226, 31526}, {5249, 20173}, {5805, 30273}, {5845, 17392}, {5852, 31350}, {5853, 29574}, {5880, 6650}, {6384, 18743}, {6600, 16412}, {6666, 17381}, {7247, 26101}, {7249, 17056}, {8049, 14746}, {9311, 17451}, {10129, 31058}, {13407, 17671}, {15254, 29595}, {15888, 26531}, {16503, 17394}, {16728, 17169}, {16830, 20135}, {17021, 18450}, {17050, 17158}, {17230, 31329}, {17266, 31317}, {17284, 24325}, {17292, 31335}, {17768, 29622}, {18139, 27476}, {20430, 31657}, {24349, 29627}, {24393, 24603}, {24427, 30663}, {27431, 27447}, {27493, 31019}, {29579, 31347}, {29581, 31323}, {30821, 32771}, {30829, 31002}

X(27475) = isogonal conjugate of X(2280)
X(27475) = isotomic conjugate of X(4384)

X(27476) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c + 2 a b^3 c + 2 b^4 c - a^3 c^2 + b^3 c^2 + 2 a b c^3 + b^2 c^3 + a c^4 + 2 b c^4 : :

X(27476) lies on these lines: {2, 742}, {75, 3936}, {312, 27493}, {321, 20435}, {740, 33122}, {2296, 27483}, {3797, 33151}, {4688, 31179}, {5249, 27478}, {5739, 27484}, {5741, 27489}, {17778, 31314}, {18139, 27475}, {19684, 31306}, {19785, 27480}, {20891, 27471}, {20892, 27488}, {24325, 33070}, {26234, 31006}, {27184, 27481}, {27495, 32782}

X(27477) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^4 b^2 + a b^5 - a^4 b c + 2 a b^4 c + 2 b^5 c - a^4 c^2 + b^4 c^2 + 2 a b c^4 + b^2 c^4 + a c^5 + 2 b c^5 : :

X(27477) lies on these lines: {2, 744}, {75, 4766}, {18697, 20236}

X(27478) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    a^2 b^2 - a b^3 + 4 a^2 b c + 4 a b^2 c + b^3 c + a^2 c^2 + 4 a b c^2 + 4 b^2 c^2 - a c^3 + b c^3 : :

X(27478) lies on these lines: {2, 726}, {10, 335}, {37, 4472}, {75, 142}, {192, 16673}, {239, 31314}, {274, 17760}, {321, 24060}, {514, 3572}, {740, 29574}, {899, 31063}, {984, 24603}, {1125, 31319}, {1266, 24357}, {1278, 5308}, {1909, 14951}, {3008, 31317}, {3661, 31329}, {3687, 27491}, {3797, 29571}, {3807, 21101}, {3993, 16826}, {4054, 27493}, {4384, 5223}, {4385, 30063}, {4688, 9055}, {4699, 4859}, {4709, 6542}, {4772, 29611}, {4821, 29621}, {4968, 30030}, {4970, 17032}, {4991, 20145}, {5249, 27476}, {8669, 16917}, {8720, 33047}, {10009, 20917}, {15497, 21062}, {16823, 17738}, {17023, 24325}, {17140, 24592}, {20131, 32921}, {20154, 32935}, {20498, 27489}, {20913, 21443}, {21078, 27492}, {22030, 22047}, {24621, 32453}, {31331, 31350}

X(27478) = complement of X(27481)

X(27479) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c - a b^3 c + 2 b^4 c - a^3 c^2 - 6 a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + 2 b c^4 : :

X(27479) lies on these lines: {2, 14439}, {75, 908}, {226, 27491}, {312, 27487}, {321, 20435}, {329, 27484}, {518, 31140}, {984, 3120}, {4713, 31993}, {5249, 20173}, {5905, 24694}, {7249, 27494}, {10707, 31178}, {15497, 21062}, {27184, 27495}, {27480, 30699}

X(27480) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    3 a^3 b + 2 a^2 b^2 + a b^3 + 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 - 4 b^2 c^2 + a c^3 - b c^3 : :

X(27480) lies on these lines: {1, 27478}, {2, 740}, {8, 27495}, {9, 192}, {37, 28634}, {75, 4470}, {145, 335}, {726, 16834}, {1278, 17014}, {2321, 27268}, {2550, 17316}, {2901, 27299}, {3210, 17027}, {3661, 3755}, {3797, 5222}, {3886, 16826}, {3912, 4780}, {3946, 4699}, {3993, 4384}, {4000, 27487}, {4085, 29593}, {4133, 24603}, {4360, 20159}, {4393, 4649}, {4704, 31323}, {4709, 17308}, {4743, 17230}, {4970, 17026}, {5853, 17389}, {17147, 25249}, {17733, 22267}, {19785, 27476}, {19791, 27491}, {20158, 31310}, {20162, 32922}, {24427, 27919}, {25241, 27472}, {27479, 30699}, {29570, 32941}, {31350, 31352}

X(27481) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (2 a^2 + a b + a c - b c) (b^2 + b c + c^2) : :

X(27481) lies on these lines: {1, 6651}, {2, 726}, {9, 192}, {10, 31329}, {37, 17339}, {38, 31028}, {75, 1213}, {190, 14621}, {335, 16593}, {440, 20254}, {518, 17389}, {536, 16590}, {597, 4370}, {740, 29617}, {742, 17333}, {984, 3661}, {1278, 31352}, {2276, 3807}, {3159, 17030}, {3161, 4704}, {3662, 27487}, {3710, 30177}, {3739, 31351}, {3799, 19586}, {3993, 4393}, {3995, 17027}, {4075, 27091}, {4392, 30967}, {4687, 31350}, {4699, 17324}, {5513, 27493}, {6544, 27486}, {7226, 31027}, {8669, 33063}, {8720, 33062}, {10459, 25270}, {16475, 29584}, {16826, 24349}, {16972, 17319}, {17032, 17165}, {17262, 20172}, {17316, 31302}, {17367, 17755}, {20132, 32935}, {20142, 32921}, {21443, 31060}, {21838, 28606}, {24068, 27255}, {24325, 29612}, {27184, 27476}, {27268, 29609}, {28582, 29622}, {31036, 32453}

X(27481) = complement of X(27494)
X(27481) = anticomplement of X(27478)

X(27482) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    2 a^3 b^3 + a^2 b^4 + a^3 b^2 c + 2 a^2 b^3 c + a^3 b c^2 - b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4 : :

X(27482) lies on these lines: {2, 714}, {37, 17339}, {75, 31344}, {3661, 4735}, {4377, 29576}, {25241, 27472}

X(27483) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (2 a b + b^2 + a c + 2 b c) (a b + 2 a c + 2 b c + c^2) : :

X(27483) lies on these lines: {2, 740}, {7, 1654}, {10, 335}, {27, 242}, {37, 1268}, {75, 1213}, {86, 239}, {192, 5936}, {273, 26023}, {310, 1921}, {673, 6651}, {675, 28841}, {871, 10009}, {903, 4688}, {984, 27494}, {1246, 27471}, {2296, 27476}, {3125, 4469}, {3661, 3826}, {3696, 16826}, {3797, 24603}, {4373, 4772}, {4384, 14621}, {4393, 5625}, {4732, 6542}, {4751, 6707}, {4835, 27447}, {6384, 19804}, {6548, 27791}, {6650, 17755}, {20090, 30712}, {21926, 26019}, {24325, 31314}, {24589, 31002}, {26626, 28626}, {27493, 31025}, {28581, 29580}, {29609, 31238}

X(27483) = isotomic conjugate of X(16826)

X(27484) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    3 a^3 b - 2 a^2 b^2 - a b^3 + 3 a^3 c + a^2 b c - 5 a b^2 c + b^3 c - 2 a^2 c^2 - 5 a b c^2 - 2 b^2 c^2 - a c^3 + b c^3 : :

X(27484) lies on these lines: {2, 210}, {7, 1654}, {8, 17755}, {9, 192}, {37, 17014}, {69, 27487}, {72, 27304}, {75, 144}, {142, 17238}, {145, 31342}, {329, 27479}, {335, 15590}, {390, 3797}, {527, 17488}, {673, 5220}, {726, 16833}, {984, 5222}, {1001, 4393}, {2550, 31329}, {3243, 16826}, {3616, 31336}, {3661, 24393}, {3691, 27288}, {3739, 5232}, {3951, 27000}, {4384, 5223}, {4740, 6172}, {4772, 20059}, {4875, 20535}, {5542, 24603}, {5698, 31310}, {5739, 27476}, {5845, 17346}, {5853, 29617}, {6008, 27855}, {6600, 16367}, {6666, 17397}, {10005, 29616}, {16552, 25242}, {16593, 17230}, {16827, 17480}, {17026, 27538}, {17134, 24435}, {17350, 20172}, {17490, 31348}, {17746, 17753}, {18230, 26626}, {20154, 32029}, {21168, 29010}, {21384, 27340}, {24592, 32937}, {24599, 31302}, {24616, 28910}, {24631, 26038}, {26063, 27471}, {27493, 31018}, {28132, 28898}

X(27485) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (b - c) (-a^2 b^2 - a^2 b c - a b^2 c - a^2 c^2 - a b c^2 + 2 b^2 c^2) : :

X(27485) lies on these lines: {37, 30835}, {75, 3835}, {192, 27138}, {514, 1921}, {872, 24749}, {4688, 31147}, {4699, 20295}, {4728, 4777}, {4751, 31286}, {4772, 26798}, {4850, 27773}, {9002, 14433}, {17458, 20906}, {20923, 20952}, {30090, 30094}, {31207, 31238}

X(27486) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (b - c) (-2 a^2 + a b + b^2 + a c + b c + c^2) : :

X(27486) lies on these lines: {2, 522}, {81, 16755}, {239, 514}, {649, 1019}, {25259, 28898}, {657, 3219}, {661, 28867}, {824, 1635}, {900, 1491}, {918, 31150}, {16892, 17494}, {1459, 17011}, {1638, 17069}, {2786, 4893}, {3210, 21225}, {3907, 13254}, {4393, 30573}, {4453, 4762}, {4786, 5011}, {4728, 30765}, {18197, 18668}, {4777, 4789}, {4897, 28902}, {4932, 4960}, {5256, 21173}, {6544, 27481}, {6546, 30519}, {6586, 28606}, {6589, 16751}, {7658, 26985}, {11125, 27344}, {14475, 24184}, {16815, 21201}, {16816, 21132}, {17924, 26023}, {19804, 20954}, {21124, 23755}, {21195, 27186}, {25009, 26732}

X(27487) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^2 b^2 + a b^3 - a^2 b c - a b^2 c + 2 b^3 c - a^2 c^2 - a b c^2 - b^2 c^2 + a c^3 + 2 b c^3 : :

X(27487) lies on these lines: {2, 742}, {37, 17266}, {69, 27484}, {75, 142}, {86, 239}, {141, 27495}, {312, 27479}, {320, 17755}, {322, 27488}, {335, 3834}, {514, 1921}, {518, 17297}, {984, 17227}, {1086, 3797}, {3008, 4751}, {3263, 3807}, {3662, 27481}, {3696, 15570}, {4000, 27480}, {4358, 27493}, {4360, 31342}, {4417, 27489}, {4648, 4699}, {4657, 31319}, {4675, 31317}, {4688, 17310}, {5224, 31322}, {10436, 20159}, {17050, 17762}, {17237, 31323}, {17244, 24357}, {17292, 25384}, {17300, 31314}, {17302, 31308}, {17322, 31336}, {17789, 20923}, {18134, 27491}, {20335, 20947}, {20955, 30030}, {24325, 32847}, {26106, 27343}, {27918, 30967}, {29607, 31238}, {31138, 31349}

X(27488) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c - a b^3 c + 2 b^4 c - a^3 c^2 + 6 a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + 2 b c^4 : :

X(27488) lies on these lines: {7, 1654}, {75, 908}, {192, 5328}, {322, 27487}, {1278, 27493}, {3306, 3739}, {4688, 31164}, {7777, 31344}, {20891, 20895}, {20892, 27476}

X(27489) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    2 a^3 b^2 - 2 a b^4 + 5 a^3 b c - 3 a^2 b^2 c - a b^3 c - b^4 c + 2 a^3 c^2 - 3 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 - a b c^3 + b^2 c^3 - 2 a c^4 - b c^4 : :

X(27489) lies on these lines: {2, 210}, {75, 908}, {239, 5289}, {312, 3807}, {3452, 20173}, {3687, 27474}, {3949, 20946}, {4417, 27487}, {4751, 30832}, {5741, 27476}, {5854, 29617}, {19804, 30985}, {20498, 27478}, {27131, 27493}

X(27490) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c + 2 a b^3 c + 2 b^4 c - a^3 c^2 + 3 a b^2 c^2 + b^3 c^2 + 2 a b c^3 + b^2 c^3 + a c^4 + 2 b c^4 : :

X(27490) lies on these lines: {37, 30832}, {75, 3936}, {86, 239}, {744, 33160}, {3687, 25361}, {4043, 27493}, {4451, 4777}, {18697, 20236}, {18805, 32861}, {20923, 20932}

X(27491) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    2 a^3 b^2 - 2 a b^4 + 5 a^3 b c + 3 a^2 b^2 c - a b^3 c - b^4 c + 2 a^3 c^2 + 3 a^2 b c^2 + b^3 c^2 - a b c^3 + b^2 c^3 - 2 a c^4 - b c^4 : :

X(27491) lies on these lines: {2, 210}, {37, 31018}, {75, 3936}, {149, 33070}, {226, 27479}, {306, 27474}, {440, 20254}, {495, 3661}, {952, 17389}, {956, 16826}, {3687, 27478}, {4664, 31179}, {5252, 6542}, {5719, 17397}, {7381, 31342}, {13257, 20430}, {18134, 27487}, {19791, 27480}, {20171, 27471}, {20173, 27493}, {30758, 31006}

X(27492) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    a^3 b^2 - a b^4 + 4 a^3 b c - 3 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 3 a^2 b c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 - a c^4 + b c^4 : :

X(27492) lies on these lines: {9, 192}, {63, 27472}, {75, 908}, {80, 4709}, {514, 4416}, {726, 5692}, {1278, 31018}, {3807, 17787}, {3949, 28974}, {3993, 5251}, {4431, 5179}, {4699, 5219}, {4740, 31142}, {4777, 22271}, {7201, 21617}, {8680, 17781}, {20882, 30006}, {21033, 26665}, {21078, 27478}

X(27493) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c + 3 a^2 b^2 c - a b^3 c + 2 b^4 c - a^3 c^2 + 3 a^2 b c^2 - 3 a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + 2 b c^4 : :

X(27493) lies on these lines: {2, 14439}, {75, 30566}, {192, 27471}, {312, 27476}, {321, 3807}, {335, 4080}, {518, 10707}, {536, 4945}, {1278, 27488}, {3696, 4767}, {3944, 31084}, {4043, 27490}, {4054, 27478}, {4358, 27487}, {4671, 27474}, {4688, 31171}, {5513, 27481}, {17484, 24712}, {17755, 30578}, {18359, 30565}, {20173, 27491}, {24325, 24709}, {26580, 27495}, {27131, 27489}, {27475, 31019}, {27483, 31025}, {27484, 31018}

X(27494) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (a b + 2 b^2 - a c + b c) (a b - a c - b c - 2 c^2) : :

X(27494) lies on these lines: {2, 726}, {7, 1278}, {8, 6650}, {37, 30598}, {75, 4377}, {86, 192}, {310, 6382}, {321, 6384}, {330, 27809}, {335, 4535}, {536, 31342}, {599, 903}, {673, 5220}, {984, 27483}, {1268, 4699}, {2296, 17147}, {3634, 31351}, {3797, 27475}, {4373, 4821}, {4393, 4649}, {4671, 31002}, {4704, 28626}, {4772, 5936}, {4788, 29585}, {7249, 27479}, {17389, 28522}, {17490, 32011}, {20055, 24692}, {20158, 32935}, {22036, 27318}, {24080, 24176}, {24325, 31319}, {27447, 27465}

X(27494) = isogonal conjugate of X(21793)
X(27494) = isotomic conjugate of X(4393)
X(27494) = anticomplement of X(27481)
X(27494) = X(19)-isoconjugate of X(23095)

X(27495) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 63

Barycentrics    (a^2 + 2 a b + 2 a c + b c) (b^2 + b c + c^2) : :

X(27495) lies on these lines: {2, 210}, {8, 27480}, {10, 335}, {37, 319}, {38, 31348}, {75, 4377}, {141, 27487}, {192, 5232}, {320, 25384}, {391, 27268}, {740, 29615}, {742, 17271}, {756, 31027}, {984, 3661}, {1100, 4687}, {1211, 30179}, {3678, 27274}, {3807, 3862}, {3842, 4649}, {3912, 31323}, {4015, 27324}, {4078, 31331}, {4698, 29592}, {4725, 16590}, {6653, 24723}, {15569, 29588}, {16503, 17260}, {16830, 20132}, {17250, 24357}, {17292, 17755}, {17308, 31317}, {17389, 17772}, {17769, 29617}, {24325, 29610}, {24349, 31347}, {26580, 27493}, {27184, 27479}, {27476, 32782}, {28605, 30638}, {29574, 31350}

leftri

Collineation mappings involving Gemini triangle 64: X(27496)-X(27503)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 64, as in centers X(27496)-X(27503). Then

m(X) = (ab-ac-bc)(ab-ac+bc)(b^2-bc+c^2)x + (ab-ac-bc)(ab-ac+bc)(ab-2ac+c^2)y : (ac-ab-bc)(ac-ab+bc)(ac-2ab+b^2)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 9, 2018)

X(27496) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    b c (-3 a + b + c) (a b - a c + b c) (-a b + a c + b c) : :

X(27496) lies on these lines: {2, 330}, {75, 4051}, {87, 1222}, {932, 9083}, {4598, 7153}, {5749, 25303}, {7155, 7320}, {7209, 27818}, {7275, 12782}, {8051, 31994}, {17754, 24524}, {27431, 30036}

X(27497) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - a b^3 - a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 - a c^3) : :

X(27497) lies on these lines:

X(27498) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (a b - b^2 - 2 a c + b c) (a b - a c + b c) (2 a b - a c - b c + c^2) : :

X(27498) lies on these lines:

X(27499) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    a (a b - a c - b c) (a b - a c + b c) (a b + b^2 + a c - 2 b c + c^2) : :

X(27499) lies on these lines:

X(27500) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^3 b^2 - a b^4 - a^3 b c + 2 a b^3 c + a^3 c^2 - b^3 c^2 + 2 a b c^3 - b^2 c^3 - a c^4) : :

X(27500) lies on these lines:

X(27501) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (a b - a c + b c) (a^2 b^2 - a b^3 - 4 a^2 b c + b^3 c + a^2 c^2 - a c^3 + b c^3) : :

X(27501) lies on these lines:

X(27502) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (2 a^2 - a b - a c + b c) (a b - a c + b c) (b^2 - b c + c^2) : :

X(27502) lies on these lines:

X(27503) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 64

Barycentrics    (a b - a c - b c) (a b - a c + b c) (2 a^3 b^3 - a^2 b^4 - a^3 b^2 c - a^3 b c^2 + b^4 c^2 + 2 a^3 c^3 - b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(27503) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 65: X(27504)-X(27549)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 65, as in centers X(27504)-X(27549). Then

m(X) = a(a+b+c)(a-b-c)x - b(a+b-c)(a-b-c) - c(a-b+c)(a-b-c)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 9, 2018)

X(27504) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 + a^5 b - a^4 b^2 + a^2 b^4 - a b^5 - b^6 + a^5 c - a^3 b^2 c - a^2 b^3 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 - 2 b^3 c^3 + a^2 c^4 + b^2 c^4 - a c^5 + b c^5 - c^6) : :

X(27504) lies on these lines: {2, 3}, {78, 28826}, {1265, 3699}, {1936, 5906}, {3193, 31034}, {5552, 27521}, {6734, 28796}, {7952, 20222}, {11500, 23541}, {27507, 27513}

X(27505) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 + 2 a^5 b + a^4 b^2 - a^2 b^4 - 2 a b^5 - b^6 + 2 a^5 c - 2 a^4 b c - 2 a^3 b^2 c + 2 a^2 b^3 c + a^4 c^2 - 2 a^3 b c^2 - 2 a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 - a^2 c^4 + b^2 c^4 - 2 a c^5 - c^6) : :

X(27505) lies on these lines:

X(27506) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 65

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 - a^3 b^2 c + 3 a^2 b^3 c - b^5 c + 2 a^4 c^2 - a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 + 3 a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - 2 a^2 c^4 - a c^5 - b c^5) : :

X(27506) lies on these lines:

X(27507) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (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(27507) lies on these lines:

X(27508) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (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(27508) lies on these lines:

X(27509) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^2 - b^2 - c^2) (a^2 + b^2 - 2 b c + c^2) : :

X(27509) lies on these lines:

X(27510) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^8 + a^7 b + a^5 b^3 - a^3 b^5 - a b^7 - b^8 + 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 + 2 b^4 c^4 - a^3 c^5 - b^3 c^5 - a c^7 + b c^7 - c^8) : :

X(27510) lies on these lines:

X(27511) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^8 + a^7 b + a^5 b^3 - a^3 b^5 - a b^7 - b^8 + a^7 c - a^4 b^3 c - a^3 b^4 c + b^7 c + a^4 b^2 c^2 - a^2 b^4 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^3 b c^4 - a^2 b^2 c^4 + a b^3 c^4 + 2 b^4 c^4 - a^3 c^5 - b^3 c^5 - a c^7 + b c^7 - c^8) : :

X(27511) lies on these lines:

X(27512) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (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(27512) lies on these lines:

X(27513) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 + a^5 b - a b^5 - b^6 + a^5 c + b^5 c - a c^5 + b c^5 - c^6) : :

X(27513) lies on these lines:

X(27514) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - b^3 c + 2 b^2 c^2 - a c^3 - b c^3) : :

X(27514) lies on these lines:

X(27515) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 b^2 - a^2 b^4 + a^3 b^2 c + a^2 b^3 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 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4) : :

X(27515) lies on these lines:

X(27516) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 - a^4 b^2 + a^2 b^4 - b^6 - 2 a^4 b c + 2 b^5 c - a^4 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 + 2 b c^5 - c^6) : :

X(27516) lies on these lines:

X(27517) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 65

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 - b^4 c + a^3 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27517) lies on these lines:

X(27518) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + a^3 b c + 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 + a b c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27518) lies on these lines:

X(27519) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (2 a^4 + a^3 b - a b^3 - 2 b^4 + a^3 c - 2 a^2 b c + 3 b^3 c - 2 b^2 c^2 - a c^3 + 3 b c^3 - 2 c^4) : :

X(27519) lies on these lines:

X(27520) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 - a^3 b + a b^3 - b^4 - a^3 c - 4 a^2 b c + 3 b^3 c - 4 b^2 c^2 + a c^3 + 3 b c^3 - c^4) : :

X(27520) lies on these lines:

X(27521) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^5 - a^3 b^2 + a^2 b^3 - b^5 - 2 a^3 b c + 2 b^4 c - a^3 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + 2 b c^4 - c^5) : :

X(27521) lies on these lines:

X(27522) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 + 2 a^3 b - 2 a b^3 - b^4 + 2 a^3 c - 2 a^2 b c + 2 b^2 c^2 - 2 a c^3 - c^4) : :

X(27522) lies on these lines:

X(27523) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^2 b - a b^2 + a^2 c + a b c - b^2 c - a c^2 - b c^2) : :

X(27523) lies on these lines:

X(27524) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 + 2 a^3 b - 2 a b^3 - b^4 + 2 a^3 c + a^2 b c + a b^2 c + a b c^2 + 2 b^2 c^2 - 2 a c^3 - c^4) : :

X(27524) lies on these lines:

X(27525) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (3 a^3 + 3 a^2 b - 3 a b^2 - 3 b^3 + 3 a^2 c - 2 a b c + 3 b^2 c - 3 a c^2 + 3 b c^2 - 3 c^3) : :

X(27525) lies on these lines:

X(27526) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 65

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(27526) lies on these lines:

X(27527) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a + b) (a + c) (b - c) (a - b - c) (a b + a c - b c) : :

X(27527) lies on these lines:

X(27528) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 65

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 - 2 a b^2 c^2 + a b c^3 - a c^4 + b c^4 - c^5) : :

X(27528) lies on these lines:

X(27529) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c + a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(27529) lies on these lines:

X(27530) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (3 a^6 + 4 a^5 b - a^4 b^2 + a^2 b^4 - 4 a b^5 - 3 b^6 + 4 a^5 c - 2 a^4 b c - 4 a^3 b^2 c + 2 b^5 c - a^4 c^2 - 4 a^3 b c^2 - 2 a^2 b^2 c^2 + 4 a b^3 c^2 + 3 b^4 c^2 + 4 a b^2 c^3 - 4 b^3 c^3 + a^2 c^4 + 3 b^2 c^4 - 4 a c^5 + 2 b c^5 - 3 c^6) : :

X(27530) lies on these lines:

X(27531) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 + a^5 b - a^4 b^2 + a^2 b^4 - a b^5 - b^6 + a^5 c - a^4 b c - 2 a^3 b^2 c + a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 - 2 a^2 b^2 c^2 + b^4 c^2 - 2 b^3 c^3 + a^2 c^4 + a b c^4 + b^2 c^4 - a c^5 + b c^5 - c^6) : :

X(27531) lies on these lines:

X(27532) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^8 + a^7 b + a^5 b^3 - a^3 b^5 - a b^7 - b^8 + a^7 c - a^4 b^3 c - a^3 b^4 c + b^7 c + 2 a^4 b^2 c^2 - 2 a^2 b^4 c^2 + a^5 c^3 - a^4 b c^3 + 4 a^2 b^3 c^3 + a b^4 c^3 - b^5 c^3 - a^3 b c^4 - 2 a^2 b^2 c^4 + a b^3 c^4 + 2 b^4 c^4 - a^3 c^5 - b^3 c^5 - a c^7 + b c^7 - c^8) : :

X(27532) lies on these lines:

X(27533) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^8 + 2 a^7 b + 2 a^6 b^2 + 2 a^5 b^3 - 2 a^3 b^5 - 2 a^2 b^6 - 2 a b^7 - b^8 + 2 a^7 c + 5 a^6 b c + a^5 b^2 c - 2 a^4 b^3 c - 3 a^2 b^5 c - 3 a b^6 c + 2 a^6 c^2 + a^5 b c^2 - 4 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 + a b^5 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 6 a^2 b^3 c^3 + 4 a b^4 c^3 + 2 a^2 b^2 c^4 + 4 a b^3 c^4 + 2 b^4 c^4 - 2 a^3 c^5 - 3 a^2 b c^5 + a b^2 c^5 - 2 a^2 c^6 - 3 a b c^6 - 2 a c^7 - c^8) : :

X(27533) lies on these lines:

X(27534) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^9 + 2 a^8 b + a^7 b^2 + a^6 b^3 + a^5 b^4 - a^4 b^5 - a^3 b^6 - a^2 b^7 - 2 a b^8 - b^9 + 2 a^8 c + 3 a^7 b c + 2 a^6 b^2 c + a^5 b^3 c - 2 a^4 b^4 c - 3 a^3 b^5 c - 2 a^2 b^6 c - a b^7 c + a^7 c^2 + 2 a^6 b c^2 - a^4 b^3 c^2 - a^3 b^4 c^2 - 2 a^2 b^5 c^2 + b^7 c^2 + a^6 c^3 + a^5 b c^3 - a^4 b^2 c^3 + 2 a^3 b^3 c^3 + 5 a^2 b^4 c^3 + a b^5 c^3 - b^6 c^3 + a^5 c^4 - 2 a^4 b c^4 - a^3 b^2 c^4 + 5 a^2 b^3 c^4 + 4 a b^4 c^4 + b^5 c^4 - a^4 c^5 - 3 a^3 b c^5 - 2 a^2 b^2 c^5 + a b^3 c^5 + b^4 c^5 - a^3 c^6 - 2 a^2 b c^6 - b^3 c^6 - a^2 c^7 - a b c^7 + b^2 c^7 - 2 a c^8 - c^9) : :

X(27534) lies on these lines:

X(27535) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^9 + a^8 b - 2 a^7 b^2 - 2 a^6 b^3 + 2 a^3 b^6 + 2 a^2 b^7 - a b^8 - b^9 + a^8 c - 3 a^7 b c - 4 a^6 b^2 c + a^5 b^3 c - 2 a^4 b^4 c - a^3 b^5 c + 4 a^2 b^6 c + 3 a b^7 c + b^8 c - 2 a^7 c^2 - 4 a^6 b c^2 + 2 a^5 b^2 c^2 + 6 a^4 b^3 c^2 - 2 a^3 b^4 c^2 - 4 a^2 b^5 c^2 + 2 a b^6 c^2 + 2 b^7 c^2 - 2 a^6 c^3 + a^5 b c^3 + 6 a^4 b^2 c^3 + 2 a^3 b^3 c^3 - 2 a^2 b^4 c^3 - 3 a b^5 c^3 - 2 b^6 c^3 - 2 a^4 b c^4 - 2 a^3 b^2 c^4 - 2 a^2 b^3 c^4 - 2 a b^4 c^4 - a^3 b c^5 - 4 a^2 b^2 c^5 - 3 a b^3 c^5 + 2 a^3 c^6 + 4 a^2 b c^6 + 2 a b^2 c^6 - 2 b^3 c^6 + 2 a^2 c^7 + 3 a b c^7 + 2 b^2 c^7 - a c^8 + b c^8 - c^9) : :

X(27535) lies on these lines:

X(27536) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (5 a^6 + 6 a^5 b - 3 a^4 b^2 + 3 a^2 b^4 - 6 a b^5 - 5 b^6 + 6 a^5 c - 2 a^4 b c - 6 a^3 b^2 c - 2 a^2 b^3 c + 4 b^5 c - 3 a^4 c^2 - 6 a^3 b c^2 - 2 a^2 b^2 c^2 + 6 a b^3 c^2 + 5 b^4 c^2 - 2 a^2 b c^3 + 6 a b^2 c^3 - 8 b^3 c^3 + 3 a^2 c^4 + 5 b^2 c^4 - 6 a c^5 + 4 b c^5 - 5 c^6) : :

X(27536) lies on these lines:

X(27537) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^6 + a^5 b - a b^5 - b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c + b^5 c - a^3 b c^2 - a^2 b^2 c^2 + a b^3 c^2 + a^2 b c^3 + a b^2 c^3 - a c^5 + b c^5 - c^6) : :

X(27537) lies on these lines:

X(27538) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a b + a c - b c) : :

X(27538) lies on these lines: {1, 4090}, {2, 38}, {7, 30758}, {8, 210}, {9, 2319}, {10, 3944}, {11, 4126}, {43, 192}, {44, 3769}, {45, 24351}, {55, 3699}, {63, 5205}, {69, 20947}, {75, 3740}, {165, 25728}, {171, 16995}, {190, 1376}, {194, 2664}, {200, 3685}, {329, 4645}, {333, 3715}, {344, 25568}, {346, 3985}, {354, 26103}, {386, 4075}, {390, 6555}, {392, 4737}, {518, 18743}, {612, 27064}, {644, 28130}, {726, 16569}, {748, 32927}, {750, 32938}, {883, 31526}, {894, 5268}, {899, 3210}, {908, 29641}, {976, 17697}, {986, 26029}, {1089, 9534}, {1329, 20487}, {1575, 21884}, {1621, 4767}, {1757, 29649}, {1961, 17379}, {1962, 25294}, {1997, 24477}, {2292, 25123}, {2550, 20716}, {3006, 27131}, {3161, 3693}, {3212, 6376}, {3240, 3995}, {3263, 21590}, {3305, 3757}, {3434, 17777}, {3452, 3705}, {3596, 7064}, {3617, 25253}, {3661, 4104}, {3662, 30791}, {3678, 10449}, {3681, 4358}, {3687, 3790}, {3703, 5233}, {3711, 3996}, {3729, 8580}, {3758, 4682}, {3836, 33101}, {3840, 30861}, {3846, 33165}, {3873, 30947}, {3912, 21060}, {3932, 4417}, {3961, 4011}, {3992, 5692}, {3994, 32860}, {4023, 6057}, {4028, 17242}, {4195, 5293}, {4318, 28996}, {4362, 17349}, {4365, 4937}, {4383, 32926}, {4384, 30393}, {4385, 5044}, {4388, 10327}, {4413, 32939}, {4415, 4429}, {4434, 7262}, {4439, 32855}, {4579, 9306}, {4640, 17336}, {4651, 4671}, {4661, 29824}, {4703, 33079}, {4704, 17592}, {4741, 33085}, {4756, 32933}, {4899, 11019}, {5218, 17611}, {5220, 14829}, {5223, 30567}, {5274, 10005}, {5297, 26223}, {5552, 7105}, {5741, 32862}, {5748, 30741}, {6327, 26792}, {6377, 20286}, {6686, 17591}, {7046, 28137}, {7085, 26264}, {7191, 26688}, {7308, 16823}, {7322, 16830}, {9350, 32845}, {9369, 19861}, {9780, 31993}, {10180, 25295}, {11680, 30566}, {12647, 21290}, {13405, 25101}, {14997, 17150}, {16602, 28582}, {16989, 26685}, {17018, 31035}, {17026, 27484}, {17061, 17352}, {17122, 32935}, {17123, 32920}, {17124, 32940}, {17125, 32923}, {17155, 24620}, {17164, 27798}, {17230, 33084}, {17232, 33064}, {17236, 33174}, {17261, 17594}, {17279, 33126}, {17314, 20693}, {17353, 29634}, {17358, 32783}, {17480, 21214}, {17597, 25531}, {17720, 33118}, {17749, 24068}, {17760, 27288}, {17792, 26069}, {17889, 21093}, {20012, 21805}, {20363, 24528}, {20683, 30830}, {21080, 28248}, {21085, 21713}, {21951, 25612}, {24703, 32850}, {24988, 33146}, {25269, 32934}, {25960, 33162}, {25961, 32856}, {26227, 27065}, {26580, 29679}, {27518, 28830}, {27523, 33299}, {27527, 30584}, {28058, 28070}, {28118, 28142}, {29687, 33065}, {30578, 33110}

X(27538) = anticomplement of X(17063)
X(27538) = X(9)-Ceva conjugate of X(8)

X(27539) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^5 + a^4 b - a b^4 - b^5 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c - 2 a^2 b c^2 + 6 a b^2 c^2 - 2 a b c^3 - a c^4 + b c^4 - c^5) : :

X(27539) lies on these lines:

X(27540) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (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(27540) lies on these lines:

X(27541) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (3 a^3 - a^2 b + a b^2 - 3 b^3 - a^2 c - 2 a b c + 3 b^2 c + a c^2 + 3 b c^2 - 3 c^3) : :

X(27541) lies on these lines:

X(27542) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^3 b c - a b^3 c + 2 b^4 c - a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - a b c^3 - b^2 c^3 + 2 b c^4 - c^5) : :

X(27542) lies on these lines:

X(27543) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^4 - b^4 - 3 a^2 b c + a b^2 c + 2 b^3 c + a b c^2 - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(27543) lies on these lines:

X(27544) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^3 b - a b^3 + a^3 c + 3 a^2 b c - a b^2 c - b^3 c - a b c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(27544) lies on these lines:

X(27545) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (b - c) (a^3 + 2 a^2 b + a b^2 + 2 a^2 c - 5 a b c - b^2 c + a c^2 - b c^2) : :

X(27545) lies on these lines:

X(27546) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^3 - b^3 - 3 a b c + 2 b^2 c + 2 b c^2 - c^3) : :

X(27546) lies on these lines:

X(27547) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 65

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(27547) lies on these lines:

X(27548) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (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(27548) lies on these lines:

X(27549) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 65

Barycentrics    (a - b - c) (a^2 + 2 a b - b^2 + 2 a c - c^2) : :

X(27549) lies on these lines:

X(27550) =  ISOGONAL CONJUGATE OF X(9203)

Barycentrics    (b^2-c^2) (-5 a^2+b^2+c^2+2 Sqrt[3] S) : :
Barycentrics    (Sqrt[3] a^2+Sqrt[3] b^2-Sqrt[3] c^2-2 S) (Sqrt[3] a^2-Sqrt[3] b^2+Sqrt[3] c^2-2 S) (5 a^2-b^2-c^2-2 Sqrt[3] S) (Sqrt[3] (b^2-c^2) (-a^2+b^2+c^2)-2 (b^2-c^2) S) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28613.

X(27550) lies on these lines: {2,9162}, {13,9180}, {14,5466}, {15,9147}, {30,511}, {98,2378}, {99,9202}, {115,11625}, {351,9194}, {619,1649}, {623,9148}, {691,23896}, {842,11613}, {1637,11627}, {5460,8371}, {5464,9168}, {5479,23283}, {5608,6109}, {5652,22689}, {5996,6114}, {6671,11176}, {7684,19912}, {8594,9485}, {9123,13304}, {9191,9205}, {14174,22687}, {14176,14181}, {14184,14187}, {14817,16220}, {15342,23895}, {25152,25172}, {25153,25174}, {25155,25176}, {25207,25210}, {25212,25216}, {25215,25229}, {25221,25231}, {25225,25233}

X(27550) = isogonal conjugate of X(9203)
X(27550) = barycentric product X(21467)*X(23871)
X(27550) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9203}, {13304, 396}, {21467, 23896}

X(27551) =  ISOGONAL CONJUGATE OF X(9202)

Barycentrics    (b^2-c^2) (-5 a^2+b^2+c^2-2 Sqrt[3] S) : :
Barycentrics    (Sqrt[3] a^2+Sqrt[3] b^2-Sqrt[3] c^2+2 S) (Sqrt[3] a^2-Sqrt[3] b^2+Sqrt[3] c^2+2 S) (5 a^2-b^2-c^2+2 Sqrt[3] S) (Sqrt[3] (b^2-c^2) (-a^2+b^2+c^2)+2 (b^2-c^2) S) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28613.

X(27551) lies on these lines: {2,9163}, {13,5466}, {14,9180}, {16,9147}, {30,511}, {98,2379}, {99,9203}, {115,11627}, {351,9195}, {618,1649}, {624,9148}, {691,23895}, {842,11612}, {1637,11625}, {5459,8371}, {5463,9168}, {5478,23284}, {5607,6108}, {5652,22687}, {5996,6115}, {6672,11176}, {7685,19912}, {8595,9485}, {9123,13305}, {9191,9204}, {14175,14177}, {14180,22689}, {14183,14185}, {14816,16220}, {15342,23896}, {25162,25171}, {25163,25179}, {25165,25181}, {25208,25209}, {25211,25213}, {25218,25230}, {25222,25232}, {25226,25234}

X(27551) = isogonal conjugate of X(9202)
X(27551) = barycentric product X(21466)*X(23870)
X(27551) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9202}, {13305, 395}, {21466, 23895}

X(27552) =  X(54)X(5498)∩X(125)X(8254)

Barycentrics    2 a^16-9 a^14 b^2+15 a^12 b^4-9 a^10 b^6-5 a^8 b^8+13 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-9 a^14 c^2+18 a^12 b^2 c^2-3 a^10 b^4 c^2-8 a^8 b^6 c^2-7 a^6 b^8 c^2+18 a^4 b^10 c^2-13 a^2 b^12 c^2+4 b^14 c^2+15 a^12 c^4-3 a^10 b^2 c^4-8 a^8 b^4 c^4-9 a^6 b^6 c^4+9 a^2 b^10 c^4-4 b^12 c^4-9 a^10 c^6-8 a^8 b^2 c^6-9 a^6 b^4 c^6-14 a^4 b^6 c^6-a^2 b^8 c^6-4 b^10 c^6-5 a^8 c^8-7 a^6 b^2 c^8-a^2 b^6 c^8+10 b^8 c^8+13 a^6 c^10+18 a^4 b^2 c^10+9 a^2 b^4 c^10-4 b^6 c^10-11 a^4 c^12-13 a^2 b^2 c^12-4 b^4 c^12+5 a^2 c^14+4 b^2 c^14-c^16 : :
X(27552) = 3 X[11245] - X[22051]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28613.

X(27552) lies on these lines: {54,5498}, {125,8254}, {3530,10610}, {5012,21230}, {10116,11802}, {11245,22051}

leftri

Collineation mappings involving Gemini triangle 66: X(27553)-X(27589)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 66, as in centers X(27553)-X(27589). Then

m(X) = (b+c)(a^2+b^2+c^2+2ab+2ac+bc)x - a(a+b)(a+c)y - a(a+b)(a+c)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 10, 2018)

X(27553) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 2 a^5 b + a^3 b^3 + a b^5 + b^6 - 2 a^5 c - a^4 b c + 2 a^3 b^2 c + a^2 b^3 c + 2 a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 + a^3 c^3 + a^2 b c^3 - a b^2 c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27553) lies on these lines: {2, 3}, {37, 8818}, {40, 125}, {210, 20653}, {498, 14873}, {1698, 2940}, {3178, 3971}, {3454, 4011}, {3695, 3952}, {3936, 19582}, {5692, 22076}, {6739, 18481}, {8286, 12701}, {8287, 24914}, {15526, 31158}, {16974, 23903}, {21075, 27572}, {21682, 23902}, {27556, 27564}

X(27554) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 3 a^5 b - 2 a^4 b^2 + a^2 b^4 + 3 a b^5 + 2 b^6 - 3 a^5 c - a^4 b c + 2 a^3 b^2 c + a b^4 c + b^5 c - 2 a^4 c^2 + 2 a^3 b c^2 + 2 a^2 b^2 c^2 - 4 a b^3 c^2 - 2 b^4 c^2 - 4 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 - 2 b^2 c^4 + 3 a c^5 + b c^5 + 2 c^6) : :

X(27554) lies on these lines:

X(27555) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^5 b - 2 a^4 b^2 - a^3 b^3 + a^2 b^4 + 2 a b^5 + b^6 - a^5 c - a^2 b^3 c + a b^4 c + b^5 c - 2 a^4 c^2 - 3 a b^3 c^2 - b^4 c^2 - a^3 c^3 - a^2 b c^3 - 3 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 - b^2 c^4 + 2 a c^5 + b c^5 + c^6) : :

X(27555) lies on these lines:

X(27555) = complement of X(37405)

X(27556) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 - 2 a^3 b - a^2 b^2 + a b^3 + b^4 - 2 a^3 c - a^2 b c - a^2 c^2 + a c^3 + c^4) : :

X(27556) lies on these lines: {2, 6}, {115, 3875}, {125, 30738}, {536, 8818}, {3178, 4078}, {4000, 20337}, {4272, 21245}, {4360, 23903}, {4361, 5949}, {4851, 8287}, {17058, 17298}, {17314, 23947}, {18755, 21287}, {20653, 21698}, {21076, 27697}, {27553, 27564}, {27563, 27567}, {27569, 27573}, {27585, 27586}

X(27557) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 - 3 a^3 b - a^2 b^2 + 3 a b^3 + 2 b^4 - 3 a^3 c + a^2 b c - a b^2 c - b^3 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + 3 a c^3 - b c^3 + 2 c^4) : :

X(27557) lies on these lines:

X(27558) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^3 - 2 a^2 b + a b^2 + 2 b^3 - 2 a^2 c + a b c + b^2 c + a c^2 + b c^2 + 2 c^3) : :

X(27558) lies on these lines:

X(27559) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 - a^3 b + a b^3 + b^4 - a^3 c + 3 a^2 b c + a b^2 c - b^3 c + a b c^2 + a c^3 - b c^3 + c^4) : :

X(27559) lies on these lines:

X(27560) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 b - a^3 b^2 + a b^4 + b^5 - a^4 c + 4 a^3 b c + a^2 b^2 c - 2 a b^3 c - a^3 c^2 + a^2 b c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 + a c^4 + c^5) : :

X(27560) lies on these lines:

X(27561) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^8 - 2 a^7 b - a^6 b^2 - a^5 b^3 + 2 a^3 b^5 + a^2 b^6 + a b^7 + b^8 - 2 a^7 c - a^6 b c + 2 a^3 b^4 c + a^2 b^5 c - a^6 c^2 + a^2 b^4 c^2 - a^5 c^3 - a b^4 c^3 + 2 a^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27561) lies on these lines:

X(27562) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^8 - 2 a^7 b - a^6 b^2 - a^5 b^3 + 2 a^3 b^5 + a^2 b^6 + a b^7 + b^8 - 2 a^7 c - a^6 b c + 2 a^3 b^4 c + a^2 b^5 c - a^6 c^2 - a^4 b^2 c^2 - a^3 b^3 c^2 + a^2 b^4 c^2 - a^5 c^3 - a^3 b^2 c^3 - a^2 b^3 c^3 - a b^4 c^3 + 2 a^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27562) lies on these lines:

X(27563) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^5 - 2 a^4 b - a^3 b^2 + a b^4 + b^5 - 2 a^4 c - a^3 b c - a^3 c^2 + a c^4 + c^5) : :

X(27563) lies on these lines:

X(27564) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 2 a^5 b - a^4 b^2 + a b^5 + b^6 - 2 a^5 c - a^4 b c - a^4 c^2 + a c^5 + c^6) : :

X(27564) lies on these lines:

X(27565) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^3 b - a^2 b^2 - a^3 c - 4 a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + b c^3) : :

X(27565) lies on these lines:

X(27566) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 b^2 - a^3 b^3 - 2 a^3 b^2 c - a^2 b^3 c - a^4 c^2 - 2 a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 + b^2 c^4) : :

X(27566) lies on these lines:

X(27567) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c)^2 (-a^4 - 2 a^3 b + a b^3 - 2 a^3 c - 2 a^2 b c + b^3 c - b^2 c^2 + a c^3 + b c^3) : :

X(27567) lies on these lines:

X(27568) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^5 - a^4 b + a^3 b^2 + b^5 - 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 - a b c^3 - b c^4 + c^5) : :

X(27568) lies on these lines:

X(27569) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    b c (b + c) (-a^2 + a b + b^2 + a c + b c + c^2) : :

X(27569) lies on these lines: {2, 37}, {10, 23928}, {142, 24058}, {238, 3702}, {313, 338}, {314, 6651}, {319, 20538}, {594, 23947}, {894, 1509}, {1089, 3178}, {1654, 17762}, {2321, 24086}, {3661, 21810}, {3662, 24077}, {3701, 3773}, {3729, 17736}, {3770, 20932}, {3912, 24050}, {3948, 18697}, {3949, 3969}, {3950, 24081}, {3970, 24092}, {4044, 20234}, {4053, 17233}, {4115, 4416}, {4125, 27589}, {4357, 24067}, {4431, 24044}, {4647, 25354}, {17231, 24076}, {17241, 24063}, {17248, 21816}, {17294, 24048}, {17363, 21839}, {23868, 32929}, {24090, 29594}, {27556, 27573}, {27557, 27558}

X(27570) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    b c (b + c) (-a^3 b - a^3 c - a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 + b^2 c^2 + b c^3) : :

X(27570) lies on these lines:

X(27571) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-3 a^3 - 6 a^2 b + a b^2 + 4 b^3 - 6 a^2 c - a b c + b^2 c + a c^2 + b c^2 + 4 c^3) : :

X(27571) lies on these lines:

X(27572) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^5 + 3 a^3 b^2 - a^2 b^3 - 2 a b^4 + b^5 + 5 a^3 b c - 3 a b^3 c - 2 b^4 c + 3 a^3 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 - 2 a c^4 - 2 b c^4 + c^5) : :

X(27572) lies on these lines:

X(27573) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 - 2 a^3 b - a^2 b^2 + a b^3 + b^4 - 2 a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4) : :

X(27573) lies on these lines:

X(27574) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b - c) (b + c) (2 a^3 b + 3 a^2 b^2 + a b^3 + 2 a^3 c + 5 a^2 b c + 2 a b^2 c + 3 a^2 c^2 + 2 a b c^2 - b^2 c^2 + a c^3) : :

X(27574) lies on these lines:

X(27575) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    b c (b^2 - c^2) (3 a^2 + 2 a b + b^2 + 2 a c + b c + c^2) : :

X(27575) lies on these lines:

X(27576) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^5 - 2 a^4 b - a^3 b^2 + a b^4 + b^5 - 2 a^4 c + a^2 b^2 c - a^3 c^2 + a^2 b c^2 + a b^2 c^2 + a c^4 + c^5) : :

X(27576) lies on these lines:

X(27577) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-2 a^3 - 4 a^2 b - a b^2 + b^3 - 4 a^2 c - 4 a b c - b^2 c - a c^2 - b c^2 + c^3) : :

X(27577) lies on these lines:

X(27578) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-3 a^6 - 7 a^5 b - 2 a^4 b^2 + 2 a^3 b^3 + a^2 b^4 + 5 a b^5 + 4 b^6 - 7 a^5 c - 3 a^4 b c + 6 a^3 b^2 c + 2 a^2 b^3 c + a b^4 c + b^5 c - 2 a^4 c^2 + 6 a^3 b c^2 + 6 a^2 b^2 c^2 - 6 a b^3 c^2 - 4 b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 - 6 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 - 4 b^2 c^4 + 5 a c^5 + b c^5 + 4 c^6) : :

X(27578) lies on these lines:

X(27579) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 2 a^5 b + a^3 b^3 + a b^5 + b^6 - 2 a^5 c + 4 a^3 b^2 c + 2 a^2 b^3 c + 4 a^3 b c^2 + 5 a^2 b^2 c^2 - b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27579) lies on these lines:

X(27580) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^8 - 2 a^7 b - a^6 b^2 - a^5 b^3 + 2 a^3 b^5 + a^2 b^6 + a b^7 + b^8 - 2 a^7 c - a^6 b c + 2 a^3 b^4 c + a^2 b^5 c - a^6 c^2 - 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + a^2 b^4 c^2 - a^5 c^3 - 2 a^3 b^2 c^3 - 2 a^2 b^3 c^3 - a b^4 c^3 + 2 a^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27580) lies on these lines:

X(27581) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^8 - 3 a^7 b - 4 a^6 b^2 - 3 a^5 b^3 + 3 a^3 b^5 + 4 a^2 b^6 + 3 a b^7 + b^8 - 3 a^7 c - 8 a^6 b c - 6 a^5 b^2 c + 3 a^3 b^4 c + 6 a^2 b^5 c + 6 a b^6 c + 2 b^7 c - 4 a^6 c^2 - 6 a^5 b c^2 + a^4 b^2 c^2 + 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + b^6 c^2 - 3 a^5 c^3 + 2 a^3 b^2 c^3 - 8 a^2 b^3 c^3 - 9 a b^4 c^3 - 2 b^5 c^3 + 3 a^3 b c^4 - 2 a^2 b^2 c^4 - 9 a b^3 c^4 - 4 b^4 c^4 + 3 a^3 c^5 + 6 a^2 b c^5 - 2 b^3 c^5 + 4 a^2 c^6 + 6 a b c^6 + b^2 c^6 + 3 a c^7 + 2 b c^7 + c^8) : :

X(27581) lies on these lines:

X(27582) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-5 a^6 - 11 a^5 b - 2 a^4 b^2 + 4 a^3 b^3 + a^2 b^4 + 7 a b^5 + 6 b^6 - 11 a^5 c - 5 a^4 b c + 10 a^3 b^2 c + 4 a^2 b^3 c + a b^4 c + b^5 c - 2 a^4 c^2 + 10 a^3 b c^2 + 10 a^2 b^2 c^2 - 8 a b^3 c^2 - 6 b^4 c^2 + 4 a^3 c^3 + 4 a^2 b c^3 - 8 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 - 6 b^2 c^4 + 7 a c^5 + b c^5 + 6 c^6) : :

X(27582) lies on these lines:

X(27583) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 3 a^5 b - 2 a^4 b^2 + a^2 b^4 + 3 a b^5 + 2 b^6 - 3 a^5 c - 3 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c + b^5 c - 2 a^4 c^2 - 2 a^3 b c^2 - 4 a^2 b^2 c^2 - 6 a b^3 c^2 - 2 b^4 c^2 - 2 a^2 b c^3 - 6 a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 - 2 b^2 c^4 + 3 a c^5 + b c^5 + 2 c^6) : :

X(27583) lies on these lines:

X(27584) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^6 - 2 a^5 b - a^4 b^2 + a b^5 + b^6 - 2 a^5 c - a^4 b c + a^3 b^2 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 - 2 a b^2 c^3 - b^3 c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27584) lies on these lines:

X(27585) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^4 - a^3 b + a b^3 + b^4 - a^3 c + 4 a^2 b c - 2 b^3 c - b^2 c^2 + a c^3 - 2 b c^3 + c^4) : :

X(27585) lies on these lines:

X(27586) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b + c) (-a^3 b - a^2 b^2 - a^3 c - 5 a^2 b c + 2 b^3 c - a^2 c^2 + b^2 c^2 + 2 b c^3) : :

X(27586) lies on these lines:

X(27587) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b^2 - c^2) (-a^4 - 2 a^3 b - 2 a^2 b^2 - a b^3 - 2 a^3 c - 2 a^2 b c + b^3 c - 2 a^2 c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(27587) lies on these lines:

X(27588) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    (b^2 - c^2) (-a^4 + a^2 b^2 + 3 a^2 b c + 2 a b^2 c + b^3 c + a^2 c^2 + 2 a b c^2 + b c^3) : :

X(27588) lies on these lines:

X(27589) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 66

Barycentrics    b c (b + c) (a^4 + a^3 b - a^2 b^2 - 2 a b^3 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 + b^2 c^2 - 2 a c^3 + b c^3) : :

X(27589) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 67: X(27590)-X(27620)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 67, as in centers X(27590)-X(27620). Then

m(X) = (2a+b+c)(a^2+b^2+c^2+2ab+2ac+bc)x + b(2a+b+c)(a+b+2c)y + c(2a+b+c)(a+2b+c)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 11, 2018)

X(27590) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + 2 a^5 b - 3 a^3 b^3 - 2 a^2 b^4 + a b^5 + b^6 + 2 a^5 c + a^4 b c - 2 a^3 b^2 c - 3 a^2 b^3 c + 2 b^5 c - 2 a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 - 3 a^3 c^3 - 3 a^2 b c^3 - a b^2 c^3 - 4 b^3 c^3 - 2 a^2 c^4 - b^2 c^4 + a c^5 + 2 b c^5 + c^6) : :

X(27590) lies on these lines:

X(27591) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - 3 a^4 b c + 2 a^3 b^2 c + 4 a^2 b^3 c - 3 a b^4 c - b^5 c + 2 a^3 b c^2 + 6 a^2 b^2 c^2 + 4 a b^3 c^2 + 4 a^2 b c^3 + 4 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 3 a b c^4 - a c^5 - b c^5) : :

X(27591) lies on these lines:

X(27592) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^5 b - 3 a^3 b^3 - a^2 b^4 + 2 a b^5 + b^6 + a^5 c + 4 a^4 b c - 4 a^3 b^2 c - 7 a^2 b^3 c + 3 a b^4 c + 3 b^5 c - 4 a^3 b c^2 - 8 a^2 b^2 c^2 - 5 a b^3 c^2 - b^4 c^2 - 3 a^3 c^3 - 7 a^2 b c^3 - 5 a b^2 c^3 - 6 b^3 c^3 - a^2 c^4 + 3 a b c^4 - b^2 c^4 + 2 a c^5 + 3 b c^5 + c^6) : :

X(27592) lies on these lines:

X(27593) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 + 2 a^3 b + a^2 b^2 + a b^3 + b^4 + 2 a^3 c + a^2 b c + 2 b^3 c + a^2 c^2 + a c^3 + 2 b c^3 + c^4) : :

X(27593) lies on these lines:

X(27594) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 3 a^2 b c + 7 a b^2 c + b^3 c - a^2 c^2 + 7 a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(27594) lies on these lines:

X(27595) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^3 - a b^2 - 9 a b c - 3 b^2 c - a c^2 - 3 b c^2) : :

X(27595) lies on these lines:

X(27596) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 + a^3 b - 2 a^2 b^2 - a b^3 + b^4 + a^3 c - 3 a^2 b c - 5 a b^2 c + b^3 c - 2 a^2 c^2 - 5 a b c^2 - 4 b^2 c^2 - a c^3 + b c^3 + c^4) : :

X(27596) lies on these lines:

X(27597) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c)^2 (a b + b^2 + a c + 4 b c + c^2) : :

X(27597) lies on these lines: {1, 2}, {4974, 27607}, {6533, 8040}, {27591, 27596}, {27605, 27606}

X(27598) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 b - a^3 b^2 - 2 a^2 b^3 + a b^4 + b^5 + a^4 c + 4 a^3 b c - 3 a^2 b^2 c + 2 a b^3 c + 2 b^4 c - a^3 c^2 - 3 a^2 b c^2 - 3 b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 - 3 b^2 c^3 + a c^4 + 2 b c^4 + c^5) : :

X(27598) lies on these lines:

X(27599) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^8 + 2 a^7 b + a^6 b^2 - a^5 b^3 - 2 a^4 b^4 - 2 a^3 b^5 - a^2 b^6 + a b^7 + b^8 + 2 a^7 c + a^6 b c - 2 a^4 b^3 c - 2 a^3 b^4 c - a^2 b^5 c + 2 b^7 c + a^6 c^2 - a^2 b^4 c^2 - a^5 c^3 - 2 a^4 b c^3 - a b^4 c^3 - 2 b^5 c^3 - 2 a^4 c^4 - 2 a^3 b c^4 - a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 - 2 b^3 c^5 - a^2 c^6 + a c^7 + 2 b c^7 + c^8) : :

X(27599) lies on these lines:

X(27600) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^8 + 2 a^7 b + a^6 b^2 - a^5 b^3 - 2 a^4 b^4 - 2 a^3 b^5 - a^2 b^6 + a b^7 + b^8 + 2 a^7 c + a^6 b c - 2 a^4 b^3 c - 2 a^3 b^4 c - a^2 b^5 c + 2 b^7 c + a^6 c^2 + a^4 b^2 c^2 + 3 a^3 b^3 c^2 + a^2 b^4 c^2 - a^5 c^3 - 2 a^4 b c^3 + 3 a^3 b^2 c^3 + 5 a^2 b^3 c^3 - a b^4 c^3 - 2 b^5 c^3 - 2 a^4 c^4 - 2 a^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 - 2 b^3 c^5 - a^2 c^6 + a c^7 + 2 b c^7 + c^8) : :

X(27600) lies on these lines:

X(27601) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^5 + 2 a^4 b + a^3 b^2 + a b^4 + b^5 + 2 a^4 c + a^3 b c + 2 b^4 c + a^3 c^2 + a c^4 + 2 b c^4 + c^5) : :

X(27601) lies on these lines:

X(27602) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + 2 a^5 b + a^4 b^2 + a b^5 + b^6 + 2 a^5 c + a^4 b c + 2 b^5 c + a^4 c^2 + a c^5 + 2 b c^5 + c^6) : :

X(27602) lies on these lines:

X(27603) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^3 b + 3 a^2 b^2 + 2 a b^3 + a^3 c + 4 a^2 b c + 5 a b^2 c + b^3 c + 3 a^2 c^2 + 5 a b c^2 + 4 b^2 c^2 + 2 a c^3 + b c^3) : :

X(27603) lies on these lines:

X(27604) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 3 a^2 b c - 3 a b^2 c - b^3 c - a^2 c^2 - 3 a b c^2 - 4 b^2 c^2 - a c^3 - b c^3) : :

X(27604) lies on these lines:

X(27605) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    b c (2 a + b + c) (3 a^2 + 5 a b + b^2 + 5 a c + b c + c^2) : :

X(27605) lies on these lines:

X(27606) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    b c (2 a + b + c) (a^3 b + 2 a^2 b^2 + a^3 c + 3 a^2 b c + 2 a b^2 c + b^3 c + 2 a^2 c^2 + 2 a b c^2 + b^2 c^2 + b c^3) : :

X(27606) lies on these lines:

X(27607) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^2 + 2 a b + b^2 + 2 a c + b c + c^2) (a^2 + a b + b^2 + a c + 3 b c + c^2) : :

X(27607) lies on these lines:

X(27608) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (3 a^3 + 4 a^2 b + 3 a b^2 + 2 b^3 + 4 a^2 c - 7 a b c + b^2 c + 3 a c^2 + b c^2 + 2 c^3) : :

X(27608) lies on these lines:

X(27609) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (b - c) (2 a + b + c) (a^2 b^2 + a b^3 + a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - b^2 c^2 + a c^3) : :

X(27609) lies on these lines:

X(27610) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    b c (b - c) (2 a + b + c) (a^2 + b^2 + b c + c^2) : :

X(27610) lies on these lines: {2, 650}

X(27611) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (3 a^6 + 5 a^5 b - 6 a^3 b^3 - 5 a^2 b^4 + a b^5 + 2 b^6 + 5 a^5 c - a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 3 a b^4 c + 3 b^5 c - 2 a^3 b c^2 + 2 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 - 6 a^3 c^3 - 2 a^2 b c^3 + 2 a b^2 c^3 - 6 b^3 c^3 - 5 a^2 c^4 - 3 a b c^4 - 2 b^2 c^4 + a c^5 + 3 b c^5 + 2 c^6) : :

X(27611) lies on these lines:

X(27612) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + 2 a^5 b - 3 a^3 b^3 - 2 a^2 b^4 + a b^5 + b^6 + 2 a^5 c - 6 a^3 b^2 c - 8 a^2 b^3 c - 2 a b^4 c + 2 b^5 c - 6 a^3 b c^2 - 13 a^2 b^2 c^2 - 8 a b^3 c^2 - b^4 c^2 - 3 a^3 c^3 - 8 a^2 b c^3 - 8 a b^2 c^3 - 4 b^3 c^3 - 2 a^2 c^4 - 2 a b c^4 - b^2 c^4 + a c^5 + 2 b c^5 + c^6) : :

X(27612) lies on these lines:

X(27613) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^8 + 2 a^7 b + a^6 b^2 - a^5 b^3 - 2 a^4 b^4 - 2 a^3 b^5 - a^2 b^6 + a b^7 + b^8 + 2 a^7 c + a^6 b c - 2 a^4 b^3 c - 2 a^3 b^4 c - a^2 b^5 c + 2 b^7 c + a^6 c^2 + 2 a^4 b^2 c^2 + 6 a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a^5 c^3 - 2 a^4 b c^3 + 6 a^3 b^2 c^3 + 10 a^2 b^3 c^3 - a b^4 c^3 - 2 b^5 c^3 - 2 a^4 c^4 - 2 a^3 b c^4 + 3 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 - 2 b^3 c^5 - a^2 c^6 + a c^7 + 2 b c^7 + c^8) : :

X(27613) lies on these lines:

X(27614) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^8 + 3 a^7 b + 2 a^6 b^2 - a^5 b^3 - 2 a^4 b^4 - 3 a^3 b^5 - 2 a^2 b^6 + a b^7 + b^8 + 3 a^7 c + 4 a^6 b c - 2 a^5 b^2 c + a^3 b^4 c - 6 a^2 b^5 c - 2 a b^6 c + 2 b^7 c + 2 a^6 c^2 - 2 a^5 b c^2 + 3 a^4 b^2 c^2 + 18 a^3 b^3 c^2 + 8 a^2 b^4 c^2 - 4 a b^5 c^2 - b^6 c^2 - a^5 c^3 + 18 a^3 b^2 c^3 + 24 a^2 b^3 c^3 + 5 a b^4 c^3 - 2 b^5 c^3 - 2 a^4 c^4 + a^3 b c^4 + 8 a^2 b^2 c^4 + 5 a b^3 c^4 - 3 a^3 c^5 - 6 a^2 b c^5 - 4 a b^2 c^5 - 2 b^3 c^5 - 2 a^2 c^6 - 2 a b c^6 - b^2 c^6 + a c^7 + 2 b c^7 + c^8) : :

X(27614) lies on these lines:

X(27615) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (5 a^6 + 9 a^5 b - 12 a^3 b^3 - 9 a^2 b^4 + 3 a b^5 + 4 b^6 + 9 a^5 c + a^4 b c - 6 a^3 b^2 c - 8 a^2 b^3 c - 3 a b^4 c + 7 b^5 c - 6 a^3 b c^2 - 2 a^2 b^2 c^2 - 4 b^4 c^2 - 12 a^3 c^3 - 8 a^2 b c^3 - 14 b^3 c^3 - 9 a^2 c^4 - 3 a b c^4 - 4 b^2 c^4 + 3 a c^5 + 7 b c^5 + 4 c^6) : :

X(27615) lies on these lines:

X(27616) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - a^4 b c + 10 a^3 b^2 c + 14 a^2 b^3 c + a b^4 c - b^5 c + 10 a^3 b c^2 + 28 a^2 b^2 c^2 + 18 a b^3 c^2 + 14 a^2 b c^3 + 18 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + a b c^4 - a c^5 - b c^5) : :

X(27616) lies on these lines:

X(27617) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^6 + 2 a^5 b + a^4 b^2 + a b^5 + b^6 + 2 a^5 c + a^4 b c + a^3 b^2 c + 2 a^2 b^3 c + 2 b^5 c + a^4 c^2 + a^3 b c^2 + 3 a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 + b^2 c^4 + a c^5 + 2 b c^5 + c^6) : :

X(27617) lies on these lines:

X(27618) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^4 + a^3 b - 2 a^2 b^2 - a b^3 + b^4 + a^3 c + 2 b^3 c - 2 a^2 c^2 - 3 b^2 c^2 - a c^3 + 2 b c^3 + c^4) : :

X(27618) lies on these lines:

X(27619) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (2 a + b + c) (a^3 b + 3 a^2 b^2 + 2 a b^3 + a^3 c + a^2 b c + 3 a^2 c^2 + 3 b^2 c^2 + 2 a c^3) : :

X(27619) lies on these lines:

X(27620) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 67

Barycentrics    (b - c) (2 a + b + c) (3 a^3 + 2 a^2 b - 2 a b^2 - b^3 + 2 a^2 c - 5 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2 - c^3) : :

X(27620) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 68: X(27621)-X(27682)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 68, as in centers X(27621)-X(27682). Then

m(X) = a(b^2+c^2+ab+ca+bc)x - ac(a+c)y - ab(a+b)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. Fixed points of m include X(2), X(36), X(238), and X(667). (Clark Kimberling, November 11, 2018)

X(27621) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^5 b+2 a^4 b^2-2 a^3 b^3-2 a^2 b^4+2 a^5 c+a^4 b c-2 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^3-2 a^2 c^4-2 a b c^4-b c^5) : :

X(27621) lies on these lines: {2, 3}, {7, 22345}, {56, 1610}, {57, 959}, {60, 5138}, {228, 5703}, {386, 1730}, {388, 23361}, {497, 23383}, {610, 1400}, {978, 1044}, {1249, 3209}, {1410, 18623}, {1829, 17080}, {3000, 15601}, {3185, 3485}, {4267, 5712}, {4293, 15654}, {4652, 28287}, {5122, 27625}, {5204, 28265}, {5745, 31339}, {7288, 20470}, {8583, 10856}, {9965, 20805}, {10476, 25941}, {16678, 30478}, {19734, 19764}, {19767, 25059}, {27632, 27669}, {27639, 27657}, {27640, 27642}

X(27622) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b+a^4 b^2-a^3 b^3-a^2 b^4+a^5 c+a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+a b^3 c^2-a^3 c^3+a^2 b c^3+a b^2 c^3+2 b^3 c^3-a^2 c^4-a b c^4-b c^5) : :

X(27622) lies on these lines: {2, 3}, {11, 23383}, {12, 23361}, {19, 216}, {35, 33109}, {46, 978}, {56, 5230}, {57, 1425}, {65, 1193}, {100, 30029}, {226, 22345}, {228, 13411}, {672, 28246}, {992, 2245}, {1155, 27627}, {1400, 2182}, {1465, 1829}, {1478, 15654}, {1730, 3216}, {1745, 26892}, {1764, 22076}, {1824, 17102}, {1951, 7119}, {3075, 26884}, {3185, 11375}, {3831, 17647}, {3915, 28353}, {4267, 5718}, {4292, 22344}, {4551, 16980}, {4999, 16678}, {5135, 23692}, {5348, 14529}, {5396, 18180}, {5433, 20470}, {5905, 20805}, {10527, 23853}, {11374, 21319}, {12609, 24169}, {15803, 27659}, {15950, 23846}, {26066, 31339}, {27385, 29967}, {27634, 27669}

X(27623) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (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(27623) lies on these lines: {1, 19282}, {2, 6}, {3, 238}, {9, 980}, {31, 28247}, {44, 21371}, {55, 16690}, {75, 2176}, {213, 10436}, {219, 21246}, {239, 20923}, {241, 30456}, {326, 16968}, {386, 19518}, {405, 3736}, {474, 5156}, {748, 2309}, {899, 2209}, {958, 2274}, {1001, 1193}, {1191, 5263}, {1269, 4713}, {1376, 1918}, {1400, 6180}, {1616, 20036}, {1722, 10441}, {1724, 18792}, {1764, 23511}, {1975, 2669}, {2277, 28287}, {2300, 4384}, {2305, 11329}, {2911, 29967}, {3008, 24220}, {3230, 3875}, {3286, 13738}, {3759, 21785}, {3879, 29968}, {4279, 17749}, {4360, 16969}, {4361, 16685}, {4833, 27647}, {5110, 16367}, {5710, 31339}, {5711, 16456}, {16345, 17123}, {16355, 17125}, {16458, 16466}, {16468, 25528}, {16483, 32941}, {16502, 29960}, {17033, 21788}, {18147, 29983}, {21857, 22370}, {22144, 30017}, {27624, 28283}, {27631, 27635}, {27636, 27663}, {27641, 27646}, {27642, 27662}, {27650, 28276}, {27656, 27672}, {27667, 28285}, {27669, 28263}, {27670, 27671}, {30022, 30940}

X(27624) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^3 b - 2 a b^3 + 2 a^3 c + a^2 b c + b^3 c - 2 b^2 c^2 - 2 a c^3 + b c^3) : :

X(27624) lies on these lines: {1, 5756}, {2, 7}, {22, 7083}, {46, 7613}, {65, 24554}, {71, 3672}, {145, 22370}, {192, 20247}, {573, 5222}, {583, 4644}, {651, 5120}, {959, 28265}, {978, 27649}, {991, 1193}, {1386, 2646}, {1723, 7291}, {1732, 7289}, {1958, 4188}, {2245, 4000}, {2260, 3945}, {2269, 17014}, {3286, 4225}, {3501, 4461}, {4019, 31130}, {5022, 6180}, {5036, 17366}, {5043, 17365}, {5753, 13731}, {7229, 16549}, {7288, 24553}, {7465, 17126}, {14636, 15937}, {17002, 26236}, {18206, 21296}, {21061, 29611}, {24471, 24635}, {24557, 25524}, {24612, 26671}, {25631, 31338}, {25895, 26689}, {27623, 28283}, {27625, 27640}, {27643, 27651}

X(27625) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^2 b + 2 a b^2 + 2 a^2 c - 3 a b c - 3 b^2 c + 2 a c^2 - 3 b c^2) : :

X(27625) lies on these lines: {1, 2}, {31, 17572}, {238, 4188}, {404, 17127}, {443, 33107}, {474, 17126}, {748, 4189}, {988, 27065}, {992, 16885}, {1064, 10303}, {1450, 5261}, {1468, 14997}, {2277, 16814}, {2975, 8572}, {3218, 11512}, {3868, 9335}, {3869, 16610}, {3876, 4392}, {3976, 4661}, {3984, 5573}, {4225, 27639}, {4255, 5284}, {4383, 5253}, {4671, 25079}, {4850, 25917}, {5044, 7226}, {5122, 27621}, {5274, 22072}, {5710, 9342}, {5711, 17535}, {10448, 17570}, {11375, 26724}, {13738, 27666}, {16466, 17531}, {16669, 28244}, {16859, 17125}, {16865, 17123}, {17164, 24620}, {17490, 25253}, {17495, 19582}, {17674, 25959}, {23536, 27131}, {24178, 31053}, {24954, 33133}, {25524, 32911}, {25591, 28605}, {25681, 33129}, {25914, 32782}, {27624, 27640}, {27628, 28271}, {27642, 27676}, {27657, 28250}, {27671, 27680}

X(27626) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b - a b^3 + a^3 c + a^2 b c - a b^2 c + b^3 c - a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(27626) lies on these lines: {1, 3688}, {2, 7}, {3, 238}, {40, 1738}, {56, 25878}, {69, 21384}, {71, 4000}, {75, 3501}, {86, 25500}, {219, 1429}, {239, 3169}, {256, 8731}, {284, 27644}, {314, 17026}, {322, 21232}, {573, 3008}, {583, 4675}, {610, 992}, {614, 3778}, {748, 4224}, {942, 984}, {960, 25887}, {1018, 17151}, {1193, 2293}, {1212, 24471}, {1334, 3672}, {1475, 3945}, {1697, 3755}, {1722, 9548}, {1756, 15803}, {1958, 21495}, {2092, 2999}, {2108, 16571}, {2171, 24554}, {2176, 28358}, {2212, 4219}, {2245, 17278}, {2260, 4648}, {2269, 5222}, {2275, 28350}, {2277, 16970}, {2287, 25940}, {2354, 7490}, {2911, 18162}, {3000, 15601}, {3094, 16968}, {3208, 3875}, {3220, 13738}, {3663, 3730}, {3664, 4253}, {3691, 5232}, {3717, 24391}, {3718, 17755}, {3821, 12514}, {3973, 21362}, {4684, 6762}, {4859, 20367}, {4901, 17751}, {5120, 7175}, {5138, 16468}, {5272, 17065}, {5709, 6211}, {5738, 26101}, {5783, 21526}, {5791, 33159}, {7174, 11518}, {8726, 13731}, {8728, 32784}, {9317, 17134}, {10383, 21321}, {10447, 29433}, {10856, 21363}, {10889, 24600}, {12437, 20036}, {12723, 24341}, {15509, 21892}, {16549, 25590}, {16552, 17272}, {17123, 25514}, {17125, 22174}, {17284, 21061}, {17298, 18206}, {21195, 22443}, {24310, 24789}, {27640, 28254}, {27641, 27670}, {27646, 27671}, {28251, 28260}

X(27627) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^2 b + a b^2 + a^2 c - 2 a b c - 2 b^2 c + a c^2 - 2 b c^2) : :

X(27627) lies on these lines: {1, 2}, {3, 748}, {11, 22072}, {12, 1450}, {21, 17123}, {31, 474}, {38, 5044}, {44, 583}, {45, 2277}, {58, 27643}, {63, 11512}, {65, 16602}, {72, 244}, {73, 5433}, {75, 25591}, {88, 11684}, {106, 5288}, {140, 1064}, {171, 17531}, {238, 404}, {321, 25079}, {392, 4642}, {405, 17125}, {496, 33136}, {602, 6911}, {631, 4300}, {740, 29982}, {750, 16408}, {896, 27657}, {902, 25440}, {908, 24178}, {956, 32577}, {960, 16610}, {968, 3646}, {982, 3876}, {988, 3305}, {1010, 32944}, {1036, 16419}, {1042, 3911}, {1043, 25531}, {1046, 27003}, {1055, 4426}, {1066, 15325}, {1155, 27622}, {1191, 4413}, {1211, 25914}, {1334, 1575}, {1376, 3915}, {1457, 24914}, {1458, 7288}, {1468, 4383}, {1475, 2238}, {1574, 3230}, {1724, 19769}, {1739, 3878}, {1740, 19278}, {1742, 15717}, {2170, 16605}, {2173, 28261}, {2230, 27634}, {2234, 15254}, {2243, 28243}, {2246, 28246}, {2274, 17259}, {2275, 3691}, {2292, 3752}, {2309, 16342}, {2347, 21892}, {2635, 5204}, {2650, 5439}, {2887, 17674}, {3000, 15601}, {3057, 4695}, {3072, 6946}, {3073, 6940}, {3120, 21616}, {3142, 7173}, {3290, 33299}, {3338, 32912}, {3452, 23536}, {3551, 27671}, {3555, 21805}, {3579, 19513}, {3670, 10176}, {3678, 3953}, {3681, 3976}, {3701, 24003}, {3736, 17557}, {3772, 24954}, {3846, 4202}, {3868, 17063}, {3869, 24174}, {3877, 24440}, {3884, 3987}, {3893, 17460}, {3951, 18193}, {4005, 21342}, {4187, 21935}, {4197, 17717}, {4225, 27666}, {4255, 4423}, {4256, 5259}, {4281, 5333}, {4297, 5400}, {4322, 4551}, {4357, 27162}, {4653, 25542}, {4766, 17670}, {4849, 17609}, {4887, 20245}, {5217, 8053}, {5247, 5253}, {5277, 21764}, {5687, 9350}, {5692, 24046}, {5711, 16862}, {5904, 17449}, {5919, 21896}, {7515, 22057}, {8012, 9367}, {8056, 12526}, {8167, 19765}, {8421, 17593}, {8728, 33105}, {9709, 16483}, {10448, 11108}, {10571, 31231}, {11362, 32486}, {11375, 17278}, {12702, 19549}, {13624, 13731}, {16062, 25960}, {16454, 25496}, {16477, 27644}, {16552, 23649}, {16948, 27660}, {17067, 21246}, {17122, 17535}, {17127, 17572}, {17239, 24668}, {17337, 24953}, {17348, 24739}, {17349, 23579}, {17490, 19582}, {18792, 22343}, {20227, 21033}, {20331, 28245}, {21075, 23675}, {22345, 28351}, {24161, 26724}, {24514, 27318}, {24789, 25681}, {24984, 26010}, {26060, 33109}, {27641, 27642}, {27655, 27661}, {28238, 28239}

X(27628) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c - b^4 c + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - b c^4) : :

X(27628) lies on these lines: {2, 11}, {31, 16056}, {46, 978}, {56, 24597}, {88, 959}, {238, 851}, {244, 942}, {291, 2282}, {444, 2355}, {474, 1036}, {659, 28283}, {748, 4192}, {899, 28256}, {992, 20331}, {1004, 7083}, {1183, 5253}, {1284, 33129}, {1400, 2246}, {1402, 26723}, {1460, 16438}, {2783, 17888}, {3120, 15507}, {3185, 24789}, {3579, 19513}, {4225, 28265}, {4359, 18235}, {4557, 17724}, {5249, 20967}, {6187, 13731}, {7465, 23868}, {7742, 9798}, {13097, 33145}, {14798, 28238}, {15253, 23067}, {15496, 28259}, {16415, 16466}, {16827, 28264}, {18785, 23988}, {21319, 33130}, {21320, 33148}, {22345, 24178}, {27625, 28271}, {27669, 28285}, {28242, 28248}, {28247, 28251}, {28253, 28274}, {31339, 33115}

X(27629) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + a^5 b c + a^4 b^2 c - a^2 b^4 c - a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 - a^2 b c^4 - a b^2 c^4 - a^2 c^5 - a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27629) lies on these lines: {2, 3}, {9571, 25645}, {20470, 26747}, {23853, 33090}

X(27630) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + a^5 b c + a^4 b^2 c - a^2 b^4 c - a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 + a b^3 c^3 - a^2 b c^4 - a b^2 c^4 - a^2 c^5 - a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27630) lies on these lines: {2, 3}, {4057, 27675}, {24436, 27679}

X(27631) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b + a^3 b^2 + a^4 c + a^3 b c - a b^3 c + a^3 c^2 - b^3 c^2 - a b c^3 - b^2 c^3) : :

X(27631) lies on these lines: {2, 31}, {978, 4225}, {993, 1193}, {1468, 19717}, {1740, 4210}, {1918, 24552}, {2209, 17135}, {3915, 17751}, {4279, 31330}, {4423, 19734}, {4426, 20965}, {5284, 16690}, {16468, 18169}, {27623, 27635}, {27636, 27666}

X(27632) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b + a^4 b^2 + a^5 c + a^4 b c - a b^4 c + a^4 c^2 - b^4 c^2 - a b c^4 - b^2 c^4) : :

X(27632) lies on these lines: {2, 32}, {1914, 29966}, {2205, 24549}, {3286, 13738}, {27621, 27669}, {27642, 27656}, {27665, 27672}

X(27633) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^2 b^2 + a b^3 - b^3 c + a^2 c^2 + a c^3 - b c^3) : :

X(27633) lies on these lines: {2, 37}, {3, 238}, {6, 20769}, {39, 4357}, {69, 2275}, {72, 4283}, {86, 16604}, {198, 4383}, {239, 21857}, {274, 25538}, {314, 26959}, {319, 17448}, {583, 4641}, {662, 1333}, {980, 17306}, {992, 2235}, {1015, 3879}, {1086, 29967}, {1107, 5224}, {1193, 1386}, {1400, 28283}, {1574, 4967}, {1755, 28260}, {2092, 17023}, {2221, 19591}, {2227, 28269}, {2234, 15254}, {2236, 28242}, {2237, 28243}, {3596, 27091}, {3662, 24598}, {3771, 24653}, {3773, 3831}, {3783, 24575}, {3834, 29981}, {3882, 20228}, {3912, 17053}, {4360, 20691}, {4446, 20358}, {4643, 5069}, {4852, 21858}, {5337, 16470}, {5439, 24923}, {5564, 21868}, {7032, 9025}, {8610, 17243}, {9367, 27420}, {16043, 30479}, {16696, 17237}, {16726, 17376}, {16975, 17270}, {17121, 24625}, {17148, 27095}, {17189, 25532}, {17277, 21892}, {17353, 21796}, {17793, 21080}, {18134, 26746}, {18139, 26747}, {18698, 24786}, {21769, 22370}, {24471, 28391}, {24739, 30939}, {24744, 33137}, {27639, 27650}

X(27634) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b^3 + a^2 b^4 + a^2 b^3 c - a b^3 c^2 - b^4 c^2 + a^3 c^3 + a^2 b c^3 - a b^2 c^3 + a^2 c^4 - b^2 c^4) : :

X(27634) lies on these lines: {2, 39}, {141, 2275}, {978, 28285}, {992, 2231}, {1193, 1386}, {2230, 27627}, {2232, 28242}, {2233, 28243}, {3286, 13738}, {16584, 20911}, {17053, 27097}, {27622, 27669}, {27656, 27662}

X(27635) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b^2 + a^2 b^3 + 2 a^3 b c + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(27635) lies on these lines: {1, 2}, {238, 4210}, {3725, 24589}, {14969, 19734}, {16604, 21753}, {27623, 27631}, {27643, 27666}, {28250, 28289}

X(27636) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b^2 + a^2 b^3 + 2 a^3 b c + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - 5 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(27636) lies on these lines: {1, 2}, {238, 4191}, {748, 1740}, {1743, 2350}, {16343, 17123}, {16571, 32930}, {18792, 27643}, {27623, 27663}, {27631, 27666}, {27641, 28269}, {29982, 32915}

X(27637) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^3 b + a^2 b^2 - a b^3 + 2 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 - 4 b^2 c^2 - a c^3 + b c^3) : :

X(27637) lies on these lines: {2, 44}, {3, 238}, {513, 27647}, {524, 29988}, {1193, 3246}, {2245, 16610}, {4273, 27644}, {24715, 28198}, {27646, 27670}, {27678, 28244}, {28252, 28256}

X(27638) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b - a^2 b^2 - 2 a b^3 + a^3 c + a^2 b c - a b^2 c + 2 b^3 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(27638) lies on these lines: {2, 45}, {3, 238}, {6, 27678}, {1001, 28288}, {3285, 27644}, {4383, 27661}, {4484, 17597}, {7232, 29964}

X(27639) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^3 b - a b^3 + a^3 c - a^2 b c - a b^2 c - b^3 c - a b c^2 + 4 b^2 c^2 - a c^3 - b c^3) : :

X(27639) lies on these lines: {2, 11}, {3, 748}, {31, 16059}, {36, 978}, {56, 32911}, {228, 5272}, {238, 4191}, {244, 20760}, {474, 32772}, {750, 16409}, {899, 23853}, {999, 1066}, {1011, 17123}, {1036, 16410}, {2209, 28360}, {2269, 25889}, {3185, 16610}, {3286, 27643}, {3306, 20967}, {3624, 19763}, {4038, 19734}, {4210, 20992}, {4225, 27625}, {4267, 5333}, {4383, 20470}, {4557, 17597}, {5204, 27645}, {5687, 32943}, {5711, 16297}, {7485, 23868}, {8168, 17751}, {11358, 32944}, {11512, 22345}, {11688, 24620}, {16058, 17125}, {16414, 16466}, {16421, 17124}, {19684, 25524}, {24436, 27680}, {27621, 27657}, {27623, 27631}, {27633, 27650}

X(27640) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^3 b + 2 a^2 b^2 + 2 a^3 c + a^2 b c - 2 a b^2 c - b^3 c + 2 a^2 c^2 - 2 a b c^2 - 4 b^2 c^2 - b c^3) : :

X(27640) lies on these lines: {2, 6}, {238, 1958}, {332, 11342}, {899, 22370}, {978, 28287}, {2239, 25571}, {3008, 29965}, {27621, 27642}, {27624, 27625}, {27626, 28254}

X(27641) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (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(27641) lies on these lines: {2, 37}, {39, 17248}, {58, 87}, {980, 17326}, {992, 28283}, {1015, 17363}, {1211, 26746}, {1400, 28254}, {1654, 2275}, {2092, 17397}, {2228, 25279}, {3123, 16571}, {3661, 17053}, {3662, 29985}, {3876, 4283}, {3963, 27091}, {4357, 24598}, {4393, 21857}, {5069, 17256}, {6376, 17148}, {8610, 17233}, {16604, 17379}, {16696, 17250}, {16726, 17361}, {17157, 17793}, {17349, 21892}, {17368, 21796}, {17393, 21858}, {17786, 27044}, {20917, 27095}, {21214, 22370}, {24653, 29846}, {25624, 31338}, {26747, 32782}, {27147, 31198}, {27623, 27646}, {27624, 27625}, {27626, 27670}, {27627, 27642}, {27636, 28269}, {27651, 27661}

X(27642) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b^3 + a^2 b^4 - a b^3 c^2 - b^4 c^2 + a^3 c^3 - a b^2 c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(27642) lies on these lines: {2, 39}, {404, 5156}, {992, 28264}, {1193, 23493}, {1575, 17033}, {6374, 19565}, {13738, 27665}, {27621, 27640}, {27623, 27662}, {27625, 27676}, {27627, 27641}, {27632, 27656}, {27669, 28243}, {28254, 28274}

X(27643) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a^2 b + a b^2 + a^2 c + a b c - 2 b^2 c + a c^2 - 2 b c^2) : :

X(27643) lies on these lines: {2, 6}, {44, 16700}, {58, 27627}, {238, 4184}, {274, 26223}, {748, 3736}, {978, 4225}, {1193, 4653}, {2176, 28605}, {2999, 25060}, {3008, 17167}, {3286, 27639}, {3786, 7191}, {3995, 33296}, {4641, 16736}, {4720, 16483}, {5208, 7292}, {5711, 17551}, {5905, 16752}, {10458, 17123}, {13588, 17127}, {14005, 16466}, {16948, 27645}, {17012, 25058}, {17020, 25059}, {17182, 26723}, {17185, 28281}, {18792, 27636}, {27064, 30599}, {27624, 27651}, {27635, 27666}

X(27644) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a b + a c - b c) : :

X(27644) lies on these lines: {1, 3728}, {2, 6}, {7, 16752}, {10, 31338}, {21, 238}, {31, 1582}, {37, 4469}, {43, 2209}, {44, 16696}, {58, 87}, {100, 715}, {110, 9082}, {144, 18600}, {171, 28248}, {192, 2176}, {213, 274}, {239, 314}, {284, 27626}, {310, 24514}, {332, 16050}, {404, 5156}, {579, 24598}, {608, 14013}, {614, 5208}, {651, 1014}, {662, 1333}, {673, 20028}, {748, 10458}, {757, 27646}, {759, 29237}, {1010, 16466}, {1043, 1191}, {1045, 3747}, {1172, 2905}, {1203, 25526}, {1434, 6180}, {1444, 1778}, {1621, 16690}, {1724, 5145}, {1743, 18186}, {1790, 27659}, {1817, 28274}, {2245, 24530}, {2274, 2975}, {2295, 28604}, {2305, 19308}, {2308, 28247}, {2323, 21246}, {2664, 20964}, {2999, 17185}, {3142, 24883}, {3193, 5230}, {3216, 4279}, {3230, 17319}, {3285, 27638}, {3286, 4225}, {3725, 11688}, {3758, 16709}, {3759, 20923}, {4000, 17139}, {4184, 17127}, {4273, 27637}, {4360, 16685}, {4384, 10455}, {4393, 21769}, {4416, 16887}, {4503, 17252}, {4594, 7104}, {4604, 27664}, {4641, 16700}, {4653, 15485}, {4658, 28650}, {4720, 32941}, {5009, 27680}, {5222, 17183}, {5256, 25058}, {5299, 29960}, {5711, 14007}, {7032, 16476}, {7083, 16876}, {7109, 17759}, {7168, 20663}, {7304, 31008}, {10025, 16750}, {14005, 31339}, {14621, 26643}, {16477, 27627}, {16669, 16726}, {16670, 18164}, {16705, 17257}, {16712, 17333}, {16744, 28366}, {17012, 25060}, {17033, 17743}, {17034, 29983}, {17121, 20228}, {17142, 32922}, {17167, 26723}, {17187, 22343}, {17189, 29967}, {17196, 17382}, {17200, 29991}, {17202, 17367}, {17212, 20980}, {17363, 33297}, {19283, 19767}, {21024, 24958}, {23092, 27527}, {23125, 27334}, {23444, 24520}, {23692, 27653}, {26223, 30599}, {30984, 32843}

X(27644) = isogonal conjugate of X(16606)
X(27644) = isotomic conjugate of complement of X(36857)
X(27644) = X(92)-isoconjugate of X(22381)

X(27645) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (4 a^2 b + 4 a b^2 + 4 a^2 c - 5 a b c - 5 b^2 c + 4 a c^2 - 5 b c^2) : :

X(27645) lies on these lines:

X(27646) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b+2 a^2 b^2+a b^3+a^3 c+a^2 b c-2 a b^2 c-b^3 c+2 a^2 c^2-2 a b c^2-3 b^2 c^2+a c^3-b c^3) : :

X(27646) lies on these lines:

X(27647) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (b - c) (a^2 b^2 + a b^3 + a^2 b c + 3 a b^2 c + b^3 c + a^2 c^2 + 3 a b c^2 + b^2 c^2 + a c^3 + b c^3) : :

X(27647) lies on these lines:

X(27648) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (b - c) (a^3 b - a b^3 + a^3 c - 2 a b^2 c - b^3 c - 2 a b c^2 - b^2 c^2 - a c^3 - b c^3) : :

X(27648) lies on these lines:

X(27649) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (4 a^5 b+4 a^4 b^2-4 a^3 b^3-4 a^2 b^4+4 a^5 c+3 a^4 b c-2 a^2 b^3 c-4 a b^4 c-b^5 c+4 a^4 c^2+4 a b^3 c^2-4 a^3 c^3-2 a^2 b c^3+4 a b^2 c^3+2 b^3 c^3-4 a^2 c^4-4 a b c^4-b c^5) : :

X(27649) lies on these lines:

X(27650) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + a^5 b c + a^4 b^2 c - a^2 b^4 c - a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - a b^4 c^2 + b^5 c^2 + 2 a b^3 c^3 - a^2 b c^4 - a b^2 c^4 - a^2 c^5 - a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27650) lies on these lines:

X(27651) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a^5 b + 2 a^4 b^2 - 2 a^2 b^4 - a b^5 + a^5 c + 3 a^4 b c - 2 a^2 b^3 c - a b^4 c - b^5 c + 2 a^4 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 - 2 a^2 c^4 - a b c^4 - a c^5 - b c^5) : :

X(27651) lies on these lines:

X(27652) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a + b) (a + c) (a^5 b + a^4 b^2 - a b^5 - b^6 + a^5 c + a^4 b c + 2 a^3 b^2 c - 3 a b^4 c - b^5 c + a^4 c^2 + 2 a^3 b c^2 + b^4 c^2 + 2 b^3 c^3 - 3 a b c^4 + b^2 c^4 - a c^5 - b c^5 - c^6) : :

X(27652) lies on these lines:

X(27653) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a+b) (a+c) (a^6 b-a^5 b^2-2 a^4 b^3+2 a^3 b^4+a^2 b^5-a b^6+a^6 c-3 a^5 b c-a^4 b^2 c+2 a^3 b^3 c-a^2 b^4 c+a b^5 c+b^6 c-a^5 c^2-a^4 b c^2+a b^4 c^2+b^5 c^2-2 a^4 c^3+2 a^3 b c^3-2 a b^3 c^3-2 b^4 c^3+2 a^3 c^4-a^2 b c^4+a b^2 c^4-2 b^3 c^4+a^2 c^5+a b c^5+b^2 c^5-a c^6+b c^6) : :

X(27653) lies on these lines:

X(27654) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (6 a^5 b + 6 a^4 b^2 - 6 a^3 b^3 - 6 a^2 b^4 + 6 a^5 c + 5 a^4 b c - 4 a^2 b^3 c - 6 a b^4 c - b^5 c + 6 a^4 c^2 + 6 a b^3 c^2 - 6 a^3 c^3 - 4 a^2 b c^3 + 6 a b^2 c^3 + 2 b^3 c^3 - 6 a^2 c^4 - 6 a b c^4 - b c^5) : :

X(27654) lies on these lines:

X(27655) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^5 b + 2 a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + 2 a^5 c + a^4 b c - 2 a b^4 c - b^5 c + 2 a^4 c^2 + 2 a^2 b^2 c^2 + 4 a b^3 c^2 - 2 a^3 c^3 + 4 a b^2 c^3 + 2 b^3 c^3 - 2 a^2 c^4 - 2 a b c^4 - b c^5) : :

X(27655) lies on these lines:

X(27656) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b+a^4 b^2-a^3 b^3-a^2 b^4+a^5 c+a^4 b c-a b^4 c+a^4 c^2+a b^3 c^2-a^3 c^3+a b^2 c^3+b^3 c^3-a^2 c^4-a b c^4) : :

X(27656) lies on these lines:

X(27657) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b+a^4 b^2-a^3 b^3-a^2 b^4+a^5 c+2 a^3 b^2 c+3 a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2-a b^3 c^2-a^3 c^3+3 a^2 b c^3-a b^2 c^3+2 b^3 c^3-a^2 c^4-a b c^4-b c^5) : :

X(27657) lies on these lines:

X(27658) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (-a^2 b^3 - a b^4 + 2 a^3 b c + b^4 c - 2 a b^2 c^2 - a^2 c^3 - a c^4 + b c^4) : :

X(27658) lies on these lines:

X(27659) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 3 a^3 b c - a b^3 c + b^4 c + a^3 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(27659) lies on these lines:

X(27660) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a^3 b + a^2 b^2 + a^3 c + a^2 b c - b^3 c + a^2 c^2 - 2 b^2 c^2 - b c^3) : :

X(27660) lies on these lines:

X(27661) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 3 a^3 b c - 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 - a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(27661) lies on these lines:

X(27662) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b + a^4 b^2 + a^5 c + a^4 b c + a^2 b^3 c - a b^4 c + a^4 c^2 - b^4 c^2 + a^2 b c^3 + b^3 c^3 - a b c^4 - b^2 c^4) : :

X(27662) lies on these lines:

X(27663) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b^3+a^3 b^4-a^4 b^2 c-a^3 b^3 c+a^2 b^4 c-a^4 b c^2-a b^4 c^2+a^4 c^3-a^3 b 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(27663) lies on these lines:

X(27664) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^4 b - 3 a^2 b^3 - 2 a b^4 + a^4 c + 3 a^3 b c - a^2 b^2 c - a b^3 c + 2 b^4 c - a^2 b c^2 + 5 a b^2 c^2 - 3 a^2 c^3 - a b c^3 - 2 a c^4 + 2 b c^4) : :

X(27664) lies on these lines:

X(27665) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a^3 b - 2 a b^3 + a^3 c - a^2 b c + b^3 c + b^2 c^2 - 2 a c^3 + b c^3) : :

X(27665) lies on these lines:

X(27666) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^3 b - a b^3 + a^3 c - a^2 b c - a b^2 c - b^3 c - a b c^2 + 5 b^2 c^2 - a c^3 - b c^3) : :

X(27666) lies on these lines:

X(27667) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b - a^3 b^3 + a^5 c - a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - a b^4 c - a^3 b c^2 - a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 - a^3 c^3 + 2 a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a b c^4 - b^2 c^4) : :

X(27667) lies on these lines:

X(27668) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a + b) (a + c) (a^5 b - a^3 b^3 + a^5 c - a^4 b c - a^3 b^2 c + a^2 b^3 c + a b^4 c - b^5 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 + 2 b^3 c^3 + a b c^4 - b c^5) : :

X(27668) lies on these lines:

X(27669) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^5 b + a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a^5 c - a b^4 c - b^5 c + a^4 c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - 2 a^2 c^4 - a b c^4 + b^2 c^4 - b c^5) : :

X(27669) lies on these lines:

X(27670) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b - a^2 b^2 - 2 a b^3 + a^3 c + a^2 b c + 2 b^3 c - a^2 c^2 - b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(27670) lies on these lines:

X(27671) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^2 b^2 + 2 a b^3 - a b^2 c - 2 b^3 c + 2 a^2 c^2 - a b c^2 - b^2 c^2 + 2 a c^3 - 2 b c^3) : :

X(27671) lies on these lines:

X(27672) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (2 a^3 b^3 + 2 a^2 b^4 + a^2 b^3 c - 2 a b^3 c^2 - 2 b^4 c^2 + 2 a^3 c^3 + a^2 b c^3 - 2 a b^2 c^3 - b^3 c^3 + 2 a^2 c^4 - 2 b^2 c^4) : :

X(27672) lies on these lines:

X(27673) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(27673) lies on these lines:

X(27674) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (b-c) (a^3 b-a b^3+a^3 c+a^2 b c-3 a b^2 c-b^3 c-3 a b c^2-a c^3-b c^3) : :

X(27674) lies on these lines:

X(27675) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (b - c) (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c - 2 b^3 c + a^2 c^2 - 2 b^2 c^2 - 2 b c^3) : :

X(27675) lies on these lines:

X(27676) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (a^3 b^3 + a^2 b^4 - 2 a^3 b^2 c - 2 a^2 b^3 c - 2 a^3 b c^2 + 3 a^2 b^2 c^2 + 2 a b^3 c^2 - b^4 c^2 + a^3 c^3 - 2 a^2 b c^3 + 2 a b^2 c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(27676) lies on these lines:

X(27677) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (b - c) (a^4 b^2 + a^3 b^3 + 2 a^4 b c + 2 a^3 b^2 c + a^2 b^3 c + a^4 c^2 + 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(27677) lies on these lines:

X(27678) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (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(27678) lies on these lines: {2, 7}, {6, 27638}, {56, 26657}, {71, 17302}, {238, 28288}, {320, 583}, {573, 17367}, {662, 1333}, {978, 4257}, {1278, 3501}, {1475, 20090}, {1716, 17127}, {2092, 17012}, {2176, 28395}, {2245, 16706}, {2260, 17300}, {3730, 17247}, {3778, 7191}, {3882, 17121}, {4253, 17364}, {4393, 22370}, {5043, 7232}, {7292, 17065}, {14964, 17200}, {16549, 17116}, {16552, 17252}, {16723, 17376}, {17123, 22174}, {17288, 18206}, {17292, 21061}, {17343, 21384}, {20228, 24625}, {21010, 25279}, {27623, 27641}, {27637, 28244}

X(27679) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (b + c) (2 a^4 + 2 a^3 b - a^2 b^2 - a b^3 + 2 a^3 c + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 - 3 b^2 c^2 - a c^3 + b c^3) : :

X(27679) lies on these lines: {2, 896}, {23, 238}, {513, 27637}, {758, 20703}, {978, 4225}, {7292, 20984}, {7449, 27659}, {24436, 27630}

X(27680) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a (-a^2 b^3 - a b^4 + 2 a^3 b c + b^4 c - 3 a b^2 c^2 - a^2 c^3 - a c^4 + b c^4) : :

X(27680) lies on these lines: {2, 38}, {3, 238}, {9, 9367}, {986, 16706}, {2292, 17383}, {3086, 24744}, {3831, 33165}, {4650, 28274}, {5009, 27644}, {5247, 22769}, {5657, 24440}, {6211, 19549}, {7262, 28242}, {7786, 17353}, {17123, 25494}, {17164, 27192}, {17278, 24174}, {24436, 27639}, {24520, 25650}, {25079, 26107}, {25253, 27011}, {25591, 26971}, {27625, 27671}, {27627, 27641}, {29960, 33087}

X(27681) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^10 b + a^9 b^2 - 2 a^8 b^3 - 2 a^7 b^4 + 2 a^4 b^7 + 2 a^3 b^8 - a^2 b^9 - a b^10 + a^10 c + a^9 b c - a^8 b^2 c - 2 a^7 b^3 c - 2 a^6 b^4 c + 2 a^4 b^6 c + 2 a^3 b^7 c + a^2 b^8 c - a b^9 c - b^10 c + a^9 c^2 - a^8 b c^2 - 2 a^7 b^2 c^2 - 2 a^5 b^4 c^2 + 2 a^3 b^6 c^2 + a b^8 c^2 + b^9 c^2 - 2 a^8 c^3 - 2 a^7 b c^3 + 2 a^5 b^3 c^3 - 2 a^3 b^5 c^3 + 2 a b^7 c^3 + 2 b^8 c^3 - 2 a^7 c^4 - 2 a^6 b c^4 - 2 a^5 b^2 c^4 - 2 b^7 c^4 - 2 a^3 b^3 c^5 - 2 a b^5 c^5 + 2 a^4 b c^6 + 2 a^3 b^2 c^6 + 2 a^4 c^7 + 2 a^3 b c^7 + 2 a b^3 c^7 - 2 b^4 c^7 + 2 a^3 c^8 + a^2 b c^8 + a b^2 c^8 + 2 b^3 c^8 - a^2 c^9 - a b c^9 + b^2 c^9 - a c^10 - b c^10) : :

X(27681) lies on these lines:

X(27682) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 68

Barycentrics    a^2 (a^10 b + a^9 b^2 - 2 a^8 b^3 - 2 a^7 b^4 + 2 a^4 b^7 + 2 a^3 b^8 - a^2 b^9 - a b^10 + a^10 c + a^9 b c - a^8 b^2 c - 2 a^7 b^3 c - 2 a^6 b^4 c + 2 a^4 b^6 c + 2 a^3 b^7 c + a^2 b^8 c - a b^9 c - b^10 c + a^9 c^2 - a^8 b c^2 - 2 a^7 b^2 c^2 - 2 a^6 b^3 c^2 - 4 a^5 b^4 c^2 + 2 a^4 b^5 c^2 + 4 a^3 b^6 c^2 + a b^8 c^2 + b^9 c^2 - 2 a^8 c^3 - 2 a^7 b c^3 - 2 a^6 b^2 c^3 + 2 a^2 b^6 c^3 + 2 a b^7 c^3 + 2 b^8 c^3 - 2 a^7 c^4 - 2 a^6 b c^4 - 4 a^5 b^2 c^4 - 2 a^2 b^5 c^4 - 2 b^7 c^4 + 2 a^4 b^2 c^5 - 2 a^2 b^4 c^5 - 2 a b^5 c^5 + 2 a^4 b c^6 + 4 a^3 b^2 c^6 + 2 a^2 b^3 c^6 + 2 a^4 c^7 + 2 a^3 b c^7 + 2 a b^3 c^7 - 2 b^4 c^7 + 2 a^3 c^8 + a^2 b c^8 + a b^2 c^8 + 2 b^3 c^8 - a^2 c^9 - a b c^9 + b^2 c^9 - a c^10 - b c^10) : :

X(27682) lies on these lines:

X(27683) = (name pending)

Barycentrics    4 a^18 -24 a^16 (b^2+c^2) +9 a^14 (7 b^4+10 b^2 c^2+7 c^4) -a^12 (91 b^6+121 b^4 c^2+121 b^2 c^4+91 c^6) +2 a^10 (34 b^8+23 b^6 c^2+24 b^4 c^4+23 b^2 c^6+34 c^8) -a^8 (4 b^10-27 b^8 c^2-13 b^6 c^4-13 b^4 c^6-27 b^2 c^8+4 c^10) -a^6 (b^2-c^2)^2 (41 b^8+61 b^6 c^2+69 b^4 c^4+61 b^2 c^6+41 c^8) +a^4 (b^2-c^2)^2 (37 b^10-14 b^8 c^2-17 b^6 c^4-17 b^4 c^6-14 b^2 c^8+37 c^10) -a^2 (b^2-c^2)^4 (14 b^8-7 b^6 c^2-9 b^4 c^4-7 b^2 c^6+14 c^8) +2 (b^2-c^2)^8 (b^2+c^2) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28615.

X(27683) lies on this line: {1503,3628}

X(27684) = X(4)X(15047)∩X(1263)X(14627)

Barycentrics    4 a^16-3 (b^2-c^2)^8-23 a^14 (b^2+c^2)+19 a^2 (b^2-c^2)^6 (b^2+c^2)+a^12 (51 b^4+66 b^2 c^2+51 c^4)-a^10 (47 b^6+43 b^4 c^2+43 b^2 c^4+47 c^6)-a^8 (5 b^8+14 b^6 c^2+6 b^4 c^4+14 b^2 c^6+5 c^8)-a^4 (b^2-c^2)^2 (47 b^8-30 b^6 c^2-45 b^4 c^4-30 b^2 c^6+47 c^8)+3 a^6 (17 b^10-13 b^8 c^2-9 b^6 c^4-9 b^4 c^6-13 b^2 c^8+17 c^10) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28615.

X(27684) lies on these lines: {4,15047}, {1263,14627}, {5944,13631}, {13163,14706}

leftri

Collineation mappings involving Gemini triangle 69: X(27685)-X(27735)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 69, as in centers X(27685)-X(27735). Then

m(X) = (b+c)(b^2+c^2-a^2-bc)x - b(a+b)(b+c)y - c(a+c)(b+c)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 12, 2018)

X(27685) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - 2 a^4 b^2 - a^3 b^3 + a b^5 + b^6 + a^4 b c - a^2 b^3 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27685) lies on these lines: {1, 125}, {2, 3}, {10, 21318}, {227, 21674}, {355, 6739}, {1478, 14873}, {1837, 8287}, {3698, 27714}, {5130, 17073}, {5587, 30436}, {8286, 11376}, {12079, 13869}, {17181, 23674}, {20277, 20278}, {27688, 27696}

X(27686) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - a^5 b - 2 a^4 b^2 - a^2 b^4 + a b^5 + 2 b^6 - a^5 c + a^4 b c + 2 a^3 b^2 c - a b^4 c - b^5 c - 2 a^4 c^2 + 2 a^3 b c^2 + 6 a^2 b^2 c^2 - 2 b^4 c^2 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 + a c^5 - b c^5 + 2 c^6) : :

X(27686) lies on these lines: {2, 3}, {8, 125}, {944, 6739}, {10590, 14873}, {17170, 23674}, {21674, 27691}, {26364, 31845}, {27704, 27706}

X(27687) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c)^2 (-a^5 + a^3 b^2 - a^2 b^3 + b^5 + a^3 b c + 2 a^2 b^2 c - a b^3 c - 2 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 - a b c^3 + b^2 c^3 - 2 b c^4 + c^5) : :

X(27687) lies on these lines: {2, 3}, {10, 125}, {12, 201}, {120, 27698}, {594, 21687}, {1385, 6739}, {1698, 1726}, {1834, 3924}, {3057, 8286}, {3454, 10176}, {3822, 22001}, {3925, 27714}, {7951, 14873}, {8287, 17606}, {10175, 30436}, {17757, 27690}, {20337, 30961}, {21029, 23897}, {21033, 23921}, {22076, 31806}, {25466, 32775}, {25623, 27688}

X(27688) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^2 b^2 + a b^3 + b^4 + a^2 b c - a^2 c^2 + a c^3 + c^4) : :

X(27688) lies on these lines: {2, 6}, {7, 20337}, {9, 16565}, {10, 7235}, {115, 3729}, {125, 29857}, {192, 23903}, {346, 23947}, {645, 25687}, {2245, 21245}, {2305, 21287}, {3923, 20546}, {3963, 27733}, {4019, 16886}, {4363, 5949}, {4416, 27970}, {4461, 23942}, {8287, 17279}, {8818, 17351}, {11104, 20558}, {17058, 17282}, {21674, 21728}, {24271, 32431}, {24311, 33100}, {25623, 27687}, {27685, 27696}, {27695, 27700}, {27705, 27709}, {27723, 27729}, {27726, 27727}, {27971, 28604}

X(27689) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^3 b - 3 a^2 b^2 + a b^3 + 2 b^4 - a^3 c + 7 a^2 b c + a b^2 c - 3 b^3 c - 3 a^2 c^2 + a b c^2 + 2 b^2 c^2 + a c^3 - 3 b c^3 + 2 c^4) : :

X(27689) lies on these lines:

X(27690) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^3 - 2 a^2 b - a b^2 + 2 b^3 - 2 a^2 c - a b c - b^2 c - a c^2 - b c^2 + 2 c^3) : :

X(27690) lies on these lines:

X(27691) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (a + b - c) (a - b + c) (b + c) (a^2 - a b - b^2 - a c - b c - c^2) : :

X(27691) lies on these lines:

X(27692) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (-a^4 b + a^3 b^2 - a b^4 + b^5 - a^4 c + a^2 b^2 c - 2 b^4 c + a^3 c^2 + a^2 b c^2 + b^3 c^2 + b^2 c^3 - a c^4 - 2 b c^4 + c^5) : :

X(27692) lies on these lines:

X(27693) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^8 - a^6 b^2 - a^5 b^3 - 2 a^4 b^4 + a^2 b^6 + a b^7 + b^8 + a^6 b c - a^2 b^5 c - a^6 c^2 + a^2 b^4 c^2 - a^5 c^3 - a b^4 c^3 - 2 a^4 c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27693) lies on these lines:

X(27694) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^8 - a^6 b^2 - a^5 b^3 - 2 a^4 b^4 + a^2 b^6 + a b^7 + b^8 + a^6 b c - a^2 b^5 c - a^6 c^2 + a^4 b^2 c^2 + a^3 b^3 c^2 + a^2 b^4 c^2 - a^5 c^3 + a^3 b^2 c^3 + a^2 b^3 c^3 - a b^4 c^3 - 2 a^4 c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27694) lies on these lines:

X(27695) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^5 - a^3 b^2 + a b^4 + b^5 + a^3 b c - a^3 c^2 + a c^4 + c^5) : :

X(27695) lies on these lines:

X(27696) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - a^4 b^2 + a b^5 + b^6 + a^4 b c - a^4 c^2 + a c^5 + c^6) : :

X(27696) lies on these lines:

X(27697) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^2 + b c) (a b + b^2 + a c + c^2) : :

X(27697) lies on these lines:

X(27698) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 b^2 + a^3 b^3 + a^2 b^3 c + a^4 c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^2 c^4) : :

X(27698) lies on these lines:

X(27699) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + b^6 - a^5 c + a^4 b c - a b^4 c - b^5 c - a^4 c^2 + a^3 c^3 - a^2 c^4 - a b c^4 - b c^5 + c^6) : :

X(27699) lies on these lines:

X(27700) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c)^2 (a^4 + a b^3 + b^3 c - b^2 c^2 + a c^3 + b c^3) : :

X(27700) lies on these lines:

X(27701) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 b + a b^4 + a^4 c - a^3 b c - a^2 b^2 c + a b^3 c + b^4 c - a^2 b c^2 - a b^2 c^2 + a b c^3 + a c^4 + b c^4) : :

X(27701) lies on these lines:

X(27702) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - 2 a^3 b - 3 a^2 b^2 + a b^3 + b^4 - 2 a^3 c + a^2 b c - 2 a b^2 c - 2 b^3 c - 3 a^2 c^2 - 2 a b c^2 + a c^3 - 2 b c^3 + c^4) : :

X(27702) lies on these lines:

X(27703) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^5 - a^4 b - a^3 b^2 + b^5 - a^4 c + a^3 b c - a b^3 c - b^4 c - a^3 c^2 - a b c^3 - b c^4 + c^5) : :

X(27703) lies on these lines:

X(27704) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^3 b - 3 a^2 b^2 + a b^3 + 2 b^4 - a^3 c + a^2 b c - a b^2 c - b^3 c - 3 a^2 c^2 - a b c^2 + a c^3 - b c^3 + 2 c^4) : :

X(27704) lies on these lines:

X(27705) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    b c (b + c) (-3 a^2 - a b + b^2 - a c - b c + c^2) : :

X(27705) lies on these lines:

X(27706) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    b c (b + c) (-a^3 b - a^3 c - 3 a^2 b c + b^3 c - b^2 c^2 + b c^3) : :

X(27706) lies on these lines:

X(27707) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 + a^3 b + a b^3 + b^4 + a^3 c + 4 a^2 b c + 2 a b^2 c + 2 a b c^2 + b^2 c^2 + a c^3 + c^4) : :

X(27707) lies on these lines:

X(27708) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (3 a^3 - 2 a^2 b - a b^2 + 4 b^3 - 2 a^2 c + a b c - b^2 c - a c^2 - b c^2 + 4 c^3) : :

X(27708) lies on these lines:

X(27709) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^2 b^2 + a b^3 + b^4 - 2 a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 - b^2 c^2 + a c^3 + b c^3 + c^4) : :

X(27709) lies on these lines:

X(27710) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (a^2 b^2 + a b^3 + a^2 b c + a^2 c^2 - b^2 c^2 + a c^3) : :

X(27710) lies on these lines:

X(27711) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (a^5 - a^3 b^2 - a^2 b^3 - a b^4 + a^3 b c - a^2 b^2 c + b^4 c - a^3 c^2 - a^2 b c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 + b c^4) : :

X(27711) lies on these lines:

X(27712) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    b c (b^2 - c^2) (a^2 + b^2 - b c + c^2) : :

X(27712) lies on these lines:

X(27713) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^5 - a^3 b^2 + a b^4 + b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 - a b^2 c^2 + a c^4 + c^5) : :

X(27713) lies on these lines:

X(27714) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (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(27714) lies on these lines:

X(27715) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (3 a^6 - a^5 b - 6 a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + 3 a b^5 + 4 b^6 - a^5 c + 3 a^4 b c + 2 a^3 b^2 c - 2 a^2 b^3 c - a b^4 c - b^5 c - 6 a^4 c^2 + 2 a^3 b c^2 + 10 a^2 b^2 c^2 - 2 a b^3 c^2 - 4 b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 4 b^2 c^4 + 3 a c^5 - b c^5 + 4 c^6) : :

X(27715) lies on these lines:

X(27716) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - 2 a^4 b^2 - a^3 b^3 + a b^5 + b^6 - 2 a^3 b^2 c - 2 a^2 b^3 c - 2 a^4 c^2 - 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 - 2 a^2 b c^3 - 2 a b^2 c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27716) lies on these lines:

X(27717) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^8 - a^6 b^2 - a^5 b^3 - 2 a^4 b^4 + a^2 b^6 + a b^7 + b^8 + a^6 b c - a^2 b^5 c - a^6 c^2 + 2 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + a^2 b^4 c^2 - a^5 c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 - 2 a^4 c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - a^2 b c^5 + a^2 c^6 + a c^7 + c^8) : :

X(27717) lies on these lines:

X(27718) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^8 + a^7 b - 2 a^6 b^2 - 3 a^5 b^3 - 2 a^4 b^4 - a^3 b^5 + 2 a^2 b^6 + 3 a b^7 + b^8 + a^7 c - 2 a^5 b^2 c - a^3 b^4 c - 2 a^2 b^5 c + 2 a b^6 c + 2 b^7 c - 2 a^6 c^2 - 2 a^5 b c^2 + 7 a^4 b^2 c^2 + 10 a^3 b^3 c^2 - 4 a b^5 c^2 - b^6 c^2 - 3 a^5 c^3 + 10 a^3 b^2 c^3 + 8 a^2 b^3 c^3 - a b^4 c^3 - 2 b^5 c^3 - 2 a^4 c^4 - a^3 b c^4 - a b^3 c^4 - a^3 c^5 - 2 a^2 b c^5 - 4 a b^2 c^5 - 2 b^3 c^5 + 2 a^2 c^6 + 2 a b c^6 - b^2 c^6 + 3 a c^7 + 2 b c^7 + c^8) : :

X(27718) lies on these lines:

X(27719) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^9 + a^8 b - a^7 b^2 - 2 a^6 b^3 - 3 a^5 b^4 - 2 a^4 b^5 + a^3 b^6 + 2 a^2 b^7 + 2 a b^8 + b^9 + a^8 c + 2 a^7 b c - a^6 b^2 c - 3 a^5 b^3 c - 2 a^4 b^4 c - 2 a^3 b^5 c + a^2 b^6 c + 3 a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 + 3 a^5 b^2 c^2 + 6 a^4 b^3 c^2 + 3 a^3 b^4 c^2 - a^2 b^5 c^2 - a b^6 c^2 - 2 a^6 c^3 - 3 a^5 b c^3 + 6 a^4 b^2 c^3 + 12 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - 3 a b^5 c^3 - 3 a^5 c^4 - 2 a^4 b c^4 + 3 a^3 b^2 c^4 + 2 a^2 b^3 c^4 - 2 a b^4 c^4 - 2 b^5 c^4 - 2 a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 - 3 a b^3 c^5 - 2 b^4 c^5 + a^3 c^6 + a^2 b c^6 - a b^2 c^6 + 2 a^2 c^7 + 3 a b c^7 + 2 a c^8 + b c^8 + c^9) : :

X(27719) lies on these lines:

X(27720) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^9 - a^7 b^2 + a^6 b^3 - a^5 b^4 - a^4 b^5 + a^3 b^6 - a^2 b^7 + b^9 + 2 a^7 b c + 2 a^6 b^2 c - a^5 b^3 c - a^4 b^4 c - a b^7 c - b^8 c - a^7 c^2 + 2 a^6 b c^2 + 3 a^5 b^2 c^2 - 5 a^4 b^3 c^2 - 3 a^3 b^4 c^2 + 4 a^2 b^5 c^2 + a b^6 c^2 - b^7 c^2 + a^6 c^3 - a^5 b c^3 - 5 a^4 b^2 c^3 - 4 a^3 b^3 c^3 - 3 a^2 b^4 c^3 + a b^5 c^3 + 3 b^6 c^3 - a^5 c^4 - a^4 b c^4 - 3 a^3 b^2 c^4 - 3 a^2 b^3 c^4 - 2 a b^4 c^4 - 2 b^5 c^4 - a^4 c^5 + 4 a^2 b^2 c^5 + a b^3 c^5 - 2 b^4 c^5 + a^3 c^6 + a b^2 c^6 + 3 b^3 c^6 - a^2 c^7 - a b c^7 - b^2 c^7 - b c^8 + c^9) : :

X(27720) lies on these lines:

X(27721) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (5 a^6 - a^5 b - 10 a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 5 a b^5 + 6 b^6 - a^5 c + 5 a^4 b c + 2 a^3 b^2 c - 4 a^2 b^3 c - a b^4 c - b^5 c - 10 a^4 c^2 + 2 a^3 b c^2 + 14 a^2 b^2 c^2 - 4 a b^3 c^2 - 6 b^4 c^2 - 4 a^3 c^3 - 4 a^2 b c^3 - 4 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 6 b^2 c^4 + 5 a c^5 - b c^5 + 6 c^6) : :

X(27721) lies on these lines:

X(27722) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - a^5 b - 2 a^4 b^2 - a^2 b^4 + a b^5 + 2 b^6 - a^5 c + 3 a^4 b c + 6 a^3 b^2 c + 2 a^2 b^3 c - a b^4 c - b^5 c - 2 a^4 c^2 + 6 a^3 b c^2 + 12 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 + a c^5 - b c^5 + 2 c^6) : :

X(27722) lies on these lines:

X(27723) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^6 - a^4 b^2 + a b^5 + b^6 + a^4 b c + a^3 b^2 c - a^4 c^2 + a^3 b c^2 + 3 a^2 b^2 c^2 - b^4 c^2 + b^3 c^3 - b^2 c^4 + a c^5 + c^6) : :

X(27723) lies on these lines:

X(27724) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (-2 a^3 b^2 - a^2 b^3 + a b^4 - a b^3 c - 2 a^3 c^2 - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 + a c^4) : :

X(27724) lies on these lines:

X(27725) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^5 - 3 a^3 b^2 - a^2 b^3 + 2 a b^4 + b^5 + a^3 b c - a b^3 c - 3 a^3 c^2 - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 + 2 a c^4 + c^5) : :

X(27725) lies on these lines:

X(27726) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^3 b - 2 a^2 b^2 + a b^3 + b^4 - a^3 c + 4 a^2 b c - 2 b^3 c - 2 a^2 c^2 + b^2 c^2 + a c^3 - 2 b c^3 + c^4) : :

X(27726) lies on these lines:

X(27727) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^3 b + a^2 b^2 + a^3 c - 3 a^2 b c + 2 b^3 c + a^2 c^2 - b^2 c^2 + 2 b c^3) : :

X(27727) lies on these lines:

X(27728) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (3 a^3 - 2 a^2 b - 4 a b^2 + b^3 - 2 a^2 c + 3 a b c + 2 b^2 c - 4 a c^2 + 2 b c^2 + c^3) : :

X(27728) lies on these lines:

X(27729) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 b^2 + a^3 b^3 - a^3 b^2 c + a^2 b^3 c + a^4 c^2 - a^3 b c^2 - 5 a^2 b^2 c^2 + a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 - b^3 c^3 + 2 b^2 c^4) : :

X(27729) lies on these lines:

X(27730) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (a^4 - 2 a^2 b^2 - a b^3 + b^3 c - 2 a^2 c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(27730) lies on these lines:

X(27731) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (a^4 - a^2 b^2 + a^2 b c + b^3 c - a^2 c^2 + b c^3) : :

X(27731) lies on these lines:

X(27732) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b - c) (b + c) (a^5 - a^3 b^2 - a^2 b^3 - a b^4 + a^3 b c + 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 + a b c^3 + b^2 c^3 - a c^4 + b c^4) : :

X(27732) lies on these lines:

X(27733) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    b c (b + c) (-a^4 + a^3 b + a^2 b^2 - 2 a b^3 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 - b^2 c^2 - 2 a c^3 + b c^3) : :

X(27733) lies on these lines:

X(27734) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^4 - a^2 b^2 + a b^3 + b^4 + 4 a^2 b c + a b^2 c - b^3 c - a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 - b c^3 + c^4) : :

X(27734) lies on these lines:

X(27735) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 69

Barycentrics    (b + c) (a^12 - 3 a^10 b^2 - a^9 b^3 + a^8 b^4 + 2 a^7 b^5 + 4 a^6 b^6 - 3 a^4 b^8 - 2 a^3 b^9 - a^2 b^10 + a b^11 + b^12 + a^10 b c - 2 a^8 b^3 c + 2 a^4 b^7 c - a^2 b^9 c - 3 a^10 c^2 + 4 a^8 b^2 c^2 + 2 a^7 b^3 c^2 + 2 a^6 b^4 c^2 - a^2 b^8 c^2 - 2 a b^9 c^2 - 2 b^10 c^2 - a^9 c^3 - 2 a^8 b c^3 + 2 a^7 b^2 c^3 - 2 a^5 b^4 c^3 + 2 a^3 b^6 c^3 + 2 a^2 b^7 c^3 - a b^8 c^3 + a^8 c^4 + 2 a^6 b^2 c^4 - 2 a^5 b^3 c^4 - 6 a^4 b^4 c^4 + 2 a^2 b^6 c^4 - b^8 c^4 + 2 a^7 c^5 - 2 a^2 b^5 c^5 + 2 a b^6 c^5 + 4 a^6 c^6 + 2 a^3 b^3 c^6 + 2 a^2 b^4 c^6 + 2 a b^5 c^6 + 4 b^6 c^6 + 2 a^4 b c^7 + 2 a^2 b^3 c^7 - 3 a^4 c^8 - a^2 b^2 c^8 - a b^3 c^8 - b^4 c^8 - 2 a^3 c^9 - a^2 b c^9 - 2 a b^2 c^9 - a^2 c^10 - 2 b^2 c^10 + a c^11 + c^12) : :

X(27735) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 70: X(27736)-X(27777)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 70, as in centers X(27736)-X(27777). Then

m(X) = (b+c)(b^2+c^2-a^2-bc)x - b(a+b)(b+c)y - c(a+c)(b+c)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 12, 2018)

X(27736) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^4 b + 2 a^3 b^2 - a^2 b^3 - 2 a b^4 + a^4 c - 3 a^3 b c + a^2 b^2 c + 3 a b^3 c - 2 b^4 c + 2 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 - a^2 c^3 + 3 a b c^3 + 2 b^2 c^3 - 2 a c^4 - 2 b c^4) : :

X(27736) lies on these lines:

X(27737) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^5 - 4 a^3 b^2 + 2 a^2 b^3 - 2 b^5 + 3 a^3 b c - 3 a^2 b^2 c - 3 a b^3 c + 3 b^4 c - 4 a^3 c^2 - 3 a^2 b c^2 + 6 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 - 3 a b c^3 - b^2 c^3 + 3 b c^4 - 2 c^5) : :

X(27737) lies on these lines:

X(27738) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^5 - a^4 b - 6 a^3 b^2 + 3 a^2 b^3 + 2 a b^4 - 2 b^5 - a^4 c + 6 a^3 b c - 4 a^2 b^2 c - 6 a b^3 c + 5 b^4 c - 6 a^3 c^2 - 4 a^2 b c^2 + 8 a b^2 c^2 - 3 b^3 c^2 + 3 a^2 c^3 - 6 a b c^3 - 3 b^2 c^3 + 2 a c^4 + 5 b c^4 - 2 c^5) : :

X(27738) lies on these lines:

X(27739) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^2 - a b - 2 b^2 - a c + 2 b c - 2 c^2) : :

X(27739) lies on these lines:

X(27740) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^4 - 2 a^2 b^2 - 2 b^4 - 5 a^2 b c - 2 a b^2 c + 3 b^3 c - 2 a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 + 3 b c^3 - 2 c^4) : :

X(27740) lies on these lines:

X(27741) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^2 - 2 b^2 + 3 b c - 2 c^2) : :

X(27741) lies on these lines:

X(27742) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^2 + a b - 2 b^2 + a c + 4 b c - 2 c^2) : :

X(27742) lies on these lines:

X(27743) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^7 b + 3 a^6 b^2 + 2 a^5 b^3 - a^3 b^5 - 3 a^2 b^6 - 2 a b^7 + a^7 c - a^6 b c + a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c + a^2 b^5 c - a b^6 c - 2 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 + 2 a^5 c^3 + 2 a^4 b c^3 + 2 a b^4 c^3 + 2 b^5 c^3 - a^3 b c^4 + a^2 b^2 c^4 + 2 a b^3 c^4 - a^3 c^5 + a^2 b c^5 + a b^2 c^5 + 2 b^3 c^5 - 3 a^2 c^6 - a b c^6 - 2 a c^7 - 2 b c^7) : :

X(27743) lies on these lines:

X(27744) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^7 b + 3 a^6 b^2 + 2 a^5 b^3 - a^3 b^5 - 3 a^2 b^6 - 2 a b^7 + a^7 c - a^6 b c + a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c + a^2 b^5 c - a b^6 c - 2 b^7 c + 3 a^6 c^2 + a^5 b c^2 - 6 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 3 a^2 b^4 c^2 + a b^5 c^2 + 2 a^5 c^3 + 2 a^4 b c^3 - 2 a^3 b^2 c^3 - 5 a^2 b^3 c^3 + 2 a b^4 c^3 + 2 b^5 c^3 - a^3 b c^4 + 3 a^2 b^2 c^4 + 2 a b^3 c^4 - a^3 c^5 + a^2 b c^5 + a b^2 c^5 + 2 b^3 c^5 - 3 a^2 c^6 - a b c^6 - 2 a c^7 - 2 b c^7) : :

X(27744) lies on these lines:

X(27745) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^4 b + a^3 b^2 - 2 a^2 b^3 - 2 a b^4 + a^4 c - a^3 b c - a b^3 c - 2 b^4 c + a^3 c^2 + b^3 c^2 - 2 a^2 c^3 - a b c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4) : :

X(27745) lies on these lines:

X(27746) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^5 b + a^4 b^2 - 2 a^2 b^4 - 2 a b^5 + a^5 c - a^4 b c - a b^4 c - 2 b^5 c + a^4 c^2 + b^4 c^2 - 2 a^2 c^4 - a b c^4 + b^2 c^4 - 2 a c^5 - 2 b c^5) : :

X(27746) lies on these lines:

X(27747) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (2 a^2 + a b - b^2 + a c + 4 b c - c^2) : :

X(27747) lies on these lines:

X(27748) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (2 a^4 b^2 + a^3 b^3 - a^2 b^4 + 3 a^2 b^3 c + 2 a^4 c^2 + 3 a b^3 c^2 - b^4 c^2 + a^3 c^3 + 3 a^2 b c^3 + 3 a b^2 c^3 + 4 b^3 c^3 - a^2 c^4 - b^2 c^4) : :

X(27748) lies on these lines:

X(27749) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^3 b^2 + a^2 b^3 - 2 a^3 b c + 3 a^2 b^2 c + 4 a b^3 c + a^3 c^2 + 3 a^2 b c^2 + b^3 c^2 + a^2 c^3 + 4 a b c^3 + b^2 c^3) : :

X(27749) lies on these lines:

X(27750) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^3 b^2 + a^2 b^3 - 6 a^3 b c + a^2 b^2 c + 6 a b^3 c + a^3 c^2 + a^2 b c^2 - 5 a b^2 c^2 + b^3 c^2 + a^2 c^3 + 6 a b c^3 + b^2 c^3) : :

X(27750) lies on these lines:

X(27751) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^2 - a b - 5 b^2 - a c + 8 b c - 5 c^2) : :

X(27751) lies on these lines:

X(27752) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (5 a^2 + a b - 4 b^2 + a c + 10 b c - 4 c^2) : :

X(27752) lies on these lines:

X(27753) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^4 + 4 a^3 b - 2 a b^3 - 2 b^4 + 4 a^3 c + 3 a^2 b c + b^3 c - 2 a c^3 + b c^3 - 2 c^4) : :

X(27753) lies on these lines:

X(27754) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (2 a^2 + a b - b^2 + a c + b c - c^2) : :

X(27754) lies on these lines:

X(27755) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (2 a^4 b^2 + a^3 b^3 - a^2 b^4 + 2 a^3 b^2 c + 2 a^2 b^3 c + 2 a^4 c^2 + 2 a^3 b c^2 - a b^3 c^2 - b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - b^2 c^4) : :

X(27755) lies on these lines:

X(27756) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^2 - 2 a b - 2 b^2 - 2 a c + b c - 2 c^2) : :

X(27756) lies on these lines:

X(27757) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^2 - b^2 + b c - c^2) : :

X(27757) lies on these lines:

X(27758) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (b - c) (a^3 b + 2 a^2 b^2 + a^3 c - a^2 b c + 2 a^2 c^2 - 2 b^2 c^2) : :

X(27758) lies on these lines:

X(27759) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^2 - 2 b^2 + 3 b c - 2 c^2) : :

X(27759) lies on these lines:

X(27760) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^2 + 3 a b - 2 b^2 + 3 a c + 6 b c - 2 c^2) : :

X(27760) lies on these lines:

X(27761) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^5 + 2 a^4 b - 4 a b^4 - 2 b^5 + 2 a^4 c - 3 a^3 b c - a^2 b^2 c + 3 a b^3 c - b^4 c - a^2 b c^2 + 2 a b^2 c^2 + 3 b^3 c^2 + 3 a b c^3 + 3 b^2 c^3 - 4 a c^4 - b c^4 - 2 c^5) : :

X(27761) lies on these lines:

X(27762) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a + b) (a - 2 b - 2 c) (a + c) (a^2 b + a b^2 - 2 b^3 + a^2 c - a b c + 2 b^2 c + a c^2 + 2 b c^2 - 2 c^3) : :

X(27762) lies on these lines:

X(27763) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^7 b + 3 a^6 b^2 + 2 a^5 b^3 - a^3 b^5 - 3 a^2 b^6 - 2 a b^7 + a^7 c - a^6 b c + a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c + a^2 b^5 c - a b^6 c - 2 b^7 c + 3 a^6 c^2 + a^5 b c^2 - 10 a^4 b^2 c^2 - 4 a^3 b^3 c^2 + 5 a^2 b^4 c^2 + a b^5 c^2 + 2 a^5 c^3 + 2 a^4 b c^3 - 4 a^3 b^2 c^3 - 10 a^2 b^3 c^3 + 2 a b^4 c^3 + 2 b^5 c^3 - a^3 b c^4 + 5 a^2 b^2 c^4 + 2 a b^3 c^4 - a^3 c^5 + a^2 b c^5 + a b^2 c^5+2 b^3 c^5 - 3 a^2 c^6 - a b c^6 - 2 a c^7 - 2 b c^7) : :

X(27763) lies on these lines:

X(27764) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a + b) (a - 2 b - 2 c) (a + c) (5 a^5 b + 2 a^4 b^2 - 4 a^3 b^3 + 2 a^2 b^4 - a b^5 - 4 b^6 + 5 a^5 c + a^4 b c - 4 a^3 b^2 c - 2 a^2 b^3 c - a b^4 c + b^5 c + 2 a^4 c^2 - 4 a^3 b c^2 - 8 a^2 b^2 c^2 + 2 a b^3 c^2 + 4 b^4 c^2 - 4 a^3 c^3 - 2 a^2 b c^3 + 2 a b^2 c^3 - 2 b^3 c^3 + 2 a^2 c^4 - a b c^4 + 4 b^2 c^4 - a c^5 + b c^5 - 4 c^6) : :

X(27764) lies on these lines:

X(27765) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a + b) (a - 2 b - 2 c) (a + c) (a^5 b + 2 a^4 b^2 - a b^5 - 2 b^6 + a^5 c + a^4 b c - 3 a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + 2 a^4 c^2 - 3 a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + 2 b^4 c^2 - a^2 b c^3 - a b^2 c^3 + 2 a b c^4 + 2 b^2 c^4 - a c^5 - 2 c^6) : :

X(27765) lies on these lines:

X(27766) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a + b) (a - 2 b - 2 c) (a + c) (3 a^5 b - 2 a^4 b^2 - 4 a^3 b^3 + 2 a^2 b^4 + a b^5 + 3 a^5 c - a^4 b c + 2 a^3 b^2 c - 5 a b^4 c + b^5 c - 2 a^4 c^2 + 2 a^3 b c^2 - 4 a^2 b^2 c^2 + 4 a b^3 c^2 - 4 a^3 c^3 + 4 a b^2 c^3 - 2 b^3 c^3 + 2 a^2 c^4 - 5 a b c^4 + a c^5 + b c^5) : :

X(27766) lies on these lines:

X(27767) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^5 + 4 a^4 b + 4 a^3 b^2 - 2 a^2 b^3 - 8 a b^4 - 2 b^5 + 4 a^4 c - 9 a^3 b c + a^2 b^2 c + 9 a b^3 c - 5 b^4 c + 4 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 + 7 b^3 c^2 - 2 a^2 c^3 + 9 a b c^3 + 7 b^2 c^3 - 8 a c^4 - 5 b c^4 - 2 c^5) : :

X(27767) lies on these lines:

X(27768) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (4 a^5 - 4 a^3 b^2 + 2 a^2 b^3 - 2 b^5 - 5 a^3 b c - 7 a^2 b^2 c + a b^3 c + 3 b^4 c - 4 a^3 c^2 - 7 a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 + a b c^3 - b^2 c^3 + 3 b c^4 - 2 c^5) : :

X(27768) lies on these lines:

X(27769) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^4 - a^3 b - a^2 b^2 - a b^3 - 2 b^4 - a^3 c + 2 b^3 c - a^2 c^2 - b^2 c^2 - a c^3 + 2 b c^3 - 2 c^4) : :

X(27769) lies on these lines:

X(27770) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^3 b^2 + a^2 b^3 - 2 a^3 b c - 3 a^2 b^2 c - 2 a b^3 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 - 2 a b c^3 + b^2 c^3) : :

X(27770) lies on these lines:

X(27771) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a + b) (a - 2 b - 2 c) (a + c) (a^3 b - 2 b^4 + a^3 c - 3 a^2 b c + b^3 c + 3 b^2 c^2 + b c^3 - 2 c^4) : :

X(27771) lies on these lines:

X(27772) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    b c (-a + 2 b + 2 c) (-2 a^2 + a b + a c + 3 b c) : :

X(27772) lies on these lines:

X(27773) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (b - c) (a^3 b + a^2 b^2 + a^3 c + a^2 c^2 - b^2 c^2) : :

X(27773) lies on these lines:

X(27774) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a-2 b-2 c) (b-c) (2 a^4-a^3 b-2 a^2 b^2+a b^3-a^3 c+7 a^2 b c+a b^2 c-b^3 c-2 a^2 c^2+a b c^2+4 b^2 c^2+a c^3-b c^3) : :

X(27774) lies on these lines:

X(27775) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (b - c) (a^3 b + 2 a^2 b^2 + a^3 c - 4 a^2 b c + 2 a^2 c^2 - 2 b^2 c^2) : :

X(27775) lies on these lines:

X(27776) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (a^2 + 2 a b + b^2 + 2 a c - b c + c^2) : :

X(27776) lies on these lines:

X(27777) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 70

Barycentrics    (a - 2 b - 2 c) (2 a^2 - b^2 + 3 b c - c^2) : :

X(27777) lies on these lines:

X(27778) =  REFLECTION OF X(11) IN X(17660)

Barycentrics    a (2 a^4 b-4 a^3 b^2+4 a b^4-2 b^5+2 a^4 c+2 a^2 b^2 c-5 a b^3 c+b^4 c-4 a^3 c^2+2 a^2 b c^2+2 a b^2 c^2+b^3 c^2-5 a b c^3+b^2 c^3+4 a c^4+b c^4-2 c^5) : :
X(27778) = 5 X[11] - 6 X[354], 5 X[100] - 3 X[4661], 9 X[354] - 10 X[5083], 3 X[11] - 4 X[5083], 5 X[1317] - 4 X[9957], 2 X[6797] - 3 X[11570], 3 X5903] - 5 X[11571], 6 X[9957] - 5 X[12758], 3 X[1317] - 2 X[12758], 3 X[354] - 5 X[17660], 2 X[5083] - 3 X[17660], 7 X[11] - 8 X[18240], 7 X[5083] - 6 X[18240], 7 X[17660] - 4 X[18240]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28619.

X(27778) lies on these lines: {11,118}, {55,13243}, {80,10404}, {100,4661}, {518,6154}, {952,5903}, {1317,2771}, {2800,12680}, {3614,12005}, {3874,12690}, {5432,13226}, {5904,9945}, {6797,11570}, {7672,11246}, {9803,12763}, {9809,13274}, {9946,14872}, {9964,10950}, {10864,13253}, {12675,12691}, {12750,16128}

X(27778) = reflection of X(i) in X(j) for these {i,j}: {11, 17660}, {5904, 9945}, {12690, 3874}, {12691, 12675}, {14872, 9946}

X(27779) = X(99)X(11593)∩X(115)X(373)

Barycentrics    a^2 (2 a^6 b^2-4 a^4 b^4+4 a^2 b^6-2 b^8+2 a^6 c^2+8 a^4 b^2 c^2-10 a^2 b^4 c^2-b^6 c^2-4 a^4 c^4-10 a^2 b^2 c^4+14 b^4 c^4+4 a^2 c^6-b^2 c^6-2 c^8) : :
X(27779) = 5 X[115] - 6 X[373], 5 X[2482] - 4 X[3819], 16 X[10219] - 15 X[14971], 2 X[12162] - 5 X[14981], 5 X[14928] - 2 X[17710]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28619.

X(27779) lies on these lines: {99,11593}, {115,373}, {543,3060}, {2482,3819}, {2936,8780}, {10219,14971}, {12162,14981}, {14928,17710}

X(27780) =  X(6)X(647)∩X(9)X(650)

Barycentrics    a^2 (a-b-c) (b-c) (a^5-2 a^4 b-2 a^3 b^2+4 a^2 b^3+a b^4-2 b^5-2 a^4 c+3 a^3 b c+a^2 b^2 c-3 a b^3 c+b^4 c-2 a^3 c^2+a^2 b c^2-2 a b^2 c^2+b^3 c^2+4 a^2 c^3-3 a b c^3+b^2 c^3+a c^4+b c^4-2 c^5) : :

See Aris Pavlakis and Peter Moses, Hyacinthos 28621.

X(27780) lies on these lines: {6,647}, {9,650}, {212,663}, {652,3217}, {654,3196}, {2423,6586}

X(27781) =  X(2)X(525)∩X(78)X(522)

Barycentrics    (a-b-c) (b-c) (a^5-2 a^4 b-2 a^3 b^2+4 a^2 b^3+a b^4-2 b^5-2 a^4 c+3 a^3 b c+a^2 b^2 c-3 a b^3 c+b^4 c-2 a^3 c^2+a^2 b c^2-2 a b^2 c^2+b^3 c^2+4 a^2 c^3-3 a b c^3+b^2 c^3+a c^4+b c^4-2 c^5) : :

See Aris Pavlakis and Peter Moses, Hyacinthos 28621.

X(27781) lies on these lines: {2,525}, {78,522}, {312,4391} ,{2401,25259}, {3239,21198}, {4130,25082}, {6332,6505}

leftri

Centers associated with Gemini triangles 11-18: X(27782)-X(27812)

rightri

These centers were contributed by Randy Hutson, November 12, 2018. Gemini triangles are introduced in the preamble just before X(24537).

X(27782) = CENTROID OF GEMINI TRIANGLE 11

Barycentrics    16 a^3 + 45 a^2 (b + c) + a (39 b^2 + 79 b c + 39 c^2) + 10 (b + c)^3 : :

X(27782) lies on the line {2, 3723}

X(27783) = CENTROID OF GEMINI TRIANGLE 12

Barycentrics    9 a^2 (b + c) + a (15 b^2 + 29 b c + 15 c^2) + 6 (b + c)^3 : :

X(27783) lies on these lines: {2, 319}, {191, 6175}

X(27784) = PERSPECTOR OF GEMINI TRIANGLE 11 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 11 AND 12

Barycentrics    a (a^2 (b + c) + 2 a (b^2 + 3 b c + c^2) + b^3 + 4 b^2 c + 4 b c^2 + c^3) : :
Trilinears    a^2 (b + c) + 2 a (b^2 + 3 b c + c^2) + b^3 + 4 b^2 c + 4 b c^2 + c^3 : :

X(27784) lies on these lines: {1, 748}, {2, 3743}, {3, 7611}, {10, 3706}, {37, 39}, {58, 1963}, {214, 10448}, {386, 2667}, {409, 4653}, {549, 8143}, {595, 1961}, {968, 25440}, {975, 5248}, {984, 3881}, {986, 3833}, {1698, 4868} et al

X(27784) = {X(2),X(27785)}-harmonic conjugate of X(3743)

X(27785) = {X(3743),X(27784)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a (a^2 (b + c) + a (2 b + c) (b + 2 c) + (b + c)^3) : :
Trilinears    a^2 (b + c) + a (2 b + c) (b + 2 c) + (b + c)^3 : :
Trilinears    (b + c) (a + b + c)^2 + a b c : :
Trilinears    4 s^2 (b + c) + a b c : :

X(27785) lies on these lines: {1, 6}, {2, 3743}, {3, 2941}, {35, 968}, {40, 25430}, {191, 940}, {386, 1962}, {595, 5311}, {612, 1995}, {1125, 26747}, {1224, 2345}, {1479, 7557}, {1698, 3931} et al

X(27786) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 11 AND 12

Barycentrics    1/((a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) (a^3 + a^2 (b + c) - a (b^2 + b c + c^2) - (b + c)^3)) : :

X(27786) lies on the line {1213, 1224}

X(27786) = isogonal conjugate of X(27787)
X(27786) = trilinear pole of line X(4988)X(8043)

X(27787) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 11 AND 12

Barycentrics    a^2 (a^2 + b^2 + c^2 + 2 a b + 2 a c + b c) (a^3 + a^2 (b + c) - a (b^2 + b c + c^2) - (b + c)^3) : :

X(27787) lies on these lines: {1, 1030}, {35, 42}, {55, 8185}, {100, 1224}, {678, 1283}, {1203, 4272}, {1724, 3795}, {1962, 3746} et al

X(27787) = isogonal conjugate of X(27786)
X(27787) = crossdifference of every pair of points on line X(4988)X(8043)

X(27788) = EIGENCENTER OF GEMINI TRIANGLE 11

Barycentrics    a (a^7 + 3 a^6 (b + c) + 3 a^5 (b + c)^2 + 2 a^4 b c (b + c) - a^3 (3 b^4 + 6 b^3 c + 8 b^2 c^2 + 6 b c^3 + 3 c^4) - a^2 (b + c) (3 b^4 + 5 b^3 c + 10 b^2 c^2 + 5 b c^3 + 3 c^4) - a (b^6 + 3 b^5 c + 8 b^4 c^2 + 13 b^3 c^3 + 8 b^2 c^4 + 3 b c^5 + c^6) - b^2 c^2 (b + c)^3) : :
Trilinears    a^7 + 3 a^6 (b + c) + 3 a^5 (b + c)^2 + 2 a^4 b c (b + c) - a^3 (3 b^4 + 6 b^3 c + 8 b^2 c^2 + 6 b c^3 + 3 c^4) - a^2 (b + c) (3 b^4 + 5 b^3 c + 10 b^2 c^2 + 5 b c^3 + 3 c^4) - a (b^6 + 3 b^5 c + 8 b^4 c^2 + 13 b^3 c^3 + 8 b^2 c^4 + 3 b c^5 + c^6) - b^2 c^2 (b + c)^3 : :

X(27788) lies on these lines: {75, 1030}, {4657, 5333}

X(27789) = ISOGONAL CONJUGATE OF X(16884)

Barycentrics    a/(3 a + 2 b + 2 c) : :
Trilinears    1/(3 a + 2 b + 2 c) : :

Let A11B11C11 be Gemini triangle 11. Let LA be the line through A11 parallel to BC, and define LB and LC cyclically. Let A'11 = LB∩LC, and define B'11 and C'11 cyclically. Triangle A'11B'11C'11' is homothetic to ABC at X(27789).

X(27789) lies on these lines: {1, 4134}, {2, 3723}, {37, 25417}, {57, 7269}, {81, 16777}, {330, 3995}, {959, 11011}, {961, 1388}, {1224, 3616}, {1255, 4383} et al

X(27789) = isogonal conjugate of X(16884)
X(27789) = anticomplement of X(28651)

X(27790) = CENTROID OF GEMINI TRIANGLE 13

Barycentrics    3 a^2 (b + c) + 2 a (3 b^2 + 8 b c + 3 c^2) + 3 (b + c) (b^2 + 3 b c + c^2) : :

X(27790) lies on these lines: {2, 594}, {10, 6534}, {5936, 19684}, {9782, 11024}

X(27791) = CENTROID OF GEMINI TRIANGLE 14

Barycentrics    3 a^2 (b + c) + a (5 b + c) (b + 5 c) + (b + c) (2 b^2 + 11 b c + 2 c^2) : :

X(27791) lies on these lines: {2, 3943}, {1647, 27812}, {4033, 24589}, {4080, 4688}, {4359, 4708}

X(27792) = PERSPECTOR OF GEMINI TRIANGLE 13 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 13 AND 14

Barycentrics    b c (b + c) (a^3 - a b^2 - a c^2 + 3 a b c + b^2 c + b c^2) : :

X(27792) lies on these lines: {2, 3770}, {75, 1211}, {76, 2051}, {85, 1228}, {226, 313}, {306, 4043}, {312, 1230}, {321, 4033}, {341, 442}, {1022, 1577}, {1269, 3687} et al

X(27792) = {X(2),X(27794)}-harmonic conjugate of X(27793)

X(27793) = PERSPECTOR OF GEMINI TRIANGLE 14 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 13 AND 14

Barycentrics    b c (b + c) (2 a^3 + a^2 b + a^2 c - a b^2 - a c^2 + 4 a b c + b^2 c + b c^2) : :

X(27793) lies on these lines: {2, 3770}, {226, 306}, {312, 26738}, {442, 4696}, {1086, 1211}, {1230, 4358} et al

X(27793) = {X(2),X(27794)}-harmonic conjugate of X(27792)

X(27794) = {X(27792),X(27793)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    b c (b + c) (3 a^3 + a^2 b + a^2 c - 2 a b^2 - 2 a c^2 + 7 a b c + 2 b^2 c + 2 b c^2) : :

X(27794) lies on these lines: {2, 3770}, {1211, 7263}, {3936, 4671}, {3963, 4080} et al

X(27794) = {X(27792),X(27793)}-harmonic conjugate of X(2)

X(27795) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 13 AND 14

Barycentrics    b c/(3 a^4 (b + c) + a^3 (b - 3 c) (3 b - c) - a^2 (b + c) (3 b^2 - b c + 3 c^2) - a (3 b^4 + 3 c^4 - 2 b c (4 b^2 - b c + 4 c^2)) - 3 b c (b - c)^2 (b + c)) : :

The perspectrix of Gemini triangles 13 and 14 passes through X(4768).

X(27795) lies on these lines: (pending)

X(27795) = isogonal conjugate of X(27796)

X(27796) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 13 AND 14

Barycentrics    a^3 (3 a^4 (b + c) + a^3 (b - 3 c) (3 b - c) - a^2 (b + c) (3 b^2 - b c + 3 c^2) - a (3 b^4 + 3 c^4 - 2 b c (4 b^2 - b c + 4 c^2)) - 3 b c (b - c)^2 (b + c)) : :

X(27796) lies on these lines: {101, 2268}, {604, 1415}

X(27796) = isogonal conjugate of X(27795)

X(27797) = ISOTOMIC CONJUGATE OF X(26860)

Barycentrics    (b + c)/(4 a + b + c) : :

Let A14B14C14 be Gemini triangle 14. Let LA be the line through A14 parallel to BC, and define LB and LC cyclically. Let A'14 = LB∩LC, and define B'14 and C'14 cyclically. Triangle A'14B'14C'14 is homothetic to ABC at X(27797).

X(27797) lies on these lines: {2, 3943}, {4, 4678}, {594, 4080}, {4024, 4049} et al

X(27797) = isotomic conjugate of X(26860)

X(27798) = CENTROID OF GEMINI TRIANGLE 16

Barycentrics    (b + c) (a^2 + 2 a b + 2 a c + 3 b c) : :

X(27798) lies on these lines: {1, 14007}, {2, 740}, {10, 12}, {42, 4457}, {75, 17038}, {312, 1698}, {321, 3842}, {333, 4697}, {512, 27799}, {523, 21204}, {537, 4981}, {714, 4688}, {804, 4763}, {812, 9148}, {966, 4703}, {982, 3728}, {1010, 5429}, {1125, 3706}, {1213, 3985}, {1268, 20947} et al

X(27798) = midpoint of X(i) and X(j) for these {i,j}: {2, 21020}, {1962, 17163}
X(27798) = reflection of X(10180) in X(2)
X(27798) = complement of X(1962)

X(27799) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 15 AND 16

Barycentrics    (b^2 - c^2) (3 a^4 - a^3 (b + c) - 2 a^2 b c - a b c (b + c) + 3 b^2 c^2) : :

X(27799) lies on these lines: {2, 27806}, {512, 27798}, {523, 10180}, {3740, 4132}

X(27799) = complement of complement of X(27806)

X(27800) = CENTER OF THE {GEMINI 15, GEMINI 16}-CIRCUMCONIC

Barycentrics    a^4 (b - c)^2 - a^3 (b + c)^3 - 2 a^2 b c (3 b^2 + 2 b c + 3 c^2) - a b c (b + c)^3 + b^2 c^2 (b - c)^2 : :

X(27800) lies on these lines: {2, 27807}, {3005, 4369}, {3925, 17761}

X(27800) = complement of complement of X(27807)

X(27801) = TRILINEAR PRODUCT OF VERTICES OF GEMINI TRIANGLE 16

Barycentrics    (b + c)/a^3 : :

X(27801) lies on these lines: {1, 4485}, {10, 1237}, {37, 308}, {72, 290}, {75, 3670}, {76, 321}, {100, 2367}, {213, 3114}, {264, 1969}, {274, 1920}, {276, 3998}, {313, 1089}, {349, 6458}, {350, 2901}, {700, 2085}, {1502, 1928}, {1978, 18359}, {2205, 3115} et al

X(27801) = isotomic conjugate of X(1333)
X(27801) = polar conjugate of X(2203)
X(27801) = trilinear pole of line X(850)X(4036)
X(27801) = trilinear product of vertices of Gemini triangle 13
X(27801) = trilinear product of vertices of Gemini triangle 14
X(27801) = trilinear product of vertices of Gemini triangle 20

X(27802) = EIGENCENTER OF GEMINI TRIANGLE 15

Barycentrics    a^2 (a^5 + a^4 (b + c) + 2 a^3 b c + 2 a^2 b c (b + c) - a (b + c)^2 (b^2 - 4 b c + c^2) - b^5 + b^4 c + 4 b^3 c^2 + 4 b^2 c^3 + b c^4 - c^5) : :

The Gemini triangle 15 is also the Gergonne line extraversion triangle (see X(10180)), and its unary cofactor triangle is the extraversion triangle of X(55). These two triangles are perspective at X(27802).

X(27802) lies on these lines: {1, 25}, {3, 37}, {8, 4239}, {55, 2915}, {56, 226}, {101, 386}, {108, 388}, {197, 3931}, {219, 10974}, {404, 17776}, {429, 1478}, {612, 8193}, {936, 17742}, {958, 4205}, {999, 1104}, {1376, 3695}, {1460, 12514}, {1486, 5266}, {1754, 26935}, {1791, 13725}, {1995, 5262} et al

X(27803) = EIGENCENTER OF GEMINI TRIANGLE 16

Barycentrics    a (a^6 (b - c)^2 (b + c) + a^5 b c (b - c)^2 + a^4 (b + c) (b^4 - 2 b^3 c + b^2 c^2 - 2 b c^3 + c^4) - a^3 b c (3 b^4 + 2 b^3 c - b^2 c^2 + 2 b c^3 + 3 c^4) - a^2 b^2 c^2 (b + c) (b^2 + c^2) - a b^3 c^3 (b^2 + c^2) - 4 b^4 c^4 (b + c)) : :
Trilinears    a^6 (b - c)^2 (b + c) + a^5 b c (b - c)^2 + a^4 (b + c) (b^4 - 2 b^3 c + b^2 c^2 - 2 b c^3 + c^4) - a^3 b c (3 b^4 + 2 b^3 c - b^2 c^2 + 2 b c^3 + 3 c^4) - a^2 b^2 c^2 (b + c) (b^2 + c^2) - a b^3 c^3 (b^2 + c^2) - 4 b^4 c^4 (b + c) : :

X(27803) lies on these lines: {1376, 3053}, {2176, 4362}

X(27804) = CENTROID OF GEMINI TRIANGLE 17

Barycentrics    (b + c) (3 a^2 + a b + a c - b c) : :

Let A'B'C' be the incentral triangle. Let A" be the reflection of A in A', and define B" and C" cyclically. X(27804) is the centroid of A"B"C".

X(27804) lies on these lines: {1, 596}, {2, 740}, {8, 3743}, {10, 8040}, {37, 3896}, {42, 3952}, {81, 4427}, {99, 6628}, {145, 2292}, {171, 4781}, {192, 714}, {306, 4356}, {519, 3989}, {758, 3241}, {846, 16704}, {968, 3187}, {984, 20011}, {1215, 21806} et al

X(27804) = reflection of X(i) in X(j) for these (i,j): (2, 1962), (17163, 2), (21020, 10180)
X(27804) = anticomplement of X(21020)

X(27805) = TRILINEAR POLE OF LINE X(8)X(192)

Barycentrics    1/((b - c)(a^2 + b c)) : :

Line X(8)X(192) is the line of the (degenerate) cross-triangle of Gemini triangles 17 and 18.

X(27805) lies on these lines: {2, 694}, {43, 4154}, {257, 4997}, {644, 4621}, {645, 3570}, {646, 3807}, {661, 799}, {893, 6651}, {1581, 24003}, {3452, 7018}, {3699, 3799}, {3888, 4598} et al

X(27805) = isogonal conjugate of X(20981)
X(27805) = isotomic conjugate of X(4369)
X(27805) = trilinear pole of line X(8)X(192)
X(27805) = X(19)-isoconjugate of X(22093)

X(27806) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 17 AND 18

Barycentrics    (b^2 - c^2) (3 a^4 - 2 a^3 (b + c) + a^2 (3 b^2 - 4 b c + 3 c^2) - 2 a b c (b + c) + 3 b^2 c^2) : :

X(27806) lies on these lines: {2, 27799}, {875, 3112}, {3681, 4132}

X(27806) = anticomplement of anticomplement of X(27799)

X(27807) = CENTER OF THE {GEMINI 17, GEMINI 18}-CIRCUMCONIC

Barycentrics    (a^2 - b c)^2 (b - c)^2 - a^2 (b + c)^4 : :

X(27807) is also the perspector of the {Gemini 17, Gemini 18}-circumconic.

Let A17B17C17 and A18B18C18 be the Gemini triangles 17 and 18, resp. Let A' be the intersection of the tangents to the {Gemini 17, Gemini 18}-circumconic at A17 and A18. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(27807). (Randy Hutson, November 30, 2018)

X(27807) lies on these lines: {2, 27800}, {38, 7192}, {239, 3294}, {350, 4651}, {870, 17165}, {873, 4576}, {1447, 3920}, {1621, 4068} et al

X(27807) = anticomplement of anticomplement of X(27800)

X(27808) = ISOTOMIC CONJUGATE OF X(3733)

Barycentrics    b^2 c^2 (b + c)/(b - c) : :

X(27808) lies on these lines: {75, 21208}, {76, 334}, {99, 8707}, {100, 839}, {313, 4013}, {321, 3125}, {646, 2397}, {668, 891}, {670, 6540}, {756, 6382}, {874, 4557}, {1016, 4574} et al

X(27808) = isotomic conjugate of X(3733)
X(27808) = trilinear product of vertices of Gemini triangle 17
X(27808) = trilinear product of vertices of Gemini triangle 18

X(27809) = EIGENCENTER OF GEMINI TRIANGLE 17

Barycentrics    (b + c)/(a (b^2 + c^2) - b c (b + c)) : :

X(27809) lies on these lines: {2, 1978}, {6, 190}, {25, 1897}, {37, 4033}, {42, 3952}, {111, 8709}, {145, 263}, {321, 3121}, {335, 812}, {694, 6542}, {726, 23579}, {727, 835}, {941, 4704}, {1171, 4632}, {1400, 4552} et al

X(27810) = EIGENCENTER OF GEMINI TRIANGLE 18

Barycentrics    a (b + c)/(a (b^3 + c^3) - b c (b^2 + c^2)) : :
Trilinears    (b + c)/(a (b^3 + c^3) - b c (b^2 + c^2)) : :

X(27810) lies on these lines: {6, 668}, {32, 100}, {213, 3952}, {1018, 1918}, {1783, 1974} et al

X(27811) = HOMOTHETIC CENTER OF GEMINI TRIANGLE 15 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 15 AND 17

Barycentrics    (b + c) (5 a^2 + 3 a b + 3 a c + b c) : :

X(27811) lies on these lines: {1, 16704}, {2, 740}, {37, 3121}, {86, 4427}, {100, 4068}, {758, 15672}, {846, 8025}, {2292, 3622} et al

X(27811) = reflection of X(27812) in X(2)

X(27812) = HOMOTHETIC CENTER OF GEMINI TRIANGLE 16 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 16 AND 18

Barycentrics    (b + c) (a^2 + 3 a b + 3 a c + 5 b c) : :

X(27812) lies on these lines: {2, 740}, {10, 3120}, {523, 6548}, {1213, 4442}, {1647, 27791}, {1654, 17491}, {2292, 9330} et al

X(27812) = reflection of X(27811) in X(2)

leftri

Collineation mappings involving Gemini triangle 71: X(27813)-X(27837)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 71, as in centers X(27813)-X(27837). Then

m(X) = a(a-3b+c)(a+b-3c)x + (a-c)(a-3b+c)(a+b-3c)y + (a-b)(a-3b+c)(a+b-3c)z : :

(Clark Kimberling, November 13, 2018)

X(27813) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^2 + a b + a c - 2 b c) : :

X(27813) lies on these lines:

X(27814) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a + b - c) (a - 3 b + c) (a - b + c) (a^3 - a b^2 - 2 a b c + b^2 c - a c^2 + b c^2) : :

X(27814) lies on these lines: {2, 27815}, {348, 27818}, {26563, 27813}, {27817, 27822}

X(27815) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a+b-3 c) (a+b-c) (a-3 b+c) (a-b+c) (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c+b^2 c-a c^2+b c^2+c^3) : :

X(27815) lies on these lines:

X(27816) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a + b - c) (a - 3 b + c) (a - b + c) (2 a^3 - a^2 b - 2 a b^2 + b^3 - a^2 c - 4 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2 + c^3) : :

X(27816) lies on these lines:

X(27817) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a+b-3 c) (a-3 b+c) (a^3+a b^2-b^2 c+a c^2-b c^2) : :

X(27817) lies on these lines:

X(27818) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a + b - c) (a - 3 b + c) (a - b + c) : :

X(27818) lies on the cubics K1011 and K1069 and these lines: {2, 16078}, {7, 145}, {8, 1358}, {85, 5226}, {279, 3008}, {348, 27814}, {673, 63626}, {738, 37789}, {1222, 16079}, {1293, 2369}, {1434, 16711}, {3160, 3445}, {3161, 56081}, {3177, 56719}, {3212, 56174}, {3241, 24796}, {3616, 24805}, {3617, 62575}, {3621, 63591}, {3663, 7320}, {3665, 52715}, {3672, 47636}, {3674, 4052}, {4051, 53538}, {4308, 7195}, {4323, 32086}, {5222, 51839}, {6172, 27834}, {6556, 7185}, {7209, 27496}, {7233, 17090}, {8051, 24175}, {10481, 10563}, {10509, 60941}, {17951, 35578}, {18230, 27819}, {23062, 45202}, {24798, 53620}, {27816, 52422}, {27826, 27827}, {36621, 64114}, {36638, 36640}, {40154, 64146}, {40617, 62403}, {40621, 62525}, {41527, 60789}, {42318, 43760}, {44301, 53645}, {56049, 56938}, {58793, 58816}, {60831, 62786}

X(27818) = reflection of X(62525) in X(40621)
X(27818) = isotomic conjugate of X(3161)
X(27818) = anticomplement of X(63620)
X(27818) = antitomic conjugate of X(62525)
X(27818) = isotomic conjugate of the anticomplement of X(4859)
X(27818) = isotomic conjugate of the complement of X(4373)
X(27818) = isotomic conjugate of the isogonal conjugate of X(40151)
X(27818) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 27828}, {62528, 4373}
X(27818) = X(i)-cross conjugate of X(j) for these (i,j): {2, 7}, {514, 53647}, {3663, 1088}, {3731, 56348}, {4373, 16078}, {4859, 2}, {8056, 4373}, {9311, 43750}, {23681, 1440}, {24175, 75}, {24177, 273}, {24199, 7249}, {24778, 8049}, {40617, 3676}, {45202, 3680}, {47444, 279}, {58794, 65173}, {63621, 36606}
X(27818) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3158}, {9, 3052}, {31, 3161}, {32, 44720}, {33, 20818}, {41, 145}, {55, 1743}, {56, 4936}, {101, 4162}, {163, 44729}, {210, 33628}, {213, 52352}, {220, 1420}, {284, 4849}, {560, 44723}, {604, 6555}, {607, 4855}, {644, 8643}, {663, 57192}, {667, 30720}, {692, 4521}, {1110, 4534}, {1253, 5435}, {1334, 16948}, {1415, 4546}, {1973, 44722}, {2149, 4953}, {2175, 18743}, {2194, 3950}, {2204, 52354}, {3063, 43290}, {3939, 4394}, {4248, 52370}, {4729, 5546}, {4939, 23990}, {4943, 34080}, {6602, 62787}, {9247, 44721}, {9406, 44727}, {14321, 65375}, {14827, 39126}, {15519, 38266}, {52353, 57657}
X(27818) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 4936}, {2, 3161}, {9, 3158}, {115, 44729}, {223, 1743}, {478, 3052}, {514, 4534}, {650, 4953}, {1015, 4162}, {1086, 4521}, {1146, 4546}, {1214, 3950}, {3160, 145}, {3161, 6555}, {3669, 40621}, {5976, 44728}, {6337, 44722}, {6374, 44723}, {6376, 44720}, {6626, 52352}, {6631, 30720}, {9410, 44727}, {10001, 43290}, {17113, 5435}, {19604, 46946}, {24151, 9}, {30471, 44725}, {30472, 44726}, {36905, 4899}, {40590, 4849}, {40593, 18743}, {40615, 3667}, {40617, 4394}, {40621, 4943}, {40622, 14321}, {56846, 4856}, {59507, 12640}, {59608, 4848}, {62565, 52354}, {62570, 52353}, {62575, 8}, {62576, 44721}
X(27818) = cevapoint of X(i) and X(j) for these (i,j): {2, 4373}, {7, 64114}, {514, 1358}, {650, 4014}, {3676, 40617}, {8056, 19604}
X(27818) = trilinear pole of line {3667, 3676}
X(27818) = barycentric product X(i)*X(j) for these {i,j}: {1, 62528}, {7, 4373}, {57, 40014}, {75, 19604}, {76, 40151}, {85, 8056}, {92, 27832}, {145, 16078}, {279, 6557}, {479, 6556}, {561, 16945}, {655, 27836}, {673, 10029}, {693, 65173}, {1088, 3680}, {1293, 52621}, {1434, 4052}, {3261, 38828}, {3445, 6063}, {3676, 53647}, {4554, 58794}, {8051, 27828}, {9311, 27829}, {15474, 27815}, {20567, 38266}, {24002, 27834}, {27813, 42304}, {27826, 64240}, {31343, 59941}, {32017, 45205}, {40617, 57578}, {56174, 57785}
X(27818) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3158}, {2, 3161}, {7, 145}, {8, 6555}, {9, 4936}, {11, 4953}, {56, 3052}, {57, 1743}, {65, 4849}, {69, 44722}, {75, 44720}, {76, 44723}, {77, 4855}, {85, 18743}, {86, 52352}, {145, 15519}, {190, 30720}, {222, 20818}, {226, 3950}, {264, 44721}, {269, 1420}, {279, 5435}, {298, 44725}, {299, 44726}, {307, 52354}, {325, 44728}, {479, 62787}, {513, 4162}, {514, 4521}, {522, 4546}, {523, 44729}, {553, 4856}, {651, 57192}, {664, 43290}, {1014, 16948}, {1086, 4534}, {1088, 39126}, {1111, 4939}, {1122, 45219}, {1293, 3939}, {1358, 3756}, {1412, 33628}, {1434, 41629}, {1440, 56940}, {1441, 52353}, {1443, 4881}, {1494, 44727}, {2415, 30731}, {3445, 55}, {3663, 12640}, {3665, 4884}, {3667, 4943}, {3668, 4848}, {3669, 4394}, {3676, 3667}, {3680, 200}, {4017, 4729}, {4052, 2321}, {4059, 4891}, {4077, 4404}, {4373, 8}, {4654, 4898}, {4859, 63620}, {4998, 44724}, {5274, 63624}, {6556, 5423}, {6557, 346}, {7178, 14321}, {7190, 4917}, {7209, 27496}, {8051, 24150}, {8056, 9}, {9436, 4899}, {10029, 3912}, {10563, 62218}, {16078, 4373}, {16079, 3445}, {16945, 31}, {19604, 1}, {22464, 51433}, {24002, 4462}, {27813, 30568}, {27815, 17776}, {27819, 55337}, {27820, 56078}, {27824, 56079}, {27827, 25082}, {27828, 8055}, {27829, 3729}, {27831, 4529}, {27832, 63}, {27834, 644}, {27835, 19582}, {27836, 3904}, {30617, 4952}, {30719, 31182}, {30725, 14425}, {31343, 4578}, {36838, 62532}, {38266, 41}, {38828, 101}, {40014, 312}, {40151, 6}, {40617, 40621}, {41003, 4918}, {43035, 53579}, {43041, 53580}, {43042, 4925}, {43924, 8643}, {43932, 51656}, {45205, 3752}, {46367, 1201}, {47444, 63621}, {47636, 2136}, {51839, 2348}, {52563, 45204}, {53545, 21950}, {53647, 3699}, {56174, 210}, {58794, 650}, {58817, 30719}, {62528, 75}, {62543, 4012}, {62780, 64736}, {62787, 6049}, {65173, 100}, {65337, 65160}
X(27818) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 27828, 27813}, {8, 1358, 63577}, {85, 62528, 40014}, {145, 63574, 7}, {4373, 19604, 7}, {27813, 27820, 2}, {43983, 52563, 7}
X(27818) = pole of line {4862, 5274} with respect to the ABCGGe
X(27818) = pole of line {3158, 4936} with respect to the Jerabek circumhyperbola of the excentral triangle
X(27818) = pole of line {3161, 44722} with respect to the Kiepert circumhyperbola of the anticomplementary triangle
X(27818) = pole of line {3676, 4943} with respect to the Steiner circumellipse

X(27819) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    a (a + b - 3 c) (a - 3 b + c) (a^2 - 2 a b + b^2 - 2 a c + c^2) : :

X(27819) lies on these lines:

X(27820) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (2 a^2 + a b - b^2 + a c - c^2) : :

X(27820) lies on these lines:

X(27821) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^4 + a b^3 - b^3 c + a c^3 - b c^3) : :

X(27821) lies on these lines:

X(27822) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^5 + a b^4 - b^4 c + a c^4 - b c^4) : :

X(27822) lies on these lines:

X(27823) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a+b-3 c) (a-3 b+c) (b+c) (2 a^2-b c) : :

X(27823) lies on these lines:

X(27824) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^3 b + a^2 b^2 + a^3 c + a^2 c^2 - 2 b^2 c^2) : :

X(27824) lies on these lines:

X(27825) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^3 - 4 a^2 b + a b^2 - 4 a^2 c + b^2 c + a c^2 + b c^2) : :

X(27825) lies on these lines:

X(27826) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (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(27826) lies on these lines:

X(27827) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    a (a + b - 3 c) (a - 3 b + c) (a b - b^2 + a c + b c - c^2) : :

X(27827) lies on these lines:

X(27828) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^2 + 2 a b + b^2 + 2 a c - 6 b c + c^2) : :

X(27828) lies on these lines:

X(27829) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a + b - c) (a - 3 b + c) (a - b + c) (a^2 - a b - a c + 2 b c) : :

X(27829) lies on these lines:

X(27830) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^3 - a^2 b + 2 a b^2 - a^2 c - a b c - b^2 c + 2 a c^2 - b c^2) : :

X(27830) lies on these lines:

X(27831) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - b - c) (b - c) (a - 3 b + c) (a^2 + b c) : :

X(27831) lies on these lines:

X(27832) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    a (a + b - 3 c) (a + b - c) (a - 3 b + c) (a - b + c) (a^2 - b^2 - c^2) : :

X(27832) lies on these lines:

X(27833) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a-b) (a+b-3 c) (a-c) (a-3 b+c) (a^3-b^3+b^2 c+b c^2-c^3) : :

X(27833) lies on these lines:

X(27834) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    a (a - b) (a + b - 3 c) (a - c) (a - 3 b + c) : :

X(27834) lies on these lines:

X(27834) = isogonal conjugate of X(4394)
X(27834) = isotomic conjugate of X(4462)

X(27835) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (a - 3 b + c) (a^2 b + a b^2 + a^2 c - a b c - b^2 c + a c^2 - b c^2) : :

X(27835) lies on these lines:

X(27836) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    (a + b - 3 c) (b - c) (a - 3 b + c) (a^2 - b^2 + b c - c^2) : :

X(27836) lies on these lines:

X(27837) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 71

Barycentrics    a (a + b - 3 c) (b - c) (a - 3 b + c) (a^2 - a b - a c + 2 b c) : :

X(27837) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 72: X(27838)-X(27865)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 72, as in centers X(27838)-X(27865). Then

m(X) = bc(a^2-bc)^2x + b^2(a^-bc)(c^2-ab)y + c^2(a^2-bc)(b^2-ac)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 13, 2018)

X(27838) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^4 + a^3 b c - a b^2 c^2 + b^3 c^2 + b^2 c^3 - a c^4) : :

X(27838) lies on these lines:

X(27839) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^3 b^5 + a b^7 - a^6 b c + a^4 b^3 c + a^4 b^2 c^2 - a b^5 c^2 - b^6 c^2 + a^4 b c^3 + 2 b^4 c^4 - a^3 c^5 - a b^2 c^5 - b^2 c^6 + a c^7) : :

X(27839) lies on these lines:

X(27840) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^3 + a b^7 - a^6 b c + a^2 b^5 c + 3 a^4 b^2 c^2 + 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 - b^6 c^2 - a^5 c^3 + 2 a^3 b^2 c^3 - 2 a^2 b^3 c^3 - a b^4 c^3 - 2 a^2 b^2 c^4 - a b^3 c^4 + 2 b^4 c^4 + a^2 b c^5 - b^2 c^6 + a c^7) : :

X(27840) lies on these lines:

X(27841) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    a (a^2 - b c) (a^4 b^3 - a^2 b^5 + a^3 b^3 c - a b^5 c - 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + 2 a b^4 c^2 - b^5 c^2 + a^4 c^3 + a^3 b c^3 - 2 a^2 b^2 c^3 + 2 a b^3 c^3 + b^4 c^3 + 2 a b^2 c^4 + b^3 c^4 - a^2 c^5 - a b c^5 - b^2 c^5) : :

X(27841) lies on these lines:

X(27842) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^5 + a^4 b c - a^2 b^2 c^2 + b^4 c^2 + b^2 c^4 - a c^5) : :

X(27842) lies on these lines:

X(27843) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^3 b^3 - a b^5 + a^4 b c - 3 a^2 b^3 c - a^2 b^2 c^2 + 3 a b^3 c^2 + b^4 c^2 + a^3 c^3 - 3 a^2 b c^3 + 3 a b^2 c^3 - 2 b^3 c^3 + b^2 c^4 - a c^5) : :

X(27843) lies on these lines:

X(27844) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^2 b^3 - a b^4 + a^3 b c - a^2 b^2 c + a b^3 c - a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 - a c^4) : :

X(27844) lies on these lines:

X(27845) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^2 b^4 + a b^5 - a^4 b c + a^3 b^2 c - a b^4 c + a^3 b c^2 + a^2 b^2 c^2 - b^4 c^2 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - b^2 c^4 + a c^5) : :

X(27845) lies on these lines:

X(27846) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    a (b - c)^2 (a^2 - b c) : :

X(27846) lies on these lines: X(27846) = isogonal conjugate of X(5378)
X(27846) = crossdifference of every pair of points on line X(100)X(649) (the tangent at X(100) to hyperbola {{A,B,C,X(81),X(100),PU(8)})}

X(27847) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^5 + a b^9 - a^8 b c + a^4 b^5 c + a^6 b^2 c^2 + a^4 b^4 c^2 - a^2 b^6 c^2 - b^8 c^2 + a^4 b^2 c^4 - a b^5 c^4 + b^6 c^4 - a^5 c^5 + a^4 b c^5 - a b^4 c^5 - a^2 b^2 c^6 + b^4 c^6 - b^2 c^8 + a c^9) : :

X(27847) lies on these lines:

X(27848) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^5 + a b^9 - a^8 b c + a^4 b^5 c + a^6 b^2 c^2 + a^4 b^4 c^2 + a^3 b^5 c^2 - a^2 b^6 c^2 - b^8 c^2 - a^4 b^3 c^3 + a^4 b^2 c^4 - a^2 b^4 c^4 - a b^5 c^4 + b^6 c^4 - a^5 c^5 + a^4 b c^5 + a^3 b^2 c^5 - a b^4 c^5 - a^2 b^2 c^6 + b^4 c^6 - b^2 c^8 + a c^9) : :

X(27848) lies on these lines:

X(27849) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^6 + a^5 b c - a^3 b^2 c^2 + b^5 c^2 + b^2 c^5 - a c^6) : :

X(27849) lies on these lines:

X(27850) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^7 + a^6 b c - a^4 b^2 c^2 + b^6 c^2 + b^2 c^6 - a c^7) : :

X(27850) lies on these lines:

X(27851) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^2 b^4 + a^3 b^2 c - a b^4 c + a^3 b c^2 + 2 b^3 c^3 - a^2 c^4 - a b c^4) : :

X(27851) lies on these lines:

X(27852) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^3 b^5 - a^4 b^3 c + a b^5 c^2 - a^4 b c^3 - 2 b^4 c^4 + a^3 c^5 + a b^2 c^5) : :

X(27852) lies on these lines:

X(27853) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    b^2 c^2 (a - b) (a - c) (a^2 - b c) : :

X(27853) lies on these lines:

X(27853) = isotomic conjugate of X(3572)

X(27854) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (b - c) (a^2 - b c) (-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(27854) lies on these lines:

X(27855) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    b c (b - c) (a^2 - b c)^2 : :

X(27855) lies on these lines:

X(27856) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^6 + a^5 b c + a^2 b^4 c - 2 a^3 b^2 c^2 + b^5 c^2 - a b^3 c^3 + a^2 b c^4 + b^2 c^5 - a c^6) : :

X(27856) lies on these lines:

X(27857) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (-a^2 + b c) (-a^5 b^3 - 2 a^3 b^5 + 3 a b^7 - 3 a^6 b c + 2 a^4 b^3 c + a^2 b^5 c + 5 a^4 b^2 c^2 + 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 - 2 a b^5 c^2 - 3 b^6 c^2 - a^5 c^3 + 2 a^4 b c^3 + 2 a^3 b^2 c^3 - 2 a^2 b^3 c^3 - a b^4 c^3 - 2 a^2 b^2 c^4 - a b^3 c^4 + 6 b^4 c^4 - 2 a^3 c^5 + a^2 b c^5 - 2 a b^2 c^5 - 3 b^2 c^6 + 3 a c^7) : :

X(27857) lies on these lines:

X(27858) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^3 b^5 - a b^7 + a^6 b c - a^4 b^3 c + a^3 b^4 c + a^2 b^5 c - 2 a^4 b^2 c^2 - a^3 b^3 c^2 + a^2 b^4 c^2 + a b^5 c^2 + b^6 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^3 b c^4 + a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 + a^2 b c^5 + a b^2 c^5 + b^2 c^6 - a c^7) : :

X(27858) lies on these lines:

X(27859) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^5 + a b^9 - a^8 b c + a^4 b^5 c + a^6 b^2 c^2 + a^4 b^4 c^2 + 2 a^3 b^5 c^2 - a^2 b^6 c^2 - b^8 c^2 - 2 a^4 b^3 c^3 + a^4 b^2 c^4 - 2 a^2 b^4 c^4 - a b^5 c^4 + b^6 c^4 - a^5 c^5 + a^4 b c^5 + 2 a^3 b^2 c^5 - a b^4 c^5 - a^2 b^2 c^6 + b^4 c^6 - b^2 c^8 + a c^9) : :

X(27859) lies on these lines:

X(27860) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^6 b^4 + a^5 b^5 - a^2 b^8 - a b^9 + a^8 b c + a^7 b^2 c + a^6 b^3 c + a^5 b^4 c - a^4 b^5 c - a^3 b^6 c - a^2 b^7 c - a b^8 c + a^7 b c^2 - 3 a^5 b^3 c^2 - 3 a^4 b^4 c^2 - a^3 b^5 c^2 + a b^7 c^2 + b^8 c^2 + a^6 b c^3 - 3 a^5 b^2 c^3 - 5 a^4 b^3 c^3 + 2 a^3 b^4 c^3 + 3 a^2 b^5 c^3 + a b^6 c^3 + b^7 c^3 + a^6 c^4 + a^5 b c^4 - 3 a^4 b^2 c^4 + 2 a^3 b^3 c^4 + 6 a^2 b^4 c^4 - b^6 c^4 + a^5 c^5 - a^4 b c^5 - a^3 b^2 c^5 + 3 a^2 b^3 c^5 - 2 b^5 c^5 - a^3 b c^6 + a b^3 c^6 - b^4 c^6 - a^2 b c^7 + a b^2 c^7 + b^3 c^7 - a^2 c^8 - a b c^8 + b^2 c^8 - a c^9) : :

X(27860) lies on these lines:

X(27861) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (a^6 b^5 + a^5 b^6 - a^2 b^9 - a b^10 + a^9 b c + a^8 b^2 c + a^6 b^4 c - a^4 b^6 c - a^2 b^8 c - a b^9 c + a^8 b c^2 - a^6 b^3 c^2 - a^5 b^4 c^2 - 4 a^4 b^5 c^2 - 2 a^3 b^6 c^2 + a^2 b^7 c^2 + a b^8 c^2 + b^9 c^2 - a^6 b^2 c^3 - a^5 b^3 c^3 - a^4 b^4 c^3 + a^2 b^6 c^3 + a b^7 c^3 + b^8 c^3 + a^6 b c^4 - a^5 b^2 c^4 - a^4 b^3 c^4 + 4 a^3 b^4 c^4 + 4 a^2 b^5 c^4 - b^7 c^4 + a^6 c^5 - 4 a^4 b^2 c^5 + 4 a^2 b^4 c^5 - b^6 c^5 + a^5 c^6 - a^4 b c^6 - 2 a^3 b^2 c^6 + a^2 b^3 c^6 - b^5 c^6 + a^2 b^2 c^7 + a b^3 c^7 - b^4 c^7 - a^2 b c^8 + a b^2 c^8 + b^3 c^8 - a^2 c^9 - a b c^9 + b^2 c^9 - a c^10) : :

X(27861) lies on these lines:

X(27862) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^7 b^4 + a^5 b^6 + a^3 b^8 - a b^10 + a^9 b c - 2 a^7 b^3 c - a^6 b^4 c - a^5 b^5 c + 2 a^3 b^7 c + a^2 b^8 c - 2 a^7 b^2 c^2 + 3 a^5 b^4 c^2 - 3 a^4 b^5 c^2 - 2 a^3 b^6 c^2 + 2 a^2 b^7 c^2 + a b^8 c^2 + b^9 c^2 - 2 a^7 b c^3 + 9 a^5 b^3 c^3 + 4 a^4 b^4 c^3 - 4 a^3 b^5 c^3 - 2 a^2 b^6 c^3 - a b^7 c^3 - a^7 c^4 - a^6 b c^4 + 3 a^5 b^2 c^4 + 4 a^4 b^3 c^4 - 2 a^3 b^4 c^4 - a^2 b^5 c^4 - 2 b^7 c^4 - a^5 b c^5 - 3 a^4 b^2 c^5 - 4 a^3 b^3 c^5 - a^2 b^4 c^5 + 2 a b^5 c^5 + b^6 c^5 + a^5 c^6 - 2 a^3 b^2 c^6 - 2 a^2 b^3 c^6 + b^5 c^6 + 2 a^3 b c^7 + 2 a^2 b^2 c^7 - a b^3 c^7 - 2 b^4 c^7 + a^3 c^8 + a^2 b c^8 + a b^2 c^8 + b^2 c^9 - a c^10) : :

X(27862) lies on these lines:

X(27863) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^3 - 4 a^3 b^5 + 5 a b^7 - 5 a^6 b c + 4 a^4 b^3 c + a^2 b^5 c + 7 a^4 b^2 c^2 + 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 - 4 a b^5 c^2 - 5 b^6 c^2 - a^5 c^3 + 4 a^4 b c^3 + 2 a^3 b^2 c^3 - 2 a^2 b^3 c^3 - a b^4 c^3 - 2 a^2 b^2 c^4 - a b^3 c^4 + 10 b^4 c^4 - 4 a^3 c^5 + a^2 b c^5 - 4 a b^2 c^5 - 5 b^2 c^6 + 5 a c^7) : :

X(27863) lies on these lines:

X(27864) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a^5 b^3 + a b^7 - a^6 b c + 2 a^3 b^4 c + 3 a^2 b^5 c + a^4 b^2 c^2 - b^6 c^2 - a^5 c^3 - 4 a^2 b^3 c^3 - 3 a b^4 c^3 + 2 a^3 b c^4 - 3 a b^3 c^4 + 2 b^4 c^4 + 3 a^2 b c^5 - b^2 c^6 + a c^7) : :

X(27864) lies on these lines:

X(27865) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 72

Barycentrics    (a^2 - b c) (-a b^7 + a^6 b c - a^4 b^2 c^2 - a^3 b^3 c^2 + a^2 b^4 c^2 + b^6 c^2 - a^3 b^2 c^3 + a^2 b^3 c^3 + a^2 b^2 c^4 - b^4 c^4 + b^2 c^6 - a c^7) : :

X(27865) lies on these lines:

X(27866) =  X(2)X(98)∩X(3)X(7731)

Barycentrics    -a^2 (a^10+2 a^6 b^2 c^2-2 a^8 (b^2+c^2)-b^2 c^2 (b^2-c^2)^2 (b^2+c^2)+2 a^4 (b^6+c^6)-a^2 (b^8-b^6 c^2+3 b^4 c^4-b^2 c^6+c^8)) : :
Barycentrics    a^2/((b^2 - c^2)^2 (a^4 - a^2 b^2 - a^2 c^2 - b^2 c^2)) : :
Barycentrics    (6 R^4-7 R^2 SB-7 R^2 SC-11 R^2 SW+2 SB SW+2 SC SW+2 SW^2) S^2 + 7 R^2 SB SC SW-2 SB SC SW^2 : :

See Alexandr Skutin and Ercole Suppa, Hyacinthos 28625.

Another construction of the conic described in Hyacinthos 28625: Let P be a point on the Euler line. Let A'B'C' be the cevian triangle of P. Let A", B", C" be the circumcircle-inverses of A', B', C', resp. Triangle A"B"C" is perspective to ABC, and the locus of the perspector, as P moves on the Euler line, is the conic with center X(27866) and perspector X(27867). This conic passes through X(3), X(6), X(24), X(60), X(143), X(1511) and X(1986). (Randy Hutson, August 19, 2019)

X(27866) lies on these lines: {2,98}, {3,7731}, {54,1511}, {140,11597}, {146,10984}, {399,15056}, {568,12228}, {569,12383}, {1112,15107}, {1176,6593}, {1216,2914}, {1539,8718}, {1614,14643}, {1986,7691}, {2888,10114}, {2931,15043}, {2979,19504}, {5157,11061}, {5422,12310}, {5504,13472}, {5900,6699}, {6030,12824}, {6636,13417}, {7485,17847}, {7488,16223}, {7503,12270}, {7512,11557}, {7514,12281}, {7592,12273}, {10117,15080}, {10203,11702}, {10721,18564}, {11746,12834}, {12121,15033}, {12893,15053}, {13434,17702}, {15051,15463}, {15140,19121}, {19126,25321}

X(27866) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3047,5642,110})

X(27867) =  ISOGONAL CONJUGATE OF X(7668)

Barycentrics    a^2*(c^2-a^2)^2*(a^2-b^2)^2*((b^2+c^2)*a^2-b^4+b^2*c^2)*((b^2+c^2)*a^2+b^2*c^2-c^4) : :

See Alexandr Skutin and Ercole Suppa, Hyacinthos 28625.

X(27867) lies on these lines: {110,9514}, {5012,5661}, {5201,23061}

X(27867) = isogonal conjugate of X(7668)
X(27867) = isotomic conjugate of X(36901)
X(27867) = trilinear pole of line X(1625)X(1634)
X(27867) = isotomic conjugate of complement of X(11794)

X(27868) =  EULER LINE INTERCEPT OF X(1154)X(14143)

Barycentrics    2 a^16-9 a^14 (b^2+c^2)-(b^2-c^2)^6 (b^2+c^2)^2+a^12 (19 b^4+30 b^2 c^2+19 c^4)+a^2 (b^2-c^2)^4 (b^6+5 b^4 c^2+5 b^2 c^4+c^6)-a^10 (29 b^6+41 b^4 c^2+41 b^2 c^4+29 c^6)+a^4 (b^2-c^2)^2 (9 b^8-4 b^6 c^2-11 b^4 c^4-4 b^2 c^6+9 c^8)+a^8 (35 b^8+20 b^6 c^2+28 b^4 c^4+20 b^2 c^6+35 c^8)+a^6 (-27 b^10+17 b^8 c^2+b^6 c^4+b^4 c^6+17 b^2 c^8-27 c^10) : :
Barycentrics    4 S^4 + (7 R^4-4 SB SC-4 R^2 SW) S^2 + 43 R^4 SB SC-36 R^2 SB SC SW+8 SB SC SW^2 : :

As a point on the Euler line, X(27868) has Shinagawa coefficients {7 R^4+4 S^2-4 R^2 SW,43 R^4-4 S^2-36 R^2 SW+8 SW^2}.

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28626.

X(27868) lies on these lines: {2,3}, {1154,14143}, {1157,20424}, {3574,6150}, {6288,14072}, {18016,18400}, {21975,23237}, {22051,25044}

X(27868) = reflection of X(i) in X(j) for these (i,j): {3,10126}, {10285,5}, {14142,140}, {20030,15335}, {20120,546}
X(27868) = X(10285)-of-Johnson-triangle
X(27868) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {15335,20030,381}

X(27869) =  X(11)X(7671)∩X(1001)X(1006)

Barycentrics    2 a^7 b-7 a^6 b^2+6 a^5 b^3+5 a^4 b^4-10 a^3 b^5+3 a^2 b^6+2 a b^7-b^8+2 a^7 c-4 a^6 b c-6 a^4 b^3 c+24 a^3 b^4 c-18 a^2 b^5 c-2 a b^6 c+4 b^7 c-7 a^6 c^2+22 a^4 b^2 c^2-14 a^3 b^3 c^2+9 a^2 b^4 c^2-6 a b^5 c^2-4 b^6 c^2+6 a^5 c^3-6 a^4 b c^3-14 a^3 b^2 c^3+12 a^2 b^3 c^3+6 a b^4 c^3-4 b^5 c^3+5 a^4 c^4+24 a^3 b c^4+9 a^2 b^2 c^4+6 a b^3 c^4+10 b^4 c^4-10 a^3 c^5-18 a^2 b c^5-6 a b^2 c^5-4 b^3 c^5+3 a^2 c^6-2 a b c^6-4 b^2 c^6+2 a c^7+4 b c^7-c^8 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28629.

X(27869) lies on these lines: {11,7671}, {1001,1006}, {5851,5886}, {5856,5901}

X(27870) =  X(11)X(8256)∩ X(355)X(528)

Barycentrics    2 a^6 b-3 a^5 b^2-3 a^4 b^3+6 a^3 b^4-3 a b^6+b^7+2 a^6 c-8 a^5 b c+15 a^4 b^2 c+2 a^3 b^3 c-16 a^2 b^4 c+6 a b^5 c-b^6 c-3 a^5 c^2+15 a^4 b c^2-36 a^3 b^2 c^2+20 a^2 b^3 c^2+3 a b^4 c^2-3 b^5 c^2-3 a^4 c^3+2 a^3 b c^3+20 a^2 b^2 c^3-12 a b^3 c^3+3 b^4 c^3+6 a^3 c^4-16 a^2 b c^4+3 a b^2 c^4+3 b^3 c^4+6 a b c^5-3 b^2 c^5-3 a c^6-b c^6+c^7 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28629.

X(27870) lies on these lines: {11,8256}, {355,528}, {1145,10826}, {2802,10943}, {3057,15842}, {3434,10953}, {3816,17622}, {3829,17619}, {3871,6224}, {3880,12616}, {3885,10949}, {5687,10043}, {8668,12114}, {10785,10912}, {10948,17652}, {12607,12672}

X(27871) =  X(115)X(8258)∩ X(3828)X(17677)

Barycentrics    2 a^7+3 a^6 b+2 a^4 b^3+2 a^3 b^4-6 a^2 b^5-6 a b^6-b^7+3 a^6 c+4 a^5 b c+4 a^4 b^2 c+4 a^3 b^3 c-4 a^2 b^4 c-10 a b^5 c-5 b^6 c+4 a^4 b c^2+6 a^3 b^2 c^2+13 a^2 b^3 c^2+8 a b^4 c^2-2 b^5 c^2+2 a^4 c^3+4 a^3 b c^3+13 a^2 b^2 c^3+24 a b^3 c^3+12 b^4 c^3+2 a^3 c^4-4 a^2 b c^4+8 a b^2 c^4+12 b^3 c^4-6 a^2 c^5-10 a b c^5-2 b^2 c^5-6 a c^6-5 b c^6-c^7 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28629.

X(27871) lies on these lines: {115,8258}, {3828,17677}

leftri

Collineation mappings involving Gemini triangle 73: X(27872)-X(27906)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 73, as in centers X(27872)-X(27906). Then

m(X) = bc(a^2-bc)^2x + b^2(a^2-bc)(c^2-ab)y + c^2(a^2-bc)(b^2-ac)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 14, 2018)

X(27872) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^4 + a^3 b c - a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4) : :

X(27872) lies on these lines: {1, 2}, {1215, 27884}, {27875, 27881}, {27876, 27879}, {27877, 27897}

X(27873) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^3 b^5 + a b^7 + a^6 b c - a^4 b^3 c - a^4 b^2 c^2 - a b^5 c^2 + b^6 c^2 - a^4 b c^3 - 2 b^4 c^4 - a^3 c^5 - a b^2 c^5 + b^2 c^6 + a c^7) : :

X(27873) lies on these lines: {2, 3}, {27876, 27885}

X(27874) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^3 + a b^7 + a^6 b c - a^2 b^5 c - 3 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 + b^6 c^2 - a^5 c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 + 2 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - a^2 b c^5 + b^2 c^6 + a c^7) : :

X(27874) lies on these lines: {2, 3}, {27879, 27880}, {27889, 27891}

X(27875) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    a (a^2 + b c) (a^4 b^3 - a^2 b^5 - a^3 b^3 c + a b^5 c + 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 2 a b^4 c^2 - b^5 c^2 + a^4 c^3 - a^3 b c^3 - 2 a^2 b^2 c^3 - 2 a b^3 c^3 + b^4 c^3 - 2 a b^2 c^4 + b^3 c^4 - a^2 c^5 + a b c^5 - b^2 c^5) : :

X(27875) lies on these lines:

X(27876) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^5 + a^4 b c - a^2 b^2 c^2 + b^4 c^2 + b^2 c^4 + a c^5) : :

X(27876) lies on these lines:

X(27877) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^3 b^3 + a b^5 + a^4 b c + a^2 b^3 c - a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 + a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + b^2 c^4 + a c^5) : :

X(27877) lies on these lines:

X(27878) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^2 b^3 + a b^4 + a^3 b c - a^2 b^2 c - a b^3 c - a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 + a c^4) : :

X(27878) lies on these lines:

X(27879) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^2 b^4 + a b^5 + a^4 b c - a^3 b^2 c - a b^4 c - a^3 b c^2 - a^2 b^2 c^2 + b^4 c^2 - 2 b^3 c^3 - a^2 c^4 - a b c^4 + b^2 c^4 + a c^5) : :

X(27879) lies on these lines:

X(27880) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    a (b + c) (a^2 + b c) (a b^2 + b^2 c + a c^2 + b c^2) : :

X(27880) lies on these lines:

X(27881) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    a (a^2 + b c) (a^3 b^3 - a^2 b^4 - 2 a^2 b^3 c + 3 a b^4 c + 4 a^2 b^2 c^2 - 4 a b^3 c^2 - b^4 c^2 + a^3 c^3 - 2 a^2 b c^3 - 4 a b^2 c^3 + 4 b^3 c^3 - a^2 c^4 + 3 a b c^4 - b^2 c^4) : :

X(27881) lies on these lines:

X(27882) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^5 + a b^9 + a^8 b c - a^4 b^5 c - a^6 b^2 c^2 - a^4 b^4 c^2 + a^2 b^6 c^2 + b^8 c^2 - a^4 b^2 c^4 - a b^5 c^4 - b^6 c^4 - a^5 c^5 - a^4 b c^5 - a b^4 c^5 + a^2 b^2 c^6 - b^4 c^6 + b^2 c^8 + a c^9) : :

X(27882) lies on these lines:

X(27883) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^5 + a b^9 + a^8 b c - a^4 b^5 c - a^6 b^2 c^2 - a^4 b^4 c^2 + a^3 b^5 c^2 + a^2 b^6 c^2 + b^8 c^2 + a^4 b^3 c^3 - a^4 b^2 c^4 + a^2 b^4 c^4 - a b^5 c^4 - b^6 c^4 - a^5 c^5 - a^4 b c^5 + a^3 b^2 c^5 - a b^4 c^5 + a^2 b^2 c^6 - b^4 c^6 + b^2 c^8 + a c^9) : :

X(27883) lies on these lines:

X(27884) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^6 + a^5 b c - a^3 b^2 c^2 + b^5 c^2 + b^2 c^5 + a c^6) : :

X(27884) lies on these lines:

X(27885) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^7 + a^6 b c - a^4 b^2 c^2 + b^6 c^2 + b^2 c^6 + a c^7) : :

X(27885) lies on these lines:

X(27886) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a^2 b^4 + a^3 b^2 c + a b^4 c + a^3 b c^2 + 2 b^3 c^3 + a^2 c^4 + a b c^4) : :

X(27886) lies on these lines:

X(27887) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a^3 b^5 + a^4 b^3 c + a b^5 c^2 + a^4 b c^3 + 2 b^4 c^4 + a^3 c^5 + a b^2 c^5) : :

X(27887) lies on these lines:

X(27888) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (b + c) (a^2 + b c) (a^2 b^4 + a^4 b c - a^2 b^3 c + a b^4 c - a^2 b c^3 + b^3 c^3 + a^2 c^4 + a b c^4) : :

X(27888) lies on these lines:

X(27889) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^3 b^3 + a b^5 + a^4 b c - a^2 b^3 c - 3 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^2 c^4 + a c^5) : :

X(27889) lies on these lines:

X(27890) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    b c (a^2 + b c) (-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(27890) lies on these lines:

X(27891) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    b^2 c^2 (a + b) (a + c) (a b + a c - b c) (a^2 + b c) : :

X(27891) lies on these lines:

X(27892) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^5 + a^4 b c - a^2 b^3 c - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + b^2 c^4 + a c^5) : :

X(27892) lies on these lines:

X(27893) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (b - c) (a^2 + b c) (-a^2 b^5 + a^5 b c - a^2 b^4 c + a b^5 c - a^3 b^2 c^2 - a^2 b^3 c^2 - a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + a b c^5) : :

X(27893) lies on these lines:

X(27894) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    b c (b - c) (a^2 + b c) (a^3 b + a^3 c - 3 a^2 b c + a b^2 c + a b c^2 + b^2 c^2) : :

X(27894) lies on these lines:

X(27895) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^6 + a^5 b c - a^2 b^4 c - 2 a^3 b^2 c^2 + b^5 c^2 - a b^3 c^3 - a^2 b c^4 + b^2 c^5 + a c^6) : :

X(27895) lies on these lines:

X(27896) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a^2 b^3 + 2 a b^4 + 2 a^3 b c + a^2 b^2 c + a b^3 c + a^2 b c^2 + 2 b^3 c^2 + a^2 c^3 + a b c^3 + 2 b^2 c^3 + 2 a c^4) : :

X(27896) lies on these lines:

X(27897) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^3 - 2 a^3 b^5 + 3 a b^7 + 3 a^6 b c - 2 a^4 b^3 c - a^2 b^5 c - 5 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 - 2 a b^5 c^2 + 3 b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 + 2 a^2 b^2 c^4 - a b^3 c^4 - 6 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 - 2 a b^2 c^5 + 3 b^2 c^6 + 3 a c^7) : :

X(27897) lies on these lines:

X(27898) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^3 b^5 + a b^7 + a^6 b c - a^4 b^3 c - a^3 b^4 c - a^2 b^5 c - 2 a^4 b^2 c^2 - a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 c^2 + b^6 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^3 b c^4 - a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 - a^3 c^5 - a^2 b c^5 - a b^2 c^5 + b^2 c^6 + a c^7) : :

X(27898) lies on these lines:

X(27899) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^5 + a b^9 + a^8 b c - a^4 b^5 c - a^6 b^2 c^2 - a^4 b^4 c^2 + 2 a^3 b^5 c^2 + a^2 b^6 c^2 + b^8 c^2 + 2 a^4 b^3 c^3 - a^4 b^2 c^4 + 2 a^2 b^4 c^4 - a b^5 c^4 - b^6 c^4 - a^5 c^5 - a^4 b c^5 + 2 a^3 b^2 c^5 - a b^4 c^5 + a^2 b^2 c^6 - b^4 c^6 + b^2 c^8 + a c^9) : :

X(27899) lies on these lines:

X(27900) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a^7 b^4 - a^5 b^6 - a^3 b^8 + a b^10 + a^9 b c + a^6 b^4 c - a^5 b^5 c - a^2 b^8 c - 2 a^7 b^2 c^2 + 2 a^6 b^3 c^2 + a^5 b^4 c^2 - 3 a^4 b^5 c^2 + 2 a^3 b^6 c^2 - a b^8 c^2 + b^9 c^2 + 2 a^6 b^2 c^3 + a^5 b^3 c^3 - 4 a^4 b^4 c^3 - 2 a^3 b^5 c^3 - a b^7 c^3 + a^7 c^4 + a^6 b c^4 + a^5 b^2 c^4 - 4 a^4 b^3 c^4 - 6 a^3 b^4 c^4 + a^2 b^5 c^4 - 2 b^7 c^4 - a^5 b c^5 - 3 a^4 b^2 c^5 - 2 a^3 b^3 c^5 + a^2 b^4 c^5 + 2 a b^5 c^5 + b^6 c^5 - a^5 c^6 + 2 a^3 b^2 c^6 + b^5 c^6 - a b^3 c^7 - 2 b^4 c^7 - a^3 c^8 - a^2 b c^8 - a b^2 c^8 + b^2 c^9 + a c^10) : :

X(27900) lies on these lines:

X(27901) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^3 - 4 a^3 b^5 + 5 a b^7 + 5 a^6 b c - 4 a^4 b^3 c - a^2 b^5 c - 7 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 - 4 a b^5 c^2 + 5 b^6 c^2 - a^5 c^3 - 4 a^4 b c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 + 2 a^2 b^2 c^4 - a b^3 c^4 - 10 b^4 c^4 - 4 a^3 c^5 - a^2 b c^5 - 4 a b^2 c^5 + 5 b^2 c^6 + 5 a c^7) : :

X(27901) lies on these lines:

X(27902) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^5 b^3 + a b^7 + a^6 b c + 2 a^3 b^4 c + a^2 b^5 c - a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 4 a^2 b^4 c^2 + b^6 c^2 - a^5 c^3 + 4 a^3 b^2 c^3 + 4 a^2 b^3 c^3 + a b^4 c^3 + 2 a^3 b c^4 + 4 a^2 b^2 c^4 + a b^3 c^4 - 2 b^4 c^4 + a^2 b c^5 + b^2 c^6 + a c^7) : :

X(27902) lies on these lines:

X(27903) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a b^7 + a^6 b c - a^4 b^2 c^2 + a^3 b^3 c^2 + a^2 b^4 c^2 + b^6 c^2 + a^3 b^2 c^3 + a^2 b^3 c^3 + a^2 b^2 c^4 - b^4 c^4 + b^2 c^6 + a c^7) : :

X(27903) lies on these lines:

X(27904) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^3 b^4 + a b^6 + a^5 b c - a^3 b^3 c - a^3 b^2 c^2 - a^2 b^3 c^2 + b^5 c^2 - a^3 b c^3 - a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + b^2 c^5 + a c^6) : :

X(27904) lies on these lines:

X(27905) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (-a^2 b^4 + a b^5 + 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 + b^4 c^2 + a^2 b c^3 + a b^2 c^3 - 3 b^3 c^3 - a^2 c^4 - a b c^4 + b^2 c^4 + a c^5) : :

X(27905) lies on these lines:

X(27906) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 73

Barycentrics    (a^2 + b c) (a^2 b^4 + a^3 b^2 c - a^2 b^3 c + a b^4 c + a^3 b c^2 - a^2 b^2 c^2 - a b^3 c^2 - a^2 b c^3 - a b^2 c^3 + 3 b^3 c^3 + a^2 c^4 + a b c^4) : :

X(27906) lies on these lines:

leftri

Collineation mappings involving Gemini triangle 74: X(27907)-X(27952)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 74, as in centers X(27907)-X(27952). Then

m(X) = bc(a^2-bc)^2x + ac(a^2-bc)(c^2-ab)y + ab(a^2-bc)(b^2-ac)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 14, 2018)

X(27907) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^5 - a b^4 - a^3 b c - a^2 b^2 c + a b^3 c - b^4 c - a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27907) lies on these lines:

X(27908) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 b^3 + 2 a^3 b^5 - a b^7 + 3 a^6 b c - 2 a^4 b^3 c - a^2 b^5 c - a^4 b^2 c^2 + b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 + 2 a^2 b^3 c^3 + a b^4 c^3 + a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 - a^2 b c^5 + b^2 c^6 - a c^7) : :

X(27908) lies on these lines:

X(27909) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 b^3 + 2 a^3 b^5 - a b^7 + 2 a^6 b c - 2 a^4 b^3 c + a^3 b^3 c^2 - a^2 b^4 c^2 + a b^5 c^2 + b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 + a^3 b^2 c^3 - a^2 b^2 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + a b^2 c^5 + b^2 c^6 - a c^7) : :

X(27909) lies on these lines:

X(27910) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^3 b^3 + 2 a^2 b^4 - a b^5 + 3 a^4 b c - 2 a^3 b^2 c - a^2 b^3 c - 2 a^3 b c^2 + a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + 2 a^2 c^4 + b^2 c^4 - a c^5) : :

X(27910) lies on these lines:

X(27911) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 b^3 + a b^4 - 3 a^3 b c + a^2 b^2 c - a b^3 c + a^2 b c^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) : :

X(27911) lies on these lines:

X(27912) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^2 + a b + b^2 + a c - 3 b c + c^2) : :

X(27912) lies on these lines: {2, 7}, {192, 24398}, {239, 3570}, {335, 26273}, {350, 27920}, {385, 27916}, {1054, 4440}, {1281, 17793}, {1647, 32843}, {3685, 24428}, {4366, 24685}, {4465, 6651}, {9259, 18159}, {17319, 24403}, {20769, 27933}, {26240, 31317}, {27908, 27913}, {27943, 27950}

X(27913) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 b^3 + a b^4 - 2 a^3 b c - b^3 c^2 + a^2 c^3 - b^2 c^3 + a c^4) : :

X(27913) lies on these lines:

X(27914) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c - a^4 b^2 c + a b^5 c - b^6 c + a^5 c^2 - a^4 b c^2 + a b^4 c^2 + b^5 c^2 - a^3 c^4 + a b^2 c^4 + a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27914) lies on these lines:

X(27915) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c - a^4 b^2 c + a^2 b^4 c + a b^5 c - b^6 c + a^5 c^2 - a^4 b c^2 - a^3 b^2 c^2 + a b^4 c^2 + b^5 c^2 - a b^3 c^3 - a^3 c^4 + a^2 b c^4 + a b^2 c^4 + a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27915) lies on these lines:

X(27916) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^4 + a b^3 + a^2 b c - 2 b^2 c^2 + a c^3) : :

X(27916) lies on these lines:

X(27917) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 + a b^4 + a^3 b c - b^3 c^2 - b^2 c^3 + a c^4) : :

X(27917) lies on these lines:

X(27918) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (b - c)^2 (a^2 - b c) : :

X(27918) lies on these lines: {1, 16377}, {2, 37}, {6, 9318}, {7, 9599}, {8, 19951}, {9, 24398}, {11, 244}, {39, 7264}, {42, 19952}, {43, 19953}, {44, 24407}, {45, 24408}, {76, 19974}, {86, 19975}, {145, 19954}, {213, 24455}, {238, 24428}, {239, 3570}, {292, 2481}, {335, 30997}, {386, 19956}, {514, 2087}, {519, 19957}, {551, 19958}, {612, 19959}, {614, 19960}, {673, 26273}, {759, 30927}, {876, 3675}, {899, 19961}, {982, 4493}, {984, 24427}, {1015, 1111}, {1125, 19962}, {1193, 19938}, {1429, 27943}, {1447, 1914}, {1500, 24786}, {1642, 3008}, {1698, 19963}, {2161, 30930}, {2170, 21138}, {2176, 24460}, {2238, 27942}, {2275, 3673}, {2295, 17048}, {2352, 16378}, {3006, 19964}, {3121, 14296}, {3125, 17761}, {3177, 24737}, {3227, 20568}, {3241, 19966}, {3244, 19967}, {3248, 23774}, {3616, 19968}, {3617, 19969}, {3621, 19970}, {3624, 19933}, {3626, 19971}, {3632, 19972}, {3634, 20000}, {3661, 19973}, {3663, 24318}, {3679, 20006}, {3721, 24172}, {3726, 20335}, {3828, 20010}, {3834, 31041}, {3934, 21021}, {3946, 25342}, {4056, 9665}, {4124, 4448}, {4363, 24629}, {4366, 4760}, {4386, 26229}, {4444, 23822}, {4465, 17755}, {4516, 21210}, {4675, 17721}, {4692, 9466}, {4738, 13466}, {4858, 6377}, {4894, 7854}, {4986, 27076}, {5475, 7272}, {6545, 8042}, {6652, 33295}, {7198, 7745}, {7796, 30155}, {9259, 9317}, {9263, 18159}, {9458, 17119}, {9780, 20001}, {14523, 17626}, {16604, 20880}, {16720, 26959}, {16726, 16727}, {17063, 24458}, {17197, 23824}, {17374, 20042}, {17392, 17450}, {17395, 17602}, {17445, 21922}, {17448, 26563}, {18895, 32020}, {19862, 20004}, {19997, 29633}, {20005, 26115}, {20172, 26240}, {20257, 21951}, {20363, 20435}, {21004, 28111}, {21022, 21684}, {21204, 21211}, {21352, 21967}, {23816, 31647}, {24330, 24631}, {24775, 24792}, {27487, 30967}, {31337, 33117}

X(27918) = crossdifference of every pair of points on line X(101)X(667) (the tangent at X(101) to hyperbola {{A,B,C,X(101),PU(9)})}

X(27919) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c)^2 (-a b + b^2 - a c + c^2) : :

X(27919) lies on these lines: {1, 2}, {291, 32029}, {350, 3570}, {740, 27942}, {1282, 17738}, {3685, 27945}, {4366, 4368}, {4375, 27855}, {4441, 9318}, {6546, 21832}, {8299, 17755}, {20769, 27934}, {24427, 27480}, {24428, 27947}

X(27920) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^3 b - a^2 b^2 - a b^3 + a^3 c + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(27920) lies on these lines:

X(27921) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-2 a^2 + 2 a b + b^2 + 2 a c - 4 b c + c^2) : :

X(27921) lies on these lines:

X(27922) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a + b - 2 c) (a - 2 b + c) (a^2 - b c) : :

X(27922) lies on these lines:

X(27923) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^3 b^3 - a b^5 + 3 a^4 b c - a^2 b^3 c - a^2 b^2 c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 + b^2 c^4 - a c^5) : :

X(27923) lies on these lines:

X(27924) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 b^4 - a^3 b^2 c - a^3 b c^2 + a^2 b^2 c^2 - b^3 c^3 + a^2 c^4) : :

X(27924) lies on these lines:

X(27925) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^3 b^5 - a^4 b^3 c - a^4 b c^3 + a^2 b^3 c^3 - b^4 c^4 + a^3 c^5) : :

X(27925) lies on these lines:

X(27926) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c)^2 (a^2 + a b - b^2 + a c - b c - c^2) : :

X(27926) lies on these lines:

X(27927) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 b^4 + a^4 b c - a^3 b^2 c + a b^4 c - a^3 b c^2 - 2 a^2 b^2 c^2 - b^3 c^3 + a^2 c^4 + a b c^4) : :

X(27927) lies on these lines:

X(27928) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 b^4 - a^4 b c - 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^2 b c^3 - a b^2 c^3 - b^3 c^3 + a^2 c^4) : :

X(27928) lies on these lines:

X(27929) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (b - c) (a^2 - b c) (-a^2 - a b + b^2 - a c + b c + c^2) : :

X(27929) lies on these lines:

X(27929) = complement of X(4444)

X(27930) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (b - c) (a^2 - b c) (a^4 + a^2 b^2 - a b^2 c + a^2 c^2 - a b c^2 - b^2 c^2) : :

X(27930) lies on these lines:

X(27931) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^4 + 2 a b^3 - 3 b^2 c^2 + 2 a c^3) : :

X(27931) lies on these lines:

X(27932) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 b^3 + 2 a^3 b^5 - a b^7 + 5 a^6 b c - 2 a^4 b^3 c - 3 a^2 b^5 c - 3 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 - 2 a b^5 c^2 + b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 6 a^2 b^3 c^3 + 3 a b^4 c^3 + 2 a^2 b^2 c^4 + 3 a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 - 3 a^2 b c^5 - 2 a b^2 c^5 + b^2 c^6 - a c^7) : :

X(27932) lies on these lines:

X(27933) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^5 - a^2 b^3 - 2 a b^4 - b^4 c + 3 a b^2 c^2 + 2 b^3 c^2 - a^2 c^3 + 2 b^2 c^3 - 2 a c^4 - b c^4) : :

X(27933) lies on these lines:

X(27934) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c - a^4 b^2 c + 2 a^2 b^4 c + a b^5 c - b^6 c + a^5 c^2 - a^4 b c^2 - 2 a^3 b^2 c^2 + a b^4 c^2 + b^5 c^2 - 2 a b^3 c^3 - a^3 c^4 + 2 a^2 b c^4 + a b^2 c^4 + a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(27934) lies on these lines:

X(27935) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^6 b^4 + 2 a^4 b^6 - a^2 b^8 + a^8 b c + 3 a^7 b^2 c - 3 a^5 b^4 c + a^4 b^5 c + a^3 b^6 c - 2 a^2 b^7 c - a b^8 c + 3 a^7 b c^2 + 2 a^6 b^2 c^2 - 4 a^5 b^3 c^2 - 2 a^4 b^4 c^2 + a^3 b^5 c^2 - 4 a^5 b^2 c^3 - 3 a^4 b^3 c^3 + 2 a^3 b^4 c^3 + 2 a^2 b^5 c^3 + 2 a b^6 c^3 + b^7 c^3 - a^6 c^4 - 3 a^5 b c^4 - 2 a^4 b^2 c^4 + 2 a^3 b^3 c^4 + 2 a^2 b^4 c^4 - a b^5 c^4 + a^4 b c^5 + a^3 b^2 c^5 + 2 a^2 b^3 c^5 - a b^4 c^5 - 2 b^5 c^5 + 2 a^4 c^6 + a^3 b c^6 + 2 a b^3 c^6 - 2 a^2 b c^7 + b^3 c^7 - a^2 c^8 - a b c^8) : :

X(27935) lies on these lines:

X(27936) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^8 + a^7 b + a^6 b^2 - a^4 b^4 + a^3 b^5 - a^2 b^6 - 2 a b^7 + a^7 c + 2 a^6 b c - a^5 b^2 c - 3 a^4 b^3 c + a^3 b^4 c + 2 a^2 b^5 c - a b^6 c - b^7 c + a^6 c^2 - a^5 b c^2 - 5 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 3 a^2 b^4 c^2 + 3 a b^5 c^2 + b^6 c^2 - 3 a^4 b c^3 - 2 a^3 b^2 c^3 + b^5 c^3 - a^4 c^4 + a^3 b c^4 + 3 a^2 b^2 c^4 - 2 b^4 c^4 + a^3 c^5 + 2 a^2 b c^5 + 3 a b^2 c^5 + b^3 c^5 - a^2 c^6 - a b c^6 + b^2 c^6 - 2 a c^7 - b c^7) : :

X(27936) lies on these lines:

X(27937) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^7 b^4 + a^6 b^5 - 2 a^5 b^6 - 2 a^4 b^7 + a^3 b^8 + a^2 b^9 + a^9 b c - 2 a^8 b^2 c - a^7 b^3 c + 4 a^6 b^4 c - 2 a^4 b^6 c - a^3 b^7 c + a b^9 c - 2 a^8 b c^2 - 4 a^7 b^2 c^2 + 3 a^5 b^4 c^2 + 2 a^4 b^5 c^2 + 2 a^3 b^6 c^2 - a b^8 c^2 - a^7 b c^3 + a^5 b^3 c^3 - a^4 b^4 c^3 + a^3 b^5 c^3 + 2 a^2 b^6 c^3 - a b^7 c^3 - b^8 c^3 + a^7 c^4 + 4 a^6 b c^4 + 3 a^5 b^2 c^4 - a^4 b^3 c^4 - 6 a^3 b^4 c^4 - 3 a^2 b^5 c^4 + a b^6 c^4 - b^7 c^4 + a^6 c^5 + 2 a^4 b^2 c^5 + a^3 b^3 c^5 - 3 a^2 b^4 c^5 + 2 b^6 c^5 - 2 a^5 c^6 - 2 a^4 b c^6 + 2 a^3 b^2 c^6 + 2 a^2 b^3 c^6 + a b^4 c^6 + 2 b^5 c^6 - 2 a^4 c^7 - a^3 b c^7 - a b^3 c^7 - b^4 c^7 + a^3 c^8 - a b^2 c^8 - b^3 c^8 + a^2 c^9 + a b c^9) : :

X(27937) lies on these lines:

X(27938) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 b^3 + 2 a^3 b^5 - a b^7 + 7 a^6 b c - 2 a^4 b^3 c - 5 a^2 b^5 c - 5 a^4 b^2 c^2 - 4 a^3 b^3 c^2 + 4 a^2 b^4 c^2 - 4 a b^5 c^2 + b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 - 4 a^3 b^2 c^3 + 10 a^2 b^3 c^3 + 5 a b^4 c^3 + 4 a^2 b^2 c^4 + 5 a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 - 5 a^2 b c^5 - 4 a b^2 c^5 + b^2 c^6 - a c^7) : :

X(27938) lies on these lines:

X(27939) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 b^3 + 2 a^3 b^5 - a b^7 + 3 a^6 b c - 2 a^4 b^3 c + 2 a^3 b^4 c + a^2 b^5 c - 3 a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 + b^6 c^2 - a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 - a b^4 c^3 + 2 a^3 b c^4 + 2 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + a^2 b c^5 + b^2 c^6 - a c^7) : :

X(27939) lies on these lines:

X(27940) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^3 b^5 + a^6 b c - a^4 b^3 c - a^2 b^5 c - a^4 b^2 c^2 - a^4 b c^3 + a^2 b^3 c^3 + a b^4 c^3 + a b^3 c^4 - b^4 c^4 + a^3 c^5 - a^2 b c^5) : :

X(27940) lies on these lines:

X(27941) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^5 - a b^4 - 2 a^2 b^2 c + 2 a b^3 c - b^4 c - 2 a^2 b c^2 + a b^2 c^2 + b^3 c^2 + 2 a b c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27941) lies on these lines:

X(27942) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^3 b - 2 a^2 b^2 - b^4 + a^3 c + a b^2 c + b^3 c - 2 a^2 c^2 + a b c^2 + b c^3 - c^4) : :

X(27942) lies on these lines:

X(27943) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    a (a^2 - 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(27943) lies on these lines:

X(27944) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^5 - a b^4 + a^3 b c - 3 a^2 b^2 c + 3 a b^3 c - b^4 c - 3 a^2 b c^2 + b^3 c^2 + 3 a b c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(27944) lies on these lines:

X(27945) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^4 + a^3 b - 2 a^2 b^2 + a b^3 - b^4 + a^3 c - 3 a^2 b c + a b^2 c + b^3 c - 2 a^2 c^2 + a b c^2 + a c^3 + b c^3 - c^4) : :

X(27945) lies on these lines:

X(27946) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (-a^5 - a^4 b + a^2 b^3 + a b^4 - a^4 c + 2 a^2 b^2 c + 2 a^2 b c^2 - a b^2 c^2 - 2 b^3 c^2 + a^2 c^3 - 2 b^2 c^3 + a c^4) : :

X(27946) lies on these lines:

X(27947) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^4 + a^3 b - 2 a^2 b^2 - a b^3 - b^4 + a^3 c - a^2 b c + a b^2 c + b^3 c - 2 a^2 c^2 + a b c^2 + 2 b^2 c^2 - a c^3 + b c^3 - c^4) : :

X(27947) lies on these lines:

X(27948) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (2 a^4 b - a^3 b^2 - a^2 b^3 + a b^4 - b^5 + 2 a^4 c - 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 - a^2 c^3 + a c^4 + b c^4 - c^5) : :

X(27948) lies on these lines:

X(27949) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (a^2 - b c) (a^2 + 2 a b - 2 b^2 + 2 a c - b c - 2 c^2) : :

X(27949) lies on these lines:

X(27950) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    a (a^2 - b c) (a^2 - b^2 + b c - c^2) : :

X(27950) lies on these lines: {1, 4475}, {2, 101}, {6, 24625}, {36, 214}, {41, 17367}, {48, 3662}, {75, 24324}, {81, 1015}, {100, 5091}, {106, 14996}, {141, 18042}, {239, 385}, {284, 17302}, {320, 7113}, {572, 6646}, {584, 17380}, {604, 17364}, {662, 1086}, {750, 3809}, {812, 4366}, {894, 18162}, {940, 9259}, {1026, 31073}, {1438, 26626}, {1790, 26840}, {1958, 7225}, {1964, 18209}, {2112, 17397}, {2174, 16706}, {2239, 11364}, {2267, 17333}, {2268, 17247}, {2278, 4389}, {2329, 17292}, {2481, 16381}, {3204, 17352}, {3219, 24036}, {3573, 8299}, {3661, 4390}, {3720, 5168}, {3960, 17191}, {3963, 18048}, {4268, 17347}, {4287, 17255}, {4579, 21320}, {4585, 17455}, {5053, 20072}, {5228, 11329}, {5905, 28922}, {6184, 21495}, {7237, 16556}, {7269, 27059}, {7291, 26639}, {8300, 27846}, {9310, 17244}, {11716, 29817}, {16367, 20672}, {17322, 25367}, {18645, 24237}, {19308, 20367}, {20818, 26657}, {21748, 28402}, {24296, 33151}, {27838, 32115}, {27912, 27943}

X(27951) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    b c (b - c) (-a^2 + b c) (-a^3 + b^3 - a b c + c^3) : :

X(27951) lies on these lines:

X(27952) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 74

Barycentrics    (b - c) (-a^2 + b c) (-a^5 b - a^3 b^3 - a^5 c + a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 + a b^2 c^3 + b^3 c^3) : :

X(27952) lies on these lines:

X(27953) =  EULER LINE INTERCEPT OF X(6102)X(6798)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^18-9 a^16 (b^2+c^2)+14 a^14 (b^2+c^2)^2+(b^2-c^2)^6 (b^6+b^4 c^2+b^2 c^4+c^6)-2 a^12 (3 b^6+14 b^4 c^2+14 b^2 c^4+3 c^6)+a^10 (-6 b^8+10 b^6 c^2+26 b^4 c^4+10 b^2 c^6-6 c^8)-2 a^2 (b^2-c^2)^4 (2 b^8-b^6 c^2-b^4 c^4-b^2 c^6+2 c^8)+a^4 (b^2-c^2)^2 (6 b^10-16 b^8 c^2-3 b^6 c^4-3 b^4 c^6-16 b^2 c^8+6 c^10)+a^8 (8 b^10-10 b^8 c^2-19 b^6 c^4-19 b^4 c^6-10 b^2 c^8+8 c^10)-2 a^6 (3 b^12-12 b^10 c^2+4 b^8 c^4-8 b^6 c^6+4 b^4 c^8-12 b^2 c^10+3 c^12)) : :
Barycentrics    (100 R^6+18 R^2 SB SC-105 R^4 SW-4 SB SC SW+36 R^2 SW^2-4 SW^3) S^2 -156 R^6 SB SC+175 R^4 SB SC SW-66 R^2 SB SC SW^2+8 SB SC SW^3 : :

As a point on the Euler line, X(27953) has Shinagawa coefficients {100 R^6 - 105 R^4 SW + 36 R^2 SW^2 - 4 SW^3, -156 R^6 + 18 R^2 S^2 + 175 R^4 SW - 4 S^2 SW - 66 R^2 SW^2 + 8 SW^3}.

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28632.

X(27953) lies on these lines: {2,3}, {6102,6798}, {8146,20414}

leftri

Collineation mappings involving Gemini triangle 75: X(27954)-X(28010)

rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : where A'B'C' = Gemini triangle 75, as in centers X(27954)-X(28010). Then

m(X) = bc(a^4-b^2c^2)x - ac(a^2+bc)(c^2+ab)y - ab(a^2+bc)(b^2+ac)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 15, 2018)

X(27954) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 - a b - b^2 - a c - b c - c^2) : :

X(27954) lies on these lines: {1, 2}, {21, 6651}, {56, 31317}, {75, 21008}, {99, 2134}, {172, 894}, {257, 26244}, {423, 11363}, {846, 17689}, {1055, 17116}, {1078, 24254}, {1215, 6645}, {1654, 17084}, {1655, 14949}, {1909, 27966}, {2329, 7061}, {3219, 17209}, {3552, 3923}, {3980, 33062}, {4011, 16914}, {4414, 25270}, {4418, 17693}, {6626, 21879}, {8782, 17760}, {13586, 24850}, {16916, 25591}, {16918, 25079}, {17673, 24161}, {17692, 32930}, {17762, 18755}, {18047, 21021}, {22061, 27987}, {24249, 31276}, {25280, 27733}, {25385, 33030}, {27955, 27972}, {27965, 27983}, {27998, 28003}

X(27955) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 - a b^4 - a^3 b c + a^2 b^2 c + a b^3 c + b^4 c + a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(27955) lies on these lines: {2, 3}, {6645, 27959}, {27954, 27972}, {27958, 27964}

X(27956) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 b^3 - 2 a^3 b^5 + a b^7 + 3 a^6 b c - 2 a^4 b^3 c - a^2 b^5 c - a^4 b^2 c^2 + b^6 c^2 + a^5 c^3 - 2 a^4 b c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 - a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 + b^2 c^6 + a c^7) : :

X(27956) lies on these lines:

X(27957) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 b^3 - 2 a^3 b^5 + a b^7 + 2 a^6 b c - 2 a^4 b^3 c - a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + a^5 c^3 - 2 a^4 b c^3 - a^3 b^2 c^3 - a^2 b^2 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - a b^2 c^5 + b^2 c^6 + a c^7) : :

X(27957) lies on these lines:

X(27958) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a + b) (a - b - c) (a + c) (a^2 + b c) : :

X(27958) lies on these lines: {1, 1178}, {2, 6}, {8, 23902}, {9, 261}, {21, 2053}, {48, 75}, {87, 741}, {99, 3729}, {110, 26227}, {172, 894}, {274, 7132}, {284, 314}, {312, 2185}, {423, 2322}, {593, 26223}, {643, 1253}, {648, 2331}, {757, 3758}, {799, 30988}, {1326, 3923}, {1419, 4573}, {1444, 27472}, {1474, 31623}, {1909, 7119}, {1931, 17350}, {2206, 14012}, {2329, 17787}, {2330, 7081}, {2640, 23944}, {3963, 18047}, {4054, 18653}, {4110, 4390}, {4416, 27691}, {4565, 28968}, {4754, 27984}, {5209, 16788}, {6626, 17257}, {7058, 11679}, {7175, 7196}, {16702, 17351}, {17279, 25536}, {18200, 28005}, {20072, 27702}, {21728, 27714}, {22065, 28287}, {24678, 32917}, {27713, 33082}, {27955, 27964}, {27969, 27998}, {27974, 27979}, {27991, 28004}, {28002, 28003}

X(27959) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 b^3 + a b^4 + 3 a^3 b c - a^2 b^2 c - a b^3 c - a^2 b c^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) : :

X(27959) lies on these lines:

X(27960) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 b^3 + a b^4 + 2 a^3 b c + b^3 c^2 + a^2 c^3 + b^2 c^3 + a c^4) : :

X(27960) lies on these lines:

X(27961) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c + a^4 b^2 c + a b^5 c + b^6 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 - b^5 c^2 - a^3 c^4 + a b^2 c^4 + a b c^5 - b^2 c^5 - a c^6 + b c^6) : :

X(27961) lies on these lines:

X(27962) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c + a^4 b^2 c - a^2 b^4 c + a b^5 c + b^6 c + a^5 c^2 + a^4 b c^2 - a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 - a b^3 c^3 - a^3 c^4 - a^2 b c^4 + a b^2 c^4 + a b c^5 - b^2 c^5 - a c^6 + b c^6) : :

X(27962) lies on these lines:

X(27963) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 - a b^3 - a^2 b c - 2 b^2 c^2 - a c^3) : :

X(27963) lies on these lines: {2, 31}, {1215, 6645}, {1920, 19574}, {7018, 19557}, {7081, 27995}, {7175, 7196}, {27964, 27967}, {27969, 27999}, {30074, 31108}

X(27964) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 - a b^4 - a^3 b c - b^3 c^2 - b^2 c^3 - a c^4) : :

X(27964) lies on these lines:

X(27965) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 b^3 - 2 a^2 b^4 + a b^5 + 3 a^4 b c - 2 a^3 b^2 c - a^2 b^3 c - 2 a^3 b c^2 + a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 + a^3 c^3 - a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 - 2 a^2 c^4 + b^2 c^4 + a c^5) : :

X(27965) lies on these lines:

X(27966) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    b c (b + c) (a^2 + b c) (a^2 + a b + b^2 + a c - b c + c^2) : :

X(27966) lies on these lines:

X(27967) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    a (a^2 + 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 + 2 b^2 c^2 + a c^3 + b c^3 - c^4) : :

X(27967) lies on these lines:

X(27968) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 b - a^2 b^2 + a^3 c - 3 a b^2 c - b^3 c - a^2 c^2 - 3 a b c^2 - b c^3) : :

X(27968) lies on these lines:

X(27969) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 b - a^2 b^2 + a b^3 + a^3 c + a^2 b c - 3 a b^2 c - b^3 c - a^2 c^2 - 3 a b c^2 + b^2 c^2 + a c^3 - b c^3) : :

X(27969) lies on these lines:

X(27970) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (2 a^3 - a b^2 + b^3 - b^2 c - a c^2 - b c^2 + c^3) : :

X(27970) lies on these lines:

X(27971) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 + a b^2 + 2 b^3 + 3 a b c + b^2 c + a c^2 + b c^2 + 2 c^3) : :

X(27971) lies on these lines:

X(27972) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 - a^3 b + a^2 b^2 - a b^3 - a^3 c - a^2 b c + 3 a b^2 c + b^3 c + a^2 c^2 + 3 a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(27972) lies on these lines:

X(27973) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 b^3 + a b^5 + 3 a^4 b c - a^2 b^3 c - a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 + a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^2 c^4 + a c^5) : :

X(27973) lies on these lines:

X(27974) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 b^4 + a^3 b^2 c + a^3 b c^2 - a^2 b^2 c^2 + b^3 c^3 + a^2 c^4) : :

X(27974) lies on these lines:

X(27975) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^3 b^5 + a^4 b^3 c + a^4 b c^3 - a^2 b^3 c^3 + b^4 c^4 + a^3 c^5) : :

X(27975) lies on these lines:

X(27976) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 + a^3 b - a^2 b^2 - 2 a b^3 + a^3 c - 2 a^2 b c - 3 a b^2 c - b^3 c - a^2 c^2 - 3 a b c^2 - 3 b^2 c^2 - 2 a c^3 - b c^3) : :

X(27976) lies on these lines:

X(27977) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 b^3 + a b^4 + 5 a^3 b c - 3 a^2 b^2 c - 3 a b^3 c - 3 a^2 b c^2 - 3 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) : :

X(27977) lies on these lines:

X(27978) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 - 3 a^3 b + 2 a^2 b^2 - 3 a b^3 - b^4 - 3 a^3 c - 3 a^2 b c + 5 a b^2 c + b^3 c + 2 a^2 c^2 + 5 a b c^2 - 4 b^2 c^2 - 3 a c^3 + b c^3 - c^4) : :

X(27978) lies on these lines:

X(27979) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^2 b^4 + a^4 b c + 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^2 b c^3 - a b^2 c^3 + b^3 c^3 + a^2 c^4) : :

X(27979) lies on these lines:

X(27980) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (b - c) (a^2 + b c) (a^4 + a^2 b^2 - a b^2 c + a^2 c^2 - a b c^2 + b^2 c^2) : :

X(27980) lies on these lines:

X(27981) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (b - c) (a^2 + b c) (-a^3 b^3 + a^2 b^4 - a^4 b c + a^2 b^3 c + 2 a^2 b^2 c^2 - a^3 c^3 + a^2 b c^3 - b^3 c^3 + a^2 c^4) : :

X(27981) lies on these lines:

X(27982) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 - b c) (a^2 + b c)^2 : :

X(27982) lies on these lines: {2, 31}, {172, 894}, {239, 1428}, {330, 604}, {1258, 2298}, {1691, 1966}, {1922, 4589}, {2330, 17752}, {4164, 27980}, {3552, 7155}, {4434, 28009}, {7081, 27997}, {8845, 17693}, {17493, 19557}, {19554, 19565}, {20964, 28008}, {27969, 27972}

X(27983) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 b^3 - 2 a^3 b^5 + a b^7 + 5 a^6 b c - 2 a^4 b^3 c - 3 a^2 b^5 c - 3 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + 2 a^2 b^4 c^2 + 2 a b^5 c^2 + b^6 c^2 + a^5 c^3 - 2 a^4 b c^3 + 2 a^3 b^2 c^3 + 6 a^2 b^3 c^3 - 3 a b^4 c^3 + 2 a^2 b^2 c^4 - 3 a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - 3 a^2 b c^5 + 2 a b^2 c^5 + b^2 c^6 + a c^7) : :

X(27983) lies on these lines:

X(27984) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 + a^2 b^3 + 2 a^2 b^2 c + 2 a b^3 c + b^4 c + 2 a^2 b c^2 + 3 a b^2 c^2 + a^2 c^3 + 2 a b c^3 + b c^4) : :

X(27984) lies on these lines:

X(27985) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^7 + a^5 b^2 - a^3 b^4 - a b^6 - a^5 b c + a^4 b^2 c - 2 a^2 b^4 c + a b^5 c + b^6 c + a^5 c^2 + a^4 b c^2 - 2 a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 - 2 a b^3 c^3 - a^3 c^4 - 2 a^2 b c^4 + a b^2 c^4 + a b c^5 - b^2 c^5 - a c^6 + b c^6) : :

X(27985) lies on these lines:

X(27986) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^6 b^4 - 2 a^4 b^6 + a^2 b^8 + a^8 b c + 3 a^7 b^2 c + 2 a^6 b^3 c - a^5 b^4 c - 3 a^4 b^5 c - 3 a^3 b^6 c + a b^8 c + 3 a^7 b c^2 + 2 a^6 b^2 c^2 - 2 a^5 b^3 c^2 - 2 a^4 b^4 c^2 - 3 a^3 b^5 c^2 + 2 a b^7 c^2 + 2 a^6 b c^3 - 2 a^5 b^2 c^3 - 3 a^4 b^3 c^3 + 2 a^3 b^4 c^3 + b^7 c^3 + a^6 c^4 - a^5 b c^4 - 2 a^4 b^2 c^4 + 2 a^3 b^3 c^4 - 2 a^2 b^4 c^4 - 3 a b^5 c^4 - 3 a^4 b c^5 - 3 a^3 b^2 c^5 - 3 a b^4 c^5 - 2 b^5 c^5 - 2 a^4 c^6 - 3 a^3 b c^6 + 2 a b^2 c^7 + b^3 c^7 + a^2 c^8 + a b c^8) : :

X(27986) lies on these lines:

X(27987) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^8 + a^7 b + a^6 b^2 + 2 a^5 b^3 - a^4 b^4 - 3 a^3 b^5 - a^2 b^6 + a^7 c + 2 a^6 b c + a^5 b^2 c - a^4 b^3 c - 3 a^3 b^4 c - 2 a^2 b^5 c + a b^6 c + b^7 c + a^6 c^2 + a^5 b c^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 + b^6 c^2 + 2 a^5 c^3 - a^4 b c^3 - 2 a^3 b^2 c^3 - 2 a b^4 c^3 - b^5 c^3 - a^4 c^4 - 3 a^3 b c^4 - a^2 b^2 c^4 - 2 a b^3 c^4 - 2 b^4 c^4 - 3 a^3 c^5 - 2 a^2 b c^5 + a b^2 c^5 - b^3 c^5 - a^2 c^6 + a b c^6 + b^2 c^6 + b c^7) : :

X(27987) lies on these lines:

X(27988) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (-a^7 b^4 - a^6 b^5 + 2 a^5 b^6 + 2 a^4 b^7 - a^3 b^8 - a^2 b^9 + a^9 b c - 2 a^8 b^2 c - 3 a^7 b^3 c + 2 a^6 b^4 c + 2 a^5 b^5 c + 2 a^4 b^6 c + a^3 b^7 c - 2 a^2 b^8 c - a b^9 c - 2 a^8 b c^2 - 4 a^7 b^2 c^2 + 3 a^5 b^4 c^2 + 2 a^4 b^5 c^2 + 2 a^3 b^6 c^2 - a b^8 c^2 - 3 a^7 b c^3 + 5 a^5 b^3 c^3 - a^4 b^4 c^3 - a^3 b^5 c^3 + 2 a^2 b^6 c^3 - a b^7 c^3 - b^8 c^3 - a^7 c^4 + 2 a^6 b c^4 + 3 a^5 b^2 c^4 - a^4 b^3 c^4 - 2 a^3 b^4 c^4 + a^2 b^5 c^4 + a b^6 c^4 - b^7 c^4 - a^6 c^5 + 2 a^5 b c^5 + 2 a^4 b^2 c^5 - a^3 b^3 c^5 + a^2 b^4 c^5 + 4 a b^5 c^5 + 2 b^6 c^5 + 2 a^5 c^6 + 2 a^4 b c^6 + 2 a^3 b^2 c^6 + 2 a^2 b^3 c^6 + a b^4 c^6 + 2 b^5 c^6 + 2 a^4 c^7 + a^3 b c^7 - a b^3 c^7 - b^4 c^7 - a^3 c^8 - 2 a^2 b c^8 - a b^2 c^8 - b^3 c^8 - a^2 c^9 - a b c^9) : :

X(27988) lies on these lines:

X(27989) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 b^3 - 2 a^3 b^5 + a b^7 + 7 a^6 b c - 2 a^4 b^3 c - 5 a^2 b^5 c - 5 a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 4 a^2 b^4 c^2 + 4 a b^5 c^2 + b^6 c^2 + a^5 c^3 - 2 a^4 b c^3 + 4 a^3 b^2 c^3 + 10 a^2 b^3 c^3 - 5 a b^4 c^3 + 4 a^2 b^2 c^4 - 5 a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - 5 a^2 b c^5 + 4 a b^2 c^5 + b^2 c^6 + a c^7) : :

X(27989) lies on these lines:

X(27990) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 b^3 - 2 a^3 b^5 + a b^7 + 3 a^6 b c - 2 a^4 b^3 c - 2 a^3 b^4 c - 3 a^2 b^5 c - 3 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + b^6 c^2 + a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 - 3 a b^4 c^3 - 2 a^3 b c^4 - 2 a^2 b^2 c^4 - 3 a b^3 c^4 - 2 b^4 c^4 - 2 a^3 c^5 - 3 a^2 b c^5 + b^2 c^6 + a c^7) : :

X(27990) lies on these lines:

X(27991) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (-a^3 b^5 + a^6 b c - a^4 b^3 c - a^2 b^5 c - a^4 b^2 c^2 - a^4 b c^3 + a^2 b^3 c^3 - a b^4 c^3 - a b^3 c^4 - b^4 c^4 - a^3 c^5 - a^2 b c^5) : :

X(27991) lies on these lines:

X(27992) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 - a b^4 - 2 a^3 b c + 2 a^2 b^2 c + 2 a b^3 c + b^4 c + 2 a^2 b c^2 + 3 a b^2 c^2 - b^3 c^2 + 2 a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(27992) lies on these lines:

X(27993) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 - a b^4 + b^4 c + a b^2 c^2 - b^3 c^2 - b^2 c^3 - a c^4 + b c^4) : :

X(27993) lies on these lines:

X(27994) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 - a b^4 + a^3 b c - a^2 b^2 c - a b^3 c + b^4 c - a^2 b c^2 - b^3 c^2 - a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(27994) lies on these lines:

X(27995) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 + a^3 b - 3 a b^3 + b^4 + a^3 c - 3 a^2 b c + a b^2 c + b^3 c + a b c^2 - 4 b^2 c^2 - 3 a c^3 + b c^3 + c^4) : :

X(27995) lies on these lines:

X(27996) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^5 + a^4 b - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c - 2 a b^3 c - 2 a^2 b c^2 - 3 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) : :

X(27996) lies on these lines:

X(27997) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + 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 - 2 b^2 c^2 - a c^3 + b c^3 + c^4) : :

X(27997) lies on these lines:

X(27998) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 b^2 - a^3 b^3 + a^2 b^4 - 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 - b^4 c^2 - a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(27998) lies on these lines:

X(27999) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (a^4 - a^3 b + a^2 b^2 - 2 a b^3 - a^3 c - 2 a^2 b c + 3 a b^2 c + b^3 c + a^2 c^2 + 3 a b c^2 - 3 b^2 c^2 - 2 a c^3 + b c^3) : :

X(27999) lies on these lines:

X(28000) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    a (a^2 + b c) (a^4 - a^3 b + a b^3 - b^4 - a^3 c + b^2 c^2 + a c^3 - c^4) : :

X(28000) lies on these lines:

This is the end of PART 14: Centers X(26001) - X(28000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)