## 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)

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

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(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) = 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(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}

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

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(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, 20991}, {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(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(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(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(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(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(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(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}

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

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(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(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(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(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}

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

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}

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

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}

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

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}

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

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(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(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(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}

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(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(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}

Endo-homothetic centers: X(26290) - X(26525)

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-INNER-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-INNER-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): (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(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(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 : :

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)

Collineation mappings involving Gemini triangle 45: X(26526) - X(26574)

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(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}

Collineation mappings involving Gemini triangle 46: X(26575) - X(26612)

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.

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) = 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)

Collineation mappings involving Gemini triangle 47: X(26621) - X(26652)

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)

Clark Kimberling, November

### 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(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}

Collineation mappings involving Gemini triangle 48: X(26653) - X(26699)

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(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}

Circumcirle-X-antipodes: X(26700) - X(26717)

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 + f w b^2 + v c^2)(a^2 g w + f w b^2 + v c^2) : :

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) = 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(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) : :

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(6), X(656))
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(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) = 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) = 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(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))

Centers associated with the Gemini triangles 1-10: X(26718) - X(26751)

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(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 = LBC, 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}

Collineation mappings involving Gemini triangle 49: X(26752) - X(26802)

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(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}

Collineation mappings involving Gemini triangle 50: X(26801) - X(26862)

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(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(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)

Endo-homothetic centers: X(26864) - X(26958)

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 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) = 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) = 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) = {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) = 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(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 INVERSE-IN-INCIRCLE

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(11027)

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)

Collineation mappings involving Gemini triangle 51: X(26959) - X(27019)

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(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(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}

Collineation mappings involving Gemini triangle 52: X(27020) - X(27081)

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}
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}

Collineation mappings involving Gemini triangle 53: X(27091) - X(27141)

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:

### 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:

### 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:

### 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 these lines:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

Collineation mappings involving Gemini triangle 54: X(27142) - X(27196)

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(27196). 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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:

### 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)

Collineation mappings involving Gemini triangle 55: X(27198) - X(27208)

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:

### 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:

### 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:

### 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:

Collineation mappings involving Gemini triangle 56: X(27210) - X(27220)

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:

### 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:

### 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:

Collineation mappings involving Gemini triangle 57: X(27221) - X(27232)

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:

### 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:

### 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:

Collineation mappings involving Gemini triangle 58: X(27233) - X(27245)

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:

### 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:

### 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}

Collineation mappings involving Gemini triangle 59: X(27248) - X(27297)

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:

### 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 +