ENCYCLOPEDIA OF TRIANGLE CENTERS Part16

Encyclopedia of Triangle Centers - ETC

This is PART 16: Centers X(30001) - X(32000)

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

X(30001) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30001) lies on these lines: {2, 12}, {4225, 24582}, {20891, 21405}, {20923, 21581}, {24586, 30031}, {24602, 30065}, {29960, 29963}, {29962, 29980}, {29968, 30008}

X(30002) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30002) lies on these lines: {2, 35}, {3831, 30000}, {20891, 21410}, {20923, 21586}, {29960, 29961}, {29968, 29993}

X(30003) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30003) lies on these lines: {2, 36}, {14349, 29989}, {20891, 21411}, {20923, 21587}, {29960, 29961}, {29991, 29993}

X(30004) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30004) lies on these lines: {2, 38}, {561, 20923}, {1959, 29960}, {4359, 29991}, {20632, 20889}, {21330, 27158}

X(30005) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30005) lies on these lines: {2, 12}, {20891, 21420}, {20923, 21594}, {24586, 30033}, {29960, 29961}, {29962, 29969}, {29964, 29975}, {29965, 29985}, {29986, 30016}, {29990, 30012}

X(30006) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 6 a^2 b^2 c^2 - 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(30006) lies on these lines: {2, 7}, {1402, 27657}, {1403, 25681}, {10473, 24982}, {16434, 27388}, {17182, 25059}, {17596, 21616}, {19649, 27401}, {20237, 20891}, {20882, 27492}, {20923, 20928}, {26723, 28289}, {29960, 29961}

X(30007) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - a^3 c^3 - 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(30007) lies on these lines: {2, 7}, {92, 20923}, {228, 19513}, {846, 21616}, {914, 29981}, {946, 32932}, {1210, 5208}, {1959, 29960}, {3687, 24220}, {3831, 11681}, {3912, 21072}, {3980, 12047}, {4224, 27401}, {4468, 30023}, {10473, 24996}, {13731, 22060}, {14206, 29982}, {14213, 20891}, {17182, 25058}, {19549, 20760}, {20881, 22014}, {20882, 21078}, {20924, 29988}, {21077, 32913}, {29966, 29992}, {29968, 29999}, {29987, 30016}

X(30008) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30008) lies on these lines: {2, 65}, {13732, 28928}, {18726, 21246}, {20891, 21422}, {20923, 21596}, {29960, 29961}, {29965, 29966}, {29968, 30001}

X(30009) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30009) lies on these lines: {2, 66}, {20891, 21423}, {20923, 21597}

X(30010) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30010) lies on these lines: {2, 32}, {18091, 21249}, {20891, 21425}, {20923, 20933}, {29964, 29983}, {29968, 29990}, {29974, 30000}

X(30011) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30011) lies on these lines: {2, 85}, {8, 3781}, {65, 20905}, {75, 21872}, {1764, 4384}, {2481, 16827}, {3912, 22020}, {16284, 20923}, {16609, 28272}, {17755, 20436}, {20880, 28287}, {21371, 30625}, {29965, 29966}, {29985, 29997}

X(30012) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30012) lies on these lines: {2, 87}, {304, 1921}, {20891, 21426}, {29964, 29977}, {29990, 30005}

X(30013) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 16 a^2 b^2 c^2 - 3 a b^3 c^2 - b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 4 b^3 c^3 + a^2 c^4 + a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30013) lies on these lines: {2, 45}, {20891, 21427}, {20923, 21600}, {29960, 30016}, {29964, 30014}

X(30014) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30014) lies on these lines: {2, 44}, {20891, 21428}, {20923, 21601}, {29964, 30013}

X(30015) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30015) lies on these lines: {2, 99}, {17886, 20891}, {20923, 20951}, {29961, 29983}, {29973, 30022}, {29974, 30000}, {29981, 29998}

X(30016) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30016) lies on these lines: {2, 11}, {857, 24602}, {1150, 29999}, {3119, 21246}, {20891, 20901}, {20923, 20940}, {21334, 26559}, {29960, 30013}, {29961, 29966}, {29968, 29993}, {29972, 29977}, {29976, 29984}, {29986, 30005}, {29987, 30007}, {30020, 30565}

X(30017) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30017) lies on these lines: {2, 101}, {20235, 29960}, {20891, 21429}, {20923, 21602}, {22144, 27623}

X(30018) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30018) lies on these lines: {2, 99}, {20891, 21431}, {20923, 21604}, {29962, 29973}, {29963, 29974}

X(30019) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30019) lies on these lines: {2, 45}, {3662, 30819}, {3762, 30026}, {4858, 18697}, {18151, 20444}, {29967, 29982}, {29978, 29988}, {30025, 30565}

X(30020) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30020) lies on these lines: {2, 900}, {3766, 29982}, {4728, 29969}, {20891, 21433}, {20923, 21606}, {30025, 30565}

X(30021) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30021) lies on these lines: {2, 187}, {14963, 29961}, {20891, 21434}, {20923, 21607}, {24622, 29989}

X(30022) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30022) lies on these lines: {1, 3596}, {2, 39}, {99, 13738}, {304, 1921}, {313, 31997}, {314, 9534}, {350, 978}, {1240, 28626}, {1509, 19734}, {1724, 5209}, {2176, 17790}, {2277, 25264}, {3142, 7752}, {3264, 17144}, {3831, 6376}, {3975, 21384}, {4225, 7782}, {4358, 28659}, {4377, 25130}, {6381, 24215}, {6383, 29967}, {8033, 19730}, {10009, 20891}, {16992, 19518}, {17050, 30090}, {17143, 32087}, {18021, 19732}, {18147, 33296}, {20924, 29988}, {22028, 31028}, {27623, 30940}, {28654, 29570}, {29964, 30000}, {29973, 30015}

X(30023) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics (b - c) (-a^3 b^2 - a^2 b^3 - 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(30023) lies on these lines: {2, 649}, {514, 27647}, {661, 29978}, {663, 25299}, {669, 28255}, {788, 25627}, {812, 27674}, {2978, 17072}, {24622, 29989}, {4468, 30007}, {4521, 21246}, {8640, 28286}, {20891, 20909}, {20923, 20952}, {21191, 30025}

X(30024) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30024) lies on these lines: {2, 650}, {29978, 30023}, {1491, 25126}, {3239, 4374}, {3907, 24666}, {4369, 4391}, {4397, 7662}, {7234, 28255}, {7650, 9508}, {20891, 21438}, {20923, 21611}, {21052, 25128}

X(30025) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30025) lies on these lines: {2, 659}, {514, 29989}, {661, 21191}, {3261, 29971}, {20891, 21439}, {20923, 21612}, {30019, 30020}

X(30026) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30026) lies on these lines: {2, 668}, {1111, 20891}, {3762, 30019}, {18159, 20923}, {22028, 30030}, {29966, 29983}

X(30027) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30027) lies on these lines: {2, 896}, {29978, 30023}, {1959, 29960}, {20891, 20904}, {20923, 20944}

X(30028) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30028) lies on these lines: {2, 38}, {75, 18055}, {85, 6385}, {1237, 18137}, {1921, 17451}, {1930, 17760}, {3721, 3739}, {3728, 17030}, {3786, 4384}, {6381, 17758}, {21330, 26959}, {21617, 27390}, {22011, 29991}, {25123, 27313}, {25295, 26964}

X(30029) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 99

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

X(30029) lies on these lines: {2, 6}, {100, 27622}, {4358, 29967}, {4645, 30944}, {24581, 24602}, {29961, 29966}

Collineation mappings involving Gemini triangle 100: X(30030)-X(30099)

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 100, as in centers X(30030)-X(30099). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 12, 2018)

X(30030) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30030) lies on these lines: {1, 2}, {6, 24735}, {9, 21281}, {38, 26562}, {65, 17755}, {76, 17050}, {85, 1921}, {142, 1909}, {226, 3975}, {344, 3208}, {350, 20257}, {958, 24586}, {1107, 20255}, {1334, 25101}, {1423, 6604}, {1573, 21240}, {1655, 3663}, {2140, 6381}, {2295, 17353}, {2887, 26558}, {2975, 24602}, {3061, 30758}, {3263, 17451}, {3550, 17696}, {3691, 4416}, {3729, 27523}, {3765, 5249}, {3780, 3879}, {3812, 24631}, {3836, 26561}, {3970, 4986}, {4051, 18156}, {4968, 27478}, {5025, 21241}, {5258, 29473}, {6376, 20335}, {7275, 17063}, {14951, 30034}, {16720, 21332}, {17143, 21071}, {17231, 25145}, {17234, 24524}, {17245, 24656}, {17738, 27000}, {17777, 26839}, {18139, 25298}, {20348, 30625}, {20436, 20905}, {20880, 20892}, {20955, 27487}, {21025, 21264}, {21226, 24215}, {21921, 26234}, {22028, 30026}, {22343, 26825}, {24190, 24214}, {30031, 30051}, {30033, 30039}, {30035, 30064}, {30042, 30055}, {30046, 30087}, {30083, 30086}

X(30031) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30031) lies on these lines: {2, 3}, {51, 26531}, {1446, 30097}, {5179, 30063}, {20235, 20892}, {20914, 30090}, {24586, 30001}, {30030, 30051}, {30034, 30043}, {30036, 30086}, {30054, 30085}

X(30032) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30032) lies on these lines: {2, 3}, {22066, 30983}, {27428, 27459}, {30037, 30038}, {30039, 30075}, {30043, 30088}, {30051, 30053}, {30052, 30054}

X(30033) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30033) lies on these lines: {2, 3}, {17864, 20892}, {20305, 22066}, {20926, 30090}, {24586, 30005}, {30030, 30039}, {30045, 30088}

X(30034) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30034) lies on these lines: {2, 6}, {313, 29979}, {1086, 17148}, {1441, 20892}, {2975, 4645}, {2995, 30035}, {3006, 17792}, {3917, 25279}, {3920, 24678}, {3948, 24220}, {4450, 17138}, {7184, 30969}, {11680, 21299}, {14951, 30030}, {18726, 20234}, {20255, 27017}, {20930, 30089}, {22370, 33113}, {27436, 27458}, {30031, 30043}, {30042, 30047}, {30046, 30075}, {30054, 30081}, {30072, 30092}, {30083, 30084}

X(30035) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics -a^3 b^2 + a b^4 + a^3 b c - a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 - a^2 b c^2 + 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(30035) lies on these lines: {2, 7}, {56, 24563}, {75, 1953}, {92, 31917}, {946, 24280}, {1334, 25601}, {1742, 20556}, {2995, 30034}, {3061, 26538}, {3262, 18161}, {3720, 20557}, {3729, 22019}, {3836, 11681}, {3869, 24325}, {3875, 17197}, {3942, 20930}, {4431, 10447}, {4772, 20535}, {7146, 26665}, {17233, 17787}, {17298, 24237}, {20435, 20892}, {22370, 27514}, {23682, 25570}, {24320, 27401}, {24334, 27059}, {27428, 27434}, {27448, 27465}, {30030, 30064}, {30038, 32092}, {30055, 30067}, {30056, 30065}

X(30036) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30036) lies on these lines: {1, 2}, {9, 17152}, {56, 24602}, {69, 28351}, {75, 17451}, {87, 26825}, {304, 2170}, {672, 21281}, {902, 17696}, {982, 26562}, {1743, 20109}, {1909, 30949}, {2140, 3761}, {2238, 24735}, {2275, 20255}, {2975, 24586}, {3061, 3263}, {3501, 27109}, {3662, 21226}, {3729, 20244}, {3760, 17761}, {3765, 30985}, {3812, 24629}, {3869, 17755}, {3933, 4904}, {3975, 30961}, {4441, 20257}, {7791, 32948}, {8666, 29473}, {9311, 18157}, {16975, 21240}, {17137, 21384}, {17155, 20590}, {17234, 25303}, {17448, 30945}, {17474, 30962}, {17760, 31130}, {18031, 27424}, {18192, 26841}, {20435, 20892}, {20594, 20923}, {20943, 30997}, {25760, 26558}, {25957, 26561}, {27431, 27496}, {30031, 30086}, {30051, 30065}, {30064, 30078}

X(30037) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30037) lies on these lines: {2, 7}, {322, 27487}, {1086, 28366}, {1329, 1463}, {1469, 24982}, {1959, 20905}, {3436, 4334}, {4416, 24237}, {7209, 27442}, {14951, 30030}, {17182, 24177}, {18208, 23677}, {20892, 20895}, {21191, 27855}, {21616, 32857}, {23682, 24230}, {24199, 24220}, {24705, 27342}, {29960, 30054}, {30032, 30038}, {30046, 30065}, {30051, 30058}

X(30038) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30038) lies on these lines: {1, 2}, {56, 24586}, {65, 24631}, {75, 3061}, {76, 27424}, {141, 17448}, {142, 31997}, {194, 3663}, {244, 26562}, {274, 1432}, {304, 10009}, {325, 17062}, {330, 3662}, {333, 1429}, {514, 27951}, {672, 17152}, {732, 1107}, {908, 30075}, {960, 17755}, {1015, 21240}, {1334, 27109}, {1423, 4416}, {1447, 17739}, {1475, 17137}, {1909, 20335}, {1930, 20892}, {1959, 4359}, {2170, 20911}, {2176, 17353}, {2227, 23473}, {2321, 17144}, {2329, 17277}, {2887, 26561}, {3846, 26558}, {3865, 27447}, {3905, 4361}, {3946, 33296}, {4352, 17304}, {4431, 17143}, {4660, 7791}, {4673, 27474}, {5249, 30077}, {5253, 24602}, {5255, 16061}, {5563, 29473}, {7018, 18299}, {7146, 19804}, {8616, 17696}, {9436, 28391}, {9575, 10436}, {14210, 30044}, {15983, 28402}, {16502, 24549}, {16604, 20255}, {16969, 17279}, {17232, 31999}, {17242, 32095}, {17245, 25130}, {17278, 24667}, {17451, 26234}, {17474, 30941}, {17753, 20348}, {17754, 21281}, {17761, 20888}, {18055, 21101}, {20530, 21025}, {20880, 30082}, {20963, 28369}, {21226, 31004}, {21412, 30074}, {24177, 24621}, {24628, 32636}, {27429, 27501}, {27430, 30998}, {30032, 30037}, {30035, 32092}, {30062, 30081}, {30065, 30086}, {30071, 30078}

X(30038) = complement of X(17752)

X(30039) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30039) lies on these lines: {2, 11}, {3119, 20258}, {3936, 30062}, {20359, 26560}, {20892, 20901}, {20940, 30090}, {30030, 30033}, {30032, 30075}, {30091, 30565}

X(30040) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30040) lies on these lines: {2, 3}

X(30041) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30041) lies on these lines: {2, 3}, {3261, 30095}

X(30042) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30042) lies on these lines: {2, 31}, {16060, 32947}, {30030, 30055}, {30034, 30047}, {30043, 30046}, {30048, 30086}

X(30043) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30043) lies on these lines: {2, 32}, {30031, 30034}, {30032, 30088}, {30042, 30046}, {30054, 30072}, {30085, 30092}

X(30044) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30044) lies on these lines: {2, 37}, {354, 25277}, {740, 28352}, {1107, 27017}, {3264, 17245}, {3596, 27147}, {3742, 25295}, {3766, 30091}, {3948, 24199}, {3971, 24182}, {3975, 26806}, {4110, 29572}, {4696, 24325}, {4709, 4742}, {4871, 22167}, {6381, 30045}, {6383, 27438}, {14210, 30038}, {14951, 30030}, {17300, 25298}, {17755, 28402}, {20348, 25731}, {20367, 25728}, {25125, 26756}, {30051, 30066}, {30056, 30089}

X(30045) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30045) lies on these lines: {2, 39}, {10, 20892}, {3264, 25102}, {3662, 6376}, {6381, 30044}, {30031, 30034}, {30033, 30088}, {30072, 30081}

X(30046) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30046) lies on these lines: {2, 41}, {30030, 30087}, {30034, 30075}, {30037, 30065}, {30042, 30043}, {30047, 30066}

X(30047) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics -a^3 b^3 + a^2 b^4 - a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - 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(30047) lies on these lines: {1, 2}, {310, 17050}, {16748, 24199}, {30034, 30042}, {30046, 30066}, {30055, 30086}

X(30048) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30048) lies on these lines: {1, 2}, {142, 30631}, {6063, 30097}, {6374, 30090}, {17149, 20335}, {17231, 25134}, {20889, 20892}, {27436, 27458}, {28351, 30941}, {30042, 30086}, {30051, 30062}

X(30049) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30049) lies on these lines: {2, 44}, {30095, 30098}, {2171, 30097}, {3262, 20892}, {4358, 30083}, {14951, 30030}, {17791, 30090}, {21191, 30094}, {30059, 30089}

X(30050) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30050) lies on these lines: {2, 45}, {14951, 30030}

X(30051) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30051) lies on these lines: {2, 11}, {25306, 26531}, {30030, 30031}, {30032, 30053}, {30034, 30042}, {30036, 30065}, {30037, 30058}, {30044, 30066}, {30048, 30062}

X(30052) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30052) lies on these lines: {2, 6}, {75, 17443}, {87, 30969}, {572, 24587}, {2269, 33113}, {2293, 5014}, {3006, 3056}, {3765, 29967}, {5283, 17202}, {7184, 25957}, {17138, 20992}, {17220, 32933}, {20258, 30059}, {20435, 20892}, {22343, 30953}, {24988, 25571}, {28388, 32859}, {30032, 30054}, {30070, 30085}

X(30053) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30053) lies on these lines: {2, 12}, {1959, 4359}, {20258, 24547}, {30030, 30033}, {30032, 30051}

X(30054) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30054) lies on these lines: {2, 39}, {330, 3596}, {334, 27424}, {1921, 7187}, {1930, 20892}, {3264, 17448}, {3963, 31997}, {16722, 26963}, {21608, 30090}, {29960, 30037}, {30031, 30085}, {30032, 30052}, {30034, 30081}, {30043, 30072}

X(30055) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30055) lies on these lines: {2, 6}, {511, 25308}, {958, 6327}, {1107, 17184}, {1230, 17173}, {3948, 17167}, {5247, 32949}, {5790, 15973}, {14206, 30076}, {20432, 30059}, {27424, 27434}, {30030, 30042}, {30035, 30067}, {30047, 30086}

X(30056) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30056) lies on these lines: {2, 6}, {511, 25279}, {1959, 27705}, {4645, 5260}, {4754, 26806}, {5297, 24678}, {5818, 15973}, {10459, 17765}, {16589, 17202}, {20880, 20892}, {21299, 33108}, {27430, 27465}, {27436, 27438}, {27444, 27458}, {30035, 30065}, {30044, 30089}, {30062, 30063}, {30069, 30082}

X(30057) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30057) lies on these lines: {1, 2}, {4392, 26562}, {17451, 31130}, {20244, 27523}, {20448, 30090}, {25958, 26558}, {30075, 30086}

X(30058) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30058) lies on these lines: {1, 2}, {30037, 30051}, {30078, 30086}

X(30059) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30059) lies on these lines: {1, 2}, {63, 21281}, {92, 17789}, {226, 3765}, {321, 17451}, {322, 27487}, {514, 850}, {726, 20590}, {730, 23682}, {742, 15985}, {908, 3975}, {1441, 20892}, {1909, 5249}, {1953, 20336}, {1959, 3263}, {2295, 5294}, {3208, 17776}, {3596, 29967}, {3686, 22008}, {3915, 11342}, {3936, 25298}, {4001, 17137}, {4383, 24735}, {4433, 28353}, {4437, 9049}, {5255, 16050}, {5258, 24632}, {5847, 28375}, {9902, 17889}, {15973, 29331}, {17050, 20913}, {17167, 28660}, {17374, 28362}, {17862, 20436}, {18134, 24524}, {20258, 30052}, {20432, 30055}, {20924, 30078}, {30049, 30089}

X(30060) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30060) lies on these lines: {2, 667}, {514, 30095}, {3766, 30093}, {3835, 27467}, {4391, 30062}, {4462, 30075}, {4932, 27451}, {23803, 27466}

X(30061) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30061) lies on these lines: {2, 650}, {513, 25299}, {514, 24622}, {659, 2517}, {784, 22223}, {812, 28398}, {3287, 26652}, {4036, 4782}, {4391, 6002}, {4411, 30090}, {4462, 7192}, {11068, 30093}, {27438, 27466}, {28399, 29362}

X(30062) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30062) lies on these lines: {2, 31}, {3936, 30039}, {4391, 30060}, {14951, 30030}, {27436, 30075}, {30038, 30081}, {30048, 30051}, {30056, 30063}

X(30063) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30063) lies on these lines: {1, 2}, {76, 24199}, {141, 25614}, {142, 6376}, {756, 26562}, {908, 30079}, {3263, 21921}, {3501, 25101}, {3663, 27269}, {3739, 21025}, {3812, 17755}, {3836, 26558}, {4357, 20255}, {4385, 27478}, {4431, 21071}, {4660, 33029}, {4967, 21024}, {5179, 30031}, {5260, 24602}, {6381, 30044}, {17050, 18140}, {17056, 25125}, {17243, 21868}, {17245, 25102}, {17669, 21241}, {17758, 30099}, {17760, 30758}, {20257, 30963}, {20888, 20892}, {27430, 27436}, {27432, 27437}, {30056, 30062}, {30065, 30073}

X(30064) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30064) lies on these lines: {2, 3}, {17920, 18659}, {30030, 30035}, {30036, 30078}

X(30065) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30065) lies on these lines: {2, 3}, {511, 26531}, {24602, 30001}, {30030, 30042}, {30035, 30056}, {30036, 30051}, {30037, 30046}, {30038, 30086}, {30063, 30073}

X(30066) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30066) lies on these lines: {2, 3}, {30044, 30051}, {30046, 30047}

X(30067) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30067) lies on these lines: {2, 3}, {20892, 30078}, {30035, 30055}

X(30068) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30068) lies on these lines: {2, 3}, {30076, 30077}

X(30069) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30069) lies on these lines: {2, 3}, {30056, 30082}

X(30070) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30070) lies on these lines: {2, 3}, {30052, 30085}

X(30071) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30071) lies on these lines: {2, 3}, {20435, 20892}, {30038, 30078}

X(30072) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30072) lies on these lines: {2, 3}, {30034, 30092}, {30043, 30054}, {30045, 30081}

X(30073) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30073) lies on these lines: {2, 36}, {4391, 30060}, {30030, 30031}, {30063, 30065}

X(30074) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30074) lies on these lines: {2, 38}, {21412, 30038}, {22167, 26815}, {27963, 31108}, {30030, 30042}

X(30075) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30075) lies on these lines: {2, 12}, {908, 30038}, {4462, 30060}, {5176, 17752}, {15985, 20245}, {26563, 30097}, {27436, 30062}, {30030, 30031}, {30032, 30039}, {30034, 30046}, {30035, 30056}, {30057, 30086}

X(30076) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30076) lies on these lines: {1, 20557}, {2, 7}, {72, 15973}, {92, 17789}, {306, 17787}, {321, 1959}, {1046, 21077}, {1469, 24997}, {2975, 32772}, {3729, 10478}, {3782, 28358}, {3869, 32771}, {4054, 17167}, {4416, 22020}, {4468, 30094}, {4656, 17182}, {5176, 33072}, {6354, 28391}, {7081, 20498}, {11681, 25957}, {12609, 19879}, {14206, 30055}, {14213, 20234}, {15985, 31993}, {17752, 17778}, {17792, 20487}, {18161, 20928}, {19514, 22345}, {20921, 30090}, {22097, 28803}, {24320, 27388}, {30030, 30031}, {30068, 30077}

X(30077) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30077) lies on these lines: {2, 58}, {5249, 30038}, {10436, 15985}, {15991, 17062}, {27436, 29967}, {30030, 30042}, {30031, 30034}, {30056, 30062}, {30068, 30076}

X(30078) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30078) lies on these lines: {2, 7}, {56, 24545}, {274, 17167}, {306, 19811}, {1469, 24996}, {1473, 27388}, {1930, 17864}, {1959, 17862}, {3663, 17182}, {5205, 20498}, {5563, 25526}, {7248, 30960}, {15973, 24474}, {17171, 31623}, {18162, 31631}, {20892, 30067}, {20924, 30059}, {21214, 24159}, {21319, 27385}, {21616, 33099}, {22020, 24237}, {26747, 28395}, {30030, 30042}, {30036, 30064}, {30038, 30071}, {30058, 30086}

X(30079) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30079) lies on these lines: {2, 65}, {908, 30063}, {1959, 4359}, {4699, 20535}, {17451, 20258}, {20435, 20892}, {30030, 30031}

X(30080) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30080) lies on these lines: {2, 66}

X(30081) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30081) lies on these lines: {2, 32}, {30034, 30054}, {30038, 30062}, {30045, 30072}

X(30082) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30082) lies on these lines: {2, 85}, {75, 20535}, {321, 1959}, {1441, 20258}, {3263, 27424}, {3869, 20348}, {4444, 4462}, {17451, 20905}, {20435, 20892}, {20880, 30038}, {24631, 24633}, {30056, 30069}

X(30083) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30083) lies on these lines: {2, 45}, {4358, 30049}, {18050, 18359}, {30030, 30086}, {30034, 30084}

X(30084) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30084) lies on these lines: {2, 44}, {30034, 30083}

X(30085) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30085) lies on these lines: {2, 99}, {30031, 30054}, {30043, 30092}, {30045, 30072}, {30052, 30070}, {30087, 30089}

X(30086) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30086) lies on these lines: {2, 11}, {30030, 30083}, {30031, 30036}, {30038, 30065}, {30042, 30048}, {30047, 30055}, {30057, 30075}, {30058, 30078}, {30089, 30091}

X(30087) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30087) lies on these lines: {2, 101}, {1111, 30097}, {30030, 30046}, {30085, 30089}

X(30088) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30088) lies on these lines: {2, 99}, {17886, 20892}, {20951, 30090}, {30032, 30043}, {30033, 30045}

X(30089) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30089) lies on these lines: {2, 45}, {20348, 27834}, {20892, 20895}, {20930, 30034}, {30044, 30056}, {30049, 30059}, {30085, 30087}, {30086, 30091}

X(30090) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics 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(30090) lies on these lines: {2, 37}, {7, 3975}, {76, 24199}, {85, 1921}, {142, 3596}, {239, 21785}, {304, 10009}, {322, 27487}, {341, 3812}, {668, 17298}, {726, 24174}, {740, 21214}, {1423, 17755}, {3264, 17234}, {3620, 25280}, {3662, 6376}, {3663, 30830}, {3765, 26806}, {3834, 30473}, {3873, 25277}, {3912, 4110}, {3963, 27147}, {4033, 17241}, {4411, 30061}, {4494, 20195}, {4673, 10179}, {6374, 30048}, {6385, 16708}, {6682, 17038}, {7017, 25993}, {9311, 18157}, {17050, 30022}, {17117, 17144}, {17300, 24524}, {17304, 18140}, {17375, 25298}, {17789, 20258}, {17791, 30049}, {18044, 27191}, {18155, 30093}, {19514, 30273}, {20352, 25279}, {20448, 30057}, {20914, 30031}, {20921, 30076}, {20926, 30033}, {20930, 30034}, {20940, 30039}, {20951, 30088}, {21606, 30091}, {21608, 30054}, {22167, 30957}, {27485, 30094}, {28361, 32915}, {28809, 31995}

X(30091) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30091) lies on these lines: {2, 900}, {3766, 30044}, {20892, 21433}, {21606, 30090}, {30039, 30565}, {30086, 30089}

X(30092) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30092) lies on these lines: {2, 39}, {57, 3975}, {85, 1921}, {99, 28348}, {304, 6382}, {350, 21214}, {728, 17787}, {1909, 3596}, {1975, 28383}, {2979, 20352}, {3263, 20436}, {3264, 17378}, {4087, 18156}, {6374, 18157}, {10009, 20892}, {16992, 19533}, {17149, 24215}, {20924, 30059}, {28365, 30940}, {30034, 30072}, {30043, 30085}

X(30093) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30093) lies on these lines: {2, 647}, {3766, 30060}, {11068, 30061}, {14837, 21438}, {18155, 30090}

X(30094) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics (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(30094) lies on these lines: {2, 649}, {513, 23301}, {514, 850}, {522, 3005}, {667, 28401}, {693, 21191}, {812, 28374}, {834, 24353}, {3569, 28478}, {3667, 5996}, {3676, 30097}, {3766, 30060}, {4147, 20983}, {4468, 30076}, {4778, 9148}, {5466, 28565}, {8639, 28373}, {9002, 15985}, {20979, 26049}, {27485, 30090}, {28372, 30049}, {30095, 30098}

X(30095) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30095) lies on these lines: {2, 659}, {514, 30060}, {693, 21191}, {3261, 30041}, {30086, 30089}

X(30096) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30096) lies on these lines: {2, 669}, {3261, 30041}, {3766, 30060}

X(30097) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30097) lies on these lines: {1, 24257}, {2, 7}, {10, 1469}, {12, 1463}, {56, 19533}, {65, 24325}, {75, 7146}, {85, 1921}, {86, 1429}, {141, 16603}, {241, 4032}, {256, 24239}, {274, 1432}, {334, 31643}, {388, 4334}, {497, 4335}, {942, 15973}, {946, 988}, {950, 15971}, {980, 3663}, {1086, 11672}, {1111, 30087}, {1125, 1284}, {1441, 20892}, {1446, 30031}, {1450, 3485}, {1458, 10106}, {1959, 26538}, {2171, 30049}, {3212, 4699}, {3649, 28389}, {3664, 24237}, {3665, 16888}, {3666, 14749}, {3676, 30094}, {3686, 15983}, {3739, 15985}, {3782, 21471}, {3912, 17787}, {3946, 17197}, {4077, 28372}, {4292, 9840}, {4298, 28386}, {4475, 23688}, {4858, 18726}, {5057, 30359}, {5228, 28365}, {5253, 24563}, {5284, 8238}, {5773, 21748}, {6063, 30048}, {7249, 27447}, {12047, 32857}, {13411, 19514}, {15844, 25365}, {17062, 26543}, {17084, 17302}, {17132, 22019}, {17300, 17752}, {17451, 20905}, {17474, 26818}, {19637, 21636}, {24208, 24224}, {25065, 29069}, {26098, 26929}, {26563, 30075}, {30044, 30056}

X(30098) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30098) lies on these lines: {2, 896}, {30049, 30095}, {16756, 17204}, {21191, 30094}, {30030, 30042}

X(30099) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 100

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

X(30099) lies on these lines: {2, 38}, {1930, 20892}, {2170, 10009}, {3739, 21951}, {9311, 18157}, {14951, 30030}, {17758, 30063}

X(30100) = EULER LINE INTERCEPT OF X(8537)X(14531)

Barycentrics a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+7 a^6 b^2 c^2-a^4 b^4 c^2-7 a^2 b^6 c^2+4 b^8 c^2+2 a^6 c^4-a^4 b^2 c^4-4 a^2 b^4 c^4-5 b^6 c^4+2 a^4 c^6-7 a^2 b^2 c^6-5 b^4 c^6-3 a^2 c^8+4 b^2 c^8+c^10) : :
Barycentrics S^2 (8 R^4 - 6 R^2 SW + SW^2) + SB SC (-16 R^4 + 9 R^2 SW - SW^2) : :

As a point on the Euler line, X(30100) has Shinagawa coefficients (2(E+2F)F, E²+EF-4F²).

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28720.

X(30100) lies on these lines: {2,3}, {8537,14531}, {11425,15872}

X(30100) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3520,7512), (378,3541,3520), (427,12225,4), (1593,7503,4), (1885,5133,4), (9818,12084,3547)

X(30101) = X(950)X(3664)∩X(3739)X(5745)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28722.

X(30101) lies on these lines: {950, 3664}, {3739, 5745}, {5088, 17175}

X(30102) = X(140)X(6709)∩X(6748)X(11245)

Barycentrics SB*SC*(S^2-4*R^2*(2*SW-SC)+SC*(2*SA+2*SB-SC)+2*SW^2)*(S^2-4*R^2*(2*SW-SB)+SB*(2*SA+2*SC-SB)+2*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28722.

X(30102) lies on these lines: {140, 6709}, {6748, 11245}

X(30102) = isogonal conjugate of X(39243)

Points Celaeno(h,i,j,k,u,v,w): X(30103)-X(30178)

Definition: Point Celaeno(h,i,j,k,u,v,w,a,b,c) = f(h,i,j,k,u,v,w,a,b,c) : f(h,i,j,k,u,v,w,b,c,a) : f(h,i,j,k,u,v,w,c,a,b) (barycentrics), where

f(h,i,j,k,u,v,w,a,b,c) = h (a^4 + b^4 + c^4) + i (a^3 b + b^3 c + c^3 a + a^3 c + b^3 a + c^3 b) + j (b^2 c^2 + c^2 a^2 + a^2 b^2) + k (a^2 b c + a b^2 c + a b c^2) + a (u (a^3 + b^3 + c^3) + v (a^2 b + b^2 c + c^2 a + a^2 c + b^2 a + c^2 b) + w a b c)

where h, i, j, k, u, v, w are real numbers, not all zero. These points lie on the line X(1)X(2). (Clark Kimberling, December 14, 2018)

X(30103) = POINT CELAENO(1,0,0,0,0,0,1)

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

X(30103) lies on these lines: {1, 2}, {11, 7819}, {12, 8361}, {35, 7807}, {36, 6656}, {55, 32954}, {56, 7866}, {57, 28773}, {58, 4766}, {75, 20267}, {101, 24995}, {172, 626}, {192, 7945}, {315, 609}, {325, 5280}, {330, 7932}, {350, 7832}, {384, 3583}, {388, 32951}, {390, 33183}, {495, 33186}, {496, 33185}, {497, 14069}, {993, 17550}, {1015, 7852}, {1038, 28405}, {1056, 32953}, {1058, 32952}, {1447, 17192}, {1468, 30816}, {1478, 14064}, {1479, 14001}, {1500, 7874}, {1909, 7828}, {1914, 6680}, {2242, 7867}, {2275, 7834}, {2276, 3788}, {2887, 29473}, {3509, 17211}, {3552, 4324}, {3585, 5025}, {3600, 33182}, {3601, 28806}, {3746, 26629}, {3760, 7795}, {3761, 3767}, {3816, 17540}, {3825, 17541}, {4293, 33180}, {4294, 33181}, {4299, 32974}, {4302, 32973}, {4316, 6655}, {4325, 7933}, {4330, 33225}, {4366, 14043}, {4396, 7794}, {4400, 7755}, {4857, 7892}, {5010, 16925}, {5204, 11287}, {5217, 11288}, {5218, 33189}, {5225, 14039}, {5229, 33285}, {5258, 26558}, {5270, 6645}, {5277, 20541}, {5294, 28763}, {5299, 7792}, {5433, 8362}, {5563, 8363}, {6284, 8369}, {7280, 7791}, {7288, 32956}, {7296, 7759}, {7354, 33184}, {7741, 7770}, {7836, 25264}, {7841, 10483}, {7873, 9341}, {7887, 7951}, {8357, 15326}, {8360, 18990}, {8364, 15325}, {8365, 15172}, {8368, 15171}, {9654, 33240}, {9655, 33241}, {9668, 33242}, {9669, 33237}, {10588, 32955}, {10589, 16045}, {10590, 33199}, {10591, 33198}, {10895, 11318}, {10896, 11286}, {14035, 18514}, {14063, 18513}, {17686, 25639}, {17694, 26582}, {17737, 20888}, {25598, 26242}, {31452, 33222}

X(30103) = {X(1),X(2)}-harmonic conjugate of X(30104)

X(30104) = POINT CELAENO(1,0,0,0,0,0,-1)

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

X(30104) lies on these lines: {1, 2}, {11, 8361}, {12, 7819}, {35, 6656}, {36, 7807}, {55, 7866}, {56, 32954}, {172, 6680}, {192, 7932}, {304, 20267}, {315, 7031}, {325, 5299}, {330, 7945}, {350, 7828}, {384, 3585}, {388, 14069}, {390, 33182}, {495, 33185}, {496, 33186}, {497, 32951}, {595, 4766}, {626, 1914}, {1003, 10483}, {1015, 7874}, {1040, 28405}, {1056, 32952}, {1058, 32953}, {1329, 17540}, {1420, 28773}, {1478, 14001}, {1479, 14064}, {1500, 7852}, {1697, 28806}, {1909, 7832}, {2241, 7867}, {2275, 3788}, {2276, 7834}, {3496, 17211}, {3552, 4316}, {3583, 5025}, {3589, 31460}, {3600, 33183}, {3746, 8363}, {3760, 3767}, {3761, 7795}, {3814, 17541}, {3822, 17686}, {3915, 30816}, {4251, 24995}, {4279, 29990}, {4293, 33181}, {4294, 33180}, {4299, 32973}, {4302, 32974}, {4324, 6655}, {4325, 33225}, {4330, 7933}, {4366, 4857}, {4396, 7755}, {4400, 7794}, {5010, 7791}, {5204, 11288}, {5217, 11287}, {5218, 32956}, {5225, 33285}, {5229, 14039}, {5248, 17550}, {5251, 26558}, {5255, 30837}, {5270, 7892}, {5280, 7792}, {5332, 7759}, {5432, 8362}, {5563, 26686}, {6284, 33184}, {6645, 14043}, {7280, 16925}, {7288, 33189}, {7354, 8369}, {7741, 7887}, {7770, 7951}, {7778, 16502}, {7797, 25264}, {7889, 31476}, {7913, 31451}, {8357, 15338}, {8360, 15171}, {8368, 18990}, {8616, 27259}, {9654, 33237}, {9655, 33242}, {9668, 33241}, {9669, 33240}, {10588, 16045}, {10589, 32955}, {10590, 33198}, {10591, 33199}, {10895, 11286}, {10896, 11318}, {14035, 18513}, {14063, 18514}, {15172, 33213}, {16706, 24786}, {17674, 24582}, {25542, 33034}, {31452, 33221}

X(30104) = {X(1),X(2)}-harmonic conjugate of X(30103)

X(30105) = POINT CELAENO(0,1,0,0,1,0,0)

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

X(30105) lies on these lines: {1, 2}, {3721, 25497}, {5264, 26562}, {11287, 17290}, {16061, 24046}, {16600, 24549}, {16887, 16968}, {16974, 21240}, {17046, 25598}, {17211, 26099}

X(30105) = {X(1),X(2)}-harmonic conjugate of X(30108)

X(30106) = POINT CELAENO(0,1,0,0,0,1,0)

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

X(30106) lies on these lines: {1, 2}, {37, 4568}, {86, 101}, {172, 17200}, {190, 16712}, {213, 16887}, {595, 16060}, {894, 17205}, {1015, 3589}, {1334, 25599}, {2295, 24170}, {3294, 16705}, {4670, 17180}, {6693, 26686}, {7757, 17354}, {14210, 21840}, {16549, 27162}, {16706, 17761}, {16720, 21802}, {21070, 33296}

X(30106) = {X(1),X(2)}-harmonic conjugate of X(30109)

X(30107) = POINT CELAENO(0,1,0,0,0,0,1)

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

X(30107) lies on these lines: {1, 2}, {6, 21240}, {58, 24586}, {75, 16600}, {76, 16706}, {169, 10436}, {673, 13740}, {894, 24190}, {964, 24596}, {993, 16060}, {1220, 17682}, {1468, 29473}, {1500, 17279}, {1573, 24735}, {1914, 25497}, {3340, 28777}, {3585, 16910}, {3589, 17750}, {3662, 17499}, {3730, 17353}, {3739, 16583}, {3759, 33297}, {3760, 27040}, {3761, 26978}, {3875, 21070}, {3946, 21071}, {3948, 32774}, {3959, 24254}, {3997, 21281}, {4000, 20888}, {4044, 19785}, {4251, 24549}, {4657, 16589}, {4687, 28594}, {4751, 16611}, {5267, 22267}, {5358, 26643}, {5902, 26562}, {6376, 17370}, {7713, 15149}, {7719, 17913}, {7866, 20544}, {9798, 16412}, {12782, 33159}, {16061, 25440}, {16605, 31238}, {16887, 21384}, {17282, 17758}, {17357, 20691}, {17366, 21024}, {17383, 27269}, {17489, 31077}, {17540, 25992}, {17671, 32773}, {17754, 24170}, {19281, 24588}, {19786, 30830}, {20606, 24220}, {20963, 30945}, {24046, 24631}, {25610, 27076}, {26035, 32092}

X(30107) = complement of X(27248)
X(30107) = {X(1),X(2)}-harmonic conjugate of X(30110)

X(30108) = POINT CELAENO(0,1,0,0,-1,0,0)

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

X(30108) lies on these lines: {1, 2}, {9, 4568}, {32, 4797}, {41, 1930}, {75, 101}, {213, 4372}, {304, 4251}, {758, 24586}, {993, 17755}, {2224, 30894}, {2251, 4376}, {2280, 14210}, {3263, 16788}, {3761, 9317}, {3868, 29473}, {3905, 16600}, {4136, 31284}, {4168, 17046}, {4386, 24254}, {4390, 4986}, {4482, 4737}, {4561, 17277}, {4680, 4766}, {5011, 24282}, {5902, 24602}, {6381, 24249}, {17141, 17736}, {17211, 26085}, {20267, 24995}, {20893, 24333}, {30878, 30920}, {30879, 30899}, {30881, 30887}, {30882, 30885}, {30883, 30916}, {30890, 30905}, {30903, 30906}, {30931, 30933}

X(30108) = {X(1),X(2)}-harmonic conjugate of X(30105)

X(30109) = POINT CELAENO(0,1,0,0,0,-1,0)

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

X(30109) lies on these lines: {1, 2}, {36, 24602}, {39, 20255}, {76, 2140}, {99, 17729}, {116, 325}, {141, 1573}, {194, 24190}, {350, 17761}, {514, 1921}, {538, 1086}, {712, 21331}, {730, 3836}, {758, 17755}, {993, 24586}, {1107, 16887}, {1909, 17758}, {1930, 17451}, {1975, 14377}, {2170, 14210}, {2321, 20174}, {2975, 29473}, {3263, 4568}, {3286, 29772}, {3294, 17152}, {3454, 26558}, {3670, 26562}, {3730, 21281}, {3761, 30949}, {3930, 4986}, {3932, 14839}, {3933, 21258}, {3934, 21025}, {3997, 17353}, {4973, 24628}, {5883, 24631}, {6381, 20335}, {7283, 27000}, {8619, 20598}, {16549, 27109}, {16552, 17137}, {16975, 30945}, {17050, 20888}, {17143, 21070}, {17277, 22008}, {17753, 27523}, {17789, 20919}, {18061, 20947}, {18089, 33072}, {18145, 30997}, {18792, 27169}, {20257, 21071}, {20271, 24166}, {20453, 27808}, {20470, 29459}, {20893, 21443}, {20923, 24220}

X(30109) = {X(1),X(2)}-harmonic conjugate of X(30106)

X(30110) = POINT CELAENO(0,1,0,0,0,0,-1)

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

X(30110) lies on these lines: {1, 2}, {9, 16887}, {31, 29473}, {39, 17279}, {57, 28777}, {101, 24549}, {172, 25497}, {213, 30945}, {304, 16600}, {344, 25092}, {595, 24586}, {1015, 24652}, {1930, 26242}, {2140, 17282}, {2176, 21240}, {3501, 24170}, {3583, 16910}, {3760, 26978}, {3761, 27040}, {4253, 17353}, {4278, 16050}, {4441, 24790}, {4687, 18157}, {4851, 20970}, {5248, 16060}, {5264, 24602}, {5267, 17696}, {5692, 26689}, {5750, 25504}, {5903, 26562}, {7866, 30825}, {8362, 16593}, {11365, 16412}, {16604, 17357}, {17358, 27318}, {17391, 27320}, {17682, 32942}, {17683, 24552}, {20271, 24254}, {24036, 25918}, {24598, 33157}, {28594, 30758}

X(30110) = complement of X(27299)
X(30110) = {X(1),X(2)}-harmonic conjugate of X(30107)

X(30111) = POINT CELAENO(0,0,1,0,1,0,0)

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

X(30111) lies on these lines: {1, 2}, {83, 3721}, {150, 17302}, {257, 5299}, {384, 3670}, {1003, 17595}, {1759, 7787}, {1966, 30892}, {2243, 12150}, {2280, 30894}, {3096, 4950}, {3125, 20179}, {3782, 8370}, {3953, 6645}, {4366, 4424}, {4799, 7812}, {5902, 14621}, {7785, 17211}, {7808, 18055}, {11321, 17054}, {16519, 18140}, {16915, 24046}

X(30111) = {X(1),X(2)}-harmonic conjugate of X(30113)

X(30112) = POINT CELAENO(0,0,1,0,0,1,0)

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

X(30112) lies on these lines: {1, 2}, {36, 14621}, {595, 17684}, {2295, 7786}, {3329, 16788}, {5264, 7824}, {5710, 11285}, {7757, 24330}, {24190, 27162}, {24358, 24508}

X(30112) = {X(1),X(2)}-harmonic conjugate of X(30114)

X(30113) = POINT CELAENO(0,0,1,0,-1,0,0)

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

X(30113) lies on these lines: {1, 2}, {32, 18055}, {36, 335}, {1078, 3721}, {1759, 7793}, {1914, 18061}, {2896, 17211}, {3670, 7824}, {3782, 8356}, {4799, 7811}, {4950, 7752}, {9310, 30894}

X(30113) = {X(1),X(2)}-harmonic conjugate of X(30111)

X(30114) = POINT CELAENO(0,0,1,0,0,-1,0)

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

X(30114) lies on these lines: {1, 2}, {6, 668}, {76, 2295}, {101, 16997}, {150, 17300}, {171, 4112}, {192, 1018}, {194, 16549}, {213, 6376}, {335, 5902}, {384, 5264}, {385, 16788}, {595, 16916}, {894, 3761}, {985, 24294}, {1655, 3730}, {1909, 17750}, {2176, 18140}, {2242, 18047}, {3159, 25270}, {3230, 30963}, {3294, 27269}, {3501, 25264}, {3997, 6381}, {4253, 21226}, {4360, 4595}, {4465, 18146}, {4562, 24281}, {4692, 31317}, {4713, 18145}, {4721, 20943}, {5280, 17743}, {5710, 7770}, {7757, 20331}, {9331, 17319}, {16466, 26687}, {17152, 26100}, {20247, 28598}, {20963, 24524}, {24190, 26978}

X(30114) = {X(1),X(2)}-harmonic conjugate of X(30112)

X(30115) = POINT CELAENO(0,0,0,1,1,0,0)

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

X(30115) lies on these lines: {1, 2}, {6, 3940}, {21, 3453}, {30, 4415}, {31, 5692}, {35, 2292}, {36, 38}, {37, 101}, {39, 16519}, {45, 16418}, {55, 1324}, {58, 72}, {63, 4257}, {86, 4561}, {98, 29067}, {100, 4424}, {142, 26728}, {171, 758}, {172, 3954}, {190, 4234}, {192, 30933}, {214, 17457}, {228, 4276}, {238, 10176}, {350, 30893}, {376, 4419}, {392, 3744}, {404, 3670}, {442, 24160}, {443, 24159}, {474, 24046}, {540, 33066}, {580, 31837}, {581, 5810}, {595, 960}, {601, 5693}, {609, 5282}, {750, 5902}, {756, 5251}, {894, 4568}, {984, 993}, {986, 25440}, {990, 6282}, {991, 18446}, {996, 4737}, {999, 3242}, {1038, 4306}, {1043, 2901}, {1046, 4067}, {1060, 5820}, {1064, 6326}, {1104, 5044}, {1390, 2224}, {1468, 5904}, {1724, 3876}, {1757, 4134}, {2172, 9310}, {2276, 30904}, {2298, 21078}, {2303, 22021}, {2329, 28594}, {2339, 3749}, {2372, 29055}, {2392, 7186}, {2647, 3947}, {3072, 31806}, {3073, 20117}, {3159, 7283}, {3191, 22022}, {3419, 17720}, {3454, 7270}, {3465, 4304}, {3576, 6211}, {3666, 4256}, {3678, 5247}, {3721, 5277}, {3735, 4386}, {3782, 11112}, {3822, 17719}, {3841, 24161}, {3869, 5264}, {3878, 5255}, {3927, 4252}, {3930, 16785}, {3953, 5253}, {4363, 19276}, {4414, 5010}, {4680, 25760}, {4692, 32927}, {4864, 5049}, {4975, 32943}, {5119, 17461}, {5280, 33299}, {5315, 17469}, {5710, 5730}, {5711, 12635}, {5713, 5761}, {5719, 17056}, {5724, 17757}, {5883, 17122}, {6381, 24291}, {7308, 16485}, {8691, 28482}, {9945, 17246}, {11813, 33106}, {13161, 17647}, {16371, 17595}, {16393, 32933}, {16408, 17054}, {17579, 33151}, {18061, 20179}, {18393, 33104}, {19287, 19765}, {22072, 33178}, {29066, 30910}, {30892, 32922}

X(30115) = {X(1),X(2)}-harmonic conjugate of X(30117)

X(30116) = POINT CELAENO(0,0,0,1,0,1,0)

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

X(30116) lies on these lines: {1, 2}, {6, 1573}, {9, 3997}, {12, 10571}, {21, 5264}, {31, 5251}, {35, 4216}, {36, 750}, {37, 517}, {38, 5902}, {40, 19262}, {45, 4274}, {55, 859}, {56, 16374}, {58, 958}, {73, 9578}, {86, 996}, {101, 5275}, {171, 993}, {181, 2099}, {226, 24806}, {355, 581}, {388, 4306}, {405, 595}, {484, 4414}, {495, 17056}, {500, 18525}, {515, 991}, {538, 4363}, {663, 4844}, {730, 5145}, {748, 5315}, {756, 5692}, {758, 984}, {940, 956}, {942, 21342}, {957, 7962}, {968, 5119}, {970, 1482}, {982, 5883}, {986, 3754}, {990, 30503}, {999, 19261}, {1001, 4279}, {1010, 19807}, {1042, 5290}, {1056, 4648}, {1064, 5587}, {1107, 4253}, {1191, 11108}, {1320, 3032}, {1376, 4256}, {1385, 19550}, {1450, 31231}, {1457, 5219}, {1464, 11237}, {1468, 5258}, {1682, 2098}, {1695, 11531}, {1697, 2654}, {1724, 5260}, {1739, 4850}, {1742, 28164}, {1785, 30687}, {1834, 31419}, {2049, 5793}, {2051, 5603}, {2092, 16777}, {2176, 16589}, {2292, 5903}, {2295, 3730}, {2335, 3577}, {2647, 4347}, {2650, 5904}, {3029, 7983}, {3030, 10700}, {3031, 7984}, {3033, 10695}, {3034, 10699}, {3052, 16418}, {3057, 6051}, {3242, 4260}, {3295, 4245}, {3303, 19241}, {3304, 19249}, {3421, 5712}, {3485, 10408}, {3501, 25092}, {3583, 33104}, {3666, 3753}, {3695, 5835}, {3723, 21857}, {3746, 19245}, {3750, 25439}, {3812, 24046}, {3814, 17717}, {3822, 33111}, {3833, 17063}, {3880, 15569}, {3884, 27784}, {3915, 5259}, {3918, 24440}, {3931, 5836}, {3992, 32931}, {4160, 4481}, {4255, 9709}, {4262, 4386}, {4270, 17275}, {4300, 5691}, {4303, 9613}, {4390, 16785}, {4423, 16483}, {4424, 28606}, {4428, 19255}, {4482, 20131}, {4680, 33072}, {4692, 32771}, {4714, 32860}, {4792, 16672}, {4868, 17592}, {5021, 31490}, {5022, 31468}, {5030, 31449}, {5080, 33112}, {5105, 17303}, {5217, 19254}, {5248, 5255}, {5276, 16788}, {5289, 9564}, {5396, 5790}, {5436, 19246}, {5563, 19292}, {5687, 19765}, {5697, 27785}, {5717, 5795}, {5718, 17757}, {5727, 14547}, {5734, 9569}, {5737, 5774}, {5756, 7174}, {6767, 19250}, {7373, 19248}, {7951, 33105}, {7982, 9548}, {8148, 9566}, {8162, 19275}, {9549, 10440}, {9552, 10944}, {9555, 10950}, {9558, 9616}, {9567, 10247}, {9840, 31785}, {10914, 19257}, {11552, 33098}, {12609, 13161}, {14839, 19258}, {16975, 24512}, {19730, 23853}, {20060, 26131}, {24159, 28629}, {24160, 28628}, {26725, 33127}, {30858, 31479}

X(30116) = {X(1),X(2)}-harmonic conjugate of X(995)

X(30117) = POINT CELAENO(0,0,0,1,-1,0,0)

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

X(30117) lies on these lines: {1, 2}, {3, 17054}, {4, 24159}, {5, 24160}, {6, 15934}, {21, 3670}, {22, 24163}, {23, 24164}, {30, 1086}, {31, 5902}, {32, 20271}, {34, 4306}, {35, 4218}, {36, 244}, {38, 5251}, {45, 16857}, {56, 11334}, {57, 4257}, {58, 942}, {65, 595}, {86, 20924}, {88, 13587}, {100, 1739}, {101, 3290}, {105, 2224}, {106, 1168}, {109, 18838}, {169, 16780}, {171, 5883}, {190, 33309}, {226, 26728}, {229, 849}, {238, 758}, {242, 514}, {320, 540}, {330, 30933}, {484, 902}, {517, 1279}, {572, 20227}, {580, 24474}, {601, 15016}, {712, 24358}, {741, 2690}, {748, 5692}, {754, 24699}, {896, 4880}, {940, 19287}, {950, 23537}, {956, 17597}, {964, 20432}, {977, 17789}, {982, 993}, {986, 5248}, {991, 18443}, {998, 2191}, {1001, 30903}, {1015, 3002}, {1054, 24168}, {1203, 2650}, {1325, 5161}, {1421, 1457}, {1443, 6549}, {1447, 21208}, {1448, 1467}, {1453, 11518}, {1455, 3660}, {1468, 18398}, {1482, 1616}, {1621, 4424}, {1724, 3868}, {1772, 32760}, {1782, 2218}, {1807, 30858}, {1834, 12433}, {1909, 30893}, {1914, 3125}, {1953, 16488}, {2099, 16483}, {2170, 16784}, {2241, 3959}, {2275, 30904}, {2292, 5259}, {2294, 16470}, {2647, 4298}, {2654, 33178}, {2975, 3953}, {3072, 31870}, {3073, 5884}, {3120, 3583}, {3218, 18173}, {3242, 9708}, {3315, 4694}, {3339, 4332}, {3419, 24789}, {3488, 4000}, {3576, 5573}, {3586, 23681}, {3666, 4653}, {3684, 16611}, {3722, 4695}, {3726, 5291}, {3744, 3753}, {3746, 4642}, {3750, 4868}, {3752, 4256}, {3754, 5255}, {3756, 15325}, {3772, 5722}, {3782, 11113}, {3812, 5266}, {3814, 17719}, {3822, 33130}, {3833, 17122}, {3871, 3987}, {3874, 5247}, {3894, 32912}, {3915, 5903}, {3976, 8666}, {3992, 32927}, {4251, 16583}, {4252, 5708}, {4253, 16968}, {4297, 24171}, {4304, 24177}, {4363, 11354}, {4641, 24473}, {4680, 25957}, {4692, 32923}, {4714, 32945}, {4973, 18201}, {5080, 33148}, {5088, 17205}, {5176, 24222}, {5263, 30892}, {5267, 24167}, {5280, 21808}, {5299, 17451}, {5305, 21049}, {5315, 5425}, {5440, 16610}, {5497, 31845}, {6176, 17053}, {6547, 18455}, {7290, 11529}, {7951, 33127}, {8578, 21201}, {8715, 24440}, {10176, 17123}, {10436, 20893}, {10572, 23536}, {10974, 12109}, {11114, 33146}, {11346, 32933}, {11359, 17290}, {13735, 32939}, {15935, 17366}, {16370, 17595}, {16487, 18421}, {16498, 17716}, {16519, 16589}, {16788, 26242}, {16974, 17750}, {17189, 17863}, {17647, 24178}, {17715, 25439}, {17724, 17757}, {17729, 24185}, {20174, 32941}, {21842, 32577}, {23407, 24464}, {24161, 25639}, {24174, 25440}, {24281, 29331}, {26725, 33105}

X(30117) = complement of X(16086)
X(30117) = {X(1),X(2)}-harmonic conjugate of X(30115)

X(30118) = POINT CELAENO(1,0,0,0,1,1,1)

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

X(30118) lies on these lines: {1, 2}, {86, 20267}, {5276, 25598}, {7819, 17602}, {17599, 32954}, {26128, 29473}

X(30118) = {X(1),X(2)}-harmonic conjugate of X(30122)

X(30119) = POINT CELAENO(1,0,0,0,-1,1,1)

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

X(30119) lies on these lines: {1, 2}, {79, 7841}, {758, 17550}, {1003, 5441}, {3336, 7791}, {3649, 33184}, {5131, 32965}, {5221, 11287}, {5902, 6656}, {5903, 26590}, {5904, 26558}, {8357, 11246}, {8360, 16137}, {8368, 15174}, {8369, 10543}, {15079, 32992}, {16118, 33017}, {18398, 26561}

X(30119) = {X(1),X(2)}-harmonic conjugate of X(30123)

X(30120) = POINT CELAENO(1,0,0,0,1,-1,1)

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

X(30120) lies on these lines: {1, 2}, {80, 7887}, {384, 18393}, {484, 16925}, {2099, 32954}, {5443, 7770}, {5444, 11285}, {5445, 33233}, {5697, 26629}, {5902, 26686}, {5903, 7807}, {7819, 15950}, {8361, 10950}, {11813, 16920}, {15228, 33235}, {21842, 26561}

X(30120) = {X(1),X(2)}-harmonic conjugate of X(30124)

X(30121) = POINT CELAENO(1,0,0,0,1,1,-1)

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

X(30121) lies on these lines: {1, 2}, {16989, 17744}, {17380, 24786}

X(30121) = {X(1),X(2)}-harmonic conjugate of X(30125)

X(30122) = POINT CELAENO(1,0,0,0,-1,-1,-1)

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

X(30122) lies on these lines: {1, 2}, {4680, 16060}, {7866, 17599}, {8361, 17602}, {17234, 24786}, {17243, 31460}, {28806, 33178}

X(30122) = {X(1),X(2)}-harmonic conjugate of X(30118)

X(30123) = POINT CELAENO(1,0,0,0,1,-1,-1)

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

X(30123) lies on these lines: {1, 2}, {79, 1003}, {3336, 16925}, {3649, 8369}, {5131, 32964}, {5221, 11288}, {5441, 7841}, {5902, 7807}, {5903, 26629}, {8360, 15174}, {8368, 16137}, {10543, 33184}, {11263, 16919}, {11321, 26725}, {15079, 33249}, {16118, 33007}, {18398, 26686}

X(30123) = {X(1),X(2)}-harmonic conjugate of X(30119)

X(30124) = POINT CELAENO(1,0,0,0,-1,1,-1)

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

X(30124) lies on these lines: {1, 2}, {80, 7770}, {484, 7791}, {2099, 7866}, {3212, 17192}, {3878, 17550}, {5025, 18393}, {5443, 7887}, {5444, 33233}, {5445, 11285}, {5692, 26558}, {5697, 26590}, {5902, 26561}, {5903, 6656}, {7819, 10950}, {8361, 15950}, {15228, 33234}, {21842, 26686}

X(30124) = {X(1),X(2)}-harmonic conjugate of X(30120)

X(30125) = POINT CELAENO(1,0,0,0,-1,-1,1)

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

X(30125) lies on these lines: {1, 2}, {4360, 20267}, {4865, 29473}, {4894, 16060}, {7774, 17744}

X(30125) = {X(1),X(2)}-harmonic conjugate of X(30121)

X(30126) = POINT CELAENO(0,1,0,0,1,1,1)

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

X(30126) lies on these lines: {1, 2}, {86, 16600}, {3509, 17200}, {8025, 20602}, {16583, 28639}, {18157, 32014}

X(30126) = {X(1),X(2)}-harmonic conjugate of X(30130)

X(30127) = POINT CELAENO(0,1,0,0,-1,1,1)

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

X(30127) lies on these lines: {1, 2}, {758, 16060}, {2650, 29473}, {5259, 26689}, {5425, 26562}, {11263, 17680}, {18755, 24254}

X(30127) = {X(1),X(2)}-harmonic conjugate of X(30131)

X(30128) = POINT CELAENO(0,1,0,0,1,-1,1)

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

X(30128) lies on these lines: {1, 2}, {36, 26562}, {2140, 24291}, {3754, 16061}, {17192, 26140}

X(30128) = {X(1),X(2)}-harmonic conjugate of X(30132)

X(30129) = POINT CELAENO(0,1,0,0,1,1,-1)

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

X(30129) lies on these lines: {1, 2}, {86, 24339}, {17200, 18164}, {17469, 29473}

X(30129) = {X(1),X(2)}-harmonic conjugate of X(30133)

X(30130) = POINT CELAENO(0,1,0,0,-1,-1,-1)

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

X(30130) lies on these lines: {1, 2}, {38, 29473}, {58, 17755}, {83, 20947}, {484, 25248}, {1089, 17686}, {1509, 18157}, {1930, 5276}, {3159, 17738}, {3263, 5280}, {3670, 24602}, {3881, 32029}, {3891, 17683}, {3932, 7819}, {4372, 5283}, {4434, 25073}, {4680, 17550}, {5007, 24358}, {5315, 26689}, {5525, 28598}, {7889, 17279}, {16519, 21240}, {17682, 32926}, {17744, 31087}, {19791, 24588}

X(30130) = {X(1),X(2)}-harmonic conjugate of X(30126)

X(30131) = POINT CELAENO(0,1,0,0,1,-1,-1)

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

X(30131) lies on these lines: {1, 2}, {35, 26562}, {4867, 26689}, {5883, 16061}, {17758, 24291}

X(30131) = {X(1),X(2)}-harmonic conjugate of X(30127)

X(30132) = POINT CELAENO(0,1,0,0,-1,1,-1)

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

X(30132) lies on these lines: {1, 2}, {214, 16061}, {3878, 16060}, {4561, 28594}, {5251, 26689}, {9351, 24357}, {16910, 18393}, {21008, 24254}

X(30132) = {X(1),X(2)}-harmonic conjugate of X(30128)

X(30133) = POINT CELAENO(0,1,0,0,-1,-1,1)

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

X(30133) lies on these lines: {1, 2}, {75, 7760}, {169, 3875}, {595, 17755}, {3263, 5299}, {3874, 32029}, {3953, 24602}, {3992, 17541}, {4360, 16600}, {4372, 16975}, {4692, 17686}, {4852, 16583}, {4894, 17550}, {5540, 17489}, {11010, 25248}, {17147, 20602}, {17277, 28594}, {17738, 24068}, {17744, 28598}, {17788, 21425}

X(30133) = {X(1),X(2)}-harmonic conjugate of X(30129)

X(30134) = POINT CELAENO(0,0,1,0,1,1,1)

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

X(30134) lies on these lines: {1, 2}, {3670, 14621}, {7770, 17599}, {7785, 17302}, {17602, 32992}, {18398, 20132}

X(30134) = {X(1),X(2)}-harmonic conjugate of X(30138)

X(30135) = POINT CELAENO(0,0,1,0,-1,1,1)

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

X(30135) lies on these lines: {1, 2}, {79, 7833}, {191, 33063}, {758, 17684}, {3336, 33004}, {3649, 8356}, {5131, 33022}, {5426, 16914}, {5441, 11361}, {5692, 33047}, {5719, 26561}, {5902, 7824}, {8354, 11544}, {8359, 16137}, {8370, 10543}, {11281, 17670}, {15015, 33062}, {15079, 16922}, {16118, 33264}

X(30135) = {X(1),X(2)}-harmonic conjugate of X(30139)

X(30136) = POINT CELAENO(0,0,1,0,1,-1,1)

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

X(30136) lies on these lines: {1, 2}, {80, 5025}, {257, 16788}, {384, 5903}, {484, 3552}, {952, 26561}, {2099, 7770}, {3754, 16915}, {3878, 16916}, {4366, 5697}, {5443, 16921}, {5444, 33015}, {5445, 7907}, {5690, 26629}, {5730, 26687}, {5902, 6645}, {6656, 10950}, {8361, 11545}, {9317, 24190}, {9620, 25264}, {15228, 33257}, {15950, 32992}, {16044, 18393}

X(30136) = {X(1),X(2)}-harmonic conjugate of X(30140)

X(30137) = POINT CELAENO(0,0,1,0,1,1,-1)

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

X(30137) lies on these lines: {1, 2}, {3953, 14621}, {17181, 17396}

X(30137) = {X(1),X(2)}-harmonic conjugate of X(30141)

X(30138) = POINT CELAENO(0,0,1,0,-1,-1,-1)

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

X(30138) lies on these lines: {1, 2}, {1089, 17688}, {6656, 17602}, {11285, 17599}, {16061, 32926}, {16600, 16997}, {16974, 18140}, {17061, 17670}, {17181, 17391}, {20947, 25497}

X(30138) = {X(1),X(2)}-harmonic conjugate of X(30134)

X(30139) = POINT CELAENO(0,0,1,0,1,-1,-1)

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

X(30139) lies on these lines: {1, 2}, {79, 11361}, {191, 16914}, {257, 16783}, {384, 5902}, {758, 16916}, {1003, 5221}, {3336, 3552}, {3649, 8370}, {3970, 17743}, {4366, 5903}, {5131, 33014}, {5426, 33063}, {5441, 7833}, {5442, 33274}, {5692, 16918}, {5883, 16915}, {6645, 18398}, {8356, 10543}, {8359, 15174}, {11246, 19687}, {12433, 26590}, {15079, 32967}, {18221, 33198}, {26725, 33045}

X(30139) = {X(1),X(2)}-harmonic conjugate of X(30135)

X(30140) = POINT CELAENO(0,0,1,0,-1,1,-1)

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

X(30140) lies on these lines: {1, 2}, {80, 16921}, {214, 16915}, {484, 33004}, {2099, 11285}, {2320, 16920}, {3878, 17684}, {5025, 5443}, {5444, 7907}, {5445, 33015}, {5901, 26590}, {5903, 7824}, {6645, 21842}, {6655, 18393}, {6656, 15950}, {10950, 32992}, {15228, 33275}

X(30140) = {X(1),X(2)}-harmonic conjugate of X(30136)

X(30141) = POINT CELAENO(0,0,1,0,-1,-1,1)

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

X(30141) lies on these lines: {1, 2}, {335, 5264}, {668, 16974}, {4692, 17688}, {7766, 17744}, {16998, 28594}, {24080, 32930}

X(30141) = {X(1),X(2)}-harmonic conjugate of X(30137)

X(30142) = POINT CELAENO(0,0,0,1,1,1,1)

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

X(30142) lies on these lines: {1, 2}, {3, 20990}, {6, 3678}, {19, 3247}, {32, 37}, {33, 4314}, {34, 3947}, {35, 28606}, {55, 2915}, {58, 984}, {72, 3745}, {79, 33151}, {81, 5904}, {191, 17126}, {197, 3295}, {226, 4347}, {442, 17602}, {474, 17599}, {595, 17716}, {750, 3670}, {756, 1724}, {758, 5711}, {940, 3874}, {942, 4682}, {964, 1089}, {990, 12512}, {1001, 27784}, {1010, 32926}, {1038, 4298}, {1060, 21620}, {1203, 3876}, {1386, 5044}, {1453, 7322}, {1621, 27785}, {2292, 5264}, {3056, 31757}, {3159, 3923}, {3242, 3881}, {3306, 24167}, {3337, 4392}, {3610, 5750}, {3666, 25440}, {3671, 8270}, {3688, 10974}, {3744, 6051}, {3772, 3841}, {3833, 17054}, {3868, 9347}, {3878, 5710}, {3891, 16454}, {3931, 8715}, {3932, 17698}, {4296, 5290}, {4353, 12436}, {4372, 25499}, {4438, 6693}, {4653, 11102}, {4680, 5051}, {4868, 5687}, {5218, 33178}, {5269, 12514}, {5275, 16600}, {5284, 31318}, {5453, 9958}, {5717, 21077}, {6763, 7226}, {8728, 17061}, {9895, 15569}, {10176, 16466}, {10618, 32148}, {16287, 16687}, {16288, 16684}, {16519, 17750}, {16589, 16974}, {16777, 18755}, {17122, 24046}, {17720, 25639}, {19762, 21010}, {24160, 33111}, {25526, 32771}, {26060, 33150}, {26131, 33153}, {30285, 31785}

X(30143) = POINT CELAENO(0,0,0,1,-1,1,1)

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

X(30143) lies on these lines: {1, 2}, {3, 5883}, {4, 11263}, {5, 11281}, {6, 25081}, {9, 4067}, {21, 5902}, {40, 3919}, {55, 3754}, {57, 5267}, {58, 18174}, {65, 5248}, {79, 11114}, {86, 17861}, {142, 17647}, {191, 16865}, {214, 25524}, {354, 8666}, {405, 758}, {409, 9275}, {474, 3833}, {535, 10404}, {942, 993}, {944, 6900}, {950, 12609}, {956, 3881}, {958, 3874}, {986, 4653}, {988, 24167}, {1001, 3878}, {1043, 28612}, {1058, 21630}, {1329, 5719}, {1385, 3742}, {1482, 3898}, {1621, 5903}, {1724, 2650}, {1837, 3822}, {2099, 3884}, {2177, 3987}, {2271, 16611}, {2476, 26725}, {2646, 5439}, {2647, 4306}, {2802, 3303}, {2886, 12433}, {2975, 18398}, {3090, 6326}, {3219, 3901}, {3306, 3612}, {3336, 4189}, {3419, 3841}, {3485, 11813}, {3488, 28629}, {3523, 5538}, {3543, 16143}, {3553, 3986}, {3560, 5884}, {3576, 6876}, {3585, 31019}, {3647, 16418}, {3649, 11113}, {3670, 10448}, {3683, 4018}, {3689, 4002}, {3694, 3723}, {3711, 4540}, {3715, 3988}, {3748, 10914}, {3753, 8715}, {3812, 24929}, {3814, 11374}, {3817, 6261}, {3825, 11375}, {3868, 5251}, {3869, 5259}, {3873, 5258}, {3889, 5288}, {3890, 11009}, {3892, 12513}, {3897, 5563}, {3918, 5687}, {4084, 5436}, {4127, 15650}, {4134, 11523}, {4256, 24174}, {4297, 6869}, {4305, 9776}, {4414, 18173}, {4423, 5730}, {4428, 12702}, {4512, 4744}, {4640, 31794}, {4867, 25542}, {4973, 5708}, {5047, 5692}, {5049, 11260}, {5083, 22759}, {5131, 17548}, {5221, 16370}, {5249, 10572}, {5260, 5904}, {5441, 17579}, {5450, 10202}, {5493, 30503}, {5496, 10180}, {5506, 17570}, {5535, 6875}, {5603, 6903}, {5691, 18444}, {5693, 6920}, {5722, 25639}, {5745, 17706}, {5836, 25439}, {5837, 14563}, {5885, 6914}, {5901, 7680}, {6265, 32557}, {6600, 10912}, {6701, 17532}, {6702, 12739}, {6796, 24299}, {6862, 10265}, {6873, 8227}, {6883, 31806}, {6884, 9803}, {6893, 21635}, {6906, 15016}, {6912, 15071}, {6913, 31803}, {7280, 27003}, {7489, 13465}, {7504, 15079}, {8261, 22937}, {10176, 11108}, {10177, 12672}, {10179, 10222}, {10246, 11500}, {10543, 11112}, {10595, 11014}, {10826, 31266}, {10884, 28164}, {11116, 15792}, {11507, 12736}, {11684, 16858}, {12005, 22758}, {12563, 12572}, {13161, 26728}, {13463, 15170}, {13746, 28619}, {14100, 17646}, {15015, 17572}, {15935, 31419}, {16126, 16859}, {16608, 17045}, {16783, 17451}, {16788, 21808}, {17010, 17700}, {17394, 17791}, {17558, 18221}, {17758, 24249}, {18446, 19925}, {18465, 28620}, {18990, 25557}, {19716, 25080}, {24477, 31458}

X(30143) = {X(1),X(2)}-harmonic conjugate of X(22836)

X(30144) = POINT CELAENO(0,0,0,1,1,-1,1)

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

X(30144) lies on these lines: {1, 2}, {3, 214}, {4, 11813}, {9, 2317}, {21, 15446}, {35, 3877}, {36, 3869}, {37, 5114}, {40, 6942}, {46, 11682}, {55, 3884}, {56, 758}, {57, 4084}, {58, 18465}, {61, 5240}, {62, 5239}, {65, 17614}, {72, 1319}, {75, 24202}, {80, 4193}, {100, 5330}, {101, 3061}, {104, 5693}, {106, 3976}, {141, 17043}, {219, 25078}, {220, 24036}, {224, 4304}, {326, 3663}, {355, 3814}, {392, 2646}, {404, 5903}, {442, 15950}, {474, 2099}, {484, 4188}, {501, 17512}, {515, 6928}, {516, 6934}, {517, 6924}, {518, 24928}, {604, 21078}, {912, 960}, {944, 6326}, {946, 6917}, {952, 1329}, {956, 1388}, {958, 10176}, {962, 5538}, {999, 3874}, {1043, 20320}, {1055, 1759}, {1259, 22767}, {1317, 21031}, {1376, 1482}, {1387, 3813}, {1389, 6946}, {1420, 4067}, {1442, 17272}, {1457, 4347}, {1483, 3820}, {1537, 11826}, {1706, 16200}, {1818, 4660}, {1837, 3825}, {2098, 2802}, {2320, 3467}, {2475, 18393}, {2476, 5443}, {2550, 10595}, {2551, 7967}, {2886, 5901}, {2975, 5692}, {3035, 5690}, {3057, 5440}, {3295, 3898}, {3304, 3881}, {3333, 12559}, {3339, 4744}, {3340, 3919}, {3419, 11376}, {3452, 5882}, {3485, 11263}, {3555, 20323}, {3576, 5267}, {3612, 5250}, {3647, 13465}, {3652, 18515}, {3681, 5288}, {3735, 21008}, {3746, 3890}, {3753, 11011}, {3817, 6867}, {3822, 11375}, {3847, 12019}, {3868, 4867}, {3876, 5258}, {3892, 7373}, {3897, 5251}, {3899, 4881}, {3916, 31165}, {3918, 4413}, {3927, 11194}, {3940, 12513}, {3953, 32577}, {4018, 32636}, {4187, 10950}, {4225, 18417}, {4255, 4868}, {4297, 6261}, {4299, 11415}, {4301, 6885}, {4317, 5905}, {4361, 18261}, {4537, 5223}, {4640, 13624}, {4647, 12081}, {4757, 5221}, {4855, 5119}, {4930, 5708}, {5044, 15178}, {5046, 6224}, {5048, 10914}, {5057, 10483}, {5086, 7741}, {5087, 18480}, {5253, 5902}, {5265, 18467}, {5270, 31053}, {5328, 5531}, {5433, 12832}, {5438, 7982}, {5445, 17566}, {5450, 5887}, {5493, 6282}, {5526, 26690}, {5603, 6901}, {5657, 11014}, {5698, 16132}, {5727, 25522}, {5783, 16777}, {5794, 5886}, {5795, 13607}, {5815, 6049}, {5836, 10222}, {5837, 10165}, {5844, 8256}, {5855, 6691}, {5881, 30827}, {5883, 25524}, {5884, 10269}, {6001, 14925}, {6256, 21635}, {6284, 10609}, {6603, 25066}, {6681, 24914}, {6874, 8227}, {6914, 26287}, {6936, 12572}, {7269, 25590}, {7504, 17057}, {7705, 31263}, {9709, 10247}, {9956, 11567}, {9957, 25439}, {10106, 18962}, {10265, 26492}, {10283, 31419}, {10284, 15813}, {10827, 30852}, {10944, 17757}, {10966, 11517}, {11010, 15015}, {11231, 33281}, {11249, 31806}, {11260, 25405}, {11512, 24168}, {11715, 18254}, {11928, 16174}, {12005, 16203}, {12114, 31803}, {12267, 12543}, {12645, 12737}, {12738, 18526}, {13384, 31435}, {14988, 32612}, {15079, 31272}, {15556, 26437}, {15558, 26358}, {15799, 17515}, {15854, 21342}, {16137, 25557}, {17439, 33299}, {17857, 28236}, {18253, 31650}, {18444, 30389}, {18481, 24703}, {20117, 22758}, {21879, 31456}, {22935, 32141}

X(30144) = complement of X(10573)
X(30144) = {X(1),X(2)}-harmonic conjugate of X(30147)

X(30145) = POINT CELAENO(0,0,0,1,1,1,-1)

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

X(30145) lies on these lines: {1, 2}, {6, 28594}, {33, 12575}, {37, 2241}, {38, 5264}, {55, 20833}, {58, 17716}, {169, 3247}, {191, 7226}, {388, 4347}, {595, 984}, {596, 3980}, {750, 3953}, {758, 5710}, {940, 3881}, {964, 4692}, {990, 5493}, {993, 5266}, {1038, 4315}, {1089, 24552}, {1191, 10176}, {1203, 3681}, {1421, 10588}, {1469, 31737}, {1486, 3295}, {1573, 16974}, {1724, 17469}, {2901, 32941}, {3052, 3647}, {3242, 3874}, {3336, 4392}, {3454, 4865}, {3555, 3745}, {3666, 8715}, {3677, 24167}, {3678, 16466}, {3723, 16583}, {3744, 5248}, {3746, 28606}, {3825, 17721}, {3876, 5315}, {3889, 9347}, {3891, 4647}, {3913, 4868}, {3923, 24068}, {3931, 25439}, {3992, 5192}, {4011, 4075}, {4015, 4383}, {4030, 13728}, {4298, 8270}, {4318, 5290}, {4682, 5045}, {4894, 5051}, {5100, 19786}, {5255, 9941}, {5506, 9330}, {5687, 17599}, {6198, 7713}, {6763, 17126}, {7174, 12514}, {7190, 10521}, {7751, 25368}, {9791, 20087}, {11263, 33144}, {11510, 16577}, {11533, 17461}, {16600, 16777}, {17061, 31419}, {17598, 24046}, {17602, 24390}, {17720, 24387}, {17725, 24160}, {28612, 32922}

X(30145) = {X(1),X(2)}-harmonic conjugate of X(30148)

X(30146) = POINT CELAENO(2,0,0,1,1,-1,-1)

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

X(30146) lies on these lines: {1, 2}, {758, 32954}

X(30147) = POINT CELAENO(0,0,0,1,-1,1,-1)

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

X(30147) lies on these lines: {1, 2}, {3, 3754}, {21, 5903}, {36, 3897}, {40, 6875}, {46, 3919}, {56, 5883}, {63, 4084}, {65, 993}, {80, 2476}, {104, 15016}, {142, 5882}, {214, 474}, {219, 25081}, {226, 18962}, {355, 3822}, {392, 11011}, {405, 2099}, {409, 17104}, {442, 10950}, {484, 4189}, {495, 11281}, {501, 11116}, {515, 6917}, {516, 6868}, {517, 5248}, {529, 6147}, {758, 958}, {942, 8666}, {944, 6901}, {946, 6928}, {952, 25466}, {956, 3874}, {1001, 1482}, {1006, 1389}, {1155, 4004}, {1159, 4757}, {1319, 5439}, {1376, 3918}, {1385, 3812}, {1478, 11263}, {1621, 5697}, {1656, 6265}, {1837, 25639}, {2098, 3898}, {2140, 24249}, {2170, 16783}, {2320, 4188}, {2478, 11813}, {2646, 3753}, {2800, 3560}, {2802, 3295}, {2975, 5902}, {3303, 11517}, {3340, 12514}, {3576, 6942}, {3579, 10107}, {3678, 9708}, {3698, 5440}, {3742, 24928}, {3746, 14923}, {3813, 12433}, {3814, 11375}, {3816, 5901}, {3824, 28204}, {3825, 5886}, {3833, 10246}, {3838, 18480}, {3841, 5794}, {3868, 5258}, {3869, 5251}, {3873, 5288}, {3876, 4867}, {3877, 5259}, {3881, 12513}, {3940, 4015}, {3968, 9709}, {3970, 4390}, {3997, 16968}, {4127, 5220}, {4187, 15950}, {4193, 5443}, {4245, 23846}, {4256, 24440}, {4297, 6934}, {4301, 6936}, {4347, 24806}, {4424, 10448}, {4670, 32047}, {4868, 19765}, {4973, 5221}, {5045, 11260}, {5046, 18393}, {5250, 25415}, {5253, 21842}, {5260, 5692}, {5261, 18467}, {5270, 31019}, {5284, 5330}, {5289, 11108}, {5426, 11010}, {5436, 7982}, {5444, 17566}, {5587, 6874}, {5603, 6902}, {5690, 6690}, {5708, 11194}, {5719, 12607}, {5722, 24387}, {5730, 10176}, {5795, 21077}, {5818, 6326}, {5836, 8715}, {5880, 18481}, {5881, 25525}, {5884, 22758}, {5885, 32153}, {6261, 6867}, {6667, 19907}, {6691, 31659}, {6701, 18525}, {6757, 7100}, {6767, 10912}, {6852, 12247}, {6885, 12436}, {6984, 18446}, {8071, 12736}, {9620, 25092}, {10222, 31837}, {10436, 24209}, {10483, 20292}, {10595, 26105}, {10827, 31266}, {10914, 25439}, {11230, 33281}, {11249, 31870}, {11715, 16203}, {12512, 30503}, {12520, 28164}, {12565, 28158}, {12702, 28443}, {12739, 15863}, {12740, 32557}, {13463, 15172}, {14563, 24391}, {14882, 19525}, {16788, 17451}, {17045, 18261}, {17825, 23112}, {18421, 31424}, {21888, 31451}

X(30147) = {X(1),X(2)}-harmonic conjugate of X(30144)

X(30148) = POINT CELAENO(0,0,0,1,-1,-1,1)

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

X(30148) lies on these lines: {1, 2}, {6, 3881}, {31, 3953}, {34, 4315}, {37, 7772}, {46, 24167}, {56, 2922}, {58, 3976}, {101, 16787}, {105, 29048}, {106, 833}, {191, 4392}, {244, 5264}, {388, 1421}, {496, 17061}, {595, 982}, {596, 3923}, {758, 1191}, {942, 4906}, {999, 11365}, {1015, 16974}, {1040, 12575}, {1058, 11677}, {1104, 8666}, {1203, 3873}, {1279, 5248}, {1386, 5045}, {1449, 22356}, {1468, 4694}, {1469, 31757}, {1616, 3884}, {2191, 4349}, {3242, 3678}, {3246, 31445}, {3303, 4868}, {3304, 27802}, {3315, 18398}, {3337, 17126}, {3361, 4318}, {3670, 3915}, {3677, 12514}, {3743, 17599}, {3744, 25440}, {3746, 4850}, {3752, 8715}, {3754, 17054}, {3772, 24387}, {3868, 5315}, {3874, 16466}, {3878, 16483}, {3890, 16489}, {4011, 24068}, {4202, 4894}, {4296, 13462}, {4692, 5192}, {5255, 24046}, {5286, 21090}, {5299, 26242}, {5710, 5883}, {6763, 17127}, {16487, 31424}, {16502, 16600}, {17721, 25639}, {17736, 21764}

X(30148) = {X(1),X(2)}-harmonic conjugate of X(30145)

X(30149) = POINT CELAENO(1,1,1,1,-1,-1,-1)

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

X(30149) lies on these lines: {1, 2}, {76, 16886}, {384, 4680}, {1089, 5025}, {1502, 1928}, {1930, 3314}, {3695, 26590}, {3703, 6656}, {3974, 14064}, {4056, 7939}, {4153, 24514}, {4366, 4894}, {4372, 7832}, {4376, 7768}, {7785, 17280}, {7796, 16720}, {7931, 20267}, {16600, 16991}, {16986, 24786}, {17550, 32862}, {20543, 20592}, {20654, 30473}, {21412, 30632}, {22025, 26763}

X(30150) = POINT CELAENO(-1,1,1,1,1,-1,-1)

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

X(30150) lies on these lines: {1, 2}, {21029, 24190}

X(30151) = POINT CELAENO(1,-1,1,1,-1,1,-1)

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

X(30151) lies on these lines: {1, 2}

X(30152) = POINT CELAENO(1,1,-1,1,-1,-1,-1)

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

X(30152) lies on these lines: {1, 2}, {1089, 16921}, {1111, 7906}, {1930, 7777}, {3314, 24786}, {3703, 32992}, {3797, 7741}, {3974, 32975}, {4056, 7941}, {4372, 7769}, {4376, 7858}, {4680, 7824}, {7786, 16886}, {7897, 17192}, {7925, 20267}, {11813, 25270}

X(30153) = POINT CELAENO(1, 1, 1, -1, -1, -1, 1)

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

X(30153) lies on these lines: {1, 2}, {76, 20483}, {3120, 24080}, {3992, 5025}, {4894, 16918}, {16991, 28594}

X(30154) = POINT CELAENO(1, 1, -1, -1, -1, -1, 1)

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

X(30154) lies on these lines: {1, 2}, {3992, 16921}, {4692, 33045}, {4894, 33047}, {7786, 20483}

X(30155) = POINT CELAENO(1, -1, 1, -1, -1, 1, 1)

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

X(30155) lies on these lines: {1, 2}, {4692, 16921}, {4894, 7824}, {7264, 7906}, {7272, 7941}, {7796, 27918}

X(30156) = POINT CELAENO(-1, 1, 1, -1, 1, -1, 1)

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

X(30156) lies on these lines: {1, 2}, {21044, 24190}

X(30157) = POINT CELAENO(-1, 1, -1, 1, 1, -1, -1)

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

X(30157) lies on these lines: {1, 2}

X(30158) = POINT CELAENO(1, -1, 1, -1, 1, -1, 1)

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

X(30158) lies on these lines: {1, 2}

X(30159) = POINT CELAENO(1, 0, 1, -1, 1, -1, 1)

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

X(30159) lies on these lines: {1, 2}, {3754, 7892}, {9351, 17322}

X(30160) = POINT CELAENO(1, -1, 0, -1, 1, -1, 1)

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

X(30160) lies on these lines: {1, 2}, {214, 17672}, {3918, 24582}, {5443, 17681}, {6710, 21921}, {17691, 18393}

X(30161) = POINT CELAENO(1, -1, 1, 0, 1, -1, 1)

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

X(30161) lies on these lines: {1, 2}

X(30162) = POINT CELAENO(-1, -1, -1, 1, 1, 1, 1)

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

X(30162) lies on these lines: {1, 2}, {3932, 17670}, {4680, 16918}

X(30163) = POINT CELAENO(1, -1, 1, 1, -1, -1, -1)

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

X(30163) lies on these lines: {1, 2}

X(30164) = POINT CELAENO(1, 1, -1, 1, -1, 1, -1)

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

X(30164) lies on these lines: {1, 2}, {80, 17688}, {3754, 16906}

X(30165) = POINT CELAENO(1, 1, -1, -1, -1, 1, 1)

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

X(30165) lies on these lines: {1, 2}, {758, 16906}, {5902, 17673}

X(30166) = POINT CELAENO(1, -1, 1, -1, -1, -1, 1)

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

X(30166) lies on these lines: {1, 2}

X(30167) = POINT CELAENO(0, 1, 1, 1, -1, -1, -1)

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

X(30167) lies on these lines: {1, 2}, {35, 3797}, {75, 5277}, {76, 4372}, {335, 29473}, {384, 1089}, {385, 1930}, {1111, 17129}, {3314, 20267}, {3695, 26629}, {3703, 7807}, {3974, 14001}, {4056, 7893}, {4376, 6179}, {4400, 20924}, {4680, 5025}, {4692, 6645}, {7760, 16720}, {7828, 16886}, {8782, 21378}, {10583, 17280}, {16060, 32926}, {16991, 25598}, {17181, 17363}

X(30168) = POINT CELAENO(0, 0, 1, 1, -1, -1, -1)

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

X(30168) lies on these lines: {1, 2}, {58, 335}, {76, 16974}, {385, 16600}, {894, 17200}, {1966, 21412}, {3159, 6651}, {3314, 25598}, {3678, 20142}, {3954, 33295}, {4680, 16906}, {6292, 16706}, {6656, 17061}, {7768, 25345}, {7839, 24036}, {16061, 32922}

X(30169) = POINT CELAENO(1, 0, 1, 1, -1, -1, -1)

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

X(30169) lies on these lines: {1, 2}, {3314, 16600}, {3662, 4153}, {4071, 24190}, {7832, 16974}, {7906, 24036}, {7914, 16706}, {7922, 25345}, {7931, 25598}, {8363, 17061}

X(30170) = POINT CELAENO(1, 0, 0, 1, -1, -1, -1)

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

X(30170) lies on these lines: {1, 2}, {325, 16600}, {3670, 4766}, {3788, 16974}, {4071, 24170}, {4109, 16887}, {7778, 25598}, {7821, 25345}, {7944, 16706}, {8361, 17061}, {21090, 25264}

X(30171) = POINT CELAENO(1, 1, 0, 1, -1, -1, -1)

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

X(30171) lies on these lines: {1, 2}, {3, 4680}, {5, 1089}, {11, 3695}, {12, 4692}, {35, 5015}, {36, 7270}, {38, 3454}, {39, 16886}, {55, 4894}, {58, 33119}, {79, 32939}, {141, 24786}, {147, 21378}, {191, 4388}, {312, 7206}, {321, 25639}, {325, 1930}, {345, 1479}, {346, 10591}, {496, 4975}, {594, 31460}, {595, 32844}, {672, 4153}, {993, 5016}, {1111, 3933}, {1203, 33071}, {1224, 19280}, {1329, 3992}, {1330, 6763}, {1574, 20483}, {1724, 4438}, {2476, 33089}, {2886, 4647}, {2887, 3670}, {2901, 32848}, {2915, 23361}, {3090, 3974}, {3314, 17192}, {3336, 4645}, {3583, 7283}, {3678, 5741}, {3685, 4857}, {3701, 3814}, {3702, 24387}, {3704, 24390}, {3710, 21616}, {3712, 15171}, {3746, 4514}, {3788, 4372}, {3822, 4968}, {3825, 4358}, {3841, 4359}, {3869, 16110}, {3874, 3936}, {3909, 23156}, {3925, 28611}, {3932, 4187}, {3966, 5791}, {4006, 4119}, {4056, 7776}, {4071, 16549}, {4109, 16552}, {4193, 32862}, {4347, 28774}, {4376, 7759}, {4385, 7951}, {4387, 9669}, {4417, 5904}, {4714, 31419}, {4865, 5264}, {5014, 8715}, {5248, 33113}, {5259, 33116}, {5300, 25440}, {6057, 7173}, {6358, 26481}, {6533, 8728}, {7764, 16720}, {7778, 20267}, {8279, 10526}, {11813, 25253}, {18134, 18398}, {21021, 31476}, {24046, 25957}, {28612, 33108}

X(30172) = POINT CELAENO(1, 1, 0, 0, -1, -1, -1)

Barycentrics -a^2 b^2 + b^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(30172) lies on these lines: {1, 2}, {5, 3932}, {9, 4153}, {21, 4680}, {35, 5300}, {58, 4438}, {75, 3841}, {79, 32933}, {191, 6327}, {312, 25639}, {346, 31418}, {442, 3703}, {595, 4865}, {984, 3454}, {993, 7270}, {1089, 2476}, {1203, 33070}, {1479, 17776}, {1621, 4894}, {1724, 33115}, {1770, 3977}, {2325, 18483}, {2886, 3695}, {2901, 33092}, {3159, 3944}, {3416, 5138}, {3596, 6757}, {3670, 25957}, {3678, 4417}, {3701, 7951}, {3704, 31419}, {3710, 12047}, {3717, 21077}, {3730, 4071}, {3743, 32773}, {3746, 5014}, {3790, 4066}, {3822, 4385}, {3825, 18743}, {3836, 4446}, {3874, 18134}, {3925, 28612}, {3936, 5904}, {3974, 6856}, {3985, 24045}, {3992, 11681}, {4197, 33089}, {4276, 11102}, {4358, 7741}, {4647, 33108}, {4671, 7206}, {5015, 5248}, {5016, 5251}, {5100, 25439}, {5264, 33072}, {5283, 16886}, {7752, 20947}, {7759, 24358}, {7808, 17279}, {8715, 32850}, {11813, 19582}, {18139, 18398}, {18393, 25253}, {21026, 24443}, {25306, 31757}, {25308, 31737}, {25440, 32851}, {26131, 33170}

X(30173) = POINT CELAENO(1, 1, 1, 0, -1, -1, -1)

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

X(30173) lies on these lines: {1, 2}, {83, 17279}, {626, 20947}, {3695, 26582}, {3703, 17670}, {3932, 6656}, {4680, 16916}, {6376, 20444}, {7768, 24358}, {16886, 18140}, {17143, 20483}, {21412, 30631}, {24614, 33113}

X(30174) = POINT CELAENO(1, 1, 1, 0, 0, -1, -1)

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

X(30174) lies on these lines: {1, 2}, {171, 16905}, {384, 25957}, {2887, 16916}, {3836, 16915}, {7892, 24602}, {11319, 31041}, {16917, 25961}, {16918, 25760}, {16920, 25959}, {16927, 32784}, {17279, 18055}, {26687, 30811}

X(30175) = POINT CELAENO(1, 1, 1, 1, 0, -1, -1)

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

X(30175) lies on these lines: {1, 2}, {31, 16905}, {41, 16991}, {384, 2887}, {3314, 24549}, {3662, 7795}, {3836, 16917}, {3846, 16918}, {6327, 16909}, {6679, 16907}, {7832, 21240}, {7892, 24586}, {7931, 17046}, {11115, 31041}, {11319, 31023}, {16906, 31237}, {16915, 25957}, {16916, 25760}, {16919, 25959}, {16920, 25958}

X(30176) = POINT CELAENO(1, 1, 1, 1, 0, 0, -1)

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

X(30176) lies on these lines: {1, 2}, {993, 16905}, {5299, 31090}, {17694, 25914}, {17698, 26561}, {26969, 27185}

X(30177) = POINT CELAENO(1, 1, 1, 1, -1, 0, -1)

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

X(30177) lies on these lines: {1, 2}, {257, 5224}, {2345, 17741}, {3061, 31090}, {3096, 24254}, {3212, 17238}, {3710, 27481}, {3923, 6655}, {3980, 17565}, {4011, 17685}, {5835, 26582}, {7270, 14621}, {7833, 24850}, {16991, 17451}, {17289, 17743}, {17669, 25591}, {25079, 33046}, {26561, 31317}

X(30178) = POINT CELAENO(1, 1, 1, 1, -1, 0, 0)

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

X(30178) lies on these lines: {1, 2}, {150, 28604}, {1921, 30893}, {4680, 14621}

X(30179) = POINT CELAENO(1, 1, 1, 1, -1, -1, 0)

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

X(30179) lies on these lines: {1, 2}, {37, 31090}, {75, 3314}, {183, 4445}, {230, 4478}, {257, 4136}, {305, 6382}, {319, 385}, {321, 1916}, {325, 594}, {335, 2887}, {384, 5015}, {894, 4071}, {1211, 27495}, {1281, 33082}, {1447, 17287}, {1909, 16886}, {2345, 7774}, {2899, 33057}, {3329, 17289}, {3701, 17669}, {3703, 3797}, {3758, 7837}, {3759, 7875}, {3773, 4518}, {4030, 26629}, {4153, 17499}, {4361, 7868}, {4363, 7788}, {4366, 4514}, {4385, 5025}, {4766, 33162}, {4865, 14621}, {4911, 7939}, {5300, 16915}, {5564, 7931}, {5687, 21485}, {5839, 16989}, {6645, 7270}, {6651, 33164}, {6653, 32932}, {7179, 7897}, {7249, 16603}, {7766, 17363}, {7792, 17362}, {8024, 18891}, {9866, 17741}, {11174, 17293}, {16986, 17228}, {16991, 26242}, {17140, 31041}, {17165, 31023}, {20132, 33073}, {20142, 33118}, {26244, 32025}, {31317, 33169}

X(30180) = POINT CELAENO(1, -1, 1, -1, 0, -1, 1)

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

X(30180) lies on these lines: {1, 2}, {25260, 25263}

Frégier points: X(30181)-X(30257)

This preamble and centers X(30181)-X(30257) were contributed by César Eliud Lozada, December 15, 2018.

Let Γ be a conic and P a fixed point on it. The sides of any right angle with vertex P cut Γ again in P' and P". Then all the lines P'P" intersect in a common point Q_Γ(P).
(See: Frégier's Theorem in MathWorld)

This section deals with some named conics in the plane of a triangle ABC. The point Q_Γ(P) is denoted here as the P-Frégier point of Γ.

It can be proved that if Γ is a rectangular hyperbola then Q_Γ(P) lies in the line at infinity. Isogonal conjugates of these points in the infinity are also included in this section.

The appearance of (i, j) in the following table means that Q_Γ(X(i)) = X(j) for the indicated conic Γ:

Γ = dual of Yff parabola:
(2, 4786), (7, 30181), (27, 30182), (75, 30183), (86, 30184), (273, 30185), (310, 30186), (335, 30187), (673, 30188), (675, 4025), (871, 30189), (903, 30190), (1088, 30191), (1268, 30192), (1659, 30193), (2296, 30194), (4373, 30195)

Γ = excentral-incentral ellipse:
(1768, 20), (5400, 30196), (16528, 30197)

Γ = Feuerbach hyperbola:
(1, 513), (4, 521), (7, 14077), (8, 30198), (9, 30199), (21, 30200), (79, 8702), (80, 513), (84, 30201), (90, 30202), (104, 30202), (256, 30203), (294, 30204), (314, 30205), (885, 30206), (941, 30207), (943, 30208)

Γ = Jerabek hyperbola:
(3, 924), (4, 520), (6, 30209), (54, 30210), (64, 30211), (65, 30212), (66, 30213), (67, 30209), (69, 20186)

Γ = Johnson circumconic:
(110, 5562), (265, 52), (1625, 30214)

Γ = Kiepert hyperbola:
(2, 1499), (4, 525), (10, 6002), (13, 523), (14, 523), (17, 30215), (18, 30216), (76, 30217), (83, 30218), (98, 525)

Γ = Kiepert parabola:
(669, 30219), (1649, 30220), (3233, 30221), (3733, 30222)

Γ = Mandart inellipse:
(11, 30223), (3271, 30224)

Γ = orthic inconic:
(125, 184)

Γ = Stammler hyperbola:
(1, 6003), (3, 523), (6, 1499), (399, 523)

Γ = Steiner circumellipse:
(99, 69), (190, 30225), (290, 30226), (648, 30227), (664, 30228)

Γ = Steiner inellipse:
(115, 6), (1015, 24289), (1084, 30229), (1086, 24281)

Γ = Thomson-Gibert-Moses hyperbola:
(2, 512), (3, 8675), (6, 30230), (110, 512), (154, 8673), (354, 30231), (392, 9013)

Γ = Yff hyperbola:
(2, 3830), (4, 3), (14163, 30232), (14164, 30233)

Γ = conic {A, B, C, X(1), X(2)}:
(1, 3669), (2, 30234), (57, 30235)

X(30181) = X(7)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics (a^2-2*(b+c)*a+b^2+4*b*c+c^2)*(b-c)*(a-c+b)*(a-b+c) : :
X(30181) = 2*X(3676)-3*X(24002)

X(30181) lies on these lines: {7,514}, {77,4449}, {522,693}, {663,7190}, {1086,30188}, {3663,21185}, {4040,4328}, {7178,28898}

X(30181) = reflection of X(30188) in X(1086)

X(30182) = X(27)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30182) lies on these lines: {2,4064}, {441,525}, {4467,8611}

X(30183) = X(75)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30183) lies on these lines: {69,4498}, {75,28470}, {513,3004}, {1086,30187}, {4106,24560}, {4357,4401}, {4481,4785}, {8643,17321}

X(30183) = reflection of X(30187) in X(1086)

X(30184) = X(86)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30184) lies on these lines: {523,4025}, {2786,4791}, {17159,21187}

X(30185) = X(273)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30185) lies on the line {521,4025}

X(30186) = X(310)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30186) lies on the line {512,4025}

X(30187) = X(335)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30187) lies on these lines: {335,28470}, {812,3776}, {1086,30183}

X(30187) = reflection of X(30183) in X(1086)

X(30188) = X(673)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30188) lies on these lines: {514,673}, {522,3912}, {650,918}, {1086,30181}, {1155,4786}, {3935,27486}

X(30188) = reflection of X(30181) in X(1086)

X(30189) = X(871)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30189) lies on the line {788,4025}

X(30190) = X(903)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics (5*a^4-5*(b+c)*a^3-(2*b^2-7*b*c+2*c^2)*a^2+(b+c)*(7*b^2-12*b*c+7*c^2)*a-b^4-5*(b-c)^2*b*c-c^4)*(b-c) : :

X(30190) lies on these lines: {320,514}, {900,4025}, {903,3667}, {1086,4786}

X(30190) = reflection of X(4786) in X(1086)

X(30191) = X(1088)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
(2*a^6-7*(b+c)*a^5+(10*b^2+21*b*c+10*c^2)*a^4-4*(b+c)*(2*b^2+3*b*c+2*c^2)*a^3+2*(b^2+b*c+c^2)*(2*b^2+b*c+2*c^2)*a^2-(b^2-c^2)^2*(b+c)*a+(b^2+c^2)*(b-c)^2*b*c)*(b-c)*(a-c+b)*(a-b+c) : :

X(30191) lies on these lines: {3900,4025}, {21195,21453}

X(30192) = X(1268)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics (13*a^4+19*(b+c)*a^3+3*(b+2*c)*(2*b+c)*a^2-(b+c)*(b^2+4*b*c+c^2)*a-(b^2+b*c+c^2)*(b^2+4*b*c+c^2))*(b-c) : :

X(30192) lies on the line {4025,4977}

X(30193) = X(1659)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

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

X(30193) lies on these lines: {}

X(30194) = X(2296)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
((b+c)^2*a^6+(b+c)*(3*b^2+4*b*c+3*c^2)*a^5+(2*b^4+2*c^4+(9*b^2+16*b*c+9*c^2)*b*c)*a^4-(b+c)*(b^4+c^4-(4*b^2+7*b*c+4*c^2)*b*c)*a^3-(b^6+c^6+(3*b^4-5*b^2*c^2+3*c^4)*b*c)*a^2-(2*b^2-3*b*c+2*c^2)*(b+c)^3*b*c*a-(b^4+c^4+(b-c)^2*b*c)*b^2*c^2)*(b-c) : :

X(30194) lies on the line {784,4025}

X(30195) = X(4373)-FRÉGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics (13*a^4-6*(b+c)*a^3-2*(2*b-c)*(b-2*c)*a^2+2*(b+c)*(7*b^2-12*b*c+7*c^2)*a-b^4-2*(5*b^2-7*b*c+5*c^2)*b*c-c^4)*(b-c) : :

X(30195) lies on the line {3667,4025}

X(30196) = X(5400)-FRÉGIER POINT OF EXCENTRAL-HEXYL ELLIPSE

Barycentrics
3*(b+c)*a^6-3*(b+c)^2*a^5-(b+c)*(6*b^2-11*b*c+6*c^2)*a^4+2*(3*b^4+3*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a^3+(b+c)*(3*b^4+3*c^4-2*(6*b^2-7*b*c+6*c^2)*b*c)*a^2-3*(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^2*(b+c)*b*c : :

X(30196) lies on these lines: {3,30197}, {4,8}, {20,3667}, {40,17780}, {499,28774}, {2800,3952}, {4427,6326}, {5690,25030}, {5693,25253}, {9803,17777}, {17164,20117}, {26364,28826}

X(30196) = reflection of X(30197) in X(3)

X(30197) = X(16528)-FRÉGIER POINT OF EXCENTRAL-HEXYL ELLIPSE

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

X(30197) lies on these lines: {3,30196}, {4,2457}, {20,145}

X(30197) = reflection of X(30196) in X(3)

X(30198) = X(8)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30198) lies on these lines: {4,23836}, {7,23819}, {30,511}, {84,23800}, {885,10307}, {905,6615}, {1052,9355}, {1339,4498}, {1357,16185}, {1769,14353}, {2254,13252}, {2401,12246}, {4905,30235}, {6705,23808}, {7659,29487}, {7971,14812}, {12114,24457}, {17410,21390}

X(30198) = circumnormal isogonal conjugate of X(8686)
X(30198) = isogonal conjugate of X(30236)

X(30199) = X(9)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30199) lies on these lines: {3,4394}, {4,4106}, {20,4380}, {30,511}, {84,23893}, {885,3427}, {3576,30234}, {6223,20297}, {6260,20314}

X(30199) = circumnormal isogonal conjugate of X(15728)
X(30199) = isogonal conjugate of X(30237)

X(30200) = X(21)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30200) lies on these lines: {30,511}, {13250,23035}, {13251,23036}

X(30200) = isogonal conjugate of X(30238)

X(30201) = X(84)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30201) lies on these lines: {30,511}, {17896,20297}

X(30201) = isogonal conjugate of X(30239)

X(30202) = X(90)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30202) lies on these lines: {30,511}, {4730,23224}

X(30202) = isogonal conjugate of X(30240)

X(30203) = X(256)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30203) lies on these lines: {1,7234}, {11,2680}, {30,511}, {649,10459}, {810,4879}, {4507,25128}, {5710,16874}

X(30203) = isogonal conjugate of X(30241)

X(30204) = X(294)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30204) lies on these lines: {30,511}, {1530,4106}

X(30204) = isogonal conjugate of X(30242)

X(30205) = X(314)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30205) lies on the line {30,511}

X(30205) = isogonal conjugate of X(30243)

X(30206) = X(885)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics
a*((b^2+6*b*c+c^2)*a^8-(3*b+c)*(b+3*c)*(b+c)*a^7+(b^4+c^4+(11*b^2+24*b*c+11*c^2)*b*c)*a^6+(b+c)*(5*b^4+5*c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^5-(5*b^4+5*c^4-(7*b^2-2*b*c+7*c^2)*b*c)*(b+c)^2*a^4-(b^2-c^2)*(b-c)*(b^4+c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^3+(3*b^6+3*c^6-(5*b^4+5*c^4+3*(b+c)^2*b*c)*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a-3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*b*c) : :

X(30206) lies on the line {30,511}

X(30206) = isogonal conjugate of X(30244)

X(30207) = X(941)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics a^2*((4*b^2+9*b*c+4*c^2)*a^2+2*(b+2*c)*(2*b+c)*(b+c)*a+(5*b^2+12*b*c+5*c^2)*b*c)*(b-c) : :

X(30207) lies on the line {30,511}

X(30207) = isogonal conjugate of X(30245)

X(30208) = X(943)-FRÉGIER POINT OF FEUERBACH HYPERBOLA

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

X(30208) lies on the line {30,511}

X(30208) = isogonal conjugate of X(30246)

X(30209) = X(6)-FRÉGIER POINT OF JERABEK HYPERBOLA

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

X(30209) lies on these lines: {3,647}, {4,850}, {6,7652}, {30,511}, {110,7482}, {111,23701}, {879,4846}, {1147,21905}, {1296,15406}, {1636,3288}, {1640,9730}, {2081,3581}, {2435,3426}, {2519,22159}, {3569,14696}, {4549,15421}, {5652,5654}, {5926,8651}, {6643,28729}, {6699,22264}, {8574,13335}, {12038,14135}, {15451,22089}, {16194,23616}, {16229,18314}, {19543,24782}

X(30209) = complementary conjugate of X(14672)
X(30209) = isogonal conjugate of X(30247)

X(30210) = X(54)-FRÉGIER POINT OF JERABEK HYPERBOLA

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

X(30210) lies on these lines: {30,511}, {110,13863}, {185,6798}, {1291,15958}, {5943,20392}, {16106,16107}

X(30210) = isogonal conjugate of X(30248)

X(30211) = X(64)-FRÉGIER POINT OF JERABEK HYPERBOLA

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

X(30211) lies on these lines: {30,511}, {2435,6391}, {14380,15316}

X(30211) = isogonal conjugate of X(30249)

X(30212) = X(65)-FRÉGIER POINT OF JERABEK HYPERBOLA

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

X(30212) lies on these lines: {3,656}, {4,7253}, {5,8062}, {30,511}, {355,4086}, {661,23090}, {7629,11248}

X(30212) = isogonal conjugate of X(30250)

X(30213) = X(66)-FRÉGIER POINT OF JERABEK HYPERBOLA

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

X(30213) lies on these lines: {30,511}, {10097,15316}

X(30213) = isogonal conjugate of X(30251)

X(30214) = X(1625)-FRÉGIER POINT OF JOHNSON CIRCUMCONIC

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

X(30214) lies on these lines: {3,6}, {525,5562}, {3150,3917}

X(30215) = X(17)-FRÉGIER POINT OF KIEPERT HYPERBOLA

Barycentrics (2*sqrt(3)*S-a^2-c^2+7*b^2)*(2*sqrt(3)*S-a^2-b^2+7*c^2)*(12*S^2+4*sqrt(3)*(7*a^2-b^2-c^2)*S+(7*a^2-b^2-c^2)^2)*(b^2-c^2) : :

X(30215) lies on these lines: {30,511}, {6137,22934}

X(30215) = isogonal conjugate of X(30252)

X(30216) = X(18)-FRÉGIER POINT OF KIEPERT HYPERBOLA

Barycentrics (-2*sqrt(3)*S-a^2-c^2+7*b^2)*(-2*sqrt(3)*S-a^2-b^2+7*c^2)*(12*S^2-4*sqrt(3)*(7*a^2-b^2-c^2)*S+(7*a^2-b^2-c^2)^2)*(b^2-c^2) : :

X(30216) lies on these lines: {30,511}, {6138,22889}

X(30216) = isogonal conjugate of X(30253)

X(30217) = X(76)-FRÉGIER POINT OF KIEPERT HYPERBOLA

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

X(30217) lies on these lines: {3,9491}, {30,511}, {5027,23301}, {5926,9494}, {9147,9210}

X(30217) = isogonal conjugate of X(30254)

X(30218) = X(83)-FRÉGIER POINT OF KIEPERT HYPERBOLA

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

X(30218) lies on these lines: {3,13511}, {30,511}, {2531,3095}, {3005,14316}, {9210,13309}, {14886,22159}

X(30218) = isogonal conjugate of X(30255)

X(30219) = X(669)-FRÉGIER POINT OF KIEPERT PARABOLA

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

X(30219) lies on these lines: {6,1649}, {323,523}, {394,8029}, {399,30220}, {511,669}, {647,1570}, {684,8675}, {1993,11123}, {1994,10190}, {2451,3231}, {3288,3906}, {8371,15066}, {9168,11004}

X(30220) = X(1649)-FRÉGIER POINT OF KIEPERT PARABOLA

Barycentrics
(16*a^10-32*(b^2+c^2)*a^8+(19*b^4+58*b^2*c^2+19*c^4)*a^6-(b^2+c^2)*(5*b^4+22*b^2*c^2+5*c^4)*a^4+(b^8+c^8+7*(b^4+c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :

X(30220) lies on these lines: {110,8371}, {399,30219}, {523,9143}, {542,1649}, {669,2930}, {690,24981}, {9168,14683}

X(30220) = midpoint of X(9168) and X(14683)
X(30220) = reflection of X(8371) in X(110)

X(30221) = X(3233)-FRÉGIER POINT OF KIEPERT PARABOLA

Barycentrics (4*a^8-4*(b^2+c^2)*a^6-(7*b^4-18*b^2*c^2+7*c^4)*a^4+10*(b^4-c^4)*(b^2-c^2)*a^2-3*(b^2-c^2)^4)*(c^2-a^2)*(a^2-b^2) : :
X(30221) = 7*X(110)-3*X(476) = 4*X(110)-3*X(3233) = 5*X(110)-3*X(7471) = X(110)+3*X(14480) = X(110)-3*X(14611) = 4*X(476)-7*X(3233) = 5*X(476)-7*X(7471) = X(476)+7*X(14480) = X(476)-7*X(14611) = 5*X(3233)-4*X(7471) = X(3233)+4*X(14480) = X(3233)-4*X(14611) = 4*X(6723)-3*X(12079) = X(7471)+5*X(14480) = X(7471)-5*X(14611) = X(10620)-3*X(14934)

X(30221) lies on these lines: {30,24981}, {110,476}, {2452,3066}, {6723,12079}, {10620,14934}

X(30221) = midpoint of X(14480) and X(14611)

X(30222) = X(3733)-FRÉGIER POINT OF KIEPERT PARABOLA

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

X(30222) lies on these lines: {517,3733}, {3737,8702}

X(30223) = X(11)-FRÉGIER POINT OF MANDART INELLIPSE

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

X(30223) lies on these lines: {1,90}, {9,55}, {11,57}, {19,1857}, {21,6514}, {31,33}, {34,774}, {36,7171}, {40,1728}, {44,7074}, {46,3583}, {56,84}, {63,497}, {65,12705}, {165,11502}, {171,9817}, {184,15503}, {212,4319}, {223,8758}, {238,1040}, {390,3219}, {405,12711}, {496,24467}, {516,1708}, {522,30224}, {610,15494}, {612,7069}, {614,7004}, {920,1479}, {950,12514}, {968,14547}, {971,1617}, {1001,10391}, {1104,1854}, {1108,2192}, {1118,1712}, {1155,10860}, {1158,1210}, {1182,4207}, {1376,15297}, {1397,10535}, {1406,2956}, {1420,10085}, {1445,3474}, {1454,10896}, {1467,7992}, {1478,18540}, {1490,1898}, {1519,3086}, {1621,10394}, {1697,3632}, {1698,10958}, {1707,1936}, {1723,2361}, {1736,8270}, {1737,3359}, {1839,1856}, {1851,2385}, {1852,7713}, {1859,12723}, {2098,3962}, {2175,11429}, {2182,37519}, {2187,2261}, {2308,4336}, {2950,12832}, {3022,11189}, {3056,5227}, {3057,12629}, {3058,3929}, {3065,7284}, {3100,17127}, {3218,5274}, {3220,10832}, {3271,11436}, {3305,5218}, {3306,10589}, {3333,3649}, {3338,18393}, {3467,7162}, {3475,8545}, {3486,5250}, {3554,19354}, {3587,4302}, {3685,3719}, {3899,7962}, {3928,11238}, {4383,9371}, {4423,17603}, {4428,15296}, {5119,5727}, {5204,9841}, {5223,10388}, {5225,7098}, {5248,10393}, {5281,27065}, {5285,10833}, {5426,13384}, {5432,7308}, {5534,11508}, {5720,8069}, {5853,20588}, {6210,10319}, {6763,10959}, {6765,26358}, {7091,7285}, {7174,24431}, {7289,12589}, {7677,11220}, {7741,17700}, {7951,17699}, {8543,11020}, {8583,22768}, {9614,12704}, {10389,15298}, {10578,29007}, {10947,24392}, {11435,21746}, {12185,24469}, {12686,18838}, {12717,24310}, {15171,26921}, {16541,24320}, {20992,23207}

X(30223) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 90, 7330), (55, 7082, 9), (3683, 14100, 55)

X(30224) = X(3271)-FRÉGIER POINT OF MANDART INELLIPSE

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

X(30224) lies on these lines: {9,4124}, {57,7336}, {522,30223}, {1738,3359}, {2808,3271}

X(30225) = X(190)-FRÉGIER POINT OF STEINER CIRCUMELLIPSE

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

X(30225) lies on these lines: {2,24262}, {8,14947}, {10,19932}, {69,514}, {145,7760}, {190,952}, {239,17740}, {315,20535}, {346,1016}, {519,1992}, {1121,3699}, {1146,4561}, {3570,6790}, {3807,21290}, {3912,5219}, {4370,24807}, {4555,29616}, {4671,6542}, {6631,17233}, {16086,29331}, {24247,24282}

X(30225) = reflection of X(24807) in X(4370)
X(30225) = anticomplement of X(24281)

X(30226) = X(290)-FRÉGIER POINT OF STEINER CIRCUMELLIPSE

Barycentrics
(b^4+3*b^2*c^2+c^4)*a^10-3*(b^2+c^2)*(b^4+c^4)*a^8+3*(b^4+b^2*c^2+c^4)*(b^4-b^2*c^2+c^4)*a^6-(b^2+c^2)*(b^4-b^2*c^2+c^4)^2*a^4-(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*b^4*c^4 : :
X(30226) = 5*X(3618)-4*X(5661)

X(30226) lies on these lines: {4,69}, {6,10684}, {99,25332}, {1992,23878}, {2549,30227}, {3618,5661}

X(30226) = anticomplement of X(34359)

X(30227) = X(648)-FRÉGIER POINT OF STEINER CIRCUMELLIPSE

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

X(30227) lies on these lines: {4,9513}, {20,18338}, {30,1351}, {69,525}, {99,1562}, {125,671}, {148,3269}, {287,543}, {376,2966}, {648,2777}, {1249,23582}, {2549,30226}, {3164,15351}, {4235,6794}, {7738,10684}, {7748,9289}, {13172,17974}, {14568,21663}

X(30227) = anticomplement of X(34360)

X(30228) = X(664)-FRÉGIER POINT OF STEINER CIRCUMELLIPSE

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

X(30228) lies on these lines: {7,3675}, {69,522}, {144,666}, {347,1275}, {527,1992}, {2481,4440}, {4357,24411}, {16091,20254}

X(30228) = anticomplement of X(34361)

X(30229) = X(1084)-FRÉGIER POINT OF STEINER INELLIPSE

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

X(30229) is the perspector of the 1st Brocard triangle and the vertex-triangle of the 3rd and 4th Brocard triangles. (Randy Hutson, October 8, 2019)

X(30229) lies on these lines: {6,512}, {538,599}, {543,694}, {574,3229}, {671,3124}, {695,7765}, {1613,5118}, {2086,14700}, {2088,3981}, {2936,20998}, {3978,7790}, {7757,18829}, {8591,9998}, {11152,11654}, {11171,12525}

X(30229) = complement of X(34341)
X(30229) = X(670)-of-1st-Brocard-triangle
X(30229) = 1st-Brocard-isogonal conjugate of X(5027)
X(30229) = 1st-Brocard-isotomic conjugate of X(804)

X(30230) = X(6)-FRÉGIER POINT OF THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics a^2*(b^2-c^2)*(a^6-(b^2+c^2)*a^4-(b^4-24*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-8*b^2*c^2+c^4)) : :

X(30230) lies on these lines: {3,8644}, {30,511}, {74,10102}, {1351,2444}, {1597,2489}, {3426,10097}, {3524,15724}, {4550,21905}, {7464,9137}, {8651,9126}, {11472,21733}

X(30230) = isogonal conjugate of X(30256)

X(30231) = X(354)-FRÉGIER POINT OF THOMSON-GIBERT-MOSES HYPERBOLA

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

X(30231) lies on these lines: {30,511}, {2254,22160}, {4170,12699}, {4729,12702}, {7216,15934}, {9404,24290}

X(30231) = isogonal conjugate of X(30257)

X(30232) = X(14163)-FRÉGIER POINT OF YFF HYPERBOLA

Barycentrics
a^16-5*(b^2+c^2)*a^14+(3*b^4+32*b^2*c^2+3*c^4)*a^12+9*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^10-(13*b^8+13*c^8-(8*b^4+83*b^2*c^2+8*c^4)*b^2*c^2)*a^8+(b^2+c^2)*(9*b^8+9*c^8+(2*b^4-41*b^2*c^2+2*c^4)*b^2*c^2)*a^6-(7*b^12+7*c^12-(12*b^8+12*c^8-(39*b^4-76*b^2*c^2+39*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(3*b^8+3*c^8-2*(3*b^4-7*b^2*c^2+3*c^4)*b^2*c^2)*a^2-4*(b^2-c^2)^6*b^2*c^2 : :
X(30232) = 3*X(381)-2*X(14164) = X(14164)-3*X(14215)

X(30232) lies on these lines: {3,8029}, {381,14164}, {523,30233}, {524,3830}

X(30232) = reflection of X(381) in X(14215)

X(30233) = X(14164)-FRÉGIER POINT OF YFF HYPERBOLA

Barycentrics 7*S^6-3*(6*R^2*(12*R^2+3*SA-5*SW)-4*SA^2+SB*SC+4*SW^2)*S^4-(9*R^2*SA*(27*R^2*(SA-SW)-SW*(-7*SW+9*SA))+SW^2*(7*R^2*SW+12*SA^2-8*SA*SW-SW^2))*S^2+(9*R^2-SW)*SB*SC*SW^3 : :
X(30233) = 3*X(381)-2*X(14163) = X(14163)-3*X(14214)

X(30233) lies on these lines: {3,8151}, {381,14163}, {523,30232}

X(30233) = reflection of X(381) in X(14214)

X(30234) = X(2)-FRÉGIER POINT OF CONIC {A, B, C, X(1), X(2)}

Barycentrics a*(b-c)*(5*a^2-b^2-c^2) : :
X(30234) = X(1)+2*X(4394) = 2*X(667)+X(905) = 5*X(667)+X(2530) = 4*X(667)-X(3803) = 5*X(905)-2*X(2530) = 2*X(905)+X(3803) = 4*X(1125)-X(4106) = X(2254)+5*X(8656) = 4*X(2530)+5*X(3803) = X(2530)-5*X(14419) = 5*X(3616)+X(4380) = X(3669)+2*X(4401) = X(3803)+4*X(14419) = X(4367)+2*X(6050) = 2*X(4794)+X(7659)

X(30234) lies on these lines: {1,4394}, {2,28475}, {3,8642}, {28,6591}, {36,238}, {110,2746}, {650,4160}, {665,29350}, {1125,4106}, {1437,1980}, {1499,4786}, {1635,14077}, {2254,8656}, {2832,3669}, {3309,8643}, {3576,30199}, {3616,4380}, {4367,6050}, {4794,7659}, {9048,16475}

X(30234) = midpoint of X(667) and X(14419)
X(30234) = reflection of X(905) in X(14419)
X(30234) = {X(667), X(905)}-harmonic conjugate of X(3803)

X(30235) = X(57)-FRÉGIER POINT OF CONIC {A, B, C, X(1), X(2)}

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

X(30235) lies on these lines: {1,650}, {56,11934}, {522,905}, {693,14986}, {999,8760}, {1125,25925}, {2826,3669}, {3086,4885}, {3616,24562}, {3622,26641}, {4294,8142}, {4905,30198}, {10529,26546}, {11019,29066}, {11193,22767}

X(30236) = ISOGONAL CONJUGATE OF X(30198)

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

X(30236) lies on the circumcircle and these lines: {3,8686}, {40,106}, {100,25737}, {104,5854}, {105,6244}, {108,23832}, {165,1477}, {953,13528}, {1292,8683}, {2077,2718}, {2291,3973}, {4571,6079}, {5537,22942}, {6282,28233}, {8059,23703}, {14110,28219}, {23981,30239}

X(30236) = reflection of X(8686) in X(3)
X(30236) = circumnormal isogonal conjugate of X(3880)
X(30236) = circumperp conjugate of X(8686)
X(30236) = isogonal conjugate of X(30198)
X(30236) = antipode of X(8686) in the circumcircle
X(30236) = trilinear pole of the line {6, 20323}

X(30237) = ISOGONAL CONJUGATE OF X(30199)

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

X(30237) lies on the circumcircle and these lines: {3,15728}, {40,2291}, {103,6282}, {104,5759}, {105,3428}, {1477,3576}, {8059,23890}, {10310,15731}

X(30237) = reflection of X(15728) in X(3)
X(30237) = circumnormal isogonal conjugate of X(15733)
X(30237) = circumperp conjugate of X(15728)
X(30237) = isogonal conjugate of X(30199)
X(30237) = antipode of X(15728) in the circumcircle

X(30238) = ISOGONAL CONJUGATE OF X(30200)

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

X(30238) lies on the circumcircle and these lines: {74,14110}, {104,12519}, {759,11012}, {5951,10310}

X(30238) = isogonal conjugate of X(30200)

X(30239) = ISOGONAL CONJUGATE OF X(30201)

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

X(30239) lies on the circumcircle and these lines: {56,1295}, {102,1420}, {104,10309}, {972,1617}, {1319,2745}, {2291,8602}, {2716,5193}, {23981,30236}

X(30239) = isogonal conjugate of X(30201)
X(30239) = trilinear pole of the line {6, 8602}

X(30240) = ISOGONAL CONJUGATE OF X(30202)

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

X(30240) lies on the circumcircle and these lines: {104,11415}, {915,11249}

X(30240) = isogonal conjugate of X(30202)

X(30241) = ISOGONAL CONJUGATE OF X(30203)

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

X(30241) lies on the circumcircle and these lines: {}

X(30241) = isogonal conjugate of X(30203)

X(30242) = ISOGONAL CONJUGATE OF X(30204)

Barycentrics
a*(c-a)*(a^9-(3*b+c)*a^8+(4*b^2-b*c-2*c^2)*a^7-2*(2*b^3-c^3-4*(b+c)*b*c)*a^6+(2*b^3-7*c^3-b*c*(7*b+8*c))*b*a^5+2*(b^4+3*c^4+2*b*c^2*(b-c))*b*a^4-(4*b^4-2*c^4+b*c*(7*b^2+12*b*c+3*c^2))*(b-c)^2*a^3+2*(2*b^5-c^5+(2*b^3+2*c^3+b*c*(2*b+c))*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(3*b^3+c^3+2*b*c*(b+2*c))*a+(b^4-c^4)*(b^2+c^2)*(b-c)^3)*(a-b)*(a^9-(b+3*c)*a^8-(2*b^2+b*c-4*c^2)*a^7+2*(b^3-2*c^3+4*(b+c)*b*c)*a^6-(7*b^3-2*c^3+b*c*(8*b+7*c))*c*a^5+2*(3*b^4+c^4-2*b^2*c*(b-c))*c*a^4+(2*b^4-4*c^4-b*c*(3*b^2+12*b*c+7*c^2))*(b-c)^2*a^3-2*(b^5-2*c^5-(2*b^3+2*c^3+b*c*(b+2*c))*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(b^3+3*c^3+2*b*c*(2*b+c))*a+(b^4-c^4)*(b^2+c^2)*(b-c)^3) : :

X(30242) lies on the circumcircle and these lines: {}

X(30242) = isogonal conjugate of X(30204)

X(30243) = ISOGONAL CONJUGATE OF X(30205)

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

X(30243) lies on the circumcircle and these lines: {}

X(30243) = isogonal conjugate of X(30205)

X(30244) = ISOGONAL CONJUGATE OF X(30206)

Barycentrics
a*((b+3*c)*a^9-(3*b^2+7*b*c+6*c^2)*a^8+(b^2+11*b*c+12*c^2)*b*a^7+(5*b^4+6*c^4-b*c*(13*b^2+10*b*c+4*c^2))*a^6-(b+c)*(5*b^4+6*c^4-b*c*(8*b^2+b*c+4*c^2))*a^5-(b-c)*(b^5+6*c^5-(6*b^3-4*c^3-b*c*(b-2*c))*b*c)*a^4+(3*b^4-4*c^4-b*c*(5*b^2-b*c+3*c^2))*(b-c)^2*b*a^3-(b^5-6*c^5-(10*b^3+6*c^3+b*c*(9*b+10*c))*b*c)*(b-c)^3*a^2-(b^4-c^4)*c*(b-c)^2*(6*b^2-b*c+3*c^2)*a-(b^2-c^2)^2*(b-c)^3*b*c^2)*((3*b+c)*a^9-(6*b^2+7*b*c+3*c^2)*a^8+(12*b^2+11*b*c+c^2)*c*a^7+(6*b^4+5*c^4-b*c*(4*b^2+10*b*c+13*c^2))*a^6-(b+c)*(6*b^4+5*c^4-b*c*(4*b^2+b*c+8*c^2))*a^5+(b-c)*(6*b^5+c^5+(4*b^3-6*c^3-b*c*(2*b-c))*b*c)*a^4-(4*b^4-3*c^4+b*c*(3*b^2-b*c+5*c^2))*(b-c)^2*c*a^3-(6*b^5-c^5+(6*b^3+10*c^3+b*c*(10*b+9*c))*b*c)*(b-c)^3*a^2+(b^4-c^4)*b*(b-c)^2*(3*b^2-b*c+6*c^2)*a+(b^2-c^2)^2*(b-c)^3*b^2*c) : :

X(30244) lies on the circumcircle and these lines: {}

X(30244) = isogonal conjugate of X(30206)

X(30245) = ISOGONAL CONJUGATE OF X(30207)

Barycentrics (c-a)*((4*b+5*c)*a^3+2*(b+2*c)*(2*b+3*c)*a^2+(b+c)*(9*b+5*c)*c*a+4*b*c^2*(b+c))*(a-b)*((5*b+4*c)*a^3+2*(3*b+2*c)*(2*b+c)*a^2+(5*b+9*c)*(b+c)*b*a+4*b^2*c*(b+c)) : :

X(30245) lies on the circumcircle and these lines: {}

X(30245) = isogonal conjugate of X(30207)

X(30246) = ISOGONAL CONJUGATE OF X(30208)

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

X(30246) lies on the circumcircle and these lines: {}

X(30246) = isogonal conjugate of X(30208)

X(30247) = ISOGONAL CONJUGATE OF X(30209)

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

X(30247) lies on the circumcircle and these lines: {2,14672}, {3,2373}, {4,111}, {24,2374}, {25,9084}, {28,9061}, {74,5486}, {98,378}, {105,4227}, {110,4235}, {186,2770}, {376,1297}, {468,10102}, {476,7482}, {523,10098}, {524,23701}, {648,691}, {675,7431}, {842,10295}, {925,11634}, {935,1632}, {1302,4230}, {1311,7436}, {1576,10423}, {2409,9064}, {2697,7464}, {3520,9076}, {3563,18533}, {4221,26703}, {4238,9058}, {4244,9107}, {4247,9083}, {4249,9057}, {5966,7576}, {7463,9056}, {7468,16167}, {7472,10420}, {7473,9060}, {11456,26717}

X(30247) = reflection of X(i) in X(j) for these (i,j): (4, 1560), (2373, 3)
X(30247) = circumnormal isogonal conjugate of X(2393)
X(30247) = circumperp conjugate of X(2373)
X(30247) = isogonal conjugate of X(30209)
X(30247) = anticomplement of X(14672)
X(30247) = antipode of X(2373) in the circumcircle
X(30247) = inverse of X(5512) in the polar circle
X(30247) = trilinear pole of the line {6, 468}
X(30247) = circumcircle-X(4)-antipode of X(111)

X(30248) = ISOGONAL CONJUGATE OF X(30210)

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

X(30248) lies on the circumcircle and these lines: {74,12254}, {427,5966}, {523,13863}, {550,18401}, {1141,3520}, {2383,6240}, {7722,15907}, {13619,14979}

X(30248) = isogonal conjugate of X(30210)
X(30248) = trilinear pole of the line {6, 13418}

X(30249) = ISOGONAL CONJUGATE OF X(30211)

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

X(30249) lies on the circumcircle and these lines: {4,5897}, {74,12250}, {5896,18213}, {6776,15324}

X(30249) = isogonal conjugate of X(30211)
X(30249) = polar conjugate of the anticomplement of X(20580)
X(30249) = polar-circle-inverse of X(35968)
X(30249) = trilinear pole of the line {6, 1885}

X(30250) = ISOGONAL CONJUGATE OF X(30212)

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

X(30250) lies on the circumcircle and these lines: {4,759}, {102,21740}, {104,7414}, {105,4231}, {110,4242}, {186,12030}, {925,13589}, {1295,3651}, {4220,26703}, {7438,9061}

X(30250) = isogonal conjugate of X(30212)

X(30251) = ISOGONAL CONJUGATE OF X(30213)

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

X(30251) lies on the circumcircle and these lines: {24,2373}, {111,3542}, {1297,18533}, {4235,13398}

X(30251) = isogonal conjugate of X(30213)

X(30252) = ISOGONAL CONJUGATE OF X(30215)

Barycentrics a^2*(2*sqrt(3)*S-a^2+7*b^2-c^2)*(2*sqrt(3)*S-a^2-b^2+7*c^2)*(c^2-a^2)*(a^2-b^2) : :

X(30252) lies on the circumcircle and these lines: {98,5487}, {107,14185}, {2380,10645}

X(30252) = isogonal conjugate of X(30215)

X(30253) = ISOGONAL CONJUGATE OF X(30216)

Barycentrics a^2*(2*sqrt(3)*S+a^2-7*b^2+c^2)*(2*sqrt(3)*S+a^2+b^2-7*c^2)*(c^2-a^2)*(a^2-b^2) : :

X(30253) lies on the circumcircle and these lines: {98,5488}, {107,14187}, {2381,10646}

X(30253) = isogonal conjugate of X(30216)

X(30254) = ISOGONAL CONJUGATE OF X(30217)

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

X(30254) lies on the circumcircle and these lines: {3,699}, {98,7751}, {729,3098}, {733,6234}

X(30254) = reflection of X(699) in X(3)
X(30254) = circumnormal isogonal conjugate of X(698)
X(30254) = circumperp conjugate of X(699)
X(30254) = isogonal conjugate of X(30217)
X(30254) = antipode of X(699) in the circumcircle

X(30255) = ISOGONAL CONJUGATE OF X(30218)

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

X(30255) lies on the circumcircle and these lines: {4,13499}, {98,8150}, {733,3398}, {755,5092}, {5188,29011}

X(30255) = reflection of X(4) in X(13499)
X(30255) = isogonal conjugate of X(30218)

X(30256) = ISOGONAL CONJUGATE OF X(30230)

Barycentrics (a^6-(b^2+7*c^2)*a^4-(b^4-24*b^2*c^2+7*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^6-(7*b^2+c^2)*a^4-(7*b^4-24*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(c^2-a^2)*(a^2-b^2) : :

X(30256) lies on the circumcircle and these lines: {3,9084}, {30,10102}, {111,376}, {378,2374}, {1302,11634}, {2373,21312}, {2770,7464}, {4221,9061}, {4235,9064}, {7472,9060}

X(30256) = reflection of X(9084) in X(3)
X(30256) = circumnormal isogonal conjugate of X(9027)
X(30256) = circumperp conjugate of X(9084)
X(30256) = isogonal conjugate of X(30230)
X(30256) = antipode of X(9084) in the circumcircle
X(30256) = trilinear pole of the line {6, 16317}

X(30257) = ISOGONAL CONJUGATE OF X(30231)

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

X(30257) lies on the circumcircle and these lines: {105,3651}, {376,759}, {1302,13589}, {4220,9061}, {4242,9064}, {7414,15344}, {7464,12030}

X(30257) = isogonal conjugate of X(30231)

X(30258) = X(3)X(6)∩X(5)X(264)

Barycentrics    a^2*((b^4+b^2*c^2+c^4)*a^4-2*(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) : :
Barycentrics    (S^2-SB*SC)*(S^2+4*R^2*SW+2*SB*SC-SW^2) : :
Trilinears    cos(A + ω') : :, where ω' = Brocard angle of orthic triangle
X(30258) = 3*X(2)-4*X(10003), 5*X(1656)-4*X(14767)

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30258) lies on these lines: {2, 1972}, {3, 6}, {4, 3164}, {5, 264}, {51, 6638}, {184, 13558}, {237, 6403}, {339, 7697}, {417, 15043}, {418, 3060}, {426, 5422}, {441, 18583}, {852, 5640}, {1073, 14489}, {1656, 14059}, {1942, 4846}, {1993, 6641}, {1994, 23606}, {5562, 17039}, {5889, 26897}, {5943, 6509}, {6375, 9243}, {6389, 14561}, {6776, 20975}, {10519, 20819}, {10796, 15013}, {11272, 28407}, {12161, 14152}, {15073, 20775}, {15526, 24206}, {20576, 28697}, {21969, 26907}

X(30258) = midpoint of X(4) and X(3164)
X(30258) = reflection of X(i) in X(j) for these (i,j): (3, 216), (264, 5)
X(30258) = anticomplement of the anticomplement of X(10003)
X(30258) = X(216)-of-X3-ABC reflections triangle
X(30258) = X(264)-of-Johnson triangle
X(30258) = X(3164)-of-Euler triangle
X(30258) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(1970)
X(30258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5093, 15905), (3, 15851, 5050), (371, 372, 1970)

X(30259) = X(3)X(6)∩X(5012)X(14652)

Barycentrics (SB+SC)*((5*R^2-2*SA-2*SW)*S^2-R^2*SA*SW) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30259) lies on these lines: {3, 6}, {5012, 14652}

X(30260) = X(3)X(6)∩X(13)X(9159)

Barycentrics (SB+SC)*(S^2+sqrt(3)*(SA-3*R^2+SW)*S+(9*R^2-SW)*SA) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30260) lies on these lines: {3, 6}, {13, 9159}, {17, 10217}, {11658, 16241}

X(30261) = X(3)X(6)∩X(14)X(9159)

Barycentrics (SB+SC)*(S^2-sqrt(3)*(SA-3*R^2+SW)*S+(9*R^2-SW)*SA) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30261) lies on these lines: {3, 6}, {14, 9159}, {18, 10218}, {11659, 16242}

X(30262) = X(3)X(6)∩X(20)X(3186)

Barycentrics a^2 (-a^2+b^2+c^2)*(2*(b^2+c^2)*a^6-(3*b^4-b^2*c^2+3*c^4)*a^4+2*(b^2+c^2)*b^2*c^2*a^2+(b^6-c^6)*(b^2-c^2)) : :
Barycentrics (S^2-SB*SC)*(S^2-16*R^2*SW-2*SB*SC+3*SW^2) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30262) lies on these lines: {3, 6}, {20, 3186}, {185, 20794}, {2790, 23240}, {7750, 14615}, {9306, 14673}, {11328, 12294}

X(30262) = midpoint of X(20) and X(3186)

X(30263) = X(20)X(2979)∩X(185)X(5667)

Barycentrics a^2 (a^16 b^4-7 a^14 b^6+21 a^12 b^8-35 a^10 b^10+35 a^8 b^12-21 a^6 b^14+7 a^4 b^16-a^2 b^18-4 a^16 b^2 c^2+14 a^14 b^4 c^2-15 a^12 b^6 c^2+4 a^10 b^8 c^2-5 a^8 b^10 c^2+14 a^6 b^12 c^2-9 a^4 b^14 c^2+b^18 c^2+a^16 c^4+14 a^14 b^2 c^4-38 a^12 b^4 c^4+39 a^10 b^6 c^4-33 a^8 b^8 c^4+20 a^6 b^10 c^4-4 a^4 b^12 c^4+7 a^2 b^14 c^4-6 b^16 c^4-7 a^14 c^6-15 a^12 b^2 c^6+39 a^10 b^4 c^6+6 a^8 b^6 c^6-13 a^6 b^8 c^6-15 a^4 b^10 c^6-11 a^2 b^12 c^6+16 b^14 c^6+21 a^12 c^8+4 a^10 b^2 c^8-33 a^8 b^4 c^8-13 a^6 b^6 c^8+42 a^4 b^8 c^8+5 a^2 b^10 c^8-26 b^12 c^8-35 a^10 c^10-5 a^8 b^2 c^10+20 a^6 b^4 c^10-15 a^4 b^6 c^10+5 a^2 b^8 c^10+30 b^10 c^10+35 a^8 c^12+14 a^6 b^2 c^12-4 a^4 b^4 c^12-11 a^2 b^6 c^12-26 b^8 c^12-21 a^6 c^14-9 a^4 b^2 c^14+7 a^2 b^4 c^14+16 b^6 c^14+7 a^4 c^16-6 b^4 c^16-a^2 c^18+b^2 c^18) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28731.

X(30263) lies on these lines: {20,2979}, {185,5667}, {577,6759}, {3087,10110}

X(30264) = REFLECTION OF X(12) IN X(3)

Barycentrics 4*a^7 - 4*a^6*b - 7*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 + a*b^6 - b^7 - 4*a^6*c + 6*a^5*b*c - 3*a^4*b^2*c - 4*a^3*b^3*c + 6*a^2*b^4*c - 2*a*b^5*c + b^6*c - 7*a^5*c^2 - 3*a^4*b*c^2 + 12*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 7*a^4*c^3 - 4*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*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 - 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7 : :
X(30264) = 4 X[3] - 3 X[21155],2 X[12] - 3 X[21155],3 X[376] - X[11491], 5 X[631] - 4 X[6668],5 X[3522] - X[20060],2 X[8068] - 3 X[21154]

X(30264) lies on these lines: {3, 12}, {4, 4999}, {11, 7491}, {20, 2894}, {30, 11012}, {40, 550}, {56, 6868}, {104, 12519}, {376, 529}, {411, 2829}, {515, 3916}, {517, 15338}, {548, 2077}, {631, 6668}, {758, 1071}, {944, 5855}, {958, 6934}, {1329, 6942}, {1350, 5849}, {1385, 3649}, {1483, 3894}, {3058, 10680}, {3522, 20060}, {3612, 5812}, {3869, 9964}, {4018, 5882}, {4189, 7680}, {4302, 22770}, {4311, 12709}, {4325, 15931}, {4428, 10597}, {5204, 6827}, {5267, 6831}, {5303, 6840}, {5426, 5901}, {5433, 6928}, {5434, 10267}, {5441, 5536}, {5584, 6948}, {5732, 5857}, {5759, 5852}, {5794, 21165}, {6244, 15696}, {6253, 22758}, {6284, 10959}, {6691, 6902}, {6825, 12943}, {6872, 22753}, {6875, 25466}, {6907, 10483}, {6910, 10894}, {6917, 24953}, {6922, 7280}, {6936, 25524}, {6954, 10895}, {6971, 7294}, {7489, 7958}, {7681, 11114}, {7965, 13743}, {8736, 22056}, {10386, 16200}, {10543, 24474}, {10902, 18990}, {11014, 28174}, {11194, 12116}, {12512, 13528}, {16370, 26332}, {17768, 21740}

X(30264) = midpoint of X(20) and X(2975)
X(30264) = reflection of X(i) and X(j) for these {i,j}: {4, 4999}, {12, 3}, {6831, 5267}, {15908, 11012}
X(30264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12, 21155}, {3, 10526, 5432}, {20, 3428, 11826}, {40, 550, 24466}, {6922, 7280, 21154}, {7491, 26286, 11}

X(30265) = REFLECTION OF X(19) IN X(3)

Barycentrics a*(a^8 - 2*a^7*b + 2*a^5*b^3 - 2*a^4*b^4 + 2*a^3*b^5 - 2*a*b^7 + b^8 - 2*a^7*c + 6*a^5*b^2*c - 6*a^3*b^4*c + 2*a*b^6*c + 6*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 + 2*a^5*c^3 - 4*a^3*b^2*c^3 + 2*a*b^4*c^3 - 2*a^4*c^4 - 6*a^3*b*c^4 + 2*a*b^3*c^4 + 6*b^4*c^4 + 2*a^3*c^5 - 2*a*b^2*c^5 + 2*a*b*c^6 - 4*b^2*c^6 - 2*a*c^7 + c^8) : :
X(30265) = 4 X[3] - 3 X[21160],2 X[19] - 3 X[21160],5 X[3522] - X[20061]

X(30265) lies on these lines: {1, 7}, {3, 19}, {4, 18589}, {22, 15931}, {33, 1214}, {78, 25252}, {103, 13397}, {109, 7070}, {152, 2822}, {165, 3101}, {204, 22119}, {219, 971}, {278, 1040}, {376, 534}, {612, 25080}, {1096, 22057}, {1295, 28291}, {1297, 6011}, {1350, 3827}, {1486, 11414}, {1490, 2324}, {1709, 2328}, {1723, 13329}, {1838, 6836}, {3152, 19860}, {3428, 21312}, {3522, 20061}, {3870, 6360}, {4219, 10319}, {5706, 9943}, {7520, 7987}, {7538, 19861}, {7991, 15954}, {8583, 27402}, {8680, 18446}, {9746, 26260}, {11471, 15951}, {12522, 22770}

X(30265) = midpoint of X(20) and X(4329)
X(30265) = reflection of X(i) and X(j) for these {i,j}: {4, 18589}, {19, 3}
X(30265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1721, 3332}, {3, 19, 21160}, {347, 3100, 1}

X(30266) = REFLECTION OF X(27) IN X(3)

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

As a point on the Euler line, X(30266) has Shinagawa coefficients (E-F+$bc$, -2E+F-2$bc$).

X(30266) lies on these lines: {2, 3}, {74, 1305}, {101, 1294}, {2690, 2693}, {5897, 26705}, {8680, 18446}, {16099, 29243}

X(30267) = REFLECTION OF X(28) IN X(3)

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

As a point on the Euler line, X(30267) has Shinagawa coefficients ((E+F)$a$+abc, (-2E+F)$a$-2abc).

X(30267) lies on these lines: {2, 3}, {74, 13397}, {100, 1294}, {108, 5897}, {347, 3295}, {942, 3100}, {1290, 2693}, {1292, 1297}, {1295, 6011}, {2691, 2697}, {3101, 3579}, {3182, 7070}, {3430, 6282}, {3871, 6360}, {4296, 24929}, {4329, 6361}, {9538, 15934}, {9643, 11518}, {12262, 14110}, {26703, 30257}

X(30268) = REFLECTION OF X(29) IN X(3)

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

As a point on the Euler line, X(30268) has Shinagawa coefficients (FS²+$bcS_BS_C$, -FS²-2$bcS_BS_C$).

X(30268) lies on these lines: {2, 3}, {74, 28788}, {109, 1294}, {2689, 2693}, {3916, 10538}, {4296, 7100}, {5897, 26704}, {6360, 12702}

X(30269) = REFLECTION OF X(31) IN X(3)

Barycentrics a^2*(a^5 + 2*a^2*b^3 - a*b^4 - 2*b^5 + 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 - a*c^4 - 2*c^5) : :
X(30269) = 3 X[549] - 2 X[20575],5 X[631] - 4 X[6679],5 X[3522] - X[20064]

X(30269) lies on these lines: {3, 31}, {4, 2887}, {20, 6327}, {40, 758}, {73, 8193}, {74, 6010}, {102, 1292}, {103, 28474}, {104, 28469}, {106, 28584}, {209, 5584}, {326, 1310}, {376, 752}, {515, 4680}, {517, 3938}, {549, 20575}, {573, 7688}, {631, 6679}, {674, 1350}, {734, 11257}, {953, 28520}, {1006, 6210}, {1066, 12410}, {1293, 28159}, {1496, 11573}, {1742, 4221}, {1973, 20727}, {2390, 10310}, {3522, 20064}, {3556, 3682}, {5603, 28885}, {6905, 20368}, {28145, 28524}, {28173, 28518}, {28291, 28900}, {28293, 28876}, {28299, 28873}, {28303, 28912}, {28467, 29372}

X(30269) = midpoint of X(20) and X(6327)
X(30269) = reflection of X(i) and X(j) for these {i,j}: {4, 2887}, {31, 3}

X(30270) = REFLECTION OF X(32) IN X(3)

Barycentrics a^2*(a^6 + a^2*b^4 - 2*b^6 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 - 2*b^2*c^4 - 2*c^6) : :
Trilinears 2 cos(A + 2ω) + cos(A - 2ω) + cos A : :
X(30270) = 4 X[5] - 5 X[7867],3 X[549] - 2 X[20576],5 X[631] - 4 X[6680],5 X[631] - 3 X[9753],5 X[3522] - X[20065],4 X[6680] - 3 X[9753],2 X[7805] - 3 X[9755],2 X[18806] - 3 X[22712]}

X(30270) lies on these lines: {3, 6}, {4, 626}, {5, 7822}, {20, 99}, {30, 7801}, {40, 760}, {76, 5999}, {98, 7751}, {103, 28469}, {114, 7888}, {194, 12203}, {262, 7808}, {376, 754}, {515, 4769}, {549, 20576}, {631, 6680}, {736, 6309}, {805, 2710}, {980, 4220}, {1078, 6194}, {1092, 8922}, {1296, 14388}, {1352, 7794}, {1503, 3933}, {1513, 3788}, {1974, 20819}, {2001, 2979}, {2353, 5562}, {2386, 21312}, {3117, 7467}, {3148, 3917}, {3425, 10323}, {3522, 13571}, {3552, 10334}, {3564, 7855}, {3934, 13860}, {5480, 7819}, {6248, 17130}, {6287, 11178}, {6660, 9306}, {6776, 7758}, {7488, 28710}, {7764, 9744}, {7768, 9863}, {7782, 22676}, {7789, 29181}, {7800, 10519}, {7805, 9755}, {7815, 18806}, {7826, 10991}, {7832, 13862}, {7889, 14561}, {7896, 9873}, {8671, 10310}, {9888, 14645}, {10358, 14881}, {14931, 20081}, {16187, 21513}, {18502, 22728}, {28295, 28563}

X(30270) = midpoint of X(20) and X(315)
X(30270) = midpoint of X(11824) and X(11825)
X(30270) = X(32)-of-circumcevian-triangle-of-X(511)
X(30270) = reflection of X(i) and X(j) for these {i,j}: {4, 626}, {32, 3}
X(30270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1350, 5188}, {3, 3095, 182}, {3, 5013, 21163}, {3, 5171, 5206}, {3, 5188, 8722}, {3, 9605, 5085}, {3, 9737, 574}, {3, 9821, 5171}, {3, 12054, 17508}, {20, 3926, 8721}, {98, 12251, 7751}, {182, 3095, 7772}, {631, 9753, 6680}, {1350, 18860, 8722}, {3098, 9737, 3}, {3926, 8721, 14981}, {5013, 5017, 13357}, {5017, 13357, 32}, {5188, 18860, 3}, {12305, 12306, 5085}, {13334, 14810, 3}, {14538, 14539, 182}

X(30271) = REFLECTION OF X(37) IN X(3)

Barycentrics a*(3*a^4*b - 2*a^2*b^3 - b^5 + 3*a^4*c + 4*a^3*b*c - 2*a^2*b^2*c - 4*a*b^3*c - b^4*c - 2*a^2*b*c^2 + 2*b^3*c^2 - 2*a^2*c^3 - 4*a*b*c^3 + 2*b^2*c^3 - b*c^4 - c^5) : :
X(30271) = 3 X[3] - X[20430],3 X[37] - 2 X[20430],3 X[165] - X[984],X[192] - 5 X[3522],8 X[548] - X[4718],4 X[550] + X[4686],5 X[631] - 4 X[4698],5 X[3091] - 7 X[4751],X[3146] - 5 X[4699],7 X[3523] - 5 X[4687],3 X[3524] - 2 X[4755],7 X[3528] - 2 X[4681],X[3529] + 4 X[4739],3 X[3576] - 2 X[15569],2 X[3842] - 3 X[10164],X[4664] - 3 X[10304],5 X[4704] - 13 X[21734],2 X[4726] + 5 X[17538],7 X[4772] + X[5059],3 X[9778] + X[24349],11 X[15717] - 7 X[27268]

X(30271) lies on these lines: {3, 37}, {4, 3739}, {20, 75}, {30, 4688}, {40, 518}, {56, 11997}, {63, 3198}, {71, 5784}, {103, 6011}, {104, 1296}, {165, 984}, {192, 3522}, {376, 536}, {515, 3696}, {516, 24325}, {517, 991}, {548, 4718}, {550, 4686}, {573, 971}, {631, 4698}, {726, 5188}, {740, 4297}, {851, 25939}, {910, 24320}, {944, 28581}, {1001, 12717}, {1009, 25887}, {1108, 3286}, {1212, 20605}, {1284, 17635}, {1764, 10167}, {1818, 21871}, {1824, 22060}, {2223, 12721}, {2691, 28838}, {2831, 16728}, {2941, 15931}, {3091, 4751}, {3146, 4699}, {3428, 21312}, {3523, 4687}, {3524, 4755}, {3528, 4681}, {3529, 4739}, {3576, 15569}, {3601, 7201}, {3752, 4192}, {3781, 21872}, {3842, 10164}, {3916, 16551}, {4259, 21866}, {4664, 10304}, {4704, 21734}, {4726, 17538}, {4772, 5059}, {5728, 20367}, {5927, 21363}, {6210, 15726}, {9778, 24349}, {10178, 20368}, {1031+0, 15624}, {10391, 24310}, {12545, 25124}, {14110, 20718}, {14872, 22271}, {15717, 27268}, {16602, 19540}, {16610, 19647}, {18607, 20243}

X(30271) = midpoint of X(20) and X(75)
X(30271) = reflection of X(i) and X(j) for these {i,j}: {4, 3739}, {37, 3}, {14872, 22271}
{X(40), X(5732)}-harmonic conjugate of X(1350)}

X(30272) = REFLECTION OF X(38) IN X(3)

Barycentrics a*(2*a^5*b + a^4*b^2 - 2*a^3*b^3 - b^6 + 2*a^5*c - 2*a*b^4*c + a^4*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 2*a*b*c^4 + b^2*c^4 - c^6) : :
X(30272) = X[631] - 4 X[6682],5 X[3522] - X[20068]

X(30272) lies on these lines: {3, 38}, {4, 1215}, {20, 17165}, {40, 758}, {42, 517}, {102, 26712}, {104, 28486}, {244, 19550}, {376, 537}, {515, 4692}, {573, 3949}, {631, 6682}, {984, 19262}, {1006, 6211}, {1350, 9020}, {3428, 12329}, {3522, 20068}, {4362, 5767}, {5690, 20653}, {5886, 29647}, {17442, 22061}, {19260, 25024}

X(30272) = midpoint of X(20) and X(17165)
X(30272) = reflection of X(i) and X(j) for these {i,j}: {4, 1215}, {38, 3}

X(30273) = REFLECTION OF X(75) IN X(3)

Barycentrics 2 a^5 b-2 a^3 b^3+2 a^5 c+a^4 b c-2 a^3 b^2 c-b^5 c-2 a^3 b c^2-2 a^3 c^3+2 b^3 c^3-b c^5 : :
X(30273) = 4 X[5] - 5 X[4687],8 X[140] - 7 X[4751],8 X[548] - X[4764],4 X[550] + X[3644],5 X[631] - 4 X[3739],X[1278] - 5 X[3522],7 X[3090] - 8 X[4698],5 X[3091] - 7 X[27268],X[3146] - 5 X[4704],7 X[3523] - 5 X[4699],3 X[3524] - 2 X[4688],7 X[3528] - 2 X[4686],X[3529] + 4 X[4681],3 X[3545] - 4 X[4755],3 X[3576] - 2 X[24325],2 X[3696] - 3 X[5657],4 X[3842] - 3 X[5587],3 X[4664] - 2 X[20430],2 X[4718] + 5 X[17538],4 X[4726] - 11 X[21735],8 X[4739] - 13 X[10299],X[4740] - 3 X[10304],7 X[4772] - 11 X[15717],5 X[4821] - 13 X[21734],3 X[5603] - 4 X[15569],3 X[5731] - X[24349],2 X[5805] - 3 X[27475],2 X[21443] - 3 X[22712]

X(30273) lies on these lines: {1, 4032}, {3, 75}, {4, 37}, {5, 4687}, {20, 192}, {30, 4664}, {40, 740}, {55, 7009}, {72, 25252}, {92, 228}, {98, 6011}, {140, 4751}, {198, 242}, {312, 4192}, {335, 29243}, {376, 536}, {411, 20171}, {515, 984}, {516, 3993}, {518, 944}, {548, 4764}, {550, 3644}, {573, 29016}, {631, 3739}, {726, 4297}, {742, 1350}, {991, 29069}, {1278, 3522}, {1742, 29057}, {2223, 4008}, {2329, 3923}, {2724, 2730}, {2805, 13199}, {3090, 4698}, {3091, 27268}, {3146, 4704}, {3190, 22001}, {3523, 4699}, {3524, 4688}, {3528, 4686}, {3529, 4681}, {3545, 4755}, {3576, 24325}, {3696, 5657}, {3842, 5587}, {3868, 25241}, {4292, 7201}, {4294, 7718}, {4358, 19647}, {4718, 17538}, {4726, 21735}, {4739, 10299}, {4740, 10304}, {4772, 15717}, {4821, 21734}, {5603, 15569}, {5731, 24349}, {5732, 24813}, {5768, 27472}, {5805, 27475}, {6210, 28850}, {6360, 17441}, {7414, 11491}, {7580, 20173}, {8680, 18446}, {9441, 24257}, {12245, 28581}, {12512, 28522}, {17479, 20243}, {18137, 19543}, {18743, 19540}, {18750, 20760}, {19513, 20923}, {19514, 30090}, {21443, 22712}

X(30273) = midpoint of X(20) and X(192)
X(30273) = reflection of X(i) and X(j) for these {i,j}: {4, 37}, {75, 3}
X(30273) = {X(944), X(5759)}-harmonic conjugate of X(6776)

Endo-homothetic centers: X(30274)-X(30434)

This preamble and centers X(30274)-X(30434) were contributed by César Eliud Lozada, December 19, 2018.

This section consists of the endo-homothetic centers of the family of triangles homothetic with the orthic triangle of a reference triangle ABC. This family is composed by the following 37 triangles:

anti-Ascella, anti-Atik, 1st anti-circumperp, anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 4th anti-Euler, anti-excenters-reflections, 2nd anti-extouch, anti-Honsberger, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, 1st anti-Sharygin, anti-tangential-midarc, anti-Ursa minor, anti-Wasat, circumorthic, Ehrmann-side, Ehrmann-vertex, 2nd Ehrmann, 2nd Euler, 1st excosine, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, Lucas antipodal tangential, Lucas(-1) antipodal tangential, orthic, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh.

For definitions and coordinates of these triangles, see the index of triangles referenced in ETC.

A table showing the endo-homothetic centers among these triangles can be seen here.

X(30274) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND EHRMANN-VERTEX

Barycentrics a*((b+c)*a^5-(b^2-b*c+c^2)*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3+(2*b^4+2*c^4-(3*b^2+2*b*c+3*c^2)*b*c)*a^2+(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(30274) = 2*X(942)+X(17603) = 4*X(942)+X(30282)

The homothetic center of these triangles is X(18386). X(30274) is their endo-homothetic center only when ABC is acute.

X(30274) lies on these lines: {1,3}, {2,18389}, {7,6840}, {80,6826}, {142,1737}, {226,6830}, {388,12005}, {443,10573}, {498,5904}, {499,26725}, {581,1393}, {631,15556}, {758,5744}, {912,5219}, {920,5259}, {938,2475}, {1056,5083}, {1210,2476}, {1439,4888}, {1478,5768}, {1858,8227}, {2800,6935}, {2801,10590}, {3085,3874}, {3485,5884}, {3487,6952}, {3583,5805}, {3585,5787}, {3586,10391}, {3754,6904}, {3868,13411}, {5432,5771}, {5439,25525}, {5443,6824}, {5445,6989}, {5691,9942}, {5692,5745}, {5693,11375}, {5728,6173}, {5883,9776}, {6224,12736}, {6245,12047}, {6879,12691}, {6972,11036}, {8068,12831}, {8261,25524}, {8727,18393}, {8728,10954}, {8729,18408}, {8731,30358}, {8732,30329}, {8733,18399}, {8734,18409}, {9579,13369}, {9613,12675}, {9614,12711}, {9655,26201}, {10039,24391}, {10449,20882}, {10527,20612}, {10855,30286}, {10896,17637}, {11020,17579}, {11219,11570}, {11237,17660}, {11571,13226}, {18410,30276}, {18411,30277}, {18422,30280}, {18423,30281}

X(30274) = reflection of X(30282) in X(17603)
X(30274) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3336, 11507), (3601, 24474, 5697), (11529, 18838, 5902)

X(30275) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 2nd EHRMANN

Barycentrics (a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a+3*(b^2-c^2)*(b-c))*(a+b-c)*(a-b+c) : :
X(30275) = X(7)+2*X(5219) = 4*X(142)-X(5744)

The homothetic center of these triangles is X(11405). X(30275) is their endo-homothetic center only when ABC is acute.

X(30275) lies on these lines: {2,7}, {3,8543}, {347,4648}, {388,25557}, {390,5603}, {443,5730}, {497,8255}, {516,30282}, {942,5261}, {948,4675}, {952,1056}, {954,6905}, {971,6844}, {997,12560}, {1441,4869}, {2095,8164}, {2099,2550}, {2801,10590}, {3090,5729}, {3091,10394}, {3485,5880}, {3601,30332}, {3753,7672}, {4552,29621}, {5177,5784}, {5220,10588}, {5274,7671}, {5308,22464}, {5542,18391}, {5686,7679}, {5698,11375}, {5714,9940}, {5728,6843}, {5735,5766}, {5762,6954}, {5779,6859}, {5780,6147}, {6987,21151}, {8544,8726}, {8727,30311}, {8728,30312}, {8729,30404}, {8731,30359}, {8733,30367}, {8734,30405}, {9814,11407}, {10569,10865}, {10707,18801}, {10855,30287}, {10857,30353}, {11023,13407}, {11025,17620}, {11518,30318}, {12573,13462}, {15726,17603}, {15803,30424}, {18443,18450}

X(30275) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 12848), (7, 142, 8732), (142, 6173, 9776)

X(30276) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(5410). X(30276) is their endo-homothetic center only when ABC is acute.

X(30276) lies on these lines: {2,7}, {3,30296}, {277,16232}, {942,30341}, {3513,24154}, {3514,24155}, {3601,30333}, {4000,13388}, {4648,13389}, {6351,8243}, {7133,10858}, {8726,30400}, {8727,30306}, {8728,30313}, {8729,30406}, {8731,30360}, {8733,30368}, {8734,30418}, {10857,30354}, {11018,30346}, {11407,30396}, {11518,30319}, {13390,13940}, {15803,30425}, {17603,30375}, {18410,30274}, {18443,18458}, {30282,30431}

X(30276) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30324), (57, 142, 30277)

X(30277) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(5411). X(30277) is their endo-homothetic center only when ABC is acute.

X(30277) lies on these lines: {2,7}, {3,30297}, {277,2362}, {942,30342}, {1659,13887}, {3514,24154}, {3601,30334}, {4000,13389}, {4648,13388}, {8726,30401}, {8727,30307}, {8728,30314}, {8729,30407}, {8731,30361}, {8733,30369}, {8734,30419}, {10855,30289}, {10857,30355}, {11018,30347}, {11407,30397}, {11518,30320}, {15803,30426}, {17603,30376}, {18411,30274}, {18443,18460}, {30282,30432}

X(30277) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30325), (57, 142, 30276)

X(30278) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND LUCAS ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19404). X(30278) is their endo-homothetic center only when ABC is acute.

X(30278) lies on these lines: {2,30302}, {57,7133}, {610,15892}, {3601,30335}, {5745,30416}, {10857,30279}, {11018,30348}, {11518,16213}

X(30279) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND LUCAS(-1) ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19405). X(30279) is their endo-homothetic center only when ABC is acute.

X(30279) lies on these lines: {2,30303}, {57,30430}, {610,15891}, {3601,30336}, {5745,30417}, {10857,30278}, {11018,30349}, {11518,16214}

X(30280) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11408). X(30280) is their endo-homothetic center only when ABC is acute.

X(30280) lies on these lines: {2,7}, {3,30300}, {277,2306}, {942,30344}, {3601,30338}, {8726,10649}, {8727,30309}, {8728,30316}, {8729,30409}, {8731,30364}, {8733,30372}, {8734,30421}, {10855,30292}, {10857,30356}, {11018,30351}, {11407,10655}, {11518,30321}, {15803,10651}, {17603,30377}, {18422,30274}, {18443,18469}, {30282,30433}

X(30280) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30327), (57, 142, 30281)

X(30281) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11409). X(30281) is their endo-homothetic center only when ABC is acute.

X(30281) lies on these lines: {2,7}, {3,30301}, {942,30345}, {3601,30339}, {8726,10650}, {8727,30310}, {8728,30317}, {8729,30410}, {8731,30365}, {8733,30373}, {8734,30422}, {10855,30293}, {10857,30357}, {11018,30352}, {11407,10656}, {11518,30322}, {15803,10652}, {17603,30378}, {18423,30274}, {18443,18471}, {30282,30434}

X(30281) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30328), (57, 142, 30280)

X(30282) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND TRINH

Barycentrics a*(5*a^3-(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a+(b^2-c^2)*(b-c)) : :
X(30282) = 2*X(942)-5*X(17603) = 4*X(942)-5*X(30274) = (r+2R)*X(1) + 4r*X(3)

The homothetic center of these triangles is X(11410). X(30282) is their endo-homothetic center only when ABC is acute.

X(30282) lies on these lines: {1,3}, {2,3586}, {7,10304}, {9,5440}, {10,4305}, {20,5226}, {21,936}, {30,5219}, {33,7501}, {63,17549}, {72,19535}, {78,3219}, {80,6174}, {100,9623}, {103,15730}, {140,9581}, {142,25055}, {200,993}, {226,376}, {284,1743}, {405,5438}, {443,1479}, {452,6700}, {474,5436}, {497,10165}, {498,5691}, {515,5218}, {516,30275}, {519,5744}, {549,5722}, {550,9579}, {551,9776}, {553,19708}, {579,16667}, {581,22072}, {610,3465}, {631,950}, {938,15717}, {944,6705}, {956,3158}, {975,7520}, {991,22350}, {997,4512}, {1012,1750}, {1125,4294}, {1210,3523}, {1335,9615}, {1387,3653}, {1453,4255}, {1490,6906}, {1698,6857}, {1699,4302}, {1708,21161}, {1724,19764}, {1817,4653}, {1876,11410}, {1914,9592}, {2975,6765}, {2999,4256}, {3062,15175}, {3085,4297}, {3086,4314}, {3216,13726}, {3306,13587}, {3452,11111}, {3486,6684}, {3487,3528}, {3488,3524}, {3522,4292}, {3583,6826}, {3584,5726}, {3585,6851}, {3616,10624}, {3632,12437}, {3633,24391}, {3679,5745}, {3811,5267}, {3871,12629}, {3876,17574}, {3916,11523}, {3928,19704}, {3929,3940}, {4031,15710}, {4114,21735}, {4224,5268}, {4258,16572}, {4276,17194}, {4293,13405}, {4295,12512}, {4299,5290}, {4324,6869}, {4330,6885}, {4652,17548}, {4654,5719}, {4853,8715}, {4866,15446}, {4882,5258}, {4995,5252}, {5013,16780}, {5044,17571}, {5248,8583}, {5251,8580}, {5259,19520}, {5281,5731}, {5313,16469}, {5414,9583}, {5432,5587}, {5435,15692}, {5437,16371}, {5439,19537}, {5441,6675}, {5444,23708}, {5705,6910}, {5714,17538}, {5715,6934}, {5720,6914}, {5727,11545}, {5732,6909}, {5768,12647}, {5886,9580}, {6245,10039}, {6284,8227}, {6666,17561}, {6824,7989}, {6872,27385}, {6875,10393}, {6950,18446}, {7308,16418}, {7675,10398}, {7741,8728}, {7951,8727}, {7972,13226}, {8729,30411}, {8731,16569}, {8732,30331}, {8733,30374}, {8734,30423}, {9578,18481}, {9582,16232}, {9588,10573}, {9624,12701}, {9668,11230}, {9785,11023}, {10058,15015}, {10164,18391}, {10386,11373}, {10391,18397}, {10543,24914}, {10590,28164}, {10591,19862}, {10855,30294}, {11112,25525}, {11375,15338}, {11491,12650}, {12433,15712}, {12526,22836}, {12572,17576}, {15326,17718}, {15677,27131}, {15688,18541}, {15705,15933}, {15935,17504}, {16056,25502}, {16132,16140}, {16342,19859}, {17284,24609}, {18540,28444}, {21483,23511}, {24604,29571}, {28452,30308}, {30276,30431}, {30277,30432}, {30280,30433}, {30281,30434}

X(30282) = reflection of X(i) in X(j) for these (i,j): (1, 13384), (30274, 17603)
X(30282) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 15803), (1, 484, 18421), (1, 7280, 3361), (484, 18421, 2093)

X(30283) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-SIDE

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-7*b*c+c^2)*a^4+2*(b+c)*(b^2-5*b*c+c^2)*a^3+(b^2-8*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2)*a-4*(b^2-c^2)^2*b*c) : :
X(30283) = 3*X(999)-2*X(22753) = X(3062)-3*X(10384) = 4*X(10269)-3*X(16417) = 3*X(19541)-4*X(22753)

The homothetic center of these triangles is X(18917). X(30283) is their endo-homothetic center only when ABC is acute.

X(30283) lies on these lines: {1,971}, {3,8}, {4,7373}, {20,8158}, {36,30286}, {40,11519}, {56,5727}, {84,9957}, {153,17556}, {355,8582}, {381,12115}, {382,12001}, {392,5779}, {496,12667}, {515,999}, {517,7171}, {519,6244}, {942,3577}, {958,12447}, {997,18227}, {1012,6767}, {1056,8727}, {1319,17604}, {1376,28236}, {1385,5720}, {1490,24928}, {1656,10785}, {1709,5919}, {2096,28174}, {2098,13253}, {2801,5289}, {2829,9668}, {3057,10085}, {3295,5882}, {3304,5691}, {3526,10786}, {3576,8580}, {3600,20420}, {3655,16418}, {3895,17613}, {3940,9954}, {4297,5853}, {4308,5809}, {4317,6253}, {4423,30291}, {4511,11678}, {5258,8273}, {5288,5584}, {5697,12767}, {5787,10106}, {5789,24987}, {5818,16863}, {5881,9709}, {5886,10863}, {5927,6913}, {6256,9669}, {6259,12053}, {6260,11373}, {6265,13227}, {6831,10805}, {6911,28224}, {6918,16203}, {6971,18545}, {7962,30304}, {8166,14986}, {9614,22792}, {9623,11227}, {9785,12246}, {9841,12629}, {9943,12448}, {9949,11496}, {9951,12737}, {10267,17571}, {10269,16417}, {10569,15934}, {10855,18443}, {10861,18444}, {10865,30284}, {10868,30285}, {11260,12520}, {11499,17573}, {11858,18448}, {11859,18456}, {11860,18454}, {12672,12684}, {12678,30384}, {15178,18761}, {16202,26321}, {17612,19860}, {18450,30287}, {18458,30288}, {18460,30289}, {18469,30292}, {18471,30293}, {18481,22770}

X(30283) = midpoint of X(i) and X(j) for these {i,j}: {944, 5768}, {7962, 30304}
X(30283) = reflection of X(i) in X(j) for these (i,j): (5720, 1385), (19541, 999)
X(30283) = X(5727)-of-2nd circumperp tangential triangle
X(30283) = X(6244)-of-inner-Garcia triangle
X(30283) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10864, 9856), (956, 5731, 3)

X(30284) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND EHRMANN-SIDE

Barycentrics a*(a^4-2*(b+c)*a^3-3*b*c*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-b*c+c^2)*(b-c)^2)*(-a+b+c) : :
X(30284) = 3*X(2646)-X(15837)

The homothetic center of these triangles is X(19129). X(30284) is their endo-homothetic center only when ABC is acute.

X(30284) lies on these lines: {1,7}, {3,7672}, {9,2320}, {33,29814}, {36,30329}, {40,11526}, {55,3218}, {78,5686}, {100,17603}, {104,2346}, {142,6224}, {355,7679}, {411,16193}, {497,29817}, {515,21617}, {517,7676}, {518,2330}, {971,8543}, {997,18230}, {999,11025}, {1001,10394}, {1040,17018}, {1156,6265}, {1319,5572}, {1385,5728}, {1445,3576}, {1467,18221}, {1482,7673}, {1602,22769}, {1617,11020}, {1621,10391}, {1864,5284}, {2094,10388}, {2099,11495}, {2801,29007}, {3057,15570}, {3243,3601}, {3616,5809}, {3826,10950}, {3870,5281}, {3935,5218}, {4420,24393}, {4666,5274}, {4861,5853}, {5173,7411}, {5223,22836}, {5563,20116}, {5727,20195}, {5775,6765}, {5886,7678}, {7191,14547}, {7671,10246}, {8232,18446}, {8238,30285}, {8387,18448}, {8388,18456}, {8389,18454}, {8732,18443}, {10865,30283}, {11496,12706}, {12114,12669}, {12730,12737}, {12740,14100}, {27542,29835}, {30330,30392}

X(30284) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 991, 4318), (1, 12520, 4323), (5731, 11037, 4293)

X(30285) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND EHRMANN-SIDE

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

The homothetic center of these triangles is X(19176). X(30285) is their endo-homothetic center only when ABC is acute.

X(30285) lies on these lines: {1,256}, {3,2292}, {21,104}, {36,30358}, {40,11533}, {182,5692}, {355,5051}, {515,4425}, {517,3920}, {846,3576}, {944,26117}, {960,7193}, {997,18235}, {999,11031}, {1319,17611}, {3145,10267}, {4199,18446}, {4511,11688}, {5289,8424}, {5492,15952}, {5693,13323}, {5731,9791}, {5886,8229}, {8238,30284}, {8249,18448}, {8250,18456}, {8425,18454}, {8731,18443}, {10868,30283}, {11496,12713}, {12114,12683}, {12737,12746}, {18450,30359}, {18458,30360}, {18460,30361}, {18469,30364}, {18471,30365}, {30363,30392}

X(30285) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8235, 9840), (1385, 9959, 21)

X(30286) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-VERTEX

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

The homothetic center of these triangles is X(18918). X(30286) is their endo-homothetic center only when ABC is acute.

X(30286) lies on these lines: {1,2}, {11,11224}, {36,30283}, {46,10864}, {65,9656}, {80,2093}, {165,5727}, {355,3339}, {484,10860}, {517,17604}, {758,11678}, {952,13462}, {1111,4902}, {1146,1743}, {1728,11010}, {1837,7991}, {2099,7988}, {2801,30287}, {3337,7091}, {3340,7989}, {3361,5881}, {3419,10398}, {3474,4848}, {3486,9588}, {3586,5759}, {3753,15587}, {3894,12736}, {4731,11018}, {5119,10384}, {5252,10980}, {5435,28236}, {5587,11545}, {5603,16236}, {5692,18227}, {5697,10866}, {5722,9819}, {5726,5790}, {5763,9581}, {5902,8581}, {5903,9856}, {5927,18397}, {7743,7982}, {7987,10950}, {8164,14563}, {8256,12625}, {10175,11041}, {10855,30274}, {10863,18393}, {10865,30329}, {10868,30358}, {11035,18398}, {11375,30315}, {11571,13227}, {11858,18399}, {11859,18409}, {11860,18408}, {18410,30288}, {18411,30289}, {18422,30292}, {18423,30293}, {24914,30389}

X(30286) = reflection of X(30294) in X(17604)
X(30286) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1210, 3632, 1), (3086, 3633, 1), (3679, 18391, 1)

X(30287) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd EHRMANN

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

The homothetic center of these triangles is X(18919). X(30287) is their endo-homothetic center only when ABC is acute.

X(30287) lies on these lines: {7,8}, {516,30294}, {527,11678}, {1156,1445}, {2801,30286}, {4326,30389}, {5435,15726}, {5927,12848}, {6172,18227}, {7671,11019}, {7677,10384}, {8543,8583}, {8544,10864}, {8545,8580}, {8582,30312}, {8732,11575}, {9814,30291}, {10855,30275}, {10860,30295}, {10863,30311}, {10866,30332}, {10868,30359}, {11035,30340}, {11519,30318}, {11858,30367}, {11859,30405}, {11860,30404}, {14986,21151}, {17615,20059}, {18450,30283}, {25722,26015}, {30290,30424}

X(30287) = {X(7), X(15587)}-harmonic conjugate of X(10865)

X(30288) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(18923). X(30288) is their endo-homothetic center only when ABC is acute.

X(30288) lies on these lines: {7,8}, {3062,7133}, {5927,30324}, {6203,8580}, {8582,30313}, {8583,30385}, {10860,30296}, {10863,30306}, {10864,30400}, {10866,30333}, {10868,30360}, {11019,30346}, {11035,30341}, {11519,30319}, {11858,30368}, {11859,30418}, {11860,30406}, {17604,30375}, {18227,30412}, {18410,30286}, {18458,30283}, {30290,30425}, {30291,30396}, {30294,30431}

X(30288) = {X(8581), X(15587)}-harmonic conjugate of X(30289)

X(30289) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(18924). X(30289) is their endo-homothetic center only when ABC is acute.

X(30289) lies on these lines: {7,8}, {3062,30355}, {5927,30325}, {6204,8580}, {8582,30314}, {8583,30386}, {10855,30277}, {10860,30297}, {10863,30307}, {10864,30401}, {10866,30334}, {11019,30347}, {11035,30342}, {11519,30320}, {11858,30369}, {11859,30419}, {11860,30407}, {17604,30376}, {18227,30413}, {18411,30286}, {18460,30283}, {30290,30426}, {30291,30397}, {30294,30432}

X(30289) = {X(8581), X(15587)}-harmonic conjugate of X(30288)

X(30290) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND KOSNITA

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

The homothetic center of these triangles is X(18925). X(30290) is their endo-homothetic center only when ABC is acute.

X(30290) lies on these lines: {1,971}, {8,79}, {10,11678}, {35,10860}, {36,8583}, {46,8580}, {55,16143}, {65,9656}, {72,4312}, {80,13227}, {90,5563}, {226,9948}, {516,10865}, {942,17604}, {946,12666}, {1698,18227}, {1858,4654}, {1898,11518}, {2093,9954}, {3339,5777}, {3633,12448}, {3671,12528}, {4292,5692}, {4303,27785}, {5223,18251}, {5290,6001}, {5691,12709}, {5694,18541}, {5696,11523}, {5697,9589}, {5714,5884}, {5784,9814}, {5902,5927}, {6765,17646}, {7741,10863}, {7951,8582}, {7972,9951}, {7991,17634}, {8545,12520}, {9949,13407}, {9952,11571}, {9961,13405}, {10392,10399}, {10394,12563}, {10624,28164}, {10855,15803}, {10868,30362}, {10883,11019}, {10948,18393}, {11519,25415}, {11522,17625}, {11858,30370}, {11859,30420}, {11860,30408}, {12565,15298}, {14872,18421}, {30287,30424}, {30288,30425}, {30289,30426}, {30292,10651}, {30293,10652}

X(30290) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9856, 30294), (8581, 10866, 11035), (10866, 11035, 1)

X(30291) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND SUBMEDIAL

Barycentrics a*(a^5-7*(b+c)*a^4+2*(5*b^2+6*b*c+5*c^2)*a^3+2*(b+c)*(b^2-10*b*c+c^2)*a^2-(11*b^2+26*b*c+11*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(5*b^2+22*b*c+5*c^2)) : :

The homothetic center of these triangles is X(18928). X(30291) is their endo-homothetic center only when ABC is acute.

X(30291) lies on these lines: {1,9947}, {8,1699}, {9,165}, {200,24644}, {210,7991}, {226,7989}, {1698,5658}, {3339,5777}, {4423,30283}, {5047,8583}, {5223,11678}, {5226,7988}, {5437,24645}, {5587,11545}, {5691,12447}, {7308,7987}, {7994,10241}, {7997,16209}, {8581,10980}, {8582,30315}, {9814,30287}, {9819,30294}, {10855,11407}, {10861,30304}, {10863,30308}, {10865,30330}, {10866,30337}, {10868,30363}, {11035,30343}, {11519,12635}, {11858,30371}, {11859,30395}, {11860,30394}, {30288,30396}, {30289,30397}, {30292,10655}, {30293,10656}

X(30291) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1750, 30393, 165), (3062, 8580, 165), (9947, 10157, 17604)

X(30292) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(18929). X(30292) is their endo-homothetic center only when ABC is acute.

X(30292) lies on these lines: {7,8}, {1653,8580}, {3062,30356}, {5927,30327}, {8523,28097}, {8582,30316}, {8583,10647}, {10855,30280}, {10860,30300}, {10863,30309}, {10864,10649}, {10866,30338}, {10868,30364}, {11019,30351}, {11035,30344}, {11519,30321}, {11859,30421}, {11860,30409}, {17604,30377}, {18227,30414}, {18422,30286}, {18469,30283}, {30290,10651}, {30291,10655}, {30294,30433}

X(30292) = {X(8581), X(15587)}-harmonic conjugate of X(30293)

X(30293) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(18930). X(30293) is their endo-homothetic center only when ABC is acute.

X(30293) lies on these lines: {7,8}, {1251,3062}, {1652,8580}, {5927,30328}, {8582,30317}, {8583,10648}, {10855,30281}, {10860,30301}, {10863,30310}, {10864,10650}, {10866,30339}, {11019,30352}, {11035,30345}, {11519,30322}, {11858,30373}, {11859,30422}, {11860,30410}, {17604,30378}, {18227,30415}, {18471,30283}, {30290,10652}, {30291,10656}, {30294,30434}

X(30293) = {X(8581), X(15587)}-harmonic conjugate of X(30292)

X(30294) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND TRINH

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

The homothetic center of these triangles is X(18931). X(30294) is their endo-homothetic center only when ABC is acute

X(30294) lies on these lines: {1,971}, {8,80}, {35,8583}, {36,10860}, {392,15587}, {516,30287}, {517,17604}, {519,11678}, {1864,11224}, {2098,7993}, {2801,4345}, {3586,5927}, {3632,12448}, {3679,18227}, {4294,12446}, {4301,7672}, {4304,10861}, {5083,15071}, {5119,8580}, {5289,5696}, {5441,16120}, {5902,11019}, {5903,9614}, {6797,9669}, {7091,7284}, {7741,8582}, {7951,10863}, {7972,13227}, {8275,18908}, {9671,25414}, {9819,30291}, {9948,12053}, {10392,18397}, {10624,12447}, {10855,30282}, {10865,30331}, {10868,30366}, {11519,30323}, {11858,30374}, {11859,30423}, {11860,30411}, {30288,30431}, {30289,30432}, {30292,30433}, {30293,30434}

X(30294) = reflection of X(30286) in X(17604)
X(30294) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9856, 30290), (9856, 10866, 1)

X(30295) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 2nd EHRMANN

Barycentrics a*(a^4-3*(b+c)*a^3+(3*b^2-b*c+3*c^2)*a^2+3*(b-c)^2*b*c-(b^2-c^2)*(b-c)*a) : :
X(30295) = 4*X(36)-3*X(7677) = 4*X(1155)-X(1156) = 2*X(4316)+X(20119) = X(12730)-4*X(21578)

The homothetic center of these triangles is X(11416). X(30295) is their endo-homothetic center only when ABC is acute.

X(30295) lies on these lines: {2,30311}, {3,8543}, {4,30312}, {7,55}, {21,5880}, {35,30424}, {36,516}, {40,8544}, {46,10394}, {56,30332}, {57,7671}, {100,527}, {142,5284}, {165,8545}, {376,390}, {404,5698}, {484,2801}, {517,14151}, {518,5183}, {528,15326}, {651,3000}, {1001,17549}, {1155,1156}, {1376,6172}, {1445,2951}, {1621,6173}, {1633,11349}, {1770,6986}, {2093,5732}, {2550,17579}, {3218,15733}, {3295,30340}, {3826,6175}, {4038,4343}, {4220,30359}, {4312,5010}, {4316,20119}, {4319,17092}, {4321,9819}, {4326,10980}, {4480,4578}, {5274,8732}, {6600,20059}, {6767,11038}, {7580,12848}, {7589,30404}, {7673,7962}, {7675,11529}, {7678,10589}, {7679,10590}, {7991,30318}, {8075,30367}, {8076,30405}, {8257,9352}, {10177,27003}, {10860,30287}, {12730,21578}, {13587,28534}, {14189,24011}, {15254,17531}

X(30295) = reflection of X(14151) in X(18450)
X(30295) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 7676, 2346), (7, 11495, 7676), (8255, 11246, 7)

X(30296) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11417). X(30283) is their endo-homothetic center only when ABC is acute.

X(30296) lies on these lines: {2,30306}, {3,30276}, {4,30313}, {7,55}, {35,30425}, {36,30431}, {40,30400}, {56,30333}, {57,30346}, {165,6203}, {484,18410}, {516,30380}, {517,18458}, {1155,30375}, {1376,30412}, {1721,6204}, {1742,5414}, {2066,9441}, {3295,30341}

X(30296) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30297), (165, 30354, 6203)

X(30297) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11418). X(30297) is their endo-homothetic center only when ABC is acute.

X(30297) lies on these lines: {2,30307}, {3,30277}, {4,30314}, {7,55}, {35,30426}, {36,30432}, {40,30401}, {56,30334}, {57,30347}, {165,6204}, {484,18411}, {516,30381}, {517,18460}, {1155,30376}, {1376,30413}, {1721,6203}, {1742,2066}, {3295,30342}, {4220,30361}, {4319,13389}, {5414,9441}, {7580,30325}, {7589,30407}, {7991,30320}, {8075,30369}, {8076,30419}, {10860,30289}

X(30297) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30296), (165, 30355, 6204)

X(30298) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a^2*((8*a^5+12*(b+c)*a^4-16*(3*b^2+5*b*c+3*c^2)*a^3+24*(b^2-c^2)*(b-c)*a^2+8*(b^2+12*b*c+c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)^3)*S-(a+b-c)*(a-b+c)*(3*a^5-13*(b+c)*a^4+2*(3*b^2-34*b*c+3*c^2)*a^3+2*(b+c)*(7*b^2+38*b*c+7*c^2)*a^2-(9*b^2+22*b*c+9*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)))*((a^3+(5*b-c)*a^2-(5*b^2+2*b*c+c^2)*a-(b-c)^3)*S-(a+b-c)*(a^4-(2*b+c)*a^3-(3*b+c)*a^2*c+(2*b^3+c^3+(b+4*c)*b*c)*a-(b-c)^3*b))*((a^3-(b-5*c)*a^2-(b^2+2*b*c+5*c^2)*a+(b-c)^3)*S-(a-b+c)*(a^4-(b+2*c)*a^3-(b+3*c)*a^2*b+(b^3+2*c^3+(4*b+c)*b*c)*a+(b-c)^3*c))*((a^3-(b+3*c)*a^2-(b+3*c)*(b-c)*a+(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(b-3*c)*(b-c)*a^2-(b^2-c^2)*(b+3*c)*a+(b^2-c^2)*(b-c)^2))*((a^3-(3*b+c)*a^2+(3*b+c)*(b-c)*a-(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(3*b-c)*(b-c)*a^2+(b^2-c^2)*(3*b+c)*a-(b^2-c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19406). X(30298) is their endo-homothetic center only when ABC is acute.

X(30298) lies on these lines: {}

X(30299) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a^2*(-(8*a^5+12*(b+c)*a^4-16*(3*b^2+5*b*c+3*c^2)*a^3+24*(b^2-c^2)*(b-c)*a^2+8*(b^2+12*b*c+c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)^3)*S-(a+b-c)*(a-b+c)*(3*a^5-13*(b+c)*a^4+2*(3*b^2-34*b*c+3*c^2)*a^3+2*(b+c)*(7*b^2+38*b*c+7*c^2)*a^2-(9*b^2+22*b*c+9*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)))*(-(a^3+(5*b-c)*a^2-(5*b^2+2*b*c+c^2)*a-(b-c)^3)*S-(a+b-c)*(a^4-(2*b+c)*a^3-(3*b+c)*a^2*c+(2*b^3+c^3+(b+4*c)*b*c)*a-(b-c)^3*b))*(-(a^3-(b-5*c)*a^2-(b^2+2*b*c+5*c^2)*a+(b-c)^3)*S-(a-b+c)*(a^4-(b+2*c)*a^3-(b+3*c)*a^2*b+(b^3+2*c^3+(4*b+c)*b*c)*a+(b-c)^3*c))*(-(a^3-(b+3*c)*a^2-(b+3*c)*(b-c)*a+(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(b-3*c)*(b-c)*a^2-(b^2-c^2)*(b+3*c)*a+(b^2-c^2)*(b-c)^2))*(-(a^3-(3*b+c)*a^2+(3*b+c)*(b-c)*a-(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(3*b-c)*(b-c)*a^2+(b^2-c^2)*(3*b+c)*a-(b^2-c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19407). X(30299) is their endo-homothetic center only when ABC is acute.

X(30299) lies on these lines: {}

X(30300) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11420). X(30300) is their endo-homothetic center only when ABC is acute.

X(30300) lies on these lines: {2,30309}, {3,30280}, {4,30316}, {7,55}, {35,10651}, {36,30433}, {40,10649}, {56,30338}, {57,30351}, {165,1653}, {484,18422}, {516,30382}, {1082,4336}, {1155,30377}, {1250,1742}, {1376,30414}, {3295,30344}, {4220,30364}, {7580,30327}, {7589,30409}, {7991,30321}, {8075,30372}, {8076,30421}, {9441,10638}, {10860,30292}

X(30300) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30301), (165, 30356, 1653)

X(30301) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11421). X(30301) is their endo-homothetic center only when ABC is acute.

X(30301) lies on these lines: {2,30310}, {3,30281}, {4,30317}, {7,55}, {35,10652}, {36,30434}, {40,10650}, {56,30339}, {57,30352}, {165,1652}, {484,18423}, {516,30383}, {517,18471}, {559,4336}, {1155,30378}, {1250,9441}, {1251,1653}, {1376,30415}, {1742,10638}, {3295,30345}, {4220,30365}, {7580,30328}, {7589,30410}, {7991,30322}, {8075,30373}, {8076,30422}, {10860,30293}

X(30301) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30300), (165, 30357, 1652)

X(30302) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND LUCAS ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19408). X(30302) is their endo-homothetic center only when ABC is acute.

X(30302) lies on these lines: {2,30278}, {7,10134}, {21,30387}, {63,30429}, {4313,30335}, {5273,30416}, {8822,10430}, {11020,30348}

X(30303) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND LUCAS(-1) ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19409). X(30303) is their endo-homothetic center only when ABC is acute.

X(30303) lies on these lines: {2,30279}, {7,10135}, {21,30388}, {63,30430}, {4313,30336}, {5273,30417}, {8822,10430}, {11020,30349}

X(30304) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND SUBMEDIAL

Barycentrics a*(a^5+(b+c)*a^4-2*(3*b^2-4*b*c+3*c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(5*b^2+2*b*c+5*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(-3*b^2-2*b*c-3*c^2)) : :
X(30304) = 3*X(57)-2*X(19541) = 3*X(165)-2*X(200) = 4*X(997)-5*X(7987) = 3*X(1699)-4*X(11019) = 3*X(1750)-4*X(19541) = 2*X(5720)-3*X(21164) = 3*X(9778)-X(20015)

The homothetic center of these triangles is X(9306). X(30304) is their endo-homothetic center only when ABC is acute.

X(30304) lies on these lines: {1,84}, {2,11407}, {3,3929}, {4,553}, {7,1699}, {9,10167}, {20,519}, {21,30363}, {40,5918}, {46,12671}, {57,971}, {63,100}, {65,10864}, {72,9841}, {223,7004}, {354,11372}, {388,9948}, {515,2093}, {516,9965}, {518,7994}, {610,8558}, {912,6282}, {936,12528}, {942,12684}, {944,14646}, {950,12246}, {997,5267}, {1210,6223}, {1490,1708}, {1728,9942}, {1765,3731}, {1776,6261}, {2808,3784}, {2951,15733}, {3057,9845}, {3158,17613}, {3219,21153}, {3333,12688}, {3339,4292}, {3361,9960}, {3522,3951}, {3576,3683}, {3586,5768}, {3679,6916}, {3742,16112}, {3868,11531}, {3874,12651}, {3911,5658}, {3928,7580}, {4197,30315}, {4297,12526}, {4304,9819}, {4313,30337}, {4350,8835}, {4654,8727}, {4866,9588}, {5220,10178}, {5249,7988}, {5272,9355}, {5273,5785}, {5400,8056}, {5437,5927}, {5587,12678}, {5720,21164}, {5735,10431}, {5779,7308}, {5784,8580}, {5787,9579}, {5851,24703}, {5882,10385}, {6173,8226}, {6245,6844}, {6259,9581}, {6260,6969}, {6883,7330}, {6950,18446}, {6993,7989}, {7291,9572}, {7489,18443}, {7962,30283}, {7996,10444}, {9778,20015}, {9949,12577}, {10122,18219}, {10202,18540}, {10270,17857}, {10384,12915}, {10861,30291}, {10883,30308}, {11020,24644}, {11036,30343}, {11520,16189}, {11888,30371}, {11889,30395}, {11890,30394}, {18389,18421}, {18444,30392}, {20307,26932}

X(30304) = midpoint of X(i) and X(j) for these {i,j}: {7991, 18452}, {9965, 10430}
X(30304) = reflection of X(i) in X(j) for these (i,j): (1750, 57), (3586, 5768), (5691, 18391), (6282, 7171), (7962, 30283), (7994, 10860)
X(30304) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10085, 15071, 1), (12675, 12705, 1)

X(30305) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY AND TRINH

Barycentrics a^4-8*a^2*b*c+2*(b+c)*a^3-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(30305) = 3*X(1)-2*X(4315) = 3*X(497)-2*X(5722) = 5*X(497)-4*X(18527) = 3*X(4293)-4*X(4315) = X(4293)-4*X(4342) = X(4315)-3*X(4342) = 3*X(5603)-2*X(22753) = 4*X(5722)-3*X(18391) = 5*X(5722)-6*X(18527) = 5*X(18391)-8*X(18527)

The homothetic center of these triangles is X(11438). X(30305) is their endo-homothetic center only when ABC is acute.

X(30305) lies on these lines: {1,7}, {2,5119}, {3,1387}, {4,1000}, {8,80}, {10,6919}, {11,5657}, {30,3476}, {35,3616}, {36,9778}, {40,3086}, {46,14986}, {55,5603}, {56,6361}, {57,28194}, {65,1058}, {145,10572}, {319,4673}, {329,519}, {355,5225}, {376,1319}, {377,3890}, {388,9957}, {392,2550}, {411,11508}, {484,5435}, {496,1788}, {497,517}, {498,6979}, {499,11010}, {515,7962}, {528,5289}, {535,3241}, {551,9776}, {631,11376}, {908,3895}, {938,5903}, {944,1317}, {946,1697}, {950,5758}, {952,9668}, {956,5698}, {958,13463}, {960,5082}, {997,17784}, {999,3474}, {1056,1836}, {1145,17556}, {1191,17366}, {1210,7991}, {1265,5100}, {1320,11114}, {1388,15338}, {1478,9812}, {1482,3486}, {1699,5726}, {1737,5274}, {1837,12245}, {2093,11019}, {2099,3058}, {2136,21075}, {2475,16155}, {2478,14923}, {2551,10914}, {2646,10595}, {2800,5768}, {3091,10039}, {3189,5730}, {3218,11240}, {3295,3485}, {3296,18490}, {3303,3487}, {3333,4031}, {3340,14563}, {3421,3880}, {3434,3877}, {3436,3885}, {3465,3938}, {3475,6767}, {3523,18220}, {3579,7288}, {3583,12647}, {3601,13464}, {3612,3622}, {3617,10826}, {3624,11024}, {3632,5815}, {3651,11510}, {3656,10385}, {3679,18228}, {3698,17559}, {3746,5703}, {3753,26105}, {3832,10827}, {3902,5739}, {3992,8055}, {4511,20075}, {4853,12572}, {4857,10573}, {4861,6872}, {5048,7967}, {5080,12648}, {5084,5836}, {5183,17728}, {5218,5886}, {5222,5315}, {5226,10056}, {5229,22793}, {5234,9874}, {5250,19843}, {5270,5561}, {5290,30337}, {5328,5541}, {5330,9963}, {5425,15933}, {5441,14450}, {5493,15803}, {5526,5838}, {5556,13602}, {5586,30343}, {5687,12732}, {5690,9669}, {5714,15888}, {5727,28234}, {5744,21630}, {5748,11813}, {5809,18397}, {5811,5881}, {5812,13600}, {5818,10896}, {5880,10179}, {5902,10580}, {5904,6764}, {6001,17642}, {6666,19855}, {6845,10957}, {6906,10966}, {6909,22767}, {6969,22835}, {7080,21616}, {7741,9780}, {7743,10589}, {7951,9779}, {7957,10866}, {7972,9809}, {8163,12246}, {8164,17605}, {8715,27383}, {9578,18483}, {9581,11362}, {9670,10950}, {9791,30366}, {9793,30374}, {9795,30423}, {9955,10588}, {10200,26062}, {10284,10526}, {10944,12953}, {11491,26358}, {11522,13411}, {11531,16236}, {11891,30411}, {12247,13274}, {12248,20586}, {12527,12629}, {12700,17622}, {12740,13199}, {14217,15558}, {15170,15934}, {15172,15935}, {21669,22759}, {24046,28016}, {26129,26364}, {26839,27253}

X(30305) = midpoint of X(7962) and X(9580)
X(30305) = reflection of X(i) in X(j) for these (i,j): (1, 4342), (2093, 11019), (3421, 24703), (3474, 999), (4293, 1), (17784, 997), (18391, 497), (30353, 5542)
X(30305) = X(200)-of-inner-Garcia-triangle
X(30305) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4302, 5731), (4345, 5731, 1), (5731, 30332, 4302)

X(30306) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11447). X(30306) is their endo-homothetic center only when ABC is acute.

X(30306) lies on these lines: {2,30296}, {4,30385}, {5,30313}, {7,11}, {12,30333}, {226,30346}, {496,30341}, {1699,6203}, {2886,30412}, {3817,30380}, {5886,18458}, {7133,8228}, {7741,30425}, {7951,30431}, {7988,30354}, {8085,30368}, {8086,30418}, {8226,30324}, {8227,30400}, {8229,30360}, {8379,30406}, {8727,30276}, {10863,30288}, {11522,30319}, {17605,30375}, {18393,18410}, {30308,30396}

X(30307) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11448). X(30307) is their endo-homothetic center only when ABC is acute.

X(30307) lies on these lines: {2,30297}, {4,30386}, {5,30314}, {7,11}, {12,30334}, {226,30347}, {496,30342}, {1699,6204}, {2886,30413}, {3817,30381}, {5886,18460}, {7741,30426}, {7951,30432}, {7988,30355}, {8085,30369}, {8086,30419}, {8226,30325}, {8227,30401}, {8229,30361}, {8379,30407}, {8727,30277}, {10863,30289}, {11522,30320}, {17605,30376}, {18393,18411}, {30308,30397}

X(30308) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND SUBMEDIAL

Barycentrics a^3+2*(b+c)*a^2+5*(b-c)^2*a-8*(b^2-c^2)*(b-c) : :
X(30308) = X(1)+4*X(381) = X(1)-16*X(9955) = 7*X(1)+8*X(18480) = X(1)+2*X(18492) = X(1)-4*X(18493) = 11*X(1)+4*X(18525) = 19*X(1)-4*X(18526) = 8*X(2)-3*X(165) = 2*X(2)+3*X(1699) = X(2)-6*X(3817) = 4*X(2)-9*X(7988) = 13*X(2)-3*X(9778) = X(2)+9*X(9779) = 7*X(2)+3*X(9812) = 11*X(2)-6*X(10164) = 7*X(2)-12*X(10171) = X(165)+4*X(1699) = X(165)-16*X(3817) = X(165)-6*X(7988) = 13*X(165)-8*X(9778) = 7*X(165)+8*X(9812) = 11*X(165)-16*X(10164) = X(381)+4*X(9955) = 7*X(381)-2*X(18480) = 11*X(381)-X(18525) = 19*X(381)+X(18526) = X(1699)+4*X(3817) = 2*X(1699)+3*X(7988) = 13*X(1699)+2*X(9778) = X(1699)-6*X(9779) = 7*X(1699)-2*X(9812) = 14*X(9955)+X(18480) = 8*X(9955)+X(18492) = 4*X(9955)-X(18493) = 4*X(18480)-7*X(18492) = 2*X(18480)+7*X(18493) = 22*X(18480)-7*X(18525) = X(18492)+2*X(18493)

The homothetic center of these triangles is X(11451). X(30308) is their endo-homothetic center only when ABC is acute.

X(30308) lies on these lines: {1,381}, {2,165}, {4,25055}, {5,3654}, {11,4654}, {12,30337}, {20,19883}, {30,7987}, {40,5055}, {376,3624}, {496,30343}, {517,19709}, {519,3091}, {528,15017}, {546,3655}, {547,12699}, {551,3839}, {946,3545}, {962,3828}, {1125,3543}, {1385,14269}, {1572,18362}, {1656,28198}, {1698,5071}, {1743,17737}, {2886,30393}, {3058,5219}, {3062,3255}, {3090,9589}, {3241,19925}, {3339,7741}, {3361,3582}, {3534,11230}, {3544,11362}, {3576,3830}, {3579,15703}, {3653,15687}, {3656,4677}, {3829,8226}, {3843,28208}, {3845,5886}, {3850,5881}, {3851,7982}, {3855,13464}, {3929,5536}, {4301,5068}, {4312,10589}, {4857,6849}, {4995,9580}, {5054,16192}, {5056,9588}, {5087,8580}, {5223,11680}, {5231,17781}, {5298,9579}, {5493,7486}, {5531,10707}, {5550,15683}, {5690,14892}, {5726,30384}, {5901,23046}, {6175,8583}, {6361,19872}, {7280,28444}, {7678,30330}, {7951,9819}, {7993,10711}, {8085,30371}, {8086,30395}, {8229,30363}, {8379,30394}, {8727,11407}, {9166,9860}, {9592,11648}, {9612,10072}, {9614,10056}, {9814,30311}, {9875,14639}, {10109,26446}, {10129,10582}, {10165,11001}, {10863,30291}, {10883,30304}, {11019,30340}, {11235,17618}, {11737,22791}, {12512,15708}, {13174,23234}, {13462,23708}, {13624,15684}, {14893,18481}, {15685,17502}, {15692,19862}, {15693,28146}, {15694,28202}, {15695,28154}, {15697,28158}, {15711,28182}, {15713,28178}, {15721,19878}, {18393,18421}, {19708,28150}, {28452,30282}, {30306,30396}, {30307,30397}, {30309,10655}, {30310,10656}

X(30308) = midpoint of X(381) and X(18493)
X(30308) = reflection of X(i) in X(j) for these (i,j): (1698, 5071), (3679, 5818), (15692, 19862), (18492, 381)
X(30308) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1699, 3817, 7988), (1699, 7988, 165), (18492, 18493, 1)

X(30309) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11452). X(30309) is their endo-homothetic center only when ABC is acute.

X(30309) lies on these lines: {2,30300}, {4,10647}, {5,30316}, {7,11}, {12,30338}, {226,30351}, {496,30344}, {1653,1699}, {2886,30414}, {3817,30382}, {5886,18469}, {7741,10651}, {7951,30433}, {7988,30356}, {8086,30421}, {8226,30327}, {8227,10649}, {8229,30364}, {8379,30409}, {8727,30280}, {10863,30292}, {11522,30321}, {17605,30377}, {18393,18422}, {30308,10655}

X(30310) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11453). X(30310) is their endo-homothetic center only when ABC is acute.

X(30310) lies on these lines: {2,30301}, {4,10648}, {5,30317}, {7,11}, {12,30339}, {226,30352}, {496,30345}, {1652,1699}, {2886,30415}, {3817,30383}, {5886,18471}, {7741,10652}, {7951,30434}, {7988,30357}, {8085,30373}, {8086,30422}, {8226,30328}, {8227,10650}, {8229,30365}, {8379,30410}, {8727,30281}, {10863,30293}, {11522,30322}, {17605,30378}, {18393,18423}, {30308,10656}

X(30311) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 2nd EHRMANN

Barycentrics (b^2-5*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+5*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :
X(30311) = 3*X(7679)-4*X(7951)

The homothetic center of these triangles is X(11443). X(30311) is their endo-homothetic center only when ABC is acute.

X(30311) lies on these lines: {2,30295}, {4,390}, {5,30312}, {7,11}, {9,5057}, {12,10248}, {142,10129}, {226,7671}, {496,30340}, {516,6932}, {527,11680}, {920,1445}, {1001,11114}, {1699,8545}, {2476,3826}, {2550,17577}, {2801,18393}, {2886,6172}, {3817,30379}, {3869,24393}, {4193,5880}, {4197,15254}, {4293,6912}, {5218,7676}, {5226,7965}, {5542,10394}, {5603,14151}, {5886,18450}, {6600,20095}, {7672,18397}, {7741,30424}, {7988,30353}, {8085,30367}, {8086,30405}, {8226,12848}, {8227,8544}, {8229,30359}, {8257,20292}, {8379,30404}, {8727,30275}, {9814,30308}, {10863,30287}, {11522,30318}, {15726,17605}

X(30311) = {X(954), X(9668)}-harmonic conjugate of X(390)

X(30312) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 2nd EHRMANN

Barycentrics ((b^2+5*b*c+c^2)*a^2-(b+c)*(2*b^2+b*c+2*c^2)*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :
X(30312) = 3*X(7678)-4*X(7741)

The homothetic center of these triangles is X(11458). X(30312) is their endo-homothetic center only when ABC is acute.

X(30312) lies on these lines: {2,8543}, {4,30295}, {5,30311}, {7,12}, {8,14151}, {9,25010}, {10,30379}, {11,30332}, {46,6991}, {56,26060}, {142,4848}, {355,18450}, {390,496}, {404,2550}, {442,12848}, {443,956}, {495,30340}, {516,6943}, {527,11681}, {528,5433}, {1001,14882}, {1155,10883}, {1156,6932}, {1210,7671}, {1329,6172}, {1445,3841}, {1698,8545}, {1737,10394}, {2346,6989}, {2476,5880}, {2801,18395}, {3671,5692}, {3679,30318}, {3925,5435}, {4193,5698}, {4294,6986}, {4308,9710}, {4318,17278}, {4323,5289}, {4429,17077}, {5051,30359}, {5273,25973}, {5587,8544}, {5729,6937}, {6049,6067}, {6901,9655}, {7951,30424}, {7989,30353}, {8087,30367}, {8088,30405}, {8270,26724}, {8382,30404}, {8582,30287}, {8728,30275}, {9814,30315}, {11495,12953}, {15726,17606}

X(30312) = {X(7), X(3826)}-harmonic conjugate of X(7679)

X(30313) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11462). X(30313) is their endo-homothetic center only when ABC is acute.

X(30313) lies on these lines: {2,30385}, {4,30296}, {5,30306}, {7,12}, {10,30380}, {11,30333}, {355,18458}, {442,30324}, {495,30341}, {1210,30346}, {1329,30412}, {1698,6203}, {3679,30319}, {5051,30360}, {5587,30400}, {7133,8230}, {7741,30431}, {7951,30425}, {7989,30354}, {8088,30418}, {8382,30406}, {8582,30288}, {8728,30276}, {17606,30375}, {18395,18410}, {30315,30396}

X(30313) = {X(12), X(3826)}-harmonic conjugate of X(30314)

X(30314) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11463). X(30314) is their endo-homothetic center only when ABC is acute.

X(30314) lies on these lines: {2,30386}, {4,30297}, {5,30307}, {7,12}, {10,30381}, {11,30334}, {355,18460}, {442,30325}, {495,30342}, {1210,30347}, {1329,30413}, {1698,6204}, {3679,30320}, {5051,30361}, {5587,30401}, {7741,30432}, {7951,30426}, {7989,30355}, {8087,30369}, {8088,30419}, {8382,30407}, {8582,30289}, {8728,30277}, {17606,30376}, {18395,18411}, {30315,30397}

X(30314) = {X(12), X(3826)}-harmonic conjugate of X(30313)

X(30315) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND SUBMEDIAL

Barycentrics a^4+3*(b+c)*a^3-3*(3*b^2+2*b*c+3*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+8*(b^2-c^2)^2 : :
X(30315) = 3*X(1)-20*X(1656) = 5*X(1)+12*X(5790) = X(1)+16*X(9956) = 13*X(1)+4*X(12645) = 24*X(2)-7*X(30389) = 8*X(4)+9*X(165) = 2*X(4)+15*X(1698) = 5*X(4)+12*X(6684) = X(4)-18*X(10175) = 3*X(165)-20*X(1698) = 3*X(165)+14*X(7989) = X(165)+16*X(10175) = 5*X(1656)+12*X(9956) = 35*X(1656)-18*X(11230) = 25*X(1698)-8*X(6684) = 10*X(1698)+7*X(7989) = 5*X(1698)+12*X(10175) = 3*X(5790)-20*X(9956) = 7*X(5790)+10*X(11230) = 2*X(6684)+15*X(10175) = 14*X(9956)+3*X(11230)

The homothetic center of these triangles is X(11465). X(30315) is their endo-homothetic center only when ABC is acute.

X(30315) lies on these lines: {1,1656}, {2,30389}, {3,19876}, {4,165}, {5,3654}, {10,5056}, {11,30337}, {12,10980}, {40,3851}, {140,5587}, {355,30392}, {442,30326}, {495,30343}, {515,3533}, {516,3854}, {519,7486}, {547,4677}, {1210,30350}, {1329,30393}, {1657,11231}, {1699,5068}, {3062,3826}, {3090,3679}, {3091,3828}, {3339,7951}, {3361,5270}, {3522,19877}, {3523,3634}, {3544,28194}, {3545,9589}, {3584,6887}, {3617,10171}, {3624,5818}, {3628,5881}, {3656,12812}, {3850,26446}, {4197,30304}, {4301,15022}, {4668,5886}, {4745,5734}, {5051,30363}, {5055,7982}, {5059,10164}, {5067,25055}, {5071,11362}, {5223,11681}, {5425,5780}, {5531,6702}, {5559,23708}, {7679,30330}, {7741,9819}, {8087,30371}, {8088,30395}, {8227,11224}, {8382,30394}, {8582,30291}, {8728,11407}, {8960,19004}, {9814,30312}, {9851,17529}, {10827,13462}, {11375,30286}, {15178,15703}, {15720,18480}, {18395,18421}, {30313,30396}, {30314,30397}, {30316,10655}, {30317,10656}

X(30315) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1698, 7989, 165), (1698, 10175, 7989)

X(30316) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11466). X(30316) is their endo-homothetic center only when ABC is acute.

X(30316) lies on these lines: {2,10647}, {4,30300}, {5,30309}, {7,12}, {10,30382}, {11,30338}, {355,18469}, {442,30327}, {495,30344}, {1210,30351}, {1329,30414}, {1653,1698}, {3679,30321}, {5051,30364}, {5587,10649}, {7741,30433}, {7951,10651}, {7989,30356}, {8088,30421}, {8382,30409}, {8582,30292}, {8728,30280}, {17606,30377}, {18395,18422}, {30315,10655}

X(30316) = {X(12), X(3826)}-harmonic conjugate of X(30317)

X(30317) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11467). X(30317) is their endo-homothetic center only when ABC is acute.

X(30317) lies on these lines: {2,10648}, {4,30301}, {5,30310}, {7,12}, {10,30383}, {11,30339}, {355,18471}, {442,30328}, {1210,30352}, {1329,30415}, {1652,1698}, {3679,30322}, {5051,30365}, {5587,10650}, {7741,30434}, {7951,10652}, {7989,30357}, {8087,30373}, {8088,30422}, {8382,30410}, {8582,30293}, {8728,30281}, {17606,30378}, {18395,18423}, {30315,10656}

X(30317) = {X(12), X(3826)}-harmonic conjugate of X(30316)

X(30318) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 2nd EHRMANN

Barycentrics a*(a^3-5*(b+c)*a^2+(7*b^2+4*b*c+7*c^2)*a-3*(b+c)*(b^2+c^2))*(a+b-c)*(a-b+c) : :
X(30318) = 4*X(56)-3*X(1445)

The homothetic center of these triangles is X(11470). X(30318) is their endo-homothetic center only when ABC is acute.

X(30318) lies on these lines: {1,651}, {7,145}, {8,30379}, {9,1404}, {40,18450}, {56,78}, {57,3935}, {63,2078}, {77,3242}, {142,5554}, {390,5882}, {516,30323}, {517,8544}, {527,11682}, {938,5261}, {1319,5220}, {1388,15254}, {1420,3984}, {1998,3873}, {2098,15726}, {2951,7673}, {3306,5083}, {3339,3874}, {3679,30312}, {3870,17625}, {3988,5223}, {4308,11523}, {4312,11280}, {4326,30337}, {4864,6180}, {5252,25557}, {5728,7373}, {5729,24928}, {5880,10944}, {6049,6172}, {7962,30332}, {7991,30295}, {8232,10392}, {8581,11011}, {8732,24391}, {9814,16189}, {9846,12560}, {10388,11220}, {11025,30343}, {11518,30275}, {11519,30287}, {11522,30311}, {11529,30340}, {11531,30353}, {11533,30359}, {11534,30367}, {11535,30404}, {11899,30405}, {25415,30424}

X(30318) = reflection of X(5729) in X(24928)
X(30318) = {X(7), X(3243)}-harmonic conjugate of X(11526)

X(30319) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11473). X(30319) is their endo-homothetic center only when ABC is acute.

X(30319) lies on these lines: {1,372}, {7,145}, {8,30380}, {40,18458}, {517,30400}, {2098,30375}, {3641,6204}, {3679,30313}, {5605,13389}, {7962,30333}, {7991,30296}, {11518,30276}, {11519,30288}, {11522,30306}, {11523,30324}, {11529,30341}, {11531,30354}, {11533,30360}, {11534,30368}, {11535,30406}, {11899,30418}, {15829,30412}, {16189,30396}, {25415,30425}, {30323,30431}

X(30319) = {X(3243), X(3340)}-harmonic conjugate of X(30320)

X(30320) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11474). X(30320) is their endo-homothetic center only when ABC is acute.

X(30320) lies on these lines: {1,371}, {7,145}, {8,30381}, {40,18460}, {517,30401}, {2098,30376}, {3640,6203}, {3679,30314}, {5604,13388}, {7962,30334}, {7991,30297}, {11518,30277}, {11519,30289}, {11522,30307}, {11523,30325}, {11529,30342}, {11531,30355}, {11533,30361}, {11534,30369}, {11535,30407}, {11899,30419}, {15829,30413}, {16189,30397}, {25415,30426}, {30323,30432}

X(30320) = {X(3243), X(3340)}-harmonic conjugate of X(30319)

X(30321) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11475). X(30321) is their endo-homothetic center only when ABC is acute.

X(30321) lies on these lines: {1,16}, {7,145}, {8,30382}, {40,18469}, {517,10649}, {2098,30377}, {3679,30316}, {7962,30338}, {7991,30300}, {11518,30280}, {11519,30292}, {11522,30309}, {11523,30327}, {11529,30344}, {11531,30356}, {11533,30364}, {11534,30372}, {11535,30409}, {11899,30421}, {15829,30414}, {16189,10655}, {25415,10651}, {30323,30433}

X(30321) = {X(3243), X(3340)}-harmonic conjugate of X(30322)

X(30322) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11476). X(30322) is their endo-homothetic center only when ABC is acute.

X(30322) lies on these lines: {1,15}, {7,145}, {8,30383}, {40,18471}, {517,10650}, {2098,30378}, {3679,30317}, {7962,30339}, {7991,30301}, {11518,30281}, {11519,30293}, {11522,30310}, {11523,30328}, {11529,30345}, {11531,30357}, {11533,30365}, {11534,30373}, {11535,30410}, {11899,30422}, {15829,30415}, {16189,10656}, {25415,10652}, {30323,30434}

X(30322) = {X(3243), X(3340)}-harmonic conjugate of X(30321)

X(30323) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND TRINH

Barycentrics a*(a^3-3*(b+c)*a^2-(b^2-8*b*c+c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(30323) = 3*X(1)-2*X(56) = 5*X(1)-4*X(24928) = 3*X(46)-4*X(56) = X(46)-4*X(2098) = 5*X(46)-8*X(24928) = X(56)-3*X(2098) = 5*X(56)-6*X(24928) = 4*X(1329)-3*X(3679) = 5*X(1698)-4*X(8256) = 3*X(3241)-X(20076) = 2*X(4848)-3*X(10072) = 8*X(6691)-9*X(25055) = 4*X(7681)-5*X(11522)

The homothetic center of these triangles is X(24). X(30323) is their endo-homothetic center only when ABC is acute.

X(30323) lies on these lines: {1,3}, {8,5187}, {9,17444}, {10,6931}, {12,3656}, {63,22837}, {72,10912}, {78,2802}, {80,3632}, {90,1320}, {145,10572}, {498,13464}, {499,11362}, {516,30318}, {519,1479}, {550,12735}, {946,6968}, {952,12701}, {997,5330}, {998,1126}, {1000,3485}, {1145,25681}, {1317,18481}, {1329,3679}, {1387,24914}, {1389,7162}, {1404,1766}, {1405,17452}, {1478,4301}, {1537,12749}, {1572,7296}, {1698,8256}, {1720,10696}, {1737,12245}, {1770,3476}, {1837,5844}, {2800,10085}, {2829,7971}, {2841,13541}, {3085,5734}, {3086,4345}, {3241,4294}, {3243,5441}, {3244,10624}, {3419,13463}, {3544,7317}, {3577,5559}, {3583,5881}, {3586,3633}, {3623,4305}, {3625,3984}, {3654,5433}, {3655,15338}, {3811,3885}, {3877,5260}, {3880,5730}, {3884,19860}, {3893,3940}, {3895,22836}, {4299,28194}, {4302,5882}, {4304,11520}, {4311,28228}, {4333,28174}, {4338,18990}, {4668,11525}, {4674,11512}, {4816,11524}, {4848,10072}, {4853,5692}, {4857,5727}, {4861,12514}, {4867,6765}, {4919,17742}, {5252,22791}, {5261,12047}, {5288,12526}, {5289,10914}, {5603,10039}, {5665,13606}, {5690,11376}, {5904,12629}, {6261,10698}, {6361,21578}, {6691,25055}, {6796,10087}, {6833,15868}, {7171,11571}, {7294,26446}, {8543,15298}, {8666,10058}, {8679,15430}, {9578,18393}, {9589,10483}, {9897,12764}, {10051,10935}, {10526,10947}, {10573,12053}, {10944,12699}, {11519,30294}, {11526,30331}, {11533,30366}, {11534,30374}, {11535,30411}, {11899,30423}, {12648,21077}, {12737,24467}, {12953,28204}, {21271,24179}, {30319,30431}, {30320,30432}, {30321,30433}, {30322,30434}

X(30323) = midpoint of X(145) and X(11415)
X(30323) = reflection of X(i) in X(j) for these (i,j): (1, 2098), (8, 21616), (46, 1), (7991, 10310), (9897, 12764), (10573, 12053), (10680, 24680)
X(30323) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 25415), (1, 11531, 5903), (3340, 7982, 11280)

X(30324) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(19355). X(30324) is their endo-homothetic center only when ABC is acute.

X(30324) lies on these lines: {1,1588}, {2,7}, {4,1123}, {5,8957}, {6,1659}, {12,14121}, {37,13390}, {65,7090}, {176,7586}, {281,13459}, {405,30385}, {442,30313}, {486,8953}, {950,30333}, {1490,30400}, {1750,30354}, {1864,30375}, {3069,13389}, {3085,6212}, {3487,30341}, {3586,30431}, {4000,8243}, {4199,30360}, {4295,6213}, {5728,30346}, {5927,30288}, {6351,13388}, {7580,30296}, {7593,30406}, {8079,30368}, {8080,30418}, {8226,30306}, {9612,30425}, {10910,18995}, {11523,30319}, {18397,18410}, {18446,18458}, {30326,30396}

X(30324) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30276), (7, 30412, 6203), (9, 226, 30325)

X(30325) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(19356). X(30325) is their endo-homothetic center only when ABC is acute.

X(30325) lies on these lines: {1,1587}, {2,7}, {4,1336}, {6,13390}, {12,7090}, {37,1659}, {65,14121}, {175,7585}, {281,13437}, {405,30386}, {442,30314}, {942,8957}, {950,30334}, {1490,30401}, {1750,30355}, {1864,30376}, {3068,13388}, {3085,6213}, {3586,30432}, {4199,30361}, {4295,6212}, {4419,8243}, {5728,30347}, {5927,30289}, {6352,13389}, {7580,30297}, {7593,30407}, {8079,30369}, {8080,30419}, {8226,30307}, {9612,30426}, {10911,18996}, {11523,30320}, {18397,18411}, {18446,18460}, {30326,30397}

X(30325) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30277), (7, 30413, 6204), (9, 226, 30324)

X(30326) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND SUBMEDIAL

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

The homothetic center of these triangles is X(10601). X(30326) is their endo-homothetic center only when ABC is acute.

X(30326) lies on these lines: {1,1864}, {2,11407}, {4,3679}, {5,4654}, {9,165}, {40,3715}, {57,5779}, {72,4915}, {200,15064}, {210,7994}, {223,7069}, {226,5817}, {329,1699}, {355,9580}, {405,9851}, {442,30315}, {920,10045}, {936,6909}, {950,30337}, {971,7308}, {1006,1490}, {1697,9947}, {1698,6260}, {1728,3361}, {1737,3339}, {1754,3973}, {1836,2093}, {2801,10582}, {3058,5881}, {3305,5732}, {3487,30343}, {3586,9819}, {3646,12680}, {3678,12651}, {3829,8226}, {3832,3951}, {3929,19541}, {4199,30363}, {5268,9355}, {5691,12572}, {5720,6914}, {5728,30350}, {5735,17781}, {5785,18228}, {6282,18540}, {6667,11219}, {6846,10072}, {6907,19875}, {6911,7330}, {7593,30394}, {7701,10270}, {7996,10445}, {8001,11379}, {8079,30371}, {8080,30395}, {8232,30330}, {9814,12848}, {10863,24477}, {10864,25917}, {11523,16189}, {12526,19925}, {18397,18421}, {18446,30392}, {21153,27065}, {30324,30396}, {30325,30397}, {30327,10655}, {30328,10656}

X(30326) = reflection of X(10857) in X(7308)
X(30326) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 30304, 11407), (9, 1750, 165), (3062, 30393, 165)

X(30327) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(19363). X(30327) is their endo-homothetic center only when ABC is acute.

X(30327) lies on these lines: {1,5334}, {2,7}, {4,1833}, {388,5240}, {405,10647}, {442,30316}, {498,3179}, {950,30338}, {1276,3085}, {1277,4295}, {1490,10649}, {1750,30356}, {1864,30377}, {2306,5714}, {3340,5245}, {3485,5239}, {3487,30344}, {3586,30433}, {4199,30364}, {5246,9578}, {5728,30351}, {5927,30292}, {7580,30300}, {7593,30409}, {8079,30372}, {8080,30421}, {8226,30309}, {9612,10651}, {11523,30321}, {18397,18422}, {18446,18469}, {30326,10655}

X(30327) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30280), (7, 30414, 1653), (9, 226, 30328)

X(30328) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(19364). X(30328) is their endo-homothetic center only when ABC is acute.

X(30328) lies on these lines: {1,5335}, {2,7}, {4,1251}, {388,5239}, {405,10648}, {442,30317}, {950,30339}, {1276,4295}, {1277,3085}, {1490,10650}, {1750,30357}, {1864,30378}, {3340,5246}, {3485,5240}, {3586,30434}, {4199,30365}, {5245,9578}, {5728,30352}, {5927,30293}, {7580,30301}, {7593,30410}, {8079,30373}, {8080,30422}, {8226,30310}, {9612,10652}, {11523,30322}, {18397,18423}, {18446,18471}, {30326,10656}

X(30328) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30281), (7, 30415, 1652), (9, 226, 30327)

X(30329) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND EHRMANN-VERTEX

Barycentrics a*((b+c)*a^4-2*(b^2+b*c+c^2)*a^3-(b+c)*b*c*a^2+2*(b^2-c^2)^2*a-(b^3+c^3)*(b-c)^2) : :
X(30329) = 3*X(1)-5*X(11025) = X(7)-3*X(5902) = 3*X(65)+X(14100) = 2*X(142)-3*X(5883) = X(3059)-3*X(3753) = 6*X(3833)-5*X(20195) = 2*X(4757)+X(5698) = 3*X(5686)-X(5904) = 3*X(5692)-5*X(18230) = X(5693)-3*X(5817) = X(5697)-3*X(8236) = 3*X(5728)-X(14100) = 3*X(5902)+X(18412) = 4*X(6666)-3*X(10176) = 3*X(7672)+5*X(11025) = X(7672)+2*X(20116) = 5*X(11025)-6*X(20116) = 3*X(11038)-5*X(18398) = 5*X(15016)-3*X(21151)

The homothetic center of these triangles is X(19130). X(30329) is their endo-homothetic center only when ABC is acute.

X(30329) lies on these lines: {1,1170}, {7,80}, {9,758}, {10,141}, {36,30284}, {40,12564}, {46,7675}, {65,516}, {72,12563}, {165,11020}, {200,3306}, {214,999}, {226,15064}, {354,3911}, {390,5903}, {484,7676}, {517,5572}, {519,15185}, {954,15556}, {971,5884}, {1001,3878}, {1156,11571}, {1159,2800}, {1737,21617}, {1837,13159}, {1858,10392}, {2093,4326}, {2099,15558}, {2550,3754}, {2802,11041}, {2807,29957}, {3059,3753}, {3085,5445}, {3243,3333}, {3339,5732}, {3487,3678}, {3555,6743}, {3833,20195}, {3868,5223}, {3919,15733}, {3946,22465}, {3956,8164}, {4295,5809}, {4312,10394}, {4343,4424}, {4757,5698}, {4860,5083}, {5045,10165}, {5173,11019}, {5493,12710}, {5686,5904}, {5690,15901}, {5691,12669}, {5692,18230}, {5693,5817}, {5696,20612}, {5697,8236}, {5708,11500}, {6666,10176}, {7678,18393}, {7679,18395}, {7680,10265}, {8232,18397}, {8238,30358}, {8387,18399}, {8388,18409}, {8389,18408}, {8732,30274}, {10090,14151}, {10164,11018}, {10398,12560}, {10578,15104}, {10865,30286}, {15006,28194}, {15016,21151}, {15570,24928}, {17706,24474}, {18421,30330}

X(30329) = midpoint of X(i) and X(j) for these {i,j}: {1, 7672}, {7, 18412}, {65, 5728}, {80, 12755}, {390, 5903}, {1156, 11571}, {3868, 5223}, {4312, 10394}, {5691, 12669}
X(30329) = reflection of X(i) in X(j) for these (i,j): (1, 20116), (2550, 3754), (3243, 3881), (3878, 1001), (5542, 942), (30331, 5572)
X(30329) = {X(5902), X(18412)}-harmonic conjugate of X(7)

X(30330) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND SUBMEDIAL

Barycentrics a*(a^4-2*(3*b^2-2*b*c+3*c^2)*a^2+8*(b^2-c^2)*(b-c)*a-(3*b^2+2*b*c+3*c^2)*(b-c)^2) : :
X(30330) = X(3062)+3*X(24645)

The homothetic center of these triangles is X(19137). X(30330) is their endo-homothetic center only when ABC is acute.

X(30330) lies on these lines: {1,6}, {3,15008}, {7,1699}, {57,2951}, {65,10384}, {144,10580}, {165,1445}, {269,21346}, {388,10392}, {390,6738}, {516,938}, {942,7992}, {971,3333}, {1125,5785}, {1156,18240}, {1479,4312}, {2093,15006}, {2310,4328}, {3059,8580}, {3174,8257}, {3303,9898}, {3306,25722}, {3361,5732}, {3600,9851}, {3829,6173}, {4321,10394}, {4907,5228}, {5045,5779}, {5437,15587}, {5542,14986}, {5691,5809}, {5817,21620}, {5833,10916}, {5850,21625}, {7672,11531}, {7675,7987}, {7677,30389}, {7678,30308}, {7679,30315}, {7988,21617}, {8056,9445}, {8236,30337}, {8238,30363}, {8255,20195}, {8387,30371}, {8388,30395}, {8389,30394}, {8545,11025}, {8732,11407}, {9819,16236}, {10389,15837}, {10582,11020}, {10865,30291}, {11038,30343}, {11526,16189}, {13405,18230}, {16112,17626}, {18421,30329}, {30284,30392}

X(30330) = reflection of X(i) in X(j) for these (i,j): (7, 15841), (15829, 1001)
X(30330) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3973, 9440), (1, 10398, 5223), (9, 5572, 1)

X(30331) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND TRINH

Barycentrics 4*a^3-5*(b+c)*a^2+2*(b-c)^2*a-(b^2-c^2)*(b-c) : :
X(30331) = 3*X(1)-X(7) = 5*X(1)-X(4312) = X(1)-3*X(8236) = 5*X(1)-3*X(11038) = 5*X(1)+X(30332) = 11*X(1)-5*X(30340) = 4*X(1)-X(30424) = X(7)+3*X(390) = 5*X(7)-3*X(4312) = 2*X(7)-3*X(5542) = X(7)-9*X(8236) = 5*X(7)-9*X(11038) = 5*X(7)+3*X(30332) = 11*X(7)-15*X(30340) = 4*X(7)-3*X(30424) = 3*X(10)-4*X(6666) = 5*X(390)+X(4312) = 2*X(390)+X(5542) = X(390)+3*X(8236) = X(1000)+3*X(3488) = 3*X(1001)-2*X(6666)

The homothetic center of these triangles is X(5092). X(30331) is their endo-homothetic center only when ABC is acute.

X(30331) lies on these lines: {1,7}, {3,21625}, {8,25101}, {9,519}, {10,1001}, {35,7677}, {36,7676}, {40,6744}, {55,3911}, {57,10385}, {65,20116}, {80,2346}, {142,214}, {144,3241}, {145,5223}, {165,10580}, {226,3058}, {319,3883}, {354,4031}, {392,3059}, {496,3826}, {497,3817}, {515,6767}, {517,5572}, {518,3244}, {673,29571}, {758,15185}, {938,21153}, {942,5493}, {944,11372}, {946,5719}, {950,954}, {971,5882}, {997,3174}, {999,11495}, {1056,28164}, {1058,1125}, {1156,7972}, {1210,3746}, {1279,3755}, {1317,2801}, {1445,5119}, {1479,3947}, {1621,4847}, {1699,10578}, {1757,4924}, {1890,6198}, {2321,4702}, {2802,10177}, {3057,5728}, {3158,20103}, {3189,12447}, {3243,5698}, {3247,5819}, {3333,12512}, {3475,9580}, {3486,10384}, {3586,8232}, {3625,6541}, {3632,5686}, {3679,18230}, {3689,5316}, {3731,5838}, {3753,12732}, {3870,21060}, {3871,8582}, {3874,12710}, {3898,15733}, {3938,4656}, {3957,17484}, {4078,17765}, {4085,16593}, {4421,6692}, {4428,5745}, {4660,21255}, {4684,17361}, {4882,5129}, {5045,10386}, {5234,6764}, {5274,10171}, {5441,16133}, {5697,7672}, {5726,19925}, {5759,7982}, {5762,24680}, {5766,10398}, {5805,13464}, {5817,5881}, {5851,12735}, {5902,11025}, {5903,7673}, {5918,10569}, {6068,25416}, {6690,24386}, {6765,18250}, {7671,18412}, {7674,9623}, {7678,7951}, {7679,7741}, {8238,30366}, {8255,20330}, {8387,30374}, {8388,30423}, {8389,30411}, {8543,10572}, {8581,10543}, {8715,9843}, {8732,30282}, {9577,29815}, {9778,10980}, {9797,11106}, {9819,16236}, {10175,18527}, {10179,15587}, {10390,18490}, {10392,10950}, {10865,30294}, {11362,12433}, {11373,15808}, {11526,30323}, {11529,28228}, {12053,15950}, {12702,17706}, {13159,16137}, {13411,23708}, {15171,21620}, {15174,15570}, {15933,18421}, {15934,28194}, {16173,20119}, {17023,20533}, {17601,24216}, {17603,18240}, {17715,17725}, {18530,26446}, {21454,30350}, {21617,30384}, {24175,29820}

X(30331) = midpoint of X(i) and X(j) for these {i,j}: {1, 390}, {80, 12730}, {145, 5223}, {944, 11372}, {1156, 7972}, {3057, 5728}, {3243, 5698}, {3632, 12630}, {4294, 12560}, {4312, 30332}, {5441, 16133}, {5697, 7672}, {5759, 7982}, {5903, 7673}, {6068, 25416}, {10624, 12573}
X(30331) = reflection of X(i) in X(j) for these (i,j): (10, 1001), (65, 20116), (2550, 1125), (3625, 24393), (5542, 1), (5805, 13464), (13159, 16137), (14563, 15935), (24393, 15254), (30329, 5572), (30424, 5542)
X(30331) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 20, 12577), (1, 10624, 3671), (11038, 30332, 4312)

X(30332) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 2nd EHRMANN

Barycentrics 7*a^3-5*(b+c)*a^2+(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30332) = 4 X[1] - 3 X[7], 2 X[1] - 3 X[390], 5 X[1] - 3 X[4312], 7 X[1] - 6 X[5542], 8 X[1] - 9 X[8236], 10 X[1] - 9 X[11038], 5 X[1] - 6 X[30331], 6 X[1] - 5 X[30340], 3 X[1] - 2 X[30424], 11 X[1] - 12 X[43179], 5 X[1] - 4 X[43180], 11 X[1] - 9 X[59372], 5 X[7] - 4 X[4312], 7 X[7] - 8 X[5542], 2 X[7] - 3 X[8236], 5 X[7] - 6 X[11038], 5 X[7] - 8 X[30331], 9 X[7] - 10 X[30340], 9 X[7] - 8 X[30424], 11 X[7] - 16 X[43179], 15 X[7] - 16 X[43180], 11 X[7] - 12 X[59372], 5 X[390] - 2 X[4312], 7 X[390] - 4 X[5542], 4 X[390] - 3 X[8236], 5 X[390] - 3 X[11038], 5 X[390] - 4 X[30331], and many others

The homothetic center of these triangles is X(11477). X(30332) is their endo-homothetic center only when ABC is acute.

Let Γ_a be the circle passing through X(7) tangent to AB and AC such that the touchpoints do not lie on the Adams circle. Define Γ_{b/sub> and Γ_c cyclically. Then X(30332) is the center of the circle tangent to and encompassing the circles Γ_a, Γ_{b/sub>c Γ_c. (Angel Montesdeoca, June 2, 2023)}}

Let Γ_a be the circle passing through X(7) and tangent to the two sides AB and AC of triangle ABC (the larger one of the two possible circles). Circle Γ_a intersects side BC at points A₁, and A₂. We define points B₁, B₂, C₁, and C₂ cyclically. Then, points A₁, A₂, B₁, B₂, C₁, and C₂ lie on one circle with center X(30332). Additionally, these points lie on one circle only for X(8) and X(7). (Miłosz Płatek, June 14, 2024)

The above-mentioned circle with center X(30332) has squared radius 4*r^2*(4 - (9*s^2) / (r + 4*R)^2). The circular hull of the 3 circles is a circle centered on X(30332) with radius of 4*r. If you have GeoGebra, you can view X(30332). (Peter Moses, July 22, 2024)

X(30332) lies on these lines: {1, 7}, {2, 9580}, {3, 53055}, {4, 5766}, {8, 190}, {9, 3617}, {10, 50836}, {11, 30312}, {12, 10248}, {40, 37787}, {44, 5838}, {45, 5819}, {55, 5226}, {56, 30295}, {65, 7671}, {100, 5328}, {142, 37267}, {144, 3621}, {145, 527}, {149, 5744}, {165, 5274}, {329, 2900}, {376, 5126}, {404, 1001}, {497, 1155}, {515, 60946}, {517, 10394}, {518, 3644}, {519, 60905}, {673, 42318}, {938, 6361}, {944, 54199}, {946, 61008}, {950, 12848}, {954, 5714}, {1000, 28160}, {1004, 1621}, {1056, 28146}, {1125, 38094}, {1159, 3488}, {1266, 39567}, {1420, 50693}, {1445, 5128}, {1447, 64308}, {1479, 5445}, {1697, 3146}, {1698, 38092}, {1699, 5281}, {1788, 9670}, {1836, 10385}, {2078, 35986}, {2098, 14151}, {2136, 36973}, {2246, 52164}, {2346, 5556}, {2478, 2550}, {2800, 64321}, {2801, 5697}, {2898, 3599}, {3057, 15726}, {3058, 3474}, {3091, 61763}, {3149, 15911}, {3161, 32850}, {3241, 28534}, {3243, 20059}, {3245, 18391}, {3254, 37299}, {3340, 60975}, {3434, 5273}, {3485, 63273}, {3486, 38454}, {3487, 10386}, {3522, 12053}, {3523, 9614}, {3524, 7743}, {3528, 11373}, {3529, 9957}, {3543, 31397}, {3579, 5704}, {3586, 59417}, {3601, 30275}, {3614, 7679}, {3616, 5880}, {3622, 6173}, {3623, 60984}, {3624, 51100}, {3625, 5223}, {3626, 5686}, {3632, 50835}, {3634, 31418}, {3817, 31508}, {3826, 7173}, {3832, 61015}, {3839, 31434}, {3869, 12536}, {3870, 64143}, {3877, 5784}, {3878, 5696}, {3883, 32087}, {3886, 32099}, {3895, 56551}, {3912, 4779}, {3923, 5772}, {4402, 62392}, {4440, 15590}, {4450, 34255}, {4454, 49466}, {4460, 51192}, {4668, 50840}, {4678, 51102}, {4701, 50834}, {4746, 50837}, {5046, 6594}, {5057, 63168}, {5059, 7320}, {5087, 62710}, {5119, 54370}, {5204, 7677}, {5218, 9779}, {5250, 60981}, {5261, 51118}, {5265, 12512}, {5423, 63139}, {5528, 20066}, {5558, 52783}, {5603, 37606}, {5657, 9668}, {5703, 12699}, {5727, 50573}, {5728, 50193}, {5729, 5759}, {5762, 8148}, {5779, 37705}, {5817, 18357}, {5825, 21168}, {5843, 61295}, {5851, 12730}, {5936, 50314}, {6006, 21105}, {6601, 62827}, {6666, 46931}, {6767, 28178}, {6872, 60997}, {6953, 38037}, {7672, 14100}, {7962, 30318}, {7987, 18220}, {8164, 51787}, {8232, 52835}, {8240, 30359}, {8241, 30367}, {8242, 30405}, {8715, 60885}, {8732, 63413}, {9578, 17578}, {9579, 60967}, {9613, 49135}, {9809, 41701}, {9814, 30337}, {9819, 28164}, {10005, 25728}, {10304, 44675}, {10543, 36971}, {10572, 54204}, {10589, 63211}, {10591, 59316}, {10866, 30287}, {10896, 19877}, {11010, 51768}, {11041, 28212}, {11220, 17642}, {11372, 29007}, {11496, 64280}, {11529, 28232}, {11662, 37739}, {11924, 30404}, {12527, 12632}, {12534, 37179}, {12647, 50864}, {12700, 59345}, {12706, 30628}, {12912, 37431}, {13279, 42842}, {13405, 50865}, {13462, 59420}, {14986, 31730}, {15006, 60939}, {15683, 60952}, {15717, 50443}, {15933, 28198}, {15934, 28216}, {16020, 24715}, {16816, 41845}, {17314, 28566}, {17538, 24928}, {17718, 64340}, {17738, 39749}, {17768, 36977}, {17781, 20015}, {17784, 18228}, {18483, 60943}, {18493, 38073}, {19862, 38052}, {19872, 38059}, {20057, 51099}, {20095, 31018}, {20533, 29579}, {21454, 64162}, {24393, 61006}, {24703, 34607}, {25557, 36005}, {26129, 64154}, {28150, 31393}, {28194, 60951}, {28610, 34611}, {28626, 50302}, {29611, 49484}, {30384, 54445}, {31663, 47743}, {31794, 63972}, {35242, 61019}, {36991, 52683}, {37000, 54051}, {37556, 60953}, {37567, 60941}, {38149, 61261}, {38316, 62778}, {41864, 60932}, {41869, 61027}, {43733, 56028}, {46873, 59584}, {46933, 60986}, {49451, 64015}, {49468, 51053}, {49515, 51052}, {51190, 64070}, {51782, 53052}, {51791, 63207}, {52769, 59325}, {56936, 64002}, {59388, 64198}, {59414, 61000}, {60993, 63208}, {62208, 62875}

X(30332) = midpoint of X(12630) and X(60957)
X(30332) = reflection of X(i) in X(j) for these {i,j}: {7, 390}, {8, 5698}, {4312, 30331}, {5696, 3878}, {7672, 14100}, {20059, 3243}, {60956, 7962}, {60957, 63975}, {60971, 3241}
X(30332) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4312, 43180}, {1, 30424, 30340}, {1, 43178, 18450}, {1, 43180, 11038}, {4, 5766, 60995}, {7, 390, 8236}, {8, 5698, 6172}, {11, 64108, 31188}, {20, 9785, 4308}, {20, 10624, 9785}, {55, 9812, 5226}, {165, 5274, 64114}, {165, 51783, 5274}, {175, 176, 1323}, {329, 20075, 64146}, {390, 11038, 30331}, {481, 482, 20121}, {497, 9778, 5435}, {962, 4294, 4313}, {962, 4313, 4323}, {1001, 59412, 60996}, {1836, 10385, 10578}, {2550, 15254, 9780}, {2550, 52653, 18230}, {3058, 3474, 10580}, {3616, 5880, 59374}, {4302, 30305, 5731}, {4312, 11038, 7}, {4312, 30331, 11038}, {4319, 12652, 4318}, {5225, 37568, 9780}, {5686, 51090, 60983}, {5698, 34711, 6068}, {5731, 30305, 4345}, {5880, 47357, 3616}, {6361, 15171, 938}, {9780, 15254, 18230}, {9780, 52653, 15254}, {10386, 48661, 3487}, {11010, 51768, 60912}, {11200, 30333, 17802}, {11200, 30334, 17805}, {12512, 51785, 5265}, {12575, 64005, 3600}, {17802, 17805, 31721}, {24703, 34607, 64083}, {30305, 63971, 60926}, {30331, 43180, 1}, {30333, 52805, 17805}, {30334, 52808, 17802}, {30340, 30424, 7}, {31601, 31602, 62788}, {34611, 44447, 36845}, {36845, 44447, 28610}, {51118, 53053, 5261}, {52805, 52808, 11200}, {60926, 63971, 7}
X(30332) = outer-Garcia-to-inner-Garcia similarity image of X(7)

X(30333) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(1151). X(30333) is their endo-homothetic center only when ABC is acute.

X(30333) lies on these lines: {1,7}, {8,7090}, {11,30313}, {12,30306}, {55,16440}, {56,30296}, {65,30346}, {144,3641}, {145,3640}, {950,30324}, {1659,9812}, {1697,6203}, {2066,30413}, {3057,6405}, {3083,17784}, {3601,30276}, {4419,5605}, {5274,5393}, {5281,5405}, {5697,18410}, {7962,30319}, {8240,30360}, {8241,30368}, {8242,30418}, {9778,13389}, {10578,13390}, {10580,13388}, {10866,30288}, {11370,17014}, {11924,30406}, {12053,30380}, {13386,14004}, {14942,30335}, {30337,30396}

X(30333) = reflection of X(i) in X(j) for these (i,j): (8, 7090), (176, 1), (8986, 4297)
X(30333) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4344, 9785, 30334), (8236, 17802, 1), (17802, 30332, 11200)

X(30334) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(1152). X(30334) is their endo-homothetic center only when ABC is acute.

X(30334) lies on these lines: {1,7}, {8,14121}, {11,30314}, {12,30307}, {55,16441}, {56,30297}, {65,30347}, {144,3640}, {145,3641}, {950,30325}, {1659,10578}, {1697,6204}, {3057,6283}, {3084,17784}, {3601,30277}, {4419,5604}, {5274,5405}, {5281,5393}, {5414,30412}, {5697,18411}, {7962,30320}, {8240,30361}, {8241,30369}, {8242,30419}, {9778,13388}, {9812,13390}, {10580,13389}, {10866,30289}, {11371,17014}, {11924,30407}, {12053,30381}, {13387,14004}, {14942,30336}, {30337,30397}

X(30334) = reflection of X(i) in X(j) for these (i,j): (8, 14121), (175, 1)
X(30334) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30333), (4344, 9785, 30333), (17805, 30332, 11200)

X(30335) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics a^2*(-a+b+c)*(S+b*(a-b+c))*(S+c*(a+b-c)) : :
Trilinears 1/(1 + sec A/2 cos B/2 cos C/2) : :

The homothetic center of these triangles is X(13021). X(30335) is their endo-homothetic center only when ABC is acute.

X(30335) lies on the cubic K632, curve Q104 and these lines: {1,16213}, {8,30416}, {33,16232}, {55,1152}, {65,30348}, {175,21453}, {200,15892}, {220,2066}, {221,2293}, {371,4845}, {1697,30429}, {1806,2328}, {3601,30278}, {4313,30302}, {14942,30333}

X(30335) = isogonal conjugate of X(176)
X(30335) = X(56)-cross conjugate of X(30336)
X(30335) = X(3207)-vertex conjugate of X(30336)
X(30335) = anticomplement of the complementary conjugate of X(7090)
X(30335) = {X(2293), X(3303)}-harmonic conjugate of X(30336)

X(30336) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics a^2*(-a+b+c)*(-S+b*(a-b+c))*(-S+c*(a+b-c)) : :
Trilinears 1/(-1 + sec A/2 cos B/2 cos C/2) : :

The homothetic center of these triangles is X(13022). X(30336) is their endo-homothetic center only when ABC is acute.

X(30336) lies on the cubic K632, curve Q104 and these lines: {1,16214}, {8,30417}, {33,2362}, {55,1151}, {65,30349}, {176,21453}, {200,15891}, {220,5414}, {221,2293}, {372,4845}, {1697,30430}, {1805,2328}, {3601,30279}, {4313,30303}, {14942,30334}

X(30336) = isogonal conjugate of X(175)
X(30336) = anticomplement of the complementary conjugate of X(14121)
X(30336) = {X(2293), X(3303)}-harmonic conjugate of X(30335)
X(30336) = X(56)-cross conjugate of X(30335)
X(30336) = X(3207)-vertex conjugate of X(30335)

X(30337) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND SUBMEDIAL

Barycentrics a*(a^3+3*(b+c)*a^2-(b^2+22*b*c+c^2)*a-3*(b^2-c^2)*(b-c)) : :

The homothetic center of these triangles is X(11479). X(30337) is their endo-homothetic center only when ABC is acute.

X(30337) lies on these lines: {1,3}, {4,11379}, {8,30393}, {9,11519}, {10,12541}, {11,30315}, {12,30308}, {145,5223}, {200,3890}, {390,3062}, {392,4882}, {496,19875}, {936,3898}, {950,30326}, {1000,3632}, {1001,3680}, {1056,9589}, {1058,3679}, {1699,5261}, {1706,10179}, {2136,8580}, {2347,3731}, {3241,12526}, {3488,3633}, {3884,6765}, {3893,7308}, {3895,8583}, {4313,30304}, {4326,30318}, {4342,11522}, {4355,28194}, {4668,5722}, {4677,15170}, {4853,5260}, {4900,7160}, {5234,12629}, {5290,30305}, {5436,10912}, {5531,15558}, {5558,11034}, {5691,12575}, {5726,9614}, {5785,12536}, {5881,15172}, {5882,7992}, {5918,12128}, {7988,10588}, {7989,10591}, {8078,10968}, {8167,11530}, {8236,30330}, {8240,30363}, {8241,30371}, {8242,30395}, {9588,14986}, {9814,30332}, {10582,14923}, {10866,30291}, {11037,28228}, {11108,11525}, {11924,30394}, {12447,12632}, {12577,20070}, {12640,26105}, {12735,12767}, {13463,25525}, {16673,17451}, {30333,30396}, {30334,30397}, {30338,10655}, {30339,10656}

X(30337) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3339, 30343), (3339, 30343, 10980), (6767, 7982, 1)

X(30338) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11480). X(30338) is their endo-homothetic center only when ABC is acute.

X(30338) lies on these lines: {1,7}, {8,5240}, {11,30316}, {12,30309}, {55,10647}, {56,30300}, {65,30351}, {559,9778}, {950,30327}, {1082,10580}, {1653,1697}, {3057,30377}, {3601,30280}, {5697,18422}, {7962,30321}, {8240,30364}, {8241,30372}, {8242,30421}, {10866,30292}, {11924,30409}, {12053,30382}, {30337,10655}

X(30338) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30339), (1, 3639, 11038), (4344, 4345, 30339)

X(30339) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11481). X(30339) is their endo-homothetic center only when ABC is acute.

X(30339) lies on these lines: {1,7}, {8,1251}, {11,30317}, {12,30310}, {55,10648}, {56,30301}, {65,30352}, {559,10580}, {950,30328}, {1082,9778}, {1652,1697}, {3057,30378}, {3601,30281}, {5697,18423}, {7962,30322}, {8240,30365}, {8241,30373}, {8242,30422}, {10866,30293}, {11924,30410}, {12053,30383}, {30337,10656}

X(30339) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30338), (1, 3638, 11038), (4344, 4345, 30338)

X(30340) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 2nd EHRMANN

Barycentrics a^3+7*(b+c)*a^2-5*(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30340) = 2*X(1)+3*X(7) = 8*X(1)-3*X(390) = 7*X(1)+3*X(4312) = X(1)-6*X(5542) = 14*X(1)-9*X(8236) = 4*X(1)-9*X(11038) = 11*X(1)-6*X(30331) = 6*X(1)-X(30332) = 3*X(1)+2*X(30424) = 4*X(7)+X(390) = 7*X(7)-2*X(4312) = X(7)+4*X(5542) = 7*X(7)+3*X(8236) = 2*X(7)+3*X(11038) = 11*X(7)+4*X(30331) = 9*X(7)+X(30332) = 9*X(7)-4*X(30424) = 7*X(390)+8*X(4312) = X(390)-16*X(5542) = 7*X(390)-12*X(8236)

The homothetic center of these triangles is X(11482). X(30340) is their endo-homothetic center only when ABC is acute.

X(30340) lies on these lines: {1,7}, {2,3715}, {8,6173}, {9,5550}, {142,5686}, {144,15254}, {145,5880}, {149,25558}, {354,5274}, {388,18221}, {495,30312}, {496,30311}, {518,3617}, {527,3616}, {528,3623}, {553,10578}, {938,18492}, {942,5261}, {971,11025}, {999,8543}, {1001,20059}, {1056,1159}, {1125,6172}, {1155,3475}, {1156,3296}, {2346,5551}, {2550,3621}, {2801,3091}, {3295,30295}, {3333,8545}, {3337,10303}, {3487,5265}, {3488,28168}, {3523,5557}, {3579,21151}, {3622,5698}, {3634,5223}, {3797,29583}, {3873,5784}, {3874,4208}, {3881,5696}, {3889,15733}, {3982,9812}, {4114,10389}, {4461,4966}, {4869,24349}, {5045,7671}, {5226,10980}, {5232,24325}, {5284,20214}, {5556,10390}, {5558,12053}, {5704,21617}, {5708,5771}, {5714,5728}, {5729,6832}, {5759,13624}, {5772,21255}, {5819,16666}, {5850,18230}, {7988,11034}, {8092,30404}, {8351,30405}, {9814,30343}, {10861,15185}, {11019,30308}, {11035,30287}, {11529,30318}, {13743,16133}, {15726,17609}, {15934,28186}, {17784,26842}, {21620,30379}

X(30340) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 30424, 30332), (7, 8236, 4312), (7, 30332, 30424)

X(30341) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(3311). X(30341) is their endo-homothetic center only when ABC is acute.

X(30341) lies on these lines: {1,7}, {142,3641}, {354,1659}, {495,30313}, {496,30306}, {942,30276}, {1125,30412}, {3008,5589}, {3295,30296}, {3296,7133}, {3333,6203}, {3475,13389}, {3487,30324}, {4667,11370}, {4675,5605}, {5045,30346}, {5393,10980}, {8092,30406}, {11035,30288}, {11043,30360}, {17609,30375}, {18398,18410}, {21620,30380}, {30343,30396}

X(30341) = midpoint of X(1) and X(1373)
X(30341) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30333, 30425), (390, 21169, 30426), (4310, 11037, 30342)

X(30342) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(3312). X(30342) is their endo-homothetic center only when ABC is acute.

X(30342) lies on these lines: {1,7}, {142,3640}, {354,13390}, {495,30314}, {496,30307}, {942,30277}, {999,30386}, {1125,30413}, {3008,5588}, {3295,30297}, {3333,6204}, {3475,13388}, {4667,11371}, {4675,5604}, {5045,30347}, {5405,10980}, {8092,30407}, {8351,30419}, {11035,30289}, {11043,30361}, {11044,30369}, {11529,30320}, {17609,30376}, {18398,18411}, {21620,30381}, {30343,30397}

X(30342) = midpoint of X(1) and X(1374)
X(30342) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30341), (7, 30334, 30426), (4310, 11037, 30341)

X(30343) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND SUBMEDIAL

Barycentrics a*(a^3+3*(b+c)*a^2-(b^2-26*b*c+c^2)*a-3*(b^2-c^2)*(b-c)) : :

The homothetic center of these triangles is X(11484). X(30343) is their endo-homothetic center only when ABC is acute.

X(30343) lies on these lines: {1,3}, {496,30308}, {738,7274}, {936,3892}, {1058,4355}, {1125,30393}, {1475,3731}, {1699,9851}, {3062,5542}, {3487,30326}, {3616,5223}, {3624,4866}, {3633,17706}, {3753,12127}, {3812,11519}, {3889,8583}, {4342,11034}, {4353,7996}, {4915,5439}, {5260,10582}, {5261,7989}, {5290,10591}, {5586,30305}, {5691,10580}, {7190,17106}, {7988,21620}, {7992,13464}, {7993,18240}, {7997,11376}, {8092,30394}, {9814,30340}, {10085,24644}, {10569,15071}, {11025,30318}, {11035,30291}, {11036,30304}, {11043,30363}, {11044,30371}, {16469,28011}, {30341,30396}, {30342,30397}, {30344,10655}, {30345,10656}

X(30343) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 999, 30389), (1, 3339, 30337), (3339, 30337, 7991)

X(30344) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11485). X(30344) is their endo-homothetic center only when ABC is acute.

X(30344) lies on these lines: {1,7}, {495,30316}, {496,30309}, {497,554}, {559,3475}, {942,30280}, {999,10647}, {1125,30414}, {1653,3333}, {3295,30300}, {3487,30327}, {5045,30351}, {8092,30409}, {11035,30292}, {11043,30364}, {11044,30372}, {11529,30321}, {17609,30377}, {18398,18422}, {21620,30382}, {30343,10655}

X(30344) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30345), (1, 10652, 390), (7, 30338, 10651)

X(30345) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11486). X(30345) is their endo-homothetic center only when ABC is acute.

X(30345) lies on these lines: {1,7}, {496,30310}, {497,1081}, {942,30281}, {999,10648}, {1082,3475}, {1125,30415}, {1251,3296}, {1652,3333}, {3295,30301}, {5045,30352}, {8092,30410}, {8351,30422}, {11035,30293}, {11043,30365}, {11044,30373}, {11529,30322}, {17609,30378}, {18398,18423}, {21620,30383}, {30343,10656}

X(30345) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30344), (1, 10651, 390), (1, 10652, 30339)

X(30346) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(3068). X(30346) is their endo-homothetic center only when ABC is acute.

X(30346) lies on these lines: {1,372}, {7,354}, {57,30296}, {65,30333}, {226,30306}, {518,30412}, {999,18458}, {1210,30313}, {1827,13390}, {3333,30400}, {4319,13389}, {5045,30341}, {5902,30431}, {8083,30406}, {10980,30354}, {11018,30276}, {11019,30288}, {11031,30360}, {11032,30368}, {11033,30418}, {18398,30425}, {30350,30396}

X(30346) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30347), (354, 30375, 7)

X(30347) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(3069). X(30347) is their endo-homothetic center only when ABC is acute.

X(30347) lies on these lines: {1,371}, {7,354}, {57,30297}, {65,30334}, {226,30307}, {518,30413}, {999,18460}, {1210,30314}, {1659,1827}, {3333,30401}, {4319,13388}, {5045,30342}, {5728,30325}, {5902,30432}, {8083,30407}, {10980,30355}, {11018,30277}, {11019,30289}, {11031,30361}, {11032,30369}, {11033,30419}, {18398,30426}, {30350,30397}

X(30347) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30346), (354, 30376, 7)

X(30348) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a*((b+c)*a^12-(3*b^2+2*b*c+3*c^2)*a^11-(b+c)*(8*b^2+21*b*c+8*c^2)*a^10+(23*b^2-30*b*c+23*c^2)*(b+c)^2*a^9+(b+c)*(13*b^4+13*c^4+(113*b^2+36*b*c+113*c^2)*b*c)*a^8-2*(b+c)*(101*b^4+101*c^4+2*(21*b^2+17*b*c+21*c^2)*b*c)*a^6*b*c-2*(27*b^6+27*c^6+(34*b^4+34*c^4-(59*b^2+68*b*c+59*c^2)*b*c)*b*c)*a^7+2*(27*b^6+27*c^6+(2*b^4+2*c^4-3*(49*b^2-36*b*c+49*c^2)*b*c)*b*c)*(b+c)^2*a^5-(b^2-c^2)*(b-c)*(13*b^6+13*c^6-(128*b^4+128*c^4+(361*b^2+360*b*c+361*c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(23*b^6+23*c^6+(74*b^4+74*c^4-(135*b^2+116*b*c+135*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(8*b^4+8*c^4-(33*b^2+94*b*c+33*c^2)*b*c)*a^2-(a+b+c)*((b+c)*a^9+7*(b+c)^2*a^8-28*(b+c)*(b^2+c^2)*a^7+4*(3*b^4+3*c^4-(17*b^2+20*b*c+17*c^2)*b*c)*a^6+2*(b+c)*(23*b^4+23*c^4+10*(4*b^2-3*b*c+4*c^2)*b*c)*a^5-2*(b^2+c^2)*(23*b^4+23*c^4-2*(14*b^2+59*b*c+14*c^2)*b*c)*a^4-4*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(19*b^2+17*b*c+19*c^2)*b*c)*a^3+4*(b^2-c^2)^2*(7*b^4+7*c^4+(b^2-24*b*c+c^2)*b*c)*a^2-(b^2-c^2)^3*(b-c)*(7*b^2-34*b*c+7*c^2)*a-(b^2-c^2)^2*(b-c)^4*(b^2+10*b*c+c^2))*S+(b^2-c^2)^4*(b-c)^2*(3*b^2+22*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3*(-b^4-c^4+(b^2+8*b*c+c^2)*b*c)) : :

The homothetic center of these triangles is X(19420). X(30348) is their endo-homothetic center only when ABC is acute.

X(30348) lies on these lines: {1,30387}, {65,30335}, {354,4328}, {518,30416}, {11018,30278}, {11020,30302}, {16213,17609}

X(30349) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a*((b+c)*a^12-(3*b^2+2*b*c+3*c^2)*a^11-(b+c)*(8*b^2+21*b*c+8*c^2)*a^10+(23*b^2-30*b*c+23*c^2)*(b+c)^2*a^9+(b+c)*(13*b^4+13*c^4+(113*b^2+36*b*c+113*c^2)*b*c)*a^8-2*(b+c)*(101*b^4+101*c^4+2*(21*b^2+17*b*c+21*c^2)*b*c)*a^6*b*c-2*(27*b^6+27*c^6+(34*b^4+34*c^4-(59*b^2+68*b*c+59*c^2)*b*c)*b*c)*a^7+2*(27*b^6+27*c^6+(2*b^4+2*c^4-3*(49*b^2-36*b*c+49*c^2)*b*c)*b*c)*(b+c)^2*a^5-(b^2-c^2)*(b-c)*(13*b^6+13*c^6-(128*b^4+128*c^4+(361*b^2+360*b*c+361*c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(23*b^6+23*c^6+(74*b^4+74*c^4-(135*b^2+116*b*c+135*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(8*b^4+8*c^4-(33*b^2+94*b*c+33*c^2)*b*c)*a^2+(a+b+c)*((b+c)*a^9+7*(b+c)^2*a^8-28*(b+c)*(b^2+c^2)*a^7+4*(3*b^4+3*c^4-(17*b^2+20*b*c+17*c^2)*b*c)*a^6+2*(b+c)*(23*b^4+23*c^4+10*(4*b^2-3*b*c+4*c^2)*b*c)*a^5-2*(b^2+c^2)*(23*b^4+23*c^4-2*(14*b^2+59*b*c+14*c^2)*b*c)*a^4-4*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(19*b^2+17*b*c+19*c^2)*b*c)*a^3+4*(b^2-c^2)^2*(7*b^4+7*c^4+(b^2-24*b*c+c^2)*b*c)*a^2-(b^2-c^2)^3*(b-c)*(7*b^2-34*b*c+7*c^2)*a-(b^2-c^2)^2*(b-c)^4*(b^2+10*b*c+c^2))*S+(b^2-c^2)^4*(b-c)^2*(3*b^2+22*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3*(-b^4-c^4+(b^2+8*b*c+c^2)*b*c)) : :

The homothetic center of these triangles is X(19421). X(30349) is their endo-homothetic center only when ABC is acute.

X(30349) lies on these lines: {1,30388}, {65,30336}, {354,4328}, {518,30417}, {11018,30279}, {11020,30303}, {16214,17609}

X(30350) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND SUBMEDIAL

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

The homothetic center of these triangles is X(7392). X(30350) is their endo-homothetic center only when ABC is acute.

X(30350) lies on these lines: {1,3}, {105,28156}, {200,9342}, {390,11034}, {497,3982}, {518,30393}, {551,5273}, {672,16673}, {1002,26102}, {1210,30315}, {1282,3315}, {1449,4906}, {1699,5542}, {1709,24645}, {1743,29820}, {2124,11029}, {3062,5572}, {3158,15570}, {3219,4666}, {3243,3742}, {3296,4355}, {3305,3873}, {3632,17706}, {3892,9623}, {3945,10520}, {4031,10385}, {4114,4312}, {4315,15933}, {4512,29817}, {4882,5439}, {4900,14563}, {5226,7988}, {5531,14151}, {5543,9533}, {5558,12577}, {5586,10624}, {5691,6744}, {5728,30326}, {6602,9327}, {7671,9814}, {7989,21620}, {7992,12005}, {8083,8090}, {8423,11033}, {8545,11025}, {9949,12563}, {11020,24644}, {11031,30363}, {11032,30371}, {11036,11522}, {11379,12688}, {17022,17450}, {21454,30331}, {24392,25557}, {30346,30396}, {30347,30397}, {30351,10655}, {30352,10656}

X(30350) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 65, 30337), (1, 5902, 9819), (354, 5173, 18398)

X(30351) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11488). X(30351) is their endo-homothetic center only when ABC is acute.

X(30351) lies on these lines: {1,16}, {7,354}, {57,30300}, {65,30338}, {226,30309}, {518,30414}, {559,4336}, {999,18469}, {1210,30316}, {5045,30344}, {5902,30433}, {10980,30356}, {11018,30280}, {11019,30292}, {11031,30364}, {11032,30372}, {18398,10651}, {30350,10655}

X(30351) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30352), (354, 30377, 7)

X(30352) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11489). X(30352) is their endo-homothetic center only when ABC is acute.

X(30352) lies on these lines: {1,15}, {7,354}, {57,30301}, {65,30339}, {226,30310}, {518,30415}, {999,18471}, {1082,4336}, {1210,30317}, {3333,10650}, {5045,30345}, {5728,30328}, {5902,30434}, {8083,30410}, {10980,30357}, {11018,30281}, {11019,30293}, {11031,30365}, {11032,30373}, {11033,30422}, {18398,10652}, {30350,10656}

X(30352) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30351), (354, 30378, 7)

X(30353) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 2nd EHRMANN

Barycentrics a*(a^4-4*(b+c)*a^3+2*(3*b^2-4*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+(b^2+10*b*c+c^2)*(b-c)^2) : :
X(30353) = 2*X(1)-3*X(4321)

The homothetic center of these triangles is X(11511). X(30353) is their endo-homothetic center only when ABC is acute.

X(30353) lies on these lines: {1,7}, {9,1155}, {57,15726}, {165,8545}, {200,527}, {535,9623}, {1156,1445}, {1418,4907}, {1699,30379}, {1750,12848}, {1836,6173}, {2093,2801}, {3174,5528}, {3339,10394}, {3935,20059}, {4654,8255}, {4853,7354}, {4860,14100}, {5128,5220}, {5698,8583}, {5766,12512}, {5784,12526}, {5880,9579}, {6172,8580}, {7671,10980}, {7677,24644}, {7987,8543}, {7988,30311}, {7989,30312}, {8089,30367}, {8090,30405}, {8245,30359}, {8423,30404}, {10857,30275}, {11224,14151}, {11372,22753}, {11531,30318}, {11662,17857}

X(30353) = reflection of X(i) in X(j) for these (i,j): (390, 4315), (11372, 22753), (30305, 5542)
X(30353) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 2951, 4326), (4312, 5732, 12560)

X(30354) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11513). X(30354) is their endo-homothetic center only when ABC is acute.

X(30354) lies on these lines: {1,7}, {9,9616}, {57,30375}, {165,6203}, {1699,30380}, {1750,30324}, {2093,18410}, {3062,7133}, {7988,30306}, {7989,30313}, {8090,30418}, {8245,30360}, {8580,30412}, {10980,30346}, {11531,30319}

X(30354) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30355), (1742, 4326, 30355), (4335, 5732, 30355)

X(30355) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(11514). X(30355) is their endo-homothetic center only when ABC is acute.

X(30355) lies on these lines: {1,7}, {57,30376}, {165,6204}, {1699,30381}, {1750,30325}, {2093,18411}, {3062,30289}, {7987,30386}, {7988,30307}, {7989,30314}, {8089,30369}, {8090,30419}, {8245,30361}, {8423,30407}, {8580,30413}, {10857,30277}, {10980,30347}, {11531,30320}, {13389,14100}

X(30355) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30354), (1742, 4326, 30354), (4335, 5732, 30354)

X(30356) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11515). X(30356) is their endo-homothetic center only when ABC is acute.

X(30356) lies on these lines: {1,7}, {57,30377}, {165,1653}, {1082,14100}, {1699,30382}, {1750,30327}, {2093,18422}, {3062,30292}, {5240,15587}, {7987,10647}, {7988,30309}, {7989,30316}, {8089,30372}, {8090,30421}, {8245,30364}, {8423,30409}, {8580,30414}, {10857,30280}, {10980,30351}, {11531,30321}

X(30356) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30357), (165, 10655, 1653), (1653, 30300, 165)

X(30357) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(11516). X(30357) is their endo-homothetic center only when ABC is acute.

X(30357) lies on these lines: {1,7}, {165,1652}, {559,14100}, {1251,3062}, {1699,30383}, {1750,30328}, {2093,18423}, {5239,15587}, {7987,10648}, {7988,30310}, {7989,30317}, {8089,30373}, {8090,30422}, {8245,30365}, {8580,30415}, {10857,30281}, {10980,30352}, {11531,30322}

X(30357) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30356), (165, 10656, 1652), (1652, 30301, 165)

X(30358) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND EHRMANN-VERTEX

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

The homothetic center of these triangles is X(19177). X(30358) is their endo-homothetic center only when ABC is acute.

X(30358) lies on these lines: {1,21}, {5,986}, {36,30285}, {46,8235}, {65,9959}, {80,256}, {484,4220}, {517,17611}, {982,15950}, {1284,5902}, {1737,4425}, {2093,8245}, {2801,30359}, {3670,5443}, {4199,18397}, {5051,18395}, {5492,24851}, {5691,12683}, {5692,18235}, {5697,8240}, {5903,9840}, {6839,24248}, {6905,17596}, {7504,24443}, {8229,18393}, {8238,30329}, {8249,18399}, {8250,18409}, {8425,18408}, {8731,30274}, {9791,18391}, {10573,26117}, {10868,30286}, {10950,24430}, {11043,18398}, {11571,13265}, {11813,24239}, {18410,30360}, {18411,30361}, {18422,30364}, {18423,30365}

X(30358) = reflection of X(30366) in X(17611)
X(30358) = {X(65), X(9959)}-harmonic conjugate of X(30362)

X(30359) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 2nd EHRMANN

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

The homothetic center of these triangles is X(19178). X(30359) is their endo-homothetic center only when ABC is acute.

X(30359) lies on these lines: {7,21}, {256,1156}, {516,30366}, {527,11688}, {846,8545}, {2801,30358}, {4199,12848}, {4220,30295}, {4335,10394}, {4425,30379}, {5051,30312}, {5057,30097}, {6172,18235}, {6912,24248}, {7671,11031}, {8229,30311}, {8235,8544}, {8240,30332}, {8245,30353}, {8249,30367}, {8250,30405}, {8425,30404}, {8731,30275}, {9814,30363}, {10868,30287}, {11043,30340}, {11533,30318}, {15726,17611}, {18450,30285}, {30362,30424}

X(30360) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(19183). X(30360) is their endo-homothetic center only when ABC is acute.

X(30360) lies on these lines: {7,21}, {256,7133}, {846,6203}, {4199,30324}, {4220,30296}, {4425,30380}, {5051,30313}, {8229,30306}, {8235,30400}, {8240,30333}, {8245,30354}, {8249,30368}, {8250,30418}, {8425,30406}, {8731,30276}, {10868,30288}, {11031,30346}, {11043,30341}, {11533,30319}, {17611,30375}, {18235,30412}, {18410,30358}, {18458,30285}, {30362,30425}, {30363,30396}, {30366,30431}

X(30361) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(19184). X(30361) is their endo-homothetic center only when ABC is acute.

X(30361) lies on these lines: {7,21}, {846,6204}, {4199,30325}, {4220,30297}, {4425,30381}, {5051,30314}, {8229,30307}, {8235,30401}, {8240,30334}, {8245,30355}, {8249,30369}, {8250,30419}, {8425,30407}, {8731,30277}, {11031,30347}, {11043,30342}, {11533,30320}, {17611,30376}, {18235,30413}, {18411,30358}, {18460,30285}, {30362,30426}, {30363,30397}, {30366,30432}

X(30362) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND KOSNITA

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

The homothetic center of these triangles is X(19185). X(30362) is their endo-homothetic center only when ABC is acute.

X(30362) lies on these lines: {1,256}, {3,3944}, {10,11688}, {12,5143}, {21,36}, {35,4220}, {46,846}, {65,9959}, {80,13265}, {404,25385}, {405,8424}, {516,8238}, {859,24161}, {942,17611}, {958,24697}, {1054,28238}, {1281,7283}, {1283,14798}, {1478,26117}, {1580,1724}, {1698,18235}, {2093,13097}, {2292,5903}, {2475,3724}, {2975,4683}, {3120,4225}, {3339,30363}, {3633,12642}, {4199,9612}, {4295,9791}, {4297,23821}, {5051,7951}, {5248,29634}, {5251,25906}, {5902,11203}, {7741,8229}, {7742,20834}, {7972,12746}, {8249,30370}, {8250,30420}, {8425,30408}, {8731,15803}, {9579,16778}, {10868,30290}, {11031,18398}, {11533,25415}, {11571,12770}, {12683,15071}, {13731,17596}, {13738,17889}, {15507,28265}, {22769,26731}, {30359,30424}, {30360,30425}, {30361,30426}, {30364,10651}, {30365,10652}

X(30362) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1284, 8240, 11043), (1284, 9840, 1), (8240, 11043, 1)

X(30363) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND SUBMEDIAL

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

The homothetic center of these triangles is X(19188). X(30363) is their endo-homothetic center only when ABC is acute.

X(30363) lies on these lines: {1,9959}, {21,30304}, {165,846}, {1284,10980}, {1699,9791}, {2292,11531}, {3339,30362}, {4199,30326}, {4425,7988}, {5051,30315}, {5223,11688}, {5691,12579}, {7987,8235}, {7991,9840}, {8229,30308}, {8238,30330}, {8240,30337}, {8249,30371}, {8250,30395}, {8425,30394}, {8731,11407}, {9814,30359}, {9819,30366}, {10868,30291}, {11031,30350}, {11043,30343}, {11533,16189}, {18235,30393}, {18421,30358}, {30285,30392}, {30360,30396}, {30361,30397}, {30364,10655}, {30365,10656}

X(30363) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (846, 8245, 165), (846, 11203, 8245)

X(30364) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(19190). X(30364) is their endo-homothetic center only when ABC is acute.

X(30364) lies on these lines: {7,21}, {846,1653}, {4199,30327}, {4220,30300}, {4425,30382}, {5051,30316}, {8229,30309}, {8235,10649}, {8240,30338}, {8245,30356}, {8249,30372}, {8250,30421}, {8425,30409}, {8731,30280}, {10868,30292}, {11031,30351}, {11043,30344}, {11098,24248}, {11533,30321}, {17611,30377}, {18235,30414}, {18422,30358}, {18469,30285}, {30362,10651}, {30363,10655}, {30366,30433}

X(30365) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(19191). X(30365) is their endo-homothetic center only when ABC is acute.

X(30365) lies on these lines: {7,21}, {256,1251}, {846,1652}, {4199,30328}, {4220,30301}, {4425,30383}, {5051,30317}, {8229,30310}, {8235,10650}, {8240,30339}, {8245,30357}, {8249,30373}, {8250,30422}, {8425,30410}, {8731,30281}, {11031,30352}, {11043,30345}, {11097,24248}, {11533,30322}, {17611,30378}, {18235,30415}, {18423,30358}, {30362,10652}, {30363,10656}, {30366,30434}

X(30366) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND TRINH

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

The homothetic center of these triangles is X(19192). X(30366) is their endo-homothetic center only when ABC is acute.

X(30366) lies on these lines: {1,256}, {10,21}, {36,4220}, {56,17722}, {516,30359}, {517,17611}, {519,11688}, {846,855}, {956,8424}, {958,7295}, {978,3612}, {993,3705}, {2292,5697}, {3057,9959}, {3145,15654}, {3586,4199}, {3632,12642}, {3679,18235}, {3920,28377}, {4294,12567}, {4425,30384}, {5051,7741}, {5143,5724}, {5259,25906}, {5902,11031}, {7419,21674}, {7951,8229}, {7972,13265}, {8238,30331}, {8249,30374}, {8250,30423}, {8425,30411}, {8731,16569}, {9791,30305}, {10624,12579}, {10868,30294}, {11113,20545}, {11533,30323}, {16975,21745}, {30360,30431}, {30361,30432}, {30364,30433}, {30365,30434}

X(30366) = reflection of X(30358) in X(17611)
X(30366) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9840, 30362), (8240, 9840, 1)

X(30367) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 2nd EHRMANN

Barycentrics a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)) : : , where
F(a,b,c) = 6*(a+b-c)*(a-b+c)*(-a+b+c)*b*c
G(a,b,c) = 2*(a+b-c)*(a-b+c)*c*(a^2+(b-2*c)*a-(2*b+c)*(b-c))
H(a,b,c) = -(a+b-c)*(a-b+c)*(-a+b+c)*(a^2-2*(b+c)*a+(b-c)^2)

The homothetic center of these triangles is X(19369). X(30367) is their endo-homothetic center only when ABC is acute.

X(30367) lies on these lines: {1,30405}, {7,1488}, {177,1156}, {188,6172}, {516,30374}, {527,11690}, {2801,18399}, {7670,8389}, {7671,11032}, {8075,30295}, {8077,8543}, {8078,8545}, {8079,12848}, {8081,8544}, {8085,30311}, {8087,30312}, {8089,30353}, {8241,30332}, {8249,30359}, {8733,30275}, {9814,30371}, {10503,15726}, {11044,30340}, {11858,30287}, {18448,18450}, {21622,30379}, {30370,30424}

X(30368) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 1st KENMOTU DIAGONALS

Barycentrics a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)) : : , where
F(a,b,c) = 4*(a+b-c)*(a-b+c)*b*c*(-a+c+b)
G(a,b,c) = -2*(a+b-c)*(a-b+c)*(2*S+(-a+c+b)*(a+b-c))*c
H(a,b,c) = (a+b-c)*(a-b+c)*(-a+c+b)*(-a^2+2*(b+c)*a+2*S-(b-c)^2)

The homothetic center of these triangles is X(2067). X(30368) is their endo-homothetic center only when ABC is acute.

X(30368) lies on these lines: {1,30418}, {7,1488}, {177,7133}, {188,30412}, {6203,8078}, {8075,30296}, {8077,30385}, {8079,30324}, {8081,30400}, {8085,30306}, {8087,30313}, {8089,30354}, {8241,30333}, {8249,30360}, {8733,30276}, {10503,30375}, {11032,30346}, {11534,30319}, {11858,30288}, {18399,18410}, {18448,18458}, {21622,30380}, {30370,30425}, {30371,30396}, {30374,30431}

X(30369) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 2nd KENMOTU DIAGONALS

Barycentrics a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)) : : , where
F(a,b,c) = 4*(a+b-c)*(a-b+c)*b*c*(-a+c+b)
G(a,b,c) = 2*(a+b-c)*(a-b+c)*(2*S-(-a+c+b)*(a+b-c))*c
H(a,b,c) = -(a+b-c)*(a-b+c)*(-a+c+b)*(2*S+a^2-2*(b+c)*a+(b-c)^2)

The homothetic center of these triangles is X(6502). X(30369) is their endo-homothetic center only when ABC is acute.

X(30369) lies on these lines: {1,30419}, {7,1488}, {188,30413}, {6204,8078}, {8075,30297}, {8077,30386}, {8079,30325}, {8081,30401}, {8085,30307}, {8087,30314}, {8089,30355}, {8241,30334}, {8249,30361}, {8733,30277}, {10503,30376}, {11032,30347}, {11044,30342}, {11534,30320}, {11858,30289}, {18399,18411}, {18448,18460}, {21622,30381}, {30370,30426}, {30371,30397}, {30374,30432}

X(30370) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND KOSNITA

Barycentrics a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+2*S^2) : : , where
F(a,b,c) = b*c*a*(-a+b+c)
G(a,b,c) = -c*(a-b+c)*(a^2+a*b+b^2-c^2)
X(30370) = 4*X(13385)-X(30423)

The homothetic center of these triangles is X(36). X(30370) is their endo-homothetic center only when ABC is acute.

X(30370) lies on these lines: {1,167}, {10,11690}, {35,8075}, {36,8077}, {46,8078}, {65,8099}, {79,15997}, {80,8103}, {188,1698}, {516,8387}, {517,10506}, {942,10503}, {1699,9836}, {2093,8101}, {3339,30371}, {3633,12643}, {4292,11888}, {4295,9793}, {5902,11192}, {5903,8093}, {7951,8087}, {7972,8097}, {8079,9612}, {8095,15071}, {8249,30362}, {8733,15803}, {10500,12813}, {11009,11013}, {11032,18398}, {11534,25415}, {11571,12771}, {11858,30290}, {12047,21622}, {30367,30424}, {30368,30425}, {30369,30426}, {30372,10651}, {30373,10652}

X(30370) = inverse of X(12908) in the incircle
X(30370) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 177, 30408), (1, 8091, 30374), (2089, 8091, 1)

X(30371) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND SUBMEDIAL

Barycentrics a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)) : : , where
F(a,b,c) = (-a+b+c)*(a^2+2*(b+c)*a-3*(b-c)^2)
G(a,b,c) = (a-b+c)*(a^2+2*(b-3*c)*a-(3*b+5*c)*(b-c))
H(a,b,c) = 4*(-a+b+c)*(a-b+c)*(a+b-c)

The homothetic center of these triangles is X(19372). X(30371) is their endo-homothetic center only when ABC is acute.

X(30371) lies on these lines: {1,8099}, {165,8075}, {177,30394}, {188,30393}, {1699,9793}, {3339,30370}, {5223,11690}, {5691,12580}, {7987,8081}, {7988,21622}, {7991,8091}, {8077,30389}, {8079,30326}, {8085,30308}, {8087,30315}, {8090,10967}, {8093,11531}, {8241,30337}, {8249,30363}, {8387,30330}, {8733,11407}, {9814,30367}, {9819,30374}, {11032,30350}, {11044,30343}, {11379,12908}, {11534,16189}, {11858,30291}, {11888,30304}, {18399,18421}, {18448,30392}, {30368,30396}, {30369,30397}, {30373,10656}

X(30371) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8078, 8089, 165), (8078, 11192, 8089)

X(30372) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND INNER TRI-EQUILATERAL

Barycentrics a*(2*sqrt(3)*(2*c*sin(B/2)+2*b*sin(C/2)+a-b-c)*S+(-4*sin(A/2)*b*c+2*c*(a+b-c)*sin(B/2)+2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2)*(-a+b+c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(7051). X(30372) is their endo-homothetic center only when ABC is acute.

X(30372) lies on these lines: {1,30421}, {7,1488}, {177,30409}, {188,30414}, {1653,8078}, {8075,30300}, {8077,10647}, {8079,30327}, {8081,10649}, {8087,30316}, {8089,30356}, {8241,30338}, {8249,30364}, {8733,30280}, {10503,30377}, {11032,30351}, {11044,30344}, {11534,30321}, {11858,30292}, {18399,18422}, {18448,18469}, {21622,30382}, {30370,10651}, {30371,10655}, {30374,30433}

X(30373) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND OUTER TRI-EQUILATERAL

Barycentrics a*(-2*sqrt(3)*(2*c*sin(B/2)+2*b*sin(C/2)+a-b-c)*S+(-4*sin(A/2)*b*c+2*c*(a+b-c)*sin(B/2)+2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2)*(-a+b+c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19373). X(30373) is their endo-homothetic center only when ABC is acute.

X(30373) lies on these lines: {1,30422}, {7,1488}, {177,1251}, {188,30415}, {1652,8078}, {8075,30301}, {8077,10648}, {8079,30328}, {8081,10650}, {8085,30310}, {8087,30317}, {8089,30357}, {8241,30339}, {8249,30365}, {8733,30281}, {10503,30378}, {11032,30352}, {11044,30345}, {11534,30322}, {11858,30293}, {18399,18423}, {18448,18471}, {21622,30383}, {30370,10652}, {30371,10656}, {30374,30434}

X(30374) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND TRINH

Barycentrics a*(3*(-a+b+c)*sin(A/2)*a*b*c-(a+b-c)*(a^2-a*c-b^2+c^2)*b*sin(C/2)-(a-b+c)*(a^2-a*b+b^2-c^2)*c*sin(B/2)+2*S^2) : :

The homothetic center of these triangles is X(35). X(30374) is their endo-homothetic center only when ABC is acute.

X(30374) lies on these lines: {1,167}, {35,8077}, {36,8075}, {80,1128}, {188,3679}, {516,30367}, {517,10503}, {519,11690}, {3057,8099}, {3586,8079}, {3632,12643}, {4294,12568}, {4304,11888}, {5119,8078}, {5441,16146}, {5691,9836}, {5697,8093}, {5902,11032}, {7741,8087}, {7951,8085}, {7972,8103}, {8249,30366}, {8387,30331}, {8733,30282}, {9793,30305}, {9819,30371}, {9957,10506}, {10500,18408}, {10624,12580}, {11534,30323}, {11858,30294}, {17641,18409}, {21622,30384}, {30368,30431}, {30369,30432}, {30372,30433}, {30373,30434}

X(30374) = reflection of X(18399) in X(10503)
X(30374) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 177, 30411), (1, 8091, 30370), (8091, 8241, 1)

X(30375) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(590). X(30375) is their endo-homothetic center only when ABC is acute.

X(30375) lies on these lines: {6,7133}, {7,354}, {11,30380}, {55,6203}, {56,30400}, {57,30354}, {210,13359}, {517,18410}, {942,30425}, {1155,30296}, {1319,18458}, {1721,13389}, {1864,30324}, {2098,30319}, {2646,30385}, {3057,6405}, {10501,30418}, {10502,30406}, {10503,30368}, {11030,21746}, {17603,30276}, {17604,30288}, {17605,30306}, {17606,30313}, {17609,30341}, {17611,30360}

X(30375) = midpoint of X(18410) and X(30431)
X(30375) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30346, 354), (354, 14100, 30376)

X(30376) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(615). X(30376) is their endo-homothetic center only when ABC is acute.

X(30376) lies on these lines: {1,30397}, {6,9043}, {7,354}, {11,30381}, {55,6204}, {56,30401}, {57,30355}, {210,13360}, {517,18411}, {942,30426}, {1155,30297}, {1319,18460}, {1721,13388}, {1864,30325}, {2098,30320}, {2646,30386}, {3057,6283}, {10501,30419}, {10502,30407}, {10503,30369}, {17603,30277}, {17604,30289}, {17605,30307}, {17606,30314}, {17609,30342}, {17611,30361}

X(30376) = midpoint of X(18411) and X(30432)
X(30376) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30347, 354), (354, 14100, 30375)

X(30377) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(23302). X(30377) is their endo-homothetic center only when ABC is acute.

X(30377) lies on these lines: {1,10655}, {7,354}, {11,30382}, {55,1653}, {56,10649}, {57,30356}, {210,30414}, {517,18422}, {1100,7127}, {1155,30300}, {1319,18469}, {1864,30327}, {2098,30321}, {2646,10647}, {3057,30338}, {10501,30421}, {10502,30409}, {10503,30372}, {17603,30280}, {17604,30292}, {17605,30309}, {17606,30316}, {17609,30344}, {17611,30364}

X(30377) = midpoint of X(18422) and X(30433)
X(30377) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30351, 354), (354, 14100, 30378)

X(30378) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(23303). X(30378) is their endo-homothetic center only when ABC is acute.

X(30378) lies on these lines: {1,10656}, {7,354}, {11,30383}, {55,1652}, {56,10650}, {210,30415}, {517,18423}, {942,10652}, {1100,1251}, {1155,30301}, {1319,18471}, {1864,30328}, {2098,30322}, {2646,10648}, {3057,30339}, {10501,30422}, {10502,30410}, {10503,30373}, {17603,30281}, {17604,30293}, {17605,30310}, {17606,30317}, {17609,30345}, {17611,30365}

X(30378) = midpoint of X(18423) and X(30434)
X(30378) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30352, 354), (354, 14100, 30377)

X(30379) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd EHRMANN

Barycentrics ((b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*(a+b-c)*(a+c-b) : :
X(30379) = X(7)+2*X(3911) = 4*X(142)-X(908) = 2*X(10427)+X(26015)

The homothetic center of these triangles is X(21639). X(30379) is their endo-homothetic center only when ABC is acute.

X(30379) lies on these lines: {2,7}, {4,8544}, {8,30318}, {10,30312}, {11,15726}, {36,516}, {56,5880}, {65,25557}, {77,3554}, {222,26723}, {241,1086}, {269,4859}, {277,4350}, {279,24181}, {354,8255}, {377,1467}, {390,3576}, {514,7216}, {515,18450}, {518,6735}, {519,14151}, {528,1319}, {651,3008}, {653,5236}, {673,909}, {946,5265}, {971,1532}, {1001,1470}, {1012,5805}, {1042,24178}, {1125,8543}, {1156,11219}, {1210,6932}, {1266,4552}, {1323,15727}, {1407,24789}, {1420,4190}, {1422,15474}, {1429,1813}, {1441,24199}, {1442,3946}, {1443,17067}, {1458,1738}, {1519,15325}, {1699,30353}, {1737,2801}, {2078,3254}, {2550,3476}, {3011,9364}, {3243,12648}, {3361,12609}, {3488,6916}, {3522,12053}, {3523,5766}, {3586,5732}, {3660,10427}, {3668,17092}, {3817,30311}, {3826,8581}, {3912,20881}, {4292,6912}, {4298,5258}, {4312,6974}, {4321,9623}, {4341,24779}, {4425,30359}, {4648,7190}, {4652,5698}, {4675,5228}, {5220,24914}, {5298,28534}, {5433,15254}, {5445,13407}, {5542,5902}, {5696,10916}, {5704,6260}, {5723,6610}, {5728,6907}, {5735,6966}, {5784,6734}, {6049,21627}, {6067,15587}, {6180,17278}, {6913,18541}, {7176,17050}, {7671,11019}, {7676,15931}, {7988,9814}, {8236,13384}, {9358,24198}, {9710,9850}, {11038,11526}, {12047,30424}, {12709,24564}, {12832,25558}, {14100,15845}, {14953,17197}, {16133,26725}, {17080,24177}, {17095,17305}, {17625,25006}, {21620,30340}, {21622,30367}, {21623,30405}, {21624,30404}, {24389,25722}, {26001,26932}

X(30379) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 8545), (7, 142, 21617), (142, 6173, 5249)

X(30380) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(21640). X(30380) is their endo-homothetic center only when ABC is acute.

X(30380) lies on these lines: {2,7}, {4,30400}, {8,30319}, {10,30313}, {11,30375}, {482,2362}, {515,18458}, {516,30296}, {1086,8243}, {1125,30385}, {1659,4000}, {1699,30354}, {1737,18410}, {3817,30306}, {4425,30360}, {4648,13390}, {5393,7133}, {7988,30396}, {11019,30288}, {12047,30425}, {12053,30333}, {13388,18589}, {21620,30341}, {21622,30368}, {21623,30418}, {21624,30406}, {30384,30431}

X(30380) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 6203), (142, 226, 30381)

X(30381) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(21641). X(30381) is their endo-homothetic center only when ABC is acute.

X(30381) lies on these lines: {2,7}, {4,30401}, {8,30320}, {10,30314}, {11,30376}, {37,8243}, {481,16232}, {515,18460}, {516,30297}, {1125,30386}, {1659,4648}, {1699,30355}, {1737,18411}, {3817,30307}, {4000,13390}, {4425,30361}, {5405,7595}, {7988,30397}, {10858,16432}, {11019,30289}, {12047,30426}, {12053,30334}, {13389,18589}, {21620,30342}, {21622,30369}, {21623,30419}, {21624,30407}, {30384,30432}

X(30381) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 6204), (142, 226, 30380)

X(30382) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND INNER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(21647). X(30382) is their endo-homothetic center only when ABC is acute.

X(30382) lies on these lines: {2,7}, {4,10649}, {8,30321}, {10,30316}, {11,30377}, {515,18469}, {516,30300}, {1125,10647}, {1699,30356}, {1737,18422}, {3639,5074}, {3817,30309}, {4292,11098}, {4425,30364}, {7988,10655}, {11019,30292}, {12047,10651}, {12053,30338}, {21620,30344}, {21622,30372}, {21623,30421}, {21624,30409}, {30384,30433}

X(30382) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 1653), (142, 226, 30383)

X(30383) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND OUTER TRI-EQUILATERAL

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

The homothetic center of these triangles is X(21648). X(30383) is their endo-homothetic center only when ABC is acute.

X(30383) lies on these lines: {2,7}, {4,10650}, {8,30322}, {10,30317}, {11,30378}, {515,18471}, {516,30301}, {1125,10648}, {1699,30357}, {1737,18423}, {3638,5074}, {3817,30310}, {4292,11097}, {4425,30365}, {7988,10656}, {11019,30293}, {12047,10652}, {12053,30339}, {21620,30345}, {21622,30373}, {21623,30422}, {21624,30410}, {30384,30434}

X(30383) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 1652), (142, 226, 30382)

X(30384) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND TRINH

Barycentrics (b+c)*a^3+(b^2-4*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(30384) = 3*X(36)-X(15228) = X(36)-3*X(16173) = X(484)-3*X(3582) = X(908)+2*X(21630) = 4*X(1387)-X(21578) = X(1512)-4*X(16174) = X(1737)-4*X(7743) = 3*X(3582)-2*X(3911) = 5*X(3616)-3*X(4881) = 2*X(5122)-3*X(5298) = X(15228)-9*X(16173) = 5*X(18493)-X(18524)

The homothetic center of these triangles is X(21663). X(30384) is their endo-homothetic center only when ABC is acute.

X(30384) lies on these lines: {1,4}, {2,5119}, {3,11376}, {5,3057}, {7,7284}, {8,5187}, {10,3877}, {11,517}, {12,9955}, {20,18220}, {21,16155}, {30,1319}, {35,404}, {36,516}, {40,499}, {46,962}, {55,5886}, {56,1770}, {57,10072}, {65,496}, {72,3813}, {79,1476}, {80,519}, {90,10529}, {142,25055}, {145,21077}, {149,4511}, {354,11551}, {355,2098}, {381,5252}, {392,2886}, {411,14798}, {442,20288}, {484,3582}, {495,5919}, {498,1697}, {528,5440}, {551,4304}, {553,11552}, {758,26015}, {912,18839}, {920,12704}, {942,17705}, {952,5048}, {956,24703}, {960,24390}, {995,3914}, {997,3434}, {999,1836}, {1000,3545}, {1012,22767}, {1145,5123}, {1149,3120}, {1155,15325}, {1158,10785}, {1201,23537}, {1210,4301}, {1279,16581}, {1317,12611}, {1329,10914}, {1385,6284}, {1388,12953}, {1420,4299}, {1447,5195}, {1482,1837}, {1484,14988}, {1512,8068}, {1532,22835}, {1537,5570}, {1702,13904}, {1703,13962}, {1706,25522}, {1727,3218}, {1728,5758}, {1739,5121}, {1743,21068}, {1858,10959}, {2077,10090}, {2099,3656}, {2170,5179}, {2446,23517}, {2447,23477}, {2476,3890}, {2646,5901}, {2800,5533}, {2802,3814}, {3058,15950}, {3061,21073}, {3085,6953}, {3091,10827}, {3149,11508}, {3245,28228}, {3295,11375}, {3303,11374}, {3338,4295}, {3419,5289}, {3452,3679}, {3560,10966}, {3576,4302}, {3579,5433}, {3601,4309}, {3612,3616}, {3622,4305}, {3624,17567}, {3632,21075}, {3636,5441}, {3649,5045}, {3667,4017}, {3671,18398}, {3674,7264}, {3698,17527}, {3702,3969}, {3746,5443}, {3748,4870}, {3753,3816}, {3755,5313}, {3817,4342}, {3822,3898}, {3825,24982}, {3838,10179}, {3841,24564}, {3869,10916}, {3878,6734}, {3880,5087}, {3884,24987}, {3885,10915}, {3899,24386}, {3902,5741}, {3912,4975}, {3918,25011}, {3936,4742}, {3940,4863}, {4187,5836}, {4292,5563}, {4293,9812}, {4297,21842}, {4310,15430}, {4311,10483}, {4316,28150}, {4317,9579}, {4329,24179}, {4424,24239}, {4425,30366}, {4653,17167}, {4679,9708}, {4847,5692}, {4858,23580}, {4861,5046}, {4872,24203}, {5010,10165}, {5074,17761}, {5086,5330}, {5122,5298}, {5126,15326}, {5131,28232}, {5183,28212}, {5219,10056}, {5248,24541}, {5250,26363}, {5253,14803}, {5258,12572}, {5274,18391}, {5288,12527}, {5316,19875}, {5432,11230}, {5542,7671}, {5587,6973}, {5657,10589}, {5687,25681}, {5690,10593}, {5726,30308}, {5727,16200}, {5806,13375}, {5844,12019}, {5887,10943}, {5902,11019}, {5905,11240}, {6147,17609}, {6238,12259}, {6265,13274}, {6361,7288}, {6684,11010}, {6767,17718}, {6841,10957}, {6844,10051}, {6886,7162}, {6941,7704}, {6985,11510}, {6988,16208}, {7173,9956}, {7354,22793}, {7373,10404}, {7680,15845}, {7681,10523}, {7727,13605}, {7965,12915}, {7972,21635}, {7982,9581}, {7988,9819}, {8069,22753}, {8071,11496}, {8196,26417}, {8203,26393}, {8256,17619}, {8715,27385}, {9589,15803}, {9598,9619}, {9599,9620}, {9668,10246}, {9709,24954}, {9779,10590}, {10050,11372}, {10052,10085}, {10073,10698}, {10176,25006}, {10310,12700}, {10527,12514}, {10543,14526}, {10679,11502}, {10680,22760}, {10738,12740}, {10742,20586}, {10944,18480}, {10948,12672}, {10950,24680}, {11020,21625}, {11035,13865}, {11362,18395}, {11499,26358}, {11715,23243}, {11720,12896}, {12678,30283}, {12702,24914}, {12735,28224}, {12737,12764}, {12743,19907}, {12848,15299}, {13374,13750}, {13624,15338}, {14100,20330}, {16153,21669}, {17188,17519}, {18650,24202}, {18651,26728}, {18976,22938}, {18990,20323}, {21617,30331}, {21622,30374}, {21623,30423}, {21624,30411}, {23340,26476}, {24046,28018}, {24159,28011}, {24160,28027}, {24474,26475}, {25405,28160}, {30380,30431}, {30381,30432}, {30382,30433}, {30383,30434}

X(30384) = midpoint of X(i) and X(j) for these {i,j}: {1, 3583}, {149, 4511}, {1320, 5176}, {2077, 14217}, {3218, 5180}, {11813, 21630}
X(30384) = reflection of X(i) in X(j) for these (i,j): (11, 7743), (484, 3911), (908, 11813), (1145, 5123), (1155, 15325), (1319, 1387), (1519, 946), (1532, 22835), (1737, 11), (6735, 3814), (11570, 5570), (15326, 5126), (17757, 5087), (21578, 1319)
X(30384) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1479, 10572), (1, 4857, 950), (1, 9614, 1479)

X(30385) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(10880). X(30385) is their endo-homothetic center only when ABC is acute.

X(30385) lies on these lines: {1,372}, {2,30313}, {3,30276}, {4,30306}, {7,21}, {25,13390}, {35,30431}, {36,30425}, {55,16440}, {105,175}, {198,13940}, {238,2067}, {405,30324}, {614,13388}, {958,30412}, {968,13389}, {999,30341}, {1125,30380}, {1385,18458}, {2646,30375}, {3576,30400}, {7587,30406}, {7588,30418}, {7987,30354}, {8077,30368}, {8583,30288}, {22756,24328}, {30389,30396}

X(30385) = {X(56), X(1001)}-harmonic conjugate of X(30386)

X(30386) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(10881). X(30386) is their endo-homothetic center only when ABC is acute.

X(30386) lies on these lines: {1,371}, {2,30314}, {3,30277}, {4,30307}, {7,21}, {25,1659}, {35,30432}, {36,8225}, {55,16441}, {105,176}, {198,13887}, {238,6502}, {405,30325}, {614,13389}, {958,30413}, {968,13388}, {999,30342}, {1125,30381}, {1385,18460}, {2646,30376}, {3576,30401}, {7587,30407}, {7588,30419}, {7987,30355}, {8077,30369}, {8583,30289}, {22757,24328}, {30389,30397}

X(30386) = {X(56), X(1001)}-harmonic conjugate of X(30385)

X(30387) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND LUCAS ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19424). X(30387) is their endo-homothetic center only when ABC is acute.

X(30387) lies on these lines: {1,30348}, {3,30278}, {21,30302}, {55,1152}, {958,30416}, {2360,8273}, {3304,16213}

X(30388) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND LUCAS(-1) ANTIPODAL TANGENTIAL

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

The homothetic center of these triangles is X(19425). X(30388) is their endo-homothetic center only when ABC is acute.

X(30388) lies on these lines: {1,30349}, {3,30279}, {21,30303}, {55,1151}, {958,30417}, {2360,8273}, {3304,16214}

X(30389) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND SUBMEDIAL

Barycentrics a*(7*a^3-3*(b+c)*a^2-(7*b^2-6*b*c+7*c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(30389) = 3*X(1)+4*X(3) = 5*X(1)+2*X(40) = 4*X(1)+3*X(165) = X(1)-8*X(1385) = 11*X(1)-4*X(1482) = X(1)+6*X(3576) = 9*X(1)-2*X(7982) = 2*X(1)+5*X(7987) = 6*X(1)+X(7991) = 5*X(1)-12*X(10246) = 19*X(1)-12*X(10247) = 10*X(1)-3*X(11224) = 8*X(1)-X(11531) = 9*X(1)-16*X(15178) = 12*X(1)-5*X(16189) = 23*X(1)-9*X(16191) = 13*X(1)-6*X(16200) = 15*X(1)-8*X(24680) = 2*X(1)-9*X(30392) = 24*X(2)-17*X(30315)

The homothetic center of these triangles is X(3090). X(30389) is their endo-homothetic center only when ABC is acute.

X(30389) lies on these lines: {1,3}, {2,30315}, {4,25055}, {5,3653}, {9,23073}, {10,10303}, {20,551}, {21,30304}, {30,9624}, {104,28148}, {140,3655}, {145,10164}, {214,936}, {355,632}, {376,9589}, {392,15071}, {405,9851}, {515,3090}, {516,3622}, {518,10541}, {519,3523}, {546,8227}, {548,3656}, {549,4677}, {572,3731}, {573,17474}, {610,22357}, {631,3679}, {944,1698}, {946,3529}, {952,14869}, {958,30393}, {960,22333}, {962,3636}, {991,1201}, {997,5234}, {1001,3062}, {1125,3091}, {1279,15839}, {1376,7990}, {1490,6920}, {1699,3146}, {1702,6453}, {1703,6454}, {1743,21748}, {1750,5436}, {1768,5250}, {2771,15039}, {2948,15034}, {2975,3984}, {3083,21565}, {3084,21568}, {3158,11260}, {3241,15717}, {3522,4301}, {3524,11362}, {3526,19876}, {3528,28194}, {3582,6825}, {3584,6891}, {3592,18992}, {3594,18991}, {3627,5886}, {3628,5587}, {3632,6684}, {3633,5657}, {3646,16860}, {3654,15712}, {3680,4421}, {3698,10156}, {3851,28208}, {3857,28186}, {3897,17531}, {3899,5884}, {3901,12005}, {3951,4511}, {4308,13405}, {4311,5290}, {4315,5703}, {4317,6987}, {4325,6868}, {4326,30287}, {4330,6948}, {4668,12108}, {4669,15708}, {4745,15721}, {4853,4855}, {4857,6850}, {4866,5258}, {4881,17572}, {4882,5440}, {5007,9619}, {5047,8583}, {5056,19883}, {5072,11230}, {5076,9955}, {5079,18480}, {5259,12114}, {5265,6738}, {5270,6827}, {5272,7963}, {5281,6049}, {5303,11682}, {5426,16143}, {5433,5727}, {5444,10827}, {5450,7992}, {5493,5734}, {5550,15022}, {5573,8572}, {5603,17538}, {5660,24954}, {5692,12675}, {5732,24644}, {5818,19872}, {5901,15704}, {6176,19646}, {6419,9583}, {6420,19004}, {6425,7968}, {6426,7969}, {6713,9897}, {6908,10072}, {6926,10056}, {6946,12650}, {6960,10199}, {6972,10197}, {6986,8666}, {7308,9845}, {7415,28619}, {7419,17194}, {7587,30394}, {7588,30395}, {7677,30330}, {7772,9592}, {7786,22650}, {7972,21154}, {7993,11715}, {7997,26321}, {8077,30371}, {8543,9814}, {9579,15950}, {9612,21578}, {9780,28236}, {9904,15021}, {10085,19526}, {11194,11523}, {11219,26066}, {11231,18526}, {11363,11403}, {11477,16475}, {11709,14094}, {11710,23235}, {11720,15054}, {12103,12699}, {12104,19919}, {12407,15027}, {13731,22392}, {15017,25522}, {15808,28164}, {15829,24645}, {16126,21161}, {16132,16138}, {16842,17614}, {16865,19861}, {18444,30144}, {19546,25502}, {19647,26102}, {24914,30286}, {30385,30396}, {30386,30397}, {10647,10655}, {10648,10656}

X(30389) = midpoint of X(1) and X(16192)
X(30389) = reflection of X(i) in X(j) for these (i,j): (7989, 3624), (9588, 3523)
X(30389) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 7991), (1, 7280, 2093), (3612, 21842, 1), (7982, 15178, 1)

X(30390) = PERSPECTOR OF THESE TRIANGLES: INNER-NAPOLEON AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics a^2*(2*sqrt(3)*(a^4-b^4-c^4)*S*a^2+a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-(b^4-c^4)*(b^2-c^2)*a^2) : :

X(30390) lies on these lines: {5,182}, {154,3129}, {184,398}, {3490,11088}, {8919,14560}

X(30390) = {X(206), X(6759)}-harmonic conjugate of X(30391)

X(30391) = PERSPECTOR OF THESE TRIANGLES: OUTER-NAPOLEON AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics a^2*(-2*sqrt(3)*(a^4-b^4-c^4)*S*a^2+a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-(b^4-c^4)*(b^2-c^2)*a^2) : :

X(30391) lies on these lines: {5,182}, {154,3130}, {184,397}, {3489,11083}, {8918,14560}

X(30391) = {X(206), X(6759)}-harmonic conjugate of X(30390)

X(30392) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE AND SUBMEDIAL

Barycentrics a*(9*a^3-5*(b+c)*a^2-(9*b^2-10*b*c+9*c^2)*a+5*(b^2-c^2)*(b-c)) : :
Trilinears 5 r + 4 R cos A : :
X(30392) = 5*X(1)+4*X(3) = 7*X(1)+2*X(40) = 2*X(1)+X(165) = X(1)+8*X(1385) = 13*X(1)-4*X(1482) = X(1)+2*X(3576) = 11*X(1)-2*X(7982) = 4*X(1)+5*X(7987) = 8*X(1)+X(7991) = X(1)-4*X(10246) = 7*X(1)-4*X(10247) = 4*X(1)-X(11224) = 10*X(1)-X(11531) = 7*X(1)-16*X(15178) = 14*X(1)-5*X(16189) = 3*X(1)-X(16191) = 11*X(1)+7*X(16192) = 5*X(1)-2*X(16200) = 7*X(1)+8*X(17502) = 17*X(1)-8*X(24680)

The homothetic center of these triangles is X(5055). X(30392) is their endo-homothetic center only when ABC is acute.

X(30392) lies on these lines: {1,3}, {2,28236}, {20,3636}, {104,28170}, {145,9588}, {214,7993}, {355,30315}, {376,28232}, {515,3545}, {519,15708}, {547,3655}, {551,1699}, {572,16673}, {631,3632}, {944,3624}, {952,3653}, {991,1149}, {997,18452}, {1125,5056}, {1698,3533}, {1702,9585}, {2320,3062}, {2801,16858}, {3083,21564}, {3084,21569}, {3091,15808}, {3241,10164}, {3244,3523}, {3524,28234}, {3616,3817}, {3622,4297}, {3626,10303}, {3633,6684}, {3654,19711}, {3656,15690}, {3679,7967}, {3845,5886}, {3850,8227}, {3853,9624}, {3897,8583}, {4423,30283}, {4511,5223}, {4512,24645}, {4677,11812}, {4915,5440}, {5041,9619}, {5102,16475}, {5234,10176}, {5400,6176}, {5426,28461}, {5436,5927}, {5531,11715}, {5603,11001}, {5657,15719}, {5734,12512}, {5790,15723}, {5881,16239}, {6429,9615}, {6431,18992}, {6432,18991}, {6433,9616}, {7990,19860}, {9589,10595}, {9618,10137}, {9814,18450}, {10167,10179}, {10283,15686}, {10304,28228}, {10442,17394}, {11379,16143}, {11714,15735}, {12767,19907}, {15717,20057}, {15726,24644}, {16859,19861}, {16864,17614}, {18444,30304}, {18446,30326}, {18448,30371}, {18454,30394}, {18456,30395}, {18458,30396}, {18460,30397}, {18469,10655}, {18471,10656}, {30284,30330}, {30285,30363}

X(30392) = reflection of X(7988) in X(25055)
X(30392) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 11531), (7982, 13624, 16192), (11531, 16189, 11278), (11531, 16200, 11224)

X(30393) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND SUBMEDIAL

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

The homothetic center of these triangles is X(17825). X(30393) is their endo-homothetic center only when ABC is acute.

X(30393) lies on these lines: {1,210}, {2,5223}, {8,30337}, {9,165}, {10,962}, {40,10157}, {43,3731}, {57,3715}, {63,9342}, {171,3973}, {188,30371}, {200,1621}, {226,1698}, {236,30394}, {392,4915}, {497,3679}, {518,30350}, {612,16469}, {756,2999}, {936,993}, {960,11531}, {984,23511}, {997,18452}, {1125,30343}, {1215,16832}, {1329,30315}, {1697,3983}, {1743,5268}, {1961,16667}, {2093,3820}, {2886,30308}, {3097,16569}, {3158,15254}, {3243,8167}, {3452,7988}, {3475,3624}, {3587,18529}, {3681,10582}, {3697,4882}, {3711,10389}, {3729,26038}, {3928,15481}, {3929,4413}, {3930,16673}, {3951,19877}, {3958,10158}, {3971,17151}, {4005,11518}, {4015,6765}, {4104,17284}, {4312,26040}, {4355,17582}, {4384,27538}, {4512,27065}, {4533,16842}, {4668,4863}, {4679,10826}, {4682,16670}, {5129,6743}, {5220,5437}, {5231,10584}, {5256,9330}, {5273,5785}, {5302,5438}, {5506,12658}, {5658,6684}, {5686,11019}, {5691,18250}, {5692,18421}, {5732,15064}, {5745,11407}, {5795,7990}, {6172,9814}, {6666,25568}, {7028,30395}, {7994,24644}, {7996,17355}, {7997,25440}, {9623,10176}, {9780,12526}, {9898,12053}, {10268,18524}, {13405,18230}, {15829,16189}, {18235,30363}, {24393,26105}, {30396,30412}, {30397,30413}, {10655,30414}, {10656,30415}

X(30393) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (210, 7308, 1), (4383, 7322, 1)

X(30394) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND SUBMEDIAL

Barycentrics a*(16*b*c*sin(A/2)+a^2+2*(b+c)*a-3*(b-c)^2) : :

The homothetic center of these triangles is X(9816). X(30394) is their endo-homothetic center only when ABC is acute.

X(30394) lies on these lines: {1,10502}, {165,173}, {177,30371}, {236,30393}, {1699,11891}, {5223,8126}, {5691,12582}, {7587,30389}, {7590,7987}, {7593,30326}, {7988,21624}, {7991,8351}, {8083,8090}, {8092,30343}, {8379,30308}, {8382,30315}, {8389,30330}, {8425,30363}, {8729,11407}, {9814,30404}, {9819,30411}, {11531,12445}, {11535,16189}, {11860,30291}, {11924,30337}, {18408,18421}, {18454,30392}, {30396,30406}, {30397,30407}, {10655,30409}, {10656,30410}

X(30394) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (173, 8423, 165), (173, 11195, 8423)

X(30395) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND SUBMEDIAL

Barycentrics a*(-16*b*c*sin(A/2)+a^2+2*(b+c)*a-3*(b-c)^2) : :

The homothetic center of these triangles is X(9817). X(30395) is their endo-homothetic center only when ABC is acute.

X(30395) lies on these lines: {1,8099}, {165,258}, {174,10980}, {1699,9795}, {3339,30420}, {5223,8125}, {5691,12581}, {7028,30393}, {7588,30389}, {7987,8082}, {7988,21623}, {7991,8092}, {8080,30326}, {8084,8089}, {8086,30308}, {8088,30315}, {8094,11531}, {8242,30337}, {8250,30363}, {8351,30343}, {8388,30330}, {8423,11033}, {8734,11407}, {9814,30405}, {9819,30423}, {11859,30291}, {11889,30304}, {11899,16189}, {18409,18421}, {18456,30392}, {30396,30418}, {30397,30419}, {10655,30421}, {10656,30422}

X(30395) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (258, 8090, 165), (258, 11217, 8090)

X(30396) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND SUBMEDIAL

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

The homothetic center of these triangles is X(10961). X(30396) is their endo-homothetic center only when ABC is acute.

X(30396) lies on these lines: {1,30375}, {7,1699}, {165,6203}, {1743,8941}, {3339,30425}, {30306,30308}, {30313,30315}, {30393,30412}

X(30396) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3062, 10980, 30397), (6203, 30354, 165)

X(30397) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND SUBMEDIAL

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

The homothetic center of these triangles is X(10963). X(30397) is their endo-homothetic center only when ABC is acute.

X(30397) lies on these lines: {1,30376}, {7,1699}, {165,6204}, {1743,8945}, {3339,30426}, {7987,30401}, {7988,30381}, {9819,30432}, {11407,30277}, {16189,30320}, {18411,18421}, {18460,30392}, {30289,30291}, {30307,30308}, {30314,30315}, {30325,30326}, {30334,30337}, {30342,30343}, {30347,30350}, {30361,30363}, {30369,30371}, {30386,30389}, {30393,30413}, {30394,30407}, {30395,30419}

X(30397) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3062, 10980, 30396), (6204, 30355, 165)

X(30398) = PERSPECTOR OF THESE TRIANGLES: INNER-SQUARES AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

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

X(30398) lies on these lines: {6,3156}, {140,141}, {157,371}, {159,8276}, {184,590}, {206,8969}, {491,5012}, {1151,12977}, {1503,23313}

X(30399) = PERSPECTOR OF THESE TRIANGLES: OUTER-SQUARES AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

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

X(30399) lies on these lines: {6,3155}, {140,141}, {157,372}, {159,8277}, {184,615}, {206,13972}, {492,5012}, {1152,13068}, {1503,23314}

X(30400) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER AND 1st KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(10897). X(30400) is their endo-homothetic center only when ABC is acute.

X(30400) lies on these lines: {1,7}, {3,6203}, {4,30380}, {40,30296}, {46,18410}, {56,30375}, {84,2067}, {517,30319}, {936,30412}, {1490,30324}, {3333,30346}, {3423,6213}, {3576,30385}, {5587,30313}, {7590,30406}, {7987,30396}, {8081,30368}, {8082,30418}, {8227,30306}, {8235,30360}, {8726,30276}, {10391,13388}, {10864,30288}

X(30400) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 30401), (990, 4292, 30401)

X(30401) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER AND 2nd KENMOTU DIAGONALS

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

The homothetic center of these triangles is X(10898). X(30401) is their endo-homothetic center only when ABC is acute.

X(30401) lies on these lines: {1,7}, {3,6204}, {4,30381}, {40,30297}, {46,18411}, {56,30376}, {84,6502}, {517,30320}, {936,30413}, {1490,30325}, {3333,30347}, {3423,6212}, {3576,30386}, {5587,30314}, {7590,30407}, {7987,30397}, {8081,30369}, {8082,30419}, {8227,30307}, {8235,30361}, {8726,30277}, {10391,13389}, {10864,30289}

X(30401) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 30400), (990, 4292, 30400)

X(30402) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

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

X(30402) lies on these lines: {3,10639}, {6,25}, {15,6759}, {16,10282}, {26,10661}, {110,10681}, {156,10662}, {182,10643}, {1498,11480}, {1503,23302}, {1614,10632}, {2917,15962}, {5321,16252}, {6000,10645}, {6353,18929}, {7051,26888}, {9306,11515}, {9833,18582}, {10192,23303}, {10535,10638}, {10536,10636}, {10539,10634}, {10540,18468}, {10633,11466}, {10646,11202}, {10657,13289}, {10664,20773}, {10682,15647}, {11206,11488}, {11421,11452}, {11475,26883}, {11476,13367}, {11481,17821}, {11485,14530}, {11486,14819}, {16808,18400}, {16966,18381}, {19190,26887}

X(30402) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8740, 21648, 6), (10533, 10534, 11243), (11408, 19364, 6)

X(30403) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

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

X(30403) lies on these lines: {3,10640}, {6,25}, {15,10282}, {16,6759}, {26,10662}, {110,10682}, {156,10661}, {182,10644}, {1250,10535}, {1498,11481}, {1503,23303}, {1614,10633}, {2917,15961}, {5318,16252}, {6000,10646}, {6353,18930}, {9306,11516}, {9833,18581}, {10192,23302}, {10536,10637}, {10539,10635}, {10540,18470}, {10632,11467}, {10645,11202}, {10658,13289}, {10663,20773}, {10681,15647}, {11206,11489}, {11420,11453}, {11475,13367}, {11476,26883}, {11480,17821}, {11485,14818}, {11486,14530}, {16809,18400}, {16967,18381}, {19191,26887}, {19373,26888}

X(30403) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 17827, 11244), (154, 17827, 6), (10533, 10534, 11244)

X(30404) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND EXTANGENTS

Barycentrics -2*(a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+3*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(8539). X(30404) is their endo-homothetic center only when ABC is acute.

X(30404) lies on these lines: {7,174}, {173,8545}, {177,1156}, {236,6172}, {516,30411}, {527,8126}, {2801,18408}, {6173,8125}, {7587,8543}, {7589,30295}, {7590,8544}, {7593,12848}, {7671,8083}, {8092,30340}, {8379,30311}, {8382,30312}, {8423,30353}, {8425,30359}, {8729,30275}, {9814,30394}, {10502,15726}, {11535,30318}, {11860,30287}, {11924,30332}, {18450,18454}, {21624,30379}, {30408,30424}

X(30404) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 8389, 8388), (174, 30405, 8388), (8389, 30405, 174)

X(30405) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND INTANGENTS

Barycentrics 2*(a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+3*(a+b-c)*(a-b+c) : :
X(30405) = 3*X(8387)-4*X(13385)

The homothetic center of these triangles is X(8540). X(30405) is their endo-homothetic center only when ABC is acute.

X(30405) lies on these lines: {1,30367}, {7,174}, {258,8545}, {516,30423}, {527,8125}, {2801,18409}, {5542,30411}, {6172,7028}, {6173,8126}, {7588,8543}, {7671,11033}, {8076,30295}, {8080,12848}, {8082,8544}, {8086,30311}, {8088,30312}, {8090,30353}, {8242,30332}, {8250,30359}, {8351,30340}, {8387,13385}, {8734,30275}, {9814,30395}, {10501,15726}, {11859,30287}, {11899,30318}, {18450,18456}, {21623,30379}, {30420,30424}

X(30405) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 174, 30404), (7, 8388, 8389), (174, 30404, 8389)

X(30406) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND 1st KENMOTU DIAGONALS

Barycentrics (-2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5415). X(30406) is their endo-homothetic center only when ABC is acute.

X(30406) lies on these lines: {7,174}, {173,6203}, {177,7133}, {236,30412}, {7014,11923}, {7587,30385}, {7590,30400}, {7593,30324}, {8083,30346}, {8092,30341}, {8379,30306}, {8382,30313}, {8425,30360}, {8729,30276}, {10502,30375}, {11535,30319}, {11860,30288}, {11924,30333}, {18408,18410}, {18454,18458}, {21624,30380}, {30394,30396}, {30408,30425}, {30411,30431}

X(30406) = {X(7), X(174)}-harmonic conjugate of X(30418)

X(30407) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND 2nd KENMOTU DIAGONALS

Barycentrics (2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5416). X(30407) is their endo-homothetic center only when ABC is acute.

X(30407) lies on these lines: {7,174}, {173,6204}, {236,30413}, {7587,30386}, {7589,30297}, {7590,30401}, {7593,30325}, {8083,30347}, {8092,30342}, {8379,30307}, {8382,30314}, {8423,30355}, {8729,30277}, {10502,30376}, {11535,30320}, {11860,30289}, {11924,30334}, {18408,18411}, {18454,18460}, {21624,30381}, {30394,30397}, {30408,30426}, {30411,30432}

X(30407) = {X(7), X(174)}-harmonic conjugate of X(30419)

X(30408) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND KOSNITA

Barycentrics 2*(a+b+c)*sin(A/2)+a : :

The homothetic center of these triangles is X(10902). X(30408) is their endo-homothetic center only when ABC is acute.

X(30408) lies on these lines: {1,167}, {10,8126}, {36,7587}, {40,8130}, {46,173}, {65,12491}, {79,1127}, {80,13267}, {236,1698}, {258,3338}, {516,8389}, {942,10502}, {1125,8125}, {1130,6724}, {2093,13098}, {3576,8129}, {3624,7028}, {3633,12646}, {3746,8076}, {4292,11890}, {4295,11891}, {5045,10501}, {5542,8388}, {5563,7588}, {5691,9837}, {5902,8094}, {5903,12445}, {6684,8128}, {7593,9612}, {7741,8379}, {7951,8382}, {7972,12748}, {8104,16173}, {8127,10165}, {8425,30362}, {11535,25415}, {11571,12774}, {11860,30290}, {12047,21624}, {12685,15071}, {13407,21623}, {30404,30424}, {30406,30425}, {30407,30426}, {30409,10651}, {30410,10652}

X(30408) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8092, 8351, 11924), (8092, 11924, 1), (30411, 30420, 1)

X(30409) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND INNER TRI-EQUILATERAL

Barycentrics (-2*sqrt(3)*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10636). X(30409) is their endo-homothetic center only when ABC is acute.

X(30409) lies on these lines: {7,174}, {173,1653}, {177,30372}, {236,30414}, {7587,10647}, {7589,30300}, {7590,10649}, {7593,30327}, {8083,30351}, {8092,30344}, {8379,30309}, {8382,30316}, {8423,30356}, {8425,30364}, {11535,30321}, {11860,30292}, {11924,30338}, {18408,18422}, {18454,18469}, {30394,10655}, {30408,10651}, {30411,30433}

X(30409) = {X(7), X(174)}-harmonic conjugate of X(30421)

X(30410) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND OUTER TRI-EQUILATERAL

Barycentrics (2*sqrt(3)*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10637). X(30410) is their endo-homothetic center only when ABC is acute.

X(30410) lies on these lines: {7,174}, {173,1652}, {177,1251}, {236,30415}, {7589,30301}, {7590,10650}, {7593,30328}, {8083,30352}, {8092,30345}, {8379,30310}, {8382,30317}, {8425,30365}, {8729,30281}, {10502,30378}, {11535,30322}, {11860,30293}, {11924,30339}, {18408,18423}, {18454,18471}, {21624,30383}, {30394,10656}, {30408,10652}, {30411,30434}

X(30410) = {X(7), X(174)}-harmonic conjugate of X(30422)

X(30411) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND TRINH

Barycentrics 2*(a+b+c)*sin(A/2)+3*a : :

The homothetic center of these triangles is X(7688). X(30411) is their endo-homothetic center only when ABC is acute.

X(30411) lies on these lines: {1,167}, {35,7587}, {36,7589}, {80,7707}, {173,5119}, {236,3679}, {354,18409}, {516,30404}, {517,10502}, {519,8126}, {551,8125}, {1699,9837}, {3057,12491}, {3586,7593}, {3632,12646}, {4294,12570}, {4304,11890}, {5049,10501}, {5441,16151}, {5542,30405}, {5697,12445}, {5902,8083}, {7028,25055}, {7741,8382}, {7951,8379}, {7972,13267}, {7982,8130}, {8094,18398}, {8100,17609}, {8128,11362}, {8389,30331}, {8425,30366}, {8729,30282}, {9819,30394}, {10500,18399}, {10624,12582}, {11535,30323}, {11860,30294}, {11891,30305}, {21624,30384}, {30406,30431}, {30407,30432}, {30409,30433}, {30410,30434}

X(30411) = reflection of X(18408) in X(10502)
X(30411) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 174, 30423), (1, 177, 30374), (174, 30423, 30420)

X(30412) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND 1st KENMOTU DIAGONALS

Barycentrics a*(-a+b+c)+S : :

The homothetic center of these triangles is X(17819). X(30412) is their endo-homothetic center only when ABC is acute.

X(30412) lies on these lines: {1,7586}, {2,7}, {6,6351}, {8,7090}, {10,1132}, {37,3069}, {44,3068}, {45,615}, {169,8231}, {188,15892}, {190,1267}, {198,16440}, {210,13359}, {219,13387}, {236,30406}, {281,1586}, {344,492}, {346,30416}, {354,13360}, {518,30346}, {590,16885}, {591,17243}, {936,30400}, {958,30385}, {962,6213}, {997,18458}, {1100,19053}, {1123,1124}, {1125,30341}, {1270,3912}, {1271,4416}, {1329,30313}, {1376,30296}, {1600,15817}, {1698,30425}, {1743,5393}, {2324,3083}, {2886,30306}, {3300,13905}, {3593,25101}, {3679,30431}, {3731,5405}, {3973,8972}, {4643,5591}, {4851,5860}, {5391,17277}, {5414,30334}, {5590,17279}, {5692,18410}, {5704,8957}, {6347,20262}, {7028,30418}, {8580,30354}, {9780,14121}, {11292,25066}, {13759,29574}, {13847,16675}, {15829,30319}, {16669,19054}, {18227,30288}, {18235,30360}, {30393,30396}

X(30412) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (329, 27382, 30413), (5279, 27540, 30413), (5282, 27547, 30413)

X(30413) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND 2nd KENMOTU DIAGONALS

Barycentrics a*(-a+b+c)-S : :

The homothetic center of these triangles is X(17820). X(30413) is their endo-homothetic center only when ABC is acute.

X(30413) lies on these lines: {1,7585}, {2,7}, {6,6352}, {8,14121}, {10,1131}, {37,3068}, {44,3069}, {45,590}, {188,30369}, {190,5391}, {198,16441}, {210,13360}, {219,13386}, {236,30407}, {281,1585}, {344,491}, {346,15891}, {354,13359}, {518,30347}, {615,16885}, {936,30401}, {938,8957}, {958,30386}, {962,6212}, {997,18460}, {1100,19054}, {1123,9646}, {1125,30342}, {1267,17277}, {1270,4416}, {1271,3912}, {1329,30314}, {1335,1336}, {1376,30297}, {1599,15817}, {1698,30426}, {1743,5405}, {1991,17243}, {2066,30333}, {2324,3084}, {2886,30307}, {3302,13963}, {3595,25101}, {3679,30432}, {3731,5393}, {3973,13941}, {4643,5590}, {4851,5861}, {5591,17279}, {5692,18411}, {6348,20262}, {7028,30419}, {7090,9780}, {8580,30355}, {11291,25066}, {13639,29574}, {13846,16675}, {15829,30320}, {16669,19053}, {18227,30289}, {18235,30361}, {30393,30397}

X(30413) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (329, 27382, 30412), (5279, 27540, 30412), (5282, 27547, 30412)

X(30414) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND INNER TRI-EQUILATERAL

Barycentrics a*(-a+b+c)+sqrt(3)*S : :

The homothetic center of these triangles is X(17826). X(30414) is their endo-homothetic center only when ABC is acute.

X(30414) lies on these lines: {2,7}, {8,5240}, {10,22237}, {37,11489}, {44,11488}, {45,23303}, {145,5245}, {219,5367}, {236,30409}, {281,471}, {302,344}, {395,16777}, {518,30351}, {936,10649}, {958,10647}, {962,1277}, {997,18469}, {1125,30344}, {1329,30316}, {1376,30300}, {1698,10651}, {2886,30309}, {3241,7026}, {3616,5239}, {3617,5246}, {3639,5199}, {3679,30433}, {5692,18422}, {7028,30421}, {8580,30356}, {9761,17243}, {11790,16808}, {16645,16675}, {16885,23302}, {18227,30292}, {18235,30364}, {30393,10655}

X(30414) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5242, 2), (3306, 27508, 30415), (5328, 5749, 30415)

X(30415) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND OUTER TRI-EQUILATERAL

Barycentrics a*(-a+b+c)-sqrt(3)*S : :

The homothetic center of these triangles is X(17827). X(30415) is their endo-homothetic center only when ABC is acute.

X(30415) lies on these lines: {2,7}, {8,1251}, {10,22235}, {37,11488}, {44,11489}, {45,23302}, {145,5246}, {188,30373}, {210,30378}, {219,5362}, {236,30410}, {281,470}, {303,344}, {396,16777}, {518,30352}, {936,10650}, {958,10648}, {962,1276}, {997,18471}, {1125,30345}, {1329,30317}, {1376,30301}, {1698,10652}, {2886,30310}, {3179,4295}, {3241,7043}, {3616,5240}, {3617,5245}, {3638,5199}, {3679,30434}, {5692,18423}, {7028,30422}, {8580,30357}, {9763,17243}, {11791,16809}, {15829,30322}, {16644,16675}, {16885,23303}, {18227,30293}, {18235,30365}, {30393,10656}

X(30415) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9, 30414), (3306, 27508, 30414), (5328, 5749, 30414)

X(30416) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics (a*(a+b+c)+S)*((a+b-c)*c+S)*((a-b+c)*b+S)*(-a+b+c) : :

The homothetic center of these triangles is X(19430). X(30416) is their endo-homothetic center only when ABC is acute.

X(30416) lies on these lines: {7,7090}, {8,30335}, {9,30429}, {346,15892}, {518,30348}, {958,30387}, {3616,16213}, {5273,30302}, {5745,30278}

X(30417) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics (a*(a+b+c)-S)*((a+b-c)*c-S)*((a-b+c)*b-S)*(-a+b+c) : :

The homothetic center of these triangles is X(19431). X(30417) is their endo-homothetic center only when ABC is acute.

X(30417) lies on these lines: {7,14121}, {8,30336}, {9,30430}, {346,15891}, {518,30349}, {958,30388}, {3616,16214}, {5273,30303}, {5745,30279}

X(30418) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND 1st KENMOTU DIAGONALS

Barycentrics (-2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(2066). X(30418) is their endo-homothetic center only when ABC is acute.

X(30418) lies on these lines: {1,30368}, {7,174}, {258,6203}, {1489,7014}, {7028,30412}, {7133,8248}, {7588,30385}, {8076,30296}, {8080,30324}, {8082,30400}, {8086,30306}, {8088,30313}, {8090,30354}, {8242,30333}, {8250,30360}, {8734,30276}, {10501,30375}, {11033,30346}, {11859,30288}, {11899,30319}, {18409,18410}, {18456,18458}, {21623,30380}, {30395,30396}, {30420,30425}, {30423,30431}

X(30418) = {X(7), X(174)}-harmonic conjugate of X(30406)

X(30419) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND 2nd KENMOTU DIAGONALS

Barycentrics (a^2-2*(b+c)*a+2*S+(b-c)^2)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5414). X(30419) is their endo-homothetic center only when ABC is acute.

X(30419) lies on these lines: {1,30369}, {7,174}, {258,6204}, {7028,30413}, {7588,30386}, {8076,30297}, {8080,30325}, {8082,30401}, {8086,30307}, {8088,30314}, {8090,30355}, {8242,30334}, {8250,30361}, {8351,30342}, {8734,30277}, {10501,30376}, {11033,30347}, {11859,30289}, {11899,30320}, {18409,18411}, {18456,18460}, {21623,30381}, {30395,30397}, {30420,30426}, {30423,30432}

X(30419) = {X(7), X(174)}-harmonic conjugate of X(30407)

X(30420) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND KOSNITA

Barycentrics    2*(a+b+c)*sin(A/2)-a : :
Barycentrics    Sin[A]*(4*Cos[B/2]*Cos[C/2] - 1) : :
Trilinears    1 - 4 cos B/2 cos C/2 : :

The homothetic center of these triangles is X(35). X(30420) is their endo-homothetic center only when ABC is acute.

In the plane of a triangle ABC, let
P = the point on segment BC such that AP bisects angle BAC, and define Q and R cyclically;
Pb = the point on AC suchb that PPc besects angle APC, and define Qb and Ra cyclically;
Pc = the point on AB such that PPc besects angle APB, and define Qc and Rb cyclically;
S = RRb∩QQc, and define T and U cyclically;
X = RRa∩QQa, and define Y and Z cyclically;
Then the lines SX, TY, UZ concur in X(30420). (It is known that the six points Rc, Qa, Pb, Rb, Ac Pc lies on an ellipse; its center is X(46329). See X(30420). (Steven Verpoort, February 17, 2024)

X(30420) lies on these lines: {1, 167}, {10, 8125}, {35, 8076}, {36, 7588}, {40, 8129}, {46, 258}, {65, 8100}, {79, 16147}, {80, 8104}, {173, 3338}, {236, 3624}, {354, 12491}, {516, 8388}, {942, 10501}, {1125, 8126}, {1130, 6732}, {1698, 7028}, {2093, 8102}, {3339, 30395}, {3576, 8130}, {3633, 12644}, {3746, 7589}, {4292, 11889}, {4295, 9795}, {4312, 45707}, {5045, 10502}, {5542, 8389}, {5557, 7707}, {5563, 7587}, {5902, 11217}, {5903, 8094}, {6684, 8127}, {6724, 10231}, {7741, 8086}, {7951, 8088}, {7972, 8098}, {8080, 9612}, {8083, 50190}, {8096, 15071}, {8128, 10165}, {8131, 37557}, {8250, 30362}, {8379, 37720}, {8382, 37719}, {8734, 15803}, {10023, 18291}, {10651, 30421}, {10652, 30422}, {11033, 18398}, {11571, 12772}, {11859, 30290}, {11899, 25415}, {12047, 21623}, {12646, 51093}, {13267, 16173}, {13407, 21624}, {18399, 53810}, {30404, 43180}, {30405, 30424}, {30418, 30425}, {30419, 30426}, {34034, 34043}, {45708, 59372}

X(30420) = reflection of X(30374) in X(1)
X(30420) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1, 174, 30408}, {1, 8092, 30423}, {1, 30408, 30411}, {65, 8100, 18409}, {174, 8092, 1}, {174, 8242, 8351}, {174, 30423, 30411}, {8092, 8351, 8242}, {8242, 8351, 1}, {30408, 30423, 1}

X(30421) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND INNER TRI-EQUILATERAL

Barycentrics (a^2-2*(b+c)*a+(b-c)^2-2*sqrt(3)*S)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10638). X(30421) is their endo-homothetic center only when ABC is acute.

X(30421) lies on these lines: {1,30372}, {7,174}, {258,1653}, {7028,30414}, {7588,10647}, {8076,30300}, {8080,30327}, {8082,10649}, {8086,30309}, {8088,30316}, {8090,30356}, {8242,30338}, {8250,30364}, {8734,30280}, {10501,30377}, {11033,30351}, {11859,30292}, {11899,30321}, {18409,18422}, {18456,18469}, {21623,30382}, {30395,10655}, {30420,10651}, {30423,30433}

X(30421) = {X(7), X(174)}-harmonic conjugate of X(30409)

X(30422) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND OUTER TRI-EQUILATERAL

Barycentrics (a^2-2*(b+c)*a+(b-c)^2+2*sqrt(3)*S)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(1250). X(30422) is their endo-homothetic center only when ABC is acute.

X(30422) lies on these lines: {1,30373}, {7,174}, {258,1652}, {7028,30415}, {7588,10648}, {8076,30301}, {8080,30328}, {8082,10650}, {8086,30310}, {8088,30317}, {8090,30357}, {8242,30339}, {8250,30365}, {8351,30345}, {8734,30281}, {10501,30378}, {11033,30352}, {11859,30293}, {11899,30322}, {18409,18423}, {18456,18471}, {21623,30383}, {30395,10656}, {30420,10652}, {30423,30434}

X(30422) = {X(7), X(174)}-harmonic conjugate of X(30410)

X(30423) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND TRINH

Barycentrics 2*(a+b+c)*sin(A/2)-3*a : :
X(30423) = 3*X(1)-2*X(13385) = 4*X(13385)-3*X(30370)

The homothetic center of these triangles is X(36). X(30423) is their endo-homothetic center only when ABC is acute.

X(30423) lies on these lines: {1,167}, {35,7588}, {36,8076}, {80,8098}, {236,25055}, {258,5119}, {354,18408}, {516,30405}, {517,10501}, {519,8125}, {551,8126}, {3057,8100}, {3586,8080}, {3632,12644}, {3679,7028}, {4294,12569}, {4304,11889}, {5049,10502}, {5441,16147}, {5542,30404}, {5697,8094}, {6732,10231}, {7741,8088}, {7951,8086}, {7972,8104}, {7982,8129}, {8127,11362}, {8250,30366}, {8388,30331}, {8734,30282}, {9795,30305}, {9819,30395}, {10624,12581}, {11859,30294}, {11899,30323}, {12445,18398}, {12491,17609}, {17641,18399}, {21623,30384}, {30418,30431}, {30419,30432}, {30421,30433}, {30422,30434}

X(30423) = incircle-inverse of X(32183)
X(30423) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8092, 30420), (8092, 8242, 1), (30411, 30420, 174)

X(30424) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND KOSNITA

Barycentrics 4*a^3+(b+c)*a^2-2*(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30424) = X(1)-3*X(7) = 5*X(1)-3*X(390) = X(1)+3*X(4312) = 2*X(1)-3*X(5542) = 11*X(1)-9*X(8236) = 7*X(1)-9*X(11038) = 4*X(1)-3*X(30331) = 3*X(1)-X(30332) = 3*X(1)-5*X(30340) = 5*X(7)-X(390) = 11*X(7)-3*X(8236) = 7*X(7)-3*X(11038) = 4*X(7)-X(30331) = 9*X(7)-X(30332) = 9*X(7)-5*X(30340) = 3*X(9)-4*X(3634) = 3*X(10)-2*X(5220) = X(390)+5*X(4312) = 2*X(390)-5*X(5542) = 11*X(390)-15*X(8236) = X(5220)-3*X(5880)

The homothetic center of these triangles is X(575). X(30424) is their endo-homothetic center only when ABC is acute.

X(30424) lies on these lines: {1,7}, {9,3634}, {10,527}, {11,4031}, {12,11662}, {35,30295}, {36,8543}, {46,3947}, {55,3982}, {57,1776}, {65,2801}, {79,1156}, {142,3647}, {144,9780}, {226,1155}, {354,4114}, {382,17706}, {515,1159}, {518,3625}, {528,3244}, {551,5126}, {553,1836}, {673,16477}, {758,5784}, {938,5586}, {942,15726}, {946,20418}, {954,5217}, {971,5884}, {1001,5267}, {1056,28228}, {1088,10136}, {1125,5698}, {1698,6172}, {1699,21454}, {1743,7613}, {2094,5231}, {2550,3626}, {3008,24695}, {3062,5556}, {3339,9814}, {3474,4654}, {3487,12512}, {3488,28158}, {3579,5762}, {3617,5223}, {3755,17365}, {3826,10592}, {3868,5696}, {3874,15733}, {3883,7321}, {3923,21255}, {3935,17483}, {3950,28526}, {4052,29649}, {4078,17767}, {4652,5550}, {4663,5845}, {4847,20292}, {4851,28557}, {5221,5729}, {5435,10171}, {5493,21620}, {5551,10390}, {5572,13369}, {5708,5805}, {5728,17637}, {5819,16670}, {5843,18357}, {5851,12019}, {5852,24393}, {5902,10394}, {5905,21060}, {6006,21201}, {6147,8255}, {6738,9579}, {6767,28232}, {7263,28570}, {7671,18398}, {7741,30311}, {7951,30312}, {9612,12848}, {9812,10980}, {11009,14151}, {11529,28164}, {12047,30379}, {12609,15823}, {12699,21625}, {12702,12872}, {14100,15009}, {14563,28160}, {15803,30275}, {15934,28150}, {15935,28154}, {16125,18482}, {17274,19868}, {17298,24280}, {17376,28530}, {18230,19872}, {18421,28236}, {20103,28609}, {20533,29601}, {21635,24465}, {25415,30318}, {30287,30290}, {30359,30362}, {30367,30370}, {30404,30408}, {30405,30420}

X(30424) = midpoint of X(i) and X(j) for these {i,j}: {7, 4312}, {3868, 5696}, {5223, 20059}
X(30424) = reflection of X(i) in X(j) for these (i,j): (10, 5880), (5542, 7), (5698, 1125), (14100, 20116), (30331, 5542)
X(30424) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30332, 30340), (4307, 4862, 4353), (30332, 30340, 1)

X(30425) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND KOSNITA

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

The homothetic center of these triangles is X(372). X(30425) is their endo-homothetic center only when ABC is acute.

X(30425) lies on these lines: {1,7}, {35,30296}, {36,30385}, {46,486}, {65,18410}, {79,7133}, {942,30375}, {1478,9907}, {1698,30412}, {1836,13388}, {3339,30396}, {3474,13390}, {7741,30306}, {7951,30313}, {9612,30324}, {11246,13389}, {12047,30380}, {13436,28849}, {15803,30276}, {18398,30346}, {25415,30319}, {30288,30290}, {30360,30362}, {30368,30370}, {30406,30408}, {30418,30420}

X(30425) = reflection of X(1) in X(481)
X(30425) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390, 30342, 1), (4292, 24248, 30426), (30333, 30341, 1)

X(30426) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND KOSNITA

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

The homothetic center of these triangles is X(371). X(30426) is their endo-homothetic center only when ABC is acute.

X(30426) lies on these lines: {1,7}, {35,30297}, {36,8225}, {46,485}, {65,18411}, {942,30376}, {1478,9906}, {1659,3474}, {1698,30413}, {1836,13389}, {3339,30397}, {7741,30307}, {7951,30314}, {9612,30325}, {11246,13388}, {12047,30381}, {13453,28849}, {15803,30277}, {18398,30347}, {25415,30320}, {30289,30290}, {30361,30362}, {30369,30370}, {30407,30408}, {30419,30420}

X(30426) = reflection of X(1) in X(482)
X(30426) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30334, 30342), (390, 30341, 1), (30334, 30342, 1)

X(30427) = PERSPECTOR OF THESE TRIANGLES: INNER-VECTEN AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics a^2*(a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2+2*(a^4-b^4-c^4)*S*a^2-(b^4-c^4)*(b^2-c^2)*a^2) : :
X(30427) = 3*X(154)+X(12970)

X(30427) lies on these lines: {5,182}, {6,8946}, {110,489}, {154,1151}, {184,3071}, {486,50004}, {3594,17843}, {5409,13056}, {8964,9306}, {8967,30398}, {8996,10666}, {12229,19148}

X(30427) = midpoint of X(8996) and X(10666)
X(30427) = {X(206), X(6759)}-harmonic conjugate of X(30428)

X(30428) = PERSPECTOR OF THESE TRIANGLES: OUTER-VECTEN AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics a^2*(a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-2*(a^4-b^4-c^4)*S*a^2-(b^4-c^4)*(b^2-c^2)*a^2) : :
X(30428) = 3*X(154)+X(12964)

X(30428) lies on these lines: {5,182}, {6,8948}, {110,490}, {154,1152}, {184,3070}, {485,30398}, {3592,17840}, {5408,13055}, {8968,23313}, {9306,13027}, {12230,19147}

X(30428) = {X(206), X(6759)}-harmonic conjugate of X(30427)

X(30429) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL AND ORTHIC

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

The homothetic center of these triangles is X(19446). X(30429) is their endo-homothetic center only when ABC is acute.

X(30429) lies on these lines: {1,30348}, {9,30416}, {40,971}, {57,7133}, {63,30302}, {1697,30335}

X(30430) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL AND ORTHIC

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

The homothetic center of these triangles is X(19447). X(30430) is their endo-homothetic center only when ABC is acute.

X(30430) lies on these lines: {1,30349}, {9,30417}, {40,971}, {57,30279}, {63,30303}, {1697,30336}

X(30431) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND TRINH

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

The homothetic center of these triangles is X(6200). X(30431) is their endo-homothetic center only when ABC is acute.

X(30431) lies on these lines: {1,7}, {36,30296}, {80,7133}, {517,18410}, {3586,30324}, {3679,30412}, {5119,6203}, {5902,30346}, {7741,30313}, {7951,30306}, {9819,30396}, {30276,30282}, {30288,30294}, {30319,30323}, {30360,30366}, {30368,30374}, {30380,30384}, {30406,30411}, {30418,30423}

X(30431) = reflection of X(18410) in X(30375)

X(30432) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND TRINH

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

The homothetic center of these triangles is X(6396). X(30432) is their endo-homothetic center only when ABC is acute.

X(30432) lies on these lines: {1,7}, {35,30386}, {36,30297}, {517,18411}, {3586,30325}, {3679,30413}, {5119,6204}, {5902,30347}, {7741,30314}, {7951,30307}, {9819,30397}, {30277,30282}, {30289,30294}, {30320,30323}, {30361,30366}, {30369,30374}, {30381,30384}, {30407,30411}, {30419,30423}

X(30432) = reflection of X(18411) in X(30376)

X(30433) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL AND TRINH

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

The homothetic center of these triangles is X(10645). X(30433) is their endo-homothetic center only when ABC is acute.

X(30433) lies on these lines: {1,7}, {14,80}, {35,10647}, {36,30300}, {517,18422}, {1653,5119}, {3586,30327}, {3679,30414}, {5902,30351}, {7741,30316}, {7951,30309}, {9819,10655}, {30280,30282}, {30292,30294}, {30321,30323}, {30364,30366}, {30372,30374}, {30382,30384}, {30409,30411}, {30421,30423}

X(30433) = reflection of X(18422) in X(30377)
X(30433) = {X(1), X(4312)}-harmonic conjugate of X(16038)

X(30434) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL AND TRINH

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

The homothetic center of these triangles is X(10646). X(30434) is their endo-homothetic center only when ABC is acute.

X(30434) lies on these lines: {1,7}, {13,80}, {35,10648}, {36,30301}, {517,18423}, {1652,5119}, {3586,30328}, {3679,30415}, {5902,30352}, {7741,30317}, {7951,30310}, {9819,10656}, {30281,30282}, {30293,30294}, {30322,30323}, {30365,30366}, {30373,30374}, {30383,30384}, {30410,30411}, {30422,30423}

X(30434) = reflection of X(18423) in X(30378)

X(30435) = ISOGONAL CONJUGATE X(18840)

Barycentrics    a^2 (3 a^2 + b^2 + c^2) : :
Barycentrics    S^2 - SB SC - 2 SB SW - 2 SC SW : :
Trilinears    2 sin A - cos A tan ω : :
Trilinears    cos A - 2 sin A cot ω : :
Trilinears    a - R cos A tan ω : :
X(30435) = X[7896]-2*X[7915]

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28746.

X(30435) lies on these lines: {2,7762}, {3,6}, {4,3172}, {5,7735}, {9,5266}, {20,1285}, {25,251}, {30,5286}, {31,218}, {41,16466}, {55,5280}, {56,609}, {69,7819}, {76,11286}, {81,11343}, {83,183}, {99,7894}, {101,1191}, {112,1593}, {115,3843}, {140,7736}, {141,14023}, {159,15257}, {169,1104}, {172,999}, {193,3933}, {194,1003}, {198,16470}, {217,19347}, {220,595}, {230,1656}, {232,3517}, {237,11402}, {248,3527}, {315,7792}, {316,7851}, {378,8778}, {381,3767}, {382,5254}, {384,7754}, {385,7770}, {393,6756}, {405,5276}, {441,11433}, {524,7795}, {550,7738}, {598,15031}, {599,7822}, {754,7784}, {940,16783}, {942,16780}, {966,17698}, {980,21509}, {988,16667}, {995,3207}, {1078,7878}, {1181,8779}, {1184,3291}, {1186,3511}, {1194,9909}, {1249,7487}, {1385,9575}, {1472,1496}, {1482,1572}, {1506,5070}, {1595,3087}, {1597,1968}, {1598,2207}, {1617,4548}, {1627,7484}, {1657,2549}, {1724,19761}, {1743,3965}, {1914,3295}, {1915,8780}, {1971,14530}, {1975,3972}, {1992,3926}, {1995,5354}, {2070,16308}, {2138,17409}, {2229,16396}, {2241,6767}, {2242,7373}, {2300,20818}, {2493,7506}, {2896,7875}, {3051,3167}, {3052,3730}, {3148,9777}, {3224,3499}, {3303,16785}, {3304,16784}, {3314,10583}, {3329,7793}, {3407,12206}, {3509,16787}, {3523,14930}, {3526,3815}, {3528,14482}, {3534,7739}, {3552,7839}, {3567,9475}, {3579,9593}, {3589,7800}, {3618,3785}, {3629,7758}, {3734,7805}, {3744,17742}, {3749,3991}, {3763,7854}, {3788,7838}, {3830,5309}, {3849,7872}, {3851,5475}, {3934,8667}, {4383,5337}, {4386,9709}, {4426,9708}, {5025,20088}, {5032,11165}, {5054,9300}, {5055,7746}, {5073,5355}, {5077,7802}, {5275,11108}, {5277,16408}, {5283,16418}, {5523,12173}, {5710,16788}, {5938,12167}, {6090,9463}, {6144,7820}, {6392,14033}, {6655,7920}, {6656,16989}, {6660,20977}, {6680,7759}, {6792,15000}, {7375,8974}, {7376,13950}, {7388,13763}, {7389,13644}, {7574,16306}, {7581,21736}, {7585,11292}, {7586,11291}, {7750,7803}, {7751,7804}, {7756,15681}, {7761,7829}, {7763,11288}, {7765,17800}, {7768,7846}, {7769,11163}, {7773,7812}, {7774,7807}, {7775,7886}, {7779,7881}, {7780,7808}, {7785,7806}, {7788,7832}, {7797,7823}, {7798,7816}, {7801,7890}, {7809,7942}, {7811,7859}, {7817,7825}, {7818,7852}, {7835,7905}, {7836,7837}, {7840,7945}, {7842,7902}, {7843,7844}, {7845,7867}, {7848,7914}, {7850,7944}, {7857,7858}, {7860,7919}, {7864,14712}, {7869,7882}, {7873,7913}, {7874,7903}, {7880,7916}, {7883,7943}, {7884,7911}, {7885,7932}, {7891,13571}, {7896,7915}, {7897,14043}, {7898,7923}, {7899,7926}, {7900,7901}, {7909,7949}, {7912,16984}, {7917,7930}, {7928,9939}, {7929,7948}, {7931,7946}, {8364,14929}, {8550,8721}, {8744,10594}, {8879,15809}, {8882,19173}, {9310,16483}, {9327,16486}, {9490,18899}, {9592,13624}, {9607,15696}, {9609,13564}, {9620,12702}, {9715,22240}, {9969,20993}, {10306,10315}, {10313,11414}, {10314,11484}, {11313,13758}, {11314,13638}, {11321,16998}, {11335,20023}, {11610,11641}, {11648,15684}, {12164,23128}, {12174,13509}, {12188,12829}, {12203,14532}, {12308,14901}, {13735,27523}, {14003,26869}, {14269,14537}, {14581,18535}, {14602,20854}, {14974,21793}, {14996,21516}, {14997,21540}, {15270,19153}, {15589,16045}, {15720,21843}, {16042,21448}, {16060,17379}, {16061,17349}, {16394,26035}, {16589,16857}, {16918,16995}, {17001,17541}, {17002,17686}, {17597,17736}, {18494,27376}, {19118,27369}, {19125,20960}, {19767,21982}, {20897,26864}

X(30435) = isogonal conjugate of X(18840)
X(30435) = midpoint of X(8396) and X(8416)
X(30435) = reflection of X(i) in X(j) for these {i,j}: {7784,7834}, {7896,7915}
X(30435) = crossdifference of every pair of points on line X(523)X(2525)
X(30435) = intersection of tangents at PU(1) to hyperbola {{X(6),PU(1),PU(2)}}
X(30435) = inverse-in-1st-Brocard-circle of X(9605)
X(30435) = inverse-in-circle-{X(371), X(372),PU(1),PU(39)} of X(1350)
X(30435) = radical center of Lucas(-4 cot ω) circles
X(30435) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7762,7776}, {2,7893,7879}, {3,6,9605}, {3,32,1384}, {3,5093,3095}, {3,9605,5024}, {3,21309,32}, {4,5304,5305}, {6,32,3}, {6,574,22246}, {6,800,15851}, {6,1333,5120}, {6,1384,5024}, {6,2220,4254}, {6,3053,39}, {6,4252,4253}, {6,4258,386}, {6,5008,21309}, {6,5013,7772}, {6,5052,5093}, {6,6423,3312}, {6,6424,3311}, {6,12963,6422}, {6,12968,6421}, {6,13345,8573}, {6,21309,1384}, {6,22331,5013}, {32,39,3053}, {32,187,22331}, {32,5007,6}, {32,5041,5023}, {32,7772,187}, {32,13356,2080}, {32,14075,7772}, {39,187,15515}, {39,3053,3}, {39,5206,15815}, {39,15515,5013}, {41,21764,16466}, {61,62,11477}, {83,6179,183}, {172,5332,16502}, {172,16502,999}, {187,5013,3}, {187,7772,5013}, {187,14075,6}, {193,14001,3933}, {230,2548,1656}, {251,5359,25}, {315,7792,7866}, {316,7856,7851}, {371,372,1350}, {384,7766,7754}, {385,7787,7770}, {574,5023,3}, {609,5299,56}, {1078,7878,11174}, {1384,5024,15655}, {1384,9605,3}, {1692,2031,2080}, {1975,7760,22253}, {2207,10311,1598}, {2242,16781,7373}, {3053,15815,5206}, {3329,7793,11285}, {3618,3785,8362}, {3629,7789,7758}, {3767,7745,381}, {3788,7838,9766}, {3793,8362,3785}, {3972,7760,1975}, {4264,5037,6}, {4383,5337,21526}, {5007,5008,32}, {5007,21309,9605}, {5013,22331,187}, {5032,19661,11165}, {5052,13357,3095}, {5085,5188,3}, {5093,13357,9605}, {5206,15815,3}, {5254,7737,382}, {5280,7031,55}, {5305,18907,4}, {5306,7745,3767}, {5319,7737,5254}, {5368,7747,5309}, {5475,7755,13881}, {5475,13881,3851}, {6421,12968,6398}, {6422,12963,6221}, {6680,7759,7778}, {7750,7803,11287}, {7768,7846,7868}, {7772,15515,39}, {7772,22331,3}, {7773,7828,11318}, {7779,7892,7881}, {7780,7808,15271}, {7785,7806,7887}, {7797,7823,7841}, {7812,7828,7773}, {7822,7826,599}, {7832,7877,7788}, {7854,7889,3763}, {8573,10317,1384}, {8743,10312,25}, {12150,14614,11286}, {16989,20065,6656}

X(30436) = X(5)X(113)∩X(10)X(27555)

Barycentrics (b+c) (-2 a^4 b^2-3 a^3 b^3+a^2 b^4+3 a b^5+b^6-a^3 b^2 c-2 a^2 b^3 c+a b^4 c+2 b^5 c-2 a^4 c^2-a^3 b c^2-2 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2-3 a^3 c^3-2 a^2 b c^3-4 a b^2 c^3-4 b^3 c^3+a^2 c^4+a b c^4-b^2 c^4+3 a c^5+2 b c^5+c^6) : :
X(30436) = 5 X[1698] - X[2940]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28747.

X(30436) lies on these lines: {5,113}, {10,27555}, {12,8287}, {115,24443}, {429,1861}, {442,22798}, {542,3615}, {857,29610}, {1495,9958}, {1698,2940}, {1737,14873}, {2392,3142}, {2899,27704}, {3013,8614}, {3136,25972}, {5044,22076}, {5221,8818}, {5587,27685}, {6723,24904}, {6739,18357}, {7173,8286}, {9780,27554}, {10175,27687}

X(30436) = X(26734)-complementary conjugate of X(3741)

X(30437) = X(7)X(2808)∩X(674)X(7671)

Barycentrics a^2*((b^2+c^2)*a^4-2*(b^3+c^3)*a^3-b^2*c^2*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4+c^4+(2*b^2+b*c+2*c^2)*b*c)*(b-c)^2) : :
X(30437) = X(7)-4*X(29957)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30437) lies on these lines: {7, 2808}, {674, 7671}, {942, 15058}, {2772, 5902}, {2836, 11188}, {5889, 10399}, {8236, 9052}, {10122, 11444}

X(30438) = X(8)X(29958)∩X(392)X(23155)

Barycentrics a^2*((b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b^4-b^2*c^2+c^4)*a-(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)) : :
X(30438) = X(8)-4*X(29958), 5*X(3616)-2*X(23154), 3*X(5640)-2*X(5902), 3*X(5640)-4*X(15049), 2*X(5693)+X(5889), 4*X(5694)-X(11412), 4*X(5883)-5*X(11451), 4*X(5884)-7*X(15043), 8*X(5885)-11*X(15024), 3*X(7998)-4*X(10176), 5*X(10574)-2*X(15071), 5*X(11444)-8*X(20117), 10*X(15016)-13*X(15028)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30438) lies on these lines: {8, 29958}, {392, 23155}, {511, 7985}, {513, 17579}, {758, 3060}, {1464, 19245}, {2392, 2979}, {2771, 5890}, {2779, 15305}, {2810, 3241}, {2836, 11188}, {2842, 5640}, {3616, 23154}, {3877, 8679}, {4511, 26892}, {5693, 5889}, {5694, 11412}, {5752, 11684}, {5883, 11451}, {5884, 15043}, {5885, 15024}, {6126, 10546}, {7998, 10176}, {10574, 15071}, {11346, 24482}, {11444, 20117}, {15016, 15028}

X(30438) = reflection of X(i) in X(j) for these (i,j): (2979, 5692), (5902, 15049), (23155, 392)
X(30438) = {X(5902), X(15049)}-harmonic conjugate of X(5640)

X(30439) = REFLECTION OF X(13) IN X(11624)

Barycentrics (SB+SC)*(2*S^2+3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : :
X(30439) = 3*X(5640)-X(16259)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30439) lies on these lines: {3, 6}, {4, 11581}, {13, 5663}, {14, 5640}, {17, 11459}, {18, 13363}, {531, 25165}, {1154, 16962}, {3411, 12006}, {3412, 5889}, {5946, 11626}, {6104, 14170}, {6780, 16637}, {7998, 16241}, {8929, 15441}, {10654, 11002}, {11455, 12816}, {13754, 16267}, {15045, 16963}, {15072, 16965}, {16261, 16808}

X(30439) = reflection of X(13) in X(11624)

X(30440) = REFLECTION OF X(13) IN X(11626)

Barycentrics (SB+SC)*(2*S^2-3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : :
X(30440) = 3*X(5640)-X(16260)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30440) lies on these lines: {3, 6}, {4, 11582}, {13, 5640}, {14, 5663}, {17, 13363}, {18, 11459}, {530, 25155}, {1154, 16963}, {3411, 5889}, {3412, 12006}, {5946, 11624}, {6105, 14169}, {6779, 16636}, {7998, 16242}, {8930, 15442}, {10653, 11002}, {11455, 12817}, {13754, 16268}, {15045, 16962}, {15072, 16964}, {16261, 16809}

X(30440) = reflection of X(13) in X(11626)

X(30441) = BARYCENTRIC PRODUCT X(99)*X(15318)

Barycentrics (SA-SB)*(SA-SC)*(SB^2-4*R^2*SB+SC*SA)*(SC^2-4*R^2*SC+SA*SB) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28750.

X(30441) lies on this line: {2404, 14570}

X(30441) = isogonal conjugate of X(30442)
X(30441) = barycentric product X(99)*X(15318)
X(30441) = barycentric quotient X(i)/X(j) for these (i,j): (99, 20477), (110, 6759)
X(30441) = trilinear product X(i)*X(j) for these {i,j}: {662, 15318}, {811, 18890}
X(30441) = trilinear quotient X(i)/X(j) for these (i,j): (662, 6759), (799, 20477)

X(30442) = X(6)X(2430)∩X(421)X(2501)

Barycentrics (SB^2-SC^2)*(SA^2-4*R^2*SA+SB*SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28750.

X(30442) lies on these lines: {6, 2430}, {421, 2501}, {647, 657}, {2485, 17434}, {3265, 10601}, {3288, 7927}, {6753, 14398}, {11792, 15140}, {15609, 23438}

X(30442) = isogonal conjugate of X(30441)

X(30443) = X(30)X(21651)∩X(51)X(5895)

Barycentrics a^2 ( a^12 (b^2+c^2)-4 a^10 (b^2-c^2)^2+5 a^8 (b^2-c^2)^2 (b^2+c^2) -40 a^6 b^2 c^2 (b^2-c^2)^2-5 a^4 (b^2-c^2)^2 (b^6-9 b^4 c^2-9 b^2 c^4+c^6) +4 a^2 (b^2-c^2)^2 (b^8-2 b^6 c^2-14 b^4 c^4-2 b^2 c^6+c^8) -(b^2-c^2)^4 (b^6+7 b^4 c^2+7 b^2 c^4+c^6)) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28754.

X(30443) lies on these lines: {30,21651}, {51,5895}, {185,1885}, {373,5893}, {1498,6090}, {1906,6247}, {2777,21649}, {3357,15030}, {3917,5894}, {4319,7355}, {4320,6285}, {5650,8567}, {5878,6816}, {6225,7386}, {10575,12362}, {10990,15105}, {12163,13093}, {12315,14641}, {13417,16879}, {14642,17807}

X(30443) = the reflection of X(i) in X(j), for these {i, j}: {5562,20427}, {11381,64}, {12315,14641}

X(30444) = EULER LINE INTERCEPT OF X(12)X(3931)

Barycentrics (b+c) (a^5 b+a^4 b^2-a b^5-b^6+a^5 c+2 a^3 b^2 c-3 a b^4 c+a^4 c^2+2 a^3 b c^2+4 a^2 b^2 c^2+4 a b^3 c^2+b^4 c^2+4 a b^2 c^3-3 a b c^4+b^2 c^4-a c^5-c^6) : :
X(30444) = 5 X[1698] - X[2941]

As a point on the Euler line, X(30444) has Shinagawa coefficients ((E+F)$a$+abc, (E+F)$a$+3abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30444) lies on these lines: {2, 3}, {12, 3931}, {115, 119}, {120, 5512}, {127, 25640}, {321, 17757}, {339, 21664}, {355, 1834}, {496, 5716}, {517, 1211}, {946, 3454}, {952, 17015}, {1213, 1766}, {1245, 21935}, {1329, 5955}, {1698, 2941}, {2345, 3820}, {3936, 5603}, {5016, 24390}, {5019, 5475}, {5706, 5810}, {5752, 5799}, {5886, 17056}, {5887, 10974}, {7682, 17052}, {7951, 17594}, {10175, 12618}, {10381, 24474}, {14672, 20621}

X(30444) = complement of X(4221)
X(30444) = midpoint of X(4) and X(4220)
X(30444) = {X(429), X(442)}-harmonic conjugate of X(21530)
X(30444) = X(i)-complementary conjugate of X(j) for these (i,j): {3420, 1125}, {9107, 8062}
X(30444) = X(9058)-Ceva conjugate of X(523)

X(30445) = EULER LINE INTERCEPT OF X(12)X(18588)

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

As a point on the Euler line, X(30445) has Shinagawa coefficients ((E+2F)S²+(E+F)$a$abc, (-E+2F)S²-(E+F)$a$abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30445) lies on these lines: {2, 3}, {12, 18588}, {115, 18591}, {119, 127}, {120, 14672}, {339, 1234}, {1060, 17720}, {3822, 18589}, {4463, 17757}, {7951, 10319}, {10202, 18635}

X(30445) = complement of X(4227)
X(30445) = X(i)-complementary conjugate of X(j) for these (i,j): {998, 942}, {9058, 8062}

X(30446) = EULER LINE INTERCEPT OF X(12)X(3743)

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

As a point on the Euler line, X(30446) has Shinagawa coefficients ((E+2F)$a$+abc, 2(E+F)$a$+5abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30446) lies on these lines: {2, 3}, {12, 3743}, {80, 1834}, {119, 137}, {1089, 3704}, {1211, 3878}, {1213, 16548}, {3454, 11813}, {5443, 17056}, {20625, 25640}

X(30446) = X(26711)-Ceva conjugate of X(523)
X(30446) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 1904, 11113}

X(30447) = EULER LINE INTERCEPT OF X(12)X(502)

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

As a point on the Euler line, X(30447) has Shinagawa coefficients ((E-2F)$a$+abc, -2(E+F)$a$-3abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30447) lies on the cubic K720 and these lines: {2, 3}, {12, 502}, {115, 3290}, {119, 3258}, {120, 5099}, {125, 517}, {515, 6739}, {523, 1577}, {758, 10693}, {1211, 10176}, {1213, 16547}, {1290, 5080}, {1737, 8287}, {3454, 25079}, {3814, 5520}, {4872, 23674}, {5074, 21253}, {8286, 30384}, {9956, 30436}, {16177, 25640}

X(30447) = complement of X(1325)
X(30447) = midpoint of X(1290) and X(5080)
X(30447) = reflection of X(5520) and X(3814)
X(30447) = circumcircle-inverse of X(2915)
X(30447) = nine point circle inverse of X(442)
X(30447) = polar circle inverse of X(28)
X(30447) = orthoptic circle of the Steiner inellipe inverse of X(4220)
X(30447) = complement of the isogonal of X(10693)
X(30447) = X(i)-complementary conjugate of X(j) for these (i,j): {2766, 8062}, {10693, 10}
X(30447) = X(i)-Ceva conjugate of X(j) for these (i,j): {1290, 523}, {5080, 758}
X(30447) = crosssum of X(184) and X(19622)
X(30447) = crossdifference of every pair of points on line {647, 1333}
X(30447) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16430, 7483}, {5, 25648, 4187}, {5, 27687, 442}, {429, 21530, 442}, {1113, 1114, 2915}, {1312, 1313, 442}, {5142, 24933, 5}, {27553, 27685, 3}, {27554, 27686, 4}, {27555, 27687, 5}, {27561, 27693, 22}, {27562, 27694, 23}, {27578, 27715, 20}, {27579, 27716, 21}, {27580, 27717, 25}, {27581, 27718, 27}, {27582, 27721, 376}, {27583, 27722, 377}, {27584, 27723, 384}

X(30448) = EULER LINE INTERCEPT OF X(12)X(986)

Barycentrics a^5*b^2 - a^3*b^4 + a^2*b^5 - b^7 + 2*a^5*b*c - a^3*b^3*c - a*b^5*c + a^5*c^2 + 2*a^3*b^2*c^2 + a^2*b^3*c^2 + 2*b^5*c^2 - a^3*b*c^3 + a^2*b^2*c^3 + 2*a*b^3*c^3 - b^4*c^3 - a^3*c^4 - b^3*c^4 + a^2*c^5 - a*b*c^5 + 2*b^2*c^5 - c^7 : :
X(30448) = 5 X[1698] - X[21375]

As a point on the Euler line, X(30448) has Shinagawa coefficients ((E+F)$a$-$aS_A$-abc, (E+F)$a$-$aS_A$+3abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28760.

X(30448) lies on these lines: {2, 3}, {12, 986}, {119, 5518}, {517, 2887}, {970, 3454}, {1698, 21375}, {3821, 3822}, {4417, 9567}, {5254, 22380}, {6211, 26446}, {7680, 29243}, {7951, 17596}, {13323, 20083}

X(30448) = {X(6881), X(30444)}-harmonic conjugate of X(5)

X(30449) = EULER LINE INTERCEPT OF X(12)X(4424)

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

As a point on the Euler line, X(30449) has Shinagawa coefficients ((E+F)$a$-$aS_A$, (E+F)$a$-$aS_A$+4abc).

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30449) lies on these lines: {2, 3}, {12, 4424}, {115, 22425}, {517, 3454}, {952, 1834}, {1211, 5690}, {1482, 3936}, {3073, 20575}, {3814, 24850}, {5510, 9955}, {5901, 17056}, {7680, 12621}, {7951, 24851}, {10974, 14988}, {12610, 21245}

X(30449) = X(15617)-complementary conjugate of X(1125)
X(30449) = {X(5), X(5499)}-harmonic conjugate of X(15973)

X(30450) = X(4)X(6754)∩X(107)X(925)

Barycentrics    a*b^3*c^3*(a^2-b^2)*(a^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*(a^4-2*b^2*a^2+(b^2-c^2)^2)*(a^4-2*c^2*a^2+(b^2-c^2)^2) : :
Barycentrics    (tan A)/(sec 2B - sec 2C) : :

Barycentrics    (sec A)/(b cos(A - B) - c cos(A - C)) : :

See César Lozada, Hyacinthos 28756.

X(30450) lies on these lines: {4, 6754}, {107, 925}, {687, 4558}, {847, 17983}, {2165, 16081}, {5392, 15466}, {6330, 20563}, {6528, 16813}, {15352, 16237}, {18817, 18883}

X(30450) = isogonal conjugate of X(30451)
X(30450) = polar conjugate of X(924)
X(30450) = trilinear pole of line X(4)X(52)

X(30451) = X(6)X(2501)∩X(184)X(669)

Barycentrics a^4*(-a^2+b^2+c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*(b^2-c^2) : :
X(30451) = 2*X(6753)-3*X(14397)

See César Lozada, Hyacinthos 28756.

X(30451) lies on these lines: {6, 2501}, {184, 669}, {520, 647}, {523, 2623}, {526, 16040}, {826, 3288}, {924, 6753}, {1181, 1499}, {1409, 7180}, {1899, 23301}, {1993, 6563}, {2451, 12077}, {5926, 19357}, {9009, 19459}, {10601, 14341}, {11422, 11450}, {13366, 21646}

X(30451) = isogonal conjugate of X(30450)
X(30451) = isotomic conjugate of polar conjugate of X(34952)
X(30451) = crossdifference of every pair of points on line X(4)X(52)

X(30452) = BARYCENTRIC PRODUCT X(13)*X(115)

Barycentrics (b - c)^2*(b + c)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S))::

X(30452) lies on the Simmons inconic with foci X(13) and X(15) and on these lines: {13, 531}, {298, 11118}, {300, 18896}, {1989, 3457}, {2381, 22510}, {3124, 30453}, {6531, 8737}, {8014, 18777}

X(30452) = X(13)-Ceva conjugate of X(20578)
X(30452) = crosspoint of X(13) and X(20578)
X(30452) = crosssum of X(15) and X(17402)
X(30452) = crossdifference of every pair of points on line {10411, 17402}
X(30452) = X(i)-isoconjugate of X(j) for these (i,j): {15, 24041}, {298, 1101}, {662, 17402}, {2151, 4590}
X(30452) = barycentric product X(i) X(j) for these {i,j}: {13, 115}, {125, 8737}, {300, 3124}, {338, 3457}, {523, 20578}, {1109, 2153}, {5995, 23105}, {6138, 10412}, {8029, 23895}, {15475, 23871}, {20579, 23283}
X(30452) = barycentric quotient X(i) / X(j) for these {i,j}: {13, 4590}, {115, 298}, {512, 17402}, {2153, 24041}, {2971, 8739}, {3124, 15}, {3457, 249}, {6138, 10411}, {8029, 23870}, {8737, 18020}, {8754, 470}, {15475, 23896}, {20578, 99}, {22260, 6137}

X(30453) = BARYCENTRIC PRODUCT X(14)*X(115)

Barycentrics (b - c)^2*(b + c)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*S*(Sqrt[3]*a^2 - S))::

X(30453) lies on the Simmons inconic with foci X(14) and X(16) and on these lines: {14, 530}, {299, 11117}, {301, 18896}, {1989, 3458}, {2380, 22511}, {3124, 30452}, {6531, 8738}, {8015, 18776}

X(30453) = X(14)-Ceva conjugate of X(20579)
X(30453) = crosspoint of X(14) and X(20579)
X(30453) = crosssum of X(16) and X(17403)
X(30453) = crossdifference of every pair of points on line {10411, 17403}
X(30453) = X(i)-isoconjugate of X(j) for these (i,j): {16, 24041}, {299, 1101}, {662, 17403}, {2152, 4590}
X(30453) = barycentric product X(i) X(j) for these {i,j}: {14, 115}, {125, 8738}, {301, 3124}, {338, 3458}, {523, 20579}, {1109, 2154}, {5994, 23105}, {6137, 10412}, {8029, 23896}, {15475, 23870}, {20578, 23284}
X(30453) = barycentric quotient X(i) / X(j) for these {i,j}: {14, 4590}, {115, 299}, {512, 17403}, {2154, 24041}, {2971, 8740}, {3124, 16}, {3458, 249}, {6137, 10411}, {8029, 23871}, {8738, 18020}, {8754, 471}, {15475, 23895}, {20579, 99}, {22260, 6138}

X(30454) = BARYCENTRIC PRODUCT X(13)*X(2482)

Barycentrics (2*a^2 - b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S))::
X(30454) = X[16256] - 3 X[16962]

X(30454) lies on the Simmons inconic with foci X(13) and X(15) and on these lines: {13, 531}, {396, 18777}, {690, 9117}, {3180, 21466}, {16256, 16962}

X(30454) = barycentric product X(i) X(j) for these {i,j}: {13, 2482}, {1649, 23895}, {2153, 24038}, {9205, 14559}
X(30454) = barycentric quotient X(i) / X(j) for these {i,j}: {1649, 23870}, {2482, 298}, {3457, 10630}, {5095, 470}

X(30455) = BARYCENTRIC PRODUCT X(14)*X(2482)

Barycentrics (2*a^2 - b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*S*(Sqrt[3]*a^2 - S))::
X(30455) = X[16255] - 3 X[16963]

X(30455) lies on the Simmons inconic with foci X(14) and X(16) and on these lines: on lines {14, 530}, {395, 18776}, {690, 9115}, {3181, 21467}, {16255, 16963}

X(30455) = barycentric product X(i) X(j) for these {i,j}: {14, 2482}, {1649, 23896}, {2154, 24038}, {9204, 14559}
X(30455) = barycentric quotient X(i) / X(j) for these {i,j}: {1649, 23871}, {2482, 299}, {3458, 10630}, {5095, 471}

X(30456) = X(9)X(223)∩X(37)X(73)

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

X(30456) lies on the cubic K1090 and these lines: {1, 5776}, {6, 19}, {9, 223}, {37, 73}, {48, 1455}, {71, 227}, {154, 204}, {198, 1035}, {213, 8898}, {219, 21147}, {225, 1901}, {241, 27623}, {278, 5746}, {573, 15498}, {579, 1465}, {581, 12664}, {604, 1104}, {610, 1394}, {651, 5279}, {828, 18591}, {910, 2199}, {965, 1038}, {966, 10361}, {1060, 5778}, {1108, 1457}, {1400, 1427}, {1402, 7083}, {1441, 5749}, {1451, 5115}, {1765, 17102}, {1766, 7078}, {1838, 5798}, {2269, 15852}, {2287, 4296}, {3209, 3556}, {3694, 4551}, {3931, 12705}, {3990, 4559}, {4343, 14100}, {5227, 9370}, {5317, 26888}, {5712, 5928}, {5930, 8804}, {14110, 22134}, {18623, 18750}

X(30456) = X(i)-Ceva conjugate of X(j) for these (i,j): {9, 1400}, {223, 73}, {1214, 65}, {5930, 3198}, {18623, 5930}
X(30456) = X(i)-isoconjugate of X(j) for these (i,j): {6, 5931}, {21, 2184}, {29, 1073}, {64, 333}, {253, 284}, {283, 459}, {314, 2155}, {1172, 19611}, {1301, 6332}, {2287, 8809}, {6514, 6526}, {8748, 15394}
X(30456) = crosspoint of X(i) and X(j) for these (i,j): {9, 27382}, {610, 1249}, {1394, 18623}
X(30456) = crosssum of X(i) and X(j) for these (i,j): {1, 5776}, {1073, 2184}
X(30456) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1841, 2262}, {37, 3330, 1903}, {1409, 1880, 65}, {1456, 2264, 608}
X(30456) = barycentric product X(i) X(j) for these {i,j}: {1, 5930}, {7, 3198}, {10, 1394}, {20, 65}, {37, 18623}, {57, 8804}, {73, 1895}, {108, 8057}, {109, 17898}, {154, 1441}, {204, 307}, {226, 610}, {306, 3213}, {651, 6587}, {934, 14308}, {1020, 14331}, {1214, 1249}, {1231, 3172}, {1400, 18750}, {1402, 14615}, {1409, 15466}, {1427, 27382}, {3344, 8807}, {3668, 7070}, {4551, 21172}, {14249, 22341}
X(30456) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 5931}, {20, 314}, {65, 253}, {73, 19611}, {154, 21}, {204, 29}, {610, 333}, {1042, 8809}, {1394, 86}, {1400, 2184}, {1402, 64}, {1409, 1073}, {1880, 459}, {3172, 1172}, {3198, 8}, {3213, 27}, {5930, 75}, {6525, 1896}, {6587, 4391}, {7070, 1043}, {7156, 2322}, {8804, 312}, {14308, 4397}, {15905, 1812}, {18623, 274}, {18750, 28660}, {21172, 18155}, {22341, 15394}

X(30457) = X(9)X(223)∩X(64)X(71)

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

The trilinear polar of X(30457) passes through X(4105).

Let A'B'C' and A"B"C" be the Hutson intouch and anti-Hutson intouch triangles, resp. Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(30457). (Randy Hutson, January 15, 2019)

X(30457) lies on the cubic K1090 and these lines: {6, 7367}, {9, 223}, {37, 2331}, {64, 71}, {253, 6559}, {345, 5931}, {393, 5514}, {480, 2318}, {579, 2338}, {728, 2324}, {2192, 7037}, {2911, 14642}, {5776, 20226}

X(30457) = isogonal conjugate of X(18623)
X(30457) = X(2184)-Ceva conjugate of X(64)
X(30457) = X(i)-cross conjugate of X(j) for these (i,j): {607, 55}, {1400, 9}
X(30457) = crosspoint of X(281) and X(8805)
X(30457) = crosssum of X(610) and X(1394)
X(30457) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18623}, {2, 1394}, {7, 610}, {20, 57}, {56, 18750}, {69, 3213}, {77, 1249}, {81, 5930}, {85, 154}, {204, 348}, {222, 1895}, {269, 27382}, {273, 15905}, {279, 7070}, {603, 15466}, {604, 14615}, {651, 21172}, {934, 14331}, {1014, 8804}, {1414, 6587}, {1434, 3198}, {2184, 7338}, {3172, 7182}, {4565, 17898}, {4637, 14308}, {6525, 7183}, {7056, 7156}, {7125, 14249}
X(30457) = barycentric product X(i) X(j) for these {i,j}: {8, 64}, {9, 2184}, {33, 19611}, {42, 5931}, {55, 253}, {200, 8809}, {219, 459}, {281, 1073}, {312, 2155}, {318, 19614}, {1259, 6526}, {1857, 15394}, {3343, 8805}, {7017, 14642}, {7068, 15384}
X(30457) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 18623}, {8, 14615}, {9, 18750}, {31, 1394}, {33, 1895}, {41, 610}, {42, 5930}, {55, 20}, {64, 7}, {154, 7338}, {220, 27382}, {253, 6063}, {281, 15466}, {459, 331}, {607, 1249}, {657, 14331}, {663, 21172}, {1073, 348}, {1253, 7070}, {1334, 8804}, {1857, 14249}, {1973, 3213}, {2155, 57}, {2175, 154}, {2184, 85}, {2212, 204}, {3709, 6587}, {4041, 17898}, {4524, 14308}, {5931, 310}, {6059, 6525}, {7070, 1097}, {8809, 1088}, {14379, 1804}, {14642, 222}, {15394, 7055}, {19611, 7182}, {19614, 77}

X(30458) = (name pending)

Barycentrics 35150 a^16-224435 a^14 b^2+621425 a^12 b^4-960475 a^10 b^6+875725 a^8 b^8-439825 a^6 b^10+77435 a^4 b^12+25375 a^2 b^14-10375 b^16-224435 a^14 c^2+868714 a^12 b^2 c^2-1207083 a^10 b^4 c^2+532540 a^8 b^6 c^2+253363 a^6 b^8 c^2-202074 a^4 b^10 c^2-91925 a^2 b^12 c^2+70900 b^14 c^2+621425 a^12 c^4-1207083 a^10 b^2 c^4+516660 a^8 b^4 c^4+87335 a^6 b^6 c^4+76038 a^4 b^8 c^4+123525 a^2 b^10 c^4-217900 b^12 c^4-960475 a^10 c^6+532540 a^8 b^2 c^6+87335 a^6 b^4 c^6+97202 a^4 b^6 c^6-56975 a^2 b^8 c^6+399500 b^10 c^6+875725 a^8 c^8+253363 a^6 b^2 c^8+76038 a^4 b^4 c^8-56975 a^2 b^6 c^8-484250 b^8 c^8-439825 a^6 c^10-202074 a^4 b^2 c^10+123525 a^2 b^4 c^10+399500 b^6 c^10+77435 a^4 c^12-91925 a^2 b^2 c^12-217900 b^4 c^12+25375 a^2 c^14+70900 b^2 c^14-10375 c^16 : :
Barycentrics 99960 S^4+S^2 (42201 R^4-158760 SB SC-38772 R^2 SW+860 SW^2)+SB SC (-2187 R^4-18468 R^2 SW+23340 SW^2) : :

As a point on the Euler line, X(30458) has Shinagawa coefficients {42201 R^4 + 99960 S^2 - 38772 R^2 SW + 860 SW^2, -3 (729 R^4 + 52920 S^2 + 6156 R^2 SW - 7780 SW^2).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28759.

X(30458) lies on this line: {2,3}

X(30459) = REFLECTION OF X(30452) IN X(396)

Barycentrics 3*(12*R^2-5*SA-2*SW)*S^2+sqrt(3)*(14*S^2+3*SA^2-18*SB*SC-3*SW^2)*S+9*SB*SC*SW : :

Centers X(30459)-X(30479) were contributed by César Eliud Lozada (December 26, 2018).

X(30459) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {2,14}, {13,1338}, {30,30460}, {395,15778}, {396,30452}, {523,30461}, {533,14921}, {9117,13305}, {16529,22738}

X(30459) = reflection of X(30452) in X(396)
X(30459) = antipode of X(30452) in the Simmons inconic with foci {X(13), X(15)}

X(30460) = REFLECTION OF X(30452) IN THE LINE X(13)X(15)

Barycentrics (SB-SC)^2*(S^2+sqrt(3)*(9*R^2-2*SW)*S-3*(3*R^2-SA)*SA) : :

X(30460) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {13,476}, {30,30459}, {115,12077}, {323,532}, {523,30452}, {3258,15610}, {11537,30454}, {12079,20578}

X(30460) = antipode of X(30461) in the Simmons inconic with foci {X(13), X(15)}

X(30461) = REFLECTION OF X(30460) IN X(396)

Barycentrics
-(2*(4*a^12-(8*(b^2+c^2))*a^10+(8*((b^2+c^2)^2-b^2*c^2))*a^8-(8*(b^6+c^6))*a^6+(2*(b^8+c^8+6*(b^2-c^2)^2*b^2*c^2))*a^4-(4*(b^2*c^2-(b^2-c^2)^2))*(b^4-c^4)*(b^2-c^2)*a^2-(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^4))*S+sqrt(3)*(4*a^14-(12*(b^2+c^2))*a^12+(8*(b^4+5*b^2*c^2+c^4))*a^10-(8*(4*b^2*c^2-(b^2-c^2)^2))*(b^2+c^2)*a^8-(12*(-b^2*c^6-b^6*c^2+b^8+c^8-3*b^4*c^4))*a^6+(4*(b^2+c^2))*((b^4-c^4)^2-b^4*c^4)*a^4-(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3*b^2*c^2) : :

X(30461) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {15,30467}, {30,30452}, {395,30465}, {523,30459}

X(30461) = antipode of X(30460) in the Simmons inconic with foci {X(13), X(15)}

X(30462) = REFLECTION OF X(30453) IN X(395)

Barycentrics 3*(12*R^2-5*SA-2*SW)*S^2-sqrt(3)*(14*S^2+3*SA^2-18*SB*SC-3*SW^2)*S+9*SB*SC*SW : :

X(30462) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {2,13}, {14,1337}, {30,30463}, {395,30453}, {523,30464}, {532,14922}, {9115,13304}, {16530,22739}

X(30462) = reflection of X(30453) in X(395)
X(30462) = antipode of X(30453) in the Simmons inconic with foci {X(14), X(16)}

X(30463) = REFLECTION OF X(30453) IN THE LINE X(14)X(16)

Barycentrics (SB-SC)^2*(S^2-sqrt(3)*(9*R^2-2*SW)*S-3*(3*R^2-SA)*SA) : :

X(30463) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {14,476}, {115,12077}, {323,533}, {523,30453}, {3258,15609}, {11549,30455}, {12079,20579}

X(30463) = antipode of X(30464) in the Simmons inconic with foci {X(14), X(16)}
X(30463) = reflection of X(30464) in X(395)

X(30464) = REFLECTION OF X(30463) IN X(395)

Barycentrics
2*(4*a^12-8*(b^2+c^2)*a^10+8*((b^2+c^2)^2-b^2*c^2)*a^8-8*(b^6+c^6)*a^6+2*(b^8+c^8+6*(b^2-c^2)^2*b^2*c^2)*a^4-4*(b^2*c^2-(b^2-c^2)^2)*(b^4-c^4)*(b^2-c^2)*a^2-(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^4)*S+sqrt(3)*(4*a^14-12*(b^2+c^2)*a^12+8*(b^4+5*b^2*c^2+c^4)*a^10-8*(4*b^2*c^2-(b^2-c^2)^2)*(b^2+c^2)*a^8-12*(-b^2*c^6-b^6*c^2+b^8+c^8-3*b^4*c^4)*a^6+4*(b^2+c^2)*((b^4-c^4)^2-b^4*c^4)*a^4-(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3*b^2*c^2) : :

X(30464) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {16,30470}, {30,30453}, {395,30463}, {396,30468}, {523,30462}

X(30464) = antipode of X(30463) in the Simmons inconic with foci {X(14), X(16)}

X(30465) = REFLECTION OF X(30454) IN X(396)

Barycentrics (S+sqrt(3)*SA)*(2*S^2+SA^2+2*SB*SC-SW^2) : :

X(30465) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {2,14}, {5,14816}, {13,5916}, {30,30466}, {110,6777}, {115,125}, {338,30453}, {395,30461}, {396,18777}, {523,30467}, {1316,22512}, {3258,15609}, {3448,6778}, {3457,22513}, {5318,8014}, {10545,16809}, {10653,11658}, {11078,23005}, {12079,20578}, {23283,30452}

X(30465) = midpoint of X(13) and X(16256)
X(30465) = reflection of X(30454) in X(396)
X(30465) = complement of X(35314)
X(30465) = antipode of X(30454) in the Simmons inconic with foci {X(13), X(15)}
X(30465) = intersection of tangents to circle X(13)X(14)X(16) at X(13) and X(14)
X(30465) = X(13)-isoconjugate of X(1101)
X(30465) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 125, 30468), (868, 1648, 30468), (13636, 13722, 30468)

X(30466) = REFLECTION OF X(30454) IN THE LINE X(13)X(15)

Barycentrics (27*R^2*(12*R^2-SA-6*SW)+6*SA^2+12*SB*SC+22*SW^2)*S^2-S*sqrt(3)*(2*S^2+SA^2+2*SB*SC-SW^2)*(9*R^2-2*SW)+3*(27*R^2-8*SW)*SB*SC*SW : :

X(30466) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {13,476}, {23,6104}, {30,30465}, {115,30469}, {396,30467}, {523,30454}, {11537,30452}, {11549,18777}

X(30466) = reflection of X(30467) in X(396)
X(30466) = antipode of X(30467) in the Simmons inconic with foci {X(13), X(15)}

X(30467) = REFLECTION OF X(30466) IN X(396)

Barycentrics
(b^2-c^2)^2*(2*(3*b^2*c^2*(b^4+c^4)+4*a^8-12*(b^2+c^2)*a^6+2*(5*b^4+8*b^2*c^2+5*c^4)*a^4-8*(b^2+c^2)*b^2*c^2*a^2-2*(b^4+c^4)^2+2*b^4*c^4)*sqrt(3)*S+12*a^10-24*(b^2+c^2)*a^8+36*b^2*c^2*a^6+12*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-3*(-2*b^4*c^4+b^6*c^2+b^2*c^6+4*b^8+4*c^8)*a^2+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

X(30467) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {15,30461}, {30,30454}, {396,30466}, {523,30465}, {23992,30470}

X(30467) = reflection of X(30466) in X(396)
X(30467) = antipode of X(30466) in the Simmons inconic with foci {X(13), X(15)}

X(30468) = REFLECTION OF X(30455) IN X(395)

Barycentrics (-S+sqrt(3)*SA)*(2*S^2+SA^2+2*SB*SC-SW^2) : :

X(30468) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {2,13}, {5,14817}, {14,5917}, {30,30469}, {110,6778}, {115,125}, {338,30452}, {395,18776}, {396,30464}, {523,30470}, {1316,22513}, {3258,15610}, {3448,6777}, {3458,22512}, {5321,8015}, {10545,16808}, {10654,11659}, {11092,23004}, {12079,20579}, {23284,30453}

X(30468) = midpoint of X(14) and X(16255)
X(30468) = reflection of X(30455) in X(395)
X(30468) = complement of X(35315)
X(30468) = intersection of tangents to circle X(13)X(14)X(15) at X(13) and X(14)
X(30468) = X(14)-isoconjugate of X(1101)
X(30468) = antipode of X(30455) in the Simmons inconic with foci {X(14), X(16)}
X(30468) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 125, 30465), (868, 1648, 30465), (13636, 13722, 30465)

X(30469) = REFLECTION OF X(30455) IN THE LINE X(14)X(16)

Barycentrics (27*R^2*(12*R^2-SA-6*SW)+6*SA^2+12*SB*SC+22*SW^2)*S^2+S*sqrt(3)*(2*S^2+SA^2+2*SB*SC-SW^2)*(9*R^2-2*SW)+3*(27*R^2-8*SW)*SB*SC*SW : :

X(30469) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {14,476}, {23,6105}, {30,30468}, {115,30466}, {395,30470}, {523,30455}, {11537,18776}, {11549,30453}

X(30469) = reflection of X(30470) in X(395)
X(30469) = antipode of X(30470) in the Simmons inconic with foci {X(14), X(16)}

X(30470) = REFLECTION OF X(30469) IN X(395)

Barycentrics
(b^2-c^2)^2*(-2*(3*b^2*c^2*(b^4+c^4)+4*a^8-12*(b^2+c^2)*a^6+2*(5*b^4+8*b^2*c^2+5*c^4)*a^4-8*(b^2+c^2)*b^2*c^2*a^2-2*(b^4+c^4)^2+2*b^4*c^4)*sqrt(3)*S+12*a^10-24*(b^2+c^2)*a^8+36*b^2*c^2*a^6+12*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-3*(-2*b^4*c^4+b^6*c^2+b^2*c^6+4*b^8+4*c^8)*a^2+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

X(30470) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {16,30464}, {30,30455}, {395,30469}, {523,30468}, {23992,30467}

X(30470) = reflection of X(30469) in X(395)
X(30470) = antipode of X(30469) in the Simmons inconic with foci {X(14), X(16)}

X(30471) = CENTER OF THE DUAL CONIC OF THE SIMMONS INCONIC WITH FOCI X(13) AND X(15)

Barycentrics (2*S+(3*a^2-b^2-c^2)*sqrt(3))*(2*S+(-a^2+b^2+c^2)*sqrt(3)) : :
X(30471) = X(616)+3*X(628) = X(616)-3*X(14145) = 3*X(630)-2*X(6669) = 4*X(6669)-3*X(22846) = 3*X(21359)-X(22849)

The perspector of this dual conic is X(298). It is a circumconic passing through X(99) and X(9198).

X(30471) lies on the cubic K341a and these lines: {2,11121}, {3,299}, {6,14972}, {13,99}, {15,298}, {18,629}, {76,16241}, {114,1080}, {302,11301}, {532,22855}, {620,22848}, {630,6669}, {1975,16644}, {3180,19780}, {3200,10411}, {3643,11132}, {5238,7796}, {5352,7768}, {5464,6298}, {7763,10654}, {7788,11480}, {7811,10645}, {7837,19781}, {8299,10648}, {9885,14904}, {11127,14921}, {11131,14922}, {11299,22861}, {21359,22849}

X(30471) = midpoint of X(628) and X(14145)
X(30471) = reflection of X(i) in X(j) for these (i,j): (298, 11133), (11121, 22847), (22846, 630), (22848, 620)
X(30471) = isotomic conjugate of the isogonal conjugate of X(3170)
X(30471) = anticomplement of X(22847)
X(30471) = complement of X(11121)
X(30471) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11121, 22847), (15, 7799, 298), (618, 11129, 298)

X(30472) = CENTER OF THE DUAL CONIC OF THE SIMMONS INCONIC WITH FOCI X(14) AND X(16)

Barycentrics (-2*S+(3*a^2-b^2-c^2)*sqrt(3))*(-2*S+(-a^2+b^2+c^2)*sqrt(3)) : :
X(30472) = X(617)+3*X(627) = X(617)-3*X(14144) = 3*X(629)-2*X(6670) = 4*X(6670)-3*X(22891) = 3*X(21360)-X(22895)

The perspector of this dual conic is X(299). It is a circumconic passing through X(99) and X(9199).

X(30472) lies on the cubic K341b and these lines: {2,11122}, {3,298}, {6,14972}, {14,99}, {16,299}, {17,630}, {76,16242}, {114,383}, {303,11302}, {533,22901}, {620,22892}, {629,6670}, {1975,16645}, {3181,19781}, {3201,10411}, {3642,11133}, {5237,7796}, {5351,7768}, {5463,6299}, {7763,10653}, {7788,11481}, {7811,10646}, {7837,19780}, {8299,10647}, {9886,14905}, {11126,14922}, {11130,14921}, {11300,22907}, {21360,22895}

X(30472) = midpoint of X(627) and X(14144)
X(30472) = reflection of X(i) in X(j) for these (i,j): (299, 11132), (11122, 22893), (22891, 629), (22892, 620)
X(30472) = isotomic conjugate of the isogonal conjugate of X(3171)
X(30472) = anticomplement of X(22893)
X(30472) = complement of X(11122)
X(30472) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11122, 22893), (16, 7799, 299), (619, 11128, 299)

X(30473) = CENTER OF THE DUAL CONIC OF THE DE LONGCHAMPS ELLIPSE

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

The perspector of this dual conic is X(76).

X(30473) lies on these lines: {2,18040}, {6,668}, {8,22289}, {9,29381}, {10,4446}, {37,2998}, {69,17790}, {75,4377}, {76,594}, {141,3596}, {190,29695}, {192,4033}, {239,18044}, {312,17229}, {313,3661}, {314,4445}, {346,26757}, {350,17299}, {536,4110}, {579,29400}, {646,17262}, {984,28593}, {1100,24524}, {1575,17149}, {1909,17303}, {2321,6381}, {2345,3770}, {3161,4391}, {3204,3570}, {3210,18136}, {3264,3662}, {3679,20174}, {3739,20917}, {3759,25298}, {3760,4007}, {3765,17289}, {3834,30090}, {3948,17233}, {3963,5224}, {3975,17279}, {4261,26752}, {4358,17240}, {4361,29802}, {4384,18065}, {4393,18046}, {4437,21933}, {4494,17272}, {4506,17345}, {4643,17787}, {4699,18143}, {5069,21226}, {5839,25278}, {6335,9308}, {6374,6386}, {6542,18147}, {7148,25625}, {9780,25457}, {13466,21796}, {16574,29511}, {16696,26042}, {16777,18140}, {16816,18073}, {17053,27076}, {17148,27044}, {17227,20892}, {17228,20891}, {17230,18137}, {17231,20923}, {17241,29982}, {17275,25280}, {17314,18135}, {17316,25660}, {17350,29423}, {17443,18055}, {17490,18739}, {18067,21101}, {19804,28633}, {20654,30149}, {21904,25287}, {24004,25269}, {25107,28244}

X(30473) = trilinear pole of the line {20909, 21260}
X(30473) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4377, 17239, 75), (6376, 17786, 37)

X(30474) = CENTER OF THE DUAL CONIC OF THE EVANS CONIC

Barycentrics (a^4-2*(b^2+c^2)*a^2+b^4+4*b^2*c^2+c^4)*(b^2-c^2) : :
X(30474) = X(850)+2*X(3265) = 2*X(850)+X(6563) = X(2525)+2*X(30476) = 4*X(3265)-X(6563) = X(5996)-3*X(9191)

The perspector of this dual conic is X(30475).

X(30474) lies on these lines: {2,525}, {22,22089}, {99,7471}, {325,523}, {647,7630}, {1499,5971}, {1637,2525}, {2373,2693}, {3566,4108}, {3906,6333}, {9146,9182}, {11163,18311}, {12079,23965}, {14417,23878}, {16063,18556}

X(30474) = midpoint of X(i) and X(j) for these {i,j}: {850, 3268}, {1637, 2525}
X(30474) = reflection of X(i) in X(j) for these (i,j): (1637, 30476), (3268, 3265), (6563, 3268)
X(30474) = isogonal conjugate of X(32738)
X(30474) = isotomic conjugate of X(1302)
X(30474) = anticomplement of X(9209)
X(30474) = crossdifference of every pair of points on line X(32)X(1495)
X(30474) = {X(850), X(3265)}-harmonic conjugate of X(6563)

X(30475) = PERSPECTOR OF THE DUAL CONIC OF THE EVANS CONIC

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

The center of this dual conic is X(30474).

X(30475) lies on these lines: {}

X(30476) = CENTER OF THE DUAL CONIC OF THE JERABEK HYPERBOLA

Barycentrics (a^4+2*b^2*c^2-(b^2+c^2)*a^2)*(b^2-c^2) : :
X(30476) = 3*X(2)+X(850) = X(669)+3*X(9148) = 3*X(1637)+X(2525) = X(2525)-3*X(30474) = X(3804)-3*X(4108) = X(6563)-3*X(14417) = X(12077)+3*X(14417)

The perspector of this dual conic is X(6331).

X(30476) lies on these lines: {2,647}, {5,30209}, {10,4524}, {125,15630}, {126,16188}, {127,16177}, {141,8675}, {306,21719}, {512,625}, {520,6130}, {523,4885}, {525,3239}, {620,22104}, {669,9148}, {804,8651}, {1637,2525}, {1649,15850}, {2451,17215}, {2485,23285}, {2489,3267}, {2501,2799}, {2528,8891}, {2793,6562}, {3589,9030}, {3739,17069}, {3766,27345}, {3767,7652}, {3788,23105}, {3804,4108}, {3906,3934}, {4077,24459}, {4139,4928}, {4369,8672}, {4374,27527}, {5907,9242}, {5972,11595}, {6363,23803}, {6563,12077}, {6723,22264}, {7777,10567}, {7886,8574}, {9404,19732}, {10097,11318}, {15143,16229}, {17478,21050}, {17899,17921}, {20907,25098}, {24353,24718}

X(30476) = midpoint of X(i) and X(j) for these {i,j}: {647, 850}, {1637, 30474}, {2485, 23285}, {2489, 3267}, {2501, 3265}, {6563, 12077}, {7624, 18312}, {16229, 22089}, {24353, 24718}
X(30476) = reflection of X(i) in X(j) for these (i,j): (6587, 14341), (22264, 6723)
X(30476) = complementary conjugate of X(15526)
X(30476) = isotomic conjugate of the isogonal conjugate of X(2451)
X(30476) = isotomic conjugate of the polar conjugate of X(16229)
X(30476) = polar conjugate of the isogonal conjugate of X(22089)
X(30476) = complement of X(647)
X(30476) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 850, 647), (2, 24622, 24782), (2, 25258, 25594)

X(30477) = PERSPECTOR OF THE DUAL CONIC OF THE YFF HYPERBOLA

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

The center of this dual conic is X(524).

X(30477) lies on the line {511,13619}

X(30478) = CENTER OF THE DUAL CONIC OF THE YIU CONIC

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

Let A_b, A_c be the touchpoints of the A-excircle and sides AB and AC, respectively, and define B_c, B_a, C_a, C_b cyclically. These six points lie on the Yiu conic, defined at X(478).
The perspector of this dual conic is X(30479).

X(30478) lies on these lines: {1,5745}, {2,12}, {3,1602}, {4,993}, {7,15823}, {8,2320}, {9,1125}, {10,631}, {11,452}, {20,2886}, {21,497}, {35,5082}, {36,443}, {40,6935}, {48,966}, {63,3485}, {65,5744}, {71,10476}, {72,16193}, {75,6337}, {104,6889}, {140,9708}, {142,3361}, {145,18231}, {154,20306}, {210,27383}, {220,24512}, {261,30479}, {281,7521}, {329,11375}, {354,960}, {355,6954}, {376,5267}, {377,19841}, {390,3813}, {404,26040}, {405,3086}, {442,4293}, {474,19855}, {496,16418}, {498,3421}, {499,5084}, {515,5705}, {517,6892}, {518,5703}, {549,9709}, {551,12559}, {936,10165}, {946,5698}, {950,5231}, {956,3085}, {962,4640}, {988,4000}, {997,6878}, {999,6675}, {1000,22837}, {1001,14986}, {1006,10785}, {1036,5324}, {1056,8666}, {1058,5248}, {1107,7735}, {1212,2275}, {1376,3523}, {1377,13935}, {1378,9540}, {1385,5791}, {1468,5712}, {1469,28275}, {1478,6856}, {1479,11111}, {1588,9678}, {1621,10529}, {1698,5795}, {1706,10164}, {1788,19860}, {1935,25885}, {2476,5229}, {2478,10589}, {3035,10303}, {3189,3601}, {3419,4305}, {3428,6847}, {3434,4189}, {3452,3624}, {3474,4652}, {3476,24987}, {3486,6734}, {3488,10916}, {3524,25440}, {3525,10805}, {3526,3820}, {3622,5289}, {3649,9965}, {3671,3928}, {3683,11376}, {3698,26062}, {3812,5435}, {3814,5067}, {3816,5129}, {3826,17580}, {3878,10595}, {3913,5281}, {3916,4295}, {3925,5204}, {4190,5303}, {4220,22654}, {4252,4307}, {4267,16713}, {4294,16370}, {4298,25525}, {4314,24392}, {4426,7736}, {4512,12053}, {4648,25500}, {4679,26129}, {4719,5222}, {4855,25006}, {5080,6933}, {5123,19877}, {5177,7354}, {5217,17784}, {5219,12527}, {5225,6872}, {5259,10072}, {5274,11106}, {5284,10586}, {5288,10056}, {5302,5550}, {5325,25055}, {5432,7080}, {5436,11019}, {5450,6916}, {5587,6927}, {5603,12514}, {5657,6977}, {5716,29639}, {5731,5794}, {5818,6880}, {5836,6966}, {6284,17576}, {6351,13902}, {6352,13959}, {6636,9712}, {6684,9623}, {6690,12513}, {6737,13384}, {6762,13405}, {6824,11249}, {6825,12667}, {6826,26286}, {6844,11827}, {6846,22753}, {6852,10532}, {6853,12115}, {6855,26332}, {6859,10526}, {6861,22765}, {6868,26470}, {6875,12116}, {6908,12114}, {6921,9780}, {6961,26446}, {6970,9956}, {6989,10269}, {8167,17554}, {8227,12572}, {9701,11003}, {9710,15717}, {10167,18251}, {10200,17559}, {10448,11269}, {10591,11113}, {10934,19528}, {11036,11281}, {11108,15325}, {11110,27509}, {11365,17560}, {13411,25568}, {14001,17030}, {15171,17571}, {15654,19262}, {16678,27621}, {17314,17733}, {18250,19862}, {23207,27407}, {24363,24432}, {24570,27339}

X(30478) = complement of X(5261)
X(30478) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2975, 388), (2, 3436, 10588), (11194, 25466, 3600)

X(30479) = PERSPECTOR OF THE DUAL CONIC OF THE YIU CONIC

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

The center of this dual conic is X(30478).

X(30479) lies on the Feuerbach hyperbola and these lines: {1,69}, {2,2221}, {4,75}, {7,4388}, {8,3718}, {9,345}, {10,989}, {21,332}, {63,2354}, {79,15434}, {80,4986}, {84,6210}, {104,1310}, {238,987}, {256,7019}, {261,30478}, {294,391}, {307,1041}, {314,497}, {319,1000}, {320,3296}, {326,1064}, {333,1172}, {464,28287}, {885,4811}, {941,5739}, {960,1264}, {966,981}, {1479,10447}, {2551,3596}, {2975,8048}, {2997,4441}, {3672,26117}, {3717,4866}, {4329,20911}, {4514,6601}, {5016,24547}, {5558,21296}, {5665,6604}, {5738,17137}, {9534,26939}, {10436,26098}, {16043,27633}, {17097,17152}, {17306,18141}, {17361,18490}

X(30479) = isogonal conjugate of X(1460)
X(30479) = isotomic conjugate of X(388)
X(30479) = anticomplement of X(34261)
X(30479) = trilinear pole of the line {650, 3910}

X(30480) = X(5)X(128)∩X(14071)X(25149)

Barycentrics (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6+2 a^4 b^2-a^2 b^4-a^2 b^3 c+b^5 c+2 a^4 c^2+a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3-a^2 c^4+b c^5) (a^6-2 a^4 b^2+a^2 b^4-a^2 b^3 c+b^5 c-2 a^4 c^2-a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3+a^2 c^4+b c^5) (-a^10 b^2+4 a^8 b^4-6 a^6 b^6+4 a^4 b^8-a^2 b^10-a^10 c^2+6 a^8 b^2 c^2-9 a^6 b^4 c^2+5 a^4 b^6 c^2-2 a^2 b^8 c^2+b^10 c^2+4 a^8 c^4-9 a^6 b^2 c^4+3 a^2 b^6 c^4-4 b^8 c^4-6 a^6 c^6+5 a^4 b^2 c^6+3 a^2 b^4 c^6+6 b^6 c^6+4 a^4 c^8-2 a^2 b^2 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : :
Barycentrics S^6+S^4 (-9 R^4-5 R^2 SB-5 R^2 SC-3 SB SC+3 R^2 SW+2 SB SW+2 SC SW)+SB SC (18 R^8-6 R^6 SW-7 R^4 SW^2+5 R^2 SW^3-SW^4)+S^2 (114 R^8-40 R^6 SB-40 R^6 SC-126 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+41 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2-R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-53 R^4+33 R^2 SW-4 SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30480) lies on these lines: {5,128}, {14071,25149}

X(30481) = X(5)X(51)∩X(110)X(1157)

Barycentrics a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^14 b^2-6 a^12 b^4+15 a^10 b^6-20 a^8 b^8+15 a^6 b^10-6 a^4 b^12+a^2 b^14+a^14 c^2-9 a^12 b^2 c^2+25 a^10 b^4 c^2-30 a^8 b^6 c^2+15 a^6 b^8 c^2-a^4 b^10 c^2-a^2 b^12 c^2-6 a^12 c^4+25 a^10 b^2 c^4-30 a^8 b^4 c^4+9 a^6 b^6 c^4+3 a^4 b^8 c^4-2 a^2 b^10 c^4+b^12 c^4+15 a^10 c^6-30 a^8 b^2 c^6+9 a^6 b^4 c^6-a^4 b^6 c^6+2 a^2 b^8 c^6-4 b^10 c^6-20 a^8 c^8+15 a^6 b^2 c^8+3 a^4 b^4 c^8+2 a^2 b^6 c^8+6 b^8 c^8+15 a^6 c^10-a^4 b^2 c^10-2 a^2 b^4 c^10-4 b^6 c^10-6 a^4 c^12-a^2 b^2 c^12+b^4 c^12+a^2 c^14) : :
Barycentrics S^4 (R^2-SB-SC+SW)+SB SC (15 R^6-18 R^4 SW+7 R^2 SW^2-SW^3)+S^2 (-17 R^6-8 R^4 SB-8 R^4 SC+SB SC (5 R^2-SW)+22 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-9 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : :

See Alexandr Skutin, Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30481) lies on these lines: {5,51}, {110,1157}, {5944,6150}

X(30482) = X(5)X(51)∩X(1510)X(6150)

Barycentrics a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10+a^10 c^2-6 a^8 b^2 c^2+9 a^6 b^4 c^2-5 a^4 b^6 c^2+2 a^2 b^8 c^2-b^10 c^2-4 a^8 c^4+9 a^6 b^2 c^4-3 a^2 b^6 c^4+4 b^8 c^4+6 a^6 c^6-5 a^4 b^2 c^6-3 a^2 b^4 c^6-6 b^6 c^6-4 a^4 c^8+2 a^2 b^2 c^8+4 b^4 c^8+a^2 c^10-b^2 c^10) : :
Barycentrics S^4 (3 R^2-SB-SC+SW)+SB SC (6 R^6-10 R^4 SW+5 R^2 SW^2-SW^3)+S^2 (-26 R^6-8 R^4 SB-8 R^4 SC+SB SC (7 R^2-SW)+30 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-11 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30482) lies on these lines: {5,51}, {1510,6150}, {12060,18350}

X(30483) = X(5)X(128)∩X(6343)X(25149)

Barycentrics (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^22 b^2-8 a^20 b^4+28 a^18 b^6-56 a^16 b^8+70 a^14 b^10-56 a^12 b^12+28 a^10 b^14-8 a^8 b^16+a^6 b^18+a^22 c^2-13 a^20 b^2 c^2+56 a^18 b^4 c^2-119 a^16 b^6 c^2+140 a^14 b^8 c^2-91 a^12 b^10 c^2+28 a^10 b^12 c^2-a^8 b^14 c^2-a^6 b^16 c^2-8 a^20 c^4+56 a^18 b^2 c^4-141 a^16 b^4 c^4+163 a^14 b^6 c^4-79 a^12 b^8 c^4-7 a^10 b^10 c^4+30 a^8 b^12 c^4-23 a^6 b^14 c^4+13 a^4 b^16 c^4-5 a^2 b^18 c^4+b^20 c^4+28 a^18 c^6-119 a^16 b^2 c^6+163 a^14 b^4 c^6-79 a^12 b^6 c^6+5 a^10 b^8 c^6-11 a^8 b^10 c^6+35 a^6 b^12 c^6-39 a^4 b^14 c^6+25 a^2 b^16 c^6-8 b^18 c^6-56 a^16 c^8+140 a^14 b^2 c^8-79 a^12 b^4 c^8+5 a^10 b^6 c^8+7 a^8 b^8 c^8-12 a^6 b^10 c^8+39 a^4 b^12 c^8-45 a^2 b^14 c^8+28 b^16 c^8+70 a^14 c^10-91 a^12 b^2 c^10-7 a^10 b^4 c^10-11 a^8 b^6 c^10-12 a^6 b^8 c^10-26 a^4 b^10 c^10+25 a^2 b^12 c^10-56 b^14 c^10-56 a^12 c^12+28 a^10 b^2 c^12+30 a^8 b^4 c^12+35 a^6 b^6 c^12+39 a^4 b^8 c^12+25 a^2 b^10 c^12+70 b^12 c^12+28 a^10 c^14-a^8 b^2 c^14-23 a^6 b^4 c^14-39 a^4 b^6 c^14-45 a^2 b^8 c^14-56 b^10 c^14-8 a^8 c^16-a^6 b^2 c^16+13 a^4 b^4 c^16+25 a^2 b^6 c^16+28 b^8 c^16+a^6 c^18-5 a^2 b^4 c^18-8 b^6 c^18+b^4 c^20) : :
Barycentrics S^6+S^4 (R^4-5 R^2 SB-5 R^2 SC-3 SB SC-R^2 SW+2 SB SW+2 SC SW)+SB SC (-27 R^8+52 R^6 SW-33 R^4 SW^2+9 R^2 SW^3-SW^4)+S^2 (69 R^8-40 R^6 SB-40 R^6 SC-68 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+15 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2+3 R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-43 R^4+29 R^2 SW-4 SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30483) lies on these lines: {5,128}, {6343,25149}

X(30484) = X(3)X(128)∩X(4)X(252)

Barycentrics 5*S^4+(R^2*(4*R^2-5*SA)+2*SA^2-7*SB*SC-SW^2)*S^2+(R^2*(20*R^2-19*SW)+5*SW^2)*SB*SC : :
X(30484) = 3*X(5)-2*X(20414)

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30484) lies on these lines: {3, 128}, {4, 252}, {5, 6150}, {30, 14143}, {550, 6247}, {930, 2888}, {933, 3462}, {1510, 11591}, {3153, 14097}, {3574, 12060}, {15619, 15704}

X(30484) = midpoint of X(15619) and X(15704)
X(30484) = anticomplement of X(33545)
X(30484) = {X(3), X(1601)}-harmonic conjugate of X(23320)
X(30484) = Napoleon-Feuerbach isogonal conjugate of X(54)

X(30485) = X(3)X(24303)∩X(61)X(16641)

Barycentrics (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2-sqrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : :
X(30485) = 2*X(13350)-3*X(14170)

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30485) lies on these lines: {3, 24303}, {61, 16461}, {616, 10409}, {1495, 13350}, {5663, 13859}

X(30486) = X(3)X(24304)∩X(62)X(16462)

Barycentrics (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2+qrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30486) lies on these lines: {3, 24304}, {62, 16462}, {617, 10410}, {1495, 13349}, {5663, 13858}

X(30487) = X(3)X(695)∩X(192)X(815)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30487) lies on these lines: {3, 695}, {192, 815}, {6310, 19548}

X(30488) = ISOGONAL CONJUGATE OF X(6031)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30488) lies on these lines: {6, 6324}, {574, 12367}, {599, 8705}

X(30488) = reflection of X(6) in X(6324)
X(30488) = anticomplement of the complementary conjugate of X(6032)
X(30488) = antigonal conjugate of the isogonal conjugate of X(5971)
X(30488) = isogonal conjugate of X(6031)

X(30489) = ISOGONAL CONJUGATE OF X(10130)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30489) lies on these lines: {6, 23}, {39, 9019}, {76, 524}, {141, 23297}, {230, 25488}, {523, 18907}, {597, 13410}, {755, 11636}, {882, 9009}, {2353, 30435}, {2393, 27375}, {2854, 5052}, {3291, 16776}, {3629, 6664}, {6698, 15820}, {7737, 11594}, {8584, 20380}, {9465, 9971}

X(30489) = isogonal conjugate of X(10130)

X(30490) = X(54)X(143)∩X(195)X(25043)

Barycentrics (SB+SC) *(2*R^2-SA-SW) *(2*SB+R^2)*(2*SC+R^2)*(S^2+SA*SB)*(S^2+SA*SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30490) lies on these lines: {54, 143}, {195, 25043}

X(30491) = X(512)X(2030)∩X(523)X(18907)

Barycentrics a^2*(b^2-c^2)*(-a^2+b^2+c^2)*(2*a^2-b^2+2*c^2)*(2*a^2+2*b^2-c^2) : :
X(30491) = 3*X(3049)-X(10097)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28769.

X(30491) lies on these lines: {512, 2030}, {523, 18907}, {647, 9517}, {1383, 2433}, {1384, 17414}, {1499, 23287}, {2501, 8599}, {2715, 11636}, {3049, 10097}, {17979, 30209}

X(30491) = isogonal conjugate of the polar conjugate of X(8599)
X(30491) = barycentric product X(i)*X(j) for these {i, j}: {3, 8599}, {125, 11636}, {525, 1383}, {598, 647}, {895, 23287}
X(30491) = barycentric quotient X(i)/X(j) for these (i, j): (3, 9146), (184, 9145), (228, 3908), (512, 5094), (525, 9464), (598, 6331), (647, 599), (669, 8541), (1383, 648)
X(30491) = trilinear product X(i)*X(j) for these {i, j}: {48, 8599}, {598, 810}, {656, 1383}
X(30491) = trilinear quotient X(i)/X(j) for these (i, j): (48, 9145), (63, 9146), (71, 3908), (598, 811), (656, 599), (661, 5094), (798, 8541), (810, 574), (1383, 162)

X(30492) = TRILINEAR POLE OF THE LINE {338, 2086}

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28769.

X(30492) lies on this line: {850, 5027}

X(30492) = trilinear pole of the line {338, 2086}

X(30493) = X(1)X(1361)∩X(7)X(286)

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

See Tran Quang Hung and Peter Moses, Hyacinthos 287771.

X(30493) lies on the cubic K714 and these lines: {1, 1361}, {7, 286}, {12, 26932}, {34, 26892}, {49, 23070}, {51, 1393}, {56, 58}, {57, 1745}, {63, 7066}, {65, 515}, {73, 22345}, {84, 7355}, {109, 23850}, {181, 1454}, {185, 7004}, {201, 3917}, {221, 22654}, {226, 14058}, {227, 8679}, {388, 26871}, {603, 1425}, {942, 1875}, {971, 1887}, {1038, 3784}, {1122, 1439}, {1214, 11573}, {1354, 1363}, {1355, 1367}, {1358, 20618}, {1400, 14597}, {1406, 18954}, {1433, 3304}, {1455, 23842}, {1469, 7289}, {1473, 19349}, {2003, 19365}, {3220, 26888}, {4014, 7702}, {4303, 22341}, {5399, 23981}, {5907, 24430}, {7352, 24467}, {9291, 18026}, {15524, 20323}, {17114, 18838}, {18915, 26929}, {26933, 26955}

X(30493) = X(18180)-Ceva conjugate of X(1393)
X(30493) = X(i)-isoconjugate of X(j) for these (i,j): {8, 2190}, {9, 275}, {33, 95}, {41, 276}, {54, 318}, {78, 8884}, {212, 8795}, {281, 2167}, {312, 8882}, {933, 4086}, {2148, 7017}, {2289, 8794}, {4041, 18831}, {8611, 16813}
X(30493) = crosspoint of X(7) and X(222)
X(30493) = crosssum of X(55) and X(281)
X(30493) = barycentric product X(i) X(j) for these {i,j}: {5, 222}, {7, 216}, {51, 348}, {53, 1804}, {56, 343}, {63, 1393}, {65, 16697}, {73, 17167}, {77, 1953}, {217, 6063}, {278, 5562}, {324, 7335}, {331, 418}, {603, 14213}, {604, 18695}, {1214, 18180}, {1397, 28706}, {1625, 17094}, {1813, 21102}, {2179, 7182}, {2181, 7183}, {3199, 7055}, {4565, 6368}, {4573, 15451}, {7069, 7177}, {7178, 23181}, {8798, 18623}, {17076, 27372}
X(30493) = barycentric quotient X(i) / X(j) for these {i,j}: {5, 7017}, {7, 276}, {51, 281}, {56, 275}, {216, 8}, {217, 55}, {222, 95}, {278, 8795}, {343, 3596}, {418, 219}, {603, 2167}, {604, 2190}, {608, 8884}, {1118, 8794}, {1393, 92}, {1397, 8882}, {1953, 318}, {2179, 33}, {3199, 1857}, {4565, 18831}, {5562, 345}, {7069, 7101}, {7335, 97}, {15451, 3700}, {16697, 314}, {18695, 28659}, {23181, 645}
X(30493) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 222, 7335}, {1425, 3937, 603}

X(30494) = PERSPECTOR OF ADAMS CIRCLE

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

Centers X(30494)-X(30503) were contributed by César Lozada, December 30, 2018. See Triangle circles at MathWorld for definitions of circles referred in these centers.

X(30494) lies on the Feuerbach hyperbola and these lines: {7,15658}, {9,6706}

X(30494) = isogonal conjugate of X(30502)
X(30494) = trilinear product of vertices of polar triangle of Adams circle

X(30495) = PERSPECTOR OF BROCARD CIRCLE

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

X(30495) is the pole, wrt the Brocard circle, of line X(669)X(688), which is the isogonal conjugate of the isotomic conjugate of the Lemoine axis, and the trilinear polar of X(32). (Randy Hutson, January 15, 2019)

X(30495) lies on these lines: {538,599}, {574,3117}, {694,3734}, {695,7751}, {2387,5028}, {2549,20021}, {3721,3760}, {3981,18546}, {7748,27366}, {11648,20859}, {14820,17130}

X(30495) = reflection of X(6195) in X(3117)
X(30495) = isogonal conjugate of X(3972)
X(30495) = anticomplement of the complementary conjugate of X(7853)
X(30495) = complement of the anticomplementary conjugate of X(7898)
X(30495) = trilinear pole of the line {888, 17414}

X(30496) = PERSPECTOR OF 2nd BROCARD CIRCLE

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

X(30496) lies on the Jerabek hyperbola and these lines: {3,3229}, {69,698}, {5254,19222}

X(30496) = isogonal conjugate of X(3552)
X(30496) = anticomplement of the complementary conjugate of X(5025)

X(30497) = PERSPECTOR OF DAO-MOSES-TELV CIRCLE

Barycentrics
(S^4+3*(16*R^2*(9*R^2-4*SW)-SB^2+7*SW^2)*S^2-3*(4*R^2-SW)*(36*R^2-8*SW+3*SB)*SB*SW)*(S^4+3*(16*R^2*(9*R^2-4*SW)-SC^2+7*SW^2)*S^2-3*(4*R^2-SW)*(36*R^2-8*SW+3*SC)*SC*SW) : :

X(30497) lies on these lines: {542,1651}, {6070,17986}

X(30498) = PERSPECTOR OF EHRMANN CIRCLE

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

X(30498) lies on these lines: {353,20251}, {574,20977}, {599,625}, {6323,11002}

X(30499) = PERSPECTOR OF GALLATLY CIRCLE

Barycentrics (SB+SC)*(S^4+(3*SW-SC)*SW*S^2-SA*SB*SW^2)*(S^4+(3*SW-SB)*SW*S^2-SW^2*SC*SA) : :

X(30499) lies on these lines: {262,20021}, {325,14994}, {511,14096}

X(30500) = PERSPECTOR OF HEXYL CIRCLE

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

X(30500) lies on the Feuerbach hyperbola and these lines: {8,6769}, {9,1012}, {1000,10388}, {1537,3254}, {2096,10307}, {4292,10309}, {7284,30304}, {12867,21669}

X(30500) = isogonal conjugate of X(30503)

X(30501) = PERSPECTOR OF LONGUET-HIGGINS CIRCLE

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

X(30501) lies on these lines: {7,5342}, {63,391}, {69,962}, {77,3616}, {938,969}, {17011,20211}

X(30501) = isotomic conjugate of X(5815)
X(30501) = cyclocevian conjugate of the isotomic conjugate of X(9874)
X(30501) = trilinear pole of the line {905, 4765}

X(30502) = ISOGONAL CONJUGATE OF X(30494)

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

X(30502) lies on these lines: {1,3}, {41,10482}, {1253,4251}, {2293,4253}, {5022,16688}, {5248,28071}, {5259,28053}, {14942,17687}

X(30503) = ISOGONAL CONJUGATE OF X(30500)

Barycentrics a*(a^6-2*(b+c)*a^5-(b+c)^2*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^2-6*b*c+c^2)*(b+c)^2*a^2-2*(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b-c)^2) : :
X(30503) = 3*X(165)-2*X(3587) = 3*X(165)+X(18421) = X(1056)-3*X(21151)

X(30503) lies on these lines: {1,3}, {4,12565}, {8,10884}, {9,3197}, {10,1490}, {19,2267}, {20,19860}, {30,2951}, {71,2324}, {73,1103}, {84,958}, {200,5657}, {207,7952}, {380,572}, {405,12705}, {515,2550}, {516,6987}, {519,3174}, {631,8583}, {912,5223}, {936,6261}, {944,4853}, {946,6865}, {952,4915}, {956,10167}, {960,7971}, {971,9708}, {990,30116}, {997,10164}, {1006,4512}, {1012,10860}, {1056,4321}, {1064,2999}, {1072,23681}, {1125,6926}, {1483,12127}, {1519,6947}, {1698,6825}, {1699,6827}, {1706,11500}, {1709,5251}, {1750,3925}, {1935,2956}, {2270,23840}, {2551,6260}, {2800,21153}, {3062,18540}, {3088,19784}, {3189,5882}, {3753,7580}, {3880,7966}, {4297,12650}, {4847,5768}, {4882,5534}, {5234,7330}, {5258,10085}, {5260,9961}, {5437,22753}, {5705,12616}, {5715,12609}, {5720,8580}, {5745,14647}, {5790,18528}, {5795,12667}, {5811,18250}, {5924,12572}, {6154,6264}, {6174,6326}, {6245,19843}, {6361,12651}, {6705,30478}, {6762,12675}, {6765,11362}, {6838,24982}, {6846,21628}, {6848,8582}, {6882,7988}, {6890,24541}, {6913,11372}, {6953,25011}, {7171,22758}, {7680,25525}, {9579,11827}, {9841,12114}, {9856,11108}, {12664,18251}, {16132,17857}

X(30503) = midpoint of X(i) and X(j) for these {i,j}: {40, 11529}, {5732, 9623}
X(30503) = reflection of X(i) in X(j) for these (i,j): (1, 18443), (3062, 18540), (6767, 1385), (11372, 6913)
X(30503) = isogonal conjugate of X(30500)
X(30503) = X(18420)-of-excentral-triangle
X(30503) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7982, 7957), (165, 2093, 40), (1040, 24806, 1)

X(30504) = X(54)X(21394)∩X(186)X(323)

Barycentrics a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^16-4 a^14 b^2+5 a^12 b^4+a^10 b^6-10 a^8 b^8+14 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-4 a^14 c^2+8 a^12 b^2 c^2-a^10 b^4 c^2-3 a^8 b^6 c^2-12 a^6 b^8 c^2+28 a^4 b^10 c^2-23 a^2 b^12 c^2+7 b^14 c^2+5 a^12 c^4-a^10 b^2 c^4-a^8 b^4 c^4-2 a^6 b^6 c^4-18 a^4 b^8 c^4+39 a^2 b^10 c^4-22 b^12 c^4+a^10 c^6-3 a^8 b^2 c^6-2 a^6 b^4 c^6+2 a^4 b^6 c^6-21 a^2 b^8 c^6+41 b^10 c^6-10 a^8 c^8-12 a^6 b^2 c^8-18 a^4 b^4 c^8-21 a^2 b^6 c^8-50 b^8 c^8+14 a^6 c^10+28 a^4 b^2 c^10+39 a^2 b^4 c^10+41 b^6 c^10-11 a^4 c^12-23 a^2 b^2 c^12-22 b^4 c^12+5 a^2 c^14+7 b^2 c^14-c^16) : :
Barycentrics (17 R^2+SB+SC-5 SW)S^4 + (-64 R^6+8 R^4 SB+8 R^4 SC+68 R^4 SW-6 R^2 SB SW-6 R^2 SC SW-25 R^2 SW^2+SB SW^2+SC SW^2+3 SW^3+SB SC (-15 R^2+5 SW))S^2 + SB SC (30 R^6-40 R^4 SW+19 R^2 SW^2-3 SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28775.

X(30504) lies on these lines: {54,21394}, {186,323}, {6150,12006}

X(30505) = X(2)X(3613)∩X(76)X(3060)

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

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30505) lies on the Kiepert hyperbola and these lines: {2, 3613}, {76, 3060}, {83, 14957}, {98, 251}, {1916, 11794}, {2052, 10550}

X(30505) = isotomic conjugate of the anticomplement of X(20965)
X(30505) = barycentric product X(i)*X(j) for these {i,j}: {83, 3613}, {308, 27375}
X(30505) = barycentric quotient X(i)/X(j) for these (i,j): (32, 3203), (82, 18042), (83, 1078), (251, 5012)
X(30505) = trilinear product X(82)*X(3613)
X(30505) = trilinear quotient X(i)/X(j) for these (i,j): (31, 3203), (82, 5012), (83, 18042)

X(30506) = EULER LINE INTERCEPT OF X(51)X(324)

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

As a point on the Euler line, X(30506) has Shinagawa coefficients (EF, 2(E+F)F+2S²).

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30506) lies on these lines: {2, 3}, {51, 324}, {53, 17500}, {110, 275}, {143, 14978}, {251, 6531}, {264, 3060}, {1629, 5012}, {2052, 5640}, {3289, 6748}, {6747, 19130}, {6749, 25051}, {8884, 13434}, {11451, 15466}, {14389, 19174}

X(30506) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 458, 14957), (4, 6819, 1370), (5, 6755, 467)
X(30506) = barycentric product X(i)*X(j) for these {i,j}: {53, 1078}, {308, 27370}, {311, 10312}, {324, 5012}, {343, 1629}
X(30506) = barycentric quotient X(i)/X(j) for these (i,j): (53, 3613), (1629, 275)
X(30506) = trilinear product X(i)*X(j) for these {i,j}: {53, 18042}, {1078, 2181}
X(30506) = trilinear quotient X(i)/X(j) for these (i,j): (1629, 2190), (2181, 27375)

X(30507) = X(3)X(21357)∩X(5)X(10721)

Barycentrics (4*SA-143*R^2+30*SW)*S^2+11*(11*R^2-2*SW)*SB*SC : :
X(30507) = X(15704)+2*X(17505)

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30507) lies on these lines: {3, 21357}, {5, 10721}, {368, 8421}, {549, 12162}, {550, 20191}, {15704, 17505}

X(30508) = ANTICOMPLEMENT OF X(13636)

Barycentrics (-(2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2-K)*(-a^2+b^2-K)*(c^2-a^2-K)*(-b^2+c^2-K) : :, where K=sqrt(SW^2-3*S^2)
Barycentrics = (b^2-c^2) (5 a^4-5 a^2 b^2+2 b^4-5 a^2 c^2+b^2 c^2+2 c^4+2 (2 a^2-b^2-c^2) Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : : (Peter Moses, January 3, 2019)

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30508) lies on the Kiepert parabola, the cubics K010, K015, K242, K408 and these lines: {2, 1340}, {99, 110}, {523, 6190}

X(30508) = isotomic conjugate of X(30509)
X(30508) = anticomplement of X(13636)
X(30508) = trilinear pole of the line {115, 2029}
X(30508) = reflection of X(2) in the line X(3413)X(9168)
X(30508) = Gibert-Simson transform of X(1379)
X(30508) = barycentric product X(i)*X(j) for these {i,j}: {670, 2029}, {2966, 14501}, {3414, 6190}
X(30508) = barycentric quotient X(i)/X(j) for these (i,j): (512, 2028), (1379, 1380), (2029, 512), (2799, 14502), (3414, 3413)
X(30508) = trilinear product X(799)*X(2029)
X(30508) = trilinear quotient X(i)/X(j) for these (i,j): (661, 2028), (2029, 798)

X(30509) = ANTICOMPLEMENT OF X(13722)

Barycentrics ((2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2+K)*(-a^2+b^2+K)*(c^2-a^2+K)*(-b^2+c^2+K) : :, where K=sqrt(SW^2-3*S^2)
Barycentrics (b^2-c^2) (5 a^4-5 a^2 b^2+2 b^4-5 a^2 c^2+b^2 c^2+2 c^4-2 (2 a^2-b^2-c^2) Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : : (Peter Moses, January 3, 2019)

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30509) lies on the Kiepert parabola, the cubics K010, K015, K242, K408 and these lines: {2, 1341}, {99, 110}, {523, 6189}

X(30509) = anticomplement of X(13722)
X(30509) = isotomic conjugate of X(30508)
X(30509) = Gibert-Simson transform of X(1380)
X(30509) = isotomic conjugate of the anticomplement of X(13636)
X(30509) = trilinear pole of the line {115, 2028}
X(30509) = reflection of X(2) in the line X(3414)X(9168)
X(30509) = barycentric product X(i)*X(j) for these {i,j}: {670, 2028}, {2966, 14502}, {3413, 6189}
X(30509) = barycentric quotient X(i)/X(j) for these (i,j): (512, 2029), (1380, 1379), (2028, 512), (2799, 14501), (3413, 3414)
X(30509) = trilinear product X(799)*X(2028)
X(30509) = trilinear quotient X(i)/X(j) for these (i,j): (661, 2029), (2028, 798)

X(30510) = X(3)X(6030)∩X(23)X(16186)

Barycentrics (SB+SC)*(SA-SB)*(SA-SC)*(SA-15*R^2+3*SW) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30510) lies on these lines: {3, 6030}, {23, 16186}, {30, 14385}, {110, 351}, {476, 1304}, {1113, 10288}, {1114, 10287}, {2070, 14670}, {2071, 7740}, {6760, 12113}, {9717, 15107}, {10130, 11058}, {12270, 14703}, {13595, 18114}, {14685, 15080}

X(30511) = X(67)X(9003)∩X(476)X(2407)

Barycentrics (SB-SC)*(4*S^2-3*R^2*(30*R^2+5*SA-17*SW)+3*SA^2-4*SB*SC-7*SW^2) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30511) lies on these lines: {67, 9003}, {476, 2407}, {523, 550}, {924, 12162}, {2528, 3313}, {9033, 12901}

X(30512) = EULER LINE INTERCEPT OF X(107)X(13398)

Barycentrics (SA-SB)*(SA-SC)*(2*S^2-(SB+SC)*(SA-6*R^2+2*SW)) : :

As a point on the Euler line, X(30512) has Shinagawa coefficients (E²+4EF+12F²-4S², -2E²-10EF-8F²+8S²).

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30512) lies on these lines: {2, 3}, {107, 13398}, {110, 925}, {476, 2407}, {691, 16167}, {1302, 3565}, {3233, 5502}, {5468, 6563}

X(30512) = isogonal conjugate of orthocenter of X(3)X(4)X(68)

X(30513) = ISOGONAL CONJUGATE OF X(1470)

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

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30513) lies on the Feuerbach hyperbola and these lines: {1, 908}, {2, 104}, {4, 5554}, {7, 5080}, {9, 6735}, {10, 90}, {21, 2551}, {65, 5555}, {80, 3434}, {84, 377}, {149, 24297}, {388, 1476}, {390, 13278}, {404, 12667}, {405, 10942}, {442, 18542}, {443, 7705}, {452, 943}, {497, 1320}, {958, 10958}, {1000, 3421}, {1001, 10956}, {1041, 1877}, {1156, 2550}, {1329, 22768}, {1389, 5046}, {1392, 4345}, {1478, 3306}, {1512, 3359}, {1519, 6957}, {1537, 6929}, {1837, 10522}, {2320, 5328}, {2346, 11239}, {2475, 10308}, {2481, 11185}, {3577, 26333}, {3680, 5727}, {3753, 18516}, {3897, 5084}, {4187, 16203}, {4190, 18491}, {5187, 10532}, {5250, 7162}, {5251, 6910}, {5559, 5692}, {6850, 25005}, {6872, 11248}, {6919, 10586}, {6930, 12775}, {6931, 7951}, {8068, 10584}, {10527, 26476}, {10596, 14497}, {10679, 11113}, {11108, 18545}, {12608, 19860}, {12647, 18254}, {20895, 30479}, {21301, 23836}

X(30513) = isogonal conjugate of X(1470)
X(30513) = trilinear pole of the line {650, 2804}
X(30513) = {X(6256), X(24982)}-harmonic conjugate of X(377)
X(30513) = barycentric product X(312)*X(998)
X(30513) = barycentric quotient X(i)/X(j) for these (i,j): (8, 17740), (9, 997), (21, 26637), (607, 11383), (650, 9001), (998, 57)
X(30513) = trilinear product X(i)*X(j) for these {i,j}: {8, 998}, {522, 9058}
X(30513) = trilinear quotient X(i)/X(j) for these (i,j): (8, 997), (29, 4227), (33, 11383), (312, 17740), (333, 26637), (522, 9001), (998, 56)

Centers associated with intriangles and extriangles: X(30514)-X(30516)

Contributed by Clark Kimberling and Peter Moses, January 1, 2019.

Following TCCT (page 196), the intriangle of a point P = p: q : r (barycentrics) is the central triangle having A-vertex

A' = 0 : b (c q + b r cos A) : c (b r + c q) cos A

and the extriangle of P is the central triangle having A-vertex

A'' = -a p (c q + b r cos A)(b r + c q cos A) : a q (c q + b r cos A)(a r + c p cos B) : a p (a r + c q cos A)(a q + b p cos C)

Properties of A'B'C' and A'''B'''C'':

1. A'B'C' is inscribed in ABC, which is inscribed in A''B''C''.

2. The locus of P for which A'B'C' is perspective to ABC is the Darboux cubic, K004; the locus of the perspector is the Thomson cubic, K002.

3. The locus of P for which A''B''C'' is perspective to ABC is the union of the circumcircle, the lines BC, CA, AB, the line at infinity, and the cubic K004. If P is on the line at infinity, then the perspector is on the cubic K162.

4. The locus of P for which A''B''C'' is perspective to the cevian triangle of P is the same as for property 3.

5. The locus of P for which A'B'C' is perspective to the anticevian triangle of P is the union of the lines BC, CA, AB, and the cubic K004.

6. A'B'C'and A''B''C'' are perspective with perspector X(6).

7. The intriangle of X(2) is perspective to the circumsymmedial triangle at X(3).

8. The intriangle of X(6) is perspective to the 1st Ehrmann triangle (see X(8537) at X(12027); to the Artzt triangle (see X(9742) at X(6776), and to the anti-Honsberger triangle at X(184).

X(30514) = ORTHOLOGIC CENTER: INTRIANGLE OF X(6) TO 4TH BROCARD TRIANGLE

Barycentrics (4 a^6-a^4 b^2-4 a^2 b^4+b^6-a^4 c^2+6 a^2 b^2 c^2-b^4 c^2-4 a^2 c^4-b^2 c^4+c^6) (5 a^10-22 a^8 b^2+14 a^6 b^4+20 a^4 b^6-19 a^2 b^8+2 b^10-22 a^8 c^2+83 a^6 b^2 c^2-48 a^4 b^4 c^2-19 a^2 b^6 c^2+14 b^8 c^2+14 a^6 c^4-48 a^4 b^2 c^4+60 a^2 b^4 c^4-16 b^6 c^4+20 a^4 c^6-19 a^2 b^2 c^6-16 b^4 c^6-19 a^2 c^8+14 b^2 c^8+2 c^10) : :

X(30514) lies on these lines: {111,381}, {3849,10295}

X(30515) = ORTHOLOGIC CENTER: INTRIANGLE OF X(6) TO CIRCIUMSYMMEDIAL TRIANGLE

Barycentrics a^2 (a^2 b^2-b^4+a^2 c^2+3 b^2 c^2-c^4) (2 a^8 b^2-2 a^6 b^4-2 a^4 b^6+2 a^2 b^8+2 a^8 c^2-29 a^6 b^2 c^2+38 a^4 b^4 c^2+a^2 b^6 c^2-4 b^8 c^2-2 a^6 c^4+38 a^4 b^2 c^4-22 a^2 b^4 c^4+4 b^6 c^4-2 a^4 c^6+a^2 b^2 c^6+4 b^4 c^6+2 a^2 c^8-4 b^2 c^8) : :

X(30515) lies on these lines: {353,10564}, {575,6235}, {8705,14867}, {9027,12505}

X(30516) = CENTROID OF INTRIANGLE OF X(6)

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

X(30516) lies on these lines: {2, 11166}, {39, 647}, {182, 9745}, {353, 6032}, {373, 10418}, {524, 10160}, {574, 13857}, {575, 6792}, {597, 6791}, {1495, 5475}, {1499, 12506}, {2502, 7603}, {3589, 16317}, {3815, 5642}, {3849, 10166}, {5913, 10168}, {7619, 9127}, {7736, 8779}, {9830, 10162}, {11550, 15880}, {12036, 12040}, {13394, 18907}

X(30516) = midpoint of X(353) and X(6032)
X(30516) = reflection of X(i) and X(j) for these {i,j}: {10163, 10160}, {13378, 10162}
X(30516) = crossdifference of every pair of points on line {23, 9871}
X(30516) = barycentric product X(i)*X(j) for these {i,j}: {599, 18907}, {5094, 13394}
X(30516) = barycentric quotient X(18907) / X(598)

X(30517) = COMPLEMENT OF X(22049)

Barycentrics SA*((-64*R^2+SA+15*SW)*S^2+4*(SB+SC)*(4*R^2-SW)*(4*SA-32*R^2+7*SW)) : :

As a point on the Euler line, X(30517) has Shinagawa coefficients (4(E-3F)F-S², -4(E-7F)F+S²).

See Tran Quang Hung and César Lozada, Hyacinthos 28777.

X(30517) lies on this line: {2, 3}

X(30517) = reflection of X(i) in X(j) for these (i,j): (16273, 15948), (18017, 3)
X(30517) = complement of X(22049)
X(30517) = {X(15948), X(16273)}-harmonic conjugate of X(376)

X(30518) = COMPLEMENT OF X(22050)

Barycentrics 216*S^4-3*(R^2*(113*R^2+44*SW)-168*SB*SC-4*SW^2)*S^2-(7*R^2*(R^2-20*SW)-92*SW^2)*SB*SC : :

As a point on the Euler line, X(30518) has Shinagawa coefficients (675E²+144EF-192F²-3456S², -2025E²-3504EF-1472F²-8064S²).

See Tran Quang Hung and César Lozada, Hyacinthos 28777.

X(30518) lies on this line: {2, 3}

X(30518) = complement of X(22050)

X(30519) = (name pending)

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

X(30519) lies on these lines: {30,511}, {39,6586}, {76,3261}, {194,21225}, {693,4931}, {1459,5145}, {3239,21212}, {3700,3776}, {3762,20906}, {3835,4120}, {3960,21348}, {4024,21116}, {4025,26248}, {4079,22037}, {4122,24720}, {4391,20908}, {4468,21196}, {4474,9902}, {4500,21104}, {4928,4944}, {6546,27486}, {7265,22043}, {14425,17069}, {20907,21443}

X(30519) = isogonal conjugate of X(30554)

X(30520) = (name pending)

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

X(30520) lies on these lines: {30,511}, {312,693}, {650,3752}, {2490,7658}, {2509,3669}, {2516,11068}, {2526,4088}, {3004,4468}, {3210,17494}, {3700,23813}, {3776,4885}, {3777,21349}, {4024,22034}, {4025,4394}, {4106,25259}, {4171,17458}, {4379,21115}, {4382,4820}, {4391,4408}, {4462,15413}, {4728,4944}, {4949,20295}, {6590,21104}, {18071,21438}

X(30520) = isogonal conjugate of X(30555)
>

X(30521) = X(2)X(11147)∩X(187)X(11258)

Barycentrics (SB+SC)*(27*S^4-9*(3*(6*SA+SW)*R^2-3*SA^2-6*SB*SC+SW^2)*S^2+(6*SA+SW)*SW^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30521) lies on these lines: {2, 11147}, {187, 11258}, {2930, 8586}, {5210, 14262}

X(30522) = ISOGONAL CONJUGATE OF X(22751)

Barycentrics (5*R^2+SA-2*SW)*S^2-5*(3*R^2-SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30522) lies on these lines: {3, 12278}, {4, 49}, {5, 13367}, {26, 12293}, {30, 511}, {68, 15138}, {110, 18403}, {125, 15646}, {143, 3575}, {186, 265}, {343, 550}, {382, 1993}, {403, 10113}, {546, 13403}, {568, 18559}, {1147, 18377}, {1495, 11563}, {1511, 2072}, {1568, 18572}, {1658, 9927}, {2070, 12902}, {2071, 12121}, {3153, 12383}, {3448, 13619}, {3618, 18420}, {3763, 7514}, {3853, 12897}, {5448, 18567}, {5449, 15331}, {5480, 19155}, {5576, 22804}, {5609, 18323}, {5654, 18568}, {5876, 14516}, {5899, 12412}, {5944, 10024}, {5946, 12022}, {6101, 12225}, {6102, 6240}, {6146, 13630}, {6241, 18565}, {6644, 18396}, {7564, 11425}, {7574, 15132}, {7577, 18430}, {7689, 18356}, {10095, 12241}, {10149, 12896}, {10151, 12140}, {10224, 12038}, {10226, 20299}, {10254, 11464}, {10255, 11449}, {10264, 21663}, {10282, 13406}, {10296, 23236}, {10610, 13160}, {10733, 14157}, {11250, 18381}, {11459, 18564}, {11591, 12605}, {12106, 18390}, {12111, 18562}, {12112, 12419}, {12118, 15139}, {12161, 12173}, {13142, 16982}, {13399, 16111}, {13490, 16657}, {14852, 18324}, {15114, 15122}, {18474, 18570}, {18945, 18952}, {19205, 19211}

X(30522) = isogonal conjugate of X(22751)

X(30523) = X(1153)X(3054)∩X(8588)X(14650)

Barycentrics (SB+SC)*(567*S^6-27*(6*(18*SA-SW)*R^2-21*SA^2+30*SB*SC+SW^2)*S^4-3*(9*SA^2+18*SB*SC+2*(9*R^2-5*SW)*SW)*SW^2*S^2+2*SW^6) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30523) lies on these lines: {1153, 3054}, {8588, 14650}

X(30523) = reflection of X(11841) in the line X(28585)X(29235)

X(30524) = CIRCUMCIRCLE INVERSE OF X(28447)

Barycentrics 3*(S^2-3*SB*SC)*a*b*c - 4*(SB+SC)*OH*SA*S : :
X(30524) = 3*X(2100)+X(7982), 3*X(2104)-5*X(11482), X(11477)+3*X(15162)

As a point on the Euler line, X(30524) has Shinagawa coefficients [-3*R+2*OH, 9*R-2*OH] .

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30524) lies on these lines: {2, 3}, {2100, 7982}, {2104, 11482}, {2574, 5609}, {11477, 15162}

X(30524) = midpoint of X(i) and X(j) for these {i,j}: {3, 15157}, {10751, 15160}
X(30524) = reflection of X(20409) in X(140)
X(30524) = circumperp conjugate of X(28448)
X(30524) = circumcircle-inverse-of X(28447)
X(30524) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15154, 15157), (1113, 1114, 28447), (1113, 15157, 3)

X(30525) = CIRCUMCIRCLE INVERSE OF X(28448)

Barycentrics 3*(S^2-3*SB*SC)*a*b*c + 4*(SB+SC)*OH*SA*S : :

As a point on the Euler line, X(30525) has Shinagawa coefficients [-3*R-2*OH, 9*R+2*OH] .

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30525) lies on these lines: {2, 3}, {2101, 7982}, {2105, 11482}, {2575, 5609}, {11477, 15163}

X(30525) = midpoint of X(i) and X(j) for these {i,j}: {3, 15156}, {10750, 15161}
X(30525) = reflection of X(20408) in X(140)
X(30525) = circumperp conjugate of X(28447)
X(30525) = circumcircle-inverse-of X(28448)
X(30525) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15155, 15156), (140, 20408, 13626), (1113, 1114, 28448)

X(30526) = TRILINEAR POLE OF THE LINE X(1510)X(10095)

Barycentrics (2*SB+R^2) *(2*SC+R^2)*(2*SB-3*R^2)*(2*SC-3*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30526) lies on these lines: {1263, 27684}, {1994, 11538}

X(30526) = trilinear pole of the line {1510, 10095}
X(30526) = barycentric quotient X(1263)/X(21230)

X(30527) = TRILINEAR POLE OF THE LINE X(30)X(54)

Barycentrics (SA-SB)*(SA-SC)*(S^2+SA*SB)*(S^2+SA*SC)*(S^2+5*SB^2)*(S^2+5*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30527) lies on these lines: {933, 4240}, {2407, 18315}, {9214, 11815}

X(30527) = trilinear pole of the line {30, 54}
X(30527) = barycentric product X(99)*X(11815)
X(30527) = barycentric quotient X(i)/X(j) for these (i,j): (110, 11591), (933, 3520)
X(30527) = trilinear product X(662)*X(11815)
X(30527) = trilinear quotient X(662)/X(11591)

X(30528) = TRILINEAR POLE OF THE LINE X(30)X(110)

Barycentrics (SA-SB)*(SA-SC)*(3*S^2-18*R^2*SB-2*SA*SC+5*SB^2)*(3*S^2-18*R^2*SC-2*SA*SB+5*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30528) lies on these lines: {249, 2407}, {250, 4240}, {376, 477}, {648, 5664}, {687, 15466}, {2421, 2436}, {14590, 23582}

X(30528) = trilinear pole of the line {30, 110}

X(30529) = TRILINEAR POLE OF THE LINE X(143)X(1510)

Barycentrics (3*S^2-SA^2)*(S^2-3*SB^2)*(S^2-3*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30529) lies on these lines: {2, 94}, {61, 8838}, {62, 8836}, {265, 3091}, {476, 5966}, {1141, 1166}, {3542, 6344}, {3832, 18300}, {3839, 18316}, {4232, 18384}, {7110, 18359}, {7533, 14356}, {7578, 9220}, {7785, 11004}, {15226, 25044}

X(30529) = polar conjugate of X(562)
X(30529) = trilinear pole of line {143, 1510}
X(30529) = barycentric product X(i)*X(j) for these {i,j}: {49, 18817}, {94, 1994}, {328, 3518}
X(30529) = barycentric quotient X(i)/X(j) for these (i,j): (4, 562), (49, 22115), (94, 11140), (143, 1154), (265, 3519), (476, 930)
X(30529) = trilinear product X(94)*X(2964)
X(30529) = trilinear quotient X(i)/X(j) for these (i,j): (92, 562), (94, 2962), (143, 2290)
X(30529) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (94, 18883, 2), (1989, 18883, 94)

X(30530) = TRILINEAR POLE OF THE LINE X(511)X(6033)

Barycentrics (SA-SB)*(SA-SC)*(5*S^4-(4*SA*SC-SB^2-SW^2)*S^2+(5*SB-4*SW)*SB*SW^2)*(5*S^4-(4*SA*SB-SC^2-SW^2)*S^2+(5*SC-4*SW)*SC*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30530) lies on this line: {5968, 7778}

X(30530) = trilinear pole of the line {511, 6033}

X(30531) = MIDPOINT OF X(5) AND X(11803)

Barycentrics 2 a^10-13 a^8 b^2+24 a^6 b^4-14 a^4 b^6-2 a^2 b^8+3 b^10-13 a^8 c^2+14 a^6 b^2 c^2+11 a^4 b^4 c^2-3 a^2 b^6 c^2-9 b^8 c^2+24 a^6 c^4+11 a^4 b^2 c^4+10 a^2 b^4 c^4+6 b^6 c^4-14 a^4 c^6-3 a^2 b^2 c^6+6 b^4 c^6-2 a^2 c^8-9 b^2 c^8+3 c^10 : :
Barycentrics (19*R^2+6*SA-10*SW)*S^2+(23*R^2-8*SW)*SB*SC : :
X(30531) = X(3)-3*X(8254), X(3)+3*X(20424), 5*X(5)-X(3519), 3*X(5)+X(15801), 3*X(54)+X(3627), 3*X(195)+5*X(3091), X(546)-3*X(3574), X(546)+3*X(22051), 5*X(546)-3*X(22804), 3*X(1209)-5*X(12812), X(1493)+3*X(3574), X(1493)-3*X(22051), 5*X(1493)+3*X(22804), 3*X(2888)-11*X(5072), 7*X(3090)-3*X(21230), X(3519)+5*X(11803), 3*X(3519)+5*X(15801), 5*X(3574)-X(22804), 3*X(11803)-X(15801), 5*X(22051)+X(22804)

See Antreas Hatzipolakis, Ercole Suppa and César Lozada, Hyacinthos 28785 and Hyacinthos 28786

X(30531) lies on these lines: {3, 8254}, {4, 17507}, {5, 1173}, {30, 12242}, {54, 3627}, {113, 137}, {143, 12010}, {195, 3091}, {539, 3850}, {1154, 3628}, {1209, 12812}, {2888, 5072}, {2914, 11801}, {2918, 17714}, {3090, 21230}, {3518, 15806}, {3525, 12307}, {3851, 11271}, {3853, 10619}, {3857, 6288}, {5076, 12254}, {5079, 12316}, {6152, 13451}, {6689, 12108}, {7691, 14869}, {10272, 25714}, {10610, 12103}, {12102, 18400}, {12325, 15022}, {12606, 14449}, {12811, 20584}, {13432, 19709}, {15425, 16337}, {24144, 27423}

X(30531) = midpoint of X(i) and X(j) for these {i,j}: {5, 11803}, {546, 1493}, {2914, 11801}, {3574, 22051}, {3853, 10619}, {8254, 20424}, {12606, 14449}
X(30531) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (546, 22051, 1493), (1493, 3574, 546)

X(30532) = X(549)X(29959)∩X(21248)X(22110)

Barycentrics 4 a^8-18 a^6 b^2+25 a^4 b^4-12 a^2 b^6+b^8-18 a^6 c^2+22 a^4 b^2 c^2+28 a^2 b^4 c^2-6 b^6 c^2+25 a^4 c^4+28 a^2 b^2 c^4+10 b^4 c^4-12 a^2 c^6-6 b^2 c^6+c^8 : :
Barycentrics 15 S^2-18 R^2 SB-18 R^2 SC-9 SB SC+24 R^2 SW+3 SB SW+3 SC SW+2 SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30532) lies on these lines: {549,29959}, {21248,22110}

X(30533) = X(5)X(10216)∩X(140)X(6153)

Barycentrics -2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^12-6 a^10 b^2+14 a^8 b^4-16 a^6 b^6+9 a^4 b^8-2 a^2 b^10-6 a^10 c^2+18 a^8 b^2 c^2-10 a^6 b^4 c^2-11 a^4 b^6 c^2+10 a^2 b^8 c^2-b^10 c^2+14 a^8 c^4-10 a^6 b^2 c^4-11 a^4 b^4 c^4-8 a^2 b^6 c^4+4 b^8 c^4-16 a^6 c^6-11 a^4 b^2 c^6-8 a^2 b^4 c^6-6 b^6 c^6+9 a^4 c^8+10 a^2 b^2 c^8+4 b^4 c^8-2 a^2 c^10-b^2 c^10) : :
Barycentrics 3 S^4 + (-5 R^4-9 R^2 SB-9 R^2 SC-SB SC+R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 + (-R^4-R^2 SW+SW^2)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30533) lies on these lines: {5,10216}, {140,6153}, {523,3628}, {1209,13856}, {9827,13467}

X(30534) = X(182)X(1992)∩X(184)X(7736)

Barycentrics 2 a^4 b^2 c^2 (a^8-5 a^6 b^2+5 a^4 b^4-a^2 b^6-5 a^6 c^2+7 a^4 b^2 c^2+15 a^2 b^4 c^2-3 b^6 c^2+5 a^4 c^4+15 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-3 b^2 c^6) : :
Barycentrics 3 S^4 + (-5 R^4-9 R^2 SB-9 R^2 SC-SB SC+R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 + (-R^4-R^2 SW+SW^2)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30534) lies on these lines: {182,1992}, {184,7736}, {575,4558}, {7709,15033}

X(30535) = ISOGONAL CONJUGATE OF X(3815)

Barycentrics a^2 (a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2) (a^4-3 a^2 b^2-2 a^2 c^2-3 b^2 c^2+c^4) : :
Barycentrics (2 SB+2 SC+SW)S^2 -SB SC SW + 2 R^2 SW^2 + SB SW^2 + SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30535) lies on these lines: {2,5034}, {25,5012}, {37,26639}, {39,2987}, {97,14965}, {111,15018}, {182,263}, {251,1692}, {575,1976}, {597,1989}, {694,5038}, {1994,3108}, {2165,3618}, {2456,11673}, {2963,3589}, {8770,10601}, {8791,14389}, {9178,21460}

X(30535) = isogonal conjugate of X(3815)
X(30535) = cevapoint of X(6) and X(182)
X(30535) = trilinear pole of line X(512)X(2080)

X(30536) = X(186)X(6152)∩X(523)X(8254)

Barycentrics -a^2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-3 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-3 b^6 c^2+2 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4+a^2 c^6+b^2 c^6-c^8) (a^8-3 a^6 b^2+2 a^4 b^4+a^2 b^6-b^8-4 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2+b^6 c^2+6 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-3 b^2 c^6+c^8) : :
Barycentrics (9 R^2-SB-SC-3 SW)S^4 + (-13 R^6-43 R^4 SB-43 R^4 SC+21 R^4 SW+30 R^2 SB SW+30 R^2 SC SW-9 R^2 SW^2-5 SB SW^2-5 SC SW^2+SW^3+SB SC (-7 R^2+3 SW))S^2 + (-R^6-R^4 SW+3 R^2 SW^2-SW^3)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30536) lies on these lines: {186,6152}, {523,8254}, {567,15620}, {1209,13856}, {6288,9221}, {13434,14979}

X(30537) = ISOGONAL CONJUGATE OF X(15018)

Barycentrics a^4 b^4 c^4 (a^4-2 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4) (a^4-5 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : :
Barycentrics 7 S^2+9 R^2 SB+9 R^2 SC-3 SB SC+3 SB SW+3 SC SW : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30537) lies on these lines: {2,13337}, {6,5054}, {37,3582}, {39,1989}, {50,251}, {111,3815}, {308,7799}, {393,13351}, {549,13338}, {597,2987}, {1383,7736}, {2165,7739}, {2963,5421}, {3108,5306}, {6128,9698}, {8749,9606}

X(30537) = isogonal conjugate of X(15018)

X(30538) = MIDPOINT OF X(1) AND X(15446)

Barycentrics a^3 b^2 (a-b-c) c^2 (a+b+c)^3 (2 a^5-a^4 b-4 a^3 b^2+2 a^2 b^3+2 a b^4-b^5-a^4 c+6 a^3 b c-2 a^2 b^2 c-5 a b^3 c+4 b^4 c-4 a^3 c^2-2 a^2 b c^2+6 a b^2 c^2-3 b^3 c^2+2 a^2 c^3-5 a b c^3-3 b^2 c^3+2 a c^4+4 b c^4-c^5) : :
Barycentrics (2 SB+2 SC+SW)S^2 - SB SC SW+2 R^2 SW^2+SB SW^2+SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30538) lies on these lines: {1,1399}, {10,2646}, {11,1385}, {55,3885}, {214,17606}, {946,1319}, {1001,10394}, {1071,11715}, {1388,12114}, {1459,24457}, {1737,26287}, {1837,15079}, {2320,3486}, {3583,11376}, {3601,11525}, {3646,13384}, {3649,24928}, {3916,5048}, {5542,20323}, {10543,10959}, {12743,24387}, {12758,26087}

X(30538) = midpoint of X(1) and X(15446)

X(30539) = X(141)X(574)∩X(525)X(11168)

Barycentrics 4 a^10-11 a^8 b^2-2 a^6 b^4+22 a^4 b^6-14 a^2 b^8+b^10-11 a^8 c^2+16 a^6 b^2 c^2+4 a^2 b^6 c^2-5 b^8 c^2-2 a^6 c^4+36 a^2 b^4 c^4+4 b^6 c^4+22 a^4 c^6+4 a^2 b^2 c^6+4 b^4 c^6-14 a^2 c^8-5 b^2 c^8+c^10 : :
Barycentrics (54 R^2+3 SB+3 SC-18 SW)S^2 + (-54 R^2+12 SW)SB SC + 18 R^2 SB SW+18 R^2 SC SW-3 SB SW^2-3 SC SW^2-2 SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30539) lies on these lines: {141,574}, {525,11168}

X(30540) = X(2)X(575)∩X(2395)X(11166)

Barycentrics 2 a^4 b^4 c^4 (4 a^10-12 a^8 b^2+14 a^6 b^4-6 a^4 b^6-12 a^8 c^2-a^6 b^2 c^2+16 a^4 b^4 c^2-11 a^2 b^6 c^2-2 b^8 c^2+14 a^6 c^4+16 a^4 b^2 c^4+22 a^2 b^4 c^4+2 b^6 c^4-6 a^4 c^6-11 a^2 b^2 c^6+2 b^4 c^6-2 b^2 c^8) : :
Barycentrics (-9 SB-9 SC-7 SW)S^4 + (3 SB SC SW+12 R^2 SW^2+3 SB SW^2+3 SC SW^2-3 SW^3)S^2 -SB SC SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30540) lies on these lines: {2,575}, {2395,11166}

X(30541) = ISOGONAL CONJUGATE OF X(7737)

Barycentrics a^2 (a^4-2 a^2 b^2+b^4-3 c^4) (a^4-3 b^4-2 a^2 c^2+c^4) : :
Barycentrics (12 R^2+SB+SC-4 SW)S^2 + (-18 R^2+4 SW)SB SC + 6 R^2 SB SW+6 R^2 SC SW-2 R^2 SW^2-SB SW^2-SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30541) lies on these lines: {3,11653}, {23,14906}, {183,525}, {297,11185}, {378,511}, {574,15066}, {599,6393}, {1078,9289}, {3455,6800}, {20977,21399}

X(30541) = isogonal conjugate of X(7737)

X(30542) = ISOGONAL CONJUGATE OF X(11002)

Barycentrics a^4 b^4 c^4 (2 a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-2 b^2 c^2) (2 a^4-2 a^2 b^2-3 a^2 c^2-2 b^2 c^2+2 c^4) : :
Barycentrics 7 S^4 + (9 R^2 SB+9 R^2 SC-3 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 + SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30542) lies on these lines: {50,5094}, {182,599}, {183,7496}, {186,2453}, {187,18575}, {7771,11643}, {7778,10130}

X(30542) = isogonal conjugate of X(11002)

Centers associated with duple triangles: X(30543)-X(30550)

Suppose that A'B'C' is a central triangle in the plane of a reference triangle ABC, and that barycentrics for A' are u : v : w. The duple of A'B'C' is here introduced as the central triangle A''B''C'' having A'' = u : w : v; for example, the duple of the excentral triangle has A'' = -a : c : b. It is easy to prove that if two triangles are perspective, then their duples are perspective. (Clark Kimberling, January 1, 2019)

Peter Moses found the following perspectivities for the duple of the excentral triangle. (January 1, 2019):

ABC (TCCT 6.1): X(76)
medial (TCCT 6.2): X(6)
excentral (TCCT 6.7): X(3509)
half altitude / mid-height (TCCT 6.38): X(6)
second Neuberg (MathWorld): X(511)
first Brocard (CTC): X(76)
anti-first-Brocard (see ETC X(5939)): X(8782)
inner inscribed squares (MathWorld): X(6)
outer inscribed squares (MathWorld): X(6)
submedial (see ETC X(9813)): X(6)
Gemini 40: X(30543)
Gemini 42: X(141)
Gemini 75: X(30544)

See also the lists at X(30545), X(30547), and X(30556).

X(30543) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-EXCENTRAL AND GEMINI 40

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

X(30543) lies on these lines: {1, 26051}, {2, 4642}, {8, 12}, {10, 26136}, {73, 4861}, {75, 17084}, {145, 26137}, {946, 16824}, {3120, 5484}, {3616, 4000}, {3878, 25446}, {4054, 9369}, {4384, 11522}, {4673, 28628}, {4714, 5443}, {5260, 17777}, {5835, 9780}, {5836, 28389}, {6533, 16173}, {11376, 19804}, {12047, 16821}, {12053, 16823}, {16817, 30384}

X(30543) = X(4673), X(28628)}-harmonic conjugate of X(29839)

X(30544) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-EXCENTRAL AND GEMINI 75

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

X(30544) lies on these lines: {75, 21008}, {239, 27995}

X(30545) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND INTOUCH

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

The A-vertex of the duple-of-anticevian-triangle-of-X(75) is A'' = -b c : a b : a c. Among its perspectivities with other triangles are these, all with perspector X(1):

ABC, media, incentral, excentral, mid-arc, 2nd midarc, 2nd circumperp, inner mixtilinears, outer mixtilinear, Andromeda, Antila, Aquilla, Caelum, innear Malfatti, outer Malfatti, inverse-of-ABC in circle, inner Yff, outer Yff, anti-Aquilla, 4th Conway, 5th Conway, inner Yff tangents, outer Yff tangents, Gemini 15. In addition to those, Peter Moses found the following perspectivities for A''B''C'' (January 5, 2019):

intouch (TCCT 6.8): X(30545)
2nd extouch (ETC X(5927)): X(30546)
2nd Conway (ETC X(9776)): X(30547)
Gemini 60: X(30548)

X(30545) lies on these lines: {1,18299}, {2,10030}, {7,350}, {33,18026}, {57,4554}, {65,18832}, {75,325}, {76,85}, {181,18057}, {194,28391}, {331,1848}, {335,1088}, {348,4352}, {497,6604}, {518,20935}, {693,3873}, {982,3663}, {1432,3978}, {1463,17082}, {1469,17149}, {1502,17786}, {1699,2481}, {2171,20567}, {3212,6376}, {3673,3944}, {4008,5988}, {4052,10029}, {4110,6382}, {4872,10446}, {5219,7243}, {6649,9316}, {7033,24524}, {7201,7205}, {8055,18135}, {16593,18045}, {17095,19786}, {17181,20256}, {21404,29641}, {22015,22019}

X(30546) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND 2ND EXTOUCH

Barycentrics 2 a^5 b^3-2 a^3 b^5+a^5 b^2 c-2 a^3 b^4 c+a b^6 c+a^5 b c^2-3 a^4 b^2 c^2-2 a^3 b^3 c^2-a b^5 c^2-b^6 c^2+2 a^5 c^3-2 a^3 b^2 c^3-2 a^3 b c^4+2 b^4 c^4-2 a^3 c^5-a b^2 c^5+a b c^6-b^2 c^6 : : X(30546) lies on these lines: {9,17793}, {72,19222}, {226,262}, {440,20254}, {1513,7179}, {5928,21334}, {7350,7413}, {20256,27184}

X(30547) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND 2ND CONWAY

Barycentrics a^4 b^2-a^2 b^4-a^4 b c-a^3 b^2 c-a^2 b^3 c+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+2 b^3 c^3-a^2 c^4+a b c^4-b^2 c^4 : : X(30547) lies on these lines: {2,7167}, {7,256}, {8,3978}, {329,1655}, {3794,9309}, {4388,17137}, {4594,8033}

X(30548) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND GEMINI 60

Barycentrics a^4 b^4-a^4 b^3 c+a^3 b^4 c-4 a^4 b^2 c^2-5 a^3 b^3 c^2-2 a^2 b^4 c^2-a^4 b c^3-5 a^3 b^2 c^3+a^2 b^3 c^3-a b^4 c^3+a^4 c^4+a^3 b c^4-2 a^2 b^2 c^4-a b^3 c^4+b^4 c^4 : : X(30548) lies on these lines: {2,22167}, {192,6378}

X(30549) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-X(3)-REFLECTION-OF-ABC AND TANGENTIAL

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

The A-vertex of the duple-of-X(3)-reflection-of-ABC is A'' = -(a^2+b^2-c^2) (a^2-b^2+c^2) : 2 c^2 (a^2+b^2-c^2) : -2 b^2 (-a^2+b^2-c^2). Among its perspectivities with other triangles are these (Peter Moses, January 5, 2019):

ABC: X(264)
medial: X(1249)
anticomplementary: X(193)
orthic: X(3193)
Macbeath: X(264)
Artzt (ETC X(9742)) : X(523)
tangential: X(30549)
anti-Ascella (ETC X(11363): X(30550)

X(30549) lies on these lines: {6,1632}, {20,64}, {154,3164}, {230,393}, {264,9747}, {1368,6389}, {2847,6716}, {3053,3186}, {3515,15653}, {8667,9909}, {8716,20794}

X(30550) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-X(3)-REFLECTION-OF-ABC AND ANTI-ASCELLA

Barycentrics 3 a^8-9 a^6 b^2+13 a^4 b^4-7 a^2 b^6-9 a^6 c^2+6 a^4 b^2 c^2-a^2 b^4 c^2+8 b^6 c^2+13 a^4 c^4-a^2 b^2 c^4-16 b^4 c^4-7 a^2 c^6+8 b^2 c^6 : :

X(30550) lies on these lines: {2,27364}, {20,3564}, {427,1007}, {2165,3054}, {2974,11404}, {7494,18287}, {8266,8667}, {14570,19118}

X(30551) = EULER LINE INTERCEPT OF X(54)X(20193)

Barycentrics 4 a^10-9 a^8 b^2+2 a^6 b^4+8 a^4 b^6-6 a^2 b^8+b^10-9 a^8 c^2+4 a^6 b^2 c^2-9 a^4 b^4 c^2+17 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-9 a^4 b^2 c^4-22 a^2 b^4 c^4+2 b^6 c^4+8 a^4 c^6+17 a^2 b^2 c^6+2 b^4 c^6-6 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics (41 R^2-10 SW)S^2 + (R^2+6 SW)SB SC : :

As a point on the Euler line, X(30551) has Shinagawa coefficients {E - 40 F, 25 E + 24 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30551) lies on these lines: {2,3}, {54,20193}, {5642,16982}

X(30551) = reflection of X(5) in X(21451)
X(30551) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,23409,5}, {5,2070,550}, {140,21308,5}, {3845,6240,3627}, {10096,13621,5}, {13595,18282,5}

X(30552) = EULER LINE INTERCEPT OF X(69)X(11440)

Barycentrics -5 a^10+9 a^8 b^2+2 a^6 b^4-10 a^4 b^6+3 a^2 b^8+b^10+9 a^8 c^2-32 a^6 b^2 c^2+18 a^4 b^4 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4+18 a^4 b^2 c^4-22 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+3 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics (10 R^2-2 SW)S^2 + (-16 R^2+3 SW)SB SC : :

As a point on the Euler line, X(30552) has Shinagawa coefficients {E - 4 F, -2 E + 6 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30552) lies on these lines: {2,3}, {69,11440}, {74,11411}, {110,6225}, {343,8567}, {394,5894}, {925,5897}, {1092,20427}, {1204,6515}, {1294,13398}, {5012,15740}, {5504,16111}, {5895,11064}, {5925,20725}, {9140,15077}, {9833,16163}, {11206,12279}, {11441,12250}, {12324,13445}, {14457,18911}, {15072,18925}

X(30552) = reflection of X(i) in X(j) for these {i,j}: {4,3548}, {3542,3}
X(30552) = anticomplement of X(37197)
X(30552) = X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,382,16238}, {3,1885,2}, {20,2071,4}, {20,3522,22}, {20,7396,5059}, {20,11413,1370}, {22,858,4232}, {550,21312,20}, {11413,16386,20}

X(30553) = (name pending)

Barycentrics -10 a^10+21 a^8 b^2-2 a^6 b^4-20 a^4 b^6+12 a^2 b^8-b^10+21 a^8 c^2-18 a^6 b^2 c^2+47 a^4 b^4 c^2-53 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4+47 a^4 b^2 c^4+82 a^2 b^4 c^4-2 b^6 c^4-20 a^4 c^6-53 a^2 b^2 c^6-2 b^4 c^6+12 a^2 c^8+3 b^2 c^8-c^10 : :
Barycentrics (115 R^2-22 SW)S^2 + (-R^2+18 SW)SB SC : :

As a point on the Euler line, X(30553) has Shinagawa coefficients {27 E - 88 F, 71 E + 72 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30553) lies on this line: {2,3}

X(30553) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {140,5899,548}, {547,13163,546}

X(30554) = ISOGONAL CONJUGATE OF X(30519)

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

X(30554) lies on the circumcircle and these lines: {6,28563}, {32,106}, {103,182}, {105,10789}, {753,4279}, {1477,12835}, {1691,2712}, {2080,2700}, {12212,28485}

X(30554) = isogonal conjugate of X(30519)
X(30554) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30519}, {513, 17230}, {9461, 20568}
X(30554) = cevapoint of X(649) and X(5332)
X(30554) = trilinear pole of line {6, 9459}
X(30554) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 30519}, {101, 17230}, {9459, 9461}

X(30555) = ISOGONAL CONJUGATE OF X(30520)

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

X(30555) lies on the circumcircle and these lines: {604,1477}, {692,6078}, {907,5546}, {5549,8695}

X(30555) = isogonal conjugate of X(30520)
X(30555) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30520}, {513, 17284}, {514, 3242}, {3669, 4901}
X(30555) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 30520}, {101, 17284}, {692, 3242}, {3939, 4901}

X(30556) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND CEVIAN OF X(13386)

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

Let A'B'C' be the duple of the half-altitude triangle, so that

A' = a^2 : a^2 - b^2 + c^2 : a^2 + b^2 - c^2.

The locus of a point P such that A'B'C' is perpsective to the cevian triangle of P is the cubic K170.
The locus of a point P such that A'B'C' is perpsective to the anticevian triangle of P is the cubic K707.
The locus of a point P such that A'B'C' is orthologic to the cevian triangle of P is the cubic K170.
The locus of a point P such that A'B'C' is orthologic to the anticevian triangle of P is the cubic K045.
The locus of a point P such that A'B'C' is paralogic to the cevian triangle of P is the cubic K211.
The locus of a point P such that A'B'C' is paralogic to the anticevian triangle of P is a cubic; see just below for an equation.
(Peter Moses, January 8, 2019)

Let f(a,b,c,x,y,z) = a^2 (b^2 - c^2) (-a^2 + b^2 + c^2) (y - z) y z. The cubic mentioned just above is given by the following equation:

f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) - 2 a^2 b^2 c^2 x y z = 0.

The following perspectivities were contributed by Peter Moses, January 5, 2019: A'B'C' is perspective to the following triangles, with perspector X(2): orthic, submedial (see (9813)), orthic-of-anticomplementary.
A'B'C' is perspective to the following triangles, with perspector X(3): medial, tangential, 1st circumperp, 2nd circumperp, outer Napoleon, inner Napoleon, outer Fermat, inner Fermat, outer Vecten, inner Vector, 1st Neuberg, 2nd Neuberg, Fuhrmann, 1st Brocard, Kosnita, McCay, Trinh, Carnot, 2nd Euler, Ara, 1st Ehrmann, Ascella, Ae, Ai, infinite altitude, anti-Hutson intouch (see X11363)), anti-incircle-circles (see X(11363)), Ehrmann side-triangle The appearance of T,i in the following list means that A'B'C' is perspective to the triangle T, and the perspector is X(i):

ABC,69
reflection of ABC in X(3), 11821
reflection of X(3) in ABC, 11850
extangents, 10319
circum-orthic, 631
inner Garcia, 11512
4th extouch, 69
2nd Ehrmann, 11511
1st Kenmotu diagonal, (see X(31)), 11513
2nd Kenmotu diagonal, 11514
inner tri-equilateral (see X(10631)), 11515
outer tri-equilateral (see X(10631)), 11516
anti-Ascella (see X(11363)), 7484
anti-Conway (see X(11363)), 182
medial-of-orthic, 511
anti-3rd Euler (see X(11363), 7998
anti-4rd Euler (see X(11363), 7999
5th mixtilinear of orthic (see X(11363)), 20
tangential-of-anticomplementary (see X(11363)), 7386
aAOS (see X(15015)), 19378
1st excosine (see X(17807)), 17811
Ehrmann vertex-triangle, 18531
anti-Atik, 69
1st anti-Sharygin, 95

X(30556) lies on the cubics K168, K199, K332 and these lines: {1,6}, {2,175}, {3,6212}, {8,7090}, {10,486}, {21,1805}, {55,7348}, {56,6204}, {63,3083}, {65,6203}, {78,2066}, {142,481}, {144,176}, {169,8225}, {188,3082}, {200,15892}, {329,1659}, {345,13425}, {348,13453}, {371,997}, {372,12514}, {388,30324}, {482,527}, {485,21616}, {517,1377}, {590,25681}, {615,26066}, {936,1702}, {993,13333}, {1123,3421}, {1152,4640}, {1267,6337}, {1329,13911}, {1336,6857}, {1372,20195}, {1374,6173}, {1378,5044}, {1385,9678}, {1806,1812}, {2067,19861}, {2362,3869}, {2551,6351}, {3070,24703}, {3071,5794}, {3084,3305}, {3086,8957}, {3452,5393}, {3485,30325}, {3616,30413}, {3681,15890}, {4517,6405}, {5250,5414}, {5405,5745}, {5409,6513}, {5438,9616}, {5698,6460}, {5784,30400}, {5837,13936}, {5880,30425}, {6172,17805}, {6352,13959}, {6561,17647}, {6700,13912}, {8945,17594}, {13941,18231}, {15587,30354}, {15891,16214}, {17768,30426}, {17802,18230}, {19029,21677}, {20059,21169}

X(30556) = isogonal conjugate of X(2362)
X(30556) = X(7347)-complementary conjugate of X(141)
X(30556) = X(15890)-Ceva conjugate of X(9)
X(30556) = X(2066)-cross conjugate of X(13389)
X(30556) = cevapoint of X(3) and X(1124)
X(30556) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2362}, {4, 2067}, {6, 1659}, {19, 13388}, {56, 7090}, {57, 7133}, {225, 1805}, {278, 5414}, {1123, 6502}, {2066, 13437}, {6213, 16232}
X(30556) = barycentric product X(i) X(j) for these {i,j}: {8, 13389}, {63, 14121}, {75, 2066}, {78, 13390}, {312, 6502}, {321, 1806}, {345, 16232}, {1267, 7133}, {1806, 321}, {2066, 75}, {2362, 13425}, {3083, 7090}, {6502, 312}, {7090, 3083}, {7133, 1267}, {13389, 8}, {13390, 78}, {13425, 2362}, {14121, 63}, {16232, 345}
X(30556) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 1659}, {3, 13388}, {6, 2362}, {9, 7090}, {48, 2067}, {55, 7133}, {212, 5414}, {605, 6502}, {1124, 13389}, {1806, 81}, {2066, 1}, {2193, 1805}, {2362, 13437}, {5414, 6213}, {6212, 13390}, {6502, 57}, {7133, 1123}, {13389, 7}, {13390, 273}, {14121, 92}, {16232, 278}
X(30556) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 18991}, {1, 5223, 3641}, {1, 16469, 11371}, {1, 19003, 1449}, {8, 30412, 7090}, {37, 7968, 1}, {63, 3083, 13389}, {1279, 5604, 1}

X(30557) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND CEVIAN OF X(13387)

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

X(30557) lies on the cubics K168, K199, K332 and these lines: {1,6}, {2,176}, {3,6213}, {8,14121}, {10,485}, {21,1806}, {55,7347}, {56,6203}, {63,2067}, {65,6204}, {78,5414}, {142,482}, {144,175}, {188,483}, {200,15891}, {329,13390}, {345,13458}, {348,13436}, {371,12514}, {372,997}, {388,30325}, {481,527}, {486,21616}, {517,1378}, {590,26066}, {615,25681}, {936,1703}, {993,13332}, {1123,6857}, {1151,4640}, {1329,13973}, {1336,3421}, {1371,20195}, {1373,6173}, {1377,5044}, {1805,1812}, {2066,5250}, {2551,6352}, {3070,5794}, {3071,24703}, {3083,3305}, {3452,5405}, {3485,30324}, {3579,9679}, {3616,30412}, {3681,15889}, {3781,7594}, {3869,16232}, {4517,6283}, {5391,6337}, {5393,5745}, {5408,6513}, {5698,6459}, {5784,30401}, {5837,13883}, {5880,30426}, {6172,17802}, {6351,13902}, {6502,19861}, {6560,17647}, {6700,13975}, {8941,17594}, {8957,18391}, {8972,18231}, {15587,30355}, {15892,16213}, {17768,30425}, {17805,18230}, {19030,21677}

X(30557) = isogonal conjugate of X(16232)
X(30557) = isotomic conjugate of the polar conjugate of X(7133)
X(30557) = X(7348)-complementary conjugate of X(141)
X(30557) = X(15889)-Ceva conjugate of X(9)
X(30557) = X(5414)-cross conjugate of X(13388)
X(30557) = cevapoint of X(3) and X(1335)
X(30557) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16232}, {4, 6502}, {6, 13390}, {19, 13389}, {56, 14121}, {225, 1806}, {278, 2066}, {1336, 2067}, {2362, 6212}, {5414, 13459}
X(30557) = barycentric product X(i) X(j) for these {i,j}: {8, 13388}, {63, 7090}, {69, 7133}, {75, 5414}, {78, 1659}, {312, 2067}, {321, 1805}, {345, 2362}, {1659, 78}, {1805, 321}, {2067, 312}, {2362, 345}, {3084, 14121}, {5414, 75}, {7090, 63}, {7133, 69}, {13388, 8}, {13458, 16232}, {14121, 3084}, {16232, 13458}
X(30557) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 13390}, {3, 13389}, {6, 16232}, {9, 14121}, {48, 6502}, {212, 2066}, {606, 2067}, {1335, 13388}, {1659, 273}, {1805, 81}, {2066, 6212}, {2067, 57}, {2193, 1806}, {2362, 278}, {5414, 1}, {6213, 1659}, {7090, 92}, {7133, 4}, {13388, 7}, {16232, 13459}

X(30558) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND ANTICEVIAN OF X(6339)

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

X(30558) lies on the cubic K168 and these lines: {2,14248}, {3,15369}, {6,6337}, {69,6342}, {439,19118}, {3926,15525}, {5395,11059}

X(30558) = X(19214)-complementary conjugate of X(141)
X(30558) = X(3)-cross conjugate of X(6337)
X(30558) = X(i)-isoconjugate of X(j) for these (i,j): {1611, 8769}, {2128, 14248}
X(30558) = barycentric product X(i) X(j) for these {i,j}: {193, 6339}, {6339, 193}
X(30558) = barycentric quotient X(i) / X(j) for these {i,j}: {193, 6392}, {3053, 1611}, {3167, 19588}, {6337, 19583}, {6339, 2996}, {10607, 6461}, {15369, 14248}

X(30559) = MIDPOINT OF X(16) AND X(5238)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28802.

X(30559) lies on these lines: {3, 6}, {17, 10616}, {30, 22891}, {533, 14144}, {617, 16530}, {2004, 3131}

X(30559) = midpoint of X(16) and X(5238)
X(30559) = reflection of X(17) in X(10616)
X(30559) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19780, 3107), (16, 21159, 5351), (182, 5206, 30560)

X(30560) = MIDPOINT OF X(15) AND X(5237)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28802.

X(30560) lies on these lines: {3, 6}, {18, 10617}, {30, 22846}, {532, 14145}, {616, 16529}, {2005, 3132}

X(30560) = midpoint of X(15) and X(5237)
X(30560) = reflection of X(18) in X(10617)
X(30560) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19781, 3106), (15, 21158, 5352), (182, 5206, 30559)

Centers associated with the Gemini triangles 1-40: X(30561)-X(30713)

These centers were contributed by Randy Hutson, January 9, 2019. The Gemini triangles are introduced in the preamble just before X(24537).

X(30561) = X(2)X(319)∩X(149)X(214)

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

Let A'₁B'₁C'₁ be as at X(25417). Triangle A'₁B'₁C'₁ is homothetic to the medial triangle at X(30561).

X(30561) lies on these lines: {2, 319}, {149, 214}, {4671, 16826}, {5333, 16777}, {25418, 30563}, {30587, 30589} et al

X(30561) = complement of X(30590)
X(30561) = {X(2), X(30562)}-harmonic conjugate of X(16777)
X(30561) = barycentric product X(30563)*X(30589)
X(30561) = barycentric quotient X(30563)/X(30590)

X(30562) = X(2)X(319)∩X(20)X(1385)

Barycentrics 4 a^3 + 9 a^2 (b + c) + a (5 b^2 + 12 b c + 5 c^2) + 3 b c (b + c) : :

Let A'₁B'₁C'₁ be as at X(25417). Triangle A'₁B'₁C'₁ is homothetic to the anticomplementary triangle at X(30562).

X(30562) lies on these lines: {1, 17163}, {2, 319}, {20, 1385}, {63, 8025}, {81, 25418}, {86, 17147}, {194, 3995}, {5625, 17135} et al

X(30562) = {X(16777), X(30561)}-harmonic conjugate of X(2)

X(30563) = X(2)X(44)∩X(45)X(4671)

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

Let A'₂B'₂C'₂ be as at X(25418). Triangle A'₂B'₂C'₂ is homothetic to the medial triangle at X(30563).

X(30563) lies on these lines: {2, 44}, {45, 4671}, {2475, 3647}, {25418, 30561} et al

X(30563) = complement of X(30589)
X(30563) = {X(2), X(30564)}-harmonic conjugate of X(89)
X(30563) = barycentric product X(30561)*X(30590)
X(30563) = barycentric quotient X(30561)/X(30589)

X(30564) = X(2)X(44)∩X(20)X(355)

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

Let A'₂B'₂C'₂ be as at X(25418). Triangle A'₂B'₂C'₂ is homothetic to the anticomplementary triangle at X(30564).

X(30564) lies on these lines: {1, 16704}, {2, 44}, {20, 355}, {45, 1150}, {81, 25418}, {194, 16816}, {213, 14996}, {333, 17147}, {1764, 3219} et al

X(30564) = anticomplement of X(30588)
X(30564) = {X(89), X(30563)}-harmonic conjugate of X(2)

X(30565) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 2 AND 30

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

Since Gemini triangles 2 and 30 are perspective (at X(63)), their side-triangle is degenerate, lying on the perspectrix, line X(514)X(661).

X(30565) lies on these lines: {2, 918}, {100, 190}, {514, 661}, {522, 14392}, {523, 4800}, {650, 4467}, {654, 3219}, {812, 4120}, {824, 4893}, {926, 3681}, {1121, 6366}, {1635, 2786} et al

X(30565) = anticomplement of X(1638)
X(30565) = isotomic conjugate of X(37143)

X(30566) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 30

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

Since the vertices of Gemini triangles 2 and 30 lie on the same conic (the circumconic centered at X(9)), their cross-triangle is degenerate, lying on line X(514)X(661).

X(30566) lies on these lines: {2, 45}, {8, 4767}, {11, 3952}, {100, 15507}, {121, 4674}, {149, 3699}, {244, 11814}, {312, 3969}, {321, 3452}, {329, 6557}, {514, 661}, {528, 17780}, {537, 1647}, {645, 24624}, {899, 4442}, {1265, 5187} et al

X(30566) = isotomic conjugate of X(37222)

X(30567) = X(1)X(2)∩X(57)X(312)

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

X(30567) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 2, of the circumconic centered at X(9).

X(30567) lies on these lines: {1, 2}, {9, 14829}, {57, 312}, {63, 4358}, {69, 1997}, {75, 5437}, {76, 7196}, {144, 8055}, {165, 3685}, {190, 3928}, {320, 28609}, {321, 3306}, {333, 7308}, {341, 6762}, {344, 5745}, {345, 3911}, {346, 5435}, {908, 21621}, {1043, 5438}, {1089, 3338}, {1150, 3305}, {1155, 4387}, {1265, 24391}, {1376, 3886}, {1453, 13741}, {1699, 4645}, {1706, 4673}, {1707, 4011}, {2050, 9856} et al

X(30567) = {X(63), X(4358)}-harmonic conjugate of X(30568)

X(30568) = X(1)X(979)∩X(9)X(312)

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

X(30568) is the perspector of Gemini triangle 2 and the tangential triangle, wrt Gemini triangle 30, of the circumconic centered at X(9).

X(30568) lies on these lines: {1, 979}, {2, 2415}, {8, 4082}, {9, 312}, {10, 2899}, {43, 4368}, {55, 4009}, {57, 190}, {63, 4358}, {75, 7308}, {86, 25430}, {165, 5205}, {192, 2999}, {200, 3685}, {210, 3886}, {226, 344}, {321, 3294}, {329, 3912}, {341, 1697}, {345, 2325}, {346, 3687}, {390, 5423}, {497, 3717}, {552, 4633}, {726, 5272}, {748, 3994}, {908, 17776}, {936, 7283}, {950, 1265}, {1001, 3967}, {1120, 3622}, {1698, 4425}, {1699, 17777}, {1743, 1999}, {1997, 3911}, {2321, 14555} et al

X(30568) = anticomplement of X(24175)
X(30568) = {X(63), X(4358)}-harmonic conjugate of X(30567)

X(30569) = X(1936)X(2342)∩X(4781)X(5744)

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

Let A₂B₂C₂ and A₃₀B₃₀C₃₀ be Gemini triangles 2 and 30, resp. X(30569) is the radical center of the circumcircles of triangles AA₂A₃₀, BB₂B₃₀ and CC₂C₃₀.

X(30569) lies on these lines: {1936, 2342}, {4781, 5744}

X(30570) = (name pending)

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

Let A₄B₄C₄ be Gemini triangle 4. Let A' be the center of conic {{A,B,C,B₄,C₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30570).

X(30570) lies on the line {3736, 30571}

X(30571) = ISOGONAL CONJUGATE OF X(4649)

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

Let A₄B₄C₄ be Gemini triangle 4. Let A' be the perspector of conic {{A,B,C,B₄,C₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30571).

X(30571) lies on these lines: {1, 1573}, {2, 740}, {28, 2201}, {37, 291}, {42, 1255}, {43, 25430}, {57, 846}, {81, 238}, {86, 4368}, {105, 8297}, {274, 350}, {277, 24161}, {278, 1874}, {330, 1655}, {538, 25055}, {551, 3227}, {984, 1002}, {985, 1001}, {1929, 8299}, {3736, 30570} et al

X(30571) = isogonal conjugate of X(4649)

X(30572) = X(523)X(656)∩X(900)X(1317)

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

X(30572) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 9, Gemini triangle 10}.

X(30572) lies on these lines: {65, 4145}, {522, 4318}, {523, 656}, {900, 1317}, {1365, 2611}, {1769, 21132} et al

X(30572) = barycentric product X(i)*X(j) for these {i,j}: {7, 4120}, {65, 3762}, {226, 900}, {519, 7178}, {523, 3911}, {1317, 4049}, {3676, 3943}
X(30572) = barycentric quotient X(i)/X(j) for these (i,j): (7, 4615), (10, 4582), (56, 4591), (57, 4622), (65, 3257), (226, 4555), (519, 645), (523, 4997), (900, 333), (3762, 314), (3911, 99), (3943, 3699), (4120, 8), (7178, 903)

X(30573) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 9

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

The side-triangle of ABC and Gemini triangle 9 is degenerate, lying on the perspectrix, line X(900)X(1317).

X(30573) lies on these lines: {1, 514}, {390, 6006}, {513, 5919}, {519, 4543}, {522, 3241}, {900, 1317}

X(30573) = tripolar centroid of X(3911)
X(30573) = barycentric product X(i)*X(j) for these {i,j}: {514, 6174}, {519, 1638}, {16704, 30574}
X(30573) = barycentric quotient X(i)/X(j) for these (i,j): (1638, 903), (6174, 190), (30574, 4080)

X(30574) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 10

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

The side-triangle of ABC and Gemini triangle 10 is degenerate, lying on the perspectrix, line X(523)X(656).

X(30574) lies on the line {523, 656}

X(30574) = tripolar centroid of X(226)
X(30574) = barycentric product X(i)*X(j) for these {i,j}: {10, 1638}, {4080, 30573}
X(30574) = barycentric quotient X(i)/X(j) for these (i,j): (1638, 86), (30573, 16704)

X(30575) = X(44)X(88)∩X(81)X(4638)

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

Let A₉B₉C₉ be Gemini triangle 9. Let A' be the center of conic {{A,B,C,B₉,C₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30575).

X(30575) lies on these lines: {44, 88}, {81, 4638}, {758, 4674}, {903, 17495}, {908, 6549}, {1318, 1870} et al

X(30575) = isotomic conjugate of X(16729)
X(30575) = barycentric product X(i)*X(j) for these {i,j}: {10, 679}, {88, 4080}, {903, 4674}
X(30575) = barycentric quotient X(i)/X(j) for these (i,j): (2, 16729), (10, 4738), (88, 16704), (679, 86), (4080, 4358), (4674, 519)

X(30576) = X(21)X(849)∩X(81)X(593)

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

Let A₁₀B₁₀C₁₀ be Gemini triangle 10. Let A' be the center of conic {{A,B,C,B₁₀,C₁₀}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30576).

X(30576) lies on these lines: {21, 849}, {58, 5303}, {60, 13624}, {81, 593}, {88, 4591}, {100, 1326}, {249, 1931}, {261, 5235}, {1019, 3960}, {1255, 2298} et al

X(30576) = barycentric product X(i)*X(j) for these {i,j}: {81, 16704}, {261, 1319}, {519, 757}
X(30576) = barycentric quotient X(i)/X(j) for these (i,j): (81, 4080), (519, 1089), (757, 903), (1319, 12), (16704, 321)

X(30577) = PERSPECTOR OF GEMINI TRIANGLE 10 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

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

X(30577) lies on these lines: {2, 45}, {10, 1054}, {56, 100}, {57, 3882}, {244, 986}, {519, 9324}, {678, 3241}, {1145, 14193}, {1155, 5211}, {1266, 3911}, {1282, 5212} et al

X(30577) = anticomplement of X(4997)
X(30577) = {X(2), X(30579)}-harmonic conjugate of X(30578)

X(30578) = PERSPECTOR OF GEMINI TRIANGLE 27 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

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

X(30578) lies on these lines: {2, 45}, {8, 80}, {11, 4756}, {312, 2895}, {320, 4358}, {528, 4152}, {908, 2325}, {1644, 9324}, {1647, 24821}, {1698, 3120}, {1997, 23958}, {2607, 4427}, {2796, 9458}, {3218, 4480}, {3699, 20095}, {3992, 5180}, {4009, 5057} et al

X(30578) = isotomic conjugate of X(8046)
X(30578) = complement of X(20092)
X(30578) = anticomplement of X(88)
X(30578) = {X(2), X(30579)}-harmonic conjugate of X(30577)

X(30579) = {X(30577), X(30578)}-HARMONIC CONJUGATE OF X(2)

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

X(30579) lies on these lines: {1, 4427}, {2, 45}, {20, 952}, {44, 17495}, {75, 16729}, {89, 192}, {537, 678}, {3977, 4887} et al

X(30579) = anticomplement of X(4080)
X(30579) = {X(30577), X(30578)}-harmonic conjugate of X(2)

X(30580) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

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

The side-triangle of Gemini triangles 10 and 27 is degenerate, lying on the perspectrix, line X(1)X(523).

X(30580) lies on these lines: {1, 523}, {99, 110}, {392, 513}, {512, 3877}, {514, 551}, {522, 3251}, {764, 4778}, {953, 2726}, {993, 4367}, {1022, 4977}, {1125, 4049} et al

X(30581) = X(58)X(5253)∩X(81)X(593)

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

Let A₁₁B₁₁C₁₁ be Gemini triangle 11. Let A' be the center of conic {{A,B,C,B₁₁,C₁₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30581).

X(30581) lies on these lines: {58, 5253}, {81, 593}, {261, 5333}, {662, 1171}, {1255, 1963} et al

X(30581) = barycentric product X(i)*X(j) for these {i,j}: {1, 30593}, {58, 16709}, {81, 8025}, {593, 4359}, {757, 1125}, {1100, 1509}
X(30581) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6538), (81, 6539), (593, 1255), (757, 1268), (1100, 594), (1125, 1089), (4359, 28654), (8025, 321), (16709, 313), (30593, 75)

X(30582) = X(1255)X(3723)∩X(3743)X(4540)

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

Let A₁₂B₁₂C₁₂ be Gemini triangle 12. Let A' be the center of conic {{A,B,C,B₁₂,C₁₂}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30582).

X(30582) lies on these lines: {1255, 3723}, {3743, 4540}, {6539, 30594}

X(30582) = barycentric product X(i)*X(j) for these {i,j}: {1, 30594}, {1255, 6539}
X(30582) = barycentric quotient X(i)/X(j) for these (i,j): (1255, 8025), (6539, 4359), (30594, 75)

X(30583) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 14

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

The side-triangle of ABC and Gemini triangle 14 is degenerate, lying on the perspectrix, line X(900)X(1145).

X(30583) lies on these lines: {1, 9260}, {2, 14421}, {8, 6161}, {10, 514}, {513, 3679}, {519, 3251}, {667, 956}, {891, 4728}, {900, 1145}, {984, 4777}, {1022, 19875}, {1647, 2087} et al

X(30583) = barycentric product X(i)*X(j) for these {i,j}: {519, 4728}, {536, 900}
X(30583) = barycentric quotient X(i)/X(j) for these (i,j): (519, 4607), (536, 4555), (900, 3227), (4728, 903)

X(30584) = X(522)X(650)∩X(3835)X(4083)

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

X(30584) is the intersection of perspectrices of [ABC and Gemini triangle 15] and [ABC and Gemini triangle 16].

X(30584) lies on these lines: {522, 650}, {1215, 3805}, {3835, 4083} et al

X(30584) = barycentric product X(i)*X(j) for these {i,j}: {192, 3907}, {522, 17752}, {894, 4147}, {3835, 7081}, {4369, 27538}
X(30584) = barycentric quotient X(i)/X(j) for these (i,j): (522, 27447), (3835, 7049), (3907, 330), (4147, 257), (7081, 4598), (17752, 664), (27538, 27805)

X(30585) = X(58)X(750)∩X(191)X(3294)

Barycentrics 2 a^6 + 7 a^5 (b + c) + 3 a^4 (3 b^2 + 7 b c + 3 c^2) + a^3 (b + c) (5 b^2 + 18 b c + 5 c^2) + a^2 (3 b^4 + 17 b^3 c + 29 b^2 c^2 + 17 b c^3 + 3 c^4) + a (b + c)^3 (2 b^2 + 3 b c + 2 c^2) + 2 b c (b + c)^2 (b^2 + c^2) : :

Let A₁₉B₁₉C₁₉ and A₂₀B₂₀C₂₀ be Gemini triangles 19 and 20, resp. X(30585) is the radical center of the circumcircles of triangles AA₁₉A₂₀, BB₁₉B₂₀ and CC₁₉C₂₀.

X(30585) lies on these lines: {58, 750}, {191, 3294}

X(30586) = (name pending)

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

Let A₁₉B₁₉C₁₉ be Gemini triangle 19. Let A' be the center of conic {{A,B,C,B₁₉,C₁₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30586).

X(30586) lies on these lines: {11599, 20536}

X(30587) = (name pending)

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

Let A₂₀B₂₀C₂₀ be Gemini triangle 20. Let A' be the center of conic {{A,B,C,B₂₀,C₂₀}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30587).

X(30587) lies on the line {30561, 30589}

X(30587) = barycentric product X(30588)*X(30589)
X(30587) = barycentric quotient X(30589)/X(5235)

X(30588) = ISOGONAL CONJUGATE OF X(4273)

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

Let A₂₀B₂₀C₂₀ be Gemini triangle 20. Let A' be the perspector of conic {{A,B,C,B₂₀,C₂₀}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30588).

X(30588) lies on these lines: {2, 44}, {4, 1385}, {10, 2650}, {37, 4080}, {76, 4358}, {86, 4604}, {94, 20565}, {98, 4588}, {142, 14554}, {321, 3943}, {661, 4049}, {671, 4597}, {908, 17758}, {1125, 14020}, {1751, 2364}, {2051, 5249}, {2163, 3624} et al

X(30588) = isogonal conjugate of X(4273)
X(30588) = isotomic conjugate of X(5235)
X(30588) = complement of X(30564)
X(30588) = barycentric product X(i)*X(j) for these {i,j}: {37, 20569}, {76, 28658}, {89, 321}, {226, 30608}, {30587, 30590}
X(30588) = barycentric quotient X(i)/X(j) for these (i,j): (2, 5235), (6, 4273), (10, 3679), (37, 45), (89, 81), (321, 4671), (20569, 274), (28658, 6), (30587, 30589), (30608, 333)

X(30589) = BARYCENTRIC QUOTIENT X(30587)/X(30588)

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

X(30589) lies on these lines: {2, 44}, {79, 2320}, {1125, 2163}, {30561, 30587} et al

X(30589) = isotomic conjugate of X(30590)
X(30589) = anticomplement of X(30563)
X(30589) = barycentric product X(i)*X(j) for these {i,j}: {89, 28605}, {2163, 30596}, {5235, 30587}
X(30589) = barycentric quotient X(i)/X(j) for these (i,j): (2, 30590), (89, 25417), (1698, 3679), (28605, 4671), (30561, 30563), (30587, 30588), (30595, 30605)

X(30590) = BARYCENTRIC QUOTIENT X(30588)/X(30587)

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

X(30590) lies on these lines: {2, 319}, {80, 4294}, {8652, 9093}

X(30590) = isotomic conjugate of X(30589)
X(30590) = anticomplement of X(30561)
X(30590) = barycentric product X(i)*X(j) for these {i,j}: {3679, 30598}
X(30590) = barycentric quotient X(i)/X(j) for these (i,j): (2, 30589), (3679, 1698), (30563, 30561), (30588, 30587), (30605, 30595)

X(30591) = X(320)X(350)∩X(523)X(1577)

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

X(30591) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 21, Gemini triangle 22}.

X(30591) lies on these lines: {320, 350}, {522, 4823}, {523, 1577}, {656, 4804}, {2517, 4777}, {2533, 4132}, {4024, 8061}, {4304, 15526}, {4977, 4983} et al

X(30591) = isotomic conjugate of X(4596)
X(30591) = anticomplement of X(8043)
X(30591) = barycentric product X(i)*X(j) for these {i,j}: {10, 4978}, {75, 4988}, {321, 4977}, {514, 4647}, {523, 4359}, {693, 1213}, {1125, 1577}
X(30591) = barycentric quotient X(i)/X(j) for these (i,j): (2, 4596), (75, 4632), (321, 6540), (523, 1255), (1125, 662), (1213, 100), (1577, 1268), (4359, 99), (4647, 190), (4977, 81), (4978, 86), (4988, 1)

X(30592) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 22

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

The side-triangle of ABC and Gemini triangle 22 is degenerate, lying on the perspectrix, line X(4977)X(4983).

X(30592) lies on these lines: {522, 2530}, {690, 6545}, {764, 4010}, {812, 14419}, {891, 4728}, {2787, 14421}, {2832, 4800} et al

X(30592) = tripolar centroid of X(4359)
X(30592) = barycentric product X(i)*X(j) for these {i,j}: {536, 4977}, {1125, 4728}
X(30592) = barycentric quotient X(i)/X(j) for these (i,j): (536, 6540), (1125, 4607), (4728, 1268), (4977, 3227)

X(30593) = X(58)X(86)∩X(81)X(17495)

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

Let A₂₁B₂₁C₂₁ be Gemini triangle 21. Let A' be the center of conic {{A,B,C,B₂₁,C₂₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30593).

X(30593) lies on these lines: {58, 86}, {81, 17495}, {190, 1963}, {1224, 1268}, {1434, 7341} et al

X(30593) = isotomic conjugate of X(6538)
X(30593) = barycentric product X(i)*X(j) for these {i,j}: {75, 30581}, {81, 16709}, {86, 8025}
X(30593) = barycentric quotient X(i)/X(j) for these (i,j): (2, 6538), (86, 6539), (1125, 594), (8025, 10), (16709, 321), (30581, 1)

X(30594) = X(1268)X(3634)∩X(6539)X(30582)

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

Let A₂₂B₂₂C₂₂ be Gemini triangle 22. Let A' be the center of conic {{A,B,C,B₂₂,C₂₂}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30594).

X(30594) lies on these lines: {1268, 3634}, {6539, 30582}

X(30594) = barycentric product X(i)*X(j) for these {i,j}: {75, 30582}, {1268, 6539}
X(30594) = barycentric quotient X(i)/X(j) for these (i,j): (1268, 8025), (6539, 1125), (30582, 1)

X(30595) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 24

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

The side-triangle of ABC and Gemini triangle 24 is degenerate, lying on the perspectrix, line X(4716)X(4802).

X(30595) lies on these lines: {351, 690}, {2786, 14431}, {4716, 4802} et al

X(30595) = tripolar centroid of X(5333)
X(30595) = barycentric product X(i)*X(j) for these {i,j}: {514, 4938}, {524, 4802}, {1698, 4750}, {30589, 30605}
X(30595) = barycentric quotient X(i)/X(j) for these (i,j): (4750, 30598), (4802, 671), (4938, 190), (30605, 30590)

X(30596) = X(10)X(75)∩X(69)X(5080)

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

Let A₂₃B₂₃C₂₃ be Gemini triangle 23. Let A' be the center of conic {{A,B,C,B₂₃,C₂₃}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30596).

X(30596) lies on these lines: {10, 75}, {69, 5080}, {86, 3761}, {192, 4377}, {264, 5342}, {310, 28650}, {311, 322}, {312, 1230}, {314, 5560}, {319, 5016}, {1964, 9902} et al

X(30596) = isotomic conjugate of isogonal conjugate of X(1698)
X(30596) = barycentric product X(i)*X(j) for these {i,j}: {75, 28605}, {76, 1698}, {30590, 30595}
X(30596) = barycentric quotient X(i)/X(j) for these (i,j): (75, 25417), (76, 30598), (1698, 6), (28605, 1), (30589, 2163), (30595, 30589)

X(30597) = TRILINEAR SQUARE OF X(25417)

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

Let A₂₄B₂₄C₂₄ be Gemini triangle 24. Let A' be the center of conic {{A,B,C,B₂₄,C₂₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30597).

X(30597) lies on these lines: {5224, 19862}, {25417, 28606}

X(30597) = trilinear square of X(25417)
X(30597) = barycentric product X(25417)*X(30598)
X(30597) = barycentric quotient X(25417)/X(1698)

X(30598) = ISOTOMIC CONJUGATE OF X(1698)

Barycentrics 1/(a + 2 b + 2 c) : :
Trilinears 1/(1 - cos 2A + 4 (1 + cos A) (cos B + cos C)) : :

Let A₂₄B₂₄C₂₄ be Gemini triangle 24. Let A' be the perspector of conic {{A,B,C,B₂₄,C₂₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30598).

X(30598) lies on these lines: {1, 1268}, {2, 319}, {7, 5550}, {10, 28650}, {69, 28626}, {75, 1125}, {86, 3624}, {310, 16709}, {335, 4422}, {551, 4464}, {675, 8652}, {748, 757}, {749, 756}, {903, 10436}, {3616, 4460}, {3622, 5564}, {3644, 4472}, {3879, 19878} et al

X(30598) = isotomic conjugate of X(1698)
X(30598) = barycentric product X(i)*X(j) for these {i,j}: {75, 25417}, {310, 28625}
X(30598) = barycentric quotient X(i)/X(j) for these (i,j): (2, 1698), (25417, 1), (28625, 42), (30590, 3679)

X(30599) = X(2)X(39)∩X(75)X(81)

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

Let A₂₁B₂₁C₂₁ and A₂₃B₂₃C₂₃ be Gemini triangles 21 and 23, resp. Let L_A be the tangent at A to conic {{A,B₂₁,C₂₁,B₂₃,C₂₃}}, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30599).

Let A₂₂B₂₂C₂₂ and A₂₄B₂₄C₂₄ be Gemini triangles 22 and 24, resp. Let A' be the center of conic {{A,B₂₂,C₂₂,B₂₄,C₂₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30599).

X(30599) lies on these lines: {2, 39}, {27, 20880}, {75, 81}, {86, 321}, {92, 14014}, {312, 5333}, {314, 8025}, {333, 2160}, {1010, 3920}, {1043, 3957}, {1255, 4043}, {1412, 6358} et al

X(30599) = barycentric product X(i)*X(j) for these {i,j}: {75, 25526}, {314, 10404}
X(30599) = barycentric quotient X(i)/X(j) for these (i,j): (10404, 65), (25526, 1)

X(30600) = X(44)X(513)∩X(4467)X(7265)

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

X(30600) is the intersection of perspectrices of [ABC and Gemini triangle 25] and [ABC and Gemini triangle 26].

X(30600) lies on these lines: {44, 513}, {4467, 7265}

X(30600) = barycentric product X(i)*X(j) for these {i,j}: {4467, 5311}, {14838, 17303}
X(30600) = barycentric quotient X(i)/X(j) for these (i,j): (5311, 6742), (17303, 15455)

X(30601) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 25

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

The side-triangle of ABC and Gemini triangle 25 is degenerate, lying on the perspectrix, line X(4467)X(7265).

X(30601) lies on the line {4467, 7265}

X(30601) = tripolar centroid of X(319)
X(30601) = barycentric product X(319)*X(514)*X(1698)*X(4725)
X(30601) = barycentric product X(319)*X(4725)*X(4802)
X(30601) = barycentric product X(1698)*X(4467)*X(4725)

X(30602) = X(57)X(267)∩X(79)X(81)

Barycentrics 1/((a^2 - b^2 - c^2 - b c) (a^3 + a^2 (b + c) - a (b^2 + b c + c^2) - (b + c) (b^2 + c^2))) : :

Let A₂₅B₂₅C₂₅ be Gemini triangle 25. Let A' be the center of conic {{A,B,C,B₂₅,C₂₅}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30602).

X(30602) lies on these lines: {1, 8818}, {2, 6757}, {57, 267}, {79, 81}, {274, 20565}, {502, 1255}, {1029, 1479} et al

X(30602) = barycentric product X(i)*X(j) for these {i,j}: {79, 1029}, {267, 30690}, {3444, 20565}
X(30602) = barycentric quotient X(i)/X(j) for these (i,j): (79, 2895), (267, 3219), (1029, 319), (3444, 35)

X(30603) = (name pending)

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

Let A₂₆B₂₆C₂₆ be Gemini triangle 26. Let A' be the center of conic {{A,B,C,B₂₆,C₂₆}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30603).

X(30603) lies on the circumconic with center X(4858) and these lines: {}

X(30604) = X(523)X(1577)∩X(4693)X(4775)

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

X(30604) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 27, Gemini triangle 28}.

X(30604) lies on these lines: {523, 1577}, {4693, 4775}, {4983, 6089} et al

X(30604) = barycentric product X(i)*X(j) for these {i,j}: {3664, 4931}, {3679, 23755}, {4777, 17056}
X(30604) = barycentric quotient X(17056)/X(4597)

X(30605) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 28

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

The side-triangle of ABC and Gemini triangle 28 is degenerate, lying on the perspectrix, line X(4693)X(4775).

X(30605) lies on these lines: {351, 690}, {514, 4010}, {900, 6161}, {918, 14421}, {2785, 14431}, {4693, 4775} et al

X(30605) = tripolar centroid of X(5235)
X(30605) = barycentric product X(i)*X(j) for these {i,j}: {514, 4933}, {524, 4777}, {3679, 4750}
X(30605) = barycentric quotient X(i)/X(j) for these (i,j): (524, 4597), (4777, 671), (4933, 190)

X(30606) = X(60)X(1043)∩X(261)X(284)

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

Let A₂₇B₂₇C₂₇ be Gemini triangle 27. Let A' be the center of conic {{A,B,C,B₂₇,C₂₇}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30606).

X(30606) lies on these lines: {60, 1043}, {81, 17495}, {261, 284}, {1509, 2325}, {3285, 4969} et al

X(30606) = barycentric product X(i)*X(j) for these {i,j}: {60, 3264}, {261, 519}, {333, 16704}, {3285, 28660}
X(30606) = barycentric quotient X(i)/X(j) for these (i,j): (60, 106), (261, 903), (333, 4080), (519, 12), (3285, 1400), (16704, 226)

X(30607) = X(89)X(4850)∩X(2320)X(3877)

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

Let A₂₈B₂₈C₂₈ be Gemini triangle 28. Let A' be the center of conic {{A,B,C,B₂₈,C₂₈}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30607).

X(30607) lies on these lines: {89, 4850}, {2320, 3877}, {3306, 4604}, {3707, 5233}

X(30607) = barycentric product X(89)*X(30608)
X(30607) = barycentric quotient X(i)/X(j) for these (i,j): (5219, 89), (30608, 4671)

X(30608) = ISOGONAL CONJUGATE OF X(1405)

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

Let A₂₈B₂₈C₂₈ be Gemini triangle 28. Let A' be the perspector of conic {{A,B,C,B₂₈,C₂₈}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30608).

X(30608) lies on these lines: {2, 44}, {8, 2320}, {9, 4997}, {75, 18359}, {85, 3911}, {92, 8756}, {312, 2325}, {333, 2364}, {1121, 4384}, {1220, 1698}, {1311, 4588}, {1732, 3306}, {2217, 5303}, {3707, 5233} et al

X(30608) = isogonal conjugate of X(1405)
X(30608) = isotomic conjugate of X(5219)
X(30608) = barycentric product X(i)*X(j) for these {i,j}: {9, 20569}, {75, 2320}, {76, 2364}, {89, 312}, {333, 30588}, {4671, 30607}
X(30608) = barycentric quotient X(i)/X(j) for these (i,j): (2, 5219), (6, 1405), (9, 45), (89, 57), (312, 4671), (333, 5235), (2320, 1), (2364, 6), (20569, 85), (30588, 226), (30607, 89)

X(30609) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 29 AND 30

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

X(30609) lies on these lines: {2, 30612}, {144, 145}, {294, 1642}

X(30609) = anticomplement of X(30612)

X(30610) = TRILINEAR POLE OF LINE X(144)X(145)

Barycentrics 1/((b - c) (a^2 - a b - a c + 2 b c)) : :

Line X(144)X(145) is the anticomplement of line X(7)X(8), and the line of the degenerate cross-triangle of Gemini triangles 29 and 30.

X(30610) lies on these lines: {2, 14936}, {100, 8641}, {650, 4554}, {1025, 4763}, {1026, 4595} et al

X(30610) = isogonal conjugate of X(20980)
X(30610) = isotomic conjugate of X(4885)
X(30610) = trilinear pole of line X(144)X(145)
X(30610) = X(19)-isoconjugate of X(22091)
X(30610) = barycentric product X(i)*X(j) for these {i,j}: {190, 9311}, {4595, 27498}
X(30610) = barycentric quotient X(i)/X(j) for these (i,j): (2, 4885), (6, 20980), (100, 1376), (190, 3729), (9311, 514)

X(30611) = CENTROID OF CROSS-TRIANGLE OF EXTOUCH AND INTOUCH TRIANGLES

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

The cross-triangle of the extouch and intouch triangles is degenerate, lying on line X(7)X(8).

X(30611) lies on these lines: {2, 30609}, {7, 8}

X(30611) = complement of X(30609)
X(30611) = anticomplement of X(30612)

X(30612) = CENTROID OF CROSS-TRIANGLE OF 1st AND 2nd ZANIAH TRIANGLES

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

The cross triangle of the 1st and 2nd Zaniah triangles is degenerate, lying on line X(1)X(6).

X(30612) lies on these lines: {2, 30609}, {1, 6}

X(30612) = complement of X(30611)

X(30613) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 29 AND 30

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

X(30613) lies on these lines: {2, 11193}, {31, 1734}, {100, 11124}, {513, 3681}, {693, 3434}, {1621, 3126} et al

X(30613) = anticomplement of X(11193)

X(30614) = X(1)X(3710)∩X(2)X(4906)

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

X(30614) is the perspector of Gemini triangle 29 and the tangential triangle, wrt Gemini triangle 29, of the {Gemini 29, Gemini 30}-circumconic.

X(30614) lies on these lines: {1, 3710}, {2, 4906}, {65, 145}, {81, 3241}, {278, 1280}, {354, 20020}, {1211, 3242}, {1357, 5222} et al

X(30614) = anticomplement of X(30615)

X(30615) = ANTICOMPLEMENT OF X(4906)

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

X(30615) is the perspector of the extouch triangle and the tangential triangle, wrt the extouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30615) lies on these lines: {2, 4906}, {8, 210}, {9, 4030}, {55, 3717}, {200, 1040}, {344, 3748}, {345, 3689}, {346, 2348}, {518, 10327}, {519, 4383}, {612, 6703}, {614, 9053}, {1211, 3679}, {1386, 20020} et al

X(30615) = complement of X(30614)
X(30615) = anticomplement of X(4906)
X(30615) = barycentric quotient X(30617)/X(279)

X(30616) = X(144)X(145)∩X(218)X(329)

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

X(30616) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 29, of the {Gemini 29, Gemini 30}-circumconic.

X(30616) lies on these lines: {2, 30617}, {8, 3732}, {63, 3730}, {144, 145}, {218, 329}, {348, 4564}, {3189, 20071} et al

X(30616) = anticomplement of X(30617)
X(30616) = barycentric product X(190)*X(28590)
X(30616) = barycentric quotient X(28590)/X(514)

X(30617) = X(7)X(8)∩X(218)X(226)

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

X(30617) is the perspector of the intouch triangle and the tangential triangle, wrt the extouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30617) lies on these lines: {1, 1565}, {2, 30616}, {7, 8}, {57, 16549}, {150, 1837}, {169, 4904}, {218, 226}, {279, 3476}, {348, 1319}, {355, 1111}, {651, 14078}, {664, 7185}, {950, 3663}, {1358, 9312}, {1420, 7181}, {1788, 3598}, {1836, 4911}, {2082, 5845}, {2099, 3674}, {2218, 3423}, {2295, 4675} et al

X(30617) = complement of X(30616)
X(30617) = anticomplement of X(30618)
X(30617) = {X(7), X(8)}-harmonic conjugate of X(7195)
X(30617) = barycentric product X(i)*X(j) for these {i,j}: {7, 17279}, {279, 30615}, {348, 5101}, {651, 4373}
X(30617) = barycentric quotient X(i)/X(j) for these (i,j): (651, 145), (4373, 4391), (5101, 281), (17279, 8), (30615, 346)

X(30618) = X(1)X(6)∩X(8)X(2348)

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

X(30618) is the perspector of the 2nd Zaniah triangle and the tangential triangle, wrt the 1st Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30618) lies on these lines: {1, 6}, {2, 30616}, {8, 2348}, {21, 7259}, {41, 3693}, {169, 5836}, {346, 2264}, {644, 3057}, {728, 3913}, {910, 3501}, {1265, 3161}, {1376, 15876}, {1385, 24036} et al

X(30618) = barycentric product X(i)*X(j) for these {i,j}: {8, 3744}, {9, 17353}
X(30618) = barycentric quotient X(i)/X(j) for these (i,j): (3744, 7), (17353, 85)

X(30619) = X(144)X(145)∩X(344)X(765)

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

X(30619) is the perspector of Gemini triangle 29 and the tangential triangle, wrt Gemini triangle 30, of the {Gemini 29, Gemini 30}-circumconic.

X(30619) lies on these lines: {2, 30620}, {77, 3870}, {100, 1037}, {144, 145}, {344, 765}, {2398, 4000} et al

X(30619) = anticomplement of X(30620)

X(30620) = X(7)X(8)∩X(200)X(8271)

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

X(30620) is the perspector of the extouch triangle and the tangential triangle, wrt the intouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30620) lies on these lines: {2, 30619}, {7, 8}, {200, 8271}, {219, 3686}, {480, 3912}, {1037, 1376} et al

X(30620) = complement of X(30619)
X(30620) = anticomplement of X(30621)
X(30620) = {X(7), X(8)}-harmonic conjugate of X(4012)
X(30620) = barycentric product X(346)*X(30623)
X(30620) = barycentric quotient X(30623)/X(279)

X(30621) = X(1)X(6)∩X(55)X(77)

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

X(30621) is the perspector of the 1st Zaniah triangle and the tangential triangle, wrt the 2nd Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30621) lies on these lines: {1, 6}, {2, 30619}, {55, 77}, {241, 1253}, {269, 11495}, {390, 1456}, {651, 14100}, {916, 14520}, {1407, 10178}, {1418, 9441}, {1419, 4326}, {1442, 2346}, {1443, 7676}, {1697, 15832}, {1742, 6610} et al

X(30621) = complement of X(30620)
X(30621) = {X(1), X(6)}-harmonic conjugate of X(5572)

X(30622) = X(144)X(3059)∩X(6172)X(6605)

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

X(30622) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 30, of the {Gemini 29, Gemini 30}-circumconic.

X(30622) lies on these lines: {2, 30623}, {144, 3059}, {6172, 6605}, {17732, 17781}

X(30622) = anticomplement of X(30623)

X(30623) = X(7)X(354)∩X(222)X(553)

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

X(30623) is the perspector of the intouch triangle and the tangential triangle, wrt the intouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30623) lies on these lines: {2, 30622}, {7, 354}, {222, 553}, {269, 1040}, {279, 3474}, {348, 3683}, {658, 17728}, {1434, 2194}, {1565, 1709} et al

X(30623) = complement of X(30622)
X(30623) = anticomplement of X(30624)
X(30623) = barycentric product X(279)*X(30620)
X(30623) = barycentric quotient X(30620)/X(346)

X(30624) = X(9)X(165)∩X(210)X(6605)

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

X(30624) is the perspector of the 2nd Zaniah triangle and the tangential triangle, wrt the 2nd Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30624) lies on these lines: {2, 30622}, {9, 165}, {210, 6605}, {3967, 6559}, {7308, 21446}, {8012, 15481}

X(30624) = complement of X(30623)

X(30625) = X(8)X(144)∩X(9)X(85)

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

X(30625) is the perspector, wrt Gemini triangle 30, of the {ABC, Gemini 30}-circumconic.

X(30625) lies on these lines: {1, 3177}, {2, 10481}, {8, 144}, {9, 85}, {10, 30694}, {40, 3732}, {63, 169}, {190, 728}, {220, 9312}, {329, 3912}, {519, 20111}, {527, 6604}, {2481, 21384}, {17732, 17781} et al

X(30625) = anticomplement of X(10481)

X(30626) = (name pending)

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

Let A₂₉B₂₉C₂₉ and A₃₀B₃₀C₃₀ be Gemini triangles 29 and 30, resp. Let E* be the {Gemini 29, Gemini 30}-circumconic. Let A' be the intersection of the tangents to E* at A₂₉ and A₃₀. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(13577). Let E₂₉ and E₃₀ be the {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic, resp. Let A" be the intersection of the tangent to E₂₉ at A₂₉ and the tangent to E₃₀ at A₃₀. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(693). The lines A'A", B'B", C'C" concur in X(30626).

X(30626) lies on the line {2, 1814}

X(30627) = EIGENCENTER OF GEMINI TRIANGLE 29

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

X(30627) lies on these lines: {6, 664}, {41, 100}, {220, 3699}, {607, 1897}, {644, 1253}, {2287, 7257} et al

X(30627) = trilinear pole of line X(9)X(8641)

X(30628) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND HONSBERGER

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

X(30628) lies on these lines: {1, 5785}, {2, 3059}, {7, 3434}, {8, 5728}, {9, 1174}, {38, 4335}, {63, 4326}, {65, 20008}, {100, 1445}, {142, 11025}, {144, 145}, {226, 10865}, {354, 15587}, {480, 3935}, {516, 3868}, {519, 18412}, {528, 12755}, {651, 8271}, {962, 971}, {1320, 2801} et al

X(30628) = anticomplement of X(3059)

X(30629) = CENTROID OF GEMINI TRIANGLE 31

Barycentrics 2 a^3 (b^3 + c^3) + 9 a^2 b^2 c^2 + 3 a b c (b^3 + c^3) + 8 b^3 c^3 : :

X(30629) lies on these lines: {2, 4495}, {3912, 26738}

X(30630) = CENTROID OF GEMINI TRIANGLE 32

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

X(30630) lies on the line {2, 30636}

X(30631) = PERSPECTOR OF GEMINI TRIANGLE 32 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 32

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

X(30631) lies on these lines: {2, 893}, {75, 3703}, {334, 561}, {350, 3914}, {1909, 5249}, {1920, 3836}, {1921, 2887}, {1965, 4645} et al

X(30631) = {X(2), X(17493)}-harmonic conjugate of X(30646)
X(30631) = {X(2), X(30632)}-harmonic conjugate of X(7018)
X(30631) = barycentric product X(76)*X(24575)
X(30631) = barycentric quotient X(24575)/X(6)

X(30632) = {X(7018), X(30631)}-HARMONIC CONJUGATE OF X(2)

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

X(30632) lies on these lines: {2, 893}, {75, 3006}, {76, 3120}, {334, 30635}, {561, 2887}, {799, 4655}, {1920, 25957}, {1921, 25760}, {1965, 6327} et al

X(30632) = {X(7018), X(30631)}-harmonic conjugate of X(2)
X(30632) = barycentric product X(76)*X(4443)
X(30632) = barycentric quotient X(4443)/X(6)

X(30633) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 31 AND 32

Barycentrics 1/(a (a^2 - b c) (a^3 b^3 + a^3 c^3 - a^2 b^2 c^2 - b^3 c^3)) : :

X(30633) lies on these lines: {292, 1966}, {334, 14603}, {698, 3862}, {726, 24576}, {1581, 1921}, {1920, 30663}, {1965, 30648} et al

X(30633) = isogonal conjugate of X(30634)

X(30634) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 31 AND 32

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

X(30634) lies on these lines: {32, 18756}, {101, 699}, {110, 727}, {239, 1281}, {825, 18893}, {1922, 1967}, {1933, 14599} et al

X(30634) = isogonal conjugate of X(30633)
X(30634) = barycentric product X(i)*X(j) for these {i,j}: {31, 19579}, {32, 19581}, {238, 18278}, {1914, 3510}
X(30634) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30633), (32, 24576), (3510, 18895), (18278, 334), (19579, 561), (19581, 1502)

X(30635) = ISOTOMIC CONJUGATE OF X(17126)

Barycentrics 1/(2 a^3 + a b c) : :
Barycentrics 1/(2 a^2 sin A + S) : :

Let A₃₁B₃₁C₃₁ be Gemini triangle 31. Let L_A be the line through A₃₁ parallel to BC, and define L_B and L_C cyclically. Let A'₃₁ = L_B∩L_C, and define B'₃₁ and C'₃₁ cyclically. Triangle A'₃₁B'₃₁C'₃₁ is homothetic to ABC at X(30635).

X(30635) lies on these lines: {2, 4495}, {334, 30632}, {2887, 30636}, {3006, 7179}, {3661, 4044}, {4388, 7357}, {4389, 4441}, {6327, 7224}, {25760, 30638} et al

X(30635) = isotomic conjugate of X(17126)
X(30635) = {X(2887), X(30637)}-harmonic conjugate of X(30636)

X(30636) = ISOTOMIC CONJUGATE OF X(17127)

Barycentrics 1/(2 a^3 - a b c) : :
Barycentrics 1/(2 a^2 sin A - S) : :

Let A₃₂B₃₂C₃₂ be Gemini triangle 32. Let L_A be the line through A₃₂ parallel to BC, and define L_B and L_C cyclically. Let A'₃₂ = L_B∩L_C, and define B'₃₂ and C'₃₂ cyclically. Triangle A'₃₂B'₃₂C'₃₂ is homothetic to ABC at X(30636).

X(30636) lies on these lines: {2, 30630}, {2887, 30635}, {3661, 20888}, {4645, 7357}, {6327, 7261}, {25957, 30638}

X(30636) = isotomic conjugate of X(17127)
X(30636) = {X(2887), X(30637)}-harmonic conjugate of X(30635)

X(30637) = {X(30635), X(30636)}-HARMONIC CONJUGATE OF X(2887)

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

X(30637) lies on these lines: {2, 561}, {1278, 6385}, {2887, 30635} et al

X(30637) = {X(2), X(561)}-harmonic conjugate of X(30638)
X(30637) = {X(30635), X(30636)}-harmonic conjugate of X(2887)
X(30637) = barycentric product X(i)*X(j) for these {i,j}: {76, 17118}, {561, 17124}
X(30637) = barycentric quotient X(i)/X(j) for these (i,j): (17118, 6), (17124, 31)

X(30638) = {X(2), X(561)}-HARMONIC CONJUGATE OF X(30637)

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

X(30638) lies on these lines: {2, 561}, {76, 27797}, {3596, 9464}, {25760, 30635}, {25957, 30636} et al

X(30638) = {X(2), X(561)}-harmonic conjugate of X(30637)
X(30638) = barycentric product X(i)*X(j) for these {i,j}: {76, 17119}, {561, 17125}
X(30638) = barycentric quotient X(i)/X(j) for these (i,j): (17119, 6), (17125, 31)

X(30639) = X(824)X(4391)∩X(2533)X(3805)

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

X(30639) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 31, Gemini triangle 32}.

X(30639) lies on these lines: {788, 21301}, {824, 4391}, {2533, 3805}, {3766, 4010} et al

X(30639) = barycentric product X(i)*X(j) for these {i,j}: {561, 30654}, {788, 14603}, {824, 1966}, {1491, 3978}, {1920, 30665}, {1921, 3805}, {3661, 14296}
X(30639) = barycentric quotient X(i)/X(j) for these (i,j): (239, 30670), (385, 1492), (788, 9468), (824, 1581), (894, 30664), (1491, 694), (1966, 4586), (3805, 292), (3978, 789), (14296, 14621), (30654, 31), (30665, 893)

X(30640) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 31

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

The side-triangle of ABC and Gemini triangle 31 is degenerate, lying on the perspectrix, line X(3766)X(4010).

X(30640) lies on the line {3766, 4010}

X(30640) = tripolar centroid of X(1921)
X(30640) = barycentric product X(i)*X(j) for these {i,j}: {716, 30665}, {17493, 30641}
X(30640) = barycentric quotient X(30641)/X(30669)

X(30641) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 32

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

The side-triangle of ABC and Gemini triangle 32 is degenerate, lying on the perspectrix, line X(2533)X(3805).

X(30641) lies on the line {2533, 3805}

X(30641) = tripolar centroid of X(1920)
X(30641) = barycentric product X(i)*X(j) for these {i,j}: {716, 3805}, {30640, 30669}
X(30641) = barycentric quotient X(30640)/X(17493)

X(30642) = X(334)X(1921)∩X(1966)X(30657)

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

Let A₃₁B₃₁C₃₁ be Gemini triangle 31. Let A' be the center of conic {{A,B,C,B₃₁,C₃₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30642).

X(30642) lies on these lines: {334, 1921}, {1966, 30657}

X(30642) = barycentric product X(i)*X(j) for these {i,j}: {334, 30669}, {561, 30657}, {1920, 30663}
X(30642) = barycentric quotient X(i)/X(j) for these (i,j): (334, 17493), (1581, 30658), (1909, 4366), (18896, 30643), (30657, 31), (30663, 893), (30669, 238)

X(30643) = X(1921)X(3846)∩X(17493)X(30658)

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

Let A₃₂B₃₂C₃₂ be Gemini triangle 32. Let A' be the center of conic {{A,B,C,B₃₂,C₃₂}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30643).

X(30643) lies on these lines: {1921, 3846}, {17493, 30658}

X(30643) = barycentric product X(i)*X(j) for these {i,j}: {561, 30658}, {7018, 17493}
X(30643) = barycentric quotient X(i)/X(j) for these (i,j): (350, 6645), (1581, 30657), (7018, 30669), (17493, 171), (18896, 30642), (30658, 31)

X(30644) = CENTROID OF GEMINI TRIANGLE 33

Barycentrics a (8 a^3 b c + 3 a^2 (b^3 + c^3) + 9 a b^2 c^2 + 2 b c (b^3 + c^3)) : :
Trilinears 8 a^3 b c + 3 a^2 (b^3 + c^3) + 9 a b^2 c^2 + 2 b c (b^3 + c^3) : :

X(30644) lies on these lines: {2, 1908}, {1015, 4850}, {1573, 25057}

X(30645) = CENTROID OF GEMINI TRIANGLE 34

Barycentrics a (8 a^3 b c - 3 a^2 (b^3 + c^3) - 7 a b^2 c^2 + 2 b c (b^3 + c^3)) : :
Trilinears 8 a^3 b c - 3 a^2 (b^3 + c^3) - 7 a b^2 c^2 + 2 b c (b^3 + c^3) : :

X(30645) lies on the line {2, 30651}

X(30646) = PERSPECTOR OF GEMINI TRIANGLE 34 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 33 AND 34

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

X(30646) lies on these lines: {2, 893}, {6, 3725}, {31, 292}, {37, 3757}, {39, 846}, {63, 2275}, {228, 1914}, {238, 9285}, {968, 2276}, {1621, 21814}, {1908, 17122}, {2176, 5364} et al

X(30646) = {X(2), X(17493)}-harmonic conjugate of X(30631)
X(30646) = {X(2), X(30647)}-harmonic conjugate of X(893)
X(30646) = barycentric product X(1)*X(24478)
X(30646) = barycentric quotient X(24478)/X(75)

X(30647) = {X(893), X(30646)}-HARMONIC CONJUGATE OF X(2)

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

X(30647) lies on these lines: {1, 8620}, {2, 893}, {6, 3121}, {31, 1501}, {32, 3724}, {39, 4414}, {55, 21814}, {292, 17126}, {750, 1908}, {1185, 3725} et al

X(30647) = {X(893), X(30646)}-harmonic conjugate of X(2)
X(30647) = barycentric product X(i)*X(j) for these {i,j}: {1, 3764}, {6, 3735}, {31, 25760}
X(30647) = barycentric quotient X(i)/X(j) for these (i,j): (3764, 75), (3735, 76), (25760, 561)

X(30648) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 33 AND 34

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

X(30648) lies on these lines: {238, 1581}, {292, 1691}, {295, 7281}, {334, 1966}, {511, 1757}, {518, 7061}, {1326, 2223}, {1755, 2076}, {1965, 30633} et al

X(30648) = isogonal conjugate of X(1281)
X(30648) = trilinear pole of line X(5029)X(30654)
X(30648) = barycentric product X(i)*X(j) for these {i,j}: {1, 24479}, {292, 7261}, {694, 7061}
X(30648) = barycentric quotient X(i)/X(j) for these (i,j): (1, 18037), (6, 1281), (292, 4645), (24479, 75), (7061, 3978), (7261, 1921)

X(30649) = EIGNENCENTER OF GEMINI TRIANGLE 34

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

X(30649) lies on these lines: {1, 1281}, {2, 3494}, {21, 3551}, {41, 43}, {87, 256}, {404, 2108}, {846, 3229}, {1740, 8424} et al

X(30649) = eigencenter of 1st Sharygin triangle

X(30650) = ISOGONAL CONJUGATE OF X(4363)

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

Let A₃₃B₃₃C₃₃ be Gemini triangle 33. Let L_A be the line through A₃₃ parallel to BC, and define L_B, L_C cyclically. Let A'₃₃ = L_B∩L_C, and define B'₃₃, C'₃₃ cyclically. Triangle A'₃₃B'₃₃C'₃₃ is homothetic to ABC at X(30650).

X(30650) lies on these lines: {2, 1908}, {44, 751}, {89, 1015}, {292, 17126}, {649, 995}, {869, 902}, {3285, 21793}, {16584, 30651} et al

X(30650) = isogonal conjugate of X(4363)
X(30650) = {X(16584), X(30652)}-harmonic conjugate of X(30651)
X(30650) = barycentric product X(1)*X(751)
X(30650) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3761), (6, 4363), (751, 75)

X(30651) = ISOGONAL CONJUGATE OF X(4361)

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

Let A₃₄B₃₄C₃₄ be Gemini triangle 34. Let L_A be the line through A₃₄ parallel to BC, and define L_B, L_C cyclically. Let A'₃₄ = L_B∩L_C, and define B'₃₄, C'₃₄ cyclically. Triangle A'₃₄B'₃₄C'₃₄ is homothetic to ABC at X(30651).

X(30651) lies on these lines: {2, 30645}, {292, 17127}, {749, 1100}, {869, 2308}, {893, 17126}, {1500, 25417}, {16584, 30650} et al

X(30651) = isogonal conjugate of X(4361)
X(30651) = {X(16584), X(30652)}-harmonic conjugate of X(30650)
X(30651) = barycentric product X(1)*X(749)
X(30651) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3760), (6, 4361), (749, 75)

X(30652) = {X(30650), X(30651)}-HARMONIC CONJUGATE OF X(16584)

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

X(30652) lies on these lines: {2, 31}, {23, 1460}, {43, 21747}, {58, 145}, {81, 3052}, {89, 354}, {109, 9105}, {181, 11002}, {346, 4275}, {387, 20066}, {595, 3622}, {601, 3522}, {602, 15717}, {982, 9340}, {1054, 3892}, {1185, 1979}, {1376, 14997}, {1397, 11003}, {1399, 3600}, {1414, 7268}, {1468, 3623}, {1621, 14996}, {1707, 3920}, {16584, 30650} et al

X(30652) = anticomplement of X(25958)
X(30652) = {X(2), X(31)}-harmonic conjugate of X(30653)
X(30652) = {X(30650), X(30651)}-harmonic conjugate of X(16584)

X(30653) = {X(2), X(31)}-HARMONIC CONJUGATE OF X(30652)

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

X(30653) lies on these lines: {1, 21747}, {2, 31}, {23, 7083}, {58, 3622}, {89, 3246}, {100, 14997}, {109, 9095}, {145, 595}, {390, 2361}, {580, 20070}, {601, 15717}, {602, 3522}, {614, 23958}, {643, 4779}, {902, 3240}, {1001, 14996}, {1191, 16948}, {1397, 9544}, {1399, 5265}, {1460, 13595} et al

X(30653) = anticomplement of X(25959)
X(30653) = {X(2), X(31)}-harmonic conjugate of X(30652)

X(30654) = X(659)X(4435)∩X(663)X(788)

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

X(30654) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 33, Gemini triangle 34}.

X(30654) lies on these lines: {659, 4435}, {663, 788}, {824, 4560}, {3287, 3805}, {5029, 30671} et al

X(30654) = barycentric product X(i)*X(j) for these {i,j}: {31, 30639}, {171, 30665}, {238, 3805}, {788, 1966}, {824, 1691}, {869, 14296}, {1491, 1580}
X(30654) = barycentric quotient X(i)/X(j) for these (i,j): (788, 1581), (1491, 1934), (1580, 789), (1691, 4586), (3805, 334), (14296, 871), (30639, 561), (30665, 7018)

X(30655) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 33

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

The side-triangle of ABC and Gemini triangle 33 is degenerate, lying on the perspectrix, line X(659)X(4435).

X(30655) lies on these lines: {659, 4435}, {824, 6546}, {4809, 14402} et al

X(30655) = reflection of X(30656) in X(14402)
X(30655) = tripolar centroid of X(238)
X(30655) = barycentric product X(i)*X(j) for these {i,j}: {752, 30665}, {2243, 4486}, {3783, 4809}, {17493, 30656}
X(30655) = barycentric quotient X(30656)/X(30669)

X(30656) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 34

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

The side-triangle of ABC and Gemini triangle 34 is degenerate, lying on the perspectrix, line X(3287)X(3805).

X(30656) lies on these lines: {3287, 3805}, {4809, 14402}

X(30656) = reflection of X(30655) in X(14402)
X(30656) = tripolar centroid of X(171)
X(30656) = barycentric product X(i)*X(j) for these {i,j}: {752, 3805}, {30655, 30669}
X(30656) = barycentric quotient X(30655)/X(17493)

X(30657) = X(238)X(292)∩X(1966)X(30642)

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

Let A₃₃B₃₃C₃₃ be Gemini triangle 33. Let A' be the center of conic {{A,B,C,B₃₃,C₃₃}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30657).

X(30657) lies on these lines: {238, 292}, {1966, 30642}, {4447, 18787}

X(30657) = barycentric product X(i)*X(j) for these {i,j}: {31, 30642}, {171, 30663}, {291, 18787}, {292, 30669}
X(30657) = barycentric quotient X(i)/X(j) for these (i,j): (172, 4366), (1911, 18786), (1581, 30643), (9468, 30658), (18787, 350), (30642, 561), (30663, 7018), (30669, 1921)

X(30658) = X(238)X(893)∩X(17493)X(30643)

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

Let A₃₄B₃₄C₃₄ be Gemini triangle 34. Let A' be the center of conic {{A,B,C,B₃₄,C₃₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30658).

X(30658) lies on these lines: {238, 893}, {17493, 30643}

X(30658) = barycentric product X(i)*X(j) for these {i,j}: {31, 30643}, {256, 18786}, {893, 17493}
X(30658) = barycentric quotient X(i)/X(j) for these (i,j): (893, 30669), (1581, 30642), (1914, 6645), (9468, 30657), (17493, 1920), (18786, 1909), (30643, 561)

X(30659) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

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

X(30659) lies on these lines: {2, 30666}, {31, 561}, {740, 3873}, {874, 4418} et al

X(30659) = reflection of X(30666) in X(2)

X(30660) = PERSPECTOR OF GEMINI TRIANGLE 31 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

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

X(30660) lies on these lines: {2, 893}, {7, 310}, {8, 3978}, {69, 4485}, {312, 18037}, {321, 1909}, {329, 26735}, {561, 4388}, {1920, 4645} et al

X(30660) = anticomplement of X(893)
X(30660) = {X(2), X(30662)}-harmonic conjugate of X(30661)
X(30660) = barycentric product X(75)*X(17739)
X(30660) = barycentric quotient X(17739)/X(2)

X(30661) = PERSPECTOR OF GEMINI TRIANGLE 33 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

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

Let A'B'C' be the medial triangle. Let A₁ and A₂ be the 1st and 2nd bicentrics of A', resp., and define B₁, B₂, C₁, C₂ cyclically. The lines AB₁, BC₁, CA₁ concur in P(8) (the 1st bicentric of X(2)). The lines AC₂, BA₂, CB₂ concur in U(8) (the 2nd bicentric of X(2)). Let A" = A₁C₂∩B₁A₂, B" = B₁A₂∩C₁B₂, C" = C₁B₂∩A₁C₂. Triangle A"B"C" is homothetic to ABC at X(893), to the medial triangle at X(1966), and to the anticomplementary triangle at X(30661).

X(30661) lies on these lines: {2, 893}, {31, 19580}, {42, 894}, {55, 192}, {63, 24579}, {81, 330}, {846, 1655}, {1908, 1920} et al

X(30661) = anticomplement of X(7018)
X(30661) = {X(2), X(30662)}-harmonic conjugate of X(30660)

X(30662) = {X(30660), X(30661)}-HARMONIC CONJUGATE OF X(2)

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

X(30662) lies on these lines: {2, 893}, {145, 740}, {722, 20064}, {3924, 4043} et al

X(30662) = anticomplement of X(17493)
X(30662) = {X(30660), X(30661)}-harmonic conjugate of X(2)

X(30663) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 31 AND 33

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

X(30663) lies on these lines: {1, 3252}, {37, 9505}, {238, 292}, {241, 1463}, {291, 518}, {334, 1921}, {335, 726}, {660, 1757}, {716, 7245}, {740, 4562}, {984, 22116}, {1581, 4645}, {1920, 30633} et al

X(30663) = isogonal conjugate of X(8300)
X(30663) = isotomic conjugate of X(39044)
X(30663) = trilinear pole of line X(876)X(2254)
X(30663) = isotomic conjugate of complement of X(30669)
X(30663) = trilinear square of X(291)
X(30663) = barycentric product X(i)*X(j) for these {i,j}: {291, 335}, {292, 334}, {660, 4444}, {876, 4562}, {893, 30642}, {1581, 30669}, {7018, 30657}
X(30663) = barycentric quotient X(i)/X(j) for these (i,j): (1, 4366), (6, 8300), (291, 239), (292, 238), (334, 1921), (335, 350), (660, 3570), (876, 812), (4444, 3766), (4562, 874), (30642, 1920), (30657, 171), (30669, 1966)

X(30664) = TRILINEAR POLE OF LINE X(6)X(291)

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

X(30664) is the intersection, other than X(789), of the circumcircle and the tangent at X(789) to hyperbola {A,B,C,X(789),PU(6)}.

X(30664) is the intersection, other than X(825), of the circumcircle and the tangent at X(825) to hyperbola {A,B,C,X(825),PU(12)}.

X(30664) lies on the circumcircle and these lines: {99, 4613}, {100, 4562}, {101, 660}, {110, 4584}, {291, 753}, {292, 743}, {334, 9075}, {335, 761}, {701, 14598}, {717, 1922}, {731, 1911}, {813, 3573}, {870, 9073}, {985, 2382}, {14621, 14665} et al

X(30664) = isogonal conjugate of X(30665)
X(30664) = trilinear pole of line X(6)X(291)
X(30664) = Ψ(X(6), X(291))
X(30664) = Λ(X(659), X(4435))
X(30664) = Λ(X(876), X(2254))
X(30664) = Λ(X(3766), X(4010))
X(30664) = barycentric product X(i)*X(j) for these {i,j}: {291, 4586}, {292, 789}, {335, 1492}, {660, 14621}, {813, 870}, {985, 4562}, {30670, 30669}
X(30664) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30665), (291, 824), (292, 1491), (660, 3661), (789, 1921), (813, 984), (894, 30639, (985, 812), (1492, 239), (4586, 350), (14621, 3766), (30670, 17493)

X(30665) = ISOGONAL CONJUGATE OF X(30664)

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

X(30665) is the infinite point of the perspectrices of every pair of {ABC, Gemini triangle 31, Gemini triangle 33}.

X(30665) lies on these lines: {30, 511}, {659, 4435}, {876, 2254}, {3766, 4010}

X(30665) = isogonal conjugate of X(30664)
X(30665) = crossdifference of every pair of points on line X(6)X(291)
X(30665) = barycentric product X(i)*X(j) for these {i,j}: {238, 824}, {239, 1491}, {350, 3250}, {893, 30639}, {2276, 3766}, {3805, 17493}
X(30665) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30664), (238, 4586), (239, 789), (869, 813), (1491, 335), (1914, 1492), (2276, 660), (3250, 291), (3805, 30669), (8632, 985), (30639, 1920), (30655, 752)

X(30666) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 32 AND 34

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

X(30666) lies on these lines: {2, 30659}, {740, 3681}, {748, 1966}, {2022, 19551} et al

X(30666) = reflection of X(30659) in X(2)

X(30667) = PERSPECTOR OF GEMINI TRIANGLE 34 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 32 AND 34

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

X(30667) lies on these lines: {1, 1655}, {2, 292}, {6, 190}, {31, 19580}, {105, 330}, {239, 672}, {894, 9359}, {1931 ,2109}, {2112, 2145}, {2113, 17493} et al

X(30667) = anticomplement of X(334)
X(30667) = {X(2), X(30668)}-harmonic conjugate of X(20345)

X(30668) = {X(20345), X(30667)}-HARMONIC CONJUGATE OF X(2)

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

X(30668) lies on these lines: {2, 292}, {144, 1278}, {145, 17794}, {17343, 20554}

X(30668) = anticomplement of X(30669)
X(30668) = anticomplement of the isotomic conjugate of X(17493)
X(30668) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 30662}, {238, 30660}, {256, 20553}, {893, 4645}, {904, 6542}, {1178, 30941}, {1967, 6653}, {3903, 21303}, {7104, 17759}, {8300, 25332}, {9468, 30669}, {14599, 30661}, {17493, 6327}, {18786, 69}, {30658, 4388}
X(30668) = X(17493)-Ceva conjugate of X(2)
X(30668) = {X(20345), X(30667)}-harmonic conjugate of X(2)

X(30669) = ISOTOMIC CONJUGATE OF X(17493)

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

X(30669) lies on these lines: {2, 292}, {7, 192}, {8, 291}, {69, 3862}, {171, 7369}, {193, 7077}, {194, 3864}, {295, 20096}, {385, 4447}, {660, 20072}, {894, 7184}, {1581, 4645}, {1655, 6625}, {1909, 7187}, {1911, 17379}, {1966, 30642}, {2295, 6645} et al

X(30669) = isotomic conjugate of X(17493)
X(30669) = complement of X(30668)
X(30669) = anticomplement of X(39044)
X(30669) = anticomplement of isotomic conjugate of X(30663)
X(30669) = areal center of cevian triangles of PU(35)
X(30669) = barycentric product X(i)*X(j) for these {i,j}: {335, 894}, {238, 30642}, {1921, 30657}
X(30669) = barycentric quotient X(i)/X(j) for these (i,j): (2, 17493), (893, 30658), (894, 239), (7018, 30643), (30641, 30640), (30642, 334), (30656, 30655), (30657, 292), (30664, 30670)

X(30670) = TRILINEAR POLE OF LINE X(6)X(256)

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

X(30670) is the intersection, other than A, B, and C, of the circumcircle and conic {{A,B,C,PU(36)}}. X(30670) is also the isogonal conjugate of X(3805), which is the infinite point of perspectrices of every pair of {ABC, Gemini triangle 32, Gemini triangle 34}.

X(30670) lies on the circumcircle and these lines: {82, 733}, {99, 7260}, {100, 27805}, {101, 3903}, {109, 1492}, {110, 4603}, {256, 753}, {257, 761}, {662, 805}, {717, 7104}, {731, 904}, {741, 985}, {743, 893}, {813, 4579}, {932, 4586}, {1916, 8301}, {1967, 8300} et al

X(30670) = isogonal conjugate of X(3805)
X(30670) = trilinear pole of line X(6)X(256)
X(30670) = Ψ(X(6), X(256))
X(30670) = Λ(X(38), X(661))
X(30670) = barycentric product X(i)*X(j) for these {i,j}: {256, 4586}, {257, 1492}, {789, 893}, {985, 27805}, {3903, 14621}, {17493, 30664}
X(30670) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3805), (239, 30639), (256, 824), (789, 1920), (893, 1491), (985, 4369), (1492, 894), (3903, 3661), (4586, 1909), (14621, 4374), (30664, 30669)

X(30671) = X(38)X(661)∩X(42)X(649)

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

X(30671) is the intersection of perspectrices of every pair of {Gemini triangles 31, 32, 33, 34}.

X(30671) lies on these lines: {38, 661}, {42, 649}, {321, 693}, {876, 2254}, {882, 3569}, {5029, 30654} et al

X(30671) = barycentric product X(i)*X(j) for these {i,j}: {6, 23596}, {291, 1491}, {292, 824}, {334, 788}, {335, 3250}, {876, 984}, {2276, 4444}, {3572, 3661}
X(30671) = barycentric quotient X(i)/X(j) for these (i,j): (291, 789), (292, 4586), (788, 238), (824, 1921), (875, 985), (876, 870), (984, 874), (1491, 350), (1911, 1492), (2276, 3570), (3250, 239), (3572, 14621), (3661, 27853), (23596, 76)

X(30672) = CENTROID OF GEMINI TRIANGLE 35

Barycentrics a^6 + 6 a^5 (b + c) - a^4 (b^2 - 32 b c + c^2) - 4 a^3 (b + c) (3 b^2 - b c + 3 c^2) - a^2 (b + c)^2 (b^2 + 30 b c + c^2) + 2 a (b - c)^2 (b + c) (3 b^2 + 4 b c + 3 c^2) + (b^2 - c^2)^2 (b^2 + c^2) : :

X(30672) lies on the line {2, 30679}

X(30673) = CENTROID OF GEMINI TRIANGLE 36

Barycentrics a^6 - 6 a^5 (b + c) - a^4 (b^2 - 32 b c + c^2) + 4 a^3 (b + c) (3 b^2 - 5 b c + 3 c^2) - a^2 (b - c)^2 (b^2 + 34 b c + c^2) - 2 a (b - c)^2 (b + c) (3 b^2 - 4 b c + 3 c^2) + (b^2 - c^2)^2 (b^2 + c^2) : :

X(30673) lies on these lines: {2, 30680}, {100, 999}, {6349, 26740}

X(30674) = PERSPECTOR OF GEMINI TRIANLGE 35 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 36

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

X(30674) lies on these lines: {2, 19}, {3, 4512}, {9, 223}, {40, 18641}, {57, 17073}, {63, 348}, {77, 6508}, {221, 960}, {1001, 1040}, {1528, 6908} et al

X(30674) = {X(2), X(30675)}-harmonic conjugate of X(10319)

X(30675) = {X(10319), X(30674)}-HARMONIC CONJUGATE OF X(2)

Barycentrics a (a^2 - b^2 - c^2) (a^6 - a^4 (b^2 + c^2) - 4 a^3 b c (b + c) - a^2 (b^2 - c^2)^2 + 4 a b c (b - c)^2 (b + c) + (b^2 - c^2)^2 (b^2 + c^2)) : :
Barycentrics cos A + sec B + sec C : :

X(30675) lies on these lines: {2, 19}, {3, 392}, {9, 6350}, {57, 6349}, {63, 77}, {343, 8897}, {1038, 3869}, {1040, 1621}, {1158, 2360} et al

X(30675) = {X(2), X(4329)}-harmonic conjugate of X(30687)
X(30675) = {X(10319), X(30674)}-harmonic conjugate of X(2)

X(30676) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 35 AND 36

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

The perspectrix of Gemini triangles 35 and 36 passes through X(663).

X(30676) lies on these lines: {6, 7131}, {9, 30701}, {57, 30705}

X(30676) = isogonal conjugate of X(30677)

X(30677) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 35 AND 36

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

X(30677) lies on these lines: {2, 7}, {19, 1611}, {169, 5272}, {444, 3291}, {614, 1184}, {2128, 21216} et al

X(30677) = isogonal conjugate of X(30676)

X(30678) = EIGENCENTER OF GEMINI TRIANGLE 36

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

X(30678) lies on these lines: {2, 294}, {57, 7075}, {100, 11329}, {3912, 9441} et al

X(30679) = X(21)X(999)∩X(78)X(4001)

Barycentrics (a^2 - b^2 - c^2)/(a^2 - b^2 - c^2 - 4 b c) : :
Barycentrics 1/(2 sec A + 1) : :

Let A₃₅B₃₅C₃₅ be Gemini triangle 35. Let L_A be the line through A₃₅ parallel to BC, and define L_B, L_C cyclically. Let A'₃₅ = L_B∩L_C, and define B'₃₅, C'₃₅ cyclically. Triangle A'₃₅B'₃₅C'₃₅ is homothetic to ABC at X(30679).

X(30679) lies on these lines: {2, 30672}, {21, 999}, {78, 4001}, {280, 6360}, {347, 7361}, {1214, 30680}, {1812, 22129}, {2339, 3218}, {3219, 7131} et al

X(30679) = isotomic conjugate of polar conjugate of X(3296)

X(30680) = X(21)X(145)∩X(78)X(3977)

Barycentrics (a^2 - b^2 - c^2)/(a^2 - b^2 - c^2 + 4 b c) : :
Barycentrics 1/(2 sec A - 1) : :

Let A₃₆B₃₆C₃₆ be Gemini triangle 36. Let L_A be the line through A₃₆ parallel to BC, and define L_B, L_C cyclically. Let A'₃₆ = L_B∩L_C, and define B'₃₆, C'₃₆ cyclically. Triangle A'₃₆B'₃₆C'₃₆ is homothetic to ABC at X(30680).

X(30680) lies on these lines: {2, 30673}, {21, 145}, {78, 3977}, {1214, 30679}, {2339, 3219}, {3218, 7131} et al

X(30680) = isotomic conjugate of polar conjugate of X(1000)

X(30681) = X(341)X(346)∩X(345)X(3694)

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

Let A₃₅B₃₅C₃₅ be Gemini triangle 35. Let A' be the center of conic {{A,B,C,B₃₅,C₃₅}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30681).

X(30681) lies on these lines: {341, 346}, {345, 3694}, {480, 5423}, {1265, 3692}, {3161, 3871} et al

X(30681) = isotomic conjugate of polar conjugate of X(5423)
X(30681) = barycentric product X(i)*X(j) for these {i,j}: {8, 1265}, {69, 5423}, {78, 341}, {305, 480}, {312, 3692}, {345, 346}
X(30681) = barycentric quotient X(i)/X(j) for these (i,j): (8, 1119), (69, 479), (78, 269), (312, 1847), (341, 273), (345, 279), (346, 278), (480, 25), (1265, 7), (3692, 57), (3926, 30682), (5423, 4)

X(30682) = X(77)X(1040)∩X(348)X(17073)

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

Let A₃₆B₃₆C₃₆ be Gemini triangle 36. Let A' be the center of conic {{A,B,C,B₃₆,C₃₆}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30682).

X(30682) lies on these lines: {7, 4626}, {77, 1040}, {279, 1418}, {348, 17073}, {479, 1014}, {910, 9533}, {934, 1486}, {1088, 1440} et al

X(30682) = isotomic conjugate of polar conjugate of X(479)
X(30682) = barycentric product X(i)*X(j) for these {i,j}: {7, 7056}, {69, 479}, {77, 1088}, {279, 348}, {4025, 4626}
X(30682) = barycentric quotient X(i)/X(j) for these (i,j): (7, 7046), (69, 5423), (77, 200), (279, 281), (348, 346), (479, 4), (1088, 318), (3926, 30681), (4025, 4163), (4626, 1897), (7056, 8)

X(30683) = CENTROID OF GEMINI TRIANGLE 37

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

X(30683) lies on these lines: {2, 7110}, {191, 6175}, {30684, 30685}

X(30683) = reflection of X(30684) in X(30685)

X(30684) = CENTROID OF GEMINI TRIANGLE 38

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

X(30684) lies on these lines: {1, 10031}, {2, 2006}, {693, 3960}, {908, 5723}, {2990, 28609}, {30683, 30685} et al

X(30684) = reflection of X(30683) in X(30685)

X(30685) = CENTROID OF MID-TRIANGLE OF GEMINI TRIANGLES 37 AND 38

Barycentrics 2 a^12 - 5 a^10 (b^2 + c^2) + 2 a^8 (b^4 + 4 b^2 c^2 + c^4) + a^6 (2 b^6 - 3 b^4 c^2 - 3 b^2 c^4 + 2 c^6) + 2 a^4 (b^8 - 3 b^6 c^2 + 3 b^4 c^4 - 3 b^2 c^6 + c^8) - a^2 (b^2 - c^2)^2 (b^2 + c^2) (5 b^4 - 9 b^2 c^2 + 5 c^4) + 2 (b^2 - c^2)^4 (b^4 + c^4) : :
Barycentrics SA^4 (SB + SC)^2 + 4 SA^3 (SB + SC) (SB^2 + SB SC + SC^2) - SA^2 (5 SB^4 + 4 SB^3 SC + 30 SB^2 SC^2 + 4 SB SC^3 + 5 SC^4) - 2 SA SB SC (SB + SC) (5 SB^2 - 12 SB SC + 5 SC^2) - SB^2 SC^2 (5 SB^2 - 22 SB SC + 5 SC^2) : :

X(30685) lies on these lines: {2, 94}, {110, 381}, {115, 14389}, {136, 7576}, {542, 13448}, {2641, 15073}, {30683, 30684} et al

X(30685) = midpoint of X(30683) and X(30684)

X(30686) = PERSPECTOR OF GEMINI TRIANGLE 37 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 37 AND 38

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

X(30686) lies on these lines: {2, 19}, {4, 12565}, {33, 2550}, {34, 28629}, {40, 406}, {65, 13567}, {85, 92}, {142, 278}, {196, 226}, {204, 4307}, {207, 388}, {329, 7079}, {451, 6197}, {516, 4183}, {946, 7498}, {1435, 9776}, {1519, 7551}, {1842, 11109}, {1859, 1861}, {1871, 8728} et al

X(30686) = polar conjugate of isogonal conjugate of X(4300)
X(30686) = {X(2), X(30687)}-harmonic conjugate of X(1848)

X(30687) = {X(1848), X(30686)}-HARMONIC CONJUGATE OF X(2)

Barycentrics (a^4 (b + c) - 2 a^2 (b^3 + c^3) - 4 a b^2 c^2 + (b - c)^2 (b + c) (b^2 + c^2))/(a^2 - b^2 - c^2) : :
Barycentrics sec A + cos B + cos C : :

X(30687) lies on these lines: {2, 19}, {29, 102}, {33, 3434}, {77, 278}, {92, 226}, {142, 17923}, {281, 908}, {406, 5250}, {442, 1871}, {516, 1013}, {962, 4194}, {1432, 16082}, {1748, 5745}, {1838, 12609}, {1844, 10916}, {1859, 2886} et al

X(30687) = polar conjugate of X(1065)
X(30687) = {X(2), X(4329)}-harmonic conjugate of X(30675)
X(30687) = {X(1848), X(30686)}-harmonic conjugate of X(2)

X(30688) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 37 AND 38

Barycentrics 1/((a^2 - b^2 - c^2) (a^2 + b^2 + c^2 - 2 b c) (a^2 b - a b^2 + a^2 c - a c^2 - a b c + b^2 c + b c^2)) : :

The perspectrix of Gemini triangles 37 and 38 passes through X(18344).

X(30688) lies on these lines: {19, 4209}, {278, 30705}, {281, 17786}

X(30688) = isogonal conjugate of X(30689)

X(30689) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 37 AND 38

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

X(30689) lies on these lines: {48, 1613}, {63, 77}, {614, 1184}

X(30689) = isogonal conjugate of X(30688)

X(30690) = ISOGONAL CONJUGATE OF X(2174)

Barycentrics    b c/(a^2 - b^2 - c^2 - b c) : :
Barycentrics    1/(1 + 2 cos A) : :
Barycentrics    1 + 2 cos B + 2 cos C + 4 cos B cos C : :

Let A₃₇B₃₇C₃₇ be Gemini triangle 37. Let L_A be the line through A₃₇ parallel to BC, and define L_B, L_C cyclically. Let A'₃₇ = L_B∩L_C, and define B'₃₇, C'₃₇ cyclically. Triangle A'₃₇B'₃₇C'₃₇ is homothetic to ABC at X(30690).

X(30690) lies on these lines: {2, 7110}, {7, 2994}, {8, 79}, {29, 1870}, {75, 3578}, {92, 445}, {94, 226}, {312, 1230}, {321, 4102}, {333, 2160}, {554, 7043}, {1081, 7026} et al

X(30690) = isogonal conjugate of X(2174)
X(30690) = isotomic conjugate of X(3219)
X(30690) = complement of anticomplementary conjugate of X(21276)
X(30690) = anticomplement of X(16585)
X(30690) = polar conjugate of X(6198)
X(30690) = trilinear pole of line X(522)X(4823) (the polar wrt polar circle of X(6198))
X(30690) = barycentric product X(i)*X(j) for these {i,j}: {1, 20565}, {75, 79}, {76, 2160}, {85, 7110}, {94, 3218}, {328, 1870}, {20932, 30602}
X(30690) = barycentric quotient X(i)/X(j) for these (i,j): (1, 35), (2, 3219), (4, 6198), (6, 2174), (75, 319), (79, 1), (85, 17095), (94, 18359), (1870, 186), (2160, 6), (3218, 323), (7110, 9), (20565, 75), (30602, 267)

X(30691) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 37

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

The side-triangle of ABC and Gemini triangle 37 is degenerate, lying on the perspectrix, line X(513)X(1835).

X(30691) lies on these lines: {65, 676}, {244, 665}, {354, 6366}, {513, 1835}, {928, 5902}, {942, 10015}, {1637, 9391} et al

X(30691) = tripolar centroid of X(278)

X(30692) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 38

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

The side-triangle of ABC and Gemini triangle 38 is degenerate, lying on the perspectrix, line X(3064)X(3700).

X(30692) lies on these lines: {926, 4120}, {2310, 3119}, {3064, 3700} et al

X(30692) = tripolar centroid of X(281)

X(30693) = ISOTOMIC CONJUGATE OF X(738)

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

Let A₃₇B₃₇C₃₇ be Gemini triangle 37. Let A' be the center of conic {{A,B,C,B₃₇,C₃₇}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30693).

X(30693) lies on these lines: {85, 17786}, {312, 2321}, {322, 4033}, {341, 346}, {594, 3959}, {646, 3718}, {2324, 3699} et al

X(30693) = isotomic conjugate of X(738)
X(30693) = barycentric product X(i)*X(j) for these {i,j}: {8, 341}, {75, 5423}, {76, 728}, {312, 346}, {646, 3239}, {3699, 4397}, {3718, 7046}
X(30693) = barycentric quotient X(i)/X(j) for these (i,j): (2, 738), (8, 269), (75, 479), (76, 23062), (312, 279), (341, 7), (346, 57), (646, 658), (728, 6), (3239, 3669), (3699, 934), (3718, 7056), (4397, 3676), (5423, 1), (7046, 34)

X(30694) = PERSPECTOR OF GEMINI TRIANGLE 37 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 37

Barycentrics a^4 - 2 a^3 (b + c) + 2 a^2 (b^2 - b c + c^2) - 2 a (b - c)^2 (b + c) + (b - c)^2 (b^2 + 4 b c + c^2) : :
Barycentrics 1/(1 + sec A) - 1/(1 + sec B) - 1/(1 + sec C) : :

X(30694) lies on these lines: {2, 85}, {4, 3732}, {8, 10025}, {10, 30625}, {63, 3691}, {92, 6392}, {100, 9305}, {120, 11681}, {144, 1654}, {145, 10405}, {193, 5942}, {329, 3661}, {346, 17786}, {1146, 6604} et al

X(30694) = anticomplement of X(348)
X(30694) = polar conjugate of X(7)-cross conjugate of X(4)
X(30694) = {X(2), X(30695)}-harmonic conjugate of X(3177)
X(30694) = {X(6392), X(21216)}-harmonic conjugate of X(30699)

X(30695) = {X(3177), X(30694)}-HARMONIC CONJUGATE OF X(2)

Barycentrics a^4 - 4 a^3 (b + c) + a^2 (6 b^2 - 4 b c + 6 c^2) - 4 a (b - c)^2 (b + c) + (b - c)^2 (b^2 + 6 b c + c^2) : :
Barycentrics tan^2(A/2) - tan^2(B/2) - tan^2(C/2) : :

X(30695) lies on these lines: {2, 85}, {8, 144}, {20, 3732}, {63, 28638}, {145, 10025}, {193, 20008}, {346, 16284}, {3621, 20111} et al

X(30695) = anticomplement of X(279)
X(30695) = {X(3177), X(30694)}-harmonic conjugate of X(2)

X(30696) = PERSPECTOR OF GEMINI TRIANGLE 38 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 38

Barycentrics a^9 - a^8 (b + c) - 2 a^7 (b^2 + b c + c^2) + 2 a^6 (b + c) (b^2 - 5 b c + c^2) + 2 a^5 b c (b^2 - 10 b c + c^2) + 18 a^4 b c (b - c)^2 (b + c) + 2 a^3 (b + c)^2 (b^4 - b^3 c + 12 b^2 c^2 - b c^3 + c^4) - 2 a^2 (b + c) (b^6 + 3 b^5 c - 13 b^4 c^2 + 2 b^3 c^3 - 13 b^2 c^4 + 3 b c^5 + c^6) - a (b - c)^2 (b + c)^4 (b^2 + c^2) + (b - c)^2 (b + c)^3 (b^4 - 2 b^3 c + 10 b^2 c^2 - 2 b c^3 + c^4) : :

X(30696) lies on these lines: {2, 800}, {278, 318}, {5809, 10453}

X(30696) = {X(2), X(30698)}-harmonic conjugate of X(30697)

X(30697) = PERSPECTOR OF GEMINI TRIANGLE 36 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 36 AND 37

Barycentrics a^8 (b + c) - a^7 (b^2 - b c + c^2) - a^6 (b + c) (3 b^2 - 5 b c + 3 c^2) + a^5 (3 b^4 - b^3 c - 12 b^2 c^2 - b c^3 + 3 c^4) + a^4 (b + c) (3 b^4 - 9 b^3 c - 4 b^2 c^2 - 9 b c^3 + 3 c^4) - a^3 (3 b^6 + b^5 c - 7 b^4 c^2 + 22 b^3 c^3 - 7 b^2 c^4 + b c^5 + 3 c^6) - a^2 (b + c)^3 (b^4 - 5 b^3 c + 4 b^2 c^2 - 5 b c^3 + c^4) + a (b^2 - c^2)^2 (b^4 + b^3 c + 8 b^2 c^2 + b c^3 + c^4) + b c (b - c)^2 (b + c)^5 : :

X(30697) lies on these lines: {2, 800}, {8, 57}, {345, 17786}

X(30697) = {X(2), X(30698)}-harmonic conjugate of X(30696)

X(30698) = {X(30696), X(30697)}-HARMONIC CONJUGATE OF X(2)

Barycentrics a^8 - 4 a^6 (b^2 + c^2) + a^4 (6 b^4 - 44 b^2 c^2 + 6 c^4) - 4 a^2 (b^6 - 9 b^4 c^2 - 9 b^2 c^4 + c^6) + (b^2 - c^2)^2 (b^4 + 14 b^2 c^2 + c^4) : :
Barycentrics 4 SA^3 (SB + SC) + SA^2 (5 SB^2 + 6 SB SC + 5 SC^2) + 2 SA SB SC (SB + SC) - 3 SB^2 SC^2 : :

X(30698) lies on these lines: {2, 800}, {69, 3146}, {75, 279}, {253, 1370}, {346, 18750}, {394, 17037}, {14360, 23974} et al

X(30698) = anticomplement of polar conjugate of isotomic conjugate of X(15740)
X(30698) = {X(30696), X(30697)}-harmonic conjugate of X(2)

X(30699) = PERSPECTOR OF GEMINI TRIANGLE 38 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 36 AND 38

Barycentrics a^3 + a^2 (b + c) + a (b^2 + c^2) + (b + c) (b^2 - 4 b c + c^2) : :
Barycentrics 1/(1 - sec A) - 1/(1 - sec B) - 1/(1 - sec C) : :

X(30699) lies on these lines: {2, 37}, {7, 1999}, {8, 3914}, {31, 24280}, {57, 1266}, {69, 3782}, {92, 6392}, {145, 388}, {149, 7391}, {193, 1839}, {225, 11851}, {226, 3875}, {239, 329}, {333, 4419}, {377, 20009}, {1722, 2899} et al

X(30699) = anticomplement of X(345)
X(30699) = polar conjugate of X(8)-cross conjugate of X(4)
X(30699) = {X(2), X(4452)}-harmonic conjugate of X(3210)
X(30699) = {X(6392), X(21216)}-harmonic conjugate of X(30694)

X(30700) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 35 AND 37

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

The side-triangle of Gemini triangles 35 and 37 is degenerate, lying on the perspectrix, line X(513)X(4468).

X(30700) lies on these lines: {2, 30704}, {210, 918}, {513, 4468}, {518, 1639}, {654, 5220}, {668, 891}, {926, 3681}, {984, 3310}, {1635, 4712}, {1638, 3740}, {3219, 6139}, {3695, 18289}, {3887, 14740} et al

X(30700) = reflection of X(30704) in X(2)

X(30701) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 35 AND 37

Barycentrics    1/(a^2 + (b - c)^2) : :
Barycentrics    1/(b c - SW) : :
Barycentrics    1/(cot^2(B/2) + cot^2(C/2)) : :

X(30701) lies on these lines: {1, 344}, {8, 105}, {28, 1043}, {57, 345}, {69, 17742}, {75, 277}, {81, 7123}, {88, 17740}, {220, 4437}, {274, 2345}, {278, 312}, {279, 304}, {281, 17786}, {291, 3976}, {306, 2333}, {321, 15474}, {668, 6554}, {959, 7672}, {961, 1037}, {985, 5255}, {1002, 3889}, {1390, 3616} et al

X(30701) = isogonal conjugate of X(16502)
X(30701) = isotomic conjugate of X(4000)
X(30701) = polar conjugate of X(1851)
X(30701) = trilinear pole of line X(513)X(4468) (the polar of X(1851) wrt polar circle)
X(30701) = barycentric product X(i)*X(j) for these {i,j}: {8, 8817}, {76, 7123}, {561, 7084}
X(30701) = barycentric quotient X(i)/X(j) for these (i,j): (1, 614), (2, 4000), (4, 1851), (6, 16502), (8, 497), (9, 2082), (55, 7083), (220, 30706), (7084, 31), (7123, 6), (8817, 7)

X(30702) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 36 AND 37

Barycentrics 1/((a^2 + b^2 + c^2 - 2 b c) (a^5 - 3 a^4 (b + c) + 2 a^3 (b + c)^2 + 2 a^2 (b - c)^2 (b + c) - a (3 b^4 + 4 b^3 c + 10 b^2 c^2 + 4 b c^3 + 3 c^4) + (b + c) (b^4 + 4 b^3 c - 2 b^2 c^2 + 4 b c^3 + c^4))) : :

X(30702) lies on these lines: {2257, 7131}, {3824, 8237}

X(30702) = isogonal conjugate of X(30703)

X(30703) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 36 AND 37

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

X(30703) lies on these lines: {57, 219}, {614, 1184}

X(30703) = isogonal conjugate of X(30702)

X(30704) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 36 AND 38

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

The side-triangle of Gemini triangles 36 and 38 is degenerate, lying on the perspectrix, line X(3900)X(4025).

X(30704) lies on these lines: {2, 30700}, {354, 918}, {518, 1638}, {891, 3227}, {926, 3873}, {982, 3310}, {1639, 3742}, {3218, 6139}, {3738, 4458}, {3900, 4025}, {4083, 4786} et al

X(30704) = reflection of X(30700) in X(2)

X(30705) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 36 AND 38

Barycentrics 1/((a - b - c)^2 (a^2 + (b - c)^2)) : :
Barycentrics 1/(tan^2(B/2) + tan^2(C/2)) : :

X(30705) lies on these lines: {7, 1037}, {9, 348}, {69, 200}, {77, 1041}, {85, 281}, {86, 4183}, {279, 304}, {1323, 21629} et al

X(30705) = isogonal conjugate of X(30706)
X(30705) = isotomic conjugate of X(6554)
X(30705) = polar conjugate of X(1863)
X(30705) = trilinear pole of line X(3900)X(4025) (the polar of X(1863) wrt polar circle)

X(30706) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 36 AND 38

Barycentrics a^2 (a - b - c)^2 (a^2 + (b - c)^2) : :
Barycentrics a^2 (tan^2(B/2) + tan^2(C/2)) : :

X(30706) lies on these lines: {2, 294}, {6, 57}, {25, 41}, {31, 1200}, {43, 170}, {55, 2195}, {100, 7123}, {198, 800}, {200, 220}, {213, 3198}, {219, 3169}, {440, 2238}, {613, 8300}, {614, 1184}, {650, 11502}, {851, 21753}, {1185, 9449}, {1202, 1471}, {1253, 8012}, {1436, 5065} et al

X(30706) = isogonal conjugate of X(30705)
X(30706) = crossdifference of every pair of points on line X(3900)X(4025)
X(30706) = barycentric product X(i)*X(j) for these {i,j}: {1, 4319}, {3, 1863}, {6, 6554}, {8, 7083}, {9, 2082}, {55, 497}, {56, 4012}, {57, 28070}, {100, 17115}, {200, 614}, {220, 4000}
X(30706) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30705), (55, 8817), (220, 30701), (497, 6063), (614, 1088), (1863, 264), (2082, 85), (4012, 3596), (4319, 75), (6554, 76), (7083, 7), (17115, 693), (28070, 312)

X(30707) = CENTROID OF GEMINI TRIANGLE 39

Barycentrics 3 a^3 - 25 a^2 (b + c) - a (43 b^2 + 82 b c + 43 c^2) - (b + c) (5 b + 3 c) (3 b + 5 c) : :

X(30707) lies on these lines: {2, 1449}, {1770, 18231}

X(30707) = anticomplement of X(28617)

X(30708) = CENTROID OF GEMINI TRIANGLE 40

Barycentrics 3 a^3 + 31 a^2 (b + c) + 13 a (b + c)^2 - (3 b - 5 c) (5 b - 3 c) (b + c) : :

X(30708) lies on these lines: {1, 6556}, {2, 1743}, {950, 18220}, {3731, 3982}, {3752, 4052}, {3817, 6686}, {3950, 4358}, {3986, 5316} et al

X(30709) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 39 AND 40

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

The side-triangle of Gemini triangles 39 and 40 is degenerate, lying on the perspectrix, line X(513)X(2517).

X(30709) lies on these lines: {2, 2787}, {4, 2775}, {8, 4010}, {80, 885}, {405, 16158}, {513, 2517}, {668, 891}, {671, 690}, {812, 14430}, {1022, 4013} et al

X(30709) = anticomplement of X(14419)

X(30710) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 39 AND 40

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

X(30710) lies on these lines: {1, 312}, {2, 1240}, {8, 181}, {28, 1791}, {57, 75}, {81, 314}, {88, 4359}, {89, 28605}, {105, 3757}, {226, 1432}, {239, 1258}, {264, 278}, {274, 1920}, {279, 6063}, {291, 3741}, {309, 1422}, {341, 7322}, {668, 1211}, {957, 3421}, {961, 4968}, {985, 4362}, {1002, 10453}, {1255, 4358}, {1402, 7081} et al

X(30710) = isogonal conjugate of X(2300)
X(30710) = isotomic conjugate of X(3666)
X(30710) = polar conjugate of X(1829)
X(30710) = X(19)-isoconjugate of X(22345)
X(30710) = trilinear pole of line X(513)X(2517) (the polar of X(1829) wrt polar circle)

X(30711) = X(2)X(1449)∩X(92)X(144)

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

Let A₃₉B₃₉C₃₉ be Gemini triangle 39. Let L_A be the line through A₃₉ parallel to BC, and define L_B, L_C cyclically. Let A'₃₉ = L_B∩L_C, and define B'₃₉, C'₃₉ cyclically. Triangle A'₃₉B'₃₉C'₃₉ is homothetic to ABC at X(30711).

X(30711) lies on these lines: {2, 1449}, {8, 4314}, {63, 10405}, {85, 4359}, {92, 144}, {145, 4981}, {312, 391}, {346, 4102}, {3663, 3943} et al

X(30711) = isotomic conjugate of anticomplement of X(5273)

X(30712) = ISOTOMIC CONJUGATE OF X(3617)

Barycentrics 1/(a - 3 b - 3 c) : :

Let A₄₀B₄₀C₄₀ be Gemini triangle 40. Let L_A be the line through A₄₀ parallel to BC, and define L_B, L_C cyclically. Let A'₄₀ = L_B∩L_C, and define B'₄₀, C'₄₀ cyclically. Triangle A'₄₀B'₄₀C_40' is homothetic to ABC at X(30712).

X(30712) lies on these lines: {1, 4373}, {2, 1743}, {7, 1420}, {27, 8025}, {69, 1268}, {75, 145}, {86, 16948}, {273, 7518}, {335, 4704}, {673, 17379}, {903, 3672}, {1215, 3633}, {1292, 3625}, {1440, 10586} et al

X(30712) = isotomic conjugate of X(3617)

X(30713) = ISOTOMIC CONJUGATE OF X(1412)

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

Let A₃₉B₃₉C₃₉ be Gemini triangle 39. Let A' be the center of conic {{A,B,C,B₃₉,C₃₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30713).

X(30713) lies on these lines: {2, 3264}, {306, 4033}, {312, 2321}, {313, 321}, {318, 341}, {333, 17787}, {349, 6358}, {561, 1233}, {1215, 4710}, {1269, 28605}, {1334, 3975} et al

X(30713) = isogonal conjugate of X(16947)
X(30713) = isotomic conjugate of X(1412)
X(30713) = barycentric product X(i)*X(j) for these {i,j}: {8, 313}, {10, 3596}, {75, 3701}, {76, 2321}, {210, 561}, {306, 7017}, {312, 321}, {318, 20336}, {333, 28654}, {341, 1441}, {346, 349}, {1334, 1502}, {4033, 4391}
X(30713) = barycentric quotient X(i)/X(j) for these (i,j): (2, 1412), (6, 16947), (8, 58), (10, 56), (75, 1014), (76, 1434), (210, 31), (306, 222), (312, 81), (313, 7), (318, 28), (321, 57), (333, 593), (341, 21), (346, 284), (349, 279), (1334, 32), (1441, 269), (2321, 6), (3701, 1), (4033, 651), (4391, 1019), (7017, 27), (20336, 77), (28654, 226)

X(30714) = MIDPOINT OF X(3) AND X(23236)

Barycentrics -4 a^10+10 a^8 (b^2+c^2)+(b^2-c^2)^4 (b^2+c^2)-7 a^6 (b^2+c^2)^2-a^2 (b^2-c^2)^2 (b^4+c^4)+a^4 (b^6+5 b^4 c^2+5 b^2 c^4+c^6) : :
X(30714) = X2*X[2]-3*X[11693], X[4]-3*X[110], 2*X[5]-3*X[5642], 3*X[74]-5*X[3522], 3*X[146]+X[5059], 3*X[265]-5*X[1656], 3*X[376]-X[15054], 3*X[381]-7*X[15039], X[382]-3*X[5655], 3*X[549]-2*X[20379], 2*X[576]-3*X[15303], 5*X[631]-3*X[9140], 5*X[632]-4*X[20396], 3*X[1495]-2*X[16619], 3*X[1568]-2*X[18572], X[3146]-3*X[10706], 2*X[3233]-X[25641], 3*X[3448]-7*X[3523], 9*X[3524]-7*X[15057], 7*X[3526]-5*X[15027], 7*X[3528]-5*X[15021], 17*X[3533]-15*X[15059], 9*X[3545]-7*X[15044], 2*X[3628]-3*X[11694], 4*X[3850]-3*X[10113], 7*X[3851]-6*X[7687], 11*X[3855]-13*X[15029], 11*X[5056]-12*X[12900], 13*X[5067]-11*X[15025], X[5073]-6*X[6053], 2*X[5446]-3*X[12824], 3*X[5891]-2*X[15738], X[7722]+X[12273], 3*X[10264]-5*X[15712], 13*X[10299]-15*X[15051], 3*X[10540]-X[18325], 5*X[11522]-6*X[11723], 3*X[11562]-2*X[13148], 3*X[11597]-2*X[12242], 3*X[11702]-2*X[11803], 3*X[11720]-2*X[13464], 2*X[11800]-3*X[16222], X[11801]-2*X[13392], X[12219]+X[15102], 2*X[12236]-3*X[16223], X[12308]+X[20127], X[12317]-3*X[15055], 2*X[14156]-X[25739], 2*X[14708]-X[21649], 9*X[14845]-8*X[15465], 5*X[15040]-3*X[15061], 3*X[16164]-2*X[16617], 3*X[16165]-2*X[16618]

Let A'B'C' be the 2nd Euler triangle. Let L, M, N be lines through A', B' and C', respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(30714); cf. X(i) for i = 74, 113, 399, 1147, 1511, 5504, 5609, 5655, 10692, 12383, 14094.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28804.

X(30714) lies on these lines: {2,11693}, {3,67}, {4,110}, {5,5642}, {20,541}, {24,12828}, {30,3292}, {52,5095}, {74,3522}, {125,128}, {146,5059}, {186,539}, {265,1656}, {376,15054}, {381,15039}, {382,5655}, {399,1498}, {549,20379}, {550,5562}, {567,25555}, {569,15118}, {576,15303}, {631,9140}, {632,20396}, {690,10992}, {1092,11750}, {1154,14448}, {1495,16619}, {1503,10564}, {1568,18572}, {1986,13431}, {2771,12757}, {2781,10625}, {2836,12675}, {2854,8550}, {2929,5898}, {2931,3515}, {3146,10706}, {3233,25641}, {3448,3523}, {3516,12168}, {3517,12310}, {3519,21394}, {3524,15057}, {3526,15027}, {3528,15021}, {3533,15059}, {3545,15044}, {3581,5965}, {3628,11694}, {3850,10113}, {3851,7687}, {3855,15029}, {4857,12896}, {5056,12900}, {5067,15025}, {5073,6053}, {5094,15115}, {5270,18968}, {5446,12824}, {5449,11449}, {5622,13336}, {5891,15738}, {7495,18475}, {7533,15033}, {7574,15139}, {7722,12273}, {8542,11179}, {8674,10993}, {8907,12893}, {9033,12790}, {9517,14900}, {9703,18388}, {9977,15037}, {10116,22467}, {10264,15712}, {10295,13754}, {10299,15051}, {10540,18325}, {10620,11850}, {11411,25712}, {11430,18553}, {11522,11723}, {11561,14049}, {11562,13148}, {11597,12242}, {11702,11803}, {11720,13464}, {11800,16222}, {11801,13392}, {12038,12827}, {12134,18488}, {12219,15102}, {12236,16223}, {12308,20127}, {12317,15055}, {13403,18350}, {14156,25739}, {14708,21649}, {14805,24206}, {14845,15465}, {15040,15061}, {15473,19504}, {16164,16617}, {16165,16618}, {16176,17834}, {18555,20771}

X(30714) = midpoint of X(i) and X(j) for these {i,j}: {3,23236}, {74,14683}, {110,12383}, {12219,15102}
X(30714) = reflection of X(i) in X(j) for these {i,j}: {4,16534}, {52,25711}, {113,110}, {125,1511}, {265,5972}, {3448,6699}, {5181,12584}, {7728,6053}, {10113,10272}, {11801,13392}, {12295,113}, {12902,7687}, {15063,5609}, {15133,15115}, {16003,3}, {21649,14708}, {25641,3233}, {25739,14156}
X(30714) = anticomplement of X(36253)
X(30714) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4,110,16534}, {4,16534,113}, {20,9143,14094}, {265,5972,23515}, {550,10990,16111}, {631,9140,20397}, {3448,15035,6699}, {9140,15020,631}, {10990,16163,550}, {12902,14643,7687}

Centers associated with line-reflected triangles: X(30715)-X(30721)

In the plane of a triangle ABC, suppose that A'B'C' is a triangle and L is a line. Let A'' be the reflection A' in L, and define B'' and C'' cyclically. The triangle A''B''C'' is here named the L-reflection of A'B'C'. If A'B'C' is a central triangle and L a central line, then A''B''C'' is a central triangle. (Clark Kimberling), January 11, 2019)

Let T denote the Euler-line-reflection of ABC. Peter Moses (January 12, 2019) found that T is perspective to the following triangles, with perspectors as indicated:

ABC: X(523)
Schroeter (anticevian triangle of X(523); see X(8286), X(10276)): X(523)
tangential: X(30715)
Macbeath: X(30716)
orthic-of-medial (anti-6th-mixtilinear; see X(11363)): X(30717)
5th Euler (see X(3758): X(30718)
circum-medial: X(23)
Gemini 44: X(23)
Gossard: X(30)
reflection of ABC in X(3): X(30)
infinite altitude: X(74)
circum-orthic: X(186)
Carnot (Johnson, the reflection of ABC in X(5)): X(30)
Kosnita: X(186)

The triangle T is also perspective to these triangles: Euler, Trinh, 2nd Euler, 5th Euler, Artzt, anti-Artzt, tangential of tangential, anti-1st-Euler, anti-Hutson intouch, anti-incircle-circles (see X(11363), orthic-of-medial, Ehrmann side-triangle.

The locus of a point P such that the cevian triangle of P is perspective to T is the cubic pK(14618,264). The locus of P such that the anticevian triangle of P is perspective to T is the cubic pK(112,648). (Peter Moses, January 13, 2019)

For the Nagel-line-reflection of ABC, see X(30719)-X(30721).

X(30715) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND TANGENTIAL

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

X(30715) lies on these lines: {6, 250}, {23, 230}, {157, 2453}, {186, 1503}, {523, 3447}, {5099, 11641}, {14729, 21006}

X(30715) = midpoint of X(3447) and X(7669)
X(30715) = reflection of X(3447) in the Euler line
X(30715) = tangential-isogonal conjugate of X(110)
X(30715) = X(338)-Ceva conjugate of X(6)
X(30715) = X(59)-of-tangential-triangle if ABC is acute

X(30716) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND MACBEATH

Barycentrics (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :
X(30716) = X[648] - 4 X[7473]

X(30716) lies on the cubic K556 and these lines: {4, 14884}, {20, 13573}, {99, 935}, {107, 476}, {110, 6368}, {112, 1287}, {242, 2074}, {250, 523}, {264, 2453}, {691, 1289}, {925, 1304}, {1286, 10423}, {1288, 10420}, {1316, 23635}, {1325, 10538}, {2409, 16237}, {4226, 14590}, {10421, 17702}, {13619, 29012}

X(30716) = reflection of X(i) and X(j) for these {i,j}: {250, 7473}, {648, 250}
X(30716) = reflection of X(250) in the Euler line
X(30716) = X(264)-Ceva conjugate of X(648)
X(30716) = X(i)-isoconjugate of X(j) for these (i,j): {656, 3447}, {810, 13485}, {4575, 6328}
X(30716) = trilinear pole of line {3448, 22146}
X(30716) = polar conjugate of isogonal conjugate of X(36830)
X(30716) = barycentric product X(i)*X(j) for these {i,j}: {162, 20941}, {648, 3448}, {811, 16562}, {6331, 7669}, {6528, 22146}, {14366, 14618}
X(30716) = barycentric quotient X(i) / X(j) for these {i,j}: {112, 3447}, {648, 13485}, {2501, 6328}, {3448, 525}, {7669, 647}, {8574, 20975}, {14366, 4558}, {16562, 656}, {20941, 14208}, {21092, 4064}, {21203, 4466}, {22146, 520}
X(30716) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {476, 7480, 107}, {935, 7482, 99}

X(30717) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND ORTHIC-OF-MEDIAL

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

X(30717) lies on these lines: {186, 249}, {523, 14060}, {858, 16320}, {3563, 16760}, {5968, 7485}

X(30718) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND 5TH EULER

Barycentrics a^12 - 2*a^10*b^2 + 2*a^8*b^4 + 6*a^6*b^6 - 5*a^4*b^8 - 4*a^2*b^10 + 2*b^12 - 2*a^10*c^2 + 2*a^8*b^2*c^2 - 8*a^6*b^4*c^2 + a^4*b^6*c^2 + 11*a^2*b^8*c^2 - 2*b^10*c^2 + 2*a^8*c^4 - 8*a^6*b^2*c^4 + 9*a^4*b^4*c^4 - 7*a^2*b^6*c^4 - 2*b^8*c^4 + 6*a^6*c^6 + a^4*b^2*c^6 - 7*a^2*b^4*c^6 + 4*b^6*c^6 - 5*a^4*c^8 + 11*a^2*b^2*c^8 - 2*b^4*c^8 - 4*a^2*c^10 - 2*b^2*c^10 + 2*c^12 : :

X(30718) lies on these lines: {2, 8877}, {23, 3849}, {99, 5189}, {111, 5099}, {112, 468}, {523, 10415}, {858, 10717}, {2453, 6032}

X(30718) = reflection of X(10415) in the Euler line

X(30719) = X(190)-CEVA CONJUGATE OF X(57)

Barycentrics (3 a-b-c) (b-c) (a+b-c) (a-b+c) : :
X(30719) = 3 X[3669] - X[7178],3 X[3676] - 2 X[7178],3 X[8643] + X[23764]

Let T be the Nagel-line-reflection of ABC. The locus of a ponit P such that T is perspective to the cevian triangle of P is the cubic pK(30719,7). (Peter Moses, January 13, 2019)

X(30719) lies on these lines: {56, 4401}, {57, 4498}, {109, 2737}, {190, 5382}, {241, 514}, {278, 1022}, {522, 4318}, {651, 25737}, {664, 1016}, {1420, 8643}, {1422, 2401}, {2403, 5435}, {3476, 28591}, {3667, 4162}, {4017, 4778}, {4077, 4801}, {4462, 4521}, {4546, 4925}, {4560, 7203}, {4905, 28292}, {6332, 21222}, {10106, 28470}, {24099, 29324}, {25576, 28846}

X(30719) = midpoint of X(6332) and X(21222)
X(30719) = reflection of X(i) and X(j) for these {i,j}: {3676, 3669}, {4462, 4521}, {4546, 4925}, {14837, 3960}, {21120, 7658}
X(30719) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2137, 150}, {8051, 21293}
X(30719) = X(i)-complementary conjugate of X(j) for these (i,j): {1415, 15347}, {30236, 141}
X(30719) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 57}, {664, 145}
X(30719) = X(i)-cross conjugate of X(j) for these (i,j): {3667, 3676}, {4394, 3667}
X(30719) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1293}, {55, 27834}, {101, 3680}, {644, 3445}, {663, 5382}, {692, 6557}, {1320, 2429}, {1415, 6556}, {3939, 8056}, {6558, 16945}
X(30719) = crosspoint of X(i) and X(j) for these (i,j): {190, 18743}, {279, 664}
X(30719) = crosssum of X(220) and X(663)
X(30719) = trilinear pole of line {2976, 3756}
X(30719) = crossdifference of every pair of points on line {55, 2347}
X(30719) = barycentric product X(i)*X(j) for these {i,j}: {7, 3667}, {57, 4462}, {85, 4394}, {145, 3676}, {279, 4521}, {479, 4546}, {514, 5435}, {658, 4534}, {664, 3756}, {693, 1420}, {934, 4939}, {1014, 4404}, {1088, 4162}, {1434, 14321}, {1743, 24002}, {2403, 3911}, {3669, 18743}, {3950, 17096}, {4077, 16948}, {4248, 17094}, {4504, 7249}, {4573, 21950}, {4626, 4953}, {4848, 7192}, {4998, 23764}, {6063, 8643}
X(30719) = barycentric quotient X(i) / X(j) for these {i,j}: {56, 1293}, {57, 27834}, {145, 3699}, {513, 3680}, {514, 6557}, {522, 6556}, {651, 5382}, {1404, 2429}, {1420, 100}, {1743, 644}, {2403, 4997}, {2441, 2316}, {2976, 5853}, {3052, 3939}, {3158, 4578}, {3161, 6558}, {3667, 8}, {3669, 8056}, {3676, 4373}, {3756, 522}, {3911, 2415}, {4162, 200}, {4394, 9}, {4404, 3701}, {4462, 312}, {4504, 7081}, {4521, 346}, {4534, 3239}, {4546, 5423}, {4729, 210}, {4848, 3952}, {4849, 4069}, {4855, 4571}, {4925, 3717}, {4939, 4397}, {4949, 4007}, {4953, 4163}, {5435, 190}, {7178, 4052}, {7200, 27831}, {8643, 55}, {14284, 6736}, {14321, 2321}, {14425, 2325}, {16948, 643}, {18211, 3737}, {18743, 646}, {20818, 4587}, {21950, 3700}, {23764, 11}

X(30720) = X(190)-CEVA CONJUGATE OF X(3699)

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

Let T be the Nagel-line-reflection of ABC. The locus of a ponit P such that T is perspective to the anticevian triangle of P is the cubic pK(30720,190). (Peter Moses, January 13, 2019)

The triangle T is perspective to the following triangles, with perspectors as indicated: ABC: X(3667)
intouch: X(30721)
Caelum (5th mixtilinear; see X(5603)): X(519)
outer Garcia: X(519)
Yff contact: X(4076)

The triangle T is also perspective to the following triangles: hexyl, 6th mixtilinear, Hutson intouch, Artzt, reflection of X(1) in sides of ABC, 3rd Conway (see X(10434), incircle-circles (see X(10434), anti-Artzt (see X(11147), and Jenkins (vertices are the centers of the Jenkings circles).

X(30720) lies on these lines: {8, 3021}, {101, 6079}, {190, 2415}, {312, 4986}, {346, 4370}, {644, 1639}, {664, 1016}, {728, 21384}, {1018, 28521}, {3161, 4534}, {4115, 4752}, {4513, 16969}

X(30720) = X(644), X(6558)}-harmonic conjugate of X(3699)
X(30720) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 3699}, {1016, 145}, {4076, 15519}, {8706, 4578}
X(30720) = X(i)-cross conjugate of X(j) for these (i,j): {4521, 3161}, {15519, 4076}
X(30720) = X(i)-isoconjugate of X(j) for these (i,j): {514, 16945}, {649, 19604}, {667, 27818}, {1357, 27834}, {3445, 3669}, {4394, 16079}
X(30720) = cevapoint of X(i) and X(j) for these (i,j): {3161, 4521}, {4546, 4936}
X(30720) = trilinear pole of line {3158, 3161}
X(30720) = crossdifference of every pair of points on line {1357, 17071}
X(30720) = barycentric product X(i)*X(j) for these {i,j}: {145, 3699}, {190, 3161}, {644, 18743}, {645, 3950}, {646, 1743}, {664, 6555}, {668, 3158}, {1016, 4521}, {3667, 4076}, {4162, 7035}, {4534, 6632}, {4546, 4998}, {4554, 4936}, {4848, 7256}, {4849, 7257}, {5435, 6558}, {8706, 12640}
X(30720) = barycentric quotient X(i) / X(j) for these {i,j}: {100, 19604}, {145, 3676}, {190, 27818}, {644, 8056}, {692, 16945}, {1293, 16079}, {1332, 27832}, {1743, 3669}, {3158, 513}, {3161, 514}, {3667, 1358}, {3699, 4373}, {3939, 3445}, {3950, 7178}, {4162, 244}, {4513, 27837}, {4521, 1086}, {4534, 6545}, {4546, 11}, {4578, 3680}, {4849, 4017}, {4936, 650}, {4953, 21132}, {6065, 1293}, {6555, 522}, {6558, 6557}, {8643, 1357}, {15519, 3667}, {16948, 7203}, {18743, 24002}

X(30721) = PERSPECTOR OF THESE TRIANGLES: NAGEL-LINE-REFLECTION OF ABC AND INTOUCH

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

X(30721) lies on these lines: {109, 765}, {190, 3667}, {513, 3699}, {519, 1738}, {2234, 5524}, {2748, 6012}, {4582, 4962}, {5205, 9362}, {5730, 6790}, {6163, 17780}

X(30721) = reflection of X(4076) in the Nagel line

X(30722) = X(241)X(514)∩X(1434)X(17906)

Barycentrics (b - c)*(-a + b - c)*(a + b - c)*(4*a + b + c) : :
X(30722) = 3 X[3669] + 2 X[3676],4 X[3669] + X[7178],7 X[3669] - 2 X[30719],8 X[3676] - 3 X[7178],7 X[3676] + 3 X[30719],4 X[3960] + X[21104],7 X[7178] + 8 X[30719]

Centers X(30722)-X(30726) are given by first barycentrics (b-c)(a-b+c)(a+b-c)(na+b+c), for n = 4, 3, 2, -2, -4 respectively; these points lie on the line X(241)X(514). See also X(30727).

X(30722) lies on these lines: {241, 514}, {1434, 17096}, {3649, 4017}

X(30722) = X(9)-isoconjugate of X(28210)
X(30722) = barycentric product X(i)*X(j) for these {i,j}: {7, 28209}, {514, 4031}, {551, 3676}, {1358, 4781}, {3669, 24589}, {4714, 7203}, {7178, 26860}, {16666, 24002}
X(30722) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 28210}, {551, 3699}, {3707, 6558}, {4031, 190}, {4781, 4076}, {7178, 27797}, {14435, 2325}, {16666, 644}, {21747, 3939}, {21806, 4069}, {22357, 4587}, {24589, 646}, {26860, 645}, {28209, 8}

X(30723) = X(241)X(514)∩X(1019)X(17096)

Barycentrics (b - c)*(-a + b - c)*(a + b - c)*(3*a + b + c) : :
X(30723) = 3 X[3669] + X[7178],3 X[3669] - X[30719],3 X[3676] - X[7178],3 X[3676] + X[30719],X[17496] + 3 X[21183]

See X(30722).

X(30723) lies on these lines: {241, 514}, {1019, 17096}, {3667, 4017}, {4040, 28225}, {4504, 28296}, {4765, 4801}, {4778, 16533}, {4978, 24002}, {7265, 22042}, {17496, 21183}, {28161, 30572}

X(30723) = midpoint of X(i) and X(j) for these {i,j}: {3669, 3676}, {4765, 4801}, {7178, 30719}
X(30723) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3669, 7178, 30719}, {3676, 30719, 7178}
X(30723) = X(i)-Ceva conjugate of X(j) for these (i,j): {664, 5586}, {4624, 7}
X(30723) = X(4790)-cross conjugate of X(4778)
X(30723) = X(i)-isoconjugate of X(j) for these (i,j): {9, 8694}, {55, 4606}, {101, 4866}, {210, 4627}, {644, 2334}, {1253, 4624}, {1334, 4614}, {3939, 25430}, {4515, 5545}
X(30723) = crosspoint of X(7) and X(4624)
X(30723) = crossdifference of every pair of points on line {55, 3217}
X(30723) = barycentric product X(i)*X(j) for these {i,j}: {7, 4778}, {57, 4801}, {85, 4790}, {269, 4811}, {279, 4765}, {514, 21454}, {693, 3361}, {1014, 4815}, {1434, 4841}, {1449, 24002}, {3616, 3676}, {3669, 19804}, {3671, 7192}, {4827, 23062}, {4830, 7233}, {5257, 17096}
X(30723) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 8694}, {57, 4606}, {279, 4624}, {391, 6558}, {513, 4866}, {1014, 4614}, {1412, 4627}, {1434, 4633}, {1449, 644}, {3361, 100}, {3616, 3699}, {3669, 25430}, {3671, 3952}, {3676, 5936}, {4512, 4578}, {4652, 4571}, {4765, 346}, {4773, 2325}, {4778, 8}, {4790, 9}, {4801, 312}, {4811, 341}, {4815, 3701}, {4818, 3790}, {4822, 210}, {4827, 728}, {4830, 3685}, {4832, 1334}, {4839, 3985}, {4841, 2321}, {4843, 4082}, {5586, 4756}, {19804, 646}, {21454, 190}

X(30724) = X(241)X(514)∩X(693)X(26732)

Barycentrics (b - c)*(-a + b - c)*(a + b - c)*(2*a + b + c) : :
X(30724) = 2 X[905] + X[21104],2 X[1019] + X[23729],4 X[2487] - X[4498],X[3669] + 2 X[3676],2 X[3669] + X[7178],5 X[3669] - 2 X[30719],4 X[3676] - X[7178],5 X[3676] + X[30719],X[4801] + 2 X[17069],X[4976] + 2 X[4978],5 X[7178] + 4 X[30719],X[21120] - 4 X[21188]

See X(30722).

X(30724) lies on these lines: {241, 514}, {693, 26732}, {900, 4017}, {918, 28779}, {1019, 23729}, {2487, 4498}, {3649, 4992}, {3910, 4453}, {4773, 29302}, {4801, 17069}, {4897, 28493}, {4927, 6002}, {4976, 4978}, {4977, 5298}, {6545, 29162}, {21183, 23880}

X(30724) = {X(3669), X(3676)}-harmonic conjugate of X(7178)
X(30724) = X(1434)-Ceva conjugate of X(1358)
X(30724) = X(4979)-cross conjugate of X(4977)
X(30724) = X(i)-isoconjugate of X(j) for these (i,j): {9, 8701}, {41, 6540}, {210, 4629}, {644, 1126}, {692, 4102}, {1171, 4069}, {1255, 3939}, {1334, 4596}, {3699, 28615}
X(30724) = crosspoint of X(3676) and X(17096)
X(30724) = crossdifference of every pair of points on line {55, 7064}
X(30724) = barycentric product X(i)*X(j) for these {i,j}: {7, 4977}, {57, 4978}, {85, 4979}, {269, 4985}, {279, 4976}, {479, 4990}, {514, 553}, {552, 6367}, {1014, 30591}, {1100, 24002}, {1125, 3676}, {1213, 17096}, {1358, 4427}, {1434, 4988}, {3649, 7192}, {3669, 4359}, {4017, 16709}, {4647, 7203}, {5298, 6548}, {7178, 8025}
X(30724) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 6540}, {56, 8701}, {514, 4102}, {553, 190}, {1014, 4596}, {1100, 644}, {1125, 3699}, {1358, 4608}, {1412, 4629}, {1434, 4632}, {1962, 4069}, {2308, 3939}, {3649, 3952}, {3669, 1255}, {3676, 1268}, {3683, 4578}, {3686, 6558}, {3916, 4571}, {4359, 646}, {4427, 4076}, {4856, 30720}, {4870, 4767}, {4976, 346}, {4977, 8}, {4978, 312}, {4979, 9}, {4983, 210}, {4984, 2325}, {4985, 341}, {4988, 2321}, {4990, 5423}, {4992, 27538}, {5298, 17780}, {6367, 6057}, {7178, 6539}, {7341, 6578}, {8025, 645}, {8663, 7064}, {16709, 7257}, {22054, 4587}, {30581, 4612}, {30591, 3701}, {30592, 4009}

X(30725) = X(1)X(2826)∩X(241)X(514)

Barycentrics (2*a - b - c)*(b - c)*(a + b - c)*(a - b + c) : :
X(30725) = 2 X[676] - 3 X[14413],3 X[1638] - 4 X[3960],3 X[1638] - 2 X[10015],3 X[1639] - 2 X[3762],3 X[3669] - 2 X[3676],4 X[3676] - 3 X[7178],X[3676] - 3 X[30719],X[4467] - 3 X[17496],X[4895] - 3 X[30573],X[7178] - 4 X[30719],3 X[14413] - X[21132]

See X(30722).

X(30725) lies on these lines: {1, 2826}, {7, 6009}, {8, 4925}, {12, 3837}, {56, 659}, {57, 21385}, {65, 891}, {88, 2403}, {190, 644}, {241, 514}, {523, 7286}, {676, 14413}, {764, 29240}, {812, 14759}, {900, 1317}, {1022, 2006}, {1086, 1358}, {1319, 1960}, {1388, 25569}, {1407, 24115}, {1639, 3762}, {2099, 21343}, {2254, 6366}, {2401, 21786}, {2821, 3057}, {2827, 15558}, {3910, 4467}, {4017, 4977}, {4391, 29005}, {4449, 6362}, {4453, 30577}, {4462, 26695}, {4504, 28487}, {4560, 18199}, {4762, 30181}, {4773, 23888}, {4897, 28468}, {4905, 28473}, {5433, 24093}, {6180, 24098}, {7288, 24128}, {9001, 14307}, {10074, 19916}, {14284, 30198}, {14425, 21129}, {23729, 29126}, {28585, 28591}

X(30725) = midpoint of X(i) and X(j) for these {i,j}: {659, 24097}, {2254, 21105}, {3904, 21222}
X(30725) = reflection of X(i) and X(j) for these {i,j}: {8, 4925}, {3669, 30719}, {3837, 24099}, {7178, 3669}, {10015, 3960}, {21120, 905}, {21129, 14425}, {21132, 676}
X(30725) = isogonal conjugate of X(5548)
X(30725) = isotomic conjugate of X(4582)
X(30725) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6079, 21286}, {8686, 150}
X(30725) = X(2743)-complementary conjugate of X(141)
X(30725) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 14027}, {655, 57}, {664, 1317}, {2006, 1086}
X(30725) = X(i)-cross conjugate of X(j) for these (i,j): {1635, 900}, {2087, 1319}, {4530, 1877}, {14027, 7}
X(30725) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5548}, {9, 901}, {31, 4582}, {41, 4555}, {55, 3257}, {88, 3939}, {100, 2316}, {101, 1320}, {106, 644}, {210, 4591}, {650, 9268}, {663, 5376}, {692, 4997}, {1022, 6065}, {1023, 1318}, {1252, 23838}, {1334, 4622}, {1417, 6558}, {2170, 6551}, {3689, 4638}, {3699, 9456}, {4571, 8752}, {4674, 5546}, {4792, 5549}
X(30725) = cevapoint of X(i) and X(j) for these (i,j): {900, 14425}, {2087, 6550}, {3310, 6085}
X(30725) = crosspoint of X(i) and X(j) for these (i,j): {514, 2401}, {655, 14628}, {2415, 4358}
X(30725) = crosssum of X(i) and X(j) for these (i,j): {101, 2427}, {2441, 9456}
X(30725) = trilinear pole of line {1647, 3259}
X(30725) = crossdifference of every pair of points on line {55, 2316}
X(30725) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3960, 10015, 1638}, {14413, 21132, 676}
X(30725) = barycentric product X(i)*X(j) for these {i,j}: {7, 900}, {44, 24002}, {57, 3762}, {85, 1635}, {86, 30572}, {269, 4768}, {279, 1639}, {331, 22086}, {479, 4528}, {514, 3911}, {519, 3676}, {658, 4530}, {664, 1647}, {693, 1319}, {1088, 4895}, {1111, 23703}, {1317, 6548}, {1358, 17780}, {1404, 3261}, {1434, 4120}, {1847, 14418}, {1877, 4025}, {1960, 6063}, {2087, 4554}, {3669, 4358}, {3943, 17096}, {3960, 14628}, {3992, 7203}, {4448, 7233}, {4453, 14584}, {4555, 14027}, {4608, 5298}, {4922, 7249}, {4998, 6550}, {7178, 16704}, {7209, 14408}, {14425, 27818}, {14427, 23062}
X(30725) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4582}, {6, 5548}, {7, 4555}, {44, 644}, {56, 901}, {57, 3257}, {59, 6551}, {109, 9268}, {244, 23838}, {513, 1320}, {514, 4997}, {519, 3699}, {649, 2316}, {651, 5376}, {900, 8}, {902, 3939}, {1014, 4622}, {1317, 17780}, {1319, 100}, {1357, 23345}, {1358, 6548}, {1404, 101}, {1412, 4591}, {1434, 4615}, {1635, 9}, {1639, 346}, {1647, 522}, {1877, 1897}, {1960, 55}, {2087, 650}, {2325, 6558}, {3251, 3689}, {3259, 2804}, {3285, 5546}, {3669, 88}, {3676, 903}, {3689, 4578}, {3762, 312}, {3911, 190}, {4017, 4674}, {4120, 2321}, {4358, 646}, {4448, 3685}, {4528, 5423}, {4530, 3239}, {4542, 4528}, {4730, 210}, {4768, 341}, {4773, 391}, {4895, 200}, {4922, 7081}, {4958, 4007}, {4984, 3686}, {4998, 6635}, {5298, 4427}, {5440, 4571}, {6544, 2325}, {6550, 11}, {7178, 4080}, {8661, 3271}, {14027, 900}, {14122, 6163}, {14407, 1334}, {14408, 3208}, {14418, 3692}, {14425, 3161}, {14427, 728}, {14429, 3710}, {14435, 3707}, {14442, 4530}, {16704, 645}, {17460, 23705}, {17780, 4076}, {21805, 4069}, {22086, 219}, {22356, 4587}, {23344, 6065}, {23345, 1318}, {23703, 765}, {23757, 6735}, {23888, 5233}, {24002, 20568}, {24188, 21132}, {24816, 23354}, {30572, 10}, {30573, 6745}, {30576, 4612}, {30583, 4009}

X(30726) = X(241)X(514)∩X(918)X(29002)

Barycentrics (4*a - b - c)*(b - c)*(a + b - c)*(a - b + c) : :
X(30726) = 5 X[3669] - 2 X[3676],4 X[3669] - X[7178],X[3669] + 2 X[30719],8 X[3676] - 5 X[7178],X[3676] + 5 X[30719],4 X[3960] - X[21120],X[7178] + 8 X[30719]

See X(30722).

X(30726) lies on these lines: {241, 514}, {918, 29002}, {4017, 28209}, {4897, 28501}, {4927, 28490}, {17496, 26732}

X(30726) = X(9)-isoconjugate of X(28218)
X(30726) = barycentric product X(i)*X(j) for these {i,j}: {7, 28217}, {3244, 3676}, {16669, 24002}
X(30726) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 28218}, {3244, 3699}, {16669, 644}, {28217, 8}

X(30727) = X(100)X(4427)∩X(644)X(1639)

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

Centers X(30727)-X(30732) are given by first barycentrics (a-b)(a-c)(a-b-c)(na+b+c), for n = 4, 3, 2, 0, -2, -4, respectively; these points lie on the line X(644)X(1639). See also X(30722).

X(30727) lies on these lines: {101, 4427}, {644, 1639}, {645, 4560}, {663, 4069}, {1023, 3952}, {4752, 17780}, {4781, 14435}

X(30727) = barycentric product X(i)*X(j) for these {i,j}: {8, 4781}, {100, 3902}, {190, 3707}, {551, 3699}, {643, 4714}, {644, 24589}, {646, 16666}, {4031, 6558}, {4076, 28209}, {7257, 21806}
X(30727) = barycentric quotient X(i)/X(j) for these {i,j}: {551, 3676}, {3707, 514}, {3902, 693}, {4714, 4077}, {4781, 7}, {6065, 28210}, {16666, 3669}, {21806, 4017}, {24589, 24002}, {26860, 17096}, {28209, 1358}

X(30728) = X(99)X(101)∩X(644)X(1639)

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

See X(30727).

X(30728) lies on these lines: {99, 101}, {644, 1639}, {1043, 27415}, {4997, 26074}, {9057, 29163}

X(30728) = X(4765)-cross conjugate of X(391)
X(30728) = X(i)-isoconjugate of X(j) for these (i,j): {1357, 4606}, {2334, 3669}, {3125, 5545}, {3248, 4624}
X(30728) = cevapoint of X(391) and X(4765),trilinear pole of line {391, 4061}
X(30728) = crossdifference of every pair of points on line {1357, 3122}
X(30728) = {X(644), X(3699)}-harmonic conjugate of X(6558)
X(30728) = barycentric product X(i)*X(j) for these {i,j}: {99, 4061}, {100, 4673}, {190, 391}, {461, 4561}, {644, 19804}, {645, 5257}, {646, 1449}, {668, 4512}, {765, 4811}, {1016, 4765}, {1978, 4258}, {3616, 3699}, {3671, 7256}, {4076, 4778}, {4571, 5342}, {4582, 4700}, {4600, 4843}, {6558, 21454}
X(30728) = barycentric quotient X(i)/X(j) for these {i,j}: {391, 514}, {461, 7649}, {644, 25430}, {1016, 4624}, {1449, 3669}, {3616, 3676}, {3699, 5936}, {3939, 2334}, {4061, 523}, {4101, 17094}, {4258, 649}, {4512, 513}, {4570, 5545}, {4578, 4866}, {4673, 693}, {4765, 1086}, {4771, 7212}, {4778, 1358}, {4811, 1111}, {4819, 30572}, {4827, 2170}, {4843, 3120}, {5257, 7178}, {6065, 8694}, {8653, 3122}, {19804, 24002}

X(30729) = X(101)X(835)∩X(644)X(1639)

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

See X(30727). X(30729) lies on these lines: {8, 4919}, {101, 835}, {644, 1639}, {645, 4612}, {1018, 17780}, {1023, 4103}, {4115, 4427}, {4723, 6603}

X(30729) = X(i)-cross conjugate of X(j) for these (i,j): {4976, 3686}, {4990, 3702}
X(30729) = X(i)-isoconjugate of X(j) for these (i,j): {604, 4608}, {1126, 3669}, {1171, 4017}, {3676, 28615}
X(30729) = cevapoint of X(i) and X(j) for these (i,j): {3686, 4976}, {3700, 4060}
X(30729) = crosspoint of X(645) and X(3699)
X(30729) = trilinear pole of line {3683, 3686}
X(30729) = barycentric product X(i)*X(j) for these {i,j}: {8, 4427}, {99, 4046}, {100, 3702}, {190, 3686}, {333, 4115}, {553, 6558}, {643, 4647}, {644, 4359}, {645, 1213}, {646, 1100}, {668, 3683}, {765, 4985}, {1016, 4976}, {1125, 3699}, {1230, 5546}, {1269, 3939}, {1962, 7257}, {3649, 7256}, {4069, 16709}, {4076, 4977}, {4582, 4969}, {4631, 21816}, {4990, 4998}, {6064, 6367}
X(30729) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4608}, {644, 1255}, {1100, 3669}, {1125, 3676}, {1213, 7178}, {1962, 4017}, {3683, 513}, {3686, 514}, {3699, 1268}, {3702, 693}, {3939, 1126}, {4046, 523}, {4076, 6540}, {4115, 226}, {4359, 24002}, {4427, 7}, {4587, 1796}, {4647, 4077}, {4856, 30719}, {4976, 1086}, {4977, 1358}, {4985, 1111}, {4990, 11}, {5546, 1171}, {6065, 8701}, {6367, 1365}, {6558, 4102}, {8025, 17096}, {20970, 7180}

X(30730) = X(2)X(4986)∩X(644)X(1639)

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

See X(30727). X(30730) lies on these lines: {2, 4986}, {8, 2170}, {10, 21950}, {101, 9059}, {190, 4606}, {346, 27546}, {514, 25272}, {644, 1639}, {646, 4526}, {1018, 3952}, {1930, 26757}, {2321, 21030}, {3212, 29715}, {3693, 4723}, {3701, 4515}, {3807, 4595}, {3950, 21041}, {4006, 17751}, {4033, 4552}, {4482, 17136}, {4560, 7257}, {4568, 21272}, {4674, 22035}, {4738, 24036}, {5546, 7256}, {17164, 21067}, {17314, 17465}, {20247, 29400}, {24403, 27076}, {24786, 27043}, {26752, 28598}

X(30730) = isotomic conjugate of X(17096)
X(30730) = X(i)-Ceva conjugate of X(j) for these (i,j): {3699, 4069}, {4033, 3952}, {4076, 6057}
X(30730) = X(i)-cross conjugate of X(j) for these (i,j): {3700, 2321}, {3709, 210}, {4041, 8}, {4069, 3952}, {6057, 4076}
X(30730) = X(i)-isoconjugate of X(j) for these (i,j): {6, 7203}, {31, 17096}, {34, 7254}, {56, 1019}, {57, 3733}, {58, 3669}, {59, 8042}, {60, 7216}, {109, 16726}, {163, 1358}, {244, 4565}, {269, 7252}, {513, 1412}, {514, 1408}, {552, 798}, {593, 4017}, {603, 17925}, {604, 7192}, {649, 1014}, {661, 7341}, {662, 1357}, {667, 1434}, {693, 16947}, {738, 21789}, {757, 7180}, {849, 7178}, {1015, 1414}, {1021, 7023}, {1106, 4560}, {1333, 3676}, {1395, 15419}, {1396, 1459}, {1397, 7199}, {1407, 3737}, {1415, 17205}, {1431, 18200}, {1435, 23189}, {1461, 18191}, {1577, 7342}, {1977, 4625}, {2185, 7250}, {2206, 24002}, {3248, 4573}, {3271, 4637}, {4620, 8027}, {7153, 16695}, {7253, 7366}
X(30730) = cevapoint of X(i) and X(j) for these (i,j): {37, 14321}, {210, 3709}, {522, 3706}, {523, 21949}, {650, 3686}, {2321, 3700}, {3239, 3965}
X(30730) = crosspoint of X(646) and X(3699)
X(30730) = trilinear pole of line {210, 2321}
X(30730) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1018, 4103, 3952}, {3699, 6558, 644}, {4103, 4169, 1018}, {4568, 23891, 21272}
X(30730) = crossdifference of every pair of points on line {1357, 8650}
X(30730) = barycentric product X(i)*X(j) for these {i,j}: {8, 3952}, {9, 4033}, {10, 3699}, {12, 7256}, {37, 646}, {55, 27808}, {75, 4069}, {99, 6057}, {100, 3701}, {101, 30713}, {190, 2321}, {210, 668}, {226, 6558}, {312, 1018}, {313, 3939}, {321, 644}, {333, 4103}, {341, 4551}, {346, 4552}, {523, 4076}, {594, 645}, {643, 1089}, {664, 4082}, {670, 7064}, {756, 7257}, {762, 4631}, {765, 4086}, {850, 6065}, {1016, 3700}, {1020, 30693}, {1334, 1978}, {1441, 4578}, {1897, 3710}, {2171, 7258}, {3596, 4557}, {3694, 6335}, {3704, 8707}, {3790, 4613}, {3943, 4582}, {3985, 4562}, {4041, 7035}, {4046, 6540}, {4052, 30720}, {4095, 27805}, {4102, 4115}, {4136, 4621}, {4169, 4997}, {4433, 4583}, {4515, 4554}, {4566, 5423}, {4574, 7017}, {5546, 28654}, {6358, 7259}, {6632, 21044}, {8706, 21031}
X(30730) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7203}, {2, 17096}, {8, 7192}, {9, 1019}, {10, 3676}, {37, 3669}, {55, 3733}, {99, 552}, {100, 1014}, {101, 1412}, {110, 7341}, {181, 7250}, {190, 1434}, {200, 3737}, {210, 513}, {219, 7254}, {220, 7252}, {281, 17925}, {312, 7199}, {321, 24002}, {341, 18155}, {345, 15419}, {346, 4560}, {480, 21789}, {512, 1357}, {522, 17205}, {523, 1358}, {594, 7178}, {643, 757}, {644, 81}, {645, 1509}, {646, 274}, {650, 16726}, {692, 1408}, {728, 1021}, {756, 4017}, {765, 1414}, {1016, 4573}, {1018, 57}, {1020, 738}, {1089, 4077}, {1252, 4565}, {1260, 23189}, {1334, 649}, {1500, 7180}, {1576, 7342}, {1783, 1396}, {2170, 8042}, {2171, 7216}, {2318, 1459}, {2321, 514}, {2329, 18200}, {3208, 18197}, {3239, 17197}, {3694, 905}, {3695, 17094}, {3699, 86}, {3700, 1086}, {3701, 693}, {3704, 3004}, {3709, 1015}, {3710, 4025}, {3711, 4833}, {3715, 4840}, {3717, 23829}, {3900, 18191}, {3939, 58}, {3950, 30719}, {3952, 7}, {3985, 812}, {4007, 4960}, {4033, 85}, {4037, 7212}, {4041, 244}, {4046, 4977}, {4061, 4778}, {4069, 1}, {4076, 99}, {4082, 522}, {4086, 1111}, {4095, 4369}, {4097, 4401}, {4103, 226}, {4111, 6372}, {4115, 553}, {4136, 3776}, {4140, 7200}, {4147, 23824}, {4162, 18211}, {4169, 3911}, {4171, 2170}, {4391, 16727}, {4433, 659}, {4513, 18199}, {4515, 650}, {4516, 764}, {4524, 3271}, {4538, 830}, {4551, 269}, {4552, 279}, {4557, 56}, {4559, 1407}, {4564, 4637}, {4566, 479}, {4571, 1444}, {4574, 222}, {4578, 21}, {4587, 1790}, {4612, 763}, {4998, 4616}, {5423, 7253}, {5546, 593}, {6057, 523}, {6065, 110}, {6066, 1576}, {6558, 333}, {6632, 4620}, {6735, 23788}, {7035, 4625}, {7064, 512}, {7081, 17212}, {7256, 261}, {7257, 873}, {7259, 2185}, {8611, 3942}, {17787, 16737}, {21044, 6545}, {21859, 1427}, {23067, 7053}, {24394, 2832}, {25268, 18600}, {27538, 17217}, {27808, 6063}, {30681, 15411}, {30713, 3261}

X(30731) = X(8)X(9)∩X(644)X(1639)

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

See X(30727). X(30731) lies on these lines: {8, 9}, {190, 6009}, {644, 1639}, {1018, 4427}, {1023, 4169}, {2415, 21129}, {2429, 14425}, {3952, 4752}

X(30731) = X(i)-Ceva conjugate of X(j) for these (i,j): {4076, 4152}, {4582, 3699}, {24004, 17780}
X(30731) = X(i)-cross conjugate of X(j) for these (i,j): {1639, 2325}, {4152, 4076}, {4528, 4723}, {4543, 8}
X(30731) = X(i)-isoconjugate of X(j) for these (i,j): {56, 1022}, {57, 23345}, {106, 3669}, {514, 1417}, {604, 6548}, {1357, 3257}, {1407, 23838}, {1408, 4049}, {1415, 6549}, {2403, 16945}, {2441, 19604}, {3676, 9456}
X(30731) = cevapoint of X(i) and X(j) for these (i,j): {44, 14425}, {1639, 2325}
X(30731) = crosspoint of X(i) and X(j) for these (i,j): {190, 6079}, {2415, 17780}, {3699, 4582}
X(30731) = crosssum of X(i) and X(j) for these (i,j): {649, 6085}, {2441, 23345}
X(30731) = trilinear pole of line {2325, 3689}
X(30731) = crossdifference of every pair of points on line {1357, 8054}
X(30731) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1023, 4169, 17780}, {6558, 30720, 644}
X(30731) = barycentric product X(i)*X(j) for these {i,j}: {8, 17780}, {9, 24004}, {44, 646}, {100, 4723}, {190, 2325}, {312, 1023}, {333, 4169}, {341, 23703}, {519, 3699}, {643, 3992}, {644, 4358}, {645, 3943}, {668, 3689}, {765, 4768}, {900, 4076}, {1016, 1639}, {2415, 3161}, {3264, 3939}, {3596, 23344}, {3911, 6558}, {4103, 30606}, {4152, 4555}, {4370, 4582}, {4528, 4998}, {4530, 6632}, {4542, 6635}, {4895, 7035}, {7257, 21805}
X(30731) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 6548}, {9, 1022}, {44, 3669}, {55, 23345}, {200, 23838}, {519, 3676}, {522, 6549}, {644, 88}, {646, 20568}, {692, 1417}, {900, 1358}, {1023, 57}, {1639, 1086}, {1960, 1357}, {2321, 4049}, {2325, 514}, {2415, 27818}, {3161, 2403}, {3689, 513}, {3699, 903}, {3711, 23352}, {3939, 106}, {3943, 7178}, {3992, 4077}, {4069, 4674}, {4076, 4555}, {4152, 900}, {4169, 226}, {4358, 24002}, {4528, 11}, {4530, 6545}, {4542, 6550}, {4543, 1647}, {4578, 1320}, {4587, 1797}, {4723, 693}, {4768, 1111}, {4873, 23598}, {4895, 244}, {5548, 2226}, {6065, 901}, {6558, 4997}, {14418, 3942}, {14427, 2170}, {16704, 17096}, {17780, 7}, {21805, 4017}, {23344, 56}, {23703, 269}, {24004, 85}

X(30732) = X(644)X(1639)∩X(649)X(1018)

Barycentrics (a - b)*(a - c)*(a - b - c)*(4*a - b - c) : :

See X(30727). X(30732) lies on these lines: {644, 1639}, {649, 1018}

X(30732) = barycentric product X(i)*X(j) for these {i,j}: {646, 16669}, {3244, 3699}, {4076, 28217}
X(30732) = barycentric quotient X(i)/X(j) for these {i,j}: {3244, 3676}, {6065, 28218}, {16669, 3669}, {28217, 1358}

X(30733) = ISOGONAL CONJUGATE OF X(28787)

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

As a point on the Euler line, X(30733) has Shinagawa coefficients {(2R + r - p)(p + r + 2 R)(2R + r), p^2 (r + R) - r (r + 2 R) (r + 3 R)}, or equivalently, (2FS²+2Fabc$a$, -(E+2F)S²-(E+F)abc$a$).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28809.

X(30733) lies on these lines: {1,2299}, {2,3}, {9,1474}, {19,5248}, {34,18593}, {37,943}, {72,2203}, {104,1301}, {105,1289}, {107,915}, {112,15344}, {158,2218}, {241,1396}, {1068,8747}, {1104,1870}, {1612,5317}, {1708,1780}, {1848,5259}, {1891,5251}, {1974,10477}, {2164,7040}, {2189,2303}, {2204,5089}, {2332,7719}, {2360,18446}, {2687,22239}, {2752,10423}, {3487,27802}, {5436,7713}, {11517,17776}, {14344,21789}, {20626,26707}

X(30733) = isogonal conjugate of X(28787)
X(30733) = barycentric product of X(i) and X(j) for these {i,j}: {27,3811}, {28,17776}, {29,1708}, {92,1780}, {286,2911}, {648,15313}, {1289,26217}, {1896,3173}, {2322,4341}, {4567,5521}
X(30733) = trilinear product of X(i) and X(j) for these {i,j}: {4,1780}, {27,2911}, {28,3811}, {162,15313}, {1172,1708}, {1474,17776}, {1896,3215}, {3173,8748}, {4183,4341}, {4570,5521}, {8747,11517}
X(30733) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,3651}, {4,3147,6889}, {4,6878,3541}, {4,7505,6829}, {21,28,4227}, {21,4233,28}, {25,405,4}, {25,468,4239}, {25,17520,28}, {28,2074,21}, {28,4183,4}, {29,13739,28}, {405,2915,440}, {440,2915,3651}, {2074,4233,4227}, {3089,6987,4}, {3575,8226,4}, {4248,17515,28}, {5047,7466,5142}, {6846,7487,4}

X(30734) = EULER LINE INTERCEPT OF X(32)X(21448)

Barycentrics a^2 (5 a^4-5 b^4+26 b^2 c^2-5 c^4) : :
Barycentrics (36 R^2-5 SW)S^2 + 5 SW SB SC : :

As a point on the Euler line, X(30734) has Shinagawa coefficients {4 E - 5 F, 5 E + 5 F)}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28811.

X(30734) lies on these lines: {2,3}, {32,21448}, {111,9605}, {373,26864}, {576,3066}, {1351,10545}, {1495,10541}, {3053,8585}, {3167,15019}, {3292,9777}, {5050,5643}, {5544,6800}, {5640,11482}, {5644,9544}, {5651,11477}, {5943,22234}, {8780,11451}, {9306,22330}, {10314,15860}, {11465,14530}, {14924,22112}, {15034,15465}

X(30734) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,5020,16042}, {3,12105,9715}, {3,16042,11284}, {1995,5020,11284}, {1995,11284,25}, {1995,16042,3}

X(30735) = (name pending)

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

X(30735) lies on these lines: {4, 512}, {325, 523}, {669, 6130}, {770, 2501}, {1995, 4108}, {3800, 18314}, {8599, 10412}, {12075, 16230}

X(30735) = isotomic conjugate of X(35575)

X(30736) = (name pending)

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

X(30736) lies on these lines: {2, 9462}, {76, 141}, {325, 523}, {524, 670}, {538, 30938}, {599, 20023}, {702, 3229}, {1084, 6379}, {1086, 18891}, {1978, 3943}, {1990, 6331}, {3314, 9464}, {3589, 9230}, {3815, 11059}, {4665, 6382}, {6383, 7263}, {18023, 18896}

X(30736) = isotomic conjugate of X(729)

X(30737) = (name pending)

Barycentrics b^2 c^2 (-2a^6 + b^6 + c^6 + a^4 b^2 + a^4 c^2 - b^4 c^2 - b^2 c^4) : :
Barycentrics (csc 2A) (sec B cos(B + ω) + sec C cos(C + ω)) : :

X(30737) lies on these lines: {2, 216}, {3, 1235}, {20, 76}, {22, 157}, {23, 30716}, {30, 339}, {66, 69}, {95, 15246}, {99, 1236}, {112, 15013}, {230, 338}, {248, 290}, {253, 305}, {276, 5481}, {314, 2897}, {315, 28706}, {316, 3153}, {317, 7391}, {322, 19799}, {325, 523}, {327, 14387}, {328, 476}, {340, 5189}, {350, 3100}, {384, 26164}, {427, 13409}, {441, 9475}, {511, 16083}, {925, 2857}, {1007, 30794}, {1078, 7488}, {1230, 3151}, {1297, 5999}, {1447, 10538}, {1494, 10989}, {1529, 16096}, {1909, 4296}, {1975, 11413}, {2207, 26226}, {2864, 26703}, {2867, 10229}, {2868, 3565}, {3101, 19810}, {3162, 28701}, {3552, 26179}, {4329, 30660}, {7386, 32000}, {7493, 26235}, {7499, 32078}, {7664, 16387}, {7750, 12225}, {7771, 10298}, {7791, 26214}, {8743, 28695}, {9464, 10513}, {11059, 30769}, {11326, 12143}, {13575, 20563}, {13854, 28412}, {14807, 15165}, {14808, 15164}, {15574, 26233}, {18135, 27505}, {18147, 27386}, {18686, 28428}, {26154, 27376}

X(30737) = isotomic conjugate of X(1297)
X(30737) = anticomplement of X(232)
X(30737) = polar conjugate of isogonal conjugate of X(441)
X(30737) = pole wrt polar circle of line X(25)X(647)

Collineation mappings involving Gemini triangle 101: X(30738)-X(30803)

Extending the preambles just before X(24537), X(26153), and X(27378), Gemini triangles A'B'C', indexed as 101 to 111, are introduced here, given by barycentrics for A', followed by the range of associated triangle centers. Each range of centers is preceded by a preamble.

Gemini triangle 101: A' = a^2 - b^2 - c^2 : 2 a^2 : 2 a ^2; range X(30738)-X(30803)
Gemini triangle 102: A' = a - b - c: 2 a : 2 a; range X(30808)-X(30869)
Gemini triangle 103: A' = - a^3 : b^3 + c^3 : b^3 + c^3; range X(30878)-X(30939)
Gemini triangle 104: A' = - b c : a(b + c) : a(b + c); range X(30942)-X(31007)
Gemini triangle 105: A' = - b - c : 2a + b + c : 2a + b + c; range X(31014)-X(31063)
Gemini triangle 106: A' = - b^2 - c^2 : 2a^2 + b^2 + c^2 : 2a^2 + b^2 + c^2; range X(31071)-X(31132)
Gemini triangle 107: A' = -1 : 2 : 2; range X(31133)-X(31181)
Gemini triangle 108: A' = 3a - b - c : 2(a - b - c) : 2(a - b - c); range X(31183)-X(31234)
Gemini triangle 109: A' = 1 : 2 : 2; range X(31235)-X(31283)
Gemini triangle 110: A' = 2 : 1 : 1; range X(31284)-X(31289)
Gemini triangle 111: A' = -3 : 1 : 1; range X(31290)-X(31205)

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 101, as in centers X(30738)-X(30803). Then

m(X) = a^3 x - (a + c) (a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. Also, m(nine-point circle) = nine-point circle. The fixed points of m are X(2) and every point on the line X(325)X(523), which is the isotomic conjugate of the circumcircle. Among the fixed points are X(i) for these i: 2, 325, 523, 684, 850 ,3260, 3265, 3266, 3267, 20735, 30736, 30737. (Clark Kimberling, January 19, 2019)

X(30738) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30738) lies on these lines: {2, 40}, {125, 27556}, {3012, 16051}, {5094, 29857}, {29826, 30739}, {30741, 30769}

X(30739) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics -a^4 b^2 + b^6 - a^4 c^2 + 8 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :
Barycentrics (12 R^2 - SW)S^2 - SB SC SW : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28990.

X(30739) lies on these lines: {2,3}, {9,26933}, {11,4319}, {12,4320}, {57,21015}, {69,26869}, {95,8901}, {120,29857}, {122,216}, {125,126}, {182,11064}, {230,5063}, {323,1353}, {325,11059}, {343,3819}, {373,5480}, {394,11245}, {495,5297}, {496,7292}, {574,24855}, {597,13857}, {599,5486}, {612,15888}, {908,25365}, {1177,15131}, {1184,5319}, {1194,9607}, {1196,7765}, {1503,5651}, {1506,15820}, {1899,15069}, {2549,16317}, {3003,3815}, {3054,9722}, {3266,3933}, {3291,5254}, {3292,8550}, {3564,15066}, {3580,7998}, {3589,22112}, {3763,5646}, {3818,16187}, {3917,13567}, {3925,23304}, {5092,5972}, {5640,21850}, {5888,15059}, {5891,16003}, {5913,9465}, {5971,14929}, {6090,6776}, {6292,15667}, {6509,26905}, {7085,20266}, {7668,11168}, {7999,26879}, {8262,16789}, {8758,29639}, {9680,18289}, {9777,18928}, {9820,13336}, {10192,22352}, {10293,10706}, {11061,15106}, {11513,13884}, {11514,13937}, {14569,15466}, {15082,24206}, {15135,22151}, {16511,19510}, {17818,26937}, {19924,20192}, {25406,26864}, {26866,27509}

X(30739) = midpoint of X(i) and X(j) for these {i,j}: {1995,16063}, {15066,18911}
X(30739) = reflection of X(i) in X(j) for these {i,j}: {10301,1995}, {16063,10300}
X(30739) = complement of X(1995)
X(30739) = complementary conjugate of X(8542)
X(30739) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,3,468}, {2,4,11284}, {2,22,6677}, {2,858,5}, {2,1368,427}, {2,1370,5020}, {2,5189,16042}, {2,7386,25}, {2,7396,7392}, {2,7484,7499}, {2,7485,6676}, {2,7496,7495}, {2,8889,7539}, {2,10300,10301}, {2,16051,5094}, {2,16063,1995}, {3,6816,1885}, {5,20,1906}, {5,858,427}, {5,1368,858}, {25,7386,7667}, {125,5650,141}, {140,5159,2}, {140,11585,7399}, {140,12362,17928}, {427,468,235}, {468,1885,25}, {632,13371,7405}, {1368,5159,11585}, {1370,5020,428}, {1594,14789,5}, {5067,15559,5}, {5092,5972,13394}, {5169,7493,11799}, {6676,7734,7485}, {6677,10691,22}, {7392,7396,5064}, {7495,7496,549}, {8889,15809,427}

X(30740) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30740) lies on these lines: {2, 7}, {5, 25583}, {8, 4561}, {75, 10589}, {348, 1329}, {1007, 3262}, {2551, 17095}, {2899, 3926}, {3007, 29857}, {3436, 17081}, {3665, 31246}, {4911, 17567}, {5804, 7380}, {6604, 25681}, {6931, 20880}, {7181, 31141}, {17170, 26364}, {30756, 30761}, {30760, 30772}

X(30741) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30741) lies on these lines: {1, 2}, {5, 2899}, {11, 344}, {38, 26132}, {69, 25613}, {125, 27704}, {341, 10588}, {345, 2886}, {497, 33116}, {518, 30828}, {908, 27549}, {962, 8229}, {1007, 3262}, {1265, 11375}, {2550, 32851}, {3161, 5510}, {3434, 33113}, {3618, 17723}, {3695, 31493}, {3701, 6933}, {3703, 31245}, {3712, 31140}, {3717, 5219}, {3756, 17265}, {3817, 30568}, {3932, 28808}, {3977, 24280}, {4385, 6856}, {4388, 5273}, {4438, 4672}, {4517, 30986}, {4645, 5744}, {4696, 10585}, {4737, 8164}, {5015, 6857}, {5094, 30787}, {5159, 13869}, {5218, 32850}, {5226, 32937}, {5261, 9369}, {5300, 6910}, {5712, 33121}, {5748, 27538}, {5846, 31187}, {7270, 30478}, {7283, 31418}, {10589, 18743}, {11491, 16434}, {11680, 17776}, {17064, 30699}, {17257, 25760}, {17740, 33108}, {18134, 24477}, {20483, 31497}, {20486, 30960}, {20556, 30943}, {21241, 24248}, {24597, 33070}, {26685, 33115}, {30738, 30769}, {30755, 30770}, {30791, 30800}, {33105, 33163}

X(30742) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30742) lies on these lines: {1, 116}, {2, 7}, {3, 24784}, {5, 20269}, {78, 17046}, {277, 3090}, {1656, 24774}, {2082, 17181}, {3012, 16051}, {3693, 30825}, {3739, 31245}, {4000, 10589}, {4904, 5886}, {5074, 5119}, {5137, 29855}, {5252, 17044}, {5587, 9317}, {7191, 18261}, {7778, 29857}, {11375, 21258}, {11681, 27006}, {13411, 26101}, {16601, 17675}, {16603, 25930}, {17062, 19861}, {17084, 26531}, {17761, 23708}, {30743, 30788}, {30750, 30770}, {30758, 30790}, {30763, 30791}, {31240, 33299}

X(30743) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30743) lies on these lines: {2, 48}, {7778, 30750}, {29857, 30773}, {30742, 30788}

X(30744) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30744) lies on these lines: {2, 3}, {6, 26913}, {69, 11216}, {110, 1853}, {125, 1993}, {155, 23294}, {373, 12058}, {394, 23293}, {599, 11416}, {1092, 32767}, {1194, 9745}, {1531, 11204}, {1568, 23329}, {1994, 26869}, {3060, 11746}, {3164, 17005}, {3167, 3448}, {3410, 6090}, {3763, 12220}, {5359, 5368}, {5972, 11550}, {6563, 8029}, {6697, 20806}, {8252, 11418}, {8253, 11417}, {9820, 11457}, {11064, 11442}, {11441, 20299}, {12827, 15113}, {14076, 15605}, {14156, 18474}, {15066, 21243}, {15534, 32244}, {16163, 18376}, {18018, 30786}, {18911, 23292}, {20243, 31245}, {22240, 31489}, {23300, 28408}, {29872, 30755}, {29873, 30756}

X(30745) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30745) lies on these lines: {2, 3}, {67, 22151}, {125, 323}, {230, 18365}, {511, 15059}, {523, 7925}, {599, 15826}, {625, 691}, {895, 19510}, {1236, 3266}, {1531, 15055}, {1568, 20417}, {1994, 26913}, {2697, 33640}, {3258, 31843}, {3292, 9140}, {3410, 23332}, {3448, 11064}, {3619, 32113}, {3763, 8705}, {5346, 5354}, {5504, 11564}, {5651, 7703}, {5971, 31655}, {6698, 10510}, {7736, 16306}, {7779, 16315}, {8288, 9225}, {10564, 14644}, {11004, 26869}, {12112, 14643}, {13857, 23061}, {14156, 25739}, {14920, 16177}, {16308, 31489}, {21680, 29653}

X(30745) = complement of X(37760)
X(30745) = radical trace of orthocentroidal circle and de Longchamps circle
X(30745) = circumcircle-inverse of X(37913)

X(30746) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30746) lies on these lines: {2, 31}, {48, 21235}, {2223, 30817}, {4892, 26267}, {7778, 30751}, {21238, 31265}, {21256, 31163}, {29857, 30759}, {29872, 30800}, {30747, 30750}, {30752, 30787}, {30758, 30784}, {31121, 32935}

X(30747) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30747) lies on these lines: {2, 32}, {22, 7842}, {23, 7825}, {122, 126}, {305, 7947}, {427, 7789}, {574, 31107}, {620, 16063}, {625, 1995}, {858, 3788}, {1194, 7851}, {3001, 5094}, {3266, 7888}, {3291, 7887}, {3314, 11056}, {3734, 5169}, {5031, 5651}, {7495, 7761}, {7784, 15822}, {7796, 19577}, {7816, 31133}, {7844, 9465}, {7874, 15820}, {7908, 9464}, {7925, 11059}, {7934, 26257}, {30746, 30750}, {30786, 30793}

X(30748) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30748) lies on these lines: {2, 37}, {10, 116}, {65, 20255}, {72, 21240}, {141, 210}, {142, 1215}, {304, 27299}, {518, 30945}, {612, 15668}, {614, 3706}, {910, 4376}, {1089, 24790}, {1104, 4372}, {1212, 16720}, {1930, 16583}, {3663, 3985}, {3701, 26978}, {3753, 24254}, {3788, 24784}, {3920, 28639}, {3930, 30821}, {3934, 24774}, {4009, 17290}, {4090, 21255}, {4113, 4445}, {4384, 16782}, {4395, 4519}, {4670, 5276}, {4851, 10327}, {4852, 7191}, {4875, 24735}, {5266, 25497}, {5275, 10436}, {5300, 26099}, {6600, 10472}, {7778, 29857}, {7795, 20269}, {16605, 20911}, {17007, 17344}, {17137, 21874}, {17152, 26689}, {17231, 20693}, {17239, 29679}, {17372, 33091}, {19846, 20267}, {20254, 30755}, {20880, 27040}, {21242, 24386}, {28633, 29667}, {29872, 30784}, {30749, 30768}, {30761, 30790}, {30949, 32931}

X(30748) = complement of X(26242)

X(30749) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30749) lies on these lines: {2, 39}, {3, 15652}, {23, 7816}, {51, 4074}, {99, 26257}, {125, 126}, {187, 26233}, {373, 24256}, {468, 7789}, {599, 11336}, {620, 7495}, {625, 5169}, {626, 858}, {1368, 21248}, {1495, 4048}, {1995, 3734}, {3001, 5094}, {3231, 14994}, {5189, 7842}, {5354, 7805}, {5971, 7804}, {7492, 32456}, {7496, 10130}, {7761, 16063}, {7784, 31152}, {7821, 15820}, {7825, 31133}, {7853, 31107}, {7908, 9745}, {7931, 9229}, {8371, 30476}, {8589, 31128}, {10162, 22110}, {16951, 33651}, {30748, 30768}

X(30750) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30750) lies on these lines: {2, 41}, {7778, 30743}, {29857, 30788}, {30742, 30770}, {30746, 30747}, {30751, 30771}

X(30751) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30751) lies on these lines: {1, 2}, {672, 4438}, {3838, 24330}, {4138, 20347}, {4441, 17064}, {7778, 30746}, {17279, 30959}, {21241, 24259}, {30750, 30771}, {30759, 30780}, {30760, 30787}, {30969, 33119}

X(30752) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30752) lies on these lines: {1, 2}, {57, 30969}, {63, 30953}, {141, 25613}, {3778, 17282}, {4713, 17605}, {7778, 30797}, {17754, 33119}, {21241, 24260}, {21264, 31245}, {30746, 30787}, {30755, 30767}

X(30753) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30753) lies on these lines: {2, 44}, {626, 24784}, {1491, 4885}, {7778, 29857}, {7862, 24774}, {30763, 30790}

X(30754) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30754) lies on these lines: {2, 45}, {4361, 5211}, {5205, 17313}, {7778, 29857}

X(30755) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30755) lies on these lines: {2, 11}, {219, 17047}, {499, 14019}, {612, 17783}, {2340, 30820}, {3290, 17064}, {3841, 16852}, {4438, 24318}, {5094, 29857}, {5275, 33111}, {7778, 30746}, {7925, 30791}, {10896, 16048}, {12953, 17522}, {16051, 30778}, {20254, 30748}, {20544, 30808}, {29872, 30744}, {29873, 31236}, {30741, 30770}, {30752, 30767}

X(30756) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30756) lies on these lines: {2, 12}, {55, 16067}, {120, 16051}, {5094, 29857}, {7778, 30743}, {29872, 31236}, {29873, 30744}, {30740, 30761}, {30762, 30787}, {30767, 30797}

X(30757) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30757) lies on these lines: {2, 7}, {77, 21244}, {78, 26012}, {241, 30826}, {5094, 29857}, {16603, 19861}, {21514, 24784}, {21807, 30800}, {25066, 30808}, {29826, 30778}, {30759, 30773}

X(30758) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30758) lies on these lines: {1, 4561}, {2, 37}, {7, 27538}, {8, 18156}, {9, 24586}, {10, 304}, {12, 85}, {69, 210}, {86, 612}, {99, 11116}, {183, 5205}, {239, 16524}, {274, 4385}, {305, 313}, {319, 10327}, {322, 325}, {341, 1909}, {480, 7081}, {518, 30962}, {614, 4360}, {894, 5275}, {960, 21281}, {1007, 3262}, {1089, 32092}, {1107, 25918}, {1215, 3718}, {1441, 6340}, {1698, 1930}, {1920, 3596}, {3061, 30030}, {3264, 4485}, {3664, 4090}, {3673, 18140}, {3679, 14210}, {3681, 30941}, {3699, 14828}, {3729, 3985}, {3753, 24282}, {3758, 5276}, {3761, 3992}, {3769, 33295}, {3807, 27475}, {3875, 5272}, {3876, 17137}, {3920, 17394}, {3926, 25583}, {3945, 6555}, {4087, 24239}, {4357, 21590}, {4386, 24358}, {4851, 20693}, {4903, 31995}, {4967, 24217}, {4968, 25585}, {5224, 18138}, {5282, 24602}, {5293, 24549}, {5880, 20716}, {6381, 20925}, {7191, 17393}, {7283, 19309}, {7778, 30763}, {8024, 30596}, {9436, 21609}, {9780, 20911}, {16020, 17158}, {16284, 25280}, {16605, 21216}, {16823, 17144}, {16830, 31997}, {17007, 17328}, {17754, 17755}, {17760, 30063}, {18135, 20880}, {18150, 31071}, {18157, 30966}, {18697, 27798}, {19591, 24511}, {20932, 29667}, {20955, 21605}, {26227, 30939}, {27491, 31006}, {28594, 30110}, {29648, 30598}, {29857, 30761}, {29966, 33299}, {30742, 30790}, {30746, 30784}, {30772, 30779}

X(30759) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30759) lies on these lines: {2, 58}, {3001, 5094}, {29857, 30746}, {30751, 30780}, {30757, 30773}, {30761, 30767}

X(30760) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30760) lies on these lines: {2, 6}, {114, 4220}, {316, 16046}, {321, 8781}, {625, 24271}, {626, 21495}, {980, 7888}, {1444, 21245}, {3788, 21511}, {5337, 7821}, {7784, 21537}, {18206, 30851}, {29857, 30746}, {29873, 30789}, {30740, 30772}, {30751, 30787}

X(30761) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30761) lies on these lines: {2, 6}, {10, 4561}, {75, 17064}, {99, 17677}, {114, 6626}, {316, 4234}, {620, 21937}, {625, 24275}, {626, 16061}, {1268, 7249}, {1447, 17273}, {3263, 29872}, {3454, 17206}, {3705, 4360}, {3788, 16060}, {4195, 7773}, {4352, 32821}, {6390, 16052}, {7081, 32025}, {7752, 13740}, {7763, 16062}, {7832, 17681}, {7912, 17688}, {9780, 17084}, {17322, 24239}, {29857, 30758}, {30740, 30756}, {30748, 30790}, {30759, 30767}, {30774, 30796}

X(30762) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30762) lies on these lines: {1, 2}, {4438, 24695}, {5846, 31232}, {8229, 20070}, {9779, 17339}, {21241, 24280}, {30756, 30787}

X(30763) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30763) lies on these lines: {1, 2}, {2899, 32972}, {3218, 31041}, {3219, 31023}, {4766, 33115}, {5513, 6651}, {7778, 30758}, {17738, 21241}, {17755, 30837}, {21026, 24602}, {24628, 31151}, {26582, 32851}, {26590, 33116}, {26629, 32850}, {26985, 30764}, {30742, 30791}, {30753, 30790}, {30782, 30793}

X(30764) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30764) lies on these lines: {2, 649}, {650, 30865}, {663, 21261}, {665, 30836}, {1459, 21262}, {4885, 30753}, {3005, 30476}, {9002, 17327}, {26985, 30763}

X(30765) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30765) lies on these lines: {2, 661}, {4885, 30753}, {2254, 3667}, {2487, 4806}, {4486, 21212}, {4728, 27486}, {5275, 18199}, {9013, 15668}, {17420, 21191}

X(30766) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30766) lies on these lines: {2, 667}, {3005, 30476}, {8642, 25686}, {30767, 30781}

X(30767) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30767) lies on these lines: {2, 31}, {7778, 29857}, {29639, 30784}, {29873, 30801}, {30752, 30755}, {30756, 30797}, {30759, 30761}, {30766, 30781}, {31121, 32938}

X(30768) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30768) lies on these lines: {1, 2}, {11, 17357}, {125, 1213}, {226, 26061}, {427, 1842}, {908, 33159}, {1738, 32779}, {2321, 33128}, {2887, 4672}, {3120, 17355}, {3663, 33161}, {3717, 32775}, {3739, 24186}, {3755, 33156}, {3821, 3977}, {3914, 5695}, {3946, 32848}, {4138, 26223}, {4357, 33115}, {5094, 8756}, {5249, 32780}, {5251, 7465}, {5745, 32781}, {6679, 28595}, {6684, 8229}, {7484, 23843}, {17353, 25760}, {21241, 24295}, {24210, 33157}, {24231, 33170}, {25527, 33163}, {30748, 30749}, {30759, 30761}, {30770, 30781}

X(30769) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics 3 a^6 - 7 a^4 b^2 - 3 a^2 b^4 + 7 b^6 - 7 a^4 c^2 + 14 a^2 b^2 c^2 - 7 b^4 c^2 - 3 a^2 c^4 - 7 b^2 c^4 + 7 c^6 : :

X(30769) lies on these lines: {2, 3}, {69, 23326}, {125, 193}, {253, 30786}, {3007, 29857}, {3818, 15431}, {5181, 15113}, {5272, 9643}, {5650, 12220}, {5921, 11064}, {6247, 32605}, {6723, 31670}, {7292, 9538}, {11002, 21847}, {11059, 30737}, {11061, 15131}, {11160, 13857}, {11441, 17822}, {11451, 12058}, {15069, 23332}, {15880, 31404}, {19121, 22112}, {19510, 23327}, {30738, 30741}

X(30770) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30770) lies on these lines: {2, 3}, {125, 15988}, {1441, 4554}, {29639, 30780}, {29857, 30746}, {30740, 30756}, {30741, 30755}, {30742, 30750}, {30768, 30781}

X(30771) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics (a^2 - b^2 - c^2) (a^4 - a^2 b^2 - 2 b^4 - a^2 c^2 + 4 b^2 c^2 - 2 c^4) : :
Barycentrics 4 - (cot ω)(tan B + tan C) : :

X(30771) lies on these lines: {2, 3}, {6, 32068}, {115, 8770}, {122, 31842}, {125, 394}, {141, 23327}, {154, 5972}, {155, 26944}, {183, 22468}, {216, 31489}, {230, 15905}, {305, 339}, {343, 14914}, {511, 26958}, {599, 11511}, {612, 18447}, {614, 18455}, {1040, 9629}, {1060, 5268}, {1062, 5272}, {1092, 12429}, {1180, 9745}, {1184, 22120}, {1196, 14961}, {1350, 6723}, {1351, 13567}, {1352, 23332}, {1353, 18950}, {1503, 8780}, {1560, 3162}, {1568, 10605}, {1578, 10576}, {1579, 10577}, {1611, 23115}, {1613, 14965}, {1619, 32125}, {1660, 15126}, {1853, 9306}, {1899, 3167}, {1993, 26869}, {2968, 30787}, {2972, 30789}, {2979, 12099}, {3564, 23291}, {3589, 15812}, {3763, 9973}, {3819, 9967}, {3917, 18438}, {3926, 6340}, {5050, 23292}, {5093, 11433}, {5644, 18583}, {6036, 13611}, {6090, 11442}, {6221, 18289}, {6390, 19583}, {6398, 18290}, {7736, 15851}, {7778, 20208}, {7999, 31807}, {8252, 11514}, {8253, 11513}, {8280, 8976}, {8281, 13951}, {8854, 13665}, {8855, 13785}, {9820, 19347}, {9927, 22808}, {10192, 21968}, {11123, 31279}, {11402, 18911}, {12160, 26879}, {12167, 26156}, {13857, 18449}, {13881, 22401}, {15066, 23293}, {15484, 15820}, {15533, 32257}, {17811, 21243}, {17814, 17822}, {18396, 18466}, {19459, 28408}, {19588, 28419}, {20254, 30748}, {21970, 33586}, {26926, 28708}, {30750, 30751}

X(30771) = complement of X(6353)

X(30772) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30772) lies on these lines: {2, 3}, {306, 4561}, {18589, 28653}, {30740, 30760}, {30758, 30779}

X(30773) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30773) lies on these lines: {2, 3}, {20336, 30786}, {29857, 30743}, {30757, 30759}

X(30774) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30774) lies on these lines: {2, 3}, {307, 30786}, {30761, 30796}

X(30775) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics 5 a^6 - 11 a^4 b^2 - 5 a^2 b^4 + 11 b^6 - 11 a^4 c^2 + 18 a^2 b^2 c^2 - 11 b^4 c^2 - 5 a^2 c^4 - 11 b^2 c^4 + 11 c^6 : :

X(30775) lies on these lines: {2, 3}, {69, 13857}, {125, 1992}, {1007, 9214}, {5032, 26869}, {11064, 11180}, {15113, 23327}

X(30776) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30776) lies on these lines: {2, 3}, {1007, 3262}, {29639, 30782}

X(30777) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30777) lies on these lines: {2, 3}, {83, 15820}, {3266, 7947}, {5971, 31076}, {7778, 30793}, {7873, 33651}, {7925, 11059}, {7931, 9229}, {17129, 19577}

X(30778) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30778) lies on these lines: {2, 12}, {11, 75}, {120, 29857}, {123, 17073}, {225, 26020}, {612, 32049}, {4187, 5121}, {5205, 21031}, {5272, 25522}, {6256, 19544}, {16051, 30755}, {21015, 33119}, {25760, 26933}, {29639, 30783}, {29826, 30757}

X(30779) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30779) lies on these lines: {2, 19}, {3, 28897}, {9, 25343}, {1368, 17073}, {3012, 16051}, {5089, 30840}, {7778, 30783}, {17134, 31044}, {20254, 30748}, {29857, 30743}, {30758, 30772}

X(30780) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30780) lies on these lines: {2, 35}, {5094, 29857}, {29639, 30770}, {30751, 30759}

X(30781) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30781) lies on these lines: {2, 36}, {5094, 29857}, {30766, 30767}, {30768, 30770}

X(30782) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30782) lies on these lines: {2, 7}, {92, 4554}, {326, 20305}, {4551, 6505}, {5227, 28755}, {9624, 16823}, {10584, 19785}, {11343, 24784}, {25083, 30808}, {29639, 30776}, {29857, 30746}, {30738, 30741}, {30758, 30772}, {30763, 30793}

X(30783) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30783) lies on these lines: {2, 65}, {241, 30960}, {1007, 3262}, {5094, 29857}, {7778, 30779}, {29639, 30778}

X(30784) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30784) lies on these lines: {2, 82}, {29639, 30767}, {29872, 30748}, {30746, 30758}

X(30785) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30785) lies on these lines: {2, 32}, {22, 7910}, {23, 7911}, {99, 31107}, {141, 4563}, {305, 7881}, {858, 7832}, {1194, 7923}, {1995, 7934}, {3266, 7909}, {3291, 7901}, {5094, 7868}, {7495, 7831}, {7748, 16276}, {7778, 11059}, {7794, 19577}, {7835, 16063}, {7853, 26257}, {7915, 15820}, {7919, 9465}, {7928, 15822}, {7931, 9229}, {14061, 26235}, {29639, 30767}

X(30786) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30786) lies on these lines: {2, 99}, {4, 10603}, {69, 125}, {95, 8901}, {183, 32216}, {253, 30769}, {264, 2970}, {287, 11064}, {305, 339}, {306, 4561}, {307, 30774}, {316, 691}, {325, 892}, {542, 10553}, {625, 17964}, {648, 1560}, {850, 9213}, {877, 16080}, {897, 28653}, {1007, 9214}, {1236, 3266}, {1368, 1799}, {1441, 4554}, {1641, 11161}, {1648, 10754}, {1650, 6394}, {3268, 5466}, {3589, 32740}, {3763, 31360}, {4576, 15059}, {5108, 8288}, {5116, 10160}, {5159, 6390}, {5468, 9140}, {5642, 14833}, {5913, 14568}, {5971, 7809}, {5972, 14928}, {6034, 32525}, {6091, 7386}, {7417, 10723}, {7496, 11643}, {7752, 14246}, {7757, 9745}, {7887, 14263}, {7931, 9229}, {8430, 30476}, {8753, 8889}, {10097, 24284}, {10163, 33273}, {10562, 31072}, {10989, 26276}, {11053, 11646}, {12036, 21356}, {14417, 14977}, {15031, 16042}, {17948, 22110}, {18018, 30744}, {20336, 30773}, {20563, 30802}, {20564, 30803}, {30747, 30793}, {30788, 30790}, {31101, 33651}, {31655, 31998}

X(30786) = isotomic conjugate of X(468)
X(30786) = complement of X(7665)
X(30786) = trilinear pole of line X(69)X(525)
X(30786) = isogonal conjugate of polar conjugate of X(18023)
X(30786) = X(92)-isoconjugate of X(14567)

X(30787) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30787) lies on these lines: {2, 11}, {858, 1290}, {1026, 30858}, {1332, 21252}, {2968, 30771}, {5094, 30741}, {7427, 10724}, {29639, 30770}, {30746, 30752}, {30751, 30760}, {30756, 30762}, {30790, 30792}

X(30788) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30788) lies on these lines: {2, 101}, {858, 2690}, {29857, 30750}, {30742, 30743}, {30786, 30790}

X(30789) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30789) lies on these lines: {2, 98}, {67, 24975}, {441, 10749}, {476, 858}, {868, 6321}, {1316, 22505}, {2407, 32244}, {2967, 5094}, {2972, 30771}, {3268, 5466}, {5159, 12079}, {6033, 15000}, {7422, 10733}, {7473, 10735}, {7703, 13860}, {7868, 12093}, {9146, 32458}, {10545, 13862}, {10748, 33184}, {16051, 31842}, {16092, 22110}, {29873, 30760}

X(30790) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30790) lies on these lines: {2, 45}, {116, 4561}, {244, 24412}, {325, 18025}, {4568, 31273}, {5205, 17297}, {7778, 30791}, {12035, 21356}, {30742, 30758}, {30748, 30761}, {30753, 30763}, {30786, 30788}, {30787, 30792}

X(30791) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30791) lies on these lines: {2, 37}, {120, 3314}, {210, 3620}, {3662, 27538}, {4090, 17298}, {5272, 17117}, {7778, 30790}, {7925, 30755}, {10327, 17373}, {17464, 29579}, {23668, 27798}, {29857, 30793}, {30741, 30800}, {30742, 30763}

X(30792) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30792) lies on these lines: {2, 900}, {120, 2977}, {523, 7625}, {2786, 25380}, {3828, 23888}, {21260, 29126}, {30787, 30790}

X(30793) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30793) lies on these lines: {2, 39}, {126, 7897}, {468, 7891}, {858, 7912}, {2972, 16051}, {3314, 30739}, {3620, 5650}, {5094, 7925}, {5206, 33651}, {7492, 31128}, {7778, 30777}, {7814, 15820}, {7840, 11336}, {7885, 31152}, {7898, 16063}, {11284, 17128}, {15822, 33022}, {29857, 30791}, {30747, 30786}, {30763, 30782}

X(30794) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30794) lies on these lines: {2, 66}, {114, 3548}, {157, 7484}, {1007, 30737}, {1368, 6389}, {1990, 3815}, {2549, 33324}

X(30795) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30795) lies on these lines: {2, 659}, {10, 21343}, {513, 24924}, {514, 31251}, {523, 7925}, {891, 1698}, {900, 27191}, {1125, 25569}, {4885, 30753}, {1656, 2826}, {1960, 3624}, {2254, 4800}, {2821, 8227}, {2977, 4927}, {3617, 25574}, {3762, 19947}, {3960, 14431}, {4010, 4928}, {4088, 14475}, {4367, 21260}, {4453, 18004}, {4728, 4810}, {4782, 31207}, {6550, 31263}, {14288, 28284}, {20317, 23765}, {20947, 21439}, {21722, 29653}, {23301, 27193}, {24533, 25511}, {24768, 25128}, {25126, 26114}, {26983, 31946}, {29362, 31209}, {30787, 30790}

X(30796) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics b c (a^4 - 2 a^3 b - 2 a^2 b^2 + 2 a b^3 + b^4 - 2 a^3 c + 10 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(30796) lies on these lines: {2, 85}, {11, 75}, {1007, 3262}, {3452, 7182}, {3673, 5121}, {5205, 16284}, {5211, 17158}, {21609, 30827}, {30761, 30774}

X(30797) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30797) lies on these lines: {2, 87}, {7778, 30752}, {29857, 30791}, {30756, 30767}

X(30798) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30798) lies on these lines: {2, 7}, {3314, 5205}, {4911, 17694}, {7778, 30758}, {14064, 25583}, {16060, 24784}, {17095, 26558}, {29857, 30791}, {30748, 30761}

X(30799) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30799) lies on these lines: {2, 896}, {4885, 30753}, {2173, 24716}, {29857, 30746}

X(30800) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30800) lies on these lines: {2, 38}, {3264, 4485}, {4650, 26232}, {7778, 29857}, {17595, 20716}, {19584, 25741}, {21807, 30757}, {26265, 32938}, {26267, 32940}, {29872, 30746}, {30741, 30791}

X(30801) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30801) lies on these lines: {2, 38}, {896, 26232}, {4011, 14439}, {4703, 31080}, {4892, 31121}, {17451, 20340}, {17872, 27102}, {21805, 32853}, {26267, 32935}, {29857, 30746}, {29873, 30767}

X(30802) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30802) lies on these lines: {2, 3}, {125, 20806}, {1993, 12585}, {20563, 30786}

X(30803) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 101

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

X(30803) lies on these lines: {2, 3}, {20564, 30786}

X(30804) = (name pending)

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

X(30804) lies on these lines: {7, 513}, {322, 20949}, {514, 661}, {522, 23819}, {3309, 20295}, {3900, 4106}, {4077, 8713}, {4509, 4811}, {7180, 28025}, {14077, 21297}, {20895, 20906}

X(30804) = isotomic conjugate of X(37223)

X(30805) = (name pending)

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

X(30805) lies on these lines: {75, 8058}, {86, 21172}, {99, 2727}, {190, 1275}, {514, 1921}, {520, 3265}, {1459, 4025}, {1919, 3798}, {3239, 18160}, {3904, 24002}, {6332, 15413}, {7649, 17215}, {10436, 21186}

X(30805) = isotomic conjugate of isogonal conjugate of X(4091)

X(30806) = (name pending)

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

X(30806) lies on these lines: {1, 26563}, {7, 8}, {76, 3702}, {78, 1446}, {100, 5088}, {145, 3673}, {150, 5176}, {218, 26653}, {220, 28961}, {279, 7080}, {304, 3701}, {321, 3761}, {341, 21605}, {348, 5552}, {350, 4742}, {355, 21285}, {404, 7176}, {498, 27187}, {514, 661}, {517, 20347}, {519, 1111}, {644, 10025}, {664, 4511}, {668, 3263}, {672, 21232}, {948, 28795}, {999, 26229}, {1055, 6647}, {1121, 17297}, {1125, 7278}, {1150, 3306}, {1212, 28742}, {1229, 17296}, {1259, 3188}, {1320, 2481}, {1323, 6745}, {1376, 7223}, {1565, 17757}, {1575, 7200}, {1739, 17205}, {1930, 4696}, {2170, 20335}, {2280, 24249}, {3035, 7181}, {3085, 25581}, {3160, 27383}, {3177, 25082}, {3244, 7264}, {3436, 17170}, {3555, 20247}, {3664, 24993}, {3665, 12607}, {3684, 9317}, {3726, 21138}, {3812, 17169}, {3877, 30946}, {3879, 17863}, {3902, 4441}, {3930, 21139}, {3991, 25237}, {4390, 24333}, {4416, 25001}, {4487, 4986}, {4642, 24214}, {4671, 29616}, {4694, 21208}, {4737, 31130}, {4872, 5080}, {4875, 6706}, {4911, 20060}, {4998, 28058}, {5057, 5195}, {5209, 16741}, {5328, 29627}, {5440, 17136}, {5736, 27401}, {6063, 18810}, {6554, 28740}, {6735, 9436}, {6736, 10481}, {6921, 17081}, {7187, 26752}, {8897, 19645}, {9780, 25585}, {10914, 20244}, {11681, 17181}, {14923, 17753}, {14951, 30030}, {15988, 17120}, {16720, 25102}, {16732, 17374}, {16823, 26239}, {17048, 17474}, {17073, 27507}, {17095, 27529}, {17158, 20050}, {17272, 24547}, {17288, 26538}, {17298, 20905}, {17347, 28974}, {17364, 26665}, {18135, 18156}, {20089, 25242}, {24215, 24443}, {25066, 27096}, {25068, 27025}, {25244, 26757}, {25728, 30625}, {25994, 27097}, {27253, 27267}, {27541, 28756}, {28980, 32008}, {29965, 29966}

X(30806) = isogonal conjugate of X(34068)
X(30806) = isotomic conjugate of X(1156)

X(30807) = ISOGONAL CONJUGATE OF X(911)

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

X(30807) lies on these lines: {2, 85}, {4, 8}, {6, 17863}, {7, 14524}, {9, 1441}, {41, 24268}, {44, 16732}, {63, 169}, {69, 1229}, {75, 144}, {100, 2724}, {145, 20173}

X(30807) = isogonal conjugate of X(911)
X(30807) = isotomic conjugate of X(36101)
X(30807) = anticomplement of X(241)
X(30807) = X(6)-isoconjugate of X(103)

Collineation mappings involving Gemini triangle 102: X(30808)-X(30869)

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), 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 102, as in centers X(30808)-X(30869). Then

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

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(514)X(661), which is the isotomic conjugate of the circumellipse {{A,B,C,X(88), X(100), X(162), X(190)}}. Among the fixed points are X(i) for these i: 2, 514, 693, 3912, 6381, 4358, 15413, 30804, 30805, 30806, 30807. (Clark Kimberling, January 19, 2019)

X(30808) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30808) lies on these lines: {2, 3}, {6, 8287}, {71, 31265}, {116, 5228}, {218, 26012}, {219, 20305}, {226, 10402}, {281, 16596}, {312, 17864}, {355, 26006}, {965, 17052}, {1826, 17073}, {1853, 17188}, {3061, 17284}, {5074, 5316}, {5226, 14256}, {5747, 18635}, {5886, 25935}, {7359, 24316}, {10478, 26958}, {11396, 26157}, {11681, 28757}, {14963, 30811}, {18743, 20926}, {18747, 28755}, {20544, 30755}, {20818, 21270}, {22005, 31993}, {22356, 31163}, {25066, 30757}, {25083, 30782}, {29627, 30834}, {30830, 30856}

X(30809) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30809) lies on these lines: {2, 3}, {69, 645}, {169, 7308}, {196, 18588}, {281, 18589}, {312, 20235}, {944, 26006}, {948, 1323}, {966, 17052}, {1214, 17916}, {1446, 5226}, {1826, 31261}, {2391, 3452}, {3008, 9581}, {3436, 28757}, {3940, 29616}, {4648, 5747}, {5044, 29611}, {5222, 5722}, {5308, 11374}, {5603, 25935}, {5719, 29624}, {5720, 25930}, {5746, 18635}, {5748, 29627}, {5819, 6666}, {8756, 31158}, {8804, 18634}, {12433, 17014}, {16593, 30826}, {18743, 20914}, {18747, 28753}, {30817, 30860}, {30825, 30847}, {30828, 30830}

X(30809) = complement of X(24604)
X(30809) = anticomplement of X(31184)

X(30810) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30810) lies on these lines: {2, 3}, {9, 141}, {71, 16608}, {72, 3912}, {142, 8804}, {216, 16595}, {219, 26130}, {226, 241}, {312, 21403}, {329, 18139}, {517, 25935}, {573, 25964}, {579, 18635}, {950, 3008}, {954, 26939}, {1214, 5236}, {1260, 28795}, {1385, 26006}, {1565, 24635}, {1855, 6708}, {1901, 17245}, {2893, 17277}, {2975, 28757}, {3002, 5718}, {3419, 4384}, {3487, 5308}, {3488, 5222}, {3586, 31183}, {4260, 5728}, {4648, 5746}, {5283, 17056}, {5436, 29598}, {5742, 17052}, {5745, 17046}, {5776, 25878}, {5798, 24220}, {6703, 25497}, {7359, 24315}, {7719, 10319}, {8053, 23305}, {11523, 29573}, {12625, 16833}, {12690, 29628}, {15650, 29579}, {16607, 29604}, {17073, 31261}, {17243, 22021}, {18250, 20106}, {18446, 25930}, {18743, 21579}, {20367, 21258}, {22018, 31993}, {25499, 25525}, {30819, 30860}

X(30810) = isotomic conjugate of cevapoint of X(2) and X(379)
X(30810) = complement of X(379)

X(30811) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30811) lies on these lines: {2, 6}, {3, 23860}, {5, 19782}, {8, 17724}, {10, 3711}, {31, 29865}, {42, 31237}, {45, 26580}, {55, 2887}, {63, 17345}, {100, 25959}, {220, 28796}, {226, 17355}, {238, 29858}, {306, 3772}, {312, 17268}, {345, 3782}, {379, 26079}, {405, 3454}, {518, 29857}, {748, 29869}, {902, 31134}, {908, 16580}, {984, 29862}, {1001, 25760}, {1086, 17740}, {1125, 17723}, {1350, 8229}, {1376, 25957}, {1386, 29855}, {1407, 28774}, {1621, 25958}, {1836, 4138}, {1999, 17386}, {2886, 33171}, {3006, 3242}, {3011, 3416}, {3052, 6327}, {3120, 5695}, {3218, 7232}, {3286, 30984}, {3306, 3834}, {3616, 17726}, {3662, 17595}, {3666, 17304}, {3681, 29873}, {3687, 24789}, {3703, 33144}, {3705, 17597}, {3706, 17064}, {3712, 24248}, {3741, 31245}, {3836, 4413}, {3844, 29828}, {3846, 4423}, {3873, 29872}, {3911, 21255}, {3912, 17720}, {3923, 4892}, {3944, 4387}, {3965, 16610}, {3977, 17276}, {4042, 33084}, {4054, 17281}, {4062, 33128}, {4201, 25663}, {4202, 4255}, {4258, 26085}, {4358, 17267}, {4361, 33077}, {4363, 31019}, {4415, 17776}, {4421, 32948}, {4422, 31018}, {4428, 32947}, {4438, 33064}, {4450, 21000}, {4649, 29856}, {4671, 17269}, {4850, 17290}, {4865, 29656}, {4966, 11269}, {4997, 30861}, {5124, 21488}, {5219, 7146}, {5220, 33065}, {5226, 31598}, {5284, 29870}, {5328, 6554}, {5725, 19869}, {5846, 26228}, {6679, 32946}, {6690, 26034}, {7081, 17783}, {8167, 25960}, {9053, 31091}, {11235, 32943}, {11287, 24296}, {11680, 33173}, {13881, 28808}, {14963, 30808}, {15523, 33127}, {16062, 19765}, {16458, 24931}, {16884, 29833}, {17061, 33088}, {17184, 33113}, {17227, 24627}, {17230, 25529}, {17258, 27184}, {17262, 32849}, {17266, 30829}, {17318, 33155}, {17351, 31164}, {17395, 19823}, {17396, 19786}, {17397, 19832}, {17599, 26128}, {17717, 29637}, {17719, 29674}, {17722, 29660}, {17725, 32847}, {17889, 33160}, {18593, 25078}, {18743, 20444}, {20343, 28798}, {21241, 31140}, {22021, 25525}, {24441, 27754}, {24620, 27191}, {24892, 33081}, {24943, 33105}, {26230, 33070}, {26687, 30174}, {28595, 29670}, {29473, 32954}, {29634, 33073}, {29638, 32844}, {29640, 32784}, {29641, 33126}, {29643, 32775}, {29658, 32846}, {29665, 33078}, {29675, 33076}, {29678, 32781}, {29681, 33075}, {29839, 32773}, {29848, 33072}, {29849, 33123}, {30816, 30821}, {30820, 30826}, {30830, 30853}, {30846, 30863}, {31053, 33157}, {31197, 31243}, {32778, 33130}, {32783, 33111}, {32848, 33143}, {32855, 33147}, {32856, 33161}, {32858, 33133}, {32862, 33153}, {33069, 33119}, {33087, 33140}, {33089, 33148}, {33092, 33152}, {33101, 33164}, {33103, 33167}, {33108, 33175}, {33146, 33168}

X(30811) = complement of X(24597)

X(30812) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30812) lies on these lines: {2, 65}, {9, 4670}, {37, 21246}, {210, 28797}, {241, 30961}, {312, 21422}, {958, 16831}, {1212, 3452}, {1319, 24612}, {1329, 3912}, {2264, 27381}, {2551, 5308}, {3061, 17284}, {3661, 5123}, {3687, 3813}, {4384, 5289}, {5087, 7377}, {5302, 29578}, {5328, 29627}, {8165, 29621}, {8287, 17231}, {15829, 16832}, {16593, 16596}, {18743, 21596}, {21233, 31238}, {30946, 31225}

X(30813) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30813) lies on these lines: {1, 2}, {57, 16593}, {226, 15490}, {277, 21096}, {312, 21436}, {344, 9436}, {728, 21258}, {1088, 4554}, {1280, 4929}, {3158, 26007}, {3689, 31203}, {3834, 24352}, {4936, 6604}, {5219, 30825}, {10025, 17298}, {14548, 17353}, {20173, 23681}, {30852, 30857}

X(30814) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30814) lies on these lines: {2, 3}, {312, 21407}, {1997, 30857}, {18743, 21583}

X(30815) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30815) lies on these lines: {2, 3}, {312, 21408}, {18743, 21584}, {26985, 30865}

X(30816) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30816) lies on these lines: {2, 31}, {41, 626}, {213, 7867}, {312, 20627}, {672, 7778}, {1106, 28773}, {1193, 7866}, {1253, 28806}, {1468, 30103}, {1959, 17284}, {2177, 26590}, {2251, 7818}, {3915, 30104}, {4153, 20267}, {4721, 7869}, {5230, 32951}, {7901, 17033}, {7931, 24514}, {9350, 26582}, {10448, 17550}, {14064, 21935}, {18743, 20641}, {21240, 31240}, {21241, 24596}, {26085, 31284}, {26561, 32577}, {30811, 30821}, {30817, 30820}, {30822, 30857}, {31031, 32941}

X(30817) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30817) lies on these lines: {2, 32}, {312, 21409}, {2175, 21235}, {2223, 30746}, {5019, 18744}, {5138, 25668}, {14827, 28804}, {14963, 30808}, {18743, 21585}, {24296, 31050}, {30809, 30860}, {30816, 30820}, {30830, 30846}, {30856, 30863}

X(30818) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30818) lies on these lines: {1, 24277}, {2, 37}, {8, 17721}, {10, 11}, {38, 3967}, {43, 3706}, {44, 1150}, {65, 3831}, {141, 908}, {190, 24627}, {210, 3741}, {226, 1122}, {257, 4997}, {354, 1215}, {518, 30942}, {519, 21870}, {726, 4003}, {740, 4519}, {756, 31241}, {899, 3696}, {936, 7532}, {940, 30567}, {960, 25591}, {984, 4009}, {1001, 29828}, {1086, 4054}, {1089, 19864}, {1104, 5192}, {1125, 17602}, {1155, 3923}, {1193, 3714}, {1211, 3452}, {1212, 27040}, {1279, 26227}, {1386, 17763}, {1698, 3987}, {1909, 18149}, {2177, 4702}, {2300, 4383}, {2887, 17605}, {2901, 20108}, {3216, 5295}, {3218, 17351}, {3230, 4384}, {3240, 28581}, {3305, 5737}, {3306, 4363}, {3661, 5233}, {3683, 4011}, {3685, 4689}, {3689, 32941}, {3702, 4646}, {3703, 24239}, {3720, 31264}, {3729, 17595}, {3740, 30959}, {3742, 30957}, {3744, 7081}, {3745, 25496}, {3748, 29670}, {3823, 33108}, {3834, 31019}, {3836, 25385}, {3838, 25957}, {3844, 5087}, {3911, 17355}, {3912, 5718}, {3932, 29639}, {3934, 17760}, {3936, 17231}, {3944, 33174}, {3971, 6682}, {3977, 17340}, {3999, 24349}, {4387, 17594}, {4392, 28582}, {4396, 4670}, {4640, 32918}, {4641, 14829}, {4643, 31018}, {4663, 32919}, {4682, 32772}, {4716, 17779}, {4849, 17135}, {4852, 17012}, {4871, 24325}, {4885, 14475}, {4891, 17018}, {4906, 32923}, {4968, 26094}, {5044, 10479}, {5057, 33086}, {5205, 5263}, {5219, 7146}, {5271, 16685}, {5278, 26688}, {5328, 29611}, {5782, 25934}, {5793, 19861}, {5835, 24982}, {6708, 30827}, {7232, 31164}, {7308, 18229}, {9599, 17275}, {10129, 25959}, {11680, 29679}, {15254, 32917}, {16669, 16704}, {16736, 30599}, {16823, 25531}, {17021, 28639}, {17022, 19701}, {17063, 31242}, {17165, 21342}, {17229, 33077}, {17230, 25102}, {17235, 33151}, {17237, 26580}, {17272, 31142}, {17345, 17484}, {17374, 31034}, {17448, 21893}, {17449, 31161}, {17717, 29674}, {17719, 29637}, {17722, 32847}, {17725, 29660}, {17777, 24723}, {21611, 27139}, {21805, 31136}, {23536, 25914}, {24217, 29659}, {24656, 29569}, {24703, 26034}, {25107, 29591}, {25125, 29593}, {25130, 29595}, {26061, 29662}, {27105, 27285}, {27131, 32782}, {27776, 31171}, {29579, 30828}, {29604, 30819}, {29668, 32920}, {29676, 33165}, {29677, 33127}, {29680, 32862}, {29687, 33105}, {30710, 32017}, {30825, 30840}, {31053, 33172}, {33078, 33107}, {33079, 33106}, {33085, 33096}, {33140, 33159}

X(30818) = complement of X(4850)
X(30818) = {X(2),X(75)}-harmonic conjugate of X(16610)

X(30819) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30819) lies on these lines: {2, 39}, {141, 8287}, {183, 11353}, {187, 11320}, {226, 17114}, {312, 21412}, {313, 17053}, {321, 1574}, {404, 24271}, {626, 26019}, {1015, 3765}, {1211, 4187}, {1575, 4044}, {1909, 25510}, {2092, 18147}, {3159, 3634}, {3596, 26107}, {3661, 27076}, {3662, 30019}, {3688, 21238}, {3734, 11329}, {3780, 29769}, {3782, 24170}, {3975, 26959}, {4260, 25688}, {4263, 26772}, {4377, 8610}, {4383, 29455}, {4384, 17475}, {6248, 19522}, {7815, 16367}, {7816, 19308}, {12263, 20340}, {14873, 20337}, {14963, 30808}, {16609, 25079}, {17023, 20530}, {17143, 26048}, {18050, 18743}, {18133, 26979}, {19786, 27324}, {21257, 21746}, {21264, 24603}, {25102, 29574}, {29604, 30818}, {30810, 30860}, {30846, 30853}

X(30820) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30820) lies on these lines: {2, 41}, {312, 21414}, {604, 28738}, {1253, 17047}, {2251, 31196}, {2340, 30755}, {4466, 17279}, {5219, 30839}, {17284, 30858}, {18743, 21589}, {21069, 24773}, {21808, 31266}, {30811, 30826}, {30816, 30817}, {30821, 30840}

X(30821) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30821) lies on these lines: {1, 2}, {312, 21415}, {350, 17283}, {672, 17279}, {673, 32943}, {902, 24586}, {1042, 28777}, {1334, 21240}, {2238, 17231}, {2276, 17267}, {3834, 24330}, {3930, 30748}, {4422, 24690}, {4441, 17282}, {17232, 24514}, {17263, 30966}, {17268, 17759}, {17353, 30941}, {17357, 24512}, {18138, 18743}, {20347, 21255}, {20533, 32948}, {21070, 24790}, {21071, 26978}, {21241, 31031}, {23649, 27109}, {24215, 26770}, {24596, 32941}, {27475, 32771}, {30811, 30816}, {30820, 30840}, {30831, 30857}, {30848, 30851}

X(30822) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30822) lies on these lines: {1, 2}, {9, 24690}, {312, 21416}, {350, 17282}, {1215, 27475}, {1575, 17267}, {1743, 30941}, {2238, 17296}, {3208, 20255}, {3305, 30965}, {3550, 24602}, {3834, 4713}, {4422, 24691}, {4441, 4859}, {4479, 27191}, {6173, 24330}, {8616, 24586}, {17265, 21264}, {17279, 17754}, {17283, 30963}, {17298, 24514}, {17311, 21904}, {17353, 30962}, {18157, 18743}, {21255, 30946}, {24215, 27523}, {24596, 32943}, {30816, 30857}, {30825, 30837}

X(30823) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30823) lies on these lines: {1, 27759}, {2, 44}, {37, 30834}, {239, 25529}, {306, 21689}, {312, 21417}, {536, 27757}, {30835, 30865}, {899, 31280}, {908, 4422}, {1155, 4892}, {3661, 27747}, {3689, 21241}, {3771, 17605}, {3838, 29846}, {3912, 5461}, {3936, 17374}, {4009, 29862}, {4053, 31266}, {4138, 5432}, {4358, 16732}, {4379, 4885}, {4384, 27739}, {4440, 32851}, {4864, 20042}, {4997, 17266}, {5087, 24709}, {5219, 7146}, {5226, 32777}, {5233, 29628}, {5241, 31211}, {5718, 17023}, {16610, 27191}, {16666, 31179}, {16669, 31229}, {17316, 17720}, {17717, 29660}, {17719, 32847}, {18743, 21591}, {24589, 27141}, {29610, 30832}, {29626, 30867}

X(30824) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30824) lies on these lines: {1, 27777}, {2, 45}, {312, 21418}, {908, 4643}, {1001, 24709}, {1211, 5748}, {3452, 19732}, {3661, 27739}, {3711, 21242}, {4384, 27747}, {4413, 24693}, {5087, 29828}, {5219, 7146}, {5241, 5328}, {5316, 31211}, {5718, 17316}, {5737, 27131}, {11238, 29670}, {17023, 17720}, {17267, 30834}, {17269, 27757}, {17271, 31056}, {17313, 26738}, {17717, 32847}, {17719, 29660}, {17775, 18141}, {17783, 32942}, {18743, 21592}, {22014, 30827}, {29610, 30867}, {29626, 30829}, {30861, 31026}, {31035, 31281}

X(30825) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30825) lies on these lines: {2, 11}, {10, 17675}, {12, 28740}, {45, 25353}, {56, 28734}, {220, 17046}, {312, 20890}, {388, 28756}, {902, 31195}, {958, 17671}, {1334, 31240}, {1388, 28969}, {1997, 17084}, {2098, 26526}, {3008, 24386}, {3061, 17284}, {3550, 31210}, {3693, 30742}, {3711, 26593}, {4258, 31284}, {5219, 30813}, {7004, 25944}, {7778, 17279}, {7866, 30110}, {10944, 28967}, {13881, 29571}, {17044, 24247}, {17265, 20530}, {18743, 20922}, {20269, 21073}, {25914, 32956}, {30809, 30847}, {30811, 30816}, {30818, 30840}, {30822, 30837}

X(30826) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30826) lies on these lines: {2, 12}, {6, 21244}, {9, 4708}, {11, 28795}, {55, 28789}, {241, 30757}, {312, 21420}, {497, 28812}, {960, 17308}, {1376, 7377}, {2099, 26575}, {3061, 17284}, {3452, 29604}, {3661, 5289}, {3763, 21246}, {3912, 25681}, {3913, 27526}, {4193, 28813}, {4384, 5123}, {5328, 28827}, {10950, 28922}, {16593, 30809}, {18743, 21594}, {20270, 21074}, {20317, 30836}, {30811, 30820}, {30833, 30857}

X(30827) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30827) lies on these lines: {1, 1329}, {2, 7}, {3, 22792}, {4, 5438}, {5, 936}, {8, 18220}, {10, 3090}, {11, 200}, {12, 8583}, {40, 1519}, {65, 31246}, {78, 4193}, {84, 6891}, {100, 9580}, {109, 25938}, {123, 1040}, {124, 25957}, {140, 31424}, {165, 3035}, {198, 19517}, {210, 5231}, {214, 10711}, {220, 31183}, {281, 20106}, {312, 646}, {346, 6557}, {354, 31249}, {392, 31434}, {404, 9579}, {474, 9612}, {496, 6765}, {497, 3158}, {498, 31435}, {518, 17626}, {529, 13462}, {590, 31438}, {631, 12572}, {946, 1706}, {950, 6919}, {958, 3624}, {960, 1698}, {997, 3814}, {1001, 8169}, {1056, 1125}, {1086, 8056}, {1210, 11523}, {1213, 15479}, {1319, 31141}, {1376, 1699}, {1420, 3436}, {1490, 6922}, {1532, 6282}, {1538, 6244}, {1656, 5044}, {1697, 5552}, {1997, 3912}, {2082, 28813}, {2098, 15347}, {2136, 7080}, {2195, 28050}, {2323, 4383}, {2324, 3772}, {2325, 8055}, {2329, 29598}, {2478, 3601}, {2550, 3817}, {2884, 16593}, {2886, 7988}, {2999, 17720}, {3036, 4677}, {3061, 17284}, {3086, 6762}, {3119, 30857}, {3220, 16434}, {3243, 11019}, {3333, 10200}, {3340, 24982}, {3361, 6691}, {3419, 17533}, {3485, 8582}, {3487, 9843}, {3526, 31445}, {3576, 6947}, {3586, 5440}, {3616, 5795}, {3628, 5791}, {3646, 10198}, {3679, 5123}, {3680, 6736}, {3681, 31272}, {3687, 4007}, {3689, 11238}, {3705, 4901}, {3742, 10569}, {3771, 11814}, {3811, 3825}, {3813, 4882}, {3815, 16517}, {3820, 5886}, {3824, 16863}, {4034, 4886}, {4069, 20487}, {4292, 17567}, {4413, 17605}, {4417, 17296}, {4422, 28657}, {4511, 5727}, {4512, 4679}, {4521, 6546}, {4652, 17566}, {4847, 10589}, {4853, 11376}, {4855, 5046}, {4859, 16602}, {4862, 25580}, {4999, 5234}, {5070, 31446}, {5084, 5436}, {5121, 5573}, {5128, 11415}, {5223, 6667}, {5250, 27529}, {5268, 17717}, {5272, 17719}, {5274, 5853}, {5290, 13370}, {5393, 19066}, {5405, 19065}, {5574, 13609}, {5687, 9614}, {5692, 31263}, {5698, 10164}, {5704, 24391}, {5709, 6959}, {5714, 12436}, {5715, 6918}, {5718, 17022}, {5720, 6882}, {5730, 17619}, {5743, 18229}, {5784, 10157}, {5794, 7989}, {5811, 6705}, {5836, 11522}, {5837, 9780}, {5855, 30286}, {5881, 30144}, {6260, 6926}, {6708, 30818}, {6734, 6931}, {6735, 7962}, {6769, 7681}, {6853, 12514}, {6958, 7330}, {6963, 18446}, {7174, 24239}, {7288, 12527}, {7322, 29639}, {8256, 11531}, {9342, 10129}, {9578, 11681}, {9613, 17614}, {9708, 11230}, {9709, 9955}, {9785, 27525}, {10382, 14022}, {10388, 15845}, {10582, 17718}, {10584, 26015}, {10585, 24564}, {11113, 30282}, {11373, 12629}, {11374, 17527}, {11682, 25005}, {11813, 31162}, {12526, 24914}, {13257, 30304}, {13405, 26105}, {13747, 15803}, {13893, 30556}, {13947, 30557}, {15733, 17604}, {16552, 19720}, {16569, 17064}, {16578, 17080}, {16594, 16596}, {16610, 23681}, {17051, 30350}, {17182, 18163}, {17647, 18492}, {17786, 32017}, {18193, 33101}, {18250, 19862}, {18743, 20928}, {21060, 24477}, {21267, 22942}, {21609, 30796}, {22014, 30824}, {25645, 25647}, {25661, 25682}, {25960, 29828}, {28827, 29596}, {30568, 32851}, {30842, 30851}, {31401, 31429}, {31442, 31455}

X(30828) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30828) lies on these lines: {2, 6}, {4, 25650}, {7, 32851}, {8, 17718}, {145, 17724}, {226, 345}, {306, 21675}, {312, 1441}, {320, 5744}, {329, 17336}, {344, 908}, {497, 29839}, {518, 30741}, {914, 1997}, {1043, 5177}, {1330, 6857}, {1434, 32831}, {1465, 28936}, {1999, 4916}, {3454, 13725}, {3475, 3705}, {3616, 17723}, {3622, 17726}, {3666, 26132}, {3687, 25525}, {3712, 24280}, {3771, 26098}, {3772, 4852}, {3911, 17298}, {3912, 5219}, {3977, 31164}, {4028, 17064}, {4035, 11679}, {4138, 17594}, {4195, 25663}, {4358, 20927}, {4514, 10578}, {4645, 5218}, {4684, 5231}, {4892, 24248}, {4997, 10405}, {5273, 33066}, {5328, 29627}, {5552, 26031}, {5703, 7270}, {5714, 7283}, {5748, 18743}, {5846, 26245}, {5905, 33113}, {6856, 10449}, {10585, 17751}, {17244, 30867}, {17316, 17720}, {17740, 27757}, {17776, 31053}, {24217, 27759}, {24296, 32986}, {25101, 31142}, {25568, 29641}, {26034, 29678}, {26228, 33070}, {26738, 32779}, {28740, 28794}, {28757, 28836}, {29579, 30818}, {29671, 33144}, {29866, 33107}, {30809, 30830}, {30844, 30856}, {31280, 33128}, {33088, 33127}, {33105, 33171}

X(30828) = anticomplement of X(31187)

X(30829) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30829) lies on these lines: {1, 1120}, {2, 37}, {8, 10179}, {10, 24217}, {45, 24627}, {57, 25728}, {81, 26688}, {85, 4554}, {100, 9095}, {190, 3306}, {304, 29596}, {320, 31018}, {333, 7308}, {341, 3616}, {354, 26103}, {518, 30947}, {551, 4737}, {693, 31992}, {748, 3769}, {750, 4676}, {756, 30957}, {908, 17234}, {940, 17120}, {975, 13741}, {984, 4871}, {1001, 5205}, {1125, 17725}, {1150, 17335}, {1215, 25502}, {1233, 18153}, {2006, 32019}, {2177, 9458}, {3218, 17336}, {3241, 21870}, {3247, 30693}, {3305, 14829}, {3452, 18134}, {3624, 4385}, {3661, 5241}, {3673, 25529}, {3681, 17145}, {3685, 4413}, {3701, 5550}, {3702, 19877}, {3706, 26038}, {3740, 10453}, {3742, 32937}, {3757, 8167}, {3812, 19582}, {3816, 29641}, {3828, 4793}, {3831, 31359}, {3842, 29827}, {3848, 3967}, {3885, 22313}, {3911, 25101}, {3912, 5233}, {3936, 17241}, {3948, 29581}, {3952, 17146}, {3971, 17063}, {3975, 5308}, {3992, 25055}, {3996, 8580}, {3999, 31302}, {4009, 24349}, {4011, 17122}, {4031, 4480}, {4078, 5121}, {4383, 17121}, {4423, 7081}, {4434, 15485}, {4514, 26105}, {4645, 4679}, {4647, 19872}, {4673, 9780}, {4714, 19876}, {4886, 34255}, {4891, 20012}, {4975, 19875}, {5084, 7270}, {5235, 30939}, {5268, 32942}, {5272, 32926}, {5300, 26127}, {5328, 29627}, {5437, 30568}, {5718, 6376}, {5748, 18750}, {5880, 17777}, {6381, 20917}, {6682, 31242}, {6686, 17592}, {7283, 16408}, {9342, 32929}, {11814, 17717}, {14206, 20921}, {14210, 17284}, {14555, 32099}, {16708, 31008}, {16729, 26860}, {16815, 17144}, {16817, 16853}, {16820, 24277}, {17012, 17393}, {17021, 17394}, {17074, 28996}, {17123, 29649}, {17124, 32930}, {17125, 17763}, {17227, 26580}, {17240, 33077}, {17261, 17595}, {17266, 30811}, {17298, 31142}, {17387, 31034}, {17721, 26139}, {17762, 29608}, {18139, 27131}, {18141, 18228}, {18146, 20569}, {18150, 26738}, {18156, 29579}, {19546, 30273}, {20930, 30834}, {21361, 29417}, {21609, 31627}, {21896, 26046}, {24150, 30712}, {24524, 29569}, {24595, 29490}, {24709, 33104}, {24988, 33134}, {25280, 29583}, {25430, 32017}, {25918, 27269}, {25960, 29687}, {27475, 31002}, {28612, 31253}, {29578, 31997}, {29626, 30824}, {29658, 31289}, {30566, 31019}, {30950, 32931}

X(30829) = complement of X(24620)
X(30829) = anticomplement of X(31197)

X(30830) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30830) lies on these lines: {1, 3975}, {2, 39}, {6, 645}, {10, 312}, {37, 3596}, {45, 17790}, {75, 4044}, {99, 11329}, {192, 21796}, {226, 29968}, {264, 281}, {313, 4687}, {314, 966}, {321, 21816}, {350, 4384}, {406, 31623}, {668, 17316}, {940, 17499}, {948, 4554}, {1078, 16367}, {1269, 4751}, {1909, 16831}, {1975, 16412}, {2275, 25510}, {3009, 7976}, {3061, 3452}, {3264, 4664}, {3501, 30568}, {3661, 4358}, {3662, 29982}, {3663, 30090}, {3687, 21071}, {3731, 17787}, {3760, 16832}, {3765, 16826}, {3770, 15668}, {3782, 24190}, {3797, 10009}, {3950, 4110}, {3963, 27268}, {3971, 12782}, {3972, 11320}, {3992, 29659}, {4261, 27111}, {4357, 20923}, {4377, 4755}, {4383, 17034}, {4415, 20255}, {4494, 16676}, {5084, 10449}, {5224, 18137}, {5337, 16997}, {5739, 33297}, {5743, 21024}, {6381, 20917}, {7017, 17916}, {7752, 26019}, {7782, 19308}, {9535, 15488}, {10472, 26045}, {11257, 19522}, {11342, 26282}, {16552, 29456}, {16915, 24271}, {17023, 30963}, {17065, 21080}, {17234, 18133}, {17243, 30473}, {17245, 18144}, {17247, 20892}, {17248, 20891}, {17263, 18044}, {17277, 18147}, {17294, 25280}, {17346, 30939}, {17352, 18046}, {17389, 25298}, {17671, 18134}, {17750, 27064}, {17758, 18136}, {19786, 30107}, {19804, 20888}, {20154, 30940}, {20544, 29641}, {20683, 27538}, {20691, 25125}, {21226, 26113}, {21240, 27184}, {24296, 24610}, {24524, 29574}, {25303, 29597}, {25683, 31120}, {29376, 29400}, {29627, 30866}, {29966, 30961}, {30808, 30856}, {30809, 30828}, {30811, 30853}, {30817, 30846}

X(30830) = complement of X(24621)
X(30830) = anticomplement of X(31198)

X(30831) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30831) lies on these lines: {2, 6}, {10, 5425}, {11, 33173}, {21, 3454}, {43, 31237}, {55, 25958}, {58, 25669}, {100, 2887}, {210, 29873}, {226, 32779}, {238, 29865}, {306, 33133}, {312, 20896}, {321, 646}, {345, 33151}, {518, 29872}, {748, 29858}, {756, 29862}, {908, 20106}, {1001, 29866}, {1330, 16948}, {1376, 25959}, {1386, 29874}, {1621, 3771}, {1698, 2650}, {1959, 17284}, {2475, 24946}, {2886, 33175}, {3006, 33126}, {3011, 33075}, {3120, 33160}, {3315, 33124}, {3416, 29665}, {3550, 31134}, {3666, 27757}, {3681, 29857}, {3687, 33129}, {3703, 33153}, {3705, 33122}, {3712, 33100}, {3772, 33077}, {3782, 33168}, {3794, 3909}, {3836, 9342}, {3840, 31272}, {3846, 5284}, {3944, 33156}, {3966, 29681}, {4046, 17070}, {4062, 33135}, {4138, 20292}, {4205, 24936}, {4415, 32849}, {4418, 4892}, {4423, 29870}, {4438, 33065}, {4649, 29863}, {4756, 33164}, {4850, 25527}, {4865, 29848}, {5051, 25650}, {5328, 30841}, {5432, 33086}, {6675, 26064}, {6679, 32843}, {6690, 33083}, {7232, 23958}, {10707, 32943}, {11680, 33171}, {14005, 24931}, {15523, 17719}, {16581, 17279}, {17061, 32842}, {17123, 29869}, {17184, 32851}, {17369, 17775}, {17602, 33093}, {17717, 24943}, {17718, 29667}, {17720, 32858}, {17722, 29686}, {17723, 29648}, {17724, 33090}, {17725, 32854}, {17740, 26132}, {18228, 31256}, {18743, 20929}, {20653, 24161}, {21020, 31280}, {21241, 32945}, {24892, 33084}, {25663, 26117}, {25960, 29642}, {26128, 29849}, {26230, 33071}, {26580, 33116}, {27096, 27119}, {27130, 27132}, {27184, 33113}, {29634, 33070}, {29656, 32844}, {29658, 32852}, {29662, 33087}, {29671, 32775}, {29678, 32784}, {29683, 32846}, {30821, 30857}, {31053, 32777}, {32778, 33127}, {32783, 33105}, {32848, 33152}, {32855, 33143}, {32856, 33167}, {33064, 33119}, {33081, 33140}, {33089, 33144}, {33101, 33161}

X(30832) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30832) lies on these lines: {2, 6}, {8, 17602}, {10, 3699}, {37, 27490}, {63, 17329}, {190, 26580}, {226, 19808}, {257, 4997}, {306, 17315}, {312, 17286}, {908, 11683}, {1010, 3454}, {1043, 5051}, {1762, 3305}, {1999, 17372}, {3218, 17273}, {3306, 17227}, {3661, 17720}, {3666, 17324}, {3687, 3946}, {3712, 9791}, {3772, 28634}, {3775, 33140}, {3807, 31025}, {3842, 29862}, {3844, 5205}, {3846, 32783}, {3966, 29634}, {3977, 17258}, {3996, 32773}, {4104, 33118}, {4205, 25650}, {4357, 32851}, {4358, 17285}, {4360, 33077}, {4389, 17740}, {4425, 33160}, {4733, 17070}, {4751, 27489}, {4850, 17305}, {4859, 19804}, {4892, 24342}, {4974, 29859}, {4981, 29872}, {5219, 16609}, {5256, 19812}, {5263, 25760}, {5316, 29596}, {5328, 28827}, {9780, 21677}, {10159, 14554}, {11110, 25645}, {14210, 17284}, {16610, 17291}, {17023, 19832}, {17160, 33155}, {17236, 17595}, {17237, 24627}, {17276, 27184}, {17290, 24620}, {17339, 32777}, {17354, 31018}, {18743, 20932}, {20106, 25072}, {21085, 33135}, {24589, 27191}, {24943, 25960}, {25531, 29637}, {26037, 31237}, {27025, 27119}, {27793, 30599}, {28808, 29611}, {29604, 30837}, {29610, 30823}, {29635, 33084}, {29645, 32861}, {29845, 33081}, {29847, 32852}, {29863, 32864}, {30843, 30854}, {32775, 32922}, {32778, 32926}

X(30833) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30833) lies on these lines: {1, 2}, {144, 4473}, {312, 21432}, {344, 17227}, {346, 1086}, {391, 17231}, {3161, 21255}, {3672, 17283}, {3834, 4454}, {3945, 17241}, {4000, 32105}, {4310, 4439}, {4346, 17264}, {4452, 17282}, {4461, 17268}, {4644, 4869}, {4665, 17265}, {4747, 17313}, {5232, 17263}, {5749, 30712}, {17339, 20059}, {18743, 21605}, {26997, 27544}, {28809, 30866}, {30826, 30857}

X(30833) = antiomplement of X(31188)

X(30834) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30834) lies on these lines: {2, 6}, {8, 5719}, {11, 29830}, {37, 30823}, {55, 21282}, {75, 27757}, {142, 24594}, {226, 3977}, {321, 31266}, {344, 30566}, {894, 26738}, {908, 25101}, {964, 25645}, {1441, 4358}, {2177, 21241}, {2476, 25650}, {2886, 21283}, {2887, 29678}, {3006, 17718}, {3011, 33070}, {3454, 16342}, {3685, 10129}, {3771, 24552}, {3838, 32929}, {3846, 29661}, {3891, 29671}, {3896, 17064}, {4080, 17262}, {4138, 32950}, {4256, 17679}, {4262, 24638}, {4359, 25525}, {4360, 25529}, {4414, 4892}, {5014, 13405}, {5226, 17776}, {5564, 33077}, {5745, 32859}, {5748, 6350}, {6327, 6690}, {6685, 31237}, {7232, 31029}, {7321, 31019}, {8229, 31670}, {9534, 31254}, {10436, 30588}, {11680, 29839}, {12635, 27690}, {14212, 30852}, {16484, 27759}, {17118, 31030}, {17135, 31245}, {17229, 27747}, {17261, 27754}, {17267, 30824}, {17273, 30991}, {17298, 24593}, {17361, 30608}, {17717, 29632}, {17719, 29643}, {17722, 29638}, {17723, 26230}, {17724, 29832}, {17726, 29831}, {17740, 31995}, {18743, 20919}, {20930, 30829}, {25385, 33156}, {25496, 29865}, {25760, 29640}, {26070, 31300}, {26128, 29688}, {29627, 30808}, {29639, 33122}, {29657, 32775}, {29664, 33126}, {29665, 33073}, {29675, 32844}, {29680, 33124}, {29681, 33071}, {29846, 33111}, {29849, 33130}, {29858, 32944}, {29862, 32931}, {29866, 32942}, {30947, 31272}, {31053, 33116}

X(30835) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30835) lies on these lines: {2, 649}, {37, 27485}, {42, 24749}, {306, 21720}, {312, 20909}, {512, 31251}, {513, 24924}, {514, 17266}, {650, 4728}, {657, 28834}, {661, 30823}, {663, 15283}, {788, 31241}, {812, 31209}, {1459, 31946}, {1635, 31287}, {1638, 14321}, {1698, 29350}, {1699, 15599}, {1960, 31149}, {30836, 30862}, {3676, 5219}, {3720, 23655}, {3763, 9002}, {3837, 4724}, {4107, 31040}, {4369, 4776}, {4380, 4763}, {4423, 23865}, {4449, 21051}, {4468, 6545}, {4474, 14431}, {4521, 6546}, {4750, 7658}, {4885, 30868}, {4893, 4928}, {5333, 18200}, {6006, 20195}, {6544, 11068}, {8227, 28292}, {8643, 21301}, {8655, 25537}, {9313, 31237}, {14426, 20983}, {14433, 17284}, {14437, 17290}, {14825, 17675}, {17155, 21197}, {17265, 21143}, {18743, 20952}, {20965, 23575}, {20966, 22445}, {21297, 27115}, {24666, 30950}, {25128, 30942}, {25143, 26037}, {25627, 31330}, {25637, 30970}, {27139, 27527}, {27193, 28758}, {28846, 31266}

X(30836) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30836) lies on these lines: {2, 667}, {312, 21440}, {514, 30865}, {665, 30764}, {3063, 21262}, {3250, 30835}, {4083, 17308}, {30837, 30849}, {18743, 21613}, {20317, 30826}

X(30837) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30837) lies on these lines: {1, 7866}, {2, 31}, {312, 20629}, {673, 21241}, {894, 7931}, {3684, 20541}, {3750, 26590}, {3912, 17719}, {5219, 7146}, {5255, 30104}, {7778, 17754}, {7867, 17750}, {14349, 30836}, {17023, 17722}, {17316, 17725}, {17755, 30763}, {18743, 20643}, {29571, 30853}, {29604, 30832}, {30822, 30825}, {31031, 32943}, {31093, 32927}

X(30838) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30838) lies on these lines: {2, 3}, {3160, 29571}, {5199, 5328}, {29627, 30852}

X(30839) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30839) lies on these lines: {2, 3}, {12, 28757}, {644, 16603}, {1014, 28755}, {1441, 6335}, {1778, 24890}, {1959, 17284}, {1997, 17084}, {2287, 17052}, {5219, 30820}, {5328, 28827}, {29571, 30848}, {29604, 30849}

X(30840) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30840) lies on these lines: {2, 3}, {5089, 30779}, {8192, 28404}, {11396, 26153}, {30818, 30825}, {30820, 30821}

X(30841) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30841) lies on these lines: {2, 3}, {306, 3699}, {653, 18588}, {5328, 30831}, {14206, 20921}

X(30842) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30842) lies on these lines: {2, 3}, {646, 20336}, {2303, 25651}, {17284, 18669}, {30827, 30851}

X(30843) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30843) lies on these lines: {2, 3}, {190, 307}, {2322, 18642}, {30832, 30854}

X(30844) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics 5 a^5 - 7 a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - a b^4 + 11 b^5 - 7 a^4 c + 8 a^2 b^2 c - b^4 c - 4 a^3 c^2 + 8 a^2 b c^2 + 2 a b^2 c^2 - 10 b^3 c^2 - 4 a^2 c^3 - 10 b^2 c^3 - a c^4 - b c^4 + 11 c^5 : :

X(30844) lies on these lines: {2, 3}, {30828, 30856}

X(30845) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30845) lies on these lines: {2, 3}, {966, 8287}, {2551, 28757}, {5328, 29627}, {29571, 30852}

X(30945) = complement of X(37657)

X(30846) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30846) lies on these lines: {2, 3}, {30811, 30863}, {30817, 30830}, {30819, 30853}

X(30847) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30847) lies on these lines: {2, 12}, {9, 141}, {11, 28797}, {21, 28813}, {55, 28795}, {312, 21405}, {594, 21233}, {960, 3912}, {1212, 3452}, {1213, 21244}, {2886, 7377}, {3008, 5795}, {3687, 4662}, {5123, 24603}, {5218, 28812}, {5289, 17316}, {5302, 29596}, {5326, 28816}, {5432, 28789}, {5737, 26036}, {5745, 6292}, {5837, 29594}, {7402, 19843}, {10944, 26621}, {10950, 28916}, {15479, 17296}, {15829, 29573}, {16435, 22654}, {16831, 25681}, {17245, 21246}, {17292, 18253}, {17308, 26066}, {18134, 18228}, {18229, 19542}, {18743, 21581}, {20917, 30854}, {24633, 26575}, {30809, 30825}

X(30848) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30848) lies on these lines: {2, 35}, {312, 21410}, {3061, 17284}, {16832, 17675}, {18743, 21586}, {29571, 30839}, {30821, 30851}

X(30849) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30849) lies on these lines: {2, 36}, {312, 21411}, {2323, 21237}, {3061, 17284}, {7308, 21372}, {14349, 30836}, {18743, 21587}, {29604, 30839}

X(30850) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30850) lies on these lines: {2, 38}, {312, 20889}, {561, 18743}, {1959, 17284}, {3934, 33299}, {3954, 31239}, {4713, 14439}, {5282, 15271}, {17026, 21805}, {25079, 28742}

X(30851) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30851) lies on these lines: {2, 58}, {284, 21245}, {312, 21421}, {857, 25645}, {1959, 17284}, {2328, 28818}, {4417, 7828}, {4653, 26601}, {7899, 14829}, {14963, 30808}, {16832, 31254}, {17669, 25665}, {18206, 30760}, {18743, 21595}, {21287, 33628}, {25658, 30906}, {29604, 30832}, {30821, 30848}, {30827, 30842}

X(30852) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30852) lies on these lines: {1, 3814}, {2, 7}, {4, 4855}, {5, 78}, {8, 5056}, {10, 6933}, {11, 3870}, {12, 19861}, {40, 5180}, {55, 5087}, {72, 1656}, {84, 6972}, {92, 4997}, {100, 1699}, {140, 4652}, {149, 3158}, {165, 5057}, {200, 7988}, {224, 6831}, {228, 19540}, {306, 8797}, {312, 14213}, {354, 17615}, {377, 6700}, {381, 5440}, {392, 31479}, {404, 9612}, {498, 5250}, {499, 21077}, {519, 23708}, {612, 17717}, {614, 17719}, {899, 17064}, {914, 1997}, {936, 2476}, {946, 5552}, {950, 5187}, {956, 11230}, {997, 7951}, {1022, 4462}, {1125, 3436}, {1145, 3656}, {1210, 6931}, {1259, 6918}, {1279, 17783}, {1319, 11236}, {1329, 11375}, {1376, 17605}, {1420, 20060}, {1490, 6943}, {1698, 3754}, {1709, 21635}, {1836, 3035}, {1959, 17284}, {2099, 5123}, {2270, 27524}, {2475, 5438}, {2478, 13411}, {2551, 24541}, {2975, 3624}, {2999, 33133}, {3090, 3984}, {3091, 27383}, {3174, 7678}, {3177, 29626}, {3340, 25005}, {3434, 3817}, {3485, 24982}, {3526, 3916}, {3576, 5080}, {3601, 5046}, {3614, 5794}, {3616, 25522}, {3628, 3951}, {3677, 33153}, {3681, 5231}, {3689, 11235}, {3697, 31493}, {3740, 31245}, {3751, 29662}, {3811, 7741}, {3812, 31246}, {3816, 4666}, {3824, 16862}, {3829, 4863}, {3838, 4413}, {3845, 9945}, {3846, 29828}, {3871, 9614}, {3872, 5886}, {3873, 18240}, {3876, 5705}, {3877, 31434}, {3895, 30384}, {3927, 5070}, {3935, 24392}, {3936, 30567}, {3940, 5055}, {4187, 11374}, {4188, 9579}, {4292, 6921}, {4312, 9352}, {4468, 6545}, {4511, 5587}, {4675, 17775}, {4679, 6690}, {4847, 10171}, {4861, 9624}, {5068, 5175}, {5086, 7989}, {5119, 11813}, {5154, 9581}, {5233, 5271}, {5253, 5290}, {5256, 17720}, {5268, 33105}, {5269, 33107}, {5272, 33127}, {5287, 5718}, {5314, 19544}, {5432, 24703}, {5573, 33148}, {5603, 6735}, {5687, 9955}, {5703, 6919}, {5709, 6949}, {5714, 17567}, {5715, 6915}, {5720, 6830}, {5722, 17533}, {5730, 9956}, {5741, 11679}, {5761, 6981}, {5902, 31263}, {6260, 6890}, {6282, 6932}, {6667, 17728}, {6684, 11415}, {6691, 10404}, {6882, 18446}, {6910, 12572}, {6922, 10884}, {6952, 7330}, {6963, 18443}, {7174, 29680}, {7231, 17292}, {7290, 29665}, {7293, 16434}, {7322, 29664}, {8580, 33108}, {9654, 17614}, {9779, 17784}, {10200, 13407}, {10527, 21075}, {10528, 12053}, {10584, 11019}, {10588, 24987}, {10589, 25568}, {10826, 22836}, {10827, 30144}, {10914, 18493}, {11114, 30282}, {11246, 31235}, {11376, 12607}, {11522, 14923}, {11683, 17371}, {11684, 19872}, {11814, 29642}, {12047, 26364}, {12527, 19862}, {12635, 17606}, {14206, 20921}, {14212, 30834}, {15015, 18513}, {15022, 20007}, {15803, 17566}, {16475, 29683}, {17174, 18163}, {17263, 31261}, {17266, 30863}, {17286, 27141}, {17341, 17785}, {17556, 24929}, {18193, 32856}, {18662, 31035}, {19549, 22345}, {19804, 20879}, {20223, 33116}, {23511, 33129}, {24612, 29603}, {24954, 25466}, {25011, 28629}, {25516, 27412}, {29571, 30845}, {29627, 30838}, {30566, 30568}, {30813, 30857}

X(30853) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30853) lies on these lines: {2, 32}, {141, 645}, {312, 21425}, {3452, 29596}, {17266, 18140}, {18743, 20933}, {29571, 30837}, {30811, 30830}, {30819, 30846}

X(30854) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30854) lies on these lines: {2, 85}, {6, 27420}, {8, 210}, {9, 75}, {55, 28058}, {57, 30625}, {63, 32024}, {69, 20946}, {86, 27384}, {92, 3305}, {144, 20905}, {169, 6996}, {219, 3759}, {220, 239}, {281, 17289}, {314, 15479}, {322, 344}, {329, 6604}, {390, 28057}, {391, 1229}, {664, 25930}, {728, 30568}, {908, 25935}, {958, 16048}, {1111, 31183}, {1146, 3661}, {1329, 20486}, {1441, 18230}, {1944, 3758}, {2324, 4360}, {2898, 31994}, {2975, 26265}, {3008, 3673}, {3061, 3452}, {3161, 20895}, {3219, 32100}, {3618, 27382}, {3739, 26059}, {4358, 29616}, {4385, 18250}, {4393, 6603}, {4657, 27547}, {4872, 5813}, {5125, 5342}, {5179, 7377}, {5199, 29604}, {5224, 20262}, {5228, 10025}, {5273, 19804}, {5328, 29627}, {5819, 7406}, {5834, 17747}, {5905, 32007}, {14942, 28043}, {15853, 27304}, {16367, 32561}, {16706, 27509}, {17227, 26932}, {17263, 20930}, {17294, 20942}, {17308, 23058}, {17321, 27508}, {17349, 20171}, {19786, 27540}, {19790, 26723}, {20917, 30847}, {26001, 33298}, {27065, 32088}, {27108, 27396}, {30832, 30843}

X(30855) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics a^3 - 3 a^2 b - 2 a b^2 + 2 b^3 - 3 a^2 c + 13 a b c - 4 b^2 c - 2 a c^2 - 4 b c^2 + 2 c^3 : :

X(30855) lies on these lines: {1, 10713}, {2, 45}, {100, 11814}, {121, 1320}, {312, 21427}, {646, 4358}, {908, 27834}, {1647, 4767}, {3699, 20042}, {4792, 25025}, {5235, 5316}, {5328, 26932}, {9458, 10707}, {17284, 30857}, {18743, 21600}, {24003, 31272}, {27130, 32911}

X(30856) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30856) lies on these lines: {2, 99}, {190, 31278}, {312, 17886}, {645, 8287}, {4049, 4997}, {15455, 18151}, {17308, 20538}, {18743, 20951}, {30808, 30830}, {30817, 30863}, {30819, 30846}, {30828, 30844}, {30857, 30866}

X(30857) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30857) lies on these lines: {2, 11}, {8, 17675}, {116, 644}, {312, 20901}, {902, 31210}, {908, 15634}, {1018, 31273}, {1023, 10708}, {1814, 5375}, {1997, 30814}, {2098, 26544}, {2975, 17671}, {3119, 30827}, {3436, 28756}, {4437, 4767}, {4997, 6548}, {8616, 31195}, {11681, 28740}, {11998, 26698}, {17284, 30855}, {18743, 20940}, {21272, 31640}, {29571, 30839}, {29627, 30808}, {30813, 30852}, {30816, 30822}, {30821, 30831}, {30826, 30833}, {30856, 30866}

X(30858) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30858) lies on these lines: {2, 101}, {312, 21429}, {1026, 30787}, {1055, 31222}, {1807, 30117}, {3939, 21252}, {4049, 4997}, {6326, 6702}, {17284, 30820}, {18743, 21602}, {21261, 28798}, {30116, 31479}

X(30859) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30859) lies on these lines: {2, 98}, {312, 21430}, {5546, 21253}, {18743, 21603}

X(30860) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30860) lies on these lines: {2, 99}, {120, 3140}, {125, 24250}, {190, 17058}, {312, 21431}, {523, 16597}, {645, 26147}, {3008, 23947}, {3814, 5520}, {3912, 10026}, {4092, 21254}, {4422, 8287}, {6537, 17292}, {8818, 17265}, {18743, 21604}, {21090, 31993}, {30809, 30817}, {30810, 30819}

X(30861) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30861) lies on these lines: {2, 37}, {8, 17460}, {145, 21870}, {306, 27130}, {518, 30948}, {908, 17232}, {982, 4903}, {1215, 26103}, {1743, 30567}, {3061, 3119}, {3617, 3893}, {3661, 5316}, {3701, 17480}, {3790, 5121}, {3840, 27538}, {3911, 17339}, {3971, 31242}, {4009, 31302}, {4051, 25615}, {4473, 5744}, {4709, 16569}, {4732, 26038}, {4734, 6686}, {4741, 31018}, {4871, 24349}, {4997, 30811}, {5219, 17266}, {5233, 16594}, {5241, 29593}, {5718, 29572}, {6557, 26132}, {10453, 21805}, {11814, 29674}, {14829, 16885}, {16816, 16969}, {17284, 30863}, {17288, 31142}, {17449, 30957}, {17450, 30947}, {17595, 25269}, {29627, 30869}, {30824, 31026}, {31017, 31056}

X(30861) = anticomplement of X(31233)

X(30862) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30862) lies on these lines: {2, 187}, {312, 21434}, {3250, 30835}, {14963, 30808}, {18743, 21607}

X(30863) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30863) lies on these lines: {2, 39}, {312, 21435}, {330, 25510}, {385, 11353}, {1211, 33046}, {3760, 17490}, {4384, 30998}, {7912, 26019}, {16412, 17128}, {17144, 25125}, {17266, 30852}, {17284, 30861}, {18743, 21608}, {24271, 33062}, {30811, 30846}, {30817, 30856}

X(30863) = anticomplement of X(31234)

X(30864) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30864) lies on these lines: {2, 647}, {312, 21437}, {512, 31003}, {1021, 18229}, {4885, 28894}, {3239, 23799}, {30835, 30836}, {3687, 21719}, {4524, 31330}, {5737, 9404}, {6332, 10015}, {17066, 24924}, {17069, 31993}, {18743, 21610}, {21348, 31250}

X(30865) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30865) lies on these lines: {2, 659}, {306, 21722}, {312, 21439}, {514, 30836}, {650, 30764}, {661, 4379}, {891, 17308}, {1960, 29603}, {3661, 21343}, {4435, 21261}, {4486, 4928}, {4997, 6548}, {17397, 25569}, {18743, 21612}, {26985, 30815}

X(30866) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30866) lies on these lines: {2, 668}, {76, 29579}, {150, 18141}, {190, 18150}, {274, 17292}, {312, 1111}, {314, 17231}, {334, 350}, {646, 1086}, {903, 24004}, {1022, 3762}, {1056, 21290}, {1909, 29596}, {3675, 4518}, {3761, 17284}, {3765, 29629}, {3834, 4506}, {4033, 27191}, {4110, 4859}, {4358, 4945}, {4738, 19875}, {4986, 19804}, {5308, 20345}, {7232, 29542}, {16704, 32012}, {17143, 17230}, {17241, 18044}, {17244, 18140}, {17265, 30473}, {17267, 18144}, {17273, 29396}, {17282, 17786}, {17283, 18040}, {17285, 18143}, {17295, 29484}, {17309, 29802}, {17787, 21255}, {18065, 20923}, {18159, 18743}, {20913, 29587}, {25298, 29607}, {28809, 30833}, {29446, 32025}, {29627, 30830}, {30856, 30857}

X(30867) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30867) lies on these lines: {2, 7}, {10, 26136}, {239, 5233}, {257, 4997}, {312, 21442}, {645, 5235}, {1999, 5741}, {2177, 25378}, {3035, 24723}, {3661, 28808}, {3757, 25960}, {3816, 33126}, {3846, 7081}, {3936, 17312}, {4201, 6700}, {4358, 17268}, {4417, 4851}, {4494, 4671}, {4554, 26563}, {5087, 5263}, {5205, 25760}, {5241, 16815}, {5718, 16826}, {11814, 29637}, {14829, 17344}, {16823, 17719}, {16830, 17717}, {17064, 26038}, {17244, 30828}, {17261, 32851}, {17266, 30811}, {17267, 17788}, {17284, 30861}, {17290, 31233}, {17310, 27739}, {17335, 31187}, {25568, 29843}, {26117, 27385}, {27141, 27757}, {27191, 31197}, {27283, 31996}, {29610, 30824}, {29626, 30823}, {30566, 32779}

X(30868) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30868) lies on these lines: {2, 896}, {306, 21729}, {312, 20904}, {661, 4379}, {1959, 17284}, {4885, 30835}, {8772, 17023}, {18743, 20944}, {21839, 31275}

X(30869) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 102

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

X(30869) lies on these lines: {2, 38}, {312, 21443}, {1920, 18743}, {3509, 15271}, {3661, 21242}, {3912, 17717}, {4051, 25102}, {4437, 29676}, {4871, 27475}, {5219, 7146}, {6381, 20917}, {7808, 16787}, {16832, 20358}, {17023, 17725}, {17316, 17722}, {17601, 17738}, {24327, 24341}, {25591, 28742}, {29627, 30861}

X(30870) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(560)X(9233)

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

X(30870) lies on these lines: {788, 17217}, {794, 30912}, {824, 1577}, {23285, 33316}, {50334, 52619}

X(30870) = isotomic conjugate of X(34069)
X(30870) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7357, 39345}, {40145, 39347}
X(30870) = X(46132)-Ceva conjugate of X(561)
X(30870) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34069}, {32, 825}, {560, 1492}, {789, 1917}, {1501, 4586}, {1980, 5384}, {8022, 33514}, {9233, 37133}, {14574, 40718}, {18892, 30664}, {18894, 37207}, {32739, 40746}
X(30870) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34069}, {824, 788}, {6374, 1492}, {6376, 825}, {19584, 32739}, {27481, 692}, {33568, 14402}, {36901, 40747}, {38995, 560}, {40619, 40746}, {55049, 1501}
X(30870) = crossdifference of every pair of points on line {560, 9233}
X(30870) = barycentric product X(i)*X(j) for these {i,j}: {561, 824}, {788, 40362}, {1491, 1502}, {1928, 3250}, {3261, 33931}, {3661, 40495}, {4122, 6385}, {4486, 44172}, {4505, 23989}, {4522, 20567}, {7357, 30872}, {8630, 40359}, {18891, 23596}, {18896, 30639}, {20948, 30966}, {30665, 44170}, {30671, 44171}, {40773, 44173}
X(30870) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34069}, {75, 825}, {76, 1492}, {561, 4586}, {693, 40746}, {788, 1501}, {824, 31}, {850, 40747}, {984, 32739}, {1491, 32}, {1502, 789}, {1928, 37133}, {1978, 5384}, {3250, 560}, {3261, 985}, {3661, 692}, {3799, 23990}, {3807, 1110}, {4122, 213}, {4475, 1919}, {4481, 2206}, {4486, 2210}, {4505, 1252}, {4522, 41}, {7179, 1415}, {8630, 9233}, {18895, 30664}, {20948, 40718}, {23596, 1911}, {27801, 4613}, {30639, 1691}, {30665, 14599}, {30671, 14598}, {30872, 6327}, {30966, 163}, {31909, 32676}, {33904, 52957}, {33931, 101}, {35519, 2344}, {40362, 46132}, {40495, 14621}, {40773, 1576}, {44170, 41072}, {44172, 37207}, {44187, 30670}, {46386, 1917}, {50549, 21751}

X(30871) = X(788)X(7357)∩X(824)X(1577)

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

X(30871) lies on these lines: {788, 7357}, {824, 1577}

X(30872) = X(46132)-CEVA CONJUGATE OF X(6327)

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

X(30872) lies on this line: {824, 1577}

X(30872) = X(46132)-Ceva conjugate of X(6327)
X(30872) = X(34069)-isoconjugate of X(40145)
X(30872) = barycentric product X(6327)*X(30870)
X(30872) = barycentric quotient X(i)/X(j) for these {i,j}: {824, 40145}, {6327, 34069}, {20444, 825}, {30870, 7357}, {30873, 32664}, {40365, 1492}

X(30873) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(560)X(7078)

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

X(30873) lies on this line: {824, 1577}

X(30873) = X(34069)-complementary conjugate of X(40368)
X(30873) = X(4586)-Ceva conjugate of X(6327)
X(30873) = crossdifference of every pair of points on line {560, 7087}
X(30873) = barycentric product X(30872)*X(32664)

X(30874) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(560)X(8630)

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

X(30874) lies on these lines: {2, 561}, {722, 52892}, {824, 1577}, {899, 35546}, {3264, 27801}

X(30874) = X(i)-isoconjugate of X(j) for these (i,j): {32, 753}, {1501, 43097}, {1980, 5386}
X(30874) = X(i)-Dao conjugate of X(j) for these (i,j): {752, 52957}, {6376, 753}
X(30874) = crossdifference of every pair of points on line {560, 8630}
X(30874) = barycentric product X(i)*X(j) for these {i,j}: {75, 35548}, {561, 752}, {1502, 2243}, {1928, 8626}, {4070, 20567}, {4144, 6385}, {4809, 6386}, {33904, 46132}, {40362, 52957}
X(30874) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 753}, {561, 43097}, {752, 31}, {1978, 5386}, {2243, 32}, {4070, 41}, {4144, 213}, {4809, 667}, {8626, 560}, {14438, 1919}, {33568, 14402}, {33904, 788}, {35548, 1}, {52957, 1501}

X(30875) = X(561)X(2887)∩X(752)X(46132)

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

X(30875) lies on these lines: {561, 2887}, {752, 46132}, {824, 1577}, {3836, 14603}, {4439, 6386}

X(30875) = X(i)-isoconjugate of X(j) for these (i,j): {32, 717}, {1501, 43095}
X(30875) = X(i)-Dao conjugate of X(j) for these (i,j): {716, 52892}, {6376, 717}
X(30875) = barycentric product X(i)*X(j) for these {i,j}: {75, 35533}, {561, 716}, {1502, 2230}, {1928, 8621}, {40362, 52892}
X(30875) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 717}, {561, 43095}, {716, 31}, {2230, 32}, {8621, 560}, {35533, 1}, {52892, 1501}

X(30876) = X(561)X(6327)∩X(824)X(1577)

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

X(30876) lies on these lines: {561, 6327}, {824, 1577}

X(30877) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(560)X(3250)

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

X(30877) lies on these lines: {2, 31}, {824, 1577}, {16584, 40379}, {32664, 33796}

X(i)-complementary conjugate of X(j) for these (i,j): {795, 4874}, {14945, 2}
X(30877) = crossdifference of every pair of points on line {560, 3250}
X(30877) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4645, 30914}, {2, 6327, 30890}

Collineation mappings involving Gemini triangle 103: X(30878)-X(30937)

m(X) = a^3 x - (a + c)(a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(824)X(30870). Among the fixed points are X(i) for these i: 2, 824, 30870, 30871, 30872, 30873, 30874, 30875, 30876, 30877. (Clark Kimberling, January 20, 2019)

X(30878) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30878) lies on these lines: {2, 48}, {75, 2908}, {2224, 30885}, {7193, 11334}, {30108, 30920}, {30882, 30894}

X(30879) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30879) lies on these lines: {2, 3}, {6, 3732}, {264, 24019}, {24263, 24511}, {30108, 30899}, {30882, 30891}, {30884, 30928}, {30904, 30927}

X(30880) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30880) lies on these lines: {2, 3}, {69, 163}, {30885, 30886}, {30887, 30900}, {30891, 30929}, {30902, 30904}

X(30881) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30881) lies on these lines: {2, 3}, {30108, 30887}, {30893, 30929}

X(30882) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30882) lies on these lines: {2, 6}, {3, 190}, {48, 313}, {76, 662}, {284, 18147}, {645, 1078}, {674, 26232}, {922, 4112}, {1269, 1958}, {2278, 3948}, {3285, 11320}, {3596, 18042}, {3765, 7113}, {5025, 24957}, {18744, 25651}, {30108, 30885}, {30878, 30894}, {30879, 30891}, {30890, 30895}, {30903, 30908}, {30924, 30933}, {30930, 30931}

X(30883) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30883) lies on these lines: {2, 7}, {8, 692}, {7359, 24612}, {16560, 28734}, {19512, 27393}, {30108, 30916}, {30884, 30902}, {30900, 30906}, {30905, 30919}

X(30884) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics 3 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(30884) lies on these lines: {1, 2}, {7, 1415}, {8369, 32933}, {20267, 21285}, {30879, 30928}, {30883, 30902}, {30899, 30917}, {30916, 30926}, {30931, 30936}

X(30885) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30885) lies on these lines: {1, 21210}, {2, 7}, {85, 1461}, {1266, 24334}, {2224, 30878}, {17023, 24315}, {17301, 24324}, {30108, 30882}, {30880, 30886}, {30894, 30917}, {30903, 30930}, {30908, 30931}

X(30886) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30886) lies on these lines: {1, 2}, {32, 4799}, {41, 25598}, {86, 163}, {2251, 25345}, {3670, 7807}, {3721, 6680}, {3782, 8369}, {3953, 26686}, {4424, 26629}, {4950, 7867}, {7846, 18055}, {11288, 17595}, {16706, 25532}, {17686, 24160}, {20267, 24549}, {30880, 30885}, {30903, 30904}, {30917, 30928}, {30923, 30926}

X(30887) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30887) lies on these lines: {2, 11}, {30108, 30881}, {30880, 30900}

X(30888) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30888) lies on these lines: {2, 3}

X(30889) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30889) lies on these lines: {2, 3}

X(30890) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30890) lies on these lines: {2, 31}, {561, 4586}, {5161, 11339}, {30108, 30905}, {30882, 30895}, {30891, 30894}, {30896, 30928}

X(30891) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30891) lies on these lines: {2, 32}, {1502, 4593}, {2172, 27801}, {30879, 30882}, {30880, 30929}, {30890, 30894}, {30904, 30924}, {30927, 30933}

X(30892) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30892) lies on these lines: {2, 37}, {10, 21210}, {190, 17681}, {274, 662}, {1730, 1760}, {1739, 4429}, {1966, 30111}, {2239, 18805}, {3589, 20234}, {3618, 20444}, {3753, 32850}, {5263, 30117}, {17366, 21442}, {17369, 20432}, {30108, 30882}, {30115, 32922}, {30893, 30915}, {30899, 30918}, {30906, 30930}

X(30893) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30893) lies on these lines: {2, 39}, {99, 4218}, {141, 21138}, {183, 11334}, {304, 18045}, {308, 4593}, {313, 4692}, {350, 30115}, {1269, 20894}, {1909, 30117}, {1921, 30178}, {16086, 17143}, {18139, 20924}, {18143, 20925}, {30879, 30882}, {30881, 30929}, {30892, 30915}, {30924, 30927}

X(30894) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30894) lies on these lines: {2, 41}, {2224, 30108}, {2280, 30111}, {9310, 30113}, {30878, 30882}, {30885, 30917}, {30890, 30891}, {30895, 30918}

X(30895) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30895) lies on these lines: {1, 2}, {310, 662}, {30882, 30890}, {30894, 30918}, {30905, 30928}

X(30896) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30896) lies on these lines: {1, 2}, {30890, 30928}, {30899, 30914}

X(30897) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30897) lies on these lines: {2, 44}, {8060, 30909}, {30108, 30882}, {30908, 30930}

X(30898) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30898) lies on these lines: {2, 45}, {30108, 30882}

X(30899) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30899) lies on these lines: {2, 11}, {30108, 30879}, {30882, 30890}, {30884, 30917}, {30892, 30918}, {30896, 30914}

X(30900) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30900) lies on these lines: {2, 12}, {30108, 30879}, {30878, 30882}, {30880, 30887}, {30883, 30906}, {30907, 30928}

X(30901) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30901) lies on these lines: {2, 7}, {101, 312}, {30108, 30879}

X(30902) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30902) lies on these lines: {2, 6}, {4, 32676}, {3285, 31015}, {4419, 24609}, {30880, 30904}, {30883, 30884}, {30922, 30927}

X(30903) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30903) lies on these lines: {1, 692}, {2, 37}, {573, 18179}, {744, 30953}, {1001, 30117}, {1279, 3877}, {1760, 2220}, {2209, 4118}, {3670, 16455}, {4422, 17540}, {5301, 16566}, {15624, 17446}, {22426, 24699}, {25065, 25523}, {30108, 30906}, {30882, 30908}, {30883, 30884}, {30885, 30930}, {30886, 30904}, {30919, 30926}

X(30904) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30904) lies on these lines: {2, 39}, {6, 163}, {574, 4218}, {2275, 30117}, {2276, 30115}, {4286, 16374}, {5277, 22380}, {30879, 30927}, {30880, 30902}, {30886, 30903}, {30891, 30924}

X(30905) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30905) lies on these lines: {2, 6}, {101, 321}, {284, 4358}, {1817, 32933}, {2360, 4696}, {5014, 17188}, {5016, 27412}, {21997, 33151}, {30108, 30890}, {30883, 30919}, {30895, 30928}

X(30906) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30906) lies on these lines: {2, 6}, {10, 692}, {21, 4422}, {44, 24632}, {190, 21997}, {284, 17279}, {857, 25659}, {1043, 17269}, {1333, 17353}, {2173, 24335}, {3285, 16050}, {3912, 4273}, {4363, 16054}, {17189, 17356}, {17369, 26643}, {21245, 25651}, {25658, 30851}, {30108, 30903}, {30883, 30900}, {30892, 30930}, {30914, 30915}

X(30907) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30907) lies on these lines: {1, 2}, {30900, 30928}

X(30908) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30908) lies on these lines: {1, 2}, {30882, 30903}, {30885, 30931}, {30897, 30930}, {30926, 30933}

X(30909) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30909) lies on these lines: {2, 649}, {8060, 30897}, {8631, 30912}, {9313, 32772}

X(30910) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30910) lies on these lines: {2, 650}, {8060, 30897}, {29066, 30115}

X(30911) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30911) lies on these lines: {2, 661}, {4728, 8062}, {8060, 30897}, {14838, 30913}

X(30912) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30912) lies on these lines: {2, 667}, {794, 30870}, {8631, 30909}

X(30913) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30913) lies on these lines: {2, 650}, {100, 32666}

X(30914) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30914) lies on these lines: {2, 31}, {4495, 19557}, {30108, 30882}, {30896, 30899}, {30906, 30915}

X(30915) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30915) lies on these lines: {1, 2}, {1213, 21138}, {30892, 30893}, {30906, 30914}

X(30916) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30916) lies on these lines: {2, 3}, {30108, 30883}, {30884, 30926}

X(30917) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics (a + b) (a + c) (a^5 - a^4 b + a^3 b^2 - a^2 b^3 - a^4 c + 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 + 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(30917) lies on these lines: {2, 3}, {1441, 32674}, {30108, 30890}, {30883, 30900}, {30884, 30899}, {30885, 30894}, {30886, 30928} {2, 3}, {1441, 32674}, {30108, 30890}, {30883, 30900}, {30884, 30899}, {30885, 30894}, {30886, 30928}

X(30918) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30918) lies on these lines: {2, 3}, {305, 662}, {653, 3162}, {30892, 30899}, {30894, 30895}

X(30919) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30919) lies on these lines: {2, 3}, {306, 692}, {30883, 30905}, {30903, 30926}

X(30920) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30920) lies on these lines: {2, 3}, {101, 20336}, {30108, 30878}

X(30921) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30921) lies on these lines: {2, 3}, {307, 1415}

X(30922) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30922) lies on these lines: {2, 3}, {30902, 30927}

X(30923) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30923) lies on these lines: {2, 3}, {30883, 30884}, {30886, 30926}

X(30924) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30924) lies on these lines: {2, 3}, {30882, 30933}, {30891, 30904}, {30893, 30927}

X(30925) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30925) lies on these lines: {2, 38}, {30108, 30890}

X(30926) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30926) lies on these lines: {2, 7}, {92, 32674}, {604, 14206}, {2187, 17884}, {30108, 30890}, {30884, 30916}, {30886, 30923}, {30903, 30919}, {30908, 30933}

X(30927) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30927) lies on these lines: {2, 99}, {110, 24281}, {163, 21138}, {523, 923}, {741, 2690}, {759, 27918}, {1086, 3109}, {2224, 30930}, {3124, 24288}, {4237, 26273}, {30879, 30904}, {30891, 30933}, {30893, 30924}, {30902, 30922}

X(30928) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30928) lies on these lines: {2, 11}, {693, 1438}, {30879, 30884}, {30886, 30917}, {30890, 30896}, {30895, 30905}, {30900, 30907}

X(30929) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30929) lies on these lines: {2, 99}, {30880, 30891}, {30881, 30893}

X(30930) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30930) lies on these lines: {2, 45}, {6, 3732}, {37, 25532}, {244, 24346}, {514, 9456}, {692, 21210}, {2161, 27918}, {2224, 30927}, {3285, 14955}, {3666, 24324}, {16561, 24398}, {17455, 24281}, {30882, 30931}, {30885, 30903}, {30892, 30906}, {30897, 30908}

X(30931) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30931) lies on these lines: {2, 37}, {330, 2224}, {30108, 30933}, {30882, 30930}, {30884, 30936}, {30885, 30908}

X(30932) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30932) lies on these lines: {2, 187}, {8631, 30909}, {30879, 30882}

X(30933) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30933) lies on these lines: {2, 39}, {192, 30115}, {330, 30117}, {385, 11334}, {30108, 30931}, {30882, 30924}, {30891, 30927}, {30908, 30926}

X(30934) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30934) lies on these lines: {2, 669}, {8631, 30909}

X(30935) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30935) lies on these lines: {2, 7}, {30108, 30931}, {30882, 30903}, {30892, 30906}

X(30936) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30936) lies on these lines: {2, 38}, {30108, 30882}, {30884, 30931}, {30886, 30903}

X(30937) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 103

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

X(30937) lies on these lines: {2, 6}, {4945, 11352}, {11320, 30566}, {11346, 31171}, {11347, 33146}, {30879, 30884}

X(30938) = (name pending)

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

X(30938) lies on these lines: {44, 799}, {274, 1107}, {320, 350}, {536, 670}, {538, 30736}, {2978, 7192}, {4623, 16702}, {4625, 6610}, {4648, 18135}, {4670, 8033}

X(30939) = ISOTOMIC CONJUJGATE OF X(4674)

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

X(30939) lies on these lines: {1, 75}, {2, 4277}, {6, 18137}, {7, 20615}, {9, 29767}, {37, 16738}, {44, 4358}, {58, 4676}, {63, 29766}, {69, 2478}, {76, 17378}, {81, 312}, {99, 2718}, {110, 2863}, {141, 17202}, {142, 29756}, {190, 18206}, {192, 16696}, {194, 18172}, {285, 309}, {310, 4479}, {313, 3879}, {319, 17751}, {320, 350}, {321, 4670}, {333, 3305}, {344, 16713}, {346, 26818}, {519, 3264}, {524, 3948}, {536, 16726}, {573, 29558}, {579, 29763}, {594, 29388}, {670, 18822}, {679, 4634}, {751, 17250}, {757, 10457}, {765, 4600}, {857, 18745}, {889, 4639}, {894, 4043}, {903, 17179}, {1100, 20891}, {1168, 4555}, {1227, 4432}, {1266, 17205}, {1269, 3664}, {1278, 16710}, {1319, 4702}, {1654, 25660}, {1992, 28809}, {2092, 26979}, {2235, 20453}, {2287, 20946}, {2321, 29705}, {2978, 23464}, {3210, 16700}, {3263, 20045}, {3286, 3685}, {3596, 17377}, {3662, 29764}, {3701, 4663}, {3729, 18164}, {3759, 20923}, {3770, 20090}, {3882, 29456}, {3912, 17197}, {3936, 17174}, {3943, 24004}, {3963, 17390}, {3992, 4753}, {3995, 26819}, {4033, 6542}, {4044, 4667}, {4385, 4658}, {4389, 16887}, {4417, 17182}, {4494, 29605}, {4506, 4727}, {4671, 26860}, {4687, 27164}, {4725, 25298}, {4751, 25508}, {4851, 18040}, {4852, 20892}, {4908, 16723}, {5235, 30829}, {5249, 19821}, {5333, 19804}, {6376, 17360}, {7081, 18185}, {9282, 17731}, {13476, 17142}, {14829, 17185}, {16574, 29746}, {16705, 17320}, {16732, 20956}, {16736, 17490}, {16749, 21605}, {16752, 24663}, {16753, 17495}, {17147, 18601}, {17167, 18134}, {17173, 18139}, {17187, 24696}, {17196, 31008}, {17227, 30965}, {17233, 29423}, {17234, 29484}, {17242, 29712}, {17243, 29396}, {17245, 29446}, {17271, 18140}, {17296, 18044}, {17297, 18150}, {17300, 18143}, {17315, 17787}, {17346, 30830}, {17348, 29982}, {17351, 22016}, {17373, 30473}, {17375, 18144}, {17386, 17786}, {17387, 20917}, {17392, 20913}, {17778, 27792}, {18136, 32863}, {18163, 30567}, {18171, 25264}, {18195, 21330}, {20171, 20444}, {20938, 25698}, {20985, 24425}, {21858, 27102}, {24524, 28660}, {24739, 27633}, {24944, 26045}, {26115, 28653}, {26227, 30758}, {26234, 29823}, {27163, 28606}

X(30939) = isotomic conjugate of X(4674)

X(30940) = (name pending)

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

X(30940) lies on these lines: {1, 75}, {6, 76}, {42, 17176}, {69, 1244}, {81, 310}, {99, 3286}, {238, 350}, {239, 1921}, {321, 16707}, {333, 17026}, {334, 32846}, {561, 3187}, {799, 16704}, {812, 4509}, {873, 8025}, {1016, 4601}, {1078, 5132}, {1150, 30964}, {1429, 10030}, {1434, 7153}, {1509, 18166}, {1909, 4649}, {1920, 1999}, {1965, 3791}, {2111, 17738}, {3112, 17150}, {3416, 33297}, {3500, 17206}, {3759, 21615}, {3760, 16468}, {3761, 28650}, {3821, 16887}, {3934, 20148}, {3948, 20142}, {4361, 10009}, {4495, 4716}, {4589, 4645}, {4639, 19565}, {4657, 17030}, {5235, 17028}, {5333, 17032}, {7018, 33135}, {7304, 18021}, {13610, 18298}, {16705, 16738}, {16712, 17301}, {16727, 16741}, {17033, 17743}, {17135, 18064}, {17141, 17142}, {17149, 32853}, {17156, 18056}, {17178, 18600}, {17277, 18046}, {17303, 25508}, {17349, 18135}, {17731, 24731}, {17737, 27273}, {18152, 32911}, {20132, 20913}, {20154, 30830}, {20158, 31060}, {27145, 27162}, {27623, 30022}, {28365, 30092}, {29383, 29388}, {29437, 29454}, {29742, 29764}, {29746, 29765}, {30631, 33132}, {30632, 33128}, {30966, 32784}

X(30941) = (name pending)

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

X(30941) lies on these lines: {1, 16705}, {2, 6}, {7, 310}, {8, 274}, {10, 17175}, {21, 3423}, {37, 24690}, {39, 18171}, {42, 3879}, {59, 4600}, {75, 3873}, {99, 840}, {110, 2862}, {142, 24592}, {145, 18600}, {150, 5209}, {213, 27097}, {239, 16752}, {261, 6061}, {291, 32846}, {304, 3868}, {319, 4651}, {320, 350}, {332, 1014}, {354, 16739}, {386, 27162}, {518, 3263}, {519, 16711}, {672, 3912}, {742, 3726}, {757, 33173}, {758, 14210}, {894, 31027}, {942, 20911}, {1009, 3933}, {1043, 1434}, {1125, 17210}, {1444, 3433}, {1458, 4684}, {1468, 24549}, {1475, 29960}, {1575, 16726}, {1743, 30822}, {1909, 17751}, {1930, 3874}, {2113, 17794}, {2140, 29742}, {2176, 20109}, {2275, 18172}, {2276, 4851}, {2280, 24586}, {2295, 26759}, {2978, 20295}, {3125, 8682}, {3241, 16712}, {3252, 3930}, {3286, 4966}, {3293, 24170}, {3446, 5078}, {3449, 29839}, {3662, 17027}, {3664, 3741}, {3679, 17180}, {3681, 30758}, {3684, 24602}, {3691, 29968}, {3693, 16728}, {3717, 14626}, {3720, 4357}, {3721, 18189}, {3780, 20255}, {3783, 18792}, {3794, 9309}, {3869, 18156}, {3952, 20947}, {3954, 25263}, {4251, 29473}, {4253, 27109}, {4368, 17770}, {4430, 31130}, {4562, 6542}, {4589, 4645}, {4615, 19634}, {4754, 21024}, {6384, 20028}, {6626, 17588}, {7179, 17177}, {7804, 14968}, {9025, 20863}, {9500, 14953}, {10436, 31330}, {10446, 20788}, {10449, 10471}, {11115, 17103}, {11269, 30984}, {12649, 16749}, {16710, 17373}, {16714, 17377}, {16737, 21302}, {16891, 33069}, {17026, 17298}, {17032, 17391}, {17034, 26978}, {17144, 20244}, {17149, 20245}, {17164, 17762}, {17167, 30985}, {17176, 32949}, {17182, 30961}, {17189, 33137}, {17197, 20335}, {17200, 29637}, {17202, 17288}, {17203, 24241}, {17270, 26037}, {17272, 26102}, {17296, 17754}, {17321, 29814}, {17353, 30821}, {17361, 30963}, {17364, 24514}, {17365, 24330}, {17376, 21264}, {17390, 25349}, {17474, 30038}, {17489, 18167}, {17499, 27040}, {17758, 29433}, {18198, 20331}, {20541, 31041}, {20963, 21240}, {21330, 24513}, {21384, 29966}, {21874, 26689}, {24697, 30571}, {26841, 26843}, {26871, 30973}, {28351, 30048}, {30969, 32919}

X(30941) = isotomic conjugate of X(13576)
X(30941) = anticomplement of X(2238)

Collineation mappings involving Gemini triangle 104: X(30942)-X(31007)

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

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(513)X(693). Among the fixed points are X(i) for these i: 2, 513, 693, 3250, 7912, 30938, 30939, 30940, 30941, 31008. (Clark Kimberling, January 20, 2019)

X(30942) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30942) lies on these lines: {1, 2}, {6, 32919}, {11, 141}, {31, 14829}, {36, 11322}, {38, 312}, {55, 32918}, {56, 16405}, {57, 4418}, {63, 32930}, {69, 32843}, {72, 25591}, {75, 244}, {76, 23473}, {81, 25496}, {86, 9345}, {87, 17178}, {100, 32941}, {149, 4660}, {171, 24552}, {226, 33069}, {238, 1150}, {310, 6384}, {320, 24725}, {321, 982}, {333, 748}, {350, 4389}, {354, 32771}, {355, 19546}, {497, 26034}, {515, 19647}, {518, 30818}, {536, 4003}, {594, 3756}, {726, 4392}, {740, 4850}, {750, 5263}, {751, 17250}, {756, 18743}, {896, 4676}, {908, 33065}, {940, 32772}, {958, 16373}, {984, 4358}, {986, 3702}, {1001, 32917}, {1086, 30629}, {1211, 3816}, {1215, 3873}, {1266, 4441}, {1376, 32945}, {1468, 13740}, {1580, 26634}, {1621, 32916}, {1654, 26139}, {1740, 27145}, {1836, 33067}, {1985, 5087}, {2049, 19726}, {2051, 10439}, {2229, 2275}, {2276, 3943}, {2280, 26244}, {2321, 17756}, {2886, 25957}, {2887, 11680}, {3120, 3662}, {3210, 4365}, {3218, 3923}, {3219, 4011}, {3242, 32927}, {3304, 5793}, {3416, 17721}, {3434, 32948}, {3454, 7741}, {3666, 32915}, {3685, 4414}, {3696, 16610}, {3706, 3752}, {3728, 20923}, {3742, 31993}, {3760, 16887}, {3769, 17469}, {3772, 33123}, {3773, 33089}, {3789, 16594}, {3821, 33134}, {3834, 21264}, {3836, 21242}, {3846, 32782}, {3876, 25079}, {3891, 17598}, {3914, 33125}, {3925, 25961}, {3936, 17717}, {3944, 17184}, {3969, 32855}, {3971, 7226}, {3976, 4968}, {3980, 27003}, {4022, 17157}, {4038, 19684}, {4054, 24231}, {4090, 4661}, {4192, 18481}, {4359, 17063}, {4363, 4860}, {4379, 24720}, {4383, 32864}, {4387, 32936}, {4388, 33080}, {4417, 33081}, {4423, 5737}, {4429, 33136}, {4438, 33157}, {4442, 33149}, {4465, 4643}, {4514, 33074}, {4642, 4673}, {4645, 33104}, {4647, 24046}, {4655, 5057}, {4683, 24703}, {4689, 4702}, {4693, 17593}, {4713, 24690}, {4759, 24616}, {4851, 17723}, {4865, 33078}, {4884, 6057}, {4892, 10129}, {4966, 5718}, {4972, 33141}, {5014, 33079}, {5192, 5247}, {5204, 16395}, {5278, 17123}, {5372, 17127}, {5695, 17595}, {5698, 30943}, {5741, 33084}, {5774, 16483}, {6075, 24250}, {6327, 33085}, {6646, 17777}, {6682, 28606}, {8054, 18194}, {8167, 19732}, {8299, 30944}, {9709, 16421}, {10886, 24220}, {16062, 19801}, {16468, 16704}, {16571, 27017}, {16706, 33128}, {17125, 17277}, {17147, 17591}, {17149, 18152}, {17183, 17272}, {17227, 24688}, {17228, 21238}, {17233, 32848}, {17237, 25378}, {17238, 21257}, {17239, 25624}, {17240, 28593}, {17279, 33115}, {17280, 33161}, {17281, 20331}, {17449, 24349}, {17596, 32929}, {17597, 32923}, {17599, 32928}, {17719, 33122}, {17720, 32775}, {17722, 32846}, {17793, 30990}, {18134, 33105}, {18139, 33111}, {18169, 27163}, {18192, 26819}, {18525, 19540}, {19804, 21020}, {19818, 23537}, {20072, 24514}, {20172, 24602}, {20284, 21327}, {21241, 25959}, {21384, 27040}, {22791, 31778}, {24165, 28605}, {24169, 33131}, {24170, 32104}, {24210, 32776}, {24217, 32784}, {24330, 24691}, {24342, 26627}, {24477, 33163}, {24756, 31207}, {25128, 30835}, {25385, 31019}, {26061, 33121}, {26064, 26127}, {26098, 32949}, {26128, 33133}, {26223, 32913}, {26840, 33098}, {27064, 32912}, {30588, 31006}, {30946, 30971}, {30956, 30994}, {30974, 30983}, {30989, 30998}, {31053, 33064}, {32773, 32781}, {32774, 33135}, {32777, 33119}, {32851, 33156}, {32852, 33071}, {32853, 32911}, {32859, 33096}, {32863, 32946}, {32950, 33095}, {33068, 33094}, {33113, 33158}, {33114, 33159}, {33124, 33127}

X(30943) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30943) lies on these lines: {2, 3}, {11, 20992}, {31, 497}, {40, 26013}, {42, 3486}, {43, 10572}, {51, 9535}, {69, 799}, {242, 24611}, {243, 1040}, {244, 28090}, {388, 10448}, {940, 5327}, {1042, 3485}, {1044, 12047}, {1249, 21148}, {1478, 29640}, {1479, 33140}, {2328, 13478}, {2550, 32917}, {3000, 30949}, {3271, 9554}, {3741, 12514}, {3869, 10453}, {4872, 30988}, {5057, 30947}, {5698, 30942}, {5712, 10458}, {6360, 20254}, {7155, 17777}, {8299, 30960}, {10478, 17194}, {17139, 30962}, {17441, 18750}, {20556, 30741}, {22060, 27339}, {24703, 30986}, {26011, 30271}, {26105, 32772}, {30954, 30996}, {30959, 30979}

X(30944) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics a (a^4 b - 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 - 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(30944) lies on these lines: {2, 3}, {11, 8053}, {31, 21321}, {35, 33140}, {36, 29640}, {42, 2646}, {43, 3612}, {46, 26102}, {51, 17194}, {55, 11269}, {63, 21319}, {65, 3720}, {216, 2331}, {226, 22060}, {228, 5745}, {238, 28289}, {672, 17603}, {1155, 30950}, {1284, 4414}, {1764, 22080}, {2223, 29639}, {2238, 2278}, {2245, 24493}, {3286, 5718}, {3941, 17723}, {4271, 18191}, {4447, 29643}, {4640, 30986}, {4645, 30029}, {4995, 15621}, {5794, 26037}, {6684, 26013}, {6690, 16678}, {8299, 30942}, {12609, 25501}, {15485, 28393}, {17074, 17975}, {17125, 28239}, {26066, 31330}, {30955, 30996}

X(30945) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30945) lies on these lines: {1, 4372}, {2, 6}, {3, 8299}, {7, 24330}, {8, 20255}, {9, 24690}, {32, 29473}, {42, 4851}, {43, 17296}, {58, 25497}, {75, 3726}, {76, 4602}, {142, 3741}, {172, 24549}, {213, 30110}, {291, 29674}, {304, 3721}, {320, 24514}, {350, 3662}, {354, 3739}, {518, 30748}, {672, 17279}, {742, 26242}, {980, 2276}, {1009, 7795}, {1086, 4441}, {1107, 29966}, {1201, 24652}, {1575, 17231}, {1716, 17306}, {1914, 24586}, {1985, 30954}, {2176, 17137}, {2227, 21330}, {2275, 29960}, {2295, 27248}, {3509, 4376}, {3720, 4657}, {3727, 18156}, {3734, 11355}, {3735, 14210}, {3745, 28639}, {3780, 27299}, {3783, 33087}, {3788, 25645}, {3817, 3840}, {3834, 21264}, {3844, 28600}, {3959, 26562}, {4000, 10453}, {4361, 17135}, {4368, 4655}, {4386, 24602}, {4445, 4651}, {4465, 30946}, {4514, 24763}, {4713, 7232}, {4799, 4872}, {4950, 7270}, {5282, 24358}, {5283, 16887}, {5347, 33173}, {5902, 24254}, {9055, 31130}, {16696, 17208}, {16706, 17027}, {16969, 17152}, {16975, 30109}, {17018, 17390}, {17026, 17282}, {17032, 17317}, {17045, 29814}, {17103, 17688}, {17143, 24190}, {17149, 20923}, {17169, 26035}, {17227, 30963}, {17233, 17759}, {17239, 26037}, {17243, 25349}, {17278, 24592}, {17284, 17754}, {17290, 29824}, {17374, 21904}, {17448, 30036}, {17756, 25350}, {18144, 18152}, {19283, 26101}, {20271, 20911}, {20331, 29579}, {20530, 30957}, {20541, 25957}, {20963, 30107}, {21071, 24214}, {24657, 33124}, {30956, 30960}, {30964, 30987}, {30978, 31000}, {30990, 30991}, {30997, 30998}

X(30946) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30946) lies on these lines: {2, 7}, {8, 76}, {10, 17753}, {41, 17691}, {42, 3672}, {43, 3663}, {69, 350}, {72, 3673}, {75, 210}, {85, 960}, {86, 4423}, {141, 4713}, {192, 3930}, {193, 17027}, {218, 17681}, {304, 19582}, {319, 4479}, {320, 4679}, {335, 26274}, {347, 25941}, {391, 24592}, {664, 5289}, {899, 4346}, {938, 10381}, {956, 24203}, {978, 24214}, {997, 5088}, {1001, 14828}, {1111, 5692}, {1193, 4352}, {1329, 33298}, {1434, 25524}, {1575, 17276}, {1699, 3741}, {1750, 10444}, {2238, 4000}, {2276, 4419}, {2345, 24330}, {2550, 3789}, {2551, 6604}, {3057, 16284}, {3061, 3177}, {3160, 10571}, {3212, 3869}, {3263, 21590}, {3475, 17321}, {3487, 16850}, {3617, 20244}, {3620, 31027}, {3664, 26102}, {3691, 27304}, {3720, 3945}, {3760, 10449}, {3783, 24248}, {3794, 9309}, {3875, 20012}, {3876, 20880}, {3877, 30806}, {3925, 5224}, {3952, 31130}, {4059, 25917}, {4192, 5658}, {4329, 33171}, {4368, 17170}, {4416, 17026}, {4460, 20011}, {4465, 30945}, {4643, 21264}, {4644, 24512}, {4651, 32087}, {4655, 17793}, {4685, 17151}, {4703, 30982}, {4741, 24712}, {4862, 16569}, {4872, 24703}, {4888, 25502}, {5046, 21285}, {5232, 17220}, {5550, 17169}, {5698, 8299}, {5736, 16343}, {5815, 13161}, {5850, 10520}, {5904, 7264}, {5936, 8049}, {6147, 16846}, {6376, 21281}, {6818, 21279}, {7176, 19861}, {8822, 13588}, {9312, 15829}, {9534, 20888}, {14009, 17139}, {14548, 26105}, {17095, 25681}, {17135, 32099}, {17137, 18135}, {17181, 21616}, {17255, 25349}, {17271, 31140}, {17288, 31028}, {17301, 21904}, {17345, 20530}, {17451, 27288}, {17697, 24549}, {21214, 24215}, {21255, 30822}, {24349, 26234}, {25242, 33299}, {25590, 30393}, {26038, 31995}, {27523, 29960}, {30812, 31225}, {30942, 30971}, {30965, 30973}

X(30947) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30947) lies on these lines: {1, 2}, {6, 25531}, {7, 1357}, {11, 17234}, {75, 4519}, {190, 4860}, {192, 244}, {312, 3742}, {320, 4679}, {333, 8167}, {341, 17609}, {354, 4009}, {377, 19801}, {518, 30829}, {740, 24620}, {942, 19582}, {944, 19546}, {964, 19726}, {1011, 5303}, {1150, 16355}, {1985, 30993}, {1997, 3475}, {2325, 17754}, {2899, 11037}, {2975, 16373}, {3210, 17063}, {3306, 3685}, {3756, 17243}, {3816, 18134}, {3836, 24217}, {3848, 19804}, {3873, 27538}, {3932, 17051}, {4003, 4664}, {4078, 24216}, {4358, 24349}, {4388, 18141}, {4392, 31035}, {4423, 14829}, {4465, 4644}, {4648, 21299}, {4684, 5316}, {4737, 5049}, {4891, 16602}, {4903, 17165}, {4966, 5233}, {5057, 30943}, {5253, 16405}, {5437, 32932}, {5687, 16421}, {5731, 19647}, {9335, 17147}, {9345, 17379}, {9779, 24220}, {10980, 30568}, {14009, 30959}, {17053, 17756}, {17124, 32943}, {17125, 17349}, {17232, 25760}, {17300, 26139}, {17317, 17723}, {17375, 26069}, {17449, 31302}, {17450, 30861}, {17490, 32915}, {17594, 27002}, {17728, 33116}, {21609, 31526}, {24709, 24725}, {24736, 33115}, {28581, 31197}, {30834, 31272}, {30971, 30985}, {30998, 31005}

X(30948) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30948) lies on these lines: {1, 2}, {11, 17232}, {69, 26139}, {192, 4003}, {244, 1278}, {312, 3999}, {346, 20331}, {350, 4346}, {518, 30861}, {3756, 17233}, {3790, 24216}, {4358, 31302}, {4373, 31002}, {4519, 4740}, {4679, 4741}, {5274, 20359}, {17349, 25531}, {20942, 21342}, {28581, 31233}, {30960, 30993}

X(30949) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30949) lies on these lines: {1, 2140}, {2, 7}, {4, 26101}, {8, 17050}, {12, 21258}, {21, 25500}, {35, 14377}, {41, 17682}, {42, 3475}, {43, 4859}, {65, 6706}, {69, 24592}, {75, 3930}, {76, 29966}, {85, 17451}, {116, 7951}, {141, 3779}, {145, 20257}, {183, 24602}, {210, 3739}, {244, 3116}, {277, 3487}, {310, 18054}, {350, 17234}, {386, 24790}, {495, 4904}, {497, 2293}, {673, 2280}, {942, 24774}, {1002, 5542}, {1086, 2276}, {1212, 4059}, {1334, 17753}, {1699, 1742}, {1743, 31200}, {1909, 30036}, {2238, 17278}, {2352, 4657}, {2476, 17046}, {3000, 30943}, {3136, 18635}, {3208, 20244}, {3240, 17067}, {3501, 28742}, {3673, 21808}, {3684, 24596}, {3741, 21255}, {3761, 30109}, {3789, 3826}, {3834, 21264}, {3838, 30959}, {3912, 4441}, {3946, 17018}, {4032, 24554}, {4361, 32923}, {4368, 11263}, {4402, 20012}, {4423, 15668}, {4479, 17241}, {4675, 24512}, {4713, 17265}, {4851, 4863}, {4869, 10453}, {5010, 17729}, {5074, 18393}, {5880, 8299}, {6384, 27431}, {6818, 15669}, {7201, 25001}, {7232, 24690}, {10129, 30993}, {11374, 20269}, {14009, 30956}, {15950, 17044}, {16054, 25940}, {17026, 17298}, {17027, 17300}, {17032, 17302}, {17135, 17296}, {17227, 30966}, {17232, 31027}, {17245, 17747}, {17279, 24330}, {17313, 31140}, {17686, 24549}, {18135, 29968}, {18152, 20923}, {20181, 32945}, {20267, 24160}, {21101, 31130}, {21620, 24181}, {24046, 24786}, {24190, 27020}, {24259, 29642}, {30748, 32931}, {30963, 30997}, {30967, 30998}, {30983, 30994}, {31063, 31077}

X(30950) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics a (a b + a c + 4 b c) : :
Trilinears 4 csc A + csc B + csc C : :

X(30950) lies on these lines:{1, 2}, {6, 9345}, {11, 2293}, {12, 4322}, {31, 4423}, {37, 244}, {38, 3742}, {44, 24512}, {45, 672}, {55, 17124}, {56, 16373}, {58, 25542}, {81, 16477}, {86, 799}, {88, 17593}, {100, 16484}, {142, 3120}, {171, 5284}, {238, 21747}, {344, 33161}, {350, 25382}, {354, 756}, {373, 20962}, {405, 19726}, {518, 17450}, {726, 31035}, {740, 24589}, {748, 940}, {750, 902}, {846, 27003}, {896, 15254}, {968, 5437}, {982, 3989}, {984, 17449}, {991, 7988}, {1010, 19801}, {1011, 5204}, {1042, 11375}, {1064, 11230}, {1086, 25422}, {1155, 30944}, {1255, 17600}, {1385, 19546}, {1458, 5219}, {1468, 11108}, {1475, 16589}, {1621, 17122}, {1742, 9779}, {1962, 3752}, {1985, 2635}, {2177, 4413}, {2229, 16604}, {2230, 20530}, {2234, 21264}, {2238, 16666}, {2239, 3246}, {2276, 8610}, {2292, 5439}, {2309, 15668}, {2310, 17603}, {2356, 5094}, {2650, 25917}, {2667, 31238}, {2901, 6533}, {3000, 30943}, {3136, 7173}, {3231, 23660}, {3247, 17756}, {3248, 32044}, {3295, 16421}, {3305, 32912}, {3306, 4414}, {3576, 19647}, {3666, 3848}, {3670, 27784}, {3740, 4883}, {3816, 33105}, {3826, 33136}, {3833, 4424}, {3846, 18139}, {3917, 20961}, {3923, 26627}, {3944, 27186}, {3970, 25089}, {3971, 17140}, {3993, 17495}, {3995, 24165}, {4009, 31161}, {4038, 32911}, {4085, 24988}, {4191, 5217}, {4192, 13624}, {4300, 8227}, {4343, 20195}, {4358, 24325}, {4359, 4365}, {4379, 4448}, {4465, 4670}, {4519, 4688}, {4648, 20978}, {4675, 4679}, {4682, 17469}, {4689, 8299}, {4893, 14421}, {4896, 20347}, {4937, 31178}, {4966, 5241}, {5225, 6817}, {5229, 6818}, {5247, 17536}, {5276, 16786}, {5283, 23649}, {5333, 17187}, {5573, 25430}, {5650, 21746}, {5743, 33081}, {6051, 24443}, {8041, 22200}, {8054, 28639}, {8543, 9364}, {9335, 17591}, {10448, 16405}, {10458, 25507}, {14009, 30980}, {14752, 27811}, {14996, 16468}, {14997, 28650}, {15485, 17126}, {15569, 16610}, {16676, 17754}, {17063, 28606}, {17234, 25760}, {17243, 32848}, {17263, 33115}, {17277, 32919}, {17278, 33128}, {17300, 32843}, {17302, 25420}, {17605, 22053}, {17777, 25421}, {18134, 25960}, {18141, 33080}, {18743, 32771}, {19804, 32915}, {21806, 31197}, {24046, 27785}, {24217, 33108}, {24487, 30963}, {24666, 30835}, {24709, 30997}, {25124, 29982}, {25557, 32856}, {25961, 32773}, {26724, 33135}, {26842, 33099}, {27065, 32913}, {30829, 32931}, {30969, 30987}, {30977, 30985}, {31260, 31880}

X(30950) = {X(1), X(2)}-harmonic conjugate of X(899)

X(30951) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30951) lies on these lines: {2, 3}, {20915, 21322}

X(30952) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30952) lies on these lines: {2, 3}, {3837, 26985}, {20916, 21323}, {31119, 31122}

X(30953) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30953) lies on these lines: {2, 31}, {42, 4865}, {63, 30752}, {226, 1469}, {256, 3662}, {291, 3705}, {350, 3944}, {672, 4438}, {744, 30903}, {1008, 29637}, {1215, 3681}, {1716, 17282}, {1836, 24259}, {2209, 17138}, {2210, 24587}, {2276, 4071}, {2309, 29981}, {3720, 26128}, {3771, 4192}, {3772, 17031}, {3783, 4417}, {3817, 3840}, {3838, 21264}, {4112, 24630}, {4118, 20444}, {4199, 29642}, {4368, 24703}, {4892, 10129}, {10453, 33144}, {10455, 17272}, {17026, 17064}, {17027, 33135}, {17135, 32920}, {17149, 21590}, {17165, 31119}, {17232, 21299}, {17759, 32855}, {17793, 30961}, {17794, 33101}, {18067, 18152}, {22343, 30052}, {24333, 24690}, {24512, 29635}, {24514, 33096}, {25527, 26102}, {26037, 28595}, {26237, 33127}, {30954, 30956}, {30957, 30993}

X(30954) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30954) lies on these lines: {2, 32}, {42, 4950}, {1985, 30945}, {3734, 11330}, {14599, 24637}, {20641, 21324}, {30943, 30996}, {30953, 30956}, {30964, 30978}, {30992, 31000}

X(30955) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30955) lies on these lines: {2, 39}, {11, 141}, {42, 25102}, {183, 11339}, {561, 21327}, {1985, 30945}, {2228, 21264}, {2230, 20530}, {2231, 24512}, {3734, 11322}, {4871, 17758}, {6384, 27158}, {8620, 21443}, {18139, 30967}, {21753, 29557}, {22199, 26973}, {30944, 30996}, {30978, 30987}

X(30956) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30956) lies on these lines: {2, 41}, {1985, 20335}, {3662, 30978}, {3840, 30972}, {14009, 30949}, {17227, 31001}, {20922, 21329}, {30942, 30994}, {30945, 30960}, {30953, 30954}

X(30957) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30957) lies on these lines: {1, 2}, {11, 25957}, {38, 18743}, {57, 32930}, {141, 25960}, {244, 312}, {321, 17063}, {333, 17125}, {350, 4398}, {354, 32931}, {497, 32948}, {561, 18149}, {748, 14829}, {750, 32942}, {756, 30829}, {908, 33069}, {940, 32944}, {942, 25591}, {982, 4358}, {1001, 32918}, {1054, 32929}, {1150, 17123}, {1376, 32943}, {1468, 13741}, {1757, 26688}, {2886, 25961}, {3218, 4011}, {3306, 4418}, {3315, 32920}, {3452, 33065}, {3681, 24003}, {3685, 27002}, {3701, 3976}, {3702, 24174}, {3703, 3756}, {3706, 16602}, {3742, 30818}, {3752, 32915}, {3816, 25760}, {3834, 17605}, {3836, 11680}, {3846, 33172}, {3848, 31993}, {3868, 25079}, {3923, 27003}, {3967, 3999}, {3971, 4392}, {3994, 20942}, {3995, 17591}, {3996, 9350}, {4009, 21342}, {4090, 4430}, {4365, 17490}, {4383, 32919}, {4387, 32845}, {4388, 26139}, {4413, 32945}, {4423, 32917}, {4465, 24691}, {4650, 24593}, {4671, 9335}, {4679, 4683}, {4821, 24182}, {4860, 32940}, {4972, 24217}, {5233, 33081}, {5263, 17124}, {5284, 32916}, {5482, 30980}, {5741, 33087}, {6384, 18152}, {11814, 27131}, {16610, 32860}, {16690, 29437}, {17122, 24552}, {17157, 18137}, {17208, 31008}, {17228, 25624}, {17232, 21257}, {17234, 33105}, {17241, 21238}, {17249, 30963}, {17265, 31245}, {17279, 17728}, {17283, 31237}, {17449, 30861}, {17595, 32936}, {17597, 32927}, {17717, 18139}, {17720, 33123}, {17721, 33072}, {17777, 33098}, {18141, 32949}, {18201, 32933}, {20530, 30945}, {20923, 21330}, {22167, 30090}, {23649, 27523}, {24169, 33134}, {24210, 33125}, {24703, 24709}, {24988, 32865}, {25385, 27186}, {26034, 26105}, {30566, 33101}, {30953, 30993}, {30959, 30969}

X(30958) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30958) lies on these lines: {2, 45}, {10, 2140}, {141, 25959}, {142, 4871}, {320, 17028}, {350, 17244}, {1266, 2276}, {3240, 4395}, {3616, 26978}, {3739, 32931}, {3834, 21264}, {3943, 4441}, {4361, 19998}, {4493, 17063}, {7263, 17756}, {17029, 17378}, {17250, 31004}, {17313, 29824}, {20925, 21332}

X(30959) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30959) lies on these lines: {2, 11}, {36, 11355}, {42, 17721}, {43, 24392}, {350, 7179}, {499, 1009}, {1985, 5087}, {2238, 9599}, {2276, 24239}, {2293, 24655}, {2486, 17290}, {3452, 3741}, {3720, 17720}, {3740, 30818}, {3817, 3840}, {3838, 30949}, {7741, 29637}, {10453, 25568}, {14008, 30965}, {14009, 30947}, {16560, 24329}, {17063, 24458}, {17183, 30966}, {17279, 30751}, {17284, 20544}, {17447, 20927}, {17602, 29814}, {17605, 30985}, {20256, 24330}, {20486, 29579}, {21264, 30972}, {24264, 24618}, {25525, 26102}, {30943, 30979}, {30957, 30969}

X(30960) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30960) lies on these lines: {2, 12}, {11, 69}, {42, 32049}, {73, 25681}, {241, 30783}, {329, 20545}, {908, 1469}, {946, 3741}, {1985, 5087}, {3816, 5712}, {3882, 9554}, {4192, 6256}, {5231, 20544}, {5836, 31330}, {5880, 30970}, {7248, 30078}, {8299, 30943}, {10886, 17272}, {10912, 17135}, {14009, 17139}, {17149, 20449}, {20486, 30741}, {20928, 21333}, {22097, 24703}, {25522, 26102}, {30945, 30956}, {30948, 30993}, {30969, 30989}

X(30961) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30961) lies on these lines: {2, 7}, {42, 25568}, {43, 22020}, {75, 21033}, {120, 21015}, {200, 13576}, {241, 30812}, {278, 25941}, {291, 33101}, {312, 561}, {322, 17452}, {350, 4417}, {1211, 2886}, {1429, 24612}, {1848, 1851}, {1985, 5087}, {2140, 16832}, {2238, 3772}, {2276, 4415}, {3661, 21029}, {3687, 4441}, {3705, 17794}, {3720, 5712}, {3740, 26037}, {3741, 3817}, {3771, 4368}, {3783, 3944}, {3869, 16609}, {3912, 18135}, {3930, 20173}, {3975, 30036}, {4193, 26012}, {4204, 32775}, {4511, 24268}, {4517, 20486}, {4766, 18134}, {4873, 22031}, {5274, 10453}, {6996, 25940}, {7146, 26563}, {7175, 24540}, {7201, 24993}, {8167, 19701}, {8299, 24703}, {11374, 16850}, {11681, 16603}, {14555, 24592}, {16833, 17761}, {17135, 24392}, {17182, 30941}, {17220, 27039}, {17272, 24220}, {17793, 30953}, {17861, 21078}, {20337, 27687}, {20978, 24669}, {28809, 29960}, {29966, 30830}, {30974, 30984}

X(30962) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 104

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(30962) lies on these lines: {1, 21281}, {2, 6}, {4, 811}, {7, 350}, {8, 25303}, {37, 24691}, {43, 3879}, {56, 6337}, {65, 18156}, {75, 354}, {142, 17026}, {192, 3726}, {194, 18827}, {274, 10449}, {304, 942}, {320, 4679}, {332, 11358}, {344, 672}, {405, 17206}, {518, 30758}, {742, 26274}, {894, 31028}, {1008, 4340}, {1009, 3926}, {1011, 1444}, {1014, 4203}, {1434, 1975}, {1475, 29966}, {1575, 4851}, {1930, 18398}, {2140, 29748}, {2276, 17316}, {2280, 24602}, {2344, 24609}, {2345, 31027}, {3263, 3873}, {3416, 28600}, {3616, 17137}, {3622, 17152}, {3664, 3840}, {3720, 17321}, {3741, 10436}, {3745, 17394}, {3912, 17754}, {4000, 17027}, {4195, 17103}, {4357, 26102}, {4360, 17597}, {4368, 24695}, {4441, 29824}, {4479, 7321}, {4592, 5398}, {4644, 24514}, {4675, 21264}, {4713, 17365}, {5902, 14210}, {7763, 25650}, {9534, 33297}, {10479, 17175}, {11269, 30969}, {11355, 32815}, {16503, 24586}, {16777, 25349}, {16783, 29473}, {17139, 30943}, {17257, 24690}, {17272, 25502}, {17298, 20335}, {17314, 17759}, {17353, 30822}, {17376, 20530}, {17377, 20012}, {17390, 25350}, {17474, 30036}, {17750, 27248}, {17758, 29455}, {18135, 18148}, {19767, 27162}, {20271, 21216}, {20331, 29583}, {20947, 32937}, {20963, 27299}, {21296, 26103}, {21384, 29968}, {26038, 32099}, {30976, 30992}, {30989, 32949}

X(30962) = anticomplement of X(37673)

X(30963) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30963) lies on these lines: {1, 668}, {2, 37}, {7, 24495}, {10, 17144}, {38, 20711}, {42, 17393}, {43, 4360}, {69, 26069}, {76, 1125}, {85, 11375}, {86, 87}, {141, 31028}, {145, 25280}, {172, 16916}, {183, 1001}, {190, 17754}, {194, 16604}, {239, 16515}, {274, 3624}, {304, 29637}, {310, 16709}, {313, 17725}, {319, 3966}, {320, 4679}, {322, 29839}, {325, 3816}, {334, 17244}, {551, 6381}, {672, 17336}, {693, 4448}, {748, 33295}, {751, 17250}, {799, 9345}, {811, 1982}, {870, 16826}, {889, 24338}, {894, 4713}, {978, 33296}, {980, 25510}, {982, 20598}, {1078, 5248}, {1107, 27269}, {1269, 25501}, {1447, 30545}, {1500, 27091}, {1621, 26232}, {1655, 2275}, {1698, 17143}, {1909, 3616}, {1914, 16997}, {1921, 4485}, {1965, 5287}, {1975, 25524}, {2238, 3759}, {2280, 3570}, {3097, 32035}, {3159, 24166}, {3212, 28389}, {3230, 30114}, {3294, 29438}, {3596, 6685}, {3622, 25303}, {3623, 25278}, {3681, 20723}, {3718, 29635}, {3720, 17149}, {3723, 30473}, {3741, 5224}, {3758, 4465}, {3761, 18145}, {3765, 29586}, {3795, 3993}, {3825, 7752}, {3840, 4357}, {3875, 16569}, {3934, 27255}, {3948, 17397}, {3957, 25297}, {3975, 26626}, {3985, 24631}, {4021, 6686}, {4087, 17592}, {4110, 17319}, {4366, 4386}, {4368, 4676}, {4384, 25427}, {4389, 4871}, {4393, 21904}, {4396, 16998}, {4406, 14437}, {4423, 16992}, {4426, 16918}, {4583, 24427}, {4675, 24510}, {5275, 20179}, {5283, 26959}, {5284, 26238}, {6646, 24691}, {7763, 10200}, {7786, 25092}, {8024, 29666}, {10198, 32832}, {10436, 25502}, {16549, 29440}, {16552, 29750}, {16589, 17030}, {16606, 32033}, {16705, 26094}, {16777, 17786}, {16969, 17752}, {17018, 25287}, {17023, 30830}, {17026, 17277}, {17032, 18040}, {17122, 24260}, {17227, 30945}, {17228, 31027}, {17234, 20335}, {17246, 25350}, {17247, 25349}, {17249, 30957}, {17271, 31137}, {17283, 30822}, {17329, 24690}, {17360, 29824}, {17361, 30941}, {17791, 29830}, {18055, 21808}, {18144, 28639}, {18152, 30596}, {18156, 20955}, {19847, 25599}, {19862, 20888}, {20170, 21857}, {20257, 30063}, {20345, 29569}, {20347, 24494}, {20913, 29612}, {20932, 24943}, {20971, 24745}, {21071, 29991}, {21219, 31999}, {21238, 25624}, {21590, 26128}, {21615, 30571}, {21868, 25109}, {23493, 24662}, {24349, 28600}, {24487, 30950}, {24577, 25666}, {25107, 32095}, {26035, 27148}, {26037, 28593}, {26100, 27097}, {30949, 30997}, {30973, 30985}, {30983, 31001}

X(30964) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30964) lies on these lines: {2, 39}, {6, 799}, {7, 1357}, {42, 17149}, {55, 18064}, {75, 30970}, {89, 32020}, {99, 11322}, {192, 1978}, {334, 29643}, {350, 4389}, {561, 3666}, {668, 3240}, {811, 1013}, {873, 19701}, {899, 6376}, {1150, 30940}, {1920, 28606}, {1921, 4850}, {1965, 17017}, {1966, 4414}, {1975, 16405}, {1985, 30992}, {2296, 3720}, {3112, 17599}, {3760, 29827}, {3761, 29825}, {3821, 30632}, {4033, 32044}, {4495, 17593}, {4871, 24214}, {6382, 17147}, {7018, 32776}, {8033, 19684}, {8620, 32453}, {10009, 17495}, {10453, 17152}, {17137, 29824}, {17139, 30943}, {17144, 31136}, {17592, 18059}, {17594, 18056}, {18052, 18739}, {24487, 30950}, {24503, 24512}, {30631, 33125}, {30945, 30987}, {30954, 30978}, {30967, 33151}

X(30965) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30965) lies on these lines: {2, 6}, {8, 16752}, {38, 4476}, {42, 32846}, {58, 29637}, {239, 21240}, {274, 3661}, {291, 15523}, {310, 321}, {314, 3662}, {350, 17184}, {672, 33157}, {1575, 16700}, {1931, 16050}, {2239, 33085}, {2276, 32858}, {3219, 24358}, {3305, 30822}, {3736, 33087}, {3741, 5208}, {3783, 33081}, {3794, 3840}, {3873, 24325}, {3912, 16887}, {3969, 17759}, {4016, 20932}, {4030, 17018}, {4184, 8299}, {4368, 4683}, {4441, 33146}, {4469, 18157}, {4658, 29633}, {4892, 10129}, {5329, 13588}, {6536, 30571}, {6542, 33296}, {10453, 19785}, {10455, 17298}, {10471, 20913}, {14008, 30959}, {16696, 17231}, {16705, 17316}, {16709, 17228}, {16712, 17310}, {16741, 30967}, {16831, 17210}, {17027, 32774}, {17031, 33123}, {17135, 24643}, {17167, 20335}, {17169, 29611}, {17175, 17308}, {17205, 29594}, {17208, 18601}, {17227, 30939}, {17284, 18206}, {17483, 24330}, {18600, 29616}, {24259, 33067}, {24514, 32859}, {24592, 26724}, {24691, 32777}, {26237, 33124}, {27184, 31008}, {30946, 30973}, {30990, 31001}

X(30966) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30966) lies on these lines: {1, 17210}, {2, 6}, {8, 16705}, {10, 274}, {21, 6626}, {37, 31027}, {38, 75}, {42, 319}, {43, 17270}, {73, 17095}, {76, 10471}, {99, 753}, {110, 9075}, {213, 27274}, {256, 314}, {261, 7305}, {307, 7196}, {315, 1008}, {320, 30969}, {337, 16720}, {388, 1434}, {594, 17759}, {672, 17289}, {749, 16709}, {751, 17250}, {756, 20947}, {894, 24690}, {984, 4476}, {1010, 17103}, {1444, 7224}, {1469, 3786}, {1509, 19856}, {1575, 16696}, {1626, 4184}, {1655, 21024}, {1698, 17175}, {2276, 3661}, {2292, 17762}, {2893, 4199}, {3263, 4981}, {3617, 18600}, {3679, 16712}, {3720, 17322}, {3736, 3775}, {3807, 3862}, {3828, 17180}, {3954, 18167}, {4360, 17135}, {4368, 24697}, {4389, 4441}, {4479, 17249}, {4576, 31117}, {4643, 24514}, {4651, 32025}, {4657, 17027}, {4690, 21904}, {4708, 30967}, {4713, 17253}, {4851, 17032}, {4872, 22097}, {4938, 17360}, {6376, 28660}, {6646, 24330}, {7241, 16748}, {9780, 17169}, {10436, 32913}, {10453, 17321}, {10455, 17272}, {14009, 17139}, {16604, 18172}, {16695, 21305}, {16706, 24592}, {16737, 17072}, {16744, 17448}, {16750, 25006}, {16819, 21240}, {17018, 17377}, {17026, 17306}, {17034, 25499}, {17179, 19875}, {17183, 30959}, {17189, 33138}, {17196, 30992}, {17227, 30949}, {17237, 21264}, {17248, 31028}, {17256, 24712}, {17263, 30821}, {17273, 20347}, {17303, 24691}, {17308, 17754}, {17320, 31136}, {18133, 18152}, {18156, 31359}, {18157, 30758}, {20331, 29591}, {20334, 20458}, {20553, 33083}, {21070, 32026}, {21263, 21763}, {25282, 31111}, {25354, 30571}, {28653, 31341}, {30940, 32784}, {30975, 30988}, {31339, 31997}

X(30966) = isotomic conjugate of isogonal conjugate of X(3736)

X(30967) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30967) lies on these lines: {1, 2}, {86, 4396}, {244, 3797}, {291, 4439}, {320, 4465}, {335, 4358}, {350, 1086}, {672, 4473}, {3218, 6651}, {3726, 20947}, {3834, 30997}, {4379, 7192}, {4366, 24602}, {4392, 27481}, {4432, 24628}, {4644, 24514}, {4708, 30966}, {16741, 30965}, {17227, 30945}, {17242, 17756}, {17264, 20331}, {18139, 30955}, {20132, 32944}, {20142, 32919}, {21025, 25303}, {26738, 31006}, {27487, 27918}, {30949, 30998}, {30964, 33151}, {30985, 31000}, {30991, 32129}, {31041, 31058}

X(30968) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30968) lies on these lines: {2, 667}, {310, 23807}, {513, 30969}, {890, 31176}, {3835, 30999}, {3837, 4379}, {4083, 31330}, {20952, 21350}, {24719, 30970}, {25126, 31207}, {26148, 27293}

X(30969) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30969) lies on these lines: {2, 31}, {42, 32850}, {57, 30752}, {87, 30052}, {244, 2227}, {291, 3006}, {310, 21415}, {320, 30966}, {350, 3120}, {513, 30968}, {518, 31330}, {672, 33115}, {851, 8299}, {1740, 29981}, {2108, 29862}, {2238, 32843}, {2276, 29643}, {3454, 19856}, {3720, 19786}, {3741, 5208}, {3783, 3936}, {3789, 33065}, {3823, 26037}, {3834, 21264}, {3912, 23682}, {4368, 5057}, {4475, 20924}, {4871, 30993}, {4892, 17793}, {7184, 30034}, {11269, 30962}, {17018, 17765}, {17027, 33128}, {17031, 33129}, {17135, 32923}, {17140, 31119}, {17754, 29857}, {17759, 32848}, {17789, 20590}, {17794, 32856}, {19791, 32915}, {20292, 24259}, {24512, 29631}, {24514, 24725}, {26237, 33130}, {30751, 33119}, {30941, 32919}, {30950, 30987}, {30957, 30959}, {30960, 30989}

X(30970) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30970) lies on these lines: {1, 2}, {11, 1213}, {31, 5737}, {37, 4519}, {38, 31993}, {75, 30964}, {86, 32919}, {210, 31264}, {238, 5235}, {244, 3728}, {320, 30966}, {321, 3989}, {333, 2308}, {573, 10886}, {594, 32848}, {672, 17369}, {726, 31025}, {748, 19732}, {756, 4009}, {902, 5263}, {958, 16405}, {993, 11322}, {1001, 16355}, {1011, 23361}, {1086, 32044}, {1107, 2229}, {1211, 33105}, {1215, 4981}, {1268, 31002}, {1468, 2049}, {1491, 4379}, {1654, 32843}, {1738, 21027}, {1962, 3706}, {1985, 2183}, {2209, 17125}, {2228, 21264}, {2260, 17303}, {2309, 27164}, {2345, 33161}, {3120, 4357}, {3218, 24342}, {3666, 21020}, {3763, 3779}, {3775, 3936}, {3842, 4358}, {3844, 21026}, {3925, 32781}, {4003, 4688}, {4026, 33136}, {4038, 5333}, {4192, 18480}, {4359, 6682}, {4365, 28606}, {4413, 20470}, {4423, 19744}, {4465, 4708}, {4643, 24725}, {4657, 33128}, {4970, 17163}, {5057, 14009}, {5224, 25760}, {5278, 25496}, {5303, 13588}, {5587, 19647}, {5880, 30960}, {6536, 24210}, {9345, 15668}, {9956, 19546}, {11339, 20172}, {16739, 21415}, {17056, 33081}, {17210, 20888}, {17275, 17723}, {17277, 32944}, {17289, 33115}, {17449, 24325}, {17495, 27812}, {19684, 32853}, {19808, 33119}, {21806, 28581}, {24719, 30968}, {25385, 26580}, {25637, 30835}, {26893, 31245}, {27773, 29350}, {32782, 33111}, {32784, 33108}, {33082, 33112}, {33083, 33109}

X(30971) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30971) lies on these lines: {2, 3}, {244, 28113}, {962, 26013}, {2308, 5274}, {4303, 26102}, {10590, 29640}, {10591, 33140}, {30942, 30946}, {30947, 30985}

X(30972) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30972) lies on these lines: {2, 3}, {305, 4602}, {3741, 21621}, {3840, 30956}, {21264, 30959}

X(30973) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30973) lies on these lines: {2, 3}, {286, 18589}, {306, 668}, {1959, 6508}, {4872, 22097}, {18651, 31008}, {26871, 30941}, {30946, 30965}, {30963, 30985}

X(30974) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30974) lies on these lines: {2, 3}, {333, 21015}, {1978, 20336}, {8822, 26933}, {30942, 30983}, {30961, 30984}

X(30975) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30975) lies on these lines: {2, 3}, {307, 314}, {20220, 31330}, {30966, 30988}

X(30976) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30976) lies on these lines: {2, 3}, {26013, 31162}, {30962, 30992}

X(30977) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30977) lies on these lines: {2, 3}, {238, 11269}, {320, 4679}, {960, 10453}, {5250, 26013}, {21616, 26102}, {25504, 25531}, {30950, 30985}

X(30978) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30978) lies on these lines: {2, 3}, {3662, 30956}, {30945, 31000}, {30954, 30964}, {30955, 30987}

X(30979) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30979) lies on these lines: {2, 12}, {42, 11260}, {896, 15254}, {1009, 29827}, {1284, 24627}, {1357, 29382}, {3741, 6684}, {3756, 16684}, {3840, 8731}, {4192, 5450}, {4447, 29639}, {8299, 30942}, {14829, 21321}, {17277, 28239}, {26102, 31435}, {30943, 30959}

X(30980) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30980) lies on these lines: {2, 35}, {5, 29632}, {11, 2330}, {36, 11330}, {1985, 5087}, {3136, 29851}, {3814, 33175}, {3840, 14008}, {5044, 25591}, {5482, 30957}, {7951, 29830}, {11680, 17123}, {14009, 30950}, {20919, 21325}, {24387, 33139}

X(30981) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30981) lies on these lines: {2, 36}, {11, 524}, {42, 5176}, {350, 14616}, {513, 30968}, {517, 31330}, {1985, 5087}, {3741, 11813}, {5057, 14009}, {5123, 26037}, {5137, 29631}, {5172, 16343}, {5520, 30996}, {20335, 30994}, {20920, 21326}, {25639, 26064}

X(30982) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics a^3 b^2 + 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(30982) lies on these lines: {1, 20917}, {2, 38}, {10, 20358}, {31, 26238}, {43, 32920}, {55, 24260}, {86, 87}, {141, 674}, {226, 1401}, {256, 26149}, {350, 1920}, {872, 29484}, {875, 4369}, {1086, 24327}, {1111, 24255}, {1575, 21101}, {1582, 18048}, {1964, 18143}, {2108, 32939}, {2239, 26237}, {2667, 29764}, {3210, 3795}, {3485, 7146}, {3739, 25120}, {3742, 20530}, {3791, 17031}, {3912, 12263}, {4011, 4713}, {4038, 20132}, {4657, 25124}, {4703, 30946}, {4865, 10453}, {4871, 25557}, {4892, 10129}, {8299, 24259}, {17000, 17123}, {17026, 32853}, {17149, 18067}, {17278, 25106}, {17297, 31137}, {17300, 21299}, {17398, 25501}, {17445, 18040}, {18140, 19567}, {18208, 29637}, {21330, 26971}, {24789, 25123}, {25277, 27192}, {25295, 27011}, {28595, 31330}

X(30983) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30983) lies on these lines: {2, 48}, {73, 24664}, {2276, 21091}, {3741, 18589}, {21231, 31330}, {22065, 29962}, {22066, 30032}, {30942, 30974}, {30945, 30956}, {30949, 30994}, {30963, 31001}

X(30984) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30984) lies on these lines: {2, 58}, {21, 29632}, {42, 7270}, {81, 2887}, {86, 25760}, {314, 3120}, {320, 30966}, {333, 25957}, {1985, 30945}, {3286, 30811}, {3736, 3936}, {3741, 11263}, {3771, 4184}, {3786, 33065}, {3821, 25060}, {3836, 5235}, {3840, 14008}, {3846, 5333}, {3868, 31330}, {4137, 20929}, {4138, 17167}, {4202, 4281}, {4278, 25645}, {4653, 29830}, {4658, 29829}, {4892, 10129}, {5208, 33069}, {8025, 25958}, {10458, 18134}, {11269, 30941}, {13588, 29846}, {16704, 25959}, {16887, 29639}, {18206, 29857}, {25058, 32776}, {25059, 33125}, {25507, 25960}, {27644, 32843}, {30599, 32778}, {30961, 30974}

X(30985) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics -a^3 b^2 + a b^4 - a^3 b c - a^2 b^2 c + a b^3 c + b^4 c - a^3 c^2 - a^2 b c^2 - 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(30985) lies on these lines: {1, 20556}, {2, 7}, {42, 19785}, {43, 23681}, {75, 3949}, {85, 1959}, {92, 18031}, {291, 33103}, {306, 4441}, {321, 20435}, {322, 17868}, {350, 18134}, {379, 20769}, {978, 16752}, {1215, 26037}, {1444, 25523}, {1836, 8299}, {1953, 20930}, {2140, 4384}, {2238, 24789}, {2239, 33127}, {2269, 27267}, {2276, 3782}, {2887, 3775}, {3720, 26098}, {3741, 4138}, {3760, 3912}, {3765, 30036}, {3771, 24259}, {3783, 17889}, {3789, 3925}, {3870, 13576}, {3948, 29966}, {4071, 32858}, {4368, 29642}, {4892, 10129}, {5333, 25496}, {5739, 24592}, {6327, 26237}, {6384, 27460}, {6821, 27391}, {7176, 24559}, {7201, 26665}, {16831, 17758}, {16834, 17761}, {17027, 17778}, {17031, 32946}, {17167, 30941}, {17182, 17194}, {17220, 22370}, {17298, 24220}, {17605, 30959}, {17789, 18055}, {17794, 29641}, {18152, 21590}, {19804, 27489}, {20498, 26038}, {20913, 21405}, {21101, 28605}, {24330, 32777}, {24586, 26243}, {28809, 29988}, {30947, 30971}, {30950, 30977}, {30963, 30973}, {30967, 31000}

X(30986) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30986) lies on these lines: {2, 65}, {5, 3741}, {43, 5730}, {72, 33140}, {244, 28366}, {320, 4679}, {392, 29640}, {405, 19715}, {518, 11269}, {896, 15254}, {1329, 26013}, {1402, 29472}, {1837, 10453}, {1985, 5087}, {3819, 4138}, {3840, 14058}, {4388, 20359}, {4417, 21334}, {4517, 30741}, {4640, 30944}, {6675, 25501}, {8167, 19727}, {14009, 17605}, {17556, 31137}, {17792, 25760}, {24703, 30943}, {25135, 26893}

X(30987) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30987) lies on these lines: {2, 32}, {141, 799}, {17456, 20934}, {30945, 30964}, {30950, 30969}, {30955, 30978}

X(30988) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30988) lies on these lines: {2, 85}, {9, 4554}, {55, 20935}, {63, 30545}, {75, 2652}, {320, 4679}, {415, 6626}, {799, 27958}, {2053, 6384}, {2481, 5231}, {3673, 33140}, {4872, 30943}, {5205, 6376}, {5744, 10030}, {5745, 6063}, {7209, 27433}, {17206, 31008}, {17336, 21580}, {18031, 30608}, {26013, 33298}, {30966, 30975}

X(30989) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30989) lies on these lines: {2, 87}, {10, 21219}, {38, 75}, {42, 25311}, {2276, 20532}, {3223, 24661}, {3720, 26143}, {3741, 17236}, {20530, 30945}, {20936, 21337}, {20943, 25122}, {25121, 26037}, {25350, 31337}, {30942, 30998}, {30960, 30969}, {30962, 32949}

X(30990) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30990) lies on these lines: {2, 45}, {10, 3799}, {908, 4871}, {1921, 1978}, {3699, 19998}, {17244, 18140}, {17465, 20937}, {17793, 30942}, {26738, 32129}, {30945, 30991}, {30965, 31001}

X(30991) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30991) lies on these lines: {2, 44}, {10, 3901}, {76, 1978}, {88, 27739}, {145, 5015}, {1150, 25529}, {1266, 33077}, {1330, 3616}, {2887, 4661}, {2895, 26132}, {3936, 4389}, {3943, 33151}, {4430, 25958}, {4432, 24710}, {4604, 25694}, {4683, 29866}, {4703, 29870}, {5233, 24183}, {17127, 29860}, {17244, 26580}, {17273, 30834}, {17274, 27757}, {17305, 31179}, {17740, 20092}, {17778, 25417}, {20073, 32849}, {20938, 21338}, {24344, 24692}, {24620, 26758}, {29579, 30578}, {29593, 31030}, {29834, 32946}, {30945, 30990}, {30967, 32129}

X(30992) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30992) lies on these lines: {2, 99}, {11, 799}, {350, 14616}, {873, 3816}, {1647, 18827}, {1985, 30964}, {2611, 20939}, {4610, 25533}, {4615, 19634}, {14008, 31008}, {17196, 30966}, {30954, 31000}, {30955, 30978}, {30962, 30976}, {30993, 31002}, {30994, 30997}

X(30993) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30993) lies on these lines: {2, 11}, {291, 1647}, {692, 29490}, {693, 18031}, {1985, 30947}, {2486, 27191}, {3573, 24618}, {3699, 17135}, {3720, 17719}, {3741, 11814}, {3794, 3840}, {3825, 29637}, {3837, 4800}, {4358, 4518}, {4368, 24709}, {4499, 24237}, {4871, 30969}, {4928, 30994}, {4997, 29824}, {10129, 30949}, {14009, 30950}, {15626, 19540}, {16850, 26127}, {17266, 20544}, {17463, 18151}, {17724, 29814}, {17793, 30942}, {17794, 30566}, {21320, 27290}, {24003, 31330}, {26102, 31266}, {30948, 30960}, {30953, 30957}, {30992, 31002}, {30995, 31001}

X(30994) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30994) lies on these lines: {2, 101}, {4928, 30993}, {20335, 30981}, {20940, 21339}, {21232, 31330}, {29481, 32739}, {30942, 30956}, {30949, 30983}, {30992, 30997}

X(30995) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30995) lies on these lines: {2, 98}, {662, 21252}, {3817, 14008}, {20941, 21340}, {30993, 31001}

X(30996) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30996) lies on these lines: {2, 99}, {799, 21341}, {3720, 4128}, {5520, 30981}, {5539, 26102}, {30943, 30954}, {30944, 30955}

X(30997) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30997) lies on these lines: {2, 45}, {75, 3807}, {99, 25532}, {100, 24447}, {101, 29479}, {244, 24413}, {291, 24193}, {320, 24712}, {334, 350}, {335, 27918}, {514, 20568}, {524, 17029}, {668, 17761}, {673, 3570}, {1111, 18061}, {1320, 6631}, {2140, 18140}, {2170, 18159}, {2238, 29590}, {3673, 18055}, {3834, 30967}, {3837, 4800}, {3840, 23821}, {4358, 27487}, {4398, 17756}, {4499, 24494}, {4643, 17028}, {4728, 31002}, {6386, 18152}, {10707, 17297}, {17205, 27195}, {17227, 24688}, {17237, 21264}, {17793, 25351}, {18047, 24203}, {18145, 30109}, {20257, 25280}, {20333, 29576}, {20943, 30036}, {24709, 30950}, {25378, 31005}, {30945, 30998}, {30949, 30963}, {30992, 30994}

X(30998) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30998) lies on these lines: {2, 37}, {8, 17793}, {11, 3314}, {39, 32107}, {56, 17128}, {76, 330}, {183, 4366}, {194, 3760}, {310, 16710}, {497, 16990}, {499, 7836}, {1479, 2896}, {1573, 18146}, {1909, 31999}, {2275, 20081}, {2309, 25528}, {3056, 3620}, {3551, 3662}, {3583, 7898}, {3720, 24661}, {3741, 17238}, {3934, 32095}, {4384, 30863}, {4393, 31026}, {4396, 7766}, {4398, 25350}, {4423, 16994}, {4713, 17350}, {4741, 24712}, {4876, 29579}, {5433, 7891}, {6284, 7904}, {6685, 17396}, {7741, 7912}, {7768, 9665}, {7779, 9599}, {7831, 9664}, {7879, 9669}, {7885, 10896}, {9331, 27020}, {9596, 33020}, {9598, 33021}, {10453, 17373}, {14823, 23485}, {16502, 17129}, {16569, 17117}, {16722, 28660}, {16738, 31008}, {16816, 17475}, {16975, 18145}, {16986, 26590}, {16995, 20179}, {16999, 20172}, {17004, 26629}, {17026, 17349}, {17030, 27269}, {17230, 20532}, {17232, 20335}, {17300, 21299}, {17448, 20943}, {17499, 29748}, {18135, 26801}, {20494, 29674}, {20888, 27318}, {21223, 26815}, {25121, 31330}, {27430, 30038}, {30942, 30989}, {30945, 30997}, {30947, 31005}, {30949, 30967}

X(30999) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(30999) lies on these lines: {2, 187}, {1985, 30945}, {3835, 30968}, {20944, 21344}

X(31000) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31000) lies on these lines: {2, 39}, {385, 11339}, {14996, 23660}, {16355, 16994}, {16405, 17128}, {17149, 17448}, {20945, 21345}, {21224, 30638}, {30942, 30989}, {30945, 30978}, {30954, 30992}, {30967, 30985}

X(31001) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31001) lies on these lines: {2, 662}, {99, 116}, {3708, 20951}, {6626, 17046}, {17227, 30956}, {17256, 24712}, {20337, 26019}, {21254, 31330}, {30963, 30983}, {30965, 30990}, {30992, 30994}, {30993, 30995}

X(31002) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31002) lies on these lines: {2, 668}, {7, 1357}, {27, 811}, {42, 32011}, {75, 244}, {86, 799}, {88, 874}, {310, 3840}, {334, 1647}, {335, 4358}, {350, 889}, {673, 4607}, {675, 898}, {739, 789}, {1268, 30970}, {2296, 26102}, {3766, 6548}, {4373, 30948}, {4671, 27494}, {4728, 30997}, {4871, 6381}, {6382, 9335}, {6384, 18152}, {7035, 24841}, {21297, 32020}, {24589, 27483}, {27475, 30829}, {30992, 30993}

X(31002) = isotomic conjugate of X(899)

X(31003) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31003) lies on these lines: {2, 669}, {512, 30864}, {663, 15283}, {3835, 30968}, {3837, 26985}, {4367, 26983}, {4379, 23818}, {4774, 21051}, {9148, 24622}, {17414, 25258}, {20953, 21351}, {20983, 25128}, {21191, 23464}, {21726, 32778}

X(31004) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31004) lies on these lines: {2, 7}, {10, 17794}, {42, 17302}, {43, 17304}, {69, 17027}, {141, 350}, {257, 18031}, {320, 24512}, {335, 26234}, {1575, 17235}, {1654, 24592}, {1655, 29960}, {1698, 24190}, {2140, 16819}, {2238, 16706}, {2239, 32775}, {2275, 28397}, {2276, 4389}, {3314, 25760}, {3620, 10453}, {3661, 4441}, {3663, 17759}, {3720, 4388}, {3727, 20955}, {3763, 4713}, {3783, 3821}, {3789, 4429}, {3952, 31077}, {4368, 17192}, {4479, 17228}, {4651, 17117}, {4699, 26037}, {8299, 24723}, {16738, 17176}, {16818, 17499}, {16827, 26978}, {16829, 17761}, {16887, 19579}, {16916, 24549}, {16990, 26034}, {16997, 24586}, {16999, 24602}, {17018, 17396}, {17026, 17272}, {17031, 33082}, {17032, 17321}, {17046, 17669}, {17135, 17287}, {17137, 26100}, {17142, 17238}, {17152, 17752}, {17169, 27148}, {17202, 17288}, {17227, 30945}, {17237, 21264}, {17250, 30958}, {17273, 24690}, {17289, 24330}, {17298, 26102}, {17382, 21904}, {17391, 29814}, {17758, 31996}, {18140, 21240}, {18152, 20891}, {21226, 30038}, {23812, 25501}, {24214, 29991}, {24259, 32783}, {26101, 33029}, {26237, 33083}, {27035, 27095}, {27269, 29966}, {30942, 30989}

X(31005) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31005) lies on these lines: {2, 38}, {43, 32923}, {171, 26238}, {1621, 24260}, {2667, 29802}, {3741, 25957}, {3795, 17495}, {3834, 21264}, {3840, 5249}, {3844, 31330}, {10453, 33072}, {12263, 17244}, {17026, 32919}, {17383, 25124}, {20335, 25760}, {20917, 21352}, {24487, 30950}, {25295, 27192}, {25378, 30997}, {26102, 32772}, {30947, 30998}

X(31006) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 104

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

X(31006) lies on these lines: {1, 30969}, {2, 6}, {291, 29643}, {350, 31019}, {1008, 26131}, {1009, 19769}, {1985, 30947}, {3720, 26128}, {4368, 24725}, {5283, 17244}, {8299, 11322}, {11358, 29839}, {17027, 33129}, {26035, 29579}, {26234, 27476}, {26738, 30967}, {27491, 30758}, {29572, 31036}, {30588, 30942}

X(31007) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics -a^9 b^3 + 2 a^7 b^5 - 2 a^3 b^9 + a b^11 - a^10 b c + a^8 b^3 c + 2 a^6 b^5 c - 2 a^4 b^7 c - a^2 b^9 c + b^11 c + 2 a^7 b^3 c^2 - 2 a b^9 c^2 - a^9 c^3 + a^8 b c^3 + 2 a^7 b^2 c^3 + 4 a^6 b^3 c^3 - 2 a^5 b^4 c^3 - 2 a^4 b^5 c^3 + 2 a^3 b^6 c^3 - a b^8 c^3 - 3 b^9 c^3 - 2 a^5 b^3 c^4 + 2 a^7 c^5 + 2 a^6 b c^5 - 2 a^4 b^3 c^5 + 2 a^2 b^5 c^5 + 2 a b^6 c^5 + 2 b^7 c^5 + 2 a^3 b^3 c^6 + 2 a b^5 c^6 - 2 a^4 b c^7 + 2 b^5 c^7 - a b^3 c^8 - 2 a^3 c^9 - a^2 b c^9 - 2 a b^2 c^9 - 3 b^3 c^9 + a c^11 + b c^11 : :

X(31007) lies on these lines: {2, 3}

X(31008) = (name pending)

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

X(31008) lies on these lines: {1, 1965}, {2, 39}, {10, 31341}, {21, 18299}, {37, 1221}, {42, 668}, {43, 6376}, {75, 17038}, {81, 799}, {86, 87}, {99, 9082}, {192, 6382}, {226, 4554}, {256, 314}, {312, 16739}, {333, 17026}, {334, 29653}, {561, 28606}, {811, 4183}, {846, 1966}, {870, 29644}, {873, 5333}, {968, 18056}, {1018, 29390}, {1423, 30545}, {1621, 18064}, {1921, 3666}, {1962, 18059}, {1975, 11358}, {1978, 3995}, {3210, 10009}, {3403, 18078}, {3736, 7168}, {3760, 10471}, {3794, 9309}, {3821, 30631}, {3840, 16887}, {3971, 23824}, {4358, 16703}, {4595, 29707}, {4685, 25280}, {5277, 16956}, {6381, 6685}, {7304, 27644}, {9791, 30660}, {10453, 33297}, {14008, 30992}, {16345, 16992}, {16552, 29557}, {16606, 16744}, {16696, 20530}, {16708, 30829}, {16738, 30998}, {17143, 31330}, {17169, 26103}, {17175, 25502}, {17196, 30939}, {17206, 30988}, {17208, 30957}, {17790, 25349}, {18036, 31623}, {18052, 18136}, {18157, 18743}, {18651, 30973}, {19565, 21827}, {20943, 29825}, {21226, 22199}, {21264, 27164}, {21877, 25264}, {23533, 30667}, {27184, 30965}, {29758, 29767}, {30632, 32776}

X(31008) = isogonal conjugate of X(21759)
X(31008) = isotomic conjugate of X(16606)
X(31008) = X(19)-isoconjugate of X(22381)

X(31009) = (name pending)

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

X(31009) lies on these lines: {86, 4977}, {514, 5195}, {4382, 7265}, {4813, 22037}

X(31010) = (name pending)

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

X(31010) lies on these lines: {10, 6367}, {514, 5134}, {523, 6541}, {1016, 4115}, {2690, 8701}, {2901, 29051}, {4382, 4813}, {4049, 6539}, {4064, 28155}, {17484, 22037}, {7649, 17734}

X(31011) = (name pending)

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

X(31011) lies on these lines: {1, 6538}, {2, 594}, {145, 996}, {319, 3995}, {346, 28615}, {514, 4382}, {941, 30590}, {1000, 3621}, {1016, 20016}, {2726, 8701}, {3943, 16704}, {4358, 4727}, {6541, 17162}, {7227, 8025}, {8046, 30579}, {17299, 31035}, {17309, 31017}, {17315, 31025}, {19741, 29588}, {20017, 31018}, {20569, 30589}, {31010, 31290}, {27797, 32014}

X(31012) = (name pending)

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

X(31012) lies on these lines: {514, 4024}, {519, 6540}, {1268, 3634}, {4632, 6629}, {4813, 23731}

X(31013) = (name pending)

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

X(31013) lies on these lines: {10, 86}, {306, 1171}, {514, 4382}, {1255, 29574}, {1909, 32018}, {2729, 8701}, {3671, 25719}, {4024, 23731}, {4062, 6629}, {4115, 20536}, {4938, 14210}, {6539, 29615}, {16826, 27081}, {24051, 24090}

Collineation mappings involving Gemini triangle 105: X(31014)-X(31063)

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

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(514)X(4024). Among the fixed points are X(i) for these i: 2, 514, 4024, 4608, 6542, 31009, 31010, 31011, 31012, 31013, 31064. (Clark Kimberling, January 21, 2019)

X(31014) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31014) lies on these lines: {2, 3}, {48, 20289}, {264, 23978}, {321, 21579}, {1901, 5740}, {3661, 27131}, {3817, 25935}, {3995, 22018}, {4417, 29616}, {5011, 29610}, {8555, 17019}, {14543, 24682}, {16713, 18747}, {17220, 20305}, {17316, 31058}, {19925, 26006}, {21011, 21271}, {21403, 28605}, {31017, 31024}, {31036, 31057}

X(31015) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31015) lies on these lines: {2, 3}, {19, 20291}, {63, 21285}, {69, 144}, {71, 21270}, {219, 20074}, {226, 3188}, {279, 1255}, {306, 12527}, {321, 20926}, {390, 5800}, {516, 25935}, {980, 26099}, {1448, 5287}, {1726, 3219}, {1901, 5736}, {3007, 31158}, {3285, 30902}, {3951, 17294}, {3970, 3995}, {4259, 10394}, {4297, 26006}, {4872, 24635}, {5074, 5088}, {5739, 26770}, {6542, 25270}, {7247, 28606}, {7713, 26169}, {8804, 18650}, {14543, 24683}, {14963, 31034}, {17134, 18589}, {17220, 26130}, {17864, 28605}, {18747, 27507}, {20289, 26063}, {21286, 22370}, {31020, 31032}, {31024, 31059}, {31031, 31039}

X(31015) = anticomplement of X(379)

X(31016) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31016) lies on these lines: {2, 3}, {2140, 29612}, {3661, 5773}, {4256, 5222}, {5228, 29624}, {10164, 25935}, {14543, 24684}, {14829, 29616}, {16826, 20367}, {17220, 25523}, {17221, 21231}, {26243, 26770}, {26790, 29592}, {31026, 31059}

X(31017) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31017) lies on these lines: {2, 6}, {8, 24159}, {31, 20290}, {76, 1978}, {145, 16062}, {306, 3663}, {319, 33129}, {320, 32779}, {321, 17229}, {495, 3617}, {518, 31079}, {908, 20248}, {986, 27558}, {1330, 11319}, {2887, 17135}, {3187, 25527}, {3218, 17288}, {3416, 20045}, {3661, 31019}, {3662, 17495}, {3703, 20068}, {3771, 33080}, {3773, 32856}, {3782, 3969}, {3821, 4062}, {3834, 24589}, {3879, 29833}, {3912, 26580}, {3923, 17491}, {3938, 28599}, {3952, 29674}, {3963, 28605}, {3995, 17242}, {4001, 20106}, {4023, 24988}, {4054, 29594}, {4358, 17231}, {4388, 33173}, {4427, 4655}, {4429, 19998}, {4645, 33175}, {4651, 25957}, {4683, 33158}, {4684, 29835}, {4720, 17678}, {4850, 17227}, {4886, 26724}, {4967, 5249}, {4972, 20011}, {5847, 26230}, {6327, 33171}, {6542, 24281}, {6646, 32849}, {7865, 24296}, {10453, 25958}, {15523, 17165}, {16347, 25650}, {17012, 17291}, {17013, 17383}, {17021, 17312}, {17025, 26150}, {17140, 32778}, {17145, 33120}, {17150, 26128}, {17154, 33089}, {17162, 33128}, {17163, 17889}, {17170, 29579}, {17233, 33151}, {17249, 28606}, {17280, 17484}, {17286, 31164}, {17298, 26627}, {17309, 31011}, {18066, 20955}, {19333, 24936}, {19785, 20017}, {21241, 31136}, {21282, 31134}, {24217, 25760}, {24627, 27757}, {24943, 32946}, {26840, 33168}, {27030, 27263}, {27072, 27096}, {27804, 32776}, {29632, 33082}, {29637, 32843}, {29822, 32784}, {29823, 33070}, {29839, 33083}, {29846, 33085}, {30861, 31056}, {31014, 31024}, {31023, 31027}, {31028, 31055}, {31032, 31033}, {31050, 31060}, {31237, 32853}, {32775, 32846}, {32777, 32859}, {32783, 32949}, {32861, 33123}, {33066, 33157}, {33067, 33160}, {33075, 33124}, {33078, 33126}

X(31018) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31018) lies on these lines: {2, 7}, {3, 13257}, {4, 3876}, {5, 15650}, {8, 80}, {10, 6871}, {11, 5220}, {20, 5720}, {37, 27491}, {44, 17720}, {45, 5718}, {69, 4358}, {72, 2478}, {78, 4304}, {100, 5698}, {119, 5657}, {145, 5815}, {190, 5233}, {191, 10940}, {200, 20075}, {210, 3434}, {220, 5723}, {238, 17725}, {306, 30568}, {312, 319}, {320, 30829}, {321, 14555}, {330, 27461}, {344, 3936}, {345, 5741}, {346, 33077}, {377, 5044}, {390, 3935}, {405, 5719}, {497, 3681}, {517, 6957}, {518, 4679}, {528, 3711}, {651, 15066}, {748, 33144}, {750, 24695}, {756, 26098}, {899, 24248}, {912, 6947}, {936, 4190}, {938, 18397}, {940, 7277}, {950, 3984}, {956, 1387}, {958, 15950}, {960, 3436}, {962, 3617}, {984, 17722}, {997, 21578}, {1000, 3421}, {1150, 28808}, {1210, 3951}, {1211, 17293}, {1265, 5016}, {1278, 27492}, {1317, 5289}, {1323, 25930}, {1478, 10176}, {1621, 25568}, {1698, 11552}, {1699, 25006}, {1757, 11269}, {1788, 11684}, {1836, 3740}, {1851, 26911}, {2550, 5057}, {2551, 3869}, {2886, 3715}, {2895, 8055}, {2899, 17751}, {2911, 32911}, {3006, 27549}, {3083, 31538}, {3084, 31539}, {3091, 5758}, {3161, 32849}, {3241, 4867}, {3339, 25011}, {3416, 4009}, {3475, 5284}, {3485, 5260}, {3487, 5047}, {3522, 6223}, {3553, 17011}, {3616, 5251}, {3621, 5727}, {3661, 5179}, {3672, 17012}, {3697, 12699}, {3717, 31091}, {3786, 14956}, {3841, 4295}, {3846, 33163}, {3868, 5084}, {3870, 21060}, {3873, 26105}, {3916, 6921}, {3927, 4187}, {3940, 11113}, {3945, 17021}, {3966, 3967}, {3971, 33088}, {3974, 33075}, {3977, 25728}, {3995, 21078}, {4000, 33151}, {4001, 30567}, {4011, 33171}, {4021, 4656}, {4023, 5695}, {4054, 4384}, {4096, 4865}, {4189, 27383}, {4197, 5714}, {4294, 4420}, {4307, 5297}, {4310, 7292}, {4315, 12527}, {4329, 17280}, {4363, 5241}, {4370, 27739}, {4383, 4415}, {4388, 10327}, {4413, 17768}, {4417, 17776}, {4419, 4850}, {4422, 30811}, {4430, 10580}, {4440, 24620}, {4473, 31056}, {4643, 30818}, {4652, 6700}, {4655, 24003}, {4662, 12701}, {4699, 27471}, {4703, 26034}, {4741, 30861}, {4756, 33089}, {4860, 5852}, {5087, 15481}, {5178, 5225}, {5180, 31160}, {5187, 6734}, {5211, 31302}, {5222, 5526}, {5223, 26015}, {5234, 24541}, {5235, 17139}, {5250, 10528}, {5290, 24564}, {5302, 11375}, {5372, 6557}, {5423, 33091}, {5443, 31458}, {5552, 12514}, {5658, 7411}, {5660, 9809}, {5703, 16865}, {5708, 17575}, {5709, 6953}, {5726, 24987}, {5731, 6326}, {5743, 7227}, {5761, 6920}, {5768, 12691}, {5770, 6963}, {5777, 6836}, {5778, 19645}, {5791, 6933}, {5795, 11682}, {5812, 6835}, {5817, 10883}, {5818, 7548}, {5927, 10431}, {6147, 16842}, {6361, 18491}, {6701, 14450}, {6763, 10200}, {6818, 20242}, {6825, 26878}, {6834, 26921}, {6865, 12528}, {6890, 7330}, {6898, 24474}, {6910, 31445}, {6928, 31835}, {6967, 24467}, {6992, 18446}, {7079, 30687}, {8165, 25005}, {9330, 33112}, {9596, 21879}, {9812, 33110}, {9963, 11114}, {10157, 18482}, {10446, 17331}, {10527, 21616}, {10587, 31435}, {10805, 31838}, {10941, 21077}, {11068, 31992}, {12526, 24982}, {12541, 20052}, {15254, 17718}, {16020, 33148}, {16058, 21319}, {16569, 33099}, {16610, 17276}, {16704, 17183}, {16713, 17174}, {16788, 26626}, {16815, 17753}, {16825, 21093}, {16862, 24470}, {17123, 33101}, {17125, 32856}, {17170, 29579}, {17316, 31034}, {17334, 17595}, {17336, 32851}, {17354, 30832}, {17358, 17481}, {17361, 18743}, {17367, 19823}, {18141, 32859}, {18250, 19860}, {18747, 32782}, {19544, 26867}, {19740, 31039}, {20017, 31011}, {20073, 30680}, {20095, 30332}, {20236, 28605}, {20292, 26040}, {22129, 25934}, {24929, 31156}, {25960, 32938}, {27049, 27096}, {27081, 31032}, {27174, 27398}, {27385, 31424}, {27484, 27493}, {31037, 31045}

X(31019) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31019) lies on these lines: {1, 149}, {2, 7}, {4, 18444}, {5, 9964}, {6, 33129}, {8, 12559}, {10, 3901}, {11, 10129}, {12, 25005}, {31, 29681}, {37, 27127}, {38, 29664}, {42, 17889}, {55, 20292}, {72, 3824}, {75, 3936}, {76, 18054}, {79, 5248}, {81, 3772}, {85, 18359}, {86, 17174}, {89, 4896}, {92, 445}, {100, 5880}, {104, 5886}, {145, 21620}, {171, 29665}, {238, 24725}, {239, 31034}, {244, 17717}, {306, 4431}, {312, 17241}, {320, 1150}, {321, 17233}, {333, 32859}, {350, 31006}, {354, 3838}, {377, 3487}, {404, 11374}, {442, 3868}, {484, 10197}, {495, 1145}, {497, 29817}, {518, 33108}, {545, 27754}, {551, 2320}, {612, 33153}, {614, 33107}, {675, 825}, {726, 29643}, {748, 33096}, {750, 17719}, {756, 33101}, {846, 29661}, {858, 25365}, {902, 29675}, {912, 6829}, {938, 6871}, {940, 33133}, {942, 2476}, {946, 3146}, {954, 1004}, {962, 10587}, {968, 33100}, {971, 10883}, {982, 29680}, {984, 32856}, {993, 26725}, {1001, 5057}, {1071, 6828}, {1086, 4850}, {1125, 16865}, {1210, 5141}, {1211, 28651}, {1215, 25957}, {1434, 27187}, {1444, 5333}, {1468, 24161}, {1490, 6894}, {1621, 1836}, {1698, 3951}, {1699, 4666}, {1738, 3240}, {1788, 10585}, {1848, 6994}, {1909, 30632}, {1962, 33154}, {2096, 6974}, {2140, 5773}, {2177, 24715}, {2478, 5714}, {2550, 3935}, {2886, 3873}, {2887, 29667}, {2895, 5271}, {2975, 10404}, {3006, 24349}, {3008, 14997}, {3011, 17126}, {3090, 5770}, {3091, 5768}, {3124, 20271}, {3136, 20256}, {3151, 17134}, {3187, 17778}, {3315, 17721}, {3419, 6175}, {3434, 3475}, {3436, 28629}, {3580, 16608}, {3585, 30143}, {3615, 11107}, {3616, 4293}, {3649, 3869}, {3661, 31017}, {3664, 14996}, {3666, 33146}, {3671, 24987}, {3679, 21027}, {3681, 3925}, {3685, 29830}, {3705, 17140}, {3720, 3944}, {3729, 32849}, {3741, 33069}, {3742, 17605}, {3750, 33094}, {3751, 33139}, {3754, 14740}, {3757, 6327}, {3771, 4418}, {3782, 17056}, {3812, 11681}, {3822, 5902}, {3832, 6260}, {3834, 30818}, {3835, 14475}, {3836, 32931}, {3841, 5904}, {3870, 33110}, {3874, 6701}, {3876, 8728}, {3889, 24390}, {3891, 33073}, {3897, 18990}, {3912, 4054}, {3914, 17018}, {3920, 26032}, {3923, 29632}, {3938, 33109}, {3946, 17013}, {3947, 24982}, {3971, 29854}, {3980, 29846}, {3995, 22019}, {4000, 17012}, {4001, 5361}, {4011, 29851}, {4080, 17244}, {4138, 25958}, {4188, 13411}, {4189, 4292}, {4190, 5703}, {4193, 5439}, {4298, 24541}, {4307, 26228}, {4313, 31295}, {4358, 17234}, {4359, 4417}, {4362, 32949}, {4363, 30811}, {4383, 26724}, {4389, 30588}, {4392, 24231}, {4414, 29640}, {4430, 4847}, {4438, 32940}, {4643, 5235}, {4644, 24597}, {4645, 26227}, {4648, 17021}, {4649, 33128}, {4655, 32917}, {4661, 25006}, {4675, 17720}, {4676, 24542}, {4865, 32923}, {4892, 24325}, {5046, 9612}, {5074, 5088}, {5176, 11237}, {5177, 11036}, {5233, 24589}, {5236, 30687}, {5252, 12531}, {5253, 11375}, {5256, 23681}, {5261, 5554}, {5262, 24159}, {5263, 33122}, {5270, 30147}, {5278, 33066}, {5284, 24703}, {5290, 19860}, {5311, 33152}, {5363, 32775}, {5432, 9352}, {5542, 26015}, {5550, 21616}, {5587, 9803}, {5603, 6925}, {5712, 17011}, {5715, 6895}, {5719, 9945}, {5722, 17577}, {5741, 19804}, {5761, 6897}, {5775, 21075}, {5777, 6991}, {5791, 31254}, {5811, 6886}, {5812, 6986}, {5883, 6702}, {6223, 10586}, {6349, 18588}, {6384, 27434}, {6685, 33125}, {6690, 11246}, {6830, 10202}, {6839, 18446}, {6840, 18443}, {6845, 13369}, {6852, 24467}, {6862, 26877}, {6870, 9799}, {6884, 7330}, {6937, 24474}, {6943, 9940}, {7191, 26098}, {7272, 24630}, {7282, 17923}, {7321, 30834}, {7354, 11281}, {7483, 24470}, {7580, 31671}, {7705, 10592}, {7988, 21635}, {8025, 17167}, {8616, 29689}, {8727, 11220}, {9347, 17602}, {9579, 15680}, {9780, 21077}, {9782, 27529}, {10106, 18467}, {10177, 30311}, {10529, 11037}, {10578, 20075}, {11018, 17616}, {11679, 32863}, {11813, 25055}, {12248, 33594}, {12436, 17572}, {12514, 14450}, {12564, 16120}, {12572, 16859}, {12690, 15935}, {12699, 33557}, {12755, 12831}, {14923, 15888}, {15674, 31424}, {15934, 17532}, {16056, 21319}, {16370, 18541}, {16412, 25593}, {16580, 17321}, {16704, 17364}, {16816, 17050}, {16825, 32843}, {17017, 33147}, {17064, 33142}, {17155, 29671}, {17165, 29641}, {17169, 17181}, {17197, 26860}, {17239, 31993}, {17275, 31143}, {17379, 29833}, {17397, 19740}, {17449, 29676}, {17450, 24217}, {17579, 24929}, {17591, 29688}, {17592, 33145}, {17594, 33102}, {17596, 29678}, {17740, 27757}, {18398, 25639}, {18481, 33592}, {19684, 19786}, {19767, 23537}, {19823, 26626}, {21020, 33084}, {21026, 31161}, {21241, 33120}, {21565, 31540}, {21568, 31541}, {23812, 29645}, {24165, 29849}, {24183, 31233}, {24184, 24620}, {24210, 29814}, {24552, 33124}, {24789, 32911}, {24892, 32913}, {25385, 30942}, {25496, 29666}, {26011, 26540}, {26128, 29648}, {26964, 27183}, {27081, 29576}, {27268, 27472}, {27475, 27493}, {27739, 31139}, {27747, 31138}, {29642, 32930}, {29651, 32947}, {29653, 32925}, {29670, 32948}, {29828, 33086}, {29839, 32929}, {29857, 33170}, {29862, 33161}, {29873, 33163}, {30566, 30829}, {31031, 31038}, {31033, 31043}, {31134, 33076}, {31237, 32780}, {31264, 33174}, {31330, 33064}, {32912, 33138}, {32914, 32946}, {32916, 33067}, {32920, 33072}, {32922, 33070}, {32933, 33116}, {32935, 33115}, {32939, 33113}

X(31020) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31020) lies on these lines: {2, 11}, {3, 27096}, {21, 27025}, {950, 26676}, {2975, 26757}, {3295, 27146}, {3661, 5773}, {3871, 26964}, {4375, 6544}, {4781, 17738}, {4855, 26653}, {7824, 26759}, {11115, 27020}, {14439, 24685}, {17136, 21232}, {17495, 26247}, {17539, 27043}, {17572, 27253}, {17588, 27026}, {17589, 27048}, {17647, 28986}, {17755, 17780}, {25440, 28742}, {26100, 31451}, {26687, 27112}, {31015, 31032}, {31093, 32851}

X(31021) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31021) lies on these lines: {2, 3}, {14543, 24686}, {16607, 18656}

X(31022) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31022) lies on these lines: {2, 3}, {693, 20950}, {14543, 24687}, {18657, 21234}, {31058, 31125}

X(31023) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31023) lies on these lines: {2, 31}, {69, 17737}, {226, 31063}, {626, 17137}, {3219, 30763}, {3314, 20347}, {3661, 31037}, {3936, 26590}, {3995, 22009}, {4071, 31087}, {4144, 25345}, {4153, 17211}, {4450, 26629}, {5025, 17751}, {5741, 26582}, {6542, 33153}, {11319, 30175}, {17165, 30179}, {20109, 24995}, {21241, 24592}, {31017, 31027}, {31024, 31033}, {31028, 31058}

X(31024) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31024) lies on these lines: {2, 32}, {4156, 25346}, {4766, 26589}, {11330, 31113}, {17138, 21235}, {20556, 31075}, {31014, 31017}, {31015, 31059}, {31023, 31033}, {31036, 31050}, {31057, 31060}

X(31025) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31025) lies on these lines: {1, 19740}, {2, 37}, {8, 31034}, {10, 3120}, {42, 4709}, {72, 3617}, {145, 5295}, {190, 5235}, {210, 22294}, {213, 14997}, {226, 6539}, {274, 1978}, {306, 4058}, {594, 3936}, {726, 30970}, {740, 21806}, {756, 27798}, {894, 16704}, {908, 4967}, {1089, 19874}, {1100, 19741}, {1150, 4363}, {1215, 4651}, {1255, 25507}, {1449, 3187}, {1654, 17484}, {1698, 3159}, {1738, 26251}, {1743, 5271}, {1824, 7378}, {1920, 16748}, {1999, 8025}, {2886, 31111}, {2901, 3616}, {3218, 17116}, {3294, 16815}, {3452, 22010}, {3661, 31017}, {3696, 19998}, {3700, 31072}, {3741, 17140}, {3775, 32856}, {3807, 30832}, {3828, 4937}, {3842, 3994}, {3954, 29591}, {3969, 17056}, {3970, 29579}, {3993, 27811}, {14475, 26985}, {4026, 4442}, {4066, 16828}, {4115, 27776}, {4125, 19870}, {4365, 27804}, {4427, 32917}, {4644, 31303}, {4647, 4868}, {4649, 17162}, {4665, 5718}, {4670, 26860}, {5219, 27141}, {5224, 33151}, {5241, 30566}, {5263, 20045}, {5278, 16885}, {5316, 22031}, {5712, 20017}, {5737, 32933}, {5750, 29833}, {6358, 18593}, {6535, 29653}, {7283, 17588}, {9330, 9780}, {14624, 20234}, {16668, 19743}, {16672, 19749}, {16884, 19684}, {17012, 17117}, {17135, 32771}, {17145, 31136}, {17146, 31178}, {17150, 32772}, {17155, 22024}, {17165, 31330}, {17244, 21070}, {17270, 31164}, {17284, 22048}, {17292, 22011}, {17315, 31011}, {17450, 24325}, {17491, 33082}, {17763, 24342}, {18359, 25244}, {20290, 33097}, {21282, 33076}, {22000, 27131}, {22008, 33077}, {22020, 24048}, {22036, 31276}, {24044, 29578}, {24165, 31241}, {25253, 31339}, {25590, 26627}, {26030, 28612}, {27005, 27070}, {27163, 30599}, {27483, 27493}, {27797, 30588}, {28599, 33109}, {29610, 31026}, {29823, 32922}, {31031, 31044}, {31079, 33108}, {31143, 32025}

X(31025) = {X(2),X(75)}-harmonic conjugate of X(17495)

X(31026) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31026) lies on these lines: {2, 39}, {183, 11320}, {308, 27809}, {313, 26971}, {314, 26756}, {321, 27044}, {1269, 27102}, {1698, 4075}, {1909, 27166}, {3760, 17147}, {3770, 26963}, {3831, 17184}, {3995, 27020}, {4393, 30998}, {4699, 17325}, {12263, 20352}, {16738, 18133}, {17128, 19308}, {17142, 21238}, {17148, 25505}, {17790, 26976}, {18044, 27261}, {18144, 27145}, {19789, 26029}, {20544, 31119}, {21022, 28597}, {29610, 31025}, {30824, 30861}, {31014, 31017}, {31016, 31059}, {31050, 31057}

X(31027) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31027) lies on these lines: {1, 2}, {37, 30966}, {38, 3797}, {69, 24514}, {75, 3726}, {141, 350}, {190, 24690}, {213, 33297}, {291, 3773}, {310, 321}, {312, 17149}, {319, 2238}, {320, 24330}, {599, 4713}, {672, 17280}, {694, 7018}, {756, 27495}, {894, 30941}, {982, 27474}, {1468, 17688}, {1575, 17229}, {1655, 21071}, {1909, 21024}, {2227, 22167}, {2276, 17233}, {2321, 17759}, {2345, 30962}, {3219, 6651}, {3620, 30946}, {3662, 4441}, {3696, 9507}, {3721, 17762}, {3873, 31317}, {3943, 25349}, {3969, 16587}, {4042, 20154}, {4366, 32943}, {4368, 33082}, {4479, 17227}, {4766, 33081}, {4883, 31306}, {4981, 31323}, {6653, 32948}, {7226, 27481}, {14621, 24552}, {16887, 21070}, {17143, 21240}, {17228, 30963}, {17231, 21264}, {17232, 30949}, {17281, 24691}, {17286, 17754}, {17288, 20347}, {17289, 24512}, {17372, 21904}, {20132, 32772}, {20142, 32864}, {20337, 23917}, {21025, 25280}, {22028, 28660}, {24190, 32104}, {24259, 33085}, {24586, 32941}, {24602, 32945}, {31017, 31023}, {31033, 31044}, {31037, 31058}

X(31028) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31028) lies on these lines: {1, 2}, {38, 27481}, {63, 6651}, {141, 30963}, {190, 24691}, {194, 21071}, {226, 17082}, {257, 18156}, {291, 3790}, {312, 335}, {320, 4713}, {350, 3662}, {354, 31317}, {672, 17339}, {894, 30962}, {982, 3797}, {1086, 4479}, {1575, 17233}, {2238, 17363}, {2276, 17242}, {3097, 6541}, {3943, 25350}, {3948, 17149}, {4366, 24586}, {4664, 25349}, {14621, 32942}, {17063, 27474}, {17144, 20255}, {17231, 20530}, {17232, 20335}, {17234, 21264}, {17248, 30966}, {17280, 17754}, {17288, 30946}, {17333, 24690}, {17364, 24514}, {17368, 24512}, {17377, 21904}, {17762, 20271}, {17793, 33087}, {20081, 24215}, {20132, 25496}, {20142, 32853}, {20337, 23918}, {21024, 31997}, {21025, 24524}, {21877, 24598}, {22028, 30022}, {24602, 32943}, {27184, 30965}, {31017, 31055}, {31023, 31058}, {31031, 31041}

X(31029) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31029) lies on these lines: {2, 44}, {1086, 3936}, {3306, 27141}, {3617, 3824}, {3661, 31017}, {7192, 30967}, {3912, 4080}, {3995, 18134}, {4358, 18150}, {4432, 31177}, {4439, 32856}, {4473, 17484}, {4671, 20917}, {4781, 24692}, {4892, 29824}, {5222, 31034}, {5249, 27791}, {7232, 30834}, {17145, 21241}, {17290, 31179}, {17491, 29632}, {17780, 31151}, {19717, 25527}, {19740, 29603}, {20058, 32850}, {20290, 33130}, {20891, 27794}, {24589, 26758}, {24594, 27739}, {24603, 27081}, {26580, 29571}, {27070, 27113}, {27776, 29626}, {29572, 31035}, {29588, 33155}, {29676, 33069}

X(31030) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31030) lies on these lines: {2, 45}, {75, 26738}, {527, 30564}, {536, 30588}, {1647, 25385}, {3596, 27794}, {3661, 31017}, {3936, 4665}, {4054, 29571}, {4407, 32856}, {4644, 16704}, {4671, 20913}, {4699, 27791}, {17118, 30834}, {17119, 31179}, {17140, 29676}, {17146, 21242}, {17780, 24693}, {19740, 29586}, {24593, 27747}, {24603, 26580}, {27812, 33065}, {29593, 30991}

X(31031) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics -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(31031) lies on these lines: {2, 11}, {5, 27096}, {312, 31093}, {321, 31121}, {496, 17672}, {908, 26593}, {3661, 27131}, {3995, 22015}, {4193, 27025}, {4766, 17135}, {5025, 26759}, {11681, 26757}, {12053, 26548}, {17046, 20244}, {17181, 25244}, {17316, 31043}, {17647, 28969}, {21073, 25237}, {21241, 30821}, {21282, 24586}, {25440, 28761}, {25639, 28742}, {28740, 31418}, {30816, 32941}, {30837, 32943}, {31015, 31039}, {31017, 31023}, {31019, 31038}, {31025, 31044}, {31028, 31041}

X(31032) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31032) lies on these lines: {2, 12}, {226, 26599}, {857, 27096}, {908, 26575}, {993, 28816}, {3219, 16549}, {3452, 26581}, {3661, 27131}, {3814, 28797}, {5046, 27526}, {5123, 24633}, {5795, 26628}, {20245, 21244}, {26563, 31121}, {26601, 27041}, {27081, 31018}, {29616, 31058}, {31015, 31020}, {31017, 31033}, {31041, 31055}

X(31033) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31033) lies on these lines: {2, 41}, {3995, 25247}, {21069, 25241}, {31017, 31032}, {31019, 31043}, {31023, 31024}, {31027, 31044}

X(31034) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31034) lies on these lines: {1, 26580}, {2, 6}, {4, 145}, {7, 17495}, {8, 31025}, {32, 21341}, {42, 4660}, {43, 32949}, {55, 20064}, {192, 17484}, {194, 21220}, {226, 3187}, {238, 29830}, {239, 31019}, {306, 17355}, {312, 17386}, {320, 4277}, {321, 17299}, {329, 3995}, {388, 20040}, {518, 29832}, {519, 4054}, {573, 3218}, {740, 24725}, {752, 2177}, {754, 24296}, {894, 33077}, {908, 3879}, {941, 6646}, {1215, 32852}, {1330, 17676}, {1351, 8229}, {1386, 29831}, {1449, 29833}, {1757, 29643}, {1836, 3896}, {1962, 4703}, {1999, 10478}, {2003, 28774}, {2271, 21997}, {2308, 3771}, {2323, 28796}, {2475, 20018}, {2550, 19998}, {3006, 3751}, {3060, 5208}, {3193, 27504}, {3210, 17483}, {3240, 4645}, {3241, 21291}, {3434, 20011}, {3662, 4270}, {3664, 26627}, {3666, 17345}, {3681, 33073}, {3758, 32779}, {3759, 33129}, {3791, 33127}, {3868, 5752}, {3873, 33071}, {3875, 31164}, {3909, 4259}, {3923, 4062}, {4028, 32929}, {4035, 5294}, {4038, 25960}, {4085, 31134}, {4101, 5717}, {4189, 20077}, {4340, 19284}, {4358, 4851}, {4360, 33151}, {4388, 17018}, {4393, 33155}, {4414, 17770}, {4427, 24695}, {4430, 29840}, {4438, 4722}, {4641, 33113}, {4644, 17740}, {4649, 25760}, {4663, 33114}, {4671, 6542}, {4672, 33156}, {4675, 24589}, {4683, 17592}, {4689, 28570}, {4734, 33102}, {4892, 33128}, {4970, 33098}, {5082, 20051}, {5120, 21488}, {5222, 31029}, {5256, 17184}, {5713, 12649}, {5847, 26227}, {5905, 17147}, {6685, 33080}, {9534, 26131}, {10453, 33107}, {11352, 18907}, {14963, 31015}, {16468, 29632}, {16475, 26230}, {16610, 17376}, {17011, 17396}, {17013, 17302}, {17014, 19823}, {17017, 33064}, {17021, 17391}, {17127, 29839}, {17135, 26098}, {17150, 33144}, {17165, 33088}, {17258, 28606}, {17268, 27064}, {17316, 31018}, {17350, 32849}, {17374, 30818}, {17387, 30829}, {17490, 26842}, {17491, 24248}, {17717, 32919}, {18668, 25241}, {19789, 20043}, {19792, 27793}, {20012, 33110}, {20234, 28605}, {20290, 26034}, {20970, 26085}, {21219, 30578}, {24349, 32842}, {25496, 33081}, {28650, 29631}, {29671, 32912}, {29821, 33069}, {29849, 32913}, {31048, 31057}, {32771, 32861}, {32772, 33084}, {32846, 32931}, {32848, 32935}, {32853, 33105}, {32855, 32940}, {32856, 32921}, {32860, 33097}, {32864, 33111}, {32915, 33096}, {32924, 33103}, {32928, 33101}, {32937, 33093}, {32938, 33092}, {32944, 33087}

X(31034) = complement of X(31303)
X(31034) = anticomplement of X(1150)

X(31035) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31035) lies on these lines: {1, 3952}, {2, 37}, {8, 9330}, {9, 16704}, {39, 27070}, {43, 27804}, {45, 1150}, {145, 392}, {194, 29595}, {210, 20011}, {354, 20068}, {537, 17146}, {551, 4937}, {726, 30950}, {748, 17150}, {750, 4427}, {756, 17135}, {894, 17021}, {899, 3993}, {975, 11115}, {984, 29824}, {1001, 20045}, {1089, 27784}, {1655, 21220}, {1698, 27812}, {1961, 32930}, {1978, 18140}, {1999, 19742}, {2177, 17780}, {3006, 4078}, {3187, 3305}, {3218, 17261}, {3230, 4393}, {3661, 17497}, {3685, 5297}, {3701, 6051}, {3717, 29835}, {3720, 3971}, {3729, 26627}, {3740, 3896}, {3743, 26030}, {3758, 26860}, {3826, 4442}, {3840, 3989}, {3891, 4423}, {3912, 26580}, {3932, 31079}, {3936, 17243}, {3943, 5241}, {3944, 29854}, {3969, 5743}, {3994, 24325}, {4009, 15569}, {4011, 5311}, {4029, 5316}, {4038, 32938}, {4054, 29571}, {4066, 25512}, {4080, 17244}, {4135, 25501}, {4277, 27036}, {4392, 30947}, {4415, 18139}, {4425, 29687}, {4552, 5219}, {4651, 32915}, {4656, 17184}, {4662, 20047}, {4676, 9347}, {4679, 33070}, {4703, 20290}, {4717, 19870}, {4759, 21747}, {5256, 26688}, {5268, 32929}, {5284, 32926}, {5287, 8025}, {5718, 30566}, {7283, 19284}, {7398, 20075}, {8543, 14594}, {9331, 30730}, {9345, 32935}, {9791, 33086}, {10180, 31264}, {14555, 20017}, {14996, 17350}, {16484, 32927}, {16685, 32911}, {16826, 19740}, {17012, 17319}, {17018, 27538}, {17019, 19717}, {17032, 31052}, {17122, 32936}, {17123, 32928}, {17124, 32934}, {17125, 32921}, {17140, 26102}, {17155, 25502}, {17163, 26037}, {17230, 27269}, {17234, 33151}, {17242, 33077}, {17292, 17489}, {17295, 31143}, {17299, 31011}, {17300, 17484}, {17310, 27776}, {17316, 31018}, {17353, 29833}, {17367, 28598}, {17394, 19741}, {17494, 31992}, {17570, 19851}, {17602, 24542}, {17722, 24709}, {17777, 33112}, {17778, 26792}, {18662, 30852}, {18668, 31045}, {19684, 30562}, {21027, 25352}, {25254, 31266}, {25257, 29599}, {25261, 29579}, {25264, 29578}, {25960, 33092}, {25961, 33154}, {26115, 27785}, {27141, 27757}, {29572, 31029}, {29814, 32937}, {29822, 32931}, {29837, 33166}, {29845, 33164}, {29851, 33152}, {30824, 31281}, {31037, 32858}

X(31035) = anticomplement of X(24589)

X(31036) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31036) lies on these lines: {1, 3159}, {2, 39}, {6, 190}, {37, 17148}, {239, 3219}, {321, 1107}, {330, 1255}, {726, 21352}, {869, 3952}, {1045, 25277}, {1212, 19791}, {2275, 27166}, {2276, 3765}, {2309, 21080}, {3009, 3971}, {3175, 17448}, {3187, 21384}, {3210, 5278}, {3596, 26764}, {4261, 26772}, {4383, 16722}, {4384, 17495}, {5069, 18147}, {6376, 27044}, {6542, 21226}, {6872, 20018}, {7754, 16367}, {7783, 19308}, {7976, 20044}, {9263, 29588}, {9534, 17676}, {10477, 31302}, {11329, 31859}, {11330, 29832}, {12782, 20352}, {14963, 31015}, {15668, 16710}, {16476, 17150}, {16696, 18137}, {16826, 19740}, {16827, 27065}, {17001, 21537}, {17184, 29960}, {17316, 31061}, {17490, 19732}, {18133, 27095}, {19281, 25242}, {19522, 32448}, {19701, 29595}, {19719, 27340}, {19722, 31999}, {19731, 32939}, {19738, 32095}, {19789, 27304}, {20045, 23407}, {20691, 25298}, {20923, 27017}, {21035, 21278}, {21412, 28605}, {24214, 29988}, {24268, 25254}, {24296, 24587}, {27481, 32453}, {29572, 31006}, {31014, 31057}, {31024, 31050}, {32911, 33296}

X(31037) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31037) lies on these lines: {2, 6}, {8, 3454}, {75, 27793}, {145, 5051}, {171, 20290}, {226, 6539}, {306, 3950}, {319, 33133}, {321, 4033}, {442, 1159}, {1230, 4671}, {1330, 11115}, {1834, 3621}, {2092, 17236}, {2245, 4741}, {2292, 27558}, {2887, 4651}, {3120, 17163}, {3219, 21362}, {3622, 4205}, {3661, 31023}, {3681, 31079}, {3687, 17184}, {3775, 33105}, {3846, 29824}, {3948, 17230}, {3952, 15523}, {3961, 28599}, {3966, 33122}, {3969, 4415}, {4046, 4442}, {4062, 4425}, {4272, 17383}, {4388, 33175}, {4392, 20966}, {4418, 17491}, {4427, 4683}, {4643, 33113}, {4703, 33156}, {4720, 17677}, {4886, 33129}, {4892, 21020}, {4972, 19998}, {4982, 29833}, {5249, 27791}, {5748, 17052}, {6535, 21093}, {6536, 27811}, {6646, 33168}, {8013, 27812}, {12245, 30449}, {15674, 25663}, {16052, 31145}, {16589, 29572}, {17020, 17291}, {17135, 25760}, {17140, 33064}, {17145, 29655}, {17147, 27184}, {17150, 32775}, {17162, 33135}, {17164, 20653}, {17165, 32778}, {17270, 31266}, {17280, 17482}, {17284, 26688}, {17288, 27003}, {17588, 25650}, {17589, 26131}, {17751, 26589}, {17780, 33079}, {19823, 20043}, {20011, 32773}, {20045, 33075}, {20068, 33089}, {21033, 27727}, {21282, 32945}, {21677, 27708}, {21805, 28595}, {25960, 33087}, {26601, 29616}, {26757, 26781}, {26791, 26793}, {27021, 27096}, {27794, 28605}, {29594, 31057}, {29823, 33071}, {29846, 33082}, {31018, 31045}, {31027, 31058}, {31035, 32858}, {32779, 33066}, {32783, 32843}

X(31038) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31038) lies on these lines: {1, 2}, {57, 20533}, {192, 9436}, {320, 24352}, {335, 1088}, {894, 14548}, {2886, 27475}, {3555, 17671}, {3662, 26590}, {3693, 17242}, {3868, 27129}, {3943, 25355}, {3970, 17181}, {4437, 18743}, {6651, 24771}, {10025, 17364}, {14828, 17391}, {16284, 21049}, {17158, 21258}, {21096, 25242}, {21795, 24635}, {31019, 31031}, {31053, 31058}

X(31039) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31039) lies on these lines: {2, 12}, {993, 28797}, {3219, 3294}, {3661, 5773}, {3911, 26599}, {5372, 29616}, {5745, 26581}, {14996, 29624}, {16050, 27163}, {18654, 21233}, {19740, 31018}, {27131, 29612}, {27187, 31121}, {31015, 31031}

X(31040) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31040) lies on these lines: {2, 667}, {3766, 31094}, {3835, 21053}, {4063, 29610}, {4083, 29593}, {4107, 30835}, {4129, 31041}

X(31041) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31041) lies on these lines: {2, 31}, {226, 31052}, {321, 18052}, {335, 31093}, {908, 31062}, {3120, 3912}, {3218, 30763}, {3454, 27026}, {3661, 31017}, {3834, 27918}, {3936, 26582}, {31040, 31051}, {4144, 25357}, {4366, 21282}, {6542, 33148}, {11115, 30175}, {11319, 30174}, {17140, 30179}, {17211, 25263}, {17316, 33155}, {17755, 21026}, {17789, 18066}, {18139, 26590}, {20541, 30941}, {27081, 29610}, {30967, 31058}, {31028, 31031}, {31032, 31055}, {31079, 31317}

X(31042) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31042) lies on these lines: {2, 3}, {169, 27065}, {193, 21221}, {281, 20061}, {312, 10405}, {321, 20914}, {346, 4150}, {948, 26738}, {1699, 25935}, {1826, 4329}, {1901, 5738}, {3661, 5179}, {4869, 18147}, {5691, 26006}, {5819, 17289}, {17316, 31053}, {20110, 21270}, {20235, 28605}, {26790, 29593}

X(31043) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31043) lies on these lines: {2, 3}, {7, 17052}, {144, 1901}, {226, 21029}, {306, 3947}, {1834, 17014}, {2345, 21245}, {3454, 29611}, {3661, 31023}, {3936, 29616}, {3995, 21073}, {5949, 20533}, {16826, 31058}, {17056, 23903}, {17316, 31031}, {18088, 18703}, {21033, 27557}, {21372, 27065}, {26582, 26772}, {27081, 31018}, {31019, 31033}

X(31044) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31044) lies on these lines: {2, 3}, {305, 1978}, {17134, 30779}, {31025, 31031}, {31027, 31033}

X(31045) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31045) lies on these lines: {2, 3}, {306, 3952}, {3995, 21062}, {5273, 17052}, {18635, 21454}, {18668, 31035}, {22076, 26911}, {27065, 27081}, {27129, 32858}, {31018, 31037}

X(31046) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31046) lies on these lines: {2, 3}, {4033, 20336}, {5813, 27081}

X(31047) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31047) lies on these lines: {2, 3}, {307, 2321}, {347, 18642}, {3672, 18635}, {18721, 27396}

X(31048) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31048) lies on these lines: {2, 3}, {31034, 31057}

X(31049) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31049) lies on these lines: {2, 3}, {144, 5738}, {306, 18250}, {966, 4422}, {3661, 21373}, {5712, 29624}, {5739, 29616}, {5905, 25261}, {16826, 31053}, {17316, 31018}

X(31050) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31050) lies on these lines: {2, 3}, {3661, 3952}, {14543, 24726}, {17316, 33153}, {24296, 30817}, {26770, 31089}, {31017, 31060}, {31024, 31036}, {31026, 31057}

X(31051) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31051) lies on these lines: {2, 36}, {908, 21044}, {3661, 27131}, {4124, 5087}, {31040, 31041}, {5748, 17316}, {17139, 21237}, {17310, 31058}, {18096, 26601}, {21372, 27065}

X(31052) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31052) lies on these lines: {2, 38}, {226, 31041}, {312, 18059}, {3661, 31023}, {3934, 17141}, {3953, 26987}, {3995, 22013}, {4090, 24592}, {5293, 16930}, {6542, 33107}, {17032, 31035}, {17142, 20148}, {17152, 21021}, {17164, 26752}, {20247, 31276}, {20335, 31077}, {21067, 28598}, {24443, 26779}, {25248, 27020}, {25958, 29593}, {26247, 27064}

X(31053) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31053) lies on these lines: {1, 5046}, {2, 7}, {4, 5761}, {5, 3868}, {6, 33133}, {8, 6871}, {10, 25958}, {11, 3873}, {12, 3869}, {21, 11374}, {31, 17719}, {37, 27180}, {38, 17717}, {42, 3944}, {43, 3120}, {46, 14450}, {55, 5057}, {65, 5123}, {72, 2476}, {75, 5741}, {78, 2475}, {79, 25440}, {81, 17174}, {85, 21579}, {92, 324}, {100, 1836}, {145, 946}, {149, 1699}, {171, 24725}, {190, 33113}, {200, 33110}, {210, 3838}, {238, 29681}, {239, 5826}, {244, 33103}, {306, 4671}, {312, 1230}, {321, 3262}, {333, 17173}, {344, 16580}, {345, 27757}, {354, 5087}, {355, 1389}, {377, 5714}, {382, 11015}, {411, 5812}, {442, 3876}, {495, 3877}, {496, 3889}, {497, 3957}, {517, 6932}, {518, 11680}, {519, 18393}, {529, 15950}, {612, 33112}, {614, 33148}, {726, 29849}, {748, 33130}, {750, 33097}, {756, 33111}, {758, 7951}, {846, 29678}, {899, 17889}, {912, 6830}, {938, 5187}, {942, 4193}, {962, 10528}, {975, 26131}, {982, 32856}, {984, 29664}, {1071, 6943}, {1125, 16859}, {1150, 33066}, {1210, 5154}, {1211, 5949}, {1215, 25760}, {1320, 3656}, {1329, 3649}, {1376, 20292}, {1392, 3244}, {1478, 4511}, {1490, 6895}, {1519, 12648}, {1621, 17718}, {1698, 11263}, {1709, 9809}, {1724, 24160}, {1757, 24892}, {1901, 27396}, {1994, 2006}, {1999, 10478}, {2051, 4080}, {2096, 6966}, {2099, 5176}, {2140, 29628}, {2177, 33095}, {2308, 29658}, {2478, 3487}, {2886, 3681}, {2887, 29679}, {2895, 11679}, {2975, 11375}, {2999, 33150}, {3006, 32937}, {3007, 6360}, {3011, 17127}, {3035, 9352}, {3052, 17783}, {3085, 11415}, {3091, 5804}, {3101, 18664}, {3146, 6260}, {3158, 20095}, {3187, 17035}, {3212, 26594}, {3240, 3914}, {3241, 30384}, {3419, 17577}, {3434, 3935}, {3436, 3485}, {3475, 29817}, {3555, 9955}, {3576, 20067}, {3585, 22836}, {3601, 15680}, {3616, 13407}, {3617, 5828}, {3622, 21620}, {3623, 12053}, {3648, 14526}, {3661, 31023}, {3666, 33151}, {3671, 24982}, {3687, 4054}, {3705, 17165}, {3729, 33168}, {3741, 33065}, {3751, 33142}, {3752, 33146}, {3771, 32930}, {3772, 32911}, {3782, 4850}, {3814, 5902}, {3817, 4430}, {3822, 5692}, {3825, 18398}, {3835, 6545}, {3840, 33069}, {3845, 12690}, {3846, 32771}, {3854, 9842}, {3871, 12699}, {3874, 7741}, {3878, 13375}, {3885, 22791}, {3890, 15888}, {3891, 33071}, {3895, 31162}, {3912, 31060}, {3920, 26098}, {3923, 29846}, {3938, 33106}, {3940, 17532}, {3947, 24987}, {3951, 5705}, {3952, 29641}, {3961, 33104}, {3967, 32862}, {3970, 24045}, {3971, 29643}, {3989, 29657}, {3994, 33092}, {3995, 17479}, {4000, 17020}, {4001, 5372}, {4011, 29632}, {4018, 9956}, {4084, 18395}, {4090, 21241}, {4138, 25959}, {4187, 6147}, {4188, 4292}, {4189, 13411}, {4190, 27383}, {4192, 21319}, {4197, 5044}, {4293, 4881}, {4295, 5552}, {4358, 18134}, {4359, 5233}, {4362, 32843}, {4383, 33129}, {4388, 26227}, {4392, 24239}, {4414, 33099}, {4415, 5718}, {4416, 5361}, {4438, 32938}, {4466, 27290}, {4655, 32918}, {4661, 4847}, {4679, 5284}, {4683, 32916}, {4703, 32917}, {4855, 9579}, {4865, 32927}, {4892, 25957}, {5025, 26589}, {5074, 29575}, {5086, 10895}, {5119, 5180}, {5121, 9335}, {5141, 6734}, {5173, 17615}, {5175, 20013}, {5208, 14008}, {5220, 31245}, {5253, 10404}, {5256, 33155}, {5260, 28628}, {5270, 30144}, {5289, 11237}, {5290, 19861}, {5425, 31160}, {5432, 17768}, {5440, 17579}, {5443, 8666}, {5603, 6957}, {5658, 10431}, {5703, 6872}, {5709, 6960}, {5712, 17019}, {5715, 6894}, {5719, 11113}, {5720, 6839}, {5730, 9654}, {5758, 6838}, {5770, 6879}, {5777, 6828}, {5811, 6837}, {5904, 25639}, {5927, 10883}, {6327, 7081}, {6350, 18588}, {6354, 26611}, {6505, 18625}, {6685, 32776}, {6700, 17572}, {6827, 18444}, {6831, 12528}, {6840, 18446}, {6853, 26921}, {6888, 7330}, {6919, 11036}, {6937, 31837}, {6941, 24474}, {6952, 24467}, {6958, 26877}, {6963, 10202}, {6971, 24475}, {6990, 14054}, {7191, 33144}, {7226, 29639}, {7678, 15185}, {8024, 21590}, {8025, 17182}, {8068, 12532}, {8727, 13257}, {9539, 16870}, {9578, 11682}, {9581, 11520}, {9780, 12609}, {9812, 20075}, {10526, 21740}, {10586, 11037}, {10826, 12559}, {11114, 24929}, {11239, 30305}, {11684, 26066}, {12527, 24541}, {12572, 16865}, {12607, 14923}, {12738, 18407}, {13747, 24470}, {14829, 32859}, {14997, 26723}, {15485, 29689}, {15733, 30311}, {15934, 17556}, {16371, 18541}, {16704, 17167}, {16826, 31049}, {17012, 19785}, {17017, 33152}, {17018, 24210}, {17054, 26729}, {17056, 17775}, {17064, 33139}, {17258, 31281}, {17303, 31247}, {17316, 31042}, {17364, 24220}, {17469, 17725}, {17594, 33100}, {17596, 33098}, {17619, 31794}, {17728, 31272}, {17758, 29581}, {17763, 32946}, {17776, 30828}, {17777, 29839}, {17778, 21221}, {18136, 18139}, {18668, 31035}, {19717, 29841}, {20014, 21627}, {20089, 29572}, {20173, 27491}, {20612, 31803}, {21060, 25006}, {21093, 29671}, {21564, 31540}, {21569, 31541}, {21805, 32865}, {21873, 28633}, {22020, 24048}, {23675, 28370}, {23806, 27115}, {23989, 30545}, {24003, 25961}, {24178, 27625}, {24325, 25960}, {24552, 33126}, {25385, 31330}, {25496, 29648}, {26128, 29666}, {27049, 27129}, {27050, 27255}, {29649, 32949}, {29662, 32913}, {29670, 32947}, {29821, 33143}, {29828, 33083}, {29857, 33166}, {29872, 33163}, {30568, 30578}, {30811, 33157}, {30818, 33172}, {30831, 32777}, {30834, 33116}, {30942, 33064}, {31038, 31058}, {31134, 33079}, {31161, 33169}, {31237, 33159}, {31264, 32784}, {32844, 32920}, {32851, 32933}, {32912, 33140}, {32926, 33070}, {32935, 33119}, {32942, 33122}

X(31054) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31054) lies on these lines: {2, 66}

X(31055) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31055) lies on these lines: {2, 87}, {3661, 4044}, {17149, 21250}, {31017, 31028}, {31032, 31041}

X(31056) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31056) lies on these lines: {2, 44}, {190, 27739}, {599, 4997}, {908, 3661}, {1086, 5233}, {1647, 33065}, {3210, 5741}, {3241, 25378}, {3617, 10129}, {3687, 4052}, {3936, 29572}, {4033, 4671}, {4080, 4740}, {4407, 17717}, {4473, 31018}, {4699, 27791}, {4781, 24710}, {6172, 26070}, {7232, 31202}, {17230, 30566}, {17271, 30824}, {29570, 31179}, {30861, 31017}

X(31057) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31057) lies on these lines: {2, 99}, {321, 21604}, {868, 5992}, {3140, 31126}, {3936, 4945}, {3995, 21090}, {4049, 4080}, {4440, 8287}, {6542, 6543}, {8818, 17300}, {16816, 23942}, {21043, 21295}, {21431, 28605}, {25269, 27704}, {27081, 27776}, {29594, 31037}, {31014, 31036}, {31024, 31060}, {31026, 31050}, {31034, 31048}, {31058, 31061}

X(31058) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 105

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

X(31058) lies on these lines: {2, 11}, {35, 28761}, {312, 31086}, {321, 21580}, {335, 4080}, {496, 26964}, {1329, 26757}, {1479, 28734}, {3452, 26593}, {3995, 22032}, {4187, 27025}, {4193, 27096}, {4358, 31093}, {4437, 30566}, {4766, 29824}, {5141, 27253}, {5291, 26145}, {7741, 28742}, {9581, 26653}, {10129, 27475}, {10572, 28969}, {10591, 28740}, {10826, 28961}, {12053, 26526}, {16826, 31043}, {17181, 25237}, {17310, 31051}, {17316, 31014}, {17539, 26686}, {17669, 26759}, {20042, 32029}, {20173, 31121}, {21044, 21272}, {21073, 25244}, {21282, 24602}, {21404, 28605}, {29616, 31032}, {30967, 31041}, {31022, 31125}, {31023, 31028}, {31027, 31037}, {31038, 31053}, {31057, 31061}

X(31059) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics (a + b) (a + c) (2 a - b - c) (a^2 + a b - b^2 + a c - b c - c^2) : :

X(31059) lies on these lines: {2, 99}, {86, 545}, {519, 902}, {662, 4473}, {1931, 6157}, {2786, 5029}, {3110, 3799}, {4236, 31073}, {4440, 25536}, {6626, 29591}, {6629, 17310}, {8025, 29580}, {11115, 16830}, {16702, 17264}, {17103, 29569}, {18827, 29570}, {24624, 26070}, {31015, 31024}, {31016, 31026}

X(31060) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics b c (-a^2 b - a^2 c + a b c + 2 b^2 c + 2 b c^2) : :

X(31060) lies on these lines: {2, 39}, {10, 28605}, {37, 30596}, {69, 21221}, {75, 4708}, {192, 313}, {239, 3760}, {312, 17230}, {314, 17343}, {321, 6376}, {330, 27166}, {350, 3765}, {385, 11320}, {668, 20055}, {1234, 27267}, {1269, 4699}, {1278, 3596}, {1909, 29570}, {1975, 19308}, {2895, 5046}, {2998, 27809}, {3009, 9902}, {3175, 25102}, {3264, 4740}, {3661, 4044}, {3761, 16826}, {3770, 17379}, {3782, 21025}, {3797, 21615}, {3912, 31053}, {3933, 26019}, {3963, 4704}, {3975, 16816}, {4043, 30473}, {4358, 20917}, {4377, 4664}, {10447, 17252}, {13108, 19522}, {17144, 25298}, {17148, 26107}, {17155, 20340}, {17157, 21257}, {17232, 18137}, {17236, 20891}, {17238, 18133}, {17268, 18065}, {17358, 18044}, {17786, 22016}, {17787, 25269}, {18743, 29599}, {19791, 25994}, {20158, 30940}, {20255, 33146}, {20486, 32862}, {20888, 29576}, {21024, 32782}, {21071, 32858}, {21443, 27481}, {24296, 24614}, {27186, 29968}, {27299, 33150}, {31017, 31050}, {31024, 31057}

X(31061) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 105

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 - b^3 c^2 + a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(31061) lies on these lines: {2, 668}, {81, 18047}, {192, 545}, {291, 23354}, {321, 7200}, {330, 17230}, {335, 812}, {519, 4674}, {594, 16710}, {730, 29824}, {1022, 4080}, {3187, 9317}, {3218, 10027}, {3778, 25284}, {3963, 17178}, {3995, 22035}, {4033, 16726}, {4465, 29570}, {4562, 6542}, {4851, 17148}, {9457, 26627}, {14839, 17154}, {17053, 26756}, {17065, 25292}, {17147, 17389}, {17316, 31036}, {18040, 26963}, {20055, 24621}, {20530, 27166}, {20532, 29593}, {21226, 29569}, {25264, 29619}, {31057, 31058}

X(31062) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics -3 a^3 b - 2 a^2 b^2 + 3 a b^3 + 4 b^4 - 3 a^3 c - 2 a^2 b c + b^3 c - 2 a^2 c^2 - 4 b^2 c^2 + 3 a c^3 + b c^3 + 4 c^4 : :

X(31062) lies on these lines: {2, 896}, {908, 31041}, {3661, 31023}, {4379, 7192}, {3995, 22038}, {29586, 33107}, {29588, 33153}

X(31063) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics a^3 b + 2 a^2 b^2 - a b^3 + a^3 c + 4 a^2 b c + 2 a b^2 c + b^3 c + 2 a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - a c^3 + b c^3 : :

X(31063) lies on these lines: {2, 38}, {226, 31023}, {321, 18138}, {894, 26247}, {899, 27478}, {976, 16930}, {1909, 18050}, {3661, 31017}, {3995, 17032}, {4671, 14210}, {6542, 33112}, {14621, 20045}, {16826, 19740}, {17029, 31314}, {17142, 20140}, {17489, 22011}, {21101, 31087}, {21416, 28605}, {24174, 26779}, {25959, 27812}, {30949, 31077}

X(31064) = (name pending)

Barycentrics (a + 2 b + c)(a + b + 2c)(2a^3 - b^3 - c^3 + a^2 b + a^2 c - a b^2 - a c^2) : :

X(31064) lies on these lines: {2, 1171}, {10, 20290}, {514, 4382}, {1126, 1330}, {1255, 17778}, {1268, 1654}, {2895, 6539}, {4024, 4608}, {16704, 20337}, {20349, 20536}

X(31065) = (name pending)

Barycentrics (b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31065) lies on these lines: {420, 2501}, {476, 7953}, {523, 2528}, {826, 14318}, {850, 7950}, {2395, 3108}, {5466, 10159}

X(31066) = (name pending)

Barycentrics (b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2)(3 a^4 + b^4 + c^4 + 3 a^2 b^2 + 3 a^2 c^2 + b^2 c^2) : :

X(31066) lies on these lines: {83, 7927}, {523, 2528}

X(31067) = (name pending)

Barycentrics (b^4 - c^4)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31067) lies on these lines: {523, 2528}, {1287, 7953}

X(31068) = (name pending)

Barycentrics (2a^2 - b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31068) lies on these lines: {2, 3108}, {193, 9516}, {523, 2528}, {2770, 7953}, {5486, 20080}, {7855, 31088}, {7905, 31078}, {7916, 31076}

X(31069) = (name pending)

Barycentrics (2a^2 - b^2 - c^2)(a^2 + 2 b^2 + c^2)(2 a^4 - b^4 - c^4 + 2 a^2 b^2 + 2 a^2 c^2 - 4 b^2 c^2) : :

X(31069) lies on these lines: (none)

X(31070) = (name pending)

Barycentrics (2 a^4 - b^4 - c^4)(a^2 + 2b^2 - c^2)(a^2 + b^2 + 2c^2) : :

X(31070) lies on these lines: (none)

Collineation mappings involving Gemini triangle 106: X(31071)-X(31132)

m(X) = - (b^2 + c^2) x + (a^2 + 2 b^2 + c^2) y + (a^2 + b^2 + 2 c^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(523)X(2528). Among the fixed points are X(i) for these i: 2, 523, 2528, 31065, 31066, 31067, 31068, 31069, 31070. (Clark Kimberling, January 21, 2019)

X(31071) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b - a b^3 + 2 b^4 - a^3 c - 2 a^2 b c - 3 b^3 c + 2 b^2 c^2 - a c^3 - 3 b c^3 + 2 c^4 : :

X(31071) lies on these lines: {2, 7}, {3263, 18040}, {3314, 31077}, {3674, 26526}, {7247, 27006}, {16063, 29823}, {17046, 20247}, {17181, 26964}, {17192, 26030}, {17302, 26133}, {17671, 25261}, {18150, 30758}, {31084, 31092}, {31087, 31129}, {31093, 31130}

X(31072) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 + 3 b^2 c^2) : :

X(31072) lies on these lines: {2, 647}, {512, 26798}, {523, 7925}, {2492, 23285}, {2501, 30474}, {2525, 9979}, {2528, 8371}, {3091, 30209}, {3268, 12077}, {3618, 9030}, {3620, 8675}, {3700, 31025}, {3906, 31276}, {5025, 10097}, {6331, 11794}, {7836, 23105}, {8552, 18314}, {8664, 9148}, {31094, 31096}, {10561, 31125}, {10562, 30786}

X(31073) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a (a^4 - a^3 b + a^2 b^2 - a b^3 - a^3 c + 2 a^2 b c - 2 a b^2 c + 2 b^3 c + a^2 c^2 - 2 a b c^2 - b^2 c^2 - a c^3 + 2 b c^3) : :

X(31073) lies on these lines: {2, 11}, {3, 24808}, {183, 31130}, {244, 3920}, {291, 750}, {612, 1054}, {659, 28602}, {678, 7292}, {899, 8300}, {1026, 27950}, {1281, 5205}, {1633, 4473}, {3218, 4712}, {3306, 9451}, {3315, 29815}, {3722, 7191}, {3799, 5091}, {4236, 31059}, {5189, 5520}, {5276, 20331}, {5297, 17593}, {7081, 20882}, {7485, 20999}, {7492, 23859}, {7496, 31079}, {7570, 31109}, {8290, 24533}, {9708, 19325}, {9709, 19314}, {10989, 31110}, {12329, 17232}, {13635, 18524}, {14829, 33091}, {15621, 31081}, {16063, 31085}, {16999, 31087}, {17339, 24309}, {29679, 32916}

X(31074) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + b^6 - a^4 c^2 + a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :

X(31074) lies on these lines: {2, 3}, {11, 9539}, {51, 26913}, {52, 23294}, {98, 12092}, {110, 11550}, {113, 11455}, {125, 3060}, {146, 23306}, {193, 11216}, {305, 7871}, {323, 11442}, {325, 13481}, {394, 3410}, {511, 23293}, {1194, 15820}, {1236, 8024}, {1272, 15356}, {1503, 9544}, {1568, 15305}, {1853, 1993}, {1899, 1994}, {2979, 7703}, {3167, 14683}, {3574, 10574}, {3580, 23332}, {3867, 26156}, {3925, 9536}, {5446, 26917}, {5448, 12290}, {5476, 12834}, {5889, 20299}, {6243, 13561}, {7605, 17825}, {8029, 23301}, {8267, 31125}, {9820, 16659}, {10169, 32220}, {10721, 33547}, {10733, 18376}, {11002, 13567}, {11003, 23292}, {11423, 18128}, {11451, 19130}, {11572, 12278}, {11671, 23319}, {13352, 25739}, {14360, 23318}, {15058, 15103}, {15072, 18388}, {15534, 25328}, {16275, 26233}, {18018, 18019}, {20061, 23305}, {23958, 26933}, {24321, 24687}, {24322, 24686}, {26881, 29012}, {32064, 32125}

X(31074) = complement of X(37913)

X(31075) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b^2 + a^2 b^3 + 2 b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + 2 c^5 : :

X(31075) lies on these lines: {2, 31}, {3314, 31084}, {4892, 26263}, {20556, 31024}, {21235, 21278}, {31079, 31089}, {31081, 31126}, {31115, 33108}

X(31076) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + a^2 b^4 + 2 b^6 - a^4 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 + 2 c^6 : :

X(31076) lies on these lines: {2, 32}, {23, 7885}, {325, 31088}, {625, 26235}, {1180, 7918}, {1502, 4609}, {3266, 7821}, {3314, 5169}, {5189, 7836}, {5254, 8267}, {5354, 7901}, {5971, 30777}, {7492, 7898}, {7496, 7925}, {7814, 15302}, {7881, 31133}, {7895, 8024}, {7916, 31068}, {7934, 9465}, {16063, 31128}

X(31077) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b + a b^3 + a^3 c + 2 a^2 b c + 3 b^3 c + a c^3 + 3 b c^3 : :

X(31077) lies on these lines: {2, 37}, {7, 17007}, {10, 17192}, {1930, 26965}, {3314, 31071}, {3620, 4661}, {3662, 17165}, {3952, 31004}, {7191, 17117}, {16815, 18785}, {16818, 25263}, {17141, 21240}, {17300, 33091}, {17304, 32925}, {17396, 29666}, {17489, 30107}, {20335, 31052}, {30949, 31063}, {31078, 31098}, {31084, 31101}, {31090, 31117}

X(31078) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b^2 + c^2) (a^4 + a^2 b^2 + a^2 c^2 + 3 b^2 c^2) : :

X(31078) lies on these lines: {2, 39}, {23, 17128}, {99, 10130}, {141, 4576}, {308, 4609}, {2528, 8371}, {2896, 5189}, {3053, 16949}, {3313, 3620}, {3314, 5169}, {3734, 26233}, {5354, 17129}, {7496, 9149}, {7570, 7925}, {7879, 31133}, {7905, 31068}, {11794, 18019}, {13410, 24256}, {16063, 16990}, {18067, 27152}, {31077, 31098}

X(31079) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^2 b - a b^2 + 2 b^3 + a^2 c + b^2 c - a c^2 + b c^2 + 2 c^3 : :

X(31079) lies on these lines: {1, 2}, {31, 28599}, {38, 28595}, {75, 18066}, {149, 17280}, {518, 31017}, {1654, 3448}, {2887, 17165}, {3264, 9464}, {3314, 31071}, {3315, 17283}, {3416, 16704}, {3662, 17154}, {3681, 31037}, {3703, 4972}, {3717, 26580}, {3773, 33136}, {3790, 33134}, {3823, 24589}, {3923, 21282}, {3932, 31035}, {3952, 25760}, {3995, 32773}, {4085, 32848}, {4388, 33166}, {4427, 4660}, {4429, 17495}, {4438, 33074}, {4514, 33157}, {4645, 33170}, {4865, 26061}, {4892, 31161}, {4980, 21949}, {5014, 32777}, {5015, 11319}, {5169, 31084}, {5300, 11115}, {5690, 8229}, {6327, 24695}, {6636, 23361}, {7496, 31073}, {7779, 24345}, {11330, 20556}, {17140, 25957}, {17145, 33087}, {17163, 32865}, {17184, 20068}, {17491, 31134}, {19717, 33073}, {19742, 33075}, {20064, 26065}, {20290, 32912}, {21026, 24325}, {21747, 28512}, {23541, 25243}, {24349, 25959}, {25958, 32937}, {27804, 33092}, {31025, 33108}, {31041, 31317}, {31075, 31089}, {31080, 31099}, {31087, 31090}, {31103, 31112}, {31237, 32920}, {32779, 32850}, {32780, 33072}, {32844, 33159}, {32947, 33164}, {32948, 33167}, {33076, 33115}, {33078, 33121}, {33079, 33119}

X(31080) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 - a^3 b + 2 a^2 b^2 - a b^3 - b^4 - a^3 c - 2 a b^2 c + b^3 c + 2 a^2 c^2 - 2 a b c^2 - a c^3 + b c^3 - c^4 : :

X(31080) lies on these lines: {2, 7}, {4, 25244}, {8, 4568}, {78, 20071}, {2478, 25237}, {3263, 20444}, {4703, 30801}, {5046, 25242}, {5084, 25261}, {7774, 29832}, {17001, 20072}, {17170, 27096}, {17744, 28734}, {20060, 27340}, {21285, 33299}, {31079, 31099}, {31085, 31090}, {31089, 31102}

X(31081) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b^2 - a^2 b^3 + 2 a b^4 - 2 a^3 b c - 2 a b^3 c + 2 b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 2 a b c^3 + b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(31081) lies on these lines: {1, 2}, {674, 17232}, {15621, 31073}, {21282, 24260}, {31075, 31126}, {31084, 31097}

X(31082) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b - a b^3 + 4 b^4 - a^3 c - 2 a^2 b c - 3 b^3 c + 4 b^2 c^2 - a c^3 - 3 b c^3 + 4 c^4 : :

X(31082) lies on these lines: {2, 44}, {3314, 31071}, {3837, 26985}, {31093, 31129}

X(31083) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b + a b^3 - b^4 + a^3 c + 2 a^2 b c + 3 b^3 c - b^2 c^2 + a c^3 + 3 b c^3 - c^4 : :

X(31083) lies on these lines: {2, 45}, {3314, 31071}, {3662, 32856}, {4699, 33108}, {5121, 24199}, {16997, 26806}

X(31084) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b^2 + a^2 b^3 - a b^4 + b^5 - b^4 c - a^3 c^2 - a b^2 c^2 + a^2 c^3 - a c^4 - b c^4 + c^5 : :

X(31084) lies on these lines: {2, 11}, {325, 31130}, {3314, 31075}, {3705, 17860}, {3846, 29679}, {3920, 17725}, {3944, 27493}, {4417, 33091}, {4699, 23305}, {5169, 31079}, {6980, 24808}, {7191, 33128}, {16063, 31108}, {17056, 29815}, {19314, 31493}, {29832, 31100}, {31071, 31092}, {31077, 31101}, {31081, 31097}

X(31085) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + b^6 + 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 + 2 a b c^4 - b^2 c^4 + c^6 : :

X(31085) lies on these lines: {2, 12}, {3314, 31112}, {4193, 5211}, {5169, 31079}, {16063, 31073}, {16067, 17757}, {31080, 31090}, {31091, 31126}

X(31086) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c + 4 a^3 b 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 + 2 c^5 : :

X(31086) lies on these lines: {2, 7}, {312, 31058}, {5169, 31079}, {7377, 25244}, {20248, 21244}, {20535, 26610}, {31103, 31113}

X(31087) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b + a b^3 + a^3 c + 2 a b^2 c - b^3 c + 2 a b c^2 + a c^3 - b c^3 : :

X(31087) lies on these lines: {1, 4568}, {2, 37}, {10, 17489}, {72, 20109}, {144, 20101}, {148, 1655}, {190, 5276}, {193, 4661}, {194, 25244}, {304, 26759}, {612, 3729}, {894, 3920}, {1390, 5263}, {1654, 33091}, {1930, 28594}, {3314, 31093}, {3501, 25248}, {3617, 21216}, {3679, 17497}, {3952, 24514}, {3954, 17137}, {3963, 8024}, {3971, 24259}, {4071, 31023}, {4365, 4431}, {4393, 16782}, {4552, 7179}, {5205, 26279}, {5275, 17262}, {7191, 17319}, {7774, 29832}, {8878, 21217}, {9055, 24512}, {10327, 17007}, {10459, 17760}, {16830, 25264}, {16998, 20045}, {16999, 31073}, {17141, 17750}, {17163, 29667}, {17248, 29679}, {17379, 29815}, {17744, 30130}, {20483, 25345}, {21101, 31063}, {25255, 29641}, {25261, 27269}, {25282, 28369}, {29823, 31088}, {31071, 31129}, {31079, 31090}, {31102, 31121}

X(31088) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b^2 + a^2 b^4 + a^4 c^2 + 4 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 : :

X(31088) lies on these lines: {2, 39}, {6, 4576}, {23, 7783}, {148, 7533}, {193, 22829}, {325, 31076}, {385, 7496}, {574, 26233}, {1995, 31859}, {3001, 7774}, {3231, 32449}, {3314, 31124}, {3613, 5169}, {5189, 7785}, {5354, 7839}, {5640, 25047}, {5969, 13410}, {5971, 26257}, {7292, 25264}, {7467, 32516}, {7570, 17005}, {7855, 31068}, {7998, 32451}, {9168, 31296}, {10754, 15019}, {11163, 14570}, {15822, 31652}, {17024, 32095}, {29815, 31999}, {29823, 31087}, {29832, 31118}

X(31089) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a b^4 + b^5 + a^3 b c + a b^3 c + b^4 c + a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 + a c^4 + b c^4 + c^5 : :

X(31089) lies on these lines: {2, 6}, {38, 3705}, {147, 4220}, {239, 24995}, {291, 2887}, {316, 24271}, {321, 1916}, {980, 7796}, {1281, 4683}, {1432, 16603}, {1655, 26601}, {1959, 3661}, {2896, 21495}, {3770, 21245}, {3948, 17669}, {4518, 15523}, {4766, 27064}, {5337, 7768}, {5988, 21085}, {7836, 21511}, {7879, 21477}, {7881, 11343}, {7891, 21508}, {7904, 21537}, {14712, 16046}, {16752, 17673}, {16886, 17789}, {19312, 26064}, {26770, 31050}, {31075, 31079}, {31080, 31102}, {31119, 31126}

X(31090) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4 : :

X(31090) lies on these lines: {2, 6}, {9, 19555}, {10, 257}, {37, 30179}, {75, 25345}, {147, 6998}, {148, 17677}, {192, 3703}, {194, 16062}, {274, 17673}, {315, 17688}, {316, 24275}, {894, 2887}, {1281, 24697}, {1447, 17252}, {1655, 5051}, {1698, 17739}, {2896, 16061}, {3061, 30177}, {3454, 17499}, {3662, 24631}, {3705, 17248}, {3791, 17363}, {3925, 11683}, {4195, 7823}, {4201, 7783}, {4234, 14712}, {4972, 17759}, {5299, 30176}, {6626, 8290}, {7179, 16609}, {7249, 27691}, {7762, 17698}, {7785, 13740}, {7813, 16712}, {7836, 16060}, {7891, 22267}, {11359, 31859}, {16915, 26085}, {17280, 26590}, {17289, 20541}, {17669, 27040}, {24514, 25760}, {29679, 31337}, {31077, 31117}, {31079, 31087}, {31080, 31085}, {31097, 31098}

X(31091) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 - a^2 b + 3 a b^2 - 3 b^3 - a^2 c - b^2 c + 3 a c^2 - b c^2 - 3 c^3 : :

X(31091) lies on these lines: {1, 2}, {144, 33650}, {149, 346}, {345, 5014}, {390, 32849}, {497, 32862}, {908, 4901}, {956, 7465}, {2550, 33089}, {3262, 31130}, {3434, 3703}, {3701, 5187}, {3717, 31018}, {3974, 11680}, {4190, 5300}, {4307, 33170}, {4310, 25959}, {4385, 6871}, {4514, 17776}, {4672, 4865}, {4720, 7474}, {5015, 6872}, {5423, 27131}, {5846, 24597}, {6057, 11235}, {6327, 20078}, {7270, 20076}, {8229, 12245}, {9053, 30811}, {11330, 26770}, {16063, 20344}, {17740, 32850}, {17784, 33168}, {19819, 21949}, {21282, 24280}, {24477, 33078}, {25592, 32818}, {26098, 33162}, {30614, 33124}, {31085, 31126}

X(31092) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b - 3 a^3 b^2 + 5 a^2 b^3 - 5 a b^4 + 2 b^5 + a^4 c + 4 a^3 b c + 2 a^2 b^2 c + 4 a b^3 c - 3 b^4 c - 3 a^3 c^2 + 2 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 + 5 a^2 c^3 + 4 a b c^3 + b^2 c^3 - 5 a c^4 - 3 b c^4 + 2 c^5 : :

X(31092) lies on these lines: {1, 2}, {21282, 24283}, {31071, 31084}, {31121, 31126}

X(31093) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b - a b^3 + 2 b^4 - a^3 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(31093) lies on these lines: {1, 2}, {312, 31031}, {335, 31041}, {693, 20950}, {3263, 4033}, {3314, 31087}, {3933, 25244}, {3936, 4437}, {3952, 4766}, {3995, 26590}, {4071, 20347}, {4358, 31058}, {9464, 31121}, {17002, 17373}, {17495, 26582}, {17738, 21282}, {20533, 32849}, {30837, 32927}, {31020, 32851}, {31071, 31130}, {31082, 31129}

X(31094) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b - c) (a^3 b + a^2 b^2 + 2 a b^3 + a^3 c - a^2 b c + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 - b^2 c^2 + 2 a c^3 - 2 b c^3) : :

X(31094) lies on these lines: {2, 649}, {693, 28894}, {3766, 31040}, {3837, 26985}, {4010, 4926}, {8665, 23301}, {9002, 17238}, {20950, 20952}, {23655, 29815}, {31096, 31132}

X(31095) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b - c) (a^4 + 2 a^2 b^2 + a b^3 + 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(31095) lies on these lines: {2, 3572}, {6084, 17494}, {3837, 30952}, {9013, 17300}

X(31096) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b - c) (2 a^3 b^2 + 2 a b^4 + 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 + 3 a b^2 c^2 - 2 b^3 c^2 + 2 a b c^3 - 2 b^2 c^3 + 2 a c^4 - 2 b c^4) : :

X(31096) lies on these lines: {2, 667}, {23301, 31072}, {31097, 31110}

X(31097) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b^2 + a^2 b^3 + 2 b^5 - 2 a^3 b c - 2 a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 + 2 c^5 : :

X(31097) lies on these lines: {2, 31}, {120, 31116}, {3314, 31071}, {29823, 31124}, {31081, 31084}, {31090, 31098}, {31096, 31110}

X(31098) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics 2 a^3 + 3 a^2 b + a b^2 + 4 b^3 + 3 a^2 c + 3 b^2 c + a c^2 + 3 b c^2 + 4 c^3 : :

X(31098) lies on these lines: {1, 2}, {3844, 16704}, {4655, 26061}, {17154, 17291}, {21282, 24295}, {25959, 26083}, {31077, 31078}, {31090, 31097}, {31100, 31110}

X(31099) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^6 + 3 a^4 b^2 - a^2 b^4 - 3 b^6 + 3 a^4 c^2 - 2 a^2 b^2 c^2 + 3 b^4 c^2 - a^2 c^4 + 3 b^2 c^4 - 3 c^6 : :

X(31099) lies on these lines: {2, 3}, {66, 193}, {125, 31670}, {253, 18019}, {323, 5921}, {576, 1899}, {1853, 6515}, {1993, 8549}, {2986, 3424}, {3060, 23291}, {3260, 9464}, {3266, 32816}, {3292, 11550}, {3313, 3620}, {3580, 23049}, {3619, 21766}, {5274, 9539}, {5297, 10590}, {6453, 8280}, {6454, 8281}, {6776, 11422}, {6800, 14927}, {7292, 10591}, {7748, 15820}, {9140, 32247}, {11002, 19161}, {11056, 14907}, {11245, 11482}, {11442, 23061}, {12384, 31127}, {14389, 25406}, {14853, 15019}, {15302, 31404}, {15801, 32337}, {18382, 23315}, {18919, 26926}, {19577, 20065}, {20061, 28604}, {21850, 26869}, {23332, 33586}, {29832, 31121}, {31079, 31080}

X(31099) = anticomplement of X(7493)

X(31100) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + b^6 - a^4 b c - a^3 b^2 c - a^2 b^3 c - 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 b c^4 - b^2 c^4 + c^6 : :

X(31100) lies on these lines: {2, 3}, {1224, 3841}, {1390, 21907}, {1441, 18019}, {3294, 24055}, {3448, 15988}, {3822, 5297}, {7191, 24387}, {7292, 25639}, {11680, 17070}, {12607, 33091}, {29823, 31109}, {29832, 31084}, {31075, 31079}, {31080, 31085}, {31098, 31110}

X(31101) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + b^6 - a^4 c^2 + 3 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :

X(31101) lies on these lines: {2, 3}, {125, 2979}, {154, 23315}, {305, 18019}, {323, 1899}, {394, 3448}, {511, 26913}, {1216, 23294}, {1568, 15072}, {1853, 3410}, {1994, 18911}, {3620, 23300}, {3917, 23293}, {5972, 26881}, {7712, 10192}, {7998, 21243}, {9544, 11064}, {10625, 26917}, {11004, 11245}, {11123, 23301}, {11206, 32125}, {11444, 20299}, {11464, 14156}, {12111, 13399}, {15051, 32743}, {15533, 25328}, {18388, 20791}, {30786, 33651}, {31077, 31084}

X(31101) = orthoptic-circle-of-Steiner-inellipse-inverse of X(34152)

X(31102) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -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^6 b c - a^5 b^2 c - a^4 b^3 c + a^3 b^4 c + a^2 b^5 c + a b^6 c + b^7 c - a^6 c^2 - a^5 b c^2 + a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 2 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 - a^4 c^4 + a^3 b c^4 + 2 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 + 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 + c^8 : :

X(31102) lies on these lines: {2, 3}, {306, 4568}, {1762, 24687}, {22000, 24055}, {31080, 31089}, {31087, 31121}

X(31103) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 106

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 - 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 + b^8 c - a^7 c^2 - a^6 b c^2 + 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 - a^6 c^3 - a^5 b c^3 + a^4 b^2 c^3 + 4 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - a b^5 c^3 - a^5 c^4 - a^4 b c^4 + 2 a^3 b^2 c^4 + 2 a^2 b^3 c^4 - 2 a b^4 c^4 - 2 b^5 c^4 - a^4 c^5 + a^3 b c^5 + 2 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^2 c^7 + a b c^7 + a c^8 + b c^8 + c^9 : :

X(31103) lies on these lines: {2, 3}, {18019, 20336}, {31079, 31112}, {31086, 31113}

X(31104) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^8 b - 2 a^4 b^5 + b^9 + 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 + 3 a^5 b^2 c^2 - 2 a^4 b^3 c^2 - 3 a^3 b^4 c^2 + 2 a^2 b^5 c^2 - b^7 c^2 + a^5 b c^3 - 2 a^4 b^2 c^3 - 4 a^3 b^3 c^3 - a^2 b^4 c^3 + a b^5 c^3 + b^6 c^3 - a^4 b c^4 - 3 a^3 b^2 c^4 - a^2 b^3 c^4 - b^5 c^4 - 2 a^4 c^5 - a^3 b c^5 + 2 a^2 b^2 c^5 + a b^3 c^5 - b^4 c^5 - a^2 b c^6 + b^3 c^6 - a b c^7 - b^2 c^7 + c^9 : :

X(31104) lies on these lines: {2, 3}, {307, 18019}

X(31105) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^6 + 5 a^4 b^2 - a^2 b^4 - 5 b^6 + 5 a^4 c^2 + 5 b^4 c^2 - a^2 c^4 + 5 b^2 c^4 - 5 c^6 : :

X(31105) lies on these lines: {2, 3}, {323, 11180}, {1007, 14360}, {1992, 3448}, {2549, 6032}, {2781, 11002}, {3014, 7774}, {3818, 13857}, {5476, 18911}, {5971, 32827}, {7703, 15360}, {9019, 21356}, {9140, 10752}, {9143, 14982}, {11056, 11057}, {11160, 15431}, {20582, 21766}, {22112, 25565}

X(31106) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 106

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 - 4 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(31106) lies on these lines: {2, 3}, {497, 28605}, {7774, 29832}, {29823, 31121}

X(31107) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b^2 + c^2) (b^4 + c^4 + a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(31107) lies on these lines: {2, 3}, {99, 30785}, {115, 26235}, {141, 4576}, {251, 16275}, {305, 7922}, {325, 31076}, {574, 30747}, {626, 3266}, {2896, 19577}, {3291, 7861}, {3314, 9464}, {3815, 31132}, {5354, 7797}, {7752, 15302}, {7761, 26233}, {7790, 9465}, {7794, 8024}, {7831, 10130}, {7853, 30749}, {7931, 14360}, {7934, 11059}, {9229, 18019}, {24322, 24726}

X(31108) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a (a^5 - a b^4 - a^2 b^2 c + a b^3 c - 2 b^4 c - a^2 b c^2 - 5 a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4 - 2 b c^4) : :

X(31108) lies on these lines: {2, 12}, {21, 5211}, {1616, 17024}, {2292, 7191}, {5205, 5258}, {7496, 31073}, {7570, 31110}, {10989, 31109}, {16063, 31084}, {16823, 25495}, {27963, 30074}, {29823, 31122

X(31109) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -2 a^4 b^2 + 2 b^6 - a^3 b^2 c + a^2 b^3 c - 2 a b^4 c - 2 a^4 c^2 - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - a b^2 c^3 - 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31109) lies on these lines: {2, 35}, {5169, 31079}, {7570, 31073}, {10989, 31108}, {29823, 31100}, {31113, 31119}

X(31110) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -2 a^4 b^2 + 2 b^6 + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 - 2 b^4 c^2 - a^2 b c^3 + a b^2 c^3 + 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31110) lies on these lines: {2, 36}, {5169, 31079}, {7570, 31108}, {10989, 31073}, {31096, 31097}, {31098, 31100}

X(31111) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b + 2 a^2 b^3 - a b^4 + 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 : :

X(31111) lies on these lines: {2, 38}, {2886, 31025}, {3112, 4583}, {7779, 33091}, {25282, 30966}, {26232, 32935}, {26250, 32940}, {31075, 31079}

X(31112) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^5 b^2 - a^4 b^3 + a^3 b^4 - a^2 b^5 + 2 b^7 - 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 - b^4 c^3 + a^3 c^4 - b^3 c^4 - a^2 c^5 - b^2 c^5 + 2 c^7 : :

X(31112) lies on these lines: {2, 48}, {3314, 31085}, {31079, 31103}

X(31113) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b^2 + a^2 b^4 + 2 a b^5 + 2 b^6 + 2 a b^4 c + 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 + b^4 c^2 + 2 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 + 2 c^6 : :

X(31113) lies on these lines: {2, 58}, {3314, 5169}, {11330, 31024}, {31075, 31079}, {31086, 31103}, {31090, 31097}, {31109, 31119}

X(31114) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 106

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 + 3 a^8 b^2 c^2 + 3 a^6 b^4 c^2 - a^4 b^6 c^2 - 2 a^2 b^8 c^2 - 2 b^10 c^2 + a^8 c^4 + 3 a^6 b^2 c^4 - 4 a^4 b^4 c^4 + 3 a^2 b^6 c^4 - b^8 c^4 + 2 a^6 c^6 - a^4 b^2 c^6 + 3 a^2 b^4 c^6 + 4 b^6 c^6 - 2 a^4 c^8 - 2 a^2 b^2 c^8 - b^4 c^8 - a^2 c^10 - 2 b^2 c^10 + c^12 : :

X(31114) lies on these lines: {2, 3}, {18019, 20564}

X(31115) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b + 2 a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c + 2 a^2 b^2 c + 2 a b^3 c + b^4 c + 2 a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - a c^4 + b c^4 : :

X(31115) lies on these lines: {2, 38}, {3314, 31071}, {3995, 17599}, {24241, 29835}, {29823, 31087}, {29832, 31130}, {31075, 33108}

X(31116) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b - 2 a^3 b^2 + a b^4 + 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 + 4 c^5 : :

X(31116) lies on these lines: {2, 896}, {120, 31097}, {3837, 26985}, {31075, 31079}

X(31117) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b + a b^3 + 2 b^4 + a^3 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(31117) lies on these lines: {2, 7}, {3264, 9464}, {3314, 31087}, {4576, 30966}, {4741, 17001}, {4911, 11115}, {4920, 17164}, {6656, 25244}, {10513, 20020}, {16347, 25581}, {17550, 25237}, {17751, 24211}, {24241, 29824}, {31077, 31090}

X(31118) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b^2 + a^2 b^4 - 2 a^4 b c - 4 a^2 b^3 c + 2 a b^4 c + a^4 c^2 + 4 a^2 b^2 c^2 - b^4 c^2 - 4 a^2 b c^3 + a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(31118) lies on these lines: {2, 668}, {17488, 26274}, {29832, 31088}, {31125, 31126}

X(31119) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(31119) lies on these lines: {1, 2}, {2239, 28599}, {3314, 31075}, {17140, 30969}, {17165, 30953}, {20544, 31026}, {21282, 24259}, {30952, 31122}, {31089, 31126}, {31109, 31113}

X(31120) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^4 b + a^3 b^2 + a^2 b^3 - b^5 + a^4 c + 2 a^2 b^2 c + a^3 c^2 + 2 a^2 b c^2 + a b^2 c^2 + a^2 c^3 - c^5 : :

X(31120) lies on these lines: {2, 6}, {316, 24296}, {1281, 24725}, {3705, 32771}, {3891, 11680}, {4376, 32851}, {4518, 29643}, {5169, 29832}, {7081, 32852}, {17367, 24995}, {24239, 33064}, {25683, 30830}

X(31121) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c - 2 a^2 b^2 c - b^4 c - a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4 - b c^4 + 2 c^5 : :

X(31121) lies on these lines: {2, 7}, {321, 31031}, {857, 25237}, {3448, 17778}, {3674, 26575}, {4892, 30801}, {9464, 31093}, {16580, 27514}, {17491, 26232}, {20173, 31058}, {20247, 26012}, {26563, 31032}, {27187, 31039}, {29823, 31106}, {29832, 31099}, {30746, 32935}, {30767, 32938}, {31075, 31079}, {31087, 31102}, {31092, 31126}

X(31122) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^5 b + 2 a^3 b^3 - 3 a b^5 + a^5 c - 2 a^4 b c + a^3 b^2 c - 3 a^2 b^3 c - b^5 c + a^3 b c^2 - 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 - 3 a c^5 - b c^5 : :

X(31122) lies on these lines: {2, 65}, {5169, 31079}, {7774, 29832}, {29823, 31108}, {30952, 31119}

X(31123) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics a^10 + 2 a^8 b^2 - a^6 b^4 - a^4 b^6 - b^10 + 2 a^8 c^2 - 2 a^4 b^4 c^2 - a^6 c^4 - 2 a^4 b^2 c^4 + b^6 c^4 - a^4 c^6 + b^4 c^6 - c^10 : :

X(31123) lies on these lines: {2, 66}, {20, 99}, {160, 3314}, {393, 3108}, {3329, 5169}, {7391, 7774}

X(31124) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b^2 + c^2) (a^4 + 3 a^2 b^2 + 2 b^4 + 3 a^2 c^2 + b^2 c^2 + 2 c^4) : :

X(31124) lies on these lines: {2, 32}, {23, 7928}, {141, 4576}, {3005, 9168}, {3266, 7849}, {3314, 31088}, {5169, 16986}, {5354, 7948}, {7496, 7931}, {7853, 26235}, {7864, 8267}, {7922, 15302}, {7937, 9465}, {8024, 14125}, {29823, 31097}

X(31125) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (a^2 + b^2 - 2 c^2) (a^2 - 2 b^2 + c^2) (b^2 + c^2) : :

X(31125) lies on these lines: {2, 99}, {39, 23297}, {66, 193}, {125, 25047}, {141, 4576}, {194, 31857}, {316, 17964}, {325, 31132}, {385, 10989}, {542, 10552}, {625, 3266}, {691, 5189}, {850, 8430}, {892, 7779}, {897, 28604}, {1370, 14908}, {1502, 4609}, {1916, 5466}, {3014, 7774}, {3613, 5169}, {3618, 32740}, {4235, 8791}, {4568, 15523}, {5025, 14263}, {5468, 11646}, {5969, 8288}, {5971, 14041}, {6031, 33264}, {6032, 7757}, {6321, 7417}, {7378, 8753}, {7496, 17006}, {7785, 14246}, {7840, 17948}, {8267, 31074}, {8801, 17983}, {8877, 8878}, {9140, 10754}, {9143, 14833}, {9191, 18007}, {9213, 31296}, {9775, 14639}, {9878, 9999}, {10416, 20063}, {10561, 31072}, {14977, 18019}, {16063, 17008}, {31022, 31058}, {31118, 31126}

X(31125) = anticomplement of X(7664)

X(31126) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^3 b^2 + a^2 b^3 - a b^4 + b^5 + a^3 b c + a b^3 c - b^4 c - a^3 c^2 - a b^2 c^2 + a^2 c^3 + a b c^3 - a c^4 - b c^4 + c^5 : :

X(31126) lies on these lines: {2, 11}, {5, 24808}, {427, 1897}, {858, 5211}, {867, 5992}, {1290, 5189}, {1479, 17522}, {3006, 4518}, {3140, 31057}, {3699, 5741}, {3705, 20237}, {3799, 24250}, {3837, 31129}, {3920, 17719}, {5169, 29832}, {5988, 25094}, {7191, 33135}, {7379, 26470}, {7427, 10738}, {9669, 16048}, {16823, 24387}, {16830, 25639}, {17724, 29815}, {17793, 25760}, {21252, 25048}, {24003, 25960}, {29823, 31100}, {31075, 31081}, {31085, 31091}, {31089, 31119}, {31092, 31121}, {31118, 31125}

X(31127) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -a^6 b^2 - a^2 b^6 + 2 b^8 - a^6 c^2 + 4 a^4 b^2 c^2 - b^6 c^2 - 2 b^4 c^4 - a^2 c^6 - b^2 c^6 + 2 c^8 : :

X(31127) lies on these lines: {2, 98}, {67, 2407}, {262, 7703}, {265, 7422}, {476, 5189}, {858, 12079}, {1550, 5999}, {1916, 5466}, {2023, 8288}, {2419, 18019}, {6636, 23181}, {7840, 16092}, {12384, 31099}

X(31128) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (2 a^2 - b^2 - c^2) (a^4 + a^2 b^2 + a^2 c^2 - 3 b^2 c^2) : :

X(31128) lies on these lines: {2, 99}, {6, 4576}, {187, 3266}, {352, 12215}, {1383, 3552}, {1649, 7711}, {1975, 20481}, {2418, 10354}, {2502, 10330}, {3291, 15301}, {3448, 3620}, {4366, 17495}, {5026, 5468}, {5099, 5189}, {5182, 9146}, {5912, 14588}, {5971, 13586}, {7492, 30793}, {7496, 9149}, {7708, 18906}, {7779, 23965}, {7836, 14357}, {7925, 10989}, {8289, 9828}, {8588, 26233}, {8589, 30749}, {9775, 21166}, {10552, 18800}, {10553, 14916}, {14424, 22105}, {16063, 31076}, {16597, 17302}, {26276, 32456}

X(31129) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics b^4 - a^2 b c + a b^2 c - 2 b^3 c + a b c^2 + b^2 c^2 - 2 b c^3 + c^4 : :

X(31129) lies on these lines: {2, 45}, {7, 16997}, {105, 5992}, {141, 3807}, {239, 24712}, {334, 3263}, {1565, 9263}, {2530, 32454}, {3314, 31130}, {3665, 21226}, {3837, 31126}, {4013, 31647}, {4442, 5211}, {4688, 25383}, {5121, 24200}, {5205, 32856}, {6386, 8024}, {31071, 31087}, {31077, 31090}, {31082, 31093}

X(31130) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics b c (a^2 - a b + 2 b^2 - a c + 2 c^2) : :

X(31130) lies on these lines: {2, 37}, {7, 10327}, {8, 150}, {69, 4661}, {85, 4696}, {86, 29815}, {105, 5695}, {145, 304}, {183, 31073}, {274, 17589}, {322, 10513}, {325, 31084}, {341, 26563}, {612, 25590}, {614, 4365}, {3006, 7179}, {3241, 14210}, {3262, 31091}, {3264, 9464}, {3314, 31129}, {3596, 8024}, {3617, 20911}, {3623, 18156}, {3663, 4082}, {3673, 3701}, {3757, 10447}, {3761, 20893}, {3875, 7191}, {3920, 10436}, {3945, 20020}, {3952, 30946}, {4019, 27624}, {4208, 4385}, {4357, 29679}, {4360, 17024}, {4363, 5276}, {4430, 30941}, {4737, 30806}, {4872, 5014}, {4911, 5300}, {4967, 29667}, {5205, 26229}, {5275, 17118}, {6646, 17007}, {7396, 20895}, {9055, 30945}, {16284, 25296}, {16823, 32104}, {16830, 32092}, {17163, 18697}, {17451, 30057}, {17489, 27299}, {17760, 30036}, {20347, 21415}, {20881, 26258}, {20898, 26034}, {20955, 25278}, {21101, 30949}, {21436, 31995}, {24209, 29857}, {24241, 33120}, {25237, 27523}, {25242, 26770}, {25592, 32831}, {27248, 28598}, {29832, 31115}, {31071, 31093}

X(31130) = anticomplement of X(26242)

X(31131) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics (b - c) (a^3 - a^2 b + 3 a b^2 - b^3 - a^2 c + a b c - 2 b^2 c + 3 a c^2 - 2 b c^2 - c^3) : :

X(31131) lies on these lines: {2, 900}, {513, 30565}, {522, 4728}, {523, 7840}, {659, 28602}, {2254, 2786}, {2526, 28894}, {2826, 10711}, {3263, 3766}, {3679, 23888}, {3837, 31126}, {4088, 28890}, {4526, 26242}, {6084, 20344}, {14432, 28521}, {21052, 28487}, {24721, 28859}

X(31132) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics -3 a^4 b^2 + a^2 b^4 + 4 b^6 - 3 a^4 c^2 - 2 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 + 4 c^6 : :

X(31132) lies on these lines: {2, 187}, {325, 31125}, {1648, 5103}, {3314, 5169}, {3815, 31107}, {7912, 31857}, {7925, 10989}, {8665, 23301}

Collineation mappings involving Gemini triangle 107: X(31133)-X(31181)

m(X) = x - 2 y - 2 z : -2 x + y - 2 z : -2 x - 2 y + z : :

and m(X) is a self-inverse mapping; indeed, m(X) = reflection of X in X(2); the fixed points are X(2) and every point on the line at infinity. (Clark Kimberling, January 22, 2019)

X(31133) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^6 + 2 a^4 b^2 - a^2 b^4 - 2 b^6 + 2 a^4 c^2 + 2 b^4 c^2 - a^2 c^4 + 2 b^2 c^4 - 2 c^6 : :

X(31133) lies on these lines: {2, 3}, {146, 3426}, {184, 11645}, {220, 24055}, {305, 7809}, {323, 18440}, {524, 11442}, {542, 1993}, {599, 2979}, {895, 8877}, {1194, 11648}, {1351, 3448}, {1494, 18018}, {1799, 11057}, {1853, 2781}, {1899, 20423}, {1992, 11216}, {2892, 18125}, {2986, 14458}, {3167, 9143}, {3260, 7788}, {3266, 7773}, {3580, 31670}, {3818, 15066}, {3819, 25561}, {3917, 11178}, {3920, 11237}, {5032, 18919}, {5297, 10895}, {5309, 5359}, {5422, 5476}, {5480, 18911}, {5654, 16658}, {6800, 29012}, {7191, 11238}, {7292, 10896}, {7703, 15107}, {7776, 9464}, {7802, 11056}, {7811, 16275}, {7816, 30747}, {7823, 19577}, {7825, 30749}, {7837, 8878}, {7865, 8891}, {7879, 31078}, {7881, 31076}, {7998, 10516}, {8029, 31176}, {9306, 13857}, {9745, 15820}, {9766, 15356}, {10559, 14833}, {10706, 11455}, {11002, 26869}, {14537, 19220}, {14836, 15437}, {15069, 23061}, {15360, 23293}, {16789, 21356}, {17810, 26913}, {18382, 32125}, {18424, 20481}, {19924, 21243}, {21766, 24206}

X(31133) = reflection of X(22) in X(2)

X(31134) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 - 2 b^3 - 2 c^3 : :

X(31134) lies on these lines: {2, 31}, {8, 32856}, {9, 21026}, {10, 24725}, {30, 30269}, {38, 17274}, {69, 33136}, {141, 33104}, {149, 33087}, {226, 33074}, {306, 28580}, {320, 33120}, {354, 31138}, {516, 33156}, {519, 3891}, {545, 3703}, {599, 674}, {722, 30666}, {734, 7757}, {744, 4664}, {758, 3679}, {766, 7818}, {846, 27754}, {896, 29857}, {902, 30811}, {903, 17155}, {1150, 21241}, {1376, 27739}, {1836, 15523}, {1853, 2390}, {2177, 3936}, {2835, 10712}, {2886, 33080}, {2895, 32865}, {3006, 4655}, {3052, 29865}, {3120, 3416}, {3434, 33081}, {3550, 30831}, {3662, 32844}, {3705, 33067}, {3720, 17313}, {3771, 4450}, {3782, 28503}, {3821, 33070}, {3838, 27747}, {3925, 17330}, {3944, 33078}, {3989, 24441}, {4085, 31034}, {4138, 33127}, {4417, 32948}, {4429, 32843}, {4514, 33069}, {4650, 29872}, {4683, 17333}, {4715, 32912}, {4865, 17184}, {4892, 26227}, {4972, 32946}, {5057, 29674}, {5741, 9350}, {5846, 33143}, {5847, 33128}, {5905, 33162}, {7232, 17449}, {7262, 29873}, {9256, 31148}, {9313, 31147}, {9812, 28885}, {10707, 31137}, {11680, 33085}, {15699, 20575}, {17017, 17382}, {17271, 31330}, {17275, 21027}, {17298, 17450}, {17310, 32915}, {17320, 32776}, {17342, 32930}, {17378, 32773}, {17469, 25527}, {17483, 33169}, {17484, 33165}, {17491, 31079}, {17601, 27757}, {17679, 23682}, {17766, 33122}, {17768, 33161}, {17770, 33114}, {17889, 33075}, {18134, 32947}, {20290, 32853}, {20292, 32778}, {21282, 31017}, {24248, 32848}, {24703, 29687}, {24715, 33077}, {24723, 29643}, {26223, 28595}, {27184, 33072}, {28599, 32920}, {29667, 33097}, {29671, 32950}, {29673, 32859}, {29679, 33096}, {29849, 33068}, {31019, 33076}, {31053, 33079}, {32782, 33109}, {32842, 33149}, {32846, 33134}, {32847, 33151}, {32850, 33065}, {32855, 33102}, {32857, 33089}, {32858, 33095}, {32861, 33131}, {32862, 33099}, {32863, 33141}, {32866, 33146}, {33066, 33117}, {33071, 33125}, {33082, 33108}, {33084, 33110}, {33088, 33145}, {33090, 33103}, {33091, 33101}, {33092, 33100}, {33093, 33154}, {33106, 33172}

X(31134) = reflection of X(31) in X(2)

X(31135) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 - a^3 b + 2 a b^3 - 2 b^4 - a^3 c + 2 b^3 c + 2 a c^3 + 2 b c^3 - 2 c^4 : :

X(31135) lies on these lines: {2, 41}, {150, 4390}, {599, 8679}, {766, 7818}, {2251, 31195}, {2389, 31140}, {2809, 3679}, {6173, 6175}, {31136, 31152}

X(31135) = reflection of X(41) in X(2)

X(31136) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^2 b - 2 a b^2 + a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 : :

X(31136) lies on these lines: {1, 2}, {38, 536}, {69, 33104}, {75, 17449}, {106, 4803}, {141, 33136}, {142, 21027}, {149, 33082}, {244, 3696}, {310, 670}, {312, 4937}, {319, 32844}, {321, 537}, {333, 32943}, {350, 17271}, {354, 4688}, {518, 31161}, {545, 24690}, {599, 674}, {672, 17281}, {748, 4042}, {788, 30640}, {902, 1150}, {1468, 16394}, {2229, 17448}, {2308, 24552}, {2813, 10708}, {2886, 33081}, {2895, 33106}, {3434, 33080}, {3550, 5372}, {3739, 17450}, {3873, 31178}, {3886, 4414}, {3896, 6682}, {3936, 21242}, {3989, 4664}, {3994, 4519}, {3996, 32918}, {4007, 17756}, {4191, 15621}, {4192, 28204}, {4428, 16343}, {4441, 17274}, {4465, 4690}, {4479, 17149}, {4660, 21283}, {4674, 4793}, {4683, 24711}, {4709, 17495}, {4715, 24330}, {4732, 24589}, {4740, 17155}, {4755, 4891}, {4863, 33074}, {5235, 16484}, {5263, 32919}, {5361, 8616}, {5881, 19647}, {6702, 21042}, {8666, 11322}, {10707, 31143}, {11194, 16395}, {11680, 33084}, {12513, 16405}, {14753, 22072}, {14829, 32945}, {16704, 21747}, {16711, 17208}, {16748, 17179}, {17144, 30964}, {17145, 31025}, {17163, 24165}, {17231, 21026}, {17320, 30966}, {17678, 19787}, {21241, 31017}, {21805, 30818}, {24259, 28562}, {25349, 28309}, {31135, 31152}, {32782, 33141}, {32863, 33109}, {32864, 32942}, {32865, 33172}, {33085, 33110}, {33087, 33108}

X(31136) = reflection of X(42) in X(2)

X(31137) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^2 b - 2 a b^2 + a^2 c + 3 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 : :

X(31137) lies on these lines: {1, 2}, {11, 33087}, {36, 16395}, {56, 16396}, {141, 24217}, {310, 17179}, {312, 537}, {314, 18173}, {319, 24760}, {350, 17274}, {354, 31178}, {497, 33085}, {536, 982}, {545, 24691}, {594, 17051}, {599, 9025}, {903, 4479}, {1054, 3886}, {1150, 15485}, {1401, 4654}, {1699, 29353}, {2321, 24216}, {2810, 31142}, {3550, 32943}, {3706, 17063}, {3742, 4688}, {3816, 33084}, {3873, 31161}, {3999, 4519}, {4671, 17449}, {4693, 17595}, {4702, 17601}, {4709, 24620}, {4713, 4715}, {4740, 24165}, {4851, 17722}, {4873, 20331}, {4891, 17592}, {4966, 17717}, {5258, 16373}, {5563, 16405}, {5695, 18201}, {5881, 19546}, {6007, 6173}, {8616, 14829}, {10707, 31134}, {16468, 32919}, {17149, 18145}, {17196, 30939}, {17232, 21241}, {17234, 21242}, {17271, 30963}, {17281, 17754}, {17297, 30982}, {17313, 21264}, {17556, 30986}, {17591, 32915}, {17678, 19803}, {17721, 32846}, {17728, 33160}, {18141, 33109}, {19540, 28204}, {24260, 28562}, {24477, 33164}, {25350, 28309}, {28650, 32944}, {31140, 31151}

X(31137) = reflection of X(43) in X(2)

X(31138) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^2 + a b - 4 b^2 + a c + 4 b c - 4 c^2 : :

X(31138) lies on these lines: {2, 44}, {7, 17231}, {37, 7232}, {141, 10022}, {142, 17330}, {335, 536}, {354, 31134}, {513, 4379}, {518, 599}, {519, 1086}, {527, 4370}, {545, 3912}, {551, 752}, {1100, 3662}, {1266, 4727}, {1757, 19876}, {3241, 4645}, {3246, 17325}, {3306, 27739}, {3631, 24199}, {3664, 17384}, {3723, 17235}, {3739, 17271}, {3763, 4888}, {3828, 3836}, {3943, 4887}, {4677, 17119}, {4681, 17312}, {4686, 17296}, {4698, 17273}, {4702, 24692}, {4718, 4862}, {4726, 17295}, {4739, 17287}, {4755, 17254}, {4852, 17375}, {4869, 17276}, {4896, 17369}, {4912, 17264}, {4966, 28580}, {4969, 17067}, {7321, 17229}, {15492, 17265}, {15533, 16833}, {16666, 17290}, {16668, 16706}, {16669, 17282}, {16671, 17356}, {16726, 17179}, {16814, 17234}, {17232, 17342}, {17236, 28639}, {17239, 26806}, {17272, 31238}, {17278, 21296}, {17348, 17361}, {17357, 17365}, {24231, 28503}, {27487, 31349}, {27747, 31019}, {31145, 32850}

X(31138) = reflection of X(44) in X(2)

X(31139) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^2 + 2 a b - 2 b^2 + 2 a c + 8 b c - 2 c^2 : :

X(31139) lies on these lines: {1, 24452}, {2, 45}, {6, 4795}, {7, 17330}, {75, 17309}, {142, 17118}, {244, 4492}, {518, 599}, {519, 4675}, {551, 4356}, {1266, 16672}, {1644, 4413}, {3218, 25057}, {3241, 17392}, {3306, 27747}, {3739, 17253}, {3763, 25590}, {4361, 17378}, {4384, 4715}, {4445, 4772}, {4659, 4908}, {4677, 17374}, {4699, 7232}, {4739, 17298}, {4751, 17255}, {4862, 31238}, {4888, 6144}, {7222, 17337}, {7263, 16777}, {7321, 17259}, {10436, 17382}, {14475, 23352}, {15534, 16833}, {15668, 17320}, {16590, 16832}, {16724, 18206}, {17116, 17265}, {17237, 19875}, {17245, 31995}, {17251, 31317}, {17262, 27147}, {17316, 28309}, {27739, 31019}, {28301, 29571}, {29595, 31332}

X(31139) = reflection of X(45) in X(2)

X(31140) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 - a^2 b + 2 a b^2 - 2 b^3 - a^2 c + 2 b^2 c + 2 a c^2 + 2 b c^2 - 2 c^3 : :

X(31140) lies on these lines: {1, 3824}, {2, 11}, {6, 33104}, {8, 10895}, {10, 4679}, {12, 5082}, {30, 3428}, {35, 31493}, {38, 4492}, {56, 11112}, {63, 28534}, {200, 17605}, {210, 381}, {226, 519}, {354, 6173}, {376, 5842}, {377, 3304}, {382, 5258}, {405, 9670}, {442, 3303}, {443, 31420}, {474, 10199}, {499, 17564}, {518, 27479}, {527, 1836}, {535, 956}, {599, 674}, {614, 21949}, {748, 19624}, {902, 31187}, {908, 3711}, {940, 33109}, {958, 11114}, {962, 21677}, {1150, 21282}, {1155, 5231}, {1388, 17647}, {1466, 10957}, {1479, 31419}, {1482, 33592}, {1698, 9669}, {1706, 17606}, {1738, 17721}, {1824, 5064}, {1853, 31163}, {2094, 11246}, {2098, 5794}, {2320, 9963}, {2389, 31135}, {2475, 9657}, {2476, 3913}, {2478, 9671}, {2807, 10710}, {3006, 5695}, {3052, 24892}, {3091, 21031}, {3120, 3242}, {3241, 3475}, {3254, 15346}, {3534, 18499}, {3545, 7680}, {3579, 18544}, {3582, 8069}, {3583, 9708}, {3614, 7080}, {3632, 9654}, {3683, 9580}, {3689, 5219}, {3698, 9581}, {3706, 17294}, {3712, 30741}, {3715, 24703}, {3744, 17064}, {3748, 25525}, {3755, 17723}, {3830, 18407}, {3838, 3870}, {3893, 9578}, {3914, 17301}, {3929, 11372}, {3935, 10129}, {3936, 21283}, {3940, 18393}, {4042, 4388}, {4294, 24953}, {4309, 6675}, {4330, 17571}, {4361, 32844}, {4363, 24712}, {4383, 32865}, {4387, 17264}, {4442, 29832}, {4654, 5173}, {4660, 21242}, {4669, 21060}, {4677, 11525}, {4688, 31152}, {4713, 17251}, {4725, 17156}, {4740, 7840}, {4857, 11108}, {4860, 5880}, {4884, 28297}, {4971, 33088}, {5054, 32613}, {5055, 10679}, {5057, 5220}, {5119, 7308}, {5172, 16371}, {5175, 10950}, {5177, 15888}, {5178, 12635}, {5187, 9711}, {5204, 10527}, {5217, 26363}, {5221, 10916}, {5251, 9668}, {5288, 9655}, {5541, 17057}, {5687, 25639}, {5737, 32947}, {5784, 18839}, {5818, 10893}, {5853, 17718}, {5855, 31145}, {6063, 18821}, {6172, 9812}, {6284, 11111}, {6871, 12607}, {6955, 20418}, {7741, 9709}, {7776, 32104}, {7841, 16829}, {8255, 10580}, {8273, 12116}, {9614, 25917}, {9956, 11928}, {10310, 26470}, {10453, 17297}, {10894, 12245}, {11194, 17579}, {11274, 20586}, {11366, 26360}, {11367, 26359}, {12950, 20306}, {13624, 18543}, {15171, 19854}, {15338, 30478}, {17070, 26228}, {17267, 21026}, {17271, 30946}, {17313, 30949}, {17330, 17747}, {17389, 33073}, {17595, 24715}, {17597, 17889}, {17728, 24386}, {19029, 31413}, {19038, 31484}, {21241, 30811}, {24929, 25055}, {29690, 33094}, {31137, 31151}, {31448, 31488}

X(31140) = reflection of X(55) in X(2)
X(31140) = {X(381),X(3679)}-harmonic conjugate of X(31141)
X(31140) = {X(1699),X(3679)}-harmonic conjugate of X(31142)

X(31141) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 + a^2 b^2 - 2 b^4 + 2 a^2 b c - 4 a b^2 c + a^2 c^2 - 4 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31141) lies on these lines: {2, 12}, {4, 21031}, {8, 10896}, {10, 1836}, {11, 3421}, {30, 10310}, {46, 3929}, {55, 11113}, {63, 5123}, {65, 28609}, {80, 3940}, {119, 3428}, {149, 8168}, {210, 381}, {376, 2829}, {377, 9656}, {405, 10197}, {474, 9657}, {480, 528}, {495, 4423}, {519, 1837}, {535, 16371}, {551, 17718}, {599, 8679}, {908, 2099}, {956, 3814}, {1089, 5827}, {1319, 30827}, {1376, 5080}, {1388, 25681}, {1478, 3820}, {1656, 5258}, {1698, 9654}, {1788, 28610}, {1826, 17281}, {1828, 5064}, {1853, 2390}, {2478, 3303}, {2841, 10713}, {3241, 25568}, {3304, 4187}, {3419, 3711}, {3452, 5252}, {3476, 5328}, {3545, 7681}, {3579, 18542}, {3584, 8069}, {3585, 9709}, {3586, 3689}, {3614, 19843}, {3632, 9669}, {3683, 31434}, {3698, 9612}, {3813, 5187}, {3847, 10529}, {3872, 5087}, {3893, 9614}, {3913, 5046}, {3925, 10590}, {3927, 18395}, {4193, 12513}, {4325, 17573}, {4421, 11114}, {4677, 30323}, {4679, 31397}, {4995, 11111}, {5044, 10827}, {5054, 32612}, {5055, 10680}, {5084, 15888}, {5176, 5289}, {5204, 26364}, {5217, 5552}, {5221, 24982}, {5251, 31479}, {5270, 16408}, {5584, 18242}, {5687, 12953}, {5726, 7308}, {5748, 15950}, {5795, 11375}, {5818, 10894}, {5854, 10707}, {6172, 6175}, {6284, 7080}, {6735, 24703}, {6736, 12701}, {6871, 9710}, {6880, 20400}, {7181, 30740}, {7773, 25280}, {7818, 13466}, {7951, 9708}, {8256, 11415}, {8582, 10404}, {9578, 25917}, {9623, 17605}, {9655, 19706}, {9956, 11929}, {10106, 24954}, {10526, 28452}, {10592, 19854}, {10893, 12245}, {12527, 24914}, {12940, 20307}, {13624, 18545}, {13897, 31453}, {17330, 26063}, {17582, 31410}, {17857, 28204}, {20323, 25522}, {24928, 25055}, {26487, 28465}, {31151, 31170}

X(31141) = reflection of X(56) in X(2)
X(31141) = {X(381),X(3679)}-harmonic conjugate of X(31140)

X(31142) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 + 2 a^2 b - a b^2 - 2 b^3 + 2 a^2 c - 6 a b c + 2 b^2 c - a c^2 + 2 b c^2 - 2 c^3 : :

X(31142) lies on these lines: {1, 4679}, {2, 7}, {11, 5223}, {30, 1750}, {72, 9581}, {78, 11015}, {80, 4677}, {119, 3654}, {165, 5660}, {191, 31423}, {200, 528}, {210, 381}, {312, 668}, {376, 5658}, {497, 519}, {518, 31146}, {535, 997}, {549, 21164}, {551, 3475}, {936, 9579}, {960, 9578}, {962, 9842}, {999, 4423}, {1329, 12526}, {1376, 28534}, {1420, 12527}, {1697, 21075}, {1706, 11415}, {1743, 17720}, {1836, 8580}, {2093, 3820}, {2095, 5055}, {2096, 3524}, {2097, 21358}, {2316, 4945}, {2478, 11523}, {2551, 3340}, {2810, 31137}, {2823, 10710}, {2835, 10712}, {2999, 4415}, {3011, 15601}, {3058, 10388}, {3175, 21078}, {3361, 24954}, {3436, 15829}, {3474, 20103}, {3485, 18250}, {3545, 7682}, {3586, 3940}, {3601, 11111}, {3646, 13407}, {3681, 10707}, {3715, 17605}, {3729, 5233}, {3731, 5718}, {3782, 23511}, {3876, 17577}, {3886, 17777}, {3951, 4193}, {3952, 4901}, {3984, 5046}, {4005, 10896}, {4007, 4671}, {4312, 4413}, {4358, 17296}, {4383, 5526}, {4416, 28808}, {4417, 17264}, {4688, 27471}, {4740, 27492}, {4848, 8165}, {4862, 16610}, {4873, 30578}, {4882, 12701}, {5044, 9612}, {5084, 11518}, {5087, 5220}, {5179, 29594}, {5193, 11194}, {5225, 6743}, {5229, 12447}, {5234, 11375}, {5241, 25590}, {5258, 9624}, {5268, 33096}, {5272, 33101}, {5290, 25917}, {5686, 9779}, {5698, 6745}, {5705, 15650}, {5712, 25430}, {5732, 13257}, {5815, 12053}, {5881, 6928}, {6326, 28459}, {6358, 20921}, {6546, 31147}, {6762, 11240}, {6893, 7982}, {6919, 24391}, {7322, 26098}, {7956, 30308}, {7960, 29571}, {7989, 21677}, {7991, 21031}, {8727, 30326}, {9954, 11238}, {10199, 25522}, {10382, 11113}, {10389, 25568}, {10588, 18249}, {11235, 17658}, {11679, 17346}, {12915, 18412}, {15492, 31187}, {15803, 17564}, {16469, 17602}, {17022, 17392}, {17272, 30818}, {17281, 17747}, {17288, 30861}, {17297, 18743}, {17298, 30829}, {18491, 28198}, {18593, 26669}, {21077, 31435}, {24712, 30566}, {25101, 30828}, {25728, 32851}, {26942, 28657}, {30567, 33066}

X(31142) = reflection of X(57) in X(2)
X(31142) = complement of X(2094)
X(31142) = X(57)-of-triangle-A'B'C' as defined at X(5658)
X(31142) = {X(1699),X(3679)}-harmonic conjugate of X(31140)

X(31143) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 + a^2 b - 2 a b^2 - 2 b^3 + a^2 c - 3 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 - 2 c^3 : :

X(31143) lies on these lines: {1, 4938}, {2, 6}, {8, 4442}, {100, 33082}, {210, 2836}, {319, 26580}, {321, 668}, {519, 4425}, {542, 4220}, {594, 17484}, {758, 3679}, {846, 4933}, {1255, 29574}, {1621, 33084}, {2796, 4683}, {3218, 17344}, {3686, 33129}, {3706, 4956}, {3773, 4756}, {3775, 32843}, {3786, 3909}, {3849, 24271}, {3920, 28538}, {4023, 33086}, {4042, 25958}, {4046, 33100}, {4062, 24697}, {4104, 33078}, {4228, 15360}, {4358, 17287}, {4416, 32779}, {4418, 28558}, {4445, 4671}, {4643, 33077}, {4690, 25383}, {4850, 17272}, {4886, 17184}, {5284, 33081}, {6172, 31153}, {7794, 21516}, {7801, 21511}, {7810, 21495}, {7854, 21540}, {8013, 33097}, {8682, 17310}, {9041, 33090}, {9342, 33085}, {9534, 17679}, {10707, 31136}, {11684, 20653}, {13745, 26064}, {17012, 17237}, {17013, 17325}, {17021, 17374}, {17273, 17495}, {17275, 31019}, {17288, 24589}, {17295, 31035}, {17332, 32849}, {17361, 26627}, {17362, 33155}, {27184, 29617}, {28562, 32945}, {31025, 32025}, {31171, 31175}

X(31143) = reflection of X(81) in X(2)

X(31144) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^2 - 3 a b - 2 b^2 - 3 a c - 3 b c - 2 c^2 : :

X(31144) lies on these lines: {2, 6}, {9, 16568}, {10, 190}, {37, 29615}, {44, 29610}, {45, 29593}, {67, 22374}, {239, 4708}, {256, 21699}, {319, 5257}, {320, 24603}, {519, 25354}, {542, 6998}, {545, 6650}, {594, 32101}, {740, 3679}, {894, 1268}, {903, 4688}, {1043, 13745}, {1494, 32040}, {1698, 3758}, {2482, 6626}, {3617, 4918}, {3686, 17322}, {3739, 17252}, {3828, 17770}, {3849, 24275}, {3986, 17315}, {4034, 17393}, {4360, 17248}, {4364, 17160}, {4384, 17250}, {4389, 4748}, {4407, 24841}, {4416, 28653}, {4422, 29591}, {4445, 27268}, {4472, 20072}, {4643, 29576}, {4687, 17270}, {4690, 16826}, {4698, 17287}, {4699, 17253}, {4725, 29580}, {4733, 9791}, {4740, 24441}, {4751, 17272}, {4755, 17310}, {4772, 17255}, {4851, 29622}, {4912, 17116}, {4967, 17132}, {4969, 25358}, {5296, 17233}, {5461, 6537}, {5463, 21898}, {5464, 21869}, {5564, 17133}, {5625, 25055}, {6172, 6175}, {6651, 16590}, {7380, 20423}, {7474, 15360}, {7801, 16060}, {7810, 16061}, {7812, 13740}, {8591, 26081}, {9939, 17688}, {10436, 17328}, {16666, 29609}, {16672, 20055}, {16815, 17237}, {16816, 17325}, {16830, 28538}, {16831, 17360}, {16832, 17227}, {16833, 17399}, {17121, 25498}, {17239, 17260}, {17247, 28634}, {17263, 31311}, {17288, 31238}, {17303, 17331}, {17308, 17335}, {17319, 28329}, {17326, 17348}, {17329, 25590}, {17332, 28604}, {17374, 29578}, {18714, 21033}, {18823, 24348}, {19875, 24342}, {22174, 24437}, {26774, 27037}, {28209, 28602}, {31155, 31169}

X(31144) = reflection of X(86) in X(2)

X(31145) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 107

X(31145) = 4 X(1) - 5 X(2)

Barycentrics 7 a - 5 b - 5 c : :

X(31145) lies on these lines: {1, 2}, {20, 28204}, {30, 12245}, {69, 903}, {75, 32093}, {76, 25296}, {100, 8168}, {144, 528}, {149, 3421}, {193, 28538}, {312, 4487}, {319, 32105}, {346, 4370}, {355, 3839}, {376, 952}, {381, 5844}, {390, 15481}, {391, 16814}, {515, 15683}, {517, 3543}, {518, 4740}, {535, 20078}, {536, 31302}, {537, 1278}, {547, 10247}, {549, 7967}, {599, 9053}, {944, 3654}, {956, 17549}, {966, 16674}, {1000, 15170}, {1043, 4921}, {1145, 10031}, {1150, 4954}, {1320, 3940}, {1385, 15708}, {1482, 3545}, {1483, 5054}, {1992, 5846}, {2136, 3929}, {2345, 16668}, {2975, 4421}, {3091, 3656}, {3146, 5881}, {3219, 3895}, {3242, 21356}, {3247, 4545}, {3295, 16858}, {3303, 16859}, {3522, 11362}, {3524, 5690}, {3653, 10303}, {3655, 5657}, {3672, 17271}, {3680, 3984}, {3681, 3880}, {3813, 5154}, {3829, 11681}, {3832, 7982}, {3845, 8148}, {3869, 3893}, {3871, 16370}, {3879, 30712}, {3890, 4662}, {3897, 33595}, {3902, 4671}, {3913, 4189}, {3945, 5564}, {4060, 5749}, {4188, 12513}, {4234, 4720}, {4346, 17360}, {4371, 4869}, {4399, 17313}, {4419, 28309}, {4428, 16865}, {4430, 24473}, {4452, 17274}, {4454, 4715}, {4460, 17270}, {4461, 17363}, {4664, 28581}, {4665, 4747}, {4709, 4821}, {4711, 5919}, {4725, 4795}, {4803, 26860}, {4863, 5176}, {4916, 28634}, {4929, 28313}, {4971, 24441}, {5055, 10595}, {5056, 10222}, {5059, 7991}, {5068, 5734}, {5071, 5790}, {5082, 20060}, {5141, 12607}, {5175, 28609}, {5178, 32049}, {5183, 17784}, {5232, 17320}, {5434, 21454}, {5601, 11208}, {5602, 11207}, {5687, 13587}, {5839, 16669}, {5853, 6172}, {5854, 10707}, {5855, 31140}, {5882, 15717}, {6361, 28208}, {6767, 17542}, {6871, 32537}, {6872, 12632}, {7270, 19819}, {7714, 12135}, {8692, 17349}, {8703, 18526}, {8715, 17548}, {9708, 16861}, {9778, 28236}, {9779, 11224}, {9965, 17579}, {10005, 17264}, {10022, 28337}, {10246, 15702}, {10283, 15703}, {10385, 10950}, {10405, 20111}, {10679, 28461}, {11001, 12702}, {11050, 11910}, {11346, 19742}, {12536, 12640}, {12630, 24393}, {13464, 15022}, {15174, 15675}, {15640, 28198}, {15672, 21677}, {15681, 28224}, {15682, 18525}, {15684, 28212}, {15697, 18481}, {15721, 26446}, {16052, 31037}, {16126, 31420}, {16675, 17314}, {16711, 33297}, {17144, 25278}, {17488, 20073}, {17678, 19789}, {24280, 28562}, {25719, 32003}, {26792, 30305}, {27065, 31393}, {28234, 31162}, {31138, 32850}

X(31145) = reflection of X(2) in X(8)
X(31145) = reflection of X(145) in X(2)
X(31145) = complement of X(20049)
X(31145) = anticomplement of X(3241)

X(31146) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 - 4 a^2 b + 5 a b^2 - 2 b^3 - 4 a^2 c - 6 a b c + 2 b^2 c + 5 a c^2 + 2 b c^2 - 2 c^3 : :

X(31146) lies on these lines: {1, 2}, {11, 3243}, {57, 528}, {149, 4312}, {354, 6173}, {497, 527}, {516, 2094}, {518, 31142}, {535, 3586}, {537, 20173}, {903, 1088}, {1004, 5563}, {1005, 8666}, {1058, 12526}, {1320, 16236}, {1699, 2801}, {1836, 3254}, {1864, 11238}, {2078, 4421}, {3058, 3928}, {3158, 6174}, {3304, 12625}, {3333, 11112}, {3434, 10980}, {3475, 24386}, {3555, 17556}, {3656, 8727}, {3677, 17301}, {3689, 31190}, {3813, 11518}, {3874, 9614}, {3881, 9612}, {3889, 5290}, {4370, 8557}, {4512, 24477}, {4654, 11235}, {4715, 24352}, {4859, 33136}, {4862, 17449}, {4863, 5437}, {4888, 33104}, {5045, 17528}, {5175, 12577}, {5249, 30350}, {5735, 10431}, {5744, 30331}, {6001, 24473}, {8568, 17314}, {9580, 28534}, {9581, 11236}, {10883, 11520}, {11523, 14022}, {15600, 17724}, {16487, 24597}, {16496, 24217}, {16750, 17179}, {17678, 19790}, {19541, 28204}, {22464, 31527}, {23681, 33141}, {24283, 28562}, {25355, 28309}

X(31146) = reflection of X(200) in X(2)

X(31147) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b - c) (-a^2 - 2 a b - 2 a c + 2 b c) : :

X(31147) lies on these lines: {2, 649}, {512, 9148}, {513, 4379}, {514, 4120}, {599, 9002}, {693, 28840}, {788, 30640}, {812, 4776}, {1635, 6008}, {1638, 28217}, {3676, 4654}, {3679, 14433}, {4762, 4940}, {4453, 28867}, {4688, 27485}, {4790, 24924}, {4810, 4948}, {4931, 28894}, {4958, 28898}, {6006, 6173}, {6545, 28846}, {6546, 31142}, {9313, 31134}, {14437, 24441}, {17313, 21143}, {28292, 31162}, {28882, 30565}

X(31147) = reflection of X(649) in X(2)

X(31148) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b - c) (2 a^2 + a b + a c + 2 b c) : :

X(31148) lies on these lines: {2, 661}, {513, 4379}, {514, 1635}, {523, 4750}, {553, 4077}, {599, 9013}, {4785, 4932}, {1638, 4977}, {1639, 28902}, {2786, 4789}, {3679, 4160}, {3805, 31161}, {4448, 14475}, {6084, 21116}, {8672, 31174}, {8774, 31164}, {9256, 31134}, {9279, 21020}, {14433, 14474}, {19346, 23864}, {28855, 30565}

X(31148) = reflection of X(661) in X(2)

X(31149) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b - c) (-a^3 - 2 a b^2 - 2 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2) : :

X(31149) lies on these lines: {2, 667}, {10, 24719}, {381, 3309}, {512, 9148}, {513, 14431}, {599, 9010}, {764, 11236}, {1698, 4782}, {1960, 30835}, {3669, 11237}, {3679, 4083}, {3835, 4775}, {4162, 11238}, {4705, 4762}, {4776, 29188}, {4785, 4834}, {5064, 18344}, {6161, 17556}, {6175, 29150}, {9320, 10708}, {14077, 30592}, {14419, 28475}, {29070, 31150}

X(31149) = reflection of X(667) in X(2)

X(31150) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b - c) (-2 a^2 + 2 a b + 2 a c + b c) : :

X(31150) lies on these lines: {2, 650}, {100, 666}, {351, 523}, {376, 8760}, {513, 14404}, {514, 1635}, {522, 14392}, {599, 9015}, {649, 28840}, {812, 4776}, {824, 6546}, {900, 14410}, {918, 27486}, {1992, 9001}, {2401, 30673}, {3241, 14077}, {3679, 29066}, {3873, 9443}, {4382, 25666}, {4448, 4664}, {4467, 4468}, {4685, 21727}, {4750, 28851}, {4786, 28878}, {4980, 21438}, {10385, 11934}, {14435, 28886}, {29070, 31149}

X(31150) = reflection of X(693) in X(2)

X(31151) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 - 2 b^3 + 3 a b c - 2 c^3 : :

X(31151) lies on these lines: {1, 3834}, {2, 31}, {7, 33165}, {10, 320}, {44, 1698}, {142, 33076}, {190, 24692}, {239, 25351}, {334, 4495}, {381, 15310}, {513, 14431}, {518, 599}, {519, 1738}, {545, 3932}, {551, 17766}, {716, 7245}, {726, 903}, {740, 17310}, {984, 17274}, {1086, 28503}, {1155, 29862}, {1279, 25055}, {1463, 11237}, {1757, 3823}, {2550, 33087}, {2796, 17264}, {3006, 18201}, {3218, 21026}, {3241, 17765}, {3246, 3624}, {3661, 24693}, {3750, 18139}, {3821, 17320}, {3826, 17330}, {3828, 17770}, {3844, 24342}, {3912, 4693}, {3923, 17342}, {3925, 33085}, {4038, 4972}, {4085, 17300}, {4370, 17768}, {4413, 27739}, {4414, 27754}, {4429, 4649}, {4432, 17266}, {4439, 4440}, {4450, 29851}, {4655, 17333}, {4660, 16484}, {4675, 29659}, {4709, 17295}, {4732, 17287}, {4892, 5205}, {5249, 33079}, {5880, 17281}, {7301, 16048}, {9780, 20072}, {10327, 33103}, {11246, 33164}, {15485, 17265}, {15668, 16801}, {16786, 29633}, {17232, 32941}, {17256, 25352}, {17487, 17767}, {17593, 29643}, {17598, 33072}, {17600, 33125}, {17780, 31029}, {18141, 33141}, {19876, 28570}, {20292, 29687}, {24169, 33073}, {24628, 30763}, {24988, 32843}, {26842, 33162}, {27186, 33074}, {29653, 33068}, {29854, 32950}, {31137, 31140}, {31141, 31170}

X(31151) = reflection of X(238) in X(2)

X(31152) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^6 + 2 a^4 b^2 - a^2 b^4 - 2 b^6 + 2 a^4 c^2 - 6 a^2 b^2 c^2 + 2 b^4 c^2 - a^2 c^4 + 2 b^2 c^4 - 2 c^6 : :

X(31152) lies on these lines: {2, 3}, {67, 15533}, {125, 1350}, {154, 5642}, {159, 32125}, {184, 13857}, {305, 670}, {394, 542}, {511, 26869}, {524, 1899}, {574, 15820}, {599, 1853}, {612, 11237}, {614, 11238}, {1184, 5309}, {1196, 11648}, {1351, 18911}, {1503, 6090}, {1533, 33534}, {1660, 15139}, {1992, 11245}, {2386, 7818}, {2790, 6054}, {2834, 10712}, {2979, 9140}, {3001, 9766}, {3162, 3163}, {3266, 7776}, {3448, 11898}, {3819, 11178}, {4688, 31140}, {5297, 9654}, {5422, 14848}, {5476, 10601}, {5650, 10516}, {7292, 9669}, {7773, 11059}, {7784, 30749}, {7885, 30793}, {8263, 21356}, {9306, 11645}, {9777, 20423}, {10510, 15534}, {10717, 10718}, {11057, 33651}, {11064, 26864}, {11123, 31176}, {11179, 11402}, {11180, 32064}, {11412, 26944}, {11550, 17811}, {11820, 32111}, {12017, 14389}, {12167, 15812}, {14580, 22401}, {15066, 18440}, {15107, 21970}, {15141, 32264}, {15360, 26913}, {15577, 23315}, {15919, 16188}, {16163, 32227}, {19924, 33586}, {20772, 26881}, {26958, 32225}, {31135, 31136}

X(31152) = reflection of X(25) in X(2)

X(31153) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^6 + 5 a^5 b + 2 a^4 b^2 - 4 a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + 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 + 2 a^4 c^2 - 4 a^3 b c^2 - 6 a^2 b^2 c^2 + 2 a b^3 c^2 + 2 b^4 c^2 - 4 a^3 c^3 - 4 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 - a c^5 - b c^5 - 2 c^6 : :

X(31153) lies on these lines: {2, 3}, {190, 306}, {545, 16099}, {2822, 10710}, {4664, 8680}, {6172, 31143}, {17078, 18651}, {25361, 31332}

X(31153) = reflection of X(27) in X(2)

X(31154) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^7 + a^6 b + 2 a^5 b^2 + 2 a^4 b^3 - a^3 b^4 - a^2 b^5 - 2 a b^6 - 2 b^7 + a^6 c + 5 a^5 b c + 2 a^4 b^2 c - 4 a^3 b^3 c - a^2 b^4 c - a b^5 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 - 6 a^3 b^2 c^2 - 6 a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 + 2 a^4 c^3 - 4 a^3 b c^3 - 6 a^2 b^2 c^3 + 2 a b^3 c^3 + 2 b^4 c^3 - a^3 c^4 - a^2 b c^4 + 2 a b^2 c^4 + 2 b^3 c^4 - a^2 c^5 - a b c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 - 2 c^7 : :

X(31154) lies on these lines: {2, 3}, {668, 1494}, {2828, 10711}, {2838, 10712}, {3679, 31158}

X(31154) = reflection of X(28) in X(2)

X(31155) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^7 - 4 a^6 b - 3 a^5 b^2 + 6 a^4 b^3 + 3 a^3 b^4 - a b^6 - 2 b^7 - 4 a^6 c - 5 a^5 b c + a^4 b^2 c + 4 a^3 b^3 c + 4 a^2 b^4 c + a b^5 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 + 3 b^5 c^2 + 6 a^4 c^3 + 4 a^3 b c^3 - 4 a^2 b^2 c^3 - 2 a b^3 c^3 + 3 a^3 c^4 + 4 a^2 b c^4 + a b^2 c^4 + a b c^5 + 3 b^2 c^5 - a c^6 - b c^6 - 2 c^7 : :

X(31155) lies on these lines: {2, 3}, {307, 319}, {2816, 10709}, {31144, 31169}

X(31155) = reflection of X(29) in X(2)

X(31156) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (a - b - c) (5 a^3 + 5 a^2 b + a b^2 + b^3 + 5 a^2 c + 4 a b c - b^2 c + a c^2 - b c^2 + c^3) : :

X(31156) lies on these lines: {1, 17781}, {2, 3}, {8, 3683}, {391, 4720}, {392, 3655}, {518, 1992}, {519, 5250}, {551, 31164}, {950, 5325}, {956, 15170}, {958, 3058}, {966, 6740}, {993, 10072}, {1001, 5434}, {1864, 3876}, {3017, 24597}, {3219, 3488}, {3305, 4304}, {3434, 5251}, {3436, 5248}, {3476, 29007}, {3486, 7082}, {3616, 4870}, {3654, 5554}, {3679, 4512}, {3868, 15933}, {3897, 20323}, {4293, 5284}, {4294, 5260}, {4428, 11239}, {4654, 5436}, {4857, 31458}, {4930, 5330}, {4995, 5552}, {5550, 17605}, {5985, 12243}, {7288, 26127}, {8167, 15326}, {9708, 20075}, {10527, 11238}, {12649, 31445}, {15175, 30513}, {15650, 20013}, {15934, 20078}, {19860, 28194}, {24929, 31018}

X(31156) = reflection of X(377) in X(2)

X(31157) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 4 a^4 - 5 a^2 b^2 + b^4 + 2 a^2 b c - 4 a b^2 c - 5 a^2 c^2 - 4 a b c^2 - 2 b^2 c^2 + c^4 : :

X(31157) lies on these lines: {2, 12}, {11, 993}, {30, 11012}, {36, 3925}, {48, 17330}, {63, 15950}, {140, 5258}, {191, 5901}, {210, 10165}, {354, 392}, {376, 5842}, {381, 5841}, {474, 31458}, {519, 2646}, {527, 4870}, {528, 4996}, {535, 17530}, {547, 31160}, {549, 952}, {599, 5849}, {956, 5432}, {1006, 20418}, {1319, 5745}, {1385, 21677}, {2099, 5744}, {2886, 15326}, {3058, 10959}, {3241, 5855}, {3304, 6857}, {3333, 3929}, {3485, 28610}, {3524, 8273}, {3649, 24541}, {3656, 12704}, {3813, 4189}, {3820, 31235}, {3829, 11114}, {3872, 13996}, {3899, 10283}, {4188, 9710}, {4293, 31245}, {4299, 31493}, {4428, 11240}, {4863, 30282}, {5010, 6154}, {5054, 31659}, {5204, 19843}, {5234, 24954}, {5251, 15325}, {5267, 15338}, {5326, 17757}, {5429, 17726}, {5563, 6675}, {5852, 6172}, {5857, 6173}, {6284, 10527}, {6856, 9657}, {6910, 11239}, {7280, 31419}, {7354, 17532}, {7483, 8666}, {9670, 17576}, {9678, 19029}, {9711, 17566}, {10072, 16418}, {10543, 10916}, {10707, 15677}, {10944, 26066}, {11111, 11238}, {11375, 28609}, {11376, 31424}, {17564, 19875}, {17573, 31494}, {17625, 25917}, {18966, 31453}, {26286, 28452}

X(31157) = reflection of X(12) in X(2)

X(31158) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^5 + 2 a^4 b - a b^4 - 2 b^5 + 2 a^4 c - 4 a^2 b^2 c + 2 b^4 c - 4 a^2 b c^2 + 2 a b^2 c^2 - a c^4 + 2 b c^4 - 2 c^5 : :

X(31158) lies on these lines: {2, 19}, {30, 30265}, {40, 4466}, {376, 516}, {549, 21160}, {599, 3827}, {1086, 12701}, {2263, 17392}, {3007, 31015}, {3058, 4319}, {3668, 4654}, {3679, 31154}, {4664, 8680}, {4688, 31140}, {8756, 30809}, {8804, 28609}, {15526, 27553}, {17320, 18655}, {24683, 26006}

X(31158) = reflection of X(19) in X(2)

X(31159) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 + a^2 b^2 - 2 b^4 - a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31159) lies on these lines: {1, 3838}, {2, 35}, {4, 5258}, {30, 11012}, {36, 11680}, {79, 10916}, {149, 3822}, {191, 22793}, {210, 381}, {442, 4857}, {519, 5086}, {528, 3584}, {529, 3585}, {547, 6174}, {551, 6175}, {599, 9047}, {956, 18513}, {958, 18514}, {1125, 11015}, {1376, 31263}, {1698, 9580}, {1706, 11010}, {1836, 4880}, {1900, 5064}, {2475, 5563}, {2476, 3746}, {2646, 11238}, {2779, 10706}, {2886, 3583}, {3419, 4867}, {3434, 7951}, {3582, 3829}, {3624, 9669}, {3632, 10895}, {3633, 9654}, {3813, 5270}, {3814, 33110}, {3830, 18761}, {3850, 21031}, {3929, 24468}, {4301, 7548}, {4309, 6856}, {4324, 4999}, {4330, 7483}, {4677, 11236}, {5055, 11849}, {5119, 17057}, {5141, 8715}, {5225, 19854}, {5315, 33106}, {5441, 24541}, {5537, 6830}, {5904, 28609}, {6871, 11239}, {6919, 31420}, {7773, 32104}, {7989, 10893}, {8227, 11928}, {9668, 31245}, {9671, 11108}, {10483, 10527}, {10894, 11531}, {11011, 11237}, {11230, 15015}, {12953, 31493}, {14041, 16829}, {14794, 16370}, {15031, 25280}, {16474, 33141}

X(31159) = reflection of X(35) in X(2)
X(31159) = {X(381),X(3679)}-harmonic conjugate of X(31160)

X(31160) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 + a^2 b^2 - 2 b^4 + a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31160) lies on these lines: {1, 5087}, {2, 36}, {4, 5537}, {5, 5258}, {9, 484}, {10, 3245}, {12, 2078}, {30, 119}, {35, 11114}, {79, 24982}, {80, 519}, {115, 2238}, {191, 9956}, {210, 381}, {498, 11111}, {513, 14431}, {515, 5660}, {527, 1737}, {528, 3583}, {529, 3582}, {546, 21031}, {547, 31157}, {599, 9037}, {958, 31262}, {1155, 1698}, {1319, 5219}, {1329, 3585}, {1376, 18513}, {1512, 28194}, {1826, 5146}, {1878, 5064}, {3035, 4316}, {3218, 6702}, {3336, 17619}, {3436, 5288}, {3545, 8166}, {3584, 11113}, {3624, 5126}, {3632, 10896}, {3633, 9669}, {3660, 5290}, {3746, 5046}, {3825, 20060}, {3828, 6175}, {3830, 18491}, {3845, 18406}, {3929, 5535}, {4187, 5270}, {4193, 5563}, {4325, 13747}, {4654, 18838}, {4677, 11235}, {4857, 12607}, {5048, 5727}, {5054, 23961}, {5055, 22765}, {5131, 19876}, {5154, 8666}, {5172, 16418}, {5179, 10712}, {5180, 31018}, {5187, 11240}, {5193, 5434}, {5425, 31053}, {5536, 7989}, {5570, 18412}, {5687, 18514}, {5902, 31164}, {5904, 10826}, {6326, 28204}, {6763, 17606}, {6788, 32856}, {7354, 17564}, {8227, 11929}, {9581, 18839}, {9655, 31246}, {9656, 16408}, {10483, 26364}, {10893, 11531}, {13466, 31173}, {15015, 28160}, {15180, 30513}, {16489, 24222}, {16490, 24217}, {16829, 33013}, {17264, 21094}, {17276, 24223}, {18397, 28609}, {22835, 30308}, {27471, 31178}

X(31160) = reflection of X(36) in X(2)
X(31160) = {X(381),X(3679)}-harmonic conjugate of X(31159)

X(31161) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^2 b - a b^2 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 : :

X(31161) lies on these lines: {1, 3994}, {2, 38}, {8, 24725}, {10, 32856}, {30, 30272}, {42, 536}, {75, 21805}, {192, 21806}, {210, 4688}, {226, 33162}, {321, 519}, {518, 31136}, {545, 24326}, {551, 3971}, {599, 9020}, {714, 1962}, {758, 3679}, {894, 32927}, {896, 26227}, {902, 17351}, {976, 16394}, {2177, 3729}, {2650, 4385}, {3112, 18822}, {3241, 32915}, {3711, 17118}, {3720, 3967}, {3722, 3923}, {3757, 32938}, {3805, 31148}, {3873, 31137}, {3930, 17281}, {4009, 30950}, {4054, 33136}, {4082, 29600}, {4090, 4359}, {4358, 17450}, {4362, 4722}, {4415, 29685}, {4418, 24345}, {4672, 20045}, {4685, 4980}, {4686, 21870}, {4740, 32860}, {4884, 29688}, {4892, 31079}, {4931, 29110}, {5434, 7211}, {5905, 33074}, {7081, 32940}, {17147, 28554}, {17354, 29638}, {17449, 30818}, {17469, 26223}, {17483, 33079}, {17484, 33076}, {17718, 33161}, {21026, 31019}, {27064, 32923}, {29584, 32928}, {29594, 33081}, {29659, 33151}, {29667, 33101}, {29670, 32933}, {29679, 33103}, {29857, 31280}, {31053, 33169}, {33090, 33096}, {33091, 33097}

X(31161) = reflection of X(38) in X(2)

X(31162) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 + 3 a^3 b + a^2 b^2 - 3 a b^3 - 2 b^4 + 3 a^3 c - 6 a^2 b c + 3 a b^2 c + a^2 c^2 + 3 a b c^2 + 4 b^2 c^2 - 3 a c^3 - 2 c^4 : :

X(31162) lies on these lines: {1, 30}, {2, 40}, {3, 9589}, {4, 519}, {5, 3654}, {8, 3839}, {10, 3545}, {11, 2093}, {20, 13464}, {46, 3582}, {55, 4870}, {57, 10072}, {63, 5180}, {65, 9614}, {84, 11240}, {98, 12258}, {114, 9881}, {140, 31425}, {145, 31673}, {165, 549}, {210, 381}, {226, 30305}, {355, 3845}, {376, 516}, {382, 10222}, {484, 23708}, {495, 9819}, {496, 3339}, {497, 11529}, {499, 5128}, {515, 3241}, {527, 11372}, {528, 1537}, {541, 33535}, {547, 7988}, {550, 30389}, {553, 3333}, {631, 5493}, {758, 24392}, {942, 9848}, {944, 15682}, {952, 11224}, {993, 28461}, {999, 4312}, {1001, 7688}, {1012, 5735}, {1056, 4342}, {1058, 3671}, {1125, 3524}, {1159, 18527}, {1385, 3534}, {1387, 13462}, {1420, 1770}, {1478, 7962}, {1479, 3340}, {1482, 3830}, {1484, 12767}, {1519, 12703}, {1572, 5309}, {1656, 9588}, {1657, 15178}, {1697, 10056}, {1698, 5055}, {1702, 32787}, {1703, 32788}, {1706, 21616}, {1902, 5064}, {2077, 16371}, {2098, 9613}, {2099, 3586}, {2136, 21077}, {2771, 3894}, {2796, 21636}, {2800, 10707}, {2802, 10711}, {2807, 14831}, {2809, 10710}, {2817, 10716}, {2948, 5655}, {2951, 20330}, {3057, 9612}, {3091, 11362}, {3146, 5734}, {3149, 4421}, {3361, 11373}, {3421, 11525}, {3428, 16418}, {3434, 18406}, {3485, 10385}, {3487, 12575}, {3577, 26333}, {3579, 3624}, {3583, 5727}, {3584, 5119}, {3585, 30323}, {3616, 10304}, {3627, 16189}, {3632, 8148}, {3633, 11278}, {3653, 5901}, {3746, 6985}, {3751, 20423}, {3817, 3828}, {3829, 6831}, {3872, 5057}, {3877, 6175}, {3895, 31053}, {3897, 15678}, {3928, 12704}, {3940, 18482}, {4007, 4717}, {4034, 32431}, {4297, 10595}, {4302, 13384}, {4309, 6869}, {4333, 21842}, {4338, 5563}, {4355, 7373}, {4428, 10902}, {4512, 15670}, {4664, 29054}, {4668, 18357}, {4669, 12245}, {4745, 5818}, {4848, 10591}, {4995, 11375}, {5048, 12943}, {5066, 5690}, {5068, 31399}, {5248, 21161}, {5288, 18761}, {5290, 9957}, {5298, 11376}, {5325, 19843}, {5330, 15679}, {5426, 28460}, {5429, 29097}, {5537, 6911}, {5541, 12611}, {5550, 15708}, {5693, 9856}, {5697, 9578}, {5715, 17532}, {5722, 18421}, {5731, 15683}, {5758, 6766}, {5762, 24644}, {5805, 6282}, {5837, 31418}, {5844, 14893}, {5847, 11180}, {5903, 9581}, {5904, 31937}, {6001, 24473}, {6321, 9875}, {6769, 12700}, {6845, 24387}, {6851, 11518}, {7171, 11552}, {7680, 17530}, {7681, 17533}, {7686, 17556}, {7739, 9575}, {7753, 9620}, {7865, 12497}, {7967, 28164}, {7978, 12407}, {7993, 16128}, {8666, 21669}, {8724, 13174}, {8960, 31440}, {9300, 9593}, {9623, 24703}, {9625, 14070}, {9626, 9909}, {9746, 28862}, {9778, 10165}, {9779, 10175}, {9785, 21620}, {9860, 11632}, {9880, 13178}, {9884, 10723}, {9897, 22938}, {9900, 25164}, {9901, 25154}, {9904, 12261}, {9956, 19709}, {9961, 12005}, {10031, 10724}, {10164, 15702}, {10246, 15681}, {10247, 15684}, {10283, 15686}, {10310, 16417}, {10444, 17320}, {10446, 17274}, {10738, 13253}, {10742, 12653}, {10914, 16616}, {11011, 12953}, {11012, 11496}, {11014, 11114}, {11112, 12651}, {11179, 16475}, {11230, 15694}, {11231, 15703}, {11235, 12672}, {11239, 12608}, {11249, 28444}, {11280, 18514}, {11415, 17781}, {11599, 12243}, {11711, 12117}, {11813, 30827}, {12019, 30286}, {12100, 16192}, {12512, 19708}, {12514, 24468}, {12526, 24390}, {12629, 13463}, {13374, 15016}, {13541, 15522}, {13600, 31822}, {13607, 15640}, {13624, 15688}, {13888, 31439}, {14093, 17502}, {14110, 17528}, {14636, 31394}, {15693, 31663}, {15709, 19862}, {15710, 15808}, {15950, 30282}, {16113, 17525}, {16191, 28224}, {16483, 23681}, {16496, 31670}, {17605, 31434}, {23058, 24045}, {24608, 26006}, {24929, 31671}, {25362, 33536}, {26013, 30976}, {28234, 31145}, {28292, 31147}

X(31162) = reflection of X(40) in X(2)
X(31162) = complement of X(34632)

X(31163) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^5 - a^3 b^2 + 2 a^2 b^3 - 2 b^5 - a^3 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - 2 c^5 : :

X(31163) lies on these lines: {2, 48}, {12, 17392}, {355, 4466}, {381, 916}, {527, 1826}, {551, 29219}, {599, 8679}, {1837, 17301}, {1853, 31140}, {2801, 5587}, {3679, 31154}, {3949, 17294}, {4664, 31175}, {8287, 9310}, {21237, 25940}, {21256, 30746}, {21298, 26232}, {22356, 30808}

X(31163) = reflection of X(48) in X(2)

X(31164) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 + 2 a^2 b - a b^2 - 2 b^3 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 - 2 c^3 : :

X(31164) lies on these lines: {1, 535}, {2, 7}, {4, 11520}, {30, 18446}, {40, 14450}, {55, 28534}, {65, 11236}, {69, 4054}, {72, 17528}, {78, 11112}, {79, 3811}, {92, 1121}, {100, 4312}, {148, 17389}, {149, 3243}, {153, 3577}, {200, 20292}, {312, 17297}, {321, 3761}, {377, 3984}, {381, 912}, {388, 11682}, {442, 3951}, {515, 3241}, {518, 27479}, {519, 1478}, {528, 1836}, {544, 16834}, {549, 21165}, {551, 31156}, {612, 33097}, {614, 33096}, {651, 30684}, {758, 3679}, {942, 17556}, {946, 11240}, {968, 33099}, {993, 5284}, {1022, 29148}, {1320, 5561}, {1449, 33155}, {1537, 3656}, {1699, 2801}, {1707, 33127}, {1743, 33129}, {1992, 9028}, {2475, 11523}, {2792, 11177}, {2999, 33146}, {3011, 24695}, {3120, 3751}, {3338, 10199}, {3339, 11681}, {3340, 20060}, {3436, 3671}, {3487, 11111}, {3534, 33595}, {3585, 12559}, {3649, 19860}, {3655, 5841}, {3677, 33107}, {3729, 3936}, {3782, 5256}, {3822, 19875}, {3824, 15650}, {3868, 9612}, {3869, 5290}, {3875, 31034}, {3889, 9614}, {3957, 9580}, {3977, 30828}, {4018, 9654}, {4084, 10827}, {4138, 33163}, {4292, 4855}, {4338, 8715}, {4355, 5253}, {4358, 17298}, {4415, 5287}, {4430, 24392}, {4652, 11374}, {4655, 29828}, {4659, 33077}, {4664, 8680}, {4666, 24703}, {4671, 17296}, {4672, 29855}, {4679, 25557}, {4688, 27488}, {4850, 4862}, {4860, 5087}, {4870, 11194}, {4872, 17378}, {4892, 29857}, {4980, 20895}, {5046, 11518}, {5080, 11529}, {5134, 29574}, {5176, 18421}, {5180, 31393}, {5223, 33108}, {5231, 10129}, {5233, 7321}, {5250, 13407}, {5269, 33153}, {5271, 17346}, {5440, 18541}, {5714, 6734}, {5715, 12528}, {5718, 17276}, {5805, 13257}, {5812, 10884}, {5902, 31160}, {6174, 11246}, {6180, 22128}, {6354, 6505}, {6545, 28846}, {6745, 30424}, {6871, 24391}, {7174, 33112}, {7232, 30818}, {7290, 33148}, {8774, 31148}, {9809, 11372}, {10404, 19861}, {10439, 23155}, {11239, 12703}, {11415, 21620}, {11679, 32859}, {12437, 31295}, {12531, 16236}, {13384, 20067}, {16475, 33143}, {16496, 33104}, {17064, 32912}, {17174, 18164}, {17251, 31993}, {17264, 18134}, {17270, 31025}, {17286, 31017}, {17351, 30811}, {17365, 17720}, {17491, 26227}, {17564, 24470}, {17594, 33098}, {17718, 17768}, {18139, 30568}, {18210, 20430}, {23681, 32911}, {24268, 29584}, {25734, 33116}, {29069, 31179}, {31254, 31446}

X(31164) = reflection of X(63) in X(2)

X(31165) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a (3 a^2 b - 3 b^3 + 3 a^2 c - 4 a b c - b^2 c - b c^2 - 3 c^3) : :

X(31165) lies on these lines: {1, 3683}, {2, 65}, {8, 3967}, {9, 2099}, {10, 3614}, {12, 5837}, {30, 5887}, {46, 16417}, {56, 3928}, {63, 1319}, {72, 519}, {78, 4421}, {144, 3476}, {191, 1385}, {210, 381}, {329, 5252}, {354, 392}, {376, 5918}, {518, 1992}, {527, 5434}, {528, 3059}, {529, 17781}, {542, 10693}, {549, 14988}, {553, 12709}, {599, 3827}, {912, 3655}, {942, 25055}, {946, 21677}, {956, 5048}, {958, 11011}, {995, 4003}, {997, 1155}, {1108, 3958}, {1122, 17274}, {1125, 4018}, {1149, 21342}, {1212, 20616}, {1376, 5183}, {1464, 25941}, {1532, 11362}, {1698, 3922}, {1858, 11111}, {1898, 11114}, {2093, 4413}, {2098, 7082}, {2262, 17330}, {2390, 3917}, {2646, 5730}, {2771, 28460}, {2778, 10706}, {2796, 17635}, {2800, 6174}, {2802, 4134}, {3243, 8162}, {3244, 4127}, {3303, 11523}, {3545, 7686}, {3555, 3884}, {3624, 31794}, {3625, 3988}, {3626, 4533}, {3634, 4004}, {3653, 31838}, {3654, 31837}, {3678, 4669}, {3681, 3880}, {3689, 3940}, {3697, 4745}, {3698, 5044}, {3714, 25253}, {3715, 9623}, {3727, 21874}, {3740, 4731}, {3753, 3828}, {3829, 6734}, {3830, 31937}, {3833, 4744}, {3868, 17609}, {3872, 5220}, {3873, 10179}, {3876, 3983}, {3893, 4677}, {3894, 5049}, {3901, 5045}, {3902, 4043}, {3913, 3984}, {3916, 30144}, {3951, 12513}, {4084, 5439}, {4301, 8226}, {4423, 11529}, {4428, 5250}, {4511, 4640}, {4661, 20049}, {4662, 14923}, {4663, 17015}, {4679, 18391}, {4688, 20718}, {4757, 19862}, {4863, 30305}, {4867, 24929}, {4921, 18178}, {5123, 27131}, {5176, 26792}, {5221, 8583}, {5223, 7962}, {5258, 7489}, {5282, 6603}, {5330, 11260}, {5584, 7971}, {5691, 31821}, {5693, 12680}, {5694, 14872}, {5745, 15950}, {5775, 10589}, {5784, 17579}, {5794, 11415}, {5806, 30308}, {5904, 9957}, {6284, 6737}, {6763, 24928}, {6913, 7982}, {7308, 18421}, {7957, 12672}, {7991, 19541}, {8261, 15671}, {9037, 30438}, {9670, 12625}, {9708, 25415}, {9780, 10107}, {9943, 10304}, {10031, 12532}, {10273, 11231}, {10944, 12527}, {11112, 17634}, {11237, 28609}, {12721, 17132}, {15254, 16861}, {15683, 15726}, {17281, 21871}, {17461, 21870}, {17525, 17637}, {17533, 17606}, {17553, 18165}, {17636, 18254}, {18191, 18417}, {18249, 24953}, {18253, 24541}, {19582, 20942}, {19861, 32636}, {21161, 21740}, {21872, 33299}, {22836, 33595}

X(31165) = reflection of X(65) in X(2)

X(31166) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 5 a^8 - 4 a^4 b^4 - b^8 - 4 a^4 c^4 + 2 b^4 c^4 - c^8 : :

X(31166) lies on these lines: {2, 66}, {6, 428}, {20, 9968}, {30, 19139}, {68, 542}, {69, 26881}, {141, 8780}, {154, 599}, {159, 524}, {315, 4577}, {381, 597}, {549, 23041}, {1352, 10540}, {1576, 8721}, {1598, 8550}, {1899, 18374}, {1992, 2393}, {2781, 13340}, {2937, 15582}, {3827, 24473}, {5064, 19125}, {5648, 32264}, {5656, 10706}, {5878, 15141}, {6593, 18404}, {6776, 14157}, {7517, 15581}, {7540, 9833}, {7714, 9969}, {9143, 11160}, {9924, 15534}, {10168, 23042}, {10192, 20582}, {11216, 20583}, {13248, 15303}, {14216, 14787}, {15305, 25406}, {15585, 22165}, {19132, 23300}, {19506, 25566}, {23325, 25565}

X(31166) = reflection of X(66) in X(2)

X(31167) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^5 + a^3 b^2 - 2 a^2 b^3 - 2 b^5 - 2 a^2 b^2 c + a^3 c^2 - 2 a^2 b c^2 + a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 - 2 c^5 : :

X(31167) lies on these lines: {2, 82}, {551, 17766}, {744, 4664}, {17264, 21083}

X(31167) = reflection of X(82) in X(2)

X(31168) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^4 - 3 a^2 b^2 - 2 b^4 - 3 a^2 c^2 - 3 b^2 c^2 - 2 c^4 : :

X(31168) lies on these lines: {2, 32}, {30, 6287}, {39, 32027}, {69, 14482}, {76, 11287}, {99, 141}, {183, 7919}, {302, 25188}, {303, 25184}, {376, 11178}, {381, 22712}, {384, 10159}, {519, 7977}, {538, 32476}, {543, 10302}, {549, 6054}, {551, 17766}, {599, 732}, {616, 23025}, {617, 23019}, {671, 7924}, {1003, 21358}, {3058, 12954}, {3329, 7848}, {3524, 12252}, {3545, 6249}, {3582, 10080}, {3584, 10064}, {3619, 14039}, {3734, 33264}, {3763, 3972}, {3830, 22803}, {3934, 7911}, {5007, 16897}, {5008, 16987}, {5055, 13111}, {5064, 12144}, {5434, 12944}, {5463, 11300}, {5464, 11299}, {6656, 14568}, {6661, 20582}, {6683, 7939}, {7760, 7854}, {7761, 11361}, {7767, 7859}, {7768, 8362}, {7770, 7936}, {7771, 7868}, {7780, 7948}, {7786, 7879}, {7790, 16990}, {7794, 33021}, {7795, 33008}, {7796, 16043}, {7799, 8359}, {7802, 14033}, {7804, 16988}, {7822, 7904}, {7824, 7849}, {7828, 13468}, {7830, 33265}, {7850, 11174}, {7853, 14046}, {7856, 32956}, {7869, 33004}, {7880, 8290}, {7884, 8667}, {7885, 31239}, {7897, 15482}, {7922, 11285}, {7934, 15271}, {7935, 18546}, {8368, 26613}, {8556, 33219}, {8703, 8725}, {9166, 9478}, {9168, 12073}, {9741, 21356}, {9903, 19875}, {10516, 22676}, {11057, 11286}, {11149, 15700}, {11185, 33210}, {11237, 18983}, {11238, 13078}, {12264, 25055}, {16712, 17297}, {19091, 32788}, {19092, 32787}, {23288, 31950}, {32832, 33285}

X(31168) = reflection of X(83) in X(2)

X(31169) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^3 b - 4 a^2 b^2 + 2 a b^3 + 2 a^3 c + a^2 b c - 2 a b^2 c - b^3 c - 4 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - b c^3 : :

X(31169) lies on these lines: {2, 85}, {9, 664}, {190, 3872}, {220, 32100}, {374, 27472}, {518, 1992}, {644, 17336}, {666, 993}, {1121, 3679}, {1334, 9311}, {3929, 16834}, {4875, 25242}, {9312, 32008}, {16284, 25082}, {17158, 25237}, {31144, 31155}

X(31169) = reflection of X(85) in X(2)

X(31170) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 b^2 - 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 - a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 + 4 a b c^3 - 2 b^2 c^3 : :

X(31170) lies on these lines: {2, 87}, {599, 9025}, {726, 3679}, {4479, 17271}, {6376, 18830}, {31141, 31151}

X(31170) = reflection of X(87) in X(2)

X(31171) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 + 3 a^2 b - 2 b^3 + 3 a^2 c - 15 a b c + 6 b^2 c + 6 b c^2 - 2 c^3 : :

X(31171) lies on these lines: {2, 45}, {9, 25057}, {81, 17195}, {100, 1644}, {519, 4767}, {551, 32944}, {599, 31172}, {668, 4358}, {1639, 6009}, {2802, 3679}, {4688, 27493}, {4756, 11814}, {11346, 30937}, {27776, 30818}, {31143, 31175}

X(31171) = reflection of X(88) in X(2)

X(31172) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 4 a^3 + 6 a^2 b - 6 a b^2 - 8 b^3 + 6 a^2 c - 15 a b c - 6 a c^2 - 8 c^3 : :

X(31172) lies on these lines: {2, 44}, {551, 33065}, {599, 31171}, {668, 4671}, {3679, 4134}, {3828, 25959}

X(31172) = reflection of X(89) in X(2)

X(31173) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^4 + a^2 b^2 - 4 b^4 + a^2 c^2 + 4 b^2 c^2 - 4 c^4 : :

X(31173) lies on these lines: {2, 187}, {4, 7801}, {5, 7810}, {30, 114}, {32, 11318}, {39, 7773}, {69, 7615}, {99, 8597}, {115, 524}, {141, 3363}, {183, 7617}, {230, 8355}, {315, 33006}, {325, 543}, {376, 7694}, {381, 511}, {382, 7888}, {385, 9166}, {512, 9148}, {530, 623}, {531, 624}, {538, 671}, {542, 15980}, {546, 7794}, {574, 5077}, {597, 1692}, {620, 8598}, {626, 8370}, {754, 5461}, {868, 1641}, {1007, 7618}, {1506, 8359}, {1570, 1992}, {2021, 11287}, {2030, 7913}, {2080, 5055}, {2548, 33190}, {2549, 9770}, {3091, 7854}, {3627, 7863}, {3734, 11317}, {3788, 33007}, {3843, 17130}, {3934, 7883}, {5007, 5025}, {5008, 7844}, {5031, 20582}, {5041, 7785}, {5064, 5140}, {5104, 21358}, {5148, 11238}, {5162, 11286}, {5184, 19875}, {5194, 11237}, {5319, 33292}, {5355, 8584}, {5971, 17964}, {5999, 11150}, {6033, 11645}, {6292, 8367}, {6324, 6787}, {6390, 15300}, {6566, 32435}, {6567, 32432}, {6655, 31652}, {6683, 7911}, {6781, 9167}, {7622, 8589}, {7745, 7852}, {7746, 23055}, {7747, 7874}, {7748, 32816}, {7752, 7833}, {7759, 14063}, {7760, 14045}, {7763, 33192}, {7764, 33229}, {7768, 32993}, {7778, 11159}, {7779, 11054}, {7780, 7860}, {7784, 31239}, {7788, 14711}, {7796, 14062}, {7799, 8591}, {7802, 33274}, {7805, 7900}, {7814, 33019}, {7816, 7870}, {7823, 7886}, {7849, 16044}, {7862, 15513}, {7867, 33237}, {7880, 11361}, {7909, 14042}, {7922, 33018}, {7925, 9855}, {7936, 33002}, {7939, 15031}, {7941, 32450}, {8357, 9698}, {8556, 9301}, {8586, 15533}, {8667, 18362}, {8859, 14061}, {8860, 10631}, {9766, 11648}, {10297, 15526}, {10488, 12151}, {10717, 10989}, {11676, 23234}, {12150, 14046}, {12156, 33288}, {13466, 31160}, {14023, 32980}, {14693, 15699}, {14929, 16509}, {15993, 19662}, {18842, 33196}, {20112, 22165}, {31417, 33202}, {31455, 33215}, {31457, 33023}

X(31173) = reflection of X(187) in X(2)
X(31173) = complement of isotomic conjugate of X(36882)

X(31174) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 + 4 b^2 c^2) : :

X(31174) lies on these lines: {2, 647}, {381, 30209}, {512, 9148}, {523, 7625}, {525, 1637}, {597, 9030}, {599, 8675}, {804, 8644}, {2489, 23285}, {2501, 2525}, {2799, 30474}, {3265, 12077}, {3906, 8371}, {4108, 32472}, {4139, 4728}, {5309, 7652}, {5642, 13480}, {6070, 15526}, {7801, 23105}, {8672, 31148}, {8704, 9191}, {10162, 18311}, {10567, 11163}, {31953, 32225}

X(31174) = reflection of X(647) in X(2)
X(31174) = complement of X(36900)

X(31175) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^5 - a^3 b^2 + 2 a^2 b^3 - 2 b^5 - 2 a^2 b^2 c - a^3 c^2 - 2 a^2 b c^2 + a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - 2 c^5 : :

X(31175) lies on these lines: {2, 662}, {86, 14061}, {190, 24086}, {319, 27559}, {671, 903}, {897, 24711}, {1121, 1494}, {1577, 14616}, {4664, 31163}, {8674, 9140}, {31143, 31171}

X(31175) = reflection of X(662) in X(2)

X(31176) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics (b^2 - c^2) (-a^4 - 2 a^2 b^2 - 2 a^2 c^2 + 2 b^2 c^2) : :

X(31176) lies on these lines: {2, 669}, {351, 32472}, {381, 1499}, {512, 9148}, {523, 7840}, {599, 9009}, {804, 5996}, {890, 30968}, {1637, 32478}, {2501, 5064}, {3005, 23878}, {3566, 14420}, {5054, 5926}, {7775, 23099}, {7841, 14824}, {8029, 31133}, {9134, 32473}, {11123, 31152}, {11182, 11186}, {15099, 31181}

X(31176) = reflection of X(669) in X(2)

X(31177) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^3 + 2 a^2 b - a b^2 - 4 b^3 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 - 4 c^3 : :

X(31177) lies on these lines: {2, 896}, {513, 4379}, {519, 4442}, {524, 3120}, {551, 4425}, {758, 3679}, {1647, 7238}, {1992, 33128}, {2650, 17677}, {2796, 3936}, {3218, 27759}, {3241, 17778}, {3722, 28562}, {4141, 4912}, {4432, 31029}, {4715, 25383}, {4945, 17960}, {5333, 17553}, {11544, 20653}, {17132, 32848}, {17484, 21026}, {17601, 27741}

X(31177) = reflection of X(896) in X(2)

X(31178) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics 2 a^2 b - a b^2 + 2 a^2 c + 3 a b c + 2 b^2 c - a c^2 + 2 b c^2 : :

X(31178) lies on these lines: {1, 536}, {2, 38}, {7, 33076}, {8, 24693}, {10, 17227}, {37, 25055}, {45, 24821}, {75, 519}, {142, 33165}, {190, 24331}, {192, 28554}, {354, 31137}, {376, 29054}, {518, 599}, {545, 24357}, {551, 726}, {678, 24344}, {740, 3241}, {742, 4795}, {870, 18822}, {1086, 29659}, {1125, 17354}, {3242, 24342}, {3475, 33160}, {3655, 29010}, {3696, 4677}, {3729, 16484}, {3739, 19875}, {3751, 16833}, {3757, 4650}, {3773, 29577}, {3790, 29582}, {3873, 31136}, {3967, 25502}, {3999, 29827}, {4003, 29825}, {4054, 24217}, {4080, 25378}, {4141, 27754}, {4407, 29576}, {4418, 17715}, {4430, 21020}, {4439, 17244}, {4649, 16834}, {4660, 7321}, {4671, 17450}, {4675, 32847}, {4687, 19883}, {4732, 4772}, {4753, 16816}, {4755, 28582}, {4777, 14421}, {4937, 30950}, {5249, 33169}, {5542, 29594}, {7262, 32940}, {9055, 10022}, {9458, 24594}, {10707, 27479}, {15485, 17351}, {16496, 25590}, {16823, 32935}, {17116, 32941}, {17146, 31025}, {17155, 17592}, {17180, 18157}, {17310, 27474}, {17369, 29660}, {17392, 28503}, {17716, 32923}, {18201, 29828}, {24231, 32784}, {24399, 24456}, {24436, 26241}, {25557, 29674}, {26627, 32927}, {26842, 33074}, {27186, 33162}, {27471, 31160}, {29584, 32921}, {29651, 32939}, {29669, 33068}, {29685, 33146}

X(31178) = reflection of X(984) in X(2)

X(31179) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^3 + 4 a^2 b + a b^2 - 2 b^3 + 4 a^2 c + b^2 c + a c^2 + b c^2 - 2 c^3 : :

X(31179) lies on these lines: {1, 25378}, {2, 6}, {239, 26738}, {381, 952}, {386, 17679}, {519, 25385}, {551, 33122}, {908, 29574}, {2177, 28562}, {2796, 24725}, {3758, 27757}, {3849, 24296}, {3923, 4933}, {4054, 17133}, {4080, 17318}, {4141, 32935}, {4358, 29573}, {4384, 30588}, {4414, 28558}, {4649, 27759}, {4664, 27491}, {4675, 24594}, {4688, 27476}, {4725, 27747}, {8229, 20423}, {9041, 29832}, {16666, 30823}, {17119, 31030}, {17126, 27741}, {17290, 31029}, {17305, 30991}, {17316, 30566}, {17677, 19767}, {17763, 27777}, {26227, 28538}, {29069, 31164}, {29570, 31056}

X(31179) = reflection of X(1150) in X(2)

X(31180) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^10 - 4 a^6 b^4 + 2 a^4 b^6 + 3 a^2 b^8 - 2 b^10 - 6 a^6 b^2 c^2 + 6 a^4 b^4 c^2 - 6 a^2 b^6 c^2 + 6 b^8 c^2 - 4 a^6 c^4 + 6 a^4 b^2 c^4 + 6 a^2 b^4 c^4 - 4 b^6 c^4 + 2 a^4 c^6 - 6 a^2 b^2 c^6 - 4 b^4 c^6 + 3 a^2 c^8 + 6 b^2 c^8 - 2 c^10 : :

X(31180) lies on these lines: {2, 3}, {394, 25739}, {542, 20806}, {1216, 27365}, {1494, 14615}, {1533, 18418}, {1568, 11456}, {1853, 11459}, {2979, 14852}, {3917, 23325}, {6391, 33565}, {7788, 28706}, {15066, 18474}, {17834, 26917}

X(31180) = reflection of X(24) in X(2)

X(31181) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics a^10 - 4 a^6 b^4 + 2 a^4 b^6 + 3 a^2 b^8 - 2 b^10 - 8 a^6 b^2 c^2 + 4 a^4 b^4 c^2 - 2 a^2 b^6 c^2 + 6 b^8 c^2 - 4 a^6 c^4 + 4 a^4 b^2 c^4 - 2 a^2 b^4 c^4 - 4 b^6 c^4 + 2 a^4 c^6 - 2 a^2 b^2 c^6 - 4 b^4 c^6 + 3 a^2 c^8 + 6 b^2 c^8 - 2 c^10 : :

X(31181) lies on these lines: {2, 3}, {539, 16266}, {1154, 1853}, {1494, 20564}, {8144, 11238}, {11237, 32047}, {11255, 15534}, {11265, 13846}, {11266, 13847}, {11267, 16644}, {11268, 16645}, {11425, 13470}, {11550, 15068}, {13391, 14852}, {13561, 17834}, {14864, 15083}, {15099, 31176}

X(31181) = reflection of X(26) in X(2)

X(31182) = (name pending)

Barycentrics (b - c) (3a - b - c)^2 : :

X(31182) lies on these lines: {10, 28296}, {101, 6079}, {241, 514}, {522, 2490}, {1635, 3239}, {2527, 4778}, {2976, 3667}, {3798, 10196}, {4546, 8643}

Collineation mappings involving Gemini triangle 108: X(31183)-X(31234)

m(X) = (3a - b - c) x - 2(a - b + c) y - 2(a + b - c) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(241)X(514). Among the fixed points are X(i) for these i: 2, 241, 514, 650, 1323, 3008, 3676,3911,30719, 31182, 31182. (Clark Kimberling, January 23, 2019)

X(31183) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^2 - 3 a b + 2 b^2 - 3 a c - 4 b c + 2 c^2 : :

X(31133) lies on these lines: {1, 2}, {6, 20195}, {7, 3973}, {9, 1086}, {40, 19512}, {44, 6173}, {57, 24796}, {142, 1743}, {144, 4902}, {165, 7397}, {218, 25525}, {220, 30827}, {241, 31184}, {277, 5745}, {344, 17151}, {391, 21255}, {484, 24590}, {599, 31243}, {673, 15485}, {948, 3911}, {1111, 30854}, {1212, 16602}, {1323, 31188}, {1449, 17245}, {1453, 17529}, {1699, 9441}, {3247, 17366}, {3305, 23681}, {3452, 24181}, {3487, 8951}, {3586, 30810}, {3589, 4798}, {3644, 31333}, {3663, 18230}, {3672, 25072}, {3729, 4473}, {3731, 4000}, {3752, 16601}, {3826, 7290}, {3875, 17263}, {3946, 16673}, {3950, 4402}, {4007, 17267}, {4034, 17231}, {4363, 6687}, {4383, 17745}, {4413, 21542}, {4419, 17067}, {4422, 4659}, {4488, 15828}, {4648, 16667}, {4665, 17279}, {4675, 16670}, {4708, 17259}, {4873, 17119}, {4887, 6172}, {4898, 17243}, {5219, 5228}, {5234, 24178}, {5251, 21514}, {5273, 24175}, {5435, 10481}, {5437, 16572}, {5584, 19517}, {5723, 31186}, {5880, 15601}, {7271, 8732}, {7274, 8232}, {7280, 11349}, {7308, 24789}, {7402, 7989}, {8616, 24596}, {10434, 28250}, {10436, 17352}, {14433, 31207}, {15251, 26446}, {16676, 17301}, {17227, 17272}, {17260, 17304}, {17265, 17296}, {17270, 17283}, {17274, 17335}, {17286, 17341}, {17298, 17349}, {17353, 25590}, {24046, 31446}, {24199, 26685}, {24635, 31226}, {30393, 33144}, {31187, 31190}, {31195, 31204}, {31205, 31233}

X(31183) = complement of X(29627)

X(31184) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 3 a^2 b^2 c - 2 b^4 c - a^3 c^2 + 3 a^2 b c^2 + 4 a b^2 c^2 - a^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31184) lies on these lines: {2, 3}, {6, 17058}, {116, 5792}, {241, 31183}, {1482, 26006}, {2391, 6692}, {3002, 31187}, {5199, 31211}, {5687, 28757}, {10246, 25935}, {16608, 20818}, {18644, 24316}, {31189, 31226}

X(31184) = complement of X(30809)

X(31185) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 7 a^5 - a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - 3 a b^4 + 5 b^5 - a^4 c + 4 a^2 b^2 c - 3 b^4 c - 4 a^3 c^2 + 4 a^2 b c^2 + 6 a b^2 c^2 - 2 b^3 c^2 - 4 a^2 c^3 - 2 b^2 c^3 - 3 a c^4 - 3 b c^4 + 5 c^5 : :

X(31185) lies on these lines: {2, 3}, {8074, 31190}, {12245, 26006}, {23058, 31211}, {31192, 31230}, {31203, 31221}, {31232, 31234}

X(31186) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 4 a^5 - 3 a^3 b^2 - 3 a^2 b^3 - a b^4 + 3 b^5 + a^2 b^2 c - b^4 c - 3 a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 3 a^2 c^3 - 2 b^2 c^3 - a c^4 - b c^4 + 3 c^5 : :

X(31186) lies on these lines: {2, 3}, {1729, 5437}, {5690, 26006}, {5723, 31183}, {5829, 20195}, {17073, 20204}, {18644, 24315}

X(31187) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31187) lies on these lines: {2, 6}, {3, 759}, {31, 31245}, {44, 5219}, {45, 17720}, {55, 24892}, {165, 21949}, {442, 4252}, {499, 25493}, {595, 31493}, {631, 5721}, {748, 2361}, {902, 31140}, {908, 16885}, {956, 24222}, {1001, 24217}, {1086, 5744}, {1104, 5705}, {1191, 26363}, {1279, 5231}, {1376, 33138}, {1465, 16602}, {1616, 10527}, {1656, 5398}, {1707, 3838}, {1714, 4255}, {1723, 5437}, {1834, 6857}, {1865, 7490}, {2049, 6693}, {2886, 3052}, {3002, 31184}, {3008, 24281}, {3011, 3242}, {3160, 5723}, {3434, 21000}, {3526, 5396}, {3624, 31503}, {3634, 5725}, {3663, 3772}, {3668, 3911}, {3681, 17783}, {3756, 16020}, {3927, 24160}, {3928, 4902}, {4042, 29846}, {4257, 17528}, {4361, 32851}, {4415, 5273}, {4421, 32865}, {4422, 28808}, {4423, 29662}, {4428, 33141}, {4640, 17064}, {4641, 31266}, {4888, 25525}, {5220, 17719}, {5230, 24953}, {5292, 6675}, {5336, 18229}, {5706, 6852}, {5724, 9780}, {5846, 30741}, {6690, 33137}, {7297, 24611}, {7988, 15601}, {8609, 25939}, {8616, 11235}, {9053, 26245}, {9708, 17734}, {11679, 17229}, {15492, 31142}, {16455, 19760}, {17070, 24248}, {17119, 17740}, {17242, 33116}, {17290, 24627}, {17335, 30867}, {17595, 33129}, {17597, 29681}, {19273, 20083}, {19765, 24883}, {19832, 25503}, {29607, 31233}, {31183, 31190}, {31195, 31199}, {31227, 31228}

X(31187) = complement of X(30828)

X(31188) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (7 a - 5 b - 5 c) (a + b - c) (a - b + c) : :

X(31188) lies on these lines: {2, 7}, {8, 1317}, {11, 30312}, {56, 17535}, {80, 5731}, {140, 938}, {241, 31197}, {348, 27814}, {499, 11010}, {551, 16236}, {631, 4313}, {632, 3487}, {942, 3533}, {948, 31202}, {962, 23708}, {1000, 26446}, {1155, 9779}, {1210, 10303}, {1323, 31183}, {1387, 4345}, {1466, 17536}, {1617, 9342}, {1698, 4315}, {1788, 4323}, {3160, 5723}, {3339, 19878}, {3475, 5326}, {3485, 7294}, {3488, 5054}, {3523, 4304}, {3525, 5703}, {3526, 5719}, {3545, 5122}, {3586, 15692}, {3600, 3634}, {3616, 11011}, {3617, 6049}, {3676, 31992}, {3711, 14151}, {3828, 13462}, {4292, 7486}, {4297, 7319}, {4298, 19872}, {4308, 5252}, {4344, 17722}, {4413, 7677}, {4552, 24620}, {5056, 15803}, {5070, 5714}, {5218, 8236}, {5222, 31201}, {5274, 10164}, {5281, 30331}, {5290, 31253}, {5432, 10580}, {5698, 6667}, {5708, 16239}, {5828, 8666}, {5933, 28626}, {6174, 12730}, {6357, 31204}, {6684, 9785}, {7173, 10248}, {7181, 31205}, {9364, 17125}, {9579, 15022}, {9581, 15717}, {9778, 10589}, {10176, 18419}, {10578, 17728}, {11539, 15934}, {12630, 26015}, {14986, 31423}, {15699, 18541}, {15702, 24929}, {15708, 30282}, {15709, 15933}, {16602, 17080}, {17090, 31228}, {18421, 19883}, {31189, 31219}

X(31188) = complement of X(30833)

X(31189) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 7 a^2 - 4 a b + 5 b^2 - 4 a c - 6 b c + 5 c^2 : :

X(31189) lies on these lines: {1, 2}, {7, 17352}, {142, 4747}, {348, 31230}, {962, 19512}, {3161, 4000}, {3752, 25082}, {3772, 6557}, {4402, 17279}, {4419, 6687}, {4440, 4488}, {4454, 17067}, {4643, 17356}, {5226, 30617}, {5260, 21529}, {5296, 17325}, {5435, 7195}, {5749, 17278}, {5936, 17371}, {7397, 9778}, {16706, 18230}, {17282, 21296}, {17283, 32099}, {17353, 31995}, {31184, 31226}, {31188, 31219}, {31203, 31213}, {31212, 31224}

X(31190) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c + 10 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31190) lies on these lines: {1, 1145}, {2, 7}, {40, 6967}, {46, 25522}, {55, 31249}, {84, 6944}, {104, 5587}, {165, 3816}, {200, 17728}, {392, 3624}, {404, 9581}, {516, 8166}, {551, 1000}, {631, 5436}, {632, 5771}, {956, 1698}, {958, 13370}, {1001, 5537}, {1108, 16602}, {1125, 3525}, {1210, 5438}, {1329, 3361}, {1376, 24392}, {1420, 24982}, {1512, 3576}, {1532, 21164}, {1699, 17613}, {1706, 3086}, {1768, 15297}, {1788, 15829}, {2136, 14986}, {2323, 25934}, {2950, 12611}, {3090, 12436}, {3158, 11019}, {3243, 6745}, {3333, 26364}, {3339, 25681}, {3526, 24474}, {3586, 16371}, {3601, 6921}, {3689, 31146}, {3754, 9624}, {3772, 8056}, {3824, 5070}, {4187, 15803}, {4193, 9579}, {4312, 5087}, {4413, 5231}, {4659, 28808}, {4850, 16578}, {4859, 25580}, {4873, 17740}, {4901, 5205}, {4924, 14469}, {5121, 7290}, {5253, 9578}, {5265, 5795}, {5432, 10582}, {5660, 25558}, {5705, 16408}, {5722, 9945}, {5727, 6224}, {5880, 6667}, {6700, 11523}, {6705, 6964}, {6717, 9817}, {6848, 9841}, {6952, 8227}, {7288, 8582}, {7681, 10270}, {8074, 31185}, {8583, 24914}, {9311, 30610}, {9613, 17619}, {9623, 15325}, {10164, 26105}, {10199, 21630}, {10826, 14800}, {11518, 27385}, {12053, 26062}, {12526, 24954}, {14475, 31207}, {17233, 30567}, {17527, 31424}, {17718, 31235}, {19804, 20237}, {19862, 28629}, {20103, 24477}, {29607, 31228}, {30337, 32157}, {31183, 31187}, {31246, 32636}

X(31190) = complement of X(5328)

X(31191) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 4 a^2 - a b + 3 b^2 - a c - 2 b c + 3 c^2 : :

X(31191) lies on these lines: {1, 2}, {6, 21255}, {37, 25097}, {57, 24797}, {142, 3589}, {169, 5437}, {190, 3663}, {344, 4021}, {516, 7397}, {527, 17290}, {597, 3834}, {599, 4700}, {946, 19512}, {958, 21529}, {993, 21514}, {1266, 17354}, {1375, 6693}, {1376, 21542}, {2321, 17357}, {2325, 17301}, {2391, 6692}, {3416, 4989}, {3452, 30618}, {3618, 3664}, {3686, 3763}, {3707, 17237}, {3752, 16600}, {3755, 4702}, {3758, 4896}, {3821, 4759}, {3836, 4349}, {3875, 4072}, {3879, 17283}, {3911, 31230}, {3946, 3950}, {3986, 4657}, {4000, 4659}, {4029, 17395}, {4058, 4361}, {4085, 16593}, {4357, 17335}, {4363, 17067}, {4364, 6687}, {4395, 17359}, {4416, 17291}, {4422, 17382}, {4431, 17358}, {4464, 17240}, {4656, 32774}, {4672, 30424}, {4690, 20582}, {4748, 17306}, {4856, 17296}, {4859, 5749}, {4869, 16667}, {4873, 28313}, {4967, 17371}, {5257, 17337}, {5267, 11343}, {5294, 24177}, {5435, 10521}, {5750, 17278}, {6329, 17376}, {6679, 10164}, {7402, 19925}, {7719, 17917}, {8074, 31185}, {12511, 16435}, {15828, 17304}, {16583, 16602}, {16609, 31221}, {16611, 31197}, {17133, 17269}, {17302, 25101}, {17321, 25072}, {17338, 17383}, {17341, 17380}, {17368, 24199}, {17392, 31243}, {20602, 27003}, {20927, 24208}, {21060, 26128}, {21526, 25440}, {24175, 24781}, {31213, 31226}, {31219, 31224}, {31233, 31234}

X(31191) = complement of X(17284)

X(31192) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 4 a^4 - 4 a^3 b + a^2 b^2 - 4 a b^3 + 3 b^4 - 4 a^3 c + 2 a^2 b c + 4 a b^2 c - 4 b^3 c + a^2 c^2 + 4 a b c^2 + 2 b^2 c^2 - 4 a c^3 - 4 b c^3 + 3 c^4 : :

X(31192) lies on these lines: {2, 11}, {1638, 6544}, {3321, 3323}, {3614, 17682}, {4152, 4437}, {4534, 17044}, {4904, 6710}, {5723, 31183}, {6284, 17675}, {15338, 17671}, {31185, 31230}

X(31193) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^4 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31193) lies on these lines: {2, 3}

X(31194) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 - a^3 b^2 c^2 - a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^4 c^3 - a^2 b^2 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31194) lies on these lines: {2, 3}

X(31195) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c - 2 b^3 c - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31195) lies on these lines: {2, 31}, {32, 31240}, {902, 30825}, {2251, 31135}, {8616, 30857}, {9350, 24582}, {24892, 26007}, {31183, 31204}, {31187, 31199}, {31200, 31226}

X(31196) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^5 - a^4 b - 2 a b^4 + 2 b^5 - a^4 c - 2 b^4 c - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31196) lies on these lines: {2, 32}, {2251, 30820}, {3002, 31184}, {5138, 24901}, {31220, 31234}

X(31197) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics a (a b + b^2 + a c - 10 b c + c^2) : :

X(31197) lies on these lines: {1, 16864}, {2, 37}, {10, 3756}, {43, 3848}, {44, 3306}, {57, 16885}, {65, 28257}, {140, 15852}, {210, 17449}, {241, 31188}, {354, 21805}, {392, 4674}, {650, 14475}, {899, 17450}, {940, 16668}, {1054, 15254}, {1086, 5316}, {1104, 16408}, {1149, 4731}, {1155, 17125}, {1279, 4413}, {1418, 3911}, {1427, 31231}, {1449, 23511}, {1743, 5437}, {2999, 16884}, {3218, 15492}, {3616, 21896}, {3624, 4646}, {3696, 4871}, {3698, 28352}, {3740, 17063}, {3742, 4849}, {3744, 9342}, {3748, 9350}, {3816, 21949}, {3826, 5121}, {3834, 5233}, {3840, 4732}, {3921, 4694}, {3931, 19878}, {4054, 16594}, {4383, 16671}, {4415, 24175}, {4868, 19862}, {4891, 26103}, {5235, 16726}, {5241, 17237}, {5524, 15570}, {14951, 16604}, {15481, 18201}, {16611, 31191}, {16814, 17595}, {16815, 17448}, {16832, 16975}, {17244, 21868}, {19546, 30271}, {21242, 25377}, {21806, 30950}, {24183, 26580}, {24774, 24775}, {27191, 30867}, {28581, 30947}, {30811, 31243}, {31183, 31187}, {31198, 31211}, {31203, 31214}, {31205, 31227}

X(31197) = complement of X(30829)

X(31198) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics a (a^2 b^2 + a b^3 - 3 a b^2 c + a^2 c^2 - 3 a b c^2 - 4 b^2 c^2 + a c^3) : :

X(31198) lies on these lines: {2, 39}, {32, 16412}, {86, 4263}, {141, 17058}, {142, 28244}, {187, 11329}, {216, 17073}, {386, 16408}, {474, 19761}, {940, 20970}, {978, 5437}, {1015, 4384}, {1125, 3752}, {1500, 16831}, {1573, 24603}, {1574, 3912}, {1575, 29571}, {2092, 15668}, {2275, 16832}, {3002, 31184}, {3008, 6692}, {3199, 15149}, {3306, 28289}, {3664, 21892}, {3729, 21826}, {3734, 11353}, {3739, 17053}, {4260, 24923}, {4688, 8610}, {4850, 29612}, {5188, 19522}, {5337, 25946}, {5737, 23447}, {5743, 16736}, {5745, 9367}, {10436, 21796}, {15513, 19308}, {16574, 28252}, {16610, 17023}, {16726, 17330}, {17143, 26113}, {20917, 27076}, {24199, 28366}, {24581, 24918}, {27147, 27641}, {31197, 31211}

X(31198) = complement of X(30830)

X(31199) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 b - 3 a^2 b^2 + 2 a b^3 + 3 a^3 c - 2 a^2 b c - 4 a b^2 c + 2 b^3 c - 3 a^2 c^2 - 4 a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(31199) lies on these lines: {1, 2}, {672, 17278}, {902, 24596}, {6687, 24330}, {31187, 31195}, {31204, 31226}

X(31200) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 b - 3 a^2 b^2 + 2 a b^3 + 3 a^3 c - a^2 b c - 3 a b^2 c + 2 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(31200) lies on these lines: {1, 2}, {57, 24800}, {672, 4859}, {673, 8616}, {1743, 30949}, {3550, 24596}, {3752, 25074}, {3973, 20347}, {4713, 6687}, {16593, 32865}, {17278, 17754}, {20195, 24512}, {31195, 31226}, {31203, 31210}

X(31201) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 6 a^3 - 3 a^2 b - 5 a b^2 + 4 b^3 - 3 a^2 c + 10 a b c - 4 b^2 c - 5 a c^2 - 4 b c^2 + 4 c^3 : :

X(31201) lies on these lines: {2, 44}, {650, 4379}, {1086, 3911}, {1279, 1647}, {3752, 8609}, {5222, 31188}, {6603, 29571}, {16602, 31224}, {29607, 31227}, {31183, 31187}

X(31202) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - 3 a^2 b - 4 a b^2 + 2 b^3 - 3 a^2 c + 20 a b c - 2 b^2 c - 4 a c^2 - 2 b c^2 + 2 c^3 : :

X(31202) lies on these lines: {2, 45}, {948, 31188}, {1647, 4413}, {7232, 31056}, {8572, 25011}, {29630, 31233}, {31183, 31187}

X(31203) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^4 - 4 a^3 b + 3 a^2 b^2 - 4 a b^3 + 2 b^4 - 4 a^3 c + 2 a^2 b c + 4 a b^2 c - 4 b^3 c + 3 a^2 c^2 + 4 a b c^2 + 4 b^2 c^2 - 4 a c^3 - 4 b c^3 + 2 c^4 : :

X(31203) lies on these lines: {2, 11}, {241, 31183}, {3008, 17728}, {3689, 30813}, {10895, 17682}, {10896, 17675}, {12953, 17671}, {31185, 31221}, {31187, 31195}, {31189, 31213}, {31197, 31214}, {31200, 31210}

X(31204) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31204) lies on these lines: {2, 6}, {21, 24880}, {58, 24902}, {88, 24175}, {100, 33138}, {442, 16948}, {1621, 24892}, {1834, 15674}, {3090, 13408}, {3526, 5453}, {4653, 15671}, {4656, 33133}, {4850, 25080}, {5273, 33151}, {5284, 33140}, {5745, 33129}, {6357, 31188}, {6675, 24883}, {6690, 33139}, {6693, 14005}, {11684, 24161}, {17070, 33100}, {17123, 31272}, {17127, 31245}, {17557, 25441}, {18593, 31231}, {31183, 31195}, {31199, 31226}

X(31205) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c - 3 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3 : :

X(31205) lies on these lines: {2, 6}, {190, 4054}, {1043, 6675}, {1732, 3306}, {2321, 33116}, {2490, 31992}, {3772, 17247}, {4085, 33138}, {4358, 31333}, {5219, 17335}, {5745, 24199}, {6693, 14007}, {7181, 31188}, {9791, 17070}, {11110, 24880}, {11679, 17240}, {24627, 27191}, {25529, 26580}, {31183, 31233}, {31197, 31227}, {31210, 31211}, {31217, 31225}

X(31206) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^11 - a^10 b - 4 a^9 b^2 - 4 a^7 b^4 + 4 a^6 b^5 + 6 a^5 b^6 - 2 a^4 b^7 + a^3 b^8 - 3 a^2 b^9 - 2 a b^10 + 2 b^11 - a^10 c + 4 a^8 b^2 c - 4 a^6 b^4 c - 2 a^4 b^6 c + 5 a^2 b^8 c - 2 b^10 c - 4 a^9 c^2 + 4 a^8 b c^2 - 2 a^7 b^2 c^2 - 2 a^6 b^3 c^2 + 2 a^5 b^4 c^2 + 2 a^4 b^5 c^2 - 2 a^3 b^6 c^2 - 2 a^2 b^7 c^2 + 6 a b^8 c^2 - 2 b^9 c^2 - 2 a^6 b^2 c^3 + 2 a^4 b^4 c^3 - 2 a^2 b^6 c^3 + 2 b^8 c^3 - 4 a^7 c^4 - 4 a^6 b c^4 + 2 a^5 b^2 c^4 + 2 a^4 b^3 c^4 + 2 a^3 b^4 c^4 + 2 a^2 b^5 c^4 - 4 a b^6 c^4 - 4 b^7 c^4 + 4 a^6 c^5 + 2 a^4 b^2 c^5 + 2 a^2 b^4 c^5 + 4 b^6 c^5 + 6 a^5 c^6 - 2 a^4 b c^6 - 2 a^3 b^2 c^6 - 2 a^2 b^3 c^6 - 4 a b^4 c^6 + 4 b^5 c^6 - 2 a^4 c^7 - 2 a^2 b^2 c^7 - 4 b^4 c^7 + a^3 c^8 + 5 a^2 b c^8 + 6 a b^2 c^8 + 2 b^3 c^8 - 3 a^2 c^9 - 2 b^2 c^9 - 2 a c^10 - 2 b c^10 + 2 c^11 : :

X(31206) lies on these lines: {2, 3}

X(31207) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (b - c) (3 a^2 - 2 a b - 2 a c + 2 b c) : :

X(31207) lies on these lines: {2, 649}, {42, 24666}, {514, 27115}, {650, 4379}, {663, 31288}, {665, 31208}, {693, 4763}, {899, 23655}, {1635, 4885}, {1638, 2490}, {1639, 2487}, {3231, 23575}, {3239, 4750}, {3624, 29350}, {3676, 6546}, {3798, 4120}, {4369, 4893}, {4394, 4728}, {4413, 23865}, {4468, 6544}, {4782, 30795}, {5235, 18200}, {6545, 11068}, {7658, 16892}, {8643, 17072}, {8655, 24533}, {14425, 21104}, {14433, 31183}, {14475, 31190}, {19804, 20909}, {24756, 30942}, {25126, 30968}, {25128, 26037}, {25301, 26038}, {27485, 31238}, {28292, 31423}

X(31208) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (b - c) (3 a^4 - a^3 b + 2 a^2 b^2 - 2 a b^3 - a^3 c + 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 a c^3 + 2 b c^3) : :

X(31208) lies on these lines: {2, 667}, {665, 31207}, {905, 31210}, {3669, 31230}, {4083, 29598}

X(31209) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (b - c) (2 a^2 - 2 a b - 2 a c + b c) : :

X(31209) lies on these lines: {2, 650}, {100, 8641}, {513, 27013}, {514, 24924}, {631, 8760}, {657, 25955}, {812, 30835}, {905, 4462}, {908, 23806}, {1635, 3835}, {1639, 17069}, {1643, 17244}, {1698, 29066}, {2490, 3004}, {2512, 4108}, {2516, 4106}, {3146, 8142}, {3310, 25084}, {3616, 14077}, {3618, 9001}, {3763, 9015}, {3776, 6546}, {3890, 9366}, {4025, 4521}, {4359, 21438}, {4893, 31207}, {4391, 14838}, {4411, 4751}, {4453, 4468}, {4687, 4777}, {4763, 4776}, {4789, 25594}, {4850, 25098}, {4932, 31286}, {4998, 5375}, {5218, 11934}, {6008, 26798}, {6050, 21301}, {6544, 16892}, {6586, 20906}, {7626, 7803}, {7650, 8043}, {10196, 21212}, {11124, 15280}, {11680, 15283}, {16751, 27527}, {17490, 25271}, {17496, 20317}, {19804, 21611}, {22383, 32911}, {29070, 31251}, {29362, 30795}, {31233, 31992}

X(31209) = complement of X(26985)
X(31209) = anticomplement of X(31250)

X(31210) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c + a^2 b c + a b^2 c - 2 b^3 c + a b c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31210) lies on these lines: {2, 31}, {902, 30857}, {905, 31208}, {3550, 30825}, {26007, 33140}, {31183, 31187}, {31200, 31203}, {31205, 31211}

X(31211) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 2 a^2 - 5 a b + b^2 - 5 a c - 6 b c + c^2 : :

X(31211) lies on these lines: {1, 2}, {9, 7222}, {39, 16602}, {45, 17132}, {75, 25072}, {142, 4643}, {516, 24693}, {536, 31285}, {597, 4758}, {966, 20195}, {1213, 17058}, {1323, 31225}, {1743, 4747}, {2140, 3452}, {2325, 4688}, {3664, 17277}, {3686, 17245}, {3707, 4675}, {3739, 4422}, {3752, 25092}, {3759, 4909}, {3842, 4353}, {3946, 4698}, {3986, 4000}, {4021, 4687}, {4029, 17119}, {4072, 32087}, {4098, 17151}, {4253, 5437}, {4357, 27191}, {4364, 17067}, {4395, 4755}, {4402, 16673}, {4416, 27147}, {4440, 17260}, {4472, 6687}, {4510, 31227}, {4699, 25101}, {4700, 17392}, {4751, 17353}, {4856, 17348}, {4859, 5296}, {4967, 17263}, {5199, 31184}, {5241, 30823}, {5257, 17278}, {5267, 11349}, {5316, 30824}, {5750, 17337}, {5806, 15489}, {6684, 19512}, {6692, 6706}, {6996, 12512}, {7397, 10164}, {7406, 28158}, {10171, 28858}, {17258, 31311}, {18230, 25590}, {23058, 31185}, {31197, 31198}, {31205, 31210}, {31213, 31222}

X(31211) = complement of X(29571)

X(31212) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 13 a^5 - 3 a^4 b - 6 a^3 b^2 - 6 a^2 b^3 - 7 a b^4 + 9 b^5 - 3 a^4 c + 10 a^2 b^2 c - 7 b^4 c - 6 a^3 c^2 + 10 a^2 b c^2 + 14 a b^2 c^2 - 2 b^3 c^2 - 6 a^2 c^3 - 2 b^2 c^3 - 7 a c^4 - 7 b c^4 + 9 c^5 : :

X(31212) lies on these lines: {2, 3}, {1323, 31183}, {31189, 31224}

X(31213) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + a^3 b c + 5 a^2 b^2 c + a b^3 c - 2 b^4 c - a^3 c^2 + 5 a^2 b c^2 + 6 a b^2 c^2 - a^2 c^3 + a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31213) lies on these lines: {2, 3}, {2287, 24884}, {7181, 31188}, {31183, 31195}, {31189, 31203}, {31191, 31226}, {31211, 31222}

X(31214) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 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 + 2 b^5 c^2 - 2 a^4 c^3 - 2 a^2 b^2 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31214) lies on these lines: {2, 3}, {31197, 31203}

X(31215) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^6 + 3 a^5 b - 2 a^4 b^2 - 4 a^3 b^3 - 3 a^2 b^4 + a b^5 + 2 b^6 + 3 a^5 c - a^4 b c - 3 a b^4 c + b^5 c - 2 a^4 c^2 + 6 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 - 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 - 2 b^2 c^4 + a c^5 + b c^5 + 2 c^6 : :

X(31215) lies on these lines: {2, 3}, {6357, 31188}, {31224, 31233}

X(31216) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^8 + 2 a^7 b + a^6 b^2 - 5 a^4 b^4 - 2 a^3 b^5 - a^2 b^6 + 2 b^8 + 2 a^7 c + 5 a^6 b c + 3 a^5 b^2 c - 4 a^4 b^3 c - 6 a^3 b^4 c - a^2 b^5 c + a b^6 c + a^6 c^2 + 3 a^5 b c^2 + 2 a^4 b^2 c^2 + a^2 b^4 c^2 + a b^5 c^2 - 4 a^4 b c^3 + 2 a^2 b^3 c^3 - 2 a b^4 c^3 - 5 a^4 c^4 - 6 a^3 b c^4 + a^2 b^2 c^4 - 2 a b^3 c^4 - 4 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 + a b^2 c^5 - a^2 c^6 + a b c^6 + 2 c^8 : :

X(31216) lies on these lines: {2, 3}, {2303, 24884}

X(31217) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^8 - 5 a^7 b - 5 a^6 b^2 + 5 a^5 b^3 + 3 a^4 b^4 + 5 a^3 b^5 - 3 a^2 b^6 - 5 a b^7 + 2 b^8 - 5 a^7 c - 7 a^6 b c + 3 a^5 b^2 c + 5 a^4 b^3 c + 5 a^3 b^4 c + 7 a^2 b^5 c - 3 a b^6 c - 5 b^7 c - 5 a^6 c^2 + 3 a^5 b c^2 + 4 a^4 b^2 c^2 - 10 a^3 b^3 c^2 + 3 a^2 b^4 c^2 + 7 a b^5 c^2 - 2 b^6 c^2 + 5 a^5 c^3 + 5 a^4 b c^3 - 10 a^3 b^2 c^3 - 14 a^2 b^3 c^3 + a b^4 c^3 + 5 b^5 c^3 + 3 a^4 c^4 + 5 a^3 b c^4 + 3 a^2 b^2 c^4 + a b^3 c^4 + 5 a^3 c^5 + 7 a^2 b c^5 + 7 a b^2 c^5 + 5 b^3 c^5 - 3 a^2 c^6 - 3 a b c^6 - 2 b^2 c^6 - 5 a c^7 - 5 b c^7 + 2 c^8 : :

X(31217) lies on these lines: {2, 3}, {31205, 31225}

X(31218) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 19 a^5 - 5 a^4 b - 8 a^3 b^2 - 8 a^2 b^3 - 11 a b^4 + 13 b^5 - 5 a^4 c + 16 a^2 b^2 c - 11 b^4 c - 8 a^3 c^2 + 16 a^2 b c^2 + 22 a b^2 c^2 - 2 b^3 c^2 - 8 a^2 c^3 - 2 b^2 c^3 - 11 a c^4 - 11 b c^4 + 13 c^5 : :

X(31218) lies on these lines: {2, 3}

X(31219) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 7 a^5 - a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - 3 a b^4 + 5 b^5 - a^4 c - 2 a^3 b c - 2 a b^3 c - 3 b^4 c - 4 a^3 c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 4 a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 - 3 a c^4 - 3 b c^4 + 5 c^5 : :

X(31219) lies on these lines: {2, 3}, {31188, 31189}, {31191, 31224}

X(31220) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^5 - a^4 b - 2 a^3 b^2 - 2 a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 2 a^2 b^2 c - 2 b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 + 3 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - b^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31220) lies on these lines: {2, 3}, {31196, 31234}

X(31221) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (a + b - c) (a - b + c) (4 a^3 + a b^2 - 3 b^3 - 2 a b c - 5 b^2 c + a c^2 - 5 b c^2 - 3 c^3) : :

X(31221) lies on these lines: {2, 12}, {1317, 3661}, {1319, 29604}, {1388, 29611}, {3911, 5244}, {5204, 7402}, {5723, 31183}, {6996, 7173}, {7377, 15326}, {10944, 17308}, {15950, 29603}, {16609, 31191}, {24914, 29598}, {31185, 31203}

X(31222) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 3 a^3 b c + 2 a b^3 c - 2 b^4 c - a^3 c^2 - a^2 c^3 + 2 a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31222) lies on these lines: {2, 36}, {241, 31183}, {905, 31208}, {1055, 30858}, {5437, 21372}, {5444, 29571}, {31211, 31213}

X(31223) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 2 a^3 b - 5 a^2 b^2 + a b^3 + 2 a^3 c + 4 a^2 b c + 3 a b^2 c + 2 b^3 c - 5 a^2 c^2 + 3 a b c^2 - 4 b^2 c^2 + a c^3 + 2 b c^3 : :

X(31223) lies on these lines: {2, 38}, {9350, 24600}, {31183, 31195}

X(31224) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c + 8 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31224) lies on these lines: {1, 6681}, {2, 7}, {46, 11813}, {56, 5123}, {78, 13747}, {84, 6979}, {88, 23681}, {516, 10584}, {614, 26231}, {1125, 25415}, {1210, 4855}, {1420, 25005}, {1621, 31249}, {1647, 3749}, {1698, 5253}, {1699, 9352}, {1788, 11682}, {1836, 6667}, {1997, 3977}, {3035, 3870}, {3333, 27529}, {3361, 11681}, {3419, 17564}, {3526, 5439}, {3586, 13587}, {3616, 11362}, {3624, 3833}, {3872, 15325}, {3873, 14740}, {3890, 9588}, {3895, 10072}, {3984, 6700}, {4187, 4652}, {4188, 9581}, {4193, 15803}, {4292, 6931}, {4383, 22128}, {4666, 5432}, {4881, 5727}, {5119, 10199}, {5122, 17556}, {5154, 9579}, {5176, 13462}, {5250, 10200}, {5433, 19860}, {5573, 29665}, {5705, 17531}, {6545, 11068}, {6684, 12703}, {6691, 19861}, {6705, 6953}, {6734, 17567}, {6910, 9843}, {6932, 21164}, {6933, 12436}, {6966, 7682}, {7288, 24982}, {7294, 28628}, {7705, 9613}, {7988, 20292}, {8056, 33129}, {11520, 27385}, {16578, 26741}, {16602, 31201}, {24582, 24600}, {25011, 30478}, {31183, 31195}, {31189, 31212}, {31191, 31219}, {31215, 31233}

X(31225) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (a + b - c) (a - b + c) (2 a^2 b - 2 a b^2 + 2 a^2 c - a b c - b^2 c - 2 a c^2 - b c^2) : :

X(31225) lies on these lines: {2, 85}, {7, 4687}, {56, 16830}, {57, 16831}, {65, 3616}, {75, 4552}, {77, 17277}, {78, 7572}, {86, 1445}, {307, 17234}, {312, 25083}, {651, 17335}, {664, 4384}, {1214, 19804}, {1323, 31211}, {1418, 4698}, {1441, 4751}, {1442, 3759}, {1952, 30608}, {2283, 23407}, {3842, 4334}, {3911, 7146}, {3912, 33298}, {4059, 5226}, {4389, 30379}, {4675, 17950}, {5228, 16826}, {5308, 6604}, {5723, 29628}, {6180, 17260}, {6378, 7209}, {8583, 31359}, {8732, 17321}, {9312, 16832}, {9436, 29571}, {16609, 31231}, {16833, 25716}, {17086, 17278}, {17263, 28739}, {17336, 28968}, {17341, 28780}, {18743, 25082}, {23151, 28920}, {25067, 26059}, {25099, 27334}, {28740, 33116}, {30812, 30946}, {31188, 31189}, {31205, 31217}

X(31226) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^4 - 4 a^3 b + 3 a^2 b^2 - 4 a b^3 + 2 b^4 - 4 a^3 c + a^2 b c + 3 a b^2 c - 4 b^3 c + 3 a^2 c^2 + 3 a b c^2 + 4 b^2 c^2 - 4 a c^3 - 4 b c^3 + 2 c^4 : :

X(31226) lies on these lines: {2, 11}, {1462, 5375}, {6548, 31227}, {24635, 31183}, {31184, 31189}, {31191, 31213}, {31195, 31200}, {31199, 31204}

X(31227) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (a - 2 b + c) (a + b - 2 c) (3 a - b - c) : :

X(31227) lies on these lines: {2, 45}, {10, 106}, {11, 14193}, {145, 3756}, {244, 24418}, {348, 27814}, {901, 9083}, {1125, 4792}, {1155, 19634}, {1168, 6681}, {1266, 4582}, {1320, 3035}, {2316, 6692}, {2403, 14425}, {3008, 4555}, {3257, 3911}, {3624, 4674}, {4013, 31253}, {4510, 31211}, {6336, 8756}, {6548, 31226}, {6667, 19636}, {9311, 30610}, {9456, 14829}, {23838, 25380}, {29607, 31201}, {31187, 31228}, {31197, 31205}

X(31228) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^2 b + 3 a b^2 + 3 a^2 c - 17 a b c + b^2 c + 3 a c^2 + b c^2 : :

X(31228) lies on these lines: {2, 37}, {5437, 17120}, {8056, 25728}, {17090, 31188}, {17121, 23511}, {29607, 31190}, {31187, 31227}

X(31229) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 3 a^3 - a b^2 + 2 b^3 - b^2 c - a c^2 - b c^2 + 2 c^3 : :

X(31229) lies on these lines: {2, 6}, {31, 21241}, {625, 5170}, {964, 24880}, {3011, 33114}, {3306, 16551}, {3759, 27757}, {3769, 29873}, {3772, 32933}, {3891, 4438}, {4257, 17679}, {4402, 17740}, {5745, 32774}, {6679, 21242}, {6693, 16454}, {11269, 24542}, {16342, 20083}, {16669, 30823}, {17120, 26738}, {17253, 30564}, {17273, 24616}, {17278, 24594}, {17282, 24593}, {17319, 27754}, {17370, 30608}, {26685, 30566}, {29658, 33115}, {29665, 33118}, {29681, 33121}, {29856, 32917}, {29858, 32919}, {29865, 32853}, {29867, 32916}, {31184, 31189}

X(31230) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (a + b - c) (a - b + c) (3 a^3 - a^2 b + 2 a b^2 - 2 b^3 - a^2 c - 2 b^2 c + 2 a c^2 - 2 b c^2 - 2 c^3) : :

X(31230) lies on these lines: {2, 12}, {11, 7397}, {57, 24798}, {65, 29598}, {241, 31183}, {348, 31189}, {499, 19512}, {599, 1404}, {604, 3763}, {1038, 23511}, {1317, 29616}, {1319, 17284}, {1388, 3912}, {1470, 21514}, {1617, 21542}, {2099, 17023}, {2285, 17384}, {3008, 24914}, {3665, 5435}, {3669, 31208}, {3911, 31191}, {5172, 21477}, {5226, 7198}, {5252, 29604}, {6996, 10896}, {7181, 31188}, {7354, 7402}, {7377, 12943}, {10944, 29611}, {12513, 28813}, {31185, 31192}

X(31231) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics (3 a - 2 b - 2 c) (a + b - c) (a - b + c) : :
Barycentrics 1 - (2 a - b - c)/(a - b - c) : :

X(31231) lies on these lines: {1, 140}, {2, 7}, {3, 3586}, {5, 9579}, {10, 1420}, {11, 165}, {12, 3361}, {36, 5587}, {40, 499}, {46, 6862}, {56, 1698}, {65, 3624}, {73, 17749}, {78, 17566}, {84, 6834}, {100, 24392}, {109, 748}, {181, 29825}, {200, 3035}, {201, 24046}, {210, 3660}, {241, 31183}, {269, 17337}, {278, 8756}, {312, 20881}, {381, 5122}, {388, 3634}, {474, 5705}, {484, 23708}, {497, 10164}, {498, 3333}, {516, 10589}, {549, 5722}, {631, 1210}, {632, 11374}, {936, 13747}, {938, 10303}, {942, 3526}, {946, 5128}, {950, 3523}, {956, 5193}, {997, 6681}, {999, 11231}, {1001, 3256}, {1038, 7561}, {1042, 28257}, {1054, 17064}, {1125, 1788}, {1155, 1699}, {1214, 16610}, {1284, 31242}, {1317, 4677}, {1319, 3679}, {1376, 2078}, {1387, 3654}, {1388, 3632}, {1404, 5361}, {1412, 5235}, {1421, 5272}, {1427, 31197}, {1429, 17284}, {1442, 17020}, {1450, 30116}, {1465, 16602}, {1466, 11108}, {1467, 5791}, {1470, 5251}, {1471, 17124}, {1617, 4413}, {1656, 9612}, {1697, 3086}, {1703, 9661}, {1706, 10527}, {1728, 6863}, {1730, 19720}, {1737, 3576}, {1768, 7082}, {1836, 7988}, {1837, 7987}, {1864, 11227}, {2003, 4383}, {2006, 8056}, {2067, 13947}, {2093, 5886}, {2099, 25055}, {2136, 10529}, {2242, 31441}, {2999, 3554}, {3011, 5573}, {3057, 9588}, {3058, 31508}, {3090, 4292}, {3158, 26015}, {3212, 29630}, {3339, 7294}, {3474, 3817}, {3485, 19862}, {3487, 3533}, {3524, 4304}, {3525, 11518}, {3579, 9614}, {3582, 5119}, {3600, 19877}, {3616, 4848}, {3655, 11545}, {3666, 26742}, {3671, 19878}, {3676, 6546}, {3681, 5083}, {3740, 17625}, {3752, 8609}, {3816, 4512}, {3828, 4315}, {3947, 31253}, {3999, 17783}, {4032, 4751}, {4187, 31424}, {4193, 4652}, {4293, 10175}, {4298, 10588}, {4299, 18492}, {4311, 5818}, {4312, 17605}, {4551, 16569}, {4554, 7243}, {4678, 6049}, {4698, 7201}, {4850, 16577}, {4855, 12625}, {4863, 6174}, {4873, 33168}, {5054, 24929}, {5123, 11194}, {5126, 5790}, {5204, 5691}, {5218, 10389}, {5223, 31235}, {5225, 12512}, {5252, 5298}, {5259, 11509}, {5265, 9780}, {5269, 24239}, {5285, 16434}, {5290, 19872}, {5326, 10980}, {5434, 5726}, {5436, 6910}, {5438, 6734}, {5442, 7741}, {5552, 6762}, {5560, 7280}, {5564, 11679}, {5657, 7962}, {5692, 18838}, {5709, 6958}, {5719, 11539}, {5881, 18395}, {5903, 9624}, {6147, 16239}, {6245, 6927}, {6284, 16192}, {6358, 19804}, {6502, 13893}, {6667, 24703}, {6690, 10582}, {6691, 8583}, {6705, 6848}, {6745, 24477}, {6838, 9841}, {6856, 12436}, {6857, 9843}, {6889, 10396}, {6907, 21164}, {6935, 7682}, {6959, 7330}, {6962, 10395}, {6963, 21165}, {7146, 29598}, {7175, 17259}, {7354, 7989}, {7991, 11376}, {8068, 31515}, {8582, 30478}, {8616, 23703}, {9312, 29628}, {9316, 17125}, {9364, 17123}, {9613, 9956}, {9669, 31663}, {10072, 31393}, {10156, 17603}, {10157, 11575}, {10165, 13384}, {10172, 10590}, {10200, 31435}, {10202, 18397}, {10270, 15908}, {10571, 27627}, {10591, 31730}, {10950, 30389}, {11502, 15931}, {11523, 27385}, {11680, 30312}, {12433, 14869}, {12514, 25522}, {12526, 25681}, {13226, 30304}, {13370, 22759}, {13388, 17803}, {13389, 17806}, {14829, 17360}, {15015, 20118}, {15104, 18839}, {15694, 15934}, {15713, 15935}, {15844, 17529}, {15950, 18421}, {16208, 26475}, {16209, 26476}, {16609, 31225}, {17719, 18193}, {17784, 24386}, {18593, 31204}, {18965, 19004}, {18966, 19003}, {19858, 26126}, {21129, 30719}, {24391, 27383}, {24880, 24882}, {24897, 24915}, {25716, 29590}, {25734, 30566}, {26740, 28606}, {30286, 30392}, {30567, 32851}, {31436, 31792}

X(31231) = complement of X(5748)
X(31231) = {X(2),X(9)}-harmonic conjugate of X(20196)
X(31231) = {X(2),X(57)}-harmonic conjugate of X(5219)

X(31232) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 7 a^3 - a^2 b - 3 a b^2 + 5 b^3 - a^2 c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 5 c^3 : :

X(31232) lies on these lines: {2, 6}, {3008, 30225}, {5273, 17258}, {5745, 17304}, {5846, 30762}, {8229, 14927}, {17070, 24280}, {17345, 26132}, {31185, 31234}, {31188, 31189}

X(31233) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 2 a^2 b + 2 a b^2 + 2 a^2 c - 7 a b c + b^2 c + 2 a c^2 + b c^2 : :

X(31233) lies on these lines: {2, 37}, {1054, 4676}, {1122, 5435}, {3306, 3758}, {3782, 27130}, {3911, 17352}, {4383, 27002}, {4389, 5316}, {4417, 21255}, {4429, 5121}, {4673, 25492}, {5219, 27191}, {5233, 17227}, {5241, 17250}, {6686, 17063}, {9311, 30610}, {11814, 33149}, {14759, 27195}, {14829, 23511}, {14997, 24593}, {17290, 30867}, {17335, 24627}, {17336, 17595}, {17721, 26073}, {22295, 29822}, {24183, 31019}, {24217, 25377}, {28257, 31359}, {28581, 30948}, {29607, 31187}, {29630, 31202}, {31183, 31205}, {31188, 31189}, {31191, 31234}, {31209, 31992}, {31215, 31224}

X(31233) = complement of X(30861)

X(31234) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics 2 a^3 b^2 + 2 a^2 b^3 - 2 a^2 b^2 c + 2 a^3 c^2 - 2 a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 + 2 a^2 c^3 + b^2 c^3 : :

X(31234) lies on these lines: {2, 39}, {83, 16412}, {99, 11353}, {3972, 11329}, {5222, 27195}, {5435, 17114}, {7976, 20340}, {14829, 17749}, {31185, 31232}, {31191, 31233}, {31196, 31220}

X(31234) = complement of X(30863)

Collineation mappings involving Gemini triangle 109: X(31235)-X(31283)

m(X) = x + 2 y + 2 z : :

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 24, 2019)

If P is not on the Euler line, then the (A,B,C,X(2); A',B',C',X(2)) collineation image of P, where A'B'C' = Gemini triangle 109 is the intersection of lines X(2)P and X(5)P', where P' =reflection of P in X(3). (Randy Hutson, October 8, 2019)

X(31235) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 4 a^3 - 4 a^2 b - 3 a b^2 + 3 b^3 - 4 a^2 c + 10 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 3 c^3 : :

X(31235) lies on these lines: {2, 11}, {5, 24466}, {10, 1317}, {12, 13747}, {30, 31263}, {104, 3525}, {119, 140}, {142, 6068}, {210, 5083}, {214, 3634}, {354, 14740}, {404, 3614}, {443, 13273}, {547, 22938}, {551, 25416}, {590, 13991}, {615, 13922}, {631, 2829}, {632, 952}, {936, 12739}, {956, 5433}, {1125, 1145}, {1320, 5550}, {1329, 17566}, {1387, 3624}, {1537, 6684}, {1656, 5840}, {1768, 7308}, {2551, 12763}, {2787, 31274}, {2800, 25917}, {2802, 3698}, {2932, 16842}, {3036, 9780}, {3243, 17728}, {3524, 10728}, {3526, 6713}, {3530, 22799}, {3533, 20418}, {3616, 5854}, {3628, 23513}, {3679, 12735}, {3763, 5848}, {3814, 15326}, {3820, 31157}, {3828, 15863}, {3911, 5850}, {4193, 15338}, {4996, 6668}, {5044, 11570}, {5047, 17100}, {5054, 10742}, {5056, 10724}, {5070, 10738}, {5084, 12764}, {5219, 24465}, {5223, 31231}, {5298, 6681}, {5316, 21635}, {5533, 31419}, {5660, 11407}, {5745, 12831}, {5851, 18230}, {5856, 20195}, {6246, 10172}, {6666, 10427}, {6691, 15888}, {6700, 12832}, {6702, 10609}, {6897, 12761}, {6921, 7354}, {6958, 18491}, {7173, 25440}, {7951, 17564}, {7972, 19875}, {8068, 8728}, {8583, 12740}, {9623, 20586}, {9708, 10074}, {9709, 10087}, {9897, 19876}, {9940, 12665}, {10058, 11108}, {10090, 16408}, {10711, 15709}, {11246, 30852}, {11539, 11698}, {11729, 26446}, {12019, 15015}, {12248, 15702}, {15017, 20196}, {17529, 20104}, {17718, 31190}, {19112, 32785}, {19113, 32786}, {19878, 32557}, {20107, 24390}, {24954, 31423}

X(31235) = complement of X(31272)

X(31236) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31236) lies on these lines: {2, 3}, {6, 23293}, {125, 5422}, {251, 6032}, {974, 15100}, {1196, 9745}, {1853, 5012}, {1899, 14389}, {1993, 5965}, {2781, 11451}, {3167, 3410}, {3448, 11402}, {3619, 16789}, {3763, 9019}, {5346, 5359}, {5640, 26958}, {5889, 15739}, {6720, 11605}, {6800, 11550}, {7603, 19220}, {9544, 18440}, {10601, 26913}, {11442, 23292}, {14852, 15033}, {15355, 31489}, {17004, 17035}, {18911, 23332}, {29872, 30756}, {29873, 30755}

X(31237) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 + 2 b^3 + 2 c^3 : :

X(31237) lies on these lines: {2, 31}, {5, 30269}, {6, 29867}, {10, 33127}, {11, 29677}, {38, 25527}, {42, 30811}, {43, 30831}, {55, 29865}, {81, 29856}, {141, 24892}, {226, 26061}, {306, 33128}, {345, 33145}, {612, 21026}, {674, 3763}, {734, 7786}, {744, 4687}, {758, 1698}, {766, 7867}, {940, 29863}, {982, 29872}, {984, 29873}, {1001, 29869}, {1125, 4680}, {1621, 29858}, {2177, 3771}, {2390, 31246}, {2835, 20196}, {2886, 24943}, {3006, 26128}, {3011, 33074}, {3120, 32777}, {3454, 19846}, {3662, 33119}, {3703, 33143}, {3705, 33123}, {3750, 29866}, {3772, 15523}, {3782, 33161}, {3821, 33113}, {3873, 29861}, {3914, 20106}, {3936, 25453}, {3944, 33157}, {4026, 29661}, {4038, 29864}, {4138, 5294}, {4417, 29850}, {4429, 29846}, {4438, 17184}, {4514, 29638}, {4649, 29868}, {4657, 29682}, {4751, 18805}, {4865, 26230}, {4892, 26223}, {5014, 29656}, {5718, 29663}, {6685, 30834}, {9256, 24924}, {9313, 30835}, {9345, 18139}, {10448, 16062}, {11680, 29637}, {16484, 29870}, {16706, 29849}, {16906, 30175}, {16908, 17033}, {17056, 29647}, {17061, 32854}, {17234, 29845}, {17283, 30957}, {17357, 17605}, {17469, 29855}, {17716, 29874}, {17719, 29679}, {17720, 29687}, {17722, 29666}, {17723, 29684}, {17725, 33091}, {17889, 32779}, {18134, 29631}, {19785, 32848}, {19786, 29643}, {21241, 24552}, {21298, 26222}, {25440, 25669}, {26037, 30832}, {26132, 32856}, {26227, 28595}, {27184, 33115}, {28606, 29862}, {29632, 32773}, {29634, 33072}, {29636, 33073}, {29641, 32775}, {29654, 33070}, {29658, 33078}, {29665, 33079}, {29667, 33130}, {29671, 32774}, {29673, 33122}, {29674, 33133}, {29681, 33076}, {29848, 32850}, {29852, 33071}, {31017, 32853}, {31019, 32780}, {31053, 33159}, {31079, 32920}, {31242, 31272}, {32776, 33116}, {32778, 33129}, {32782, 33138}, {32783, 33108}, {32849, 33154}, {32851, 33125}, {32855, 33150}, {32858, 33135}, {32862, 33152}, {32865, 33175}, {33064, 33114}, {33065, 33118}, {33069, 33121}, {33077, 33132}, {33081, 33137}, {33084, 33139}, {33087, 33142}, {33089, 33147}, {33092, 33155}, {33101, 33166}, {33103, 33170}, {33117, 33126}, {33120, 33124}, {33131, 33160}, {33134, 33158}, {33136, 33171}, {33140, 33172}, {33141, 33173}, {33144, 33162}, {33146, 33167}, {33148, 33169}, {33149, 33168}, {33151, 33164}, {33153, 33165}

X(31238) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 3 a b + 3 a c + 4 b c : :

X(31238) lies on these lines: {2, 37}, {5, 30271}, {6, 16832}, {10, 4966}, {44, 10436}, {45, 25590}, {57, 19744}, {86, 16666}, {141, 24603}, {142, 1213}, {210, 13476}, {239, 28639}, {354, 22271}, {391, 30712}, {518, 1698}, {551, 4732}, {594, 29571}, {632, 29010}, {726, 31239}, {740, 19862}, {894, 15492}, {966, 4675}, {1086, 5257}, {1100, 4384}, {1104, 16458}, {1125, 3696}, {1268, 17292}, {1418, 17077}, {1449, 31312}, {1654, 17376}, {2667, 30950}, {2805, 31272}, {3008, 17398}, {3247, 17119}, {3525, 30273}, {3616, 28581}, {3618, 4798}, {3624, 15569}, {3634, 24325}, {3661, 28633}, {3662, 4708}, {3664, 17330}, {3679, 17311}, {3686, 17392}, {3707, 7277}, {3723, 4361}, {3731, 17118}, {3826, 14019}, {3834, 5224}, {3986, 17246}, {4060, 29606}, {4360, 29578}, {4363, 16814}, {4364, 24199}, {4371, 29624}, {4399, 29574}, {4405, 4464}, {4411, 31287}, {4413, 15624}, {4470, 18230}, {4472, 17353}, {4648, 17275}, {4659, 16675}, {4670, 16669}, {4690, 17300}, {4709, 19883}, {4715, 17331}, {4725, 17391}, {4738, 19870}, {4777, 31250}, {4828, 27115}, {4849, 26037}, {4852, 16826}, {4859, 17325}, {4862, 31139}, {4889, 29617}, {4967, 17243}, {4981, 17146}, {5070, 20430}, {5222, 28626}, {5296, 17276}, {5308, 17299}, {5564, 29569}, {5750, 17337}, {6173, 17253}, {6666, 17369}, {6683, 21443}, {6687, 17368}, {6707, 17023}, {7227, 25101}, {14210, 16818}, {15254, 24342}, {16590, 17332}, {16605, 30107}, {16672, 17151}, {16726, 27164}, {16816, 17394}, {16819, 17448}, {16833, 16884}, {17014, 28641}, {17054, 19859}, {17229, 17244}, {17233, 29581}, {17234, 17239}, {17235, 17248}, {17240, 29599}, {17241, 29593}, {17251, 17298}, {17256, 17345}, {17260, 17351}, {17265, 17308}, {17270, 17313}, {17272, 31138}, {17282, 17327}, {17283, 29610}, {17285, 29626}, {17288, 31144}, {17312, 32025}, {17316, 28634}, {17317, 17372}, {17326, 27191}, {17340, 25072}, {17380, 29612}, {17381, 29628}, {17393, 29595}, {17529, 19857}, {20718, 25917}, {21233, 30812}, {21868, 27255}, {24773, 24784}, {24774, 24778}, {25501, 27798}, {25538, 27111}, {26626, 28640}, {27483, 29609}, {27485, 31207}, {27487, 29607}, {28329, 29622}, {28635, 29616}, {29592, 31342}, {31245, 31255}, {31319, 31351}, {31323, 31335}

X(31238) = complement of X(4687)

X(31239) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 3 a^2 b^2 + 3 a^2 c^2 + 4 b^2 c^2 : :

X(31239) lies on these lines: {2, 39}, {5, 5188}, {6, 24861}, {32, 15271}, {83, 5008}, {115, 8362}, {140, 6248}, {141, 1506}, {183, 5007}, {187, 7770}, {230, 7889}, {262, 5067}, {316, 33020}, {381, 7935}, {384, 15513}, {511, 1656}, {547, 14881}, {548, 22681}, {590, 13983}, {597, 5368}, {599, 7903}, {615, 8992}, {625, 3096}, {626, 7603}, {632, 2782}, {726, 31238}, {730, 19862}, {1078, 7804}, {1574, 21264}, {1698, 14839}, {1916, 14047}, {1975, 15482}, {2021, 32954}, {2023, 33186}, {2548, 7845}, {2549, 32960}, {2896, 7843}, {3054, 33185}, {3090, 22712}, {3095, 5070}, {3202, 5651}, {3329, 7805}, {3525, 11257}, {3526, 7697}, {3589, 7755}, {3618, 5346}, {3619, 32975}, {3634, 12263}, {3734, 11285}, {3785, 5395}, {3815, 7794}, {3832, 22676}, {3849, 7904}, {3906, 31277}, {3917, 27375}, {3933, 9698}, {3954, 30850}, {4403, 25918}, {4698, 21443}, {5013, 17130}, {5041, 7751}, {5055, 9821}, {5092, 24273}, {5206, 11286}, {5215, 33237}, {5326, 13077}, {5475, 7800}, {5550, 7976}, {5976, 6722}, {6194, 7486}, {6337, 31457}, {6704, 7792}, {7294, 18982}, {7735, 18841}, {7736, 7855}, {7745, 7810}, {7748, 16043}, {7749, 7819}, {7750, 14537}, {7752, 7849}, {7753, 7767}, {7756, 8359}, {7759, 16990}, {7761, 16924}, {7773, 7865}, {7775, 7879}, {7777, 7895}, {7784, 31173}, {7785, 7848}, {7812, 14762}, {7813, 31406}, {7816, 7824}, {7830, 8370}, {7831, 7842}, {7835, 33015}, {7846, 17004}, {7857, 16895}, {7858, 7882}, {7861, 7876}, {7862, 7868}, {7864, 32457}, {7885, 31168}, {7887, 7914}, {7888, 31489}, {7890, 9300}, {7899, 16922}, {7909, 17005}, {7910, 33018}, {7911, 33013}, {7913, 13881}, {7916, 11163}, {7919, 16897}, {7924, 15031}, {7931, 10159}, {7934, 33002}, {7937, 32966}, {7941, 32027}, {7944, 16988}, {7948, 14061}, {8167, 12338}, {8361, 24256}, {8556, 30435}, {9605, 17131}, {9624, 22697}, {10007, 12815}, {10124, 32516}, {10513, 31407}, {13330, 21358}, {13354, 24206}, {14907, 33269}, {15699, 32521}, {17006, 19694}, {17030, 27076}, {18424, 32974}, {18806, 33249}, {19089, 32785}, {19090, 32786}, {20970, 29455}, {21843, 33198}, {25303, 26959}, {31450, 32817}, {32456, 33004}

X(31239) = complement of X(7786)

X(31240) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c - 2 b^3 c - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31240) lies on these lines: {2, 41}, {32, 31195}, {766, 7867}, {1334, 30825}, {1698, 2809}, {2389, 31245}, {3763, 8679}, {17062, 28734}, {17278, 28096}, {20195, 31254}, {20269, 21029}, {21240, 30816}, {30742, 33299}, {31241, 31255}

X(31241) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^2 b + 2 a b^2 + a^2 c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(31241) lies on these lines: {1, 2}, {38, 28582}, {141, 33105}, {244, 31993}, {312, 3989}, {321, 6682}, {333, 32944}, {354, 14973}, {518, 31264}, {674, 3763}, {748, 5737}, {756, 30818}, {788, 30835}, {902, 24552}, {1150, 2308}, {2227, 21264}, {2239, 28566}, {2530, 4379}, {2813, 31273}, {2886, 32781}, {3666, 4365}, {3739, 21330}, {3752, 21020}, {3775, 5741}, {4192, 28160}, {4413, 15621}, {4418, 24627}, {4912, 24330}, {4921, 16477}, {4972, 21242}, {5224, 25960}, {5235, 17123}, {5263, 32918}, {5361, 16468}, {5718, 33081}, {9345, 19701}, {11680, 32784}, {14829, 32772}, {14969, 19747}, {16738, 22343}, {17125, 19732}, {17184, 25385}, {17237, 17605}, {17239, 25618}, {17289, 33119}, {17303, 17728}, {17449, 32771}, {17591, 28605}, {17717, 32782}, {17722, 33075}, {17723, 32852}, {23649, 26035}, {24165, 31025}, {24259, 28550}, {24342, 27003}, {24589, 27798}, {24690, 28333}, {25349, 28297}, {26034, 33104}, {26098, 33080}, {31240, 31255}, {31247, 31272}, {32917, 32942}, {33082, 33107}, {33083, 33106}, {33085, 33112}, {33086, 33109}, {33108, 33174}, {33111, 33172}

X(31242) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^2 b + 2 a b^2 + a^2 c - 5 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(31242) lies on these lines: {1, 2}, {312, 28516}, {982, 3967}, {1284, 31231}, {1401, 5219}, {2810, 20196}, {3752, 28484}, {3756, 33169}, {3763, 9025}, {3816, 33174}, {3923, 27002}, {3971, 30861}, {4358, 17591}, {4713, 4912}, {6007, 20195}, {6682, 30829}, {11814, 27184}, {15485, 32918}, {17063, 30818}, {17728, 33159}, {19540, 28160}, {24260, 28550}, {24691, 28333}, {24709, 32950}, {25350, 28297}, {25531, 32916}, {28808, 33147}, {31237, 31272}, {31245, 31252}

X(31243) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 2 a^2 - 3 a b + 4 b^2 - 3 a c - 4 b c + 4 c^2 : :

X(31243) lies on these lines: {2, 44}, {8, 3823}, {37, 17265}, {142, 17357}, {513, 24924}, {518, 1698}, {536, 17266}, {599, 31183}, {752, 19862}, {1086, 2325}, {1100, 17234}, {1125, 1279}, {1266, 4908}, {3008, 4969}, {3244, 17366}, {3246, 3624}, {3622, 32850}, {3633, 17311}, {3662, 16814}, {3707, 17337}, {3723, 16706}, {3739, 17283}, {3742, 29861}, {3912, 4395}, {3943, 17067}, {4357, 31285}, {4370, 4887}, {4422, 4480}, {4473, 4912}, {4686, 4859}, {4688, 17284}, {4690, 29628}, {4698, 17291}, {4702, 25351}, {4725, 29590}, {4726, 17268}, {4739, 17285}, {4755, 17305}, {4852, 17241}, {7232, 15492}, {16610, 27757}, {16666, 17313}, {16668, 17300}, {16669, 17298}, {16671, 17352}, {16832, 21358}, {17232, 17348}, {17235, 17263}, {17244, 17382}, {17297, 29607}, {17301, 29627}, {17338, 17345}, {17341, 17351}, {17359, 29629}, {17370, 28639}, {17385, 27147}, {17392, 31191}, {17395, 29600}, {17399, 29599}, {20582, 24603}, {24894, 25645}, {25961, 29638}, {30811, 31197}

X(31244) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^2 - 6 a b + 2 b^2 - 6 a c - 8 b c + 2 c^2 : :

X(31244) lies on these lines: {2, 45}, {8, 17245}, {142, 17253}, {320, 17259}, {518, 1698}, {599, 16832}, {1125, 3755}, {2325, 17118}, {3306, 5043}, {3622, 17366}, {3707, 4675}, {3739, 17267}, {4361, 29569}, {4384, 4725}, {4395, 16777}, {4405, 17316}, {4648, 4969}, {4688, 4873}, {4751, 17265}, {6687, 10436}, {7222, 31722}, {7263, 16677}, {15668, 17367}, {16675, 24199}, {16815, 17313}, {17119, 17133}, {17269, 29626}, {17318, 29581}, {18493, 29309}, {19862, 28580}, {21358, 24603}

X(31245) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31245) lies on these lines: {2, 11}, {3, 18407}, {4, 24953}, {5, 3428}, {6, 24892}, {9, 17605}, {10, 2099}, {12, 3421}, {21, 12953}, {31, 31187}, {36, 17528}, {56, 442}, {63, 3838}, {65, 5705}, {142, 17728}, {144, 33558}, {210, 5173}, {226, 5850}, {354, 5231}, {377, 4999}, {381, 5251}, {405, 10896}, {443, 5433}, {474, 3841}, {498, 31419}, {499, 8728}, {517, 1656}, {518, 31266}, {631, 5842}, {674, 3763}, {908, 3715}, {940, 33111}, {956, 3822}, {958, 2476}, {993, 12943}, {1125, 3419}, {1150, 20290}, {1329, 6933}, {1377, 13954}, {1378, 13897}, {1479, 6675}, {1482, 11218}, {1699, 3683}, {1824, 5094}, {1836, 5745}, {1853, 10537}, {2098, 24987}, {2361, 25885}, {2389, 31240}, {2551, 3614}, {2900, 10582}, {2975, 9657}, {3052, 33104}, {3090, 7680}, {3219, 10129}, {3242, 29690}, {3303, 10198}, {3304, 10527}, {3305, 5087}, {3338, 3824}, {3485, 21677}, {3526, 32613}, {3583, 16418}, {3616, 31254}, {3617, 5855}, {3624, 24929}, {3646, 5119}, {3666, 17064}, {3679, 31479}, {3703, 30741}, {3711, 25006}, {3739, 30742}, {3740, 30852}, {3741, 30811}, {3742, 16465}, {3748, 24392}, {3771, 21242}, {3772, 17599}, {3817, 4679}, {3825, 16842}, {3846, 19732}, {3961, 17783}, {4003, 23681}, {4042, 4417}, {4197, 25524}, {4208, 7288}, {4293, 31157}, {4361, 29849}, {4363, 25366}, {4383, 17717}, {4387, 33116}, {4438, 25385}, {4699, 7925}, {4847, 17718}, {4860, 5249}, {4863, 13405}, {5054, 18499}, {5070, 10679}, {5084, 7173}, {5141, 5260}, {5175, 10543}, {5177, 7354}, {5217, 7483}, {5220, 31053}, {5221, 12609}, {5225, 17558}, {5248, 9670}, {5258, 9654}, {5259, 9669}, {5316, 10171}, {5552, 6668}, {5584, 6831}, {5695, 33113}, {5718, 33137}, {5737, 25760}, {5744, 11246}, {5789, 15071}, {5790, 6326}, {5791, 12047}, {5794, 24541}, {6182, 31250}, {6253, 6988}, {6284, 6857}, {6734, 28628}, {6824, 15908}, {6829, 22753}, {6832, 7681}, {6842, 18516}, {6852, 11496}, {6853, 11500}, {6862, 10310}, {6867, 11827}, {6874, 10894}, {6889, 8273}, {6892, 11826}, {6920, 10893}, {6937, 12114}, {7294, 17567}, {7308, 7988}, {7504, 9780}, {7741, 11108}, {7887, 16819}, {7951, 9708}, {8069, 16408}, {9656, 31458}, {9668, 31159}, {10200, 17529}, {10585, 12607}, {10588, 21031}, {10591, 16845}, {11269, 17056}, {11281, 12649}, {11366, 26359}, {11367, 26360}, {11415, 18253}, {13955, 31473}, {13958, 31413}, {15668, 29845}, {15733, 20195}, {15934, 26725}, {16466, 24880}, {16502, 31488}, {16777, 29682}, {17070, 19785}, {17127, 31204}, {17135, 30834}, {17259, 25960}, {17265, 30957}, {17307, 25631}, {17566, 26060}, {17595, 17889}, {17597, 29676}, {19037, 31484}, {19544, 20989}, {19701, 29635}, {19744, 21912}, {19875, 25415}, {19877, 26129}, {20182, 29657}, {20243, 30744}, {20988, 25514}, {21241, 32916}, {21264, 30752}, {21920, 32778}, {24239, 24789}, {26686, 33028}, {26893, 30970}, {29640, 33141}, {29664, 33133}, {29678, 33136}, {29680, 33129}, {29688, 33128}, {31238, 31255}, {31242, 31252}

X(31246) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - 3 a^2 b^2 + 2 b^4 + 2 a^2 b c + 4 a b^2 c - 3 a^2 c^2 + 4 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31246) lies on these lines: {2, 12}, {5, 4413}, {10, 2098}, {11, 5082}, {46, 7308}, {55, 4187}, {65, 30827}, {100, 9670}, {145, 33559}, {404, 12943}, {443, 3614}, {474, 3814}, {498, 4423}, {499, 3820}, {517, 1656}, {631, 2829}, {908, 5221}, {936, 17606}, {997, 17619}, {1001, 27529}, {1317, 24558}, {1376, 4193}, {1377, 13898}, {1378, 13955}, {1399, 25938}, {1788, 5328}, {1828, 5094}, {1837, 6700}, {2099, 24982}, {2390, 31237}, {2478, 3035}, {2550, 7173}, {2886, 6931}, {3057, 25522}, {3086, 21031}, {3090, 3925}, {3242, 28096}, {3303, 3816}, {3304, 10200}, {3428, 6959}, {3434, 3847}, {3452, 24914}, {3526, 5251}, {3585, 16417}, {3617, 5854}, {3624, 24928}, {3628, 19854}, {3634, 5316}, {3649, 5748}, {3665, 30740}, {3683, 31423}, {3711, 10916}, {3763, 8679}, {3812, 30852}, {3813, 10584}, {3822, 16862}, {3825, 5687}, {3826, 6933}, {3983, 5231}, {4294, 6174}, {4299, 17564}, {4428, 26127}, {4679, 6684}, {4860, 21077}, {5056, 26040}, {5067, 19855}, {5070, 10680}, {5084, 5432}, {5123, 19861}, {5141, 9342}, {5172, 25875}, {5204, 13747}, {5289, 25005}, {5326, 6857}, {5584, 6834}, {6284, 6919}, {6667, 9711}, {6692, 10404}, {6863, 32554}, {6882, 18517}, {6946, 10894}, {6963, 11500}, {6967, 18242}, {6970, 11827}, {6973, 11826}, {6975, 11496}, {6981, 15908}, {6983, 7680}, {7354, 17567}, {7504, 19877}, {7741, 9709}, {7951, 16408}, {8069, 11108}, {8162, 10528}, {8256, 9780}, {8582, 11375}, {9373, 31250}, {9655, 31160}, {9843, 17718}, {10198, 17575}, {10483, 17573}, {10786, 20400}, {11501, 25893}, {12953, 17556}, {13897, 31473}, {16209, 22792}, {17101, 22102}, {17609, 31249}, {17728, 21075}, {17768, 18230}, {19875, 30323}, {25011, 28628}, {26590, 33053}, {31190, 32636}, {31252, 31270}

X(31247) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 + a^2 b + 2 a b^2 + 2 b^3 + a^2 c + 5 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 + 2 c^3 : :

X(31247) lies on these lines: {2, 6}, {10, 33133}, {21, 24931}, {274, 27793}, {306, 1255}, {758, 1698}, {1014, 24912}, {2836, 15059}, {3452, 7110}, {3454, 14005}, {3775, 29845}, {3920, 4914}, {4042, 29864}, {4220, 29012}, {4364, 33168}, {4418, 28546}, {4886, 29833}, {5284, 32783}, {6536, 33160}, {7359, 18228}, {7504, 18417}, {8013, 33135}, {8682, 17266}, {9342, 32781}, {16948, 26064}, {17020, 17384}, {17237, 27003}, {17248, 33113}, {17303, 31053}, {17369, 26792}, {17371, 26688}, {17514, 24936}, {17557, 25645}, {17946, 27076}, {18230, 31256}, {19808, 26580}, {19822, 33151}, {19827, 26223}, {19856, 33105}, {24076, 31993}, {27792, 30599}, {29604, 30839}, {29667, 30615}, {31241, 31272}, {31271, 31278}

X(31248) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^2 + 5 a b + 2 b^2 + 5 a c + 5 b c + 2 c^2 : :

X(31248) lies on these lines: {2, 6}, {10, 17315}, {37, 1268}, {190, 5257}, {344, 19877}, {594, 32089}, {740, 1698}, {1043, 17514}, {1434, 24909}, {1449, 30598}, {1738, 3634}, {2092, 24944}, {2796, 14061}, {3624, 3759}, {3739, 17324}, {3946, 17322}, {3948, 25457}, {4360, 28634}, {4422, 6650}, {4445, 29595}, {4595, 19870}, {4698, 17285}, {4708, 17273}, {4733, 9780}, {4751, 4859}, {4798, 17331}, {4835, 27805}, {4967, 28313}, {6542, 32101}, {6651, 17385}, {6998, 29012}, {10436, 17329}, {16589, 24530}, {16815, 25498}, {16826, 17372}, {16831, 17295}, {16832, 17400}, {17116, 28322}, {17239, 29578}, {17248, 17276}, {17275, 29612}, {17293, 31308}, {17302, 25358}, {17303, 17339}, {17308, 31336}, {17319, 28633}, {17326, 27191}, {17342, 19876}, {17348, 29609}, {17353, 24697}, {17362, 29592}, {17363, 28640}, {17768, 18230}, {17770, 31252}, {18714, 21921}, {22174, 24450}, {24342, 28546}, {25534, 27154}, {31258, 31269}

X(31249) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - 3 a b^2 + 2 b^3 + 10 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31249) lies on these lines: {1, 2}, {9, 17728}, {11, 5437}, {55, 31190}, {57, 3816}, {142, 10589}, {354, 30827}, {461, 1877}, {497, 6692}, {518, 20196}, {908, 10980}, {942, 25522}, {1001, 3256}, {1004, 14803}, {1470, 13615}, {1621, 31224}, {1699, 3306}, {1864, 25525}, {2478, 3361}, {2801, 15017}, {3035, 10389}, {3333, 4187}, {3359, 28174}, {3601, 6691}, {3671, 26129}, {3677, 3756}, {3742, 5219}, {3817, 9776}, {3825, 9612}, {3911, 4512}, {3914, 8056}, {3928, 4679}, {4193, 5290}, {4298, 6919}, {4312, 27003}, {4413, 24392}, {4652, 26127}, {4654, 5087}, {4860, 28609}, {4912, 24352}, {5249, 7988}, {5253, 5691}, {5316, 24477}, {5433, 5436}, {5439, 6001}, {5542, 5748}, {5573, 17720}, {6173, 17605}, {6769, 6967}, {6926, 12651}, {7741, 16154}, {7956, 10860}, {9581, 25524}, {10269, 19541}, {10270, 10531}, {10585, 18452}, {10591, 12436}, {11518, 25681}, {11523, 24954}, {12575, 26062}, {15733, 20195}, {16417, 18527}, {17063, 23681}, {17272, 25960}, {17573, 31795}, {17609, 31246}, {20173, 28516}, {21164, 26333}, {24283, 28550}, {24386, 26040}, {25355, 28297}

X(31250) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics (b - c) (3 a^2 - 3 a b - 3 a c + 4 b c) : :

X(31250) lies on these lines: {2, 650}, {499, 30235}, {513, 24924}, {523, 31277}, {654, 25924}, {905, 4791}, {1635, 23813}, {1638, 3239}, {1639, 3676}, {1656, 8760}, {1698, 14077}, {1938, 25917}, {2516, 4382}, {3523, 8142}, {3669, 3762}, {3698, 9366}, {3700, 7658}, {3763, 9001}, {3925, 15280}, {4025, 4944}, {4928, 31286}, {4162, 17072}, {4394, 4728}, {4411, 4698}, {4423, 8641}, {4521, 21104}, {4777, 31238}, {4820, 17069}, {4927, 11068}, {6008, 27013}, {6182, 31245}, {7626, 7874}, {7655, 8062}, {8678, 31251}, {9373, 31246}, {17720, 24793}, {18743, 21438}, {19862, 29066}, {21348, 30864}

X(31250) = complement of X(31209)

X(31251) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics (b - c) (a^3 - 2 a b^2 - 2 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2) : :

X(31251) lies on these lines: {2, 667}, {512, 30835}, {513, 31252}, {514, 30795}, {764, 20317}, {905, 14431}, {1656, 3309}, {1698, 4083}, {3763, 9010}, {3835, 4834}, {4462, 19947}, {4782, 19872}, {5094, 18344}, {8678, 31250}, {9320, 31273}, {18140, 23807}, {20947, 21440}, {29070, 31209}, {29150, 31254}

X(31252) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 + 2 b^3 - 5 a b c + 2 c^3 : :

X(31252) lies on these lines: {1, 3823}, {2, 31}, {10, 17283}, {44, 19872}, {142, 33159}, {344, 33149}, {513, 31251}, {518, 1698}, {726, 27191}, {740, 17266}, {984, 17282}, {1125, 32850}, {1279, 3624}, {1656, 15310}, {1738, 4693}, {1757, 3834}, {3008, 32846}, {3616, 17765}, {3634, 17307}, {3679, 4864}, {3685, 25351}, {3750, 29851}, {3821, 17263}, {3826, 29637}, {3842, 17291}, {3912, 4716}, {3923, 17341}, {4038, 29850}, {4413, 29858}, {4422, 32857}, {4429, 16484}, {4473, 17767}, {4649, 17234}, {4655, 17338}, {4974, 29607}, {6666, 24697}, {7292, 21026}, {9342, 29865}, {9350, 29866}, {16477, 17352}, {16610, 29862}, {16786, 17398}, {17245, 29633}, {17278, 29674}, {17313, 28650}, {17337, 33082}, {17357, 24342}, {17600, 29854}, {17766, 19862}, {17770, 31248}, {17772, 29590}, {18201, 33115}, {21241, 25531}, {24988, 29632}, {26724, 29687}, {31242, 31245}, {31246, 31270}

X(31253) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 6 a + 7 b + 7 c : :

X(31253) lies on these lines: {1, 2}, {5, 12512}, {35, 17534}, {55, 16856}, {140, 10172}, {165, 7486}, {373, 31737}, {515, 632}, {516, 1656}, {547, 18483}, {631, 18492}, {726, 31238}, {756, 24167}, {942, 3988}, {946, 5070}, {950, 5326}, {993, 16862}, {1213, 16671}, {1266, 28653}, {1376, 16855}, {3090, 10164}, {3091, 28158}, {3525, 4297}, {3526, 10175}, {3533, 5587}, {3579, 15699}, {3628, 6684}, {3739, 28516}, {3740, 4547}, {3742, 4015}, {3812, 4757}, {3814, 17529}, {3817, 5067}, {3825, 3826}, {3833, 4127}, {3834, 17771}, {3841, 17527}, {3842, 28582}, {3848, 3881}, {3874, 4533}, {3878, 3922}, {3892, 3983}, {3898, 4002}, {3946, 4535}, {3947, 31231}, {3956, 5045}, {3962, 5883}, {4004, 25917}, {4005, 5439}, {4013, 31227}, {4018, 10176}, {4066, 19804}, {4292, 5442}, {4413, 16854}, {4464, 32089}, {4472, 4912}, {4687, 28522}, {4698, 28484}, {4700, 17398}, {4708, 28333}, {5010, 17570}, {5054, 31673}, {5055, 31730}, {5068, 16192}, {5217, 19536}, {5226, 5586}, {5248, 16853}, {5251, 17535}, {5259, 9342}, {5267, 17531}, {5290, 31188}, {5493, 7988}, {5731, 30315}, {5750, 15492}, {5850, 20195}, {6666, 6701}, {6668, 6692}, {6688, 31757}, {7294, 10106}, {7989, 10303}, {8167, 8715}, {8227, 28228}, {9591, 16042}, {9956, 16239}, {10124, 13624}, {10165, 18525}, {11231, 22791}, {11539, 18480}, {11695, 31752}, {12577, 31479}, {12699, 15703}, {12812, 28146}, {12815, 17355}, {13883, 32790}, {13936, 32789}, {13996, 32557}, {15712, 28172}, {16677, 17303}, {16842, 25440}, {17385, 28557}, {17575, 25639}, {17770, 31248}, {19886, 19981}, {19933, 19954}, {24295, 28550}, {25072, 33149}, {25358, 28297}, {28232, 31447}, {28236, 31399}, {28566, 31289}, {28612, 30829}, {31254, 31263}

X(31253) = complement of X(19862)

X(31254) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - 3 a^2 b^2 + 2 b^4 - 5 a^2 b c - 5 a b^2 c - 3 a^2 c^2 - 5 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31254) lies on these lines: {2, 3}, {8, 11281}, {10, 5425}, {58, 24902}, {81, 24880}, {100, 3841}, {191, 3305}, {758, 1698}, {908, 3634}, {1376, 31660}, {1788, 3649}, {1834, 24936}, {2287, 24937}, {2771, 15059}, {2795, 14061}, {3035, 11604}, {3218, 3824}, {3454, 5235}, {3616, 31245}, {3622, 31493}, {3647, 19872}, {3822, 5260}, {3826, 27529}, {3868, 25525}, {3871, 10198}, {3936, 25446}, {3984, 16126}, {4720, 25650}, {5276, 5346}, {5284, 25639}, {5333, 25441}, {5362, 16960}, {5367, 16961}, {5432, 26060}, {5441, 19878}, {5660, 8582}, {5791, 31019}, {5965, 15988}, {6598, 12437}, {7677, 26481}, {9534, 30834}, {9780, 21677}, {10585, 19855}, {11231, 16139}, {11544, 31888}, {14450, 18228}, {16133, 30312}, {16137, 31479}, {16832, 30851}, {17056, 24883}, {17768, 18230}, {19862, 31262}, {20195, 31240}, {21077, 32635}, {21674, 24161}, {24987, 28234}, {25459, 26109}, {26446, 33592}, {27577, 33135}, {29150, 31251}, {31164, 31446}, {31253, 31263}

X(31254) = complement of X(15674)

X(31255) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 + 10 a^2 b^2 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31255) lies on these lines: {2, 3}, {125, 17811}, {141, 10602}, {394, 5965}, {614, 9627}, {1184, 5346}, {1660, 1853}, {1899, 6090}, {2386, 7867}, {2393, 3763}, {3167, 18911}, {3619, 8263}, {3796, 5972}, {3917, 26958}, {11064, 11402}, {12834, 14848}, {13857, 15004}, {14984, 15059}, {15066, 26913}, {15812, 19118}, {20266, 21015}, {31238, 31245}, {31240, 31241}

X(31255) = {X(2),X(3)}-harmonic conjugate of X(37453)

X(31256) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^6 - 3 a^5 b - 2 a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - a b^5 + 2 b^6 - 3 a^5 c - 3 a^4 b c + 4 a^3 b^2 c + 4 a^2 b^3 c - a b^4 c - b^5 c - 2 a^4 c^2 + 4 a^3 b c^2 + 10 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 + 4 a^3 c^3 + 4 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 - a c^5 - b c^5 + 2 c^6 : :

X(31256) lies on these lines: {2, 3}, {908, 25361}, {1762, 3305}, {4422, 16099}, {4687, 8680}, {18228, 30831}, {18230, 31247}

X(31257) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^7 + a^6 b - 2 a^5 b^2 - 2 a^4 b^3 - a^3 b^4 - a^2 b^5 + 2 a b^6 + 2 b^7 + a^6 c - 3 a^5 b c - 2 a^4 b^2 c + 4 a^3 b^3 c - a^2 b^4 c - a b^5 c + 2 b^6 c - 2 a^5 c^2 - 2 a^4 b c^2 + 10 a^3 b^2 c^2 + 10 a^2 b^3 c^2 - 2 a b^4 c^2 - 2 b^5 c^2 - 2 a^4 c^3 + 4 a^3 b c^3 + 10 a^2 b^2 c^3 + 2 a b^3 c^3 - 2 b^4 c^3 - a^3 c^4 - a^2 b c^4 - 2 a b^2 c^4 - 2 b^3 c^4 - a^2 c^5 - a b c^5 - 2 b^2 c^5 + 2 a c^6 + 2 b c^6 + 2 c^7 : :

X(31257) lies on these lines: {2, 3}, {1698, 31261}, {16100, 27076}

X(31258) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^7 + 4 a^6 b + a^5 b^2 - 6 a^4 b^3 - 5 a^3 b^4 + 3 a b^6 + 2 b^7 + 4 a^6 c + 3 a^5 b c - 3 a^4 b^2 c - 4 a^3 b^3 c - 4 a^2 b^4 c + a b^5 c + 3 b^6 c + a^5 c^2 - 3 a^4 b c^2 + 2 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 6 a^4 c^3 - 4 a^3 b c^3 + 4 a^2 b^2 c^3 - 2 a b^3 c^3 - 4 b^4 c^3 - 5 a^3 c^4 - 4 a^2 b c^4 - 3 a b^2 c^4 - 4 b^3 c^4 + a b c^5 - b^2 c^5 + 3 a c^6 + 3 b c^6 + 2 c^7 : :

X(31258) lies on these lines: {2, 3}, {78, 17295}, {31248, 31269}

X(31259) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 3 a^4 - 4 a^2 b^2 + b^4 - 10 a^2 b c - 10 a b^2 c - 4 a^2 c^2 - 10 a b c^2 - 2 b^2 c^2 + c^4 : :

X(31259) lies on these lines: {2, 3}, {8, 3748}, {69, 17201}, {78, 6666}, {518, 3616}, {551, 3984}, {908, 3624}, {1125, 3305}, {1621, 19855}, {3434, 5259}, {3487, 27065}, {3646, 24541}, {3828, 31452}, {4423, 10527}, {5218, 17606}, {5226, 7288}, {5250, 28228}, {5284, 19843}, {5302, 5550}, {5346, 16589}, {8167, 24953}, {9708, 10587}, {11415, 15254}, {16817, 17776}, {17337, 19765}, {19854, 24387}, {19860, 28234}, {19862, 31266}, {25055, 31458}, {25086, 26258}, {25542, 26363}

X(31260) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 4 a^4 - 7 a^2 b^2 + 3 b^4 - 2 a^2 b c - 4 a b^2 c - 7 a^2 c^2 - 4 a b c^2 - 6 b^2 c^2 + 3 c^4 : :

X(31260) lies on these lines: {2, 12}, {5, 30264}, {10, 5326}, {11, 5248}, {21, 7173}, {30, 31262}, {140, 3925}, {631, 5842}, {632, 952}, {758, 5439}, {993, 3614}, {1213, 2317}, {1358, 27187}, {1656, 5841}, {2476, 15326}, {3525, 4413}, {3526, 19854}, {3533, 19855}, {3616, 5855}, {3624, 11529}, {3628, 5251}, {3634, 17614}, {3763, 5849}, {3825, 15670}, {3826, 17566}, {3847, 16865}, {4995, 24390}, {4996, 5047}, {5204, 6856}, {5267, 17530}, {5432, 5687}, {5852, 18230}, {5857, 20195}, {6174, 31419}, {6253, 6954}, {6284, 6910}, {6675, 25542}, {6681, 17529}, {6763, 7308}, {6852, 7958}, {6863, 18761}, {6888, 7965}, {8068, 17527}, {11375, 12526}, {11682, 15950}, {15338, 25639}, {15674, 31272}, {17575, 20107}, {17757, 20104}, {30950, 31880}

X(31261) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^5 - 2 a^4 b - a b^4 + 2 b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c + 4 a^2 b c^2 + 2 a b^2 c^2 - a c^4 - 2 b c^4 + 2 c^5 : :

X(31261) lies on these lines: {2, 19}, {5, 30265}, {9, 4466}, {11, 4319}, {46, 24884}, {71, 18634}, {140, 21160}, {516, 631}, {1486, 4423}, {1698, 31257}, {1826, 30809}, {2263, 11375}, {3624, 7523}, {3668, 5219}, {3763, 3827}, {4687, 8680}, {5227, 28757}, {7573, 17400}, {8804, 25525}, {11677, 26105}, {17073, 30810}, {17263, 30852}, {17322, 18655}, {20196, 20200}, {25361, 27413}, {26006, 26130}, {31238, 31245}

X(31262) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - 3 a^2 b^2 + 2 b^4 - a^2 b c - 2 a b^2 c - 3 a^2 c^2 - 2 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31262) lies on these lines: {1, 17057}, {2, 35}, {5, 5251}, {10, 5443}, {12, 5288}, {30, 31260}, {36, 2476}, {80, 24541}, {100, 20104}, {142, 14526}, {191, 17605}, {405, 14794}, {442, 6691}, {498, 5082}, {499, 6856}, {517, 1656}, {958, 31160}, {993, 5141}, {1125, 5086}, {1203, 17717}, {1210, 26725}, {1900, 5094}, {2646, 3624}, {3090, 19854}, {3336, 3838}, {3436, 5258}, {3582, 25466}, {3583, 7483}, {3584, 6668}, {3585, 4999}, {3628, 3925}, {3632, 31479}, {3679, 11011}, {3746, 11680}, {3763, 9047}, {3822, 5563}, {4413, 5070}, {4857, 6690}, {4867, 11375}, {5219, 5904}, {5267, 17577}, {5444, 17647}, {5692, 5705}, {5790, 33281}, {6667, 17529}, {6675, 7173}, {6701, 27003}, {6853, 15931}, {6863, 18517}, {6871, 10483}, {7280, 17532}, {7308, 24468}, {7486, 19855}, {8728, 26476}, {10399, 25525}, {11010, 31435}, {11015, 19878}, {11280, 15829}, {16370, 18514}, {16819, 32967}, {17123, 24902}, {17531, 20107}, {18393, 26066}, {18398, 31266}, {19862, 31254}, {25431, 29682}

X(31263) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - 3 a^2 b^2 + 2 b^4 + a^2 b c + 2 a b^2 c - 3 a^2 c^2 + 2 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31263) lies on these lines: {1, 5123}, {2, 36}, {5, 2077}, {10, 5330}, {12, 5193}, {30, 31235}, {35, 4193}, {78, 15079}, {484, 5087}, {498, 26105}, {499, 3421}, {513, 31251}, {517, 1656}, {519, 31272}, {547, 3925}, {908, 4880}, {1125, 5176}, {1155, 19872}, {1319, 3624}, {1329, 5258}, {1376, 31159}, {1699, 13528}, {1737, 4867}, {1878, 5094}, {2078, 17527}, {2975, 20107}, {3035, 3583}, {3090, 5537}, {3245, 3634}, {3434, 6931}, {3526, 23961}, {3582, 6667}, {3584, 3816}, {3585, 13747}, {3679, 5048}, {3683, 10225}, {3746, 3825}, {3763, 9037}, {3838, 27247}, {3847, 4857}, {4187, 5259}, {4413, 5055}, {4511, 6702}, {5010, 17556}, {5047, 20104}, {5067, 19854}, {5070, 22765}, {5154, 25440}, {5172, 11108}, {5219, 18838}, {5270, 6691}, {5443, 24982}, {5445, 21616}, {5535, 7308}, {5541, 7743}, {5563, 11681}, {5570, 5904}, {5692, 30827}, {5902, 30852}, {6001, 15017}, {6550, 30795}, {6668, 17575}, {6735, 16173}, {6921, 10483}, {6958, 18516}, {6963, 15931}, {6971, 18407}, {7705, 30144}, {7988, 22835}, {11009, 25005}, {16371, 18513}, {16819, 16922}, {18395, 25681}, {25055, 25405}, {31253, 31254}

X(31264) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 2 a^2 b + a b^2 + 2 a^2 c + 2 b^2 c + a c^2 + 2 b c^2 : :

X(31264) lies on these lines: {2, 38}, {5, 30272}, {10, 5741}, {11, 29685}, {31, 29828}, {42, 3706}, {43, 21020}, {210, 30970}, {226, 32781}, {312, 1962}, {321, 4970}, {518, 31241}, {714, 4687}, {750, 26243}, {758, 1698}, {894, 32918}, {896, 26223}, {899, 31993}, {1125, 3701}, {1150, 4722}, {1329, 27714}, {2887, 26251}, {3666, 28555}, {3703, 29688}, {3720, 30818}, {3722, 24552}, {3740, 22275}, {3757, 32944}, {3763, 9020}, {3772, 29663}, {3805, 24924}, {3822, 19867}, {3840, 17450}, {3873, 29827}, {3891, 29650}, {3932, 29682}, {3967, 3989}, {3994, 28606}, {4023, 8013}, {4054, 33145}, {4090, 4981}, {4459, 5432}, {4671, 17592}, {4972, 25385}, {5718, 15523}, {6686, 24589}, {7081, 32772}, {9345, 30567}, {10180, 31035}, {10436, 17124}, {11680, 29659}, {17056, 29687}, {17061, 29684}, {17243, 21713}, {17279, 29661}, {17289, 29846}, {17303, 21033}, {17357, 29869}, {17381, 29847}, {17469, 25496}, {17717, 29667}, {17718, 24943}, {17720, 29647}, {17722, 33090}, {17723, 32854}, {17724, 29686}, {17725, 29648}, {17774, 33120}, {19684, 29649}, {21026, 29679}, {21805, 31330}, {21806, 32915}, {21951, 25629}, {24627, 32940}, {24697, 26792}, {24725, 26034}, {25124, 27261}, {26098, 33074}, {27064, 32917}, {29633, 33133}, {29639, 33162}, {29640, 33157}, {29657, 32862}, {29664, 33165}, {29678, 32777}, {29680, 33169}, {31019, 33174}, {31053, 32784}, {33076, 33107}, {33079, 33112}, {33083, 33096}, {33086, 33097}

X(31265) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^5 - a^3 b^2 - 2 a^2 b^3 + 2 b^5 - a^3 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 2 c^5 : :

X(31265) lies on these lines: {2, 48}, {71, 30808}, {916, 1656}, {1698, 31257}, {2268, 8287}, {2801, 20195}, {3763, 8679}, {4687, 31278}, {7951, 24884}, {17073, 21011}, {19862, 29219}, {21236, 25940}, {21238, 30746}

X(31266) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - 2 a^2 b - a b^2 + 2 b^3 - 2 a^2 c - 2 b^2 c - a c^2 - 2 b c^2 + 2 c^3 : :

X(31266) lies on these lines: {1, 2476}, {2, 7}, {5, 18446}, {10, 3984}, {11, 4666}, {12, 19860}, {21, 9612}, {42, 17064}, {46, 11263}, {55, 3838}, {78, 442}, {81, 26738}, {84, 6888}, {140, 21165}, {149, 10389}, {165, 20292}, {200, 33108}, {306, 21675}, {321, 30834}, {345, 4054}, {377, 4855}, {388, 24541}, {411, 5715}, {443, 27385}, {469, 5307}, {474, 3824}, {495, 3872}, {498, 12609}, {515, 3091}, {518, 31245}, {551, 23708}, {612, 17719}, {614, 17717}, {758, 1698}, {857, 16831}, {912, 1656}, {936, 4197}, {940, 22128}, {946, 6838}, {950, 6871}, {968, 3944}, {993, 3624}, {1001, 17605}, {1125, 1478}, {1210, 6933}, {1215, 29857}, {1490, 6828}, {1532, 5886}, {1621, 1699}, {1707, 24725}, {1750, 10883}, {1836, 6690}, {1837, 11281}, {2475, 3601}, {2801, 15017}, {2886, 3870}, {2887, 29828}, {2975, 5290}, {2999, 33129}, {3011, 26098}, {3120, 17594}, {3158, 33110}, {3173, 10601}, {3419, 5719}, {3434, 13405}, {3436, 3947}, {3475, 26015}, {3485, 11682}, {3487, 6734}, {3555, 31493}, {3576, 6840}, {3586, 17577}, {3618, 9028}, {3649, 26066}, {3677, 29680}, {3729, 33113}, {3748, 11235}, {3749, 33104}, {3751, 24892}, {3753, 31479}, {3761, 27793}, {3771, 25385}, {3772, 5256}, {3868, 5705}, {3873, 5231}, {3890, 11522}, {3895, 10056}, {3897, 9613}, {3936, 11679}, {3951, 5791}, {3957, 24392}, {4053, 30823}, {4138, 26034}, {4189, 9579}, {4208, 27383}, {4292, 6910}, {4384, 5741}, {4417, 4886}, {4423, 5087}, {4512, 5057}, {4641, 31187}, {4652, 7483}, {4659, 33168}, {4687, 8680}, {4850, 23681}, {4870, 5289}, {4892, 32916}, {4917, 5082}, {4999, 10404}, {5046, 5436}, {5119, 10197}, {5129, 5550}, {5141, 9581}, {5176, 5726}, {5177, 5703}, {5250, 10198}, {5269, 29665}, {5287, 5949}, {5425, 17057}, {5426, 18513}, {5432, 5880}, {5440, 17528}, {5709, 6853}, {5714, 6857}, {5720, 6829}, {5722, 17530}, {6245, 6860}, {6260, 6837}, {6513, 7363}, {6667, 12831}, {6668, 24914}, {6701, 25440}, {6735, 8164}, {6830, 18443}, {6831, 10884}, {6852, 7330}, {6921, 12436}, {6931, 9843}, {6939, 12115}, {6943, 8726}, {6992, 10165}, {7174, 29664}, {7290, 29681}, {7504, 18389}, {7951, 26725}, {7988, 10582}, {8774, 24924}, {10171, 10584}, {10527, 21620}, {10587, 12053}, {10588, 24982}, {10826, 30143}, {10827, 30147}, {11230, 22758}, {11375, 19861}, {13407, 26363}, {13478, 30588}, {16496, 29690}, {16586, 28606}, {16826, 24268}, {17011, 18261}, {17061, 17723}, {17173, 18163}, {17266, 31276}, {17270, 31037}, {17304, 29069}, {17532, 24929}, {17579, 30282}, {17603, 17616}, {17728, 25557}, {18139, 30567}, {18229, 32782}, {18398, 31262}, {18588, 30675}, {19854, 21077}, {19862, 31259}, {20254, 21807}, {21221, 26109}, {21241, 29670}, {21808, 30820}, {21907, 25430}, {23511, 26724}, {24630, 29603}, {25006, 25568}, {25254, 31035}, {25496, 29855}, {25524, 25875}, {26102, 30993}, {26128, 29826}, {26227, 28599}, {27757, 28605}, {28846, 30835}, {29639, 33144}, {29657, 33152}, {29675, 33106}, {29688, 33143}

X(31267) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 3 a^8 - 4 a^4 b^4 + b^8 - 4 a^4 c^4 - 2 b^4 c^4 + c^8 : :

X(31267) lies on these lines: {2, 66}, {5, 23041}, {6, 468}, {69, 19122}, {140, 19149}, {141, 19153}, {154, 23300}, {159, 3589}, {597, 15585}, {1177, 5972}, {1352, 6639}, {1503, 1656}, {2070, 31670}, {2393, 3618}, {3098, 10182}, {3147, 19161}, {3313, 7493}, {3517, 5480}, {3763, 19132}, {3827, 5439}, {3867, 15448}, {5878, 15578}, {6329, 11216}, {6353, 9969}, {6776, 26917}, {7506, 14561}, {7800, 15257}, {9833, 20300}, {9968, 10303}, {10020, 19139}, {11202, 19130}, {13339, 14216}, {14805, 18382}, {15812, 19127}, {16195, 29181}, {19121, 28408}, {23042, 24206}

X(31268) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 + 5 a^2 b^2 + 2 b^4 + 5 a^2 c^2 + 5 b^2 c^2 + 2 c^4 : :

X(31268) lies on these lines: {2, 32}, {3, 22803}, {5, 12122}, {10, 7977}, {39, 10159}, {99, 8362}, {140, 6287}, {141, 7905}, {187, 16896}, {549, 8725}, {590, 19092}, {615, 19091}, {620, 11606}, {631, 29012}, {732, 3763}, {1125, 12783}, {3090, 6249}, {3329, 7882}, {3525, 12252}, {3619, 7796}, {3624, 12264}, {3934, 7923}, {4413, 12339}, {5070, 13111}, {5094, 12144}, {5305, 7859}, {5432, 12954}, {5433, 12944}, {6656, 15031}, {6683, 7909}, {7484, 9918}, {7748, 7876}, {7760, 16986}, {7770, 7910}, {7780, 16987}, {7802, 16045}, {7822, 7891}, {7832, 12055}, {7868, 31467}, {7915, 8290}, {7917, 11174}, {7919, 13881}, {7943, 15271}, {7948, 14061}, {8993, 32785}, {11285, 24273}, {11307, 33386}, {11308, 33387}, {12795, 15184}, {12934, 24953}, {13112, 26364}, {13113, 26363}, {17766, 19862}, {29596, 32851}

X(31269) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 2 a^3 b - 4 a^2 b^2 + 2 a b^3 + 2 a^3 c - a^2 b c - 2 a b^2 c + b^3 c - 4 a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 + 2 a c^3 + b c^3 : :

X(31269) lies on these lines: {2, 85}, {9, 10012}, {75, 25082}, {78, 17277}, {220, 32088}, {518, 3616}, {1475, 27475}, {1698, 28850}, {3693, 27304}, {4698, 24652}, {7308, 16831}, {10025, 32100}, {14828, 16572}, {16284, 28742}, {17263, 28740}, {17687, 31637}, {17747, 27183}, {18743, 28797}, {19804, 25083}, {31248, 31258}

X(31270) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 b^2 + 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 - a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 - 4 a b c^3 + 2 b^2 c^3 : :

X(31270) lies on these lines: {1, 27095}, {2, 87}, {726, 1698}, {3763, 9025}, {16495, 17265}, {17307, 29827}, {17327, 21264}, {25140, 25535}, {31246, 31252}

X(31271) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - 5 a^2 b - 4 a b^2 + 2 b^3 - 5 a^2 c + 25 a b c - 6 b^2 c - 4 a c^2 - 6 b c^2 + 2 c^3 : :

X(31271) lies on these lines: {2, 45}, {1698, 2802}, {4767, 25377}, {31247, 31278}

X(31272) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c + 5 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31272) lies on these lines: {1, 6702}, {2, 11}, {3, 10724}, {4, 6713}, {5, 104}, {8, 1387}, {10, 1320}, {20, 21154}, {21, 3825}, {35, 20107}, {40, 16174}, {56, 5154}, {80, 1125}, {88, 3120}, {119, 3090}, {140, 10738}, {141, 10755}, {142, 1156}, {145, 3036}, {153, 5056}, {214, 3624}, {381, 10728}, {392, 6797}, {404, 7741}, {405, 4996}, {474, 17100}, {485, 19081}, {496, 27529}, {498, 5533}, {499, 2975}, {519, 31263}, {547, 11698}, {551, 7972}, {590, 19113}, {615, 19112}, {620, 10769}, {631, 5840}, {693, 10006}, {900, 27191}, {908, 5850}, {942, 12532}, {952, 1656}, {1145, 9780}, {1255, 29688}, {1317, 3622}, {1479, 17566}, {1484, 3628}, {1587, 13977}, {1588, 13913}, {1647, 3315}, {1698, 2802}, {1699, 9352}, {1768, 3306}, {1862, 5094}, {2320, 12747}, {2475, 6691}, {2476, 10200}, {2771, 5439}, {2787, 14061}, {2800, 8227}, {2801, 15017}, {2805, 31238}, {2829, 3091}, {2932, 16408}, {3045, 9306}, {3065, 6701}, {3086, 6931}, {3218, 5087}, {3254, 6666}, {3485, 12832}, {3523, 24466}, {3525, 13199}, {3533, 10993}, {3545, 12248}, {3560, 18861}, {3576, 6246}, {3582, 3814}, {3583, 6681}, {3614, 12763}, {3617, 5854}, {3618, 5848}, {3626, 26726}, {3634, 21630}, {3681, 30827}, {3742, 17660}, {3756, 33148}, {3763, 9024}, {3812, 17638}, {3817, 20292}, {3837, 13266}, {3840, 30831}, {3847, 5046}, {3851, 22799}, {3868, 18254}, {3869, 12736}, {3873, 18240}, {3887, 31273}, {3911, 5057}, {3934, 32454}, {3952, 4997}, {4187, 5260}, {4188, 10896}, {4756, 30566}, {4861, 17619}, {4945, 21093}, {5055, 10711}, {5070, 12331}, {5080, 15325}, {5083, 5219}, {5121, 33129}, {5141, 25524}, {5178, 6700}, {5187, 7288}, {5231, 14740}, {5249, 10171}, {5298, 20067}, {5330, 18395}, {5435, 24465}, {5531, 10582}, {5550, 6224}, {5587, 11715}, {5817, 13257}, {5856, 18230}, {5883, 11571}, {5886, 10698}, {5901, 19914}, {5972, 10778}, {6036, 10768}, {6265, 11230}, {6366, 31640}, {6675, 11604}, {6684, 14217}, {6699, 10767}, {6710, 10770}, {6711, 10771}, {6712, 10772}, {6715, 10774}, {6716, 10775}, {6717, 10776}, {6718, 10777}, {6719, 10779}, {6720, 10780}, {6915, 18406}, {6921, 10591}, {6941, 26492}, {6952, 12775}, {6961, 10598}, {6972, 7681}, {6978, 32554}, {6981, 10785}, {7483, 26127}, {7484, 13222}, {7486, 10585}, {7808, 13194}, {7914, 13235}, {7951, 10074}, {8104, 8126}, {8125, 13267}, {8674, 15059}, {8697, 19631}, {8983, 19077}, {8988, 18992}, {9624, 25485}, {9776, 9809}, {9897, 10031}, {9955, 12515}, {9956, 12737}, {9963, 15015}, {10157, 17661}, {10165, 12119}, {10175, 12751}, {10586, 10588}, {10593, 13747}, {10759, 14561}, {11375, 20118}, {11376, 25005}, {11684, 21616}, {11729, 12247}, {11814, 33115}, {12528, 15528}, {12653, 19875}, {12735, 31479}, {12746, 19864}, {12758, 23708}, {12761, 13729}, {13268, 15184}, {13272, 24953}, {13278, 26364}, {13279, 26363}, {13532, 29008}, {13922, 32785}, {13971, 19078}, {13976, 18991}, {13991, 32786}, {14923, 15558}, {15079, 30144}, {15343, 25529}, {15674, 31260}, {17123, 31204}, {17531, 25639}, {17605, 27003}, {17681, 28761}, {17728, 31053}, {19862, 31254}, {19877, 31493}, {22102, 31512}, {24003, 30855}, {24387, 25438}, {24624, 25533}, {24712, 25342}, {28808, 33089}, {29662, 32911}, {30834, 30947}, {31237, 31242}, {31241, 31247}

X(31272) = anticomplement of X(31235)

X(31273) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c + a^2 b c + 2 a b^2 c - 2 b^3 c + 2 a b c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31273) lies on these lines: {2, 101}, {3, 10725}, {4, 6712}, {5, 103}, {8, 11726}, {10, 10695}, {118, 3090}, {140, 10739}, {141, 10756}, {152, 5056}, {381, 10727}, {514, 31640}, {1018, 30857}, {1656, 2808}, {1698, 2809}, {2774, 15059}, {2784, 19862}, {2786, 14061}, {2801, 20195}, {2810, 3763}, {2813, 31241}, {3035, 10770}, {3046, 9306}, {3533, 33520}, {3624, 11712}, {3730, 17675}, {3887, 31272}, {4568, 30790}, {5055, 10710}, {5068, 33521}, {5094, 5185}, {5587, 11714}, {5886, 10697}, {9320, 31251}, {10758, 14561}, {10772, 23513}, {17671, 24047}

X(31274) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 4 a^4 - 4 a^2 b^2 + 3 b^4 - 4 a^2 c^2 - 2 b^2 c^2 + 3 c^4 : :
Trilinears (sin A - sin(A + 2ω)) (2 sin B sin C - sin(B + 2ω) sin C - sin B sin(C + 2ω)) - e^2 sin(A + ω) (sin(B + ω) sin C + sin B sin(C + ω)) : :

X(31274) lies on these lines: {2, 99}, {3, 6721}, {30, 31275}, {32, 1007}, {98, 3525}, {114, 140}, {141, 5477}, {183, 3788}, {230, 7813}, {315, 33262}, {542, 3763}, {547, 22515}, {549, 7853}, {590, 13989}, {615, 8997}, {625, 6781}, {626, 7771}, {631, 2794}, {632, 2782}, {754, 7925}, {1153, 25486}, {1506, 7804}, {1569, 3934}, {1656, 23698}, {1916, 14067}, {2023, 33185}, {2548, 33203}, {2787, 31235}, {3023, 5326}, {3027, 7294}, {3054, 9466}, {3055, 8368}, {3090, 21166}, {3329, 6680}, {3455, 16419}, {3523, 7935}, {3524, 10722}, {3526, 6036}, {3530, 22505}, {3533, 11623}, {3589, 22848}, {3618, 14645}, {3619, 5182}, {3624, 11725}, {3628, 23514}, {3634, 11711}, {3767, 32959}, {5054, 6033}, {5056, 10723}, {5067, 14639}, {5070, 6321}, {5149, 11285}, {5206, 32989}, {5215, 7845}, {5319, 32835}, {5475, 11288}, {5550, 7983}, {5939, 7915}, {5972, 15357}, {5976, 6683}, {6054, 15709}, {6055, 11539}, {6704, 14043}, {7603, 8369}, {7746, 7863}, {7747, 7862}, {7755, 7763}, {7756, 7887}, {7764, 7766}, {7765, 7886}, {7772, 32829}, {7778, 7810}, {7795, 32977}, {7815, 10352}, {7816, 33249}, {7818, 21843}, {7821, 14929}, {7825, 32964}, {7826, 7888}, {7830, 7899}, {7832, 16923}, {7870, 17004}, {7872, 33248}, {7889, 31455}, {7908, 17008}, {7914, 33001}, {7930, 33015}, {7934, 33274}, {8167, 13173}, {8352, 10150}, {8588, 33216}, {8589, 33184}, {8781, 32839}, {9862, 15702}, {9880, 15092}, {10011, 10256}, {10190, 19598}, {11123, 12076}, {11540, 26614}, {11632, 15723}, {11724, 26446}, {11812, 22566}, {13188, 20398}, {14064, 15515}, {14148, 14568}, {14535, 31489}, {18800, 20582}, {19108, 32785}, {19109, 32786}, {31401, 33189}, {31415, 33191}, {32456, 33228}

X(31274) = complement of X(14061)
X(31274) = QA-P35 center (1st Penta Point) of quadrangle ABCX(2)

X(31275) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 2 a^4 - 3 a^2 b^2 + 4 b^4 - 3 a^2 c^2 - 4 b^2 c^2 + 4 c^4 : :

X(31275) lies on these lines: {2, 187}, {5, 7820}, {30, 31274}, {39, 7844}, {69, 1570}, {115, 6390}, {140, 13449}, {141, 5107}, {230, 7845}, {325, 6722}, {511, 1656}, {512, 30835}, {538, 7925}, {574, 11318}, {620, 33228}, {626, 33249}, {641, 3070}, {642, 3071}, {671, 15301}, {1007, 5309}, {1506, 1692}, {2021, 7866}, {2031, 7818}, {2072, 16760}, {2080, 5070}, {2482, 8355}, {2548, 32955}, {3054, 7810}, {3055, 8360}, {3090, 7822}, {3526, 7935}, {3628, 6292}, {3629, 7755}, {3788, 11185}, {3934, 7899}, {5007, 7752}, {5008, 7775}, {5041, 7828}, {5094, 5140}, {5099, 5159}, {6144, 7903}, {6683, 7901}, {6721, 15980}, {7622, 15602}, {7748, 32972}, {7749, 7873}, {7763, 33277}, {7769, 7861}, {7777, 7817}, {7778, 9466}, {7780, 7850}, {7795, 32988}, {7799, 32457}, {7800, 32976}, {7805, 7814}, {7808, 33218}, {7813, 14971}, {7815, 10631}, {7816, 7940}, {7825, 15513}, {7834, 33248}, {7841, 8589}, {7842, 7907}, {7843, 7857}, {7848, 17004}, {7883, 17006}, {7888, 13881}, {7889, 33186}, {7908, 14711}, {7911, 16923}, {7913, 31489}, {7915, 16921}, {7919, 17005}, {7930, 33002}, {7944, 16922}, {8352, 22247}, {8586, 21358}, {9167, 32459}, {9301, 15703}, {14041, 32456}, {14064, 31455}, {15482, 33219}, {18424, 32984}, {18584, 33237}, {21839, 30868}, {31401, 33199}, {31457, 32839}

X(31275) = intersection of Brocard axes of outer and inner Vecten triangles

X(31276) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^2 b^2 + a^2 c^2 + 3 b^2 c^2 : :

X(31276) lies on these lines: {1, 30998}, {2, 39}, {3, 17128}, {4, 2896}, {5, 3314}, {6, 17129}, {8, 12263}, {20, 6248}, {30, 7904}, {69, 7785}, {83, 7751}, {99, 7815}, {115, 3096}, {140, 7709}, {141, 5025}, {148, 5976}, {183, 384}, {192, 3760}, {193, 14994}, {230, 7892}, {262, 5056}, {308, 2998}, {315, 16044}, {316, 7854}, {325, 16921}, {330, 3761}, {381, 7879}, {385, 7770}, {511, 3091}, {524, 7921}, {599, 7773}, {625, 7922}, {626, 32966}, {631, 2782}, {671, 7872}, {698, 3763}, {726, 1698}, {730, 3616}, {732, 3618}, {736, 19689}, {1007, 32999}, {1078, 3552}, {1107, 20943}, {1125, 9902}, {1269, 26042}, {1506, 7796}, {1621, 12338}, {1656, 7881}, {1916, 14064}, {1975, 7824}, {1995, 9917}, {2023, 33248}, {2548, 7779}, {2549, 33021}, {3085, 10079}, {3086, 10063}, {3090, 3095}, {3094, 3619}, {3097, 3634}, {3104, 18581}, {3105, 18582}, {3146, 5188}, {3329, 7754}, {3523, 11257}, {3525, 11171}, {3526, 32448}, {3545, 14881}, {3589, 7920}, {3617, 14839}, {3622, 7976}, {3628, 32447}, {3662, 3831}, {3691, 17028}, {3785, 14035}, {3815, 7906}, {3850, 22728}, {3854, 22682}, {3906, 31072}, {3933, 7777}, {3972, 7780}, {4441, 26752}, {4479, 20691}, {4688, 25109}, {4721, 17350}, {5052, 20080}, {5054, 32516}, {5059, 22676}, {5067, 11272}, {5070, 32520}, {5218, 13077}, {5253, 22779}, {5254, 7876}, {5277, 16913}, {5305, 7875}, {5355, 6704}, {5475, 7768}, {5921, 13354}, {6048, 17117}, {6179, 7804}, {6292, 7790}, {6337, 33001}, {6353, 12143}, {6376, 21264}, {6381, 17030}, {6655, 7800}, {6656, 16986}, {6658, 14907}, {7187, 20925}, {7288, 18982}, {7585, 8992}, {7586, 13983}, {7603, 7814}, {7617, 10302}, {7735, 9983}, {7736, 13571}, {7745, 7893}, {7747, 7811}, {7748, 7831}, {7749, 7835}, {7750, 11361}, {7752, 7794}, {7753, 7877}, {7755, 7846}, {7760, 7808}, {7761, 33019}, {7767, 7823}, {7771, 7816}, {7774, 32968}, {7775, 7917}, {7778, 32967}, {7782, 33022}, {7783, 11285}, {7784, 14041}, {7788, 7941}, {7789, 7907}, {7792, 16895}, {7802, 7810}, {7805, 7878}, {7806, 7819}, {7807, 17004}, {7809, 7896}, {7812, 7826}, {7818, 32027}, {7820, 7857}, {7821, 33010}, {7825, 7883}, {7830, 33264}, {7833, 32819}, {7839, 11174}, {7841, 7928}, {7842, 7936}, {7843, 7850}, {7844, 7944}, {7848, 7860}, {7849, 7934}, {7855, 7858}, {7856, 7889}, {7861, 7937}, {7862, 7909}, {7864, 8362}, {7865, 7911}, {7867, 14061}, {7868, 7901}, {7869, 7899}, {7882, 7926}, {7887, 7931}, {7914, 7919}, {7923, 10335}, {7935, 18546}, {7947, 16922}, {8177, 24273}, {8290, 12188}, {8359, 32480}, {8556, 13586}, {8591, 32822}, {8610, 26107}, {8859, 33237}, {9780, 12782}, {10303, 13334}, {10479, 17238}, {10584, 12923}, {10585, 12933}, {10586, 13109}, {10587, 13110}, {10588, 12837}, {10589, 12836}, {11108, 16994}, {11168, 33274}, {11321, 16999}, {13468, 14036}, {14001, 17008}, {14023, 20088}, {14034, 15598}, {15589, 20065}, {16045, 16989}, {16918, 16992}, {16975, 21219}, {16984, 33217}, {16997, 17686}, {16998, 17541}, {17006, 33233}, {17048, 31317}, {17144, 25102}, {17266, 31266}, {17284, 17760}, {17349, 29433}, {17382, 25629}, {17499, 29455}, {18144, 18148}, {19312, 19768}, {20247, 31052}, {20530, 31997}, {20888, 27091}, {21299, 23634}, {21356, 33006}, {22036, 31025}, {24249, 27954}, {24327, 24717}, {24351, 24688}, {27044, 28605}, {27076, 32035}, {27268, 32453}, {31404, 33261}, {31489, 32821}, {32006, 33016}, {32815, 32965}, {32816, 32962}, {32817, 32978}, {32818, 32975}, {32826, 32997}, {32827, 32995}

X(31276) = anticomplement of X(7786)

X(31277) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics (b^2 - c^2) (3 a^4 - 3 a^2 b^2 - 3 a^2 c^2 + 4 b^2 c^2) : :

X(31277) lies on these lines: {2, 647}, {512, 30835}, {523, 31250}, {1637, 3265}, {1656, 30209}, {2501, 14417}, {2525, 6587}, {3763, 8675}, {3906, 31239}, {7652, 7746}, {8651, 9148}, {8672, 24924}, {9404, 19744}

X(31278) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^5 - a^3 b^2 - 2 a^2 b^3 + 2 b^5 + 2 a^2 b^2 c - a^3 c^2 + 2 a^2 b c^2 + a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 2 c^5 : :

X(31278) lies on these lines: {2, 662}, {190, 30856}, {643, 24955}, {2786, 14061}, {4687, 31265}, {4858, 15455}, {8674, 15059}, {17058, 24957}, {31247, 31271}

X(31279) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics (b^2 - c^2) (a^4 - 2 a^2 b^2 - 2 a^2 c^2 + 2 b^2 c^2) : :

X(31279) lies on these lines: {2, 669}, {125, 30451}, {512, 30835}, {523, 7925}, {647, 7631}, {850, 17414}, {1499, 1656}, {2501, 5094}, {2528, 30474}, {3005, 30476}, {3526, 5926}, {3763, 9009}, {5996, 8665}, {6563, 8029}, {6587, 14420}, {7862, 23099}, {7887, 14824}, {8639, 21260}, {11123, 30771}, {12075, 14417}, {14424, 33294}, {20983, 25637}, {21051, 26983}, {21726, 29653}, {22089, 33314}

X(31280) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics 2 a^3 - 2 a^2 b - a b^2 + 4 b^3 - 2 a^2 c - 2 b^2 c - a c^2 - 2 b c^2 + 4 c^3 : :

X(31280) lies on these lines: {2, 896}, {8, 24161}, {513, 24924}, {758, 1698}, {899, 30823}, {1125, 5051}, {3120, 3712}, {3722, 21241}, {3838, 29865}, {3936, 4938}, {3994, 29862}, {4062, 17070}, {4442, 4933}, {6701, 25669}, {7292, 27759}, {10129, 29858}, {17605, 29869}, {17719, 21026}, {17763, 25529}, {17785, 32930}, {21020, 30831}, {26738, 29856}, {29857, 31161}, {30828, 33128}

X(31281) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^3 - 4 a^2 b - 3 a b^2 + 2 b^3 - 4 a^2 c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3 : :

X(31281) lies on these lines: {2, 6}, {952, 1656}, {1621, 19540}, {3306, 30588}, {3891, 29688}, {4054, 28301}, {4414, 28546}, {4660, 33105}, {17244, 24277}, {17258, 31053}, {17304, 29069}, {17355, 33113}, {17396, 33133}, {24552, 29678}, {24627, 26738}, {30824, 31035}

X(31282) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^10 - 4 a^8 b^2 + 4 a^6 b^4 + 2 a^4 b^6 - 5 a^2 b^8 + 2 b^10 - 4 a^8 c^2 + 10 a^6 b^2 c^2 - 10 a^4 b^4 c^2 + 10 a^2 b^6 c^2 - 6 b^8 c^2 + 4 a^6 c^4 - 10 a^4 b^2 c^4 - 10 a^2 b^4 c^4 + 4 b^6 c^4 + 2 a^4 c^6 + 10 a^2 b^2 c^6 + 4 b^4 c^6 - 5 a^2 c^8 - 6 b^2 c^8 + 2 c^10 : :

X(31282) lies on these lines: {2, 3}, {155, 26913}, {394, 26917}, {5449, 15066}, {5651, 32767}, {5965, 20806}, {5972, 9707}, {9820, 18911}, {11064, 18912}, {11412, 26958}, {11444, 15059}, {17814, 23294}

X(31283) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics a^10 - 4 a^8 b^2 + 4 a^6 b^4 + 2 a^4 b^6 - 5 a^2 b^8 + 2 b^10 - 4 a^8 c^2 + 8 a^6 b^2 c^2 - 4 a^4 b^4 c^2 + 6 a^2 b^6 c^2 - 6 b^8 c^2 + 4 a^6 c^4 - 4 a^4 b^2 c^4 - 2 a^2 b^4 c^4 + 4 b^6 c^4 + 2 a^4 c^6 + 6 a^2 b^2 c^6 + 4 b^4 c^6 - 5 a^2 c^8 - 6 b^2 c^8 + 2 c^10 : :

X(31283) lies on these lines: {2, 3}, {125, 12161}, {143, 26958}, {155, 13561}, {156, 1853}, {599, 9972}, {1147, 32767}, {3167, 18356}, {3567, 15059}, {5050, 8254}, {5448, 23329}, {5449, 16266}, {5965, 8548}, {6697, 19139}, {8252, 11266}, {8253, 11265}, {9820, 23332}, {12038, 23325}, {14644, 25487}, {15806, 19347}, {18445, 23294}, {18952, 23292}, {20299, 32139}, {22051, 32334}

Collineation mappings involving Gemini triangle 110: X(31284)-X(31289)

m(X) = 2 x + y + z : : = complement(complement(X))

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 25, 2019)

See the preamble just before X(6666).

X(31284) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics 2 a^4 - 2 a^3 b - a b^3 + b^4 - 2 a^3 c - b^3 c - a c^3 - b c^3 + c^4 : :

X(31284) lies on these lines: {2, 41}, {101, 17062}, {766, 6679}, {1125, 2809}, {1930, 4070}, {2280, 28734}, {2389, 6690}, {3589, 6691}, {4136, 30108}, {4258, 30825}, {6666, 6675}, {6677, 6685}, {7807, 17353}, {9317, 27068}, {16888, 25582}, {17050, 26007}, {20269, 24333}, {24318, 24784}, {26085, 30816}

X(31284) = complement of X(17046)

X(31285) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics 2 a^2 - 6 a b + b^2 - 6 a c - 4 b c + c^2 : :

X(31285) lies on these lines: {2, 45}, {8, 17243}, {37, 4395}, {320, 17245}, {518, 1125}, {524, 3707}, {528, 25352}, {536, 31211}, {597, 16831}, {1213, 17263}, {1654, 31311}, {1698, 4026}, {2325, 3739}, {3008, 4755}, {3244, 17348}, {3634, 28580}, {3731, 7263}, {3943, 16815}, {3986, 17356}, {4029, 28309}, {4357, 31243}, {4384, 4405}, {4399, 4727}, {4480, 16814}, {4665, 4873}, {4675, 28333}, {4687, 17045}, {4690, 29600}, {4691, 17229}, {4708, 20582}, {4725, 29606}, {4751, 17340}, {4969, 17277}, {5241, 27757}, {5296, 17265}, {6329, 28639}, {6707, 17353}, {7227, 25101}, {9041, 24331}, {15668, 18230}, {17244, 17330}, {17251, 29627}, {17256, 29626}, {17316, 28337}, {17334, 27147}, {17335, 17392}, {17338, 17398}, {17346, 29599}, {17366, 27268}, {17385, 25354}, {17395, 29628}, {28604, 31333}

X(31285) = complement of X(34824)
X(31285) = complement of complement of X(45)

X(31286) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics (b - c) (2 a^2 - a b - a c + b c) : :

X(31286) lies on these lines: {1, 24666}, {2, 649}, {10, 7234}, {43, 23655}, {101, 4998}, {241, 514}, {333, 18200}, {512, 31288}, {513, 6687}, {522, 4874}, {659, 24720}, {667, 17072}, {788, 6685}, {802, 17066}, {812, 4394}, {824, 17069}, {918, 2487}, {1125, 4507}, {1376, 23865}, {1447, 21195}, {1575, 9294}, {1613, 23575}, {1635, 24924}, {1639, 4897}, {1698, 25637}, {2516, 4762}, {2520, 10006}, {2786, 3239}, {3249, 27091}, {3589, 9002}, {3667, 3716}, {3733, 20316}, {3837, 4782}, {3840, 24756}, {4025, 26248}, {4063, 26114}, {4106, 31250}, {4147, 4367}, {4359, 20909}, {4468, 10196}, {4521, 5745}, {4598, 8709}, {4751, 27485}, {6006, 6666}, {6050, 29051}, {6139, 15283}, {6373, 25142}, {6679, 9313}, {6684, 28292}, {6710, 22102}, {8639, 28255}, {8640, 25537}, {8643, 21302}, {10164, 15599}, {14296, 20907}, {14425, 28851}, {14433, 29628}, {16468, 23568}, {16569, 24749}, {17352, 21143}, {17353, 21211}, {17761, 24192}, {18197, 27014}, {19804, 20952}, {20979, 28758}, {21260, 25126}, {21343, 24768}, {21348, 23886}, {21894, 29512}, {22224, 23572}, {25577, 28743}, {27195, 32016}

X(31286) = complement of X(3835)

X(31287) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics (b - c) (3 a^2 - 3 a b - 3 a c + 2 b c) : :

X(31287) lies on these lines: {2, 650}, {4, 8142}, {140, 8760}, {513, 6687}, {514, 2490}, {523, 14341}, {652, 25955}, {812, 2516}, {905, 3762}, {918, 4521}, {1125, 14077}, {1376, 8641}, {1635, 4106}, {1638, 4468}, {1639, 4025}, {1938, 3812}, {2487, 28846}, {2886, 15283}, {3035, 10006}, {3085, 30235}, {3239, 17069}, {3452, 23806}, {3589, 9001}, {3634, 29066}, {3752, 25098}, {3776, 10196}, {3798, 14321}, {3816, 15280}, {3835, 4394}, {3848, 9443}, {4147, 24756}, {4380, 27138}, {4383, 22383}, {4411, 31238}, {4698, 4777}, {4790, 27013}, {4791, 14838}, {4928, 23813}, {5432, 11934}, {6050, 21260}, {6182, 6690}, {6586, 24782}, {6589, 25084}, {6666, 23808}, {6691, 9373}, {7626, 7834}, {8678, 31288}, {9010, 25142}, {9034, 14298}, {11068, 14425}, {17278, 24793}, {19804, 21438}, {21212, 30520}, {21894, 27045}, {24620, 25271}, {28743, 30610}

X(31287) = complement of X(4885)

X(31288) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics (b - c) (2 a^3 - a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

X(31288) lies on these lines: {2, 667}, {140, 3309}, {468, 10006}, {512, 31286}, {513, 6681}, {659, 23815}, {663, 31207}, {784, 4874}, {1125, 4083}, {1946, 15283}, {1960, 17072}, {2490, 29288}, {3239, 29090}, {3589, 9010}, {3624, 4063}, {3669, 5433}, {3837, 4401}, {4162, 5432}, {4391, 14419}, {4448, 4905}, {4782, 19862}, {4885, 6050}, {4999, 20317}, {6004, 25380}, {6161, 17566}, {6675, 29150}, {6691, 19947}, {6710, 9320}, {8637, 25537}, {8678, 31287}, {14837, 29094}, {21188, 29102}

X(31288) = complement of X(21260)

X(31289) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics 2 a^3 + b^3 - 4 a b c + c^3 : :

X(31289) lies on these lines: {1, 17263}, {2, 31}, {10, 1279}, {44, 5257}, {140, 15310}, {142, 4672}, {210, 29672}, {344, 32921}, {513, 6681}, {516, 19512}, {518, 1125}, {551, 4864}, {726, 4422}, {740, 3008}, {765, 3582}, {899, 24542}, {902, 24988}, {966, 16786}, {984, 17338}, {993, 19267}, {1001, 4085}, {1086, 17767}, {1463, 5433}, {1471, 28741}, {1698, 4894}, {1738, 4432}, {1757, 3624}, {2325, 28516}, {3011, 24003}, {3246, 3634}, {3305, 26128}, {3681, 29853}, {3683, 24169}, {3685, 29607}, {3739, 24295}, {3740, 29656}, {3773, 16825}, {3775, 17277}, {3821, 15254}, {3834, 6693}, {3912, 4974}, {3923, 17278}, {3932, 17769}, {3993, 17366}, {4011, 24789}, {4361, 4527}, {4383, 29642}, {4395, 28522}, {4423, 25453}, {4429, 15485}, {4438, 5272}, {4439, 32922}, {4655, 17282}, {4684, 4753}, {4687, 29646}, {4716, 29590}, {4759, 17768}, {4991, 17390}, {5233, 29858}, {5278, 29677}, {5284, 29850}, {5741, 29869}, {6686, 6690}, {7292, 33115}, {8167, 29635}, {10200, 23693}, {16468, 17234}, {16477, 17300}, {16801, 19856}, {16823, 33159}, {17067, 28526}, {17248, 25539}, {17266, 32846}, {17283, 33082}, {17291, 24697}, {17341, 29674}, {17349, 33087}, {17353, 24325}, {17384, 25354}, {19847, 26979}, {25531, 33140}, {26685, 32935}, {26688, 33127}, {26724, 32930}, {27065, 33123}, {27191, 32857}, {28566, 31253}, {29658, 30829}, {29820, 33118}, {29851, 32911}

X(31289) = complement of X(3836)

Collineation mappings involving Gemini triangle 111: X(31290)-X(31305)

m(X) = 3 x - y - z : : = anticomplement(anticomplement(X))

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 25, 2019)

See the preambles just before X(6666) and X(31281).

X(31290) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics (b - c) (a^2 + 3 a b + 3 a c + b c) : :

X(31290) lies on these lines: {2, 3572}, {145, 4160}, {193, 9013}, {513, 17494}, {514, 20295}, {523, 14779}, {1654, 28209}, {3805, 20068}, {4024, 4382}, {4468, 4778}, {4560, 15309}, {5134, 5195}, {4977, 18004}, {5466, 6625}, {6994, 17926}, {8774, 20078}, {9256, 20064}, {17154, 21220}, {20974, 26846}, {20982, 26845}, {26853, 31301}, {24719, 28195}, {25666, 27167}, {27647, 29402}

X(31290) = anticomplement of X(7192)
X(31290) = anticomplementary conjugate of anticomplement of X(4557)

X(31291) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics (b - c) (-3 a^3 - a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

X(31291) lies on these lines: {2, 667}, {8, 4063}, {20, 3309}, {145, 4083}, {193, 9010}, {390, 4162}, {512, 14712}, {513, 17496}, {649, 28470}, {830, 4560}, {1912, 20109}, {3600, 3669}, {3617, 4782}, {3622, 24719}, {3803, 4391}, {3900, 4380}, {4198, 17924}, {4293, 4905}, {4705, 26777}, {6050, 27115}, {6161, 15680}, {6995, 18344}, {8655, 27293}, {8656, 27138}, {8678, 17494}, {9320, 20096}, {21003, 26778}, {21645, 31383}

X(31291) = anticomplement of X(21301)

X(31292) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^6 + 5 a^5 b + a^4 b^2 - 2 a^3 b^3 - 3 a^2 b^4 - 3 a b^5 - b^6 + 5 a^5 c + 5 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 3 a b^4 c - 3 b^5 c + a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 + 6 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 + 6 a b^2 c^3 + 6 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 + b^2 c^4 - 3 a c^5 - 3 b c^5 - c^6 : :

X(31292) lies on these lines: {2, 3}, {1278, 8680}, {1762, 9536}, {3187, 20077}, {17220, 17483}, {20059, 20086}, {25361, 31334}

X(31292) = anticomplement of X(3151)

X(31293) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^7 + 3 a^6 b + a^5 b^2 + a^4 b^3 - 3 a^3 b^4 - 3 a^2 b^5 - a b^6 - b^7 + 3 a^6 c + 5 a^5 b c + a^4 b^2 c - 2 a^3 b^3 c - 3 a^2 b^4 c - 3 a b^5 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 + a^4 c^3 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 + 6 a b^3 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 - 3 a^2 c^5 - 3 a b c^5 + b^2 c^5 - a c^6 - b c^6 - c^7 : :

X(31293) lies on these lines: {2, 3}, {8, 9536}, {145, 20061}, {1891, 3101}, {2321, 5279}, {2838, 20097}, {3704, 17784}, {3743, 4294}, {3868, 20086}, {4292, 5262}, {4313, 9539}, {9965, 20077}

X(31293) = anticomplement of anticomplement of X(28)

X(31294) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^7 - 2 a^6 b - 4 a^5 b^2 + 3 a^4 b^3 - a^3 b^4 + 2 a b^6 - b^7 - 2 a^6 c - 5 a^5 b c - 2 a^4 b^2 c + 2 a^3 b^3 c + 2 a^2 b^4 c + 3 a b^5 c + 2 b^6 c - 4 a^5 c^2 - 2 a^4 b c^2 + 6 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 2 a b^4 c^2 + 4 b^5 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 - 5 b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - 2 a b^2 c^4 - 5 b^3 c^4 + 3 a b c^5 + 4 b^2 c^5 + 2 a c^6 + 2 b c^6 - c^7 : :

X(31294) lies on these lines: {2, 3}, {962, 2816}, {3621, 20110}, {6385, 17233}, {12164, 12754}, {12649, 20077}, {20008, 20086}, {20082, 20089}

X(31294) = anticomplement of X(3152)

X(31295) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 5 a^4 - 2 a^2 b^2 - 3 b^4 + 2 a^2 b c + 2 a b^2 c - 2 a^2 c^2 + 2 a b c^2 + 6 b^2 c^2 - 3 c^4 : :

X(31295) lies on these lines: {2, 3}, {8, 1770}, {10, 4333}, {100, 5229}, {145, 4295}, {149, 3600}, {388, 20075}, {518, 1278}, {519, 4338}, {528, 9657}, {1478, 8715}, {1479, 10586}, {2996, 17001}, {3218, 5175}, {3434, 7354}, {3436, 12943}, {3474, 5086}, {3476, 3623}, {3486, 20292}, {3487, 11015}, {3585, 5552}, {3622, 4305}, {4292, 12649}, {4293, 10529}, {4294, 10587}, {4299, 10527}, {4313, 31019}, {4316, 26363}, {4317, 11240}, {4324, 10198}, {4678, 5176}, {5217, 10585}, {5225, 5253}, {5250, 28150}, {5258, 10483}, {5261, 11501}, {5270, 11239}, {5302, 18231}, {5554, 5691}, {5708, 12690}, {5840, 10532}, {5905, 9579}, {9613, 12648}, {10200, 18514}, {10599, 26285}, {10894, 24466}, {11415, 17647}, {12437, 31164}, {14927, 15988}, {17484, 20007}, {17784, 20060}, {18135, 32826}, {18513, 26364}, {19860, 28164}, {20019, 20086}

X(31295) = anticomplement of X(6872)

X(31296) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics (b^2 - c^2) (-a^4 + a^2 b^2 + a^2 c^2 + b^2 c^2) : :
Barycentrics csc A sin(B - C) - csc B sin(C - A) - csc C sin(A - B) : :

The trilinear polar of X(31296) passes through X(7668) and the complement of X(1625).

X(31296) lies on these lines: {2, 647}, {20, 30209}, {23, 385}, {69, 9030}, {75, 16751}, {110, 9514}, {148, 14731}, {193, 8675}, {194, 3906}, {323, 401}, {512, 14712}, {671, 10562}, {804, 3005}, {826, 5027}, {879, 11003}, {1021, 3187}, {2394, 7578}, {2525, 3268}, {2966, 23357}, {3164, 7712}, {3448, 14721}, {3569, 13306}, {3700, 3995}, {4108, 8651}, {4524, 19998}, {4560, 17161}, {6753, 14618}, {7473, 14611}, {7797, 8574}, {7833, 10097}, {7927, 14318}, {9168, 31088}, {9213, 31125}, {9404, 19742}, {9979, 12077}, {11127, 23870}, {14570, 14999}, {17414, 23301}, {18155, 28606}, {21053, 27730}, {21828, 26983}

X(31296) = isotomic conjugate of X(11794)
X(31296) = anticomplement of X(850)
X(31296) = pole of Brocard axis wrt Steiner circumellipse
X(31296) = crosssum of Kiepert hyperbola intercepts of Lemoine axis
X(31296) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(34235)

X(31297) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^5 - 3 a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 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 - c^5 : :

X(31297) lies on these lines: {2, 662}, {7, 4393}, {1030, 1654}, {1278, 20074}, {2640, 23943}, {2786, 20094}, {3210, 20086}, {4299, 20077}, {5698, 15680}, {6542, 16548}, {8674, 14683}, {17467, 24711}, {17740, 24616}, {19642, 26141}, {19785, 21907}

X(31297) = anticomplement of X(21221)

X(31298) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics a^2 b^2 - 5 a^2 b c + 3 a b^2 c + a^2 c^2 + 3 a b c^2 - 3 b^2 c^2 : :

X(31298) lies on these lines: {2, 668}, {145, 17794}, {291, 3617}, {330, 25278}, {537, 1278}, {2276, 21226}, {2787, 20094}, {2810, 20080}, {3621, 14839}, {3622, 17793}

X(31298) = anticomplement of X(9263)

X(31299) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics (b^2 - c^2) (-3 a^4 - a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(31299) lies on these lines: {2, 669}, {20, 1499}, {193, 9009}, {512, 14712}, {523, 20063}, {804, 8664}, {850, 32472}, {2501, 6995}, {3523, 5926}, {5996, 8651}, {14824, 33260}, {17135, 18197}, {18106, 26846}, {20088, 23099}, {21646, 31383}

X(31300) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^2 - a b - b^2 - a c + 3 b c - c^2 : :

X(31300) lies on these lines: {2, 7}, {6, 4398}, {8, 17770}, {20, 29369}, {44, 7321}, {75, 20072}, {86, 17334}, {145, 726}, {190, 17243}, {192, 4644}, {193, 742}, {319, 4715}, {320, 17231}, {346, 17375}, {391, 4772}, {545, 4360}, {594, 28333}, {732, 30662}, {903, 17366}, {983, 17126}, {1100, 4912}, {1266, 17121}, {1268, 10022}, {1654, 4363}, {1743, 29590}, {1964, 24722}, {2325, 17312}, {2345, 4741}, {3009, 7240}, {3161, 29572}, {3617, 33082}, {3621, 5847}, {3629, 17160}, {3663, 17120}, {3664, 4480}, {3723, 4796}, {3729, 6542}, {3758, 17276}, {3927, 26051}, {3945, 4704}, {3973, 29628}, {4098, 29625}, {4307, 31302}, {4388, 32940}, {4416, 17116}, {4419, 17379}, {4461, 20055}, {4464, 28301}, {4473, 17234}, {4488, 17316}, {4499, 21746}, {4643, 28604}, {4645, 32935}, {4649, 17767}, {4659, 17363}, {4667, 17319}, {4670, 17258}, {4675, 17336}, {4699, 7222}, {4740, 5839}, {4795, 17394}, {4860, 26139}, {4862, 17367}, {4888, 17244}, {4896, 25101}, {5263, 5852}, {5843, 13727}, {5845, 6653}, {7227, 17271}, {7229, 29593}, {7232, 17354}, {7238, 17283}, {13610, 20351}, {16816, 31995}, {17118, 17346}, {17141, 20088}, {17165, 20056}, {17230, 21296}, {17247, 29586}, {17255, 17381}, {17262, 17378}, {17264, 17376}, {17272, 29591}, {17273, 17369}, {17281, 17361}, {17288, 17355}, {17289, 17345}, {17297, 17340}, {17298, 17339}, {17303, 17329}, {17331, 25590}, {17491, 33170}, {17778, 32933}, {24349, 24695}, {24451, 25291}, {26070, 30834}, {26076, 27095}, {28558, 33076}, {29583, 32093}, {29837, 33099}

X(31300) = anticomplement of X(6646)

X(31301) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 6 a^3 + a^2 b - 3 a b^2 - 2 b^3 + a^2 c + b^2 c - 3 a c^2 + b c^2 - 2 c^3 : :

X(31301) lies on these lines: {2, 896}, {145, 758}, {513, 4380}, {527, 20045}, {2796, 17162}, {4062, 4427}, {4442, 4831}, {6361, 20047}, {9791, 26860}, {17960, 24200}, {21282, 28508}, {24280, 31303}, {30579, 32842}

X(31301) = anticomplement of X(17491)

X(31302) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics a^2 b - 3 a b^2 + a^2 c - a b c + b^2 c - 3 a c^2 + b c^2 : :

X(31302) lies on these lines: {1, 4704}, {2, 38}, {8, 726}, {10, 4772}, {37, 3622}, {72, 20036}, {75, 3617}, {144, 145}, {190, 3242}, {210, 17490}, {239, 5223}, {329, 29840}, {341, 20892}, {536, 31145}, {740, 3621}, {894, 7174}, {960, 17480}, {986, 25277}, {1001, 24841}, {1266, 24393}, {1279, 17336}, {1463, 25279}, {2292, 25295}, {2550, 4440}, {3146, 29054}, {3210, 3681}, {3241, 3993}, {3416, 4741}, {3632, 28522}, {3644, 20054}, {3662, 3717}, {3663, 4899}, {3670, 26029}, {3685, 16496}, {3696, 4740}, {3699, 17595}, {3728, 17164}, {3751, 4393}, {3790, 17230}, {3797, 29616}, {3840, 4903}, {3883, 17333}, {3923, 24821}, {3932, 17232}, {3976, 22220}, {3995, 4430}, {3999, 30829}, {4009, 30861}, {4022, 27291}, {4078, 29572}, {4090, 17591}, {4307, 31300}, {4353, 17367}, {4358, 30948}, {4417, 4884}, {4419, 9055}, {4439, 33087}, {4656, 29843}, {4661, 17147}, {4678, 4821}, {4684, 17242}, {4901, 17274}, {5211, 31018}, {5220, 17349}, {5542, 17244}, {5846, 17347}, {5850, 17364}, {5904, 20018}, {9053, 17334}, {10327, 26840}, {10449, 24068}, {10453, 32925}, {10477, 31036}, {12782, 26778}, {15590, 18230}, {15680, 20035}, {17276, 32850}, {17316, 27481}, {17449, 30947}, {17459, 24528}, {17484, 29832}, {17751, 21080}, {18743, 21342}, {20020, 20078}, {20052, 28516}, {21093, 29676}, {24165, 26038}, {24599, 27484}, {26065, 29838}, {29610, 31347}, {30615, 33068}

X(31302) = anticomplement of X(24349)

X(31303) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^3 + 2 a^2 b - 2 a b^2 - b^3 + 2 a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 - c^3 : :

X(31303) lies on these lines: {2, 6}, {20, 952}, {63, 20017}, {752, 21283}, {991, 3935}, {3187, 3663}, {3218, 17363}, {3219, 17242}, {3617, 4340}, {3686, 26627}, {4641, 17229}, {4644, 31025}, {4671, 20072}, {4741, 33155}, {4888, 5271}, {5839, 17495}, {5847, 29832}, {6327, 32853}, {8229, 11898}, {10446, 17484}, {17021, 17331}, {17135, 20064}, {17147, 20046}, {17162, 24248}, {17272, 29833}, {17360, 32779}, {17361, 33129}, {17373, 32849}, {20078, 29069}, {20290, 33137}, {24217, 32919}, {24280, 31301}, {29829, 33082}

X(31303) = anticomplement of X(31034)

X(31304) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^10 - 5 a^8 b^2 - 2 a^6 b^4 + 6 a^4 b^6 - a^2 b^8 - b^10 - 5 a^8 c^2 + 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 + 6 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(31304) lies on these lines: {2, 3}, {49, 31815}, {193, 32354}, {576, 10619}, {1204, 29012}, {1238, 32840}, {1352, 7691}, {1994, 18925}, {3060, 19467}, {3581, 32140}, {4294, 9539}, {4549, 15058}, {4846, 8718}, {5422, 11745}, {5654, 26882}, {5889, 9833}, {6800, 12233}, {9544, 32379}, {11750, 18912}, {12111, 31383}, {12163, 16659}, {12278, 15107}, {14516, 17834}, {17845, 33586}, {18382, 32345}, {20806, 29181}, {27082, 28708}

X(31304) = anticomplement of X(37444)
X(31304) = anticomplement of anticomplement of X(24)
X(31304) = anticomplement of isogonal conjugate of X(34438)

X(31305) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics 3 a^10 - 5 a^8 b^2 - 2 a^6 b^4 + 6 a^4 b^6 - a^2 b^8 - b^10 - 5 a^8 c^2 - 4 a^6 b^2 c^2 + 2 a^4 b^4 c^2 + 4 a^2 b^6 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 + 6 a^4 c^6 + 4 a^2 b^2 c^6 - 2 b^4 c^6 - a^2 c^8 + 3 b^2 c^8 - c^10 : :

X(31305) lies on these lines: {2, 3}, {52, 6776}, {69, 12134}, {155, 11206}, {193, 6243}, {311, 3785}, {390, 8144}, {393, 10316}, {497, 9645}, {511, 5596}, {569, 1176}, {578, 31670}, {1216, 14826}, {1351, 19119}, {1352, 13419}, {1503, 11411}, {2931, 13203}, {3600, 32047}, {5032, 11255}, {5562, 31383}, {5891, 11821}, {6146, 33586}, {7585, 11265}, {7586, 11266}, {8141, 17784}, {8745, 23115}, {10117, 12319}, {11426, 21850}, {11750, 18945}, {12163, 12324}, {12250, 14915}, {12359, 32064}, {13346, 29317}, {13353, 19154}, {14216, 29012}, {14927, 18909}, {19149, 29181}, {19347, 31802}

X(31305) = anticomplement of X(14790)

Perspectors involving Gemini triangles 1 to 111: X(31306)-X(31352)

This preamble and centers X(31306)-X(31352) were contributed by César Eliud Lozada, January 26, 2019.

The perspector of Gemini triangles i and j is X(2) for i, j ∈ {1, 2, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111}

Also, the appearance of (i, j, k) in the following list means that the perspector of Gemini triangles i and j is X(k):
(1, 8, 21), (1, 25, 1125), (1, 29, 1), (1, 63, 31306), (2, 7, 81), (2, 15, 28606), (2, 30, 63), (3, 4, 86), (3, 5, 2), (3, 6, 1), (3, 7, 1), (3, 16, 2), (3, 18, 2), (3, 19, 86), (3, 23, 1125), (3, 25, 86), (3, 26, 2), (3, 40, 3616), (3, 62, 31307), (3, 63, 31308), (3, 73, 31309), (3, 104, 3720), (4, 5, 9), (4, 6, 192), (4, 8, 9), (4, 17, 192), (4, 19, 86), (4, 25, 86), (4, 63, 31310), (4, 107, 190), (4, 109, 31311), (4, 110, 31312), (4, 111, 31313), (5, 8, 9), (5, 16, 2), (5, 18, 2), (5, 26, 2), (5, 110, 1), (6, 7, 1), (6, 16, 8), (6, 17, 192), (6, 23, 10), (6, 40, 8), (6, 63, 31314), (6, 64, 31315), (6, 104, 42), (7, 15, 2), (7, 16, 24174), (7, 17, 2), (7, 21, 24174), (7, 23, 24161), (7, 25, 2), (7, 30, 88), (7, 60, 330), (7, 110, 4859), (8, 18, 8), (8, 29, 1320), (8, 71, 31316), (9, 63, 31317), (11, 15, 31318), (11, 63, 31319), (12, 15, 31320), (12, 26, 31321), (13, 25, 1), (13, 29, 8), (13, 63, 31322), (14, 63, 31323), (15, 17, 2), (15, 19, 37), (15, 25, 2), (15, 60, 192), (15, 61, 31324), (15, 65, 31325), (15, 108, 31326), (16, 18, 2), (16, 21, 24174), (16, 22, 31327), (16, 23, 1), (16, 26, 2), (16, 40, 8), (16, 62, 31328), (16, 63, 31329), (16, 104, 31330), (17, 25, 2), (17, 105, 3995), (17, 111, 145), (18, 26, 2), (18, 111, 1278), (19, 25, 86), (19, 63, 31331), (19, 107, 31332), (19, 109, 31333), (19, 110, 142), (19, 111, 31334), (20, 63, 29576), (23, 63, 31335), (25, 29, 10), (25, 63, 31336), (26, 32, 5224), (26, 39, 9780), (26, 57, 31337), (26, 68, 31338), (26, 80, 31339), (26, 81, 31340), (26, 104, 31341), (28, 63, 239), (29, 63, 31342), (29, 71, 31343), (29, 111, 20059), (30, 100, 20348), (30, 111, 3621), (31, 63, 31344), (32, 62, 31345), (38, 63, 31346), (40, 63, 31347), (62, 64, 2), (62, 100, 27424), (63, 104, 31348), (63, 107, 31349), (63, 109, 31350), (63, 110, 31351), (63, 111, 31352)

X(31306) = PERSPECTOR OF THESE TRIANGLES: GEMINI 1 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+2*(b^2+4*b*c+c^2)*a^2+(b+c)*(b^2+4*b*c+c^2)*a+2*(b^2+b*c+c^2)*b*c : :
X(31306) = 3*X(2)+X(31314) = X(319)-7*X(4751) = X(1100)+2*X(3739) = 2*X(17239)-5*X(31238)

X(31306) lies on these lines: {1,27474}, {2,210}, {37,17339}, {75,4470}, {86,239}, {319,4648}, {335,29614}, {872,28254}, {984,29603}, {1125,17755}, {3666,31348}, {3696,4393}, {3775,24603}, {3797,15569}, {4359,18157}, {4384,4649}, {4670,14621}, {4698,29609}, {4699,17014}, {4725,31351}, {4755,31349}, {4883,31027}, {5045,27274}, {10436,20172}, {17023,24325}, {17234,17239}, {17348,20159}, {17769,29574}, {19684,27476}, {28581,29584}

X(31306) = midpoint of X(27495) and X(31314)
X(31306) = complement of X(27495)
X(31306) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27474, 31342), (2, 27484, 31322), (2, 31314, 27495)

X(31307) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 62

Barycentrics
((b+c)^3*a^7+(2*b^2-b*c+2*c^2)*(b+c)^2*a^6+3*(b^3+c^3)*(b^2+b*c+c^2)*a^5+(2*b^6+2*c^6+(b^4+c^4+(4*b^2+9*b*c+4*c^2)*b*c)*b*c)*a^4+(b+c)*(b^6+c^6+(5*b^2-6*b*c+5*c^2)*b^2*c^2)*a^3+(b^6+c^6+(b+c)^2*b^2*c^2)*b*c*a^2-(b^3+c^3)*(b^2+b*c+c^2)*b^2*c^2*a-(b^2+b*c+c^2)^2*b^3*c^3)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31307) lies on these lines: {24661,27444}, {24669,27429}

X(31308) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+(5*b^2+11*b*c+5*c^2)*a^2+(b+c)*(b^2+4*b*c+c^2)*a-(b^2+b*c+c^2)*b*c : :
X(31308) = 3*X(2)-4*X(31336) = 4*X(37)-X(1654) = 2*X(86)+X(192) = 4*X(1213)-7*X(27268) = 2*X(3993)+X(24342) = 5*X(4699)-8*X(6707) = 5*X(4704)+X(20090) = 4*X(5625)-X(24349)

X(31308) lies on these lines: {1,6651}, {2,740}, {37,319}, {75,28640}, {86,192}, {144,1959}, {524,17488}, {726,29580}, {984,29588}, {1001,4393}, {1125,31335}, {1213,17233}, {3616,31347}, {3661,25354}, {3720,31348}, {3797,15569}, {3993,16826}, {3995,19565}, {4664,4795}, {4699,6707}, {5625,24349}, {5880,6650}, {9791,17316}, {17293,31248}, {17302,27487}, {17499,24051}, {20016,31323}

X(31308) = reflection of X(i) in X(j) for these (i,j): (27483, 31336), (31310, 6651)
X(31308) = anticomplement of X(27483)
X(31308) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27481, 31314), (27474, 31319, 2), (27483, 31336, 2)

X(31309) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 73

Barycentrics
a*((b^6+c^6-(b^2-c^2)^2*b*c)*a^6+(b+c)*(2*b^2-3*b*c+2*c^2)*b^2*c^2*a^5-(b^2-c^2)^2*b^2*c^2*a^4+(b+c)*(3*b^2-5*b*c+3*c^2)*b^3*c^3*a^3-(b^2-4*b*c+c^2)*b^4*c^4*a^2+(b^2-c^2)*(b-c)*b^4*c^4*a-(b-c)^2*b^5*c^5) : :

X(31309) lies on these lines: {86,24662}, {24656,27880}

X(31310) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 63

Barycentrics
3*(b+c)*a^5+(4*b^2+13*b*c+4*c^2)*a^4-(b+c)*(5*b^2-4*b*c+5*c^2)*a^3-(4*b^4+4*c^4+(11*b^2+12*b*c+11*c^2)*b*c)*a^2-(b+c)*(b^2+b*c+c^2)*(b^2-3*b*c+c^2)*a+(b^2+b*c+c^2)*(b^2+5*b*c+c^2)*b*c : :
X(31310) = X(6650)-4*X(17755)

X(31310) lies on these lines: {1,6651}, {86,27949}, {190,20142}, {335,31336}, {740,31349}, {5698,27484}, {6650,17755}, {20158,27480}

X(31310) = reflection of X(i) in X(j) for these (i,j): (335, 31336), (6650, 27483), (27483, 17755), (31308, 6651)

X(31311) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 109

Barycentrics 3*a^2-7*(b+c)*a-5*b*c : :

X(31311) lies on these lines: {1,872}, {2,7232}, {10,31333}, {45,4772}, {86,16669}, {190,3739}, {192,16675}, {239,31350}, {1268,4422}, {1654,31285}, {2476,3826}, {3731,4764}, {3758,31312}, {4360,16674}, {4678,17233}, {4686,16815}

X(31312) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 110

Barycentrics 3*a^2+7*(b+c)*a+8*b*c : :
X(31312) = 3*X(2)+X(30712) = 5*X(1698)-2*X(15593)

X(31312) lies on these lines: {1,3696}, {2,1743}, {86,16667}, {142,3624}, {165,24220}, {190,3731}, {192,16673}, {239,31313}, {1125,4779}, {1418,25086}, {1449,31238}, {1698,4648}, {2999,5333}, {3247,4686}, {3634,4869}, {3636,4402}, {3729,29578}, {3758,31311}, {3946,28641}, {3973,4670}, {4000,25055}, {4364,4902}, {4677,17390}, {4699,29597}, {4751,16833}, {4772,16826}, {4816,4916}, {4821,29595}, {4888,5257}, {4898,29624}, {4967,29602}, {5234,21246}, {6707,17306}, {8056,10455}, {17175,18186}, {17296,19875}, {17304,29612}, {17313,19876}, {19701,23511}

X(31312) = complement of the complement of X(30712)
X(31312) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (86, 16832, 16667), (16831, 25590, 16673)

X(31313) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 111

Barycentrics 8*a^2+7*(b+c)*a+3*b*c : :
X(31313) = 3*X(2)-4*X(30598)

X(31313) lies on these lines: {1,1278}, {2,319}, {9,29570}, {86,4772}, {190,4704}, {239,31312}, {1386,20080}, {1449,29595}, {3616,17343}, {3622,5625}, {3636,17364}, {4670,4788}, {4821,17393}, {4909,17397}, {5698,11038}, {14996,21769}, {16667,16826}, {16669,27268}, {16673,17350}, {17018,24661}, {17300,26104}, {17349,31311}, {17375,29586}

X(31313) = anticomplement of the anticomplement of X(30598)
X(31313) = {X(25417), X(30562)}-harmonic conjugate of X(2)

X(31314) = PERSPECTOR OF THESE TRIANGLES: GEMINI 6 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+(b^2+7*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2+b*c+c^2)*b*c : :
X(31314) = 3*X(2)-4*X(31306) = X(192)-4*X(1100) = 2*X(319)-5*X(4699) = 2*X(4649)+X(24349)

X(31314) lies on these lines: {1,6651}, {2,210}, {8,31329}, {10,31335}, {42,31348}, {75,20016}, {192,1100}, {239,27478}, {319,4675}, {726,29584}, {984,29586}, {3797,29588}, {3807,24512}, {4393,4649}, {4740,4795}, {6542,27474}, {17029,31063}, {17300,27487}, {17755,29569}, {17778,27476}, {20072,24357}, {24325,27483}, {29592,31323}

X(31314) = reflection of X(27495) in X(31306)
X(31314) = anticomplement of X(27495)
X(31314) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27481, 31308), (27495, 31306, 2)

X(31315) = PERSPECTOR OF THESE TRIANGLES: GEMINI 6 AND GEMINI 64

Barycentrics
((b-c)*a-b*c)*((b-c)*a+b*c)*((b+c)^3*a^7-9*(b+c)^2*b*c*a^6-(b+c)*(b^4+c^4-(6*b^2+23*b*c+6*c^2)*b*c)*a^5+(2*b^6+2*c^6-(9*b^4+9*c^4-(10*b^2-67*b*c+10*c^2)*b*c)*b*c)*a^4-(b+c)*(b^6+c^6-(10*b^4+10*c^4-(35*b^2-64*b*c+35*c^2)*b*c)*b*c)*a^3-(b^6+c^6+(2*b^4+2*c^4-3*(11*b^2-18*b*c+11*c^2)*b*c)*b*c)*b*c*a^2+(b+c)*(b^4+c^4-(10*b^2-13*b*c+10*c^2)*b*c)*b^2*c^2*a+(b^4+b^2*c^2+c^4)*b^3*c^3) : :

X(31315) lies on these lines: {}

X(31316) = PERSPECTOR OF THESE TRIANGLES: GEMINI 8 AND GEMINI 71

Barycentrics a*(a^2-3*(2*b-c)*a+b^2+3*b*c-2*c^2)*(a+b-3*c)*(a^2+3*(b-2*c)*a-2*b^2+3*b*c+c^2)*(a-3*b+c)*(-a+b+c) : :

X(31316) lies on the Feuerbach hyperbola and these lines: {1,27834}, {9,31343}, {2505,23836}

X(31316) = trilinear pole of the line {650, 3680}

X(31317) = PERSPECTOR OF THESE TRIANGLES: GEMINI 9 AND GEMINI 63

Barycentrics (b+c)*a^3+2*b*c*a^2+(b+c)*b*c*a+(b^2+b*c+c^2)*b*c : :
X(31317) = 5*X(4699)-X(4741)

X(31317) lies on these lines: {1,3797}, {2,38}, {6,75}, {7,1654}, {37,17339}, {56,27954}, {190,24357}, {192,5749}, {321,17027}, {354,31028}, {518,3661}, {726,17023}, {740,4393}, {871,3978}, {1001,6651}, {1278,17014}, {1757,4384}, {1921,3765}, {3008,27478}, {3252,19584}, {3662,3739}, {3666,16606}, {3696,28538}, {3873,31027}, {3923,4366}, {4372,17103}, {4645,24693}, {4675,27487}, {4687,29609}, {4688,4715}, {4692,30114}, {4732,31329}, {4751,17291}, {4850,31348}, {4968,17033}, {4974,20158}, {5220,16815}, {5695,20162}, {6542,27474}, {6645,16822}, {9055,17369}, {9318,27931}, {16584,28606}, {16825,20142}, {16975,31344}, {17048,31276}, {17141,26035}, {17251,31139}, {17266,27475}, {17308,27495}, {20678,26241}, {24046,27324}, {24231,24603}, {24327,27913}, {24331,27949}, {24346,27941}, {24514,26234}, {25253,26807}, {26240,27912}, {26561,30177}, {31041,31079}

X(31317) = midpoint of X(75) and X(3758)
X(31317) = reflection of X(17237) in X(3739)
X(31317) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24349, 335), (1215, 24631, 2), (17755, 24325, 2)

X(31318) = PERSPECTOR OF THESE TRIANGLES: GEMINI 11 AND GEMINI 15

Barycentrics a*((b+c)*a^2+(2*b^2+7*b*c+2*c^2)*a+(b+c)*(b^2+4*b*c+c^2)) : :

X(31318) lies on these lines: {1,210}, {2,3743}, {6,5506}, {35,1486}, {36,27802}, {37,3624}, {45,6763}, {192,6533}, {312,25512}, {474,4436}, {764,14349}, {975,5259}, {1125,3971}, {1203,5287}, {1698,4646}, {1962,17749}, {2650,5692}, {3338,3731}, {3616,3952}, {3666,31320}, {3670,25502}, {3678,29814}, {3720,5904}, {3746,5268}, {4011,25526}, {4673,19870}, {4687,19863}, {4850,19878}, {4868,19877}, {4975,19853}, {5054,8143}, {5284,30142}, {5312,15569}, {5563,25579}, {5697,22300}, {5747,24933}, {7294,26742}, {7611,19514}, {12047,29571}, {16296,20326}, {17263,19846}, {18398,26102}, {19862,28606}, {19883,27782}, {24945,25079}

X(31318) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 25430, 25431), (3646, 25430, 1)

X(31319) = PERSPECTOR OF THESE TRIANGLES: GEMINI 11 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+2*(b+2*c)*(2*b+c)*a^2+(b+2*c)*(2*b+c)*(b+c)*a+(b^2+b*c+c^2)*b*c : :
X(31319) = 5*X(4687)+X(17393)

X(31319) lies on these lines: {1,27495}, {2,740}, {37,17339}, {75,29609}, {192,31347}, {239,4687}, {335,3616}, {984,29586}, {1001,14621}, {1125,27478}, {3661,15569}, {3797,29603}, {3842,4393}, {4698,17233}, {5625,20145}, {16929,17733}, {18230,26626}, {24325,27494}, {28566,29622}, {28606,31348}, {31238,31351}

X(31319) = complement of X(31329)
X(31319) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 27480, 27483), (2, 31308, 27474)

X(31320) = PERSPECTOR OF THESE TRIANGLES: GEMINI 12 AND GEMINI 15

Barycentrics a*(3*(b+c)*a^2+(6*b^2+11*b*c+6*c^2)*a+(b+c)*(3*b^2+4*b*c+3*c^2)) : :

X(31320) lies on these lines: {1,3683}, {8,3743}, {10,27783}, {35,27802}, {37,1698}, {46,16673}, {79,2335}, {191,16777}, {594,31321}, {1089,4704}, {1125,24165}, {1486,3746}, {1962,5904}, {3666,31318}, {27268,28611}

X(31320) = {X(46), X(16673)}-harmonic conjugate of X(25431)

X(31321) = PERSPECTOR OF THESE TRIANGLES: GEMINI 12 AND GEMINI 26

Barycentrics 3*(b+c)*a^3+(8*b^2+15*b*c+8*c^2)*a^2+(b+c)*(7*b^2+12*b*c+7*c^2)*a+2*(b+c)^4 : :

X(31321) lies on these lines: {6,1698}, {10,4970}, {46,3929}, {594,31320}, {1330,9780}, {3828,27783}

X(31322) = PERSPECTOR OF THESE TRIANGLES: GEMINI 13 AND GEMINI 63

Barycentrics (4*b^2+7*b*c+4*c^2)*a^2+2*(b+c)*(b^2+4*b*c+c^2)*a+(b^2+4*b*c+c^2)*b*c : :
X(31322) = 5*X(4687)-2*X(16777)

X(31322) lies on these lines: {1,31336}, {2,210}, {8,31342}, {10,27474}, {37,27480}, {75,1213}, {239,4687}, {984,24603}, {1698,17755}, {3739,31347}, {3797,31329}, {3842,4384}, {3876,27156}, {4751,17291}, {4971,31350}, {5224,27487}, {14621,17335}, {16826,20156}, {16830,20154}, {16972,17277}, {17337,17397}, {19804,31348}, {31349,31351}

X(31322) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 27484, 31306), (2, 27495, 27475)

X(31323) = PERSPECTOR OF THESE TRIANGLES: GEMINI 14 AND GEMINI 63

Barycentrics (2*b^2+3*b*c+2*c^2)*a^2+(b+c)*(b^2+3*b*c+c^2)*a+b^2*c^2 : :

X(31323) lies on these lines: {1,16912}, {2,38}, {9,14621}, {10,3797}, {37,239}, {45,20172}, {75,1213}, {190,25384}, {192,5296}, {518,16826}, {726,24603}, {742,17256}, {1757,20132}, {3509,16993}, {3589,4687}, {3661,3932}, {3681,17032}, {3739,17305}, {3912,27495}, {4359,18152}, {4389,25357}, {4393,4974}, {4451,17038}, {4664,28309}, {4698,29609}, {4704,27480}, {4981,31027}, {5220,20131}, {5263,6651}, {7384,29054}, {15569,29584}, {17143,21816}, {17237,27487}, {18230,26626}, {20016,31308}, {20343,25145}, {24589,31348}, {27474,29593}, {27475,29581}, {29592,31314}, {31238,31335}

X(31323) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 984, 335), (3842, 17755, 2)

X(31324) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 61

Barycentrics
a*((b+c)*a^6-(2*b^2+7*b*c+2*c^2)*a^5-(b^3+c^3)*a^4+2*(b+2*c)*(2*b+c)*(b^2+c^2)*a^3-(b+c)*(b^4+c^4-2*(b^2+5*b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*(2*b^2+3*b*c+2*c^2)*a+(b^2-c^2)^2*(b+c)*(b^2-3*b*c+c^2)) : :

X(31324) lies on these lines: {1,2287}, {2,5831}, {37,5703}, {281,1895}, {938,966}, {2285,6986}, {3998,7229}, {10445,21617}, {13411,17355}, {20223,27404}

X(31325) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 65

Barycentrics a^5-3*(b+c)*a^4+2*(b^2+c^2)*a^3+2*(b+c)*(b^2-8*b*c+c^2)*a^2-3*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c) : :

X(31325) lies on these lines: {1,346}, {2,20270}, {37,6554}, {938,2321}, {1766,3600}, {2171,11036}, {2345,14986}, {3085,25081}, {3161,9369}, {3247,5703}, {3672,4552}, {3945,28968}, {4072,6744}, {8165,21074}, {9785,10445}, {9819,10443}, {13405,16673}

X(31326) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 108

Barycentrics 3*a^3-8*(b+c)*a^2-3*(3*b^2-2*b*c+3*c^2)*a+2*(b^2-c^2)*(b-c) : :

X(31326) lies on these lines: {1,3523}, {226,4862}, {1018,5437}, {1699,17593}, {3586,13442}, {3633,14829}, {3666,26742}, {3752,16601}, {4384,14949}, {4850,25080}, {4888,21454}, {5256,5483}, {6692,16673}, {8056,24181}, {17760,29598}

X(31327) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 22

Barycentrics (b^2+b*c+c^2)*a^2+(b+c)*(b^2+5*b*c+c^2)*a+2*(b+c)^2*b*c : :

X(31327) lies on these lines: {1,3696}, {8,2650}, {10,312}, {40,5788}, {72,3679}, {75,24214}, {304,4967}, {740,19853}, {764,4802}, {958,4436}, {982,24176}, {984,4647}, {986,31330}, {1046,4042}, {1698,4646}, {3617,3952}, {3702,26037}, {3741,24174}, {3976,4359}, {4197,21027}, {4714,10479}, {4732,9534}, {5082,20539}, {5255,5271}, {5263,16478}, {16819,27474}, {16828,22316}, {17063,28611}, {17592,19858}, {25079,26038}

X(31327) = {X(4714), X(10479)}-harmonic conjugate of X(24440)

X(31328) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 62

Barycentrics ((b^2+c^2)*(b+c)^3*a^5+(b^4+6*b^2*c^2+c^4)*b*c*a^4+2*(b+c)*(3*b^2-b*c+3*c^2)*b^2*c^2*a^3+4*b^4*c^4*a^2+(b+c)*b^4*c^4*a-b^5*c^5)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31328) lies on these lines: {3741,27436}, {17792,27429}, {25121,27444}, {25124,27447}

X(31329) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 63

Barycentrics (b^2+4*b*c+c^2)*a^2+2*(b^2+4*b*c+c^2)*(b+c)*a+(4*b^2+7*b*c+4*c^2)*b*c : :
X(31329) = 4*X(3739)-X(17393)

X(31329) lies on these lines: {1,31335}, {2,740}, {8,31314}, {10,27481}, {75,4377}, {1921,28605}, {2550,27484}, {3661,27478}, {3696,4393}, {3739,17393}, {3797,31322}, {4648,4699}, {4709,17397}, {4732,31317}, {17230,27475}, {20055,24325}, {20145,24342}, {29612,31351}, {31330,31348}

X(31329) = anticomplement of X(31319)
X(31329) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 31347, 31314), (27474, 27483, 2)

X(31330) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 104

Barycentrics (b^2+b*c+c^2)*a+b*c*(b+c) : :

X(31330) lies on these lines: {1,2}, {6,4042}, {9,3588}, {11,5743}, {31,333}, {37,3706}, {38,75}, {45,4387}, {55,5737}, {63,4418}, {141,3779}, {149,4368}, {171,1150}, {192,3989}, {238,5278}, {244,19804}, {274,17208}, {312,756}, {321,984}, {350,5224}, {354,3739}, {355,4192}, {497,966}, {517,30981}, {518,30969}, {594,2276}, {649,25128}, {672,2345}, {726,7226}, {740,28606}, {748,2209}, {750,14829}, {851,5794}, {908,4104}, {941,5257}, {956,11358}, {958,1011}, {960,1985}, {964,5247}, {968,3886}, {982,4359}, {986,31327}, {993,4184}, {1001,19732}, {1008,5015}, {1010,1468}, {1043,10448}, {1107,21877}, {1211,2886}, {1215,3681}, {1376,4191}, {1458,27339}, {1573,21838}, {1621,5235}, {1654,4388}, {1740,16738}, {1836,4643}, {1861,4196}, {2049,19714}, {2051,10886}, {2177,3996}, {2238,3966}, {2550,6817}, {2551,6818}, {2887,3775}, {2975,13588}, {3056,15985}, {3097,6539}, {3120,17794}, {3169,24392}, {3218,3980}, {3219,3923}, {3295,16345}, {3303,16355}, {3419,4199}, {3662,21027}, {3666,3696}, {3670,28612}, {3740,30818}, {3742,3846}, {3758,4722}, {3790,6535}, {3816,5241}, {3826,25961}, {3844,31005}, {3847,11680}, {3868,30984}, {3869,14009}, {3873,24325}, {3878,14008}, {3896,17592}, {3911,16878}, {3914,4357}, {3944,26580}, {3971,4671}, {4008,11031}, {4011,27065}, {4041,21259}, {4083,30968}, {4113,4849}, {4147,4893}, {4210,25440}, {4279,27631}, {4307,14552}, {4361,17599}, {4363,24690}, {4364,4854}, {4379,17072}, {4392,24165}, {4423,17259}, {4479,17250}, {4524,30864}, {4649,19684}, {4655,20292}, {4660,24259}, {4665,4884}, {4688,21342}, {4698,4891}, {4703,5057}, {4709,4970}, {4713,17251}, {4716,17600}, {4732,4850}, {4733,24643}, {4751,17450}, {4804,24718}, {5044,25591}, {5232,17220}, {5258,11322}, {5361,17126}, {5739,26098}, {5741,17717}, {5790,19540}, {5791,8731}, {5793,16405}, {5835,21677}, {5836,30960}, {6376,18152}, {6382,19562}, {6384,7148}, {6536,17248}, {7396,18659}, {8692,19751}, {9552,10475}, {9708,16058}, {9709,16059}, {10436,30941}, {10439,24220}, {10456,20245}, {10458,27164}, {10472,10473}, {10478,29311}, {12514,14956}, {16062,19787}, {16468,19742}, {16684,17293}, {16748,16887}, {16975,23632}, {17063,24589}, {17065,23444}, {17140,25294}, {17142,17238}, {17143,31008}, {17147,17163}, {17165,31025}, {17178,25528}, {17184,17889}, {17228,21026}, {17237,21949}, {17239,21264}, {17271,31134}, {17272,20347}, {17289,26061}, {17303,24512}, {17495,17591}, {18792,27163}, {20220,30975}, {20594,22230}, {21231,30983}, {21232,30994}, {21238,25624}, {21241,25958}, {21254,31001}, {21384,26035}, {21727,25667}, {21805,31264}, {23791,26824}, {24003,30993}, {24046,28611}, {24686,24694}, {25121,30998}, {25301,27345}, {25385,31053}, {25627,30835}, {26066,30944}, {28595,30982}, {31329,31348}

X(31330) = midpoint of X(7226) and X(28605)
X(31330) = isotomic conjugate of X(2296)
X(31330) = complement of X(17018)
X(31330) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 8, 42), (2, 29839, 29661), (899, 31241, 2), (3661, 29641, 15523)

X(31331) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 63

Barycentrics
(5*b^2+8*b*c+5*c^2)*a^4+2*(b+c)*(7*b^2+13*b*c+7*c^2)*a^3+(7*b^4+7*c^4+(38*b^2+63*b*c+38*c^2)*b*c)*a^2+(b+c)*(b^4+c^4+(b+4*c)*(4*b+c)*b*c)*a-(b^4+c^4+(3*b^2+b*c+3*c^2)*b*c)*b*c : :
X(31331) = X(27481)+2*X(31336)

X(31331) lies on these lines: {86,27949}, {1001,4393}, {3797,24603}, {4078,27495}, {9055,31332}, {24325,27481}, {27478,31350}

X(31332) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 107

Barycentrics 5*a^2+13*(b+c)*a+2*b^2+b*c+2*c^2 : :
X(31332) = 5*X(17319)+4*X(28633) = 4*X(17319)+5*X(31248)

These triangles are triple-perspective

X(31332) lies on these lines: {2,3943}, {37,24625}, {86,545}, {190,551}, {519,25354}, {523,27811}, {903,16826}, {1001,6172}, {1022,28840}, {3247,17240}, {4370,29586}, {4664,24325}, {4715,29580}, {9055,31331}, {10022,29592}, {16590,29584}, {16673,17342}, {16777,17271}, {17045,31333}, {17319,28633}, {24441,29570}, {25361,31153}

X(31333) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 109

Barycentrics 3*a^2-5*(b+c)*a+2*b^2-b*c+2*c^2 : :

X(31333) lies on these lines: {2,4398}, {8,344}, {9,17241}, {10,31311}, {37,29630}, {45,17236}, {86,4422}, {142,190}, {144,17234}, {551,17353}, {908,25361}, {1268,17359}, {1738,3634}, {3624,4687}, {3629,29589}, {3644,31183}, {3731,17305}, {3973,17387}, {4358,31205}, {4360,17338}, {4370,26806}, {4431,6666}, {4473,17245}, {4681,29607}, {6687,17319}, {15492,17312}, {16668,29625}, {16669,29575}, {16676,17370}, {16677,17383}, {16706,25600}, {16814,17266}, {16885,29572}, {17045,31332}, {17160,17337}, {17239,17260}, {17248,17279}, {17261,27191}, {17267,17271}, {17295,17335}, {17342,19875}, {17347,29627}, {17351,29626}, {17368,28640}, {27037,27164}, {27073,27111}, {28604,31285}, {29609,31350}

X(31333) = These triangles are triple-perspective.
X(31333) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (344, 18230, 17233), (17233, 18230, 17277)

X(31334) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 111

Barycentrics 15*a^2+17*(b+c)*a+3*b^2+9*b*c+3*c^2 : :

These triangles are triple-perspective.

X(31334) lies on these lines: {2,4445}, {142,29586}, {368,24656}, {1001,20059}, {1278,3622}, {1449,29592}, {3616,20090}, {17014,31352}, {25361,31292}

X(31335) = PERSPECTOR OF THESE TRIANGLES: GEMINI 23 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+(4*b^2+13*b*c+4*c^2)*a^2+(b+c)*(2*b^2+11*b*c+2*c^2)*a+(4*b^2+7*b*c+4*c^2)*b*c : :

X(31335) lies on these lines: {1,31329}, {2,726}, {75,29609}, {86,239}, {142,17238}, {1125,31308}, {3797,31336}, {3946,4699}, {4384,20145}, {4698,31350}, {4751,17291}, {16826,27474}, {17292,27475}, {24325,27495}, {29580,31342}, {31238,31323}

X(31335) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31347, 27481), (3739, 31306, 27483), (27483, 31306, 239)

X(31336) = PERSPECTOR OF THESE TRIANGLES: GEMINI 25 AND GEMINI 63

Barycentrics (a^2+2*(b+c)*a+b*c)*(3*(b+c)*a+b^2+4*b*c+c^2) : :
X(31336) = 3*X(2)+X(31308) = X(37)+2*X(6707) = X(86)+5*X(4687) = X(1213)-4*X(4698) = 2*X(3842)+X(5625) = X(4733)+2*X(15569) = X(27481)-3*X(31331)

X(31336) lies on these lines: {1,31322}, {2,740}, {9,86}, {10,31342}, {37,4472}, {335,31310}, {524,16590}, {1125,17755}, {1213,3912}, {1654,5308}, {3161,27268}, {3797,31335}, {3842,4649}, {4370,4755}, {4733,15569}, {4789,6544}, {6651,29578}, {17308,31248}, {17322,27487}, {24325,27481}, {25354,29571}, {29592,31314}

X(31336) = midpoint of X(i) and X(j) for these {i,j}: {335, 31310}, {27483, 31308}
X(31336) = complement of X(27483)
X(31336) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31308, 27483), (16826, 20142, 5625)

X(31337) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 57

Barycentrics (b^3+c^3)*a^2-(b^2+b*c+c^2)*b*c*a+(b+c)*b^2*c^2 : :

X(31337) lies on these lines: {2,18170}, {10,75}, {1213,25624}, {1964,27091}, {3122,17786}, {3661,21238}, {3688,27076}, {3778,30473}, {17065,18040}, {17072,21143}, {17233,21257}, {17234,20340}, {17248,28593}, {18044,24478}, {20352,27095}, {21278,27044}, {24517,29712}, {25292,26963}, {25350,30989}, {29679,31090}

X(31337) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 25121, 5224), (20340, 25140, 17234)

X(31338) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 68

Barycentrics
(b+c)*(b^2+3*b*c+c^2)*a^5+(b^2+b*c+c^2)*(3*b^2+7*b*c+3*c^2)*a^4+(b+c)*(3*b^4+3*c^4+4*(2*b^2+3*b*c+2*c^2)*b*c)*a^3+(b^2+b*c+c^2)*(b^2+3*b*c+c^2)^2*a^2+2*(b^2+b*c+c^2)*(b+c)^3*b*c*a+(b+c)^4*b^2*c^2 : :

X(31338) lies on the line {25631,27624}

X(31339) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 80

Barycentrics (b^2+3*b*c+c^2)*a^2+(b+c)^3*a+(b+c)^2*b*c : :

X(31339) lies on these lines: {1,2}, {5,25960}, {9,26035}, {12,5743}, {31,1010}, {55,19283}, {56,5737}, {65,3739}, {75,2292}, {171,16454}, {226,959}, {238,964}, {333,1468}, {355,13731}, {377,28287}, {388,966}, {442,25760}, {748,13740}, {756,4385}, {958,13738}, {984,4968}, {986,4359}, {992,2295}, {993,4225}, {1042,27339}, {1211,25466}, {1213,2277}, {1215,3876}, {1220,17277}, {1329,5241}, {1334,2345}, {1402,9552}, {1469,15985}, {1655,17248}, {1837,21321}, {1909,5224}, {2049,16466}, {2051,10887}, {2274,27164}, {2352,28265}, {2476,3847}, {2551,28270}, {2886,3142}, {2887,4197}, {2901,27785}, {2975,5235}, {3295,19282}, {3780,10371}, {3846,5836}, {3868,24325}, {3877,14011}, {3915,5263}, {4042,19730}, {4147,17166}, {4357,23536}, {4388,26051}, {4418,12514}, {4424,28612}, {4429,24438}, {4438,28267}, {4474,20316}, {4643,4754}, {4761,27647}, {5192,17123}, {5247,5278}, {5252,24735}, {5295,6051}, {5296,27523}, {5363,16424}, {5484,26044}, {5687,19518}, {5710,27623}, {5711,16458}, {5745,27621}, {5791,28258}, {5837,21246}, {6210,15971}, {6533,24046}, {8476,9666}, {8728,25957}, {9565,22076}, {10106,16878}, {10436,17137}, {10448,11110}, {10456,17183}, {10472,10480}, {11109,25885}, {12435,24220}, {12526,20245}, {13741,17125}, {14005,27644}, {14621,16926}, {14636,18481}, {17239,24656}, {17306,26978}, {17337,25992}, {17529,25961}, {17533,21935}, {19513,26446}, {19804,24443}, {21422,24547}, {24174,24589}, {25253,31025}, {25591,25917}, {25624,27641}, {26040,28272}, {26061,28242}, {26063,28266}, {26066,27622}, {26077,28279}, {27648,29066}

X(31339) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1698, 19858), (2, 20036, 3616), (978, 1698, 2)

X(31340) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 81

Barycentrics
(b+c)*(b^2+3*b*c+c^2)*a^5+(3*b^4+3*c^4+(5*b^2+3*b*c+5*c^2)*b*c)*a^4+(b+c)*(3*b^4+3*c^4-(b^2-8*b*c+c^2)*b*c)*a^3+(b^2+b*c+c^2)*(b^4+c^4+2*(b^2+3*b*c+c^2)*b*c)*a^2+2*(b^3+c^3)*(b+c)^2*b*c*a+(b+c)^4*b^2*c^2 : :

X(31340) lies on these lines: {10,1423}, {25624,28366}

X(31341) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 104

Barycentrics (b+c)*(b^2+3*b*c+c^2)*a^3+(b^4+c^4+(8*b^2+13*b*c+8*c^2)*b*c)*a^2+2*(b+c)*(b^2+3*b*c+c^2)*b*c*a+(b^2+b*c+c^2)*b^2*c^2 : :

X(31341) lies on these lines: {2,3780}, {10,31008}, {3925,5224}, {17149,26037}, {21238,25624}, {28653,30966}

X(31342) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+2*(b+2*c)*(2*b+c)*a^2-(b^2-c^2)*(b-c)*a-2*(b^2+b*c+c^2)*b*c : :
X(31342) = 3*X(17389)+X(27481) = X(27478)-3*X(29574)

X(31342) lies on these lines: {1,27474}, {2,4891}, {8,31322}, {10,31336}, {37,319}, {75,29585}, {145,27484}, {192,20059}, {335,29619}, {354,31348}, {518,17389}, {536,27494}, {740,27478}, {984,29605}, {2550,17316}, {3661,15569}, {3696,16826}, {3706,17032}, {3739,17393}, {3797,29588}, {3846,29612}, {3886,20131}, {3912,4085}, {4360,27487}, {4663,6651}, {4664,11160}, {4698,17240}, {4702,14621}, {7381,27491}, {29580,31335}, {29592,31238}

X(31342) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27474, 31306), (6542, 31308, 27495), (27495, 31308, 37)

X(31343) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND GEMINI 71

Barycentrics a*(-a+b+c)*(-a+c)*(a-3*b+c)*(a-b)*(a+b-3*c) : :

X(31343) lies on these lines: {8,1120}, {9,31316}, {78,1320}, {100,1293}, {200,244}, {404,16945}, {517,1339}, {519,22942}, {1897,17780}, {2415,3952}, {3158,24151}, {3699,25268}, {5524,17958}, {6556,7080}, {6557,14942}, {8706,30610}, {30198,30236}

X(31343) = trilinear pole of the line {9, 3057}

X(31344) = PERSPECTOR OF THESE TRIANGLES: GEMINI 31 AND GEMINI 63

Barycentrics (b^2+b*c+c^2)*((2*b^2+b*c+2*c^2)*a^4+(b+c)*(b^2+3*b*c+c^2)*a^3+(b^2+7*b*c+c^2)*b*c*a^2+2*(b+c)*b^2*c^2*a-b^3*c^3) : :

X(31344) lies on these lines: {75,27482}, {871,10009}, {984,3661}, {7777,27488}

X(31345) = PERSPECTOR OF THESE TRIANGLES: GEMINI 32 AND GEMINI 62

Barycentrics
((b^3-c^3)*(b-c)*b*c*a^6+(b+c)*(b^6+c^6-2*(b^3-c^3)*(b-c)*b*c)*a^5-(b^6+c^6-(2*b^4+2*c^4-(b^2+5*b*c+c^2)*b*c)*b*c)*b*c*a^4+(b+c)*(b^4+c^4-(4*b^2-9*b*c+4*c^2)*b*c)*b^2*c^2*a^3+(b^2-7*b*c+c^2)*b^4*c^4*a^2+2*(b+c)*b^5*c^5*a-(b^2+b*c+c^2)*b^5*c^5)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31345) lies on the line {76,27447}

X(31346) = PERSPECTOR OF THESE TRIANGLES: GEMINI 38 AND GEMINI 63

Barycentrics 2*(b^2+b*c+c^2)*a^4-3*(b^3+c^3)*a^3-3*(b^2+b*c+c^2)*b*c*a^2+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a-(b^3-c^3)*(b-c)*b*c : :

X(31346) lies on these lines: {9,192}, {19,27472}, {273,26023}, {7777,27488}

X(31347) = PERSPECTOR OF THESE TRIANGLES: GEMINI 40 AND GEMINI 63

Barycentrics 3*(b+c)*a^3+(2*b^2+11*b*c+2*c^2)*a^2+(b+c)*(b^2+10*b*c+c^2)*a+(5*b^2+8*b*c+5*c^2)*b*c : :

X(31347) lies on these lines: {2,726}, {7,1654}, {8,31314}, {75,4470}, {192,31319}, {239,4772}, {1278,4021}, {1992,4688}, {3616,31308}, {3739,31322}, {4393,24342}, {4704,29609}, {17236,24199}, {17316,24325}, {24349,27495}, {27475,29579}, {29610,31302}

X(31347) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4699, 27484, 27483), (27481, 31335, 2), (27483, 31317, 27484)

X(31348) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 104

Barycentrics (2*b^2-b*c+2*c^2)*a^2+(b^2-c^2)*(b-c)*a-(b^2+b*c+c^2)*b*c : :

X(31348) lies on these lines: {2,726}, {38,27495}, {42,31314}, {75,27482}, {239,514}, {244,3797}, {310,1921}, {354,31342}, {596,27020}, {982,27474}, {1266,24318}, {1575,3807}, {2228,24413}, {3210,17027}, {3661,4392}, {3666,31306}, {3720,31308}, {4850,31317}, {6381,29576}, {6542,17449}, {6651,7292}, {8720,17692}, {16819,24176}, {17011,17187}, {17490,27484}, {19804,31322}, {21214,25270}, {24166,25264}, {24589,31323}, {28606,31319}, {31329,31330}

X(31348) = reflection of X(3807) in X(1575)
X(31348) = {X(244), X(3797)}-harmonic conjugate of X(30967)

X(31349) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 107

Barycentrics 2*(b+c)*a^3-(2*b^2-b*c+2*c^2)*a^2-(b+c)*(b^2+b*c+c^2)*a+(2*b^2+b*c+2*c^2)*b*c : :
X(31349) = X(335)-4*X(17755) = 3*X(27487)-2*X(31138)

X(31349) lies on these lines: {1,27949}, {2,38}, {37,24625}, {44,190}, {75,545}, {518,3799}, {519,3797}, {528,29617}, {551,31331}, {597,4370}, {673,15481}, {740,31310}, {903,4688}, {1086,17250}, {1386,6651}, {3679,7924}, {3994,17029}, {4366,16468}, {4384,24821}, {4393,4432}, {4422,17397}, {4437,29577}, {4473,26626}, {4740,6172}, {4755,31306}, {9041,17389}, {11329,24826}, {16367,24820}, {16593,29582}, {16826,24841}, {16833,17738}, {27487,31138}, {31322,31351}

X(31349) = midpoint of X(4740) and X(17487)
X(31349) = reflection of X(i) in X(j) for these (i,j): (2, 17755), (335, 2), (903, 4688), (4664, 4370)

X(31350) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 109

Barycentrics 6*(b+c)*a^3+(14*b^2+29*b*c+14*c^2)*a^2+(b+c)*(7*b^2+19*b*c+7*c^2)*a+(b+2*c)*(2*b+c)*b*c : :
X(31350) = 2*X(37)+X(1268)

X(31350) lies on these lines: {2,28516}, {37,1268}, {75,31351}, {239,31311}, {335,31310}, {4687,27481}, {4971,31322}, {5852,27475}, {15481,16826}, {18230,26626}, {27478,31331}, {27480,31352}, {27495,29574}, {29609,31333}

X(31351) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 110

Barycentrics 6*(b+c)*a^3+2*(7*b^2+19*b*c+7*c^2)*a^2+(b+c)*(7*b^2+37*b*c+7*c^2)*a+(11*b^2+23*b*c+11*c^2)*b*c : :
X(31351) = 7*X(4751)-X(30598)

X(31351) lies on these lines: {2,28522}, {75,31350}, {142,17238}, {239,31312}, {3739,27481}, {4725,31306}, {4751,6707}, {29612,31329}, {31238,31319}, {31322,31349}

X(31352) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 111

Barycentrics 9*(b+c)*a^3-(14*b^2+23*b*c+14*c^2)*a^2-(b+c)*(7*b^2+52*b*c+7*c^2)*a-(11*b^2+38*b*c+11*c^2)*b*c : :

X(31352) lies on these lines: {2,4891}, {239,31312}, {1278,27481}, {4678,27475}, {4704,31322}, {4772,20059}, {17014,31334}, {20090,27483}, {27480,31350}

Conics associated to pairs of orthologic or parallelogic triangles: X(31353)-X(31375)

This preamble and centers X(31353)-X(31375) were contributed by César Eliud Lozada, January 27, 2019.

I) Conics associated to a pair of orthologic triangles:

Let T'=A'B'C' and T"=A"B"C" be two orthologic triangles. Denote a'_b and a'_c the perpendicular lines from A' to A"C" and A"B", respectively, and build b'_c, b'_a, c'_a,c'_b cyclically. Then these six lines are tangent to a conic Φ'_t, here named the orthologic tangential-conic T' to T".
Swap T' and T", repeat the above construction and name the respective lines a"_b, a"_c, b"_c, b"_a, c"_a, c"_b. Then these six lines are tangent to another conic Φ"_t, here named the orthologic tangential-conic T" to T'.
Let A'_b=a'_b ∩ B'C' and A'_c=a'_c ∩ B'C' and define B'_c, B'_a, C'_a, C'_b cyclically. Then these six points lie on a conic Φ'_p, here named the orthologic conic T' to T".
Swap T' and T", repeat the above construction and name the respective points A"_b, A"_c, B"_c, B"_a, C"_a, C"_b. Then these six points lie on another conic Φ"_p, here named the orthologic conic T" to T'.

II) Conics associated to a pair of parallelogic triangles:

Let T'=A'B'C' and T"=A"B"C" be two parallelogic triangles. Denote a'_b and a'_c the parallel lines from A' to A"C" and A"B", respectively, and build b'_c, b'_a, c'_a,c'_b cyclically. Then these six lines are tangent to a conic Ψ'_t, here named the parallelogic tangential-conic T' to T".
Swap T' and T", repeat the above construction and name the respective lines a"_b, a"_c, b"_c, b"_a, c"_a, c"_b. Then these six lines are tangent to another conic Ψ"_t, here named the parallelogic tangential-conic T" to T'.
Let A'_b=a'_b ∩ B'C' and A'_c=a'_c ∩ B'C' and define B'_c, B'_a, C'_a, C'_b cyclically. Then these six points lie on a conic Ψ'_p, here named the parallelogic conic T' to T".
Swap T' and T", repeat the above construction and name the respective points A"_b, A"_c, B"_c, B"_a, C"_a, C"_b. Then these six points lie on another conic Ψ"_p, here named the parallelogic conic T" to T'.

If T' and T" are orthologic (parallelogic) triangles and P is the orthologic (parallelogic) center T' to T", then the orthologic (parallelogic) conic T' to T" does not depend on T". Moreover, the orthologic conic T' to T" and the parallelogic conic T' to T" coincide. Therefore, when T' are P are given, a more convenient name for this conic Φ'_p= Ψ'_p is P-orthoparallelogic conic of T'. A similar coincidence occurs for the tangential conics Φ'_t= Ψ'_t and therefore a better name for this conic is P-orthoparallelogic tangential-conic of T'.

If P = x:y:z (barycentrics) then the center of the P-orthoparallelogic conic of ABC is:

O′_p = x^2*(y+z)*((y+z)*(x^4-(y-z)^2*y*z)-x*(y^2+z^2)*(-x^2+y^2+z^2)-(y^3+z^3)*x^2) : :

and its perspector is:

Q′_p = x*(y+z)*F(x,y,z)*F(x,z,y) : :, where F(x,y,z) = (x^2*(2*x*y+x*z+2*z^2)-(y-z)*(2*y*((x+z)*y+z^2)+x*z*(y+z)))

The center of the P-orthoparallelogic tangential-conic of ABC is O'_t=complement-of-P and its perspector is:

Q′_t = (x*(2*z+y)+y*(y+z))*(x*(2*y+z)+z*(y+z)) : :

The appearance of (i, j) in the following list means that X(j)=center of the X(i)-orthoparallelogic conic of ABC:
(1,3588), (2,2), (3,31353), (4,17807), (5,31354), (6,31355), (7,7955), (30,3163), (511,11672), (512,1084), (513,1015), (514,1086), (515,23986), (516,23972), (517,23980), (518,6184), (519,4370), (522,1146), (523,115), (524,2482), (525,15526), (526,18334), (536,13466), (542,23967), (690,23992), (726,20532), (826,15449), (1503,23976), (3566,15525), (3906,17416) , (7927,15527), (17430,17429)

The appearance of (i, j) in the following list means that X(j)=perspector of the X(i)-orthoparallelogic conic of ABC:
(1,31356), (2,2), (3,31357), (4,17808), (6,31358)

The appearance of (i, j) in the following list means that X(j)=perspector of the X(i)-orthoparallelogic tangential-conic of ABC:
(1,31359), (2,2), (3,13599), (4,15740), (5,22268), (6,31360), (8,7320), (20,31361), (69,17040), (75,17038), (95,17041), (99,9293), (264,17039), (550,3521), (668,9267), (3244,5559), (3629,13622)

Note: For P in the infinity, O′_p lies in the Steiner inellipse, Q′_p = P and Q′_t = P.

X(31353) = CENTER OF THE X(3)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics SA^2*(SB+SC)^2*(S^2+SB*SC)*((SA-2*R^2)*S^2+SA*(8*(2*R^2-SW)*R^2+SW^2)) : :

X(31353) lies on these lines: {185,216}, {217,31357}

X(31354) = CENTER OF THE X(5)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics (3*S^2-SB*SC)*(4*S^2+(SB+SC)*(4*R^2-3*SA-SW))*(S^4-(R^2*(12*R^2+8*SA-7*SW)-2*SA^2+SW^2)*S^2+(4*R^2-SW)*(4*R^2-SA)*R^2*SA) : :

X(31354) lies on these lines: {}

X(31355) = CENTER OF THE X(6)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics (SA+SW)*(SB+SC)^2*(S^4*SA-(2*(SW^2+2*SB*SC)*R^2+(SB+SC)*(SA^2-SB*SC))*S^2+SB*SC*SW^3) : :

X(31355) lies on these lines: {39,6467}, {69,15270}, {3051,14820}, {7794,23208}, {20775,31358}

X(31356) = PERSPECTOR OF THE X(1)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics a*(b+c)*((2*b+c)*a^3+2*c^2*a^2-(b-c)*(2*b^2+b*c+c^2)*a-2*(b^2-c^2)*b*c)*((b+2*c)*a^3+2*b^2*a^2+(b-c)*(b^2+b*c+2*c^2)*a+2*(b^2-c^2)*b*c) : :

X(31356) lies on these lines: {8,10435}, {42,10474}, {55,10448}, {210,22299}, {1334,3588}

X(31357) = PERSPECTOR OF THE X(3)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics SA^2*(SB+SC)*(S^2+SB*SC)*(S^2+16*R^2*(2*R^2-SW)-SB^2+2*SW^2)*(S^2+16*R^2*(2*R^2-SW)-SC^2+2*SW^2) : :

X(31357) lies on these lines: {51,8799}, {184,26897}, {217,31353}

X(31358) = PERSPECTOR OF THE X(6)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics (SA^2-SW^2)*((8*R^2*SB-(2*SB-SW)*SW)*S^2-(2*SW-SB)*SB*SW^2)*((8*R^2*SC-(2*SC-SW)*SW)*S^2-(2*SW-SC)*SC*SW^2) : :

X(31358) lies on the line {20775,31355}

X(31359) = PERSPECTOR OF THE X(1)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics ((b+2*c)*a+b^2+b*c)*((2*b+c)*a+b*c+c^2) : :

X(31359) lies on these lines: {1,333}, {2,65}, {8,37}, {9,1220}, {10,312}, {19,29}, {21,1610}, {31,1098}, {45,5793}, {75,2292}, {82,3915}, {85,3668}, {86,969}, {92,225}, {145,4981}, {257,17248}, {341,756}, {377,24723}, {388,17257}, {392,25490}, {517,19853}, {740,17038}, {759,931}, {964,4676}, {968,1043}, {984,7275}, {986,19804}, {994,3878}, {997,19270}, {1010,12514}, {1213,3959}, {1621,2218}, {1654,10371}, {1697,14942}, {1698,4674}, {1836,26051}, {1938,25511}, {2176,5275}, {2214,2303}, {2652,5794}, {3085,28807}, {3486,13736}, {3617,3714}, {3646,25531}, {3679,4102}, {3683,4195}, {3868,13476}, {3876,26115}, {3931,9534}, {4181,20682}, {4386,28631}, {4389,23536}, {4511,16342}, {4518,21711}, {4642,26037}, {5233,5530}, {5257,5837}, {5278,17016}, {5730,16343}, {5902,25512}, {5903,16828}, {6051,10449}, {6557,9780}, {6646,10404}, {6682,21214}, {8421,24463}, {10405,27288}, {10436,12526}, {12709,27339}, {15254,17697}, {16062,23604}, {16824,19732}, {17149,18298}, {17250,20955}, {17260,17743}, {17338,25992}, {18359,23541}, {19874,22299}, {21674,25760}, {24627,25524}, {25466,27184}

X(31359) = isogonal conjugate of X(1468)
X(31359) = isotomic conjugate of X(10436)
X(31359) = polar conjugate of X(5307)
X(31359) = trilinear pole of the line {522, 661}
X(31359) = {X(1), X(2258)}-harmonic conjugate of X(5331)

X(31360) = PERSPECTOR OF THE X(6)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics (SA*SB+SW^2)*(SA*SC+SW^2) : :

X(31360) lies on these lines: {2,1843}, {6,1799}, {39,69}, {76,23642}, {95,11285}, {141,305}, {264,6656}, {287,19459}, {306,21035}, {2373,26206}, {3589,30489}, {3619,6340}, {7876,9229}, {7950,14977}, {8797,14064}

X(31360) = isotomic conjugate of X(7770)
X(31360) = trilinear pole of the line {525, 3005}

X(31361) = PERSPECTOR OF THE X(20)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics (S^2-2*(16*R^2-SB-3*SW)*SB)*(S^2-2*(-3*SW+16*R^2-SC)*SC) : :

The trilinear polar of X(31361) passes through X(6587). (Randy Hutson, March 21, 2019)

X(31361) lies on these lines: {4,14572}, {20,5893}, {253,5895}, {1249,3146}, {3091,6716}, {3543,14249}

X(31361) = isogonal conjugate of X(8567)
X(31361) = trilinear pole of radical axis of circumcircle and midheight circle
X(31361) = polar conjugate of isotomic conjugate of X(37878)

X(31362) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: EULER TO ABC

Barycentrics SB*SC*((2*R^2*(12*R^2-SA-5*SW)+SW^2)*S^2-(16*R^2*(4*R^2-SW)+SW^2)*(8*R^2*SW+SB*SC-2*SW^2)) : :
X(31362) = 5*X(3091)-X(31369)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31362) lies on these lines: {3,15259}, {4,17807}, {5,31367}, {3091,31369}, {5480,13488}, {10002,18537}

X(31362) = midpoint of X(4) and X(17807)
X(31362) = reflection of X(31367) in X(5)

X(31363) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: EULER TO ABC

Barycentrics (S^2-2*(4*R^2-SW)*SB)*(S^2-2*(4*R^2-SW)*SC) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(15740)

X(31363) lies on the Kiepert hyperbola and these lines: {2,9786}, {5,459}, {20,275}, {1498,3424}, {2052,3091}, {3316,6810}, {3317,6809}, {3832,8796}, {5056,16080}, {7395,18841}, {7399,18840}

X(31363) = antigonal conjugate of the antitomic conjugate of X(31363)
X(31363) = antitomic conjugate of the antigonal conjugate of X(31363)
X(31363) = isogonal conjugate of X(11425)

X(31364) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics (S^2-SB*SC)*(S^4+(2*(SA-3*SW)*R^2+SB*SC+SW^2)*S^2+SB*SC*(8*R^2*(2*R^2-SW)+SW^2)) : :

The center of the reciprocal orthologic conic of these triangles is X(31353)

X(31364) lies on these lines: {6,14773}, {185,216}, {570,15231}, {1885,3003}, {6467,23195}

X(31365) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics
SA*(S^4-(4*R^2*(24*R^2-15*SW-SC)-9*SC^2+8*SW^2)*S^2+(32*R^2*(3*R^2-2*SW)+9*SW^2)*(SA+SC)*(SB+SC))*(S^4-(4*R^2*(24*R^2-15*SW-SB)-9*SB^2+8*SW^2)*S^2+(32*R^2*(3*R^2-2*SW)+9*SW^2)*(SA+SB)*(SB+SC)) : :

The perspector of the reciprocal orthologic conic of these triangles is X(31357)

X(31365) lies on these lines: {}

X(31366) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics SB^2*SC^2*((4*R^2-SB)*S^2+(8*R^2-SW)*SA*SC)*((4*R^2-SC)*S^2+(8*R^2-SW)*SA*SB) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(13599)

X(31366) lies on the Jerabek hyperbola and the line {2052,14457}

X(31367) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: MEDIAL TO ABC

Barycentrics SA*(6*R^2*S^4+(32*R^4*(8*R^2-SA-3*SW)+2*(7*SW^2+3*SA^2-SB*SC)*R^2-SW^3)*S^2-(8*R^2-SW)^2*SB*SC*SW) : :
X(31367) = 3*X(2)+X(31369)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31367) lies on these lines: {2,17807}, {5,31362}, {141,5894}, {5020,15259}, {13567,31368}

X(31367) = midpoint of X(17807) and X(31369)
X(31367) = reflection of X(31362) in X(5)
X(31367) = complement of X(17807)
X(31367) = {X(2), X(31369)}-harmonic conjugate of X(17807)

X(31368) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: MEDIAL TO ABC

Barycentrics SA*((8*R^2-3*SW)*S^2-32*R^2*SB*(4*R^2-SW)+SW*(2*SA*SC-SB^2))*((8*R^2-3*SW)*S^2-32*R^2*SC*(4*R^2-SW)+SW*(2*SA*SB-SC^2)) : :

The perspector of the reciprocal orthologic conic of these triangles is X(17808)

X(31368) lies on these lines: {1368,2883}, {13567,31367}

X(31369) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics 2*(10*R^2-3*SA-2*SW)*R^2*S^2-(256*R^6-32*R^4*(SA+3*SW)+2*R^2*(3*SA^2-3*SB*SC+7*SW^2)-SW^3)*SA : :
X(31369) = 3*X(2)-4*X(31367) = 5*X(3091)-4*X(31362)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31369) lies on these lines: {2,17807}, {253,6995}, {1370,14615}, {1995,15259}, {3091,31362}

X(31369) = reflection of X(17807) in X(31367)
X(31369) = anticomplement of X(17807)
X(31369) = {X(17807), X(31367)}-harmonic conjugate of X(2)

X(31370) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics
(2*(62*R^2-7*SW)*R^2*S^2-32*R^4*(8*R^2*SB+3*SA*SC-4*SB^2)+2*R^2*SW*(SA*SC-8*SB^2)+SA*SC*SW^2)*(2*(62*R^2-7*SW)*R^2*S^2-32*R^4*(8*SC*R^2+3*SA*SB-4*SC^2)+2*R^2*SW*(SA*SB-8*SC^2)+SA*SB*SW^2) : :

The perspector of the reciprocal orthologic conic of these triangles is X(17808)

X(31370) lies on these lines: {}

X(31371) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics SA*(3*S^2-4*(SA+SC)*(SB+SC))*(3*S^2-4*(SB+SC)*(SA+SB)) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(15740).

X(31371) lies on these lines: {2,3532}, {4,15010}, {6,3146}, {20,14528}, {54,3529}, {64,3091}, {65,5225}, {66,6225}, {74,3090}, {185,15077}, {265,18909}, {546,3426}, {1899,15749}, {3431,17538}, {3525,11270}, {3527,3627}, {3531,5076}, {3544,13452}, {3618,5895}, {6145,12324}, {7394,16620}, {11433,22466}, {11541,13472}, {14861,18531}, {14865,18532}, {15022,15751}, {18918,21400}

X(31371) = isogonal conjugate of X(3516)
X(31371) = isotomic conjugate of the anticomplement of X(15851)

X(31372) = CENTER OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: ABC TO 1st ANTI-BROCARD

Barycentrics 3*a^8-6*(b^2+c^2)*a^6-3*(b^4-8*b^2*c^2+c^4)*a^4+6*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-b^8-(2*b^4-9*b^2*c^2+2*c^4)*b^2*c^2-c^8 : :

The center of the reciprocal parallelogic conic of these triangles is X(31374).

X(31372) lies on these lines: {148,31373}, {523,20094}, {858,7779}

X(31372) = anticomplementary conjugate of the anticomplement of X(20998)
X(31372) = isotomic conjugate of X(31373)
X(31372) = anticomplement of the antitomic conjugate of X(4590)
X(31372) = anticomplement of the cyclocevian conjugate of X(670)
X(31372) = anticomplement of X(35511)
X(31372) = anticomplement of the isogonal conjugate of X(20998)
X(31372) = anticomplement of the isotomic conjugate of X(148)

X(31373) = PERSPECTOR OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: ABC TO 1st ANTI-BROCARD

Barycentrics
(a^8+2*(b^2-3*c^2)*a^6-3*(3*b^4-4*b^2*c^2-c^4)*a^4+2*(b^6+3*c^6+6*(b^2-2*c^2)*b^2*c^2)*a^2+b^8-3*(2*b^4-b^2*c^2-2*c^4)*b^2*c^2-3*c^8)*(a^8-2*(3*b^2-c^2)*a^6+3*(b^4+4*b^2*c^2-3*c^4)*a^4+2*(3*b^6+c^6-6*(2*b^2-c^2)*b^2*c^2)*a^2-3*b^8+3*(2*b^4+b^2*c^2-2*c^4)*b^2*c^2+c^8) : :

The perspector of the reciprocal parallelogic conic of these triangles is X(31375)

X(31373) lies on the line {148,31372}

X(31373) = isotomic conjugate of X(31372)

X(31374) = CENTER OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: 1st ANTI-BROCARD TO ABC

Barycentrics
3*a^12-5*(b^2+c^2)*a^10-5*(2*b^2-c^2)*(b^2-2*c^2)*a^8+(b^2+c^2)*(7*b^4-4*b^2*c^2+7*c^4)*a^6+(14*b^8+14*c^8-3*(11*b^4-b^2*c^2+11*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(5*b^8+5*c^8+3*(2*b^4-11*b^2*c^2+2*c^4)*b^2*c^2)*a^2-(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*(b^4-b^2*c^2-c^4)*(b^4+b^2*c^2-c^4) : :

The center of the reciprocal parallelogic conic of these triangles is X(31372)

X(31374) lies on the line {2396,20094}

X(31375) = PERSPECTOR OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: 1st ANTI-BROCARD TO ABC

Barycentrics
(3*S^6-(384*R^2-8*SC-69*SW)*SW*S^4+(64*(12*SA*SB-3*SC^2+SW^2)*R^2-(144*SA*SB-16*SC^2+15*SW^2)*SW)*SW*S^2+(8*SC-SW)*SW^5)*(3*S^6-(384*R^2-8*SB-69*SW)*SW*S^4+(64*(12*SA*SC-3*SB^2+SW^2)*R^2-(144*SA*SC-16*SB^2+15*SW^2)*SW)*SW*S^2+(8*SB-SW)*SW^5) : :

The perspector of the reciprocal parallelogic conic of these triangles is X(31373)

X(31375) lies on these lines: {}

X(31376) = COMPLEMENT OF X(252)

Barycentrics (S^2+SB*SC)*(3*S^2-SA^2)*(4*S^2+(SB+SC)*(2*R^2-SB-SC)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31376) lies on these lines: {2, 252}, {3, 24573}, {5, 128}, {140, 6150}, {233, 5421}, {1209, 1493}, {2072, 10600}, {5501, 6592}, {7575, 15848}, {13372, 21975}

X(31376) = midpoint of X(3) and X(24573)
X(31376) = reflection of X(i) in X(j) for these (i,j): (5, 23281), (23280, 3628)
X(31376) = complement of X(252)
X(31376) = complementary conjugate of X(32142)
X(31376) = {X(5), X(15345)}-harmonic conjugate of X(137)

X(31377) = COMPLEMENT OF X(6526)

Barycentrics (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(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)^2 : :
Barycentrics SA^2*(S^2-2*SB*SC)*(2*S^2-(SB+SC)*(8*R^2-SB-SC)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31377) lies on these lines: {2, 1105}, {3, 1661}, {4, 12096}, {20, 122}, {131, 3548}, {140, 20207}, {216, 631}, {1073, 6696}, {3546, 10600}, {5895, 27089}, {6225, 11589}, {6389, 16196}, {6523, 6716}, {6760, 14216}, {11457, 14385}

X(31377) = complement of X(6526)
X(31377) = complementary conjugate of complement of X(35602)

X(31378) = COMPLEMENT OF X(5627)

Barycentrics (S^2-3*SB*SC)*(3*SA^2-S^2)*(4*S^2-3*(SB+SC)*(6*R^2-SB-SC)) : :
X(31378) = 4*X(140)-X(6070), 3*X(549)+X(18285), 5*X(631)+X(14480), 2*X(1511)+X(3258), X(1553)-4*X(10272), 2*X(3154)+X(30714), 2*X(5972)+X(14934), 4*X(5972)-X(25641), 2*X(6699)+X(14611), X(10564)+2*X(16319), X(14508)+5*X(20125), 2*X(14934)+X(25641), 5*X(15034)+X(17511), 5*X(15040)+X(20957)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

Another construction is given by Elias M. Hagos and Peter Moses: euclid 1779.

X(31378) lies on the cubics K515, K900 and these lines: {2, 5627}, {30, 113}, {128, 6760}, {140, 6070}, {186, 14920}, {476, 1138}, {541, 15468}, {549, 18285}, {631, 14480}, {3003, 3163}, {3154, 30714}, {5972, 14934}, {6699, 14611}, {14508, 20125}, {14993, 22104}, {15034, 17511}, {15040, 20957}

X(31378) = midpoint of X(476) and X(1138)
X(31378) = reflection of X(14993) in X(22104)
X(31378) = complement of X(5627)
X(31378) = complementary conjugate of X(20304)
X(31378) = {X(5972), X(14934)}-harmonic conjugate of X(25641)

X(31379) = COMPLEMENT OF X(25641)

Barycentrics S^4-(3*R^2*(90*R^2+3*SA-40*SW)-2*SA^2-SB*SC+13*SW^2)*S^2+(18*R^2-5*SW)*(9*R^2-SW)*SB*SC : :
X(31379) = 3*X(2)+X(477), 3*X(3)+X(20957), X(476)-5*X(631), 5*X(632)-X(18319), X(1553)-3*X(14643), 5*X(3091)-X(14989), 3*X(3258)-X(20957), 7*X(3523)+X(14731), X(6070)-3*X(15061), X(11749)+7*X(14869), 3*X(15035)+X(17511)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31379) lies on these lines: {2, 477}, {3, 3258}, {30, 5972}, {125, 14934}, {140, 16168}, {476, 631}, {523, 6699}, {620, 15122}, {632, 18319}, {1511, 16340}, {1553, 14643}, {3091, 14989}, {3154, 17702}, {3523, 14731}, {5446, 12052}, {6070, 15061}, {10625, 16978}, {11749, 14869}, {12079, 20397}, {14611, 16003}, {14915, 16319}, {15035, 17511}, {15088, 21316}

X(31379) = midpoint of X(i) and X(j) for these {i,j}: {3, 3258}, {125, 14934}, {477, 25641}, {1511, 16340}, {10625, 16978}, {14611, 16003}
X(31379) = reflection of X(i) in X(j) for these (i,j): (5446, 12052), (12079, 20397), (21316, 15088), (22104, 140)
X(31379) = complement of X(25641)
X(31379) = {X(2), X(477)}-harmonic conjugate of X(25641)

X(31380) = COMPLEMENT OF X(5513)

Barycentrics 2*a^6-2*(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+(b+c)*(5*b^2-6*b*c+5*c^2)*a^3-(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^2-(b^4-c^4)*(b-c)*a+(b^4+c^4+(b-c)^2*b*c)*(b-c)^2 : :
X(31380) = 3*X(2)+X(675)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31380) lies on these lines: {2, 101}, {3, 25642}, {142, 6718}, {1054, 4859}, {3035, 3739}, {3315, 8458}, {5432, 6025}, {5972, 6707}, {6678, 6720}, {10165, 15746}, {16056, 25468}

X(31380) = midpoint of X(i) and X(j) for these {i,j}: {3, 25642}, {675, 5513}
X(31380) = complement of X(5513)
X(31380) = orthoptic circle of Steiner inellipse-inverse-of X(150)
X(31380) = {X(2), X(675)}-harmonic conjugate of X(5513)
X(31380) = centroid of {A,B,C,X(675)}

X(31381) = X(3)X(49)∩X(24)X(157)

Barycentrics (S^2-SB*SC)*(S^2+2*R^2*(4*R^2+SA-4*SW)+SW^2) : :

See Nguyen Dang Khoa and César Lozada, Hyacinthos 28830.

X(31381) lies on these lines: {3, 49}, {4, 19172}, {24, 157}, {25, 8887}, {26, 2351}, {68, 23181}, {418, 6146}, {578, 16035}, {973, 23635}, {1624, 7505}, {2917, 7669}, {2937, 13558}, {3133, 12134}, {3135, 9833}, {6776, 26876}, {7512, 8266}, {7592, 15231}, {8553, 17849}, {9715, 15512}, {18925, 26874}

X(31381) = X(73)-of-tangential-triangle if ABC is acute

X(31382) = X(20)X(64)∩X(154)X(160)

Barycentrics (SB+SC)*(3*S^4-(4*R^2*(8*R^2+4*SA-3*SW)-4*SA^2+2*SB*SC+SW^2)*S^2+2*(4*R^2-SW)*SB*SC*SW) : :
X(31382) = 3*X(154)-4*X(160)

See Nguyen Dang Khoa and César Lozada, Hyacinthos 28830.

X(31382) lies on these lines: {6, 1987}, {20, 64}, {154, 160}, {6748, 17810}, {18445, 22552}

X(31383) = MARTA POINT

Barycentrics 3a^6-a^4(b^2+c^2)-a^2(b^2-c^2)^2-(b^2-c^2)^2(b^2+c^2) : :

X(31383) = 3 X[25] - 2 X[13567],5 X[25] - 3 X[26869],3 X[1899] - 4 X[13567],5 X[1899] - 6 X[26869],10 X[13567] - 9 X[26869]

See Angel Montesdeoca, Hyacinthos 28832 and Centro ortológico y punto fijo de una afinidad.

X(31383) lies on these lines: {2, 1495}, {3, 16655}, {4, 54}, {5, 3796}, {6, 428}, {20, 3917}, {22, 1352}, {23, 11442}, {24, 14216}, {25, 1503}, {30, 394}, {51, 6776}, {66, 20987}, {68, 7517}, {69, 16276}, {110, 7391}, {125, 6353}, {154, 427}, {155, 7553}, {182, 6997}, {185, 7487}, {193, 21969}, {343, 9909}, {373, 7398}, {378, 16658}, {382, 3167}, {393, 8779}, {462, 5868}, {463, 5869}, {468, 1853}, {511, 7500}, {542, 6515}, {1181, 6756}, {1204, 12324}, {1370, 9306}, {1498, 3575}, {1501, 3767}, {1514, 3830}, {1593, 16621}, {1595, 19357}, {1596, 18396}, {1597, 16654}, {1598, 6146}, {1843, 5596}, {1885, 15811}, {1971, 22075}, {1994, 20423}, {2165, 2980}, {2182, 5101}, {2883, 12173}, {2979, 20062}, {3051, 3331}, {3060, 7519}, {3091, 11572}, {3146, 3292}, {3147, 20299}, {3148, 8721}, {3426, 3534}, {3515, 6247}, {3518, 11457}, {3541, 10282}, {3542, 18381}, {3867, 19125}, {4232, 23291}, {4549, 18435}, {5012, 7394}, {5064, 23292}, {5094, 10192}, {5133, 6800}, {5198, 12241}, {5200, 5871}, {5310, 12588}, {5320, 5800}, {5322, 12589}, {5422, 11179}, {5480, 11402}, {5562, 31305}, {5651, 7386}, {5654, 10540}, {5878, 6240}, {6000, 18533}, {6053, 10706}, {6243, 9936}, {6623, 13851}, {6696, 15750}, {7387, 12134}, {7392, 25406}, {7401, 10984}, {7408, 13366}, {7493, 21243}, {7499, 10516}, {7507, 16252}, {7540, 18445}, {7576, 11456}, {7667, 17811}, {7714, 11433}, {7715, 18914}, {7716, 26926}, {7795, 10328}, {8550, 9777}, {8780, 11064}, {9707, 15559}, {9714, 12359}, {9934, 12140}, {10151, 18405}, {10301, 11245}, {10535, 11393}, {10539, 14790}, {11392, 26888}, {11403, 16656}, {12111, 31304}, {12112, 18559}, {12174, 13568}, {12290, 20427}, {13399, 18931}, {13417, 14683}, {13595, 18911}, {14227, 19219}, {14912, 15004}, {15152, 18386}, {15448, 23332}, {15583, 19118}, {16105, 23236}, {16319, 18870}, {16657, 18535}, {17907, 19558}, {18374, 23327}, {18378, 25738}, {21645, 31291}, {21646, 31299}, {22135, 27376}

X(31383) = reflection of X(i) in X(j) for these (i,j): {1370,9306}, {1899,25}, {18396,1596}
X(31383) = crosssum of X(2972) and X(8673)
X(31383) = crossdifference of every pair of points on the line {9210, 17434}
X(31383) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 9833, 19467}, {4, 11206, 184}, {4, 18925, 11424}, {20, 14826, 3917}, {24, 14216, 26937}, {24, 16659, 14216}, {1495, 11550, 2}, {5012, 7394, 14561}, {5064, 26864, 23292}, {6759, 13419, 4}, {6776, 6995, 51}, {9909, 18440, 343}, {10301, 11245, 17810}, {15811, 17845, 1885}

X(31384) = REFLECTION OF X(4) IN X(431)

Barycentrics a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^8-a^7 b-3 a^6 b^2+3 a^5 b^3+3 a^4 b^4-3 a^3 b^5-a^2 b^6+a b^7-a^7 c-a^6 b c+a^5 b^2 c+a^4 b^3 c+a^3 b^4 c+a^2 b^5 c-a b^6 c-b^7 c-3 a^6 c^2+a^5 b c^2+4 a^4 b^2 c^2-a^2 b^4 c^2-a b^5 c^2+3 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+3 a^4 c^4+a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-3 a^3 c^5+a^2 b c^5-a b^2 c^5+b^3 c^5-a^2 c^6-a b c^6+a c^7-b c^7) : :
Barycentrics (8 a R^4-8 b R^4-4 a R^2 SB+4 b R^2 SB-4 a R^2 SC+4 c R^2 SC-2 a R^2 SW+6 b R^2 SW+a SB SW-b SB SW+a SC SW-c SC SW-b SW^2) S^2 +(-8 R^3 SB SC+2 R SB SC SW) S +2 b R^2 SB SC^2-2 c R^2 SB SC^2-2 b R^2 SB SC SW-b SB SC^2 SW+c SB SC^2 SW+b SB SC SW^2 : :

As a point on the Euler line, X(31384) has Shinagawa coefficients {(p - r - 2 R) (r + R) (p + r + 2 R), r^3 + 6 r^2 R + 10 r R^2 + 6 R^3 - p^2 (r + 2 R)}, or, equivalently, (-4FS²-2Fabc$a$, 2(E+2F)S²+(E+4F)abc$a$).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28835.

X(31384) lies on these lines: {2,3}, {100,1299}, {108,1300}, {3563,26706}

X(31384) = reflection of X(4) in X(431)
X(31384) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,21}, {4,3147,6824}, {4,6853,1594}, {4,6874,7547}, {4,6876,378}, {4,6988,3541}, {4,7505,6828}, {25,6985,4}, {3575,6842,4}, {3651,4231,7414}, {6838,7487,4}

X(31385) = EULER LINE INTERCEPT OF X(108)X(1299)

Barycentrics a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^11-3 a^9 b^2+2 a^7 b^4+2 a^5 b^6-3 a^3 b^8+a b^10+a^9 b c-a^8 b^2 c-4 a^7 b^3 c+4 a^6 b^4 c+6 a^5 b^5 c-6 a^4 b^6 c-4 a^3 b^7 c+4 a^2 b^8 c+a b^9 c-b^10 c-3 a^9 c^2-a^8 b c^2+6 a^7 b^2 c^2+2 a^6 b^3 c^2-4 a^5 b^4 c^2+2 a^3 b^6 c^2-2 a^2 b^7 c^2-a b^8 c^2+b^9 c^2-4 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^3-6 a^2 b^6 c^3+2 b^8 c^3+2 a^7 c^4+4 a^6 b c^4-4 a^5 b^2 c^4+2 a^4 b^3 c^4+2 a^3 b^4 c^4+4 a^2 b^5 c^4-2 b^7 c^4+6 a^5 b c^5+4 a^2 b^4 c^5-2 a b^5 c^5+2 a^5 c^6-6 a^4 b c^6+2 a^3 b^2 c^6-6 a^2 b^3 c^6-4 a^3 b c^7-2 a^2 b^2 c^7-2 b^4 c^7-3 a^3 c^8+4 a^2 b c^8-a b^2 c^8+2 b^3 c^8+a b c^9+b^2 c^9+a c^10-b c^10) : :

As a point on the Euler line, X(31385) has Shinagawa coefficients {(p - r - 2 R) (p + r + 2 R) (p^2 - r^2 - 2 r R - 2 R^2), -p^4 - r^4 - 8 r^3 R - 22 r^2 R^2 - 28 r R^3 - 12 R^4 + 2 p^2 (r + R) (r + 3 R)}, or, equivalently, (-2(E+2F)F$a$-4Fabc, (E²+2EF+4F²)$a$+2Eabc).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28835.

X(31385) lies on these lines: {2,3}, {108,1299}

X(31385) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,16049}, {24,403,28}

X(31386) = X(3)X(20032)∩X(4)X(20034)

Barycentrics (Sin[B] Tan[A] Sqrt[Tan[B]]-Sin[A] Sqrt[Tan[A]] Tan[B]) (Sin[2 A] Sin[C] (Sin[C]-Sqrt[Tan[A] Tan[B]]) Tan[C]-Sin[A] Sin[2 C] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]]))- (Sin[C] Tan[A] Sqrt[Tan[C]]-Sin[A] Sqrt[Tan[A]] Tan[C]) (Sin[2 A] Sin[B] (Sin[B]-Sqrt[Tan[A] Tan[C]]) Tan[B]-Sin[A] Sin[2 B] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]])) : :
Barycentrics a SB SC (4 a (b^2-c^2) S^2 SA+b (a^6-2 a^4 b^2+a^2 b^4-5 a^4 c^2+b^4 c^2+5 a^2 c^4-c^6) Sqrt[SA SB]-c (a^6-5 a^4 b^2+5 a^2 b^4-b^6-2 a^4 c^2+a^2 c^4+b^2 c^4) Sqrt[SA SC]-4 a b c (b^2-c^2) SA Sqrt[SB SC]) : :

See Kadir Altintas and Peter Moses, Hyacinthos 28838.

X(31386) lies on the cubic K742 and these lines: {3,20032}, {4,20034}, {5,20033}

X(31387) = X(3)X(915)∩X(4)X(8)

Barycentrics (a^6-(b+c)^2*a^4-(b^2+c^2)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2*(b-c)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics SB*SC*(b*c+2*R^2-SW) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28843.

X(31387) lies on these lines: {3, 915}, {4, 8}, {11, 7040}, {24, 278}, {52, 5905}, {68, 2994}, {242, 3542}, {281, 1594}, {513, 5553}, {1068, 11398}, {1118, 1870}, {2973, 17170}, {3147, 17923}, {7510, 11396}, {10018, 17917}, {17516, 21664}

X(31388) = X(3)X(54)∩X(4)X(11197)

Barycentrics a^2 (a^2-b^2-c^2)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (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) : :
Barycentrics S^4 +(16 R^4+8 R^2 SB+8 R^2 SC-SB SC-16 R^2 SW-2 SB SW-2 SC SW+3 SW^2) S^2-32 R^4 SB SC+20 R^2 SB SC SW-3 SB SC SW^2 : :
X(31388) = 2*X[4]-3*X[11197], 7*X[3090]-6*X[14635], 5*X[3091]-6*X[10184], 7*X[3523]-6*X[12012]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31388) lies on these lines: {3,54}, {4,11197}, {5,14918}, {30,14978}, {160,6293}, {185,20775}, {216,14531}, {417,3917}, {418,5562}, {511,26897}, {577,8565}, {852,11793}, {1216,2972}, {3090,14635}, {3091,10184}, {3523,12012}, {3575,26166}, {6368,14329}, {6638,11444}, {6641,17834}, {11413,23709}, {15905,26216}, {18564,25043}

X(31388) = barycentric product of X(i) and X(j) for these {i,j}: {343, 13367}, {394, 3574}, {418, 26166}, {5562, 23292}
X(31388) = trilinear product of X(i) and X(j) for these {i,j}: {255, 3574}, {418, 17859}, {418, 17859}
X(31388) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11412,13409}, {97,7691,3}

X(31389) = X(2)X(22269)∩X(5)X(23607)

Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^4 b^2 c^2-3 a^2 b^4 c^2+6 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4-10 b^4 c^4+3 a^2 c^6+6 b^2 c^6-c^8) : :
Barycentrics 3 S^4 + (16 R^2 SB+16 R^2 SC+9 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 -16 R^4 SB SC+SB SC SW^2 : :
X(31389) = 6*X[2]-5*X[22269], 7*X[3090]-6*X[12013]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31389) lies on these lines: {2,22269}, {5,23607}, {381,2888}, {1209,11197}, {3078,14978}, {3090,12013}

X(31390) = REFLECTION OF X(23642) IN X(6)

Barycentrics a^2 (b^2+c^2) (2 a^4+a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : :
Barycentrics (8 R^2 SB+8 R^2 SC-4 R^2 SW-3 SB SW-3 SC SW+2 SW^2) S^2 -2 SB SC SW^2-SB SW^3-SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31390) lies on these lines: {6,22}, {32,23635}, {69,8878}, {141,23297}, {193,732}, {217,5052}, {511,8152}, {688,22260}, {1843,3051}, {3124,9969}, {3172,12167}, {3231,9822}, {3313,8041}, {3618,13410}, {9971,16285}, {11574,20965}

X(31390) = reflection of X(23642) in X(6)
X(31390) = barycentric product of X(i) and X(j) for these {i,j}: {31, 23665}, {39, 7745}, {427, 21637}, {1843, 6676}, {2084, 18063}
X(31390) = trilinear product of X(i) and X(j) for these {i,j}: {32, 23665}, {32, 23665}, {688, 18063}, {688, 18063}, {1964, 7745}, {1964, 7745}, {17442, 21637}, {17442, 21637}
X(31390) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,23642,11205}

X(31391) = MIDPOINT OF X(20059) AND X(25722)

Barycentrics a (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+6 a^2 b c-3 a b^2 c-4 b^3 c-3 a^2 c^2-3 a b c^2+10 b^2 c^2+3 a c^3-4 b c^3-c^4) : :
X(31391) = 4*X[7]-3*X[354], 2*X[144]-3*X[210], 2*X[960]-3*X[10861], 4*X[15008]-5*X[18398]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31391) lies on these lines: {4,10307}, {7,354}, {9,1155}, {37,3000}, {46,5779}, {55,2951}, {56,11372}, {57,3062}, {65,971}, {142,17605}, {144,210}, {226,5918}, {279,10939}, {513,2262}, {516,3057}, {518,1278}, {527,3059}, {960,10861}, {962,9850}, {990,1456}, {1001,8544}, {1122,4014}, {1418,2310}, {1420,24644}, {1445,16112}, {1721,6180}, {1770,5762}, {1864,11246}, {2646,5732}, {2801,17636}, {3304,10384}, {3600,10866}, {3698,5229}, {3748,4326}, {3983,5128}, {4292,12688}, {4295,12680}, {4298,9848}, {4321,20323}, {4336,6610}, {4887,14523}, {4907,7271}, {5221,10398}, {5226,10178}, {5728,17637}, {5784,17768}, {5805,7702}, {5817,24914}, {5843,14872}, {7988,11575}, {8545,11495}, {10442,21334}, {11375,21151}, {12764,18482}, {15008,18398}, {15185,17660}

X(31391) = midpoint of X(20059 ) and X(25722)
X(31391) = reflection of X(i) in X(j) for these {i,j}: {65,4312}, {144,15587}, {3057,8581}, {3059,17668}, {14100,7}
X(31391) = X(193)-of-intouch-triangle
X(31391) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {7,14100,354}, {144,15587,210}, {4014,12723,1122}, {7354,17634,3057}, {8545,11495,15837}

X(31392) = REFLECTION OF X(5) IN X(14051)

Barycentrics (a^4+(b^2-c^2)^2-a^2 (b^2+2 c^2)) (a^12-(b^2-c^2)^6-a^10 (4 b^2+3 c^2)+a^8 (5 b^4+5 b^2 c^2+3 c^4)-a^6 (3 b^4 c^2+2 b^2 c^4)+a^2 (b^2-c^2)^2 (4 b^6-6 b^4 c^2-3 b^2 c^4+3 c^6)+a^4 (-5 b^8+9 b^6 c^2+b^4 c^4+4 b^2 c^6-3 c^8)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28850.

Let A' be the reflection of A in BC. Let A_b be the intersection of AC and the perpendicular bisector of A'B, and let A_c be the intersection of AB and the perpendicular bisector of A'C. Let A'' be the orthogonal projection of A' on A_bA_c, and define B'' and C'' cyclically. The lines AA''', BB'', CC'' concur in X(31392). (Angel Montesdeoca, July 16, 2020)

X(31392) lies on these lines: {4,93}, {5,930}, {17,8173}, {18,8172}, {30,252}, {54,1263}, {143,11538}, {195,20414}, {265,6798}, {550,1487}, {1157,28237}, {1879,2937}, {2070,21394}, {3459,6150}, {11671,15345}, {20030,20424}, {20413,27090}, {21230,24306}

X(31392) = reflection of X(i) in X(j) for these {i, j}: {5,14051}, {930,19268}
X(31392) = isogonal conjugate of X(18212)

Centroids of curvatures: X(31393)-X(31502)

This preamble and centers X(31393)-X(31502) were contributed by César Eliud Lozada, February 08, 2019.

Let (O₁), (O₂), (O₃) be three circles with distinct and non-collinear centers. Denote I₁, I₂, I₃ the internal centers of similitude of {(O₂), (O₃)}, {(O₃), (O₁)} and {(O₁), (O₂)} , respectively. Then the lines O₁I₁, O₂I₂, O₃I₃ concur.

Assume that exact trilinear coordinates of the centers are O_i = ( U_i, V_i, W_i ) and their radius are R_i (i = 1..3). If M is the trilinear matrix of the centers then the given point of intersection Q is:

Q = M^T.| ρ₁ ρ₂ ρ₃ |^T, where ρ_i = 1/R_i, i.e., ρ_i is the curvature of the circle (O_i).

In fewer words:

Q = ∑ ρ_iU_i : ∑ ρ_iV_i : ∑ ρ_iW_i, i=1..3

Then, for obvious reasons, the point Q is here named here the centroid of curvatures of the circles (O₁), (O₂), (O₃).

Suppose a fourth circle (O₄) is added to the above configuration and let Q_ℓ be the centroid of curvatures of (O_i), (O_j), (O_k). Then the lines O_iQ_i concur at:

Q = ∑ ρ_iU_i : ∑ ρ_iV_i : ∑ ρ_iW_i, i=1..4

The last extension provides a geometrical recursive construction of the centroid of curvatures of any number n ≥ 3 of circles with distinct and not collinear centers.

The appearance of (C, n) in the following list means that X(n) is the centroid of curvatures of the three circles C:
(excircles, 1), (excosine, 6), (Johnson, 381), (Lucas(+1), 6221), (Lucas(+1) secondary, 3311), (Lucas(-1), 6398), (Lucas(-1) secondary, 3312), (Malfatti, 31495), (McCay, 7611), (mixtilinear incircles, 31393), (mixtilinear excircles, 1), (Neuberg, 31394), (2nd Neuberg, 31395), (Soddy, 7) (Yff internal, 10056), (Yff external, 10072).

The appearance of (C₁,C₂,C₃,n) in the following list means that X(n) is the centroid of curvatures of the three circles C₁, C₂, C₃:
(anticomplementary, Bevan, circumcircle, 6684), (anticomplementary, Bevan, half-Moses, 31396), (anticomplementary, Bevan, incircle, 31397), (anticomplementary, Bevan, 2nd Lemoine, 13883), (anticomplementary, Bevan, Moses, 31398), (anticomplementary, Bevan, nine-points, 10175), (anticomplementary, Bevan, Stammler, 26446), (anticomplementary, Bevan, Steiner, 31399), (anticomplementary, circumcircle, half-Moses, 31400), (anticomplementary, circumcircle, incircle, 3085), (anticomplementary, circumcircle, 2nd Lemoine, 3068), (anticomplementary, circumcircle, Moses, 31401), (anticomplementary, circumcircle, Spieker, 19843), (anticomplementary, half-Moses, incircle, 31402), (anticomplementary, half-Moses, 2nd Lemoine, 31403), (anticomplementary, half-Moses, nine-points, 31404), (anticomplementary, half-Moses, Spieker, 31405), (anticomplementary, half-Moses, Stammler, 31406), (anticomplementary, half-Moses, Steiner, 31407), (anticomplementary, incircle, 2nd Lemoine, 31408), (anticomplementary, incircle, Moses, 31409), (anticomplementary, incircle, nine-points, 10590), (anticomplementary, incircle, Spieker, 443), (anticomplementary, incircle, Stammler, 495), (anticomplementary, incircle, Steiner, 31410), (anticomplementary, 2nd Lemoine, Moses, 31411), (anticomplementary, 2nd Lemoine, nine-points, 31412), (anticomplementary, 2nd Lemoine, Spieker, 31413), (anticomplementary, 2nd Lemoine, Stammler, 7583), (anticomplementary, 2nd Lemoine, Steiner, 31414), (anticomplementary, Moses, nine-points, 31415), (anticomplementary, Moses, Spieker, 31416), (anticomplementary, Moses, Stammler, 3815), (anticomplementary, Moses, Steiner, 31417), (anticomplementary, nine-points, Spieker, 31418), (anticomplementary, Spieker, Stammler, 31419), (anticomplementary, Spieker, Steiner, 31420), (Bevan, circumcircle, half-Moses, 31421), (Bevan, circumcircle, 2nd Lemoine, 9616), (Bevan, circumcircle, Moses, 31422), (Bevan, circumcircle, nine-points, 31423), (Bevan, circumcircle, Spieker, 31424), (Bevan, circumcircle, Steiner, 31425), (Bevan, half-Moses, incircle, 31426), (Bevan, half-Moses, 2nd Lemoine, 31427), (Bevan, half-Moses, nine-points, 31428), (Bevan, half-Moses, Spieker, 31429), (Bevan, half-Moses, Stammler, 31430), (Bevan, half-Moses, Steiner, 31431), (Bevan, incircle, 2nd Lemoine, 31432), (Bevan, incircle, Moses, 31433), (Bevan, incircle, nine-points, 31434), (Bevan, incircle, Spieker, 31435), (Bevan, incircle, Steiner, 31436), (Bevan, 2nd Lemoine, Moses, 31437), (Bevan, 2nd Lemoine, nine-points, 13893), (Bevan, 2nd Lemoine, Spieker, 31438), (Bevan, 2nd Lemoine, Stammler, 31439), (Bevan, 2nd Lemoine, Steiner, 31440), (Bevan, Moses, nine-points, 31441), (Bevan, Moses, Spieker, 31442), (Bevan, Moses, Stammler, 31443), (Bevan, Moses, Steiner, 31444), (Bevan, nine-points, Spieker, 5705), (Bevan, nine-points, Stammler, 11231), (Bevan, Spieker, Stammler, 31445), (Bevan, Spieker, Steiner, 31446), (Bevan, Stammler, Steiner, 31447), (circumcircle, half-Moses, incircle, 31448), (circumcircle, half-Moses, nine-points, 31401), (circumcircle, half-Moses, Spieker, 31449), (circumcircle, half-Moses, Steiner, 31450), (circumcircle, incircle, 2nd Lemoine, 2066), (circumcircle, incircle, Moses, 31451), (circumcircle, incircle, nine-points, 498), (circumcircle, incircle, Spieker, 405), (circumcircle, incircle, Steiner, 31452), (circumcircle, 2nd Lemoine, nine-points, 590), (circumcircle, 2nd Lemoine, Spieker, 31453), (circumcircle, 2nd Lemoine, Steiner, 31454), (circumcircle, Moses, nine-points, 31455), (circumcircle, Moses, Spieker, 31456), (circumcircle, Moses, Steiner, 31457), (circumcircle, nine-points, Spieker, 26363), (circumcircle, Spieker, Steiner, 31458), (half-Moses, incircle, 2nd Lemoine, 31459), (half-Moses, incircle, nine-points, 31460), (half-Moses, incircle, Spieker, 5283), (half-Moses, incircle, Stammler, 31461), (half-Moses, incircle, Steiner, 31462), (half-Moses, 2nd Lemoine, nine-points, 31463), (half-Moses, 2nd Lemoine, Spieker, 31464), (half-Moses, 2nd Lemoine, Steiner, 31465), (half-Moses, nine-points, Spieker, 31466), (half-Moses, nine-points, Stammler, 31467), (half-Moses, Spieker, Stammler, 31468), (half-Moses, Spieker, Steiner, 31469), (half-Moses, Stammler, Steiner, 31470), (incircle, 2nd Lemoine, Moses, 31471), (incircle, 2nd Lemoine, nine-points, 31472), (incircle, 2nd Lemoine, Spieker, 31473), (incircle, 2nd Lemoine, Stammler, 31474), (incircle, 2nd Lemoine, Steiner, 31475), (incircle, Moses, nine-points, 31476), (incircle, Moses, Spieker, 16589), (incircle, Moses, Stammler, 31477), (incircle, Moses, Steiner, 31478), (incircle, nine-points, Spieker, 442), (incircle, nine-points, Stammler, 31479), (incircle, Spieker, Stammler, 11108), (incircle, Spieker, Steiner, 17529), (incircle, Stammler, Steiner, 31480), (2nd Lemoine, Moses, nine-points, 31481), (2nd Lemoine, Moses, Spieker, 31482), (2nd Lemoine, Moses, Steiner, 31483), (2nd Lemoine, nine-points, Spieker, 31484), (2nd Lemoine, nine-points, Stammler, 8976), (2nd Lemoine, Spieker, Stammler, 31485), (2nd Lemoine, Spieker, Steiner, 31486), (2nd Lemoine, Stammler, Steiner, 31487), (Moses, nine-points, Spieker, 31488), (Moses, nine-points, Stammler, 31489), (Moses, Spieker, Stammler, 31490), (Moses, Spieker, Steiner, 31491), (Moses, Stammler, Steiner, 31492), (nine-points, Spieker, Stammler, 31493), (Spieker, Stammler, Steiner, 31494).

The appearance of (C, n) in the following list means that X(n) is the centroid of curvatures of the circles { C, circumcircle, incircle, nine-points }:
(Apollonius, 31496), (extangents, 31498), (half-Moses, 31497), (2nd Johnson-Yff, 55), (2nd Lemoine, 9646), (radical circle of Lucas circles, 31499), (inner Lucas, 31500), (Moses, 31501), (Parry, 31502), (Spieker, 2)

For definitions of all these circles, see Triangle circles at Wolfram Mathworld. Note: Only some few circles in this list are treated here.

X(31393) = CENTROID OF CURVATURES OF THE MIXTILINEAR INCIRCLES

Barycentrics    a*(a^3+(b+c)*a^2-(b^2+10*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 10*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :
Trilinears    cos A - cos B - cos C + 5 : :
X(31393) = 5*X(1)-2*X(1159) = 3*X(1)-2*X(15934) = 3*X(1)-X(18421) = 3*X(40)-4*X(3587) = X(1000)+2*X(30331) = 4*X(1001)-X(11525) = X(4900)-6*X(16857) = 2*X(7966)+X(11372) = 2*X(14563)-3*X(15933) = 4*X(15935)-X(16236)
X(31393) = (4*R - r)*X[1] + 2*r*X[3], 4*R*X[1] + r*X[40], 5*R*X[1] + (r - R)*X[46], 6*R*X[1] + (r - 2*R)*X[57], 8*R*X[3] + (r - 4*R)*X[40], 9*R*X[36] + (r - 5*R)*X[484], 5*r*X[40] - 4*(r - R)*X[46], 3*r*X[40] - 2*(r - 2*R)*X[57], r*(r + 6*R)*X[40] + 8*R^2*X[65], (r - 8*R)*X[40] + 12*R*X[165]

Let A'B'C' be the mixtilinear incentral triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(31393). (Randy Hutson, March 21, 2019)

X(31393) lies on these lines {1, 3}, {2, 3895}, {4, 12575}, {7, 28194}, {8, 3305}, {9, 519}, {10, 1058}, {11, 31434}, {12, 9614}, {21, 36846}, {63, 3241}, {78, 3890}, {84, 4313}, {90, 13606}, {145, 3219}, {164, 32183}, {191, 15174}, {200, 392}, {226, 30305}, {355, 15172}, {376, 4315}, {380, 2256}, {381, 5726}, {388, 10624}, {390, 515}, {405, 4853}, {495, 1699}, {496, 1698}, {497, 5587}, {498, 50443}, {516, 1056}, {551, 5437}, {758, 3243}, {908, 11239}, {936, 3913}, {938, 11362}, {943, 3680}, {944, 4314}, {946, 5226}, {950, 5881}, {952, 10384}, {956, 4512}, {958, 12629}, {960, 6765}, {962, 21620}, {997, 3158}, {1001, 3880}, {1015, 9574}, {1125, 1706}, {1158, 13607}, {1334, 16572}, {1335, 31432}, {1376, 10179}, {1387, 5541}, {1419, 1480}, {1445, 14563}, {1449, 2267}, {1453, 3915}, {1478, 9580}, {1479, 9578}, {1490, 45776}, {1500, 9575}, {1512, 10596}, {1616, 4646}, {1621, 3872}, {1656, 50444}, {1702, 3298}, {1703, 3297}, {1728, 37724}, {1768, 12735}, {1837, 45081}, {1953, 2270}, {2275, 31426}, {2346, 3577}, {2800, 7675}, {2802, 38316}, {2809, 7174}, {2999, 16483}, {3058, 3586}, {3085, 6964}, {3086, 31423}, {3208, 36479}, {3244, 6762}, {3306, 38314}, {3421, 40998}, {3452, 34619}, {3476, 4304}, {3487, 4301}, {3555, 12526}, {3584, 23708}, {3600, 31730}, {3624, 11373}, {3632, 37730}, {3633, 37739}, {3634, 47743}, {3655, 7171}, {3656, 5719}, {3671, 41870}, {3679, 4863}, {3740, 8168}, {3752, 16486}, {3753, 10582}, {3811, 3884}, {3813, 5705}, {3817, 8164}, {3869, 41863}, {3870, 3877}, {3871, 19861}, {3878, 11523}, {3885, 19860}, {3902, 5271}, {3928, 51071}, {3929, 37728}, {3982, 4295}, {4255, 45219}, {4271, 48846}, {4294, 10106}, {4298, 6361}, {4307, 28881}, {4309, 45287}, {4312, 28174}, {4326, 6001}, {4342, 5603}, {4420, 4917}, {4642, 28011}, {4653, 18163}, {4737, 30568}, {4857, 10827}, {4859, 20328}, {4866, 45830}, {4882, 5044}, {4900, 16857}, {4915, 9708}, {5084, 6736}, {5180, 31164}, {5218, 44675}, {5219, 10056}, {5227, 49681}, {5234, 11519}, {5261, 18483}, {5274, 10175}, {5281, 10165}, {5290, 12699}, {5438, 8715}, {5493, 12577}, {5542, 28228}, {5552, 25522}, {5559, 7162}, {5657, 11019}, {5687, 8583}, {5691, 15171}, {5703, 13464}, {5727, 12647}, {5731, 10860}, {5745, 34625}, {5774, 35613}, {5777, 9848}, {5790, 18527}, {5844, 8275}, {6198, 7713}, {6284, 9613}, {6326, 15558}, {6684, 14986}, {6738, 12245}, {7190, 23839}, {7284, 13602}, {7289, 49465}, {7290, 40091}, {7330, 37727}, {7686, 12260}, {7701, 10543}, {7743, 7988}, {7989, 9669}, {7995, 12680}, {8077, 10968}, {8236, 28234}, {9581, 10039}, {9592, 31477}, {9593, 16781}, {9612, 12701}, {9616, 35768}, {9624, 13411}, {9851, 12684}, {10072, 31231}, {10197, 21630}, {10198, 49600}, {10386, 18481}, {10528, 41012}, {10572, 37709}, {10573, 37723}, {10695, 34925}, {10914, 25893}, {11037, 20070}, {11041, 43179}, {11218, 11374}, {11496, 12650}, {12127, 31445}, {12128, 31805}, {12407, 46687}, {12513, 31424}, {12758, 37736}, {12844, 31766}, {13463, 28628}, {15935, 16236}, {16485, 49487}, {16574, 48858}, {17019, 39592}, {17306, 48803}, {17448, 31429}, {17754, 48830}, {17802, 34495}, {17805, 34494}, {18525, 31795}, {18528, 30294}, {18530, 30286}, {18540, 28204}, {18541, 28198}, {18743, 51284}, {18991, 35808}, {18992, 35809}, {19004, 31474}, {19843, 21627}, {21454, 34632}, {21625, 43174}, {24590, 29624}, {27065, 31145}, {28038, 34937}, {28150, 30332}, {30117, 35227}, {30223, 37740}, {30275, 38036}, {30827, 45701}, {31165, 41711}, {31188, 38068}, {31436, 37722}, {33903, 40214}, {34231, 40971}, {36973, 50836}, {39779, 43166}, {40131, 48856}, {42871, 44663}, {46917, 48696}
X(31393) lies on these lines {1, 3}, {2, 3895}, {4, 12575}, {7, 28194}, {8, 3305}, {9, 519}, {10, 1058}, {11, 31434}, {12, 9614}, {21, 36846}, {63, 3241}, {78, 3890}, {84, 4313}, {90, 13606}, {145, 3219}, {164, 32183}, {191, 15174}, {200, 392}, {226, 30305}, {355, 15172}, {376, 4315}, {380, 2256}, {381, 5726}, {388, 10624}, {390, 515}, {405, 4853}, {495, 1699}, {496, 1698}, {497, 5587}, {498, 50443}, {516, 1056}, {551, 5437}, {758, 3243}, {908, 11239}, {936, 3913}, {938, 11362}, {943, 3680}, {944, 4314}, {946, 5226}, {950, 5881}, {952, 10384}, {956, 4512}, {958, 12629}, {960, 6765}, {962, 21620}, {997, 3158}, {1001, 3880}, {1015, 9574}, {1125, 1706}, {1158, 13607}, {1334, 16572}, {1335, 31432}, {1376, 10179}, {1387, 5541}, {1419, 1480}, {1445, 14563}, {1449, 2267}, {1453, 3915}, {1478, 9580}, {1479, 9578}, {1490, 45776}, {1500, 9575}, {1512, 10596}, {1616, 4646}, {1621, 3872}, {1656, 50444}, {1702, 3298}, {1703, 3297}, {1728, 37724}, {1768, 12735}, {1837, 45081}, {1953, 2270}, {2275, 31426}, {2346, 3577}, {2800, 7675}, {2802, 38316}, {2809, 7174}, {2999, 16483}, {3058, 3586}, {3085, 6964}, {3086, 31423}, {3208, 36479}, {3244, 6762}, {3306, 38314}, {3421, 40998}, {3452, 34619}, {3476, 4304}, {3487, 4301}, {3555, 12526}, {3584, 23708}, {3600, 31730}, {3624, 11373}, {3632, 37730}, {3633, 37739}, {3634, 47743}, {3655, 7171}, {3656, 5719}, {3671, 41870}, {3679, 4863}, {3740, 8168}, {3752, 16486}, {3753, 10582}, {3811, 3884}, {3813, 5705}, {3817, 8164}, {3869, 41863}, {3870, 3877}, {3871, 19861}, {3878, 11523}, {3885, 19860}, {3902, 5271}, {3928, 51071}, {3929, 37728}, {3982, 4295}, {4255, 45219}, {4271, 48846}, {4294, 10106}, {4298, 6361}, {4307, 28881}, {4309, 45287}, {4312, 28174}, {4326, 6001}, {4342, 5603}, {4420, 4917}, {4642, 28011}, {4653, 18163}, {4737, 30568}, {4857, 10827}, {4859, 20328}, {4866, 45830}, {4882, 5044}, {4900, 16857}, {4915, 9708}, {5084, 6736}, {5180, 31164}, {5218, 44675}, {5219, 10056}, {5227, 49681}, {5234, 11519}, {5261, 18483}, {5274, 10175}, {5281, 10165}, {5290, 12699}, {5438, 8715}, {5493, 12577}, {5542, 28228}, {5552, 25522}, {5559, 7162}, {5657, 11019}, {5687, 8583}, {5691, 15171}, {5703, 13464}, {5727, 12647}, {5731, 10860}, {5745, 34625}, {5774, 35613}, {5777, 9848}, {5790, 18527}, {5844, 8275}, {6198, 7713}, {6284, 9613}, {6326, 15558}, {6684, 14986}, {6738, 12245}, {7190, 23839}, {7284, 13602}, {7289, 49465}, {7290, 40091}, {7330, 37727}, {7686, 12260}, {7701, 10543}, {7743, 7988}, {7989, 9669}, {7995, 12680}, {8077, 10968}, {8236, 28234}, {9581, 10039}, {9592, 31477}, {9593, 16781}, {9612, 12701}, {9616, 35768}, {9624, 13411}, {9851, 12684}, {10072, 31231}, {10197, 21630}, {10198, 49600}, {10386, 18481}, {10528, 41012}, {10572, 37709}, {10573, 37723}, {10695, 34925}, {10914, 25893}, {11037, 20070}, {11041, 43179}, {11218, 11374}, {11496, 12650}, {12127, 31445}, {12128, 31805}, {12407, 46687}, {12513, 31424}, {12758, 37736}, {12844, 31766}, {13463, 28628}, {15935, 16236}, {16485, 49487}, {16574, 48858}, {17019, 39592}, {17306, 48803}, {17448, 31429}, {17754, 48830}, {17802, 34495}, {17805, 34494}, {18525, 31795}, {18528, 30294}, {18530, 30286}, {18540, 28204}, {18541, 28198}, {18743, 51284}, {18991, 35808}, {18992, 35809}, {19004, 31474}, {19843, 21627}, {21454, 34632}, {21625, 43174}, {24590, 29624}, {27065, 31145}, {28038, 34937}, {28150, 30332}, {30117, 35227}, {30223, 37740}, {30275, 38036}, {30827, 45701}, {31165, 41711}, {31188, 38068}, {31436, 37722}, {33903, 40214}, {34231, 40971}, {36973, 50836}, {39779, 43166}, {40131, 48856}, {42871, 44663}, {46917, 48696}

X(31393) = midpoint of X(i) and X(j) for these {i,j}: {1, 9819}, {1000, 3488}
X(31393) = reflection of X(i) in X(j) for these (i,j): (1, 6767), (3488, 30331), (4915, 9708), (9623, 1001), (11525, 9623), (11529, 1), (18421, 15934)
X(31393) = (Aquila)-isogonal conjugate of-X(11034)
X(31393) = X(1597)-of-excentral triangle
X(31393) = X(9818)-of-Hutson intouch triangle
X(31393) = X(9819)-of-anti-Aquila triangle
X(31393) = X(11529)-of-5th mixtilinear triangle
X(31393) = X(18494)-of-excenters-reflections triangle
X(31393) = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X(40)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35, 1420}, {1, 40, 3333}, {1, 55, 3576}, {1, 165, 999}, {1, 1697, 40}, {1, 2093, 354}, {1, 3057, 7982}, {1, 3339, 5045}, {1, 3361, 7373}, {1, 3746, 3601}, {1, 4424, 3677}, {1, 5119, 57}, {1, 5255, 37554}, {1, 5697, 3340}, {1, 5902, 44841}, {1, 5903, 11518}, {1, 7962, 16200}, {1, 7987, 24928}, {1, 7991, 942}, {1, 10388, 37569}, {1, 10980, 5049}, {1, 11010, 3338}, {1, 11224, 50194}, {1, 15803, 3304}, {1, 18421, 15934}, {1, 30282, 1319}, {1, 30337, 9957}, {1, 30392, 25405}, {1, 31508, 13462}, {1, 37563, 46}, {1, 37610, 5269}, {3, 31792, 1}, {35, 13370, 3}, {40, 7982, 6766}, {55, 1319, 30282}, {55, 1470, 35}, {55, 5919, 1}, {57, 1697, 5119}, {57, 5119, 40}, {84, 5882, 9845}, {226, 30305, 31162}, {354, 8162, 1}, {388, 10624, 41869}, {497, 31397, 5587}, {944, 12705, 10864}, {997, 25439, 3158}, {1015, 31433, 9574}, {1319, 30282, 3576}, {1385, 49163, 37560}, {1479, 9578, 18492}, {1697, 37556, 1}, {2098, 37080, 1}, {2099, 3303, 3748}, {2099, 3748, 1}, {3057, 3303, 1}, {3057, 3748, 2099}, {3058, 5252, 3586}, {3085, 12053, 8227}, {3244, 12514, 6762}, {3295, 9957, 1}, {3304, 37568, 15803}, {3338, 11010, 5128}, {3476, 4304, 50811}, {3476, 10385, 4304}, {3576, 12703, 40}, {3579, 7373, 3361}, {3587, 15934, 57}, {3811, 3884, 15829}, {3898, 25439, 997}, {4342, 13405, 5603}, {5045, 12702, 3339}, {5049, 36279, 10980}, {5128, 11010, 40}, {5219, 30384, 38021}, {6767, 9819, 11529}, {7743, 31479, 7988}, {7962, 10389, 1}, {7991, 37551, 40}, {10056, 30384, 5219}, {11362, 40270, 938}, {12701, 15888, 9612}, {13462, 31508, 3}, {15934, 18421, 11529}, {18421, 30282, 30503}, {25405, 37606, 30392}, {37542, 37548, 1}, {37560, 49163, 40}, {37709, 41864, 10572}, {45711, 45712, 9957}

X(31394) = CENTROID OF CURVATURES OF THE 1st NEUBERG CIRCLES

Barycentrics a*((b+c)*a^4-b*c*a^3-(b^3+c^3)*a^2-(b^2+c^2)*b*c*a+(b^2-c^2)*(b-c)*b*c) : :
X(31394) = 2*X(37)-3*X(7611) = X(1742)-3*X(3576) = 3*X(5603)-X(10446) = 3*X(5886)-2*X(24220) = 3*X(7611)-X(31395)
X(31394) = N_A/R_A + N_B/R_B + N_C/R_C, where N_A, N_B, N_C are the centers of the 1st Neuberg circles, and R_A, R_B, R_C are their radii

X(31394) lies on these lines: {1,256}, {3,142}, {4,17913}, {5,4026}, {6,15953}, {9,15507}, {21,17202}, {25,30687}, {37,517}, {38,11203}, {40,13731}, {55,17720}, {75,2783}, {98,30670}, {104,13396}, {165,25502}, {182,238}, {226,23853}, {228,3434}, {344,5657}, {390,7390}, {528,15624}, {575,16468}, {576,4649}, {631,25492}, {956,4416}, {991,1279}, {993,2792}, {995,28358}, {999,3664}, {1006,29105}, {1064,4116}, {1361,24806}, {1402,26098}, {1403,24239}, {1482,29311}, {1503,15976}, {1621,4220}, {1699,4192}, {1742,3576}, {1757,7609}, {1836,16678}, {2187,25885}, {2788,4455}, {2886,3185}, {3073,13323}, {3624,19514}, {3705,11688}, {3753,25099}, {3817,19540}, {3870,21319}, {3932,5690}, {4078,11362}, {4184,17174}, {4423,16434}, {4512,8731}, {4847,20760}, {4851,15571}, {5092,15485}, {5143,17717}, {5259,13732}, {5263,6998}, {5272,28364}, {5284,19649}, {5603,10446}, {5794,23846}, {6842,15666}, {7988,19546}, {8227,19513}, {8299,9746}, {8666,17770}, {8692,12017}, {9779,19647}, {9955,19543}, {10246,29353}, {10527,22345}, {10882,11522}, {10902,19548}, {11230,17384}, {11231,17357}, {11248,19547}, {13624,29229}, {13724,19860}, {14636,31162}, {16419,25893}, {16589,20606}, {16825,24257}, {17279,26446}, {17594,21321}, {20368,26102}, {37519,25514}, {23844,26066}, {24325,29057}, {24331,24728}, {24357,29069}, {24541,28348}, {29073,30273}

X(31394) = midpoint of X(1) and X(6210)
X(31394) = reflection of X(i) in X(j) for these (i,j): (991, 1385), (31395, 37)
X(31394) = X(6210)-of-anti-Aquila triangle
X(31394) = X(30258)-of-2nd circumperp triangle
X(31394) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1756, 1469), (7611, 31395, 37)

X(31395) = CENTROID OF CURVATURES OF THE 2nd NEUBERG CIRCLES

Barycentrics a*(b*c*a^3-(b+c)*(b^2+b*c+c^2)*a^2+(b^2+c^2)*b*c*a+(b^2-c^2)*(b^3-c^3)) : :
X(31395) = 4*X(37)-3*X(7611) = 3*X(7611)-2*X(31394)
X(31395) = N'_A/R'_A + N'_B/R'_B + N'_C/R'_C, where N'_A, N'_B, N'_C are the centers of the 2nd Neuberg circles, and R'_A, R'_B, R'_C are their radii

X(31395) lies on these lines: {1,182}, {3,20990}, {4,29073}, {5,3932}, {37,517}, {192,2783}, {262,4518}, {344,5603}, {355,29016}, {392,25099}, {511,984}, {516,20430}, {576,1757}, {946,4078}, {1001,1482}, {1279,24680}, {1284,5903}, {1351,5220}, {1486,10679}, {2177,21326}, {2810,18161}, {2937,11849}, {3098,18788}, {3434,21807}, {3870,21318}, {4026,5690}, {4657,26446}, {5657,17321}, {5886,17279}, {5988,12837}, {8931,12497}, {11230,17357}, {11231,17384}, {13405,20254}, {14853,27549}, {16434,17599}, {16577,23853}, {16593,20330}, {17594,21333}, {24206,29674}

X(31395) = reflection of X(31394) in X(37)
X(31395) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6211, 182), (37, 31394, 7611)

X(31396) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, HALF-MOSES}

Barycentrics (b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :

X(31396) lies on these lines: {1,31400}, {2,9593}, {4,9574}, {6,6684}, {8,9592}, {10,39}, {20,31421}, {30,31430}, {32,10164}, {37,9843}, {40,7736}, {57,31402}, {226,31460}, {355,5024}, {497,31426}, {515,5013}, {516,1571}, {517,31406}, {519,9619}, {574,4297}, {946,3815}, {950,31448}, {1125,9620}, {1210,2276}, {1500,11019}, {1506,3817}, {1588,31427}, {1698,5286}, {1699,31404}, {1703,31403}, {1706,31405}, {2275,31397}, {2549,19925}, {2551,31429}, {3501,24239}, {3634,3767}, {3828,7739}, {3947,31476}, {4292,9596}, {4298,31409}, {4301,9698}, {4314,31451}, {5218,16780}, {5254,10175}, {5283,8582}, {5305,11231}, {5530,17754}, {5587,7738}, {5657,9575}, {5722,31461}, {5795,31449}, {5886,31467}, {6421,13883}, {6422,13936}, {6734,17756}, {6736,16975}, {7735,31423}, {7737,12512}, {7745,31443}, {9589,31407}, {9599,10624}, {9605,26446}, {9606,11362}, {9956,15048}, {10172,13881}, {12571,31415}, {12575,31433}, {18250,31442}, {19862,31455}

X(31396) = midpoint of X(1571) and X(2548)
X(31396) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31398, 10), (9593, 31428, 2)

X(31397) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, INCIRCLE}

Barycentrics (b+c)*a^3-(b^2+6*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(31397) = X(1)-3*X(10056) = 3*X(2)+X(12648) = X(8)+3*X(11239) = X(1836)-3*X(11237) = X(2099)-3*X(17718) = X(4304)+2*X(5252) = 3*X(10956)-X(12831)

X(31397) lies on these lines: {1,2}, {3,4311}, {4,1697}, {5,7743}, {7,2093}, {9,3421}, {11,5919}, {12,946}, {20,9613}, {35,4297}, {36,4315}, {37,1146}, {38,1735}, {40,388}, {46,4298}, {55,515}, {56,6684}, {57,1056}, {65,11362}, {72,5837}, {80,2346}, {92,1785}, {119,15558}, {140,24928}, {142,1145}, {153,29007}, {165,4293}, {226,495}, {255,5264}, {322,4357}, {355,950}, {390,3586}, {392,3452}, {405,5795}, {442,10914}, {443,1706}, {496,9956}, {497,5587}, {516,1478}, {518,8255}, {529,4640}, {549,5126}, {553,3654}, {595,3074}, {611,5847}, {631,1420}, {726,10063}, {908,3877}, {942,4848}, {944,3601}, {952,24929}, {954,3419}, {956,5745}, {960,12607}, {962,5261}, {993,8069}, {999,3911}, {1000,1512}, {1006,2078}, {1015,31398}, {1058,5818}, {1064,4551}, {1074,1111}, {1124,13936}, {1155,5434}, {1167,1220}, {1317,10265}, {1319,5432}, {1335,13883}, {1479,6957}, {1482,11374}, {1496,1771}, {1497,1724}, {1572,31409}, {1588,31432}, {1621,5176}, {1656,11373}, {1699,5726}, {1703,31408}, {1739,24175}, {1770,5270}, {1788,3333}, {1836,11237}, {1837,3303}, {1864,18908}, {2067,13912}, {2098,11375}, {2099,17718}, {2136,5082}, {2269,10445}, {2275,31396}, {2292,17874}, {2476,3885}, {2549,31433}, {2550,24409}, {2551,31435}, {2646,5882}, {2784,10053}, {2796,10054}, {2800,10956}, {2801,18801}, {2802,3822}, {2886,3880}, {3091,9614}, {3245,30424}, {3297,13973}, {3298,13911}, {3304,24914}, {3338,12577}, {3340,3487}, {3361,9588}, {3434,3895}, {3436,5250}, {3475,11529}, {3476,3576}, {3485,7982}, {3486,5881}, {3488,5727}, {3523,4308}, {3543,30332}, {3555,24391}, {3579,18990}, {3600,15803}, {3612,6966}, {3671,5903}, {3672,24213}, {3680,6856}, {3681,18397}, {3692,17355}, {3695,10396}, {3710,4696}, {3717,4737}, {3744,5724}, {3746,4314}, {3812,8256}, {3814,3898}, {3816,5123}, {3817,4342}, {3820,5316}, {3833,18240}, {3873,30274}, {3874,13750}, {3878,10954}, {3884,21616}, {3890,11681}, {3913,5794}, {3947,4301}, {4002,17529}, {4294,5691}, {4295,5290}, {4299,12512}, {4302,28164}, {4642,23536}, {4652,20076}, {4679,31141}, {4711,5572}, {4999,11260}, {5010,21578}, {5048,15950}, {5056,7320}, {5083,10202}, {5183,11246}, {5225,18492}, {5248,11508}, {5281,5731}, {5296,31325}, {5433,20323}, {5450,22759}, {5534,10393}, {5542,5902}, {5559,11009}, {5570,5883}, {5686,10398}, {5692,21060}, {5710,5717}, {5719,5844}, {5722,5790}, {5728,24393}, {5817,10384}, {5828,8165}, {5836,25466}, {5886,31479}, {6260,12672}, {6361,9579}, {6502,13975}, {6706,24181}, {6796,11501}, {6842,23340}, {6847,12650}, {6974,28236}, {7288,31423}, {7738,31426}, {7967,13384}, {7989,10591}, {8068,21630}, {8071,25440}, {8227,10588}, {8275,11224}, {8666,22766}, {8715,11507}, {8983,9646}, {9575,31402}, {9589,31410}, {9654,12699}, {9803,30284}, {9948,12680}, {9955,10592}, {10057,10087}, {10058,12749}, {10064,17766}, {10122,12670}, {10171,23708}, {10523,25639}, {10895,12701}, {10955,20117}, {11023,11024}, {11231,15325}, {11236,24703}, {12019,15170}, {12115,12686}, {12513,26066}, {12514,12527}, {12608,26482}, {12619,12735}, {12667,12705}, {12711,14872}, {12758,21635}, {12763,16140}, {12943,28150}, {15171,18480}, {15172,18357}, {16120,17646}, {16777,24005}, {17527,20789}, {17606,31399}, {21031,25917}, {21290,25101}, {21627,24390}, {23529,24010}, {23675,24171}, {24178,24440}, {24215,24464}

X(31397) = midpoint of X(i) and X(j) for these {i,j}: {1, 12647}, {8, 3870}, {55, 5252}, {1478, 5119}, {2292, 17874}, {3434, 3895}, {10057, 10087}, {10058, 12749}
X(31397) = reflection of X(i) in X(j) for these (i,j): (1, 13405), (226, 495), (956, 5745), (4304, 55), (4847, 10)
X(31397) = complement of the complement of X(12648)
X(31397) = X(10)-of-inner-Yff triangle
X(31397) = X(946)-of-1st Johnson-Yff triangle
X(31397) = X(1210)-of-inner-Yff tangents triangle
X(31397) = X(4847)-of-outer-Garcia triangle
X(31397) = X(12647)-of-anti-Aquila triangle
X(31397) = X(13405)-of-Aquila triangle
X(31397) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 10915, 6736), (145, 5703, 1), (18395, 21625, 1210)

X(31398) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, MOSES}

Barycentrics (b+c)*a^3-(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :

X(31398) lies on these lines: {1,31401}, {2,9620}, {4,1571}, {6,26446}, {8,9619}, {10,39}, {20,31422}, {32,6684}, {40,2548}, {46,9596}, {57,31409}, {65,31460}, {115,10175}, {165,7737}, {187,10164}, {226,31476}, {230,11231}, {355,5013}, {497,31433}, {515,574}, {516,5475}, {517,3815}, {946,1506}, {950,31451}, {962,31404}, {980,25007}, {1015,31397}, {1210,1500}, {1482,31467}, {1504,13936}, {1505,13883}, {1572,5657}, {1588,31437}, {1698,3767}, {1699,31415}, {1703,31411}, {1706,31416}, {1737,2276}, {1788,31402}, {1837,31448}, {2242,3911}, {2275,10039}, {2549,5587}, {2551,31442}, {3055,11230}, {3579,7745}, {3634,7746}, {3679,9592}, {3817,7603}, {4292,9650}, {5024,5790}, {5034,5847}, {5058,13912}, {5062,13975}, {5119,9599}, {5164,10440}, {5254,9956}, {5280,5445}, {5283,24982}, {5286,9780}, {5530,17750}, {5690,31406}, {5691,31421}, {5722,31477}, {5795,31456}, {5818,7738}, {5836,31466}, {5881,31450}, {5886,31489}, {6421,13911}, {6422,13973}, {6735,16975}, {7739,19875}, {7748,19925}, {7765,31399}, {8582,16589}, {9581,31426}, {9589,31417}, {9597,10827}, {9598,10826}, {9665,10624}, {9698,11362}, {10915,17448}, {10916,20691}, {15815,18481}, {18480,31430}, {21965,25066}, {26100,26676}

X(31398) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31428, 31401), (9620, 31441, 2)

X(31399) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, STEINER}

Barycentrics 3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+5*(b^2-c^2)^2 : :
X(31399) = 3*X(1)-13*X(5067) = X(1)-6*X(10172) = 9*X(2)+X(5881) = 6*X(2)-X(5882) = X(3)-6*X(3828) = 3*X(4)+7*X(9588) = X(4)+9*X(19875) = 2*X(5)+3*X(10) = 8*X(5)-3*X(946) = 14*X(5)-9*X(3817) = 6*X(5)-X(4301) = 7*X(5)+3*X(5690) = 11*X(5)-6*X(9955) = X(5)-6*X(9956) = 4*X(5)-9*X(10175) = 4*X(5)+X(11362) = 13*X(5)-3*X(22791) = 4*X(10)+X(946) = 7*X(10)+3*X(3817) = 9*X(10)+X(4301) = 7*X(10)-2*X(5690) = 11*X(10)+4*X(9955) = X(10)+4*X(9956) = 2*X(10)+3*X(10175) = 6*X(10)-X(11362) = 13*X(10)+2*X(22791) = 7*X(946)-12*X(3817) = 13*X(5067)-18*X(10172) = 2*X(5881)+3*X(5882)

X(31399) lies on these lines: {1,5067}, {2,5881}, {3,3828}, {4,9588}, {5,10}, {8,7486}, {20,5587}, {30,31447}, {40,3832}, {57,31410}, {119,3841}, {226,18395}, {355,3526}, {382,19925}, {497,31436}, {515,631}, {516,3843}, {519,1656}, {546,5493}, {547,4669}, {548,10164}, {551,3628}, {632,28204}, {950,31452}, {952,19862}, {1125,5070}, {1210,15888}, {1385,16239}, {1482,4691}, {1483,15808}, {1512,6845}, {1572,31417}, {1588,31440}, {1703,31414}, {1706,31420}, {1737,18398}, {2549,31444}, {2551,31446}, {2800,3698}, {3090,3679}, {3091,28194}, {3244,11230}, {3525,19876}, {3528,5691}, {3530,4297}, {3533,30389}, {3545,7991}, {3576,19877}, {3579,3853}, {3617,5734}, {3624,13607}, {3625,5901}, {3626,5886}, {3636,12645}, {3654,3851}, {3656,5079}, {3671,10592}, {3753,20117}, {3855,5657}, {3856,22793}, {3858,28198}, {3859,28174}, {3911,4317}, {3918,5887}, {3919,5694}, {4015,24474}, {4197,15016}, {4292,9656}, {4309,10826}, {4314,12019}, {4342,10593}, {4413,5450}, {4668,10595}, {4678,16200}, {4701,10247}, {4731,12672}, {4745,5055}, {4848,7951}, {5056,7982}, {5068,31162}, {5071,11522}, {5258,6946}, {5325,6917}, {5722,31480}, {5745,6885}, {5795,6970}, {6256,26040}, {6260,6937}, {6738,31479}, {6861,12437}, {6881,24391}, {7705,24987}, {7738,31431}, {7765,31398}, {7988,12245}, {8582,10265}, {9575,31407}, {9607,31396}, {9657,24914}, {9671,10624}, {9948,18242}, {10246,19878}, {10915,24386}, {11374,14563}, {12005,18908}, {12368,15057}, {12512,17800}, {12571,12702}, {12616,12671}, {12640,24387}, {13605,20396}, {15178,19883}, {15696,28164}, {15712,28208}, {15717,31423}, {17575,17619}, {17578,18492}, {17606,31397}, {28236,31253}, {31441,31450}

X(31399) = midpoint of X(i) and X(j) for these {i,j}: {1698, 5818}, {3617, 8227}, {4668, 10595}
X(31399) = reflection of X(16189) in X(13464)
X(31399) = X(3091)-of-4th Euler triangle
X(31399) = X(3843)-of-Wasat triangle
X(31399) = X(15040)-of-K798i triangle
X(31399) = X(17538)-of-3rd Euler triangle
X(31399) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 10, 11362), (5, 11362, 946), (10, 3817, 5690)

X(31400) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, CIRCUMCIRCLE, HALF-MOSES}

Barycentrics a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2 : :
Barycentrics 2 csc^2 A - 2 csc^2 B - 2 csc^2 C - 1 : :

X(31400) lies on these lines: {1,31396}, {2,39}, {3,7736}, {4,3815}, {5,5024}, {6,631}, {8,9619}, {10,9592}, {20,574}, {24,3087}, {32,3523}, {56,31402}, {69,11285}, {115,5056}, {140,7735}, {187,15717}, {193,1078}, {217,18913}, {230,3525}, {232,3088}, {325,16043}, {372,31403}, {376,7745}, {388,31460}, {390,31451}, {393,570}, {439,3972}, {496,31461}, {497,31448}, {516,31421}, {549,30435}, {550,15484}, {597,10542}, {946,9574}, {962,1571}, {1007,6656}, {1058,31477}, {1125,9593}, {1285,5023}, {1376,31405}, {1384,3530}, {1504,7586}, {1505,7585}, {1506,2549}, {1575,19843}, {1587,31463}, {1656,15048}, {2241,5281}, {2242,5265}, {2275,3085}, {2276,3086}, {2550,31466}, {2551,31449}, {3053,3524}, {3055,5067}, {3068,6421}, {3069,6422}, {3090,5254}, {3094,14853}, {3146,5475}, {3329,16925}, {3452,31429}, {3518,9609}, {3522,7737}, {3526,5305}, {3533,14482}, {3543,7756}, {3547,14961}, {3589,14069}, {3600,31409}, {3616,9620}, {3618,7807}, {3619,7881}, {3620,7796}, {3785,7774}, {3820,31468}, {3832,7748}, {4293,9596}, {4294,9599}, {4301,31431}, {5007,14930}, {5063,10312}, {5068,7603}, {5110,5802}, {5206,15692}, {5261,31476}, {5275,17567}, {5276,6921}, {5304,7772}, {5306,15702}, {5319,7616}, {6337,7770}, {6361,31443}, {6459,9600}, {6684,9575}, {6700,16517}, {7080,16975}, {7400,22401}, {7486,7765}, {7512,9608}, {7667,15437}, {7750,11163}, {7753,10304}, {7758,7815}, {7764,7800}, {7776,8359}, {7777,7791}, {7783,16924}, {7789,16045}, {7833,23334}, {7839,17008}, {7854,10513}, {7858,14907}, {7864,17005}, {7891,16898}, {7905,20080}, {7906,16990}, {7907,16989}, {7920,16923}, {8367,11165}, {8589,21734}, {8889,27376}, {9597,10590}, {9598,10591}, {9744,9873}, {9778,31422}, {9780,31441}, {9785,31433}, {10200,25092}, {10527,17756}, {11174,14001}, {11307,11489}, {11308,11488}, {12053,31426}, {12699,31430}, {14535,19697}, {16784,31452}, {17578,31417}, {18228,31442}

X(31400) = complement of X(32834)
X(31400) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 39, 5286), (6683, 7795, 2), (7763, 7786, 2)

X(31401) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, CIRCUMCIRCLE, MOSES}

Barycentrics a^4-4*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31401) lies on these lines: {1,31398}, {2,39}, {3,2548}, {4,574}, {5,2549}, {6,140}, {10,9619}, {20,5475}, {30,15815}, {32,631}, {35,9599}, {36,9596}, {37,10200}, {56,31409}, {69,5034}, {83,16925}, {99,16924}, {114,10356}, {115,3090}, {183,7758}, {187,3523}, {193,7780}, {216,3546}, {217,26937}, {230,3526}, {232,3541}, {315,7777}, {325,7800}, {372,31411}, {376,7747}, {388,31476}, {496,31477}, {497,31451}, {498,2275}, {499,2276}, {516,31422}, {549,3053}, {566,18281}, {590,6421}, {615,6422}, {620,7808}, {626,1007}, {632,5305}, {858,15880}, {946,1571}, {1015,3085}, {1078,7774}, {1107,26364}, {1125,9620}, {1153,5032}, {1329,31449}, {1376,31416}, {1384,15720}, {1500,3086}, {1504,3069}, {1505,3068}, {1569,14651}, {1572,6684}, {1574,19843}, {1575,26363}, {1587,31481}, {1656,3055}, {1698,9592}, {1699,31421}, {2023,15561}, {2165,13351}, {2241,5218}, {2242,7288}, {2493,14787}, {2550,31488}, {2551,31456}, {3071,9600}, {3088,3199}, {3091,7603}, {3094,11272}, {3147,10311}, {3329,7907}, {3452,31442}, {3517,6748}, {3518,9700}, {3522,8589}, {3524,5206}, {3525,5368}, {3528,6781}, {3530,5023}, {3533,7755}, {3547,22401}, {3549,14961}, {3589,6387}, {3618,5028}, {3619,7869}, {3620,7895}, {3624,9593}, {3628,13881}, {3734,6337}, {3785,7759}, {3820,31490}, {3850,18584}, {4045,7862}, {4293,9650}, {4294,9665}, {4301,31444}, {5007,10303}, {5025,17005}, {5038,10104}, {5041,5304}, {5052,10519}, {5054,9300}, {5058,9540}, {5059,15602}, {5062,13935}, {5067,7765}, {5068,18424}, {5070,9607}, {5071,11648}, {5094,27376}, {5111,14693}, {5210,15712}, {5275,13747}, {5276,17566}, {5277,6921}, {5306,15694}, {5503,9167}, {5552,16975}, {6292,7888}, {6389,28407}, {6459,9674}, {6722,7902}, {7506,9609}, {7512,9699}, {7525,15109}, {7542,23115}, {7618,8370}, {7741,9598}, {7752,7791}, {7760,17008}, {7762,11163}, {7767,9766}, {7771,7858}, {7773,8356}, {7775,7830}, {7778,8362}, {7782,14035}, {7783,11185}, {7784,8359}, {7785,14907}, {7789,15491}, {7796,16990}, {7806,16923}, {7807,11174}, {7810,7903}, {7812,8182}, {7814,7831}, {7820,16045}, {7835,16898}, {7839,17004}, {7847,14063}, {7855,15589}, {7857,16989}, {7872,16041}, {7876,7925}, {7889,14069}, {7947,16986}, {7951,9597}, {8227,9574}, {8367,12040}, {8588,10299}, {8721,13860}, {8889,27371}, {9575,31423}, {9651,10590}, {9664,10591}, {9725,9726}, {9771,11318}, {9955,31430}, {10024,15075}, {10198,16604}, {10304,14537}, {11432,22270}, {12053,31433}, {12108,22331}, {12699,31443}, {14067,16987}, {15513,15717}, {30827,31429}

X(31401) = complement of X(32828)
X(31401) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31428, 31398), (2, 3926, 3934), (2, 7763, 7795)

X(31402) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, INCIRCLE}

Barycentrics a^4+4*(b^2+b*c+c^2)*a^2-(b^2-c^2)^2 : :

X(31402) lies on these lines: {1,7736}, {2,31460}, {4,2276}, {6,3085}, {8,31405}, {9,5530}, {11,31404}, {12,5286}, {20,31448}, {30,31461}, {32,5218}, {37,5084}, {39,388}, {43,26036}, {56,31400}, {57,31396}, {69,27020}, {172,631}, {192,16924}, {217,18922}, {226,9593}, {377,17756}, {443,1575}, {495,9605}, {497,1500}, {498,5280}, {499,16785}, {516,31426}, {999,31406}, {1056,2275}, {1058,9599}, {1107,3421}, {1335,31403}, {1478,7738}, {1506,10589}, {1571,3474}, {1574,26040}, {1588,31459}, {1788,31398}, {2241,31478}, {2242,7288}, {2549,5229}, {2551,5283}, {3086,3815}, {3087,11398}, {3476,9619}, {3485,9620}, {3501,26098}, {3767,10588}, {3911,31428}, {4292,9574}, {4293,5013}, {4294,7745}, {4426,6857}, {5024,18990}, {5082,20691}, {5225,5475}, {5254,10590}, {5276,5552}, {5291,30478}, {5299,10056}, {5305,31479}, {5712,17750}, {5725,25066}, {6421,31408}, {6933,17737}, {7031,31452}, {7737,31451}, {7753,10385}, {9300,16781}, {9592,10106}, {9607,31410}, {9654,15048}, {12527,31429}, {13405,16780}, {15171,15484}, {15325,31467}, {16517,21075}, {19767,26074}, {21956,31418}

X(31402) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31409, 388), (2276, 9596, 4)

X(31403) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, 2nd LEMOINE}

Barycentrics a^4+4*S*a^2+4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31403) lies on these lines: {2,6}, {4,6422}, {20,31465}, {32,9540}, {39,1587}, {372,31400}, {376,9600}, {393,3127}, {485,5286}, {486,31404}, {488,18993}, {493,6806}, {497,31459}, {516,31427}, {631,6423}, {1164,3087}, {1335,31402}, {1378,31405}, {1504,1588}, {1575,31413}, {1703,31396}, {2275,31408}, {2549,23249}, {2551,31464}, {3070,7738}, {3103,21737}, {3156,19006}, {3312,31406}, {3399,14244}, {3523,12968}, {3767,31481}, {5013,6460}, {5058,31483}, {5062,13935}, {5254,31412}, {5280,13904}, {5299,13905}, {5305,8976}, {5475,23259}, {6221,18907}, {6420,19103}, {6421,7581}, {6459,7745}, {7583,9605}, {7737,9541}, {7748,23253}, {8981,30435}, {9575,13883}, {9607,31414}, {10577,19102}, {13665,15048}, {13966,31467}, {13975,31428}

X(31403) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 590, 7735), (3068, 5591, 590), (3595, 8974, 590)

X(31404) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, NINE-POINTS}

Barycentrics a^4+6*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :

X(31404) lies on these lines: {2,32}, {4,3815}, {5,5286}, {6,3090}, {11,31402}, {20,5475}, {30,31467}, {39,3091}, {50,7558}, {115,5068}, {140,15484}, {187,10303}, {193,7858}, {217,23291}, {230,5067}, {381,7738}, {390,9665}, {393,1594}, {486,31403}, {497,31460}, {516,31428}, {546,5024}, {574,3146}, {631,5023}, {632,1384}, {962,31398}, {1007,7770}, {1015,5261}, {1285,3533}, {1329,31405}, {1500,5274}, {1571,9812}, {1572,9780}, {1573,8165}, {1575,31418}, {1587,12969}, {1588,12962}, {1656,7735}, {1699,31396}, {2207,8889}, {2275,10590}, {2276,10591}, {2549,3832}, {2551,31466}, {2996,7757}, {3053,3055}, {3085,9599}, {3086,9596}, {3087,3542}, {3199,7378}, {3522,7747}, {3523,7737}, {3526,18907}, {3529,11742}, {3544,18584}, {3545,5254}, {3600,9650}, {3613,8801}, {3618,7887}, {3620,5052}, {3628,30435}, {3767,5041}, {3817,9593}, {3839,7748}, {3851,15048}, {3855,9606}, {3861,31470}, {3926,7777}, {3933,9770}, {3972,5395}, {5034,5921}, {5038,6776}, {5055,5305}, {5058,8972}, {5062,13941}, {5071,9300}, {5225,31448}, {5276,6931}, {5283,6919}, {5304,7486}, {5319,14930}, {6337,8370}, {6421,31412}, {6781,21734}, {7484,15437}, {7585,31481}, {7586,31411}, {7756,17578}, {7759,15589}, {7772,15022}, {7773,16043}, {7774,16921}, {7778,16045}, {7784,15491}, {7789,11184}, {7861,8176}, {7903,10513}, {7921,16922}, {7925,16898}, {7941,16990}, {7949,11160}, {8164,16781}, {9574,18483}, {9575,10175}, {9592,19925}, {11174,14064}, {12108,15655}, {14537,15692}, {14986,31409}, {15302,31099}, {15880,30769}, {16925,17005}

X(31404) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7785, 3785), (1506, 2548, 2)

X(31405) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, SPIEKER}

Barycentrics a^4+4*(b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31405) lies on these lines: {1,26036}, {2,31466}, {4,1107}, {6,19843}, {8,31402}, {10,7736}, {20,31449}, {30,31468}, {32,30478}, {37,1058}, {39,2550}, {69,17030}, {388,16975}, {443,2275}, {497,5283}, {516,31429}, {631,4386}, {946,16517}, {966,2300}, {1056,17448}, {1329,31404}, {1376,31400}, {1378,31403}, {1573,2548}, {1588,31464}, {1706,31396}, {1914,6857}, {2082,29639}, {2276,5082}, {2280,28246}, {2886,5286}, {3086,5275}, {3421,9596}, {3487,16973}, {3767,31488}, {3926,20172}, {4307,5021}, {5084,9599}, {5254,31418}, {5276,10527}, {5277,7288}, {5299,19854}, {5305,31493}, {5698,31442}, {5712,20963}, {6421,31413}, {7080,31460}, {7735,26363}, {7737,31456}, {7745,31490}, {8817,26134}, {9605,31419}, {9607,31420}, {9711,31407}, {14001,20179}, {16604,17582}, {17784,31448}, {21384,26098}, {21921,28074}

X(31405) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31416, 2550), (1573, 2548, 2551)

X(31406) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, STAMMLER}

Barycentrics 5*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31406) lies on these lines: {2,3933}, {3,7736}, {4,5024}, {5,39}, {6,140}, {20,15484}, {30,2548}, {32,549}, {83,5503}, {141,6683}, {187,15712}, {230,632}, {232,1595}, {315,8359}, {325,3096}, {355,9592}, {381,7738}, {382,31407}, {495,2275}, {496,2276}, {497,31461}, {516,31430}, {517,31396}, {524,7815}, {546,2549}, {547,7739}, {548,7737}, {550,574}, {570,23335}, {597,6680}, {631,30435}, {952,9619}, {999,31402}, {1007,7866}, {1107,3820}, {1285,15717}, {1353,5034}, {1384,3523}, {1504,19116}, {1505,19117}, {1570,14693}, {1571,28174}, {1575,31419}, {1656,5286}, {2551,31468}, {3053,3530}, {3054,7755}, {3055,7746}, {3068,11316}, {3069,11315}, {3087,3517}, {3094,21850}, {3312,31403}, {3329,7807}, {3525,5304}, {3526,7735}, {3589,3788}, {3618,6393}, {3627,5475}, {3628,3767}, {3629,7780}, {3631,7916}, {3665,24786}, {3816,25092}, {3845,7748}, {3850,31415}, {3861,31417}, {3934,15491}, {5007,14869}, {5023,12100}, {5028,20576}, {5041,5306}, {5067,14482}, {5120,19547}, {5206,31457}, {5276,13747}, {5277,17564}, {5280,5433}, {5283,17527}, {5299,5432}, {5309,15699}, {5319,16239}, {5690,31398}, {5886,9593}, {5901,9620}, {6337,11286}, {6390,7770}, {6421,7583}, {6422,7584}, {6431,19104}, {6432,19103}, {6656,7777}, {6661,7891}, {6676,23115}, {6704,7880}, {6748,7715}, {6823,14961}, {7502,9608}, {7542,22120}, {7747,15704}, {7750,7858}, {7752,7918}, {7753,8703}, {7758,15271}, {7759,14929}, {7762,7824}, {7763,7819}, {7767,7774}, {7769,7792}, {7773,8357}, {7776,16043}, {7778,8364}, {7783,8370}, {7785,8356}, {7789,7808}, {7797,17005}, {7800,9766}, {7803,8361}, {7805,13468}, {7813,31239}, {7817,9771}, {7867,22110}, {7894,22329}, {7925,8363}, {8360,11184}, {8770,13361}, {8962,15235}, {9574,12699}, {9575,26446}, {9596,18990}, {9599,15171}, {9709,31405}, {10299,15655}, {10303,14930}, {10336,16987}, {10386,31451}, {12108,21843}, {13846,19105}, {13847,19102}, {14537,15686}, {14910,30537}, {15172,31477}, {15302,30739}, {15513,17504}, {15720,21309}, {17756,24390}, {24239,25066}, {25068,29639}

X(31406) = midpoint of X(2548) and X(5013)
X(31406) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9605, 5305), (9605, 31467, 2)

X(31407) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, STEINER}

Barycentrics 3*a^4+14*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31407) lies on these lines: {2,5007}, {4,9606}, {5,5286}, {6,3316}, {20,574}, {30,31470}, {39,3832}, {382,31406}, {497,31462}, {516,31431}, {548,15484}, {631,1285}, {1506,5319}, {1575,31420}, {1588,31465}, {1906,15433}, {2275,31410}, {2551,31469}, {3090,9300}, {3091,7765}, {3523,7753}, {3526,21309}, {3528,7745}, {3530,15655}, {3843,7738}, {3853,5024}, {3855,9607}, {5056,7772}, {5068,7739}, {5070,7735}, {5309,15022}, {5475,17578}, {6421,31414}, {7737,21734}, {7746,14930}, {7777,16898}, {9575,31399}, {9589,31396}, {9711,31405}, {10513,31239}, {11184,14069}, {15513,15717}, {15702,22331}, {16239,30435}

X(31407) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 9698, 31400), (2548, 9698, 20), (7736, 31404, 5286)

X(31408) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, 2nd LEMOINE}

Barycentrics ((-a+b+c)*(a+b+c)*a^2+(a^2+(b+c)^2)*S)*(a-b+c)*(a+b-c) : :

X(31408) lies on these lines: {1,1587}, {2,6502}, {4,1124}, {6,388}, {7,16232}, {8,2362}, {11,31412}, {12,3069}, {20,2066}, {30,31474}, {36,9540}, {55,6460}, {56,3068}, {57,13883}, {65,19066}, {176,3177}, {189,13389}, {226,18992}, {329,1659}, {371,4293}, {372,3085}, {443,1378}, {485,3086}, {486,10590}, {495,3312}, {496,13665}, {497,3070}, {498,13935}, {516,31432}, {590,7288}, {605,1935}, {615,10588}, {631,9646}, {999,7583}, {1015,31411}, {1056,1335}, {1058,23267}, {1131,5274}, {1152,5218}, {1319,13902}, {1377,3421}, {1420,8983}, {1478,1588}, {1479,23249}, {1505,31409}, {1702,4292}, {1703,31397}, {1788,13911}, {2067,3600}, {2275,31403}, {2549,31471}, {2551,31473}, {3071,5229}, {3072,3076}, {3304,19030}, {3311,18990}, {3476,7969}, {3485,7968}, {3529,9660}, {3583,23253}, {3585,23259}, {3614,13955}, {3911,13893}, {4294,6560}, {4299,9541}, {4311,9583}, {5083,19078}, {5204,13901}, {5219,13971}, {5225,23251}, {5252,19065}, {5261,7586}, {5265,8972}, {5290,19003}, {5433,13897}, {5434,18996}, {5563,13904}, {6421,31402}, {6459,7354}, {6564,10591}, {7584,9654}, {7738,31459}, {7951,13962}, {8976,15325}, {9578,13936}, {9661,13886}, {10106,18991}, {10895,19029}, {10956,19112}, {11237,19027}, {11375,13959}, {12527,31438}, {13966,31479}, {13975,31434}, {15888,19037}, {18924,19349}, {19013,19026}

X(31408) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12, 18995, 3069), (6502, 31472, 2)

X(31409) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, MOSES}

Barycentrics a^4+2*(b+c)^2*a^2-(b^2-c^2)^2 : :

X(31409) lies on these lines: {1,2548}, {2,2242}, {4,1500}, {6,495}, {8,31416}, {11,31415}, {12,3767}, {20,31451}, {30,31477}, {32,3085}, {37,5725}, {39,388}, {55,7737}, {56,31401}, {57,31398}, {115,10590}, {172,498}, {187,5218}, {192,11185}, {226,9620}, {230,31479}, {443,1574}, {497,5475}, {516,31433}, {529,31449}, {574,4293}, {609,3584}, {999,3815}, {1015,1056}, {1058,9665}, {1335,31411}, {1478,2276}, {1505,31408}, {1506,3086}, {1571,4292}, {1572,31397}, {1573,3421}, {1588,31471}, {1909,7758}, {1914,10056}, {2551,16589}, {3295,7745}, {3361,31428}, {3436,5283}, {3583,9331}, {3585,9598}, {3600,31400}, {3911,31441}, {4294,7747}, {4298,31396}, {4317,31450}, {4426,10198}, {5013,18990}, {5229,7748}, {5254,9654}, {5261,5286}, {5270,9597}, {5275,17757}, {5277,5552}, {5280,5319}, {5290,9593}, {5432,21843}, {6645,7763}, {6767,15484}, {7288,31455}, {7354,31448}, {7603,10589}, {7735,8164}, {7739,11237}, {7746,10588}, {7765,31410}, {7800,27020}, {7951,16785}, {9579,31426}, {9619,10106}, {9655,31461}, {9657,31462}, {10385,14537}, {10592,13881}, {12513,31466}, {12527,31442}, {14986,31404}, {15325,31489}, {17532,21956}

X(31409) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9596, 2548), (2242, 31476, 2)

X(31410) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, STEINER}

Barycentrics 3*a^4+2*(b^2+6*b*c+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31410) lies on these lines: {1,3832}, {2,4317}, {4,3058}, {5,388}, {7,10827}, {8,31420}, {10,9965}, {12,631}, {20,35}, {30,31480}, {56,5067}, {57,31399}, {80,11036}, {226,5881}, {382,495}, {443,9711}, {497,3843}, {498,4325}, {516,31436}, {529,6856}, {548,5218}, {1015,31417}, {1056,3855}, {1058,9671}, {1335,31414}, {1588,31475}, {2275,31407}, {2549,31478}, {2551,17529}, {3090,5434}, {3146,4330}, {3295,3853}, {3304,3545}, {3421,9710}, {3436,4197}, {3475,18480}, {3522,3584}, {3526,10588}, {3528,7354}, {3530,31479}, {3543,3746}, {3582,15022}, {3585,4309}, {3600,7486}, {3627,10385}, {3822,31458}, {3856,9669}, {3861,5225}, {3947,9613}, {4292,5726}, {4295,9578}, {4299,21734}, {4301,9612}, {4338,10039}, {4995,17538}, {5056,5563}, {5068,10072}, {5070,7288}, {5252,5714}, {5290,18391}, {5734,12047}, {5818,10404}, {6845,12115}, {6896,10711}, {7738,31462}, {7765,31409}, {9589,31397}, {9607,31402}, {9624,10106}, {10197,17576}, {10826,11037}, {11271,12946}, {12527,31446}, {17552,25466}, {17582,31141}, {18395,21454}, {19843,20060}, {31450,31476}

X(31410) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9656, 11237, 15888), (9656, 15888, 4)

X(31411) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, MOSES}

Barycentrics a^4+4*S*a^2+2*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31411) lies on these lines: {2,5062}, {4,1504}, {5,6}, {20,31483}, {32,638}, {39,1587}, {76,13707}, {115,31412}, {172,13904}, {187,9540}, {230,8976}, {371,7737}, {372,31401}, {376,9674}, {491,7795}, {497,31471}, {516,31437}, {574,6460}, {590,6423}, {1015,31408}, {1335,31409}, {1378,31416}, {1384,13903}, {1505,7581}, {1506,3069}, {1572,13883}, {1574,31413}, {1588,5475}, {1703,31398}, {1914,13905}, {2549,3070}, {2551,31482}, {3053,8981}, {3299,9599}, {3301,9596}, {3311,7745}, {3312,3815}, {5058,7585}, {5254,13665}, {5418,12968}, {6228,13879}, {6275,13648}, {6395,31467}, {6417,15484}, {6459,7747}, {6561,12962}, {7586,31404}, {7735,13886}, {7738,23267}, {7748,23249}, {7753,19054}, {7765,31414}, {9605,18512}, {13935,31455}, {13966,31489}, {13975,31441}, {19037,31460}

X(31411) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 485, 3767), (486, 19103, 6), (5062, 31481, 2)

X(31412) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, NINE-POINTS}

Barycentrics a^4+4*S*a^2+2*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(31412) = X(6449)-3*X(8976) = 2*X(6449)-3*X(9540) = 2*X(6449)+3*X(23253) = 2*X(8976)+X(23253)

X(31412) lies on these lines: {2,490}, {3,18538}, {4,371}, {5,1587}, {6,3091}, {11,31408}, {20,590}, {30,6449}, {114,13773}, {115,31411}, {140,6456}, {372,3090}, {376,1327}, {378,8276}, {381,1588}, {382,6407}, {486,3545}, {487,7620}, {488,26361}, {492,12222}, {497,31472}, {515,13902}, {516,13893}, {546,3311}, {550,6496}, {615,5056}, {626,638}, {631,6560}, {632,6450}, {637,5861}, {640,7375}, {642,26620}, {946,19066}, {962,13911}, {1124,10591}, {1132,3854}, {1151,2671}, {1270,23311}, {1329,31413}, {1335,10590}, {1378,31418}, {1505,31415}, {1593,13889}, {1656,13935}, {1699,13883}, {1702,18483}, {1703,10175}, {2066,5225}, {2067,5229}, {2549,31481}, {2551,31484}, {3071,3832}, {3092,6623}, {3127,13051}, {3297,5274}, {3298,5261}, {3365,18582}, {3390,18581}, {3520,9682}, {3523,8253}, {3525,6396}, {3529,6200}, {3533,6487}, {3536,8968}, {3543,6429}, {3544,6420}, {3564,26468}, {3583,13905}, {3585,13904}, {3590,5059}, {3594,13941}, {3614,19037}, {3627,6221}, {3628,6398}, {3817,18992}, {3830,13903}, {3839,6470}, {3850,13785}, {3851,6501}, {3855,6565}, {3857,6427}, {3861,31487}, {4293,9661}, {4294,9646}, {5055,13966}, {5066,19116}, {5067,5420}, {5068,7586}, {5071,10577}, {5072,6418}, {5079,6395}, {5254,31403}, {5410,23047}, {5413,6622}, {5414,10588}, {5448,19062}, {5475,26463}, {5587,19065}, {5591,7389}, {5640,12240}, {5691,8983}, {5893,19088}, {5895,8991}, {6253,13896}, {6256,13906}, {6284,13897}, {6353,8280}, {6410,10303}, {6419,23273}, {6421,31404}, {6428,12811}, {6447,12102}, {6451,12103}, {6452,14869}, {6455,15704}, {6475,15703}, {6497,12108}, {6502,10589}, {6919,31473}, {7173,18995}, {7354,13898}, {7374,13638}, {7486,8252}, {7687,19111}, {7714,18289}, {7728,13915}, {7738,31463}, {7988,13971}, {7989,13936}, {8227,13959}, {8854,8889}, {8980,10722}, {8994,10721}, {8997,10723}, {8998,10733}, {9582,28150}, {9615,28164}, {9975,14853}, {10195,10299}, {10724,13922}, {10728,13913}, {10735,13923}, {10895,19030}, {10896,19028}, {11291,26362}, {11479,19006}, {12111,12239}, {12173,13884}, {12297,13882}, {12510,22646}, {12943,18965}, {12953,13901}, {17578,31454}, {18945,19355}, {18991,19925}, {19055,23514}, {19059,23515}, {19081,23513}, {19087,23332}, {22554,22618}

X(31412) = midpoint of X(9540) and X(23253)
X(31412) = reflection of X(9540) in X(8976)
X(31412) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1131, 3070), (2, 3070, 6460)

X(31413) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, SPIEKER}

Barycentrics 4*S*a^2+(a+b+c)*(a^3-(b+c)*a^2+(b+c)^2*a-(b^2-c^2)*(b-c)) : :

X(31413) lies on these lines: {2,5414}, {4,1377}, {6,2550}, {8,2362}, {10,1587}, {20,31453}, {30,31485}, {372,19843}, {376,9678}, {443,1335}, {486,31418}, {497,31473}, {516,31438}, {958,6460}, {962,30556}, {1124,5082}, {1131,8165}, {1152,30478}, {1329,31412}, {1376,3068}, {1378,7581}, {1505,31416}, {1574,31411}, {1575,31403}, {1706,13883}, {2066,17784}, {2067,6904}, {2549,31482}, {2551,3070}, {2886,3069}, {3312,31419}, {3820,13665}, {3925,19037}, {4413,19030}, {5438,8983}, {5705,13975}, {5794,19065}, {5836,19066}, {6421,31405}, {7080,31472}, {7090,27382}, {7583,9709}, {7738,31464}, {8987,10270}, {9540,25440}, {9661,17567}, {9711,31414}, {13935,26363}, {13958,31245}, {13966,31493}, {19029,31140}, {19087,20306}

X(31414) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, STEINER}

Barycentrics 3*a^4+12*S*a^2+2*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(31414) = 3*X(6447)-5*X(31487)

X(31414) lies on these lines: {2,6426}, {4,1327}, {5,1587}, {6,1131}, {20,1151}, {30,6447}, {371,23269}, {372,5067}, {376,8960}, {382,6199}, {485,631}, {486,6436}, {497,31475}, {516,31440}, {548,6451}, {590,15717}, {1335,31410}, {1378,31420}, {1505,31417}, {1588,3843}, {1703,31399}, {2549,31483}, {2551,31486}, {3090,10194}, {3091,19053}, {3311,3853}, {3522,13846}, {3526,6408}, {3528,6560}, {3530,6497}, {3533,6454}, {3543,3592}, {3545,6420}, {3594,5056}, {3845,6427}, {3850,6428}, {3855,6564}, {3856,6499}, {3859,19116}, {3861,19117}, {4301,19066}, {4325,13904}, {4330,13905}, {5059,6425}, {5070,13935}, {5590,12222}, {6250,10784}, {6398,16239}, {6417,23263}, {6421,31407}, {6453,11001}, {6470,7585}, {6489,10303}, {6522,11539}, {6776,26330}, {7738,31465}, {7745,26463}, {7765,31411}, {7863,11291}, {8972,21734}, {8981,15696}, {9541,17800}, {9589,13883}, {9607,31403}, {9657,19030}, {9670,19028}, {9711,31413}, {10195,15702}, {12322,26339}, {13847,15022}, {31450,31481}

X(31414) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1327, 6419, 4), (1587, 13665, 31412), (1587, 31412, 3069)

X(31415) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, NINE-POINTS}

Barycentrics a^4+4*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(31415) = X(11742)-9*X(31489)

X(31415) lies on these lines: {2,187}, {3,3055}, {4,574}, {5,6}, {11,31409}, {20,8589}, {30,11742}, {32,3090}, {39,3091}, {69,7775}, {115,3545}, {140,5210}, {141,11173}, {230,5055}, {315,16921}, {381,2549}, {497,31476}, {498,10987}, {516,31441}, {546,5013}, {547,18907}, {549,5585}, {575,14162}, {590,8375}, {615,8376}, {620,14033}, {626,3619}, {631,7747}, {632,5023}, {1007,3734}, {1015,10590}, {1329,31416}, {1384,1656}, {1500,10591}, {1505,31412}, {1571,18483}, {1572,10175}, {1574,31418}, {1588,31481}, {1699,31398}, {1992,7617}, {2241,10588}, {2242,10589}, {2551,31488}, {3053,3628}, {3085,9665}, {3086,9650}, {3524,6781}, {3525,5206}, {3526,15655}, {3529,15515}, {3542,10985}, {3543,15602}, {3544,7772}, {3547,22052}, {3589,11318}, {3618,7844}, {3620,3934}, {3627,15815}, {3631,7776}, {3763,8367}, {3785,7843}, {3793,7610}, {3817,9620}, {3832,7748}, {3843,31450}, {3850,31406}, {3851,5254}, {3855,7738}, {3857,22332}, {3861,31492}, {4045,16041}, {5007,15022}, {5008,5056}, {5033,7808}, {5066,15048}, {5067,7749}, {5068,5286}, {5071,7735}, {5072,9605}, {5079,30435}, {5107,14853}, {5187,5283}, {5225,31451}, {5277,6931}, {5355,18362}, {5461,5477}, {6565,31463}, {6639,18472}, {6643,10979}, {6919,16589}, {7505,10986}, {7577,8744}, {7615,11163}, {7618,11317}, {7694,13860}, {7741,9596}, {7751,11008}, {7752,7795}, {7755,14075}, {7759,20080}, {7763,16044}, {7769,14035}, {7773,7800}, {7777,11185}, {7782,14068}, {7785,14023}, {7786,14063}, {7809,16990}, {7812,17008}, {7813,9770}, {7823,16922}, {7825,16043}, {7845,15589}, {7862,14001}, {7867,16045}, {7899,16898}, {7940,14037}, {7951,9599}, {8556,14929}, {8586,20423}, {9112,16268}, {9113,16267}, {9300,19709}, {9619,19925}, {9743,22682}, {9754,10788}, {9771,11159}, {10153,18842}, {10254,22121}, {10303,15513}, {10592,16781}, {10896,31460}, {11284,24855}, {11287,15491}, {11305,23303}, {11306,23302}, {11361,17005}, {11623,14912}, {11646,22566}, {12571,31396}, {14061,16989}, {14790,14806}, {15534,16509}, {15603,15694}, {17578,31457}

X(31415) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5475, 7737), (2, 7737, 21843), (11614, 14537, 187)

X(31416) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, SPIEKER}

Barycentrics a^4+2*(b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31416) lies on these lines: {2,2241}, {4,1573}, {6,31419}, {8,31409}, {10,1572}, {20,31456}, {30,31490}, {32,19843}, {39,2550}, {75,7758}, {115,31418}, {187,30478}, {230,31493}, {377,16975}, {443,1015}, {497,16589}, {516,31442}, {958,7737}, {1107,2549}, {1329,31415}, {1378,31411}, {1500,5082}, {1505,31413}, {1574,7736}, {1588,31482}, {1698,9599}, {1706,31398}, {1914,19854}, {2551,5475}, {2886,3767}, {3421,9650}, {3434,5283}, {3679,9596}, {3815,9709}, {3933,20181}, {4340,9346}, {4386,26363}, {4999,21843}, {5084,9665}, {5275,24390}, {5277,10527}, {7080,31476}, {7745,9708}, {7765,31420}, {7795,20172}, {7800,17030}, {8728,16781}, {9711,31417}, {12609,16973}, {17784,31451}

X(31416) = {X(2550), X(31405)}-harmonic conjugate of X(39)

X(31417) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, STEINER}

Barycentrics 3*a^4+8*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31417) lies on these lines: {2,7843}, {4,9698}, {5,6}, {20,5475}, {30,31492}, {32,5067}, {39,3832}, {381,9607}, {382,3815}, {497,31478}, {516,31444}, {631,1506}, {1015,31410}, {1505,31414}, {1572,31399}, {1574,31420}, {1588,31483}, {2549,3843}, {3053,16239}, {3090,7753}, {3091,7739}, {3523,14537}, {3526,7745}, {3528,7747}, {3530,31489}, {3545,7772}, {3845,22332}, {3851,9300}, {3853,5013}, {3855,7736}, {3859,15048}, {3861,31406}, {5007,5056}, {5068,5309}, {5070,15484}, {5071,7755}, {5079,5306}, {7486,7603}, {7752,16898}, {7759,11160}, {7775,21356}, {7795,7814}, {7796,16924}, {9589,31398}, {9670,31460}, {9671,31462}, {9711,31416}, {9770,17130}, {14023,16921}, {15717,31455}, {17578,31400}, {17800,31467}

X(31417) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2548, 31415, 3767), (5475, 31404, 31401)

X(31418) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, NINE-POINTS, SPIEKER}

Barycentrics a^4+2*(b^2+c^2)*a^2+4*(b+c)*b*c*a-3*(b^2-c^2)^2 : :

X(31418) lies on these lines: {1,5175}, {2,35}, {4,958}, {5,2550}, {7,10916}, {8,6871}, {9,18483}, {10,962}, {11,443}, {12,5082}, {20,5267}, {30,30478}, {40,6844}, {55,6856}, {79,9965}, {100,6933}, {115,31416}, {346,30172}, {355,6982}, {376,4999}, {377,3086}, {381,2551}, {387,26098}, {388,17532}, {390,10198}, {405,5225}, {442,497}, {452,3583}, {474,10589}, {486,31413}, {496,17528}, {498,17784}, {499,6904}, {516,5705}, {517,6867}, {518,5714}, {519,4323}, {546,9708}, {936,3817}, {938,12609}, {946,6843}, {956,5229}, {993,3146}, {997,6993}, {1000,13463}, {1056,3813}, {1058,11235}, {1125,4208}, {1329,3545}, {1376,3090}, {1378,31412}, {1478,5288}, {1574,31415}, {1575,31404}, {1588,31484}, {1698,6919}, {1706,10175}, {1770,5744}, {2475,4293}, {2476,3085}, {2478,19855}, {2549,31488}, {3035,5067}, {3189,11374}, {3333,24386}, {3419,3485}, {3421,10895}, {3436,17577}, {3487,3838}, {3488,28628}, {3616,17647}, {3617,4015}, {3622,26725}, {3634,30332}, {3811,5226}, {3814,5068}, {3820,3851}, {3826,17559}, {3829,25524}, {3855,9710}, {3861,31494}, {3913,8164}, {3925,5084}, {3947,6765}, {4187,26040}, {4292,5231}, {4295,6734}, {4305,24541}, {4307,5292}, {4340,11269}, {4413,7173}, {4421,6668}, {4847,9612}, {5051,19866}, {5056,26364}, {5141,5552}, {5178,10129}, {5187,9780}, {5254,31405}, {5587,9842}, {5603,5794}, {5687,10588}, {5696,30275}, {5698,5791}, {5704,15299}, {5722,28629}, {5734,16206}, {5784,13374}, {5795,18492}, {5810,21293}, {5818,5836}, {5837,31162}, {5840,6892}, {5842,6988}, {5883,12446}, {6284,6857}, {6361,26066}, {6675,9668}, {6700,7988}, {6824,10525}, {6829,10531}, {6842,18518}, {6846,26333}, {6850,26470}, {6855,11496}, {6859,11248}, {6861,10738}, {6864,7681}, {6866,12699}, {6869,18407}, {6870,9812}, {6881,11928}, {6923,26321}, {6935,11826}, {6937,12116}, {6939,10893}, {6951,10785}, {6956,10310}, {6973,9956}, {7080,7951}, {7283,30741}, {7288,11112}, {7613,24046}, {7738,31466}, {8728,9669}, {9623,19925}, {9671,17552}, {9779,21616}, {9940,17668}, {10200,17580}, {10584,17531}, {10593,16408}, {10599,12245}, {11036,11263}, {11111,12953}, {12529,13750}, {14064,20172}, {14986,24387}, {16041,26558}, {17578,31458}, {21620,24392}, {21956,31402}, {23259,31453}, {24898,30653}, {24987,30305}, {28740,31031}

X(31418) = reflection of X(30478) in X(31493)
X(31418) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 2886, 19843), (8, 6871, 10590)

X(31419) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, SPIEKER, STAMMLER}

Barycentrics (b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(31419) = 3*X(2)+X(5082) = 3*X(10)-X(5837) = X(388)-3*X(17528) = X(1697)-5*X(1698) = X(3340)+3*X(3679) = 7*X(3624)-3*X(10389) = 3*X(3742)-2*X(16216) = X(4294)-3*X(16418)

X(31419) lies on these lines: {1,3925}, {2,496}, {3,1602}, {4,9708}, {5,10}, {6,31416}, {8,442}, {9,12699}, {11,1697}, {12,3340}, {20,31494}, {21,20066}, {30,958}, {36,17563}, {40,5791}, {55,6675}, {72,25006}, {75,3933}, {84,5833}, {100,7483}, {119,11530}, {140,1376}, {141,2140}, {142,5045}, {145,4197}, {149,5047}, {200,11374}, {210,12047}, {281,15763}, {355,1490}, {377,956}, {381,2551}, {382,31420}, {388,17528}, {390,16845}, {405,3434}, {427,29667}, {443,999}, {452,9668}, {474,10527}, {497,11108}, {498,31245}, {499,4413}, {516,31445}, {518,6147}, {519,3841}, {528,5248}, {548,31458}, {549,4999}, {550,993}, {596,7263}, {632,1484}, {908,3697}, {912,18251}, {936,5886}, {942,4847}, {952,5794}, {962,8226}, {984,21926}, {997,5901}, {1056,4208}, {1107,15048}, {1125,3813}, {1377,7584}, {1378,7583}, {1387,19861}, {1479,31140}, {1482,6881}, {1532,5818}, {1573,5254}, {1574,3815}, {1575,31406}, {1588,31485}, {1595,1861}, {1706,5705}, {1714,5710}, {1731,17369}, {1737,3698}, {1834,30116}, {2078,5433}, {2476,3617}, {2549,31490}, {2975,11112}, {3036,11698}, {3058,5259}, {3086,16408}, {3242,24159}, {3293,5718}, {3312,31413}, {3333,6067}, {3419,19860}, {3421,5177}, {3428,20420}, {3436,17532}, {3485,3940}, {3545,8165}, {3555,5249}, {3579,5745}, {3612,9945}, {3616,17529}, {3624,10389}, {3626,3822}, {3628,26364}, {3632,15888}, {3634,3816}, {3649,5904}, {3656,15829}, {3695,29641}, {3703,4647}, {3704,30172}, {3742,16216}, {3753,6734}, {3757,5100}, {3811,5719}, {3812,10916}, {3824,21620}, {3825,3828}, {3831,21242}, {3838,4662}, {3881,25557}, {3884,13463}, {3889,27186}, {3897,10609}, {3913,10198}, {3927,4295}, {3961,24161}, {3983,17605}, {3996,25650}, {4026,19858}, {4187,9780}, {4205,19853}, {4294,16418}, {4323,4930}, {4426,18907}, {4514,16817}, {4714,30171}, {4731,17606}, {4848,15844}, {4972,13728}, {5015,16824}, {5084,9669}, {5250,14022}, {5251,6284}, {5253,26060}, {5258,7354}, {5260,11113}, {5263,17698}, {5265,30312}, {5267,8703}, {5273,6361}, {5274,17559}, {5283,21956}, {5284,17590}, {5288,5434}, {5325,28198}, {5432,26475}, {5439,26015}, {5440,24541}, {5484,17678}, {5533,31235}, {5554,11545}, {5587,15908}, {5657,6831}, {5708,24477}, {5714,5815}, {5787,30503}, {5789,14647}, {5790,6842}, {5795,18480}, {5880,24470}, {5903,21677}, {5905,11544}, {6000,20306}, {6048,17717}, {6174,31260}, {6175,20060}, {6244,6847}, {6260,9947}, {6347,15235}, {6348,15236}, {6585,6924}, {6690,8715}, {6700,11230}, {6763,11246}, {6765,25525}, {6824,10306}, {6826,22770}, {6829,12245}, {6841,12702}, {6845,18231}, {6856,7080}, {6857,17784}, {6861,10679}, {7270,16821}, {7280,31157}, {7288,16417}, {7308,9614}, {7502,9712}, {7535,12410}, {7738,31468}, {7741,19875}, {7795,20181}, {7819,20172}, {7951,21031}, {7958,11522}, {7965,9589}, {8227,8580}, {8362,17030}, {8583,11373}, {9605,31405}, {9607,31491}, {9843,24386}, {10283,30144}, {10707,26127}, {10883,20070}, {10914,24987}, {10957,24914}, {11019,16201}, {11681,17530}, {12447,13464}, {12514,28174}, {12558,28228}, {12572,22793}, {12675,15587}, {13576,16850}, {14019,16823}, {14986,17582}, {15674,20095}, {15935,30143}, {16160,18253}, {16415,23853}, {16829,26561}, {16853,26105}, {17061,30145}, {17355,18257}, {17575,19877}, {17670,26801}, {17687,20533}, {18249,28194}, {18250,18483}, {20299,20307}, {20790,21625}, {21949,23537}, {23708,24954}, {24174,29676}, {24443,29690}, {24988,26094}, {25917,30384}, {26321,28458}

X(31419) = midpoint of X(i) and X(j) for these {i,j}: {3295, 5082}, {3927, 4295}
X(31419) = reflection of X(i) in X(j) for these (i,j): (6147, 12609), (10267, 140), (10386, 5248), (21620, 3824), (25466, 3841)
X(31419) = complement of X(3295)
X(31419) = X(1593)-of-4th Euler triangle
X(31419) = X(1595)-of-Wasat triangle
X(31419) = X(11414)-of-3rd Euler triangle
X(31419) = excentral-to-4th-Euler similarity image of X(1697)
X(31419) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3925, 8728), (2, 5082, 3295), (2, 24390, 496)

X(31420) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, SPIEKER, STEINER}

Barycentrics 3*a^4+2*(b^2+c^2)*a^2+12*(b+c)*b*c*a-5*(b^2-c^2)^2 : :

X(31420) lies on these lines: {2,4309}, {4,9710}, {5,2550}, {8,31410}, {10,3832}, {20,993}, {30,31494}, {382,31419}, {390,3841}, {443,31140}, {497,17529}, {516,31446}, {548,30478}, {631,2886}, {1376,5067}, {1378,31414}, {1574,31417}, {1575,31407}, {1588,31486}, {2549,31491}, {2551,3843}, {3421,9656}, {3434,4197}, {3530,31493}, {3853,9708}, {3855,9711}, {3925,9670}, {4330,19854}, {5082,15888}, {5084,9671}, {5758,11362}, {6919,31159}, {7288,17583}, {7486,25639}, {7738,31469}, {7765,31416}, {9607,31405}, {11235,17582}, {15717,26363}, {16126,31145}, {17575,26040}, {17580,24387}, {17784,31452}, {31450,31488}

X(31420) = {X(3925), X(9670)}-harmonic conjugate of X(17552)

X(31421) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, HALF-MOSES}

Barycentrics a*(3*a^3+(b+c)*a^2-(7*b^2+2*b*c+7*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31421) lies on these lines: {1,574}, {3,9574}, {4,31428}, {6,9582}, {20,31396}, {32,16192}, {39,165}, {40,5013}, {56,31426}, {57,31448}, {372,31427}, {516,31400}, {936,21879}, {1376,31429}, {1500,3361}, {1574,5234}, {1575,31424}, {1698,2549}, {1699,31401}, {1706,31449}, {2276,15803}, {3333,31477}

X(31421) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9574, 9593), (3, 31430, 9574), (574, 1571, 1)

X(31422) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, MOSES}

Barycentrics a*(3*a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31422) lies on these lines: {3,1571}, {4,31441}, {6,31430}, {20,31398}, {32,9574}, {39,165}, {40,574}, {56,31433}, {57,31451}, {115,31423}, {187,9593}, {372,31437}, {516,31401}, {517,15815}, {1155,31448}, {1376,31442}, {1500,15803}, {1505,9616}, {1572,3579}, {1574,31424}, {1698,7748}, {1699,31455}, {1706,31456}, {2242,31426}, {2549,6684}

X(31422) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1571, 9620), (3, 31443, 1571)

X(31423) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, NINE-POINTS}

Barycentrics 3*a^4+(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31423) = X(1)-8*X(140) = 3*X(1)+4*X(5690) = X(1)+6*X(26446) = 6*X(2)+X(40) = 9*X(2)-2*X(946) = 15*X(2)-X(962) = 3*X(2)+4*X(6684) = 12*X(2)-5*X(8227) = 8*X(2)-X(31162) = 3*X(40)+4*X(946) = 5*X(40)+2*X(962) = X(40)-8*X(6684) = 2*X(40)+5*X(8227) = 9*X(40)-2*X(20070) = 4*X(40)+3*X(31162) = 6*X(140)+X(5690) = 4*X(140)+3*X(26446) = 10*X(946)-3*X(962) = X(946)+6*X(6684) = 8*X(946)-15*X(8227) = 6*X(946)+X(20070) = 16*X(946)-9*X(31162) = 9*X(962)+5*X(20070) = 8*X(962)-15*X(31162) = 2*X(5690)-9*X(26446) = 10*X(8227)-3*X(31162)

X(31423) lies on these lines: {1,140}, {2,40}, {3,1698}, {4,3634}, {5,165}, {6,31428}, {8,10165}, {9,2252}, {10,631}, {12,15803}, {20,10175}, {30,7989}, {32,31441}, {35,6883}, {36,9578}, {46,5219}, {56,31434}, {57,498}, {63,27529}, {72,15016}, {115,31422}, {119,16209}, {191,31142}, {210,9940}, {230,9593}, {355,549}, {371,13947}, {372,13893}, {376,19925}, {405,2077}, {411,9342}, {442,16113}, {451,1753}, {474,11012}, {486,9616}, {495,3361}, {499,1697}, {515,3523}, {516,3090}, {517,3526}, {519,15702}, {550,30315}, {551,12245}, {580,750}, {581,899}, {590,1703}, {615,1702}, {632,5886}, {936,3035}, {952,14869}, {970,29825}, {993,6940}, {1006,25440}, {1040,9610}, {1064,17749}, {1125,3525}, {1155,9612}, {1158,3305}, {1210,5218}, {1329,31424}, {1376,5705}, {1385,3679}, {1420,10039}, {1482,15694}, {1483,3653}, {1512,6977}, {1571,7746}, {1572,31455}, {1656,1699}, {1706,6967}, {1737,3601}, {1788,11529}, {1837,30282}, {2093,5326}, {2948,15061}, {3068,13975}, {3069,13912}, {3071,9582}, {3085,3333}, {3086,31393}, {3091,10172}, {3149,7688}, {3158,10916}, {3336,4654}, {3338,3584}, {3339,11374}, {3359,6863}, {3428,16408}, {3524,3828}, {3528,28164}, {3530,18481}, {3533,5603}, {3541,7713}, {3586,5217}, {3587,6861}, {3612,5727}, {3616,11362}, {3617,5882}, {3622,28234}, {3626,7967}, {3628,7988}, {3632,10246}, {3633,15178}, {3654,5901}, {3655,11812}, {3656,10124}, {3681,12005}, {3683,31246}, {3697,12675}, {3767,9574}, {3814,6937}, {3817,5067}, {3820,5234}, {3822,21165}, {3826,6831}, {3832,28150}, {3841,6830}, {3844,5085}, {3851,28146}, {3876,5884}, {3901,5885}, {3928,21077}, {4292,10588}, {4298,8164}, {4652,11681}, {4668,30392}, {4999,9623}, {5010,10826}, {5012,9622}, {5044,5693}, {5049,31480}, {5122,9654}, {5128,12047}, {5204,9613}, {5231,5687}, {5254,31421}, {5258,10269}, {5260,5450}, {5281,5704}, {5288,16203}, {5290,31479}, {5418,18991}, {5420,18992}, {5435,21620}, {5437,10198}, {5535,25525}, {5703,18221}, {5708,18217}, {5720,16132}, {5726,18990}, {5744,21075}, {5771,17718}, {5790,13624}, {5791,8580}, {5904,10202}, {5919,31436}, {5955,18229}, {6264,6713}, {6667,14217}, {6705,10864}, {7280,10827}, {7288,31397}, {7294,11376}, {7735,31396}, {7741,9580}, {7749,9620}, {7815,12197}, {7951,9579}, {8715,24392}, {8981,19004}, {9540,13936}, {9575,31401}, {9583,13973}, {9819,11373}, {9860,15561}, {9902,11171}, {9904,14643}, {10200,12703}, {10222,11415}, {10283,16189}, {10389,31452}, {10573,13384}, {10589,10624}, {11010,23708}, {11227,14872}, {12407,15035}, {12515,15017}, {12619,15015}, {12751,21154}, {12779,23328}, {12782,15819}, {13881,31443}, {13883,13935}, {13951,31439}, {13966,19003}, {14986,31188}, {15805,16472}, {16150,22937}, {17286,24257}, {24954,31235}, {26878,28609}, {28292,31207}

X(31423) = midpoint of X(i) and X(j) for these {i,j}: {3523, 9780}, {3624, 9588}, {7989, 16192}
X(31423) = reflection of X(i) in X(j) for these (i,j): (3624, 3526), (9624, 3624)
X(31423) = X(15056)-of-hexyl triangle
X(31423) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 26446, 1), (5432, 24914, 1)

X(31424) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, SPIEKER}

Barycentrics a*(3*a^3+(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a-(b+c)^3) : :

X(31424) lies on these lines: {1,21}, {2,4292}, {3,9}, {4,5705}, {6,31429}, {7,1125}, {8,3977}, {10,20}, {30,5791}, {32,16517}, {33,22361}, {35,200}, {36,8583}, {37,4252}, {40,958}, {44,4255}, {46,5251}, {55,6765}, {56,3683}, {57,405}, {72,3601}, {78,3219}, {90,6512}, {140,30827}, {142,16845}, {144,5703}, {197,15592}, {210,5217}, {214,13243}, {223,1935}, {224,18232}, {226,6857}, {238,988}, {326,1098}, {329,13411}, {372,31438}, {376,5325}, {377,1698}, {379,16832}, {380,4267}, {386,1743}, {392,1420}, {404,3305}, {411,1750}, {442,9579}, {452,1210}, {474,7308}, {515,10268}, {516,5833}, {519,4313}, {527,3487}, {551,11036}, {553,17561}, {572,15479}, {580,1741}, {631,2096}, {908,6910}, {938,11106}, {942,3928}, {944,5837}, {946,5698}, {950,11111}, {956,1697}, {975,3731}, {978,27640}, {991,3682}, {997,5267}, {1001,3333}, {1006,8726}, {1010,19859}, {1011,22345}, {1158,30503}, {1214,1394}, {1247,8769}, {1329,31423}, {1376,5302}, {1378,9616}, {1385,15829}, {1453,3666}, {1466,16293}, {1479,5231}, {1572,31456}, {1574,31422}, {1575,31421}, {1699,6837}, {1703,31453}, {1706,3579}, {1709,12565}, {1722,17596}, {1724,2999}, {1728,11344}, {1729,3496}, {1770,19854}, {1836,24953}, {2093,19860}, {2551,6256}, {2886,16113}, {2951,12511}, {3085,12527}, {3194,8765}, {3218,16865}, {3295,6762}, {3306,5047}, {3338,5259}, {3423,17742}, {3428,12705}, {3436,31434}, {3488,24391}, {3522,10430}, {3523,6700}, {3555,10389}, {3560,5709}, {3586,6734}, {3612,5692}, {3616,9965}, {3624,5249}, {3634,4208}, {3646,15254}, {3712,10371}, {3753,5128}, {3811,5223}, {3814,4197}, {3876,4855}, {3911,5084}, {3923,10444}, {3927,11523}, {3973,4256}, {3984,17574}, {4015,9859}, {4187,31231}, {4188,27065}, {4294,4847}, {4297,9799}, {4305,6737}, {4312,12609}, {4421,4662}, {4426,9593}, {4641,19765}, {4654,15670}, {4668,11015}, {4679,5433}, {4853,5119}, {4862,24159}, {4882,8715}, {4999,8227}, {5122,16408}, {5129,5435}, {5204,25917}, {5218,21075}, {5219,7483}, {5247,17594}, {5266,7174}, {5268,26264}, {5272,25494}, {5281,5815}, {5285,13730}, {5290,10198}, {5316,17567}, {5328,10303}, {5437,11108}, {5440,19535}, {5584,10860}, {5587,26066}, {5657,5795}, {5708,16866}, {5715,6824}, {5730,13384}, {5794,18253}, {5832,12699}, {5904,16465}, {6061,17104}, {6173,24470}, {6245,6987}, {6260,6988}, {6282,6906}, {6666,17582}, {6675,25525}, {6692,17559}, {6705,6865}, {6839,7989}, {6875,18446}, {6884,7988}, {6914,26921}, {6950,26878}, {6985,18540}, {6986,10857}, {7193,13323}, {7283,11679}, {7411,8580}, {7992,12520}, {8171,11035}, {8720,16825}, {8822,10436}, {9575,31449}, {9580,24390}, {9581,11113}, {9583,30557}, {9589,31458}, {9613,24987}, {9614,10527}, {9678,18991}, {9711,31425}, {9963,15863}, {10085,15931}, {10164,18250}, {10176,11220}, {10434,10463}, {10479,14058}, {10882,15654}, {10883,25639}, {11374,28609}, {11376,31157}, {11415,24541}, {11512,17123}, {11518,19526}, {12513,31393}, {12579,29635}, {12650,22758}, {13369,28466}, {13738,22060}, {13747,20196}, {14021,17284}, {15015,18254}, {15171,24392}, {15674,31019}, {15676,17483}, {16020,24171}, {16058,23085}, {16209,26364}, {16287,23206}, {16289,16574}, {16347,26223}, {16552,19763}, {16850,17754}, {16859,27003}, {17064,24851}, {17272,18650}, {17306,17698}, {17527,31190}, {17579,19875}, {17580,18230}, {17697,24627}, {17768,28628}, {18229,19645}, {18421,30147}, {18443,24467}, {18444,30144}, {18655,25590}, {19278,27064}, {21982,25083}, {22076,26892}, {22793,31493}, {25513,27339}, {26357,30223}, {27385,31018}

X(31424) = reflection of X(i) in X(j) for these (i,j): (5290, 10198), (5715, 6824)
X(31424) = X(18925)-of-2nd circumperp triangle
X(31424) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16570, 1046), (968, 1468, 1), (3901, 5426, 1)

X(31425) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, STEINER}

Barycentrics 9*a^4+3*(b+c)*a^3-(11*b^2+6*b*c+11*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31425) = 3*X(1)-16*X(3530) = 10*X(3)+3*X(3679) = 12*X(3)+X(5881) = 6*X(3)+7*X(9588) = 3*X(3)+10*X(31447) = 4*X(5)+9*X(165) = 6*X(10)+7*X(3528) = 4*X(20)+9*X(5587) = X(20)+12*X(6684) = 3*X(20)+10*X(31399) = 3*X(40)+10*X(631) = 5*X(40)+8*X(1125) = 9*X(40)+4*X(4301) = 7*X(40)+6*X(5603) = 6*X(40)+7*X(9624) = X(40)+12*X(10164) = 6*X(165)+7*X(31423) = 25*X(631)-12*X(1125) = 15*X(631)-2*X(4301) = 20*X(631)-7*X(9624) = 5*X(631)-18*X(10164) = 18*X(1125)-5*X(4301) = 28*X(1125)-15*X(5603) = 2*X(1125)-15*X(10164) = 18*X(3679)-5*X(5881) = 3*X(5587)-16*X(6684) = X(5603)-14*X(10164) = X(5881)-14*X(9588) = 20*X(6684)-7*X(9780) = 18*X(6684)-5*X(31399) = 7*X(9588)-20*X(31447)

X(31425) lies on these lines: {1,3530}, {3,3679}, {5,165}, {6,31431}, {10,3528}, {20,5587}, {32,31444}, {40,631}, {56,31436}, {57,31452}, {140,31162}, {145,3576}, {372,31440}, {382,1698}, {515,21734}, {516,5067}, {519,10299}, {546,19876}, {548,16192}, {549,7991}, {550,19875}, {944,4746}, {1376,31446}, {1572,31457}, {1699,5070}, {1703,31454}, {1706,31458}, {3523,7982}, {3525,5493}, {3526,3579}, {3529,3828}, {3625,5657}, {3627,30315}, {3632,17502}, {3634,3855}, {3653,16189}, {3654,15712}, {3656,12108}, {3843,11231}, {3853,7989}, {4325,9578}, {4330,9581}, {4338,5219}, {4512,17575}, {4669,15715}, {4677,17504}, {4745,15710}, {5054,11522}, {5079,28202}, {5319,9574}, {5691,15696}, {5734,10165}, {5882,15692}, {7486,9778}, {7765,31422}, {9575,31450}, {9607,31421}, {9657,31434}, {9680,18991}, {9711,31424}, {9956,17800}, {10175,17578}, {10303,28194}, {12512,18492}, {12699,16239}, {15693,24680}, {15720,25055}, {15803,15888}, {19872,22793}, {19877,28150}

X(31425) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9588, 5881), (3, 31447, 9588)

X(31426) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, INCIRCLE}

Barycentrics a*(a^3+(b+c)*a^2-(5*b^2+6*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31426) lies on these lines: {1,5013}, {8,31429}, {9,6048}, {11,31428}, {39,1697}, {40,2276}, {55,9593}, {56,31421}, {57,1500}, {497,31396}, {516,31402}, {517,31461}, {574,1420}, {988,3208}, {999,31430}, {1335,31427}, {1453,17735}, {1574,7308}, {1575,31435}, {1699,31460}, {1703,31459}, {1706,5283}, {2136,16975}, {2242,31422}, {2548,9580}, {2549,9578}, {2999,14974}, {3057,9592}, {3338,9331}, {3501,17594}, {3601,9620}, {3815,9614}, {4255,21872}, {4853,31449}, {5024,9957}, {5119,9575}, {5250,17756}, {5254,31434}, {5587,9598}, {5687,16517}, {5705,21956}, {6421,31432}, {7736,10624}, {7738,31397}, {7743,31467}, {7962,9619}, {9579,31409}, {9581,31398}, {9589,31462}, {9607,31436}, {12053,31400}, {15803,31443}

X(31426) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31433, 1697), (55, 9593, 16780)

X(31427) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, 2nd LEMOINE}

Barycentrics a*(a^3+(b+c)*a^2-4*S*a-(5*b^2+2*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31427) lies on these lines: {6,165}, {32,9582}, {39,1702}, {40,6422}, {57,31459}, {371,9593}, {372,31421}, {486,31428}, {516,31403}, {1335,31426}, {1378,31429}, {1504,1571}, {1575,31438}, {1588,31396}, {1699,31463}, {1706,31464}, {2275,31432}, {3312,31430}

X(31427) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31437, 1702), (1504, 1571, 1703)

X(31428) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, NINE-POINTS}

Barycentrics a^4+(b+c)*a^3-(7*b^2+2*b*c+7*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31428) lies on these lines: {1,31398}, {2,9593}, {4,31421}, {5,9574}, {6,31423}, {10,9592}, {11,31426}, {39,1698}, {40,3815}, {57,31460}, {165,2548}, {381,31430}, {486,31427}, {516,31404}, {517,31467}, {574,5691}, {1329,31429}, {1506,1571}, {1572,9588}, {1575,5705}, {1703,31463}, {1706,31466}, {2275,31434}, {2549,7989}, {3361,31409}, {3624,9620}, {3634,5286}, {3679,9619}, {3911,31402}, {5013,5587}, {5024,9956}, {5290,31476}, {5432,16780}, {5475,31422}, {5881,31492}, {6421,13893}, {6422,13947}, {6684,7736}, {7738,10175}, {7739,19876}, {7746,19872}, {8227,31489}, {9575,26446}, {9581,31448}, {9589,31444}, {9596,15803}, {9605,11231}, {13975,31403}, {16517,26364}

X(31428) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31396, 9593), (31398, 31401, 1)

X(31429) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, SPIEKER}

Barycentrics a*(a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a-(b+c)^3) : :

X(31429) lies on these lines: {1,5021}, {3,16517}, {6,31424}, {8,31426}, {9,39}, {10,7738}, {21,16780}, {37,3333}, {40,1107}, {57,5283}, {200,31448}, {516,31405}, {517,31468}, {574,5438}, {936,5013}, {958,9593}, {960,9592}, {968,1475}, {975,5030}, {1329,31428}, {1376,31421}, {1378,31427}, {1500,6762}, {1571,1573}, {1697,16975}, {1699,31466}, {1703,31464}, {2082,4414}, {2136,31433}, {2275,31435}, {2551,31396}, {3158,31451}, {3452,31400}, {3646,16604}, {4652,5276}, {5024,5044}, {5254,5705}, {5275,15803}, {5286,5745}, {5437,16589}, {5791,15048}, {6421,31438}, {6765,31477}, {7736,12572}, {9575,12514}, {9589,31469}, {9607,31446}, {9619,15829}, {9620,31456}, {9623,31490}, {9709,31430}, {9711,31431}, {12527,31402}, {17448,31393}, {24627,27523}

X(31429) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31442, 9), (1571, 1573, 1706)

X(31430) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, STAMMLER}

Barycentrics a*(2*a^3+(b+c)*a^2-2*(3*b^2+b*c+3*c^2)*a-(b^2-c^2)*(b-c)) : :
X(31430) = 3*X(1571)+X(9619) = 3*X(5013)-X(9619)

X(31430) lies on these lines: {3,9574}, {6,31422}, {30,31396}, {39,3579}, {40,5024}, {57,31461}, {165,9605}, {381,31428}, {382,31431}, {516,31406}, {517,1571}, {574,1385}, {942,31448}, {999,31426}, {1384,16192}, {1572,22332}, {1575,31445}, {1699,31467}, {1706,31468}, {2548,28146}, {2549,9956}, {3312,31427}

X(31430) = midpoint of X(1571) and X(5013)
X(31430) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31443, 3579), (9574, 31421, 3)

X(31431) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, STEINER}

Barycentrics 3*a^4+3*(b+c)*a^3-(17*b^2+6*b*c+17*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31431) lies on these lines: {1,31450}, {5,9574}, {6,31425}, {20,31396}, {39,9588}, {40,9606}, {57,31462}, {382,31430}, {516,31407}, {517,31470}, {631,9593}, {1571,9589}, {1575,31446}, {1698,7765}, {1703,31465}, {1706,31469}, {2275,31436}, {4301,31400}, {5013,5881}, {6421,31440}, {7738,31399}, {9592,11362}, {9605,31447}, {9620,31457}, {9624,31492}, {9711,31429}

X(31431) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31444, 9588), (1571, 9698, 9589)

X(31432) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, 2nd LEMOINE}

Barycentrics a*(4*S*a+(a^2+2*(b+c)*a+(b-c)^2)*(-a+b+c)) : :

X(31432) lies on these lines: {1,371}, {6,1697}, {8,31438}, {11,13893}, {36,9582}, {40,1124}, {55,18992}, {56,9616}, {57,3297}, {165,6502}, {200,15892}, {485,9614}, {486,31434}, {497,13883}, {516,31408}, {517,31474}, {944,19068}, {950,19066}, {999,31439}, {1015,31437}, {1151,1420}, {1319,9615}, {1335,31393}, {1378,31435}, {1505,31433}, {1572,31471}, {1587,10624}, {1588,31397}, {1699,31472}, {1703,3299}, {1706,31473}, {2275,31427}, {2362,7991}, {3057,18991}, {3068,12053}, {3070,9580}, {3071,9578}, {3086,13912}, {3311,9957}, {3601,7968}, {4311,9541}, {4853,31453}, {4866,7133}, {5126,6449}, {5218,13971}, {5414,19003}, {5919,18996}, {6221,24928}, {6421,31426}, {6459,10106}, {6561,9613}, {7585,9785}, {7743,8976}, {7962,7969}, {8227,9646}, {8981,11373}, {9575,31459}, {9581,13911}, {9589,31475}, {9819,19004}, {11376,13901}, {12701,19028}, {13905,30384}, {13947,19029}, {15558,19113}

X(31432) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3057, 19038, 18991), (3299, 5119, 1703)

X(31433) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, MOSES}

Barycentrics a*(a^3+(b+c)*a^2-3*(b+c)^2*a-(b^2-c^2)*(b-c)) : :

X(31433) lies on these lines: {1,574}, {8,31442}, {11,31441}, {39,1697}, {40,1500}, {55,9620}, {56,31422}, {115,31434}, {165,2242}, {484,9331}, {497,31398}, {516,31409}, {517,31477}, {999,31443}, {1015,9574}, {1335,31437}, {1505,31432}, {1506,9614}, {1572,2276}, {1574,31435}, {1699,31476}, {1703,31471}, {1706,16589}, {2136,31429}, {2241,9593}, {2271,21872}, {2548,10624}, {2549,31397}, {3057,9619}, {3880,31449}, {3895,16975}, {4646,14974}, {4853,31456}, {4868,16972}, {5013,9957}, {5475,9580}, {5587,9664}, {7743,31489}, {7748,9578}, {7756,9613}, {7765,31436}, {9589,31478}, {9592,9819}, {9598,10039}, {9785,31400}, {12053,31401}, {12514,20691}, {12575,31396}, {12701,31460}, {15815,24928}, {16973,25439}

X(31433) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1697, 31426, 39), (9574, 31393, 1015)

X(31434) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, NINE-POINTS}

Barycentrics a^4+(b+c)*a^3-3*(b+c)^2*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31434) lies on these lines: {1,2}, {3,9578}, {5,1697}, {9,17757}, {11,31393}, {12,40}, {35,1012}, {46,5290}, {55,3586}, {56,31423}, {57,495}, {115,31433}, {140,1420}, {165,1478}, {191,16152}, {210,18397}, {226,2093}, {281,1785}, {355,3601}, {380,26063}, {381,9580}, {388,6684}, {392,30827}, {442,1706}, {484,4312}, {486,31432}, {497,10175}, {515,5218}, {516,10590}, {517,5219}, {518,30274}, {631,10106}, {946,6969}, {950,5818}, {952,13384}, {984,1735}, {999,11231}, {1001,5123}, {1015,31441}, {1056,3911}, {1124,13947}, {1155,11237}, {1329,31435}, {1335,13893}, {1467,6989}, {1479,6939}, {1490,10786}, {1532,1699}, {1572,31476}, {1656,9957}, {1703,31472}, {1728,21031}, {1739,4859}, {1788,21620}, {2078,6883}, {2098,9624}, {2136,24390}, {2275,31428}, {2646,5881}, {3057,8227}, {3074,5264}, {3090,12053}, {3091,10624}, {3097,10063}, {3158,3419}, {3247,21933}, {3295,9581}, {3303,17606}, {3333,15888}, {3336,4355}, {3338,5445}, {3339,13407}, {3340,5690}, {3421,5745}, {3436,31424}, {3476,10165}, {3485,11362}, {3487,4848}, {3523,4311}, {3526,24928}, {3576,5252}, {3579,9579}, {3583,6957}, {3585,6925}, {3614,12701}, {3628,11373}, {3681,18389}, {3683,31141}, {3746,10826}, {3753,25525}, {3817,30305}, {3820,7308}, {3839,30332}, {3877,30852}, {3885,7504}, {3895,11680}, {3921,5728}, {3947,4295}, {3983,10399}, {4292,5261}, {4293,10164}, {4299,16192}, {4304,5281}, {4308,10303}, {4342,10171}, {4540,12564}, {4640,11236}, {4652,20060}, {5010,6909}, {5054,5126}, {5055,7743}, {5056,9785}, {5250,11681}, {5251,8069}, {5254,31426}, {5258,22766}, {5259,11508}, {5269,5725}, {5443,30323}, {5541,8068}, {5697,7686}, {5722,10389}, {5727,5790}, {5795,6857}, {5815,18231}, {5832,5856}, {5886,7962}, {5904,13750}, {6284,18492}, {6600,17057}, {6763,17700}, {6833,12650}, {6932,9589}, {7330,10942}, {7741,15845}, {7982,11375}, {7988,9819}, {7991,12047}, {8256,28628}, {9575,31460}, {9597,31421}, {9646,18991}, {9651,31422}, {9657,31425}, {10057,15015}, {10172,10589}, {10391,18908}, {10572,31452}, {10591,12575}, {10592,12699}, {10902,11501}, {11529,17718}, {12260,15079}, {12436,26062}, {12526,21077}, {12607,26066}, {12705,18242}, {12758,15017}, {13905,19004}, {13963,19003}, {13975,31408}, {14647,30304}, {15950,16200}, {16601,23058}, {17605,31162}

X(31434) = reflection of X(i) in X(j) for these (i,j): (5219, 31479), (30282, 5218)
X(31434) = X(5231)-of-inner-Yff triangle
X(31434) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1737, 10056, 1), (6734, 10528, 6765), (10915, 26363, 4853)

X(31435) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, SPIEKER}

Barycentrics a*(a^3+(b+c)*a^2-(b^2+6*b*c+c^2)*a-(b+c)^3) : :

X(31435) lies on these lines: {1,6}, {2,40}, {3,4512}, {8,3305}, {10,497}, {11,5705}, {12,4679}, {19,7498}, {20,11372}, {21,84}, {35,5438}, {38,28011}, {46,3624}, {55,936}, {56,3683}, {57,1125}, {63,3333}, {65,4423}, {78,1621}, {140,3359}, {142,4295}, {144,11037}, {145,27065}, {165,474}, {169,19868}, {191,3338}, {200,3295}, {210,3303}, {329,21620}, {377,24564}, {380,965}, {388,12572}, {406,1848}, {442,1699}, {443,516}, {452,515}, {496,5231}, {498,30827}, {517,11108}, {551,3929}, {581,25941}, {595,975}, {602,2328}, {612,3915}, {614,2292}, {758,11518}, {846,988}, {859,10882}, {942,10582}, {968,1193}, {978,17594}, {986,5272}, {993,1420}, {997,3601}, {999,31445}, {1009,2944}, {1010,12717}, {1015,31442}, {1056,12527}, {1058,4847}, {1158,10165}, {1210,26105}, {1329,31434}, {1378,31432}, {1385,7330}, {1467,12709}, {1482,16857}, {1490,13615}, {1512,6898}, {1519,6889}, {1572,16589}, {1574,31433}, {1575,31426}, {1698,1706}, {1703,31473}, {1709,7987}, {1722,17123}, {1730,16844}, {1754,19523}, {1764,16343}, {1788,9843}, {2093,3812}, {2136,3679}, {2270,5257}, {2275,31429}, {2478,5587}, {2550,10624}, {2551,31397}, {2646,30223}, {2821,25926}, {2950,21154}, {2999,3931}, {3057,9623}, {3085,3452}, {3086,5745}, {3149,10268}, {3158,3746}, {3219,3622}, {3306,5550}, {3339,5439}, {3340,3878}, {3361,3916}, {3421,18250}, {3474,12436}, {3488,6737}, {3579,16408}, {3586,5794}, {3587,8728}, {3600,8545}, {3617,3895}, {3652,3653}, {3670,5573}, {3680,7162}, {3697,4882}, {3702,5271}, {3740,3913}, {3749,5293}, {3753,7991}, {3757,19582}, {3811,10176}, {3816,26066}, {3817,6856}, {3868,4666}, {3869,5284}, {3870,3876}, {3873,3951}, {3877,5047}, {3884,7962}, {3885,11525}, {3886,9534}, {3897,16858}, {3925,12701}, {3927,5045}, {4292,5698}, {4308,29007}, {4314,10384}, {4357,17170}, {4385,30568}, {4414,28352}, {4640,15803}, {4642,17125}, {4652,5253}, {4659,24424}, {4673,17277}, {4853,9708}, {4915,30337}, {5128,19862}, {5218,6700}, {5219,10198}, {5249,11415}, {5255,5268}, {5273,14986}, {5285,11365}, {5327,25526}, {5432,24954}, {5534,16202}, {5584,21153}, {5686,6764}, {5687,8580}, {5704,18231}, {5709,5886}, {5711,17022}, {5720,10267}, {5731,10864}, {5732,8273}, {5785,14100}, {5799,21363}, {5837,18391}, {5881,7966}, {5887,18443}, {5901,26921}, {5903,25542}, {5919,12629}, {6001,8726}, {6282,11496}, {6666,19855}, {6690,25681}, {6743,30331}, {7091,13462}, {7171,13624}, {7743,31493}, {7992,10167}, {9612,24703}, {9624,12704}, {9711,31436}, {9785,18230}, {9819,10914}, {10085,30389}, {10200,31231}, {10436,17753}, {10439,18180}, {10476,17185}, {10587,31018}, {10595,26878}, {10866,15837}, {11010,31262}, {11019,18249}, {11235,17619}, {11376,24953}, {11512,17596}, {12047,25525}, {12053,19843}, {12608,15239}, {12672,30503}, {12703,24982}, {13384,30144}, {13407,28609}, {16878,19762}, {17306,18589}, {18164,28619}, {18228,21075}, {18481,18540}, {19854,30384}, {19919,24467}, {20196,26364}, {21077,31142}, {24178,24248}, {24627,26093}, {25430,27784}, {25591,29828}, {25924,28292}, {26102,30979}, {26128,28039}

X(31435) = X(10982)-of-excentral triangle
X(31435) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5223, 3555), (1, 5904, 3243), (10179, 12513, 1)

X(31436) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, STEINER}

Barycentrics 3*a^4+3*(b+c)*a^3-(5*b^2+18*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31436) lies on these lines: {1,631}, {5,1697}, {8,31446}, {20,9613}, {40,10404}, {55,5881}, {56,31425}, {165,4317}, {382,9578}, {405,3679}, {497,31399}, {498,9819}, {499,30337}, {516,31410}, {517,31480}, {999,31447}, {1015,31444}, {1145,5436}, {1335,31440}, {1420,3530}, {1572,31478}, {1698,3816}, {1703,31475}, {1706,17529}, {2275,31431}, {3057,9624}, {3085,4301}, {3526,9957}, {3528,10106}, {3584,6834}, {3586,4309}, {3624,5836}, {3654,11518}, {3680,7483}, {3832,10624}, {3843,9580}, {3895,5705}, {4311,21734}, {4330,5691}, {4338,5290}, {4512,10915}, {4853,31458}, {4857,6898}, {5067,12053}, {5119,9589}, {5587,9670}, {5690,10389}, {5734,13411}, {5919,31423}, {6857,12640}, {6908,7991}, {7320,10303}, {7486,9785}, {7765,31433}, {9575,31462}, {9607,31426}, {9711,31435}, {11373,16239}, {11525,24953}, {13747,25055}, {17559,19875}

X(31436) = {X(1697), X(31434)}-harmonic conjugate of X(9614)

X(31437) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, MOSES}

Barycentrics a*(a^3+(b+c)*a^2-4*S*a-(3*b^2+2*b*c+3*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31437) lies on these lines: {6,1571}, {32,9616}, {39,1702}, {40,1504}, {57,31471}, {115,13893}, {165,5062}, {187,9582}, {371,9620}, {372,31422}, {486,31441}, {516,31411}, {574,18992}, {1015,31432}, {1335,31433}, {1378,31442}, {1505,9574}, {1572,6422}, {1574,31438}, {1588,31398}, {1699,31481}, {1706,31482}, {2549,13883}, {3312,31443}

X(31437) = {X(1702), X(31427)}-harmonic conjugate of X(39)

X(31438) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, SPIEKER}

Barycentrics a*((a+b+c)*S+(a-b+c)*(a+b-c)*a)*(-a+b+c) : :

X(31438) lies on these lines: {1,6}, {3,19068}, {8,31432}, {10,1588}, {40,1377}, {57,31473}, {200,2066}, {210,19038}, {371,936}, {372,31424}, {391,30412}, {486,5705}, {590,30827}, {997,9583}, {1151,5438}, {1329,13893}, {1376,9616}, {1572,31482}, {1574,31437}, {1575,31427}, {1587,12572}, {1703,12514}, {2362,12526}, {2551,13883}, {3068,3452}, {3069,5745}, {3311,5044}, {3312,31445}, {3686,7090}, {5273,7586}, {5325,19053}, {5328,8972}, {5791,7584}, {5795,19066}, {5837,19065}, {6421,31429}, {6700,9540}, {7330,19067}, {7585,18228}, {9582,25440}, {9709,31439}, {9711,31440}, {12527,31408}, {13947,26066}, {13971,30478}

X(31438) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 1449, 30557), (30556, 31453, 1)

X(31439) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, STAMMLER}

Barycentrics a*(2*a^3+(b+c)*a^2-4*S*a-2*(b^2+b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(31439) = 3*X(371)-X(7969)

X(31439) lies on these lines: {1,6221}, {3,1702}, {5,13912}, {6,1571}, {30,13883}, {40,3311}, {46,19038}, {57,31474}, {165,3312}, {355,6459}, {371,517}, {381,13893}, {382,31440}, {485,22793}, {486,11231}, {516,7583}, {549,13971}, {582,605}, {590,9955}, {942,2066}, {999,31432}, {1151,1385}, {1155,3299}, {1378,31445}, {1482,9583}, {1588,26446}, {1698,13785}, {1699,8976}, {1703,6417}, {1706,31485}, {1770,19028}, {1836,13905}, {1902,10880}, {2067,9957}, {3068,12699}, {3070,28146}, {3071,9956}, {3316,9779}, {3576,6449}, {3622,9542}, {3634,18762}, {3654,19065}, {3656,13902}, {5119,18996}, {5122,6502}, {5418,11230}, {5886,9540}, {6199,12702}, {6200,7968}, {6361,7585}, {6398,19003}, {6407,9615}, {6409,17502}, {6421,31430}, {6425,24680}, {6437,11278}, {6447,7982}, {6450,16192}, {6455,7987}, {6519,9618}, {6561,13911}, {6684,7584}, {7581,9778}, {7743,9661}, {8983,22791}, {8988,22938}, {8994,12261}, {9541,18481}, {9584,30389}, {9589,31487}, {9605,31427}, {9648,15950}, {9679,30556}, {9709,31438}, {9780,23273}, {9812,13886}, {10164,13966}, {10819,11699}, {12047,13901}, {12515,19113}, {12611,13922}, {12701,13904}, {12778,19060}, {13888,31162}, {13947,18510}, {13951,31423}, {13975,19116}, {16232,24929}, {18483,18538}, {18965,30384}

X(31439) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1702, 9582, 18992), (1702, 9616, 3), (9582, 18992, 3)

X(31440) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, STEINER}

Barycentrics 3*a^4-12*S*a^2+3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31440) lies on these lines: {1,31454}, {5,1702}, {6,9588}, {20,9616}, {57,31475}, {371,5881}, {372,31425}, {382,31439}, {516,31414}, {517,31487}, {548,9582}, {631,13912}, {1335,31436}, {1378,31446}, {1505,31444}, {1572,31483}, {1588,31399}, {1706,31486}, {3068,4301}, {3312,31447}, {3576,9680}, {3592,3679}, {5731,9692}, {5734,8983}, {6421,31431}, {6447,28204}, {7765,31437}, {8960,31162}, {9575,31465}, {9607,31427}, {9615,19066}, {9711,31438}, {11362,18991}, {11522,13846}

X(31441) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, NINE-POINTS}

Barycentrics a^4+(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31441) lies on these lines: {2,9620}, {4,31422}, {5,1571}, {6,11231}, {10,9619}, {11,31433}, {32,31423}, {39,1698}, {40,1506}, {57,31476}, {115,9574}, {165,5475}, {381,31443}, {486,31437}, {516,31415}, {574,5587}, {1015,31434}, {1329,31442}, {1505,13893}, {1572,3815}, {1574,5705}, {1699,7603}, {1703,31481}, {1706,31488}, {2242,31231}, {2548,6684}, {2549,10175}, {3055,5886}, {3634,3767}, {3911,31409}, {5013,9956}, {5123,31449}, {5286,19877}, {5309,19876}, {7737,10164}, {7746,9593}, {7748,7989}, {7756,18492}, {9575,9698}, {9581,31451}, {9592,19875}, {9650,15803}, {9780,31400}, {13975,31411}, {15815,18480}, {17606,31448}, {24914,31460}, {31399,31450}

X(31441) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31398, 9620), (10, 31401, 9619)

X(31442) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, SPIEKER}

Barycentrics a*(a^3+(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a-(b+c)^3) : :

X(31442) lies on these lines: {1,9346}, {6,31445}, {8,31433}, {9,39}, {10,1571}, {32,16517}, {37,5021}, {40,1573}, {45,5022}, {57,16589}, {63,5283}, {115,5705}, {200,31451}, {210,31448}, {516,31416}, {517,31490}, {574,936}, {846,21384}, {958,9620}, {960,9619}, {968,20963}, {997,21879}, {1015,31435}, {1107,1572}, {1329,31441}, {1376,31422}, {1378,31437}, {1505,31438}, {1574,9574}, {1699,31488}, {1703,31482}, {2241,4512}, {2548,12572}, {2551,31398}, {3452,31401}, {3691,4414}, {3767,5745}, {3916,5275}, {4652,5277}, {5013,5044}, {5234,9593}, {5248,16973}, {5250,16975}, {5254,5791}, {5273,5286}, {5325,7739}, {5698,31405}, {7765,31446}, {8580,31421}, {9589,31491}, {9709,31443}, {12527,31409}, {18228,31400}, {18250,31396}, {24703,31466}, {30827,31455}

X(31442) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 31429, 39), (16517, 31424, 32)

X(31443) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, STAMMLER}

Barycentrics a*(2*a^3+(b+c)*a^2-2*(2*b^2+b*c+2*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31443) lies on these lines: {1,15815}, {3,1571}, {6,165}, {37,17122}, {39,3579}, {40,5013}, {46,31448}, {57,31477}, {115,11231}, {230,10164}, {381,31441}, {382,31444}, {516,3815}, {517,574}, {942,31451}, {999,31433}, {1100,3550}, {1155,2276}, {1386,5116}, {1506,22793}, {1572,5024}, {1574,31445}, {1575,4640}, {1699,31489}, {1706,31490}, {1770,31460}, {2108,15254}, {2242,5122}, {2549,26446}, {3053,9593}, {3055,3817}, {3097,4663}, {3312,31437}, {4095,8720}, {4421,16973}

X(31443) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (165, 9574, 6), (1571, 31422, 3)

X(31444) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, STEINER}

Barycentrics 3*a^4+3*(b+c)*a^3-(11*b^2+6*b*c+11*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31444) lies on these lines: {1,31457}, {5,1571}, {6,31447}, {20,31398}, {32,31425}, {39,9588}, {40,9698}, {57,31478}, {382,31443}, {516,31417}, {517,31492}, {574,5881}, {631,9620}, {1015,31436}, {1505,31440}, {1572,9606}, {1574,31446}, {1703,31483}, {1706,31491}, {2549,31399}, {4301,31401}, {5319,6684}, {7765,9574}, {9589,31428}, {9607,26446}, {9619,11362}, {9711,31442}

X(31444) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9588, 31431, 39), (11362, 31450, 9619)

X(31445) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, SPIEKER, STAMMLER}

Barycentrics    a*(2*a^3+(b+c)*a^2-2*(b^2+b*c+c^2)*a-(b+c)^3) : :
Trilinears    R cos A + s cot(A/2) : :
Trilinears    4 s^2 (a - b - c) + a (a^2 - b^2 - c^2) : :
X(31445) = X(1)+3*X(3929) = X(1)-3*X(16418) = X(8)+3*X(11111) = X(10)-3*X(5325) = 5*X(1698)-X(9579) = 5*X(1698)-3*X(17528) = X(3295)-3*X(4512) = 5*X(3616)-9*X(17561) = 7*X(3624)-3*X(4654) = X(3927)-3*X(3929) = X(3927)+3*X(16418) = X(6850)-3*X(26446) = X(9579)-3*X(17528) = 2*X(13624)-3*X(28466)

X(31445) lies on these lines: {1,3683}, {2,3824}, {3,9}, {4,5273}, {5,5745}, {6,31442}, {7,16845}, {8,11111}, {10,30}, {21,72}, {28,1868}, {35,210}, {36,25917}, {37,58}, {40,5234}, {44,386}, {45,975}, {57,11108}, {63,405}, {65,191}, {78,16370}, {100,3697}, {140,3452}, {142,24470}, {144,3487}, {165,9709}, {226,6675}, {228,17524}, {329,6857}, {333,5295}, {345,5814}, {354,5259}, {355,6868}, {381,5705}, {382,31446}, {392,2975}, {404,27065}, {411,5927}, {452,5722}, {474,3305}, {484,3698}, {495,12527}, {499,4679}, {500,3682}, {515,18249}, {516,31419}, {517,958}, {518,5248}, {527,1125}, {549,6700}, {579,16848}, {580,1212}, {631,18228}, {672,16850}, {846,3931}, {894,11110}, {908,7483}, {912,960}, {946,5762}, {952,5837}, {956,5250}, {984,5266}, {997,13624}, {999,31435}, {1001,5045}, {1155,1698}, {1214,1935}, {1329,11231}, {1378,31439}, {1437,26885}, {1468,6051}, {1572,31490}, {1574,31443}, {1575,31430}, {1621,3555}, {1699,31493}, {1703,31485}, {1707,5711}, {1709,5584}, {1724,3666}, {1770,3925}, {1776,12711}, {1836,19854}, {1867,14016}, {1998,13615}, {2003,22136}, {2074,11363}, {2355,19822}, {2551,6850}, {2646,5692}, {2886,22793}, {3057,5258}, {3074,17102}, {3149,10157}, {3191,17194}, {3218,5047}, {3246,30148}, {3293,4689}, {3295,4512}, {3296,3616}, {3306,16842}, {3312,31438}, {3337,25542}, {3338,4423}, {3361,3646}, {3419,6872}, {3428,9856}, {3474,19855}, {3488,11106}, {3525,5328}, {3526,30827}, {3601,3940}, {3624,4654}, {3648,20292}, {3678,15481}, {3715,5217}, {3739,14377}, {3740,25440}, {3753,5260}, {3811,5220}, {3820,6684}, {3868,16865}, {3876,4189}, {3878,24680}, {3884,11260}, {3899,11011}, {3911,17527}, {3928,5708}, {3951,19526}, {3976,15485}, {4018,11684}, {4255,16885}, {4256,15492}, {4257,16814}, {4292,8728}, {4309,4863}, {4357,17698}, {4420,4533}, {4662,8715}, {4847,15171}, {4855,19535}, {4973,19862}, {4999,11230}, {5084,5744}, {5123,10225}, {5126,19861}, {5231,9669}, {5267,10176}, {5282,16601}, {5285,20831}, {5288,5919}, {5289,15178}, {5294,13728}, {5314,20833}, {5435,17559}, {5436,15934}, {5437,16853}, {5690,5795}, {5698,12699}, {5703,6172}, {5709,5806}, {5787,6987}, {5794,28160}, {5805,6846}, {5811,6988}, {5812,6824}, {5818,18231}, {5843,10165}, {5853,10386}, {5886,30478}, {5956,21892}, {6048,17601}, {6259,6908}, {6666,12436}, {6690,21077}, {6734,11113}, {6762,6767}, {6883,9940}, {6906,26878}, {6907,22792}, {6910,31018}, {6917,9956}, {6986,10167}, {7066,20122}, {7069,22361}, {7082,26357}, {7085,13730}, {7308,15803}, {7489,24474}, {7688,7701}, {7743,10527}, {8273,10085}, {9275,21873}, {9352,19877}, {9575,31468}, {9589,31494}, {9605,31429}, {9623,12702}, {9678,30557}, {9711,31447}, {9776,17552}, {9947,11500}, {9955,24703}, {10246,15829}, {10436,16844}, {10572,21677}, {10855,16410}, {10902,14872}, {10916,18527}, {11015,15677}, {11491,18908}, {11517,20835}, {11520,19538}, {12047,24953}, {12127,31393}, {12433,24391}, {12511,15726}, {12609,17768}, {12649,31156}, {12680,15931}, {13723,25083}, {13725,26065}, {13741,24627}, {15670,17781}, {15674,17484}, {16058,20805}, {16091,17095}, {16113,18406}, {16138,17668}, {16192,30393}, {16286,23169}, {16287,22345}, {16288,16574}, {16342,26223}, {16374,22344}, {16453,22060}, {16517,30435}, {16846,17754}, {16858,24473}, {17276,24159}, {17536,27003}, {17570,23958}, {17582,18230}, {18254,22935}, {19270,27064}, {19277,19859}, {19518,21371}

X(31445) = midpoint of X(i) and X(j) for these {i,j}: {1, 3927}, {3, 7330}, {355, 6868}, {958, 12514}, {3560, 26921}, {3929, 16418}
X(31445) = reflection of X(i) in X(j) for these (i,j): (6147, 1125), (6917, 9956)
X(31445) = anticomplement of X(3824)
X(31445) = X(3927)-of-anti-Aquila triangle
X(31445) = X(12161)-of-2nd Zaniah triangle
X(31445) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3929, 3927), (3, 9, 5044), (37,58,37593), (3927, 16418, 1)

X(31446) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, SPIEKER, STEINER}

Barycentrics 3*a^4+3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b+c)^3*a+2*(b^2-c^2)^2 : :

X(31446) lies on these lines: {1,24597}, {2,3951}, {4,5325}, {5,9}, {7,3634}, {8,31436}, {10,20}, {21,3679}, {40,9710}, {57,17529}, {63,1698}, {84,26446}, {191,4338}, {200,31452}, {377,19875}, {382,31445}, {442,3929}, {516,31420}, {517,31494}, {519,17558}, {631,936}, {958,5881}, {960,9624}, {1259,5251}, {1376,31425}, {1378,31440}, {1572,31491}, {1574,31444}, {1575,31431}, {1703,31486}, {2551,31399}, {3219,9612}, {3452,5067}, {3526,5044}, {3530,5438}, {3617,4304}, {3624,3868}, {3626,4313}, {3654,11530}, {3681,10122}, {3683,9670}, {3697,10391}, {3729,25446}, {3731,5292}, {3828,4208}, {3832,12572}, {3841,4312}, {3876,18389}, {3927,25525}, {3928,8728}, {4292,9780}, {4301,18249}, {4309,4512}, {4669,12536}, {5070,30827}, {5223,10198}, {5302,5587}, {5319,16517}, {5708,20195}, {5732,6684}, {6245,21153}, {6675,11523}, {6769,9623}, {6837,7991}, {6838,30326}, {6884,11522}, {6993,30315}, {7308,17575}, {7486,18228}, {7765,31442}, {9575,31469}, {9589,10883}, {9607,31429}, {9708,11248}, {9709,31447}, {9711,26066}, {9799,10164}, {9843,18230}, {9960,15064}, {9965,19877}, {11036,19862}, {11520,25055}, {12526,19854}, {12527,31410}, {12625,16418}, {16845,24391}, {24046,31183}, {26364,30393}, {31164,31254}

X(31446) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5791, 5705), (10, 5273, 31424)

X(31447) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, STAMMLER, STEINER}

Barycentrics 6*a^4+3*(b+c)*a^3-2*(4*b^2+3*b*c+4*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31447) = 7*X(3)+3*X(3679) = 9*X(3)+X(5881) = 3*X(3)+7*X(9588) = 3*X(3)-13*X(31425) = 2*X(5)+3*X(3579) = 7*X(5)-12*X(3634) = X(5)-6*X(6684) = 13*X(5)-18*X(10172) = 4*X(5)-9*X(11231) = 17*X(5)-12*X(12571) = 11*X(5)-6*X(18483) = 8*X(5)-3*X(22793) = 7*X(3579)+8*X(3634) = X(3579)+4*X(6684) = 2*X(3579)+3*X(11231) = 11*X(3579)+4*X(18483) = 4*X(3579)+X(22793) = 2*X(3634)-7*X(6684) = 17*X(3634)-7*X(12571) = 22*X(3634)-7*X(18483) = 27*X(3679)-7*X(5881) = 13*X(6684)-3*X(10172) = 8*X(6684)-3*X(11231) = 17*X(6684)-2*X(12571) = 11*X(6684)-X(18483) = 7*X(9588)+13*X(31425)

X(31447) lies on these lines: {3,3679}, {5,516}, {6,31444}, {10,548}, {20,5818}, {30,31399}, {40,3526}, {57,31480}, {140,4301}, {165,382}, {355,3528}, {517,631}, {519,15712}, {549,24680}, {551,12108}, {632,28194}, {942,31452}, {946,16239}, {999,31436}, {1385,3244}, {1572,31492}, {1656,28198}, {1657,19875}, {1698,3843}, {1703,31487}, {1706,31494}, {3091,28202}, {3312,31440}, {3522,28208}, {3523,3654}, {3621,5657}, {3627,3828}, {3628,5493}, {3655,10299}, {3656,10303}, {3855,9778}, {3859,28178}, {3861,10175}, {4309,18527}, {4317,5122}, {4330,5445}, {4669,14891}, {4677,15706}, {4701,5690}, {5054,7991}, {5067,12699}, {5070,9589}, {5072,19876}, {5442,5919}, {5587,17800}, {5790,16192}, {5882,12100}, {6361,7486}, {7765,31443}, {7982,15720}, {9575,31470}, {9605,31431}, {9607,31430}, {9624,12702}, {9681,13973}, {9709,31446}, {9711,31445}, {10165,11278}, {11522,15694}, {12778,15057}, {13464,14869}, {15696,28160}, {17552,26062}, {17578,28154}, {17583,24987}, {18481,21734}, {19862,28212}, {28232,31253}

X(31447) = midpoint of X(40) and X(18493)
X(31447) = reflection of X(i) in X(j) for these (i,j): (18480, 5818), (18492, 9956)
X(31447) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3579, 6684, 11231), (3579, 11231, 22793), (9588, 31425, 3)

X(31448) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, INCIRCLE}

Barycentrics a^2*(a^2-3*b^2-2*b*c-3*c^2) : :

X(31448) lies on these lines: {1,5013}, {3,172}, {4,31460}, {5,9598}, {6,35}, {8,31449}, {12,2549}, {20,31402}, {21,17756}, {30,9596}, {32,5217}, {36,15815}, {37,474}, {39,55}, {42,5021}, {46,31443}, {56,574}, {57,31421}, {65,1571}, {183,25264}, {192,7824}, {200,31429}, {210,31442}, {213,4255}, {218,18755}, {219,5110}, {350,11285}, {372,31459}, {405,1575}, {495,9597}, {497,31400}, {498,5254}, {609,5023}, {611,3094}, {672,2271}, {942,31430}, {950,31396}, {956,20691}, {1015,3303}, {1107,5687}, {1155,31422}, {1193,14974}, {1376,5283}, {1384,7296}, {1475,2177}, {1479,3815}, {1504,19037}, {1505,19038}, {1506,9664}, {1697,9592}, {1837,31398}, {1914,9605}, {1975,27020}, {2023,10086}, {2066,6421}, {2067,9600}, {2242,5204}, {2275,3295}, {2334,9346}, {2345,19270}, {2548,6284}, {2646,9620}, {3053,5010}, {3057,9619}, {3085,7738}, {3269,19349}, {3434,31466}, {3496,17601}, {3601,9593}, {3666,21477}, {3679,31490}, {3730,4256}, {3746,16781}, {3760,15271}, {3767,5432}, {3913,16975}, {4261,16287}, {4294,7736}, {4302,7745}, {4309,9606}, {4354,9595}, {4400,22253}, {4413,16589}, {4426,16370}, {4995,7739}, {5022,20963}, {5038,10801}, {5218,5286}, {5225,31404}, {5275,25092}, {5414,6422}, {5475,12953}, {5563,9331}, {6181,6554}, {7280,16785}, {7354,31409}, {7737,15338}, {7741,31489}, {7748,10895}, {7756,9650}, {7763,26590}, {7803,26629}, {8588,9341}, {9581,31428}, {9599,15171}, {9651,11237}, {9657,31478}, {9665,9670}, {9669,31467}, {11507,13006}, {15668,25599}, {16343,17303}, {16466,17735}, {17606,31441}, {17684,17759}, {17750,19765}, {17784,31405}, {18995,31471}, {21859,22759}, {21956,26363}, {25520,26042}, {31140,31488}

X(31448) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31461, 2276), (5013, 31477, 1)

X(31449) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, SPIEKER}

Barycentrics a*(a^3-3*(b^2+c^2)*a-2*(b+c)*b*c) : :

X(31449) lies on these lines: {1,5021}, {3,1107}, {4,31466}, {6,993}, {8,31448}, {9,9592}, {10,5013}, {20,31405}, {36,5275}, {37,999}, {39,958}, {45,9259}, {55,11998}, {56,5283}, {99,20172}, {372,31464}, {405,2275}, {442,9597}, {519,31477}, {529,31409}, {574,1376}, {956,2276}, {960,9619}, {988,16583}, {1001,1015}, {1003,20179}, {1146,6181}, {1329,31401}, {1475,10448}, {1500,12513}, {1571,5836}, {1572,4640}, {1575,5024}, {1682,23630}, {1706,31421}, {1914,16370}, {1975,17030}, {2170,4414}, {2242,11194}, {2271,21384}, {2549,2886}, {2551,31400}, {3053,5267}, {3125,17595}, {3295,17448}, {3436,31460}, {3576,16517}, {3767,4999}, {3814,31489}, {3880,31433}, {3913,31451}, {4426,9605}, {4853,31426}, {5022,17750}, {5030,30116}, {5123,31441}, {5204,5277}, {5248,16781}, {5254,26363}, {5286,30478}, {5795,31396}, {6376,11285}, {6381,15271}, {6421,31453}, {7738,19843}, {7748,31488}, {7763,26558}, {7786,26687}, {8666,25092}, {8716,20181}, {9574,9623}, {9575,31424}, {9598,24390}, {9599,11113}, {9607,31458}, {9664,11235}, {9711,31450}, {11108,16604}, {11236,31476}, {15482,27076}, {15815,25440}, {16589,25524}, {16973,24929}, {17684,21226}, {18967,20616}, {20691,31461}, {24215,25500}

X(31449) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31468, 1107), (5013, 31490, 10)

X(31450) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, STEINER}

Barycentrics 3*a^4-10*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31450) lies on these lines: {1,31431}, {2,7765}, {3,9300}, {5,2549}, {6,3530}, {20,574}, {32,14930}, {39,631}, {56,31462}, {115,7486}, {140,7739}, {372,31465}, {382,3815}, {384,7618}, {548,7737}, {1352,12055}, {1376,31469}, {1506,3832}, {1571,4301}, {1575,31458}, {2275,31452}, {3054,3526}, {3522,7753}, {3523,7772}, {3524,5007}, {3525,5309}, {3528,7736}, {3843,31415}, {3855,7748}, {3926,15482}, {4317,31409}, {4325,9596}, {4330,9599}, {5056,11648}, {5067,7738}, {5070,5254}, {5306,15720}, {5881,31398}, {6337,6683}, {6421,31454}, {7622,7829}, {7745,15696}, {7755,10303}, {7756,17578}, {7758,7824}, {7763,7876}, {7786,16898}, {7791,7814}, {7796,7800}, {7849,16043}, {7873,9770}, {7891,16896}, {8357,11184}, {8364,12040}, {9574,9624}, {9575,31425}, {9588,9592}, {9589,31421}, {9605,21843}, {9619,11362}, {9657,31460}, {9711,31449}, {12100,22331}, {14537,17538}, {15048,16239}, {15515,21734}, {31399,31441}, {31410,31476}, {31414,31481}, {31420,31488}

X(31450) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31470, 9606), (5, 31492, 31401), (5013, 31401, 2549)

X(31451) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, MOSES}

Barycentrics a^2*(a^2-2*b^2-2*b*c-2*c^2) : :

X(31451) lies on these lines: {1,574}, {3,1500}, {4,31476}, {5,9664}, {6,24047}, {8,31456}, {11,31455}, {12,7748}, {20,31409}, {30,9650}, {32,35}, {36,15515}, {37,25440}, {39,55}, {57,31422}, {100,5283}, {101,28509}, {115,498}, {172,5010}, {187,5217}, {192,1078}, {200,31442}, {350,7815}, {372,31471}, {386,17735}, {390,31400}, {405,1574}, {495,9651}, {497,31401}, {528,31466}, {942,31443}, {950,31398}, {993,20691}, {999,15815}, {1015,3295}, {1107,8715}, {1376,16589}, {1478,7756}, {1479,1506}, {1504,5414}, {1505,2066}, {1569,10053}, {1573,5687}, {1575,5248}, {1697,9619}, {1909,7781}, {1914,7772}, {2067,9674}, {2176,4256}, {2177,20963}, {2275,3746}, {2548,4294}, {2549,3085}, {3055,10593}, {3056,5034}, {3158,31429}, {3298,9600}, {3434,31488}, {3584,11648}, {3601,9620}, {3614,18424}, {3730,18755}, {3734,27020}, {3767,5218}, {3788,26590}, {3815,9665}, {3871,16975}, {3913,31449}, {3954,4414}, {4189,5291}, {4255,14974}, {4302,7747}, {4309,9599}, {4314,31396}, {4354,9636}, {4366,7786}, {4386,25092}, {4995,5309}, {5023,9341}, {5024,16781}, {5062,31459}, {5204,8589}, {5225,31415}, {5281,5286}, {5299,10987}, {5432,7746}, {5475,6284}, {6645,7782}, {7280,9331}, {7483,21956}, {7603,10896}, {7737,31402}, {7751,25264}, {7765,31452}, {7834,26629}, {7913,30104}, {8588,16785}, {9351,21008}, {9581,31441}, {9597,10056}, {9669,31489}, {10386,31406}, {15482,26959}, {16783,20331}, {17143,17684}, {17448,25439}, {17784,31416}, {21888,30147}, {26100,31020}

X(31451) = isogonal conjugate of the isotomic conjugate of X(4445)
X(31451) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1500, 2242), (3, 31477, 1500)

X(31452) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, STEINER}

Barycentrics 3*a^4-2*(2*b^2+3*b*c+2*c^2)*a^2+(b^2-c^2)^2 : :

X(31452) lies on these lines: {1,631}, {2,3746}, {3,4317}, {4,3584}, {5,55}, {6,31462}, {8,31458}, {11,5070}, {12,382}, {20,35}, {30,9656}, {32,31478}, {36,15717}, {40,5761}, {46,13405}, {56,3530}, {57,31425}, {63,7162}, {79,9778}, {80,4313}, {100,4197}, {140,3303}, {165,13407}, {191,25568}, {200,31446}, {226,4338}, {372,31475}, {376,5270}, {377,10197}, {388,3528}, {390,7486}, {405,9711}, {442,4421}, {484,3487}, {495,548}, {496,16239}, {497,5067}, {499,3295}, {519,6910}, {529,19535}, {549,3304}, {550,11237}, {551,6921}, {612,7493}, {632,15170}, {938,5445}, {942,31447}, {943,6937}, {993,10528}, {1001,17575}, {1015,31457}, {1056,7280}, {1253,5733}, {1335,31454}, {1376,17529}, {1621,26364}, {1656,3058}, {1697,6970}, {1698,17552}, {1907,11398}, {2066,13963}, {2067,9680}, {2177,5292}, {2241,9698}, {2275,31450}, {2276,5319}, {2548,10987}, {2646,12647}, {3090,4857}, {3241,5559}, {3336,3475}, {3338,10164}, {3411,7005}, {3412,7006}, {3485,11010}, {3488,18395}, {3523,5563}, {3525,3582}, {3579,17718}, {3583,3855}, {3585,8164}, {3601,5881}, {3612,6966}, {3614,9668}, {3616,3833}, {3628,11238}, {3632,30478}, {3679,6857}, {3689,5791}, {3828,31259}, {3832,4294}, {3843,6284}, {3853,10895}, {3861,10592}, {3871,26363}, {3913,7483}, {4187,4428}, {4293,21734}, {4301,5119}, {4304,10827}, {4314,10826}, {4324,5229}, {4354,9644}, {5128,11551}, {5175,17057}, {5188,22729}, {5248,5552}, {5251,7080}, {5298,15720}, {5414,13905}, {5433,6767}, {5443,30305}, {5537,6908}, {5550,9802}, {5687,6690}, {5697,5734}, {5703,5903}, {5790,10543}, {5882,6977}, {6174,16408}, {6681,10586}, {6845,11491}, {6880,13464}, {6927,11522}, {6954,7982}, {6988,7991}, {7031,31402}, {7354,15696}, {7765,31451}, {8666,11239}, {9589,12047}, {9607,31448}, {9654,15338}, {10053,14981}, {10054,10992}, {10065,15063}, {10070,20398}, {10088,16003}, {10389,31423}, {10572,31434}, {10573,24929}, {10578,18398}, {10590,17578}, {11849,26487}, {12575,23708}, {12607,16370}, {12904,20396}, {13901,31487}, {13958,31474}, {15079,19877}, {15621,16455}, {16418,21031}, {16781,31492}, {16784,31400}, {16845,19875}, {16898,27020}, {17567,25055}, {17784,31420}, {20075,25639}, {21155,22770}, {24953,31494}

X(31452) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15888, 4317), (3, 31480, 15888), (15888, 31480, 10056)

X(31453) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, 2nd LEMOINE, SPIEKER}

Barycentrics a*(2*S*a+(-a+b+c)*(a^2+(b+c)*a+2*b*c)) : :

X(31453) lies on these lines: {1,6}, {2,2067}, {3,1377}, {4,31484}, {8,2066}, {10,371}, {20,31413}, {21,5414}, {32,31482}, {56,31473}, {63,2362}, {333,7090}, {372,993}, {486,26363}, {590,1329}, {615,4999}, {936,9583}, {1151,1376}, {1220,14121}, {1378,3311}, {1505,31456}, {1573,5058}, {1588,19843}, {1702,9623}, {1703,31424}, {1706,9616}, {1861,11473}, {2550,6459}, {2551,3068}, {2886,3071}, {2975,6502}, {3069,30478}, {3436,31472}, {3452,8983}, {3814,10576}, {3820,8981}, {4187,9661}, {4853,31432}, {5415,19065}, {5418,26364}, {5438,9615}, {6200,25440}, {6221,9679}, {6429,9689}, {8165,8972}, {8991,20307}, {9646,17757}, {13897,31141}, {13901,21031}, {18966,31157}

X(31453) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31438, 30556), (9, 18991, 30557)

X(31454) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, 2nd LEMOINE, STEINER}

Barycentrics 3*a^4-6*S*a^2-4*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31454) lies on these lines: {1,31440}, {2,3591}, {3,9680}, {4,6425}, {5,371}, {6,631}, {20,1151}, {30,6453}, {32,31483}, {56,31475}, {140,6419}, {230,12962}, {372,3530}, {381,6447}, {382,485}, {395,2042}, {396,2041}, {486,5070}, {548,6200}, {549,6420}, {615,3311}, {1131,9543}, {1152,7585}, {1327,5073}, {1328,5072}, {1335,31452}, {1376,31486}, {1378,31458}, {1505,31457}, {1587,3528}, {1588,5067}, {1657,6519}, {1702,9624}, {1703,31425}, {1906,11473}, {1907,5412}, {1991,11292}, {2066,18965}, {2067,13901}, {2548,8375}, {3069,6431}, {3316,23259}, {3364,3411}, {3389,3412}, {3523,3594}, {3524,6426}, {3525,13847}, {3832,6437}, {3843,6561}, {3853,6564}, {3855,23261}, {3861,18538}, {4301,8983}, {4309,13904}, {4317,13905}, {5024,19105}, {5054,6427}, {5055,10195}, {5058,9698}, {5319,6422}, {5420,6417}, {5734,13902}, {5881,9583}, {6247,11241}, {6278,13882}, {6396,19117}, {6407,13665}, {6410,7581}, {6411,6460}, {6421,31450}, {6428,15720}, {6429,9541}, {6433,23267}, {6441,13941}, {6449,6560}, {6454,15712}, {6455,18512}, {6468,23249}, {6470,7582}, {6522,15700}, {7584,16239}, {7969,11362}, {8276,9714}, {8909,9936}, {8991,12964}, {8994,16003}, {8997,14981}, {8998,15063}, {9588,18991}, {9589,9616}, {9648,19030}, {9656,13897}, {9657,31472}, {9663,19028}, {9671,13898}, {9711,31453}, {10141,11001}, {10147,17538}, {10303,19053}, {10819,23236}, {11294,13637}, {11541,14241}, {12239,14531}, {12305,26516}, {15057,19111}, {15765,16963}, {16962,18585}, {17578,31412}

X(31454) = midpoint of X(6453) and X(8960)
X(31454) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371, 590, 3071), (371, 8981, 590), (6425, 13846, 4)

X(31455) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, NINE-POINTS}

Barycentrics a^4-3*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31455) lies on these lines: {2,39}, {3,1506}, {4,7603}, {5,574}, {6,3411}, {11,31451}, {20,8589}, {30,15515}, {32,140}, {35,9665}, {36,9650}, {56,31476}, {83,7907}, {99,16921}, {115,1656}, {141,5034}, {183,7764}, {187,631}, {216,3548}, {230,632}, {325,7815}, {372,31481}, {381,7756}, {498,1015}, {499,1500}, {547,11648}, {549,5206}, {577,7542}, {590,1505}, {615,1504}, {620,7770}, {625,7791}, {626,11285}, {1007,7800}, {1078,7759}, {1329,31456}, {1376,31488}, {1570,3618}, {1571,8227}, {1572,31423}, {1573,26364}, {1574,26363}, {1698,9619}, {1699,31422}, {2241,5432}, {2242,5433}, {2549,3090}, {2896,7814}, {2937,15109}, {3035,31466}, {3053,5054}, {3054,5305}, {3071,9674}, {3096,7925}, {3199,3541}, {3329,7857}, {3523,7737}, {3524,14537}, {3525,5007}, {3530,8588}, {3533,5041}, {3549,22401}, {3589,5028}, {3624,9620}, {3628,5254}, {3785,7845}, {3818,5116}, {3832,15602}, {3843,18584}, {4045,7887}, {5023,15484}, {5024,5070}, {5058,5418}, {5062,5420}, {5067,7738}, {5277,17566}, {5306,10124}, {5319,11614}, {5471,22236}, {5472,22238}, {5569,7812}, {5972,14901}, {6292,7778}, {6390,17130}, {6421,8253}, {6422,8252}, {6639,14961}, {6656,7862}, {6680,11174}, {6722,7851}, {7230,28808}, {7288,31409}, {7506,9700}, {7516,9608}, {7608,11170}, {7622,8370}, {7741,9664}, {7750,7775}, {7752,7761}, {7755,9605}, {7760,17004}, {7767,7903}, {7771,7785}, {7773,7830}, {7774,7780}, {7776,7810}, {7782,16044}, {7783,16922}, {7787,10631}, {7793,7858}, {7794,15271}, {7804,16925}, {7805,17008}, {7807,7808}, {7809,7904}, {7811,7941}, {7816,16924}, {7819,15491}, {7825,8356}, {7826,9766}, {7831,7912}, {7833,8176}, {7838,11163}, {7839,17006}, {7843,14907}, {7853,16043}, {7864,14061}, {7867,8362}, {7876,7899}, {7889,31274}, {7890,8667}, {7895,16990}, {7909,16986}, {7913,8361}, {7935,8359}, {7951,9651}, {7988,31421}, {8366,22247}, {8960,12969}, {9167,16508}, {9300,11539}, {9603,13353}, {9955,31443}, {10018,10311}, {10303,21843}, {10314,16238}, {13935,31411}, {14064,31275}, {14869,18907}, {15694,30435}, {15699,18362}, {15717,31417}, {16975,27529}, {17683,24918}, {19102,31465}, {19862,31396}, {20107,25092}, {30827,31442}

X(31455) = complement of X(32832)
X(31455) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3788, 7822), (2, 7763, 3934), (3934, 7763, 7801)

X(31456) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, SPIEKER}

Barycentrics a*(a^3-2*(b^2+c^2)*a-2*(b+c)*b*c) : :

X(31456) lies on these lines: {1,9346}, {3,1573}, {4,31488}, {6,31468}, {8,31451}, {9,9619}, {10,574}, {20,31416}, {21,2241}, {32,993}, {37,5042}, {39,958}, {45,5053}, {56,16589}, {75,7781}, {115,26363}, {372,31482}, {405,1015}, {442,9651}, {668,17684}, {956,1500}, {1329,31455}, {1505,31453}, {1571,9623}, {1572,31424}, {1574,5013}, {1706,31422}, {1759,21332}, {2242,2975}, {2275,5251}, {2276,5258}, {2549,19843}, {2551,31401}, {2886,7748}, {3294,9351}, {3436,31476}, {3734,17030}, {3767,30478}, {3788,26558}, {4386,5206}, {4426,7772}, {4853,31433}, {4877,21769}, {4999,7746}, {5058,9678}, {5062,31464}, {5234,9592}, {5475,31466}, {5795,31398}, {6376,7815}, {6683,26687}, {7737,31405}, {7765,31458}, {7816,20172}, {7935,20541}, {9597,19854}, {9620,31429}, {9664,24390}, {9665,11113}, {9709,15815}, {9711,31457}, {10448,20963}, {11285,27076}, {15482,27091}, {15515,25440}, {16418,16781}, {17130,21264}, {21879,30144}

X(31456) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31490, 1573), (21, 16975, 2241)

X(31457) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, STEINER}

Barycentrics 3*a^4-7*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31457) lies on these lines: {1,31444}, {3,7753}, {5,574}, {6,31470}, {20,5475}, {32,3530}, {39,631}, {56,31478}, {115,5070}, {140,5309}, {187,15717}, {372,31483}, {382,1506}, {548,3815}, {549,7772}, {1015,31452}, {1376,31491}, {1505,31454}, {1571,9624}, {1572,31425}, {1574,31458}, {2242,31462}, {2548,3528}, {2549,5067}, {3522,14537}, {3523,5007}, {3526,5013}, {3628,11648}, {3788,7876}, {3832,7603}, {3843,7756}, {4325,9650}, {4330,9665}, {5024,7749}, {5041,21843}, {5054,7755}, {5058,9680}, {5062,31465}, {5206,31406}, {5254,16239}, {5306,12108}, {5569,6179}, {6337,31239}, {6683,16898}, {7622,7807}, {7736,15513}, {7737,21734}, {7739,10303}, {7747,15696}, {7759,9939}, {7761,7814}, {7763,7849}, {7769,7933}, {7796,7824}, {7801,11285}, {7822,15482}, {7835,16896}, {7888,8359}, {9300,15712}, {9588,9619}, {9589,31422}, {9620,31431}, {9657,31476}, {9711,31456}, {11742,17800}, {17578,31415}

X(31457) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31492, 9698), (574, 31455, 7748)

X(31458) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, SPIEKER, STEINER}

Barycentrics 3*a^4-4*(b^2+c^2)*a^2-6*(b+c)*b*c*a+(b^2-c^2)^2 : :

X(31458) lies on these lines: {1,24597}, {2,5258}, {3,9710}, {5,958}, {6,31469}, {8,31452}, {9,9624}, {10,631}, {20,993}, {21,4309}, {32,31491}, {56,17529}, {372,31486}, {377,4325}, {382,2886}, {442,9657}, {452,24387}, {474,31157}, {499,5260}, {519,6857}, {528,17571}, {535,5177}, {548,31419}, {551,11523}, {956,10198}, {962,3647}, {1107,5319}, {1125,3475}, {1329,5070}, {1376,3530}, {1378,31454}, {1482,18253}, {1574,31457}, {1575,31450}, {1706,31425}, {2550,3528}, {2551,5067}, {2975,4197}, {3058,19526}, {3241,15674}, {3303,15670}, {3434,4330}, {3526,4999}, {3612,25006}, {3616,3881}, {3626,5218}, {3634,4315}, {3679,6910}, {3754,5744}, {3813,16418}, {3814,7486}, {3820,16239}, {3822,31410}, {3828,17567}, {3832,25639}, {3841,4293}, {3843,31493}, {3878,5273}, {3968,26062}, {4015,27383}, {4301,12514}, {4853,31436}, {4857,31156}, {5047,10072}, {5204,17583}, {5234,21616}, {5251,10527}, {5259,10529}, {5298,16862}, {5302,5886}, {5325,13464}, {5443,31018}, {5534,10165}, {5705,6962}, {5745,6892}, {5791,24299}, {5795,6970}, {6675,12513}, {6690,31480}, {6921,19875}, {6966,9588}, {6974,7991}, {7765,31456}, {8728,11194}, {9589,31424}, {9607,31449}, {9656,31245}, {9670,24390}, {9671,11113}, {10199,17559}, {10200,17575}, {10586,25542}, {11373,15254}, {12577,19862}, {15717,25440}, {15862,20050}, {16898,17030}, {17578,31418}, {24477,30143}, {25055,31259}

X(31458) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31494, 9710), (956, 24953, 10198)

X(31459) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, 2nd LEMOINE}

Barycentrics a^2*(b^2+b*c+c^2+S) : :

X(31459) lies on these lines: {1,6422}, {6,31}, {8,31464}, {9,8941}, {11,31463}, {35,6423}, {36,9600}, {37,493}, {39,1124}, {57,31427}, {172,1151}, {372,31448}, {486,31460}, {491,26590}, {497,31403}, {1335,1500}, {1378,5283}, {1575,31473}, {1588,31402}, {1703,31426}, {2275,3297}, {3070,9598}, {3071,9596}, {3299,6421}, {3312,31461}, {3767,9646}, {5013,6502}, {5062,31451}, {5217,12968}, {5277,9679}, {5280,6424}, {5291,9678}, {7737,9660}, {7738,31408}, {9575,31432}, {9605,31474}, {12962,18996}, {21956,31484}

X(31459) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 31477, 5414), (39, 31471, 1124)

X(31460) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, NINE-POINTS}

Barycentrics (3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2 : :

X(31460) lies on these lines: {1,3815}, {2,31402}, {3,9596}, {4,31448}, {5,2276}, {6,498}, {8,31466}, {11,1500}, {12,39}, {32,5432}, {35,7745}, {37,4187}, {55,2548}, {56,31401}, {57,31428}, {65,31398}, {83,26629}, {115,3614}, {140,172}, {192,16921}, {217,26956}, {226,31396}, {230,5280}, {325,27020}, {381,9598}, {388,31400}, {442,1575}, {486,31459}, {495,2275}, {497,31404}, {499,31489}, {574,7354}, {594,30171}, {999,31467}, {1015,9698}, {1086,24786}, {1107,17757}, {1329,5283}, {1335,31463}, {1478,5013}, {1479,31477}, {1504,19027}, {1505,19028}, {1571,1836}, {1573,21031}, {1574,3925}, {1699,31426}, {1770,31443}, {2242,5433}, {2476,17756}, {2549,10895}, {3035,5277}, {3055,16785}, {3058,9665}, {3085,7736}, {3295,9599}, {3436,31449}, {3501,17717}, {3584,5299}, {3589,30104}, {3814,25092}, {4299,15815}, {4317,31492}, {4426,7483}, {4995,7753}, {4999,5291}, {5024,9597}, {5058,13901}, {5062,13958}, {5217,7737}, {5219,9593}, {5252,9619}, {5254,7951}, {5275,26364}, {5286,10588}, {5326,7749}, {5475,6284}, {5718,17750}, {7080,31405}, {7173,7603}, {7504,17737}, {7738,10590}, {7747,15338}, {7752,26590}, {7769,26686}, {7786,26561}, {9560,10406}, {9574,9612}, {9575,31434}, {9578,9592}, {9579,31421}, {9605,31479}, {9620,11375}, {9657,31450}, {9670,31417}, {10056,16781}, {10592,15048}, {10896,31415}, {10955,11998}, {10957,21859}, {12607,16975}, {12701,31433}, {17243,30122}, {19029,31471}, {19030,31481}, {19037,31411}, {20691,24390}, {21956,25639}, {24914,31441}

X(31460) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31476, 12), (1500, 1506, 11)

X(31461) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, STAMMLER}

Barycentrics a^2*(a^2-5*b^2-4*b*c-5*c^2) : :

X(31461) lies on these lines: {1,5024}, {3,172}, {6,24047}, {8,31468}, {11,31467}, {30,31402}, {35,30435}, {37,16408}, {39,3295}, {55,5299}, {57,31430}, {192,11285}, {220,4256}, {381,9598}, {382,9596}, {405,17756}, {495,7738}, {496,31400}, {497,31406}, {517,31426}, {942,9574}, {968,25068}, {988,3991}, {999,1500}, {1015,22332}, {1376,25092}, {1384,5217}, {1575,11108}, {2242,15815}, {2275,6767}, {2345,19273}, {2548,9668}, {2549,9654}, {3085,15048}, {3304,9331}, {3312,31459}, {3666,21526}, {3730,4255}, {3815,9669}, {3970,17595}, {4261,16286}, {4426,17571}, {5110,20818}, {5204,16785}, {5218,5305}, {5254,31479}, {5283,9709}, {5722,31396}, {6284,15484}, {6421,31474}, {7736,15171}, {8162,9336}, {8572,9327}, {9592,9957}, {9593,24929}, {9607,31480}, {9651,31478}, {9655,31409}, {10987,22246}, {16457,17303}, {16549,19765}, {17594,25066}, {20691,31449}, {21956,31493}

X(31461) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31477, 3295), (2276, 31448, 3)

X(31462) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, STEINER}

Barycentrics (7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2 : :

X(31462) lies on these lines: {1,9606}, {5,2276}, {6,31452}, {8,31469}, {12,7765}, {20,31402}, {37,17575}, {39,15888}, {56,31450}, {57,31431}, {172,3530}, {382,9596}, {497,31407}, {999,31470}, {1335,31465}, {1575,17529}, {2242,31457}, {2548,9670}, {2549,9656}, {3746,9300}, {3843,9598}, {4197,17756}, {4309,31477}, {4317,5013}, {4330,7745}, {4995,5007}, {5283,9711}, {6421,31475}, {7738,31410}, {7814,26590}, {9575,31436}, {9589,31426}, {9657,31409}, {9671,31417}

X(31462) = {X(39), X(31478)}-harmonic conjugate of X(15888)

X(31463) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, NINE-POINTS}

Barycentrics -(b^2-c^2)^2+2*S*a^2+3*(b^2+c^2)*a^2 : :

X(31463) lies on these lines: {2,6}, {5,6422}, {11,31459}, {30,9600}, {32,5418}, {39,485}, {53,3127}, {140,6423}, {371,2548}, {372,31401}, {486,1504}, {493,1592}, {574,6560}, {631,12968}, {1151,7745}, {1329,31464}, {1335,31460}, {1378,31466}, {1505,9698}, {1575,31484}, {1587,31400}, {1588,12962}, {1699,31427}, {1703,31428}, {2066,9599}, {2067,9596}, {2275,31472}, {2549,6564}, {3070,5013}, {3094,6813}, {3312,31467}, {3767,10576}, {5024,13665}, {5062,5420}, {5200,6748}, {5309,13711}, {5475,6561}, {6200,7737}, {6221,15484}, {6421,7583}, {6424,8981}, {6565,31415}, {7581,12969}, {7738,31412}, {7747,9674}, {7753,9675}, {7755,10195}, {7756,22644}, {8976,9605}, {9540,12963}, {9575,13893}, {13889,19448}, {15048,18538}

X(31463) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31403, 6), (6, 8253, 230), (3068, 7736, 6)

X(31464) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, SPIEKER}

Barycentrics a*((b+c)*b*c+S*a+(b^2+c^2)*a) : :

X(31464) lies on these lines: {1,6}, {8,31459}, {10,6422}, {32,9678}, {39,1377}, {368,27343}, {372,31449}, {486,31466}, {491,26558}, {493,6348}, {993,6423}, {1151,4386}, {1329,31463}, {1378,1504}, {1588,31405}, {1703,31429}, {1706,31427}, {2067,5275}, {2275,31473}, {2551,31403}, {3312,31468}, {5062,31456}, {5254,31484}, {7738,31413}, {9600,25440}, {9605,31485}, {9607,31486}, {9711,31465}

X(31464) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31482, 1377), (1504, 1573, 1378)

X(31465) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, STEINER}

Barycentrics -(b^2-c^2)^2+6*S*a^2+7*(b^2+c^2)*a^2 : :

X(31465) lies on these lines: {5,6422}, {6,631}, {20,31403}, {32,9680}, {39,31483}, {372,31450}, {485,7765}, {491,7876}, {548,9600}, {1335,31462}, {1378,31469}, {1504,9698}, {1575,31486}, {1588,31407}, {1703,31431}, {1991,16043}, {2275,31475}, {3312,31470}, {3530,6423}, {3592,9300}, {5062,31457}, {7375,13846}, {7736,12962}, {7738,31414}, {7739,8960}, {9575,31440}, {9589,31427}, {9605,31487}, {9711,31464}, {12968,15717}, {19102,31455}

X(31466) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, NINE-POINTS, SPIEKER}

Barycentrics 3*(b^2+c^2)*a^2+2*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31466) lies on these lines: {2,31405}, {4,31449}, {5,1107}, {6,26363}, {8,31460}, {10,3815}, {11,5283}, {12,16975}, {32,4999}, {37,496}, {39,2886}, {140,4386}, {325,17030}, {381,31468}, {405,9599}, {442,2275}, {486,31464}, {495,17448}, {499,5275}, {528,31451}, {529,9650}, {956,9596}, {958,2548}, {993,7745}, {1015,25466}, {1329,1506}, {1376,31401}, {1378,31463}, {1500,3813}, {1572,26066}, {1574,9698}, {1575,31406}, {1699,31429}, {1706,31428}, {1914,7483}, {2241,6690}, {2276,24390}, {2300,5742}, {2550,31400}, {2551,31404}, {3035,31455}, {3434,31448}, {3816,16589}, {3933,21264}, {4884,22036}, {5021,26098}, {5254,25639}, {5277,5433}, {5475,31456}, {5626,23112}, {5705,9575}, {5718,20963}, {5794,9619}, {5836,31398}, {6421,31484}, {7736,19843}, {7738,31418}, {7752,26558}, {7763,20172}, {7786,26582}, {7807,20179}, {8227,16517}, {8362,20541}, {8728,16604}, {9597,17532}, {9605,31493}, {9709,31467}, {9711,31491}, {10198,16781}, {11374,16973}, {12513,31409}, {12607,31476}, {16583,24239}, {17045,25598}, {17717,21384}, {21921,28096}, {24387,25092}, {24703,31442}, {26364,31489}

X(31466) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31488, 2886), (1506, 1573, 1329)

X(31467) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, NINE-POINTS, STAMMLER}

Barycentrics a^4-7*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31467) lies on these lines: {2,3933}, {3,2548}, {5,5024}, {6,3411}, {11,31461}, {30,31404}, {32,5054}, {39,1656}, {83,11288}, {115,5079}, {140,7736}, {183,7905}, {381,1506}, {382,574}, {517,31428}, {590,13934}, {615,13882}, {626,11184}, {631,1384}, {632,7735}, {999,31460}, {1007,8362}, {1078,11163}, {1329,31468}, {1482,31398}, {1575,31493}, {1657,5475}, {1699,31430}, {1975,11165}, {2275,31479}, {2549,3851}, {2896,7776}, {3053,15720}, {3054,5319}, {3055,3767}, {3090,15048}, {3312,31463}, {3523,18907}, {3530,15655}, {3533,5304}, {3618,10008}, {3628,5286}, {3763,7888}, {3788,6704}, {3843,31415}, {5023,7753}, {5034,11898}, {5041,15723}, {5055,5254}, {5072,7603}, {5076,7756}, {5094,15302}, {5790,9619}, {5886,31396}, {6395,31411}, {6421,8976}, {6422,13951}, {6683,7778}, {7542,15905}, {7608,11257}, {7610,7805}, {7739,15703}, {7743,31426}, {7747,15696}, {7752,11287}, {7764,15271}, {7769,7846}, {7770,7891}, {7773,7910}, {7782,11159}, {7784,15482}, {7786,7866}, {7795,15491}, {7815,7882}, {7824,7900}, {7855,8556}, {7868,31268}, {7887,7923}, {9300,15694}, {9574,9955}, {9575,11231}, {9592,9956}, {9593,11230}, {9669,31448}, {9709,31466}, {10542,25555}, {13966,31403}, {14001,14535}, {14093,14537}, {15325,31402}, {15513,15700}, {17800,31417}, {18512,31481}, {22793,31421}

X(31467) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31406, 9605), (3815, 31401, 3)

X(31468) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, SPIEKER, STAMMLER}

Barycentrics a*(a^3-5*(b^2+c^2)*a-4*(b+c)*b*c) : :

X(31468) lies on these lines: {1,4520}, {3,1107}, {6,31456}, {8,31461}, {10,5024}, {30,31405}, {37,7373}, {39,9708}, {381,31466}, {382,31469}, {517,31429}, {958,9605}, {993,30435}, {999,5283}, {1329,31467}, {1385,16517}, {1573,5013}, {1574,22332}, {1706,31430}, {1914,17571}, {2275,11108}, {2551,31406}, {3312,31464}, {3820,31400}, {5022,30116}, {5044,9592}, {5254,31493}, {5305,30478}, {6421,31485}, {6767,17448}, {7738,31419}, {7781,20181}, {9575,31445}, {9597,17528}, {9607,31494}, {9711,31470}, {12513,25092}, {15048,19843}, {16345,23632}, {16604,16853}

X(31468) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31490, 9708), (1107, 31449, 3)

X(31469) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, SPIEKER, STEINER}

Barycentrics 7*(b^2+c^2)*a^2+6*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31469) lies on these lines: {5,1107}, {6,31458}, {8,31462}, {10,9606}, {20,31405}, {39,9710}, {382,31468}, {1376,31450}, {1378,31465}, {1573,9698}, {1706,31431}, {2275,17529}, {2551,31407}, {2886,7765}, {3530,4386}, {6421,31486}, {7738,31420}, {7814,26558}, {9575,31446}, {9589,31429}, {9605,31494}, {9624,16517}, {9709,31470}, {15888,16975}

X(31469) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31491, 9710), (1573, 9698, 9711)

X(31470) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, STAMMLER, STEINER}

Barycentrics 3*a^4-17*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31470) lies on these lines: {3,9300}, {5,5024}, {6,31457}, {20,15484}, {30,31407}, {39,3526}, {382,5013}, {517,31431}, {548,7736}, {631,5304}, {999,31462}, {1384,15717}, {1575,31494}, {1656,7765}, {2275,31480}, {2548,17800}, {3312,31465}, {3530,30435}, {3815,3843}, {3861,31404}, {5007,15693}, {5067,15048}, {5070,9607}, {5286,16239}, {6421,31487}, {7772,15720}, {7814,11287}, {9575,31447}, {9589,31430}, {9709,31469}, {9711,31468}, {10335,16987}, {12040,14069}, {15700,22331}, {18907,21734}

X(31470) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5024, 31400, 31467), (9606, 31450, 3)

X(31471) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, MOSES}

Barycentrics a^2*(2*S+(b+c)^2) : :

X(31471) lies on these lines: {1,1504}, {6,595}, {8,31482}, {11,31481}, {32,2066}, {36,9674}, {39,1124}, {55,5062}, {57,31437}, {115,31472}, {172,9675}, {371,2242}, {372,31451}, {486,31476}, {497,31411}, {574,6502}, {1015,3297}, {1378,16589}, {1505,2276}, {1572,31432}, {1574,31473}, {1588,31409}, {1703,31433}, {2549,31408}, {3070,9664}, {3071,9650}, {3312,31477}, {5058,19038}, {7746,9646}, {18995,31448}, {19029,31460}

X(31471) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1124, 31459, 39), (3297, 6422, 1015)

X(31472) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, NINE-POINTS}

Barycentrics 2*S*a^2+(b+c)^2*(a-b+c)*(a+b-c) : :

X(31472) lies on these lines: {1,485}, {2,6502}, {3,9646}, {4,2066}, {5,1124}, {6,12}, {8,31484}, {10,2362}, {11,3297}, {35,6560}, {36,5418}, {55,3070}, {56,590}, {57,13893}, {65,13911}, {92,1585}, {115,31471}, {226,13883}, {371,1478}, {372,498}, {381,31474}, {382,9660}, {388,2067}, {390,1131}, {442,1378}, {486,3299}, {491,1909}, {495,1335}, {496,18538}, {497,31412}, {499,10576}, {615,18995}, {908,30556}, {999,8976}, {1015,31481}, {1056,13886}, {1151,7354}, {1152,5432}, {1329,31473}, {1377,17757}, {1479,6564}, {1505,31476}, {1587,3085}, {1588,10590}, {1699,31432}, {1702,9612}, {1703,31434}, {1773,6204}, {2275,31463}, {3069,10588}, {3071,10895}, {3103,10063}, {3295,13665}, {3298,15888}, {3304,13898}, {3311,9654}, {3312,31479}, {3436,31453}, {3476,13902}, {3485,19066}, {3585,6561}, {3594,13958}, {3600,8972}, {3614,19029}, {4292,13912}, {4293,9540}, {4294,23249}, {4299,6200}, {4325,9680}, {5058,9650}, {5083,8988}, {5218,6460}, {5229,6459}, {5252,7969}, {5254,31459}, {5261,7585}, {5412,11392}, {5433,8253}, {5434,13846}, {5726,19004}, {6221,9647}, {6284,23251}, {6348,13389}, {6409,15326}, {6420,13963}, {6429,9649}, {7080,31413}, {7581,8164}, {7584,10592}, {7968,11375}, {8252,18966}, {8831,26040}, {8909,18970}, {8960,13904}, {8981,18990}, {8983,10106}, {9578,18991}, {9579,9616}, {9583,9613}, {9657,31454}, {9679,11112}, {10055,10665}, {10577,13962}, {10819,18968}, {11237,18996}, {13882,18988}, {18513,22615}, {19048,26482}, {19050,26481}, {24987,30557}

X(31472) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31408, 6502), (12, 19028, 6)

X(31473) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, SPIEKER}

Barycentrics a*((a+b+c)*b*c+S*a) : :

X(31473) lies on these lines: {1,1377}, {2,6}, {8,3297}, {9,13389}, {10,1124}, {11,31484}, {21,1152}, {36,9678}, {37,494}, {56,31453}, {57,31438}, {65,6203}, {220,30412}, {371,474}, {372,405}, {377,3071}, {404,1151}, {406,3093}, {442,486}, {443,1588}, {452,6460}, {475,3092}, {485,4187}, {497,31413}, {517,8231}, {572,16433}, {573,16432}, {958,6502}, {960,2362}, {999,31485}, {1001,5414}, {1015,31482}, {1030,1600}, {1100,3084}, {1125,1335}, {1172,3536}, {1329,31472}, {1376,2066}, {1378,1698}, {1505,16589}, {1574,31471}, {1575,31459}, {1587,5084}, {1599,5124}, {1703,31435}, {1706,31432}, {2067,25524}, {2256,6351}, {2264,7348}, {2275,31464}, {2475,23261}, {2478,3070}, {2551,31408}, {3232,31221}, {3298,3616}, {3301,3624}, {3311,16408}, {3312,11108}, {3592,17531}, {3594,5047}, {3812,16232}, {3925,19029}, {4188,6409}, {4189,6410}, {4413,19038}, {4423,19037}, {5046,23251}, {5069,8962}, {5277,6424}, {5283,6421}, {5416,13940}, {5418,13747}, {5420,7483}, {6200,16371}, {6221,16417}, {6347,17275}, {6348,17303}, {6395,16857}, {6396,16370}, {6398,16418}, {6411,13587}, {6412,17549}, {6417,16863}, {6418,16853}, {6419,16862}, {6420,16842}, {6425,17572}, {6426,16865}, {6428,16855}, {6431,17535}, {6432,17536}, {6438,16858}, {6449,17573}, {6450,17571}, {6454,19526}, {6459,6904}, {6471,17570}, {6485,19539}, {6560,11113}, {6561,11112}, {6564,17556}, {6565,17532}, {6675,13966}, {6857,13935}, {6919,31412}, {7581,17559}, {7582,17582}, {7583,17527}, {7584,8728}, {7968,19860}, {7969,19861}, {8582,13883}, {8583,18991}, {9540,17567}, {9646,26364}, {9661,10200}, {9681,17583}, {9711,31475}, {11513,25947}, {11514,25907}, {13785,17528}, {13887,25893}, {13893,26459}, {13897,31246}, {13911,19047}, {13947,26464}, {13955,31245}, {13962,19854}, {13973,19050}, {18966,24953}, {19023,25973}, {19049,24541}, {25917,30557}

X(31473) = {X(1698), X(3299)}-harmonic conjugate of X(1378)

X(31474) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, STAMMLER}

Barycentrics a^2*(-a^2+b^2+4*b*c+c^2+4*S) : :

X(31474) lies on these lines: {1,3311}, {3,1124}, {6,595}, {8,31485}, {11,8976}, {12,13785}, {30,31408}, {35,6398}, {36,6449}, {55,3299}, {56,6221}, {57,31439}, {371,999}, {381,31472}, {382,31475}, {390,7581}, {485,9669}, {486,31479}, {495,1588}, {496,3068}, {497,7583}, {498,13951}, {499,13901}, {517,31432}, {942,1702}, {1058,7585}, {1335,6417}, {1378,11108}, {1387,13902}, {1479,13665}, {1504,16781}, {1505,31477}, {1587,15171}, {1656,9646}, {1657,9660}, {2067,6199}, {2362,12702}, {3070,9668}, {3071,9654}, {3085,7584}, {3086,8981}, {3298,6419}, {3301,3303}, {3584,13954}, {3746,6428}, {3940,30556}, {5010,6456}, {5122,9582}, {5126,9615}, {5204,6455}, {5217,6450}, {5218,13966}, {5261,23273}, {5274,13886}, {5298,9648}, {5410,6198}, {5414,6418}, {5432,13962}, {5563,6447}, {5722,13883}, {6421,31461}, {6451,7280}, {6459,18990}, {6561,9655}, {7741,13897}, {8983,11373}, {9540,15325}, {9583,24928}, {9605,31459}, {9661,13903}, {9679,16417}, {9709,31473}, {9957,18991}, {10056,19027}, {10072,18965}, {10588,18762}, {10591,18538}, {12735,19082}, {13904,31487}, {13958,31452}, {15170,19054}, {15172,19117}, {15934,16232}, {18992,24929}, {18999,26464}, {19000,26459}, {19004,31393}

X(31474) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 19038, 3311), (1124, 2066, 3)

X(31475) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, STEINER}

Barycentrics -(b^2-c^2)^2+6*S*a^2+(b^2+6*b*c+c^2)*a^2 : :

X(31475) lies on these lines: {5,1124}, {6,15888}, {8,31486}, {20,2066}, {36,9680}, {56,31454}, {57,31440}, {371,4317}, {372,31452}, {382,31474}, {497,31414}, {631,6502}, {999,31487}, {1378,17529}, {1505,31478}, {1588,31410}, {1703,31436}, {2275,31465}, {2362,11362}, {3070,9670}, {3071,9656}, {3297,19028}, {3312,31480}, {3526,9646}, {3592,5434}, {4325,9681}, {4330,6560}, {4995,6426}, {6420,10056}, {6421,31462}, {8960,10072}, {9589,31432}, {9657,19038}, {9660,17800}, {9679,17583}, {9711,31473}, {16785,19105}

X(31476) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, NINE-POINTS}

Barycentrics 2*(b^2+b*c+c^2)*a^2-(b^2-c^2)^2 : :

X(31476) lies on these lines: {1,1506}, {2,2242}, {3,9650}, {4,31451}, {5,1500}, {6,17734}, {8,31488}, {11,7603}, {12,39}, {32,498}, {35,7747}, {37,3814}, {55,5475}, {56,31455}, {57,31441}, {115,2276}, {172,7749}, {187,5432}, {226,31398}, {381,9664}, {388,31401}, {442,1574}, {486,31471}, {495,1015}, {497,31415}, {574,1478}, {625,26590}, {626,27020}, {999,31489}, {1213,10469}, {1329,16589}, {1335,31481}, {1505,31472}, {1571,9612}, {1572,31434}, {1573,17757}, {1575,3822}, {1699,31433}, {1909,7764}, {1914,3584}, {2241,2548}, {2275,9698}, {2549,10590}, {3295,9665}, {3436,31456}, {3585,7756}, {3761,7813}, {3767,10588}, {3947,31396}, {4299,15515}, {4400,7890}, {4995,14537}, {5010,6781}, {5013,9651}, {5034,12588}, {5058,9646}, {5218,7737}, {5219,9620}, {5254,10592}, {5261,31400}, {5277,27529}, {5280,7755}, {5283,11681}, {5290,31428}, {5472,7127}, {5726,9592}, {6645,7769}, {6683,26561}, {7080,31416}, {7736,8164}, {7748,10895}, {7804,26629}, {7889,30104}, {8589,15326}, {9578,9619}, {9579,31422}, {9599,10056}, {9655,15815}, {9657,31457}, {11236,31449}, {12607,31466}, {17530,21956}, {20691,25639}, {21021,30171}, {31410,31450}

X(31476) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31409, 2242), (12, 31460, 39)

X(31477) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, STAMMLER}

Barycentrics a^2*(a^2-3*b^2-4*b*c-3*c^2) : :

X(31477) lies on these lines: {1,5013}, {2,20181}, {3,1500}, {6,31}, {8,31490}, {11,31489}, {12,9598}, {30,31409}, {35,609}, {36,9331}, {37,1376}, {39,3295}, {45,3693}, {56,15815}, {57,31443}, {100,5275}, {115,31479}, {172,5023}, {183,192}, {220,18755}, {230,5218}, {232,7071}, {345,594}, {346,26244}, {350,15271}, {381,9664}, {382,9650}, {386,14974}, {390,7736}, {495,2549}, {496,31401}, {497,3815}, {498,13881}, {517,31433}, {519,31449}, {574,999}, {942,1571}, {958,20691}, {967,1796}, {1001,1575}, {1015,5024}, {1058,31400}, {1100,3749}, {1107,3913}, {1479,31460}, {1506,9669}, {1574,11108}, {1621,17756}, {2176,4255}, {2223,16523}, {2241,9605}, {2256,5110}, {2271,3730}, {2275,3303}, {2286,21794}, {2295,19765}, {2548,15171}, {2933,3207}, {3055,10589}, {3058,9599}, {3085,5254}, {3158,16517}, {3247,17122}, {3306,3666}, {3312,31471}, {3333,31421}, {3509,17601}, {3726,17595}, {3744,16884}, {3750,17754}, {3930,4414}, {4294,7745}, {4309,31462}, {4366,11174}, {4386,4421}, {4646,16968}, {4704,16999}, {5010,5210}, {5021,24047}, {5045,31430}, {5220,20693}, {5281,7735}, {5283,5687}, {5291,16370}, {5475,9668}, {6284,9596}, {6765,31429}, {7748,9654}, {7756,9655}, {7765,31480}, {7778,26590}, {8715,25092}, {9300,10385}, {9592,31393}, {9597,15888}, {9619,9957}, {9620,24929}, {9709,16589}, {14829,17314}, {15172,31406}, {16992,17759}, {17118,24326}, {17450,17599}

X(31477) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31448, 5013), (1500, 31451, 3)

X(31478) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, STEINER}

Barycentrics 2*(2*b^2+3*b*c+2*c^2)*a^2-(b^2-c^2)^2 : :

X(31478) lies on these lines: {1,9698}, {5,1500}, {6,31480}, {8,31491}, {20,31409}, {32,31452}, {39,15888}, {57,31444}, {382,9650}, {495,9607}, {497,31417}, {574,4317}, {631,2242}, {999,31492}, {1015,9606}, {1335,31483}, {1505,31475}, {1572,31436}, {1574,17529}, {2241,31402}, {2276,7765}, {2549,31410}, {3085,5319}, {3584,7755}, {3746,7753}, {3843,9664}, {4309,9596}, {4330,7747}, {5475,9670}, {7748,9656}, {7749,16785}, {7772,10056}, {9589,31433}, {9651,31461}, {9657,31448}, {9711,16589}

X(31478) = {X(15888), X(31462)}-harmonic conjugate of X(39)

X(31479) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, NINE-POINTS, STAMMLER}

Barycentrics a^4-(3*b^2+4*b*c+3*c^2)*a^2+2*(b^2-c^2)^2 : :

X(31479) lies on these lines: {1,1656}, {2,495}, {3,12}, {4,5281}, {5,497}, {6,17734}, {8,31493}, {10,3940}, {11,5055}, {30,5218}, {35,382}, {36,5054}, {55,381}, {56,3526}, {57,11231}, {100,17532}, {115,31477}, {119,6913}, {140,388}, {145,7504}, {202,16645}, {203,16644}, {210,942}, {226,26446}, {230,31409}, {355,13411}, {390,3545}, {392,30852}, {403,7071}, {405,11681}, {442,1260}, {474,27529}, {486,31474}, {496,3090}, {517,5219}, {546,4294}, {547,10589}, {549,4293}, {550,5229}, {631,5261}, {632,7288}, {954,6829}, {993,11236}, {1001,3814}, {1015,31489}, {1058,5056}, {1060,5268}, {1068,7140}, {1124,13951}, {1155,18541}, {1329,10198}, {1335,8976}, {1376,3822}, {1385,9578}, {1479,3614}, {1482,10039}, {1500,13881}, {1506,16781}, {1621,17556}, {1657,3585}, {1697,9955}, {1737,15934}, {1788,6147}, {1870,5094}, {1914,15484}, {2066,13785}, {2098,5443}, {2275,31467}, {2330,18440}, {2476,5687}, {2551,6675}, {2646,10827}, {3035,16417}, {3053,9650}, {3057,18493}, {3058,19709}, {3086,3628}, {3091,15171}, {3167,10055}, {3297,10577}, {3298,10576}, {3299,13954}, {3301,13897}, {3303,5079}, {3311,9646}, {3312,31472}, {3434,17530}, {3436,7483}, {3485,5690}, {3486,18357}, {3487,9780}, {3488,12019}, {3517,11392}, {3525,3600}, {3530,31410}, {3533,5265}, {3534,5010}, {3576,5726}, {3579,9612}, {3601,18480}, {3624,24928}, {3632,31262}, {3634,21620}, {3679,31245}, {3746,5072}, {3748,18530}, {3753,31266}, {3830,4302}, {3843,6284}, {3850,5225}, {3871,5141}, {3913,25639}, {3920,7539}, {3927,21077}, {3947,6684}, {4245,19721}, {4316,15688}, {4417,5774}, {4870,25415}, {5050,12588}, {5066,10385}, {5067,14986}, {5071,5274}, {5073,15338}, {5080,16370}, {5119,17605}, {5204,5270}, {5221,5445}, {5226,5657}, {5251,31141}, {5254,31461}, {5290,31423}, {5305,31402}, {5326,5434}, {5414,13665}, {5587,24929}, {5703,5818}, {5708,13407}, {5719,18391}, {5722,10175}, {5789,14872}, {5791,21075}, {5886,31397}, {5919,23708}, {6199,13901}, {6244,6907}, {6395,13958}, {6417,13905}, {6418,13963}, {6455,9647}, {6668,12607}, {6690,16418}, {6738,31399}, {6796,10894}, {6831,10786}, {6842,10306}, {6856,7080}, {6861,10321}, {6862,10942}, {6863,22770}, {6931,10587}, {6933,10528}, {6969,7956}, {6971,16202}, {6980,10679}, {7082,17699}, {7280,9657}, {7489,8069}, {7506,10831}, {7545,9673}, {7571,29815}, {7680,19541}, {7743,7988}, {7866,27020}, {8068,12331}, {8165,16845}, {8227,9957}, {8540,14848}, {8666,20104}, {9596,30435}, {9605,31460}, {9613,13624}, {9651,15815}, {9656,10483}, {9658,13564}, {10072,15703}, {10172,11019}, {10247,12647}, {10320,10954}, {10387,19130}, {10389,18527}, {10742,28444}, {10980,19876}, {11235,25439}, {11286,26629}, {11318,26590}, {11499,15865}, {11529,19875}, {12047,12702}, {12373,15041}, {12577,31253}, {12735,31272}, {13384,28204}, {13465,14526}, {13743,18542}, {13903,18996}, {13961,18995}, {13966,31408}, {15040,18968}, {16137,31254}, {16203,26482}, {16408,25466}, {18510,19038}, {18512,19037}, {18545,22768}, {19322,26231}, {19349,26944}, {19854,21031}, {21696,27577}, {25055,25405}, {28160,30282}, {30116,30858}

X(31479) = midpoint of X(i) and X(j) for these {i,j}: {5218, 10590}, {5219, 31434}
X(31479) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 495, 999), (2, 8164, 495), (2, 17757, 9708)

X(31480) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, STAMMLER, STEINER}

Barycentrics 3*a^4-(5*b^2+12*b*c+5*c^2)*a^2+2*(b^2-c^2)^2 : :

X(31480) lies on these lines: {1,3526}, {3,4317}, {5,497}, {6,31478}, {8,31494}, {12,3843}, {20,495}, {30,31410}, {35,9657}, {55,382}, {57,31447}, {381,3746}, {388,548}, {390,3855}, {496,5067}, {498,5070}, {517,31436}, {546,10385}, {631,999}, {942,9588}, {954,6937}, {1015,31492}, {1056,15717}, {1058,7486}, {1335,31487}, {1656,3303}, {1657,11237}, {2275,31470}, {2894,4197}, {3058,3851}, {3086,16239}, {3090,15170}, {3304,5054}, {3312,31475}, {3528,5281}, {3530,5218}, {3534,5270}, {3624,3893}, {3832,8164}, {3853,4294}, {3856,5225}, {3861,10386}, {3913,10197}, {4301,11374}, {4325,5217}, {4857,5072}, {5049,31423}, {5079,11238}, {5432,7373}, {5552,17575}, {5563,15720}, {5722,31399}, {5881,24929}, {6690,31458}, {7080,17552}, {7483,11239}, {7765,31477}, {7951,9671}, {8715,17528}, {9605,31462}, {9607,31461}, {9624,9957}, {9698,16781}, {9708,10528}, {9709,17529}, {9710,10198}, {9711,11108}, {9956,10389}, {11362,13405}, {12607,16418}, {12702,17718}, {17606,18530}

X(31480) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10056, 31452, 15888), (15888, 31452, 3)

X(31481) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, NINE-POINTS}

Barycentrics -(b^2-c^2)^2+2*S*a^2+2*(b^2+c^2)*a^2 : :

X(31481) lies on these lines: {2,5062}, {5,1504}, {6,17}, {11,31471}, {30,9674}, {32,590}, {39,485}, {115,6422}, {187,5418}, {371,5475}, {372,31455}, {486,7603}, {491,3934}, {574,3070}, {615,6118}, {1015,31472}, {1151,7747}, {1329,31482}, {1335,31476}, {1378,31488}, {1505,3815}, {1572,13893}, {1574,31484}, {1587,31401}, {1588,31415}, {1699,31437}, {1703,31441}, {2066,9665}, {2067,9650}, {2241,9646}, {2242,9661}, {2548,3068}, {2549,31412}, {3055,13966}, {3312,31489}, {3316,7735}, {3767,31403}, {5013,13665}, {5254,18538}, {6409,6781}, {6421,9698}, {6423,7749}, {6424,7753}, {6564,7748}, {6565,12962}, {7736,13886}, {7737,9540}, {7745,8981}, {7756,9600}, {9596,13904}, {9599,13905}, {13903,15484}, {18512,31467}, {19030,31460}, {31414,31450}

X(31481) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31411, 5062), (6, 10576, 7746)

X(31482) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, SPIEKER}

Barycentrics a*(2*(b+c)*b*c+2*S*a+(b^2+c^2)*a) : :

X(31482) lies on these lines: {6,1573}, {8,31471}, {10,1504}, {32,31453}, {39,1377}, {115,31484}, {187,9678}, {372,31456}, {486,31488}, {958,5062}, {1015,31473}, {1107,1505}, {1329,31481}, {1335,16589}, {1572,31438}, {1574,6422}, {1588,31416}, {1703,31442}, {1706,31437}, {2549,31413}, {2551,31411}, {3312,31490}, {4386,9675}, {7765,31486}, {9674,25440}, {9711,31483}

X(31482) = {X(1377), X(31464)}-harmonic conjugate of X(39)

X(31483) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, STEINER}

Barycentrics -(b^2-c^2)^2+6*S*a^2+4*(b^2+c^2)*a^2 : :

X(31483) lies on these lines: {5,1504}, {6,3411}, {20,31411}, {32,31454}, {39,31465}, {187,9680}, {372,31457}, {491,7849}, {548,9674}, {631,5062}, {1335,31478}, {1378,31491}, {1505,9606}, {1572,31440}, {1574,31486}, {1588,31417}, {1703,31444}, {1991,7854}, {2549,31414}, {3068,5319}, {3312,31492}, {3592,7753}, {5058,31403}, {5309,8960}, {5475,12962}, {6422,7765}, {7583,9607}, {7755,13846}, {7817,13637}, {9589,31437}, {9711,31482}

X(31484) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, NINE-POINTS, SPIEKER}

Barycentrics 2*S*a^2+(a+b+c)*((b^2+c^2)*a-(b+c)*(b-c)^2) : :

X(31484) lies on these lines: {2,5414}, {4,31453}, {5,1377}, {6,2886}, {8,31472}, {10,485}, {11,31473}, {30,9678}, {75,491}, {115,31482}, {142,5393}, {372,26363}, {377,2067}, {381,31485}, {442,1335}, {474,9661}, {486,25639}, {590,1376}, {946,30556}, {958,3070}, {993,6560}, {1124,24390}, {1152,4999}, {1378,7583}, {1505,31488}, {1574,31481}, {1575,31463}, {1587,19843}, {1588,31418}, {1699,31438}, {1703,5705}, {1706,13893}, {2066,3434}, {2362,6734}, {2550,3068}, {2551,31412}, {3035,8253}, {3297,3813}, {3298,25466}, {3312,31493}, {3820,18538}, {3925,19030}, {5254,31464}, {5418,25440}, {5687,9646}, {5709,6213}, {6502,10527}, {8981,9679}, {10576,26364}, {19038,31140}, {21956,31459}

X(31484) = {X(7583), X(31419)}-harmonic conjugate of X(1378)

X(31485) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, SPIEKER, STAMMLER}

Barycentrics a*(a^3-4*(b+c)*b*c-4*S*a-(b^2+c^2)*a) : :

X(31485) lies on these lines: {3,1377}, {6,1573}, {8,31474}, {10,3311}, {30,31413}, {371,9709}, {381,31484}, {382,31486}, {486,31493}, {517,31438}, {958,3312}, {993,6398}, {999,31473}, {1329,8976}, {1335,11108}, {1376,6221}, {1378,6417}, {1482,30556}, {1505,31490}, {1588,31419}, {1698,18996}, {1703,31445}, {1706,31439}, {2067,16408}, {2362,3927}, {2551,7583}, {2886,13785}, {3068,3820}, {3679,19038}, {5044,18991}, {5251,19037}, {5258,18995}, {5267,6456}, {5414,16418}, {5791,13936}, {6421,31468}, {6449,25440}, {7584,19843}, {8165,13886}, {9605,31464}, {9711,31487}, {13905,21031}, {13951,26363}, {13963,24953}, {13966,30478}, {19027,19854}

X(31485) = {X(1377), X(31453)}-harmonic conjugate of X(3)

X(31486) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, SPIEKER, STEINER}

Barycentrics 6*S*a^2+6*(b+c)*b*c*a+(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31486) lies on these lines: {5,1377}, {6,9710}, {8,31475}, {20,31413}, {372,31458}, {382,31485}, {548,9678}, {1335,17529}, {1376,31454}, {1505,31491}, {1574,31483}, {1575,31465}, {1588,31420}, {1703,31446}, {1706,31440}, {2551,31414}, {3312,31494}, {9607,31464}, {9680,25440}

X(31487) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, STAMMLER, STEINER}

Barycentrics 3*a^4-12*S*a^2-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(31487) = 3*X(6447)+2*X(31414)

X(31487) lies on these lines: {2,6427}, {3,9680}, {5,1588}, {6,3411}, {20,6221}, {30,6447}, {140,6428}, {371,382}, {376,9692}, {381,3592}, {485,3843}, {517,31440}, {548,1587}, {550,6519}, {590,5070}, {615,6500}, {631,3312}, {632,19053}, {999,31475}, {1151,15696}, {1335,31480}, {1505,31492}, {1656,6419}, {1657,6425}, {1703,31447}, {3069,16239}, {3070,9681}, {3316,18762}, {3523,6448}, {3524,6522}, {3528,6455}, {3530,6398}, {3534,6453}, {3544,3590}, {3594,15720}, {3832,13886}, {3853,6459}, {3855,18538}, {3856,23259}, {3861,31412}, {4309,19030}, {4317,19028}, {5054,6420}, {5067,7584}, {5410,15559}, {5418,6418}, {5420,6501}, {6407,6560}, {6421,31470}, {6426,15693}, {6450,7581}, {6451,6460}, {6470,6565}, {6496,21734}, {7486,7582}, {9589,31439}, {9605,31465}, {9624,13888}, {9709,31486}, {9711,31485}, {10195,15703}, {11314,13637}, {13901,31452}, {13904,31474}, {13905,15888}, {19111,20379}

X(31487) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1588, 3068, 13925), (3068, 3311, 8976), (3311, 8976, 13785)

X(31488) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, NINE-POINTS, SPIEKER}

Barycentrics 2*(b^2+c^2)*a^2+2*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31488) lies on these lines: {2,2241}, {4,31456}, {5,1573}, {6,31493}, {8,31476}, {10,1506}, {11,16589}, {32,26363}, {37,24387}, {39,2886}, {75,7764}, {115,1107}, {187,4999}, {381,31490}, {405,9665}, {442,1015}, {486,31482}, {625,26558}, {626,17030}, {956,9650}, {958,5475}, {993,7747}, {1329,7603}, {1376,31455}, {1378,31481}, {1500,24390}, {1505,31484}, {1572,5705}, {1574,3815}, {1575,9698}, {1699,31442}, {1706,31441}, {2242,10527}, {2476,16975}, {2548,19843}, {2549,31418}, {2550,31401}, {2551,31415}, {3434,31451}, {3767,31405}, {3788,20172}, {3822,17448}, {3841,16604}, {3847,6537}, {4386,7749}, {4426,7753}, {5267,6781}, {5283,11680}, {6155,29688}, {6292,20541}, {6547,17062}, {6680,20179}, {6683,26582}, {7737,30478}, {7748,31449}, {7794,21264}, {7813,20888}, {9599,19854}, {9651,17532}, {9709,31489}, {11813,21879}, {31140,31448}, {31420,31450}

X(31488) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1107, 25639, 115), (2886, 31466, 39)

X(31489) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, NINE-POINTS, STAMMLER}

Barycentrics a^4-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(31489) = X(11742)+8*X(31415)

X(31489) lies on these lines: {2,6}, {3,1506}, {4,8719}, {5,2549}, {11,31477}, {22,15109}, {30,11742}, {32,3526}, {39,1656}, {53,8889}, {98,11669}, {111,11639}, {114,10516}, {115,5024}, {140,2548}, {187,5054}, {194,16922}, {216,30771}, {232,566}, {262,7608}, {381,574}, {498,16781}, {499,31460}, {547,15048}, {549,5210}, {569,9603}, {620,11286}, {625,11287}, {631,5023}, {999,31476}, {1015,31479}, {1030,16434}, {1078,7926}, {1196,5421}, {1285,15709}, {1329,31490}, {1368,15880}, {1384,7753}, {1504,13951}, {1505,8976}, {1571,9955}, {1572,11231}, {1574,31493}, {1609,16419}, {1657,15515}, {1699,31443}, {1975,16921}, {1995,9609}, {2165,11548}, {2453,16316}, {2493,15302}, {3090,5254}, {3291,13337}, {3312,31481}, {3363,7618}, {3524,5585}, {3525,22331}, {3530,31417}, {3534,8589}, {3614,9597}, {3628,3767}, {3814,31449}, {3843,7756}, {3851,7748}, {3934,7908}, {4045,11318}, {5008,15723}, {5038,15069}, {5050,5477}, {5056,7738}, {5063,10314}, {5067,5286}, {5070,5355}, {5077,8176}, {5107,14848}, {5116,13860}, {5124,19544}, {5159,16303}, {5206,15720}, {5309,15703}, {5432,9599}, {5433,9596}, {5471,11485}, {5472,11486}, {5544,6388}, {5651,30516}, {5886,31398}, {6143,8743}, {6353,6748}, {6421,10576}, {6422,10577}, {6459,9601}, {6565,9600}, {6639,23115}, {6683,7862}, {6811,23261}, {6813,23251}, {7173,9598}, {7484,8553}, {7486,9607}, {7509,9608}, {7539,13351}, {7622,11159}, {7739,15699}, {7741,31448}, {7743,31433}, {7749,30435}, {7752,7784}, {7769,7770}, {7773,7824}, {7776,7815}, {7786,7887}, {7787,16923}, {7804,11288}, {7814,7879}, {7867,13357}, {7874,13356}, {7888,31239}, {7913,31275}, {7988,9574}, {8227,31428}, {8588,14537}, {8716,11185}, {8770,9722}, {9112,16963}, {9113,16962}, {9306,9604}, {9619,9956}, {9620,11230}, {9669,31451}, {9709,31488}, {9744,9756}, {9769,16176}, {10011,14561}, {10194,19102}, {10195,19105}, {10418,11284}, {11637,15546}, {13330,15819}, {13966,31411}, {14494,14853}, {14535,31274}, {15325,31409}, {15355,31236}, {15655,15701}, {16308,30745}, {16777,24239}, {18424,19709}, {21448,30537}, {22240,30744}, {22793,31422}, {26364,31466}

X(31489) = complement of X(34229)
X(31489) = complement of the isotomic conjugate of X(14494)
X(31489) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,37637), (2, 1007, 141), (7610, 11163, 15534), (7774, 8667, 6144)

X(31490) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, SPIEKER, STAMMLER}

Barycentrics a*(a^3-3*(b^2+c^2)*a-4*(b+c)*b*c) : :

X(31490) lies on these lines: {1,6}, {3,1573}, {8,31477}, {10,5013}, {30,31416}, {39,9708}, {115,31493}, {230,30478}, {381,31488}, {382,31491}, {517,31442}, {574,9709}, {846,4051}, {988,16605}, {993,3053}, {999,16589}, {1015,11108}, {1030,22654}, {1329,31489}, {1376,15815}, {1505,31485}, {1572,31445}, {1574,5024}, {1575,22332}, {1706,31443}, {1975,20181}, {2241,16418}, {2549,31419}, {2551,3815}, {2975,5275}, {3312,31482}, {3679,31448}, {3780,19765}, {3820,31401}, {3925,9597}, {4386,5023}, {5021,30116}, {5044,9619}, {5210,5267}, {5254,19843}, {5737,24359}, {6376,15271}, {7745,31405}, {7765,31494}, {7778,26558}, {9336,25542}, {9623,31429}, {9651,17528}, {9711,31492}, {13881,26363}, {16992,21226}, {17595,21951}

X(31490) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (405, 16975, 16781), (958, 1107, 6)

X(31491) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, SPIEKER, STEINER}

Barycentrics 4*(b^2+c^2)*a^2+6*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31491) lies on these lines: {5,1573}, {6,31494}, {8,31478}, {10,9698}, {20,31416}, {32,31458}, {39,9710}, {382,31490}, {1015,17529}, {1107,7765}, {1376,31457}, {1378,31483}, {1505,31486}, {1572,31446}, {1574,9606}, {1706,31444}, {2549,31420}, {4197,16975}, {5319,19843}, {9589,31442}, {9607,31419}, {9709,31492}, {9711,31466}

X(31491) = {X(9710), X(31469)}-harmonic conjugate of X(39)

X(31492) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, STAMMLER, STEINER}

Barycentrics 3*a^4-11*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31492) lies on these lines: {2,9607}, {3,7753}, {5,2549}, {6,631}, {20,3815}, {30,31417}, {39,3526}, {140,5319}, {382,574}, {517,31444}, {548,2548}, {599,7796}, {632,7739}, {999,31478}, {1015,31480}, {1078,6144}, {1505,31487}, {1506,3843}, {1572,31447}, {1574,31494}, {1656,18362}, {3053,3530}, {3055,7486}, {3312,31483}, {3523,9300}, {3528,7745}, {3763,7763}, {3767,16239}, {3861,31415}, {4317,31460}, {5007,15720}, {5023,7736}, {5024,5070}, {5054,7772}, {5067,5254}, {5079,11648}, {5306,10303}, {5475,17800}, {5881,31428}, {6337,15491}, {7748,18584}, {7755,15694}, {7778,7876}, {7784,7814}, {7786,24256}, {7791,11184}, {7824,7946}, {7849,15482}, {9589,31443}, {9608,15109}, {9624,31431}, {9709,31491}, {9711,31490}, {10516,12055}, {11165,17130}, {11307,16645}, {11308,16644}, {15484,15515}, {16781,31452}

X(31492) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 31450, 5013), (9698, 31457, 3), (31401, 31450, 5)

X(31493) = CENTROID OF CURVATURES OF THESE CIRCLES: {NINE-POINTS, SPIEKER, STAMMLER}

Barycentrics a^4-3*(b^2+c^2)*a^2-4*(b+c)*b*c*a+2*(b^2-c^2)^2 : :

X(31493) lies on these lines: {1,31245}, {2,496}, {3,2886}, {5,2551}, {6,31488}, {8,31479}, {9,9955}, {10,1482}, {11,11108}, {21,9668}, {30,30478}, {35,31140}, {56,17528}, {115,31490}, {140,2550}, {230,31416}, {381,958}, {382,993}, {405,9669}, {442,999}, {443,15325}, {474,26060}, {486,31485}, {495,6856}, {497,6675}, {499,3925}, {517,5705}, {595,31187}, {936,11230}, {942,5231}, {946,5791}, {956,2476}, {960,18493}, {1001,24387}, {1125,12437}, {1191,24880}, {1329,5055}, {1376,3526}, {1377,13951}, {1378,8976}, {1479,16418}, {1573,13881}, {1574,31489}, {1575,31467}, {1617,10957}, {1698,3057}, {1699,31445}, {1706,11231}, {2975,9655}, {3086,8728}, {3090,3820}, {3242,24160}, {3312,31484}, {3333,3824}, {3419,24541}, {3421,10592}, {3434,7483}, {3436,17530}, {3452,12864}, {3530,31420}, {3534,5267}, {3555,31266}, {3617,7504}, {3622,31254}, {3656,5837}, {3679,11011}, {3695,30741}, {3697,30852}, {3788,20181}, {3813,6767}, {3814,5079}, {3816,16853}, {3822,12513}, {3826,10200}, {3829,16857}, {3843,31458}, {3927,12047}, {3940,11375}, {4299,31157}, {4413,11508}, {4426,15484}, {4847,11374}, {5044,8227}, {5045,25525}, {5054,25440}, {5070,9710}, {5071,8165}, {5084,10593}, {5177,18990}, {5248,11235}, {5251,10896}, {5254,31468}, {5258,10895}, {5259,11238}, {5260,17556}, {5274,16845}, {5288,11237}, {5305,31405}, {5433,16417}, {5436,18527}, {5708,12609}, {5745,12699}, {5779,12608}, {5784,13373}, {5789,6001}, {5794,10246}, {5827,16821}, {6147,24477}, {6244,6833}, {6284,17571}, {6846,7956}, {6857,15171}, {6862,10306}, {6933,17757}, {7373,25466}, {7489,11928}, {7539,29667}, {7680,8158}, {7743,31435}, {7866,17030}, {9605,31466}, {9623,9956}, {9965,11544}, {10202,18251}, {10584,17575}, {10589,17527}, {10916,15934}, {11318,26558}, {12702,26066}, {12953,31159}, {13785,31453}, {13966,31413}, {16466,24892}, {17542,26127}, {19314,31084}, {19875,30323}, {19877,31272}, {21956,31461}, {22793,31424}, {23708,25917}, {24161,29676}

X(31493) = midpoint of X(30478) and X(31418)
X(31493) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24390, 3295), (2, 31419, 9709)

X(31494) = CENTROID OF CURVATURES OF THESE CIRCLES: {SPIEKER, STAMMLER, STEINER}

Barycentrics 3*a^4-5*(b^2+c^2)*a^2-12*(b+c)*b*c*a+2*(b^2-c^2)^2 : :

X(31494) lies on these lines: {3,9710}, {5,2551}, {6,31491}, {8,31480}, {10,3526}, {20,31419}, {30,31420}, {382,958}, {517,31446}, {548,2550}, {631,9709}, {956,4197}, {993,15696}, {999,17529}, {1574,31492}, {1575,31470}, {1698,20323}, {1706,31447}, {2886,3843}, {3312,31486}, {3530,30478}, {3813,16857}, {3820,5067}, {3861,31418}, {3925,4317}, {4309,16418}, {5044,9624}, {5070,9711}, {5251,9670}, {5258,9657}, {5260,9669}, {5791,11362}, {7765,31490}, {9589,31445}, {9605,31469}, {9607,31468}, {10072,16855}, {10527,17575}, {11240,17590}, {15888,19854}, {17573,31157}, {24953,31452}

X(31494) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9708, 19843, 31493), (9710, 31458, 3)

X(31495) = CENTROID OF CURVATURES OF THE MALFATTI CIRCLES

Barycentrics    Sin[A] (1+Cos[B/2]) (1+Cos[C/2]) (1+Cos[A/2]+2 (1+Cos[B/2]) (1+Cos[C/2])) : :
Trilinears    1 + (2 sec(A/4) cos(B/4) cos(C/4))^2 : :
Trilinears    1 + 2(1 + cos B/2)(1 + cos C/2)/(1 + cos A/2) : :
Trilinears    (2*y*z + x)*y*z : :, where x = 1+cos(A/2) = 2*cos(A/4)^2, y= 1+cos(B/2) = 2*cos(B/4)^2, z = 1+cos(C/2) = 2* cos(C/4)^2"

Note: With the above notation, the A-Malfatti circle has center A'=(2*y*z - x)/x : 1 : 1 (trilinears) and radius r_a=|S*x/(a*(2*y*z-x)+(b+c)*x)| (César Lozada, February 08, 2019)

X(31495) lies on these lines: {1,179}, {483,1127}

X(31495) = {X(1), X(179)}-harmonic conjugate of X(1142)

X(31496) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, APOLLONIUS}

Barycentrics (-a+b+c)*((b+c)*a^5+(3*b^2+5*b*c+3*c^2)*a^4+2*(b+c)^3*a^3-(b^2-4*b*c+c^2)*(b+c)^2*a^2-(b^2-c^2)^2*b*c-(b^2-c^2)^2*(b+c)*a) : :

X(31496) lies on these lines: {2,1682}, {3,9552}, {5,9554}, {10,55}, {12,573}, {181,3085}, {386,5432}, {498,970}, {1686,9646}, {1695,5219}, {2092,5230}, {3057,19858}, {3687,26066}, {5530,5711}, {5552,9564}, {9535,10407}, {9547,31497}, {9548,31434}, {9557,31499}, {9559,31500}, {9561,31501}, {9566,31479}, {10479,21321}, {19721,22299}

X(31496) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 55, 9555), (12, 573, 9553)

X(31497) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, HALF-MOSES}

Barycentrics a^4-2*(2*b^2+b*c+2*c^2)*a^2+(b^2-c^2)^2 : :

X(31497) lies on these lines: {1,31398}, {2,37}, {3,9596}, {5,9598}, {6,5432}, {11,31477}, {12,5013}, {35,2548}, {36,31409}, {39,498}, {55,3815}, {172,631}, {499,1500}, {574,1478}, {609,21843}, {615,31459}, {1015,10056}, {1107,5552}, {1479,1506}, {1504,13963}, {1505,13905}, {1571,12047}, {1574,19854}, {1656,31461}, {1737,31441}, {1770,31422}, {1836,31443}, {1914,5218}, {2241,9698}, {2275,3085}, {2549,7951}, {3035,5275}, {3295,31467}, {3526,31462}, {3582,9331}, {3583,31415}, {4294,31404}, {4299,9650}, {4302,5475}, {4309,9665}, {4317,31457}, {4330,31417}, {4426,6910}, {5010,7737}, {5217,7745}, {5219,9574}, {5281,10987}, {5283,26364}, {5414,31463}, {5687,31466}, {6421,9646}, {6424,31499}, {7354,15815}, {7603,9664}, {7738,10588}, {7763,27020}, {8227,31426}, {9547,31496}, {9592,31434}, {9612,31421}, {9619,10039}, {10527,20691}, {10528,17448}, {11174,26629}, {13411,31396}, {13962,31471}, {15888,31492}, {17757,31449}, {20483,30741}, {21031,31490}

X(31497) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31460, 9596), (5, 31448, 9598)

X(31498) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, EXTANGENTS}

Barycentrics
a*(a^7+(b+c)*a^6-(b^2+b*c+c^2)*a^5-(b+c)*(b^2+3*b*c+c^2)*a^4-(b^2+b*c+c^2)*(b+c)^2*a^3-(b+c)*(b^4+c^4-(3*b^2+4*b*c+3*c^2)*b*c)*a^2+(b^2+c^2)*(b+c)^4*a+(b^4-c^4)*(b^2-c^2)*(b+c))*(a-b+c)*(a+b-c) : :

X(31498) lies on these lines: {1,3}, {12,3101}, {498,8141}, {5218,9537}, {5219,9573}, {5432,6197}, {9536,10588}

X(31499) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, RADICAL CIRCLE OF LUCAS CIRCLES}

Barycentrics 3*a^4-2*S*a^2-2*(2*b^2+b*c+2*c^2)*a^2+(b^2-c^2)^2 : :

X(31499) lies on these lines: {2,9679}, {3,9646}, {5,9660}, {12,6200}, {35,590}, {55,5418}, {140,2066}, {371,5432}, {372,13901}, {485,5217}, {498,1151}, {549,6502}, {631,1124}, {1152,13905}, {1335,5218}, {1378,6910}, {1478,6409}, {1479,8253}, {3070,5010}, {3298,31452}, {3301,31454}, {3524,31408}, {3592,13963}, {3612,13911}, {4299,6411}, {4995,18965}, {5054,31474}, {5219,9582}, {5326,10577}, {5414,8981}, {5420,19038}, {5552,9678}, {6284,10576}, {6396,19028}, {6419,13958}, {6424,31497}, {6449,31479}, {6451,9655}, {6453,31500}, {6455,9654}, {6484,9649}, {6560,13897}, {6564,15338}, {8965,9371}, {9541,10588}, {9557,31496}, {9615,31434}, {9675,31501}, {12963,31460}, {13893,30282}

X(31499) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12, 6200, 9647), (9648, 19027, 371)

X(31500) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, LUCAS INNER}

Barycentrics 8*a^4-8*S*a^2-(9*b^2+2*b*c+9*c^2)*a^2+(b^2-c^2)^2 : :

X(31500) lies on these lines: {2,9689}, {3,9648}, {5,9662}, {12,1151}, {55,9663}, {371,13958}, {498,6407}, {1478,6445}, {2066,5298}, {4324,13925}, {5218,9542}, {5219,9584}, {5418,7173}, {5432,6221}, {5552,9688}, {6200,13901}, {6284,9540}, {6409,19028}, {6425,19027}, {6447,13963}, {6449,7354}, {6453,31499}, {6455,13905}, {6484,9647}, {8981,15338}, {9543,10588}, {9559,31496}, {9615,10944}, {9616,15950}, {9617,31434}, {9685,31501}, {9690,31479}

X(31500) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12, 1151, 9649), (6200, 13901, 15326)

X(31501) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, MOSES}

Barycentrics a^4-(3*b^2+2*b*c+3*c^2)*a^2+(b^2-c^2)^2 : :

X(31501) lies on these lines: {2,1500}, {3,9650}, {5,9664}, {12,574}, {32,5432}, {35,5475}, {39,498}, {55,1506}, {115,31448}, {140,2242}, {187,9596}, {496,3055}, {615,31471}, {631,31409}, {1015,3085}, {1479,7603}, {1505,9646}, {1571,5219}, {1573,5552}, {1656,31477}, {2241,3815}, {2275,3584}, {2276,7746}, {2548,5218}, {2549,10588}, {3295,31489}, {3526,31478}, {3788,27020}, {4294,31415}, {4299,8589}, {5013,31479}, {5217,7747}, {5281,31404}, {5283,27529}, {5414,31481}, {5687,31488}, {7354,15515}, {7748,7951}, {7756,10895}, {7862,26590}, {8227,31433}, {9341,21843}, {9561,31496}, {9599,31452}, {9612,31422}, {9619,31434}, {9654,15815}, {9675,31499}, {9685,31500}, {10197,16604}, {13411,31398}, {13881,31461}, {15482,26561}, {16589,26364}, {16781,31467}, {17757,31456}

X(31501) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31476, 9650), (5, 31451, 9664)

X(31502) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, NINE-POINTS, PARRY}

Barycentrics 2*(b^2-c^2)*(2*a^2-b^2-c^2)*S*a^2-(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)*(a^4-2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^2) : :

X(31502) lies on the line {351,498}

X(31503) = X(1)X(3052)∩X(10)X(4035)

Barycentrics a (3 a+3 b-c) (b+c) (3 a-b+3 c),b (3 a+3 b-c) (a+c) (-a+3 b+3 c) : :
Barycentrics S^4 + (16 R^4+8 R^2 SB+8 R^2 SC-SB SC-16 R^2 SW-2 SB SW-2 SC SW+3 SW^2)S^2-32 R^4 SB SC+20 R^2 SB SC SW-3 SB SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31503) lies on these lines: {1,3052}, {10,4035}, {19,1100}, {37,2650}, {42,3922}, {75,145}, {225,3649}, {267,5425}, {518,17038}, {596,3635}, {759,4658}, {897,17016}, {942,994}, {3057,13476}, {3924,16666}, {3931,4757}, {4004,4646}, {4802,21105}, {4864,23051}, {6737,17392}, {11518,15509}

X(31503) = barycentric product of X(i) and X(j) for these {i,j}: {1577, 28162}
X(31503) = trilinear product of X(i) and X(j) for these {i,j}: {523, 28162}, {523, 28162}

X(31504) = ISOGONAL CONJUGATE OF X(19169)

Barycentrics a^2 (a^2-b^2-c^2) (3 a^4-6 a^2 b^2+3 b^4-2 a^2 c^2-2 b^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^4-2 a^2 b^2-b^4-6 a^2 c^2-2 b^2 c^2+3 c^4) : :
Barycentrics 3 S^4 + (32 R^4+12 R^2 SB+12 R^2 SC-3 SB SC-32 R^2 SW-3 SB SW-3 SC SW+6 SW^2)S^2 -64 R^4 SB SC+40 R^2 SB SC SW-6 SB SC SW^2 : :
X(31504) = 4*X[5]-3*X[8799], 5*X[631]-3*X[13599]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31504) lies on these lines: {3,13382}, {5,8799}, {20,264}, {216,14531}, {418,8798}, {511,17039}, {548,6662}, {631,13599}, {3528,15318}

X(31504) = isogonal conjugate of X(19169)
X(31504) = barycentric product of X(i) and X(j) for these {i,j}: {343, 14528}

X(31505) = X(4)X(95)∩X(1656)X(22268)

Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-4 a^2 b^2+3 b^4-2 a^2 c^2-4 b^2 c^2+c^4) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-4 a^2 c^2-4 b^2 c^2+3 c^4) : :
Barycentrics 6 S^4 + (12 R^2 SB+12 R^2 SC+10 SB SC-12 R^2 SW-3 SB SW-3 SC SW+3 SW^2)S^2 -16 R^4 SB SC+SB SC SW^2 : :
X(31505) = 5*X[1656]-3*X[22268]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31505) lies on these lines: {4,95}, {1656,22268}, {3519,3527}, {3854,11282}, {5056,6750}, {5068,15319}

X(31505) = barycentric product of X(i) and X(j) for these {i,j}: {233, 8797}

X(31506) = X(6)X(9909)∩X(76)X(193)

Barycentrics a^2 (3 a^2+3 b^2-c^2) (b^2+c^2) (3 a^2-b^2+3 c^2) : :
Barycentrics (16 R^2 SB+16 R^2 SC-8 R^2 SW-6 SB SW-6 SC SW+3 SW^2)S^2 -3 SB SC SW^2-3 SB SW^3-3 SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31506) lies on these lines: {6,9909}, {76,193}, {141,3787}, {511,17042}, {2353,5007}, {6467,27375}

X(31506) = barycentric product of X(i) and X(j) for these {i,j}: {39, 5395}
X(31506) = trilinear product of X(i) and X(j) for these {i,j}: {1964, 5395}, {1964, 5395}

X(31507) = X(4)X(5586)∩X(8)X(4312)

Barycentrics a (a^2-2 a b+b^2+6 a c+6 b c-7 c^2) (a^2+6 a b-7 b^2-2 a c+6 b c+c^2) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31507) lies on the Feuerbach hyperbola and these lines: {4,5586}, {8,4312}, {9,8169}, {516,7320}, {1000,28194}, {1699,10307}, {2346,2951}, {3927,4866}, {7285,15299}, {10390,15726}

X(31507) = isogonal conjugate of X(31508)

X(31508) = ISOGONAL CONJUGATE OF X(31507)

Barycentrics a (7 a^2-6 a b-b^2-6 a c+2 b c-c^2) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31508) lies on these lines: {1,3}, {9,4421}, {10,11106}, {30,5726}, {100,3305}, {200,3219}, {390,10164}, {516,5226}, {902,2999}, {910,3731}, {950,9588}, {993,4915}, {1190,5526}, {1200,3730}, {1323,3599}, {1479,17559}, {1698,4294}, {1699,5218}, {2177,9340}, {2951,7676}, {2975,11519}, {3052,16469}, {3062,15837}, {3158,4640}, {3474,3982}, {3475,4114}, {3523,12575}, {3583,6939}, {3586,19875}, {3624,10624}, {3632,4305}, {3679,4304}, {3689,3929}, {3752,16487}, {3895,17549}, {3911,10385}, {4189,4853}, {4312,9778}, {4315,10304}, {4428,5437}, {4882,8715}, {4995,5219}, {5231,20075}, {5234,5687}, {5267,12629}, {5312,10460}, {5432,7988}, {5493,5703}, {5974,11995}, {6174,20196}, {6284,7989}, {6736,17576}, {7290,21000}, {7322,15430}, {7951,7965}, {7967,8275}, {8164,28150}, {8616,23511}, {8666,12127}, {8833,9582}, {8917,10482}, {9578,15338}, {9589,13411}, {10391,15104}, {10591,19872}, {10860,15298}, {16140,16143}, {18524,18529}, {19541,24644}

X(31508) = isogonal conjugate of X(31507)
X(31508) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,13370,7280}, {40,24929,18421}, {55,165,1}, {55,1155,10389}, {100,4512,8580}, {165,10980,1155}, {1155,10389,10980}, {1697,5217,7987}, {1697,7987,1}, {2646,11531,1}, {3158,4640,5223}, {3295,3361,1}, {3576,9819,1}, {3601,7991,1}, {3746,15803,1}, {5432,9580,7988}, {9778,13405,4312}, {10389,10980,1}, {11224,13384,1}, {18421,24929,1}

X(31509) = X(1)X(3848)∩X(4)X(3625)

Barycentrics a (a+b-7 c) (a-b-c) (a-7 b+c) : :
X(31509) = 5*X[3617]-3*X[7320], 14*X[14150]-15*X[16853]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28852.

X(31509) lies on these lines: {1,3848}, {4,3625}, {7,3621}, {9,3893}, {21,2136}, {79,3632}, {80,4816}, {84,12702}, {519,3296}, {1000,3626}, {1392,4420}, {1706,15179}, {3577,11278}, {3617,7320}, {3679,13606}, {3880,4866}, {4677,5560}, {4778,23836}, {4900,15829}, {5221,7091}, {5558,18221}, {6762,7284}, {7982,16615}, {11256,15015}, {12629,16417}, {13602,17575}, {14150,16853}

X(31510) = MIDPOINT OF X(107) AND X(1304)

Barycentrics (a^2-b^2) (a^2-c^2) (2 a^14-2 a^12 (b^2+c^2)+9 a^8 (b^2-c^2)^2 (b^2+c^2)-8 a^4 (b^2-c^2)^4 (b^2+c^2)+(b^2-c^2)^6 (b^2+c^2)+a^10 (-7 b^4+16 b^2 c^2-7 c^4)+4 a^6 (b^2-c^2)^2 (b^4-5 b^2 c^2+c^4)+a^2 (b^2-c^2)^4 (b^4+8 b^2 c^2+c^4)) : :

As a point on the Euler line, X(31510) has Shinagawa coefficients (-2FS²+9EF²-18F³, (E-2F)S²-4E²F+10EF²+14F³).

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28858.

X(31510) lies on these lines: {2,3}, {107,523}, {110,8057}, {112,9209}, {250,3233}, {476,1301}, {935,9064}, {1289,9060}, {1302,10423}, {1552,2777}, {2693,23239}, {6070,17986}, {6530,16319}, {6587,23964}, {6716,16177}, {10420,30249}, {16166,20626}

X(31510) = midpoint of X(107) and X(1304)
X(31510) = reflection of X(i) in X(j), for these {i, j}: {1552,18809}, {16177,6716}, {16386,27089}

X(31511) = X(1)X(3)∩X(109)X(13138)

Barycentrics a (a-b) (a^2-(b-c)^2) (a-c) (2 a^7-a^6 (b+c)+a^4 (b-c)^2 (b+c)-(b-c)^2 (b+c)^5-6 a^3 (b^2-c^2)^2+4 a (b^2-c^2)^2 (b^2+c^2)+a^2 (b-c)^2 (b^3+7 b^2 c+7 b c^2+c^3)) : :

See Tran Quang Hung and Angel Montesdeoca, AdGeom 5142.

X(31511) lies on these lines: {1,3}, {109,13138}

X(31512) = X(4)X(8)∩X(11)X(901)

Barycentrics a^6+6 a^4 b c-2 a^5 (b+c)-2 a^3 b c (b+c)-(b-c)^4 (b+c)^2+a^2 b c (3 b^2-5 b c+3 c^2)+a (b-c)^2 (2 b^3-3 b^2 c-3 b c^2+2 c^3) : :

See Tran Quang Hung and Angel Montesdeoca, AdGeom 5144.

X(31512) lies on these lines: {4,8}, {11,901}, {100,3259}, {149,513}, {497,14115}, {528,14513}, {953,5840}, {2829,14511}, {3025,13274}, {3583,10774}, {4380,17036}, {5854,17101}, {6075,10707}, {6789,11813}, {8047,23813}, {13273,13756}

X(31512) = reflection of X(i) in X(j) for these {i, j}: {100,3259}, {901,11}

X(31513) = X(4)X(69)∩X(99)X(2679)

Barycentrics a^2 (a^8 b^2 c^2-2 a^2 b^4 c^4 (b^2+c^2)-a^6 (b^6+b^4 c^2+b^2 c^4+c^6)+a^4 (b^8+4 b^4 c^4+c^8)-b^2 c^2 (b^8-3 b^6 c^2+3 b^4 c^4-3 b^2 c^6+c^8)) : :

See Tran Quang Hung and Angel Montesdeoca, AdGeom 5144.

X(31513) lies on these lines: {4,69}, {99,2679}, {115,805}, {148,512}, {543,14509}, {671,6071}, {2549,14113}, {2698,23698}, {2794,14510}, {14061,22103}

X(31513) = reflection of X(i) in X(j) for these {i, j}: {99,2679}, {805,115}

X(31514) = X(1)X(4015)∩X(106)X(5297)

Barycentrics a*(a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3 + 3*a^2*c + a*b*c + 8*b^2*c + 4*a*c^2 + 8*b*c^2 + 2*c^3) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28859.

X(31514) lies on these lines: {1, 4015}, {106, 5297}, {519, 25354}, {551, 3699}, {1022, 1390}, {3242, 4260}, {3722, 4653}, {4669, 17600}, {4752, 16521}

X(31515) = X(1)X(3)∩X(63)X(10090)

Barycentrics a (3 a^6-2 a^5 b-7 a^4 b^2+4 a^3 b^3+5 a^2 b^4-2 a b^5-b^6-2 a^5 c+6 a^4 b c-8 a^2 b^3 c+2 a b^4 c+2 b^5 c-7 a^4 c^2+10 a^2 b^2 c^2+b^4 c^2+4 a^3 c^3-8 a^2 b c^3-4 b^3 c^3+5 a^2 c^4+2 a b c^4+b^2 c^4-2 a c^5+2 b c^5-c^6) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31515) lies on these lines: {1,3}, {63,10090}, {499,4652}, {5533,9580}, {8070,9579}

X(31515) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,17437,1}, {36,5697,3576}, {36,5902,1420}, {57,14793,1}, {1470,11249,36}, {8071,17700,1}

X(31516) = X(1)X(4)∩X(1359)X(13462)

Barycentrics a^7-2 a^6 b-a^5 b^2-a^3 b^4+6 a^2 b^5+a b^6-4 b^7-2 a^6 c+2 a^5 b c+4 a^3 b^3 c-2 a^2 b^4 c-6 a b^5 c+4 b^6 c-a^5 c^2-6 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+12 b^5 c^2+4 a^3 b c^3-4 a^2 b^2 c^3+12 a b^3 c^3-12 b^4 c^3-a^3 c^4-2 a^2 b c^4-a b^2 c^4-12 b^3 c^4+6 a^2 c^5-6 a b c^5+12 b^2 c^5+a c^6+4 b c^6-4 c^7 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31516) lies on these lines: {1,4}, {1359,13462}, {3624,24565}

X(31516) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1699,1785,1}

X(31517) = X(1)X(5)∩X(3583)X(6950)

Barycentrics 2 a^5 b^2-2 a^4 b^3-4 a^3 b^4+4 a^2 b^5+2 a b^6-2 b^7-2 a^5 b c+6 a^3 b^3 c-2 a^2 b^4 c-4 a b^5 c+2 b^6 c+2 a^5 c^2-5 a^3 b^2 c^2-2 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2-2 a^4 c^3+6 a^3 b c^3-2 a^2 b^2 c^3+8 a b^3 c^3-6 b^4 c^3-4 a^3 c^4-2 a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+4 a^2 c^5-4 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 : :
Barycentrics (3 a R^2-4 b R^2+2 a SB-2 b SB+c SB+2 a SC+b SC-2 c SC-4 a SW+b SW)S^2 +6 R S^3+2 R S SB SC+b SB SC^2-c SB SC^2-b SB SC SW : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31517) lies on these lines: {1,5}, {3583,6950}, {10896,14792}, {17566,25639}

X(31517) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,1317,7951}

X(31518) = (name pending)

Barycentrics a (3 a^6-2 a^5 b-3 a^4 b^2+4 a^3 b^3-7 a^2 b^4+6 a b^5-b^6-2 a^5 c+6 a^4 b c-8 a^3 b^2 c+8 a^2 b^3 c-6 a b^4 c+2 b^5 c-3 a^4 c^2-8 a^3 b c^2-2 a^2 b^2 c^2-3 b^4 c^2+4 a^3 c^3+8 a^2 b c^3+4 b^3 c^3-7 a^2 c^4-6 a b c^4-3 b^2 c^4+6 a c^5+2 b c^5-c^6) : :
Barycentrics (6 a R^2-2 a SW+2 b SW-c SW)S^2 + (2 R SB SW+2 R SC SW+2 R SW^2) S +2 a SB SC SW+b SB SC SW-2 c SB SC SW+3 b SC^2 SW-3 c SC^2 SW-2 b SB SW^2+c SB SW^2-2 b SC SW^2+c SC SW^2-a SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31518) lies on this line: {1,6}

X(31519) = (name pending)

Barycentrics a (3 a^6-10 a^5 b+5 a^4 b^2+20 a^3 b^3-35 a^2 b^4+22 a b^5-5 b^6-10 a^5 c+30 a^4 b c-28 a^3 b^2 c+20 a^2 b^3 c-26 a b^4 c+14 b^5 c+5 a^4 c^2-28 a^3 b c^2+14 a^2 b^2 c^2+4 a b^3 c^2-11 b^4 c^2+20 a^3 c^3+20 a^2 b c^3+4 a b^2 c^3+4 b^3 c^3-35 a^2 c^4-26 a b c^4-11 b^2 c^4+22 a c^5+14 b c^5-5 c^6) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31519) lies on this line: {1,6}

X(31520) = X(1)X(2)∩X(381)X(18201)

Barycentrics a^3 b-3 a^2 b^2-2 a b^3+2 b^4+a^3 c+3 a^2 b c+a b^2 c-3 a^2 c^2+a b c^2-4 b^2 c^2-2 a c^3+2 c^4 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28866.

X(31520) lies on these lines: {1,2}, {381,18201}, {3953,15079}, {3976,17606}, {7179,24240}, {10175,24216}, {11231,17715}, {17065,24802}, {17601,18527}, {20118,24806}, {21949,24174}

X(31521) = COMPLEMENT OF X(15435)

Barycentrics a^2 (a^6+a^4 b^2-a^2 b^4-b^6+a^4 c^2-10 a^2 b^2 c^2-7 b^4 c^2-a^2 c^4-7 b^2 c^4-c^6) : :
Barycentrics (2 R^2 SB+2 R^2 SC+4 R^2 SW+SW^2)S^2 -SB SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28867.

X(31521) lies on the Stammler hyperbola and these lines: {1,12329}, {2,159}, {3,3589}, {6,3917}, {25,2916}, {125,2930}, {140,9937}, {141,16419}, {155,182}, {195,5050}, {206,1498}, {399,12017}, {511,15805}, {610,2297}, {936,22769}, {1351,15047}, {1486,29598}, {1593,16936}, {1609,14096}, {1843,22112}, {2918,6642}, {3313,10601}, {3360,5116}, {3564,13154}, {3618,7485}, {5092,9818}, {5157,19153}, {5898,15694}, {7503,15740}, {7509,11821}, {7866,23333}, {9969,17825}, {11284,20987}, {11479,16656}, {12085,17508}, {15141,26206}

X(31521) = complement of X(15435)

X(31522) = X(11)X(523)∩X(30)X(1317)

Barycentrics (a - b - c)*(b - c)^2*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)^2 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28869.

X(31522) lies on the incircle and these lines: {11, 523}, {30, 1317}, {55, 1290}, {56, 2687}, {513, 3024}, {517, 3028}, {1319, 1354}, {1360, 2078}, {3021, 5160}, {3319, 11011}, {4854, 6023}, {9957, 13756}

X(31522) = reflection of X(3024) in the X(1)X(3) line

X(31523) = X(1)X(8674)∩X(104)X(7978)

Barycentrics a*(2*a^9 - 2*a^8*b - 5*a^7*b^2 + 4*a^6*b^3 + 3*a^5*b^4 + a^3*b^6 - 4*a^2*b^7 - a*b^8 + 2*b^9 - 2*a^8*c + 8*a^7*b*c - 8*a^5*b^3*c + 4*a^4*b^4*c - 7*a^3*b^5*c + a^2*b^6*c + 7*a*b^7*c - 3*b^8*c - 5*a^7*c^2 + 8*a^5*b^2*c^2 - 6*a^4*b^3*c^2 - 4*a^3*b^4*c^2 + 11*a^2*b^5*c^2 + a*b^6*c^2 - 5*b^7*c^2 + 4*a^6*c^3 - 8*a^5*b*c^3 - 6*a^4*b^2*c^3 + 22*a^3*b^3*c^3 - 8*a^2*b^4*c^3 - 7*a*b^5*c^3 + 5*b^6*c^3 + 3*a^5*c^4 + 4*a^4*b*c^4 - 4*a^3*b^2*c^4 - 8*a^2*b^3*c^4 + b^5*c^4 - 7*a^3*b*c^5 + 11*a^2*b^2*c^5 - 7*a*b^3*c^5 + b^4*c^5 + a^3*c^6 + a^2*b*c^6 + a*b^2*c^6 + 5*b^3*c^6 - 4*a^2*c^7 + 7*a*b*c^7 - 5*b^2*c^7 - a*c^8 - 3*b*c^8 + 2*c^9) : :
X(31523) = X[13211] - 3 X[16173]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28869.

X(31523) lies on these lines: {1, 8674}, {104, 7978}, {110, 1320}, {113, 952}, {119, 11723}, {125, 1387}, {944, 10767}, {1145, 5972}, {2771, 7984}, {2776, 13868}, {2802, 11720}, {10738, 12898}, {13211, 16173}

X(31523) = midpoint of X(i) and X(j) for these {i,j}: {104, 7978}, {110, 1320}, {944, 10767}, {10738, 12898}
X(31523) = reflection of X(i) in X(j) for these {i,j}: {119, 11723}, {125, 1387}, {1145, 5972}

X(31524) = X(11)X(30)∩X(12)X(2222)

Barycentrics (a + b - c)*(a - b + c)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c - a^2*c^2 + 2*b^2*c^2 + a*c^3 - c^4)^2 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28869.

X(31524) lies on the incircle and these lines: {11, 30}, {12, 2222}, {55, 2687}, {56, 1290}, {65, 23341}, {513, 3028}, {517, 3024}, {523, 1317}, {942, 3025}, {1319, 1365}, {1325, 5172}, {1358, 1443}, {2646, 3326}, {3318, 10149}

X(31524) = reflection of X(5520) in X(3109)
X(31524) = reflection of X(3028) in the X(1)X(3) line

X(31525) = X(1)X(8674)∩X(11)X(11735)

Barycentrics a*(2*a^9 - 2*a^8*b - 3*a^7*b^2 + 4*a^6*b^3 - 3*a^5*b^4 + 7*a^3*b^6 - 4*a^2*b^7 - 3*a*b^8 + 2*b^9 - 2*a^8*c + 4*a^7*b*c - 4*a^5*b^3*c + 4*a^4*b^4*c - 5*a^3*b^5*c - a^2*b^6*c + 5*a*b^7*c - b^8*c - 3*a^7*c^2 + 12*a^5*b^2*c^2 - 6*a^4*b^3*c^2 - 8*a^3*b^4*c^2 + 9*a^2*b^5*c^2 - a*b^6*c^2 - 3*b^7*c^2 + 4*a^6*c^3 - 4*a^5*b*c^3 - 6*a^4*b^2*c^3 + 14*a^3*b^3*c^3 - 4*a^2*b^4*c^3 - 5*a*b^5*c^3 + 3*b^6*c^3 - 3*a^5*c^4 + 4*a^4*b*c^4 - 8*a^3*b^2*c^4 - 4*a^2*b^3*c^4 + 8*a*b^4*c^4 - b^5*c^4 - 5*a^3*b*c^5 + 9*a^2*b^2*c^5 - 5*a*b^3*c^5 - b^4*c^5 + 7*a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + 3*b^3*c^6 - 4*a^2*c^7 + 5*a*b*c^7 - 3*b^2*c^7 - 3*a*c^8 - b*c^8 + 2*c^9) : :
X(31525) = 3 X[5603] - X[10767],3 X[15061] - X[19914]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28869.

X(31525) lies on these lines: {1, 8674}, {11, 11735}, {21, 104}, {74, 10698}, {100, 7984}, {113, 11729}, {125, 952}, {214, 16598}, {1537, 2777}, {2773, 13868}, {2800, 11709}, {5603, 10767}, {5663, 19907}, {6224, 10778}, {7972, 13211}, {9140, 10031}, {10693, 17660}, {15061, 19914}

X(31525) = midpoint of X(i) and X(j) for these {i,j}: {74, 10698}, {100, 7984}, {6224, 10778}, {7972, 13211}, {9140, 10031}, {10693, 17660}
X(31525) = reflection of X(i) in X(j) for these {i,j}: {11, 11735}, {113, 11729}

X(31526) = X(1)X(9446)∩X(2)X(3119)

Barycentrics (a + b - c)*(a - b + c)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28873.

X(31526) lies on these lines: {1, 9446}, {2, 3119}, {7, 354}, {55, 658}, {57, 14189}, {77, 614}, {85, 3742}, {192, 6168}, {241, 2275}, {279, 9445}, {347, 23668}, {348, 24477}, {390, 9533}, {664, 1376}, {883, 27538}, {1442, 20277}, {1462, 2162}, {1996, 3475}, {3158, 25716}, {3599, 8236}, {3673, 17626}, {3676, 23655}, {4124, 17090}, {4388, 7055}, {4566, 5281}, {5226, 27475}, {5274, 10004}, {5437, 9312}, {7176, 24268}, {7182, 10453}, {17084, 18633}, {21609, 30947}

X(31526) = X(1)-Ceva conjugate of X(7)
X(31526) = X(1742)-cross conjugate of X(3177)
X(31526) = crosspoint of X(1) and X(1742)
X(31526) = barycentric product X(i) X(j) for these {i,j}: {7, 3177}, {57, 20935}, {85, 1742}, {331, 20793}, {664, 21195}, {1434, 21084}, {6063, 20995}
X(31526) = barycentric quotient X(i) / X(j) for these {i,j}: {1742, 9}, {3177, 8}, {20793, 219}, {20935, 312}, {20995, 55}, {21084, 2321}, {21195, 522}, {21856, 210}
X(31526) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {354, 1088, 7}, {479, 10580, 7}, {497, 7056, 7}, {5572, 23062, 7}

X(31527) = X(2)X(3160)∩X(7)X(354)

Barycentrics (a + b - c)*(a - b + c)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 4*a^3*c - 4*a^2*b*c + 4*a*b^2*c + 4*b^3*c + 6*a^2*c^2 + 4*a*b*c^2 - 10*b^2*c^2 - 4*a*c^3 + 4*b*c^3 + c^4) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28873.

X(31527) lies on the cubic K200 and these lines: {2, 3160}, {7, 354}, {77, 10582}, {144, 15913}, {200, 25718}, {279, 11019}, {347, 4847}, {390, 3599}, {516, 9533}, {658, 9778}, {934, 7580}, {1699, 10004}, {1996, 10578}, {2124, 30695}, {2951, 17113}, {3817, 15511}, {4452, 9436}, {4554, 5423}, {5435, 14189}, {8236, 9446}, {22464, 31146}

X(31527) = anticomplement of X(19605)
X(31527) = anticomplement of the isogonal of X(1419)
X(31527) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 10405}, {56, 20059}, {57, 9812}, {109, 3239}, {144, 3436}, {165, 329}, {1419, 8}, {3160, 69}, {3207, 144}, {7339, 658}, {9533, 3434}, {16284, 21286}, {17106, 7}
X(31527) = X(8)-Ceva conjugate of X(7)
X(31527) = X(i)-cross conjugate of X(j) for these (i,j): {2124, 17113}, {2951, 30695}
X(31527) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2125}, {55, 8917}
X(31527) = cevapoint of X(i) and X(j) for these (i,j): {2124, 2951}, {3160, 15913}
X(31527) = barycentric product X(i) X(j) for these {i,j}: {7, 30695}, {8, 17113}, {75, 2124}, {85, 2951}, {1615, 6063}, {4554, 17427}
X(31527) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 2125}, {57, 8917}, {1615, 55}, {2124, 1}, {2951, 9}, {17113, 7}, {17427, 650}, {30695, 8}
X(31527) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {479, 497, 7}, {1088, 10580, 7}, {7056, 9812, 7}

Triangles associated to Soddy circles: X(31528)-X(31605)

This preamble and centers X(31528)-X(31605) were contributed by César Eliud Lozada, February 19, 2019.

Let A'B'C' be the intouch triangle of ABC and (S_a), (S_b), (S_c) their Soddy circles. Let A_oB_oC_o be the outer-Soddy triangle of ABC (touchpoints of Soddy circles and the circle internally tangent to them) and and A_iB_iC_i the inner-Soddy triangle of ABC (touchpoints of Soddy circles and the circle externally tangent to them). Then quadrilaterals B'C'C_oB_o and B'C'B_iC_i are cyclic. (See Antreas Hatzipolakis, Hyacinthos #28871)

The circles (O_a) and (I_a) circumscribing B'C'C_oB_o and B'C'C_iB_i, respectively, will be named here the 2nd A-outer-Soddy circle and 2nd A-inner-Soddy circle, respectively and triangles O_aO_bO_c and I_aI_bI_c are called here the 2nd outer-Soddy triangle and 2nd inner-Soddy triangle, respectively. Barycentric coordinates of A-vertices are:

O_a = (-a+b+c)*a-2*S : b*(-a+b+c) : c*(-a+b+c)

I_a = (-a+b+c)*a+2*S : b*(-a+b+c) : c*(-a+b+c)

Perspective triangles to O_aO_bO_c and perspectors: (0 means a not calculated center)

(ABC, 1), (Andromeda, 1), (anti-Aquila, 1), (Antlia, 1), (Aquila, 1), (BCI, 174), (2nd circumperp, 1), (4th Conway, 1), (5th Conway, 1), (excenters-midpoints, 1), (excenters-reflections, 1), (excentral, 1), (3rd extouch, 31528), (4th extouch, 31530), (5th extouch, 31532), (incentral, 1), (intouch, 481), (inverse-in-incircle, 1), (Malfatti, 1), (medial, 31534), (midarc, 1), (2nd midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear, 1), (7th mixtilinear, 31536), (inner-Soddy, 31538), (outer-Soddy, 481), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1)

Perspective triangles to I_aI_bI_c and perspectors: (0 means a not calculated center)

(ABC, 1), (Andromeda, 1), (anti-Aquila, 1), (Antlia, 1), (Aquila, 1), (2nd circumperp, 1), (4th Conway, 1), (5th Conway, 1), (excenters-midpoints, 1), (excenters-reflections, 1), (excentral, 1), (3rd extouch, 31529), (4th extouch, 31531), (5th extouch, 31533), (incentral, 1), (intouch, 482), (inverse-in-incircle, 1), (Malfatti, 1), (medial, 31535), (midarc, 1), (2nd midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear, 1), (7th mixtilinear, 31537), (1st Pamfilos-Zhou, 0), (inner-Soddy, 482), (outer-Soddy, 31539), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1)

Orthologic triangles to O_aO_bO_c and orthologic centers: (0 means a not calculated center)

(4th anti-tri-squares, 31582, 0), (Ascella, 1, 31540), (Atik, 1, 31542), (1st circumperp, 1, 31544), (2nd circumperp, 1, 31546), (inner-Conway, 1, 31547), (Conway, 1, 31549), (2nd Conway, 1, 31551), (3rd Conway, 1, 31553), (3rd Euler, 1, 31555), (4th Euler, 1, 31557), (excenters-reflections, 1, 31559), (excentral, 1, 6212), (2nd extouch, 1, 31561), (hexyl, 1, 31563), (Honsberger, 1, 31565), (inner-Hutson, 1, 0), (Hutson intouch, 1, 31567), (outer-Hutson, 1, 0), (incircle-circles, 1, 31569), (intouch, 1, 481), (inverse-in-incircle, 1, 31571), (6th mixtilinear, 1, 31573), (2nd Pamfilos-Zhou, 1, 31575), (1st Sharygin, 1, 31576), (outer-Soddy, 175, 481), (tangential-midarc, 1, 31578), (2nd tangential-midarc, 1, 31580), (3rd tri-squares, 31582, 31584), (Ursa-major, 1, 31586), (Ursa-minor, 1, 31588), (outer-Vecten, 31582, 6212), (Wasat, 1, 31590), (Yff central, 1, 31592), (2nd Zaniah, 1, 31594)

Orthologic triangles to I_aI_bI_c and orthologic centers: (0 means a not calculated center)

(3rd anti-tri-squares, 31583, 0), (Ascella, 1, 31541), (Atik, 1, 31543), (1st circumperp, 1, 31545), (2nd circumperp, 1, 8225), (inner-Conway, 1, 31548), (Conway, 1, 31550), (2nd Conway, 1, 31552), (3rd Conway, 1, 31554), (3rd Euler, 1, 31556), (4th Euler, 1, 31558), (excenters-reflections, 1, 31560), (excentral, 1, 6213), (2nd extouch, 1, 31562), (hexyl, 1, 31564), (Honsberger, 1, 31566), (inner-Hutson, 1, 0), (Hutson intouch, 1, 31568), (outer-Hutson, 1, 0), (incircle-circles, 1, 31570), (intouch, 1, 482), (inverse-in-incircle, 1, 31572), (6th mixtilinear, 1, 31574), (2nd Pamfilos-Zhou, 1, 8225), (1st Sharygin, 1, 31577), (inner-Soddy, 176, 482), (tangential-midarc, 1, 31579), (2nd tangential-midarc, 1, 31581), (4th tri-squares, 31583, 31585), (Ursa-major, 1, 31587), (Ursa-minor, 1, 31589), (inner-Vecten, 31583, 6213), (Wasat, 1, 31591), (Yff central, 1, 31593), (2nd Zaniah, 1, 31595)

Peter Moses found that vertices B_o, C_o, B_i, C_i are concyclic, and similarly the other quartets of vertices (see Hyacinthos #28873). Denote Q_a the center of B_oC_oB_iC_i and cyclically Q_b and Q_c. The triangle Q_aQ_bQ_c will be named here the Soddy triangle of ABC and has first vertex Q_a = -1/(-a+b+c) : 1/(a-b+c) : 1/(a+b-c).

The appearance of (T, n) in the following list means that triangles T and Q_aQ_bQ_c are perspective with perspector X(n):
(ABC, 7), (Ascella, 14256), (Conway, 7), (2nd Conway, 7), (extouch, 31527), (3rd extouch, 5932), (4th extouch, 5933), (5th extouch, 8), (2nd Hatzipolakis, 31598), (Honsberger, 7), (Hutson extouch, 7), (incentral, 31526), (intouch, 7), (Lemoine, 31599), (Macbeath, 31600), (medial, 3160), (7th mixtilinear, 15913), (orthic, 2898), (inner-Soddy, 31601), (outer-Soddy, 31602), (Steiner, 31603), (symmedial, 31604), (Yff contact, 31605), (1st Zaniah, 7)

The appearance of (T, i, j) in the following list means that triangles T and Q_aQ_bQ_c are orthologic with orthologic centers X(i) and X(j): (inner-Soddy, 176, 1), (outer-Soddy, 175, 1)

X(31528) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND 3rd EXTOUCH

Barycentrics (a^6+2*(b+c)*a^5-(b+c)^2*a^4-4*(b^3+c^3)*a^3-(b^2-c^2)^2*a^2+2*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b+c)^2)*a+2*(a+b+c)*(a^2+b^2-c^2)*(a^2+c^2-b^2)*S : :

X(31528) lies on these lines: {1,4}, {175,5932}, {481,1439}, {482,10400}, {5405,11347}, {5929,31530}, {7078,31561}, {8808,13388}, {10364,31531}, {10903,31536}, {10904,31538}, {18641,31534}, {20262,30556}

X(31528) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4, 31529), (1439, 10905, 481)

X(31529) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND 3rd EXTOUCH

Barycentrics (a^6+2*(b+c)*a^5-(b+c)^2*a^4-4*(b^3+c^3)*a^3-(b^2-c^2)^2*a^2+2*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b+c)^2)*a-2*(a+b+c)*(a^2+b^2-c^2)*(a^2+c^2-b^2)*S : :

X(31529) lies on these lines: {1,4}, {176,5932}, {481,10400}, {482,1439}, {5393,11347}, {5929,31531}, {7078,31562}, {8808,13389}, {10364,31530}, {10903,31537}, {10905,31539}, {18641,31535}, {20262,30557}

X(31529) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4, 31528), (1439, 10904, 482)

X(31530) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND 4th EXTOUCH

Barycentrics (a^3+(b+c)*a^2+(b^2+4*b*c+c^2)*a+(b+c)*(b^2+c^2))*a-2*(-a^2+b^2+c^2)*S : :

X(31530) lies on these lines: {1,69}, {65,481}, {175,5933}, {488,17594}, {492,5530}, {638,24210}, {940,1124}, {1211,5393}, {1685,13388}, {5929,31528}, {10364,31529}, {10372,31533}, {10906,31536}

X(31530) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 69, 31531), (65, 10908, 481)

X(31531) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND 4th EXTOUCH

Barycentrics (a^3+(b+c)*a^2+(b^2+4*b*c+c^2)*a+(b+c)*(b^2+c^2))*a+2*(-a^2+b^2+c^2)*S : :

X(31531) lies on these lines: {1,69}, {65,482}, {176,5933}, {487,17594}, {491,5530}, {637,24210}, {940,1335}, {1211,5405}, {1686,13389}, {5929,31529}, {10364,31528}, {10372,31532}, {10906,31537}, {10908,31539}

X(31531) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 69, 31530), (65, 10907, 482)

X(31532) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND 5th EXTOUCH

Barycentrics (2*(a^2+(b+c)^2)*S-(-a+b+c)*(a+b+c)^2*a)*(a-b+c)*(a+b-c) : :

X(31532) lies on these lines: {1,4}, {8,175}, {10,13388}, {12,5393}, {56,5405}, {65,481}, {482,10404}, {1335,9370}, {1449,31408}, {3084,3436}, {3146,30334}, {3303,31568}, {4298,13389}, {5252,12949}, {6203,13936}, {6284,31567}, {6738,31569}, {10372,31531}, {10909,31536}, {10910,31538}, {11518,30342}, {12527,30557}, {18989,31583}, {18991,30325}

X(31532) = reflection of X(31533) in X(5930)
X(31532) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 388, 31533), (1, 5290, 1659)

X(31533) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND 5th EXTOUCH

Barycentrics (-2*(a^2+(b+c)^2)*S-(-a+b+c)*(a+b+c)^2*a)*(a-b+c)*(a+b-c) : :

X(31533) lies on these lines: {1,4}, {8,176}, {9,31408}, {10,13389}, {12,5405}, {56,5393}, {65,482}, {481,8243}, {1124,9370}, {3083,3436}, {3146,30333}, {3303,31567}, {4298,13388}, {5252,12948}, {6204,13883}, {6284,31568}, {6738,31570}, {8965,31561}, {10372,31530}, {10909,31537}, {10911,31539}, {11518,30341}, {12527,30556}, {18988,31582}, {18992,30324}

X(31533) = reflection of X(31532) in X(5930)
X(31533) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 388, 31532), (1, 5290, 13390)

X(31534) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND MEDIAL

Barycentrics (-2*S+(a+b-c)*(a-b+c))*((-a+b+c)*a-2*S) : :
X(31534) = 3*X(2)+X(175) = 5*X(3616)-X(30334)

X(31534) lies on these lines: {2,175}, {3,142}, {37,31582}, {56,30381}, {86,1805}, {348,13436}, {481,10253}, {615,16605}, {1086,31583}, {1589,6349}, {3008,13971}, {3084,6505}, {3160,7090}, {3485,30277}, {3616,30334}, {3664,8983}, {3752,5405}, {5393,17056}, {7288,30276}, {11375,30380}, {18641,31528}

X(31534) = midpoint of X(i) and X(j) for these {i,j}: {175, 14121}, {3160, 7090}
X(31534) = complement of X(14121)
X(31534) = Kosnita(X(175), X(2)) point
X(31534) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 175, 14121), (3, 17073, 31535), (4657, 25524, 31535)

X(31535) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND MEDIAL

Barycentrics (2*S+(a+b-c)*(a-b+c))*((-a+b+c)*a+2*S) : :
X(31535) = 3*X(2)+X(176) = 3*X(1699)+X(8986) = 5*X(3616)-X(30333)

X(31535) lies on these lines: {2,176}, {3,142}, {4,8984}, {56,30380}, {86,1806}, {348,13453}, {482,10252}, {590,16605}, {1086,31582}, {1590,6349}, {1699,8986}, {3008,8983}, {3083,6505}, {3160,14121}, {3485,30276}, {3616,30333}, {3664,13971}, {3752,5393}, {5405,17056}, {7288,30277}, {11375,30381}, {18641,31529}

X(31535) = midpoint of X(i) and X(j) for these {i,j}: {4, 8984}, {176, 7090}, {3160, 14121}
X(31535) = complement of X(7090)
X(31535) = Kosnita(X(176), X(2)) point
X(31535) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 176, 7090), (3, 17073, 31534), (4657, 25524, 31534)

X(31536) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND 7th MIXTILINEAR

Barycentrics
a*(a^6-6*(b+c)*a^5+(15*b^2-14*b*c+15*c^2)*a^4-20*(b^2-c^2)*(b-c)*a^3+5*(b-c)^2*(b+3*c)*(3*b+c)*a^2-2*(b^2-c^2)*(b-c)*(3*b+c)*(b+3*c)*a-4*(a^2-2*(b-c)*a+(b+3*c)*(b-c))*(a^2+2*(b-c)*a-(3*b+c)*(b-c))*S+(b-c)^6) : :

X(31536) lies on these lines: {1,971}, {175,15913}, {481,9533}, {10903,31528}, {10906,31530}, {10909,31532}, {10972,31538}

X(31536) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3062, 31537), (9533, 10973, 481)

X(31537) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND 7th MIXTILINEAR

Barycentrics
a*(a^6-6*(b+c)*a^5+(15*b^2-14*b*c+15*c^2)*a^4-20*(b^2-c^2)*(b-c)*a^3+5*(b-c)^2*(b+3*c)*(3*b+c)*a^2-2*(b^2-c^2)*(b-c)*(3*b+c)*(b+3*c)*a+4*(a^2-2*(b-c)*a+(b+3*c)*(b-c))*(a^2+2*(b-c)*a-(3*b+c)*(b-c))*S+(b-c)^6) : :

X(31537) lies on these lines: {1,971}, {176,15913}, {482,9533}, {10903,31529}, {10906,31531}, {10909,31533}, {10973,31539}

X(31537) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3062, 31536), (9533, 10972, 482)

X(31538) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-SODDY AND INNER-SODDY

Barycentrics (2*S+3*(-a+b+c)*a)*(a-b+c)*(a+b-c) : :
Trilinears 3 + 2 sec A/2 cos B/2 cos C/2 : :
X(31538) = 3*X(1)-X(30431)

Let B' and C' be the orthogonal projections of the incenter of ABC on AC and AB, respectively. Then the quadrangle BCB'C' is tangential. Let (Oa) be its incircle and define (Ob) and (Oc) cyclically. Then X(31538) is the centroid of curvatures of (Oa), (Ob), and (Oc). (Randy Hutson, February 12, 2019). These circles are the companion incircles used in Elkies's construction of X(176).

X(31538) lies on these lines: {1,7}, {226,1328}, {1317,22107}, {1659,5219}, {3083,31018}, {3911,5393}, {4031,13388}, {5252,12948}, {6666,30557}, {7277,7968}, {7969,17366}, {10904,31528}, {10910,31532}, {10972,31536}, {13453,25716}, {14760,30346}

X(31538) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 31539, 481), (482, 31539, 7), (17802, 21169, 1374)

X(31539) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-SODDY AND OUTER-SODDY

Barycentrics (-2*S+3*(-a+b+c)*a)*(a-b+c)*(a+b-c) : :
Trilinears 3 - 2 sec A/2 cos B/2 cos C/2 : :
X(31539) = 3*X(1)-X(30432)

Let B' and C' be the orthogonal projections of the A-excenter of ABC on AC and AB, respectively. Then the quadrangle BCB'C' is tangential. Let (O'a) be its incircle and define (O'b) and (O'c) cyclically. Then X(31539) is the centroid of curvatures of (O'a), (O'b), and (O'c). (Randy Hutson, February 12, 2019). These circles are the companion incircles used in Elkies's construction of X(176).

X(31539) lies on these lines: {1,7}, {226,1327}, {1317,22106}, {3084,31018}, {3911,5405}, {4031,13389}, {5219,5393}, {5252,12949}, {6666,30556}, {7277,7969}, {7968,17366}, {10905,31529}, {10908,31531}, {10911,31533}, {10973,31537}, {13436,25716}, {14760,30347}

X(31539) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17803, 175), (175, 17802, 1), (17802, 17803, 481)

X(31540) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO 2nd OUTER-SODDY

Barycentrics -((b+c)*a-(b-c)^2)*S+(-a^2+b^2+c^2)*a^2 : :
X(31540) = 3*X(2)+X(31549)

The reciprocal orthologic center of these triangles is X(1)

X(31540) lies on these lines: {1,30277}, {2,31549}, {3,142}, {57,481}, {226,16433}, {371,3664}, {372,3008}, {487,17298}, {488,4384}, {641,5745}, {908,21553}, {1151,4675}, {1152,17278}, {3311,4667}, {3452,21547}, {3601,31567}, {4292,30381}, {4357,21909}, {5249,16441}, {5316,21546}, {5744,31547}, {8726,31563}, {8727,31555}, {8728,31557}, {8729,31592}, {8731,31576}, {8732,31565}, {8733,31578}, {8734,31580}, {9776,31551}, {10436,11292}, {10855,31542}, {10856,31553}, {10857,31573}, {11018,31571}, {11291,17282}, {11518,31559}, {13411,30380}, {15803,30276}, {17248,21991}, {17603,31588}, {17612,31586}, {21564,31053}, {21565,31019}, {21566,27186}

X(31540) = midpoint of X(i) and X(j) for these {i,j}: {481, 6212}, {31549, 31561}
X(31540) = complement of X(31561)
X(31540) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31549, 31561), (3, 142, 31541)

X(31541) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO 2nd INNER-SODDY

Barycentrics ((b+c)*a-(b-c)^2)*S+(-a^2+b^2+c^2)*a^2 : :
X(31541) = 3*X(2)+X(31550)

The reciprocal orthologic center of these triangles is X(1)

X(31541) lies on these lines: {1,30276}, {2,31550}, {3,142}, {57,482}, {226,16432}, {371,3008}, {372,3664}, {487,4384}, {488,17298}, {642,5745}, {908,21492}, {942,31570}, {1151,17278}, {1152,4675}, {3312,4667}, {3452,21548}, {3601,31568}, {4292,30380}, {5249,16440}, {5316,21549}, {5744,31548}, {8726,31564}, {8727,31556}, {8728,31558}, {8729,31593}, {8731,31577}, {8732,31566}, {8733,31579}, {8734,31581}, {9776,31552}, {10436,11291}, {10855,31543}, {10856,31554}, {10857,31574}, {11018,31572}, {11292,17282}, {11518,31560}, {13411,30381}, {15803,30277}, {17603,31589}, {17612,31587}, {21567,27186}, {21568,31019}, {21569,31053}

X(31541) = midpoint of X(i) and X(j) for these {i,j}: {482, 6213}, {31550, 31562}
X(31541) = complement of X(31562)
X(31541) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31550, 31562), (3, 142, 31540)

X(31542) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 2nd OUTER-SODDY

Barycentrics
a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a+(-2*(b+c)*a^3+6*(b^2+c^2)*a^2-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a+2*(b^2+4*b*c+c^2)*(b-c)^2)*S-(b^2-c^2)^2*(b+c)^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(31542) lies on these lines: {1,30289}, {8,637}, {481,8581}, {516,960}, {3062,31573}, {5927,31561}, {6212,8580}, {8582,31557}, {8583,31546}, {10855,31540}, {10860,31544}, {10861,31549}, {10862,31553}, {10863,31555}, {10864,31563}, {10865,31565}, {10866,31567}, {10868,31576}, {11019,31571}, {11035,31569}, {11519,31559}, {11678,31547}, {11858,31578}, {11859,31580}, {11860,31592}, {17604,31588}, {18227,31594}, {30288,30290}

X(31542) = midpoint of X(31561) and X(31586)
X(31542) = reflection of X(31571) in X(31590)
X(31542) = {X(9856), X(15587)}-harmonic conjugate of X(31543)

X(31543) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 2nd INNER-SODDY

Barycentrics
a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(-2*(b+c)*a^3+6*(b^2+c^2)*a^2-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a+2*(b^2+4*b*c+c^2)*(b-c)^2)*S-(b^2-c^2)^2*(b+c)^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(31543) lies on these lines: {1,30288}, {8,638}, {482,8581}, {516,960}, {3062,31574}, {5927,31562}, {6213,8580}, {8225,8583}, {8582,31558}, {10855,31541}, {10860,31545}, {10861,31550}, {10862,31554}, {10863,31556}, {10864,31564}, {10865,31566}, {10866,31568}, {10868,31577}, {11019,31572}, {11035,31570}, {11519,31560}, {11678,31548}, {11858,31579}, {11859,31581}, {11860,31593}, {17604,31589}, {18227,31595}, {30289,30290}

X(31543) = midpoint of X(31562) and X(31587)
X(31543) = reflection of X(31572) in X(31591)
X(31543) = {X(9856), X(15587)}-harmonic conjugate of X(31542)

X(31544) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 2nd OUTER-SODDY

Barycentrics a*(2*(-a^2+b^2+c^2)*S*a-(a+b+c)*(a^4-3*(b+c)*a^3+3*(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b-c)^2*b*c)) : :
X(31544) = 3*X(165)-X(6212) = 3*X(165)+X(31573) = 3*X(9778)+X(31551)

The reciprocal orthologic center of these triangles is X(1)

X(31544) lies on these lines: {1,30297}, {2,31555}, {3,142}, {4,31557}, {35,30296}, {40,3640}, {55,481}, {56,31567}, {57,31571}, {100,31547}, {165,6212}, {371,1742}, {372,9441}, {1155,31588}, {1376,31594}, {3295,31569}, {4220,31576}, {7411,31549}, {7580,31561}, {7589,31592}, {7676,31565}, {7991,31559}, {8075,31578}, {8076,31580}, {9778,31551}, {10434,31553}, {10860,31542}, {17613,31586}

X(31544) = midpoint of X(i) and X(j) for these {i,j}: {40, 31563}, {6212, 31573}, {7991, 31559}
X(31544) = reflection of X(i) in X(j) for these (i,j): (4, 31557), (31546, 3)
X(31544) = anticomplement of X(31555)
X(31544) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11495, 31545), (12512, 24309, 31545)

X(31545) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO 2nd INNER-SODDY

Barycentrics a*(-2*(-a^2+b^2+c^2)*S*a-(a+b+c)*(a^4-3*(b+c)*a^3+3*(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b-c)^2*b*c)) : :
X(31545) = 3*X(165)-X(6213) = 3*X(165)+X(31574) = 3*X(9778)+X(31552)

The reciprocal orthologic center of these triangles is X(1)

X(31545) lies on these lines: {1,30296}, {2,31556}, {3,142}, {4,31558}, {35,30297}, {40,3641}, {55,482}, {56,31568}, {57,31572}, {100,31548}, {165,6213}, {371,9441}, {372,1742}, {1155,31589}, {1376,31595}, {3295,31570}, {4220,31577}, {7411,31550}, {7580,31562}, {7589,31593}, {7676,31566}, {7991,31560}, {8075,31579}, {8076,31581}, {9778,31552}, {10434,31554}, {10860,31543}, {17613,31587}

X(31545) = midpoint of X(i) and X(j) for these {i,j}: {40, 31564}, {6213, 31574}, {7991, 31560}
X(31545) = reflection of X(i) in X(j) for these (i,j): (4, 31558), (8225, 3)
X(31545) = anticomplement of X(31556)
X(31545) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11495, 31544), (12512, 24309, 31544)

X(31546) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO 2nd OUTER-SODDY

Barycentrics a*(2*(-a^2+b^2+c^2)*S*a+(a+b-c)*(a^2-(b+c)*a-2*b*c)*(a-b+c)*(-a+b+c)) : :
X(31546) = 3*X(1)-X(31559) = 4*X(1125)-X(31575) = 3*X(3576)-X(31563) = 5*X(3616)-X(31551) = 3*X(6212)+X(31559) = 5*X(7987)-X(31573)

The reciprocal orthologic center of these triangles is X(1)

X(31546) lies on these lines: {1,371}, {2,31557}, {3,142}, {4,31555}, {21,31549}, {36,30385}, {55,5393}, {56,481}, {100,21553}, {169,30557}, {238,372}, {405,31561}, {642,8299}, {958,31594}, {993,22624}, {999,31569}, {1376,21547}, {1621,16441}, {2646,31588}, {2975,31547}, {3576,31563}, {3616,31551}, {4254,13887}, {4413,21546}, {4421,21557}, {4423,16432}, {4428,21558}, {4649,6419}, {5120,13940}, {5263,21909}, {5284,16440}, {6136,14121}, {6210,11825}, {6396,15485}, {6398,8692}, {6420,16468}, {7587,31592}, {7588,31580}, {7677,31565}, {7987,31573}, {8077,31578}, {8167,21548}, {8583,31542}, {10882,31553}, {17614,31586}

X(31546) = midpoint of X(1) and X(6212)
X(31546) = reflection of X(i) in X(j) for these (i,j): (4, 31555), (31544, 3), (31575, 31590), (31590, 1125)
X(31546) = anticomplement of X(31557)
X(31546) = {X(3), X(1001)}-harmonic conjugate of X(8225)

X(31547) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 2nd OUTER-SODDY

Barycentrics 2*(-a+b+c)*S+(a+b+c)*(3*a^2-2*(b+c)*a-(b-c)^2) : :
X(31547) = 3*X(2)-4*X(31594) = 5*X(3616)-4*X(31569) = 3*X(3873)-4*X(31571)

The reciprocal orthologic center of these triangles is X(1)

X(31547) lies on these lines: {2,481}, {8,144}, {9,31565}, {10,30425}, {63,488}, {78,31563}, {100,31544}, {145,31567}, {175,30413}, {200,31573}, {329,31551}, {348,13453}, {518,31588}, {585,3177}, {908,31590}, {2975,31546}, {3436,12787}, {3616,31569}, {3873,31571}, {4419,19066}, {5744,31540}, {8125,31580}, {8126,31592}, {11678,31542}, {11679,31553}, {11680,31555}, {11681,31557}, {11682,31559}, {11687,31575}, {11688,31576}, {11690,31578}, {13387,17781}, {17615,31586}, {27541,30412}

X(31547) = reflection of X(i) in X(j) for these (i,j): (145, 31567), (481, 31594), (30425, 10), (31548, 30625), (31549, 6212), (31551, 31561), (31565, 9)
X(31547) = anticomplement of X(481)
X(31547) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 144, 31548), (481, 31594, 2), (3729, 12527, 31548)

X(31548) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO 2nd INNER-SODDY

Barycentrics -2*(-a+b+c)*S+(a+b+c)*(3*a^2-2*(b+c)*a-(b-c)^2) : :
X(31548) = 3*X(2)-4*X(31595) = 5*X(3616)-4*X(31570) = 3*X(3873)-4*X(31572)

The reciprocal orthologic center of these triangles is X(1)

X(31548) lies on these lines: {2,482}, {8,144}, {9,31566}, {10,30426}, {63,487}, {78,31564}, {100,31545}, {145,31568}, {176,30412}, {200,31574}, {329,31552}, {348,13436}, {518,31589}, {586,3177}, {908,31591}, {2975,8225}, {3436,12788}, {3616,31570}, {3873,31572}, {4419,19065}, {5744,31541}, {8125,31581}, {8126,31593}, {11678,31543}, {11679,31554}, {11680,31556}, {11681,31558}, {11682,31560}, {11688,31577}, {11690,31579}, {13386,17781}, {17615,31587}, {27541,30413}

X(31548) = reflection of X(i) in X(j) for these (i,j): (145, 31568), (482, 31595), (30426, 10), (31547, 30625), (31550, 6213), (31552, 31562), (31566, 9)
X(31548) = anticomplement of X(482)
X(31548) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 144, 31547), (482, 31595, 2), (3729, 12527, 31547)

X(31549) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO 2nd OUTER-SODDY

Barycentrics 3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2+2*S*(a+b-c)*(a-b+c) : :
X(31549) = 3*X(2)-4*X(31540)

The reciprocal orthologic center of these triangles is X(1)

X(31549) lies on these lines: {1,7}, {2,31540}, {21,31546}, {63,488}, {75,490}, {320,489}, {894,11294}, {3662,11293}, {4000,6460}, {4197,31557}, {4644,6459}, {5249,31590}, {5273,31594}, {5391,12323}, {7411,31544}, {10391,31588}, {10861,31542}, {10883,31555}, {10885,31575}, {11020,31571}, {11520,31559}, {11888,31578}, {11889,31580}, {11890,31592}, {17616,31586}

X(31549) = reflection of X(i) in X(j) for these (i,j): (31547, 6212), (31551, 481), (31561, 31540)
X(31549) = anticomplement of X(31561)
X(31549) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 20, 31550), (4292, 10444, 31550), (10884, 18650, 31550)

X(31550) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO 2nd INNER-SODDY

Barycentrics 3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2-2*S*(a+b-c)*(a-b+c) : :
X(31550) = 3*X(2)-4*X(31541)

The reciprocal orthologic center of these triangles is X(1)

X(31550) lies on these lines: {1,7}, {2,31541}, {21,7595}, {63,487}, {75,489}, {320,490}, {894,11293}, {1267,12322}, {3662,11294}, {4000,6459}, {4197,31558}, {4644,6460}, {5249,31591}, {5273,31595}, {7411,31545}, {8233,11291}, {10391,31589}, {10861,31543}, {10883,31556}, {11020,31572}, {11520,31560}, {11888,31579}, {11889,31581}, {11890,31593}, {17616,31587}

X(31550) = reflection of X(i) in X(j) for these (i,j): (31548, 6213), (31552, 482), (31562, 31541)
X(31550) = anticomplement of X(31562)
X(31550) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 20, 31549), (4292, 10444, 31549), (10884, 18650, 31549)

X(31551) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO 2nd OUTER-SODDY

Barycentrics a^4+2*(b+c)*a^3-4*b*c*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2-2*S*(a+b-c)*(a-b+c) : :
X(31551) = 3*X(2)-4*X(31590) = X(8)-4*X(31575) = 5*X(3616)-4*X(31546) = 3*X(9778)-4*X(31544) = 9*X(9779)-8*X(31555) = 7*X(9780)-8*X(31557)

The reciprocal orthologic center of these triangles is X(1)

X(31551) lies on these lines: {1,7}, {2,6212}, {8,637}, {145,31559}, {169,8231}, {329,31547}, {487,11415}, {497,31588}, {3068,31584}, {3434,31586}, {3616,31546}, {5905,12222}, {9776,31540}, {9778,31544}, {9779,31555}, {9780,31557}, {9791,31576}, {9793,31578}, {9795,31580}, {10580,31571}, {11891,31592}, {18228,31594}

X(31551) = reflection of X(i) in X(j) for these (i,j): (20, 31563), (145, 31559), (6212, 31590), (31547, 31561), (31549, 481), (31552, 17170)
X(31551) = anticomplementary conjugate of X(31552)
X(31551) = anticomplement of X(6212)
X(31551) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 962, 31552), (20, 4329, 31552), (4295, 10446, 31552)

X(31552) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO 2nd INNER-SODDY

Barycentrics a^4+2*(b+c)*a^3-4*b*c*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2+2*S*(a+b-c)*(a-b+c) : :
X(31552) = 3*X(2)-4*X(31591) = 5*X(3616)-4*X(8225) = 3*X(9778)-4*X(31545) = 9*X(9779)-8*X(31556) = 7*X(9780)-8*X(31558)

The reciprocal orthologic center of these triangles is X(1)

X(31552) lies on these lines: {1,7}, {2,6213}, {8,638}, {145,31560}, {169,30413}, {329,31548}, {488,11415}, {497,31589}, {3069,31585}, {3434,31587}, {3616,8225}, {5905,12221}, {9776,31541}, {9778,31545}, {9779,31556}, {9780,31558}, {9791,31577}, {9793,31579}, {9795,31581}, {10580,31572}, {11891,31593}, {18228,31595}

X(31552) = reflection of X(i) in X(j) for these (i,j): (20, 31564), (145, 31560), (6213, 31591), (31548, 31562), (31550, 482), (31551, 17170)
X(31552) = anticomplementary conjugate of X(31551)
X(31552) = anticomplement of X(6213)
X(31552) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 962, 31551), (20, 4329, 31551), (30426, 31554, 31550)

X(31553) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 2nd OUTER-SODDY

Barycentrics 8*S^3*a-(a+b+c)^2*(a^5+4*(b+c)*a^4+2*(b-c)^2*a^3-4*(b^3+c^3)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)*b*c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31553) lies on these lines: {1,7}, {637,5691}, {1764,6212}, {10434,31544}, {10473,31588}, {10478,31590}, {10856,31540}, {10862,31542}, {10882,31546}, {10886,31555}, {10887,31557}, {10888,31561}, {10891,31575}, {10892,31576}, {11021,31571}, {11521,31559}, {11679,31547}, {11894,31578}, {11895,31580}, {11896,31592}, {17617,31586}, {18229,31594}

X(31553) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10442, 31554), (10444, 12545, 31554), (31549, 31551, 30425)

X(31554) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 2nd INNER-SODDY

Barycentrics -8*S^3*a-(a+b+c)^2*(a^5+4*(b+c)*a^4+2*(b-c)^2*a^3-4*(b^3+c^3)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)*b*c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31554) lies on these lines: {1,7}, {638,5691}, {1764,6213}, {8225,10882}, {10434,31545}, {10473,31589}, {10478,31591}, {10856,31541}, {10862,31543}, {10886,31556}, {10887,31558}, {10888,31562}, {10892,31577}, {11021,31572}, {11521,31560}, {11679,31548}, {11894,31579}, {11895,31581}, {11896,31593}, {17617,31587}, {18229,31595}

X(31554) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10442, 31553), (10444, 12545, 31553), (31550, 31552, 30426)

X(31555) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 2nd OUTER-SODDY

Barycentrics 2*((b^2+c^2)*a^2-(b^2-c^2)^2)*S+(a+b+c)*((b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3) : :
X(31555) = 3*X(1699)+X(6212) = 3*X(3817)-X(31590) = 9*X(7988)-X(31573) = 5*X(8227)-X(31563) = 9*X(9779)-X(31551) = 5*X(11522)-X(31559)

The reciprocal orthologic center of these triangles is X(1)

X(31555) lies on these lines: {1,30307}, {2,31544}, {4,31546}, {5,516}, {11,481}, {12,31567}, {226,31571}, {496,31569}, {1699,6212}, {2886,31594}, {3817,31590}, {7678,31565}, {7741,30306}, {7988,31573}, {8085,31578}, {8086,31580}, {8226,31561}, {8227,31563}, {8228,31575}, {8229,31576}, {8379,31592}, {8727,31540}, {9779,31551}, {10863,31542}, {10883,31549}, {10886,31553}, {11522,31559}, {11680,31547}, {17605,31588}, {17618,31586}

X(31555) = midpoint of X(4) and X(31546)
X(31555) = reflection of X(31557) in X(5)
X(31555) = complement of X(31544)

X(31556) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO 2nd INNER-SODDY

Barycentrics -2*((b^2+c^2)*a^2-(b^2-c^2)^2)*S+(a+b+c)*((b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3) : :
X(31556) = 3*X(1699)+X(6213) = 3*X(3817)-X(31591) = 9*X(7988)-X(31574) = 5*X(8227)-X(31564) = 9*X(9779)-X(31552) = 5*X(11522)-X(31560)

The reciprocal orthologic center of these triangles is X(1)

X(31556) lies on these lines: {1,30306}, {2,31545}, {4,8225}, {5,516}, {11,482}, {12,31568}, {226,31572}, {496,31570}, {1699,6213}, {2886,31595}, {3817,31591}, {5511,7596}, {7678,31566}, {7741,30307}, {7988,31574}, {8085,31579}, {8086,31581}, {8226,31562}, {8227,31564}, {8229,31577}, {8379,31593}, {8727,31541}, {9779,31552}, {10863,31543}, {10883,31550}, {10886,31554}, {11522,31560}, {11680,31548}, {17605,31589}, {17618,31587}

X(31556) = midpoint of X(4) and X(8225)
X(31556) = reflection of X(31558) in X(5)
X(31556) = complement of X(31545)

X(31557) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 2nd OUTER-SODDY

Barycentrics (a+b+c)*((b^2+c^2)*a^2-(b^2-c^2)^2)-2*S*((b^2+4*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(31557) = 5*X(1698)-X(6212) = 5*X(1698)+X(31575) = 3*X(3679)+X(31559) = 3*X(5587)+X(31563) = 7*X(7989)+X(31573) = 7*X(9780)+X(31551)

The reciprocal orthologic center of these triangles is X(1)

X(31557) lies on these lines: {1,30314}, {2,31546}, {4,31544}, {5,516}, {10,639}, {11,31567}, {12,481}, {442,31561}, {495,31569}, {640,3836}, {1210,31571}, {1329,31594}, {1698,6212}, {3679,31559}, {4197,31549}, {5051,31576}, {5587,31563}, {7951,30313}, {7989,31573}, {8087,31578}, {8088,31580}, {8382,31592}, {8582,31542}, {8728,31540}, {9780,31551}, {10887,31553}, {11681,31547}, {17606,31588}, {17619,31586}

X(31557) = midpoint of X(i) and X(j) for these {i,j}: {4, 31544}, {10, 31590}, {6212, 31575}
X(31557) = reflection of X(31555) in X(5)
X(31557) = complement of X(31546)
X(31557) = {X(5), X(3826)}-harmonic conjugate of X(31558)

X(31558) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO 2nd INNER-SODDY

Barycentrics (a+b+c)*((b^2+c^2)*a^2-(b^2-c^2)^2)+2*S*((b^2+4*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(31558) = 5*X(1698)-X(6213) = 3*X(3679)+X(31560) = 3*X(5587)+X(31564) = 7*X(7989)+X(31574) = 7*X(9780)+X(31552)

The reciprocal orthologic center of these triangles is X(1)

X(31558) lies on these lines: {1,30313}, {2,8225}, {4,31545}, {5,516}, {10,640}, {11,31568}, {12,482}, {442,31562}, {495,31570}, {639,3836}, {1210,31572}, {1329,31595}, {1698,6213}, {3679,31560}, {4197,31550}, {5051,31577}, {5587,31564}, {7679,31566}, {7951,30314}, {7989,31574}, {8087,31579}, {8088,31581}, {8382,31593}, {8582,31543}, {8728,31541}, {9780,31552}, {10887,31554}, {11681,31548}, {17606,31589}, {17619,31587}

X(31558) = midpoint of X(i) and X(j) for these {i,j}: {4, 31545}, {10, 31591}
X(31558) = reflection of X(31556) in X(5)
X(31558) = complement of X(8225)
X(31558) = {X(5), X(3826)}-harmonic conjugate of X(31557)

X(31559) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 2nd OUTER-SODDY

Barycentrics a*((a+b+c)*(a^3-3*(b+c)*a^2-(b^2-6*b*c+c^2)*a+3*(b^2-c^2)*(b-c))-2*S*(a^2-4*(b+c)*a+3*b^2-2*b*c+3*c^2)) : :
X(31559) = 3*X(1)-2*X(31546) = X(3633)+2*X(31575) = 3*X(3679)-4*X(31557) = 3*X(6212)-4*X(31546) = 5*X(11522)-4*X(31555)

The reciprocal orthologic center of these triangles is X(1)

X(31559) lies on these lines: {1,371}, {8,31590}, {145,31551}, {481,3340}, {516,944}, {517,31563}, {2098,31588}, {2809,3640}, {3633,11532}, {3679,31557}, {7962,31567}, {7980,30323}, {7991,31544}, {10912,31586}, {11518,31540}, {11519,31542}, {11520,31549}, {11521,31553}, {11522,31555}, {11523,31561}, {11526,31565}, {11529,31569}, {11531,31573}, {11533,31576}, {11534,31578}, {11535,31592}, {11682,31547}, {11899,31580}, {15829,31594}, {25415,30319}

X(31559) = midpoint of X(i) and X(j) for these {i,j}: {145, 31551}, {11531, 31573}
X(31559) = reflection of X(i) in X(j) for these (i,j): (8, 31590), (6212, 1), (7991, 31544)
X(31559) = {X(3243), X(7982)}-harmonic conjugate of X(31560)

X(31560) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 2nd INNER-SODDY

Barycentrics a*((a+b+c)*(a^3-3*(b+c)*a^2-(b^2-6*b*c+c^2)*a+3*(b^2-c^2)*(b-c))+2*S*(a^2-4*(b+c)*a+3*b^2-2*b*c+3*c^2)) : :
X(31560) = 3*X(1)-2*X(8225) = 3*X(3679)-4*X(31558) = 3*X(6213)-4*X(8225) = 5*X(11522)-4*X(31556)

The reciprocal orthologic center of these triangles is X(1)

X(31560) lies on these lines: {1,372}, {8,31591}, {145,31552}, {482,3340}, {516,944}, {517,31564}, {2098,31589}, {2809,3641}, {3679,31558}, {7962,31568}, {7981,30323}, {7991,31545}, {10912,31587}, {11518,31541}, {11519,31543}, {11520,31550}, {11521,31554}, {11522,31556}, {11523,31562}, {11526,31566}, {11529,31570}, {11531,31574}, {11533,31577}, {11534,31579}, {11535,31593}, {11682,31548}, {11899,31581}, {15829,31595}, {25415,30320}

X(31560) = midpoint of X(i) and X(j) for these {i,j}: {145, 31552}, {11531, 31574}
X(31560) = reflection of X(i) in X(j) for these (i,j): (8, 31591), (6213, 1), (7991, 31545)
X(31560) = {X(3243), X(7982)}-harmonic conjugate of X(31559)

X(31561) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 2nd OUTER-SODDY

Barycentrics -2*(-a+b+c)*S*a+(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(31561) lies on the cubic K122 and these lines: {1,1587}, {2,31540}, {4,9}, {20,30413}, {37,3070}, {44,3071}, {45,23251}, {57,8957}, {226,481}, {329,31547}, {344,12323}, {372,5405}, {405,31546}, {442,31557}, {482,8953}, {515,30557}, {637,4416}, {638,3912}, {946,30556}, {950,31567}, {1210,6203}, {1335,9370}, {1449,7581}, {1490,31563}, {1588,1743}, {1750,31573}, {1864,31588}, {2045,5243}, {2046,5242}, {2048,5745}, {3091,30412}, {3247,23267}, {3487,31569}, {3731,23249}, {3973,23259}, {4199,31576}, {4292,6204}, {4357,7389}, {5728,31571}, {5927,31542}, {6351,31412}, {6352,6460}, {7078,31528}, {7133,31472}, {7375,17306}, {7388,17353}, {7580,31544}, {7582,16670}, {7593,31592}, {8079,31578}, {8080,31580}, {8226,31555}, {8232,31565}, {8233,31575}, {8965,31533}, {9612,30324}, {10888,31553}, {11523,31559}, {12222,17316}, {16676,23269}, {16885,23261}

X(31561) = midpoint of X(31547) and X(31551)
X(31561) = reflection of X(i) in X(j) for these (i,j): (481, 31590), (6212, 31594), (31549, 31540), (31586, 31542)
X(31561) = anticomplement of X(31540)
X(31561) = complement of X(31549)
X(31561) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31549, 31540), (4, 9, 31562), (10445, 12572, 31562)

X(31562) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO 2nd INNER-SODDY

Barycentrics 2*(-a+b+c)*S*a+(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(31562) lies on the cubic K122 and these lines: {1,1588}, {2,31541}, {4,9}, {20,30412}, {37,3071}, {44,3070}, {45,23261}, {226,482}, {329,31548}, {344,12322}, {371,5393}, {405,8225}, {442,31558}, {515,30556}, {637,3912}, {638,4416}, {946,30557}, {950,31568}, {1124,9370}, {1210,6204}, {1449,7582}, {1490,31564}, {1587,1743}, {1750,31574}, {1864,31589}, {2045,5242}, {2046,5243}, {2048,3452}, {3091,30413}, {3247,23273}, {3487,31570}, {3731,23259}, {3973,23249}, {4199,31577}, {4292,6203}, {4357,7388}, {5728,31572}, {5927,31543}, {6351,6459}, {7078,31529}, {7376,17306}, {7389,17353}, {7580,31545}, {7581,16670}, {7593,31593}, {8079,31579}, {8080,31581}, {8226,31556}, {8232,31566}, {8957,9581}, {8978,31538}, {9612,30325}, {10888,31554}, {11523,31560}, {12221,17316}, {16676,23275}, {16885,23251}

X(31562) = midpoint of X(31548) and X(31552)
X(31562) = reflection of X(i) in X(j) for these (i,j): (482, 31591), (6213, 31595), (31550, 31541), (31587, 31543)
X(31562) = anticomplement of X(31541)
X(31562) = complement of X(31550)
X(31562) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31550, 31541), (4, 9, 31561), (10445, 12572, 31561)

X(31563) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 2nd OUTER-SODDY

Barycentrics a*((a+b+c)*(a^5-3*(b+c)*a^4+2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)^2*(b+c))-8*S^3) : :
X(31563) = 3*X(3576)-2*X(31546) = 3*X(5587)-4*X(31557) = 5*X(8227)-4*X(31555)

The reciprocal orthologic center of these triangles is X(1)

X(31563) lies on these lines: {1,7}, {3,6212}, {4,31590}, {40,3640}, {56,31588}, {78,31547}, {517,31559}, {936,31594}, {971,30557}, {1490,31561}, {3070,31584}, {3333,31571}, {3576,31546}, {5587,31557}, {7580,13388}, {7590,31592}, {8081,31578}, {8082,31580}, {8227,31555}, {8234,31575}, {8235,31576}, {8726,31540}, {10167,13389}, {10864,31542}, {12114,31586}, {12257,18446}

X(31563) = midpoint of X(i) and X(j) for these {i,j}: {1, 31573}, {20, 31551}
X(31563) = reflection of X(i) in X(j) for these (i,j): (4, 31590), (40, 31544), (6212, 3)
X(31563) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 31564), (20, 30265, 31564), (990, 4297, 31564)

X(31564) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 2nd INNER-SODDY

Barycentrics a*((a+b+c)*(a^5-3*(b+c)*a^4+2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)^2*(b+c))+8*S^3) : :
X(31564) = 3*X(3576)-2*X(8225) = 3*X(5587)-4*X(31558) = 5*X(8227)-4*X(31556)

The reciprocal orthologic center of these triangles is X(1)

X(31564) lies on these lines: {1,7}, {3,6213}, {4,31591}, {40,3641}, {56,31589}, {78,31548}, {517,31560}, {936,31595}, {971,30556}, {1490,31562}, {3071,31585}, {3333,31572}, {3576,8225}, {5587,31558}, {7580,13389}, {7590,31593}, {8081,31579}, {8082,31581}, {8227,31556}, {8235,31577}, {8726,31541}, {10167,13388}, {10864,31543}, {12114,31587}, {12256,18446}

X(31564) = midpoint of X(i) and X(j) for these {i,j}: {1, 31574}, {20, 31552}
X(31564) = reflection of X(i) in X(j) for these (i,j): (4, 31591), (40, 31545), (6213, 3)
X(31564) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 31563), (20, 30265, 31563), (990, 4297, 31563)

X(31565) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 2nd OUTER-SODDY

Barycentrics (2*(a^2-2*(b+c)*a+(b-c)^2)*S+(-a+b+c)^2*(3*a^2+b^2-2*b*c+c^2))*(a-b+c)*(a+b-c) : :
X(31565) = 5*X(11025)-4*X(31571) = 5*X(18230)-4*X(31594)

The reciprocal orthologic center of these triangles is X(1)

X(31565) lies on these lines: {1,7}, {9,31547}, {673,13390}, {1445,6212}, {5572,31588}, {7676,31544}, {7677,31546}, {7678,31555}, {8232,31561}, {8237,31575}, {8238,31576}, {8387,31578}, {8388,31580}, {8389,31592}, {8732,31540}, {10865,31542}, {11025,31571}, {11526,31559}, {17620,31586}, {18230,31594}, {21617,31590}

X(31565) = reflection of X(i) in X(j) for these (i,j): (7, 481), (31547, 9), (31588, 5572)
X(31565) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 17802, 11038), (481, 31567, 10481), (10481, 31567, 176)

X(31566) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 2nd INNER-SODDY

Barycentrics (-2*(a^2-2*(b+c)*a+(b-c)^2)*S+(-a+b+c)^2*(3*a^2+b^2-2*b*c+c^2))*(a-b+c)*(a+b-c) : :
X(31566) = 5*X(11025)-4*X(31572) = 5*X(18230)-4*X(31595)

The reciprocal orthologic center of these triangles is X(1)

X(31566) lies on these lines: {1,7}, {9,31548}, {673,1659}, {1445,6213}, {5572,31589}, {7676,31545}, {7677,8225}, {7678,31556}, {7679,31558}, {8232,31562}, {8238,31577}, {8387,31579}, {8389,31593}, {8732,31541}, {10865,31543}, {11025,31572}, {17620,31587}, {18230,31595}, {21617,31591}

X(31566) = reflection of X(i) in X(j) for these (i,j): (7, 482), (31548, 9), (31589, 5572)
X(31566) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 390, 31565), (482, 31568, 10481), (10481, 31568, 175)

X(31567) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 2nd OUTER-SODDY

Barycentrics -2*S*a+(-a+b+c)*(3*a^2+(b-c)^2) : :
X(31567) = 3*X(1)-X(30425) = 3*X(1)-2*X(31569) = 3*X(481)-2*X(30425) = 3*X(481)-4*X(31569)

The reciprocal orthologic center of these triangles is X(1)

X(31567) lies on these lines: {1,7}, {8,31594}, {11,31557}, {12,31555}, {55,5393}, {56,31544}, {65,31571}, {145,31547}, {497,5405}, {950,31561}, {1659,10389}, {1697,6212}, {3021,22106}, {3057,31588}, {3084,20075}, {3303,31533}, {3640,5698}, {5853,30557}

X(31567) = midpoint of X(i) and X(j) for these {i,j}: {145, 31547}, {3057, 31588}
X(31567) = reflection of X(i) in X(j) for these (i,j): (8, 31594), (65, 31571), (481, 1), (30425, 31569)
X(31567) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 31568), (390, 30334, 1), (30331, 30432, 31538)

X(31568) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO 2nd INNER-SODDY

Barycentrics 2*S*a+(-a+b+c)*(3*a^2+(b-c)^2) : :
X(31568) = 3*X(1)-X(30426) = 3*X(1)-2*X(31570) = 3*X(482)-2*X(30426) = 3*X(482)-4*X(31570)

The reciprocal orthologic center of these triangles is X(1)

X(31568) lies on these lines: {1,7}, {8,31595}, {11,31558}, {12,31556}, {55,5405}, {56,31545}, {65,31572}, {145,31548}, {497,5393}, {950,31562}, {1659,9580}, {1697,6213}, {3021,22107}, {3057,31589}, {3083,20075}, {3303,31532}, {3601,31541}, {3641,5698}, {5853,30556}, {6284,31533}, {7962,31560}, {8240,31577}, {8241,31579}, {8242,31581}, {10389,13390}, {10866,31543}, {11924,31593}, {12053,31591}, {17622,31587}

X(31568) = midpoint of X(i) and X(j) for these {i,j}: {145, 31548}, {3057, 31589}
X(31568) = reflection of X(i) in X(j) for these (i,j): (8, 31595), (65, 31572), (482, 1), (30426, 31570)
X(31568) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 31567), (390, 30333, 1), (30331, 30431, 31539)

X(31569) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 2nd OUTER-SODDY

Barycentrics -4*S*a+3*(b+c)*a^2-2*(b-c)^2*a-(b-c)*(b^2-c^2) : :
X(31569) = 3*X(1)+X(30425) = 3*X(1)-X(31567) = 3*X(481)-X(30425) = 3*X(481)+X(31567) = 5*X(3616)-X(31547) = 5*X(17609)-X(31588)

The reciprocal orthologic center of these triangles is X(1)

X(31569) lies on these lines: {1,7}, {10,3640}, {495,31557}, {496,31555}, {950,10911}, {999,31546}, {1125,30556}, {3295,31544}, {3333,6212}, {3487,31561}, {3616,31547}, {5045,31571}, {5850,30557}, {6738,31532}, {8092,31592}, {8351,31580}, {8945,24175}, {11019,13390}, {11035,31542}, {11042,31575}, {11043,31576}, {11044,31578}, {11529,31559}, {13388,13405}, {17609,31588}, {17624,31586}, {21620,31590}

X(31569) = midpoint of X(i) and X(j) for these {i,j}: {1, 481}, {30425, 31567}
X(31569) = reflection of X(i) in X(j) for these (i,j): (31571, 5045), (31594, 1125)
X(31569) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (481, 31567, 30425), (4353, 12577, 31570), (11038, 17802, 1)

X(31570) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 2nd INNER-SODDY

Barycentrics 4*S*a+3*(b+c)*a^2-2*(b-c)^2*a-(b-c)*(b^2-c^2) : :
X(31570) = 3*X(1)+X(30426) = 3*X(1)-X(31568) = 3*X(482)-X(30426) = 3*X(482)+X(31568) = 5*X(3616)-X(31548) = 5*X(17609)-X(31589)

The reciprocal orthologic center of these triangles is X(1)

X(31570) lies on these lines: {1,7}, {10,3641}, {495,31558}, {496,31556}, {942,31541}, {950,10910}, {1125,30557}, {1659,11019}, {3295,31545}, {3333,6213}, {3487,31562}, {3616,31548}, {5850,30556}, {6738,31533}, {8092,31593}, {8351,31581}, {8941,24175}, {8953,31594}, {11035,31543}, {11043,31577}, {11044,31579}, {11529,31560}, {13389,13405}, {17609,31589}, {17624,31587}, {21620,31591}

X(31570) = midpoint of X(i) and X(j) for these {i,j}: {1, 482}, {30426, 31568}
X(31570) = reflection of X(31595) in X(1125)
X(31570) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (482, 31568, 30426), (4353, 12577, 31569), (21169, 30333, 4312)

X(31571) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO 2nd OUTER-SODDY

Barycentrics a*(2*((b+c)*a^2+2*b*c*a-(b-c)*(b^2-c^2))*S+(a+b+c)*((b+c)*a^3-3*(b^2+c^2)*a^2+3*(b-c)*(b^2-c^2)*a-(b^2+c^2)*(b-c)^2)) : :
X(31571) = 3*X(354)-X(481) = 3*X(354)+X(31588) = 3*X(3873)+X(31547) = 5*X(11025)-X(31565) = 5*X(18398)-X(30425)

The reciprocal orthologic center of these triangles is X(1)

X(31571) lies on these lines: {1,371}, {57,31544}, {65,31567}, {226,31555}, {354,481}, {482,30376}, {516,942}, {518,31594}, {1210,31557}, {3333,31563}, {3873,31547}, {5045,31569}, {5728,31561}, {8083,31592}, {10580,31551}, {10980,31573}, {11018,31540}, {11019,31542}, {11020,31549}, {11021,31553}, {11025,31565}, {11030,31575}, {11031,31576}, {11033,31580}, {17626,31586}, {18398,30346}

X(31571) = midpoint of X(i) and X(j) for these {i,j}: {65, 31567}, {481, 31588}
X(31571) = reflection of X(i) in X(j) for these (i,j): (31542, 31590), (31569, 5045)
X(31571) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 31588, 481), (942, 5572, 31572)

X(31572) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO 2nd INNER-SODDY

Barycentrics a*(-2*((b+c)*a^2+2*b*c*a-(b-c)*(b^2-c^2))*S+(a+b+c)*((b+c)*a^3-3*(b^2+c^2)*a^2+3*(b-c)*(b^2-c^2)*a-(b^2+c^2)*(b-c)^2)) : :
X(31572) = 3*X(354)-X(482) = 3*X(354)+X(31589) = 3*X(3873)+X(31548) = 5*X(18398)-X(30426)

The reciprocal orthologic center of these triangles is X(1)

X(31572) lies on these lines: {1,372}, {57,31545}, {65,31568}, {226,31556}, {354,482}, {481,30375}, {516,942}, {518,31595}, {1210,31558}, {3333,31564}, {3873,31548}, {5728,31562}, {10580,31552}, {10980,31574}, {11018,31541}, {11019,31543}, {11020,31550}, {11021,31554}, {11031,31577}, {11032,31579}, {17626,31587}, {18398,30347}

X(31572) = midpoint of X(i) and X(j) for these {i,j}: {65, 31568}, {482, 31589}
X(31572) = reflection of X(31543) in X(31591)
X(31572) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 31589, 482), (942, 5572, 31571)

X(31573) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO 2nd OUTER-SODDY

Barycentrics a*(-8*S^3+(a+b+c)^2*(a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b-c)*(b^2-c^2)*a+(b^2+6*b*c+c^2)*(b-c)^2)) : :
X(31573) = 3*X(165)-2*X(6212) = 3*X(165)-4*X(31544) = 3*X(1699)-4*X(31590) = 5*X(7987)-4*X(31546) = 9*X(7988)-8*X(31555) = 7*X(7989)-8*X(31557)

The reciprocal orthologic center of these triangles is X(1)

X(31573) lies on these lines: {1,7}, {57,31588}, {165,6212}, {200,31547}, {1699,31590}, {1709,31586}, {1750,31561}, {3062,31542}, {5918,13389}, {7987,31546}, {7988,31555}, {7989,31557}, {8089,31578}, {8090,31580}, {8244,31575}, {8245,31576}, {8423,31592}, {8580,31594}, {10857,31540}, {10980,31571}, {11495,30556}, {11531,31559}, {15726,30557}

X(31573) = reflection of X(i) in X(j) for these (i,j): (1, 31563), (6212, 31544), (11531, 31559)
X(31573) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 31574), (20, 1721, 31574), (1742, 12565, 31574)

X(31574) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO 2nd INNER-SODDY

Barycentrics a*(8*S^3+(a+b+c)^2*(a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b-c)*(b^2-c^2)*a+(b^2+6*b*c+c^2)*(b-c)^2)) : :
X(31574) = 3*X(165)-2*X(6213) = 3*X(165)-4*X(31545) = 3*X(1699)-4*X(31591) = 5*X(7987)-4*X(8225) = 9*X(7988)-8*X(31556) = 7*X(7989)-8*X(31558)

The reciprocal orthologic center of these triangles is X(1)

X(31574) lies on these lines: {1,7}, {57,31589}, {165,6213}, {200,31548}, {1699,31591}, {1709,31587}, {1750,31562}, {3062,31543}, {5918,13388}, {7987,8225}, {7988,31556}, {7989,31558}, {8089,31579}, {8090,31581}, {8245,31577}, {8423,31593}, {8580,31595}, {10857,31541}, {10980,31572}, {11495,30557}, {11531,31560}, {15726,30556}

X(31574) = reflection of X(i) in X(j) for these (i,j): (1, 31564), (6213, 31545), (11531, 31560)
X(31574) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 31573), (20, 1721, 31573), (1742, 12565, 31573)

X(31575) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 2nd OUTER-SODDY

Barycentrics -2*(a^3+2*b*c*a-(b-c)*(b^2-c^2))*S+(a+b+c)*(a^4+(b+c)*a^3-2*b*c*a^2-(b-c)*(b^2-c^2)*a-(b^2-c^2)^2) : :
X(31575) = X(8)+3*X(31551) = 4*X(1125)-3*X(31546) = 2*X(1125)-3*X(31590) = 5*X(1698)-3*X(6212) = 5*X(1698)-6*X(31557) = X(3633)-3*X(31559)

The reciprocal orthologic center of these triangles is X(1)

X(31575) lies on these lines: {3,142}, {4,11211}, {8,637}, {79,7133}, {481,8243}, {1698,6212}, {1836,5393}, {3633,11532}, {8228,31555}, {8233,31561}, {8234,31563}, {8237,31565}, {8239,31567}, {8244,31573}, {8246,31576}, {8247,31578}, {8248,31580}, {10885,31549}, {10891,31553}, {11030,31571}, {11042,31569}, {11687,31547}, {11996,31592}, {12490,18480}, {17610,31588}, {17627,31586}, {18234,31594}

X(31575) = reflection of X(i) in X(j) for these (i,j): (6212, 31557), (31546, 31590)
X(31575) = {X(12578), X(12610)}-harmonic conjugate of X(8225)

X(31576) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 2nd OUTER-SODDY

Barycentrics a*(-2*S*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c)+(b+c)*a^5+(b^2+c^2)*a^4-(b^3+c^3)*a^3-(b^2+c^2)*(b^2+b*c+c^2)*a^2-(b+c)*(b^2+c^2)*b*c*a+(b^2-c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31576) lies on these lines: {1,30361}, {21,31546}, {481,1284}, {516,9840}, {846,6212}, {4199,31561}, {4220,31544}, {4425,31590}, {5051,31557}, {8229,31555}, {8235,31563}, {8238,31565}, {8240,31567}, {8245,31573}, {8246,31575}, {8249,31578}, {8250,31580}, {8425,31592}, {8731,31540}, {9791,31551}, {10868,31542}, {10892,31553}, {11031,31571}, {11043,31569}, {11533,31559}, {11688,31547}, {17611,31588}, {17628,31586}, {18235,31594}, {30360,30362}

X(31577) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO 2nd INNER-SODDY

Barycentrics a*(2*S*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c)+(b+c)*a^5+(b^2+c^2)*a^4-(b^3+c^3)*a^3-(b^2+c^2)*(b^2+b*c+c^2)*a^2-(b+c)*(b^2+c^2)*b*c*a+(b^2-c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31577) lies on these lines: {1,30360}, {21,7595}, {482,1284}, {516,9840}, {846,6213}, {4199,31562}, {4220,31545}, {4425,31591}, {5051,31558}, {8229,31556}, {8235,31564}, {8238,31566}, {8240,31568}, {8245,31574}, {8249,31579}, {8250,31581}, {8425,31593}, {8731,31541}, {9791,31552}, {10868,31543}, {10892,31554}, {11031,31572}, {11043,31570}, {11533,31560}, {11688,31548}, {17611,31589}, {17628,31587}, {18235,31595}, {30361,30362}

X(31578) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 2nd OUTER-SODDY

Barycentrics a*(2*(-a*(-a+b+c)+2*S)*sin(A/2)*b*c-(2*(a+b-c)*S-(a-b+c)*(a^2+b^2-c^2))*sin(B/2)*c-(2*(a-b+c)*S-(a+b-c)*(a^2+c^2-b^2))*sin(C/2)*b-S*(2*S+a^2-2*(b+c)*a+(b-c)^2)) : :

The reciprocal orthologic center of these triangles is X(1)

X(31578) lies on these lines: {1,30369}, {177,31592}, {188,31594}, {481,2089}, {516,8091}, {6212,8078}, {8075,31544}, {8077,31546}, {8079,31561}, {8081,31563}, {8085,31555}, {8087,31557}, {8089,31573}, {8241,31567}, {8247,31575}, {8249,31576}, {8387,31565}, {8733,31540}, {9793,31551}, {10503,31588}, {11032,31571}, {11044,31569}, {11534,31559}, {11690,31547}, {11858,31542}, {11888,31549}, {11894,31553}, {17629,31586}, {21622,31590}, {30368,30370}

X(31579) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 2nd INNER-SODDY

Barycentrics a*(2*(-a*(-a+b+c)-2*S)*sin(A/2)*b*c-(-2*(a+b-c)*S-(a-b+c)*(a^2+b^2-c^2))*sin(B/2)*c-(-2*(a-b+c)*S-(a+b-c)*(a^2+c^2-b^2))*sin(C/2)*b+S*(-2*S+a^2-2*(b+c)*a+(b-c)^2)) : :

The reciprocal orthologic center of these triangles is X(1)

X(31579) lies on these lines: {1,30368}, {177,31593}, {188,31595}, {482,2089}, {516,8091}, {6213,8078}, {8075,31545}, {8077,8225}, {8079,31562}, {8081,31564}, {8085,31556}, {8087,31558}, {8089,31574}, {8241,31568}, {8249,31577}, {8387,31566}, {8733,31541}, {9793,31552}, {10503,31589}, {11032,31572}, {11044,31570}, {11534,31560}, {11690,31548}, {11858,31543}, {11888,31550}, {11894,31554}, {17629,31587}, {21622,31591}, {30369,30370}

X(31580) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 2nd OUTER-SODDY

Barycentrics (a+b+c)*(a^2-2*a*b-2*c*a+(b-c)^2+2*S)*sin(A/2)+a^3+(b+c)*a^2-((b-c)^2+2*S)*a-(b^2-c^2)*(b-c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31580) lies on these lines: {1,30369}, {174,481}, {258,6212}, {516,8092}, {1374,30407}, {7028,31594}, {7588,31546}, {8076,31544}, {8080,31561}, {8082,31563}, {8086,31555}, {8088,31557}, {8090,31573}, {8125,31547}, {8242,31567}, {8248,31575}, {8250,31576}, {8351,31569}, {8388,31565}, {8734,31540}, {9795,31551}, {10481,31593}, {10501,31588}, {11033,31571}, {11859,31542}, {11889,31549}, {11895,31553}, {11899,31559}, {17630,31586}, {21623,31590}, {30418,30420}

X(31580) = {X(174), X(481)}-harmonic conjugate of X(31592)

X(31581) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 2nd INNER-SODDY

Barycentrics (a+b+c)*(a^2-2*a*b-2*c*a+(b-c)^2-2*S)*sin(A/2)+a^3+(b+c)*a^2-((b-c)^2-2*S)*a-(b^2-c^2)*(b-c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31581) lies on these lines: {1,30368}, {174,482}, {258,6213}, {516,8092}, {1373,30406}, {7028,31595}, {7588,8225}, {8076,31545}, {8080,31562}, {8082,31564}, {8086,31556}, {8088,31558}, {8090,31574}, {8125,31548}, {8242,31568}, {8250,31577}, {8351,31570}, {8734,31541}, {9795,31552}, {10481,31592}, {10501,31589}, {11033,31572}, {11859,31543}, {11889,31550}, {11895,31554}, {11899,31560}, {17630,31587}, {21623,31591}, {30419,30420}

X(31581) = {X(174), X(482)}-harmonic conjugate of X(31593)

X(31582) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-SODDY TO 3rd TRI-SQUARES

Barycentrics -2*(a^2+2*(b+c)*a+b^2+c^2)*S+(a^3-(b+c)*a^2-3*(b^2+c^2)*a-(b+c)*(b^2+c^2))*a : :

The reciprocal orthologic center of these triangles is X(31584)

X(31582) lies on these lines: {1,488}, {10,5391}, {56,482}, {175,5261}, {226,481}, {527,8983}, {641,3666}, {988,1125}, {1086,31535}, {3946,13971}, {5252,12949}, {6462,17257}, {18988,31533}

X(31582) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 8243, 482), (988, 17321, 31583), (1125, 3663, 31583)

X(31583) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-SODDY TO 4th TRI-SQUARES

Barycentrics 2*(a^2+2*(b+c)*a+b^2+c^2)*S+(a^3-(b+c)*a^2-3*(b^2+c^2)*a-(b+c)*(b^2+c^2))*a : :

The reciprocal orthologic center of these triangles is X(31585)

X(31583) lies on these lines: {1,487}, {10,1267}, {37,31535}, {56,481}, {176,5261}, {226,482}, {527,13971}, {642,3666}, {988,1125}, {1086,31534}, {3946,8983}, {5252,12948}, {6463,17257}, {8243,11375}, {18989,31532}

X(31583) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (988, 17321, 31582), (1125, 3663, 31582)

X(31584) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 2nd OUTER-SODDY

Barycentrics 2*(3*a^4-(b+c)*a^3+4*(b^2-3*b*c+c^2)*a^2-(b+c)^3*a-(b^2-c^2)^2)*S+(-a+b+c)*(a+b+c)*(a^4-3*(b+c)*a^3-2*(b-c)^2*a^2-(b^2-c^2)*(b-c)*a-(3*b^2-2*b*c+3*c^2)*(b-c)^2) : :

The reciprocal orthologic center of these triangles is X(31582)

X(31584) lies on these lines: {6,31590}, {516,1151}, {590,6212}, {3068,31551}, {3070,31563}, {17278,31585}

X(31585) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 2nd INNER-SODDY

Barycentrics -2*(3*a^4-(b+c)*a^3+4*(b^2-3*b*c+c^2)*a^2-(b+c)^3*a-(b^2-c^2)^2)*S+(-a+b+c)*(a+b+c)*(a^4-3*(b+c)*a^3-2*(b-c)^2*a^2-(b^2-c^2)*(b-c)*a-(3*b^2-2*b*c+3*c^2)*(b-c)^2) : :

The reciprocal orthologic center of these triangles is X(31583)

X(31585) lies on these lines: {6,31591}, {516,1152}, {615,6213}, {3069,31552}, {3071,31564}, {13911,31558}, {17278,31584}

X(31586) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MAJOR TO 2nd OUTER-SODDY

Barycentrics
a*((b+c)*a^5-(b+c)^2*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)-2*((b+c)*a^3-(3*b^2-2*b*c+3*c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^2+4*b*c+c^2)*(b-c)^2)*S) : :

The reciprocal orthologic center of these triangles is X(1)

X(31586) lies on these lines: {11,31588}, {481,17625}, {516,3878}, {1376,6212}, {1709,31573}, {3434,31551}, {5927,31542}, {10912,31559}, {12114,31563}, {17612,31540}, {17613,31544}, {17614,31546}, {17615,31547}, {17616,31549}, {17617,31553}, {17618,31555}, {17619,31557}, {17620,31565}, {17622,31567}, {17624,31569}, {17626,31571}, {17627,31575}, {17628,31576}, {17629,31578}, {17630,31580}, {17631,31592}, {18236,31594}

X(31586) = reflection of X(i) in X(j) for these (i,j): (31561, 31542), (31588, 31590)
X(31586) = {X(12672), X(17668)}-harmonic conjugate of X(31587)

X(31587) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MAJOR TO 2nd INNER-SODDY

Barycentrics
a*((b+c)*a^5-(b+c)^2*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)+2*((b+c)*a^3-(3*b^2-2*b*c+3*c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^2+4*b*c+c^2)*(b-c)^2)*S) : :

The reciprocal orthologic center of these triangles is X(1)

X(31587) lies on these lines: {11,31589}, {482,17625}, {516,3878}, {1376,6213}, {1709,31574}, {3434,31552}, {5927,31543}, {8225,17614}, {10912,31560}, {12114,31564}, {17612,31541}, {17613,31545}, {17615,31548}, {17616,31550}, {17617,31554}, {17618,31556}, {17619,31558}, {17620,31566}, {17622,31568}, {17624,31570}, {17626,31572}, {17628,31577}, {17629,31579}, {17630,31581}, {17631,31593}, {18236,31595}

X(31587) = reflection of X(i) in X(j) for these (i,j): (31562, 31543), (31589, 31591)
X(31587) = {X(12672), X(17668)}-harmonic conjugate of X(31586)

X(31588) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MINOR TO 2nd OUTER-SODDY

Barycentrics a*(-2*(b+c)*(a-b+c)*(a+b-c)*S+(-a+b+c)*(a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :
X(31588) = 3*X(210)-4*X(31594) = 3*X(354)-2*X(481) = 3*X(354)-4*X(31571) = 5*X(17609)-4*X(31569)

The reciprocal orthologic center of these triangles is X(1)

X(31588) lies on these lines: {1,30376}, {11,31586}, {55,6212}, {56,31563}, {57,31573}, {65,516}, {175,30347}, {210,31594}, {354,481}, {497,31551}, {518,31547}, {942,30375}, {1155,31544}, {1864,31561}, {2646,31546}, {3057,31567}, {4319,16232}, {5572,31565}, {10391,31549}, {10473,31553}, {10501,31580}, {10502,31592}, {10503,31578}, {17603,31540}, {17604,31542}, {17605,31555}, {17606,31557}, {17609,31569}, {17610,31575}, {17611,31576}

X(31588) = reflection of X(i) in X(j) for these (i,j): (481, 31571), (3057, 31567), (30425, 942), (31565, 5572), (31586, 31590)
X(31588) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 14100, 31589), (950, 12723, 31589), (12711, 21746, 31589)

X(31589) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MINOR TO 2nd INNER-SODDY

Barycentrics a*(2*(b+c)*(a-b+c)*(a+b-c)*S+(-a+b+c)*(a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :
X(31589) = 3*X(210)-4*X(31595) = 3*X(354)-2*X(482) = 3*X(354)-4*X(31572) = 5*X(17609)-4*X(31570)

The reciprocal orthologic center of these triangles is X(1)

X(31589) lies on these lines: {1,30375}, {11,31587}, {55,6213}, {56,31564}, {57,31574}, {65,516}, {176,30346}, {210,31595}, {354,482}, {497,31552}, {518,31548}, {942,30376}, {1155,31545}, {1864,31562}, {2098,31560}, {2362,4319}, {2646,8225}, {3057,31568}, {5572,31566}, {9042,30335}, {10473,31554}, {10501,31581}, {10502,31593}, {10503,31579}, {17603,31541}, {17604,31543}, {17605,31556}, {17606,31558}, {17609,31570}, {17611,31577}

X(31589) = reflection of X(i) in X(j) for these (i,j): (482, 31572), (3057, 31568), (30426, 942), (31566, 5572), (31587, 31591)
X(31589) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 14100, 31588), (950, 12723, 31588), (12711, 21746, 31588)

X(31590) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 2nd OUTER-SODDY

Barycentrics -2*((b+c)*a-b^2+2*b*c-c^2)*S+(b+c)*a^3+(b-c)^2*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(31590) = 3*X(2)+X(31551) = 2*X(1125)+X(31575) = 3*X(1699)+X(31573) = 3*X(3817)-2*X(31555)

The reciprocal orthologic center of these triangles is X(1)

X(31590) lies on these lines: {1,30381}, {2,6212}, {3,142}, {4,31563}, {6,31584}, {8,31559}, {10,639}, {11,31586}, {226,481}, {640,4138}, {642,21616}, {908,31547}, {1699,31573}, {1805,17197}, {3086,30276}, {3452,31594}, {3817,31555}, {4295,30277}, {4425,31576}, {5249,31549}, {5272,8947}, {6213,17170}, {8243,8965}, {10478,31553}, {11019,31542}, {12047,30380}, {12053,31567}, {12268,17761}, {21617,31565}, {21620,31569}, {21622,31578}, {21623,31580}, {21624,31592}

X(31590) = midpoint of X(i) and X(j) for these {i,j}: {4, 31563}, {8, 31559}, {481, 31561}, {6212, 31551}, {6213, 17170}, {31542, 31571}, {31546, 31575}, {31586, 31588}
X(31590) = reflection of X(i) in X(j) for these (i,j): (10, 31557), (31546, 1125)
X(31590) = complementary conjugate of X(31591)
X(31590) = complement of X(6212)
X(31590) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31551, 6212), (3, 18589, 31591), (1125, 12610, 31591)

X(31591) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 2nd INNER-SODDY

Barycentrics 2*((b+c)*a-b^2+2*b*c-c^2)*S+(b+c)*a^3+(b-c)^2*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(31591) = 3*X(2)+X(31552) = 3*X(1699)+X(31574) = 3*X(3817)-2*X(31556)

The reciprocal orthologic center of these triangles is X(1)

X(31591) lies on these lines: {1,30380}, {2,6213}, {3,142}, {4,31564}, {6,31585}, {8,31560}, {10,640}, {11,31587}, {226,482}, {639,4138}, {641,21616}, {908,31548}, {1699,31574}, {1806,17197}, {3086,30277}, {3452,31595}, {3817,31556}, {4295,30276}, {4425,31577}, {5249,31550}, {5272,8949}, {6212,17170}, {10478,31554}, {11019,31543}, {12047,30381}, {12053,31568}, {12269,17761}, {21617,31566}, {21620,31570}, {21622,31579}, {21623,31581}, {21624,31593}

X(31591) = midpoint of X(i) and X(j) for these {i,j}: {4, 31564}, {8, 31560}, {482, 31562}, {6212, 17170}, {6213, 31552}, {31543, 31572}, {31587, 31589}
X(31591) = reflection of X(i) in X(j) for these (i,j): (10, 31558), (8225, 1125)
X(31591) = complementary conjugate of X(31590)
X(31591) = complement of X(6213)
X(31591) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31552, 6213), (3, 18589, 31590), (1125, 12610, 31590)

X(31592) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 2nd OUTER-SODDY

Barycentrics -(a+b+c)*(2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-2*S*a+(a+b+c)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31592) lies on these lines: {1,30407}, {173,6212}, {174,481}, {177,31578}, {236,31594}, {516,8351}, {7589,31544}, {7590,31563}, {7593,31561}, {8083,31571}, {8092,31569}, {8126,31547}, {8379,31555}, {8382,31557}, {8389,31565}, {8423,31573}, {8425,31576}, {10481,31581}, {10502,31588}, {11535,31559}, {11860,31542}, {11890,31549}, {11891,31551}, {11896,31553}, {11924,31567}, {11996,31575}, {17631,31586}, {21624,31590}, {30406,30408}

X(31592) = {X(174), X(481)}-harmonic conjugate of X(31580)

X(31593) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 2nd INNER-SODDY

Barycentrics -(a+b+c)*(-2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+2*S*a+(a+b+c)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(1)

X(31593) lies on these lines: {1,30406}, {173,6213}, {174,482}, {177,31579}, {236,31595}, {516,8351}, {1373,30418}, {7587,8225}, {7589,31545}, {7590,31564}, {7593,31562}, {8092,31570}, {8126,31548}, {8379,31556}, {8382,31558}, {8389,31566}, {8423,31574}, {8425,31577}, {8729,31541}, {10481,31580}, {10502,31589}, {11535,31560}, {11860,31543}, {11891,31552}, {11896,31554}, {11924,31568}, {17631,31587}, {21624,31591}, {30407,30408}

X(31593) = {X(174), X(482)}-harmonic conjugate of X(31581)

X(31594) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 2nd OUTER-SODDY

Barycentrics -(b+c)*S+(a+b+c)*(-a+b+c)*a : :
X(31594) = 3*X(2)+X(31547) = 3*X(210)+X(31588) = 5*X(1698)-X(30425) = 5*X(18230)-X(31565)

The reciprocal orthologic center of these triangles is X(1)

X(31594) lies on these lines: {1,7585}, {2,481}, {4,9}, {8,31567}, {37,13883}, {44,13936}, {45,13911}, {188,31578}, {210,31588}, {220,1378}, {236,31592}, {482,30625}, {518,31571}, {519,30557}, {641,5745}, {936,31563}, {942,13359}, {958,31546}, {1125,30556}, {1329,31557}, {1376,31544}, {1698,30412}, {2886,31555}, {3219,6348}, {3452,31590}, {3671,30325}, {3932,5689}, {3947,30324}, {4298,6204}, {5273,31549}, {6347,27065}, {6351,13893}, {6352,18992}, {7028,31580}, {8580,31573}, {8953,31570}, {8957,11019}, {13973,16885}, {15829,31559}, {18227,31542}, {18228,31551}, {18229,31553}, {18230,31565}, {18234,31575}, {18235,31576}, {18236,31586}

X(31594) = midpoint of X(i) and X(j) for these {i,j}: {8, 31567}, {481, 31547}, {482, 30625}, {6212, 31561}
X(31594) = reflection of X(31569) in X(1125)
X(31594) = complement of X(481)
X(31594) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31547, 481), (9, 10, 31595), (17355, 18250, 31595)

X(31595) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 2nd INNER-SODDY

Barycentrics (b+c)*S+(a+b+c)*(-a+b+c)*a : :
X(31595) = 3*X(2)+X(31548) = 3*X(210)+X(31589) = 5*X(1698)-X(30426) = 5*X(18230)-X(31566)

The reciprocal orthologic center of these triangles is X(1)

X(31595) lies on these lines: {1,7586}, {2,482}, {4,9}, {8,31568}, {37,13936}, {44,13883}, {45,13973}, {210,31589}, {220,1377}, {236,31593}, {481,30625}, {518,31572}, {519,30556}, {642,5745}, {936,31564}, {942,13360}, {1125,30557}, {1329,31558}, {1376,31545}, {1698,30413}, {2886,31556}, {3219,6347}, {3452,31591}, {3671,30324}, {3932,5688}, {3947,30325}, {4298,6203}, {5273,31550}, {6348,27065}, {6351,18991}, {6352,13947}, {7028,31581}, {13911,16885}, {15829,31560}, {18227,31543}, {18228,31552}, {18230,31566}, {18235,31577}, {18236,31587}

X(31595) = midpoint of X(i) and X(j) for these {i,j}: {8, 31568}, {481, 30625}, {482, 31548}, {6213, 31562}
X(31595) = reflection of X(31570) in X(1125)
X(31595) = complement of X(482)
X(31595) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31548, 482), (9, 10, 31594), (17355, 18250, 31594)

X(31596) = PARALLELOGIC CENTERS OF THESE TRIANGLES: 2nd SHARYGIN TO 2nd OUTER-SODDY

Barycentrics a*(b-c)*(a^4-2*(b+c)*a^3+(b^2-b*c+c^2)*a^2+(b^2+c^2)*b*c+2*(a^2-b*c)*S) : :

The reciprocal parallelogic center of these triangles is X(1)

X(31596) lies on the line {514,659}

X(31597) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 2nd INNER-SODDY

Barycentrics a*(b-c)*(a^4-2*(b+c)*a^3+(b^2-b*c+c^2)*a^2+(b^2+c^2)*b*c-2*(a^2-b*c)*S) : :

The reciprocal orthologic center of these triangles is X(1)

X(31597) lies on the line {514,659}

X(31598) = PERSPECTOR OF THESE TRIANGLES: SODDY AND 2nd HATZIPOLAKIS

Barycentrics (a^3+(b+c)*a^2+(b^2+c^2)*a+(b+c)*(b^2-4*b*c+c^2))*(a-b+c)*(a+b-c) : :

X(31598) lies on these lines: {7,8}, {77,3924}, {192,28015}, {226,17286}, {239,28079}, {273,1851}, {347,1447}, {1014,17568}, {3938,7190}, {5226,30811}, {5435,7365}, {6046,17093}, {7176,19851}, {8897,28039}, {17170,17861}, {17316,28081}, {17895,21279}, {28016,28113}

X(31598) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 3212, 5933), (75, 7195, 7)

X(31599) = PERSPECTOR OF THESE TRIANGLES: SODDY AND LEMOINE

Barycentrics (3*a^4-4*(b+c)*a^3+2*(2*b^2-b*c+2*c^2)*a^2-2*(b+c)*(2*b^2-3*b*c+2*c^2)*a+(b^2+6*b*c+c^2)*(b-c)^2)*(a-b+c)*(a+b-c) : :

X(31599) lies on these lines: {857,948}, {17074,31527}

X(31600) = PERSPECTOR OF THESE TRIANGLES: SODDY AND MACBEATH

Barycentrics (a^6-(3*b^2-2*b*c+3*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^2-c^2)*(b-c)*b*c*a-(b^4-c^4)*(b^2-c^2))*(a-b+c)*(a+b-c) : :

X(31600) lies on these lines: {7,104}, {75,280}, {77,4357}, {222,17043}, {273,7040}, {1804,4329}, {3160,5932}, {6349,18623}

X(31601) = PERSPECTOR OF THESE TRIANGLES: SODDY AND INNER-SODDY

Barycentrics (3*S+2*(-a+b+c)*a)*(a-b+c)*(a+b-c) : :

X(31601) lies on these lines: {1,7}, {8,10910}, {1659,5435}, {5226,13389}, {5932,10904}, {5933,10907}, {10972,15913}

X(31601) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7, 31602), (1, 482, 21169), (1, 21169, 7)

X(31602) = PERSPECTOR OF THESE TRIANGLES: SODDY AND OUTER-SODDY

Barycentrics (-3*S+2*(-a+b+c)*a)*(a-b+c)*(a+b-c) : :

X(31602) lies on these lines: {1,7}, {8,10911}, {5226,13388}, {5435,13390}, {5932,10905}, {5933,10908}, {10973,15913}

X(31602) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (481, 31539, 1374), (1374, 31539, 176), (21171, 31539, 1)

X(31603) = PERSPECTOR OF THESE TRIANGLES: SODDY AND STEINER

Barycentrics (b-c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+b^3+c^3)*(a-b+c)*(a+b-c) : :

X(31603) lies on these lines: {7,4897}, {77,7203}, {347,6563}, {522,693}, {1638,28834}, {2487,5435}, {3669,3904}, {4367,8638}, {4560,7178}, {4573,24041}, {5226,14321}, {7180,17080}, {7192,17094}

X(31603) = {X(3676), X(4025)}-harmonic conjugate of X(24002)

X(31604) = PERSPECTOR OF THESE TRIANGLES: SODDY AND SYMMEDIAL

Barycentrics ((b^2+c^2)*a^4-2*(b^3+c^3)*a^3+(b^4-b^2*c^2+c^4)*a^2-(b-c)^2*b^2*c^2)*(a-b+c)*(a+b-c) : :

X(31604) lies on these lines: {7,2481}, {77,614}, {927,2175}, {1445,2082}, {2898,5929}, {6063,17049}

X(31605) = PERSPECTOR OF THESE TRIANGLES: SODDY AND YFF CONTACT

Barycentrics (b-c)*(a^2-2*(b+c)*a+b^2+c^2)*(a-b+c)*(a+b-c) : :
X(31605) = 4*X(3676)-X(30181)

X(31605) lies on these lines: {7,3667}, {57,4786}, {77,23465}, {241,30188}, {273,7649}, {347,20294}, {514,657}, {522,693}, {664,765}, {918,3669}, {934,2730}, {1440,2400}, {1445,2402}, {2898,6545}, {4468,24562}, {4762,7178}, {4905,26721}, {7216,28478}, {7253,17096}, {7265,22042}, {20520,21186}, {21174,23792}, {21182,21185}, {28898,30724}

X(31605) = midpoint of X(657) and X(23748)
X(31605) = reflection of X(i) in X(j) for these (i,j): (24002, 3676), (30181, 24002)
X(31605) = {X(21182), X(23798)}-harmonic conjugate of X(21185)

X(31606) = COMPLEMENT OF X(12506)

Barycentrics -4 a^10+13 a^8 (b^2+c^2)-3 a^6 (5 b^4-4 b^2 c^2+5 c^4)-6 (b^2-c^2)^2 (b^6-2 b^4 c^2-2 b^2 c^4+c^6)-a^4 (7 b^6+33 b^4 c^2+33 b^2 c^4+7 c^6)+a^2 (19 b^8-32 b^6 c^2-6 b^4 c^4-32 b^2 c^6+19 c^8) : :
Barycentrics (54 R^2-21 SW)S^4 +(54 R^2 SB SC-9 SB SC SW+3 SB SW^2+3 SC SW^2-4 SW^3)S^2 -2 SB SC SW^3 : :
X(31606) = 3*X[2]+X[12505], X[3]-3*X[10163], X[4]+3*X[9829], 5*X[1656]-3*X[10162], 7*X[3090]-3*X[6032], 5*X[3091]+3*X[6031], 4*X[3628]-3*X[10173]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28875

X(31606) lies on these lines: {2,12505}, {3,10163}, {4,9829}, {5,3849}, {1656,10162}, {3090,6032}, {3091,6031}, {3628,10173}, {3934,8704}, {7550,14682}

X(31606) = complement of X(12506)
X(31606) = midpoint of X(i) and X(j) for these {i,j}: {3,14866}, {12505,12506}
X(31606) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,12505,12506}, {10163,14866,3}

X(31607) = (name pending)

Barycentrics a^20 (b^2+c^2)-(b^2-c^2)^10 (b^2+c^2)-2 a^18 (4 b^4+5 b^2 c^2+4 c^4)+3 a^16 (9 b^6+11 b^4 c^2+11 b^2 c^4+9 c^6)+2 a^2 (b^2-c^2)^6 (4 b^8-2 b^6 c^2-5 b^4 c^4-2 b^2 c^6+4 c^8)-4 a^14 (12 b^8+10 b^6 c^2+9 b^4 c^4+10 b^2 c^6+12 c^8)+2 a^10 b^2 c^2 (25 b^8+13 b^6 c^2+11 b^4 c^4+13 b^2 c^6+25 c^8)-a^4 (b^2-c^2)^4 (27 b^10-7 b^8 c^2-29 b^6 c^4-29 b^4 c^6-7 b^2 c^8+27 c^10)+a^12 (42 b^10-8 b^8 c^2-17 b^6 c^4-17 b^4 c^6-8 b^2 c^8+42 c^10)+12 a^6 (b^2-c^2)^2 (4 b^12-b^10 c^2-3 b^8 c^4-4 b^6 c^6-3 b^4 c^8-b^2 c^10+4 c^12)+a^8 (-42 b^14+10 b^12 c^2+28 b^10 c^4+22 b^8 c^6+22 b^6 c^8+28 b^4 c^10+10 b^2 c^12-42 c^14) : :
Barycentrics (13 R^2-4 SW)S^4 + (-R^6-5 R^4 SB-5 R^4 SC+17 R^2 SB SC+2 R^4 SW+2 R^2 SB SW+2 R^2 SC SW-4 SB SC SW-R^2 SW^2)S^2 + R^6 SB SC -R^2 SB SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28875

X(31607) lies on this line: {5,252}

X(31607) = complement of complement of X(34794)

X(31608) = MIDPOINT OF X(3) AND X(14867)

Barycentrics -4 a^10+15 a^8 (b^2+c^2)+3 a^2 (b^2-c^2)^2 (2 b^4+b^2 c^2+2 c^4)-4 a^6 (5 b^4+2 b^2 c^2+5 c^4)-(b^2-c^2)^2 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6)+a^4 (5 b^6-21 b^4 c^2-21 b^2 c^4+5 c^6) : :
Barycentrics (54 R^2-9 SB-9 SC-18 SW)S^4 + (54 R^2 SB SC-3 SB SW^2-3 SC SW^2)S^2 -2 SB SC SW^3 : :
X(31608) = X[3]-3*X[10166], X[4]+3*X[353], 2*X[140]-3*X[10160]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28875

X(31608) lies on these lines: {3,10166}, {4,353}, {5,9830}, {6,22100}, {39,11615}, {83,6233}, {140,10160}, {1499,12506}, {1506,12494}, {8705,15074}

X(31608) = midpoint of X(3) and X(14867)
X(31608) = midpoint of X(5) and X(5)-of-circumsymmedial-triangle
X(31608) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {10166,14867,3}

X(31609) = X(6)X(5888)∩X(32)X(353)

Barycentrics a^2 (5 a^4+13 a^2 b^2+2 b^4+13 a^2 c^2+b^2 c^2+2 c^4) : :
Barycentrics (18 R^2-3 SB-3 SC+3 SW)S^2 -3 SB SC SW-10 SB SW^2-10 SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28876

X(31609) lies on these lines: {6,5888}, {32,353}, {51,5354}, {110,5008}, {111,12212}, {251,2502}, {352,5007}, {5039,22111}, {5359,11173}, {5526,21747}, {21309,26864}

Points associated with cubics pK(U',U'): X(31610)-X(31643)

This preamble and centers X(31610)-X(31643) were contributed by Clark Kimberling and Peter Moses, February 20, 2019.

Suppose that U = u : v : w (barycentrics) is a point in the plane of a triangle ABC. The self-inverse Gemini triangle based on U is defined as the triangle with vertices

A' = - u : v + w : v + w, B' = - v : w + u : w + u, C' = - w : u + v : u + v .

The triangle A'B'C' is perspective to the cevian triangle of a point X = x : y : z if and only if X lies on the cubic

v w (y - z) (y z - x y - x z) + w u (z - x )(z x - y z - y x) + u v (x - y)(x y - z x - z y) = 0,

this being, Bernard Gibert's notation ( Parallel Tripolars Cubics) the cubic pK(U',U'), where U' = isotomic conjugate of the crosspoint of X(2) and U; that is, U' = 1/(v+w) : 1/(w+u) : 1/(u+v). Points on this cubic, which we denote by Γ(U), include the following:

1 : 1 : 1 = X(2) = centroid
v w : w u : u v = isotomic conjugate of U
1/(v + w ) : 1/(w + u) : 1/(u + v) = isotomic conjugate of the complement of U
u/(v + w ) : v/(w + u) : w/(u + v)
v w/(v + w ) : w u/(w + u) : u v/(u + v)
(-u + v + w)/(v + w ) : (u - v + w)/w + u) : (u + v - w)/(u + v)
u - d : v - d : w - d, where d = sqrt(v w + w u + u v)
u + d : v + d : w + d, where d = sqrt(v w + w u + u v)

Let t1 = u^2(v+w) + v^2(w+u) + w^2(u+v) + uvw, t2 = (-u+v+w)(u-v+w)(u+v-w), and t = sqrt(t1/t2). Then two more points on the cubic are

u/(-u + v + w) - t : v(u - v + w) - t : w(u + v - w) - t
u/(-u + v + w) + t : v(u - v + w) + t : w(u + v + w) - t

If U is a triangle center, then A'B'C' is a central triangle and those ten points on Γ(U) are triangle centers. Note that the points involving square roots are nonreal for some choices of U and a,b,c.

Let DEF be the cevian triangle of v w : w u : u v, so that D = 0 : 1/v : 1/w, E = 1/u : 0 : 1/w, F = 1/u : 1/v : 0. The perspector of A'B'C' and DEF is the point

u(-u^2 + v^2 + w^2) : v(u^2 - v^2 + w^2) : w(u^2 + v^2 - w^2) = U*-Ceva conjugate of U, where U* = isotomic conjugate of U.

The appearance of

i	j(1), j(2), ..., j(n)	k	Kxxx

in the following table means that points on Γ(X(i)) include X(j(1)), X(j(2)), ..., x(j(n)), that X(k) is the perspector of A'B'C' and DEF, and that Γ(X(i)) = Kxxx .

1	2, 75, 85, 86, 274, 333, 348	x	x
3	2, 75, 85, 86, 274, 333, 348	x	x
4	2, 76, 95, 264, 275, 276, 213, 7763	x	K184
5	2, 69, 75, 76, 85, 264, 312, 15466	x	x
6	2, 76, 83, 264, 308, 1799, 17907	x	x
7	2, 8, 9, 1088, 75, 31627	x	x
8	2, 7, 75, 85, 4146, 18743, 27818, 27828	x	x
9	2, 85, 142, 6605, 31618, 21453	x	x
10	2, 86, 274, 1255, 1268, 18140	x	x
11	2, 4998, 3035, 31611, 31619, 31628	x	x
12	2, 261, 4999, 31612, 31620, 31629	x	x
19	2, 86, 274, 304, 30701, 30705	x	x
30	2, 1494, 30, 1, 31621, 2	x	x
39	2, 308, 3934, 31613, 31622, 31630	x	x
63	2, 92, 226, 1812, 31623, 31631	x	x
64	2, 76, 264, 801, 3926, 14615	x	x
65	2, 76, 86, 264, 274, 314, 1240, 14534, 14829, 30710	x	x
66	2, 76, 264, 305, 315, 1502	x	x
67	2, 76, 264, 316, 671, 3266, 18023, 30786	x	x
68	2, 76, 264, 317, 2052, 18027	x	x
69	2, 4, 76, 264, 491, 492	x	x
74	2, 76, 264, 2986, 3260, 7799	x	x
75	1, 2, 86, 87, 274, 31008	x	x
79	2, 75, 85, 319, 321, 1268, 4102	x	x
80	2, 75, 85, 320, 903, 4358, 4997, 20568	x	x
99	2, 523, 115, 31614, 999, 31632	x	x
100	2, 693, 11, 31615, 4554, 31633	x	x
101	2, 3261, 116, 31616, 31624, 31634	x	x
256	2, 37, 75, 85, 1221, 1909	x	x
257	1, 2, 10, 87, 894, 1220, 14534, 30710	x	x
265	2, 76, 264, 340, 1494, 16080	x	x
290	2, 76, 98, 264, 287, 290, 511, 3978, 16089	x	x
314	2, 65, 75, 85, 1220, 7196	x	x
330	1, 2, 87, 192, 366	x	K101
513	2, 668, 513, 2, 31625, 2	x	x
514	2, 190, 514, 2, 1016, 2	x	x
523	2, 99, 4590, 6189, 6190, 14089	x	x
596	2, 81, 86, 274, 1509, 4360	x	x
941	2, 75, 85, 1218, 2296, 4687	x	x
1000	2, 75, 85, 20569, 30608, 30829	x	x
1120	2, 7, 903, 1266, 3911, 27818, 31227	x	x
1581	2, 86, 274, 292, 334, 1966	x	x
2184	2, 312, 333, 345, 18750, 27398	x	x
2481	2, 75, 85, 350, 518, 673, 2481, 14942	x	x
2992	2, 13, 76, 264, 298, 300, 303, 621	x	K342a
2993	2, 14, 76, 264, 299, 301, 302, 622	x	K342b
2996	2, 4, 193	x	K181
2997	2, 27, 29, 75, 85, 286, 3868, 18147	x	x
2998	1, 2, 6, 43, 87, 194, 3224, 15963, 15964, 15965, 15966, 15967, 15968	x	K102
3225	2, 6, 385, 698, 3224, 3225	x	x
3226	1, 2, 87, 239, 726, 3226	x	x
3346	2, 69, 345, 348, 3926, 6527	x	x
3668	2, 69, 86, 274, 1043, 1257	x	x
3762	2, 190, 1016, 3257, 4997, 24624	x	x
5486	2, 76, 264, 598, 11059, 11185	x	x
6333	2, 685, 6330, 6331, 9476, 18020	x	x
6339	2, 4, 393, 1123, 1336, 3068, 3069, 6353, 6392, 13429, 13440	x	K1046
6504	2, 92, 1585, 1586, 2052, 6515	x	x
6553	2, 7,279, 4452, 5435, 18886, 21456, 27818	x	x
6601	2, 75, 85, 1088, 6604, 20946	x	x
7261	2, 10, 75, 85, 334, 335, 1581, 1920, 1921, 3912, 4518, 4645, 20947, 27436, 30663	x	K868
7319	2, 75, 85, 4373, 6557, 20942, 21296	x	x
8044	2, 10, 76, 264, 306, 313, 1330	x	x
9229	2, 6, 141, 384, 1241, 3224	x	x
9285	2, 333, 893, 1965, 7018, 7033, 31008	x	x
9289	2, 3, 4, 801, 1105, 9308	x	x
9311	1, 2, 8, 87, 1222, 3729	x	x
10015	2, 4997, 5376, 6335, 13136, 16082	x	x
11117	2, 14, 532, 11092, 11117, 11119	x	x
11118	2, 13, 533, 11078, 11118, 11120	x	x
13476	2, 86, 274, 310, 17143, 29767	x	x
13582	2, 94, 324, 8836, 8838, 11078, 11092, 18359, 30690	x	x
14970	2, 83, 308, 694, 732, 1916, 9477, 14970	x	x
15318	2, 69, 95, 276, 394, 20477	x	x
15321	2, 76, 264, 7768, 8024, 10159	x	x
15412	2, 249, 662, 2185, 4564, 8115, 8116, 14570, 18315	x	x
18827	2, 86, 274, 291, 335, 740, 17731, 18827	x	x
19222	2, 6, 76, 183, 264, 870, 3114, 3224, 3407, 7033, 14382, 18906	x	K1014
20029	2, 76, 264, 3596, 4417, 5224	x	x
21739	2, 321, 4358, 17484, 18359, 24624	x	x
30725	2, 190, 1016, 4582, 5376, 31227	x	x
31359	2, 86, 274, 5936, 10436, 25430, 27813	x	x

X(31610) = X(2)X(10979)∩X(5)X(23607)

Barycentrics (a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :

X(31610) lies on these lines: {2, 10979}, {5, 23607}, {68, 1173}, {288, 30529}, {5068, 18855}

X(31611) = POINT ELTANIN 1

Barycentrics (b - c)^2*(-a + b + c)*(a^3 - a^2*b - 2*a*b^2 + 2*b^3 - a^2*c + 4*a*b*c - 2*b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 4*a*b*c - b^2*c - 2*a*c^2 - 2*b*c^2 + 2*c^3) : :

X(31611) lies on these lines: {2, 31628}

X(31612) = POINT ELTANIN 1

Barycentrics (a + b - c)^2*(a - b + c)^2*(b + c)^2*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 4*a*b*c + b^2*c - 2*a*c^2 - 2*b*c^2 - 2*c^3)*(a^3 + a^2*b - 2*a*b^2 - 2*b^3 - a^2*c - 4*a*b*c - 2*b^2*c - a*c^2 + b*c^2 + c^3) : :

X(31612) lies on these lines: {2, 31629}

X(31613) = X(32)X(5012)∩X(881)X(8664)

Barycentrics a^2*(b^2 + c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :

X(31613) lies on these lines: {32, 5012}, {881, 8664}, {1634, 20965}

X(31613) = isogonal conjugate of X(18092)

X(31614) = X(2)X(4590)∩X(99)X(11123)

Barycentrics (a - b)^3*(a + b)^3*(a - c)^3*(a + c)^3 : :

X(31614) lies on these lines: {2, 4590}, {99, 11123}, {249, 1501}, {2396, 4235}, {4600, 21085}, {6353, 18020}, {9182, 10278}, {10190, 14588}

X(31614) = isogonal conjugate of X(22260)
X(31614) = isotomic conjugate of X(8029)
X(31614) = barycentric cube of X(99)

X(31615) = X(2)X(1252)∩X(55)X(5377)

Barycentrics a*(a - b)^3*(a - c)^3*(a + b - c)*(a - b + c) : :

X(31615) lies on these lines: {2, 1252}, {55, 5377}, {57, 4564}, {59, 1397}, {100, 11124}, {200, 765}, {333, 4567}, {345, 1016}, {1025, 4585}, {1026, 23703}, {1110, 3550}, {1275, 17093}

X(31615) = isotomic conjugate of Danneels point of X(693)

X(31616) = (name pending)

Barycentrics a^2*(a - b)^3*(a - c)^3*(a^2 + a*b + b^2 - a*c - b*c)*(a^2 - a*b + a*c - b*c + c^2) : :

X(31616) lies on these lines: {2, 23990}

X(31617) = X(5)X(95)∩X(140)X(18831)

Barycentrics (a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :

X(31617) lies on these lines: {5, 95}, {140, 18831}, {276, 324}, {288, 343}

X(31617) = isotomic conjugate of X(233)

X(31618) = X(7)X(20683)∩X(9)X(85)

Barycentrics b*(-a + b - c)*(a + b - c)*c*(a^2 - 2*a*b + b^2 - a*c - b*c)*(-a^2 + a*b + 2*a*c + b*c - c^2) : :
Barycentrics 1/(sec^2(B/2) + sec^2(C/2)) : :

X(31618) lies on these lines: {7, 20683}, {9, 85}, {75, 200}, {76, 346}, {142, 4569}, {274, 1170}, {281, 331}, {286, 4183}, {344, 6063}, {1441, 2346}, {4554, 17263}, {6605, 10025}

X(31618) = isogonal conjugate of X(20229)
X(31618) = isotomic conjugate of X(1212)
X(31618) = polar conjugate of X(1827)
X(31618) = trilinear pole of line X(693)X(3900)
X(31618) = X(19)-isoconjugate of X(22079)

X(31619) = (name pending)

Barycentrics (a - b)^2*(a - c)^2*(a + b - c)*(a - b + c)*(a^3 - a^2*b - 2*a*b^2 + 2*b^3 - a^2*c + 4*a*b*c - 2*b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 4*a*b*c - b^2*c - 2*a*c^2 - 2*b*c^2 + 2*c^3) : :

X(31619) lies on these lines: {11, 4998}

X(31619) = isotomic conjugate of isogonal conjugate of X(38809)

X(31620) = (name pending)

Barycentrics (a + b)^2*(a + c)^2*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 4*a*b*c + b^2*c - 2*a*c^2 - 2*b*c^2 - 2*c^3)*(a^3 + a^2*b - 2*a*b^2 - 2*b^3 - a^2*c - 4*a*b*c - 2*b^2*c - a*c^2 + b*c^2 + c^3) : :

X(31620) lies on these lines: {12, 261}

X(31621) = X(30)X(340)∩X(5627)X(5641)

Barycentrics (a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)^2*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)^2 : :
Barycentrics (csc^2 A)/(cos A - 2 cos B cos C)^2 : :

X(31621) lies on these lines: {30, 340}, {5627, 5641}, {7799, 11064}

X(31621) = isogonal conjugate of X(9408)
X(31621) = isotomic conjugate of X(3163)
X(31621) = polar conjugate of X(16240)
X(31621) = cevapoint of X(2) and X(1494)
X(31621) = X(2)-cross conjugate of X(1494)
X(31621) = barycentric square of X(1494)
X(31621) = trilinear pole of line X(1494)X(3268) (the tangent to the Steiner circumellipse at X(1494))

X(31622) = X(32)X(26192)∩X(39)X(308)

Barycentrics b^2*(a^2 + b^2)*c^2*(a^2 + c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :

X(31622) lies on these lines: {32, 26192}, {39, 308}, {83, 3051}, {9466, 9495}

X(31622) = isotomic conjugate of complement of X(308)
X(31622) = isotomic conjugate of crosspoint of X(2) and X(39)
X(31622) = isotomic conjugate of crosssum of X(6) and X(83)
X(31622) = isotomic conjugate of polar conjugate of isogonal conjugate of X(23210)
X(31622) = X(19)-isoconjugate of X(23210)

X(31623) = X(4)X(216)∩X(4)X(970)

Barycentrics    b*(a + b)*c*(a + c)*(-a + b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :
Barycentrics    (sec A)/(cos B + cos C) : :
Barycentrics    (tan A)/((b + c) (1 - cos A)) : :
Barycentrics    1/(sec B + sec C) : :

X(31623) lies on these lines: {2, 216}, {4, 970}, {19, 27}, {21, 243}, {28, 1791}, {29, 33}, {53, 5743}, {58, 20320}, {76, 19793}, {81, 648}, {86, 309}, {99, 20624}, {107, 4228}, {158, 11110}, {226, 1947}, {273, 19804}, {274, 278}, {281, 345}, {297, 1211}, {314, 1172}, {317, 5739}, {321, 2287}, {340, 2895}, {391, 6994}, {406, 30830}, {422, 2203}, {445, 3936}, {458, 4383}, {469, 5233}, {940, 9308}, {1010, 8747}, {1093, 6824}, {1474, 27958}, {1848, 7018}, {1948, 5745}, {1982, 4653}, {1990, 6703}, {2299, 14006}, {2328, 15146}, {2339, 8748}, {2417, 15411}, {3260, 28754}, {4194, 28809}, {4886, 5081}, {6335, 20566}, {6513, 20570}, {6708, 25515}, {6837, 14249}, {6852, 13450}, {7131, 16054}, {7140, 17987}, {15014, 24271}, {16700, 16759}, {16713, 30599}, {17056, 25986}, {17102, 25517}, {17171, 30078}, {17188, 24026}, {17923, 20565}, {18036, 31008}, {18591, 26053}, {18603, 26011}, {20477, 21482}

X(31623) = isogonal conjugate of X(1409)
X(31623) = isotomic conjugate of X(1214)
X(31623) = complement of X(18667)
X(31623) = anticomplement of X(18592)
X(31623) = polar conjugate of X(65)
X(31623) = pole wrt polar circle of trilinear polar of X(65) (line X(647)X(661))
X(31623) = trilinear pole of line X(521)X(1948)
X(31623) = trilinear product X(2)*X(29)
X(31623) = perspector of ABC and orthoanticevian triangle of X(314)
X(31623) = X(19)-isoconjugate of X(22341)

X(31624) = X(99)X(4249)∩X(101)X(3261)

Barycentrics b^2*(-a + b)*(a - c)*c^2*(a^2 + a*b + b^2 - a*c - b*c)*(-a^2 + a*b - a*c + b*c - c^2) : :

X(31624) lies on these lines: {99, 4249}, {101, 3261}

X(31624) = isotomic conjugate of X(6586)

X(31625) = X(6)X(5383)∩X(190)X(20979)

Barycentrics (a - b)^2*b^2*(a - c)^2*c^2 : :

X(31625) lies on the hyperbola {{A,B,C,PU(41)}} and on these lines: {6, 5383}, {190, 20979}, {350, 899}, {513, 668}, {523, 4583}, {536, 1921}, {646, 6386}, {670, 4033}, {765, 5388}, {1016, 4601}, {1909, 25382}, {1978, 3261}, {3596, 4076}, {4057, 8709}, {6376, 24338}, {7209, 17786}, {15742, 16085}, {24004, 27853}, {24289, 30473}

X(31625) = isogonal conjugate of X(1977)
X(31625) = isotomic conjugate of X(1015)
X(31625) = cevapoint of X(2) and X(668)
X(31625) = cevapoint of Steiner circumellipse intercepts of line X(2)X(37)
X(31625) = X(2)-cross conjugate of X(668)
X(31625) = barycentric square of X(668)
X(31625) = trilinear pole of line X(668)X(891) (the tangent to the Steiner circumellipse at X(668))
X(31625) = X(19)-isoconjugate of X(22096)

X(31626) = X(2)X(10979)∩X(3)X(143)

Barycentrics a^2*(a^2 - b^2 - c^2)*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :
Barycentrics (cos A)/(cos A + 2 sin B sin C) : :

X(31626) lies on the hyperbola {{A,B,C,X(2), X(3)}} and on these lines: {2, 10979}, {3, 143}, {4, 22270}, {5, 12013}, {97, 216}, {140, 14938}, {276, 324}, {465, 8836}, {466, 8838}, {1214, 27003}, {1217, 3523}, {1297, 15246}, {1487, 27361}, {3346, 15717}, {4993, 8613}, {5481, 6636}, {6641, 26881}, {7484, 14489}, {11433, 14806}, {12100, 18317}, {13445, 26897}, {17974, 22352}, {18475, 20574}

X(31626) = isogonal conjugate of X(6748)
X(31626) = cevapoint of X(3) and X(216)
X(31626) = X(92)-isoconjugate of X(13366)

X(31627) = X(2)X(85)∩X(7)X(3742)

Barycentrics b*c*(-a + b - c)*(a + b - c)*(-3*a^2 + 2*a*b + b^2 + 2*a*c - 2*b*c + c^2) : :

X(31627) lies on these lines: {2, 85}, {7, 3742}, {8, 31527}, {9, 23062}, {63, 658}, {144, 9533}, {200, 664}, {312, 4554}, {329, 7056}, {479, 18228}, {518, 31526}, {982, 3663}, {1001, 9446}, {1323, 20103}, {1376, 14189}, {2550, 2898}, {3160, 16284}, {4514, 8817}, {4666, 21453}, {4679, 30623}, {5088, 19541}, {6063, 19804}, {6516, 7411}, {6604, 10580}, {7271, 25496}, {8056, 27829}, {8580, 9312}, {8727, 17181}, {12652, 14942}, {18698, 27798}, {21609, 30829}

X(31627) = isotomic conjugate of X(19605)

X(31628) = X(1376)X(14947)∩X(5377)X(11124)

Barycentrics a*(a - b)*(a - c)*(a^3 - a^2*b - 2*a*b^2 + 2*b^3 - a^2*c + 4*a*b*c - 2*b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 4*a*b*c - b^2*c - 2*a*c^2 - 2*b*c^2 + 2*c^3) : :

X(31628) lies on these lines: {1376, 14947}, {5377, 11124}

X(31629) = (name pending)

Barycentrics a*(a + b - c)*(a - b + c)*(a^3 - a*b^2 + a*b*c - b^2*c - a*c^2 - b*c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 4*a*b*c + b^2*c - 2*a*c^2 - 2*b*c^2 - 2*c^3)*(a^3 + a^2*b - 2*a*b^2 - 2*b^3 - a^2*c - 4*a*b*c - 2*b^2*c - a*c^2 + b*c^2 + c^3) : :

X(31639) lies on these lines: (none)

X(31630) = X(76)X(8041)∩X(83)X(3051)

Barycentrics b^2*c^2*(2*a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :

X(31630) lies on these lines: {76, 8041}, {83, 3051}, {308, 30505}, {1627, 3407}, {3978, 10159}, {18840, 20023}

X(31631) = X(2)X(6)∩X(4)X(14868)

Barycentrics (a + b)*(a - b - c)*(a + 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(31631) lies on these lines: {2, 6}, {4, 14868}, {21, 22768}, {27, 662}, {226, 6514}, {283, 27385}, {284, 17182}, {312, 332}, {329, 1444}, {811, 8777}, {908, 1790}, {997, 1010}, {1043, 3615}, {1412, 18645}, {1792, 27383}, {1800, 3559}, {1817, 17139}, {1944, 7106}, {2178, 5905}, {2327, 5249}, {3175, 16759}, {3193, 5552}, {3485, 17518}, {3782, 21008}, {4997, 19607}, {5208, 25533}, {5327, 13588}, {6505, 20930}, {6513, 20570}, {14547, 27401}, {17095, 17206}, {18162, 30078}, {19796, 24203}, {25082, 28950}, {28922, 28934}, {28936, 28944}

X(31632) = X(2)X(4590)∩X(25)X(18020)

Barycentrics (a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2*(a^4 - a^2*b^2 - b^4 - a^2*c^2 + 3*b^2*c^2 - c^4) : :

X(31632) lies on these lines: {2, 4590}, {25, 18020}, {99, 9293}, {249, 1915}, {892, 8029}, {4600, 8013}, {9182, 10189}, {11634, 17941}, {20094, 31373}

X(31633) = X(2)X(1252)∩X(226)X(4564)

Barycentrics (a - b)^2*(a - c)^2*(a + b - c)*(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(31633) lies on these lines: {2, 1252}, {226, 4564}, {278, 5376}, {312, 1016}, {497, 5377}, {765, 4847}, {1088, 1275}

X(31634) = (name pending)

Barycentrics (a - b)^2*(a - c)^2*(a^2 + a*b + b^2 - a*c - b*c)*(a^2 - a*b + a*c - b*c + c^2)*(a^4 - a^3*b + a*b^3 - b^4 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a*b*c^2 + a*c^3 + b*c^3 - c^4) : :

X(31634) lies on these lines: {2, 23990}

X(31635) = X(2)X(248)∩X(3)X(76)

Barycentrics (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^4-a^2 b^2-b^2 c^2+c^4) : :

X(31635) lies on these lines: {2,248}, {3,76}, {69,1976}, {95,141}, {157,264}, {230,297}, {317,571}, {325,14601}, {491,26920}, {492,8911}, {631,17974}, {1147,7763}, {1300,22456}, {6699,11653}, {6720,11610}, {7807,14600}

X(31635) = cevapoint of X(157) and X(230)
X(31635) = trilinear pole of line {1993, 6563}
X(31635) = X(i)-isoconjugate of X(j) for these (i,j): {91, 237}, {232, 1820}, {240, 2351}, {1755, 2165}, {5392, 9417}, {9418, 20571}
X(31635) = barycentric product X(i)*X(j) for these {i,j}: {98, 7763}, {287, 317}, {290, 1993}, {336, 1748}, {571, 18024}, {2966, 6563}, {6394, 11547}, {9723, 16081}
X(31635) = barycentric quotient X(i) / X(j) for these {i,j}: {24, 232}, {47, 1755}, {98, 2165}, {248, 2351}, {287, 68}, {290, 5392}, {293, 1820}, {317, 297}, {571, 237}, {924, 3569}, {1147, 3289}, {1748, 240}, {1821, 91}, {1993, 511}, {2966, 925}, {6531, 14593}, {6563, 2799}, {6753, 17994}, {7763, 325}, {11547, 6530}, {16081, 847}, {18883, 14356}, {22456, 30450}
X(31635) = {X(98), X(6394)}-harmonic conjugate of X(290)

X(31636) = X(66)X(1755)∩X(511)X(2156)

Barycentrics (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-b^4-c^4) (a^4-a^2 b^2-b^2 c^2+c^4) : :

X(31636) lies on these lines: {2,248}, {4,17974}, {5,83}, {6,264}, {76,23128}, {113,11653}, {127,315}, {206,17907}, {297,1971}, {316,2715}, {325,441}, {569,7803}, {877,10766}, {1976,3618}, {2422,24284}, {6656,14600}, {7790,14355}, {10349,14382}

X(31636) = isotomic conjugate of X(34138)
X(31636) = isotomic conjugate of the isogonal conjugate of X(11610)
X(31636) = X(18024)-Ceva conjugate of X(98)
X(31636) = X(i)-isoconjugate of X(j) for these (i,j): {66, 1755}, {511, 2156}, {1959, 2353}, {9417, 18018}
X(31636) = cevapoint of X(441) and X(23128)
X(31636) = barycentric product X(i)*X(j) for these {i,j}: {22, 290}, {76, 11610}, {98, 315}, {206, 18024}, {287, 17907}, {1760, 1821}, {1910, 20641}, {8673, 22456}, {16081, 20806},barycentric quotient X(i) / X(j) for these {i,j}: {22, 511}, {98, 66}, {206, 237}, {287, 14376}, {290, 18018}, {315, 325}, {685, 1289}, {1760, 1959}, {1910, 2156}, {1976, 2353}, {2172, 1755}, {2485, 3569}, {4611, 2421}, {6531, 13854}, {8673, 684}, {8743, 232}, {10316, 3289}, {11610, 6}, {17409, 2211}, {17453, 9417}, {17907, 297}, {20968, 9418}
X(31636) = {X(287), X(6531)}-harmonic conjugate of X(290)

X(31637) = X(1)X(85)∩X(2)X(294)

Barycentrics (a^2+b^2-a c-b c) (a^2-b^2-c^2) (a^2-a b-b c+c^2) : :

X(31637) lies on the conic {{A,B,C,X(1), X(3) and these lines: {1,85}, {2,294}, {3,348}, {7,1037}, {29,811}, {33,21609}, {69,219}, {78,304}, {86,142}, {102,927}, {105,1036}, {282,309}, {283,4592}, {337,25083}, {377,13576}, {666,2338}, {951,3674}, {1027,3716}, {1040,7182}, {1057,1387}, {1462,3945}, {1897,20940}, {2195,4357}, {2359,18650}, {4872,6996}, {5845,20672}, {9318,17464}, {9436,9441}, {16831,18785}, {17181,21554}, {17687,31269}, {20341,26012}, {20769,26006}

X(31637) = isogonal conjugate of X(2356)
X(31637) = isotomic conjugate of X(1861)
X(31637) = X(18031)-Ceva conjugate of X(673)
X(31637) = X(i)-cross conjugate of X(j) for these (i,j): {1818, 63}, {20731, 3}, {20769, 17206}, {26006, 348}
X(31637) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2356}, {4, 2223}, {6, 5089}, {19, 672}, {25, 518}, {28, 20683}, {31, 1861}, {33, 1458}, {34, 2340}, {41, 5236}, {55, 1876}, {92, 9454}, {108, 926}, {112, 24290}, {213, 15149}, {241, 607}, {264, 9455}, {393, 20752}, {512, 4238}, {608, 3693}, {665, 1783}, {1096, 1818}, {1395, 3717}, {1474, 3930}, {1824, 3286}, {1973, 3912}, {1974, 3263}, {2201, 3252}, {2203, 3932}, {2207, 25083}, {2212, 9436}, {2254, 8750}, {2283, 18344}, {2284, 6591}, {2333, 18206}, {5338, 14626}, {6184, 8751}, {7115, 17435}, {8638, 18026}, {8752, 14439}, {15344, 20455}
X(31637) = cevapoint of X(i) and X(j) for these (i,j): {2, 3100}, {3, 20744}, {63, 1818}
X(31637) = trilinear pole of line {63, 652}
X(31637) = barycentric product X(i)*X(j) for these {i,j}: {3, 18031}, {63, 2481}, {69, 673}, {75, 1814}, {105, 304}, {294, 7182}, {305, 1438}, {337, 6654}, {348, 14942}, {666, 4025}, {799, 10099}, {927, 6332}, {1462, 3718}, {4554, 23696}, {6559, 7056}, {13576, 17206}
X(31637) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 5089}, {2, 1861}, {3, 672}, {6, 2356}, {7, 5236}, {48, 2223}, {57, 1876}, {63, 518}, {69, 3912}, {71, 20683}, {72, 3930}, {77, 241}, {78, 3693}, {86, 15149}, {105, 19}, {184, 9454}, {219, 2340}, {222, 1458}, {255, 20752}, {294, 33}, {295, 3252}, {304, 3263}, {306, 3932}, {326, 25083}, {345, 3717}, {348, 9436}, {394, 1818}, {525, 4088}, {652, 926}, {656, 24290}, {662, 4238}, {666, 1897}, {673, 4}, {885, 3064}, {905, 2254}, {919, 8750}, {927, 653}, {1024, 18344}, {1027, 6591}, {1331, 2284}, {1332, 1026}, {1416, 608}, {1438, 25}, {1444, 18206}, {1459, 665}, {1462, 34}, {1790, 3286}, {1813, 2283}, {1814, 1}, {1818, 6184}, {2195, 607}, {2481, 92}, {3942, 3675}, {4001, 4966}, {4025, 918}, {5440, 14439}, {6516, 1025}, {6559, 7046}, {6654, 242}, {7004, 17435}, {8751, 1096}, {9247, 9455}, {10099, 661}, {13576, 1826}, {14942, 281}, {15419, 23829}, {17206, 30941}, {18031, 264}, {18785, 1824}, {20769, 8299}, {20780, 20662}, {23601, 4319}, {23696, 650}, {25083, 4712}, {28071, 7079}

X(31638) = X(2)X(294)∩X(8)X(105)

Barycentrics (a^2+b^2-a c-b c) (a^2-2 a b+b^2-2 a c+c^2) (a^2-a b-b c+c^2) : :

X(31638) lies on the cubic K697 and these lines: {2,294}, {8,105}, {9,75}, {10,10482}, {85,169}, {218,4904}, {344,6600}, {666,3008}, {927,8074}, {1027,25380}, {1416,5247}, {1438,3684}, {1814,3618}, {2195,17353}, {2478,13576}, {3759,8271}, {11343,15288}, {18230,28071}

X(31638) = cevapoint of X(169) and X(3008)
X(31638) = trilinear pole of line {2402, 3870}
X(31638) = X(i)-isoconjugate of X(j) for these (i,j): {277, 2223}, {513, 2428}, {665, 1292}, {667, 2414}, {672, 2191}, {2340, 17107}
X(31638) = barycentric product X(i)*X(j) for these {i,j}: {190, 2402}, {218, 18031}, {294, 21609}, {344, 673}, {666, 4468}, {1978, 2440}, {2481, 3870}, {6559, 17093}, {6604, 14942}
X(31638) = barycentric quotient X(i) / X(j) for these {i,j}: {101, 2428}, {105, 2191}, {190, 2414}, {218, 672}, {344, 3912}, {673, 277}, {1445, 241}, {1462, 17107}, {1617, 1458}, {2402, 514}, {2440, 649}, {3309, 2254}, {3870, 518}, {3991, 3930}, {4468, 918}, {4878, 20683}, {6600, 2340}, {6604, 9436}, {7719, 5089}, {14942, 6601}, {21059, 2223}
X(31638) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {673, 6559, 2481}, {4384, 18785, 2481}

X(31639) = X(2)X(670)∩X(308)X(6375)

Barycentrics 2 a^4 b^4-3 a^4 b^2 c^2-a^2 b^4 c^2+2 a^4 c^4-a^2 b^2 c^4+b^4 c^4 : :
X(31639) = 6 X[2] - X[670],3 X[2] + 2 X[1084],4 X[2] + X[3228],9 X[2] + X[25054],X[670] + 4 X[1084],2 X[670] + 3 X[3228],3 X[670] + 2 X[25054],X[694] + 4 X[3589],3 X[694] + 2 X[25327],8 X[1084] - 3 X[3228],6 X[1084] - X[25054],9 X[3228] - 4 X[25054],6 X[3589] - X[25327],7 X[3619] + 3 X[25318],4 X[6677] + X[16098],4 X[6719] + X[14948]

X(31639) lies on these lines: {2, 670}, {308, 6375}, {694, 3589}, {804, 14061}, {3619, 25318}, {5969, 7786}, {6677, 16098}, {6719, 14948}

X(31639) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1084, 670}, {670, 1084, 3228}

X(31640) = X(2)X(664)∩X(10)X(10482)

Barycentrics a^4-a^3 b+a^2 b^2-3 a b^3+2 b^4-a^3 c-a^2 b c+3 a b^2 c-b^3 c+a^2 c^2+3 a b c^2-2 b^2 c^2-3 a c^3-b c^3+2 c^4 : :
X(31640) = 6 X[2] - X[664],4 X[2] + X[1121],3 X[2] + 2 X[1146],9 X[2] - 4 X[17044],4 X[10] + X[14942],4 X[116] + X[3732],2 X[664] + 3 X[1121],X[664] + 4 X[1146],3 X[664] - 8 X[17044],3 X[1121] - 8 X[1146],9 X[1121] + 16 X[17044],3 X[1146] + 2 X[17044],X[1952] + 4 X[6708],X[4872] + 4 X[8074],4 X[5199] + X[9436],4 X[6712] + X[18328]

X(31640) lies on these lines: {2, 664}, {10, 10482}, {85, 23058}, {116, 3732}, {333, 4592}, {514, 31273}, {528, 18230}, {673, 21044}, {918, 27191}, {1698, 28850}, {1952, 6708}, {2785, 14061}, {4437, 27546}, {4872, 8074}, {4997, 18061}, {5199, 9436}, {6366, 31272}, {6604, 27541}, {6712, 18328}, {21272, 30857}, {25005, 27132}, {26532, 27068}

X(31640) = isotomic conjugate of X(36956)
{X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1146, 664}, {664, 1146, 1121}

X(31641) = (name pending)

Barycentrics a^2*b^2 - 2*a^2*b*c + a^2*c^2 - b^2*c^2 - Sqrt[-(a^4*b^4) + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 - b^4*c^4] : :

X(31641) lies on the cubic K101 and the line {2, 330}

X(31642) = (name pending)

Barycentrics a^2*b^2 - 2*a^2*b*c + a^2*c^2 - b^2*c^2 + Sqrt[-(a^4*b^4) + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 - b^4*c^4] : :

X(31642) lies on the cubic K101 and the line {2, 330}

X(31643) = X(7)X(76)∩X(57)X(752)

Barycentrics b*(-a + b - c)*(a + b - c)*c*(a^2 + b^2 + a*c + b*c)*(a^2 + a*b + b*c + c^2) : :

X(31643) lies on these lines: {7, 76}, {57, 75}, {65, 314}, {85, 269}, {264, 14257}, {274, 961}, {286, 1396}, {331, 1119}, {334, 30097}, {388, 3596}, {767, 8687}, {1221, 1447}, {1462, 2298}, {3339, 10447}, {3668, 7196}, {6385, 7205}, {6604, 8814}, {8707, 15728}, {24547, 24627}

X(31643) = isogonal conjugate of X(20967)
X(31643) = isotomic conjugate of X(960)
X(31643) = isotomic conjugate of the anticomplement of X(3812)
X(31643) = isotomic conjugate of the complement of X(65)
X(31643) = X(i)-cross conjugate of X(j) for these (i,j): {850, 18026}, {1220, 30710}, {3812, 2}, {4374, 4569}, {4581, 6648}, {7178, 4554}, {7192, 664}, {24993, 75}
X(31643) = cevapoint of X(i) and X(j) for these (i,j): {2, 65}, {7, 1441}, {85, 7196}, {7187, 7248}
X(31643) = trilinear pole of line {693, 3669}
X(31643) = barycentric product X(i)*X(j) for these {i,j}: {7, 30710}, {57, 1240}, {76, 961}, {85, 1220}, {331, 1791}, {349, 2363}, {693, 6648}, {1441, 14534}, {2298, 6063}, {4554, 4581}, {8707, 24002}, {15420, 18026}
X(31643) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 2269}, {2, 960}, {3, 22074}, {6, 20967}, {7, 3666}, {8, 3965}, {10, 21033}, {12, 21810}, {34, 2354}, {56, 2300}, {57, 1193}, {65, 2092}, {75, 3687}, {77, 22097}, {81, 4267}, {85, 4357}, {86, 17185}, {222, 22345}, {226, 2292}, {273, 1848}, {278, 1829}, {279, 24471}, {321, 3704}, {349, 18697}, {514, 17420}, {664, 3882}, {693, 3910}, {894, 18235}, {961, 6}, {1088, 3674}, {1169, 2194}, {1214, 22076}, {1220, 9}, {1240, 312}, {1400, 3725}, {1441, 1211}, {1791, 219}, {1798, 2193}, {2298, 55}, {2359, 212}, {2363, 284}, {2995, 19608}, {3666, 1682}, {3669, 6371}, {4077, 21124}, {4581, 650}, {6063, 20911}, {6358, 20653}, {6648, 100}, {7176, 28369}, {7223, 4503}, {8687, 692}, {8707, 644}, {14534, 21}, {14624, 210}, {15420, 521}, {21454, 4719}, {24002, 3004}, {30710, 8}
X(31643) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20967}, {6, 2269}, {9, 2300}, {19, 22074}, {21, 3725}, {31, 960}, {32, 3687}, {33, 22345}, {41, 3666}, {42, 4267}, {55, 1193}, {212, 1829}, {213, 17185}, {219, 2354}, {284, 2092}, {604, 3965}, {607, 22097}, {692, 17420}, {904, 18235}, {1253, 24471}, {1333, 21033}, {2150, 21810}, {2175, 4357}, {2194, 2292}, {2206, 3704}, {2299, 22076}, {3063, 3882}, {3674, 14827}, {3939, 6371}, {9447, 20911}

Points associated with cubics pK(U*Y,U): X(31644)-X(31648)

This preamble and centers X(31644)-X(31648) were contributed by Clark Kimberling and Peter Moses, February 20, 2019.

Suppose that U = u : v : w (barycentrics) is a point in the plane of a triangle ABC. The self-inverse Gemini triangle based on U is defined (above, in the preamble just before X(31610) as the triangle with vertices

A' = - u : v + w : v + w, B' = - v : w + u : w + u, C' = - w : u + v : u + v .

The triangle A'B'C' is perspective to the anticevian triangle of a point X = x : y : z if and only if X lies on the cubic

(y - z) (y z u^2 + x^2 v w) + (z - x) (z x v^2 + y^2 w u) + (x - y) (x y w^2 + z^2 u v) = 0,

this being, in Bernard Gibert's notation (see Special Isocubics in the Triangle Plane (Section 1.3)) the cubic pK(U*Y, U), where

Y = 1/(v + w) : 1/(w + u) : 1/(u + v) = cevapoint of X(2) and U, and * = barycentric product.

Points on this cubic, which we denote by Γ*(U), include the following:

1 : 1 : 1 = X(2) = centroid
u : v : w = U
1/(v + w ) : 1/(w + u) : 1/(u + v) = cevapoint of X(2) and U = isotomic conjugate of the complement of U
u/(v + w ) : v/(w + u) : w/(u + v)

The appearance of

i	j(1), j(2), ..., j(n)	k	Kxxx

in the following table means that points on Γ*(X(i)) include X(j(1)), X(j(2)), ..., x(j(n)) and that Γ*(X(i)) = Kxxx .

1	1, 2, 7, 21, 29, 77, 81, 86	K317
3	2, 3, 54, 69, 95, 96, 97, 9723	K646
4	2, 4, 92, 253, 264, 273, 318, 342, 2052, 7020, 14249	K647
6	2, 4, 6, 83, 251, 1176, 1342, 1343, 8743	K644
8	2, 8, 75, 312, 556, 4373	K317
19	2, 19, 27, 28, 2322, 7219	x
40	2, 8, 40, 63, 7013, 7080	x
63	2, 63, 333, 1812, 2994, 7347, 7348	x
65	2, 7, 57, 65, 961, 1441, 2995, 14257	x
72	2, 8, 72, 78, 321, 943, 2997, 20336	x
113	2, 4, 113, 403, 3260, 14920	x
115	2, 115, 523, 31644	x
265	2, 94, 265, 328, 5627, 14254	x
523	2, 99, 523, 1113, 1114, 3413, 3414, 6189, 6190, 22339, 22340, 30508, 30509	K242
525	2, 525, 648, 2479, 2480, 2592, 2593, 8115, 8116	x
659	2, 100, 105, 659, 3570, 8709, 18793	x
684	2, 325, 684, 2421, 3265, 14941, 23181	x
690	2, 99, 523, 524, 671, 690, 892, 5466, 5468	x
887	2, 887, 1634, 3231, 9491, 23342	x
888	2, 512, 538, 670, 886, 888, 3228	x
891	2, 513, 536, 668, 889, 891, 3227	x
895	2, 671, 895, 10422, 14246, 30786	x
900	2, 190, 514, 519, 900, 903, 4555, 6548, 17780	x
1015	2, 1015, 513, 31645	x
1084	2, 1084, 512, 31646	x
1086	2, 1086, 514, 31647	x
1146	2, 1146, 522, 31648	x
1993	2, 275, 1585, 1586, 1993, 13579	x
2254	2, 291, 1025, 2254, 3676, 4444, 4551, 9436	x
2574	2, 4, 69, 1113, 2574, 2592, 8115, 15164, 22339	x
2575	2, 4, 69, 1114, 2575, 2593, 8116, 15165, 22340	x
3146	2, 253, 3146, 14572	K347
3569	2, 297, 694, 2501, 3569, 4230	x
3762	2, 3762, 4033, 4358, 4391, 18359, 24004	x
3768	2, 899, 1018, 3768, 20979, 23891	x
3904	2, 190, 3904, 4585, 4997, 13136	x
6148	2, 298, 299, 6148, 11128, 11129	x
6366	2, 522, 527, 664, 1121, 6366	x
8053	2, 86, 2141, 4184, 8053, 29767	x
9033	2, 30, 525, 648, 1494, 4240, 9033, 16077	x
10015	2, 88, 653, 655, 10015, 17924	x
14273	2, 112, 468, 2374, 4235, 14273	x
17475	1, 2, 6, 239, 4366, 17475	x

X(31644) = X(23)X(230)∩X(115)X(523)

Barycentrics (b^2 - c^2)^2*(a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :
X(31644) = 3 X[115] - X[23991],7 X[115] - X[23992],3 X[671] + X[4590],7 X[23991] - 3 X[23992]

X(31644) lies on these lines: {23, 230}, {115, 523}, {148, 14588}, {338, 3267}, {524, 5207}, {671, 4590}, {14113, 20188}

X(31644) = midpoint of X(148) and X(14588)
X(31644) = X(99)-Ceva conjugate of X(8029)
X(31644) = crosspoint of X(39022) and X(39023)
X(31644) = polar conjugate of isotomic conjugate of X(34953)
X(31644) = barycentric product X(i)*X(j) for these {i,j}: {115, 14061}, {2970, 14060}
X(31644) = barycentric quotient X(i) / X(j) for these {i,j}: {8754, 14052}, {14061, 4590}, {19598, 14588}

X(31645) = X(513)X(1015)∩X(536)X(19565)

Barycentrics a^2*(b - c)^2*(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(31645) lies on these lines: {513, 1015}, {536, 19565}, {1575, 19998}, {3248, 20979}, {7199, 16726}, {9336, 24338}

X(31645) = X(668)-Ceva conjugate of X(8027)
X(31645) = barycentric product X(1015)*X(27195)

X(31646) = X(512)X(1084)∩X(538)X(19566)

Barycentrics a^4*(b - c)^2*(b + c)^2*(2*a^4*b^4 - 3*a^4*b^2*c^2 - a^2*b^4*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 + b^4*c^4)::

X(31646) lies on these lines: {512, 1084}, {538, 19566}, {9427, 9491}

X(31646) = X(670)-Ceva conjugate of X(23610)

X(31647) = X(244)X(4905)∩X(514)X(1086)

Barycentrics (b - c)^2*(a^2 - a*b + 2*b^2 - a*c - 3*b*c + 2*c^2)::
3 X[903] + X[1016],3 X[1086] - X[6547],2 X[1086] + X[6549],2 X[6547] + 3 X[6549]

X(31647) lies on these lines: {244, 4905}, {514, 1086}, {519, 4645}, {903, 1016}, {1019, 17205}, {1111, 4391}, {1125, 24428}, {1358, 30719}, {3008, 3218}, {4013, 31129}, {17461, 25031}, {21188, 21208}, {23816, 27918}

X(31647) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 6545}, {903, 20042}
X(31647) = barycentric product X(i)*X(j) for these {i,j}: {1086, 27191}, {1111, 3315}, {6549, 20042}
X(31647) = barycentric quotient X(i) / X(j) for these {i,j}: {3315, 765}, {27191, 1016}

X(31648) = X(522)X(1146)∩X(1121)X(1275)

Barycentrics (b - c)^2*(-a + b + c)^2*(a^4 - a^3*b + a^2*b^2 - 3*a*b^3 + 2*b^4 - a^3*c - a^2*b*c + 3*a*b^2*c - b^3*c + a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 - 3*a*c^3 - b*c^3 + 2*c^4)::
X(31648) = 3 X[1121] + X[1275]

X(31648) lies on these lines: {522, 1146}, {1121, 1275}, {1566, 28147}, {4858, 24002}
X(31648) = X(664)-Ceva conjugate of X(23615)

X(31649) = EULER LINE INTERCEPT OF X(1)X(3652)

Barycentrics a (-2 a^6+2 a^5 (b+c)+3 b c (b^2-c^2)^2+4 a^4 (b^2-b c+c^2)-4 a^3 (b^3+c^3)+a^2 (-2 b^4+b^3 c-4 b^2 c^2+b c^3-2 c^4)+2 a (b^5-b^4 c-b c^4+c^5)) : :
Barycentrics R S^3 + (-8 a R^2+8 b R^2+2 a SB-2 b SB+2 a SC-2 c SC-2 b SW)S^2 -7 R S SB SC-2 b SB SC^2+2 c SB SC^2+2 b SB SC SW : :
X(31649) = X[1]+X[3652], 3*X[191]+X[7982], X[355]+X[5441], X[1385]+X[26202], X[1482]+X[11684], X[3065]+X[6265], 5*X[3616]-X[16116], X[3648]+3*X[5603], 3*X[5426]+X[7701], X[5690]-2*X[18253], X[5887]+X[17637], 2*X[6701]-3*X[11230], X[7991]-3*X[16139], 5*X[8227]-X[16118], 2*X[10122]-X[24475], X[12699]+X[16113], X[16132]+X[16138], X[16150]-5*X[18493]

As a point on the Euler line, X(31649) has Shinagawa coefficients {4 r + 3 R, -4 r + 3 R}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28880.

X(31649) lies on these lines: {1,3652}, {2,3}, {35,18357}, {56,11544}, {79,5427}, {191,7982}, {355,5441}, {515,22798}, {517,3647}, {758,10222}, {952,3746}, {993,22791}, {1125,12611}, {1385,26202}, {1482,11684}, {1483,3303}, {1621,26321}, {2771,5609}, {2975,3650}, {3065,6265}, {3304,10283}, {3579,3918}, {3616,16116}, {3648,5603}, {3649,5563}, {4265,18358}, {4653,5453}, {5267,9955}, {5298,27197}, {5426,7701}, {5690,18253}, {5887,17637}, {6246,9956}, {6701,11230}, {7173,14792}, {7991,16139}, {8227,16118}, {10058,11698}, {10122,24475}, {10175,26086}, {10593,14793}, {10902,28186}, {11375,16152}, {11376,16153}, {11518,24467}, {11545,14882}, {12047,18977}, {12699,16113}, {13391,22076}, {15446,15950}, {16132,16138}, {16150,18493}, {17768,20330}, {22938,25639}

X(31649) = midpoint of X(i) and X(j) for these {i,j}: {1,3652}, {3,21669}, {21,13743}, {355,5441}, {381,15678}, {1385,26202}, {1482,11684}, {3065,6265}, {5887,17637}, {16132,16138}, {28453,28461}
X(31649) = reflection of X(i) in X(j) for these {i,j}: {3,12104}, {5,16617}, {442,10021}, {549,15673}, {1483,15174}, {3649,5901}, {5428,21}, {5499,6675}, {5690,18253}, {16125,9955}, {19919,22936}, {24475,10122}
X(31649) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,21,12104}, {3,1656,17572}, {3,5047,140}, {3,6912,546}, {3,6920,3628}, {3,7489,5047}, {3,12104,5428}, {3,13743,21669}, {21,405,15673}, {21,3560,16617}, {21,3651,28443}, {21,5428,28463}, {21,15678,4189}, {21,21669,3}, {21,28461,13743}, {1012,19526,3}, {3560,6914,5}, {5047,6906,3}, {6906,7489,140}, {6913,6924,5}, {6914,16617,5428}, {6950,17572,3}, {13743,28453,21}

X(31650) = EULER LINE INTERCEPT OF X(495)X(5427)

Barycentrics 4 a^7-4 a^6 b-9 a^5 b^2+9 a^4 b^3+6 a^3 b^4-6 a^2 b^5-a b^6+b^7-4 a^6 c+a^4 b^2 c+3 a^3 b^3 c+4 a^2 b^4 c-3 a b^5 c-b^6 c-9 a^5 c^2+a^4 b c^2+8 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+9 a^4 c^3+3 a^3 b c^3+2 a^2 b^2 c^3+6 a b^3 c^3+3 b^4 c^3+6 a^3 c^4+4 a^2 b c^4+a b^2 c^4+3 b^3 c^4-6 a^2 c^5-3 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : :
Barycentrics 3 R S^3 + (12 a R^2-12 b R^2-5 a SB+5 b SB-5 a SC+5 c SC+2 a SW+3 b SW) S^2 +3 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : :
X(31650) = 2*X[1125]+X[22937], X[1483]+2*X[21677], 7*X[3624]-X[16159], X[5426]+X[26446], X[10264]+2*X[16164]

As a point on the Euler line, X(31650) has Shinagawa coefficients {10 r + 13 R, -3 (2 r + R)}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28882.

X(31650) lies on these lines: {2,3}, {495,5427}, {551,6583}, {1125,22937}, {1483,21677}, {1749,5444}, {2771,10165}, {3337,5298}, {3624,16159}, {3656,24468}, {4860,16137}, {5251,11698}, {5426,26446}, {5441,12019}, {5536,5901}, {10264,16164}, {10543,18395}, {10573,15174}, {15254,22936}

X(31650) = midpoint of X(i) and X(j) for these {i,j}: {2,28443}, {5054,15672}, {5426,26446}, {15670,28465}
X(31650) = reflection of X(549) in X(28465)
X(31650) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10021,16160}, {3,15674,10021}, {21,140,5499}, {549,632,17564}, {631,13743,11277}, {631,15676,13743}, {3523,16117,11276}, {5428,6675,5}

X(31651) = EULER LINE INTERCEPT OF X(79)X(5719)

Barycentrics 4 a^7-4 a^6 b-7 a^5 b^2+7 a^4 b^3+2 a^3 b^4-2 a^2 b^5+a b^6-b^7-4 a^6 c-8 a^5 b c-a^4 b^2 c+5 a^3 b^3 c+4 a^2 b^4 c+3 a b^5 c+b^6 c-7 a^5 c^2-a^4 b c^2+8 a^3 b^2 c^2-2 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+7 a^4 c^3+5 a^3 b c^3-2 a^2 b^2 c^3-6 a b^3 c^3-3 b^4 c^3+2 a^3 c^4+4 a^2 b c^4-a b^2 c^4-3 b^3 c^4-2 a^2 c^5+3 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
Barycentrics 5 R S^3 + (20 a R^2-20 b R^2-3 a SB+3 b SB-3 a SC+3 c SC-2 a SW+5 b SW)S^2 -11 R S SB SC+5 b SB SC^2-5 c SB SC^2-5 b SB SC SW : :
X(31651) = 2*X[12512]-X[22937]

As a point on the Euler line, X(31651) has Shinagawa coefficients 6 r + 11 R, -10 r - 21 R}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28882.

X(31651) lies on these lines: {{2,3}, {79,5719}, {2771,4067}, {3337,5441}, {3648,3940}, {4297,6583}, {5761,16150}, {10123,24929}, {10386,16137}, {10543,15935}, {11263,28146}, {11374,16118}, {12512,22937}, {16113,24466}, {18481,24468}

X(31651) = reflection of X(i) in X(j) for these {i,j}: {4,11277}, {21,548}, {3627,442}, {16160,3}, {22937,12512}, {28460,15690}
X(31651) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,382,6884}, {3,6894,140}

X(31652) = X(3)X(6)∩X(20)X(14537)

Barycentrics a^2 (-2 a^2+5 (b^2+c^2)) : :
Barycentrics 7 S^2-7 SB SC+3 SB SW+3 SC SW : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28884.

X(31652) lies on these lines: {3,6}, {20,14537}, {30,9698}, {99,6683}, {115,3628}, {140,7765}, {230,12108}, {232,14865}, {538,7824}, {546,1506}, {548,9300}, {549,7755}, {550,7753}, {620,7852}, {625,7847}, {631,5309}, {632,5254}, {671,16922}, {940,21538}, {1015,3746}, {1194,7496}, {1500,5563}, {1571,7982}, {1656,11648}, {1975,15482}, {2482,7819}, {2548,3529}, {2549,3090}, {3055,12812}, {3091,7603}, {3146,5475}, {3292,8041}, {3520,14581}, {3523,7739}, {3524,5319}, {3525,7738}, {3544,18424}, {3627,3815}, {3767,10303}, {3849,7858}, {3934,7783}, {4045,7874}, {4383,21517}, {5070,18362}, {5215,7827}, {5283,17572}, {5306,15712}, {5346,21843}, {6292,6390}, {6337,7822}, {7618,14001}, {7622,7902}, {7736,17538}, {7745,12103}, {7747,15704}, {7749,14869}, {7757,7780}, {7763,7853}, {7764,7873}, {7769,7861}, {7771,7805}, {7777,7842}, {7781,9466}, {7782,7804}, {7786,7816}, {7791,7821}, {7794,8359}, {7799,7849}, {7801,16043}, {7817,7907}, {7830,7845}, {7831,7895}, {7833,7843}, {7848,7906}, {7854,15810}, {7863,8362}, {7864,7886}, {7876,7880}, {7882,7904}, {7888,11287}, {7891,7915}, {7991,9619}, {8367,15300}, {8716,17130}, {12815,16239}, {14002,15302}, {14901,15034}, {15301,17128}, {16021,16022}, {16589,17531}

X(31652) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,39,5007}, {3,5007,187}, {3,9605,22331}, {3,22331,5206}, {3,22332,7772}, {6,574,15602}, {6,15515,15513}, {32,574,15815}, {32,5024,39}, {32,15815,8589}, {39,187,5041}, {39,5008,9605}, {39,8589,32}, {39,15513,6}, {39,15602,15513}, {549,9607,7755}, {550,9606,7753}, {574,5013,39}, {574,5024,8589}, {5013,15815,5024}, {5024,15815,32}, {5206,9605,5008}, {7764,8356,7873}, {7772,22332,39}, {7781,11285,9466}, {15513,15602,15515}

X(31653) = COMPLEMENT OF X(13395)

Barycentrics (-a+b+c) (b-c)^2 (a^2-b^2-c^2) (a^4-b^4+2 a^2 b c+2 a b^2 c+2 a b c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-4 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) : :
X(31653) = 3*X[2]-X[13395]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28887.

X(31653) lies on the nine-point circle and these lines: {2,13395}, {119,5778}, {120,26066}, {122,26933}, {125,6506}, {127,26932}, {1146,5521}, {3270,15607}, {5798,25640}

X(31653) = complement of X(13395)
X(31653) = complementary conjugate of isogonal conjugate of X(13395)
X(31653) = barycentric pproduct of X(377) and X(26956)
X(31653) = Kirikami six-circles image of X(63)
X(31653) = crosssum of circumcircle intercepts of line X(3)X(19)
X(31653) = orthopole of line X(3)X(19)
X(31653) = center of hyperbola {{A,B,C,X(4),X(63)}}

X(31654) = COMPLEMENT OF X(6082)

Barycentrics (b-c)^2 (b+c)^2 (-5 a^2+b^2+c^2) (-2 a^2+b^2+c^2) (a^2+b^2-3 b c+c^2) (a^2+b^2+3 b c+c^2) : :
Barycentrics (972 R^4+81 R^2 SB+81 R^2 SC-243 R^2 SW+9 SW^2)S^4 + (-81 R^2 SB SC SW-54 R^2 SB SW^2-54 R^2 SC SW^2+9 SB SC SW^2-36 R^2 SW^3+7 SW^4)S^2 + 3 SB SC SW^4+2 SB SW^5+2 SC SW^5 : :
X(31654) = 3*X[2]-X[6082], X[4]+X[6093], 2*X[5]-X[6092], X[14360]+X[14515]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28887.

X(31654) lies on the nine-point circle and these lines: {2,6082}, {4,6093}, {5,6092}, {114,5913}, {126,524}, {1499,2686}, {1648,5099}, {6792,16188}, {8176,9169}, {8288,20383}, {14360,14515}

X(31654) = midpoint of X(4) and X(6093)
X(31654) = reflection of X(6092) in X(5)
X(31654) = complement of X(6082)
X(31654) = complementary conjugate of X(6088)
X(31654) = crosssum of X(111) and X(1296)
X(31654) = crosspoint of X(524) and X(1499)
X(31654) = nine-point-circle-antipode of X(6092)
X(31654) = perspector of circumconic centered at X(9125)
X(31654) = barycentric quotient of X(9125) and X(6082)
X(31654) = reflection of X(i) in X(j) for these {i,j}: {6076,5512}, {6077,126}, {6092,5}, {6791,14858}

X(31655) = COMPLEMENT OF X(2770)

Barycentrics (-a^4 b^2+b^6-a^4 c^2+4 a^2 b^2 c^2-2 b^4 c^2-2 b^2 c^4+c^6) (2 a^6-2 a^4 b^2-3 a^2 b^4+b^6-2 a^4 c^2+8 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4-b^2 c^4+c^6) : :
Barycentrics (324 R^4-99 R^2 SW+7 SW^2)S^4 + (-27 R^2 SB SC SW-9 R^2 SB SW^2-9 R^2 SC SW^2+3 SB SC SW^2-6 R^2 SW^3+2 SB SW^3+2 SC SW^3+SW^4)S^2 + SB SC SW^4 : :
X(31655) = X[4]+X[2696], 2*X[6698]-X[16339]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28887.

X(31655) lies on the nine-point circle and these lines: {2,691}, {4,2696}, {30,5512}, {113,1499}, {115,858}, {125,524}, {126,523}, {127,5159}, {148,15398}, {373,2679}, {468,5139}, {1560,2489}, {2072,14672}, {2453,11336}, {3815,9193}, {5094,16221}, {5476,9169}, {5912,23991}, {6698,16339}, {7472,10418}, {8371,16188}, {9127,12494}, {11594,13994}, {14568,20389}, {16051,16177}

X(31655) = midpoint of X(858) and X(5913)
X(31655) = reflection of X(16339) in X(6698)
X(31655) = complement of X(2770)
X(31655) = complementary conjugate of X(2854)
X(31655) = perspector of circumconic centered at X(10418)
X(31655) = barycentric quotient of X(10418) and X(2770)
X(31655) = reflection of X(126) in Euler line
X(31655) = X(110)-of-5th-Euler-triangle
X(31655) = X(1296)-of-reflection-of-Euler-triangle-in-Euler-line
X(31655) = inverse-in-polar-circle of Ψ(X(3), X(111))
X(31655) = inverse-in-orthoptic-circle-of-Steiner-inellipe of X(691)
X(31655) = orthopole of PU(63)
X(31655) = crosssum of circumcircle intercepts of line PU(63) (line X(3)X(351))

X(31656) = MIDPOINT OF X(382) AND X(13512)

Barycentrics    a^16-3 a^14 b^2+a^12 b^4+6 a^10 b^6-10 a^8 b^8+9 a^6 b^10-7 a^4 b^12+4 a^2 b^14-b^16-3 a^14 c^2+4 a^12 b^2 c^2+4 a^10 b^4 c^2-8 a^8 b^6 c^2+11 a^4 b^10 c^2-13 a^2 b^12 c^2+5 b^14 c^2+a^12 c^4+4 a^10 b^2 c^4-3 a^8 b^4 c^4-7 a^4 b^8 c^4+15 a^2 b^10 c^4-10 b^12 c^4+6 a^10 c^6-8 a^8 b^2 c^6+6 a^4 b^6 c^6-6 a^2 b^8 c^6+11 b^10 c^6-10 a^8 c^8-7 a^4 b^4 c^8-6 a^2 b^6 c^8-10 b^8 c^8+9 a^6 c^10+11 a^4 b^2 c^10+15 a^2 b^4 c^10+11 b^6 c^10-7 a^4 c^12-13 a^2 b^2 c^12-10 b^4 c^12+4 a^2 c^14+5 b^2 c^14-c^16 : :
Barycentrics    S^4 + (3 R^4-5 R^2 SB-5 R^2 SC+SB SC+R^2 SW+2 SB SW+2 SC SW-SW^2)S^2 -27 R^4 SB SC+23 R^2 SB SC SW-5 SB SC SW^2 : :
Trilinears    (6*cos(2*A)+4*cos(4*A)+5)*cos(B-C)-2*(cos(A)+cos(3*A))*cos(2*(B-C))+(2*cos(2*A)+1)*cos(3*(B-C))-cos(5*A)-3*cos(A)-cos(3*A) : :
X(31656) = X[3]-2*X[128], 3*X[4]-X[11671], 2*X[137]-3*X[381], 2*X[140]-3*X[23237], X[382]+X[13512], 2*X[546]-X[1263], X[550]-2*X[6592], X[3627]+X[14073], 4*X[3850]-3*X[25147], 7*X[3851]-6*X[23516], 2*X[3853]+X[23238], 3*X[11459]-X[13504], X[12111]+X[13505], X[12254]-2*X[14071]

See Tran Quang Hung, Ercole Suppa and César Lozada AdGeom 5175, AdGeom 5176.

X(31656) lies on the circumcircle of Johnson triangle and these lines: {3,128}, {4,11671}, {5,49}, {30,930}, {137,381}, {140,23237}, {382,13512}, {523,14980}, {539,16337}, {546,1263}, {550,6592}, {1209,11016}, {1478,3327}, {1479,7159}, {2070,23181}, {2072,23319}, {3153,20957}, {3627,14073}, {3652,10747}, {3850,25147}, {3851,23516}, {3853,23238}, {11459,13504}, {12111,13505}, {12254,14071}, {13556,22823}, {14050,18439}, {14142,21230}, {16336,18400}

X(31656) = midpoint of X(i) and X(j) for these {i,j}: {382, 13512}, {3627, 14073}, {12111, 13505}
X(31656) = reflection of X(i) in X(j) for these (i,j): (3, 128), (550, 6592), (930, 14072), (1141, 5), (1263, 546), (12254, 14071), (14674, 14769), (19553, 16337)
X(31656) = reflection of X(265) in the line X(4)X(14050)
X(31656) = (Ehrmann-side)-isogonal conjugate of-X(1510)
X(31656) = (Johnson)-isogonal conjugate of-X(1154)

X(31657) = COMPLEMENT OF X(5779)

Barycentrics 2 a^5 b-3 a^4 b^2-2 a^3 b^3+4 a^2 b^4-b^6+2 a^5 c+4 a^4 b c-2 a^3 b^2 c-6 a^2 b^3 c+2 b^5 c-3 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^3-6 a^2 b c^3-4 b^3 c^3+4 a^2 c^4+b^2 c^4+2 b c^5-c^6 : :
X(31657) = 3*X[2]-X[5779], X[3]+X[7], X[5]-2*X[142], X[9]-2*X[140], X[144]-5*X[631], 2*X[548]+X[5735], 5*X[632]-4*X[6666], X[1482]-3*X[11038], 5*X[1656]-3*X[5817], X[2951]+X[12699], X[3062]-5*X[8227], 7*X[3523]+X[20059], 7*X[3526]-5*X[18230], 4*X[3530]-3*X[21153], 3*X[3576]+X[4312], X[3627]-2*X[18482], 4*X[3628]-5*X[20195], 3*X[5054]-X[6172], X[5223]-3*X[26446], X[5572]-2*X[13373], X[6916]+X[15934], X[6987]+X[18541], 7*X[9624]-3*X[24644], X[12675]+X[15587], 5*X[15016]-X[18412], X[15937]+X[15970], X[20430]-3*X[27475]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28890.

X(31657) lies on these lines: {2,5779}, {3,7}, {5,142}, {9,140}, {30,5732}, {40,5586}, {55,24465}, {144,631}, {182,5845}, {226,11227}, {390,6948}, {442,10861}, {495,8581}, {496,14100}, {516,550}, {517,5542}, {518,5690}, {527,549}, {548,5735}, {632,6666}, {952,2550}, {990,4675}, {991,1086}, {1001,6914}, {1071,8728}, {1482,11038}, {1483,5853}, {1656,5817}, {2096,16418}, {2346,11849}, {2801,3826}, {2808,20328}, {2951,12699}, {3062,8227}, {3243,5844}, {3452,10156}, {3475,6244}, {3517,7717}, {3523,20059}, {3526,18230}, {3530,21153}, {3576,4312}, {3627,18482}, {3628,20195}, {3649,7987}, {3742,7956}, {3834,12618}, {4326,15172}, {4654,10857}, {5054,6172}, {5219,11407}, {5220,26487}, {5223,26446}, {5249,8727}, {5428,17768}, {5572,13373}, {5696,6067}, {5708,6908}, {5728,6907}, {5729,6863}, {5768,17528}, {5785,5791}, {5811,16853}, {5850,6684}, {5851,6713}, {5886,7171}, {6825,8732}, {6842,10394}, {6846,12684}, {6850,12433}, {6881,12669}, {6891,8232}, {6916,15934}, {6922,21617}, {6954,12848}, {6955,9945}, {6987,18541}, {7263,29016}, {7411,26842}, {8158,11037}, {8226,11220}, {9624,24644}, {9776,19541}, {10004,28344}, {10267,11495}, {10384,11373}, {10884,20420}, {10943,17668}, {11019,15008}, {11112,18444}, {11246,15931}, {12528,17529}, {12675,15587}, {13329,17365}, {13727,26806}, {15016,18412}, {15299,15325}, {15726,27869}, {15937,15970}, {16112,26492}, {18857,28534}, {20430,27475}, {24390,25722}

X(31657) = complement of X(5779)
X(31657) = midpoint of X(i) in X(j) for these {i,j}: {3,7}, {5732,5805}, {6916,15934}, {6987,18541}, {12675,15587}, {15937,15970}
X(31657) = reflection of X(i) in X(j) for these {i,j}: {5,142}, {9,140}, {3627,18482}, {5572,13373}, {20330,25557}, {22791,20330}
X(31657) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,6147,5763}, {7,21151,3}, {3523,20059,21168}, {3824,6245,5}, {5249,10167,8727}, {5732,6173,5805}, {11220,27186,8226}

X(31658) = COMPLEMENT OF X(5805)

Barycentrics a (2 a^5-3 a^4 b-2 a^3 b^2+4 a^2 b^3-b^5-3 a^4 c-2 a^3 b c+4 a^2 b^2 c+2 a b^3 c-b^4 c-2 a^3 c^2+4 a^2 b c^2-4 a b^2 c^2+2 b^3 c^2+4 a^2 c^3+2 a b c^3+2 b^2 c^3-b c^4-c^5) : :
X(31658) = 3*X[2]+X[5759], X[3]+X[9], X[4]-5*X[18230], X[7]-5*X[631], X[20]+3*X[5817], 2*X[140]-X[142], X[144]+7*X[3523], X[390]+3*X[5657], X[944]+3*X[5686], X[3062]+7*X[16192], X[3243]-3*X[10246], 3*X[3524]+X[6172], 7*X[3526]-X[5735], X[3587]+X[6913], 3*X[5054]-X[6173], X[5220]+2*X[13624], X[6068]+3*X[21154], 3*X[8236]+X[12245]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28890.

X(31658) lies on these lines: {1,15837}, {2,5759}, {3,9}, {4,18230}, {5,516}, {7,631}, {20,5817}, {35,14100}, {36,8581}, {37,13329}, {40,5806}, {44,991}, {45,990}, {55,15299}, {56,15298}, {57,10156}, {63,11227}, {72,6986}, {140,142}, {144,3523}, {165,3683}, {182,518}, {210,5531}, {390,5657}, {495,12573}, {517,1001}, {527,549}, {528,12619}, {942,954}, {944,5686}, {952,6594}, {962,17552}, {1006,5728}, {1071,26878}, {1125,5763}, {1155,4312}, {1214,2954}, {1538,4679}, {1699,7964}, {1708,11018}, {2550,6827}, {2646,18412}, {2801,15481}, {2808,28345}, {3035,3452}, {3059,10902}, {3062,16192}, {3088,7717}, {3219,10167}, {3243,10246}, {3305,7580}, {3419,6992}, {3524,6172}, {3526,5735}, {3530,5843}, {3576,3940}, {3587,6913}, {3601,10398}, {3748,15104}, {3824,5812}, {3927,8726}, {3929,10857}, {4188,10861}, {4297,5302}, {4304,10392}, {4422,12618}, {4512,6244}, {5054,6173}, {5122,8545}, {5220,13624}, {5259,7957}, {5542,5719}, {5584,9856}, {5690,5853}, {5698,6825}, {5729,7675}, {5791,6865}, {5832,6958}, {5856,6713}, {5901,14150}, {5927,7411}, {6068,21154}, {6282,16418}, {6600,10267}, {6875,10394}, {6887,12699}, {6940,29007}, {6985,11495}, {7677,24928}, {8071,15518}, {8128,8389}, {8236,12245}, {9581,9588}, {9709,10268}, {9940,26921}, {10225,28534}, {10386,15006}, {10884,15650}, {11230,25379}, {11362,12433}, {12260,20790}, {12572,22792}, {13727,17260}, {15296,26286}, {15297,26285}, {15587,18233}, {15935,28234}, {17348,29016}, {17768,22937}

X(31658) = complement of X(5805)
X(31658) = midpoint of X(i) in X(j) for these {i,j}: {3,9}, {3587,6913}, {5732,5779}, {5759,5805}
X(31658) = reflection of X(i) in X(j) for these {i,j}: {5,6666}, {142,140}, {18482,5}, {20330,1125}
X(31658) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,5759,5805}, {3,5779,5732}, {9,5438,5785}, {9,5732,5779}, {9,21153,3}, {40,11108,5806}, {144,3523,21151}, {165,7308,19541}, {631,21168,7}, {954,1445,942}, {3305,7580,10157}, {5812,6989,3824}, {7411,27065,5927}

X(31659) = COMPLEMENT OF X(26470)

Barycentrics 2 a^7-2 a^6 b-5 a^5 b^2+5 a^4 b^3+4 a^3 b^4-4 a^2 b^5-a b^6+b^7-2 a^6 c+3 a^4 b^2 c-2 a^3 b^3 c+2 a b^5 c-b^6 c-5 a^5 c^2+3 a^4 b c^2+4 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+5 a^4 c^3-2 a^3 b c^3+4 a^2 b^2 c^3-4 a b^3 c^3+3 b^4 c^3+4 a^3 c^4+a b^2 c^4+3 b^3 c^4-4 a^2 c^5+2 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : :
Barycentrics 2 R S^2 + (3 a SB-3 b SB-c SB+3 a SC-b SC-3 c SC-2 a SW)S + 2 R SB SC : :
X(31659) = X[3]+X[12], 5*X[631]-X[2975], 7*X[3523]+X[20060], 3*X[3584]+X[11012], 3*X[4995]-X[11849]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28890.

X(31659) lies on these lines: {2,10267}, {3,12}, {5,5248}, {10,140}, {21,119}, {35,5840}, {55,6863}, {78,26446}, {100,6853}, {182,5849}, {355,7483}, {495,15865}, {499,10959}, {517,13411}, {529,549}, {631,2975}, {758,6684}, {993,10942}, {1001,6959}, {1006,27529}, {1621,6949}, {1737,24299}, {3085,6954}, {3523,20060}, {3526,19854}, {3530,23961}, {3584,11012}, {3652,13257}, {3884,11729}, {3898,5901}, {3911,13373}, {4309,11928}, {4995,11849}, {4996,6940}, {5217,6923}, {5218,6825}, {5433,10246}, {5554,17566}, {5690,5855}, {5790,24953}, {5885,15556}, {6284,6980}, {6675,9956}, {6745,9940}, {6824,18491}, {6862,11500}, {6868,10588}, {6875,11681}, {6882,10902}, {6883,26364}, {6889,11517}, {6892,18761}, {6907,26285}, {6910,10786}, {6911,10198}, {6914,18242}, {6924,25466}, {6934,10585}, {6998,26231}, {7491,7951}, {8070,14795}, {10056,10680}, {10225,11277}, {12639,22937}, {13226,26201}, {13747,19860}, {15178,15325}, {15867,16203}, {15888,22765}, {16617,19925}

X(31659) = complement of X(26470)
X(31659) = midpoint of X(i) in X(j) for these {i,j}: {3,12}, {35,6842}, {11491,26470}, {11849,15908}
X(31659) = reflection of X(i) in X(j) for these {i,j}: {5,6668}, {4999,140}
X(31659) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,11491,26470}, {12,21155,3}, {140,1385,6713}, {3085,6954,11249}, {4995,15908,11849}, {5218,6825,11248}, {6910,10786,22758}

X(31660) = MIDPOINT OF X(21) AND X(3871)

Barycentrics a^2 (a^5-b^5-b^4 c+b^3 c^2+b^2 c^3-b c^4-c^5-a^4 (b+c)-2 a^3 (b^2+b c+c^2)+a (b^2+b c+c^2)^2+2 a^2 (b^3+b^2 c+b c^2+c^3)) : :
Barycentrics 2 R S^2 + (5 a R^2-b R^2-c R^2-a SB+b SB+c SB-a SC+b SC+c SC)S -2 R SB SC : :

See Tran Quang Hung and Ercole Suppa, AdGeom 5181.

X(31660) lies on these lines: {1,27086}, {8,21}, {30,11491}, {35,758}, {100,442}, {191,3811}, {404,11281}, {1392,6767}, {1621,6675}, {2475,3085}, {2646,4996}, {2802,3746}, {3241,10267}, {3647,3935}, {3649,14882}, {3651,5758}, {3874,14799}, {3957,14798}, {4421,6175}, {4428,15671}, {5010,16126}, {5428,5844}, {5703,11507}, {7676,17768}, {9780,11517}, {10528,15680}, {11510,20057}, {12867,13615}, {13607,17009}, {15674,19843}, {25440,26725}

X(31660) = midpoint of X(21) and X(3871)
X(31660) = reflection of X(24390) in X(6675)}

X(31661) = (name pending)

Barycentrics 77 a^10-205 a^8 b^2+101 a^6 b^4+116 a^4 b^6-124 a^2 b^8+35 b^10-205 a^8 c^2+107 a^6 b^2 c^2+90 a^4 b^4 c^2+47 a^2 b^6 c^2-91 b^8 c^2+101 a^6 c^4+90 a^4 b^2 c^4+90 a^2 b^4 c^4+56 b^6 c^4+116 a^4 c^6+47 a^2 b^2 c^6+56 b^4 c^6-124 a^2 c^8-91 b^2 c^8+35 c^10 : :
Barycentrics (1188 R^2-27 SB-27 SC-414 SW)S^4 + (-324 R^2 SB SC+162 SB SC SW-9 SB SW^2-9 SC SW^2+16 SW^3)S^2 +12 SB SC SW^3 : :

See Tran Quang Hung and Ercole Suppa, AdGeom 5182.

X(31661) lies on this line: {15694,24206}

X(31662) = MIDPOINT OF X(1385) AND X(3576)

Barycentrics a (12 a^3-5 a^2 b-12 a b^2+5 b^3-5 a^2 c+10 a b c-5 b^2 c-12 a c^2-5 b c^2+5 c^3) : :
X(31662) = 5*X[1]+7*X[3], 5*X[355]-17*X[3533], X[548]+2*X[3636], 5*X[551]+X[15686], 5*X[1125]-2*X[3850], X[3244]+5*X[15712], 49*X[3523]-X[20054], 2*X[3530]+X[13607], X[3543]-5*X[5886], X[3545]-5*X[3653], X[3626]-4*X[12108], X[3627]-7*X[15808], X[3655]+X[11231], 5*X[3817]-3*X[3845], 7*X[3832]+5*X[18481], 2*X[3853]-5*X[9955], 11*X[5056]-5*X[18480], X[5059]+5*X[22793], 7*X[5657]+X[20049], 5*X[7967]+11*X[15719], 5*X[9812]+3*X[11001], 5*X[9956]-8*X[16239], 5*X[10164]-7*X[19711], 5*X[10165]-3*X[11539], 9*X[15708]-5*X[26446]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28895.

X(31662) lies on these lines: {1,3}, {104,28180}, {140,28236}, {214,3740}, {355,3533}, {515,547}, {516,15690}, {548,3636}, {551,15686}, {952,4745}, {1125,3850}, {2771,28463}, {3244,15712}, {3523,20054}, {3530,13607}, {3543,5886}, {3545,3653}, {3626,12108}, {3627,15808}, {3655,11231}, {3817,3845}, {3832,18481}, {3853,9955}, {5056,18480}, {5059,22793}, {5603,28202}, {5657,20049}, {5901,28150}, {7967,15719}, {9812,11001}, {9956,16239}, {10164,19711}, {10165,11539}, {10283,28198}, {12100,28234}, {13464,28216}, {15708,26446}, {17536,17614}, {19538,19861}

X(31662) = midpoint of X(i) and X(j) for these {i,j}: {1385,3576}, {3655,11231}, {10246,17502}
X(31662) = reflection of X(13624) in X(576)
X(31662) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10246,16200}, {3,11531,3579}, {1385,13624,15178}, {1385,17502,10246}, {3576,10246,17502}, {3653,5731,11230}

X(31663) = COMPLEMENT OF X(22793)

Barycentrics    a (4 a^3+a^2 b-4 a b^2-b^3+a^2 c-2 a b c+b^2 c-4 a c^2+b c^2-c^3) : :
Trilinears    4 cos A - cos B - cos C + 1 : :
Trilinears    r - 5 R cos A : :

X(31663) = X[1] - 5 X[3],3 X[1] + 5 X[40],X[1] + 15 X[165],3 X[1] - 5 X[1385],9 X[1] - 5 X[1482],7 X[1] - 15 X[3576],X[1] + 5 X[3579],13 X[1] - 5 X[7982],9 X[1] - 25 X[7987],11 X[1] + 5 X[7991],17 X[1] - 5 X[8148],7 X[1] - 5 X[10222],11 X[1] - 15 X[10246],19 X[1] - 15 X[10247],31 X[1] - 15 X[11224],11 X[1] - 5 X[11278],21 X[1] - 5 X[11531],7 X[1] + 5 X[12702],2 X[1] - 5 X[13624],4 X[1] - 5 X[15178],41 X[1] - 25 X[16189],77 X[1] - 45 X[16191],3 X[1] - 35 X[16192],23 X[1] - 15 X[16200],X[1] - 3 X[17502],3 X[3] + X[40],X[3] + 3 X[165],3 X[3] - X[1385],9 X[3] - X[1482],7 X[3] - 3 X[3576],13 X[3] - X[7982],9 X[3] - 5 X[7987],11 X[3] + X[7991],17 X[3] - X[8148],7 X[3] - X[10222],11 X[3] - 3 X[10246],19 X[3] - 3 X[10247],31 X[3] - 3 X[11224],11 X[3] - X[11278],21 X[3] - X[11531],7 X[3] + X[12702],4 X[3] - X[15178],41 X[3] - 5 X[16189],77 X[3] - 9 X[16191],3 X[3] - 7 X[16192],23 X[3] - 3 X[16200],5 X[3] - 3 X[17502],X[4] - 3 X[11231],5 X[4] - 13 X[19877],X[5] - 3 X[10164],X[8] + 7 X[3528],3 X[20] + 5 X[5818],X[20] + 3 X[26446],X[20] + 5 X[31447],X[40] - 9 X[165],3 X[40] + X[1482],7 X[40] + 9 X[3576],X[40] - 3 X[3579],13 X[40] + 3 X[7982],3 X[40] + 5 X[7987],11 X[40] - 3 X[7991],17 X[40] + 3 X[8148],7 X[40] + 3 X[10222],11 X[40] + 9 X[10246],19 X[40] + 9 X[10247],31 X[40] + 9 X[11224],11 X[40] + 3 X[11278],7 X[40] + X[11531],7 X[40] - 3 X[12702],2 X[40] + 3 X[13624],4 X[40] + 3 X[15178],41 X[40] + 15 X[16189],77 X[40] + 27 X[16191],X[40] + 7 X[16192],23 X[40] + 9 X[16200],5 X[40] + 9 X[17502],5 X[140] - 4 X[19878],9 X[165] + X[1385],27 X[165] + X[1482],7 X[165] + X[3576],3 X[165] - X[3579],39 X[165] + X[7982],27 X[165] + 5 X[7987],33 X[165] - X[7991],51 X[165] + X[8148],21 X[165] + X[10222],11 X[165] + X[10246],19 X[165] + X[10247],31 X[165] + X[11224],33 X[165] + X[11278],63 X[165] + X[11531],21 X[165] - X[12702],6 X[165] + X[13624],12 X[165] + X[15178],123 X[165] + 5 X[16189],77 X[165] + 3 X[16191],9 X[165] + 7 X[16192],23 X[165] + X[16200],5 X[165] + X[17502],X[355] + 3 X[376],3 X[381] - 7 X[31423],X[382] - 5 X[1698],X[382] - 13 X[31425],3 X[547] - 2 X[12571],5 X[548] + 2 X[4691],3 X[549] - X[946],5 X[549] - 3 X[19883],X[551] - 3 X[17504],5 X[631] + 3 X[9778],5 X[631] - 3 X[11230],5 X[631] - X[12699],5 X[632] - 3 X[3817],X[944] + 3 X[3654],X[944] - 9 X[10304],5 X[944] + 3 X[31145],5 X[946] - 9 X[19883],X[962] - 9 X[3524],3 X[1385] - X[1482],7 X[1385] - 9 X[3576],X[1385] + 3 X[3579],13 X[1385] - 3 X[7982],3 X[1385] - 5 X[7987],11 X[1385] + 3 X[7991],17 X[1385] - 3 X[8148],7 X[1385] - 3 X[10222],11 X[1385] - 9 X[10246],19 X[1385] - 9 X[10247],31 X[1385] - 9 X[11224],11 X[1385] - 3 X[11278],7 X[1385] - X[11531],7 X[1385] + 3 X[12702],2 X[1385] - 3 X[13624],4 X[1385] - 3 X[15178],41 X[1385] - 15 X[16189],77 X[1385] - 27 X[16191],X[1385] - 7 X[16192],23 X[1385] - 9 X[16200],5 X[1385] - 9 X[17502],X[1386] - 3 X[17508],7 X[1482] - 27 X[3576],X[1482] + 9 X[3579],13 X[1482] - 9 X[7982],X[1482] - 5 X[7987],11 X[1482] + 9 X[7991],17 X[1482] - 9 X[8148],7 X[1482] - 9 X[10222],11 X[1482] - 27 X[10246],19 X[1482] - 27 X[10247],31 X[1482] - 27 X[11224],11 X[1482] - 9 X[11278],7 X[1482] - 3 X[11531],7 X[1482] + 9 X[12702],2 X[1482] - 9 X[13624],4 X[1482] - 9 X[15178],41 X[1482] - 45 X[16189],77 X[1482] - 81 X[16191],X[1482] - 21 X[16192],23 X[1482] - 27 X[16200],5 X[1482] - 27 X[17502],X[1657] + 3 X[5587],5 X[1698] - 13 X[31425],3 X[1699] - 7 X[3526],X[2100] + 3 X[28448],X[2101] + 3 X[28447],X[2948] + 3 X[15041],X[3241] - 9 X[15710],25 X[3522] + 7 X[4678],5 X[3522] + 3 X[5657],5 X[3522] - X[18481],7 X[3523] - 3 X[5886],7 X[3523] + X[6361],11 X[3525] - 3 X[9812],X[3529] + 7 X[9780],17 X[3533] - 9 X[9779],3 X[3534] + X[5691],3 X[3576] + 7 X[3579],39 X[3576] - 7 X[7982],27 X[3576] - 35 X[7987],33 X[3576] + 7 X[7991],51 X[3576] - 7 X[8148],3 X[3576] - X[10222],11 X[3576] - 7 X[10246],19 X[3576] - 7 X[10247],31 X[3576] - 7 X[11224],33 X[3576] - 7 X[11278],9 X[3576] - X[11531],3 X[3576] + X[12702],6 X[3576] - 7 X[13624],12 X[3576] - 7 X[15178],123 X[3576] - 35 X[16189],11 X[3576] - 3 X[16191],9 X[3576] - 49 X[16192],23 X[3576] - 7 X[16200],5 X[3576] - 7 X[17502],13 X[3579] + X[7982],9 X[3579] + 5 X[7987],11 X[3579] - X[7991],17 X[3579] + X[8148],7 X[3579] + X[10222],11 X[3579] + 3 X[10246],19 X[3579] + 3 X[10247],31 X[3579] + 3 X[11224],11 X[3579] + X[11278],21 X[3579] + X[11531],7 X[3579] - X[12702],2 X[3579] + X[13624],4 X[3579] + X[15178],41 X[3579] + 5 X[16189],77 X[3579] + 9 X[16191],3 X[3579] + 7 X[16192],23 X[3579] + 3 X[16200],5 X[3579] + 3 X[17502],5 X[3616] - 13 X[10299],7 X[3624] - 11 X[15720],X[3627] - 3 X[10175],5 X[3627] - 19 X[22266],3 X[3651] + X[3652],X[3652] - 3 X[22937],9 X[3653] - 5 X[10595],3 X[3653] - 7 X[15698],X[3654] + 3 X[10304],5 X[3654] - X[31145],3 X[3655] + X[12245],X[3655] - 5 X[19708],X[3656] - 5 X[15692],3 X[3656] + X[20070],X[3679] + 3 X[15688],3 X[3828] - 5 X[6684],6 X[3828] - 5 X[9956],3 X[3828] + 5 X[12512],9 X[3828] - 5 X[19925],3 X[3830] - 7 X[7989],2 X[3850] - 3 X[10172],5 X[4297] + 3 X[4669],X[4297] - 3 X[8703],3 X[4669] - 5 X[5690],X[4669] + 5 X[8703],7 X[4678] - 15 X[5657],7 X[4678] + 5 X[18481],9 X[5054] - 5 X[8227],15 X[5071] - 7 X[10248],X[5073] - 5 X[18492],13 X[5079] - 17 X[19872],X[5493] + 3 X[10165],X[5493] + 5 X[15712],3 X[5603] - 11 X[15717],3 X[5657] + X[18481],X[5690] + 3 X[8703],15 X[5731] + X[20053],3 X[5731] - 11 X[21735],3 X[5790] - 7 X[9588],3 X[5790] + 5 X[15696],5 X[5818] - 3 X[18480],5 X[5818] - 9 X[26446],X[5818] - 3 X[31447],3 X[5886] + X[6361],X[5901] - 3 X[12100],3 X[6684] - X[19925],X[7957] + 3 X[10202],9 X[7982] - 65 X[7987],11 X[7982] + 13 X[7991],17 X[7982] - 13 X[8148],7 X[7982] - 13 X[10222],11 X[7982] - 39 X[10246],19 X[7982] - 39 X[10247],31 X[7982] - 39 X[11224],11 X[7982] - 13 X[11278],21 X[7982] - 13 X[11531],7 X[7982] + 13 X[12702],2 X[7982] - 13 X[13624],4 X[7982] - 13 X[15178],41 X[7982] - 65 X[16189],77 X[7982] - 117 X[16191],3 X[7982] - 91 X[16192],23 X[7982] - 39 X[16200],5 X[7982] - 39 X[17502],55 X[7987] + 9 X[7991],85 X[7987] - 9 X[8148],35 X[7987] - 9 X[10222],55 X[7987] - 27 X[10246],95 X[7987] - 27 X[10247],155 X[7987] - 27 X[11224],55 X[7987] - 9 X[11278],35 X[7987] - 3 X[11531],35 X[7987] + 9 X[12702],10 X[7987] - 9 X[13624],20 X[7987] - 9 X[15178],41 X[7987] - 9 X[16189],385 X[7987] - 81 X[16191],5 X[7987] - 21 X[16192],115 X[7987] - 27 X[16200],25 X[7987] - 27 X[17502],17 X[7991] + 11 X[8148],7 X[7991] + 11 X[10222],X[7991] + 3 X[10246],19 X[7991] + 33 X[10247],31 X[7991] + 33 X[11224],21 X[7991] + 11 X[11531],7 X[7991] - 11 X[12702],2 X[7991] + 11 X[13624],4 X[7991] + 11 X[15178],41 X[7991] + 55 X[16189],7 X[7991] + 9 X[16191],3 X[7991] + 77 X[16192],23 X[7991] + 33 X[16200],5 X[7991] + 33 X[17502],7 X[8148] - 17 X[10222],11 X[8148] - 51 X[10246],19 X[8148] - 51 X[10247],31 X[8148] - 51 X[11224],11 X[8148] - 17 X[11278],21 X[8148] - 17 X[11531],7 X[8148] + 17 X[12702],2 X[8148] - 17 X[13624],4 X[8148] - 17 X[15178],41 X[8148] - 85 X[16189],77 X[8148] - 153 X[16191],3 X[8148] - 119 X[16192],23 X[8148] - 51 X[16200],5 X[8148] - 51 X[17502],7 X[9588] + 5 X[15696],X[9589] - 5 X[18493],3 X[9778] + X[12699],5 X[9955] - 8 X[19878],X[9956] + 2 X[12512],3 X[9956] - 2 X[19925],3 X[10165] - 5 X[15712],3 X[10165] - X[22791],3 X[10171] - 4 X[16239],15 X[10175] - 19 X[22266],3 X[10178] - X[13369],11 X[10222] - 21 X[10246],19 X[10222] - 21 X[10247],31 X[10222] - 21 X[11224],11 X[10222] - 7 X[11278],3 X[10222] - X[11531],2 X[10222] - 7 X[13624],4 X[10222] - 7 X[15178],41 X[10222] - 35 X[16189],11 X[10222] - 9 X[16191],3 X[10222] - 49 X[16192],23 X[10222] - 21 X[16200],5 X[10222] - 21 X[17502],19 X[10246] - 11 X[10247],31 X[10246] - 11 X[11224],3 X[10246] - X[11278],63 X[10246] - 11 X[11531],21 X[10246] + 11 X[12702],6 X[10246] - 11 X[13624],12 X[10246] - 11 X[15178],123 X[10246] - 55 X[16189],7 X[10246] - 3 X[16191],9 X[10246] - 77 X[16192],23 X[10246] - 11 X[16200],5 X[10246] - 11 X[17502],31 X[10247] - 19 X[11224],33 X[10247] - 19 X[11278],63 X[10247] - 19 X[11531],21 X[10247] + 19 X[12702],6 X[10247] - 19 X[13624],12 X[10247] - 19 X[15178],123 X[10247] - 95 X[16189],77 X[10247] - 57 X[16191],9 X[10247] - 133 X[16192],23 X[10247] - 19 X[16200],5 X[10247] - 19 X[17502],15 X[10304] + X[31145],5 X[10595] - 21 X[15698],3 X[11204] - X[12262],33 X[11224] - 31 X[11278],63 X[11224] - 31 X[11531],21 X[11224] + 31 X[12702],6 X[11224] - 31 X[13624],12 X[11224] - 31 X[15178],123 X[11224] - 155 X[16189],77 X[11224] - 93 X[16191],9 X[11224] - 217 X[16192],23 X[11224] - 31 X[16200],5 X[11224] - 31 X[17502],3 X[11230] - X[12699],15 X[11231] - 13 X[19877],21 X[11278] - 11 X[11531],7 X[11278] + 11 X[12702],2 X[11278] - 11 X[13624],4 X[11278] - 11 X[15178],41 X[11278] - 55 X[16189],7 X[11278] - 9 X[16191],3 X[11278] - 77 X[16192],23 X[11278] - 33 X[16200],5 X[11278] - 33 X[17502],X[11531] + 3 X[12702],2 X[11531] - 21 X[13624],4 X[11531] - 21 X[15178],41 X[11531] - 105 X[16189],11 X[11531] - 27 X[16191],X[11531] - 49 X[16192],23 X[11531] - 63 X[16200],5 X[11531] - 63 X[17502],X[11699] - 3 X[15035],X[11826] + 3 X[28459],X[11827] + 3 X[28458],X[12245] + 15 X[19708],3 X[12512] + X[19925],X[12645] + 15 X[14093],2 X[12702] + 7 X[13624],4 X[12702] + 7 X[15178],41 X[12702] + 35 X[16189],11 X[12702] + 9 X[16191],3 X[12702] + 49 X[16192],23 X[12702] + 21 X[16200],5 X[12702] + 21 X[17502],X[12778] + 3 X[15055],X[13607] - 6 X[15759],41 X[13624] - 10 X[16189],77 X[13624] - 18 X[16191],3 X[13624] - 14 X[16192],23 X[13624] - 6 X[16200],5 X[13624] - 6 X[17502],3 X[14269] - 7 X[19876],7 X[14869] - 5 X[19862],41 X[15178] - 20 X[16189],77 X[15178] - 36 X[16191],3 X[15178] - 28 X[16192],23 X[15178] - 12 X[16200],5 X[15178] - 12 X[17502],X[15681] + 3 X[19875],15 X[15692] + X[20070],5 X[15693] - X[31162],7 X[15700] - 3 X[25055],5 X[15712] - X[22791],385 X[16189] - 369 X[16191],15 X[16189] - 287 X[16192],115 X[16189] - 123 X[16200],25 X[16189] - 123 X[17502],27 X[16191] - 539 X[16192],69 X[16191] - 77 X[16200],15 X[16191] - 77 X[17502],161 X[16192] - 9 X[16200],35 X[16192] - 9 X[17502],5 X[16200] - 23 X[17502],X[18480] - 3 X[26446],X[18480] - 5 X[31447],X[20014] - 65 X[21734],X[20053] + 55 X[21735],3 X[26446] - 5 X[31447].

See Kadir Altintas and Ercole Suppa, Hyacinthos 28895.

X(31663) lies on these lines: {1,3}, {2,22793}, {4,11231}, {5,10164}, {8,3528}, {10,550}, {20,5818}, {21,14496}, {30,3828}, {72,11270}, {100,3916}, {104,8698}, {140,516}, {186,1902}, {355,376}, {382,1698}, {392,4188}, {411,17613}, {515,548}, {518,14810}, {546,3634}, {547,12571}, {549,946}, {551,17504}, {572,16668}, {573,16669}, {582,601}, {631,9778}, {632,3817}, {910,24047}, {944,3654}, {952,4701}, {962,3524}, {971,6796}, {1125,3530}, {1384,9593}, {1386,17508}, {1445,15008}, {1571,3053}, {1572,15815}, {1657,5587}, {1699,3526}, {1702,6398}, {1703,6221}, {1737,15338}, {1766,16675}, {1770,5432}, {1829,3520}, {2771,9943}, {2807,5447}, {2948,15041}, {3241,15710}, {3311,9582}, {3312,9616}, {3474,11374}, {3522,4678}, {3523,5886}, {3525,9812}, {3529,9780}, {3533,9779}, {3534,5691}, {3616,10299}, {3624,15720}, {3626,28224}, {3627,10175}, {3628,18483}, {3647,3740}, {3651,3652}, {3653,10595}, {3655,12245}, {3656,15692}, {3679,15688}, {3683,5506}, {3689,6763}, {3702,4781}, {3753,4189}, {3824,6690}, {3827,15578}, {3830,7989}, {3844,29012}, {3850,10172}, {3853,28158}, {3911,15171}, {4257,4646}, {4294,18527}, {4297,4669}, {4302,24914}, {4309,17728}, {4324,5445}, {4333,10895}, {4512,16408}, {4640,5044}, {4652,5687}, {4857,5442}, {4995,13407}, {5023,9620}, {5054,8227}, {5071,10248}, {5073,18492}, {5079,19872}, {5250,16371}, {5267,5836}, {5399,22053}, {5428,7686}, {5433,7743}, {5439,9352}, {5493,10165}, {5731,20053}, {5790,9588}, {5887,6876}, {5901,12100}, {6451,9615}, {6455,9583}, {6693,29032}, {6706,17729}, {6737,9945}, {6836,18407}, {6865,10525}, {6905,9856}, {6916,10526}, {6918,21153}, {6942,12672}, {6985,11495}, {8583,17573}, {9589,18493}, {9590,13564}, {9947,11499}, {10039,15326}, {10171,16239}, {10178,13369}, {10386,11019}, {10624,15325}, {11111,26062}, {11114,17619}, {11204,12262}, {11699,15035}, {11826,28459}, {11827,28458}, {12085,15592}, {12103,18357}, {12108,28216}, {12515,22935}, {12619,24466}, {12645,14093}, {12778,15055}, {13405,24470}, {13464,28212}, {13587,17614}, {13607,15759}, {14269,19876}, {14869,19862}, {15586,16814}, {15681,19875}, {15700,25055}, {16117,22936}, {17229,29093}, {19513,28257}, {19535,19860}, {19537,19861}, {19543,29229}, {20014,21734}

X(31663) = midpoint of X(i) and X(j) for these {i,j}: {3,3579}, {10,550}, {20,18480}, {40,1385}, {2077,10225}, {3651,22937}, {4297,5690}, {5493,22791}, {6684,12512}, {7991,11278}, {9778,11230}, {10222,12702}, {12103,18357}, {12515,22935}, {12619,24466}, {13528,23961}, {16117,22936}
X(31663) = reflection of X(i) in X(j) for these {i,j}: {546,3634}, {1125,3530}, {6583,9940}, {9955,140}, {9956,6684}, {13624,3}, {15178,13624}, {18483,3628}
X(31663) = complement of X(22793)
X(31663)= X(27065)-Ceva conjugate of X(1100)
X(31663) = excentral-isogonal conjugate of X(32632)
X(31663) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1, 3, 17502}, {3, 40, 1385}, {3, 165, 3579}, {3, 1482, 7987}, {3, 6244, 10267}, {3, 10679, 8273}, {3, 11012, 23961}, {3, 11849, 15931}, {3, 12702, 3576}, {6, 31422, 31430}, {20, 26446, 18480}, {35, 1155, 942}, {40, 3576, 11531}, {40, 7987, 1482}, {40, 11531, 12702}, {40, 16192, 3}, {46, 5217, 24929}, {165, 16192, 40}, {631, 9778, 12699}, {631, 12699, 11230}, {1385, 3579, 40}, {1482, 7987, 1385}, {3057, 7280, 5126}, {3522, 5657, 18481}, {3523, 6361, 5886}, {3576, 12702, 10222}, {3576, 16191, 10246}, {4640, 25440, 5044}, {5119, 5204, 24928}, {5493, 10165, 22791}, {5690, 8703, 4297}, {7991, 10246, 11278}, {10222, 11278, 16191}, {15712, 22791, 10165}, {18480, 31447, 26446}

X(31663) = centroid of X(3) and the excenters
X(31663) = X(3628)-of-excentral-triangle

X(31664) = (name pending)

Barycentrics S(30 R^2 S^2 - 18 R^2 SB SC - 7 S^2 SW + 5 SB SC SW) + (3 SB SC - S^2)T : :, where T^2 = SA SB SC SW

As a point on the Euler line, X(31664) has Shinagawa coefficients (E S - 14 F S - 2 T, E S + 10 F S + 6 T).

See Kadir Altintas and Francisco Javier García Capitán, AdGeom 5185.

X(31664) lies on this line: {2,3}

X(31665) = (name pending)

Barycentrics S(30 R^2 S^2 - 18 R^2 SB SC - 7 S^2 SW + 5 SB SC SW) - (3 SB SC - S^2)T : :, where T^2 = SA SB SC SW

As a point on the Euler line, X(31665) has Shinagawa coefficients (E S - 14 F S + 2 T, E S + 10 F S - 6 T).

See Kadir Altintas and Francisco Javier García Capitán, AdGeom 5185.

X(31665) lies on this line: {2,3}

X(31666) = MIDPOINT OF X(11522) AND X(15696)

Barycentrics a (10 a^3-3 a^2 b-10 a b^2+3 b^3-3 a^2 c+6 a b c-3 b^2 c-10 a c^2-3 b c^2+3 c^3) : :
X(31666) = 3*X[1]+7*X[3], 3*X[10]-8*X[12108], X[20]+9*X[3653], 3*X[355]-13*X[10303], 2*X[546]+3*X[4297], 2*X[548]+3*X[551], 7*X[549]-2*X[4745], 3*X[946]+2*X[12103], 6*X[1125]-X[3627], X[1657]+9*X[25055], 7*X[3090]+3*X[18481], X[3146]-6*X[9955], 7*X[3523]+3*X[3655], 11*X[3525]+9*X[5731], 7*X[3528]+3*X[3656], X[3529]+9*X[5886], 4*X[3530]+X[5882], 3*X[3534]+7*X[9624], 3*X[3616]+X[17538], 21*X[3624]-11*X[5072], 4*X[3628]-9*X[10165], 3*X[3654]-13*X[10299], 9*X[3817]-4*X[12102], 4*X[3850]-9*X[19883], X[4677]-11*X[15718], 2*X[4746]-7*X[6684], X[5076]-3*X[8227], 13*X[5079]-3*X[5691], 21*X[5657]-X[20054], X[5734]+3*X[19708], X[5881]-11*X[15720], 3*X[8703]+2*X[13464], X[9589]+9*X[15688], 3*X[10283]+2*X[12512], 9*X[11231]-14*X[14869], X[11362]-6*X[12100], X[11522]+X[15696], 8*X[12104]-3*X[22936], 2*X[12812]-3*X[19862], 2*X[15704]+3*X[22793], 77*X[15717]+3*X[20049]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28895.

X(31666) lies on these lines: {1,3}, {10,12108}, {20,3653}, {104,28214}, {355,10303}, {392,17574}, {515,632}, {519,15712}, {546,4297}, {548,551}, {549,4745}, {572,15492}, {631,28204}, {946,12103}, {1125,3627}, {1656,28208}, {1657,25055}, {2771,15034}, {3090,18481}, {3091,28160}, {3146,9955}, {3522,28198}, {3523,3655}, {3525,5731}, {3528,3656}, {3529,5886}, {3530,5882}, {3534,9624}, {3616,17538}, {3624,5072}, {3628,10165}, {3654,10299}, {3817,12102}, {3850,19883}, {4677,15718}, {4746,6684}, {4881,5047}, {5076,8227}, {5079,5691}, {5657,20054}, {5734,19708}, {5881,15720}, {6428,9583}, {8703,13464}, {9589,15688}, {9619,22331}, {10283,12512}, {11231,14869}, {11362,12100}, {11363,14865}, {11522,15696}, {12104,22936}, {12812,19862}, {15704,22793}, {15717,20049}, {16865,17614}, {18493,28154}

X(31666) = midpoint of X(11522) and X(15696)
X(31666) = reflection of X(7987) in X(13624)
X(31666) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,1385,10222}, {3,10222,3579}, {3,10246,7991}, {1385,11278,10246}, {1385,13624,17502}, {1385,17502,3579}, {3576,13624,1385}, {10222,17502,3}, {13151,26287,1385}

X(31667) = X(522)X(905)∩X(900)X(5901)

Barycentrics a (b+c-a) (b-c) (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^2 b^2 c-3 b^4 c-2 a^3 c^2+2 a^2 b c^2-a b^2 c^2+2 b^3 c^2-2 a^2 c^3+2 b^2 c^3+a c^4-3 b c^4+c^5) : :

See Tran Quang Hung and Ercole Suppa, AdGeom 5197.

X(31667) lies on these lines: {522,905}, {900,5901}, {3738,14315}

X(31668) = (name pending)

Barycentrics a (a+b) (a+c) ((b-c)^6 (b+c)^7+(b^2-c^2)^4 (b^4-b^3 c+8 b^2 c^2-b c^3+c^4) a-(b-c)^2 (b+c)^3 (4 b^6-2 b^5 c+11 b^4 c^2-14 b^3 c^3+11 b^2 c^4-2 b c^5+4 c^6) a^2-(b^2-c^2)^2 (4 b^6-7 b^5 c-3 b^4 c^2+7 b^3 c^3-3 b^2 c^4-7 b c^5+4 c^6) a^3+(5 b^9-3 b^8 c+15 b^7 c^2+31 b^6 c^3-18 b^5 c^4-18 b^4 c^5+31 b^3 c^6+15 b^2 c^7-3 b c^8+5 c^9) a^4+(5 b^8-18 b^7 c-43 b^6 c^2-16 b^5 c^3+14 b^4 c^4-16 b^3 c^5-43 b^2 c^6-18 b c^7+5 c^8) a^5+b c (12 b^5+3 b^4 c-19 b^3 c^2-19 b^2 c^3+3 b c^4+12 c^5) a^6+b c (22 b^4+41 b^3 c+47 b^2 c^2+41 b c^3+22 c^4) a^7+(-5 b^5-13 b^4 c-17 b^3 c^2-17 b^2 c^3-13 b c^4-5 c^5) a^8+(-5 b^4-13 b^3 c-17 b^2 c^2-13 b c^3-5 c^4) a^9+2 (2 b^3+3 b^2 c+3 b c^2+2 c^3) a^10+(4 b^2+3 b c+4 c^2) a^11+(-b-c) a^12-a^13) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28896.

X(31668) lies on this line: {10,21}

X(31669) = EULER LINE INTERCEPT OF X(1000)X(5424)

Barycentrics 7 a^7-7 a^6 b-15 a^5 b^2+15 a^4 b^3+9 a^3 b^4-9 a^2 b^5-a b^6+b^7-7 a^6 c+a^5 b c+a^4 b^2 c+4 a^3 b^3 c+7 a^2 b^4 c-5 a b^5 c-b^6 c-15 a^5 c^2+a^4 b c^2+14 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+15 a^4 c^3+4 a^3 b c^3+2 a^2 b^2 c^3+10 a b^3 c^3+3 b^4 c^3+9 a^3 c^4+7 a^2 b c^4+a b^2 c^4+3 b^3 c^4-9 a^2 c^5-5 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : :
Barycentrics (12 a R^2-12 b R^2-4 a SB+4 b SB-4 a SC+4 c SC+a SW+3 b SW)S^2 + 2 R S^3+3 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : :
X(31669) = 5*X[3616]+4*X[22937], 2*X[5426]+X[5657], 11*X[5550]-2*X[16159], 5*X[10595]+4*X[16139], X[12317]+8*X[16164]

As a point on the Euler line, X(31669) has Shinagawa coefficients (16S²+10abc$a$, -12S²-3abc$a$).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28899.

X(31669) lies on these lines: {2,3}, {1000,5424}, {1056,5427}, {3616,22937}, {5426,5657}, {5550,16159}, {10595,16139}, {12317,16164}, {21165,25055}

X(31669) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,376,6901}, {2,6875,376}, {2,6928,5071}, {2,15692,6924}, {3524,17561,3545}, {5428,15674,4}, {15671,15672,17561}, {21161,28443,6875}

X(31670) = ANTICOMPLEMENT OF X(3098)

Barycentrics 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 : :
Barycentrics (SB+SC-SW)S^2 + 3 SB SC SW : :
X(31670) = 3*X[2]-2*X[3098], 3*X[3]-4*X[3589], 3*X[4]-X[69], 2*X[5]-X[1350], X[20]-2*X[182], X[67]-2*X[10113], 2*X[141]-3*X[381], 3*X[376]-5*X[3618], X[382]+X[1351], 4*X[546]-3*X[10516], 2*X[550]-3*X[5085], 4*X[575]-X[3529], 2*X[576]+X[3146], 2*X[597]-X[3534], X[599]-2*X[3845], 5*X[631]-4*X[14810], X[895]+X[10721], 2*X[1353]-3*X[5102], 2*X[1386]-X[18481], X[1657]-3*X[5050], X[2892]-2*X[19506], X[3242]-2*X[22791], X[3416]-2*X[18480], 5*X[3522]-6*X[17508], 7*X[3526]-6*X[21167], 9*X[3545]-7*X[3619], 5*X[3620]-9*X[3839], 4*X[3631]-9*X[14269], 4*X[3853]-X[15069], 3*X[5032]+X[15640], 4*X[5066]-3*X[21358], X[5073]+3*X[5093], 5*X[5076]-X[11898], 4*X[5097]-3*X[14912], 3*X[5621]-2*X[14677], 3*X[6034]-2*X[12042], 8*X[6329]-9*X[14848], 2*X[6593]-X[12121], 2*X[9880]-X[19905], 4*X[10168]-3*X[10304], 7*X[10541]-4*X[12103], X[10620]-2*X[25328], X[10723]+X[10753], X[10724]+X[10759], X[10725]+X[10758], X[10726]+X[10764], X[10727]+X[10756], X[10728]+X[10755], X[10732]+X[10757], 3*X[11180]-X[20080], 5*X[11482]-4*X[12007], X[11541]+8*X[22330], X[11646]-2*X[22515], 4*X[12101]-X[15533], 3*X[12177]-2*X[14928], X[12244]-3*X[25320], X[12254]-2*X[19150], X[12383]-2*X[19140], 2*X[15118]-X[16111], 5*X[17538]-8*X[20190], 5*X[19709]-4*X[20582], 3*X[21356]-4*X[25561], X[22677]-2*X[22682], 3*X[23049]-2*X[23300]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28903.

X(31670) lies on these lines: {2,3098}, {3,3589}, {4,69}, {5,1350}, {6,30}, {20,182}, {22,14389}, {25,11064}, {51,1370}, {52,14216}, {66,265}, {67,10113}, {68,10263}, {110,7519}, {141,381}, {146,148}, {159,18534}, {184,7500}, {206,13352}, {263,14957}, {303,383}, {343,5064}, {376,3618}, {382,1351}, {394,428}, {495,10387}, {518,12699}, {524,3830}, {546,10516}, {550,5085}, {575,3529}, {576,3146}, {597,3534}, {599,3845}, {611,6284}, {613,7354}, {631,14810}, {698,7758}, {895,10721}, {1176,15033}, {1353,5102}, {1368,17810}, {1386,18481}, {1428,4299}, {1469,1479}, {1478,3056}, {1539,9973}, {1568,28419}, {1595,17834}, {1596,7716}, {1597,3867}, {1657,5050}, {1899,3060}, {1974,15462}, {1992,11645}, {2330,4302}, {2548,3094}, {2777,11579}, {2810,10741}, {2854,7728}, {2892,19506}, {2979,7394}, {3091,7938}, {3095,8721}, {3242,22791}, {3313,18420}, {3416,18480}, {3448,16981}, {3522,17508}, {3526,21167}, {3545,3619}, {3564,3627}, {3583,12589}, {3585,12588}, {3593,7374}, {3595,7000}, {3620,3839}, {3631,14269}, {3767,5017}, {3819,7392}, {3853,15069}, {3917,6997}, {4232,5972}, {4260,6851}, {5012,20062}, {5028,7747}, {5032,15640}, {5033,6781}, {5034,7756}, {5039,5286}, {5052,7748}, {5066,21358}, {5073,5093}, {5076,11898}, {5080,25304}, {5097,14912}, {5138,6869}, {5189,11002}, {5227,18540}, {5319,12212}, {5446,14790}, {5486,8705}, {5596,18400}, {5621,14677}, {5640,16063} ,{5654,7530}, {5722,24471}, {5728,24701}, {5846,18525}, {5921,5965}, {5943,7386}, {5969,6033}, {5999,7806}, {6034,12042}, {6036,9752}, {6193,13419}, {6201,21737}, {6329,14848}, {6393,7773}, {6515,11550}, {6593,12121}, {6643,10110}, {6803,13348}, {6995,9306}, {7378,21243}, {7401,15644}, {7408,14826}, {7470,7803}, {7487,13346}, {7517,15577}, {7528,10625}, {7553,9833}, {7576,20806}, {7667,10601}, {7694,22728}, {7703,15360}, {7731,12317}, {7800,9821}, {7931,13862}, {8148,9053}, {8177,9301}, {9019,18438}, {9024,10742}, {9037,12586}, {9047,12587}, {9308,16264}, {9739,21736}, {9822,18537}, {9873,20065}, {9880,19905}, {9909,23292}, {9969,15812}, {9970,17702}, {10168,10304}, {10192,20850}, {10210,16771}, {10541,12103}, {10620,25328}, {10691,17825}, {10723,10753}, {10724,10759}, {10725,10758}, {10726,10764}, {10727,10756}, {10728,10755}, {10732,10757}, {11003,20063}, {11114,15988}, {11180,20080}, {11433,21849}, {11482,12007}, {11487,15606}, {11541,22330}, {11646,22515}, {11663,15800}, {11800,13203}, {11807,12319}, {11818,13391}, {11819,12118}, {12101,15533}, {12160,16655}, {12164,16621}, {12177,14928}, {12244,25320}, {12254,19150}, {12383,19140}, {13634,17381}, {13635,17352}, {13857,26255}, {14965,15075}, {15118,16111}, {15311,15583}, {15435,18489}, {16625,18909}, {17532,26543}, {17538,20190}, {18390,21851}, {18396,26926}, {18559,22151}, {19709,20582}, {21356,25561}, {22677,22682}, {23049,23300}, {24248,29301}, {24695,29097}

X(31670) = anticomplement of X(3098)
X(31670) = midpoint of X(895) and X(10721)
X(31670) = reflection of X(i) in X(j) for these {i,j}: {3,5480}, {6,21850}, {20,182}, {66,18382}, {67,10113}, {69,3818}, {376,5476}, {550,18583}, {599,3845}, {1350,5}, {1352,4}, {2892,19506}, {3094,14881}, {3098,19130}, {3242,22791}, {3416,18480}, {3534,597}, {6776,576}, {9821,24256}, {10620,25328}, {11179,20423}, {11646,22515}, {12118,19139}, {12121,6593}, {12254,19150}, {12383,19140}, {16111,15118}, {18481,1386}, {19161,5446}, {19905,9880}, {22677,22682}
X(31670) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,6147,5763}, {7,21151,3}, {3523,20059,21168}, {3824,6245,5}, {5249,10167,8727}, {5732,6173,5805}, {11220,27186,8226}
X(31670) = isotomic conjugate of isogonal conjugate of X(20897)
X(31670) = barycentric product of X(i) and X(j) for these {i,j}: {76, 20897}
X(31670) = barycentric quotient of X(i) and X(j) for these {i,j}: {20897, 6}
X(31670) = trilinear product of X(i) and X(j) for these {i,j}: {75, 20897}, {75, 20897}
X(31670) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,5480,14561}, {4,69,3818}, {6,21850,20423}, {20,14853,182}, {69,3818,1352}, {376,3618,5092}, {550,18583,5085}, {1531,1843,3818}, {3060,7391,1899}, {3091,10519,24206}, {3098,19130,2}, {5092,5476,3618}, {5189,11002,18911}, {11550,21969,6515}

X(31671) = X(3)X(142)∩X(4)X(144)

Barycentrics 3 a^6-4 a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4+2 a b^5-2 b^6-4 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+4 b^5 c-2 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2-4 a b^3 c^2+2 b^4 c^2+2 a^3 c^3-2 a^2 b c^3-4 a b^2 c^3-8 b^3 c^3+a^2 c^4+2 a b c^4+2 b^2 c^4+2 a c^5+4 b c^5-2 c^6 : :
X(31671) = 3*X[3]-4*X[142], 3*X[4]-X[144], 2*X[5]-X[5759], 6*X[5] - 5*X[18230], 2*X[9]-3*X[381], 2*X[142] - 3*X[5805], 2*X[144] - 3*X[5779], 3* X[381] - 4*X[18482], 4*X[546]-3*X[5817], 2*X[550]-3*X[21151], X[673]-2*X[24827], 4*X[1001] - 5*X[18493], X[1156]-2*X[22938], 5*X[1656] - 4* X[31658], X[1657]-2*X[5732], 2*X[2550]-X[12702], 5*X[3091]-3*X[21168], 2*X[3243]-X[18526], 2*X[3254]-X[12773], 7*X[3526]-6*X[21153], X[3534]-2*X[6173], 3*X[3656] - 2*X[30331], 2*X[3845]-X[6172], 9*X[5054]-10*X[20195], 9*X[5055]-8*X[6666], X[5223]-2*X[18480], 2*X[5542]-X[18481], 3*X[5686]-4*X[18357], 3*X[5759] - 5*X[18230], 3*X[10246]-4*X[20330], X[12669]-2*X[24475]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28903.

X(31671) lies on these lines: {3,142}, {4,144}, {5,5759}, {7,30}, {9,381}, {20,31657}, {57,11238}, {382,971}, {390,6869}, {480,18491}, {517,3059}, {518,18345}, {527,3830}, {528,4930}, {546,5817}, {550,21151}, {673,24827}, {908,1260}, {942,4312}, {954,6985}, {962,5730}, {1156,22938}, {1537,9945}, {1596,7717}, {1656,31658}, {1657,5732}, {1699,3683}, {2095,10738}, {2550,12702}, {2951,18443}, {3019,16884}, {3091,21168}, {3243,18526}, {3254,12773}, {3428,15909}, {3526,21153}, {3534,6173}, {3543,12690}, {3587,28198}, {3627,5843}, {3656,30331}, {3845,6172}, {5054,20195}, {5055,6666}, {5223,18480}, {5542,18481}, {5572,18530}, {5686,18357}, {5698,6841}, {5708,6851}, {5709,5789}, {5744,8727}, {5791,18483}, {5833,31445}, {5853,8148}, {5856,10742}, {6361,8728}, {6600,18524}, {6826,28174}, {7373,12573}, {7580,31019}, {8581,9655}, {9580,11018}, {9654,15298}, {9669,15299}, {10246,20330}, {12669,24475}, {12953,17637}, {13727,17236}, {15733,18499}, {15837,31479}, {18527,30330}, {24391,28646}, {24929,31162}

X(31671) = reflection of X(i) in X(j) for these {i,j}: {3, 5805}, {9, 18482}, {20, 31657}, {390, 22791}, {673, 24827}, {1156, 22938}, {1657, 5732}, {3534, 6173}, {5223, 18480}, {5759, 5}, {5779, 4}, {6172, 3845}, {12669, 24475}, {12702, 2550}, {12773, 3254}, {18481, 5542}, {18526, 3243}
X(31671) ={X(9), X(18482)}-harmonic conjugate of X(381)

X(31672) = MIDPOINT OF X(382) AND X(5779)

Barycentrics a^6-3 a^5 b+3 a^4 b^2-3 a^2 b^4+3 a b^5-b^6-3 a^5 c-4 a^4 b c+2 a^2 b^3 c+3 a b^4 c+2 b^5 c+3 a^4 c^2+2 a^2 b^2 c^2-6 a b^3 c^2+b^4 c^2+2 a^2 b c^3-6 a b^2 c^3-4 b^3 c^3-3 a^2 c^4+3 a b c^4+b^2 c^4+3 a c^5+2 b c^5-c^6 : :
X(31672) = 3*X[3]-4*X[6666], 3*X[4]-X[7], 2*X[5]-X[5732], X[20]-3*X[5817], 2*X[142]-3*X[381], 3*X[376]-5*X[18230], 2*X[550]-3*X[21153], 2*X[1001]-X[18481], X[1156]+X[10728], 3*X[1699]-2*X[20330], X[2550]-2*X[18480], X[2951]-3*X[5587], 5*X[3091]-3*X[21151], X[3146]+X[5759], X[3243]-2*X[22791], X[3254]-2*X[22938], 2*X[3845]-X[6173], 4*X[3853]-X[5735], X[5528]-2*X[11698], X[5542]-2*X[18483], 3*X[5686]-X[6361], X[6172]+X[15682], 2*X[11495]-3*X[26446]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28903.

X(31672) lies on these lines: {3,6666}, {4,7}, {5,5732}, {9,30}, {20,5817}, {33,6357}, {80,2093}, {142,381}, {144,3419}, {355,382}, {376,18230}, {495,4326}, {496,4321}, {515,6767}, {518,12699}, {527,3830}, {550,21153}, {990,17366}, {1001,18481}, {1156,10728}, {1478,14100}, {1479,8581}, {1490,5719}, {1697,5252}, {1699,20330}, {1728,3358}, {1750,5219}, {1836,18412}, {1837,4312}, {2550,18480}, {2801,10738}, {2951,5587}, {3091,21151}, {3146,5759}, {3174,18528}, {3243,22791}, {3254,22938}, {3627,5762}, {3820,18529}, {3845,6173}, {3853,5735}, {3911,19541}, {4292,10392}, {4302,15837}, {4335,5725}, {5046,10861}, {5080,25722}, {5528,11698}, {5542,18483}, {5561,15909}, {5686,6361}, {5853,18525}, {5927,10431}, {6172,15682}, {6923,15726}, {6930,28160}, {7354,15299}, {7675,11374}, {7678,18450}, {8232,24929}, {8257,28452}, {9579,10398}, {9613,10384}, {9655,12573}, {10430,11227}, {10947,17642}, {11495,26446}, {12618,17293}, {13727,17368}, {16152,18513}, {17668,18516}

X(31672) = midpoint of X(i) and X(j) for these {i,j}: {382,5779}, {1156,10728}, {3146,5759}, {5691,11372}, {6172,15682}
X(31672) = reflection of X(i) in X(j) for these {i,j}: {7,18482}, {2550,18480}, {3243,22791}, {3254,22938}, {5528,11698}, {5542,18483}, {5732,5}, {5805,4}, {6173,3845}, {12702,24393}, {18481,1001}
X(31672) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,7,18482}, {4,9799,5806}, {7,18482,5805}

X(31673) = MIDPOINT OF X(4) AND X(5691)

Barycentrics 4 a^4-a^3 b-a^2 b^2+a b^3-3 b^4-a^3 c+2 a^2 b c-a b^2 c-a^2 c^2-a b c^2+6 b^2 c^2+a c^3-3 c^4 : :
Trilinears r - 6 R cos B cos C : :
X(31673) = X[1]-3*X[4], 3*X[2]-5*X[18492], 3*X[3]-4*X[3634], 2*X[5]-X[4297], X[8]+3*X[3543], X[20]-3*X[5587], X[40]+X[3146], X[146]+X[12407], 3*X[165]-X[3529], X[355]+X[382], 3*X[376]-5*X[1698], 3*X[381]-2*X[1125], 2*X[546]-X[1385], 2*X[548]-3*X[11231], X[550]-2*X[9956], X[551]-2*X[3845], 5*X[631]-7*X[7989], X[942]-2*X[16616], 3*X[962]+X[3621], X[1482]-5*X[5076], X[1657]-2*X[12512], 7*X[3090]-5*X[7987], 5*X[3091]-3*X[3576], X[3244]-6*X[15687], X[3534]-2*X[3828], 9*X[3545]-7*X[3624], 5*X[3616]-9*X[3839], 4*X[3628]-3*X[17502], 2*X[3635]-3*X[3656], 4*X[3636]-3*X[3655], 3*X[3654]-4*X[4691], 3*X[3679]-X[6361], 7*X[3832]-3*X[5731], 5*X[3843]-3*X[5886], 4*X[3850]-3*X[11230], 7*X[3851]-6*X[10171], 11*X[3855]-9*X[7988], 2*X[3861]-X[5901], 5*X[4816]+3*X[9589], 9*X[5055]-8*X[19878], 4*X[5066]-3*X[19883], X[5073]+3*X[5790], X[5493]-2*X[5690], 2*X[5806]-X[12675], 3*X[5883]-2*X[13369], 3*X[5927]-X[14110], 2*X[7687]-X[11709], X[7957]-3*X[18908], X[7982]-3*X[9812], 7*X[9624]-9*X[9779], X[9864]+X[10723], 2*X[10113]-X[13605], X[10222]-4*X[12102], 9*X[10304]-13*X[19877], X[10721]+X[13211], X[10722]+X[13178], X[10724]+X[12751], X[10733]+X[12368], X[10735]+X[12784], X[11001]-3*X[19875], 3*X[11220]-5*X[15016], X[11599]-2*X[22515], 2*X[12005]-X[12680], 4*X[12009]-3*X[26201], X[12747]+X[16128], 7*X[16192]-5*X[17538], 5*X[19708]-7*X[19876], X[21630]-2*X[22938], X[21635]-2*X[22799], X[21636]-2*X[22505]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28903.

X(31673) lies on these lines: {1,4}, {2,18492}, {3,3634}, {5,4297}, {7,17706}, {8,3543}, {10,30}, {11,4311}, {12,4304}, {20,5587}, {35,21669}, {40,3146}, {44,10445}, {65,16006}, {79,16615}, {80,1770}, {84,7319}, {104,17501}, {140,28190}, {146,12407}, {165,3529}, {307,18661}, {355,382}, {376,1698}, {378,8185}, {381,1125}, {495,4314}, {496,4315}, {517,3625}, {519,3830}, {535,10916}, {546,1385}, {548,11231}, {550,9956}, {551,3845}, {631,7989}, {936,18529}, {942,16616}, {952,3853}, {962,3621}, {971,5884}, {993,6985}, {1012,5217}, {1155,12616}, {1158,5128}, {1159,6259}, {1210,7354}, {1420,10591}, {1482,5076}, {1503,4663}, {1532,7173}, {1597,9798}, {1657,12512}, {1737,10483}, {1826,2173}, {1837,4292}, {1872,2817}, {2349,2816}, {2475,18406}, {2771,4084}, {2784,6321}, {2800,12688}, {2801,24474}, {2807,13474}, {2829,6245}, {3090,7987}, {3091,3576}, {3149,5204}, {3244,15687}, {3419,12527}, {3452,17647}, {3520,9590}, {3534,3828}, {3545,3624}, {3555,12690}, {3577,5556}, {3601,10590}, {3614,6831}, {3616,3839}, {3628,17502}, {3635,3656}, {3636,3655}, {3651,5251}, {3654,4691}, {3671,11544}, {3679,6361}, {3755,29040}, {3811,18528}, {3822,6841}, {3832,5731}, {3843,5886}, {3850,11230}, {3851,10171}, {3855,7988}, {3861,5901}, {3911,4299}, {3947,24929}, {4293,9581}, {4294,9578}, {4295,5727}, {4298,5722}, {4302,10827}, {4305,5219}, {4325,12248}, {4420,5080}, {4667,13408}, {4669,28198}, {4816,9589}, {5055,19878}, {5066,19883}, {5073,5790}, {5086,11684}, {5126,10593}, {5195,25719}, {5252,10624}, {5260,7688}, {5316,6903}, {5493,5690}, {5542,12433}, {5708,5787}, {5745,6869}, {5794,12572}, {5806,12675}, {5842,21628}, {5883,13369}, {5927,14110}, {6845,7951}, {6912,10902}, {6920,15931}, {6928,9842}, {6996,29596}, {6999,16815}, {7384,29578}, {7406,29579}, {7491,12617}, {7687,11709}, {7741,21578}, {7957,18908}, {7982,9812}, {7991,28232}, {8582,11112}, {8666,18519}, {8715,18518}, {8727,10592}, {9579,18391}, {9624,9779}, {9626,14118}, {9654,13405}, {9656,17718}, {9668,12575}, {9864,10723}, {10113,13605}, {10151,11363}, {10222,12102}, {10304,19877}, {10721,13211}, {10722,13178}, {10724,12751}, {10733,12368}, {10735,12784}, {10742,12437}, {10863,13729}, {10895,13411}, {11001,19875}, {11019,18990}, {11114,24987}, {11220,15016}, {11365,18535}, {11372,29007}, {11456,16473}, {11552,16116}, {11599,22515}, {12005,12680}, {12009,26201}, {12114,19541}, {12135,13473}, {12514,18540}, {12618,29024}, {12640,18499}, {12747,16128}, {13746,18653}, {15326,17606}, {16192,17538}, {16948,24624}, {17577,24541}, {17579,24982}, {17679,25967}, {18491,25440}, {18527,21625}, {19708,19876}, {21630,22938}, {21635,22799}, {21636,22505}

X(31673) = midpoint of X(i) and X(j) for these {i,j}: {4,5691}, {40,3146}, {80,10728}, {146,12407}, {355,382}, {962,5881}, {3654,15684}, {3679,15682}, {9589,12245}, {9864,10723}, {10721,13211}, {10722,13178}, {10724,12751}, {10726,13532}, {10733,12368}, {10735,12784}, {12699,18525}, {12747,16128}
X(31673) = reflection of X(i) in X(j) for these {i,j}: {1,18483}, {3,19925}, {10,18480}, {20,6684}, {550,9956}, {551,3845}, {942,16616}, {944,13464}, {946,4}, {1385,546}, {1657,12512}, {3244,22791}, {3452,18516}, {3534,3828}, {3579,18357}, {4297,5}, {4301,22793}, {5493,5690}, {5542,18482}, {5882,946}, {5884,7686}, {5901,3861}, {10265,6246}, {11362,355}, {11599,22515}, {11709,7687}, {12437,21077}, {12675,5806}, {12680,12005}, {12702,3626}, {13605,10113}, {14110,20117}, {18481,1125}, {21630,22938}, {21635,22799}, {21636,22505}, {22793,3853}
X(31673) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,4,18483}, {1,18483,946}, {3,19925,10175}, {4,944,1699}, {4,12667,26332}, {5,4297,10165}, {5,13624,19862}, {20,5587,6684}, {80,1770,4848}, {355,12702,3626}, {381,18481,1125}, {546,1385,3817}, {550,9956,10164}, {631,7989,10172}, {944,1699,13464}, {950,1478,21620}, {1657,26446,12512}, {1699,13464,946}, {1837,12943,4292}, {3529,5818,165}, {3579,18357,10}, {3579,18480,18357}, {3585,10572,226}, {3626,12702,11362}, {3655,18493,3636}, {3656,18526,3635}, {3830,18525,12699}, {3832,5731,8227}, {3843,5886,12571}, {4297,19862,13624}, {4299,10826,3911}, {5252,12953,10624}, {5722,9655,4298}, {5927,14110,20117}, {6985,18761,993}, {13624,19862,10165}

X(31674) = MIDPOINT OF X(6343) AND X(11671)

Barycentrics 2 a^22-15 a^20 b^2+52 a^18 b^4-115 a^16 b^6+188 a^14 b^8-238 a^12 b^10+224 a^10 b^12-142 a^8 b^14+50 a^6 b^16-3 a^4 b^18-4 a^2 b^20+b^22-15 a^20 c^2+82 a^18 b^2 c^2-192 a^16 b^4 c^2+249 a^14 b^6 c^2-165 a^12 b^8 c^2-31 a^10 b^10 c^2+173 a^8 b^12 c^2-133 a^6 b^14 c^2+16 a^4 b^16 c^2+25 a^2 b^18 c^2-9 b^20 c^2+52 a^18 c^4-192 a^16 b^2 c^4+284 a^14 b^4 c^4-218 a^12 b^6 c^4+108 a^10 b^8 c^4-121 a^8 b^10 c^4+176 a^6 b^12 c^4-72 a^4 b^14 c^4-52 a^2 b^16 c^4+35 b^18 c^4-115 a^16 c^6+249 a^14 b^2 c^6-218 a^12 b^4 c^6+100 a^10 b^6 c^6+9 a^8 b^8 c^6-112 a^6 b^10 c^6+150 a^4 b^12 c^6+12 a^2 b^14 c^6-75 b^16 c^6+188 a^14 c^8-165 a^12 b^2 c^8+108 a^10 b^4 c^8+9 a^8 b^6 c^8+38 a^6 b^8 c^8-91 a^4 b^10 c^8+120 a^2 b^12 c^8+90 b^14 c^8-238 a^12 c^10-31 a^10 b^2 c^10-121 a^8 b^4 c^10-112 a^6 b^6 c^10-91 a^4 b^8 c^10-202 a^2 b^10 c^10-42 b^12 c^10+224 a^10 c^12+173 a^8 b^2 c^12+176 a^6 b^4 c^12+150 a^4 b^6 c^12+120 a^2 b^8 c^12-42 b^10 c^12-142 a^8 c^14-133 a^6 b^2 c^14-72 a^4 b^4 c^14+12 a^2 b^6 c^14+90 b^8 c^14+50 a^6 c^16+16 a^4 b^2 c^16-52 a^2 b^4 c^16-75 b^6 c^16-3 a^4 c^18+25 a^2 b^2 c^18+35 b^4 c^18-4 a^2 c^20-9 b^2 c^20+c^22 : :
X(31674) = X[2888] - 3 X[25147]

See Seiichi Kirikami and Peter Moses, Hyacinthos 28904.

X(31674) lies on the curve Q093 and these lines: {4,195}, {30,27246}, {54,16766}, {1154,12026}, {1157,27196}, {2888,25147}, {3459,21230}, {6343,11671}, {6592,8254}, {7691,16768}, {14072,16762}, {22051,25150}

X(31674) = midpoint of X(6343) and X(11671)
X(31674) = reflection of X(6592) in X(8254)

X(31675) = REFLECTION OF X(12026) IN X(54)

Barycentrics (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) (3 a^12-12 a^10 b^2+19 a^8 b^4-16 a^6 b^6+9 a^4 b^8-4 a^2 b^10+b^12-12 a^10 c^2+20 a^8 b^2 c^2-4 a^6 b^4 c^2-10 a^4 b^6 c^2+12 a^2 b^8 c^2-6 b^10 c^2+19 a^8 c^4-4 a^6 b^2 c^4+5 a^4 b^4 c^4-8 a^2 b^6 c^4+15 b^8 c^4-16 a^6 c^6-10 a^4 b^2 c^6-8 a^2 b^4 c^6-20 b^6 c^6+9 a^4 c^8+12 a^2 b^2 c^8+15 b^4 c^8-4 a^2 c^10-6 b^2 c^10+c^12) : :

See Seiichi Kirikami and Peter Moses, Hyacinthos 28904.

X(31675) lies on these lines: {5,49}, {195,6343}, {5898,14367}, {6150,6592}, {20414,22051}

X(31675) = reflection of X(12026) in X(54)

X(31676) = X(3)X(125)∩X(25)X(6344)

Barycentrics a^2SA (SA^2 + 5S^2)/(4SA^2-b^2c^2) : :
Barycentrics (3 Sin[2 A] - Sin[4 A]) / (2 + 4 Cos[2 A]) : : (Peter Moses, March 24, 2019)

See Nguyễn Danh Lân and Angel Montesdeoca, Hyacinthos 28906, HG080319, AoPS.

X(31676) lies on these lines: {3, 125}, {25, 6344}, {94, 7517}, {476, 5899}, {1141, 11815}, {1593, 23956}, {1989, 8573}, {11060, 21309}, {11141, 21310}, {11142, 21311}, {11816, 18378}, {13861, 30529}, {14356, 21308}

X(31676) = barycentric quotient X(10095) / X(14918)

X(31677) = (name pending)

Barycentrics 2 a^10-4 a^9 (b+c)+a (b-c)^4 (b+c)^3 (b^2-3 b c+c^2)-(b^2-c^2)^4 (b^2-b c+c^2)-a^8 (3 b^2-8 b c+3 c^2)+2 a^2 b c (b^2-c^2)^2 (3 b^2-4 b c+3 c^2)-a^7 (-11 b^3+3 b^2 c+3 b c^2-11 c^3)-a^6 (2 b^4+9 b^3 c-16 b^2 c^2+9 b c^3+2 c^4)-a^5 (9 b^5-13 b^4 c+10 b^3 c^2+10 b^2 c^3-13 b c^4+9 c^5)-a^4 (-4 b^6+6 b^5 c+8 b^4 c^2-22 b^3 c^3+8 b^2 c^4+6 b c^5-4 c^6)+a^3 (b^7-2 b^6 c+12 b^5 c^2-9 b^4 c^3-9 b^3 c^4+12 b^2 c^5-2 b c^6+c^7) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28909.

X(31677) lies on this line: {6713, 31678)

X(31678) = (name pending)

Barycentrics 4 a^8 b c-b c (b^2-c^2)^4+3 a^2 b c (b^2-c^2)^2 (b^2-b c+c^2)-a^4 b (b-c)^2 c (b^2+5 b c+c^2)-a^6 b c (5 b^2-6 b c+5 c^2)+a^7 (b^3-3 b^2 c-3 b c^2+c^3)-a (b-c)^2 (b+c)^3 (b^4+b^3 c-3 b^2 c^2+b c^3+c^4)-a^5 (3 b^5-6 b^4 c+b^3 c^2+b^2 c^3-6 b c^4+3 c^5)-a^3 (-3 b^7+b^6 c+2 b^4 c^3+2 b^3 c^4+b c^6-3 c^7) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28909.

X(31678) lies on this line: {6713, 31677)

X(31679) = X(3)X(10)∩X(4)X(1647)

Barycentrics a^6 b+3 a^5 b^2-a^4 b^3-4 a^3 b^4+a^2 b^5+a b^6-b^7+a^6 c-10 a^5 b c+2 a^4 b^2 c+8 a^3 b^3 c-3 a^2 b^4 c+2 a b^5 c+3 a^5 c^2+2 a^4 b c^2-8 a^3 b^2 c^2+2 a^2 b^3 c^2-a b^4 c^2+2 b^5 c^2-a^4 c^3+8 a^3 b c^3+2 a^2 b^2 c^3-4 a b^3 c^3-b^4 c^3-4 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+2 b^2 c^5+a c^6-c^7 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28914.

X(31679) lies on these lines: {3,10}, {4,1647}, {5,25377}, {946,3667}, {2827,25437}, {3976,30384}

X(31680) = X(1)X(2)∩X(522)X(946)

Barycentrics a^6 b+a^5 b^2-3 a^4 b^3-2 a^3 b^4+3 a^2 b^5+a b^6-b^7+a^6 c-6 a^5 b c+4 a^4 b^2 c+6 a^3 b^3 c-5 a^2 b^4 c+a^5 c^2+4 a^4 b c^2-8 a^3 b^2 c^2+2 a^2 b^3 c^2-a b^4 c^2+2 b^5 c^2-3 a^4 c^3+6 a^3 b c^3+2 a^2 b^2 c^3-b^4 c^3-2 a^3 c^4-5 a^2 b c^4-a b^2 c^4-b^3 c^4+3 a^2 c^5+2 b^2 c^5+a c^6-c^7 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28914.

X(31680) lies on these lines: {1,2}, {522,946}, {3738,25437}

X(31681) = EULER LINE INTERCEPT OF X(2100)X(5886)

Barycentrics OH*(3*S^2-SB*SC)-2*R*(S^2-3*SB*SC) : :
X(31681) = X(2100)+3*X(5886), X(2102)-3*X(10283), 3*X(14561)+X(15162)

As a point on the Euler line, X(31681) has Shinagawa coefficients (3OH-2R, -OH+6R).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28916.

X(31681) lies on these lines: {2, 3}, {2100, 5886}, {2102, 10283}, {2574, 10272}, {12041, 14500}, {14374, 15806}, {14561, 15162}

X(31681) = midpoint of X(i) and X(j) for these {i,j}: {550, 10751}, {12041, 14500}
X(31681) = reflection of X(13627) in X(10124)
X(31681) = complement of the complement of X(15154)

X(31682) = EULER LINE INTERCEPT OF X(2101)X(5886)

Barycentrics OH*(3*S^2-SB*SC)+2*R*(S^2-3*SB*SC) : :
X(31682) = X(2101)+3*X(5886), X(2103)-3*X(10283), 3*X(14561)+X(15163)

As a point on the Euler line, X(31682) has Shinagawa coefficients (3OH+2R, -OH-6R).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28916.

X(31682) lies on these lines: {2, 3}, {2101, 5886}, {2103, 10283}, {2575, 10272}, {12041, 14499}, {14375, 15806}, {14561, 15163}

X(31682) = midpoint of X(i) and X(j) for these {i,j}: {550, 10750}, {12041, 14499}
X(31682) = reflection of X(13626) in X(10124)
X(31682) = complement of the complement of X(15155)

3rd and 4th isodynamic-Dao triangles and related centers: X(31683)-X(31720)

This preamble and centers X(31683)-X(31720) were contributed by César Eliud Lozada, March 15, 2019.

Let A_oB_oC_o be the orthic triangle of ABC. Denote by A', B', C' the 1st isodynamic points X(15) of AB_oC_o, BC_oA_o and CA_oB_o, respectively, and A", B", C" the 2nd isodynamic points X(16) of AB_oC_o, BC_oA_o and CA_oB_o , respectively. Then the triangles A'B'C' and A"B"C" are equilateral. (Dao Thanh Oai, pers. comm., March 12, 2019).

The triangles A'B'C' and A"B"C" are here named here 3rd isodynamic-Dao and 4th isodynamic-Dao triangles, respectively. (See also triangles A*B*C* and A**B**C** constructed in Hyacinthos 28021 [Aug 11, 2018, Tran Quang Hung and Randy Hutson]). These triangles have the following properties and relations:

3rd isodynamic-Dao triangle :

A-vertex = (sqrt(3)*(S^2+SB*SC)+2*(SB+SC)*S)/SA : sqrt(3)*SC+S : sqrt(3)*SB+S (barycentrics)

Squared-side = S^2/(SW-sqrt(3)*S)

Center = X(13)
Perspective triangles and perspectors:
(ABC, 13), (anti-Euler, 31683), (Euler, 115), (5th Euler, 6108), (2nd excosine, 31685), (outer-Fermat, 13), (3rd Fermat-Dao, 15), (7th Fermat-Dao, 30), (11th Fermat-Dao, 6), (15th Fermat-Dao, 13), (1st isodynamic-Dao, 13), (4th isodynamic-Dao, 115), (1st Lemoine-Dao, 4), (2nd Lemoine-Dao, 13), (outer-Napoleon, 18582), (orthic, 31687), (inner-squares, 31689), (outer-squares, 31692)
Orthologic triangles and orthologic centers:
(ABC, 5318, 13), (ABC-X3 reflections, 5318, 5473), (anti-Aquila, 5318, 11705), (anti-Ara, 5318, 12142), (anti-Artzt, 31693, 12155), (1st anti-Brocard, 115, 5979), (5th anti-Brocard, 5318, 12205), (6th anti-Brocard, 115, 12214), (2nd anti-circumperp-tangential, 5318, 18974), (anti-Euler, 5318, 6770), (anti-inner-Grebe, 5318, 19073), (anti-outer-Grebe, 5318, 19074), (anti-Mandart-incircle, 5318, 12337), (anti-McCay, 31695, 8595), (3rd anti-tri-squares, 31697, 22601), (4th anti-tri-squares, 31699, 22630), (anticomplementary, 5318, 616), (Aquila, 5318, 9901), (Ara, 5318, 9916), (Artzt, 31693, 9762), (1st Auriga, 5318, 12472), (2nd Auriga, 5318, 12473), (1st Brocard-reflected, 31701, 22687), (1st Brocard, 115, 3643), (5th Brocard, 5318, 9982), (6th Brocard, 115, 9989), (2nd circumperp tangential, 5318, 22773), (Ehrmann-mid, 5318, 22796), (Euler, 5318, 5478), (inner-Fermat, 31703, 616), (outer-Fermat, 31705, 13), (2nd Fermat-Dao, 31707, 25207), (3rd Fermat-Dao, 13, 16267), (4th Fermat-Dao, 31709, 5469), (6th Fermat-Dao, 31707, 25152), (7th Fermat-Dao, 13, 396), (8th Fermat-Dao, 31709, 115), (10th Fermat-Dao, 31707, 25153), (11th Fermat-Dao, 13, 381), (12th Fermat-Dao, 31709, 25154), (14th Fermat-Dao, 31707, 25155), (15th Fermat-Dao, 13, 13), (16th Fermat-Dao, 31709, 25156), (outer-Garcia, 5318, 12781), (Gossard, 5318, 12793), (inner-Grebe, 5318, 6270), (outer-Grebe, 5318, 6268), (1st isodynamic-Dao, 13, 13), (2nd isodynamic-Dao, 31709, 22998), (4th isodynamic-Dao, 31709, 31710), (Johnson, 5318, 5617), (inner-Johnson, 5318, 12922), (outer-Johnson, 5318, 12932), (1st Johnson-Yff, 5318, 12942), (2nd Johnson-Yff, 5318, 12952), (1st Lemoine-Dao, 13, 10654), (2nd Lemoine-Dao, 31709, 13), (inner-Le Viet An, 31707, 14181), (Lucas homothetic, 5318, 12990), (Lucas(-1) homothetic, 5318, 12991), (Mandart-incircle, 5318, 13076), (McCay, 31695, 13084), (medial, 5318, 618), (5th mixtilinear, 5318, 7975), (inner-Napoleon, 31709, 5463), (outer-Napoleon, 13, 2), (1st Neuberg, 31711, 6582), (2nd Neuberg, 31713, 6298), (1st tri-squares-central, 31715, 13705), (2nd tri-squares-central, 31717, 13825), (3rd tri-squares-central, 5318, 13917), (4th tri-squares-central, 5318, 13982), (1st tri-squares, 31693, 13646), (2nd tri-squares, 31693, 13765), (3rd tri-squares, 31699, 13876), (4th tri-squares, 31697, 13929), (inner-Vecten, 31697, 6302), (outer-Vecten, 31699, 6306), (X3-ABC reflections, 5318, 13103), (inner-Yff, 5318, 10062), (outer-Yff, 5318, 10078), (inner-Yff tangents, 5318, 13105), (outer-Yff tangents, 5318, 13107)
Parallelogic triangles and parallelogic centers:
(2nd Fermat-Dao, 31719, 25216), (4th Fermat-Dao, 6783, 16530), (6th Fermat-Dao, 31719, 25229), (8th Fermat-Dao, 6783, 9115), (10th Fermat-Dao, 31719, 25231), (12th Fermat-Dao, 6783, 5617), (14th Fermat-Dao, 31719, 25233), (16th Fermat-Dao, 6783, 25235), (2nd isodynamic-Dao, 6783, 23005), (4th isodynamic-Dao, 6783, 6782), (2nd Lemoine-Dao, 6783, 23006), (inner-Le Viet An, 31719, 14187), (inner-Napoleon, 6783, 13), (1st Parry, 5318, 13305), (2nd Parry, 5318, 9200)
Cyclologic triangles and cyclologic centers: (orthic, 6111, 115)
4th isodynamic-Dao triangle:

A-vertex = (sqrt(3)*(S^2+SB*SC)-2*(SB+SC)*S)/SA : sqrt(3)*SC-S : sqrt(3)*SB-S (barycentrics)

Squared-side = S^2/(SW+sqrt(3)*S)

Center = X(14)
Perspective triangles and perspectors:
(ABC, 14), (anti-Euler, 31684), (Euler, 115), (5th Euler, 6109), (2nd excosine, 31686), (inner-Fermat, 14), (4th Fermat-Dao, 16), (8th Fermat-Dao, 30), (12th Fermat-Dao, 6), (16th Fermat-Dao, 14), (2nd isodynamic-Dao, 14), (3rd isodynamic-Dao, 115), (1st Lemoine-Dao, 14), (2nd Lemoine-Dao, 4), (inner-Napoleon, 18581), (orthic, 31688), (inner-squares, 31691), (outer-squares, 31690)
Orthologic triangles and orthologic centers (0 means a center not calculated):
(ABC, 5321, 14), (ABC-X3 reflections, 5321, 5474), (anti-Aquila, 5321, 11706), (anti-Ara, 5321, 12141), (anti-Artzt, 31694, 12154), (1st anti-Brocard, 115, 5978), (5th anti-Brocard, 5321, 12204), (6th anti-Brocard, 115, 12213), (2nd anti-circumperp-tangential, 5321, 18975), (anti-Euler, 5321, 6773), (anti-inner-Grebe, 5321, 19075), (anti-outer-Grebe, 5321, 19076), (anti-Mandart-incircle, 5321, 12336), (anti-McCay, 31696, 8594), (3rd anti-tri-squares, 31698, 22603), (4th anti-tri-squares, 31700, 22632), (anticomplementary, 5321, 617), (Aquila, 5321, 9900), (Ara, 5321, 9915), (Artzt, 31694, 9760), (1st Auriga, 5321, 12470), (2nd Auriga, 5321, 12471), (1st Brocard-reflected, 31702, 22689), (1st Brocard, 115, 3642), (5th Brocard, 5321, 9981), (6th Brocard, 115, 9988), (circumnormal, 0, 13594), (2nd circumperp tangential, 5321, 22774), (circumtangential, 0, 13593), (Ehrmann-mid, 5321, 22797), (Euler, 5321, 5479), (inner-Fermat, 31706, 14), (outer-Fermat, 31704, 617), (1st Fermat-Dao, 31708, 25208), (3rd Fermat-Dao, 31710, 5470), (4th Fermat-Dao, 14, 16268), (5th Fermat-Dao, 31708, 25162), (7th Fermat-Dao, 31710, 115), (8th Fermat-Dao, 14, 395), (9th Fermat-Dao, 31708, 25163), (11th Fermat-Dao, 31710, 25164), (12th Fermat-Dao, 14, 381), (13th Fermat-Dao, 31708, 25165), (15th Fermat-Dao, 31710, 25166), (16th Fermat-Dao, 14, 14), (outer-Garcia, 5321, 12780), (Gossard, 5321, 12792), (inner-Grebe, 5321, 6271), (outer-Grebe, 5321, 6269), (1st isodynamic-Dao, 31710, 22997), (2nd isodynamic-Dao, 14, 14), (3rd isodynamic-Dao, 31710, 31709), (Johnson, 5321, 5613), (inner-Johnson, 5321, 12921), (outer-Johnson, 5321, 12931), (1st Johnson-Yff, 5321, 12941), (2nd Johnson-Yff, 5321, 12951), (1st Lemoine-Dao, 31710, 14), (2nd Lemoine-Dao, 14, 10653), (outer-Le Viet An, 31708, 14177), (Lucas homothetic, 5321, 12988), (Lucas(-1) homothetic, 5321, 12989), (Mandart-incircle, 5321, 13075), (McCay, 31696, 13083), (medial, 5321, 619), (5th mixtilinear, 5321, 7974), (1st Morley-midpoint, 0, 0), (2nd Morley-midpoint, 0, 0), (3rd Morley-midpoint, 0, 0), (1st Morley, 0, 0), (2nd Morley, 0, 0), (3rd Morley, 0, 0), (inner-Napoleon, 14, 2), (outer-Napoleon, 31710, 5464), (1st Neuberg, 31712, 6295), (2nd Neuberg, 31714, 6299), (Roussel, 0, 0), (Stammler, 0, 8009), (1st tri-squares-central, 31718, 13703), (2nd tri-squares-central, 31716, 13823), (3rd tri-squares-central, 5321, 13916), (4th tri-squares-central, 5321, 13981), (1st tri-squares, 31694, 13645), (2nd tri-squares, 31694, 13764), (3rd tri-squares, 31700, 13875), (4th tri-squares, 31698, 13928), (inner-Vecten, 31698, 6303), (outer-Vecten, 31700, 6307), (X3-ABC reflections, 5321, 13102), (inner-Yff, 5321, 10061), (outer-Yff, 5321, 10077), (inner-Yff tangents, 5321, 13104), (outer-Yff tangents, 5321, 13106)
Parallelogic triangles and parallelogic centers (0 means a center not calculated):
(circumnormal, 0, 13593), (circumtangential, 0, 13594), (1st Fermat-Dao, 31720, 25213), (3rd Fermat-Dao, 6782, 16529), (5th Fermat-Dao, 31720, 25230), (7th Fermat-Dao, 6782, 9117), (9th Fermat-Dao, 31720, 25232), (11th Fermat-Dao, 6782, 5613), (13th Fermat-Dao, 31720, 25234), (15th Fermat-Dao, 6782, 25236), (1st isodynamic-Dao, 6782, 23004), (3rd isodynamic-Dao, 6782, 6783), (1st Lemoine-Dao, 6782, 23013), (outer-Le Viet An, 31720, 14185), (1st Morley-midpoint, 0, 0), (2nd Morley-midpoint, 0, 0), (3rd Morley-midpoint, 0, 0), (1st Morley, 0, 0), (2nd Morley, 0, 0), (3rd Morley, 0, 0), (outer-Napoleon, 6782, 14), (1st Parry, 5321, 13304), (2nd Parry, 5321, 9201), (Roussel, 0, 0), (Stammler, 0, 8008)
Cyclologic triangles and cyclologic centers: (orthic, 6110, 115)

X(31683) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND ANTI-EULER

Barycentrics -2*(15*a^4+6*(b^2+c^2)*a^2-13*(b^2-c^2)^2)*S+3*(a^6-5*(b^2+c^2)*a^4+3*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*sqrt(3) : :

X(31683) lies on these lines: {4,9112}, {13,376}, {115,5335}, {148,31695}, {298,31693}, {5318,6770}, {5340,22531}, {5344,22513}, {6778,12243}, {14853,31684}, {15484,16940}

X(31684) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND ANTI-EULER

Barycentrics 2*(15*a^4+6*(b^2+c^2)*a^2-13*(b^2-c^2)^2)*S+3*(a^6-5*(b^2+c^2)*a^4+3*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*sqrt(3) : :

X(31684) lies on these lines: {4,9113}, {14,376}, {115,5334}, {148,31696}, {299,31694}, {5321,6773}, {5339,22532}, {5343,22512}, {6777,12243}, {14853,31683}, {15484,16941}

X(31685) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND 2nd EXCOSINE

Barycentrics S^4-(64*R^4-2*SA^2+5*SB*SC-2*SW^2)*S^2+sqrt(3)*((16*R^2+SB+SC)*S^2-4*(4*R^2*(4*R^2+2*SA-SW)-SA^2+SB*SC)*(SB+SC))*S-4*(4*R^2*(12*R^2-5*SW)+SW^2)*SB*SC : :

X(31685) lies on these lines: {13,64}, {115,31686}, {1033,9112}, {6108,17830}, {6525,31687}, {6770,17829}, {22838,31689}, {22839,31692}

X(31686) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND 2nd EXCOSINE

Barycentrics S^4-(64*R^4-2*SA^2+5*SB*SC-2*SW^2)*S^2-sqrt(3)*((16*R^2+SB+SC)*S^2-4*(4*R^2*(4*R^2+2*SA-SW)-SA^2+SB*SC)*(SB+SC))*S-4*(4*R^2*(12*R^2-5*SW)+SW^2)*SB*SC : :

X(31686) lies on these lines: {14,64}, {115,31685}, {1033,9113}, {6109,17830}, {6525,31688}, {6773,17829}, {22838,31691}, {22839,31690}

X(31687) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND ORTHIC

Barycentrics SB*SC*(9*S^2-sqrt(3)*(12*R^2-SA-5*SW)*S-3*SB*SC) : :

X(31687) lies on these lines: {4,13}, {14,20774}, {25,6115}, {53,5472}, {115,6748}, {371,31689}, {372,31692}, {393,9112}, {427,6108}, {462,12142}, {470,618}, {471,6669}, {472,5459}, {473,530}, {624,16249}, {1585,6302}, {1586,6306}, {1597,22513}, {1986,16538}, {2902,5962}, {5318,6000}, {6525,31685}, {6772,18494}, {16001,16625}

X(31687) = polar-circle-inverse of X(38943)

X(31688) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND ORTHIC

Barycentrics SB*SC*(9*S^2+sqrt(3)*(12*R^2-SA-5*SW)*S-3*SB*SC) : :

X(31688) lies on these lines: {4,14}, {13,20774}, {25,6114}, {53,5471}, {115,6748}, {371,31691}, {372,31690}, {393,9113}, {427,6109}, {463,12141}, {470,6670}, {471,619}, {472,531}, {473,5460}, {623,16250}, {1585,6303}, {1586,6307}, {1597,22512}, {1986,16539}, {2903,5962}, {5321,6000}, {6525,31686}, {6775,18494}, {16002,16625}

X(31688) = polar-circle-inverse of X(38944)

X(31689) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND INNER-SQUARES

Barycentrics sqrt(3)*(sqrt(3)*SA-(2+sqrt(3))*SW+4*R^2)*S^2-(6*S^2-(12*R^2+3*SA-(3+sqrt(3))*SW)*(SB+SC))*S+4*sqrt(3)*(3*R^2-SW)*SB*SC : :

X(31689) lies on these lines: {6,31692}, {13,485}, {115,31691}, {371,31687}, {5394,8164}, {6108,8280}, {6115,8854}, {6302,8968}, {22838,31685}

X(31690) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND OUTER-SQUARES

Barycentrics sqrt(3)*(sqrt(3)*SA-(2+sqrt(3))*SW+4*R^2)*S^2+(6*S^2-(12*R^2+3*SA-(3+sqrt(3))*SW)*(SB+SC))*S+4*sqrt(3)*(3*R^2-SW)*SB*SC : :

X(31690) lies on these lines: {6,31691}, {14,486}, {115,31692}, {372,31688}, {6109,8281}, {6114,8855}, {22839,31686}

X(31691) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND INNER-SQUARES

Barycentrics -sqrt(3)*(-sqrt(3)*SA-(2-sqrt(3))*SW+4*R^2)*S^2-(6*S^2-(12*R^2+3*SA-(-sqrt(3)+3)*SW)*(SB+SC))*S-4*sqrt(3)*(3*R^2-SW)*SB*SC : :

X(31691) lies on these lines: {6,31690}, {14,485}, {115,31689}, {371,31688}, {6109,8280}, {6114,8854}, {6303,8968}, {22838,31686}

X(31692) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND OUTER-SQUARES

Barycentrics -sqrt(3)*(-sqrt(3)*SA-(2-sqrt(3))*SW+4*R^2)*S^2+(6*S^2-(12*R^2+3*SA-(-sqrt(3)+3)*SW)*(SB+SC))*S-4*sqrt(3)*(3*R^2-SW)*SB*SC : :

X(31692) lies on these lines: {6,31689}, {13,486}, {115,31690}, {372,31687}, {6108,8281}, {6115,8855}, {22839,31685}

X(31693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO ANTI-ARTZT

Barycentrics 2*(a^2+b^2+c^2)*S+sqrt(3)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2) : :
X(31693) = 3*X(13)+X(22493) = 3*X(13)-X(22495) = X(15)-3*X(22489) = X(621)+2*X(11542) = 2*X(623)+X(5318)

As a point on the Euler line, X(31693) has Shinagawa coefficients (2(E+F)+3^1/2S, 3(3^1/2)S).

The reciprocal orthologic center of these triangles is X(12155)

X(31693) lies on these lines: {2,3}, {6,22491}, {13,524}, {14,597}, {15,22489}, {115,396}, {141,16808}, {298,31683}, {395,5475}, {530,623}, {543,6115}, {599,22492}, {621,11542}, {671,5978}, {3589,16809}, {3815,6775}, {3849,6108}, {5349,6694}, {5461,6109}, {5464,20112}, {5617,12155}, {6295,22893}, {6772,22576}, {7615,9763}, {8584,22496}, {9761,10653}, {9762,22110}, {9771,9886}, {11645,22796}, {12154,25164}, {13083,18424}, {22165,22494}, {22797,25561}

X(31693) = midpoint of X(i) and X(j) for these {i,j}: {671, 5978}, {5464, 23004}, {22493, 22495}
X(31693) = reflection of X(i) in X(j) for these (i,j): (396, 5459), (6109, 5461)
X(31693) = inverse of X(11295) in the orthocentroidal circle
X(31693) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11295), (2, 381, 31694), (3830, 16041, 31694)

X(31694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO ANTI-ARTZT

Barycentrics -2*(a^2+b^2+c^2)*S+sqrt(3)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2) : :
X(31694) = 3*X(14)+X(22494) = 3*X(14)-X(22496) = X(16)-3*X(22490) = X(622)+2*X(11543) = 2*X(624)+X(5321)

As a point on the Euler line, X(31694) has Shinagawa coefficients (2(E+F)-3^1/2S, -3(3^1/2)S).

The reciprocal orthologic center of these triangles is X(12154)

X(31694) lies on these lines: {2,3}, {6,22492}, {13,597}, {14,524}, {16,22490}, {115,395}, {141,16809}, {299,31684}, {396,5475}, {531,624}, {543,6114}, {599,22491}, {622,11543}, {671,5979}, {3589,16808}, {3815,6772}, {3849,6109}, {5350,6695}, {5461,6108}, {5463,20112}, {5613,12154}, {6582,22847}, {6775,22575}, {7615,9761}, {8584,22495}, {9760,22110}, {9763,10654}, {9771,9885}, {11645,22797}, {12155,25154}, {13084,18424}, {22165,22493}, {22796,25561}

X(31694) = midpoint of X(i) and X(j) for these {i,j}: {671, 5979}, {5463, 23005}, {22494, 22496}
X(31694) = reflection of X(i) in X(j) for these (i,j): (395, 5460), (6108, 5461)
X(31694) = inverse of X(11296) in the orthocentroidal circle
X(31694) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11296), (2, 381, 31693), (3830, 16041, 31693)

X(31695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO ANTI-MCCAY

Barycentrics -4*(a^2-2*b^2+c^2)*(a^2+b^2-2*c^2)*S+(2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :
X(31695) = 3*X(13)-X(5464) = 5*X(13)-X(9114) = 3*X(13)+X(22577) = X(15)-3*X(22571) = 3*X(115)-2*X(5460) = X(616)-3*X(9166) = 2*X(618)-3*X(14971) = 2*X(620)-3*X(22489) = 2*X(5461)-3*X(5470) = X(5463)-3*X(5470) = 5*X(5464)-3*X(9114) = 4*X(6669)-3*X(9167) = 3*X(9114)+5*X(22577) = X(9116)-3*X(22489) = X(10992)-4*X(20415)

The reciprocal orthologic center of these triangles is X(8595)

X(31695) lies on these lines: {4,542}, {6,22575}, {13,543}, {15,22571}, {30,22573}, {115,395}, {148,31683}, {524,31709}, {531,5318}, {616,9166}, {618,14971}, {619,15300}, {620,9116}, {630,2482}, {5461,5463}, {6115,9771}, {6669,9167}, {6775,22492}, {9886,18582}, {10992,20415}, {11632,13103}, {22495,23004}

X(31695) = midpoint of X(i) and X(j) for these {i,j}: {5464, 22577}, {11632, 13103}, {22495, 23004}
X(31695) = reflection of X(i) in X(j) for these (i,j): (2482, 5459), (5463, 5461), (9116, 620), (15300, 619), (31696, 671)
X(31695) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 22577, 5464), (9116, 22489, 620)

X(31696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO ANTI-MCCAY

Barycentrics 4*(a^2-2*b^2+c^2)*(a^2+b^2-2*c^2)*S+(2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :
X(31696) = 3*X(14)-X(5463) = 5*X(14)-X(9116) = 3*X(14)+X(22578) = X(16)-3*X(22572) = 3*X(115)-2*X(5459) = X(617)-3*X(9166) = 2*X(619)-3*X(14971) = 2*X(620)-3*X(22490) = 2*X(5461)-3*X(5469) = 5*X(5463)-3*X(9116) = X(5464)-3*X(5469) = 4*X(6670)-3*X(9167) = X(9114)-3*X(22490) = 3*X(9116)+5*X(22578) = X(10992)-4*X(20416)

The reciprocal orthologic center of these triangles is X(12154)

X(31696) lies on these lines: {4,542}, {6,22576}, {14,543}, {16,22572}, {30,22574}, {115,396}, {148,31684}, {524,31710}, {530,5321}, {617,9166}, {618,15300}, {619,14971}, {620,9114}, {629,2482}, {5461,5464}, {6114,9771}, {6670,9167}, {6772,22491}, {9885,18581}, {10992,20416}, {11632,13102}, {22496,23005}

X(31696) = midpoint of X(i) and X(j) for these {i,j}: {5463, 22578}, {11632, 13102}, {22496, 23005}
X(31696) = reflection of X(i) in X(j) for these (i,j): (2482, 5460), (5464, 5461), (9114, 620), (15300, 618), (31695, 671)
X(31696) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 22578, 5463), (9114, 22490, 620)

X(31697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics 9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)-2*S*((2*(2*a^2+b^2+c^2)*S-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+3*a^2*(2*a^2+b^2+c^2)-9*(b^2-c^2)^2) : :
X(31697) = 3*X(13)+X(22607) = X(15)-3*X(22602)

The reciprocal orthologic center of these triangles is X(22601)

X(31697) lies on these lines: {4,372}, {6,22605}, {13,22607}, {15,22602}, {30,13929}, {530,31717}, {939,1144}, {6300,18582}

X(31697) = midpoint of X(22607) and X(22609)
X(31697) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 22607, 22609), (486, 6565, 31698)

X(31698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics 9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)-2*S*(-(2*(2*a^2+b^2+c^2)*S-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+3*a^2*(2*a^2+b^2+c^2)-9*(b^2-c^2)^2) : :
X(31698) = 3*X(14)+X(22608) = X(16)-3*X(22604)

The reciprocal orthologic center of these triangles is X(22603)

X(31698) lies on these lines: {4,372}, {6,22606}, {14,22608}, {16,22604}, {30,13928}, {531,31716}, {6301,18581}

X(31698) = midpoint of X(22608) and X(22610)
X(31698) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 22608, 22610), (486, 6565, 31697)

X(31699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics 9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)+2*S*(-(-2*(2*a^2+b^2+c^2)*S-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+3*a^2*(2*a^2+b^2+c^2)-9*(b^2-c^2)^2) : :
X(31699) = 3*X(13)+X(22636) = X(15)-3*X(22631)

The reciprocal orthologic center of these triangles is X(22630)

X(31699) lies on these lines: {4,371}, {6,22634}, {13,22636}, {15,22631}, {30,13876}, {530,31715}, {6304,18582}

X(31699) = midpoint of X(22636) and X(22638)
X(31699) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 22636, 22638), (485, 6564, 31700)

X(31700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics 9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)+2*S*((-2*(2*a^2+b^2+c^2)*S-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+3*a^2*(2*a^2+b^2+c^2)-9*(b^2-c^2)^2) : :
X(31700) = 3*X(14)+X(22637) = X(16)-3*X(22633)

The reciprocal orthologic center of these triangles is X(22632)

X(31700) lies on these lines: {4,371}, {6,22635}, {14,22637}, {16,22633}, {30,13875}, {531,31718}, {6305,18581}

X(31700) = midpoint of X(22637) and X(22639)
X(31700) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 22637, 22639), (485, 6564, 31699)

X(31701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 1st BROCARD-REFLECTED

Barycentrics 2*sqrt(3)*(b^2+c^2)*(c^2+a^2+b^2)*S*a^2+3*(b^2+c^2)*a^6+2*(b^4+4*b^2*c^2+c^4)*a^4-4*(b^2-c^2)^2*b^2*c^2-5*(b^4-c^4)*(b^2-c^2)*a^2 : :
X(31701) = 3*X(13)+X(22695) = 3*X(13)-X(22701) = X(15)-3*X(22688)

The reciprocal orthologic center of these triangles is X(22687)

X(31701) lies on these lines: {4,39}, {13,511}, {15,22688}, {30,22691}, {115,7684}, {381,3107}, {623,31713}, {2782,31709}, {3094,22694}, {3104,5340}, {3105,7697}, {18582,22715}

X(31701) = midpoint of X(22695) and X(22701)
X(31701) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 22682, 31702), (262, 5475, 31702), (262, 22708, 39)

X(31702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 1st BROCARD-REFLECTED

Barycentrics -2*sqrt(3)*(b^2+c^2)*(c^2+a^2+b^2)*S*a^2+3*(b^2+c^2)*a^6+2*(b^4+4*b^2*c^2+c^4)*a^4-4*(b^2-c^2)^2*b^2*c^2-5*(b^4-c^4)*(b^2-c^2)*a^2 : :
X(31702) = 3*X(14)+X(22696) = 3*X(14)-X(22702) = X(16)-3*X(22690)

The reciprocal orthologic center of these triangles is X(22689)

X(31702) lies on these lines: {4,39}, {6,22694}, {14,511}, {16,22690}, {30,22692}, {115,7685}, {381,3106}, {624,31714}, {2782,31710}, {3094,22693}, {3104,7697}, {3105,5339}, {18581,22714}

X(31702) = midpoint of X(22696) and X(22702)
X(31702) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 22682, 31701), (262, 22707, 39)

X(31703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO INNER-FERMAT

Barycentrics (a^4+(b^2+c^2)*a^2-2*b^4-2*b^2*c^2-2*c^4)*a^2-2*sqrt(3)*(a^4-2*(b^2-c^2)^2)*S : :
X(31703) = 3*X(13)+X(22849) = 3*X(13)-X(22855) = X(15)-3*X(22846) = 3*X(18)+X(19106)

The reciprocal orthologic center of these triangles is X(616)

X(31703) lies on these lines: {4,16}, {6,16628}, {13,533}, {15,115}, {30,10617}, {623,11133}, {628,18582}, {630,16966}, {3098,31704}, {3105,16627}, {5335,22114}, {5965,22900}, {8260,11543}, {9115,22796}, {10645,22843}, {11082,19295}, {19780,22511}, {22869,22907}

X(31703) = midpoint of X(22849) and X(22855)
X(31703) = reflection of X(11133) in X(623)
X(31703) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 16809, 31706), (18, 22862, 16), (22831, 31706, 16809)

X(31704) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO OUTER-FERMAT

Barycentrics (a^4+(b^2+c^2)*a^2-2*b^4-2*b^2*c^2-2*c^4)*a^2+2*sqrt(3)*(a^4-2*(b^2-c^2)^2)*S : :
X(31704) = 3*X(14)+X(22895) = 3*X(14)-X(22901) = X(16)-3*X(22891) = 3*X(17)+X(19107)

The reciprocal orthologic center of these triangles is X(617)

X(31704) lies on these lines: {4,15}, {6,16629}, {14,532}, {16,115}, {30,10616}, {624,11132}, {627,18581}, {629,16967}, {3098,31703}, {3104,16626}, {5334,22113}, {5965,22856}, {8259,11542}, {9117,22797}, {10646,22890}, {11087,19294}, {19781,22510}, {22861,22914}

X(31704) = midpoint of X(22895) and X(22901)
X(31704) = reflection of X(11132) in X(624)
X(31704) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 16808, 31705), (17, 22906, 15), (22832, 31705, 16808)

X(31705) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO OUTER-FERMAT

Barycentrics -2*sqrt(3)*(2*a^4+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2)*S+2*a^6-7*(b^2+c^2)*a^4-3*(b^2+c^2)*b^2*c^2+2*(b^4-8*b^2*c^2+c^4)*a^2+3*b^6+3*c^6 : :
X(31705) = 3*X(13)+X(22894) = 3*X(13)-X(22900) = X(15)-3*X(17) = 2*X(15)-3*X(14138) = 3*X(10611)-2*X(11542)

The reciprocal orthologic center of these triangles is X(13)

X(31705) lies on these lines: {4,15}, {6,16626}, {13,298}, {16,629}, {30,22892}, {115,6783}, {546,10613}, {621,16634}, {627,5335}, {1692,31706}, {5318,6115}, {5321,8259}, {5978,11122}, {6109,22797}, {6671,11299}, {6673,11307}, {11132,11303}, {20080,22113}

X(31705) = midpoint of X(i) and X(j) for these {i,j}: {5978, 11122}, {22894, 22900}
X(31705) = reflection of X(i) in X(j) for these (i,j): (6109, 22893), (14138, 17)
X(31705) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 16808, 31704), (16808, 31704, 22832), (18582, 22907, 17)

X(31706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO INNER-FERMAT

Barycentrics 2*sqrt(3)*(2*a^4+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2)*S+2*a^6-7*(b^2+c^2)*a^4-3*(b^2+c^2)*b^2*c^2+2*(b^4-8*b^2*c^2+c^4)*a^2+3*b^6+3*c^6 : :
X(31706) = 3*X(14)+X(22850) = 3*X(14)-X(22856) = X(16)-3*X(18) = 2*X(16)-3*X(14139) = 3*X(10612)-2*X(11543)

The reciprocal orthologic center of these triangles is X(14)

X(31706) lies on these lines: {4,16}, {6,16627}, {14,299}, {15,630}, {30,22848}, {115,6782}, {546,10614}, {622,16635}, {628,5334}, {1692,31705}, {5318,8260}, {5321,6114}, {5979,11121}, {6108,22796}, {6672,11300}, {6674,11308}, {11133,11304}, {20080,22114}

X(31706) = midpoint of X(i) and X(j) for these {i,j}: {5979, 11121}, {22850, 22856}
X(31706) = reflection of X(i) in X(j) for these (i,j): (6108, 22847), (14139, 18)
X(31706) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 16809, 31703), (16809, 31703, 22831), (18581, 22861, 18)

X(31707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 2nd FERMAT-DAO

Barycentrics (SB+SC)*(3*S^4-(18*SW*R^2-SA^2-2*SB*SC-5*SW^2)*S^2-2*sqrt(3)*S*(2*S^2*SW+(2*SA^2-(SW+9*SA)*R^2)*SW)-(6*R^2-3*SA+2*SW)*SA*SW^2) : :
X(31707) = 3*X(13)-X(22999) = X(15)-3*X(25151)

The reciprocal orthologic center of these triangles is X(25207)

X(31707) lies on these lines: {4,23017}, {6,25180}, {13,22999}, {15,25151}, {30,25178}, {462,6111}, {511,6783}, {512,31709}, {10653,14188}, {11542,25220}, {14182,18582}

X(31707) = midpoint of X(i) and X(j) for these {i,j}: {13, 23007}, {22999, 25182}
X(31707) = reflection of X(i) in X(j) for these (i,j): (25220, 11542), (31719, 13)
X(31707) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 25182, 22999), (22999, 23007, 25182)

X(31708) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 1st FERMAT-DAO

Barycentrics (SB+SC)*(3*S^4-(18*SW*R^2-SA^2-2*SB*SC-5*SW^2)*S^2+2*sqrt(3)*S*(2*S^2*SW+(2*SA^2-(SW+9*SA)*R^2)*SW)-(6*R^2-3*SA+2*SW)*SA*SW^2) : :
X(31708) = 3*X(14)-X(23008) = X(16)-3*X(25161)

The reciprocal orthologic center of these triangles is X(25208)

X(31708) lies on these lines: {4,23023}, {6,25175}, {14,23008}, {16,25161}, {30,25173}, {463,6110}, {511,6782}, {512,31710}, {10654,14186}, {11543,25219}, {14178,18581}

X(31708) = midpoint of X(i) and X(j) for these {i,j}: {14, 23014}, {23008, 25177}
X(31708) = reflection of X(i) in X(j) for these (i,j): (25219, 11543), (31720, 14)
X(31708) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 25177, 23008), (23008, 23014, 25177)

X(31709) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 4th FERMAT-DAO

Barycentrics S^2*SW+sqrt(3)*S*(6*S^2-(SB+SC)*(6*SA+SW))-3*SW*SB*SC : :
X(31709) = 3*X(13)-X(22997) = 3*X(13)+X(25166) = 5*X(13)-X(25236) = X(15)-3*X(5470) = X(1080)-3*X(14639) = 3*X(5469)+X(19106) = X(5979)-3*X(14041) = X(6780)-3*X(16267) = 3*X(6783)-2*X(22997) = X(6783)+2*X(23004) = 3*X(6783)+2*X(25166) = 5*X(6783)-2*X(25236) = X(22997)+3*X(23004) = 5*X(22997)-3*X(25236) = 3*X(23004)-X(25166) = 5*X(23004)+X(25236) = 5*X(25166)+3*X(25236)

The reciprocal orthologic center of these triangles is X(5469)

X(31709) lies on these lines: {2,11154}, {4,14}, {6,25164}, {13,531}, {15,5470}, {16,5460}, {30,115}, {148,5978}, {316,532}, {381,6114}, {396,20415}, {397,16002}, {511,25222}, {512,31707}, {524,31695}, {530,8352}, {542,5318}, {543,6115}, {619,11303}, {623,11133}, {1080,14639}, {2782,31701}, {3642,11185}, {3643,7841}, {3830,22512}, {5032,5335}, {5321,11645}, {5464,18582}, {5469,19106}, {5969,31711}, {5979,14041}, {6321,6772}, {6670,11304}, {6774,21402}, {6778,12243}, {6780,16267}, {9117,11542}, {11632,22513}, {11646,25154}

X(31709) = midpoint of X(i) and X(j) for these {i,j}: {13, 23004}, {148, 5978}, {22997, 25166}
X(31709) = reflection of X(i) in X(j) for these (i,j): (6109, 115), (6783, 13), (9117, 11542)
X(31709) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 25166, 22997), (115, 9880, 31710), (22997, 23004, 25166)
X(31709) = X(16)-pedal-to-X(15)-pedal similarity image of X(13)

X(31710) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 3rd FERMAT-DAO

Barycentrics S^2*SW-sqrt(3)*S*(6*S^2-(SB+SC)*(6*SA+SW))-3*SW*SB*SC : :
X(31710) = 3*X(14)-X(22998) = 3*X(14)+X(25156) = 5*X(14)-X(25235) = X(16)-3*X(5469) = X(383)-3*X(14639) = 3*X(5470)+X(19107) = X(5978)-3*X(14041) = X(6779)-3*X(16268) = 3*X(6782)-2*X(22998) = X(6782)+2*X(23005) = 3*X(6782)+2*X(25156) = 5*X(6782)-2*X(25235) = X(22998)+3*X(23005) = 5*X(22998)-3*X(25235) = 3*X(23005)-X(25156) = 5*X(23005)+X(25235) = 5*X(25156)+3*X(25235)

The reciprocal orthologic center of these triangles is X(5470)

X(31710) lies on these lines: {2,11153}, {4,13}, {6,25154}, {14,530}, {15,5459}, {16,5469}, {30,115}, {148,5979}, {316,533}, {381,6115}, {383,14639}, {395,20416}, {398,16001}, {511,25221}, {512,31708}, {524,31696}, {531,8352}, {542,5321}, {543,6114}, {618,11304}, {624,11132}, {2782,31702}, {3642,7841}, {3643,11185}, {3830,22513}, {5032,5334}, {5318,11645}, {5463,18581}, {5470,19107}, {5969,31712}, {5978,14041}, {6321,6775}, {6669,11303}, {6771,21401}, {6777,12243}, {6779,16268}, {9115,11543}, {11632,22512}, {11646,25164}

X(31710) = midpoint of X(i) and X(j) for these {i,j}: {14, 23005}, {148, 5979}, {22998, 25156}
X(31710) = reflection of X(i) in X(j) for these (i,j): (6108, 115), (6782, 14), (9115, 11543)
X(31710) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 25156, 22998), (115, 9880, 31709), (22998, 23005, 25156)
X(31710) = X(15)-pedal-to-X(16)-pedal similarity image of X(14)

X(31711) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 1st NEUBERG

Barycentrics 3*S^4+3*(SB*SC+SA^2)*S^2-sqrt(3)*S*(S^2*(SA-3*SW)-(3*SA-SW)*SA*SW)+3*SB*SC*SW^2 : :
X(31711) = 3*X(13)-X(23000) = 3*X(13)+X(25199) = X(15)-3*X(25157)

The reciprocal orthologic center of these triangles is X(6582)

X(31711) lies on these lines: {4,69}, {6,25191}, {13,538}, {15,3734}, {30,25183}, {115,623}, {3106,7844}, {3643,9466}, {5969,31709}, {6581,18582}, {6671,7820}

X(31711) = midpoint of X(23000) and X(25199)
X(31711) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 25199, 23000), (76, 3818, 31712)

X(31712) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 1st NEUBERG

Barycentrics 3*S^4+3*(SB*SC+SA^2)*S^2+sqrt(3)*S*(S^2*(SA-3*SW)-(3*SA-SW)*SA*SW)+3*SB*SC*SW^2 : :
X(31712) = 3*X(14)-X(23009) = 3*X(14)+X(25203) = X(16)-3*X(25167)

The reciprocal orthologic center of these triangles is X(6295)

X(31712) lies on these lines: {4,69}, {6,25195}, {14,538}, {16,3734}, {30,25187}, {115,624}, {3107,7844}, {3642,9466}, {5969,31710}, {6294,18581}, {6672,7820}

X(31712) = midpoint of X(23009) and X(25203)
X(31712) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 25203, 23009), (76, 3818, 31711)

X(31713) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 2nd NEUBERG

Barycentrics 3*S^4+3*(SA^2-3*SB*SC-2*SW^2)*S^2+sqrt(3)*S*(SA+SW)*(3*SA*SW-4*SW^2+S^2)-15*SB*SC*SW^2 : :
X(31713) = 3*X(13)-X(23001) = 3*X(13)+X(25200) = X(15)-3*X(25158)

The reciprocal orthologic center of these triangles is X(6298)

X(31713) lies on these lines: {4,83}, {6,25192}, {13,754}, {15,25158}, {30,25184}, {623,31701}, {624,6292}, {4045,19106}, {6296,18582}

X(31713) = midpoint of X(23001) and X(25200)
X(31713) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 25200, 23001), (83, 19130, 31714)

X(31714) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 2nd NEUBERG

Barycentrics 3*S^4+3*(SA^2-3*SB*SC-2*SW^2)*S^2-sqrt(3)*S*(SA+SW)*(3*SA*SW-4*SW^2+S^2)-15*SB*SC*SW^2 : :
X(31714) = 3*X(14)-X(23010) = 3*X(14)+X(25204) = X(16)-3*X(25168)

The reciprocal orthologic center of these triangles is X(6299)

X(31714) lies on these lines: {4,83}, {6,25196}, {14,754}, {16,25168}, {30,25188}, {623,6292}, {624,31702}, {4045,19107}, {6297,18581}

X(31714) = midpoint of X(23010) and X(25204)
X(31714) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 25204, 23010), (83, 19130, 31713)

X(31715) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics
-2*sqrt(3)*((3*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)+10*a^4+(b^2+c^2)*a^2-11*(b^2-c^2)^2)*S+3*(b^6+c^6)*sqrt(3)-(9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2+3*(b^2+c^2)*b^2*c^2)*sqrt(3)-4*S^2*(10*a^2+b^2+c^2) : :
X(31715) = 3*X(13)-X(23002) = X(15)-3*X(25159)

The reciprocal orthologic center of these triangles is X(13705)

X(31715) lies on these lines: {4,1327}, {6,25193}, {13,23002}, {15,25159}, {30,25185}, {530,31699}, {5394,31533}, {13706,18582}

X(31715) = midpoint of X(23002) and X(25201)
X(31715) = {X(13), X(25201)}-harmonic conjugate of X(23002)

X(31716) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
2*sqrt(3)*((3*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)+10*a^4+(b^2+c^2)*a^2-11*(b^2-c^2)^2)*S+3*(b^6+c^6)*sqrt(3)-(9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2+3*(b^2+c^2)*b^2*c^2)*sqrt(3)-4*S^2*(10*a^2+b^2+c^2) : :
X(31716) = 3*X(14)-X(23012) = X(16)-3*X(25170)

The reciprocal orthologic center of these triangles is X(13823)

X(31716) lies on these lines: {4,1328}, {6,25198}, {14,23012}, {16,25170}, {30,25190}, {531,31698}, {13824,18581}

X(31716) = midpoint of X(23012) and X(25206)
X(31716) = {X(14), X(25206)}-harmonic conjugate of X(23012)

X(31717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics
-2*sqrt(3)*(-(3*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)+10*a^4+(b^2+c^2)*a^2-11*(b^2-c^2)^2)*S-3*(b^6+c^6)*sqrt(3)+(9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2+3*(b^2+c^2)*b^2*c^2)*sqrt(3)-4*S^2*(10*a^2+b^2+c^2) : :
X(31717) = 3*X(13)-X(23003) = X(15)-3*X(25160)

The reciprocal orthologic center of these triangles is X(13825)

X(31717) lies on these lines: {4,1328}, {6,25194}, {13,23003}, {15,25160}, {30,25186}, {530,31697}, {13826,18582}

X(31717) = midpoint of X(23003) and X(25202)
X(31717) = {X(13), X(25202)}-harmonic conjugate of X(23003)

X(31718) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics
2*sqrt(3)*(-(3*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)+10*a^4+(b^2+c^2)*a^2-11*(b^2-c^2)^2)*S-3*(b^6+c^6)*sqrt(3)+(9*(b^2+c^2)*a^4-6*(b^2-c^2)^2*a^2+3*(b^2+c^2)*b^2*c^2)*sqrt(3)-4*S^2*(10*a^2+b^2+c^2) : :
X(31718) = 3*X(14)-X(23011) = X(16)-3*X(25169)

The reciprocal orthologic center of these triangles is X(13703)

X(31718) lies on these lines: {4,1327}, {6,25197}, {14,23011}, {16,25169}, {30,25189}, {531,31700}, {13704,18581}

X(31718) = midpoint of X(23011) and X(25205)
X(31718) = {X(14), X(25205)}-harmonic conjugate of X(23011)

X(31719) = PARALLELOGIC CENTER OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO TO 2nd FERMAT-DAO

Barycentrics (SB+SC)*(9*S^4-(SW^2-11*SA^2+2*SB*SC+6*(6*SA+SW)*R^2)*S^2+2*sqrt(3)*S*(2*S^2*(3*R^2-SW)+SW*(-2*SA^2+(3*SA+SW)*R^2))-(6*R^2-SA-2*SW)*SA*SW^2) : :
X(31719) = 3*X(13)-X(23007) = X(15)-3*X(25217)

The reciprocal parallelogic center of these triangles is X(25216)

X(31719) lies on these lines: {4,23022}, {6,25224}, {13,22999}, {15,25217}, {30,25220}, {511,25222}, {512,6783}, {10653,14182}, {11542,25178}, {14188,18582}

X(31719) = midpoint of X(i) and X(j) for these {i,j}: {13, 22999}, {23007, 25228}
X(31719) = reflection of X(i) in X(j) for these (i,j): (25178, 11542), (31707, 13)
X(31719) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 25228, 23007), (22999, 23007, 25228)

X(31720) = PARALLELOGIC CENTER OF THESE TRIANGLES: 4th ISODYNAMIC-DAO TO 1st FERMAT-DAO

Barycentrics (SB+SC)*(9*S^4-(SW^2-11*SA^2+2*SB*SC+6*(6*SA+SW)*R^2)*S^2-2*sqrt(3)*S*(2*S^2*(3*R^2-SW)+SW*(-2*SA^2+(3*SA+SW)*R^2))-(6*R^2-SA-2*SW)*SA*SW^2) : :
X(31720) = 3*X(14)-X(23014) = X(16)-3*X(25214)

The reciprocal parallelogic center of these triangles is X(25213)

X(31720) lies on these lines: {4,23028}, {6,25223}, {14,23008}, {16,25214}, {30,25219}, {511,25221}, {512,6782}, {10654,14178}, {11543,25173}, {14186,18581}

X(31720) = midpoint of X(i) and X(j) for these {i,j}: {14, 23008}, {23014, 25227}
X(31720) = reflection of X(i) in X(j) for these (i,j): (25173, 11543), (31708, 14)
X(31720) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 25227, 23014), (23008, 23014, 25227)

X(31721) = X(1)X(7) ∩ X(145)X(25723)

Barycentrics (11*a^2-10*(b+c)*a-(b-c)^2)*(a-b+c)*(a+b-c) : :

Let (I) be the incircle of ABC and B', C' the orthogonal projections of I on AC, AB, respectively. Denote B", C" the points other than B',C' where lines BB', CC' cut (I), respectively. The tangents to (I) at B" and C" cuts BC at A_b and A_c, respectively. Build B_c, B_a, C_a, C_b cyclically. Then these six points lie on a conic with perspector X(7) and center X(31721). (César E. Lozada, March 18, 2019)

A_b is the center of the circle that is the inverse-in-B-Soddy-circle of the C-Soddy-circle. A_c is the center of the circle that is the inverse-in-C-Soddy-circle of the B-Soddy-circle. B_c, B_a, C_a, C_b can be constructed cyclically. (Randy Hutson, March 21, 2019)

X(31721) lies on these lines: {1,7}, {145,25723}, {348,20057}, {934,6767}, {3599,3748}, {3616,25718}, {3622,25716}, {3911,17014}, {5219,29624}, {5222,31188}, {5308,5723}, {6172,6603}, {17095,20050}

X(31721) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3160, 5543), (175, 176, 30424), (17802, 17805, 30332)

X(31722) = X(2)X(4480) ∩ X(8)X(9)

Barycentrics (-11*a+b+c)*(-a+b+c) : :

Let (I_a) be the A-excircle of ABC and A'_b, A'_c the orthogonal projections of I_a on AC, AB, respectively. Denote A"_b, A"_c the points other than A'_b, A'_c where BA'_b, CA'_c cut (I_a). The tangents to (I_a) at A"_b, A"_c cut BC at A_b and A_c, respectively. Build B_c, B_a, C_a, C_b cyclically. Then these six points lie on a conic with perspector X(8) and center X(31722). (César E. Lozada, March 18, 2019)

X(31722) lies on these lines: {2,4480}, {8,9}, {44,3241}, {45,3616}, {320,6172}, {958,8834}, {1743,3623}, {3244,3973}, {3622,3731}, {3950,20014}, {4419,6687}, {4473,29611}, {4488,17336}, {4727,20053}, {4969,20050}, {5273,6557}, {5296,17369}, {5749,16814}, {7222,31244}, {9780,24723}, {16670,20057}, {17383,26685}, {21296,25101}

X(31722) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (346, 391, 4060), (346, 3707, 8), (391, 4873, 8)

X(31723) = EULER LINE INTERCEPT OF X(66)X(265)

Barycentrics a^10-(b^2+c^2)*a^8-2*(b^4+b^2*c^2+c^4)*a^6+2*(b^6+c^6)*a^4+(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics R^2*S^2+(3*R^2-2*SW)*SB*SC : :
X(31723) = 3*X(14561)-2*X(19127), 3*X(14643)-2*X(16165)

As a point on the Euler line, X(31723) has Shinagawa coefficients (-E, 5*E+8*F).

See César Lozada, Hyacinthos 28921.

X(31723) lies on these lines: {2, 3}, {49, 9833}, {52, 18381}, {66, 265}, {68, 6243}, {115, 571}, {143, 18912}, {317, 339}, {399, 12319}, {497, 9642}, {511, 18474}, {568, 1899}, {570, 5475}, {578, 11750}, {1154, 11442}, {1352, 9019}, {1503, 18445}, {3060, 25739}, {3313, 3818}, {3521, 14542}, {3567, 18952}, {4857, 9644}, {5157, 19130}, {5448, 26883}, {5512, 15563}, {5654, 10540}, {5946, 18911}, {6033, 13556}, {6102, 11457}, {6776, 15087}, {7747, 19220}, {9927, 11572}, {10316, 27371}, {10539, 13419}, {10620, 13203}, {11392, 18447}, {11393, 18455}, {11433, 13321}, {11550, 13754}, {12295, 19506}, {13202, 19479}, {13352, 18400}, {13598, 18383}, {14516, 16266}, {14561, 19127}, {14643, 16165}, {14983, 19160}, {16655, 22660}, {18388, 29012}, {18390, 19161}, {18439, 22661}, {21850, 26926}, {22120, 27376}

X(31723) = midpoint of X(4) and X(7391)
X(31723) = reflection of X(i) in X(j) for these (i, j): (3, 427), (20, 18570), (22, 5), (7555, 13413), (12083, 15760), (14983, 19160)
X(31723) = anticomplement of X(7502)
X(31723) = orthocentroidal circle-inverse-of X(11818)
X(31723) = X(37569)-of-orthic-triangle if ABC is acute
X(31723) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 12083, 15760), (381, 382, 18534), (382, 7517, 7553), (12225, 15559, 7526)

X(31724) = EULER LINE INTERCEPT OF X(49)X(18400)

Barycentrics a^10-(b^2+c^2)*a^8-(2*b^4-b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(b^4+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics R^2*S^2+(11*R^2-4*SW)*SB*SC : :

As a point on the Euler line, X(31724) has Shinagawa coefficients (-E, 5*E+16*F).

See César Lozada, Hyacinthos 28921.

X(31724) lies on these lines: {2, 3}, {49, 18400}, {50, 1879}, {52, 265}, {67, 17505}, {70, 21400}, {113, 13419}, {343, 12307}, {399, 22660}, {567, 3574}, {1141, 12092}, {1351, 18382}, {1352, 12061}, {1479, 9642}, {1568, 18350}, {1994, 20424}, {3060, 18394}, {3521, 10575}, {3581, 5449}, {3583, 9630}, {5012, 13470}, {5446, 13851}, {5448, 10540}, {5562, 6288}, {6102, 25739}, {6146, 15087}, {6241, 15134}, {6242, 12219}, {6243, 9927}, {6247, 10620}, {7728, 11381}, {10263, 18379}, {10733, 15132}, {11550, 18439}, {11572, 13754}, {11591, 22804}, {11750, 18388}, {12022, 14627}, {12062, 18387}, {12164, 22661}, {13556, 22823}, {15107, 23330}, {15432, 18428}, {18436, 18474}

X(31724) = reflection of X(i) in X(j) for these (i, j): (3, 1594), (2937, 10024), (7488, 5), (10024, 23047)
X(31724) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 382, 7517), (3627, 18567, 4), (10750, 10751, 2070)
X(31724) = orthocenter of cross-triangle of Ehrmann vertex-triangle and Ehrmann side-triangle

X(31725) = EULER LINE INTERCEPT OF X(52)X(22802)

Barycentrics a^10-(b^2+c^2)*a^8-2*(b^4-5*b^2*c^2+c^4)*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics R^2*S^2-(13*R^2-2*SW)*SB*SC : :

As a point on the Euler line, X(31725) has Shinagawa coefficients (E, -5*E+8*F).

See César Lozada, Hyacinthos 28921.

X(31725) lies on these lines: {2, 3}, {52, 22802}, {68, 18439}, {113, 13346}, {184, 12897}, {265, 14216}, {388, 9642}, {399, 6193}, {1514, 22660}, {1533, 21659}, {2883, 18445}, {3521, 14457}, {5270, 9644}, {6000, 25738}, {6225, 18917}, {6243, 7728}, {9927, 11381}, {10540, 12118}, {10575, 18390}, {10620, 12250}, {11456, 12370}, {11663, 31670}, {12121, 20771}, {13445, 26917}, {13474, 18474}, {13491, 18912}, {13556, 22337}, {15041, 18933}, {15063, 15083}, {15072, 18952}, {17702, 26883}, {22661, 31383}

X(31725) = midpoint of X(382) and X(7517)
X(31725) = reflection of X(i) in X(j) for these (i,j): (3, 235), (11413, 5), (12121, 20771), (18404, 4), (31180, 3845)
X(31725) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3146, 18569), (3627, 7553, 382), (5073, 18403, 14790)

X(31726) = EULER LINE INTERCEPT OF X(113)X(22115)

Barycentrics a^10-(b^2+c^2)*a^8-(2*b^4-7*b^2*c^2+2*c^4)*a^6+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics R^2*S^2-(21*R^2-4*SW)*SB*SC : :
X(31726) = X(399)-4*X(1514), X(3581)+2*X(13202), X(13445)-3*X(14644)

As a point on the Euler line, X(31726) has Shinagawa coefficients (E, -5*E+16*F).

See César Lozada, Hyacinthos 28921.

X(31726) lies on these lines: {2, 3}, {113, 22115}, {195, 5893}, {265, 6000}, {389, 3521}, {399, 1514}, {539, 15063}, {1154, 1539}, {1478, 9642}, {1495, 19479}, {3581, 13202}, {3585, 9627}, {4846, 16227}, {5878, 25738}, {6760, 18809}, {7728, 13417}, {7747, 18373}, {8718, 13470}, {9927, 18439}, {10113, 17854}, {10540, 17702}, {10620, 15311}, {10733, 14157}, {11455, 18392}, {11550, 18430}, {12295, 15089}, {13434, 15807}, {13445, 14644}, {13851, 14915}, {20127, 21663}, {20957, 22337}, {22816, 22951}

X(31726) = midpoint of X(i) and X(j) for these {i, j}: {382, 2070}, {3146, 13619}, {10733, 14157}, {18325, 18403}
X(31726) = reflection of X(i) in X(j) for these (i, j): (3, 403), (20, 15646), (186, 11563), (389, 13446), (858, 23323), (2070, 11799), (2071, 5), (2072, 10151), (6760, 18809), (7574, 18403), (11563, 11558), (13619, 7575), (18323, 13473), (18403, 4), (18859, 2072), (20127, 21663), (22115, 113), (25739, 10113)
X(31726) = 2nd Droz-Farny circle-inverse-of X(5)
X(31726) = polar circle-inverse-of X(18560)
X(31726) = Stammler circle-inverse-of X(7517)
X(31726) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3146, 18377), (4, 18325, 7574), (15761, 18560, 3)

Orthopolar circles: X(31727)-X(31840)

This preamble and centers X(31727)-X(31840) were contributed by César Eliud Lozada, March 22, 2019.

Let T'=A'B'C' and T"=A"B"C" be two triangles. Then the orthopoles of the sidelines of T' with respect to T" and the orthopoles of the sidelines of T" with respect to T' lie on an ellipse having for center the midpoint of the orthocenters of T' and T". When T' and T" are inscribed in concentric circles, the ellipse is a circle. Also, under certain conditions, the orthopoles may be coincident or collinear. [R. Goormaghtigh: On pairs of triangles. American Mathematical Monthly , Vol. 57, No. 3 (Mar., 1950), pp. 150-153].

The ellipse just described is here named the orthopolar ellipse or orthopolar circle of T' and T".

This section deals with pairs of triangles inscribed in concentric circles. The appearance of (T', T", i) in the following lists means that X(i) is the center of the orthopolar circle of triangles T' and T".

Pairs of triangles with equal circumcircles:

(ABC, ABC-X3 reflections, 3), (ABC, 1st anti-circumperp, 5562), (ABC, circummedial, 14866), (ABC, circumnormal, 5), (ABC, circumorthic, 52), (ABC, 1st circumperp, 10), (ABC, 2nd circumperp, 946), (ABC, circumsymmedial, 14867), (ABC, circumtangential, 5), (ABC-X3 reflections, 1st anti-circumperp, 10625), (ABC-X3 reflections, circummedial, 31729), (ABC-X3 reflections, circumnormal, 550), (ABC-X3 reflections, circumorthic, 185), (ABC-X3 reflections, 1st circumperp, 31730), (ABC-X3 reflections, 2nd circumperp, 4297), (ABC-X3 reflections, circumsymmedial, 31731), (ABC-X3 reflections, circumtangential, 550), (2nd anti-circumperp-tangential, Hutson intouch, 10106), (2nd anti-circumperp-tangential, intouch, 4292), (2nd anti-circumperp-tangential, Mandart-incircle, 1), (2nd anti-circumperp-tangential, midarc, 31735), (2nd anti-circumperp-tangential, 2nd midarc, 31734), (1st anti-circumperp, circummedial, 31736), (1st anti-circumperp, circumnormal, 6101), (1st anti-circumperp, circumorthic, 3), (1st anti-circumperp, 1st circumperp, 31737), (1st anti-circumperp, 2nd circumperp, 31738), (1st anti-circumperp, circumsymmedial, 31739), (1st anti-circumperp, circumtangential, 6101), (anti-incircle-circles, Ara, 155), (Aquila, excentral, 4), (1st Brocard, 2nd Brocard, 31742), (4th Brocard, orthocentroidal, 31743), (circummedial, circumnormal, 31744), (circummedial, circumorthic, 31745), (circummedial, 1st circumperp, 31746), (circummedial, 2nd circumperp, 31747), (circummedial, circumsymmedial, 31748), (circummedial, circumtangential, 31744), (circumnormal, circumorthic, 6102), (circumnormal, 1st circumperp, 3579), (circumnormal, 2nd circumperp, 1385), (circumnormal, circumsymmedial, 31727), (circumnormal, circumtangential, 3), (circumorthic, 1st circumperp, 31728), (circumorthic, 2nd circumperp, 31732), (circumorthic, circumsymmedial, 31733), (circumorthic, circumtangential, 6102), (1st circumperp, 2nd circumperp, 3), (1st circumperp, circumsymmedial, 31740), (1st circumperp, circumtangential, 3579), (2nd circumperp, circumsymmedial, 31741), (2nd circumperp, circumtangential, 1385), (circumsymmedial, circumtangential, 31727), (Ehrmann-side, Johnson, 5562), (Euler, 2nd Euler, 5907), (Euler, 3rd Euler, 18483), (Euler, 4th Euler, 19925), (Euler, 5th Euler, 31749), (Euler, Feuerbach, 31750), (Euler, medial, 5), (Euler, orthic, 5446), (2nd Euler, 3rd Euler, 31751), (2nd Euler, 4th Euler, 31752), (2nd Euler, 5th Euler, 31753), (2nd Euler, Feuerbach, 31754), (2nd Euler, medial, 1216), (2nd Euler, orthic, 5), (3rd Euler, 4th Euler, 5), (3rd Euler, 5th Euler, 31755), (3rd Euler, Feuerbach, 31756), (3rd Euler, medial, 1125), (3rd Euler, orthic, 31757), (4th Euler, 5th Euler, 31758), (4th Euler, Feuerbach, 31759), (4th Euler, medial, 6684), (4th Euler, orthic, 31760), (5th Euler, Feuerbach, 31761), (5th Euler, medial, 31762), (5th Euler, orthic, 31763), (Feuerbach, medial, 31764), (Feuerbach, orthic, 31765), (Hutson intouch, intouch, 1), (Hutson intouch, Mandart-incircle, 10624), (Hutson intouch, midarc, 31766), (Hutson intouch, 2nd midarc, 31767), (intouch, Mandart-incircle, 950), (intouch, midarc, 31768), (intouch, 2nd midarc, 5571), (Mandart-incircle, midarc, 31769), (Mandart-incircle, 2nd midarc, 31770), (medial, orthic, 389), (midarc, 2nd midarc, 1), (1st orthosymmedial, 2nd orthosymmedial, 31771), (1st Parry, 2nd Parry, 351), (1st Parry, 3rd Parry, 31772), (2nd Parry, 3rd Parry, 31773), (Stammler, X3-ABC reflections, 4).

Pairs of triangles with distinct but concentric circumcircles:

(ABC, anti-Ascella, 31802), (ABC, Ascella, 5806), (ABC, inner-Garcia, 31803), (ABC, Stammler, 5), (ABC, X3-ABC reflections, 3627), (ABC-X3 reflections, anti-Ascella, 31804), (ABC-X3 reflections, Ascella, 31805), (ABC-X3 reflections, inner-Garcia, 31806), (ABC-X3 reflections, Stammler, 550), (ABC-X3 reflections, X3-ABC reflections, 5), (anti-Ara, anticomplementary, 31829), (anti-Ara, Ehrmann-side, 31831), (anti-Ara, 2nd extouch, 31832), (anti-Ara, Johnson, 31833), (anti-Ascella, 1st anti-circumperp, 31807), (anti-Ascella, Ascella, 31808), (anti-Ascella, circummedial, 31809), (anti-Ascella, circumnormal, 12161), (anti-Ascella, circumorthic, 31810), (anti-Ascella, 1st circumperp, 31811), (anti-Ascella, 2nd circumperp, 31812), (anti-Ascella, circumsymmedial, 31813), (anti-Ascella, circumtangential, 12161), (anti-Ascella, inner-Garcia, 31814), (anti-Ascella, Stammler, 12161), (anti-Ascella, X3-ABC reflections, 31815), (6th anti-Brocard, 1st Brocard, 114), (6th anti-Brocard, 2nd Brocard, 31839), (2nd anti-circumperp-tangential, 3rd Conway, 31774), (2nd anti-circumperp-tangential, hexyl, 31775), (2nd anti-circumperp-tangential, Hutson intouch, 10106), (2nd anti-circumperp-tangential, incircle-circles, 31776), (2nd anti-circumperp-tangential, intouch, 4292), (2nd anti-circumperp-tangential, Mandart-incircle, 1), (2nd anti-circumperp-tangential, midarc, 31735), (2nd anti-circumperp-tangential, 2nd midarc, 31734), (2nd anti-circumperp-tangential, 6th mixtilinear, 31777), (1st anti-circumperp, Ascella, 31816), (1st anti-circumperp, inner-Garcia, 31817), (1st anti-circumperp, Stammler, 6101), (1st anti-circumperp, X3-ABC reflections, 5876), (anti-Euler, Conway, 24474), (anticomplementary, Ehrmann-mid, 548), (anticomplementary, Ehrmann-side, 6101), (anticomplementary, 2nd extouch, 31793), (anticomplementary, Johnson, 550), (Ascella, circummedial, 31818), (Ascella, circumnormal, 9940), (Ascella, circumorthic, 31819), (Ascella, 1st circumperp, 31787), (Ascella, 2nd circumperp, 5045), (Ascella, circumsymmedial, 31820), (Ascella, circumtangential, 9940), (Ascella, inner-Garcia, 31821), (Ascella, Stammler, 9940), (Ascella, X3-ABC reflections, 31822), (4th Brocard, 11th Fermat-Dao, 31840), (4th Brocard, 12th Fermat-Dao, 31840), (circummedial, inner-Garcia, 31823), (circummedial, Stammler, 31744), (circummedial, X3-ABC reflections, 31824), (circumnormal, inner-Garcia, 5694), (circumnormal, Stammler, 3), (circumnormal, X3-ABC reflections, 4), (circumorthic, inner-Garcia, 31825), (circumorthic, Stammler, 6102), (circumorthic, X3-ABC reflections, 10263), (1st circumperp, inner-Garcia, 72), (1st circumperp, Stammler, 3579), (1st circumperp, X3-ABC reflections, 18480), (2nd circumperp, inner-Garcia, 5887), (2nd circumperp, Stammler, 1385), (2nd circumperp, X3-ABC reflections, 22793), (circumsymmedial, inner-Garcia, 31826), (circumsymmedial, Stammler, 31727), (circumsymmedial, X3-ABC reflections, 31827), (circumtangential, inner-Garcia, 5694), (circumtangential, Stammler, 3), (circumtangential, X3-ABC reflections, 4), (3rd Conway, hexyl, 31778), (3rd Conway, Hutson intouch, 31779), (3rd Conway, incircle-circles, 31780), (3rd Conway, intouch, 31781), (3rd Conway, Mandart-incircle, 31782), (3rd Conway, midarc, 31783), (3rd Conway, 2nd midarc, 31784), (3rd Conway, 6th mixtilinear, 31785), (Ehrmann-mid, Ehrmann-side, 31834), (Ehrmann-mid, 2nd extouch, 31835), (Ehrmann-mid, Johnson, 140), (Ehrmann-side, 2nd extouch, 31836), (2nd extouch, Johnson, 31837), (11th Fermat-Dao, 12th Fermat-Dao, 381), (11th Fermat-Dao, orthocentroidal, 5946), (12th Fermat-Dao, orthocentroidal, 5946), (13th Fermat-Dao, 14th Fermat-Dao, 5640), (15th Fermat-Dao, 1st isodynamic-Dao, 13), (15th Fermat-Dao, 3rd isodynamic-Dao, 13), (16th Fermat-Dao, 2nd isodynamic-Dao, 14), (16th Fermat-Dao, 4th isodynamic-Dao, 14), (Fuhrmann, outer-Garcia, 3), (inner-Garcia, Stammler, 5694), (inner-Garcia, X3-ABC reflections, 31828), (hexyl, Hutson intouch, 31786), (hexyl, incircle-circles, 31787), (hexyl, intouch, 31788), (hexyl, Mandart-incircle, 31789), (hexyl, midarc, 31790), (hexyl, 2nd midarc, 31791), (hexyl, 6th mixtilinear, 12702), (Hutson intouch, incircle-circles, 31792), (Hutson intouch, intouch, 1), (Hutson intouch, Mandart-incircle, 10624), (Hutson intouch, midarc, 31766), (Hutson intouch, 2nd midarc, 31767), (Hutson intouch, 6th mixtilinear, 31793), (incircle-circles, intouch, 31794), (incircle-circles, Mandart-incircle, 31795), (incircle-circles, midarc, 12813), (incircle-circles, 2nd midarc, 31796), (incircle-circles, 6th mixtilinear, 31797), (intouch, Mandart-incircle, 950), (intouch, midarc, 31768), (intouch, 2nd midarc, 5571), (intouch, 6th mixtilinear, 31798), (1st isodynamic-Dao, 3rd isodynamic-Dao, 13), (2nd isodynamic-Dao, 4th isodynamic-Dao, 14), (K798i, 2nd Zaniah, 31838), (Mandart-incircle, midarc, 31769), (Mandart-incircle, 2nd midarc, 31770), (Mandart-incircle, 6th mixtilinear, 31799), (midarc, 2nd midarc, 1), (midarc, 6th mixtilinear, 31800), (2nd midarc, 6th mixtilinear, 31801), (inner-Napoleon, outer-Napoleon, 2)

Note: 3rd and 4th mixtilinear triangles are not included because of uninteresting coordinates.

X(31727) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMNORMAL AND CIRCUMSYMMEDIAL

Barycentrics a^2*(8*a^8-26*(b^2+c^2)*a^6+6*(5*b^4+2*b^2*c^2+5*c^4)*a^4-(b^2+c^2)*(10*b^4-37*b^2*c^2+10*c^4)*a^2-2*b^8+(7*b^4-18*b^2*c^2+7*c^4)*b^2*c^2-2*c^8) : :
X(31727) = X(3)-3*X(353) = 2*X(140)-3*X(10166) = 5*X(632)-6*X(10160) = 3*X(9128)-2*X(12105)

X(31727) lies on these lines: {3,352}, {6,30521}, {30,14867}, {140,10166}, {517,31740}, {575,28662}, {576,8705}, {632,10160}, {1154,31733}, {3398,6233}, {7545,11226}, {9128,12105}, {14730,22234}

X(31727) = midpoint of X(i) and X(j) for these {i,j}: {14867, 31731}, {31733, 31739}, {31740, 31741}
X(31727) = X(5)-of-circumsymmedial-triangle

X(31728) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMORTHIC AND 1st CIRCUMPERP

Barycentrics a^2*((b^2+c^2)*a^6-3*(b^4+c^4)*a^4+(b+c)*b^2*c^2*a^3+(3*b^4+3*c^4+(6*b^2+7*b*c+6*c^2)*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*b^2*c^2*a-(b^6-c^6)*(b^2-c^2)) : :
X(31728) = X(1)-3*X(5890) = 3*X(51)-2*X(18483) = 3*X(165)-X(11412) = 3*X(185)+X(16980) = 3*X(568)-X(12699) = 3*X(568)-2*X(31757) = 2*X(1125)-3*X(9730) = 2*X(1216)-3*X(10164) = 5*X(1698)-3*X(11459) = 3*X(1699)-5*X(3567) = 3*X(3576)-5*X(10574) = 7*X(3624)-9*X(15045) = 4*X(3634)-3*X(5891) = 3*X(3817)-4*X(5462) = 3*X(5587)-X(12111) = X(5882)-4*X(13382) = 6*X(5892)-5*X(19862) = 2*X(5907)-3*X(10175) = 3*X(5946)-2*X(9955) = 9*X(7988)-11*X(15024)

X(31728) lies on these lines: {1,5890}, {2,31751}, {3,31738}, {4,31760}, {40,5889}, {51,18483}, {52,516}, {65,2779}, {143,22793}, {165,11412}, {185,515}, {378,16473}, {389,946}, {511,31730}, {517,6102}, {568,12699}, {974,11709}, {1125,9730}, {1154,3579}, {1216,10164}, {1385,13630}, {1614,9590}, {1698,11459}, {1699,3567}, {2781,4663}, {2948,12284}, {3576,10574}, {3624,15045}, {3634,5891}, {3817,5462}, {5562,6684}, {5587,12111}, {5663,18480}, {5691,6241}, {5876,9956}, {5882,13382}, {5892,19862}, {5907,10175}, {5946,9955}, {6000,31673}, {6101,31663}, {7722,13211}, {7731,9904}, {7988,15024}, {7989,15058}, {8185,11456}, {8227,15043}, {9572,11460}, {9577,11461}, {9587,11464}, {9621,11449}, {9729,10165}, {10263,28146}, {10575,28164}, {10625,12512}, {11230,12006}, {11231,11591}, {11444,31423}, {11561,11699}, {11720,14708}, {11806,13605}, {12162,19925}, {12270,12407}, {13369,23156}, {13491,28160}, {14449,28178}, {14831,28194}, {15305,18492}, {18436,26446}

X(31728) = midpoint of X(i) and X(j) for these {i,j}: {40, 5889}, {2948, 12284}, {5691, 6241}, {7722, 13211}, {7731, 9904}, {12270, 12407}
X(31728) = reflection of X(i) in X(j) for these (i,j): (4, 31760), (946, 389), (1385, 13630), (5562, 6684), (5876, 9956), (6101, 31663), (10625, 12512), (11699, 11561), (11709, 974), (11720, 14708), (12162, 19925), (12699, 31757), (13605, 11806), (18436, 31752), (22793, 143), (23156, 13369), (31732, 6102), (31737, 3579), (31738, 3)
X(31728) = anticomplement of X(31751)
X(31728) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (568, 12699, 31757), (18436, 26446, 31752)

X(31729) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND CIRCUMMEDIAL

Barycentrics 6*SW*S^4-(54*R^2*SA*(SA-SW)-SW*(-SW^2+27*SA^2-30*SW*SA))*S^2+2*SB*SC*SW^3 : :
X(31729) = 3*X(3)-2*X(31762) = X(4)-3*X(9829) = 2*X(5)-3*X(10163) = X(20)+3*X(6031) = 4*X(140)-3*X(10162) = 5*X(631)-3*X(6032) = 7*X(3526)-6*X(10173) = 3*X(3917)-2*X(31753) = 3*X(6031)-X(12505) = 3*X(9730)-2*X(31763) = 3*X(9829)-2*X(31606) = 3*X(10164)-2*X(31758) = 3*X(10165)-2*X(31755) = 3*X(12506)-4*X(31762)

X(31729) lies on these lines: {2,31749}, {3,3849}, {4,9829}, {5,10163}, {20,6031}, {30,14866}, {140,10162}, {511,31745}, {515,31746}, {516,31747}, {631,6032}, {3526,10173}, {3917,31753}, {5188,8704}, {9730,31763}, {10164,31758}, {10165,31755}, {13754,31736}

X(31729) = midpoint of X(20) and X(12505)
X(31729) = reflection of X(i) in X(j) for these (i,j): (4, 31606), (12506, 3), (14866, 31744)
X(31729) = anticomplement of X(31749)
X(31729) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 9829, 31606), (20, 6031, 12505)

X(31730) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND 1st CIRCUMPERP

Barycentrics 4*a^4+(b+c)*a^3-(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(31730) = X(1)-3*X(376) = 3*X(3)-2*X(1125) = 5*X(3)-3*X(5886) = 4*X(3)-3*X(10165) = 3*X(3)-X(12699) = 9*X(3)-5*X(18493) = X(4)-3*X(165) = 3*X(4)-5*X(1698) = 5*X(4)-7*X(7989) = 2*X(4)-3*X(10175) = 11*X(4)-17*X(30315) = 9*X(165)-5*X(1698) = 3*X(165)-2*X(6684) = 15*X(165)-7*X(7989) = 33*X(165)-17*X(30315) = 3*X(376)+X(6361) = 3*X(946)-4*X(1125) = 5*X(946)-6*X(5886) = 2*X(946)-3*X(10165) = X(946)-4*X(12512) = 3*X(946)-2*X(12699) = 9*X(946)-10*X(18493) = 10*X(1125)-9*X(5886) = 8*X(1125)-9*X(10165) = X(1125)-3*X(12512) = 6*X(1125)-5*X(18493) = 5*X(1698)-6*X(6684) = 10*X(1698)-9*X(10175) = X(5880)-3*X(11495) = 4*X(5886)-5*X(10165) = 10*X(6684)-7*X(7989) = 4*X(6684)-3*X(10175) = 22*X(6684)-17*X(30315) = 14*X(7989)-15*X(10175)

X(31730) lies on these lines: {1,376}, {2,18483}, {3,142}, {4,165}, {5,10164}, {8,20}, {10,30}, {21,7688}, {35,79}, {36,12053}, {46,950}, {55,4292}, {56,10624}, {57,4294}, {58,3755}, {65,4304}, {72,3650}, {99,29039}, {100,16113}, {109,1294}, {140,3817}, {186,9591}, {212,1777}, {307,20291}, {329,16127}, {355,1657}, {381,3634}, {382,19925}, {390,3333}, {411,2077}, {443,4512}, {484,4324}, {496,5122}, {497,15803}, {500,4667}, {511,31728}, {517,550}, {519,3534}, {527,3811}, {535,10915}, {546,11231}, {548,1385}, {549,9955}, {551,8703}, {573,3707}, {580,9441}, {581,1742}, {601,1754}, {631,1699}, {726,9821}, {758,12437}, {901,2695}, {902,23536}, {929,2738}, {936,5698}, {942,4314}, {944,3633}, {952,12103}, {962,3522}, {970,15310}, {993,16004}, {999,12575}, {1012,5584}, {1058,3361}, {1062,4347}, {1155,1210}, {1279,24171}, {1292,26702}, {1376,12572}, {1420,30305}, {1478,4333}, {1479,3911}, {1482,15696}, {1490,2951}, {1519,6942}, {1571,7737}, {1587,9616}, {1656,12571}, {1697,4293}, {1702,6460}, {1703,6459}, {1761,2321}, {1766,2325}, {1768,13199}, {1788,3586}, {1836,5217}, {2093,3486}, {2096,7994}, {2100,15161}, {2101,15160}, {2548,31422}, {2550,31424}, {2771,4067}, {2792,3430}, {2796,8669}, {2800,10609}, {2801,10993}, {2807,15644}, {2915,21062}, {2948,12244}, {3057,4311}, {3062,21168}, {3068,9582}, {3070,13912}, {3071,13975}, {3073,13329}, {3085,9579}, {3086,9580}, {3091,10172}, {3146,5587}, {3149,4679}, {3218,20066}, {3219,7701}, {3241,15697}, {3295,4298}, {3336,4330}, {3338,4309}, {3339,3488}, {3340,4305}, {3359,6868}, {3428,5450}, {3434,4652}, {3452,6985}, {3485,30282}, {3487,4312}, {3509,21096}, {3523,8227}, {3524,3624}, {3525,7988}, {3526,10171}, {3528,5603}, {3529,5657}, {3530,11230}, {3543,9780}, {3550,13161}, {3583,6903}, {3584,16118}, {3587,6869}, {3600,31393}, {3601,4295}, {3616,10304}, {3617,15683}, {3623,5731}, {3625,15686}, {3626,3654}, {3627,9956}, {3636,3656}, {3646,17580}, {3648,4420}, {3653,14093}, {3655,8148}, {3663,5266}, {3671,16137}, {3679,11001}, {3710,4427}, {3828,3830}, {3832,31425}, {3839,19877}, {3841,6841}, {3853,31447}, {3895,20076}, {3916,4847}, {3917,31751}, {3919,8261}, {3927,6743}, {3977,5300}, {4031,18398}, {4190,5250}, {4278,17197}, {4299,5119}, {4313,11529}, {4315,9957}, {4316,11010}, {4342,24928}, {4669,19710}, {4745,15685}, {4855,11415}, {4857,5131}, {5010,6876}, {5044,15587}, {5045,10386}, {5054,19878}, {5055,31253}, {5056,10248}, {5071,19872}, {5128,18391}, {5183,10950}, {5204,12701}, {5218,9612}, {5229,31434}, {5231,6705}, {5251,21669}, {5285,30267}, {5290,31508}, {5303,6909}, {5530,17601}, {5536,26877}, {5537,11491}, {5538,21740}, {5541,12248}, {5542,24470}, {5550,15692}, {5560,18395}, {5687,12527}, {5690,15704}, {5693,9961}, {5697,21578}, {5708,6744}, {5717,17594}, {5719,13159}, {5732,6769}, {5745,6851}, {5752,29353}, {5777,15726}, {5790,17800}, {5818,9588}, {5837,17647}, {5840,10265}, {5842,6245}, {5853,7171}, {5894,6001}, {5901,17502}, {5925,12779}, {6068,18239}, {6147,8255}, {6200,8983}, {6244,11500}, {6253,7964}, {6260,6745}, {6261,6282}, {6396,13971}, {6560,13883}, {6561,13936}, {6666,6849}, {6690,16125}, {6700,24703}, {6767,12577}, {6864,21153}, {6865,10270}, {6881,12558}, {6904,31435}, {6915,10863}, {6916,10268}, {6928,24042}, {6996,24588}, {6999,17292}, {7280,30384}, {7288,9614}, {7354,31397}, {7359,8804}, {7411,10902}, {7736,31421}, {7745,31396}, {7747,31398}, {7967,11531}, {7992,14646}, {8185,12082}, {8193,21312}, {8582,11113}, {8616,24178}, {8666,21627}, {8720,8725}, {8726,12651}, {9524,21180}, {9541,18991}, {9590,12088}, {9624,21734}, {9625,22467}, {9670,17728}, {9709,18250}, {9730,31757}, {9779,10303}, {9860,13172}, {9862,13174}, {9904,12383}, {9940,10178}, {10039,10483}, {10058,17009}, {10167,12005}, {10445,17369}, {10476,29747}, {10531,16209}, {10532,16208}, {10591,31231}, {10595,30389}, {10741,28346}, {11019,15171}, {11024,11106}, {11114,24982}, {11227,13374}, {11249,12522}, {11278,15690}, {11346,25967}, {11413,15177}, {11522,21735}, {11599,12042}, {11826,12616}, {11827,13528}, {12041,13605}, {12100,19883}, {12114,12857}, {12162,31752}, {12253,13221}, {12408,13200}, {12618,29050}, {12688,20117}, {12696,16190}, {12778,20127}, {12953,24914}, {13754,31737}, {13893,23249}, {13947,23259}, {14986,30332}, {15172,21625}, {15489,29349}, {15682,19875}, {15702,30308}, {16117,21077}, {16174,21154}, {17549,24541}, {17579,24987}, {17733,28580}, {17768,28645}, {19708,25055}, {20420,21628}, {21629,29291}, {24680,28212}, {29054,30271}

X(31730) = midpoint of X(i) and X(j) for these {i,j}: {1, 6361}, {20, 40}, {355, 1657}, {944, 7991}, {1768, 13199}, {2096, 7994}, {2948, 12244}, {2951, 5759}, {3529, 5691}, {3654, 15681}, {3679, 11001}, {4297, 5493}, {5541, 12248}, {5690, 15704}, {5693, 9961}, {5925, 12779}, {7982, 20070}, {9860, 13172}, {9862, 13174}, {9904, 12383}, {12253, 13221}, {12408, 13200}, {12702, 18481}, {12778, 20127}, {20066, 24468}
X(31730) = reflection of X(i) in X(j) for these (i,j): (3, 12512), (4, 6684), (5, 31663), (10, 3579), (382, 19925), (551, 8703), (946, 3), (962, 13464), (1385, 548), (3627, 9956), (3830, 3828), (3874, 13369), (4297, 550), (4301, 1385), (5882, 4297), (5884, 9943), (6260, 6796), (7982, 13607), (9948, 1158), (10175, 165), (10741, 28346), (11362, 40), (11599, 12042), (12162, 31752), (12688, 20117), (12699, 1125), (13605, 12041), (18525, 3626), (21627, 8666), (22791, 13624), (22793, 140), (31673, 10), (31738, 15644)
X(31730) = anticomplement of X(18483)
X(31730) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 946, 10165), (3, 12699, 1125), (376, 6361, 1)

X(31731) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND CIRCUMSYMMEDIAL

Barycentrics 8*a^10-21*(b^2+c^2)*a^8+(10*b^4+13*b^2*c^2+10*c^4)*a^6+(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*(b^2-2*c^2)*(2*b^2-c^2) : :
X(31731) = X(4)-3*X(353) = 2*X(5)-3*X(10166) = 3*X(353)-2*X(31608) = 5*X(1656)-6*X(10160) = 3*X(9128)-2*X(16619) = 3*X(30516)-2*X(31749)

X(31731) lies on these lines: {3,9830}, {4,353}, {5,10166}, {30,14867}, {511,31733}, {515,31740}, {516,31741}, {568,8550}, {1656,10160}, {9128,16619}, {11645,12506}, {13192,22100}, {13754,31739}, {18440,31742}, {30516,31749}

X(31731) = reflection of X(i) in X(j) for these (i,j): (4, 31608), (14867, 31727), (18440, 31742)
X(31731) = {X(4), X(353)}-harmonic conjugate of X(31608)

X(31732) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMORTHIC AND 2nd CIRCUMPERP

Barycentrics a^2*((b^2+c^2)*a^6-3*(b^4+c^4)*a^4-(b+c)*b^2*c^2*a^3+(3*b^4+3*c^4-(6*b^2-7*b*c+6*c^2)*b*c)*(b+c)^2*a^2+(b^2-c^2)*(b-c)*b^2*c^2*a-(b^6-c^6)*(b^2-c^2)) : :
X(31732) = X(40)-3*X(5890) = 3*X(51)-2*X(19925) = 3*X(165)-5*X(10574) = X(355)-3*X(568) = 3*X(568)-2*X(31760) = 2*X(1216)-3*X(10165) = 5*X(1698)-7*X(15043) = 3*X(1699)-X(12111) = 3*X(2979)-5*X(7987) = 3*X(3060)-X(5691) = 5*X(3567)-3*X(5587) = 3*X(3576)-X(11412) = 7*X(3624)-5*X(11444) = 3*X(3817)-2*X(5907) = 2*X(3828)-3*X(16226) = 4*X(5462)-3*X(10175) = X(5493)-4*X(13382) = 9*X(5640)-7*X(7989) = 2*X(5777)-3*X(15049) = 3*X(5886)-X(18436) = 3*X(5886)-2*X(31751)

For these triangles the orthopoles are concident.

X(31732) lies on these lines: {1,5889}, {2,31752}, {3,31737}, {4,31757}, {10,389}, {40,5890}, {51,19925}, {52,515}, {103,29091}, {143,18480}, {165,10574}, {185,516}, {355,568}, {511,4297}, {517,6102}, {519,14831}, {573,2304}, {946,12259}, {1125,5562}, {1154,1385}, {1216,10165}, {1614,9625}, {1698,15043}, {1699,12111}, {2772,12688}, {2779,24474}, {2781,12262}, {2807,4301}, {2818,4084}, {2979,7987}, {3060,5691}, {3567,5587}, {3576,11412}, {3579,13630}, {3624,11444}, {3817,5907}, {3828,16226}, {5446,31673}, {5462,10175}, {5493,13382}, {5640,7989}, {5663,22793}, {5777,15049}, {5847,19161}, {5876,9955}, {5886,18436}, {5946,9956}, {6101,13624}, {6243,18481}, {6684,9730}, {7503,16472}, {7592,15177}, {7988,15056}, {8227,11459}, {9586,11449}, {9611,11461}, {9622,11464}, {9729,10164}, {9781,18492}, {10263,28160}, {10575,28150}, {10627,17502}, {10628,13605}, {11230,11591}, {11231,12006}, {11365,12164}, {11793,19862}, {12162,18483}, {12239,13883}, {12240,13936}, {12571,15030}, {12675,23156}, {13491,28146}, {14449,28186}, {15045,31423}, {16192,20791}, {16881,18357}, {16980,28236}

X(31732) = midpoint of X(i) and X(j) for these {i,j}: {1, 5889}, {6243, 18481}
X(31732) = reflection of X(i) in X(j) for these (i,j): (4, 31757), (10, 389), (355, 31760), (3579, 13630), (5562, 1125), (5876, 9955), (6101, 13624), (12162, 18483), (18357, 16881), (18436, 31751), (18480, 143), (23156, 12675), (31673, 5446), (31728, 6102), (31737, 3), (31738, 1385)
X(31732) = anticomplement of X(31752)
X(31732) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355, 568, 31760), (5886, 18436, 31751)

X(31733) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMORTHIC AND CIRCUMSYMMEDIAL

Barycentrics (SB+SC)*((36*R^2-9*SW)*S^4+(6*R^2*(18*R^2*SA-21*SA*SW-SW^2)+(9*SA^2+18*SA*SW-SW^2)*SW)*S^2-(4*R^2-SA-2*SW)*SA*SW^3) : :
X(31733) = 3*X(353)-X(11412) = 2*X(1216)-3*X(10166) = 3*X(30516)-2*X(31753)

X(31733) lies on these lines: {52,9830}, {353,11412}, {511,31731}, {1154,31727}, {1216,10166}, {5562,31608}, {13754,14867}, {30516,31753}

X(31733) = reflection of X(i) in X(j) for these (i,j): (5562, 31608), (31739, 31727)

X(31734) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 2nd MIDARC

Barycentrics
((-a+b+c)*(2*a^2+b^2+2*b*c+c^2)*sin(A/2)+(2*a^3-(2*b-c)*a^2+(b+c)*b*a-(b+c)*(b^2-c^2))*sin(B/2)+(2*a^3+(b-2*c)*a^2+(b+c)*c*a+(b+c)*(b^2-c^2))*sin(C/2))*(a-b+c)*(a+b-c) : :
X(31734) = X(177)-3*X(5434) = X(6284)-3*X(11234)

X(31734) lies on these lines: {1,31769}, {30,31770}, {56,12622}, {177,5434}, {388,12523}, {515,5571}, {516,31767}, {529,18258}, {1478,12614}, {4293,12518}, {4298,31768}, {4312,11528}, {4315,21633}, {6284,11234}, {7354,8422}, {18990,31735}

X(31734) = midpoint of X(7354) and X(8422)
X(31734) = reflection of X(i) in X(j) for these (i,j): (31735, 18990), (31768, 4298), (31769, 1)

X(31735) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND MIDARC

Barycentrics
(-a+b+c)*(2*a^4-(b-c)^2*a^2-(b^2-c^2)^2)*sin(A/2)+(2*a^3+(2*b-c)*a^2+(b+c)*b*a+(b+c)*(b^2-c^2))*(a-b+c)^2*sin(B/2)+(2*a^3-(b-2*c)*a^2+(b+c)*c*a-(b+c)*(b^2-c^2))*(a+b-c)^2*sin(C/2) : :
X(31735) = 3*X(5434)-X(8422) = X(6284)-3*X(11191)

X(31735) lies on these lines: {1,31770}, {30,12908}, {56,12614}, {177,7354}, {388,12518}, {515,31768}, {516,31766}, {1478,12622}, {4293,12523}, {4298,5571}, {4312,12656}, {5434,8422}, {6284,11191}, {12813,28160}, {18990,31734}

X(31735) = midpoint of X(177) and X(7354)
X(31735) = reflection of X(i) in X(j) for these (i,j): (5571, 4298), (31734, 18990), (31770, 1)

X(31736) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND CIRCUMMEDIAL

Barycentrics (SB+SC)*(3*SW*S^4-(216*SA*R^4-6*(9*SA^2+18*SA*SW+SW^2)*R^2+(15*SA^2+18*SA*SW+SW^2)*SW)*S^2+(8*R^2-3*SA-2*SW)*SA*SW^3) : :
X(31736) = 2*X(389)-3*X(10163) = 3*X(3917)-2*X(31762) = X(5889)-3*X(9829) = 3*X(5891)-2*X(31749) = 3*X(6032)-5*X(11444) = 3*X(10162)-4*X(11793) = 3*X(23039)-2*X(31753) = 3*X(31743)-4*X(31762)

X(31736) lies on these lines: {2,31763}, {52,31606}, {389,10163}, {511,14866}, {1154,31744}, {1216,12506}, {3849,5562}, {3917,31743}, {5889,9829}, {5891,31749}, {6032,11444}, {10162,11793}, {11412,12505}, {13754,31729}, {23039,31753}

X(31736) = midpoint of X(11412) and X(12505)
X(31736) = reflection of X(i) in X(j) for these (i,j): (52, 31606), (12506, 1216), (31743, 3917), (31745, 31744)
X(31736) = anticomplement of X(31763)

X(31737) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 1st CIRCUMPERP

Barycentrics a^2*((b^2+c^2)*a^3+(b+c)*(b^2+c^2)*a^2-(b^4+c^4)*a-(b^3+c^3)*(b^2+b*c+c^2)) : :
X(31737) = X(1)-3*X(2979) = 5*X(10)-4*X(23841) = 3*X(51)-4*X(3634) = 2*X(143)-3*X(11231) = 3*X(165)-X(5889) = 9*X(373)-10*X(31253) = 2*X(389)-3*X(10164) = 2*X(1125)-3*X(3917) = 5*X(1698)-3*X(3060) = 3*X(1699)-5*X(11444) = 5*X(3567)-7*X(31423) = 7*X(3624)-9*X(7998) = 3*X(3817)-4*X(11793) = 6*X(3819)-5*X(19862) = 3*X(4134)-2*X(29958) = X(4301)-4*X(15606) = 4*X(5044)-3*X(15049) = 2*X(5446)-3*X(10175) = 4*X(5447)-3*X(10165) = 9*X(5650)-8*X(19878)

For these triangles the orthopoles are concident.

X(31737) lies on these lines: {1,2979}, {2,31757}, {3,31732}, {4,31752}, {10,511}, {40,11412}, {51,3634}, {52,6684}, {58,7186}, {72,2392}, {101,29092}, {110,9591}, {143,11231}, {165,5889}, {185,12512}, {373,31253}, {389,10164}, {515,10625}, {516,5562}, {517,6101}, {518,23156}, {595,3792}, {674,3874}, {942,9047}, {946,1216}, {1125,3917}, {1154,3579}, {1385,10627}, {1469,30145}, {1698,3060}, {1699,11444}, {2807,5493}, {2842,3962}, {2948,13201}, {3313,5847}, {3555,23157}, {3567,31423}, {3624,7998}, {3626,16980}, {3817,11793}, {3819,19862}, {3828,21969}, {3841,18180}, {3888,5508}, {4134,29958}, {4297,15644}, {4301,15606}, {5044,15049}, {5446,10175}, {5447,10165}, {5650,19878}, {5752,25440}, {5876,28146}, {5891,18483}, {6102,31663}, {6243,26446}, {7485,16472}, {7999,8227}, {9573,11445}, {9576,11446}, {9904,12273}, {9955,15067}, {9956,10263}, {10574,16192}, {11002,19877}, {11381,28158}, {11451,19872}, {11591,22793}, {12162,28150}, {12699,23039}, {13340,18481}, {13391,18480}, {13754,31730}, {14964,20727}, {20872,22136}, {25308,30172}

X(31737) = midpoint of X(i) and X(j) for these {i,j}: {40, 11412}, {2948, 13201}, {9904, 12273}
X(31737) = reflection of X(i) in X(j) for these (i,j): (4, 31752), (52, 6684), (185, 12512), (946, 1216), (1385, 10627), (3555, 23157), (3874, 11573), (4297, 15644), (6102, 31663), (6243, 31760), (10263, 9956), (12699, 31751), (16980, 3626), (21969, 3828), (22793, 11591), (31728, 3579), (31732, 3), (31738, 6101)
X(31737) = anticomplement of X(31757)
X(31737) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6243, 26446, 31760), (12699, 23039, 31751)

X(31738) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 2nd CIRCUMPERP

Barycentrics a^2*((b^2+c^2)*a^6-(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b+c)*b^2*c^2*a^3+(3*b^6+3*c^6+2*(b^2-b*c+c^2)*b^2*c^2)*a^2-(b^2-c^2)*(b-c)*b^2*c^2*a-(b^6-c^6)*(b^2-c^2)) : :
X(31738) = X(40)-3*X(2979) = 2*X(143)-3*X(11230) = X(355)-3*X(23039) = 2*X(389)-3*X(10165) = 5*X(1698)-7*X(7999) = 3*X(3060)-5*X(8227) = 5*X(3567)-7*X(3624) = 3*X(3576)-X(5889) = 3*X(3817)-2*X(5446) = 3*X(3917)-2*X(6684) = 4*X(5447)-3*X(10164) = 4*X(5462)-5*X(19862) = 3*X(5587)-5*X(11444) = X(5691)-3*X(11459) = 3*X(5886)-X(6243) = 3*X(5886)-2*X(31757) = 3*X(5890)-5*X(7987) = 3*X(5891)-2*X(19925) = 9*X(7988)-7*X(9781) = 3*X(23039)-2*X(31752)

X(31738) lies on these lines: {1,11412}, {2,31760}, {3,31728}, {4,31751}, {10,1216}, {40,2979}, {52,1125}, {110,9626}, {143,11230}, {355,23039}, {389,10165}, {511,946}, {515,5562}, {516,10625}, {517,6101}, {580,3792}, {912,23156}, {1154,1385}, {1698,7999}, {2392,5887}, {2779,14110}, {2807,15644}, {3060,8227}, {3567,3624}, {3576,5889}, {3579,10627}, {3817,5446}, {3917,6684}, {4297,13754}, {5447,10164}, {5462,19862}, {5587,11444}, {5691,11459}, {5777,9037}, {5876,28160}, {5884,11573}, {5886,6243}, {5890,7987}, {5891,19925}, {5907,31673}, {6102,13624}, {7509,16473}, {7988,9781}, {7998,31423}, {9610,11446}, {9955,10263}, {9956,15067}, {10175,11793}, {11362,15606}, {11381,28172}, {11591,18480}, {12162,28164}, {13391,22793}, {13630,17502}, {15056,18492}, {18436,18481}, {23841,31399}

X(31738) = midpoint of X(i) and X(j) for these {i,j}: {1, 11412}, {18436, 18481}
X(31738) = reflection of X(i) in X(j) for these (i,j): (4, 31751), (10, 1216), (52, 1125), (355, 31752), (3579, 10627), (5884, 11573), (6102, 13624), (6243, 31757), (10263, 9955), (18480, 11591), (31673, 5907), (31728, 3), (31730, 15644), (31732, 1385), (31737, 6101)
X(31738) = anticomplement of X(31760)
X(31738) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355, 23039, 31752), (5886, 6243, 31757)

X(31739) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND CIRCUMSYMMEDIAL

Barycentrics (SB+SC)*(9*(-SW+2*R^2)*S^4+(216*SA*R^4-6*(9*SA^2+15*SA*SW-SW^2)*R^2+(9*SA^2+18*SA*SW-SW^2)*SW)*S^2-(8*R^2-SA-2*SW)*SA*SW^3) : :
X(31739) = 3*X(353)-X(5889) = 2*X(389)-3*X(10166) = 3*X(30516)-2*X(31763)

X(31739) lies on these lines: {52,31608}, {353,5889}, {389,10166}, {511,14867}, {1154,31727}, {1843,31742}, {5562,9830}, {8705,15030}, {13754,31731}, {30516,31763}

X(31739) = reflection of X(i) in X(j) for these (i,j): (52, 31608), (1843, 31742), (31733, 31727)

X(31740) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st CIRCUMPERP AND CIRCUMSYMMEDIAL

Barycentrics a*(8*a^6+12*(b+c)*a^5-6*(b^2+c^2)*a^4-12*(b+c)*(b^2+c^2)*a^3-3*(b+c)*(2*b^4+b^2*c^2+2*c^4)*a-2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)) : :
X(31740) = X(1)-3*X(353) = 2*X(1125)-3*X(10166) = 6*X(10160)-5*X(19862) = 3*X(30516)-2*X(31755)

X(31740) lies on these lines: {1,353}, {10,9830}, {515,31731}, {516,14867}, {517,31727}, {946,31608}, {1125,10166}, {4663,8705}, {6233,12197}, {10160,19862}, {30516,31755}

X(31740) = reflection of X(i) in X(j) for these (i,j): (946, 31608), (31741, 31727)

X(31741) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd CIRCUMPERP AND CIRCUMSYMMEDIAL

Barycentrics
a*(8*a^9-4*(b+c)*a^8-2*(13*b^2-4*b*c+13*c^2)*a^7+2*(b+c)*(5*b^2-4*b*c+5*c^2)*a^6+3*(b+c)*(4*b^2-3*b*c+4*c^2)*a^4*b*c+6*(5*b^4+5*c^4-2*(b^2-b*c+c^2)*b*c)*a^5-(10*b^6+10*c^6+3*(4*b^4+4*c^4-(9*b^2-2*b*c+9*c^2)*b*c)*b*c)*a^3-(b+c)*(10*b^6+10*c^6-3*(4*b^4+4*c^4-(b-c)^2*b*c)*b*c)*a^2-(2*b^6+2*c^6-3*(4*b^2+3*b*c+4*c^2)*(b-c)^2*b*c)*(b+c)^2*a+2*(b^4-c^4)*(b-c)*(b^2-2*c^2)*(2*b^2-c^2)) : :
X(31741) = X(40)-3*X(353) = 2*X(6684)-3*X(10166) = 3*X(30516)-2*X(31758)

X(31741) lies on these lines: {10,31608}, {40,353}, {515,14867}, {516,31731}, {517,31727}, {946,9830}, {6233,12194}, {6684,10166}, {30516,31758}

X(31741) = reflection of X(i) in X(j) for these (i,j): (10, 31608), (31740, 31727)

X(31742) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st BROCARD AND 2nd BROCARD

Barycentrics 216*(4*R^2-SW)*(SA+SW)*S^4+8*(108*R^2*SA*(SA-SW)-SW*(SW^2+18*SA^2-17*SA*SW))*SW*S^2-32*SB*SC*SW^4 : :
X(31742) = 2*X(182)-3*X(10160) = 3*X(353)+X(5921) = X(6776)-3*X(10166)

X(31742) lies on these lines: {69,14867}, {182,10160}, {353,5921}, {1352,7618}, {1843,31739}, {3564,31608}, {6776,10166}, {8705,9967}, {18440,31731}

X(31742) = midpoint of X(i) and X(j) for these {i,j}: {69, 14867}, {1843, 31739}, {18440, 31731}
X(31742) = X(14866)-of-1st-Brocard-triangle

X(31743) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 4th BROCARD AND ORTHOCENTROIDAL

Barycentrics (SB+SC)*(9*(12*R^2-SW)*S^4+(24*R^2-5*SA-2*SW)*SA*SW^3-3*(6*R^2-SW)*(36*R^2*SA-9*SA^2+6*SB*SC-SW^2)*S^2) : :
X(31743) = 3*X(6032)-X(15305) = X(12505)-3*X(15045) = X(31736)-4*X(31762)

X(31743) lies on these lines: {3060,31763}, {3917,31736}, {5943,14866}, {6032,15305}, {12505,15045}, {12506,13754}, {14682,15053}, {16194,31749}

X(31743) = reflection of X(i) in X(j) for these (i,j): (3060, 31763), (3917, 31762), (14866, 5943), (16194, 31749), (31736, 3917)

X(31744) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND CIRCUMNORMAL

Barycentrics 6*(9*R^2-2*SW)*S^4-3*(9*(SB+SC)*R^2+SW^2)*SA*S^2-SB*SC*SW^3 : :
X(31744) = X(3)-3*X(9829) = X(4)+3*X(6031) = 3*X(5)-2*X(31749) = 2*X(140)-3*X(10163) = 3*X(549)-2*X(31762) = 5*X(1656)-3*X(6032) = 4*X(3628)-3*X(10162) = 3*X(5946)-2*X(31763) = 3*X(9829)+X(12505) = 3*X(10163)-X(12506) = 3*X(11230)-2*X(31755) = 3*X(11231)-2*X(31758) = 3*X(15067)-2*X(31753) = 3*X(31606)-X(31749)

X(31744) lies on these lines: {3,9829}, {4,6031}, {5,3849}, {30,14866}, {140,10163}, {517,31746}, {524,31608}, {549,31762}, {1154,31736}, {1656,6032}, {3628,10162}, {5946,31763}, {11230,31755}, {11231,31758}, {11594,16619}, {15067,31753}

X(31744) = midpoint of X(i) and X(j) for these {i,j}: {3, 12505}, {14866, 31729}, {31736, 31745}, {31746, 31747}
X(31744) = reflection of X(i) in X(j) for these (i,j): (5, 31606), (12506, 140)
X(31744) = anticomplement of X(32156)
X(31744) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9829, 12505, 3), (10163, 12506, 140)
X(31744) = X(5)-of-circummedial-triangle

X(31745) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND CIRCUMORTHIC

Barycentrics (SB+SC)*((54*R^2-15*SW)*S^4+3*(36*SA*R^4-2*(18*SA-SW)*SW*R^2+(SA^2+6*SA*SW-SW^2)*SW)*S^2-(4*R^2+SA-2*SW)*SA*SW^3) : :
X(31745) = 3*X(51)-2*X(31749) = 3*X(568)-2*X(31763) = 2*X(1216)-3*X(10163) = 5*X(3567)-3*X(6032) = 4*X(5462)-3*X(10162) = 3*X(9730)-2*X(31762) = 3*X(9829)-X(11412)

X(31745) lies on these lines: {2,31753}, {51,31749}, {52,3849}, {389,12506}, {511,31729}, {568,31763}, {1154,31736}, {1216,10163}, {3567,6032}, {5462,10162}, {5562,31606}, {5889,12505}, {9730,31762}, {9829,11412}, {13754,14866}

X(31745) = midpoint of X(5889) and X(12505)
X(31745) = reflection of X(i) in X(j) for these (i,j): (5562, 31606), (12506, 389), (31736, 31744)
X(31745) = anticomplement of X(31753)

X(31746) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND 1st CIRCUMPERP

Barycentrics 4*a^7+8*(b+c)*a^6+3*(b^2+c^2)*a^5-3*(b+c)*(b^2+c^2)*a^4-3*(b^4+c^4)*a^3-3*(b+c)*(3*b^4+b^2*c^2+3*c^4)*a^2-2*(b^6+c^6)*a+2*(b+c)*(b^2+c^2)*(b^4-4*b^2*c^2+c^4) : :
X(31746) = X(1)-3*X(9829) = X(8)+3*X(6031) = 2*X(1125)-3*X(10163) = 5*X(1698)-3*X(6032) = 4*X(3634)-3*X(10162) = 3*X(10164)-2*X(31762) = 3*X(10175)-2*X(31749) = 3*X(26446)-2*X(31758)

X(31746) lies on these lines: {1,9829}, {2,31755}, {8,6031}, {10,3849}, {40,12505}, {515,31729}, {516,14866}, {517,31744}, {946,31606}, {1125,10163}, {1698,6032}, {3634,10162}, {6684,12506}, {10164,31762}, {10175,31749}, {26446,31758}

X(31746) = midpoint of X(40) and X(12505)
X(31746) = reflection of X(i) in X(j) for these (i,j): (946, 31606), (12506, 6684), (31747, 31744)
X(31746) = anticomplement of X(31755)

X(31747) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND 2nd CIRCUMPERP

Barycentrics
4*a^10-4*(b+c)*a^9-(9*b^2-8*b*c+9*c^2)*a^8+2*(b+c)*(5*b^2-4*b*c+5*c^2)*a^7+(5*b^4+5*c^4-4*(3*b^2+4*b*c+3*c^2)*b*c)*a^6+3*(b+c)*(4*b^2-3*b*c+4*c^2)*b*c*a^5+(7*b^6+7*c^6-6*(2*b^4+2*c^4-(3*b^2-b*c+3*c^2)*b*c)*b*c)*a^4-(b+c)*(10*b^6+10*c^6-3*(4*b^4+4*c^4-(b-c)^2*b*c)*b*c)*a^3-(9*b^8+9*c^8-(8*b^6+8*c^6+3*(7*b^4+7*c^4-4*(b^2+c^2)*b*c)*b*c)*b*c)*a^2+2*(b^4-c^4)*(b-c)*(b^2-2*c^2)*(2*b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(2*b^4-8*b^2*c^2+2*c^4) : :
X(31747) = X(40)-3*X(9829) = X(962)+3*X(6031) = 3*X(3817)-2*X(31749) = 3*X(5886)-2*X(31755) = 3*X(6032)-5*X(8227) = 2*X(6684)-3*X(10163) = 3*X(10165)-2*X(31762)

X(31747) lies on these lines: {1,12505}, {2,31758}, {10,31606}, {40,9829}, {515,14866}, {516,31729}, {517,31744}, {946,3849}, {962,6031}, {1125,12506}, {3817,31749}, {5886,31755}, {6032,8227}, {6684,10163}, {8704,12263}, {10165,31762}

X(31747) = midpoint of X(1) and X(12505)
X(31747) = reflection of X(i) in X(j) for these (i,j): (10, 31606), (12506, 1125), (31746, 31744)
X(31747) = anticomplement of X(31758)

X(31748) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND CIRCUMSYMMEDIAL

Barycentrics 3*(36*R^2-7*SW+3*SA)*S^4-9*(12*R^2-SW)*(SB+SC)*SA*S^2-4*SB*SC*SW^3 : :
X(31748) = 4*X(5)-3*X(13378) = 3*X(10166)-2*X(31762) = 2*X(12506)-3*X(30516) = 3*X(30516)-4*X(31608)

X(31748) lies on these lines: {4,575}, {5,13378}, {597,14856}, {1499,12506}, {3849,14867}, {9830,14866}, {10166,31762}

X(31748) = reflection of X(12506) in X(31608)
X(31748) = {X(12506), X(31608)}-harmonic conjugate of X(30516)

X(31749) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: EULER AND 5th EULER

Barycentrics 6*(18*R^2-7*SW)*S^4-3*(SB+SC)*(18*R^2*SA-9*SA*SW+SW^2)*S^2-2*SB*SC*SW^3 : :
X(31749) = X(3)-3*X(10162) = X(4)+3*X(6032) = 3*X(5)-X(31744) = 3*X(51)-X(31745) = 2*X(140)-3*X(10173) = 3*X(381)-X(14866) = 5*X(1656)-3*X(10163) = 7*X(3090)-3*X(9829) = 5*X(3091)-X(12505) = 3*X(3817)-X(31747) = 11*X(5056)-3*X(6031) = 3*X(5891)-X(31736) = 3*X(6032)-X(12506) = 3*X(10175)-X(31746) = 3*X(30516)-X(31731) = 3*X(31606)-2*X(31744)

X(31749) lies on these lines: {2,31729}, {3,10162}, {4,6032}, {5,3849}, {30,31762}, {51,31745}, {140,10173}, {381,14866}, {511,31753}, {515,31755}, {516,31758}, {1656,10163}, {3090,9829}, {3091,12505}, {3817,31747}, {5056,6031}, {5891,31736}, {10175,31746}, {11645,31608}, {13754,31763}, {16194,31743}, {30516,31731}

X(31749) = midpoint of X(i) and X(j) for these {i,j}: {4, 12506}, {16194, 31743}
X(31749) = reflection of X(31606) in X(5)
X(31749) = complement of X(31729)
X(31749) = {X(4), X(6032)}-harmonic conjugate of X(12506)

X(31750) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: EULER AND FEUERBACH

Barycentrics
2*a^13+2*(b+c)*a^12-7*(b^2+c^2)*a^11-(b+c)*(7*b^2+2*b*c+7*c^2)*a^10+(5*b^4+5*c^4-2*(3*b^2-2*b*c+3*c^2)*b*c)*a^9+(b+c)*(5*b^4+8*b^2*c^2+5*c^4)*a^8+(10*b^6+10*c^6+(16*b^4+16*c^4+(5*b^2+16*b*c+5*c^2)*b*c)*b*c)*a^7+(b+c)*(10*b^6+10*c^6+(12*b^4+12*c^4+(b+3*c)*(3*b+c)*b*c)*b*c)*a^6-2*(10*b^8+10*c^8+(6*b^6+6*c^6-(14*b^4+14*c^4+(13*b^2+8*b*c+13*c^2)*b*c)*b*c)*b*c)*a^5-4*(b^2-c^2)^2*(b+c)*(5*b^4+5*c^4+2*(2*b^2+3*b*c+2*c^2)*b*c)*a^4+(b^2-c^2)^2*(13*b^6+13*c^6-2*(13*b^2+15*b*c+13*c^2)*b^2*c^2)*a^3+(b^2-c^2)^3*(b-c)*(13*b^4+13*c^4+(32*b^2+39*b*c+32*c^2)*b*c)*a^2-(b^2-c^2)^4*(b+c)^2*(3*b^2-8*b*c+3*c^2)*a-3*(b^2-c^2)^6*(b+c) : :

X(31750) lies on these lines: {3,5947}, {4,5948}, {30,31764}, {511,31754}, {515,31756}, {2829,10277}, {6713,10276}, {13754,31765}

X(31750) = midpoint of X(4) and X(5948)
X(31750) = reflection of X(6713) in X(10276)

X(31751) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd EULER AND 3rd EULER

Barycentrics a^2*((b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4+(b+c)*b^2*c^2*a^3+(3*b^6+3*c^6+2*(b^2-b*c+c^2)*b^2*c^2)*a^2-(b^2-c^2)*(b-c)*b^2*c^2*a-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(31751) = X(1)+3*X(11459) = X(10)-3*X(5891) = X(40)-5*X(11444) = X(52)-3*X(3817) = 3*X(165)-7*X(7999) = X(185)-3*X(10165) = 3*X(1699)+X(11412) = 5*X(3567)-9*X(7988) = 3*X(3576)+X(12111) = X(3579)-3*X(15067) = 7*X(3624)-3*X(5890) = 2*X(3634)-3*X(10170) = 3*X(3917)-X(31730) = 2*X(5462)-3*X(10171) = 3*X(5587)-7*X(15056) = X(5691)-5*X(15058) = 3*X(5886)+X(18436) = 3*X(5886)-X(31732) = X(5889)-5*X(8227) = 3*X(5892)-4*X(19878)

X(31751) lies on these lines: {1,11459}, {2,31728}, {4,31738}, {5,31760}, {10,5891}, {40,11444}, {52,3817}, {165,7999}, {185,10165}, {355,18330}, {511,18483}, {515,5907}, {516,1216}, {517,11591}, {916,12005}, {946,5562}, {960,2779}, {1125,13754}, {1154,9955}, {1385,5876}, {1699,11412}, {2772,13369}, {2807,6684}, {3567,7988}, {3576,12111}, {3579,15067}, {3624,5890}, {3634,10170}, {3917,31730}, {4297,12162}, {5446,12571}, {5447,12512}, {5462,10171}, {5587,15056}, {5663,13624}, {5691,15058}, {5886,18436}, {5889,8227}, {5892,19878}, {6101,22793}, {6102,11230}, {6241,7987}, {7691,9625}, {7723,11720}, {9730,19862}, {9956,14128}, {10627,28146}, {11709,12825}, {12699,23039}, {13474,28172}, {13491,17502}, {15030,31673}, {15060,18480}, {15644,28150}, {18435,18481}

X(31751) = midpoint of X(i) and X(j) for these {i,j}: {4, 31738}, {946, 5562}, {1385, 5876}, {4297, 12162}, {6101, 22793}, {7723, 11720}, {11709, 12825}, {12699, 31737}, {18436, 31732}
X(31751) = reflection of X(i) in X(j) for these (i,j): (5446, 12571), (6684, 11793), (9956, 14128), (12512, 5447), (31752, 11591), (31757, 9955), (31760, 5)
X(31751) = complement of X(31728)
X(31751) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5886, 18436, 31732), (12699, 23039, 31737)

X(31752) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd EULER AND 4th EULER

Barycentrics a^2*((b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4-(b+c)*b^2*c^2*a^3+(3*b^6+3*c^6+2*(b^2+b*c+c^2)*b^2*c^2)*a^2+(b^2-c^2)*(b-c)*b^2*c^2*a-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(31752) = X(1)-5*X(11444) = X(40)+3*X(11459) = X(52)-3*X(10175) = 3*X(165)+X(12111) = X(185)-3*X(10164) = X(355)+3*X(23039) = X(946)-3*X(5891) = X(1385)-3*X(15067) = 5*X(1698)-X(5889) = 3*X(1699)-7*X(15056) = 3*X(2979)+X(5691) = 3*X(3060)-7*X(7989) = 3*X(3576)-7*X(7999) = 3*X(3917)-X(4297) = 2*X(5462)-3*X(10172) = 3*X(5587)+X(11412) = 3*X(5890)-7*X(31423) = X(6102)-3*X(11231) = 5*X(7987)-9*X(7998) = 3*X(23039)-X(31738)

For these triangles the orthopoles are concident.

X(31752) lies on these lines: {1,11444}, {2,31732}, {4,31737}, {5,31757}, {10,5562}, {40,11459}, {52,10175}, {165,12111}, {185,10164}, {355,23039}, {389,3634}, {511,19925}, {515,1216}, {516,5907}, {517,11591}, {946,5891}, {1125,11793}, {1154,9956}, {1385,15067}, {1698,5889}, {1699,15056}, {2392,5777}, {2772,9943}, {2801,11573}, {2979,5691}, {3060,7989}, {3576,7999}, {3579,5876}, {3917,4297}, {5462,10172}, {5587,11412}, {5663,31663}, {5806,9047}, {5890,31423}, {6000,12512}, {6101,18480}, {6102,11231}, {6684,13754}, {7691,9590}, {7987,7998}, {9037,9947}, {9955,14128}, {10625,31673}, {10627,28160}, {11695,31253}, {12162,31730}, {13474,28158}, {14872,23156}, {15028,19872}, {15060,22793}, {15072,16192}, {15644,28164}, {18436,26446}

X(31752) = midpoint of X(i) and X(j) for these {i,j}: {4, 31737}, {10, 5562}, {355, 31738}, {3579, 5876}, {6101, 18480}, {10625, 31673}, {12162, 31730}, {14872, 23156}, {18436, 31728}
X(31752) = reflection of X(i) in X(j) for these (i,j): (389, 3634), (1125, 11793), (9955, 14128), (31751, 11591), (31757, 5), (31760, 9956)
X(31752) = complement of X(31732)
X(31752) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355, 23039, 31738), (18436, 26446, 31728)

X(31753) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd EULER AND 5th EULER

Barycentrics (SB+SC)*(3*SW*S^4+(108*SA*R^4-6*(9*SA^2+18*SA*SW-SW^2)*R^2+(21*SA^2+18*SA*SW-SW^2)*SW)*S^2-(4*R^2-SA-2*SW)*SA*SW^3) : :
X(31753) = X(52)-3*X(10162) = 3*X(3917)-X(31729) = 2*X(5462)-3*X(10173) = 3*X(5891)-X(14866) = 3*X(6032)+X(11412) = 7*X(7999)-3*X(9829) = 5*X(11444)-X(12505) = 3*X(15067)-X(31744) = 3*X(23039)-X(31736) = 3*X(30516)-X(31733)

X(31753) lies on these lines: {2,31745}, {52,10162}, {511,31749}, {1154,31763}, {1216,3849}, {3917,31729}, {5462,10173}, {5562,12506}, {5891,14866}, {6032,11412}, {7999,9829}, {11444,12505}, {11793,31606}, {13754,31762}, {15067,31744}, {23039,31736}, {30516,31733}

X(31753) = midpoint of X(5562) and X(12506)
X(31753) = reflection of X(31606) in X(11793)
X(31753) = complement of X(31745)

X(31754) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd EULER AND FEUERBACH

Barycentrics
a^2*((b^2+c^2)*a^15+(b+c)*(b^2+c^2)*a^14-(7*b^4+7*c^4+(b^2+10*b*c+c^2)*b*c)*a^13-(b+c)*(7*b^4+7*c^4+2*(b^2+5*b*c+c^2)*b*c)*a^12+(21*b^6+21*c^6+(4*b^4+4*c^4+(29*b^2+6*b*c+29*c^2)*b*c)*b*c)*a^11+(b+c)*(21*b^6+21*c^6+(10*b^4+10*c^4+(33*b^2+14*b*c+33*c^2)*b*c)*b*c)*a^10-(35*b^8+35*c^8+(5*b^6+5*c^6+(31*b^4+28*b^2*c^2+31*c^4)*b*c)*b*c)*a^9-(b+c)*(35*b^8+35*c^8+(20*b^6+20*c^6+(49*b^4+49*c^4+26*(b+c)^2*b*c)*b*c)*b*c)*a^8+(35*b^10+35*c^10-(b^6+c^6+(25*b^4+25*c^4+(17*b^2+36*b*c+17*c^2)*b*c)*b*c)*b^2*c^2)*a^7+(b+c)*(35*b^10+35*c^10+(20*b^8+20*c^8+(31*b^6+31*c^6+(14*b^4+14*c^4+5*(5*b^2+2*b*c+5*c^2)*b*c)*b*c)*b*c)*b*c)*a^6-(21*b^12+21*c^12-(5*b^10+5*c^10+(28*b^8+28*c^8+(30*b^6+30*c^6+(35*b^4+35*c^4+(39*b^2+28*b*c+39*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)*(b-c)*(21*b^10+21*c^10+(52*b^8+52*c^8+(83*b^6+83*c^6+2*(55*b^4+55*c^4+(63*b^2+64*b*c+63*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+(b^2-c^2)^2*(7*b^10+7*c^10-(4*b^6+4*c^6-(b^4+c^4-(15*b^2+2*b*c+15*c^2)*b*c)*b*c)*(b+c)^2*b*c)*a^3+(b^3-c^3)*(b^2-c^2)^3*(7*b^6+7*c^6+(9*b^4+9*c^4+7*(2*b^2+3*b*c+2*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^4*(b+c)^2*(b^6+c^6-(3*b^4+3*c^4-4*(b-c)^2*b*c)*b*c)*a-(b^2-c^2)^6*(b+c)*(b^4+3*b^2*c^2+c^4)) : :

X(31754) lies on these lines: {52,5947}, {505,8528}, {511,31750}, {1154,31765}, {5562,5948}, {13754,31764}

X(31754) = midpoint of X(5562) and X(5948)

X(31755) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd EULER AND 5th EULER

Barycentrics
4*a^7-15*(b^2+c^2)*a^5-9*(b^2+c^2)*(b+c)*a^4-3*(3*b^4+2*b^2*c^2+3*c^4)*a^3-3*(b^3+c^3)*(b^2+b*c+c^2)*a^2+2*(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4)*a+6*(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(31755) = X(1)+3*X(6032) = X(10)-3*X(10162) = 7*X(3624)-3*X(9829) = 2*X(3634)-3*X(10173) = 3*X(3817)-X(14866) = 11*X(5550)-3*X(6031) = 3*X(5886)-X(31747) = 5*X(8227)-X(12505) = 3*X(10163)-5*X(19862) = 3*X(10165)-X(31729) = 3*X(11230)-X(31744) = 3*X(30516)-X(31740)

X(31755) lies on these lines: {1,6032}, {2,31746}, {10,10162}, {515,31749}, {516,31762}, {517,31758}, {946,12506}, {1125,3849}, {3624,9829}, {3634,10173}, {3817,14866}, {5550,6031}, {5886,31747}, {8227,12505}, {10163,19862}, {10165,31729}, {11230,31744}, {30516,31740}

X(31755) = midpoint of X(946) and X(12506)
X(31755) = complement of X(31746)

X(31756) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd EULER AND FEUERBACH

Barycentrics
(b+c)*a^9+4*(b^2+b*c+c^2)*a^8+(b+c)*(2*b^2+3*b*c+2*c^2)*a^7-2*(5*b^4+5*c^4+2*(2*b^2+b*c+2*c^2)*b*c)*a^6-(b+c)*(12*b^4+12*c^4+(13*b^2+7*b*c+13*c^2)*b*c)*a^5+2*(3*b^6+3*c^6-(4*b^4+4*c^4+3*(5*b^2+6*b*c+5*c^2)*b*c)*b*c)*a^4+(b+c)*(14*b^6+14*c^6+(9*b^4+9*c^4-(17*b^2+27*b*c+17*c^2)*b*c)*b*c)*a^3+2*(b^2-c^2)^2*(b+c)^2*(b^2+6*b*c+c^2)*a^2-(b^2-c^2)^2*(b+c)^3*(5*b^2-11*b*c+5*c^2)*a-2*(b^2-c^2)^4*(b+c)^2 : :
X(31756) = X(10)-3*X(5947) = 3*X(10281)-X(12619)

X(31756) lies on these lines: {10,5947}, {515,31750}, {516,31764}, {517,31759}, {946,5948}, {2800,10277}, {6702,10276}, {10208,17605}, {10209,11263}, {10281,12619}

X(31756) = midpoint of X(i) and X(j) for these {i,j}: {946, 5948}, {10209, 11263}
X(31756) = reflection of X(6702) in X(10276)

X(31757) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd EULER AND ORTHIC

Barycentrics a^2*((b^2+c^2)*a^3+(b+c)*(b^2+c^2)*a^2-(b^2-c^2)^2*a-(b+c)*(b^4-b^2*c^2+c^4)) : :
X(31757) = X(1)+3*X(3060) = X(8)-9*X(11002) = X(10)-3*X(51) = X(40)-5*X(3567) = X(72)-3*X(15049) = 3*X(165)-7*X(15043) = 3*X(354)-X(23156) = 3*X(375)-2*X(4015) = 3*X(568)+X(12699) = 3*X(568)-X(31728) = 5*X(1698)-9*X(5640) = 3*X(1699)+X(5889) = 3*X(2979)-7*X(3624) = X(3579)-3*X(5946) = 7*X(3622)+9*X(16981) = 2*X(3634)-3*X(5943) = 3*X(3817)-X(5562) = 3*X(3819)-4*X(19878) = X(3901)+3*X(30438) = 3*X(3917)-5*X(19862)

For these triangles the orthopoles are concident.

X(31757) lies on these lines: {1,3060}, {2,31737}, {4,31732}, {5,31752}, {8,11002}, {10,51}, {22,16472}, {40,3567}, {52,946}, {54,9625}, {72,15049}, {143,517}, {165,15043}, {354,23156}, {375,4015}, {386,23414}, {389,516}, {511,1125}, {515,5446}, {519,21849}, {551,21969}, {568,12699}, {674,3678}, {730,27375}, {942,2392}, {1112,1829}, {1154,9955}, {1351,11365}, {1385,10263}, {1469,30148}, {1698,5640}, {1699,5889}, {1902,6746}, {2390,4757}, {2807,16625}, {2979,3624}, {3056,30142}, {3244,16980}, {3271,10974}, {3293,20962}, {3579,5946}, {3622,16981}, {3626,23841}, {3634,5943}, {3817,5562}, {3819,19878}, {3881,8679}, {3901,30438}, {3917,19862}, {4347,19366}, {5012,9591}, {5044,9047}, {5045,9037}, {5248,5752}, {5462,6684}, {5482,6681}, {5587,9781}, {5847,9969}, {5886,6243}, {5901,14449}, {5907,12571}, {6101,11230}, {6102,22793}, {6688,31253}, {7988,11444}, {8193,9777}, {8227,11412}, {9587,11422}, {9622,26882}, {9729,12512}, {9730,31730}, {9911,11432}, {9956,10095}, {10110,19925}, {10165,10625}, {10171,11793}, {11231,15026}, {12006,31663}, {12702,13321}, {13391,13624}, {13417,13605}, {13451,18357}, {13598,28164}, {13630,28146}, {13754,18483}, {15024,31423}, {15178,16982}, {16881,28174}, {18180,25639}, {25306,30172}

X(31757) = midpoint of X(i) and X(j) for these {i,j}: {4, 31732}, {52, 946}, {551, 21969}, {1385, 10263}, {3244, 16980}, {5901, 14449}, {6102, 22793}, {6243, 31738}, {12699, 31728}, {13417, 13605}
X(31757) = reflection of X(i) in X(j) for these (i,j): (3626, 23841), (5907, 12571), (6684, 5462), (9956, 10095), (12512, 9729), (19925, 10110), (23157, 5045), (31663, 12006), (31751, 9955), (31752, 5), (31760, 143)
X(31757) = complement of X(31737)
X(31757) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (568, 12699, 31728), (5886, 6243, 31738)

X(31758) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 4th EULER AND 5th EULER

Barycentrics
4*a^10+4*(b+c)*a^9-(19*b^2+8*b*c+19*c^2)*a^8-2*(b+c)*(5*b^2-4*b*c+5*c^2)*a^7+3*(5*b^4+5*c^4+2*(2*b^2+7*b*c+2*c^2)*b*c)*a^6-3*(b+c)*(4*b^2-3*b*c+4*c^2)*b*c*a^5+(13*b^6+13*c^6+6*(2*b^4+2*c^4+(3*b^2+b*c+3*c^2)*b*c)*b*c)*a^4+(b+c)*(10*b^6+10*c^6-3*(4*b^4+4*c^4-(b-c)^2*b*c)*b*c)*a^3-(19*b^6+19*c^6-6*(5*b^4+5*c^4-3*(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*(b+c)^2*a^2-2*(b^4-c^4)*(b-c)*(b^2-2*c^2)*(2*b^2-c^2)*a+6*(b^4-c^4)*(b^2-c^2)^3 : :
X(31758) = X(40)+3*X(6032) = X(946)-3*X(10162) = 5*X(1698)-X(12505) = 3*X(9829)-7*X(31423) = 3*X(10164)-X(31729) = 3*X(10175)-X(14866) = 3*X(11231)-X(31744) = 3*X(26446)-X(31746) = 3*X(30516)-X(31741)

X(31758) lies on these lines: {2,31747}, {10,12506}, {40,6032}, {515,31762}, {516,31749}, {517,31755}, {946,10162}, {1698,12505}, {3634,31606}, {3849,6684}, {9829,31423}, {10164,31729}, {10175,14866}, {11231,31744}, {26446,31746}, {30516,31741}

X(31758) = midpoint of X(10) and X(12506)
X(31758) = reflection of X(31606) in X(3634)
X(31758) = complement of X(31747)

X(31759) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 4th EULER AND FEUERBACH

Barycentrics
(b+c)*a^12-(b+c)^2*a^11-(b+c)*(7*b^2+3*b*c+7*c^2)*a^10+(5*b^4+5*c^4+7*(b^2+c^2)*b*c)*a^9+(b+c)*(20*b^4+20*c^4+(12*b^2+23*b*c+12*c^2)*b*c)*a^8-(10*b^6+10*c^6+(8*b^4+8*c^4-3*(5*b^2+6*b*c+5*c^2)*b*c)*b*c)*a^7-(b+c)*(30*b^6+30*c^6+(18*b^4+18*c^4+(16*b^2+7*b*c+16*c^2)*b*c)*b*c)*a^6+(10*b^8+10*c^8+(2*b^6+2*c^6-(34*b^4+34*c^4+(41*b^2+34*b*c+41*c^2)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)*(b-c)*(25*b^6+25*c^6+(62*b^4+62*c^4+(78*b^2+77*b*c+78*c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(5*b^6+5*c^6-(2*b^4+2*c^4+(18*b^2+17*b*c+18*c^2)*b*c)*b*c)*a^3-(b^2-c^2)^4*(b+c)*(11*b^2+3*b*c+11*c^2)*a^2+(b^2-c^2)^4*(b+c)^2*(b^2-3*b*c+c^2)*a+2*(b^2-c^2)^6*(b+c) : :

X(31759) lies on these lines: {10,5948}, {515,31764}, {517,31756}, {946,5947}, {2802,10277}, {10276,16174}

X(31759) = midpoint of X(10) and X(5948)
X(31759) = reflection of X(16174) in X(10276)

X(31760) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 4th EULER AND ORTHIC

Barycentrics
a^2*((b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4+(b+c)*b^2*c^2*a^3+(3*b^4+3*c^4+(6*b^2+7*b*c+6*c^2)*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*b^2*c^2*a-(b^4-b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(31760) = X(1)-5*X(3567) = X(40)+3*X(3060) = 3*X(51)-X(946) = X(355)+3*X(568) = 3*X(568)-X(31732) = X(962)-9*X(11002) = X(1385)-3*X(5946) = X(1482)-9*X(13321) = 5*X(1698)-X(11412) = 3*X(1699)-7*X(9781) = 3*X(2979)-7*X(31423) = 3*X(3576)-7*X(15043) = 7*X(3624)-11*X(15024) = X(4297)-3*X(9730) = X(5562)-3*X(10175) = 3*X(5587)+X(5889) = 9*X(5640)-5*X(8227) = X(5691)+3*X(5890) = X(5887)-3*X(15049) = X(6101)-3*X(11231)

X(31760) lies on these lines: {1,3567}, {2,31738}, {4,31728}, {5,31751}, {10,52}, {24,16473}, {40,3060}, {51,946}, {54,9590}, {143,517}, {185,31673}, {355,568}, {389,515}, {511,6684}, {516,5446}, {952,16881}, {962,11002}, {1112,1902}, {1125,5462}, {1154,9956}, {1216,3634}, {1385,5946}, {1482,13321}, {1698,11412}, {1699,9781}, {1829,6746}, {2771,13358}, {2779,7686}, {2807,10110}, {2979,31423}, {3576,15043}, {3579,10263}, {3624,15024}, {4297,9730}, {5012,9626}, {5562,10175}, {5587,5889}, {5640,8227}, {5691,5890}, {5882,16980}, {5887,15049}, {6101,11231}, {6102,18480}, {6243,26446}, {7592,8185}, {7730,9905}, {7987,15045}, {7989,11459}, {8679,12005}, {9037,9940}, {9587,26882}, {9622,11422}, {9798,11432}, {9955,10095}, {10164,10625}, {10172,11793}, {10202,23156}, {11230,15026}, {11720,16222}, {11722,16224}, {12006,13624}, {12111,18492}, {13373,23157}, {13391,31663}, {13598,28150}, {13630,28160}, {13754,19925}, {14531,31399}, {16625,23841}, {21849,28194}

X(31760) = midpoint of X(i) and X(j) for these {i,j}: {4, 31728}, {10, 52}, {185, 31673}, {355, 31732}, {3579, 10263}, {5882, 16980}, {6102, 18480}, {6243, 31737}, {16625, 23841}
X(31760) = reflection of X(i) in X(j) for these (i,j): (1125, 5462), (1216, 3634), (9955, 10095), (13624, 12006), (18483, 10110), (23157, 13373), (31751, 5), (31752, 9956), (31757, 143)
X(31760) = complement of X(31738)
X(31760) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355, 568, 31732), (6243, 26446, 31737)

X(31761) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 5th EULER AND FEUERBACH

Barycentrics
4*a^19+4*(b+c)*a^18-(35*b^2+12*b*c+35*c^2)*a^17-(b+c)*(35*b^2+16*b*c+35*c^2)*a^16+(115*b^4+115*c^4+7*(9*b^2+26*b*c+9*c^2)*b*c)*a^15+(b+c)*(115*b^4+115*c^4+2*(45*b^2+107*b*c+45*c^2)*b*c)*a^14-(177*b^6+177*c^6+(100*b^4+100*c^4+(251*b^2+82*b*c+251*c^2)*b*c)*b*c)*a^13-(b+c)*(177*b^6+177*c^6+(162*b^4+162*c^4+(399*b^2+286*b*c+399*c^2)*b*c)*b*c)*a^12+(99*b^8+99*c^8+(21*b^6+21*c^6-(3*b^4+3*c^4+2*(139*b^2+152*b*c+139*c^2)*b*c)*b*c)*b*c)*a^11+(b+c)*(99*b^8+99*c^8+(72*b^6+72*c^6+(183*b^4+183*c^4+2*(61*b^2+92*b*c+61*c^2)*b*c)*b*c)*b*c)*a^10+(81*b^10+81*c^10+(90*b^8+90*c^8+(153*b^6+153*c^6+(225*b^4+225*c^4+(457*b^2+696*b*c+457*c^2)*b*c)*b*c)*b*c)*b*c)*a^9+(b+c)*(81*b^10+81*c^10+(108*b^8+108*c^8+(165*b^6+165*c^6+(110*b^4+110*c^4+(171*b^2+302*b*c+171*c^2)*b*c)*b*c)*b*c)*b*c)*a^8-(175*b^12+175*c^12+(87*b^10+87*c^10-(132*b^8+132*c^8+(296*b^6+296*c^6+(257*b^4+257*c^4+(277*b^2+412*b*c+277*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^7-(b+c)*(175*b^12+175*c^12+(150*b^10+150*c^10+(48*b^8+48*c^8-(68*b^6+68*c^6+(93*b^4+93*c^4+2*(87*b^2+158*b*c+87*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^6+(125*b^14+125*c^14+(24*b^12+24*c^12-(349*b^10+349*c^10+(293*b^8+293*c^8+2*(68*b^6+68*c^6-(5*b^4+5*c^4+(212*b^2+315*b*c+212*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^2*(b+c)*(125*b^10+125*c^10+(70*b^8+70*c^8+(33*b^6+33*c^6-(20*b^4+20*c^4+(157*b^2+186*b*c+157*c^2)*b*c)*b*c)*b*c)*b*c)*a^4-(b^4-c^4)*(b^2-c^2)*(43*b^10+43*c^10-(3*b^8+3*c^8+(174*b^6+174*c^6+(66*b^4+66*c^4-(127*b^2+108*b*c+127*c^2)*b*c)*b*c)*b*c)*b*c)*a^3-(b^2-c^2)^3*(b-c)*(43*b^10+43*c^10+(98*b^8+98*c^8+(60*b^6+60*c^6-(10*b^4+10*c^4+(107*b^2+192*b*c+107*c^2)*b*c)*b*c)*b*c)*b*c)*a^2+2*(b^4-c^4)*(b^2-c^2)^3*(b+c)^2*(3*b^6+3*c^6-(7*b^4+7*c^4+(3*b^2-13*b*c+3*c^2)*b*c)*b*c)*a+(b^2-c^2)^8*(b+c)*(6*b^2+6*c^2) : :

X(31761) lies on these lines: {3849,31764}, {5947,14866}, {5948,12506}

X(31761) = midpoint of X(5948) and X(12506)

X(31762) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 5th EULER AND MEDIAL

Barycentrics 6*SW*S^4-(108*R^2*SA*(SB+SC)+SW*(27*SA^2-24*SA*SW-5*SW^2))*S^2-4*SB*SC*SW^3 : :
X(31762) = 3*X(3)-X(31729) = X(4)-3*X(10162) = 2*X(5)-3*X(10173) = X(20)+3*X(6032) = 3*X(549)-X(31744) = 5*X(631)-3*X(10163) = 5*X(631)-X(12505) = 7*X(3523)-3*X(9829) = 3*X(3917)-X(31736) = 3*X(6031)-11*X(15717) = 3*X(9730)-X(31745) = 3*X(10163)-X(12505) = 3*X(10164)-X(31746) = 3*X(10165)-X(31747) = 3*X(10166)-X(31748) = 3*X(12506)+X(31729) = X(14867)-3*X(30516) = X(31736)+3*X(31743)

X(31762) lies on these lines: {2,14866}, {3,3849}, {4,10162}, {5,10173}, {20,6032}, {30,31749}, {140,31606}, {511,31763}, {515,31758}, {516,31755}, {549,31744}, {631,10163}, {3523,9829}, {3917,31736}, {5569,13608}, {6031,15717}, {8704,13334}, {9730,31745}, {10164,31746}, {10165,31747}, {10166,31748}, {13754,31753}, {14867,30516}

X(31762) = midpoint of X(i) and X(j) for these {i,j}: {3, 12506}, {3917, 31743}
X(31762) = reflection of X(31606) in X(140)
X(31762) = complement of X(14866)
X(31762) = {X(631), X(12505)}-harmonic conjugate of X(10163)

X(31763) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 5th EULER AND ORTHIC

Barycentrics (SB+SC)*(3*(18*R^2-7*SW)*S^4+(216*SA*R^4-6*(18*SA+SW)*SW*R^2-(3*SA^2-18*SA*SW+SW^2)*SW)*S^2-(8*R^2-SA-2*SW)*SA*SW^3) : :
X(31763) = 3*X(51)-X(14866) = 3*X(568)-X(31745) = 5*X(3567)-X(12505) = X(5562)-3*X(10162) = X(5889)+3*X(6032) = 3*X(5946)-X(31744) = 3*X(9730)-X(31729) = 3*X(9829)-7*X(15043) = 3*X(10173)-2*X(11793) = 3*X(30516)-X(31739)

X(31763) lies on these lines: {2,31736}, {51,14866}, {52,12506}, {389,3849}, {511,31762}, {568,31745}, {1154,31753}, {3060,31743}, {3567,12505}, {5462,31606}, {5562,10162}, {5889,6032}, {5946,31744}, {9730,31729}, {9829,15043}, {10173,11793}, {13754,31749}, {30516,31739}

X(31763) = midpoint of X(i) and X(j) for these {i,j}: {52, 12506}, {3060, 31743}
X(31763) = reflection of X(31606) in X(5462)
X(31763) = complement of X(31736)

X(31764) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: FEUERBACH AND MEDIAL

Barycentrics
2*a^13+2*(b+c)*a^12-(11*b^2+4*b*c+11*c^2)*a^11-(b+c)*(11*b^2+6*b*c+11*c^2)*a^10+(25*b^4+25*c^4+4*(3*b^2+5*b*c+3*c^2)*b*c)*a^9+(b+c)*(25*b^4+25*c^4+4*(5*b^2+8*b*c+5*c^2)*b*c)*a^8-(30*b^6+30*c^6+(12*b^4+12*c^4-(9*b^2+28*b*c+9*c^2)*b*c)*b*c)*a^7-(b+c)*(30*b^6+30*c^6+(24*b^4+24*c^4+(21*b^2+10*b*c+21*c^2)*b*c)*b*c)*a^6+4*(5*b^8+5*c^8+(b^6+c^6-2*(5*b^4+5*c^4+2*(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*a^5+4*(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(13*b^4+13*c^4+17*(b^2+b*c+c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(7*b^6+7*c^6-2*(7*b^2+4*b*c+7*c^2)*b^2*c^2)*a^3-(b^2-c^2)^3*(b-c)*(7*b^4+7*c^4+(16*b^2+17*b*c+16*c^2)*b*c)*a^2+(b^2-c^2)^6*a+(b^2-c^2)^6*(b+c) : :

X(31764) lies on these lines: {3,5948}, {4,5947}, {30,31750}, {511,31765}, {515,31759}, {516,31756}, {3651,10209}, {3849,31761}, {5432,10208}, {5840,10277}, {10225,11277}, {10281,22938}, {13754,31754}

X(31764) = midpoint of X(i) and X(j) for these {i,j}: {3, 5948}, {3651, 10209}

X(31765) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: FEUERBACH AND ORTHIC

Barycentrics
a^2*((b^2+c^2)*a^15+(b+c)*(b^2+c^2)*a^14-(7*b^4+7*c^4+(b^2+10*b*c+c^2)*b*c)*a^13-(b+c)*(7*b^4+7*c^4+2*(b^2+5*b*c+c^2)*b*c)*a^12+(21*b^6+21*c^6+(4*b^4+4*c^4+(25*b^2+2*b*c+25*c^2)*b*c)*b*c)*a^11+(b+c)*(21*b^6+21*c^6+(10*b^4+10*c^4+(29*b^2+10*b*c+29*c^2)*b*c)*b*c)*a^10-(35*b^8+35*c^8+(5*b^6+5*c^6+(11*b^4+11*c^4-6*(3*b^2-2*b*c+3*c^2)*b*c)*b*c)*b*c)*a^9-(b+c)*(35*b^8+35*c^8+(20*b^6+20*c^6+(29*b^4+29*c^4+2*(3*b^2+14*b*c+3*c^2)*b*c)*b*c)*b*c)*a^8+(35*b^10+35*c^10-(41*b^6+41*c^6+(53*b^4+53*c^4+(13*b^2+24*b*c+13*c^2)*b*c)*b*c)*b^2*c^2)*a^7+(b+c)*(35*b^10+35*c^10+(20*b^8+20*c^8-(9*b^6+9*c^6+(22*b^4+22*c^4-(b^2-10*b*c+c^2)*b*c)*b*c)*b*c)*b*c)*a^6-(21*b^12+21*c^12-(5*b^10+5*c^10+(68*b^8+68*c^8+(46*b^6+46*c^6-(33*b^4+33*c^4+5*(7*b^2+4*b*c+7*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)*(b-c)*(21*b^10+21*c^10+(52*b^8+52*c^8+(43*b^6+43*c^6+2*(b^4+c^4-(9*b^2+10*b*c+9*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+(b^2-c^2)^2*(7*b^10+7*c^10-(4*b^8+4*c^8+(27*b^6+27*c^6+(17*b^4+17*c^4-(9*b^2+4*b*c+9*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(7*b^8+7*c^8+(16*b^6+16*c^6+(10*b^4+10*c^4-(4*b^2+7*b*c+4*c^2)*b*c)*b*c)*b*c)*a^2-(b^2-c^2)^4*(b+c)^2*(b^6+c^6-(3*b^4-2*b^2*c^2+3*c^4)*b*c)*a-(b^2-c^2)^6*(b+c)*(b^4-b^2*c^2+c^4)) : :

X(31765) lies on these lines: {52,5948}, {511,31764}, {1154,31754}, {5562,5947}, {13754,31750}

X(31765) = midpoint of X(52) and X(5948)

X(31766) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HUTSON INTOUCH AND MIDARC

Barycentrics a*(-((b+c)*a+(b-c)^2)*(-a+b+c)^2*sin(A/2)+(a-b+c)*(a^2-6*c*a+c^2-b^2)*sin(B/2)*b+(a+b-c)*(a^2-6*b*a+b^2-c^2)*sin(C/2)*c) : :
X(31766) = X(65)-3*X(11191) = 5*X(3890)-X(11691) = 3*X(5919)-X(8422) = 2*X(12813)-3*X(12908) = 4*X(12813)-3*X(31768)

X(31766) lies on these lines: {1,164}, {65,11191}, {167,30337}, {177,3057}, {515,31769}, {516,31735}, {517,12813}, {1697,12518}, {3890,11691}, {5919,8422}, {9957,12580}, {10506,10508}, {10967,10968}, {12053,12614}, {12575,31770}, {12622,31397}, {12844,31393}

X(31766) = midpoint of X(177) and X(3057)
X(31766) = reflection of X(i) in X(j) for these (i,j): (5571, 1), (31767, 9957), (31770, 12575)

X(31767) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HUTSON INTOUCH AND 2nd MIDARC

Barycentrics -a*(-a+b+c)*((-(b+c)*a^2+8*b*c*a+(b^2-c^2)*(b-c))*sin(A/2)+(a-b+c)*((b+2*c)*a+b^2-3*b*c+2*c^2)*sin(B/2)+(a+b-c)*((2*b+c)*a+2*b^2-3*b*c+c^2)*sin(C/2)) : :
X(31767) = X(65)-3*X(11234) = X(177)-3*X(5919) = 3*X(3057)+X(17641) = 3*X(8422)-X(17641)

X(31767) lies on these lines: {1,168}, {65,11234}, {164,9819}, {177,5919}, {515,31770}, {516,31734}, {517,5571}, {3057,8422}, {3880,18258}, {4342,21633}, {6018,10504}, {9957,12580}, {12053,12622}, {12575,31769}, {12614,31397}

X(31767) = midpoint of X(3057) and X(8422)
X(31767) = reflection of X(i) in X(j) for these (i,j): (31766, 9957), (31768, 1), (31769, 12575)

X(31768) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INTOUCH AND MIDARC

Barycentrics a*(-(b+c)*(a-b-c)*sin(A/2)+((b+2*c)*a-b^2+b*c+2*c^2)*sin(B/2)+((2*b+c)*a+2*b^2+b*c-c^2)*sin(C/2))*(a-b+c)*(a+b-c) : :
X(31768) = 3*X(354)-X(8422) = X(3057)-3*X(11191) = 3*X(11234)-5*X(17609) = 4*X(12813)-X(31766)

X(31768) lies on these lines: {1,168}, {57,12523}, {65,177}, {164,3339}, {226,12622}, {354,8422}, {515,31735}, {516,31769}, {517,12813}, {942,5571}, {1210,12614}, {1357,10505}, {3057,11191}, {3671,21633}, {3812,18258}, {4298,31734}, {8094,13092}, {11234,17609}, {11529,12844}, {12656,18421}

X(31768) = midpoint of X(65) and X(177)
X(31768) = reflection of X(i) in X(j) for these (i,j): (5571, 942), (31734, 4298), (31767, 1)

X(31769) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: MANDART-INCIRCLE AND MIDARC

Barycentrics
(-a+b+c)*((-2*a^4+(b+c)^2*a^2+(b^2-c^2)^2)*sin(A/2)+(a-b+c)*(2*a^3+(2*b+c)*a^2+(b-c)*b*a+(b^2-c^2)*(b-c))*sin(B/2)+(a+b-c)*(2*a^3+(b+2*c)*a^2-(b-c)*c*a+(b^2-c^2)*(b-c))*sin(C/2)) : :
X(31769) = 3*X(3058)-X(8422) = X(7354)-3*X(11191)

X(31769) lies on these lines: {1,31734}, {30,12908}, {55,12622}, {177,6284}, {497,12523}, {515,31766}, {516,31768}, {1479,12614}, {3058,8422}, {4294,12518}, {4314,21633}, {7354,11191}, {12575,31767}, {12813,28146}

X(31769) = midpoint of X(177) and X(6284)
X(31769) = reflection of X(i) in X(j) for these (i,j): (31734, 1), (31767, 12575)

X(31770) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: MANDART-INCIRCLE AND 2nd MIDARC

Barycentrics
(2*a^2+b^2-2*b*c+c^2)*(-a+b+c)^2*sin(A/2)-(a-b+c)*(2*a^3-(2*b+c)*a^2+(b-c)*b*a-(b^2-c^2)*(b-c))*sin(B/2)-(a+b-c)*(2*a^3-(b+2*c)*a^2-(b-c)*c*a-(b^2-c^2)*(b-c))*sin(C/2) : :
X(31770) = X(177)-3*X(3058) = X(7354)-3*X(11234)

X(31770) lies on these lines: {1,31735}, {30,31734}, {55,12614}, {177,3058}, {497,12518}, {515,31767}, {516,5571}, {528,18258}, {1479,12622}, {4294,12523}, {6284,8422}, {7354,11234}, {12575,31766}, {12908,15172}

X(31770) = midpoint of X(6284) and X(8422)
X(31770) = reflection of X(i) in X(j) for these (i,j): (12908, 15172), (31735, 1), (31766, 12575)

X(31771) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ORTHOSYMMEDIAL AND 2nd ORTHOSYMMEDIAL

Barycentrics (SB+SC)*(3*(12*R^2-5*SW)*(3*R^2-SW)*S^4-SW*(SW^2*(15*SA-4*SW)+216*R^4*SA-SW*(117*SA-10*SW)*R^2)*S^2+(8*R^2+SA-3*SW)*SA*SW^4) : :
X(31771) = 3*X(11226)-X(12294)

X(31771) lies on the line {11226,12294}

X(31772) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st PARRY AND 3rd PARRY

Barycentrics (3*SA-SW)*(27*S^4-3*(7*SW^2-9*SA^2+18*SB*SC+36*R^2*SA)*S^2-(9*SA-10*SW)*SA*SW^2)*(SB-SC) : :

X(31772) lies on these lines: {351,8704}, {523,31773}, {3849,13306}

X(31773) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd PARRY AND 3rd PARRY

Barycentrics (SB-SC)*(27*SW*S^4+(54*(3*SA+2*SW)*(SB+SC)*R^2+(54*SA^2-45*SA*SW-20*SW^2)*SW)*S^2+(SA^2+5*SB*SC-2*SW^2)*SW^3) : :

X(31773) lies on these lines: {523,31772}, {8704,9979}

X(31774) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 3rd CONWAY

Barycentrics 2*a^7-2*(b+c)*a^6-7*(b^2+c^2)*a^5-(b+c)*(b^2+4*b*c+c^2)*a^4+2*(2*b^4+2*c^4-(b-c)^2*b*c)*a^3+2*(2*b^2-3*b*c+2*c^2)*(b+c)^3*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :
X(31774) = X(6284)-3*X(10439)

X(31774) lies on these lines: {3,4340}, {30,10441}, {140,25526}, {511,20420}, {515,31781}, {516,31779}, {517,4292}, {1724,19512}, {6284,10439}, {6996,20077}, {7354,12435}, {9579,12555}, {12545,29207}, {28160,31780}

X(31774) = midpoint of X(7354) and X(12435)
X(31774) = reflection of X(31782) in X(10441)

X(31775) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND HEXYL

Barycentrics
2*a^7-2*(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5+(b+c)*(3*b^2-4*b*c+3*c^2)*a^4-2*(3*b^2-2*b*c+3*c^2)*b*c*a^3+2*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(31775) = 3*X(165)+X(10483) = 3*X(165)-X(11827) = 3*X(3576)-X(6284) = 3*X(5434)-X(7982) = 3*X(10202)-2*X(12433) = 3*X(10246)-2*X(15172) = 3*X(15170)-4*X(15178)

As a point on the Euler line, X(31775) has Shinagawa coefficients (2*R-r, -2*R+3*r)

X(31775) lies on these lines: {1,11826}, {2,3}, {7,1482}, {8,2096}, {10,2829}, {12,2077}, {35,24466}, {36,15908}, {40,5252}, {63,5690}, {79,5538}, {84,355}, {104,24390}, {165,10483}, {388,10306}, {495,11248}, {496,10269}, {515,5836}, {516,3884}, {517,4292}, {528,5882}, {529,11362}, {950,9940}, {1125,22835}, {1158,5794}, {1259,10942}, {1350,5820}, {1376,6256}, {1385,1387}, {1478,10310}, {1519,17614}, {1770,14110}, {2883,14925}, {2886,5450}, {2894,12773}, {3428,4299}, {3576,6284}, {3868,5844}, {3916,5771}, {4293,22770}, {4297,5842}, {4313,10246}, {4316,30264}, {4324,15931}, {5123,6684}, {5249,5901}, {5270,5537}, {5434,7982}, {5559,7991}, {5587,10270}, {5687,12115}, {5720,6259}, {5732,5832}, {5780,5811}, {5784,5887}, {5787,7171}, {5812,6282}, {5881,30304}, {5885,10122}, {6001,17647}, {6244,9655}, {6253,30503}, {7675,31657}, {7987,23708}, {9581,21164}, {9799,18525}, {9965,12245}, {10039,13528}, {10200,10893}, {10202,12433}, {10247,11036}, {10386,16202}, {10593,26492}, {10609,21740}, {10826,16209}, {10902,15338}, {10957,11012}, {11015,18444}, {11220,28224}, {12678,17857}, {13243,19914}, {15170,15178}, {15325,15866}, {16004,31798}, {16465,24475}, {17768,31806}, {18242,25440}, {20418,24387}, {22758,31419}, {24470,24474}, {25524,26333}, {26086,31659}, {26446,31424}, {28160,31787}

X(31775) = midpoint of X(i) and X(j) for these {i,j}: {1, 11826}, {40, 7354}, {1770, 14110}, {10483, 11827}, {18990, 31777}
X(31775) = reflection of X(i) in X(j) for these (i,j): (950, 9940), (15171, 1385), (24474, 24470), (31789, 3), (31798, 16004)
X(31775) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 6827), (4, 6904, 6918), (4, 6964, 381)

X(31776) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND INCIRCLE-CIRCLES

Barycentrics 4*a^4+(b+c)*a^3-2*(b^2-3*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(31776) = 3*X(354)+X(10483) = 3*X(942)-X(10572) = X(1770)+3*X(5434) = X(3555)+3*X(17579) = 3*X(4292)+X(10106) = 4*X(4298)-X(31795) = 3*X(5049)-X(6284) = 3*X(5434)-X(9957) = 3*X(7354)+X(10572) = X(10106)-3*X(18990) = 3*X(11227)-X(11827)

X(31776) lies on these lines: {1,1657}, {3,5290}, {7,18481}, {12,5122}, {30,4298}, {46,9657}, {56,9955}, {57,9655}, {65,28204}, {79,1319}, {226,13624}, {354,10483}, {381,3361}, {382,3333}, {388,3579}, {495,31663}, {515,24470}, {516,31792}, {517,4292}, {535,3812}, {548,13405}, {549,3947}, {550,21620}, {553,28208}, {942,7354}, {950,28168}, {993,3824}, {999,9579}, {1155,5270}, {1385,3485}, {1478,9956}, {1482,4312}, {1770,5434}, {1836,4317}, {2098,4338}, {2646,4325}, {2829,5806}, {3303,4333}, {3338,12943}, {3339,18525}, {3476,11278}, {3529,11037}, {3555,17579}, {3600,12699}, {3627,11019}, {3649,21578}, {3656,4308}, {3748,4324}, {3884,28534}, {3982,16137}, {4295,24680}, {4297,6147}, {4299,10404}, {4311,15178}, {4314,15704}, {4315,22791}, {4355,15934}, {5049,6284}, {5126,12047}, {5563,7743}, {5691,5708}, {5882,30424}, {6738,28186}, {6744,28172}, {9612,11230}, {9654,11231}, {10578,17538}, {10624,28202}, {11011,11552}, {11227,11827}, {11374,17502}, {12433,28164}, {12575,28178}, {12577,15172}, {13407,15326}, {13462,18493}, {15171,28154}, {17563,21075}, {18421,18526}, {22753,22792}, {31409,31430}, {31434,31447}

X(31776) = midpoint of X(i) and X(j) for these {i,j}: {942, 7354}, {1770, 9957}, {4292, 18990}
X(31776) = reflection of X(i) in X(j) for these (i,j): (5045, 4298), (15172, 12577), (31794, 24470), (31795, 5045)
X(31776) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57, 9655, 18480), (382, 3333, 18527)

X(31777) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 6th MIXTILINEAR

Barycentrics 2*a^7-2*(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5+3*(b+c)*(b^2+c^2)*a^4-2*(3*b^2+2*b*c+3*c^2)*b*c*a^3-2*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(31777) = 3*X(40)-X(11827) = 3*X(165)-X(6284) = X(962)-3*X(11112) = 4*X(1385)-3*X(15170) = X(1482)-3*X(28458) = 3*X(3058)-5*X(7987) = 3*X(11826)+X(11827) = 2*X(11826)+X(31799) = 2*X(11827)-3*X(31799) = 3*X(17579)+X(20070)

X(31777) lies on these lines: {1,31657}, {3,496}, {4,3820}, {5,4413}, {8,12246}, {12,5537}, {20,956}, {30,40}, {140,2077}, {165,6284}, {200,6259}, {474,7956}, {495,6850}, {515,31798}, {516,960}, {517,4292}, {528,4297}, {548,11012}, {550,3428}, {944,12541}, {950,31787}, {952,3893}, {962,11112}, {1012,31419}, {1385,15170}, {1482,28458}, {1538,6700}, {1699,24954}, {1753,6756}, {1770,5762}, {1836,5763}, {2829,11362}, {3058,7987}, {3295,6916}, {3651,13199}, {4420,13257}, {4863,10085}, {5175,9778}, {5687,6925}, {5787,10860}, {5805,12651}, {5843,5904}, {6734,17613}, {6909,24390}, {6935,31493}, {7354,7991}, {7994,9579}, {8728,11496}, {10916,13226}, {17527,26333}, {17563,22753}, {17579,20070}, {21075,22792}

X(31777) = midpoint of X(i) and X(j) for these {i,j}: {40, 11826}, {1770, 7957}, {7354, 7991}
X(31777) = reflection of X(i) in X(j) for these (i,j): (950, 31787), (15171, 3), (18990, 31775), (31799, 40)
X(31777) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2077, 15908, 140), (6850, 10306, 495)

X(31778) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND HEXYL

Barycentrics a*((b^2+b*c+c^2)*a^4+(b^3+c^3)*a^3-(b^2-c^2)^2*a^2-(b^2-c^2)^2*b*c-(b^3+c^3)*(b^2+c^2)*a) : :
X(31778) = 2*X(970)-3*X(26446)

X(31778) lies on these lines: {1,3}, {4,6327}, {5,25760}, {10,5752}, {43,5754}, {72,5774}, {355,511}, {382,15310}, {388,11573}, {392,19273}, {572,2214}, {601,10457}, {970,26446}, {1064,19543}, {1193,19550}, {2049,3753}, {2050,12672}, {2262,5783}, {2650,30272}, {2818,5812}, {3784,18990}, {3877,19270}, {4972,5797}, {5492,20430}, {5657,26115}, {5690,15973}, {5955,9565}, {7119,20739}, {7413,26227}, {11362,29311}, {12514,22299}, {12699,15488}, {14988,17763}, {18180,19259}, {18493,29827}, {22791,30942}, {29353,31673}, {29828,31837}

X(31778) = midpoint of X(i) and X(j) for these {i,j}: {40, 12435}, {10441, 31785}
X(31778) = reflection of X(i) in X(j) for these (i,j): (5752, 10), (12699, 15488)

X(31779) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND HUTSON INTOUCH

Barycentrics a*((b+c)*a^5+3*(b^2+c^2)*a^4+2*(b+c)*(b^2+3*b*c+c^2)*a^3-2*(b^4+c^4-(b^2-4*b*c+c^2)*b*c)*a^2-(b+c)*(3*b^4+3*c^4+2*(3*b^2-5*b*c+3*c^2)*b*c)*a-(b^2-c^2)^2*(b+c)^2) : :
X(31779) = X(65)-3*X(10439)

X(31779) lies on these lines: {1,3}, {511,9856}, {515,31782}, {516,31774}, {960,29311}, {962,21296}, {2262,15829}, {4673,10446}, {5886,19866}

X(31779) = midpoint of X(3057) and X(12435)
X(31779) = reflection of X(31781) in X(10441)

X(31780) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND INCIRCLE-CIRCLES

Barycentrics
a*((b+c)*a^5-3*(3*b^2+4*b*c+3*c^2)*a^4-2*(b+c)*(5*b^2-3*b*c+5*c^2)*a^3+2*(5*b^4+5*c^4+(b^2-4*b*c+c^2)*b*c)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*(3*b^2+2*b*c+3*c^2)*a-(b^2-10*b*c+c^2)*(b^2-c^2)^2) : :

X(31780) lies on these lines: {1,3}, {28146,31782}, {28160,31774}

X(31780) = midpoint of X(i) and X(j) for these {i,j}: {942, 12435}, {10441, 31781}

X(31781) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND INTOUCH

Barycentrics a*((b+c)*a^5-(5*b^2+8*b*c+5*c^2)*a^4-6*(b^3+c^3)*a^3+2*(3*b^2-5*b*c+3*c^2)*(b+c)^2*a^2+(b+c)*(b^2+c^2)*(5*b^2-6*b*c+5*c^2)*a-(b^2-6*b*c+c^2)*(b^2-c^2)^2) : :

X(31781) lies on these lines: {1,3}, {515,31774}, {516,31782}, {5836,29311}, {9856,15488}

X(31781) = midpoint of X(65) and X(12435)
X(31781) = reflection of X(i) in X(j) for these (i,j): (9856, 15488), (10441, 31780), (31779, 10441)

X(31782) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND MANDART-INCIRCLE

Barycentrics
2*a^7-2*(b+c)*a^6+(b^2+8*b*c+c^2)*a^5+(b+c)*(7*b^2-4*b*c+7*c^2)*a^4-2*(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^3-2*(b+c)*(2*b^4+2*c^4-(b-c)^2*b*c)*a^2+(b^2-c^2)^2*(b^2-6*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(31782) = X(7354)-3*X(10439)

X(31782) lies on these lines: {30,10441}, {389,517}, {515,31779}, {516,31781}, {1482,5716}, {1872,1891}, {6284,12435}, {7354,10439}, {15488,20420}, {28146,31780}

X(31782) = midpoint of X(6284) and X(12435)
X(31782) = reflection of X(i) in X(j) for these (i,j): (20420, 15488), (31774, 10441)

X(31783) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND MIDARC

Barycentrics
a*(2*(-a+b+c)*((2*b^2+3*b*c+2*c^2)*a^4+(b+c)*(2*b^2-b*c+2*c^2)*a^3-(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^2-(b+c)*(2*b^4+2*c^4-(b-c)^2*b*c)*a-2*(b^2-c^2)^2*b*c)*sin(A/2)-(a-b+c)*(a^5*c-(4*b^2+5*b*c+5*c^2)*a^4-2*(2*b^3+3*c^3+(2*b+c)*b*c)*a^3+2*(2*b^4-b^2*c^2+3*c^4)*a^2+(b+c)*(4*b^4+5*c^4-(b^2-3*b*c+3*c^2)*b*c)*a+(b^2-c^2)^2*(5*b-c)*c)*sin(B/2)-(a+b-c)*(a^5*b-(5*b^2+5*b*c+4*c^2)*a^4-2*(3*b^3+2*c^3+(b+2*c)*b*c)*a^3+2*(3*b^4-b^2*c^2+2*c^4)*a^2+(b+c)*(5*b^4+4*c^4-(3*b^2-3*b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b-5*c)*b)*sin(C/2)) : :

X(31783) lies on these lines: {177,12435}, {517,12813}, {8422,10439}, {10441,31784}

X(31783) = midpoint of X(177) and X(12435)
X(31783) = reflection of X(31784) in X(10441)

X(31784) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND 2nd MIDARC

Barycentrics
a*(-2*(-a+b+c)*((2*b^2+b*c+2*c^2)*a^3-(2*b^4+2*c^4-(b-c)^2*b*c)*a-2*(b^2-c^2)*(b-c)*b*c)*sin(A/2)-(a-b+c)*(a^4*c+2*(2*b^2+b*c+c^2)*a^3-2*(b-c)*a^2*b*c-2*(2*b^4+c^4-(b^2-b*c-c^2)*b*c)*a-(b^2-c^2)*(b-c)*(3*b+c)*c)*sin(B/2)-(a+b-c)*(a^4*b+2*(b^2+b*c+2*c^2)*a^3+2*(b-c)*a^2*b*c-2*(b^4+2*c^4+(b^2+b*c-c^2)*b*c)*a-(b^2-c^2)*(b-c)*(b+3*c)*b)*sin(C/2)) : :
X(31784) = X(177)-3*X(10439)

X(31784) lies on these lines: {177,10439}, {517,5571}, {8422,12435}, {10441,31783}, {18258,29311}

X(31784) = midpoint of X(8422) and X(12435)
X(31784) = reflection of X(31783) in X(10441)

X(31785) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 3rd CONWAY AND 6th MIXTILINEAR

Barycentrics a*((b^2+b*c+c^2)*a^4+(b^2-c^2)*(b-c)*a^3-(b^4-4*b^2*c^2+c^4)*a^2-(b^2-c^2)^2*b*c-(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a) : :
X(31785) = 2*X(970)-3*X(5657)

X(31785) lies on these lines: {1,3}, {4,7017}, {8,511}, {9,20719}, {51,5554}, {213,20606}, {355,4680}, {573,2295}, {595,13732}, {946,3831}, {957,17567}, {962,15488}, {970,5657}, {995,19514}, {1191,16434}, {2262,5782}, {2292,31395}, {2818,14216}, {3146,29349}, {3421,29958}, {3781,5837}, {3784,10106}, {3794,11115}, {3840,4301}, {3869,7081}, {3937,20076}, {5690,5752}, {5691,15310}, {5793,5836}, {5886,19864}, {6243,19914}, {9840,30116}, {10446,21281}, {18180,19531}, {20070,29309}, {30142,30285}

X(31785) = midpoint of X(7991) and X(12435)
X(31785) = reflection of X(i) in X(j) for these (i,j): (962, 15488), (5752, 5690), (10441, 31778)

X(31786) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND HUTSON INTOUCH

Barycentrics a*((b+c)*a^5-(b^2+8*b*c+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^4+c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31786) = 5*X(3)-X(25413) = 3*X(3)-2*X(31787) = X(4)-3*X(392) = X(65)-3*X(3576) = 3*X(65)-5*X(15016) = 3*X(165)+X(5697) = 3*X(210)-X(5881) = 3*X(551)-2*X(13374) = 3*X(960)-2*X(20117) = X(962)-5*X(3890) = 3*X(1699)-2*X(31822) = X(3633)+3*X(15104) = 3*X(3817)-2*X(16616) = 5*X(3876)-3*X(18908) = 3*X(3899)+X(15071) = 3*X(5587)-5*X(25917) = 3*X(5692)-X(14872) = X(5693)-3*X(31165) = 3*X(5777)-4*X(20117) = 3*X(10157)-2*X(18480) = X(12680)+3*X(31165)

X(31786) lies on these lines: {1,3}, {4,392}, {9,12650}, {72,944}, {104,3916}, {210,5881}, {219,2261}, {355,2551}, {388,5812}, {515,960}, {518,5882}, {551,13374}, {573,1108}, {758,12675}, {946,3838}, {952,6737}, {962,3890}, {971,5698}, {997,11500}, {1056,5758}, {1125,7686}, {1490,15829}, {1699,31822}, {1766,2256}, {2771,12757}, {2800,9943}, {2802,20418}, {3419,12116}, {3427,5787}, {3633,15104}, {3817,16616}, {3876,18908}, {3878,4297}, {3880,11362}, {3899,15071}, {3927,30283}, {3940,5534}, {4640,5450}, {4679,10157}, {4847,5690}, {4999,5836}, {5289,6261}, {5440,11491}, {5587,25917}, {5692,14872}, {5693,12680}, {5730,18446}, {5732,7971}, {5806,5886}, {5837,6245}, {5909,5930}, {6256,24703}, {6765,7966}, {9947,18525}, {11715,17009}, {12114,12514}, {12119,17638}, {13369,14988}, {18242,21616}, {22758,31445}

X(31786) = midpoint of X(i) and X(j) for these {i,j}: {1, 14110}, {40, 3057}, {72, 944}, {3878, 4297}, {5693, 12680}, {5882, 31806}, {5887, 18481}, {7957, 7982}, {9957, 31793}, {12119, 17638}, {12702, 23340}
X(31786) = reflection of X(i) in X(j) for these (i,j): (65, 9940), (355, 5044), (942, 1385), (1482, 31792), (5777, 960), (5836, 6684), (7686, 1125), (13600, 9957), (18525, 9947), (24474, 5045), (31788, 3), (31798, 3579)
X(31786) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1617, 24928), (3428, 7742, 26286), (10246, 24474, 5045)

X(31787) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND INCIRCLE-CIRCLES

Barycentrics a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^4+c^4-(3*b^2+4*b*c+3*c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31787) = X(1)-3*X(11227) = 7*X(3)+X(25413) = 3*X(3)-X(31786) = X(8)+3*X(10167) = 3*X(40)+5*X(15016) = X(65)+3*X(165) = 3*X(165)-X(31793) = 3*X(210)-7*X(9588) = 3*X(210)+X(15071) = 3*X(354)+X(7991) = X(960)-3*X(10164) = X(962)-5*X(5439) = 2*X(1125)-3*X(10156) = 5*X(1698)-3*X(10157) = 5*X(1698)-X(12688) = 5*X(3617)+3*X(11220) = 3*X(3679)+X(12680) = 7*X(9588)+X(15071) = 2*X(9943)+X(9947) = 3*X(10157)-X(12688)

X(31787) lies on these lines: {1,3}, {2,9856}, {4,11024}, {8,10167}, {10,971}, {84,9708}, {210,9588}, {355,9799}, {515,31805}, {950,31777}, {960,10164}, {962,5439}, {991,4646}, {1000,17624}, {1125,10156}, {1158,31445}, {1376,12520}, {1490,9709}, {1698,10157}, {1706,5732}, {1742,24440}, {1788,12711}, {2550,5787}, {2551,6259}, {2771,20417}, {2801,4662}, {3617,11220}, {3679,12680}, {3697,12528}, {3698,5691}, {3740,31803}, {3754,12512}, {3820,6260}, {3824,7680}, {4297,5836}, {4853,30283}, {4915,9845}, {5044,6696}, {5658,5777}, {5690,13369}, {5779,7992}, {5791,14647}, {5927,9780}, {6223,10307}, {6245,31419}, {7080,9954}, {7686,31730}, {9623,9841}, {11362,12675}, {18251,26066}

X(31787) = midpoint of X(i) and X(j) for these {i,j}: {1, 31798}, {3, 31788}, {10, 9943}, {40, 942}, {65, 31793}, {950, 31777}, {3754, 12512}, {4297, 5836}, {5045, 31797}, {5690, 13369}, {7686, 31730}, {11362, 12675}, {13145, 31663}
X(31787) = reflection of X(i) in X(j) for these (i,j): (5044, 6684), (5045, 9940), (9947, 10), (31792, 1385), (31821, 5044)
X(31787) = complement of X(9856)
X(31787) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 165, 31793), (942, 16201, 5045), (3295, 8726, 1385)

X(31788) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND INTOUCH

Barycentrics a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^2-3*b*c+c^2)*(b+c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31788) = 3*X(3)+X(25413) = X(4)-3*X(3753) = 5*X(10)-3*X(15064) = 3*X(10)-X(31803) = 3*X(40)-X(7957) = 3*X(65)+X(7957) = X(72)-3*X(5657) = 3*X(165)+X(5903) = 3*X(165)-X(14110) = 3*X(210)-X(5693) = 3*X(354)-X(7982) = 3*X(354)-5*X(15016) = 3*X(392)-5*X(631) = 3*X(549)-2*X(31838) = X(944)-3*X(10167) = 5*X(3617)-X(12528) = 5*X(3617)-3*X(18908) = 5*X(5777)-6*X(15064) = 3*X(5777)-2*X(31803) = 3*X(10167)+X(10914) = X(12528)-3*X(18908) = 9*X(15064)-5*X(31803)

X(31788) lies on these lines: {1,3}, {2,12672}, {4,3753}, {5,1538}, {10,5777}, {72,5657}, {84,9623}, {100,17654}, {210,5693}, {355,971}, {392,631}, {497,12700}, {515,5836}, {516,3754}, {518,5884}, {519,12675}, {549,31838}, {573,21866}, {581,4646}, {912,5690}, {936,7971}, {944,10167}, {946,3812}, {952,13369}, {958,1158}, {960,2800}, {962,6865}, {1012,19860}, {1210,15845}, {1329,12608}, {1376,6261}, {1413,21147}, {1490,1706}, {1519,4187}, {1532,24982}, {1737,15908}, {1753,1905}, {1768,5258}, {1770,11827}, {1829,6197}, {1902,7412}, {2771,12665}, {2801,3626}, {2818,5909}, {2886,12616}, {3085,12709}, {3555,12245}, {3617,12528}, {3679,14872}, {3698,5587}, {3740,20117}, {3869,27383}, {3880,5882}, {3901,15104}, {3918,19925}, {3922,31822}, {3925,9956}, {4002,5818}, {4301,13374}, {4303,24028}, {5044,5887}, {5082,5768}, {5178,9803}, {5440,21740}, {5554,6925}, {5687,18446}, {5720,9709}, {5780,8580}, {5790,9947}, {5806,12699}, {5814,18909}, {5828,11678}, {5854,15528}, {5880,26332}, {5881,12680}, {5886,28629}, {6259,10309}, {6745,14988}, {6796,15813}, {6864,11024}, {6882,9955}, {6906,17613}, {6913,12705}, {6927,26062}, {6928,22793}, {6932,25005}, {6958,11230}, {6975,17618}, {7330,9708}, {7680,12609}, {9841,12650}, {9845,11525}, {10107,31730}, {11500,12520}, {12119,17636}, {14647,19843}

X(31788) = midpoint of X(i) and X(j) for these {i,j}: {40, 65}, {100, 17654}, {942, 31798}, {944, 10914}, {1770, 11827}, {3555, 12245}, {5836, 9943}, {5881, 12680}, {5884, 11362}, {5903, 14110}, {12119, 17636}, {12667, 17649}, {12702, 24474}, {14872, 15071}, {31794, 31797}
X(31788) = reflection of X(i) in X(j) for these (i,j): (1, 9940), (3, 31787), (946, 3812), (960, 6684), (1482, 5045), (4301, 13374), (5777, 10), (5887, 5044), (7686, 3754), (9856, 5), (9957, 1385), (12699, 5806), (12702, 31797), (13600, 1), (19925, 3918), (23340, 31792), (24474, 31794), (24680, 13373), (25405, 18856), (31786, 3), (31793, 3579)
X(31788) = complement of X(12672)
X(31788) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7982, 7994), (6769, 11529, 1482), (7982, 15016, 354)

X(31789) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND MANDART-INCIRCLE

Barycentrics 2*a^7-2*(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+2*(b+c)^2*b*c*a^3+2*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(31789) = 3*X(165)-X(11826) = 3*X(3058)-X(7982) = 3*X(3576)-X(7354) = 3*X(5587)-X(6253) = 5*X(7987)-X(10483) = 3*X(10202)-2*X(24470) = 3*X(11246)-5*X(15016) = 3*X(15170)-2*X(24680)

As a point on the Euler line, X(31789) has Shinagawa coefficients (2*R+r, -2*R-3*r)

X(31789) lies on these lines: {1,5812}, {2,3}, {9,355}, {10,5842}, {11,11012}, {12,10902}, {36,30264}, {40,1728}, {72,952}, {165,5445}, {214,2829}, {226,1385}, {329,944}, {389,517}, {495,10267}, {496,11249}, {497,22770}, {515,960}, {516,3754}, {528,11362}, {529,5882}, {572,1901}, {580,1834}, {938,2095}, {954,16202}, {956,12116}, {1001,26332}, {1329,6796}, {1479,3428}, {1482,3488}, {1484,13279}, {1490,18481}, {1617,10629}, {1713,5755}, {1724,5721}, {1750,4679}, {1753,15941}, {1852,1882}, {1864,10572}, {2077,10958}, {2949,6598}, {3058,7982}, {3419,5690}, {3487,4308}, {3564,10477}, {3576,7354}, {3579,5840}, {3583,15908}, {3585,15931}, {3655,28609}, {3820,11499}, {3885,5844}, {4292,9940}, {4294,10306}, {4302,10310}, {4324,24466}, {4330,5537}, {4640,12616}, {5175,5657}, {5248,7680}, {5436,5715}, {5441,5538}, {5444,7987}, {5535,16113}, {5584,12953}, {5587,6253}, {5709,5722}, {5719,24299}, {5728,5762}, {5729,5759}, {5771,6734}, {5780,18228}, {5787,7330}, {5884,17768}, {5887,12664}, {5924,7971}, {5927,28186}, {6261,24703}, {6748,18591}, {8070,14794}, {8273,12943}, {8544,31657}, {8726,9579}, {9580,12700}, {9803,11684}, {9841,15239}, {10198,10894}, {10202,24470}, {10386,10679}, {10592,26487}, {10827,16208}, {10950,18397}, {11246,15016}, {11491,17757}, {12575,13600}, {12680,12757}, {15170,24680}, {15254,31673}, {15325,26286}, {15911,22793}, {25413,28212}, {28146,31787}

X(31789) = midpoint of X(i) and X(j) for these {i,j}: {1, 11827}, {3, 7491}, {40, 6284}, {10572, 14110}, {15171, 31799}
X(31789) = reflection of X(i) in X(j) for these (i,j): (1482, 15172), (4292, 9940), (5777, 12572), (13600, 12575), (18990, 1385), (20420, 5), (24474, 12433), (31775, 3), (31777, 3579)
X(31789) = anticomplement of X(37281)
X(31789) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 6850), (6834, 17556, 5), (6942, 6963, 13747)

X(31790) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND MIDARC

Barycentrics
a*(2*(-a+b+c)*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*sin(A/2)*a*b*c+(a+b-c)*(a^5-(b-3*c)*a^4-2*(b-c)*a^3*b+2*(b^3-2*c^3-(2*b+c)*b*c)*a^2+(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^2*(b-c))*sin(C/2)*b+(a-b+c)*(a^5+(3*b-c)*a^4+2*(b-c)*a^3*c-2*(2*b^3-c^3+(b+2*c)*b*c)*a^2-(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^2*(b-c))*sin(B/2)*c) : :
X(31790) = 3*X(3576)-X(8422) = X(7982)-3*X(11191)

X(31790) lies on these lines: {3,12443}, {40,177}, {517,12813}, {3576,8422}, {3579,31801}, {5571,9940}, {6684,18258}, {6907,12622}, {6908,12694}, {6922,12614}, {7982,11191}

X(31790) = midpoint of X(40) and X(177)
X(31790) = reflection of X(i) in X(j) for these (i,j): (5571, 9940), (31791, 3), (31801, 3579)

X(31791) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HEXYL AND 2nd MIDARC

Barycentrics
a*(2*(-a+b+c)*(3*a^3-(b+c)*a^2-(3*b^2-2*b*c+3*c^2)*a+(b^2-c^2)*(b-c))*sin(A/2)*a*b*c-(a+b-c)*(a^5-(b+5*c)*a^4-2*(b-c)*a^3*b+2*(b^3+2*c^3+(2*b-c)*b*c)*a^2+(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^2*(b-c))*sin(C/2)*b-(a-b+c)*(a^5-(5*b+c)*a^4+2*(b-c)*a^3*c+2*(2*b^3+c^3-(b-2*c)*b*c)*a^2-(b^2-c^2)*(b-c)^2*a+(b^2-c^2)^2*(b-c))*sin(B/2)*c) : :
X(31791) = X(177)-3*X(3576) = 3*X(5731)+X(11691) = X(7982)-3*X(11234)

X(31791) lies on these lines: {3,12443}, {40,8422}, {177,3576}, {515,18258}, {517,5571}, {1385,12908}, {3579,31800}, {5731,11691}, {6907,12614}, {6922,12622}, {6987,12694}, {7982,11234}, {9940,31768}

X(31791) = midpoint of X(40) and X(8422)
X(31791) = reflection of X(i) in X(j) for these (i,j): (12908, 1385), (31768, 9940), (31790, 3), (31800, 3579)

X(31792) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HUTSON INTOUCH AND INCIRCLE-CIRCLES

Barycentrics a*(-10*a*b*c+(b+c)*a^2-(b^2-c^2)*(b-c)) : :
X(31792) = 5*X(1)-X(65) = 7*X(1)-3*X(354) = 3*X(1)-X(942) = 3*X(1)+X(3057) = 5*X(1)-3*X(5049) = 7*X(1)+X(5697) = 11*X(1)-3*X(5902) = 9*X(1)-X(5903) = X(1)+3*X(5919) = 9*X(1)-5*X(17609) = 13*X(1)-5*X(18398) = 4*X(1)-X(31794) = 5*X(8)-9*X(3921) = X(10)-3*X(10179) = 7*X(65)-15*X(354) = 3*X(65)-5*X(942) = X(72)+3*X(3241) = X(72)-5*X(3890) = X(145)+3*X(392) = 3*X(145)+5*X(3876) = 5*X(145)+7*X(4533) = 9*X(392)-5*X(3876) = 15*X(392)-7*X(4533) = 3*X(3241)+5*X(3890)

X(31792) lies on these lines: {1,3}, {8,3921}, {10,10179}, {12,7743}, {30,10105}, {72,3241}, {145,392}, {226,31822}, {355,1058}, {388,22793}, {390,18481}, {474,3895}, {495,9955}, {496,9956}, {497,18480}, {515,15172}, {516,31776}, {518,3884}, {519,4015}, {550,4315}, {551,5836}, {910,9327}, {938,1000}, {944,9856}, {950,15170}, {952,9947}, {960,3244}, {971,5882}, {1056,9785}, {1125,3880}, {1279,15955}, {1387,13411}, {1483,31821}, {1698,3893}, {2136,9709}, {2771,12735}, {2802,3636}, {3058,26088}, {3085,11230}, {3086,11231}, {3486,17622}, {3487,3656}, {3488,5777}, {3555,3623}, {3616,10914}, {3621,3697}, {3622,3753}, {3625,3740}, {3632,25917}, {3646,4915}, {3655,4313}, {3683,5288}, {3698,25055}, {3816,10915}, {3848,3918}, {3851,5726}, {3869,20057}, {3871,17614}, {3889,4018}, {3957,5330}, {3983,4677}, {4002,5550}, {4292,28198}, {4297,10386}, {4298,28174}, {4301,6147}, {4314,11035}, {4342,21620}, {4853,11108}, {5248,11260}, {5434,28202}, {5439,14923}, {5559,13602}, {5572,31806}, {5690,11019}, {5719,5806}, {5790,13867}, {5844,6738}, {5881,10157}, {5886,6964}, {5887,8236}, {5901,13405}, {6001,13607}, {6284,28168}, {6736,17527}, {6744,28234}, {7354,28154}, {7967,12672}, {8091,10968}, {8167,14150}, {9578,9669}, {9580,9655}, {9589,18541}, {9613,9668}, {9614,9654}, {9708,12629}, {9843,12640}, {9845,12684}, {10056,11376}, {10106,15171}, {10459,17460}, {10578,10595}, {10580,12245}, {10624,18990}, {10827,11238}, {10866,18526}, {10912,12260}, {11362,21625}, {12513,31445}, {12577,24470}, {12609,13463}, {12645,18530}, {12705,30283}, {12908,31767}, {14986,26446}, {15888,30384}, {15935,31837}, {21627,31419}, {31231,31436}, {31430,31433}

X(31792) = midpoint of X(i) and X(j) for these {i,j}: {1, 9957}, {3, 13600}, {942, 3057}, {944, 9856}, {960, 3244}, {1482, 31786}, {7982, 31793}, {10106, 15171}, {10624, 18990}, {12735, 15558}, {12908, 31767}, {23340, 31788}
X(31792) = reflection of X(i) in X(j) for these (i,j): (3812, 3636), (5045, 1), (5806, 13464), (9940, 15178), (24470, 12577), (31787, 1385), (31794, 5045), (31795, 15172), (31797, 3)
X(31792) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 15934), (5903, 17609, 942), (10246, 23340, 31788)

X(31793) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: HUTSON INTOUCH AND 6th MIXTILINEAR

Barycentrics a*((b+c)*a^5-(b^2+8*b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^4+c^4+3*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31793) = 3*X(3)-2*X(9940) = 5*X(3)-3*X(10202) = 4*X(3)-3*X(11227) = 3*X(3)-X(24474) = 2*X(4)-3*X(10157) = 3*X(20)+X(12528) = 3*X(40)-2*X(31797) = X(65)-3*X(165) = 3*X(72)-X(12528) = 3*X(165)-2*X(31787) = 3*X(210)-X(5691) = 3*X(210)-2*X(9947) = 3*X(353)-2*X(31820) = 3*X(354)-5*X(7987) = 3*X(381)-2*X(31822) = 3*X(392)-X(962) = 3*X(1699)-5*X(25917) = 4*X(5044)-3*X(10157) = 3*X(5777)-4*X(31835) = 2*X(31835)-3*X(31837)

Let A'B'C' be the Hutson-extouch triangle. Let L_A be the tangent to the A-excircle at A', and define L _B and L _C cyclically. Let A" = L_B∩L _C, B" = L _C∩L_A, C" = L_A∩L _B. Then triangle A"B"C" is homothetic to ABC and the orthic-of-intouch triangle at X(57), and X(31793) = X(942)-of- A"B"C". (Randy Hutson, March 28, 2019)

X(31793) lies on these lines: {1,3}, {2,5806}, {4,5044}, {5,5316}, {8,5787}, {10,8727}, {20,72}, {28,1902}, {30,5777}, {78,7580}, {84,3927}, {142,4301}, {210,5691}, {220,1766}, {226,5763}, {329,6259}, {353,31820}, {355,6851}, {381,31822}, {386,15852}, {392,443}, {405,31658}, {411,5440}, {515,6743}, {516,960}, {518,4297}, {548,13369}, {550,912}, {573,1212}, {610,21871}, {758,9943}, {936,19541}, {946,8728}, {1012,31445}, {1104,13329}, {1295,15439}, {1394,22117}, {1439,3160}, {1465,22072}, {1490,3940}, {1538,21616}, {1699,25917}, {1753,7497}, {1829,4219}, {1858,15338}, {2771,16111}, {2778,15647}, {2800,9942}, {2802,13226}, {2979,31816}, {3008,15489}, {3146,3876}, {3182,7955}, {3243,12120}, {3309,5592}, {3419,6836}, {3474,12709}, {3522,3868}, {3523,5439}, {3532,28787}, {3555,5731}, {3600,11035}, {3678,28164}, {3698,9588}, {3730,15853}, {3740,19925}, {3753,6857}, {3812,10164}, {3827,12262}, {3869,9778}, {3877,6904}, {3878,5493}, {3880,24391}, {3884,12436}, {3890,9776}, {3916,6909}, {3962,5918}, {4292,5762}, {4313,5728}, {4315,12128}, {5223,10864}, {5250,19520}, {5265,17626}, {5436,21153}, {5657,5791}, {5663,31836}, {5692,12688}, {5715,17528}, {5722,6865}, {5732,11523}, {5744,14923}, {5745,5836}, {5758,6916}, {5768,6764}, {5812,6850}, {5886,6989}, {5887,6869}, {5890,31819}, {5894,6001}, {5904,12680}, {6245,11362}, {6361,12672}, {6675,6684}, {6762,30283}, {6824,19855}, {6826,12699}, {6835,18482}, {6841,9956}, {6861,11231}, {6881,7958}, {6908,11374}, {6925,22792}, {7074,21147}, {7673,8732}, {7682,17527}, {8703,24475}, {9047,31732}, {9589,31671}, {9829,31818}, {9840,16601}, {9954,12527}, {10165,13374}, {10175,16616}, {10304,24473}, {10391,15556}, {10860,12526}, {11036,21151}, {11347,25930}, {11402,31808}, {11495,12520}, {11573,13348}, {11826,28146}, {11827,28160}, {12109,17704}, {12443,31800}, {12511,22836}, {12629,12842}, {12651,31435}, {13730,26935}, {15488,29571}, {15726,31803}, {20117,28150}, {21669,26878}, {22791,31838}, {28198,28452}, {30290,31391}

X(31793) = midpoint of X(i) and X(j) for these {i,j}: {1, 7957}, {20, 72}, {40, 14110}, {3057, 7991}, {3878, 5493}, {3962, 15071}, {5904, 12680}, {6361, 12672}, {31730, 31806}
X(31793) = reflection of X(i) in X(j) for these (i,j): (4, 5044), (65, 31787), (942, 3), (5691, 9947), (5777, 31837), (7686, 6684), (7982, 31792), (9856, 960), (9943, 12512), (9957, 31786), (11573, 13348), (12109, 17704), (12688, 31821), (13369, 548), (22791, 31838), (24474, 9940), (31788, 3579), (31798, 40)
X(31793) = anticomplement of X(5806)
X(31793) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7982, 8726, 15934), (9940, 24474, 942), (10857, 11531, 11518)

X(31794) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INCIRCLE-CIRCLES AND INTOUCH

Barycentrics a*(2*a*b*c+3*(b+c)*a^2-3*(b^2-c^2)*(b-c)) : :
Trilinears 1 + 3 cos B + 3 cos C : :
X(31794) = X(1)+3*X(65) = 5*X(1)-9*X(354) = X(1)-3*X(942) = 7*X(1)-3*X(3057) = 2*X(1)-3*X(5045) = 7*X(1)-9*X(5049) = 11*X(1)-3*X(5697) = X(1)-9*X(5902) = 5*X(1)+3*X(5903) = 13*X(1)-9*X(5919) = 5*X(1)-3*X(9957) = 11*X(1)-15*X(17609) = 7*X(1)-15*X(18398) = 4*X(1)-3*X(31792) = 3*X(2)+X(4018) = 5*X(65)+3*X(354) = 7*X(65)+X(3057) = 2*X(65)+X(5045) = 7*X(65)+3*X(5049) = 11*X(65)+X(5697)

X(31794) lies on these lines: {1,3}, {2,4018}, {4,5556}, {5,3671}, {7,355}, {8,4004}, {10,3824}, {12,11551}, {30,6738}, {72,5775}, {79,17501}, {210,3901}, {226,9956}, {382,4312}, {495,4848}, {515,24470}, {516,12433}, {518,3626}, {519,10107}, {553,18990}, {758,3634}, {912,9947}, {938,12699}, {944,21454}, {950,28146}, {952,4298}, {960,4084}, {971,5884}, {1042,5396}, {1111,4955}, {1125,4757}, {1210,9955}, {1376,12559}, {1483,4315}, {1698,3962}, {1736,5492}, {1737,3614}, {1770,28154}, {1788,11231}, {1844,1888}, {1864,31828}, {2771,7687}, {2800,13374}, {3306,5730}, {3485,11230}, {3487,26446}, {3488,18221}, {3555,3621}, {3600,11041}, {3617,3753}, {3624,31165}, {3625,3874}, {3647,8261}, {3656,14986}, {3679,3922}, {3681,4002}, {3698,5904}, {3740,4067}, {3742,3878}, {3828,4127}, {3869,5439}, {3873,10914}, {3880,3881}, {3897,23958}, {3918,4662}, {3982,11545}, {4005,19875}, {4031,4311}, {4292,28160}, {4295,5225}, {4297,14563}, {4314,15935}, {4338,12953}, {4355,5881}, {4640,30143}, {4654,9654}, {4896,24471}, {5218,31447}, {5229,18391}, {5290,5790}, {5493,10386}, {5542,11362}, {5586,18525}, {5657,11036}, {5665,6913}, {5687,11520}, {5690,21620}, {5691,18541}, {5692,19872}, {5693,10157}, {5694,6858}, {5704,5887}, {5714,5777}, {5719,6684}, {5727,9655}, {5791,28629}, {5806,5893}, {6284,28202}, {6744,15172}, {6797,11570}, {7173,12047}, {7354,28208}, {7672,30340}, {8094,12491}, {8100,12445}, {9709,11523}, {10404,10573}, {10572,11246}, {11019,22791}, {11037,12245}, {11108,12526}, {12575,28212}, {12577,28234}, {13411,16137}, {15009,28174}, {15171,28198}, {16672,21853}, {17614,27003}, {17619,31053}, {21746,29229}, {21848,24248}, {24391,31419}

X(31794) = midpoint of X(i) and X(j) for these {i,j}: {65, 942}, {960, 4084}, {1125, 4757}, {3742, 4744}, {3874, 5836}, {5884, 7686}, {5903, 9957}, {6797, 11570}, {13600, 25413}, {21848, 24248}, {24474, 31788}
X(31794) = reflection of X(i) in X(j) for these (i,j): (4662, 3918), (5044, 3812), (5045, 942), (9940, 5885), (12433, 17706), (15172, 6744), (31776, 24470), (31792, 5045), (31795, 12433), (31797, 31788), (31821, 5)
X(31794) = complement of the complement of X(4018)
X(31794) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 46, 5217), (46, 24929, 31663), (7982, 10980, 7373)

X(31795) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INCIRCLE-CIRCLES AND MANDART-INCIRCLE

Barycentrics 4*a^4-(b+c)*a^3-2*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(31795) = 3*X(942)-X(1770) = 5*X(942)-3*X(11246) = 3*X(950)+X(10624) = X(1770)+3*X(6284) = 5*X(1770)-9*X(11246) = 3*X(3058)-X(9957) = 3*X(3058)+X(10572) = 9*X(3058)-X(10944) = 2*X(4298)-3*X(5045) = 4*X(4298)-3*X(31776) = 5*X(6284)+3*X(11246) = 3*X(9957)-X(10944) = 3*X(10572)+X(10944) = X(10624)-3*X(15171) = 3*X(12433)-2*X(17706) = 4*X(17706)-3*X(31794)

X(31795) lies on these lines: {1,382}, {3,18527}, {4,10578}, {5,4314}, {10,10386}, {30,4298}, {55,9956}, {65,28198}, {355,390}, {389,517}, {496,4304}, {497,1385}, {515,15172}, {516,12433}, {519,4536}, {546,13405}, {550,11019}, {942,1770}, {952,12575}, {1058,4308}, {1155,4330}, {1159,9589}, {1210,31663}, {1319,5441}, {1479,9955}, {1483,4342}, {1657,3333}, {1788,3579}, {1837,4309}, {2646,4857}, {3057,5694}, {3058,9957}, {3086,17502}, {3295,3586}, {3361,3534}, {5231,17571}, {11015,17614}, {12690,24987}

X(31795) = midpoint of X(i) and X(j) for these {i,j}: {942, 6284}, {950, 15171}, {9957, 10572}
X(31795) = reflection of X(i) in X(j) for these (i,j): (31776, 5045), (31792, 15172), (31794, 12433)
X(31795) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9668, 22793), (496, 4304, 13624)

X(31796) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INCIRCLE-CIRCLES AND 2nd MIDARC

Barycentrics a*((-a+b+c)*(10*a*b*c+(b+c)*a^2-(b^2-c^2)*(b-c))*sin(A/2)+(a-b+c)*(6*a*b*c+(b-c)*a^2-(b+c)*(b^2-c^2))*sin(B/2)-(a+b-c)*(-6*a*b*c+(b-c)*a^2-(b+c)*(b^2-c^2))*sin(C/2)) : :
X(31796) = 3*X(1)+X(17641) = X(177)-3*X(5049) = 3*X(5571)+X(31767) = X(9957)-3*X(11234)

X(31796) lies on these lines: {1,8099}, {164,6767}, {177,5049}, {517,5571}, {942,8422}, {1159,11528}, {5045,12813}, {7373,12844}, {9957,11234}, {10508,18409}, {28146,31770}, {28160,31734}

X(31796) = midpoint of X(942) and X(8422)
X(31796) = reflection of X(12813) in X(5045)

X(31797) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INCIRCLE-CIRCLES AND 6th MIXTILINEAR

Barycentrics a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2-7*b*c+c^2)*a^3+2*(b^4+c^4-3*(b^2+4*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-12*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31797) = 3*X(3)-X(13600) = 5*X(40)-X(14110) = 3*X(40)-X(31793) = 3*X(165)-X(9957) = X(3244)-3*X(10178) = X(3632)+3*X(5918) = 3*X(3654)-X(5777) = 5*X(3698)-X(9589) = 3*X(3753)+X(20070) = 7*X(4002)-3*X(9812) = 3*X(5657)-X(9856) = 3*X(9778)+X(10914) = 3*X(10156)-2*X(13464) = 3*X(31821)-4*X(31835)

X(31797) lies on these lines: {1,3}, {519,31805}, {962,17559}, {971,11362}, {1000,12128}, {3244,10178}, {3626,15726}, {3632,5918}, {3654,5777}, {3698,9589}, {3753,5129}, {3812,28228}, {3880,12512}, {4002,9812}, {5493,5836}, {5657,9856}, {5690,9947}, {5806,28194}, {6939,12699}, {6964,26446}, {9778,10914}, {10156,13464}, {11826,28168}, {11827,28154}, {16616,28232}, {28146,31799}, {28160,31777}, {28198,31822}, {31821,31835}

X(31797) = midpoint of X(i) and X(j) for these {i,j}: {40, 31798}, {942, 7991}, {5493, 5836}, {12702, 31788}
X(31797) = reflection of X(i) in X(j) for these (i,j): (5045, 31787), (9947, 5690), (31792, 3), (31794, 31788)

X(31798) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INTOUCH AND 6th MIXTILINEAR

Barycentrics a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2-5*b*c+c^2)*a^3+2*(b^4+c^4-(b^2+8*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(31798) = 2*X(1)-3*X(11227) = 3*X(3)-X(23340) = 3*X(8)+X(9961) = 4*X(10)-3*X(10157) = 3*X(40)-X(14110) = X(145)-3*X(10167) = 3*X(165)-X(3057) = 3*X(210)-2*X(31821) = 3*X(354)-X(11531) = X(962)-3*X(3753) = 3*X(1699)-5*X(3698) = 5*X(3091)-7*X(4002) = 5*X(3522)-X(3885) = 5*X(3616)-6*X(10156) = 5*X(3617)-3*X(5927) = X(3621)+3*X(11220) = 3*X(3654)-X(5887) = 3*X(3679)-2*X(9947) = 3*X(3679)-X(12688) = 3*X(3753)-2*X(5806) = 2*X(9856)-3*X(10157)

X(31798) lies on these lines: {1,3}, {8,971}, {10,9842}, {20,10914}, {72,5658}, {145,10167}, {210,31821}, {238,2943}, {355,17650}, {392,17567}, {452,5759}, {515,31777}, {516,5795}, {518,12640}, {519,9943}, {573,21872}, {944,12632}, {946,17527}, {960,20103}, {962,3753}, {1329,1538}, {1699,3698}, {1706,19541}, {1902,4222}, {2136,5732}, {2802,12512}, {2975,17613}, {3091,4002}, {3421,6259}, {3436,22792}, {3522,3885}, {3616,10156}, {3617,5927}, {3621,11220}, {3632,12680}, {3654,5887}, {3679,9947}, {3754,28228}, {3812,4301}, {3880,4297}, {3893,5918}, {3913,12520}, {3940,7971}, {3962,15104}, {3968,12571}, {4662,31803}, {4731,7989}, {4853,10860}, {4915,10864}, {5044,5657}, {5082,5787}, {5250,31658}, {5690,5777}, {5844,13369}, {6001,11362}, {6736,9954}, {6827,12700}, {6893,12699}, {6922,7743}, {6926,11373}, {6944,26446}, {7288,17622}, {7686,28194}, {7956,8582}, {9578,17634}, {9708,12705}, {9710,12617}, {9778,14923}, {9800,31672}, {9841,12629}, {9845,11519}, {9956,15908}, {11826,28160}, {11827,28146}, {12675,28234}, {16004,31775}

X(31798) = midpoint of X(i) and X(j) for these {i,j}: {20, 10914}, {65, 7991}, {3632, 12680}, {5903, 7957}
X(31798) = reflection of X(i) in X(j) for these (i,j): (1, 31787), (40, 31797), (942, 31788), (944, 31805), (962, 5806), (1482, 9940), (4301, 3812), (5777, 5690), (7982, 5045), (9856, 10), (9957, 3), (11278, 13373), (12672, 5044), (12688, 9947), (13600, 1385), (31775, 16004), (31786, 3579), (31793, 40), (31803, 4662)
X(31798) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1420, 3057, 20789), (1482, 9940, 5049), (11575, 20789, 1420)

X(31799) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: MANDART-INCIRCLE AND 6th MIXTILINEAR

Barycentrics 2*a^7-2*(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-8*b*c+3*c^2)*a^4+2*(b^2+6*b*c+c^2)*b*c*a^3+6*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(31799) = 3*X(40)-X(11826) = 3*X(165)-X(7354) = X(962)-3*X(11113) = 2*X(1482)-3*X(15170) = X(1482)-3*X(28459) = 3*X(3058)-X(11531) = 3*X(3679)-X(6253) = 2*X(4298)-3*X(11227) = 3*X(5434)-5*X(7987) = 3*X(10157)-4*X(18250) = 3*X(11114)+X(20070) = X(11826)+3*X(11827) = 2*X(11826)-3*X(31777) = 2*X(11827)+X(31777)

X(31799) lies on these lines: {1,5763}, {3,388}, {4,9708}, {5,3428}, {8,5759}, {10,20420}, {20,3421}, {30,40}, {65,5762}, {140,11012}, {165,7354}, {389,517}, {411,17757}, {496,6827}, {497,8158}, {515,6743}, {516,5795}, {529,4297}, {535,12512}, {546,15908}, {548,2077}, {550,10310}, {952,14110}, {956,6836}, {958,8727}, {962,11113}, {971,12527}, {999,6865}, {1478,5584}, {1482,15170}, {1753,13488}, {2478,7956}, {2551,19541}, {2807,22299}, {2829,31730}, {3058,11531}, {3149,3820}, {3295,6987}, {3436,7580}, {4292,31787}, {4298,11227}, {5260,8226}, {5302,12617}, {5434,7987}, {5534,6282}, {5537,15338}, {5727,6284}, {5842,11362}, {5843,15071}, {6259,12565}, {6675,7680}, {6825,10592}, {6840,24390}, {6844,31493}, {6868,10306}, {6907,10526}, {6908,9654}, {6916,9655}, {6922,11249}, {6928,11928}, {6988,31479}, {7411,20060}, {7491,12702}, {7957,10572}, {7982,15172}, {8728,26332}, {9856,12572}, {10157,18250}, {10404,31657}, {11114,20070}, {12103,24466}, {15888,15931}, {17527,22753}, {28146,31797}

X(31799) = midpoint of X(i) and X(j) for these {i,j}: {40, 11827}, {6284, 7991}, {7491, 12702}, {7957, 10572}
X(31799) = reflection of X(i) in X(j) for these (i,j): (4292, 31787), (7982, 15172), (9856, 12572), (15170, 28459), (15171, 31789), (18990, 3), (20420, 10), (31777, 40)
X(31799) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2077, 30264, 548), (6827, 22770, 496)

X(31800) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: MIDARC AND 6th MIXTILINEAR

Barycentrics
a*(2*(-a+b+c)*(a^3+3*(b+c)*a^2-(b^2+6*b*c+c^2)*a-3*(b^2-c^2)*(b-c))*sin(A/2)*a*b*c+(a+b-c)*(a^5-(b-3*c)*a^4-2*(b^2-3*b*c-2*c^2)*a^3+2*(b+c)*(b^2-3*b*c-2*c^2)*a^2+(b^2-c^2)*(b-c)*(b-5*c)*a-(b^2-c^2)^2*(b-c))*sin(C/2)*b+(a-b+c)*(a^5+(3*b-c)*a^4+2*(2*b^2+3*b*c-c^2)*a^3-2*(b+c)*(2*b^2+3*b*c-c^2)*a^2-(b^2-c^2)*(b-c)*(5*b-c)*a+(b^2-c^2)^2*(b-c))*sin(B/2)*c) : :
X(31800) = 3*X(165)-X(8422) = 5*X(7987)-3*X(11234) = 3*X(11191)-X(11531)

X(31800) lies on these lines: {40,164}, {165,8422}, {177,7991}, {517,12813}, {3579,31791}, {5571,31787}, {6244,12523}, {7987,11234}, {11191,11531}, {12443,31793}

X(31800) = midpoint of X(177) and X(7991)
X(31800) = reflection of X(i) in X(j) for these (i,j): (5571, 31787), (31791, 3579), (31801, 40)

X(31801) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd MIDARC AND 6th MIXTILINEAR

Barycentrics
a*(2*(-a+b+c)*(3*a^2-2*(b+c)*a-(b-c)^2)*a*b*c*sin(A/2)-(a-b+c)*(a^4-2*(3*b+c)*a^3+2*(b+3*c)*a^2*b+2*(b^2-c^2)*(b-c)*a+(b^2-c^2)*(b-c)^2)*c*sin(B/2)-(a+b-c)*(a^4-2*(b+3*c)*a^3+2*(3*b+c)*a^2*c+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)*(b-c)^2)*b*sin(C/2)) : :
X(31801) = 3*X(165)-X(177) = 5*X(7987)-3*X(11191) = 3*X(9778)+X(11691) = 3*X(11234)-X(11531)

X(31801) lies on these lines: {3,12908}, {40,164}, {165,177}, {516,18258}, {517,5571}, {3579,31790}, {5759,12694}, {6244,12518}, {7987,11191}, {7991,8422}, {9778,11691}, {11234,11531}, {31768,31787}

X(31801) = midpoint of X(7991) and X(8422)
X(31801) = reflection of X(i) in X(j) for these (i,j): (12908, 3), (31768, 31787), (31790, 3579), (31800, 40)

X(31802) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC AND ANTI-ASCELLA

Barycentrics ((b^2+c^2)*a^2-b^4+2*b^2*c^2-c^4)*(5*a^6-9*(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(31802) = X(20)-3*X(11402) = X(31804)+2*X(31815)

X(31802) lies on these lines: {3,11427}, {4,193}, {5,51}, {6,12362}, {20,11402}, {30,1181}, {155,6756}, {235,3060}, {389,1368}, {394,9825}, {427,5889}, {428,11441}, {511,6823}, {515,31812}, {516,31811}, {550,10984}, {568,11585}, {576,12241}, {758,31814}, {971,31808}, {1192,16976}, {1353,6146}, {1498,31670}, {1595,13754}, {1596,5446}, {1598,12166}, {1907,12111}, {1992,18945}, {1993,3575}, {1994,12225}, {2393,2883}, {3167,7487}, {3527,18537}, {3618,11821}, {3627,16655}, {3629,10112}, {3849,31809}, {5448,21847}, {5480,5907}, {5654,21841}, {6102,23335}, {6193,18494}, {6243,15760}, {6515,7507}, {6643,11432}, {6676,17834}, {6816,9777}, {7395,18583}, {7399,11412}, {7403,18436}, {7405,23039}, {7553,18445}, {7691,14389}, {7715,10539}, {9306,11745}, {9786,16196}, {9815,17811}, {9830,31813}, {11479,14853}, {11576,27365}, {11818,31831}, {13292,18569}, {13346,13568}, {13488,19458}, {13567,16625}, {13595,22550}, {14449,15761}, {14516,15801}, {14790,18914}, {15043,30739}, {15873,21849}, {16266,31833}, {19347,31305}

X(31802) = midpoint of X(i) and X(j) for these {i,j}: {4, 12160}, {12161, 31815}
X(31802) = reflection of X(i) in X(j) for these (i,j): (6823, 12233), (31804, 12161)
X(31802) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 193, 12429), (4, 1351, 13142)

X(31803) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC AND INNER-GARCIA

Barycentrics a*((b+c)*a^5-(b^2+c^2)*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^2+(b^3+c^3)*(b-c)^2*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :
X(31803) = 2*X(3)-3*X(10176) = 4*X(5)-3*X(5883) = 3*X(5)-2*X(5885) = 2*X(10)-3*X(15064) = 3*X(10)-2*X(31788) = X(20)-3*X(5692) = X(65)-3*X(5927) = 3*X(72)-X(7957) = 3*X(165)-5*X(3876) = 3*X(165)-X(9961) = 5*X(3876)-X(9961) = 4*X(5777)-3*X(15064) = 3*X(5777)-X(31788) = 3*X(5883)-2*X(5884) = 9*X(5883)-8*X(5885) = 3*X(5884)-4*X(5885) = 3*X(5927)-2*X(19925) = X(7957)+3*X(12688) = 3*X(10176)-4*X(20117) = 9*X(15064)-4*X(31788) = X(31806)+2*X(31828)

X(31803) lies on these lines: {1,651}, {2,15071}, {3,3647}, {4,758}, {5,2771}, {7,30290}, {8,12059}, {9,12520}, {10,5777}, {12,13257}, {20,5692}, {30,5694}, {40,3678}, {65,5927}, {72,516}, {78,1709}, {79,6839}, {84,997}, {118,1425}, {165,3876}, {185,2772}, {191,411}, {214,5450}, {226,1858}, {355,2800}, {377,16120}, {389,15049}, {392,12680}, {404,1768}, {511,31817}, {515,3878}, {517,3625}, {518,4301}, {519,12672}, {551,12675}, {572,8235}, {581,3743}, {912,946}, {916,31732}, {936,7992}, {942,3817}, {944,3884}, {950,1898}, {952,10284}, {958,5779}, {960,971}, {962,5904}, {993,6261}, {1006,16132}, {1012,22836}, {1042,1736}, {1158,5720}, {1490,10268}, {1519,24387}, {1699,3868}, {1750,12526}, {1765,25078}, {1836,15556}, {1864,6738}, {2392,5562}, {2475,9809}, {2550,12446}, {2779,12162}, {2802,5881}, {2842,15030}, {3057,28236}, {3090,3833}, {3091,5902}, {3336,6915}, {3419,12679}, {3452,9948}, {3487,12564}, {3564,31814}, {3579,31835}, {3626,18908}, {3649,8226}, {3671,30329}, {3681,7991}, {3740,31787}, {3754,5587}, {3811,12705}, {3812,10157}, {3814,12616}, {3849,31823}, {3869,5691}, {3871,5531}, {3873,11522}, {3881,5603}, {3892,13464}, {3898,5882}, {3918,5818}, {3988,6361}, {4015,5657}, {4084,7686}, {4134,5493}, {4295,12432}, {4420,5537}, {4662,31798}, {4757,18492}, {4973,24467}, {5044,9943}, {5083,11376}, {5248,18446}, {5252,13227}, {5253,13243}, {5259,18444}, {5396,5492}, {5400,24443}, {5439,10171}, {5443,9964}, {5534,25439}, {5660,27529}, {5696,20007}, {5728,12563}, {5784,12447}, {5787,24703}, {5794,6259}, {5817,28629}, {5836,9947}, {5886,12005}, {6126,15052}, {6245,21616}, {6265,26321}, {6326,6906}, {6691,13226}, {6700,17649}, {6701,6829}, {6736,17615}, {6835,13159}, {6837,10122}, {6850,16127}, {6884,26725}, {6888,15096}, {6894,14450}, {6901,16116}, {6913,30143}, {7411,16143}, {7508,22936}, {7688,26878}, {7741,11570}, {7987,11220}, {8581,12577}, {8583,30304}, {8715,17857}, {9803,13729}, {9830,31826}, {9940,19862}, {9949,21060}, {9955,24475}, {9960,31424}, {10085,19861}, {10165,13369}, {10167,25917}, {10470,11203}, {10571,24430}, {10826,12736}, {10864,15829}, {10944,17638}, {11036,20116}, {11112,17653}, {11372,11523}, {11849,12738}, {12047,18389}, {12111,30438}, {12114,30144}, {12572,12664}, {12608,25639}, {12635,16112}, {12711,13405}, {13754,31825}, {14110,28164}, {14647,26364}, {14988,18480}, {15104,20070}, {15726,31793}, {17768,20420}, {18483,24474}, {20612,31053}, {24644,30628}, {31730,31837}

X(31803) = midpoint of X(i) and X(j) for these {i,j}: {1, 12528}, {4, 5693}, {72, 12688}, {84, 12666}, {962, 5904}, {3869, 5691}, {5694, 31828}, {12672, 14872}, {17638, 17661}
X(31803) = reflection of X(i) in X(j) for these (i,j): (3, 20117), (10, 5777), (40, 3678), (65, 19925), (944, 3884), (960, 31821), (3579, 31835), (3874, 946), (3878, 5887), (4084, 7686), (4297, 960), (4301, 9856), (5836, 9947), (5884, 5), (9943, 5044), (24474, 18483), (24475, 9955), (31730, 31837), (31798, 4662), (31806, 5694)
X(31803) = complement of X(15071)
X(31803) = X(1)-of-X(4)-Brocard-triangle
X(31803) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 20117, 10176), (5, 5884, 5883)

X(31804) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND ANTI-ASCELLA

Barycentrics (-a^2+b^2+c^2)*(4*a^8-7*(b^2+c^2)*a^6+3*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(31804) = X(4)-3*X(11402) = 3*X(3796)-2*X(16197) = 3*X(12161)-X(31815) = X(12167)-3*X(14912) = 3*X(31802)-2*X(31815)

X(31804) lies on these lines: {3,69}, {4,11402}, {5,156}, {6,6756}, {20,12160}, {24,11245}, {26,13292}, {30,1181}, {49,11585}, {51,7715}, {52,1353}, {54,427}, {68,6676}, {98,19179}, {125,632}, {140,1899}, {154,21841}, {155,12362}, {185,550}, {217,18907}, {235,1614}, {287,8362}, {381,18945}, {389,2393}, {468,9707}, {511,31810}, {515,31811}, {516,31812}, {546,18396}, {548,10605}, {549,13367}, {567,7403}, {578,1503}, {631,26944}, {858,9545}, {1147,1368}, {1199,7576}, {1204,8703}, {1351,19119}, {1498,13488}, {1596,6759}, {1598,11206}, {1885,11456}, {1906,14157}, {1907,15033}, {1994,19362}, {2072,9704}, {2883,13403}, {2888,7495}, {3089,14530}, {3147,26869}, {3167,6643}, {3270,10386}, {3311,18924}, {3312,18923}, {3515,18916}, {3517,11433}, {3526,23291}, {3527,6995}, {3530,26937}, {3542,26864}, {3547,12429}, {3567,11387}, {3575,7592}, {3627,21659}, {3796,16197}, {3851,18918}, {3858,13851}, {5012,7399}, {5050,7401}, {5480,13419}, {5622,23236}, {6240,12254}, {6243,15073}, {6247,11430}, {6515,9715}, {7387,13142}, {7404,18440}, {7405,13353}, {7487,11432}, {7514,31831}, {7528,18583}, {7540,14627}, {7542,25738}, {7553,21850}, {7583,19355}, {7584,19356}, {9781,10301}, {9908,15818}, {10116,12359}, {10282,13567}, {10938,12111}, {10982,31383}, {11003,13160}, {11424,16655}, {11425,14216}, {11464,26879}, {11485,18930}, {11486,18929}, {11542,19363}, {11543,19364}, {11745,12007}, {12038,18128}, {12118,19458}, {12233,18400}, {12605,18445}, {13754,31807}, {14070,18951}, {14786,18358}, {15171,19354}, {16238,18952}, {16252,18390}, {16270,30714}, {16657,26883}, {18381,23292}, {18990,19349}, {19116,21641}, {19117,21640}, {23195,31381}

X(31804) = midpoint of X(i) and X(j) for these {i,j}: {20, 12160}, {1181, 19467}, {6776, 19459}, {12118, 19458}
X(31804) = reflection of X(i) in X(j) for these (i,j): (1595, 578), (31802, 12161)
X(31804) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6776, 18914), (6776, 18925, 3)

X(31805) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND ASCELLA

Barycentrics a*((b+c)*a^5-(b^2-12*b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^4+c^4-5*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*(b+c)^2) : :
X(31805) = X(1)+3*X(5918) = 3*X(3)-X(5777) = X(4)-3*X(11227) = 2*X(5)-3*X(10156) = X(10)-3*X(10178) = X(20)+3*X(10167) = X(72)-5*X(3522) = X(72)+3*X(11220) = 3*X(165)+X(12680) = 3*X(210)-7*X(16192) = 3*X(392)+X(9961) = 3*X(550)+X(24475) = 5*X(631)-3*X(10157) = X(942)-3*X(10167) = X(962)-3*X(5049) = 5*X(3522)+3*X(11220) = 3*X(5044)-2*X(5777) = 3*X(5806)-2*X(31822) = 3*X(9940)-X(31822) = 3*X(13369)-X(24475)

X(31805) lies on these lines: {1,5918}, {3,9}, {4,11227}, {5,10156}, {10,10178}, {20,942}, {30,5806}, {72,3522}, {165,12680}, {210,16192}, {392,9961}, {405,10855}, {411,5122}, {443,10430}, {452,17612}, {511,31819}, {515,31787}, {516,5045}, {517,550}, {518,12512}, {519,31797}, {548,912}, {631,10157}, {916,13348}, {943,6909}, {944,12632}, {946,31657}, {962,3296}, {999,12565}, {1125,15726}, {1210,11575}, {1385,11496}, {1657,10202}, {1709,8273}, {1750,16408}, {1770,16193}, {2951,3333}, {3062,3646}, {3146,5439}, {3295,10860}, {3361,14100}, {3523,5927}, {3534,24474}, {3555,9778}, {3576,9856}, {3579,24467}, {3655,13600}, {3660,6284}, {3812,28164}, {3824,8727}, {3848,12571}, {3876,21734}, {3916,7411}, {3927,30304}, {4189,17616}, {4292,11018}, {4294,12915}, {4298,16201}, {4316,13750}, {4324,5570}, {5435,9844}, {5542,20790}, {5584,10085}, {5731,9957}, {5787,6916}, {6259,6865}, {6684,9947}, {6763,7964}, {6827,22792}, {6914,13624}, {6948,18481}, {7987,12688}, {8703,31837}, {9579,17603}, {9589,17609}, {9708,10864}, {9848,13462}, {10304,12528}, {10306,12631}, {10624,16215}, {10857,11108}, {10884,24929}, {11495,12516}, {12128,31393}, {12511,15733}, {13373,28146}, {13374,28150}, {13384,17634}, {13754,31816}, {16616,28172}, {17668,30478}

X(31805) = midpoint of X(i) and X(j) for these {i,j}: {20, 942}, {550, 13369}, {944, 31798}, {4297, 9943}, {12675, 31730}, {18481, 31788}
X(31805) = reflection of X(i) in X(j) for these (i,j): (5044, 3), (5806, 9940), (9947, 6684)
X(31805) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 84, 31445), (3, 7330, 31658), (5732, 9841, 3)

X(31806) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND INNER-GARCIA

Barycentrics a*((b+c)*a^5-(b^2+4*b*c+c^2)*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(2*b^4+2*c^4+(3*b^2-2*b*c+3*c^2)*b*c)*a^2+(b^3+c^3)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(31806) = X(4)-3*X(5692) = 2*X(5)-3*X(10176) = 3*X(72)-X(14872) = 4*X(140)-3*X(5883) = 3*X(376)-X(15071) = 3*X(392)-2*X(13464) = 3*X(549)-2*X(5885) = 3*X(551)-4*X(31838) = 5*X(631)-3*X(5902) = 2*X(942)-3*X(10165) = 6*X(3740)-5*X(31399) = 4*X(5044)-3*X(10175) = 3*X(5657)-X(5903) = 3*X(5692)-2*X(20117) = 3*X(5694)-X(31828) = X(5697)+3*X(15104) = 2*X(7686)-3*X(10175) = X(12245)-3*X(15104) = 3*X(14110)+X(14872) = 3*X(31803)-2*X(31828)

X(31806) lies on these lines: {1,201}, {3,758}, {4,5692}, {5,10}, {8,6840}, {20,3648}, {30,5694}, {40,78}, {63,5450}, {65,5432}, {72,515}, {79,6951}, {104,6763}, {140,5883}, {191,5538}, {214,26286}, {255,11700}, {329,6256}, {355,3678}, {376,15071}, {392,13464}, {404,5535}, {497,5697}, {498,3485}, {511,31825}, {516,5887}, {518,5882}, {519,28459}, {549,5885}, {550,2771}, {551,31838}, {631,5442}, {912,4067}, {942,10165}, {944,5904}, {962,6894}, {993,26921}, {997,5709}, {1001,1482}, {1125,24474}, {1158,6282}, {1385,3874}, {1735,22072}, {1765,3958}, {1844,7531}, {1858,4304}, {1938,15584}, {2095,25524}, {2392,10625}, {2646,18389}, {2689,2744}, {2772,10575}, {2778,2883}, {2779,5562}, {2801,4127}, {2802,19914}, {3057,28234}, {3072,30115}, {3336,6940}, {3428,5730}, {3486,18397}, {3523,15016}, {3526,3833}, {3555,13607}, {3576,3868}, {3579,14988}, {3587,12520}, {3612,21165}, {3647,6914}, {3654,25413}, {3681,5881}, {3754,26446}, {3876,5587}, {3877,5047}, {3881,10246}, {3890,16200}, {3892,15178}, {3894,30389}, {3898,24680}, {3899,7991}, {3901,7987}, {3927,12114}, {3940,11500}, {3984,17857}, {3988,18525}, {4005,18908}, {4015,5790}, {4084,10164}, {4511,11012}, {4867,21740}, {4880,26877}, {4999,5771}, {5223,12650}, {5251,26878}, {5289,22770}, {5446,15049}, {5506,11531}, {5552,6960}, {5572,31792}, {5603,5659}, {5755,25078}, {5758,26332}, {5761,10198}, {5777,31673}, {5794,5812}, {5894,6001}, {6244,18237}, {6831,21677}, {6864,16134}, {6883,30143}, {6909,11684}, {6917,16125}, {6922,10265}, {7280,11570}, {7411,16132}, {7508,22937}, {7957,12672}, {8127,18409}, {8128,18408}, {8666,11715}, {9614,12758}, {10157,16616}, {10609,30264}, {11249,30144}, {11499,12702}, {12119,12532}, {12359,13605}, {12559,18443}, {12688,28150}, {12704,19861}, {12736,24914}, {13624,24475}, {13754,31817}, {15064,18480}, {15558,30323}, {17768,31775}, {18412,21168}, {18589,18638}, {22076,27687}, {30329,31658}

X(31806) = midpoint of X(i) and X(j) for these {i,j}: {20, 5693}, {40, 3869}, {72, 14110}, {944, 5904}, {4067, 4297}, {5697, 12245}, {7957, 12672}, {12119, 12532}
X(31806) = reflection of X(i) in X(j) for these (i,j): (4, 20117), (10, 31837), (65, 6684), (355, 3678), (946, 960), (1482, 3884), (3555, 13607), (3868, 12005), (3874, 1385), (5882, 31786), (5884, 3), (7686, 5044), (18480, 31835), (24474, 1125), (24475, 13624), (30329, 31658), (31673, 5777), (31730, 31793), (31803, 5694)
X(31806) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5692, 20117), (5044, 7686, 10175)

X(31807) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND 1st ANTI-CIRCUMPERP

Barycentrics (S^2-SB*SC)*(S^2+2*R^2*(8*R^2-2*SA-5*SW)+SA^2-2*SB*SC+2*SW^2) : :
X(31807) = X(5889)-3*X(11402)

X(31807) lies on these lines: {2,6746}, {3,54}, {4,18438}, {52,6676}, {143,3549}, {155,15818}, {511,6823}, {1199,19129}, {1216,1368}, {2392,31814}, {2393,5907}, {3522,13148}, {3546,13416}, {3547,6243}, {3564,4173}, {3917,16196}, {5876,12605}, {5946,7542}, {6639,15026}, {6643,23039}, {7395,11255}, {7999,30771}, {8550,11574}, {10263,15760}, {10625,31829}, {11479,12167}, {11585,15067}, {11591,18531}, {11898,12271}, {12164,19459}, {12429,15073}, {13391,31815}, {13754,31804}, {15060,18404}, {16659,18563}

X(31807) = midpoint of X(11412) and X(12160)
X(31807) = reflection of X(31810) in X(12161)
X(31807) = {X(5562), X(9967)}-harmonic conjugate of X(12362)

X(31808) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND ASCELLA

Barycentrics
a*((b+c)*a^11-(b^2-4*b*c+c^2)*a^10-(b+c)*(3*b^2-2*b*c+3*c^2)*a^9+(3*b^4+3*c^4-2*(13*b^2-b*c+13*c^2)*b*c)*a^8+2*(b^4-c^4)*(b-c)*a^7-2*(b^6+c^6-(26*b^4+26*c^4+(b^2+16*b*c+c^2)*b*c)*b*c)*a^6+2*(b+c)*(b^2+c^2)*(b^4+c^4)*a^5-2*(b^2+c^2)*(b^6+c^6+(20*b^4+20*c^4+(b^2-28*b*c+c^2)*b*c)*b*c)*a^4-(b^4-c^4)*(b-c)*(3*b^4+3*c^4+2*(b^2+b*c+c^2)*b*c)*a^3+(b^2-c^2)^2*(b+c)^4*(3*b^2-4*b*c+3*c^2)*a^2+(b^4-c^4)^2*(b^2-c^2)*(b-c)*a-(b^4-c^4)*(b^2-c^2)^3*(b-c)^2) : :
X(31808) = 3*X(11402)-X(31793)

X(31808) lies on these lines: {517,12161}, {942,12160}, {971,31802}, {1154,9940}, {3564,5806}, {11402,31793}

X(31808) = midpoint of X(942) and X(12160)

X(31809) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND CIRCUMMEDIAL

Barycentrics 18*S^6-6*(36*R^4+(9*SA-28*SW)*R^2-3*SA^2+3*SB*SC+5*SW^2)*S^4+(216*R^4*SA*(SB+SC)+R^2*SW*(135*SA^2-129*SA*SW-2*SW^2)-SW^2*SA*(18*SA-17*SW))*S^2+(8*R^2-3*SW)*SB*SC*SW^3 : :

X(31809) lies on these lines: {1154,31736}, {3564,14866}, {3849,31802}, {12160,12505}

X(31809) = midpoint of X(12160) and X(12505)

X(31810) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND CIRCUMORTHIC

Barycentrics (SB+SC)*((SB+SC)*S^2-(2*R^2*(4*R^2-5*SW)+SA^2-SB*SC+2*SW^2)*SA) : :
X(31810) = 3*X(11402)-X(11412)

X(31810) lies on these lines: {3,54}, {52,1843}, {141,389}, {143,6515}, {193,6243}, {511,31804}, {568,7401}, {1595,13754}, {3580,15026}, {5663,31815}, {5876,7403}, {5946,7405}, {7404,18436}, {7553,10263}, {11591,14786}

X(31810) = midpoint of X(5889) and X(12160)
X(31810) = reflection of X(31807) in X(12161)

X(31811) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND 1st CIRCUMPERP

Barycentrics a*(2*a^6+3*(b+c)*a^5-3*(b^2+c^2)*a^4-4*(b+c)*(b^2+c^2)*a^3+2*(b^4+c^4)*a^2+(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2)) : :
X(31811) = X(1)-3*X(11402)

X(31811) lies on these lines: {1,11402}, {10,3564}, {40,12160}, {515,31804}, {516,31802}, {517,12161}, {1154,3579}, {1829,16473}, {2393,4663}, {3751,9798}, {6001,31814}, {7713,12167}, {9928,19458}, {28146,31815}

X(31811) = midpoint of X(i) and X(j) for these {i,j}: {40, 12160}, {3751, 19459}, {9928, 19458}
X(31811) = reflection of X(31812) in X(12161)

X(31812) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND 2nd CIRCUMPERP

Barycentrics
a*(2*a^9-(b+c)*a^8-2*(4*b^2-b*c+4*c^2)*a^7+2*(b^3+c^3)*a^6+2*(b+c)*(b^2+c^2)*a^4*b*c+2*(6*b^4+6*c^4-(b^2-4*b*c+c^2)*b*c)*a^5-2*(b^2+c^2)*(4*b^2-7*b*c+4*c^2)*(b+c)^2*a^3-2*(b^3-c^3)*(b^4-c^4)*a^2+2*(b^2-c^2)^2*(b+c)*(b^3+c^3)*a+(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
X(31812) = X(40)-3*X(11402)

X(31812) lies on these lines: {1,12160}, {40,11402}, {515,31802}, {516,31804}, {517,12161}, {912,31814}, {946,3564}, {1154,1385}, {9911,19459}, {11365,12166}, {28160,31815}

X(31812) = midpoint of X(1) and X(12160)
X(31812) = reflection of X(31811) in X(12161)

X(31813) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND CIRCUMSYMMEDIAL

Barycentrics (SB+SC)*((36*R^2-9*SW)*S^4+(432*SA*R^4-6*(18*SA^2+39*SA*SW-2*SW^2)*R^2+(3*SA+2*SW)*(15*SA-4*SW)*SW)*S^2-2*(8*R^2-SA-3*SW)*SA*SW^3) : :

X(31813) lies on these lines: {1154,31727}, {3564,14867}, {9830,31802}

X(31814) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND INNER-GARCIA

Barycentrics
a*((b+c)*a^11-(b+c)^2*a^10-3*(b^3+c^3)*a^9+(3*b^4+3*c^4+8*(b^2+c^2)*b*c)*a^8+2*(b+c)*(b^4+c^4-3*(b^2-b*c+c^2)*b*c)*a^7-2*(b^6+c^6+2*(b^2+b*c+c^2)*(3*b^2-4*b*c+3*c^2)*b*c)*a^6+2*(b^6+c^6)*(b+c)*a^5-2*(b^2+c^2)*(b^4+c^4-(b+2*c)*(2*b+c)*b*c)*(b-c)^2*a^4-3*(b^8-c^8)*(b-c)*a^3+(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*(b^4+c^4)*a^2+(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*a-(b^4-c^4)^2*(b^2-c^2)^2) : :
X(31814) = 3*X(11402)-X(15071)

X(31814) lies on these lines: {517,31815}, {758,31802}, {912,31812}, {1154,5694}, {2392,31807}, {2771,12161}, {3564,31803}, {5693,12160}, {6001,31811}, {11402,15071}

X(31814) = midpoint of X(5693) and X(12160)

X(31815) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ASCELLA AND X3-ABC REFLECTIONS

Barycentrics a^10+(b^2+c^2)*a^8-4*(2*b^4+b^2*c^2+2*c^4)*a^6+2*(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^4-(b^4+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(31815) = X(1657)-3*X(11402) = 3*X(12161)-2*X(31804) = 3*X(31802)-X(31804)

X(31815) lies on these lines: {3,14389}, {4,93}, {5,17810}, {30,1181}, {49,31304}, {52,18569}, {143,18531}, {155,11819}, {343,7564}, {382,12160}, {389,14791}, {394,31830}, {517,31814}, {550,11425}, {568,18952}, {632,9815}, {1192,15122}, {1351,12370}, {1370,13630}, {1493,18925}, {1657,11402}, {2393,22802}, {3060,18404}, {3546,15741}, {3564,3627}, {3575,16266}, {4846,15704}, {5562,11818}, {5663,31810}, {5925,18431}, {5946,6643}, {6101,18420}, {6102,14790}, {6756,15068}, {6815,10627}, {6816,10095}, {6997,14128}, {7401,15067}, {7502,20424}, {7528,11591}, {7530,22660}, {7540,11441}, {7544,23039}, {7574,18912}, {7706,15644}, {9833,17824}, {10610,11427}, {11004,12254}, {13391,31807}, {13419,15083}, {13490,17814}, {14449,18377}, {14531,18474}, {18568,21969}, {28146,31811}, {28160,31812}

X(31815) = midpoint of X(382) and X(12160)
X(31815) = reflection of X(12161) in X(31802)

X(31816) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND ASCELLA

Barycentrics
a^2*(4*(b^2+c^2)*a^6-(b+c)*b*c*a^5-(12*b^4+12*c^4-(b^2-12*b*c+c^2)*b*c)*a^4+2*(b^3+c^3)*b*c*a^3+2*(6*b^6+6*c^6-(b^4+c^4-3*(b^2+c^2)*b*c)*b*c)*a^2-(b^4-c^4)*(b-c)*b*c*a-(b^2-c^2)^2*(4*b^4+4*c^4-(b^2-6*b*c+c^2)*b*c)) : :
X(31816) = 2*X(389)-3*X(10156) = 3*X(2979)-X(31793) = X(5777)-3*X(23039) = X(5889)-3*X(11227) = 3*X(10157)-5*X(11444)

X(31816) lies on these lines: {389,10156}, {511,5806}, {517,6101}, {942,11412}, {971,5562}, {1154,9940}, {1216,5044}, {2392,31821}, {2979,31793}, {5777,23039}, {5889,11227}, {9037,9947}, {10157,11444}, {13391,31822}, {13754,31805}

X(31816) = midpoint of X(942) and X(11412)
X(31816) = reflection of X(i) in X(j) for these (i,j): (5044, 1216), (9947, 31752), (31819, 9940)

X(31817) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND INNER-GARCIA

Barycentrics
a^2*((b^2+c^2)*a^6+(b+c)*b*c*a^5-(b^2+b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*b*c*a^3+(b^2+b*c+c^2)*(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^2+(b^3+c^3)*(b-c)^2*b*c*a-(b^2-c^2)^2*(b^4+c^4+(b^2+b*c+c^2)*b*c)) : :
X(31817) = 2*X(52)-3*X(15049) = 2*X(389)-3*X(10176) = 3*X(2979)-X(15071) = 3*X(5692)-X(5889) = 3*X(5883)-4*X(11793) = 2*X(5885)-3*X(15067) = 3*X(5902)-5*X(11444) = 7*X(7999)-5*X(15016) = 3*X(15049)-4*X(20117)

X(31817) lies on these lines: {20,2772}, {52,15049}, {65,31752}, {389,10176}, {511,31803}, {517,5876}, {674,4301}, {758,5562}, {912,23156}, {916,4297}, {960,31732}, {1154,5694}, {1216,5884}, {2392,5693}, {2771,6101}, {2779,18436}, {2979,15071}, {5692,5889}, {5883,11793}, {5885,15067}, {5902,11444}, {6001,31737}, {7999,15016}, {9047,9856}, {13391,31828}, {13754,31806}, {24474,31751}, {31728,31837}

X(31817) = midpoint of X(5693) and X(11412)
X(31817) = reflection of X(i) in X(j) for these (i,j): (52, 20117), (65, 31752), (5884, 1216), (23156, 31738), (24474, 31751), (31728, 31837), (31732, 960), (31825, 5694)
X(31817) = {X(52), X(20117)}-harmonic conjugate of X(15049)

X(31818) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ASCELLA AND CIRCUMMEDIAL

Barycentrics
a*(4*(b+c)*a^11-4*(b^2+c^2)*a^10-2*(b+c)*(7*b^2-4*b*c+7*c^2)*a^9+2*(7*b^4+7*c^4-2*(b^2-3*b*c+c^2)*b*c)*a^8+(b+c)*(10*b^4+10*c^4-(20*b^2-29*b*c+20*c^2)*b*c)*a^7-(10*b^6+10*c^6-(40*b^4+40*c^4+(b^2-92*b*c+c^2)*b*c)*b*c)*a^6+2*(b+c)*(5*b^6+5*c^6-3*(b^2-3*b*c+c^2)*b^2*c^2)*a^5-2*(5*b^8+5*c^8+(6*b^6+6*c^6+(2*b^4+2*c^4-3*(15*b^2-4*b*c+15*c^2)*b*c)*b*c)*b*c)*a^4-(b+c)*(14*b^6+14*c^6-(20*b^4+20*c^4+(3*b^2-14*b*c+3*c^2)*b*c)*b*c)*(b^2+c^2)*a^3+(14*b^10+14*c^10-(40*b^8+40*c^8+(15*b^6+15*c^6-(98*b^4+98*c^4-(7*b^2+36*b*c+7*c^2)*b*c)*b*c)*b*c)*b*c)*a^2+2*(b^4-c^4)*(b^2+c^2)*(b-c)*(b^2-2*c^2)*(2*b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(-4*b^6-4*c^6+2*(8*b^4+8*c^4+(3*b^2-26*b*c+3*c^2)*b*c)*b*c)) : :
X(31818) = 3*X(9829)-X(31793) = 3*X(10156)-2*X(31762)

X(31818) lies on these lines: {517,31744}, {942,12505}, {971,14866}, {3849,5806}, {5044,31606}, {9829,31793}, {10156,31762}

X(31818) = midpoint of X(942) and X(12505)
X(31818) = reflection of X(5044) in X(31606)

X(31819) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ASCELLA AND CIRCUMORTHIC

Barycentrics
a^2*(4*(b^2+c^2)*a^6+(b+c)*b*c*a^5-(4*b^2-5*b*c+4*c^2)*(3*b^2+4*b*c+3*c^2)*a^4-2*(b^3+c^3)*b*c*a^3+2*(6*b^6+6*c^6+(b^4+c^4-3*(b^2+c^2)*b*c)*b*c)*a^2+(b^4-c^4)*(b-c)*b*c*a-(b^2-c^2)^2*(4*b^4+4*c^4+(b+c)^2*b*c)) : :
X(31819) = 3*X(568)-X(5777) = 2*X(1216)-3*X(10156) = 5*X(3567)-3*X(10157) = 3*X(5890)-X(31793) = 3*X(5946)-X(31836) = 3*X(11227)-X(11412)

X(31819) lies on these lines: {52,971}, {389,5044}, {511,31805}, {517,6102}, {568,5777}, {916,16625}, {942,5889}, {1154,9940}, {1216,10156}, {3567,10157}, {5663,31822}, {5806,13754}, {5890,31793}, {5946,31836}, {9947,31760}, {11227,11412}

X(31819) = midpoint of X(942) and X(5889)
X(31819) = reflection of X(i) in X(j) for these (i,j): (5044, 389), (9947, 31760), (31816, 9940)

X(31820) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ASCELLA AND CIRCUMSYMMEDIAL

Barycentrics
a*(4*(b+c)*a^11-4*(b^2-4*b*c+c^2)*a^10-2*(b+c)*(7*b^2-4*b*c+7*c^2)*a^9+2*(7*b^4+7*c^4-6*(6*b^2-b*c+6*c^2)*b*c)*a^8+(b+c)*(10*b^4+10*c^4-(20*b^2-29*b*c+20*c^2)*b*c)*a^7-(10*b^6+10*c^6-(140*b^4+140*c^4+(b^2+20*b*c+c^2)*b*c)*b*c)*a^6+2*(b+c)*(5*b^6+5*c^6-3*(b^2-3*b*c+c^2)*b^2*c^2)*a^5-2*(5*b^8+5*c^8+(40*b^6+40*c^6+(2*b^4+2*c^4-3*(21*b^2-4*b*c+21*c^2)*b*c)*b*c)*b*c)*a^4-(b+c)*(14*b^6+14*c^6-(20*b^4+20*c^4+(3*b^2-14*b*c+3*c^2)*b*c)*b*c)*(b^2+c^2)*a^3+(14*b^10+14*c^10-(12*b^8+12*c^8+(15*b^6+15*c^6-(42*b^4+42*c^4-(7*b^2+108*b*c+7*c^2)*b*c)*b*c)*b*c)*b*c)*a^2+2*(b^4-c^4)*(b^2+c^2)*(b-c)*(b^2-2*c^2)*(2*b^2-c^2)*a-2*(b^4-c^4)*(b^2-c^2)*(b-c)^2*(b^2-2*c^2)*(2*b^2-c^2)) : :
X(31820) = 3*X(353)-X(31793)

X(31820) lies on these lines: {353,31793}, {517,31727}, {971,14867}, {5044,31608}, {5806,9830}

X(31820) = reflection of X(5044) in X(31608)

X(31821) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ASCELLA AND INNER-GARCIA

Barycentrics
a*(3*(b+c)*a^5-(3*b^2+4*b*c+3*c^2)*a^4-2*(b+c)*(3*b^2-5*b*c+3*c^2)*a^3+2*(3*b^4+3*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(3*b^2-4*b*c+3*c^2)*a-(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2) : :
X(31821) = 3*X(65)-7*X(7989) = X(65)-3*X(10157) = 3*X(72)+X(962) = 3*X(210)-X(31798) = X(355)-3*X(5777) = X(355)+3*X(5887) = 2*X(355)-3*X(9947) = 3*X(392)+X(12528) = 3*X(942)-5*X(8227) = 3*X(960)-X(4297) = X(962)-3*X(9856) = X(3869)+3*X(5927) = X(4297)+3*X(31803) = 3*X(5045)-4*X(5901) = 3*X(5693)+5*X(8227) = 3*X(5694)+X(22793) = 3*X(5806)-4*X(12571) = 2*X(5887)+X(9947) = 7*X(7989)-9*X(10157) = X(12245)+3*X(12672)

X(31821) lies on these lines: {1,5779}, {3,7992}, {4,8}, {5,3671}, {65,7989}, {210,31798}, {392,11106}, {758,5806}, {912,5045}, {942,5693}, {952,12575}, {960,971}, {1385,7330}, {1483,31792}, {1538,6734}, {1699,3962}, {1858,11018}, {2392,31816}, {2771,5972}, {3091,4018}, {3340,30326}, {3359,5780}, {3579,5720}, {3652,21161}, {3940,12705}, {3988,28228}, {4005,7991}, {4298,5843}, {5044,6001}, {5223,8158}, {5691,31165}, {5692,12688}, {5763,21628}, {5794,22792}, {5836,15064}, {5881,8275}, {5886,11036}, {6261,31445}, {7956,24391}, {7958,11551}, {7971,9708}, {9848,9957}, {9943,10176}, {10884,13624}, {11227,15071}, {12432,14988}, {12520,31658}, {12526,19541}, {13257,24987}, {15178,22758}, {31797,31835}

X(31821) = midpoint of X(i) and X(j) for these {i,j}: {72, 9856}, {942, 5693}, {960, 31803}, {5777, 5887}, {9957, 14872}, {12688, 31793}
X(31821) = reflection of X(i) in X(j) for these (i,j): (5044, 20117), (9947, 5777), (31787, 5044), (31794, 5)
X(31821) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5692, 12688, 31793), (15071, 25917, 11227)

X(31822) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ASCELLA AND X3-ABC REFLECTIONS

Barycentrics a*((b+c)*a^5-(b^2+12*b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^3+c^3)*(b+c)*a^2+(b^4-c^4)*(b-c)*a-(b^2-10*b*c+c^2)*(b^2-c^2)^2) : :
X(31822) = 5*X(4)-X(72) = 3*X(4)-X(5777) = 7*X(4)-3*X(5927) = 5*X(40)-9*X(4731) = 3*X(72)-5*X(5777) = 7*X(72)-15*X(5927) = 3*X(381)-X(31793) = 2*X(548)-3*X(10156) = 5*X(946)-3*X(10179) = X(1657)-3*X(11227) = 3*X(1699)-X(31786) = X(3529)-5*X(5439) = 3*X(3627)+X(24475) = 3*X(3830)+X(24474) = 5*X(3843)-3*X(10157) = 3*X(3845)-X(31837) = 2*X(3918)-5*X(16616) = 7*X(5777)-9*X(5927) = 3*X(5806)-X(31805) = 3*X(9940)-2*X(31805)

X(31822) lies on these lines: {4,8}, {30,5806}, {40,4731}, {226,31792}, {381,31793}, {382,942}, {405,31663}, {516,3918}, {546,5044}, {548,10156}, {912,3853}, {946,10179}, {950,5045}, {971,3627}, {1482,1750}, {1490,24680}, {1657,11227}, {1699,31786}, {3057,9656}, {3529,5439}, {3579,6913}, {3660,10483}, {3812,28150}, {3830,24474}, {3843,10157}, {3845,31837}, {3919,7686}, {3922,31788}, {3956,19925}, {4757,6001}, {5073,10202}, {5436,17502}, {5663,31819}, {6260,22791}, {6846,11231}, {6907,9955}, {6908,11230}, {6917,18482}, {7580,13624}, {7957,18492}, {9612,9957}, {9655,12915}, {11113,28202}, {12572,28174}, {13373,28168}, {13374,28164}, {13391,31816}, {13600,31162}, {14893,31835}, {16201,31795}, {28146,31787}, {28198,31797}

X(31822) = midpoint of X(382) and X(942)
X(31822) = reflection of X(i) in X(j) for these (i,j): (5044, 546), (9940, 5806)

X(31823) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND INNER-GARCIA

Barycentrics
a*(4*(b+c)*a^11-4*(b^2+b*c+c^2)*a^10-2*(b+c)*(7*b^2-6*b*c+7*c^2)*a^9+(14*b^4+14*c^4+(9*b^2+4*b*c+9*c^2)*b*c)*a^8+(b+c)*(10*b^4+10*c^4-(30*b^2-37*b*c+30*c^2)*b*c)*a^7-(10*b^4+10*c^4-(15*b^2-7*b*c+15*c^2)*b*c)*(b+c)^2*a^6+(b+c)*(10*b^6+10*c^6-9*(2*b^2-3*b*c+2*c^2)*b^2*c^2)*a^5-(10*b^8+10*c^8+(7*b^6+7*c^6-2*(4*b^4+4*c^4-9*(b^2+b*c+c^2)*b*c)*b*c)*b*c)*a^4-(b+c)*(14*b^8+14*c^8-(30*b^6+30*c^6-(23*b^4+23*c^4-9*(b^2+c^2)*b*c)*b*c)*b*c)*a^3+(14*b^10+14*c^10+(9*b^8+9*c^8-(23*b^6+23*c^6+(21*b^4+21*c^4-5*(b^2+c^2)*b*c)*b*c)*b*c)*b*c)*a^2+2*(b^3+c^3)*(b-c)^2*(b^2-2*c^2)*(b^2+c^2)*(2*b^2-c^2)*a-2*(b^4-c^4)*(b^2-c^2)*(b+c)^2*(2*b^4+2*c^4-(3*b^2-b*c+3*c^2)*b*c)) : :
X(31823) = 3*X(9829)-X(15071) = 3*X(10176)-2*X(31762) = 3*X(15049)-2*X(31763)

X(31823) lies on these lines: {517,31824}, {758,14866}, {912,31747}, {2392,31736}, {2771,31744}, {3849,31803}, {5693,12505}, {5884,31606}, {6001,31746}, {9829,15071}, {10176,31762}, {12506,20117}, {15049,31763}

X(31823) = midpoint of X(5693) and X(12505)
X(31823) = reflection of X(i) in X(j) for these (i,j): (5884, 31606), (12506, 20117)

X(31824) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMMEDIAL AND X3-ABC REFLECTIONS

Barycentrics 6*(27*R^2-8*SW)*S^4+(189*R^2*SA*(SA-SW)-3*SA*SW*(-17*SW+18*SA)+2*SW^3)*S^2-7*SB*SC*SW^3 : :
X(31824) = 7*X(5)-6*X(10173) = 3*X(5)-2*X(31762) = 2*X(548)-3*X(10163) = X(1657)-3*X(9829) = 5*X(3843)-3*X(6032) = 3*X(3845)-2*X(31749) = 4*X(3850)-3*X(10162) = 9*X(10173)-7*X(31762) = 3*X(14866)-X(31729) = 3*X(15060)-2*X(31753) = 2*X(31729)-3*X(31744) = 4*X(31749)-3*X(31840)

X(31824) lies on these lines: {5,10173}, {30,14866}, {382,12505}, {517,31823}, {546,12506}, {548,10163}, {550,31606}, {1657,9829}, {3627,3849}, {3830,14262}, {3843,6032}, {3845,31749}, {3850,10162}, {5663,31745}, {13391,31736}, {15060,31753}, {28146,31746}, {28160,31747}

X(31824) = midpoint of X(382) and X(12505)
X(31824) = reflection of X(i) in X(j) for these (i,j): (550, 31606), (12506, 546), (31744, 14866), (31840, 3845)

X(31825) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMORTHIC AND INNER-GARCIA

Barycentrics
a^2*((b^2+c^2)*a^6-(b+c)*b*c*a^5-(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^4+(b+c)*(2*b^2-3*b*c+2*c^2)*b*c*a^3+(3*b^4+3*c^4+(4*b^2+3*b*c+4*c^2)*b*c)*(b-c)^2*a^2-(b^3+c^3)*(b-c)^2*b*c*a-(b^2-c^2)^2*(b^4+c^4-(b^2-b*c+c^2)*b*c)) : :
X(31825) = 2*X(5)-3*X(15049) = 6*X(375)-5*X(31399) = 2*X(1216)-3*X(10176) = 5*X(3567)-3*X(5902) = 4*X(5462)-3*X(5883) = 3*X(5692)-X(11412) = X(5693)-3*X(30438) = 2*X(5885)-3*X(5946) = X(5889)+3*X(30438) = 3*X(5890)-X(15071) = 3*X(10165)-2*X(11573) = 3*X(10246)-2*X(23157) = 5*X(15016)-7*X(15043)

X(31825) lies on these lines: {3,2392}, {4,2779}, {5,15049}, {52,758}, {65,31760}, {375,31399}, {389,5884}, {511,31806}, {517,10263}, {568,2842}, {912,31732}, {960,31738}, {1154,5694}, {1216,10176}, {1385,23156}, {2771,6102}, {3567,5902}, {5450,26892}, {5462,5883}, {5562,20117}, {5663,31828}, {5692,11412}, {5693,5889}, {5882,8679}, {5885,5946}, {5890,15071}, {6001,31728}, {9037,31786}, {10165,11573}, {10246,23157}, {12005,23154}, {13754,31803}, {15016,15043}, {24474,31757}, {31737,31837}

X(31825) = midpoint of X(5693) and X(5889)
X(31825) = reflection of X(i) in X(j) for these (i,j): (65, 31760), (5562, 20117), (5884, 389), (23154, 12005), (23156, 1385), (24474, 31757), (31737, 31837), (31738, 960), (31817, 5694)
X(31825) = {X(5889), X(30438)}-harmonic conjugate of X(5693)

X(31826) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND INNER-GARCIA

Barycentrics
a*(4*(b+c)*a^11-4*(b+c)^2*a^10-2*(b+c)*(7*b^2-6*b*c+7*c^2)*a^9+2*(7*b^4+7*c^4+(13*b^2+2*b*c+13*c^2)*b*c)*a^8+(b+c)*(10*b^4+10*c^4-(30*b^2-37*b*c+30*c^2)*b*c)*a^7-(10*b^6+10*c^6+(2*b^2-3*b*c+2*c^2)*(15*b^2+16*b*c+15*c^2)*b*c)*a^6+(b+c)*(10*b^6+10*c^6-9*(2*b^2-3*b*c+2*c^2)*b^2*c^2)*a^5-(10*b^8+10*c^8-(10*b^6+10*c^6+(8*b^4+8*c^4-9*(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*a^4-(b+c)*(14*b^8+14*c^8-(30*b^6+30*c^6-(23*b^4+23*c^4-9*(b^2+c^2)*b*c)*b*c)*b*c)*a^3+(14*b^8+14*c^8+(30*b^6+30*c^6+(23*b^4+23*c^4+9*(b^2+c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^2+2*(b^3+c^3)*(b-c)^2*(b^2+c^2)*(b^2-2*c^2)*(2*b^2-c^2)*a-2*(b^4-c^4)^2*(2*b^2-c^2)*(b^2-2*c^2)) : :
X(31826) = 3*X(353)-X(15071)

X(31826) lies on these lines: {353,15071}, {517,31827}, {758,14867}, {912,31741}, {2392,31739}, {2771,31727}, {5884,31608}, {6001,31740}, {9830,31803}

X(31826) = reflection of X(5884) in X(31608)

X(31827) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND X3-ABC REFLECTIONS

Barycentrics 9*(18*R^2-5*SW+SA)*S^4-3*(63*(SB+SC)*R^2+SW*(15*SA-16*SW))*SA*S^2-7*SB*SC*SW^3 : :
X(31827) = 3*X(353)-X(1657) = 2*X(548)-3*X(10166) = 6*X(10160)-5*X(15712) = 3*X(14867)-X(31731) = 3*X(31727)-2*X(31731)

X(31827) lies on these lines: {30,14867}, {353,1657}, {517,31826}, {548,10166}, {550,31608}, {3627,9830}, {5663,31733}, {10160,15712}, {13391,31739}, {28146,31740}, {28160,31741}

X(31827) = reflection of X(i) in X(j) for these (i,j): (550, 31608), (31727, 14867)

X(31828) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: INNER-GARCIA AND X3-ABC REFLECTIONS

Barycentrics a*((b+c)*a^5-(b-c)^2*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+2*(b^4-b^2*c^2+c^4)*a^2+(b^3+c^3)*(b-c)^2*a-(b^2-c^2)^2*(b+c)^2) : :
X(31828) = 7*X(5)-6*X(3833) = 3*X(381)-2*X(5885) = 3*X(381)-X(15071) = 2*X(548)-3*X(10176) = X(1657)-3*X(5692) = 3*X(1699)-2*X(6583) = 6*X(3740)-5*X(31447) = 5*X(3843)-3*X(5902) = 4*X(3850)-3*X(5883) = 7*X(3851)-5*X(15016) = 3*X(5587)-2*X(13145) = 3*X(5694)-2*X(31806) = 3*X(5886)-2*X(26201) = 3*X(5927)-2*X(9956) = 2*X(9943)-3*X(11231) = X(9961)-3*X(26446) = 4*X(10107)-7*X(18480) = 3*X(10247)-4*X(26088) = 3*X(12688)+X(14872) = 3*X(31803)-X(31806)

X(31828) lies on these lines: {2,17653}, {3,7701}, {4,2771}, {5,3833}, {30,5694}, {65,18513}, {72,28146}, {381,5885}, {382,517}, {411,3652}, {546,5884}, {548,10176}, {550,20117}, {581,8143}, {758,3627}, {912,22793}, {942,1898}, {971,1001}, {1158,10225}, {1482,7993}, {1657,5692}, {1699,6583}, {1709,26285}, {1864,31794}, {2392,5876}, {2772,6102}, {2801,22791}, {3065,14804}, {3579,5777}, {3740,31447}, {3843,5902}, {3850,5883}, {3851,15016}, {3878,28186}, {5587,13145}, {5658,26487}, {5663,31825}, {5787,12666}, {5886,26201}, {5887,28160}, {5927,6937}, {6001,10107}, {6894,16116}, {6906,16138}, {6909,10308}, {6914,26202}, {6917,16127}, {7489,16132}, {7957,28202}, {8728,16120}, {9856,24680}, {9943,11231}, {9961,26446}, {10247,26088}, {10269,12684}, {10284,12672}, {11230,13369}, {12290,30438}, {12528,12699}, {12680,15178}, {13391,31817}, {13630,15049}, {14110,28168}, {14988,31673}, {15726,31837}, {15934,30290}, {18483,24475}, {19526,31666}, {31730,31835}

X(31828) = midpoint of X(i) and X(j) for these {i,j}: {382, 5693}, {5787, 12666}, {12528, 12699}
X(31828) = reflection of X(i) in X(j) for these (i,j): (550, 20117), (3579, 5777), (5694, 31803), (5884, 546), (10284, 12672), (12680, 15178), (15071, 5885), (24475, 18483), (24680, 9856), (31730, 31835)
X(31828) = {X(381), X(15071)}-harmonic conjugate of X(5885)

X(31829) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ARA AND ANTICOMPLEMENTARY

Barycentrics (-a^2+b^2+c^2)*(2*a^8-(b^2+c^2)*a^6-(3*b^4-14*b^2*c^2+3*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(31829) = 3*X(5892)-X(12897) = 2*X(10116)-3*X(18914) = 5*X(10574)-3*X(11245) = X(13403)-3*X(16836) = X(14516)+3*X(15072)

As a point on the Euler line, X(31829) has Shinagawa coefficients (E-F, -E+3*F)

X(31829) lies on these lines: {2,3}, {64,1352}, {141,5894}, {155,4846}, {185,3564}, {343,1204}, {389,13142}, {511,13568}, {971,31832}, {974,22663}, {1038,6284}, {1040,7354}, {1060,15171}, {1062,18990}, {1105,6530}, {1503,14913}, {1578,6561}, {1579,6560}, {2777,11793}, {2883,9306}, {4292,9944}, {5158,9607}, {5663,31831}, {5878,17814}, {5892,12897}, {5895,17811}, {5907,15311}, {6225,14826}, {6391,6776}, {6696,21243}, {7745,22401}, {7750,14615}, {8548,19467}, {9729,12241}, {10116,18914}, {10574,11245}, {10575,12134}, {10625,31807}, {11424,18583}, {11574,29181}, {11745,13598}, {12118,19458}, {12233,13346}, {12324,18440}, {12429,18909}, {13292,13630}, {13403,16836}, {14389,18466}, {14516,15072}, {15048,15075}, {15172,18447}, {15583,19126}, {16270,17702}, {18907,23115}, {19460,22647}, {22581,22968}

X(31829) = midpoint of X(i) and X(j) for these {i,j}: {20, 3575}, {1657, 7553}, {10575, 12134}, {11819, 15704}
X(31829) = reflection of X(i) in X(j) for these (i,j): (4, 9825), (6756, 31833), (12241, 9729), (12362, 3), (13142, 389), (13292, 13630), (13488, 5), (13598, 11745)
X(31829) = complement of X(1885)
X(31829) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5073, 18536), (1593, 6815, 5), (3146, 7398, 4)

X(31830) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ARA AND EHRMANN-MID

Barycentrics 2*a^10-3*(b^2+c^2)*a^8-2*(b^2-c^2)^2*a^6+2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+2*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(31830) = 3*X(51)-X(12370) = 3*X(389)-X(10116) = 3*X(568)+X(14516) = 9*X(5640)-X(12289) = 3*X(5946)-X(6146) = 7*X(9781)+X(12278) = 3*X(13363)-X(13470)

As a point on the Euler line, X(31830) has Shinagawa coefficients (E-4*F, 5*E+12*F)

X(31830) lies on these lines: {2,3}, {51,12370}, {143,11745}, {156,12233}, {389,10116}, {394,31815}, {539,16625}, {568,14516}, {973,12236}, {1503,13630}, {2965,16310}, {3580,6288}, {3818,7689}, {5462,18400}, {5640,12289}, {5663,13568}, {5946,6146}, {6102,12134}, {6329,13566}, {6759,7706}, {9781,12278}, {10095,12241}, {10110,17702}, {12140,14708}, {12293,17810}, {13363,13470}, {13491,16655}, {15012,18128}, {17704,17712}, {18488,21663}

X(31830) = midpoint of X(i) and X(j) for these {i,j}: {3, 11819}, {5, 3575}, {550, 7553}, {6102, 12134}, {6756, 31833}, {12140, 14708}, {13491, 16655}
X(31830) = reflection of X(i) in X(j) for these (i,j): (140, 9825), (143, 11745), (3850, 13163), (12241, 10095), (12362, 3628), (13292, 16881), (13488, 3861), (17712, 17704), (18128, 15012)
X(31830) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6644, 13371), (4, 7506, 5), (5, 3627, 18404)

X(31831) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ARA AND EHRMANN-SIDE

Barycentrics 2*a^10-7*(b^2+c^2)*a^8+2*(5*b^4+4*b^2*c^2+5*c^4)*a^6-4*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^4+4*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(31831) = 3*X(51)-4*X(23411) = 2*X(389)-3*X(10127) = 3*X(428)-X(6243) = 5*X(1656)-3*X(11245) = X(1885)-3*X(18435) = 3*X(2979)+X(16659) = 3*X(5891)-X(6146) = 3*X(5907)-X(13403) = X(10116)-3*X(10170) = 6*X(10128)-5*X(15026) = 9*X(11459)-X(12289) = 3*X(11459)-X(12605) = 3*X(11459)+X(14516) = 3*X(11591)-X(13470) = 3*X(12022)-7*X(15056) = X(12289)-3*X(12605) = X(12289)+3*X(14516) = 3*X(12362)-2*X(13470) = X(12370)-3*X(15060) = X(12421)-3*X(14852)

X(31831) lies on these lines: {5,6}, {30,5562}, {51,23411}, {64,550}, {69,7387}, {110,7542}, {140,184}, {156,6676}, {323,15559}, {343,10539}, {389,10127}, {394,23335}, {403,2888}, {428,6243}, {468,18350}, {524,5446}, {539,12241}, {542,11793}, {546,13142}, {1154,6756}, {1216,1503}, {1511,3530}, {1594,3410}, {1595,16266}, {1598,11898}, {1656,11245}, {1843,10263}, {1885,18435}, {1993,7403}, {2979,16659}, {3548,6090}, {3574,3850}, {3575,18436}, {3628,15806}, {4549,17845}, {5020,18951}, {5159,13561}, {5663,31829}, {5891,6146}, {5907,13403}, {5921,6643}, {5965,10110}, {6102,9825}, {6193,9818}, {6288,23047}, {6515,7529}, {6642,11411}, {6644,9908}, {6759,16618}, {6776,7393}, {7399,18445}, {7405,7592}, {7514,31804}, {7526,12301}, {7528,12160}, {7553,11412}, {9306,12359}, {9820,21243}, {10024,25740}, {10116,10170}, {10128,15026}, {11402,14786}, {11441,15760}, {11442,11585}, {11457,15066}, {11459,12289}, {11577,15738}, {11591,12362}, {11818,31802}, {12010,17713}, {12022,15056}, {12164,18420}, {12233,15083}, {12370,15060}, {13568,13754}, {14790,18440}, {15606,29012}, {16881,23410}, {19347,26926}

X(31831) = midpoint of X(i) and X(j) for these {i,j}: {3575, 18436}, {5562, 12134}, {7553, 11412}, {10625, 16655}, {12605, 14516}
X(31831) = reflection of X(i) in X(j) for these (i,j): (6102, 9825), (12362, 11591), (13142, 546), (13292, 5), (18914, 140)
X(31831) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (68, 17814, 5), (155, 1352, 5), (15069, 17814, 68)

X(31832) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ARA AND 2nd EXTOUCH

Barycentrics
2*a^10-3*(b^2+c^2)*a^8-2*(b+c)*b*c*a^7-2*(b^2-3*b*c+c^2)*(b+c)^2*a^6+4*(b^3+c^3)*b*c*a^5+4*(b^4+c^4-3*(b^2-b*c+c^2)*b*c)*(b+c)^2*a^4-2*(b^4-c^4)*(b-c)*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)^3 : :
X(31832) = X(1885)-3*X(5927) = 5*X(3876)-X(12225)

X(31832) lies on these lines: {4,27410}, {30,5777}, {72,3575}, {517,6756}, {912,31833}, {916,13568}, {942,9825}, {971,31829}, {1885,5927}, {3876,12225}, {5044,12362}, {5810,11827}, {5812,7511}, {5849,31732}, {10445,20420}, {29291,31777}

X(31832) = midpoint of X(72) and X(3575)
X(31832) = reflection of X(i) in X(j) for these (i,j): (942, 9825), (12362, 5044)

X(31833) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: ANTI-ARA AND JOHNSON

Barycentrics 2*a^10-3*(b^2+c^2)*a^8-2*(b^4-4*b^2*c^2+c^4)*a^6+4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3 : :
X(31833) = 3*X(2)+X(6240) = 3*X(389)-X(10112) = 5*X(389)-3*X(11225) = 3*X(5890)+X(14516) = 3*X(5943)-X(13403) = 3*X(5946)-X(12370) = X(6146)-3*X(9730) = 5*X(10112)-9*X(11225) = 2*X(10112)-3*X(13292) = 6*X(11225)-5*X(13292) = 3*X(12022)+X(12278) = 3*X(12022)-7*X(15043) = X(12278)+7*X(15043) = X(12279)+3*X(16658) = X(12289)-9*X(15045) = 2*X(13568)+X(31831) = 3*X(15072)+X(16659)

As a point on the Euler line, X(31833) has Shinagawa coefficients (E-2*F, E+6*F)

X(31833) lies on these lines: {2,3}, {6,12118}, {68,9786}, {143,13142}, {185,12134}, {389,10112}, {542,13382}, {912,31832}, {1147,7706}, {1199,12383}, {1352,12163}, {1498,4846}, {3357,3818}, {3564,6102}, {5446,11745}, {5462,9826}, {5890,14516}, {5943,13403}, {5946,12370}, {6146,9730}, {7748,10314}, {7830,14767}, {8883,19176}, {9306,22660}, {9729,9827}, {9820,18388}, {9822,29012}, {9927,13567}, {10111,14708}, {10483,19372}, {10575,16655}, {10643,19106}, {10644,19107}, {10937,11188}, {11436,18970}, {11438,12359}, {11472,20427}, {12006,30522}, {12022,12278}, {12038,23292}, {12161,14542}, {12236,22530}, {12279,16658}, {12289,15045}, {12428,19366}, {12429,18951}, {13568,13754}, {13630,18914}, {14915,16621}, {15053,26879}, {15072,16659}, {15807,18874}, {16266,31802}

X(31833) = midpoint of X(i) and X(j) for these {i,j}: {3, 3575}, {20, 7553}, {185, 12134}, {550, 11819}, {6240, 12605}, {6756, 31829}, {10575, 16655}
X(31833) = reflection of X(i) in X(j) for these (i,j): (5, 9825), (3861, 13163), (5446, 11745), (6756, 31830), (12241, 5462), (12362, 140), (13142, 143), (13292, 389), (13488, 546), (15807, 18874), (18914, 13630)
X(31833) = complement of X(12605)
X(31833) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 1370), (4, 6642, 5), (24, 15760, 13383)

X(31834) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: EHRMANN-MID AND EHRMANN-SIDE

Barycentrics a^2*(2*(b^2+c^2)*a^6-2*(3*b^4+b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(6*b^4-5*b^2*c^2+6*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :
X(31834) = 5*X(5)-3*X(568) = 7*X(5)-5*X(3567) = 11*X(5)-9*X(5640) = 3*X(5)-X(5889) = X(5)-3*X(11459) = 5*X(5)-7*X(15056) = 3*X(5)-2*X(16881) = 11*X(568)-15*X(5640) = 9*X(568)-5*X(5889) = X(568)-5*X(11459) = 3*X(568)-7*X(15056) = 9*X(568)-10*X(16881) = 3*X(568)+5*X(18436) = 15*X(3567)-7*X(5889) = 15*X(3567)-14*X(16881) = 5*X(3567)+7*X(18436) = 3*X(5562)-X(6101) = 5*X(5562)-X(10625) = 7*X(5562)+X(11381) = 3*X(5562)+X(12162) = 27*X(5640)-11*X(5889) = 3*X(5640)-11*X(11459) = 9*X(5640)+11*X(18436) = 3*X(5876)+X(6101) = 5*X(5876)+X(10625) = 7*X(5876)-X(11381) = 3*X(5876)-X(12162) = 5*X(6101)-3*X(10625) = 7*X(6101)+3*X(11381) = 7*X(10625)+5*X(11381) = 3*X(10625)+5*X(12162) = 3*X(11381)-7*X(12162)

X(31834) lies on these lines: {3,9544}, {5,568}, {23,12307}, {30,5562}, {51,12811}, {52,3850}, {140,9729}, {143,5066}, {185,3530}, {323,14130}, {343,13406}, {389,547}, {394,11250}, {399,7512}, {428,12300}, {511,3853}, {542,13470}, {546,1154}, {548,1216}, {549,11444}, {550,12111}, {632,5890}, {1092,10226}, {1658,15068}, {2888,18403}, {2979,15704}, {3060,3858}, {3627,11412}, {3628,5891}, {3845,6243}, {3856,14531}, {3859,10110}, {3861,10263}, {3917,13491}, {5068,13321}, {5133,20424}, {5462,12812}, {5498,11064}, {5901,31751}, {6000,10627}, {6241,8703}, {6759,7555}, {7502,11441}, {7514,12164}, {7575,18350}, {7691,10540}, {7723,12605}, {7999,15712}, {9730,16239}, {10020,10272}, {10024,21230}, {10109,14831}, {10170,12006}, {10539,12107}, {10574,14869}, {10628,16252}, {11562,13392}, {12101,13598}, {12106,17814}, {12279,15686}, {12290,13340}, {12358,16196}, {13348,15690}, {13364,16625}, {14641,15691}, {14892,18874}, {14915,15606}, {15043,15699}, {15052,18378}, {15426,22051}, {17714,18451}, {18356,18531}, {22949,30507}, {28178,31737}, {28186,31738}

X(31834) = midpoint of X(i) and X(j) for these {i,j}: {5, 18436}, {550, 12111}, {3627, 11412}, {5562, 5876}, {6101, 12162}, {15704, 18439}
X(31834) = reflection of X(i) in X(j) for these (i,j): (52, 3850), (140, 11591), (185, 3530), (389, 14128), (546, 5907), (548, 1216), (5889, 16881), (5901, 31751), (6102, 3628), (10263, 3861), (11562, 13392), (12103, 10627), (13451, 15060), (13630, 11793), (14449, 546), (14831, 10109)
X(31834) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5889, 16881), (568, 15056, 5), (11459, 18436, 5)

X(31835) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: EHRMANN-MID AND 2nd EXTOUCH

Barycentrics a*((b+c)*a^5-(b+c)^2*a^4-2*(b^3+c^3)*a^3+(2*b^4+2*c^4+3*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b-c)*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :
X(31835) = X(3)-5*X(3876) = 7*X(3)-3*X(11220) = 3*X(3)+X(12528) = 3*X(5)-X(24474) = 3*X(72)+X(24474) = 3*X(140)-2*X(9940) = 7*X(140)-6*X(10156) = 3*X(210)-X(5690) = 3*X(210)+X(5887) = X(355)+3*X(5692) = 3*X(392)-X(1483) = 5*X(632)-3*X(10202) = X(946)+3*X(4134) = 15*X(3876)+X(12528) = 3*X(5044)-X(9940) = 7*X(5044)-3*X(10156) = 3*X(5777)+X(31793) = 7*X(9940)-9*X(10156) = 9*X(11220)+7*X(12528) = X(31793)-3*X(31837)

X(31835) lies on these lines: {2,24475}, {3,3219}, {4,26792}, {5,72}, {8,6929}, {10,5694}, {30,5777}, {63,6924}, {65,10592}, {78,6914}, {119,21677}, {140,912}, {144,6885}, {210,5690}, {329,6917}, {355,5692}, {392,1483}, {517,546}, {518,5901}, {548,971}, {632,10202}, {758,9956}, {916,13630}, {936,24467}, {942,3628}, {946,4134}, {952,960}, {1125,8254}, {1216,29958}, {1385,5302}, {1482,3681}, {1656,3868}, {1807,3074}, {2077,3652}, {2771,6684}, {2800,4015}, {2801,13624}, {3530,13369}, {3555,10283}, {3560,3940}, {3579,31803}, {3617,6982}, {3625,10284}, {3627,5927}, {3634,5885}, {3786,15952}, {3817,4537}, {3850,10157}, {3869,5790}, {3874,11230}, {3877,12645}, {3927,5780}, {3988,9955}, {4002,10273}, {4005,22791}, {4067,10175}, {4084,31399}, {4420,11849}, {4533,12672}, {4661,10595}, {5066,5806}, {5178,10738}, {5220,11249}, {5258,6265}, {5267,22935}, {5428,12691}, {5693,26446}, {5720,26921}, {5844,13600}, {5884,11231}, {5886,5904}, {6901,17484}, {6928,31018}, {6930,20007}, {6971,27131}, {6976,20013}, {7508,31445}, {9021,24206}, {9856,28212}, {10167,15712}, {10902,12738}, {11227,12108}, {11374,18397}, {14893,31822}, {15064,18480}, {15699,24473}, {16208,17857}, {18253,31659}, {28224,31786}, {31730,31828}, {31797,31821}

X(31835) = midpoint of X(i) and X(j) for these {i,j}: {5, 72}, {10, 5694}, {1216, 29958}, {3579, 31803}, {3625, 10284}, {3678, 20117}, {5690, 5887}, {5777, 31837}, {18480, 31806}, {31730, 31828}
X(31835) = reflection of X(i) in X(j) for these (i,j): (140, 5044), (942, 3628), (5885, 3634), (13369, 3530)
X(31835) = complement of X(24475)
X(31835) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (210, 5887, 5690), (3927, 5780, 6911)

X(31836) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: EHRMANN-SIDE AND 2nd EXTOUCH

Barycentrics a^2*(2*(b^2+c^2)*a^4+(b+c)*b*c*a^3-(4*b^4+4*c^4+(b^2+c^2)*b*c)*a^2-(b^2-c^2)*(b-c)*b*c*a+(2*b^2+b*c+2*c^2)*(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(31836) = 2*X(143)-3*X(10157) = 5*X(5562)-X(23154) = 2*X(5806)-3*X(15060) = 3*X(5927)-X(6243) = 3*X(5946)-2*X(31819) = 2*X(9940)-3*X(15067) = 3*X(10202)-5*X(11444) = 3*X(11459)-X(24474)

X(31836) lies on these lines: {3,23165}, {72,18436}, {143,10157}, {517,5876}, {912,5562}, {916,1216}, {942,11591}, {971,6101}, {1154,5777}, {2808,15606}, {5044,6102}, {5663,31793}, {5806,15060}, {5927,6243}, {5946,31819}, {9047,22793}, {9940,15067}, {10202,11444}, {11459,24474}, {13754,31837}

X(31836) = midpoint of X(72) and X(18436)
X(31836) = reflection of X(i) in X(j) for these (i,j): (942, 11591), (6102, 5044), (13369, 1216)

X(31837) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 2nd EXTOUCH AND JOHNSON

Barycentrics a*((b+c)*a^3-(b^2+4*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(31837) = 5*X(3)-3*X(10167) = X(4)-5*X(3876) = 3*X(5)-2*X(5806) = X(40)+3*X(5692) = X(65)-3*X(26446) = 5*X(72)+3*X(10167) = 2*X(72)+X(13369) = 3*X(165)+X(5693) = 3*X(210)-X(355) = 3*X(210)+X(14110) = 3*X(376)+X(12528) = X(382)-3*X(5927) = X(946)-3*X(10176) = 7*X(960)-2*X(26200) = 3*X(3740)-X(7686) = 3*X(3740)-2*X(9956) = 3*X(5044)-X(5806) = 3*X(5692)-X(5887) = 6*X(10167)-5*X(13369) = X(31793)+2*X(31835)

X(31837) lies on these lines: {1,6883}, {2,5761}, {3,63}, {4,3876}, {5,10}, {8,6827}, {9,3560}, {21,26878}, {30,5777}, {35,1858}, {40,5692}, {65,498}, {125,21530}, {140,942}, {165,5693}, {182,518}, {191,2077}, {201,22350}, {210,355}, {329,6850}, {376,12528}, {382,5927}, {392,1482}, {515,3678}, {546,10157}, {549,9940}, {550,971}, {580,30115}, {602,976}, {631,3868}, {758,6684}, {908,6842}, {936,5709}, {944,3681}, {952,6737}, {962,6849}, {975,5707}, {997,11249}, {1006,14054}, {1038,3157}, {1060,7078}, {1490,3587}, {1766,5778}, {1794,1807}, {1872,7524}, {1898,4302}, {1902,15763}, {2095,16408}, {2771,9943}, {2779,31752}, {2801,3988}, {3057,5722}, {3072,5293}, {3218,6940}, {3219,6906}, {3357,3579}, {3359,12526}, {3419,6928}, {3487,6989}, {3526,5439}, {3528,11220}, {3530,11227}, {3555,10246}, {3576,5904}, {3601,18397}, {3617,6844}, {3654,31165}, {3697,5790}, {3742,6583}, {3811,10267}, {3812,11231}, {3845,31822}, {3869,5552}, {3870,16202}, {3874,10165}, {3877,5084}, {3884,28234}, {3901,15016}, {4005,14872}, {4067,5884}, {4134,4297}, {4420,11491}, {4533,18525}, {4640,14454}, {4847,10943}, {5054,24473}, {5219,5903}, {5220,12114}, {5250,10679}, {5273,6892}, {5328,6981}, {5432,13750}, {5433,5570}, {5443,5659}, {5447,11573}, {5499,12444}, {5603,6887}, {5697,9581}, {5744,6961}, {5747,21853}, {5758,6826}, {5759,6869}, {5770,6926}, {5780,12672}, {5791,6862}, {5794,10526}, {5812,6917}, {5818,6866}, {5840,18254}, {5844,9957}, {5886,19854}, {5902,31423}, {5905,6897}, {6244,17650}, {6282,7330}, {6666,13464}, {6734,6882}, {6745,14988}, {6858,9780}, {6893,18228}, {6914,31445}, {6920,27065}, {6937,31053}, {6941,27131}, {6947,12649}, {6954,27383}, {6986,13151}, {6987,20007}, {6992,20013}, {7308,7982}, {7508,10391}, {7957,12699}, {8703,31805}, {9856,28174}, {10039,13375}, {10156,14869}, {10320,24914}, {10525,24703}, {10595,17552}, {10610,12675}, {10680,19861}, {10693,12778}, {10942,21075}, {11230,13374}, {11248,12514}, {11373,17642}, {11523,18443}, {11695,12109}, {13754,31836}, {15064,31673}, {15644,29958}, {15726,31828}, {15935,31792}, {17573,17612}, {17614,22765}, {17781,28458}, {24680,30147}, {29828,31778}, {31728,31817}, {31730,31803}, {31737,31825}

X(31837) = midpoint of X(i) and X(j) for these {i,j}: {3, 72}, {10, 31806}, {40, 5887}, {355, 14110}, {3579, 5694}, {3654, 31165}, {3878, 11362}, {4067, 5884}, {5777, 31793}, {7957, 12699}, {10693, 12778}, {12245, 23340}, {12672, 12702}, {14872, 18481}, {15644, 29958}, {17781, 28458}, {31728, 31817}, {31730, 31803}, {31737, 31825}
X(31837) = reflection of X(i) in X(j) for these (i,j): (1, 31838), (5, 5044), (942, 140), (3874, 13373), (5777, 31835), (7686, 9956), (9943, 31663), (10391, 7508), (11573, 5447), (12109, 11695), (12675, 13624), (13369, 3), (24475, 9940)
X(31837) = complement of X(24474)
X(31837) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3927, 24467), (4855, 21165, 3)

X(31838) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: K798I AND 2nd ZANIAH

Barycentrics a*((b+c)*a^5-(b^2+8*b*c+c^2)*a^4-2*(b+c)*(b^2-3*b*c+c^2)*a^3+2*(b^4+c^4+4*(b^2-b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
X(31838) = X(3)+3*X(392) = 3*X(3)+X(12672) = X(72)+3*X(10246) = X(355)-5*X(25917) = 9*X(392)-X(12672) = 3*X(551)+X(31806) = 3*X(960)-X(5694) = 3*X(960)+X(12675) = 3*X(1385)+X(5694) = 3*X(1385)-X(12675) = X(3057)+3*X(26446) = 3*X(3576)+X(5887) = 3*X(3576)-X(13369) = 9*X(3576)-X(15071) = 5*X(3616)-X(24474) = 3*X(3653)+X(31165) = 3*X(3655)+X(14872) = 3*X(3656)+X(7957) = 3*X(5887)+X(15071) = 3*X(13369)-X(15071)

X(31838) lies on these lines: {1,6883}, {3,392}, {10,10943}, {63,16203}, {72,10246}, {78,16202}, {355,25917}, {499,3057}, {518,575}, {551,31806}, {758,13373}, {912,960}, {952,5044}, {997,10267}, {999,26921}, {1210,5690}, {2975,24927}, {3452,10942}, {3576,5887}, {3616,24474}, {3653,31165}, {3655,14872}, {3656,7957}, {3678,13607}, {3697,12645}, {3869,10202}, {3876,7967}, {3878,10165}, {3890,5657}, {3898,11362}, {3899,15016}, {4511,24299}, {5267,18857}, {5693,30389}, {5697,31423}, {5836,11231}, {5844,6738}, {5882,10176}, {5886,14110}, {6001,10282}, {7686,11230}, {9940,14988}, {9943,17502}, {9947,28224}, {10179,24680}, {10199,10284}, {10269,12514}, {10805,31018}, {13151,21740}, {15829,18443}, {18253,20418}, {18526,18908}, {22791,31793}, {25681,26487}, {26066,26492}, {26201,31662}

X(31838) = midpoint of X(i) and X(j) for these {i,j}: {1, 31837}, {960, 1385}, {3678, 13607}, {5690, 9957}, {5694, 12675}, {5887, 13369}, {22791, 31793}
X(31838) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (960, 12675, 5694), (1385, 5694, 12675)

X(31839) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 6th ANTI-BROCARD AND 2nd BROCARD

Barycentrics 9*(3*SA+24*R^2-7*SW)*S^6-2*(54*R^2*(SA+SW)-18*SA^2-9*SB*SC-8*SW^2)*SW*S^4+(108*R^2*SA*(SB+SC)+SW*(18*SA^2-17*SA*SW+SW^2))*SW^2*S^2+4*SB*SC*SW^5 : :
X(31839) = X(98)-3*X(10166) = X(147)+3*X(353) = 2*X(6036)-3*X(10160)

X(31839) lies on these lines: {98,10166}, {99,14867}, {114,9771}, {147,353}, {542,31742}, {2782,31608}, {6033,31731}, {6036,10160}, {21636,31740}

X(31839) = midpoint of X(i) and X(j) for these {i,j}: {99, 14867}, {6033, 31731}, {21636, 31740}

X(31840) = CENTER OF THE ORTHOPOLAR CIRCLE OF THESE TRIANGLES: 4th BROCARD AND 11th FERMAT-DAO

Barycentrics 18*(3*R^2-2*SW)*S^4+(81*R^2*SA*(SB+SC)+(9*SA-8*SW)*SW^2)*S^2+3*SB*SC*SW^3 : :
X(31840) = X(381)-3*X(6032) = 2*X(547)-3*X(10162) = 3*X(5055)-X(12505) = 3*X(6031)-7*X(15702) = 3*X(9829)-5*X(15694) = 4*X(10124)-3*X(10163) = 3*X(15699)-2*X(31606) = 4*X(31749)-X(31824)

X(31840) lies on these lines: {2,31744}, {30,12506}, {111,381}, {547,10162}, {549,3849}, {3845,31749}, {5055,12505}, {5066,14866}, {5475,30523}, {5663,31743}, {6031,15702}, {8176,14650}, {8703,31762}, {9829,15694}, {10124,10163}, {12100,31729}, {15699,31606}

X(31840) = reflection of X(i) in X(j) for these (i,j): (3845, 31749), (8703, 31762), (14866, 5066), (31729, 12100), (31744, 2), (31824, 3845)

X(31841) = COMPLEMENT OF X(953)

Barycentrics (2 a^4-2 a^3 b-a^2 b^2+2 a b^3-b^4-2 a^3 c+4 a^2 b c-2 a b^2 c-a^2 c^2-2 a b c^2+2 b^2 c^2+2 a c^3-c^4) (a^4 b^2-2 a^2 b^4+b^6-2 a^3 b^2 c+2 a^2 b^3 c+2 a b^4 c-2 b^5 c+a^4 c^2-2 a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+2 a^2 b c^3-2 a b^2 c^3+4 b^3 c^3-2 a^2 c^4+2 a b c^4-b^2 c^4-2 b c^5+c^6) : :

See Ioannis Panakis and Peter Moses, Hyacinthos 28929.

X(31841) lies on the nine-point circle and these lines: {2, 953}, {3, 2222}, {4, 901}, {5, 3259}, {11, 517}, {12, 3025}, {115, 2245}, {119, 513}, {121, 20316}, {123, 5123}, {124, 3814}, {136, 860}, {952, 6073}, {1210, 24201}, {1482, 5516}, {1772, 24639}, {1878, 5521}, {5190, 8756}, {5520, 9956}, {7649, 20619}, {7741, 23153}, {15611, 17734}, {15612, 26446}, {17606, 23152}

X(31841) = midpoint of X(4) and X(901)
X(31841) = reflection of X(i) and X(j) for these {i,j}: {3, 22102}, {3259, 5}
X(31841) = complement of X(953)
X(31841) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 952}, {952, 10}, {2265, 2}
X(31841) = X(4)-Ceva conjugate of X(952)
X(31841) = center of hyperbola {{A,B,C,X(4), X(901)}}
X(31841) = Λ(X(1), X(5)) wrt orthic triangle

X(31842) = COMPLEMENT OF X(3563)

Barycentrics (a^2-b^2-c^2) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-4 a^4 b^2 c^2+3 a^2 b^4 c^2-4 b^6 c^2-a^4 c^4+3 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8) : :

See Ioannis Panakis and Peter Moses, Hyacinthos 28929.

X(31842) lies on the nine-point circle and these lines: {2, 136}, {3, 115}, {4, 3565}, {5, 5139}, {22, 15241}, {114, 2974}, {122, 30771}, {125, 343}, {127, 11585}, {135, 427}, {137, 6676}, {468, 16178}, {858, 16221}, {1560, 2967}, {2072, 5099}, {2453, 6721}, {3258, 5159}, {3546, 28438}, {3549, 14669}, {5512, 15760}, {5522, 7386}, {10691, 11792}, {14672, 18531}, {16051, 30789}

X(31842) = complement of X(3563)
X(31842) = midpoint of X(4) and X(3565)
X(31842) = Inverse of X(925) in the orthoptic circle of the Steiner inellipse
X(31842) = complement of the isogonal of X(3564)
X(31842) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3564}, {230, 226}, {293, 6036}, {460, 24005}, {656, 868}, {1692, 16583}, {1733, 5}, {3564, 10}, {4226, 8062}, {4575, 6132}, {8772, 6}, {17462, 15595}
X(31842) = X(4)-Ceva conjugate of X(3564)

X(31843) = X(2)X(137)∩X(115)X(140)

Barycentrics (2 a^6-4 a^4 b^2+3 a^2 b^4-b^6-4 a^4 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-6 a^4 b^2 c^2+4 a^2 b^4 c^2-5 b^6 c^2-a^4 c^4+4 a^2 b^2 c^4+8 b^4 c^4-a^2 c^6-5 b^2 c^6+c^8) : :

See Ioannis Panakis and Peter Moses, Hyacinthos 28929.

X(31843) lies on the nine-point circle and these lines: {2, 137}, {115, 140}, {125, 3819}, {1368, 20625}, {1594, 5139}, {3258, 30745}

X(31843) = complement of X(5966)
X(31843) = orthic-isogonal conjugate of X(5965)
X(31843) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 5965}, {5965, 10}
X(31843) = X(4)-Ceva conjugate of X(5965)
X(31843) = inverse of X(930) in the orthoptic circle of the Steiner inellipse

X(31844) = COMPLEMENT OF X(2291)

Barycentrics (2 a^2-a b-b^2-a c+2 b c-c^2) (a^2 b^2-2 a b^3+b^4+a b^2 c+b^3 c+a^2 c^2+a b c^2-4 b^2 c^2-2 a c^3+b c^3+c^4) : :

See Ioannis Panakis and Peter Moses, Hyacinthos 28929.

X(31844) lies on the nine-point circle and these lines: {2, 2291}, {4, 28291}, {11, 142}, {115, 15903}, {116, 2886}, {120, 4928}, {123, 18589}, {124, 141}, {125, 17052}, {1125, 15746}, {1566, 16593}, {3259, 20335}, {3452, 5514}, {3817, 5511}, {4728, 5513}, {5074, 5087}, {5274, 26140}, {5510, 24220}, {5518, 20528}, {5886, 25642}, {25532, 26105}

X(31844) = complement of X(2291)
X(31844) = reflection of X(15746) and X(1125)
X(31844) = Inverse of X(9086) in the orthoptic circle of the Steiner inellipse
X(31844) = complement of the isogonal of X(527)
X(31844) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 527}, {2, 5087}, {9, 5199}, {527, 10}, {651, 6366}, {1055, 37}, {1155, 2}, {1323, 142}, {1638, 11}, {6174, 16594}, {6366, 26932}, {6510, 3}, {6603, 9}, {6610, 1}, {6745, 3452}, {14392, 13609}, {14413, 1086}, {14414, 16596}, {15730, 6594}, {23346, 905}, {23710, 226}, {23890, 522}, {24685, 17793}, {30574, 8287}, {30806, 141}
X(31844) = X(4)-Ceva conjugate of X(527)

X(31845) = COMPLEMENT OF X(759)

Barycentrics (b+c)^2 (-a^2+b^2-b c+c^2) (a^3+b^3-a b c-b^2 c-b c^2+c^3) : :
X(31845) = 5 X[1698] - X[21381]

See Ioannis Panakis and Peter Moses, Hyacinthos 28929.

X(31845) lies on the nine-point circle and these lines: {2, 759}, {4, 6011}, {5, 25652}, {10, 125}, {11, 214}, {12, 1365}, {37, 115}, {116, 3739}, {123, 21530}, {124, 960}, {127, 18589}, {338, 15065}, {429, 5521}, {860, 13999}, {1211, 15614}, {1283, 5051}, {1511, 5499}, {1656, 14663}, {1698, 21381}, {2679, 19563}, {2802, 8286}, {3259, 11813}, {3634, 5993}, {3814, 5520}, {3936, 4867}, {4197, 19642}, {5074, 20529}, {5099, 16597}, {5497, 30117}, {5510, 9955}, {5511, 30444}, {5515, 16586}, {6702, 8287}, {8728, 25448}, {21232, 21253}, {24220, 30448}, {26364, 27686}

X(31845) = complement of X(759)
X(31845) = inverse of X(3031) in the Spieker radical circle
X(31845) = inverse of X(9070) in the orthoptic circle of the Steiner inellipse
X(31845) = complement of the isogonal of X(758)
X(31845) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 758}, {10, 3814}, {36, 1125}, {37, 908}, {42, 44}, {55, 7359}, {65, 1737}, {109, 21180}, {110, 6370}, {320, 3741}, {321, 21237}, {526, 6741}, {654, 4858}, {758, 10}, {860, 5}, {1443, 3742}, {1464, 1}, {1835, 1210}, {1870, 942}, {1983, 14838}, {2245, 2}, {2323, 5745}, {2594, 6149}, {2610, 8287}, {3028, 6739}, {3218, 3739}, {3724, 37}, {3936, 141}, {3960, 17761}, {4053, 1211}, {4242, 8062}, {4282, 16579}, {4511, 960}, {4551, 3738}, {4557, 1639}, {4559, 10015}, {4585, 4369}, {4674, 6702}, {4707, 116}, {6370, 125}, {6739, 113}, {6742, 526}, {7113, 3666}, {8818, 3580}, {17078, 17050}, {18593, 142}, {20924, 21240}, {21828, 1086}, {23493, 21331}
X(31845) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 758}, {3952, 6370}

X(31846) = X(324)X(547)∩X(343)X(15699)

Barycentrics 1/(5 a^8-17 a^6 (b^2+c^2)+a^4 (21 b^4+17 b^2 c^2+21 c^4)-11 a^2 (b^2-c^2)^2 (b^2+c^2)+(b^2-c^2)^2 (2 b^4-7 b^2 c^2+2 c^4)) : :

See Ioannis Panakis, Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28926.

X(31846) lies on these lines: {324,547}, {343,15699}, {5055,31610}

X(31847) = X(3)X(513)∩X(4)X(8)

Barycentrics a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-2 a^6 b^2 c+a^5 b^3 c+5 a^4 b^4 c-2 a^3 b^5 c-4 a^2 b^6 c+a b^7 c+b^8 c+a^7 c^2-2 a^6 b c^2+2 a^5 b^2 c^2-a^4 b^3 c^2-5 a^3 b^4 c^2+4 a^2 b^5 c^2+2 a b^6 c^2-b^7 c^2+a^5 b c^3-a^4 b^2 c^3+4 a^3 b^3 c^3-a b^5 c^3-3 b^6 c^3-3 a^5 c^4+5 a^4 b c^4-5 a^3 b^2 c^4-2 a b^4 c^4+3 b^5 c^4-2 a^3 b c^5+4 a^2 b^2 c^5-a b^3 c^5+3 b^4 c^5+3 a^3 c^6-4 a^2 b c^6+2 a b^2 c^6-3 b^3 c^6+a b c^7-b^2 c^7-a c^8+b c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31847) lies on these lines: {3,513}, {4,8}, {36,1935}, {119,2818}, {499,14115}, {953,2975}, {1329,31841}, {2771,18341}, {2779,21635}, {2842,10265}, {3025,5433}, {3649,5462}, {3874,13753}, {3937,6713}, {5687,15632}, {6075,23154}, {6830,30438}, {8674,18342}, {10575,18243}, {10993,29349}, {11491,14513}, {15635,26492}, {26470,29958}

X(31848) = X(3)X(512)∩X(4)X(69)

Barycentrics a^2 (a^8 b^4-3 a^6 b^6+3 a^4 b^8-a^2 b^10-a^6 b^4 c^2+a^4 b^6 c^2-a^2 b^8 c^2+b^10 c^2+a^8 c^4-a^6 b^2 c^4+2 a^4 b^4 c^4-4 b^8 c^4-3 a^6 c^6+a^4 b^2 c^6+6 b^6 c^6+3 a^4 c^8-a^2 b^2 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31848) lies on the circle O(3,4) and these lines: {3,512}, {4,69}, {52,7843}, {140,3111}, {185,18347}, {1078,2698}, {1216,7873}, {1975,15631}, {2387,15980}, {3091,6785}, {4173,6071}, {4230,9306}, {5025,13137}, {5462,15544}, {6102,15536}, {6784,20398}, {7746,14113}, {7936,7999}

X(31848) = X(99)-of-X(4)-Brocard-triangle

X(31849) = X(1)X(3)∩X(4)X(513)

Barycentrics a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-2 a^6 b^2 c+3 a^5 b^3 c+3 a^4 b^4 c-6 a^3 b^5 c+3 a b^7 c-b^8 c+a^7 c^2-2 a^6 b c^2+2 a^5 b^2 c^2-3 a^4 b^3 c^2+a^3 b^4 c^2+4 a^2 b^5 c^2-4 a b^6 c^2+b^7 c^2+3 a^5 b c^3-3 a^4 b^2 c^3+4 a^3 b^3 c^3-4 a^2 b^4 c^3-3 a b^5 c^3+3 b^6 c^3-3 a^5 c^4+3 a^4 b c^4+a^3 b^2 c^4-4 a^2 b^3 c^4+10 a b^4 c^4-3 b^5 c^4-6 a^3 b c^5+4 a^2 b^2 c^5-3 a b^3 c^5-3 b^4 c^5+3 a^3 c^6-4 a b^2 c^6+3 b^3 c^6+3 a b c^7+b^2 c^7-a c^8-b c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31849) lies on these lines: {1,3}, {4,513}, {11,2818}, {59,3562}, {1046,2957}, {1512,8679}, {1519,2390}, {1537,2841}, {2771,18342}, {2779,10265}, {2817,12736}, {2829,3937}, {2842,21635}, {3025,7354}, {3109,18180}, {3259,7681}, {5552,15632}, {5884,13868}, {6073,12607}, {6945,30438}, {8674,18341}, {11793,21677}, {12114,15635}, {15608,31841}, {18242,23154}

X(31850) = X(3)X(6)∩X(4)X(512)

Barycentrics a^2 (a^8 b^4-3 a^6 b^6+3 a^4 b^8-a^2 b^10+a^6 b^4 c^2-3 a^4 b^6 c^2+3 a^2 b^8 c^2-b^10 c^2+a^8 c^4+a^6 b^2 c^4+2 a^4 b^4 c^4-2 a^2 b^6 c^4+2 b^8 c^4-3 a^6 c^6-3 a^4 b^2 c^6-2 a^2 b^4 c^6-2 b^6 c^6+3 a^4 c^8+3 a^2 b^2 c^8+2 b^4 c^8-a^2 c^10-b^2 c^10) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31850) lies on orthosymmedial circle, the circle O(3,4), and on these lines: {3,6}, {4,512}, {51,1316}, {184,21525}, {185,18338}, {381,18321}, {1513,2387}, {2698,12110}, {2715,10312}, {3060,4226}, {3091,6787}, {3150,13567}, {3849,12508}, {5562,7780}, {5889,6179}, {5890,7422}, {6072,7764}, {6784,11623}, {6786,20399}, {7752,12833}, {7760,14510}, {7763,15631}, {7878,15043}, {8704,13239}, {12162,18348}, {15026,15536}, {15030,15098}

X(31850) = midpoint of polar circle intercepts of Brocard axis
X(31850) = orthogonal projection of X(4) on Brocard axis
X(31850) = inverse-in-polar-circle of X(14618)
X(31850) = inverse-in-circle-{{X(371), X(372),PU(1),PU(39)}} of X(14966)
X(31850) = {X(371), X(372)}-harmonic conjugate of X(14966)
X(31850) = intersection, other than X(6), of Brocard axis and orthosymmedial circle
X(31850) = intersection of the Brocard axes of ABC and the 2nd orthosymmedial triangle
X(31850) = X(98)-of-1st-orthosymmedial-triangle
X(31850) = X(1380)-of-2nd-orthosymmedial-triangle
X(31850) = X(99)-of-X(4)-Brocard-triangle
X(31850) = crossdifference of every pair of points on line X(523)X(3289)

X(31851) = X(3)X(142)∩X(4)X(514)

Barycentrics 2 a^8-2 a^7 b+a^6 b^2-3 a^5 b^3-2 a^4 b^4+4 a^3 b^5+a^2 b^6+a b^7-2 b^8-2 a^7 c+3 a^5 b^2 c+3 a^4 b^3 c-6 a^2 b^5 c-a b^6 c+3 b^7 c+a^6 c^2+3 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^2-3 a b^5 c^2+2 b^6 c^2-3 a^5 c^3+3 a^4 b c^3-4 a^3 b^2 c^3+4 a^2 b^3 c^3+3 a b^4 c^3-3 b^5 c^3-2 a^4 c^4+3 a^2 b^2 c^4+3 a b^3 c^4+4 a^3 c^5-6 a^2 b c^5-3 a b^2 c^5-3 b^3 c^5+a^2 c^6-a b c^6+2 b^2 c^6+a c^7+3 b c^7-2 c^8 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31851) lies on these lines: {3,142}, {4,514}, {664,10725}, {3843,18329}, {10767,18341}

X(31852) = X(3)X(514)∩X(4)X(9)

Barycentrics 2 a^8-2 a^7 b-a^6 b^2-a^5 b^3+4 a^3 b^5-a^2 b^6-a b^7-2 a^7 c+5 a^5 b^2 c+a^4 b^3 c-4 a^3 b^4 c-2 a^2 b^5 c+a b^6 c+b^7 c-a^6 c^2+5 a^5 b c^2-6 a^4 b^2 c^2+a^2 b^4 c^2+3 a b^5 c^2-2 b^6 c^2-a^5 c^3+a^4 b c^3+4 a^2 b^3 c^3-3 a b^4 c^3-b^5 c^3-4 a^3 b c^4+a^2 b^2 c^4-3 a b^3 c^4+4 b^4 c^4+4 a^3 c^5-2 a^2 b c^5+3 a b^2 c^5-b^3 c^5-a^2 c^6+a b c^6-2 b^2 c^6-a c^7+b c^7 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31852) lies on these lines: {3,514}, {4,9}, {20,18328}, {103,3732}, {191,2958}, {220,3234}, {348,14116}, {1565,6712}, {2328,4241}, {2724,24047}, {5532,6284}, {10164,24980}, {15634,17170}

X(31853) = X(3)X(3667)∩X(4)X(519)

Barycentrics 2 a^7-4 a^6 b+3 a^5 b^2+12 a^4 b^3-8 a^3 b^4-8 a^2 b^5+3 a b^6-4 a^6 c-6 a^4 b^2 c-9 a^3 b^3 c+13 a^2 b^4 c+9 a b^5 c-3 b^6 c+3 a^5 c^2-6 a^4 b c^2+14 a^3 b^2 c^2-a^2 b^3 c^2-3 a b^4 c^2-3 b^5 c^2+12 a^4 c^3-9 a^3 b c^3-a^2 b^2 c^3-18 a b^3 c^3+6 b^4 c^3-8 a^3 c^4+13 a^2 b c^4-3 a b^2 c^4+6 b^3 c^4-8 a^2 c^5+9 a b c^5-3 b^2 c^5+3 a c^6-3 b c^6 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31853) lies on these lines: {3,3667}, {4,519}, {3091,6788}, {3146,6790}, {7991,30196}, {18341,21635}

X(31854) = X(3)X(690)∩X(4)X(542)

Barycentrics 2 a^14-6 a^12 b^2+12 a^10 b^4-21 a^8 b^6+21 a^6 b^8-9 a^4 b^10+a^2 b^12-6 a^12 c^2+4 a^10 b^2 c^2+5 a^8 b^4 c^2-6 a^6 b^6 c^2-2 a^4 b^8 c^2+8 a^2 b^10 c^2-3 b^12 c^2+12 a^10 c^4+5 a^8 b^2 c^4-12 a^6 b^4 c^4+9 a^4 b^6 c^4-25 a^2 b^8 c^4+9 b^10 c^4-21 a^8 c^6-6 a^6 b^2 c^6+9 a^4 b^4 c^6+32 a^2 b^6 c^6-6 b^8 c^6+21 a^6 c^8-2 a^4 b^2 c^8-25 a^2 b^4 c^8-6 b^6 c^8-9 a^4 c^10+8 a^2 b^2 c^10+9 b^4 c^10+a^2 c^12-3 b^2 c^12 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28936.

X(31854) lies on these lines: {3,690}, {4,542}, {5,5465}, {98,15054}, {99,15034}, {110,23235}, {125,18347}, {541,10991}, {543,30714}, {631,11006}, {2782,5609}, {3090,18331}, {3091,11005}, {6055,20417}, {11623,11656}, {11638,25555}, {13188,15039}, {14639,15044}, {14981,16534}, {15027,15359}, {15357,20397}

X(31855) = MIDPOINT OF X(4695) AND X(21805)

Barycentrics a (b+c) (a^2+a b+a c-3 b c) : :
X(31855) = X[1] - 4 X[899],3 X[1] - 4 X[1149],2 X[10] + X[19998],3 X[899] - X[1149],X[4674] + 2 X[21805]

X(31855) lies on these lines: {1,2}, {9,20973}, {37,3921}, {71,3973}, {72,3987}, {209,2093}, {210,4424}, {213,21868}, {313,17151}, {484,513}, {517,5400}, {518,1739}, {740,3992}, {748,25439}, {756,3956}, {758,4674}, {872,4732}, {1018,2238}, {1215,4714}, {1464,4551}, {1574,3780}, {1724,3052}, {2234,4753}, {2292,4015}, {2650,3918}, {3125,20693}, {3294,20691}, {3678,4642}, {3697,4646}, {3725,4457}, {3743,4540}, {3753,4849}, {3880,22306}, {3930,16611}, {3931,3983}, {3970,16605}, {4006,16583}, {4125,4365}, {4568,17497}, {4694,16610}, {4731,21870}, {4866,15315}, {4880,22323}, {4975,24003}, {5248,17782}, {5692,22325}, {5904,24440}, {8168,16483}, {9349,28650}, {16474,17122}, {17160,18145}, {17449,24168}, {21061,21857}, {21839,21888}

X(31855) = midpoint of X(4695) and X(21805)
X(31855) = reflection of X(i) and X(j) for these {i,j}: {4674, 4695}, {4694, 16610}, {4975, 24003}, {17449, 24168}

X(31856) = X(2)X(3)∩X(3631)X(23326)

Barycentrics 5 a^6-10 a^4 b^2-5 a^2 b^4+10 b^6-10 a^4 c^2+22 a^2 b^2 c^2-10 b^4 c^2-5 a^2 c^4-10 b^2 c^4+10 c^6 : :

As a point on the Euler line, X(31856) has Shinagawa coefficients (3E-15F, -5E-5F).

X(31856) lies on these lines: {2,3}, {3631,23326}, {6723,21970}, {8549,15113}

X(31857) = X(2)X(3)∩X(39)X(6032)

Barycentrics 2 a^4 b^2-2 b^6+2 a^4 c^2-a^2 b^2 c^2+2 b^4 c^2+2 b^2 c^4-2 c^6 : :

As a point on the Euler line, X(31857) has Shinagawa coefficients (-E+8F, 8E+8F).

X(31857) lies on these lines: {2,3}, {39,6032}, {125,11002}, {194,31125}, {323,15069}, {511,7703}, {542,9716}, {568,20379}, {576,9140}, {1383,7747}, {1853,1994}, {1992,16176}, {2892,23327}, {3266,7814}, {3448,9976}, {3580,16981}, {5466,23301}, {5971,7773}, {5996,23105}, {6031,7802}, {6698,9971}, {7699,14915}, {7712,29012}, {7763,14360}, {7775,14246}, {7786,23297}, {7796,9464}, {7912,31132}, {8288,13330}, {9465,15820}, {9544,11550}, {9873,9999}, {11061,15140}, {11188,19510}, {11438,15057}, {11580,13881}, {13857,18553}, {20301,20423}

X(31857) = orthoptic circle of the Steiner inellipe inverse of X(18571)

X(31858) = X(2)X(14270)∩X(5)X(5465)

Barycentrics 3 a^10 b^6-9 a^8 b^8+9 a^6 b^10-3 a^4 b^12-2 a^12 b^2 c^2+a^10 b^4 c^2+4 a^8 b^6 c^2-2 a^6 b^8 c^2-5 a^4 b^10 c^2+4 a^2 b^12 c^2+a^10 b^2 c^4-2 a^8 b^4 c^4-3 a^6 b^6 c^4+13 a^4 b^8 c^4-8 a^2 b^10 c^4-3 b^12 c^4+3 a^10 c^6+4 a^8 b^2 c^6-3 a^6 b^4 c^6-12 a^4 b^6 c^6+4 a^2 b^8 c^6+12 b^10 c^6-9 a^8 c^8-2 a^6 b^2 c^8+13 a^4 b^4 c^8+4 a^2 b^6 c^8-18 b^8 c^8+9 a^6 c^10-5 a^4 b^2 c^10-8 a^2 b^4 c^10+12 b^6 c^10-3 a^4 c^12+4 a^2 b^2 c^12-3 b^4 c^12 : :

X(31858) lies on these lines: {2,14270}, {5,5465}, {7579,14995}

X(31859) = REFLECTION OF X(183) IN X(574)

Barycentrics a^4 - 3*a^2*b^2 - 3*a^2*c^2 + 2*b^2*c^2 : :

X(31859) lies on these lines: {2, 2418}, {3, 194}, {4, 10983}, {6, 99}, {20, 7762}, {30, 7774}, {39, 1975}, {56, 25264}, {69, 8356}, {75, 31449}, {76, 5013}, {148, 381}, {183, 538}, {187, 7798}, {192, 999}, {193, 376}, {262, 23235}, {316, 9766}, {325, 2549}, {330, 3295}, {378, 9308}, {382, 7785}, {384, 9605}, {474, 1655}, {524, 14907}, {543, 5475}, {549, 17008}, {550, 20065}, {599, 7831}, {620, 5309}, {625, 11648}, {631, 6392}, {671, 11184}, {956, 17759}, {1078, 15815}, {1180, 11324}, {1181, 9289}, {1351, 11676}, {1384, 7766}, {1569, 5989}, {1657, 7823}, {1909, 31448}, {1916, 11170}, {1992, 8598}, {1995, 31088}, {2396, 5108}, {2452, 7472}, {2482, 5355}, {2782, 13860}, {2996, 3090}, {3053, 7760}, {3117, 11333}, {3146, 15428}, {3164, 21312}, {3314, 11287}, {3329, 11286}, {3534, 7837}, {3552, 7839}, {3618, 6661}, {3788, 7765}, {3793, 8703}, {3815, 11185}, {3926, 6656}, {3933, 7791}, {4045, 7801}, {4229, 20018}, {4234, 17379}, {4255, 17499}, {4352, 16061}, {5022, 17034}, {5023, 6179}, {5054, 17004}, {5055, 17005}, {5077, 7840}, {5093, 10788}, {5206, 7805}, {5254, 7763}, {5286, 6337}, {5305, 16925}, {5687, 21226}, {5939, 9888}, {5976, 15483}, {6462, 7581}, {6463, 7582}, {6655, 7776}, {6658, 7921}, {6683, 17130}, {7283, 25918}, {7485, 8267}, {7603, 18546}, {7610, 11054}, {7618, 21843}, {7622, 8860}, {7736, 8370}, {7739, 7792}, {7748, 7764}, {7750, 7758}, {7756, 7759}, {7761, 7788}, {7769, 13881}, {7771, 8667}, {7772, 7816}, {7778, 7790}, {7779, 7833}, {7780, 15515}, {7784, 7796}, {7786, 22332}, {7789, 7803}, {7797, 7891}, {7802, 7905}, {7804, 15301}, {7806, 11288}, {7815, 31652}, {7821, 7872}, {7824, 20081}, {7827, 7835}, {7830, 7855}, {7834, 7863}, {7836, 7864}, {7842, 7903}, {7853, 7908}, {7861, 7888}, {7870, 7919}, {7871, 7911}, {7873, 7916}, {7874, 7902}, {7880, 7913}, {7895, 7935}, {7897, 7924}, {7909, 7918}, {7910, 7917}, {7923, 7945}, {7925, 11318}, {7933, 7947}, {8352, 9770}, {8354, 14929}, {8359, 16990}, {8369, 16989}, {8591, 11159}, {8719, 11477}, {9259, 17262}, {9466, 15482}, {11108, 27318}, {11152, 14931}, {11164, 15300}, {11329, 31036}, {11353, 24598}, {11359, 31090}, {11361, 15484}, {11482, 22521}, {13587, 17001}, {14039, 14482}, {15668, 16712}, {15685, 19569}, {15694, 17006}, {15822, 19568}, {16370, 16998}, {16371, 16997}, {16408, 27269}, {16417, 16999}, {16418, 17000}, {16921, 31467}, {16924, 31406}, {17002, 17549}, {17129, 20105}, {18533, 27377}
X(31859) = 3 X[183] - 2 X[17131],3 X[574] - X[17131],2 X[5475] - 3 X[11163],4 X[5475] - 3 X[11317]

X(31859) = reflection of X(i) and X(j) for these {i,j}: {183, 574}, {11185, 3815}, {11317, 11163}
X(31859) = anticomplement of the isotomic of X(11169)
X(31859) = X(11169)-anticomplementary conjugate of X(6327)
X(31859) = X(11169)-Ceva conjugate of X(2)
X(31859) = crossdifference of every pair of points on line {888, 8644}
X(31859) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 194, 7754}, {3, 22253, 385}, {6, 99, 1003}, {6, 8716, 99}, {39, 1975, 7770}, {39, 3734, 11174}, {39, 7781, 1975}, {76, 5013, 11285}, {99, 7757, 6}, {148, 7777, 381}, {187, 7798, 14614}, {194, 385, 22253}, {194, 7783, 3}, {325, 2549, 7841}, {385, 22253, 7754}, {1975, 11174, 3734}, {3552, 7839, 30435}, {3734, 11174, 7770}, {3788, 7765, 7851}, {3926, 6656, 7881}, {3926, 7738, 6656}, {3933, 7791, 7879}, {4045, 7801, 7868}, {4045, 14148, 7801}, {5254, 7763, 7887}, {5286, 6337, 7807}, {6390, 15048, 2}, {6655, 7906, 7776}, {7748, 7764, 7773}, {7757, 8716, 1003}, {7760, 7782, 3053}, {7761, 7813, 7788}, {7766, 13586, 1384}, {7789, 9607, 7803}, {7790, 7799, 7778}, {7796, 7847, 7784}, {7836, 7864, 7866}

X(31860) = X(3)X(14924)∩X(6)X(25)

Barycentrics a^2 (5 a^4+2 a^2 b^2-7 b^4+2 a^2 c^2+14 b^2 c^2-7 c^4) : :

X(31860) lies on these lines: {3,14924}, {4,8567}, {6,25}, {22,10545}, {23,3066}, {26,18874}, {64,1598}, {74,14490}, {110,5102}, {323,11477}, {394,10546}, {428,26958}, {1192,5198}, {1350,1995}, {1498,13382}, {1620,11403}, {1853,6995}, {3098,5020}, {3129,11481}, {3130,11480}, {3148,5210}, {3155,6412}, {3156,6411}, {3431,3518}, {3517,11430}, {3619,7398}, {3630,14826}, {3796,15018}, {3818,21970}, {4232,5480}, {5013,20897}, {5092,9909}, {5422,7712}, {5544,17508}, {5646,11284}, {5650,30734}, {5943,12017}, {6241,9786}, {7408,23332}, {7687,18494}, {7714,13567}, {8780,21849}, {10110,17821}, {10601,15080}, {10606,18535}, {10752,17847}, {10982,11464}, {11064,26255}, {11591,13861}, {13595,15066}, {14853,15448}, {15873,17845}, {19219,23249}

X(31860) = isogonal conjugate of the isotomic conjugate of X(3839)
X(31860) = X(3531)-Ceva conjugate of X(6)
X(31860) = barycentric product X(6)*X(3839)
X(31860) = barycentric quotient X(3839)/X(76)
X(31860) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 26864, 17809}, {23, 3066, 5085}, {25, 17810, 154}, {51, 26864, 6}, {17809, 17810, 51}

X(31861) = MIDPOINT OF X(6) AND X(11472)

Barycentrics a^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+8 a^4 b^2 c^2-6 b^6 c^2+14 b^4 c^4+2 a^2 c^6-6 b^2 c^6-c^8) : :
X(31861) = X[3426] + 3 X[5050] X[4846] - 3 X[14561] X[11820] - 5 X[12017]

As a point on the Euler line, X(31861) has Shinagawa coefficients (E+4F, 5E-4F).

X(31861) is the center of circle {{X(6), X(1344), X(1345), X(11472),PU(4)}}. This circle is the orthocentroidal-circle-inverse of the nine-point circle, and also the locus of symmedian points of triangles in a MacBeath porism (triangles that share circumcircle and MacBeath inconic with ABC). (Randy Hutson, March 28, 2019)

X(31861) lies on the cubic K582 and these lines: {2, 3}, {6, 5663}, {54, 11439}, {64, 13630}, {74, 5640}, {110, 16261}, {143, 12163}, {182, 14915}, {184, 16194}, {394, 15060}, {511, 4550}, {541, 5476}, {542, 13233}, {567, 11456}, {569, 11381}, {575, 6000}, {576, 8548}, {1154, 9972}, {1177, 10249}, {1503, 8546}, {1539, 19457}, {1993, 18435}, {2453, 16168}, {2777, 7706}, {2782, 15928}, {2935, 20304}, {2936, 22566}, {3066, 12041}, {3292, 13352}, {3357, 5462}, {3426, 5050}, {3527, 16881}, {3567, 15062}, {3818, 12584}, {4549, 31670}, {4846, 14561}, {5012, 11455}, {5092, 8717}, {5448, 9938}, {5609, 18451}, {5643, 13445}, {5651, 10564}, {5890, 15019}, {5907, 16266}, {5946, 10605}, {6102, 10982}, {6247, 18952}, {6800, 14805}, {7592, 18439}, {7624, 30209}, {7689, 10110}, {7703, 14644}, {9175, 30230}, {9775, 12027}, {9781, 11440}, {9786, 10095}, {10264, 26869}, {10545, 15055}, {10546, 15035}, {10606, 13364}, {11003, 12112}, {11178, 19510}, {11422, 14094}, {11424, 12161}, {11459, 23061}, {11820, 12017}, {11935, 18551}, {12133, 12228}, {12290, 13434}, {13293, 15113}, {14581, 15860}, {15021, 15053}, {15027, 15121}, {15126, 23325}, {15311, 15579}, {15581, 18400}, {20397, 23329}

X(31861) = midpoint of X(6) and X(11472)
X(31861) = midpoint of X(1344) and X(1345)
X(31861) = reflection of X(i) and X(j) for these {i,j}: {7706, 19130}, {8717, 5092}
X(31861) = crossdifference of every pair of points on line {647, 9003}
X(31861) = orthocentroidal-circle-inverse of X(11799)
X(31861) = harmonic center of circumcircle and orthocentroidal circle
X(31861) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 11799}, {2, 7464, 3}, {3, 4, 7530}, {3, 25, 7575}, {3, 381, 1995}, {3, 382, 12082}, {3, 546, 13861}, {3, 1995, 6644}, {3, 3843, 7545}, {3, 7387, 7555}, {3, 7517, 7556}, {3, 7527, 7526}, {3, 7530, 26}, {3, 7545, 24}, {3, 7550, 7516}, {3, 7575, 18324}, {3, 11403, 3627}, {3, 12083, 7492}, {4, 20, 7540}, {4, 376, 7519}, {4, 5169, 381}, {4, 7526, 26}, {4, 7527, 3}, {4, 7565, 3843}, {4, 10296, 3830}, {4, 14118, 7517}, {5, 1593, 12084}, {5, 3541, 31283}, {5, 15122, 2}, {20, 7550, 3}, {25, 18570, 18324}, {376, 7496, 3}, {378, 381, 6644}, {378, 1995, 3}, {381, 5094, 5}, {549, 16619, 7493}, {1346, 1347, 2072}, {1593, 5094, 378}, {1658, 3861, 1598}, {3090, 12086, 3}, {3091, 14865, 3}, {3520, 3832, 7506}, {3541, 7577, 5094}, {3843, 14130, 24}, {3845, 18570, 25}, {3850, 11250, 6642}, {5169, 7577, 5576}, {7502, 15687, 18534}, {7503, 12082, 3}, {7516, 7540, 26}, {7526, 7530, 3}, {7545, 14130, 3}, {7556, 14118, 3}, {7575, 18570, 3}, {11422, 14094, 18445}, {11422, 15305, 14094}, {11424, 12162, 12161}, {11479, 12085, 140}, {13352, 15030, 15068}, {14094, 15033, 11422}, {15019, 15054, 5890}, {15033, 15305, 18445}

X(31862) = X(2)X(1340)∩X(6)X(13)

Barycentrics (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4) + Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6): :

X(31862) lies on the orthocentroidal circle, the cubics K025, K492, K591, K800, K1092, and on these lines: {2, 1340}, {4, 3413}, {5, 14630}, {6, 13}, {30, 1380}, {230, 1379}, {316, 6189}, {1341, 2549}, {1346, 13870}, {2039, 6039}, {2040, 6190}, {2469, 13415}, {2470, 13414}, {2558, 12203}, {3557, 3767}, {5254, 14631}, {6142, 9158}, {6792, 13722}

X(31862) = reflection of X(i) and X(j) for these {i,j}: {6039, 2039}, {6190, 2040}, (31863, 381)
X(31862) = orthocentroidal-circkle-antipode of X(31863)
X(31862) = X(1380)-of-orthocentroidal-triangle
X(31862) = X(1380)-of-4th-Brocard-triangle
X(31862) = {X(6), X(115)}-harmonic conjugate of X(31863)
X(31862) = {X(13), X(14)}-harmonic conjugate of X(31863)
X(31862) = psi-transform of X(13636)
X(31862) = crossdifference of every pair of points on line {526, 5638}

X(31863) = X(2)X(1341)∩X(6)X(13)

Barycentrics (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4) - Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6): :

X(31863) lies on the orthocentroidal circle, the cubics K025, K492, K591, K800, K1092, and on these lines: {2, 1341}, {4, 3414}, {5, 14631}, {6, 13}, {30, 1379}, {230, 1380}, {316, 6190}, {1340, 2549}, {1347, 13870}, {2039, 6189}, {2040, 6040}, {2469, 13414}, {2470, 13415}, {2559, 12203}, {3558, 3767}, {5254, 14630}, {6141, 9158}, {6792, 13636}

X(31863) = reflection of X(i) and X(j) for these {i,j}: {6040, 2040}, {6189, 2039}, {31862, 381}
X(31863) = psi-transform of X(13722)
X(31863) = crossdifference of every pair of points on line {526, 5639}
X(31863) = antipode of X(31862) in orthocentroidal circle
X(31863) = X(1379)-of-orthocentroidal-triangle
X(31863) = X(1379)-of-4th-Brocard-triangle
X(31863) = {X(6), X(115)}-harmonic conjugate of X(31862)
X(31863) = {X(13), X(14)}-harmonic conjugate of X(31862)

X(31864) = X(3)X(18336)∩X(4)X(5965)

Barycentrics 2 a^14 - 10 a^12 (b^2 + c^2) + a^10 (34 b^4 + 8 b^2 c^2 + 34 c^4) - a^8 (65 b^6 + 3 b^4 c^2 + 3 b^2 c^4 + 65 c^6) + a^6 (61 b^8 - 4 b^6 c^2 - 4 b^2 c^6 + 61 c^8) - a^4 (b^2 + c^2) (25 b^8 - 13 b^6 c^2 - 6 b^4 c^4 - 13 b^2 c^6 + 25 c^8) + a^2 (b^2 - c^2)^2 (3 b^8 + 32 b^6 c^2 - 22 b^4 c^4 + 32 b^2 c^6 + 3 c^8) - 5 b^2 c^2 (b^2 - c^2)^4 (b^2 + c^2) : :

See Antreas Hatzipolakis and Randy Hutson, Hyacinthos 28938.

X(31864) lies on these lines: {3,18336}, {4,5965}, {3545,18335}

X(31865) = X(3)X(667)∩X(4)X(518)

Barycentrics a (a^8 (b^2 + c^2) - a^7 (3 b^3 + b^2 c + b c^2 + 3 c^3) + a^6 (b^4 - b^3 c + 6 b^2 c^2 - b c^3 + c^4) + a^5 (5 b^5 + 5 b^4 c - 4 b^3 c^2 - 4 b^2 c^3 + 5 b c^4 + 5 c^5) - a^4 (5 b^6 + b^5 c + b^4 c^2 + 6 b^3 c^3 + b^2 c^4 + b c^5 + 5 c^6) - a^3 (b^7 + 3 b^6 c + b^5 c^2 - 9 b^4 c^3 - 9 b^3 c^4 + b^2 c^5 + 3 b c^6 + c^7) + a^2 (b - c)^2 (3 b^6 + 7 b^5 c + 15 b^4 c^2 + 18 b^3 c^3 + 15 b^2 c^4 + 7 b c^5 + 3 c^6) - a (b - c)^2 (b^7 + 3 b^6 c + 7 b^5 c^2 + b^4 c^3 + b^3 c^4 + 7 b^2 c^5 + 3 b c^6 + c^7) + b c (b - c)^4 (b + c)^2 (b^2 + c^2)) : :

See Antreas Hatzipolakis and Randy Hutson, Hyacinthos 28938.

X(31865) lies on these lines: {3,667}, {4,518}, {6835,18343}, {15030,31849}, {31803,31851}

X(31866) = X(1)X(4)∩X(3)X(522)

Barycentrics 2 a^10 - 2 a^9 (b + c) - a^8 (3 b^2 - 4 b c + 3 c^2) + a^7 (b + c) (5 b^2 - 6 b c + 5 c^2) - a^6 (3 b^4 + 5 b^3 c - 12 b^2 c^2 + 5 b c^3 + 3 c^4) - 3 a^5 (b - c)^4 (b + c) + a^4 (b - c)^2 (7 b^4 + 11 b^3 c + 4 b^2 c^2 + 11 b c^3 + 7 c^4) - a^3 (b - c)^2 (b + c) (b^4 + 8 b^3 c - 2 b^2 c^2 + 8 b c^3 + c^4) - a^2 (b - c)^4 (b + c)^2 (3 b^2 + b c + 3 c^2) + a (b - c)^4 (b + c)^5 - b c (b^2 - c^2)^4 : :

See Antreas Hatzipolakis and Randy Hutson, Hyacinthos 28938.

X(31866) lies on the circle O(1,3) and these lines: {1,4}, {2,18339}, {3,522}, {8,15633}, {102,1897}, {117,15252}, {124,952}, {2222,11491}, {2360,7452}, {2968,6711}, {3326,6284}, {5494,10265}, {5884,31849}, {6796,23981}, {11719,24030}

X(31866) = midpoint of circumcircle intercepts of line X(1)X(4)
X(31866) = X(102)-of-X(1)-Brocard-triangle

X(31867) = X(4)X(19172)∩X(5)X(5944)

Barycentrics a^12 b^4-5 a^10 b^6+10 a^8 b^8-10 a^6 b^10+5 a^4 b^12-a^2 b^14-2 a^12 b^2 c^2+a^10 b^4 c^2+3 a^8 b^6 c^2+6 a^6 b^8 c^2-16 a^4 b^10 c^2+9 a^2 b^12 c^2-b^14 c^2+a^12 c^4+a^10 b^2 c^4-2 a^8 b^4 c^4+4 a^6 b^6 c^4+11 a^4 b^8 c^4-21 a^2 b^10 c^4+6 b^12 c^4-5 a^10 c^6+3 a^8 b^2 c^6+4 a^6 b^4 c^6+13 a^2 b^8 c^6-15 b^10 c^6+10 a^8 c^8+6 a^6 b^2 c^8+11 a^4 b^4 c^8+13 a^2 b^6 c^8+20 b^8 c^8-10 a^6 c^10-16 a^4 b^2 c^10-21 a^2 b^4 c^10-15 b^6 c^10+5 a^4 c^12+9 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14 : :
Barycentrics S^4 + (16 R^4-2 R^2 SB-2 R^2 SC+SB SC-8 R^2 SW+SW^2)S^2 -16 R^4 SB SC+4 R^2 SB SC SW+SB SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

X(31867) lies on these lines: {4,19172}, {5,5944}, {389,7668}, {523,14978}, {578,3613}, {1510,18488}, {3574,8901}, {7404,15270}

X(31868) = X(5)X(27684)∩X(2883)X(3845)

Barycentrics 4 a^16-22 a^14 b^2+45 a^12 b^4-31 a^10 b^6-30 a^8 b^8+76 a^6 b^10-63 a^4 b^12+25 a^2 b^14-4 b^16-22 a^14 c^2+58 a^12 b^2 c^2-27 a^10 b^4 c^2-21 a^8 b^6 c^2-56 a^6 b^8 c^2+156 a^4 b^10 c^2-119 a^2 b^12 c^2+31 b^14 c^2+45 a^12 c^4-27 a^10 b^2 c^4-18 a^8 b^4 c^4-20 a^6 b^6 c^4-81 a^4 b^8 c^4+207 a^2 b^10 c^4-106 b^12 c^4-31 a^10 c^6-21 a^8 b^2 c^6-20 a^6 b^4 c^6-24 a^4 b^6 c^6-113 a^2 b^8 c^6+209 b^10 c^6-30 a^8 c^8-56 a^6 b^2 c^8-81 a^4 b^4 c^8-113 a^2 b^6 c^8-260 b^8 c^8+76 a^6 c^10+156 a^4 b^2 c^10+207 a^2 b^4 c^10+209 b^6 c^10-63 a^4 c^12-119 a^2 b^2 c^12-106 b^4 c^12+25 a^2 c^14+31 b^2 c^14-4 c^16 : :
Barycentrics 3 S^4 + (-16 R^4-10 R^2 SB-10 R^2 SC+27 SB SC+8 R^2 SW+4 SB SW+4 SC SW-SW^2)S^2 +16 R^4 SB SC-12 R^2 SB SC SW-SB SC SW^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

Let DEF be the cevian triangle of X(5). Let ℓ_b be the perpendicular to BE at E, and let ℓ_c be the perpendicular to CF at F. Let A' = ℓ_b∩ℓ_c, and define B' and C' cyclically. The triangles ABC and A'B'C' are orthologic (by construction), and the orthologic center of A'B'C' respect to ABC is X(31868). (Angel Montesdeoca, March 13, 2021)

X(31868) lies on these lines: {5,27684}, {2883,3845}

X(31869) = REFLECTION OF X(14134) IN X(140)

Barycentrics 3 a^6 b^4-2 a^4 b^6-a^2 b^8+2 a^6 b^2 c^2+6 a^4 b^4 c^2+a^2 b^6 c^2-b^8 c^2+3 a^6 c^4+6 a^4 b^2 c^4+b^6 c^4-2 a^4 c^6+a^2 b^2 c^6+b^4 c^6-a^2 c^8-b^2 c^8 : :
Barycentrics (2 R^2+SB+SC)S^4 + (-14 R^2 SB SC+2 SB SC SW+SB SW^2+SC SW^2)S^2 2 SB SC SW^3 : :
X(31869) = 2*X[140]-X[14134]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

X(31869) lies on these lines: {5,6310}, {30,14133}, {140,14134}, {575,3627}

X(31869) = reflection of X(14134) in X(140)

X(31870) = MIDPOINT OF X(4) AND X(5884)

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-2 a^4 b c+a^3 b^2 c-a^2 b^3 c-2 a b^4 c+3 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^2-2 a^3 c^3-a^2 b c^3+a b^2 c^3-6 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5+3 b c^5-c^6) : :
Barycentrics 2 R S^3 + (12 a R^2-12 b R^2+2 a SB+2 b SB-c SB+2 a SC-b SC+2 c SC-4 a SW+3 b SW)S^2 + 10 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : :
X(31870) = X[3]-3*X[5883], X[20]-5*X[15016], X[72]-3*X[10175], 3*X[354]-X[5882], X[355]+X[3874], X[944]-5*X[18398], 5*X[1656]-3*X[10176], 7*X[3090]-3*X[5692], 5*X[3091]-X[5693], X[3678]-2*X[9956], 3*X[3753]-X[11362], 3*X[3817]+X[4084], X[3868]+3*X[5587], X[3869]-5*X[8227], 3*X[3873]+X[5881], 3*X[3877]-7*X[9624], X[3878]-3*X[5886], X[3901]+7*X[7989], 3*X[3919]+X[4301], X[4297]-3*X[10202], X[4757]+2*X[9955], 2*X[5044]-3*X[10172], 2*X[5045]-X[13607], 5*X[5439]-3*X[10165], 5*X[5818]-X[5904], X[12528]-5*X[18492], X[14872]+3*X[24473], X[14923]+3*X[16200]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

X(31870) lies on these lines: {1,1389}, {3,5883}, {4,79}, {5,758}, {7,6256}, {10,6881}, {11,65}, {20,15016}, {21,5535}, {30,5885}, {40,1621}, {57,5450}, {72,10175}, {104,3337}, {140,517}, {191,6920}, {354,5882}, {355,3874}, {389,2779}, {499,1788}, {515,942}, {546,2771}, {912,19925}, {944,18398}, {950,13750}, {952,3881}, {958,2095}, {971,16616}, {1012,5221}, {1158,3339}, {1376,1482}, {1476,3333}, {1512,13407}, {1532,3649}, {1656,10176}, {1735,2654}, {1737,15556}, {1768,21669}, {1837,18389}, {1845,1940}, {2392,5446}, {2778,6696}, {2801,18480}, {2802,10222}, {2829,24470}, {3075,11700}, {3090,5692}, {3091,5693}, {3109,18180}, {3336,6906}, {3585,6246}, {3647,7489}, {3656,10199}, {3671,7682}, {3678,9956}, {3753,11362}, {3817,4084}, {3826,3918}, {3868,5587}, {3869,8227}, {3873,5881}, {3877,9624}, {3878,5886}, {3901,7989}, {3919,4301}, {4292,18838}, {4295,5804}, {4297,10202}, {4757,9955}, {5044,10172}, {5045,13607}, {5218,5697}, {5330,7982}, {5425,21740}, {5439,10165}, {5538,6940}, {5563,11715}, {5570,10106}, {5708,12114}, {5720,12559}, {5734,26062}, {5755,25081}, {5761,26364}, {5806,5893} {5818,5904}, {5836,28234}, {5842,12433}, {6147,18242}, {6261,11529}, {6326,6915}, {6842,11263}, {6853,26725}, {6894,9803}, {6911,22836}, {6918,12635}, {7680,12432}, {8070,12047}, {9943,28150}, {10044,10051}, {10532,10573}, {10597,12647}, {10980,12650}, {11009,25485}, {11246,18977}, {11500,15934}, {11520,17857}, {12245,26040}, {12528,18492}, {12704,19860}, {13145,28174}, {13369,28164}, {13729,14450}, {14872,24473}, {14923,16200}, {18240,24928}, {18391,26332}, {26201,28186

X(31870) = midpoint of X(i) and X(j) for these {i,j}: {4,5884}, {10,24474}, {65,946}, {355,3874}, {942,7686}, {4084,5887}, {6246,11570}, {18480,24475}
X(31870) = reflection of X(i) in X(j) for these {i,j}: {3678,9956}, {3881,6583}, {3884,5901}, {5690,3918}, {6684,3812}, {12005,942}, {13464,13374}, {13607,5045}, {18483,5806}, {20117,5}
X(31870) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,5902,5884}, {946,10265,6831}, {3671,7682,12608}, {3817,4084,5887}, {4295,5804,26333}, {5439,14110,10165}

X(31871) = MIDPOINT OF X(10) AND X(12688)

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+2 a^4 b c+a^3 b^2 c+a^2 b^3 c-2 a b^4 c-3 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^2-2 a^3 c^3+a^2 b c^3+a b^2 c^3+6 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5-3 b c^5-c^6) : :
Barycentrics 2 R S^3 - (12 a R^2-12 b R^2+2 a SB+2 b SB-c SB+2 a SC-b SC+2 c SC-4 a SW+3 b SW)S^2 +10 R S SB SC-3 b SB SC^2+3 c SB SC^2+3 b SB SC SW : :
X(31871) = 4*X[5]-3*X[3833], X[10]-3*X[5927], X[20]-3*X[10176], X[40]-2*X[4015], X[185]-3*X[15049], 3*X[210]-X[5493], 3*X[381]-X[5884], 3*X[551]-X[12680], X[942]-2*X[12571], X[1071]-3*X[3817], 5*X[1698]-X[9961], 3*X[1699]-X[3874], 5*X[3091]-3*X[5883], X[3146]+3*X[5692], 9*X[3545]-5*X[15016], 7*X[3624]-3*X[11220], X[3626]-2*X[9947], X[3627]+X[5694], 2*X[3634]-X[9943], 3*X[3681]+X[9589], 7*X[3832]-3*X[5902], 2*X[3850]-X[5885], X[3878]+X[5691], 3*X[3892]-5*X[11522], 2*X[3918]-3*X[5587], 3*X[4134]-X[7957], X[4301]+X[14872], 4*X[4540]-3*X[5657], 4*X[4547]-X[6361], X[4757]-2*X[7686], 2*X[5044]-X[12512], X[5904]+3*X[9812], 9*X[9779]-5*X[18398], 2*X[9940]-3*X[10171], 2*X[9955]-X[12005], 3*X[10167]-5*X[19862], 3*X[11227]-4*X[19878], X[17661]+X[21630]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

X(31871) lies on these lines: {2,16120}, {3,16112}, {4,758}, {5,3833}, {9,12511}, {10,5927}, {20,10176}, {30,20117}, {40,4015}, {58,9355}, {79,6894}, {185,15049}, {210,5493}, {226,1898}, {381,5884}, {389,2772}, {411,3647}, {515,3884}, {516,3678}, {517,3853}, {519,9856}, {546,2771}, {551,12680}, {912,18483}, {936,3062}, {942,12571}, {946,2801}, {952,26200}, {960,28164}, {971,1125}, {991,27784}, {1071,3817}, {1490,5248}, {1698,9961}, {1699,3874}, {1709,25440}, {1750,12514}, {1768,6915}, {1836,12432}, {1864,3671}, {2392,5907}, {2800,18480}, {2802,12672}, {3091,5883}, {3146,5692}, {3159,28850}, {3244,13227}, {3337,13243}, {3434,12059}, {3545,15016}, {3624,11220}, {3626,9947}, {3627,5694}, {3634,9943}, {3681,9589}, {3754,6001}, {3811,11372}, {3822,6260}, {3825,6245}, {3832,5902}, {3850,5885}, {3878,5691}, {3892,11522}, {3918,5587}, {3947,12711}, {4134,7957}, {4187,17653}, {4301,14872}, {4540,5657}, {4547,6361}, {4757,7686}, {4882,11678}, {5044,12512}, {5658,10198}, {5851,24470}, {5904,9812}, {6147,20116}, {6261,18540}, {6326,21669}, {6681,6705}, {6702,12616}, {6826,16127}, {6831,21635}, {6905,7701}, {6920,16132}, {6940,10308}, {6986,16143}, {8226,11263}, {8581,21625}, {8715,12705}, {9779,18398}, {9842,9948}, {9940,10171}, {9955,12005}, {10122,10883}, {10167,19862}, {11227,19878}, {12609,12664}, {12684,25524}, {12691,18406}, {17661,21630}, {18250,18251}, {21077,21628}

X(31871) = midpoint of X(i) and X(j) for these {i,j}: {10,12688}, {3627,5694}, {3874,12528}, {3878,5691}, {4301,14872}, {9949,17646}, {17661,21630}
X(31871) = reflection of X(i) in X(j) for these {i,j}: {40,4015}, {942,12571}, {3626,9947}, {3678,5777}, {3754,19925}, {3881,946}, {4757,7686}, {5885,3850}, {9943,3634}, {12005,9955}, {12512,5044}, {12564,12558}
X(31871) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {40,15064,4015}, {1699,12528,3874}, {3091,15071,5883}, {5927,12688,10}, {6260,12617,3822}, {6894,9809,79}, {9943,10157,3634}

X(31872) = X(4)X(3617)∩X(381)X(19722)

Barycentrics 2 a^7+5 a^6 b+3 a^5 b^2+a^4 b^3+a^3 b^4-4 a^2 b^5-6 a b^6-2 b^7+5 a^6 c+8 a^5 b c+4 a^4 b^2 c+4 a^3 b^3 c-3 a^2 b^4 c-12 a b^5 c-6 b^6 c+3 a^5 c^2+4 a^4 b c^2+6 a^3 b^2 c^2+13 a^2 b^3 c^2+6 a b^4 c^2-2 b^5 c^2+a^4 c^3+4 a^3 b c^3+13 a^2 b^2 c^3+24 a b^3 c^3+10 b^4 c^3+a^3 c^4-3 a^2 b c^4+6 a b^2 c^4+10 b^3 c^4-4 a^2 c^5-12 a b c^5-2 b^2 c^5-6 a c^6-6 b c^6-2 c^7 : :
Barycentrics 8 R S^3 - (28 a R^2-28 b R^2+3 a SB+b SB+2 c SB+3 a SC+2 b SC+c SC-11 a SW+4 b SW-3 c SW)S^2 + 40 R S SB SC-7 b SB SC^2+7 c SB SC^2+13 a SB SC SW+20 b SB SC SW+13 c SB SC SW : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28939.

X(31872) lies on these lines: {4,3617}, {381,19722}

X(31873) = X(125)X(546)∩X(133)X(1594)

Barycentrics 2 a^14 (b^2+c^2) +a^12 (-6 b^4+8 b^2 c^2-6 c^4) +a^10 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) +a^8 (15 b^8-44 b^6 c^2+62 b^4 c^4-44 b^2 c^6+15 c^8) -a^6 (b^2-c^2)^2 (20 b^6+b^4 c^2+b^2 c^4+20 c^6) +a^4 (b^2-c^2)^2 (8 b^8+31 b^6 c^2-42 b^4 c^4+31 b^2 c^6+8 c^8) +a^2 (b^2-c^2)^4 (b^6-14 b^4 c^2-14 b^2 c^4+c^6) -(b^2-c^2)^6 (b^4+5 b^2 c^2+c^4) : :

See Vu Thanh Tung and Angel Montesdeoca, Hyacinthos 28942.

X(31873) lies on these lines: {125, 546}, {133, 1594}

X(31874) = X(1553)X(14094)∩X(5627)X(14611)

Barycentrics 3 a^16-10 a^14 b^2+7 a^12 b^4+12 a^10 b^6-25 a^8 b^8+22 a^6 b^10-15 a^4 b^12+8 a^2 b^14-2 b^16-10 a^14 c^2+36 a^12 b^2 c^2-42 a^10 b^4 c^2+17 a^8 b^6 c^2-15 a^6 b^8 c^2+33 a^4 b^10 c^2-25 a^2 b^12 c^2+6 b^14 c^2+7 a^12 c^4-42 a^10 b^2 c^4+51 a^8 b^4 c^4-11 a^6 b^6 c^4-36 a^4 b^8 c^4+27 a^2 b^10 c^4+4 b^12 c^4+12 a^10 c^6+17 a^8 b^2 c^6-11 a^6 b^4 c^6+36 a^4 b^6 c^6-10 a^2 b^8 c^6-38 b^10 c^6-25 a^8 c^8-15 a^6 b^2 c^8-36 a^4 b^4 c^8-10 a^2 b^6 c^8+60 b^8 c^8+22 a^6 c^10+33 a^4 b^2 c^10+27 a^2 b^4 c^10-38 b^6 c^10-15 a^4 c^12-25 a^2 b^2 c^12+4 b^4 c^12+8 a^2 c^14+6 b^2 c^14-2 c^16 : :
Barycentrics 2 S^4 + (-63 R^2 SB-63 R^2 SC-6 SB SC+33 R^2 SW+14 SB SW+14 SC SW-8 SW^2)S^2 -162 R^4 SB SC+99 R^2 SB SC SW-12 SB SC SW^2 : :
X(31874) = 4*X[1553]-3*X[14094], 3*X[5627]-2*X[14611], 4*X[6070]-3*X[15035], 2*X[14480]-3*X[14644]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28946.

X(31874) lies on these lines: {1553,14094}, {5627,14611}, {6070,15035}, {14480,14644}

X(31875) = (name pending)

Barycentrics a^2 (2 a^12 b^4-6 a^10 b^6+8 a^8 b^8-8 a^6 b^10+6 a^4 b^12-2 a^2 b^14-a^12 b^2 c^2-4 a^10 b^4 c^2+5 a^8 b^6 c^2-3 a^6 b^8 c^2-2 a^4 b^10 c^2+5 a^2 b^12 c^2+2 a^12 c^4-4 a^10 b^2 c^4+24 a^8 b^4 c^4-19 a^6 b^6 c^4+15 a^4 b^8 c^4-15 a^2 b^10 c^4+3 b^12 c^4-6 a^10 c^6+5 a^8 b^2 c^6-19 a^6 b^4 c^6-3 a^4 b^6 c^6+8 a^2 b^8 c^6-11 b^10 c^6+8 a^8 c^8-3 a^6 b^2 c^8+15 a^4 b^4 c^8+8 a^2 b^6 c^8+16 b^8 c^8-8 a^6 c^10-2 a^4 b^2 c^10-15 a^2 b^4 c^10-11 b^6 c^10+6 a^4 c^12+5 a^2 b^2 c^12+3 b^4 c^12-2 a^2 c^14) : :
Barycentrics (3 R^2+2 SB+2 SC-2 SW)S^6 +(-21 R^2 SB SW-21 R^2 SC SW+2 SB SC SW+5 R^2 SW^2)S^4 + (21 R^2 SB SC SW^2+15 R^2 SB SW^3+15 R^2 SC SW^3-10 R^2 SW^4-2 SB SW^4-2 SC SW^4+2 SW^5)S^2 + R^2 SB SC SW^4-2 SB SC SW^5 : :
X(31875) = 3*X[6054]-4*X[13137]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28946.

X(31875) lies on this line: {6054,13137}

X(31876) = REFLECTION OF X(14731) IN X(24981)

Barycentrics 5 a^12-10 a^10 b^2+4 a^8 b^4+a^4 b^8+2 a^2 b^10-2 b^12-10 a^10 c^2+22 a^8 b^2 c^2-10 a^6 b^4 c^2-a^4 b^6 c^2-7 a^2 b^8 c^2+6 b^10 c^2+4 a^8 c^4-10 a^6 b^2 c^4+5 a^4 b^4 c^4+5 a^2 b^6 c^4-6 b^8 c^4-a^4 b^2 c^6+5 a^2 b^4 c^6+4 b^6 c^6+a^4 c^8-7 a^2 b^2 c^8-6 b^4 c^8+2 a^2 c^10+6 b^2 c^10-2 c^12 : :
Barycentrics 2 S^4+ (-54 R^4+27 R^2 SB+27 R^2 SC-6 SB SC+27 R^2 SW-6 SB SW-6 SC SW-4 SW^2)S^2 -63 R^2 SB SC SW+16 SB SC SW^2 : :
X(31876) = 5*X[110]-4*X[3258], 4*X[7471]-3*X[9140], 16*X[12068]-15*X[15059], X[14731]-2*X[24981], 5*X[15034]-4*X[16340]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28946.

X(31876) lies on these lines: {30,14094}, {110,3258}, {7471,9140}, {12068,15059}, {14731,24981}, {15034,16340}

X(31876) = reflection of X(14731) in X(24981)

X(31877) = (name pending)

Barycentrics a^2 (2 a^5 b^2-2 a^4 b^3-4 a^3 b^4+4 a^2 b^5+2 a b^6-2 b^7+a^5 b c-8 a^4 b^2 c+12 a^3 b^3 c+2 a^2 b^4 c-13 a b^5 c+6 b^6 c+2 a^5 c^2-8 a^4 b c^2+14 a^3 b^2 c^2-16 a^2 b^3 c^2+11 a b^4 c^2-3 b^5 c^2-2 a^4 c^3+12 a^3 b c^3-16 a^2 b^2 c^3+5 a b^3 c^3-b^4 c^3-4 a^3 c^4+2 a^2 b c^4+11 a b^2 c^4-b^3 c^4+4 a^2 c^5-13 a b c^5-3 b^2 c^5+2 a c^6+6 b c^6-2 c^7) : :
X(31877) = 5*X[100]-4*X[3025]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28946.

X(31877) lies on this line: {100,3025}

X(31878) = (name pending)

Barycentrics a^2 (2 a^8 b^4-6 a^6 b^6+6 a^4 b^8-2 a^2 b^10+a^8 b^2 c^2-4 a^6 b^4 c^2+2 a^4 b^6 c^2-a^2 b^8 c^2+2 a^8 c^4-4 a^6 b^2 c^4+14 a^4 b^4 c^4-7 a^2 b^6 c^4+3 b^8 c^4-6 a^6 c^6+2 a^4 b^2 c^6-7 a^2 b^4 c^6-b^6 c^6+6 a^4 c^8-a^2 b^2 c^8+3 b^4 c^8-2 a^2 c^10) : :
X(31878) = 3*X[671]-4*X[12833]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28946.

X(31878) lies on this line: {671,12833}

X(31879) = MIDPOINT OF X(15345) AND X(20120)

Barycentrics 2*a^16 - 11*a^14*b^2 + 21*a^12*b^4 - 9*a^10*b^6 - 25*a^8*b^8 + 43*a^6*b^10 - 29*a^4*b^12 + 9*a^2*b^14 - b^16 - 11*a^14*c^2 + 26*a^12*b^2*c^2 - 5*a^10*b^4*c^2 - 16*a^8*b^6*c^2 - 27*a^6*b^8*c^2 + 68*a^4*b^10*c^2 - 45*a^2*b^12*c^2 + 10*b^14*c^2 + 21*a^12*c^4 - 5*a^10*b^2*c^4 - 20*a^8*b^4*c^4 - 7*a^6*b^6*c^4 - 30*a^4*b^8*c^4 + 81*a^2*b^10*c^4 - 40*b^12*c^4 - 9*a^10*c^6 - 16*a^8*b^2*c^6 - 7*a^6*b^4*c^6 - 18*a^4*b^6*c^6 - 45*a^2*b^8*c^6 + 86*b^10*c^6 - 25*a^8*c^8 - 27*a^6*b^2*c^8 - 30*a^4*b^4*c^8 - 45*a^2*b^6*c^8 - 110*b^8*c^8 + 43*a^6*c^10 + 68*a^4*b^2*c^10 + 81*a^2*b^4*c^10 + 86*b^6*c^10 - 29*a^4*c^12 - 45*a^2*b^2*c^12 - 40*b^4*c^12 + 9*a^2*c^14 + 10*b^2*c^14 - c^16 : :
X(31879) = 3 X[5] - X[14143],3 X[5066] + X[23337]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28951.

X(31879) lies on these lines: {5, 252}, {30, 13856}, {113, 137}, {140, 15425}, {1154, 5501}, {5066, 20413}, {10272, 13362}, {11803, 14051}, {15345, 20120}, {18807, 19939}, {20030, 25150}

X(31879) = midpoint of X(15345) and X(20120)
X(31879) = reflection of X(140) and X(15425)
X(31879) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 1157, 23280}, {546, 22051, 137}

X(31880) = X(1)X(21)∩X(12)X(42)

Barycentrics a (b+c) (a^5-2 a^3 b^2+a b^4-a^3 b 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-b^3 c^2-b^2 c^3+a c^4+b c^4) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28952.

X(31880) lies on these lines: {1,21}, {12,42}, {65,3724}, {78,21020}, {581,24725}, {756,3191}, {899,6668}, {2177,11491}, {2294,9310}, {2646,20718}, {2658,2667}, {3120,11553}, {3720,4999}, {3725,3924}, {4343,5857}, {4647,22836}, {5855,10459}, {17018,20060}

X(31880) = {X(i), X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,5496,12081}

X(31881) = X(3)X(1093)∩X(577)X(1204)

Barycentrics a^2 (a^2-b^2-c^2)^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) (a^16-4 a^14 b^2+6 a^12 b^4-5 a^10 b^6+5 a^8 b^8-6 a^6 b^10+4 a^4 b^12-a^2 b^14-4 a^14 c^2+15 a^12 b^2 c^2-17 a^10 b^4 c^2+a^8 b^6 c^2+10 a^6 b^8 c^2-7 a^4 b^10 c^2+3 a^2 b^12 c^2-b^14 c^2+6 a^12 c^4-17 a^10 b^2 c^4+20 a^8 b^4 c^4-4 a^6 b^6 c^4-8 a^4 b^8 c^4-3 a^2 b^10 c^4+6 b^12 c^4-5 a^10 c^6+a^8 b^2 c^6-4 a^6 b^4 c^6+22 a^4 b^6 c^6+a^2 b^8 c^6-15 b^10 c^6+5 a^8 c^8+10 a^6 b^2 c^8-8 a^4 b^4 c^8+a^2 b^6 c^8+20 b^8 c^8-6 a^6 c^10-7 a^4 b^2 c^10-3 a^2 b^4 c^10-15 b^6 c^10+4 a^4 c^12+3 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14) : :
Barycentrics S^6 + (-SB SC+8 R^2 SW-2 SW^2)S^4 + (-1536 R^8+64 R^6 SB+64 R^6 SC+16 R^4 SB SC+1344 R^6 SW-32 R^4 SB SW-32 R^4 SC SW-12 R^2 SB SC SW-432 R^4 SW^2+4 R^2 SB SW^2+4 R^2 SC SW^2+2 SB SC SW^2+60 R^2 SW^3-3 SW^4)S^2 +2048 R^8 SB SC-1664 R^6 SB SC SW+496 R^4 SB SC SW^2-64 R^2 SB SC SW^3+3 SB SC SW^4 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28952.

X(31881) lies on these lines: {3,1093}, {577,1204}

X(31882) = (name pending)

Barycentrics (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)^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) (a^20-4 a^18 b^2+a^16 b^4+20 a^14 b^6-42 a^12 b^8+28 a^10 b^10+14 a^8 b^12-36 a^6 b^14+25 a^4 b^16-8 a^2 b^18+b^20-4 a^18 c^2+8 a^16 b^2 c^2+10 a^14 b^4 c^2-25 a^12 b^6 c^2-19 a^10 b^8 c^2+64 a^8 b^10 c^2-28 a^6 b^12 c^2-25 a^4 b^14 c^2+25 a^2 b^16 c^2-6 b^18 c^2+a^16 c^4+10 a^14 b^2 c^4-5 a^12 b^4 c^4-51 a^10 b^6 c^4+41 a^8 b^8 c^4+52 a^6 b^10 c^4-55 a^4 b^12 c^4-3 a^2 b^14 c^4+10 b^16 c^4+20 a^14 c^6-25 a^12 b^2 c^6-51 a^10 b^4 c^6+50 a^8 b^6 c^6+12 a^6 b^8 c^6+45 a^4 b^10 c^6-61 a^2 b^12 c^6+10 b^14 c^6-42 a^12 c^8-19 a^10 b^2 c^8+41 a^8 b^4 c^8+12 a^6 b^6 c^8+20 a^4 b^8 c^8+47 a^2 b^10 c^8-59 b^12 c^8+28 a^10 c^10+64 a^8 b^2 c^10+52 a^6 b^4 c^10+45 a^4 b^6 c^10+47 a^2 b^8 c^10+88 b^10 c^10+14 a^8 c^12-28 a^6 b^2 c^12-55 a^4 b^4 c^12-61 a^2 b^6 c^12-59 b^8 c^12-36 a^6 c^14-25 a^4 b^2 c^14-3 a^2 b^4 c^14+10 b^6 c^14+25 a^4 c^16+25 a^2 b^2 c^16+10 b^4 c^16-8 a^2 c^18-6 b^2 c^18+c^20) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28952.

X(31882) lies on this line: {5,8884}

X(31883) = (name pending)

Barycentrics a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+4 a^4 b^4-a^2 b^6-4 a^6 c^2+7 a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+4 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6) : :
Barycentrics (8 R^2-5 SB-5 SC)S^4 + (-64 R^4 SB-64 R^4 SC-4 R^2 SB SC-16 R^4 SW+32 R^2 SB SW+32 R^2 SC SW+4 R^2 SW^2-6 SB SW^2-6 SC SW^2)S^2 + 4 R^2 SB SW^3+4 R^2 SC SW^3-SB SW^4-SC SW^4 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28952.

X(31883) lies on this line: {6,95}

X(31884) = MIDPOINT OF X(376) AND X(10519)

Barycentrics a^2*(3*a^4 + 2*a^2*b^2 - 5*b^4 + 2*a^2*c^2 - 6*b^2*c^2 - 5*c^4) : :
Trilinears 4 cos A - sin A tan ω : :
X(31884) = 4 X[3] - X[6],5 X[3] - 2 X[182],13 X[3] - 4 X[575],11 X[3] - 2 X[576],2 X[3] + X[1350],7 X[3] - X[1351],X[3] + 2 X[3098],3 X[3] - X[5050],7 X[3] - 4 X[5092],5 X[3] - X[5093],19 X[3] - 4 X[5097],6 X[3] - X[5102],2 X[4] - 5 X[3763],5 X[6] - 8 X[182],13 X[6] - 16 X[575],11 X[6] - 8 X[576],X[6] + 2 X[1350],7 X[6] - 4 X[1351],X[6] + 8 X[3098],3 X[6] - 4 X[5050],7 X[6] - 16 X[5092],5 X[6] - 4 X[5093],19 X[6] - 16 X[5097],3 X[6] - 2 X[5102],X[20] + 2 X[141],2 X[40] + X[3242],X[64] + 2 X[159],2 X[66] + X[17845],X[67] + 2 X[16163],X[69] + 5 X[3522],2 X[74] + X[2930],4 X[140] - X[31670],13 X[182] - 10 X[575],11 X[182] - 5 X[576],4 X[182] + 5 X[1350],14 X[182] - 5 X[1351],X[182] + 5 X[3098],6 X[182] - 5 X[5050],4 X[182] - 5 X[5085],7 X[182] - 10 X[5092],19 X[182] - 10 X[5097],12 X[182] - 5 X[5102],X[193] - 13 X[21734],2 X[376] + X[599],5 X[376] + X[11180],X[382] - 4 X[24206],8 X[548] + X[15069],2 X[550] + X[1352],22 X[575] - 13 X[576],8 X[575] + 13 X[1350],28 X[575] - 13 X[1351],2 X[575] + 13 X[3098],12 X[575] - 13 X[5050],8 X[575] - 13 X[5085],7 X[575] - 13 X[5092],20 X[575] - 13 X[5093],19 X[575] - 13 X[5097],24 X[575] - 13 X[5102],4 X[576] + 11 X[1350],14 X[576] - 11 X[1351],X[576] + 11 X[3098],6 X[576] - 11 X[5050],4 X[576] - 11 X[5085],7 X[576] - 22 X[5092],10 X[576] - 11 X[5093],19 X[576] - 22 X[5097],12 X[576] - 11 X[5102],2 X[597] - 5 X[15692],5 X[599] - 2 X[11180],5 X[631] - 2 X[5480],2 X[1113] + X[15163],2 X[1114] + X[15162],7 X[1350] + 2 X[1351],X[1350] - 4 X[3098],3 X[1350] + 2 X[5050],7 X[1350] + 8 X[5092],5 X[1350] + 2 X[5093],19 X[1350] + 8 X[5097],3 X[1350] + X[5102],X[1351] + 14 X[3098],3 X[1351] - 7 X[5050],2 X[1351] - 7 X[5085],X[1351] - 4 X[5092],5 X[1351] - 7 X[5093],19 X[1351] - 28 X[5097],6 X[1351] - 7 X[5102],2 X[1386] - 5 X[7987],X[1498] - 4 X[15577],X[1657] + 2 X[3818],X[2097] + 2 X[6282],5 X[2916] - 2 X[8718],X[3094] + 2 X[5188],6 X[3098] + X[5050],4 X[3098] + X[5085],7 X[3098] + 2 X[5092],10 X[3098] + X[5093],19 X[3098] + 2 X[5097],12 X[3098] + X[5102],X[3146] - 7 X[3619],X[3416] + 2 X[4297],7 X[3523] - 4 X[3589],3 X[3524] - X[14853],7 X[3526] - 4 X[19130],7 X[3528] - X[6776],4 X[3530] - X[21850],X[3543] - 4 X[20582],5 X[3618] - 11 X[15717],5 X[3620] + X[14927],4 X[3631] - X[5921],2 X[3734] + X[14532],X[3751] - 7 X[16192],4 X[3844] - X[5691],2 X[5050] - 3 X[5085],7 X[5050] - 12 X[5092],5 X[5050] - 3 X[5093],19 X[5050] - 12 X[5097],7 X[5085] - 8 X[5092],5 X[5085] - 2 X[5093],19 X[5085] - 8 X[5097],3 X[5085] - X[5102],20 X[5092] - 7 X[5093],19 X[5092] - 7 X[5097],24 X[5092] - 7 X[5102],19 X[5093] - 20 X[5097],6 X[5093] - 5 X[5102],24 X[5097] - 19 X[5102],2 X[5476] - 5 X[15693],X[5695] + 2 X[24728],X[5894] + 2 X[15585],X[6144] - 4 X[8550],X[6144] - 22 X[21735],2 X[6593] - 5 X[15051],4 X[6698] - X[10733],X[8549] - 4 X[15578],2 X[8550] - 11 X[21735],5 X[8567] + X[9924],8 X[8703] + X[15533],4 X[10168] - 7 X[15700],3 X[10304] - X[25406],2 X[10516] - 3 X[21358],5 X[10519] - X[11180],X[10620] + 2 X[12584],X[10752] - 7 X[15036],2 X[11178] + X[15681],4 X[12041] - X[16010],4 X[12100] - X[20423],2 X[12122] + X[24273],2 X[12383] + X[25335],2 X[14718] + X[15588],X[14848] - 3 X[15706],X[14912] - 5 X[19708],X[14982] + 2 X[16111],5 X[15040] - 2 X[19140],13 X[15042] - 4 X[25556],X[15534] - 10 X[19708],X[15684] - 4 X[25561],5 X[15696] + X[18440],X[15704] + 2 X[18358],5 X[15712] - 2 X[18583],11 X[15723] - 8 X[25565],5 X[17821] - 2 X[19149],X[24476] + 2 X[31793]

X(31884) lies on these lines: {2, 21167}, {3, 6}, {4, 3763}, {20, 141}, {22, 7998}, {23, 21766}, {25, 5650}, {30, 10516}, {40, 3242}, {56, 10387}, {64, 159}, {66, 17845}, {67, 16163}, {69, 3522}, {74, 2930}, {140, 31670}, {154, 3917}, {165, 518}, {193, 21734}, {206, 15748}, {220, 3220}, {373, 7484}, {376, 599}, {381, 29317}, {382, 24206}, {394, 6636}, {524, 10304}, {542, 15041}, {548, 15069}, {549, 14561}, {550, 1352}, {597, 15692}, {611, 5010}, {613, 7280}, {631, 5480}, {698, 6194}, {732, 8716}, {1003, 22676}, {1113, 15163}, {1114, 15162}, {1204, 19459}, {1386, 7987}, {1407, 5285}, {1469, 5217}, {1498, 2916}, {1593, 7716}, {1657, 3818}, {1818, 3207}, {1843, 3516}, {1853, 7667}, {1974, 15750}, {2071, 8705}, {2077, 12594}, {2097, 6282}, {2781, 15035}, {2854, 5621}, {2883, 11821}, {2979, 3796}, {3056, 5204}, {3066, 15107}, {3146, 3619}, {3416, 4297}, {3515, 12294}, {3523, 3589}, {3524, 14853}, {3526, 19130}, {3528, 6776}, {3530, 21850}, {3534, 29012}, {3543, 20582}, {3564, 8703}, {3601, 24471}, {3618, 15717}, {3620, 14927}, {3631, 5921}, {3734, 14532}, {3751, 16192}, {3819, 9909}, {3844, 5691}, {4190, 26543}, {5020, 15082}, {5054, 19924}, {5227, 9841}, {5303, 25304}, {5406, 13616}, {5407, 13617}, {5422, 16981}, {5476, 15693}, {5584, 9049}, {5640, 7485}, {5646, 11284}, {5695, 24728}, {5731, 5846}, {5894, 15585}, {5965, 14093}, {5999, 15271}, {6144, 8550}, {6393, 14907}, {6593, 15051}, {6698, 10733}, {7379, 17327}, {7385, 17265}, {7386, 26958}, {7387, 10170}, {7390, 17245}, {7467, 21001}, {7492, 15066}, {7503, 16776}, {7512, 17821}, {8549, 15578}, {8556, 9756}, {8567, 9924}, {9026, 12329}, {9027, 21663}, {10168, 15700}, {10601, 11002}, {10620, 12584}, {10752, 15036}, {11012, 12595}, {11178, 15681}, {11188, 11413}, {11410, 19124}, {11414, 15030}, {11645, 15689}, {12041, 16010}, {12082, 16261}, {12100, 20423}, {12122, 24273}, {12383, 25335}, {12588, 15326}, {12589, 15338}, {14718, 15588}, {14848, 15706}, {14912, 15534}, {14915, 19596}, {14982, 16111}, {15040, 19140}, {15042, 25556}, {15067, 17814}, {15684, 25561}, {15696, 18440}, {15704, 18358}, {15712, 18583}, {15723, 25565}, {15988, 17548}, {17809, 22352}, {21737, 23251}, {24476, 31793}

X(31884) = midpoint of X(i) and X(j) for these {i,j}: {376, 10519}, {1350, 5085}
X(31884) = reflection of X(i) and X(j) for these {i,j}: {2, 21167}, {6, 5085}, {599, 10519}, {5085, 3}, {5093, 182}, {5102, 5050}, {5621, 15055}, {14561, 549}, {15534, 14912}
X(31884) = Schoutte-circle-inverse of X(21309)
X(31884) = isogonal conjugate of the isotomic conjugate of X(10513)
X(31884) = Thomson-isogonal conjugate of X(14482)
X(31884) = barycentric product X(6)*X(10513)
X(31884) = barycentric quotient X(10513)/X(76)
X(31884) = X(1350)-Gibert-Moses centroid
X(31884) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1350, 6}, {3, 1351, 5092}, {3, 3098, 1350}, {3, 3430, 4255}, {3, 5050, 17508}, {3, 5188, 3053}, {3, 8722, 5210}, {3, 11824, 1151}, {3, 11825, 1152}, {3, 12305, 6410}, {3, 12306, 6409}, {3, 14538, 11481}, {3, 14539, 11480}, {3, 14540, 22238}, {3, 14541, 22236}, {3, 15644, 11425}, {3, 30270, 5013}, {15, 16, 21309}, {182, 11477, 6}, {1151, 1152, 32}, {3053, 3094, 6}, {3053, 5013, 13357}, {3098, 14810, 3}, {5013, 5017, 6}, {5050, 5102, 6}, {5050, 17508, 5085}, {5085, 5102, 5050}, {5092, 22330, 182}, {6409, 6410, 5023}, {6411, 6412, 8588}, {9756, 22712, 8556}, {10323, 15577, 2916}, {11425, 19161, 6}, {11480, 11481, 187}, {22236, 22238, 5007}

X(31885) = X(32)X(184)∩X(3292)X(21309)

Barycentrics a^4*(5*a^2 + 11*(b^2 + c^2)) : :

X(31885) lies on these lines {32, 184}, {3292, 21309}, {5039, 22112}

X(31885) = isogonal conjugate of the isotomic conjugate of X(22246)
X(31885) = barycentric product X(6)*X(22246)
X(31885) = barycentric quotient X(22246)/X(76)

X(31886) = X(6)X(20218)∩X(20)X(64)

Barycentrics a^10-13 a^8 b^2+26 a^6 b^4-10 a^4 b^6-11 a^2 b^8+7 b^10-13 a^8 c^2-36 a^6 b^2 c^2+10 a^4 b^4 c^2+12 a^2 b^6 c^2+27 b^8 c^2+26 a^6 c^4+10 a^4 b^2 c^4-2 a^2 b^4 c^4-34 b^6 c^4-10 a^4 c^6+12 a^2 b^2 c^6-34 b^4 c^6-11 a^2 c^8+27 b^2 c^8+7 c^10 : :
X(31886) = 2 X[6] + X[20218],X[69] - 4 X[253],5 X[3618] - 2 X[17037]

X(31886) lies on the cubic K1094 and these lines: {6,20218}, {20,64}, {3618,17037}

X(31887) = X(69)X(17037)∩X(141)X(20218)

Barycentrics 9 a^10+15 a^8 b^2-38 a^6 b^4-2 a^4 b^6+13 a^2 b^8+3 b^10+15 a^8 c^2+28 a^6 b^2 c^2+2 a^4 b^4 c^2-20 a^2 b^6 c^2-25 b^8 c^2-38 a^6 c^4+2 a^4 b^2 c^4+14 a^2 b^4 c^4+22 b^6 c^4-2 a^4 c^6-20 a^2 b^2 c^6+22 b^4 c^6+13 a^2 c^8-25 b^2 c^8+3 c^10 : :
X(31887) = X[69] + 2 X[17037],4 X[141] - X[20218],4 X[253] - 7 X[3619],8 X[1249] - 5 X[3618]

X(31887) lies on the cubic K1094 and these lines: {69,17037}, {141,20218}, {253,3619}, {264,1249}, {648,14927}, {1503,1992}

X(31888) = X(2)X(191)∩X(4)X(13465)

Barycentrics 3 a^4+4 a^3 b-2 a^2 b^2-4 a b^3-b^4+4 a^3 c-a^2 b c-a b^2 c-2 a^2 c^2-a b c^2+2 b^2 c^2-4 a c^3-c^4 : :
X(31888) = 3 X[2] - 4 X[191],9 X[2] - 8 X[11263],8 X[21] - 7 X[3622],5 X[21] - 4 X[16137],X[145] - 4 X[3648],5 X[145] - 8 X[5441],3 X[191] - 2 X[11263],4 X[2475] - 5 X[3617],3 X[2475] - 4 X[21677],5 X[3091] - 4 X[16159],5 X[3522] - 4 X[16132],7 X[3523] - 8 X[22937],5 X[3617] - 8 X[11684],5 X[3617] - 2 X[20084],15 X[3617] - 16 X[21677],7 X[3622] - 16 X[3650],35 X[3622] - 32 X[16137],5 X[3623] - 4 X[16126],8 X[3647] - 7 X[15676],5 X[3648] - 2 X[5441],4 X[3649] - 5 X[15674],5 X[3650] - 2 X[16137],7 X[4678] - 4 X[16118],4 X[5441] - 5 X[15680],3 X[5603] - 4 X[22936],3 X[9778] - 2 X[16143],5 X[10595] - 6 X[28453],4 X[11263] - 3 X[14450],8 X[11281] - 9 X[15672],4 X[11544] - 5 X[31254],4 X[11684] - X[20084],3 X[11684] - 2 X[21677],3 X[12535] - 2 X[12682],4 X[12682] - 3 X[12849],3 X[20084] - 8 X[21677]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28959.

X(31888) lies on these lines: {2, 191}, {4, 13465}, {8, 12535}, {20, 2771}, {21, 999}, {30, 12245}, {46, 26792}, {79, 10590}, {144, 1654}, {145, 758}, {390, 17637}, {962, 7701}, {1749, 3086}, {3091, 16159}, {3120, 24898}, {3189, 20066}, {3522, 16132}, {3523, 22937}, {3623, 16126}, {3647, 15676}, {3649, 7288}, {3869, 20067}, {3873, 28646}, {4127, 15228}, {4295, 18259}, {4661, 6361}, {4678, 16118}, {5178, 28534}, {5180, 6763}, {5603, 22936}, {5904, 20095}, {5905, 12913}, {9778, 16143}, {10032, 15677}, {10056, 18244}, {10528, 20214}, {10587, 20059}, {10595, 28453}, {10786, 16116}, {11041, 13100}, {11281, 15672}, {11544, 31254}, {12514, 17483}, {17484, 21077}

X(31888) = reflection of X(i) and X(j) for these {i,j}: {4, 13465}, {21, 3650}, {145, 15680}, {962, 7701}, {2475, 11684}, {12849, 12535}, {14450, 191}, {15677, 10032}, {15680, 3648}, {16116, 16139}, {20084, 2475}
X(31888) = anticomplement of X(14450)
X(31888) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {191, 14450, 2}, {11684, 20084, 3617}

X(31889) = (name pending)

Barycentrics (a-b-c)^3 (b-c)^2 (2 a^3+a^2 b-2 a b^2-b^3+a^2 c+2 a b c+b^2 c-2 a c^2+b c^2-c^3)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4)^2 : :

See L. A. Emelianov, Kadir Altintas and Ercole Suppa, Hyacinthos 28961.

X(31889) lies on the incirle and these lines: {}

X(31890) = X(11)X(24185)∩X(55)X(15322)

Barycentrics (a - b - c) (b - c)^2 (b + c)^2 (3 a + b + c)^2 : :

See L. A. Emelianov, Kadir Altintas and Ercole Suppa, Hyacinthos 28961.

X(31890) lies on the incircle and these lines: {11,24185}, {55,15322}, {1358,4934}

X(31890) = barycentric product of X(i) and X(j) for these {i,j}: {4841, 4843}
X(31890) = barycentric quotient of X(i) and X(j) for these {i,j}: {4832, 5545}, {4843, 4633}, {8653, 4627}

X(31891) = (name pending)

Barycentrics (a-b-c)^5 (b-c)^6 (2 a^2-a b-b^2-a c+2 b c-c^2)^2 : :

See L. A. Emelianov, Kadir Altintas and Ercole Suppa, Hyacinthos 28961.

X(31891) lies on the incircle and this line: {3022,11193}

X(31892) = X(55)X(28847)∩X(56)X(28848)

Barycentrics a^2 (a-b-c) (b-c)^2 (a^2-b^2-c^2)^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^3-a^2 b+a b^2-b^3-a^2 c+2 a b c+b^2 c+a c^2+b c^2-c^3)^2 : :

See L. A. Emelianov, Kadir Altintas and Ercole Suppa, Hyacinthos 28961.

X(31892) lies on the incircle and these lines: {55,28847}, {56,28848}, {1362,6180}, {3271,15615}, {3323,4014}

X(31893) = X(1360)X(6284)∩X(1362)X(12680)

Barycentrics 4 (a-b-c)^3 (b-c)^2 (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3)^2 (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4)^2 : :

See L. A. Emelianov, Kadir Altintas and Ercole Suppa, Hyacinthos 28961.

X(31893) lies on the incircle and these lines: {1360,6284}, {1362,12680}, {3022,4081}

X(31893) = barycentric product of X(i) and X(j) for these {i,j}: {1097,3119}, {1146,6060}, {7338,23970}
X(31893) = barycentric quotient of X(i) and X(j) for these {i,j}: {6060,1275}, {7338,23586}
X(31893) = trilinear product of X(i) and X(j) for these {i,j}: {1097,3022}, {1097,3022}, {2310,6060} ,{2310,6060}, {7338,24010}, {7338,24010}
X(31893) = trilinear quotient of X(i) and X(j) for these {i,j}: {6060,7045}, {7338,24013}

X(31894) = X(4)X(7)∩X(3900)X(17069)

Barycentrics 2 a (a^8 b-10 a^6 b^3+16 a^5 b^4-16 a^3 b^6+10 a^2 b^7-b^9+a^8 c-2 a^7 b c+8 a^6 b^2 c-10 a^5 b^3 c-14 a^4 b^4 c+26 a^3 b^5 c-14 a b^7 c+5 b^8 c+8 a^6 b c^2-8 a^5 b^2 c^2+14 a^4 b^3 c^2-8 a^3 b^4 c^2-36 a^2 b^5 c^2+32 a b^6 c^2-2 b^7 c^2-10 a^6 c^3-10 a^5 b c^3+14 a^4 b^2 c^3-4 a^3 b^3 c^3+26 a^2 b^4 c^3-2 a b^5 c^3-14 b^6 c^3+16 a^5 c^4-14 a^4 b c^4-8 a^3 b^2 c^4+26 a^2 b^3 c^4-32 a b^4 c^4+12 b^5 c^4+26 a^3 b c^5-36 a^2 b^2 c^5-2 a b^3 c^5+12 b^4 c^5-16 a^3 c^6+32 a b^2 c^6-14 b^3 c^6+10 a^2 c^7-14 a b c^7-2 b^2 c^7+5 b c^8-c^9) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28962.

X(31894) lies on these lines: {4,7}, {3900,17069}

X(31895) = X(4)X(8)∩X(36)X(24320)

Barycentrics 2 a (a+b+c) (a^5 b-3 a^4 b^2-4 a^3 b^3+4 a^2 b^4+3 a b^5-b^6+a^5 c+4 a^3 b^2 c+4 a^2 b^3 c-5 a b^4 c-4 b^5 c-3 a^4 c^2+4 a^3 b c^2-4 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2-4 a^3 c^3+4 a^2 b c^3-2 a b^2 c^3+8 b^3 c^3+4 a^2 c^4-5 a b c^4+b^2 c^4+3 a c^5-4 b c^5-c^6) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28962.

X(31895) lies on these lines: {4,8}, {36,24320}, {513,20315}, {3660,28036}, {3814,4138}, {5123,20306}

X(31896) = X(4)X(9)∩X(514)X(28984)

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) (2 a^6-a^5 b-a^4 b^2-2 a^3 b^3+3 a b^5-b^6-a^5 c-2 a^4 b c+4 a^3 b^2 c-3 a b^4 c+2 b^5 c-a^4 c^2+4 a^3 b c^2+b^4 c^2-2 a^3 c^3-4 b^3 c^3-3 a b c^4+b^2 c^4+3 a c^5+2 b c^5-c^6) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28962.

X(31896) lies on these lines: {4,9}, {514,28984}, {1323,24635}, {4295,27541}, {4512,28118}, {5088,5273}

X(31897) = X(4)X(9)∩X(101)X(29219)

Barycentrics a^5 b-a^3 b^3-a^2 b^4+b^6+a^5 c-2 a^4 b c-a^3 b^2 c+a^2 b^3 c+b^5 c-a^3 b c^2+4 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+b c^5+c^6 : :
X(31897) = 5*X[1698]-X[5018]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28962.

X(31897) lies on these lines: {4,9}, {101,29219}, {514,20315}, {1146,29016}, {1698,5018}, {3332,27541}, {5044,5074}, {5224,7112}

X(31898) = X(395)X(1506)∩X(530)X(15544)

Barycentrics 6*(9*R^2-3*SA+SW)*S^2-sqrt(3)*(4*S^2+3*(9*SA-11*SW)*R^2+3*SA^2-6*SB*SC-SW^2)*S-3*(SB+SC)*(3*R^2*(3*SA-2*SW)-SA^2+SB*SC) : :

See Vu Thanh Tung and César Lozada, Hyacinthos 28963.

X(31898) lies on these lines: {395, 1506}, {530, 15544}, {8838, 9112}

X(31899) = X(396)X(1506)∩X(531)X(15544)

Barycentrics 6*(9*R^2-3*SA+SW)*S^2+sqrt(3)*(4*S^2+3*(9*SA-11*SW)*R^2+3*SA^2-6*SB*SC-SW^2)*S-3*(SB+SC)*(3*R^2*(3*SA-2*SW)-SA^2+SB*SC) : :

See Vu Thanh Tung and César Lozada, Hyacinthos 28963.

X(31899) lies on these lines: {396, 1506}, {531, 15544}, {8466, 16397}, {8836, 9113}

X(31900) = X(2)X(3)∩X(19)X(596)

Barycentrics (a + b)*(a + c)*(2*a + b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31900) has Shinagawa coefficients (-4F$a$, $a$³+2(E+F)$a$+2abc).

X(31900) lies on these lines: {2, 3}, {19, 596}, {58, 4000}, {107, 28173}, {112, 9108}, {242, 17729}, {284, 3487}, {1125, 1839}, {1172, 3296}, {1396, 5228}, {1780, 3474}, {1829, 5773}, {2140, 17171}, {2194, 4295}, {2328, 6361}, {2355, 3916}, {2360, 5603}

X(31900) = polar conjugate of X(6539)

X(31901) = X(2)X(3)∩X(19)X(6763)

Barycentrics (a + b)*(a + c)*(3*a + 2*b + 2*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31901) has Shinagawa coefficients (-3F$a$, $a$³ +(E+F)$a$+abc).

X(31901) lies on these lines: {2, 3}, {19, 6763}, {107, 28197}, {1172, 5557}, {1396, 2906}, {1871, 26201}, {2360, 11522}

X(31902) = X(2)X(3)∩X(19)X(191)

Barycentrics (a + b)*(a + c)*(a + 2*b + 2*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31902) has Shinagawa coefficients (-F$a$, $a$³-(E+F)$a$-abc).

X(31902) lies on these lines: {2, 3}, {19, 191}, {58, 9579}, {79, 1172}, {92, 11684}, {107, 28145}, {112, 9103}, {225, 1396}, {278, 3649}, {281, 18253}, {284, 9612}, {758, 5307}, {971, 18180}, {1698, 4877}, {1699, 2360}, {1714, 1778}, {1848, 11263}, {1859, 17637}, {1871, 2771}, {1896, 10308}, {2322, 3650}, {3824, 5333}, {3927, 28605}, {4654, 4658}, {5016, 19848}, {5235, 31445}, {6197, 16139}, {17188, 18483}, {17924, 29150}

X(31902) = polar conjugate of isotomic conjugate of X(5333)

X(31903) = X(2)X(3)∩X(58)X(24177)

Barycentrics (a + b)*(a + c)*(3*a + b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31903) has Shinagawa coefficients (-6F$a$, $a$³+4(E+F)$a$+4abc).

X(31903) lies on these lines: {2, 3}, {58, 24177}, {107, 28193}, {112, 9105}, {270, 757}, {1172, 5558}, {1890, 16706}, {2163, 8747}, {2328, 5493}, {2360, 13464}, {4652, 5338}

X(31904) = X(2)X(3)∩X(19)X(648)

Barycentrics (a + b)*(a + c)*(a^2 + 2*a*b + 2*a*c + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31904) has Shinagawa coefficients (2(E+F)F-3F$a$², 2S²+3(E+F)$a$²+6abc$a$).

X(31904) lies on these lines: {2, 3}, {19, 648}, {86, 24683}, {107, 28842}, {1172, 1963}, {18653, 25935}

X(31905) = X(2)X(3)∩X(86)X(28106)

Barycentrics (a + b)*(a + c)*(a^2 - b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31905) has Shinagawa coefficients (2(E+F)F+F$a$²,2S² -(E+F)$a$²+2abc$a$).

X(31905) lies on these lines: {2, 3}, {86, 28106}, {92, 1973}, {107, 12032}, {112, 9073}, {239, 242}, {264, 18048}, {286, 870}, {350, 2201}, {648, 18822}, {1333, 28087}, {1851, 26626}, {5317, 28090}, {7033, 31623}, {7192, 7254}

X(31905) = polar conjugate of the isogonal conjugate of X(5009)
X(31905) = polar conjugate of the isotomic conjugate of X(33295)

X(31906) = X(2)X(3)∩X(19)X(3875)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 + a*b + b^2 + a*c + c^2) : :

As a point on the Euler line, X(31906) has Shinagawa coefficients (2(E+F)F+F$a$², -2(E+F)$a$²-abc$a$).

X(31906) lies on these lines: {2, 3}, {19, 3875}, {1396, 1434}, {1890, 17023}, {4384, 5338}

X(31907) = X(2)X(3)∩X(107)X(28861)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 + 2*a*b + b^2 + 2*a*c + b*c + c^2) : :

As a point on the Euler line, X(31907) has Shinagawa coefficients (2(E+F)F+3F$a$², -2S²-5(E+F)$a$²-6abc$a$).

X(31907) lies on these lines: {2, 3}, {107, 28861}

X(31908) = X(2)X(3)∩X(81)X(7224)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 + b^2 - b*c + c^2) : :

As a point on the Euler line, X(31908) has Shinagawa coefficients (6(E+F)F+F$a$², 2S²-3(E+F)$a$²+2abc$a$).

X(31908) lies on these lines: {2, 3}, {107, 28884}

X(31909) = X(2)X(3)∩X(81)X(7224)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + b*c + c^2) : :

As a point on the Euler line, X(31909) has Shinagawa coefficients (2(E+F)F-F$a$², -2S²-(E+F)$a$²-2abc$a$).

X(31909) lies on these lines: {2, 3}, {81, 7224}, {86, 26130}, {92, 304}, {107, 28844}, {112, 9075}, {270, 27660}, {278, 17084}, {286, 334}, {1172, 2905}, {1400, 7282}, {1848, 7018}, {3661, 3781}, {4269, 20305}, {5090, 17033}, {5342, 29966}, {7009, 16826}, {7102, 17316}, {17167, 25935}

X(31910) = X(2)X(3)∩X(19)X(4360)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 + 2*a*b + b^2 + 2*a*c + c^2) : :

As a point on the Euler line, X(31910) has Shinagawa coefficients (2(E+F)F+2F$a$², -3(E+F)$a$²-2abc$a$).

X(31910) lies on these lines: {2, 3}, {19, 4360}, {239, 2355}, {1839, 17322}

X(31911) = (name pending)

Barycentrics (a + b)*(a + c)*(2*a^2 + 3*a*b + 3*a*c + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31911) has Shinagawa coefficients (2(E+F)F-5F$a$², 2S²+5(E+F)$a$²+8abc$a$).

X(31911) lies on these lines: {2, 3}

X(31912) = X(2)X(3)∩X(648)X(1474)

Barycentrics (a + b)*(a + c)*(2*a^2 + a*b + a*c - b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31912) has Shinagawa coefficients (2(E+F)F+3F$a$², 2S²-3(E+F)$a$²).

X(31912) lies on these lines: {2, 3}, {648, 1474}, {4393, 23095}

X(31913) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 + 3*a*b + b^2 + 3*a*c + b*c + c^2) : :

As a point on the Euler line, X(31913) has Shinagawa coefficients (2(E+F)F+5F$a$², -2S²-7(E+F)$a$²-8abc$a$).

X(31912) lies on these lines: {2, 3}

X(31914) = X(2)X(3)∩X(107)X(28857)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 + 3*a*b - b^2 + 3*a*c + b*c - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31914) has Shinagawa coefficients (6(E+F)F-5F$a$², 2S²+3(E+F)$a$²+8abc$a$).

X(31914) lies on these lines: {2, 3}, {107, 28857}

X(31915) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 + a*b + b^2 + a*c - b*c + c^2) : :

As a point on the Euler line, X(31915) has Shinagawa coefficients (6(E+F)F+3F$a$², 2S²-5(E+F)$a$²).

X(31915) lies on these lines: {2, 3}

X(31916) = X(2)X(3)∩X(6331)X(6336)

Barycentrics (a + b)*(a + c)*(a*b - 2*b^2 + a*c - b*c - 2*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31916) has Shinagawa coefficients (6(E+F)F-3F$a$², -2S²-(E+F)$a$²).

X(31916) lies on these lines: {2, 3}, {6331, 6336}, {8756, 17171}

X(31917) = X(4)X(17300)∩X(7)X(27)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 - b*c + c^2) : :

X(31917) lies on these lines: {4, 17300}, {7, 27}, {86, 28106}, {92, 30035}, {286, 334}, {469, 18139}, {2354, 17923}, {3662, 3784}, {15149, 27334}, {16747, 20435}, {26109, 28100}

X(31918) = X(2)X(3)∩X(19)X(17151)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 + a*b + 2*b^2 + a*c + 2*c^2) : :

As a point on the Euler line, X(31918) has Shinagawa coefficients (4(E+F)F+F$a$², -3(E+F)$a$²-abc$a$).

X(31918) lies on these lines: {2, 3}, {19, 17151}, {278, 7198}, {1396, 10481}

X(31919) = X(2)X(3)∩X(19)X(18206)

Barycentrics (a + b)*(a + c)*(a^2 + 3*a*b + 3*a*c + 2*b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31919) has Shinagawa coefficients (2(E+F)F-2F$a$², 2S²+2(E+F)$a$²+5abc$a$).

X(31919) lies on these lines: {2, 3}, {19, 18206}, {278, 1434}

X(31920) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 + 2*a*b + 2*b^2 + 2*a*c + b*c + 2*c^2) : :

As a point on the Euler line, X(31920) has Shinagawa coefficients (6(E+F)F+3F$a$², -2S²-7(E+F)$a$²-6abc$a$).

X(31920) lies on these lines: {2, 3}

X(31921) = X(2)X(3)∩X(107)X(28892)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - 2*a*b + 2*b^2 - 2*a*c - b*c + 2*c^2) : :

As a point on the Euler line, X(31921) has Shinagawa coefficients (10(E+F)F-3F$a$², 2S²-(E+F)$a$²+6abc$a$).

X(31921) lies on these lines: {2, 3}, {107, 28892}

X(31922) = X(2)X(3)∩X(1172)X(18166)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(3*a*b - b^2 + 3*a*c + 2*b*c - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31922) has Shinagawa coefficients (4(E+F)F-2F$a$², 2S²+(E+F)$a$²+5abc$a$).

X(31922) lies on these lines: {2, 3}, {1172, 18166}, {5307, 17911}

X(31923) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - 3*a*b + 2*b^2 - 3*a*c - 2*b*c + 2*c^2) : :

As a point on the Euler line, X(31923) has Shinagawa coefficients (6(E+F)F-2F$a$², 2S²+5abc$a$).

X(31923) lies on these lines: {2, 3}

X(31924) = X(2)X(3)∩X(4967)X(8756)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^2 - a*b + 2*b^2 - a*c - 2*b*c + 2*c^2) : :

As a point on the Euler line, X(31924) has Shinagawa coefficients (6(E+F)F, 2S²-2(E+F)$a$²+3abc$a$).

X(31924) lies on these lines: {2, 3}, {4967, 8756}

X(31925) = X(2)X(3)∩X(19)X(25590)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - a*b + 2*b^2 - a*c + 2*c^2) : :

As a point on the Euler line, X(31925) has Shinagawa coefficients (4(E+F)F-F$a$², -(E+F)$a$²+abc$a$).

X(31925) lies on these lines: {2, 3}, {19, 25590}, {278, 3665}, {1172, 17171}, {1396, 5236}, {16747, 31623}

X(31926) = X(2)X(3)∩X(81)X(3673)

Barycentrics (a + b)*(a + c)*(a^2 - a*b - a*c - 2*b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31926) has Shinagawa coefficients (2(E+F)F, 2S²+3abc$a$).

X(31926) lies on these lines: {2, 3}, {81, 3673}, {286, 648}, {1014, 26856}, {1396, 1509}, {1839, 4357}, {1973, 5307}, {4441, 23151}

X(31927) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^2 + 3*a*b + 2*b^2 + 3*a*c + b*c + 2*c^2) : :

As a point on the Euler line, X(31927) has Shinagawa coefficients (6(E+F)F+5F$a$², -2S² -9(E+F)$a$²-8abc$a$).

X(31927) lies on these lines: {2, 3}

X(31928) = (name pending)

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^2 + a*b + 2*b^2 + a*c - b*c + 2*c^2) : :

As a point on the Euler line, X(31928) has Shinagawa coefficients (10(E+F)F+3F$a$², 2S² -7(E+F)$a$²).

X(31928) lies on these lines: {2, 3}

X(31929) = (name pending)

Barycentrics (a + b)*(a + c)*(3*a*b - 2*b^2 + 3*a*c + b*c - 2*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(31929) has Shinagawa coefficients (10(E+F)F-5F$a$²,2S² +(E+F)$a$²+8abc$a$).

X(31929) lies on these lines: {2, 3}

X(31930) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMNORMAL AND 1st MORLEY-MIDPOINT

Barycentrics a*((-(2*cos(A/3)*cos(C/3)+cos(B/3))*cos(2*B/3)*sin(2*A/3-2*C/3)*sin(B/3+π/3)+(cos(C/3)+2*cos(A/3)*cos(B/3))*cos(2*C/3)*sin(2*A/3-2*B/3)*sin(C/3+π/3))*cos(A)-(2*cos(C/3)*cos(B/3)+cos(A/3))*(-cos(2*B/3)*sin(2*A/3-2*C/3)*sin(B/3+π/3)*cos(B)+cos(2*C/3)*sin(2*A/3-2*B/3)*sin(C/3+π/3)*cos(C))) : :
Trilinears cos(A)*((2*u*w+v)*y+(2*u*v+w)*z)-(u+2*v*w)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3) : :
x : y : z = sin(2*B/3-2*C/3)*cos(2*A/3)*sin(A/3+π/3) : :

Centers X(31930)-X(31935) were contributed by César E. Lozada, April 6, 2019.

X(31924) lies on these lines: {3,356}, {3604,6125}

X(31931) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMNORMAL AND 2nd MORLEY-MIDPOINT

Barycentrics a*(((2*cos(A/3+π/3)*cos(C/3+π/3)-cos(B/3+π/3))*sin(2*B/3+π/6)*sin(2*A/3-2*C/3)*cos(B/3+π/6)-(2*cos(A/3+π/3)*cos(B/3+π/3)-cos(C/3+π/3))*sin(2*C/3+π/6)*sin(2*A/3-2*B/3)*cos(C/3+π/6))*cos(A)-(2*cos(B/3+π/3)*cos(C/3+π/3)-cos(A/3+π/3))*(sin(2*B/3+π/6)*sin(2*A/3-2*C/3)*cos(B/3+π/6)*cos(B)-sin(2*C/3+π/6)*sin(2*A/3-2*B/3)*cos(C/3+π/6)*cos(C))) : :
Trilinears cos(A)*((2*u*w+v)*y+(2*u*v+w)*z)-(u+2*v*w)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3+4*π/3) : :
x : y : z = sin(2*B/3-2*C/3)*cos(2*A/3+2*π/3)*sin(A/3+2*π/3) : :

X(31931) lies on these lines: {3,3276}, {3602,6124}

X(31932) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMNORMAL AND 3rd MORLEY-MIDPOINT

Barycentrics a*((-(2*sin(A/3+π/6)*sin(C/3+π/6)-sin(B/3+π/6))*sin(2*A/3-2*C/3)*cos(2*B/3+π/3)*sin(B/3)+(2*sin(A/3+π/6)*sin(B/3+π/6)-sin(C/3+π/6))*sin(2*A/3-2*B/3)*cos(2*C/3+π/3)*sin(C/3))*cos(A)+(2*sin(B/3+π/6)*sin(C/3+π/6)-sin(A/3+π/6))*sin(2*A/3-2*C/3)*cos(2*B/3+π/3)*sin(B/3)*cos(B)-(2*sin(B/3+π/6)*sin(C/3+π/6)-sin(A/3+π/6))*sin(2*A/3-2*B/3)*cos(2*C/3+π/3)*sin(C/3)*cos(C)) : :
Trilinears cos(A)*((2*u*w-v)*y+(2*u*v-w)*z)-(2*v*w-u)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3-π/3) : :
x : y : z = sin(2*B/3-2*C/3)*cos(2*A/3+π/3)*sin(A/3) : :

X(31932) lies on these lines: {3,3277}, {3603,6123}

X(31933) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMTANGENTIAL AND 1st MORLEY-MIDPOINT

Barycentrics a*((-(2*cos(A/3)*cos(C/3)+cos(B/3))*sin(-C/3+A/3)*(cos(B/3)+cos(-C/3+A/3)*cos(2*B/3))*sin(B/3+π/3)+(cos(C/3)+2*cos(A/3)*cos(B/3))*sin(A/3-B/3)*(cos(C/3)+cos(A/3-B/3)*cos(2*C/3))*sin(C/3+π/3))*cos(A)+(2*cos(C/3)*cos(B/3)+cos(A/3))*sin(-C/3+A/3)*(cos(B/3)+cos(-C/3+A/3)*cos(2*B/3))*sin(B/3+π/3)*cos(B)-(2*cos(C/3)*cos(B/3)+cos(A/3))*sin(A/3-B/3)*(cos(C/3)+cos(A/3-B/3)*cos(2*C/3))*sin(C/3+π/3)*cos(C)) : :
Trilinears ((2*u*w+v)*y+(2*u*v+w)*z)*cos(A)-(u+2*v*w)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3) : :
x : y : z = sin(B/3-C/3)*(cos(A/3)+cos(B/3-C/3)*cos(2*A/3))*sin(A/3+π/3) : :

X(31933) lies on these lines: {3,356}, {3604,6122}

X(31993) = isotomic conjugate of X(37870)

X(31934) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMTANGENTIAL AND 2nd MORLEY-MIDPOINT

Barycentrics a*((-(2*cos(A/3+π/3)*cos(C/3+π/3)-cos(B/3+π/3))*sin(-C/3+A/3)*(cos(B/3+π/3)+sin(2*B/3+π/6)*cos(-C/3+A/3))*cos(B/3+π/6)+(2*cos(A/3+π/3)*cos(B/3+π/3)-cos(C/3+π/3))*sin(A/3-B/3)*(cos(C/3+π/3)+sin(2*C/3+π/6)*cos(A/3-B/3))*cos(C/3+π/6))*cos(A)+(cos(A/3+π/3)-2*cos(B/3+π/3)*cos(C/3+π/3))*(-(cos(B/3+π/3)+sin(2*B/3+π/6)*cos(-C/3+A/3))*sin(-C/3+A/3)*cos(B/3+π/6)*cos(B)+(cos(C/3+π/3)+sin(2*C/3+π/6)*cos(A/3-B/3))*sin(A/3-B/3)*cos(C/3+π/6)*cos(C))) : :
Trilinears ((2*u*w-v)*y+(2*u*v-w)*z)*cos(A)+(u-2*v*w)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3+π/3) : :
x : y : z = sin(B/3-C/3)*(cos(A/3+π/3)-cos(B/3-C/3)*cos(2*A/3+2*π/3))*sin(A/3+2*π/3) : :

X(31934) lies on these lines: {3,3276}, {3602,6121}

X(31935) = HOMOTHETIC CENTER OF THESE EQUILATERAL TRIANGLES: CIRCUMTANGENTIAL AND 3rd MORLEY-MIDPOINT

Barycentrics a*((-(2*sin(A/3+π/6)*sin(C/3+π/6)-sin(B/3+π/6))*sin(-C/3+A/3)*(sin(B/3+π/6)+cos(-C/3+A/3)*cos(2*B/3+π/3))*sin(B/3)+(2*sin(A/3+π/6)*sin(B/3+π/6)-sin(C/3+π/6))*sin(A/3-B/3)*(sin(C/3+π/6)+cos(A/3-B/3)*cos(2*C/3+π/3))*sin(C/3))*cos(A)-(2*sin(B/3+π/6)*sin(C/3+π/6)-sin(A/3+π/6))*(-sin(-C/3+A/3)*(sin(B/3+π/6)+cos(-C/3+A/3)*cos(2*B/3+π/3))*sin(B/3)*cos(B)+sin(A/3-B/3)*(sin(C/3+π/6)+cos(A/3-B/3)*cos(2*C/3+π/3))*sin(C/3)*cos(C))) : :
Trilinears ((2*u*w+v)*y+(2*u*v+w)*z)*cos(A)-(2*v*w+u)*(y*cos(B)+z*cos(C)) : : , where:
u : v : w = cos(A/3+2*π/3) : :
x : y : z = sin(B/3-C/3)*(cos(A/3+2*π/3)-cos(B/3-C/3)*cos(2*A/3+π/3))*sin(A/3) : :

X(31935) lies on these lines: {3,3277}, {3603,6120}

X(31936) = X(4)X(1001)∩X(5)X(6261)

Barycentrics a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7+2 a^5 b c-3 a^4 b^2 c-6 a^3 b^3 c+2 a^2 b^4 c+4 a b^5 c+b^6 c+a^5 c^2-3 a^4 b c^2-4 a^3 b^2 c^2-8 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3-6 a^3 b c^3-8 a^2 b^2 c^3-8 a b^3 c^3-3 b^4 c^3-2 a^3 c^4+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5+4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28967.

X(31936) lies on these lines: {4,1001}, {5,6261} ,{8,12}, {11,11281}, {21,12615}, {142,9948}, {411,6690}, {431,1848}, {442,960}, {497,6871}, {946,3822}, {1125,6841}, {1329,6856}, {1858,5249}, {3816,6828}, {3826,11024}, {3829,15933}, {3847,6873}, {3869,3925}, {3957,5086}, {4197,11415}, {5141,10958}, {5880,8728}, {5887,6881}, {6668,6853}, {6691,6852}, {6824,25524}, {6825,7680}, {6867,18242}, {6870,26105}, {6985,10198}, {6991,28629}, {10590,15843}

X(31936) = {X(2476), X(3485)}-harmonic conjugate of X(2886)

X(31937) = X(1)X(1898)∩X(3)X(1709)

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+2 a^3 b^2 c+2 a^2 b^3 c-3 a b^4 c-2 b^5 c-a^4 c^2+2 a^3 b c^2-4 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^3 c^3+2 a^2 b c^3+2 a b^2 c^3+4 b^3 c^3+2 a^2 c^4-3 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6) : :
X(31937) = 5 X[3] - 3 X[5918],3 X[3] - 5 X[25917],3 X[4] + X[3869],3 X[5] - 2 X[3812],X[65] - 3 X[381],3 X[210] - X[12702],3 X[354] - 5 X[18493],X[355] - 3 X[5927],3 X[355] - X[10914],3 X[392] - X[18481],5 X[631] - X[9961],3 X[946] - X[3874],X[1071] - 3 X[5886],9 X[1699] - X[3901],3 X[1699] + X[5693],3 X[1699] - X[24474],X[1770] - 3 X[28452],4 X[3530] - 3 X[10178],X[3555] - 3 X[3656],3 X[3654] - 5 X[3697],3 X[3817] - X[5884],3 X[3845] - 2 X[16616],X[3869] - 3 X[5887],X[3874] + 3 X[31803],5 X[3876] - X[6361],X[3884] + 3 X[31871],5 X[3889] - 9 X[5603],5 X[3889] + 3 X[12528],X[3901] + 3 X[5693],X[3901] - 3 X[24474],3 X[5603] + X[12528],X[5696] + 3 X[11372],3 X[5886] - 2 X[13373],X[5903] - 5 X[18492],X[5904] + 3 X[31162],3 X[5918] + 5 X[12688],9 X[5918] - 25 X[25917],3 X[5919] - X[18526],9 X[5927] - X[10914],3 X[5927] + X[12672],9 X[7988] - 5 X[15016],5 X[8227] - 3 X[10202],5 X[8227] - X[15071],2 X[9940] - 3 X[11230],2 X[9956] - 3 X[10157],3 X[10157] - X[31788],3 X[10176] - X[31730],3 X[10202] - X[15071],3 X[10246] - X[12680],X[10914] + 3 X[12672],3 X[11231] - 2 X[31787],X[11362] - 3 X[15064],3 X[12688] + 5 X[25917],3 X[15049] - X[31728],3 X[15178] - 2 X[26089],3 X[17502] - 2 X[31805],X[22793] + 2 X[31821]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28967.

X(31937) lies on these lines: {1,1898}, {3,1709}, {4,8}, {5,3812}, {11,113}, {21,4881}, {30,960}, {36,7701}, {40,18491}, {56,90}, {65,381}, {84,10269}, {140,9943}, {210,12702}, {224,1012}, {354,18493}, {382,14110}, {392,6872}, {411,17613}, {500,6051}, {515,3884}, {516,20117}, {518,21850}, {546,7686}, {550,15726}, {631,9961}, {758,18483}, {912,946}, {971,1001}, {1071,5886}, {1125,13369}, {1156,15179}, {1158,6911}, {1319,26321}, {1376,3579}, {1456,18447}, {1482,14872}, {1490,10267}, {1519,26470}, {1539,2778}, {1697,18528}, {1699,3901}, {1770,28452}, {2392,31751}, {2476,7703}, {2646,13743}, {2800,19925}, {2801,13464}, {3057,9668}, {3058,9957}, {3333,30290}, {3359,7995}, {3485,5045}, {3486,17622}, {3487,16216}, {3530,10178}, {3555,3656}, {3556,9818}, {3583,16155}, {3612,28444}, {3652,3916}, {3654,3697}, {3666,5492}, {3678,28194}, {3753,6871}, {3817,5884}, {3818,3827}, {3830,31165}, {3845,16616}, {3876,6361}, {3878,31673}, {3889,5603}, {3925,6842}, {4295,6849}, {4297,31838}, {4511,21669}, {4870,17637}, {5119,18518}, {5252,10043}, {5259,13151}, {5439,10584}, {5450,18857}, {5453,15569}, {5696,11372}, {5698,6869}, {5719,12710}, {5720,11248}, {5722,12709}, {5779,22770}, {5780,6244}, {5806,10893}, {5836,18357}, {5881,23340}, {5885,6828}, {5901,12675}, {5903,18492}, {5904,31162}, {5919,18526}, {6000,9895}, {6583,10883}, {6705,6713}, {6767,9848}, {6824,9940}, {6825,11231}, {6838,26446}, {6857,17612}, {6866,31794}, {6870,10598}, {6873,10129}, {6883,12520}, {6900,20292}, {6912,15178}, {6913,12664}, {6923,12679}, {6944,14647}, {7171,8583}, {7330,11249}, {7373,8581}, {7491,28160}, {7548,17654}, {7728,10693}, {7741,18838}, {7743,10948}, {7988,15016}, {7991,18529}, {8227,10202}, {8543,17620}, {10057,10742}, {10085,16203}, {10176,31730}, {10246,12680}, {10394,15008}, {10532,10941}, {10679,17857}, {10827,18542}, {10912,11278}, {10915,27870}, {10949,30384}, {11114,28208}, {11362,15064}, {11374,12711}, {11699,12889}, {11826,28146}, {12446,16004}, {12571,31870}, {12737,17661}, {12773,20323}, {13374,24475}, {13600,26200}, {13750,17605}, {15049,31728}, {15842,21616}, {17502,31805}, {18243,25466}, {22753,24467}, {22760,24928}, {28174,31835}

X(31937) = midpoint of X(i) and X(j) for these {i,j}: {3, 12688}, {4, 5887}, {72, 12699}, {355, 12672}, {382, 14110}, {946, 31803}, {1385, 31828}, {1482, 14872}, {3057, 18525}, {3830, 31165}, {3878, 31673}, {5693, 24474}, {5694, 22793}, {5777, 9856}, {5881, 23340}, {7728, 10693}, {10742, 17638}, {12737, 17661}, {16138, 17653}
X(31937) =reflection of X(i) and X(j) for these {i,j}: {942, 9955}, {1071, 13373}, {3579, 5044}, {4297, 31838}, {5694, 31821}, {5836, 18357}, {7686, 546}, {9943, 140}, {12675, 5901}, {13369, 1125}, {13600, 26200}, {24475, 13374}, {31788, 9956}, {31837, 20117}, {31870, 12571}
X(31937) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 18540, 18761}, {4, 11415, 12699}, {355, 12699, 3434}, {355, 18516, 18480}, {1071, 5886, 13373}, {1699, 5693, 24474}, {1858, 12047, 942}, {3560, 6261, 1385}, {5259, 16132, 13151}, {5720, 12705, 11248}, {5927, 12672, 355}, {6841, 12047, 9955}, {6985, 12514, 3579}, {8227, 15071, 10202}, {10157, 31788, 9956}, {11373, 17625, 5045}, {12608, 12617, 5}, {17614, 17653, 17616}, {18480, 22793, 18407}

X(31938) = MIDPOINT OF X(79) AND X(5904)

Barycentrics a (a-b-c) (a^2-b^2-b c-c^2) (a^2 b-b^3+a^2 c+2 a b c+b^2 c+b c^2-c^3) : :
X(31938) = 3 X[21] - 5 X[3876],3 X[210] - X[17637],3 X[210] - 2 X[18253],3 X[392] - 2 X[15174],3 X[442] - 2 X[942],3 X[3681] - X[11684],X[3868] - 3 X[6175],4 X[4662] - 3 X[21677],4 X[5044] - 3 X[15670],X[5441] - 3 X[5692],6 X[11281] - 5 X[17609]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28967.

X(31938) lies on the Mandart hyperbola and these lines: {2,10122}, {8,79}, {9,21}, {10,3580}, {30,72}, {35,3219}, {40,3681}, {63,3651}, {69,23581}, {144,3648}, {191,200}, {210,17637}, {319,340}, {392,15174}, {442,942}, {445,1844}, {500,16585}, {518,3649}, {527,10123}, {908,6841}, {960,10543}, {1071,5771}, {1145,2771}, {3059,17768}, {3146,31803}, {3555,16137}, {3652,11248}, {3868,4654}, {3872,8000}, {3874,6701}, {3927,16117}, {3940,13743}, {3984,21669}, {4511,5259}, {4662,6735}, {4847,11263}, {4853,16126}, {4855,21161}, {5044,15670}, {5086,18406}, {5178,16125}, {5223,16143}, {5428,5440}, {5441,5692}, {5687,16139}, {5693,6925}, {6675,27385}, {6912,20117}, {10176,16865}, {10527,26725}, {11281,17609}, {12691,31837}, {14740,15227}, {15674,27383}, {15680,20007}, {17653,17658}, {18259,31660}, {20084,20214}, {31649,31835}

X(31938) = anticomplement of X(10122)
X(31938) = midpoint of X(79) and X(5904)
X(31938) = reflection of X(i) and X(j) for these {i,j}: {3555, 16137}, {3647, 3678}, {3874, 6701}, {10543, 960}, {14450, 16120}, {17637, 18253}, {31649, 31835}
X(31938) = X(8)-Ceva conjugate of X(6734)
X(31938) = X(2160)-isoconjugate of X(2982)
X(31938) = crosspoint of X(8) and X(4420)
X(31938) = barycentric product X(i) X(j) for these {i,j}: {8, 16585}, {78, 445}, {312, 500}, {345, 1844}, {3219, 6734}, {4420, 5249}
X(31938) = barycentric quotient X(i) / X(j) for these {i,j}: {35, 2982}, {445, 273}, {500, 57}, {1844, 278}, {6734, 30690}, {14547, 2160}, {16585, 7}
X(31938) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {210, 17637, 18253}, {2475, 5905, 79}, {3681, 12528, 3951}

X(31939) = X(3)X(13)∩X(62)X(1337)

Barycentrics (SB+SC)*((15*R^2-2*SA+3*SW)*S^2+sqrt(3)*(S^2+SA^2+4*SB*SC-(21*SA-3*SW)*R^2)*S-3*(R^2-SA+2*SW)*SA*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28969.

X(31939) lies on these lines: {3, 13}, {62, 1337}

X(31939) = midpoint of X(8172) and X(16965)
X(31939) = circumcircle-inverse of X(36782)

X(31940) = X(3)X(14)∩X(61)X(1338)

Barycentrics (SB+SC)*((15*R^2-2*SA+3*SW)*S^2-sqrt(3)*(S^2+SA^2+4*SB*SC-(21*SA-3*SW)*R^2)*S-3*(R^2-SA+2*SW)*SA*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28969.

X(31940) lies on these lines: {3, 14}, {61, 1338}

X(31940) = midpoint of X(8173) and X(16964)

X(31941) = X(110)X(924)∩X(476)X(12092)

Barycentrics (SB+SC)*(SA-SB)*(SA-SC)*(S^2-27*R^4-2*R^2*(3*SA-8*SW)+SA^2-2*SB*SC-2*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28969.

X(31941) lies on these lines: {110, 924}, {476, 12092}, {933, 1291}, {2071, 10628}

X(31942) = X(4)X(253)∩X(25)X(64)

Barycentrics a^2*(a^2-b^2+c^2)*(a^4+2*(b^2-c^2)*a^2-(b^2-c^2)*(3*b^2+c^2))^2*(a^2+b^2-c^2)*(a^4-2*(b^2-c^2)*a^2+(b^2-c^2)*(b^2+3*c^2))^2 : :
Barycentrics SB*SC*(SB+SC)*(S^2+4*(4*R^2-SW)*(SA+SC))*(S^2+4*(4*R^2-SW)*(SA+SB)) : :

See Vu Thanh Tung and César Lozada, Hyacinthos 28972.

X(31942) lies on these lines: {4, 253}, {25, 64}, {185, 3343}, {235, 459}, {1073, 1593}, {1301, 3515}, {3516, 14379}, {5907, 15394}, {6524, 6526}, {6622, 14572}, {8798, 11403}, {11589, 15750}

X(31942) = {X(64), X(28785)}-harmonic conjugate of X(1204)

X(31943) = X(2)X(3349)∩X(4)X(3344)

Barycentrics (S^2-2*(4*R^2-SB)*SB)*(3*S^2-16*R^2*(4*R^2+SB-2*SW)+4*SB^2-2*SC*SA-4*SW^2)*(S^2-2*(4*R^2-SC)*SC)*(3*S^2-16*R^2*(4*R^2+SC-2*SW)-4*SW^2-2*SA*SB+4*SC^2) : :

See Vu Thanh Tung and César Lozada, Hyacinthos 28972.

X(31943) lies on Kiepert hyperbola and these lines: {2, 3349}, {4, 3344}, {459, 3346}

X(31943) = isogonal conjugate of X(31944)
X(31943) = X(2), X(14365)}-harmonic conjugate of X(3349)

X(31944) = X(3)X(6)∩X(1498)X(3343)

Barycentrics (SB+SC)*(S^4-2*SB*SC*(S^2+SA^2))*(S^6-2*SB*SC*(S^4+2*(S^2-SB*SC)*SA^2)) : :

See Vu Thanh Tung and César Lozada, Hyacinthos 28972.

X(31944) lies on these lines: {3, 6}, {1498, 3343}, {2060, 3350}, {6621, 13567}

X(31944) = isogonal conjugate of X(31943)

X(31945) = MIDPOINT OF X(110) AND X(3154)

Barycentrics (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^8 - 3*a^6*b^2 - 2*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - 3*a^6*c^2 + 10*a^4*b^2*c^2 - 6*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 6*a^2*b^2*c^4 + 2*b^4*c^4 + 5*a^2*c^6 + b^2*c^6 - 2*c^8) : :
X(31945) = 3 X[2] + X[14611],X[113] + 3 X[31378],X[3233] - 3 X[5642],X[3258] + 3 X[5642],3 X[5972] - X[22104],3 X[7471] + X[14731],3 X[12068] - 2 X[22104],3 X[14643] + X[14934]

Let L_A, L_B, L_C be the lines through A, B, C, resp. parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in L_A, L_B, L_C, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. X(31945) is the center of inverse similitude of ABC and A'B'C'; see Hyacinthos #16741/16782, September 2008. (Randy Hutson, June 7, 2019)

For another construction see Antreas Hatzipolakis and Peter Moses, Euclid 2455 .

X(31945) lies on the cubic K1095 and these lines: {2, 9717}, {5, 15454}, {30, 113}, {110, 3154}, {114, 5159}, {140, 14670}, {230, 3163}, {468, 2967}, {523, 5972}, {1649, 5664}, {6070, 30221}, {7471, 14731}, {8780, 18870}, {9209, 23589}, {9820, 14894}, {12052, 14984}, {14643, 14934}, {16534, 31379}

X(31945) = midpoint of X(i) and X(j) for these {i,j}: {110, 3154}, {3233, 3258}, {6070, 30221}, {11064, 16319}, {12079, 14611}, {16534, 31379}
X(31945) = reflection of X(12068) and X(5972)
X(31945) = complement of X(12079)
X(31945) = Thomson isogonal conjugate of X(15055)
X(31945) = X(i)-complementary conjugate of X(j) for these (i,j): {1101, 30}, {1495, 24040}, {2407, 21253}, {2420, 8287}, {4575, 1650}, {9406, 23991}, {23357, 18593}, {23995, 3003}
X(31945) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 30}, {5972, 402}
X(31945) = X(2159)-isoconjugate of X(12066),barycentric quotient X(i) / X(j) for these {i,j}: {30, 12066}, {1637, 12065}
X(31945) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14611, 12079}, {3258, 5642, 3233}

X(31946) = MIDPOINT OF X(3835) AND X(20316)

Barycentrics (b^2 - c^2)*(a^2*b + a*b^2 + a^2*c - b^2*c + a*c^2 - b*c^2) : :
X(31946) = X[1459] - 5 X[30835],X[4036] - 3 X[14431],2 X[8062] - 3 X[24959],X[20293] + 7 X[27138]

X(31946) lies on these lines: {2, 3733}, {5, 6003}, {10, 4132}, {12, 4017}, {37, 21099}, {125, 3259}, {141, 27854}, {429, 24006}, {442, 28217}, {512, 20546}, {513, 3814}, {523, 1577}, {594, 21055}, {656, 900}, {659, 27045}, {661, 1639}, {798, 1213}, {834, 3835}, {1211, 14434}, {1459, 30835}, {3454, 6363}, {3948, 20953}, {4057, 21301}, {4064, 21121}, {4079, 21053}, {4806, 23301}, {5224, 17217}, {6002, 24920}, {6587, 14321}, {7668, 8287}, {8062, 9013}, {9276, 14873}, {14288, 17420}, {15313, 17115}, {17398, 20981}, {18004, 27710}, {20293, 27138}, {21124, 23282}, {23815, 28195}, {26983, 30795}

X(31946) = midpoint of X(i) and X(j) for these {i,j}: {3835, 20316}, {4057, 21301}, {4064, 21121}, {4705, 30591}, {14288, 17420}, {21124, 23282}
X(31946) = reflection of X(4086) and X(21714)
X(31946) = complement of X(3733)
X(31946) = medial-isogonal conjugate of X(244)
X(31946) = X(i)-Ceva conjugate of X(j) for these (i,j): {513, 523}, {4129, 661}, {21260, 23301}
X(31946) = X(1897)-isoconjugate of X(15409)
X(31946) = crosspoint of X(2) and X(27808)
X(31946) = crossdifference of every pair of points on line {595, 1333}
X(31946) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 244}, {2, 17761}, {9, 4858}, {10, 11}, {12, 8286}, {37, 1086}, {42, 1015}, {65, 3756}, {72, 2968}, {100, 1125}, {101, 3666}, {181, 16613}, {190, 3739}, {210, 1146}, {213, 6377}, {226, 4904}, {313, 21252}, {321, 116}, {594, 8287}, {643, 4999}, {644, 5745}, {645, 21233}, {646, 21246}, {651, 3946}, {660, 740}, {661, 6547}, {662, 17045}, {664, 3742}, {668, 3741}, {756, 115}, {762, 6627}, {765, 523}, {872, 1084}, {943, 24235}, {983, 21210}, {1016, 4369}, {1018, 2}, {1020, 4000}, {1026, 8299}, {1042, 17071}, {1089, 125}, {1110, 647}, {1252, 14838}, {1255, 24185}, {1402, 16614}, {1441, 17059}, {1500, 16592}, {1897, 942}, {1978, 21240}, {2171, 17058}, {2292, 15611}, {2298, 24195}, {2321, 26932}, {2748, 3712}, {3257, 4395}, {3293, 8054}, {3678, 6741}, {3690, 16573}, {3694, 16596}, {3699, 960}, {3701, 124}, {3710, 123}, {3903, 6682}, {3949, 15526}, {3952, 10}, {3971, 5518}, {3992, 3259}, {4033, 141}, {4069, 9}, {4082, 5514}, {4103, 1211}, {4125, 15614}, {4169, 16594}, {4515, 13609}, {4551, 1}, {4552, 142}, {4554, 17050}, {4557, 37}, {4559, 3752}, {4564, 17069}, {4566, 11019}, {4567, 21196}, {4574, 1214}, {4605, 18635}, {4613, 24325}, {4674, 1647}, {4695, 5516}, {5377, 4458}, {5378, 9508}, {5379, 21187}, {5382, 2487}, {5546, 16579}, {6540, 27798}, {7035, 512}, {7161, 1109}, {8694, 6051}, {8701, 3743}, {8708, 10180}, {14624, 17197}, {15742, 8062}, {18098, 21208}, {18785, 27918}, {18793, 27846}, {21075, 7358}, {21859, 17056}, {23067, 17102}, {27808, 2887}, {28654, 21253}, {30730, 3452}
X(31946) = barycentric product X(i)*X(j) for these {i,j}: {86, 21720}, {514, 3159}, {523, 17147}, {661, 18133}, {693, 21858}, {850, 16685}, {1577, 3216}, {14618, 22458}
X(31946) = barycentric quotient X(i)/X(j) for these {i,j}: {3159, 190}, {3216, 662}, {16685, 110}, {17147, 99}, {18133, 799}, {21720, 10}, {21858, 100}, {22383, 15409}, {22458, 4558}
{X(21055), X(21834)}-harmonic conjugate of X(594)

X(31947) = MIDPOINT OF X(656) AND X(2605)

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(31947) = X[3733] - 3 X[14419]

X(31947) lies on these lines: {1, 8702}, {2, 4036}, {11, 3258}, {36, 238}, {522, 1125}, {523, 8043}, {650, 4802}, {656, 2605}, {657, 2260}, {1015, 23992}, {1193, 1459}, {1649, 3666}, {3005, 4132}, {3669, 28195}, {3900, 18455}, {3908, 26698}, {3960, 4977}, {4010, 16751}, {4140, 17303}, {4560, 30591}, {4777, 6129}, {4782, 28374}, {4824, 24948}, {5029, 8061}, {6370, 21187}, {7662, 16757}, {16755, 17322}, {24920, 29298}

X(31947) = midpoint of X(i) and X(j) for these {i,j}: {656, 2605}, {2530, 4057}, {4560, 30591}, {4840, 4983}
X(31947) = reflection of X(8043) and X(14838)
X(31947) = complement of X(4036)
X(31947) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 513}, {5606, 1}, {7372, 514}, {14838, 650}
X(31947) = X(i)-isoconjugate of X(j) for these (i,j): {100, 267}, {101, 1029}, {110, 502}, {190, 3444}, {662, 21353}
X(31947) = crosspoint of X(1) and X(13486)
X(31947) = crosssum of X(6) and X(4705)
X(31947) = crossdifference of every pair of points on line {35, 37}
X(31947) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 6627}, {58, 125}, {60, 124}, {81, 21253}, {110, 3454}, {163, 1211}, {249, 3835}, {250, 20316}, {593, 116}, {662, 21245}, {667, 24040}, {757, 21252}, {849, 11}, {1101, 513}, {1333, 8287}, {1408, 8286}, {1576, 1213}, {1790, 127}, {1919, 23991}, {2150, 26932}, {2206, 115}, {4556, 141}, {4565, 17052}, {4575, 21530}, {4590, 21262}, {4610, 626}, {4612, 21244}, {4623, 21235}, {4636, 1329}, {7341, 17059}, {7342, 3756}, {14574, 21838}, {16947, 17058}, {17940, 20546}, {23357, 514}, {23963, 6586}, {23995, 650}, {24041, 21260}
X(31947) = barycentric product X(i)*X(j) for these {i,j}: {1, 21192}, {191, 514}, {451, 905}, {501, 1577}, {513, 2895}, {525, 2906}, {649, 20932}, {693, 1030}, {1019, 21081}, {4391, 8614}, {7192, 21873}, {17924, 22136}
X(31947) = barycentric quotient X(i) / X(j) for these {i,j}: {191, 190}, {451, 6335}, {501, 662}, {512, 21353}, {513, 1029}, {649, 267}, {661, 502}, {667, 3444}, {1030, 100}, {2895, 668}, {2906, 648}, {8614, 651}, {20932, 1978}, {21081, 4033}, {21192, 75}, {21873, 3952}, {22136, 1332}

X(31948) = X(1)X(3520)∩X(4)X(145)

Barycentrics a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^5-2 a^4 b-2 a^3 b^2+4 a^2 b^3+a b^4-2 b^5-2 a^4 c+4 a^3 b c-4 a b^3 c+2 b^4 c-2 a^3 c^2+a b^2 c^2+4 a^2 c^3-4 a b c^3+a c^4+2 b c^4-2 c^5) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28980.

X(31948) lies on these lines: {1,3520}, {4,145}, {8,16868}, {24,8148}, {40,17506}, {108,2099}, {186,517}, {378,10247}, {403,5844}, {915,8652}, {1483,18560}, {1829,26863}, {1830,1870}, {1845,6198}, {2807,7722}, {3518,7982}, {5603,7577}, {5690,14940}, {5901,6143}, {7505,12245}, {10222,14865}, {10295,28212}, {12702,21844}, {13596,16200}, {13619,28174}

X(31949) = MIDPOINT OF X(9168) AND X(31065)

Barycentrics (b^2-c^2) (7 a^4+11 a^2 b^2+10 b^4+11 a^2 c^2+23 b^2 c^2+10 c^4) : :

X(31949) lies on these lines: {523,7840}, {9168,31065}

X(31950) = X(2)X(17436)∩X(30)X(511)

Barycentrics (b^2-c^2) (-a^4+4 a^2 b^2+2 b^4+4 a^2 c^2+7 b^2 c^2+2 c^4) : :

X(31950) lies on these lines: {2,17436}, {30,511}, {23288,31168}

X(31951) = X(39)X(6323)∩X(111)X(12212)

Barycentrics a^2 (a^2-b^2) (a^2-c^2) (2 a^4+7 a^2 b^2+2 b^4+4 a^2 c^2+4 b^2 c^2-c^4) (2 a^4+4 a^2 b^2-b^4+7 a^2 c^2+4 b^2 c^2+2 c^4) : :

X(31951) lies on the circumcircle and these lines: {39,6323}, {111,12212}, {2076,9831}, {14388,14810}

X(31952) = X(2)X(3)∩X(185)X(3095)

Barycentrics a^2 (a^8 b^2-3 a^6 b^4+3 a^4 b^6-a^2 b^8+a^8 c^2+a^6 b^2 c^2+4 a^4 b^4 c^2-3 a^2 b^6 c^2-3 b^8 c^2-3 a^6 c^4+4 a^4 b^2 c^4+3 b^6 c^4+3 a^4 c^6-3 a^2 b^2 c^6+3 b^4 c^6-a^2 c^8-3 b^2 c^8) : :

As a point on the Euler line, X(31952) has Shinagawa coefficients (E²-F²-S², -3E²-2EF+F²+S²).

X(31952) lies on these lines: {2,3}, {185,3095}, {1503,20794}, {1843,30262}, {3398,11424}, {3491,30270}, {5921,22152}, {8549,22143}, {8841,9737}, {14927,20775}

{X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 4, 11328}, {3, 11414, 21512}, {3148, 11413, 3}, {3522, 14096, 3}

X(31953) = X(98)X(804)∩X(99)X(107)

Barycentrics (b^2-c^2) (-a^2 b^2+b^4-a^2 c^2+c^4) (a^8-a^6 b^2-a^6 c^2+a^4 b^2 c^2+b^6 c^2-2 b^4 c^4+b^2 c^6) : :
X(31953) = 2 X[8552] - 3 X[15561]

X(31953) lies on these lines: {98,804}, {99,107}, {114,132}, {115,125}, {262,14223}, {446,23105}, {523,1513}, {526,6785}, {2794,9409}, {6033,9517}, {7684,23871}, {7685,23870}, {8552,15561}, {9033,12181}, {13558,21525}

X(39153) = reflection of X(i) and X(j) for these {i,j}: {98, 6130}, {684, 114}
X(39153) = {X(132), X(16230)}-harmonic conjugate of X(17994)
X(39153) = Dao-Moses-Telv circle inverse of X(3569)
X(39153) = crossdifference of every pair of points on line {110, 248}
X(31953) = orthocenter of X(98)X(114)X(115)
X(39153) = barycentric product X(1316)*X(2799)
X(39153) = barycentric quotient X(i)/X(j) for these {i,j}: {1316, 2966}, {3569, 9513}

X(31954) = X(3)X(15461)∩X(6)X(74)

Barycentrics (a^2-b^2-c^2) (a^2 J (a^6+a^4 b^2-5 a^2 b^4+3 b^6+a^4 c^2+10 a^2 b^2 c^2-3 b^4 c^2-5 a^2 c^4-3 b^2 c^4+3 c^6-(a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) J)-4 (b^2-c^2)^2 S^2) : :
X(31954) = 2 SW X[6] + (J - 3) J R^2 X[74]

X(31954) lies on the cubic K1096 and these lines: {3,15461}, {4,2574}, {6,74}, {125,1347}, {185,13415}, {511,1113}, {1112,1344}, {1345,5663}, {2575,6776}, {6785,8426}, {8115,13754}

X(39154) = reflection of X(25408) and X(3)

X(31955) = X(3)X(15460)∩X(6)X(74)

Barycentrics (a^2-b^2-c^2) (a^2 J (a^6+a^4 b^2-5 a^2 b^4+3 b^6+a^4 c^2+10 a^2 b^2 c^2-3 b^4 c^2-5 a^2 c^4-3 b^2 c^4+3 c^6+(a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) J)+4 (b^2-c^2)^2 S^2) : :
X(31955) = 2 SW X[6] + (J + 3) J R^2 X[74]

X(31955) lies on the cubic K1096 and these lines: {3,15460}, {4,2575}, {6,74}, {125,1346}, {185,13414}, {511,1114}, {1112,1345}, {1344,5663}, {2574,6776}, {6785,8427}, {8116,13754}

X(31955) = reflection of X(25407) and X(3)

X(31956) = X(64)X(28783)∩X(154)X(1033)

Barycentrics a^2 (a^12+6 a^10 b^2-29 a^8 b^4+36 a^6 b^6-9 a^4 b^8-10 a^2 b^10+5 b^12-6 a^10 c^2+14 a^8 b^2 c^2+4 a^6 b^4 c^2-36 a^4 b^6 c^2+34 a^2 b^8 c^2-10 b^10 c^2+15 a^8 c^4-20 a^6 b^2 c^4+50 a^4 b^4 c^4-36 a^2 b^6 c^4-9 b^8 c^4-20 a^6 c^6-20 a^4 b^2 c^6+4 a^2 b^4 c^6+36 b^6 c^6+15 a^4 c^8+14 a^2 b^2 c^8-29 b^4 c^8-6 a^2 c^10+6 b^2 c^10+c^12) (a^12-6 a^10 b^2+15 a^8 b^4-20 a^6 b^6+15 a^4 b^8-6 a^2 b^10+b^12+6 a^10 c^2+14 a^8 b^2 c^2-20 a^6 b^4 c^2-20 a^4 b^6 c^2+14 a^2 b^8 c^2+6 b^10 c^2-29 a^8 c^4+4 a^6 b^2 c^4+50 a^4 b^4 c^4+4 a^2 b^6 c^4-29 b^8 c^4+36 a^6 c^6-36 a^4 b^2 c^6-36 a^2 b^4 c^6+36 b^6 c^6-9 a^4 c^8+34 a^2 b^2 c^8-9 b^4 c^8-10 a^2 c^10-10 b^2 c^10+5 c^12) : :
Barycentrics (8 R^2-SB-SC-2 SW)S^4 + (512 R^6-128 R^4 SB-128 R^4 SC-256 R^4 SW+48 R^2 SB SW+48 R^2 SC SW+2 SB SC SW+32 R^2 SW^2-4 SB SW^2-4 SC SW^2)S^2 + 4096 R^8 SB+4096 R^8 SC+1024 R^6 SB SC-3072 R^6 SB SW-3072 R^6 SC SW-384 R^4 SB SC SW+768 R^4 SB SW^2+768 R^4 SC SW^2+32 R^2 SB SC SW^2-64 R^2 SB SW^3-64 R^2 SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28981.

X(31956) lies on these lines: {64,28783}, {154,1033}, {1073,14481}, {1249,3349}, {1498,3348}, {3197,8803}

X(31956) = barycentric product of X(i) and X(j) for these {i,j}: {4,3348}, {64,14365}, {253,28781}, {3346,3349}
X(31956) = barycentric quotient of X(i) and X(j) for these {i,j}: {25,3183}, {64,14362}, {154,2060}, {1042,8812}, {3348,69}, {3349,6527}, {14365,14615}, {28781,20}
X(31956) = trilinear product of X(i) and X(j) for these {i,j}: {19,3348}, {2155,14365}, {2184,28781}
X(31956) = trilinear quotient of X(i) and X(j) for these {i,j}: {1427,8812}, {14365,18750}

Orthocenters of central not-equilateral triangles: X(31957)-X(31991)

This preamble and centers X(31957)-X(31991) were contributed by César Eliud Lozada, April 14, 2019.

The appearance of (T, n) in the following list means that the orthocenter of triangle T is X(n):

(ABC, 4), (ABC-X3 reflections, 20), (anti-Aquila, 946), (anti-Ara, 3575), (anti-Artzt, 1992), (anti-Ascella, 12160), (anti-Atik, 6643), (1st anti-Brocard, 147), (4th anti-Brocard, 1296), (5th anti-Brocard, 12110), (6th anti-Brocard, 12177), (2nd anti-circumperp-tangential, 7354), (1st anti-circumperp, 11412), (anti-Conway, 12161), (2nd anti-Conway, 5), (anti-Euler, 4), (3rd anti-Euler, 5889), (4th anti-Euler, 11412), (anti-excenters-reflections, 12162), (2nd anti-extouch, 3), (anti-inner-Grebe, 1588), (anti-outer-Grebe, 1587), (anti-Honsberger, 19139), (anti-Hutson intouch, 12163), (anti-incircle-circles, 12164), (anti-inverse-in-incircle, 11411), (anti-Mandart-incircle, 11500), (anti-McCay, 8591), (6th anti-mixtilinear, 1216), (anti-orthocentroidal, 74), (1st anti-orthosymmedial, 1297), (1st anti-Sharygin, 19194), (anti-tangential-midarc, 7352), (3rd anti-tri-squares, 22617), (4th anti-tri-squares, 22646), (anti-Ursa minor, 12359), (anti-Wasat, 5562), (antiAOA, 15133), (anticomplementary, 20), (AOA, 15115), (Apus, 31984), (Aquila, 5691), (Ara, 3), (Aries, 12420), (Artzt, 9770), (Ascella, 942), (Atik, 8), (1st Auriga, 9834), (2nd Auriga, 9835), (Ayme, 9958), (BCI, 31957), (1st Brocard-reflected, 31958), (1st Brocard, 1352), (2nd Brocard, 31959), (3rd Brocard, 31960), (4th Brocard, 31961), (5th Brocard, 9873), (6th Brocard, 9863), (circummedial, 12505), (circumorthic, 5889), (2nd circumperp tangential, 12114), (1st circumperp, 40), (2nd circumperp, 1), (circumsymmedial, 31962), (inner-Conway, 3869), (Conway, 3868), (2nd Conway, 8), (3rd Conway, 12435), (4th Conway, 31963), (5th Conway, 31964), (Ehrmann-cross, 30), (Ehrmann-mid, 5), (Ehrmann-side, 18436), (Ehrmann-vertex, 9927), (1st Ehrmann, 8542), (2nd Ehrmann, 8548), (Euler, 4), (2nd Euler, 5562), (3rd Euler, 946), (4th Euler, 10), (5th Euler, 12506), (excenters-midpoints, 12640), (excenters-reflections, 1), (excentral, 1), (1st excosine, 17834), (2nd excosine, 17850), (extangents, 6237), (extouch, 18239), (2nd extouch, 72), (3rd extouch, 31965), (4th extouch, 31966), (5th extouch, 31967), (inner-Fermat, 22114), (outer-Fermat, 22113), (Feuerbach, 5948), (Fuhrmann, 1), (2nd Fuhrmann, 7701), (inner-Garcia, 5693), (outer-Garcia, 40), (Garcia-reflection, 3680), (Gossard, 12113), (inner-Grebe, 5871), (outer-Grebe, 5870), (3rd Hatzipolakis, 31985), (hexyl, 40), (Honsberger, 7672), (Hutson extouch, 31968), (inner-Hutson, 9805), (Hutson intouch, 3057), (outer-Hutson, 9806), (1st Hyacinth, 389), (2nd Hyacinth, 12421), (incentral, 500), (incircle-circles, 942), (intangents, 6238), (intouch, 65), (inverse-in-incircle, 1), (Johnson, 3), (inner-Johnson, 12114), (outer-Johnson, 11500), (1st Johnson-Yff, 55), (2nd Johnson-Yff, 56), (K798e, 22936), (K798i, 1385), (1st Kenmotu diagonals, 10665), (2nd Kenmotu diagonals, 10666), (Kosnita, 1147), (Lemoine, 31969), (Lucas antipodal, 31970), (Lucas antipodal tangents, 18939), (Lucas central, 31972), (Lucas homothetic, 9838), (Lucas tangents, 31974), (Lucas(-1) antipodal, 31971), (Lucas(-1) antipodal tangents, 18940), (Lucas(-1) central, 31973), (Lucas(-1) homothetic, 9839), (Lucas(-1) tangents, 31975), (Macbeath, 31976), (Mandart-excircles, 31977), (Mandart-incircle, 6284), (McCay, 7618), (medial, 3), (midarc, 177), (2nd midarc, 8422), (midheight, 31978), (mixtilinear, 7961), (2nd mixtilinear, 7960), (3rd mixtilinear, 31979), (4th mixtilinear, 31980), (5th mixtilinear, 944), (6th mixtilinear, 7991), (1st Neuberg, 31981), (2nd Neuberg, 31982), (orthic, 52), (orthocentroidal, 5890), (1st orthosymmedial, 19161), (2nd orthosymmedial, 31983), (2nd Pamfilos-Zhou, 9808), (1st Parry, 9131), (2nd Parry, 9979), (3rd Parry, 31986), (Pelletier, 1), (reflection, 13423), (2nd Schiffler, 12641), (Schroeter, 5), (1st Sharygin, 2292), (2nd Sharygin, 2254), (2nd inner-Soddy, 482), (2nd outer-Soddy, 481), (inner-squares, 31987), (outer-squares, 31988), (Steiner, 3), (submedial, 5462), (symmedial, 31989), (tangential, 155), (tangential-midarc, 8093), (2nd tangential-midarc, 8094), (inner tri-equilateral, 10661), (outer tri-equilateral, 10662), (1st tri-squares-central, 13662), (2nd tri-squares-central, 13782), (3rd tri-squares-central, 485), (4th tri-squares-central, 486), (1st tri-squares, 13639), (2nd tri-squares, 13759), (3rd tri-squares, 13882), (4th tri-squares, 13934), (Trinh, 7689), (Ursa-major, 10914), (Ursa-minor, 3057), (inner-Vecten, 486), (outer-Vecten, 485), (Wasat, 10), (X-parabola-tangential, 31990), (X3-ABC reflections, 382), (Yff central, 12445), (Yff contact, 8), (inner-Yff, 1478), (outer-Yff, 1479), (inner-Yff tangents, 12115), (outer-Yff tangents, 12116), (Yiu, 31991), (1st Zaniah, 18238), (2nd Zaniah, 960)

X(31957) = ORTHOCENTER OF THE BCI TRIANGLE

Barycentrics F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : : , where
F(a,b,c) = -2* (2*a^2+7*(b+c)*a+(b+c)^2)*S+(-a+b+c)*(a^3-(b+c)*a^2-3*(b+c)^2*a-(b^2-c^2)*(b-c))
G(a,b,c) = 2*a*((a-3*b-3*c)*S-(a-b+c)*(3*a^2+(5*b+6*c)*a+(b+c)*(2*b+c)))
H(a,b,c) = -2*a*(7*a+3*b+3*c)*S-(b+c)*(a+b-c)*(-a+b+c)*(a-b+c)

X(31957) lies on the line {1,483}

X(31958) = ORTHOCENTER OF THE 1st BROCARD-REFLECTED TRIANGLE

Barycentrics 3*(SB+SC)*S^4+(SA^2+2*SB*SC+2*SW^2)*SW*S^2-SB*SC*SW^3 : :
X(31958) = 2*X(182)+X(18906) = X(194)-4*X(575) = 2*X(262)-3*X(14561) = 2*X(576)+X(12251) = X(1352)-4*X(24256) = 2*X(8550)+X(13108)

X(31958) lies on the cubic K906 and these lines: {2,51}, {6,22525}, {83,576}, {99,182}, {140,8179}, {194,575}, {698,5050}, {1351,14535}, {1352,6033}, {2548,13330}, {2782,11179}, {3398,31981}, {3589,10256}, {3734,12177}, {5054,7606}, {5480,22728}, {5652,8704}, {5969,7618}, {7846,25555}, {8550,13108}, {9744,9772}

X(31958) = midpoint of X(i) and X(j) for these {i,j}: {7709, 18906}, {22486, 22712}
X(31958) = reflection of X(i) in X(j) for these (i,j): (1352, 7697), (7697, 24256), (7709, 182), (22677, 15819), (22728, 5480)
X(31958) = anticomplement of X(11261)
X(31958) = 1st-Brocard-reflected-isogonal conjugate of X(11261)
X(31958) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22714, 22715, 2), (22726, 22727, 262)

X(31959) = ORTHOCENTER OF THE 2nd BROCARD TRIANGLE

Barycentrics 9*(6*R^2-SW)*(SA+SW)*S^4-(SB+SC)*(54*R^2*SA+SW^2)*SW*S^2-2*SB*SC*SW^4 : :
X(31959) = 2*X(182)-3*X(10166) = 3*X(353)-X(6776)

X(31959) lies on these lines: {6,22100}, {69,31962}, {126,12494}, {182,9169}, {353,6776}, {511,14867}, {1352,7618}, {1503,31731}, {3564,31727}, {5480,8705}, {5512,16511}, {5847,31741}, {12177,31839}

X(31959) = midpoint of X(69) and X(31962)
X(31959) = reflection of X(i) in X(j) for these (i,j): (6, 31608), (1352, 31742), (12177, 31839)
X(31959) = 2nd-Brocard-isogonal conjugate of X(182)

X(31960) = ORTHOCENTER OF THE 3rd BROCARD TRIANGLE

Barycentrics
(b^4+b^2*c^2+c^4)*(b^2+c^2)*b^2*c^2*a^18-(b^12+c^12+5*(b^4+c^4+(b^2+b*c+c^2)*b*c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)*b^2*c^2)*a^16+(b^2+c^2)*(b^12+7*b^6*c^6+c^12)*a^14-(b^16+c^16+(3*b^12+3*c^12+(2*b^8+2*c^8+3*(b^2-c^2)^2*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12+(b^2+c^2)*(b^16+c^16-(b^12+c^12-2*(b^2+b*c+c^2)^2*(b^2-b*c+c^2)^2*b^2*c^2)*b^2*c^2)*a^10-2*(b^4+b^2*c^2+c^4)*(b^8+c^8+(2*b^4-3*b^2*c^2+2*c^4)*b^2*c^2)*b^4*c^4*a^8+3*(b^2+c^2)*b^10*c^10*a^6-(b^12+c^12-(b^4-3*b^2*c^2+c^4)*(2*b^4+3*b^2*c^2+2*c^4)*b^2*c^2)*b^6*c^6*a^4+3*(b^4-c^4)*(b^2-c^2)*b^10*c^10*a^2+(b^2-c^2)^2*b^12*c^12 : :

X(31960) lies on the line {262,6234}

X(31961) = ORTHOCENTER OF THE 4th BROCARD TRIANGLE

Barycentrics 54*R^2*S^4-(162*R^2*SA*(SB+SC)+9*SA*SW*(3*SA-2*SW)-7*SW^3)*S^2-6*SB*SC*SW^3 : :
X(31961) = 5*X(2)-4*X(31606) = 2*X(381)-3*X(6032) = 4*X(549)-3*X(9829) = 3*X(3524)-4*X(31762) = 3*X(3545)-2*X(14866) = 3*X(3839)-4*X(31749) = 3*X(5054)-2*X(31744) = 5*X(5071)-6*X(10162) = 3*X(6031)-5*X(15692) = 3*X(6032)-4*X(31840) = 6*X(10163)-7*X(15702) = 3*X(10304)-2*X(31729) = X(12505)-4*X(12506) = 5*X(12505)-8*X(31606) = 5*X(12506)-2*X(31606) = 3*X(14269)-2*X(31824) = 3*X(19875)-4*X(31758) = 3*X(25055)-2*X(31747)

X(31961) lies on these lines: {2,12505}, {111,381}, {376,3849}, {378,16010}, {549,9829}, {1003,6232}, {1296,11165}, {2408,7757}, {3524,31762}, {3545,14866}, {3839,31749}, {5054,31744}, {5071,10162}, {5890,31743}, {6031,15692}, {6093,13241}, {6644,14682}, {8176,14654}, {10163,15702}, {10304,31729}, {14269,31824}, {19875,31758}, {25055,31747}, {30230,30515}

Let (O_A) be the circumcircle of BCX(2), and define (O_B) and (O_C) cyclically. Let A' be the intersection, other than X(2), of (O_A) and AX(2), and define B' and C' cyclically. Let A" be the antipode of A' in (O_A), and define B" and C" cyclically. X(31961) is the centroid of triangle A"B"C". (Randy Hutson, June 7, 2019)

X(31961) = reflection of X(i) in X(j) for these (i,j): (2, 12506), (381, 31840), (5890, 31743), (6236, 30514), (12505, 2)
X(31961) = 4th-Brocard-isogonal conjugate of X(381)
X(31961) = {X(381), X(31840)}-harmonic conjugate of X(6032)

X(31962) = ORTHOCENTER OF THE CIRCUMSYMMEDIAL TRIANGLE

Barycentrics (SB+SC)*(9*S^4-3*(9*SA*(4*R^2-SA)+2*SW^2)*S^2+4*SA*SW^3) : :
X(31962) = 3*X(2)-4*X(31608) = 2*X(3)-3*X(353) = 3*X(353)-4*X(31727) = 5*X(631)-6*X(10166) = 11*X(3525)-12*X(10160) = 4*X(11615)-3*X(13242)

X(31962) lies on these lines: {1,31741}, {2,31608}, {3,352}, {4,9830}, {6,9871}, {20,31731}, {32,6233}, {40,31740}, {69,31959}, {382,31827}, {575,6235}, {576,10765}, {631,10166}, {942,31820}, {3525,10160}, {5693,31826}, {5889,31733}, {6323,9737}, {7752,12494}, {8705,11456}, {11412,31739}, {11615,13242}, {12160,31813}, {12505,31748}

X(31962) = reflection of X(i) in X(j) for these (i,j): (1, 31741), (3, 31727), (4, 14867), (20, 31731), (40, 31740), (69, 31959), (382, 31827), (942, 31820), (5693, 31826), (5889, 31733), (6235, 30515), (11412, 31739), (12160, 31813), (12505, 31748)
X(31962) = anticomplement of the anticomplement of X(31608)
X(31962) = circumsymmedial-isogonal conjugate of X(3)
X(31962) = {X(3), X(31727)}-harmonic conjugate of X(353)

X(31963) = ORTHOCENTER OF THE 4th CONWAY TRIANGLE

Barycentrics
(4*b^2+7*b*c+4*c^2)*a^8+4*(b+c)*(3*b^2+5*b*c+3*c^2)*a^7+(12*b^4+12*c^4+(57*b^2+88*b*c+57*c^2)*b*c)*a^6+2*(b+c)*(19*b^2+36*b*c+19*c^2)*b*c*a^5-3*(4*b^4+4*c^4-(3*b^2+16*b*c+3*c^2)*b*c)*(b+c)^2*a^4-4*(b+c)*(3*b^6+3*c^6+(7*b^4+7*c^4-(2*b^2+11*b*c+2*c^2)*b*c)*b*c)*a^3-(4*b^2-7*b*c+4*c^2)*(b^2+4*b*c+c^2)*(b+c)^4*a^2-2*(3*b^4-4*b^2*c^2+3*c^4)*(b+c)^3*b*c*a-2*(b^2-c^2)^2*(b+c)^2*b^2*c^2 : :

X(31963) lies on these lines: {2285,12514}, {4295,10449}

X(31964) = ORTHOCENTER OF THE 5th CONWAY TRIANGLE

Barycentrics (b+c)*(a^6+(b+c)*a^5-(b-c)^2*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3-(2*b^4+2*c^4+3*(b^2+b*c+c^2)*b*c)*a^2-(b+c)*(b^2+c^2)*(b^2+b*c+c^2)*a-(b^3+c^3)*(b+c)*b*c) : :

X(31964) lies on these lines: {1,1330}, {10,1400}, {314,24851}, {3336,3980}, {3741,4292}, {4368,12572}, {4655,10441}, {10449,24248}

X(31965) = ORTHOCENTER OF THE 3rd EXTOUCH TRIANGLE

Barycentrics
a*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*((b+c)*a^11+(3*b^2+2*b*c+3*c^2)*a^10-(b+c)*(b^2+c^2)*a^9-(11*b^4+11*c^4+6*(b^2+b*c+c^2)*b*c)*a^8-6*(b^2-c^2)^2*(b+c)*a^7+2*(7*b^6+7*c^6+(2*b^4+2*c^4+(b^2-4*b*c+c^2)*b*c)*b*c)*a^6+2*(b^2-c^2)*(b-c)*(7*b^4+7*c^4+2*(7*b^2+3*b*c+7*c^2)*b*c)*a^5-2*(b^4-c^4)*(b^2-c^2)*(3*b^2-2*b*c+3*c^2)*a^4-(b^2-c^2)*(b-c)*(11*b^6+11*c^6+(22*b^4+22*c^4+5*(b^2+4*b*c+c^2)*b*c)*b*c)*a^3-(b^2-c^2)^4*(b^2+6*b*c+c^2)*a^2+3*(b^4-c^4)*(b^2-c^2)^3*(b+c)*a+(b^4-c^4)*(b^2-c^2)^3*(b+c)^2) : :
X(31965) = 3*X(11212)-2*X(11254)

X(31965) lies on these lines: {40,221}, {517,10365}, {946,10373}, {11212,11254}

X(31965) = 3rd-extouch-isogonal conjugate of X(11254)

X(31966) = ORTHOCENTER OF THE 4th EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^14+4*b*c*a^13-3*(b+c)*(b^2-4*b*c+c^2)*a^12+4*(b^2+8*b*c+c^2)*b*c*a^11+(b+c)*(b^4+c^4-2*(6*b^2-41*b*c+6*c^2)*b*c)*a^10-16*(b^4+c^4-3*(b^2+4*b*c+c^2)*b*c)*b*c*a^9+(b+c)*(5*b^6+5*c^6-(24*b^4+24*c^4+(9*b^2-304*b*c+9*c^2)*b*c)*b*c)*a^8-8*(b^6+c^6+(4*b^4+4*c^4-(17*b^2+48*b*c+17*c^2)*b*c)*b*c)*b*c*a^7-(b+c)*(5*b^8+5*c^8-2*(12*b^6+12*c^6-(10*b^4+10*c^4+(28*b^2-121*b*c+28*c^2)*b*c)*b*c)*b*c)*a^6+4*(5*b^6+5*c^6-(2*b^4+2*c^4+(13*b^2-4*b*c+13*c^2)*b*c)*b*c)*(b+c)^2*b*c*a^5-(b+c)*(b^10+c^10-(12*b^8+12*c^8+(11*b^6+11*c^6-2*(88*b^4+88*c^4+(37*b^2-164*b*c+37*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+4*(b^2-c^2)^2*(b^6+c^6-(16*b^4+16*c^4+5*(13*b^2+16*b*c+13*c^2)*b*c)*b*c)*b*c*a^3+(b^2-c^2)^2*(b+c)*(3*b^8+3*c^8-2*(6*b^6+6*c^6+(28*b^4+28*c^4+3*(14*b^2+25*b*c+14*c^2)*b*c)*b*c)*b*c)*a^2-8*(b^4-c^4)^2*(b^2+c^2)*(b+c)^2*b*c*a-(b^4-c^4)^3*(b^2-c^2)*(b+c)) : :

X(31966) lies on these lines: {940,12711}, {1211,12688}

X(31967) = ORTHOCENTER OF THE 5th EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^10+4*b*c*a^9-(b+c)*(b^2-4*b*c+c^2)*a^8-8*(b^2+c^2)*b*c*a^7-2*(b+c)*(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^6-16*(b^2-3*b*c+c^2)*b^2*c^2*a^5+2*(b^2-c^2)*(b-c)*(b^4+6*b^2*c^2+c^4)*a^4+8*(b^4+c^4+4*(b^2+b*c+c^2)*b*c)*(b-c)^2*b*c*a^3+(b^2-c^2)^2*(b+c)*(b^4+c^4+2*(2*b^2-9*b*c+2*c^2)*b*c)*a^2-4*(b^2-c^2)^4*b*c*a-(b^4-c^4)*(b^2-c^2)^3*(b+c)) : :

X(31967) lies on the line {56,990}

X(31968) = ORTHOCENTER OF THE HUTSON EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^8-2*(b-c)^2*a^7-2*(b+c)*(b^2+10*b*c+c^2)*a^6+2*(3*b^4+3*c^4+2*(2*b^2-15*b*c+2*c^2)*b*c)*a^5+8*(b+c)*(5*b^2+9*b*c+5*c^2)*b*c*a^4-2*(3*b^6+3*c^6+(14*b^4+14*c^4-(23*b^2+116*b*c+23*c^2)*b*c)*b*c)*a^3+2*(b+c)*(b^6+c^6-(10*b^4+10*c^4+(37*b^2+4*b*c+37*c^2)*b*c)*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4+2*(4*b^2+5*b*c+4*c^2)*b*c)*a-(b^2-c^2)^4*(b+c)) : :
X(31968) = 3*X(354)-2*X(12731)

X(31968) lies on these lines: {9,3295}, {20,12537}, {65,12854}, {354,12731}, {1210,12620}, {3748,12521}, {9804,11024}, {12439,15998}, {12654,15934}

X(31968) = reflection of X(i) in X(j) for these (i,j): (9804, 18241), (12670, 12658), (12692, 12631), (15998, 12439)
X(31968) = Hutson-extouch-isogonal conjugate of X(32051)

X(31969) = ORTHOCENTER OF THE LEMOINE TRIANGLE

Barycentrics (5*a^6-10*(b^2+c^2)*a^4+(4*b^4-7*b^2*c^2+4*c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*(2*a^4+5*(b^2+c^2)*a^2-3*(b^2-c^2)^2) : :

X(31969) lies on this line: {3,5476}

X(31969) = Lemoine-isogonal conjugate of X(8145)

X(31970) = ORTHOCENTER OF THE LUCAS ANTIPODAL TRIANGLE

Barycentrics 2*S^4+2*(2*(2*SA-SW)*R^2-3*SB*SC-SA^2)*S^2-((2*SA-SW)*(2*S^2-SW^2)-8*R^2*SB*SC)*S+2*SB*SC*SW^2 : :

X(31970) lies on this line: {492,6337}

X(31970) = Lucas-antipodal-isogonal conjugate of X(32052)

X(31971) = ORTHOCENTER OF THE LUCAS(-1) ANTIPODAL TRIANGLE

Barycentrics 2*S^4+2*(2*(2*SA-SW)*R^2-3*SB*SC-SA^2)*S^2+((2*SA-SW)*(2*S^2-SW^2)-8*R^2*SB*SC)*S+2*SB*SC*SW^2 : :

X(31971) lies on this line: {491,6337}

X(31971) = Lucas(-1)-antipodal-isogonal conjugate of X(32053)

X(31972) = ORTHOCENTER OF THE LUCAS CENTRAL TRIANGLE

Barycentrics (SB+SC)*((2*R^2-4*SA-5*SW)*S^2-(8*S^2-SA*(4*R^2-5*SA)-3*SB*SC+SW^2)*S-SA^2*SW) : :
X(31972) = 2*X(8155)-3*X(11198)

X(31972) lies on these lines: {6,8910}, {493,1151}, {8155,11198}, {8939,12962}, {13882,23311}

X(31972) = Lucas-central-isogonal conjugate of X(8155)

X(31973) = ORTHOCENTER OF THE LUCAS(-1) CENTRAL TRIANGLE

Barycentrics (SB+SC)*((2*R^2-4*SA-5*SW)*S^2+(8*S^2-SA*(4*R^2-5*SA)-3*SB*SC+SW^2)*S-SA^2*SW) : :

X(31973) lies on these lines: {494,1152}, {8943,12969}, {13934,23312}

X(31973) = Lucas(-1)-central-isogonal conjugate of X(8156)

X(31974) = ORTHOCENTER OF THE LUCAS TANGENTS TRIANGLE

Barycentrics (SB+SC)*(21*S^4+SA*SW^3+(8*SA*R^2+13*SA^2-15*SB*SC+6*SW^2)*S^2+S*(2*(9*SW+2*R^2+5*SA)*S^2+SW*(7*SA^2-8*SB*SC+SW^2))) : :
X(31974) = 2*X(1151)-3*X(11199)

X(31974) lies on this line: {1151,8155}

X(31974) = Lucas-tangents-isogonal conjugate of X(1151)

X(31975) = ORTHOCENTER OF THE LUCAS(-1) TANGENTS TRIANGLE

Barycentrics (SB+SC)*(21*S^4+SA*SW^3+(8*SA*R^2+13*SA^2-15*SB*SC+6*SW^2)*S^2-S*(2*(9*SW+2*R^2+5*SA)*S^2+SW*(7*SA^2-8*SB*SC+SW^2))) : :

X(31975) lies on this line: {1152,8156}

X(31975) = Lucas(-1)-tangents-isogonal conjugate of X(1152)

X(31976) = ORTHOCENTER OF THE MACBEATH TRIANGLE

Barycentrics ((2*R^2-SA)*S^2+(8*R^2-3*SW)*SB*SC)*(8*R^2+SA-3*SW) : :
X(31976) = 2*X(8146)-3*X(11197)

X(31976) lies on these lines: {3,161}, {4,8154}, {311,12225}, {1594,31867}, {8146,11197}

X(31976) = isogonal conjugate of X(14118)
X(31976) = MacBeath-isogonal conjugate of X(8146)

X(31977) = ORTHOCENTER OF THE MANDART-EXCIRCLES TRIANGLE

Barycentrics
2*a^13-(7*b^2+6*b*c+7*c^2)*a^11-(b+c)*(b^2-10*b*c+c^2)*a^10+(7*b^2+2*b*c+7*c^2)*(b+c)^2*a^9+(b+c)*(3*b^4+3*c^4-2*(13*b^2+5*b*c+13*c^2)*b*c)*a^8+2*(b^6+c^6-(8*b^4+8*c^4+3*(3*b^2-8*b*c+3*c^2)*b*c)*b*c)*a^7-2*(b+c)*(b^6+c^6-(8*b^4+8*c^4+(5*b^2+24*b*c+5*c^2)*b*c)*b*c)*a^6-4*(2*b^8+2*c^8-(3*b^6+3*c^6-(2*b^4+2*c^4+(7*b^2+32*b*c+7*c^2)*b*c)*b*c)*b*c)*a^5-2*(b+c)*(b^8+c^8-2*(2*b^6+2*c^6-5*(2*b-c)*(b-2*c)*b^2*c^2)*b*c)*a^4+(b^2-c^2)^2*(b-c)^2*(5*b^4+22*b^2*c^2+5*c^4)*a^3+(b^2-c^2)^3*(b-c)*(3*b^4+3*c^4-2*(2*b^2+3*b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^4*(b-c)^4*a-(b^2-c^2)^5*(b-c)*(b^2+c^2) : :

X(31977) lies on the line {12659,12699}

X(31977) = Mandart-excircles-isogonal conjugate of X(32054)

X(31978) = ORTHOCENTER OF THE MIDHEIGHT TRIANGLE

Barycentrics (SB+SC)*(-2*R^2*S^2+(8*R^2-SW)*(8*R^2-SA-SW)*SA) : :
X(31978) = 3*X(51)-X(5895) = 3*X(51)+X(30443) = 3*X(64)+X(6293) = X(64)+3*X(7729) = X(64)+2*X(22967) = 3*X(185)-X(6293) = X(185)-3*X(7729) = 3*X(1853)-X(11381) = 3*X(3917)-5*X(8567) = 2*X(5447)-3*X(11204) = X(5562)-3*X(10606) = X(5878)-3*X(9730) = 3*X(5890)+X(12250) = 2*X(5893)-3*X(5943) = X(6225)-5*X(10574) = X(6293)-9*X(7729) = X(6293)-6*X(22967) = 3*X(7729)-2*X(22967) = X(9833)-3*X(14855) = X(12324)+3*X(15072)

X(31978) lies on these lines: {3,1660}, {5,2883}, {6,64}, {20,2393}, {30,12235}, {51,5895}, {52,20427}, {66,6815}, {125,22970}, {389,13488}, {511,5894}, {1503,14913}, {1853,11381}, {1885,22530}, {2777,5446}, {2807,12262}, {3357,9938}, {3541,6241}, {3548,12162}, {3917,8567}, {5447,11204}, {5462,22802}, {5562,10606}, {5663,15115}, {5878,9730}, {5890,11431}, {5893,5943}, {5907,6696}, {6225,10574}, {7687,13474}, {9786,9914}, {9833,14855}, {9924,16936}, {9969,13568}, {10170,25563}, {10192,17704}, {10575,14216}, {11793,23328}, {12085,19458}, {13382,15105}, {14641,18400}, {14915,18381}, {16252,16836}

X(31978) = midpoint of X(i) and X(j) for these {i,j}: {52, 20427}, {64, 185}, {5895, 30443}, {10575, 14216}
X(31978) = reflection of X(i) in X(j) for these (i,j): (185, 22967), (2883, 9729), (5907, 6696), (22802, 5462)
X(31978) = complement of X(36982)
X(31978) = midheight-isogonal conjugate of X(5893)
X(31978) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64, 7729, 185), (185, 7729, 22967)

X(31979) = ORTHOCENTER OF THE 3rd MIXTILINEAR TRIANGLE

Barycentrics
a*(4*a^9-12*(b+c)*a^8+63*b*c*a^7+(b+c)*(34*b^2-125*b*c+34*c^2)*a^6-3*(8*b^4+8*c^4+(17*b^2-106*b*c+17*c^2)*b*c)*a^5-(b+c)*(30*b^4+30*c^4-(249*b^2-526*b*c+249*c^2)*b*c)*a^4+(32*b^6+32*c^6-(79*b^4+79*c^4+2*(112*b^2-327*b*c+112*c^2)*b*c)*b*c)*a^3+(b+c)*(6*b^6+6*c^6-(115*b^4+115*c^4-2*(241*b^2-381*b*c+241*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(12*b^4+12*c^4-(67*b^2-118*b*c+67*c^2)*b*c)*a+(b^2-c^2)^3*(b-c)*(2*b-c)*(b-2*c)) : :

X(31979) lies on these lines: {}

X(31979) = 3rd-mixtilinear-isogonal conjugate of X(3)

X(31980) = ORTHOCENTER OF THE 4th MIXTILINEAR TRIANGLE

Barycentrics a*(4*a^6-3*(4*b^2+3*b*c+4*c^2)*a^4-2*(b^3+c^3)*a^3+6*(3*b^4+3*c^4+(3*b^2-2*b*c+3*c^2)*b*c)*a^2-6*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a-(b+2*c)*(2*b+c)*(b-c)^4) : :

X(31980) lies on these lines: {}

X(31980) = 4th-mixtilinear-isogonal conjugate of X(3)

X(31981) = ORTHOCENTER OF THE 1st NEUBERG TRIANGLE

Barycentrics (b^2+c^2)*a^6-b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^2+c^2)^2*b^2*c^2 : :
X(31981) = 2*X(39)-3*X(13085) = 3*X(6194)-4*X(7780) = X(6309)-3*X(13085) = 3*X(7709)-2*X(7781) = 5*X(7786)-3*X(9764)

X(31981) lies on the cubic K906 and these lines: {2,39}, {3,698}, {4,736}, {32,18906}, {98,7751}, {315,1916}, {511,14023}, {732,1352}, {754,9873}, {2353,3504}, {2782,8721}, {3094,7800}, {3098,6308}, {3398,31958}, {5152,7793}, {5969,9821}, {6194,7780}, {7709,7781}, {7735,18806}, {7787,10000}, {7824,10335}, {7826,14645}, {11155,15515}

X(31981) = reflection of X(i) in X(j) for these (i,j): (6309, 39), (7758, 3095), (12251, 7751), (18768, 76)
X(31981) = circumtangential isogonal conjugate of X(32)
X(31981) = anticomplement of X(8149)
X(31981) = 1st-Neuberg-isogonal conjugate of X(8149)
X(31981) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6294, 6581, 2), (6309, 13085, 39), (6314, 6318, 76), (22868, 22913, 20081)

X(31982) = ORTHOCENTER OF THE 2nd NEUBERG TRIANGLE

Barycentrics a^8+(b^2+c^2)*a^6+(2*b^2+c^2)*(b^2+2*c^2)*a^4-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^6+c^6)*(b^2+c^2) : :
X(31982) = X(83)-3*X(9765) = 4*X(6287)+X(7758) = 4*X(6292)-3*X(13086) = 2*X(6308)-3*X(13086) = 9*X(7618)-4*X(8725) = 3*X(7775)-X(18548) = 6*X(7775)-X(20088) = 6*X(9765)-X(18769)

X(31982) lies on the cubic K906 and these lines: {2,32}, {4,8149}, {39,5207}, {147,7764}, {194,11606}, {262,7759}, {732,1352}, {3767,9478}, {3818,6309}, {5188,7843}, {6660,9918}, {7618,8725}, {7763,8290}, {7795,24273}, {8177,14023}, {8721,9737}, {9744,12252}, {9990,14907}

X(31982) = reflection of X(i) in X(j) for these (i,j): (6308, 6292), (18769, 83), (20088, 18548)
X(31982) = anticomplement of X(8150)
X(31982) = 2nd-Neuberg-isogonal conjugate of X(8150)
X(31982) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2896, 9866, 315), (6296, 6297, 2), (6292, 6308, 13086), (7752, 7785, 2548)

X(31983) = ORTHOCENTER OF THE 2nd ORTHOSYMMEDIAL TRIANGLE

Barycentrics (SB+SC)*(6*(3*R^2-SW)^2*S^4-(2*R^2*(54*SA*R^2-27*SA*SW+4*SW^2)+3*(2*SA-SW)*SW^2)*SW*S^2+(4*R^2+SA-2*SW)*SA*SW^4) : :
X(31983) = 2*X(5480)-3*X(11226)

X(31983) lies on these lines: {353,15577}, {5480,11226}, {6403,8705}, {9076,12507}, {19161,31771}

X(31983) = reflection of X(19161) in X(31771)
X(31983) = 2nd-orthosymmedial-isogonal conjugate of X(5480)

X(31984) = ORTHOCENTER OF THE APUS TRIANGLE

Barycentrics
a^2*(a^11+(b+c)*a^10-(5*b^2+6*b*c+5*c^2)*a^9-(b+c)*(5*b^2+14*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4+4*(b^2+c^2)*b*c)*a^7+2*(b+c)*(5*b^4+5*c^4+4*(5*b^2+4*b*c+5*c^2)*b*c)*a^6-2*(5*b^6+5*c^6-(6*b^4+6*c^4+(33*b^2+52*b*c+33*c^2)*b*c)*b*c)*a^5-2*(b+c)*(5*b^6+5*c^6+(18*b^4+18*c^4+(11*b^2-8*b*c+11*c^2)*b*c)*b*c)*a^4+(5*b^6+5*c^6-(34*b^4+34*c^4+(25*b^2+36*b*c+25*c^2)*b*c)*b*c)*(b+c)^2*a^3+(b+c)*(5*b^8+5*c^8+2*(4*b^6+4*c^6-(4*b^4+4*c^4+(24*b^2+29*b*c+24*c^2)*b*c)*b*c)*b*c)*a^2-(b^2-c^2)^2*(b+c)^2*(b^4+c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a-(b^2-c^2)^5*(b-c)) : :

X(31984) lies on these lines: {1593,1871}, {3295,8573}

X(31985) = ORTHOCENTER OF THE 3rd HATZIPOLAKIS TRIANGLE

Barycentrics (128*R^6-2*(83*SA-33*SW)*R^4+(63*SA-38*SW)*SW*R^2-2*(3*SA-2*SW)*SW^2)*S^2+(16*R^2-3*SW)*(R^2*(32*R^2-17*SW)+2*SW^2)*SB*SC : :

X(31985) lies on these lines: {6,17837}, {8254,12900}, {10112,23308}, {10116,22816}, {15033,22533}

X(31985) = midpoint of X(i) and X(j) for these {i,j}: {10112, 23308}, {10116, 22816}, {22953, 22968}
X(31985) = Hatzipolakis-Moses-isogonal conjugate of X(11262)

X(31986) = ORTHOCENTER OF THE 3rd PARRY TRIANGLE

Barycentrics a^2*(2*a^8-13*(b^2+c^2)*a^6+3*(4*b^4+5*b^2*c^2+4*c^4)*a^4+(b^2+c^2)*(17*b^4-65*b^2*c^2+17*c^4)*a^2-10*b^8+(32*b^4+3*b^2*c^2+32*c^4)*b^2*c^2-10*c^8)*(b^2-c^2) : :

X(31986) lies on these lines: {8704,9979}, {9131,31772}

X(31986) = reflection of X(i) in X(j) for these (i,j): (9131, 31772), (9979, 31773)
X(31986) = 3rd-Parry-isogonal conjugate of X(351)

X(31987) = ORTHOCENTER OF THE INNER-SQUARES TRIANGLE

Barycentrics a^10-(5*b^4-4*b^2*c^2+5*c^4)*a^6+2*(b^2-c^2)^2*b^2*c^2*a^2+(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4)*a^4-4*(a^4+2*(b^2+c^2)*a^2-(b^2-c^2)^2)*S^3-(b^4-c^4)*(b^2-c^2)^3 : :

X(31987) lies on these lines: {4,8956}, {20,485}, {371,19042}, {2548,31988}, {7374,8854}, {8970,21736}, {9838,13004}

X(31988) = ORTHOCENTER OF THE OUTER-SQUARES TRIANGLE

Barycentrics a^10-(5*b^4-4*b^2*c^2+5*c^4)*a^6+2*(b^2-c^2)^2*a^2*b^2*c^2+(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4)*a^4+4*(a^4+2*(b^2+c^2)*a^2-(b^2-c^2)^2)*S^3-(b^4-c^4)*(b^2-c^2)^3 : :

X(31988) lies on these lines: {20,486}, {372,19041}, {2548,31987}, {7000,8855}

X(31989) = ORTHOCENTER OF THE SYMMEDIAL TRIANGLE

Barycentrics a^2*(a^4-2*(b^2+c^2)*a^2+b^2*c^2)*((b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4) : :
X(31989) = 2*X(3)-3*X(14133) = 4*X(3)-3*X(14134) = 2*X(546)-3*X(31869) = 2*X(8152)-3*X(11205)

X(31989) lies on these lines: {3,695}, {6,8601}, {546,31869}, {576,3146}, {1493,21905}, {3051,14135}, {8152,11205}, {14068,15004}

X(31989) = reflection of X(14134) in X(14133)
X(31989) = symmedial-isogonal conjugate of X(8152)

X(31990) = ORTHOCENTER OF THE X-PARABOLA-TANGENTIAL TRIANGLE

Barycentrics 2*a^16-6*(b^2+c^2)*a^14+3*(b^4+8*b^2*c^2+c^4)*a^12+5*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^10-(b^8+c^8+16*(b^4-4*b^2*c^2+c^4)*b^2*c^2)*a^8-2*(b^2+c^2)*(4*b^8+4*c^8-(22*b^4-39*b^2*c^2+22*c^4)*b^2*c^2)*a^6+(5*b^12+5*c^12-8*(b^8+c^8+(3*b^4-7*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(b^8+c^8-(9*b^4-17*b^2*c^2+9*c^4)*b^2*c^2)*a^2-(b^4+b^2*c^2+c^4)*(b^2-c^2)^6 : :

X(31990) lies on this line: {30,10279}

X(31990) = X-parabola-tangential-isogonal conjugate of X(32061)

X(31991) = ORTHOCENTER OF THE YIU TRIANGLE

Barycentrics a^16-5*(b^2+c^2)*a^14+(11*b^4+16*b^2*c^2+11*c^4)*a^12-2*(b^2+c^2)*(7*b^4+2*b^2*c^2+7*c^4)*a^10+(10*b^8+10*c^8+3*(2*b^4+3*b^2*c^2+2*c^4)*b^2*c^2)*a^8-(b^6+c^6)*(b^4+b^2*c^2+c^4)*a^6-(5*b^8+5*c^8-2*(b^4+c^4)*b^2*c^2)*(b^2-c^2)^2*a^4+4*(b^6+c^6)*(b^2-c^2)^4*a^2-(b^4+c^4)*(b^2-c^2)^6 : :

As a point on the Euler line, X(31991) has Shinagawa coefficients (3E²+16F²+16S², 15E²+48EF+48F²-80S²).

X(31991) lies on this line: {2,3}

X(31991) = reflection of X(i) in X(j) for these (i,j): (3, 14106), (13150, 5)
X(31991) = Yiu-isogonal conjugate of X(8154)

X(31992) = REFLECTION OF X(2) IN X(6544)

Barycentrics (b-c) (5 a^2-5 a b+2 b^2-5 a c+b c+2 c^2) : :
X(31992) = 5 X[2] - 2 X[6545],X[2] + 2 X[6546],X[2] - 4 X[10196],7 X[2] - 4 X[21204],4 X[1639] - X[21297],5 X[3617] + 4 X[5592],X[4453] - 4 X[14425],4 X[4468] + 5 X[27013],4 X[4500] + 5 X[17494],16 X[4521] - 7 X[27138],5 X[6544] - X[6545],4 X[6544] - X[6548],3 X[6544] - X[14475],7 X[6544] - 2 X[21204],X[6545] + 5 X[6546],4 X[6545] - 5 X[6548],X[6545] - 10 X[10196],3 X[6545] - 5 X[14475],7 X[6545] - 10 X[21204],4 X[6546] + X[6548],X[6546] + 2 X[10196],3 X[6546] + X[14475],7 X[6546] + 2 X[21204],X[6548] - 8 X[10196],3 X[6548] - 4 X[14475],7 X[6548] - 8 X[21204],6 X[10196] - X[14475],7 X[10196] - X[21204],8 X[11068] + X[20295],2 X[14442] + X[17487],7 X[14475] - 6 X[21204],X[17161] - 10 X[26777]

X(31992) lies on these lines: {2,514}, {100,190}, {523,27811}, {650,4850}, {661,27081}, {693,30829}, {1027,5297}, {1635,31349}, {1639,6009}, {2490,31205}, {2786,14435}, {3617,5592}, {3676,31188}, {4448,4664}, {4453,14425}, {4468,5744}, {4500,17494}, {4521,5328}, {5235,7192}, {6006,6172}, {7662,28151}, {11068,20295}, {14442,17487}, {17161,26777}, {24124,27929}, {25057,28910}, {28209,28602}, {28886,30564}, {31209,31233}

X(31992) = midpoint of X(6544) and X(6546)
X(31992) = reflection of X(i) and X(j) for these {i,j}: {2, 6544}, {6544, 10196}, {6548, 2}
X(31992) = anticomplement of X(14475)
X(31992) = X(2384)-anticomplementary conjugate of X(149)
X(31992) = crosssum of X(649) and X(8649)
X(31992) = crossdifference of every pair of points on line {902, 1015}
X(31992) = {X(6546), X(10196)}-harmonic conjugate of X(2)

X(31993) = MIDPOINT OF X(28605) AND X(28606)

Barycentrics (b + c) (a (a + b + c) + 2 b c) : :
X(31993) = 3 X[2] + X[28605] = (r^2 - s^2) X[10] - r (2 r + R) X[12]

X(31993) lies on these lines: {1,2049}, {2,37}, {6,5271}, {8,5712}, {9,1730}, {10,12}, {38,30970}, {40,10888}, {42,3696}, {44,5278}, {45,19744}, {57,18229}, {63,4363}, {81,4670}, {86,1999}, {92,1841}, {141,5249}, {171,24342}, {172,26643}, {213,4383}, {220,27413}, {223,9623}, {228,1376}, {241,27339}, {244,31241}, {274,1920}, {278,475}, {306,594}, {307,6354}, {319,17778}, {329,966}, {333,894}, {354,3741}, {427,1824}, {498,5955}, {517,10478}, {518,30969}, {756,3967}, {899,31264}, {908,5743}, {936,3191}, {940,3713}, {942,10479}, {958,1867}, {960,31339}, {964,1104}, {968,5695}, {975,16458}, {986,1698}, {1089,16828}, {1100,3187}, {1125,2901}, {1155,3980}, {1212,26035}, {1213,4054}, {1214,6358}, {1220,16824}, {1279,24552}, {1332,25897}, {1403,4413}, {1427,1441}, {1449,19722}, {1743,19723}, {1791,17518}, {1962,4365}, {2352,11358}, {2550,3198}, {2895,4690}, {3120,20716}, {3159,3634}, {3219,5235}, {3247,19749}, {3294,7308}, {3305,17259}, {3452,5241}, {3588,22097}, {3589,26723}, {3661,18134}, {3679,4113}, {3683,3923}, {3687,4967}, {3689,29670}, {3700,30476}, {3701,19874}, {3725,28248}, {3740,26037}, {3742,30942}, {3744,3757}, {3745,4362}, {3748,29651}, {3751,4042}, {3773,29653}, {3782,4357}, {3823,29679}, {3834,27186}, {3838,25760}, {3842,3971}, {3844,25957}, {3846,17605}, {3848,30957}, {3875,20182}, {3896,17163}, {3914,4026}, {3931,4647}, {3934,22036}, {3936,20483}, {3954,17308}, {3963,30713}, {3966,26098}, {3970,17284}, {3973,19751}, {3985,4656}, {3992,19870}, {3993,10180}, {4001,17365}, {4003,6682}, {4024,4885}, {4028,4046}, {4044,16589}, {4053,30823}, {4078,6057}, {4205,19857}, {4361,5256}, {4371,20043}, {4377,28654}, {4385,19853}, {4418,4640}, {4426,19281}, {4463,29667}, {4472,6703}, {4643,5905}, {4644,14552}, {4646,26115}, {4648,34255}, {4651,4849}, {4654,17272}, {4682,17763}, {4685,4732}, {4706,29825}, {4713,27479}, {4852,17011}, {4865,4914}, {4883,10453}, {4891,29814}, {4903,19877}, {4928,22043}, {4944,7265}, {4972,21949}, {4981,17165}, {5087,25960}, {5219,21078}, {5224,27184}, {5226,5936}, {5251,24271}, {5273,7229}, {5287,15668}, {5294,17369}, {5316,22019}, {5333,17019}, {5341,21376}, {5739,17275}, {5745,22001}, {5793,19860}, {5835,24987}, {5880,26034}, {6533,19864}, {6539,30588}, {6542,24656}, {6589,17894}, {6706,29611}, {7222,9965}, {7226,28582}, {7270,26051}, {7283,11110}, {7321,26840}, {7965,21629}, {8226,12618}, {8609,19720}, {9148,17989}, {9596,14555}, {9780,27538}, {13728,23537}, {13740,16817}, {15985,30076}, {16592,20496}, {16593,22032}, {16594,22031}, {16666,19717}, {16667,19739}, {16668,19738}, {16669,19742}, {16670,19750}, {16696,30599}, {16699,31623}, {16726,27163}, {16825,25496}, {16826,25507}, {17018,28581}, {17027,31306}, {17063,29827}, {17066,22044}, {17069,30864}, {17140,21342}, {17184,17237}, {17231,18139}, {17239,31019}, {17251,31164}, {17256,26044}, {17277,27064}, {17306,23681}, {17332,17781}, {17345,17483}, {17789,27319}, {18138,20911}, {18228,28657}, {19285,27802}, {20684,21923}, {20917,22028}, {20952,27345}, {21070,29571}, {21090,30860}, {21242,29655}, {21422,28797}, {21611,26049}, {21807,30757}, {21857,27042}, {21873,28633}, {21879,29576}, {21884,25614}, {21935,27714}, {22005,30808}, {22011,24774}, {22014,30824}, {22021,25525}, {22048,29596}, {24076,31247}, {24160,24931}, {24776,24784}, {25102,29593}, {25130,29570}, {28108,30617}

X(31993) = midpoint of X(28605) and X(28606)
X(31993) = complement of X(28606)
X(31993) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 75, 3666}, {2, 321, 37}, {2, 4359, 3752}, {2, 4699, 19804}, {2, 4772, 17490}, {2, 19785, 4657}, {2, 19804, 16610}, {2, 19825, 17740}, {2, 24589, 16602}, {2, 26724, 17356}, {2, 28604, 19808}, {2, 28605, 28606}, {2, 30699, 17321}, {2, 31025, 321}, {10, 226, 1211}, {10, 1215, 210}, {37, 321, 3175}, {37, 22034, 3995}, {42, 21020, 3696}, {65, 210, 22275}, {321, 3995, 22034}, {333, 894, 4641}, {594, 17056, 306}, {1215, 27798, 10}, {2345, 26011, 30818}, {3187, 19684, 1100}, {3741, 24325, 354}, {3752, 4688, 4359}, {3757, 5263, 3744}, {3772, 17303, 2}, {3846, 25385, 17605}, {3995, 22034, 3175}, {4363, 5737, 63}, {4751, 18743, 2}, {4980, 17147, 4686}, {5278, 26223, 44}, {5333, 17019, 28639}, {6682, 24165, 4003}, {10436, 11679, 940}, {17163, 29822, 3896}, {18229, 25590, 57}
X(31993) = complement of the isogonal of X(2214)
X(31993) = isotomic conjugate of the polar conjugate of X(1867)
X(31993) = X(i)-complementary conjugate of X(j) for these (i,j): {649, 5515}, {835, 3835}, {2214, 10}
X(31993) = X(5936)-Ceva conjugate of X(10)
X(31993) = X(i)-isoconjugate of X(j) for these (i,j): {6, 5331}, {58, 941}, {81, 2258}, {284, 959}, {649, 931}, {1333, 31359}
X(31993) = crossdifference of every pair of points on line {667, 7252}
X(31993) = barycentric product X(i) X(j) for these {i,j}: {7, 3714}, {10, 10436}, {69, 1867}, {226, 11679}, {306, 5307}, {313, 1468}, {321, 940}, {349, 2268}, {668, 8672}, {958, 1441}, {1446, 3713}, {4185, 20336}, {4552, 23880}, {5019, 27801}, {6386, 8639}
X(31993) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 5331}, {10, 31359}, {37, 941}, {42, 2258}, {65, 959}, {100, 931}, {940, 81}, {958, 21}, {1468, 58}, {1867, 4}, {2268, 284}, {3713, 2287}, {3714, 8}, {4185, 28}, {5019, 1333}, {5307, 27}, {8639, 667}, {8672, 513}, {10436, 86}, {10472, 27164}, {11679, 333}, {17418, 3737}, {23880, 4560}

Points I-Caph: X(31994)-X(32000)

This preamble and centers X(31994)-X(32000) were contributed by Clark Kimberling and Peter Moses, April 15, 2019.

Suppose that P = p:q:r (barycentrics). The Point I-Caph of P is the point given by

2qr + (p+q+r)p : 2rp + (p+q+r)q : 2pq + (p+q+r)r.

See sections below for Points II-Caph, III-Caph, and IV-Caph. The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P. (The name Caph is that of a a star, also known as Beta Cassiopeiae.)

The appearance of {i,j} in the following list means that X(j) = Point I-Caph of X(i):

{1,10436}, {2,2}, {6,7770}, {10,28653}, {30,1494}, {190,6631}, {511,290}, {512,670}, {513,668}, {514,190}, {516,18025}, {517,18816}, {518,2481}, {519,903}, {520,6528}, {521,18026}, {522,664}, {523,99}, {524,671}, {525,648}, {527,1121}, {528,18821}, {532,11117}, {533,11118}, {536,3227}, {537,18822}, {538,3228}, {542,5641}, {543,18823}, {664,10001}, {668,9296}, {670,9428}, {690,892}, {696,18824}, {698,3225}, {712,18825}, {714,18826}, {726,3226}, {732,14970}, {740,18827}, {758,14616}, {782,18828}, {804,18829}, {812,4562}, {824,4586}, {826,4577}, {888,886}, {891,889}, {900,4555}, {903,9460}, {918,666}, {1494,9410}, {2574,15164}, {2575,15165}, {2799,2966}, {3244,7321}, {3413,6190}, {3414,6189}, {3900,4569}, {3910,6648}, {4083,18830}, {4777,4597}, {4977,6540}, {6362,6606}, {6368,18831}, {6550,6635}, {9033,16077}, {23870,23895}, {23871,23896}

X(31994) = POINT I-CAPH OF X(7)

Barycentrics (a + b - c)*(a - b + c)*(a^2 - 2*a*b + b^2 - 2*a*c + 6*b*c + c^2) : :

X(31994) lies on these lines: {1, 25718}, {2, 3160}, {7, 8}, {9, 30695}, {10, 279}, {72, 23839}, {145, 5543}, {226, 29616}, {348, 9780}, {443, 14256}, {474, 934}, {479, 26040}, {664, 3616}, {728, 8545}, {948, 29611}, {1323, 1698}, {1447, 4308}, {1565, 5818}, {1788, 7223}, {1997, 17084}, {2481, 7320}, {2898, 30854}, {3188, 5273}, {3361, 7268}, {3598, 10106}, {3617, 9436}, {3622, 25716}, {3668, 5232}, {3674, 5261}, {3679, 10481}, {3879, 20008}, {3912, 5226}, {3945, 6738}, {4350, 9623}, {4384, 5435}, {4460, 7190}, {4470, 6610}, {4566, 11024}, {5084, 28644}, {5296, 27288}, {5423, 6063}, {6172, 30625}, {6180, 7229}, {6556, 7185}, {7196, 26038}, {8051, 27496}, {9446, 28057}, {14189, 18230}, {17095, 19877}

X(31994) = {X(7), X(8)}-harmonic conjugate of X(32003)
X(31994) = {X(85), X(6604)}-harmonic conjugate of X(32086)

X(31995) = POINT I-CAPH OF X(8)

Barycentrics a^2 - b^2 + 6*b*c - c^2 : :
Trilinears 2 sec^2(A/2) + csc^2(A/2) : :

X(31995)lies on these lines: {1, 4452}, {2, 2415}, {6, 4402}, {7, 8}, {9, 4454}, {10, 4862}, {77, 4861}, {78, 4328}, {81, 19819}, {86, 16711}, {142, 346}, {144, 4384}, {145, 3664}, {189, 14213}, {190, 18230}, {192, 5308}, {193, 17117}, {200, 7274}, {269, 3872}, {274, 17183}, {314, 5558}, {318, 1119}, {321, 9776}, {326, 7269}, {329, 4359}, {330, 27497}, {333, 28610}, {347, 10538}, {348, 30543}, {350, 26103}, {391, 527}, {479, 6063}, {519, 4888}, {524, 4371}, {536, 4648}, {545, 17259}, {644, 25878}, {894, 5222}, {903, 5224}, {966, 4688}, {1086, 2345}, {1111, 21290}, {1150, 2094}, {1222, 16079}, {1266, 3616}, {1278, 17316}, {1449, 4747}, {1699, 9950}, {1743, 24599}, {2321, 4869}, {2325, 20195}, {3026, 6018}, {3187, 19826}, {3241, 3875}, {3242, 15590}, {3501, 28351}, {3589, 4000}, {3598, 7081}, {3617, 4887}, {3621, 4896}, {3622, 4021}, {3623, 30712}, {3629, 4361}, {3662, 29611}, {3673, 24993}, {3679, 4902}, {3739, 4419}, {3757, 9778}, {3879, 20050}, {3886, 11038}, {3912, 4461}, {3923, 16020}, {3950, 29621}, {4346, 4357}, {4360, 20057}, {4385, 11024}, {4389, 19877}, {4398, 5550}, {4416, 20059}, {4429, 5772}, {4431, 17298}, {4440, 4699}, {4445, 7238}, {4460, 17160}, {4470, 4657}, {4472, 17323}, {4511, 7190}, {4643, 4739}, {4665, 7232}, {4675, 4686}, {4684, 30340}, {4726, 4851}, {4740, 17300}, {4748, 17255}, {4764, 17317}, {4772, 6646}, {4788, 29569}, {4821, 6542}, {4853, 7271}, {4898, 28313}, {4903, 30758}, {4916, 4971}, {4967, 5232}, {5271, 9965}, {5423, 26040}, {5554, 24209}, {5556, 30479}, {5839, 6144}, {6172, 17277}, {7227, 17290}, {8583, 28661}, {9801, 9812}, {10442, 20070}, {10446, 20244}, {10447, 10453}, {10455, 18600}, {10520, 30567}, {10527, 22464}, {11115, 17189}, {11679, 21454}, {16816, 31300}, {16823, 24280}, {17140, 21283}, {17164, 18698}, {17184, 19825}, {17245, 31139}, {17251, 28635}, {17260, 20073}, {17345, 28634}, {17353, 31189}, {17384, 26039}, {17385, 26104}, {17740, 30834}, {17753, 20245}, {18228, 19804}, {19824, 29833}, {20080, 29617}, {21436, 31130}, {24181, 25966}, {24203, 24540}, {24213, 24982}, {24336, 28015}, {24554, 25083}, {25521, 27514}, {25728, 31722}, {26038, 30946}, {27544, 28742}, {28605, 34255}, {28809, 30090}

X(31995) = isotomic conjugate of X(7320)
X(31995) = anticomplement of X(3731)
X(31995) = {X(7), X(8)}-harmonic conjugate of X(21296)
X(31995) = {X(69), X(75)}-harmonic conjugate of X(32087)

X(31996) = POINT I-CAPH OF X(37)

Barycentrics a^2*b^2 + 3*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2 : :

X(31996) lies on these lines: {1, 2}, {35, 16917}, {37, 274}, {86, 213}, {194, 27268}, {238, 20133}, {313, 24944}, {330, 27298}, {384, 5259}, {668, 24656}, {894, 3294}, {1001, 11321}, {1107, 4698}, {1191, 20135}, {1203, 20132}, {1573, 25303}, {1616, 20157}, {1743, 20146}, {1909, 16589}, {2176, 15668}, {2223, 16060}, {3585, 17685}, {3739, 17143}, {3761, 27269}, {3822, 17669}, {3879, 26045}, {3986, 24215}, {4366, 16911}, {4423, 7770}, {4687, 5283}, {4751, 17144}, {5248, 16915}, {5251, 6645}, {5257, 26110}, {5280, 17000}, {5284, 17686}, {5315, 20137}, {6675, 26686}, {6690, 17694}, {8728, 26590}, {16466, 20131}, {16468, 20134}, {16470, 20136}, {16474, 20138}, {16475, 20139}, {16476, 20140}, {16477, 20141}, {16483, 20152}, {16514, 17398}, {16552, 17260}, {16584, 18298}, {16842, 26687}, {16918, 25542}, {17277, 20963}, {17302, 24790}, {17322, 25499}, {17326, 27047}, {17529, 26582}, {17745, 20147}, {17758, 31004}, {25086, 25994}, {27164, 28639}, {27283, 30867}

X(31997) = POINT I-CAPH OF X(75)

Barycentrics b*c*(2*a^2 + a*b + a*c + b*c) : :

X(31997) lies on these lines: {1, 75}, {2, 330}, {6, 16827}, {8, 25303}, {10, 24524}, {31, 873}, {37, 194}, {39, 27255}, {56, 85}, {76, 1125}, {87, 1221}, {92, 423}, {99, 5248}, {142, 30038}, {172, 16998}, {183, 20449}, {213, 3758}, {239, 940}, {257, 20271}, {292, 25918}, {312, 16826}, {313, 30022}, {319, 9534}, {321, 29570}, {322, 16824}, {325, 25466}, {350, 3616}, {551, 4479}, {662, 2304}, {668, 1698}, {894, 2176}, {1001, 1975}, {1015, 17030}, {1575, 24656}, {1914, 16915}, {1999, 19715}, {2277, 26110}, {2345, 24654}, {3294, 17336}, {3622, 4441}, {3624, 3761}, {3666, 24621}, {3673, 24331}, {3739, 17448}, {3757, 21010}, {3759, 20963}, {3760, 25055}, {3822, 7752}, {3873, 16707}, {3945, 20036}, {3948, 29612}, {3963, 30054}, {4110, 28604}, {4352, 17321}, {4357, 24215}, {4358, 29595}, {4359, 4393}, {4363, 16969}, {4378, 20949}, {4384, 5437}, {4385, 20947}, {4386, 16917}, {4389, 24214}, {4400, 16997}, {4426, 6645}, {4657, 24731}, {4664, 25264}, {4687, 5283}, {4751, 16819}, {4754, 24514}, {4798, 17790}, {5088, 13730}, {5333, 28621}, {5550, 18135}, {6381, 19862}, {6533, 7278}, {6626, 8033}, {7196, 17081}, {7200, 30667}, {7321, 17753}, {7757, 25092}, {7763, 10198}, {8024, 29648}, {9367, 26068}, {9780, 25280}, {11321, 16502}, {15668, 20923}, {16549, 29383}, {16552, 17335}, {16678, 16681}, {16684, 23393}, {16738, 18172}, {16739, 20985}, {16744, 18832}, {16781, 20172}, {16816, 24589}, {16817, 19285}, {16818, 17370}, {16825, 20955}, {16830, 30758}, {16831, 18743}, {16884, 20174}, {16974, 17789}, {17033, 24512}, {17084, 30545}, {17109, 27922}, {17137, 17169}, {17234, 29960}, {17275, 25457}, {17277, 21384}, {17289, 27248}, {17303, 17786}, {17397, 20913}, {17400, 18143}, {17474, 24592}, {18133, 25512}, {18144, 25498}, {18146, 19883}, {18827, 31359}, {19853, 28653}, {20257, 24199}, {20530, 31276}, {21024, 31028}, {21590, 25496}, {23572, 25127}, {25287, 26037}, {25660, 28640}, {26035, 27097}, {26100, 27148}, {26115, 27162}, {28605, 30562}, {29578, 30829}, {29746, 29773}, {30596, 30598}, {30966, 31339}

X(31998) = POINT I-CAPH OF X(99)

Barycentrics (a - b)*(a + b)*(a - c)*(a + c)*(a^4 - a^2*b^2 - b^4 - a^2*c^2 + 3*b^2*c^2 - c^4) : :

X(31998) lies on these lines: {2, 31372}, {99, 523}, {320, 17731}, {325, 5159}, {385, 3291}, {513, 17930}, {520, 12833}, {521, 17931}, {524, 5103}, {648, 2489}, {670, 23285}, {805, 3221}, {1509, 24345}, {2142, 7760}, {2407, 17941}, {3222, 9091}, {4563, 14607}, {7779, 19577}, {8029, 31614}, {8057, 17932}, {10278, 31632}, {14061, 23991}, {30786, 31655}

X(31998) = isotomic conjugate of X(9293)
X(31998) = complement of X(35511)
X(31998) = complementary conjugate of complement of X(20998)
X(31998) = X(2)-Ceva conjugate of X(99)
X(31998) = perspector of circumconic centered at X(99)
X(31998) = center of hyperbola {{A,B,C,X(892),X(4590),PU(179)}} (the locus of trilinear poles of lines passing through X(99))
X(31998) = crosssum of PU(105)
X(31998) = crosspoint of PU(179)

X(31999) = POINT I-CAPH OF X(192)

Barycentrics -(a^2*b^2) + 6*a^2*b*c - a^2*c^2 + b^2*c^2 : :

X(31999) lies on these lines: {1, 87}, {2, 17448}, {86, 16722}, {239, 5437}, {385, 3304}, {519, 27318}, {551, 27269}, {940, 4393}, {1015, 7786}, {1107, 27268}, {1909, 30998}, {3227, 5283}, {3303, 7783}, {3616, 9263}, {3622, 21226}, {3623, 17759}, {4317, 14712}, {4740, 17144}, {5434, 7823}, {5563, 7793}, {6645, 16781}, {7222, 24761}, {7777, 15888}, {7787, 16784}, {9336, 27020}, {12513, 17000}, {16884, 20168}, {16969, 17350}, {17232, 30038}, {19285, 19851}, {19722, 31036}, {21219, 30963}, {21223, 29814}, {24621, 29584}, {29815, 31088}

X(32000) = POINT I-CAPH OF X(4)

Barycentrics (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 6*b^2*c^2 + c^4) : :
Barycentrics SB*SC*(SA^2 + b^2*c^2) : :

X(32000) lies on these lines: {2, 253}, {3, 6527}, {4, 69}, {7, 318}, {8, 273}, {25, 15589}, {53, 599}, {75, 1119}, {92, 1229}, {95, 3524}, {112, 14039}, {141, 393}, {183, 6353}, {193, 458}, {276, 19166}, {278, 11679}, {281, 3912}, {290, 17040}, {297, 3620}, {309, 17170}, {324, 6820}, {325, 8889}, {339, 18537}, {376, 20477}, {378, 3964}, {391, 26003}, {491, 3535}, {492, 3536}, {524, 3087}, {648, 3618}, {1150, 10432}, {1270, 1586}, {1271, 1585}, {1494, 5071}, {1785, 17272}, {1897, 3672}, {1990, 3763}, {2897, 6836}, {3520, 9723}, {3619, 17907}, {3867, 8801}, {3945, 11109}, {4176, 18022}, {4357, 7952}, {5085, 15258}, {5232, 17555}, {6144, 6749}, {6515, 6819}, {6618, 14826}, {6995, 30698}, {7120, 25940}, {7282, 21296}, {7378, 10513}, {7386, 30737}, {7487, 7767}, {7490, 14829}, {8743, 16045}, {10002, 10516}, {10996, 26166}, {14929, 18494}, {15262, 26206}, {18141, 31623}, {18678, 25527}, {20080, 27377}

X(32000) = isotomic conjugate of X(15740)
X(32000) = polar conjugate of isogonal conjugate of X(17811)
X(32000) = {X(4), X(69)}-harmonic conjugate of X(32001)

This is the end of PART 16: Centers X(30001) - X(32000)

PART 1:	Introduction and Centers X(1) - X(1000)	PART 2:	Centers X(1001) - X(3000)	PART 3:	Centers X(3001) - X(5000)
PART 4:	Centers X(5001) - X(7000)	PART 5:	Centers X(7001) - X(10000)	PART 6:	Centers X(10001) - X(12000)
PART 7:	Centers X(12001) - X(14000)	PART 8:	Centers X(14001) - X(16000)	PART 9:	Centers X(16001) - X(18000)
PART 10:	Centers X(18001) - X(20000)	PART 11:	Centers X(20001) - X(22000)	PART 12:	Centers X(22001) - X(24000)
PART 13:	Centers X(24001) - X(26000)	PART 14:	Centers X(26001) - X(28000)	PART 15:	Centers X(28001) - X(30000)
PART 16:	Centers X(30001) - X(32000)	PART 17:	Centers X(32001) - X(34000)	PART 18:	Centers X(34001) - X(36000)
PART 19:	Centers X(36001) - X(38000)	PART 20:	Centers X(38001) - X(40000)	PART 21:	Centers X(40001) - X(42000)
PART 22:	Centers X(42001) - X(44000)	PART 23:	Centers X(44001) - X(46000)	PART 24:	Centers X(46001) - X(48000)
PART 25:	Centers X(48001) - X(50000)	PART 26:	Centers X(50001) - X(52000)	PART 27:	Centers X(52001) - X(54000)
PART 28:	Centers X(54001) - X(56000)	PART 29:	Centers X(56001) - X(58000)	PART 30:	Centers X(58001) - X(60000)
PART 31:	Centers X(60001) - X(62000)	PART 32:	Centers X(62001) - X(64000)	PART 33:	Centers X(64001) - X(66000)
PART 34:	Centers X(66001) - X(68000)	PART 35:	Centers X(68001) - X(70000)	PART 36:	Centers X(70001) - X(72000)