ENCYCLOPEDIA OF TRIANGLE CENTERS Part11

Encyclopedia of Triangle Centers - ETC

This is PART 11: Centers X(20001) - X(22000)

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(20001) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(1698)

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

X(20001) lies on these lines: {10, 75}, {519, 19933}

X(20021) = anticomplement of X(36213)

X(20002) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3214)

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

X(20002) lies on these lines: {10, 75}

X(20003) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3241)

Barycentrics -a^2 b^3 + 5 a^3 b c + a b^3 c + 5 b^4 c - 13 a b^2 c^2 - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + 5 b c^4 : :

X(20003) lies on these lines: {10, 75}, {5550, 19933}

X(20004) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3244)

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

X(20004) lies on these lines: {10, 75}, {19883, 19933}

X(20005) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3293)

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

X(20005) lies on these lines: {10, 75}

X(20006) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3617)

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

X(20006) lies on these lines: {10, 75}, {1698, 19887}

X(20007) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(936), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20007) lies on these lines: {1, 2}, {4, 3940}, {7, 11523}, {9, 4313}, {20, 72}, {21, 1260}, {29, 4720}, {55, 1183}, {63, 3522}, {69, 279}, {100, 5584}, {210, 3486}, {218, 4195}, {220, 346}, {280, 7538}, {307, 3160}, {318, 4671}, {329, 3146}, {341, 6555}, {376, 3927}, {390, 960}, {391, 1212}, {411, 5687}, {443, 11036}, {452, 3876}, {480, 958}, {515, 5815}, {518, 3600}, {908, 3832}, {942, 17580}, {948, 7270}, {950, 12536}, {952, 6865}, {956, 6986}, {959, 3779}, {965, 17314}, {1010, 5765}, {1145, 6962}, {1170, 1219}, {1229, 4673}, {1259, 4189}, {1320, 15998}, {1445, 4308}, {1446, 16284}, {1482, 6864}, {1610, 12329}, {1837, 8165}, {2550, 12635}, {2894, 6826}, {2895, 3152}, {2975, 8273}, {3057, 12632}, {3091, 3419}, {3149, 8158}, {3158, 5837}, {3219, 17576}, {3421, 6836}, {3434, 6894}, {3436, 6895}, {3452, 12625}, {3474, 3962}, {3487, 3824}, {3488, 5044}, {3523, 5440}, {3528, 9945}, {3555, 12128}, {3601, 5273}, {3681, 12125}, {3693, 15853}, {3711, 10950}, {3715, 10543}, {3812, 18221}, {3813, 18220}, {3868, 6904}, {3869, 7957}, {3871, 11344}, {3916, 10304}, {3998, 14552}, {4190, 9965}, {4292, 20059}, {4293, 5904}, {4294, 5692}, {4297, 5223}, {4661, 20076}, {4855, 5744}, {5068, 5748}, {5082, 5730}, {5086, 6870}, {5218, 18231}, {5261, 5794}, {5328, 9581}, {5435, 5438}, {5436, 18230}, {5690, 6988}, {5731, 9845}, {5761, 6843}, {5775, 6684}, {5780, 6939}, {5790, 6855}, {5844, 6918}, {5853, 9785}, {6224, 14740}, {6282, 9799}, {6828, 17757}, {6922, 12645}, {7958, 11680}, {7962, 12541}, {9776, 11520}, {9778, 12526}, {10465, 12126}, {11024, 11529}, {11111, 15650}, {11851, 17490}, {12433, 17559}, {13736, 16601}, {15683, 17781}, {15934, 17582}, {15935, 16853}, {17158, 17863}, {17644, 17658}

X(20007) = complement of X(20008)
X(20007) = anticomplement of X(938)

X(20008) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(938), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20008) lies on these lines: {1, 2}, {7, 12625}, {57, 12536}, {144, 950}, {307, 4460}, {518, 12125}, {960, 13867}, {971, 3146}, {1445, 2136}, {1446, 17158}, {1483, 6988}, {2550, 18221}, {3218, 9841}, {3419, 11036}, {3488, 11106}, {3681, 18247}, {3854, 9842}, {3869, 9848}, {3873, 9850}, {3889, 12128}, {4208, 15934}, {4452, 6604}, {5059, 9965}, {5129, 12433}, {5175, 5665}, {5274, 12635}, {5435, 12437}, {5794, 11038}, {5837, 8236}, {5844, 6865}, {5905, 17578}, {6855, 10247}, {6864, 12645}, {9846, 15185}, {9859, 16465}, {11024, 17706}, {15935, 16845}, {16284, 17863}

X(20008) = anticomplement of X(20007)

X(20009) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(975), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 3 a^4 + 2 a^3 b + 2 a^2 b^2 + 2 a b^3 - b^4 + 2 a^3 c + 6 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(20009) lies on these lines: {1, 2}, {6, 1265}, {20, 192}, {37, 13736}, {55, 1791}, {69, 15882}, {72, 193}, {144, 20077}, {304, 3945}, {312, 5716}, {344, 1104}, {346, 2298}, {942, 11851}, {1043, 2303}, {1220, 3974}, {1257, 5738}, {1824, 4198}, {3210, 6904}, {3672, 4201}, {3685, 4339}, {3871, 11337}, {3879, 11523}, {3995, 6872}, {4190, 17147}, {4313, 5279}, {5844, 19547}, {11036, 17300}, {11106, 17742}, {17490, 17580}

X(20009) = anticomplement of anticomplement of X(975)

X(20010) = (P(10), U(10), X(1), X(75); P(10), U(10), X(75), X(1)) COLLINEATION IMAGE OF X(3634)

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

X(20010) lies on these lines: {10, 75}, {1125, 4169}, {3244, 19933}

X(20011) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20011) lies on these lines: {1, 2}, {55, 16704}, {81, 3996}, {192, 4661}, {193, 674}, {518, 3896}, {524, 4450}, {740, 17165}, {982, 17145}, {1621, 19742}, {2238, 17388}, {2813, 20096}, {3210, 4430}, {3681, 3995}, {3722, 3791}, {3755, 17184}, {3780, 7109}, {3871, 4184}, {3873, 17495}, {3891, 13576}, {4113, 15569}, {4192, 5844}, {4358, 4849}, {4392, 4734}, {4650, 4781}, {5263, 19717}, {7500, 20071}, {16714, 17377}, {20066, 20077}, {20086, 20095}

X(20011) = anticomplement of X(17135)

X(20012) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20012) lies on these lines: {1, 2}, {6, 3996}, {38, 4734}, {144, 6007}, {192, 3681}, {193, 3779}, {312, 4849}, {346, 7109}, {518, 3210}, {672, 3169}, {1011, 3871}, {1278, 17165}, {1621, 17349}, {2209, 17127}, {2238, 17314}, {2276, 5839}, {2550, 17778}, {2810, 9965}, {3689, 3769}, {3703, 4819}, {3728, 4704}, {3744, 3759}, {3873, 17490}, {4184, 16704}, {4192, 12245}, {4373, 8049}, {4430, 17495}, {4661, 17147}, {4713, 4971}, {4753, 7262}, {4974, 17715}, {5687, 13588}, {5844, 19540}, {20064, 20095}

X(20012) = anticomplement of X(10453)

X(20013) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(78), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20013) lies on these lines: {1, 2}, {20, 912}, {63, 12437}, {72, 6872}, {144, 15680}, {329, 12536}, {346, 2911}, {355, 6870}, {377, 6147}, {411, 12245}, {518, 20076}, {908, 12625}, {943, 16865}, {950, 3984}, {952, 6836}, {965, 17388}, {1482, 6835}, {1792, 16704}, {2287, 17314}, {2478, 3940}, {2800, 20095}, {3091, 5761}, {3146, 5758}, {3149, 5844}, {3189, 3869}, {3219, 4313}, {3419, 6871}, {3434, 12635}, {3485, 5178}, {3486, 3681}, {3488, 3876}, {3522, 3587}, {3523, 13151}, {3529, 9963}, {3600, 4430}, {3868, 4190}, {3951, 4304}, {4067, 4302}, {5059, 20070}, {5220, 10543}, {5738, 17377}, {5853, 11682}, {5905, 9579}, {6831, 12645}, {6986, 7967}, {6991, 10595}, {11851, 17495}, {15935, 16842}, {20080, 20082}

X(20013) = anticomplement of X(12649)

X(20014) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 8r - 5 R sin B sin C : :
Barycentrics 11 a - 5 b - 5 c : :
X(20014) = 16 X(1) - 15 X(2)

X(20014) lies on these lines: {1, 2}, {20, 5844}, {144, 12630}, {193, 9053}, {320, 4452}, {341, 4935}, {346, 4727}, {355, 3854}, {391, 16675}, {517, 5059}, {518, 4788}, {952, 3146}, {1317, 5265}, {1482, 3832}, {1483, 3523}, {2136, 3218}, {3057, 4661}, {3091, 12645}, {3189, 5183}, {3522, 12245}, {3543, 8148}, {3650, 15680}, {3672, 17360}, {3871, 17548}, {3873, 3893}, {4371, 4889}, {4395, 4869}, {4405, 4648}, {4430, 14923}, {4454, 4971}, {4969, 16885}, {4982, 5749}, {5056, 10247}, {5068, 18493}, {5253, 8168}, {5303, 12513}, {5839, 16814}, {5846, 20080}, {5853, 20059}, {5854, 12632}, {5855, 20075}, {5905, 12541}, {6767, 16859}, {7270, 19824}, {7408, 12135}, {7409, 11396}, {7492, 8192}, {7967, 15717}, {9779, 16189}, {10595, 15022}, {12571, 16191}, {16668, 17299}, {16674, 17362}, {16980, 16981}

X(20014) = anticomplement of X(3621)

X(20015) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20015) lies on these lines: {1, 2}, {144, 4661}, {329, 5853}, {390, 1864}, {516, 20070}, {518, 3474}, {1621, 5686}, {2801, 20078}, {3158, 5744}, {3243, 9776}, {3305, 8236}, {3555, 6904}, {3693, 5839}, {3869, 12632}, {3889, 17580}, {3984, 9785}, {4297, 18452}, {4358, 6555}, {4430, 7672}, {4863, 17605}, {5084, 18530}, {5178, 5261}, {5220, 10385}, {5809, 10388}, {5844, 19541}, {7057, 11686}, {7580, 12245}, {7994, 10430}, {8727, 12645}, {10005, 17776}, {11682, 12541}, {12630, 18228}, {14548, 17377}

X(20015) = anticomplement of X(36845)
X(20015) = anticomplement of anticomplement of X(200)

X(20016) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20016) lies on these lines: {1, 2}, {75, 20090}, {86, 4399}, {144, 4788}, {190, 4969}, {192, 5839}, {193, 742}, {319, 4852}, {320, 4725}, {391, 4704}, {514, 14779}, {524, 4440}, {536, 20072}, {952, 6999}, {1100, 5564}, {1482, 7384}, {1654, 4360}, {1931, 6630}, {2321, 17121}, {2895, 17152}, {3672, 17343}, {3686, 4464}, {3759, 17280}, {3765, 17144}, {3875, 6646}, {3879, 17117}, {3943, 4473}, {3945, 4772}, {3946, 17287}, {4000, 17373}, {4007, 17368}, {4021, 17252}, {4034, 17248}, {4346, 11160}, {4361, 17300}, {4371, 4699}, {4395, 17297}, {4405, 17392}, {4431, 4856}, {4445, 17380}, {4452, 20080}, {4460, 17257}, {4470, 17379}, {4478, 17307}, {4527, 16477}, {4644, 4740}, {4645, 4716}, {4690, 17320}, {4727, 17264}, {4758, 4967}, {4889, 17317}, {4910, 17275}, {5844, 6996}, {5846, 6653}, {7377, 12645}, {8682, 17141}, {16706, 17372}, {17119, 17378}, {17151, 17364}, {17270, 17396}, {17271, 17395}, {17277, 17388}, {17278, 17386}, {17295, 17366}, {17301, 17360}, {17309, 17352}, {17314, 17349}, {17315, 17348}, {17318, 17346}

X(20017) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(306), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20017) lies on these lines: {1, 2}, {6, 3969}, {31, 17772}, {69, 9022}, {81, 17377}, {192, 2895}, {321, 17299}, {345, 16704}, {594, 19684}, {740, 6327}, {952, 19645}, {1043, 17587}, {1211, 17388}, {1278, 17483}, {2345, 19717}, {2897, 6360}, {3101, 20074}, {3175, 4727}, {3210, 17373}, {3219, 17363}, {3416, 3896}, {3666, 17372}, {3782, 4971}, {3875, 17184}, {3945, 19825}, {3995, 5739}, {4359, 4851}, {4361, 18139}, {4383, 17309}, {4641, 4725}, {4886, 17315}, {5014, 19791}, {5278, 17362}, {5749, 19743}, {5839, 17776}, {5844, 19542}, {5847, 20064}, {6539, 19740}, {8025, 19822}, {9028, 20078}, {17369, 19738}, {17386, 19804}

X(20017) = complement of X(20046)
X(20017) = anticomplement of X(3187)
X(20017) = polar conjugate of X(36613)

X(20018) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(386), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20018) lies on these lines: {1, 2}, {6, 1043}, {20, 185}, {69, 4201}, {72, 192}, {213, 346}, {274, 3945}, {333, 19765}, {341, 4849}, {377, 17778}, {391, 941}, {405, 17349}, {443, 17300}, {579, 3169}, {942, 17490}, {964, 4720}, {986, 4734}, {1010, 17379}, {1104, 3759}, {1107, 5839}, {1150, 19278}, {1208, 5731}, {1265, 2176}, {1453, 17121}, {1654, 13725}, {1834, 4417}, {2209, 5247}, {2269, 4313}, {2271, 19312}, {2895, 17676}, {3189, 3779}, {3210, 3868}, {3295, 16289}, {3522, 18206}, {3555, 17480}, {3869, 3896}, {3871, 16452}, {3875, 11523}, {3996, 5710}, {4189, 16704}, {4255, 14829}, {4261, 17448}, {4292, 17364}, {4340, 20090}, {4402, 17050}, {4452, 17753}, {5132, 12513}, {5361, 16347}, {5844, 19543}, {5933, 7176}, {6762, 16574}, {6767, 16288}, {7283, 17350}, {7379, 7774}, {7513, 9308}, {7754, 13727}, {9535, 10454}, {11106, 16552}, {13728, 17238}, {14996, 19284}, {16865, 19742}, {16969, 17388}, {20064, 20066}

X(20018) = anticomplement of X(10449)

X(20019) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(387), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20019) lies on these lines: {1, 2}, {193, 1503}, {579, 2136}, {3868, 4452}, {4402, 11518}, {5059, 20077}, {5746, 12625}, {16704, 17576}

X(20019) = anticomplement of anticomplement of X(387)

X(20020) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(612), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20020) lies on these lines: {1, 2}, {22, 3871}, {144, 20064}, {149, 7394}, {192, 7500}, {193, 4661}, {390, 3995}, {940, 9053}, {1483, 16434}, {2550, 3891}, {3744, 17776}, {4220, 12245}, {4307, 17165}, {4419, 4450}, {4696, 5716}, {4901, 5294}, {4972, 19823}, {5275, 17388}, {5276, 17314}, {5322, 8715}, {5686, 19742}, {5739, 5846}, {5844, 19544}, {7391, 20060}, {7967, 19649}, {9965, 20068}, {17147, 17784}, {20062, 20066}, {20078, 20101}

X(20020) = anticomplement of anticomplement of X(612)

X(20021) = X(2)X(98)∩X(4)X(263)

Barycentrics (b^2 + c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4) : :
X(20021) = X[193] - 3 X[25317], 5 X[3618] - 3 X[25314]

Let A'B'C' be the triangle whose barycentric vertex matrix is the sum of the matrices for the 3rd and 4th Euler triangles, so that A' = b^2 + c^2 : c^2 - a^2 : b^2 - a^2. Then A'B'C' is the complement of the tangential triangle (or tangential-of-medial triangle), and A'B'C' is also the reflection of the Kosnita triangle in X(140). X(20021) is the eigencenter of A'B'C'. (Randy Hutson, July 31 2018)

X(20021) lies on the cubics K267 and K1000, and on these lines: {2, 98}, {4, 263}, {6, 3613}, {66, 248}, {67, 526}, {69, 290}, {141, 1634}, {193, 25317}, {211, 27366}, {237, 1503}, {262, 30499}, {338, 2871}, {343, 7467}, {420, 685}, {427, 3051}, {511, 14957}, {524, 25324}, {599, 36822}, {660, 1821}, {694, 804}, {732, 46161}, {826, 46157}, {1613, 1853}, {1843, 46151}, {2396, 40708}, {2548, 10014}, {2549, 30495}, {2715, 9076}, {2782, 36790}, {2896, 8870}, {2966, 43098}, {3087, 6531}, {3094, 39906}, {3404, 15523}, {3564, 21531}, {3618, 25314}, {3917, 4576}, {5207, 20022}, {5475, 43950}, {6394, 15812}, {7736, 11175}, {7779, 36897}, {8050, 20290}, {9019, 35362}, {9463, 45096}, {11245, 20965}, {11257, 42313}, {11328, 18440}, {13137, 33873}, {14003, 34118}, {14265, 34507}, {14424, 46147}, {16063, 34095}, {18553, 34236}, {19558, 21458}, {20026, 34536}, {31125, 36827}, {32140, 37466}, {34214, 41520}, {35366, 46156}, {40847, 40858}

X(20021) = midpoint of X(69) and X(25051)
X(20021) = reflection of X(i) in X(j) for these {i,j}: {6, 7668}, {1634, 141}
X(20021) = isotomic conjugate of X(20022)
X(20021) = complement of X(25046)
X(20021) = anticomplement of X(36213)
X(20021) = psi-transform of X(36183)
X(20021) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1581, 147}, {1821, 25332}, {1910, 8782}, {1967, 39355}, {15391, 6360}, {34238, 192}, {36897, 8}, {39291, 7192}, {43763, 25046}
X(20021) = X(i)-Ceva conjugate of X(j) for these (i,j): {2715, 879}, {43187, 2395}
X(20021) = X(8623)-cross conjugate of X(2)
X(20021) = cevapoint of X(i) and X(j) for these (i,j): {141, 732}, {688, 41178}, {804, 7668}
X(20021) = crosspoint of X(98) and X(290)
X(20021) = crosssum of X(237) and X(511)
X(20021) = trilinear pole of line {39, 826}
X(20021) = crossdifference of every pair of points on line {3569, 36213}
X(20021) = X(i)-line conjugate of X(j) for these (i,j): {694, 3569}, {804, 3569}, {2395, 3569}, {11646, 3569}, {19637, 3569}
X(20021) = X(98)-daleth conjugate of X(2)
X(20021) = X(14957)-of-1st-Brocard-triangle
X(20021) = X(16549)-zayin conjugate of X(1755)
X(20021) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3405}, {31, 20022}, {82, 511}, {83, 1755}, {232, 34055}, {237, 3112}, {240, 1176}, {251, 1959}, {308, 9417}, {2491, 4593}, {2799, 34072}, {3569, 4599}, {9418, 18833}, {10547, 40703}, {14966, 18070}, {17209, 18098}, {36213, 43763}
X(20021) = barycentric product X(i)*X(j) for these {i,j}: {38, 1821}, {39, 290}, {75, 3404}, {98, 141}, {248, 1235}, {287, 427}, {293, 20883}, {336, 17442}, {685, 2525}, {732, 36897}, {826, 2966}, {879, 41676}, {1634, 43665}, {1910, 1930}, {1976, 8024}, {2395, 4576}, {2715, 23285}, {3005, 43187}, {3051, 18024}, {3665, 15628}, {3917, 16081}, {3933, 6531}, {5967, 31125}, {6394, 27376}, {7813, 9154}, {8061, 36036}, {15412, 35362}, {34238, 35540}, {36824, 37858}
X(20021) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3405}, {2, 20022}, {38, 1959}, {39, 511}, {98, 83}, {141, 325}, {248, 1176}, {287, 1799}, {290, 308}, {293, 34055}, {427, 297}, {685, 42396}, {688, 2491}, {732, 5976}, {826, 2799}, {879, 4580}, {1235, 44132}, {1401, 43034}, {1634, 2421}, {1821, 3112}, {1843, 232}, {1910, 82}, {1923, 9417}, {1964, 1755}, {1976, 251}, {2422, 18105}, {2525, 6333}, {2715, 827}, {2966, 4577}, {3005, 3569}, {3051, 237}, {3404, 1}, {3917, 36212}, {3933, 6393}, {4553, 42717}, {4576, 2396}, {6531, 32085}, {8623, 36213}, {8861, 8928}, {14600, 10547}, {14617, 8840}, {16081, 46104}, {17187, 17209}, {17442, 240}, {17974, 28724}, {18024, 40016}, {20775, 3289}, {20883, 40703}, {21814, 5360}, {27369, 2211}, {27371, 39569}, {27376, 6530}, {33299, 44694}, {34238, 733}, {35325, 4230}, {35362, 14570}, {36036, 4593}, {36084, 4599}, {36897, 14970}, {39291, 41209}, {39691, 868}, {41178, 2679}, {41331, 9418}, {41676, 877}, {43187, 689}, {46147, 35910}, {46154, 5968}
X(20021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 25046, 36213}, {98, 287, 1976}, {98, 11653, 35912}, {287, 1976, 5967}

X(20022) = X(2)X(32)∩X(4)X(18022)

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

X(20022) lies on the cubics K267 and K1000, and on these lines: {2, 32}, {4, 18022}, {6, 16890}, {69, 263}, {76, 3060}, {81, 18096}, {141, 18092}, {237, 325}, {264, 10550}, {297, 2211}, {316, 512}, {333, 18703}, {420, 18020}, {689, 2698}, {827, 2857}, {1501, 10349}, {2396, 14251}, {3051, 7762}, {3112, 4388}, {3117, 7759}, {3229, 7845}, {4577, 5641}, {6656, 14822}, {7750, 14096}, {7776, 11328}, {7779, 14970}, {9308, 10549}, {10330, 14958}

X(20022) = isotomic conjugate of X(20021)
X(20022) = anticomplement X(8623)
X(20022) = cevapoint of X(325) and X(511)
X(20022) = crosspoint of X(83) and X(14970)
X(20022) = trilinear pole of line {2491, 2799}
X(20022) = crossdifference of every pair of points on line {3005, 3051}
X(20022) = crosssum of X(39) and X(8623)
X(20022) = X(83)-daleth conjugate of X(2)
X(20022) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 8782}, {733, 192}, {1581, 2896}, {1934, 1369}, {14970, 8}
X(20022) = X(3569)-cross conjugate of X(2396)
X(20022) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3404}, {38, 1976}, {39, 1910}, {98, 1964}, {248, 17442}, {290, 1923}, {293, 1843}, {1821, 3051}, {1930, 14601}, {2084, 2966}, {2715, 8061}, {4020, 6531}
X(20022) = X(83)-Hirst inverse of X(1799)
X(20022) = X(7762)-line conjugate of X(3051)
X(20022) = barycentric product X(i)*X(j) for these {i,j}: {75, 3405}, {83, 325}, {297, 1799}, {308, 511}, {689, 3569}, {877, 4580}, {1755, 18833}, {1959, 3112}, {2799, 4577}, {5976, 14970}
X(20022) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3404}, {82, 1910}, {83, 98}, {232, 1843}, {237, 3051}, {240, 17442}, {251, 1976}, {297, 427}, {308, 290}, {325, 141}, {511, 39}, {827, 2715}, {1176, 248}, {1755, 1964}, {1799, 287}, {1959, 38}, {2396, 4576}, {2421, 1634}, {2491, 688}, {2799, 826}, {3112, 1821}, {3405, 1}, {3569, 3005}, {4577, 2966}, {4580, 879}, {5976, 732}, {6333, 2525}, {6393, 3933}, {8840, 14617}, {8928, 8861}, {9417, 1923}, {10547, 14600}, {17209, 17187}, {18105, 2422}
X(20022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 17500, 308), (316, 3978, 14957)

X(20023) = ISOTOMIC CONJUGATE OF X(263)

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

X(20023) lies on the cubics K267 and K1037, and one these lines: {2, 39}, {4, 18022}, {8, 18891}, {69, 290}, {83, 10014}, {183, 14096}, {193, 9230}, {237, 1975}, {263, 18906}, {308, 3618}, {315, 2387}, {327, 1007}, {561, 3212}, {592, 10359}, {1799, 12203}, {3051, 7754}, {3114, 7766}, {3231, 11333}, {3620, 6374}, {5921, 8920}, {5984, 8783}, {6620, 17984}, {7751, 8623}, {7897, 18896}

X(20023) = isotomic conjugate of X(263)
X(20023) = anticomplement X[3117]
X(20023) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3113, 2}, {3114, 8}, {3407, 192}, {9063, 17217}, {18898, 17486}
X(20023) = X(14994)-cross conjugate of X(183)
X(20023) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3402}, {31, 263}, {32, 2186}, {262, 560}, {327, 1917}
X(20023) = barycentric product X(i)*X(j) for these {i,j}: {75, 3403}, {76, 183}, {182, 1502}, {305, 458}, {308, 14994}, {3288, 4609}, {3978, 8842}
X(20023) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3402}, {2, 263}, {75, 2186}, {76, 262}, {182, 32}, {183, 6}, {458, 25}, {1502, 327}, {3288, 669}, {3403, 1}, {6784, 1084}, {8842, 694}, {10311, 1974}, {14096, 3051}, {14994, 39}, {15819, 5052}
X(20023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 3978, 2), (7754, 11338, 3051)

X(20024) = X(2)X(3398)∩X(69)X(18896)

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

X(20024) lies on the cubic K267 and these lines: {2, 3398}, {69, 18896}

X(20024) = X(6)-isoconjugate of X(3409)
X(20024) = barycentric product X(i)*X(j) for these {i,j}: {75, 3408}, {3314, 3406}
X(20024) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3409}, {3094, 3095}, {3406, 3407}, {3408, 1}

X(20025) = X(2)X(1501)∩X(69)X(3114)

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

X(20025) lies on the cubic K267 and these lines: {2, 1501}, {69, 3114}, {263, 18906}

X(20025) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3408}, {3116, 3406}
X(20025) = barycentric product X(i)*X(j) for these {i,j}: {75, 3409}, {3095, 3114}
X(20025) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3408}, {3095, 3094}, {3407, 3406}, {3409, 1}

X(20026) = X(2)X(3095)∩X(69)X(3095)

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

X(20026) lies on the cubic K267 and these lines: {2, 3095}, {69, 3114}, {3400, 4039}

X(20026) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3401}, {1581, 3398}
X(20026) = barycentric product X(i)*X(j) for these {i,j}: {75, 3400}, {385, 3399}
X(20026) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3401}, {1691, 3398}, {3399, 1916}, {3400, 1}

X(20027) = X(6)X(3400)∩X(1933)X(3399)

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

X(20027) lies on the cubic K267 and these lines: {2, 694}, {69, 18896}, {334, 1431}, {384, 14822}, {3618, 9468}, {6234, 14853}, {7018, 7077}

X(20027) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3400}, {1933, 3399}
X(20027) = barycentric product X(i)*X(j) for these {i,j}: {75, 3401}, {3398, 18896}
X(20027) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3400}, {1916, 3399}, {3398, 1691}, {3401, 1}

X(20028) = ISOTOMIC CONJUGATE OF X(17751)

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

X(20028) lies on the conic {{A,B,C,X(2)X(7)}}, cubics K254 and K267, and on these lines: {2, 573}, {7, 10571}, {75, 3869}, {86, 4225}, {314, 1240}, {675, 17189}, {1122, 16727}, {1400, 17197}, {4373, 17753}, {14621, 14953}

X(20028) = isotomic conjugate of X(17751)
X(20028) = X(2363)-anticomplementary conjugate of X(1764)
X(20028) = X(i)-cross conjugate of X(j) for these (i,j): {65, 81}, {1193, 2}, {3752, 274}
X(20028) = X(i)-isoconjugate of X(j) for these (i,j): {31, 17751}, {37, 572}, {42, 2975}, {58, 14973}, {213, 14829}, {228, 11109}, {1334, 17074}
X(20028) = cevapoint of X(i) and X(j) for these (i,j): {124, 3910}, {513, 17197}, {1086, 6371}
X(20028) = trilinear pole of line {514, 6589}
X(20028) = barycentric product X(86)*X(2051)
X(20028) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17751}, {27, 11109}, {37, 14973}, {58, 572}, {81, 2975}, {86, 14829}, {1014, 17074}, {2051, 10}, {7192, 17496}, {18191, 11998}

X(20029) = ISOGONAL CONJUGATE OF X(11337)

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

The trilinear polar of X(20029) passes through X(647). (Randy Hutson, July 31 2018)

See César Lozada, Hyacinthos 27810.

X(20029) lies on the Jerabek hyperbola, the cubic K321 and these lines: {2, 1798}, {3, 1211}, {6, 429}, {12, 478}, {65, 1899}, {68, 10441}, {69, 1228}, {72, 5928}, {73, 10372}, {1439, 10361}, {11442, 18123}

X(20029) = isogonal conjugate of X(11337)
X(20029) = perspector of Yiu conic

X(20030) = 26TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^16-13 a^14 b^2+31 a^12 b^4-29 a^10 b^6-5 a^8 b^8+33 a^6 b^10-27 a^4 b^12+9 a^2 b^14-b^16-13 a^14 c^2+38 a^12 b^2 c^2-29 a^10 b^4 c^2+4 a^8 b^6 c^2-35 a^6 b^8 c^2+74 a^4 b^10 c^2-51 a^2 b^12 c^2+12 b^14 c^2+31 a^12 c^4-29 a^10 b^2 c^4-4 a^8 b^4 c^4-7 a^6 b^6 c^4-38 a^4 b^8 c^4+99 a^2 b^10 c^4-52 b^12 c^4-29 a^10 c^6+4 a^8 b^2 c^6-7 a^6 b^4 c^6-18 a^4 b^6 c^6-57 a^2 b^8 c^6+116 b^10 c^6-5 a^8 c^8-35 a^6 b^2 c^8-38 a^4 b^4 c^8-57 a^2 b^6 c^8-150 b^8 c^8+33 a^6 c^10+74 a^4 b^2 c^10+99 a^2 b^4 c^10+116 b^6 c^10-27 a^4 c^12-51 a^2 b^2 c^12-52 b^4 c^12+9 a^2 c^14+12 b^2 c^14-c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27812.

X(20030) lies on these lines: {2,3}, {6346,11801}, {14051, 16337}, {14072,15307}, {15425, 18016}

X(20030) = midpoint of X(i) and X(j) for these {i,j}: {4,10285}, {3627,14142}
X(20030) = reflection of X(i) in X(j) for these {i,j}: {140,15957}, {548,15327}, {10126,5}, {10205,10289}, {15334, 13469}, {18016,15425}
X(20030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 10205, 10289), (10205, 10289, 10126), (13469, 15334, 2)

X(20031) = X(98)X(6530)∩X(107)X(685)

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

See Angel Montesdeoca, HG260618.

X(20031) lies on these lines: {98,6530}, {107,685}, {112,2966}, {648,17932}, {2422,2442}, {14273,15459}

X(20032) = (name pending)

Barycentrics a^2 (2 b c - Sqrt[(a^2 + b^2 - c^2) (a^2 - b^2 + c^2)]) / Sqrt[-a^2 + b^2 + c^2] : :

Let ABC be an acute triangle, and let
A'B'C' = medial triangle
A''B''C'' = orthic triangle
U = circumcircle
O_A = circle through A' and A'' tangent to U on the negative side of line BC; define O_B and O_C cyclically
Then X(20032 is the radical center of the circles O_A, O_B, O_C.

See Kadir Altintas and Peter Moses, X(20032). See also X(20033) and X(20034).

Note that X(20032)-X(20034) are real if and only if ABC is an acute triangle.

X(20032) lies on the cubics K171 and K969, and also on these lines: {2, 20033}, {3, 31386}, {25, 20034}, {5374, 19588}

X(20033) = TOUCHPOINT OF THE NINE-POINT CIRCLE AND ALTINTAS CIRCLE

Barycentrics    b^2 (p - q) + c^2 (p - r) + Sqrt[T] : : where T = t(a,b,c) + t(b,c,a) + t(c,a,b), t = 2 b^2 c^2 (p - q) (p - r) + a^4 (q - r)^2, and p : q : r are barycentrics for X(20032)
Barycentrics    SB*SC*(((b^2+3*c^2)*(a^2-b^2)*a^2+(b^4-c^4)*c^2)*S*b*sqrt(tan(B))-((3*b^2+c^2)*(a^2-c^2)*a^2-(b^4-c^4)*b^2)*S*c*sqrt(tan(C))+(b^2-c^2)*(-a^2+b^2+c^2)*(a^2*b*c*sqrt(tan(B)*tan(C))-(b^2+c^2)*S)*a*sqrt(tan(A))) : :
Barycentrics    SB*SC*(SA*(SB-SC)*(b*sqrt(SB)-c*sqrt(SC))-a*sqrt(SA)*(S^2+SB*SC-2*b*c*sqrt(SB*SC))) : :       (César Lozada, Dec 1, 2022.)

The circle externally tangent to O_A, O_B, O_C constructed at X(20032), and here named the Altintas circle,, is tangent to the nine-point circle, and X(20033) is the touchpoint. See X(20032)

If you have GeoGebra, you can view X(20032) . In the sketch, X(20033) is labeled F_x.

X(20033) lies on the nine-point-circle and these lines: {2, 20032}, {5, 31386}, {427, 20034}

X(20034) = ISOGONAL CONJUGATE OF X(5374)

Barycentrics a / Sqrt[-a^2+b^2+c^2] : :
Barycentrics Sin[A] Sqrt[Tan[A]] : :

X(20034) lies on the curves Q066, Q103, K163, K233, K535, K539, and K701, and also on these lines: {4, 31386}, {25, 20032}, {427, 20033}

X(20034) = isogonal conjugate of X(5374)
X(20034) = barycentric product X(4)X(5374)
X(20034) = barycentric quotient X(i)/X(j) for these {i, j}: {6, 5374}, {5374, 69}

X(20035) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(976), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20035) lies on these lines: {1, 2}, {20, 20068}, {192, 20071}, {3189, 17147}, {4190, 17154}, {4310, 17690}, {5844, 19548}

X(20035) = anticomplement of anticomplement of X(976)

X(20036) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(978), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 3 a^3 b + 2 a^2 b^2 - a b^3 + 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(20036) lies on these lines: {1, 2}, {20, 15310}, {65, 17490}, {144, 194}, {192, 960}, {193, 330}, {346, 2176}, {391, 1107}, {518, 17480}, {958, 17349}, {992, 16969}, {1043, 1191}, {1183, 2975}, {1219, 1258}, {1400, 4308}, {2277, 5839}, {3210, 3869}, {3875, 15829}, {3890, 3896}, {4051, 4771}, {4190, 20101}, {4195, 16466}, {4225, 16704}, {4293, 20077}, {4298, 17364}, {4323, 4402}, {4373, 17753}, {5484, 5739}, {5844, 19549}, {6767, 19518}, {7967, 13731}, {12245, 19513}

X(20036) = anticomplement of anticomplement of X(978)

X(20037) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(995), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20037) lies on these lines: {1, 2}, {192, 3877}, {193, 2810}, {346, 2300}, {390, 6007}, {517, 3210}, {956, 19260}, {1056, 17778}, {1191, 17697}, {1319, 3769}, {1401, 3600}, {1469, 3476}, {3868, 17480}, {3875, 7962}, {4293, 20101}, {4352, 17152}, {4452, 10446}, {5844, 19550}, {6767, 19259}, {7967, 19262}, {20064, 20067}, {20076, 20077}

X(20037) = anticomplement of anticomplement of X(995)

X(20038) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1026), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20038) lies on these lines: {1, 2}, {3887, 20095}

X(20038) = anticomplement of anticomplement of X(1026)

X(20039) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1149), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20039) lies on these lines: {1, 2}, {193, 9039}, {517, 17154}, {1120, 16704}, {3880, 17495}, {3995, 5919}, {5844, 19335}, {20067, 20098}

X(20039) = anticomplement of anticomplement of X(1193)

X(20040) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1193), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20040) lies on these lines: {1, 2}, {20, 20064}, {31, 17539}, {65, 17495}, {193, 8679}, {944, 5752}, {952, 5754}, {958, 19742}, {959, 3476}, {960, 3995}, {992, 17388}, {1483, 13731}, {2274, 17178}, {2282, 6553}, {2392, 20067}, {2650, 17140}, {2895, 5484}, {2975, 4267}, {3057, 3896}, {3868, 17154}, {3869, 17147}, {3875, 11682}, {3891, 12635}, {4277, 5839}, {4430, 17480}, {4452, 17220}, {4645, 17690}, {4706, 10107}, {5711, 19284}, {5844, 19513}, {6767, 19283}, {11319, 16466}, {16685, 17314}, {17137, 18600}

X(20040) = anticomplement of X(17751)

X(20041) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1201), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20041) lies on these lines: {1, 2}, {193, 9026}, {1482, 15971}, {1483, 9840}, {2098, 3891}, {2390, 20064}, {2842, 14683}, {3057, 17147}, {3242, 15983}, {3869, 20068}, {3890, 3995}, {4442, 13463}, {5844, 19514}, {12513, 16704}, {14923, 17495}, {17154, 17480}

X(20041) = anticomplement of anticomplement of X(1201)

X(20042) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1647), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20042) lies on these lines: {1, 2}, {149, 900}, {497, 20068}, {1120, 12531}, {3976, 17690}, {4080, 10707}, {5844, 19515}, {20085, 20098}

X(20042) = complement of X(20058)
X(20042) = anticomplement of X(17780)

X(20043) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(2999), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20043) lies on these lines: {1, 2}, {144, 17147}, {149, 7381}, {193, 3210}, {329, 3875}, {345, 3759}, {390, 3896}, {3666, 5839}, {3672, 5739}, {3879, 9776}, {3929, 4700}, {3945, 4359}, {4360, 14555}, {4361, 5712}, {4373, 17483}, {4383, 17314}, {4402, 5249}, {4452, 5905}, {4460, 18228}, {4470, 19722}, {4886, 17321}, {5844, 19517}, {7382, 20060}, {12245, 16435}, {17350, 20083}, {17377, 18141}, {19717, 19825}

X(20043) = anticomplement of X(34255)
X(20043) = anticomplement of anticomplement of X(2999)

X(20044) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3009), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20044) lies on these lines: {1, 2}, {512, 14712}, {5844, 19522}, {20064, 20102}

X(20044) = anticomplement of anticomplement of X(3009)

X(20045) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3011), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

Let P and U be the circumcircle intercepts of the Nagel line. Then X(20045) = {P,U}-harmonic conjugate of X(2).

X(20045) lies on these lines: {1, 2}, {23, 385}, {31, 17165}, {55, 3891}, {63, 20068}, {100, 17495}, {105, 15571}, {171, 17140}, {192, 17002}, {238, 3952}, {244, 4434}, {321, 3744}, {518, 16704}, {528, 4442}, {536, 4760}, {537, 896}, {726, 902}, {740, 3722}, {752, 17491}, {952, 8229}, {1104, 4696}, {1150, 3242}, {1215, 17469}, {1279, 4358}, {1621, 3995}, {1918, 17142}, {2078, 4552}, {2094, 15590}, {3120, 17766}, {3218, 17154}, {3246, 4009}, {3550, 17155}, {3681, 19742}, {3745, 8025}, {3769, 3873}, {3772, 5014}, {3782, 4450}, {3871, 7465}, {3936, 5846}, {3994, 4432}, {4030, 4972}, {4080, 5057}, {4385, 11319}, {4968, 5266}, {5255, 17164}, {5905, 20064}, {7677, 14594}, {17483, 20101}, {17784, 19789}

X(20045) = anticomplement of X(3006)

X(20046) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3187), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20046) lies on these lines: {1, 2}, {193, 9022}, {740, 20064}, {1278, 20086}, {3578, 17318}, {3995, 5839}, {5278, 17388}, {5844, 19645}, {6327, 17772}, {17314, 19742}

X(20046) = anticomplement of X(20017)

X(20047) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3214), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20047) lies on these lines: {1, 2}, {2334, 8025}, {3913, 16704}, {5844, 19646}

X(20047) = anticomplement of anticomplement of X(3214)

X(20048) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3240), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20048) lies on these lines: {1, 2}, {193, 9024}, {3873, 4706}, {3896, 4661}, {5844, 19647}, {17784, 20086}

X(20048) = anticomplement of anticomplement of X(3240)

X(20049) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3241), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 12 r - 7 R sin B sin C : :
Barycentrics 17 a - 7 b - 7 c : :
X(20049) = 24 X(1) - 21 X(2)

X(20049) lies on these lines: 1, 2}, {193, 9041}, {346, 16671}, {376, 5844}, {391, 16677}, {517, 15683}, {527, 12630}, {528, 20059}, {537, 4788}, {903, 4452}, {952, 3543}, {1320, 4930}, {1482, 3839}, {1483, 3524}, {1992, 9053}, {3545, 12645}, {3656, 3832}, {3880, 4430}, {4370, 17314}, {4460, 17274}, {5068, 10222}, {5071, 10247}, {5690, 15708}, {5846, 11160}, {6767, 16861}, {7967, 15692}, {7982, 17578}, {8148, 15682}, {9945, 10031}, {10032, 15680}, {10246, 15721}, {10304, 12245}, {11001, 18526}, {12513, 17548}, {12702, 15697}, {17678, 19824}

X(20049) = anticomplement of X(31145)
X(20049) = anticomplement of anticomplement of X(3241)

X(20050) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3244), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 5 r - 3 R sin B sin C : :
Barycentrics 7 a - 3 b - 3 c : :
X(20050) = 10 X(1) - 9 X(2)

X(20050) lies on these lines: {1, 2}, {4, 11278}, {7, 10944}, {44, 3161}, {45, 5839}, {55, 17574}, {69, 4460}, {100, 19537}, {193, 4488}, {341, 4742}, {346, 16670}, {355, 3855}, {377, 19820}, {382, 952}, {391, 16676}, {517, 3529}, {518, 3644}, {546, 1482}, {550, 944}, {956, 17571}, {964, 19739}, {1150, 19289}, {1317, 1788}, {1320, 7319}, {1392, 5748}, {1475, 4050}, {1483, 3530}, {1621, 16866}, {1837, 4345}, {2136, 5128}, {2320, 5775}, {2899, 8834}, {2975, 19535}, {3091, 16200}, {3146, 11531}, {3189, 5854}, {3242, 3631}, {3295, 19526}, {3303, 17543}, {3304, 8168}, {3340, 3982}, {3476, 5221}, {3488, 15650}, {3523, 13607}, {3528, 3579}, {3555, 14923}, {3576, 4917}, {3600, 4031}, {3614, 3813}, {3629, 9053}, {3654, 15715}, {3655, 15710}, {3672, 4464}, {3680, 5556}, {3681, 9957}, {3851, 5603}, {3868, 3880}, {3871, 5217}, {3873, 10914}, {3875, 4346}, {3876, 5919}, {3878, 4661}, {3889, 5836}, {3895, 6762}, {3913, 5204}, {3951, 9819}, {4003, 4734}, {4189, 5288}, {4298, 16236}, {4299, 20095}, {4301, 10248}, {4314, 8275}, {4323, 5252}, {4358, 4935}, {4361, 4916}, {4371, 17390}, {4402, 4851}, {4419, 4725}, {4430, 5903}, {4452, 4887}, {4454, 17133}, {4644, 4971}, {4648, 4889}, {4869, 17067}, {4896, 17151}, {4910, 17372}, {5079, 5818}, {5080, 5225}, {5175, 5714}, {5226, 11011}, {5260, 6767}, {5296, 16672}, {5558, 18221}, {5687, 17573}, {5690, 15720}, {5708, 17563}, {5749, 16666}, {5881, 18483}, {5936, 17394}, {6361, 15681}, {7173, 12607}, {7270, 19830}, {7718, 10301}, {7967, 10299}, {7982, 9812}, {8236, 15254}, {9708, 17545}, {9782, 11037}, {9785, 10950}, {10031, 13996}, {10246, 14869}, {10592, 11680}, {10593, 11681}, {11034, 12577}, {11737, 18493}, {12019, 12531}, {12541, 14450}, {12653, 20085}, {15687, 18525}, {15733, 17648}, {16189, 19925}, {17315, 18230}

X(20050) = homothetic center of Caelum triangle and mid-triangle of medial and anticomplementary triangles
X(20050) = complement of X(20054)
X(20050) = anticomplement of X(3632)

X(20051) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3293), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20051) lies on these lines: {1, 2}, {1126, 19743}, {3295, 19742}, {3555, 17495}, {3871, 16704}, {3996, 11115}, {4043, 4696}, {4954, 16397}, {5844, 19648}, {20077, 20095}

X(20051) = anticomplement of anticomplement of X(3293)

X(20052) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3617), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 8 r - 7 R sin B sin C : :
Barycentrics 9 a - 7 b - 7 c : :
X(20052) = 16 X(1) - 21 X(2)

X(20052) lies on these lines: {1, 2}, {20, 12645}, {319, 4452}, {346, 15492}, {391, 3943}, {517, 17578}, {518, 4821}, {952, 3522}, {956, 17548}, {1482, 5068}, {1483, 10303}, {2136, 3219}, {2975, 8168}, {3091, 5844}, {3146, 12245}, {3620, 9053}, {3681, 3893}, {3834, 4371}, {3839, 8148}, {3873, 3922}, {3877, 4533}, {3880, 4005}, {3962, 4661}, {3988, 5697}, {4007, 4700}, {4029, 4034}, {4399, 4869}, {4430, 5836}, {4461, 20072}, {4487, 4673}, {4720, 17539}, {4725, 4747}, {5059, 6361}, {5690, 15717}, {5790, 15022}, {5839, 16671}, {6767, 17570}, {6926, 19914}, {7270, 19826}, {7486, 10247}, {9708, 17544}, {10304, 18526}, {12531, 13996}, {12702, 15683}, {15174, 15676}, {16677, 17314}

X(20052) = anticomplement of X(3623)

X(20053) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3625), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 7 r - 5 R sin B sin C : :
Barycentrics 9 a - 5 b - 5 c : :
X(20053) = 14 X(1) - 15 X(2)

X(20053) lies on these lines: {1, 2}, {319, 4460}, {346, 4700}, {391, 4029}, {404, 8168}, {518, 4764}, {548, 944}, {740, 9338}, {952, 1657}, {962, 3627}, {1392, 5828}, {1482, 3850}, {1483, 12108}, {3161, 3943}, {3295, 19538}, {3630, 9053}, {3832, 11224}, {3834, 4402}, {3843, 12645}, {3868, 3893}, {3873, 4004}, {3877, 4005}, {3880, 3962}, {4018, 14923}, {4127, 4661}, {4399, 4916}, {4409, 9041}, {4464, 5232}, {4488, 17765}, {4533, 9957}, {4737, 4935}, {4803, 17589}, {5072, 5603}, {5296, 16674}, {5657, 15712}, {5734, 9955}, {5749, 16668}, {5790, 12812}, {5846, 6144}, {5854, 9802}, {6224, 13996}, {7270, 19831}, {8162, 17536}, {9778, 12245}, {11015, 12536}, {11520, 11525}, {12630, 15481}, {12702, 15686}, {14892, 18493}, {15689, 18526}, {16669, 17299}, {16675, 17362}, {16814, 17314}, {17363, 20073}

X(20053) = anticomplement of X(3633)

X(20054) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3632), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 10 r - 7 R sin B sin C : :
Barycentrics 13 a - 7 b - 7 c : :
X(20054) = 20 X(1) - 21 X(2)

X(20054) lies on these lines: {1, 2}, {312, 4935}, {320, 4373}, {346, 4969}, {382, 5844}, {546, 12645}, {550, 12245}, {952, 3529}, {956, 17574}, {1482, 3855}, {1483, 15720}, {3530, 7967}, {3839, 11278}, {3871, 19535}, {3885, 4661}, {3913, 5303}, {4007, 4982}, {4430, 10914}, {4454, 4725}, {4727, 5839}, {5068, 16200}, {5079, 10595}, {5288, 17548}, {5846, 11008}, {5854, 20085}, {6767, 17545}, {8148, 15687}, {11531, 17578}, {12632, 20066}, {16671, 17299}, {16677, 17362}

X(20054) = anticomplement of X(20050)

X(20055) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(3661), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20055) lies on these lines: {1, 2}, {7, 4821}, {63, 4050}, {69, 1278}, {75, 17372}, {192, 319}, {320, 4740}, {536, 4741}, {594, 17377}, {599, 17160}, {712, 20081}, {894, 4007}, {1654, 4704}, {2321, 17350}, {3208, 3219}, {3295, 19237}, {3631, 4398}, {3644, 17344}, {3686, 17242}, {3739, 17386}, {3758, 4725}, {3759, 17229}, {3765, 4671}, {3875, 17236}, {3879, 4060}, {3943, 17346}, {3950, 17331}, {4034, 17260}, {4360, 4445}, {4361, 17232}, {4365, 9902}, {4389, 4971}, {4399, 17234}, {4422, 17233}, {4431, 17364}, {4454, 11160}, {4461, 20080}, {4464, 17396}, {4478, 5224}, {4664, 4690}, {4665, 17378}, {4681, 17328}, {4686, 17361}, {4688, 17387}, {4699, 4851}, {4718, 17329}, {4747, 20090}, {4764, 17345}, {4772, 17300}, {4788, 6646}, {4852, 17228}, {4889, 17394}, {4967, 17391}, {4969, 17354}, {5687, 19308}, {5839, 17280}, {5844, 7377}, {6996, 12645}, {6999, 12245}, {17117, 17296}, {17119, 17297}, {17121, 17286}, {17151, 17288}, {17239, 17393}, {17240, 17348}, {17270, 17319}, {17271, 17318}, {17275, 17315}, {17277, 17309}

X(20055) = anticomplement of X(4393)

X(20056) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(7081), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20056) lies on these lines: {1, 2}, {346, 1914}, {385, 3996}, {983, 17127}, {1278, 17784}, {2783, 5984}, {3056, 3681}, {3744, 17280}, {3749, 3790}, {4030, 4854}, {9053, 14829}, {10389, 17242}, {17165, 20101}

X(20056) = anticomplement of anticomplement of X(7081)

X(20057) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(15808), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Trilinears 5 r - 9 R sin B sin C : :
Barycentrics 9 a - b - c : :
X(20057) = 10 X(1) - 3 X(2)

X(20057) lies on these lines: {1, 2}, {7, 1392}, {20, 13607}, {65, 6049}, {86, 4460}, {100, 7373}, {104, 12000}, {354, 3885}, {376, 11278}, {377, 19830}, {382, 944}, {515, 10248}, {517, 3528}, {546, 1483}, {550, 1482}, {664, 5543}, {952, 3851}, {956, 16866}, {958, 17543}, {962, 3529}, {964, 19747}, {999, 19537}, {1058, 5080}, {1266, 3945}, {1317, 3485}, {1320, 5558}, {1385, 10299}, {1388, 5435}, {1449, 3161}, {1621, 19526}, {2098, 4313}, {2099, 4308}, {2320, 7320}, {2975, 6767}, {3057, 3889}, {3242, 3629}, {3247, 4700}, {3295, 19535}, {3303, 17574}, {3304, 3871}, {3340, 4031}, {3476, 3649}, {3486, 4345}, {3522, 11531}, {3530, 10246}, {3555, 3890}, {3579, 15710}, {3648, 15174}, {3672, 15600}, {3723, 5296}, {3748, 3897}, {3855, 9955}, {3868, 5919}, {3873, 4018}, {3876, 10179}, {3877, 3962}, {3878, 4430}, {3880, 3922}, {3892, 4757}, {3898, 4127}, {3943, 5749}, {3987, 9335}, {3988, 4661}, {4004, 5045}, {4297, 16189}, {4309, 20067}, {4315, 5586}, {4317, 20066}, {4344, 15590}, {4734, 4883}, {4916, 17045}, {4982, 16676}, {5048, 5180}, {5079, 5901}, {5226, 10944}, {5260, 17545}, {5288, 16865}, {5441, 14450}, {5657, 15720}, {5698, 5852}, {5727, 18220}, {5748, 12433}, {5818, 10283}, {5844, 14869}, {6224, 12735}, {7982, 9778}, {8148, 15688}, {8162, 12513}, {9654, 10707}, {9779, 13464}, {10301, 11396}, {10394, 17622}, {11491, 12001}, {12245, 15178}, {13624, 15715}, {14269, 18526}, {15934, 17563}, {17319, 20073}

X(20057) = anticomplement of anticomplement of X(15808)

X(20058) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(17780), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20058) lies on these lines: {1, 2}, {900, 20095}, {4952, 17147}

X(20058) = anticomplement of X(20042)

X(20059) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20059) lies on these lines: {2, 7}, {4, 5843}, {6, 4346}, {8, 4312}, {20, 5762}, {69, 4454}, {72, 10861}, {145, 516}, {149, 5851}, {190, 4869}, {193, 4440}, {239, 4373}, {279, 6603}, {320, 346}, {390, 2098}, {391, 17347}, {518, 1278}, {528, 20049}, {545, 17314}, {954, 4189}, {962, 10864}, {971, 3146}, {1100, 3672}, {1320, 10307}, {1743, 4887}, {1757, 7613}, {1992, 4398}, {1999, 10442}, {2345, 17345}, {2550, 4678}, {2801, 20085}, {2951, 3870}, {3008, 4902}, {3059, 4661}, {3062, 9812}, {3091, 5779}, {3161, 4480}, {3474, 3689}, {3522, 5759}, {3543, 12690}, {3600, 5289}, {3617, 5223}, {3622, 5542}, {3663, 16667}, {3664, 16673}, {3681, 15587}, {3711, 11246}, {3731, 4896}, {3832, 5805}, {3869, 8581}, {3873, 14100}, {3912, 4488}, {3927, 4208}, {3945, 4419}, {3951, 5785}, {3957, 4326}, {4000, 16669}, {4060, 4659}, {4292, 20007}, {4293, 4867}, {4335, 17018}, {4363, 5232}, {4430, 15726}, {4470, 17253}, {4643, 7222}, {4648, 16675}, {4715, 5839}, {4718, 4916}, {4747, 17321}, {4851, 4912}, {4862, 5222}, {4880, 10590}, {4888, 5308}, {5068, 5817}, {5187, 5729}, {5221, 8165}, {5586, 18250}, {5686, 5880}, {5698, 11038}, {5735, 12649}, {5758, 7171}, {5819, 20072}, {5853, 20014}, {5856, 7674}, {6147, 17558}, {7229, 17272}, {9589, 9797}, {10056, 16558}, {11008, 17160}, {11036, 11106}, {17147, 18663}, {17183, 17207}, {17300, 20073}

X(20059) = isotomic conjugate of X(36605)
X(20059) = anticomplement of X(144)

X(20060) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20060) lies on these lines: {1, 5046}, {2, 12}, {3, 20067}, {4, 145}, {7, 5554}, {8, 79}, {10, 3218}, {20, 5841}, {21, 495}, {30, 3871}, {35, 535}, {55, 15680}, {63, 9578}, {65, 5176}, {78, 9613}, {80, 3874}, {100, 7354}, {104, 6972}, {119, 6979}, {144, 5857}, {193, 5849}, {355, 2888}, {377, 3421}, {385, 20102}, {390, 10965}, {404, 17757}, {452, 10587}, {497, 3623}, {515, 6895}, {518, 5086}, {519, 3585}, {908, 10106}, {944, 6840}, {950, 3957}, {956, 2476}, {962, 6256}, {993, 14804}, {999, 4193}, {1056, 2478}, {1376, 9657}, {1479, 3241}, {1621, 15888}, {1770, 10915}, {1836, 14923}, {1837, 3873}, {1993, 9370}, {2550, 4678}, {3006, 9369}, {3057, 5057}, {3085, 4189}, {3086, 5154}, {3091, 10529}, {3146, 5842}, {3219, 12527}, {3244, 3583}, {3295, 11114}, {3434, 3621}, {3555, 18480}, {3584, 5267}, {3633, 18513}, {3635, 4857}, {3681, 5794}, {3814, 5563}, {3820, 17531}, {3822, 5258}, {3832, 10893}, {3869, 5252}, {3870, 5691}, {3872, 9612}, {3885, 12699}, {3889, 5722}, {3897, 11374}, {3913, 12943}, {4188, 4293}, {4190, 7080}, {4197, 9708}, {4292, 6735}, {4294, 11239}, {4308, 5748}, {4311, 4881}, {4420, 17647}, {4430, 6894}, {4696, 7270}, {4737, 5300}, {4757, 15863}, {4861, 12047}, {4973, 5445}, {5141, 10527}, {5180, 5697}, {5187, 14986}, {5218, 17548}, {5249, 5795}, {5274, 10959}, {5290, 19860}, {5303, 5432}, {5541, 16118}, {5603, 13729}, {5687, 9655}, {5690, 6951}, {5727, 11520}, {5790, 6901}, {5837, 17781}, {5901, 6965}, {6827, 10805}, {6830, 11929}, {6844, 10524}, {6845, 18519}, {6848, 10530}, {6872, 11508}, {6893, 10597}, {6900, 18357}, {6902, 10246}, {6905, 10942}, {6910, 8164}, {6919, 10586}, {6923, 12245}, {6928, 7967}, {6929, 10595}, {6941, 10680}, {6960, 11249}, {6963, 16203}, {6985, 18545}, {7373, 17556}, {7382, 20043}, {7391, 20020}, {7394, 19993}, {7504, 10592}, {7785, 9263}, {7951, 8666}, {8256, 11246}, {8715, 10483}, {9342, 9711}, {9597, 17756}, {9650, 16975}, {9897, 11604}, {9961, 12678}, {10056, 15677}, {10057, 12532}, {10198, 15674}, {10591, 11240}, {10711, 13279}, {10728, 13278}, {10895, 11680}, {11260, 17605}, {12531, 13273}, {12702, 20084}, {13161, 17016}, {15971, 20101}

X(20060) = anticomplement of X(2975)
X(20060) = perspector of ABC and reflection of medial triangle in X(12)

X(20061) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20061) lies on these lines: {2, 19}, {8, 144}, {23, 1486}, {145, 20074}, {192, 7500}, {193, 3827}, {347, 1172}, {1278, 8680}, {1766, 5813}, {2263, 17016}, {3187, 4452}, {3434, 11683}, {3617, 10251}, {3920, 4319}, {4463, 17784}, {5802, 12848}, {7391, 11677}

X(20061) = anticomplement of X(4329)

X(20062) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20062) lies on these lines: {2, 3}, {193, 9019}, {251, 2549}, {323, 11206}, {612, 4324}, {614, 4316}, {1180, 7737}, {1899, 15107}, {2781, 14683}, {3424, 11140}, {3920, 4302}, {4293, 17024}, {4299, 7191}, {4549, 11455}, {5310, 10483}, {5971, 19583}, {6515, 14927}, {7802, 16276}, {8267, 20065}, {19993, 20067}, {20020, 20066}, {20083, 20095}

X(20062) = anticomplement of X(7391)

X(20063) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20063) lies on these lines: {2, 3}, {193, 8705}, {251, 7765}, {390, 5160}, {511, 14683}, {1369, 16276}, {3448, 15107}, {3600, 7286}, {3920, 4330}, {4316, 7292}, {4317, 17024}, {4324, 5297}, {4325, 7191}, {5032, 15826}, {5092, 7605}, {6776, 16981}, {8591, 13574}, {9019, 11061}, {9143, 19924}, {14712, 20099}

X(20063) = anticomplement of X(5189)
X(20063) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(3628)
X(20063) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(5)

X(20064) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20064) lies on these lines: {2, 31}, {6, 4450}, {20, 20040}, {144, 20020}, {145, 758}, {193, 674}, {209, 17784}, {516, 3187}, {734, 20081}, {740, 20046}, {744, 1278}, {766, 19994}, {896, 4865}, {1707, 3006}, {2308, 4660}, {2390, 20041}, {2550, 19742}, {2835, 9965}, {3052, 3936}, {3434, 16704}, {3474, 17495}, {3617, 4680}, {3769, 5057}, {3891, 17768}, {3923, 6535}, {3938, 17770}, {3957, 17364}, {3995, 5698}, {4641, 5014}, {4655, 17469}, {4661, 20072}, {4683, 17716}, {4772, 18805}, {5847, 20017}, {5905, 20045}, {20012, 20095}, {20018, 20066}, {20037, 20067}, {20044, 20102}

X(20064) = anticomplement of X(6327)

X(20065) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20065) lies on these lines: {2, 32}, {3, 7762}, {4, 385}, {5, 3793}, {6, 7750}, {20, 185}, {30, 7754}, {39, 14907}, {69, 384}, {76, 7737}, {99, 7758}, {141, 16898}, {145, 760}, {148, 2794}, {183, 7745}, {187, 7759}, {192, 4294}, {217, 1993}, {230, 7773}, {297, 3172}, {316, 3767}, {317, 1968}, {325, 3053}, {330, 4293}, {371, 638}, {372, 637}, {376, 7783}, {377, 16998}, {401, 6515}, {443, 17000}, {491, 12963}, {492, 12968}, {524, 1975}, {574, 7838}, {620, 7903}, {631, 7777}, {736, 6658}, {746, 1278}, {766, 19994}, {966, 17688}, {1003, 3933}, {1007, 7907}, {1285, 3314}, {1352, 12110}, {1384, 7776}, {1654, 4195}, {1655, 6872}, {1916, 9862}, {1992, 7738}, {2243, 4950}, {2386, 7500}, {2475, 17002}, {2478, 16997}, {2549, 7760}, {2996, 3543}, {3090, 17004}, {3091, 9753}, {3329, 7904}, {3425, 7488}, {3491, 14826}, {3522, 13571}, {3523, 13335}, {3525, 17005}, {3552, 3926}, {3575, 9308}, {3617, 4769}, {3618, 7876}, {3619, 16895}, {3734, 7826}, {3788, 7845}, {3849, 7748}, {3852, 5596}, {3972, 7768}, {4201, 4340}, {4339, 17257}, {4352, 20090}, {5007, 7761}, {5008, 7834}, {5023, 9766}, {5025, 7735}, {5046, 17001}, {5067, 17006}, {5084, 16999}, {5171, 9744}, {5206, 7764}, {5254, 14614}, {5286, 6655}, {5304, 7797}, {5305, 7841}, {5306, 7851}, {5309, 7842}, {5319, 7790}, {5346, 7861}, {5355, 7872}, {5368, 7902}, {5475, 7780}, {6337, 7906}, {6656, 16989}, {6781, 7781}, {7736, 7824}, {7739, 7847}, {7746, 7843}, {7747, 7751}, {7755, 7825}, {7756, 7798}, {7767, 7770}, {7769, 7926}, {7771, 7858}, {7772, 7830}, {7782, 7905}, {7784, 7792}, {7788, 7789}, {7799, 7949}, {7801, 7882}, {7804, 7854}, {7806, 7885}, {7816, 7855}, {7819, 7879}, {7820, 7896}, {7822, 7848}, {7827, 7910}, {7828, 7860}, {7829, 7935}, {7831, 7878}, {7832, 7850}, {7835, 7917}, {7836, 7946}, {7840, 7891}, {7856, 7911}, {7859, 7936}, {7863, 7916}, {7875, 7928}, {7881, 8369}, {7892, 7939}, {7920, 7924}, {7931, 14069}, {8267, 20062}, {8356, 9605}, {8366, 19661}, {8591, 14645}, {9263, 20076}, {9983, 18906}, {9988, 10653}, {9989, 10654}, {11319, 17007}, {11361, 17129}, {11610, 13219}, {14033, 17128}, {16045, 16986}, {16991, 17526}, {16995, 17685}, {17300, 17691}, {17481, 18656}

X(20065) = anticomplement of X(315)
X(20065) = polar conjugate of isogonal conjugate of X(23163)
X(20065) = {X(7737),X(14023)}-harmonic conjugate of X(76)

X(20066) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20066) lies on these lines: {2, 35}, {3, 149}, {4, 11849}, {8, 191}, {10, 4330}, {20, 145}, {30, 3871}, {55, 2475}, {100, 1329}, {193, 9047}, {390, 2646}, {404, 15171}, {496, 13587}, {497, 4188}, {519, 4324}, {528, 2975}, {958, 15677}, {1043, 4450}, {1478, 13100}, {1770, 17483}, {1900, 6995}, {2550, 16865}, {3058, 5253}, {3146, 6256}, {3241, 4299}, {3244, 4316}, {3295, 17579}, {3434, 4189}, {3522, 10529}, {3600, 11011}, {3616, 4309}, {3617, 5086}, {3623, 4293}, {3635, 4325}, {3648, 5904}, {3811, 17484}, {3813, 5303}, {3874, 15228}, {3957, 4292}, {4193, 9668}, {4317, 20057}, {4421, 11681}, {4640, 5178}, {4855, 9580}, {4881, 12053}, {5080, 8715}, {5141, 5218}, {5154, 5225}, {5217, 11680}, {5281, 6871}, {5330, 10609}, {5433, 10707}, {5687, 11114}, {5697, 6224}, {5731, 11014}, {5840, 11491}, {5842, 6895}, {5905, 20084}, {6653, 17692}, {6840, 11248}, {6845, 18499}, {6894, 11496}, {6949, 10738}, {6960, 10525}, {9669, 17566}, {9778, 12649}, {10386, 11112}, {10527, 14794}, {10724, 18242}, {11330, 19763}, {11499, 13729}, {11604, 14795}, {12248, 18526}, {12632, 20054}, {14712, 20102}, {15676, 19854}, {20011, 20077}, {20018, 20064}, {20020, 20062}

X(20066) = anticomplement of anticomplement of X(35)
X(20066) = anticomplement of isogonal conjugate of X(34441)
X(20066) = anticomplement of isotomic conjugate of isogonal conjugate of X(20988)
X(20066) = anticomplement of polar conjugate of isogonal conjugate of X(22122)

X(20067) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20067) lies on these lines: {1, 5180}, {2, 36}, {3, 20060}, {8, 484}, {10, 4325}, {12, 5303}, {20, 145}, {21, 18990}, {30, 149}, {56, 5046}, {80, 4973}, {100, 529}, {104, 5841}, {153, 6905}, {193, 9037}, {194, 20102}, {388, 4189}, {390, 5048}, {404, 3820}, {452, 5126}, {495, 17549}, {513, 17496}, {515, 3218}, {519, 4316}, {550, 3871}, {758, 6224}, {908, 4881}, {956, 17579}, {999, 11114}, {1155, 3617}, {1319, 3485}, {1621, 5434}, {1770, 4861}, {1878, 6995}, {2077, 3522}, {2078, 10587}, {2392, 20040}, {2475, 2886}, {2476, 9655}, {2551, 17572}, {2802, 15228}, {3085, 17548}, {3146, 10529}, {3241, 4302}, {3244, 4324}, {3245, 3621}, {3436, 4188}, {3616, 4317}, {3623, 4294}, {3635, 4330}, {3648, 3878}, {3881, 5441}, {3957, 4304}, {4198, 5146}, {4297, 5538}, {4309, 20057}, {4511, 17484}, {4652, 9613}, {5078, 5484}, {5122, 5791}, {5141, 5229}, {5154, 7288}, {5183, 17784}, {5187, 5265}, {5193, 10586}, {5204, 11681}, {5267, 5270}, {5536, 12649}, {5657, 10225}, {5731, 5905}, {5844, 13199}, {6895, 12114}, {6972, 10526}, {6992, 18857}, {8666, 10483}, {9263, 14712}, {9778, 12648}, {11194, 11680}, {13587, 17757}, {19993, 20062}, {20037, 20064}, {20039, 20098}

X(20067) = anticomplement of X(5080)

X(20068) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20068) lies on these lines: {2, 38}, {20, 20035}, {63, 20045}, {144, 19993}, {145, 758}, {192, 4430}, {193, 9020}, {497, 20042}, {518, 3896}, {714, 1278}, {726, 4365}, {1227, 4346}, {3006, 4138}, {3210, 4661}, {3617, 4692}, {3681, 17495}, {3720, 17146}, {3869, 20041}, {3873, 3995}, {3891, 16704}, {3936, 4884}, {3938, 4427}, {3971, 17449}, {4080, 11680}, {4450, 9053}, {4651, 17155}, {4865, 17491}, {5014, 17276}, {9965, 20020}, {17024, 17350}

X(20068) = anticomplement of X(17165)

X(20069) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(1961), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20069) lies on these lines: {1, 2}, {192, 20101}, {199, 3871}, {3744, 17315}, {3891, 17300}, {3996, 17388}, {4038, 17769}, {5844, 19516}, {17165, 20090}

X(20069) = anticomplement of anticomplement of X(1961)

X(20070) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20070) lies on these lines: {1, 3522}, {2, 40}, {3, 3622}, {4, 3617}, {7, 1697}, {8, 144}, {10, 3832}, {20, 145}, {23, 9911}, {30, 12245}, {46, 14986}, {57, 9785}, {65, 390}, {78, 7994}, {149, 6836}, {165, 3616}, {355, 3543}, {376, 1482}, {392, 17580}, {404, 6244}, {411, 10306}, {452, 5759}, {484, 3086}, {515, 3621}, {518, 9961}, {519, 15683}, {527, 2136}, {548, 10247}, {550, 7967}, {551, 15705}, {938, 2093}, {950, 12848}, {952, 3529}, {1000, 18990}, {1131, 13911}, {1132, 13973}, {1155, 5265}, {1159, 10386}, {1385, 10304}, {1420, 4345}, {1479, 3245}, {1483, 3534}, {1490, 3935}, {1537, 6927}, {1621, 5584}, {1657, 5844}, {1698, 9779}, {1699, 5068}, {1706, 18228}, {1768, 9802}, {1788, 5183}, {1836, 5261}, {1902, 6995}, {2094, 9841}, {2550, 6894}, {2800, 20013}, {2886, 18231}, {3057, 3474}, {3085, 11010}, {3091, 5657}, {3161, 10443}, {3219, 12705}, {3241, 4297}, {3295, 7411}, {3303, 11038}, {3339, 10580}, {3340, 4313}, {3359, 10586}, {3361, 4342}, {3428, 4189}, {3434, 6895}, {3485, 5281}, {3523, 3579}, {3524, 5901}, {3525, 18493}, {3528, 10246}, {3576, 5734}, {3601, 4323}, {3623, 5731}, {3651, 10679}, {3654, 3839}, {3655, 15697}, {3656, 15692}, {3671, 10578}, {3672, 5710}, {3681, 12688}, {3746, 12511}, {3753, 5129}, {3757, 12544}, {3817, 9588}, {3854, 18483}, {3869, 7957}, {3870, 12565}, {3871, 7580}, {3873, 9943}, {3876, 9856}, {3877, 6904}, {3889, 10167}, {3895, 20059}, {3915, 9441}, {3957, 10884}, {4188, 10310}, {4190, 14110}, {4293, 5697}, {4294, 5903}, {4295, 5119}, {4298, 9819}, {4300, 17018}, {4308, 7962}, {4314, 18421}, {4452, 10444}, {4661, 12528}, {4678, 11362}, {4848, 9580}, {5082, 10431}, {5128, 5435}, {5141, 15908}, {5180, 5552}, {5536, 11240}, {5541, 9809}, {5550, 10164}, {5698, 5836}, {5704, 9614}, {5709, 10529}, {5758, 10528}, {5840, 20085}, {5846, 14927}, {5881, 20052}, {5886, 10303}, {6001, 20015}, {6223, 20214}, {6762, 12541}, {6764, 10430}, {6766, 10860}, {6876, 11849}, {6925, 20060}, {7080, 11415}, {7288, 18220}, {7486, 9955}, {7672, 12711}, {7965, 9710}, {7973, 11206}, {8193, 14118}, {8236, 11518}, {9543, 9583}, {9593, 14930}, {9798, 12087}, {9799, 9804}, {10178, 17609}, {10248, 19925}, {10465, 20037}, {10591, 15079}, {11012, 17548}, {11106, 12651}, {11224, 20057}, {11239, 14450}, {11413, 12410}, {11496, 16865}, {12115, 20084}, {12571, 19875}, {12703, 17483}, {12717, 17350}, {15680, 16113}, {15704, 18526}

X(20070) = reflection of X(145) in X(20)
X(20070) = anticomplement of X(962)

X(20071) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20071) lies on these lines: {2, 41}, {144, 15680}, {145, 2809}, {192, 20035}, {193, 8679}, {766, 19994}, {2389, 20075}, {7500, 20011}

X(20071) = anticomplement of X(21285)
X(20071) = anticomplement of anticomplement of X(41)

X(20072) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20072) lies on these lines: {1, 17333}, {2, 44}, {6, 4389}, {7, 1405}, {8, 752}, {9, 17244}, {10, 894}, {37, 20090}, {45, 17378}, {69, 17230}, {72, 20077}, {86, 7277}, {144, 145}, {190, 524}, {238, 1468}, {239, 527}, {319, 17351}, {330, 957}, {344, 17375}, {346, 17373}, {385, 4831}, {391, 4699}, {513, 4380}, {519, 4480}, {536, 20016}, {540, 16086}, {545, 4969}, {597, 17305}, {599, 17354}, {651, 17950}, {903, 4395}, {1100, 17258}, {1278, 5839}, {1449, 17247}, {1743, 3662}, {1992, 4393}, {1999, 17781}, {2183, 3218}, {2325, 17310}, {2345, 17343}, {3180, 19551}, {3181, 7126}, {3210, 20078}, {3219, 17778}, {3246, 3622}, {3257, 4080}, {3589, 17273}, {3618, 17236}, {3620, 17358}, {3624, 17248}, {3629, 4360}, {3630, 17295}, {3631, 17285}, {3632, 3729}, {3663, 17121}, {3664, 17260}, {3681, 20101}, {3686, 17116}, {3707, 16815}, {3731, 17391}, {3759, 17276}, {3836, 19877}, {3879, 4029}, {3912, 4473}, {3973, 17298}, {3995, 20086}, {4062, 9395}, {4144, 7779}, {4357, 17120}, {4363, 17346}, {4371, 4821}, {4422, 17297}, {4431, 4701}, {4454, 4740}, {4461, 20052}, {4488, 17765}, {4499, 6007}, {4585, 17796}, {4649, 9791}, {4657, 17329}, {4661, 20064}, {4667, 16826}, {4683, 4722}, {4716, 17767}, {4772, 7222}, {4851, 17336}, {5032, 17014}, {5749, 17238}, {5750, 17252}, {5819, 20059}, {6144, 17262}, {6172, 17316}, {6361, 15310}, {6763, 13571}, {7232, 17352}, {7321, 17348}, {8584, 17395}, {9965, 17490}, {10025, 17036}, {10436, 17331}, {11008, 17314}, {15492, 17263}, {15533, 17269}, {15534, 17318}, {16666, 17320}, {16667, 17396}, {16669, 16706}, {16670, 17274}, {16671, 17235}, {16814, 17317}, {16885, 17234}, {17023, 17254}, {17253, 17381}, {17264, 17374}, {17271, 17369}, {17272, 17368}, {17277, 17365}, {17279, 17361}, {17281, 17360}, {17287, 17355}, {17288, 17353}, {17289, 17344}, {17296, 17339}, {17303, 17328}, {17483, 19742}, {17495, 20092}

X(20072) = anticomplement of X(320)
X(20072) = polar conjugate of isogonal conjugate of X(23166)

X(20073) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20073) lies on these lines: {1, 4480}, {2, 45}, {7, 17244}, {8, 17333}, {9, 1266}, {10, 2996}, {69, 3943}, {144, 145}, {239, 6172}, {344, 3834}, {346, 3620}, {391, 1278}, {452, 11851}, {527, 4029}, {894, 3616}, {1654, 4461}, {1992, 17318}, {2325, 17274}, {2345, 17250}, {3161, 3662}, {3618, 17246}, {3619, 17255}, {3622, 4676}, {3632, 4416}, {3644, 5839}, {3672, 17350}, {3875, 4700}, {3945, 4704}, {3995, 20078}, {4000, 17336}, {4371, 4764}, {4384, 17132}, {4393, 5032}, {4431, 4668}, {4452, 17349}, {4644, 4664}, {4675, 4912}, {4687, 7222}, {5296, 17116}, {5739, 20083}, {5749, 17247}, {6542, 11160}, {7229, 17248}, {11008, 17388}, {17300, 20059}, {17314, 17347}, {17319, 20057}, {17321, 17351}, {17363, 20053}, {17778, 20070}

X(20073) = anticomplement of anticomplement of X(45)

X(20074) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20074) lies on these lines: {2, 48}, {20, 916}, {144, 2801}, {145, 20061}, {193, 8679}, {3101, 20017}, {9028, 17134}

X(20074) = anticomplement of X(21270)
X(20074) = anticomplement of anticomplement of X(48)

X(20075) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20075) lies on these lines: {1, 4190}, {2, 11}, {3, 10529}, {4, 3871}, {7, 3957}, {8, 90}, {10, 4309}, {20, 145}, {21, 5082}, {35, 10527}, {36, 11240}, {40, 11920}, {63, 5853}, {69, 4450}, {78, 10624}, {144, 4661}, {192, 7500}, {193, 674}, {329, 2900}, {345, 5014}, {377, 3295}, {404, 1058}, {405, 10386}, {452, 3419}, {474, 15172}, {496, 6921}, {515, 3895}, {516, 3870}, {518, 20078}, {519, 4302}, {908, 3158}, {950, 5554}, {952, 6938}, {962, 6261}, {1000, 9963}, {1056, 17579}, {1320, 6948}, {1329, 9670}, {1478, 11239}, {1479, 3814}, {1482, 6934}, {1824, 3995}, {1998, 7994}, {2099, 3600}, {2389, 20071}, {2475, 10629}, {2478, 3820}, {3085, 6871}, {3146, 5842}, {3189, 3869}, {3210, 19993}, {3218, 9778}, {3241, 4293}, {3244, 4299}, {3421, 11114}, {3428, 3522}, {3436, 3913}, {3474, 3873}, {3486, 14923}, {3550, 11269}, {3586, 6735}, {3621, 11684}, {3622, 6904}, {3632, 4330}, {3633, 4324}, {3635, 4317}, {3681, 5698}, {3685, 10327}, {3744, 19785}, {3748, 5880}, {3749, 3914}, {3811, 11415}, {3813, 5217}, {3832, 7680}, {3839, 18407}, {3872, 4304}, {3898, 9951}, {3996, 5739}, {4030, 5695}, {4188, 8069}, {4305, 4861}, {4307, 17018}, {4314, 19860}, {4339, 17016}, {4344, 17011}, {4514, 17740}, {4640, 4863}, {4671, 7172}, {4855, 12053}, {5046, 7080}, {5172, 5265}, {5225, 11681}, {5249, 10389}, {5657, 6992}, {5690, 6936}, {5790, 6976}, {5840, 12115}, {5855, 20014}, {6182, 17494}, {6601, 7676}, {6767, 11112}, {6833, 11849}, {6836, 10306}, {6838, 11491}, {6868, 12245}, {6885, 10595}, {6890, 10530}, {6897, 16202}, {6911, 10596}, {6929, 12331}, {6931, 9669}, {6953, 10531}, {6955, 10246}, {6968, 10738}, {6977, 10943}, {8164, 17577}, {8236, 9776}, {9668, 17757}, {10269, 10993}, {10524, 10525}, {10597, 12000}, {11682, 12437}, {12410, 16049}, {12513, 15338}, {12575, 19861}, {12607, 12953}, {13243, 14646}

X(20075) = anticomplement of X(3434)
X(20075) = isogonal conjugate of X(38269)
X(20075) = {X(390),X(17784)}-harmonic conjugate of X(2)

X(20076) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20076) lies on these lines: {1, 5905}, {2, 12}, {3, 10528}, {4, 10529}, {8, 46}, {10, 4317}, {20, 145}, {21, 1056}, {35, 11239}, {36, 5552}, {40, 11919}, {57, 5554}, {63, 5837}, {78, 4311}, {104, 6890}, {149, 2829}, {153, 6848}, {193, 8679}, {329, 4308}, {376, 3871}, {377, 956}, {390, 2098}, {404, 3421}, {452, 3487}, {495, 6910}, {515, 12649}, {518, 20013}, {519, 4299}, {527, 11682}, {535, 1479}, {908, 1420}, {952, 6934}, {998, 5262}, {999, 2478}, {1058, 11114}, {1320, 12248}, {1385, 6992}, {1478, 6871}, {1482, 6938}, {1727, 10043}, {1788, 5176}, {1828, 6995}, {1836, 11260}, {1999, 10465}, {2390, 20041}, {2841, 20098}, {2886, 9657}, {3086, 5080}, {3241, 4294}, {3244, 4302}, {3306, 5795}, {3434, 7354}, {3474, 14923}, {3476, 3869}, {3486, 3873}, {3488, 3889}, {3522, 10310}, {3560, 10597}, {3616, 13407}, {3617, 6904}, {3621, 17784}, {3632, 4325}, {3633, 4316}, {3635, 4309}, {3813, 12943}, {3832, 7681}, {3870, 4297}, {3872, 4292}, {3890, 5698}, {3913, 15326}, {3957, 4313}, {4188, 7080}, {4189, 8069}, {4295, 4861}, {4298, 19860}, {4315, 12527}, {4661, 20007}, {4666, 12577}, {4678, 8256}, {5046, 10629}, {5082, 17579}, {5204, 12607}, {5218, 5303}, {5229, 11680}, {5267, 10056}, {5281, 17548}, {5690, 6955}, {5841, 12116}, {5854, 12632}, {5901, 6976}, {6734, 9613}, {6735, 15803}, {6837, 10532}, {6838, 10530}, {6868, 7967}, {6879, 11929}, {6880, 10942}, {6921, 17757}, {6930, 10595}, {6931, 15325}, {6933, 9654}, {6936, 10246}, {6947, 16203}, {6948, 12245}, {7373, 11113}, {7491, 10806}, {7500, 17480}, {8192, 16049}, {9263, 20065}, {9369, 10327}, {9373, 17494}, {10524, 10526}, {10596, 12001}, {11851, 17154}, {15829, 17781}, {17437, 18391}, {20037, 20077}

X(20076) = isogonal conjugate of X(38273)
X(20076) = anticomplement of X(3436)

X(20077) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20077) lies on these lines: {1, 6646}, {2, 58}, {6, 4201}, {7, 19851}, {8, 1046}, {20, 185}, {21, 17778}, {69, 4195}, {72, 20072}, {144, 20009}, {145, 758}, {239, 4292}, {320, 1104}, {385, 7379}, {390, 10544}, {405, 17300}, {443, 17349}, {452, 3794}, {524, 1043}, {962, 2792}, {1010, 1654}, {1453, 3662}, {1468, 4388}, {2392, 20040}, {2475, 16704}, {2842, 14683}, {2895, 11115}, {3430, 3522}, {3616, 5429}, {3832, 7683}, {3888, 10822}, {3936, 16948}, {3945, 13736}, {4190, 10974}, {4252, 4417}, {4293, 20036}, {4296, 17950}, {4641, 7270}, {4645, 5247}, {5059, 20019}, {6542, 7283}, {7762, 13727}, {8258, 9780}, {13725, 17379}, {13742, 17232}, {17206, 19312}, {20011, 20066}, {20037, 20076}, {20051, 20095}, {20096, 20102}

X(20077) = anticomplement of X(1330)

X(20078) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20078) lies on these lines: {2, 7}, {8, 1770}, {20, 912}, {55, 5852}, {69, 3969}, {72, 4190}, {81, 4419}, {90, 10529}, {145, 758}, {193, 17147}, {320, 17776}, {323, 347}, {377, 3927}, {388, 11684}, {390, 4430}, {515, 3621}, {518, 20075}, {940, 17334}, {943, 4189}, {993, 3622}, {1278, 8680}, {1478, 3617}, {1654, 19825}, {2095, 6957}, {2801, 20015}, {2975, 18967}, {3011, 16570}, {3210, 20072}, {3295, 3650}, {3434, 17768}, {3436, 18961}, {3474, 3681}, {3476, 3869}, {3488, 3868}, {3522, 18446}, {3586, 12649}, {3729, 4001}, {3870, 5850}, {3873, 5698}, {3935, 9778}, {3951, 4292}, {3995, 20073}, {4067, 4299}, {4307, 7226}, {4310, 17127}, {4395, 19750}, {4440, 19789}, {4454, 14552}, {4641, 17276}, {4643, 19822}, {4661, 17784}, {5220, 11246}, {5554, 12527}, {5739, 17347}, {5759, 11220}, {5762, 10431}, {5843, 7580}, {5904, 15228}, {6512, 6516}, {6763, 10527}, {7263, 19723}, {9028, 20017}, {9963, 11001}, {10032, 10385}, {10587, 12514}, {11036, 16865}, {14450, 19843}, {17328, 19797}, {17329, 19808}, {20020, 20101}

X(20078) = anticomplement of X(5905)
X(20078) = isotomic conjugate of polar conjugate of X(38295)
X(20078) = X(19)-isoconjugate of X(38248)
X(20078) = polar conjugate of X(36610)

X(20079) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20079) lies on these lines: {2, 66}, {6, 7378}, {20, 64}, {141, 11206}, {154, 3619}, {159, 3620}, {193, 7391}, {378, 19459}, {427, 19119}, {578, 3088}, {1352, 7400}, {1843, 6000}, {1853, 3618}, {1992, 15583}, {2393, 20080}, {2892, 14683}, {3091, 19149}, {3564, 12320}, {3818, 5656}, {5169, 15431}, {6293, 15741}, {7386, 13562}, {7408, 9969}, {8889, 19125}, {9833, 10519}, {12085, 19588}, {12294, 18945}, {14826, 15812}, {14853, 18381}, {14912, 15559}

X(20079) = anticomplement of X(5596)

X(20080) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 5 a^2 - 3 b^2 - 3 c^2 : :
Barycentrics 3 cot A - cot B - cot C : :

X(20080) lies on these lines: {2, 6}, {4, 11898}, {7, 17117}, {8, 17116}, {20, 3564}, {22, 19588}, {76, 18845}, {111, 6339}, {144, 6542}, {145, 5847}, {148, 14645}, {187, 439}, {253, 401}, {315, 6392}, {319, 4644}, {320, 5839}, {340, 393}, {344, 15492}, {346, 17373}, {487, 6396}, {488, 6200}, {511, 3146}, {518, 1278}, {542, 15683}, {574, 3785}, {576, 15022}, {631, 1353}, {633, 5335}, {634, 5334}, {637, 12222}, {638, 12221}, {698, 19691}, {742, 4788}, {1204, 3098}, {1351, 3091}, {1352, 3832}, {1384, 3933}, {1503, 5059}, {1843, 7408}, {2345, 17360}, {2393, 20079}, {2979, 6467}, {3060, 14913}, {3089, 15068}, {3090, 5093}, {3161, 17310}, {3241, 17247}, {3247, 3879}, {3292, 19122}, {3313, 9027}, {3410, 7409}, {3416, 4678}, {3523, 7906}, {3541, 12325}, {3543, 18440}, {3616, 17252}, {3617, 3751}, {3622, 16491}, {3672, 4741}, {3723, 4643}, {3731, 4416}, {3767, 7882}, {3854, 5480}, {3912, 3973}, {4000, 17361}, {4371, 7321}, {4419, 17377}, {4445, 7277}, {4452, 20016}, {4461, 20055}, {4664, 4916}, {4667, 17270}, {4700, 17282}, {4715, 17299}, {4725, 17276}, {4748, 17394}, {4851, 16814}, {4856, 17304}, {4966, 8692}, {4969, 7232}, {5008, 7795}, {5024, 7767}, {5033, 7793}, {5050, 10303}, {5056, 7941}, {5068, 14853}, {5092, 10519}, {5107, 7946}, {5207, 8586}, {5210, 6337}, {5222, 17288}, {5286, 7768}, {5296, 17391}, {5308, 17331}, {5319, 7896}, {5505, 18124}, {5564, 7222}, {5596, 14683}, {5749, 17287}, {5846, 20014}, {5848, 20095}, {5984, 14931}, {6172, 17242}, {6199, 11292}, {6390, 15655}, {6391, 7396}, {6395, 11291}, {6636, 19459}, {7378, 12167}, {7400, 15032}, {7486, 18583}, {7739, 7848}, {7800, 7890}, {7813, 8588}, {7845, 18424}, {8681, 12058}, {8741, 19779}, {8742, 19778}, {9028, 20017}, {9544, 19121}, {9545, 19131}, {10112, 11821}, {10625, 12283}, {11003, 19126}, {11173, 14035}, {11179, 15705}, {11188, 16981}, {11245, 17040}, {11574, 15531}, {12219, 14984}, {14068, 18906}, {14531, 15741}, {15069, 17578}, {16063, 18935}, {16674, 17390}, {16677, 17332}, {17014, 17236}, {17312, 18230}, {17314, 17347}, {17321, 17344}, {20013, 20082}

X(20080) = reflection of X(11008) in X(6)
X(20080) = isogonal conjugate of X(36616)
X(20080) = isotomic conjugate of X(38259)
X(20080) = trilinear product X(i)*X(j) for these {i,j}: {2, 16570}, {75, 5023}
X(20080) = polar conjugate of X(36611)
X(20080) = anticomplement of X(193)
X(20080) = isotomic conjugate of isogonal conjugate of X(5023)

X(20081) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20081) lies on these lines: {2, 39}, {3, 17129}, {4, 7779}, {5, 7906}, {6, 17128}, {8, 726}, {20, 2782}, {21, 16996}, {23, 9917}, {30, 7893}, {69, 698}, {83, 7798}, {85, 3797}, {99, 5206}, {115, 7796}, {141, 7864}, {145, 730}, {148, 315}, {183, 7783}, {192, 1909}, {193, 732}, {230, 7891}, {262, 5068}, {312, 7187}, {316, 7855}, {330, 350}, {381, 7941}, {384, 7754}, {385, 1975}, {390, 13077}, {511, 3146}, {524, 7823}, {543, 7802}, {599, 7928}, {625, 7871}, {671, 7825}, {712, 20055}, {734, 20064}, {736, 6658}, {1078, 7781}, {1916, 2996}, {2548, 13571}, {2549, 2896}, {3091, 3095}, {3094, 3620}, {3096, 7765}, {3097, 9780}, {3104, 5334}, {3105, 5335}, {3210, 3765}, {3314, 5254}, {3522, 6194}, {3523, 7709}, {3600, 18982}, {3617, 12782}, {3621, 14839}, {3622, 12263}, {3623, 7976}, {3691, 16816}, {3729, 17752}, {3734, 7760}, {3770, 4277}, {3832, 6248}, {3839, 14881}, {3933, 5025}, {3972, 7805}, {3975, 17490}, {4479, 17448}, {4754, 17379}, {4941, 7275}, {5056, 7697}, {5261, 12837}, {5274, 12836}, {5276, 16913}, {5304, 14037}, {5305, 7892}, {5319, 10583}, {5355, 7846}, {5475, 7905}, {5905, 6542}, {5969, 8596}, {6179, 7816}, {6337, 17008}, {6390, 7907}, {6995, 12143}, {7486, 11272}, {7738, 10335}, {7745, 7837}, {7747, 7877}, {7748, 7768}, {7749, 14148}, {7752, 7813}, {7755, 7835}, {7756, 7811}, {7758, 7785}, {7762, 11361}, {7767, 7833}, {7770, 7839}, {7773, 7840}, {7774, 16044}, {7775, 15031}, {7776, 14041}, {7780, 7782}, {7788, 7885}, {7789, 7806}, {7790, 7794}, {7804, 7894}, {7809, 7916}, {7812, 7890}, {7819, 7920}, {7820, 7856}, {7841, 7939}, {7842, 7850}, {7843, 7949}, {7844, 7909}, {7847, 7854}, {7848, 7910}, {7849, 7918}, {7851, 7931}, {7857, 7863}, {7860, 7882}, {7861, 7922}, {7868, 7923}, {7869, 7919}, {7872, 7883}, {7876, 15048}, {7879, 7924}, {7881, 7901}, {7887, 7947}, {7888, 14061}, {7895, 7934}, {7896, 7911}, {7899, 7908}, {7902, 7944}, {7903, 18546}, {7921, 8370}, {7925, 13881}, {9870, 16055}, {10079, 14986}, {10303, 11171}, {14023, 14712}, {14034, 18907}, {14929, 19695}, {15301, 15513}, {16914, 16998}, {16915, 16995}, {16989, 19689}, {17001, 17693}, {17002, 17692}, {17033, 17350}, {17316, 17760}

X(20081) = isogonal conjugate of X(36615)
X(20081) = isotomic conjugate of X(38262)
X(20081) = isotomic conjugate of isogonal conjugate of X(21001)
X(20081) = crossdifference of every pair of points on line X(669)X(23472)
X(20081) = isotomic conjugate of the anticomplement of X(32746)
X(20081) = complement of X(20105)
X(20081) = anticomplement of X(194)
X(20081) = polar conjugate of isogonal conjugate of X(22152)

X(20082) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(77), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20082) lies on these lines: {2, 77}, {144, 4552}, {145, 516}, {3879, 5905}, {6360, 20078}, {9965, 18668}, {20013, 20080}, {20089, 20090}

X(20082) = anticomplement of X(5942)

X(20083) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(387), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20083) lies on these lines: {1, 2}, {3, 6693}, {5, 182}, {6, 3454}, {19, 17904}, {58, 16062}, {197, 16414}, {315, 17200}, {442, 1751}, {579, 1761}, {1724, 5051}, {1834, 17698}, {3812, 9895}, {3814, 5137}, {3824, 4670}, {4085, 8715}, {4153, 16972}, {4193, 17188}, {4197, 9275}, {4201, 4257}, {4252, 11359}, {4267, 19258}, {4655, 5165}, {4657, 5791}, {4658, 18134}, {4894, 17469}, {4972, 5264}, {5248, 6679}, {5708, 17290}, {6703, 8728}, {16908, 20132}

X(20083) = complement of complement of X(387)

X(20084) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(79), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20084) lies on these lines: {1, 5180}, {2, 79}, {4, 16150}, {8, 16118}, {20, 16116}, {21, 11544}, {23, 16119}, {30, 145}, {144, 1654}, {191, 9780}, {329, 10123}, {390, 16142}, {758, 3621}, {962, 6264}, {1770, 4420}, {2771, 20085}, {2894, 12849}, {3091, 3652}, {3218, 7701}, {3522, 16113}, {3600, 18977}, {3622, 3649}, {3623, 5441}, {3650, 6175}, {3832, 16125}, {5046, 5221}, {5261, 16140}, {5274, 16141}, {5550, 11263}, {5556, 15910}, {5905, 20066}, {6995, 16114}, {9965, 10308}, {10032, 18253}, {12702, 20060}, {13465, 18357}, {14986, 16153}, {15678, 16137}

X(20084) = anticomplement of X(3648)

X(20085) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(80), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20085) lies on these lines: {2, 80}, {4, 145}, {8, 191}, {11, 3622}, {20, 12247}, {23, 9912}, {100, 958}, {144, 528}, {355, 6888}, {390, 12743}, {515, 3218}, {519, 5180}, {944, 6960}, {1145, 9963}, {1317, 10707}, {1387, 10031}, {1478, 11604}, {1484, 6980}, {2475, 10950}, {2771, 20084}, {2800, 3146}, {2801, 20059}, {2802, 3621}, {3091, 6265}, {3241, 18393}, {3306, 5727}, {3522, 12119}, {3523, 12619}, {3600, 18976}, {3623, 7972}, {3689, 5176}, {3832, 6246}, {3839, 12611}, {3871, 13743}, {3895, 5881}, {4678, 15863}, {4867, 5080}, {5046, 5289}, {5261, 12739}, {5274, 12740}, {5691, 9809}, {5731, 10265}, {5854, 20054}, {6910, 10609}, {6914, 12331}, {6933, 12019}, {6995, 12137}, {9780, 15015}, {9812, 13253}, {10073, 14986}, {10528, 12751}, {11545, 13587}, {12653, 20050}, {13199, 19914}, {20042, 20098}

X(20085) = anticomplement of X(6224)

X(20086) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20086) lies on these lines: {2, 6}, {145, 758}, {1255, 17332}, {1278, 20046}, {1353, 19649}, {1999, 17484}, {2836, 4430}, {3187, 17364}, {3210, 20092}, {3219, 3879}, {3995, 20072}, {4001, 17011}, {4416, 17019}, {5337, 7890}, {6629, 17190}, {8682, 17141}, {9965, 18668}, {17316, 17744}, {17784, 20048}, {20011, 20095}

X(20086) = anticomplement of X(2895)

X(20087) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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(20087) lies on these lines: {2, 82}, {8, 1757}, {744, 1278}, {8272, 17150}

X(20087) = anticomplement of X(21289)
X(20087) = anticomplement of anticomplement of X(82)

X(20088) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20088) lies on these lines: {2, 32}, {3, 7921}, {4, 5984}, {6, 6655}, {8, 1757}, {20, 3095}, {23, 9918}, {30, 7839}, {39, 14712}, {61, 622}, {62, 621}, {99, 7838}, {147, 12110}, {148, 7747}, {187, 7858}, {193, 732}, {194, 6658}, {217, 1994}, {316, 5007}, {384, 3933}, {385, 7745}, {390, 13078}, {524, 17128}, {576, 3146}, {881, 9498}, {1003, 7906}, {1285, 16925}, {1384, 7907}, {1975, 7837}, {1992, 8596}, {2478, 16995}, {2542, 3557}, {2543, 3558}, {2549, 19691}, {3053, 7777}, {3091, 6287}, {3314, 19689}, {3329, 7750}, {3522, 9737}, {3552, 6337}, {3589, 7928}, {3600, 18983}, {3617, 12783}, {3622, 12264}, {3623, 7977}, {3734, 7877}, {3788, 7926}, {3832, 6249}, {3849, 5041}, {3972, 7759}, {4393, 5905}, {5008, 7828}, {5023, 11163}, {5207, 12212}, {5261, 12944}, {5274, 12954}, {5276, 17685}, {5304, 14063}, {5305, 14041}, {5395, 15589}, {5475, 6179}, {6392, 14068}, {6995, 12144}, {7735, 9478}, {7739, 19569}, {7748, 7894}, {7754, 11361}, {7758, 19693}, {7761, 7878}, {7768, 7804}, {7770, 7893}, {7772, 7802}, {7773, 7806}, {7776, 7892}, {7784, 7875}, {7789, 7840}, {7791, 9990}, {7792, 7885}, {7795, 7946}, {7801, 7949}, {7803, 7898}, {7805, 14537}, {7807, 7941}, {7816, 7905}, {7819, 7939}, {7820, 7917}, {7822, 7850}, {7825, 7856}, {7827, 7842}, {7829, 7911}, {7832, 7845}, {7833, 9605}, {7834, 7860}, {7835, 7903}, {7841, 7920}, {7859, 7873}, {7879, 16895}, {7881, 14036}, {7891, 9766}, {7897, 14001}, {7902, 14075}, {7904, 11174}, {7933, 16989}, {7947, 8369}, {8370, 17129}, {9751, 15717}, {9862, 14881}, {10080, 14986}, {15484, 16921}

X(20088) = anticomplement of X(2896)

X(20089) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20089) lies on these lines: {2, 85}, {514, 17753}, {518, 1278}, {672, 17090}, {3218, 16816}, {3732, 4209}, {4331, 6646}, {5905, 6542}, {9312, 10025}

X(20089) = anticomplement of X(3177)

X(20090) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20090) lies on these lines: {1, 6646}, {2, 6}, {7, 4393}, {9, 17391}, {37, 20072}, {42, 7184}, {44, 17317}, {75, 20016}, {142, 17121}, {144, 1959}, {145, 740}, {190, 7277}, {192, 4644}, {239, 3664}, {319, 4670}, {320, 1100}, {390, 2098}, {519, 17116}, {527, 17319}, {894, 2321}, {1014, 19308}, {1125, 17252}, {1330, 4658}, {1351, 7385}, {1442, 17950}, {1449, 3662}, {1743, 17244}, {1999, 4054}, {2269, 3218}, {2293, 3957}, {2345, 17373}, {2796, 3241}, {3056, 3873}, {3161, 6651}, {3247, 17333}, {3564, 7379}, {3572, 20100}, {3622, 5625}, {3723, 4715}, {3729, 4898}, {3758, 4851}, {3759, 4675}, {3834, 16668}, {3882, 18164}, {3912, 17120}, {3986, 4416}, {4034, 10436}, {4085, 4645}, {4098, 17261}, {4201, 7893}, {4340, 20018}, {4352, 20065}, {4360, 4440}, {4363, 17377}, {4389, 16884}, {4454, 4788}, {4473, 17243}, {4545, 4967}, {4643, 17394}, {4657, 17361}, {4678, 4733}, {4699, 5839}, {4725, 5564}, {4740, 7222}, {4741, 17321}, {4747, 20055}, {4795, 17299}, {4796, 4889}, {4852, 7321}, {4888, 16834}, {5021, 17695}, {5749, 17230}, {5750, 17287}, {7194, 9277}, {7202, 18714}, {7232, 17380}, {7380, 11898}, {16666, 16706}, {16667, 17298}, {16669, 17263}, {16670, 17338}, {16777, 17347}, {16831, 17331}, {17018, 20101}, {17023, 17288}, {17045, 17273}, {17165, 20069}, {17206, 17689}, {17220, 17483}, {17272, 17397}, {17274, 17396}, {17276, 17393}, {17279, 17387}, {17281, 17386}, {17289, 17374}, {17295, 17369}, {17296, 17368}, {17303, 17360}, {17310, 17355}, {17311, 17354}, {17312, 17353}, {17315, 17351}, {17320, 17345}, {17322, 17344}

X(20090) = anticomplement of X(1654)

X(20091) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20091) lies on these lines: {2, 87}, {145, 726}, {193, 3779}, {4293, 20036}

X(20091) = anticomplement of anticomplement of X(87)

X(20092) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20092) lies on these lines: {2, 45}, {145, 2802}, {193, 20093}, {1266, 3218}, {3210, 20086}, {3616, 4427}, {17495, 20072}

X(20092) = anticomplement of X(30578)

X(20093) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 12 a^3 + 8 a^2 b - 8 a b^2 - 4 b^3 + 8 a^2 c + 5 a b c - 8 a c^2 - 4 c^3 : :

X(20093) lies on these lines: {2, 44}, {145, 3901}, {193, 20092}

X(20093) = anticomplement of anticomplement of X(89)

X(20094) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20094) lies on these lines: {2, 99}, {4, 13188}, {6, 19686}, {8, 13174}, {20, 2782}, {23, 13175}, {30, 7779}, {98, 3522}, {114, 3832}, {147, 3146}, {187, 19570}, {193, 5969}, {194, 6658}, {315, 19691}, {376, 12188}, {382, 7906}, {384, 15048}, {390, 3023}, {538, 14712}, {542, 15683}, {550, 17129}, {616, 6777}, {617, 6778}, {690, 14683}, {1285, 7766}, {1657, 7893}, {1916, 5395}, {1975, 3314}, {2786, 20096}, {2787, 20095}, {2794, 5059}, {2795, 15680}, {2796, 3241}, {2896, 7756}, {3027, 3600}, {3091, 6321}, {3523, 14651}, {3543, 6033}, {3545, 12355}, {3552, 5989}, {3616, 11599}, {3617, 13178}, {3620, 11646}, {3622, 11711}, {3623, 7983}, {3627, 7941}, {3815, 7783}, {3839, 8724}, {5056, 15092}, {5068, 14639}, {5186, 6995}, {5254, 16984}, {5261, 13182}, {5274, 13183}, {5281, 15452}, {5304, 8289}, {5985, 17576}, {6055, 15705}, {6390, 14041}, {7738, 8290}, {7747, 13571}, {7748, 7836}, {7762, 19696}, {7765, 10583}, {7777, 8716}, {7781, 7785}, {7787, 19693}, {7791, 11606}, {7795, 19690}, {7797, 7816}, {7799, 15301}, {7803, 19692}, {7809, 14148}, {7839, 19687}, {7864, 19689}, {7939, 19695}, {9293, 11123}, {9605, 14034}, {9778, 9860}, {9830, 11160}, {9861, 12087}, {10089, 14986}, {10304, 12042}, {10353, 14031}, {10723, 14981}, {10754, 14928}, {11121, 11489}, {11122, 11488}, {11177, 12117}, {11361, 15484}, {11632, 15692}, {14033, 14482}, {14144, 18582}, {14145, 18581}, {14830, 15697}, {14850, 15081}

X(20094) = anticomplement of X(148)
X(20094) = complement of X(35369)
X(20094) = inverse-in-Steiner-circumellipse of X(620)
X(20094) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(126)
X(20094) = {X(99),X(671)}-harmonic conjugate of X(620)

X(20095) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20095) lies on these lines: {1, 9782}, {2, 11}, {4, 11698}, {7, 5528}, {8, 191}, {20, 952}, {23, 13222}, {40, 9803}, {80, 3617}, {104, 3522}, {119, 3832}, {145, 2802}, {153, 3146}, {192, 2805}, {193, 9024}, {214, 3622}, {376, 12773}, {452, 12690}, {495, 2475}, {516, 3935}, {519, 4316}, {631, 1484}, {678, 17719}, {900, 20058}, {944, 17654}, {962, 6326}, {1058, 17572}, {1145, 4678}, {1317, 3600}, {1320, 3296}, {1768, 9778}, {1862, 6995}, {2094, 12630}, {2771, 6361}, {2783, 5984}, {2787, 20094}, {2800, 20013}, {2801, 20015}, {2829, 5059}, {2895, 3996}, {2932, 4188}, {2950, 9799}, {3091, 10738}, {3218, 5853}, {3241, 12653}, {3474, 4430}, {3486, 17636}, {3488, 6797}, {3543, 10742}, {3616, 15015}, {3625, 4324}, {3626, 4330}, {3689, 5057}, {3870, 4312}, {3887, 20038}, {3913, 12943}, {3957, 5542}, {4189, 5082}, {4297, 7993}, {4299, 20050}, {4309, 9780}, {4344, 17013}, {4917, 9579}, {4996, 17548}, {5046, 5687}, {5047, 10386}, {5261, 13273}, {5493, 12767}, {5731, 6264}, {5848, 20080}, {5854, 12632}, {5856, 7674}, {6839, 10679}, {6868, 19914}, {6885, 19907}, {6888, 11849}, {6895, 10306}, {6904, 9945}, {7951, 8715}, {8674, 14683}, {9779, 15017}, {9913, 12087}, {10090, 14986}, {10465, 12550}, {10529, 17100}, {10724, 17578}, {10914, 11015}, {12531, 13996}, {13146, 14450}, {15172, 17531}, {17777, 17780}, {20011, 20086}, {20012, 20064}, {20051, 20077}, {20062, 20083}

X(20095) = anticomplement of X(149)
X(20095) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(120)

X(20096) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20096) lies on these lines: {2, 101}, {7, 9317}, {8, 1281}, {20, 2808}, {69, 18047}, {103, 3522}, {118, 3832}, {144, 2801}, {145, 2809}, {152, 3146}, {193, 2810}, {390, 3022}, {515, 10025}, {664, 5845}, {944, 3177}, {952, 3732}, {976, 7281}, {1362, 3600}, {2774, 14683}, {2786, 20094}, {2813, 20011}, {3091, 10739}, {3259, 17036}, {3543, 10741}, {3622, 11712}, {3623, 10695}, {3887, 20038}, {4209, 6604}, {4393, 5813}, {4568, 6790}, {4644, 7200}, {4872, 6603}, {5185, 6995}, {6542, 7291}, {9028, 17950}, {10725, 17578}, {20077, 20102}

X(20096) = anticomplement of X(150)
X(20096) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(5513)

X(20097) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(105), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20097) lies on these lines: {2, 11}, {8, 5540}, {145, 2809}, {1280, 5845}, {1292, 3522}, {1358, 3600}, {1897, 6995}, {2788, 5984}, {2795, 15680}, {2832, 20098}, {2834, 7500}, {2835, 9965}, {2836, 4430}, {2837, 20099}, {3091, 10743}, {3219, 3883}, {3543, 15521}, {3622, 11716}, {3623, 10699}, {3832, 5511}, {4344, 17024}, {10729, 17578}

X(20097) = anticomplement of X(20344)
X(20097) = anticomplement of anticomplement of X(105)
X(20097) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(6667)
X(20097) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(11)

X(20098) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(106), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 3 a^4 - 3 a^3 b - 4 a^2 b^2 + a b^3 - b^4 - 3 a^3 c + 15 a^2 b c - 5 a b^2 c + b^3 c - 4 a^2 c^2 - 5 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 - c^4 : :

X(20098) lies on these lines: {1, 17777}, {2, 106}, {8, 1054}, {145, 2802}, {193, 2810}, {390, 6018}, {528, 1120}, {1145, 14193}, {1293, 3522}, {1320, 4440}, {1357, 3600}, {2789, 5984}, {2796, 3241}, {2832, 20097}, {2841, 20076}, {2842, 14683}, {2843, 20099}, {3091, 10744}, {3543, 15522}, {3616, 11814}, {3618, 18047}, {3622, 11717}, {3623, 10700}, {3832, 5510}, {4402, 9317}, {7200, 7222}, {10730, 17578}, {20039, 20067}, {20042, 20085}

X(20098) = anticomplement of X(21290)
X(20098) = anticomplement of anticomplement of X(106)

X(20099) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(111), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20099) lies on these lines: {2, 99}, {4, 11258}, {20, 14654}, {23, 5938}, {30, 9870}, {192, 2805}, {193, 2854}, {390, 6019}, {1296, 3522}, {2793, 5984}, {2813, 20011}, {2837, 20097}, {2843, 20098}, {2996, 10511}, {3091, 10748}, {3325, 3600}, {3523, 14650}, {3618, 10330}, {3622, 11721}, {3623, 10704}, {3832, 5512}, {5485, 10355}, {7492, 14652}, {7533, 7777}, {9143, 10552}, {10304, 14666}, {10734, 17578}, {14712, 20063}

X(20099) = anticomplement of X(14360)
X(20099) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(6722)
X(20099) = inverse-in-orthoptic-circle-of-Steiner-circumellipse of X(115)

X(20100) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(163), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20100) lies on these lines: {2, 798}, {192, 4132}, {513, 4380}, {3572, 20090}, {3733, 17379}, {4374, 4832}, {4498, 4785}

X(20100) = anticomplement of X(21294)
X(20100) = anticomplement of anticomplement of X(163)

X(20101) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20101) lies on these lines: {2, 31}, {8, 1046}, {20, 145}, {55, 17778}, {57, 5211}, {81, 4450}, {192, 20069}, {193, 3779}, {320, 3744}, {516, 1999}, {524, 3996}, {1330, 5264}, {1621, 17300}, {1742, 3870}, {1836, 3769}, {2792, 5905}, {3052, 18134}, {3210, 3474}, {3617, 5300}, {3681, 20072}, {3891, 4440}, {3920, 6646}, {3961, 17770}, {4190, 20036}, {4293, 20037}, {4650, 4865}, {4655, 17716}, {5311, 9791}, {5484, 5710}, {8270, 17950}, {8272, 17150}, {10327, 17350}, {15971, 20060}, {17018, 20090}, {17165, 20056}, {17483, 20045}, {20011, 20086}, {20020, 20078}

X(20110) = isotomic conjugate of polar conjugate of X(38300)
X(20101) = anticomplement of X(4388)

X(20102) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20102) lies on these lines: {2, 172}, {145, 760}, {149, 7823}, {192, 15680}, {193, 8679}, {194, 20067}, {385, 20060}, {388, 17002}, {3436, 17001}, {14712, 20066}, {20044, 20064}, {20077, 20096}

X(20102) = anticomplement of anticomplement of X(172)

X(20103) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20103) lies on these lines: {1, 2}, {3, 18250}, {9, 2272}, {55, 5316}, {57, 5850}, {165, 18228}, {210, 3911}, {226, 4413}, {404, 12527}, {443, 3947}, {474, 4298}, {497, 20196}, {515, 3820}, {516, 1376}, {518, 6692}, {946, 9709}, {971, 18227}, {1329, 8727}, {1699, 5328}, {1706, 4301}, {1864, 5432}, {1997, 3886}, {2550, 3817}, {2551, 4297}, {2801, 3035}, {2886, 10171}, {3361, 5815}, {3421, 4315}, {3523, 5234}, {3664, 17122}, {3683, 6174}, {3697, 13747}, {3711, 17728}, {3752, 4353}, {3781, 10440}, {3812, 12563}, {3814, 8226}, {3816, 5853}, {3844, 20201}, {3914, 9350}, {3956, 6681}, {3983, 5433}, {4082, 17740}, {4310, 8056}, {4314, 5084}, {4662, 6691}, {4731, 15950}, {5044, 6001}, {5218, 7308}, {5223, 5435}, {5249, 9342}, {5273, 5785}, {5290, 17580}, {5437, 5542}, {5687, 12575}, {5691, 8165}, {5784, 15064}, {5795, 9711}, {5811, 10270}, {6554, 19605}, {6666, 6690}, {6769, 6964}, {7580, 12512}, {9352, 17781}, {9708, 10165}, {10157, 15587}, {11814, 14942}

X(20103) = complement of X(11019)

X(20104) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(498), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20104) lies on these lines: {1, 2}, {35, 7504}, {140, 3822}, {442, 5326}, {516, 6863}, {535, 10592}, {632, 6681}, {1001, 5070}, {1656, 5248}, {3035, 3841}, {3526, 11929}, {3628, 3825}, {3754, 11231}, {3814, 7483}, {3817, 6949}, {3847, 15699}, {3884, 11230}, {3919, 5445}, {4297, 6952}, {5010, 5141}, {5267, 7951}, {6691, 16239}, {6825, 12512}, {6834, 12571}, {6853, 10164}, {6862, 19925}, {6959, 10171}, {17548, 18513}

X(20104) = complement of complement of X(498)

X(20105) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20105) lies on these lines: {2, 39}, {145, 726}, {148, 7758}, {193, 698}, {385, 5023}, {511, 5059}, {543, 7877}, {671, 7903}, {730, 3621}, {732, 20080}, {736, 19691}, {1384, 3552}, {1975, 7766}, {2549, 7929}, {2782, 3146}, {3091, 13108}, {3095, 3832}, {3522, 12251}, {3617, 9902}, {3854, 6248}, {3933, 7933}, {4678, 12782}, {5254, 7897}, {7408, 12143}, {7709, 15717}, {7748, 7946}, {7756, 9939}, {7765, 7938}, {7781, 7793}, {7787, 7798}, {7813, 7912}, {7855, 7898}, {7857, 14148}, {7917, 11648}, {9607, 16986}, {10335, 15589}, {11185, 13571}, {14031, 18906}, {14839, 20014}, {20016, 20078}, {20065, 20094}

X(20105) = anticomplement of X(20081)
X(20105) = isotomic conjugate of isogonal conjugate of X(36650)
X(20105) = anticomplementary conjugate of anticomplement of X(36615)
X(20105) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(32530)

X(20106) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(306), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20106) lies on these lines: {1, 2}, {6, 4035}, {141, 5745}, {226, 17355}, {312, 17861}, {345, 3663}, {440, 3454}, {516, 2887}, {1211, 2348}, {2321, 3772}, {2325, 4415}, {3452, 17279}, {3664, 18134}, {3694, 3752}, {3782, 17132}, {3844, 6690}, {3923, 4138}, {3936, 5294}, {3977, 17184}, {4021, 19786}, {4417, 17353}, {4643, 5325}, {4656, 17776}, {5273, 17272}, {5717, 17698}, {5743, 6666}, {5750, 17056}, {5847, 6679}, {6692, 16608}, {6708, 14767}, {11500, 19517}

X(20106) = complement of complement of X(306)

X(20107) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(499), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20107) lies on these lines: {1, 2}, {5, 6681}, {140, 3825}, {214, 17606}, {516, 6958}, {549, 3847}, {632, 3816}, {1385, 6702}, {3526, 5248}, {3628, 3822}, {3754, 11230}, {3814, 5433}, {3817, 6952}, {3884, 11231}, {3919, 5443}, {4187, 7294}, {4193, 5267}, {4297, 6949}, {5057, 5442}, {5154, 7280}, {6690, 16239}, {6833, 12571}, {6861, 12436}, {6862, 10171}, {6891, 12512}, {6959, 19925}, {7741, 17566}

X(20107) = complement of complement of X(499)

X(20108) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(386), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE, A''B''C'' = MEDIAL TRIANGLE

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

X(20108) lies on these lines: {1, 2}, {140, 143}, {474, 14554}, {516, 19543}, {596, 1215}, {993, 16302}, {1001, 16291}, {1078, 17200}, {1764, 9569}, {2051, 19513}, {2277, 10469}, {3159, 3666}, {3678, 6682}, {3993, 18137}, {4021, 18147}, {4256, 13740}, {4261, 17355}, {4263, 17398}, {4653, 13741}, {5248, 16286}, {12436, 16415}, {15668, 16863}, {16862, 19701}

X(20108) = complement of complement of X(386)

X(20109) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20109) lies on these lines: {2, 213}, {6, 17152}, {8, 17499}, {144, 145}, {194, 20040}, {239, 20244}, {758, 17489}, {766, 19994}, {2388, 20011}, {4416, 10459}, {5369, 20101}, {5903, 17497}, {9263, 20041}, {20016, 20088}, {20077, 20096}

X(20109) = anticomplement of X(17137)

X(20110) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(219), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

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

X(20110) lies on these lines: {2, 219}, {8, 3332}, {20, 916}, {63, 3879}, {144, 145}, {329, 3187}, {517, 20061}, {2256, 5738}, {2807, 20096}, {3872, 4416}, {3957, 8271}, {4329, 9028}, {5942, 10025}, {6360, 20078}, {10587, 17379}, {20015, 20212}

X(20110) = anticomplement of anticomplement of X(219)

X(20111) = (A',B',C',X(2); A'',B'',C'',X(2)) COLLINEATION IMAGE OF X(220), WHERE A'B'C' = MEDIAL TRIANGLE, A''B''C'' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 3 a^4 - 6 a^3 b + 2 a^2 b^2 + 2 a b^3 - b^4 - 6 a^3 c + 6 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(20111) = 3 X(2) - 4 X(220)

X(20111) lies on these lines: {2, 220}, {8, 10025}, {20, 2808}, {63, 1334}, {69, 4513}, {144, 145}, {239, 329}, {348, 6603}, {391, 17152}, {527, 9312}, {544, 17732}, {2389, 20071}, {2996, 3436}, {3218, 7131}, {3729, 6737}, {3732, 12245}, {3912, 4936}, {4416, 4853}, {5744, 17244}, {17257, 19860}

X(20111) = anticomplement of X(6604)

X(20112) = X(11295)X(16635)∩X(11296)X(16634)

Barycentrics 4*a^4+5*(b^2+c^2)*a^2+26*b^2* c^2-11*c^4-11*b^4 : :
X(20112) = 3*X(5)-X(12040), 5*X(3091)-X(9770), 3*X(3545)+X(7620), 9*X(3545)-X(9741), 3*X(3545)-X(11184), 7*X(3832)+X(9740), 2*X(3845)+X(13468), 4*X(3850)-X(7775), 8*X(3856)+X(7751), 10*X(3859)-X(7759), X(5569)-3*X(7617), 2*X(5569)-3*X(15597), 3*X(7620)+X(9741), X(9741)-3*X(11184), 3*X(9771)-2*X(12040)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27817.

Let A'B'C' be the antipedal triangle of X(2) wrt medial triangle. Then X(20112) = X(5)-of-A'B'C'. (Randy Hutson, July 31 2018)

Let Na be the nine-point center of BCX(2), and define Nb and Nc cyclically. Then X(20112) = X(3)-of-NaNbNc. (Randy Hutson, July 31 2018)

X(20112) lies on these lines: {2, 11147}, {4, 7610}, {5, 543}, {30, 5569}, {115, 597}, {141, 18424}, {230, 11317}, {381, 524}, {547, 7622}, {598, 5306}, {671, 3815}, {1153, 8703}, {3054, 8598}, {3091, 9770}, {3545, 7620}, {3734, 8355}, {3830, 8182}, {3832, 9740}, {3845, 3849}, {3850, 7775}, {3856, 7751}, {3859, 7759}, {5055, 7618}, {5066, 8176}, {5475, 8584}, {5485, 9766}, {5512, 11569}, {6321, 19911}, {6791, 13378}, {7619, 15699}, {7828, 8370}, {8352, 11168}, {8369, 14971}, {8550, 11632}, {8596, 17005}, {9877, 14639}, {9993, 11167}, {11165, 19709}, {11295, 16635}, {11296, 16634}

X(20112) = midpoint of X(i) and X(j) for these {i,j}: {4, 7610}, {381, 7615}, {3830, 8182}, {3845, 16509}, {5485, 9766}, {6321, 19911}
X(20112) = X(7610)-of-Euler triangle
X(20112) = X(9771)-of-Johnson triangle
X(20112) = QA-P15 (OrthoCenter of the Morley Triangle) of quadrangle ABCX(2)
X(20112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 3363, 597), (3545, 7620, 11184)

X(20113) = MIDPOINT OF X(3818) AND X(8546)

Barycentrics (b^4+10*b^2*c^2+c^4)*a^4+3*(b^ 2+c^2)*b^2*c^2*a^2-(b^4-c^4)^2 : :
X(20113) = 3*X(381)+X(8547)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27817.

Let Na be the nine-point center of BCX(6), and define Nb and Nc cyclically. Then X(20113) = X(3)-of-NaNbNc. (Randy Hutson, July 31 2018)

X(20113) lies on these lines: {5, 2854}, {67, 15018}, {69, 7605}, {125, 597}, {141, 373}, {381, 8547}, {524, 547}, {3589, 5159}, {3818, 8546}, {4045, 11594}, {5480, 9019}, {5640, 8262}, {6593, 14389}, {7533, 12367}, {8705, 16511}, {14561, 18449}

X(20113) = midpoint of X(3818) and X(8546)

X(20114) = X(1)X(3659)∩X(40)X(18291)

Trilinears (b-c)*(-a+b+c)*(6*a^2-3*(b+c)* a-(b-c)^2)*b*c*sin(A/2)-(a-b+ c)*(a^3+(5*b-c)*a^2-(4*b^2+2* b*c+c^2)*a+c^2*(b+c))*a*c*sin( B/2)+(a+b-c)*(a^3-(b-5*c)*a^2- (b^2+2*b*c+4*c^2)*a+b^2*(b+c)) *a*b*sin(C/2)-(b-c)*(-a+b+c)*( a^2+a*b+a*c-b*c)*(a+b-c)*(a-b+ c) : :

See César Lozada, Hyacinthos 27827.

X(20114) lies on the Bevan circle and these lines: {1, 3659}, {40, 18291}, {57, 12809}, {164, 10215}, {188, 5541}, {258, 363}

X(20114) = X(108)-of-excentral triangle
X(20114) = X(123)-of-6th mixtilinear triangle
X(20114) = X(10731)-of-excenters- reflections triangle
X(20114) = X(10746)-of-hexyl triangle

X(20115) = (name pending)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27820.

X(20115) lies on this line: {10095, 15345}

X(20116) = X(1)X(1170)∩X(7)X(79)

Barycentrics a (-a^4 (b+c)+5 a^2 b c (b+c)+2 a^3 (b^2+c^2)-2 a (b-c)^2 (b^2+3 b c+c^2)+(b-c)^2 (b^3+c^3)) : :
X(20116) = (2r+7R) X(7) - (2r+3 R) X(79)

See Kadir Altintas and Angel Montesdeoca, ADGEOM 4766.

X(20116) lies on these lines: {1,1170}, {7,79}, {9,3874}, {10,15185}, {11,118}, {142,3841}, {390,5902}, {516,942}, {518,1125}, {758,1001}, {971,12005}, {997,3243}, {1699,12669}, {2550,5883}, {2802,14563}, {3059,5439}, {3086,11038}, {3305,3873}, {3336,7676}, {3338,7675}, {3670,4343}, {3754,5853}, {3826,3833}, {3889,8583}, {4312,7671}, {4349,14523}, {5425,12758}, {5708,11495}, {5732,10980}, {5762,6583}, {5903,8236}, {5904,18230}, {6684,16216}, {8232,10399}, {12755,16173}, {13751,13995}, {15008,15726}, {15299,18389}

X(20116) = midpoint of X(i) and X(j) for these (i, j): {9,3874}, {10,15185}, {942,5572}, {5542,5728}
X(20116) = reflection of X(3678) in X(6666)

X(20117) = X(1)X(6920)∩X(2)X(5693)

Barycentrics a (a^5 (b+c)-a^4 (b+c)^2-(b^2-c^2)^2 (b^2+b c+c^2)+a^3 (-2 b^3+b^2 c+b c^2-2 c^3)+a (b-c)^2 (b^3+c^3)+a^2 (2 b^4+3 b^3 c-2 b^2 c^2+3 b c^3+2 c^4)) : :
X(20117) = (2r+3R) X(8) - (2r-5R) X(13729)

See Kadir Altintas and Angel Montesdeoca, ADGEOM 4766.

X(20117) lies on these lines: {1,6920}, {2,5693}, {3,3647}, {4,5692}, {5,758}, {8,13729}, {9,1630}, {10,119}, {21,6326}, {40,3876}, {52,15049}, {65,3614}, {72,946}, {79,6901}, {140,2771}, {191,6905}, {210,11362}, {355,3878}, {392,5882}, {515,960}, {517,546}, {518,13464}, {631,15071}, {912,1125}, {936,1158}, {952,3884}, {956,12059}, {997,5450}, {1071,10165}, {1216,2392}, {1339,10247}, {1376,5780}, {1385,2801}, {1656,5883}, {1768,6940}, {1844,7551}, {1858,13411}, {1898,4304}, {1935,11700}, {2778,5893}, {2779,5907}, {2842,10170}, {3057,18908}, {3090,5902}, {3219,11012}, {3336,6946}, {3428,15650}, {3452,12616}, {3485,18397}, {3576,12528}, {3628,3833}, {3681,7982}, {3743,5396}, {3754,3838}, {3812,10172}, {3817,4067}, {3868,8227}, {3869,5587}, {3873,9624}, {3874,5886}, {3877,5881}, {3881,5901}, {3901,7988}, {3940,11496}, {3951,12704}, {4005,13865}, {4015,5690}, {4127,9955}, {4134,4301}, {4187,10265}, {4661,5734}, {5044,6001}, {5086,6246}, {5225,5697}, {5250,17857}, {5251,12691}, {5535,6915}, {5603,5904}, {5719,12564}, {5720,6796}, {5770,10200}, {5779,12114}, {5811,6256}, {5818,5903}, {5927,14110}, {6839,16125}, {6881,11263}, {6909,7701}, {6913,12635}, {6916,16127}, {6986,16132}, {7686,10157}, {10914,14740}, {11112,16120}, {11375,18389}, {11715,12665}, {12047,15556}, {12736,17606}

X(20117) = midpoint of X(i) and X(j) for these (i, j): {5,5694}, {10,5887}, {72,946}}, {355,3878}, {960,5777}}, {5693,5884}, {5882,14872}, {11362,12672}, {11715,12665}
X(20117) = reflection of X(i) in X(j) for these (i,j): {3754,9956}, {3881,5901}, {5690,4015}, {5885,3628}, {6684,5044}, {12005,1125}
X(20117) = complement of X(5884)
X(20117) = X(1)-of-X(5)-Brocard triangle

X(20118) = X(1)X(12619)∩X(2)X(12739)

Barycentrics (a+b-c) (a-b+c) (a^4 (b+c)+a^2 b c (b+c)+2 a (b^2-c^2)^2-2 a^3 (b^2+c^2)-(b-c)^2 (b+c)^3) : :
X(20118) = 2r X(11) - R X(65)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27840

X(20118) lies on these lines: {1,12619}, {2,12739}, {3,10073}, {5,11570}, {8,11256}, {10,1317}, {11,65}, {12,5083}, {40,13274}, {46,10738}, {56,80}, {57,13273}, {104,1837}, {119,14872}, {149,1788}, {214,5433}, {355,10074}, {496,12758}, {499,6265}, {517,5533}, {942,8068}, {952,1319}, {999,10057}, {1145,3893}, {1155,5840}, {1387,11011}, {1388,7972}, {1411,6788}, {1420,9897}, {1470,11219}, {1479,12515}, {1768,9581}, {2099,16173}, {2646,6713}, {2771,18838}, {3035,6734}, {3086,12247}, {3911,5427}, {4187,18254}, {4193,12532}, {5204,12119}, {5587,12763}, {5704,9803}, {5722,10058}, {5790,12749}, {5885,8070}, {6224,7288}, {6246,7354}, {6667,11281}, {7702,16128}, {7741,11571}, {8988,19028}, {10039,12735}, {10395,13257}, {10573,12737}, {10698,11376}, {10742,10826}, {10944,15863}, {10950,11715}, {10958,15528}, {11510,12331}, {13976,19027}, {14027,16338}, {18995,19078}, {18996,19077}

X(20119) = X(2)X(11)∩X(80)X(516)

Barycentrics (-a+b+c)*(3*a^5-3*(b+c)*a^4-( b^2-5*b*c+c^2)*a^3+(b^2-c^2)*( b-c)*a^2-(2*b^2+3*b*c+2*c^2)*( b-c)^2*a+2*(b^2-c^2)*(b-c)^3) : :
X(20119) = 2*X(1317)-3*X(11038), 4*X(1387)-3*X(8236), 4*X(3036)-3*X(5686), 4*X(5542)-3*X(14151), 3*X(5686)-2*X(6068), 2*X(7972)-3*X(14151)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27847.

X(20119) lies on these lines: {1, 12730}, {2, 11}, {7, 952}, {8, 5856}, {65, 12755}, {80, 516}, {518, 12531}, {954, 12331}, {971, 17654}, {1317, 11038}, {1320, 3254}, {1387, 8236}, {2346, 10087}, {2801, 4312}, {2802, 8275}, {3036, 5686}, {4321, 7993}, {5223, 15863}, {5425, 5542}, {5528, 7675}, {5531, 12560}, {5657, 9668}, {5727, 10394}, {5728, 6797}, {5759, 5825}, {5762, 19914}, {5805, 10698}, {5809, 12690}, {5851, 12943}, {6173, 10031}, {6224, 10427}, {7673, 12758}, {7676, 10058}, {7677, 10090}, {7679, 8068}, {7951, 8543}, {10728, 12247}

X(20119) = reflection of X(i) in X(j) for these (i,j): (1320, 3254), (5223, 15863), (5728, 6797), (6224, 10427), (7673, 12758)
X(20119) = X(9970)-of-Honsberger-triangle
X(20119) = X(12730)-of-5th-mixtilinear-triangle
X(20119) = reflection of X(i) in the line X(j)X(k) for these (i,j,k): (100, 2550, 2826), (390, 11, 1111), (1156, 80, 514)
X(20119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3036, 6068, 5686), (5542, 7972, 14151)

X(20120) = 27TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^16-13 a^14 b^2+33 a^12 b^4-39 a^10 b^6+15 a^8 b^8+13 a^6 b^10-17 a^4 b^12+7 a^2 b^14-b^16-13 a^14 c^2+42 a^12 b^2 c^2-43 a^10 b^4 c^2+14 a^8 b^6 c^2-19 a^6 b^8 c^2+46 a^4 b^10 c^2-37 a^2 b^12 c^2+10 b^14 c^2+33 a^12 c^4-43 a^10 b^2 c^4+8 a^8 b^4 c^4-3 a^6 b^6 c^4-24 a^4 b^8 c^4+69 a^2 b^10 c^4-40 b^12 c^4-39 a^10 c^6+14 a^8 b^2 c^6-3 a^6 b^4 c^6-10 a^4 b^6 c^6-39 a^2 b^8 c^6+86 b^10 c^6+15 a^8 c^8-19 a^6 b^2 c^8-24 a^4 b^4 c^8-39 a^2 b^6 c^8-110 b^8 c^8+13 a^6 c^10+46 a^4 b^2 c^10+69 a^2 b^4 c^10+86 b^6 c^10-17 a^4 c^12-37 a^2 b^2 c^12-40 b^4 c^12+7 a^2 c^14+10 b^2 c^14-c^16 : :
X(20120) = 3 X[5] - 2 X[10126], 4 X[10126] - 3 X[10205], 5 X[10205] - 8 X[10289], 5 X[10126] - 6 X[10289], 5 X[5] - 4 X[10289], 7 X[3526] - 8 X[12056], 3 X[3] - 4 X[15327], 3 X[5501] - 2 X[15327], 5 X[632] - 4 X[15334], 5 X[3858] - 4 X[15335], 8 X[12057] - 9 X[15699], 3 X[2] - 4 X[15957], 3 X[10205] - 8 X[19940], 3 X[10289] - 5 X[19940], 3 X[5] - 4 X[19940], 2 X[10289] - 5 X[20030], X[10205] - 4 X[20030], X[10126] - 3 X[20030], 2 X[19940] - 3 X[20030]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27865.

X(20120) lies on this line: {2, 3}

X(20120) = reflection of X(i) in X(j) for these {i,j}: {3, 5501}, {5, 20030}, {10126, 19940}, {10205, 5}, {14142, 10285}
X(20120) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10126, 19940, 5), (10126, 20030, 19940)

X(20121) = X(1)X(7)∩X(948)X(4031)

Barycentrics (a+b-c) (a-b+c) (a^2-5 a (b+c)+4 (b-c)^2) : :
X(20121) = 3s^2 X(1) - 4(r+4R)^2 X(7)

See Kadir Altintas and Angel Montesdeoca, ADGEOM 4791.

X(20121) lies on these lines: {1,7}, {948,4031}, {1358,18421}, {4495,7195}, {4859,7960}

X(20122) = X(25)X(222)∩X(30)X(65)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27867.

X(20122) lies on these lines: {3, 11429}, {25, 222}, {30, 65}, {51, 1465}, {57, 5751}, {60, 2075}, {73, 859}, {81, 108}, {225, 18180}, {226, 18165}, {389, 17102}, {511, 1214}, {603, 11334}, {651, 4228}, {942, 1875}, {971, 1859}, {1401, 3660}, {1469, 1617}, {1708, 4259}, {2003, 2194}, {3060, 17080}, {3185, 8679}, {3560, 7352}, {3784, 19544}, {4303, 7420}, {13730, 19349}

X(20123) = X(30)X(146)∩X(74)X(18317)

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

See Angel Montesdeoca, ADGEOM 4801.

X(20123) lies on these lines: {30,146}, {74,18317}, {265,14919}, {1294,14677}, {1494,10264}, {1511,3163}, {6699,8552}, {10272,14920}, {16163,19223}

X(20124) = EULER LINE INTERCEPT OF X(3258)X(18285)

Trilinears 3*(14*cos(2*A)-2*cos(4*A)+15)* cos(B-C)+36*(2*cos(A)+cos(3*A) )*cos(2*(B-C))-2*(11*cos(2*A)+ 7)*cos(3*(B-C))+4*cos(5*A)- 116*cos(A)-32*cos(3*A) : :
Barycentrics 8*S^4+3*(3*R^2*(111*R^2-44*SW) -8*SB*SC+12*SW^2)*S^2-9*(9*R^ 4+12*R^2*SW-4*SW^2)*SB*SC : :
X(20124)= 4*X(3258)-X(18285), 2*X(5627)+X(11749)

See Antreas Hatzipolakis, César Lozada and Angel Montesdeoca Hyacinthos 27871 and Hyacinthos 27872.

X(20124) lies on these lines: {2, 3}, {3258, 18285}, {5627, 11749}.

X(20125) = X(2)X(399)∩ X(4)X(110)

Barycentrics 2*(6*R^2+SA-2*SW)*S^2+9*R^2* SB*SC : :
X(20125) = 3*X(2)+2*X(399), 9*X(2)-4*X(10264), 3*X(2)-8*X(10272), 6*X(2)-X(12317), 3*X(3)-8*X(13392), X(4)+4*X(110), 3*X(4)-8*X(113), 9*X(4)-4*X(10733), 13*X(4)-8*X(12295), 3*X(4)+2*X(12383), X(4)-16*X(16534), 3*X(110)+2*X(113), 9*X(110)+X(10733), 13*X(110)+2*X(12295), 6*X(110)-X(12383), X(110)+4*X(16534), 6*X(113)-X(10733), 13*X(113)-3*X(12295), 4*X(113)+X(12383), X(113)-6*X(16534), 3*X(399)+2*X(10264), X(399)+4*X(10272), 4*X(399)+X(12317), 6*X(5654)-X(12319), X(10264)-6*X(10272), 8*X(10264)-3*X(12317), 16*X(10272)-X(12317), 13*X(10733)-18*X(12295), 2*X(10733)+3*X(12383)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27873.

X(20125) lies on these lines: {2, 399}, {3, 13392}, {4, 110}, {5, 14683}, {8, 11699}, {69, 19140}, {74, 3524}, {125, 5067}, {140, 12308}, {146, 376}, {184, 18933}, {186, 12168}, {193, 2914}, {265, 3545}, {468, 12165}, {541, 15051}, {542, 3618}, {631, 5663}, {1056, 10091}, {1058, 10088}, {1539, 15682}, {1986, 6353}, {2771, 3616}, {2777, 15034}, {2930, 14853}, {2935, 5656}, {2948, 5603}, {3068, 12376}, {3069, 12375}, {3089, 19504}, {3090, 3448}, {3146, 15039}, {3147, 7722}, {3522, 15040}, {3523, 10620}, {3525, 5972}, {3528, 12244}, {3529, 7728}, {3533, 15061}, {3542, 18947}, {3544, 14644}, {3832, 12902}, {5068, 11801}, {5218, 7727}, {5891, 15100}, {5898, 7545}, {5907, 15102}, {6126, 10072}, {6225, 13293}, {6593, 14912}, {6699, 15702}, {6759, 13203}, {7288, 19470}, {7343, 10056}, {7493, 12219}, {7494, 12358}, {7552, 15068}, {7577, 18440}, {7687, 11427}, {7736, 14901}, {7967, 11720}, {8780, 18559}, {9140, 12900}, {9544, 11597}, {10192, 17835}, {10299, 12041}, {10304, 11694}, {10601, 10821}, {10657, 11489}, {10658, 11488}, {10706, 11001}, {10990, 15036}, {11064, 12112}, {11433, 12227}, {11441, 14940}, {11557, 12273}, {12228, 18537}, {12284, 16223}, {15020, 16111}, {15032, 17838}, {15041, 15717}, {16252, 17847}, {18445, 18932}

X(20125) = reflection of X(3522) in X(15040)
X(20125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 399, 12317), (113, 12383, 4), (146, 1511, 376), (399, 10272, 2), (1511, 5655, 146), (3448, 14643, 3090), (5609, 14643, 3448), (5642, 6053, 74), (11801, 15046, 5068), (12244, 15035, 3528), (15035, 15063, 12244)

X(20126) = X(3)X(67)∩X(30)X(74)

Barycentrics (27*R^2-3*SA-4*SW)*S^2-3*(9*R^ 2-SW)*SB*SC : :

X(20126) = = 4*X(2)-3*X(14643), 2*X(2)-3*X(15061), X(3)+2*X(16003), 2*X(4)-5*X(15027), 2*X(5)+X(15054), 2*X(74)+X(265), X(74)+2*X(10264), 5*X(74)+X(10733), 5*X(74)-2*X(14677), X(265)-4*X(10264), 5*X(265)-2*X(10733), 5*X(265)+4*X(14677), 2*X(5655)-3*X(14643), X(5655)-3*X(15061), 2*X(8724)-3*X(14850), 5*X(9140)-X(10733), 5*X(9140)+2*X(14677), 10*X(10264)-X(10733), 5*X(10264)+X(14677), X(10733)+2*X(14677)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27882.

Let P be a point on the Jerabek hyperbola. Let A' be the orthocenter of BCP, and define B' and C' cyclically. The locus of the centroid of A'B'C' as P varies is a rectangular hyperbola, H, centered at X(125), and passing through X(2), X(381), X(1853), X(9140), X(20126). X(20126) is the point on H when P = X(67), and is the antipode in H of X(381). (Randy Hutson, July 31 2018)

X(20126) lies on these lines: {2, 5655}, {3, 67}, {4, 15027}, {5, 10706}, {23, 15361}, {30, 74}, {110, 549}, {113, 5055}, {125, 381}, {140, 14094}, {146, 3545}, {210, 2771}, {376, 3448}, {382, 10990}, {399, 5054}, {523, 14851}, {524, 11579}, {539, 12901}, {547, 15059}, {548, 13393}, {550, 15021}, {567, 5622}, {568, 2781}, {597, 9970}, {631, 5609}, {671, 15535}, {690, 11632}, {804, 19902}, {1511, 3524}, {1539, 3839}, {1597, 12828}, {1656, 15063}, {1853, 2777}, {1989, 2088}, {2070, 13399}, {2782, 11006}, {3028, 10056}, {3058, 10065}, {3521, 13561}, {3526, 16534}, {3530, 15034}, {3534, 15041}, {3541, 13148}, {3543, 10113}, {3564, 13169}, {3582, 7727}, {3584, 19470}, {3653, 11720}, {3655, 11709}, {3796, 15693}, {3845, 14644}, {3850, 15025}, {3853, 15044}, {4870, 11670}, {4995, 10088}, {5050, 15303}, {5298, 10091}, {5434, 10081}, {5972, 12308}, {6055, 18332}, {6247, 7540}, {6723, 15703}, {7552, 13491}, {7687, 14269}, {7689, 15133}, {7722, 15101}, {8703, 15055}, {9033, 18317}, {9904, 12261}, {10168, 19140}, {10201, 17854}, {10272, 11539}, {10293, 11799}, {10304, 12383}, {10657, 16241}, {10658, 16242}, {10721, 11801}, {11177, 18331}, {11178, 14982}, {11179, 14805}, {11251, 16080}, {11557, 16226}, {11559, 11744}, {11658, 18776}, {11659, 18777}, {11693, 15707}, {11806, 14831}, {12100, 15035}, {12164, 15115}, {12295, 15684}, {12902, 15681}, {13171, 14070}, {13339, 15462}, {13340, 14984}, {13754, 13857}, {14666, 19905}, {14683, 15692}, {14848, 15118}, {14891, 15036}, {15020, 15712}, {15040, 15700}, {15051, 17504}, {15106, 18445}, {15128, 16270}, {15688, 16163}, {15738, 18439}, {16219, 18400}, {18128, 18364}

X(20126) = midpoint of X(i) and X(j) for these {i,j}: {74, 9140}, {376, 3448}, {3543, 12244}, {11177, 18331}, {12902, 15681}

X(20126) = reflection of X(i) in X(j) for these (i,j): (23, 15361), (110, 549), (376, 12041), (381, 125), (399, 5642), (671, 15535), (3543, 10113), (3655, 11709), (10721, 15687)

X(20126) = X(10706)-of-Johnson-triangle

X(20126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5655, 14643), (74, 10264, 265), (74, 10733, 14677), (125, 10620, 7728), (399, 5054, 5642), (3448, 12041, 12121), (3524, 9143, 1511), (3524, 12317, 9143), (5642, 6699, 5054), (5655, 15061, 2), (14094, 15057, 140)

X(20127) = X(3)X(113)∩X(30)X(74)

Barycentrics (21*R^2-SA-4*SW)*S^2-9*(5*R^2- SW)*SB*SC : :

= 3*X(3)-2*X(113), 5*X(3)-4*X(5972), 4*X(3)-3*X(14643), 2*X(4)-3*X(15061), 5*X(113)-6*X(5972), 4*X(113)-3*X(7728), 8*X(113)-9*X(14643), X(113)-3*X(16111), 8*X(5972)-5*X(7728), 16*X(5972)-15*X(14643), 2*X(5972)-5*X(16111), 2*X(7728)-3*X(14643), X(7728)-4*X(16111), 4*X(12041)-3*X(15061), 3*X(14643)-8*X(16111)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27882.

X(20127) lies on these lines: {2, 1539}, {3, 113}, {4, 12041}, {5, 10721}, {20, 5663}, {30, 74}, {35, 12373}, {36, 12374}, {52, 17855}, {67, 11559}, {110, 550}, {125, 382}, {146, 376}, {381, 6699}, {394, 399}, {542, 15681}, {546, 15059}, {548, 13392}, {567, 15472}, {568, 974}, {1533, 2070}, {1597, 15473}, {1657, 10620}, {2771, 3962}, {2781, 17710}, {2931, 12083}, {2937, 12893}, {3028, 4302}, {3098, 14982}, {3146, 10113}, {3357, 18565}, {3448, 3529}, {3520, 3521}, {3543, 15081}, {3579, 12368}, {3627, 14644}, {3830, 7687}, {3832, 15088}, {3851, 6723}, {3853, 15057}, {4316, 7727}, {4324, 19470}, {4846, 14805}, {5054, 12900}, {5073, 12295}, {5504, 10293}, {5609, 17538}, {5642, 15688}, {5878, 15647}, {5894, 6288}, {6033, 14850}, {6053, 15689}, {6241, 13201}, {6284, 10081}, {6321, 14849}, {6449, 8998}, {6450, 13990}, {6560, 19052}, {6561, 19051}, {6781, 14901}, {7354, 10065}, {7574, 19479}, {7722, 13491}, {7723, 18439}, {7731, 15072}, {8703, 10272}, {8717, 19381}, {8994, 13665}, {9730, 11807}, {9934, 10540}, {9976, 19924}, {10088, 15338}, {10091, 15326}, {10118, 18447}, {10152, 11251}, {10254, 11204}, {10483, 12903}, {10575, 10628}, {10723, 15535}, {11001, 12317}, {11709, 12699}, {12085, 13171}, {12102, 15025}, {12103, 14094}, {12108, 15029}, {12133, 18533}, {12279, 12281}, {12302, 19908}, {12358, 14826}, {12898, 18481}, {12901, 18859}, {12902, 16003}, {13785, 13969}, {14093, 15042}, {15040, 16534}, {15046, 15720}, {15054, 15704}, {15063, 15696}, {15138, 18441}, {16105, 16222}, {17812, 18451}, {18455, 19505}

X(20127) = midpoint of X(i) and X(j) for these {i,j}: {20, 12244}, {1657, 10620}, {3448, 3529}, {6241, 13201}, {12279, 12281}, {12902, 17800}
X(20127) = reflection of X(i) in X(j) for these (i,j): (3, 16111), (4, 12041), (52, 17855), (110, 550), (146, 1511), (3146, 10113), (5073, 12295), (5878, 15647), (7722, 13491), (10723, 15535), (12898, 18481), (12902, 16003)
X(20127) = anticomplement of X(1539)
X(20127) = X(7728)-of-ABC-X(3) reflections triangle
X(20127) = X(10721)-of-Johnson triangle
X(20127) = X(16111)-of-X(3)-ABC reflections triangle
X(20127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 14643), (4, 12041, 15061), (74, 10733, 10264), (146, 376, 1511), (146, 1511, 5655), (382, 15041, 125), (399, 3534, 16163), (6699, 13202, 381), (8703, 10272, 15051), (10264, 10733, 265), (10706, 15051, 10272), (10721, 15055, 5)

X(20128) = 28TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (5 a^12-12 a^10 b^2+20 a^6 b^6-15 a^4 b^8+2 b^12-12 a^10 c^2+33 a^8 b^2 c^2-27 a^6 b^4 c^2-3 a^4 b^6 c^2+15 a^2 b^8 c^2-6 b^10 c^2-27 a^6 b^2 c^4+36 a^4 b^4 c^4-15 a^2 b^6 c^4+6 b^8 c^4+20 a^6 c^6-3 a^4 b^2 c^6-15 a^2 b^4 c^6-4 b^6 c^6-15 a^4 c^8+15 a^2 b^2 c^8+6 b^4 c^8-6 b^2 c^10+2 c^12) : :
X(20128) = 2 X[3081] + X[3534], 2 X[1650] - 3 X[5054], 4 X[402] - 3 X[5055], 2 X[549] - 3 X[11845], X[11050] - 3 X[11845], 5 X[381] - 6 X[11897], 5 X[1651] - 3 X[11897], 4 X[11897] - 5 X[11911], 2 X[381] - 3 X[11911], 4 X[1651] - 3 X[11911], 4 X[11049] - 5 X[15694], X[3] - 4 X[15774], 5 X[14093] - 6 X[16190], 3 X[14269] - 2 X[18507], 2 X[4240] + X[18508]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27891.

X(20128) lies on these lines: {2,3}, {2420,3163}, {12347,13188}, {12355,13179}

X(20128) = midpoint of X(3081) and X(12113)
X(20128) = reflection of X(i) in X(j) for these {i,j}: {381, 1651}, {3534, 12113}, {3830, 11251}, {11050, 549}, {12355, 13179}, {13188, 12347}
X(20128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 1651, 11911), (11050, 11845, 549)

X(20129) = X(30)X(553)∩X(35)X(186)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27901.

X(20129) lies on these lines: {30,553}, {35,186}, {516,15904}, {523,2488}, {1155,5160}

X(20130) = X(1)X(4)∩X(124)X(5853)

Barycentrics (a-b-c) (2 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+b^8-a^7 c-2 a^6 b c+a^5 b^2 c+2 a^4 b^3 c-7 a^3 b^4 c+2 a^2 b^5 c+7 a b^6 c-2 b^7 c-a^6 c^2+a^5 b c^2+2 a^4 b^2 c^2+6 a^3 b^3 c^2+7 a^2 b^4 c^2-15 a b^5 c^2+a^5 c^3+2 a^4 b c^3+6 a^3 b^2 c^3-20 a^2 b^3 c^3+9 a b^4 c^3+2 b^5 c^3-3 a^4 c^4-7 a^3 b c^4+7 a^2 b^2 c^4+9 a b^3 c^4-2 b^4 c^4+a^3 c^5+2 a^2 b c^5-15 a b^2 c^5+2 b^3 c^5+a^2 c^6+7 a b c^6-a c^7-2 b c^7+c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27901.

X(20130) lies on these lines: {1,4}, {124,5853}, {522,17115}

X(20131) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(8)

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

X(20131 lies on these lines: {1, 11321}, {2, 6}, {55, 17032}, {238, 16831}, {1001, 14621}, {1003, 4653}, {1509, 5021}, {3286, 16367}, {3797, 4363}, {4038, 17026}, {4255, 16917}, {4360, 20181}, {4384, 4649}, {4850, 20166}, {5132, 11329}, {5263, 17316}, {16915, 19765}, {17394, 20179}, {20161, 20178}

X(20132) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(10)

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

X(20132) lies on these lines: {1, 335}, {2, 6}, {8, 16926}, {10, 16928}, {31, 17032}, {39, 1509}, {145, 16930}, {238, 16826}, {239, 4649}, {386, 16917}, {551, 16801}, {894, 3797}, {956, 19230}, {958, 19232}, {1001, 19237}, {1100, 20179}, {1125, 16929}, {1724, 16912}, {3242, 19236}, {3552, 19765}, {3616, 16927}, {3622, 16931}, {3666, 20166}, {4340, 7791}, {4393, 20172}, {4658, 17034}, {4672, 6651}, {5132, 19308}, {5254, 6625}, {5263, 6542}, {7783, 17103}, {16468, 16831}, {16474, 16829}, {16483, 19227}, {16908, 20083}, {16915, 19767}, {19791, 20167}

X(20133) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(42)

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

X(20133) lies on these lines: {1, 19231}, {2, 6}, {37, 20167}, {75, 20164}, {292, 4687}, {384, 1001}, {1964, 16826}, {4359, 20166}, {4649, 16819}, {5132, 16917}, {5263, 16926}, {6645, 19232}, {14621, 16690}, {15485, 19227}, {16484, 19228}, {17119, 20175}, {20174, 20178}

X(20134) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(43)

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

X(20134) lies on these lines: {2, 6}, {75, 20175}, {1001, 6645}, {2309, 17032}, {3210, 20166}, {7032, 16826}, {20164, 20178}

X(20135) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(145)

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

X(20135) lies on these lines: {2, 6}, {673, 3616}, {1001, 2223}, {3797, 17118}, {4649, 16832}, {5132, 16412}, {5263, 5308}, {5283, 16728}, {16777, 20181}, {16826, 20172}

X(20136) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(306)

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

X(20136) lies on these lines: {2, 6}, {37, 384}, {346, 16930}, {1125, 16800}, {2345, 16926}, {4261, 16917}, {4649, 16817}, {4687, 19224}, {16928, 17303}, {20170, 20172}

X(20137) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(519)

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

X(20137) lies on these lines: {1, 16911}, {2, 6}, {1125, 16801}, {2177, 17032}, {2666, 16610}, {3752, 20166}, {3797, 17116}, {4256, 16917}, {4366, 16484}, {4649, 16815}, {7839, 17175}, {9345, 17028}, {14621, 15485}, {16474, 16819}, {16490, 16829}, {17394, 20180}

X(20138) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(551)

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

X(20138) lies on these lines: {2, 6}, {10, 16801}, {238, 16815}, {239, 16484}, {1001, 16816}, {1724, 16911}, {3797, 17117}, {4257, 16917}, {4366, 4384}, {5315, 16819}, {8692, 20172}, {14621, 16832}, {16489, 16829}, {17028, 17125}, {17348, 20180}

X(20139) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(612)

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

X(20139) lies on these lines: {2, 6}, {75, 20167}, {192, 20164}, {3751, 16819}, {16496, 16829}, {20165, 20168}

X(20140) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(869)

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

X(20140) lies on these lines: {2, 6}, {37, 870}, {321, 20164}, {1001, 16916}, {1621, 16955}, {1918, 14621}, {2663, 4384}, {3286, 17684}, {3666, 20167}, {4649, 17030}, {5132, 16915}, {9534, 19231}, {17160, 20175}

X(20141) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(899)

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

X(20141) lies on these lines: {2, 6}, {3248, 16826}, {3797, 20178}, {4649, 16829}, {17118, 20175}, {20165, 20167}

X(20142) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(1125)

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

X(20142) lies on these lines: {1, 16912}, {2, 6}, {37, 20166}, {58, 16917}, {238, 239}, {306, 16800}, {335, 1757}, {384, 1724}, {519, 16801}, {673, 6650}, {748, 17027}, {1001, 4393}, {1203, 16819}, {1330, 17673}, {1714, 5025}, {1834, 17685}, {1911, 2664}, {1931, 2669}, {2999, 4835}, {3008, 17770}, {3187, 16690}, {3216, 7824}, {3286, 19308}, {3454, 16908}, {3759, 20161}, {3842, 4649}, {3932, 6542}, {4093, 17011}, {4384, 14621}, {4733, 5263}, {5222, 9791}, {5247, 6645}, {5315, 16829}, {7839, 16552}, {9534, 17688}, {15485, 16834}, {16477, 16815}, {16503, 17121}, {16816, 20172}, {16918, 17034}, {16948, 17693}, {17348, 20179}

X(20143) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(1149)

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

X(20143) lies on these lines: {2, 6}

X(20144) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(1647)

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

X(20144) lies on these lines: {2, 6}, {384, 16500}, {4366, 16494}, {4649, 16820}, {14621, 16495}

X(20145) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(1698)

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

X(20145) lies on these lines: {1, 16914}, {2, 6}, {2308, 17032}, {3552, 19767}, {3758, 3797}, {4307, 6653}, {4340, 17565}, {4393, 4649}, {5263, 20055}, {16468, 16826}

X(20146) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(2999)

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

X(20146) lies on these lines: {2, 6}, {37, 330}, {75, 20168}, {1278, 20170}, {4254, 16917}, {4704, 17148}, {4772, 20174}, {16667, 16819}

X(20147) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3008)

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

X(20147) lies on these lines: {1, 7839}, {2, 6}, {75, 20180}, {238, 17120}, {384, 16783}, {537, 16484}, {894, 4366}, {1001, 17350}, {1278, 20162}, {1509, 5007}, {4251, 16917}, {4670, 20179}, {4754, 17128}, {6625, 7745}, {14621, 16779}, {16552, 16912}, {16911, 17175}, {16918, 17499}

X(20148) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3009)

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

X(20148) lies on these lines: {2, 6}, {55, 16957}, {239, 872}, {308, 3948}, {312, 20164}, {384, 5132}, {1001, 16918}, {3286, 7824}, {3752, 20167}, {4366, 18082}, {16826, 18170}, {16831, 18194}, {17318, 20175}

X(20149) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3011)

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

X(20149) lies on these lines: {2, 6}, {4366, 16792}, {5135, 16917}, {14621, 16793}

X(20150) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3187)

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

X(20150) lies on these lines: {1, 20174}, {2, 6}, {1125, 5156}, {1509, 5115}, {3286, 16289}, {3616, 16451}, {8053, 19340}, {16574, 16831}

X(20151) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3240)

Barycentrics 2 a^3 b^3 + 3 a^4 b c + 9 a^3 b^2 c + 6 a^2 b^3 c + 9 a^3 b c^2 + 12 a^2 b^2 c^2 + 6 a b^3 c^2 + 2 a^3 c^3 + 6 a^2 b c^3 + 6 a b^2 c^3 + 2 b^3 c^3 : :

X(20151) lies on these lines: {2, 6}, {75, 20178}

X(20152) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3241)

Barycentrics a^4 + 2 a^3 b + 7 a^2 b^2 + 2 a^3 c + 14 a^2 b c + 12 a b^2 c + 7 a^2 c^2 + 12 a b c^2 + 6 b^2 c^2 : :

X(20152) lies on these lines: {2, 6}, {3624, 16801}, {8692, 14621}, {16484, 16831}, {16826, 20162}

X(20153) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3244)

Barycentrics a^4 + 2 a^3 b + 6 a^2 b^2 + 2 a^3 c + 12 a^2 b c + 10 a b^2 c + 6 a^2 c^2 + 10 a b c^2 + 5 b^2 c^2 : :

X(20153) lies on these lines: {2, 6}, {4366, 16831}, {16610, 20166}, {16801, 19883}

X(20154) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3616)

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

X(20154) lies on these lines: {2, 6}, {238, 4384}, {239, 1001}, {335, 5220}, {673, 5698}, {958, 16827}, {1724, 11321}, {3008, 3821}, {3286, 11329}, {3679, 16801}, {3797, 4361}, {4252, 16917}, {4366, 16816}, {4423, 17027}, {4649, 16831}, {5132, 16367}, {5271, 16690}, {5695, 6651}, {11108, 17034}, {11285, 17749}, {14621, 16815}, {15485, 16833}, {16466, 16819}, {16468, 16832}, {16483, 16829}, {16484, 16834}, {16825, 17755}, {17026, 17123}

X(20155) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3617)

Barycentrics 3 a^4 + 6 a^3 b + 7 a^2 b^2 + 6 a^3 c + 14 a^2 b c + 8 a b^2 c + 7 a^2 c^2 + 8 a b c^2 + 4 b^2 c^2 : :

X(20155) lies on these lines: {2, 6}, {1509, 5022}, {3052, 17032}, {4649, 16833}

X(20156) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3622)

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

X(20156) lies on these lines: {2, 6}, {238, 16832}, {1001, 3696}, {1191, 16819}, {3286, 16412}, {3797, 17119}, {4482, 9708}, {8167, 17026}, {16484, 16833}, {16486, 16829}, {16801, 19875}, {16815, 20172}, {16816, 20162}

X(20157) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3623)

Barycentrics a^4 + 2 a^3 b + 9 a^2 b^2 + 2 a^3 c + 18 a^2 b c + 16 a b^2 c + 9 a^2 c^2 + 16 a b c^2 + 8 b^2 c^2 : :

X(20157) lies on these lines: {2, 6}

X(20158) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3624)

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

X(20158) lies on these lines: {2, 6}, {238, 4393}, {239, 3923}, {387, 17685}, {1724, 16914}, {3241, 16801}, {3759, 3797}, {14621, 16477}

X(20159) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3661)

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

X(20159) lies on these lines: {1, 20179}, {2, 6}, {37, 14621}, {238, 4687}, {673, 17380}, {985, 1918}, {1509, 4253}, {1914, 17032}, {3759, 4649}, {3972, 4653}, {4360, 20172}, {4366, 16777}, {5263, 17233}, {16503, 17394}, {16884, 20180}, {17160, 20181}

X(20160) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(15808)

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

X(20160) lies on these lines: {2, 6}, {4366, 16833}, {4669, 16801}

X(20161) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(3)

Barycentrics 2 a^4 b^3 + 2 a^3 b^4 + 2 a^5 b c + 9 a^4 b^2 c + 11 a^3 b^3 c + 3 a^2 b^4 c - a b^5 c + 9 a^4 b c^2 + 17 a^3 b^2 c^2 + 8 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + 2 a^4 c^3 + 11 a^3 b c^3 + 8 a^2 b^2 c^3 - b^4 c^3 + 2 a^3 c^4 + 3 a^2 b c^4 - a b^2 c^4 - b^3 c^4 - a b c^5 - b^2 c^5 : :

X(20161) lies on these lines: {2, 20168}, {75, 20166}, {3759, 20142}, {19791, 20170}, {20131, 20178}

X(20162) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(7)

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

X(20162) lies on these lines: {1, 11321}, {2, 3996}, {6, 190}, {55, 17027}, {75, 20250}, {86, 20181}, {238, 16834}, {239, 1001}, {673, 17316}, {1278, 20147}, {3295, 17034}, {3303, 17033}, {3750, 17026}, {3875, 16503}, {4384, 16484}, {4452, 17379}, {15668, 17380}, {16816, 20156}, {16826, 20152}, {17233, 17259}, {17277, 17314}, {17393, 20179}

X(20163) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(11)

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

X(20163) lies on these lines: {6, 20178}

X(20164) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(31)

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

X(20164) lies on these lines: {2, 20165}, {6, 3797}, {75, 20133}, {192, 20139}, {239, 16690}, {312, 20148}, {321, 20140}, {536, 20167}, {20134, 20178}

X(20165) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(38)

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

X(20165) lies on these lines: {2, 20164}, {20139, 20168}, {20141, 20167}

X(20166) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(39)

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

X(20166) lies on these lines: {6, 3219}, {37, 20142}, {75, 20161}, {81, 1918}, {3210, 20134}, {3666, 20132}, {3752, 20137}, {3774, 16826}, {4359, 20133}, {4850, 20131}, {16610, 20153}

X(20167) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(210)

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

X(20167) lies on these lines: {37, 20133}, {75, 20139}, {536, 20164}, {3666, 20140}, {3752, 20148}, {19791, 20132}, {20141, 20165}

X(20168) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(57)

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

X(20168) lies on these lines: {2, 20161}, {6, 190}, {75, 20146}, {86, 17490}, {87, 4970}, {194, 1449}, {330, 1100}, {1740, 4734}, {2269, 17027}, {3187, 17349}, {3210, 17011}, {3905, 16503}, {4699, 20174}, {4740, 20176}, {20139, 20165}

X(20169) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(58)

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

X(20169) lies on these lines: {2, 20164}, {75, 20161}, {238, 239}, {1918, 3187}

X(20170) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(63)

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

X(20170) lies on these lines: {2, 20164}, {6, 190}, {37, 17144}, {71, 17027}, {86, 3210}, {194, 1100}, {330, 16884}, {1001, 19851}, {1278, 20146}, {1655, 5839}, {1740, 4970}, {3995, 17349}, {15668, 17490}, {17018, 17142}, {17147, 17379}, {19791, 20161}, {20136, 20172}

X(20171) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(219)

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

X(20171) lies on these lines: {1, 20236}, {2, 37}, {78, 740}, {85, 17300}, {92, 1999}, {219, 239}, {273, 335}, {304, 20234}, {322, 6542}, {518, 1837}, {726, 1210}, {984, 6734}, {1111, 17298}, {1441, 17316}, {1446, 2996}, {1814, 2995}, {1827, 16465}, {3262, 17314}, {3662, 3673}, {3685, 4008}, {3718, 3948}, {3759, 18151}, {3875, 4858}, {3912, 17861}, {3993, 13411}, {4851, 16732}, {4872, 17481}, {4957, 17388}, {5905, 5928}, {6603, 17158}, {7264, 17304}, {16284, 17373}, {16749, 18157}, {16831, 18698}, {17386, 17791}

X(20172) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(69)

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

X(20172) lies on these lines: {1, 11321}, {2, 11}, {3, 17030}, {6, 75}, {8, 17686}, {10, 7770}, {19, 11341}, {56, 16915}, {81, 16748}, {86, 4000}, {171, 17026}, {183, 4386}, {238, 4384}, {274, 16502}, {335, 3242}, {350, 5275}, {384, 958}, {405, 16819}, {458, 1861}, {499, 17694}, {940, 17027}, {956, 16829}, {984, 17738}, {993, 1003}, {1107, 1975}, {1191, 16827}, {1329, 16924}, {1573, 3734}, {1574, 7808}, {1575, 11174}, {1738, 17023}, {1914, 16992}, {2345, 17277}, {2975, 16919}, {3797, 5695}, {3923, 17755}, {4360, 20159}, {4393, 20132}, {4441, 5276}, {4649, 16834}, {4699, 17000}, {4999, 16925}, {5204, 17693}, {5217, 17684}, {5228, 10030}, {5260, 16920}, {5710, 17033}, {5711, 17034}, {5819, 17257}, {6645, 12513}, {8053, 16367}, {8692, 20138}, {9708, 11286}, {9710, 16898}, {9780, 17541}, {10436, 16503}, {10896, 17669}, {14001, 19843}, {14377, 16887}, {14953, 16738}, {15485, 16832}, {15668, 16706}, {16468, 16833}, {16484, 16831}, {16815, 20156}, {16816, 20142}, {16826, 20135}, {17011, 19719}, {17014, 17379}, {17259, 17289}, {20136, 20170}

X(20173) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(220)

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

X(20173) lies on these lines: {2, 37}, {33, 92}, {85, 17316}, {190, 8557}, {200, 740}, {220, 239}, {322, 17314}, {329, 497}, {335, 1088}, {341, 1834}, {726, 11019}, {984, 4656}, {1921, 18153}, {1999, 10025}, {2324, 3875}, {2911, 3759}, {3208, 16609}, {3553, 4360}, {3673, 3912}, {3696, 3974}, {3950, 17861}, {3993, 13405}, {4044, 4385}, {4514, 5739}, {4872, 5905}, {4952, 20015}, {6542, 16284}, {7264, 17284}, {16673, 18698}, {16750, 18157}, {17789, 18156}

X(20174) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(81)

Barycentrics b c (-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(20174) lies on these lines: {1, 20150}, {2, 20164}, {6, 75}, {8, 9049}, {37, 16819}, {76, 17275}, {86, 4359}, {238, 4647}, {274, 1100}, {312, 17259}, {314, 3739}, {321, 17277}, {350, 1213}, {966, 4441}, {1001, 16817}, {1211, 19787}, {1269, 1654}, {1909, 17362}, {1930, 16503}, {2300, 10471}, {3686, 3770}, {3761, 4034}, {3963, 5564}, {4043, 17260}, {4261, 17030}, {4388, 15320}, {4651, 17142}, {4665, 17787}, {4688, 20176}, {4699, 20168}, {4772, 20146}, {4967, 17790}, {5262, 5263}, {15668, 19804}, {16685, 16827}, {16690, 16825}, {16738, 17495}, {16777, 17144}, {16815, 18137}, {17270, 18144}, {17287, 18143}, {20133, 20178}

X(20175) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(87)

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

X(20175) lies on these lines: {2, 594}, {75, 20134}, {4393, 18170}, {17118, 20141}, {17119, 20133}, {17160, 20140}, {17318, 20148}

X(20176) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(88)

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

X(20176) lies on these lines: {2, 20178}, {6, 4664}, {75, 20177}, {1100, 3227}, {4688, 20174}, {4740, 20168}

X(20177) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(89)

Barycentrics 2 a^3 b^2 + 2 a^2 b^3 + 13 a^3 b c + 4 a^2 b^2 c - 2 a b^3 c + 2 a^3 c^2 + 4 a^2 b c^2 - 8 a b^2 c^2 - 4 b^3 c^2 + 2 a^2 c^3 - 2 a b c^3 - 4 b^2 c^3 : :

X(20177) lies on these lines: {6, 536}, {75, 20176}

X(20178) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(100)

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

X(20178) lies on these lines: {2, 20176}, {6, 20163}, {75, 20151}, {3797, 20141}, {20131, 20161}, {20133, 20174}, {20134, 20164}

X(20179) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(141)

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

X(20179) lies on these lines: {1, 20159}, {2, 1914}, {6, 75}, {10, 82}, {32, 17030}, {37, 4366}, {86, 142}, {274, 5299}, {330, 16913}, {350, 5276}, {384, 1107}, {976, 18055}, {993, 3972}, {1001, 4687}, {1100, 20132}, {1376, 11174}, {1475, 17103}, {1573, 7804}, {1575, 3329}, {1740, 17795}, {1760, 3496}, {1909, 17686}, {1911, 18170}, {2275, 16915}, {2345, 17349}, {2550, 3618}, {2886, 7792}, {3739, 17000}, {4000, 17379}, {4426, 7787}, {4429, 17381}, {4670, 20147}, {5280, 17143}, {5291, 16829}, {5332, 16998}, {5819, 17321}, {6376, 7770}, {6645, 17448}, {9454, 18042}, {10436, 16779}, {11321, 16502}, {15668, 17370}, {16604, 16917}, {17259, 17371}, {17348, 20142}, {17393, 20162}, {17394, 20131}

X(20180) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(142)

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

X(20180) lies on these lines: {6, 190}, {42, 3329}, {75, 20147}, {86, 17366}, {145, 1001}, {238, 17121}, {239, 16503}, {385, 2280}, {673, 17300}, {1100, 20132}, {1449, 14621}, {1475, 7783}, {3684, 16999}, {3720, 16993}, {3759, 20142}, {3972, 9346}, {6542, 17243}, {7806, 11269}, {16484, 17260}, {16779, 16834}, {16884, 20159}, {17014, 17379}, {17230, 17259}, {17348, 20138}, {17394, 20137}

X(20181) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(193)

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

X(20181) lies on these lines: {6, 75}, {86, 20162}, {141, 2550}, {274, 16781}, {673, 2345}, {1001, 3739}, {1376, 15271}, {1738, 4657}, {2886, 7778}, {3616, 4000}, {3734, 9708}, {3934, 9709}, {4360, 20131}, {4366, 4699}, {4386, 8667}, {4429, 17327}, {4441, 5275}, {4772, 17000}, {5013, 17030}, {5819, 17332}, {7789, 19843}, {11321, 17143}, {16777, 20135}, {17160, 20159}

X(20182) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(194)

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

X(20182) lies on these lines: {1, 3}, {2, 594}, {6, 3219}, {37, 3305}, {42, 3789}, {45, 17013}, {63, 1100}, {75, 19701}, {81, 16884}, {86, 3210}, {219, 16579}, {226, 4021}, {239, 19732}, {306, 4657}, {312, 17319}, {321, 17318}, {333, 4393}, {497, 17726}, {614, 15569}, {894, 19722}, {966, 20043}, {968, 1386}, {1001, 1962}, {1211, 17321}, {1376, 5311}, {1407, 1442}, {1427, 7190}, {1449, 4641}, {1953, 15509}, {1961, 4413}, {1999, 17393}, {2895, 17253}, {2999, 3247}, {3187, 5737}, {3242, 17018}, {3589, 17776}, {3663, 3982}, {3664, 4114}, {3672, 3782}, {3683, 16475}, {3723, 3752}, {3743, 16466}, {3758, 19739}, {3759, 19723}, {3929, 16667}, {3989, 5220}, {3993, 4387}, {4359, 15668}, {4363, 17147}, {4364, 5739}, {4389, 17778}, {4414, 9340}, {4419, 20214}, {4428, 17469}, {4445, 20017}, {4472, 19825}, {4850, 17019}, {4852, 5271}, {4886, 17248}, {5226, 5718}, {5249, 17301}, {5262, 19728}, {5284, 17025}, {5905, 17246}, {6703, 17740}, {7269, 17080}, {7277, 20078}, {7308, 16673}, {10180, 16825}, {11238, 17722}, {14997, 16677}, {16610, 17022}, {16672, 17012}, {16826, 17144}, {17056, 17395}, {17116, 19746}, {17117, 19749}, {17121, 19750}, {17184, 17323}, {17290, 18139}, {17302, 18134}, {17396, 19786}, {17397, 19808}, {17398, 19822}, {17602, 17783}

X(20183) = X(9)X(362)∩X(10)X(164)

Barycentrics a (a^5 - a^4 (b + c) - 2 a^3 (b^2 + 10 b c + c^2) + 2 a^2 (b^3 + 7 b^2 c + 7 b c^2 + c^3) + a (b - c)^2 (b^2 + 6 b c + c^2) - (b - c)^4 (b + c) + 2 Sqrt[ b c (a + b - c) (a - b + c)] (-5 a^2 (b + c) + (b - c)^2 (b + c) + 4 a (b^2 + b c + c^2)) - 2 Sqrt[-a c (a - b - c) (a + b - c) ] (a^3 + a^2 (4 b + c) + c (b^2 - c^2) - a (5 b^2 + 4 b c + c^2)) - 2 Sqrt[ a b (a - b + c) (-a + b + c)] (a^3 - b^3 + b c^2 + a^2 (b + 4 c) - a (b^2 + 4 b c + 5 c^2))) : :
Barycentrics (1 + Cos[A]) Csc[B/2] Csc[C/2] - 2 (Csc[B/2] + Csc[C/2]) Sin[A/2] : : (Peter Moses, July 17, 2018)

See Angel Montesdeoca, HG110718.

Let A'B'C' be the excentral triangle. Let A" be the Gergonne point of triangle A'BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(20183). (Randy Hutson, July 31 2018)

X(20183) lies on these lines: {9,362}, {40,164}, {57,173}, {63,16017}, {165,3659}, {168,7028}, {504,3928}, {1489,3645}, {1697,8078}

X(20183) = X(7048)-Ceva conjugate of X(1)
X(20183) = cevapoint of X(i) and X(j) for these (i,j): {164, 168}
X(20183) = X(55)-of-excentral-triangle
X(20183) = X(3434)-of-first-circumperp-triangle
X(20183) = X(i)-aleph conjugate of X(j) for these (i,j): {188, 166}, {7028, 167}
X(20183) = X(7028)-beth conjugate of X(8138)
X(20183) = X(i)-zayin conjugate of X(j) for these (i,j): {188, 9}, {8422, 173}
X(20183) = barycentric product X(i)*X(j) for these {i,j}: {7048, 13443}
X(20183) = barycentric quotient X(i)/X(j) for these {i,j}: {13443, 7057}, {16011, 8372}
X(20183) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (173, 258, 16015)

X(20184) = CIRCUMNORMAL ISOGONAL CONJUGATE OF X(2383)

Barycentrics (SB-SC)*(2*SA-SB-SC-2*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20184) lies on these lines: {30, 511}, {6247, 15543}, {8562, 14862}, {9123, 13223}, {9185, 13224}, {15099, 15451}, {15328, 16835}, {15475, 18381}

X(20184) = isogonal conjugate of X(20185)
X(20184) = circumnormal isogonal conjugate of X(2383)

X(20185) = CIRCUMPERP CONJUGATE OF X(2383)

Barycentrics (SB+SC)*(SA-SB)*(SA-SC) *(3*SB-SW-2*R^2)*(3*SC-SW-2*R^ 2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20185) lies on the circumcircle and these lines: {3, 2383}, {20, 1141}, {22, 5966}, {74, 10625}, {550, 1300}, {1299, 3520}, {2071, 14979}, {2374, 7495}, {3563, 6636}, {11413, 18401}

X(20185) = isogonal conjugate of X(20184)
X(20185) = circumperp conjugate of X(2383)
X(20185) = circumcircle-antipode of X(2383)
X(20185) = trilinear pole of the line {6, 1493}

X(20186) = CIRCUMNORMAL ISOGONAL CONJUGATE OF X(2374)

Barycentrics (SB2-SC^2)*(2*S^2*(-SW+6*R^2) -SA^2*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20186) lies on these lines: {3, 8651}, {30, 511}, {64, 10097}, {1593, 6753}, {2444, 17813}, {5652, 5656}

X(20186) = isogonal conjugate of X(20187)
X(20186) = circumnormal isogonal conjugate of X(2374)

X(20187) = CIRCUMPERP CONJUGATE OF X(2374)

Barycentrics (SA-SB)*(SA-SC)*(2*(6*R^2-SW)* S^2-SB^2*SW)*(2*(6*R^2-SW)*S^2 -SC^2*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20187) lies on the circumcircle and these lines: {3, 2374}, {20, 111}, {22, 9084}, {107, 11634}, {376, 3563}, {858, 10102}, {1301, 4235}, {1304, 7472}, {2071, 2770}, {2373, 11413}, {4221, 15344}, {4226, 9064}, {4229, 9085}, {4236, 9107}, {9061, 16049}

X(20187) = isogonal conjugate of X(20186)
X(20187) = circumperp conjugate of X(2374)
X(20187) = circumcircle-antipode of X(2374)

X(20188) = CIRCUMNORMAL ISOGONAL CONJUGATE OF X(13597)

Barycentrics (SB^2-SC^2)*(SA^2+5*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20188) lies on these lines: {30, 511}, {2492, 3050}, {2611, 3025}, {2623, 14397}, {3005, 6132}, {8562, 14809}, {14861, 15328}

X(20188) = isogonal conjugate of X(20189)
X(20188) = incentral isogonal conjugate of X(11)
X(20188) = circumnormal isogonal conjugate of X(13597)
X(20188) = complementary conjugate of X(11792)

X(20189) = CIRCUMPERP CONJUGATE OF X(13597)

Barycentrics (SA-SB)*(SA-SC)*(5*S^2+SB^2)*( 5*S^2+SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27908.

X(20189) lies on the circumcircle and these lines: {2, 11792}, {3, 13597}, {74, 548}, {98, 15246}, {111, 3055}, {428, 3563}, {842, 20063}, {901, 5957}, {930, 1634}, {1290, 5959}, {1291, 14480}, {1300, 14865}, {1370, 13508}, {2703, 5958}, {3628, 11703}, {4226, 7953}, {10330, 10425}, {11414, 13507}, {14141, 14979}

X(20189) = anticomplement of X(11792)
X(20189) = circumperp conjugate of X(13597)
X(20189) = circumcircle-antipode of X(13597)
X(20189) = trilinear pole of the line {6, 3411}
X(20189) = isogonal conjugate of X(20188)

X(20190) = MIDPOINT OF X(3) AND X(575)

Trilinears    5 cos A + 3 sin A tan ω : :
Trilinears    3 sin A + 5 cos A cot ω : :
Trilinears   3a + 10R cos A cot ω : :
Barycentrics    a^2*(4*a^4-3*(b^2+c^2)*a^2-8* b^2*c^2-c^4-b^4) : :
X(20190) = 5*X(3)+3*X(6), X(3)+3*X(182), 3*X(3)+X(576), 11*X(3)-3*X(1350), 13*X(3)+3*X(1351), 7*X(3)-3*X(3098), 7*X(3)+9*X(5050), X(3)-9*X(5085), X(3)-3*X(5092), 7*X(3)+3*X(5097), X(3)+7*X(10541), 7*X(3)+X(11477), 11*X(3)+5*X(11482), X(3)+15*X(12017), 5*X(3)-3*X(14810), 4*X(3)+3*X(15516), 5*X(3)-9*X(17508), X(6)-5*X(182), 3*X(6)-5*X(575)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27912.

X(20190) lies on the these lines: {2, 18553}, {3, 6}, {5, 10168}, {20, 5476}, {23, 5643}, {51, 7492}, {110, 15082}, {140, 542}, {141, 14869}, {385, 9751}, {524, 3530}, {546, 3589}, {548, 19924}, {549, 8550}, {550, 597}, {599, 15720}, {1176, 16835}, {1352, 3525}, {1428, 3746}, {1495, 16042}, {1503, 3628}, {1992, 10299}, {1995, 6688}, {2330, 5563}, {3090, 3818}, {3146, 14561}, {3292, 3819}, {3357, 19153}, {3523, 11160}, {3526, 11178}, {3529, 3618}, {3564, 12108}, {3627, 19130}, {3796, 10219}, {3917, 11422}, {3934, 5026}, {4663, 17502}, {5054, 15069}, {5182, 7824}, {5462, 7555}, {5480, 15704}, {5609, 10170}, {5622, 12584}, {5650, 11003}, {5892, 7575}, {5907, 7550}, {6000, 15579}, {6636, 15019}, {6759, 10249}, {6776, 7945}, {6800, 12045}, {7859, 12252}, {7998, 9716}, {8537, 17506}, {8546, 8681}, {8549, 11202}, {8584, 17504}, {9019, 12006}, {9970, 15021}, {9976, 15035}, {10282, 15581}, {10594, 19124}, {11579, 15034}, {11649, 18571}, {11695, 12106}, {12103, 18583}, {12105, 13363}, {13171, 13402}, {13366, 15246}, {13474, 18374}, {14865, 19128}, {15054, 19140}, {15533, 15707}, {15534, 15700}, {15646, 15826}

X(20190) = midpoint of X(i) and X(j) for these {i,j}: {3, 575}, {5907, 12220} X(20190) = reflection of X(3) in X(5157) X(20190) = complement of X(18553) X(20190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 182, 575), (3, 5050, 11477), (3, 10541, 182), (3, 11477, 3098), (3, 11482, 1350), (3, 12017, 10541), (6, 17508, 14810), (182, 3098, 5050), (182, 5085, 5092), (182, 17508, 6), (575, 5092, 3), (3098, 5050, 5097), (5085, 10541, 3), (5085, 12017, 182), (5092, 14810, 17508), (8160, 8161, 11171)

X(20191) = MIDPOINT OF X(3) AND X(5449)

Barycentrics (2*a^8-2*(b^2+c^2)*a^6-(3*b^4- 4*b^2*c^2+3*c^4)*a^4+4*(b^4-c^ 4)*(b^2-c^2)*a^2-(b^2-c^2)^4)* (a^2-b^2-c^2) : :
X(20191) = 3*X(2)+X(7689), 3*X(3)+X(9927), 7*X(3)+X(12293), 5*X(3)+3*X(14852), X(68)+7*X(3523), X(155)-9*X(5054), X(156)-3*X(10182), 3*X(5449)-X(9927), 7*X(5449)-X(12293), 5*X(5449)-3*X(14852), 7*X(9927)-3*X(12293), 5*X(9927)-9*X(14852), X(12893)+3*X(15061), 3*X(12893)+X(15133), 9*X(15061)-X(15133)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27912.

X(20191) lies on the these lines: {2, 5448}, {3, 125}, {52, 11660}, {68, 3523}, {113, 11440}, {155, 5054}, {156, 10182}, {539, 549}, {578, 18580}, {631, 1147}, {1151, 13970}, {1152, 13909}, {1154, 5498}, {1192, 7706}, {1204, 6639}, {1216, 10257}, {1614, 16003}, {2777, 13406}, {2888, 15035}, {3357, 10201}, {3518, 18488}, {3520, 12897}, {3524, 12118}, {3525, 5654}, {3564, 12108}, {3917, 12606}, {5504, 13418}, {5562, 14156}, {5663, 10125}, {5876, 5972}, {5944, 10264}, {6000, 10020}, {6368, 11273}, {6689, 9730}, {6696, 13383}, {7487, 15431}, {7502, 17712}, {7568, 16836}, {7575, 13419}, {7756, 8571}, {9140, 12254}, {10018, 12162}, {10116, 13367}, {10164, 12259}, {10193, 11250}, {10274, 15132}, {10298, 11750}, {10303, 15083}, {11454, 16868}, {12111, 16534}, {12429, 15693}, {13353, 15136}, {13561, 15331}, {15059, 19479}, {15122, 15644}, {15332, 18379}, {18128, 18475}, {18324, 18381}

X(20191) = midpoint of X(i) and X(j) for these {i,j}: {3, 5449}, {6696, 13383}, {13561, 15331}, {15332, 18379}
X(20191) = complement of X(5448)
X(20191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7689, 5448), (549, 12359, 12038), (11440, 14940, 113)

X(20192) = X(2)X(1350)∩X(25)X(11179)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27919.

X(20192) lies on the these lines: {2, 1350}, {25, 11179}, {51, 5642}, {110, 8584}, {125, 3845}, {141, 10545}, {343, 11178}, {373, 549}, {468, 5476}, {524, 1995}, {597, 5640}, {3124, 5306}, {3629, 10546}, {4240, 6749}, {5544, 15693}, {5943, 10168}, {6388, 14537}, {6791, 18907}, {8550, 14002}, {9140, 13567}, {9143, 13595}, {10301, 11645}

X(20192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (597, 7426, 13394), (5640, 7426, 597)

X(20193) = X(4)X(14677)∩X(5)X(7691)

Barycentrics 2*a^10-3*(b^2+c^2)*a^8-2*(2*b^ 4+b^2*c^2+2*c^4)*a^6+(b^2+c^2) *(10*b^4-21*b^2*c^2+10*c^4)*a^ 4-(b^2-c^2)^2*(6*b^4-7*b^2*c^2 +6*c^4)*a^2+(b^4-c^4)*(b^2-c^2 )^3 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27919.

X(20193) lies on the these lines: {4, 14677}, {5, 7691}, {51, 15806}, {140, 13598}, {143, 10272}, {1531, 3850}, {1656, 7693}, {8254, 10095}, {12106, 12118}, {13362, 14051}

X(20193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (143, 10272, 11803), (10095, 10096, 8254)

X(20194) = X(6)X(376)∩X(32)X(8550)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27919.

X(20194) lies on the these lines: {6, 376}, {32, 8550}, {141, 8368}, {524, 1384}, {597, 2030}

X(20194) = {X(2030), X(18907)}-harmonic conjugate of X(597)

X(20195) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(9)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1698), (2,2), (3,1656), (4,631), (6,3763), (9, 20195), (57, 20196), (223, 20197), (282,20198), (1073, 20199), (1249, 20200)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 1 : 2 : 2

m(-a : b : c) = a - 2b - 2c : 2a - b - 2c : 2a - 2b - c

m(a : b cos C : c cos B) = 6 a^2 : 7 a^2 - b^2 + c^2 : 7 a^2 + b^2 - c^2

m(A') = a^2 + 3 a b + 2 b^2 + 3 a c - 4 b c + 2 c^2 : 2 a^2 + 3 a b + b^2 + 4 a c - 3 b c + 2 c^2 : 2 a^2 + 4 a b + 2 b^2 + 3 a c - 3 b c + c^2

m(A'') = a^3 + 2 a^2 b - a b^2 - 2 b^3 + 2 a^2 c + 10 a b c + 2 b^2 c - a c^2 + 2 b c^2 - 2 c^3 : 2 a^3 + a^2 b - 2 a b^2 - b^3 + 2 a^2 c + 10 a b c + 2 b^2 c - 2 a c^2 + b c^2 - 2 c^3 : 2 a^3 + 2 a^2 b - 2 a b^2 - 2 b^3 + a^2 c + 10 a b c + b^2 c - 2 a c^2 + 2 b c^2 - c^3,

where

-a : b : c = A-excenter

a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

A' = -a (b + c + a) : b (c - a - b) : c (-a + b - c), the A-vertex of the extraversion triangle of X(9)

A'' = -a /(b + c + a) : b/(c - a - b) : c/(-a + b - c), the A-vertex of the extraversion triangle of X(57)

If P = x : y : z (barycentrics), then m(P) = x + 2y + 2z : 2x + y + 2z : 2x + 2y + z, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(2)); m maps the line X(1)X(2) onto itself and maps the Euler line onto itself.

Let f(a,b,c,x,y,z) = 6 (b^2-c^2) x^3+(19 a^2+4 b^2+2 c^2) y^2 z-(19 a^2+2 b^2+4 c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20195) lies on these lines: {1, 3826}, {2, 7}, {3, 18482}, {5, 5732}, {10, 3243}, {11, 4326}, {12, 4321}, {35, 474}, {37, 4859}, {44, 4888}, {45, 4862}, {75, 4873}, {84, 6887}, {85, 10012}, {140, 5805}, {141, 16832}, {200, 6067}, {319, 4034}, {344, 4659}, {390, 5550}, {443, 4304}, {516, 631}, {518, 1698}, {632, 5762}, {673, 17370}, {936, 5719}, {954, 16862}, {971, 1656}, {1058, 1125}, {1086, 3731}, {1268, 4751}, {1375, 6707}, {1449, 3008}, {1699, 8167}, {1743, 4675}, {1890, 7521}, {2951, 7988}, {3035, 3254}, {3059, 5231}, {3158, 6601}, {3174, 3925}, {3247, 4000}, {3358, 6861}, {3475, 10390}, {3525, 5759}, {3526, 5735}, {3576, 6854}, {3616, 5853}, {3632, 15570}, {3634, 5542}, {3646, 12609}, {3663, 16676}, {3664, 16670}, {3672, 17067}, {3679, 17231}, {3686, 4869}, {3729, 17263}, {3739, 17265}, {3742, 15185}, {3824, 16853}, {3834, 17259}, {3848, 5572}, {3875, 17244}, {3912, 4007}, {3946, 5308}, {3973, 17365}, {4029, 4452}, {4197, 7675}, {4292, 17552}, {4312, 15254}, {4355, 5302}, {4413, 6600}, {4464, 17316}, {4657, 16593}, {4687, 17304}, {4688, 17267}, {4698, 17290}, {4699, 17266}, {4739, 17269}, {4755, 17323}, {4772, 17268}, {4798, 5845}, {4851, 16833}, {4902, 17334}, {5047, 9579}, {5067, 5817}, {5070, 5779}, {5220, 19872}, {5268, 17725}, {5272, 17722}, {5274, 15006}, {5284, 7676}, {5358, 17581}, {5686, 19877}, {5698, 19878}, {5722, 8728}, {5833, 16863}, {5851, 15017}, {5880, 7483}, {6667, 10427}, {6762, 19855}, {6833, 11372}, {6846, 9841}, {7227, 17279}, {7288, 12573}, {7504, 10861}, {7679, 9578}, {7958, 12565}, {8226, 10857}, {8583, 15950}, {9612, 16842}, {9780, 11038}, {10177, 15587}, {11375, 12560}, {12436, 16845}, {15668, 16503}, {16667, 17392}, {16673, 17301}, {16706, 16831}, {16815, 17232}, {16816, 17312}, {16834, 17317}, {17151, 17243}, {17241, 17294}, {17277, 17298}, {17313, 17348}, {20197, 20199}

X(20195) = complement of X(18230)

X(20196) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(57)

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

See X(20195)

X(20196) lies on these lines: {1, 3820}, {2, 7}, {5, 6282}, {10, 7962}, {11, 8580}, {40, 6944}, {84, 6967}, {165, 4679}, {200, 3816}, {210, 12915}, {354, 9954}, {474, 9579}, {497, 20103}, {498, 3646}, {517, 1656}, {936, 3419}, {958, 5193}, {997, 5727}, {999, 3624}, {1125, 3421}, {1329, 8583}, {1376, 9580}, {1420, 2551}, {1699, 4413}, {1997, 3687}, {2093, 5445}, {2095, 5070}, {2096, 3525}, {2478, 5438}, {3035, 4512}, {3090, 7682}, {3340, 8582}, {3359, 6863}, {3554, 17022}, {3601, 5084}, {3677, 5121}, {3740, 5231}, {3763, 20197}, {3782, 8056}, {3925, 7956}, {4853, 9711}, {4997, 19804}, {5176, 19861}, {5223, 17728}, {5234, 5433}, {5241, 18229}, {5251, 10269}, {5587, 6882}, {5741, 17296}, {6745, 10389}, {6889, 15239}, {7288, 18250}, {8165, 10106}, {9612, 16408}, {9614, 9709}, {9624, 11218}, {9843, 11518}, {12572, 17567}, {13411, 17559}, {17625, 18227}

X(20196) = {X(2),X(9)}-harmonic conjugate of X(31231)

X(20197) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(223)

Barycentrics a^6 + 3 a^5 b - 6 a^3 b^3 - 3 a^2 b^4 + 3 a b^5 + 2 b^6 + 3 a^5 c - 12 a^4 b c + 6 a^3 b^2 c + 8 a^2 b^3 c - 9 a b^4 c + 4 b^5 c + 6 a^3 b c^2 - 10 a^2 b^2 c^2 + 6 a b^3 c^2 - 2 b^4 c^2 - 6 a^3 c^3 + 8 a^2 b c^3 + 6 a b^2 c^3 - 8 b^3 c^3 - 3 a^2 c^4 - 9 a b c^4 - 2 b^2 c^4 + 3 a c^5 + 4 b c^5 + 2 c^6 : :

See X(20195)

X(20197) lies on these lines: {2, 77}, {515, 631}, {3763, 20196}, {5705, 7515}, {20195, 20199}

X(20198) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(282)

Barycentrics a^8 + a^7 b - 5 a^6 b^2 - 3 a^5 b^3 + 9 a^4 b^4 + 3 a^3 b^5 - 7 a^2 b^6 - a b^7 + 2 b^8 + a^7 c + 18 a^6 b c - 5 a^5 b^2 c - 12 a^4 b^3 c - 17 a^3 b^4 c - 6 a^2 b^5 c + 21 a b^6 c - 5 a^6 c^2 - 5 a^5 b c^2 + 6 a^4 b^2 c^2 + 14 a^3 b^3 c^2 + 7 a^2 b^4 c^2 - 9 a b^5 c^2 - 8 b^6 c^2 - 3 a^5 c^3 - 12 a^4 b c^3 + 14 a^3 b^2 c^3 + 12 a^2 b^3 c^3 - 11 a b^4 c^3 + 9 a^4 c^4 - 17 a^3 b c^4 + 7 a^2 b^2 c^4 - 11 a b^3 c^4 + 12 b^4 c^4 + 3 a^3 c^5 - 6 a^2 b c^5 - 9 a b^2 c^5 - 7 a^2 c^6 + 21 a b c^6 - 8 b^2 c^6 - a c^7 + 2 c^8 : :

See X(20195)

X(20198) lies on these lines: {2, 77}, {971, 1656}, {1698, 20200}

X(20199) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(1073)

Barycentrics a^12 + 5 a^10 b^2 - 28 a^8 b^4 + 42 a^6 b^6 - 23 a^4 b^8 + a^2 b^10 + 2 b^12 + 5 a^10 c^2 + 40 a^8 b^2 c^2 - 42 a^6 b^4 c^2 - 52 a^4 b^6 c^2 + 45 a^2 b^8 c^2 + 4 b^10 c^2 - 28 a^8 c^4 - 42 a^6 b^2 c^4 + 150 a^4 b^4 c^4 - 46 a^2 b^6 c^4 - 34 b^8 c^4 + 42 a^6 c^6 - 52 a^4 b^2 c^6 - 46 a^2 b^4 c^6 + 56 b^6 c^6 - 23 a^4 c^8 + 45 a^2 b^2 c^8 - 34 b^4 c^8 + a^2 c^10 + 4 b^2 c^10 + 2 c^12 : :

See X(20195)

X(20199) lies on these lines: {2, 253}, {1656, 6000}, {20195, 20197}

X(20200) = (X(1), X(2), X(3), X(6); X(1698), X(2), X(1656), X(3763)) COLLINEATION IMAGE OF X(1249)

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

See X(20195)

X(20200) lies on these lines: {2, 253}, {95, 3619}, {631, 1503}, {1656, 15312}, {1698, 20198}, {3090, 10002}, {3533, 15258}, {5071, 9530}, {6330, 8797}

X(20200) = complement of polar conjugate of X(35515)

X(20201) = (X(1), X(2), X(3), X(6); X(1125), X(2), X(140), X(3589)) COLLINEATION IMAGE OF X(223)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1125), (2,2), (3,140), (4,5), (6,3589), (9,6666), (57,6692), (223,20201), (282,20202), (1073,20203), (1249,20204)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 2 : 1 : 1
m(-a : b : c) = 2a - b - c: a - 2b - c : a - b - 2c
m(a : b cos C : c cos B) = 6 a^2 : 5 a^2 - b^2 + c^2 : 5 a^2 + b^2 - c^2
m(A') = 2 a^2 + 3 a b + b^2 + 3 a c - 2 b c + c^2 : a^2 + 3 a b + 2 b^2 + 2 a c - 3 b c + c^2 : a^2 + 2 a b + b^2 + 3 a c - 3 b c + 2 c^2,

where

If P = x : y : z (barycentrics), then m(P) = 2x + y + z : x + 2y + z : x + y + 2z, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(2)); m maps the line X(1)X(2) onto itself and maps the Euler line onto itself.

Let f(a,b,c,x,y,z) = 3 (b^2-c^2) x^3+(13 a^2+b^2+2 c^2) y^2 z-(13 a^2+2 b^2+c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20201) lies on these lines: {2, 77}, {5, 515}, {142, 15509}, {3452, 17073}, {3589, 6692}, {3844, 20103}, {5219, 17917}, {6666, 20203}

X(20201) = complement of X(20205)

X(20202) = (X(1), X(2), X(3), X(6); X(1125), X(2), X(140), X(3589)) COLLINEATION IMAGE OF X(282)

Barycentrics 2 a^8 - a^7 b - 7 a^6 b^2 + 3 a^5 b^3 + 9 a^4 b^4 - 3 a^3 b^5 - 5 a^2 b^6 + a b^7 + b^8 - a^7 c + 18 a^6 b c - 7 a^5 b^2 c - 12 a^4 b^3 c - 7 a^3 b^4 c - 6 a^2 b^5 c + 15 a b^6 c - 7 a^6 c^2 - 7 a^5 b c^2 + 6 a^4 b^2 c^2 + 10 a^3 b^3 c^2 + 5 a^2 b^4 c^2 - 3 a b^5 c^2 - 4 b^6 c^2 + 3 a^5 c^3 - 12 a^4 b c^3 + 10 a^3 b^2 c^3 + 12 a^2 b^3 c^3 - 13 a b^4 c^3 + 9 a^4 c^4 - 7 a^3 b c^4 + 5 a^2 b^2 c^4 - 13 a b^3 c^4 + 6 b^4 c^4 - 3 a^3 c^5 - 6 a^2 b c^5 - 3 a b^2 c^5 - 5 a^2 c^6 + 15 a b c^6 - 4 b^2 c^6 + a c^7 + c^8 : :

See X(20201).

X(20202) lies on these lines: {2, 77}, {140, 971}, {1125, 20204}, {6700, 17279}

X(20202) = complement of X(20206)

X(20203) = (X(1), X(2), X(3), X(6); X(1125), X(2), X(140), X(3589)) COLLINEATION IMAGE OF X(1073)

Barycentrics 2 a^12 + a^10 b^2 - 23 a^8 b^4 + 42 a^6 b^6 - 28 a^4 b^8 + 5 a^2 b^10 + b^12 + a^10 c^2 + 38 a^8 b^2 c^2 - 42 a^6 b^4 c^2 - 32 a^4 b^6 c^2 + 33 a^2 b^8 c^2 + 2 b^10 c^2 - 23 a^8 c^4 - 42 a^6 b^2 c^4 + 120 a^4 b^4 c^4 - 38 a^2 b^6 c^4 - 17 b^8 c^4 + 42 a^6 c^6 - 32 a^4 b^2 c^6 - 38 a^2 b^4 c^6 + 28 b^6 c^6 - 28 a^4 c^8 + 33 a^2 b^2 c^8 - 17 b^4 c^8 + 5 a^2 c^10 + 2 b^2 c^10 + c^12 : :

See X(20201).

X(20203) lies on these lines: {2, 253}, {140, 6000}, {6666, 20201}

X(20203) = complement of X(20207)

X(20204) = (X(1), X(2), X(3), X(6); X(1125), X(2), X(140), X(3589)) COLLINEATION IMAGE OF X(1249)

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

See X(20201).

X(20204) lies on these lines: {2, 253}, {3, 10002}, {5, 182}, {95, 6330}, {140, 15274}, {441, 17907}, {549, 6720}, {1125, 20202}, {1656, 15258}, {3628, 15576}, {6677, 15594}, {6692, 14743}, {7405, 19176}, {15252, 17279}

X(20204) = complement of X(20208)

X(20205) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(223)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002), which is the complement of K002. The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

{2, 77}, {3, 10}, {5, 20210}, {57, 281}, {124, 3817}, {141, 3452}, {142, 6708}, {946, 5908}, {1146, 3752}, {1210, 1453}, {1386, 11019}, {1764, 8804}, {4183, 5324}, {4847, 7070}, {5325, 17359}, {6609, 6692}, {6703, 9843}, {6847, 9120}, {7365, 18634}

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = b + c : c - a : b - a
m(a : b cos C : c cos B) = 2 a^2 : 3 a^2 - b^2 + c^2 : 3a^2 + b^2 - c^2
m(A') = a(b + c) + (b - c)^2 : b(a - c) + (a + c) ^2 : c(a - b) + (a + b)^2
m(A'') = (a + b + c)(ab + ac - (b - c)^2) : (a + b - c)(bc - ba + (a + c)^2)) : (a - b + c)(cb - ca + (a + b)^2),

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle
A' = -a (b + c + a) : b (c - a - b) : c (-a + b - c), the A-vertex of the extraversion triangle of X(9)
A'' = -a/(b + c + a) : b/(c - a - b) : c/(-a + b - c), the A-vertex of the extraversion triangle of X(57)

If P = x : y : z (barycentrics), then m(P) = y + z : z + x : x + y, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(2)); m maps the line X(1)X(2) onto itself and maps the Euler line onto itself.

Let f(a,b,c,x,y,z) = (b^2-c^2) x^3+(3 a^2+b^2) y^2 z-(3 a^2+c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20205) lies on these lines:

X(20205) = complement of X(223)
X(20205) = anticomplement of X(20201)
X(20205) = complementary conjugate of X(20206)

X(20206) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(282)

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

See X(20205).

X(20206) lies on these lines: {2, 77}, {5, 142}, {10, 17073}, {117, 18589}, {141, 20209}, {1125, 15836}, {1210, 1861}, {11019, 16608}

X(20206) = complement of X(282)
X(20206) = anticomplement of X(20202)
X(20206) = complementary conjugate of X(20205)

X(20207) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(1073)

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

See X(20205).

X(20207) lies on these lines: {2, 253}, {3, 6523}, {5, 2883}, {10, 20209}, {132, 1368}, {142, 6708}, {6716, 10192}, {7526, 15874}

X(20207) = complement of X(1073)
X(20207) = anticomplement of X(20203)
X(20207) = complementary conjugate of X(20208)

X(20208) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(1249)

Barycentrics (a^2 - b^2 - c^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) : :
Barycentrics (tan A)(tan B + tan C) - (tan B - tan C)^2 : :

See X(20205).

X(20208) lies on these lines: {2, 253}, {3, 66}, {5, 6523}, {6, 15526}, {10, 17073}, {30, 16253}, {69, 441}, {127, 133}, {140, 15258}, {142, 20210}, {216, 3763}, {264, 6330}, {281, 16596}, {343, 6617}, {458, 17035}, {577, 599}, {999, 16608}, {1656, 14059}, {2345, 6356}, {2968, 4000}, {2972, 5094}, {3164, 11331}, {3184, 10606}, {3452, 17279}, {3526, 6709}, {3589, 15851}, {5020, 15259}, {5054, 6760}, {5922, 14379}, {7400, 18840}, {7539, 13409}, {12167, 14003}, {17102, 17306}

X(20208) = complement of X(1249)
X(20208) = anticomplement of X(20204)
X(20208) = complementary conjugate of X(20207)
X(20208) = isotomic conjugate of polar conjugate of X(1853)

X(20209) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(3341)

Barycentrics (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2 - c^3) (a^9 b - 3 a^8 b^2 + 8 a^6 b^4 - 6 a^5 b^5 - 6 a^4 b^6 + 8 a^3 b^7 - 3 a b^9 + b^10 + a^9 c + 6 a^8 b c - 16 a^6 b^3 c - 6 a^5 b^4 c + 12 a^4 b^5 c + 8 a^3 b^6 c - 3 a b^8 c - 2 b^9 c - 3 a^8 c^2 + 16 a^6 b^2 c^2 + 12 a^5 b^3 c^2 - 10 a^4 b^4 c^2 - 8 a^3 b^5 c^2 - 4 a b^7 c^2 - 3 b^8 c^2 - 16 a^6 b c^3 + 12 a^5 b^2 c^3 + 8 a^4 b^3 c^3 - 8 a^3 b^4 c^3 - 4 a b^6 c^3 + 8 b^7 c^3 + 8 a^6 c^4 - 6 a^5 b c^4 - 10 a^4 b^2 c^4 - 8 a^3 b^3 c^4 + 14 a b^5 c^4 + 2 b^6 c^4 - 6 a^5 c^5 + 12 a^4 b c^5 - 8 a^3 b^2 c^5 + 14 a b^4 c^5 - 12 b^5 c^5 - 6 a^4 c^6 + 8 a^3 b c^6 - 4 a b^3 c^6 + 2 b^4 c^6 + 8 a^3 c^7 - 4 a b^2 c^7 + 8 b^3 c^7 - 3 a b c^8 - 3 b^2 c^8 - 3 a c^9 - 2 b c^9 + c^10) : :
Barycentrics (a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*((b+c)*a^9-3*(b-c)^2*a^8+8*(b^2+c^2)*(b-c)^2*a^6-6*(b^2-c^2)^2*(b+c)*a^5-2*(3*b^4+2*b^2*c^2+3*c^4)*(b-c)^2*a^4+8*(b^4-c^4)*(b^2-c^2)*(b+c)*a^3-(b^2-c^2)^2*(b+c)*(b^2+3*c^2)*(3*b^2+c^2)*a+(b^2-c^2)^4*(b-c)^2) : :

See X(20205).

X(20209) lies on these lines: {2, 271}, {3, 3452}, {10, 20207}, {141, 20206}, {936, 7952}, {1210, 7358}, {5911, 6245}

X(20209) = complement of X(3341)
X(20209) = complementary conjugate of X(20210)

X(20210) = (X(1), X(2), X(3), X(6); X(10), X(2), X(5), X(141)) COLLINEATION IMAGE OF X(3342)

Barycentrics (a^6 - 2 a^5 b - a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - 2 a b^5 + b^6 - 2 a^5 c - 2 a^4 b c + 2 a b^4 c + 2 b^5 c - a^4 c^2 + 2 a^2 b^2 c^2 - b^4 c^2 + 4 a^3 c^3 - 4 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 + c^6) (a^9 b + 3 a^8 b^2 - 8 a^6 b^4 - 6 a^5 b^5 + 6 a^4 b^6 + 8 a^3 b^7 - 3 a b^9 - b^10 + a^9 c - 6 a^8 b c + 16 a^6 b^3 c - 6 a^5 b^4 c - 12 a^4 b^5 c + 8 a^3 b^6 c - 3 a b^8 c + 2 b^9 c + 3 a^8 c^2 - 16 a^6 b^2 c^2 + 12 a^5 b^3 c^2 + 10 a^4 b^4 c^2 - 8 a^3 b^5 c^2 - 4 a b^7 c^2 + 3 b^8 c^2 + 16 a^6 b c^3 + 12 a^5 b^2 c^3 - 8 a^4 b^3 c^3 - 8 a^3 b^4 c^3 - 4 a b^6 c^3 - 8 b^7 c^3 - 8 a^6 c^4 - 6 a^5 b c^4 + 10 a^4 b^2 c^4 - 8 a^3 b^3 c^4 + 14 a b^5 c^4 - 2 b^6 c^4 - 6 a^5 c^5 - 12 a^4 b c^5 - 8 a^3 b^2 c^5 + 14 a b^4 c^5 + 12 b^5 c^5 + 6 a^4 c^6 + 8 a^3 b c^6 - 4 a b^3 c^6 - 2 b^4 c^6 + 8 a^3 c^7 - 4 a b^2 c^7 - 8 b^3 c^7 - 3 a b c^8 + 3 b^2 c^8 - 3 a c^9 + 2 b c^9 - c^10) : :

See X(20205).

X(20210) lies on these lines: {2, 271}, {5, 20205}, {142, 20208}

X(20210) = complement of X(3342)
X(20210) = complementary conjugate of X(20209)

X(20211) = (X(1), X(2), X(3), X(6); X(145), X(2), X(20), X(193)) COLLINEATION IMAGE OF X(223)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002), which is the anticomplement of K007. The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,145), (2,2), (3,20), (4,3146), (6,193), (9,144), (57,9965), (223,20211), (282,20212), (1073,20213), (1249,17037)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 3 : -1 : -1
m(-a : b : c) = 3a + b + c : -a - 3b + c : -a + b - 3
m(a : b cos C : c cos B) = a^2 : b^2 - c^2 : c^2 - b^2
m(A') = 3 a^2 + 2 a b - b^2 + 2 a c + 2 b c - c^2 : -a^2 + 2 a b + 3 b^2 - 2 a c - 2 b c - c^2 : -a^2 - 2 a b - b^2 + 2 a c - 2 b c + 3 c^2
m(A'') = 3 a^3 - a^2 b - 3 a b^2 + b^3 - a^2 c + 2 a b c - b^2 c - 3 a c^2 - b c^2 + c^3 : -a^3 + 3 a^2 b + a b^2 - 3 b^3 - a^2 c + 2 a b c - b^2 c + a c^2 + 3 b c^2 + c^3 : -a^3 - a^2 b + a b^2 + b^3 + 3 a^2 c + 2 a b c + 3 b^2 c + a c^2 - b c^2 - 3 c^3,

where

If P = x : y : z (barycentrics), then m(P) = 3x - y - z : - x + 3y - z : - x - y + 3z, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(2)); m maps the line X(1)X(2) onto itself and maps the Euler line onto itself.

Let f(a,b,c,x,y,z) = 2 (b^2-c^2) x^3-(3 a^2+b^2-3 c^2) y^2 z+(3 a^2-3 b^2+c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20211) lies on these lines: {2, 77}, {144, 6360}, {145, 515}, {175, 13387}, {176, 13386}, {193, 3210}, {279, 11433}, {391, 18607}, {1895, 5342}, {2895, 3152}, {5923, 6260}, {9799, 15237}

X(20211) = complement of X(20215)
X(20211) = anticomplement of X(22654)
X(20211) = anticomplementary conjugate of X(21279)

X(20212) = (X(1), X(2), X(3), X(6); X(145), X(2), X(20), X(193)) COLLINEATION IMAGE OF X(282)

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

See X(20211).

X(20212) lies on these lines: {2, 77}, {20, 72}, {29, 10405}, {145, 17037}, {3345, 9799}, {20015, 20110}

X(20212) = complement of X(20216)
X(20212) = anticomplement of X(5932)
X(20212) = anticomplementary conjugate of anticomplement of X(7037)

X(20213) = (X(1), X(2), X(3), X(6); X(145), X(2), X(20), X(193)) COLLINEATION IMAGE OF X(1073)

Barycentrics 3 a^12 - 6 a^10 b^2 - 7 a^8 b^4 + 28 a^6 b^6 - 27 a^4 b^8 + 10 a^2 b^10 - b^12 - 6 a^10 c^2 + 22 a^8 b^2 c^2 - 28 a^6 b^4 c^2 + 12 a^4 b^6 c^2 + 2 a^2 b^8 c^2 - 2 b^10 c^2 - 7 a^8 c^4 - 28 a^6 b^2 c^4 + 30 a^4 b^4 c^4 - 12 a^2 b^6 c^4 + 17 b^8 c^4 + 28 a^6 c^6 + 12 a^4 b^2 c^6 - 12 a^2 b^4 c^6 - 28 b^6 c^6 - 27 a^4 c^8 + 2 a^2 b^2 c^8 + 17 b^4 c^8 + 10 a^2 c^10 - 2 b^2 c^10 - c^12 : :

See X(20211).

X(20213) lies on these lines: {2, 253}, {20, 2979}, {144, 6360}, {394, 6527}, {3346, 12324}

X(20213) = complement of X(20217)
X(20213) = anticomplement of X(14361)

X(20214) = (X(1), X(2), X(3), X(6); X(3621), X(2), X(3146), X(20080)) COLLINEATION IMAGE OF X(57)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,3621), (2,2), (3,3146), (4,5059), (6,20080), (9,20059), (57,20214), (223,20215), (282,20216), (1073,20217), (1249,20215))

Nine more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -5 : 3 : 3
m(-a : b : c) = 5a + 3b + 3c : -3a - 5b + 3c : -3a + 3b - 5c
m(a : b cos C : c cos B) = a^2 : a^2 - 2b^2 + 2c^2 : a^2 + 2b^2 - 2c^2,

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

If P = x : y : z (barycentrics), then m(P) = 5x - 3y - 3z : - 3x + 5y - 3z : - 3x - 3y + 5z, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(2)); m maps the line X(1)X(2) onto itself and maps the Euler line onto itself.

Let f(a,b,c,x,y,z) = 6 (b^2-c^2) x^3-(7 a^2+9 b^2-15 c^2) y^2 z+(7 a^2-15 b^2+9 c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20214) lies on these lines: {2, 7}, {145, 10624}, {306, 4488}, {452, 15934}, {484, 7080}, {497, 5852}, {516, 20015}, {517, 3146}, {999, 16865}, {1255, 3945}, {2093, 3617}, {2095, 3091}, {2096, 3522}, {2097, 3620}, {2475, 3421}, {2895, 4461}, {3623, 6872}, {3748, 5698}, {3854, 7682}, {3897, 11106}, {3927, 5177}, {3935, 7994}, {4419, 20182}, {4430, 10394}, {4454, 5739}, {5059, 20013}, {5261, 11684}, {5712, 17334}, {5924, 9799}, {6223, 20070}, {12527, 18421}, {14552, 17347}, {17768, 17784}, {17778, 20073}, {20017, 20218}, {20080, 20215}

X(20214) = anticomplement of X(9965)

X(20215) = (X(1), X(2), X(3), X(6); X(3621), X(2), X(3146), X(20080)) COLLINEATION IMAGE OF X(223)

Barycentrics 5 a^6 + 2 a^5 b - 13 a^4 b^2 - 4 a^3 b^3 + 11 a^2 b^4 + 2 a b^5 - 3 b^6 + 2 a^5 c + 18 a^4 b c + 4 a^3 b^2 c - 12 a^2 b^3 c - 6 a b^4 c - 6 b^5 c - 13 a^4 c^2 + 4 a^3 b c^2 + 2 a^2 b^2 c^2 + 4 a b^3 c^2 + 3 b^4 c^2 - 4 a^3 c^3 - 12 a^2 b c^3 + 4 a b^2 c^3 + 12 b^3 c^3 + 11 a^2 c^4 - 6 a b c^4 + 3 b^2 c^4 + 2 a c^5 - 6 b c^5 - 3 c^6 : :

See X(20214).

X(20215) lies on these lines: {2, 77}, {515, 3621}, {5923, 6223}, {20008, 20086}, {20059, 20217}, {20080, 20214}

X(20215) = anticomplement of X(20211)

X(20216) = (X(1), X(2), X(3), X(6); X(3621), X(2), X(3146), X(20080)) COLLINEATION IMAGE OF X(282)

Barycentrics 5 a^8 - 8 a^7 b - 12 a^6 b^2 + 24 a^5 b^3 + 6 a^4 b^4 - 24 a^3 b^5 + 4 a^2 b^6 + 8 a b^7 - 3 b^8 - 8 a^7 c + 12 a^6 b c - 12 a^5 b^2 c - 8 a^4 b^3 c + 32 a^3 b^4 c - 4 a^2 b^5 c - 12 a b^6 c - 12 a^6 c^2 - 12 a^5 b c^2 + 4 a^4 b^2 c^2 - 8 a^3 b^3 c^2 - 4 a^2 b^4 c^2 + 20 a b^5 c^2 + 12 b^6 c^2 + 24 a^5 c^3 - 8 a^4 b c^3 - 8 a^3 b^2 c^3 + 8 a^2 b^3 c^3 - 16 a b^4 c^3 + 6 a^4 c^4 + 32 a^3 b c^4 - 4 a^2 b^2 c^4 - 16 a b^3 c^4 - 18 b^4 c^4 - 24 a^3 c^5 - 4 a^2 b c^5 + 20 a b^2 c^5 + 4 a^2 c^6 - 12 a b c^6 + 12 b^2 c^6 + 8 a c^7 - 3 c^8 : :

See X(20214).

X(20216) lies on these lines: {2, 77}, {971, 3146}, {3621, 20218}
X(20216) = anticomplement of X(20212)

X(20217) = (X(1), X(2), X(3), X(6); X(3621), X(2), X(3146), X(20080)) COLLINEATION IMAGE OF X(1073)

Barycentrics 5 a^12 - 14 a^10 b^2 + 3 a^8 b^4 + 28 a^6 b^6 - 37 a^4 b^8 + 18 a^2 b^10 - 3 b^12 - 14 a^10 c^2 + 18 a^8 b^2 c^2 - 28 a^6 b^4 c^2 + 52 a^4 b^6 c^2 - 22 a^2 b^8 c^2 - 6 b^10 c^2 + 3 a^8 c^4 - 28 a^6 b^2 c^4 - 30 a^4 b^4 c^4 + 4 a^2 b^6 c^4 + 51 b^8 c^4 + 28 a^6 c^6 + 52 a^4 b^2 c^6 + 4 a^2 b^4 c^6 - 84 b^6 c^6 - 37 a^4 c^8 - 22 a^2 b^2 c^8 + 51 b^4 c^8 + 18 a^2 c^10 - 6 b^2 c^10 - 3 c^12 : :

See X(20214).

X(20217) lies on these lines: {2, 253}, {3146, 5889}, {5068, 15319}, {20059, 20215}

X(20217) = anticomplement of X(20213)

X(20218) = (X(1), X(2), X(3), X(6); X(3621), X(2), X(3146), X(20080)) COLLINEATION IMAGE OF X(1249)

Barycentrics 9 a^8 - 4 a^6 b^2 - 26 a^4 b^4 + 28 a^2 b^6 - 7 b^8 - 4 a^6 c^2 + 52 a^4 b^2 c^2 - 28 a^2 b^4 c^2 - 20 b^6 c^2 - 26 a^4 c^4 - 28 a^2 b^2 c^4 + 54 b^4 c^4 + 28 a^2 c^6 - 20 b^2 c^6 - 7 c^8 : :

See X(20214).

X(20218) lies on these lines: {2, 253}, {1503, 5059}, {3146, 15312}, {3621, 20216}, {3854, 10002}, {5922, 6225}, {20017, 20214}

X(20218) = anticomplement of X(17037)

X(20219) = X(35)X(2291)∩X(65)X(15728)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27920.

X(20219) lies on the circumcircle and these lines: {35, 2291}, {65, 15728}, {103, 7688}, {2742, 4557}, {3579, 15731}

X(20219) = X(9076)-of-2nd-circumperp-triangle

X(20220) = (X(1), X(2), X(3), X(6); X(75), X(1), X(17220), X(17135)) COLLINEATION IMAGE OF X(223)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,75), (2,1), (3,17220), (4,17134), (6,17135), ((9,3873), (57,3869), (223,2020), (282,20221)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(-a : b : c) = bc+ca+ab : -ab : -ac
m(a : b cos C : c cos B) = 2 a^3 - a^2 b - b^3 - a^2 c + b^2 c + b c^2 - c^3 : 3 a^2 b - 2 a b^2 - b^3 + a^2 c - b^2 c + 2 a c^2 + b c^2 + c^3 : a^2 b + 2 a b^2 + b^3 + 3 a^2 c + b^2 c - 2 a c^2 - b c^2 - c^3
m(A') = -a b c : c (a^2 + a b - b^2 + a c + b c) : b (a^2 + a b + a c + b c - c^2)
m(A'') = a b c (3 a + b + c) : -c (-a^3 - 2 a b^2 - b^3 + a b c + a c^2 + b c^2) : -b (-a^3 + a b^2 + a b c + b^2 c - 2 a c^2 - c^3),

where

If P = x : y : z (barycentrics), then m(P) = (b+c)x - (a+c)y - (a+b)z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(1)), and m maps the line X(1)X(2) to X(1)X(75), the Euler line to the Soddy line, and the line X(2)X(6) to the line X(1)X(2).

Let f(a,b,c,x,y,z) = a^2 (b-c) (b+c)^2 x^3-b (a+c) (3 a^2 b+a b^2+2 a^2 c+a b c+b^2 c-2 a c^2-2 b c^2) y^2 z+(a+b) c (2 a^2 b-2 a b^2+3 a^2 c+a b c-2 b^2 c+a c^2+b c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) - 2 (a-b) (a-c) (b-c) (a b+a c+b c) x y z = 0. (Peter Moses, July 31, 2018)

X(20220) lies on these lines: {1, 29}, {2, 227}, {8, 3427}, {75, 1444}, {189, 962}, {312, 3436}, {318, 515}, {329, 3702}, {341, 5176}, {946, 5342}, {1420, 4858}, {3600, 17862}, {3869, 4673}, {4293, 17869}, {4297, 17860}, {6350, 19843}, {10538, 12114}, {11681, 18743}, {11682, 14206}

X(20221) = (X(1), X(2), X(3), X(6); X(75), X(1), X(17220), X(17135)) COLLINEATION IMAGE OF X(282)

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

See X(20220).

X(20221) lies on these lines: {1, 29}, {3873, 17220}, {6223, 9812}, {14544, 18750}

X(20220) = anticomplement of X(227)
X(20220) = anticomplementary conjugate of anticomplement of X(285)

X(20222) = (X(1), X(2), X(3), X(6); X(1), X(17147), X(3868), X(75)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1), (2,17147), (3,3868), (4,20222), (6,75), (9,3875), (57,63), (223,20223)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1

m(-a : b : c) = a+b+c : -c : -b

m(a : b cos C : c cos B) = (a + b + c) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : (a^3 b + a^2 b^2 - a b^3 - b^4 - a^3 c + a b^2 c - a^2 c^2 + 3 a b c^2 + a c^3 + c^4) : - (a^3 b + a^2 b^2 - a b^3 - b^4 - a^3 c - 3a b^2 c - a^2 c^2 - a b c^2 + a c^3 + c^4)

m(A') = a^2 - b^2 - c^2 : c (a + b + c) : b (a + b + c)

m(A'') = a^3 + a^2 b + a b^2 + b^3 + a^2 c - b^2 c + a c^2 - b c^2 + c^3 : 2 a^2 b + 2 a b^2 - a^2 c + b^2 c - 2 a c^2 - c^3 : -a^2 b - 2 a b^2 - b^3 + 2 a^2 c + 2 a c^2 + b c^2

where

If P = x : y : z (barycentrics), then m(P) = -bc(b+c)x + ac(a+c)y + ab(a+b)z : : , and m is the collineation indicated by (A,B,C,X(1); m(A), m(B), m(C), X(1)).

Let f(a,b,c,x,y,z) = a (b-c) (b+c)^2 (a+b+c) x^3-(a+c) (2 a^3 b+2 a^2 b^2-a^3 c+a b^2 c+a b c^2-b^2 c^2+a c^3-b c^3) y^2 z-(a+b) (a^3 b-a b^3-2 a^3 c-a b^2 c+b^3 c-2 a^2 c^2-a b c^2+b^2 c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) - 2 (a-b) (a-c) (b-c) (a+b+c)^2 x y z = 0. (Peter Moses, July 31, 2018)

X(20222) lies on these lines: {1, 18662}, {2, 1068}, {20, 145}, {22, 20045}, {75, 280}, {78, 4552}, {255, 14544}, {318, 17080}, {411, 1897}, {2406, 7114}, {3164, 7520}, {3875, 17134}, {4296, 4861}, {7538, 9538}

X(20222) = anticomplement of isogonal conjugate of X(1437)
X(20222) = anticomplement of isotomic conjugate of X(1444)
X(20222) = anticomplement of polar conjugate of X(81)
X(20222) = anticomplement of anticomplement of X(37565)
X(20222) = anticomplementary conjugate of anticomplement of X(1437)

X(20223) = (X(1), X(2), X(3), X(6); X(1), X(17147), X(3868), X(75)) COLLINEATION IMAGE OF X(223)

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

See X(20222).

X(20223) lies on these lines: {1, 18662}, {19, 27}, {57, 17862}, {77, 6360}, {189, 9965}, {280, 962}, {329, 3687}, {345, 908}, {347, 18652}, {1043, 11682}, {1708, 4858}, {3101, 10444}, {3262, 3719}, {5249, 6350}, {5942, 20078}

X(20224) = (X(1), X(2), X(3), X(6); X(1), X(43), X(46), X(9)) COLLINEATION IMAGE OF X(282)

Barycentrics a (a^7 + 3 a^6 b + a^5 b^2 - 5 a^4 b^3 - 5 a^3 b^4 + a^2 b^5 + 3 a b^6 + b^7 + 3 a^6 c - 10 a^5 b c + 5 a^4 b^2 c + 4 a^3 b^3 c - 3 a^2 b^4 c + 6 a b^5 c - 5 b^6 c + a^5 c^2 + 5 a^4 b c^2 + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 3 a b^4 c^2 - 7 b^5 c^2 - 5 a^4 c^3 + 4 a^3 b c^3 + 2 a^2 b^2 c^3 - 12 a b^3 c^3 + 11 b^4 c^3 - 5 a^3 c^4 - 3 a^2 b c^4 - 3 a b^2 c^4 + 11 b^3 c^4 + a^2 c^5 + 6 a b c^5 - 7 b^2 c^5 + 3 a c^6 - 5 b c^6 + c^7) : :

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1), (2,43), (3,46), (4,1745), (6,9), (9,1743), (57,165), (223,1750), (282,20224), (1073,20225), (1249,20226)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -a : b : c (A-vertex of excentral triangle)
m(-a : b : c) = 3a : -b : -c
m(a : b cos C : c cos B) = a (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : -b (-a^2 b + b^3 + a^2 c - 2 a b c + b^2 c - b c^2 - c^3) : -c (a^2 b - b^3 - a^2 c - 2 a b c - b^2 c + b c^2 + c^3)
m(A') = a (a - b - c) : -b (-a + b - 3 c) : c(a + 3 b - c)
m(A'') = a (a^2 + 2 a b + b^2 + 2 a c - 2 b c + c^2) : b (a^2 + 2 a b + b^2 - 2 a c + 2 b c - 3 c^2) : c (a^2 - 2 a b - 3 b^2 + 2 a c + 2 b c + c^2)

where

If P = x : y : z (barycentrics), then m(P) = a(-bcx + cay + abc) : : , and m is the collineation indicated by (A,B,C,X(1); m(A), m(B), m(C), X(1)).

Let f(a,b,c,x,y,z) = b^2 (b-c) c^2 x^3-a^2 b (2 a-c) c y^2 z+a^2 (2 a-b) b c y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20224) lies on these lines: {1, 281}, {9, 1720}, {43, 1721}, {46, 1743}, {610, 1783}, {1394, 9376}, {1465, 1723}, {1722, 5018}, {1767, 1781}

X(20225) = (X(1), X(2), X(3), X(6); X(1), X(43), X(46), X(9)) COLLINEATION IMAGE OF X(1073)

Barycentrics a (a^11 - 3 a^10 b - a^9 b^2 + 11 a^8 b^3 - 6 a^7 b^4 - 14 a^6 b^5 + 14 a^5 b^6 + 6 a^4 b^7 - 11 a^3 b^8 + a^2 b^9 + 3 a b^10 - b^11 - 3 a^10 c - 7 a^8 b^2 c + 10 a^6 b^4 c + 10 a^4 b^6 c - 7 a^2 b^8 c - 3 b^10 c - a^9 c^2 - 7 a^8 b c^2 + 12 a^7 b^2 c^2 + 4 a^6 b^3 c^2 - 14 a^5 b^4 c^2 + 14 a^4 b^5 c^2 - 4 a^3 b^6 c^2 - 12 a^2 b^7 c^2 + 7 a b^8 c^2 + b^9 c^2 + 11 a^8 c^3 + 4 a^6 b^2 c^3 - 30 a^4 b^4 c^3 + 4 a^2 b^6 c^3 + 11 b^8 c^3 - 6 a^7 c^4 + 10 a^6 b c^4 - 14 a^5 b^2 c^4 - 30 a^4 b^3 c^4 + 30 a^3 b^4 c^4 + 14 a^2 b^5 c^4 - 10 a b^6 c^4 + 6 b^7 c^4 - 14 a^6 c^5 + 14 a^4 b^2 c^5 + 14 a^2 b^4 c^5 - 14 b^6 c^5 + 14 a^5 c^6 + 10 a^4 b c^6 - 4 a^3 b^2 c^6 + 4 a^2 b^3 c^6 - 10 a b^4 c^6 - 14 b^5 c^6 + 6 a^4 c^7 - 12 a^2 b^2 c^7 + 6 b^4 c^7 - 11 a^3 c^8 - 7 a^2 b c^8 + 7 a b^2 c^8 + 11 b^3 c^8 + a^2 c^9 + b^2 c^9 + 3 a c^10 - 3 b c^10 - c^11) : :

See X(20224).

X(20225) lies on these lines: {1, 1073}, {43, 20226}, {46, 3182}, {1712, 1714}, {1723, 1779}, {1743, 1750}

X(20226) = (X(1), X(2), X(3), X(6); X(1), X(43), X(46), X(9)) COLLINEATION IMAGE OF X(1249)

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

See X(20224).

X(20226) lies on these lines: {1, 281}, {9, 1745}, {43, 20225}, {71, 165}, {219, 3362}, {2324, 3465}, {5657, 9121}

X(20227) = (X(1), X(2), X(3), X(6); X(16583), X(1196), X(3767), X(6)) COLLINEATION IMAGE OF X(9)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,16583), (2,1196), (3,3767), (4,39), (6,6), (9,20227), (57,20310), (223,20311), (282,20312)

Nine more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = a (b + c) (a^2 + b^2 - 2 b c + c^2) : b (a - c) c (a^2 + b^2 + 2 a c + c^2) : c (a - b) (a^2 + 2 a b + b^2 + c^2)
m(a : b cos C : c cos B) = a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (b^2 + c^2) : b^2 (a^4 - b^4 - 2 a^2 c^2 + 4 b^2 c^2 + c^4) : c^2 (a^4 - 2 a^2 b^2 + b^4 + 4 b^2 c^2 - c^4)

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

If P = x : y : z (barycentrics), then m(P) = a^2 c^2 (a^2 - b^2 +c^2)y + a^2 b^2(a^2 + b^2 - c^2) z : : , and m is the collineation indicated by (A,B,C,X(6); m(A), m(B), m(C), X(6)).

Let f(a,b,c,x,y,z) = (b^2-c^2) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2+4 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4+c^6) x^3-(a^8+2 a^6 b^2-2 a^4 b^4-2 a^2 b^6+b^8+a^6 c^2-7 a^4 b^2 c^2+5 a^2 b^4 c^2+b^6 c^2-3 a^4 c^4+10 a^2 b^2 c^4-3 b^4 c^4-a^2 c^6-b^2 c^6+2 c^8) y^2 z+(a^8+a^6 b^2-3 a^4 b^4-a^2 b^6+2 b^8+2 a^6 c^2-7 a^4 b^2 c^2+10 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+5 a^2 b^2 c^4-3 b^4 c^4-2 a^2 c^6+b^2 c^6+c^8) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 4 (a-b) (a+b) (a-c) (b-c) (a+c) (b+c) (a^2+b^2+c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20227) lies on these lines: {2, 3718}, {6, 169}, {9, 982}, {19, 16502}, {37, 39}, {56, 5336}, {63, 19724}, {65, 2300}, {72, 992}, {142, 2092}, {229, 1169}, {244, 1400}, {284, 16716}, {604, 1880}, {610, 16780}, {614, 2285}, {910, 16946}, {941, 4850}, {1015, 1108}, {1086, 12610}, {1104, 5019}, {1149, 17452}, {1201, 2171}, {1575, 3694}, {1781, 5299}, {2256, 9620}, {2262, 3125}, {2298, 7191}, {3554, 14571}, {3666, 10436}, {3686, 16605}, {3713, 17597}, {3836, 17748}, {3965, 16610}, {5120, 16968}, {5257, 6682}, {16488, 16548}

X(20228) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(9)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,213), (2,3051), (3,32), (4,217), (6,6), (9,20228), (57,20229), (223,20230), (282,20231), (1073,20232), (1249,20233)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b^2 : c^2
m(-a : b : c) = a^3 (b + c) : b^3 (a - c) : c^3 (a - b)
m(a : b cos C : c cos B) = a^2 (a^2 b^2 - b^4 + a^2 c^2 + 2 b^2 c^2 - c^4) : -b^4 (-a^2 + b^2 - 3 c^2), -c^4 (-a^2 - 3 b^2 + c^2)
m(A') = a^3 (a b - b^2 + a c + 2 b c - c^2), -b^3 (-a^2 + a b - 2 a c - b c - c^2), -c^3 (-a^2 - 2 a b - b^2 + a c - b c)
m(A'') = a^3 (a + b + c) (a b + b^2 + a c - 2 b c + c^2) : b^3 (a + b - c) (a^2 + a b + 2 a c - b c + c^2) : c^3 (a - b + c) (a^2 + 2 a b + b^2 + a c - b c),

where

If P = x : y : z (barycentrics), then m(P) = a^4 (c^2 y + b^2 z) : b^4 (c^2 x + a^2 z) : c^4 (b^2 x + a^2 y) , and m is the collineation indicated by (A,B,C,X(6); m(A), m(B), m(C), X(6)).

Let f(a,b,c,x,y,z) = b^6 c^6 (b^2-c^2) x^3-a^6 b^2 c^2 (c^2 (a^2+b^2+2 c^2) y^2 z-b^2 (a^2+2 b^2+c^2) y z^2), where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20228) lies on these lines: {1, 6}, {31, 5042}, {32, 604}, {39, 2269}, {48, 2251}, {109, 3451}, {163, 1333}, {217, 1404}, {572, 1914}, {573, 2275}, {992, 3686}, {1015, 1400}, {1193, 4263}, {1197, 2280}, {1201, 2347}, {1213, 19864}, {1403, 9315}, {1431, 1438}, {1572, 2285}, {1918, 3248}, {2183, 17053}, {2209, 2223}, {2220, 7113}, {2238, 3840}, {2241, 2268}, {2260, 4274}, {2262, 3125}, {2277, 4266}, {2288, 2317}, {3051, 9449}, {3752, 18163}, {3780, 4856}, {4268, 5301}, {4503, 4657}, {5816, 9599}, {20230, 20232}

X(20228) = isogonal conjugate of X(32017)
X(20228) = crosspoint of X(6) and X(604)
X(20228) = crosssum of X(2) and X(312)
X(20228) = crossdifference of every pair of points on line X(513)X(4397)
X(20228) = isogonal conjugate of polar conjugate of X(1828)
X(20228) = polar conjugate of isotomic conjugate of X(22344)

X(20229) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(57)

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

See X(20228).

X(20229) lies on these lines: {6, 57}, {31, 32}, {42, 1200}, {81, 294}, {109, 1174}, {218, 1707}, {220, 4512}, {584, 3990}, {607, 2355}, {651, 9446}, {1202, 1458}, {1212, 17194}, {1402, 9454}, {1409, 20233}, {1613, 16782}, {2170, 16971}, {2293, 8012}, {3051, 9449}, {14547, 16588}

X(20229) = isogonal conjugate of X(31618)
X(20229) = crossdifference of every pair of points on line X(693)X(3900)
X(20229) = isogonal conjugate of polar conjugate of X(1827)
X(20229) = polar conjugate of isotomic conjugate of X(22079)

X(20230) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(223)

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

See X(20228).

X(20230) lies on these lines: {6, 57}, {32, 7118}, {212, 16283}, {213, 217}, {3051, 20231}, {20228, 20232}

X(20231) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(282)

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

See X(20228).

X(20231) lies on these lines: {6, 282}, {32, 604}, {213, 20233}, {3051, 20230}

X(20232) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(1073)

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

See X(20228).

X(20232) lies on these lines: {6, 1073}, {25, 32}, {3051, 20233}, {3767, 6619}, {20228, 20230}

X(20233) = (X(1), X(2), X(3), X(6); X(213), X(3051), X(32), X(6)) COLLINEATION IMAGE OF X(1249)

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

See X(20228).

X(20233) lies on these lines: {4, 6}, {32, 14642}, {213, 20231}, {800, 3269}, {1409, 20229}, {2965, 18877}, {3051, 20232}, {5065, 14585}

X(20234) = CROSSSUM OF X(31) AND X(1501)

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

X(20234) lies on these lines: {1,4812}, {2,2064}, {6,75}, {226,306}, {304,20171}, {312,17268}, {313,16732}, {315,17481}, {560,4381}, {696,4118}, {744,1918}, {760,17138}, {1227,17345}, {1229,17878}, {1281,8857}, {1916,18895}, {1930,17760}, {2887,7237}, {3094,3662}, {4019,16609}, {4136,16888}, {4150,16580}, {4178,17047}, {4385,9654}, {4647,4709}, {9229,18891}, {14963,17864}

X(20234) = X(75)-Ceva conjugate of X(3778)
X(20234) = X(16886)-cross conjugate of X(2887)
X(20234) = cevapoint of X(2887) and X(4136)
X(20234) = crosspoint of X(75) and X(1502)
X(20234) = crossdifference of every pair of points on line {788, 8636}
X(20234) = crosssum of X(31) and X(1501)
X(20234) = X(i)-isoconjugate of X(j) for these (i,j): {213, 7305}, {983, 1333}, {2194, 7132}, {2206, 17743}, {3736, 18898}, {7252, 8685}
X(20234) = barycentric product X(i)*X(j) for these {i,j}: {75, 2887}, {76, 3721}, {85, 4136}, {274, 16886}, {310, 7237}, {312, 16888}, {313, 982}, {321, 3662}, {349, 3061}, {561, 3778}, {668, 3801}, {850, 3888}, {1237, 3865}, {1441, 3705}, {1502, 16584}, {1930, 16889}, {3261, 7239}, {3701, 7185}, {3776, 4033}, {18895, 18904}
X(20234) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 983}, {86, 7305}, {226, 7132}, {313, 7033}, {321, 17743}, {693, 7255}, {982, 58}, {2275, 1333}, {2887, 1}, {3056, 2194}, {3061, 284}, {3662, 81}, {3705, 21}, {3721, 6}, {3776, 1019}, {3777, 3733}, {3778, 31}, {3784, 1437}, {3794, 60}, {3801, 513}, {3810, 3737}, {3865, 1178}, {3888, 110}, {4033, 4621}, {4073, 2328}, {4136, 9}, {4531, 2175}, {4551, 8685}, {6385, 7307}, {7032, 2206}, {7185, 1014}, {7186, 17104}, {7237, 42}, {7239, 101}, {7248, 1408}, {8022, 1917}, {16584, 32}, {16886, 37}, {16888, 57}, {16889, 82}, {17415, 8630}, {18904, 1914}, {18905, 172}

X(20235) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1930), (2,321), (3,17864), (4,20235), (6,20234), (9,20236), (57,20237), (223,20238), (282,20239), (1073,20240), (1249,20241)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : c : b
m(-a : b : c) = -b c (b^2 + c^2) : c a (a - c) (a + c) : a b (a - b) (a + b)
m(a : b cos C : c cos B) = b c (b + c) (a^2 + b^2 - 2 b c + c^2) : c a (2 a^3 + a^2 c - b^2 c + c^3) : a b (2 a^3 + a^2 b + b^3 - b c^2)
m(A') = b c (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : c a (a + c) (a^2 + a b - b c + c^2) : a b (a + b) c^2 (a^2 + b^2 + a c - b c)
m(A'') = -b c (a + b + c) (-a b^2 + b^3 - b^2 c - a c^2 - b c^2 + c^3) : c a (a + b - c) (a + c) (a^2 - a b + b c + c^2) : a b (a + b) (a - b + c) (a^2 + b^2 - a c + b c)

where

If P = x : y : z (barycentrics), then m(P) = b c (b y + c z) : c a (c z + a x) : a b (a x + b y ), and m is the collineation indicated by (A,B,C,X(75); m(A), m(B), m(C), X(75)).

Let f(a,b,c,x,y,z) = (a^7 b+a^4 b^4-a^7 c+a^3 b^4 c-a^4 c^4-a^3 b c^4) x^3+b^2 c (b+c) (3 a^4+a b^3-a b^2 c+b^3 c+a b c^2-b^2 c^2-a c^3+b c^3) y^2 z-b c^2 (b+c) (3 a^4-a b^3+a b^2 c+b^3 c-a b c^2-b^2 c^2+a c^3+b c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 a b c (a-b) (a-c) (b-c) (a^2+a b+b^2+a c+b c+c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20235) lies on these lines: {4, 75}, {10, 307}, {63, 169}, {72, 1231}, {85, 443}, {321, 857}, {333, 16747}, {1722, 17861}, {1829, 18656}, {1930, 3687}, {2333, 8680}, {3673, 4000}, {4223, 16817}, {5179, 20236}, {7386, 17170}, {7490, 18750}, {7713, 18655}, {9798, 17134}, {16605, 16732}, {20234, 20241}

X(20236) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(9)

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

See X(20235).

X(20236) lies on these lines: {1, 20171}, {2, 17861}, {8, 6894}, {9, 75}, {63, 1746}, {80, 4692}, {85, 17298}, {92, 11679}, {141, 16732}, {312, 5219}, {314, 17788}, {321, 908}, {322, 17294}, {594, 4957}, {1089, 3790}, {1111, 3662}, {1150, 14206}, {1229, 1441}, {1234, 17052}, {1733, 3923}, {1930, 17760}, {2321, 3262}, {2911, 4361}, {3419, 15906}, {3553, 3875}, {3673, 17304}, {3992, 17057}, {4056, 17481}, {4385, 5587}, {4647, 5692}, {4812, 16788}, {5015, 18406}, {5179, 20235}, {5251, 16817}, {5526, 17117}, {6996, 11683}, {7264, 7797}, {7278, 17391}, {8680, 16574}, {16888, 17046}, {17023, 17863}, {17284, 17885}, {17295, 17791}, {17864, 20239}, {20238, 20240}

X(20237) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(57)

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

See X(20235).

X(20237) lies on these lines: {8, 79}, {57, 75}, {92, 3729}, {200, 17890}, {226, 3262}, {312, 646}, {321, 908}, {1733, 4362}, {1930, 17864}, {3706, 18839}, {3891, 17884}, {3912, 17862}, {4424, 6735}, {4970, 6745}, {17182, 17452}, {20234, 20238}

X(20238) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(223)

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

See X(20235).

X(20238) lies on these lines: {2, 17880}, {75, 223}, {92, 10888}, {226, 17858}, {318, 9612}, {321, 20239}, {1745, 4647}, {1763, 11679}, {1930, 3687}, {4417, 18695}, {18690, 19684}, {20234, 20237}, {20236, 20240}

X(20239) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(282)

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

See X(20235).

X(20239) lies on these lines: {75, 282}, {321, 20238}, {346, 347}, {1930, 20241}, {17864, 20236}

X(20240) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(1073)

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

See X(20235).

X(20240) lies on these lines: {75, 1073}, {321, 20241}, {20236, 20238}

X(20241) = (X(1), X(2), X(3), X(6); X(1930), X(321), X(17864), X(20234)) COLLINEATION IMAGE OF X(1249)

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

See X(20235).

X(20241) lies on these lines: {75, 1249}, {321, 20240}, {1930, 20239}, {20234, 20235}

X(20242) = X(1)X(17167)∩X(2,228)

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

X(20242) lies on these lines: {1, 17167}, {2, 228}, {4, 912}, {63, 14956}, {75, 1370}, {147, 149}, {314, 17135}, {315, 766}, {1331, 1746}, {1621, 7474}, {1998, 10888}, {4210, 17077}

X(20242) = anticomplement of X(228)

X(20243) = (X(1), X(2), X(3), X(6); X(17135), X(75), X(20242), X(17137)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,17135), (2,75), (3,20242), (4,20243), (6,17137), (9,20244), (57,20245), (223,20246)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1

m(-a : b : c) = a^2 b + a b^2 + a^2 c + b^2 c + a c^2 + b c^2 : -a^2 b - a b^2 - a^2 c - b^2 c + a c^2 + b c^2 : -a^2 b + a b^2 - a^2 c + b^2 c - a c^2 - b c^2

m(a : b cos C : c cos B) = a (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : -a^3 b + a b^3 - 3 a^3 c + a b^2 c + 2 b^3 c - a b c^2 - a c^3 - 2 b c^3 : -3 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

m(A') = a^3 b - a b^3 + a^3 c + 2 a^2 b c - b^3 c + 2 b^2 c^2 - a c^3 - b c^3 : -a^3 b + a b^3 - a^3 c - 2 a^2 b c + b^3 c - 2 a^2 c^2 - a c^3 - b c^3 : -a^3 b - 2 a^2 b^2 - a b^3 - a^3 c - 2 a^2 b c - b^3 c + a c^3 + b c^3

where

If P = x : y : z (barycentrics), then m(P) = a(b + c)x - b(c + a)y - c(a + b)z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(75)).

Let f(a,b,c,x,y,z) = a^5 (b-c) (b+c)^2 x^3+b (a+c) (a^4 b^2-a^2 b^4+2 a^4 b c+a^2 b^3 c+a b^4 c+a^4 c^2+a b^3 c^2+2 b^4 c^2-a^2 c^4-2 a b c^4-b^2 c^4) y^2 z-(a+b) c (a^4 b^2-a^2 b^4+2 a^4 b c-2 a b^4 c+a^4 c^2-b^4 c^2+a^2 b c^3+a b^2 c^3-a^2 c^4+a b c^4+2 b^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (a b+a c+b c) (a^2 b+a b^2+a^2 c+a b c+b^2 c+a c^2+b c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20243) lies on these lines: {1, 1719}, {2, 1824}, {3, 11401}, {19, 4228}, {20, 145}, {22, 55}, {75, 1370}, {92, 14956}, {100, 4123}, {103, 13397}, {110, 10537}, {345, 4463}, {347, 5173}, {674, 12220}, {858, 2886}, {1043, 3869}, {1829, 6872}, {1871, 6837}, {1872, 6838}, {1900, 6871}, {2099, 4296}, {2915, 8144}, {3060, 10394}, {3153, 18407}, {3428, 11413}, {3873, 4360}, {3914, 15076}, {5842, 12225}, {5905, 17441}, {6182, 17161}, {6198, 11337}, {6327, 12530}, {7465, 10319}, {7520, 9538}, {9960, 12111}, {9961, 12279}, {10679, 11414}, {17135, 20246}, {17140, 18659}

X(20243) = anticomplement of X(1824)

X(20244) = (X(1), X(2), X(3), X(6); X(17135), X(75), X(20242), X(17137)) COLLINEATION IMAGE OF X(9)

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

See X(20243).

X(20244) lies on these lines: {1, 17136}, {2, 1334}, {7, 145}, {8, 3761}, {65, 20247}, {69, 9049}, {75, 3869}, {85, 14923}, {239, 20109}, {310, 2388}, {644, 17682}, {1018, 2140}, {2262, 20248}, {2389, 3434}, {3663, 10459}, {3739, 4520}, {3754, 7264}, {3873, 17158}, {3880, 4059}, {4441, 17751}, {4861, 5088}, {7223, 10912}, {16549, 17761}, {17134, 18654}, {17140, 18659}, {17483, 20016}

X(20244) = anticomplement of X(1334)

X(20245) = (X(1), X(2), X(3), X(6); X(17135), X(75), X(20242), X(17137)) COLLINEATION IMAGE OF X(57)

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

See X(20243).

X(20245) lies on these lines: {1, 17183}, {2, 7}, {8, 10435}, {69, 313}, {75, 3869}, {78, 10444}, {86, 2975}, {200, 10442}, {304, 18659}, {314, 17135}, {319, 5176}, {320, 18133}, {326, 17134}, {978, 4862}, {992, 1086}, {1193, 3663}, {1370, 6327}, {1760, 14543}, {1764, 3588}, {1930, 18656}, {1958, 14953}, {2277, 17276}, {2385, 4329}, {2852, 14360}, {3664, 12527}, {3718, 3952}, {3765, 15983}, {3870, 10889}, {3875, 11682}, {4225, 8822}, {4452, 20036}, {5224, 11681}, {5554, 5933}, {17137, 20246}, {17863, 20247}, {18658, 18695}

X(20245) = anticomplement of X(1400)
X(20245) = isotomic conjugate of isogonal conjugate of X(23361)
X(20245) = polar conjugate of isogonal conjugate of X(23131)

X(20246) = (X(1), X(2), X(3), X(6); X(17135), X(75), X(20242), X(17137)) COLLINEATION IMAGE OF X(223)

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

See X(20243).

X(20246) lies on these lines: {189, 9965}, {2995, 17220}, {17135, 20243}, {17137, 20245}

X(20247) = (X(1), X(2), X(4), X(6); X(17165), X(6), X(22), X(76)) COLLINEATION IMAGE OF X(9)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,17165), (2,6), (3,3060), (4,22), (6,76), (9,20247), (57,20248), (223,20249)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1

m(-a : b : c) = a^2 b + a b^2 + a^2 c + b^2 c + a c^2 + b c^2 : -a^2 b - a b^2 + a^2 c + b^2 c - a c^2 - b c^2 : a^2 b - a b^2 - a^2 c - b^2 c - a c^2 + b c^2

m(a : b cos C : c cos B) = 2 a^4 - a^2 b^2 - b^4 - a^2 c^2 + 2 b^2 c^2 - c^4 : a^2 b^2 - b^4 + 3 a^2 c^2 + c^4 : 3 a^2 b^2 + b^4 + a^2 c^2 - c^4

m(A') = a^3 b - a b^3 + a^3 c - 2 a^2 b c - b^3 c + 2 b^2 c^2 - a c^3 - b c^3 : -a^3 b + a b^3 + a^3 c + 2 a b^2 c - b^3 c + 2 a^2 c^2 + a c^3 + b c^3 : a^3 b + 2 a^2 b^2 + a b^3 - a^3 c + b^3 c + 2 a b c^2 + a c^3 - b c^3

where

If P = x : y : z (barycentrics), then m(P) = (b^2 + c^2)x - (a^2 + c^2)y - (a^2 + b^2)z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(6)).

Let f(a,b,c,x,y,z) = a^2 (b-c) (b+c) (b^2+c^2)^2 x^3-(a^2+c^2) (2 a^4 b^2+2 a^2 b^4-a^4 c^2+a^2 b^2 c^2+a^2 c^4-b^2 c^4) y^2 z-(a^2+b^2) (a^4 b^2-a^2 b^4-2 a^4 c^2-a^2 b^2 c^2+b^4 c^2-2 a^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) - 2 (a-b) (a+b) (a-c) (b-c) (a+c) (b+c) (a^2+b^2+c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20247) lies on these lines: {2, 17048}, {6, 20248}, {7, 2475}, {22, 17150}, {56, 17136}, {65, 20244}, {75, 17751}, {76, 17141}, {85, 3873}, {145, 3212}, {758, 7264}, {982, 18600}, {1111, 3874}, {3673, 3868}, {3892, 7278}, {3952, 18135}, {4352, 4392}, {4441, 17164}, {4566, 6604}, {5086, 7247}, {5208, 16749}, {6737, 10520}, {14923, 17158}, {17034, 17489}, {17169, 18398}, {17863, 20245}

X(20247) = anticomplement of X(33299)

X(20248) = (X(1), X(2), X(4), X(6); X(17165), X(6), X(22), X(76)) COLLINEATION IMAGE OF X(57)

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

See X(20247).

X(20248) lies on these lines: {6, 20247}, {63, 17495}, {76, 20249}, {144, 1278}, {198, 17136}, {329, 2893}, {2262, 20244}, {3060, 17165}

X(20249) = (X(1), X(2), X(4), X(6); X(17165), X(6), X(22), X(76)) COLLINEATION IMAGE OF X(223)

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

See X(20247).

X(20249) lies on these lines: {22, 17165}, {76, 20248}

X(20250) = (X(1), X(2), X(6), X(75); X(2), X(6), X(75), X(1)) COLLINEATION IMAGE OF X(650)

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

X(20250) lies on this line: {536, 20141}

X(20251) = ISOGONAL CONJUGATE OF X(7603)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27926.

X(20251) lies on these lines: {574, 11004}, {8541, 19128}

X(20251) = isogonal conjugate of X(7603)

X(20252) = MIDPOINT OF X(5) AND X(13)

Barycentrics 3*(2*SA-3*SW)*S^2-3*SB*SC*SW- sqrt(3)*S*(5*S^2+9*SB*SC) : :
X(20252) = 3*X(2)+X(13103), 3*X(5)-X(5617), 3*X(13)+X(5617), 3*X(381)+X(6770), 3*X(549)-X(5473), X(616)-5*X(1656), 2*X(3628)+X(16001), 3*X(5459)+X(5478), 3*X(5459)-X(6771), X(5463)-3*X(15699), 3*X(5470)+X(5613), 3*X(5886)+X(9901), X(7975)-3*X(10283)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27926.

X(20252) lies on these lines: {2, 13103}, {5, 13}, {30, 5459}, {115, 6783}, {140, 6669}, {381, 6770}, {495, 10078}, {496, 10062}, {530, 547}, {542, 5066}, {549, 5473}, {616, 1656}, {618, 3628}, {952, 11705}, {2549, 18582}, {3055, 6115}, {5463, 15699}, {5470, 5613}, {5472, 11543}, {5886, 9901}, {7975, 10283}, {9916, 13861}, {10592, 12942}, {10593, 12952}, {15325, 18974}, {19073, 19117}, {19074, 19116}

X(20252) = midpoint of X(i) and X(j) for these {i,j}: {5, 13}, {618, 16001}
X(20252) = reflection of X(i) in X(j) for these (i,j): (140, 6669), (618, 3628)
X(20252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 10611, 14136), (5459, 5478, 6771), (10611, 14136, 11542)

X(20253) = MIDPOINT OF X(5) AND X(14)

Barycentrics 3*(2*SA-3*SW)*S^2-3*SB*SC*SW+ sqrt(3)*S*(5*S^2+9*SB*SC) : :
X(20253) = 3*X(2)+X(13102), 3*X(5)-X(5613), 3*X(14)+X(5613), 3*X(381)+X(6773), 3*X(549)-X(5474), X(617)-5*X(1656), 2*X(3628)+X(16002), 3*X(5460)+X(5479), 3*X(5460)-X(6774), X(5464)-3*X(15699), 3*X(5469)+X(5617), 3*X(5886)+X(9900), X(7974)-3*X(10283)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27926.

X(20253) lies on these lines: {2, 13102}, {5, 14}, {30, 5460}, {115, 6782}, {140, 6670}, {381, 6773}, {495, 10077}, {496, 10061}, {531, 547}, {542, 5066}, {549, 5474}, {617, 1656}, {619, 3628}, {952, 11706}, {2549, 18581}, {3055, 6114}, {5464, 15699}, {5469, 5617}, {5471, 11542}, {5886, 9900}, {7974, 10283}, {9915, 13861}, {10592, 12941}, {10593, 12951}, {15325, 18975}, {19075, 19117}, {19076, 19116}

X(20253) = midpoint of X(i) and X(j) for these {i,j}: {5, 14}, {619, 16002}
X(20253) = reflection of X(i) in X(j) for these (i,j): (140, 6670), (619, 3628)
X(20253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 10612, 14137), (5460, 5479, 6774), (10612, 14137, 11543)

X(20254) = X(1)X(3)∩X(2)X(17972)

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

X(20254) lies on these lines: {1, 3}, {2, 17927}, {63, 17972}, {75, 20256}, {304, 7019}, {851, 20243}, {1368, 2968}, {1465, 19540}, {1565, 7182}, {3771, 17793}, {3784, 7004}, {3840, 20259}, {3955, 20277}, {4192, 17080}, {4516, 17064}, {6198, 19548}, {8555, 13323}, {17063, 20275}, {17185, 18175}, {20255, 20261}

X(20255) = X21)X(1258)∩X(10)X(141)

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

X(20254) lies on these lines: {2, 1258}, {10, 141}, {75, 20271}, {76, 1086}, {116, 626}, {244, 7148}, {304, 3959}, {742, 16583}, {1500, 17243}, {1573, 16887}, {1930, 3125}, {2140, 3934}, {2238, 17137}, {3263, 3721}, {3501, 17279}, {3589, 17750}, {3661, 19804}, {3662, 6376}, {3730, 4422}, {3734, 14377}, {3752, 3912}, {3782, 3948}, {3831, 17050}, {3840, 20257}, {3924, 4372}, {3932, 12782}, {4361, 10449}, {4364, 16589}, {4713, 17753}, {4950, 5300}, {5031, 17047}, {7816, 17729}, {10436, 15985}, {16720, 17451}, {17034, 17366}, {17365, 17499}, {18157, 18189}, {20254, 20261}, {20258, 20259}

X(20255) = complement of X(2176)
X(20255) = complementary conjugate of X(6376)
X(20255) = isotomic conjugate of isogonal conjugate of X(22199)
X(20255) = polar conjugate of isogonal conjugate of X(22413)

X(20256) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(3)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,3840), (2,75), (3,20256), (4,20254), (6,20255), (9,20257), (57,20258), (223,20259), (282,20260), (1249,20261)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = a b^2 - 2 a b c + b^2 c + a c^2 + b c^2 : -(a - c) (a b + a c + b c) : -(a - b) (a b + a c + b c)
m(a : b cos C : c cos B) = 2 a^3 (b^3 + a b c - b^2 c - b c^2 + c^3) : a^2 (a^3 b + a b^3 + 3 a^3 c - a^2 b c - a b^2 c - b^3 c - a b c^2 + a c^3 + b c^3) : a^2 (3 a^3 b + a b^3 + a^3 c - a^2 b c - a b^2 c + b^3 c - a b c^2 + a c^3 - b c^3)
m(A') = a^2 b^2 + a b^3 - 2 a^2 b c + b^3 c + a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3 : a^3 b + a^2 b^2 + a^3 c + 2 a^2 c^2 - 2 a b c^2 - b^2 c^2 + a c^3 + b c^3 : a^3 b + 2 a^2 b^2 + a b^3 + a^3 c - 2 a b^2 c + b^3 c + a^2 c^2 - b^2 c^2

where

If P = x : y : z (barycentrics), then m(P) = (b c + a b - a c)y + (b c + a c - a b)z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(75)).

Let f(a,b,c,x,y,z) = a (b-c) (a b+a c+b c) (a^3 b+a^3 c-3 a^2 b c+a b^2 c+a b c^2+b^2 c^2) x^3+c (3 a^4 b^3+a^3 b^4+6 a^4 b^2 c-8 a^3 b^3 c+2 a^2 b^4 c+3 a^4 b c^2-6 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2+2 a^3 b c^3-2 a b^3 c^3-a^3 c^4-a^2 b c^4+a b^2 c^4+b^3 c^4) y^2 z-b (-a^3 b^4+3 a^4 b^2 c+2 a^3 b^3 c-a^2 b^4 c+6 a^4 b c^2-6 a^3 b^2 c^2+a b^4 c^2+3 a^4 c^3-8 a^3 b c^3+2 a^2 b^2 c^3-2 a b^3 c^3+b^4 c^3+a^3 c^4+2 a^2 b c^4+a b^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (a b+a c+b c) (a^2 b+a b^2+a^2 c-a b c+b^2 c+a c^2+b c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20256) lies on these lines: {1, 15973}, {5, 226}, {11, 982}, {75, 20254}, {141, 9017}, {851, 20242}, {1565, 6063}, {1985, 5905}, {3142, 3868}, {3816, 4364}, {3820, 4104}, {3840, 20258}, {8727, 9436}, {10886, 10980}, {14008, 17483}, {14213, 18210}, {20257, 20260}

X(20257) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(9)

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

X(20257) lies on these lines: {1, 142}, {2, 3208}, {10, 3934}, {75, 3061}, {85, 4051}, {194, 1266}, {226, 239}, {274, 17197}, {330, 7185}, {519, 2140}, {527, 17753}, {672, 20244}, {673, 2329}, {908, 16816}, {946, 16825}, {1086, 17448}, {1107, 3663}, {2176, 3008}, {2321, 17143}, {2886, 17062}, {3244, 17758}, {3452, 4384}, {3753, 17048}, {3840, 20255}, {3880, 6706}, {3912, 17144}, {4323, 4402}, {4361, 12635}, {4393, 5249}, {4861, 9317}, {4904, 17046}, {5257, 16819}, {5316, 16815}, {6647, 11260}, {8666, 14377}, {12053, 16823}, {14951, 18159}, {16969, 17278}, {17205, 18172}, {20256, 20260}

X(20258) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(57)

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

X(20258) lies on these lines: {2, 7}, {10, 511}, {39, 3663}, {75, 3061}, {86, 2329}, {314, 646}, {946, 3923}, {1045, 3755}, {1329, 3836}, {1334, 17183}, {1738, 16571}, {2345, 10456}, {3501, 10446}, {3664, 17750}, {3685, 12053}, {3705, 7155}, {3840, 20256}, {3879, 17752}, {3946, 5105}, {4858, 20234}, {20255, 20259}

X(20258) = complement of X(1423)
X(20258) = complementary conjugate of X(20528)
X(20258) = polar conjugate of isogonal conjugate of X(20732)

X(20259) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(223)

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

X(20259) lies on these lines: {10, 1352}, {57, 281}, {75, 20260}, {894, 20262}, {3840, 20254}, {20255, 20258}

X(20260) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(282)

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

X(20260) lies on these lines: {75, 20259}, {223, 239}, {1210, 1861}, {2808, 6260}, {3840, 20261}, {20256, 20257}

X(20261) = (X(1), X(2), X(4), X(6); X(3840), X(75), X(20254), X(20255)) COLLINEATION IMAGE OF X(1249)

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

X(20261) lies on these lines: {3840, 20260}, {18634, 20208}, {20254, 20255}

X(20262) = (X(1), X(2), X(4), X(6); X(226), X(6), X(2), X(5)) COLLINEATION IMAGE OF X(57)

Barycentrics    (a - b - c) (a^3 b + a^2 b^2 - a b^3 - b^4 + a^3 c - 2 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - a c^3 - c^4) : :
Barycentrics    cos B tan B/2 + cos C tan C/2 : :
Barycentrics    b/(1 + sec B) + c/(1 + sec C) : :

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002), which is the anticomplement of K099. The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,226), (2,6), (3,13567), (4,2), (6,5), (9,1210), (57,20262), (223,20263), (282,20264), (1073,20265), (1249,4)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = (a + b - c) (a - b + c) (b + c) : (a - c) (a - b + c) (a + b + c) : (a - b) (a + b - c) (a + b + c)
m(a : b cos C : c cos B) = 2 a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) : -a^2 (a - b - c) (a + b - c) (a - b + c) (a + b + c) : -a^2 (a - b - c) (a + b - c) (a - b + c) (a + b + c)
m(A') = -a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c + 2 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4 : a^4 + a^3 b - a^2 b^2 - a b^3 + a^2 b c - 2 a b^2 c + b^3 c - 2 a^2 c^2 - a b c^2 - b^2 c^2 - b c^3 + c^4 : a^4 - 2 a^2 b^2 + b^4 + a^3 c + a^2 b c - a b^2 c - b^3 c - a^2 c^2 - 2 a b c^2 - b^2 c^2 - a c^3 + b c^3

where

If P = x : y : z (barycentrics), then m(P) = (a^2 - b^2 + c^2)y + (a^2 + b^2 - c^2)z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(6)); m maps the line X(4)X(6) onto the Euler line and maps the Euler line onto the line X(2)X(6).

Let f(a,b,c,x,y,z) = (a-b-c) (b-c) (a+b-c) (a-b+c) (b+c) (a+b+c) (a^2-b^2-c^2) x^3-(3 a^8-6 a^6 b^2+4 a^4 b^4-2 a^2 b^6+b^8-5 a^6 c^2+7 a^4 b^2 c^2+a^2 b^4 c^2-3 b^6 c^2+a^4 c^4+3 b^4 c^4+a^2 c^6-b^2 c^6) y^2 z+(3 a^8-5 a^6 b^2+a^4 b^4+a^2 b^6-6 a^6 c^2+7 a^4 b^2 c^2-b^6 c^2+4 a^4 c^4+a^2 b^2 c^4+3 b^4 c^4-2 a^2 c^6-3 b^2 c^6+c^8) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20262) lies on these lines: {2, 77}, {4, 9}, {5, 5908}, {6, 1210}, {8, 2324}, {37, 1146}, {117, 374}, {142, 1439}, {198, 515}, {219, 3686}, {220, 17275}, {226, 6708}, {346, 6735}, {391, 6734}, {442, 10380}, {443, 3182}, {461, 7070}, {478, 2122}, {527, 10400}, {572, 14058}, {610, 6245}, {894, 20259}, {946, 2262}, {948, 18634}, {950, 4254}, {958, 13737}, {993, 15817}, {1211, 3452}, {1212, 1213}, {1329, 3844}, {1436, 6705}, {1449, 11019}, {1604, 12114}, {1609, 17010}, {1696, 5252}, {1737, 1743}, {1741, 4292}, {1903, 6260}, {1944, 4416}, {2178, 4311}, {2182, 12616}, {2321, 3965}, {2323, 10916}, {3041, 17049}, {3555, 11022}, {3663, 4858}, {3707, 7359}, {3731, 10039}, {3911, 5120}, {3925, 10374}, {3950, 10915}, {3973, 18395}, {5257, 5930}, {5745, 11347}, {5795, 10367}, {6603, 17362}, {7003, 7952}, {10368, 12527}, {10479, 15479}

X(20262) = complement of X(77)
X(20262) = isotomic conjugate of polar conjugate of X(1856)
X(20262) = complementary conjugate of X(34822)

X(20263) = (X(1), X(2), X(4), X(6); X(226), X(6), X(2), X(5)) COLLINEATION IMAGE OF X(223)

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

See X(20262).

X(20263) lies on these lines: {2, 7}, {4, 282}, {5, 5908}, {6, 20264}, {268, 4292}, {281, 946}, {610, 6260}, {1210, 9119}, {1439, 20206}, {1838, 20226}, {1903, 6245}, {2262, 5514}, {2270, 6848}, {3668, 16596}, {10400, 20202}

X(20264) = (X(1), X(2), X(4), X(6); X(226), X(6), X(2), X(5)) COLLINEATION IMAGE OF X(282)

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

See X(20262).

X(20264) lies on these lines: {1, 4}, {2, 271}, {5, 5911}, {6, 20263}, {10, 7358}, {1097, 4417}, {1210, 13567}, {18635, 20206}

X(20265) = (X(1), X(2), X(4), X(6); X(226), X(6), X(2), X(5)) COLLINEATION IMAGE OF X(1073)

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

See X(20262).

X(20265) lies on these lines: {2, 1032}, {4, 6}, {5, 5910}, {1033, 15311}, {1210, 9119}, {2130, 14092}, {13567, 20207}, {14642, 16318}

X(20266) = X(2)X(7)∩X(84,3089)

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

X(20266) lies on these lines: {2, 7}, {84, 3089}, {116, 13478}, {222, 13567}, {278, 4858}, {281, 7365}, {406, 4292}, {468, 1473}, {475, 1210}, {498, 3075}, {499, 1725}, {940, 16608}, {1375, 15509}, {2003, 11433}, {3086, 5573}, {3220, 6353}, {3546, 5709}, {3666, 17073}, {3752, 20269}, {3812, 19784}, {4000, 17917}, {4194, 9579}, {4200, 9581}, {5285, 7386}, {7293, 7493}, {16578, 17776}, {17043, 20182}, {17234, 19795}, {17595, 18644}, {18214, 18636}

X(20267) = X(1)X(17046)∩X(2,1930)

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

X(20267) lies on these lines: {1, 17046}, {2, 1930}, {32, 4056}, {116, 3924}, {172, 7272}, {183, 17192}, {499, 1733}, {609, 4911}, {626, 4372}, {1089, 7795}, {1111, 3767}, {1759, 4920}, {3120, 14377}, {3403, 16706}, {3665, 5305}, {3772, 20269}, {3915, 5074}, {4376, 6680}, {4657, 19864}, {4872, 7031}, {5280, 7179}, {5299, 17181}, {7834, 16720}, {7867, 16886}

X(20268) = (X(1), X(2), X(4), X(6); X(3772), X(4000), X(20266), X(20267)) COLLINEATION IMAGE OF X(3)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,3772), (2,4000), (3,20268), (4,20266), (6,20267), (9,20269), (57,20270)

Nine more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a : a - c : a - b
m(-a : b : c) = -a^3 + b^3 - b^2 c - b c^2 + c^3 : -a^3 + b^3 + a^2 c - a c^2 + c^3 : -a^3 + a^2 b - a b^2 + b^3 + c^3
m(a : b cos C : c cos B) = 2 a^4 + a^2 b^2 + b^4 - 2 a^2 b c + a^2 c^2 - 2 b^2 c^2 + c^4 : 2 a^4 + a^2 b^2 + b^4 - 3 a^3 c + a b^2 c + a^2 c^2 - 2 b^2 c^2 - a c^3 + c^4 : 2 a^4 - 3 a^3 b + a^2 b^2 - a b^3 + b^4 + a^2 c^2 + a b c^2 - 2 b^2 c^2 + c^4,

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

If P = x : y : z (barycentrics), then m(P) = a^2 x + (b - c)(b y - c z ) : : , and m is the collineation indicated by (A',B',C',X(75); m(A'), m(B'), m(C'), X(1)), where A' = 0 : c : b, and B' and C' are defined cyclically.

Let f(a,b,c,x,y,z) = a^2 (b-c) (a^8-a^7 b-2 a^6 b^2+a^5 b^3+2 a^4 b^4+a^3 b^5-2 a^2 b^6-a b^7+b^8-a^7 c+3 a^6 b c-a^5 b^2 c+a^4 b^3 c-3 a^3 b^4 c-3 a^2 b^5 c+5 a b^6 c-b^7 c-2 a^6 c^2-a^5 b c^2+9 a^4 b^2 c^2-10 a^3 b^3 c^2+4 a^2 b^4 c^2+3 a b^5 c^2-3 b^6 c^2+a^5 c^3+a^4 b c^3-10 a^3 b^2 c^3+12 a^2 b^3 c^3-3 a b^4 c^3-b^5 c^3+2 a^4 c^4-3 a^3 b c^4+4 a^2 b^2 c^4-3 a b^3 c^4+a^3 c^5-3 a^2 b c^5+3 a b^2 c^5-b^3 c^5-2 a^2 c^6+5 a b c^6-3 b^2 c^6-a c^7-b c^7+c^8) x^3-b (a^10-a^9 b-2 a^8 b^2+2 a^7 b^3+2 a^6 b^4-2 a^5 b^5-2 a^4 b^6+2 a^3 b^7+a^2 b^8-a b^9-4 a^9 c+11 a^8 b c-15 a^6 b^3 c+7 a^4 b^5 c+8 a^3 b^6 c-5 a^2 b^7 c-4 a b^8 c+2 b^9 c-14 a^7 b c^2+24 a^6 b^2 c^2+7 a^5 b^3 c^2-3 a^4 b^4 c^2-14 a^3 b^5 c^2-12 a^2 b^6 c^2+13 a b^7 c^2-b^8 c^2+6 a^7 c^3-10 a^6 b c^3-14 a^5 b^2 c^3+7 a^4 b^3 c^3-10 a^3 b^4 c^3+24 a^2 b^5 c^3+2 a b^6 c^3-5 b^7 c^3-a^6 c^4+13 a^5 b c^4-20 a^4 b^2 c^4+28 a^3 b^3 c^4-11 a^2 b^4 c^4-9 a b^5 c^4-4 a^5 c^5+12 a^4 b c^5-16 a^3 b^2 c^5+3 a^2 b^3 c^5-4 a b^4 c^5+5 b^5 c^5-a^4 c^6-4 a^3 b c^6+14 a^2 b^2 c^6-9 a b^3 c^6+4 b^4 c^6+6 a^3 c^7-14 a^2 b c^7+10 a b^2 c^7-3 b^3 c^7+6 a b c^8-4 b^2 c^8-4 a c^9+b c^9+c^10) y^2 z+c (a^10-4 a^9 b+6 a^7 b^3-a^6 b^4-4 a^5 b^5-a^4 b^6+6 a^3 b^7-4 a b^9+b^10-a^9 c+11 a^8 b c-14 a^7 b^2 c-10 a^6 b^3 c+13 a^5 b^4 c+12 a^4 b^5 c-4 a^3 b^6 c-14 a^2 b^7 c+6 a b^8 c+b^9 c-2 a^8 c^2+24 a^6 b^2 c^2-14 a^5 b^3 c^2-20 a^4 b^4 c^2-16 a^3 b^5 c^2+14 a^2 b^6 c^2+10 a b^7 c^2-4 b^8 c^2+2 a^7 c^3-15 a^6 b c^3+7 a^5 b^2 c^3+7 a^4 b^3 c^3+28 a^3 b^4 c^3+3 a^2 b^5 c^3-9 a b^6 c^3-3 b^7 c^3+2 a^6 c^4-3 a^4 b^2 c^4-10 a^3 b^3 c^4-11 a^2 b^4 c^4-4 a b^5 c^4+4 b^6 c^4-2 a^5 c^5+7 a^4 b c^5-14 a^3 b^2 c^5+24 a^2 b^3 c^5-9 a b^4 c^5+5 b^5 c^5-2 a^4 c^6+8 a^3 b c^6-12 a^2 b^2 c^6+2 a b^3 c^6+2 a^3 c^7-5 a^2 b c^7+13 a b^2 c^7-5 b^3 c^7+a^2 c^8-4 a b c^8-b^2 c^8-a c^9+2 b c^9) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8+5 a^6 b c+a^5 b^2 c-6 a^4 b^3 c-6 a^3 b^4 c+a^2 b^5 c+5 a b^6 c-2 a^6 c^2+a^5 b c^2+8 a^4 b^2 c^2+2 a^3 b^3 c^2+8 a^2 b^4 c^2+a b^5 c^2-2 b^6 c^2-6 a^4 b c^3+2 a^3 b^2 c^3+2 a^2 b^3 c^3-6 a b^4 c^3+2 a^4 c^4-6 a^3 b c^4+8 a^2 b^2 c^4-6 a b^3 c^4+2 b^4 c^4+a^2 b c^5+a b^2 c^5-2 a^2 c^6+5 a b c^6-2 b^2 c^6+c^8) x y z = 0. (Peter Moses, July 31, 2018)

X(20268) lies on these lines: {2, 2006}, {57, 1748}, {3772, 20270}, {4000, 17917}, {5437, 16706}

X(20269) = (X(1), X(2), X(4), X(6); X(3772), X(4000), X(20266), X(20267)) COLLINEATION IMAGE OF X(9)

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

X(20269) lies on these lines: {1, 4904}, {2, 277}, {57, 1375}, {116, 1837}, {142, 474}, {169, 3665}, {218, 9436}, {355, 9317}, {498, 6706}, {673, 17181}, {905, 2275}, {1565, 2082}, {1836, 14377}, {2140, 11375}, {3086, 4000}, {3419, 17046}, {3624, 4657}, {3732, 7185}, {3739, 19854}, {3752, 20266}, {3772, 20267}, {4209, 4911}, {4675, 5277}, {5074, 12701}, {5249, 16412}, {7179, 17682}, {11376, 17761}, {16458, 19758}, {17718, 17758}

X(20270) = (X(1), X(2), X(4), X(6); X(3772), X(4000), X(20266), X(20267)) COLLINEATION IMAGE OF X(57)

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

(20270) lies on these lines: {1, 141}, {56, 12610}, {499, 3739}, {1210, 3946}, {1737, 4361}, {3086, 4000}, {3673, 17086}, {3772, 20268}, {4402, 5704}, {4852, 10573}, {4904, 18634}, {5749, 7961}, {10039, 17327}, {10072, 17382}, {12647, 17239}

X(20271) = X(1)X(1929)∩X(2,3721)

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

X(20271) lies on these lines: {1, 1929}, {2, 3721}, {6, 169}, {8, 3726}, {37, 986}, {46, 17735}, {57, 16968}, {65, 2176}, {75, 20255}, {86, 18189}, {142, 3094}, {171, 16974}, {172, 3924}, {213, 5902}, {244, 2275}, {335, 6376}, {517, 16969}, {518, 16605}, {762, 19875}, {982, 1107}, {1086, 3673}, {1125, 3735}, {1698, 3954}, {1739, 3970}, {2087, 9336}, {2160, 5301}, {2238, 3868}, {2241, 5011}, {2271, 15934}, {2277, 2294}, {3061, 16604}, {3230, 5903}, {3339, 16970}, {3509, 4426}, {3616, 3727}, {3666, 19730}, {3670, 5283}, {3780, 3873}, {3836, 4136}, {3874, 16611}, {3931, 16777}, {3953, 16975}, {3976, 17448}, {3981, 5249}, {3999, 4875}, {5021, 5708}, {5275, 16519}, {5291, 17736}, {5573, 9575}, {5883, 16600}, {15668, 18179}, {16716, 18165}, {16726, 18176}, {17065, 18904}, {17175, 18167}, {20272, 20276}, {20274, 20275}

X(20272) = X(244)X(18671)∩X(499,3708)

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

X(20272) lies on these lines: {244, 18671}, {499, 3708}, {3085, 17471}, {17063, 20273}, {20271, 20276}

X(20273) = X(2)X(18671)∩X(499,18669)

Barycentrics a (a^4 b^2 - b^6 - 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(20273) lies on these lines: {2, 18671}, {499, 18669}, {1953, 14986}, {2083, 3075}, {3061, 3840}, {3086, 17442}, {16604, 20275}, {17063, 20272}, {17181, 18730}

X(20274) = X(2)X(4118)∩X(6,4475)

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

X(20274) lies on these lines: {2, 4118}, {6, 4475}, {75, 18208}, {86, 18190}, {244, 1953}, {894, 18168}, {982, 17445}, {1921, 18069}, {1964, 7146}, {2643, 4000}, {3248, 18161}, {3758, 18207}, {3763, 7237}, {4657, 17470}, {5272, 18713}, {17063, 17472}, {20271, 20275}

X(20275) = (X(1), X(2), X(4), X(6); X(20271), X(17063), X(20273), X(20274)) COLLINEATION IMAGE OF X(9)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1, 20271), (2,17063), (3,20272), (6,20274), (9,20275), (57,20276)

Nine more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -bc : b(a - c) : c(a - b)
m(-a : b : c) = -a (b^3 + a b c - b^2 c - b c^2 + c^3) : b (a^3 - a^2 c + a b c + a c^2 - c^3) : c (-a^3 + a^2 b - a b^2 + b^3 - a b c)
m(a : b cos C : c cos B) = a (a^2 b^2 + b^4 - 4 a^2 b c + a^2 c^2 - 2 b^2 c^2 + c^4) : b (2 a^4 - 4 a^3 c + a^2 c^2 - b^2 c^2 + c^4) : c (2 a^4 - 4 a^3 b + a^2 b^2 + b^4 - b^2 c^2),

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

If P = x : y : z (barycentrics), then m(P) = a(-b c x + (b - c)(b y - c z)) : : , and m is the collineation indicated by (A',B',C',X(75); m(A'), m(B'), m(C'), X(75), where A' = 0 : c : b, and B' and C' are defined cyclically.

Let f(a,b,c,x,y,z) = b^2 (b-c) c^2 (2 a^10 b^2-3 a^9 b^3+a^8 b^4-4 a^10 b c+3 a^9 b^2 c+3 a^8 b^3 c-2 a^7 b^4 c-a^6 b^5 c+2 a^5 b^6 c-a^4 b^7 c+2 a^10 c^2+3 a^9 b c^2-10 a^8 b^2 c^2+5 a^7 b^3 c^2+3 a^5 b^5 c^2-7 a^4 b^6 c^2+4 a^3 b^7 c^2-3 a^9 c^3+3 a^8 b c^3+5 a^7 b^2 c^3-5 a^6 b^3 c^3+a^5 b^4 c^3-6 a^4 b^5 c^3+12 a^3 b^6 c^3-6 a^2 b^7 c^3+a^8 c^4-2 a^7 b c^4+a^5 b^3 c^4-5 a^4 b^4 c^4+11 a^3 b^5 c^4-12 a^2 b^6 c^4+4 a b^7 c^4-a^6 b c^5+3 a^5 b^2 c^5-6 a^4 b^3 c^5+11 a^3 b^4 c^5-12 a^2 b^5 c^5+6 a b^6 c^5-b^7 c^5+2 a^5 b c^6-7 a^4 b^2 c^6+12 a^3 b^3 c^6-12 a^2 b^4 c^6+6 a b^5 c^6-b^6 c^6-a^4 b c^7+4 a^3 b^2 c^7-6 a^2 b^3 c^7+4 a b^4 c^7-b^5 c^7) x^3-a^2 c (-2 a^5 b^9+3 a^4 b^10+3 a^8 b^5 c-4 a^7 b^6 c+a^6 b^7 c+3 a^5 b^8 c+5 a^4 b^9 c-12 a^3 b^10 c-12 a^8 b^4 c^2+10 a^7 b^5 c^2+3 a^6 b^6 c^2-15 a^4 b^8 c^2+18 a^2 b^10 c^2+18 a^8 b^3 c^3-10 a^7 b^4 c^3-13 a^6 b^5 c^3+3 a^5 b^6 c^3-2 a^4 b^7 c^3+27 a^3 b^8 c^3-12 a^2 b^9 c^3-12 a b^10 c^3-12 a^8 b^2 c^4+4 a^7 b^3 c^4+17 a^6 b^4 c^4-2 a^5 b^5 c^4+2 a^4 b^6 c^4-4 a^3 b^7 c^4-19 a^2 b^8 c^4+14 a b^9 c^4+3 b^10 c^4+3 a^8 b c^5+2 a^7 b^2 c^5-12 a^6 b^3 c^5+3 a^5 b^4 c^5+6 a^4 b^5 c^5-13 a^3 b^6 c^5+12 a^2 b^7 c^5+2 a b^8 c^5-5 b^9 c^5-2 a^7 b c^6+4 a^6 b^2 c^6-2 a^5 b^3 c^6+3 a^4 b^4 c^6-6 a^3 b^5 c^6+7 a^2 b^6 c^6-6 a b^7 c^6+2 b^8 c^6+4 a^5 b^2 c^7-14 a^4 b^3 c^7+15 a^3 b^4 c^7-5 a^2 b^5 c^7-a b^6 c^7+b^7 c^7-4 a^4 b^2 c^8+14 a^3 b^3 c^8-18 a^2 b^4 c^8+10 a b^5 c^8-2 b^6 c^8-a^5 c^9+7 a^4 b c^9-16 a^3 b^2 c^9+16 a^2 b^3 c^9-7 a b^4 c^9+b^5 c^9) y^2 z+a^2 b (-a^5 b^9+3 a^8 b^5 c-2 a^7 b^6 c+7 a^4 b^9 c-12 a^8 b^4 c^2+2 a^7 b^5 c^2+4 a^6 b^6 c^2+4 a^5 b^7 c^2-4 a^4 b^8 c^2-16 a^3 b^9 c^2+18 a^8 b^3 c^3+4 a^7 b^4 c^3-12 a^6 b^5 c^3-2 a^5 b^6 c^3-14 a^4 b^7 c^3+14 a^3 b^8 c^3+16 a^2 b^9 c^3-12 a^8 b^2 c^4-10 a^7 b^3 c^4+17 a^6 b^4 c^4+3 a^5 b^5 c^4+3 a^4 b^6 c^4+15 a^3 b^7 c^4-18 a^2 b^8 c^4-7 a b^9 c^4+3 a^8 b c^5+10 a^7 b^2 c^5-13 a^6 b^3 c^5-2 a^5 b^4 c^5+6 a^4 b^5 c^5-6 a^3 b^6 c^5-5 a^2 b^7 c^5+10 a b^8 c^5+b^9 c^5-4 a^7 b c^6+3 a^6 b^2 c^6+3 a^5 b^3 c^6+2 a^4 b^4 c^6-13 a^3 b^5 c^6+7 a^2 b^6 c^6-a b^7 c^6-2 b^8 c^6+a^6 b c^7-2 a^4 b^3 c^7-4 a^3 b^4 c^7+12 a^2 b^5 c^7-6 a b^6 c^7+b^7 c^7+3 a^5 b c^8-15 a^4 b^2 c^8+27 a^3 b^3 c^8-19 a^2 b^4 c^8+2 a b^5 c^8+2 b^6 c^8-2 a^5 c^9+5 a^4 b c^9-12 a^2 b^3 c^9+14 a b^4 c^9-5 b^5 c^9+3 a^4 c^10-12 a^3 b c^10+18 a^2 b^2 c^10-12 a b^3 c^10+3 b^4 c^10) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) - 2 a (a-b) b (a-c) (b-c) c (a^7 b^4+a^4 b^7-6 a^7 b^3 c-a^6 b^4 c-a^5 b^5 c-a^4 b^6 c-6 a^3 b^7 c+10 a^7 b^2 c^2+a^6 b^3 c^2-a^5 b^4 c^2-a^4 b^5 c^2+a^3 b^6 c^2+10 a^2 b^7 c^2-6 a^7 b c^3+a^6 b^2 c^3+6 a^5 b^3 c^3+2 a^4 b^4 c^3+6 a^3 b^5 c^3+a^2 b^6 c^3-6 a b^7 c^3+a^7 c^4-a^6 b c^4-a^5 b^2 c^4+2 a^4 b^3 c^4+2 a^3 b^4 c^4-a^2 b^5 c^4-a b^6 c^4+b^7 c^4-a^5 b c^5-a^4 b^2 c^5+6 a^3 b^3 c^5-a^2 b^4 c^5-a b^5 c^5-a^4 b c^6+a^3 b^2 c^6+a^2 b^3 c^6-a b^4 c^6+a^4 c^7-6 a^3 b c^7+10 a^2 b^2 c^7-6 a b^3 c^7+b^4 c^7) x y z = 0. (Peter Moses, July 31, 2018)

X(20275) lies on these lines: {1, 16422}, {2, 17447}, {9, 3675}, {244, 17452}, {4516, 4859}, {16604, 20273}, {17063, 20254}, {17278, 17463}, {20271, 20274}

X(20276) = (X(1), X(2), X(4), X(6); X(20271), X(17063), X(20273), X(20274)) COLLINEATION IMAGE OF X(57)

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

See X(20275).

X(20276) lies on these lines: {17063, 20254}, {20271, 20272}

X(20277) = X(1)X(4)^∩X(2)X(14544)

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

X(20277) lies on these lines: {1, 4}, {2, 14544}, {3, 7100}, {6, 18675}, {31, 8758}, {48, 354}, {55, 6611}, {65, 7114}, {77, 1040}, {184, 18210}, {201, 7078}, {212, 1214}, {222, 7004}, {603, 17102}, {614, 3554}, {836, 3720}, {912, 18477}, {940, 8766}, {1001, 6508}, {1062, 4303}, {1473, 3942}, {1754, 18593}, {1818, 6505}, {1836, 4336}, {1899, 4466}, {1936, 17080}, {2187, 3827}, {2188, 17603}, {2286, 17599}, {2658, 3924}, {3474, 5018}, {3955, 20254}, {4224, 18161}, {4332, 7138}, {5311, 17718}, {5452, 9502}, {6357, 8727}, {20278, 20280}

X(20278) = X(1)X(5136)^∩X(48)X(3721)

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

X(20278) lies on these lines: {1, 5136}, {48, 3721}, {20277, 20280}

X(20279) = (name pending)

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

X(20279) lies on these lines: {1, 20281}

X(20280) = (X(1), X(2), X(3), X(6); X(1), X(20277), X(20278), X(20279)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1, 1), (2,20277), (3,20278), (4,20280), (6,20279), (9,20281), (57,20282)

Nine more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a^3 (a^2 - b^2 - c^2) : b(a^2 - c^2)(a^2 - b^2 + c^2) : c(a^2 - b^2)(a^2 + b^2 - c^2)
m(-a : b : c) = a (a^2 - b^2 - c^2) (a^4 + b^4 - 2 b^2 c^2 + c^4) : -b (-a^2 + b^2 - c^2) (a^4 + b^4 - c^4) : -c (-a^2 - b^2 + c^2) (a^4 - b^4 + c^4)
m(a : b cos C : c cos B) = a (a^2 b^2 + b^4 - 4 a^2 b c + a^2 c^2 - 2 b^2 c^2 + c^4) : b (2 a^4 - 4 a^3 c + a^2 c^2 - b^2 c^2 + c^4) : c (2 a^4 - 4 a^3 b + a^2 b^2 + b^4 - b^2 c^2)

where

-a : b : c = A-excenter
a : b cos C : c cos B = 2 a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2 = A-vertex of half-altitude triangle

If P = x : y : z (barycentrics), then m(P) = a(a^2 - b^2 - c^2)(a^3 x - (b^2 - c^2)(b y - c z)) : : , and m is the collineation indicated by (A',B',C',X(1); m(A'), m(B'), m(C'), X(1)), where A' = 0 : c : b, and B' and C' are defined cyclically, and m(A') = 0 : b : c..

Let f(a,b,c,x,y,z) = a b (b-c) c (a^2-b^2-c^2) (a^5-a^3 b^2-a^2 b^3+b^5-a^3 b c-a^2 b^2 c+a b^3 c+b^4 c-a^3 c^2-a^2 b c^2+2 a b^2 c^2-a^2 c^3+a b c^3+b c^4+c^5) x^3-a (a^10-2 a^8 b^2+a^6 b^4-a^5 b^5+2 a^3 b^7-a b^9+2 a^8 b c-2 a^6 b^3 c-2 a^2 b^7 c+2 b^9 c-a^8 c^2+4 a^6 b^2 c^2+a^5 b^3 c^2-3 a^4 b^4 c^2-4 a^3 b^5 c^2+3 a b^7 c^2-2 a^6 b c^3+3 a^4 b^3 c^3+4 a^2 b^5 c^3-5 b^7 c^3+4 a^3 b^3 c^4-a^2 b^4 c^4-a b^5 c^4-2 b^6 c^4-2 a^2 b^3 c^5+4 b^5 c^5-2 a^3 b c^6+2 a^2 b^2 c^6-3 a b^3 c^6+5 b^4 c^6-b^3 c^7-a^2 c^8+2 a b c^8-4 b^2 c^8+c^10) y^2 z+a (a^10-a^8 b^2-a^2 b^8+b^10+2 a^8 b c-2 a^6 b^3 c-2 a^3 b^6 c+2 a b^8 c-2 a^8 c^2+4 a^6 b^2 c^2+2 a^2 b^6 c^2-4 b^8 c^2-2 a^6 b c^3+a^5 b^2 c^3+3 a^4 b^3 c^3+4 a^3 b^4 c^3-2 a^2 b^5 c^3-3 a b^6 c^3-b^7 c^3+a^6 c^4-3 a^4 b^2 c^4-a^2 b^4 c^4+5 b^6 c^4-a^5 c^5-4 a^3 b^2 c^5+4 a^2 b^3 c^5-a b^4 c^5+4 b^5 c^5-2 b^4 c^6+2 a^3 c^7-2 a^2 b c^7+3 a b^2 c^7-5 b^3 c^7-a c^9+2 b c^9) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (a^8+a^7 b-a^6 b^2-a^5 b^3-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+2 a^3 b^3 c^2+a^2 b^4 c^2-a b^5 c^2-b^6 c^2-a^5 c^3-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^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-b^2 c^6+a c^7+b c^7+c^8) x y z = 0. (Peter Moses, July 31, 2018)

X(20280) lies on these lines: {1, 406}, {66, 73}, {3057, 3938}, {20277, 20278}

X(20281) = (X(1), X(2), X(3), X(6); X(1), X(20277), X(20278), X(20279)) COLLINEATION IMAGE OF X(9)

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

See X(20280).

X(20281) lies on these lines: {1, 20279}, {48, 16973}, {66, 73}, {836, 3720}

X(20282) = (X(1), X(2), X(3), X(6); X(1), X(20277), X(20278), X(20279)) COLLINEATION IMAGE OF X(57)

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

See X(20280).

X(20282) lies on these lines: {1, 5136}, {836, 3720}

X(20283) = (name pending)

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

X(20283) lies on these lines: {1, 20285}, {144, 145}, {3231, 10987}

X(20284) = X(1)X(893)∩X(2)X(37)

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

X(20284) lies on these lines: {1, 893}, {2, 37}, {31, 19561}, {39, 17591}, {42, 19586}, {43, 6377}, {48, 1613}, {55, 3009}, {57, 292}, {172, 11328}, {237, 2352}, {694, 1469}, {982, 2275}, {1403, 2176}, {1740, 8844}, {1908, 17716}, {2056, 12835}, {2229, 17155}, {3051, 5332}, {3056, 3116}, {3121, 3873}, {3231, 10987}, {3662, 18905}, {3705, 18904}, {4116, 8619}, {6384, 19565}, {7075, 17475}, {17149, 19581}

X(20284) = isogonal conjugate of isotomic conjugate of X(33890)

X(20285) = (X(1), X(2), X(3), X(6); X(192), X(1), X(20283), X(20284)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,192), (2,1), (3,20283), (4,20285), (6,20285), (9,20286), (57,20287)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a b c (a b + a c - b c) : b^2 (a - c)(a b - a c + b c) : c^2 (a - b)( a c - a b + b c)
m(-a : b : c) = a b c (a b + a c - b c) : b (2 a b - a c - 2 b c) (a b - a c + b c) : c (-a b + 2 a c - 2 b c) (-a b + a c + b c)
m(a : b cos C : c cos B) = a^2 (a b + a c - b c) (a^2 b^2 - b^4 - 2 a^2 b c + a^2 c^2 - c^4) : -b (a b - a c + b c) (-a^3 b^2 + a b^4 + 3 a^2 b^2 c - b^4 c + a^3 c^2 - 2 a b^2 c^2 + b^2 c^3 - a c^4) : -c (-a b + a c + b c) (a^3 b^2 - a b^4 - a^3 c^2 + 3 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a c^4 - b c^4)
m(A') = a (a b + a c - b c) (2 a b^2 - 3 a b c + b^2 c + 2 a c^2 + b c^2), b (a b - a c + b c) (2 a b^2 + a^2 c + a b c - 2 b^2 c - a c^2), c (-a b + a c + b c) (a^2 b - a b^2 + a b c + 2 a c^2 - 2 b c^2)
m(A'') = a (a b + b^2 - 2 a c - b c) (a b + a c - b c) (2 a b - a c + b c - c^2) : b (a b - a c + b c) (2 a^2 b^2 + 2 a b^3 - a^3 c - 3 a b^2 c - 2 b^3 c - 2 a^2 c^2 + 2 b^2 c^2 - a c^3) : c (-a b + a c + b c) (-a^3 b - 2 a^2 b^2 - a b^3 + 2 a^2 c^2 - 3 a b c^2 + 2 b^2 c^2 + 2 a c^3 - 2 b c^3)

where

If P = x : y : z (barycentrics), then m(P) = a (a b + a c - b c)(b^2 c^2 x + a^2 (b - c)(c y - b z)), and m is the collineation indicated by (A',B',C',X(2); m(A'), m(B'), m(C'), X(1)).

Let f(a,b,c,x,y,z) = b^3 (b-c) c^3 (a b+a c-b c) (a^2-a b-a c+2 b c) x^3+a^3 c (a^3 b^4-5 a^3 b^3 c+2 a^2 b^4 c+6 a^3 b^2 c^2-a^2 b^3 c^2-2 a b^4 c^2-4 a^3 b c^3-2 a^2 b^2 c^3+5 a b^3 c^3-b^4 c^3+2 a^3 c^4-2 a b^2 c^4) y^2 z-a^3 b (2 a^3 b^4-4 a^3 b^3 c+6 a^3 b^2 c^2-2 a^2 b^3 c^2-2 a b^4 c^2-5 a^3 b c^3-a^2 b^2 c^3+5 a b^3 c^3+a^3 c^4+2 a^2 b c^4-2 a b^2 c^4-b^3 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 4 a^2 (a-b) b^2 (a-c) (b-c) c^2 (a b+a c+b c) x y z = 0. (Peter Moses, July 31, 2018)

X(20285) lies on these lines: {1, 20283}, {192, 3100}

X(20286) = (X(1), X(2), X(3), X(6); X(192), X(1), X(20283), X(20284)) COLLINEATION IMAGE OF X(9)

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

See X(20285).

X(20286) lies on these lines: {1, 9315}, {2, 37}

X(20287) = (X(1), X(2), X(3), X(6); X(192), X(1), X(20283), X(20284)) COLLINEATION IMAGE OF X(57)

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

See X(20285).

X(20287) lies on these lines: {1, 9315}, {144, 145}

X(20288) = X(11)X(8261)∩X(496)X(2486)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27935.

X(20288 ) lies on these lines: {11, 8261}, {496, 2486}, {758, 9955}, {950, 3838}, {2475, 11376}

X(20289) = (name pending)

Barycentrics 2 a^5 + 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(20289) lies on these lines: {4, 916}, {7, 3585}, {10, 20291}, {307, 18661}, {515, 17221}, {1441, 18480}, {1826, 14543}, {2893, 5080}, {4566, 7282}, {5229, 5738}, {5691, 17134}, {5736, 10895}, {5740, 7354}, {14953, 20305}, {18650, 19925}

X(20290) = (name pending)

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

X(20290) lies on these lines: {2, 2308}, {8, 3901}, {69, 674}, {306, 4427}, {319, 17163}, {320, 17140}, {321, 17491}, {524, 4972}, {1330, 2392}, {2887, 16704}, {2895, 4645}, {3006, 4001}, {3416, 17165}, {3448, 20351}, {3578, 3925}, {3873, 17361}, {3914, 17162}, {3936, 6690}, {3969, 17768}, {3995, 4683}, {4514, 17145}, {4655, 17147}, {4660, 20011}, {4741, 7226}, {4981, 17344}, {5284, 17297}, {5847, 17150}, {7191, 17288}, {7768, 20556}, {8050, 20021}, {15523, 17770}, {17137, 20352}

X(20291) = (X(1), X(2), X(3), X(6); X(319), X(10), X(20289), X(20290)) COLLINEATION IMAGE OF X(4)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,319), (2,10), (3,20289), (4,20291), (6,20290), (9,20292), (57,20320)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(-a : b : c) = a^2 + b^2 + c^2 + a b + c a + a b : -a^2 - a b - b^2 + a^2 : - a^2 - a c - c^2 + b^2
m(a : b cos C : c cos B) = 2 a^3 - a^2 b - b^3 - a^2 c + b^2 c + b c^2 - c^3 : -(4 a^3 + a^2 b - 2 a b^2 - 3 b^3 + 3 a^2 c - 3 b^2 c + 2 a c^2 + 3 b c^2 + 3 c^3) : a^2 (4 a^3 + 3 a^2 b + 2 a b^2 + 3 b^3 + a^2 c + 3 b^2 c - 2 a c^2 - 3 b c^2 - 3 c^3)
m(A') = a^2 (a b + a c - b c) (a^2 b^2 - b^4 - 2 a^2 b c + a^2 c^2 - c^4) : -b (a b - a c + b c) (-a^3 b^2 + a b^4 + 3 a^2 b^2 c - b^4 c + a^3 c^2 - 2 a b^2 c^2 + b^2 c^3 - a c^4) : -c (-a b + a c + b c) (a^3 b^2 - a b^4 - a^3 c^2 + 3 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a c^4 - b c^4)
m(A') = a^3 + a^2 b - a b^2 - b^3 + a^2 c + a b c + b^2 c - a c^2 + b c^2 - c^3 : -a^3 - a^2 b + a b^2 + b^3 - 2 a^2 c - a b c - 2 a c^2 - c^3 : -a^3 - 2 a^2 b - 2 a b^2 - b^3 - a^2 c - a b c + a c^2 + c^3
m(A'') = a^4 - 2 a^2 b^2 + b^4 - a^2 b c - 3 a b^2 c - 2 a^2 c^2 - 3 a b c^2 - 2 b^2 c^2 + c^4 : -a^4 + 2 a^2 b^2 - b^4 - a^3 c - 2 a^2 b c - b^3 c - a b c^2 + a c^3 + b c^3 + c^4 : -a^4 - a^3 b + a b^3 + b^4 - 2 a^2 b c - a b^2 c + b^3 c + 2 a^2 c^2 - b c^3 - c^4

where

If P = x : y : z (barycentrics), then m(P) = (2a + b + c)x - (a + 2b + c)y - (a + b + 2c)z, and m is the collineation indicated by (A',B',C',X(2); m(A'), m(B'), m(C'), X(10)).

Let f(a,b,c,x,y,z) = a^2 (b-c) (2 a+b+c)^2 x^3+(a+2 b+c) (4 a^4+4 a^3 b+a^2 b^2+a b^3+2 b^4+4 a^3 c+2 a^2 b c+3 a b^2 c+7 b^3 c-2 a b c^2+2 b^2 c^2-4 a c^3-4 b c^3-4 c^4) y^2 z-(a+b+2 c) (4 a^4+4 a^3 b-4 a b^3-4 b^4+4 a^3 c+2 a^2 b c-2 a b^2 c-4 b^3 c+a^2 c^2+3 a b c^2+2 b^2 c^2+a c^3+7 b c^3+2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 4 a^2 (a-b) b^2 (a-c) (b-c) c^2 (a b+a c+b c) x y z = 0. (Peter Moses, July 31, 2018)

X(20291) lies on these lines: {1, 7}, {2, 1839}, {10, 20289}, {22, 8053}, {30, 1441}, {71, 1654}, {74, 1305}, {86, 15320}, {319, 11684}, {674, 12220}, {916, 11412}, {1155, 5740}, {1836, 5736}, {2772, 12219}, {2897, 3648}, {3474, 5738}, {6284, 17863}, {8804, 14543}, {9028, 20017}, {9961, 12111}, {14953, 18589}

X(20292) = (X(1), X(2), X(3), X(6); X(319), X(10), X(20289), X(20290)) COLLINEATION IMAGE OF X(9)

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

See X(20291).

X(20292) lies on these lines: {1, 11015}, {2, 1155}, {4, 9961}, {7, 3434}, {8, 4018}, {10, 79}, {21, 1770}, {31, 17889}, {35, 11263}, {46, 2476}, {57, 11680}, {63, 4312}, {65, 2475}, {72, 14450}, {75, 6327}, {80, 3919}, {81, 3914}, {86, 15320}, {100, 226}, {142, 5284}, {145, 10404}, {149, 354}, {171, 3120}, {191, 3841}, {210, 17484}, {319, 17163}, {320, 17135}, {321, 4645}, {376, 3616}, {377, 3869}, {388, 7702}, {392, 5180}, {404, 12047}, {443, 11415}, {484, 3822}, {516, 1621}, {517, 6951}, {518, 17483}, {528, 3957}, {693, 7196}, {750, 3944}, {758, 11552}, {894, 4972}, {908, 20103}, {946, 5253}, {956, 18541}, {962, 3890}, {1086, 7191}, {1111, 17884}, {1125, 17549}, {1158, 6828}, {1441, 16091}, {1478, 5176}, {1633, 4228}, {1699, 3306}, {1709, 10883}, {1737, 17577}, {1788, 6871}, {1999, 4442}, {2185, 5196}, {2550, 3681}, {2886, 3218}, {2887, 4418}, {2895, 3696}, {2975, 4292}, {3091, 14647}, {3219, 3925}, {3286, 17173}, {3452, 9342}, {3475, 20075}, {3485, 4190}, {3579, 16159}, {3583, 5883}, {3585, 3754}, {3622, 12701}, {3685, 18139}, {3753, 5080}, {3757, 4450}, {3772, 17126}, {3782, 3920}, {3812, 5046}, {3832, 12679}, {3868, 5178}, {3870, 4654}, {3871, 13407}, {3896, 17778}, {3897, 4299}, {3982, 5853}, {4004, 18480}, {4188, 11375}, {4197, 4338}, {4307, 19785}, {4359, 4388}, {4363, 4799}, {4415, 5297}, {4420, 11544}, {4430, 4863}, {4511, 11112}, {4514, 7321}, {4651, 17491}, {4666, 6173}, {4854, 17019}, {4860, 11235}, {4861, 18990}, {4865, 17155}, {4881, 15950}, {4911, 20556}, {4981, 6646}, {5229, 5554}, {5263, 17184}, {5302, 20084}, {5325, 10032}, {5439, 9782}, {5552, 5714}, {5603, 6948}, {5686, 20214}, {5734, 12700}, {5805, 9776}, {5812, 10585}, {5832, 9965}, {5836, 20060}, {5842, 18444}, {5886, 6950}, {5887, 6901}, {5927, 9809}, {6001, 6839}, {6894, 12688}, {6895, 9943}, {6915, 12608}, {6945, 12686}, {7226, 17276}, {7247, 20244}, {7270, 17164}, {7548, 12616}, {9579, 19860}, {9612, 11681}, {9779, 10584}, {10273, 12247}, {10572, 16154}, {10707, 11019}, {11281, 15338}, {12432, 16120}, {15679, 16152}, {17150, 19796}

X(20292) = anticomplement of X(3683)

X(20293) = (name pending)

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

X(20293) lies on these lines: {2, 1459}, {8, 522}, {69, 3261}, {340, 520}, {391, 657}, {513, 4397}, {514, 16086}, {521, 1948}, {656, 17496}, {693, 20297}, {834, 20295}, {966, 6586}, {2517, 9001}, {3699, 8050}, {3738, 4086}, {3762, 6003}, {3900, 4811}, {3907, 17420}, {3945, 17215}, {4036, 17751}, {4147, 17418}, {4163, 4778}, {4791, 10449}, {4963, 4977}, {7649, 9031}

X(20293) = anticomplement of X(1459)

X(20294) = ANTICOMPLEMENT OF X(7649)

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

X(20294) lies on these lines: {2, 7649}, {20, 3667}, {22, 4057}, {280, 18220}, {325, 523}, {513, 20296}, {514, 16086}, {521, 3904}, {522, 663}, {953, 2370}, {1305, 4561}, {3091, 16231}, {3699, 14513}, {3810, 17420}, {4571, 13397}, {4811, 6362}, {6129, 16757}, {14429, 20316}, {20295, 20298}

X(20294) = anticomplement of X(7649)
X(20294) = pole wrt de Longchamps circle of Nagel line

X(20295) = (name pending)

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

X(20295) lies on these lines: {2, 649}, {7, 3676}, {69, 9002}, {316, 512}, {320, 350}, {329, 4468}, {514, 4024}, {522, 17161}, {523, 4810}, {650, 4380}, {659, 4806}, {661, 812}, {786, 4826}, {788, 17135}, {802, 4502}, {804, 8663}, {830, 4170}, {834, 20293}, {885, 2520}, {900, 3004}, {901, 4998}, {1019, 17174}, {1836, 8049}, {1978, 8050}, {2786, 16892}, {3261, 17159}, {3309, 17896}, {3667, 4025}, {3837, 4784}, {4063, 4129}, {4367, 4992}, {4369, 4728}, {4379, 4932}, {4425, 17193}, {4453, 4897}, {4462, 8712}, {4521, 18228}, {4560, 14349}, {4651, 9400}, {4786, 7658}, {4790, 4885}, {4927, 4943}, {4978, 15309}, {6002, 17496}, {6327, 9313}, {6545, 17483}, {8025, 18200}, {9778, 15599}, {20294, 20298}

X(20296) = (X(1), X(3), X(4), X(6); X(513), X(20293), X(20294), X(20295)) COLLINEATION IMAGE OF X(223)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,513), (3,20293), (4,20294), (6,20295), (223,20296), (282,20297), (1249,20298)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(-a : b : c) = 0 : (b - a) c: (a - c) b
m(a : b cos C : c cos B) = (3 a - b - c) (b - c) (a + b + c) : -(a + b - c) (2 a^2 - a b + b^2 - a c - c^2) : (a - b + c) (2 a^2 - a b - b^2 - a c + c^2)
m(A') = (b - c) (-a^2 - a b - a c + b c) : a b (a + b - 2 c) : -a c (a - 2 b + c)
m(A'') = (b - c) (a^3 - a b^2 + a b c + b^2 c - a c^2 + b c^2) : -a b (a^2 - b^2 + a c + b c - 2 c^2) : a c (a^2 + a b - 2 b^2 + b c - c^2)

where

If P = x : y : z (barycentrics), then m(P) = (b - c)x - (c - a)y - (a - b)z, and m is the collineation indicated by (A',B',C',X(2); m(A'), m(B'), m(C'), X(514)). Note that the points on the line X(2)X(7) are not in the domain of m.

Let f(a,b,c,x,y,z) = a^2 (2 a-b-c) (b-c)^2 x^3-b (a-c)^2 (a b-3 b^2+2 a c) y^2 z-(a-b)^2 c (2 a b+a c-3 c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a^3 b^2+a^2 b^3-4 a^3 b c+2 a^2 b^2 c-4 a b^3 c+a^3 c^2+2 a^2 b c^2+2 a b^2 c^2+b^3 c^2+a^2 c^3-4 a b c^3+b^2 c^3) x y z = 0. (Peter Moses, July 31, 2018)

X(20296) lies on these lines: {2, 20318}, {63, 905}, {144, 17496}, {329, 4391}, {513, 20294}, {514, 20297}, {1332, 3257}, {1734, 12526}, {3869, 3900}, {4131, 6332}, {4462, 8712}

X(20297) = (X(1), X(3), X(4), X(6); X(513), X(20293), X(20294), X(20295)) COLLINEATION IMAGE OF X(282)

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

See X(20296).

X(20297) lies on these lines: {2, 20314}, {513, 20298}, {514, 20296}, {521, 4025}, {693, 20293}

X(20298) = (X(1), X(3), X(4), X(6); X(513), X(20293), X(20294), X(20295)) COLLINEATION IMAGE OF X(1249)

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

See X(20296).

X(20298) lies on these lines: {2, 20319}, {20, 4025}, {513, 20297}, {652, 3101}, {901, 1305}, {20294, 20295}

X(20299) = COMPLEMENT OF X(6759)

Barycentrics (b^4-4*b^2*c^2+c^4)*a^6-3*(b^ 4-c^4)*(b^2-c^2)*a^4+(b^2-c^2) ^2*(3*b^4+4*b^2*c^2+3*c^4)*a^ 2-(b^4-c^4)*(b^2-c^2)^3 : :
X(20299) = 3*X(2)+X(14216), X(3)+3*X(1853), 2*X(3)-3*X(10193), 5*X(3)-X(17845), 3*X(5)-X(2883), X(20)-3*X(11204), 2*X(1209)-3*X(14076), 2*X(1853)+X(10193), 15*X(1853)+X(17845), 3*X(1853)-X(18381), X(2883)+3*X(6247), 2*X(6696)+X(18383), 15*X(10193)-2*X(17845), 3*X(10193)+2*X(18381), X(17845)+5*X(18381)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

X(20299) lies on these lines: {2, 6759}, {3, 161}, {4, 74}, {5, 2883}, {6, 19361}, {20, 11204}, {24, 11550}, {30, 5449}, {51, 15559}, {64, 381}, {66, 182}, {68, 13346}, {113, 7729}, {122, 14059}, {140, 1503}, {143, 2781}, {154, 3526}, {184, 11457}, {185, 1594}, {195, 17823}, {235, 13474}, {265, 13293}, {343, 15644}, {378, 13403}, {382, 10606}, {389, 427}, {403, 11381}, {468, 16655}, {511, 12235}, {523, 6662}, {539, 18356}, {542, 1147}, {546, 15311}, {576, 18951}, {578, 1899}, {631, 9833}, {632, 10192}, {858, 5562}, {1092, 11442}, {1116, 20184}, {1181, 5094}, {1192, 18494}, {1352, 3546}, {1368, 11793}, {1495, 10018}, {1498, 1656}, {1514, 10019}, {1568, 12111}, {1593, 15121}, {1595, 10110}, {1614, 6143}, {1657, 8567}, {1658, 20191}, {1971, 7749}, {2072, 12162}, {2393, 5447}, {2818, 15666}, {2979, 12226}, {3090, 12324}, {3091, 5878}, {3153, 11440}, {3484, 6801}, {3516, 18396}, {3525, 11206}, {3527, 16623}, {3545, 6225}, {3548, 9306}, {3574, 5890}, {3627, 5894}, {3628, 16252}, {3818, 6642}, {3830, 5925}, {3832, 12250}, {3841, 6001}, {3843, 5895}, {3850, 5893}, {3851, 13093}, {3858, 15105}, {5012, 10274}, {5054, 17821}, {5055, 12315}, {5056, 5656}, {5169, 15043}, {5448, 5663}, {5462, 19130}, {5576, 9730}, {5627, 13489}, {5907, 11585}, {5943, 7403}, {5965, 16266}, {5972, 6640}, {6102, 10115}, {6146, 11430}, {6240, 11572}, {6241, 7577}, {6285, 7741}, {6958, 14925}, {7355, 7951}, {7399, 16836}, {7507, 10605}, {7525, 15578}, {7552, 8718}, {7564, 7706}, {7592, 12242}, {7689, 18569}, {7703, 10574}, {7973, 18493}, {8889, 18909}, {9927, 12084}, {9934, 15059}, {10024, 10575}, {10060, 10896}, {10076, 10895}, {10112, 13352}, {10113, 11598}, {10114, 15463}, {10257, 12134}, {10576, 12964}, {10577, 12970}, {10675, 16966}, {10676, 16967}, {11250, 17702}, {11424, 18912}, {11455, 11704}, {11745, 16198}, {12085, 14852}, {12106, 15579}, {12233, 13382}, {12234, 15135}, {12262, 18480}, {12278, 16163}, {12290, 16868}, {13289, 15061}, {13371, 13754}, {13665, 19087}, {13785, 19088}, {13851, 18560}, {14157, 14940}, {14915, 15761}, {15113, 16534}, {15805, 19149}, {16111, 18430}, {17814, 17822}

X(20299) = midpoint of X(i) and X(j) for these {i,j}: {3, 18381}, {4, 3357}, {5, 6247}, {66, 182}, {68, 13346}, {265, 13293}, {3627, 5894}, {7689, 18569}, {9927, 12084}, {10113, 11598}, {12262, 18480}
X(20299) = reflection of X(i) in X(j) for these (i,j): (1498, 14862), (1658, 20191), (5448, 10224)
X(20299) = complement of X(6759)
X(20299) = complementary conjugate of X(14363)
X(20299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14216, 6759), (3, 1853, 18381), (24, 11550, 13419), (140, 10282, 10182), (185, 1594, 18388), (631, 9833, 11202), (1595, 13567, 10110), (1899, 3541, 578), (6640, 10539, 5972), (8567, 18405, 1657), (10018, 16659, 1495), (10255, 18439, 113)

X(20300) = COMPLEMENT OF X(15577)

Barycentrics (R^2*(SA-5*SW)+SW^2)*S^2-(2*R^ 2-SW)*SB*SC*SW : :
X(20300) = X(66)+3*X(14561), X(159)-5*X(1656), 2*X(546)+X(15579), 6*X(547)-X(15580), X(1353)-3*X(10169), 3*X(1853)+X(19149), 7*X(3090)-X(15581), 4*X(3628)-X(15582), X(8549)+3*X(10516), 3*X(11216)+X(11898)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

X(20300) lies on these lines: {2, 161}, {3, 18382}, {5, 182}, {6, 70}, {30, 15578}, {51, 125}, {66, 5576}, {141, 1209}, {159, 1656}, {403, 10249}, {511, 5449}, {546, 15579}, {547, 15580}, {858, 1350}, {1352, 2072}, {1353, 10169}, {1595, 6696}, {1853, 5133}, {1907, 5894}, {3090, 15581}, {3564, 10224}, {3628, 15582}, {3827, 9956}, {3845, 19506}, {5085, 13160}, {5092, 18383}, {5462, 19130}, {6247, 7403}, {6776, 7577}, {7706, 15311}, {8549, 10516}, {10255, 18440}, {11216, 11898}, {14389, 15139}

X(20300) = midpoint of X(i) and X(j) for these {i,j}: {3, 18382}, {5092, 18383}
X(20300)= complement of X(15577)

X(20301) = COMPLEMENT OF X(12584)

Barycentrics (3*R^2*(3*SA-8*SW)+4*SW^2)*S^ 2-(9*R^2-4*SW)*SB*SC*SW : :
X(20301) = X(67)-5*X(15027), 3*X(381)+X(16010), X(382)+3*X(5621), X(576)+5*X(15027), X(1352)-5*X(15081), 5*X(1656)-X(2930), X(3098)-3*X(15061), X(3448)+3*X(14561), X(3818)-3*X(14644), 3*X(5085)+X(12902), 3*X(5476)-X(9970), 3*X(9140)+X(9970), X(9976)+5*X(15081), X(11579)+3*X(14644), 3*X(14561)-X(19140)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

X(20301) lies on these lines: {2, 12584}, {5, 542}, {6, 7579}, {67, 576}, {125, 511}, {143, 2781}, {182, 265}, {381, 16010}, {382, 5621}, {389, 16003}, {1177, 18381}, {1209, 5181}, {1352, 9976}, {1503, 11801}, {1594, 5095}, {1656, 2930}, {2836, 9956}, {2854, 16511}, {3098, 15061}, {3448, 14561}, {3818, 11579}, {5085, 12902}, {5092, 17702}, {5169, 5476}, {5449, 6698}, {5480, 10264}, {5663, 19130}, {5965, 11804}, {6034, 15545}, {6699, 14810}, {8262, 11649}, {8681, 15123}, {11061, 18912}, {11482, 16176}, {11645, 11799}, {12121, 17508}, {14763, 15516}, {14789, 15462}

X(20301) = midpoint of X(i) and X(j) for these {i,j}: {67, 576}, {182, 265}, {1177, 18381}, {1352, 9976}, {3448, 19140}, {3818, 11579}, {5480, 10264}
X(20301) = complement of X(12584)
X(20301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3448, 14561, 19140), (11579, 14644, 3818)

X(20302) = COMPLEMENT OF X(9932)

Barycentrics SA*(2*(SA-3*R^2)*S^2-(SB+SC)*( 10*R^4-R^2*(7*SA+6*SW)+2*SA^2- 2*SB*SC+SW^2)) : :
X(20302) = 3*X(381)+X(12301), 5*X(1656)-X(9937), 7*X(3090)+X(12318), 9*X(5055)-X(12309), 3*X(14852)+X(15316)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

In the plane of a triangle ABC, let
A'B'C' = Kiepert triangle with base angle π/4
Oa = circle with center A' and pass-through points B and C
Ab = point of intersection, other than C, of Oa and AC
Ac = point of intersection, other than B, of Ob and AB
Oab = center of circle {{A,B,Ab}}
Oac = center of circle {{A,B,Ac}}
A2 = BAb ∩A'Oab, and define B2 and C2 cyclically
A3 = CAc ∩A'Oac, and define B3 and C3 cyclically.
The six points A2, A3, B2, B3, C2, C3 lies on a conic whose center is X(20303). A barycentric equation for this conic follows:

(a^8-2 a^4 (3 b^4+2 b^2 c^2+3 c^4)+8 a^2 (b^6+c^6)-(b^2-c^2)^2 (3 b^4+2 b^2 c^2+3 c^4)) x^2 + 2 (a^8-(b^2-c^2)^4-2 a^6 (b^2+c^2)+2 a^2 (b^2-c^2)^2 (b^2+c^2)) y z + (cyclic) = 0.

See X(20303). (Angel Montesdeoca, October 24, 2023)

X(20302) lies on these lines: {2, 9932}, {4, 9938}, {5, 578}, {68, 2072}, {125, 5562}, {155, 1594}, {381, 12301}, {403, 12293}, {427, 12162}, {550, 12901}, {858, 12163}, {1656, 9937}, {2931, 10018}, {3090, 12318}, {3564, 10224}, {5055, 12309}, {5449, 11793}, {5576, 5654}, {7488, 12319}, {7505, 8907}, {7741, 9931}, {7951, 19471}, {10024, 12118}, {10255, 12429}, {10282, 15761}, {10576, 12424}, {10577, 12425}, {10659, 16966}, {10660, 16967}, {13367, 15760}, {13371, 13754}, {14852, 15316}, {18569, 19908}

X(20302) = midpoint of X(i) and X(j) for these {i,j}: {4, 9938}, {18569, 19908}
X(20302) = complement of X(9932)

X(20303) = COMPLEMENT OF X(8907)

Barycentrics SA*(2*(R^2-SA)*S^2+(SB+SC)*(4* R^4-R^2*(3*SA+4*SW)+2*SA^2-2* SB*SC+SW^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

Let A'B'C' be the orthic triangle. X(20303) is the radical center of the tangential circles of AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)

X(20303) lies on these lines: {2, 8907}, {5, 156}, {6, 70}, {52, 125}, {68, 2072}, {161, 7505}, {185, 18488}, {235, 7687}, {343, 1216}, {389, 427}, {858, 17834}, {973, 13567}, {974, 6247}, {1181, 5133}, {1853, 19360}, {1899, 5576}, {2917, 10018}, {3090, 15435}, {5169, 18909}, {6240, 19457}, {6689, 7405}, {10224, 13292}, {10605, 15559}, {11596, 15319}, {11750, 13851}, {13160, 18396}, {14788, 19357}

X(20303) = complement of X(8907)

X(20304) = COMPLEMENT OF X(1511)

Barycentrics (21*R^2-SA-4*SW)*S^2+3*(3*R^2- SW)*SB*SC : :
X(20304) = 3*X(2)+X(265), 9*X(2)-X(12383), 3*X(2)+5*X(15081), 3*X(3)+X(10733), X(3)+3*X(14644), X(3)+11*X(15025), 5*X(3)+7*X(15044), X(3)-5*X(15059), 3*X(265)+X(12383), X(265)-5*X(15081), 3*X(1511)-X(12383), X(1511)+5*X(15081), 3*X(10113)-X(10733), X(10113)-3*X(14644), X(10113)-11*X(15025), 5*X(10113)-7*X(15044), X(10113)+5*X(15059), X(10733)-9*X(14644), X(10733)+15*X(15059), X(12383)+15*X(15081), 3*X(14644)-11*X(15025), 15*X(14644)-7*X(15044), 3*X(14644)+5*X(15059), 11*X(15025)+5*X(15059)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27938.

X(20304) lies on these lines: {2, 265}, {3, 10113}, {4, 12041}, {5, 113}, {6, 13915}, {10, 12261}, {30, 6699}, {67, 14561}, {74, 381}, {110, 1656}, {114, 15535}, {140, 6723}, {141, 14984}, {143, 10224}, {146, 3545}, {147, 14849}, {148, 14850}, {156, 13198}, {382, 15055}, {399, 5055}, {403, 12133}, {468, 12140}, {498, 12904}, {499, 12903}, {511, 6698}, {539, 19481}, {541, 5066}, {542, 547}, {546, 2777}, {549, 16163}, {550, 12295}, {567, 3043}, {568, 12219}, {631, 12121}, {952, 11735}, {1001, 12334}, {1078, 12201}, {1112, 1594}, {1154, 2072}, {1209, 11804}, {1216, 11800}, {1533, 11563}, {1698, 12778}, {1853, 9934}, {1986, 5946}, {2771, 3812}, {2781, 6697}, {2782, 15359}, {2854, 16511}, {2931, 7514}, {3028, 7951}, {3047, 18350}, {3068, 19051}, {3069, 19052}, {3090, 3448}, {3091, 7728}, {3154, 16168}, {3526, 12902}, {3564, 15118}, {3616, 12898}, {3624, 12407}, {3627, 16111}, {3628, 5972}, {3832, 12244}, {3843, 10721}, {3845, 13202}, {3851, 10620}, {3858, 10990}, {5020, 12412}, {5054, 15051}, {5071, 5655}, {5072, 15054}, {5079, 14094}, {5094, 15472}, {5432, 12896}, {5433, 18968}, {5449, 11591}, {5462, 10628}, {5498, 13403}, {5504, 14852}, {5576, 18874}, {5622, 18440}, {5640, 7579}, {5642, 15699}, {5790, 7984}, {5886, 13211}, {5907, 11806}, {5943, 11557}, {5944, 11565}, {6053, 10109}, {6102, 7723}, {6644, 19457}, {7393, 12310}, {7486, 14683}, {7529, 13171}, {7846, 12501}, {7978, 18493}, {8252, 10820}, {8253, 10819}, {8976, 19111}, {9781, 13201}, {9820, 11264}, {9976, 11178}, {10020, 13470}, {10065, 10896}, {10081, 10895}, {10117, 13861}, {10175, 13605}, {10280, 12064}, {10516, 11579}, {10627, 11585}, {10706, 19709}, {10989, 15362}, {11230, 11720}, {11451, 15100}, {11465, 15102}, {11709, 18480}, {12106, 13289}, {12270, 15045}, {12281, 15043}, {12284, 15056}, {12292, 13491}, {12308, 15046}, {12790, 15183}, {12825, 15060}, {13169, 14848}, {13371, 15465}, {13448, 18781}, {13665, 19059}, {13785, 19060}, {13851, 15646}, {13951, 19110}, {14061, 18332}, {14731, 14993}, {15026, 16222}, {15036, 15720}, {15042, 15701}, {15068, 19456}, {15647, 18381}, {18420, 18933}

X(20304) = midpoint of X(i) and X(j) for these {i,j}: {3, 10113}, {4, 12041}, {5, 125}, {10, 12261}, {74, 1539}, {114, 15535}, {140, 11801}, {550, 12295}, {1209, 11804}, {1216, 11800}, {3627, 16111}, {5907, 11806}, {6102, 7723}, {11709, 18480}, {12292, 13491}, {13851, 15646}, {15647, 18381}
X(20304) = reflection of X(i) in X(j) for these (i,j): (5, 15088), (140, 6723), (143, 11746), (1112, 10095), (10627, 13416)
X(20304) = complement of X(1511)
X(20304) = nine-point circle-inverse-of X(10264)
X(20304) = X(1539)-of-Ehrmann mid triangle
X(20304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 265, 1511), (2, 15081, 265), (3, 14644, 10113), (4, 15061, 12041), (5, 10264, 113), (74, 381, 1539), (113, 125, 10264), (547, 10272, 12900), (3090, 15027, 5609), (3448, 14643, 5609), (11561, 13363, 9826), (12099, 12358, 12236), (13915, 13979, 6), (14643, 15027, 3448), (14644, 15059, 3), (15025, 15059, 14644)

X(20305) = (X(1), X(2), X(4), X(6); X(141), X(10), X(18589), X(2887)) COLLINEATION IMAGE OF X(3)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,141), (2,10), (3,20305), (4,18589), (6,2887), (9,2886), (57,1329), (223,20306), (282,20307), (1073,20308), (1249,20309)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = b^2 + c^2 : c^2 - a^2 : b^2 - a^2
m(a : b cos C : c cos B) = (b + c) (a^2 + b^2 - 2 b c + c^2) : 2 a^3 + a^2 c - b^2 c + c^3 : 2 a^3 + a^2 b + b^3 - b c^2
m(A') = a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3 : (a + c) (a^2 + a b - b c + c^2) : (a + b) (a^2 + b^2 + a c - b c)
m(A'') = (a + b + c) (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : (a + b - c) (a + c) (a^2 - a b + b c + c^2) : (a + b) (a - b + c) (a^2 + b^2 - a c + b c)

where

If P = x : y : z (barycentrics), then m(P) = b y + c z : c z + a x : a x + b y = complementary conjugate of ax : by : cz, and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(10)).

Let f(a,b,c,x,y,z) = (a+b) (b-c) (a+c) (a^2-a b+b^2-a c+b c+c^2) x^3+(b+c) (3 a^4+a b^3-a b^2 c+b^3 c+a b c^2-b^2 c^2-a c^3+b c^3) y^2 z-(b+c) (3 a^4-a b^3+a b^2 c+b^3 c-a b c^2-b^2 c^2+a c^3+b c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (a^2+a b+b^2+a c+b c+c^2) x y z = 0. (Peter Moses, July 31, 2018)

X(20305) lies on these lines: {2, 48}, {5, 916}, {10, 4523}, {12, 18635}, {37, 8287}, {63, 18747}, {71, 857}, {92, 18749}, {116, 119}, {141, 1329}, {150, 18162}, {226, 7363}, {307, 1826}, {355, 17073}, {442, 15669}, {952, 17043}, {1441, 4466}, {2260, 5740}, {2886, 20307}, {3740, 17239}, {4019, 4150}, {4648, 10588}, {4657, 17062}, {4920, 18179}, {5587, 18634}, {5788, 15668}, {8062, 9253}, {14953, 20289}, {16580, 16609}, {16732, 16888}, {17047, 20544}, {17181, 18161}, {18357, 18644}

X(20305) = isotomic conjugate of isogonal conjugate of X(23619)
X(20305) = polar conjugate of isogonal conjugate of X(22069)
X(20305) = complement of X(48)

X(20306) = (X(1), X(2), X(4), X(6); X(141), X(10), X(18589), X(2887)) COLLINEATION IMAGE OF X(223)

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

See X(20305).

X(20306) lies on these lines: {2, 221}, {5, 117}, {8, 1854}, {10, 5777}, {30, 10570}, {64, 2550}, {65, 13567}, {141, 960}, {281, 1901}, {343, 3869}, {515, 19904}, {946, 5908}, {958, 1503}, {966, 3197}, {1146, 3959}, {1329, 2390}, {1376, 6696}, {1498, 19843}, {1853, 2551}, {1861, 12688}, {2883, 2886}, {3671, 16608}, {3820, 20299}, {3925, 7355}, {4999, 10192}, {6708, 12609}, {7686, 15873}, {7959, 12324}, {8251, 12514}, {9708, 14216}, {12359, 14988}

X(20306) = complement of X(221)

X(20307) = (X(1), X(2), X(4), X(6); X(141), X(10), X(18589), X(2887)) COLLINEATION IMAGE OF X(282)

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

See X(20305).

X(20307) lies on these lines: {2, 2192}, {10, 5777}, {64, 2551}, {141, 20309}, {200, 223}, {221, 7080}, {860, 1834}, {958, 6696}, {997, 15836}, {1329, 2883}, {1376, 1503}, {1853, 2550}, {1861, 1864}, {2886, 20305}, {3035, 10192}, {3820, 6000}, {5795, 12262}, {6225, 8165}, {9709, 14216}, {11019, 16608}, {15583, 17792}

X(20307) = complement of X(2192)

X(20308) = (X(1), X(2), X(4), X(6); X(141), X(10), X(18589), X(2887)) COLLINEATION IMAGE OF X(1073)

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

See X(20305).

X(20308) lies on these lines: {2, 19614}, {10, 20309}, {1249, 17904}, {2883, 2886}

X(20309) = (X(1), X(2), X(4), X(6); X(141), X(10), X(18589), X(2887)) COLLINEATION IMAGE OF X(1249)

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

See X(20305).

X(20309) lies on these lines: {2, 204}, {3, 20106}, {10, 20308}, {123, 20205}, {141, 20307}, {1329, 1368}, {2887, 18589}

X(20310) = (X(1), X(2), X(4), X(6); X(16583), X(1196), X(39), X(6)) COLLINEATION IMAGE OF X(57)

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

See X(20227).

X(20310) lies on these lines: 1, 5574}, {2, 7182}, {6, 20311}, {9, 171}, {37, 800}, {42, 3119}, {354, 17435}, {756, 8012}, {1196, 20227}, {1200, 2310}, {1212, 3452}, {1864, 20229}, {3767, 3772}, {6181, 17594}

X(20311) = (X(1), X(2), X(4), X(6); X(16583), X(1196), X(39), X(6)) COLLINEATION IMAGE OF X(223)

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

See X(20227).

X(20311) lies on these lines: {6, 20310}, {39, 1212}, {1196, 20312}, {20227, 20313}

X(20312) = (X(1), X(2), X(4), X(6); X(16583), X(1196), X(39), X(6)) COLLINEATION IMAGE OF X(282)

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

See X(20227).

X(20312) lies on these lines: {800, 1108}, {1196, 20311}, {3767, 20227}, {4008, 7952}

X(20313) = (X(1), X(2), X(4), X(6); X(16583), X(1196), X(39), X(6)) COLLINEATION IMAGE OF X(1073)

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

See X(20227).

X(20313) lies on these lines: {232, 800}, {3344, 5065}, {3767, 20207}, {6525, 20232}, {20227, 20311}

X(20314) = (X(1), X(4), X(9), X(57); X(513), X(20315), X(4885), X(20317)) COLLINEATION IMAGE OF X(282)

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

See X(20316).

X(20314) lies on these lines: {2, 20297}, {513, 20319}, {514, 20318}, {521, 7658}, {4885, 20316}

X(20315) = (name pending)

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

X(20315) lies on these lines: {2, 7649}, {3, 3667}, {5, 16231}, {513, 20318}, {514, 20316}, {522, 8062}, {523, 4885}, {656, 6332}, {1459, 9031}, {3239, 6586}, {3835, 20319}, {4025, 4064}

X(20316) = (X(1), X(4), X(9), X(57); X(513), X(20315), X(4885), X(20317)) COLLINEATION IMAGE OF X(3)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,513), (2,514), (3,20316), (4,20315), (9,4885), (57,20317), (223,20318), (282,20314), (1249,20319)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(-a : b : c) = a (b - c) : a b - 2 a c + b c : 2 a b - a c - b c
m(a : b cos C : c cos B) = -(b - c) (-a^2 - 2 a b + b^2 - 2 a c + 2 b c + c^2) : -a^3 - a^2 b + a b^2 - b^3 + 2 a^2 c - a c^2 + b c^2 : a^3 - 2 a^2 b + a b^2 + a^2 c - b^2 c - a c^2 + c^3
m(A') = (c - b) (-a^2 - a b - a c + 2 b c) : -b (a^2 + a b - 2 a c + b c - c^2) : c (a^2 - 2 a b - b^2 + a c + b c)
m(A'') = (b - c) (a + b + c) (a^2 - a b - a c + 2 b c) : b (a + b - c) (-a^2 + a b - 2 a c + b c + c^2) : c (a - b + c) (a^2 + 2 a b - b^2 - a c - b c)

where

If P = x : y : z (barycentrics), then m(P) = (a - c)y - (a - b) z : : , and m is the collineation indicated by (A,B,C,X(2); m(A), m(B), m(C), X(514)).

Let f(a,b,c,x,y,z) = (a-b) (a-c) (a b^2-4 a b c+b^2 c+a c^2+b c^2) x^3-(b-c) (a^3 b-3 a^2 b^2+a^3 c+3 a^2 b c-5 a^2 c^2+4 a b c^2-b^2 c^2) y^2 z+(b-c) (a^3 b-5 a^2 b^2+a^3 c+3 a^2 b c+4 a b^2 c-3 a^2 c^2-b^2 c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y)+ 2 (a^3 b^2+a^2 b^3-4 a^3 b c+2 a^2 b^2 c-4 a b^3 c+a^3 c^2+2 a^2 b c^2+2 a b^2 c^2+b^3 c^2+a^2 c^3-4 a b c^3+b^2 c^3) x y z = 0. (Peter Moses, July 31, 2018)

X(20316) lies on these lines: {2, 1459}, {10, 522}, {69, 17215}, {141, 9000}, {513, 3823}, {514, 20315}, {520, 6130}, {521, 8062}, {523, 4147}, {656, 4391}, {657, 966}, {834, 3835}, {1213, 6586}, {1734, 4985}, {1769, 4397}, {2517, 17420}, {3261, 5224}, {3716, 15313}, {4017, 14430}, {4025, 18160}, {4041, 7650}, {4163, 7661}, {4885, 20314}, {14429, 20294}

X(20317) = (name pending)

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

See X(20316).

X(20317) lies on these lines: {2, 3669}, {8, 4162}, {9, 4063}, {10, 3309}, {281, 17924}, {513, 3823}, {514, 4521}, {650, 3975}, {663, 14430}, {667, 958}, {905, 3762}, {918, 14837}, {960, 4083}, {1577, 4762}, {1639, 6332}, {1698, 4905}, {2787, 6050}, {3041, 9320}, {3239, 3910}, {3716, 3900}, {3835, 8712}, {4106, 4498}, {4129, 4940}, {4394, 6002}, {4468, 7178}, {4481, 4960}, {4490, 7662}, {4782, 5302}, {11068, 20319}

X(20318) = (X(1), X(4), X(9), X(57); X(513), X(20315), X(4885), X(20317)) COLLINEATION IMAGE OF X(223)

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

See X(20316).

X(20318) lies on these lines: {2, 20296}, {9, 905}, {513, 20315}, {514, 20314}, {960, 3900}, {3835, 8712}, {4391, 18228}, {6332, 14298}

X(20319) = (X(1), X(4), X(9), X(57); X(513), X(20315), X(4885), X(20317)) COLLINEATION IMAGE OF X(1249)

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

See X(20316).

X(20319) lies on these lines: {2, 20298}, {3, 7658}, {440, 3239}, {464, 4025}, {513, 20314}, {652, 10319}, {3835, 20315}, {11068, 20317}

X(20320) = (X(1), X(2), X(3), X(6); X(17861), X(1), X(75), X(17871)) COLLINEATION IMAGE OF X(57)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,17861), (2,1), (3,75), (4,18691), (6,17871), (9,17860), (1073,158)

Twelve more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a b c : c (a^2 - c2) : b(a^2 - b^2)
m(-a : b : c) = -b c (-a^3 + b^3 - b^2 c - b c^2 + c^3) : a c (a^3 - b^3 + a^2 c - a c^2 - c^3) : a b (a^3 + a^2 b - a b^2 - b^3 - c^3)
m(a : b cos C : c cos B) = 2 b c (a^4 + b^4 - 2 b^2 c^2 + c^4) : a c (a^2 + b^2 - c^2)^2 : a b (a^2 - b^2 + c^2)^2
m(A') = b c (a + b + c) (a^3 + b^3 - b^2 c - b c^2 + c^3) : a (a + b - c) c (a^3 + b^3 + a^2 c - a c^2 - c^3) : a b (a - b + c) (a^3 + a^2 b - a b^2 - b^3 + c^3)

where

If P = x : y : z (barycentrics), then m(P) = b c (a^2 x + (b^2 - c^2)(y - z)) : : , and m is the collineation indicated by (D,E,F,X(2); D',E',F', X(1)), where D = 0 : 1 : 1 and D' = 0 : c : b.

Let f(a,b,c,x,y,z) = a^5 (b-c) (b+c) (a^2-b^2-c^2)^2 x^3-b^2 c (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8-a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+2 a^2 b^2 c^4-a^2 c^6-b^2 c^6+c^8) y^2 z+b c^2 (a^8-a^6 b^2-a^2 b^6+b^8-2 a^6 c^2+a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2+2 a^4 c^4+a^2 b^2 c^4-2 a^2 c^6-b^2 c^6+c^8) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20320) lies on these lines: {1, 17860}, {4, 2823}, {10, 75}, {46, 20223}, {92, 4292}, {280, 3086}, {312, 6700}, {318, 1210}, {321, 936}, {377, 14213}, {443, 6358}, {1068, 20266}, {4066, 20103}, {4188, 18359}, {4311, 20220}, {4359, 5705}, {4968, 9623}, {13532, 18961}, {17871, 20321}

X(20321) = (X(1), X(2), X(3), X(6); X(17861), X(1), X(75), X(17871)) COLLINEATION IMAGE OF X(223)

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

See X(20320).

X(20321) lies on these lines: {10, 158}, {347, 18698}, {17861, 17869}, {17871, 20320}

X(20322) = (X(1), X(2), X(3), X(6); X(17861), X(1), X(75), X(17871)) COLLINEATION IMAGE OF X(1249)

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

See X(20320).

X(20322) lies on these lines: {1, 29}, {17871, 18691}

X(20323) = (X(1), X(2), X(4), X(6); X(1100), X(1), X(1953), X(17469)) COLLINEATION IMAGE OF X(57)

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

Let m denote the indicated collineation, which maps the Thomson cubic, K002, onto a cubic, m(K002). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K002:

(1,1100), (2,1), (3,17438), (4,1953), (6,17469), (9,3748), (57,,20323), (223,20324), (1249,20325)

Fifteen more points on m(K002) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 2a : b : c
m(-a : b : c) = a (b + c - 2 a) : b (c - a + 2b) : c (b - a + 2a)
m(a : b cos C : c cos B) = 6 a^3 : b (5 a^2 + b^2 - c^2) : c (5 a^2 - b^2 + c^2)
m(A') = a (2 a^2 + 3 a b + b^2 + 3 a c - 2 b c + c^2) : b (a^2 + 3 a b + 2 b^2 + 2 a c - 3 b c + c^2) : c (a^2 + 2 a b + b^2 + 3 a c - 3 b c + 2 c^2)
m(A'') = a (2 a^3 + a^2 b - 2 a b^2 - b^3 + a^2 c + 8 a b c + b^2 c - 2 a c^2 + b c^2 - c^3) : -b (-a^3 - 2 a^2 b + a b^2 + 2 b^3 - a^2 c - 8 a b c - b^2 c + a c^2 - 2 b c^2 + c^3) : -c (-a^3 - a^2 b + a b^2 + b^3 - 2 a^2 c - 8 a b c - 2 b^2 c + a c^2 - b c^2 + 2 c^3)

where

If P = x : y : z (barycentrics), then m(P) = a (2 x + y + z) : : , and m is the collineation indicated by (D, E, F,X(2); D', E' F',X(1), where D' = -1 : 1 : 1 and D' = 0 : c : b.

Let f(a,b,c,x,y,z) = 3 b^3 (b-c) c^3 (b+c) x^3+a^3 b c^2 (13 a^2+b^2+2 c^2) y^2 z-a^3 b^2 c (13 a^2+2 b^2+c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K002) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20323) lies on these lines: {1, 3}, {2, 11260}, {8, 17728}, {11, 10106}, {21, 10179}, {104, 15179}, {106, 15955}, {145, 3689}, {210, 12513}, {214, 3635}, {355, 10072}, {388, 6957}, {392, 8666}, {404, 3880}, {497, 4308}, {519, 17614}, {614, 3445}, {937, 16485}, {946, 5434}, {1056, 6898}, {1100, 2183}, {1104, 1149}, {1125, 15888}, {1210, 10944}, {1222, 5205}, {1376, 3893}, {1387, 12047}, {1475, 6603}, {1478, 11373}, {1537, 3649}, {1699, 9657}, {1836, 3600}, {1837, 3476}, {1858, 12740}, {1887, 15500}, {2320, 5558}, {2348, 9310}, {2551, 3616}, {2650, 17476}, {2842, 11717}, {2975, 3683}, {2999, 15839}, {3058, 4297}, {3086, 5252}, {3244, 5440}, {3306, 3922}, {3475, 3622}, {3487, 6976}, {3582, 9956}, {3585, 7743}, {3636, 12572}, {3698, 3872}, {3812, 4861}, {3884, 3916}, {3890, 4640}, {3962, 5289}, {4009, 9369}, {4293, 12701}, {4301, 11246}, {4311, 6284}, {4315, 7354}, {4317, 12699}, {4413, 4853}, {4646, 15854}, {4719, 17015}, {4731, 16408}, {4870, 5901}, {5044, 5288}, {5087, 20060}, {5250, 11194}, {5253, 5836}, {5258, 5506}, {5270, 9955}, {5298, 6684}, {5603, 10404}, {5691, 11238}, {5698, 11038}, {5794, 10529}, {6049, 10580}, {6691, 6735}, {7677, 15837}, {8227, 11237}, {9327, 16601}, {9613, 10896}, {9614, 12943}, {10039, 15325}, {10074, 17638}, {10199, 17619}, {10624, 15326}, {10915, 13747}, {10950, 11019}, {11189, 12262}, {12575, 15338}, {12675, 17637}, {17439, 17474}, {17469, 20324}, {17636, 20586}

X(20324) = (X(1), X(2), X(4), X(6); X(1100), X(1), X(1953), X(17469)) COLLINEATION IMAGE OF X(223)

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

See X(20323).

X(20324) lies on these lines: {1, 84}, {1100, 1953}, {1389, 1870}, {3057, 20277}, {17469, 20323}

X(20325) = (X(1), X(2), X(4), X(6); X(1100), X(1), X(1953), X(17469)) COLLINEATION IMAGE OF X(1249)

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

See X(20323).

X(20325) lies on these lines: {1, 204}, {1953, 2312}, {2167, 8767}

X(20326) = X(30)X(12525)∩X(511)X(3845)

Barycentrics a^2*(4*(b^4-b^2*c^2+c^4)*a^4+( 11*b^4-30*b^2*c^2+11*c^4)*b^2* c^2-(b^2+c^2)*(2*b^2-3*b*c+2* c^2)*(2*b^2+3*b*c+2*c^2)*a^2) : :
X(20326) = 5*X(3858)+4*X(6310)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27948.

X(20326) lies on these lines: {30, 12525}, {511, 3845}, {512, 16509}, {3363, 5640}, {3858, 6310}

X(20326) = reflection of X(3845) in the line X(512)X(20112)

X(20327) = X(30)X(5447)∩X(1154)X(10285)

Barycentrics (SB+SC)*(9*S^4+(2*R^2*(3*R^2- 19*SA-SW)+11*SA^2-2*SB*SC-SW^ 2)*S^2-(2*R^4*(2*R^2-11*SA-2* SW)+2*R^2*SW*(SW+7*SA)-SA*SW^ 2-2*SW^3)*SA) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27948.

X(20327) lies on these lines: {30, 5447}, {1154, 10285}, {5501, 10095}

X(20328) = X(2)X(1565)∩X(5)X(6706)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27948.

X(20328) lies on these lines: {2, 1565}, {5, 6706}, {142, 517}, {277, 3295}, {2809, 3826}, {4000, 6767}, {4648, 15934}, {6147, 17758}

X(20329) = MIDPOINT OF X(3) AND X(3346)

Barycentrics 2*S^4-(16*R^2*(16*R^2+SA-6*SW) -5*SA^2+SA*SW+8*SW^2)*S^2+4*( 4*R^2-SW)*(16*R^2-SW)*SB*SC : :
X(20329) =3*X(3)-X(3183) = X(3183)+3*X(3346)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27948.

X(20329) lies on these lines: {3, 1033}, {5, 20203}, {140, 6523}, {550, 6759}, {1503, 16273}

X(20330) = MIDPOINT OF X(1) AND X(5805)

Barycentrics 2*(b+c)*a^5-(3*b^2+4*b*c+3*c^ 2)*a^4-2*(b^2-c^2)*(b-c)*a^3+ 2*(2*b^2+b*c+2*c^2)*(b-c)^2*a^ 2+4*(b^2-c^2)*(b-c)*b*c*a-(b^ 2-c^2)^2*(b-c)^2 : :
X(20330) = X(4)+3*X(11038) = X(7)+3*X(5603) = X(9)-3*X(5886) = X(390)-5*X(10595) = 7*X(3090)-3*X(5686) = 5*X(3616)-X(5759) = X(5223)-5*X(8227) = X(5779)-5*X(18493) = 2*X(6666)-3*X(11230) = X(11372)-5*X(11522)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27948.

X(20330) lies on these lines: {1, 5805}, {4, 11038}, {5, 518}, {6, 15251}, {7, 104}, {9, 5886}, {11, 18412}, {142, 517}, {226, 1538}, {354, 8727}, {355, 3243}, {390, 6934}, {495, 1512}, {496, 5728}, {515, 15935}, {516, 550}, {528, 19907}, {946, 971}, {1001, 5762}, {1125, 5763}, {1420, 4312}, {1445, 15325}, {1482, 2550}, {1483, 15570}, {1484, 2801}, {1503, 15939}, {2346, 6905}, {3090, 5686}, {3254, 6265}, {3333, 3358}, {3475, 19541}, {3616, 5759}, {3649, 10085}, {3656, 6173}, {3826, 3918}, {3873, 8226}, {4301, 10179}, {4860, 13226}, {5223, 8227}, {5432, 11218}, {5732, 12699}, {5779, 18493}, {5804, 9654}, {5809, 9669}, {5833, 15829}, {5856, 11729}, {5904, 7958}, {6244, 9776}, {6600, 6911}, {6601, 6826}, {6666, 11230}, {6675, 12704}, {7675, 15171}, {7680, 10265}, {8581, 12047}, {9779, 13257}, {9942, 16216}, {11375, 15298}, {11376, 15299}, {12116, 15911} X(20330) = midpoint of X(i) and X(j) for these {i,j}: {1, 5805}, {355, 3243}, {946, 5542}, {1482, 2550}, {3254, 6265}, {3656, 6173}, {5732, 12699}

X(20330) = reflection of X(i) in X(j) for these (i,j): (1001, 5901), (1483, 15570)
X(20330) = X(5805)-of-anti-Aquila-triangle
X(20330) = X(18440)-of-3rd Euler-triangle
X(20330) = X(19139)-of-incircle-circles-triangle

X(20331) = X(1)X(39)∩X(2)X(45)

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

X(20331) lies on the Brocard quartic Q143 and these lines: {1, 39}, {2, 45}, {6, 100}, {9, 1054}, {37, 244}, {42, 678}, {43, 4274}, {75, 17028}, {105, 6016}, {106, 4752}, {513, 649}, {574, 16788}, {644, 9259}, {1023, 8649}, {1100, 3722}, {1281, 2023}, {1334, 16604}, {1574, 16552}, {1914, 16786}, {2087, 2802}, {2325, 4871}, {3315, 16777}, {3550, 5332}, {3693, 3726}, {3780, 4253}, {4595, 9263}, {4969, 19998}, {5030, 5291}, {5264, 7772}, {5297, 16521}, {8297, 16468}, {8300, 16477}, {10987, 16779}, {15447, 15990}, {17029, 17160}

X(20331) = barycentric product X(1)*X(537)
X(20331) = barycentric quotient X(i)/X(j) for these (i,j): (1, 18822), (31, 2382), (537, 75)
X(20331) = trilinear product X(6)*X(537)
X(20331) = trilinear quotient X(i)/X(j) for these (i,j): (2, 18822), (6, 2382), (537, 2)
X(20331) = (1st circumperp)-isotomic conjugate of-X(2382)
X(20331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 16549, 2295), (44, 899, 2238), (44, 1155, 2243), (44, 1575, 899), (244, 14439, 37), (672, 899, 44), (672, 1575, 2238), (2229, 2245, 2238), (17029, 17759, 17160)

X(20332) = (name pending)

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

X(20332) lies on the cubic K155 and these lines: {2, 1977}, {6, 190}, {31, 43}, {75, 20639}, {81, 799}, {105, 2144}, {162, 2203}, {238, 660}, {239, 20669}, {256, 8843}, {604, 651}, {608, 653}, {658, 1407}, {662, 1333}, {739, 4607}, {823, 5317}, {1922, 6652}, {1979, 4383}, {3257, 9456}, {7121, 14823}

X(20332) = isogonal conjugate of X(1575)
X(20332) = complement of X(20355)
X(20332) = anticomplement of X(20343)
X(20332) = trilinear pole of line X(1)X(667)
X(20332) = X(2)-isoconjugate of X(3009)
X(20332) = X(92)-isoconjugate of X(20777)
X(20332) = eigencenter of Gemini triangle 30

X(20333) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(292)

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

Let m denote the indicated collineation, which maps the 2nd equal areas cubic, K155, onto a cubic, m(K155), which is the complement of K002, as at X(20205). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K155:

(1,10), (2,2), (6,141), (31,2887), (105,120), (238,3836), (292,20333), (365,20334), (672,20335), (1423,20336), (1931,20337), (2053,20338), (2054,20339), (3009,20340), (2112,20341), (2144,20342)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(A') = (b' + c')(b - b'c' + c) : (a' + c')(a - a' c' + c) : ((a' + b')(a - a' b' + b), where a' = a^1/2, and b' and c' are defined cyclically
m(A1) = 2 b c : -a^2 + b c : -a^2 + b c
m(A2) = (b + c) (b^2 - b c + c^2) : c (-a b + c^2) : b (b^2 - a c)
m(A3) = (b + c) (a b + a c + b c) : -a^3 - a^2 b - a^2 c + a b c + a c^2 + b c^2 : -a^3 - a^2 b + a b^2 - a^2 c + a b c + b^2 c
m(A4) = -(a + b + c) (b^2 + c^2) : a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3 : a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c
m(A5) = a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3 : a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3 : a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c
m(A6) = (a + b + c) (a^2 b^2 - a b^3 - b^3 c + a^2 c^2 + 2 b^2 c^2 - a c^3 - b c^3) : (a^2 + b^2 - a c - b c) (a^3 - a^2 b - a b c + 2 a c^2 + b c^2 + c^3) : (a^3 + 2 a b^2 + b^3 - a^2 c - a b c + b^2 c) (a^2 - a b - b c + c^2),

where the 6 triangles A1B1C1 to A6B6C6 are given by A-vertices (found using the method described in the preamble just before X(2106)) as follows:

A1 = -a^2 : b c : b c
A2 = -a b c : b^3 : c^3
A3 = -a^2 (a + b + c) : b (b c + c a + a b) : c (b c + c a + a b)
A4 = -a/(a + b + c) : b^2/(b c + c a + a b) : c^2/(b c + c a + a b)
A5 = a^2 (a + b + c) : b (a^2 + b^2 - a c - b c) : c (a^2 + c^2 - a b - b c)
A6 = a/(a + b + c) : b^2/(a^2 + b^2 - a c - b c) : c^2/(a^2 + c^2 - a b - b c)

X(20333) lies on these lines: {2, 292}, {10, 3934}, {116, 3454}, {141, 9016}, {668, 19974}, {1086, 1213}, {1329, 20255}, {1921, 14603}, {2887, 20341}, {3846, 19563}, {3912, 20530}, {18205, 20457}, {20335, 20340}, {20356, 20484}, {20540, 20541}

X(20333) = complement of X(292)
X(20333) = complementary conjugate of X(3912)

X(20334) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : :
Barycentrics b^(3/2) + c^(3/2) : :

See X(20333).

X(20334) lies on these lines: {2, 365}, {3661, 20357}, {4180, 20527}
X(20334) = complement of X(365)
X(20334) = complementary conjugate of X(20527)

X(20335) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(672)

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

See X(20333).

X(20335) lies on these lines: {2, 7}, {5, 17046}, {8, 20257}, {10, 2140}, {12, 17062}, {42, 3946}, {43, 4000}, {69, 17026}, {85, 3061}, {116, 3814}, {120, 3836}, {141, 674}, {295, 9470}, {334, 350}, {513, 3716}, {516, 8299}, {519, 17761}, {673, 3684}, {899, 17067}, {942, 17048}, {960, 6706}, {1009, 4292}, {1086, 1575}, {1125, 16850}, {1215, 3739}, {1266, 17759}, {1319, 6647}, {1738, 3783}, {1921, 18275}, {2238, 3008}, {2239, 3011}, {2276, 3663}, {2321, 4441}, {3136, 17052}, {3501, 17753}, {3717, 17794}, {3771, 12610}, {3817, 3840}, {3831, 3934}, {3879, 17027}, {3970, 7264}, {4051, 16284}, {4119, 4437}, {4361, 4685}, {4372, 8669}, {4465, 19593}, {4479, 17233}, {4511, 9317}, {4713, 17279}, {4851, 4865}, {4859, 16569}, {4869, 5274}, {4887, 20331}, {4904, 17757}, {5074, 11813}, {5847, 17031}, {5853, 13576}, {6823, 18639}, {8167, 15668}, {9320, 17072}, {10453, 17296}, {14828, 16503}, {15669, 16058}, {16593, 17747}, {17090, 20535}, {17192, 17211}, {17760, 18055}, {18067, 18144}, {20333, 20340}, {20358, 20486}, {20448, 20593}

X(20335) = complement of X(672)
X(20335) = complementary conjugate of X(16593)

X(20336) = ISOTOMIC CONJUGATE OF X(28)

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

See X(20333).

X(20336) lies on the hyperbola {{A,B,C,X(2),X(69)}} and these lines: {2,37}, {10,18697}, {12,313}, {28,7283}, {69,72}, {71,4019}, {95,7523}, {100,2373}, {190,5279}, {201,307}, {228,1799}, {253,322}, {264,1969}, {286,2064}, {287,336}, {306,3610}, {314,943}, {332,1807}, {668,1494}, {857,4150}, {894,2303}, {975,10436}, {1001,3702}, {1089,4078}, {1444,1791}, {1930,4357}, {2893,16086}, {2901,3875}, {3159,3663}, {3668,4082}, {3695,20235}, {3879,14210}, {3936,16580}, {3948,20234}, {3967,7211}, {3975,17788}, {4064,15413}, {4329,4463}, {4360,5262}, {4647,19857}, {5295,5722}, {6330,6335}, {6386,18024}, {9022,16685}, {9229,9239}, {16817,17143}, {17144,19851}, {18156,20009}

X(20336) = isogonal conjugate of X(2203)
X(20336) = isotomic conjugate of X(28)
X(20336) = X(15408)-complementary conjugate of X(1125)
X(20336) = X(i)-Ceva conjugate of X(j) for these (i,j): {304, 306}, {1978, 15416}, {3596, 313}, {4601, 1332}
X(20336) = X(i)-cross conjugate of X(j) for these (i,j): {72, 321}, {306, 1231}, {3695, 306}, {4466, 15413}, {18210, 525}
X(20336) = X(i)-beth conjugate of X(j) for these (i,j): {645, 5279}, {3596, 1441}, {3701, 4078}
X(20336) = X(1)-zayin conjugate of X(2203)
X(20336) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2203}, {4, 2206}, {6, 1474}, {19, 1333}, {21, 1395}, {25, 58}, {27, 32}, {28, 31}, {29, 1397}, {33, 1408}, {34, 2194}, {41, 1396}, {48, 5317}, {56, 2299}, {57, 2204}, {81, 1973}, {86, 1974}, {112, 649}, {162, 667}, {163, 6591}, {184, 8747}, {250, 3122}, {270, 1402}, {281, 16947}, {283, 7337}, {284, 608}, {286, 560}, {593, 2333}, {604, 1172}, {607, 1412}, {648, 1919}, {811, 1980}, {849, 1824}, {1014, 2212}, {1096, 1437}, {1106, 4183}, {1169, 2354}, {1398, 2328}, {1400, 2189}, {1407, 2332}, {1472, 4206}, {1576, 7649}, {1790, 2207}, {1880, 2150}, {2201, 18268}, {2208, 3194}, {2360, 7151}, {2489, 4556}, {3248, 5379}, {3285, 8752}, {3733, 8750}, {4211, 7084}, {13854, 17186}
X(20336) = X(2)-Hirst inverse of X(16085)
X(20336) = cevapoint of X(i) and X(j) for these (i,j): {72, 3998}, {306, 3710}, {525, 18210}, {4064, 4466} X(20336) = crosspoint of X(i) and X(j) for these (i,j): {304, 305}, {3596, 3718}, {4601, 6386}
X(20336) = trilinear pole of line {525, 14208}
X(20336) = crosssum of X(i) and X(j) for these (i,j): {1395, 1397}, {1973, 1974}, {1980, 3121}
X(20336) = barycentric product X(i)X(j) for these {i,j}: {8, 1231}, {10, 304}, {37, 305}, {63, 313}, {69, 321}, {71, 561}, {72, 76}, {75, 306}, {78, 349}, {85, 3710}, {100, 3267}, {125, 4601}, {190, 14208}, {226, 3718}, {228, 1502}, {264, 3998}, {274, 3695}, {307, 312}, {310, 3949}, {332, 6358}, {337, 3948}, {339, 4567}, {345, 1441}, {348, 3701}, {525, 668}, {646, 17094}, {647, 6386}, {656, 1978}, {799, 4064}, {850, 1332}, {1089, 17206}, {1214, 3596}, {1228, 1791}, {1265, 1446}, {1577, 4561}, {1928, 2200}, {1969, 3682}, {2321, 7182}, {3265, 6335}, {3690, 6385}, {3694, 6063}, {3952, 15413}, {3963, 7019}, {3990, 18022}, {4019, 7018}, {4025, 4033}, {4036, 4563}, {4466, 7035}, {4566, 15416}, {4572, 8611}
X(20336) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1474}, {2, 28}, {3, 1333}, {4, 5317}, {6, 2203}, {7, 1396}, {8, 1172}, {9, 2299}, {10, 19}, {12, 1880}, {21, 2189}, {37, 25}, {42, 1973}, {48, 2206}, {55, 2204}, {63, 58}, {65, 608}, {69, 81}, {71, 31}, {72, 6}, {73, 604}, {75, 27}, {76, 286}, {77, 1412}, {78, 284}, {92, 8747}, {100, 112}, {125, 3125}, {190, 162}, {200, 2332}, {201, 1400}, {210, 607}, {213, 1974}, {219, 2194}, {222, 1408}, {226, 34}, {227, 3209}, {228, 32}, {283, 2150}, {295, 18268}, {304, 86}, {305, 274}, {306, 1}, {307, 57}, {312, 29}, {313, 92}, {318, 8748}, {321, 4}, {326, 1790}, {329, 3194}, {332, 2185}, {333, 270}, {339, 16732}, {341, 2322}, {343, 18180}, {344, 4233}, {345, 21}, {346, 4183}, {348, 1014}, {349, 273}, {394, 1437}, {440, 1104}, {442, 1841}, {521, 7252}, {523, 6591}, {525, 513}, {594, 1824}, {603, 16947}, {656, 649}, {668, 648}, {693, 17925}, {740, 2201}, {756, 2333}, {810, 1919}, {850, 17924}, {905, 3733}, {906, 1576}, {1016, 5379}, {1018, 8750}, {1043, 2326}, {1089, 1826}, {1211, 1829}, {1213, 2355}, {1214, 56}, {1215, 7119}, {1231, 7}, {1238, 16698}, {1259, 2193}, {1264, 1812}, {1265, 2287}, {1331, 163}, {1332, 110}, {1334, 2212}, {1368, 16716}, {1400, 1395}, {1409, 1397}, {1427, 1398}, {1439, 1407}, {1441, 278}, {1444, 593}, {1446, 1119}, {1565, 16726}, {1577, 7649}, {1790, 849}, {1791, 1169}, {1792, 7054}, {1812, 60}, {1824, 2207}, {1826, 1096}, {1880, 7337}, {1903, 7151}, {1930, 17171}, {1978, 811}, {2197, 1402}, {2200, 560}, {2292, 2354}, {2318, 41}, {2321, 33}, {2345, 4206}, {2397, 4246}, {2525, 2530}, {2895, 2906}, {3049, 1980}, {3175, 4186}, {3198, 3172}, {3263, 15149}, {3265, 905}, {3267, 693}, {3610, 612}, {3668, 1435}, {3682, 48}, {3690, 213}, {3692, 2328}, {3694, 55}, {3695, 37}, {3700, 18344}, {3701, 281}, {3708, 3122}, {3710, 9}, {3718, 333}, {3719, 283}, {3797, 17569}, {3926, 1444}, {3930, 2356}, {3932, 5089}, {3933, 16696}, {3936, 1870}, {3940, 4273}, {3948, 242}, {3949, 42}, {3952, 1783}, {3954, 1843}, {3958, 2308}, {3963, 7009}, {3964, 18604}, {3969, 6198}, {3975, 14024}, {3990, 184}, {3992, 8756}, {3995, 4222}, {3998, 3}, {4000, 4211}, {4019, 171}, {4025, 1019}, {4033, 1897}, {4036, 2501}, {4037, 862}, {4043, 14004}, {4055, 9247}, {4064, 661}, {4082, 7079}, {4086, 3064}, {4101, 1449}, {4121, 18167}, {4131, 7254}, {4143, 4131}, {4158, 3990}, {4397, 17926}, {4415, 1828}, {4463, 8743}, {4466, 244}, {4515, 7071}, {4552, 108}, {4561, 662}, {4567, 250}, {4571, 5546}, {4574, 692}, {4580, 18108}, {4592, 4556}, {4601, 18020}, {4647, 1839}, {4674, 8752}, {4705, 2489}, {4850, 4247}, {5257, 5338}, {5360, 2211}, {5440, 3285}, {5930, 3213}, {6332, 3737}, {6335, 107}, {6354, 1426}, {6356, 1427}, {6358, 225}, {6386, 6331}, {6390, 16702}, {6516, 4565}, {7017, 1896}, {7066, 1409}, {7182, 1434}, {8024, 16747}, {8611, 663}, {8680, 1430}, {8804, 204}, {9033, 14399}, {9723, 18605}, {11611, 17981}, {13576, 8751}, {14208, 514}, {14417, 14419}, {14429, 1635}, {15377, 18757}, {15413, 7192}, {15416, 7253}, {15523, 17442}, {15526, 18210}, {16732, 2969}, {17094, 3669}, {17206, 757}, {17216, 3942}, {17441, 16502}, {17740, 4227}, {17757, 14571}, {17759, 15148}, {17762, 2905}, {17787, 14006}, {17790, 422}, {17879, 4466}, {17880, 17197}, {17977, 5006}, {18210, 1015}, {18589, 614}, {18695, 17167}, {18697, 1848}, {18743, 4248}, {19799, 1010}, {20235, 4000}
X(20336) = pole wrt polar circle of trilinear polar of X(5317) (line X(667)X(6591))
X(20336) = polar conjugate of X(5317)
X(20336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 18147, 17863), (304, 3718, 69), (312, 19804, 19814), (313, 349, 1234), (1441, 3701, 313), (3610, 18589, 306), (4358, 17863, 18147)

X(20337) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(1931)

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

See X(20333).

X(20337) lies on these lines: {2, 1931}, {10, 12}, {115, 3912}, {141, 5949}, {239, 10026}, {325, 18827}, {334, 3948}, {894, 1213}, {1230, 1237}, {3834, 8287}, {3836, 20339}, {3936, 6542}, {4129, 4369}, {5025, 18134}, {8818, 17279}, {16826, 17056}, {20360, 20488}

X(20337) = complement of X(1931)
X(20337) = complementary conjugate of X(20529)
X(20337) = polar conjugate of isogonal conjugate of X(20733)

X(20338) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(2053)

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

See X(20333).

X(20338) lies on these lines: {2, 2053}, {12, 85}, {1329, 3836}, {2886, 17062}, {2887, 20255}, {14823, 17717}, {17046, 20547}, {20361, 20489}

X(20338) = complement of X(2053)
X(20338) = complementary conjugate of X(3061)
X(20338) = isotomic conjugate of isogonal conjugate of X(20462)
X(20338) = polar conjugate of isogonal conjugate of X(20734)

X(20339) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(2054)

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

See X(20333).

X(20339) lies on these lines: {2, 846}, {141, 20341}, {3739, 20529}, {3741, 20548}, {3836, 20337}, {20362, 20490}

X(20339) = complement of X(2054)
X(20339) = complementary conjugate of X(10026)
X(20339) = isotomic conjugate of isogonal conjugate of X(20463)
X(20339) = polar conjugate of isogonal conjugate of X(20735)

X(20340) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(3009)

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

See X(20333).

X(20340) lies on these lines: {1, 2}, {75, 4941}, {291, 3975}, {512, 625}, {515, 19522}, {726, 3948}, {982, 6376}, {1921, 19567}, {2887, 20255}, {3122, 3264}, {3501, 4011}, {3596, 17065}, {3836, 20343}, {3971, 12782}, {17793, 20358}, {20333, 20335}, {20341, 20541}, {20363, 20491}

X(20340) = complement of X(3009)
X(20340) = complementary conjugate of X(20532)
X(20340) = isotomic conjugate of isogonal conjugate of X(20464)
X(20340) = polar conjugate of isogonal conjugate of X(20736)

X(20341) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(2112)

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

See X(20333).

X(20341) lies on these lines: {2, 2112}, {10, 116}, {141, 20339}, {2887, 20333}, {3496, 17671}, {3836, 9470}, {16830, 17062}, {20258, 20343}, {20340, 20541}, {20364, 20492}

X(20341) = complement of X(2112)
X(20341) = isotomic conjugate of isogonal conjugate of X(20465)
X(20341) = polar conjugate of isogonal conjugate of X(20737)

X(20342) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(2144)

Barycentrics a (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20333).

X(20342) lies on these lines: {2, 2113}, {120, 20343}, {20365, 20493}

X(20342) = complement of X(2144)

X(20343) = (X(1), X(2), X(6), X(105); X(10), X(2), X(141), X(120)) COLLINEATION IMAGE OF X(20332)

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

See X(20333).

X(20343) lies on these lines: {2, 1977}, {11, 2887}, {75, 141}, {120, 20342}, {1211, 16592}, {3836, 20340}, {20258, 20341}, {20366, 20494}

X(20343) = complement of X(20332)
X(20343) = complementary conjugate of X(20530)
X(20343) = isotomic conjugate of isogonal conjugate of X(20467)
X(20343) = polar conjugate of isogonal conjugate of X(20738)

X(20344) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(105)

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

Let m denote the indicated collineation, which maps the 2nd equal areas cubic, K155, onto a cubic, m(K155), which is K007. The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K155:

(1,8), (2,2), (6,69), (31,6327), (105,20344), (238,5645), (292,20345), (365,20346), (672,20347), (1423,20348), (1931,20349), (2053,20350), (2054,20351), (3009,20352), (2112,20353), (2144,20354), (20332,20355)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(A') = a^(3/2) + b^(3/2) + c^(3/2) : -a^(3/2) - b^(3/2) + c^(3/2) : -a^(3/2) + b^(3/2) - c^(3/2)
m(A1) = a^2 + 2 b c : -a^2 : -a^2
m(A2) = b^3 + a b c + c^3 : -b^3 - a b c + c^3 : b^3 - a b c - c^3
m(A3) = -a^3 - a^2 b - a b^2 - a^2 c - 2 a b c - b^2 c - a c^2 - b c^2 : a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2 : a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2
m(A4) = a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3 : -a^2 b - a b^2 - b^3 - a^2 c - a b c - b^2 c + a c^2 + b c^2 + c^3 : -a^2 b + a b^2 + b^3 - a^2 c - a b c + b^2 c - a c^2 - b c^2 - c^3
m(A5) = a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3 : -a^3 + b^3 - 2 a^2 c - b^2 c + b c^2 - c^3 : -a^3 - 2 a^2 b - b^3 + b^2 c - b c^2 + c^3
m(A6) = a^5 - a^4 b - a^2 b^3 + a b^4 - a^4 c - a^3 b c + a b^3 c + b^4 c - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + a c^4 + b c^4 : -a^5 + a^4 b + a^2 b^3 - a b^4 + a^4 c + a^3 b c - a b^3 c - b^4 c - 2 a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 + 3 a b c^3 + b^2 c^3 + a c^4 + b c^4 : -a^5 + a^4 b - 2 a^3 b^2 + a^2 b^3 + a b^4 + a^4 c + a^3 b c - 2 a^2 b^2 c + 3 a b^3 c + b^4 c - 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 - a b c^3 - b^2 c^3 - a c^4 - b c^4,

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = -x + y + z : x - y + z : x + y - z : : , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(2)), where A'B'C' = anticomplementary triangle.

Let f(a,b,c,x,y,z) = a^3 (b-c) (b^2+b c+c^2) x^3-b c (a^4+a b^3-b^2 c^2-a c^3) y^2 z+b c (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20344) lies on these lines: {2, 11}, {4, 10743}, {8, 150}, {10, 5540}, {20, 1292}, {145, 10699}, {146, 2775}, {147, 2788}, {148, 1655}, {151, 2814}, {152, 2820}, {153, 2826}, {193, 10760}, {329, 2835}, {346, 11677}, {388, 1358}, {612, 3120}, {1370, 2834}, {2551, 3039}, {2836, 2895}, {2837, 14360}, {2838, 5300}, {2968, 7386}, {3034, 9534}, {3091, 5511}, {3146, 10729}, {3263, 4872}, {3616, 11716}, {3622, 11730}, {3755, 7191}, {3920, 5249}, {4009, 5057}, {4368, 4660}, {4645, 17794}, {5800, 20020}, {6078, 14506}, {7427, 13199}, {8055, 9519}, {9523, 12384}, {14839, 18343}, {16550, 20495}, {17522, 20066}, {20060, 20089}, {20354, 20355}

X(20344) = isogonal conjugate of X(34183)
X(20344) = complement of X(20097)
X(20344) = anticomplement of X(105)
X(20344) = anticomplementary conjugate of X(518)
X(20344) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(100)
X(20344) = de-Longchamps-circle-inverse of X(26703)
X(20344) = 1st-Brocard-to-ABC similarity image of X(18343)

X(20345) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(292)

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

See X(20344).

X(20345) lies on these lines: {2, 292}, {7, 4572}, {8, 76}, {69, 9016}, {75, 1654}, {150, 1330}, {350, 6542}, {561, 4388}, {670, 19643}, {752, 4495}, {1909, 5484}, {1921, 2113}, {1978, 17777}, {3263, 3975}, {6327, 20353}, {17738, 20496}, {20347, 20352}, {20552, 20553}

X(20345) = isotomic conjugate of X(2113)
X(20345) = anticomplement of X(292)
X(20345) = anticomplementary conjugate of X(6542)
X(20345) = perspector of Gemini triangle 32 and cross-triangle of Gemini triangles 32 and 34
X(20345) = polar conjugate of isogonal conjugate of X(20742)

X(20346) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(365)

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

See X(20344).

X(20346) lies on these lines: {2, 365}, {510, 20497}

X(20346) = anticomplement of X(365)
X(20346) = anticomplementary conjugate of X(20534)

X(20347) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(672)

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

See X(20344).

X(20347) lies on these lines: {2, 7}, {8, 3761}, {42, 3663}, {43, 4862}, {69, 674}, {75, 3681}, {76, 17137}, {85, 3869}, {86, 5284}, {150, 5080}, {310, 20028}, {320, 350}, {518, 13576}, {758, 1111}, {899, 4887}, {960, 4059}, {1086, 2238}, {1125, 17169}, {1193, 18600}, {1266, 19998}, {1434, 5253}, {1621, 14828}, {1909, 17152}, {2140, 16552}, {2276, 17276}, {2481, 20556}, {2890, 2893}, {3240, 4346}, {3263, 3952}, {3294, 17758}, {3436, 6604}, {3664, 3720}, {3672, 17018}, {3673, 3868}, {3691, 17050}, {3789, 5880}, {3812, 4955}, {3874, 7264}, {3875, 20011}, {3884, 7278}, {4184, 8822}, {4440, 17759}, {4452, 20012}, {4465, 7238}, {4479, 17361}, {4511, 5088}, {4645, 17794}, {4713, 7232}, {4902, 16569}, {5180, 5195}, {5259, 17201}, {5289, 7223}, {6147, 16850}, {6647, 17439}, {7269, 18654}, {8299, 17768}, {9312, 11682}, {9812, 10439}, {11415, 17170}, {14923, 16284}, {17027, 17364}, {17031, 17770}, {17032, 17247}, {17033, 20109}, {20089, 20535}, {20345, 20352}

X(20347) = isotomic conjugate of isogonal conjugate of X(20470)
X(20347) = isotomic conjugate of anticomplement of X(39046)
X(20347) = anticomplement of X(672)
X(20347) = anticomplementary conjugate of X(20533)

X(20348) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(1423)

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

See X(20344).

X(20348) lies on these lines: {2, 7}, {8, 511}, {69, 17786}, {75, 2262}, {192, 1959}, {1201, 4310}, {1278, 20535}, {3436, 4645}, {3729, 10446}, {3927, 15973}, {4307, 10459}, {4363, 11683}, {4454, 17220}, {5069, 17276}, {7155, 10453}, {7779, 20537}, {17364, 17752}, {20353, 20355}, {20368, 20498}

X(20348) = anticomplement of X(1423)
X(20348) = anticomplementary conjugate of X(20537)

X(20349) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(1931)

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

See X(20344).

X(20349) lies on these lines: {2, 1931}, {8, 79}, {76, 1029}, {148, 6542}, {1577, 7192}, {1654, 4363}, {4645, 20351}, {17484, 17789}, {20016, 20536}, {20369, 20499}

X(20349) = isotomic conjugate of isogonal conjugate of X(20472)
X(20349) = isotomic conjugate of anticomplement of X(39042)
X(20349) = anticomplement of X(1931)
X(20349) = anticomplementary conjugate of X(20538)

X(20350) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(2053)

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

See X(20344).

X(20350) lies on these lines: {2, 2053}, {377, 2227}, {1370, 17149}, {3434, 20537}, {3436, 4645}, {6327, 20352}, {20060, 20089}, {20370, 20503}

X(20350) = anticomplement of X(2053)
X(20350) = anticomplementary conjugate of X(20535)

X(20351) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(2054)

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

See X(20344).

X(20351) lies on these lines: {2, 846}, {69, 20353}, {75, 20538}, {3448, 20290}, {4576, 17135}, {4645, 20349}, {20371, 20500}

X(20351) = anticomplement of X(2054)
X(20351) = anticomplementary conjugate of X(20536)

X(20352) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(3009)

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

See X(20344).

X(20352) lies on these lines: {1, 2}, {192, 3764}, {313, 17142}, {316, 512}, {674, 3264}, {952, 19522}, {1909, 17140}, {2295, 18091}, {3051, 3780}, {3765, 17165}, {3948, 14839}, {3952, 3975}, {3963, 17049}, {4446, 17148}, {4645, 20355}, {6327, 20350}, {7270, 19816}, {8622, 16704}, {9902, 17155}, {17137, 20290}, {20023, 20244}, {20345, 20347}, {20353, 20553}, {20372, 20501}

X(20352) = anticomplement of X(3009)
X(20352) = anticomplementary conjugate of X(39354)

X(20353) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(2112)

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

See X(20344).

X(20353) lies on these lines: {2, 2112}, {8, 150}, {69, 20351}, {334, 7357}, {4495, 4645}, {6327, 20345}, {20348, 20355}, {20352, 20553}, {20373, 20502}

X(20353) = anticomplement of X(2112)

X(20354) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(2144)

Barycentrics a (-2 a^8 b^6 + 2 a^5 b^9 + a^9 b^4 c - a^3 b^10 c + 8 a^7 b^5 c^2 -10 a^4 b^8 c^2 - 5 a^8 b^3 c^3 + 7 a^5 b^6 c^3 + 7 a^2 b^9 c^3 + a^9 b c^4 - 17 a^6 b^4 c^4 + 3 a^3 b^7 c^4 - b^10 c^4 + 8 a^7 b^2 c^5 + 3 a^4 b^5 c^5 - 4 a b^8 c^5 - 2 a^8 c^6 + 7 a^5 b^3 c^6 - 5 a^2 b^6 c^6 + 3 a^3 b^4 c^7 + 4 b^7 c^7 - 10 a^4 b^2 c^8 - 4 a b^5 c^8 + 2 a^5 c^9 + 7 a^2 b^3 c^9 - a^3 b c^10 - b^4 c^10 : :

See X(20344).

X(20354) lies on these lines: {2, 2113}, {20344, 20355}

X(20354) = anticomplement of X(2144)

X(20355) = (X(1), X(2), X(6), X(31); X(8), X(2), X(69), X(6327)) COLLINEATION IMAGE OF X(20332)

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

See X(20344).

X(20355) lies on these lines: {2, 1977}, {69, 1278}, {149, 6327}, {4645, 20352}, {20344, 20354}, {20348, 20353}

X(20355) = anticomplement of X(20332)
X(20355) = anticomplementary conjugate of X(350)

X(20356) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(292)

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

Let m denote the indicated collineation, which maps the 2nd equal areas cubic, K155, onto a cubic, m(K155). The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K155:

(1,37), (2,1), (6,38), (31,3721), (105,17464), (238,3726), (292,20356), (365,20357), (672,20358), (1423,20359), (1931,20360), (2053,20361), (2054,20362), (3009,20363), (2112,20364), (2144,20365), (20332,20366)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b : c
m(A') = -a (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c), b (Sqrt[a] - Sqrt[c]) (a + Sqrt[a] Sqrt[c] + c), (Sqrt[a] - Sqrt[b]) (a + Sqrt[a] Sqrt[b] + b) c
m(A1) = -2 a b c : -b (-a^2 + b c) : -c (-a^2 + b c)
m(A2) = a (b + c) (b^2 - b c + c^2) : -b c (a b - c^2) : b c (b^2 - a c)
m(A3) = a (b + c) (a b + a c + b c) : -b (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : -c (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = a (a + b + c) (b^2 + c^2) : -b (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : -c (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = -a (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : -b (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : -c (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = a (a + b + c) (a^2 b^2 - a b^3 - b^3 c + a^2 c^2 + 2 b^2 c^2 - a c^3 - b c^3) : -b (a^2 + b^2 - a c - b c) (-a^3 + a^2 b + a b c - 2 a c^2 - b c^2 - c^3) : -c (-a^3 - 2 a b^2 - b^3 + a^2 c + a b c - b^2 c) (a^2 - a b - b c + c^2),

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a(y + z) : b(z + x) : c(x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(2)), where A'B'C' = incentral triangle.

Let f(a,b,c,x,y,z) = b^2 (b-c) c^2 (a^2-b c) (b^2+b c+c^2) x^3-a^2 c (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z+a^2 b (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, July 31, 2018)

X(20356) lies on these lines: {1, 335}, {37, 17445}, {38, 20591}, {244, 1962}, {350, 1926}, {354, 17459}, {3721, 20364}, {3938, 6654}, {4016, 4022}, {20333, 20484}, {20358, 20363}, {20589, 20590}

X(20357) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(365)

Barycentrics a (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : :

See X(20356).

X(20357) lies on these lines: {1, 510}, {76, 18297}, {984, 20458}, {3661, 20334}

X(20358) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(672)

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

See X(20356).

X(20358) lies on these lines: {1, 3}, {2, 4517}, {7, 3056}, {38, 1107}, {72, 16825}, {142, 3688}, {175, 6405}, {176, 6283}, {210, 4384}, {239, 335}, {244, 3009}, {320, 9025}, {330, 20535}, {350, 19567}, {497, 6604}, {511, 1463}, {527, 3271}, {614, 2176}, {649, 4083}, {651, 8540}, {674, 1086}, {869, 3752}, {960, 16823}, {1002, 17014}, {1100, 13476}, {1266, 6007}, {1279, 3747}, {1469, 4310}, {1738, 9052}, {1836, 4911}, {1920, 3706}, {1953, 17447}, {2262, 16973}, {2295, 3720}, {2348, 18785}, {2481, 18033}, {2664, 16610}, {3290, 16514}, {3309, 4077}, {3662, 17792}, {3681, 16816}, {3726, 17464}, {3740, 16815}, {3742, 16826}, {3779, 4000}, {3799, 17266}, {3812, 16830}, {3834, 4553}, {3873, 4393}, {3912, 14839}, {3925, 17050}, {3975, 17794}, {4014, 4887}, {4021, 4890}, {4022, 17445}, {4298, 10544}, {4357, 17049}, {4395, 9054}, {4641, 16476}, {4847, 20257}, {4969, 9038}, {5572, 11997}, {5728, 12721}, {6384, 10453}, {6666, 7064}, {7238, 9024}, {9309, 20059}, {11851, 20036}, {12109, 13161}, {12723, 14523}, {14100, 17635}, {14267, 20556}, {17444, 17463}, {17793, 20340}, {18179, 18183}, {18191, 18206}, {20335, 20486}, {20356, 20363}

X(20359) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(1423)

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

See X(20356).

X(20359) lies on these lines: {1, 3}, {2, 3056}, {10, 10544}, {11, 2887}, {37, 1755}, {72, 8669}, {81, 2330}, {210, 333}, {312, 7155}, {321, 4459}, {497, 4645}, {518, 3769}, {613, 16434}, {1428, 19649}, {1463, 3784}, {1837, 7270}, {1964, 3752}, {2262, 4386}, {2319, 3061}, {3011, 3917}, {3271, 3452}, {3688, 5745}, {3693, 7075}, {3726, 17452}, {3831, 17606}, {3893, 3996}, {3944, 15310}, {4310, 7248}, {4417, 9025}, {4517, 5273}, {4640, 11688}, {4682, 18165}, {5325, 7064}, {5432, 6685}, {5918, 17635}, {7004, 20364}, {7186, 17719}, {7220, 17063}, {10167, 12721}, {17448, 20594}, {20258, 20487}

X(20360) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(1931)

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

See X(20356).

X(20360) lies on these lines: {1, 1326}, {37, 65}, {171, 1962}, {335, 740}, {518, 2643}, {758, 1757}, {942, 17470}, {2611, 17449}, {2650, 4649}, {3726, 20362}, {3812, 6042}, {3963, 4647}, {3999, 17476}, {4132, 4367}, {5202, 7193}, {16598, 18201}, {20337, 20488}

X(20361) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(2053)

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

See X(20356).

X(20361) lies on these lines: {1, 20370}, {7, 192}, {2294, 20284}, {3721, 17470}, {3726, 17452}, {17447, 20596}, {20338, 20489}

X(20362) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(2054)

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

See X(20356).

X(20362) lies on these lines: {1, 1929}, {38, 20364}, {1107, 20597}, {3726, 20360}, {4966, 10026}, {17149, 17778}, {20339, 20490}

X(20363) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(3009)

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

See X(20356).

X(20363) lies on these lines: {1, 6}, {39, 3993}, {75, 16604}, {192, 2275}, {292, 3685}, {312, 16606}, {350, 19565}, {536, 9296}, {726, 1015}, {740, 1575}, {798, 4083}, {982, 17459}, {1574, 4709}, {1921, 20530}, {3121, 4358}, {3721, 17470}, {3726, 20366}, {3912, 18904}, {4871, 6377}, {16742, 18157}, {17475, 20459}, {20340, 20491}, {20356, 20358}, {20364, 20590}

X(20364) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(2112)

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

See X(20356).

X(20364) lies on these lines: {1, 3506}, {37, 17447}, {38, 20362}, {3721, 20356}, {7004, 20359}, {20341, 20492}, {20363, 20590}

X(20365) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(2144)

Barycentrics a^2 (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20356).

X(20365) lies on these lines: {1, 18783}, {17464, 20366}, {20342, 20493}

X(20366) = (X(1), X(2), X(6), X(31); X(37), X(1), X(38), X(3721)) COLLINEATION IMAGE OF X(20332)

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

See X(20356).

X(20366) lies on these lines: {1, 727}, {2, 38}, {2170, 3721}, {2292, 4128}, {2611, 4137}, {3726, 20363}, {7004, 20359}, {17464, 20365}, {20343, 20494}

X(20367) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(672)

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

Let m denote the indicated collineation, which maps the 2nd equal areas cubic, K155, onto a cubic, m(K155), which is K343. The appearance of (i,j) in the following list means that m(X(i)) = X(j), where X(i) is on K155:

(1,9), (2,1), (6,63), (31,1759), (105,16550), (238,3509), (292,17738), (365,510), (672,20367), (1423,20368), (1931,20369), (2053,20370), (2054,20371), (3009,20372), (2112,20373), (2144,20375), (20332, 20375)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -a : b : c
m(A') = a (a^(3/2) + b^(3/2) + c^(3/2)) : -b (a^(3/2) + b^(3/2) - c^(3/2)) : -c (a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = -a^2 - 2 b c : a b : a c
m(A2) = a (b^3 + a b c + c^3) : -b (b^3 + a b c - c^3) : -c (-b^3 + a b c + c^3)
m(A3) = -a (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : b (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : c (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = a (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) : -b (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) : -c (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)
m(A5) = a (a^5 - a^4 b - a^2 b^3 + a b^4 - a^4 c - a^3 b c + a b^3 c + b^4 c - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + a c^4 + b c^4), -b (a^5 - a^4 b - a^2 b^3 + a b^4 - a^4 c - a^3 b c + a b^3 c + b^4 c + 2 a^3 c^2 + 2 a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 3 a b c^3 - b^2 c^3 - a c^4 - b c^4), -c (a^5 - a^4 b + 2 a^3 b^2 - a^2 b^3 - a b^4 - a^4 c - a^3 b c + 2 a^2 b^2 c - 3 a b^3 c - b^4 c + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 + a b c^3 + b^2 c^3 + a c^4 + b c^4),

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a(-x + y + z) : b(x - y + z) : c(x + y - z) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(1)), where A'B'C' = excentral triangle.

Let f(a,b,c,x,y,z) = a b (b-c) c (b^2+b c+c^2) x^3-a c (a^4+a b^3-b^2 c^2-a c^3) y^2 z+a b (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20367) lies on these lines: {1, 3}, {2, 2140}, {7, 573}, {9, 3739}, {19, 16551}, {63, 169}, {71, 142}, {75, 16574}, {101, 11349}, {116, 857}, {150, 6999}, {213, 3752}, {239, 514}, {244, 3747}, {307, 12610}, {320, 3882}, {527, 2183}, {579, 4000}, {583, 17366}, {672, 3008}, {673, 20605}, {1018, 3912}, {1054, 2664}, {1086, 2245}, {1100, 18164}, {1111, 16609}, {1400, 3663}, {1423, 4862}, {1445, 1766}, {1462, 13329}, {1765, 18655}, {2260, 3946}, {2269, 3664}, {2270, 5781}, {2481, 6996}, {3191, 16453}, {3219, 16815}, {3306, 6205}, {3501, 17284}, {3509, 16550}, {3928, 5792}, {4253, 5222}, {4266, 4644}, {4271, 17365}, {4292, 15970}, {4312, 6210}, {4447, 14839}, {4650, 16476}, {4858, 8680}, {5290, 9548}, {5745, 17050}, {11024, 19853}, {11343, 16788}, {12717, 15299}, {14829, 17143}, {16548, 16560}, {17077, 17220}, {17175, 17185}, {17738, 20372}

X(20367) = X(241)-of-tangential-of-excentral-triangle

X(20368) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(1423)

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

See X(20365).

X(20368) lies on these lines: {1, 3}, {2, 6210}, {4, 3831}, {9, 1755}, {31, 19649}, {43, 511}, {63, 6194}, {238, 16434}, {516, 3840}, {537, 3928}, {573, 6685}, {750, 4220}, {975, 8235}, {978, 19514}, {1350, 1376}, {1695, 15489}, {1698, 15973}, {1730, 11358}, {1742, 4192}, {1766, 3509}, {1768, 20373}, {2050, 11372}, {2319, 20606}, {2941, 20369}, {3305, 7609}, {3646, 19273}, {3736, 18163}, {3753, 19530}, {3794, 13588}, {5250, 19278}, {5437, 9746}, {6684, 9548}, {8227, 19864}, {10860, 12717}, {15310, 19540}, {17122, 19544}, {20348, 20498}

X(20369) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(1931)

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

See X(20365).

X(20369) lies on these lines: {1, 1326}, {9, 46}, {75, 267}, {484, 4645}, {523, 1019}, {750, 846}, {1719, 11679}, {1757, 2640}, {2941, 20368}, {3336, 3821}, {3337, 17302}, {3509, 4037}, {20349, 20499}

X(20369) = isogonal conjugate of isotomic conjugate of X(20450)

X(20370) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(2053)

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

See X(20365).

X(20370) lies on these lines: {1, 20361}, {194, 5088}, {1759, 16566}, {1766, 3509}, {16551, 20608}, {20350, 20503}

X(20371) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(2054)

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

See X(20365).

X(20371) lies on these lines: {1, 1929}, {63, 20373}, {3509, 4037}, {16552, 20609}, {20351, 20500}

X(20372) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(3009)

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

See X(20365).

X(20372) lies on these lines: {1, 6}, {672, 4368}, {726, 20459}, {798, 812}, {1018, 3685}, {1759, 16566}, {2225, 4358}, {2235, 18792}, {3509, 20375}, {3923, 16549}, {4011, 5364}, {4253, 17350}, {5150, 19554}, {16574, 18046}, {17738, 20367}, {20352, 20501}, {20373, 20602}

X(20373) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(2112)

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

See X(20365).

X(20373) lies on these lines: {1, 3506}, {9, 141}, {63, 20371}, {335, 7096}, {1759, 3760}, {1768, 20368}, {3509, 4396}, {20353, 20502}, {20372, 20602}

X(20374) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(2144)

Barycentrics a^2 (-2 a^8 b^6 + 2 a^5 b^9 + a^9 b^4 c - a^3 b^10 c + 8 a^7 b^5 c^2 - 10 a^4 b^8 c^2 - 5 a^8 b^3 c^3 + 7 a^5 b^6 c^3 + 7 a^2 b^9 c^3 + a^9 b c^4 - 17 a^6 b^4 c^4 + 3 a^3 b^7 c^4 - b^10 c^4 + 8 a^7 b^2 c^5 + 3 a^4 b^5 c^5 - 4 a b^8 c^5 - 2 a^8 c^6 + 7 a^5 b^3 c^6 - 5 a^2 b^6 c^6 + 3 a^3 b^4 c^7 + 4 b^7 c^7 - 10 a^4 b^2 c^8 - 4 a b^5 c^8 + 2 a^5 c^9 + 7 a^2 b^3 c^9 - a^3 b c^10 - b^4 c^10) : :

See X(20365).

X(20374) lies on these lines: {1, 18783}, {16550, 20375}

X(20375) = (X(1), X(2), X(6), X(31); X(9), X(1), X(63), X(1759)) COLLINEATION IMAGE OF X(20332)

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

See X(20365).

X(20375) lies on these lines: {1, 727}, {63, 1054}, {165, 15323}, {191, 5539}, {1698, 5518}, {1759, 5540}, {1768, 20368}, {3509, 20372}, {16550, 20374}

X(20376) = X(5)X(13289)∩X(54)X(67)

Barycentrics (R^2*(64*R^2+SA-41*SW)+6*SW^2) *S^2-2*(R^2*(16*R^2-9*SW)+SW^ 2)*SB*SC : :
X(20376) = 5*X(631)-X(2917) = 9*X(5054)-X(9920) = 2*X(6689)+X(6696)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27952.

X(20376) lies on these lines: {5, 13289}, {54, 67}, {140, 13470}, {511, 11262}, {631, 2917}, {1209, 10257}, {3541, 15578}, {3574, 13568}, {5054, 9920}, {5498, 12038}, {6689, 6696}, {6699, 11802}, {8254, 10628}, {11574, 16196}

X(20376) = complement of the complement of X(32345)

X(20377) = X(5)X(25560)∩X(13)X(627)

Barycentrics 6*a^6 - 14*a^4*b^2 + 3*a^2*b^4 + 5*b^6 - 14*a^4*c^2 - 32*a^2*b^2*c^2 - 5*b^4*c^2 + 3*a^2*c^4 - 5*b^2*c^4 + 5*c^6 - 2*Sqrt[3]*(2*a^4 + 6*a^2*b^2 - 5*b^4 + 6*a^2*c^2 + 12*b^2*c^2 - 5*c^4)*S : :

X(20377) lies on these lines: {5, 25560}, {13, 627}, {17, 671}, {115, 8259}, {140, 6669}, {530, 629}, {6694, 22832}, {10611, 33465}, {11305, 22737}, {11602, 32552}, {20415, 33560}

X(20378) = X(5)X(25559)∩X(14)X(628)

Barycentrics 6*a^6 - 14*a^4*b^2 + 3*a^2*b^4 + 5*b^6 - 14*a^4*c^2 - 32*a^2*b^2*c^2 - 5*b^4*c^2 + 3*a^2*c^4 - 5*b^2*c^4 + 5*c^6 + 2*Sqrt[3]*(2*a^4 + 6*a^2*b^2 - 5*b^4 + 6*a^2*c^2 + 12*b^2*c^2 - 5*c^4)*S : :

X(20378) lies on these lines: {5, 25559}, {14, 628}, {18, 671}, {115, 8260}, {140, 6670}, {531, 630}, {6695, 22831}, {10612, 33464}, {11306, 22736}, {11603, 32553}, {20416, 33561}

X(20379) = COMPLEMENT OF X(5609)

Barycentrics (27*R^2-3*SA-4*SW)*S^2-(9*R^2+ SW)*SB*SC : :
X(20369) = X(3)+3*X(9140) = 3*X(3)-7*X(15057) = X(4)-5*X(15027) = X(4)+3*X(20126) = 5*X(5)-3*X(113) = X(5)-3*X(125) = X(5)+3*X(10264) = 3*X(5)-X(15063) = 5*X(5)-6*X(15088) = 2*X(5)-3*X(20304) = X(113)-5*X(125) = X(113)+5*X(10264) = 9*X(113)-5*X(15063) = 3*X(113)+5*X(16003) = 2*X(113)-5*X(20304) = 9*X(125)-X(15063) = 5*X(125)-2*X(15088) = 3*X(125)+X(16003) = 9*X(9140)+7*X(15057) = 9*X(10264)+X(15063) = 5*X(10264)+2*X(15088) = 5*X(15027)+3*X(20126)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27952.

X(20379) lies on these lines: {2, 5609}, {3, 9140}, {4, 15027}, {5, 113}, {20, 265}, {26, 5621}, {30, 15153}, {67, 18281}, {74, 382}, {110, 3526}, {140, 542}, {143, 2781}, {146, 3855}, {381, 15054}, {399, 5070}, {541, 546}, {548, 13470}, {631, 1511}, {632, 5642}, {858, 1154}, {1112, 15559}, {1204, 18379}, {1539, 3843}, {1594, 13148}, {1595, 12828}, {1656, 14094}, {1657, 15021}, {1899, 18580}, {1986, 15101}, {2777, 3853}, {3090, 5655}, {3525, 9143}, {3528, 12121}, {3530, 6699}, {3564, 15115}, {3627, 10990}, {3628, 13393}, {3830, 15044}, {3832, 7728}, {3851, 10706}, {3856, 13566}, {3861, 7687}, {4301, 12261}, {4309, 12904}, {4317, 12903}, {5054, 15034}, {5067, 12317}, {5169, 5946}, {5498, 10116}, {5622, 15132}, {5890, 7579}, {5972, 16239}, {6070, 16340}, {6723, 10272}, {9588, 12778}, {9657, 10081}, {9670, 10065}, {9714, 13171}, {9729, 13565}, {10095, 12099}, {10627, 12359}, {10733, 15041}, {11362, 13605}, {11563, 13399}, {11579, 15069}, {11645, 12105}, {11693, 15713}, {12079, 16168}, {12161, 15106}, {12244, 17578}, {12295, 14677}, {12383, 15717}, {12824, 15026}, {12902, 15055}, {13567, 15465}, {14849, 18331}, {15020, 15720}, {15039, 15694}, {15357, 15535}

X(20379) = midpoint of X(i) and X(j) for these {i,j}: {5, 16003}, {74, 10113}, {1539, 10620}, {1986, 15101}, {3627, 10990}, {3628, 13393}, {6070, 16340}, {11563, 13399}, {12295, 14677}, {15357, 15535}
X(20379) = complement of X(5609)
X(20379) = X(13451)-of-anti-orthocentroidal triangle
X(20379) = reflection of X(113) in the line X(526)X(15088)
X(20379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 10264, 16003), (125, 16003, 5), (3448, 15061, 1511), (10620, 14644, 1539), (10706, 15025, 3851), (13561, 16270, 20304), (15027, 20126, 4)

X(20380) = LEMOINE INELLIPSE ANTIPODE OF X(8288)

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

Centers X(20380)-X(20390) were contributed by César Lozada, July 23, 2018.

X(20380) lies on the Lemoine inellipse, the conic {A, B, C, X(67), X(524)} and these lines: {2,67}, {6,598}, {524,7664}, {597,8288}, {690,15303}, {1383,1992}, {5032,20099}, {5182,11636}, {9855,10510}

X(20380) = antipode of X(8288) in the Lemoine inellipse
X(20380) = barycentric product X(598)*X(2482)
X(20380) = barycentric quotient X(i)/X(j) for these (i,j): (1383, 10630), (1649, 3906), (2482, 599)

X(20381) = X(524)X(8288) ∩ X(7841)X(14246)

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

X(20381) lies on the Lemoine inellipse and these lines: {524,8288}, {1499,20380}, {7841,14246}

X(20381) = antipode of X(20382) in the Lemoine inellipse

X(20382) = LEMOINE INELLIPSE ANTIPODE OF X(20381)

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

X(20382) lies on the Lemoine inellipse and these lines: {524,7664}, {597,20381}, {598,843}, {599,6082}, {1499,8288}, {1648,5099}, {3124,8599}, {6791,12073}

X(20382) = antipode of X(20381) in the Lemoine inellipse

X(20383) = 4th INTERSECTION OF LEMOINE INELLIPSE AND 3rd LEMOINE CIRCLE

Barycentrics (b^2-c^2)^2*(13*a^6+9*(b^2+c^2)*a^4-3*(b^4+11*b^2*c^2+c^4)*a^2+(b^2+c^2)^3)^2*(a^2-2*b^2-2*c^2) : :

Lemoine inellipse and 3rd Lemoine circle pass both through the vertices of the Lemoine triangle (cevian triangle of X(598)). X(20383) is their 4th intersection.

X(20383) lies on the 3rd Lemoine circle, the Lemoine inellipse and the line {597,20384}

X(20384) = LEMOINE INELLIPSE ANTIPODE OF X(20383)

Barycentrics (2*a^2+2*b^2-c^2)*(2*a^2-b^2+2*c^2)*(2*a^8-(b^2+c^2)*a^6+3*(3*b^4-20*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(11*b^4+19*b^2*c^2+11*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2+c^2)^2)^2 : :

X(20384) lies on the Lemoine inellipse and the line {597,20383}

X(20384) = antipode of X(20383) in the Lemoine inellipse

X(20385) = X(524)X(20383) ∩ X(1499)X(20384)

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

X(20385) lies on the Lemoine inellipse and these lines: {524,20383}, {597,20386}, {1499,20384}, {12073,20381}

X(20386) = LEMOINE INELLIPSE ANTIPODE OF X(20385)

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

X(20386) lies on the Lemoine inellipse and these lines: {524,20384}, {597,20385}, {1499,20383}, {6791,12073}

X(20386) = antipode of X(20385) in the Lemoine inellipse

X(20387) = 3rd LEMOINE CIRCLE ANTIPODE OF X(115)

Barycentrics (2*a^6+4*(b^2+c^2)*a^4-(3*b^4+4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(6*a^8-7*(b^2+c^2)*a^6-2*(2*b^4+23*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(7*b^4+13*b^2*c^2+7*c^4)*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^4+6*b^2*c^2+c^4)) : :

X(20387) lies on the 3rd Lemoine circle and these lines: {114,547}, {115,8145}

X(20387) = reflection of X(115) in X(8145)
X(20387) = antipode of X(115) in the 3rd Lemoine circle

X(20388) = 3rd LEMOINE CIRCLE ANTIPODE OF X(20383)

Barycentrics (2*a^10-(b^2+c^2)*a^8+2*(8*b^4-37*b^2*c^2+8*c^4)*a^6+(b^2+c^2)*(8*b^4+25*b^2*c^2+8*c^4)*a^4-(10*b^8+10*c^8-b^2*c^2*(17*b^4-54*b^2*c^2+17*c^4))*a^2+(b^4-c^4)^2*(b^2+c^2))*(3*a^12-11*(b^2+c^2)*a^10+(21*b^4-22*b^2*c^2+21*c^4)*a^8-(b^2+c^2)*(4*b^4-75*b^2*c^2+4*c^4)*a^6-(23*b^8+23*c^8+b^2*c^2*(143*b^4-93*b^2*c^2+143*c^4))*a^4+(b^2+c^2)*(15*b^8+15*c^8+2*b^2*c^2*(10*b^4-13*b^2*c^2+10*c^4))*a^2-(b^8+c^8+2*b^2*c^2*(4*b^4-11*b^2*c^2+4*c^4))*(b^2+c^2)^2) : :

X(20388) lies on the 3rd Lemoine circle and the line {8145,20383}

X(20388) = reflection of X(20383) in X(8145)
X(20388) = antipode of X(20383) in the 3rd Lemoine circle

X(20389) = COMPLEMENT OF X(12074)

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

X(20389) lies on the 3rd Lemoine circle, the nine-point circle and these lines: {2,12074}, {113,20301}, {114,547}, {126,3934}, {3258,17436}, {5099,7668}, {6092,20388}, {7711,7859}, {7797,11638}

X(20389) = complement of X(12074)

X(20390) = 3rd LEMOINE CIRCLE ANTIPODE OF X(20389)

Barycentrics (2*a^8+9*(b^2+c^2)*a^6-2*(2*b^4-19*b^2*c^2+2*c^4)*a^4-(b^2+c^2)*(9*b^4+7*b^2*c^2+9*c^4)*a^2+(2*b^4+13*b^2*c^2+2*c^4)*(b^2-c^2)^2)*(6*a^12+8*(b^2+c^2)*a^10-2*(11*b^4+40*b^2*c^2+11*c^4)*a^8-(b^2+c^2)*(7*b^4+94*b^2*c^2+7*c^4)*a^6+(17*b^8+17*c^8+4*b^2*c^2*(20*b^4+21*b^2*c^2+20*c^4))*a^4-(b^2+c^2)*(b^8+c^8+b^2*c^2*(b^4-24*b^2*c^2+c^4))*a^2-(b^4+5*b^2*c^2+c^4)*(b^4-c^4)^2) : :

X(20390) lies on the 3rd Lemoine circle and these lines: {115,5066}, {8145,20389}

X(20390) = antipode of X(20389) in the 3rd Lemoine circle

X(20391) = MIDPOINT OF X(6696) AND X(10024)

Barycentrics 2*S^4+(160*R^4-7*R^2*SA-81*R^2 *SW+2*SA^2-2*SB*SC+10*SW^2)*S^ 2-2*(16*R^2-5*SW)*R^2*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27960.

X(20391) lies on these lines: {30, 20191}, {125, 20376}, {140, 14076}, {597, 15047}, {1503, 7542}, {2937, 15578}, {3520, 15081}, {5893, 15062}, {6696, 10024}, {10628, 12006}

X(20391) = midpoint of X(6696) and X(10024)

X(20392) = 29TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^18-7 a^16 b^2+5 a^14 b^4+10 a^12 b^6-19 a^10 b^8+8 a^8 b^10+7 a^6 b^12-10 a^4 b^14+5 a^2 b^16-b^18-7 a^16 c^2+18 a^14 b^2 c^2-12 a^12 b^4 c^2-2 a^10 b^6 c^2+6 a^8 b^8 c^2-17 a^6 b^10 c^2+33 a^4 b^12 c^2-27 a^2 b^14 c^2+8 b^16 c^2+5 a^14 c^4-12 a^12 b^2 c^4+12 a^10 b^4 c^4-5 a^8 b^6 c^4+4 a^6 b^8 c^4-36 a^4 b^10 c^4+57 a^2 b^12 c^4-25 b^14 c^4+10 a^12 c^6-2 a^10 b^2 c^6-5 a^8 b^4 c^6+12 a^6 b^6 c^6+13 a^4 b^8 c^6-65 a^2 b^10 c^6+37 b^12 c^6-19 a^10 c^8+6 a^8 b^2 c^8+4 a^6 b^4 c^8+13 a^4 b^6 c^8+60 a^2 b^8 c^8-19 b^10 c^8+8 a^8 c^10-17 a^6 b^2 c^10-36 a^4 b^4 c^10-65 a^2 b^6 c^10-19 b^8 c^10+7 a^6 c^12+33 a^4 b^2 c^12+57 a^2 b^4 c^12+37 b^6 c^12-10 a^4 c^14-27 a^2 b^2 c^14-25 b^4 c^14+5 a^2 c^16+8 b^2 c^16-c^18 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27962.

X(20392) lies on this line: {2,3}

X(20393) = MIDPOINT OF X(5) AND X(1138)

Barycentrics 16*S^4-(27*R^2*(25*R^2+4*SA- 12*SW)-24*SA^2+36*SW^2)*S^2+3* (27*R^4-4*SW^2)*SB*SC : :

X(20393) = X(3448)-3*X(20124) = 2*X(13392)+X(14731)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20393) lies on these lines: {5, 1138}, {30, 110}, {3448, 20124}, {3628, 14993}, {13392, 14731}

X(20393) = midpoint of X(5) and X(1138)

X(20394) = X(17)X(299)∩X(61)X(5459)

Barycentrics (9*SA-22*SW)*S^2+sqrt(3)*(5* SA^2-6*SB*SC-7*SW^2)*S-7*SB* SC*SW : :
Barycentrics -2*sqrt(3)*(2*a^4+6*(b^2+c^2)* a^2+4*b^2*c^2-c^4-b^4)*S+3*a^ 2*(b^4+c^4)+6*a^6-14*(b^2+c^2) *a^4-32*b^2*c^2*a^2+5*(b^4-c^ 4)*(b^2-c^2) : :

X(20394) = X(61)+3*X(5459) = X(635)-3*X(6669) = X(635)+3*X(14136)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20394) lies on these lines: {17, 299}, {61, 5459}, {115, 8259}, {397, 530}, {546, 20252}

X(20395) = X(18)X(298)∩X(62)X(5460)

Barycentrics (9*SA-22*SW)*S^2-sqrt(3)*(5* SA^2-6*SB*SC-7*SW^2)*S-7*SB* SC*SW : :
Barycentrics 2*sqrt(3)*(2*a^4+6*(b^2+c^2)* a^2+4*b^2*c^2-c^4-b^4)*S+3*a^ 2*(b^4+c^4)+6*a^6-14*(b^2+c^2) *a^4-32*b^2*c^2*a^2+5*(b^4-c^ 4)*(b^2-c^2) : :

X(20395) = X(62)+3*X(5460) = X(636)-3*X(6670) = X(636)+3*X(14137)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20395) lies on these lines: {18, 298}, {62, 5460}, {115, 8260}, {398, 531}, {546, 20253}

X(20396) = MIDPOINT OF X(5) AND X(20379)

Barycentrics (45*R^2-3*SA-8*SW)*S^2+(9*R^2- 5*SW)*SB*SC : :

X(20396) = 3*X(2)+5*X(15027), 7*X(5)-3*X(113), X(5)+3*X(125), 5*X(5)+3*X(10264), 5*X(5)-X(15063), 2*X(5)-3*X(15088), 3*X(5)+X(16003), X(5)-3*X(20304), X(113)+7*X(125), 5*X(113)+7*X(10264), 15*X(113)-7*X(15063), 2*X(113)-7*X(15088), 9*X(113)+7*X(16003), X(113)-7*X(20304), 3*X(113)+7*X(20379), 5*X(125)-X(10264), 15*X(125)+X(15063), 2*X(125)+X(15088), 9*X(125)-X(16003)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20396) lies on these lines: {2, 15027}, {5, 113}, {20, 10113}, {74, 3843}, {110, 5070}, {143, 12099}, {156, 5622}, {265, 631}, {381, 15025}, {382, 12041}, {541, 3850}, {542, 3628}, {547, 16534}, {548, 6699}, {858, 13391}, {1511, 3526}, {1539, 3832}, {1656, 5609}, {1657, 15044}, {2777, 3861}, {2781, 10095}, {3091, 20126}, {3448, 5067}, {3530, 17702}, {3830, 15021}, {3845, 10990}, {3851, 15054}, {3853, 7687}, {3855, 7728}, {5055, 14094}, {5056, 5655}, {5072, 10706}, {5169, 13364}, {5449, 6698}, {5621, 13861}, {5946, 7579}, {6723, 16239}, {7486, 14643}, {7577, 13148}, {9656, 10081}, {9671, 10065}, {10109, 13393}, {10125, 11565}, {10733, 15696}, {11362, 12261}, {11482, 13169}, {11704, 13491}, {12121, 15717}, {12358, 13358}, {13371, 16982}, {14981, 15535}, {15020, 15694}, {15023, 15707}, {15055, 17800}, {15101, 16222}, {15153, 16531}, {17578, 20127}

X(20396) = midpoint of X(i) and X(j) for these {i,j}: {5, 20379}, {12358, 13358}, {15153, 16531}
X(20396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 125, 20379), (5, 10264, 15063), (382, 15057, 12041), (1656, 9140, 5609), (14644, 15057, 382), (15061, 15081, 10113), (20304, 20379, 5)

X(20397) = X(3)X(125)∩X(74)X(3091)

Barycentrics (18*R^2-3*SA-SW)*S^2-(18*R^2+ 9*SA-SW)*SB*SC : :

X(20397) = 9*X(2)-X(14094), 3*X(2)+X(16003), X(3)+3*X(125), 5*X(3)+3*X(265), X(3)-3*X(6699), 11*X(3)-3*X(12121), 13*X(3)+3*X(12902), 3*X(3)+5*X(15027), X(3)-9*X(15061), 7*X(3)-3*X(16163), 5*X(125)-X(265), 11*X(125)+X(12121), 13*X(125)-X(12902), 9*X(125)-5*X(15027), X(125)+3*X(15061), 7*X(125)+X(16163), X(265)+5*X(6699), 11*X(265)+5*X(12121), 13*X(265)-5*X(12902), X(265)+15*X(15061), 7*X(265)+5*X(16163), X(14094)+3*X(16003), X(14094)-3*X(16534)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20397) lies on these lines: {2, 14094}, {3, 125}, {4, 15021}, {5, 541}, {74, 3091}, {110, 3525}, {113, 3090}, {140, 542}, {146, 15022}, {182, 15132}, {381, 10990}, {539, 10257}, {546, 2777}, {576, 15118}, {631, 9140}, {632, 5609}, {895, 3546}, {1147, 5622}, {1511, 14869}, {1539, 3857}, {1656, 15063}, {2781, 5462}, {3146, 14644}, {3292, 14156}, {3448, 10303}, {3524, 15023}, {3526, 5642}, {3529, 12295}, {3541, 12828}, {3592, 8994}, {3594, 13969}, {3627, 7687}, {3628, 5663}, {5056, 10706}, {5070, 5655}, {5072, 7728}, {5076, 20127}, {5079, 10620}, {5159, 13754}, {5446, 12099}, {5447, 14984}, {5621, 6642}, {7555, 17712}, {10113, 15704}, {10124, 13393}, {10628, 15012}, {10733, 17538}, {11693, 15702}, {11801, 12103}, {11806, 12358}, {12106, 15579}, {12359, 15115}, {12811, 15088}, {13202, 15041}, {17853, 18439}, {18400, 18571}

X(20397) = midpoint of X(i) and X(j) for these {i,j}: {140, 20379}, {11806, 12358}, {12359, 15115}
X(20397) = complement of X(16534)
X(20397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 16003, 16534), (125, 15061, 6699), (632, 5609, 5972), (632, 10264, 5609), (1656, 20126, 15063), (3090, 15054, 113), (3448, 10303, 15034), (3529, 15044, 12295), (3529, 15081, 15044), (15021, 15025, 4), (15025, 15057, 15021), (15044, 15055, 3529), (15054, 15059, 3090), (15055, 15081, 12295)

X(20398) = X(3)X(115)∩X(98)X(3091)

Barycentrics 12*S^4+(6*SA^2-3*SA*SW-4*SW^2) *S^2+SB*SC*SW^2 : :

X(20398) = X(3)+3*X(115), X(3)-3*X(6036), 5*X(3)+3*X(6321), X(4)+3*X(6055), X(4)-9*X(9166), X(5)-3*X(5461), X(20)+3*X(9880), 3*X(98)+5*X(3091), 3*X(99)-11*X(3525), 3*X(114)-7*X(3090), X(114)-5*X(14061), X(114)+3*X(14651), 5*X(115)-X(6321), 7*X(3090)-15*X(14061), 7*X(3090)+9*X(14651), 3*X(5461)+X(11623), X(5609)+3*X(15535), 5*X(6036)+X(6321), X(6055)+3*X(9166), 5*X(14061)+3*X(14651)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20398) lies on these lines: {3, 115}, {4, 6055}, {5, 542}, {20, 9880}, {98, 3091}, {99, 3525}, {114, 3090}, {125, 18347}, {140, 543}, {147, 15022}, {148, 10303}, {381, 10991}, {546, 2794}, {576, 3767}, {620, 632}, {625, 5965}, {631, 671}, {1656, 11632}, {2165, 11511}, {2482, 3526}, {2782, 3628}, {3146, 14639}, {3455, 7506}, {3592, 8980}, {3594, 13967}, {3627, 12042}, {3843, 14830}, {5056, 6054}, {5067, 12243}, {5068, 11177}, {5070, 8724}, {5072, 6033}, {5079, 12188}, {5465, 16003}, {5663, 11554}, {6034, 11477}, {6771, 10654}, {6774, 10653}, {7607, 7833}, {7617, 10168}, {10723, 17538}, {11005, 15025}, {11318, 19662}, {11362, 12258}, {12117, 15717}, {12811, 15092}, {14060, 14669}, {14160, 18907}, {14162, 15516}, {15027, 18332}, {15300, 15694}

X(20398) = midpoint of X(5) and X(11623)
X(20398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (631, 671, 10992), (1656, 11632, 14981), (5461, 11623, 5), (14061, 14651, 114), (14971, 14981, 1656)

X(20399) = X(3)X(114)∩X(99)X(3091)

Barycentrics 6*S^4-(3*SA^2+SW^2)*S^2-2*SB* SC*SW^2 : :

X(20399) = 3*X(2)+X(14981), X(3)+3*X(114), X(3)-3*X(620), 5*X(3)+3*X(6033), X(3)-9*X(15561), X(4)+3*X(2482), 5*X(4)+3*X(12117), 3*X(98)-11*X(3525), 3*X(99)+5*X(3091), 5*X(114)-X(6033), X(114)+3*X(15561), 5*X(620)+X(6033), X(620)-3*X(15561), 5*X(2482)-X(12117), X(6033)+15*X(15561)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20399) lies on these lines: {2, 11623}, {3, 114}, {4, 2482}, {5, 543}, {98, 3525}, {99, 3091}, {115, 3090}, {140, 542}, {147, 10303}, {148, 15022}, {381, 10992}, {538, 10011}, {575, 6680}, {576, 7764}, {631, 6054}, {632, 6036}, {671, 5056}, {1656, 5461}, {2782, 3628}, {2936, 7395}, {3455, 7509}, {3526, 6055}, {3544, 14639}, {3545, 15300}, {3592, 8997}, {3594, 13989}, {3851, 9880}, {5067, 14971}, {5068, 8591}, {5070, 11632}, {5072, 6321}, {5079, 13188}, {5965, 14693}, {6248, 15850}, {7486, 9166}, {7619, 11178}, {7786, 9772}, {7815, 12177}, {9754, 17131}, {9881, 11522}, {10516, 14928}, {10722, 17538}, {11005, 15034}, {12042, 14869}, {14094, 15357}, {14830, 15720}, {15069, 18800}

X(20399) = complement of X(11623)
X(20399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14981, 11623), (114, 15561, 620), (631, 6054, 10991), (9167, 10991, 631)

X(20400) = X(3)X(119)∩X(100)X(3091)

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

X(20400) = X(3)+3*X(119), X(3)-3*X(3035), 5*X(3)+3*X(10742), X(4)+3*X(6174), 3*X(11)-7*X(3090), 3*X(100)+5*X(3091), 3*X(104)-11*X(3525), 5*X(119)-X(10742), 3*X(149)-19*X(15022), 3*X(153)+13*X(10303), 3*X(381)+X(10993), 5*X(631)+3*X(10711), 5*X(632)-3*X(6713), 5*X(632)+3*X(11698), 5*X(3035)+X(10742)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20400) lies on these lines: {3, 119}, {4, 6174}, {5, 528}, {11, 1058}, {12, 6946}, {100, 3091}, {104, 3525}, {149, 15022}, {153, 10303}, {381, 10993}, {546, 5840}, {631, 10711}, {632, 6713}, {952, 1125}, {1145, 7982}, {1376, 6982}, {1532, 5537}, {1537, 7991}, {1698, 5660}, {2551, 12762}, {2800, 5044}, {2801, 3634}, {3036, 6265}, {3304, 10956}, {3544, 6154}, {3592, 13922}, {3594, 13991}, {3614, 14882}, {3814, 5842}, {3913, 6981}, {4421, 6973}, {4995, 6965}, {5056, 10707}, {5072, 10738}, {5079, 12331}, {5552, 7681}, {5587, 10609}, {5603, 13996}, {5704, 14151}, {5851, 11231}, {5854, 10915}, {6691, 10942}, {6826, 13272}, {6863, 9711}, {6959, 12607}, {6970, 11236}, {10175, 12019}, {10728, 17538}, {10778, 15025}, {13145, 20117}

X(20400) = midpoint of X(i) and X(j) for these {i,j}: {3036, 6265}, {6246, 9945}

X(20401) = X(3)X(118)∩X(101)X(3091)

Barycentrics 2 a^8-2 a^7 b-8 a^6 b^2+9 a^5 b^3+5 a^4 b^4-4 a^3 b^5-2 a^2 b^6-3 a b^7+3 b^8-2 a^7 c+2 a^6 b c+7 a^5 b^2 c-7 a^4 b^3 c-8 a^3 b^4 c+8 a^2 b^5 c+3 a b^6 c-3 b^7 c-8 a^6 c^2+7 a^5 b c^2-12 a^4 b^2 c^2+12 a^3 b^3 c^2-2 a^2 b^4 c^2+9 a b^5 c^2-6 b^6 c^2+9 a^5 c^3-7 a^4 b c^3+12 a^3 b^2 c^3-8 a^2 b^3 c^3-9 a b^4 c^3+3 b^5 c^3+5 a^4 c^4-8 a^3 b c^4-2 a^2 b^2 c^4-9 a b^3 c^4+6 b^4 c^4-4 a^3 c^5+8 a^2 b c^5+9 a b^2 c^5+3 b^3 c^5-2 a^2 c^6+3 a b c^6-6 b^2 c^6-3 a c^7-3 b c^7+3 c^8 : :

X(20401) = X(3)+3*X(118), X(3)-3*X(6710), 5*X(3)+3*X(10741), 3*X(101)+5*X(3091), 3*X(103)-11*X(3525), 3*X(116)-7*X(3090), 5*X(118)-X(10741), 3*X(150)-19*X(15022), 3*X(152)+13*X(10303), 5*X(631)+3*X(10710), 5*X(632)-3*X(6712), 11*X(5056)-3*X(10708), 11*X(5072)-3*X(10739), 5*X(5818)+3*X(15735), 5*X(6710)+X(10741)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27964.

X(20401) lies on these lines: {3, 118}, {5, 544}, {101, 3091}, {103, 3525}, {116, 3090}, {150, 15022}, {152, 10303}, {631, 10710}, {632, 6712}, {2801, 13373}, {2808, 3628}, {4845, 10588}, {5056, 10708}, {5072, 10739}, {5818, 15735}, {10727, 17538}

X(20402) = X(1843)X(5895)∩X(3051)X(20232)

Barycentrics (S^2-SB*SC)*(5*S^2+16*(SA+SB)* R^2-4*SA*SB+5*SC^2-4*SW^2)*(5* S^2+16*(SA+SC)*R^2-4*SA*SC+5* SB^2-4*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27967.

X(20402) lies on these lines: {1843, 5895}, {3051, 20232}

X(20403) = (name pending)

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

X(20404) = ISOGONAL CONJUGATE OF X(20403)

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

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Fermat axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, 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. Also, X(20404) = X(2)-of-A'B'C'. (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, August 29, 2018)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27967.

X(20404) lies on the circumcircle and these lines: {74, 14830}, {110, 1649}, {111, 1648}, {476, 804}, {477, 2782}, {526, 805}, {542, 842}, {690, 691}, {729, 14901}, {2698, 5663}, {2857, 5939}, {11636, 15342}

X(20404) = isogonal conjugate of X(20403)
X(20404) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (110, 3, 11006), (476, 3, 76), (477, 3, 804), (691, 3, 67), (805, 3, 74), (842, 3, 6334) , (2698, 3, 14933)
X(20404) = reflection of X(476) in line X(3)X(76)
X(20404) = anticomplement of X(35582)

X(20405) = EULER LINE INTERCEPT OF X(15162)X(17810)

Barycentrics (SW-4*|OH|*R)*S^2-3*SB*SC*SW : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27974.

X(20405) lies on these lines: {2, 3}, {15162, 17810}

X(20405) = {X(1113), X(1312)}-harmonic conjugate of X(468)

X(20406) = EULER LINE INTERCEPT OF X(15163)X(17810)

Barycentrics (SW+4*|OH|*R)*S^2-3*SB*SC*SW : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27974.

X(20406) lies on these lines: {2, 3}, {15163, 17810}

X(20406) = {X(1114), X(1313)}-harmonic conjugate of X(468)

X(20407) = (name pending)

Barycentrics 25*S^4+27*(3*R^2*(36*R^2-13*SW )-SB*SC+3*SW^2)*S^2-27*(27*R^ 2-7*SW)*SB*SC*SW : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27974.

X(20407) lies on this line: {2, 3}

X(20408) = COMPLEMENT OF X(15156)

Barycentrics (|OH|-3*R)*S^2+(|OH|+9*R)*SB*SC : :
Barycentrics 3 (a^2 (2 a^2-b^2-c^2)-(b^2-c^2)^2)+(a^2 (b^2+c^2)-(b^2-c^2)^2) J : :
X(20408) = (|OH|-3*R)*X(3)+(|OH|+3*R)*X(4)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27975.

X(20408) lies on the Steiner circle and these lines: {2, 3}, {2100, 9589}, {2101, 9588}, {2102, 5734}, {2574, 14500}, {2575, 16003}

X(20408) = complement of X(15156)
X(20408) = {X(20),X(858)}-harmonic conjugate of X(20409)
X(20408) = {X(10720), X(15157)}-harmonic conjugate of X(4)

X(20409) = COMPLEMENT OF X(15157)

Barycentrics (|OH|+3*R)*S^2+(|OH|-9*R)*SB*SC : :
Barycentrics 3 (a^2 (2 a^2-b^2-c^2)-(b^2-c^2)^2)-(a^2 (b^2+c^2)-(b^2-c^2)^2) J : :

X(20409) = (|OH|+3*R)*X(3)+(|OH|-3*R)*X(4)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27975.

X(20409) lies on the Steiner circle and these lines: {2, 3}, {2100, 9588}, {2101, 9589}, {2103, 5734}, {2574, 16003}, {2575, 14499}

X(20409) = complement of X(15157)
X(20409) = {X(20),X(858)}-harmonic conjugate of X(20408)
X(20409) = {X(10719), X(15156)}-harmonic conjugate of X(4)

X(20410) = X(4)X(67)∩X(25)X(111)

Barycentrics SB*SC*(SB+SC)*(3*S^2+(3*SA-4* SW)*SA)*((6*R^2-SW)*S^2-SB*SC* SW) : :

X(20410) = X(112)+2*X(13166)

See César Lozada, Hyacinthos 27984.

X(20410) lies on the cubic K475 and these lines: {4, 67}, {24, 14649}, {25, 111}, {127, 5133}, {132, 403}, {381, 10718}, {648, 16175}, {1842, 2844}, {1859, 11988}, {1995, 18876}, {2393, 5523}, {2794, 7576}, {7394, 13219}, {8743, 19153}, {8744, 18374}, {9517, 9979}, {10735, 18494}, {10749, 11818}, {12145, 18386}, {18403, 19160}

X(20410) = polar-circle-inverse of X(67)
X(20410) = reflection of X(67) in the line X(9517) X(18310)
X(20410) = barycentric product X(i)*X(j) for these {i,j}: {23, 5523}, {316, 14580}, {858, 8744}, {1560, 14246}
X(20410) = orthic-isogonal conjugate of-X(5523)

X(20411) = X(16)X(186)∩X(389)X(397)

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

See César Lozada, Hyacinthos 27984.

X(20411) lies on the cubic K050 and these lines: {4, 11600}, {16, 186}, {51, 6117}, {53, 1263}, {389, 397}, {1154, 6116}, {1986, 16538}, {2383, 2902}

X(20411) = polar circle-inverse-of X(11600)
X(20411) = X(13)-of-orthic triangle, if ABC is obtuse
X(20411) = X(14)-of-orthic triangle, if ABC is acute
X(20411) = X(616)-of-2nd anti-Conway triangle
X(20411) = X(5473)-of-2nd Euler triangle

X(20412) = X(15)X(186)∩X(389)X(398)

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

See César Lozada, Hyacinthos 27984.

X(20412) lies on the cubic K050 and these lines: {4, 11601}, {15, 186}, {51, 6116}, {53, 1263}, {389, 398}, {1154, 6117}, {1986, 16539}, {2383, 2903}

X(20412) = polar circle-inverse-of X(11601)
X(20412) = X(14)-of-orthic triangle, if ABC is obtuse
X(20412) = X(13)-of-orthic triangle, if ABC is acute
X(20412) = X(617)-of-2nd anti-Conway triangle
X(20412) = X(5474)-of-2nd Euler triangle

X(20413) = (name pending)

Barycentrics 2 a^16-15 a^14 b^2+49 a^12 b^4-95 a^10 b^6+125 a^8 b^8-117 a^6 b^10+75 a^4 b^12-29 a^2 b^14+5 b^16-15 a^14 c^2+66 a^12 b^2 c^2-103 a^10 b^4 c^2+40 a^8 b^6 c^2+99 a^6 b^8 c^2-182 a^4 b^10 c^2+131 a^2 b^12 c^2-36 b^14 c^2+49 a^12 c^4-103 a^10 b^2 c^4+48 a^8 b^4 c^4-9 a^6 b^6 c^4+118 a^4 b^8 c^4-219 a^2 b^10 c^4+116 b^12 c^4-95 a^10 c^6+40 a^8 b^2 c^6-9 a^6 b^4 c^6-22 a^4 b^6 c^6+117 a^2 b^8 c^6-220 b^10 c^6+125 a^8 c^8+99 a^6 b^2 c^8+118 a^4 b^4 c^8+117 a^2 b^6 c^8+270 b^8 c^8-117 a^6 c^10-182 a^4 b^2 c^10-219 a^2 b^4 c^10-220 b^6 c^10+75 a^4 c^12+131 a^2 b^2 c^12+116 b^4 c^12-29 a^2 c^14-36 b^2 c^14+5 c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27986.

X(20413) lies on these lines: {5, 195}, {137, 140}, {547, 13856}, {10096, 15226}, {10615, 14051}

X(20413) = midpoint of X(5) and X(3459)

X(20414) = REFLECTION OF X(10126) IN X(15425)

Barycentrics 2*S^4-(R^2*(20*R^2+5*SA-18*SW) -2*SA^2+10*SB*SC+4*SW^2)*S^2-( R^2*(4*R^2+SW)-2*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27992.

X(20414) lies on these lines: {3, 15307}, {4, 54}, {5, 6150}, {30, 13856}, {546, 12026}, {1154, 10285}, {1209, 19552}, {1263, 1493}, {1503, 15557}, {1510, 10095}, {10126, 15425}, {13431, 18370}

X(20414) = reflection of X(10126) in X(15425)

X(20415) = X(3)X(13)∩X(61)X(115)

Barycentrics (3*SA-4*SW)*S^2-2*sqrt(3)*S*( S^2+SB*SC)+SW*SB*SC : :
X(20415) = X(3)+3*X(13), 7*X(3)-3*X(5473), X(3)-3*X(6771), 5*X(3)+3*X(13103), X(5)-3*X(5459), 3*X(115)-X(16002), X(546)-3*X(20252), 3*X(616)-11*X(3525), 3*X(618)-5*X(632), 7*X(3090)-3*X(5617), 5*X(3091)+3*X(6770), 7*X(3526)-3*X(5463), X(3627)-3*X(5478), 2*X(3628)-3*X(6669), X(5611)-5*X(16960)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27993.

X(20415) lies on these lines: {3, 13}, {5, 542}, {61, 115}, {62, 1506}, {83, 16627}, {140, 530}, {182, 18582}, {397, 6108}, {511, 11542}, {546, 20252}, {616, 3525}, {618, 632}, {620, 630}, {623, 5965}, {3090, 5617}, {3091, 6770}, {3303, 10078}, {3304, 10062}, {3412, 5470}, {3518, 12142}, {3526, 5463}, {3627, 5478}, {3628, 6669}, {5318, 13350}, {5335, 9736}, {5611, 16960}, {6036, 6115}, {6427, 19073}, {6428, 19074}, {9735, 11488}, {10657, 15037}, {11543, 15516}, {14651, 16626}

X(20415) = midpoint of X(i) and X(j) for these {i,j}: {3, 16001}, {5318, 13350}
X(20415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13, 16001), (5, 575, 20416), (6771, 16001, 3)

X(20416) = X(3)X(14)∩X(62)X(115)

Barycentrics (3*SA-4*SW)*S^2+2*sqrt(3)*S*( S^2+SB*SC)+SW*SB*SC : :
X(20416) = X(3)+3*X(14), 7*X(3)-3*X(5474), X(3)-3*X(6774), 5*X(3)+3*X(13102), X(5)-3*X(5460), 3*X(115)-X(16001), X(546)-3*X(20253), 3*X(617)-11*X(3525), 3*X(619)-5*X(632), 7*X(3090)-3*X(5613), 5*X(3091)+3*X(6773), 7*X(3526)-3*X(5464), X(3627)-3*X(5479), 2*X(3628)-3*X(6670), X(5615)-5*X(16961)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27993.

X(20416) lies on these lines: {3, 14}, {5, 542}, {61, 1506}, {62, 115}, {83, 16626}, {140, 531}, {182, 18581}, {398, 6109}, {511, 11543}, {546, 20253}, {617, 3525}, {619, 632}, {620, 629}, {624, 5965}, {3090, 5613}, {3091, 6773}, {3303, 10077}, {3304, 10061}, {3411, 5469}, {3518, 12141}, {3526, 5464}, {3627, 5479}, {3628, 6670}, {5321, 13349}, {5334, 9735}, {5615, 16961}, {6036, 6114}, {6427, 19075}, {6428, 19076}, {9736, 11489}, {10658, 15037}, {11542, 15516}, {14651, 16627}

X(20416) = midpoint of X(i) and X(j) for these {i,j}: {3, 16002}, {5321, 13349}
X(20416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14, 16002), (5, 575, 20415), (6774, 16002, 3)

X(20417) = X(3)X(67)∩X(4)X(74)

Barycentrics 3*(12*R^2-SA-2*SW)*S^2-(36*R^ 2-5*SW)*SB*SC : :
X(20417) = 3*X(2)+X(15054), 3*X(2)-7*X(15057), X(3)+3*X(20126), X(4)+3*X(74), X(4)-3*X(125), 2*X(4)-3*X(7687), 7*X(4)-3*X(10721), 5*X(4)+3*X(12244), 5*X(4)-3*X(13202), 5*X(4)-9*X(14644), 7*X(4)-15*X(15081), X(67)+3*X(5621), 2*X(74)+X(7687), 7*X(74)+X(10721), 3*X(74)-X(10990), 5*X(74)-X(12244), 5*X(74)+X(13202), 5*X(74)+3*X(14644), 7*X(74)+5*X(15081), 7*X(125)-X(10721), 3*X(125)+X(10990), X(10991)+3*X(15357), X(15054)+7*X(15057), 7*X(15057)-X(15063), X(16003)-3*X(20126)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27993.

X(20417) lies on these lines: {2, 15054}, {3, 67}, {4, 74}, {5, 541}, {20, 9140}, {30, 15153}, {64, 10293}, {110, 3523}, {113, 1656}, {140, 5663}, {146, 5056}, {184, 10193}, {185, 10294}, {186, 13399}, {265, 1657}, {382, 15027}, {389, 2781}, {399, 15720}, {468, 6000}, {549, 5609}, {550, 10264}, {578, 5095}, {631, 5642}, {690, 6130}, {974, 10628}, {1181, 15106}, {1511, 15712}, {1539, 3858}, {1593, 12828}, {1899, 11204}, {3090, 10706}, {3269, 6103}, {3448, 3522}, {3515, 13171}, {3516, 13293}, {3517, 10117}, {3524, 15034}, {3526, 5655}, {3543, 15044}, {3832, 15025}, {3850, 20304}, {3851, 7728}, {5059, 10733}, {5073, 12295}, {5094, 10605}, {5493, 13605}, {5882, 11709}, {5890, 14448}, {5965, 10564}, {6241, 17853}, {6247, 13419}, {7486, 15029}, {7533, 15053}, {7689, 14791}, {8550, 11430}, {9143, 15020}, {9904, 11522}, {9938, 12901}, {10018, 14862}, {10110, 12099}, {10112, 11250}, {10113, 14677}, {10116, 10226}, {10182, 11456}, {10295, 18400}, {10299, 12317}, {10606, 18390}, {10745, 13611}, {11270, 11564}, {11425, 16176}, {11693, 15039}, {11735, 13464}, {11746, 11807}, {12227, 17847}, {13358, 13421}, {13393, 15605}, {13431, 15089}, {13491, 20191}, {13754, 15115}, {14683, 15051}, {14915, 16619}, {14984, 15644}, {15126, 15311}, {16219, 18381}

X(20417) = midpoint of X(i) and X(j) for these {i,j}: {3, 16003}, {4, 10990}, {113, 10620}, {186, 13399}, {265, 16111}, {3448, 16163}, {10113, 14677}
X(20417) = reflection of X(i) in X(j) for these (i,j): (113, 6723), (389, 16270)
X(20417) = complement of X(15063)
X(20417) = X(10990)-of-Euler-triangle
X(20417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 15054, 15063), (3, 20126, 16003), (4, 74, 10990), (74, 14644, 12244), (113, 15061, 6723), (125, 10990, 4), (125, 13202, 14644), (140, 16534, 5972), (265, 15041, 16111), (631, 14094, 5642), (6699, 16534, 140), (9140, 15021, 20), (10620, 15061, 113), (12244, 14644, 13202), (15054, 15057, 2)

X(20418) = X(4)X(11)∩X(10)X(140)

Barycentrics 2*a^7-2*(b+c)*a^6-(5*b^2-12*b* c+5*c^2)*a^5+5*(b^2-c^2)*(b-c) *a^4+4*(b^2-b*c+c^2)*(b-c)^2* a^3-4*(b^3+c^3)*(b-c)^2*a^2-( b^4-c^4)*(b^2-c^2)*a+(b^2-c^2) ^3*(b-c) : :
X(20418) = X(1)+3*X(11219), 3*X(3)-X(10993), X(4)-3*X(11), X(4)+3*X(104), 7*X(4)-3*X(10728), 5*X(4)+3*X(12248), 7*X(11)-X(10728), 5*X(11)+X(12248), 7*X(104)+X(10728), 5*X(104)-X(12248), 4*X(140)-3*X(3035), 2*X(140)-3*X(6713), X(5882)+3*X(10265), X(5882)-3*X(11715), 5*X(10728)+7*X(12248)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27993.

X(20418) lies on these lines: {1, 11219}, {3, 528}, {4, 11}, {5, 10199}, {10, 140}, {20, 10707}, {36, 5842}, {100, 3523}, {119, 1656}, {149, 3522}, {153, 5056}, {355, 6691}, {388, 12762}, {496, 5450}, {499, 18242}, {515, 5126}, {529, 6882}, {550, 1484}, {631, 6174}, {676, 2826}, {942, 1387}, {944, 5433}, {999, 7680}, {1012, 10072}, {1020, 3333}, {1125, 2801}, {1145, 4853}, {1158, 11373}, {1317, 12247}, {1532, 3582}, {1537, 1768}, {1657, 10738}, {2771, 11281}, {2787, 11623}, {2886, 10269}, {3058, 6950}, {3090, 10711}, {3303, 6977}, {3304, 6833}, {3419, 3576}, {3428, 13279}, {3600, 10894}, {3612, 12750}, {3624, 5660}, {3660, 6001}, {3829, 6923}, {3851, 10742}, {3913, 6961}, {4857, 5533}, {4860, 5603}, {5059, 10724}, {5083, 16193}, {5204, 12116}, {5217, 10806}, {5270, 8068}, {5289, 5770}, {5298, 6905}, {5432, 7967}, {5434, 6830}, {5563, 6831}, {5657, 13996}, {5703, 14151}, {5848, 8550}, {5851, 5886}, {5854, 12737}, {5881, 13747}, {5884, 5901}, {6154, 10299}, {6326, 8583}, {6690, 10246}, {6827, 11194}, {6879, 11237}, {6891, 12513}, {6922, 8666}, {6938, 11238}, {6948, 11235}, {6952, 15888}, {6958, 12607}, {6966, 11240}, {6978, 11236}, {7288, 11500}, {7742, 10090}, {8071, 10058}, {9952, 12735}, {10074, 15844}, {10085, 18243}, {10310, 10529}, {11375, 12831}, {11496, 14986}, {11570, 12709}, {12119, 12690}, {12331, 15720}, {12691, 17660}, {12758, 17622}

X(20418) = reflection of X(119) in X(6667)
X(20418) = complement of X(37725)
X(20418) = X(11799)-of-K798i-triangle
X(20418) = X(15133)-of-inverse-in- incircle-triangle
X(20418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1768, 16173, 1537), (3086, 12114, 7681)

X(20419) = X(1807)X(11012)∩X(7100)X(10902)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27995.

X(20419) lies on these lines: {1807, 11012}, {7100, 10902}

X(20420) = EULER LINE INTERCEPT OF X(1)X(5805)

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(20420) = 3*X(1699)-X(6284), 3*X(3058)-5*X(11522), 3*X(5587)-X(11827), 3*X(5603)-2*X(15172), 3*X(10157)-2*X(12572), 3*X(11227)-4*X(12436), 3*X(11246)-X(15071), 4*X(13464)-3*X(15170)

As a point on the Euler line, X(20420) center has Shinagawa coefficients (r, -4*R-3*r).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27995.

X(20420) lies on these lines: {1, 5805}, {2, 3}, {57, 1837}, {72, 5762}, {78, 5763}, {142, 4297}, {225, 15252}, {355, 5709}, {495, 11500}, {515, 942}, {516, 960}, {517, 5907}, {528, 4301}, {944, 11037}, {946, 4314}, {950, 5806}, {952, 3555}, {962, 5730}, {971, 4292}, {1466, 12943}, {1479, 7956}, {1503, 4260}, {1512, 5771}, {1538, 18483}, {1699, 3601}, {1728, 3358}, {1750, 6259}, {1770, 12688}, {2095, 18525}, {2096, 12684}, {2829, 6245}, {2968, 5174}, {3058, 11522}, {3587, 12705}, {3833, 11227}, {3940, 5758}, {4304, 18482}, {4640, 12617}, {4848, 9952}, {5082, 8158}, {5122, 6705}, {5138, 5480}, {5234, 5587}, {5259, 7958}, {5396, 13408}, {5434, 11518}, {5603, 15172}, {5708, 5768}, {5715, 11374}, {5720, 5812}, {5729, 12246}, {5735, 11523}, {5744, 5789}, {5745, 19925}, {5755, 16552}, {5840, 9945}, {5843, 12528}, {5880, 12520}, {6147, 18446}, {6282, 11826}, {6796, 7680}, {7965, 15338}, {9623, 12120}, {9655, 12667}, {9947, 12527}, {10106, 11035}, {10157, 12572}, {10483, 10826}, {10526, 18491}, {10592, 10894}, {11012, 18406}, {11246, 15071}, {11249, 18517}, {12651, 12699}, {13464, 15170}, {18443, 18481}

X(20420) = midpoint of X(i) and X(j) for these {i,j}: {1, 6253}, {1770, 12688}
X(20420) = reflection of X(950) in X(5806)
X(20420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 8727), (3, 382, 6851), (4, 1532, 546), (4, 3149, 5), (4, 6848, 381), (4, 6905, 6831), (4, 6927, 6844), (4, 6934, 1012), (4, 6942, 6845), (21, 6894, 8226), (1012, 6934, 550), (6831, 6905, 140), (6834, 6847, 6861), (6844, 6927, 1656), (6849, 6868, 6913), (6851, 6885, 3)

X(20421) = ISOGONAL CONJUGATE OF X(3830)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27995.

X(20421) lies on the Jerabek hyperbola and these lines: {2, 18550}, {64, 12112}, {68, 3528}, {69, 19708}, {74, 11202}, {186, 3426}, {248, 8588}, {265, 376}, {378, 3531}, {631, 3521}, {895, 3098}, {1173, 11438}, {1495, 11738}, {3520, 3527}, {3524, 4846}, {3529, 17505}, {3532, 11456}, {6415, 6451}, {6416, 6452}, {7712, 12041}, {8617, 9210}, {8717, 15055}, {10298, 11559}, {10299, 14861}, {11204, 13603}, {11270, 11464}, {11430, 13472}, {11468, 13452}, {13418, 18909}, {13619, 18434}, {13623, 15698}, {14528, 15032}, {14530, 17506}

X(20421) = isogonal conjugate of X(3830)

X(20422) = EULER LINE INTERCEPT OF X(136)X(155)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27995.

X(20422) lies on these lines: {2, 3}, {136, 155}, {254, 3564}, {5962, 12429}, {12359, 14593}

X(20423) = X(2)X(51)∩X(6)X(30)

Barycentrics a^6-7*(b^2+c^2)*a^4+(5*b^4-6* b^2*c^2+5*c^4)*a^2+(b^4-c^4)*( b^2-c^2) : :
X(20423) = 5*X(2)-3*X(10519), 2*X(2)-3*X(14561), X(2)-3*X(14853), X(3)-3*X(14848), X(4)+2*X(576), 2*X(5)+X(11477), X(20)-4*X(575), X(69)-3*X(3545), X(69)-4*X(19130), 2*X(597)-3*X(14848), 3*X(3545)-2*X(11178), 3*X(3545)-4*X(19130), 10*X(5476)-3*X(10519), 4*X(5476)-3*X(14561), 2*X(5476)-3*X(14853), 2*X(10519)-5*X(14561), X(10519)-5*X(14853)

See César Lozada, Hyacinthos 27996.

X(20423) lies on the cubic K582 and these lines: {2, 51}, {3, 597}, {4, 542}, {5, 599}, {6, 30}, {20, 575}, {69, 1568}, {114, 9770}, {115, 19905}, {141, 5055}, {143, 18281}, {146, 9976}, {182, 376}, {193, 3818}, {230, 11173}, {381, 524}, {382, 8550}, {518, 3656}, {541, 11579}, {543, 12177}, {546, 15069}, {548, 10541}, {549, 1350}, {567, 19127}, {568, 2781}, {611, 3058}, {613, 5434}, {1353, 15687}, {1370, 15004}, {1386, 3655}, {1469, 10072}, {1478, 8540}, {1479, 19369}, {1482, 9041}, {1499, 9178}, {1503, 3830}, {1513, 11163}, {1570, 14537}, {2080, 8182}, {2104, 10720}, {2105, 10719}, {2777, 10250}, {2854, 5655}, {3056, 10056}, {3070, 9975}, {3071, 9974}, {3088, 16625}, {3091, 7946}, {3095, 5969}, {3098, 3524}, {3522, 20190}, {3534, 5050}, {3543, 5032}, {3564, 3845}, {3589, 5054}, {3629, 14269}, {3763, 15699}, {3767, 6034}, {3832, 18553}, {4663, 12699}, {5028, 7753}, {5052, 5309}, {5066, 10516}, {5085, 8703}, {5092, 10304}, {5107, 5475}, {5182, 12110}, {5485, 14485}, {5486, 11799}, {5648, 5654}, {5878, 8549}, {5921, 18392}, {6054, 7774}, {6055, 7735}, {6243, 14787}, {6321, 9830}, {6329, 12017}, {6811, 13757}, {6813, 13637}, {7000, 13639}, {7374, 13759}, {7500, 13366}, {7519, 11422}, {7540, 9833}, {7583, 13662}, {7584, 13782}, {7766, 11177}, {7775, 14645}, {7840, 13862}, {8262, 15362}, {9023, 19912}, {9140, 10752}, {9143, 11004}, {9740, 11167}, {9813, 18537}, {10169, 10249}, {10707, 10759}, {10708, 10758}, {10709, 10764}, {10710, 10756}, {10711, 10755}, {10716, 10757}, {10989, 18911}, {11064, 20192}, {11161, 14639}, {11663, 15073}, {11694, 12106}, {12007, 15684}, {12156, 14912}, {12294, 14831}, {12355, 13111}, {12584, 14002}, {13169, 14644}, {13352, 15462}, {13482, 19128}, {13490, 19139}, {14810, 15692}, {15019, 16063}, {15303, 17702}, {15361, 18580}, {15516, 15683}, {15582, 18378}, {15826, 18325}, {17508, 19708}

X(20423) = reflection of X(i) in X(j) for these (i,j): (2, 5476), (3, 597), (69, 11178), (381, 5480), (549, 18583), (3098, 10168), (9143, 19140)
X(20423) = X(597)-of-X3-ABC-reflections-triangle
X(20423) = X(599)-of-Johnson-triangle
X(20423) = X(1992)-of-Euler-triangle
X(20423) = X(7737)-of-Artzt-triangle
X(20423) = X(11579)-of-orthocentroidal-triangle
X(20423) = reflection of X(i) in the line X(j)X(k) for these (i,j,k): (2, 512, 5476), (376, 182, 12073), (381, 1499, 5480), (1352, 381, 8371), (1992, 576, 690)
X(20423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5476, 14561), (2, 14853, 5476), (3, 14848, 597), (69, 3545, 11178), (193, 3839, 11180), (382, 11482, 8550), (1351, 5480, 1352), (3098, 10168, 3524), (3524, 3618, 10168), (3543, 5032, 6776), (3839, 11180, 3818), (10653, 10654, 7737), (11178, 19130, 3545), (13352, 19136, 15462)

X(20424) = X(5)X(51)∩X(30)X(54)

Barycentrics ((b^2+c^2)*a^2-(b^2-c^2)^2)*( 3*a^6-5*(b^2+c^2)*a^4+(b^4+b^ 2*c^2+c^4)*a^2+(b^4-c^4)*(b^2- c^2)) : :
X(20424) = X(4)+2*X(11803), 3*X(5)-2*X(1209), 5*X(5)-4*X(13565), 3*X(381)-X(2888), 3*X(381)+X(12316), 2*X(546)+X(15801), 3*X(549)-4*X(6689), X(550)-4*X(12242), X(1209)-3*X(3574), 5*X(1209)-6*X(13565), 2*X(1493)+X(3627), 5*X(3091)-X(12325), X(3519)-4*X(3850), 5*X(3574)-2*X(13565), 5*X(3843)+X(11271)

Let Na be the reflection of X(5) in the A-altitude. Define Nb and Nc cyclically; then X(20424) = X(54)-of-NaNbNc. (Randy Hutson, August 29, 2018)

See César Lozada, Hyacinthos 27996.

X(20424) lies on these lines: {2, 12307}, {3, 8254}, {4, 195}, {5, 51}, {11, 7356}, {12, 6286}, {30, 54}, {140, 3581}, {235, 6152}, {381, 2888}, {382, 12254}, {403, 6242}, {427, 12300}, {495, 13079}, {496, 18984}, {539, 3845}, {546, 6288}, {549, 6689}, {550, 10610}, {568, 10224}, {576, 12899}, {1199, 7574}, {1493, 2883}, {1503, 19150}, {1531, 15807}, {1596, 11576}, {2070, 15806}, {2072, 16881}, {3060, 13406}, {3070, 12971}, {3071, 12965}, {3091, 12325}, {3153, 14627}, {3519, 3850}, {3843, 11271}, {3858, 5480}, {5318, 10678}, {5321, 10677}, {5446, 11563}, {5448, 11808}, {5946, 11802}, {6102, 10115}, {6145, 13292}, {6153, 10110}, {6823, 12363}, {7530, 9920}, {7564, 12160}, {9905, 12699}, {10024, 14449}, {10066, 15171}, {10082, 18990}, {10203, 13434}, {10263, 18388}, {11264, 11572}, {11702, 17702}, {12002, 13433}, {12161, 17824}, {12234, 12241}, {12295, 14049}, {12606, 15760}, {12785, 18357}, {13366, 13470}, {13561, 14831}, {14076, 16625}

X(20424) = midpoint of X(i) and X(j) for these {i,j}: {4, 195}, {382, 12254}, {9905, 12699}, {12295, 14049}
X(20424) = reflection of X(i) in X(j) for these (i,j): (3, 8254), (5, 3574), (550, 10610), (6102, 10115), (6153, 10110), (12785, 18357)
X(20424) = complement of X(12307)
X(20424) = X(195)-of-Euler-triangle
X(20424) = X(2888)-of-Ehrmann-mid-triangle
X(20424) = X(8254)-of-X3-ABC-reflections-triangle
X(20424) = reflection of X(5) in the line X(1510)X(3574)
X(20424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 12316, 2888), (1568, 10095, 5)

X(20425) = X(13)X(511)∩X(30)X(5611)

Barycentrics 3*(2*SA-SW)*S^2-3*SW*SB*SC- sqrt(3)*S*(S^2+3*SB*SC) : :
X(20425) = X(5473)-3*X(16962), X(5474)-3*X(16529), 2*X(6771)-3*X(16267), 2*X(13350)-3*X(16962), X(14538)-3*X(16267)

See César Lozada, Hyacinthos 27996.

X(20425) lies on these lines: {2, 5615}, {3, 396}, {4, 3180}, {5, 298}, {13, 511}, {14, 576}, {15, 5472}, {30, 5611}, {62, 6774}, {230, 11486}, {381, 524}, {385, 1080}, {530, 2080}, {531, 6321}, {532, 5617}, {533, 5478}, {546, 16628}, {3095, 5613}, {5340, 5865}, {5473, 13350}, {5474, 16529}, {6771, 14538}, {7737, 11485}, {9736, 16241}, {10788, 11299}, {11303, 12251}

X(20425) = midpoint of X(4) and X(3180)
X(20425) = reflection of X(i) in X(j) for these (i,j): (3, 396), (5473, 13350)
X(20425) = X(298)-of-Johnson triangle
X(20425) = X(396)-of-X3-ABC reflections triangle
X(20425) = X(3180)-of-Euler triangle
X(20425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 1351, 20426), (5473, 16962, 13350), (14538, 16267, 6771)

X(20426) = X(14)X(511)∩X(30)X(5615)

Barycentrics 3*(2*SA-SW)*S^2-3*SW*SB*SC+ sqrt(3)*S*(S^2+3*SB*SC) : :
X(20426) = X(5473)-3*X(16530), X(5474)-3*X(16963), 2*X(6774)-3*X(16268), 2*X(13349)-3*X(16963), X(14539)-3*X(16268)

See César Lozada, Hyacinthos 27996.

X(20426) lies on these lines: {2, 5611}, {3, 395}, {4, 3181}, {5, 299}, {13, 576}, {14, 511}, {16, 5471}, {30, 5615}, {61, 6771}, {230, 11485}, {381, 524}, {383, 385}, {530, 6321}, {531, 2080}, {532, 5479}, {533, 5613}, {546, 16629}, {3095, 5617}, {5339, 5864}, {5473, 16530}, {5474, 13349}, {6774, 14539}, {7737, 11486}, {9735, 16242}, {10788, 11300}, {11304, 12251}

X(20426) = midpoint of X(4) and X(3181)
X(20426) = reflection of X(i) in X(j) for these (i,j): (3, 395), (5474, 13349)
X(20426) = X(299)-of-Johnson triangle
X(20426) = X(395)-of-X3-ABC reflections triangle
X(20426) = X(3181)-of-Euler triangle
X(20426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 1351, 20425), (5474, 16963, 13349), (14539, 16268, 6774)

X(20427) = X(20)X(2979)∩X(30)X(64)

Barycentrics (12*R^2-SA-2*SW)*S^2-(28*R^2- 5*SW)*SB*SC : :
X(20427) = 3*X(3)-2*X(2883), 7*X(3)-6*X(10192), 5*X(3)-4*X(16252), 3*X(4)-4*X(20299), 4*X(2883)-3*X(5878), X(2883)-3*X(5894), 7*X(2883)-9*X(10192), 5*X(2883)-6*X(16252), 3*X(3357)-2*X(20299), X(5878)-4*X(5894), 7*X(5878)-12*X(10192), 5*X(5878)-8*X(16252), 7*X(5894)-3*X(10192), 5*X(5894)-2*X(16252), 15*X(10192)-14*X(16252)

See César Lozada, Hyacinthos 27996.

X(20427) lies on the cubic K928 and these lines: {2, 18504}, {3, 1661}, {4, 74}, {5, 5895}, {20, 2979}, {30, 64}, {35, 12940}, {36, 12950}, {140, 8567}, {146, 13293}, {154, 548}, {376, 6225}, {381, 6696}, {382, 6247}, {541, 1147}, {550, 1498}, {631, 11204}, {1192, 1596}, {1503, 1657}, {1597, 13568}, {1656, 5893}, {1853, 3627}, {1885, 10605}, {1899, 18560}, {2935, 5654}, {3146, 18381}, {3184, 14379}, {3426, 16621}, {3522, 5656}, {3525, 10193}, {3528, 11202}, {3529, 12324}, {3534, 12315}, {3543, 18383}, {3548, 15125}, {3579, 12779}, {4299, 6285}, {4302, 7355}, {4317, 11189}, {4846, 7526}, {5663, 12118}, {6102, 7729}, {6241, 19467}, {6284, 10076}, {6293, 13491}, {6640, 7728}, {7354, 10060}, {7487, 13474}, {7505, 11468}, {7723, 18439}, {8703, 17821}, {8778, 15341}, {8991, 13665}, {9681, 11241}, {9786, 13488}, {9908, 9938}, {9934, 10539}, {9967, 10575}, {10117, 14677}, {10182, 10299}, {11206, 17538}, {11381, 18533}, {11441, 16386}, {11744, 12041}, {12262, 12699}, {12364, 16266}, {12897, 18951}, {13403, 18909}, {13785, 13980}, {14363, 16253}, {14791, 15138}, {15704, 17845}, {17578, 18376}, {18390, 18913}, {18431, 51002}

X(20427) = midpoint of X(i) and X(j) for these {i,j}: {20, 12250}, {3529, 12324}
X(20427) = reflection of X(i) in X(j) for these (i,j): (3, 5894), (4, 3357), (146, 13293), (382, 6247), (3146, 18381), (6293, 13491), (9934, 16111), (10117, 14677), (11744, 12041)
X(20427) = X(5878)-of-ABC-X3 reflections triangle
X(20427) = X(5894)-of-X3-ABC reflections triangle
X(20427) = X(5895)-of-Johnson triangle
X(20427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (376, 6225, 6759), (3522, 5656, 10282), (5895, 10606, 5)

X(20428) = ANTICOMPLEMENT OF X(13350)

Barycentrics S^2*SA+SB*SC*(SW+2*sqrt(3)*S) : :
X(20428) = 3*X(381)-X(5611), 3*X(381)-2*X(7684), 5*X(1656)-4*X(6671), 3*X(5886)-2*X(11707), 3*X(16267)-4*X(20252), X(3)-2*X(623), X(4)+X(621), 2*X(5)-X(15), X(2080)-2*X(7685)

See César Lozada, Hyacinthos 27997.

X(20428) lies on these lines: {2, 13350}, {3, 623}, {4, 69}, {5, 15}, {13, 11602}, {14, 10796}, {30, 5463}, {114, 1080}, {137, 11127}, {182, 11303}, {187, 18581}, {302, 9736}, {381, 531}, {397, 1353}, {398, 18583}, {470, 5972}, {473, 9306}, {532, 13103}, {533, 5478}, {546, 16627}, {1656, 6671}, {2080, 7685}, {3564, 5318}, {5334, 14561}, {5640, 16771}, {5872, 16965}, {5886, 11707}, {5965, 16001}, {6298, 13188}, {6774, 11300}, {10613, 10654}, {13111, 16628}, {14138, 16644}, {16002, 19130}, {16267, 20252}, {18424, 18582}

X(20428) = midpoint of X(4) and X(621)
X(20428) = reflection of X(i) in X(j) for these (i,j): (3, 623), (15, 5), (2080, 7685)
X(20428) = complement of X(36993)
X(20428) = anticomplement of X(13350)
X(20428) = X(15)-of-Johnson-triangle
X(20428) = X(621)-of-Euler-triangle
X(20428) = X(623)-of-X3-ABC-reflections-triangle
X(20428) = X(5611)-of-Ehrmann-mid-triangle
X(20428) = {X(381), X(5611)}-harmonic conjugate of X(7684)
X(20428) = {X(4),X(1352)}-harmonic conjugate of X(20429)

X(20429) = ANTICOMPLEMENT OF X(13349)

Barycentrics S^2*SA+SB*SC*(SW-2*sqrt(3)*S) : :
X(20429) = 3*X(381)-X(5615), 3*X(381)-2*X(7685), 5*X(1656)-4*X(6672), 3*X(5886)-2*X(11708), 3*X(16268)-4*X(20253), X(3)-2*X(624), X(4)+X(622), 2*X(5)-X(16), X(2080)-2*X(7684)

See César Lozada, Hyacinthos 27997.

X(20429) lies on these lines: {2, 13349}, {3, 624}, {4, 69}, {5, 16}, {13, 10796}, {14, 11603}, {30, 5464}, {114, 383}, {137, 11126}, {182, 11304}, {187, 18582}, {303, 9735}, {381, 530}, {397, 18583}, {398, 1353}, {471, 5972}, {472, 9306}, {532, 5479}, {533, 13102}, {546, 16626}, {1656, 6672}, {2080, 7684}, {3564, 5321}, {5335, 14561}, {5640, 16770}, {5873, 16964}, {5886, 11708}, {5965, 16002}, {6299, 13188}, {6771, 11299}, {10614, 10653}, {13111, 16629}, {14139, 16645}, {16001, 19130}, {16268, 20253}, {18424, 18581}

X(20429) = midpoint of X(4) and X(622)
X(20429) = reflection of X(i) in X(j) for these (i,j): (3, 624), (16, 5), (2080, 7684)
X(20429) = complement of X(36995)
X(20429) = anticomplement of X(13349)
X(20429) = X(16)-of-Johnson triangle
X(20429) = X(622)-of-Euler-triangle
X(20429) = X(624)-of-X3-ABC reflections-triangle
X(20429) = X(5615)-of-Ehrmann-mid-triangle
X(20429) = {X(4),X(1352)}-harmonic conjugate of X(20428)
X(20429) = {X(381), X(5615)}-harmonic conjugate of X(7685)

X(20430) = MIDPOINT OF X(4) AND X(192)

Trilinears a b c SA - (b + c) (S^2 + SB SC) : :
Barycentrics a*(b*c*a^3+(b+c)*(b^2+c^2)*a^ 2-(b^2+c^2)*b*c*a-(b^2-c^2)^2* (b+c)) : :
X(20430) = X(20)-5*X(4704), 4*X(140)-5*X(4687), X(382)+4*X(4681), 4*X(546)+X(3644), X(1278)-5*X(3091), 5*X(1656)-4*X(3739), 7*X(3090)-5*X(4699), 7*X(3526)-8*X(4698), 3*X(3545)-X(4740), 8*X(3628)-7*X(4751), 2*X(3696)-3*X(5790), 7*X(3832)+X(4788), 5*X(3843)+2*X(4718), 8*X(3850)-X(4764), 7*X(3851)-2*X(4686), X(3)-2*X(37), X(4)+X(192), 2*X(5)-X(75)

See César Lozada, Hyacinthos 27997.

X(20430) lies on these lines: {3, 37}, {4, 192}, {5, 75}, {20, 4704}, {30, 4664}, {40, 8245}, {140, 4687}, {226, 20254}, {355, 740}, {381, 536}, {382, 4681}, {515, 3993}, {517, 984}, {518, 1351}, {537, 3656}, {546, 3644}, {726, 946}, {742, 1352}, {942, 7201}, {1278, 3091}, {1656, 3739}, {2805, 11258}, {3090, 4699}, {3295, 11997}, {3526, 4698}, {3545, 4740}, {3628, 4751}, {3666, 19540}, {3696, 5790}, {3751, 4516}, {3797, 7377}, {3832, 4788}, {3843, 4718}, {3850, 4764}, {3851, 4686}, {4385, 4451}, {4688, 5055}, {4726, 5072}, {4739, 5079}, {4755, 5054}, {4772, 5056}, {4821, 5068}, {4850, 19546}, {5480, 9055}, {6831, 20171}, {8727, 20173}, {9548, 17038}, {10246, 15569}, {10679, 18534}, {11849, 15624}

X(20430) = midpoint of X(4) and X(192)
X(20430) = reflection of X(3) in X(37)
X(20430) = reflection of X(75) in X(5)
X(20430) = X(192)-of-Euler-triangle
X(20430) = X(75)-of-Johnson-triangle
X(20430) = {X(1482), X(5779)}-harmonic conjugate of X(1351)

X(20431) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(105)

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

(1,321), (2,75), (6,1930), (31,20234), (105,20431), (238,20432), (292,20433), (365,20434), (672,20435), (1423,20436), (1931,20437), (2053,20438), (2054,20439), (3009,20440), (2112, 20441), (2144,20442), (20332,20553)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : c : b
m(A') = -b c (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : c a (Sqrt[a] - Sqrt[c]) (a + Sqrt[a] Sqrt[c] + c) : a b(Sqrt[a] - Sqrt[b]) (a + Sqrt[a] Sqrt[b] + b)
m(A1) = -2 b^2 c^2 : a c (a^2 - b c) : a b (a^2 - b c)
m(A2) = -b c (b + c) (b^2 - b c + c^2) : a c (a b - c^2) : a b (a c - b^2)
m(A3) = - b c (b + c) (a b + a c + b c) : a c (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : a b (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = -b c (a + b + c) (b^2 + c^2) : a c (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : a b (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = b c (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : a c (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : a b (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = -b c (a + b + c) (-a^2 b^2 + a b^3 + b^3 c - a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : a c (a^2 + b^2 - a c - b c) (a^3 - a^2 b - a b c + 2 a c^2 + b c^2 + c^3) : a b (a^3 + 2 a b^2 + b^3 - a^2 c - a b c + b^2 c) (a^2 - a b - b c + c^2),

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = b c (y + z) : c a (z + x) : a b (x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(2)), where A'B'C' = incentral triangle.

Let f(a,b,c,x,y,z) = a^4 (b-c) (a^2-b c) (b^2+b c+c^2) x^3-b^2 c (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z+b c^2 (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20431) lies on these lines: {9, 75}, {321, 20441}, {1109, 4712}, {4431, 18674}, {4568, 17877}, {6063, 6358}, {20432, 20435}, {20442, 20443}

X(20432) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(238)

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

See X(20431).

X(20432) lies on these lines: {6, 75}, {312, 17266}, {321, 1930}, {377, 16086}, {514, 17894}, {519, 2650}, {696, 2643}, {1281, 6660}, {3008, 4359}, {3263, 3948}, {3264, 16732}, {3888, 9017}, {3954, 17184}, {4385, 17528}, {4812, 10436}, {5300, 5904}, {17886, 20634}, {18891, 18895}, {20237, 20436}, {20431, 20435}, {20437, 20439}, {20440, 20443}

X(20433) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(292)

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

See X(20431).

X(20433) lies on these lines: {75, 291}, {1111, 4647}, {1930, 20630}, {20234, 20441}, {20435, 20440}, {20628, 20629}

X(20434) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(365)

Barycentrics b c (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : :

See X(20431).

X(20434) lies on these lines: {75, 366}

X(20435) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(672)

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

See X(20431).

X(20435) lies on these lines: {7, 8}, {76, 3790}, {192, 3673}, {514, 4374}, {726, 1111}, {1233, 3703}, {1921, 3263}, {1930, 17760}, {2350, 4359}, {2481, 3685}, {3705, 6063}, {3739, 16720}, {3963, 4431}, {3967, 18142}, {3993, 7264}, {20431, 20432}, {20433, 20440}

X(20436) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(1423)

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

See X(20431).

X(20436) lies on these lines: {7, 8}, {76, 3705}, {274, 7081}, {321, 1959}, {3701, 4518}, {3761, 17866}, {4073, 17157}, {17880, 20441}, {20237, 20432}

X(20437) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(1931)

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

See X(20431).

X(20437) lies on these lines: {75, 8033}, {226, 306}, {1109, 3263}, {1909, 4647}, {20432, 20439}

X(20438) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(2053)

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

See X(20431).

X(20438) lies on these lines: {75, 2319}, {6063, 6358}, {20234, 20440}, {20237, 20432}

X(20439) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(2054)

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

See X(20431).

X(20439) lies on these lines: {75, 1654}, {1930, 20441}, {20432, 20437}

X(20440) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(3009)

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

See X(20431).

X(20440) lies on these lines: {2, 37}, {335, 4087}, {661, 17893}, {3662, 6382}, {18275, 18891}, {20234, 20438}, {20432, 20443}, {20433, 20435}, {20441, 20629}

X(20441) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(2112)

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

See X(20431).

X(20441) lies on these lines: {75, 1281}, {321, 20431}, {1930, 20439}, {17880, 20436}, {20234, 20433}, {20440, 20629}

X(20442) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(2144)

Barycentrics (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20431).

X(20442) lies on these lines: {20431, 20443}

X(20443) = (X(1), X(2), X(6), X(31); X(321), X(75), X(1930), X(20234)) COLLINEATION IMAGE OF X(20332)

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

See X(20431).

X(20443) lies on these lines: {75, 87}, {76, 334}, {4128, 4647}, {4858, 20234}, {17880, 20436}, {20431, 20442}, {20432, 20440}

X(20444) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(31)

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

(1,312), (2,75), (6,304), (31,20444), (238,17789), (292,20446), (365,20447), (672,20448), (1423,20449), (1931,20450), (2053,20451), (2054,20452), (3009,20453), (2112,20454)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1/a : 1/b : 1/c
m(A') = - b c (a^(3/2) + b^(3/2) + c^(3/2)) : c a (a^(3/2) + b^(3/2) - c^(3/2)) : a b (a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = b c (a^2 + 2 b c) : -a^3 c : -a^3 b
m(A2) = - b c (b^3 + a b c + c^3) : c a (b^3 + a b c - c^3) : a b (-b^3 + a b c + c^3)
m(A3) = - b c (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : c a (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : a b (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = -b c (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3) : a c (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c - a c^2 - b c^2 - c^3) : a b (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)
m(A5) = - b c (a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3) : c a (a^3 - b^3 + 2 a^2 c + b^2 c - b c^2 + c^3) : a b (a^3 + 2 a^2 b + b^3 - b^2 c + b c^2 - c^3),

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = b c (-x + y + z) : c a (- y + z + x) : a b (-x + x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(2)), where A'B'C' = incentral triangle.

Let f(a,b,c,x,y,z) = a^6 (b-c) (b^2+b c+c^2) x^3-b^3 c^2 (a^4+a b^3-b^2 c^2-a c^3) y^2 z+ b^2 c^3 (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20444) lies on these lines: {6, 75}, {85, 18143}, {304, 18137}, {312, 1230}, {321, 17299}, {322, 4033}, {744, 2209}, {1100, 4812}, {2064, 4417}, {2210, 4412}, {4381, 7122}, {14963, 18050}, {16580, 18744}, {17786, 17791}, {18051, 20644}, {18138, 20641}, {20446, 20454}, {20451, 20453}, {20650, 20652}

X(20444) = isotomic conjugate of X(7096)

X(20445) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(105)

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

See X(20444).

X(20445) lies on these lines: {9, 75}, {149, 321}, {312, 8024}, {322, 4033}, {2345, 16732}, {7112, 20643}, {17789, 20448}

X(20446) = (name pending)

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

See X(20444).

X(20446) lies on these lines: {75, 291}, {76, 4485}, {304, 18050}, {312, 561}, {1502, 17788}, {17789, 18891}, {20444, 20454}, {20448, 20453}, {20642, 20643}

X(20447) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(365)

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

See X(20444).

X(20447) lies on these lines: {75, 366}

X(20448) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(672)

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

See X(20444).

X(20448) lies on these lines: {7, 8}, {76, 4043}, {304, 18137}, {306, 1233}, {312, 18142}, {514, 1921}, {561, 18138}, {740, 1111}, {3739, 4875}, {4359, 16708}, {4417, 6063}, {16727, 17495}, {17789, 20445}, {18031, 20646}, {18045, 18134}, {20335, 20593}, {20446, 20453}

X(20449) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(1423)

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

See X(20444).

X(20449) lies on these lines: {7, 8}, {76, 946}, {312, 1959}, {315, 6256}, {325, 1329}, {3761, 9612}, {7788, 11236}, {10912, 17144}, {17789, 20451}

X(20450) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(1931)

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

See X(20444).

X(20450) lies on these lines: {75, 8033}, {312, 1230}, {850, 7199}, {3761, 17762}, {17789, 20452}

X(20450) = isotomic conjugate of isogonal conjugate of X(20369)

X(20451) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(2053)

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

See X(20444).

X(20451) lies on these lines: {75, 2319}, {17789, 20449}, {20444, 20453}

X(20452) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(2054)

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

See X(20444).

X(20452) lies on these lines: {75, 1654}, {304, 20454}, {17789, 20450}, {18137, 20650}

X(20453) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(3009)

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

See X(20444).

X(20453) lies on these lines: {2, 37}, {76, 18050}, {190, 20610}, {661, 786}, {1215, 17445}, {1920, 18143}, {3735, 6376}, {4033, 4087}, {4485, 18040}, {17789, 20644}, {18051, 18138}, {20444, 20451}, {20446, 20448}, {20454, 20643}

X(20454) = (X(1), X(2), X(6), X(292); X(312), X(75), X(304), X(20446)) COLLINEATION IMAGE OF X(2112)

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

See X(20444).

X(20454) lies on these lines: {75, 1281}, {304, 20452}, {312, 8024}, {20444, 20446}, {20453, 20643}

X(20455) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(105)

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

(1,42), (2,6), (6,39), (31,3778), (105,20455), (238,20456), (292,20457), (365,20458), (672,20459), (1423,20460), (1931,20461), (2053,20462), (2054,20463), (3009,20464), (2112,20465)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b^2 : c^2
m(A') = a^2 (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : -b^2 (Sqrt[a] - Sqrt[c]) (a + Sqrt[a] Sqrt[c] + c) : -(Sqrt[a] - Sqrt[b]) (a + Sqrt[a] Sqrt[b] + b) c^2
m(A1) = 2 a^2 b c : b^2 (-a^2 + b c) : c^2 (-a^2 + b c)
m(A2) = a^2 (b + c) (b^2 - b c + c^2) : -b^2 c (a b - c^2) : b c^2 (b^2 - a c)
m(A3) = -a^2 (b + c) (a b + a c + b c) : b^2 (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : c^2 (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = a^2 (a + b + c) (b^2 + c^2) : -b^2 (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : -c^2 (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = -a^2 (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : -b^2 (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : -c^2 (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = a^2 (a + b + c) (a^2 b^2 - a b^3 - b^3 c + a^2 c^2 + 2 b^2 c^2 - a c^3 - b c^3) : -b^2 (a^2 + b^2 - a c - b c) (-a^3 + a^2 b + a b c - 2 a c^2 - b c^2 - c^3) : -c^2 (-a^3 - 2 a b^2 - b^3 + a^2 c + a b c - b^2 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (y + z) : b^2 (z + x) : c^2 (x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(6)), where A'B'C' = m(ABC).

Let f(a,b,c,x,y,z) = b^5 (b-c) c^5 (a^2-b c) (b^2+b c+c^2) x^3-a^5 b c^3 (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z+a^5 b^3 c (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20455) lies on these lines: {6, 692}, {42, 20465}, {43, 57}, {100, 2991}, {193, 3888}, {209, 4028}, {511, 9441}, {518, 3717}, {579, 3939}, {1814, 5091}, {2347, 3778}, {2835, 3755}, {3034, 17205}, {3169, 3174}, {3688, 16973}, {3802, 9052}, {4014, 5845}, {4517, 16496}, {5580, 15615}, {6373, 8659}, {9032, 14404}, {20456, 20459}, {20466, 20467}

X(20455) = crosssum of X(2) and X(105)
X(20455) = crosspoint of X(6) and X(518)
X(20455) = polar conjugate of isotomic conjugate of X(20728)

X(20456) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(238)

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

See X(20455).

X(20456) lies on these lines: {6, 560}, {31, 4253}, {36, 386}, {38, 17023}, {39, 42}, {44, 3122}, {75, 749}, {210, 16604}, {239, 291}, {244, 3008}, {256, 17120}, {524, 2228}, {527, 3123}, {579, 2209}, {583, 1918}, {672, 3747}, {674, 3248}, {756, 1125}, {869, 2275}, {982, 17367}, {984, 17397}, {1015, 3009}, {2239, 18206}, {2308, 11205}, {3589, 4022}, {3728, 5750}, {3758, 4443}, {3759, 4446}, {3779, 7032}, {4283, 4649}, {4380, 4905}, {4393, 12782}, {4433, 20331}, {4735, 16666}, {12263, 17165}, {16707, 16887}, {17065, 17349}, {18038, 19560}, {20455, 20459}, {20460, 20462}, {20461, 20463}, {20464, 20467}, {20669, 20670}

X(20457) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(292)

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

See X(20455).

X(20457) lies on these lines: {6, 291}, {42, 20663}, {213, 14839}, {239, 3978}, {668, 17033}, {672, 20669}, {1015, 1193}, {2238, 4974}, {3778, 20465}, {4368, 17475}, {18205, 20333}, {20459, 20464}

X(20458) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(365)

Barycentrics a^2 (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : :

See X(20455).

X(20458) lies on these lines: {6, 20469}, {75, 366}, {256, 2069}, {291, 2068}, {984, 20357}

X(20459) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(672)

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

See X(20455).

X(20459) lies on these lines: {6, 41}, {9, 16823}, {39, 2309}, {57, 20665}, {105, 238}, {239, 19565}, {244, 2225}, {330, 16827}, {573, 16779}, {614, 5364}, {667, 6373}, {673, 10030}, {726, 20372}, {910, 8850}, {1001, 1334}, {1170, 1432}, {1438, 13329}, {1445, 2082}, {1909, 3691}, {2110, 2340}, {2112, 7193}, {2209, 16502}, {2269, 16503}, {2279, 16469}, {2308, 2350}, {3730, 15485}, {4253, 5144}, {4279, 5299}, {4498, 6084}, {4649, 17474}, {5701, 20593}, {14964, 18792}, {17475, 20363}, {20455, 20456}, {20457, 20464}

X(20460) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(1423)

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

See X(20455).

X(20460) lies on these lines: {1, 20665}, {6, 41}, {8, 2319}, {21, 644}, {42, 237}, {291, 8848}, {511, 20667}, {1201, 3051}, {2053, 3056}, {2082, 3509}, {2170, 3721}, {7117, 20465}, {7991, 9315}, {20456, 20462}

X(20460) = polar conjugate of isotomic conjugate of X(20732)

X(20461) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(1931)

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

See X(20455).

X(20461) lies on these lines: {1, 9560}, {6, 2248}, {39, 2653}, {42, 181}, {111, 6083}, {115, 1737}, {291, 1757}, {511, 741}, {517, 16613}, {519, 5213}, {579, 3981}, {672, 3124}, {1015, 5164}, {1169, 1171}, {1213, 1215}, {1914, 20666}, {2245, 17735}, {3721, 16589}, {20456, 20463}

X(20461) = polar conjugate of isotomic conjugate of X(20733)

X(20462) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(2053)

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

See X(20455).

X(20462) lies on these lines: {6, 20473}, {43, 57}, {3778, 20464}, {20456, 20460}

X(20462) = isogonal conjugate of isotomic conjugate of X(20338)
X(20462) = polar conjugate of isotomic conjugate of X(20734)

X(20463) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(2054)

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

See X(20455).

X(20463) lies on these lines: {6, 2054}, {39, 20465}, {194, 1046}, {291, 8935}, {20456, 20461}

X(20463) = isogonal conjugate of isotomic conjugate of X(20339)
X(20463) = polar conjugate of isotomic conjugate of X(20735)

X(20464) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(3009)

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

See X(20455).

X(20464) lies on these lines: {2, 18194}, {6, 31}, {43, 7032}, {239, 3510}, {291, 8851}, {350, 3226}, {669, 2451}, {899, 9362}, {1575, 3248}, {1911, 3684}, {2227, 9025}, {2238, 3009}, {2998, 17157}, {3720, 18170}, {3778, 20462}, {4704, 17018}, {20456, 20467}, {20457, 20459}

X(20464) = isogonal conjugate of isotomic conjugate of X(20340)
X(20464) = polar conjugate of isotomic conjugate of X(20736)

X(20465) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(2112)

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

See X(20455).

X(20465) lies on these lines: {6, 8852}, {39, 20463}, {42, 20455}, {291, 9472}, {3778, 20457}, {7117, 20460}

X(20465) = isogonal conjugate of isotomic conjugate of X(20341)
X(20465) = polar conjugate of isotomic conjugate of X(20737)

X(20466) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(2144)

Barycentrics a^3 (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20455).

X(20466) lies on these lines: {20455, 20467}

X(20467) = (X(1), X(2), X(6), X(31); X(42), X(6), X(39), X(3778)) COLLINEATION IMAGE OF X(20332)

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

See X(20455).

X(20467) lies on these lines: {1, 39}, {1575, 20532}, {3271, 3778}, {7117, 20460}, {20455, 20466}, {20456, 20464}

X(20467) = isogonal conjugate of isotomic conjugate of X(20343)
X(20467) = polar conjugate of isotomic conjugate of X(20738)

X(20468) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(105)

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

(1,55), (2,6), (6,3), (31,1631), (105,20468), (365,20469), (672,20470), (1423,20471), (1931,20472), (2053,20473), (2054,20474), (3009,20475), (2112,20476)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a^2 : -b^2 : -c^2
m(A') = a^2 (a^(3/2) + b^(3/2) + c^(3/2)) : -b^2 (a^(3/2) + b^(3/2) - c^(3/2)) : -c^2 (a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = -a^2 - 2 b c : b^2 : c^2
m(A2) = a^2 (b^3 + a b c + c^3) : -b^2 (b^3 + a b c - c^3) : -c^2 (-b^3 + a b c + c^3)
m(A3) = -a^2 (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : b^2 (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : c^2 (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = a^2 (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) : -b^2 (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) : -c^2 (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)
m(A5) = a^2 (a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3) : b^2 (-a^3 + b^3 - 2 a^2 c - b^2 c + b c^2 - c^3) : c^2 (-a^3 - 2 a^2 b - b^3 + b^2 c - b c^2 + c^3)
m(A6) = a^2 (a^5 - a^4 b - a^2 b^3 + a b^4 - a^4 c - a^3 b c + a b^3 c + b^4 c - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + a c^4 + b c^4) : -b^2 (a^5 - a^4 b - a^2 b^3 + a b^4 - a^4 c - a^3 b c + a b^3 c + b^4 c + 2 a^3 c^2 + 2 a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 3 a b c^3 - b^2 c^3 - a c^4 - b c^4) : -c^2 (a^5 - a^4 b + 2 a^3 b^2 - a^2 b^3 - a b^4 - a^4 c - a^3 b c + 2 a^2 b^2 c - 3 a b^3 c - b^4 c + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 + a b c^3 + b^2 c^3 + a c^4 + b c^4)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (-x + y + z) : b^2 (-y + z + x) : c^2 (-z + x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(6)), where A' = m(A).

Let f(a,b,c,x,y,z) = b^3 (b-c) c^3 (b^2+b c+c^2) x^3-a^3 c^2 (a^4+a b^3-b^2 c^2-a c^3) y^2 z+a^3 b^2 (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20468) lies on these lines: {6, 692}, {38, 55}, {100, 4437}, {159, 3197}, {197, 1615}, {198, 480}, {518, 3220}, {926, 8659}, {1030, 2870}, {1350, 2807}, {1357, 1460}, {1633, 5845}, {2110, 5096}, {2930, 8674}, {3270, 10387}

X(20469) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(365)

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

See X(20468).

X(20469) lies on these lines: {6, 20458}, {366, 1631}, {2068, 4497}, {2069, 4471}

X(20470) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(672)

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

See X(20468).

X(20470) lies on these lines: {1, 5132}, {2, 16678}, {3, 142}, {6, 41}, {10, 16414}, {11, 851}, {25, 1626}, {35, 16484}, {36, 238}, {37, 12721}, {55, 750}, {57, 3185}, {86, 4225}, {105, 16693}, {197, 1617}, {228, 354}, {241, 3827}, {244, 3724}, {404, 5263}, {474, 19863}, {518, 4557}, {602, 14529}, {614, 2352}, {669, 2487}, {673, 7677}, {674, 1818}, {692, 13329}, {740, 15571}, {855, 15326}, {958, 17259}, {993, 4245}, {995, 5156}, {1011, 4423}, {1054, 5143}, {1086, 1284}, {1201, 1918}, {1279, 2223}, {1376, 3741}, {1386, 16679}, {1402, 3752}, {1473, 15494}, {1621, 4210}, {1698, 16297}, {1699, 7416}, {2110, 5096}, {2283, 4318}, {2340, 9049}, {2886, 16056}, {2933, 10835}, {2975, 17277}, {2999, 16878}, {3000, 5204}, {3149, 15622}, {3242, 4022}, {3246, 16694}, {3576, 7420}, {3616, 16451}, {3624, 16287}, {3685, 4436}, {3816, 4192}, {3825, 19648}, {3847, 19646}, {3941, 7290}, {4038, 18185}, {4068, 15569}, {4184, 5284}, {4267, 18166}, {4293, 19256}, {4316, 13744}, {4366, 19308}, {4447, 5846}, {4649, 5563}, {5251, 19241}, {5259, 17524}, {5272, 16778}, {5303, 7419}, {5437, 10434}, {5550, 16452}, {5701, 20605}, {6645, 20148}, {6667, 19546}, {6685, 19342}, {6691, 19513}, {7191, 16687}, {7280, 7428}, {7354, 13724}, {8167, 16058}, {8301, 17031}, {9342, 16057}, {10013, 19760}, {10200, 19543}, {11329, 20172}, {15507, 17768}, {16020, 18610}, {16286, 19862}, {16291, 19878}, {16684, 16823}

X(20470) = isogonal conjugate of isotomic conjugate of X(20347)
X(20470) = isogonal conjugate of anticomplement of X(39046)
X(20470) = anticomplement of complementary conjugate of X(39046)
X(20470) = crossdifference of every pair of points on line X(37)X(522)
X(20470) = polar conjugate of isotomic conjugate of X(20744)

X(20471) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(1423)

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

See X(20468).

X(20471) lies on these lines: {3, 2329}, {6, 41}, {55, 237}, {101, 15654}, {197, 17798}, {280, 1436}, {910, 17448}, {1610, 8301}, {1755, 2176}, {2076, 20676}, {2933, 20476}, {5204, 20331}, {14829, 15509}, {16969, 20674}

X(20472) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(1931)

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

See X(20468).

X(20472) lies on these lines: {2, 1029}, {6, 2248}, {55, 199}, {661, 3733}, {958, 19329}, {2242, 18755}, {2702, 20675}, {2915, 16974}, {3509, 4053}, {9509, 17798}

X(20472) = isogonal conjugate of isotomic conjugate of X(20349)
X(20472) = isogonal conjugate of anticomplement of X(39042)

X(20473) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(2053)

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

See X(20468).

X(20473) lies on these lines: {6, 20462}, {55, 17459}, {159, 1740}, {197, 17798}, {1486, 20676}, {1631, 20475}

X(20474) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(2054)

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

See X(20468).

X(20474) lies on these lines: {1, 20677}, {3, 2784}, {6, 2054}, {1634, 8053}, {3511, 20475}, {9509, 17798}

X(20475) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(3009)

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

See X(20468).

X(2047562) lies on these lines: {6, 31}, {10, 16683}, {23, 385}, {100, 16693}, {171, 16679}, {183, 18613}, {238, 18793}, {404, 16691}, {519, 8618}, {1631, 20473}, {1634, 17731}, {1755, 9016}, {3511, 20474}, {3550, 3941}, {3684, 4557}, {3750, 4068}, {5030, 8671}, {8266, 16678}, {8301, 17031}, {8844, 9055}, {17448, 18758}, {18092, 18093}

X(20476) = (X(1), X(2), X(6), X(31); X(55), X(6), X(3), X(1631)) COLLINEATION IMAGE OF X(2112)

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

See X(20468).

X(20476) lies on these lines: {3, 2784}, {6, 8852}, {38, 55}, {291, 4497}, {753, 7236}, {1283, 17592}, {1631, 4361}, {2933, 20471}, {4443, 4471}, {7281, 16560}

X(20477) = X(2)X(53)∩X(3)X(95)

Barycentrics a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+2*b^2*c^2*(b^2-c^2)^2 : :
X(20477) = 3 X(2) - 2 X(53)

Let A'B'C' be the tangential triangle. Let La be the reflection of line B'C' in the perpedicular bisector of BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is also the tangential triangle of the dual-of-orthic triangle, and X(20477) = X(7)-of-A"B"C". (Randy Hutson, August 29, 2018)

See Angel Montesdeoca, HG220718.

X(20477) lies on these lines: {2,53}, {3,95}, {6,401}, {20,64}, {22,157}, {30,317}, {75,10538}, {76,11414}, {97,19212}, {99,1294}, {159,1632}, {216,458}, {286,1012}, {297,6389}, {302,19772}, {303,19773}, {325,1370}, {338,8553}, {339,12083}, {340,1657}, {394,8613}, {441,17907}, {511,6751}, {577,9308}, {648,15905}, {925,2373}, {940,18667}, {1007,7396}, {1078,9715}, {1235,10323}, {1238,7788}, {1272,13219}, {1305,2370}, {1494,15681}, {1993,19180}, {2871,12220}, {3151,4417}, {3153,18380}, {3186,11676}, {3260,9723}, {6515,18953}, {6617,15466}, {6638,16089}, {7560,14829}, {9307,10602}, {10313,14614}, {10979,14767}, {11257,19459}, {11412,19206}, {19121,19156}

X(20477) = isogonal conjugate of X(32319)
X(20477) = isotomic conjugate of X(15318)
X(20477) = anticomplement of X(53)
X(20477) = X(6)-of-dual-of-orthic-triangle

X(20478) = EULER LINE INTERCEPT OF X(578)X(14374)

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

See Angel Montesdeoca, HG220718.

X(20478) lies on these lines: {2,3}, {578,14374}, {1147,2574}, {2575,3357}, {6102,13414}

X(20478) = {X(3),X(4)}-harmonic conjugate of X(20479)

X(20479) = EULER LINE INTERCEPT OF X(578)X(14375)

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

See Angel Montesdeoca, HG220718.

X(20479) lies on these lines: {2,3}, {578,14375}, {1147,2575}, {2574,3357}, {6102,13415}

X(20479) = {X(3),X(4)}-harmonic conjugate of X(20478)

X(20480) = X(110)X(382)∩X(476)X(15646)

Barycentrics 4 a^16 - 10 a^14 (b^2 + c^2) + a^12 (-6 b^4 + 49 b^2 c^2 - 6 c^4) + 5 a^10 (8 b^6 - 11 b^4 c^2 - 11 b^2 c^4 + 8 c^6) - a^8 (40 b^8 + 23 b^6 c^2 - 135 b^4 c^4 + 23 b^2 c^6 + 40 c^8) + 3 a^6 (b^2 - c^2)^2 (2 b^6 + 25 b^4 c^2 + 25 b^2 c^4 + 2 c^6) + a^4 (b^2 - c^2)^2 (10 b^8 - 6 b^6 c^2 - 57 b^4 c^4 - 6 b^2 c^6 + 10 c^8) - 2 a^2 (b^2 - c^2)^4 (2 b^6 + 5 b^4 c^2 + 5 b^2 c^4 + 2 c^6) - 4 b^2 c^2 (b^2 - c^2)^6 : :

See Angel Montesdeoca, HG220718 and Hyacinthos 27999.

X(20480) lies on these lines: {110,382}, {476,15646}

X(20481) = X(2)X(6)∩X(3)X(111)

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

See Angel Montesdeoca, HG030818.

X(20481) lies on these lines: {2,6}, {3,111}, {22,8588}, {23,5210}, {25,15655}, {187,1995}, {353,5085}, {399,9759}, {549,16317}, {574,3291}, {647,9175}, {1350,13192}, {1383,3053}, {1384,11284}, {2030,5651}, {2502,6800}, {5023,14002}, {5024,9465}, {5055,6032}, {5569,9172}, {5585,7492}, {5640,11173}, {7485,8589}, {8716,9870}, {8770,15246}, {9225,10485}

X(20482) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(105)

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

(1,594), (2,10), (6,15523), (31,16886), (105,20482), (238,20483), (292,20484), (365,20485), (672,20486), (1423,20487), (1931,20488), (2053,20489), (2054,20490), (3009,20491), (2112,20492), (2144,20493), (20332,20494)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : a + c : a + b
m(A') = (Sqrt[b] + Sqrt[c]) (b + c) (b - Sqrt[b] Sqrt[c] + c) : -(Sqrt[a] - Sqrt[c]) (a + c) (a + Sqrt[a] Sqrt[c] + c) : -(Sqrt[a] - Sqrt[b]) (a + b) (a + Sqrt[a] Sqrt[b] + b)
m(A1) = 2 b c (b + c) : -(a + c) (a^2 - b c) : -(a + b) (a^2 - b c)
m(A2) = (b + c)^2 (b^2 - b c + c^2) : c (a + c) (-a b + c^2) : b (a + b) (b^2 - a c)
m(A3) = (b + c)^2 (a b + a c + b c) : -(a + c) (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : -(a + b) (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = -(b + c) (a + b + c) (b^2 + c^2) : (a + c) (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : (a + b) (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = (b + c) (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : (a + c) (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : (a + b) (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = -(b + c) (a + b + c) (-a^2 b^2 + a b^3 + b^3 c - a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : (a + c) (a^2 + b^2 - a c - b c) (a^3 - a^2 b - a b c + 2 a c^2 + b c^2 + c^3) : (a + b) (a^3 + 2 a b^2 + b^3 - a^2 c - a b c + b^2 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b + c)(y + z) : (c + a)(z + x): (a + b)(x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(10)), where A' = m(A).

X(20482) lies on these lines: {10, 1018}, {120, 17464}, {594, 20492}, {1441, 16603}, {20483, 20486}, {20493, 20494}, {20653, 20656}

X(20483) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(238)

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

See X(20482).

Let f(a,b,c,x,y,z) = a (a+b)^3 (b-c) (a+c)^3 (a^2-b c) (b^2+b c+c^2) x^3-(a+b)^2 (a+c) (b+c)^3 (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z+(a+b) (a+c)^2 (b+c)^3 (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20483) lies on these lines: {10, 213}, {37, 4972}, {115, 3992}, {594, 2294}, {661, 20659}, {762, 3454}, {1575, 3006}, {3263, 20541}, {3290, 3823}, {3726, 3836}, {3932, 4037}, {4426, 5300}, {8013, 10026}, {20482, 20486}, {20487, 20489}, {20488, 20490}, {20491, 20494}

X(20484) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(292)

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

See X(20482).

X(20484) lies on these lines: {10, 20496}, {594, 2486}, {1211, 3120}, {16886, 20492}, {20333, 20356}, {20486, 20491}, {20654, 20658}

X(20485) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[b] + Sqrt[c]) (b + c) (b - Sqrt[b] Sqrt[c] + c) : :

See X(20482).

X(20485) lies on these lines: {10, 20497}, {3661, 20334}

X(20486) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(672)

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

See X(20482).

X(20486) lies on these lines: {2, 4447}, {10, 12}, {11, 3912}, {76, 3703}, {325, 334}, {354, 17048}, {661, 2533}, {1086, 2228}, {1500, 4854}, {1836, 3501}, {2486, 3943}, {2886, 3661}, {3136, 15523}, {3507, 17719}, {3782, 12782}, {3816, 17244}, {3836, 20340}, {3932, 3948}, {4044, 6057}, {4433, 13576}, {4518, 17789}, {8299, 20556}, {11680, 17230}, {16587, 16589}, {20335, 20358}, {20482, 20483}, {20484, 20491}

X(20487) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(1423)

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

See X(20482).

X(20487) lies on these lines: {10, 12}, {11, 312}, {3452, 7064}, {4193, 4903}, {16569, 17719}, {20258, 20359}, {20483, 20489}, {20492, 20494}

X(20488) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(1931)

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

See X(20482).

X(20488) lies on these lines: {10, 894}, {12, 594}, {1215, 8013}, {1220, 1268}, {2533, 4977}, {4062, 17719}, {20337, 20360}, {20483, 20490}, {20491, 20658}

X(20489) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(2053)

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

See X(20482).

X(20489) lies on these lines: {10, 20503}, {1441, 16603}, {16886, 20491}, {20338, 20361}, {20483, 20487}

X(20490) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(2054)

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

See X(20482).

X(20490) lies on these lines: {10, 115}, {594, 20531}, {15523, 20492}, {20339, 20362}, {20483, 20488}

X(20491) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(3009)

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

See X(20482).

X(20491) lies on these lines: {10, 37}, {141, 10009}, {1086, 20549}, {3836, 20532}, {16886, 20489}, {17786, 20271}, {20340, 20363}, {20483, 20494}, {20484, 20486}, {20488, 20658}

X(20492) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(2112)

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

See X(20482).

X(20492) lies on these lines: {10, 4154}, {594, 20482}, {15523, 20490}, {16886, 20484}, {20341, 20364}, {20487, 20494}

X(20493) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(2144)

Barycentrics a (b + c) (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20482).

X(20493) lies on these lines: {20342, 20365}, {20482, 20494}

X(20494) = (X(1), X(2), X(6), X(31); X(594), X(10), X(15523), X(16886)) COLLINEATION IMAGE OF X(20332)

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

See X(20482).

X(20494) lies on these lines: {10, 18793}, {321, 2887}, {20343, 20366}, {20482, 20493}, {20483, 20491}, {20487, 20492}

X(20495) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(105)

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

(1,2321), (2,10), (6,306), (31,4153), (105,20495), (238,4071), (292,20496), (365,20497), (1423,20498), (1931,20499), (2053,20581), (2054,20500), (3009,20501), (2112,20502)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - b - c : a + c : a + b
m(A') = (b + c) (a^(3/2) + b^(3/2) + c^(3/2)) : -(a + c) (a^(3/2) + b^(3/2) - c^(3/2)) : -(a + b) (a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = (b + c) (a^2 + 2 b c) : -a^2 (a + c) : -a^2 (a + b)
m(A2) = (b + c) (b^3 + a b c + c^3) : -(a + c) (b^3 + a b c - c^3) : -(a + b) (-b^3 + a b c + c^3)
m(A3) = (b + c) (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : -(a + c) (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : -(a + b) (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = (b + c) (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3) : -(a + c) (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c - a c^2 - b c^2 - c^3) : -(a + b) (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)
m(A5) = (b + c) (-a^3 + b^3 - 2 a b c - b^2 c - b c^2 + c^3) : (a + c) (a^3 - b^3 + 2 a^2 c + b^2 c - b c^2 + c^3) : (a + b) (a^3 + 2 a^2 b + b^3 - b^2 c + b c^2 - c^3),

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b + c)(-x + y + z) : (c + a)(-y + z + x): (a + b)(-z + x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(10)), where A' = m(A).

Let f(a,b,c,x,y,z) = a^3 (a+b)^3 (b-c) (a+c)^3 (b^2+b c+c^2) x^3-b (a+b)^2 c (a+c) (b+c)^3 (a^4+a b^3-b^2 c^2-a c^3) y^2 z+b (a+b) c (a+c)^2 (b+c)^3 (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20495) lies on these lines: {10, 1018}, {2321, 18589}, {4103, 4153}, {16550, 20344}

X(20496) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(292)

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

See X(20495).

X(20496) lies on these lines: {10, 20484}, {313, 2321}, {321, 1109}, {668, 18037}, {1921, 3912}, {3596, 17788}, {3948, 6541}, {4153, 20502}, {17738, 20345}

X(20497) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(365)

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

See X(20495).

X(20497) lies on these lines: {10, 20485}, {510, 20346}

X(20498) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(1423)

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

See X(20495).

X(20498) lies on these lines: {10, 12}, {329, 6210}, {518, 20545}, {908, 3705}, {946, 4385}, {984, 3452}, {4071, 20503}, {4859, 16569}, {20348, 20368}

X(20499) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(1931)

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

See X(20495).

X(20499) lies on these lines: {10, 894}, {313, 502}, {514, 4036}, {2321, 4053}, {4071, 20500}, {20349, 20369}

X(20500) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(2054)

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

See X(20495).

X(20500) lies on these lines: {10, 115}, {306, 20502}, {321, 20636}, {4071, 20499}, {4568, 18035}, {20351, 20371}

X(20501) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(3009)

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

See X(20495).

X(20501) lies on these lines: {10, 37}, {536, 20549}, {3963, 17867}, {4079, 4129}, {4153, 20503}, {10009, 17233}, {20352, 20372}

X(20502) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(2112)

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

See X(20495).

X(20502) lies on these lines: {10, 4154}, {306, 20500}, {2321, 18589}, {4153, 20496}, {20353, 20373}

X(20503) = (X(1), X(2), X(6), X(31); X(2321), X(10), X(306), X(4153)) COLLINEATION IMAGE OF X(2144)

Barycentrics a*(b + c)*(-2*a^8*b^6 + 2*a^5*b^9 + a^9*b^4*c - a^3*b^10*c + 8*a^7*b^5*c^2 - 10*a^4*b^8*c^2 - 5*a^8*b^3*c^3 + 7*a^5*b^6*c^3 + 7*a^2*b^9*c^3 + a^9*b*c^4 - 17*a^6*b^4*c^4 + 3*a^3*b^7*c^4 - b^10*c^4 + 8*a^7*b^2*c^5 + 3*a^4*b^5*c^5 - 4*a*b^8*c^5 - 2*a^8*c^6 + 7*a^5*b^3*c^6 - 5*a^2*b^6*c^6 + 3*a^3*b^4*c^7 + 4*b^7*c^7 - 10*a^4*b^2*c^8 - 4*a*b^5*c^8 + 2*a^5*c^9 + 7*a^2*b^3*c^9 - a^3*b*c^10 - b^4*c^10) : :

X(20503) lies on these lines: {10, 20493}, {20354, 20374}

X(20504) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(105)

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

(1,523), (2,514), (6,16892), (31,3801), (105,20504), (238,4791), (292,20505), (365,20506), (672,20607), (1423,20508), (1931,20509), (2053,20510), (2054,20511), (3009,20512), (2112,20513), (2144,20514), (20332,20515)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : a - c : -a + b
m(A') = -(Sqrt[b] - Sqrt[c]) (Sqrt[b] + Sqrt[c])^2 (b - Sqrt[b] Sqrt[c] + c) : -(Sqrt[a] - Sqrt[c])^2 (Sqrt[a] + Sqrt[c]) (a + Sqrt[a] Sqrt[c] + c) : (Sqrt[a] - Sqrt[b])^2 (Sqrt[a] + Sqrt[b]) (a + Sqrt[a] Sqrt[b] + b)
m(A1) = -2 b (b - c) c : -(a - c) (a^2 - b c) : (a - b) (a^2 - b c)
m(A2) = -(b - c) (b + c) (b^2 - b c + c^2) : -(a - c) c (a b - c^2) : (a - b) b (-b^2 + a c)
m(A3) = (b - c) (b + c) (a b + a c + b c) : (a - c) (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : -(a - b) (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = (b - c) (a + b + c) (b^2 + c^2) : (a - c) (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : -(a - b) (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = (-b + c) (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : (a - c) (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : -(a - b) (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = (b - c) (a + b + c) (-a^2 b^2 + a b^3 + b^3 c - a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : (a - c) (a^2 + b^2 - a c - b c) (a^3 - a^2 b - a b c + 2 a c^2 + b c^2 + c^3) : -(a - b) (a^3 + 2 a b^2 + b^3 - a^2 c - a b c + b^2 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b - c)(y + z) : (c - a)(z + x): (a - b)(x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(514)), where A' = m(A).

Let f(a,b,c,x,y,z) = a (a-b)^2 (a-c)^2 (a^2-b c) (b^2+b c+c^2) x^3-(a-b) (b-c)^2 (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z-(a-c) (b-c)^2 (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20504) lies on these lines: {1, 514}, {523, 20513}, {1441, 20510}, {2191, 7649}, {20514, 20515}

X(20505) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(292)

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

See X(20504).

X(20505) lies on these lines: {514, 3572}, {523, 3728}, {3004, 4988}, {3801, 20513}, {20507, 20512}

X(20506) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[b] - Sqrt[c]) (Sqrt[b] + Sqrt[c])^2 (b - Sqrt[b] Sqrt[c] + c) : :

See X(20504).

X(20506) lies on these lines: {514, 20519}

X(20507) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(672)

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

See X(20504).

X(20507) lies on these lines: {241, 514}, {244, 4124}, {335, 918}, {3801, 16892}, {20505, 20512}

X(20508) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(1423)

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

See X(20504).

X(20508) lies on these lines: {241, 514}, {312, 3700}, {1278, 4467}, {3287, 7192}, {17069, 17490}, {20513, 20515}

X(20509) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(1931)

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

See X(20504).

X(20509) lies on these lines: {148, 150}, {514, 17212}, {523, 656}, {4369, 4988}, {16892, 17422}

X(20510) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(2053)

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

See X(20504).

X(20510) lies on these lines: {514, 20523}, {1441, 20504}, {3801, 20512}

X(20511) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(2054)

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

See X(20504).

X(20511) lies on these lines: {514, 1125}, {16892, 20513}

X(20512) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(3009)

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

See X(20504).

X(20512) lies on these lines: {30, 511}, {3801, 20510}, {20505, 20507}

X(20512) = isogonal conjugate of X(20640)

X(20513) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(2112)

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

See X(20504).

X(20513) lies on these lines: {514, 20526}, {523, 20504}, {3801, 20505}, {16892, 20511}, {20508, 20515}

X(20514) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(2144)

Barycentrics a (b - c) (a^2 - b c) (a^6 b^6 - a^3 b^9 - a^4 b^7 c - a^5 b^5 c^2 + 7 a^2 b^8 c^2 - 2 a^6 b^3 c^3 - 9 a^3 b^6 c^3 - b^9 c^3 + 12 a^4 b^4 c^4 - 5 a b^7 c^4 - a^5 b^2 c^5 + 6 a^2 b^5 c^5 + a^6 c^6 - 9 a^3 b^3 c^6 + 4 b^6 c^6 - a^4 b c^7 - 5 a b^4 c^7 + 7 a^2 b^2 c^8 - a^3 c^9 - b^3 c^9) : :

See X(20504).

X(20514) lies on these lines: {20504, 20515}

X(20515) = (X(1), X(2), X(6), X(31); X(523), X(514), X(16892), X(3801)) COLLINEATION IMAGE OF X(20332)

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

See X(20504).

X(20515) lies on these lines: {321, 693}, {330, 514}, {20504, 20514}, {20508, 20513}

X(20516) = (name pending)

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

See X(20517).

X(20516) lies on these lines: {1, 514}, {10, 4163}, {281, 7649}, {522, 3663}, {1734, 3670}, {2812, 11028}, {4025, 4392}, {4458, 19965}, {13259, 14430}

X(20517) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(31)

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

(1,522), (2,514), (6,4025), (31,20517), (105,20516), (238,4458), (292,20518), (365,20519), (672,20520), (1423,20521), (1931,20522), (2053,20523), (2054,20524), (3009,20525), (2112,20526)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - b + c : - a + c : a - b
m(A') = (c - b)(a^(3/2) + b^(3/2) + c^(3/2)) : (c - a)(a^(3/2) + b^(3/2) - c^(3/2)) : (a - b)(a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = (c - b) (a^2 + 2 b c) : -a^2 (a - c) : a^2 (a - b)
m(A2) = (c - b) (b^3 + a b c + c^3) : (c - a) (b^3 + a b c - c^3) : (a - b) (-b^3 + a b c + c^3)
m(A3) = (b - c) (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : (a - c) (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : (b - a) (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = (b - c) (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3) : (a - c) (a^2 b + a b^2 + b^3 + a^2 c + a b c + b^2 c - a c^2 - b c^2 - c^3) : -(a - b) (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)
m(A5) = -(-b + c) (a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3) : (a - c) (a^3 - b^3 + 2 a^2 c + b^2 c - b c^2 + c^3) : -(a - b) (a^3 + 2 a^2 b + b^3 - b^2 c + b c^2 - c^3),

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b - c)(-x + y + z) : (c - a)(-y + z + x): (a - b)(-z + x + y) , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(514)), where A' = m(A).

Let f(a,b,c,x,y,z) = a^3 (a-b)^2 (a-c)^2 (b^2+b c+c^2) x^3-(a-b) b (b-c)^2 c (a^4+a b^3-b^2 c^2-a c^3) y^2 z-b (a-c) (b-c)^2 c (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20517) lies on these lines: {10, 4163}, {514, 659}, {522, 4823}, {525, 676}, {663, 4707}, {826, 4874}, {1125, 6332}, {3810, 3960}, {4025, 8714}, {4453, 4905}, {7649, 14618}, {20518, 20526}, {20523, 20525}

X(20518) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(292)

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

See X(20517).

X(20518) lies on these lines: {75, 522}, {514, 3572}, {693, 4359}, {2786, 3766}, {3676, 6063}, {13246, 14296}, {20517, 20526}, {20520, 20525}

X(20519) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(365)

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

See X(20517).

X(20519) lies on these lines: {514, 20506}

X(20520) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(672)

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

See X(20517).

X(20520) lies on these lines: {10, 13259}, {241, 514}, {244, 1111}, {522, 4411}, {693, 1734}, {1769, 3663}, {2785, 15903}, {3664, 3738}, {4025, 8714}, {4458, 19965}, {4674, 6548}, {20518, 20525}

X(20521) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(1423)

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

See X(20517).

X(20521) lies on these lines: {241, 514}, {4458, 20523}

X(20522) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(1931)

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

See X(20517).

X(20522) lies on these lines: {514, 17212}, {522, 4823}, {4458, 20524}, {16732, 17205}

X(20523) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(2053)

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

See X(20517).

X(20523) lies on these lines: {514, 20510}, {3551, 3667}, {4458, 20521}, {20517, 20525}

X(20524) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(2054)

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

See X(20517).

X(20524) lies on these lines: {514, 1125}, {693, 4425}, {3741, 4025}, {4458, 20522}, {8714, 16887}

X(20525) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(3009)

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

See X(20517).

X(20525) lies on these lines: {30, 511}, {3310, 6685}, {20517, 20523}, {20518, 20520}

X(20525) = isogonal conjugate of X(20696)

X(20526) = (X(1), X(2), X(6), X(105); X(522), X(514), X(4025), X(20516)) COLLINEATION IMAGE OF X(2112)

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

See X(20517).

X(20526) lies on these lines: {514, 20513}, {522, 3663}, {3741, 4025}, {20517, 20518}

X(20527) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(365)

Barycentrics b^(1/2) + c^(1/2) : :

(1,1), (2,37), (6,10), (31,141), (105,16593), (292,17793), (365,20527), (1423,3061), (1931,10026), (2053,20528), (2054,20529), (3009,20530), (2112,20531)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(A') = Sqrt[b] + Sqrt[c] : -Sqrt[a] + Sqrt[c] : -Sqrt[a] + Sqrt[b]
m(A1) = b + c : -a + b : -a + c
m(A2) = b^2 + c^2 : c (c - b) : b (b - c)
m(A3) = 2 (a b + a c + b c) : -a^2 + b c : -a^2 + b c
m(A4) = (b + c) (a + b + c) : c^2 - a b : b^2 - a c
m(A5) = 2 a^2 - a b + b^2 - a c - 2 b c + c^2 : 2 a^2 + a c - b c + c^2 : 2 a^2 + a b + b^2 - b c
m(A6) = a (a + b + c) (a b - b^2 + a c - c^2) : (a^2 + b^2 - a c - b c) (a^2 - a b + a c + 2 c^2) : (a^2 + a b + 2 b^2 - a c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = y/b + z/c : z/c + x/a : x/a + y/b , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(2)), where A' = medial triangle.

Let f(a,b,c,x,y,z) = (b-c) (a^2-b c) x^3+(-a^2 b+a b^2-2 a^2 c+a c^2+b c^2) y^2 z+(2 a^2 b-a b^2+a^2 c-b^2 c-a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20527) lies on these lines: {2, 366}, {75, 18297}, {86, 20664}, {4180, 20334}

X(20527) = complement of X(366)
X(20527) = complementary conjugate of X(20334)

X(20528) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(2053)

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

See X(20527).

X(20528) lies on these lines: {2, 2319}, {141, 3816}, {142, 17063}, {226, 335}, {325, 20258}, {2884, 12607}, {2886, 17062}, {3061, 3452}, {3835, 3971}, {17052, 20547}, {18589, 20254}

X(20528) = complement of X(2319)
X(20528) = complementary conjugate of X(20258)

X(20529) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(2054)

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

See X(20527).

X(20529) lies on these lines: {2, 9278}, {10, 20531}, {37, 86}, {620, 1125}, {3739, 20339}, {3834, 16597}, {3912, 10026}, {8287, 17239}, {19563, 20530}

X(20529) = complement of X(9278)
X(20529) = complementary conjugate of X(20337)

X(20530) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(3009)

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

See X(20527).

X(20530) lies on these lines: {2, 37}, {11, 20541}, {43, 4852}, {76, 16604}, {141, 3816}, {172, 17541}, {244, 20598}, {513, 3716}, {518, 17793}, {620, 6681}, {626, 3825}, {672, 4465}, {730, 1125}, {1001, 15271}, {1015, 6381}, {1107, 18140}, {1921, 20363}, {2275, 18135}, {3741, 17239}, {3836, 20531}, {3912, 20333}, {3946, 6686}, {4361, 16569}, {4384, 16515}, {4562, 17266}, {4713, 17351}, {5248, 7815}, {5332, 17001}, {6376, 17448}, {6384, 18144}, {6685, 17045}, {6691, 7789}, {7795, 10200}, {10453, 17372}, {16525, 17026}, {16999, 20179}, {19563, 20529}d

X(20530) = isotomic conjugate of antitomic conjugate of X(38247)
X(20530) = complement of X(1575)
X(20530) = complementary conjugate of X(20343)

X(20531) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(2112)

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

See X(20527).

X(20531) lies on these lines: {2, 11}, {10, 20529}, {12, 664}, {37, 5988}, {116, 14839}, {141, 17793}, {325, 3932}, {594, 20490}, {760, 5074}, {857, 4447}, {918, 3837}, {1146, 1329}, {1961, 17056}, {3509, 17747}, {3813, 16825}, {3836, 20530}, {3844, 3846}, {4553, 8287}, {5311, 17724}

X(20531) = complement of X(8301)
X(20531) = complementary conjugate of X(9470)

X(20532) = (X(1), X(2), X(6), X(31); X(2), X(37), X(10), X(141)) COLLINEATION IMAGE OF X(20332)

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

See X(20527).

X(20532) lies on these lines: {2, 3226}, {10, 1015}, {75, 141}, {115, 3454}, {1084, 1213}, {1146, 1329}, {1575, 20467}, {1977, 8050}, {2885, 3815}, {3768, 4370}, {3836, 20491}, {3912, 20333}, {6374, 6386}, {17293, 20139}

X(20532) = complement of X(3226)
X(20532) = complementary conjugate of X(20340)

X(20533) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(105)

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

(1,2), (2,192), (6,8), (31,69), (105,20533), (365,20534), (1423,20535), (1931,20536), (2053,20537), (2054,20538), (3009,350), (2112,25539)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(A') = Sqrt[a] + Sqrt[b] + Sqrt[c] : -Sqrt[a] - Sqrt[b] + Sqrt[c] : -Sqrt[a] + Sqrt[b] - Sqrt[c]
m(A1) = a + b + c : -a + b - c : -a - b + c
m(A2) = b^2 + b c + c^2 : -b^2 - b c + c^2 : b^2 - b c - c^2
m(A3) = -a^2 - 3 a b - 3 a c - 2 b c : a (a + b + c) : a (a + b + c)
m(A4) = 2 a b + b^2 + 2 a c + 3 b c + c^2 : -2 a b - b^2 - b c + c^2 : b^2 - 2 a c - b c - c^2
m(A5) = a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2 : (a - b + c) (a + b + c) : (a + b - c) (a + b + c)
m(A6) = (a^2 - b c) (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) : -a^4 + 2 a^3 b - a^2 b^2 + a^2 b c - 4 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 + 2 a c^3 + 3 b c^3 : -a^4 - a^2 b^2 + 2 a b^3 + 2 a^3 c + a^2 b c + 2 a b^2 c + 3 b^3 c - a^2 c^2 - 4 a b c^2 - 2 b^2 c^2 - b c^3

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = -x/a + y/b + z/c : -y/b + z/c + x/a : -z/c + x/a + y/b , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(2)), where A' = anticomplementary triangle.

Let f(a,b,c,x,y,z) = a (b-c) (a+b+c) x^3+(-a^2 b-a b^2-a^2 c+a c^2+2 b c^2) y^2 z+(a^2 b-a b^2+a^2 c-2 b^2 c+a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 7, 2018)

X(20533) lies on these lines: {2, 11}, {7, 192}, {8, 17755}, {9, 1654}, {63, 2890}, {69, 144}, {71, 16560}, {142, 16826}, {150, 1018}, {239, 5853}, {312, 18037}, {344, 5819}, {516, 3685}, {518, 2113}, {527, 17310}, {544, 4752}, {644, 20096}, {908, 14732}, {966, 4422}, {1086, 3672}, {1818, 3100}, {2796, 4312}, {3119, 18228}, {3243, 17389}, {3693, 4872}, {3790, 5223}, {3991, 4911}, {4294, 17691}, {4660, 17284}, {5687, 17671}, {5698, 17230}, {5880, 6650}, {6172, 17488}, {7291, 20601}, {9055, 17314}, {11997, 15587}, {14100, 17792}, {15171, 17681}, {17358, 18230}, {17375, 20059}

X(20533) = anticomplement of X(673)
X(20533) = anticomplementary conjugate of X(20347)

X(20534) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(365)

Barycentrics Sqrt[a] - Sqrt[b] - Sqrt[c] : :

See X(20527).

X(20534) lies on these lines: {2, 366}, {7, 4180}, {8, 18297}, {144, 4182}, {510, 5011}

X(20534) = anticomplement of X(366)
X(20534) = anticomplementary conjugate of X(20346)

X(20535) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(1423)

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

See X(20527).

X(20535) lies on these lines: {2, 257}, {8, 7985}, {63, 4393}, {144, 145}, {312, 10405}, {329, 6542}, {330, 20358}, {894, 3340}, {908, 17230}, {1278, 20348}, {3436, 20539}, {3666, 17014}, {3672, 3727}, {3729, 11531}, {3732, 5730}, {3735, 4352}, {10025, 11682}, {11683, 17379}, {17090, 20335}, {17489, 20037}, {20089, 20347}

X(20535) = anticomplement of X(3212)
X(20535) = anticomplementary conjugate of X(20350)

X(20536) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(1931)

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

See X(20527).

X(20536) lies on these lines: {2, 6}, {8, 6625}, {148, 519}, {523, 4963}, {540, 14712}, {4037, 6542}, {4062, 9395}, {11104, 20077}, {11599, 20558}, {13174, 17770}, {20016, 20349}

X(20536) = anticomplement of X(17731)
X(20536) = anticomplementary conjugate of X(20351)
X(20536) = anticomplementary isotomic conjugate of X(13174)

X(20537) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(2053)

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

See X(20527).

X(20537) lies on these lines: {2, 2319}, {8, 3978}, {69, 350}, {329, 6542}, {1432, 17778}, {2893, 20559}, {3434, 20350}, {3888, 17082}, {7779, 20348}

X(20537) = anticomplement of X(2319)
X(20537) = anticomplementary conjugate of X(20348)

X(20538) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(2054)

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

See X(20527).

X(20538) lies on these lines: {1, 99}, {2, 9278}, {8, 7261}, {75, 20351}, {192, 4644}, {4037, 6542}

X(20538) = anticomplement of X(9278)
X(20538) = anticomplementary conjugate of X(20349)

X(20539) = (X(1), X(2), X(6), X(31); X(2), X(192), X(8), X(69)) COLLINEATION IMAGE OF X(2112)

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

See X(20527).

X(20539) lies on these lines: {2, 11}, {4, 1840}, {8, 7261}, {69, 17794}, {150, 14839}, {192, 5992}, {312, 3416}, {350, 4645}, {388, 664}, {516, 3509}, {518, 4872}, {760, 5195}, {1146, 2551}, {1478, 9875}, {1836, 20173}, {2784, 4919}, {2802, 10770}, {2893, 3688}, {3120, 5311}, {3436, 20535}, {3696, 4514}, {3952, 3974}, {4294, 13723}, {4447, 6999}, {5282, 5698}, {17778, 20069}

X(20540) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(105)

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

(1,141), (2,10), (6,2887), (31,626), (105,20540), (238,20541), (292,20542), (365,20543), (672,20544), (1423,20545), (1931,20546), (2053,20547), (2054,20548), (3009,20549), (2112,20550), (20332, 20551)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : 1 : 1
m(A') = -(Sqrt[b] + Sqrt[c]) (b^2 - b^(3/2) Sqrt[c] + b c - Sqrt[b] c^(3/2) + c^2) : (Sqrt[a] - Sqrt[c]) (a^2 + a^(3/2) Sqrt[c] + a c + Sqrt[a] c^(3/2) + c^2) : (Sqrt[a] - Sqrt[b]) (a^2 + a^(3/2) Sqrt[b] + a b + Sqrt[a] b^(3/2) + b^2)
m(A1) = b c (b + c) : (-a^3 + b c^2) : (-a^3 + b^2 c)
m(A2) = (b^4 + c^4) : c (c^3 - a^2 b) : b (b^3 - a^2 c)
m(A3) = (a b + a c + b c) (b^2 + c^2) : -a^4 - a^3 b - a^3 c + a b c^2 + a c^3 + b c^3 : -a^4 - a^3 b + a b^3 - a^3 c + a b^2 c + b^3 c
m(A4) = -(b + c) (a + b + c) (b^2 - b c + c^2) : a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4 : a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c
m(A5) = a^2 b^2 + b^4 - a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4 : -(-a^4 - a^3 b - a^3 c - a^2 c^2 + a b c^2 + b c^3 - c^4) : a^4 + a^3 b + a^2 b^2 + b^4 + a^3 c - a b^2 c - b^3 c
m(A6) = (a + b + c) (a^2 b^3 - a b^4 - b^4 c + b^3 c^2 + a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : (a^2 + b^2 - a c - b c) (a^4 - a^3 b - a^2 b c + a^2 c^2 + a c^3 + b c^3 + c^4) : (a^4 + a^2 b^2 + a b^3 + b^4 - a^3 c - a^2 b c + b^3 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = by + cz : cz + ax : ax + by, and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(2)), where A' = 0 : 1 : 1.

Let (b-c) (a^2-b c) (-a^3 b^3-a^2 b^4+a^4 b c-a^3 b^2 c-a^2 b^3 c-a b^4 c-a^3 b c^2-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+b^3 c^3-a^2 c^4-a b c^4) x^3+(-3 a^5 b^4-a^4 b^5+3 a^6 b^2 c+a^2 b^6 c+3 a^6 b c^2+a b^6 c^2-3 a^5 c^4-b^5 c^4+a^4 c^5-b^4 c^5-a^2 b c^6+a b^2 c^6) y^2 z+(3 a^5 b^4-a^4 b^5-3 a^6 b^2 c+a^2 b^6 c-3 a^6 b c^2-a b^6 c^2+3 a^5 c^4+b^5 c^4+a^4 c^5+b^4 c^5-a^2 b c^6-a b^2 c^6) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (-b^2+a c) (a^2-b c) (a b-c^2) x y z = 0. (Peter Moses, August 7, 2018)

X(20540) lies on these lines: {2, 1438}, {10, 116}, {141, 2876}, {626, 1329}, {1213, 6666}, {5248, 8299}, {20333, 20541}

X(20541) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(238)

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

See X(20540).

X(20541) lies on these lines: {2, 1914}, {10, 626}, {11, 20530}, {75, 3314}, {115, 6381}, {120, 3823}, {141, 674}, {315, 4426}, {325, 1575}, {625, 3814}, {742, 4071}, {834, 3835}, {908, 1211}, {958, 7784}, {993, 7761}, {1107, 6656}, {1376, 7778}, {1573, 7853}, {1574, 7821}, {1999, 3772}, {2238, 4766}, {3096, 17030}, {3263, 20483}, {3739, 3925}, {3924, 4950}, {3954, 17211}, {4119, 9055}, {4372, 5300}, {4396, 17737}, {4643, 10025}, {4799, 5282}, {4805, 16788}, {5025, 6376}, {5267, 7830}, {7868, 20172}, {16604, 17670}, {17046, 20255}, {17061, 17390}, {17064, 17296}, {20333, 20540}, {20340, 20341}, {20545, 20547}, {20546, 20548}, {20549, 20551}

X(20541) = complement of X(1914)
X(20541) = complementary conjugate of X(17755)

X(20542) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(292)

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

See X(20540).

X(20542) lies on these lines: {2, 1911}, {10, 6656}, {11, 1211}, {141, 9016}, {626, 20550}, {3836, 20340}, {20544, 20549}

X(20542) = complement of X(1911)
X(20542) = complementary conjugate of X(1575)

X(20543) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[b] + Sqrt[c])(b^2 - b^(3/2) Sqrt[c] + b c - Sqrt[b] c^(3/2) + c^2) : :

See X(20540).

X(20543) lies on these lines: {2, 18753}

X(20544) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(672)

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

See X(20540).

X(20544) lies on these lines: {2, 2223}, {5, 10}, {11, 3912}, {36, 16377}, {76, 3705}, {512, 625}, {536, 2486}, {626, 766}, {760, 16609}, {1699, 3501}, {2548, 17750}, {3006, 3948}, {3035, 19512}, {3262, 4516}, {3661, 11680}, {3703, 4044}, {3739, 18252}, {3741, 3934}, {3742, 17758}, {3944, 12782}, {4138, 20256}, {9025, 17197}, {17047, 20305}, {18208, 19950}, {20333, 20540}, {20542, 20549}

X(20544) = complement of X(2223)
X(20544) = complementary conjugate of X(6184)

X(20545) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(1423)

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

See X(20540).

X(20545) lies on these lines: {2, 1284}, {5, 10}, {11, 312}, {37, 3815}, {75, 20276}, {124, 20550}, {518, 20498}, {958, 13740}, {978, 3772}, {995, 17061}, {3035, 3185}, {3714, 3813}, {3840, 20256}, {4193, 19582}, {4999, 17698}, {16569, 17064}, {17070, 17749}, {20541, 20547}

X(20545) = complement of X(1403)

X(20546) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(1931)

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

See X(20540).

X(20546) lies on these lines: {2, 1326}, {5, 141}, {10, 14873}, {115, 726}, {121, 5099}, {187, 17698}, {316, 13740}, {1213, 20666}, {1698, 2959}, {2679, 20551}, {3836, 8287}, {5145, 7752}, {20541, 20548}

X(20546) = complement of X(1326)

X(20547) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(2053)

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

See X(20540).

X(20547) lies on these lines: {2, 20559}, {626, 20549}, {16603, 18896}, {17046, 20338}, {17052, 20528}, {20541, 20545}

X(20548) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(2054)

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

See X(20540).

X(20548) lies on these lines: {2, 20560}, {10, 20529}, {2887, 20550}, {3741, 20339}, {20541, 20546}

X(20549) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(3009)

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

See X(20540).

X(20549) lies on these lines: {2, 20561}, {10, 141}, {536, 20501}, {626, 20547}, {1086, 20491}, {3662, 10009}, {5224, 17030}, {20541, 20551}, {20542, 20544}

X(20549) = complement of X(21760)

X(20550) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(2112)

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

See X(20540).

X(20550) lies on these lines: {2, 20562}, {124, 20545}, {141, 2876}, {626, 20542}, {2887, 20548}

X(20551) = (X(1), X(2), X(6), X(31); X(141), X(10), X(2887), X(626)) COLLINEATION IMAGE OF X(20332)

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

See X(20540).

X(20551) lies on these lines: {2, 727}, {10, 5518}, {11, 2887}, {115, 3454}, {116, 626}, {124, 20545}, {2679, 20546}, {20541, 20549}

X(20552) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(105)

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

(1,69), (2,8), (6,6327), (31,315), (105,20552), (238,20553), (292,20554), (365,20555), (672,20556), (1423,20557), (1931,20558), (2053,20559), (2054,20560), (3009,20561), (2112,20562)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1 : 1 : 1
m(A') = a^(5/2) + b^(5/2) + c^(5/2) : - a^(5/2) - b^(5/2) + c^(5/2)) : - a^(5/2) + b^(5/2) - c^(5/2)
m(A1) = a^3 + b^2 c + b c^2 : - a^3 - b^2 c + b c^2 : - a^3 + b^2 c - b c^2
m(A2) = b^4 + a^2 b c + c^4 : - b^4 - a^2 b c + c^4 : b^4 - a^2 b c - c^4
m(A3) = a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 : - (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c - a b c^2 - a c^3 - b c^3) : - (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c + a b c^2 + a c^3 + b c^3)
m(A4) = a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c + a c^3 + b c^3 + c^4 : -a^3 b - a b^3 - b^4 - a^3 c - a^2 b c - b^3 c + a c^3 + b c^3 + c^4 : -a^3 b + a b^3 + b^4 - a^3 c - a^2 b c + b^3 c - a c^3 - b c^3 - c^4
m(A5) = a^4 + 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 c^3 - c^4 : - (a^4 + 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 c^3 + c^4) : - (a^4 + 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 c^3 - c^4)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = - ax + by + cz : ax - by + cz : ax + by - cz, and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(2)), where A' = -1 : 1 : 1.

Let a^5 (b-c) (-b^3+a b c-b^2 c-b c^2-c^3) x^3+b^2 (-a^5 b^2+a^4 b^3+a^6 c-a^2 b^4 c+2 a b^4 c^2-2 b^3 c^4+b^2 c^5-a c^6) y^2 z-c^2 (a^6 b-a b^6-a^5 c^2+b^5 c^2+a^4 c^3-2 b^4 c^3-a^2 b c^4+2 a b^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (-b^2+a c) (a^2-b c) (a b-c^2) x y z = 0. (Peter Moses, August 7, 2018)

X(20552) lies on these lines: {2, 1438}, {8, 150}, {9, 1654}, {69, 2876}, {315, 668}, {2481, 3434}, {7185, 9312}, {20345, 20553}

X(20553) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(238)

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

See X(20552).

X(20553) lies on these lines: {2, 1914}, {8, 315}, {69, 674}, {75, 1369}, {86, 4972}, {100, 325}, {149, 350}, {183, 11680}, {304, 5300}, {316, 668}, {319, 321}, {385, 17737}, {754, 5291}, {834, 20293}, {883, 16091}, {1479, 18135}, {1909, 2475}, {2975, 7750}, {3263, 4872}, {3583, 6381}, {3684, 4766}, {3785, 10527}, {3879, 3914}, {3891, 17377}, {3959, 4950}, {4589, 4645}, {4911, 5100}, {5046, 6376}, {5195, 16086}, {5687, 7776}, {6653, 7779}, {7261, 20022}, {7761, 16975}, {7768, 17143}, {7773, 11681}, {7774, 17756}, {20345, 20552}, {20352, 20353}, {20557, 20559}, {20558, 20560}

X(20553) = isotomic conjugate of isogonal conjugate of X(20872)
X(20553) = isotomic conjugate of anticomplement of X(39029)
X(20553) = anticomplement of X(1914)
X(20553) = anticomplementary conjugate of X(33888)

X(20554) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(292)

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

See X(20552).

X(20554) lies on these lines: {2, 1911}, {8, 6653}, {69, 9016}, {149, 2895}, {315, 20562}, {4645, 20352}, {20556, 20561}

X(20554) = anticomplement of X(1911)
X(20554) = anticomplementary conjugate of X(17759)

X(20555) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(365)

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

See X(20552).

X(20555) lies on these lines: {2, 18753}

X(20556) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(672)

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

See X(20552).

X(20556) lies on these lines: {2, 2223}, {4, 8}, {11, 4447}, {36, 16376}, {75, 12530}, {76, 17135}, {100, 6996}, {149, 6542}, {239, 13576}, {315, 766}, {316, 512}, {497, 17316}, {528, 4433}, {674, 17139}, {908, 2340}, {1909, 4514}, {2295, 7745}, {2481, 20347}, {2975, 13727}, {3006, 14956}, {3208, 9580}, {3673, 3873}, {3780, 5254}, {4039, 17766}, {4911, 20292}, {4972, 6656}, {5284, 17681}, {7377, 11680}, {7406, 17784}, {7768, 20290}, {8299, 20486}, {13740, 19874}, {14267, 20358}, {16381, 17798}, {20345, 20552}, {20554, 20561}

X(20556) = anticomplement of X(2223)
X(20556) = isotomic conjugate of isogonal conjugate of X(20875)
X(20556) = anticomplementary conjugate of X(39350)

X(20557) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(1423)

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

See X(20552).

X(20557) lies on these lines: {2, 1284}, {4, 8}, {43, 908}, {63, 3741}, {69, 4485}, {192, 497}, {1193, 19785}, {1469, 5905}, {1836, 17792}, {2292, 2478}, {2975, 4195}, {3891, 20037}, {4441, 20245}, {4972, 11681}, {6818, 17777}, {6872, 8240}, {7155, 10453}, {20553, 20559}

X(20557) = anticomplement of X(1403)

X(20558) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(1931)

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

See X(20552).

X(20558) lies on these lines: {2, 1326}, {4, 69}, {10, 2959}, {148, 726}, {966, 20666}, {1029, 17165}, {1213, 20675}, {1654, 3923}, {3944, 17778}, {4195, 14712}, {5145, 7785}, {11599, 20536}, {20553, 20560}

X(20558) = anticomplement of X(1326)
X(20558) = anticomplementary conjugate of X(13174)

X(20559) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(2053)

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

See X(20552).

X(20559) lies on these lines: {2, 20547}, {315, 20561}, {2893, 20537}, {20553, 20557}

X(20560) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(2054)

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

See X(20552).

X(20560) lies on these lines: {2, 20548}, {8, 7261}, {4576, 17135}, {6327, 20562}, {20553, 20558}

X(20561) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(3009)

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

See X(20552).

X(20561) lies on these lines: {2, 20549}, {7, 8}, {315, 20559}, {788, 17217}, {3783, 18792}, {20554, 20556}

X(20561) = anticomplement of X(21760)

X(20562) = (X(1), X(2), X(6), X(31); X(69), X(8), X(6327), X(315)) COLLINEATION IMAGE OF X(2112)

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

See X(20552).

X(20562) lies on these lines: {2, 20550}, {69, 2876}, {315, 20554}, {6327, 20560}

X(20563) = ISOTOMIC CONJUGATE OF X(24)

Barycentrics b^2*c^2*(-a^2 + b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics Cot[A] Sec[2 A] : :

X(20563) lies on these lines: {2, 311}, {68, 69}, {76, 95}, {253, 3260}, {264, 847}, {305, 1238}, {317, 5962}, {325, 18018}, {339, 3964}, {925, 2373}, {1494, 14615}, {1799, 2351}, {5866, 18354}

X(20563) = isotomic conjugate of X(24)
X(20563) = polar conjugate of X(8745)
X(20563) = X(i)-cross conjugate of X(j) for these (i,j): {68, 5392}, {338, 3267}, {394, 76}, {11585, 2}
X(20563) = X(i)-isoconjugate of X(j) for these (i,j): {19, 571}, {24, 31}, {25, 47}, {32, 1748}, {48, 8745}, {163, 6753}, {317, 560}, {393, 563}, {1096, 1147}, {1973, 1993}, {2148, 14576}, {2180, 8882}, {2333, 18605}, {9247, 11547}
X(20563) = cevapoint of X(i) and X(j) for these (i,j): {339, 3265}, {394, 16391}
X(20563) = barycentric product X(i)*X(j) for these {i,j}: {68, 76}, {69, 5392}, {91, 304}, {305, 2165}, {561, 1820}, {847, 3926}, {925, 3267}, {1502, 2351}, {16391, 18027}
X(20563) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 24}, {3, 571}, {4, 8745}, {5, 14576}, {63, 47}, {68, 6}, {69, 1993}, {75, 1748}, {76, 317}, {91, 19}, {96, 8882}, {255, 563}, {264, 11547}, {305, 7763}, {311, 467}, {328, 18883}, {338, 136}, {343, 52}, {394, 1147}, {485, 5412}, {486, 5413}, {523, 6753}, {525, 924}, {847, 393}, {925, 112}, {1444, 18605}, {1820, 31}, {2165, 25}, {2351, 32}, {3267, 6563}, {3926, 9723}, {5392, 4}, {6563, 15423}, {9033, 14397}, {11090, 372}, {11091, 371}, {11140, 14111}, {13430, 1599}, {13441, 1600}, {14593, 2207}, {16391, 577}

X(20564) = ISOTOMIC CONJUGATE OF X(26)

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

X(20564) lies on these lines: {69, 70}, {95, 7516}, {264, 5576}, {1288, 2373}, {7512, 18354}

X(20564) = isotomic conjugate of X(26)
X(20564) = polar conjugate of X(8746)
X(20564) = X(i)-cross conjugate of X(j) for these (i,j): {1993, 76}, {13371, 2}
X(20564) = X(i)-isoconjugate of X(j) for these (i,j): {26, 31}, {48, 8746}
X(20564) = cevapoint of X(i) and X(j) for these (i,j): {2, 14790}, {339, 6563}
X(20564) = barycentric product X(i)*X(j) for these {i,j}: {70, 76}, {561, 2158}, {1288, 3267}
X(20564) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 26}, {4, 8746}, {70, 6}, {1288, 112}, {2158, 31}

X(20565) = ISOTOMIC CONJUGATE OF X(35)

Barycentrics b^2*c^2*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2) : :
Barycentrics 1/(Sin[A]+Sin[2 A]) : :

X(20565) lies on these lines: {75, 3260}, {79, 314}, {312, 1230}, {319, 349}, {328, 1441}, {3596, 6757}, {7110, 15455}, {7321, 18816}

X(20565) = isotomic conjugate of X(35)
X(20565) = X(4359)-cross conjugate of X(76)
X(20565) = X(i)-isoconjugate of X(j) for these (i,j): {3, 14975}, {6, 2174}, {31, 35}, {32, 3219}, {41, 2003}, {42, 17104}, {50, 2161}, {55, 1399}, {184, 6198}, {319, 560}, {692, 2605}, {1397, 4420}, {1415, 9404}, {1442, 2175}, {2194, 2594}, {2206, 3678}, {2477, 7073}, {6149, 6187}, {9447, 17095}, {18359, 19627}
X(20565) = trilinear pole of line {4391, 4707}
X(20565) = barycentric product X(i)*X(j) for these {i,j}: {76, 79}, {94, 320}, {274, 6757}, {310, 8818}, {328, 17923}, {349, 3615}, {561, 2160}, {693, 15455}, {1502, 6186}, {1969, 7100}, {3261, 6742}, {6063, 7110}
X(20565) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2174}, {2, 35}, {7, 2003}, {19, 14975}, {36, 50}, {57, 1399}, {75, 3219}, {76, 319}, {79, 6}, {81, 17104}, {85, 1442}, {92, 6198}, {94, 80}, {226, 2594}, {312, 4420}, {313, 3969}, {320, 323}, {321, 3678}, {331, 7282}, {514, 2605}, {522, 9404}, {554, 2307}, {693, 14838}, {850, 7265}, {1111, 7202}, {1125, 17454}, {1269, 3578}, {1441, 16577}, {1789, 2193}, {1989, 6187}, {2003, 2477}, {2160, 31}, {2166, 2161}, {3218, 6149}, {3261, 4467}, {3615, 284}, {3662, 7186}, {4359, 3647}, {4707, 526}, {5249, 500}, {6063, 17095}, {6186, 32}, {6742, 101}, {6757, 37}, {7073, 41}, {7100, 48}, {7110, 55}, {8818, 42}, {13486, 163}, {14844, 18755}, {15455, 100}, {16709, 17190}, {16732, 2611}, {17095, 7279}, {17923, 186}

X(20566) = ISOTOMIC CONJUGATE OF X(36)

Barycentrics b^2*c^2*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :
Barycentrics 1/(Sin[A]-Sin[2 A]) : :

X(20566) lies on these lines: {75, 311}, {76, 1227}, {80, 313}, {312, 3969}, {320, 18816}, {328, 1441}, {759, 839}, {1226, 7321}, {1807, 18147}, {3260, 17791}, {3596, 15065}

X(20566) = isotomic conjugate of X(36)
X(20566) = X(3596)-beth conjugate of X(668)
X(20566) = X(i)-cross conjugate of X(j) for these (i,j): {3262, 75}, {3814, 2}, {4358, 76}, {15065, 18359}
X(20566) = X(i)-isoconjugate of X(j) for these (i,j): {6, 7113}, {31, 36}, {32, 3218}, {50, 2160}, {56, 2361}, {58, 3724}, {109, 8648}, {184, 1870}, {215, 1411}, {320, 560}, {604, 2323}, {649, 1983}, {654, 1415}, {758, 2206}, {902, 16944}, {1333, 2245}, {1397, 4511}, {1400, 4282}, {1443, 2175}, {1464, 2194}, {1919, 4585}, {6149, 6186}, {9247, 17923}, {9447, 17078}, {9456, 17455}, {13486, 14270}
X(20566) = cevapoint of X(i) and X(j) for these (i,j): {2, 5080}, {10, 908}, {75, 17791}, {313, 3264}, {3219, 4511}
X(20566) = trilinear pole of line {321, 4391}
X(20566) = barycentric product X(i)*X(j) for these {i,j}: {75, 18359}, {76, 80}, {94, 319}, {274, 15065}, {312, 18815}, {321, 14616}, {349, 6740}, {561, 2161}, {1502, 6187}, {1807, 1969}, {2006, 3596}
X(20566) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7113}, {2, 36}, {8, 2323}, {9, 2361}, {10, 2245}, {21, 4282}, {35, 50}, {37, 3724}, {75, 3218}, {76, 320}, {80, 6}, {85, 1443}, {88, 16944}, {92, 1870}, {94, 79}, {100, 1983}, {226, 1464}, {264, 17923}, {312, 4511}, {313, 3936}, {319, 323}, {321, 758}, {519, 17455}, {522, 654}, {650, 8648}, {655, 109}, {668, 4585}, {693, 3960}, {759, 1333}, {850, 4707}, {1089, 4053}, {1168, 9456}, {1411, 604}, {1441, 18593}, {1793, 2193}, {1807, 48}, {1989, 6186}, {2006, 56}, {2161, 31}, {2166, 2160}, {2222, 1415}, {2323, 215}, {2341, 2194}, {3219, 6149}, {3261, 4453}, {3262, 16586}, {3661, 3792}, {4036, 2610}, {4358, 214}, {4359, 4973}, {4391, 3738}, {4671, 4867}, {6063, 17078}, {6187, 32}, {6335, 4242}, {6740, 284}, {7017, 5081}, {7026, 7127}, {7265, 526}, {14584, 1404}, {14616, 81}, {14628, 1319}, {15065, 37}, {17484, 6126}, {18359, 1}, {18743, 4881}, {18815, 57}

X(20567) = ISOTOMIC CONJUGATE OF X(41)

Barycentrics b^3* c^3 (a - b + c)*(a + b - c)
Barycentrics Csc[A]^2/(1+Cos[A]) : :

X(20567) lies on these lines: {7, 871}, {75, 4572}, {76, 1229}, {77, 4625}, {85, 6385}, {273, 310}, {274, 16743}, {305, 561}, {349, 1502}, {1088, 1240}

X(20567) = isogonal conjugate of X(9447)
X(20567) = isotomic conjugate of X(41)
X(20567) = polar conjugate of X(2212)
X(20567) = X(670)-beth conjugate of X(75)
X(20567) = complement of polar conjugate of isogonal conjugate of X(23175)
X(20567) = anticomplement of polar conjugate of isogonal conjugate of X(23211)
X(20567) = X(i)-cross conjugate of X(j) for these (i,j): {76, 561}, {349, 6063}, {1233, 76}, {3261, 4572}, {16888, 7}, {17046, 2}, {20236, 75}
X(20567) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9447}, {2, 9448}, {6, 2175}, {8, 1501}, {9, 560}, {21, 2205}, {31, 41}, {32, 55}, {33, 9247}, {48, 2212}, {56, 14827}, {60, 7109}, {184, 607}, {212, 1973}, {213, 2194}, {219, 1974}, {220, 1397}, {228, 2204}, {249, 7063}, {281, 14575}, {284, 1918}, {294, 9455}, {312, 1917}, {577, 6059}, {604, 1253}, {643, 1924}, {644, 1980}, {645, 9426}, {669, 5546}, {692, 3063}, {872, 2150}, {919, 8638}, {1015, 6066}, {1106, 6602}, {1334, 2206}, {1395, 1802}, {1415, 8641}, {1576, 3709}, {1857, 14585}, {1914, 18265}, {1919, 3939}, {1977, 6065}, {2187, 7118}, {2195, 9454}, {2200, 2299}, {2207, 6056}, {2316, 9459}, {2330, 7104}, {2344, 18900}, {2353, 4548}, {3449, 9449}, {3596, 9233}, {3684, 14598}, {3685, 18897}, {3700, 14574}, {3712, 19626}, {3975, 18893}, {4518, 18894}, {4876, 18892}, {5547, 14567}, {6064, 9427}, {7077, 14599}
X(20567) = cevapoint of X(i) and X(j) for these (i,j): {7, 17075}, {57, 7210}, {76, 6063}, {85, 7182}, {1111, 3776}
X(20567) = barycentric product X(i)*X(j) for these {i,j}: {7, 561}, {56, 1928}, {57, 1502}, {75, 6063}, {76, 85}, {77, 18022}, {226, 6385}, {264, 7182}, {273, 305}, {274, 349}, {304, 331}, {310, 1441}, {334, 18033}, {348, 1969}, {670, 4077}, {693, 4572}, {850, 4625}, {871, 7179}, {1088, 3596}, {3261, 4554}, {3665, 18833}, {3676, 6386}, {4017, 4609}, {4602, 7178}, {6382, 7209}, {7018, 7205}, {7034, 7185}, {7183, 18027}, {7233, 18891}, {10030, 18895}
X(20567) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2175}, {2, 41}, {4, 2212}, {6, 9447}, {7, 31}, {8, 1253}, {9, 14827}, {12, 872}, {27, 2204}, {31, 9448}, {34, 1974}, {56, 560}, {57, 32}, {65, 1918}, {69, 212}, {75, 55}, {76, 9}, {77, 184}, {85, 6}, {86, 2194}, {92, 607}, {142, 20229}, {158, 6059}, {189, 7118}, {222, 9247}, {226, 213}, {241, 9454}, {264, 33}, {269, 1397}, {273, 25}, {274, 284}, {278, 1973}, {279, 604}, {286, 2299}, {291, 18265}, {304, 219}, {305, 78}, {307, 228}, {309, 2192}, {310, 21}, {311, 7069}, {312, 220}, {313, 210}, {314, 2328}, {318, 7071}, {320, 2361}, {321, 1334}, {322, 7074}, {326, 6056}, {331, 19}, {334, 7077}, {341, 480}, {342, 3195}, {345, 1802}, {346, 6602}, {347, 2187}, {348, 48}, {349, 37}, {479, 1106}, {514, 3063}, {522, 8641}, {552, 849}, {561, 8}, {603, 14575}, {604, 1501}, {658, 1415}, {664, 692}, {668, 3939}, {670, 643}, {693, 663}, {765, 6066}, {799, 5546}, {850, 4041}, {873, 60}, {1014, 2206}, {1088, 56}, {1089, 7064}, {1111, 3271}, {1119, 1395}, {1121, 18889}, {1214, 2200}, {1226, 1864}, {1229, 8012}, {1231, 71}, {1233, 1212}, {1269, 3683}, {1275, 2149}, {1319, 9459}, {1358, 3248}, {1397, 1917}, {1400, 2205}, {1401, 1923}, {1414, 1576}, {1428, 18892}, {1429, 14599}, {1432, 7104}, {1434, 1333}, {1440, 2208}, {1441, 42}, {1446, 1400}, {1447, 2210}, {1458, 9455}, {1469, 18900}, {1502, 312}, {1509, 2150}, {1577, 3709}, {1760, 4548}, {1847, 608}, {1909, 2330}, {1920, 2329}, {1921, 3684}, {1928, 3596}, {1930, 3688}, {1969, 281}, {1978, 644}, {2171, 7109}, {2254, 8638}, {2481, 2195}, {2643, 7063}, {2887, 4531}, {3212, 2209}, {3261, 650}, {3263, 2340}, {3264, 3689}, {3267, 8611}, {3596, 200}, {3665, 1964}, {3668, 1402}, {3669, 1919}, {3673, 7083}, {3674, 2300}, {3718, 1260}, {3729, 16283}, {3911, 2251}, {3926, 2289}, {4017, 669}, {4025, 1946}, {4077, 512}, {4086, 4524}, {4391, 657}, {4397, 4105}, {4453, 8648}, {4554, 101}, {4569, 109}, {4572, 100}, {4573, 163}, {4602, 645}, {4609, 7257}, {4623, 4636}, {4625, 110}, {4635, 4565}, {4815, 8653}, {4858, 14936}, {4998, 1110}, {5018, 18262}, {6063, 1}, {6357, 9406}, {6358, 1500}, {6374, 7075}, {6382, 3208}, {6383, 2319}, {6384, 2053}, {6385, 333}, {6386, 3699}, {7017, 7079}, {7035, 6065}, {7055, 255}, {7056, 603}, {7125, 14585}, {7176, 7122}, {7178, 798}, {7179, 869}, {7180, 1924}, {7181, 922}, {7182, 3}, {7183, 577}, {7185, 7032}, {7196, 172}, {7199, 7252}, {7205, 171}, {7209, 2162}, {7210, 206}, {7217, 2085}, {7233, 1911}, {7243, 2241}, {7249, 904}, {7282, 14975}, {7340, 1101}, {8817, 7084}, {9436, 2223}, {10030, 1914}, {13436, 606}, {13453, 605}, {14256, 2199}, {14615, 7070}, {15413, 652}, {15466, 7156}, {15467, 2218}, {16603, 3774}, {16739, 4267}, {16888, 16584}, {17076, 2172}, {17078, 7113}, {17094, 810}, {17095, 2174}, {17206, 2193}, {17451, 9449}, {17880, 3270}, {18021, 1098}, {18022, 318}, {18026, 8750}, {18031, 294}, {18033, 238}, {18036, 7281}, {18135, 3217}, {18160, 9404}, {18815, 6187}, {18816, 2342}, {18891, 3685}, {18895, 4876}, {19804, 4258}, {20236, 16588}, {20336, 2318}
X(20567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7205, 18033, 7)

X(20568) = ISOTOMIC CONJUGATE OF X(44)

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

X(20568) lies on these lines: {2, 4403}, {75, 537}, {76, 1978}, {80, 320}, {85, 4554}, {88, 274}, {106, 789}, {214, 4597}, {286, 811}, {291, 19957}, {334, 4013}, {767, 901}, {1320, 2481}, {1966, 17960}, {3766, 6548}, {4358, 4945}, {4593, 9456}, {4602, 6385}, {4792, 17143}, {14210, 18032}

X(20568) = isogonal conjugate of X(2251)
X(20568) = isotomic conjugate of X(44)
X(20568) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 2251}, {1740, 44}
X(20568) = X(i)-cross conjugate of X(j) for these (i,j): {3262, 6063}, {3762, 668}, {3834, 2}, {4080, 903}, {4358, 75}, {18359, 18816}
X(20568) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2251}, {2, 9459}, {6, 902}, {31, 44}, {32, 519}, {41, 1319}, {42, 3285}, {55, 1404}, {101, 1960}, {106, 1017}, {110, 14407}, {163, 4730}, {184, 8756}, {560, 4358}, {604, 3689}, {667, 1023}, {678, 9456}, {692, 1635}, {1110, 2087}, {1397, 2325}, {1415, 4895}, {1492, 14436}, {1501, 3264}, {1576, 4120}, {1918, 16704}, {1919, 17780}, {1922, 4432}, {1973, 5440}, {1974, 3977}, {2175, 3911}, {2206, 3943}, {2429, 8643}, {4434, 7104}, {6187, 17455}
X(20568) = cevapoint of X(i) and X(j) for these (i,j): {2, 320}, {75, 4358}, {903, 4997}, {1086, 4927}, {1111, 3762}, {4049, 6549}
X(20568) = trilinear pole of line {75, 693}
X(20568) = complement of polar conjugate of isogonal conjugate of X(23178)
X(20568) = anticomplement of polar conjugate of isogonal conjugate of X(23214)
X(20568) = X(19)-isoconjugate of X(23202)
X(20568) = barycentric product X(i)*X(j) for these {i,j}: {75, 903}, {76, 88}, {85, 4997}, {106, 561}, {274, 4080}, {304, 6336}, {310, 4674}, {523, 4634}, {668, 6548}, {679, 3264}, {693, 4555}, {799, 4049}, {850, 4622}, {873, 4013}, {1022, 1978}, {1320, 6063}, {1502, 9456}, {1577, 4615}, {1797, 1969}, {3257, 3261}, {6549, 7035}
X(20568) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 902}, {2, 44}, {6, 2251}, {7, 1319}, {8, 3689}, {31, 9459}, {44, 1017}, {57, 1404}, {69, 5440}, {75, 519}, {76, 4358}, {81, 3285}, {85, 3911}, {88, 6}, {92, 8756}, {106, 31}, {190, 1023}, {273, 1877}, {274, 16704}, {304, 3977}, {312, 2325}, {313, 3992}, {320, 214}, {321, 3943}, {350, 4432}, {513, 1960}, {514, 1635}, {519, 678}, {522, 4895}, {523, 4730}, {561, 3264}, {661, 14407}, {668, 17780}, {679, 106}, {693, 900}, {764, 8661}, {900, 3251}, {901, 692}, {903, 1}, {1022, 649}, {1086, 2087}, {1111, 1647}, {1168, 6187}, {1266, 17460}, {1269, 4975}, {1320, 55}, {1417, 1397}, {1577, 4120}, {1797, 48}, {1909, 4434}, {2226, 9456}, {2316, 41}, {2403, 4394}, {3218, 17455}, {3239, 14427}, {3250, 14436}, {3257, 101}, {3261, 3762}, {3262, 1145}, {3264, 4738}, {3596, 4723}, {3762, 6544}, {3766, 4448}, {3835, 14408}, {3912, 14439}, {4013, 756}, {4033, 4169}, {4049, 661}, {4080, 37}, {4358, 4370}, {4359, 4969}, {4374, 4922}, {4391, 1639}, {4397, 4528}, {4441, 4702}, {4462, 14425}, {4510, 750}, {4555, 100}, {4582, 644}, {4591, 163}, {4615, 662}, {4618, 901}, {4622, 110}, {4634, 99}, {4671, 4908}, {4674, 42}, {4723, 4152}, {4728, 14437}, {4738, 8028}, {4768, 4543}, {4792, 2177}, {4801, 4773}, {4823, 4958}, {4858, 4530}, {4945, 45}, {4978, 4984}, {4997, 9}, {5376, 1252}, {6332, 14418}, {6336, 19}, {6548, 513}, {6549, 244}, {8752, 1973}, {9268, 1110}, {9456, 32}, {9460, 9324}, {14208, 14429}, {17089, 14122}, {17960, 5168}, {18815, 14584}, {18821, 14191}, {19804, 4700}
X(20568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1111, 18159, 668)

X(20569) = ISOTOMIC CONJUGATE OF X(45)

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

X(20569) lies on these lines: {2, 4403}, {75, 519}, {76, 4358}, {85, 3911}, {89, 274}, {767, 4588}, {870, 2163}, {903, 17461}, {1016, 4363}, {2320, 2481}, {6063, 14628}

X(20569) = isotomic conjugate of X(45)
X(20569) = isotomic of the isogonal of X(89)
X(20569) = cevapoint of X(693) and X(4957)
X(20569) = X(i)-cross conjugate of X(j) for these (i,j): {4957, 693}, {5718, 7}
X(20569) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2177}, {31, 45}, {32, 3679}, {41, 2099}, {42, 4273}, {55, 1405}, {101, 4775}, {163, 4770}, {213, 4653}, {560, 4671}, {604, 3711}, {667, 4752}, {692, 4893}, {1397, 4873}, {1415, 4814}, {1576, 4931}, {1918, 5235}, {1919, 4767}, {1922, 4693}, {1973, 3940}, {2175, 5219}, {2251, 4792}, {4945, 9459}
X(20569) = trilinear pole of line {693, 900}
X(20569) = complement of polar conjugate of isogonal conjugate of X(23179)
X(20569) = anticomplement of polar conjugate of isogonal conjugate of X(23215)
X(20569) = barycentric product X(i)*X(j) for these {i,j}: {76, 89}, {561, 2163}, {693, 4597}, {2320, 6063}, {3261, 4604}
X(20569) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2177}, {2, 45}, {7, 2099}, {8, 3711}, {57, 1405}, {69, 3940}, {75, 3679}, {76, 4671}, {81, 4273}, {85, 5219}, {86, 4653}, {89, 6}, {190, 4752}, {274, 5235}, {312, 4873}, {313, 4125}, {314, 4720}, {320, 4867}, {350, 4693}, {513, 4775}, {514, 4893}, {522, 4814}, {523, 4770}, {668, 4767}, {693, 4777}, {903, 4792}, {982, 4787}, {1269, 4717}, {1577, 4931}, {2163, 31}, {2320, 55}, {2364, 41}, {3261, 4791}, {3766, 4800}, {4358, 4908}, {4374, 4774}, {4389, 17461}, {4391, 4944}, {4406, 4844}, {4588, 692}, {4597, 100}, {4604, 101}, {4777, 4825}, {5385, 1252}, {6381, 4937}, {7192, 4833}, {14210, 4933}

X(20570) = ISOTOMIC CONJUGATE OF X(46)

Barycentrics b*c*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + b^2*c - a*c^2 - b*c^2 - c^3)*(-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(20570) lies on these lines: {75, 7318}, {90, 314}, {309, 320}, {312, 319}, {7040, 18147}

X(20570) = isotomic conjugate of X(46)
X(20570) = X(69)-cross conjugate of X(75)
X(20570) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2178}, {25, 3157}, {31, 46}, {32, 5905}, {55, 1406}, {184, 1068}, {1397, 5552}, {1402, 3193}, {1973, 6505}, {2207, 6511}
X(20570) = cevapoint of X(i) and X(j) for these (i,j): {2, 11415}, {90, 6513}, {514, 17888}, {693, 17877}, {14208, 17886}
X(20570) = trilinear pole of line {4391, 4467}
X(20570) = trilinear product of PU(129)
X(20570) = barycentric product X(i)*X(j) for these {i,j}: {75, 2994}, {76, 90}, {264, 6513}, {304, 7040}, {312, 7318}, {561, 2164}, {1069, 1969}
X(20570) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2178}, {2, 46}, {57, 1406}, {63, 3157}, {69, 6505}, {75, 5905}, {90, 6}, {92, 1068}, {312, 5552}, {326, 6511}, {333, 3193}, {1069, 48}, {1812, 1800}, {2164, 31}, {2994, 1}, {5905, 1079}, {6512, 255}, {6513, 3}, {7040, 19}, {7042, 2164}, {7072, 41}, {7318, 57}, {7363, 1254}

X(20571) = ISOTOMIC CONJUGATE OF X(47)

Barycentrics b^3*c^3*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics Csc[A] Sec[2 A] : :

X(20571) lies on these lines: {75, 91}, {92, 18041}, {321, 5392}, {1760, 1820}, {2168, 18042}

X(20571) = isotomic conjugate of X(47)
X(20571) = X(i)-cross conjugate of X(j) for these (i,j): {63, 1969}, {18695, 75}
X(20571) = X(i)-isoconjugate of X(j) for these (i,j): {6, 571}, {19, 563}, {24, 184}, {25, 1147}, {31, 47}, {32, 1993}, {213, 18605}, {317, 14575}, {577, 8745}, {924, 1576}, {1501, 7763}, {1748, 9247}, {1974, 9723}, {2148, 2180}, {5412, 8911}, {6563, 14574}, {11547, 14585}, {14533, 14576}, {18883, 19627} X(20571) = cevapoint of X(4) and X(18682)
X(20571) = barycentric product X(i)*X(j) for these {i,j}: {68, 1969}, {75, 5392}, {76, 91}, {304, 847}, {561, 2165}, {1820, 18022}
X(20571) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 571}, {2, 47}, {3, 563}, {5, 2180}, {63, 1147}, {68, 48}, {75, 1993}, {86, 18605}, {91, 6}, {92, 24}, {96, 2148}, {158, 8745}, {264, 1748}, {304, 9723}, {561, 7763}, {847, 19}, {925, 163}, {1577, 924}, {1820, 184}, {1969, 317}, {2165, 31}, {2351, 9247}, {5392, 1}, {14213, 52}, {14593, 1973}, {16391, 4100}

X(20572) = ISOTOMIC CONJUGATE OF X(49)

Barycentrics b^4*c^4*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics Csc[A] Sec[3 A] : :

X(20572) lies on these lines: {93, 264}, {311, 18817}, {317, 562}, {340, 3519}, {2052, 11140}, {2963, 16081}

X(20572) = isotomic conjugate of X(49)
X(20572) = isotomic of the isogonal of X(93)
X(20572) = polar conjugate of X(2965)
X(20572) = X(i)-isoconjugate of X(j) for these (i,j): {31, 49}, {48, 2965}, {184, 2964}, {1994, 9247}
X(20572) = cevapoint of X(93) and X(11140)
X(20572) = barycentric product X(i)*X(j) for these {i,j}: {76, 93}, {264, 11140}, {1969, 2962}, {2963, 18022}, {3519, 18027}
X(20572) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 49}, {4, 2965}, {92, 2964}, {93, 6}, {252, 14533}, {264, 1994}, {324, 143}, {562, 50}, {2052, 3518}, {2962, 48}, {2963, 184}, {3519, 577}, {11140, 3}, {13450, 14577}, {14111, 571}, {14618, 1510}, {18022, 7769}, {19552, 9380}

X(20573) = ISOTOMIC CONJUGATE OF X(50)

Barycentrics b^4*c^4*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2) : :
Barycentrics Csc[A] Csc[3 A] : :

X(20573) lies on these lines: {76, 94}, {264, 328}, {265, 290}, {276, 6331}, {300, 623}, {301, 624}, {308, 1989}, {476, 2367}, {3114, 11060}, {11057, 18316}, {14254, 14387}

X(20573) = isogonal conjugate of X(19627)
X(20573) = isotomic conjugate of X(50)
X(20573) = X(i)-cross conjugate of X(j) for these (i,j): {94, 18817}, {3260, 18022}
X(20573) = X(i)-isoconjugate of X(j) for these (i,j): {1, 19627}, {31, 50}, {32, 6149}, {163, 14270}, {186, 9247}, {323, 560}, {810, 14591}, {1576, 2624}, {1917, 7799}, {1924, 10411}, {9406, 14385}, {9417, 14355}
X(20573) = cevapoint of X(94) and X(328)
X(20573) = trilinear pole of line {311, 850}
X(20573) = barycentric product X(i)*X(j) for these {i,j}: {69, 18817}, {76, 94}, {264, 328}, {265, 18022}, {300, 301}, {305, 6344}, {561, 2166}, {670, 10412}, {1502, 1989}, {4609, 15475}, {6331, 14592}, {14356, 18024} X(20573) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 50}, {6, 19627}, {75, 6149}, {76, 323}, {94, 6}, {264, 186}, {265, 184}, {290, 14355}, {300, 16}, {301, 15}, {311, 1154}, {324, 11062}, {328, 3}, {338, 2088}, {339, 16186}, {340, 3043}, {476, 1576}, {523, 14270}, {648, 14591}, {670, 10411}, {850, 526}, {1494, 14385}, {1502, 7799}, {1577, 2624}, {1989, 32}, {2166, 31}, {3260, 1511}, {3267, 8552}, {6331, 14590}, {6344, 25}, {6757, 3724}, {8836, 11136}, {8838, 11135}, {10412, 512}, {11060, 1501}, {14213, 2290}, {14254, 1495}, {14356, 237}, {14560, 14574}, {14582, 3049}, {14583, 9407}, {14592, 647}, {14616, 17104}, {15455, 1983}, {15475, 669}, {16770, 11134}, {16771, 11137}, {18022, 340}, {18027, 14165}, {18314, 2081}, {18359, 2174}, {18384, 1974}, {18557, 1636}, {18815, 1399}, {18817, 4}, {18883, 571}

X(20574) = ISOGONAL CONJUGATE OF X(14978)

Barycentrics SA*(SB+SC)^2*(S^2+SA*SB)*(3*S^ 2-SA*SB)*(S^2+SA*SC)*(3*S^2- SA*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28005.

X(20574) lies on the these lines: {49, 418}, {51, 54}, {185, 18212}, {217, 2965}, {1141, 1487}, {3432, 19468}

X(20574) = isogonal conjugate of X(14978)
X(20574) = X(92)-isoconjugate of X(233)
X(20574) = barycentric product X(i)*X(j) for these {i,j}: {3, 288}, {97, 1173}
X(20574) = barycentric quotient X(i)/X(j) for these (i,j): (97, 1232), (184, 233), (217, 3078), (288, 264), (1173, 324)
X(20574) = trilinear product X(i)*X(j) for these {i,j}: {48, 288}, {1173, 2169}
X(20574) = trilinear quotient X(i)/X(j) for these (i,j): (48, 233), (288, 92)
= {X(54), X(1173)}-harmonic conjugate of X(288)

X(20575) = MIDPOINT OF X(5) AND X(31)

Barycentrics 2*a^7-5*(b^2+c^2)*a^5+3*(b^2- c^2)^2*a^3-(b^3+c^3)*(b^2+c^2) *a^2+(b^3+c^3)*(b^2-c^2)^2 : :
X(20575) = 5*X(1656)-X(6327), 7*X(3090)+X(20064)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28005.

X(20575) lies on the these lines: {5, 31}, {140, 6679}, {547, 752}, {674, 18583}, {758, 5901}, {1656, 6327}, {2887, 3628}, {3090, 20064}

X(20575) = midpoint of X(5) and X(31)
X(20575) = reflection of X(i) in X(j) for these (i,j): (140, 6679), (2887, 3628)
X(20575) = reflection of X(140) in the line X(834)X(6679)

X(20576) = MIDPOINT OF X(5) AND X(32)

Barycentrics 2*a^8-5*(b^2+c^2)*a^6+3*(b^2- c^2)^2*a^4-(b^2+c^2)*(b^4+c^4) *a^2+(b^4+c^4)*(b^2-c^2)^2 : :
X(20576) = X(3)+3*X(9753), X(315)-5*X(1656), 7*X(3090)+X(20065), X(5017)+3*X(14561), X(7818)-3*X(15699)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28005.

X(20576) lies on the these lines: {3, 7792}, {4, 7806}, {5, 32}, {30, 7817}, {114, 5007}, {140, 143}, {262, 7857}, {315, 1656}, {546, 2794}, {547, 754}, {576, 3788}, {626, 3628}, {631, 7875}, {760, 5901}, {1513, 3398}, {2080, 6656}, {2782, 5305}, {3090, 17004}, {3095, 7807}, {3425, 7506}, {5017, 14561}, {5025, 10788}, {5097, 7764}, {5171, 7834}, {5368, 14981}, {6055, 6249}, {6248, 7755}, {6321, 19687}, {7709, 7920}, {7789, 18806}, {7797, 11676}, {7818, 15699}, {7828, 12110}, {7829, 13334}, {7856, 11257}, {7889, 15819}, {7892, 12251}, {10983, 11288}, {12106, 18121}, {16285, 19139}

X(20576) = midpoint of X(5) and X(32)
X(20576) = reflection of X(i) in X(j) for these (i,j): (140, 6680), (626, 3628)
X(20576) = reflection of X(140) in the line X(512)X(6680)
X(20576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 18583, 11272), (7746, 10358, 5), (7828, 12110, 15980), (11272, 14693, 140)

X(20577) = X(52)X(512)∩X(216)X(18311)

Barycentrics (SB-SC)*(S^2+SB*SC)*(3*S^2-SA^ 2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28006.

X(20577) lies on the these lines: {52, 512}, {216, 18311}, {324, 14592}, {523, 2070}, {525, 15340}, {1994, 2413}, {12077, 18314}

X(20577) = crossdifference of every pair of points on line X(570)X(8603)

X(20578) = ISOGONAL CONJUGATE OF X(17402)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28006.

X(20578) lies on the these lines: {13, 5466}, {51, 512}, {395, 523}, {462, 2501}, {476, 5995}, {892, 9206}, {1637, 6137}, {2395, 3457}, {5471, 12077}, {8737, 18808}

X(20578) = isogonal conjugate of X(17402)

X(20579) = ISOGONAL CONJUGATE OF X(17403)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28006.

X(20579) lies on the these lines: {14, 5466}, {51, 512}, {396, 523}, {463, 2501}, {476, 5994}, {892, 9207}, {1637, 6138}, {2395, 3458}, {5472, 12077}, {8738, 18808}, {11092, 14447}

X(20579) = isogonal conjugate of X(17403)

X(20580) = X(20)X(14343)∩X(394)X(2416)

Barycentrics (-a^2+b^2+c^2)^2*(b^2-c^2)*(3* a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^ 2) : :
Barycentrics (SB-SC)*SA^2*(S^2-2*SB*SC) : :
X(20580) = X(20)+2*X(14343)

See Tran Quang Hung and César Lozada, Hyacinthos 28006.

X(20580) lies on the these lines: {20, 14343}, {394, 2416}, {441, 525}, {523, 2071}, {684, 3566}, {4143, 14638}, {5664, 6503}, {8057, 15427}

X(20581) = X(1741)X(8758)∩X(2331)X(7649)

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

See Angel Montesdeoca, HG040818.

X(20581) lies on these lines: {1741,8758}, {2331,7649}

X(20582) = MIDPOINT OF X(2) AND X(141)

Barycentrics 2*a^2+5*b^2+5*c^2 : :
X(20582) = 5*X(2)-X(6), 7*X(2)+X(69), 17*X(2)-X(193), 3*X(2)+X(599), 9*X(2)-X(1992), 13*X(2)-5*X(3618), X(2)+7*X(3619), 11*X(2)+5*X(3620), 11*X(2)-X(3629), 13*X(2)+X(3630), 4*X(2)+X(3631), X(2)-5*X(3763), 19*X(2)-3*X(5032), 7*X(2)-2*X(6329), 7*X(2)-X(8584), 15*X(2)+X(11160), 11*X(2)+X(15533)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28007.

X(20582) lies on the these lines: {2, 6}, {10, 9041}, {30, 14810}, {37, 17225}, {140, 542}, {182, 11539}, {376, 10516}, {511, 547}, {518, 3828}, {519, 3844}, {538, 10007}, {545, 17359}, {549, 1503}, {551, 5846}, {575, 16239}, {594, 17291}, {620, 9830}, {626, 8367}, {635, 5460}, {636, 5459}, {671, 6656}, {698, 9466}, {742, 4755}, {1086, 17292}, {1350, 3545}, {1352, 5054}, {1386, 19883}, {1656, 20423}, {2482, 6292}, {2781, 10170}, {2854, 15082}, {3096, 8370}, {3098, 3845}, {3525, 15069}, {3526, 8550}, {3530, 18553}, {3564, 10124}, {3661, 4395}, {3662, 7227}, {3679, 9053}, {3818, 8703}, {3819, 9019}, {3834, 4472}, {3917, 16776}, {3934, 5461}, {3943, 17305}, {4265, 16858}, {4364, 16676}, {4370, 17254}, {4399, 16706}, {4422, 17237}, {4478, 17228}, {4665, 17290}, {4688, 9055}, {4912, 17355}, {4971, 17382}, {5026, 9167}, {5050, 15723}, {5055, 5480}, {5066, 19924}, {5067, 11477}, {5070, 14848}, {5071, 10519}, {5085, 11180}, {5092, 11812}, {5206, 7822}, {5237, 5463}, {5238, 5464}, {5349, 11304}, {5350, 11303}, {5476, 15699}, {5646, 8547}, {5650, 8705}, {5651, 19127}, {5888, 12367}, {6034, 7944}, {6173, 10022}, {6697, 7734}, {6776, 15709}, {7228, 17289}, {7238, 17227}, {7263, 17293}, {7516, 15582}, {7745, 7883}, {7801, 8362}, {7810, 7819}, {7817, 8364}, {7820, 15810}, {7831, 8598}, {7835, 11149}, {7844, 16509}, {7874, 8787}, {7915, 8365}, {7998, 9971}, {8262, 13857}, {8288, 20385}, {8361, 12815}, {8596, 17128}, {9939, 16895}, {10109, 19130}, {10302, 11054}, {11179, 15694}, {11645, 12100}, {12040, 15482}, {14561, 15703}, {14927, 15705}, {15246, 19596}, {15561, 19905}, {15701, 18440}, {16187, 19136}, {16673, 17243}, {17045, 17231}, {17132, 17235}, {17133, 17229}, {17230, 17395}, {17236, 17340}, {17246, 17285}, {17332, 17357}, {17334, 17358}, {17362, 17370}, {17365, 17371}, {17383, 17388}, {17384, 17390}, {20382, 20384}

X(20582) = midpoint of X(i) and X(j) for these {i,j}: {2, 141}, {549, 11178}, {597, 599}, {620, 19662}, {3098, 3845}, {3818, 8703}, {3917, 16776}, {8262, 13857}
X(20582) = reflection of X(5092) in X(11812)
X(20582) = complement of X(597)
X(20582) = X(140)-of-anti-Artzt triangle
X(20582) = X(5461)-of-1st Brocard triangle
X(20582) = X(19662)-of-McCay triangle
X(20582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 599, 597), (2, 7778, 9771), (2, 11184, 15491), (2, 15271, 15597), (6, 599, 11160), (141, 597, 599), (141, 3589, 3631), (141, 3629, 3620), (3619, 3763, 141), (9761, 9763, 9740), (11180, 15702, 5085), (13637, 13757, 14614), (17227, 17369, 7238), (17228, 17366, 4478)

X(20583) = REFLECTION OF X(2) IN X(6329)

Barycentrics 14*a^2-b^2-c^2 : :
X(20583) = X(2)-5*X(6), 13*X(2)-5*X(69), 7*X(2)-5*X(141), 11*X(2)+5*X(193), 3*X(2)-5*X(597), 9*X(2)-5*X(599), 3*X(2)+5*X(1992), 4*X(2)-5*X(3589), 19*X(2)-5*X(3630), X(2)+15*X(5032), X(2)+5*X(8584), 7*X(2)+X(11008), 21*X(2)-5*X(11160), 17*X(2)-5*X(15533), 7*X(2)+5*X(15534), 13*X(6)-X(69), 7*X(6)-X(141)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28007.

X(20583) lies on the these lines: {2, 6}, {30, 5097}, {182, 17504}, {376, 5102}, {382, 8550}, {542, 546}, {547, 5965}, {550, 576}, {575, 3530}, {671, 7745}, {1350, 15710}, {1351, 15688}, {1353, 5476}, {1503, 15520}, {2482, 5007}, {3244, 4432}, {3528, 11477}, {3544, 15069}, {3564, 11737}, {3851, 14848}, {4399, 17120}, {4472, 4700}, {4686, 17225}, {5041, 8359}, {5050, 15700}, {5085, 15715}, {5093, 11179}, {5305, 5461}, {5480, 14269}, {6154, 8539}, {7228, 17121}, {7805, 8367}, {7838, 8360}, {7839, 8591}, {7894, 8370}, {8541, 10301}, {10488, 14042}, {10706, 16657}, {12150, 13196}, {15484, 20112}, {16668, 17332}, {16671, 17390}, {20380, 20386}, {20381, 20383}

X(20583) = midpoint of X(i) and X(j) for these {i,j}: {2, 3629}, {1353, 5476}
X(20583) = reflection of X(2) in X(6329)
X(20583) = complement of isotomic conjugate of X(33698)
X(20583) = X(546)-of-anti-Artzt triangle
X(20583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1992, 597), (6, 3629, 6329), (6, 5032, 8584), (141, 3629, 11008), (597, 8584, 1992), (1992, 11160, 15534), (3629, 6329, 3631), (3631, 6329, 3589), (13639, 19053, 13783), (13664, 13784, 2), (13759, 19054, 13663)

X(20584) = MIDPOINT OF X(546) AND X(1209)

Barycentrics (17*R^2-2*SA-6*SW)*S^2+(29*R^ 2-12*SW)*SB*SC : :
Barycentrics 2*a^10-(b^2+c^2)*a^8-2*(2*b^4- 3*b^2*c^2+2*c^4)*a^6-(b^2+c^2) *(2*b^4+11*b^2*c^2+2*c^4)*a^4+ (b^2-c^2)^2*(10*b^4+13*b^2*c^ 2+10*c^4)*a^2-5*(b^4-c^4)*(b^ 2-c^2)^3 : :
X(20584) = 5*X(5)-X(54), 3*X(5)+X(6288), 3*X(5)-X(8254), 3*X(54)+5*X(6288), 3*X(54)-5*X(8254), X(195)-9*X(3545), 3*X(547)-X(10610), X(2888)+7*X(3851), 7*X(2888)+X(13432), 15*X(3091)+X(12325), 5*X(3091)-X(20424), X(3574)-3*X(5066), 7*X(3851)-X(11803), 7*X(11803)-X(13432), X(12325)+3*X(20424)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28007.

X(20584) lies on the these lines: {5, 49}, {30, 13565}, {140, 11572}, {195, 3545}, {539, 11737}, {546, 1209}, {547, 10610}, {1154, 3850}, {2888, 3851}, {3091, 12325}, {3574, 5066}, {3628, 18400}, {3832, 12307}, {3845, 7691}, {5055, 12254}, {10019, 12300}, {10115, 13364}, {10592, 12956}, {10593, 12946}, {11591, 11808}, {13365, 13754}

X(20584) = midpoint of X(i) and X(j) for these {i,j}: {546, 1209}, {2888, 11803}, {11591, 11808}
X(20584) = {X(5), X(6288)}-harmonic conjugate of X(8254)

X(20585) = X(5)X(49)∩X(195)X(376)

Barycentrics (35*R^2+10*SA-18*SW)*S^2-(25* R^2-12*SW)*SB*SC : :
X(20585) = X(5)-5*X(54), 9*X(5)-5*X(6288), 3*X(5)-5*X(8254), 9*X(54)-X(6288), 3*X(54)-X(8254), 5*X(195)+3*X(376), 5*X(1493)+X(12103), X(1657)+5*X(11803), X(3146)-5*X(20424), 21*X(3523)-5*X(12325), 5*X(3574)-3*X(14893), 3*X(3830)+5*X(12254), X(6288)-3*X(8254), 5*X(10610)-3*X(12100)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28007.

X(20585) lies on the these lines: {5, 49}, {195, 376}, {539, 10124}, {1154, 13348}, {1493, 12103}, {1657, 11803}, {3146, 20424}, {3523, 12325}, {3530, 5965}, {3574, 14893}, {3830, 12254}, {10610, 12100}, {12102, 18400}

X(20585) = midpoint of X(i) and X(j) for these {i,j}: {546, 1209}, {2888, 11803}, {11591, 11808}
X(20585) = {X(5), X(6288)}-harmonic conjugate of X(8254)

X(20586) = MIDPOINT OF X(10085) AND X(13253)

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

See Tran Quang Hung and César Lozada, Hyacinthos 28008.

X(20586) lies on the these lines: {1, 5}, {8, 11256}, {34, 5151}, {55, 11715}, {56, 2802}, {57, 12653}, {65, 1320}, {100, 1319}, {104, 3057}, {109, 10700}, {149, 3476}, {214, 1388}, {515, 13274}, {517, 10074}, {944, 12743}, {946, 12763}, {1385, 10087}, {1420, 5541}, {1470, 13205}, {1482, 11570}, {1537, 12679}, {1768, 7962}, {2098, 2800}, {2099, 3892}, {2829, 12701}, {3036, 19861}, {3304, 12736}, {3885, 17100}, {3968, 4413}, {4308, 9802}, {4345, 9809}, {5048, 6001}, {5330, 12532}, {5697, 12515}, {5854, 12832}, {6224, 18467}, {7354, 14217}, {8581, 14151}, {9957, 10058}, {10085, 13253}, {10106, 13273}, {10895, 16174}, {12053, 12764}, {12619, 12647}, {12758, 12773}, {17636, 20323}

X(20586) = midpoint of X(10085) and X(13253)
X(20586) = X(12751) of 2nd Johnson-Yff triangle
X(20586) = reflection of X(80) in the line X(496)X(900)
X(20586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1317, 12739), (1, 6264, 11), (1, 7972, 6265), (149, 3476, 18976), (1387, 10956, 11375), (5048, 17660, 10698)

X(20587) = (name pending)

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

See Angel Montesdeoca, HG040818.

X(20587) lies on this line: {3523,3620}

X(20588) = X(1)X(1167)∩X(2)X(15298)

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

See Angel Montesdeoca, HG040818.

X(20588) lies on these lines: {1,1167}, {2,15298}, {8,90}, {9,497}, {10,10629}, {36,78}, {40,2123}, {46,7080}, {55,17658}, {57,6745}, {63,100}, {72,3428}, {191,4882}, {210,11502}, {354,8257}, {518,1260}, {944,6737}, {1040,3939}, {1158,5687}, {1259,3811}, {1478,6735}, {1490,12059}, {1709,17784}, {1741,3694}, {3190,3751}, {3305,5231}, {3717,3719}, {3729,17860}, {3870,18412}, {3872,5251}, {3885,4853}, {4430,4511}, {4863,7082}, {5176,5691}, {5552,13407}, {6600,10391}

X(20588) = extouch-isogonal conjugate of X(3059)

X(20589) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(105)

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

(1,38), (2,37), (6,3721), (31,41178), (105,20589), (238,20590), (292,20591), (365,20592), (672,20593), (1423,20594), (1931,20595), (2053,20596), (2054,20597), (3009,20598), (2112,20599), (20332,20600)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b : c
m(A') = a (Sqrt[b] + Sqrt[c]) (b^2 - b^(3/2) Sqrt[c] + b c - Sqrt[b] c^(3/2) + c^2) : -b (Sqrt[a] - Sqrt[c]) (a^2 + a^(3/2) Sqrt[c] + a c + Sqrt[a] c^(3/2) + c^2) : -(Sqrt[a] - Sqrt[b]) (a^2 + a^(3/2) Sqrt[b] + a b + Sqrt[a] b^(3/2) + b^2) c
m(A1) = a b c (b + c) : b (-a^3 + b c^2) : c (-a^3 + b^2 c)
m(A2) = a (b^4 + c^4) : -b c (a^2 b - c^3) : b c (b^3 - a^2 c)
m(A3) = a (a b + a c + b c) (b^2 + c^2) : -b (a^4 + a^3 b + a^3 c - a b c^2 - a c^3 - b c^3) : -c (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c -b^3 c)
m(A4) = -a (b + c) (a + b + c) (b^2 - b c + c^2) : b (a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4) : c (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c)
m(A5) = a (a^2 b^2 + b^4 - a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : b (a^4 + a^3 b + a^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : c (a^4 + a^3 b + a^2 b^2 + b^4 + a^3 c - a b^2 c - b^3 c)
m(A6) = a (a + b + c) (a^2 b^3 - a b^4 - b^4 c + b^3 c^2 + a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : -b (a^2 + b^2 - a c - b c) (-a^4 + a^3 b + a^2 b c - a^2 c^2 - a c^3 - b c^3 - c^4) : -c (-a^4 - a^2 b^2 - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = y/c + z/b : z/a + x/c : x/b + y/a, and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(1)), where A' = 0 : b : c.

Let b^3 (b-c) c^3 (a^2-b c) (-a^3 b^3-a^2 b^4+a^4 b c-a^3 b^2 c-a^2 b^3 c-a b^4 c-a^3 b c^2-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+b^3 c^3-a^2 c^4-a b c^4) x^3+a^3 b c^2 (-3 a^5 b^4-a^4 b^5+3 a^6 b^2 c+a^2 b^6 c+3 a^6 b c^2+a b^6 c^2-3 a^5 c^4-b^5 c^4+a^4 c^5-b^4 c^5-a^2 b c^6+a b^2 c^6) y^2 z-a^3 b^2 c (-3 a^5 b^4+a^4 b^5+3 a^6 b^2 c-a^2 b^6 c+3 a^6 b c^2+a b^6 c^2-3 a^5 c^4-b^5 c^4-a^4 c^5-b^4 c^5+a^2 b c^6+a b^2 c^6) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 a^2 (a-b) b^2 (a-c) (b-c) c^2 (-b^2+a c) (a^2-b c) (a b-c^2) x y z = 0. (Peter Moses, August 7, 2018)

X(20589) lies on these lines: {1, 20601}, {37, 17447}, {38, 20599}, {1962, 2611}, {3675, 16593}, {4118, 17452}, {17279, 20275}, {17459, 20596}, {20356, 20590}

X(20590) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(238)

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

See X(20589).

X(20590) lies on these lines: {1, 2210}, {2, 18208}, {37, 4118}, {38, 1107}, {517, 2292}, {536, 2643}, {612, 18788}, {760, 3747}, {1278, 17891}, {1959, 3009}, {2309, 17446}, {3720, 17456}, {3868, 3924}, {3912, 4475}, {4357, 7237}, {4414, 9441}, {16735, 17187}, {17234, 18168}, {17279, 20274}, {17300, 18207}, {17443, 17445}, {17444, 17472}, {20356, 20589}, {20363, 20364}, {20594, 20596}, {20595, 20597}, {20598, 20600}

X(20591) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(292)

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

See X(20589).

X(20591) lies on these lines: {1, 1922}, {37, 8299}, {38, 20356}, {337, 2275}, {1107, 2170}, {2276, 4518}, {3094, 4493}, {3726, 20363}, {3802, 5283}, {4118, 20599}, {20593, 20598}

X(20592) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(365)

Barycentrics a (Sqrt[b] + Sqrt[c]) (b^2 - b^(3/2) Sqrt[c] + b c - Sqrt[b] c^(3/2) + c^2) : :

See X(20589).

X(20592) lies on these lines: {1, 20604}

X(20593) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(672)

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

See X(20589).

X(20593) lies on these lines: {1, 9454}, {37, 1953}, {75, 3061}, {518, 2170}, {798, 4083}, {1107, 17445}, {1921, 18061}, {2228, 18904}, {3721, 4022}, {5701, 20459}, {13476, 17474}, {20271, 20274}, {20335, 20448}, {20356, 20589}, {20591, 20598}

X(20594) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(1423)

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

See X(20589).

X(20594) lies on these lines: {1, 20606}, {8, 2170}, {37, 1953}, {55, 16689}, {2098, 16969}, {3056, 3728}, {3959, 20284}, {7148, 12836}, {17448, 20359}, {20590, 20596}, {20599, 20600}

X(20595) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(1931)

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

See X(20589).

X(20595) lies on these lines: {1, 20607}, {38, 1755}, {1575, 2643}, {2611, 3726}, {20590, 20597}

X(20596) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(2053)

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

See X(20589).

X(20596) lies on these lines: {1, 20608}, {192, 3434}, {1934, 4518}, {4118, 20598}, {17447, 20361}, {17459, 20589}, {20590, 20594}

X(20597) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(2054)

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

See X(20589).

X(20597) lies on these lines: {1, 20609}, {1107, 20362}, {3721, 20599}, {20590, 20595}

X(20598) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(3009)

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

See X(20589).

X(20598) lies on these lines: {1, 20610}, {37, 38}, {244, 20530}, {4118, 20596}, {20590, 20600}, {20591, 20593}

X(20599) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(2112)

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

See X(20589).

X(20599) lies on these lines: {1, 20611}, {38, 20589}, {3721, 20597}, {4118, 20591}, {20594, 20600}

X(20600) = (X(1), X(2), X(6), X(31); X(38), X(37), X(3721), X(4118)) COLLINEATION IMAGE OF X(20332)

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

See X(20589).

X(20600) lies on these lines: {2170, 3721}, {2643, 3728}, {4118, 17463}, {20590, 20598}, {20594, 20599}

X(20601) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(105)

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

(1,63), (2,9), (6,1759), (31,1760), (105,20601), (238,20602), (292,20603), (365,20604), (672,20605), (1423,20606), (1931,20607), (2053,20608), (2054,20609), (3009,20610), (2112,20611)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -a : b : c
m(A') = a (a^(5/2) + b^(5/2) + c^(5/2)) : -b (a^(5/2) + b^(5/2) - c^(5/2)) : -c (a^(5/2) - b^(5/2) + c^(5/2))
m(A1) = -a (a^3 + b^2 c + b c^2), b (a^3 + b^2 c - b c^2), c (a^3 - b^2 c + b c^2)
m(A2) = a (b^4 + a^2 b c + c^4) : -b (b^4 + a^2 b c - c^4) : -c (-b^4 + a^2 b c + c^4)
m(A3) = -a (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3) : b (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c - a b c^2 - a c^3 - b c^3) : c (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c + a b c^2 + a c^3 + b c^3)
m(A4) = a (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c + a c^3 + b c^3 + c^4) : -b (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c - a c^3 - b c^3 - c^4) : -c (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c + a c^3 + b c^3 + c^4)
m(A5) = a (a^4 + 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 c^3 - c^4) : b (-a^4 - 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 c^3 - c^4) : c (-a^4 - 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 c^3 + c^4)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a(-ax + by + cz) : b(ax - by + cz): c(ax + by - cz), and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(1)), where A' = -a : b : c.

Let a^3 b (b-c) c (-b^3+a b c-b^2 c-b c^2-c^3) x^3+a b (-a^5 b^2+a^4 b^3+a^6 c-a^2 b^4 c+2 a b^4 c^2-2 b^3 c^4+b^2 c^5-a c^6) y^2 z-a c (a^6 b-a b^6-a^5 c^2+b^5 c^2+a^4 c^3-2 b^4 c^3-a^2 b c^4+2 a b^2 c^4) y z^2 , where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (-b^2+a c) (a^2-b c) (a b-c^2) x y z = 0. (Peter Moses, August 7, 2018)

X(20601) lies on these lines: {1, 20589}, {9, 141}, {55, 846}, {63, 15487}, {169, 673}, {190, 1760}, {971, 6211}, {1761, 16565}, {2195, 2809}, {4437, 17742}, {7291, 20533}, {16557, 20608}, {17738, 20602}

X(20602) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(238)

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

See X(20601).

X(20602) lies on these lines: {1, 2210}, {2, 16555}, {9, 1760}, {10, 191}, {19, 3729}, {63, 169}, {75, 16547}, {81, 16600}, {101, 1959}, {190, 16548}, {239, 5540}, {536, 7297}, {666, 2311}, {894, 1781}, {1026, 18788}, {1748, 7719}, {1762, 17739}, {2082, 16834}, {2664, 17799}, {3008, 3218}, {3405, 3508}, {3661, 17744}, {3912, 7291}, {3929, 19797}, {4063, 4380}, {4416, 5279}, {4641, 16583}, {4852, 7300}, {5081, 7713}, {5341, 17351}, {5525, 6542}, {6763, 16825}, {7289, 17298}, {17294, 17742}, {17738, 20601}, {20372, 20373}, {20606, 20608}, {20607, 20609}

X(20602) = isogonal conjugate of isotomic conjugate of X(20643)

X(20603) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(292)

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

See X(20601).

X(20603) lies on these lines: {1, 1922}, {9, 2108}, {63, 17026}, {191, 2795}, {846, 18794}, {1760, 20611}, {3099, 3508}, {3509, 20372}, {20605, 20610}

X(20604) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(365)

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

See X(20601).

X(20604) lies on these lines: {1, 20592}

X(20605) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(672)

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

See X(20601).

X(20605) lies on these lines: {1, 9454}, {4, 9}, {75, 16552}, {238, 18785}, {672, 1738}, {673, 20367}, {798, 812}, {1757, 5540}, {1759, 1760}, {2082, 3751}, {3294, 5263}, {3509, 17031}, {3886, 17742}, {4000, 4253}, {4192, 16588}, {4429, 16549}, {5701, 20470}, {10436, 16818}, {16600, 17446}, {17738, 20601}, {20603, 20610}

X(20606) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(1423)

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

See X(20601).

X(20606) lies on these lines: {1, 20594}, {3, 1107}, {4, 9}, {43, 2082}, {63, 3765}, {147, 18596}, {386, 9575}, {517, 2176}, {1764, 4384}, {1914, 13732}, {2275, 19514}, {2300, 10441}, {2319, 20368}, {5540, 6048}, {7991, 18785}, {8245, 17038}, {10476, 16825}, {16434, 16502}, {20602, 20608}

X(20607) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(1931)

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

See X(20601).

X(20607) lies on these lines: {1, 20595}, {19, 27}, {267, 16549}, {846, 4386}, {1046, 3959}, {1247, 4426}, {1575, 2640}, {2959, 17735}, {20602, 20609}

X(20608) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(2053)

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

See X(20601).

X(20608) lies on these lines: {1, 20596}, {1760, 20610}, {16551, 20370}, {16557, 20601}, {20602, 20606}

X(20609) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(2054)

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

See X(20601).

X(20609) lies on these lines: {1, 20597}, {9, 1654}, {1759, 20611}, {16552, 20371}, {20602, 20607}

X(20610) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(3009)

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

See X(20601).

X(20610) lies on these lines: {1, 20598}, {2, 7}, {190, 20453}, {802, 18197}, {1755, 17755}, {1760, 20608}, {3271, 8844}, {16514, 18206}, {20603, 20605}

X(20611) = (X(1), X(2), X(6), X(31); X(63), X(9), X(1759), X(1760)) COLLINEATION IMAGE OF X(2112)

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

See X(20601).

X(20611) lies on these lines: {1, 20599}, {63, 15487}, {1759, 20609}, {1760, 20603}

X(20612) = MIDPOINT OF X(3868) AND X(3871)

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

X(20612) lies on these lines: {8,18389}, {10,3580}, {35,758}, {40,3868}, {65,16465}, {100,15556}, {145,2802}, {214,14804}, {908,1858}, {1046,1331}, {1825,1897}, {1998,3339}, {2801,20060}, {3340,3873}, {3555,5844}, {3562,12016}, {3601,3869}, {3925,8261}, {5552,18397}, {5554,18412}, {5902,12649}, {5904,10528}, {5905,15071}, {6734,13750}, {10529,18398}, {11248,12515}, {11571,16126}, {12435,20243}

X(20612) = midpoint of X(3868) and X(3871)
X(20612) = reflection of X(6734) in X(13750)
X(20612) = X(643)-beth conjugate of X(15556)

X(20613) = X(7)-CEVA CONJUGATE OF X(652)

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

X(20613) lies on the cubic K1058 and these lines: {6, 19}, {12, 208}, {33, 7337}, {37, 3209}, {56, 5089}, {57, 5236}, {85, 653}, {108, 17916}, {169, 1783}, {196, 948}, {281, 388}, {1452, 2333}, {1826, 11392}, {1891, 5727}, {14257, 17905}

X(20613) = X(4025)-zayin conjugate of X(652)
X(20613) = X(7)-Ceva conjugate of X(33)
X(20613) = barycentric product X(i)*X(j) for these {i,j}: {4, 8270}, {34, 10327}, {273, 12329}, {278, 17742}, {653, 2509}, {1041, 11677}
X(20613) = barycentric quotient X(i)/X(j) for these {i,j}: {2509, 6332}, {8270, 69}, {10327, 3718}, {12329, 78}, {17742, 345}

X(20614) = X(12)X(116)∩X(65)X(1418)

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

X(20614) lies on the cubic K1058 and these lines: {12, 116}, {65, 1418}, {354, 7264}

X(20614) = barycentric product X(13476)X(17077)
X(20614) = barycentric quotient X(i)/X(j) for these {i,j}: {16552, 3996}, {17077, 17143}

X(20615) = ISOGONAL CONJUGATE OF X(3871)

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

X(20615) lies on the cubic K1058 and these lines: {1, 16528}, {12, 121}, {44, 583}, {65, 519}, {404, 765}, {1042, 1319}, {1417, 5253}, {1426, 1877}, {5252, 8050}, {7248, 10404}

X(20615) = isogonal conjugate of X(3871)
X(20615) = X(i)-cross conjugate of X(j) for these (i,j): {2171, 57}, {3649, 65}
X(20615) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3871}, {8, 595}, {21, 3293}, {41, 18140}, {55, 4360}, {60, 4075}, {78, 4222}, {284, 3995}, {312, 2220}, {643, 4132}, {644, 4063}, {3699, 4057}, {3939, 20295}, {4076, 8054}, {4129, 5546}, {4587, 17922}
X(20615) = cevapoint of X(1357) and X(4017)
X(20615) = trilinear pole of line {1635, 7180}
X(20615) = barycentric product X(i)*X(j) for these {i,j}: {57, 596}, {3669, 8050}
X(20615) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3871}, {7, 18140}, {57, 4360}, {65, 3995}, {596, 312}, {604, 595}, {608, 4222}, {1397, 2220}, {1400, 3293}, {2171, 4075}, {3669, 20295}, {4017, 4129}, {7180, 4132}, {8050, 646}, {15222, 105}
X(20615) = {X(5434),X(17114)}-harmonic conjugate of X(65)

X(20616) = X(7)-CEVA CONJUGATE OF X(181)

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

X(20616) lies on the cubic K1058 and these lines: {1, 4559}, {12, 115}, {37, 65}, {39, 15950}, {85, 4552}, {241, 553}, {950, 5724}, {1107, 11011}, {1825, 5089}, {2099, 5283}, {2276, 11375}, {3175, 3991}, {4032, 4059}, {5277, 14882}, {9331, 9578}, {15556, 16601}

X(20616) = X(7)-Ceva conjugate of X(181)
X(20616) = X(i)-isoconjugate of X(j) for these (i,j): {60, 17758}, {261, 2350}, {2185, 13476}
X(20616) = barycentric product X(i)*X(j) for these {i,j}: {12, 1621}, {65, 4651}, {181, 17143}, {201, 14004}, {226, 3294}, {1254, 3996}, {1400, 4043}, {2171, 17277}, {4151, 4551}, {4251, 6358}
X(20616) = barycentric quotient X(i)/X(j) for these {i,j}: {181, 13476}, {1621, 261}, {2171, 17758}, {3294, 333}, {4151, 18155}, {4251, 2185}, {4651, 314}, {17143, 18021}
X(20616) = {X(5434),X(17114)}-harmonic conjugate of X(65)
X(20616) = {X(5434),X(17114)}-harmonic conjugate of X(65)
{X(1334),X(2171)}-harmonic conjugate of X(15443)
{X(1334),X(2171)}-harmonic conjugate of X(15443)

X(20617) = X(7)-CEVA CONJUGATE OF X(6354)

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

X(20617) lies on the cubic K1058 and these lines: {1, 15622}, {12, 125}, {42, 65}, {85, 4566}, {226, 15267}, {515, 942}, {1426, 1882}, {1439, 15232}, {1876, 1888}, {2647, 18165}, {3649, 17705}, {6354, 7143}, {10441, 15832}

X(20617) = midpoint of X(65) and X(73)
X(20617) = X(7)-Ceva conjugate of X(6354)
X(20617) = X(i)-isoconjugate of X(j) for these (i,j): {2051, 7054}, {2328, 20028}
X(20617) = crosspoint of X(7) and X(17074)
X(20617) = barycentric product X(i)*X(j) for these {i,j}: {12, 17074}, {279, 14973}, {1254, 14829}, {1427, 17751}, {2975, 6354}
X(20617) = barycentric quotient X(i)/X(j) for these {i,j}: {572, 1098}, {1254, 2051}, {1427, 20028}, {2975, 7058}, {14973, 346}, {17074, 261}
X(20617) = {X(5434),X(17114)}-harmonic conjugate of X(65)
X(20617) = {X(1334),X(2171)}-harmonic conjugate of X(15443)

X(20618) = X(1)X(5894)∩X(12)X(1367)

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

X(20618) lies on the cubic K1058 and these lines: {1, 5894}, {12, 1367}, {65, 1439}, {85, 1952}, {201, 6356}, {222, 279}, {278, 14256}, {347, 20070}, {1071, 10481}

X(20618) = X(6354)-cross conjugate of X(6355)
X(20618) = X(i)-isoconjugate of X(j) for these (i,j): {19, 6061}, {21, 2332}, {33, 7054}, {55, 2326}, {60, 7079}, {200, 2189}, {220, 270}, {250, 3119}, {284, 4183}, {607, 1098}, {1043, 2204}, {1172, 2328}, {2150, 7046}, {2185, 7071}, {2194, 2322}, {2212, 7058}, {2287, 2299}
X(20618) = barycentric product X(i)*X(j) for these {i,j}: {7, 6356}, {12, 7056}, {69, 6046}, {201, 1088}, {304, 7147}, {305, 7143}, {307, 3668}, {339, 7339}, {347, 6355}, {348, 6354}, {479, 3695}, {1214, 1446}, {1231, 1427}, {1254, 7182}, {1425, 6063}, {1439, 1441}, {4064, 4626}, {4566, 17094}, {6358, 7177}
X(20618) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6061}, {12, 7046}, {57, 2326}, {65, 4183}, {73, 2328}, {77, 1098}, {125, 4081}, {181, 7071}, {201, 200}, {222, 7054}, {226, 2322}, {269, 270}, {307, 1043}, {348, 7058}, {1042, 2299}, {1214, 2287}, {1254, 33}, {1367, 2968}, {1400, 2332}, {1407, 2189}, {1410, 2194}, {1425, 55}, {1427, 1172}, {1439, 21}, {2171, 7079}, {2197, 220}, {3668, 29}, {3690, 480}, {3695, 5423}, {3708, 3119}, {3949, 728}, {4064, 4163}, {6046, 4}, {6354, 281}, {6355, 280}, {6356, 8}, {6358, 7101}, {7053, 60}, {7056, 261}, {7066, 1260}, {7099, 2150}, {7138, 212}, {7143, 25}, {7147, 19}, {7177, 2185}, {7178, 17926}, {7314, 7140}, {7339, 250}, {10376, 4206}, {13853, 7003}, {17094, 7253}

X(20619) = POLAR-CIRCLE-INVERSE OF X(106)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20619) lies on the nine-point circle and these lines: {1, 124}, {2, 2370}, {4, 106}, {11, 1883}, {37, 5514}, {53, 5190}, {116, 4000}, {117, 1769}, {121, 4768}, {123, 4187}, {125, 1834}, {127, 16052}, {225, 5521}, {1319, 1846}

X(20619) = complement of X(2370)
X(20619) = orthoptic circle of Steiner inellipse-inverse-of X(9088)
X(20619) = polar circle-inverse-of X(106)
X(20619) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (117, 5, 11727), (121, 5, 6715)

X(20620) = POLAR-CIRCLE-INVERSE OF X(109)

Barycentrics ((b+c)*a^4-b*c*a^3-(b+c)*(2*b^ 2-b*c+2*c^2)*a^2+(b+c)^2*b*c* a+(b^3-c^3)*(b^2-c^2))*(-a+b+ c)*(b-c)^2*(a^2+b^2-c^2)*(a^2- b^2+c^2) : :
Barycentrics (tan A)(cos B - cos C)[(tan A)(cos B - cos C) - (tan B)(cos C - cos A) - (tan C)(cos A - cos B)] : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20620) lies on the nine-point circle and these lines: {4, 109}, {33, 118}, {116, 7004}, {121, 11105}, {122, 3137}, {123, 14010}, {2969, 3259}, {5514, 8735}, {7649, 15608}, {13999, 16228}

X(20620) = polar circle-inverse-of X(109)
X(20620) = reflection of X(124) in the line X(5)X(6718)
X(20620) = center of hyperbola {{A,B,C,X(4),X(29)}} (the locus of trilinear poles of lines passing through X(3064))
X(20620) = perspector of circumconic centered at X(3064)
X(20620) = crosssum of circumcircle intercepts of line X(3)X(73)
X(20620) = X(2)-Ceva conjugate of X(3064)
X(20620) = orthopole of line X(3)X(73)
X(20620) = Kirikami-six-circles image of X(29)

X(20621) = POLAR-CIRCLE-INVERSE OF X(105)

Barycentrics (a^2-b^2+c^2)*(a^2+b^2-c^2)*(( b+c)*a^4-2*b*c*a^3-(b^4-c^4)*( b-c))*((b+c)*a-b^2-c^2) : :
Barycentrics (tan A)[b/(c + a - b sec B) + c/(a + b - c sec C)] : :

X(20621) is the touchpoint, other than X(11), of the line through X(676) tangent to the nine-point circle. (Randy Hutson, August 29, 2018)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20621) lies on the nine-point circle and these lines: {2, 108}, {4, 105}, {11, 33}, {12, 208}, {25, 5521}, {115, 429}, {116, 1210}, {122, 18592}, {124, 226}, {125, 15904}, {127, 442}, {225, 5190}, {431, 5139}, {468, 5520}, {1368, 15252}, {1566, 5089}, {1595, 15251}, {1785, 15612}, {3011, 13999}, {5230, 5517}, {15253, 15809}

X(20621) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (11, 5, 676), (105, 676, 6714), (120, 5, 6714)
X(20621) = orthoptic-circle-of-Steiner-inellipse-inverse of X(108)
X(20621) = polar circle-inverse-of X(105)
X(20621) = nine-point-circle intercept, other than X(11), of circle {{X(11),X(105),X(108)}}

X(20622) = POLAR-CIRCLE-INVERSE OF X(103)

Barycentrics ((b^2+c^2)*a^5-(b^3+c^3)*a^4- 2*(b^2-c^2)^2*a^3+2*(b^3-c^3)* (b^2-c^2)*a^2+(b^4-c^4)*(b^2- c^2)*a+(b^2-c^2)*(b-c)*(-b^4- c^4-(b^2+4*b*c+c^2)*b*c))*(2* a^3-(b+c)*a^2-(b^2-c^2)*(b-c)) *(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
Barycentrics f(a,b,c) (f(a,b,c) - f(b,c,a) - f(c,a,b)) : :, where f(a,b,c) = a(tan A)(a^2 - b^2 cos C - c^2 cos B)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20622) lies on the nine-point circle and these lines: {4, 103}, {11, 1427}, {122, 3136}, {123, 8226}, {124, 1699}, {125, 430}, {235, 5190}, {1855, 5514}

X(20622) = polar circle-inverse-of X(103)
X(20622) = reflection of X(118) in the line X(5)X(6712)
X(20622) = perspector of circumconic centered at X(1886)
X(20622) = center of circumconic that is locus of trilinear poles of lines passing through X(1886)
X(20622) = X(2)-Ceva conjugate of X(1886)

X(20623) = POLAR-CIRCLE-INVERSE OF X(20624)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20623) lies on the nine-point circle and these lines: {6, 11}, {9, 123}, {19, 5521}, {33, 13529}, {116, 226}, {120, 1639}, {122, 18591}, {124, 20262}, {127, 1211}, {136, 1865}, {1699, 5511}, {3925, 5514}, {5099, 5164}, {5179, 15612}

X(20623) = polar circle-inverse-of X(20624)
X(20623) = reflection of X(119) in the line X(5)X(5848)
X(20623) = orthopole of PU(125)
X(20623) = crosssum of circumcircle-intercepts of line PU(125) (line X(3)X(650))
X(20623) = Kirikami six circles image of X(651)

X(20624) = X(19)X(109)∩X(33)X(101)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20624) lies on the circumcircle and these lines: {19, 109}, {33, 101}, {100, 281}, {108, 393}, {110, 1172}, {243, 929}, {278, 934}, {1310, 8777}, {2202, 8776}, {2722, 5523}, {7129, 8059}

X(20624) = trilinear pole of the line {6, 18344}
X(20624) = polar circle-inverse-of X(20623)
X(20624) = polar conjugate of isotomic conjugate of X(8759)
X(20624) = X(63)-isoconjugate of X(8758)
X(20624) = Ψ(X(i), X(j)) for these (i,j): (3, 650), (6, 18344)

X(20625) = POLAR-CIRCLE-INVERSE OF X(20626)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20625) lies on the MacBeath circle, the nine-point circle and these lines: {2, 933}, {3, 128}, {4, 18401}, {5, 18402}, {23, 14918}, {113, 1209}, {114, 6676}, {115, 8902}, {131, 10600}, {132, 5133}, {133, 546}, {136, 15526}, {233, 1560}, {2072, 16336}, {2972, 3258}, {5576, 10214}, {6639, 8157}, {8439, 8798}, {11563, 18809}

X(20625) = midpoint of X(4) and X(18401)
X(20625) = complement of X(933)
X(20625) = circumcircle-inverse-of X(15959)
X(20625) = antipode of X(18402) in the nine-point circle
X(20625) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(2),X(5)}}
X(20625) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (115, 5, 1576), (122, 5, 107), (125, 5, 11557), (128, 5, 14225), (136, 5, 5961), (137, 5, 11701)

X(20626) = X(4)X(18401)∩X(24)X(1141)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28010.

X(20626) lies on the circumcircle and these lines: {4, 18401}, {24, 1141}, {74, 6145}, {110, 16039}, {186, 18284}, {403, 14979}, {427, 1297}, {550, 5897}, {827, 2409}, {1294, 3520}, {1298, 6403}, {2383, 3542}, {2693, 13619}, {5966, 6353}, {7473, 11635}

X(20626) = trilinear pole of the line {6, 3574}
X(20626) = polar circle-inverse of X(20625)
X(20626) = Ψ(X(6), X(3574))

X(20627) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(31)

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

(1,1930), (2,321), (6,20234), (31,20627), (105,20628), (238,20629), (292,20630), (365,20631), (672,20632), (1423,20633), (2053,20635), (2054,20636), (3009,20637), (2112,20638), (20332,20639)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : c : b
m(A') = - b c (Sqrt[b] + Sqrt[c]) (b^2 - b^(3/2) Sqrt[c] + b c - Sqrt[b] c^(3/2) + c^2) : c a (Sqrt[a] - Sqrt[c]) (a^2 + a^(3/2) Sqrt[c] + a c + Sqrt[a] c^(3/2) + c^2) : a b (Sqrt[a] - Sqrt[b]) (a^2 + a^(3/2) Sqrt[b] + a b + Sqrt[a] b^(3/2) + b^2)
m(A1) = - b^2 c^2 (b + c) : c a (a^3 - b c^2) : a b (a^3 - b^2 c)
m(A2) = - b c (b^4 + c^4) : c a (a^2 b - c^3) : a b (-b^3 + a^2 c)
m(A3) = - b c (a b + a c + b c) (b^2 + c^2) : c a (a^4 + a^3 b + a^3 c - a b c^2 - a c^3 - b c^3) : a b (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c)
m(A4) = -b c (b + c) (a + b + c) (b^2 - b c + c^2) : a c (a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4) : a b (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c)
m(A5) = b c (a^2 b^2 + b^4 - a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : c a (a^4 + a^3 b + a^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : a b (a^4 + a^3 b + a^2 b^2 + b^4 + a^3 c - a b^2 c - b^3 c)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = b c (b y + c a) : c a (c z + a x) : a b (a x + b y), and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(75)), where A' = 0 : c : b = 0 : 1/b : 1/c.

X(20627) lies on these lines: {31, 75}, {304, 17871}, {321, 4766}, {561, 1109}, {746, 2205}, {1930, 1959}, {4118, 16891}, {4121, 4178}, {14210, 17884}, {17870, 17883}, {17881, 17890}, {20234, 20632}, {20237, 20628}, {20630, 20638}, {20635, 20637}

X(20628) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(105)

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

See X(20627).

X(20628) lies on these lines: {75, 105}, {321, 20431}, {1930, 20638}, {4647, 10624}, {5278, 17755}, {20237, 20627}, {20433, 20629}

X(20627) = isotomic conjugate of isogonal conjugate of X(4118)

X(20629) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(238)

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

See X(20627).

X(20629) lies on these lines: {75, 238}, {321, 4766}, {349, 9238}, {1930, 17760}, {3262, 18697}, {3263, 5988}, {4362, 20641}, {7237, 17211}, {20433, 20628}, {20440, 20441}, {20633, 20635}, {20634, 20636}, {20637, 20639}

X(20630) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(292)

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

See X(20627).

X(20630) lies on these lines: {75, 292}, {1930, 20433}, {4858, 18697}, {20432, 20440}, {20627, 20638}, {20632, 20637}

X(20631) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(365)

Barycentrics b*c*(b^(1/2)+c^(1/2))*(b^2-b^(3/2)*c^(1/2)+b*c-b^(1/2)*c^(3/2)+c^2) : :

See X(20627).

X(20631) lies on these lines: (none yet)

X(20632) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(672)

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

See X(20627).

X(20632) lies on these lines: {75, 672}, {76, 4165}, {313, 20659}, {321, 908}, {661, 17893}, {3262, 3930}, {3263, 4858}, {17871, 20171}, {20234, 20627}, {20433, 20628}, {20630, 20637}

X(20633) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(1423)

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

See X(20627).

X(20633) lies on these lines: {8, 17153}, {75, 1423}, {321, 908}, {3210, 17861}, {3596, 4858}, {20629, 20635}, {20638, 20639}

X(20634) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(1931)

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

See X(20627).

X(20634) lies on these lines: {75, 1931}, {321, 4109}, {1930, 1959}, {17886, 20432}, {20629, 20636}

X(20635) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(2053)

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

See X(20627).

X(20635) lies on these lines: {75, 2053}, {20627, 20637}, {20629, 20633}

X(20636) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(2054)

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

See X(20627).

X(20636) lies on these lines: {75, 2054}, {321, 20500}, {20234, 20638}, {20629, 20634}

X(20637) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(3009)

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

See X(20627).

X(20637) lies on these lines: {75, 3009}, {321, 1930}, {20627, 20635}, {20629, 20639}, {20630, 20632}

X(20638) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(2112)

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

See X(20627).

X(20638) lies on these lines: {75, 2112}, {1930, 20628}, {20234, 20636}, {20627, 20630}, {20633, 20639}

X(20639) = (X(1), X(2), X(6), X(75); X(1930), X(321), X(20234), X(75)) COLLINEATION IMAGE OF X(20332)

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

See X(20627).

X(20639) lies on these lines: {75, 20332}, {4858, 20234}, {20629, 20637}, {20633, 20638}

X(20640) = ISOGONAL CONJUGATE OF X(20512)

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

X(20640) lies on the circumcircle and these lines: {513, 20340}, {514, 20363}

X(20640) = isogonal conjugate of X(20512)
X(20640) = X(i)-isoconjugate of X(j) for these (i,j): {513, 20340}, {514, 20363}
X(20640) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 20340}, {692, 20363}

X(20641) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(31)

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

(1,304), (2,312), (6,20444), (31,20641), (105,20642), (238,20643), (292,20644), (365,20645), (672,20646), (1423,20647), (1931,20648), (2053,20649), (2054,20650), (3009,20651), (2112,20652)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -1/a : 1/b : 1/c
m(A') = -a^2 b^3 c^3 (a^(5/2) + b^(5/2) + c^(5/2)) : a^3 b^2 c^3 (a^(5/2) + b^(5/2) - c^(5/2)) : a^3 b^3 c^2 (a^(5/2) - b^(5/2) + c^(5/2))
m(A1) = - b c (a^3 + b^2 c + b c^2) : c a (a^3 + b^2 c - b c^2) : a b (a^3 - b^2 c + b c^2)
m(A2) = - b c (b^4 + a^2 b c + c^4) : c a (b^4 + a^2 b c - c^4) : c a (-b^4 + a^2 b c + c^4)
m(A3) = - b c (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3) : c a (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c - a b c^2 - a c^3 - b c^3) : a b (a^4 + a^3 b - a b^3 + a^3 c - a ^2 c - b^3 c + a b c^2 + a c^3 + b c^3)
m(A4) = -b c (b + c) (a + b + c) (b^2 - b c + c^2) : a c (a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4) : a b (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c)
m(A5) = -b c (a^4 + 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 c^3 - c^4) : c a (a^4 + 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 c^3 + c^4) : a b (a^4 + 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 c^3 - c^4)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = b c (- a x + b y + c z) : c a (a x - b y + c z ) : a b (a x + b y - c z), and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(75)), where A' = -1/a : 1/b : 1/c.

Let f(a,b,c,x,y,z) = a^8 (b-c) (-b^3+a b c-b^2 c-b c^2-c^3) x^3+b^4 c (-a^5 b^2+a^4 b^3+a^6 c-a^2 b^4 c+2 a b^4 c^2-2 b^3 c^4+b^2 c^5-a c^6) y^2 z-b c^4 (a^6 b-a b^6-a^5 c^2+b^5 c^2+a^4 c^3-2 b^4 c^3-a^2 b c^4+2 a b^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 a (a-b) b (a-c) (b-c) c (-b^2+a c) (a^2-b c) (a b-c^2) x y z = 0. (Peter Moses, August 8, 2018)

X(20641) lies on these lines: {31, 75}, {92, 304}, {305, 2064}, {312, 4766}, {315, 4463}, {326, 1096}, {799, 18750}, {1966, 17871}, {1978, 20642}, {3403, 14213}, {4362, 20629}, {14210, 18056}, {18064, 18156}, {18138, 20444}, {20644, 20652}, {20649, 20651}

X(20642) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(105)

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

See X(20641).

X(20642) lies on these lines: {8, 7261}, {75, 105}, {304, 20652}, {312, 8024}, {1978, 20641}, {20446, 20643}

X(20643) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(238)

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

See X(20641).

X(20643) lies on these lines: {75, 238}, {76, 18744}, {304, 18137}, {312, 4766}, {313, 502}, {1921, 18151}, {4639, 17789}, {7112, 20445}, {14349, 18081}, {20446, 20642}, {20453, 20454}, {20647, 20649}, {20648, 20650}

X(20643) = isotomic conjugate of isogonal conjugate of X(20602)

X(20644) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(292)

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

See X(20641).

X(20644) lies on these lines: {75, 292}, {304, 18050}, {17788, 18137}, {17789, 20453}, {18051, 20444}, {20641, 20652}, {20646, 20651}

X(20645) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(365)

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

See X(20641).

X(20645) lies on these lines: {75, 365}

X(20646) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(672)

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

See X(20641).

X(20646) lies on these lines: {75, 672}, {92, 264}, {561, 18137}, {661, 786}, {3262, 3693}, {18031, 20448}, {18138, 20444}, {20446, 20642}, {20644, 20651}

X(20647) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(1423)

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

See X(20641).

X(20647) lies on these lines: {69, 3765}, {75, 1423}, {76, 12610}, {92, 264}, {192, 3262}, {20643, 20649}

X(20648) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(1931)

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

See X(20641).

X(20648) lies on these lines: {75, 1931}, {92, 304}, {3765, 17762}, {20643, 20650}

X(20649) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(2053)

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

See X(20641).

X(20649) lies on these lines: {75, 2053}, {20641, 20651}, {20643, 20647}

X(20650) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(2054)

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

See X(20641).

X(20650) lies on these lines: {75, 2054}, {312, 18035}, {18137, 20452}, {20444, 20652}, {20643, 20648}

X(20651) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(3009)

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

See X(20641).

X(20651) lies on these lines: {75, 3009}, {76, 85}, {772, 3250}, {3797, 18157}, {18051, 18137}, {20641, 20649}, {20644, 20646}

X(20652) = (X(1), X(2), X(6), X(75); X(304), X(312), X(20444), X(75)) COLLINEATION IMAGE OF X(2112)

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

See X(20641).

X(20652) lies on these lines: {75, 2112}, {304, 20642}, {20444, 20650}, {20641, 20644}

X(20653) = (name pending

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

See X(20655).

X(20653) lies on these lines:

X(20654) = (name pending

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

See X(20655).

X(20654) lies on these lines:

X(20655) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(31)

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

(1,20653), (2,8013), (6,20654), (31,(20655), (105,20656), (238,20657), (292,20658), (672,20659), (1423,20660), (1931,20661), (2053,20662), (2054,20663), (3009,20664), (2112,20665), (2054,20679)

Fifteen-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : a + c : a + b
m(A1) = b c (b + c)^2 (2 a + b + c) : -(a + c)^2 (a^3 + a^2 b - b^2 c - b c^2) : -(a + b)^2 (a^3 + a^2 c - b^2 c - b c^2)
m(A2) = -(b + c)^2 (a b^3 + b^4 + a c^3 + c^4) : -c (a + c)^2 (-a^2 b - a b^2 + b c^2 + c^3) : -b (a + b)^2 (b^3 - a^2 c + b^2 c - a c^2)
m(A3) = (b + c)^2 (a b + a c + b c) (a b + b^2 + a c + c^2) : -(a + c)^2 (a^4 + 2 a^3 b + a^2 b^2 + a^3 c + a^2 b c - a b^2 c - 2 a b c^2 - b^2 c^2 - a c^3 - b c^3) : -(a + b)^2 (a^4 + a^3 b - a b^3 + 2 a^3 c + a^2 b c - 2 a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b^2 c^2)
m(A4) = (b + c)^2 (a + b + c) (a b^2 + b^3 + a c^2 + c^3) : -(a + c)^2 (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c - a b c^2 - b^2 c^2 - a c^3 - 2 b c^3 - c^4) : -(a + b)^2 (a^3 b - a b^3 - b^4 + a^3 c + 2 a^2 b c - a b^2 c - 2 b^3 c + a^2 c^2 + a b c^2 - b^2 c^2)

where A1, A2, A3, A4 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b + c)^2 (a y + b y + a z + c z) : : , and m is the collineation indicated by (A,B,C,X(10); A'B'C',X(10)), where A' = 0 : a + c : a + b.

Let f(a,b,c,x,y,z) = (a+b)^3 (b-c) (a+c)^3 (a^6 b^2+3 a^5 b^3+3 a^4 b^4+a^3 b^5+4 a^5 b^2 c+7 a^4 b^3 c+2 a^3 b^4 c-3 a^2 b^5 c-2 a b^6 c+a^6 c^2+4 a^5 b c^2+6 a^4 b^2 c^2+a^3 b^3 c^2-7 a^2 b^4 c^2-5 a b^5 c^2+3 a^5 c^3+7 a^4 b c^3+a^3 b^2 c^3-8 a^2 b^3 c^3-6 a b^4 c^3+b^5 c^3+3 a^4 c^4+2 a^3 b c^4-7 a^2 b^2 c^4-6 a b^3 c^4+2 b^4 c^4+a^3 c^5-3 a^2 b c^5-5 a b^2 c^5+b^3 c^5-2 a b c^6) x^3-(a+b)^2 (a+c) (b+c)^3 (-3 a^6 b^3-7 a^5 b^4-5 a^4 b^5-a^3 b^6+6 a^7 b c+9 a^6 b^2 c-a^5 b^3 c-8 a^4 b^4 c-3 a^3 b^5 c+3 a^2 b^6 c+2 a b^7 c+9 a^6 b c^2+12 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+3 a b^6 c^2-3 a^6 c^3+a^5 b c^3+3 a^4 b^2 c^3+a^2 b^4 c^3-a b^5 c^3-b^6 c^3-5 a^5 c^4-4 a^4 b c^4-a^3 b^2 c^4+a^2 b^3 c^4-4 a b^4 c^4-3 b^5 c^4-a^4 c^5-3 a^3 b c^5-a b^3 c^5-3 b^4 c^5+a^3 c^6-a^2 b c^6+a b^2 c^6-b^3 c^6) y^2 z+(a+b) (a+c)^2 (b+c)^3 (-3 a^6 b^3-5 a^5 b^4-a^4 b^5+a^3 b^6+6 a^7 b c+9 a^6 b^2 c+a^5 b^3 c-4 a^4 b^4 c-3 a^3 b^5 c-a^2 b^6 c+9 a^6 b c^2+12 a^5 b^2 c^2+3 a^4 b^3 c^2-a^3 b^4 c^2+a b^6 c^2-3 a^6 c^3-a^5 b c^3+3 a^4 b^2 c^3+a^2 b^4 c^3-a b^5 c^3-b^6 c^3-7 a^5 c^4-8 a^4 b c^4+a^3 b^2 c^4+a^2 b^3 c^4-4 a b^4 c^4-3 b^5 c^4-5 a^4 c^5-3 a^3 b c^5+4 a^2 b^2 c^5-a b^3 c^5-3 b^4 c^5-a^3 c^6+3 a^2 b c^6+3 a b^2 c^6-b^3 c^6+2 a b c^7) y z^2 , where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 a (a-b) b (a-c) (b-c) c (-b^2+a c) (a^2-b c) (a b-c^2) x y z 2 (a-b) (a+b)^2 (a-c) (b-c) (a+c)^2 (b+c)^2 (a+b+c) (a^3 b^2+a^2 b^3+a^3 c^2+b^3 c^2+a^2 c^3+b^2 c^3) x y z = 0. (Peter Moses, August 8, 2018)

X(20655) lies on these lines: {10, 31}, {3613, 15523}, {8013, 20657}, {17757, 20653}

X(20656) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(105)

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

See X(20655).

X(20656) lies on these lines: {10, 105}, {20482, 20653}, {20657, 20659}

X(20657) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(238)

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

See X(20655).

X(20657) lies on these lines: {10, 82}, {12, 594}, {2886, 15523}, {3178, 4360}, {4062, 17724}, {8013, 20655}, {20656, 20659}, {20661, 20679}

X(20658) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(292)

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

See X(20655).

X(20658) lies on these lines: {10, 292}, {594, 6543}, {20484, 20654}, {20488, 20491}

X(20659) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(672)

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

See X(20655).

X(20659) lies on these lines: {10, 672}, {210, 8013}, {313, 20632}, {661, 20483}, {3613, 15523}, {20656, 20657}

X(20660) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(1423)

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

See X(20655).

X(20660) lies on these lines: {10, 1423}, {210, 8013}

X(20661) = (X(1), X(2), X(6), X(2054); X(20653), X(8013), X(20654), X(20679)) COLLINEATION IMAGE OF X(1931)

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

See X(20655).

X(20661) lies on these lines: {10, 1931}, {17757, 20653}, {20657, 20679}

X(20662) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(105)

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

(1,6), (2,213), (6,42), (31,39), (105,20662), (238,673), (292,20663), (365,20664), (672,223), (1423,20665), (1931,20666), (2053,20667), (2054,20668), (3009,20669), (2112,20670), (20332,20671)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b^2 : c^2
m(A') = a^2 (Sqrt[b] + Sqrt[c]) : - b^2 (Sqrt[a] - Sqrt[c]) : - c^2 (Sqrt[a] - Sqrt[b])
m(A1) = a^2 (b + c) : b^2 (b - a) : c^2 (c - a)
m(A2) = a^2 (b^2 + c^2), b^2 c (c - b), b c^2 (b - c)
m(A3) = 2 a^2 (a b + a c + b c) : b^2 (-a^2 + b c) : c^2 (-a^2 + b c)
m(A4) = a^2 (b + c) (a + b + c) : b^2 (c^2 - a b) : c^2 (b^2 - a c)
m(A5) = -a^2 (2 a^2 - a b + b^2 - a c - 2 b c + c^2) : b^2 (-2 a^2 - a c + b c - c^2) : c^2 (-2 a^2 - a b - b^2 + b c)
m(A6) = a^3 (a + b + c) (a b - b^2 + a c - c^2) : -b^2 (a^2 + b^2 - a c - b c) (-a^2 + a b - a c - 2 c^2) : -c^2 (-a^2 - a b - 2 b^2 + a c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^3 (c y + b z) : : , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(6)), where A' = 0 : b^2 : c^2.

Let f(a,b,c,x,y,z) = b^6 (b-c) c^6 (a^2-b c) x^3-a^6 b^2 c^4 (a^2 b-a b^2+2 a^2 c-a c^2-b c^2) y^2 z+a^6 b^4 c^2 (2 a^2 b-a b^2+a^2 c-b^2 c-a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 8, 2018)

X(20662) lies on these lines: {6, 692}, {39, 41}, {56, 101}, {213, 1015}, {292, 2279}, {294, 5091}, {672, 2223}, {1017, 1055}, {1026, 19593}, {1423, 1743}, {1642, 17464}, {2348, 3008}, {3675, 9502}, {3768, 8658}, {9321, 15615}

X(20663) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(292)

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

See X(20662).

X(20663) lies on these lines: {6, 292}, {31, 43}, {39, 20670}, {42, 20457}, {238, 239}, {748, 17026}, {899, 8622}, {1197, 1977}, {1691, 2210}, {1740, 3888}, {1923, 3216}, {2092, 2309}, {2209, 3169}, {2223, 20669}, {2876, 3778}, {4455, 8632}, {16476, 20158}, {20455, 20456}

X(20664) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(365)

Barycentrics a^2 (Sqrt[b] + Sqrt[c]) : :

See X(20662).

X(20664) lies on these lines: {1, 364}, {6, 18753}, {86, 20527}, {1220, 4181}

X(20665) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(1423)

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

See X(20662).

X(20665) lies on these lines: {1, 20460}, {2, 7167}, {6, 893}, {8, 3495}, {9, 2319}, {31, 32}, {38, 2170}, {42, 263}, {43, 165}, {55, 7077}, {57, 20459}, {63, 194}, {184, 18038}, {190, 7033}, {292, 2162}, {894, 19591}, {1334, 4512}, {1397, 19554}, {2112, 9306}, {2275, 7248}, {2311, 2344}, {2361, 9447}, {3061, 3794}, {3116, 7032}, {3185, 9454}, {3271, 16588}, {3507, 3730}, {3685, 7075}, {4020, 16502}, {4253, 17795}, {4362, 5282}, {11031, 17451}, {14936, 20670}

X(20665) = isogonal conjugate of isotomic conjugate of X(3061)

X(20666) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(1931)

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

See X(20662).

X(20666) lies on these lines: {3, 6}, {9, 2959}, {35, 2653}, {115, 516}, {512, 798}, {672, 20668}, {902, 3124}, {966, 20558}, {1155, 16592}, {1213, 20546}, {1500, 2670}, {1914, 20461}, {2054, 3747}, {2108, 2238}, {2702, 17735}, {3496, 5184}, {10026, 17770}

X(20667) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(2053)

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

See X(20662).

X(20667) lies on these lines: {6, 20676}, {39, 20669}, {43, 165}, {292, 694}, {511, 20460}, {1334, 1655}, {3501, 17350}

X(20668) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(2054)

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

See X(20662).

X(20668) lies on these lines: {6, 20677}, {42, 20670}, {58, 101}, {672, 20666}, {1757, 8298}, {2308, 5147}

X(20669) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(3009)

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

See X(20662).

X(20669) lies on these lines: {1, 6}, {39, 20667}, {239, 20332}, {667, 6373}, {672, 20457}, {899, 1977}, {1575, 18793}, {2162, 16569}, {2223, 20663}, {2235, 4974}, {2308, 8622}, {8054, 8620}, {20456, 20670}

X(20669) = isogonal conjugate of antitomic conjugate of X(38247)

X(20670) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(2112)

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

See X(20662).

X(20670) lies on these lines: {6, 692}, {39, 20663}, {42, 20668}, {44, 9018}, {109, 181}, {291, 2144}, {511, 1757}, {665, 3572}, {984, 2810}, {1469, 2114}, {1654, 3888}, {3688, 9016}, {14936, 20665}, {20456, 20669}

X(20671) = (X(1), X(2), X(6), X(31); X(6), X(213), X(42), X(39)) COLLINEATION IMAGE OF X(20332)

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

See X(20662).

X(20671) lies on the Brocard inellipse and these lines: {1, 39}, {6, 727}, {32, 8671}, {42, 1977}, {43, 16557}, {76, 1574}, {194, 668}, {537, 3774}, {538, 13466}, {672, 20457}, {726, 1575}, {1569, 2787}, {2092, 3029}, {2810, 3094}, {14936, 20665}, {17756, 17794}

X(20671) = refection of X(1015) in X(39)
X(20671) = antipode of X(1015) in Brocard inellipse
X(20671) = barycentric square of X(1575)

X(20672) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(105)

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

(1,6), (2,2176), (6,55), (31,3), (105,20672), (238,17735), (292,2110), (365,20673), (1423,20674), (1931,20675), (2053,20676), (2054,20677), (3009,238), (2112,20678)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = -a^2 : b^2 : c^2
m(A') = a^2 (Sqrt[a] + Sqrt[b] + Sqrt[c]) : - b^2 (Sqrt[a] + Sqrt[b] - Sqrt[c]) : - c^2 (Sqrt[a] - Sqrt[b] + Sqrt[c])
m(A1) = a^2 (a + b + c) : b^2 (-a + b - c) : c^2 (-a - b + c)
m(A2) = a^2 (b^2 + b c + c^2) : -b^2 (b^2 + b c - c^2) : -c^2 (-b^2 + b c + c^2)
m(A3) = -a (a^2 + 3 a b + 3 a c + 2 b c) : b^2 (a + b + c) : c^2 (a + b + c)
m(A4) = a^2 (2 a b + b^2 + 2 a c + 3 b c + c^2) : -b^2 (2 a b + b^2 + b c - c^2) : -c^2 (-b^2 + 2 a c + b c + c^2)
m(A5) = a^2 (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) : -b^2 (-a + b - c) (a + b + c) : (a + b - c) c^2 (a + b + c)
m(A6) = a^2 (a^2 - b c) (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) : -b^2 (a^4 - 2 a^3 b + a^2 b^2 - a^2 b c + 4 a b^2 c + b^3 c + a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 3 b c^3) : -c^2 (a^4 + a^2 b^2 - 2 a b^3 - 2 a^3 c - a^2 b c - 2 a b^2 c - 3 b^3 c + a^2 c^2 + 4 a b c^2 + 2 b^2 c^2 + b c^3)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (- x/a + b/y + c/z) : b^2 (x/a - b/y + c/z) : c^2 (x/a + b/y - c/z), and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(6)), where A' = a^2 : b^2 : c^2.

Let f(a,b,c,x,y,z) = b^5 (b-c) c^5 (a+b+c) x^3-a^5 b c^3 (a^2 b+a b^2+a^2 c-a c^2-2 b c^2) y^2 z+a^5 b^3 c (a^2 b-a b^2+a^2 c-2 b^2 c+a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 8, 2018)

X(20672) lies on these lines: {3, 101}, {6, 692}, {9, 8245}, {37, 16560}, {41, 2276}, {56, 292}, {672, 2112}, {910, 9441}, {1015, 1191}, {1642, 16550}, {1914, 8647}, {2223, 16514}, {2279, 17962}, {3196, 8658}, {5091, 18785}, {6996, 17747}, {9508, 9509}, {16549, 19329}, {16777, 17463}

X(20673) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(365)

Barycentrics a^2 (Sqrt[a] - Sqrt[b] - Sqrt[c]) : :

See X(20672).

X(20673) lies on these lines: {6, 18753}, {55, 365}

X(20674) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(1423)

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

See X(20672).

X(20674) lies on these lines: {6, 893}, {9, 165}, {55, 19586}, {197, 20678}, {198, 17735}, {649, 4191}, {1615, 2110}, {2176, 2223}, {4650, 5022}, {16969, 20471}

X(20675) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(1931)

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

See X(20672).

X(20675) lies on these lines: {3, 6}, {37, 2959}, {55, 2248}, {238, 9509}, {1213, 20558}, {2702, 20472}, {14712, 17688}, {17735, 20677}

X(20676) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(2053)

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

See X(20672).

X(20676) lies on these lines: {3, 238}, {6, 20667}, {55, 192}, {198, 17735}, {1486, 20473}, {2053, 17792}, {2076, 20471}, {3010, 3556}

X(20677) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(2054)

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

See X(20672).

X(20677) lies on these lines: {1, 20474}, {6, 20668}, {31, 110}, {55, 846}, {2108, 4455}, {2276, 2503}, {3571, 8300}, {4650, 16575}, {8301, 13174}, {17735, 20675}

X(20678) = (X(1), X(2), X(6), X(31); X(6), X(2176), X(55), X(3)) COLLINEATION IMAGE OF X(2112)

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

See X(20672).

X(20678) lies on these lines: {3, 2110}, {6, 692}, {25, 5364}, {55, 846}, {56, 2114}, {109, 1460}, {197, 20674}, {238, 17798}, {2223, 3220}, {8301, 17755}, {10828, 14974}

X(20679) = (name pending)

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

See X(20655).

X(20679) lies on these lines: {10, 2054}, {15523, 20490}, {20657, 20661}

X(20680) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(105)

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

(1,37), (2,1500), (6,756), (31,3954), (105,20680), (238,3930), (292,20681), (365,20683), (672,20683), (1423,20684), (1931,20685), (2053,20686), (2054,20687), (3009,20688), (2112,20689), (20332,20690)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b(a + c) : c(b + a)
m(A') = -a (Sqrt[b] + Sqrt[c]) (b + c) : b (Sqrt[a] - Sqrt[c]) (a + c) : (Sqrt[a] - Sqrt[b]) (a + b) c
m(A1) = a (b + c)^2 : b (b - a) (c + a) : c (c - a)(b + a)
m(A2) = a (b + c) (b^2 + c^2) : b c (c - b) (a + c) : b c (b - c) (a + b)
m(A3) = 2 a (b + c) (a b + a c + b c) : - b (a + c) (a^2 - b c) : - c(a + b) (a^2 - b c)
m(A4) = a (b + c)^2 (a + b + c) : b (a + c) (c^2 - a b), c (a + b) (b^2 -a c)
m(A5) = -a (b + c) (2 a^2 - a b + b^2 - a c - 2 b c + c^2) : b (a + c) (-2 a^2 - a c + b c - c^2) : -(a + b) c (2 a^2 + a b + b^2 - b c)
m(A6) = a^2 (b + c) (a + b + c) (a b - b^2 + a c - c^2) : -b (a + c) (a^2 + b^2 - a c - b c) (-a^2 + a b - a c - 2 c^2) : (a + b) c (a^2 + a b + 2 b^2 - a c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (b + c) (c y + b z ) : : , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(37)), where A' = 0 : b (a + c) : c (b + a).

Let f(a,b,c,x,y,z) = b^3 (a+b)^3 (b-c) c^3 (a+c)^3 (a^2-b c) x^3-a^3 b (a+b)^2 c^2 (a+c) (b+c)^3 (a^2 b-a b^2+2 a^2 c-a c^2-b c^2) y^2 z+a^3 b^2 (a+b) c (a+c)^2 (b+c)^3 (2 a^2 b-a b^2+a^2 c-b^2 c-a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20680) lies on these lines: {37, 4068}, {55, 16550}, {65, 1018}, {1334, 3954}, {1500, 3125}, {3295, 5540}, {3675, 6184}, {3930, 20683}

X(20681) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(292)

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

See X(20680).

X(20681) lies on these lines: {1, 6651}, {37, 3122}, {42, 3952}, {190, 1911}, {192, 869}, {612, 1281}, {726, 3009}, {740, 3948}, {756, 20704}, {1962, 3121}, {3728, 4516}, {3954, 20689}, {4037, 4093}, {4094, 4155}, {17475, 20663}, {17755, 20356}, {20363, 20456}, {20683, 20688}, {20702, 20703}

X(20682) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(365)

Barycentrics a (Sqrt[b] + Sqrt[c]) (b + c) : :

See X(20680).

X(20682) lies on these lines: {1, 364}, {10, 4179}, {37, 20695}, {75, 18297}

X(20683) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(672)

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

See X(20680).

X(20683) lies on these lines: {1, 4517}, {6, 3688}, {9, 3779}, {10, 12}, {31, 5007}, {37, 4890}, {39, 869}, {42, 213}, {43, 12782}, {44, 674}, {55, 218}, {71, 4878}, {100, 2711}, {101, 17798}, {187, 18266}, {190, 6007}, {200, 3501}, {238, 9052}, {239, 14839}, {241, 1362}, {291, 2664}, {511, 1757}, {512, 661}, {518, 3717}, {524, 4553}, {527, 4014}, {594, 4111}, {612, 17750}, {650, 9320}, {668, 3978}, {672, 2223}, {692, 17796}, {740, 20694}, {756, 3954}, {872, 2092}, {908, 20544}, {984, 4260}, {1002, 5308}, {1015, 3009}, {1017, 8626}, {1018, 4433}, {1026, 4447}, {1279, 9049}, {1402, 2318}, {1458, 14626}, {1463, 5850}, {1469, 5223}, {1631, 3204}, {1743, 3056}, {1931, 3110}, {2175, 2911}, {2238, 18785}, {2245, 4557}, {2808, 9441}, {2810, 3792}, {3008, 20358}, {3059, 12723}, {3303, 16466}, {3661, 3681}, {3751, 3781}, {3789, 17308}, {3799, 6542}, {3873, 17244}, {3888, 20072}, {3930, 20680}, {3948, 3952}, {3967, 4044}, {3974, 10449}, {4259, 5220}, {4416, 17792}, {4422, 9054}, {4531, 4849}, {4661, 17230}, {5179, 5532}, {5247, 10544}, {6376, 18045}, {7322, 11518}, {9038, 17374}, {13476, 17245}, {17049, 17277}, {20681, 20688}, {20692, 20693}

X(20684) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(1423)

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

See X(20680).

X(20684) lies on these lines: {10, 20707}, {37, 20697}, {41, 3190}, {42, 213}, {55, 9447}, {226, 20706}, {306, 3948}, {1196, 3009}, {2170, 4847}, {2886, 20593}, {2887, 7239}, {3056, 20665}, {3061, 3705}, {3094, 20284}, {3688, 16588}, {3721, 18905}, {3778, 16584}, {3930, 3950}, {4876, 7081}, {5285, 19554}, {10544, 20460}, {20689, 20690}

X(20685) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(1931)

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

See X(20680).

X(20685) lies on these lines: {1, 6}, {2238, 20708}, {3125, 20360}, {3930, 20687}, {4079, 4155}, {5277, 13610}, {9509, 20369}

X(20686) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(2053)

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

See X(20680).

X(20686) lies on these lines: {37, 20699}, {3930, 3950}, {3954, 20688}

X(20687) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(2054)

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

See X(20680).

X(20687) lies on these lines: {1, 39}, {37, 20700}, {756, 20689}, {3930, 20685}, {6541, 20693}, {16589, 20710}

X(20688) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(3009)

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

See X(20680).

X(20688) lies on these lines: {10, 37}, {39, 192}, {512, 20706}, {538, 19565}, {726, 1015}, {3121, 3994}, {3175, 16584}, {3840, 17459}, {3912, 20343}, {3930, 20690}, {3954, 20686}, {4135, 16606}, {4358, 6377}, {4526, 6165}, {4704, 5283}, {4854, 16587}, {6378, 7230}, {17475, 20372}, {20681, 20683}, {20689, 20703}

X(20689) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(2112)

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

See X(20680).

X(20689) lies on these lines: {37, 4068}, {756, 20687}, {3675, 20364}, {3954, 20681}, {20684, 20690}, {20688, 20703}

X(20690) = (X(1), X(2), X(6), X(31); X(37), X(1500), X(756), X(3954)) COLLINEATION IMAGE OF X(20332)

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

See X(20680).

X(20690) lies on these lines: {10, 762}, {37, 18793}, {756, 3121}, {3930, 20688}, {20684, 20689}

X(20691) = (name pending)

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

See X(20692).

X(20691) lies on these lines: {1, 1575}, {2, 17144}, {6, 979}, {8, 1107}, {10, 37}, {32, 8715}, {35, 5291}, {39, 519}, {42, 2229}, {43, 2176}, {44, 3730}, {55, 4426}, {65, 20692}, {72, 20693}, {76, 536}, {100, 172}, {101, 1939}, {145, 2275}, {192, 4110}, {213, 1018}, {518, 3094}, {528, 7745}, {535, 7756}, {574, 8666}, {672, 3780}, {978, 16969}, {980, 17294}, {1015, 3244}, {1045, 17792}, {1100, 5105}, {1125, 1574}, {1334, 2238}, {1475, 20331}, {1573, 3626}, {1698, 9331}, {1909, 17759}, {1914, 3871}, {2136, 9575}, {2277, 17314}, {2329, 3507}, {3053, 4421}, {3125, 3970}, {3159, 4103}, {3175, 3948}, {3216, 3230}, {3434, 9596}, {3436, 9598}, {3632, 16975}, {3661, 3666}, {3679, 5283}, {3701, 4037}, {3721, 3930}, {3752, 3912}, {3811, 9620}, {3813, 3815}, {3914, 20486}, {3954, 4006}, {3961, 16519}, {3992, 4099}, {4028, 18905}, {4125, 7230}, {4261, 10449}, {4263, 17355}, {4277, 17281}, {4386, 5687}, {4531, 4849}, {4595, 16742}, {4718, 6381}, {4734, 20284}, {4850, 17230}, {4852, 17034}, {4882, 16517}, {4970, 17459}, {5013, 12513}, {5247, 17735}, {5254, 12607}, {5277, 16785}, {6184, 12640}, {6603, 9367}, {6762, 9574}, {6765, 9593}, {9592, 12629}, {11194, 15815}, {16610, 17244}, {16696, 17372}, {17351, 17499}

X(20691) = complement of X(17144)

X(20692) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(105)

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

(1,37), (2,20602), (6,210), (31,72), (105,20692), (238,20693), (292,20694), (365,20695), (672,20683), (1423,20697), (1931,20698), (2053,20699), (2054,20700), (3009,740), (2112,20701)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - a(b + c) : b(a + c) : c(b + a)
m(A') = a (Sqrt[a] + Sqrt[b] + Sqrt[c]) (b + c) : - b (Sqrt[a] + Sqrt[b] - Sqrt[c]) (a + c), - c (a + b) (Sqrt[a] - Sqrt[b] + Sqrt[c])
m(A1) = a (b + c) (a + b + c) : b (- a + b - c) (a + c) : c (- a + c - b) (a + b)
m(A2) = a (b + c) (b^2 + b c + c^2) : b (a + c) (c^2 - b c - b^2) : c (a + b) (b^2 - b c - c^2)
m(A3) = - (b + c) (a^2 + 3 a b + 3 a c + 2 b c), b (a + c) (a + b + c), c (a + b) (a + b + c)
m(A4) = a (b + c) (2 a b + b^2 + 2 a c + 3 b c + c^2) : -b (a + c) (2 a b + b^2 + b c - c^2) : - c (a + b) (-b^2 + 2 a c + b c + c^2)
m(A5) = a (b + c) (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) : -b (-a + b - c) (a + c) (a + b + c) : c (a + b) (a + b - c) (a + b + c)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a (b + c) (-x/a + y/b + c/z) : : , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(37)), where A' = -a (c + a) : b (a + c) : c (b + a).

Let f(a,b,c,x,y,z) = b^2 (a+b)^3 (b-c) c^2 (a+c)^3 (a+b+c) x^3-a^2 (a+b)^2 c (a+c) (b+c)^3 (a^2 b+a b^2+a^2 c-a c^2-2 b c^2) y^2 z+a^2 b (a+b) (a+c)^2 (b+c)^3 (a^2 b-a b^2+a^2 c-2 b^2 c+a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

(20692) lies on these lines: {37, 4068}, {65, 20691}, {72, 1018}, {910, 16550}, {1282, 20672}, {1334, 2503}, {2809, 6184}, {3125, 4646}, {3746, 5540}, {3930, 4433}, {20683, 20693}

X(20693) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(238)

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

See X(20692).

X(20693) lies on these lines: {1, 762}, {6, 3961}, {8, 9596}, {37, 42}, {44, 765}, {72, 20691}, {172, 4420}, {200, 4386}, {213, 4006}, {291, 518}, {312, 17299}, {319, 7779}, {519, 4103}, {594, 1215}, {661, 4132}, {740, 20716}, {899, 3726}, {1100, 3920}, {1255, 3723}, {1500, 3678}, {1574, 3874}, {1757, 8298}, {2276, 3681}, {2321, 4090}, {3214, 3721}, {3293, 3954}, {3509, 5524}, {3555, 16604}, {3711, 5275}, {3811, 4426}, {3936, 20483}, {3943, 3985}, {3952, 4037}, {4009, 4727}, {4015, 16589}, {4053, 20708}, {4568, 8682}, {4661, 17756}, {5529, 9259}, {5839, 20056}, {6048, 20271}, {6541, 20687}, {6542, 20529}, {9278, 20715}, {20683, 20692}, {20697, 20699}, {20698, 20700}

X(20694) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(292)

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

See X(20692).

X(20694) lies on these lines: {37, 3122}, {42, 2107}, {72, 20701}, {75, 4517}, {190, 7077}, {210, 321}, {335, 3799}, {518, 2113}, {740, 20683}, {14839, 17755}, {20714, 20715}

X(20695) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(365)

Barycentrics a (Sqrt[a] - Sqrt[b] - Sqrt[c]) (b + c) : :

See X(20692).

X(20695) lies on these lines: {37, 20682}, {210, 4179}, {364, 20673}

X(20696) = ISOGONAL CONJUGATE OF X(20525)

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

X(20696) lies on the circumcircle and these lines: {595, 14665}, {689, 4600}

X(20696) = isogonal conjugate of X(20525)
X(20696) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20525}, {513, 20352}, {514, 20372}
X(20696) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 20352}, {692, 20372}

X(20697) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(1423)

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

See X(20692).

X(20697) lies on these lines: {37, 20684}, {2321, 3967}, {3094, 3752}, {3740, 19584}, {4531, 4849}, {6184, 10440}, {20693, 20699}

X(20698) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(1931)

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

See X(20692).

X(20698) lies on these lines: {1, 6}, {2238, 20360}, {20693, 20700}

X(20699) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(2053)

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

See X(20692).

X(20699) lies on these lines: {37, 20686}, {72, 740}, {20693, 20697}

X(20700) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(2054)

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

See X(20692).

X(20700) lies on these lines: {10, 20529}, {37, 20687}, {42, 81}, {210, 3773}, {4155, 18004}, {10026, 20720}, {20693, 20698}

X(20701) = (X(1), X(2), X(6), X(31); X(37), X(20691), X(210), X(72)) COLLINEATION IMAGE OF X(2112)

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

See X(20692).

X(20701) lies on these lines: {37, 4068}, {72, 20694}, {210, 3773}, {740, 20715}, {2809, 20455}

X(20702) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(105)

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

(1,756), (2,37), (6,3954), (31,7237), (105,20702), (238,20703), (292,20704), (365,20705), (672,20706), (1423,20707), (1931,20708), (2053,20709), (2054,20710), (3009,20711), (2112,20712)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b (c + a) : c(a + b)
m(A') = a (Sqrt[b] + Sqrt[c]) (b + c) (b - Sqrt[b] Sqrt[c] + c) : - b (Sqrt[a] - Sqrt[c]) (a + c) (a + Sqrt[a] Sqrt[c] + c) : -c (Sqrt[a] - Sqrt[b]) (a + b) (a + Sqrt[a] Sqrt[b] + b)
m(A1) = 2 a b c (b + c), - b (a + c) (a^2 - b c), - c (a + b) (a^2 - b c)
m(A2) = a (b + c)^2 (b^2 - b c + c^2) : b c (a + c) (c^2 - a b) : b (a + b) c (b^2 - a c)
m(A3) = a (b + c)^2 (a b + a c + b c) : - b (a + c) (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) : - (a + b) c (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c)
m(A4) = a (b + c) (a + b + c) (b^2 + c^2) : -b (a + c) (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) : - c (a + b) (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c)
m(A5) = a (b + c) (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : b (a + c) (a^3 + a^2 b + 2 a^2 c - a b c - b c^2 + c^3) : c (a + b) (a^3 + 2 a^2 b + b^3 + a^2 c - a b c - b^2 c)
m(A6) = a (b + c) (a + b + c) (a^2 b^2 - a b^3 - b^3 c + a^2 c^2 + 2 b^2 c^2 - a c^3 - b c^3) : -b (a + c) (a^2 + b^2 - a c - b c) (-a^3 + a^2 b + a b c - 2 a c^2 - b c^2 - c^3) : c (a + b) (a^3 + 2 a b^2 + b^3 - a^2 c - a b c + b^2 c) (a^2 - a b - b c + c^2)

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a (b + c) (y + z) : : , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(37)), where A' = 0 : a + c : a + b.

Let f(a,b,c,x,y,z) = b^2 (a+b)^3 (b-c) c^2 (a+c)^3 (a^2-b c) (b^2+b c+c^2) x^3-a^2 (a+b)^2 c (a+c) (b+c)^3 (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z+a^2 b (a+b) (a+c)^2 (b+c)^3 (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20702) lies on these lines: {37, 4068}, {120, 20431}, {226, 3971}, {756, 20712}, {3675, 16593}, {16550, 20678}, {17464, 20455}, {20681, 20703}

X(20703) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(238)

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

See X(20702).

X(20703) lies on these lines: {37, 1918}, {42, 3970}, {756, 3954}, {1961, 17799}, {2643, 3943}, {3726, 20456}, {3836, 20432}, {3912, 4475}, {4118, 17243}, {4647, 6535}, {17241, 18168}, {17266, 18208}, {17267, 20274}, {17312, 18207}, {20681, 20702}, {20688, 20689}, {20707, 20709}, {20708, 20710}

X(20704) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(292)

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

See X(20702).

X(20704) lies on these lines: {37, 4368}, {756, 20681}, {2292, 3125}, {3930, 20688}, {5283, 18061}, {7237, 20712}, {17793, 20591}, {20333, 20433}, {20356, 20457}, {20706, 20711}

X(20705) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(365)

Barycentrics a (b + c) (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) : :

See X(20702).

X(20705) lies on these lines: {37, 20717}, {984, 20357}, {3773, 20485}, {20334, 20434}, {20527, 20592}

X(20706) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(672)

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

See X(20702).

X(20706) lies on these lines: {37, 65}, {75, 18055}, {226, 20684}, {244, 8620}, {335, 1959}, {390, 17452}, {512, 20688}, {518, 2170}, {740, 3930}, {984, 17451}, {1237, 4043}, {3726, 20363}, {3728, 3954}, {3774, 4642}, {3970, 3993}, {4006, 4709}, {4137, 17456}, {4645, 4876}, {4892, 7239}, {20335, 20435}, {20358, 20459}, {20681, 20702}, {20704, 20711}

X(20707) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(1423)

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

See X(20702).

X(20707) lies on these lines: {8, 2170}, {10, 20684}, {37, 65}, {756, 5360}, {1959, 17752}, {3208, 17452}, {5836, 20593}, {16886, 20494}, {20258, 20436}, {20359, 20460}, {20703, 20709}

X(20708) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(1931)

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

See X(20702).

X(20708) lies on these lines: {37, 171}, {181, 756}, {295, 14196}, {1255, 2298}, {2238, 20685}, {4037, 4071}, {4053, 20693}, {4526, 4979}, {10026, 20595}, {20337, 20437}, {20360, 20461}, {20703, 20710}

X(20709) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(2053)

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

See X(20702).

X(20709) lies on these lines: {37, 20721}, {226, 3971}, {7237, 20711}, {20338, 20438}, {20361, 20462}, {20528, 20596}, {20703, 20707}

X(20710) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(2054)

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

See X(20702).

X(20710) lies on these lines: {37, 2054}, {1655, 6625}, {3954, 20712}, {16589, 20687}, {20339, 20439}, {20362, 20463}, {20529, 20597}, {20703, 20708}

X(20711) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(3009)

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

See X(20702).

X(20711) lies on these lines: {37, 42}, {244, 20530}, {3116, 18743}, {3123, 3263}, {3701, 7148}, {3778, 3971}, {7237, 20709}, {20340, 20440}, {20363, 20464}, {20704, 20706}

X(20712) = (X(1), X(2), X(6), X(31); X(756), X(37), X(3954), X(7237)) COLLINEATION IMAGE OF X(2112)

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

See X(20702).

X(20712) lies on these lines: {37, 20724}, {756, 20702}, {3954, 20710}, {7237, 20704}, {20341, 20441}, {20364, 20465}, {20531, 20599}

X(20713) = (name pending)

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

See X(20714).

X(20713) lies on these lines: {37, 1918}, {42, 4016}, {72, 3696}, {141, 760}, {210, 8013}, {517, 3818}, {518, 4523}, {692, 5279}, {758, 4085}, {1234, 4463}, {1631, 1759}, {1824, 14973}, {2239, 4118}, {3061, 3941}, {3663, 9020}, {3681, 5564}, {3970, 4068}, {4716, 5904}, {6327, 20444}, {20716, 20724}, {20721, 20723}

X(20714) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(105)

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

(1,210), (2,37), (6,72), (31,20713), (105,20714), (238,20715), (292,20716), (365,20717), (672,20718), (1423,20719), (1931,20720), (2053,20721), (2054,20722), (3009,20723), (2112,20724)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = a (b + c) : -b (a + c) : - c (a + b)
m(A') = a (b + c) (a^(3/2) + b^(3/2) + c^(3/2)) : -b (a + c) (a^(3/2) + b^(3/2) - c^(3/2)) : - c (a + b) (a^(3/2) - b^(3/2) + c^(3/2))
m(A1) = (b + c) (a^2 + 2 b c) : -a b (a + c) : -a c (a + b)
m(A2) = a (b + c) (b^3 + a b c + c^3) : -b (a + c) (b^3 + a b c - c^3) : - c (a + b) (-b^3 + a b c + c^3)
m(A3) = -a (b + c) (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) : b (a + c) (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) : c (a + b) (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2)
m(A4) = a (b + c) (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) : -b (a + c) (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) : - c (a + b) (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)
m(A5) = a (b + c) (a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3) : b (a + c) (-a^3 + b^3 - 2 a^2 c - b^2 c + b c^2 - c^3) : - c (a + b) (a^3 + 2 a^2 b + b^3 - b^2 c + b c^2 - c^3)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a (b + c) (- x + y + z) : : , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(37)), where A' = - b - c : a + c : a + b.

Let f(a,b,c,x,y,z) = a b (a+b)^3 (b-c) c (a+c)^3 (b^2+b c+c^2) x^3-a (a+b)^2 c (a+c) (b+c)^3 (a^4+a b^3-b^2 c^2-a c^3) y^2 z+a b (a+b) (a+c)^2 (b+c)^3 (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20714) lies on these lines: {37, 4068}, {210, 15523}, {2809, 16593}, {16550, 20468}, {20344, 20445}, {20694, 20715}

X(20715) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(238)

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

See X(20714).

X(20715) lies on these lines: {10, 12}, {37, 1918}, {42, 2240}, {55, 3496}, {171, 17799}, {239, 335}, {354, 17023}, {512, 3700}, {740, 20701}, {756, 2295}, {760, 3912}, {960, 16830}, {982, 17795}, {1086, 9020}, {1824, 1840}, {1959, 4447}, {2239, 20590}, {2330, 5279}, {3509, 17798}, {3683, 16601}, {3689, 5011}, {3744, 12194}, {3869, 4517}, {3930, 4433}, {3948, 20716}, {4053, 4557}, {4645, 17789}, {5044, 19856}, {9278, 20693}, {17770, 20670}, {20694, 20714}, {20719, 20721}, {20720, 20722}

X(20716) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(292)

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

See X(20714).

X(20716) lies on these lines: {10, 115}, {37, 4368}, {149, 3706}, {190, 1281}, {210, 321}, {312, 3416}, {335, 9507}, {350, 518}, {668, 18035}, {740, 20693}, {760, 6381}, {804, 18004}, {984, 4713}, {1215, 4026}, {2795, 4568}, {3755, 4090}, {3932, 3985}, {3948, 20715}, {4009, 5057}, {8301, 17738}, {9470, 18034}, {17205, 19895}, {20345, 20446}, {20713, 20724}, {20718, 20723}

X(20717) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(365)

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

See X(20714).

X(20717) lies on these lines: {37, 20705}, {510, 20469}, {20346, 20447}

X(20718) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(672)

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

See X(20714).

X(20718) lies on these lines: {1, 3286}, {8, 3770}, {10, 15281}, {30, 511}, {37, 65}, {40, 15624}, {72, 3696}, {75, 3869}, {181, 4415}, {191, 18180}, {209, 3914}, {226, 15282}, {321, 14973}, {354, 1962}, {672, 20593}, {846, 18165}, {872, 4642}, {896, 18191}, {942, 3743}, {960, 3739}, {984, 1756}, {1046, 18178}, {1155, 3724}, {1279, 3747}, {1319, 12081}, {1385, 5496}, {1469, 17276}, {1829, 1839}, {1858, 14053}, {1959, 16728}, {2262, 3958}, {2293, 2650}, {3678, 4732}, {3681, 17163}, {3690, 3925}, {3704, 10381}, {3725, 3752}, {3742, 10180}, {3753, 19870}, {3754, 3842}, {3781, 5880}, {3812, 4698}, {3874, 4065}, {3909, 17491}, {3917, 11246}, {3928, 10439}, {3993, 4084}, {4043, 17751}, {4067, 4709}, {4553, 4645}, {5091, 5096}, {5695, 10477}, {7235, 16732}, {7957, 18673}, {11684, 18722}, {14752, 17449}, {20347, 20448}, {20367, 20470}, {20694, 20714}, {20716, 20723}

X(20719) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(1423)

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

See X(20714).

X(20719) lies on these lines: {37, 65}, {72, 4095}, {210, 5360}, {517, 3061}, {672, 20594}, {2262, 2345}, {2276, 3057}, {16969, 20358}, {20348, 20449}, {20368, 20471}, {20715, 20721}

X(20720) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(1931)

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

See X(20714).

X(20720) lies on these lines: {37, 171}, {210, 8013}, {319, 321}, {513, 4024}, {10026, 20700}, {20349, 20450}, {20369, 20472}, {20715, 20722}

X(20721) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(2053)

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

See X(20714).

X(20721) lies on these lines: {37, 20709}, {20350, 20451}, {20370, 20473}, {20713, 20723}, {20715, 20719}

X(20722) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(2054)

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

See X(20714).

X(20722) lies on these lines: {10, 20529}, {37, 2054}, {72, 20724}, {319, 4553}, {20351, 20452}, {20371, 20474}, {20715, 20720}

X(20723) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(3009)

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

See X(20714).

X(20723) lies on these lines: {37, 42}, {518, 17793}, {899, 20598}, {2388, 4103}, {4010, 4036}, {20352, 20453}, {20372, 20475}, {20713, 20721}, {20716, 20718}

X(20724) = (X(1), X(2), X(6), X(31); X(210), X(37), X(72), X(20713)) COLLINEATION IMAGE OF X(2112)

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

See X(20714).

X(20724) lies on these lines: {37, 20712}, {72, 20722}, {210, 15523}, {20353, 20454}, {20373, 20476}, {20713, 20716}

X(20725) = X(20)X(64)∩X(30)X(125)

Barycentrics (36*R^2-7*SW)*(S^2-2*SB*SC)-S^ 2*SA : :
X(20725) = X(110)-3*X(16386), 4*X(6723)-3*X(10151), X(12112)-5*X(17538)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28015.

X(20725) lies on these lines: {3, 1514}, {20, 64}, {30, 125}, {110, 15311}, {550, 10539}, {1499, 6333}, {1657, 12359}, {2071, 10117}, {2420, 15341}, {2777, 11064}, {3534, 4549}, {3564, 10990}, {5159, 13202}, {5504, 10293}, {5876, 12103}, {6723, 10151}, {12112, 17538}, {12358, 14915}

X(20725) = isogonal conjugate of X(20726)
X(20725) = X(1514)-of-ABC-X(3)-reflections-triangle

X(20726) = ISOGONAL CONJUGATE OF X(20725)

Barycentrics (SB+SC)*((36*R^2-7*SW)*(S^2-2* SA*SB)-S^2*SC)*((36*R^2-7*SW)* (S^2-2*SA*SC)-S^2*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28015.

X(20726) lies on this line: {154, 15035

X(20726) = isogonal conjugate of X(20725)

X(20727) = (name pending)

Barycentrics (b + c) (b^2 - b c + c^2) sin 2A : :

See X(20728).

X(20727) lies on these lines: {3, 9247}, {10, 14963}, {71, 73}, {2275, 3056}, {2887, 3061}, {3399, 7594}, {3721, 18905}, {3784, 20783}, {3917, 20731}, {3949, 4101}, {4020, 11573}, {4136, 7239}, {7117, 20738}, {20730, 20737}, {20734, 20736}, {20819, 20823}, {20827, 20829}

X(20727) = isogonal conjugate of polar conjugate of X(2887)
X(20727) = isotomic conjugate of polar conjugate of X(3778)
X(20727) = {X(71),X(73)}-harmonic conjugate of X(22061)

X(20728) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(105)

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

(1,71), (2,3), (6,3917), (31,20727), (105,20728), (238,20729), (292,20730), (672,20731), (1423,20732), (1931,20733), (2053,20734), (2054,20735), (3009,20736), (2112,20737), (20332,20738)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : sin 2B : sin 2C
m(A') = - (Sqrt[b] + Sqrt[c]) (b - Sqrt[b] Sqrt[c] + c) sin 2A : (Sqrt[a] - Sqrt[c]) (a + Sqrt[a] Sqrt[c] + c) sin 2B : (Sqrt[a] - Sqrt[b]) (a + Sqrt[a] Sqrt[b] + b) sin 2C
m(A1) = 2 b c sin 2A : (a^2 - b c) sin 2B : (a^2 -b c) sin 2C
m(A2) = (b + c) (b^2 - b c + c^2) sin 2A: c (c^2 - a b) sin 2B : b (b^2 - a c) sin 2C
m(A3) = (b + c) (a b + a c + b c) sin 2A : - (a^3 + a^2 b + a^2 c - a b c - a c^2 - b c^2) sin 2B : - (a^3 + a^2 b - a b^2 + a^2 c - a b c - b^2 c) sin 2C
m(A4) = - (a + b + c) (b^2 + c^2) sin 2A : (a^2 b + a^2 c + a b c - a c^2 - b c^2 - c^3) sin 2B : (a^2 b - a b^2 - b^3 + a^2 c + a b c - b^2 c) sin 2C

where A1, A2, A3, A4 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (y + z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(3)), where A' = 0 : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^5 (b-c) c^5 (a^2-b c) (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 (b^2+b c+c^2) x^3+a^5 b c^3 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (-2 a^3 b^3+3 a^4 b c+a b^4 c-a^3 c^3-b^3 c^3) y^2 z-a^5 b^3 c (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (-a^3 b^3+3 a^4 b c-2 a^3 c^3-b^3 c^3+a b c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20728) lies on these lines: {3, 906}, {39, 2280}, {71, 20737}, {77, 2197}, {665, 1642}, {1292, 8751}, {2293, 3778}, {18591, 20776}, {20729, 20731}

X(20728) = isogonal conjugate of polar conjugate of X(120)
X(20728) = isotomic conjugate of polar conjugate of X(20455)

X(20729) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(238)

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

See X(20728).

X(20729) lies on these lines: {3, 9247}, {48, 7293}, {71, 3917}, {3937, 20785}, {3949, 4001}, {20728, 20731}, {20732, 20734}, {20733, 20735}, {20736, 20738}, {20757, 20758}

X(20730) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(292)

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

See X(20728).

X(20730) lies on these lines: {3, 20742}, {71, 20750}, {1818, 20757}, {3917, 20822}, {3937, 20778}, {20727, 20737}, {20731, 20736}, {20820, 20821}

X(20731) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(672)

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

See X(20728).

X(20731) lies on these lines: {3, 73}, {77, 20753}, {1401, 18758}, {1458, 17798}, {1814, 1818}, {2635, 6996}, {2654, 13727}, {3917, 20727}, {3937, 20777}, {20728, 20729}, {20730, 20736}

X(20732) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(1423)

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

See X(20728).

X(20732) lies on these lines: {3, 73}, {63, 20753}, {71, 3289}, {1364, 20737}, {1808, 1812}, {1936, 4203}, {2654, 4195}, {5145, 14547}, {20729, 20734}

X(20732) = isogonal conjugate of polar conjugate of X(20258)
X(20732) = isotomic conjugate of polar conjugate of X(20460)

X(20733) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(1931)

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

See X(20728).

X(20733) lies on these lines: {3, 20746}, {71, 73}, {125, 914}, {7193, 20754}, {20729, 20735}

X(20733) = isogonal conjugate of polar conjugate of X(20337)
X(20733) = isotomic conjugate of polar conjugate of X(20461)

X(20734) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(2053)

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

See X(20728).

X(20734) lies on these lines: {77, 2197}, {18591, 20783}, {20727, 20736}, {20729, 20732}

X(20734) = isogonal conjugate of polar conjugate of X(20338)
X(20734) = isotomic conjugate of polar conjugate of X(20462)

X(20735) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(2054)

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

See X(20728).

X(20735) lies on these lines: {3, 17972}, {3917, 20737}, {20729, 20733}

X(20735) = isogonal conjugate of polar conjugate of X(20339)
X(20735) = isotomic conjugate of polar conjugate of X(20463)

X(20736) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(3009)

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

See X(20728).

X(20736) lies on these lines: {3, 48}, {2524, 3049}, {20727, 20734}, {20729, 20738}, {20730, 20731}, {20737, 20821}

X(20736) = isogonal conjugate of polar conjugate of X(20340)
X(20736) = isotomic conjugate of polar conjugate of X(20464)

X(20737) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(2112)

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

See X(20728).

X(20737) lies on these lines: {3, 20748}, {71, 20728}, {1364, 20732}, {3917, 20735}, {20727, 20730}, {20736, 20821}

X(20737) = isogonal conjugate of polar conjugate of X(20341)
X(20737) = isotomic conjugate of polar conjugate of X(20465)

X(20738) = (X(1), X(2), X(6), X(31); X(71), X(3), X(3917), X(20727)) COLLINEATION IMAGE OF X(20332)

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

See X(20728).

X(20738) lies on these lines: {63, 295}, {1364, 20732}, {7117, 20727}, {20729, 20736}

X(20738) = isogonal conjugate of polar conjugate of X(20343)
X(20738) = isotomic conjugate of polar conjugate of X(20467)

X(20739) = (name pending)

Barycentrics (a^3 - b^3 - c^3) sin 2A : :

See X(20728).

X(20739) lies on these lines: {3, 9247}, {6, 10}, {48, 11573}, {219, 3157}, {394, 4001}, {4456, 14529}, {4574, 7078}, {6327, 17904}, {20742, 20748}, {20806, 20811}, {20815, 20817}

X(20739) = isogonal conjugate of polar conjugate of X(6327)
X(20739) = isotomic conjugate of polar conjugate of X(1631)
X(20739) = X(19)-isoconjugate of X(7357)
X(20739) = X(92)-isoconjugate of X(7087)

X(20740) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(105)

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

(1,219), (2,3), (6,394), (31,20739), (105,20740), (238,20741), (292,20742), (365,20743), (672,20744), (1423,20745), (1931,20746), (3009,20747), (2112,20748)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - sin 2A : sin 2B : sini 2C
m(A') = - (a^(3/2) + b^(3/2) + c^(3/2)) sin 2A : (a^(3/2) + b^(3/2) - c^(3/2)) sin 2B : (a^(3/2) - b^(3/2) + c^(3/2)) sin 2C
m(A1) = - (a^2 + 2 b c) sin 2A : sin 2B : sin 2C
m(A2) = - (b^3 + a b c + c^3) sin 2A: (b^3 + a b c - c^3) sin 2B : (c^3 + a b c - b^3) sin 2C
m(A3) = (a^3 + a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2) sin 2A : - (a^3 + a^2 b + a b^2 + a^2 c + b^2 c - a c^2 - b c^2) sin 2B : - (a^3 + a^2 b - a b^2 + a^2 c - b^2 c + a c^2 + b c^2) sin 2C
m(A4) = - (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) sin 2A : (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) sin 2B : (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) sin 2C
m(A5) = (a^3 - b^3 + 2 a b c + b^2 c + b c^2 - c^3) sin 2A : (-a^3 + b^3 - 2 a^2 c - b^2 c + b c^2 - c^3) sin 2B : (-a^3 - 2 a^2 b - b^3 + b^2 c - b c^2 + c^3) sin 2C

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (-x + y + z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(3)), where A' = - sin 2A : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^3 (b-c) c^3 (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 (b^2+b c+c^2) x^3+a^3 c^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (a^4+a b^3-b^2 c^2-a c^3) y^2 z-a^3 b^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (a^4-a b^3-b^2 c^2+a c^3) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20740) lies on these lines: {3, 906}, {6, 3939}, {219, 20748}, {4574, 7078}, {9605, 13006}, {20741, 20744}

X(20741) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(238)

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

See X(20740).

X(20741) lies on these lines: {3, 9247}, {6, 43}, {48, 3784}, {63, 77}, {71, 3955}, {295, 17972}, {521, 2522}, {579, 1397}, {1324, 14963}, {1758, 2323}, {1818, 20761}, {2201, 15310}, {3509, 5018}, {17798, 18262}, {20740, 20744}, {20747, 20809}, {20769, 20770}

X(20741) = isogonal conjugate of polar conjugate of X(4645)
X(20741) = isotomic conjugate of polar conjugate of X(17798)
X(20741) = X(19)-isoconjugate of X(7261)
X(20741) = X(92)-isoconjugate of X(8852)

X(20742) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(292)

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

See X(20740).

X(20742) lies on these lines: {3, 20730}, {6, 17755}, {63, 17972}, {69, 219}, {394, 20809}, {17976, 20769}, {20739, 20748}, {20744, 20747}, {20807, 20808}

X(20742) = isogonal conjugate of polar conjugate of X(20345)
X(20742) = isotomic conjugate of polar conjugate of X(8301)
X(20742) = X(19)-isoconjugate of X(2113)

X(20743) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(365)

Barycentrics (a^(3/2) - b^(3/2) - c^(3/2)) sin 2A : :

See X(20740).

X(20743) lies on these lines: (none)

X(20744) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(672)

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

See X(20740).

X(20744) lies on these lines: {3, 73}, {6, 142}, {394, 4001}, {651, 6996}, {905, 4131}, {1814, 20811}, {3562, 13727}, {20740, 20741}, {20742, 20747}

X(20744) = isotomic conjugate of polar conjugate of X(20470)

X(20745) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(1423)

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

See X(20740).

X(20745) lies on these lines: {3, 73}, {219, 3289}, {3562, 4195}

X(20746) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(1931)

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

See X(20740).

X(20746) lies on these lines: {3, 20733}, {219, 3157}, {656, 7254}, {17972, 20800}

X(20747) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(3009)

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

See X(20740).

X(20747) lies on these lines: {3, 48}, {6, 726}, {525, 3049}, {20741, 20809}, {20742, 20744}, {20748, 20808}

X(20748) = (X(1), X(2), X(6), X(31); X(219), X(3), X(394), X(20739)) COLLINEATION IMAGE OF X(2112)

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

See X(20740).

X(20748) lies on these lines: {3, 20737}, {219, 20740}, {20739, 20742}, {20747, 20808}

X(20749) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(105)

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

(1,3), (2,228), (6,71), (31,3917), (105,20749), (238,1818), (292,20750), (365,20751), (672,20752), (1423,20753), (1931,20754), (2053,20755), (2054,20756), (3009, 20757), (2112,20758), (20332,20759)

Twenty-four more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : sin 2B : sini 2C
m(A') = - (Sqrt[b] + Sqrt[c]) sin 2A : (Sqrt[a] - Sqrt[c]) sin 2B : (Sqrt[a] - Sqrt[b]) sin 2C
m(A1) = - (b + c) sin 2A : (a - b) sin2B : (a - c) sin 2C
m(A2) = - (b^2 + c^2) sin 2A : c (b - c) sin 2B : b (c - b) sin 2C
m(A3) = - 2 (a b + a c + b c) sin 2A : (a^2 - b c) sin 2B : (a^2 - b c) sin 2C
m(A4) = (b + c) (a + b + c) sin 2A : (c^2 - a b) sin 2B : c^2 (b^2 - a c) sin 2C
m(A5) = - (2 a^2 - a b + b^2 - a c - 2 b c + c^2) sin 2A : (-2 a^2 - a c + b c - c^2) sin 2B : (-2 a^2 - a b - b^2 + b c) sin 2C
m(A6) = - a (a + b + c) (a b - b^2 + a c - c^2) sin 2A : (a^2 + b^2 - a c - b c) (-a^2 + a b - a c - 2 c^2) sin 2B : (-a^2 - a b - 2 b^2 + a c) (a^2 - a b - b c + c^2) sin 2C

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (c y + b z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(2); A'B'C',X(3)), where A' = 0 : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^6 (b-c) c^6 (a^2-b c) (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 x^3+a^6 b^2 c^4 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (a^2 b-a b^2+2 a^2 c-a c^2-b c^2) y^2 z-a^6 b^4 c^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (2 a^2 b-a b^2+a^2 c-b^2 c-a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20749) lies on these lines: {3, 906}, {212, 1364}, {222, 1331}, {228, 3937}, {1279, 3021}, {1818, 20752}

X(20749) = isogonal conjugate of polar conjugate of X(16593)

X(20750) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(292)

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

See X(20749).

X(20750) lies on these lines: {3, 295}, {48, 1332}, {71, 20730}, {2638, 20753}, {3917, 20758}, {7117, 20822}, {8850, 17475}, {20728, 20729}, {20752, 20757}, {20777, 20785}

X(20750) = isogonal conjugate of polar conjugate of X(17793)

X(20751) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[b] + Sqrt[c]) sin 2A : :

See X(20749).

X(20751) lies on these lines: {3, 20763}, {27, 20527}, {57, 367}, {58, 20664}

X(20752) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(672)

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

See X(20749).

X(20752) lies on these lines: {6, 354}, {41, 1496}, {48, 184}, {63, 77}, {71, 3917}, {101, 1951}, {112, 2749}, {213, 1468}, {295, 7193}, {518, 5089}, {520, 647}, {603, 1802}, {604, 5364}, {651, 10025}, {672, 1362}, {968, 2256}, {1455, 4559}, {1818, 20749}, {2196, 20777}, {2200, 4020}, {2223, 9455}, {2280, 20229}, {2284, 3693}, {2300, 3051}, {2323, 3509}, {2333, 16980}, {3063, 5098}, {3167, 20760}, {3230, 17439}, {4574, 5440}, {7078, 7124}, {17976, 20761}, {20750, 20757}, {20762, 20769}

X(20752) = X(19)-isoconjugate of X(2481)
X(20752) = X(92)-isoconjugate of X(105)
X(20752) = crossdifference of every pair of points on line X(4)X(885)
X(20752) = isogonal conjugate of polar conjugate of X(518)
X(20752) = isotomic conjugate of polar conjugate of X(2223)

X(20753) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(1423)

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

See X(20749).

X(20753) lies on these lines: {3, 7015}, {48, 184}, {55, 1964}, {63, 20732}, {77, 20731}, {326, 1040}, {560, 2361}, {869, 2330}, {1818, 4855}, {2274, 2646}, {2275, 3056}, {2638, 20750}, {3270, 20758}

X(20753) = X(92)-isoconjugate of X(7132)
X(20753) = isogonal conjugate of polar conjugate of X(3061)

X(20754) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(1931)

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

See X(20749).

X(20754) lies on these lines: {3, 49}, {647, 810}, {1818, 20756}, {3690, 15377}, {7193, 20733}

X(20754) = isogonal conjugate of polar conjugate of X(10026)

X(20755) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(2053)

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

See X(20749).

X(20755) lies on these lines: {3, 20767}, {73, 295}, {1818, 4855}, {3917, 20757}

X(20756) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(2054)

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

See X(20749).

X(20756) lies on these lines: {3, 20768}, {71, 20758}, {228, 295}, {1818, 20754}

X(20757) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(3009)

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

See X(20749).

X(20757) lies on these lines: {3, 63}, {1818, 20730}, {3917, 20755}, {7117, 20821}, {20729, 20758}, {20750, 20752}

X(20758) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(2112)

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

See X(20749).

X(20758) lies on these lines: {3, 906}, {71, 20756}, {1813, 2197}, {3270, 20753}, {3917, 20750}, {20729, 20757}

X(20759) = (X(1), X(2), X(6), X(31); X(3), X(228), X(71), X(3917)) COLLINEATION IMAGE OF X(20332)

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

See X(20749).

X(20759) lies on these lines: {63, 295}, {71, 20787}, {1818, 20730}, {3270, 20753}

X(20760) = (name pending)

Barycentrics (a b + a c - b c) sin 2A: :

See X(20761).

X(20760) lies on these lines: {2, 20256}, {3, 63}, {6, 9017}, {7, 16056}, {8, 9840}, {9, 16058}, {41, 11328}, {43, 1403}, {48, 3955}, {55, 846}, {57, 16059}, {100, 11689}, {101, 9306}, {184, 1331}, {198, 3509}, {212, 7193}, {218, 5364}, {219, 7015}, {222, 295}, {329, 4192}, {394, 17976}, {497, 15507}, {511, 3190}, {518, 3185}, {851, 5905}, {859, 5208}, {894, 11358}, {908, 19540}, {968, 3295}, {1011, 3219}, {1215, 1376}, {1402, 3751}, {1707, 2223}, {1818, 3784}, {1824, 20430}, {1985, 20242}, {2200, 3504}, {2318, 3781}, {2352, 4641}, {2783, 17860}, {3157, 7016}, {3167, 20752}, {3173, 17975}, {3191, 10441}, {3218, 4191}, {3306, 16409}, {3868, 13738}, {4199, 17257}, {4203, 17350}, {4245, 15934}, {4385, 5687}, {4640, 15624}, {5223, 10434}, {5273, 8731}, {5437, 16421}, {5708, 16414}, {5748, 19546}, {6147, 16415}, {6745, 20498}, {7078, 20803}, {7124, 20812}, {7580, 10025}, {9318, 16379}, {12649, 13724}, {15650, 16287}, {16777, 18185}, {17441, 20254}

X(20760) = isogonal conjugate of polar conjugate of X(192)
X(20760) = isotomic conjugate of polar conjugate of X(2176)
X(20760) = crossdifference of every pair of points on line X(814)X(6591)
X(20760) = X(19)-isoconjugate of X(330)
X(20760) = X(92)-isoconjugate of X(2162)
X(20760) = {X(23161),X(23162)}-harmonic conjugate of X(23158)

X(20761) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(105)

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

(1,3), (2,20760), (6,219), (31,394), (105,20761), (238,17976), (292,20762), (365,20763), (672,20753), (1423,20765), (1931,20766), (2053,20767), (2054,20768), (3009, 20769), (2112,20770)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - sin 2A : sin 2B : sini 2C
m(A') = - (Sqrt[a] + Sqrt[b] + Sqrt[c]) sin 2A : (Sqrt[a] + Sqrt[b] - Sqrt[c]) sin 2B : (Sqrt[a] - Sqrt[b] + Sqrt[c]) sin 2C
m(A1) = - (a + b + c) sin 2A : (a - b + c) sin 2B : (a + b - c) sin 2C
m(A2) = - (b^2 + b c + c^2) sin 2A : (b^2 + b c - c^2) sin 2B : (c^2 + b c - b^2) sin 2C
m(A3) = (a^2 + 3 a b + 3 a c + 2 b c) sin 2A : - b c (a + b + c) sin 2B : - b c (a + b + c) sin 2C
m(A4) = - (2 a b + b^2 + 2 a c + 3 b c + c^2) sin 2A : (2 a b + b^2 + b c - c^2) sin 2B : (-b^2 + 2 a c + b c + c^2) sin 2C
m(A5) = - (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) sin 2A : (-a + b - c) (a + b + c) sin 2B : (-a + c - b) (a + b + c) sin 2C
m(A6) = - (a^2 - b c) (a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) sin 2A : (a^4 - 2 a^3 b + a^2 b^2 - a^2 b c + 4 a b^2 c + b^3 c + a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 3 b c^3) sin 2B : (a^4 + a^2 b^2 - 2 a b^3 - 2 a^3 c - a^2 b c - 2 a b^2 c - 3 b^3 c + a^2 c^2 + 4 a b c^2 + 2 b^2 c^2 + b c^3) sin 2C

where A1, A2, A3, A4, A5, A6 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (-x/a + y/b + z/c) sin 2A : : , and m is the collineation indicated by (A,B,C,X(1); A'B'C',X(3)), where A' = -sin 2A : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^5 (b-c) c^5 (a+b+c) (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 x^3+a^5 b c^3 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (a^2 b+a b^2+a^2 c-a c^2-2 b c^2) y^2 z-a^5 b^3 c (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (a^2 b-a b^2+a^2 c-2 b^2 c+a c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 11, 2018)

X(20761) lies on these lines: {3, 906}, {43, 8298}, {48, 20786}, {212, 3781}, {219, 20778}, {222, 295}, {394, 1260}, {991, 14827}, {1282, 2114}, {1818, 20741}, {17976, 20752}

X(20762) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(292)

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

See X(20749).

X(20762) lies on these lines: {3, 295}, {69, 219}, {394, 20770}, {2284, 20672}, {20740, 20741}, {20752, 20769}, {20785, 20796}

X(20763) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(365)

Barycentrics (Sqrt[a] - Sqrt[b] - Sqrt[c]) sin 2A : :

See X(20749).

X(20763) lies on thei line: {3, 20751}

X(20764) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(4)

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

See X(20749).

X(20764) lies on these lines: {1, 3}, {8, 856}, {109, 6759}, {155, 17975}, {222, 20803}, {243, 6985}, {255, 8763}, {273, 3149}, {296, 1069}, {495, 18641}, {653, 1075}, {851, 1068}, {1092, 1813}, {1148, 1816}, {1410, 18446}, {1870, 13738}, {2055, 7335}, {2655, 8757}, {3157, 7016}, {3487, 6349}, {6056, 6760}, {6638, 20805}, {7049, 8762}, {7066, 14059}, {7515, 15325}, {8555, 19763}, {11374, 17073}, {11700, 15654}, {15905, 20818}

X(20765) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(1423)

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

See X(20749).

X(20765) lies on these lines:

X(20766) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(1931)

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

See X(20749).

X(20766) lies on these lines: {3, 49}, {17976, 20768}

X(20767) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(2053)

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

See X(20749).

X(20767) lies on these lines: {3, 20755}, {219, 12215}, {222, 20801}, {394, 7124}, {7078, 17976}

X(20768) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(2054)

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

See X(20749).

X(20768) lies on these lines: {3, 20756}, {48, 4558}, {219, 20770}, {17976, 20766}

X(20769) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(3009)

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

See X(20749).

X(20769) lies on these lines: {1, 19310}, {2, 41}, {3, 63}, {7, 1958}, {9, 16367}, {36, 18206}, {48, 69}, {57, 11329}, {75, 18162}, {81, 1193}, {100, 2340}, {101, 2862}, {141, 2174}, {171, 869}, {193, 604}, {213, 5337}, {222, 7364}, {238, 2210}, {239, 385}, {241, 17966}, {284, 4357}, {297, 2202}, {306, 1799}, {319, 18042}, {320, 662}, {326, 7289}, {350, 2201}, {394, 7124}, {518, 17798}, {524, 7113}, {572, 4416}, {584, 4657}, {905, 4131}, {908, 6996}, {936, 19314}, {942, 19329}, {1580, 3783}, {1790, 4001}, {1812, 7116}, {1814, 1818}, {1959, 4511}, {2185, 6626}, {2187, 10565}, {2196, 20785}, {2239, 18266}, {2268, 17257}, {2271, 5256}, {2278, 4643}, {2323, 3882}, {2327, 18650}, {2329, 3661}, {3204, 17279}, {3218, 19308}, {3220, 16876}, {3306, 16412}, {3570, 3975}, {3666, 16519}, {3693, 20672}, {4251, 17023}, {4289, 17325}, {4303, 14868}, {5249, 16054}, {5294, 16061}, {6734, 6998}, {7120, 9308}, {7175, 17364}, {7269, 17868}, {8301, 20358}, {9310, 17316}, {9318, 16381}, {14953, 20347}, {1650 3, 17397}, {16738, 18724}, {16788, 17308}, {17976, 20742}, {20741, 20770}, {20752, 20762}

X(20769) = isogonal conjugate of polar conjugate of X(350)
X(20769) = isotomic conjugate of polar conjugate of X(238)
X(20769) = X(19)-isoconjugate of X(291)
X(20769) = X(92)-isoconjugate of X(1911)
X(20769) = crossdifference of every pair of points on line X(1824)X(6591) (the line through the polar conjugates of PU(10))

X(20770) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(2112)

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

See X(20749).

X(20770) lies on these lines: {3, 906}, {6, 292}, {219, 20768}, {394, 20762}, {1813, 2286}, {20741, 20769}

X(20771) = MIDPOINT OF X(24) AND X(110)

Barycentrics (SB+SC)*(S^2-3*SB*SC)*((14*R^ 2-3*SW)*S^2-(6*R^2-SW)*SA^2) : :
X(20771) = X(74)-3*X(15078), X(11413)-3*X(15035), 3*X(14643)-X(18404)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28019.

X(20771 ) lies on these lines: {3, 9934}, {24, 110}, {25, 5504}, {30, 113}, {49, 16222}, {74, 15078}, {125, 12134}, {156, 14708}, {159, 15462}, {186, 12825}, {235, 17702}, {399, 8780}, {974, 6644}, {1112, 1147}, {1204, 5663}, {1899, 12419}, {1974, 14984}, {3043, 12824}, {5609, 11562}, {5972, 10282}, {6593, 11597}, {6642, 13198}, {7506, 11746}, {7723, 18350}, {9306, 12358}, {10111, 13567}, {11413, 15035}, {12041, 12162}, {12106, 12236}, {12133, 12901}, {12900, 18475}, {14643, 18404}, {15063, 17701}, {18474, 20304}

X(20771) = midpoint of X(24) and X(110)
X(20771) = reflection of X(125) in X(16238)
X(20771) = center of the circle through {X(24), X(110), X(1301), X(7471)}
X(20771) = {X(9306), X(13289)}-harmonic conjugate of X(12358)

X(20772) = MIDPOINT OF X(25) AND X(110)

Barycentrics (SB+SC)*(S^2-3*SB*SC)*((24*R^ 2-5*SW)*S^2-SA^2*SW) : :
X(20772) = X(5609)+2*X(12106), 3*X(14643)-X(18531)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28019.

X(20772 ) lies on these lines: {25, 110}, {30, 113}, {125, 6677}, {154, 15462}, {468, 12827}, {542, 13567}, {1092, 16105}, {1368, 5972}, {1595, 15115}, {1596, 17702}, {1625, 2502}, {1660, 15580}, {1995, 12099}, {2063, 9909}, {2393, 6593}, {2781, 9306}, {2854, 19136}, {3564, 12828}, {5020, 5622}, {5609, 6102}, {5651, 12041}, {5663, 6644}, {6642, 16270}, {7529, 15465}, {9140, 10546}, {10117, 13416}, {10294, 15741}, {10601, 13198}, {11441, 13148}, {14643, 18531}, {15087, 16222}, {15116, 16977}

X(20772) = midpoint of X(25) and X(110)
X(20772) = reflection of X(i) in X(j) for these (i,j): (125, 6677), (1368, 5972)
X(20772) = center of the circle through {X(25), X(110), X(7471), X(9064)}
X(20772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1495, 5642, 16165), (5642, 16165, 1511)

X(20773) = MIDPOINT OF X(26) AND X(110)

Barycentrics (SB+SC)*(S^2-3*SB*SC)*((13*R^ 2-3*SW)*S^2-(3*R^2-SW)*SA^2) : :
X(20773) = X(74)-3*X(18324), 3*X(154)+X(2931), 9*X(154)-X(17838), X(265)-3*X(10201), X(399)+3*X(14070), 3*X(2931)+X(17838), X(5609)+2*X(12107), 3*X(11202)-X(12901), X(12084)-3*X(15035), X(12085)-5*X(15040), X(12302)-5*X(17821), X(12596)-3*X(19153), 3*X(14643)-X(18569)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28019.

X(20773 ) lies on these lines: {3, 12292}, {5, 12140}, {23, 3043}, {24, 14708}, {25, 12228}, {26, 110}, {30, 113}, {74, 18324}, {125, 10020}, {154, 2931}, {159, 19138}, {184, 12236}, {206, 14984}, {265, 10201}, {399, 14070}, {1658, 5663}, {1986, 2070}, {2854, 19154}, {3518, 16222}, {5609, 12107}, {5944, 10113}, {5972, 13371}, {7488, 7723}, {7502, 12358}, {7517, 15463}, {7525, 13416}, {7530, 15472}, {7556, 12219}, {7687, 18475}, {9714, 19504}, {10117, 15132}, {10226, 15030}, {10282, 15761}, {10533, 12892}, {10534, 12891}, {10540, 12825}, {10575, 12041}, {10733, 11464}, {11202, 12901}, {11597, 12824}, {12084, 15035}, {12085, 15040}, {12133, 18570}, {12295, 13367}, {12302, 17821}, {12596, 19153}, {14643, 18569}, {16111, 17701}

X(20773) = midpoint of X(i) and X(j) for these {i,j}: {26, 110}, {159, 19138}, {10117, 15132}
X(20773) = reflection of X(125) in X(10020)
X(20773) = center of the circle through {X(26), X(110), X(7471)}

X(20774) = X(4)X(542)∩X(25)X(6054)

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

See Tran Quang Hung and Randy Hutson, Hyacinthos 28021.

X(20774) lies on these lines: {4,542}, {25,6054}, {98,275}, {99,317}, {107,11005}, {114,6353}, {115,3087}, {147,6995}, {250,403}, {297,5182}, {393,5477}, {1304,16933}, {1596,6033}, {1632,5877}, {2782,18494}, {2790,5186}, {3088,10991}, {5984,7409}, {6034,6749}, {6055,8889}, {6748,11646}, {7378,11177}, {7487,14981}, {7577,14061}, {14639,18386}

X(20774) = perspector of ABC and cross-triangle of 3rd and 4th isodynamic-Dao triangles

X(20775) = (name pending)

Trilinears    sin 2A sin(A +ω) : :
Barycentrics    a^2 (b^2 + c^2) sin 2A : :
Barycentrics    a^4 (b^2 + c^2) (b^2 + c^2 - a^2) : :

See X(20776).

X(20775) lies on these lines: {3, 69}, {6, 160}, {22, 7774}, {25, 3087}, {39, 1843}, {48, 2196}, {71, 20777}, {95, 98}, {99, 9230}, {141, 1634}, {159, 3148}, {184, 418}, {206, 5063}, {216, 6467}, {217, 4173}, {228, 20785}, {264, 11257}, {311, 2782}, {317, 9744}, {325, 7467}, {417, 13367}, {427, 16030}, {524, 8266}, {570, 2393}, {1176, 4558}, {1974, 5065}, {3001, 17710}, {3135, 11402}, {3186, 7709}, {3269, 6751}, {3491, 14133}, {3618, 11328}, {3629, 5201}, {3796, 10607}, {5013, 9924}, {5106, 6375}, {5305, 11360}, {5421, 9969}, {6636, 7779}, {7485, 16990}, {7669, 15109}, {7738, 11325}, {9233, 14567}, {9973, 13351}, {11171, 11188}, {11574, 20819}, {13188, 18354}, {13334, 14913}, {15143, 17907}, {15905, 19125}, {16872, 17798}, {17423, 18475}, {20780, 20781}

X(20775) = crosssum of X(4) and X(264)
X(20775) = crosspoint of X(3) and X(184)
X(20775) = X(83)-isoconjugate of X(92)
X(20775) = crossdifference of every pair of points on line X(2489)X(4580)
X(20775) = isogonal conjugate of polar conjugate of X(39)

X(20776) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(105)

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

(1,228), (2,20775), (6,3), (31,71), (105,20776), (238,20777), (292,20778), (365,20779), (672,20780), (1423,20781), (1931,20782), (2053,20783), (2054,20784, (3009, 20785), (2112,20786), (20332,20787)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : sin 2B : sini 2C
m(A') = - a^(1/2) (Sqrt[b] + Sqrt[c]) sin 2A : b^(1/2) (Sqrt[a] - Sqrt[c]) sin 2B : c^(1/2)(Sqrt[a] - Sqrt[b]) sin 2C
m(A1) = (b^2 + c^2) sin 2A : b (b - c) sin 2B : c (c - b) (a^2 + b^2 - c^2) sin 2C
m(A2) = a (b + c) sin 2A : c (a - b) sin 2B : b (a - c) sin 2C
m(A3) = (b + c)(a b + c a + b c) sin 2A : b (c^2 - a b) sin 2B : c (b^2 - a c) sin 2C
m(A4) = 2 a (a + b + c) sin 2A : (a^2 - b c) sin 2B : (a^2 - b c) sin 2C
m(A5) = - a (a b - b^2 + a c - c^2) sin 2A : b (-a^2 + a b - a c - 2 c^2) sin 2B : c (-a^2 - a b - 2 b^2 + a c) sin 2C

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (c^2 y + b^2 z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(6); A'B'C',X(3)), where A' = 0 : sin 2B : sin 2C.

X(20776) lies on these lines: {3, 1814}, {48, 2196}, {71, 3270}, {228, 3937}, {583, 3779}, {2223, 20455}, {18591, 20728}, {20777, 20780}

X(20776) = X(92)-isoconjugate of X(6185)
X(20776) = isogonal conjugate of polar conjugate of X(6184)

X(20777) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(238)

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

See X(20776).

X(20777) lies on these lines: {3, 63}, {31, 18758}, {55, 17448}, {71, 20775}, {184, 15373}, {212, 20781}, {237, 672}, {810, 822}, {854, 899}, {908, 19522}, {2196, 20752}, {3937, 20731}, {20750, 20785}, {20776, 20780}, {20782, 20784}

X(20777) = isogonal conjugate of polar conjugate of X(1575)
X(20777) = isotomic conjugate of polar conjugate of X(21760)
X(20777) = X(19)-isoconjugate of X(32020)
X(20777) = X(92)-isoconjugate of X(20332)

X(20778) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(292)

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

See X(20776).

X(20778) lies on these lines: {3, 295}, {63, 212}, {71, 11574}, {219, 20761}, {603, 6517}, {846, 16579}, {1282, 3939}, {1818, 20749}, {3509, 13329}, {3937, 20730}, {20780, 20785}

X(20778) = isogonal conjugate of polar conjugate of X(17755)

X(20779) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(365)

Barycentrics a^(1/2) (Sqrt[b] + Sqrt[c]) sin 2A : :

See X(20776).

X(20779) lies on these lines:

X(20780) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(672)

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

See X(20776).

X(20780) lies on these lines: {3, 20798}, {56, 20673}, {104, 4180}

X(20780) = isogonal conjugate of polar conjugate of X(3008)

X(20781) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(1423)

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

See X(20776).

X(20781) lies on these lines: {3, 48}, {36, 1618}, {42, 1626}, {212, 1473}, {238, 1633}, {672, 5096}, {692, 1458}, {971, 2265}, {991, 2317}, {1279, 8647}, {1459, 1946}, {1471, 1486}, {2183, 3220}, {2261, 5732}, {2267, 5085}, {2269, 4265}, {3100, 16560}, {14547, 16064}, {20775, 20781}, {20776, 20777}, {20778, 20785}

X(20782) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(1931)

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

See X(20776).

X(20782) lies on these lines: {3, 20800}, {71, 228}, {20777, 20784}

X(20783) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(2053)

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

See X(20776).

X(20783) lies on these lines: {3, 7116}, {63, 69}, {212, 20777}, {222, 2196}, {3094, 20665}, {3784, 20727}, {18591, 20734}

X(20784) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(2054)

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

See X(20776).

X(20784) lies on these lines: {3, 17972}, {71, 4558}, {20777, 20782}

X(20785) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(3009)

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

See X(20776).

X(20785) lies on these lines: {6, 20284}, {9, 3840}, {48, 3955}, {63, 69}, {72, 4020}, {219, 20801}, {228, 20775}, {295, 1818}, {518, 1755}, {520, 647}, {1282, 2272}, {1463, 1575}, {2179, 3555}, {2183, 3509}, {2196, 20769}, {3169, 16557}, {3912, 20610}, {3937, 20729}, {7193, 20797}, {8608, 9026}, {8844, 9025}, {16973, 20665}, {17976, 20804}, {20750, 20777}, {20762, 20796}, {20778, 20780}

X(20785) = isotomic conjugate of polar conjugate of X(3009)
X(20785) = X(19)-isoconjugate of X(3226)

X(20786) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(2112)

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

See X(20776).

X(20786) lies on these lines: {3, 17972}, {48, 20761}, {71, 11574}, {228, 3937}, {295, 1818}, {1331, 5314}, {1797, 1810}, {3781, 20797}, {20781, 20787}

X(20787) = (X(1), X(2), X(6), X(31); X(228), X X(20775), X(3), X(71)) COLLINEATION IMAGE OF X(20332)

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

See X(20776).

X(20787) lies on these lines: {3, 4561}, {71, 20759}, {20750, 20777}, {20781, 20786}

X(20788) = X(1)X(3)∩X(511)X(10478)

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

Let A'B'C' be the 3rd Conway triangle of ABC. Let A* = X(11)-of-AB'C' and define B* and C* cyclically. The Euler lines of AB*C*, BC*A*, CA*B* concur in X(20788). (César Lozada, August 11, 2018)

X(20788) lies on these lines: {1, 3}, {511, 10478}, {5208, 19645}, {10454, 15488}, {16343, 18180}

X(20788) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1764, 10439, 10441), (1764, 10470, 165)

X(20789) = X(1)X(3)∩X(1329)X(10179)

Barycentrics a*((b+c)*a^5-(b^2+12*b*c+c^2)*a^4-2*(b+c)*(b^2-7*b*c+c^2)*a^3+2*(b^4+c^4+5*(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(20789) = X(56)+3*X(5919), X(1329)-3*X(10179)

Let A'B'C' be the Hutson-intouch triangle of ABC. Let A* = X(11)-of-AB'C' and define B* and C* cyclically. The Euler lines of AB*C*, BC*A*, CA*B* concur in X(20789). (César Lozada, Aug 11, 2018)

X(20789) lies on these lines: {1, 3}, {1329, 10179}, {1476, 17613}, {3476, 9856}, {3623, 5728}, {3880, 6691}, {5795, 18227}, {5854, 6738}, {6049, 10167}, {6944, 11373}, {8256, 11019}, {9947, 10944}, {10107, 18240}, {10914, 17567}, {12128, 12709}, {14923, 17626}

X(20789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3057, 12915), (1420, 1697, 165), (1697, 5919, 9957), (3476, 17622, 9856)

X(20790) = X(1)X(3)∩X(3555)X(17558)

Barycentrics a*((b+c)*a^5-(b^2-12*b*c+c^2)*a^4-2*(b+c)*(b^2+5*b*c+c^2)*a^3+2*(b^4+c^4-b*c*(7*b^2+20*b*c+7*c^2))*a^2+(b^2-c^2)*(b-c)*(b^2+12*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :

Let A'B'C' be the incircle-circles triangle of ABC. Let A* = X(11)-of-AB'C', and define B* and C* cyclically. The Euler lines of AB*C*, BC*A*, CA*B* concur in X(20790). (César Lozada, August 11, 2018)

X(20790) lies on these lines: {1, 3}, {3555, 17558}, {4666, 19521}, {5777, 15008}, {6846, 9947}, {6849, 18527}, {10578, 16845}, {10580, 17582}

X(20790) = X(16197)-of-incircle-circles-triangle
X(20790) = X(16198)-of-inverse-in-incircle-triangle
X(20790) = X(16201)-of-anti-Aquila-triangle
X(20790) = {X(3333), X(5049)}-harmonic conjugate of X(5045)

X(20791) = X(3)X(54)∩X(20)X(51)

Barycentrics a^2*((b^2+c^2)*a^6-(3*b^4-7*b^2*c^2+3*c^4)*a^4+3*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(20791) = 5*X(2)-2*X(15030), 2*X(2)+X(15072), 4*X(2)-X(15305), X(2)-4*X(16836), 4*X(3)-X(2979), 8*X(3)+X(5889), 2*X(3)+X(5890), 11*X(3)-2*X(6101), 7*X(3)+2*X(6102), 4*X(3)+5*X(10574), 13*X(3)-4*X(10627), 10*X(3)-X(11412), 5*X(3)+4*X(13630), 2*X(2979)+X(5889), X(2979)+2*X(5890), 11*X(2979)-8*X(6101), 7*X(2979)+8*X(6102), X(2979)+5*X(10574), 13*X(2979)-16*X(10627), 5*X(2979)-2*X(11412), X(5889)-4*X(5890), 7*X(5889)-16*X(6102), X(5889)-10*X(10574), 5*X(5889)+4*X(11412), 4*X(15030)+5*X(15072), 8*X(15030)-5*X(15305), X(15030)-10*X(16836), 2*X(15072)+X(15305), X(15072)+8*X(16836), X(15305)-16*X(16836)

Let A'B'C' be the ABC-X3-reflections-triangle of ABC. Let A* = X(51)-of-AB'C', and define B* and C* cyclically. The Euler lines of AB*C*, BC*A*, CA*B* concur in X(20791). (César Lozada, August 11, 2018)

X(20791) lies on these lines: {2, 5656}, {3, 54}, {4, 5892}, {5, 11455}, {20, 51}, {22, 15053}, {30, 5640}, {52, 3528}, {74, 7514}, {140, 6241}, {143, 15696}, {154, 17928}, {182, 2071}, {185, 3523}, {186, 15080}, {373, 3839}, {376, 3060}, {382, 13364}, {389, 3522}, {511, 5032}, {546, 11465}, {549, 11459}, {550, 3567}, {568, 8703}, {631, 5891}, {632, 18439}, {974, 12273}, {1204, 13347}, {1216, 10299}, {1656, 11017}, {1657, 9781}, {2781, 5085}, {3090, 10575}, {3091, 6688}, {3520, 13336}, {3524, 7998}, {3525, 12162}, {3526, 13491}, {3529, 5462}, {3530, 7999}, {3534, 5946}, {3537, 6515}, {3543, 5943}, {3545, 14915}, {3796, 15078}, {3830, 13363}, {3832, 11695}, {3917, 15692}, {5054, 5663}, {5055, 16261}, {5056, 11381}, {5059, 10110}, {5068, 13474}, {5071, 16194}, {5073, 15026}, {5446, 17538}, {5562, 15717}, {5650, 15708}, {5876, 15720}, {5907, 10303}, {5972, 17853}, {6030, 14070}, {6636, 11438}, {6699, 12270}, {6816, 15740}, {7395, 15062}, {7485, 10605}, {7503, 10606}, {7506, 8718}, {7509, 11440}, {7729, 10192}, {7738, 15575}, {10095, 17800}, {10127, 16658}, {10170, 15702}, {10546, 14157}, {10733, 12099}, {10984, 11202}, {10996, 18950}, {11002, 16226}, {11179, 15531}, {11410, 12017}, {11413, 13434}, {13201, 14708}, {13321, 15689}, {13339, 18570}, {13391, 15688}, {13451, 19710}, {15060, 15694}, {15067, 15693}, {15712, 18436}, {16227, 16386}

X(20791) = midpoint of X(13321) and X(15689)
X(20791) = reflection of X(i) in X(j) for these (i,j): (4, 14845), (11002, 16226)
X(20791) = X(3524)-of-circumorthic-triangle
X(20791) = X(3545)-of-3rd anti-Euler-triangle
X(20791) = X(5055)-of-4th anti-Euler-triangle
X(20791) = X(10304)-of-1st anti-circumperp-triangle
X(20791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 15072, 15305), (3, 5890, 2979), (3, 10574, 5889), (3, 13630, 11412), (4, 5892, 11451), (20, 9729, 15043), (140, 6241, 15056), (185, 3523, 11444), (185, 17704, 3523), (376, 9730, 3060), (974, 15051, 12273), (2979, 5890, 5889), (2979, 10574, 5890), (5892, 11451, 15028), (5892, 14855, 4)

X(20792) = X(3)X(95)∩X(20)X(53)

Barycentrics 3*a^12-7*(b^2+c^2)*a^10+2*(3*b^4+7*b^2*c^2+3*c^4)*a^8-6*(b^2+c^2)*(b^4+c^4)*a^6+(b^2-c^2)^2*(7*b^4+10*b^2*c^2+7*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-2*(b^2-c^2)^4*b^2*c^2 : :
X(20792) = 4*X(3)-X(20477), X(20)+2*X(53), 4*X(140)-X(18437), X(6751)-4*X(9729)

Let A'B'C' be the ABC-X3-reflections-triangle of ABC. Let A* =X(53)-of-AB'C', and define B* and C* cyclically. The Euler lines of AB*C*, BC*A*, CA*B* concur in X(20792). (César Lozada, Auguse 11, 2018)

X(20792) lies on these lines: {{2, 154}, {3, 95}, {20, 53}, {140, 18437}, {157, 17928}, {6751, 9729}, {13860, 19124}

X(20792) = X(5085)-of-circumorthic-triangle

X(20793) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(7)

Barycentrics (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) sin 2A : :

X(20793) lies on these lines: {3, 77}, {219, 20795}, {255, 7193}, {861, 5942}, {3167, 20752}

X(20793) = isogonal conjugate of polar conjugate of X(3177)
X(20793) = isotomic conjugate of polar conjugate of X(20995)

X(20794) = X(3)X(69)∩X(6)X(694)

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

Let T = anticevian triangle of X(3). Then X(20794) is the perspector, with respect to T, of the pivotal conic of the conico-pivotal cubic cK(#3,X(394). (Angel Montesdeoca, May 4, 2019)

X(20794) lies on these lines: {3, 69}, {6, 694}, {22, 7779}, {25, 7774}, {39, 14913}, {48, 3955}, {95, 10104}, {159, 6660}, {160, 524}, {184, 3504}, {193, 237}, {194, 3186}, {216, 8681}, {219, 20796}, {255, 7193}, {264, 2782}, {311, 13108}, {1249, 15143}, {1424, 1740}, {1843, 3095}, {1975, 9230}, {3167, 3289}, {3620, 14096}, {4558, 14575}, {5020, 7736}, {5065, 9306}, {5201, 6144}, {5943, 13341}, {7484, 16990}, {7758, 9917}, {9155, 15531}, {11257, 14615}, {20795, 20818}

X(20794) = isogonal conjugate of polar conjugate of X(194)
X(20794) = isotomic conjugate of polar conjugate of X(1613)
X(20794) = X(19)-isoconjugate of X(2998)
X(20794) = X(92)-isoconjugate of X(3224)

X(20795) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(105)

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

(1,20760), (2,20794), (6,3), (31,219), (105,20795), (238,20796), (292,20797), (365,20798), (672,7913), (1423,20799), (1931,20800), (2053,20801), (2054,20802), (3009, 20785), (2112,20804)

Twenty-one more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - sin 2A : sin 2B : sin 2C
m(A') = (u + v + w) sin 2A : (- u - v + w) sin 2B : (- u + v - w) sin 2C, where (u, v, w) = (1/sqrt(a), 1/sqrt(b), 1/sqrt(c))
m(A1) = (b^2 + b c + c^2) sin 2A : (b^2 - b c - c^2) sin 2B : (-b^2 - b c + c^2) sini 2C
m(A2) = - (a b + a c + b c) sin 2A : (a b - a c + b c) sin 2B : (-a b + a c + b c) sin 2C
m(A3) = (a b^2 + 3 a b c + 2 b^2 c + a c^2 + 2 b c^2) sin 2A : (a b^2 - a b c - a c^2 - 2 b c^2) sin 2B : (-a b^2 - a b c - 2 b^2 c + a c^2) sin 2C
m(A4) = - (2 a^2 + 3 a b + 3 a c + b c) sin 2A : (a b + a c + b c) sin 2B : (a b + a c + b c) sin 2C
m(A5) = - (a - b - c) (a b + a c + b c) sin 2A : (-a^2 b + a b^2 + a^2 c - a b c + b^2 c - a c^2 - 3 b c^2) sin 2B : (a^2 b - a b^2 - a^2 c - a b c - 3 b^2 c + a c^2 + b c^2) sin 2C

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (-x/a^2 + y/b^2 + z/c^2) sin 2A : : , and m is the collineation indicated by (A,B,C,X(6); A'B'C',X(3)), where A' = - sin 2A : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^6 (b-c) c^6 (a b+a c+b c) (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 x^3+a^6 b^2 c^4 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (2 a^2 b+a b^2-b^2 c-a c^2-b c^2) y^2 z-a^6 b^4 c^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (-a b^2+2 a^2 c-b^2 c+a c^2-b c^2) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) = 0. (Peter Moses, August 13, 2018)

X(20795) lies on these lines: {3, 1814}, {219, 20793}, {222, 295}, {7193, 20796}, {20794, 20818}

X(20795) = isogonal conjugate of polar conjugate of X(39350)
X(20795) = isotomic conjugate of polar conjugate of crosspoint of PU(97)

X(20796) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(238)

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

See X(20795).

X(20796) lies on these lines: {3, 63}, {57, 16420}, {218, 11328}, {219, 20794}, {329, 19545}, {1634, 17796}, {1783, 15143}, {2200, 3955}, {3219, 16372}, {7193, 20795}, {15148, 17759}, {20762, 20785}, {20799, 20801}, {20800, 20802}

X(20796) = isogonal conjugate of polar conjugate of X(17759)
X(20796) = isotomic conjugate of polar conjugate of X(21788)

X(20797) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(292)

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

See X(20795).

X(20797) lies on these lines: {3, 295}, {63, 17972}, {184, 1331}, {219, 20804}, {3561, 20799}, {3781, 20786}, {7193, 20785}, {17976, 20752}

X(20797) = isogonal conjugate of polar conjugate of X(33888)
X(20797) = X(92)-isoconjugate of X(2109)

X(20798) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(365)

Barycentrics (- 1/Sqrt[a] + 1/Sqrt[b] + 1/Sqrt[c]) sin 2A : :

See X(20795).

X(20798) lies on these lines: {3, 20779}

X(20799) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(1423)

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

See X(20795).

X(20799) lies on these lines: {3, 20781}, {255, 7193}, {3561, 20797}, {7078, 20804}, {20796, 20801}

X(20800) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(1931)

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

See X(20795).

X(20800) lies on these lines: {3, 20782}, {219, 7015}, {17972, 20746}, {20796, 20802}

X(20801) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(2053)

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

See X(20795).

X(20801) lies on these lines: {3, 7116}, {219, 20785}, {222, 20767}, {20796, 20799}

X(20802) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(2054)

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

See X(20795).

X(20802) lies on these lines: {3, 17972}, {48, 4558}, {20796, 20800}

X(20803) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(12)

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

X(20803) lies on these lines: {1, 859}, {3, 201}, {56, 12005}, {222, 20764}, {228, 17102}, {3075, 18162}, {7078, 20760}

X(20803) = isogonal conjugate of polar conjugate of X(18662)
X(20803) = isotomic conjugate of polar conjugate of X(21770)

X(20804) = (X(1), X(2), X(6), X(31); X(20760), X(20794), X(3), X(219)) COLLINEATION IMAGE OF X(2112)

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

See X(20795).

X(20804) lies on these lines: {3, 17972}, {109, 3033}, {219, 20797}, {222, 295}, {1331, 7085}, {7078, 20799}, {17976, 20785}

X(20805) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(8)

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

X(20805) lies on these lines: {3, 63}, {56, 1046}, {194, 3732}, {219, 4020}, {255, 7193}, {329, 19513}, {474, 894}, {758, 15654}, {855, 12649}, {908, 19549}, {1403, 5247}, {1409, 20818}, {3086, 15507}, {3218, 13738}, {3682, 3784}, {4245, 5708}, {4737, 5687}, {5744, 13731}, {6638, 20764}, {15650, 16374}, {18732, 20254}, {19514, 20348}

X(20805) = isogonal conjugate of polar conjugate of X(3210)
X(20805) = isotomic conjugate of polar conjugate of X(21769)

X(20806) = (name pending)

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

See X(20807).

Let A'B'C' be the cevian triangle of X(22). Let A" be the inverse-in-circumcircle of A', and define B", C" cyclically. The lines AA", BB", CC" concur in X(20806). (Randy Hutson, August 29, 2018)

X(20806) lies on the conic {{X(3),X(6),X(24),X(60),X(143),X(1511),X(1986)}} and these lines: {2, 6}, {3, 1176}, {20, 19149}, {22, 206}, {24, 511}, {51, 19137}, {66, 858}, {110, 159}, {143, 1351}, {146, 17812}, {155, 6643}, {157, 3001}, {182, 5562}, {184, 11574}, {219, 20808}, {287, 20563}, {297, 8745}, {311, 458}, {315, 8743}, {427, 13562}, {577, 9723}, {648, 14615}, {651, 18629}, {895, 6391}, {1147, 9967}, {1216, 19131}, {1264, 1332}, {1350, 7488}, {1352, 1594}, {1370, 5596}, {1498, 14927}, {1503, 11441}, {1511, 10752}, {1568, 3818}, {1760, 7210}, {1843, 9306}, {1986, 15462}, {1995, 9969}, {2979, 19121}, {3092, 12322}, {3093, 12323}, {3167, 19459}, {3193, 5800}, {3260, 9308}, {3292, 6467}, {3564, 11585}, {3917, 19126}, {3964, 4558}, {5050, 7393}, {5157, 7485}, {5408, 11513}, {5409, 8911}, {5480, 7544}, {5622, 12358}, {5651, 9822}, {5907, 19124}, {5921, 8549}, {6090, 11188}, {6101, 19154}, {7386, 19119}, {7396, 20079}, {7401, 14853}, {7405, 18583}, {8541, 14913}, {8548, 11898}, {9973, 10510}, {10602, 19588}, {10627, 19155}, {11061, 17847}, {11412, 19128}, {11416, 12272}, {12294, 13346}, {14561, 14788}, {14570, 20477}, {15068, 18440}, {16163, 19140}, {17928, 19161}, {20739, 20811}, {20809, 20817}, {20814, 20816

X(20806) = isogonal conjugate of X(13854)
X(20806) = isotomic conjugate of polar conjugate of X(22)
X(20806) = X(19)-isoconjugate of X(66)
X(20806) = X(92)-isoconjugate of X(2353)

X(20807) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(105)

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

(1,394), (2,219), (6,20739), (31,20806), (105,20807), (238,20808), (292,20809), (365,20810), (672,20811), (1423,20812), (1931,20813), (2053,20814), (2054,20815), (3009, 20816), (2112,20817)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - sin 2A : sin 2B : sin 2C
m(A') = - (a^(5/2) + b^(5/2) + c^(5/2)) sin 2A : (a^(5/2) + b^(5/2) - c^(5/2)) sin 2A : (a^(5/2) - b^(5/2) + c^(5/2)) sin 2A :
m(A1) = (a^3 + b^2 c + b c^2) sin 2A : (b^2 - b c - c^2) sin 2B : (-b^2 - b c + c^2) sin 2C
m(A2) = - (a b + a c + b c) sin 2A : (a^3 + b^2 c - b c^2) sin 2B : (a^3 - b^2 c + b c^2) sin 2C
m(A3) = - (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3) sin 2A : (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c - a b c^2 - a c^3 - b c^3) sin 2B : (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c + a b c^2 + a c^3 + b c^3) sin 2C
m(A4) = - (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c + a c^3 + b c^3 + c^4) sin 2A : (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c - a c^3 - b c^3 - c^4) sin 2B : (a b + a c + b c) sin 2C

where A1, A2, A3, A4 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (-a x + b y + c z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(3)), where A' = - sin 2A : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = a b^2 (b-c) c^2 (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 (-b^3+a b c-b^2 c-b c^2-c^3) x^3-a^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (-a^5 b^2+a^4 b^3+a^6 c-a^2 b^4 c+2 a b^4 c^2-2 b^3 c^4+b^2 c^5-a c^6) y^2 z+a^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (a^6 b-a b^6-a^5 c^2+b^5 c^2+a^4 c^3-2 b^4 c^3-a^2 b c^4+2 a b^2 c^4) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 (a-b) (a-c) (b-c) (-b^2+a c) (a^2-b c) (a b-c^2) (a^2-b^2-c^2)^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 x y z= 0. (Peter Moses, August 13, 2018)

X(20807) lies on these lines: {3, 20820}, {6, 16593}, {212, 3781}, {219, 20740}, {394, 20817}, {1264, 1332}, {1814, 17170}, {20742, 20808}

X(20808) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(238)

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

See X(20807).

X(20808) lies on these lines: {3, 20821}, {6, 3879}, {72, 18447}, {218, 1993}, {219, 20806}, {394, 4001}, {4511, 16466}, {20742, 20807}, {20747, 20748}, {20812, 20814}, {20813, 20815}

X(20809) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(292)

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

See X(20807).

X(20809) lies on these lines: {3, 20822}, {6, 17793}, {219, 20797}, {394, 20742}, {20741, 20747}, {20806, 20817}, {20811, 20816}

X(20810) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(365)

Barycentrics (a^(5/2) - b^(5/2) - c^(5/2)) sin 2A : :

See X(20807).

X(20810) lies on these lines: {6, 20527}

X(20811) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(672)

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

See X(20807).

X(20811) lies on these lines: {1, 6}, {3, 20823}, {525, 3049}, {1814, 20744}, {20739, 20806}, {20742, 20807}, {20809, 20816}

X(20812) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(1423)

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

See X(20807).

X(20812) lies on these lines: {1, 6}, {3, 20824}, {7124, 20760}, {20808, 20814}

X(20813) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(1931)

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

See X(20807).

X(20813) lies on these lines: {2, 6}, {3, 20825}, {20808, 20815}

X(20814) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(2053)

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

See X(20807).

X(20814) lies on these lines: {3, 20826}, {6, 20528}, {20806, 20816}, {20808, 20812}

X(20815) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(2054)

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

See X(20807).

X(20815) lies on these lines: {3, 20827}, {6, 20529}, {219, 20768}, {20739, 20817}, {20808, 20813}

X(20816) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(3009)

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

See X(20807).

X(20816) lies on these lines: {3, 20828}, {6, 20530}, {63, 77}, {20806, 20814}, {20809, 20811}

X(20817) = (X(1), X(2), X(6), X(31); X(394), X(219), X(20739), X(20806)) COLLINEATION IMAGE OF X(2112)

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

See X(20807).

X(20817) lies on these lines: {3, 20829}, {6, 20531}, {394, 20807}, {20739, 20815}, {20806, 20809}

X(20818) = (X(1), X(2), X(6), X(31); X(3), X(20760), X(219), X(394)) COLLINEATION IMAGE OF X(9)

Barycentrics (3 a - b - c) sin 2A : :

X(20818) lies on these lines: {1, 2264}, {3, 48}, {6, 101}, {9, 1385}, {19, 1482}, {145, 4248}, {154, 3190}, {169, 1100}, {184, 1260}, {198, 2323}, {218, 604}, {220, 572}, {281, 952}, {282, 5534}, {284, 2256}, {380, 9957}, {496, 5802}, {517, 610}, {573, 3207}, {692, 6600}, {857, 20074}, {956, 2287}, {965, 9708}, {1319, 1723}, {1404, 3217}, {1409, 20805}, {1420, 1743}, {1436, 11248}, {1437, 2327}, {1449, 5045}, {1766, 6603}, {1781, 2099}, {1813, 7053}, {1826, 18525}, {1901, 9655}, {1953, 10247}, {2172, 12410}, {2173, 8148}, {2174, 4254}, {2273, 9605}, {2302, 16202}, {2329, 5783}, {2911, 5120}, {3167, 20752}, {3173, 7011}, {3197, 6759}, {3692, 5440}, {3731, 13384}, {3940, 5227}, {5279, 5730}, {5746, 18990}, {5747, 9654}, {5819, 20330}, {6510, 7289}, {7359, 18526}, {7982, 18594}, {8804, 18481}, {9028, 17073}, {15831, 18446}, {15905, 20764}, {20794, 20795}

X(20818) = isogonal conjugate of polar conjugate of X(145)
X(20818) = isotomic conjugate of polar conjugate of X(3052)
X(20818) = X(19)-isoconjugate of X(4373)
X(20818) = X(92)-isoconjugate of X(3445)

X(20819) = (name pending)

Barycentrics (b^4 + c^4) sin 2A : :

See X(20820).

X(20819) lies on these lines: {3, 1176}, {71, 20821}, {141, 3001}, {160, 9155}, {216, 3289}, {237, 3313}, {570, 14096}, {1634, 17710}, {2967, 17907}, {2972, 6389}, {3095, 3618}, {8041, 18899}, {8265, 16717}, {11574, 20775}, {14881, 17500}, {20727, 20823}, {20822, 20829}, {20826, 20828}

X(20820) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(105)

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

(1,3917), (2,71), (6,2020727), (31,20819), (105,20820), (238,20821), (292,20822), (672,20823), (1423,20824), (1931,20825), (2053,20826), (2054,20827), (3009, 20828), (2112,20829), (203232,20830)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : sin 2B : sin 2C
m(A1) = b c (b + c) sin 2A : (-a^3 + b c^2) sin 2B : (-a^3 + b^2 c) sin 2C
m(A2) = (b^4 + c^4) sin 2A : c (c^3 - a^2 b) sin 2B : (b ( b^3 - a^ c) sin 2C
m(A3) = - (a b + a c + b c) (b^2 + c^2) sin 2A : (a^4 + a^3 b + a^3 c - a b c^2 - a c^3 - b c^3) sin 2B : (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c) sin 2C
m(A4) = - (b + c) (a + b + c) (b^2 - b c + c^2) sin 2A : (a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4) sin 2B : (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c) sin 2C
m(A5) = (a^2 b^2 + b^4 - a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) sin 2A : (a^4 + a^3 b + a^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) sin 2B : (a^4 + a^3 b + a^2 b^2 + b^4 + a^3 c - a b^2 c - b^3 c) sin 2C

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = (b y + c z) sin 2A : : , and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(3)), where A' = 0 : sin 2B : sin 2C.

Let f(a,b,c,x,y,z) = b^6 (b-c) c^6 (a^2-b c) (a^2+b^2-c^2)^3 (a^2-b^2+c^2)^3 (-a^3 b^3-a^2 b^4+a^4 b c-a^3 b^2 c-a^2 b^3 c-a b^4 c-a^3 b c^2-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+b^3 c^3-a^2 c^4-a b c^4) x^3-a^6 b^2 c^4 (a^2-b^2-c^2)^3 (a^2+b^2-c^2)^2 (a^2-b^2+c^2) (-3 a^5 b^4-a^4 b^5+3 a^6 b^2 c+a^2 b^6 c+3 a^6 b c^2+a b^6 c^2-3 a^5 c^4-b^5 c^4+a^4 c^5-b^4 c^5-a^2 b c^6+a b^2 c^6) y^2 z+a^6 b^4 c^2 (a^2-b^2-c^2)^3 (a^2+b^2-c^2) (a^2-b^2+c^2)^2 (-3 a^5 b^4+a^4 b^5+3 a^6 b^2 c-a^2 b^6 c+3 a^6 b c^2+a b^6 c^2-3 a^5 c^4-b^5 c^4-a^4 c^5-b^4 c^5+a^2 b c^6+a b^2 c^6) y z^2, where x : y : z are barycentyrics of a variable point; then m(K155) is given by f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,x,z,y) + 2 a^4 (a-b) b^4 (a-c) (b-c) c^4 (-b^2+a c) (a^2-b c) (a b-c^2) (a^2-b^2-c^2)^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 x y z = 0. (Peter Moses, August 13, 2018)

X(20820) lies on these lines: {3, 20807}, {71, 20728}, {3917, 20829}, {20730, 20821}

X(20821) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(238)

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

See X(20820).

X(20821) lies on these lines: {3, 20808}, {71, 20819}, {3917, 20727}, {7117, 20757}, {20730, 20820}, {20736, 20737}, {20824, 20826}, {20825, 20827}, {20828, 20830}

X(20822) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(292)

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

See X(20820).

X(20822) lies on these lines: {3, 20809}, {71, 11574}, {3917, 20730}, {7117, 20750}, {20729, 20736}, {20819, 20829}, {20823, 20828}

X(20823) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(672)

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

See X(20820).

X(20823) lies on these lines: {3, 20811}, {71, 216}, {1818, 7117}, {2524, 3049}, {20727, 20819}, {20730, 20820}, {20822, 20828}

X(20824) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(1423)

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

See X(20820).

X(20824) lies on these lines: {3, 20812}, {71, 216}, {78, 7117}, {20821, 20826}, {20829, 20830}

X(20825) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(1931)

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

See X(20820).

X(20825) lies on these lines: {3, 20813}, {216, 3289}, {20821, 20827}

X(20826) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(2053)

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

See X(20820).

X(20826) lies on these lines: {3, 20814}, {20819, 20828}, {20821, 20824}

X(20827) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(2054)

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

See X(20820).

X(20827) lies on these lines: {3, 20815}, {71, 20756}, {20727, 20829}, {20821, 20825}

X(20828) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(3009)

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

See X(20820).

X(20828) lies on these lines: {3, 20816}, {71, 3917}, {20819, 20826}, {20821, 20830}, {20822, 20823}

X(20829) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(2112)

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

See X(20820).

X(20829) lies on these lines: {3, 20817}, {3917, 20820}, {20727, 20827}, {20819, 20822}, {20824, 20830}

X(20830) = (X(1), X(2), X(6), X(31); X(3917), X(71), X(20727), X(20819)) COLLINEATION IMAGE OF X(20332)

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

See X(20820).

X(20830) lies on these lines: {3837, 20532}, {7117, 20727}, {20821, 20828}, {20824, 20829}

X(20831) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(191)

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

Suppose that X is a point in the plane of a triangle ABC. Let m(x) be the image of X under the collineation (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)). If X is on the line X(1)X(21), then m(X) is on the Euler line. The appearance of (i,j) in the following list means that m(X(i) = X(j):

(1,3), (21,3145), (31,22), (38,2), (47,26), (58,2915), (63,25), (81,199), (191,20831), (255,24), (283,20832), (595,20833), (758,859), (774,20), (846,20834), (896,23), (920,7387), (968,20835), (993,11334), (1046,20836), (1496,17928), (1497,10323), (1580,6660), (1621,16064), (1707,9909), (1725,30), (3561,20837), (3562,20838), (3573,20839), (3647,20840), (8616,20841), (8666,20842), (5330,20843), (5429,20844), (5208,20845), (10448,20846), (10457,20847), (10458,20848), (11533,20849), (12514,13730), (16570,20850), (16948,20851), (17185,20852), (17194,20853), (17469,6636), (17799,20854), (18169,20855), (18192,20856), (18206,20857), (18756,20858), (18477,378)

X(20831) lies on these lines:

X(20832) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(283)

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

See X(20831).

X(20832) lies on these lines: {2, 3}, {19, 1030}, {35, 1824}, {56, 1835}, {65, 11363}, {225, 5172}, {232, 2204}, {498, 11391}, {607, 18755}, {608, 2305}, {993, 5130}, {1474, 2245}, {1825, 14882}, {1829, 2646}, {1843, 5135}, {1844, 11399}, {1974, 4259}, {2203, 10974}, {3295, 11401}, {3612, 7713}, {5089, 5277}, {10835, 16541}

X(20832) = isogonal conjugate of X(18123)

X(20833) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(595)

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

See X(20831).

X(20833) lies on these lines: {1, 2916}, {2, 3}, {36, 2920}, {56, 2922}, {58, 5347}, {72, 3220}, {942, 7293}, {956, 8193}, {988, 7298}, {1030, 5283}, {1125, 20988}, {1376, 8185}, {1610, 10609}, {1724, 5096}, {2077, 9626}, {2921, 8069}, {2932, 2933}, {3556, 5730}, {3703, 5687}, {3916, 5285}, {5204, 14667}, {5266, 5322}, {7295, 16466}, {12114, 15177}

X(20834) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(846)

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

See X(20831).

X(20834) lies on these lines: {2, 3}, {55, 846}, {256, 7083}, {511, 2328}, {991, 9306}, {1001, 1626}, {1284, 1617}, {1423, 8616}, {1486, 8424}, {1495, 1790}, {1621, 9791}, {2292, 3295}, {2822, 14673}, {3303, 11533}, {3423, 11031}, {4389, 16099}, {5248, 12579}, {5329, 20992}, {5943, 13329}, {7193, 14547}, {7961, 13097}, {8245, 15931}, {9798, 12567}, {9959, 10267}, {16678, 20988}, {17975, 20122}

X(20835) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(968)

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

See X(20831).

X(20835) lies on these lines: {1, 18607}, {2, 3}, {7, 1617}, {35, 200}, {36, 10582}, {55, 63}, {56, 4666}, {84, 10902}, {100, 5273}, {154, 1790}, {159, 1626}, {224, 960}, {394, 991}, {572, 3796}, {954, 5905}, {967, 4252}, {993, 4304}, {1001, 1836}, {1071, 10267}, {1074, 1860}, {1078, 18153}, {1260, 3219}, {1444, 14548}, {1486, 16678}, {1709, 4512}, {1754, 17194}, {2975, 4313}, {3218, 11020}, {3295, 3868}, {3303, 11520}, {3681, 6600}, {3683, 5784}, {3838, 4423}, {3871, 20015}, {3877, 18444}, {4292, 5248}, {4414, 11031}, {5010, 8580}, {5217, 5302}, {5250, 6001}, {5267, 8071}, {5435, 15804}, {5584, 19860}, {7054, 15905}, {7676, 17784}, {8069, 13405}, {8273, 19861}, {8822, 14828}, {10601, 13329}, {14988, 16202}, {18603, 19765}, {19718, 19759}, {19790, 19841}

X(20836) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(1046)

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

See X(20831).

X(20836) lies on these lines: {2, 3}, {56, 5429}, {283, 1495}, {511, 2360}, {958, 1631}, {999, 1036}, {1962, 3295}, {2178, 16974}, {2816, 14673}, {4057, 5592}, {4298, 5144}, {5247, 17798}, {13558, 14663}, {14815, 16466}

X(20837) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(3561)

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

See X(20831).

X(20837) lies on these lines: {2, 3}, {36, 1426}, {55, 1825}, {1824, 10902}, {1845, 11398}, {11363, 14110}

X(20838) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(3562)

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

See X(20831).

X(20838) lies on these lines: {2, 3}, {36, 1410}, {56, 20277}, {185, 2360}, {580, 13367}, {947, 16980}, {1035, 5204}, {1622, 8192}, {1631, 5584}, {6197, 21318}, {15622, 20989}

X(20839) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(3573)

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

See X(20831).

X(20839) lies on these lines: {2, 3}, {8638, 20999}, {16686, 16873}

X(20840) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(3647)

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

See X(20831).

X(20840) lies on these lines: {2, 3}, {500, 1495}

X(20841) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(8616)

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

See X(20831).

X(20841) lies on these lines: {2, 3}, {1283, 1403}, {1626, 20872}, {2223, 5345}, {2328, 3098}, {3220, 20760}

X(20842) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(8666)

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

See X(20831).

X(20842) lies on these lines: {1, 2933}, {2, 3}, {56, 1324}, {88, 20999}, {197, 8071}, {386, 1437}, {1470, 9798}, {1626, 7280}, {1737, 2217}, {1993, 5754}, {2360, 4256}, {3420, 12410}, {15654, 20989}

X(20843) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(5330)

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

See X(20831).

X(20843) lies on this line: {2, 3}

X(20844) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(5429)

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

See X(20831).

X(20844) lies on these lines: {2, 3}, {3796, 9567}

X(20845) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(5208)

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

See X(20831).

X(20845) lies on these lines: {2, 3}, {171, 228}, {198, 1755}, {5156, 5320}

X(20846) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(10448)

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

See X(20831).

X(20846) lies on these lines: {2, 3}, {6, 7054}, {35, 78}, {55, 3869}, {63, 10393}, {157, 8053}, {283, 581}, {386, 1780}, {580, 5422}, {936, 5010}, {938, 8071}, {958, 5086}, {965, 1030}, {993, 6734}, {1210, 5267}, {1259, 3219}, {1444, 5738}, {1470, 5303}, {1617, 3622}, {1621, 3485}, {1792, 5739}, {1858, 4640}, {2975, 3486}, {3616, 7742}, {3871, 20013}, {3876, 11517}, {3877, 10267}, {3890, 11510}, {3897, 11249}, {4652, 10399}, {4881, 8273}, {5057, 5172}, {5248, 12047}, {5250, 6261}, {5330, 16202}, {5703, 8069}, {5705, 14794}, {5736, 17139}, {7098, 11509}, {7677, 10586}, {15931, 19861}, {19716, 19759}, {19788, 19841}

X(20847) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(10457)

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

See X(20831).

X(20847) lies on this line: {2, 3}

X(20848) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(10458)

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

See X(20831).

X(20848) lies on these lines: {2, 3}, {1030, 17735}

X(20849) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(11533)

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

See X(20831).

X(20849) lies on these lines: {2, 3}, {56, 846}, {999, 2292}, {1283, 5217}, {3304, 11533}, {5253, 9791}, {9959, 10269}

X(20850) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(16570)

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

See X(20831).

X(20850) lies on these lines: {2, 3}, {52, 14530}, {154, 1351}, {159, 3629}, {182, 5644}, {184, 5093}, {187, 8770}, {511, 8780}, {1196, 1384}, {1495, 3167}, {2056, 11173}, {2936, 13175}, {3244, 9798}, {3531, 14805}, {3632, 8185}, {3636, 11365}, {3920, 9642}, {5050, 17810}, {5943, 12017}, {6154, 13222}, {6221, 8854}, {6398, 8855}, {6428, 20197}, {6800, 9777}, {7716, 9813}, {8192, 20057}, {9157, 13310}, {9673, 16541}, {10311, 15851}, {11008, 19588}, {11416, 19118}, {11482, 17809}, {13598, 17821}, {13665, 18289}, {13785, 18290}

X(20850) = circumcircle-inverse of X(37911)

X(20851) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(16948)

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

See X(20831).

X(20851) lies on these lines: {2, 3}, {1030, 1696}, {1324, 9591}, {1495, 3430}

X(20852) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(17185)

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

See X(20831).

X(20852) lies on these lines: {2, 3}, {197, 1030}, {1460, 1486}

X(20853) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(17194)

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

See X(20831).

X(20853) lies on these lines: {2, 3}, {1030, 1604}

X(20854) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(17799)

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

See X(20831).

X(20854) lies on these lines: {2, 3}, {32, 9918}, {669, 3800}, {1495, 9301}, {2076, 3229}, {2080, 14673}, {5201, 19596}, {5943, 12054}, {8623, 20998}, {9306, 9821}

X(20855) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(18169)

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

See X(20831).

X(20855) lies on these lines: {2, 3}, {5201, 18185}

X(20856) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(18192)

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

See X(20831).

X(20856) lies on this line: {2, 3}

X(20857) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(18206)

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

See X(20831).

X(20857) lies on these lines: {2, 3}, {55, 5277}, {197, 3207}, {650, 667}, {1030, 1486}, {2223, 3290}, {2305, 7083}

X(20858) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(18756)

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

See X(20831).

X(20858) lies on this line: {2, 3}

X(20859) = X(2)X(694)∩X(6)X(22)

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

See X(20860).

X(20859) lies on these lines: {2, 694}, {6, 22}, {23, 1915}, {39, 51}, {42, 8628}, {76, 19562}, {141, 6664}, {154, 10542}, {184, 5028}, {323, 2056}, {343, 5254}, {511, 1194}, {626, 3118}, {689, 711}, {695, 6655}, {698, 8024}, {732, 8267}, {1184, 1350}, {1185, 4259}, {1196, 3231}, {1570, 13366}, {1613, 2979}, {1627, 2076}, {1691, 6636}, {1899, 2549}, {1994, 5111}, {2502, 9306}, {3155, 6421}, {3156, 6422}, {3291, 3819}, {3410, 11646}, {3721, 17184}, {3778, 8629}, {5013, 10601}, {5017, 5359}, {5034, 15004}, {5104, 5354}, {6656, 14820}, {7738, 11433}, {7748, 11550}, {7760, 13511}, {7998, 21001}, {8216, 8219}, {8265, 16717}, {8620, 20684}, {10328, 16276}, {11002, 13331}, {11451, 15302}, {12055, 15018}, {12963, 13617}, {12968, 13616}, {18203, 21324}, {20862, 20869}, {20866, 20868}, {20870, 20974}

X(20859) = isogonal conjugate of isotomic conjugate of X(626)
X(20859) = isotomic conjugate of X(38830)

X(20860) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(105)

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

(1,39), (2,42), (6,3778), (31,20859), (105,20860), (238,20861), (292,20862), (672,20863), (1423,20864), (1931,20865), (2053,20866), (2054,20867), (3009, 20868), (2112,20869), (203232,20870)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = 0 : b^2 : c^2
m(A1) = a^2 b c (b + c) : b^2 (-a^3 + b c^2) : c^2 (-a^3 + b^2 c)
m(A2) = a^2 (b^4 + c^4) : b^2 c (c^3 - a^2 b) : b c^2 (b^3 - a^2 c)
m(A3) = a^2 (a b + a c + b c) (b^2 + c^2) : -b^2 (a^4 + a^3 b + a^3 c - a b c^2 - a c^3 - b c^3) : -c^2 (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c)
m(A4) = a^2 (b + c) (a + b + c) (b^2 - b c + c^2) : -b^2 (a^3 b + a^3 c + a^2 b c - a c^3 - b c^3 - c^4) : -c^2 (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c)
m(A5) = -a^2 (a^2 b^2 + b^4 - a b^2 c - b^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : -b^2 (a^4 + a^3 b + a^3 c + a^2 c^2 - a b c^2 - b c^3 + c^4) : -c^2 (a^4 + a^3 b + a^2 b^2 + b^4 + a^3 c - a b^2 c - b^3 c)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (b y + c z) : : , and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(6)), where A' = 0 : b^2 : c^2.

X(20860) lies on these lines: {6, 20871}, {39, 20869}, {42, 20455}, {672, 20778}, {3930, 4966}, {20457, 20861}, {20662, 20970}, {20866, 20971}

X(20861) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(238)

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

See X(20860).

X(20861) lies on these lines: {1, 3094}, {6, 7295}, {39, 2309}, {42, 8628}, {43, 3981}, {213, 4260}, {899, 3124}, {1194, 1197}, {1570, 20958}, {1738, 3125}, {2092, 2183}, {3096, 3662}, {3123, 20706}, {3271, 20669}, {3720, 8041}, {3721, 3821}, {3726, 4071}, {3954, 4357}, {8637, 20979}, {20457, 20860}, {20464, 20465}, {20864, 20866}, {20865, 20867}, {20868, 20870}, {20961, 20965}, {20962, 20977}, {20963, 20969}

X(20862) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(292)

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

See X(20860).

X(20862) lies on these lines: {6, 14598}, {39, 20457}, {42, 1194}, {386, 8300}, {869, 4876}, {2092, 2309}, {20456, 20464}, {20859, 20869}, {20863, 20868}

X(20863) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(672)

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

See X(20860).

X(20863) lies on these lines: {2, 3056}, {6, 9455}, {42, 51}, {669, 2451}, {672, 3271}, {674, 2238}, {766, 3125}, {2309, 20965}, {3726, 9016}, {3778, 8629}, {4531, 17451}, {5369, 16583}, {20347, 20358}, {20457, 20860}, {20862, 20868}

X(20864) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(1423)

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

See X(20860).

X(20864) lies on these lines: {6, 20876}, {9, 3056}, {42, 51}, {213, 5052}, {20861, 20866}, {20869, 20870}

X(20865) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(1931)

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

See X(20860).

X(20865) lies on these lines: {6, 20877}, {39, 51}, {42, 2653}, {3009, 3124}, {3747, 5164}, {5168, 20754}, {20456, 20982}, {20861, 20867}

X(20866) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(2053)

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

See X(20860).

X(20866) lies on these lines: {43, 169}, {1432, 1916}, {20859, 20868}, {20860, 20971}, {20861, 20864}

X(20867) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(2054)

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

See X(20860).

X(20867) lies on these lines: {42, 20668}, {2309, 20463}, {3778, 20869}, {20861, 20865}

X(20868) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(3009)

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

See X(20860).

X(20868) lies on these lines: {6, 20878}, {39, 42}, {1125, 21327}, {2275, 4393}, {4359, 16604}, {8630, 20983}, {20859, 20866}, {20861, 20870}, {20862, 20863}

X(20869) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(2112)

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

See X(20860).

X(20869) lies on these lines: {39, 20860}, {3778, 20867}, {20859, 20862}, {20864, 20870}

X(20870) = (X(1), X(2), X(6), X(31); X(39), X(42), X(3778), X(20859)) COLLINEATION IMAGE OF X(20332)

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

See X(20860).

X(20870) lies on these lines: {3009, 20759}, {3124, 20671}, {3271, 3778}, {20859, 20974}, {20861, 20868}, {20864, 20869}

X(20871) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(105)

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

(1,39), (2,42), (6,3778), (31,20859), (105,20860), (238,20861), (292,20862), (672,20863), (1423,20864), (1931,20865), (2053,20866), (2054,20867), (3009, 20868), (2112,20869), (203232,20870)

Eighteen more points on m(K155) are given as vertices of central triangles, represented here by A-vertices:

m(A) = - a^2 : b^2 : c^2
m(A') = a^2 (a^(5/2) + b^(5/2) + c^(5/2)) : -b^2 (a^(5/2) + b^(5/2) - c^(5/2)) : -c^2 (a^(5/2) - b^(5/2) + c^(5/2))
m(A1) = -a^2 (a^3 + b^2 c + b c^2) : b^2 (a^3 + b^2 c - b c^2) : c^2 (a^3 - b^2 c + b c^2)
m(A2) = a^2 (b^4 + a^2 b c + c^4) : -b^2 (b^4 + a^2 b c - c^4) : -c^2 (-b^4 + a^2 b c + c^4)
m(A3) = -a^2 (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3) : b^2 (a^4 + a^3 b + a b^3 + a^3 c + a b^2 c + b^3 c - a b c^2 - a c^3 - b c^3) : c^2 (a^4 + a^3 b - a b^3 + a^3 c - a b^2 c - b^3 c + a b c^2 + a c^3 + b c^3)
m(A4) = a^2 (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c + a c^3 + b c^3 + c^4) : -b^2 (a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + b^3 c - a c^3 - b c^3 - c^4) : -c^2 (a^3 b - a b^3 - b^4 + a^3 c + a^2 b c - b^3 c + a c^3 + b c^3 + c^4)
m(A5) = a^2 (a^4 + 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 c^3 - c^4) : b^2 (-a^4 - 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 c^3 - c^4) : c^2 (-a^4 - 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 c^3 + c^4)

where A1, A2, A3, A4, A5 are defined at X(20333).

If P = x : y : z (barycentrics), then m(P) = a^2 (- a x + b y + c z) : : , and m is the collineation indicated by (A,B,C,X(75); A'B'C',X(6)), where A' = 0 : b^2 : c^2.

X(20871) lies on these lines: {3, 8299}, {6, 20860}, {22, 100}, {31, 20786}, {38, 55}, {41, 2276}, {105, 1486}, {1282, 3220}, {1283, 17594}, {4712, 12329}, {7248, 9316}, {7295, 8300}, {8301, 20872}

X(20871) = isogonal conjugate of isotomic conjugate of X(20552)

X(20872) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(238)

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

See X(20871).

X(20872) lies on these lines: {1, 2916}, {2, 20988}, {3, 142}, {6, 7295}, {10, 20831}, {21, 4026}, {22, 55}, {23, 100}, {24, 10310}, {25, 1376}, {26, 11248}, {31, 5347}, {35, 37}, {36, 1279}, {56, 4318}, {197, 4421}, {238, 5096}, {354, 7293}, {511, 692}, {518, 3220}, {522, 1324}, {674, 7193}, {902, 5078}, {958, 8193}, {1012, 15177}, {1155, 14667}, {1283, 5143}, {1284, 5172}, {1329, 4222}, {1621, 6636}, {1626, 20841}, {1633, 17768}, {1770, 16580}, {1995, 4413}, {2077, 9625}, {2175, 4259}, {2323, 9047}, {2886, 4224}, {2931, 12327}, {2933, 20876}, {2937, 11849}, {3052, 5329}, {3145, 8424}, {3286, 16876}, {3556, 12635}, {3666, 5310}, {3683, 5314}, {3744, 5322}, {3749, 5345}, {3816, 19649}, {3826, 4223}, {3913, 9798}, {3925, 4228}, {4220, 6690}, {4423, 7485}, {4429, 17522}, {4640, 5285}, {4689, 7302}, {5217, 11337}, {5220, 12329}, {5259, 17384}, {5284, 15246}, {5537, 9590}, {5687, 8185}, {5899, 18524}, {7298, 17594}, {7301, 16468}, {7387, 11500}, {7484, 8167}, {7517, 11499}, {7530, 18491}, {8301, 20871}, {9658, 11501}, {9673, 11502}, {9712, 14017}, {10117, 13204}, {11349, 16593}, {11491, 12088}, {12340, 19165}, {12410, 12513}, {15228, 16581}, {15338, 16049}, {15577, 18621}, {16064, 16678}, {20475, 20476}

X(20872) = isogonal conjugate of isotomic conjugate of X(20553)
X(20872) = isogonal conjugate of anticomplement of X(39029)
X(20872) = anticomplement of complementary conjugate of X(39029)

X(20873) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(292)

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

See X(20871).

X(20873) lies on these lines: {3, 8301}, {6, 14598}, {55, 16515}, {1030, 2110}, {1604, 20876}, {1631, 8266}, {2196, 9016}, {2915, 16681}, {16683, 19329}, {16693, 19308}, {17798, 20475}, {20875, 20878}

X(20874) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(365)

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

See X(20871).

X(20874) lies on these lines: (none)

X(20875) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(672)

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

See X(20871).

X(20875) lies on these lines: {2, 8053}, {3, 9305}, {6, 9455}, {19, 25}, {22, 1602}, {23, 385}, {105, 16693}, {517, 16680}, {614, 3941}, {859, 8618}, {2223, 3290}, {3263, 4436}, {3920, 4068}, {4224, 16872}, {5272, 16688}, {6636, 16994}, {7191, 16679}, {7292, 16694}, {8301, 20871}, {13595, 16993}, {16686, 17735}, {20873, 20878}

X(20875) = isogonal conjugate of isotomic conjugate of X(20556)

X(20876) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(1423)

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

See X(20871).

X(20876) lies on these lines: {3, 3923}, {6, 20864}, {19, 25}, {48, 2309}, {159, 16559}, {1284, 13738}, {1604, 20873}, {2053, 20471}, {2176, 5017}, {2183, 2209}, {2933, 20872}, {16434, 20545}

X(20877) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(1931)

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

See X(20871).

X(20877) lies on these lines: {2, 3}, {6, 20865}, {110, 20766}, {1495, 20754}, {2054, 20675}, {3009, 20998}, {3444, 20990}, {17798, 21004}

X(20878) = (X(1), X(2), X(6), X(31); X(3), X(55), X(1631), X(22)) COLLINEATION IMAGE OF X(3009)

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

See X(20871).

X(20878) lies on these lines: {1, 3}, {6, 20868}, {11, 19522}, {100, 20352}, {237, 8299}, {518, 20777}, {814, 7255}, {1755, 20750}, {3226, 20475}, {4366, 19308}, {8053, 8266}, {20873, 20875}

X(20879) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (2 a^4 - 3 a^2 b^2 + b^4 - 3 a^2 c^2 - 2 b^2 c^2 + c^4) : :
Barycentrics 2 Cos[A] + Cos[B - C] : :

X(20879) lies on these lines: {8, 5884}, {19, 27}, {38, 1733}, {321, 20881}, {908, 4359}, {1150, 20237}, {1232, 21012}, {2975, 4647}, {3218, 6358}, {3219, 4858}, {3262, 4001}, {4714, 5176}, {6507, 18695}, {16585, 18662}, {17168, 17438}, {20889, 20903}

X(20879) = isotomic conjugate of isogonal conjugate of X(17438)
X(20879) = trilinear product X(2)*X(140)

X(20880) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20880) lies on these lines: {2, 277}, {7, 8}, {10, 1111}, {21, 99}, {27, 16747}, {63, 169}, {72, 20347}, {76, 3263}, {142, 1229}, {150, 5086}, {273, 4200}, {304, 3702}, {318, 1847}, {321, 1930}, {348, 10527}, {354, 16708}, {404, 1447}, {517, 20244}, {664, 4861}, {728, 4659}, {894, 17686}, {942, 20247}, {964, 10436}, {1010, 16749}, {1071, 15970}, {1086, 3721}, {1125, 7264}, {1233, 3925}, {1269, 20336}, {1385, 17136}, {1446, 6734}, {2170, 20257}, {2292, 3663}, {2329, 9317}, {2475, 4911}, {2476, 7179}, {2886, 3665}, {2975, 5088}, {3006, 3933}, {3120, 4920}, {3241, 17158}, {3244, 7278}, {3419, 21285}, {3434, 17170}, {3598, 6904}, {3693, 6706}, {3740, 18142}, {3757, 7411}, {3760, 4358}, {3761, 4696}, {3869, 17753}, {3872, 4350}, {3902, 17143}, {3953, 17205}, {4198, 5342}, {4202, 4357}, {4208, 4385}, {4688, 16732}, {4723, 20925}, {4742, 18156}, {4847, 10481}, {4980, 17294}, {5208, 10471}, {5273, 19804}, {5308, 20173}, {6646, 17680}, {7200, 17448}, {7223, 12513}, {10914, 21272}, {1105 7, 17579}, {11680, 17181}, {16465, 17140}, {16720, 21264}, {16727, 16887}, {17046, 21029}, {17050, 17451}, {17116, 17741}, {17274, 17679}, {17862, 20890}, {17864, 20901}, {20236, 20905}, {20433, 20899}, {20906, 21129}

X(20880) = isotomic conjugate of X(2346)
X(20880) = complement of X(25237)
X(20880) = anticomplement of X(16601)

X(20881) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20881) lies on these lines: {8, 2801}, {9, 75}, {63, 20237}, {321, 20879}, {527, 3262}, {536, 8609}, {545, 16732}, {573, 20633}, {726, 1733}, {1089, 5445}, {1111, 4440}, {1227, 4033}, {2397, 16578}, {2786, 3762}, {3554, 3875}, {3904, 18689}, {3928, 20928}, {4416, 20895}, {4459, 14839}, {4647, 5258}, {4947, 19950}, {4997, 19804}, {14206, 20887}, {17116, 18698}, {17439, 18645}

X(20882) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20882) lies on these lines: {8, 2891}, {36, 4647}, {57, 75}, {63, 1746}, {321, 20879}, {333, 4858}, {1150, 14213}, {1733, 3741}, {1790, 17880}, {3670, 6734}, {3687, 3936}, {3929, 20927}, {4673, 13384}, {5231, 17591}, {5295, 10202}, {14206, 20886}, {17185, 21233}, {17304, 19788}, {17440, 18646}

X(20883) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20883) lies on these lines: {19, 27}, {47, 1733}, {48, 17859}, {82, 162}, {278, 2345}, {281, 4000}, {321, 1848}, {427, 3703}, {607, 4361}, {608, 4363}, {1089, 5142}, {1109, 17872}, {1235, 21016}, {1441, 5236}, {1478, 1845}, {1740, 17901}, {1821, 2148}, {1826, 20236}, {1838, 1861}, {1891, 4968}, {1928, 1969}, {1930, 16747}, {1953, 17858}, {1959, 17865}, {3064, 20909}, {3739, 5089}, {7079, 17681}, {11677, 17860}, {17289, 17923}, {17880, 18161}

X(20883) = isotomic conjugate of X(34055)
X(20883) = pole wrt polar circle of trilinear polar of X(82) (line X(661)X(830))
X(20883) = polar conjugate of X(82)

X(20884) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20884) lies on these lines: {19, 27}, {824, 1577}, {1236, 21017}, {1733, 6149}, {1930, 18717}, {1959, 20902}, {2173, 17882}, {2234, 17901}, {4118, 17900}, {4647, 18719}, {17172, 18669}, {17858, 18041}, {17859, 18042}, {17865, 18672}, {18049, 18693}, {18691, 18713}, {18692, 18714}, {18694, 18715}, {18695, 18716}, {18696, 18718}, {18697, 18720}, {18698, 18721}, {18699, 18722}, {18747, 20236}

X(20885) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3095), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20885) lies on these lines: {2, 3}, {51, 3094}, {154, 21001}, {160, 230}, {184, 1613}, {263, 9777}, {305, 5976}, {385, 20794}, {1184, 3117}, {1196, 2021}, {1634, 8667}, {1993, 11673}, {2187, 3009}, {2548, 10790}, {3051, 11402}, {5201, 9766}, {7735, 20775}, {7763, 9917}, {7778, 8266}, {18371, 19153}

X(20886) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20886) lies on these lines: {75, 3219}, {92, 1150}, {321, 908}, {1109, 3741}, {1211, 4957}, {1441, 18139}, {3262, 3969}, {4358, 6358}, {4359, 4858}, {14206, 20882}, {16732, 17184}, {17173, 17443}, {20629, 20889}, {20891, 20896}

X(20887) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20887) lies on these lines: {8, 6917}, {75, 1150}, {321, 908}, {514, 17894}, {527, 3578}, {726, 1109}, {3262, 3936}, {3891, 17871}, {3911, 4359}, {4358, 4858}, {5014, 17860}, {5739, 17484}, {14206, 20881}, {17174, 17444}, {17862, 18139}, {20435, 20901}, {20629, 20904}

X(20888) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20888) lies on these lines: {1, 4441}, {2, 3760}, {7, 10447}, {8, 3761}, {10, 75}, {39, 21264}, {85, 3671}, {99, 5267}, {142, 21071}, {194, 17030}, {239, 17499}, {274, 350}, {310, 3741}, {314, 3664}, {321, 1930}, {334, 6538}, {349, 9436}, {519, 1909}, {524, 4410}, {536, 1500}, {538, 1107}, {551, 4479}, {609, 16919}, {668, 3626}, {894, 17034}, {993, 1975}, {1086, 21024}, {1089, 3263}, {1111, 4647}, {1235, 1861}, {1574, 9466}, {1575, 3934}, {1655, 16819}, {1698, 18135}, {1848, 16747}, {2238, 4721}, {2886, 3933}, {3120, 17211}, {3244, 17144}, {3501, 4659}, {3634, 18140}, {3661, 17184}, {3686, 3770}, {3687, 17866}, {3702, 14210}, {3706, 4059}, {3720, 16748}, {3729, 3730}, {3734, 4426}, {3739, 16589}, {3828, 18145}, {3948, 4359}, {3963, 4431}, {3997, 17033}, {4058, 17786}, {4253, 17026}, {4363, 17750}, {4377, 4665}, {4386, 7751}, {4396, 5277}, {4671, 17244}, {4686, 20691}, {4696, 4986}, {4717, 17762}, {4754, 20963}, {4791, 20907}, {4980, 20889}, {4999, 6390}, {5011, 17739}, {5179, 20235}, {5248, 16992}, {5280, 17686}, {6063, 10481}, {7031, 17002}, {7195, 7243}, {7263, 20255}, {7754, 20172}, {7760, 20179}, {7794, 20541}, {9238, 17046}, {10030, 10521}, {14994, 17792}, {16705, 19863}, {16829, 21226}, {20518, 21201}

X(20888) = complement of X(25264)
X(20888) = anticomplement of X(25092)

X(20889) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20889) lies on these lines: {31, 3403}, {38, 75}, {63, 16564}, {244, 1920}, {321, 20433}, {756, 1921}, {896, 1965}, {1109, 20898}, {1930, 1959}, {1966, 3112}, {1969, 2181}, {3920, 4495}, {3994, 18152}, {4374, 8042}, {4683, 20345}, {4980, 20888}, {7191, 7244}, {17176, 17445}, {20629, 20886}, {20632, 20891}, {20879, 20903}, {20934, 20944}

X(20889) = isotomic conjugate of isogonal conjugate of X(17445)

X(20890) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20890) lies on these lines: {55, 75}, {304, 17887}, {321, 20431}, {1930, 17864}, {4388, 7112}, {7081, 20940}, {7217, 17047}, {14213, 20435}, {17177, 17447}, {17862, 20880}, {20234, 20627}, {20438, 20635}

X(20890) = complement of X(25246)

X(20891) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20891) lies on these lines: {2, 37}, {8, 3781}, {38, 21080}, {69, 3765}, {76, 3662}, {141, 313}, {142, 20913}, {239, 314}, {257, 1921}, {320, 3770}, {335, 1240}, {518, 4696}, {561, 6374}, {594, 3264}, {668, 17287}, {714, 4022}, {726, 1089}, {740, 1193}, {942, 4385}, {982, 17157}, {984, 3701}, {1086, 1269}, {1107, 16738}, {1230, 17184}, {1441, 7146}, {1654, 3975}, {1909, 17300}, {1930, 17760}, {2228, 21238}, {3057, 3696}, {3230, 16827}, {3250, 20906}, {3596, 3661}, {3663, 4044}, {3728, 3741}, {3729, 20367}, {3760, 17304}, {3761, 17298}, {3763, 18044}, {3834, 18143}, {3840, 21330}, {3912, 3963}, {3948, 4357}, {4033, 17229}, {4361, 16685}, {4377, 17231}, {4384, 10447}, {4494, 17286}, {4673, 20036}, {4858, 18697}, {5739, 21279}, {6376, 17238}, {6385, 16703}, {10471, 16819}, {17142, 20358}, {17227, 18144}, {17230, 17786}, {17232, 20917}, {17237, 18133}, {17326, 18140}, {17792, 21278}, {20245, 20248}, {20439, 20636}, {20632, 20889}, {20886, 20896}

X(20892) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20892) lies on these lines: {2, 37}, {7, 3765}, {141, 3264}, {142, 3963}, {239, 20228}, {313, 1086}, {314, 17117}, {561, 6383}, {646, 17268}, {668, 17288}, {726, 3701}, {740, 1201}, {850, 20508}, {1269, 7263}, {1930, 20899}, {3123, 21257}, {3596, 3662}, {3663, 3948}, {3696, 3893}, {3702, 21214}, {3770, 7321}, {3834, 18040}, {3902, 4709}, {3975, 6646}, {4033, 17231}, {4110, 17230}, {4377, 18143}, {4494, 17282}, {4609, 6385}, {4858, 20234}, {6376, 17236}, {10009, 20911}, {16722, 17178}, {17232, 17786}, {17235, 18133}, {17290, 18044}, {17324, 18140}, {17792, 20352}, {20236, 20432}, {20436, 20633}, {20906, 21123}

X(20892) = isotomic conjugate of isogonal conjugate of X(17448)

X(20893) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20893) lies on these lines: {7, 16086}, {75, 519}, {76, 4125}, {85, 4737}, {274, 17200}, {321, 1930}, {514, 4374}, {712, 1086}, {1111, 3263}, {1281, 5144}, {2140, 17760}, {3008, 17789}, {3262, 20900}, {3264, 6549}, {3626, 20955}, {3673, 18743}, {3760, 17266}, {4487, 4986}, {4568, 20335}, {4714, 20911}, {4742, 14210}, {17179, 17449}, {17886, 20437}

X(20894) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20894) lies on these lines: {75, 537}, {76, 3992}, {85, 18421}, {274, 7264}, {321, 1930}, {1266, 4424}, {3125, 7263}, {3263, 4125}, {3673, 16832}, {3760, 18743}, {4441, 14210}, {4479, 4975}, {7278, 17144}, {17180, 17450}

X(20895) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20895) lies on these lines: {2, 20270}, {7, 8}, {72, 21273}, {78, 3875}, {86, 4861}, {200, 17151}, {306, 17862}, {312, 5328}, {314, 1320}, {321, 908}, {347, 2123}, {517, 20245}, {536, 3965}, {1229, 2321}, {1385, 18654}, {2170, 20258}, {3057, 17183}, {3263, 3705}, {3264, 20336}, {3306, 4359}, {3596, 3701}, {3663, 4642}, {3672, 7080}, {3673, 4452}, {3713, 4361}, {3718, 4723}, {3872, 10436}, {3895, 10889}, {3912, 20905}, {3998, 18662}, {4021, 6745}, {4345, 4673}, {4357, 6735}, {4360, 4511}, {4416, 20881}, {4420, 17160}, {4686, 16732}, {4712, 21084}, {4847, 17874}, {4882, 17885}, {4967, 5740}, {5552, 17321}, {10446, 14923}, {10447, 11521}, {12610, 21074}, {17452, 21246}, {17658, 20347}, {17880, 20900}, {19809, 20929}, {20234, 20431}, {21030, 21244}

X(20895) = isotomic conjugate of X(1476)
X(20895) = {X(7),X(8)}-harmonic conjugate of isotomic conjugate of X(5555)

X(20896) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20896) lies on these lines: {75, 81}, {226, 306}, {278, 17740}, {388, 17164}, {519, 2650}, {740, 3891}, {1109, 20639}, {1230, 16732}, {1930, 1959}, {2294, 18139}, {2887, 4137}, {3262, 19835}, {4016, 17184}, {4358, 20106}, {4359, 18698}, {17778, 20017}, {20046, 20090}, {20433, 20901}, {20886, 20891}

X(20897) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3098), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20897) lies on these lines: {2, 3}, {32, 1495}, {51, 7772}, {154, 15257}, {157, 11063}, {184, 5007}, {1843, 5158}, {1974, 3284}, {2351, 3456}, {3053, 5191}, {3398, 6800}, {5188, 5651}, {8541, 15860}, {9821, 15066}

X(20897) = isogonal conjugate of isotomic conjugate of X(31670)

X(20898) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20898) lies on these lines: {31, 75}, {38, 1930}, {63, 16545}, {756, 3263}, {1109, 20889}, {3008, 4359}, {3219, 17755}, {16707, 17200}, {17193, 17457}, {21037, 21248}

X(20898) = isotomic conjugate of isogonal conjugate of X(17457)

X(20899) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20899) lies on these lines: {8, 20350}, {75, 330}, {76, 321}, {1930, 20892}, {20237, 20432}, {20433, 20880}, {20628, 20635}

X(20899) = isotomic conjugate of isogonal conjugate of X(17459)

X(20900) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20900) lies on these lines: {75, 537}, {321, 3452}, {646, 18743}, {1266, 4695}, {3262, 20893}, {3264, 3992}, {3762, 14442}, {3875, 4561}, {17195, 17460}, {17880, 20895}, {17886, 18697}

X(20900) = isotomic conjugate of isogonal conjugate of X(17460)

X(20901) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20901) lies on these lines: {75, 100}, {321, 20431}, {693, 15634}, {824, 21339}, {1109, 1111}, {3119, 4858}, {4467, 7004}, {5057, 7112}, {7046, 13577}, {17198, 17463}, {17864, 20880}, {17878, 20902}, {20433, 20896}, {20435, 20887}, {20627, 20639}

X(20901) = isotomic conjugate of isogonal conjugate of X(17463)

X(20902) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20902) lies on these lines: {2, 16599}, {19, 18691}, {48, 75}, {92, 823}, {125, 7068}, {321, 21091}, {339, 21046}, {523, 4081}, {1109, 2632}, {1733, 8766}, {1930, 18671}, {1953, 17858}, {1959, 20884}, {2173, 18699}, {2286, 17118}, {2294, 18692}, {3708, 4466}, {3942, 17880}, {4431, 18674}, {4605, 6358}, {4647, 18673}, {4858, 17761}, {6508, 14213}, {7124, 17119}, {17438, 17859}, {17442, 18693}, {17446, 17900}, {17878, 20901}, {18669, 18694}, {18670, 18695}, {18672, 18696}, {18675, 18698}, {18722, 20916}, {21252, 21340}

X(20902) = anticomplement of X(16599)
X(20902) = pole wrt polar circle of trilinear polar of X(24000) (line X(162)X(163))
X(20902) = polar conjugate of X(24000)
X(20902) = isotomic conjugate of isogonal conjugate of X(3708)
X(20902) = isotomic conjugate of polar conjugate of X(1109)
X(20902) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 23964}, {4, 23357}, {19, 1101}, , {48, 24000}, {92, 23995}
X(20902) = trilinear product X(i)*X(j) for these {i,j}: {2, 125}, {63, 1109}, {523, 525}, {656, 1577}

X(20903) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20903) lies on these lines: {63, 20627}, {75, 799}, {9396, 17882}, {14206, 20904}, {17199, 17467}, {20879, 20889}

X(20904) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20904) lies on these lines: {75, 896}, {661, 17893}, {1930, 1959}, {14206, 20903}, {14210, 17897}, {17204, 17472}, {17871, 18156}, {20629, 20887}

X(20905) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20905) lies on these lines: {2, 37}, {7, 14524}, {85, 10004}, {92, 1119}, {142, 1441}, {322, 4869}, {857, 12610}, {894, 1462}, {1125, 17869}, {1446, 21258}, {3187, 17811}, {3262, 17234}, {3662, 17435}, {3701, 8582}, {3702, 8583}, {3886, 19861}, {3912, 20895}, {4008, 16020}, {4859, 17861}, {5905, 18928}, {7205, 18031}, {7321, 18151}, {10582, 17860}, {13567, 17184}, {17023, 18690}, {20236, 20880}

X(20905) = complement of X(25243)

X(20906) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20906) lies on these lines: {2, 21348}, {75, 513}, {239, 3063}, {274, 4378}, {321, 4079}, {325, 523}, {514, 4374}, {522, 3766}, {650, 14296}, {661, 17893}, {824, 4391}, {894, 20980}, {918, 4462}, {1441, 20504}, {3250, 20891}, {3762, 21130}, {3777, 20512}, {4083, 17217}, {4086, 4509}, {4361, 21007}, {4406, 4977}, {4408, 4777}, {4411, 4802}, {4449, 17215}, {4762, 20950}, {4775, 17143}, {7628, 21183}, {14349, 20629}, {20880, 21129}, {20892, 21123}, {21055, 21262}

X(20906) = isotomic conjugate of X(932)
X(20906) = anticomplement of X(21348)

X(20907) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20907) lies on these lines: {1, 17215}, {7, 20293}, {75, 522}, {514, 4374}, {523, 4411}, {657, 4384}, {693, 17894}, {850, 4025}, {900, 4408}, {1459, 10436}, {1577, 17893}, {2517, 4509}, {3667, 3766}, {3739, 6586}, {4086, 15413}, {4357, 20316}, {4406, 4778}, {4449, 17218}, {4699, 21225}, {4791, 20888}, {6005, 17159}, {6590, 20909}, {17066, 21348}, {21178, 21180}

X(20908) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20908) lies on these lines: {75, 812}, {312, 4928}, {321, 4728}, {514, 4374}, {523, 2530}, {693, 4838}, {764, 20512}, {786, 4481}, {824, 1577}, {850, 16892}, {918, 1086}, {1635, 4359}, {2786, 3766}, {4155, 4647}, {4369, 20952}, {4444, 18895}, {4763, 19804}, {14838, 21225}

X(20909) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20909) lies on these lines: {75, 649}, {244, 21197}, {321, 3835}, {514, 17894}, {661, 17893}, {693, 4838}, {824, 850}, {3064, 20883}, {3250, 20637}, {3261, 4024}, {3676, 6358}, {4382, 20950}, {4408, 4820}, {4468, 14213}, {4521, 4858}, {4785, 4980}, {4813, 20949}, {6590, 20907}

X(20910) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20910) lies on these lines: {75, 798}, {313, 21055}, {661, 17893}, {824, 1577}, {14207, 14213}

X(20911) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20911) lies on these lines: {2, 304}, {7, 8}, {10, 1930}, {22, 18616}, {37, 17489}, {41, 16822}, {76, 321}, {81, 239}, {86, 5262}, {141, 3721}, {257, 1921}, {279, 7182}, {312, 18135}, {314, 17863}, {315, 5016}, {333, 7291}, {348, 17080}, {350, 3702}, {517, 17152}, {668, 4696}, {693, 18015}, {742, 2295}, {976, 3905}, {1089, 6381}, {1111, 4647}, {1125, 14210}, {1211, 1228}, {1254, 9436}, {1446, 6063}, {1575, 16720}, {1655, 3797}, {2082, 4384}, {2292, 4357}, {3125, 21240}, {3210, 4352}, {3218, 17206}, {3454, 17211}, {3616, 18156}, {3626, 4986}, {3666, 16705}, {3670, 16887}, {3673, 4441}, {3674, 3687}, {3691, 17755}, {3701, 6376}, {3718, 5232}, {3739, 17497}, {3902, 17144}, {3926, 17740}, {4109, 4766}, {4167, 17062}, {4320, 9312}, {4358, 17292}, {4372, 4386}, {4376, 4426}, {4714, 20893}, {5015, 20553}, {5178, 20552}, {5222, 19804}, {5224, 20336}, {5813, 14555}, {6734, 17875}, {8024, 19835}, {9534, 20235}, {10009, 20892}, {10447, 17861}, {10471, 17866}, {15523, 20590}, {16886, 20541}, {17135, 20247}, {17164, 20347}, {17322, 20932}, {17495, 18600}, {21024, 21138}

X(20911) = isotomic conjugate of X(2298)

X(20912) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20912) lies on these lines: {75, 524}, {226, 306}, {316, 17482}, {514, 4374}, {3262, 20432}, {3948, 16732}, {16581, 21094}, {21048, 21256}

X(20913) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

X(20913) lies on these lines: {2, 39}, {10, 38}, {37, 1269}, {44, 4410}, {75, 141}, {81, 17034}, {85, 5244}, {142, 20891}, {183, 11329}, {239, 1909}, {241, 349}, {312, 17244}, {313, 3739}, {314, 17300}, {319, 20174}, {321, 1930}, {350, 16826}, {377, 5208}, {379, 1150}, {730, 21352}, {940, 19281}, {964, 10458}, {1078, 19308}, {1213, 18133}, {1235, 15149}, {1500, 17147}, {1920, 18891}, {1975, 16367}, {1999, 19787}, {3009, 12263}, {3264, 4377}, {3596, 4699}, {3687, 20436}, {3688, 17142}, {3734, 11320}, {3760, 16831}, {3761, 3765}, {3770, 17277}, {3975, 16815}, {4043, 17243}, {4044, 4358}, {4385, 8728}, {4441, 17316}, {5224, 18144}, {6376, 18136}, {6542, 17143}, {7081, 16056}, {10447, 17298}, {12782, 17155}, {15488, 15971}, {15668, 18147}, {16709, 17398}, {16823, 16850}, {16994, 19224}, {17031, 20985}, {17049, 21278}, {17116, 17787}, {17144, 17389}, {17165, 20683}, {17245, 18137}, {17303, 18044}

X(20914) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20914) lies on these lines: {2, 85}, {4, 75}, {76, 5179}, {92, 349}, {169, 1760}, {253, 322}, {304, 4417}, {312, 857}, {329, 1231}, {379, 19804}, {1441, 2551}, {3732, 18596}, {4687, 18721}, {5142, 18738}, {5813, 14555}, {6376, 18749}

X(20914) = isotomic conjugate of X(7097)

X(20915) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20915) lies on these lines: {19, 27}, {304, 18715}, {1959, 18672}, {2964, 4008}

X(20916) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (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(20916) lies on these lines: {19, 27}, {1577, 20950}, {1959, 20941}, {18722, 20902}

X(20917) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20917) lies on these lines: {2, 330}, {7, 17787}, {8, 20358}, {10, 982}, {37, 18144}, {48, 18048}, {75, 141}, {76, 85}, {86, 18044}, {142, 3596}, {274, 17308}, {313, 17234}, {314, 17296}, {321, 17230}, {327, 2186}, {350, 17316}, {388, 18141}, {646, 4659}, {668, 4384}, {870, 16826}, {940, 19806}, {1269, 17233}, {1959, 18055}, {1999, 19803}, {3210, 20691}, {3419, 7270}, {3705, 20486}, {3761, 17284}, {3770, 17279}, {3834, 4377}, {3948, 17056}, {4043, 17240}, {4385, 18208}, {4410, 17359}, {4417, 20449}, {4445, 20174}, {4479, 17310}, {4494, 6173}, {4517, 17794}, {4675, 17790}, {4687, 18133}, {5236, 7017}, {5308, 18135}, {5834, 16284}, {6542, 17144}, {7247, 19807}, {10436, 18065}, {16831, 18140}, {17063, 20340}, {17143, 17294}, {17232, 20891}, {17241, 18137}, {17317, 18147}, {17394, 18046}, {17788, 20930}, {19584, 21101}

X(20918) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3285), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20918) lies on these lines: {2, 3}, {522, 1324}, {1030, 16675}

X(20919) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20919) lies on these lines: {2, 16732}, {75, 3219}, {92, 264}, {190, 14213}, {321, 4886}, {333, 20236}, {561, 20643}, {1233, 7112}, {3112, 6654}, {4676, 17871}, {5745, 19804}, {14206, 14829}, {18137, 20929}, {18142, 20940}, {18152, 20944}

X(20920) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20920) lies on these lines: {75, 1150}, {92, 264}, {190, 14206}, {319, 321}, {333, 6358}, {514, 17789}, {1109, 17763}, {1978, 20643}, {3769, 17871}, {3911, 19804}, {3936, 17791}, {4358, 18151}, {14213, 14829}, {20448, 20940}

X(20921) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b*c*(-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) : :

X(20921) lies on these lines: {2, 85}, {7, 18928}, {63, 17277}, {75, 329}, {78, 5342}, {92, 264}, {144, 4359}, {189, 18141}, {190, 20223}, {342, 15466}, {1441, 18228}, {1763, 6996}, {2999, 3673}, {3436, 6997}, {3618, 19802}, {3869, 4651}, {4664, 18662}, {5222, 19790}, {5748, 18743}, {6818, 17441}, {7360, 19541}, {10446, 14557}, {11678, 17165}, {17158, 20043}, {17791, 20942}, {18134, 20946}

X(20922) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20922) lies on these lines: {55, 75}, {85, 18045}, {92, 20448}, {304, 20926}, {312, 8024}, {4417, 7112}, {18138, 20444}, {18142, 20927}, {20451, 20649}

X(20923) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20923) lies on these lines: {2, 37}, {69, 3975}, {76, 142}, {85, 6385}, {141, 6376}, {244, 17157}, {304, 1921}, {313, 17234}, {314, 4384}, {325, 21239}, {341, 518}, {668, 17296}, {740, 978}, {982, 21080}, {984, 3831}, {1909, 4648}, {3264, 4110}, {3596, 3912}, {3662, 3948}, {3696, 4673}, {3718, 17755}, {3760, 4859}, {3765, 17300}, {3770, 4675}, {3834, 18144}, {3963, 17244}, {4033, 17240}, {4361, 16827}, {6381, 21255}, {6383, 20335}, {10447, 16832}, {10472, 16819}, {16817, 19282}, {17227, 18133}, {17241, 18040}, {17283, 18044}, {17306, 18140}, {17789, 20927}, {18151, 20444}, {20449, 20647}, {20719, 21281}

X(20923) = isotomic conjugate of isogonal conjugate of X(21384)

X(20924) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20924) lies on these lines: {10, 20955}, {39, 7187}, {69, 16086}, {75, 519}, {76, 85}, {81, 239}, {99, 5088}, {142, 17788}, {257, 21240}, {279, 3926}, {315, 17170}, {316, 4872}, {320, 758}, {321, 17310}, {325, 1565}, {335, 712}, {345, 17079}, {348, 7763}, {350, 1111}, {514, 1921}, {538, 3797}, {668, 3263}, {760, 4645}, {766, 3888}, {1016, 1275}, {1269, 20932}, {1447, 5977}, {1909, 1930}, {2795, 3685}, {3262, 4555}, {3264, 17791}, {3596, 20930}, {3662, 3735}, {3666, 16712}, {3673, 18156}, {3739, 16724}, {3766, 6550}, {3834, 21331}, {3902, 17143}, {4358, 4945}, {4562, 18895}, {4673, 12563}, {4717, 17762}, {4812, 17391}, {5249, 20929}, {6381, 18159}, {6542, 20432}, {7270, 7768}, {7752, 17181}, {7769, 17095}, {7799, 17078}, {16711, 17495}, {17234, 20444}, {17266, 18140}, {17300, 20234}, {18061, 20335}, {18146, 18743}, {20450, 20951}

X(20924) = isotomic conjugate of X(2161)

X(20925) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20925) lies on these lines: {69, 3419}, {75, 537}, {76, 85}, {183, 5088}, {274, 8056}, {320, 1478}, {1269, 20930}, {1847, 1969}, {1909, 3673}, {3729, 21232}, {3760, 4975}, {3765, 4359}, {3902, 4441}, {4403, 9466}, {4680, 17360}, {4723, 20880}, {4872, 11185}, {16284, 17143}, {18143, 20927}, {18145, 18743}, {18146, 20569}

X(20925) = isotomic conjugate of polar conjugate of X(3294)

X(20926) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20926) lies on these lines: {3, 75}, {55, 17887}, {85, 18359}, {304, 20922}, {312, 857}, {664, 11109}, {4872, 17492}, {5074, 7112}, {6376, 20951}, {14963, 18050}

X(20926) = isogonal conjugate of X(7139)
X(20926) = isotomic conjugate of X(7094)

X(20927) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20927) lies on these lines: {2, 15474}, {6, 20171}, {9, 75}, {69, 1229}, {76, 5179}, {80, 4737}, {85, 17234}, {92, 264}, {239, 2911}, {304, 18137}, {321, 14555}, {341, 5587}, {344, 1441}, {346, 3262}, {857, 1234}, {1111, 17282}, {1479, 4523}, {1760, 6996}, {3553, 4360}, {3618, 17863}, {3673, 7803}, {3718, 5816}, {3912, 20930}, {3929, 20882}, {4008, 4676}, {4043, 21078}, {4123, 14004}, {4957, 17340}, {5219, 18044}, {5342, 7270}, {5747, 18147}, {6376, 20547}, {14829, 18750}, {16284, 17295}, {16732, 17279}, {17240, 17791}, {17353, 17861}, {17789, 20923}, {17862, 18928}, {18031, 20642}, {18142, 20922}, {18143, 20925}, {18152, 20641}

X(20927) = isotomic conjugate of anticomplement of complementary conjugate of X(34847)

X(20928) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20928) lies on these lines: {2, 3262}, {57, 75}, {69, 321}, {92, 264}, {190, 3719}, {226, 20930}, {304, 20922}, {318, 7270}, {1897, 4123}, {1978, 20641}, {2064, 14615}, {3769, 4008}, {3827, 20557}, {3928, 20881}, {4673, 7982}, {5928, 21277}, {10453, 18839}, {16284, 17294}, {17763, 17871}, {17789, 20449}, {17862, 18141}, {18151, 20942}, {18747, 21062}

X(20929) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20929) lies on these lines: {2, 17788}, {75, 81}, {92, 304}, {306, 2064}, {312, 1230}, {321, 1909}, {322, 19799}, {345, 6360}, {1441, 19810}, {1999, 20234}, {3262, 19811}, {4463, 7270}, {5249, 20924}, {18134, 18714}, {18137, 20919}, {18138, 20446}, {19809, 20895}

X(20930) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20930) lies on these lines: {7, 8}, {63, 21231}, {86, 998}, {92, 914}, {226, 20928}, {264, 20570}, {273, 664}, {304, 313}, {309, 20566}, {312, 1230}, {314, 17098}, {326, 9312}, {355, 21276}, {1111, 3875}, {1150, 19804}, {1269, 20925}, {3007, 18133}, {3596, 20924}, {3673, 4360}, {3758, 15988}, {3761, 18697}, {3879, 17861}, {3912, 20927}, {3957, 17393}, {4654, 20237}, {4851, 16732}, {4858, 17298}, {5307, 8897}, {5736, 17394}, {5748, 18743}, {6350, 18750}, {6376, 18749}, {7146, 20647}, {8257, 17277}, {10587, 17321}, {16817, 19285}, {17220, 21272}, {17241, 18151}, {17270, 18698}, {17296, 20236}, {17315, 20173}, {17786, 17789}, {17788, 20917}, {18147, 18156}, {18589, 18747}, {20347, 21271}, {20945, 20947}

X(20930) = isotomic conjugate of X(90)

X(20931) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20931) lies on these lines: {1, 29}, {75, 2172}, {304, 1760}, {610, 1930}, {18049, 18058}

X(20932) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics 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(20932) lies on these lines: {1, 75}, {69, 2836}, {306, 18720}, {312, 1230}, {313, 502}, {319, 20336}, {321, 17315}, {322, 3260}, {1269, 20924}, {3263, 5564}, {3596, 20937}, {3718, 17360}, {3912, 18714}, {4043, 17789}, {15523, 20947}, {17233, 20445}, {17322, 20911}, {17788, 18137}, {18133, 20955}, {20538, 21289}

X(20932) = isotomic conjugate of X(267)

X(20933) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20933) lies on these lines: {2, 16720}, {75, 83}, {76, 18744}, {1031, 17788}, {6376, 20444}, {18050, 20951}

X(20934) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20934) lies on these lines: {2, 16720}, {31, 75}, {69, 3974}, {304, 9239}, {561, 17957}, {1930, 1965}, {1966, 17884}, {3403, 17890}, {15523, 20955}, {17135, 17762}, {18133, 18138}, {20889, 20944}

X(20935) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics 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) : :

X(20935) lies on these lines: {8, 2898}, {69, 350}, {75, 1088}, {85, 2886}, {200, 4554}, {312, 4437}, {319, 16090}, {322, 325}, {673, 2319}, {693, 3681}, {948, 1909}, {2550, 7196}, {3061, 3452}, {18056, 19806}

X(20936) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20936) lies on these lines: {75, 330}, {85, 20446}, {304, 1921}, {312, 17230}, {698, 17762}, {6376, 20532}, {17760, 17786}, {17789, 20449}, {20642, 20649}

X(20937) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20937) lies on these lines: {75, 537}, {304, 20938}, {312, 3969}, {3262, 4723}, {3264, 17791}, {3596, 20932}, {16284, 18816}, {20445, 20496}

X(20938) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20938) lies on these lines: {75, 519}, {304, 20937}, {312, 17791}

X(20939) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20939) lies on these lines: {75, 799}, {92, 1934}, {561, 17957}, {1821, 18750}, {1966, 14206}, {3120, 18032}, {4671, 17762}, {14212, 18056}, {17777, 20538}, {18060, 20941}, {18066, 18159}, {18149, 18151}, {20450, 20947}, {20641, 20945}

X(20940) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20940) lies on these lines: {75, 100}, {85, 18359}, {92, 18031}, {304, 1978}, {312, 8024}, {321, 20533}, {908, 7112}, {7081, 20890}, {18066, 18159}, {18138, 20446}, {18142, 20919}, {20448, 20920}

X(20941) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-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(20941) lies on these lines: {48, 75}, {897, 17876}, {1930, 16563}, {1959, 20916}, {2349, 18750}, {17233, 20445}, {18060, 20939}, {18061, 18151}

X(20942) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20942) lies on these lines: {2, 37}, {69, 8055}, {190, 3928}, {226, 17241}, {304, 18145}, {319, 18228}, {329, 17361}, {341, 519}, {518, 4903}, {551, 4385}, {3241, 3701}, {3452, 17233}, {3679, 4673}, {3685, 4421}, {3769, 4011}, {3790, 3816}, {3829, 3932}, {3912, 20943}, {3929, 14829}, {3992, 4677}, {4009, 10453}, {4135, 17063}, {4387, 5205}, {4415, 17227}, {4417, 17240}, {4428, 7081}, {4647, 19876}, {4656, 17249}, {4737, 4975}, {7283, 16371}, {11679, 17335}, {16817, 19536}, {16833, 17144}, {17791, 20921}, {18151, 20928}

X(20942) = isotomic conjugate of X(36603)

X(20943) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20943) lies on these lines: {1, 18145}, {8, 4479}, {10, 75}, {85, 20947}, {145, 350}, {304, 18159}, {312, 17230}, {330, 20530}, {561, 3994}, {668, 3632}, {1125, 18146}, {1575, 20081}, {1909, 3616}, {3624, 3761}, {3644, 20691}, {3661, 4415}, {3662, 21025}, {3814, 7796}, {3834, 18144}, {3912, 20942}, {3943, 17786}, {3948, 17056}, {4080, 18066}, {4386, 17128}, {4400, 16916}, {4426, 17129}, {4441, 4678}, {4668, 17143}, {4713, 17752}, {4871, 6384}, {4892, 18067}, {4903, 17090}, {5087, 20449}, {9466, 17030}, {9902, 17793}, {10449, 17360}, {17149, 18152}, {17228, 21024}, {17240, 21071}, {17342, 18073}, {17448, 21219}, {17787, 20073}

X(20943) = isotomic conjugate of X(36598)
X(20943) = anticomplement of X(32005)

X(20944) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20944) lies on these lines: {75, 896}, {92, 304}, {661, 786}, {799, 14206}, {1109, 1966}, {1580, 3112}, {1965, 20627}, {1978, 20643}, {14210, 18075}, {18152, 20919}, {20889, 20934}

X(20945) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20945) lies on these lines: {31, 18075}, {38, 75}, {312, 17230}, {1707, 1966}, {1920, 6376}, {1921, 6384}, {1965, 18056}, {3706, 4479}, {17472, 18069}, {20641, 20939}, {20930, 20947}

X(20945) = isotomic conjugate of X(38275)

X(20946) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20946) lies on these lines: {2, 37}, {85, 17234}, {190, 1445}, {273, 6335}, {282, 309}, {322, 3912}, {341, 938}, {936, 3886}, {1210, 3717}, {1332, 3759}, {1998, 3699}, {3673, 17282}, {3948, 18635}, {7283, 16410}, {17241, 18151}, {18134, 20921}, {18136, 18751}, {18141, 18750}

X(20947) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20947) lies on these lines: {2, 37}, {7, 4903}, {10, 17762}, {85, 20943}, {86, 1215}, {190, 3509}, {210, 319}, {257, 21025}, {274, 1089}, {304, 6376}, {313, 1920}, {320, 4009}, {325, 3932}, {334, 3948}, {341, 18156}, {668, 3992}, {693, 4036}, {765, 4600}, {985, 4676}, {1111, 18145}, {1655, 16720}, {1757, 17731}, {1909, 3701}, {1921, 20446}, {1926, 6386}, {1930, 18140}, {2227, 20711}, {3264, 4087}, {3596, 4485}, {3685, 8301}, {3706, 5564}, {3807, 3930}, {3879, 4090}, {4075, 16887}, {4372, 16916}, {4376, 16997}, {4553, 20723}, {4583, 18157}, {4645, 20716}, {4975, 4986}, {5282, 17336}, {5311, 17394}, {6381, 18159}, {6541, 18035}, {6542, 20529}, {6651, 17735}, {7283, 19329}, {15523, 20932}, {16825, 17144}, {16886, 17669}, {17017, 17393}, {18133, 18138}, {18151, 20646}, {19582, 21281}, {20450, 20939}, {20538, 20722}, {20930, 20945}

X(20948) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20948) is the trilinear pole of line X(1109)X(21207), which is the tangent to the inellipse that is the trilinear square of the de Longchamps line, at X(1109) (the trilinear square of X(523)). (Randy Hutson, October 15, 2018)

X(20948): Let P1 and P2 be the two points on the de Longchamps line whose trilinear polars are parallel to the de Longchamps line. P1 and P2 lie on the Kiepert hyperbola and circle {{X(2), X(98), X(99)}}. X(20948) is the trilinear product P1*P2. (Randy Hutson, October 15, 2018)

X(20948) lies on these lines: {75, 656}, {76, 18160}, {92, 14209}, {313, 4086}, {661, 786}, {792, 8630}, {824, 1577}, {850, 4036}, {4043, 4171}, {8062, 18147}, {14208, 18076}

X(20948) = isotomic conjugate of X(163)
X(20948) = crossdifference of every pair of points on line X(560)X(1917)
X(20948) = polar conjugate of X(32676)

X(20949) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20949) lies on these lines: {75, 513}, {239, 21007}, {312, 4776}, {514, 1921}, {523, 3766}, {661, 786}, {693, 4036}, {3063, 3759}, {3250, 18080}, {3758, 20980}, {3762, 4509}, {3835, 17458}, {4043, 4079}, {4132, 20295}, {4374, 4977}, {4391, 18158}, {4406, 4778}, {4462, 15413}, {4687, 21348}, {4775, 17144}, {4813, 20909}, {14349, 18081}, {15419, 21222}, {16709, 17212}, {16755, 17496}, {21051, 21349}, {21055, 21261}

X(20950) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20950) lies on these lines: {75, 812}, {312, 4728}, {321, 21297}, {335, 918}, {514, 1921}, {693, 20952}, {824, 20954}, {850, 18071}, {1019, 10566}, {1577, 20916}, {1635, 19804}, {2517, 4801}, {3762, 18150}, {3837, 4518}, {4033, 4568}, {4382, 20909}, {4391, 4408}, {4762, 20906}, {4928, 18743}, {7200, 16726}, {16892, 18155}

X(20951) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20951) lies on these lines: {8, 7261}, {75, 99}, {3596, 20932}, {6376, 20926}, {17789, 20648}, {18050, 20933}, {18061, 18151}, {18066, 18159}, {20450, 20924}

X(20951) = isotomic conjugate of X(39137)

X(20952) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20952) lies on these lines: {75, 649}, {92, 3064}, {312, 3835}, {321, 20295}, {514, 17789}, {650, 14296}, {661, 786}, {693, 20950}, {772, 3250}, {824, 18155}, {3261, 6590}, {3700, 3766}, {4024, 20954}, {4369, 20908}

X(20953) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20953) lies on these lines: {75, 798}, {92, 14207}, {313, 21099}, {321, 4132}, {661, 786}, {1577, 20916}

X(20954) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20954) lies on these lines: {37, 21225}, {75, 522}, {86, 21173}, {319, 20293}, {320, 350}, {514, 4079}, {523, 3766}, {649, 18154}, {657, 17335}, {788, 18081}, {798, 812}, {824, 20950}, {900, 4374}, {1459, 17394}, {2533, 9400}, {3667, 4406}, {3673, 21182}, {4024, 20952}, {4130, 4391}, {4382, 18071}, {4408, 4777}, {4411, 4926}, {4455, 18077}, {4687, 6586}, {4785, 4823}, {4791, 6376}, {4985, 18160}, {16709, 16755}, {18072, 18133}, {20955, 21132}, {21178, 21185}, {21179, 21205}

X(20955) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20955) lies on these lines: {7, 8}, {10, 20924}, {76, 4485}, {141, 257}, {274, 17731}, {304, 6376}, {312, 17230}, {668, 1930}, {742, 17752}, {1111, 17143}, {1575, 7187}, {3626, 20893}, {3661, 17789}, {3662, 3959}, {3673, 17144}, {4417, 7146}, {4479, 4673}, {4760, 17692}, {4812, 17373}, {7182, 9364}, {8682, 17034}, {14210, 18140}, {14829, 16609}, {15523, 20934}, {16816, 19804}, {17152, 21272}, {17228, 20444}, {17287, 20234}, {18133, 20932}, {20954, 21132}

X(20956) = (A,B,C,X(2); A',B',C',X(75)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(20956) lies on these lines: {75, 524}, {257, 17227}, {312, 1230}, {514, 1921}, {4033, 17789}, {4359, 16704}, {4892, 17472}, {16581, 18745}, {17788, 18143}, {17790, 17953}

X(20957) = MIDPOINT OF X(4) AND X(14731)

Barycentrics a^16-3 a^14 b^2+3 a^12 b^4-4 a^10 b^6+10 a^8 b^8-11 a^6 b^10+3 a^4 b^12+2 a^2 b^14-b^16-3 a^14 c^2+8 a^12 b^2 c^2-4 a^10 b^4 c^2-10 a^8 b^6 c^2+14 a^6 b^8 c^2+a^4 b^10 c^2-11 a^2 b^12 c^2+5 b^14 c^2+3 a^12 c^4-4 a^10 b^2 c^4+9 a^8 b^4 c^4-4 a^6 b^6 c^4-15 a^4 b^8 c^4+21 a^2 b^10 c^4-10 b^12 c^4-4 a^10 c^6-10 a^8 b^2 c^6-4 a^6 b^4 c^6+22 a^4 b^6 c^6-12 a^2 b^8 c^6+11 b^10 c^6+10 a^8 c^8+14 a^6 b^2 c^8-15 a^4 b^4 c^8-12 a^2 b^6 c^8-10 b^8 c^8-11 a^6 c^10+a^4 b^2 c^10+21 a^2 b^4 c^10+11 b^6 c^10+3 a^4 c^12-11 a^2 b^2 c^12-10 b^4 c^12+2 a^2 c^14+5 b^2 c^14-c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28036.

X(20957) lies on these lines: {3,3258}, {4,14670}, {5,476}, {30,110}, {74,16340}, {265,523}, {381,2453}, {546,18319}, {1624,2070}, {3001,6033}, {3154,15061}, {3470,3627}, {5627,11801}, {5663,17511}, {6243,16978}, {6787,9996}, {7471,14643}, {11799,12918}, {12028,14674}, {12030,13743}, {12079,15027}, {12091,18404}, {14851,20127}, {15112,18377}, {18697,20897}, {20641,20918}

X(20957) = midpoint of X(i) and X(j) for these {i,j}: {4, 14731}, {3627, 11749}
X(20957) = reflection of X(i) in X(j) for these {i,j}: {3, 3258}, {74, 16340}, {476, 5}, {6243, 16978}, {12121, 14934}, {14989, 3627}, {18319, 546}
X(20957) = reflection of X(265) in the Euler line

X(20958) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20958) lies on these lines: {1, 575}, {6, 692}, {42, 13366}, {181, 215}, {651, 4014}, {899, 3292}, {1404, 2223}, {1495, 20962}, {1570, 20861}, {1960, 9262}, {2265, 4516}, {2323, 20683}, {3035, 18645}, {3157, 17114}, {3240, 11422}, {4557, 17455}, {4579, 14839}, {5007, 18758}, {5040, 20976}

X(20958) = crosssum of X(2) and X(11)
X(20958) = crosspoint of X(6) and X(59)
X(20958) = crossdifference of every pair of points on line X(918)X(4440)
X(20958) = isogonal conjugate of isotomic conjugate of X(3035)
X(20958) = polar conjugate of isotomic conjugate of X(22055)

X(20959) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20959) lies on these lines: {6, 181}, {42, 13366}, {43, 575}, {215, 2194}, {1197, 1692}, {1402, 2317}, {1495, 20961}, {2308, 3724}, {3292, 3720}, {4999, 18646}, {11422, 17018}

X(20959) = crosssum of X(2) and X(12)
X(20959) = crosspoint of X(6) and X(60)
X(20959) = isogonal conjugate of isotomic conjugate of X(4999)
X(20959) = polar conjugate of isotomic conjugate of X(22056)

X(20960) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3313), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20960) lies on these lines: {2, 3}, {32, 206}, {39, 160}, {211, 3202}, {682, 1384}, {1501, 14820}, {1627, 8793}, {1634, 7758}, {1974, 10316}, {2353, 20987}, {3053, 20993}, {3933, 9917}, {5201, 14023}, {7767, 13562}, {7800, 8266}, {9605, 20775}, {10790, 18907}, {13236, 14691}, {18374, 20968}

X(20961) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20961) lies on these lines: {1, 3060}, {6, 20988}, {42, 51}, {43, 5640}, {181, 902}, {511, 3720}, {674, 756}, {748, 4259}, {899, 5943}, {1197, 3124}, {1495, 20959}, {2308, 3271}, {2309, 20966}, {3056, 5311}, {3724, 14547}, {3792, 5284}, {4332, 19366}, {4336, 11436}, {4883, 9037}, {6186, 7113}, {11002, 17018}, {11451, 16569}, {20861, 20965}

X(20961) = isogonal conjugate of isotomic conjugate of X(25639)
X(20961) = polar conjugate of isotomic conjugate of X(22058)
X(20961) = {X(42),X(51)}-harmonic conjugate of X(20962)

X(20962) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20962) lies on these lines: {1, 5640}, {6, 20989}, {42, 51}, {43, 3060}, {181, 2308}, {244, 8679}, {375, 756}, {511, 899}, {667, 20456}, {902, 3271}, {1201, 16980}, {1495, 20958}, {1736, 2611}, {1739, 2392}, {2183, 3724}, {2361, 6187}, {2810, 17449}, {2979, 16569}, {3240, 11002}, {3720, 5943}, {3814, 17174}, {3836, 3909}, {4424, 15049}, {6373, 8661}, {9037, 16610}, {20459, 20974}, {20861, 20977}

X(20962) = isogonal conjugate of isotomic conjugate of X(3814)
X(20962) = polar conjugate of isotomic conjugate of X(22059)
X(20962) = {X(42),X(51)}-harmonic conjugate of X(20961)

X(20963) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20963) lies on these lines: {1, 6}, {8, 17750}, {10, 3780}, {31, 2241}, {32, 1468}, {36, 18755}, {39, 42}, {41, 2242}, {55, 5021}, {56, 2271}, {58, 1914}, {76, 17027}, {81, 239}, {86, 16819}, {141, 16818}, {172, 2251}, {194, 712}, {292, 1126}, {350, 4721}, {354, 16583}, {384, 20180}, {386, 2275}, {519, 2295}, {524, 4503}, {604, 2304}, {672, 1500}, {758, 3727}, {894, 17143}, {940, 4384}, {942, 3125}, {980, 5256}, {1015, 1193}, {1125, 2238}, {1258, 3227}, {1400, 4263}, {1409, 4332}, {1574, 3214}, {1575, 3293}, {1909, 17034}, {2082, 20229}, {2092, 2260}, {2170, 2650}, {2268, 5042}, {2269, 10544}, {2276, 4253}, {2277, 4270}, {2279, 2334}, {2303, 16817}, {2308, 3747}, {2653, 20982}, {3009, 20965}, {3051, 21352}, {3063, 4378}, {3216, 16604}, {3244, 3997}, {3290, 5045}, {3303, 14974}, {3664, 17050}, {3666, 6155}, {3684, 5277}, {3691, 3720}, {3721, 3874}, {3726, 3881}, {3735, 3868}, {3739, 17175}, {3746, 17735}, {3758, 17144}, {3767, 11269}, {3959, 5902}, {4065, 17475}, {4273, 7113}, {4352, 17014}, {4383, 16831}, {4502, 4775}, {4559, 11011}, {4644, 17753}, {4667, 20257}, {4754, 20888}, {5007, 8624}, {5115, 5301}, {5202, 20976}, {5271, 19714}, {5276, 16823}, {5439, 16605}, {5563, 21008}, {5839, 9534}, {10452, 15985}, {14597, 18732}, {14996, 16816}, {16549, 20691}, {16827, 17121}, {16828, 17398}, {17451, 20752}, {18163, 20367}, {18398, 20271}, {20459, 20964}, {20463, 20668}, {20861, 20969}

X(20963) = isogonal conjugate of X(32009)
X(20963) = crosssum of X(2) and X(37)
X(20963) = crosspoint of X(6) and X(81)
X(20963) = polar conjugate of isotomic conjugate of X(22060)
X(20963) = isogonal conjugate of isotomic conjugate of X(3739)

X(20964) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20964) lies on these lines: {1, 1258}, {2, 17153}, {6, 292}, {9, 31}, {37, 1918}, {38, 16574}, {42, 181}, {58, 1757}, {78, 1468}, {81, 2663}, {100, 1045}, {171, 385}, {172, 1691}, {213, 6378}, {238, 3842}, {284, 18266}, {314, 17763}, {750, 10436}, {757, 765}, {760, 17470}, {798, 18105}, {813, 5970}, {976, 10477}, {983, 985}, {984, 5156}, {1089, 3923}, {1237, 1966}, {1420, 1471}, {1500, 2670}, {2210, 19133}, {2223, 2309}, {2239, 4357}, {2245, 21035}, {2260, 20456}, {2300, 3009}, {2650, 15556}, {3685, 5255}, {3688, 4274}, {3728, 21061}, {3952, 17126}, {3963, 4039}, {4579, 18787}, {4687, 16690}, {16476, 17349}, {18082, 21238}, {20459, 20963}

X(20964) = isogonal conjugate of X(32010)
X(20964) = complement of X(17153)
X(20964) = crosspoint of X(6) and X(82)
X(20964) = crosssum of X(2) and X(38)
X(20964) = crossdifference of every pair of points on line X(812)X(4560)
X(20964) = isogonal conjugate of isotomic conjugate of X(1215)
X(20964) = polar conjugate of isotomic conjugate of X(22061)

X(20965) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20965) is the perspector of the symmedial triangle and the tangential triangle, wrt the medial triangle, of the bicevian conic of X(2) and X(6). (Randy Hutson, August 29, 2018)

X(20965) lies on these lines: {2, 6}, {23, 10329}, {25, 11175}, {32, 10014}, {39, 51}, {42, 20457}, {83, 1207}, {110, 14153}, {171, 1977}, {182, 1501}, {184, 5034}, {187, 14765}, {213, 21352}, {217, 1899}, {238, 7109}, {251, 1691}, {263, 9777}, {373, 1196}, {427, 2211}, {511, 8041}, {694, 3108}, {899, 1197}, {1180, 3981}, {1186, 7808}, {1194, 3124}, {1627, 12212}, {1915, 5012}, {2076, 15246}, {2235, 4359}, {2295, 18091}, {2308, 8622}, {2309, 20863}, {2451, 10567}, {2970, 14768}, {2979, 13330}, {3009, 20963}, {3060, 3094}, {3117, 7772}, {3229, 5041}, {3266, 4074}, {3291, 6688}, {3331, 5475}, {3787, 5650}, {3917, 5052}, {3934, 17176}, {3995, 17475}, {4048, 16932}, {4121, 7764}, {5007, 8623}, {5017, 7485}, {5028, 15004}, {5106, 14990}, {5111, 11673}, {5116, 6636}, {5135, 5371}, {6593, 17413}, {6656, 14822}, {6676, 14965}, {7745, 14957}, {7770, 20023}, {7878, 9490}, {8789, 8842}, {8881, 12050}, {9465, 11451}, {9605, 11328}, {10328, 12215}, {10339, 16897}, {11245, 20021}, {12055, 15107}, {13341, 19032}, {13366, 20976}, {20372, 21327}, {20861, 20961}

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

X(20966) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20966) lies on these lines: {2, 4476}, {6, 199}, {31, 2245}, {38, 1211}, {39, 51}, {42, 181}, {201, 1834}, {210, 4735}, {244, 17056}, {256, 4418}, {325, 16891}, {386, 4225}, {511, 17187}, {726, 1230}, {756, 1213}, {982, 3936}, {986, 5051}, {1193, 10974}, {1724, 17521}, {1865, 2181}, {1962, 3122}, {2238, 5282}, {2251, 20456}, {2292, 4205}, {2309, 20961}, {3060, 5145}, {3120, 3136}, {3124, 20671}, {3454, 3670}, {3728, 8013}, {3909, 18601}, {3948, 12782}, {4137, 18202}, {4199, 4414}, {4215, 4261}, {6535, 21024}, {14992, 20683}, {20457, 20974}

X(20966) = isogonal conjugate of isotomic conjugate of X(3454)
X(20966) = polar conjugate of isotomic conjugate of X(22073)

X(20967) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20967) lies on these lines: {1, 20760}, {3, 1707}, {6, 1402}, {9, 55}, {21, 1183}, {31, 32}, {42, 51}, {43, 6210}, {48, 1397}, {65, 1730}, {181, 2183}, {198, 1460}, {212, 2175}, {239, 11688}, {931, 14534}, {960, 4267}, {968, 2082}, {988, 20805}, {1036, 1259}, {1400, 10460}, {1403, 2999}, {1621, 4981}, {1743, 10434}, {1962, 2170}, {2092, 2354}, {2150, 2193}, {2174, 20986}, {2300, 3725}, {2308, 3724}, {2318, 3688}, {2646, 17194}, {3011, 21319}, {3052, 15624}, {3056, 3190}, {3198, 12723}, {3271, 14547}, {3687, 18235}, {3794, 4511}, {3871, 4723}, {4640, 5132}, {4641, 16678}, {4849, 15621}, {5745, 21321}, {6737, 8240}, {12514, 19763}, {15569, 18185}, {18163, 21334}, {20460, 20985}

X(20967) = isogonal conjugate of X(31643)
X(20967) = isogonal conjugate of isotomic conjugate of X(960)
X(20967) = polar conjugate of isotomic conjugate of X(22074)

X(20968) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20968) lies on these lines: {3, 1177}, {6, 2353}, {32, 682}, {51, 5007}, {76, 827}, {206, 10316}, {217, 2909}, {864, 14602}, {1180, 14885}, {2387, 18796}, {3202, 9419}, {3398, 14561}, {7737, 11380}, {8743, 11610}, {10317, 15270}, {18374, 20960}

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

X(20969) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20969) lies on these lines: {6, 20994}, {38, 7794}, {72, 4735}, {213, 3778}, {826, 4041}, {2308, 11205}, {3954, 21035}, {4484, 16466}, {14992, 20683}, {17192, 18183}, {20861, 20963}

X(20969) = isogonal conjugate of isotomic conjugate of complement of X(82)
X(20969) = isogonal conjugate of isotomic conjugate of X(21249)
X(20969) = polar conjugate of isotomic conjugate of X(22077)

X(20970) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20970) lies on these lines: {1, 1573}, {3, 6}, {37, 3678}, {41, 6186}, {42, 213}, {43, 1574}, {56, 9346}, {81, 5277}, {101, 1126}, {115, 118}, {172, 20461}, {387, 3767}, {512, 18001}, {519, 21024}, {524, 16887}, {538, 17499}, {661, 1643}, {798, 3249}, {810, 10581}, {978, 1449}, {1015, 1193}, {1017, 3124}, {1100, 1125}, {1201, 16971}, {1203, 1914}, {1211, 17023}, {1509, 16917}, {2140, 17366}, {2229, 3240}, {2240, 17012}, {2241, 16466}, {2251, 20456}, {2275, 5313}, {2276, 5312}, {2292, 6155}, {2295, 3293}, {2332, 14581}, {2334, 9351}, {2503, 2653}, {2650, 3125}, {2901, 7230}, {3008, 17056}, {3192, 3199}, {3454, 10026}, {3759, 17030}, {3811, 16972}, {3934, 17034}, {3936, 17367}, {3948, 4393}, {3997, 20691}, {4065, 4115}, {4205, 6537}, {5283, 19767}, {5292, 7746}, {5795, 6603}, {7410, 7735}, {8649, 16474}, {9427, 20671}, {16604, 16666}, {17275, 19858}, {20662, 20860}

X(20970) = isogonal conjugate of X(32014)
X(20970) = complement of X(33297)
X(20970) = crosspoint of X(6) and X(42)
X(20970) = crosssum of X(2) and X(86)
X(20970) = isogonal conjugate of isotomic conjugate of X(1213)
X(20970) = isogonal conjugate of polar conjugate of X(430)
X(20970) = trilinear pole wrt symmedial triangle of antiorthic axis
X(20970) = polar conjugate of isotomic conjugate of X(22080)

X(20971) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20971) lies on these lines: {1, 2}, {6, 7121}, {39, 20667}, {192, 14823}, {213, 20671}, {1475, 20457}, {2275, 20464}, {18758, 20663}, {20456, 20460}, {20860, 20866}

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

X(20972) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20972) lies on these lines: {6, 101}, {39, 20973}, {42, 3271}, {43, 4274}, {44, 519}, {906, 16946}, {1018, 1743}, {1401, 1405}, {2092, 20982}, {2347, 7117}, {3052, 3939}, {3707, 3840}, {3768, 8658}, {9283, 16554}, {16594, 17195}

X(20972) = isogonal conjugate of polar conjugate of X(5151)
X(20972) = isogonal conjugate of isotomic conjugate of X(16594)
X(20972) = polar conjugate of isotomic conjugate of X(22082)

X(20973) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20973) lies on these lines: {6, 36}, {37, 3169}, {39, 20972}, {42, 2183}, {43, 44}, {45, 3679}, {100, 751}, {750, 9349}, {995, 4266}, {1405, 4273}, {2316, 4256}, {2347, 4261}, {4286, 16670}, {4715, 17595}

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

X(20974) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20974) lies on these lines: {6, 20999}, {11, 661}, {42, 20455}, {116, 17198}, {181, 2350}, {649, 3937}, {1015, 3124}, {1331, 10756}, {2170, 2969}, {2225, 8679}, {3271, 8645}, {17435, 18210}, {20457, 20966}, {20459, 20962}, {20859, 20870}

X(20974) = isogonal conjugate of isotomic conjugate of X(116)
X(20974) = crosssum of X(2) and X(101)
X(20974) = crosspoint of X(6) and X(514)
X(20974) = polar conjugate of isotomic conjugate of X(22084)

X(20975) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20975) lies on these lines: {2, 16098}, {3, 895}, {4, 1942}, {6, 157}, {25, 8749}, {32, 3455}, {39, 682}, {69, 20819}, {98, 648}, {115, 2971}, {122, 125}, {184, 5158}, {216, 6467}, {237, 2393}, {246, 526}, {264, 9307}, {338, 523}, {351, 865}, {524, 3001}, {542, 18114}, {656, 3942}, {661, 2310}, {800, 1843}, {842, 9139}, {1316, 1632}, {1624, 11746}, {1634, 2854}, {1818, 20733}, {1899, 13409}, {1992, 3095}, {2023, 14772}, {2092, 20455}, {2407, 12042}, {2782, 14570}, {2970, 8901}, {3005, 8288}, {3014, 18122}, {3053, 17813}, {3269, 9409}, {3271, 20982}, {3675, 17058}, {3964, 6391}, {4092, 4705}, {4551, 6044}, {5024, 10765}, {5095, 10991}, {5201, 9019}, {6746, 16035}, {8266, 17710}, {8287, 17463}, {8573, 12167}, {9178, 14998}, {10745, 18933}, {11188, 11328}, {11579, 14687}, {13198, 13558}, {14270, 17423}, {15000, 15118}, {15851, 19125}, {18591, 20728}, {20759, 20830}, {20785, 20825}

X(20975) = isogonal conjugate of X(18020)
X(20975) = crosssum of X(2) and X(110)
X(20975) = crosspoint of X(6) and X(523)
X(20975) = crossdifference of every pair of points on line X(99)X(112)
X(20975) = pole wrt polar circle of line X(99)X X(20975) = isotomic conjugate of polar conjugate of X(3124)
X(20975) = polar conjugate of isotomic conjugate of X(3269)
X(20975) = X(19)-isoconjugate of X(4590)
X(20975) = X(63)-isoconjugate of X(23582)
(107)
X(20975) = X(92)-isoconjugate of X(249)

X(20976) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a^4 - 2 a^2 b^2 + b^4 - 2 a^2 c^2 + c^4) : :
Barycemtrics    a^2 (4 SA a^2 - b^4 - c^4) : :
Barycentrics    (csc^2(B - C)) (csc^2 B csc^2(A - C) + csc^2 C csc^2(A - B)) : :

X(20976) lies on these lines: {2, 12829}, {6, 110}, {23, 5111}, {32, 9155}, {39, 5191}, {125, 5477}, {141, 10552}, {184, 5028}, {323, 1691}, {351, 10567}, {511, 8627}, {620, 17199}, {1495, 1570}, {1501, 1993}, {1648, 5972}, {1692, 3231}, {1915, 1994}, {1976, 11402}, {2308, 5147}, {3094, 11003}, {3266, 13196}, {3269, 13198}, {3311, 7598}, {3312, 7599}, {3448, 8288}, {3569, 14397}, {3629, 7664}, {3981, 9544}, {4563, 5182}, {4576, 5026}, {5007, 5106}, {5008, 9486}, {5012, 5116}, {5027, 14778}, {5040, 20958}, {5162, 14602}, {5202, 20963}, {5642, 6388}, {5969, 10330}, {6034, 9143}, {6409, 7601}, {6410, 7602}, {9463, 9716}, {11004, 13330}, {11205, 14153}, {11646, 14683}, {13303, 14396}, {13366, 20965}, {15107, 15514}

X(20976) = isogonal conjugate of isotomic conjugate of X(620)
X(20976) = polar conjugate of isotomic conjugate of X(22085)
X(20976) = crosspoint of X(6) and X(249)
X(20976) = crosssum of X(2) and X(115)
X(20976) = crossdifference of every pair of points on line X(148)X(690)

X(20977) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20977) lies on these lines: {6, 23}, {39, 51}, {110, 5111}, {111, 8586}, {323, 15514}, {511, 3124}, {625, 17204}, {669, 2451}, {671, 8785}, {858, 1648}, {1112, 2211}, {1495, 1570}, {1691, 15107}, {1692, 8627}, {2493, 3289}, {2502, 3292}, {2549, 16311}, {3051, 3060}, {3094, 5640}, {3266, 5969}, {5038, 15019}, {5116, 15018}, {5189, 6792}, {5943, 8041}, {6034, 15360}, {8352, 14263}, {9463, 16981}, {10330, 13196}, {10542, 17810}, {10601, 15815}, {20861, 20962}

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

X(20978) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20978) lies on these lines: {1, 144}, {6, 31}, {58, 13404}, {105, 7175}, {244, 1418}, {269, 479}, {604, 7083}, {991, 1193}, {1042, 1104}, {1108, 2310}, {1149, 16487}, {1191, 4322}, {1201, 1419}, {1279, 20323}, {1400, 3271}, {1404, 2175}, {1449, 4343}, {1453, 4300}, {1742, 5222}, {1743, 2340}, {2170, 12723}, {2183, 3941}, {2195, 3451}, {2223, 2347}, {2263, 3924}, {3000, 4000}, {3009, 16970}, {3554, 4336}, {3720, 3945}, {4266, 16688}, {4335, 17014}, {4344, 10459}, {4878, 16669}

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

X(20979) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20979) lies on these lines: {2, 21191}, {6, 1919}, {37, 17458}, {44, 513}, {86, 18196}, {239, 9294}, {512, 4502}, {514, 19565}, {573, 3667}, {663, 788}, {665, 6363}, {667, 6373}, {669, 2451}, {802, 3766}, {812, 4391}, {813, 6163}, {834, 4079}, {1400, 3572}, {1459, 5029}, {1475, 8656}, {1654, 21304}, {2309, 8630}, {3063, 8632}, {3249, 4253}, {3250, 3709}, {3835, 17217}, {4063, 4785}, {4083, 14408}, {4139, 4526}, {4379, 7199}, {4382, 18071}, {4491, 21007}, {5224, 21262}, {5383, 8709}, {6586, 9002}, {8637, 20861}, {8643, 9010}, {20316, 21053}

X(20979) = isogonal conjugate of X(4598)
X(20979) = anticomplement of X(21191)
X(20979) = polar conjugate of isotomic conjugate of X(22090)

X(20980) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20980) lies on these lines: {6, 513}, {9, 21348}, {81, 4776}, {213, 4378}, {512, 1570}, {514, 3287}, {649, 6363}, {650, 9364}, {651, 666}, {652, 7180}, {657, 665}, {667, 6373}, {668, 5383}, {894, 20906}, {900, 4501}, {1459, 3709}, {1919, 3768}, {2483, 9002}, {2484, 6371}, {2509, 9001}, {3049, 8672}, {3667, 4435}, {3758, 20949}, {4502, 4775}, {4885, 17218}, {8540, 9320}, {8646, 20983}

X(20980) = isogonal conjugate of X(30610)
X(20980) = polar conjugate of isotomic conjugate of X(22091)
X(20980) = crossdifference of every pair of points on line X(144)X(145) (the line of the degenerate cross-triangle of Gemini triangles 29 and 30)

X(20981) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20981) lies on these lines: {6, 798}, {101, 6163}, {385, 4369}, {513, 1919}, {572, 6003}, {604, 4017}, {649, 854}, {661, 3737}, {662, 4590}, {663, 4502}, {667, 6373}, {741, 5970}, {1019, 1924}, {1024, 3451}, {1100, 4132}, {1459, 2484}, {2451, 8639}, {2483, 21123}, {2605, 4079}, {3248, 18105}, {3261, 4508}, {3287, 3805}, {3407, 4444}, {3709, 5029}, {3768, 4057}, {4107, 4374}, {4140, 4922}, {4504, 4529}, {5750, 21099}, {8060, 18160}, {8061, 9013}, {17217, 17379}, {17303, 21055}, {21261, 21304}

X(20981) = isogonal conjugate of X(27805)
X(20981) = polar conjugate of isotomic conjugate of X(22093)
X(20981) = crossdifference of every pair of points on line X(8)X(192) (the line of the degenerate cross-triangle of Gemini triangles 17 and 18)

X(20982) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20982) lies on these lines: {6, 163}, {101, 2503}, {115, 661}, {672, 5164}, {1015, 3124}, {2087, 16613}, {2088, 2624}, {2092, 20972}, {2161, 3013}, {2238, 14963}, {2653, 20963}, {3125, 16592}, {3269, 14936}, {3271, 20975}, {3942, 17058}, {5213, 20331}, {7202, 8287}, {8818, 18393}, {20456, 20865}, {20662, 20860}

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

X(20983) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20983) lies on these lines: {6, 1980}, {42, 8640}, {51, 8642}, {512, 4813}, {513, 4380}, {649, 6373}, {650, 9010}, {661, 788}, {667, 20456}, {669, 2451}, {838, 4705}, {891, 4382}, {984, 21350}, {3004, 9040}, {4083, 20295}, {4394, 8027}, {4502, 8663}, {4507, 4979}, {4524, 6363}, {8630, 20868}, {8646, 20980}

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

X(20984) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20984) lies on these lines: {6, 922}, {42, 181}, {667, 6373}, {2183, 20456}, {2245, 3122}, {3123, 20367}, {3764, 20985}, {4787, 21010}, {4892, 18201}, {8540, 8586}

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

X(20985) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(20985) lies on these lines: {1, 21}, {2, 16476}, {6, 292}, {39, 42}, {82, 757}, {171, 239}, {172, 2210}, {213, 2308}, {238, 16826}, {583, 21035}, {748, 16831}, {750, 4384}, {756, 16552}, {902, 16971}, {940, 16354}, {980, 17017}, {1100, 1918}, {1107, 3745}, {1449, 2209}, {2177, 5030}, {2239, 17023}, {2260, 3778}, {2667, 8053}, {2668, 6626}, {3122, 4749}, {3764, 20984}, {4251, 18266}, {4393, 4781}, {4418, 17143}, {5247, 16830}, {5283, 5311}, {6533, 16825}, {7122, 19133}, {8622, 21352}, {14996, 16497}, {16684, 18166}, {16689, 17798}, {16690, 17394}, {16815, 17122}, {16832, 17124}, {17031, 20913}, {20460, 20967}

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

X(20986) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20986) lies on these lines: {1, 1437}, {6, 181}, {19, 2203}, {31, 48}, {42, 2317}, {51, 20989}, {54, 11491}, {55, 184}, {60, 1610}, {100, 5012}, {110, 1621}, {154, 1486}, {171, 5135}, {182, 1376}, {198, 4275}, {212, 15624}, {227, 19365}, {228, 2148}, {386, 2933}, {518, 3955}, {567, 18524}, {569, 11499}, {572, 9562}, {578, 11500}, {595, 17104}, {612, 2261}, {674, 5285}, {756, 2265}, {958, 13323}, {982, 5197}, {991, 1626}, {1001, 9306}, {1036, 3435}, {1147, 10267}, {1324, 5396}, {1402, 7113}, {1408, 1468}, {1428, 3752}, {1495, 20988}, {1630, 16679}, {1660, 18621}, {1790, 16678}, {1962, 17438}, {1977, 5371}, {1980, 4394}, {2003, 8679}, {2162, 20995}, {2174, 20967}, {2175, 3052}, {2182, 3745}, {2183, 2308}, {2304, 9247}, {2328, 8053}, {3198, 11428}, {3703, 17977}, {3914, 5137}, {4259, 5329}, {4423, 5651}, {4640, 7193}, {5138, 15509}, {6759, 11496}, {7074, 17809}, {9563, 15792}, {10310, 10984}, {11688, 18042}

X(20986) = isogonal conjugate of isotomic conjugate of X(2975)
X(20986) = polar conjugate of isotomic conjugate of X(22118)

X(20987) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20987) lies on these lines: {3, 2916}, {4, 15577}, {6, 25}, {22, 141}, {23, 69}, {24, 1503}, {26, 1352}, {66, 21213}, {67, 10117}, {113, 18534}, {157, 237}, {160, 3148}, {182, 7506}, {211, 5017}, {403, 18382}, {511, 7517}, {518, 8185}, {599, 9909}, {1216, 1350}, {1351, 18378}, {1469, 9658}, {1486, 16777}, {1609, 7669}, {1995, 3589}, {2070, 18440}, {2076, 5167}, {2353, 20960}, {2930, 6144}, {2931, 14982}, {3016, 11641}, {3056, 9673}, {3098, 5891}, {3242, 9798}, {3313, 9306}, {3518, 6776}, {3542, 20303}, {3618, 13595}, {3619, 6636}, {3711, 12329}, {3827, 11363}, {3867, 10192}, {4265, 13730}, {5050, 13621}, {5085, 6642}, {5157, 9822}, {5480, 10594}, {5621, 13289}, {5654, 7530}, {5800, 17562}, {6697, 11550}, {6759, 19161}, {6800, 16776}, {7494, 15435}, {7502, 18358}, {7505, 20300}, {8550, 15580}, {9019, 20806}, {9714, 15069}, {9920, 12242}, {10323, 16621}, {10387, 10833}, {10519, 12088}, {11188, 19121}, {11414, 15030}, {12140, 19457}, {134 74, 14810}, {13562, 16789}, {13861, 14561}, {15462, 20773}

X(20987) = isogonal conjugate of isotomic conjugate of X(7391)
X(20987) = polar conjugate of isotomic conjugate of X(22120)

X(20988) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20988) lies on these lines: {1, 20831}, {2, 20872}, {3, 1699}, {6, 20961}, {11, 4224}, {12, 4222}, {19, 25}, {22, 1001}, {23, 1621}, {24, 11496}, {28, 6284}, {31, 3122}, {51, 692}, {56, 1448}, {100, 13595}, {161, 18621}, {199, 8053}, {210, 17744}, {238, 5347}, {354, 3220}, {667, 11193}, {748, 5096}, {940, 7295}, {1011, 1631}, {1125, 20833}, {1279, 5322}, {1376, 1995}, {1473, 4860}, {1495, 20986}, {1610, 10543}, {1633, 11246}, {1829, 9627}, {1836, 14667}, {1953, 7073}, {2886, 4228}, {2915, 5248}, {3295, 8185}, {3303, 9798}, {3683, 5285}, {3715, 12329}, {3720, 4265}, {3742, 7293}, {3925, 4223}, {4185, 12953}, {4186, 10895}, {4219, 7965}, {4294, 17562}, {4413, 5020}, {5078, 8616}, {5172, 11334}, {5284, 6636}, {5314, 15254}, {5584, 9911}, {6642, 10310}, {6913, 15177}, {7083, 16470}, {7485, 8167}, {7506, 11248}, {7517, 10267}, {7545, 18524}, {8273, 11414}, {9342, 16042}, {9658, 11510}, {10594, 11500}, {11499, 13861}, {11849, 13621}, {16064, 20470}, {16372, 16681}, {16678, 20834}, {18613, 20999}

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

X(20989) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20989) lies on these lines: {3, 1698}, {6, 20962}, {10, 2915}, {12, 28}, {19, 25}, {22, 1376}, {23, 100}, {24, 11500}, {26, 11499}, {35, 20831}, {36, 16610}, {42, 2174}, {43, 5347}, {51, 20986}, {56, 998}, {111, 919}, {181, 2194}, {199, 21011}, {209, 10536}, {210, 5285}, {238, 5078}, {650, 667}, {692, 1495}, {750, 4265}, {759, 859}, {899, 5096}, {902, 16686}, {958, 11337}, {1001, 1995}, {1155, 3220}, {1325, 5080}, {1466, 18954}, {1610, 10950}, {1621, 13595}, {1626, 4191}, {2070, 18524}, {2183, 2361}, {2360, 2594}, {2551, 7520}, {3085, 17562}, {3303, 11365}, {3304, 8192}, {3518, 11491}, {3689, 5525}, {3711, 12329}, {3715, 7085}, {3724, 6187}, {3740, 5314}, {3752, 5322}, {3826, 7465}, {3925, 4220}, {4026, 4239}, {4185, 10895}, {4186, 12953}, {4222, 6284}, {4224, 5432}, {4228, 6690}, {4383, 5329}, {4423, 5020}, {5061, 18191}, {5217, 13730}, {5363, 16468}, {6253, 7412}, {6644, 15943}, {7387, 10310}, {7506, 10267}, {7517, 11248}, {8661, 21003}, {9342, 15246}, {9627, 11363}, {9639, 20243}, {9658, 11509}, {9659, 10830}, {10594, 11496}, {11502, 15509}, {11849, 18378}, {15622, 20838}, {15654, 20842}, {20470, 20999}

X(20989) = isogonal conjugate of isotomic conjugate of X(5080)
X(20989) = polar conjugate of isotomic conjugate of X(22123)
X(20989) = Stevanovic-circle-inverse of X(32758)

X(20990) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(20990) lies on these lines: {1, 5132}, {2, 16684}, {6, 292}, {7, 2283}, {9, 3941}, {32, 16683}, {37, 2223}, {45, 16694}, {55, 199}, {56, 976}, {100, 4360}, {141, 4447}, {171, 18162}, {192, 4436}, {198, 4471}, {214, 999}, {228, 3745}, {474, 6533}, {583, 20683}, {612, 2352}, {674, 1400}, {851, 17602}, {983, 3733}, {984, 3286}, {1009, 3932}, {1376, 4361}, {1402, 15621}, {1475, 4878}, {1696, 21002}, {1918, 3009}, {2174, 19133}, {2245, 3688}, {2260, 2340}, {2298, 16872}, {2330, 7113}, {3185, 5269}, {3244, 4097}, {3444, 20877}, {3731, 16688}, {3744, 18613}, {3920, 16678}, {4191, 17599}, {4286, 21035}, {4386, 20475}, {4433, 17388}, {5276, 16693}, {8299, 17243}, {9310, 21059}, {16056, 17061}, {17365, 21320}

X(20990) = isogonal conjugate of isotomic conjugate of X(17165)
X(20990) = polar conjugate of isotomic conjugate of X(22164)

X(20991) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 2*a^3*c - 4*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - c^4) : :

X(20991) lies on the cubic K180 and these lines: 1, 37260}, {3, 4512}, {6, 2187}, {19, 25}, {28, 11496}, {31, 56}, {40, 13737}, {42, 17810}, {51, 4285}, {84, 963}, {165, 37269}, {184, 5115}, {204, 3209}, {497, 15509}, {516, 11347}, {859, 2328}, {902, 33589}, {958, 28376}, {1001, 4224}, {1350, 25941}, {1376, 33849}, {1402, 7083}, {1456, 6611}, {1503, 28379}, {1617, 3220}, {1621, 37254}, {1661, 2352}, {1884, 36999}, {1962, 3303}, {2177, 31860}, {2178, 3052}, {2182, 30223}, {2183, 7074}, {2195, 33581}, {2208, 375191}, {2308, 17809}, {3475, 24328}, {3925, 37367}, {3941, 23204}, {4191, 5646}, {4196, 7965}, {4222, 11500}, {4413, 37366}, {5217, 37259}, {5248, 37052}, {5250, 37250}, {5285, 36641}, {5584, 13738}, {6253, 28076}, {6353, 25968}, {7053, 34033}, {7151, 32674}, {8185, 11508}, {9778, 11349}, {9911, 10310}, {10267, 20831}, {10434, 13615}, {11350, 35258}, {11495, 37262}, {11688, 26241}, {12953, 37226}, {15803, 34498}, {18613, 22769}, {19297, 21000}, {19649, 25893}, {20368, 25934}, {21628, 37046}, {22654, 23372}, {23853, 24320}, {24309, 37270}, {25514, 31394}, {31730, 37273}

X(20991) = isogonal conjugate of the isotomic conjugate of X(962)
X(20991) = isogonal conjugate of isotomic conjugate of X(962)
X(20991) = polar conjugate of the isotomic conjugate of X(22124)
X(20991) = X(i)-Ceva conjugate of X(j) for these (i,j): {{84, 6}, {962, 22124}}
X(20991) = X(75)-isoconjugate of X(963)
X(20991) = crosspoint of X(7115) and X(8059)
X(20991) = crosssum of X(i) and X(j) for these (i,j): {{2, 20070}, {8058, 26932}}
X(20991) = crossdifference of every pair of points on line {905, 3239}
X(20991) = barycentric product X(i)*X(j) for these {i,j}: {{1, 2270}, {4, 22124}, {6, 962}, {31, 20921}, {56, 27508}, {58, 21068}, {101, 7661}}
X(20991) = barycentric quotient X(i)/X(j) for these {i,j}: {{32, 963}, {962, 76}, {2270, 75}, {7661, 3261}, {20921, 561}, {21068, 313}, {22124, 69}, {27508, 3596}}
X(20991) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{55, 1696, 612}, {55, 15494, 198}, {84, 1622, 963}, {1486, 3185, 55}, {3556, 23383, 56}, {20988, 20989, 9673}}

X(20992) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b - a b^2 + a^2 c + a b c - b^2 c - a c^2 - b c^2) : :

X(20992) lies on these lines: {1, 20793}, {3, 238}, {6, 31}, {7, 21}, {9, 2223}, {35, 16468}, {36, 15485}, {37, 3941}, {44, 15624}, {45, 16694}, {69, 8299}, {87, 8616}, {100, 17349}, {144, 21320}, {171, 16058}, {190, 7155}, {198, 2110}, {220, 9454}, {241, 20275}, {344, 4447}, {572, 2175}, {573, 3271}, {595, 5145}, {673, 11495}, {692, 4268}, {748, 4191}, {750, 16373}, {958, 4195}, {964, 19874}, {999, 16484}, {1030, 4471}, {1045, 16476}, {1108, 11997}, {1191, 2274}, {1376, 4203}, {1397, 2328}, {1402, 4512}, {1403, 4640}, {1460, 13615}, {1475, 4343}, {1486, 17798}, {1617, 7175}, {1621, 17379}, {1631, 5124}, {1964, 2176}, {2053, 20471}, {2305, 16372}, {2352, 3683}, {3000, 5204}, {3295, 4649}, {3304, 10448}, {3688, 3730}, {3736, 16466}, {3747, 7032}, {4068, 16884}, {4097, 4700}, {4184, 17127}, {4215, 15494}, {4216, 8692}, {4261, 4749}, {4279, 11490}, {4361, 4436}, {4363, 16684}, {4413, 16405}, {4423, 15668}, {4433, 5839}, {4557, 16885}, {5120, 16503}, {5132, 5217}, {5156, 16287}, {5248, 19762}, {5329, 20834}, {7262, 20760}, {10458, 18166}, {10473, 17194}, {14621, 16367}, {16059, 17123}, {16678, 16690}, {16679, 16777}

X(20992) = isogonal conjugate of isotomic conjugate of X(10453)
X(20992) = isogonal conjugate of polar conjugate of X(17920)
X(20992) = polar conjugate of isotomic conjugate of X(22127)

X(20993) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (3 a^8 - 2 a^4 b^4 - b^8 - 2 a^4 c^4 + 2 b^4 c^4 - c^8) : :

X(20993) lies on these lines: {3, 206}, {5, 7694}, {6, 2353}, {22, 8793}, {25, 32}, {26, 19165}, {39, 19125}, {157, 15257}, {159, 10316}, {1181, 2909}, {3053, 20960}, {3202, 19357}, {3785, 7493}, {5188, 9715}, {6676, 7800}, {7539, 7889}, {7795, 13562}, {9918, 14673}, {14023, 15594}

X(20993) = isogonal conjugate of isotomic conjugate of X(5596)
X(20993) = trilinear pole, wrt tangential triangle, of de Longchamps line
X(20993) = polar conjugate of isotomic conjugate of X(22135)

X(20994) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 + a^3 b^2 - a^2 b^3 - b^5 - a^2 b^2 c + a^3 c^2 - a^2 b c^2 + a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - c^5) : :

X(20994) lies on these lines: {1, 2916}, {6, 20969}, {31, 2275}, {191, 20677}, {831, 7794}, {1631, 2176}, {14370, 21035}

X(20994) = isogonal conjugate of isotomic conjugate of X(21249)
X(20994) = polar conjugate of isotomic conjugate of X(22137)

X(20995) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (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(20995) lies on these lines: {3, 9241}, {6, 57}, {41, 9316}, {48, 1613}, {101, 16283}, {109, 14827}, {198, 1755}, {218, 4650}, {220, 4640}, {294, 17074}, {971, 20310}, {991, 16588}, {1200, 1458}, {1403, 9454}, {1436, 5110}, {2162, 20986}, {2176, 2223}, {2225, 4191}, {2272, 2276}, {2284, 4421}, {5275, 5781}, {6180, 9446}

X(20995) = isogonal conjugate of isotomic conjugate of X(3177)
X(20995) = polar conjugate of isotomic conjugate of X(20793)

X(20996) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 b^2 - a^2 b^3 - 2 a^3 b c + a^2 b^2 c + 2 a b^3 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3) : :

X(20996) lies on these lines: {3, 238}, {6, 7121}, {43, 17105}, {55, 3009}, {56, 664}, {197, 17798}, {614, 20757}, {995, 11490}, {1575, 2053}

X(20997) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (4 a^3 + 4 a^2 b - 4 a b^2 - 4 b^3 + 4 a^2 c - 5 a b c - 4 a c^2 - 4 c^3) : :

X(20997) lies on these lines: {3, 3196}, {6, 36}, {45, 5010}, {55, 16672}, {1030, 16675}

X(20998) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^4 - a^2 b^2 - b^4 - a^2 c^2 + 3 b^2 c^2 - c^4) : :

X(20998) lies on these lines: {2, 4048}, {3, 5106}, {6, 110}, {15, 14704}, {16, 14705}, {22, 21001}, {23, 2076}, {25, 694}, {31, 2054}, {51, 2056}, {55, 5147}, {100, 9509}, {115, 5972}, {125, 10418}, {141, 7665}, {148, 11053}, {154, 1976}, {199, 16365}, {230, 15448}, {247, 15131}, {323, 15514}, {351, 2872}, {373, 5038}, {511, 9225}, {512, 9217}, {542, 6388}, {574, 16187}, {647, 13558}, {902, 20675}, {1030, 5163}, {1151, 7598}, {1152, 7599}, {1196, 1915}, {1495, 1691}, {1648, 3448}, {1979, 5040}, {2079, 3569}, {2176, 5202}, {2177, 5168}, {2305, 8775}, {2641, 16575}, {3009, 20877}, {3051, 13595}, {3053, 5191}, {3094, 5651}, {3121, 17962}, {3229, 5162}, {3292, 5111}, {3444, 16685}, {3629, 10553}, {3763, 7664}, {3981, 5028}, {4563, 5969}, {4576, 5108}, {5013, 9155}, {5027, 9431}, {5029, 9259}, {5210, 9486}, {5642, 6034}, {5943, 14153}, {6429, 7601}, {6430, 7602}, {6719, 14928}, {6792, 14683}, {7492, 8617}, {8178, 11052}, {8288, 15059}, {8623, 20854}, {9169, 10488}, {9463, 14002}, {9924, 10836}, {15145, 17984}, {17735, 20472}

X(20998) = isogonal conjugate of X(35511)
X(20998) = isogonal conjugate of isotomic conjugate of X(148)
X(20998) = anticomplement of complementary conjugate of X(31998)
X(20998) = crosssum of X(i) and X(j) for these {i,j}: {2, 20094}, {523, 23991}, {524, 23992}
X(20998) = crosspoint of X(111) and X(34539)
X(20998) = crossdifference of every pair of points on line X(620)X(690) (the orthic axis of the 1st Brocard triangle)
X(20998) = trilinear pole, wrt tangential triangle, of Brocard axis
X(20998) = polar conjugate of isotomic conjugate of X(22143)
X(20998) = crosspoint of PU(105)

X(20999) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^4 - a^3 b + a b^3 - b^4 - a^3 c + a^2 b c - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3 - c^4) : :

X(20999) lies on these lines: {1, 1283}, {3, 8}, {6, 20974}, {21, 5484}, {22, 6360}, {25, 105}, {31, 1469}, {36, 1054}, {38, 55}, {56, 244}, {88, 20842}, {109, 3937}, {149, 13589}, {197, 4191}, {198, 2246}, {199, 8301}, {228, 1282}, {291, 5329}, {388, 13733}, {580, 16980}, {659, 14667}, {678, 5217}, {947, 13367}, {999, 3315}, {1011, 8299}, {1331, 2810}, {1364, 2342}, {1402, 5322}, {1621, 9791}, {1622, 3515}, {1769, 4491}, {2078, 3220}, {2222, 6075}, {2361, 8679}, {2659, 14024}, {2930, 8053}, {2933, 5204}, {3129, 10648}, {3130, 10647}, {3286, 5078}, {3436, 13732}, {3446, 3733}, {3556, 11510}, {4458, 13558}, {4712, 7085}, {5029, 9259}, {5285, 9451}, {7669, 16873}, {7742, 9798}, {8638, 20839}, {9780, 16422}, {10016, 21119}, {10527, 19548}, {11337, 19851}, {12248, 14127}, {13576, 16378}, {18613, 20988}, {20470, 20989}

X(20999) = isogonal conjugate of isotomic conjugate of X(150)
X(20999) = polar conjugate of isotomic conjugate of X(22145)
X(20999) = anticomplement of complementary conjugate of X(39026)
X(20999) = pole wrt circumcircle of Feuerbach tangent line (line X(11)X(244))

X(21000) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (5 a - 3 b - 3 c) : :

X(21000) lies on these lines: {1, 19535}, {3, 1616}, {6, 31}, {35, 1191}, {36, 16486}, {44, 3158}, {45, 4512}, {165, 1279}, {171, 4428}, {197, 16686}, {200, 16885}, {220, 2251}, {238, 4421}, {518, 16570}, {595, 4255}, {612, 16675}, {968, 16672}, {1001, 3550}, {1086, 9778}, {1376, 8616}, {1407, 2078}, {2223, 5023}, {2241, 5022}, {2328, 3285}, {3207, 8647}, {3242, 3749}, {3295, 4252}, {3445, 5204}, {3579, 17054}, {3750, 9332}, {3757, 17118}, {3915, 5217}, {3928, 4864}, {4257, 6767}, {4258, 14974}, {4370, 5423}, {5010, 16483}, {5057, 17783}, {5210, 9259}, {5269, 16777}, {5292, 10386}, {7172, 17340}, {8692, 16569}, {9668, 17734}, {10578, 17365}, {19750, 19998}

X(21000) = isogonal conjugate of X(36606)
X(21000) = crosspoint of X(1252) and X(1293)
X(21000) = crosssum of X(1086) and X(3667)
X(21000) = crossdifference of every pair of points on line X(514)X(2490)
X(21000) = polar conjugate of isotomic conjugate of X(22147)

X(21001) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b^2 + a^2 c^2 - 3 b^2 c^2) : :

X(21001) lies on these lines: {2, 6}, {3, 3229}, {22, 20998}, {25, 2076}, {55, 3009}, {57, 16514}, {76, 11333}, {154, 20885}, {171, 2176}, {182, 2056}, {184, 9225}, {237, 5023}, {238, 2162}, {420, 2207}, {694, 1350}, {1194, 5650}, {1196, 3094}, {1206, 9345}, {1575, 7075}, {1691, 9306}, {1915, 5651}, {2178, 17735}, {2235, 18743}, {2319, 21214}, {2979, 3124}, {3052, 8622}, {3053, 8601}, {3117, 5013}, {3224, 7793}, {3230, 3550}, {3290, 20359}, {3291, 3917}, {3499, 11285}, {3666, 16515}, {3752, 16525}, {3787, 5943}, {3959, 21334}, {5017, 5020}, {5052, 6688}, {5106, 5210}, {5116, 7484}, {5638, 21036}, {5639, 21032}, {7485, 10329}, {7998, 20859}, {8041, 9465}, {11205, 15302}, {12212, 16187}, {13331, 15082}, {14096, 15815}, {17475, 17490}

X(21001) = isogonal conjugate of X(38262)
X(21001) = crosspoint of X(3222) and X(34537)
X(21001) = crosssum of X(i) and X(j) for these {i,j}: {2, 20105}, {1084, 3221}
X(21001) = crossdifference of every pair of points on line X(512)X(31286)
X(21001) = isogonal conjugate of isotomic conjugate of X(20081)
X(21001) = isogonal conjugate of the anticomplement of X(32746)
X(21001) = polar conjugate of isotomic conjugate of X(22152)

X(21002) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 - 3 a^2 b + 3 a b^2 - b^3 - 3 a^2 c - 2 a b c + b^2 c + 3 a c^2 + b c^2 - c^3) : :

X(21002) lies on these lines: {3, 7290}, {6, 31}, {21, 4344}, {35, 16469}, {36, 15287}, {44, 480}, {56, 1279}, {198, 2223}, {221, 1458}, {269, 1617}, {595, 991}, {604, 8647}, {608, 8750}, {934, 2377}, {958, 4339}, {1001, 4307}, {1035, 1456}, {1042, 1616}, {1104, 5584}, {1108, 4319}, {1191, 4300}, {1419, 2078}, {1436, 2195}, {1466, 1471}, {1612, 3332}, {1621, 3945}, {1631, 5204}, {1661, 2352}, {1696, 20990}, {1723, 15733}, {1743, 6600}, {2178, 16686}, {2257, 4326}, {2650, 4068}, {3174, 16572}, {3304, 16679}, {4000, 11495}, {4349, 5248}, {4413, 17337}, {4423, 17245}, {5222, 7676}, {5269, 13615}, {7368, 14974}, {8557, 14100}, {8614, 11510}, {10310, 13329}

X(21002) = isogonal conjugate of isotomic conjugate of X(36845)
X(21002) = polar conjugate of isotomic conjugate of X(22153)

X(21003) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^3 + a b^2 - a b c - b^2 c + a c^2 - b c^2) : :

X(21003) lies on these lines: {3, 19903}, {6, 6373}, {36, 238}, {55, 890}, {512, 2076}, {649, 21005}, {659, 918}, {663, 9313}, {665, 20678}, {788, 21007}, {926, 8659}, {1473, 2504}, {1635, 8650}, {1911, 3572}, {1960, 9259}, {2483, 17990}, {2484, 14407}, {2509, 6050}, {3063, 9010}, {4790, 8646}, {4979, 8635}, {5040, 8027}, {5096, 6165}, {7192, 18108}, {8661, 20989}

X(21003) = isogonal conjugate of isotomic conjugate of anticomplement of X(659)
X(21003) = polar conjugate of isotomic conjugate of X(22155)

X(21004) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5) : :

X(21004) lies on these lines: {3, 9509}, {6, 163}, {9, 1030}, {35, 20677}, {36, 20472}, {41, 2276}, {115, 759}, {1324, 17735}, {2503, 5546}, {2915, 4426}, {3568, 21051}, {4455, 7669}, {5029, 9259}, {5127, 5164}, {9696, 17104}, {17798, 20877}

X(21004) = isogonal conjugate of isotomic conjugate of X(21221)
X(21004) = isogonal conjugate of anticomplement of X(39054)
X(21004) = crossdifference of every pair of points on line X(4458)X(6370)
X(21004) = polar conjugate of isotomic conjugate of X(22156)

X(21005) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^3 + a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21005) lies on these lines: {6, 1980}, {23, 385}, {25, 884}, {55, 8640}, {197, 4394}, {513, 5078}, {649, 21003}, {650, 667}, {661, 8635}, {788, 7252}, {814, 7255}, {1491, 3733}, {2512, 8639}, {3004, 4367}, {4383, 20473}, {4401, 11068}

X(21005) = isogonal conjugate of isotomic conjugate of X(21301)
X(21005) = polar conjugate of isotomic conjugate of X(22157)

X(21006) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (b + c) (a^4 + a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(21006) lies on these lines: {3, 9489}, {6, 3221}, {23, 385}, {25, 2489}, {159, 924}, {351, 2514}, {512, 2076}, {688, 3050}, {804, 5152}, {1634, 14588}, {1995, 15724}, {2485, 8651}, {3049, 9009}, {4840, 16874}, {9491, 13586}, {16692, 16695}

X(21006) = isogonal conjugate of isotomic conjugate of anticomplement of X(669)
X(21006) = isogonal conjugate of anticomplement of X(38996)
X(21006) = crossdifference of every pair of points on line X(39)X(698)
X(21006) = polar conjugate of isotomic conjugate of X(22159)
X(21006) = pole of line X(2)X(39) wrt circumcircle

X(21007) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^2 - a b - a c - b c) : :

X(21007) lies on these lines: {6, 513}, {239, 20949}, {512, 1691}, {523, 4435}, {649, 834}, {650, 15313}, {663, 6586}, {665, 2605}, {788, 21003}, {798, 4057}, {889, 5383}, {900, 3287}, {919, 1618}, {1980, 16874}, {2176, 4775}, {2509, 3309}, {3250, 16685}, {4164, 9400}, {4361, 20906}, {4383, 4776}, {4491, 20979}, {4501, 4777}, {4507, 8633}, {4932, 18199}, {6371, 8659}, {8640, 16692}, {9015, 15413}, {16777, 21348}, {17212, 18166}, {18106, 20295}

X(21007) = isogonal conjugate of isotomic conjugate of X(17494)
X(21007) = polar conjugate of isotomic conjugate of X(22160)
X(21007) = crossdifference of every pair of points on line X(10)X(141) (the complement of line X(1)X(6))

X(21008) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 - a b - b^2 - a c + b c - c^2) : :

X(21008) lies on these lines: {1, 1929}, {3, 2176}, {6, 41}, {32, 995}, {35, 3230}, {36, 213}, {37, 5110}, {39, 101}, {45, 5782}, {55, 3009}, {169, 9619}, {187, 595}, {190, 7783}, {214, 16600}, {220, 5013}, {284, 17053}, {386, 2242}, {404, 2295}, {501, 5006}, {574, 3730}, {869, 17798}, {904, 1964}, {978, 4426}, {999, 2271}, {1015, 4251}, {1030, 16685}, {1191, 3053}, {1201, 1914}, {1385, 16583}, {1429, 3752}, {1434, 17365}, {1500, 4256}, {1575, 2329}, {1740, 8424}, {1975, 4713}, {2238, 2975}, {2241, 4262}, {2251, 5299}, {2276, 9310}, {2646, 3290}, {3052, 5023}, {3204, 5069}, {3216, 5291}, {3570, 21226}, {3576, 16968}, {3684, 17448}, {3721, 4511}, {4210, 7109}, {4258, 16781}, {4286, 17796}, {4628, 9481}, {4642, 17439}, {5134, 7756}, {5563, 20963}, {7749, 17734}, {7987, 16970}, {14260, 17969}, {16777, 19765}

X(21008) = isogonal conjugate of isotomic conjugate of X(6646)
X(21008) = polar conjugate of isotomic conjugate of X(22161)

X(21009) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (2 a^3 + a^2 b - a b^2 - 2 b^3 + a^2 c + b^2 c - a c^2 + b c^2 - 2 c^3) : :

X(21009) lies on these lines: {3, 7611}, {6, 922}, {36, 238}, {55, 199}, {56, 4471}, {198, 4497}, {674, 1055}, {2174, 8539}, {2486, 14953}, {3122, 3285}, {4038, 18173}, {4436, 19308}, {4516, 15586}, {4557, 17798}, {4890, 17454}, {5547, 9142}, {7113, 8540}, {16686, 16694}

X(21009) = isogonal conjugate of isotomic conjugate of X(17491)
X(21009) = polar conjugate of isotomic conjugate of X(22162)

X(21010) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b - a b^2 + a^2 c + a b c + b^2 c - a c^2 + b c^2) : :

X(21010) lies on these lines: {1, 3}, {2, 4447}, {6, 292}, {11, 7377}, {12, 7380}, {25, 16974}, {31, 172}, {37, 3941}, {42, 2275}, {48, 19133}, {71, 16516}, {100, 4393}, {145, 4433}, {181, 12836}, {183, 870}, {198, 16972}, {239, 1376}, {348, 3475}, {388, 7379}, {474, 16825}, {497, 6999}, {579, 3688}, {604, 2330}, {612, 1107}, {672, 4517}, {750, 21352}, {958, 16830}, {985, 11328}, {1001, 14621}, {1011, 5311}, {1088, 7176}, {1100, 15624}, {1284, 4307}, {1397, 10799}, {1400, 3056}, {1405, 8540}, {1475, 2340}, {1696, 16970}, {1914, 16524}, {1961, 16058}, {2260, 3779}, {2276, 16523}, {2308, 7296}, {2664, 4383}, {3052, 3747}, {3061, 20715}, {3185, 20471}, {3247, 16688}, {3715, 16552}, {4191, 17017}, {4253, 20683}, {4362, 11358}, {4384, 4413}, {4423, 16831}, {4436, 17318}, {4471, 19297}, {4644, 21320}, {4787, 20984}, {7085, 16519}, {7198, 11246}, {8053, 16777}, {8299, 17316}, {8301, 11329}, {9440, 20793}, {10578, 17081}, {15668, 16684}, {16405, 17763}, {16515, 17735 }, {16672, 16694}, {17754, 19584}

X(21010) = isogonal conjugate of isotomic conjugate of X(24349)
X(21010) = polar conjugate of isotomic conjugate of X(22163)

X(21011) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + b^4 - a^2 c^2 - 2 b^2 c^2 + c^4) : :
Trilinears (b + c) cos(B - C) : :

X(21011) lies on these lines: {2, 17221}, {4, 9}, {5, 1953}, {12, 2294}, {37, 21044}, {42, 2165}, {48, 355}, {53, 2181}, {80, 284}, {101, 1141}, {199, 20989}, {201, 1865}, {219, 5790}, {306, 8797}, {311, 14213}, {313, 327}, {579, 18395}, {594, 21018}, {857, 21231}, {952, 17438}, {1018, 21065}, {1108, 17606}, {1251, 11082}, {1441, 4466}, {1737, 2260}, {1761, 5080}, {1903, 3698}, {2173, 18357}, {2267, 17303}, {2980, 21034}, {3136, 21028}, {3613, 15523}, {3949, 17757}, {4024, 10412}, {4628, 18082}, {5239, 11099}, {5240, 11100}, {5747, 10573}, {17751, 21076}, {20486, 21023}, {21035, 21043}, {21061, 21066}

X(21011) = barycentric product X(5)*X(10)

X(21012) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^4 - 3 a^2 b^2 + b^4 - 3 a^2 c^2 - 2 b^2 c^2 + c^4) : :

X(21012) lies on these lines: {4, 9}, {101, 13597}, {140, 17438}, {594, 21013}, {1232, 20879}, {1953, 5690}, {2260, 10039}, {2267, 17275}, {3958, 17757}, {4062, 15464}, {4466, 21231}, {5742, 8256}, {21022, 21047}

X(21013) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^3 - 2 a^2 b - a b^2 + b^3 - 2 a^2 c + 4 a b c - b^2 c - a c^2 - b c^2 + c^3) : :

X(21013) lies on these lines: {2, 4919}, {10, 1018}, {71, 21030}, {594, 21012}, {672, 6735}, {966, 16561}, {1145, 2170}, {1146, 14439}, {1475, 10915}, {3035, 17439}, {5282, 5657}, {8256, 17451}

X(21014) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c) (2 a^3 + 2 a^2 b - a b^2 - b^3 + 2 a^2 c + 4 a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21014) lies on these lines: {9, 11604}, {10, 1400}, {48, 17275}, {71, 21029}, {594, 21012}, {1213, 2294}, {1405, 5831}, {2269, 6734}, {3724, 8013}, {4999, 17440}, {5742, 17451}

X(21015) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 + b^2 + c^2) (a^2 + b^2 - 2 b c + c^2) : :

X(21015) lies on these lines: {2, 7085}, {5, 3305}, {9, 427}, {10, 429}, {11, 212}, {12, 201}, {40, 235}, {63, 1368}, {71, 1213}, {125, 3690}, {219, 1899}, {220, 1853}, {228, 440}, {343, 3781}, {468, 5285}, {516, 2355}, {594, 21028}, {858, 3219}, {1473, 7386}, {1851, 6554}, {1883, 12572}, {1985, 2886}, {2323, 11245}, {2550, 4207}, {3220, 7667}, {3914, 16583}, {3955, 11064}, {4026, 4204}, {4679, 17111}, {5314, 6676}, {6358, 7140}, {7293, 10691}, {15523, 21033}, {17441, 18589}, {21020, 21029}

X(21015) = barycentric product X(i)*X(j) for these {i,j}: {10, 18589}, {306, 3914}, {594, 17170}, {3695, 4000}

X(21016) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 + b^2 - c^2) (a^2 + b^2 - c^2) (b^2 + c^2) : :

X(21016) lies on these lines: {4, 9}, {42, 13854}, {427, 15523}, {429, 21029}, {860, 1840}, {1235, 20883}, {1969, 18022}, {1973, 5090}, {4466, 16607}, {5307, 17308}, {16583, 21044}, {20235, 20305}

X(21016) = polar conjugate of isotomic conjugate of X(15523)
X(21016) = barycentric product X(10)*X(427)

X(21017) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 b^2 + b^6 - a^4 c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6) : :

X(21017) lies on these lines: {4, 9}, {101, 2697}, {850, 1577}, {858, 18669}, {1236, 20884}, {4062, 10415}, {4466, 21234}, {16607, 20235}, {16611, 21044}

X(21018) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + b^4 - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + c^4) : :

X(21018) lies on these lines: {10, 21065}, {115, 4016}, {594, 21011}, {758, 8818}, {1213, 21044}, {2160, 2475}, {2174, 5086}, {2294, 5949}, {3728, 21043}, {8287, 18698}, {16732, 17052}, {20654, 21024}

X(21019) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + b^4 + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + c^4) : :

X(21019) lies on these lines: {10, 2245}, {594, 21011}, {661, 20483}, {3262, 21237}, {3814, 17444}, {3943, 21044}, {4053, 17757}, {5053, 17303}, {5123, 8609}, {5176, 7113}, {20486, 21045}

X(21020) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a b + a c + 2 b c) : :

X(21020) lies on these lines: {1, 4720}, {2, 740}, {8, 2650}, {10, 321}, {31, 5271}, {37, 4365}, {38, 75}, {42, 3696}, {63, 1719}, {210, 20718}, {244, 3741}, {274, 18059}, {333, 896}, {354, 4688}, {442, 20653}, {512, 14433}, {523, 6545}, {536, 3989}, {594, 2294}, {690, 14430}, {718, 16829}, {726, 4980}, {748, 3747}, {750, 11679}, {758, 3679}, {804, 1635}, {846, 5235}, {894, 4722}, {899, 3725}, {997, 12081}, {1045, 10458}, {1109, 4712}, {1150, 3980}, {1211, 3120}, {1213, 4037}, {1215, 4651}, {1376, 3724}, {1654, 4683}, {1698, 3743}, {1733, 11031}, {1836, 3958}, {2234, 17187}, {2643, 4137}, {2667, 3706}, {2783, 11203}, {2887, 20360}, {2901, 16828}, {3136, 21033}, {3263, 4967}, {3578, 17770}, {3617, 17164}, {3634, 4065}, {3703, 4665}, {3722, 3757}, {3775, 17184}, {3842, 3995}, {3896, 4709}, {3923, 5278}, {3932, 6535}, {3936, 21085}, {4042, 4363}, {4046, 4062}, {4054, 4104}, {4068, 4423}, {4093, 21264}, {4155, 4728}, {4201, 15349}, {4361, 17017}, {4387, 17259}, {4414, 5737}, {4457, 19998}, {465 0, 5361}, {4672, 19742}, {4697, 16704}, {4699, 10453}, {4716, 17011}, {4739, 7449}, {4763, 9147}, {4847, 17874}, {4931, 10278}, {5260, 12567}, {5263, 17469}, {5695, 19732}, {6186, 19329}, {6682, 17495}, {8025, 17162}, {10436, 17156}, {16454, 17733}, {17119, 17599}, {20486, 21021}, {20711, 21238}, {21015, 21029}

X(21020) = homothetic center of Gemini triangle 18 and cross-triangle of Gemini triangles 16 and 18
X(21020) = barycentric product X(10)*X(3739)

X(21021) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^2 + b c) : :

X(21021) lies on these lines: {8, 9596}, {10, 762}, {12, 594}, {37, 3701}, {39, 4692}, {172, 7081}, {321, 20691}, {442, 20483}, {756, 7148}, {894, 4400}, {984, 7242}, {1089, 1500}, {1107, 4696}, {1215, 2295}, {1237, 3963}, {1575, 4968}, {1909, 7187}, {2276, 4385}, {3726, 3831}, {3930, 21024}, {3992, 16589}, {4030, 7745}, {4680, 9650}, {4894, 5475}, {6645, 7267}, {7264, 9466}, {7272, 7854}, {20486, 21020}

X(21021) = barycentric product X(10)*X(1215)

X(21022) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b^2 + a^2 c^2 + 2 b^2 c^2) : :

X(21022) lies on these lines: {10, 75}, {594, 2486}, {2533, 21143}, {3122, 3963}, {3613, 15523}, {3778, 4377}, {3934, 17445}, {4039, 18082}, {21012, 21047}, {21038, 21043}, {21083, 21094}

X(21023) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b^3 + b^4 - b^3 c - a c^3 - b c^3 + c^4) : :

X(21023) lies on these lines: {10, 21069}, {594, 20482}, {1930, 4178}, {15523, 21028}, {16886, 16894}, {17046, 17447}, {17047, 20236}, {20486, 21011}

X(21024) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a b^2 + b^2 c + a c^2 + b c^2) : :

X(21024) lies on these lines: {6, 10449}, {8, 2176}, {10, 37}, {75, 20255}, {76, 141}, {115, 3454}, {257, 312}, {314, 15985}, {321, 1237}, {407, 1840}, {442, 19584}, {519, 20970}, {524, 17499}, {538, 16887}, {762, 3992}, {960, 1146}, {1043, 18755}, {1086, 20888}, {1089, 3954}, {1107, 3741}, {1575, 3831}, {1654, 17685}, {1901, 10381}, {2229, 2275}, {2245, 3501}, {2292, 4037}, {2295, 17751}, {3061, 20545}, {3125, 4647}, {3136, 15523}, {3159, 7230}, {3169, 17275}, {3589, 17034}, {3679, 4050}, {3702, 3727}, {3725, 4046}, {3726, 4968}, {3730, 17340}, {3780, 17135}, {3840, 16604}, {3912, 17056}, {3930, 21021}, {3936, 17230}, {4044, 4415}, {4266, 17330}, {4272, 17299}, {4362, 16974}, {5105, 17398}, {5283, 10479}, {5774, 14974}, {6535, 20966}, {11679, 16968}, {14973, 20683}, {16918, 17277}, {17128, 19667}, {17228, 20943}, {17369, 17750}, {18035, 18277}, {20653, 20658}, {20654, 21018}, {20911, 21138}

X(21024) = complement of X(33296)
X(21024) = barycentric product X(10)*X(3741)

X(21025) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a b^2 - 2 a b c + b^2 c + a c^2 + b c^2) : :

X(21025) lies on these lines: {8, 16969}, {10, 37}, {11, 20594}, {76, 1086}, {116, 7794}, {141, 6376}, {257, 20947}, {312, 3959}, {1089, 3125}, {1107, 3831}, {1146, 1329}, {1930, 21138}, {2084, 21051}, {2140, 9466}, {2238, 17751}, {3122, 7148}, {3501, 17340}, {3661, 5743}, {3662, 20943}, {3701, 3721}, {3703, 20284}, {3726, 4696}, {3727, 4358}, {3730, 4370}, {3840, 17448}, {3948, 4415}, {3954, 3992}, {4037, 4642}, {4385, 20271}, {4465, 17152}, {4713, 21281}, {5233, 16594}, {6381, 21240}, {7277, 17499}, {10449, 17362}, {14377, 17130}, {15523, 21040}, {16886, 20494}, {17747, 20719}, {20483, 21029}

X(21025) = complement of X(34063)
X(21025) = barycentric product X(10)*X(3840)

X(21026) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b + 2 b^2 - a c - 2 b c + 2 c^2) : :

X(21026) lies on these lines: {2, 4434}, {10, 2650}, {38, 3662}, {75, 18054}, {244, 3006}, {594, 2294}, {661, 2533}, {756, 2887}, {896, 4645}, {899, 3823}, {1213, 4144}, {1962, 4972}, {3120, 3932}, {3703, 7263}, {3834, 17449}, {3914, 3950}, {3952, 4892}, {3992, 4013}, {4062, 4819}, {4071, 5257}, {4358, 21241}, {4432, 21282}, {4442, 6541}, {17234, 17450}, {17757, 21041}, {20488, 21054}

X(21027) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-2 a b + b^2 - 2 a c - 4 b c + c^2) : :

X(21027) lies on these lines: {10, 3120}, {38, 7263}, {594, 2294}, {1647, 21242}, {2887, 8013}, {3696, 4062}, {3822, 21042}, {3841, 20653}, {3842, 4442}, {3914, 5257}, {3936, 4732}, {3950, 4365}, {4819, 17056}

X(21028) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b+c)^2*((b^2-b*c+c^2)*a^2-(b^3-c^3)*(b-c)) : :

X(21028) lies on these lines: {10, 228}, {12, 7363}, {125, 6358}, {594, 21015}, {756, 21054}, {3136, 21011}, {3925, 21045}, {14213, 21243}, {15523, 21023}, {16886, 20655}, {20305, 21318}, {21091, 21319}

X(21029) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b^2 + b^3 - b^2 c - a c^2 - b c^2 + c^3) : :

X(21029) lies on these lines: {1, 17737}, {4, 5282}, {8, 4109}, {9, 5046}, {10, 1018}, {12, 3930}, {38, 5254}, {41, 3419}, {71, 21014}, {115, 3954}, {191, 5134}, {321, 4136}, {355, 4390}, {429, 21016}, {594, 21011}, {672, 6734}, {976, 3767}, {1055, 17647}, {1475, 10916}, {1647, 16604}, {2329, 5086}, {2475, 3509}, {2886, 17451}, {3061, 11680}, {3120, 3721}, {3136, 15523}, {3684, 5178}, {3691, 5179}, {3822, 3970}, {3925, 21049}, {4071, 17751}, {4119, 4696}, {4414, 9598}, {5794, 9310}, {17046, 20880}, {17048, 17672}, {17739, 20553}, {20483, 21025}, {20653, 20659}, {21015, 21020}

X(21030) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c) (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : :

X(21030) lies on these lines: {10, 1400}, {71, 21013}, {210, 8013}, {594, 21011}, {1329, 17452}, {2171, 17757}, {2269, 6735}, {2321, 21044}, {15523, 21023}, {20483, 20487}, {20895, 21244}

X(21031) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21031) lies on these lines: {1, 3820}, {2, 3304}, {3, 6174}, {5, 3656}, {8, 11}, {10, 12}, {21, 4995}, {40, 12679}, {55, 452}, {56, 3421}, {78, 10950}, {100, 15338}, {119, 5690}, {120, 3314}, {140, 5258}, {145, 3816}, {191, 11698}, {200, 1837}, {220, 5514}, {341, 3703}, {354, 8582}, {377, 11236}, {388, 4413}, {392, 10915}, {404, 529}, {443, 11237}, {474, 5434}, {495, 1698}, {496, 3632}, {497, 8165}, {498, 9708}, {519, 4187}, {528, 5046}, {551, 17575}, {594, 21011}, {908, 5836}, {936, 5252}, {950, 3689}, {956, 5433}, {958, 5432}, {960, 6735}, {997, 10944}, {1001, 10528}, {1145, 3878}, {1155, 12527}, {1319, 6700}, {1376, 3436}, {1377, 19028}, {1378, 19027}, {1478, 9709}, {1532, 11362}, {1697, 4679}, {1706, 1836}, {1727, 16154}, {1788, 5815}, {1834, 2318}, {1904, 17281}, {2476, 9710}, {2478, 3058}, {2550, 10895}, {2646, 5795}, {2886, 3614}, {2899, 4387}, {2975, 3035}, {3057, 3452}, {3061, 4534}, {3085, 16845}, {3303, 5084}, {3584, 6675}, {3625, 3825}, {3626, 3814}, {3683, 18250}, {3695, 3992}, {3701, 3704}, {3714, 4046}, {3826, 5686}, {3828, 17529}, {3829, 5154}, {3869, 8256}, {3893, 12053}, {3929, 9588}, {3930, 21049}, {3940, 10573}, {3971, 4918}, {4317, 16417}, {4415, 4642}, {4421, 6872}, {4646, 4854}, {4662, 5123}, {4668, 7741}, {4669, 17533}, {4678, 11680}, {4745, 17530}, {4847, 17606}, {4853, 11376}, {4860, 9780}, {4863, 4882}, {4866, 5557}, {4999, 5326}, {5044, 10039}, {5082, 10896}, {5187, 11235}, {5220, 10940}, {5260, 6690}, {5288, 15325}, {5293, 5724}, {5298, 8666}, {5554, 12635}, {5587, 6769}, {5657, 18242}, {5687, 6154}, {5697, 13996}, {5791, 10954}, {5818, 6990}, {5828, 15844}, {5881, 6922}, {6762, 17728}, {6904, 9657}, {6919, 11238}, {6921, 11194}, {6949, 20400}, {7355, 20307}, {7681, 12245}, {7794, 13466}, {7956, 11531}, {7958, 10175}, {7965, 19925}, {8164, 19855}, {8167, 10587}, {8580, 9578}, {8715, 11113}, {9565, 10406}, {9623, 11375}, {9843, 17609}, {10056, 11108}, {10106, 20103}, {10391, 18247}, {10916, 17619}, {11499, 11827}, {12616, 18908}, {12953, 17784}, {13724, 15621}, {16160, 18357}, {16886, 20482}

X(21031) = barycentric product X(10)*X(3452)

X(21032) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3558), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^4 - b^4 - c^4 + (- a^2 + b^2 + c^2) Sqrt[a^4 +b^4 + c^4 - a^2 b^2 - a^2 c^2 - b^2 c^2]) : :

X(21032) lies on these lines: {2, 3}, {51, 1341}, {154, 5638}, {184, 1379}, {3557, 13366}, {5639, 21001}, {14631, 15004}

X(21033) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (a - b - c) (b + c) (a b + b^2 + a c + c^2) : :

X(21033) lies on these lines: {8, 17452}, {9, 21}, {10, 2171}, {37, 42}, {38, 2277}, {72, 1400}, {101, 2359}, {198, 199}, {346, 3985}, {391, 3061}, {429, 20653}, {573, 5692}, {594, 21011}, {604, 997}, {896, 2305}, {936, 2285}, {960, 2269}, {966, 17451}, {1213, 2294}, {1229, 3452}, {1334, 3694}, {1654, 1959}, {1743, 5429}, {1766, 2960}, {1953, 17275}, {2092, 2292}, {2170, 3686}, {2245, 3958}, {2298, 5293}, {2321, 3701}, {3136, 21020}, {3169, 3877}, {3678, 21061}, {3942, 17344}, {3950, 4006}, {3970, 3986}, {4069, 4538}, {4111, 4516}, {4511, 17440}, {4866, 16673}, {5227, 9310}, {5232, 7146}, {5777, 15979}, {6007, 10868}, {15523, 21015}

X(21033) = barycentric product X(10)*X(960)

X(21034) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a^4 (b + c) (a^4 - b^4 - c^4) : :

X(21034) lies on these lines: {10, 16277}, {22, 2172}, {25, 41}, {48, 10829}, {101, 306}, {206, 4548}, {1918, 18892}, {2980, 21011}

X(21034) = barycentric product X(10)*X(206)

X(21035) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a^2 (b + c) (b^2 + c^2) : :

Let L be the isogonal conjugate of the isotomic conjugate of the Nagel line (i.e., line X(6)X(31)).
Let M be the isotomic conjugate of the isogonal conjugate of the Nagel line (i.e., line X(10)X(75)).
Then X(21035) = L∩M. (Randy Hutson, July 11, 2019)

X(21035) lies on these lines: {1, 4283}, {2, 4446}, {6, 31}, {9, 3764}, {10, 75}, {35, 5009}, {37, 3122}, {38, 141}, {39, 1964}, {86, 291}, {100, 745}, {101, 755}, {190, 256}, {191, 1045}, {192, 4443}, {244, 17245}, {560, 2273}, {583, 20985}, {594, 3728}, {714, 3963}, {756, 1213}, {869, 4261}, {872, 2092}, {882, 4079}, {982, 17234}, {1030, 18266}, {1100, 20456}, {1500, 2667}, {1654, 4651}, {1740, 3097}, {2200, 2353}, {2245, 20964}, {2274, 3781}, {2277, 4517}, {2292, 4026}, {3123, 17246}, {3214, 5220}, {3670, 3836}, {3690, 3725}, {3912, 4022}, {3948, 21238}, {3954, 20969}, {3971, 21257}, {4016, 7237}, {4085, 4424}, {4286, 20990}, {4392, 17232}, {4415, 20487}, {4484, 16777}, {4553, 16696}, {4642, 21039}, {4685, 17346}, {4687, 17065}, {5069, 7032}, {7226, 17238}, {14370, 20994}, {14839, 17445}, {16556, 17596}, {17243, 21330}, {18179, 20590}, {21011, 21043}

X(21035) = complement of X(17142)
X(21035) = crossdifference of every pair of points on line X(514)X(1919)
X(21035) = crosssum of X(58) and X(86)
X(21035) = crosspoint of X(10) and X(42)
X(21035) = trilinear pole of line X(2084)X(3005)
X(21035) = barycentric product X(10)*X(39)

X(21036) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3557), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^4 - b^4 - c^4 + (a^2 - b^2 - c^2) Sqrt[a^4 +b^4 + c^4 - a^2 b^2 - a^2 c^2 - b^2 c^2]) : :

X(21036) lies on these lines: {2, 3}, {51, 1340}, {154, 5639}, {184, 1380}, {3558, 13366}, {5638, 21001}, {14630, 15004}

X(21037) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (b^2 + c^2) (a^2 + b^2 - b c + c^2) : :

X(21037) lies on these lines: {10, 18098}, {756, 16886}, {1213, 20483}, {4071, 5276}, {4972, 16600}, {17456, 21249}, {20898, 21248}

X(21038) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (b^2 + c^2) (2 a^2 + b^2 + c^2) : :

X(21038) lies on these lines: {10, 82}, {38, 141}, {71, 15321}, {1213, 20483}, {2292, 3932}, {3589, 4030}, {6292, 17457}, {21022, 21043}

X(21038) = barycentric product X(10)*X(6292)

X(21039) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (a - b - c) (b + c) (a b - b^2 + a c + 2 b c - c^2) : :

X(21039) lies on these lines: {2, 21346}, {9, 294}, {10, 307}, {37, 42}, {38, 4000}, {75, 4712}, {220, 4336}, {227, 3983}, {244, 17278}, {391, 4073}, {612, 2257}, {984, 3672}, {1212, 2293}, {1229, 4847}, {1827, 8012}, {2170, 3688}, {2171, 20683}, {2292, 3755}, {3000, 15587}, {3242, 3924}, {3715, 7069}, {3779, 17451}, {3958, 20713}, {4328, 5223}, {4343, 16601}, {4516, 7064}, {4517, 17452}, {4642, 21035}, {7308, 18216}

X(21039) = barycentric product X(10)*X(1212)

X(21040) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b - a c + b c) (-a b^2 + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(21040) lies on these lines: {2, 3226}, {10, 16606}, {210, 20721}, {313, 321}, {2887, 20491}, {3925, 20484}, {6382, 21250}, {15523, 21025}, {20483, 20487}

X(21041) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (2 a - b - c) (b + c) (a b + b^2 + a c - 4 b c + c^2) : :

X(21041) lies on these lines: {2, 9457}, {8, 11814}, {10, 3120}, {80, 10713}, {121, 1647}, {214, 1644}, {594, 21044}, {668, 17213}, {1317, 12035}, {9458, 21290}, {15523, 21042}, {16594, 17460}, {17757, 21026}, {20653, 21054}

X(21042) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - 2 b - 2 c) (b + c) (a b + b^2 + a c - b c + c^2) : :

X(21042) lies on these lines: {10, 2650}, {321, 4013}, {594, 17757}, {1698, 16474}, {1739, 3662}, {2099, 3679}, {3822, 21027}, {4793, 21251}, {15523, 21041}, {17239, 17313}

X(21043) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c)^3 : :

X(21043) lies on these lines: {10, 190}, {42, 1989}, {58, 502}, {115, 2643}, {313, 1934}, {338, 1109}, {594, 6543}, {662, 13178}, {1826, 1918}, {3120, 18004}, {3122, 21044}, {3728, 21018}, {3932, 20488}, {4036, 16732}, {4039, 4156}, {4516, 4705}, {17719, 21098}, {21011, 21035}, {21022, 21038}

X(21043) = barycentric product X(10)*X(115)

X(21044) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b - c)^2 (b + c) : :

X(21044) lies on these lines: {2, 9317}, {5, 17451}, {8, 4919}, {9, 11604}, {10, 1018}, {11, 1146}, {12, 21049}, {37, 21011}, {41, 1837}, {80, 101}, {115, 125}, {116, 1111}, {150, 9318}, {169, 10826}, {226, 4566}, {257, 17669}, {284, 7110}, {312, 4165}, {355, 9310}, {484, 5134}, {515, 1055}, {594, 21041}, {672, 1737}, {857, 16609}, {952, 17439}, {1015, 1647}, {1109, 3708}, {1210, 1475}, {1212, 17606}, {1213, 21018}, {1229, 21244}, {1400, 1826}, {1565, 21139}, {1566, 5532}, {1577, 21207}, {1884, 2312}, {2082, 9581}, {2246, 12019}, {2280, 5722}, {2310, 8735}, {2321, 21030}, {2345, 16561}, {2347, 20262}, {2486, 21045}, {3061, 4193}, {3121, 16613}, {3122, 21043}, {3496, 5046}, {3509, 5080}, {3583, 5011}, {3691, 6734}, {3693, 5123}, {3701, 4136}, {3702, 4167}, {3730, 18395}, {3767, 3924}, {3930, 17757}, {3943, 21019}, {4092, 4516}, {4109, 17751}, {4119, 4723}, {4466, 8287}, {5060, 7424}, {5517, 5521}, {5540, 10773}, {7200, 17213}, {9956, 16601}, {15523, 20684}, {16583, 21016}, {16611, 21017}, {16886, 20494}, {20653, 20658}, {21046, 21054}

X(21044) = isotomic conjugate of X(4620)
X(21044) = complement of X(17136)
X(21044) = crosspoint, wrt medial triangle, of X(115) and X(1146)
X(21044) = crossdifference of every pair of points on line X(109)X(110)
X(21044) = barycentric product X(10)*X(11)

X(21045) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c) (-a b + b^2 - a c + b c + c^2) : :

X(21045) lies on these lines: {10, 4557}, {116, 17463}, {125, 136}, {523, 4466}, {594, 20482}, {1826, 15320}, {2486, 21044}, {3120, 18004}, {3925, 21028}, {4092, 16732}, {4516, 8287}, {4858, 21252}, {6741, 18210}, {17886, 21340}, {20484, 20654}, {20486, 21019}

X(21046) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c)^3 (-a^2 + b^2 + c^2) : :

X(21046) lies on these lines: {10, 98}, {42, 8791}, {71, 265}, {115, 2643}, {125, 3708}, {163, 13211}, {306, 4561}, {339, 20902}, {4024, 12079}, {16886, 20658}, {17886, 21253}, {20482, 20653}, {21044, 21054}

X(21046) = X(27)-isoconjugate of X(1101)
X(21046) = barycentric product X(10)*X(125)

X(21047) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^4 - 2 a^2 b^2 + b^4 - 2 a^2 c^2 + c^4) : :

X(21047) lies on these lines: {10, 190}, {71, 16894}, {620, 17467}, {4062, 9164}, {9293, 9396}, {17768, 20488}, {21012, 21022}

X(21048) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + 2 b^4 - a^2 c^2 - 2 b^2 c^2 + 2 c^4) : :

X(21048) lies on these lines: {10, 598}, {625, 17472}, {3613, 15523}, {4079, 20491}, {20912, 21256}

X(21049) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b - 2 a b^2 + b^3 + a^2 c + 4 a b c - b^2 c - 2 a c^2 - b c^2 + c^3) : :

X(21049) lies on these lines: {1, 1146}, {6, 938}, {10, 37}, {11, 17451}, {12, 21044}, {65, 17747}, {169, 5722}, {220, 18391}, {857, 5244}, {910, 950}, {942, 5179}, {1086, 3673}, {1100, 5199}, {1210, 1212}, {1426, 1826}, {1446, 18635}, {1737, 16601}, {1759, 11113}, {2256, 17362}, {2275, 3756}, {3061, 3816}, {3207, 3486}, {3208, 8256}, {3488, 4258}, {3721, 4415}, {3753, 21073}, {3754, 21090}, {3925, 21029}, {3930, 21031}, {3970, 17757}, {4251, 12433}, {4437, 6376}, {4513, 5554}, {4882, 17299}, {4904, 7264}, {5011, 15171}, {5286, 17054}, {6706, 17245}, {6765, 17388}, {9310, 10950}, {17060, 21138}

X(21050) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (-b + c) (b + c)^2 (a^4 - a^2 b^2 - a^2 c^2 + 2 b^2 c^2) : :

X(21050) lies on these lines: {4024, 21052}, {4079, 20491}, {17899, 21259}

X(21051) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a b + a c - b c) : :

X(21051) lies on these lines: {2, 4367}, {8, 4879}, {10, 512}, {12, 7178}, {141, 9040}, {513, 3823}, {514, 3837}, {523, 1577}, {525, 18004}, {650, 814}, {659, 21301}, {661, 2533}, {693, 4490}, {784, 4791}, {804, 3709}, {900, 1734}, {1019, 1698}, {1491, 4391}, {1826, 16229}, {2084, 21025}, {2530, 3762}, {2787, 14838}, {3566, 14321}, {3568, 21004}, {3777, 4462}, {3801, 4088}, {3835, 4083}, {4010, 4041}, {4079, 20491}, {4122, 21124}, {4151, 4770}, {4170, 4730}, {4761, 4983}, {4784, 9780}, {4874, 8678}, {5518, 21138}, {5996, 8034}, {6002, 9508}, {9276, 10278}, {20949, 21349}

X(21052) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^2 - a b - a c + 2 b c) : :

X(21052) lies on these lines: {2, 3907}, {10, 1577}, {42, 17478}, {512, 14431}, {514, 14430}, {523, 14429}, {656, 4036}, {661, 2533}, {693, 4147}, {814, 1635}, {905, 4474}, {1698, 14838}, {1734, 4791}, {2254, 4391}, {2517, 17420}, {3566, 4120}, {3716, 21302}, {4010, 4729}, {4017, 4086}, {4024, 21050}, {4083, 4728}, {4088, 7178}, {4129, 4761}, {4170, 4807}, {4449, 4885}, {4560, 9780}, {4724, 20317}, {4931, 10278}

X(21053) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2)(a b^2 - b^2 c + a c^2 - b c^2) : :

X(21053) lies on these lines: {2, 4107}, {115, 125}, {121, 5513}, {523, 21055}, {649, 17072}, {661, 2533}, {850, 1577}, {1213, 14407}, {3005, 4705}, {3250, 21260}, {3261, 21262}, {3766, 21261}, {3837, 20532}, {4036, 8061}, {4375, 21303}, {20316, 20979}

X(21054) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2)^2 (-a^2 + b^2 + b c + c^2) : :

X(21054) lies on these lines: {2, 2606}, {10, 21}, {11, 5952}, {79, 502}, {125, 1109}, {388, 9405}, {756, 21028}, {2607, 3448}, {2611, 6741}, {3120, 18004}, {4705, 18210}, {6186, 15168}, {8013, 20656}, {15523, 20531}, {20488, 21026}, {20494, 20655}, {20653, 21041}, {21044, 21046}

X(21055) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21055) lies on these lines: {10, 798}, {313, 20910}, {523, 21053}, {661, 20483}, {4079, 20491}, {4086, 8061}, {4129, 4826}, {17303, 20981}, {17458, 21260}, {20906, 21262}, {20949, 21261}

X(21056) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(21056) lies on these lines: {850, 1577}, {4079, 20491}

X(21057) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 + 2 b^2 - 3 b c + 2 c^2) : :

X(21057) lies on these lines: {2, 7267}, {12, 594}, {44, 966}, {115, 4037}, {316, 4760}, {325, 7200}, {625, 14210}, {661, 2533}, {17316, 17720}, {17757, 20483}

X(21058) = X(57)-CEVA CONJUGATE OF X(25)

Barycentrics a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-2 a^5 b+a^4 b^2-a^2 b^4+2 a b^5-b^6-2 a^5 c-2 a^4 b c+2 a^3 b^2 c+2 a^2 b^3 c+a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2+2 a^2 b c^3-2 a b^2 c^3-a^2 c^4+b^2 c^4+2 a c^5-c^6) : :

X(21058) lies on these lines: {19, 614}, {31, 607}, {55, 17409}, {1783, 17784}, {2332, 7169}, {3101, 8743}, {5452, 8750}

X(21058) = X(57)-Ceva conjugate of X(25)
X(21058) = barycentric product X(4)*X(18621)
X(21058) = barycentric quotient X(18621)/X(69)

X(21059) = X(57)-CEVA CONJUGATE OF X(41)

Barycentrics a^3 (a^2-2 a b+b^2-2 a c+c^2) : :

X(21059) lies on these lines: {6,31}, {19,2195}, {32,1802}, {40,595}, {41,15624}, {44,3059}, {48,2175}, {56,20780}, {57,2191}, {65,1279}, {109,269}, {171,4648}, {218,4878}, {238,2550}, {572,16688}, {580,6769}, {601,991}, {603,1458}, {604,692}, {748,3925}, {750,17245}, {995,7688}, {1104,7957}, {1191,5584}, {1400,1486}, {1418,9316}, {1423,1633}, {1496,4252}, {1582,3550}, {1743,3174}, {2093,16487}, {2176,3010}, {2183,7083}, {2187,2352}, {2260,10934}, {2294,3747}, {2303,2328}, {2340,2911}, {3072,3332}, {3189,5247}, {3217,4557}, {3752,7964}, {3945,17126}, {4000,9441}, {4319,8557}, {4336,8609}, {4644,9440}, {8551,16283}, {11406,14975}, {15287,16483}, {17127,17784}

X(21059) = X(i)-Ceva conjugate of X(j) for these (i,j): {57, 41}, {1037, 48}
X(21059) = crosspoint of X(i) and X(j) for these (i,j): {57, 4350}, {109, 1110}, {218, 1617}, {3870, 7719}
X(21059) = crosssum of X(i) and X(j) for these (i,j): {277, 6601}, {522, 1111}, {1086, 6362}
X(21059) = X(i)-isoconjugate of X(j) for these (i,j): {2, 277}, {7, 6601}, {75, 2191}, {312, 17107}, {693, 1292}, {13577, 14268}
X(21059) = barycentric product X(i)*X(j) for these {i,j}: {1, 218}, {3, 7719}, {6, 3870}, {9, 1617}, {31, 344}, {41, 6604}, {55, 1445}, {57, 6600}, {58, 3991}, {71, 4233}, {81, 4878}, {101, 3309}, {190, 8642}, {220, 4350}, {692, 4468}, {1026, 2440}, {1110, 4904}, {1174, 15185}, {1253, 17093}
X(21059) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 277}, {32, 2191}, {41, 6601}, {218, 75}, {344, 561}, {1397, 17107}, {1445, 6063}, {1617, 85}, {3309, 3261}, {3870, 76}, {3991, 313}, {4878, 321}, {6600, 312}, {6604, 20567}, {7719, 264}, {8642, 514}, {15185, 1233}
X(21059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 19624, 1253), (31, 1253, 6), (218, 6600, 4878), (595, 13329, 7290), (692, 3941, 604), (1400, 8647, 1486), (1471, 3915, 1279), (2175, 2223, 48)

X(21060) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-3 a^2 + 2 a b + b^2 + 2 a c - 2 b c + c^2) : :

X(21060) lies on these lines: {1, 5129}, {2, 5223}, {4, 6743}, {7, 8580}, {8, 1699}, {9, 13405}, {10, 12}, {40, 5658}, {42, 4356}, {43, 3663}, {57, 5850}, {63, 6745}, {78, 4297}, {100, 10032}, {142, 3740}, {144, 165}, {200, 329}, {306, 3952}, {321, 4061}, {354, 5316}, {388, 12447}, {480, 7580}, {497, 519}, {515, 3940}, {518, 3452}, {527, 1376}, {551, 956}, {553, 4413}, {612, 4349}, {908, 3681}, {936, 4298}, {946, 10157}, {962, 4882}, {997, 4315}, {1125, 3475}, {1210, 5904}, {1836, 3711}, {2318, 4551}, {2321, 3967}, {2551, 6738}, {2999, 4353}, {3059, 5927}, {3085, 18249}, {3158, 5698}, {3210, 5212}, {3244, 4679}, {3436, 3984}, {3625, 4863}, {3664, 5268}, {3679, 10590}, {3701, 4101}, {3705, 4899}, {3715, 17718}, {3717, 4417}, {3755, 4415}, {3811, 4314}, {3868, 8582}, {3869, 6736}, {3874, 9843}, {3927, 6684}, {3929, 5218}, {3930, 3950}, {3932, 4035}, {3949, 8804}, {3951, 5552}, {3965, 10443}, {3986, 21061}, {4052, 4685}, {4054, 4651}, {4058, 21074}, {4078, 4096}, {4133, 4135}, {4355, 17580}, {4416, 7081}, {4667, 4682}, {4684, 18743}, {5084, 6744}, {5220, 5745}, {5226, 5686}, {5231, 5748}, {5234, 5703}, {5281, 6172}, {5325, 6690}, {5493, 5687}, {5691, 20007}, {5712, 7322}, {5763, 9947}, {5775, 11551}, {5795, 12635}, {5811, 6769}, {5837, 12607}, {6765, 12575}, {7074, 16870}, {7080, 12526}, {8583, 12577}, {10324, 16284}, {11362, 18242}, {12053, 17604}

X(21060) = barycentric product X(10)*X(144)

X(21061) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (a^3 - a b^2 + a b c - b^2 c - a c^2 - b c^2) : :
Trilinears cot(A - U) : :, where cot U = cot(A/2) cot(B/2) cot(C/2)

X(21061) lies on these lines: {1, 6}, {2, 10468}, {3, 3713}, {8, 573}, {10, 1400}, {19, 1759}, {40, 5295}, {56, 5783}, {57, 18229}, {58, 2298}, {63, 321}, {71, 1018}, {75, 16574}, {78, 10470}, {101, 2287}, {144, 10446}, {190, 314}, {200, 228}, {210, 1402}, {307, 1020}, {319, 3882}, {329, 10478}, {346, 3730}, {519, 2269}, {572, 2975}, {579, 2345}, {583, 17369}, {594, 2245}, {604, 8666}, {672, 3741}, {758, 2171}, {798, 4404}, {894, 10455}, {966, 3421}, {992, 17053}, {993, 2268}, {1213, 17757}, {1334, 3950}, {1423, 17272}, {1709, 7996}, {1710, 1761}, {1730, 5271}, {1781, 3509}, {1824, 12549}, {1999, 3219}, {2092, 3293}, {2183, 3686}, {2260, 5750}, {2277, 3216}, {2901, 12514}, {3159, 17733}, {3161, 10453}, {3169, 3632}, {3175, 3929}, {3218, 17116}, {3436, 5816}, {3678, 21033}, {3694, 4006}, {3728, 20964}, {3780, 4263}, {3869, 11521}, {3878, 17452}, {3912, 10452}, {3927, 10441}, {3953, 20227}, {3962, 10474}, {3986, 21060}, {4032, 18698}, {4253, 5749}, {4266, 5839}, {4271, 17362}, {4362, 5282}, {4363, 10472}, {4416, 15983}, {4670, 18164}, {4847, 10445}, {5120, 5782}, {5231, 10886}, {5257, 21075}, {5279, 10461}, {5296, 5815}, {8804, 21073}, {12435, 12526}, {18785, 21084}, {21011, 21066}

X(21061) = barycentric product X(10)*X(2975)

X(21062) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^5 - a^4 b + a b^4 + b^5 - a^4 c + 2 a^2 b^2 c - b^4 c + 2 a^2 b c^2 - 2 a b^2 c^2 + a c^4 - b c^4 + c^5) : :

X(21062) lies on these lines: {2, 1766}, {9, 1848}, {10, 429}, {25, 516}, {37, 226}, {40, 406}, {92, 5179}, {219, 5928}, {306, 21078}, {321, 857}, {329, 17742}, {405, 946}, {427, 12618}, {517, 13567}, {908, 17776}, {990, 7386}, {1104, 12053}, {1763, 4329}, {1826, 6358}, {2321, 21072}, {3178, 3971}, {3294, 4456}, {3695, 4082}, {4425, 12567}, {5249, 14021}, {5905, 18651}, {6836, 16388}, {16050, 17182}, {18747, 20928}

X(21063) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^6 - a^4 b^2 + a^2 b^4 + b^6 - a^4 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 + c^6) : :

X(21063) lies on these lines: {4, 9}, {306, 4174}, {4066, 21073}, {7391, 16545}, {8680, 16607}

X(21064) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-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(21064) lies on these lines: {4, 9}, {5189, 16546}, {8680, 21234}

X(21065) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + b^4 + a^2 b c - a b^2 c - a b c^2 - 2 b^2 c^2 + c^4) : :

X(21065) lies on these lines: {9, 5046}, {10, 21018}, {37, 115}, {100, 7110}, {594, 4015}, {1018, 21011}, {1089, 1826}, {1761, 5134}, {3686, 5179}, {4043, 21094}, {4044, 4150}, {5051, 5257}, {5750, 13740}, {21070, 21076}

X(21066) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + b^4 - a^2 b c + a b^2 c + a b c^2 - 2 b^2 c^2 + c^4) : :

X(21066) lies on these lines: {10, 2245}, {281, 11392}, {502, 594}, {661, 4071}, {908, 18359}, {1089, 1826}, {1220, 5053}, {2323, 5176}, {2325, 5179}, {3814, 8609}, {3943, 21090}, {3984, 4007}, {4033, 21094}, {4092, 20715}, {5080, 16548}, {11813, 17444}, {21011, 21061}

X(21067) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b - a b^2 + a^2 c + b^2 c - a c^2 + b c^2) : :

X(21067) lies on these lines: {10, 762}, {37, 4075}, {274, 3807}, {321, 4006}, {514, 17760}, {519, 7753}, {594, 3454}, {596, 1575}, {726, 3774}, {758, 4095}, {1089, 3930}, {1334, 4115}, {1500, 3159}, {1909, 4568}, {2321, 4053}, {3263, 17758}, {3294, 3952}, {3701, 3970}, {3934, 9055}, {3943, 7230}, {3967, 3991}, {3994, 4099}, {4125, 21071}, {4710, 20501}, {16549, 17165}

X(21068) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b+c)*(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) : :

X(21068) lies on these lines: {4, 2324}, {6, 12053}, {9, 946}, {37, 226}, {71, 3294}, {198, 516}, {346, 908}, {517, 20262}, {950, 3553}, {962, 2270}, {1089, 1826}, {1696, 1836}, {1778, 17197}, {2262, 4301}, {2345, 3452}, {3731, 12047}, {3950, 21077}, {3986, 12609}, {4254, 10624}, {5316, 17303}, {13407, 16673}

X(21069) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + a^3 b - a b^3 + b^4 + a^3 c - b^3 c - a c^3 - b c^3 + c^4) : :

X(21069) lies on these lines: {10, 21023}, {37, 17052}, {306, 21072}, {857, 21078}, {2321, 18589}, {4150, 4153}, {5074, 7112}, {16551, 21285}

X(21070) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b + a b^2 - a^2 c + b^2 c + a c^2 + b c^2) : :

X(21070) lies on these lines: {8, 3294}, {10, 37}, {69, 17732}, {72, 2809}, {76, 4043}, {101, 1043}, {169, 3886}, {213, 519}, {306, 1230}, {321, 1930}, {346, 3730}, {536, 21240}, {596, 3726}, {1018, 17751}, {1089, 3930}, {1330, 5134}, {2140, 4441}, {2292, 4099}, {3159, 3954}, {3501, 4873}, {3661, 3995}, {3678, 3985}, {3701, 4006}, {3706, 16601}, {3948, 3969}, {4066, 21101}, {4109, 21081}, {4253, 10453}, {4568, 18035}, {5179, 21078}, {6057, 20683}, {6542, 17499}, {16552, 17135}, {17034, 17280}, {17281, 17750}, {21065, 21076}

X(21071) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b + a b^2 - a^2 c - a b c + b^2 c + a c^2 + b c^2) : :

X(21071) lies on these lines: {9, 10449}, {10, 37}, {39, 3840}, {72, 3985}, {76, 85}, {142, 20888}, {306, 3948}, {346, 1400}, {519, 2176}, {536, 20255}, {1089, 3970}, {1334, 17751}, {1770, 4987}, {2276, 3831}, {2325, 3730}, {3496, 3685}, {3509, 7283}, {3661, 21216}, {3663, 21240}, {3691, 17135}, {3701, 3930}, {3702, 17451}, {3721, 4037}, {3741, 5283}, {3760, 20335}, {3767, 3771}, {3879, 17499}, {3954, 3971}, {3992, 4006}, {4067, 4115}, {4071, 21073}, {4082, 20683}, {4099, 4424}, {4101, 19582}, {4125, 21067}, {4135, 7230}, {4153, 21090}, {4441, 17050}, {6376, 17233}, {8804, 10381}, {10445, 15488}, {16968, 17733}, {17034, 17353}, {17240, 20943}, {17355, 17750}

X(21071) = barycentric product X(10)*X(10453)

X(21072) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b+c)*(a^5-(b^2+c^2)*a^3+(b^3+c^3)*a^2-(b^2-c^2)*(b^3-c^3)) : :

X(21072) lies on these lines: {10, 228}, {226, 4605}, {306, 21069}, {429, 2901}, {440, 594}, {516, 11550}, {857, 3969}, {1726, 21270}, {2321, 21062}, {3971, 21098}, {4153, 4177}, {16577, 20305}

X(21073) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 + a^2 b - a b^2 + b^3 + a^2 c - b^2 c - a c^2 - b c^2 + c^3) : :

X(21073) lies on these lines: {1, 5286}, {4, 17742}, {5, 3693}, {8, 5179}, {9, 1479}, {10, 1018}, {12, 3991}, {37, 442}, {63, 17732}, {72, 17747}, {75, 17671}, {142, 7264}, {169, 3434}, {200, 4207}, {220, 3419}, {226, 3970}, {306, 1230}, {312, 7377}, {321, 857}, {346, 3091}, {355, 4513}, {516, 1759}, {527, 4056}, {594, 3697}, {644, 5086}, {672, 10916}, {728, 5587}, {950, 16788}, {1089, 1826}, {1146, 10914}, {1210, 16549}, {1229, 12610}, {1714, 16970}, {1737, 3501}, {1766, 6836}, {1770, 3509}, {1802, 1855}, {2329, 10572}, {2345, 5084}, {2886, 16601}, {3178, 3947}, {3208, 10039}, {3290, 14019}, {3583, 17744}, {3585, 5525}, {3686, 4894}, {3692, 5816}, {3730, 6734}, {3744, 5305}, {3753, 21049}, {3760, 3912}, {3914, 16600}, {3930, 21077}, {4043, 4150}, {4066, 21063}, {4071, 21071}, {4292, 17736}, {4515, 17757}, {4847, 16552}, {5082, 6554}, {7283, 7379}, {8804, 21061}, {9310, 17647}, {9605, 17721}, {16842, 17303}, {17281, 17556}

X(21074) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + b^4 - 2 a^2 b c + 2 a b^2 c + 2 a b c^2 - 2 b^2 c^2 + c^4) : :

X(21074) lies on these lines: {9, 10039}, {10, 1400}, {37, 17757}, {72, 594}, {200, 7102}, {281, 17742}, {306, 21069}, {346, 5179}, {355, 3713}, {393, 2324}, {573, 6735}, {956, 17303}, {1018, 8804}, {1089, 1826}, {1766, 3436}, {2171, 21077}, {2269, 10915}, {2345, 3421}, {4033, 4150}, {4058, 21060}, {4071, 20498}, {4072, 21090}, {4404, 21099}, {5252, 5783}, {5296, 5828}, {5730, 17299}, {6736, 10445}, {12610, 20895}

X(21075) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 - a^2 b + a b^2 + b^3 - a^2 c + 2 a b c - b^2 c + a c^2 - b c^2 + c^3) : :

X(21075) lies on these lines: {1, 2551}, {2, 3333}, {3, 6745}, {4, 200}, {5, 4847}, {8, 908}, {9, 3085}, {10, 12}, {40, 329}, {42, 3191}, {43, 13161}, {46, 527}, {55, 12572}, {56, 6700}, {63, 5552}, {78, 515}, {100, 16113}, {119, 14740}, {142, 1698}, {219, 20263}, {225, 2318}, {306, 857}, {341, 4417}, {354, 9843}, {355, 3940}, {388, 936}, {405, 13405}, {406, 7079}, {443, 5290}, {474, 4298}, {480, 516}, {495, 5044}, {497, 6765}, {498, 5745}, {517, 6736}, {518, 1210}, {519, 1837}, {529, 4311}, {612, 5717}, {912, 12059}, {938, 8165}, {942, 3820}, {950, 3811}, {956, 1125}, {958, 13411}, {960, 12607}, {984, 5530}, {997, 10106}, {1056, 8583}, {1089, 1826}, {1103, 2324}, {1259, 6796}, {1260, 11500}, {1376, 4292}, {1479, 5853}, {1519, 12245}, {1699, 4882}, {1706, 4295}, {1737, 5904}, {1738, 6048}, {1757, 20258}, {1782, 6211}, {1834, 4849}, {1848, 3974}, {2057, 6256}, {2096, 10270}, {2478, 3870}, {2550, 9612}, {2886, 4662}, {2975, 10165}, {3086, 6762}, {3090, 5231}, {3158, 4294}, {3178, 4078}, {3189, 3586}, {3214, 3914}, {3293, 3755}, {3303, 4679}, {3338, 6692}, {3361, 17567}, {3419, 6743}, {3434, 18483}, {3485, 9623}, {3555, 4187}, {3584, 5325}, {3610, 21076}, {3617, 5828}, {3625, 11813}, {3634, 4860}, {3679, 12047}, {3681, 6734}, {3682, 4551}, {3687, 4385}, {3689, 6284}, {3694, 8804}, {3695, 4082}, {3699, 7270}, {3704, 3967}, {3710, 3952}, {3711, 10895}, {3751, 21246}, {3812, 9711}, {3813, 5087}, {3814, 10916}, {3838, 9710}, {3869, 6735}, {3872, 13464}, {3878, 10915}, {3912, 17671}, {3913, 10624}, {3916, 10164}, {3930, 21096}, {3931, 4656}, {3932, 18589}, {3935, 5046}, {3965, 10445}, {3976, 5121}, {3992, 4035}, {4061, 5295}, {4101, 17751}, {4103, 4153}, {4125, 21081}, {4253, 8568}, {4293, 5438}, {4297, 5440}, {4301, 10914}, {4314, 11113}, {4315, 17614}, {4413, 10404}, {4415, 4646}, {4420, 5080}, {4511, 5882}, {4515, 17747}, {4668, 18393}, {4696, 5741}, {4853, 5603}, {4863, 10896}, {4866, 6856}, {4915, 11522}, {5045, 17527}, {5129, 10578}, {5175, 18492}, {5219, 19843}, {5234, 6857}, {5249, 9780}, {5250, 10528}, {5257, 21061}, {5274, 6764}, {5302, 6690}, {5328, 14986}, {5439, 5542}, {5534, 6827}, {5657, 12526}, {5658, 12565}, {5692, 5837}, {5697, 12640}, {5705, 10588}, {5709, 20588}, {5748, 8227}, {5794, 11236}, {5811, 12705}, {6223, 10860}, {6244, 6259}, {6245, 14872}, {6282, 12667}, {6831, 18908}, {6851, 18528}, {7078, 20264}, {7081, 7379}, {7682, 17658}, {8727, 9947}, {9708, 11374}, {9948, 11678}, {10572, 12437}, {10582, 17559}, {11520, 17706}, {11523, 18391}, {12616, 17615}, {12617, 15064}, {13227, 18239}, {15481, 18253}, {15650, 18249}

X(21075) = barycentric product X(10)*X(329)

X(21076) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 - a^3 b + a b^3 + b^4 - a^3 c - a^2 b c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 + c^4) : :

X(21076) lies on these lines: {6, 10}, {37, 3178}, {92, 264}, {594, 1215}, {1330, 1761}, {1901, 3704}, {2160, 4987}, {2294, 3936}, {2321, 4053}, {2345, 20653}, {2385, 8804}, {3610, 21075}, {4062, 17314}, {4261, 17748}, {7119, 7270}, {17751, 21011}, {20496, 21091}, {21065, 21070}

X(21076) = barycentric product X(10)*X(1330)

X(21077) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 - a^2 b + a b^2 + b^3 - a^2 c - b^2 c + a c^2 - b c^2 + c^3) : :
Barycentrics b + c - (a + b + c) (cos B + cos C) : :

X(21077) lies on these lines: {1, 908}, {2, 3338}, {4, 2900}, {5, 518}, {8, 6871}, {9, 10198}, {10, 12}, {11, 3555}, {35, 16154}, {40, 10786}, {46, 5552}, {63, 498}, {78, 1478}, {100, 1770}, {101, 7119}, {142, 3634}, {191, 3584}, {200, 9612}, {214, 4311}, {225, 3191}, {306, 1089}, {329, 3085}, {354, 4187}, {355, 381}, {386, 13161}, {388, 997}, {392, 15888}, {405, 17718}, {430, 1867}, {474, 10404}, {495, 960}, {496, 5087}, {515, 10526}, {516, 5812}, {517, 10915}, {527, 6684}, {529, 1385}, {535, 4297}, {551, 20323}, {726, 12610}, {912, 12616}, {920, 18232}, {936, 5290}, {942, 1329}, {950, 10953}, {956, 11375}, {958, 999}, {993, 12527}, {1004, 1259}, {1079, 6505}, {1210, 3814}, {1330, 7081}, {1479, 3870}, {1519, 7982}, {1537, 2802}, {1698, 5249}, {1699, 6765}, {1724, 3011}, {1737, 3868}, {1836, 5687}, {1838, 3190}, {1901, 3694}, {2171, 21074}, {2321, 4053}, {2475, 4420}, {2476, 3681}, {2548, 16973}, {2550, 5714}, {2551, 3487}, {2784, 12183}, {2796, 12349}, {2801, 6245}, {3057, 10955}, {3086, 5748}, {3120, 3214}, {3178, 3971}, {3244, 5048}, {3293, 3914}, {3333, 10200}, {3419, 10895}, {3421, 3485}, {3475, 5084}, {3579, 17768}, {3632, 18393}, {3695, 3967}, {3701, 3936}, {3740, 8728}, {3742, 17527}, {3743, 4656}, {3751, 5292}, {3812, 3820}, {3813, 9955}, {3816, 5045}, {3817, 18908}, {3824, 3826}, {3825, 3881}, {3838, 4662}, {3869, 10039}, {3871, 5057}, {3873, 4193}, {3878, 10954}, {3879, 21277}, {3880, 16616}, {3901, 18395}, {3913, 12699}, {3916, 5432}, {3930, 21073}, {3931, 4415}, {3932, 16580}, {3940, 5794}, {3950, 21068}, {3991, 17747}, {4013, 15232}, {4035, 4125}, {4054, 4647}, {4075, 4078}, {4109, 21101}, {4295, 7080}, {4298, 6700}, {4299, 4855}, {4325, 15015}, {4363, 5955}, {4385, 4417}, {4430, 5154}, {4511, 20060}, {4658, 17182}, {4661, 5141}, {4880, 5445}, {4968, 5741}, {5080, 10572}, {5119, 10528}, {5178, 17577}, {5223, 5705}, {5226, 5815}, {5227, 5747}, {5247, 17719}, {5248, 12572}, {5250, 10056}, {5252, 5730}, {5288, 5443}, {5302, 6675}, {5316, 17590}, {5328, 11037}, {5434, 17614}, {5440, 7354}, {5542, 9843}, {5587, 10599}, {5777, 7680}, {5810, 5847}, {5852, 11231}, {5853, 18483}, {5854, 11278}, {5880, 9709}, {5883, 8582}, {5886, 12001}, {5901, 11260}, {5903, 6735}, {5904, 6734}, {6048, 17889}, {6068, 11662}, {6541, 21095}, {6762, 8227}, {6764, 9779}, {6831, 14872}, {6834, 12704}, {6890, 10085}, {6922, 12675}, {8165, 11036}, {8258, 20258}, {10106, 18962}, {10395, 14054}, {10477, 19754}, {10524, 10826}, {10742, 12437}, {11522, 12629}, {11682, 12647}, {11684, 14526}, {12059, 18389}, {12436, 20103}, {12559, 18391}, {12625, 18492}, {12688, 13257}, {12934, 17766}, {13205, 16128}, {16478, 17725}, {21090, 21096}

X(21077) = barycentric product X(10)*X(5905)

X(21078) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (a^2 b - b^3 + a^2 c - a b c - c^3) : :

X(21078) lies on these lines: {1, 6}, {8, 5816}, {10, 2171}, {78, 1766}, {101, 2327}, {190, 332}, {198, 1759}, {200, 1824}, {226, 18698}, {306, 21062}, {321, 908}, {517, 3965}, {519, 17452}, {572, 4511}, {573, 3869}, {594, 17757}, {758, 1400}, {857, 21069}, {997, 2285}, {1018, 3694}, {1089, 1826}, {1744, 3509}, {1953, 3686}, {1959, 4416}, {2092, 4424}, {2269, 3878}, {2277, 3670}, {2294, 5257}, {2345, 5747}, {3169, 5697}, {3421, 17314}, {3713, 3940}, {3727, 4263}, {3930, 3950}, {3953, 17053}, {4043, 20927}, {4149, 12329}, {4557, 20713}, {4643, 18726}, {5179, 21070}, {5295, 5587}, {5822, 5839}, {5831, 11375}, {7146, 17272}, {16566, 20769}, {17233, 18747}, {17256, 18714}, {17330, 17443}, {17346, 18041}, {17362, 17444}, {20498, 21101}

X(21079) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-3 a^8 + 2 a^4 b^4 + b^8 + 2 a^4 c^4 - 2 b^4 c^4 + c^8) : :

X(21079) lies on these lines: {10, 16277}, {33, 42}, {306, 1763}, {5596, 16544}

X(21080) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(21080) lies on these lines: {2, 17157}, {10, 75}, {37, 714}, {38, 20891}, {42, 192}, {71, 4039}, {72, 740}, {190, 1918}, {194, 1740}, {306, 3797}, {321, 3728}, {522, 4097}, {536, 4685}, {698, 17792}, {730, 3688}, {872, 4090}, {982, 20923}, {1278, 4651}, {1826, 21089}, {2273, 4112}, {2667, 3159}, {3009, 17148}, {3186, 7075}, {3778, 3948}, {3840, 4022}, {3950, 21100}, {4028, 8804}, {4043, 4135}, {4358, 21330}, {4446, 20340}, {4699, 17155}, {4718, 4946}, {4735, 21238}, {4788, 19998}, {6374, 17149}, {8680, 20721}, {17142, 21352}

X(21080) = complement of X(17157)
X(21080) = barycentric product X(10)*X(194)

X(21081) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (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(21081) lies on these lines: {1, 2}, {37, 6537}, {72, 7068}, {191, 2895}, {261, 319}, {442, 4046}, {502, 1089}, {542, 3579}, {662, 2126}, {740, 3454}, {758, 3704}, {993, 10371}, {1211, 3743}, {2321, 4053}, {3416, 8715}, {3647, 3712}, {3678, 3695}, {3696, 3841}, {3701, 21087}, {3702, 11813}, {3710, 4134}, {3822, 5295}, {3932, 4015}, {3936, 4647}, {3952, 7206}, {4006, 20495}, {4035, 12609}, {4065, 4425}, {4067, 4101}, {4075, 6541}, {4109, 21070}, {4125, 21075}, {4717, 12047}, {4851, 5955}, {5248, 5814}, {17299, 20654}

X(21081) = barycentric product X(10)*X(2895)

X(21082) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^5 - a^3 b^2 + a^2 b^3 + b^5 + a^2 b^2 c - a^3 c^2 + a^2 b c^2 - a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + c^5) : :

X(21082) lies on these lines: {10, 18098}, {37, 744}, {251, 17766}, {321, 17873}, {2321, 4177}, {3971, 4153}, {4388, 17744}, {16555, 21289}

X(21083) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + a^2 b^2 + b^4 + a^2 c^2 + b^2 c^2 + c^4) : :

X(21083) lies on these lines: {10, 82}, {37, 744}, {72, 3773}, {306, 3797}, {313, 21089}, {1031, 4388}, {2175, 9857}, {2896, 16556}, {3159, 6541}, {21022, 21094}

X(21084) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (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) : :

X(21084) lies on these lines: {10, 307}, {72, 740}, {726, 5223}, {1742, 3177}, {3178, 4078}, {3930, 3950}, {4712, 20895}, {4847, 20236}, {18785, 21061}

X(21085) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 + a b + b^2 + a c + b c + c^2) : :

X(21085) lies on these lines: {1, 2}, {37, 9281}, {69, 3980}, {100, 8935}, {171, 319}, {210, 3773}, {226, 7235}, {238, 4886}, {313, 4087}, {321, 1109}, {333, 8298}, {502, 6538}, {524, 4697}, {594, 1215}, {740, 1211}, {756, 3969}, {846, 1654}, {894, 20536}, {896, 3578}, {1213, 4771}, {1376, 4445}, {1914, 3686}, {2321, 3971}, {2784, 4220}, {2796, 4683}, {2887, 3696}, {2895, 4418}, {3120, 17163}, {3175, 4527}, {3666, 3775}, {3678, 3690}, {3706, 3846}, {3740, 17229}, {3745, 17772}, {3791, 17362}, {3914, 4709}, {3923, 5739}, {3925, 4732}, {3936, 21020}, {3952, 6535}, {3967, 21089}, {4011, 14555}, {4042, 4438}, {4058, 21060}, {4096, 4535}, {4133, 4656}, {4199, 4433}, {4357, 4970}, {4399, 17061}, {4434, 4478}, {4640, 4690}, {4649, 19808}, {4682, 17372}, {4703, 5695}, {4716, 19786}, {4733, 17056}, {4734, 17238}, {4914, 17765}, {5224, 17592}, {7262, 17346}, {17270, 17594}

X(21085) = barycentric product X(10)*X(1654)

X(21086) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 b^2 + a^2 b^3 + 2 a^3 b c - a^2 b^2 c - 2 a b^3 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3) : :

X(21086) lies on these lines: {10, 16606}, {226, 20496}, {306, 3948}, {2321, 4135}, {3971, 20690}, {4071, 20498}

X(21087) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 - a^2 b + a b^2 + b^3 - a^2 c + 5 a b c - 3 b^2 c + a c^2 - 3 b c^2 + c^3) : :

X(21087) lies on these lines: {10, 3120}, {80, 4767}, {121, 537}, {306, 21088}, {519, 13541}, {2321, 4103}, {3701, 21081}, {3936, 3992}, {4152, 12690}, {4723, 11813}, {9457, 14028}

X(21088) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-4 a^3 - 4 a^2 b + 4 a b^2 + 4 b^3 - 4 a^2 c + 5 a b c + 4 a c^2 + 4 c^3) : :

X(21088) lies on these lines: {10, 2650}, {306, 21087}, {391, 10197}, {519, 21251}, {551, 5233}, {4417, 4669}

X(21089) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + a^2 b^2 + b^4 + a^2 c^2 - 3 b^2 c^2 + c^4) : :

X(21089) lies on these lines: {10, 190}, {37, 6543}, {148, 2640}, {313, 21083}, {502, 3159}, {645, 13178}, {1826, 21080}, {2643, 11599}, {3967, 21085}, {4150, 21095}, {6370, 12078}, {6541, 20499}, {21090, 21100}

X(21090) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (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(21090) lies on these lines: {10, 1018}, {37, 115}, {80, 644}, {149, 5540}, {321, 17886}, {502, 6543}, {519, 4919}, {594, 3956}, {758, 17747}, {1146, 2802}, {1826, 3950}, {2250, 8804}, {2321, 4103}, {2795, 20531}, {3509, 5134}, {3693, 3814}, {3754, 21049}, {3943, 21066}, {4072, 21074}, {4109, 21070}, {4120, 21093}, {4153, 21071}, {5046, 17744}, {5080, 5525}, {16561, 17355}, {21077, 21096}, {21089, 21100}, {21092, 21098}

X(21090) = barycentric product X(10)*X(149)

X(21091) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + a^3 b - a b^3 + b^4 + a^3 c - a^2 b c + a b^2 c - b^3 c + a b c^2 - a c^3 - b c^3 + c^4) : :

X(21091) lies on these lines: {10, 4557}, {37, 8287}, {116, 16578}, {150, 16560}, {226, 4605}, {306, 4033}, {321, 20902}, {594, 20692}, {692, 2784}, {908, 18151}, {2321, 18589}, {3912, 17790}, {4466, 4552}, {6370, 12078}, {17243, 21239}, {20496, 21076}, {21028, 21319}

X(21092) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-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(21092) lies on these lines: {10, 98}, {306, 4568}, {1826, 6344}, {3448, 16562}, {3708, 13605}, {4006, 20495}, {5546, 13211}, {21090, 21098}

X(21093) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 + a b + b^2 + a c - 3 b c + c^2) : :

X(21093) lies on these lines: {1, 17777}, {10, 3120}, {11, 537}, {37, 16592}, {100, 2796}, {149, 519}, {190, 17719}, {226, 3971}, {244, 11814}, {306, 4135}, {321, 1109}, {329, 4362}, {527, 4396}, {545, 3035}, {726, 908}, {740, 4819}, {1054, 4440}, {1155, 17767}, {1215, 4026}, {1647, 17154}, {1699, 4929}, {2887, 3967}, {3159, 3178}, {3689, 17764}, {3717, 21241}, {3741, 17794}, {3836, 4009}, {3914, 4090}, {3925, 4096}, {3932, 4892}, {3936, 3994}, {3993, 12080}, {3995, 6758}, {4010, 21100}, {4052, 4685}, {4075, 11263}, {4082, 4138}, {4120, 21090}, {4368, 4656}, {4432, 17724}, {4434, 17768}, {4672, 17602}, {4676, 17725}, {4780, 4946}, {5057, 17766}, {5992, 7081}, {6370, 12078}, {6381, 18066}, {6543, 6627}, {17484, 17763}

X(21093) = barycentric product X(10)*X(4440)

X(21094) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 + b^4 - b^2 c^2 + c^4) : :

X(21094) lies on these lines: {10, 598}, {75, 18745}, {92, 264}, {190, 5641}, {316, 16568}, {447, 4570}, {3912, 18073}, {4033, 21066}, {4039, 4156}, {4043, 21065}, {4079, 4129}, {16581, 20912}, {17280, 20654}, {17735, 19732}, {17861, 18744}, {21022, 21083}

X(21094) = barycentric product X(10)*X(316)

X(21095) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 - a^2 c^2 + 3 b^2 c^2) : :

X(21095) lies on these lines: {10, 75}, {714, 21257}, {2321, 4135}, {3963, 3971}, {4111, 4709}, {4150, 21089}, {6541, 21077}, {16571, 20081}, {17157, 20340}

X(21096) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 + 3 a^2 b - 3 a b^2 + b^3 + 3 a^2 c + 2 a b c - b^2 c - 3 a c^2 - b c^2 + c^3) : :

X(21096) lies on these lines: {10, 37}, {101, 12437}, {169, 5853}, {201, 4099}, {220, 519}, {226, 3970}, {346, 938}, {527, 17732}, {536, 21258}, {728, 18391}, {950, 17742}, {1018, 4848}, {1210, 3693}, {2324, 6554}, {2325, 8557}, {3208, 11362}, {3673, 3912}, {3913, 8074}, {3930, 21075}, {4035, 4153}, {4847, 16601}, {5525, 10572}, {6706, 17243}, {6744, 17355}, {21077, 21090}

X(21097) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3592), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2*((3*a^2-3*b^2-3*c^2)*S-(a^2+b^2-c^2)*(a^2-b^2+c^2)) : :

X(21097) lies on these lines: {2, 3}, {39, 20197}, {51, 6398}, {154, 6200}, {184, 6221}, {371, 17809}, {494, 8400}, {1160, 5406}, {1161, 5408}, {1495, 6451}, {3311, 10133}, {3312, 15004}, {6199, 11402}, {6395, 9777}, {6396, 17810}, {6449, 10132}, {8903, 8939}

X(21098) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^5 + a^3 b^2 - a^2 b^3 + b^5 + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + c^5) : :

X(21098) lies on these lines: {10, 21}, {37, 6627}, {210, 3773}, {2796, 9140}, {3701, 21081}, {3936, 20499}, {3971, 21072}, {6370, 12078}, {6742, 14844}, {8287, 16598}, {17719, 21043}, {21090, 21092}

X(21099) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^3 + a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21099) lies on these lines: {10, 798}, {313, 20953}, {594, 4132}, {661, 4071}, {804, 3709}, {3661, 17217}, {3733, 17303}, {4079, 4129}, {4404, 21074}, {5750, 20981}, {21260, 21348}

X(21100) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + 3 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 + b^2 c^2) : :

X(21100) lies on these lines: {10, 3122}, {37, 1084}, {291, 646}, {714, 3943}, {726, 4684}, {3950, 21080}, {3971, 4029}, {4010, 21093}, {4110, 17065}, {9263, 9359}, {18150, 19945}, {21089, 21090}

X(21101) = (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b - a b^2 + a^2 c + a b c + b^2 c - a c^2 + b c^2) : :

X(21101) lies on these lines: {9, 3757}, {10, 762}, {12, 4136}, {37, 714}, {65, 4095}, {75, 20335}, {142, 3263}, {172, 8669}, {226, 306}, {594, 2887}, {672, 17165}, {726, 2276}, {756, 3778}, {1089, 3970}, {1909, 4876}, {2238, 4090}, {2886, 4119}, {3509, 7081}, {3681, 3686}, {3726, 3840}, {3950, 4037}, {3994, 4029}, {4006, 4647}, {4058, 4138}, {4066, 21070}, {4109, 21077}, {4167, 12607}, {4685, 20693}, {4696, 17451}, {4771, 4849}, {4797, 17351}, {4865, 17299}, {6382, 17786}, {9055, 21264}, {16604, 20467}, {17155, 17756}, {17754, 19587}, {19584, 20917}, {20498, 21078}

X(21102) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 - b^4 + a^2 c^2 + 2 b^2 c^2 - c^4) : :

X(21102) lies on these lines: {1, 21179}, {65, 513}, {242, 514}, {523, 1769}, {656, 10015}, {2081, 2600}, {2517, 3810}, {2618, 6369}, {3904, 8062}, {4064, 4391}, {4802, 6129}, {4988, 6589}, {14429, 20294}, {14874, 21201}, {16892, 21110}, {17496, 21187}, {20507, 21114}, {21123, 21131}, {21173, 21180}

X(21103) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2 a^4 - 3 a^2 b^2 + b^4 - 3 a^2 c^2 - 2 b^2 c^2 + c^4) : :

X(21103) lies on these lines: {242, 514}, {523, 2650}, {1769, 4977}, {2457, 3960}, {2605, 21132}, {3904, 4064}, {14429, 20293}, {21113, 21135}

X(21104) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a b - b^2 + a c + 2 b c - c^2) : :

X(21104) lies on these lines: {81, 6654}, {241, 514}, {354, 2488}, {513, 11934}, {523, 2254}, {525, 4978}, {649, 6084}, {658, 9358}, {661, 6545}, {676, 1459}, {693, 918}, {812, 4897}, {900, 4382}, {1635, 2487}, {1639, 4468}, {2490, 6546}, {2499, 6372}, {2512, 2530}, {3798, 4773}, {3835, 4927}, {3910, 4801}, {4025, 4762}, {4105, 6366}, {4162, 8713}, {4374, 18071}, {4380, 6009}, {4453, 17069}, {4728, 14321}, {6362, 6608}, {14324, 21348}, {17422, 21141}, {20505, 21128}, {21117, 21133}

X(21105) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2 a^3 - 2 a^2 b - a b^2 + b^3 - 2 a^2 c + 4 a b c - b^2 c - a c^2 - b c^2 + c^3) : :

X(21105) lies on these lines: {1, 514}, {145, 522}, {227, 3669}, {513, 3057}, {523, 2650}, {1459, 3924}, {1647, 4124}, {2170, 14825}, {2254, 6366}, {2403, 4778}, {2785, 21222}, {2826, 4895}, {2899, 6332}, {3160, 3676}, {3624, 21198}, {3632, 4543}, {3762, 14432}, {3904, 4088}, {6161, 6550}, {10015, 14413}, {11125, 21112}

X(21106) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a - b - c) (2 a^3 + 2 a^2 b - a b^2 - b^3 + 2 a^2 c + 4 a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21106) lies on these lines: {514, 4581}, {522, 4959}, {523, 2650}, {650, 2457}, {1459, 21118}, {3737, 21132}, {4977, 17420}, {4985, 14432}, {11125, 21111}

X(21107) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b^2 - c^2) (-a^2 + b^2 + c^2) (a^2 + b^2 - 2 b c + c^2) : :

X(21107) lies on these lines: {514, 6591}, {523, 21117}, {525, 14208}, {647, 1214}, {652, 2504}, {661, 6587}, {676, 1459}, {2501, 4077}, {3700, 4415}, {4468, 10015}, {16892, 17420}, {17069, 17595}

X(21108) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (b^2 + c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) : :

X(21108) lies on these lines: {242, 514}, {1828, 18344}

X(21109) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 b^2 - b^6 + a^4 c^2 - 2 a^2 b^2 c^2 + b^4 c^2 + b^2 c^4 - c^6) : :

X(21109) lies on these lines: {242, 514}, {1111, 3120}, {2832, 21132}

X(21110) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (b^4 + c^4) : :

X(21110) lies on these lines: {514, 1919}, {1459, 21135}, {3261, 21131}, {3801, 21114}, {16892, 21102}

X(21111) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 - b^4 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - c^4) : :

X(21111) lies on these lines: {514, 2605}, {523, 1769}, {4977, 21132}, {4979, 17422}, {4985, 6370}, {11125, 21106}

X(21112) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 - b^4 - a b^2 c + a^2 c^2 - a b c^2 + 2 b^2 c^2 - c^4) : :

X(21112) lies on these lines: {244, 21142}, {514, 21180}, {523, 1769}, {900, 21132}, {3762, 6370}, {11125, 21105}, {20507, 21133}

X(21113) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 + a^2 c^2 + 2 b^2 c^2) : :

X(21113) lies on these lines: {514, 1921}, {523, 3728}, {786, 3766}, {2483, 4508}, {3250, 4408}, {3837, 4824}, {4107, 10566}, {4374, 21143}, {14407, 21225}, {16892, 21102}, {21103, 21135}, {21126, 21131}, {21194, 21205}

X(21114) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a b^3 - b^4 + b^3 c + a c^3 + b c^3 - c^4) : :

X(21114) lies on these lines: {514, 3063}, {523, 20504}, {3287, 21202}, {3801, 21110}, {16892, 21117}, {20507, 21102}

X(21115) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a b - 2 b^2 + a c + 2 b c - 2 c^2) : :

X(21115) lies on these lines: {244, 4124}, {514, 1635}, {523, 2254}, {661, 3776}, {693, 4931}, {918, 4120}, {1638, 6546}, {1639, 14475}, {3762, 4049}, {4382, 4926}, {4458, 4778}, {4750, 4773}, {4809, 4977}, {4928, 6548}, {4958, 21297}, {4984, 6009}, {10015, 21129}, {20509, 21141}

X(21116) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2 a b - b^2 + 2 a c + 4 b c - c^2) : :

X(21116) lies on these lines: {2, 514}, {523, 2254}, {661, 4927}, {693, 4120}, {918, 4931}, {3667, 4382}, {3776, 4988}, {4750, 4762}

X(21117) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics ((b^2-b*c+c^2)*a^2-(b^3-c^3)*(b-c))*(b^2-c^2) : :

X(21117) lies on these lines: {514, 21184}, {523, 21107}, {661, 21141}, {2501, 17094}, {2525, 4086}, {2799, 14208}, {4077, 12077}, {16892, 21114}, {21104, 21133}

X(21118) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : :

X(21118) lies on these lines: {1, 514}, {522, 17950}, {523, 1769}, {693, 3810}, {784, 21124}, {1459, 21106}, {2254, 6362}, {3737, 21179}, {3777, 6545}, {3801, 16892}, {3910, 4804}, {4041, 10015}, {4088, 4391}, {4142, 4560}, {4458, 17496}, {4707, 8714}, {5075, 17494}, {7649, 17418}

X(21119) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a - b - c) (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : :

X(21119) lies on these lines: {8, 522}, {514, 4581}, {523, 1769}, {650, 4802}, {1459, 3924}, {2804, 6615}, {3810, 4397}, {4017, 10015}, {4036, 15523}, {4147, 20294}, {4449, 7649}, {10016, 20999}, {16892, 21114}, {20508, 20510}

X(21120) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a - b - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21120) lies on these lines: {241, 514}, {333, 4560}, {522, 4546}, {523, 1769}, {525, 3762}, {652, 4498}, {654, 4063}, {657, 6084}, {663, 6366}, {676, 4449}, {918, 4462}, {1211, 1577}, {1639, 6332}, {1734, 2826}, {3700, 3910}, {3801, 20504}, {3810, 4147}, {4041, 6362}, {4504, 13246}, {4534, 21138}, {4977, 17418}, {6615, 14284}, {8712, 14298}, {14077, 21185}, {17069, 17496}

X(21121) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b^2 - c^2) (a b^2 + b^3 + a c^2 + c^3) : :

X(21121) lies on these lines: {513, 4707}, {514, 3733}, {523, 656}, {826, 4036}, {1635, 2527}, {4057, 4142}, {4088, 21125}, {7336, 16732}, {16892, 21102}, {20505, 21133}

X(21122) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics a^4 (b - c) (a^4 - b^4 - c^4) : :

X(21122) lies on these lines: {31, 652}, {58, 4025}, {514, 21190}

X(21123) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics a^2 (b - c) (b^2 + c^2) : :

X(21123) lies on these lines: {37, 513}, {75, 18080}, {86, 10566}, {514, 1921}, {522, 17458}, {649, 834}, {661, 1639}, {663, 9313}, {665, 798}, {688, 3005}, {786, 4374}, {900, 21834}, {2084, 2530}, {2483, 20981}, {2605, 8632}, {3063, 20228}, {3709, 3768}, {3805, 21349}, {4057, 5029}, {4486, 18160}, {6084, 21127}, {6586, 9002}, {8714, 21836}, {14825, 17192}, {16892, 21126}, {20892, 20906}, {21102, 21131}, {21173, 21389}

X(21124) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b^2 - c^2) (a b + b^2 + a c + c^2) : :

X(21124) lies on these lines: {2, 8045}, {239, 514}, {522, 21301}, {523, 656}, {525, 661}, {690, 4983}, {784, 21118}, {824, 4391}, {826, 4088}, {850, 1577}, {905, 21828}, {3004, 3910}, {3125, 6547}, {3566, 4822}, {3776, 4801}, {3800, 4729}, {3810, 4818}, {4049, 6539}, {4120, 4129}, {4122, 21051}, {4379, 21188}, {4458, 17166}, {4467, 6002}, {4730, 7927}, {4770, 4808}, {4978, 6545}, {6590, 14837}, {7216, 17094}, {20504, 21134}, {21125, 21727}, {21129, 21141}

X(21125) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b^4 - c^4) (a^2 + b^2 - b c + c^2) : :

X(21125) lies on these lines: {38, 16892}, {514, 18108}, {661, 3801}, {4088, 21121}, {4977, 21126}, {21124, 21727}

X(21126) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (b^2 + c^2) (2 a^2 + b^2 + c^2) : :

X(21126) lies on these lines: {514, 1919}, {918, 4079}, {4977, 21125}, {16892, 21123}, {21113, 21131}

X(21127) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics a (b - c) (a - b - c) (a b - b^2 + a c + 2 b c - c^2) : :

X(21127) lies on these lines: {44, 513}, {284, 1024}, {514, 7216}, {522, 4171}, {523, 2294}, {663, 6182}, {665, 4017}, {926, 4041}, {1459, 17412}, {1769, 6586}, {2170, 3328}, {2488, 6607}, {2820, 4040}, {3239, 4811}, {3667, 14282}, {3709, 6615}, {4148, 20293}, {4501, 4895}, {4778, 14330}, {4976, 8611}, {6084, 21123}, {6362, 14283}, {14413, 17425}

X(21128) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a b - a c + b c) (-a b^2 + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(21128) lies on these lines: {325, 523}, {514, 21197}, {3776, 20512}, {20505, 21104}, {20508, 20510}

X(21128) = isotomic conjugate of X(35572)

X(21129) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2 a - b - c) (a b + b^2 + a c - 4 b c + c^2) : :

X(21129) lies on these lines: {2, 514}, {8, 3667}, {63, 4498}, {321, 4462}, {523, 2292}, {900, 4543}, {918, 14442}, {1145, 2826}, {1281, 2789}, {3578, 6002}, {3669, 16602}, {3762, 4120}, {4370, 6084}, {4778, 11530}, {4927, 16594}, {10015, 21115}, {16892, 21130}, {20880, 20906}, {21124, 21141}

X(21130) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a - 2 b - 2 c) (a b + b^2 + a c - b c + c^2) : :

X(21130) lies on these lines: {514, 1635}, {523, 10015}, {693, 4049}, {3679, 4777}, {3762, 20906}, {4791, 4931}, {5902, 9001}, {16892, 21129}

X(21131) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^3 (b + c)^2 : :

X(21131) lies on these lines: {86, 514}, {523, 1213}, {594, 4024}, {649, 2160}, {661, 2294}, {918, 20509}, {1474, 1919}, {1577, 18697}, {1648, 8029}, {2171, 4079}, {2533, 21922}, {2642, 6089}, {3125, 14442}, {3261, 21110}, {4107, 21205}, {4988, 6544}, {6545, 21133}, {6627, 12078}, {21102, 21123}, {21113, 21126}, {21132, 21143}, {21832, 22108}

X(21132) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^3 (a - b - c) : :

X(21132) lies on these lines: {1, 514}, {8, 522}, {34, 7649}, {65, 513}, {279, 3676}, {523, 2292}, {650, 1212}, {659, 8648}, {676, 14413}, {764, 1647}, {900, 21112}, {1024, 2082}, {1577, 3454}, {1698, 21198}, {1828, 18344}, {2254, 2826}, {2605, 21103}, {2832, 21109}, {3667, 5691}, {3701, 3810}, {3716, 3904}, {3737, 21106}, {3762, 4088}, {3801, 20515}, {3893, 3900}, {3924, 8578}, {3954, 4024}, {4041, 6362}, {4142, 17496}, {4163, 6556}, {4458, 21222}, {4508, 4750}, {4530, 14393}, {4778, 14812}, {4895, 6366}, {4977, 21111}, {7004, 15914}, {10006, 14475}, {14825, 17451}, {18328, 18343}, {20954, 20955}, {21131, 21143}, {21134, 21141}

X(21133) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^3 (a b - b^2 + a c - b c - c^2) : :

X(21133) lies on these lines: {6, 514}, {513, 12723}, {522, 17276}, {523, 20504}, {1880, 3669}, {2424, 7649}, {3239, 17267}, {4025, 4361}, {6545, 21131}, {20505, 21121}, {20507, 21112}, {21104, 21117}

X(21134) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^3 (b + c)^2 (-a^2 + b^2 + c^2) : :

X(21134) lies on these lines: {58, 514}, {523, 1834}, {810, 2658}, {1459, 7100}, {3695, 4064}, {4025, 17206}, {4079, 21799}, {7649, 8747}, {20504, 21124}, {21132, 21141}

X(21135) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2 a^4 - 2 a^2 b^2 + b^4 - 2 a^2 c^2 + c^4) : :

X(21135) lies on these lines: {86, 514}, {1459, 21110}, {4024, 17390}, {11125, 21136}, {21103, 21113}

X(21136) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 - 2 b^4 + a^2 c^2 + 2 b^2 c^2 - 2 c^4) : :

X(21136) lies on these lines: {514, 21205}, {3122, 20512}, {11125, 21135}, {16892, 21102}

X(21137) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (b + c) (a^4 - a^2 b^2 - a^2 c^2 + 2 b^2 c^2) : :

X(21137) lies on these lines:

X(21138) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a b + a c - b c) : :

X(21138) lies on these lines:

X(21138) = isotomic conjugate of X(5383)
X(21138) = complement of X(33946)

X(21139) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a^2 - a b - a c + 2 b c) : :

X(21139) lies on these lines:

X(21140) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a b^2 - b^2 c + a c^2 - b c^2) : :

X(21140) lies on these lines:

X(21141) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^3 (b + c) (-a^2 + b^2 + b c + c^2) : :

X(21141) lies on these lines:

X(21142) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21142) lies on these lines:

X(21143) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics a^2 (b - c)^3 : :

X(21143) lies on these lines:

X(21143) = isotomic conjugate of isogonal conjugate of X(3249)

X(21144) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (b + c) (a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(21144) lies on these lines:

X(21145) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b^2 - c^2) (a^2 - 2 b^2 + 3 b c - 2 c^2) : :

X(21145) lies on these lines:

X(21146) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) : :

X(21146) lies on these lines:

X(21147) = X(75)-CEVA CONJUGATE OF X(57)

Barycentrics a*(a + b - c)*(a - b + c)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(21147) lies on the cubic K619 and these lines: {1, 4}, {3, 227}, {6, 8898}, {7, 17016}, {8, 1943}, {10, 1038}, {12, 975}, {40, 109}, {46, 603}, {56, 998}, {57, 961}, {63, 12089}, {65, 222}, {72, 9370}, {77, 1441}, {78, 4551}, {145, 4318}, {175, 9789}, {221, 517}, {304, 664}, {341, 14594}, {355, 1060}, {478, 1766}, {519, 4347}, {610, 2331}, {612, 9578}, {614, 1420}, {651, 3869}, {912, 19471}, {958, 1214}, {971, 1854}, {982, 9363}, {990, 7354}, {1035, 3428}, {1040, 4297}, {1062, 18481}, {1103, 6282}, {1104, 1617}, {1125, 19372}, {1319, 8283}, {1393, 3338}, {1398, 8192}, {1419, 2263}, {1422, 9623}, {1425, 16980}, {1456, 3057}, {1458, 3924}, {1610, 1763}, {1708, 5247}, {1722, 3911}, {1880, 2286}, {1935, 12514}, {2000, 5086}, {2807, 7355}, {2975, 17080}, {3600, 5262}, {3660, 17054}, {3680, 9372}, {3751, 15556}, {3872, 4968}, {3920, 18624}, {4308, 7191}, {4351, 10573}, {4642, 9316}, {5252, 6357}, {5484, 17086}, {5725, 15844}, {5887, 8757}, {6180, 12709}, {6796, 11700}, {7004, 10085}, {7078, 14110}, {7971, 10703}, {9817, 19925}, {11109, 20220}, {12114, 17102}, {18447, 18525}, {18596, 20613}

X(21147) = X(75)-Ceva conjugate of X(57)
X(21147) = X(197)-cross conjugate of X(1766)
X(21147) = crosspoint of X(i) and X(j) for these (i,j): {75, 20928}, {664, 7128}
X(21147) = X(i)-beth conjugate of X(j) for these (i,j): {1, 1406}, {100, 78}
X(21147) = X(i)-isoconjugate of X(j) for these (i,j): {8, 3435}, {55, 8048}, {2968, 15385}
X(21147) = cevapoint of X(i) and X(j) for these (i,j): {73, 12089}, {197, 478}
X(21147) = barycentric product X(i)*X(j) for these {i,j}: {7, 1766}, {56, 20928}, {57, 3436}, {63, 14257}, {75, 478}, {85, 197}, {123, 7128}, {205, 6063}, {226, 16049}, {304, 17408}, {664, 6588}
X(21147) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 8048}, {197, 9}, {205, 55}, {478, 1}, {604, 3435}, {1766, 8}, {3436, 312}, {6588, 522}, {14257, 92}, {16049, 333}, {17408, 19}, {20928, 3596}
X(21147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 223, 10571), (1, 1745, 6261), (1, 5691, 33), (8, 4296, 8270), (40, 1394, 109), (227, 1455, 3), (944, 1870, 1), (958, 15832, 1214), (1254, 1468, 57)

X(21148) = X(1)-CEVA CONJUGATE OF X(25)

Barycentrics a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^5 + a^4*b - a*b^4 - b^5 + a^4*c - 2*a^2*b^2*c + b^4*c - 2*a^2*b*c^2 + 2*a*b^2*c^2 - a*c^4 + b*c^4 - c^5) : :

X(21148) lies on these lines: {1, 2138}, {6, 1854}, {19, 614}, {42, 3195}, {204, 4319}, {346, 1783}, {604, 608}, {857, 18683}, {1880, 2207}, {4329, 17903}, {8750, 12329}

X(21148) = isogonal conjugate of isotomic conjugate of X(17903)
X(21148) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 25}, {17903, 3556}
X(21148) = crosspoint of X(1) and X(1763)
X(21148) = crosssum of X(1) and X(7097)
X(21148) = X(112)-beth conjugate of X(478)
X(21148) = X(i)-isoconjugate of X(j) for these (i,j): {63, 7219}, {69, 7097}, {304, 7169}
X(21148) = barycentric product X(i)*X(j) for these {i,j}: {4, 3556}, {6, 17903}, {19, 1763}, {25, 4329}, {1039, 8900}, {1973, 20914}
X(21148) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 7219}, {1763, 304}, {1973, 7097}, {1974, 7169}, {3556, 69}, {4329, 305}, {17903, 76}

X(21149) = (name pending)

Barycentrics a^2*(a^8 - 2*a^7*b - 2*a^5*b^3 + 10*a^4*b^4 - 6*a^3*b^5 - 8*a^2*b^6 + 10*a*b^7 - 3*b^8 - 2*a^7*c + 4*a^6*b*c - 6*a^4*b^3*c + 6*a^3*b^4*c - 4*a*b^6*c + 2*b^7*c - 2*a^5*c^3 - 6*a^4*b*c^3 - 16*a^2*b^3*c^3 - 6*a*b^4*c^3 + 30*b^5*c^3 + 10*a^4*c^4 + 6*a^3*b*c^4 - 6*a*b^3*c^4 - 58*b^4*c^4 - 6*a^3*c^5 + 30*b^3*c^5 - 8*a^2*c^6 - 4*a*b*c^6 + 10*a*c^7 + 2*b*c^7 - 3*c^8) : :

X(21149) lies on these lines: (none)

X(21150) = X(55)X(1149)∩X(220)X(2183)

Barycentrics a^2*(a^5 - a^4*b + 4*a^2*b^3 - a*b^4 - 3*b^5 - a^4*c + 4*a^3*b*c - 6*a^2*b^2*c - 4*a*b^3*c + 7*b^4*c - 6*a^2*b*c^2 + 18*a*b^2*c^2 - 12*b^3*c^2 + 4*a^2*c^3 - 4*a*b*c^3 - 12*b^2*c^3 - a*c^4 + 7*b*c^4 - 3*c^5) : :

X(21150) lies on these lines: {55, 1149}, {220, 2183}, {513, 956}, {517, 997}, {2099, 3938}, {3478, 16610}

X(21151) = X(2)X(971)∩X(3)X(7)

Barycentrics a^6 - 4*a^5*b + 3*a^4*b^2 + 4*a^3*b^3 - 5*a^2*b^4 + b^6 - 4*a^5*c - 6*a^4*b*c + 4*a^3*b^2*c + 8*a^2*b^3*c - 2*b^5*c + 3*a^4*c^2 + 4*a^3*b*c^2 - 6*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 - b^2*c^4 - 2*b*c^5 + c^6 : :
X(21151) = 2 X[3] + X[7], X[4] - 4 X[142], 2 X[9] - 5 X[631], X[390] - 4 X[1385], X[944] + 2 X[2550], 2 X[946] + X[2951], X[144] - 7 X[3523], X[3062] - 7 X[3624], X[40] + 2 X[5542], 2 X[142] + X[5732], X[4] + 2 X[5732], 7 X[3528] + 2 X[5735], 4 X[3] - X[5759], 2 X[7] + X[5759], 4 X[140] - X[5779], X[20] + 2 X[5805], 8 X[3826] - 5 X[5818], 4 X[549] - X[6172], X[376] + 2 X[6173], 11 X[3525] - 8 X[6666], X[5223] - 4 X[6684], X[1156] - 4 X[6713], X[3488] + 2 X[6916], X[4312] + 5 X[7987], X[3428] + 2 X[8255], X[5728] - 4 X[9940], X[104] + 2 X[10427], 4 X[1125] - X[11372], X[6361] - 4 X[11495], 2 X[3243] + X[12245], 4 X[1483] - X[12630], X[3059] + 2 X[12675], 2 X[3254] + X[13199], X[12669] - 4 X[13369], 5 X[11025] - 8 X[13373], 2 X[13159] + X[16113], 8 X[140] - 5 X[18230], 2 X[5779] - 5 X[18230], X[3488] - 4 X[18443], X[6916] + 2 X[18443], X[3146] - 4 X[18482], 11 X[15717] + X[20059], 7 X[3090] - 10 X[20195], X[962] - 4 X[20330]

X(21151) lies on these lines: {2, 971}, {3, 7}, {4, 142}, {9, 631}, {20, 5805}, {24, 7717}, {40, 5542}, {84, 16845}, {104, 10427}, {140, 5779}, {144, 3523}, {165, 553}, {226, 10857}, {376, 516}, {390, 1385}, {405, 12246}, {443, 10884}, {517, 11038}, {518, 5657}, {527, 3524}, {549, 5843}, {572, 5819}, {944, 2550}, {946, 2951}, {962, 20330}, {990, 4648}, {991, 4000}, {1001, 6906}, {1006, 2096}, {1056, 4321}, {1058, 4326}, {1125, 9841}, {1156, 6713}, {1445, 6988}, {1483, 12630}, {1490, 17582}, {1788, 18412}, {2346, 11248}, {3059, 12675}, {3062, 3624}, {3085, 8581}, {3086, 14100}, {3090, 20195}, {3146, 18482}, {3243, 12245}, {3254, 13199}, {3332, 4675}, {3358, 6857}, {3428, 8255}, {3474, 15931}, {3485, 4312}, {3488, 6916}, {3525, 6666}, {3528, 5735}, {3826, 5818}, {3911, 10398}, {3928, 5850}, {4208, 5787}, {4295, 8273}, {4512, 14646}, {4644, 13329}, {5085, 5845}, {5129, 6259}, {5218, 15298}, {5223, 6684}, {5698, 6875}, {5714, 6865}, {5728, 6908}, {5731, 11112}, {5745, 5785}, {5784, 6889}, {5809, 6907}, {5832, 6955}, {5853, 7967}, {5880, 6934}, {6223, 11108}, {6244, 10578}, {6260, 17559}, {6361, 11495}, {6825, 10394}, {6880, 8257}, {6926, 8232}, {6948, 13151}, {6989, 12669}, {7288, 15299}, {7580, 9776}, {7676, 10267}, {7677, 10269}, {8129, 8389}, {8130, 8388}, {8226, 10430}, {8236, 10246}, {8728, 9799}, {10785, 17668}, {11025, 13373}, {12680, 19855}, {13159, 16113}, {15587, 19843}, {15717, 20059}

X(21151) = reflection of X(i) in X(j) for these {i,j}: {5817, 2}, {8236, 10246}
X(21151) = anticomplement of X(38108)
X(21151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7, 5759), (140, 5779, 18230), (142, 5732, 4), (6916, 18443, 3488)

X(21152) = X(3)X(6)∩X(804)X(11620)

Barycentrics (SB+SC)*((21*R^2-2*SA-4*SW)*S^ 4+(5*R^2*(SW+3*SA)-2*SA^2+2* SB*SC-2*SW^2)*SW*S^2-R^2*SA* SW^3) : :

See Le Viet An and César Lozada, Hyacinthos 28052.

X(21152) lies on the these lines: {3, 6}, {804, 11620}

X(21152) = 1st-Lemoine circle-inverse of X(2674)
X(21152) = center of circle {{X(5),X(115),PU(1)}}

Gibert-Moses Centroids: X(21153)-X(21168)

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC, not on one of the sidelines BC, CA, AB. Let pK(K, P) denote the pivotal isocubic with pole K and pivot P (see Bernard Gibert's Notations.

The cubic pK(K,P) intersects the circumcircle in A, B, C and three other points, Q1, Q2, Q3. The centroid of {Q1, Q2, Q3} is the point, here named the P-Gibert-Moses centroid, given by the combo G(P) = 2*X(3) + P and

G(P) = (b² + c² - a²⁾(3a⁴ + b⁴ + c⁴ - 4 a² b² - 4 a² c² - 2 b² c²)p - 2 a² (q + r) : :

These centroids and formulas were contributed by Peter Moses, August 20, 2018.

X(21153) = X(9)-GIBERT-MOSES CENTROID

Barycentrics a*(a - b - c)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*(b*(-a + b - c) + c*(-a - b + c)) : :
X(21153) = 2 X[3] + X[9], 2 X[142] - 5 X[631], X[40] + 2 X[1001], 4 X[1385] - X[3243], X[1490] + 2 X[3358], X[7] - 7 X[3523], 4 X[3] - X[5732], 2 X[9] + X[5732], 4 X[142] - X[5735], 10 X[631] - X[5735], 2 X[142] + X[5759], 5 X[631] + X[5759], X[5735] + 2 X[5759], 5 X[9] - 2 X[5779], 5 X[3] + X[5779], 5 X[5732] + 4 X[5779], 4 X[140] - X[5805], 4 X[549] - X[6173], X[104] + 2 X[6594], X[4] - 4 X[6666], X[2550] - 4 X[6684], X[3254] - 4 X[6713], 4 X[6600] - X[6765], X[3587] + 2 X[6883], X[5223] + 5 X[7987], X[6282] + 2 X[8257], X[11372] + 2 X[11495], 5 X[3876] + X[12669], X[11372] - 4 X[15254], X[11495] + 2 X[15254], X[6172] + 5 X[15692], X[144] + 11 X[15717], X[2951] - 7 X[16192], X[20] + 5 X[18230], 5 X[1656] - 2 X[18482], 8 X[140] - 5 X[20195], 2 X[5805] - 5 X[20195], 3 X[3524] - X[21151]

X(21153) lies on these lines: {1, 1170}, {2, 165}, {3, 9}, {4, 6666}, {7, 3523}, {20, 18230}, {35, 4326}, {36, 4321}, {40, 1001}, {46, 12560}, {56, 15837}, {57, 954}, {63, 10857}, {78, 5223}, {104, 6594}, {140, 5805}, {142, 631}, {144, 4652}, {200, 15931}, {376, 5817}, {380, 5132}, {390, 1210}, {411, 2951}, {480, 8273}, {518, 3576}, {527, 3524}, {549, 5762}, {990, 3731}, {991, 1743}, {1006, 6282}, {1385, 3243}, {1621, 7994}, {1656, 18482}, {1698, 6836}, {1708, 10383}, {1750, 3305}, {1754, 17022}, {2550, 5705}, {2801, 15015}, {3085, 12573}, {3149, 11372}, {3174, 10902}, {3218, 11407}, {3254, 6713}, {3361, 5542}, {3586, 6992}, {3587, 6883}, {3601, 5728}, {3612, 18412}, {3683, 10860}, {3826, 6831}, {3876, 12669}, {3928, 11227}, {3929, 10167}, {4292, 8232}, {4297, 5234}, {4304, 5809}, {4423, 7964}, {5204, 8581}, {5217, 14100}, {5259, 12651}, {5302, 10864}, {5584, 16410}, {5657, 5853}, {5686, 5731}, {5698, 6988}, {5715, 6989}, {5766, 8732}, {5833, 6926}, {5843, 12100}, {6172, 15692}, {6600, 6765}, {6734, 9588}, {6895, 7989}, {7308, 7580}, {7675, 10398}, {10856, 19649}, {10861, 13587}, {12652, 15485}, {14793, 15518}

X(21153) = midpoint of X(i) and X(j) for these {i,j}: {376, 5817}, {5686, 5731}
X(21153) = X(9)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K351
X(21153) = crossdifference of every pair of points on line {6129, 21127}
X(21153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9, 5732), (35, 15299, 4326), (36, 15298, 4321), (140, 5805, 20195), (142, 5759, 5735), (631, 5759, 142), (3305, 7411, 1750), (6684, 6865, 5705), (11495, 15254, 11372)

X(21154) = X(11)-GIBERT-MOSES CENTROID

Barycentrics (b - c)^2*(-a + b + c)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*((a - b)^2*(a + b - c) + (-a + c)^2*(a - b + c)) : :
X(21154) = 2 X[3] + X[11], X[119] - 4 X[140], X[104] + 5 X[631], X[1317] - 4 X[1385], X[40] + 2 X[1387], 4 X[1125] - X[1537], 5 X[631] - 2 X[3035], X[104] + 2 X[3035], X[944] + 2 X[3036], X[100] - 7 X[3523], 16 X[3530] - X[6154], 4 X[549] - X[6174], X[4] - 4 X[6667], X[1532] - 4 X[6681], X[1145] - 4 X[6684], X[4297] + 2 X[6702], X[11] - 4 X[6713], X[3] + 2 X[6713], X[80] + 5 X[7987], X[153] - 13 X[10303], 2 X[10265] + X[10609], 5 X[3522] + X[10724], 7 X[3090] - X[10728], 5 X[11] - 2 X[10738], 10 X[6713] - X[10738], 5 X[3] + X[10738], 7 X[3526] - X[10742], 2 X[1484] + X[10993], 4 X[9940] - X[11570], 2 X[6684] + X[11715], X[1145] + 2 X[11715], 2 X[12019] + X[12119], 11 X[3525] + X[12248], 2 X[11729] + X[12515], 4 X[5044] - X[12665], 13 X[10299] - X[13199], X[6326] + 2 X[13226], X[12619] + 2 X[13624], 2 X[12737] + X[13996], 2 X[12736] + X[14110], X[11698] - 7 X[14869], X[10778] + 5 X[15051], X[2077] + 2 X[15325], 2 X[6882] + X[15326], X[72] + 2 X[15528], X[10707] + 5 X[15692], X[10711] - 7 X[15702], X[10993] - 10 X[15712], X[1484] + 5 X[15712], X[149] + 11 X[15717], X[12773] + 11 X[15720], X[442] + 2 X[17009], X[1071] + 2 X[18254], X[153] - 4 X[20400], 13 X[10303] - 4 X[20400], X[100] + 2 X[20418], 7 X[3523] + 2 X[20418]

X(21154) lies on these lines: {2, 2829}, {3, 11}, {4, 6667}, {40, 1387}, {56, 6961}, {72, 15528}, {80, 7987}, {100, 3523}, {104, 631}, {119, 140}, {149, 15717}, {153, 10303}, {165, 16173}, {371, 13977}, {372, 13913}, {392, 2800}, {442, 17009}, {517, 5298}, {528, 3524}, {549, 952}, {944, 3036}, {1001, 12775}, {1006, 18861}, {1071, 18254}, {1125, 1537}, {1145, 6684}, {1317, 1385}, {1484, 10993}, {1532, 6681}, {2077, 15325}, {2478, 12761}, {2646, 12832}, {2771, 11227}, {2802, 10164}, {3090, 10728}, {3333, 10075}, {3522, 10724}, {3525, 12248}, {3526, 10742}, {3530, 6154}, {3816, 6950}, {4297, 6702}, {4861, 18802}, {4995, 10246}, {4996, 6986}, {4999, 6940}, {5010, 5533}, {5044, 12665}, {5085, 5848}, {5204, 6891}, {5432, 10269}, {5450, 13747}, {5587, 17564}, {5657, 5854}, {5842, 13587}, {5851, 21151}, {6326, 8726}, {6691, 6906}, {6827, 13273}, {6842, 7294}, {6850, 12764}, {6882, 15326}, {6921, 12114}, {6922, 7280}, {6958, 7354}, {6978, 12943}, {7288, 10310}, {9540, 19081}, {9615, 19077}, {9940, 11570}, {10265, 10609}, {10299, 13199}, {10707, 15692}, {10711, 15702}, {10778, 15051}, {11698, 14869}, {11729, 12515}, {12019, 12119}, {12619, 13624}, {12736, 14110}, {12737, 13996}, {12739, 18443}, {12773, 15720}, {13935, 19082}, {17566, 18242}

X(21154) = midpoint of X(i) and X(j) for these {i,j}: {165, 16173}, {11219, 15015}
X(21154) = X(11)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 499, 11826), (3, 5433, 15908), (3, 6713, 11), (104, 631, 3035), (549, 3576, 21155), (5204, 6891, 11827), (6684, 11715, 1145)

X(21155) = X(12)-GIBERT-MOSES CENTROID

Barycentrics (a + b - c)*(a - b + c)*(b + c)^2*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*((a + b - c)*(a + c)^2*(-a + b + c) + (a + b)^2*(a - b + c)*(-a + b + c)) : :
X(21155) = 2 X[3] + X[12], X[2975] - 7 X[3523], 5 X[631] - 2 X[4999], X[4] - 4 X[6668], 5 X[631] + X[11491], 2 X[4999] + X[11491], 2 X[6842] + X[15338], 2 X[35] + X[15908], 11 X[15717] + X[20060]

X(21155) lies on these lines: {2, 5842}, {3, 12}, {4, 6668}, {35, 15908}, {55, 6954}, {119, 7508}, {140, 3925}, {517, 4995}, {529, 3524}, {549, 952}, {631, 1376}, {758, 10164}, {1001, 6880}, {1006, 3035}, {1329, 6875}, {2829, 17549}, {2975, 3523}, {3428, 5218}, {3614, 7491}, {3753, 10165}, {4189, 18242}, {4423, 6970}, {5010, 6907}, {5085, 5849}, {5217, 6825}, {5298, 10246}, {5326, 6882}, {5433, 10267}, {5440, 6684}, {5535, 5719}, {5657, 5855}, {5852, 21151}, {6253, 6862}, {6256, 19535}, {6284, 6863}, {6690, 6905}, {6796, 7483}, {6842, 15338}, {6910, 11500}, {6961, 8273}, {6962, 11496}, {6988, 10310}, {10175, 15670}, {15717, 20060}

X(21155) = X(12)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 498, 11827), (549, 3576, 21154), (631, 11491, 4999), (5217, 6825, 11826)

X(21156) = X(13)-GIBERT-MOSES CENTROID

Barycentrics Sqrt[3]*a^2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 - 6*b^2*c^2 - c^4) + 2*(5*a^4 - 7*a^2*b^2 + 2*b^4 - 7*a^2*c^2 - 4*b^2*c^2 + 2*c^4)*S : :
X(21156) = 2 X[3] + X[13], X[98] + 2 X[619], 2 X[618] - 5 X[631], X[616] - 7 X[3523], X[376] + 2 X[5459], 4 X[549] - X[5463], 4 X[3] - X[5473], 2 X[13] + X[5473], 2 X[115] + X[5474], X[20] + 2 X[5478], 4 X[140] - X[5617], X[14] - 4 X[6036], X[5464] + 2 X[6055], X[4] - 4 X[6669], X[1080] - 4 X[6671], 2 X[618] + X[6770], 5 X[631] + X[6770], X[13] - 4 X[6771], X[3] + 2 X[6771], X[5473] + 8 X[6771], 4 X[6774] - X[6777], 4 X[1385] - X[7975], 5 X[7987] + X[9901], X[9114] + 2 X[11632], X[40] + 2 X[11705], X[5613] + 2 X[12042], 2 X[11710] + X[12780], 4 X[6684] - X[12781], 5 X[13] - 2 X[13103], 10 X[6771] - X[13103], 5 X[3] + X[13103], 5 X[5473] + 4 X[13103], X[6779] - 4 X[13349], 2 X[5479] - 5 X[14061], 2 X[396] + X[14538], 2 X[6108] + X[14539], 2 X[14136] + X[14540], 7 X[13103] - 10 X[16001], 7 X[13] - 4 X[16001], 7 X[6771] - X[16001], 7 X[3] + 2 X[16001], 7 X[5473] + 8 X[16001], 2 X[15929] + X[18863], X[550] + 2 X[20252], 5 X[16001] - 14 X[20415], 5 X[13] - 8 X[20415], X[13103] - 4 X[20415], 5 X[6771] - 2 X[20415], 5 X[3] + 4 X[20415], 5 X[5473] + 16 X[20415], 5 X[16960] - 2 X[20425]

X(21156) lies on these lines: {2, 9749}, {3, 13}, {4, 6669}, {14, 6036}, {15, 230}, {16, 9112}, {20, 5478}, {35, 10078}, {36, 10062}, {40, 11705}, {62, 9606}, {98, 619}, {115, 5474}, {140, 5617}, {182, 16242}, {371, 19073}, {372, 19074}, {376, 5459}, {396, 14538}, {511, 16962}, {530, 3524}, {542, 5054}, {549, 5463}, {550, 20252}, {616, 3523}, {617, 5982}, {618, 631}, {1080, 6671}, {1385, 7975}, {1587, 13917}, {1588, 13982}, {2794, 11297}, {3412, 5864}, {3515, 12142}, {5050, 16963}, {5171, 12205}, {5204, 18974}, {5217, 13076}, {5432, 12942}, {5433, 12952}, {5464, 6055}, {5472, 11481}, {5479, 14061}, {5613, 12042}, {5868, 11309}, {6108, 14539}, {6684, 12781}, {6774, 6777}, {6779, 13349}, {7987, 9901}, {8980, 19076}, {9114, 11632}, {9751, 9762}, {10267, 13107}, {10269, 13105}, {11710, 12780}, {13967, 19075}, {14136, 14540}, {14541, 16772}, {16960, 20425}

X(21156) = X(13)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13, 5473), (3, 6771, 13), (631, 6770, 618), (5054, 5085, 21157), (5340, 10611, 13), (13103, 20415, 13)

X(21157) = X(14)-GIBERT-MOSES CENTROID

Barycentrics Sqrt[3]*a^2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 - 6*b^2*c^2 - c^4) - 2*(5*a^4 - 7*a^2*b^2 + 2*b^4 - 7*a^2*c^2 - 4*b^2*c^2 + 2*c^4)*S : :
X(21157) = 2 X[3] + X[14], X[98] + 2 X[618], 2 X[619] - 5 X[631], X[617] - 7 X[3523], X[376] + 2 X[5460], 4 X[549] - X[5464], 2 X[115] + X[5473], 4 X[3] - X[5474], 2 X[14] + X[5474], X[20] + 2 X[5479], 4 X[140] - X[5613], X[13] - 4 X[6036], X[5463] + 2 X[6055], X[4] - 4 X[6670], X[383] - 4 X[6672], 2 X[619] + X[6773], 5 X[631] + X[6773], X[14] - 4 X[6774], X[3] + 2 X[6774], X[5474] + 8 X[6774], 4 X[6771] - X[6778], 4 X[1385] - X[7974], 5 X[7987] + X[9900], X[9116] + 2 X[11632], X[40] + 2 X[11706], X[5617] + 2 X[12042], 4 X[6684] - X[12780], 2 X[11710] + X[12781], 5 X[14] - 2 X[13102], 10 X[6774] - X[13102], 5 X[3] + X[13102], 5 X[5474] + 4 X[13102], X[6780] - 4 X[13350], 2 X[5478] - 5 X[14061], 2 X[6109] + X[14538], 2 X[395] + X[14539], 2 X[14137] + X[14541], 7 X[13102] - 10 X[16002], 7 X[14] - 4 X[16002], 7 X[6774] - X[16002], 7 X[3] + 2 X[16002], 7 X[5474] + 8 X[16002], 2 X[15930] + X[18864], X[550] + 2 X[20253], 5 X[16002] - 14 X[20416], 5 X[14] - 8 X[20416], X[13102] - 4 X[20416], 5 X[6774] - 2 X[20416], 5 X[3] + 4 X[20416], 5 X[5474] + 16 X[20416], 5 X[16961] - 2 X[20426]

X(21157) lies on these lines: {2, 9750}, {3, 14}, {4, 6670}, {13, 6036}, {15, 9113}, {16, 230}, {20, 5479}, {35, 10077}, {36, 10061}, {40, 11706}, {61, 9606}, {98, 618}, {115, 5473}, {140, 5613}, {182, 16241}, {371, 19075}, {372, 19076}, {376, 5460}, {383, 6672}, {395, 14539}, {511, 16963}, {531, 3524}, {542, 5054}, {549, 5464}, {550, 20253}, {616, 5983}, {617, 3523}, {619, 631}, {1385, 7974}, {1587, 13916}, {1588, 13981}, {2794, 11298}, {3411, 5865}, {3515, 12141}, {5050, 16962}, {5171, 12204}, {5204, 18975}, {5217, 13075}, {5432, 12941}, {5433, 12951}, {5463, 6055}, {5471, 11480}, {5478, 14061}, {5617, 12042}, {5869, 11310}, {6109, 14538}, {6684, 12780}, {6771, 6778}, {6780, 13350}, {7987, 9900}, {8980, 19074}, {9116, 11632}, {9751, 9760}, {10267, 13106}, {10269, 13104}, {11710, 12781}, {13967, 19073}, {14137, 14541}, {14540, 16773}, {16961, 20426}

X(21157) = X(14)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14, 5474), (3, 6774, 14), (631, 6773, 619), (5054, 5085, 21156), (5339, 10612, 14), (13102, 20416, 14)

X(21158) = X(15)-GIBERT-MOSES CENTROID

Barycentrics a^2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 - 6*b^2*c^2 - c^4 + 6*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :
X(21158) = 2 X[3] + X[15], 2 X[623] - 5 X[631], X[621] - 7 X[3523], 2 X[396] + X[5473], 5 X[15] - 2 X[5611], 5 X[3] + X[5611], X[5474] + 2 X[6109], X[4] - 4 X[6671], X[20] + 2 X[7684], 4 X[9126] - X[9162], X[40] + 2 X[11707], 4 X[140] - X[20428]

X(21158) lies on these lines: {2, 16652}, {3, 6}, {4, 6671}, {20, 7684}, {40, 11707}, {140, 20428}, {373, 3132}, {396, 5473}, {531, 3524}, {621, 3523}, {623, 631}, {3564, 5463}, {5474, 6109}, {6104, 16461}, {9126, 9162}, {11146, 14170}

X(21158) = X(15)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15, 14538), (3, 182, 10646), (3, 5085, 21159), (3, 5238, 14541), (3, 11480, 14539), (3, 13350, 15), (187, 11480, 15)

X(21159) = X(16)-GIBERT-MOSES CENTROID

Barycentrics a^2*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 - 6*b^2*c^2 - c^4 - 6*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :
X(21159) = 2 X[3] + X[16], 2 X[624] - 5 X[631], X[622] - 7 X[3523], 2 X[395] + X[5474], 5 X[16] - 2 X[5615], 5 X[3] + X[5615], X[5473] + 2 X[6108], X[4] - 4 X[6672], X[20] + 2 X[7685], 4 X[9126] - X[9163], X[40] + 2 X[11708], X[5615] - 10 X[13349], X[16] - 4 X[13349], X[3] + 2 X[13349], 2 X[187] + X[14538], 4 X[3] - X[14539], 2 X[16] + X[14539], 8 X[13349] + X[14539], 4 X[5615] + 5 X[14539], 2 X[10614] + X[14541], 4 X[140] - X[20429]

X(21159) lies on these lines: {2, 16653}, {3, 6}, {4, 6672}, {20, 7685}, {40, 11708}, {140, 20429}, {373, 3131}, {395, 5474}, {530, 3524}, {622, 3523}, {624, 631}, {3564, 5464}, {5473, 6108}, {6105, 16462}, {9126, 9163}, {11145, 14169}

X(21159) = X(16)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 16, 14539), (3, 182, 10645), (3, 5237, 14540), (3, 5085, 21158), (3, 11481, 14538), (3, 13349, 16), (187, 11481, 16)

X(21160) = X(19)-GIBERT-MOSES CENTROID

Barycentrics a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*(b*(a^2 + b^2 - c^2)*(-a^2 + b^2 + c^2) + c*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)) : :
X(21160) = 2 X[3] + X[19], 7 X[3523] - X[4329], 5 X[631] - 2 X[18589], 11 X[15717] + X[20061]

X(21160) lies on these lines: {2, 165}, {3, 19}, {35, 4319}, {46, 2263}, {515, 15940}, {534, 3524}, {631, 18589}, {990, 1781}, {1486, 10310}, {1844, 9643}, {2939, 10884}, {3523, 4329}, {3668, 15803}, {3827, 5085}, {4219, 9816}, {5338, 7580}, {5732, 18594}, {6803, 11677}, {7535, 11471}, {15717, 20061}

X(21160) = X(19)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K1039.

X(21161) = X(21)-GIBERT-MOSES CENTROID

Barycentrics a*(a + b)*(a - b - c)*(a + c)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*(b*(a + b)*(-a + b - c)*(b + c) + c*(a + c)*(-a - b + c)*(b + c)) : :
X(21161) = 2 X[3] + X[21], 2 X[442] - 5 X[631], X[2475] - 7 X[3523], 4 X[3] - X[3651], 2 X[21] + X[3651], X[21] - 4 X[5428], X[3] + 2 X[5428], X[3651] + 8 X[5428], 4 X[3530] - X[5499], 4 X[549] - X[6175], X[4] - 4 X[6675], X[191] + 5 X[7987], 4 X[6713] - X[11604], X[11684] + 8 X[13624], 2 X[8261] + X[14110], 2 X[11263] + X[16113], 2 X[3647] + X[16132], 2 X[1385] + X[16139], X[10308] + 2 X[16143], X[74] + 2 X[16164], X[104] - 4 X[17009]

X(21161) lies on these lines: {2, 3}, {35, 4848}, {36, 553}, {55, 5427}, {74, 16164}, {104, 15931}, {165, 5426}, {191, 7987}, {500, 16948}, {519, 10902}, {551, 11012}, {758, 3576}, {970, 16226}, {997, 3647}, {1385, 16139}, {1621, 3656}, {1737, 5441}, {2094, 10269}, {2771, 10167}, {2975, 3655}, {3218, 13151}, {3241, 10267}, {3428, 4428}, {3579, 4004}, {3582, 14794}, {3601, 10122}, {3649, 5204}, {3679, 11491}, {3916, 4511}, {3929, 18446}, {4421, 5657}, {4995, 5172}, {5217, 10543}, {5267, 18249}, {5424, 5425}, {6713, 11604}, {6796, 19875}, {7701, 9841}, {8261, 14110}, {10308, 16143}, {11263, 16113}, {17768, 21151}

X(21161 = midpoint of X(165) and X(5426)
X(21161) = X(21)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21161 = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 21, 3651), (3, 405, 6876), (3, 549, 13587), (3, 859, 7430), (3, 1006, 6905), (3, 5428, 21), (3, 6875, 6906), (3, 6914, 7411), (3, 6986, 6940), (3, 7508, 6909), (3, 16370, 376), (3, 17524, 7421), (20, 15674, 6841), (21, 13587, 6175), (376, 6875, 16370), (376, 16370, 6906), (381, 16858, 6920), (411, 16858, 381), (549, 13587, 6940), (631, 6987, 6830), (1006, 6946, 6883), (3149, 16857, 5071), (6915, 17547, 547), (6936, 6988, 6941), (6942, 15702, 16417), (6954, 6992, 6963), (6986, 13587, 549), (6992, 15717, 6954), (12104, 13743, 21), (15670, 16370, 21)

X(21162) = X(27)-GIBERT-MOSES CENTROID

Barycentrics (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*((a + b)*(b + c)*(a^2 + b^2 - c^2)*(-a^2 + b^2 + c^2) + (a + c)*(b + c)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)) : :
X(21162) = 2 X[3] + X[27], 2 X[440] - 5 X[631], X[3151] - 7 X[3523], X[4] - 4 X[6678]

X(21162) lies on this line: {2, 3}

X(21162) = X(27)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K109

X(21163) = X(39)-GIBERT-MOSES CENTROID

Barycentrics a^2*(b^2 + c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*((a^2 + b^2)*c^2 + b^2*(a^2 + c^2)) : :
X(21163) = 2 X[3] + X[39], 5 X[39] - 2 X[3095], 5 X[3] + X[3095], X[76] - 7 X[3523], 5 X[631] - 2 X[3934], 4 X[182] - X[5052], 4 X[3] - X[5188], 2 X[39] + X[5188], 4 X[3095] + 5 X[5188], 4 X[140] - X[6248], X[4] - 4 X[6683], 3 X[5054] - X[7697], 3 X[3524] + X[7709], X[20] + 5 X[7786], 4 X[549] - X[9466], 7 X[5188] - 4 X[9821], 7 X[3] - X[9821], 7 X[39] + 2 X[9821], 7 X[3095] + 5 X[9821], 5 X[631] + X[11257], 2 X[3934] + X[11257], X[550] + 2 X[11272], 13 X[10299] - X[12251], 5 X[7987] + X[12782], 2 X[548] + X[14881], X[6194] - 5 X[15692], X[7757] + 5 X[15692], X[14711] - 10 X[15693], X[194] + 11 X[15717]

X(21163) lies on these lines: {2, 9743}, {3, 6}, {4, 6683}, {20, 7786}, {76, 3523}, {98, 15483}, {114, 7853}, {140, 6248}, {141, 14981}, {147, 7831}, {194, 15717}, {237, 373}, {262, 376}, {538, 3524}, {542, 15810}, {548, 14881}, {549, 2482}, {550, 11272}, {631, 3934}, {730, 10164}, {1153, 19911}, {1503, 8359}, {1513, 4045}, {2794, 8356}, {3202, 10984}, {3455, 18475}, {3564, 7810}, {3576, 14839}, {3793, 12007}, {5054, 7697}, {5182, 11155}, {5650, 9155}, {6194, 7757}, {6390, 14994}, {7603, 15980}, {7761, 9744}, {7804, 11676}, {7824, 12203}, {7987, 12782}, {8704, 18311}, {8719, 11286}, {8721, 16043}, {9751, 9888}, {9890, 14651}, {10299, 12251}, {11653, 15035}, {13860, 15482}, {14711, 15693}

X(21163) = midpoint of X(i) and X(j) for these {i,j}: {262, 376}, {6194, 7757}
X(21163) = reflection of X(i) in X(j) for these {i,j}: {9466, 15819}, {15819, 549}
X(21163) = X(39)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K326
X(21163) = isogonal conjugate of X(14485)
X(21163) = Brocard-circle-inverse of X(8722)
X(21163) = barycentric quotient X(6)/X(14485)
X(21163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 8722), (3, 39, 5188), (3, 182, 187), (3, 574, 18860), (3, 5024, 1350), (3, 12054, 13335), (3, 13334, 39), (3, 13335, 15513), (39, 187, 5052), (574, 2021, 39), (575, 2080, 5008), (631, 11257, 3934), (1670, 1671, 11477), (1689, 1690, 5024), (5013, 13357, 39), (8160, 8161, 575), (9155, 14096, 5650), (9734, 17508, 3)

X(21164) = X(57)-GIBERT-MOSES CENTROID

Barycentrics a*(a + b - c)*(a - b + c)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*(b*(a + b - c)*(-a + b + c) + c*(a - b + c)*(-a + b + c)) : :
X(21164) = 2 X[3] + X[57], X[40] + 2 X[999], 5 X[57] - 2 X[2095], 5 X[3] + X[2095], 5 X[631] + X[2096], X[1] + 2 X[3359], 5 X[631] - 2 X[3452], X[2096] + 2 X[3452], X[329] - 7 X[3523], 4 X[3] - X[6282], 2 X[57] + X[6282], 4 X[2095] + 5 X[6282], X[3421] - 4 X[6684], X[4] - 4 X[6692], X[1750] - 4 X[6911], X[3586] + 2 X[6948], 2 X[6911] + X[7171], X[1750] + 2 X[7171], X[20] + 2 X[7682], 4 X[1385] - X[7962], X[2093] + 5 X[7987], X[5732] + 2 X[8257], X[1] - 4 X[10269], X[3359] + 2 X[10269], X[2094] + 5 X[15692], X[9965] + 11 X[15717], X[7994] - 7 X[16192], 2 X[12675] + X[17658], 7 X[7989] - 4 X[18516], 8 X[140] - 5 X[20196]

X(21164) lies on these lines: {1, 3}, {4, 6692}, {20, 7682}, {84, 474}, {104, 9623}, {140, 20196}, {142, 1519}, {329, 3523}, {374, 1436}, {404, 1490}, {405, 15239}, {443, 6256}, {527, 3524}, {631, 2096}, {936, 6940}, {971, 16417}, {1012, 5437}, {1071, 5438}, {1158, 8583}, {1750, 6911}, {2057, 3681}, {2094, 15692}, {3149, 9841}, {3306, 6909}, {3421, 6684}, {3586, 6948}, {3624, 6892}, {3679, 5770}, {3817, 6847}, {3820, 5234}, {3911, 6916}, {4188, 10884}, {4292, 6926}, {4512, 10165}, {5249, 6966}, {5691, 6885}, {5705, 6897}, {5715, 6890}, {5732, 6905}, {5744, 6735}, {5745, 12115}, {5787, 17563}, {5924, 6910}, {6001, 17612}, {6245, 6904}, {6260, 17567}, {6857, 12608}, {6891, 9612}, {6922, 9579}, {7971, 17614}, {7989, 18516}, {9858, 14872}, {9965, 15717}, {10156, 16418}, {10167, 16371}, {12675, 17658}, {12751, 13226}

X(21164) = X(57)-Gibert-Moses centroid; see the preamble just before X(21153)
X(21164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16209, 10270), (3, 57, 6282), (3, 9940, 3601), (3, 11227, 3576), (631, 2096, 3452), (3359, 10269, 1), (3576, 11227, 8726), (6911, 7171, 1750)

X(21165) = X(63)-GIBERT-MOSES CENTROID

Barycentrics a*(a^2 - b^2 - c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*a^2*(-a^2 + b^2 + c^2)*(b*(-a^2 + b^2 - c^2) + c*(-a^2 - b^2 + c^2)) : :
X(21165) = 2 X[3] + X[63], 2 X[226] - 5 X[631], X[40] + 2 X[993], X[3428] + 2 X[4640], X[4] - 4 X[5745], X[3419] - 4 X[5771], 7 X[3523] - X[5905], X[1478] - 4 X[6684], 2 X[11608] + X[13172], 4 X[3] - X[18446], 2 X[63] + X[18446], 11 X[15717] + X[20078]

X(21165) lies on these lines: {1, 6875}, {3, 63}, {4, 5705}, {9, 6905}, {10, 6934}, {21, 5709}, {35, 15104}, {40, 993}, {55, 12464}, {57, 1006}, {84, 3651}, {142, 6878}, {165, 376}, {191, 6261}, {226, 631}, {411, 7330}, {517, 16370}, {527, 3524}, {553, 10165}, {602, 988}, {758, 3576}, {908, 6954}, {936, 6942}, {944, 10268}, {1064, 1707}, {1210, 6936}, {1478, 6684}, {1490, 6876}, {1519, 5698}, {1698, 6901}, {2949, 10393}, {3149, 10157}, {3218, 18443}, {3219, 5720}, {3305, 6911}, {3306, 6883}, {3419, 5771}, {3428, 4640}, {3452, 6880}, {3523, 5905}, {3587, 6909}, {3601, 18389}, {3911, 6947}, {4292, 6889}, {4512, 5603}, {5122, 8545}, {5234, 5818}, {5248, 12704}, {5250, 11249}, {5256, 5398}, {5307, 7554}, {5715, 6852}, {5744, 6987}, {5759, 6935}, {5804, 11106}, {5812, 7483}, {5841, 11112}, {5882, 16208}, {5886, 15670}, {6282, 6950}, {6361, 12864}, {6705, 6899}, {6734, 6868}, {6834, 12572}, {6853, 9612}, {6937, 9579}, {6946, 7308}, {6976, 7682}, {7171, 7411}, {9028, 10519}, {10270, 12115}, {10786, 12527}, {11012, 12514}, {11608, 13172}, {15717, 20078}

X(21165) = X(63)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K343
X(21165) = {X(3),X(63)}-harmonic conjugate of X(18446)

X(21166) = X(99)-GIBERT-MOSES CENTROID

Barycentrics 2*a^2*(b - c)^2*(b + c)^2*(-a^2 + b^2 + c^2) - (a^2 - b^2)*(a^2 - c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(21166) = 4 X[3] - X[98], 2 X[3] + X[99], X[98] + 2 X[99], X[20] + 2 X[114], X[4] - 4 X[620], 2 X[115] - 5 X[631], 4 X[549] - X[671], X[376] + 2 X[2482], X[147] + 5 X[3522], X[148] - 7 X[3523], X[1350] + 2 X[5026], 2 X[619] + X[5473], 2 X[618] + X[5474], 2 X[550] + X[6033], X[148] - 4 X[6036], 7 X[3523] - 4 X[6036], 4 X[2482] - X[6054], 2 X[376] + X[6054], 4 X[140] - X[6321], 5 X[3091] - 8 X[6721], 11 X[3525] - 8 X[6722], X[842] + 2 X[7472], X[5503] - 4 X[7618], 2 X[40] + X[7970], 4 X[1385] - X[7983], 2 X[6055] + X[8591], 2 X[8703] + X[8724], 7 X[3528] - X[9862], 2 X[4297] + X[9864], 5 X[2] - 2 X[9880], 2 X[3654] + X[9884], 4 X[114] - X[10722], 2 X[20] + X[10722], 4 X[5] - X[10723], 4 X[5026] - X[10753], 2 X[1350] + X[10753], 4 X[182] - X[10754], 4 X[6713] - X[10769], 2 X[6036] + X[10992], X[148] + 2 X[10992], 7 X[3523] + 2 X[10992], 2 X[5976] + X[11257], 13 X[10299] - 4 X[11623], 5 X[7987] - 2 X[11710], X[7970] - 4 X[11711], X[40] + 2 X[11711], X[962] - 4 X[11724], 5 X[98] - 8 X[12042], 5 X[3] - 2 X[12042], 5 X[99] + 4 X[12042], X[11632] - 4 X[12100], 2 X[2] + X[12117], 4 X[9880] + 5 X[12117], 2 X[8290] + X[12122], 2 X[3098] + X[12177], 14 X[12042] - 5 X[12188], 7 X[98] - 4 X[12188], 7 X[3] - X[12188], 7 X[99] + 2 X[12188], 2 X[1569] + X[12251], 2 X[115] + X[13172], 5 X[631] + X[13172], 5 X[7987] + X[13174], 2 X[11710] + X[13174], 4 X[6684] - X[13178], 5 X[99] - 2 X[13188], 5 X[3] + X[13188], 2 X[12042] + X[13188], 5 X[98] + 4 X[13188], 5 X[12188] + 7 X[13188], X[1916] - 4 X[13334], 5 X[7925] - 2 X[13449], 8 X[140] - 5 X[14061], 2 X[6321] - 5 X[14061], 4 X[9880] - 5 X[14639], 3 X[3524] - X[14651], 7 X[3528] + 2 X[14981], X[9862] + 2 X[14981], 11 X[5070] - 8 X[15092], X[12243] + 2 X[15300], 4 X[1511] - X[15342], X[12383] + 2 X[15357], 2 X[6055] - 5 X[15692], X[8591] + 5 X[15692], X[12243] - 7 X[15698], 2 X[15300] + 7 X[15698], X[12355] - 7 X[15701], 4 X[5461] - 7 X[15702], 2 X[14971] - 3 X[15709], X[11005] + 2 X[16163], X[9860] - 7 X[16192], X[11676] + 2 X[18860], 11 X[15717] + X[20094], X[3529] + 8 X[20399]

X(21166) lies on these lines: {2, 9734}, {3, 76}, {4, 620}, {5, 10723}, {20, 114}, {30, 10242}, {35, 10089}, {36, 10086}, {40, 7970}, {56, 15452}, {115, 631}, {140, 6321}, {147, 3522}, {148, 3523}, {182, 10754}, {262, 1003}, {371, 19108}, {372, 19109}, {376, 2482}, {511, 5182}, {542, 10304}, {543, 3524}, {549, 671}, {550, 6033}, {618, 5474}, {619, 5473}, {690, 15035}, {842, 7472}, {962, 11724}, {1151, 19056}, {1152, 19055}, {1350, 5026}, {1385, 7983}, {1511, 15342}, {1569, 5206}, {1587, 8997}, {1588, 13989}, {1916, 13334}, {2023, 15815}, {2077, 12189}, {3023, 5217}, {3027, 5204}, {3091, 6721}, {3098, 12177}, {3515, 5186}, {3516, 12131}, {3525, 6722}, {3526, 7918}, {3528, 9862}, {3529, 20399}, {3545, 9167}, {3552, 9737}, {3654, 9884}, {4027, 5171}, {4297, 9864}, {4558, 11596}, {5010, 10053}, {5023, 12829}, {5054, 9166}, {5070, 15092}, {5085, 5969}, {5149, 11676}, {5432, 13182}, {5433, 13183}, {5461, 15702}, {5503, 7618}, {5985, 17548}, {6055, 8591}, {6684, 13178}, {6713, 10769}, {7280, 10069}, {7783, 13335}, {7925, 13449}, {7987, 11710}, {8290, 12122}, {8703, 8724}, {8716, 9755}, {9307, 10607}, {9751, 9888}, {9860, 16192}, {10267, 13190}, {10269, 13189}, {10299, 11623}, {11005, 16163}, {11012, 12190}, {11632, 12100}, {12184, 15326}, {12185, 15338}, {12243, 15300}, {12355, 15701}, {12383, 15357}, {14645, 14912}, {14971, 15709}, {15717, 20094}

X(21166) = midpoint of X(12117) and X(14639)
X(21166) = reflection of X(i) in X(j) for these {i,j}: {3545, 9167}, {9166, 5054}, {14639, 2}
X(21166) = X(99)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K035
X(21166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 99, 98), (3, 13188, 12042), (20, 114, 10722), (40, 11711, 7970), (140, 6321, 14061), (148, 3523, 6036), (376, 2482, 6054), (631, 13172, 115), (1350, 5026, 10753), (3552, 9737, 12110), (6036, 10992, 148), (7987, 13174, 11710), (8591, 15692, 6055)

X(21167) = X(141)-GIBERT-MOSES CENTROID

Barycentrics 2*a^2*(-a^2 + b^2 + c^2)*(2*a^2 + b^2 + c^2) - (b^2 + c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(21167) = 2 X[3] + X[141], 4 X[549] - X[597], 5 X[631] + X[1350], 5 X[141] - 2 X[1352], 5 X[3] + X[1352], 2 X[140] + X[3098], X[6] - 7 X[3523], X[182] - 4 X[3530], 5 X[631] - 2 X[3589], X[1350] + 2 X[3589], 5 X[3522] + 7 X[3619], 4 X[182] - X[3629], 16 X[3530] - X[3629], X[20] + 5 X[3763], 2 X[548] + X[3818], 2 X[3844] + X[4297], 3 X[3524] - X[5085], X[3630] + 8 X[5092], 4 X[140] - X[5480], 2 X[3098] + X[5480], X[159] + 2 X[6696], 2 X[3631] + X[6776], X[3416] + 5 X[7987], 4 X[5092] - X[8550], X[3630] + 2 X[8550], X[5188] + 2 X[10007], 5 X[597] - 8 X[10168], 5 X[549] - 2 X[10168], X[6776] - 13 X[10299], 2 X[3631] + 13 X[10299], 3 X[3524] + X[10519], X[193] - 7 X[10541], 4 X[6329] - X[11477], X[5476] - 4 X[11812], 2 X[12007] - 5 X[12017], X[9969] + 2 X[13348], 3 X[5054] - X[14561], X[5] + 2 X[14810], 5 X[631] - X[14853], 9 X[3524] - X[14912], 3 X[5085] - X[14912], 3 X[10519] + X[14912], X[67] + 5 X[15051], X[6247] + 2 X[15577], X[599] + 5 X[15692], X[8584] - 10 X[15693], X[5050] - 5 X[15693], X[11179] - 7 X[15700], X[5093] - 9 X[15707], X[8550] - 10 X[15712], 2 X[5092] - 5 X[15712], X[11180] + 11 X[15715], X[69] + 11 X[15717], X[5102] - 11 X[15719], 2 X[6698] + X[16163], 11 X[1352] - 5 X[18440], 11 X[141] - 2 X[18440], 11 X[3] + X[18440], 4 X[12108] - X[18583], 5 X[632] - 2 X[19130], X[1353] - 4 X[20190], 7 X[15701] - X[20423], X[376] + 2 X[20582]

X(21167) lies on these lines: {3, 66}, {5, 14810}, {6, 3523}, {20, 3763}, {67, 15051}, {69, 15717}, {98, 15598}, {140, 3098}, {182, 3530}, {193, 10541}, {343, 15246}, {376, 10516}, {511, 549}, {518, 10164}, {524, 3524}, {542, 17504}, {548, 3818}, {599, 15692}, {631, 1350}, {632, 19130}, {698, 13468}, {1353, 20190}, {2076, 3815}, {3416, 7987}, {3522, 3619}, {3564, 12100}, {3576, 5846}, {3630, 5092}, {3631, 6776}, {3819, 10192}, {3844, 4297}, {4265, 6986}, {5050, 8584}, {5054, 14561}, {5093, 15707}, {5102, 15719}, {5188, 10007}, {5306, 6194}, {5476, 11812}, {5657, 9053}, {5743, 19649}, {6329, 11477}, {6393, 7771}, {6698, 16163}, {7288, 10387}, {7390, 17265}, {7393, 15873}, {7485, 13567}, {7509, 16657}, {7998, 13394}, {9606, 12212}, {9969, 13348}, {10323, 16654}, {11179, 15700}, {11180, 15715}, {11539, 19924}, {12007, 12017}, {12108, 18583}, {15701, 20423}

X(21167) = midpoint of X(i) and X(j) for these {i,j}: {376, 10516}, {1350, 14853}, {5085, 10519}
X(21167) = reflection of X(i) in X(j) for these {i,j}: {8584, 5050}, {10516, 20582}, {14853, 3589}, {17508, 12100}
X(21167) = X(141)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K655
X(21167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 3098, 5480), (631, 1350, 3589), (3524, 10519, 5085)

X(21168) = X(144)-GIBERT-MOSES CENTROID

Barycentrics 2*a^2*(2*a^2 - 2*b^2 + 4*b*c - 2*c^2)*(-a^2 + b^2 + c^2) - (-3*a^2 + 2*a*b + b^2 + 2*a*c - 2*b*c + c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(21168) = X[4] - 4 X[9], 2 X[3] + X[144], 2 X[7] - 5 X[631], 8 X[142] - 11 X[3525], X[944] + 2 X[5223], 7 X[3528] - 4 X[5732], 13 X[5067] - 4 X[5735], 2 X[9] + X[5759], X[4] + 2 X[5759], X[20] + 2 X[5779], 7 X[3090] - 4 X[5805], X[104] + 2 X[6068], X[376] + 2 X[6172], 13 X[5067] - 16 X[6666], X[5735] - 4 X[6666], X[4312] - 4 X[6684], 8 X[1001] - 5 X[10595], X[6361] + 2 X[11372], 2 X[390] + X[12245], 2 X[1156] + X[13199], 4 X[6173] - 7 X[15702], 7 X[3090] - 10 X[18230], 2 X[5805] - 5 X[18230], 11 X[3855] - 8 X[18482], 7 X[3523] - X[20059], 3 X[3524] - 2 X[21151]

X(21168) lies on these lines: {2, 5762}, {3, 144}, {4, 9}, {7, 631}, {20, 5779}, {45, 3332}, {72, 5731}, {104, 6068}, {142, 3525}, {165, 5658}, {218, 11200}, {376, 971}, {390, 5729}, {405, 8158}, {480, 11491}, {518, 7967}, {527, 3524}, {943, 12260}, {944, 5223}, {952, 6987}, {954, 999}, {1001, 10595}, {1056, 15298}, {1058, 15299}, {1156, 13199}, {3090, 5805}, {3361, 3487}, {3488, 9819}, {3523, 20059}, {3528, 5732}, {3576, 5850}, {3579, 5811}, {3855, 18482}, {4312, 5714}, {4419, 13329}, {4679, 8166}, {5067, 5735}, {5709, 17559}, {5715, 10172}, {5728, 5766}, {5758, 5886}, {5763, 17558}, {5770, 6865}, {5812, 11231}, {5832, 6879}, {5845, 10519}, {5927, 9778}, {6173, 15702}, {6889, 8232}, {6937, 7679}, {6967, 8732}, {10385, 15104}, {10392, 11362}, {10396, 12842}

X(21168) = midpoint of X(5759) and X(5817)
X(21168) = reflection of X(i) in X(j) for these {i,j}: {4, 5817}, {5817, 9}
X(21168) = anticomplement of X(38107)
X(21168) = X(144)-Gibert-Moses centroid; see the preamble just before X(21153); the cubic is K1044
X(21168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5759, 4), (5805, 18230, 3090)

X(21169) = {X(7), X(482)}-HARMONIC CONJUGATE OF X(176)

Trilinears 1 + 3 sec A/2 cos B/2 cos C/2 : :
Barycentrics (a+b-c)*(a-b+c)*(3*S+a*(-a+b+c)) : :

See Tran Quang Hung and César Lozada, ADGEOM 4898: "About Equal Detour Point".

X(21169) lies on the these lines: {1, 7}, {226, 3591}, {8965, 17092}

X(21169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 176, 175), (7, 482, 176), (7, 17802, 1374), (7, 17805, 481), (175, 17804, 176), (176, 482, 17804), (481, 482, 1371), (481, 1371, 17805), (482, 1373, 7), (1371, 17805, 176), (1374, 17802, 17801), (17801, 17802, 175)

X(21170) = {X(7), X(1373)}-HARMONIC CONJUGATE OF X(21169)

Barycentrics (a+b-c)*(a-b+c)*(5*S+a*(-a+b+c)) : :

See Tran Quang Hung and César Lozada, ADGEOM 4898: "About Equal Detour Point".

X(21170) lies on the these lines: {1, 7}

X(21170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17804, 176), (7, 482, 175), (7, 17804, 17801), (7, 17805, 1374), (175, 482, 176), (176, 17801, 1), (481, 482, 17806), (482, 1374, 17805), (1374, 17805, 175)

X(21171) = {X(7), X(21170)}-HARMONIC CONJUGATE OF X(1373)

Barycentrics (a+b-c)*(a-b+c)*(6*S+a*(-a+b+c)) : :

See Tran Quang Hung and César Lozada, ADGEOM 4898: "About Equal Detour Point".

X(21171) lies on the these lines: {1, 7}, {226, 10194}, {553, 5393}, {3982, 13389}, {4114, 13388}, {4654, 5405}, {5589, 7613}

X(21171) = {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): (7, 176, 1374), (7, 482, 481), (7, 1373, 482), (17802, 17804, 17806)

X(21172) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (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(21172) lies on these lines: {1, 8058}, {86, 21178}, {106, 2734}, {242, 514}, {513, 676}, {521, 14837}, {522, 905}, {656, 7658}, {1519, 1769}, {1638, 7655}, {2191, 20516}, {2424, 21202}, {2509, 3239}, {3798, 21191}, {4025, 7253}, {4765, 6589}, {6003, 21188}, {6586, 14282}, {6587, 8057}, {9031, 20316}, {21173, 21185}

X(21173) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics a (b - c) (a^3 - a b^2 + a b c - b^2 c - a c^2 - b c^2) : :

X(21173) lies on these lines: {1, 522}, {9, 6586}, {10, 20293}, {36, 238}, {86, 20954}, {87, 1027}, {514, 4581}, {521, 1734}, {650, 9364}, {652, 2655}, {656, 1955}, {657, 1743}, {659, 6363}, {663, 3667}, {665, 3287}, {693, 17218}, {784, 5214}, {894, 21225}, {900, 2605}, {1024, 16779}, {1052, 9359}, {1698, 20316}, {2254, 6003}, {2457, 4778}, {2827, 6615}, {3216, 14812}, {3261, 10436}, {3751, 9000}, {3960, 4017}, {4063, 6371}, {4724, 4932}, {4977, 10015}, {4985, 8062}, {7253, 8714}, {11125, 21179}, {14838, 17420}, {21102, 21180}, {21123, 21389}, {21172, 21185}

X(21174) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^5 - a^4 b + a b^4 + b^5 - a^4 c + 2 a^2 b^2 c - b^4 c + 2 a^2 b c^2 - 2 a b^2 c^2 + a c^4 - b c^4 + c^5) : :

X(21174) lies on these lines: {513, 676}, {514, 6591}, {522, 21184}, {525, 3239}, {693, 21185}, {2522, 7658}, {3835, 21188}, {4025, 21189}, {4077, 7649}, {6129, 17094}, {17896, 21186}

X(21175) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^6 + a^4 b^2 - a^2 b^4 - b^6 + a^4 c^2 + b^4 c^2 - a^2 c^4 + b^2 c^4 - c^6) : :

X(21175) lies on these lines: {242, 514}, {4823, 21185}

X(21176) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (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(21176) lies on these lines: {242, 514}

X(21177) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(3001), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^6 b^2 - a^2 b^6 + a^6 c^2 + b^6 c^2 - 2 b^4 c^4 - a^2 c^6 + b^2 c^6) : :

X(21177) lies on these lines: {2, 3}, {6, 2387}, {160, 2549}, {512, 1691}, {538, 1634}, {754, 5201}, {1625, 5167}, {2386, 3003}, {3094, 9971}, {3767, 15270}, {7761, 8266}, {7776, 9917}, {9149, 14568}, {15048, 20775}, {20794, 22253}

X(21178) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - b^4 - c^4) : :

X(21178) lies on these lines: {75, 21186}, {86, 21172}, {514, 1919}, {2485, 16757}, {2517, 4467}, {3261, 4025}, {4360, 8058}, {11125, 17215}, {20517, 21182}, {20907, 21180}, {20954, 21185}, {21200, 21206}

X(21179) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - b^4 - a^2 b c + a b^2 c + a b c^2 + 2 b^2 c^2 - c^4) : :

X(21179) lies on these lines: {1, 21102}, {82, 10566}, {240, 522}, {513, 942}, {514, 2605}, {3737, 21118}, {8714, 21187}, {11125, 21173}, {20954, 21205}

X(21180) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - b^4 + a^2 b c - a b^2 c - a b c^2 + 2 b^2 c^2 - c^4) : :

X(21180) lies on these lines: {1, 11125}, {65, 2773}, {75, 21205}, {107, 109}, {240, 522}, {244, 1109}, {514, 21112}, {523, 8043}, {673, 897}, {676, 2804}, {900, 12019}, {1698, 14429}, {2501, 16612}, {3738, 10015}, {5620, 10265}, {6089, 9508}, {6797, 8674}, {18314, 21187}, {20520, 21202}, {20907, 21178}, {21102, 21173}

X(21180) = midpoint of X(2588) and X(2589)
X(21180) = trilinear pole of line X(48)X(1030)
X(21180) = center of circle {{X(11),X(36),X(65),X(80),X(108),X(759),X(1354),X(1845),X(2588),X(2589)}}

X(21181) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (2*a^3+(b+c)*a^2-(b^2+c^2)*a-(b+c)*(2*b^2-3*b*c+2*c^2)) : :

X(21181) lies on these lines: {10, 4458}, {214, 6366}, {244, 1111}, {514, 14419}, {522, 4823}, {551, 2785}, {690, 12258}, {1125, 4707}, {2787, 4049}, {3634, 4088}

X(21182) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - a^3 b + a b^3 - b^4 - a^3 c + b^3 c + a c^3 + b c^3 - c^4) : :

X(21182) lies on these lines: {514, 3063}, {522, 3663}, {850, 4025}, {3261, 17861}, {3673, 20954}, {7649, 20520}, {20517, 21178}

X(21183) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 - b^2 + 4 b c - c^2) : :

X(21183) lies on these lines: {2, 514}, {513, 4927}, {522, 693}, {663, 4666}, {812, 4786}, {918, 4944}, {1565, 6075}, {1638, 4762}, {1639, 4468}, {3004, 4802}, {3667, 21297}, {3776, 6590}, {3798, 4382}, {3870, 4449}, {4040, 10582}, {4801, 14837}, {4874, 4977}, {4978, 21188}, {7192, 17218}, {7628, 20906}, {7658, 17494}, {11019, 21185}, {16737, 16750}

X(21184) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^5-(b^2+c^2)*a^3+(b^3+c^3)*a^2-(b^2-c^2)*(b^3-c^3)) : :

X(21184) lies on these lines: {514, 21117}, {522, 21174}, {850, 4025}, {1734, 4467}, {2501, 2504}, {3668, 3676}, {3835, 21209}

X(21185) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^3 - a^2 b + a b^2 - b^3 - a^2 c + b^2 c + a c^2 + b c^2 - c^3) : :

X(21185) lies on these lines: {1, 514}, {240, 522}, {513, 5570}, {676, 905}, {693, 21174}, {2254, 21188}, {2826, 3669}, {3309, 7178}, {3583, 3667}, {3676, 4905}, {3679, 4546}, {3900, 10015}, {4025, 8714}, {4147, 10039}, {4391, 13259}, {4823, 21175}, {5583, 9521}, {7264, 19594}, {11019, 21183}, {14077, 21120}, {18391, 21302}, {20954, 21178}, {21172, 21173}

X(21186) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^4 + b^4 - 2 a^2 b c + 2 a b^2 c + 2 a b c^2 - 2 b^2 c^2 + c^4) : :

X(21186) lies on these lines: {1, 8058}, {75, 21178}, {240, 522}, {513, 10015}, {514, 4581}, {523, 905}, {850, 4025}, {1638, 14353}, {2501, 2522}, {2523, 12077}, {2804, 6129}, {3064, 16612}, {3667, 21301}, {4017, 21188}, {4397, 17899}, {4458, 20521}, {4859, 20516}, {4962, 21201}, {7658, 21196}, {17896, 21174}

X(21187) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^4 - a^3 b + a b^3 + b^4 - a^3 c - a^2 b c + a b^2 c + b^3 c + a b c^2 + a c^3 + b c^3 + c^4) : :

X(21187) lies on these lines: {2, 4064}, {513, 4142}, {514, 3733}, {521, 942}, {522, 4823}, {523, 2487}, {525, 8062}, {648, 2633}, {1125, 5664}, {2485, 16612}, {2605, 2785}, {3261, 4025}, {3666, 6129}, {3670, 21189}, {3737, 4707}, {4057, 13246}, {4359, 4397}, {7253, 11125}, {7658, 20315}, {8714, 21179}, {14837, 20316}, {17496, 21102}, {18314, 21180}, {20518, 21202}

X(21188) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21188) lies on these lines: {241, 514}, {522, 4823}, {523, 14353}, {525, 4885}, {676, 3309}, {942, 8676}, {1459, 2457}, {1577, 4025}, {2254, 21185}, {2504, 3798}, {2786, 21206}, {3667, 7661}, {3835, 21174}, {4000, 17922}, {4017, 21186}, {4049, 13478}, {4379, 21124}, {4391, 4453}, {4458, 17072}, {4498, 6545}, {4707, 6332}, {4794, 8713}, {4978, 21183}, {6003, 21172}, {6245, 21201}, {20205, 21198}, {21204, 21621}

X(21189) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics a (b - c) (a^2 b - b^3 + a^2 c - a b c - c^3) : :
Trilinears b^2 (cos A - cos B) - c^2 (cos A - cos C) : :

X(21189) lies on these lines: {1, 521}, {9, 2509}, {10, 4397}, {36, 238}, {90, 11512}, {240, 522}, {514, 4017}, {523, 10015}, {650, 1758}, {652, 16612}, {663, 6003}, {1021, 6591}, {1054, 2957}, {1459, 3738}, {2254, 3667}, {3309, 7655}, {3669, 14353}, {3670, 21187}, {3731, 4130}, {3777, 6363}, {4025, 21174}, {4086, 20316}, {4147, 4404}, {4357, 15413}, {4811, 8714}, {4926, 15079}, {5214, 7662}, {14838, 17418}

X(21189) = isogonal conjugate of X(36050)
X(21189) = crossdifference of every pair of points on line X(37)X(48)

X(21190) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-3 a^8 + 2 a^4 b^4 + b^8 + 2 a^4 c^4 - 2 b^4 c^4 + c^8) : :

X(21190) lies on these lines: {514, 21122}, {649, 7649}

X(21191) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(21191) lies on these lines: {2, 20979}, {75, 17458}, {86, 1919}, {141, 21262}, {513, 3716}, {514, 1921}, {649, 17204}, {665, 802}, {786, 4411}, {788, 17072}, {812, 905}, {1459, 4107}, {3250, 4374}, {3667, 21211}, {3798, 21172}, {3912, 9294}, {4025, 21194}, {4057, 15668}, {4079, 4406}, {4486, 15413}, {4786, 17167}, {6371, 17066}, {6373, 21260}, {7649, 17171}, {10436, 21389}, {17300, 21304}, {18208, 20523}

X(21191) = complement of X(20979)
X(21191) = complementary conjugate of complement of X(4598)

X(21192) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (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(21192) lies on these lines: {2, 7265}, {239, 514}, {522, 4823}, {525, 14838}, {826, 9508}, {905, 3666}, {942, 3900}, {1577, 4467}, {1734, 3670}, {2786, 4129}, {3910, 3960}, {4142, 8714}, {4151, 4458}, {4359, 4391}, {4453, 4978}, {4791, 14837}, {4897, 15309}, {5664, 5745}

X(21193) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^5 - a^3 b^2 + a^2 b^3 + b^5 + a^2 b^2 c - a^3 c^2 + a^2 b c^2 - a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + c^5) : :

X(21193) lies on these lines: {513, 3776}, {514, 18108}, {826, 4142}, {3239, 22011}, {3835, 20517}

X(21194) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^4 + a^2 b^2 + b^4 + a^2 c^2 + b^2 c^2 + c^4) : :

X(21194) lies on these lines: {513, 3776}, {514, 1919}, {824, 905}, {3261, 21200}, {4025, 21191}, {8061, 21212}, {17176, 17215}, {21113, 21205}, {22011, 22042

X(21195) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (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) : :

X(21195) lies on these lines: {7, 657}, {87, 1027}, {142, 522}, {514, 7216}, {812, 905}, {1086, 6586}, {1447, 21390}, {1459, 4000}, {2254, 3667}, {3739, 20316}

X(21196) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^2 + a b + b^2 + a c + b c + c^2) : :

X(21196) lies on these lines: {2, 4024}, {38, 21727}, {99, 2644}, {116, 3258}, {239, 514}, {513, 4818}, {522, 1491}, {523, 2487}, {647, 8045}, {650, 824}, {661, 2786}, {693, 4359}, {784, 4142}, {812, 3004}, {942, 14077}, {1125, 1649}, {1577, 17899}, {3005, 4151}, {3676, 7212}, {3739, 4500}, {3776, 4762}, {4379, 20522}, {4830, 4977}, {4841, 4897}, {4926, 4940}, {7649, 17921}, {7658, 21186}, {8043, 8060}, {16751, 21828}, {16755, 18200}, {18154, 19804}

X(21197) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^3 b^2 + a^2 b^3 + 2 a^3 b c - a^2 b^2 c - 2 a b^3 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3) : :

X(21197) lies on these lines: {244, 20909}, {514, 21128}, {522, 3837}, {3676, 6063}, {3766, 4025}, {4458, 20521}

X(21198) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^3 - a^2 b + a b^2 + b^3 - a^2 c + 5 a b c - 3 b^2 c + a c^2 - 3 b c^2 + c^3) : :

X(21198) lies on these lines: {2, 514}, {10, 522}, {169, 21389}, {1387, 6366}, {1639, 10015}, {1698, 21132}, {2397, 4169}, {2401, 7658}, {3624, 21105}, {3667, 5587}, {3679, 4543}, {3762, 4453}, {4025, 21199}, {4359, 4391}, {5376, 6633}, {6381, 20518}, {6708, 20317}, {13259, 14430}, {20205, 21188}

X(21199) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-4 a^3 - 4 a^2 b + 4 a b^2 + 4 b^3 - 4 a^2 c + 5 a b c + 4 a c^2 + 4 c^3) : :

X(21199) lies on these lines: {514, 1635}, {522, 14315}, {4025, 21198}

X(21200) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^4 + a^2 b^2 + b^4 + a^2 c^2 - 3 b^2 c^2 + c^4) : :

X(21200) lies on these lines: {86, 514}, {513, 18014}, {523, 6707}, {2786, 20522}, {3261, 21194}, {3739, 4500}, {4107, 11125}, {7649, 17171}, {10278, 11053}, {21178, 21206}, {21201, 21211}, {21202, 21204}

X(21201) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (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(21201) lies on these lines: {1, 514}, {4, 2457}, {10, 522}, {169, 1024}, {277, 7658}, {513, 942}, {523, 3743}, {650, 16601}, {676, 2826}, {900, 12019}, {1577, 5051}, {1647, 21204}, {3309, 7686}, {3676, 10481}, {3887, 10015}, {4057, 20831}, {4142, 8714}, {4391, 4696}, {4543, 4668}, {4560, 17588}, {4962, 21186}, {6245, 21188}, {6362, 6675}, {6550, 14028}, {10395, 14837}, {14874, 21102}, {20518, 20888}, {21200, 21211}, {21203, 21209}

X(21202) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - a^3 b + a b^3 - b^4 - a^3 c + a^2 b c - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3 - c^4) : :

X(21202) lies on these lines: {6, 514}, {75, 4025}, {273, 7649}, {522, 3663}, {693, 17863}, {1442, 1459}, {2424, 21172}, {3239, 17279}, {3287, 21114}, {3668, 3676}, {3835, 16580}, {7658, 17278}, {20518, 21187}, {20520, 21180}, {21200, 21204}

X(21203) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-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(21203) lies on these lines: {58, 514}, {1734, 3670}, {4025, 16887}, {21201, 21209}

X(21204) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 - a b - b^2 - a c + 3 b c - c^2) : :

X(21204) lies on these lines: {2, 514}, {226, 3676}, {513, 3742}, {693, 4359}, {812, 1638}, {918, 4928}, {1027, 5272}, {1125, 5592}, {1647, 21201}, {1699, 3667}, {1920, 3261}, {2786, 4453}, {2789, 14413}, {3776, 4885}, {3837, 4458}, {3961, 4449}, {4369, 6703}, {4375, 4932}, {4448, 4778}, {4750, 21297}, {4763, 6084}, {6631, 6634}, {6632, 6633}, {7658, 8056}, {21188, 21621}, {21200, 21202}

X(21204) = complement of X(6546)
X(21204) = centroid of triangle A'B'C' as defined at X(5997)
X(21204) = centroid of vertex-triangle of Gemini triangles 7 and 8

X(21205) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^4 - b^4 + b^2 c^2 - c^4) : :

X(21205) lies on these lines: {75, 21180}, {86, 11125}, {514, 21136}, {2492, 7664}, {3122, 20525}, {3261, 4025}, {4107, 21131}, {20954, 21179}, {21113, 21194}

X(21206) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 + a^2 c^2 - 3 b^2 c^2) : :

X(21206) lies on these lines: {514, 1921}, {522, 3837}, {802, 17066}, {2786, 21188}, {3835, 4374}, {4107, 17215}, {21178, 21200}

X(21207) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics b^2 c^2 (b - c)^2 (b + c) : :

X(21207) lies on these lines: {4, 2825}, {10, 349}, {75, 5620}, {76, 4485}, {85, 11263}, {86, 20565}, {115, 127}, {116, 2973}, {264, 17861}, {313, 4013}, {331, 20320}, {514, 21137}, {905, 16759}, {1109, 17879}, {1111, 3120}, {1234, 18697}, {1365, 1367}, {1441, 3822}, {1446, 17869}, {1577, 21044}, {3122, 20525}, {3678, 21403}, {3754, 17867}, {4368, 18031

X(21207) = isotomic conjugate of X(4570)
X(21207) = trilinear product of extraversions of X(10)

X(21208) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a^2 + a b + a c - b c) : :

X(21208) lies on these lines: {2, 4568}, {10, 4986}, {76, 596}, {106, 664}, {115, 116}, {150, 6788}, {244, 1111}, {274, 6532}, {514, 1015}, {712, 20530}, {1125, 5976}, {1227, 4357}, {1358, 3676}, {1565, 3756}, {2140, 20271}, {2973, 7649}, {3122, 20525}, {3125, 17761}, {3216, 20247}, {3825, 4920}, {3934, 22011}, {4075, 18140}, {4089, 17213}, {4103, 9055}, {4115, 4465}, {4561, 6789}, {4904, 14837}, {6534, 18145}, {7208, 21139}, {16600, 17048}

X(21208) = isotomic conjugate of isogonal conjugate of X(8054)

X(21209) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (-a^5 + a^3 b^2 - a^2 b^3 + b^5 + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + c^5) : :

X(21209) lies on these lines: {81, 514}, {522, 4425}, {650, 824}, {2610, 2786}, {3835, 21184}, {4359, 4391}, {4453, 20522}, {7192, 17422}, {21200, 21202}, {21201, 21203}

X(21210) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c)^2 (a^3 + a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21210) lies on these lines: {244, 1109}, {514, 3248}, {522, 3123}, {653, 1416}, {1086, 21252}, {2643, 17761}, {3122, 20525}, {3271, 3808}, {4475, 17197}, {4494, 17155}

X(21211) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 b^2 - 3 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21211) lies on these lines: {75, 514}, {142, 3835}, {513, 3739}, {522, 876}, {649, 894}, {1019, 10455}, {3572, 9294}, {3667, 21191}, {4375, 10436}, {4687, 14437}, {4699, 14433}, {4932, 10472}, {21200, 21201}

X(21212) = (A,B,C,X(2); A',B',C',X(514)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics (b - c) (a^2 - a b - b^2 - a c + b c - c^2) : :

X(21212) lies on these lines: {2, 16892}, {241, 514}, {513, 13246}, {522, 3837}, {661, 4453}, {693, 4359}, {812, 17069}, {824, 4885}, {1491, 4458}, {2530, 4142}, {2786, 3835}, {3700, 4928}, {3798, 4785}, {4024, 14475}, {4467, 4728}, {4739, 4777}, {4750, 20295}, {4818, 7662}, {4927, 4976}, {6545, 17494}, {8061, 21194}, {18071, 19804}

X(21213) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(10316), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^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 - 2 a^2 b^2 c^2 + b^4 c^2 - a^2 c^4 + b^2 c^4 + c^6) : :

X(21213) lies on these lines: {2, 3}, {6, 22075}, {66, 20987}, {98, 1288}, {161, 1899}, {184, 15577}, {232, 571}, {570, 9609}, {1112, 15462}, {1180, 10312}, {1609, 16318}, {1611, 2079}, {1843, 5157}, {1974, 3313}, {2916, 7716}, {2917, 9786}, {3053, 3162}, {3060, 19128}, {3092, 9683}, {3425, 19189}, {5012, 6403}, {5889, 8907}, {7295, 14975}, {8193, 12135}, {10985, 14806}, {19118, 22151}

X(21213) = isogonal conjugate of X(18124)
X(21213) = circumcircle-inverse of X(37981)
X(21213) = polar-circle-inverse of complement of X(37978)
X(21213) = Euler line intercept, other than X(26), of circle {{X(26),PU(4)}}

X(21214) = X(1)X(2) ∩ X(56)X(87)

Barycentrics a*((b+c)*(a^2-b*c)+(b^2-3*b*c+c^2)*a) : :

X(21214) = X(1)+2*X(17749), X(7991)+2*X(14261)

See K1063

X(21214) lies on the cubics K078 and K1063, and one these lines: {1,2}, {3,8616}, {9,1050}, {21,3551}, {31,5253}, {36,1044}, {40,1054}, {41,16779}, {56,87}, {65,17063}, {72,3976}, {106,979}, {171,1191}, {244,3869}, {269,17081}, {361,1130}, {392,986}, {404,3550}, {474,5255}, {500,3653}, {503,12523}, {726,19582}, {748,2975}, {846,988}, {902,4188}, {946,17889}, {958,17123}, {960,982}, {999,5247}, {1001,1740}, {1042,5265}, {1046,3338}, {1376,1616}, {1432,18786}, {1450,3485}, {1457,7288}, {1468,16468}, {1475,1743}, {1575,3208}, {1695,10476}, {1699,15971}, {1707,3361}, {1716,7290}, {1724,5563}, {1738,12053}, {1739,5697}, {1742,7963}, {2176,16604}, {2292,17591}, {2319,21001}, {3057,16610}, {3061,3290}, {3073,10269}, {3230,3501}, {3304,4383}, {3445,12513}, {3510,14823}, {3576,9840}, {3680,13541}, {3684,16781}, {3685,16571}, {3702,20892}, {3749,5438}, {3750,4255}, {3885,4695}, {3890,4642}, {3953,5692}, {3962,3999}, {4051,16605}, {4426,9259}, {4646,10179}, {4694,5904}, {4719,17592}, {4859,11522}, {4888,17169}, {5250,17596}, {5260,17125}, {5284,10448}, {5289,17054}, {5573,15829}, {5710,17122}, {5836,16602}, {5886,15973}, {7991,8056}, {8688,11194}, {9355,10085}, {11110,18169}, {12526,18193}, {16752,17182}, {17187,17588}

X(21214) = midpoint of X(1) and X(6048)
X(21214) = Gibert-circumtangential conjugate of X(3550)
X(21214) = circumtangential-isogonal conjugate of X(8715)
X(21214) = X(1075)-of-2nd-circumperp-triangle
X(21214) = X(6048)-of-anti-Aquila-triangle
X(21214) = X(14059)-of-hexyl-triangle
X(21214) = X(14249)-of-excentral-triangle

X(21215) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^9 b - 2 a^5 b^5 + a b^9 + a^9 c + a^8 b c - a b^8 c - b^9 c - 2 a^5 c^5 + 2 b^5 c^5 - a b c^8 + a c^9 - b c^9 : :

X(21215) lies on these lines:

X(21215) = isotomic conjugate of isogonal conjugate of X(21774)
X(21215) = anticomplement of isogonal conjugate of X(2172)
X(21215) = anticomplement of isotomic conjugate of X(1760)
X(21215) = anticomplement of anticomplement of X(16582)
X(21215) = anticomplement of X(6)-isoconjugate of X(22)
X(21215) = anticomplementary conjugate of X(17492)
X(21215) = polar conjugate of isogonal conjugate of X(23074)

X(21216) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^3 b + a b^3 + a^3 c + a^2 b c - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3 : :

X(21216) lies on these lines:

X(21216) = isotomic conjugate of isogonal conjugate of X(21775)
X(21216) = anticomplement of X(304)
X(21216) = anticomplementary conjugate of anticomplement of X(1973)
X(21216) = polar conjugate of isogonal conjugate of X(23075)
X(21216) = isogonal conjugate of X(63)-cross conjugate of X(6)
X(21216) = {X(3177),X(3210)}-harmonic conjugate of X(194)

X(21217) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics -a^3 b^3 + a^4 b c - a^3 b^2 c + a^2 b^3 c - a b^4 c - a^3 b c^2 - a b^3 c^2 - a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4 : :

X(21217) lies on these lines:

X(21217) = isotomic conjugate of isogonal conjugate of X(21777)
X(21217) = polar conjugate of isogonal conjugate of X(23077)
X(21217) = anticomplement of X(3112)
X(21217) = anticomplementary conjugate of X(21278)

X(21218) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 - 2 a^3 c^3 + 2 b^3 c^3 + a^2 c^4 - b^2 c^4 : :

X(21218) lies on these lines:

X(21218) = isotomic conjugate of isogonal conjugate of X(21778)
X(21218) = polar conjugate of isogonal conjugate of X(23078)
X(21218) = anticomplement of X(6063)
X(21218) = anticomplementary conjugate of X(21280)

X(21219) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^2 b^2 - 2 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - 3 b^2 c^2 : :

X(21219) lies on these lines:

X(21219) = isotomic conjugate of isogonal conjugate of X(21780)
X(21219) = polar conjugate of isogonal conjugate of X(23080)
X(21219) = anticomplement of X(330)
X(21219) = anticomplementary conjugate of X(21281)

X(21220) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics -a^3 b^3 + a^4 b c + a^3 b^2 c - a^2 b^3 c - a b^4 c + a^3 b c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a b c^4 : :

X(21220) lies on these lines:

X(21220) = isotomic conjugate of isogonal conjugate of X(21783)
X(21220) = polar conjugate of isogonal conjugate of X(23083)
X(21220) = anticomplement of X(799)
X(21220) = anticomplementary conjugate of X(17217)

X(21221) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5 : :

X(21221) lies on these lines:

X(21221) = isotomic conjugate of isogonal conjugate of X(21004)
X(21221) = isotomic conjugate of anticomplement of X(39054)
X(21221) = complement of X(31297)
X(21221) = anticomplement of X(662)
X(21221) = anticomplementary conjugate of X(7192)
X(21221) = polar conjugate of isogonal conjugate of X(22156)

X(21222) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^3 + a b^2 - 3 a b c + b^2 c + a c^2 + b c^2) : :

X(21222) lies on these lines:

X(21222) = isotomic conjugate of isogonal conjugate of X(21786)
X(21222) = polar conjugate of isogonal conjugate of X(23087)
X(21222) = anticomplement of X(3762)
X(21222) = anticomplementary conjugate of anticomplement of X(32665)

X(21223) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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 - b^3 c^3 : :

X(21223) lies on these lines:

X(21223) = anticomplement of X(17149)
X(21223) = isotomic conjugate of isogonal conjugate of X(21787)
X(21223) = polar conjugate of isogonal conjugate of X(23088)
X(21223) = anticomplementary conjugate of anticomplement of X(34248)

X(21224) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^3 b^3 - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3 : :

X(21224) lies on these lines:

X(21224) = isotomic conjugate of isogonal conjugate of X(21790)
X(21224) = polar conjugate of isogonal conjugate of X(23091)
X(21224) = anticomplement of X(1978)
X(21224) = anticomplementary conjugate of X(21304)

X(21225) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^3 b + a^2 b^2 - a^3 c + a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21225) lies on these lines:

X(21225) = isotomic conjugate of isogonal conjugate of X(21791)
X(21225) = polar conjugate of isogonal conjugate of X(23093)
X(21225) = anticomplement of X(3261)
X(21225) = anticomplementary conjugate of anticomplement of X(32739)

X(21226) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^2 b^2 - a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2 : :

X(21226) lies on these lines:

X(21226) = isotomic conjugate of isogonal conjugate of X(21792)
X(21226) = polar conjugate of isogonal conjugate of X(23094)
X(21226) = anticomplement of X(1909)
X(21226) = anticomplementary conjugate of anticomplement of X(904)

X(21227) = (name pending)

Barycentrics a^8 b^2-3 a^6 b^4+3 a^4 b^6-a^2 b^8-2 a^7 b^2 c+8 a^6 b^3 c+8 a^5 b^4 c-12 a^4 b^5 c-6 a^3 b^6 c+4 a^2 b^7 c+a^8 c^2-2 a^7 b c^2-9 a^6 b^2 c^2-6 a^5 b^3 c^2+14 a^3 b^5 c^2+7 a^2 b^6 c^2-6 a b^7 c^2+b^8 c^2+8 a^6 b c^3-6 a^5 b^2 c^3+14 a^4 b^3 c^3-8 a^3 b^4 c^3-4 a^2 b^5 c^3-6 a b^6 c^3+2 b^7 c^3-3 a^6 c^4+8 a^5 b c^4-8 a^3 b^3 c^4-12 a^2 b^4 c^4+12 a b^5 c^4-b^6 c^4-12 a^4 b c^5+14 a^3 b^2 c^5-4 a^2 b^3 c^5+12 a b^4 c^5-4 b^5 c^5+3 a^4 c^6-6 a^3 b c^6+7 a^2 b^2 c^6-6 a b^3 c^6-b^4 c^6+4 a^2 b c^7-6 a b^2 c^7+2 b^3 c^7-a^2 c^8+b^2 c^8::

X(21227) lies on the cubic K1063 and the line {8,2841}

X(21228) = REFLECTION OF X(1745) IN X(3)

Barycentrics a (a^8 b+a^7 b^2-3 a^6 b^3-3 a^5 b^4+3 a^4 b^5+3 a^3 b^6-a^2 b^7-a b^8+a^8 c-5 a^7 b c+2 a^6 b^2 c+9 a^5 b^3 c-6 a^4 b^4 c-3 a^3 b^5 c+2 a^2 b^6 c-a b^7 c+b^8 c+a^7 c^2+2 a^6 b c^2-8 a^5 b^2 c^2+3 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^2-3 a^6 c^3+9 a^5 b c^3+3 a^4 b^2 c^3-10 a^3 b^3 c^3+3 a^2 b^4 c^3+a b^5 c^3-3 b^6 c^3-3 a^5 c^4-6 a^4 b c^4+5 a^3 b^2 c^4+3 a^2 b^3 c^4-2 a b^4 c^4+3 b^5 c^4+3 a^4 c^5-3 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+2 a^2 b c^6+2 a b^2 c^6-3 b^3 c^6-a^2 c^7-a b c^7-b^2 c^7-a c^8+b c^8)::
X(21228) = 3 X[3576] - 2 X[10571]

X(21228) lies on the cubic K1063 and these lines: {1,1361}, {3,1745}, {4,14058}, {8,20}, {102,5450}, {603,7412}, {1394,3576}, {3220,6210}, {7335,14925}

X(21228) = reflection of X(i) in X(j) for these {i,j}: {{4, 14058}, {1745, 3}}.

X(21229) = X(10)X(3667)∩X(40)X(1054)

Barycentrics a (2 a^6 b^3+3 a^5 b^4-2 a^4 b^5-4 a^3 b^6+a b^8-a^6 b^2 c-11 a^5 b^3 c-4 a^4 b^4 c+16 a^3 b^5 c+5 a^2 b^6 c-5 a b^7 c-a^6 b c^2+2 a^5 b^2 c^2+25 a^4 b^3 c^2-16 a^3 b^4 c^2-25 a^2 b^5 c^2+12 a b^6 c^2-b^7 c^2+2 a^6 c^3-11 a^5 b c^3+25 a^4 b^2 c^3-42 a^3 b^3 c^3+36 a^2 b^4 c^3+3 a b^5 c^3-b^6 c^3+3 a^5 c^4-4 a^4 b c^4-16 a^3 b^2 c^4+36 a^2 b^3 c^4-30 a b^4 c^4+2 b^5 c^4-2 a^4 c^5+16 a^3 b c^5-25 a^2 b^2 c^5+3 a b^3 c^5+2 b^4 c^5-4 a^3 c^6+5 a^2 b c^6+12 a b^2 c^6-b^3 c^6-5 a b c^7-b^2 c^7+a c^8)::

X(21229) lies on the cubic K1063 and these lines: {10,3667}, {40,1054}, {2743,12029}

X(21230) = COMPLEMENT OF X(195)

Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) : :
X(21230) = 5 X[632] - 2 X[1493], 2 X[140] + X[3519], 3 X[5] - 2 X[3574], 3 X[1209] - X[3574], 5 X[632] - 4 X[6689], 3 X[3] - X[12254], 3 X[2888] + X[12254], 3 X[2] + X[12325], 2 X[8254] + X[12325], 3 X[51] - 4 X[13365], 3 X[5] - 4 X[13565], 3 X[1209] - 2 X[13565], 4 X[3574] - 3 X[20424], 8 X[13565] - 3 X[20424], 4 X[1209] - X[20424]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28057.

Let U be the circle with center X(5) and pass-through point A, and let meets the perpendicular bisector of segment BC in two points: one of them is the U-antipode of A and the other we denote by A'. Define B' and C' cyclically. Then X(21230) is the circumcenter of triangle A'B'C'. (Angel Montesdeoca, March 27, 2019)

X(21230) lies on these lines: {2,195}, {3,2888}, {4,12307}, {5,51}, {10,5885}, {11,6286}, {12,7356}, {30,6288}, {54,140}, {93,14106}, {110,10203}, {141,575}, {235,12300}

X(21230) = midpoint of X(i) and X(j) for these {i,j}: {3,2888}, {4,12307}, {54,3519}, {195,12325}, {3448,5898}, {6101,13368}, {6288,7691}
X(21230) = reflection of X(i) in X(j) for these {i,j}: {5,1209}, {54,140}, {195,8254}, {1493,6689}, {3574,13565}, {6102,11802}, {10263,11808}, {11702,5972}, {12316,11803}, {14071,13372}, {15800,546}, {19150,3589}, {20424,5}
X(21230) = anticomplement X(8254)
X(21230) = complement X(195)
X(21230) = X(3459)-complementary conjugate of X(10)
X(21230) = X(2148)-isoconjugate of X(11538)
X(21230) = crosspoint of X(95) and X(11140)
X(21230) = crosssum of X(51) and X(2965)
X(21230) = X(12307)-of-Euler-triangle
X(21230) = X(16117)-of-orthic-triangle
X(21230) = barycentric product X(i)*X(j) for these {i,j}: {5, 15108}, {311, 15109}, {343, 6143}, {11140, 15345}
X(21230) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 11538}, {6143, 275}, {15108, 95}, {15109, 54}, {15345, 1994}
X(21230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 195, 8254), (2, 12325, 195), (1209, 3574, 13565), (3574, 13565, 5)

X(21231) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 - 2 b^2 c^2 + b c^3) : :

X(21231) lies on these lines: {2, 1953}, {10, 4523}, {37, 16609}, {63, 20930}, {71, 1441}, {72, 15669}, {140, 17043}, {141, 21232}, {142, 3754}, {226, 21853}, {594, 20692}, {828, 1214}, {857, 21011}, {1215, 21865}, {2245, 4032}, {3739, 5836}, {3913, 4361}, {3946, 4868}, {4000, 4642}, {4466, 21012}, {4674, 4859}, {5690, 16608}, {7278, 18164}, {8062, 9249}, {9710, 21239}, {16580, 16603}, {16713, 21272}, {17045, 17048}, {21238, 21254}

X(21231) = isotomic conjugate of isogonal conjugate of X(23621)
X(21231) = polar conjugate of isogonal conjugate of X(22394)
X(21231) = complement of X(1953)
X(21231) = complementary conjugate of complement of X(2167)

X(21232) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(21232) lies on these lines: {1, 17048}, {2, 2170}, {9, 3732}, {10, 116}, {36, 6647}, {37, 21138}, {75, 4595}, {85, 3501}, {100, 9317}, {141, 21231}, {142, 1145}, {190, 18159}, {517, 20335}, {620, 4369}, {644, 9318}, {668, 17755}, {891, 4928}, {1018, 1111}, {1086, 21888}, {1146, 16593}, {1930, 4095}, {2087, 14759}, {2802, 17761}, {3035, 6366}, {3039, 4422}, {3208, 3673}, {3452, 5514}, {3729, 20925}, {3752, 16614}, {3754, 17758}, {3814, 5074}, {3912, 5977}, {3985, 6381}, {4361, 8168}, {5836, 6706}, {7200, 20331}, {8256, 21258}, {10015, 16578}, {10039, 17062}, {10914, 20257}, {14439, 21139}, {16284, 21384}, {18589, 21244}

X(21232) = isotomic conjugate of isogonal conjugate of X(23622)
X(21232) = polar conjugate of isogonal conjugate of X(22399)
X(21232) = complement of X(2170)
X(21232) = complementary conjugate of complement of X(4564)

X(21233) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a - b - c) (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c - b^3 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - b c^3) : :

X(21233) lies on these lines: {2, 2171}, {9, 75}, {10, 8230}, {65, 142}, {141, 21231}, {958, 4361}, {960, 3739}, {986, 4000}, {1215, 9564}, {1229, 1334}, {2170, 16713}, {3210, 4402}, {3452, 5241}, {3666, 3946}, {3691, 21422}, {4032, 16574}, {4357, 16609}, {4395, 18253}, {4657, 17048}, {4999, 17045}, {5289, 15668}, {7208, 18186}, {17046, 18589}, {17185, 20882}, {18698, 20367}

X(21233) = isotomic conjugate of isogonal conjugate of X(23623)
X(21233) = polar conjugate of isogonal conjugate of X(22400)
X(21233) = complement of X(2171)
X(21233) = complementary conjugate of complement of X(2185)

X(21234) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (-a^4 b^2 + b^6 + a^4 b c - b^5 c - a^4 c^2 + a^2 b^2 c^2 + b^4 c^2 - 2 b^3 c^3 + b^2 c^4 - b c^5 + c^6) : :

X(21234) lies on these lines: {2, 21274}, {10, 4523}, {830, 4369}, {858, 18637}, {4466, 21017}, {8287, 16611}, {8680, 21064}

X(21234) = isotomic conjugate of isogonal conjugate of X(23625)
X(21234) = polar conjugate of isogonal conjugate of X(22403)
X(21234) = complement of isogonal conjugate of isotomic conjugate of X(16568)
X(21234) = complement of complement of X(21274)
X(21234) = complementary conjugate of complement of isotomic conjugate of X(16568)

X(21235) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (b^4 - b^3 c + b^2 c^2 - b c^3 + c^4) : :

X(21235) is the center of the inellipse that is the trilinear square of the de Longchamps line. The Brianchon point (perspector) of this inellipse is X(1928). (Randy Hutson, October 15, 2018)

X(21235) lies on these lines: {2, 560}, {10, 16580}, {11, 6028}, {31, 18744}, {141, 9018}, {626, 17047}, {696, 4178}, {744, 4150}, {1760, 4837}, {2880, 21244}, {2887, 20305}, {4412, 4769}, {8287, 21257}, {16894, 20234}, {18589, 21254}

X(21235) = isogonal conjugate of X(38827)
X(21235) = isotomic conjugate of isogonal conjugate of X(23626)
X(21235) = complement of X(560)
X(21235) = polar conjugate of isogonal conjugate of X(22404)
X(21235) = complementary conjugate of X(16584)
X(21235) = crosssum of X(6) and X(1917)
X(21235) = crosspoint of X(2) and X(1928)
X(21235) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 38827}, {1501, 38812}
X(21235) = trilinear product X(i)*X(j) for these {i,j}: {2, 21324}, {6, 21409}, {10, 18168}, {38, 18090}, {75, 23626}
X(21235) = trilinear quotient X(i)/X(j) for these (i, j): (1, 38827), (1502, 38812), (18090, 82), (18168, 58), (21324, 6), (21409, 2), (23626, 31)
X(21235) = barycentric product X(i)*X(j) for these {i,j}: {1, 21409}, {75, 21324}, {76, 23626}, {141, 18090}, {321, 18168}
X(21235) = barycentric quotient X(i)/X(j) for these (i, j): (6, 38827), (561, 38812), (18090, 83), (18168, 81), (21324, 1), (21409, 75), (23626, 6)

X(21236) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (-a^2 b^3 + b^5 - a b^3 c - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 + c^5) : :

X(21236) lies on these lines: {2, 2174}, {10, 22287}, {116, 3739}, {141, 1329}, {3711, 4445}, {4357, 8287}, {4648, 10585}, {5249, 5949}, {16603, 17243}, {17045, 17062}, {21240, 21245}

X(21236) = isotomic conjugate of isogonal conjugate of X(23627)
X(21236) = polar conjugate of isogonal conjugate of X(22405)
X(21236) = complement of X(2174)
X(21236) = complementary conjugate of X(16585)

X(21237) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics -a^2 b^3 + b^5 + a b^3 c - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + c^5 : :

X(21237) lies on these lines: {2, 7113}, {10, 22288}, {116, 3834}, {141, 1329}, {834, 3835}, {3262, 21019}, {3739, 9956}, {3912, 8287}, {4364, 16603}, {5044, 5074}, {6668, 6707}, {20544, 21252}

X(21237) = isotomic conjugate of isogonal conjugate of X(23628)
X(21237) = polar conjugate of isogonal conjugate of X(22406)
X(21237) = complement of X(7113)
X(21237) = complementary conjugate of X(16586)

X(21238) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (a^2 b^2 - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21238) lies on these lines: {2, 1964}, {10, 37}, {38, 18133}, {75, 3123}, {76, 4446}, {141, 9016}, {244, 18143}, {256, 17790}, {291, 3770}, {313, 714}, {744, 4019}, {982, 18144}, {1215, 20723}, {1237, 16889}, {1631, 4112}, {2228, 20891}, {2887, 20305}, {3122, 3963}, {3596, 4443}, {3739, 20340}, {3741, 17239}, {3831, 3844}, {3840, 17231}, {3934, 17049}, {3948, 21035}, {4033, 22167}, {4359, 20598}, {4735, 21080}, {6385, 19567}, {8287, 19563}, {16525, 17275}, {16584, 21878}, {17277, 20663}, {17445, 20352}, {18040, 21330}, {18082, 20964}, {18194, 18793}, {20711, 21020}, {21231, 21254}, {21835, 22186}

X(21238) = isotomic conjugate of isogonal conjugate of X(23629)
X(21238) = polar conjugate of isogonal conjugate of X(22409)
X(21238) = complement of X(1964)
X(21238) = complementary conjugate of X(16587)

X(21239) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics -a^3 b^2 - a^2 b^3 + a b^4 + b^5 + a^2 b^2 c - 2 a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 + c^5 : :

X(21239) lies on these lines: {2, 198}, {5, 142}, {10, 22290}, {11, 4000}, {12, 4648}, {57, 1901}, {116, 117}, {123, 20208}, {141, 1329}, {241, 1826}, {325, 20923}, {427, 1827}, {496, 3946}, {857, 17077}, {1146, 18161}, {1368, 2886}, {1387, 18261}, {2883, 21258}, {3142, 15844}, {3813, 4361}, {3814, 21255}, {3816, 4657}, {3847, 17290}, {4187, 17306}, {4851, 12607}, {4859, 7741}, {4869, 11681}, {5786, 15668}, {6708, 21621}, {6831, 18634}, {7146, 21933}, {7359, 16551}, {7681, 12610}, {9710, 21231}, {9711, 17239}, {10593, 17067}, {17243, 21091}, {17296, 17757}

X(21239) = isotomic conjugate of isogonal conjugate of X(23630)
X(21239) = polar conjugate of isogonal conjugate of X(22410)
X(21239) = complement of X(198)
X(21239) = complementary conjugate of X(223)

X(21240) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics a b^3 + b^3 c + a c^3 + b c^3 : :

X(21240) lies on these lines: {1, 4372}, {2, 213}, {10, 141}, {11, 6022}, {76, 3662}, {116, 3454}, {257, 20924}, {304, 3735}, {320, 17499}, {536, 21070}, {626, 766}, {712, 1930}, {742, 16600}, {1086, 20888}, {1107, 16887}, {1500, 3666}, {1759, 4376}, {1912, 21260}, {2140, 21264}, {2388, 3741}, {3123, 22189}, {3125, 20911}, {3230, 17152}, {3263, 3954}, {3501, 17284}, {3661, 4359}, {3663, 21071}, {3664, 15985}, {3670, 12782}, {3727, 14210}, {3730, 17279}, {3782, 4044}, {3831, 3934}, {3948, 17184}, {4000, 10449}, {4056, 4799}, {4357, 16589}, {4680, 4950}, {4721, 20347}, {4805, 21285}, {5711, 15668}, {6376, 17227}, {6381, 21025}, {12610, 15488}, {16703, 18167}, {16706, 17034}, {17208, 18171}, {17230, 17495}, {17231, 20691}, {20892, 22028}, {21236, 21245}

X(21240) = complement of X(213)
X(21240) = isotomic conjugate of isogonal conjugate of X(23632)
X(21240) = polar conjugate of isogonal conjugate of X(22412)
X(21240) = complementary conjugate of X(16589)

X(21241) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics -a b^2 + 2 b^3 - b^2 c - a c^2 - b c^2 + 2 c^3 : :

X(21241) lies on these lines: {2, 902}, {10, 908}, {11, 3836}, {121, 3259}, {125, 20546}, {126, 20339}, {141, 674}, {512, 625}, {516, 8229}, {518, 4892}, {519, 3936}, {626, 17050}, {726, 3006}, {1125, 4202}, {1215, 3838}, {1699, 4011}, {1836, 4438}, {2796, 3977}, {3008, 4766}, {3011, 17766}, {3434, 3771}, {3705, 7897}, {3712, 17764}, {3717, 21093}, {3772, 4865}, {3823, 5087}, {3840, 11680}, {3846, 3925}, {3914, 4970}, {3944, 3971}, {4085, 5718}, {4138, 4847}, {4358, 21026}, {4362, 17064}, {4417, 4685}, {4429, 17717}, {4576, 17204}, {4972, 6685}, {5300, 8669}, {5846, 17070}, {7239, 20593}, {16706, 17722}, {17724, 17765}

X(21241) = complement of X(902)
X(21241) = isotomic conjugate of isogonal conjugate of X(23633)
X(21241) = polar conjugate of isogonal conjugate of X(22414)
X(21241) = complementary conjugate of X(4370)

X(21242) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (-2 a b^2 + b^3 - 2 b^2 c - 2 a c^2 - 2 b c^2 + c^3) : :

X(21242) lies on these lines: {2, 2177}, {8, 17717}, {10, 11}, {141, 674}, {239, 17722}, {519, 5718}, {537, 4054}, {752, 1150}, {1215, 4847}, {1647, 21027}, {1699, 4703}, {3006, 3773}, {3626, 4023}, {3679, 5233}, {3705, 7777}, {3817, 4104}, {3826, 4871}, {3829, 5743}, {3840, 3925}, {3846, 11680}, {3914, 6682}, {4414, 17764}, {4439, 4671}, {4519, 6541}, {4865, 11679}, {5235, 10707}, {5737, 11235}, {11238, 19732}, {16592, 17448}, {16825, 17721}

X(21242) = isotomic conjugate of isogonal conjugate of X(23634)
X(21242) = polar conjugate of isogonal conjugate of X(22415)
X(21242) = complement of X(2177)
X(21242) = complementary conjugate of X(16590)

X(21243) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :

X(21243) lies on these lines: {2, 98}, {3, 161}, {4, 14860}, {5, 389}, {6, 8280}, {10, 22296}, {22, 11550}, {25, 3818}, {26, 13419}, {51, 3580}, {52, 5576}, {66, 19126}, {68, 578}, {69, 8541}, {76, 5117}, {113, 10254}, {115, 3981}, {129, 20625}, {140, 13561}, {141, 1368}, {154, 18440}, {185, 13160}, {297, 6248}, {324, 6747}, {343, 427}, {381, 17810}, {382, 18442}, {394, 5094}, {403, 15030}, {429, 15488}, {468, 18553}, {472, 20428}, {473, 20429}, {575, 11245}, {576, 6515}, {626, 2387}, {858, 3917}, {1216, 13371}, {1370, 3098}, {1503, 6676}, {1533, 11455}, {1568, 7577}, {1594, 5562}, {1595, 13598}, {1656, 17814}, {1993, 5965}, {2072, 5891}, {2875, 2886}, {2883, 16254}, {2887, 17047}, {2979, 7703}, {3060, 5169}, {3090, 18916}, {3167, 15069}, {3455, 7820}, {3541, 13346}, {3547, 14216}, {3549, 6759}, {3574, 5889}, {3589, 11548}, {3618, 18950}, {3763, 16419}, {3787, 15820}, {3934, 14917}, {4121, 8024}, {4175, 9464}, {4550, 7687}, {5020, 10516}, {5092, 7499}, {5142, 10441}, {5476, 9777}, {5480, 21849}, {6000, 15760}, {6101, 13368}, {6242, 11412}, {6247, 6823}, {6639, 10539}, {6677, 18358}, {6689, 10116}, {7383, 13347}, {7396, 10519}, {7399, 9729}, {7403, 10110}, {7405, 11695}, {7526, 9927}, {7539, 10601}, {7542, 10282}, {7552, 14157}, {7558, 10984}, {7667, 14810}, {7749, 19627}, {7826, 15573}, {8041, 8288}, {8542, 15116}, {9738, 11091}, {9739, 11090}, {9818, 14852}, {9977, 15135}, {10024, 12162}, {10114, 12228}, {10224, 11591}, {11284, 15121}, {11425, 12429}, {11433, 14561}, {11438, 18420}, {11572, 12225}, {11585, 11793}, {11799, 16194}, {12161, 12242}, {12605, 18383}, {13366, 14389}, {13367, 14516}, {13399, 15072}, {13565, 13630}, {14118, 21659}, {14213, 21028}, {14789, 15045}, {15058, 16868}, {15107, 18427}, {15605, 16266}, {17702, 18570}, {17889, 19950}, {18430, 18564}, {19360, 19460}

X(21243) = isotomic conjugate of isogonal conjugate of X(23635)
X(21243) = polar conjugate of isogonal conjugate of X(22416)
X(21243) = complement of X(184)
X(21243) = complementary conjugate of X(216)
X(21243) = X(427)-of-1st-Brocard-triangle

X(21244) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a - b - c) (a b^3 + b^4 - b^3 c + a c^3 - b c^3 + c^4) : :

X(21244) lies on these lines: {2, 604}, {9, 1760}, {10, 8230}, {12, 142}, {116, 21255}, {141, 1329}, {313, 4858}, {958, 17327}, {960, 17239}, {1211, 3452}, {1229, 21044}, {2880, 21235}, {2887, 17047}, {3036, 4399}, {3739, 5123}, {4357, 16603}, {4417, 17296}, {4445, 5289}, {6376, 20647}, {8287, 17231}, {16608, 20335}, {17062, 17306}, {18589, 21232}, {20258, 20341}, {20541, 20545}, {20895, 21030}

X(21244) = isotomic conjugate of isogonal conjugate of X(23637)
X(21244) = polar conjugate of isogonal conjugate of X(22418)
X(21244) = complement of X(604)
X(21244) = complementary conjugate of X(3752)

X(21245) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (a b^3 + b^4 + a c^3 + c^4) : :

X(21245) lies on these lines: {2, 1333}, {5, 141}, {10, 20713}, {37, 4150}, {119, 127}, {325, 16696}, {442, 3739}, {857, 17279}, {908, 1211}, {1213, 17353}, {1228, 16886}, {1234, 16732}, {1834, 4852}, {1901, 17351}, {2887, 20305}, {3136, 21264}, {3314, 18144}, {3834, 18635}, {3936, 4851}, {4137, 20655}, {4657, 5051}, {7237, 16894}, {7778, 19544}, {8287, 20343}, {16052, 17382}, {17211, 18179}, {18697, 20654}, {20486, 21249}, {20542, 21252}, {21236, 21240}

X(21245) = isotomic conjugate of isogonal conjugate of X(23639)
X(21245) = polar conjugate of isogonal conjugate of X(22420)
X(21245) = complement of X(1333)
X(21245) = complementary conjugate of X(3666)

X(21246) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a - b - c) (a^2 b^2 + a b^3 + b^3 c + a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21246) lies on these lines: {2, 7}, {10, 3781}, {123, 18642}, {124, 126}, {141, 1329}, {304, 1921}, {314, 3687}, {958, 15668}, {960, 3739}, {978, 4000}, {992, 3008}, {1193, 3946}, {1210, 10477}, {1368, 2887}, {2269, 17183}, {2277, 3663}, {2321, 4494}, {2385, 6389}, {2551, 4648}, {3036, 4478}, {3142, 17052}, {3664, 4503}, {3686, 17197}, {3691, 16713}, {3751, 21075}, {3786, 6734}, {3886, 12053}, {3923, 12610}, {3944, 16571}, {4361, 5289}, {4858, 18697}, {4869, 8165}, {4999, 6707}, {5123, 17239}, {6700, 19513}, {9245, 17072}, {10456, 10478}, {12572, 13731}, {15479, 16832}, {15829, 20257}, {16591, 16596}, {17452, 20895}, {17758, 18250}, {20545, 21264}

X(21246) = isotomic conjugate of isogonal conjugate of X(23640)
X(21246) = polar conjugate of isogonal conjugate of X(22421)
X(21246) = complement of X(1400)
X(21246) = complementary conjugate of X(17056)

X(21247) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b+c) (-a^4+b^4+c^4) (a^4+b^4-2 b^3 c+2 b^2 c^2-2 b c^3+c^4) : :

X(21247) lies on these lines: {2, 2156}, {10, 22302}, {206, 17068}, {226, 16582}, {1215, 20305}, {16580, 16584}, {16591, 18588}

X(21247) = isotomic conjugate of isogonal conjugate of X(23641)
X(21247) = polar conjugate of isogonal conjugate of X(22422)
X(21247) = complement of X(2156)
X(21247) = complementary conjugate of X(16580)

X(21248) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b^2 + c^2) (a^2 b^2 + b^4 + a^2 c^2 + c^4) : :

X(21248) lies on these lines: {2, 32}, {10, 22303}, {22, 7761}, {25, 7784}, {127, 6676}, {141, 427}, {305, 3314}, {316, 16950}, {384, 16275}, {620, 15246}, {670, 1241}, {858, 7849}, {1180, 4045}, {1184, 7866}, {1194, 6656}, {1196, 7853}, {1370, 7795}, {3005, 11123}, {3094, 4121}, {3620, 6338}, {3734, 7391}, {3739, 3925}, {3788, 7485}, {3933, 19568}, {3934, 5133}, {5359, 7834}, {6389, 7386}, {6636, 7830}, {6655, 16276}, {7394, 7825}, {7484, 7778}, {7539, 15271}, {7667, 7789}, {7747, 16932}, {7765, 8267}, {7794, 8024}, {7820, 16949}, {7832, 16951}, {7868, 11324}, {7869, 16063}, {8041, 16893}, {15437, 16045}, {17055, 17797}, {20898, 21037}

X(21248) = isotomic conjugate of isogonal conjugate of X(23642)
X(21248) = polar conjugate of isogonal conjugate of X(22424)
X(21248) = complement of X(251)
X(21248) = complementary conjugate of X(3589)

X(21249) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (b^2 + c^2) (a^2 + b^2 - b c + c^2) : :

X(21249) lies on these lines: {2, 82}, {10, 16580}, {37, 744}, {141, 3688}, {960, 3844}, {1125, 1279}, {3454, 3932}, {3739, 3925}, {3846, 17357}, {4412, 9857}, {4514, 4972}, {4679, 17279}, {8287, 19563}, {17192, 18183}, {17456, 21037}, {20486, 21245}

X(21249) = isotomic conjugate of isogonal conjugate of X(20969)
X(21249) = polar conjugate of isogonal conjugate of X(22077)
X(21249) = complement of X(82)
X(21249) = complementary conjugate of X(1215)

X(21250) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a b + a c - b c) (a b^3 - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(21250) lies on these lines: {2, 1977}, {9, 20608}, {10, 22305}, {141, 3816}, {257, 312}, {2886, 20542}, {2887, 20255}, {3925, 5518}, {6382, 21040}, {20541, 20545}

X(21250) = isotomic conjugate of isogonal conjugate of X(23643)
X(21250) = polar conjugate of isogonal conjugate of X(22427)
X(21250) = complement of X(2162)
X(21250) = complementary conjugate of X(75)

X(21251) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a - 2 b - 2 c) (2 a b^2 + 2 b^3 - b^2 c + 2 a c^2 - b c^2 + 2 c^3) : :

X(21251) lies on these lines: {2, 2163}, {10, 908}, {121, 2887}, {141, 3814}, {519, 21088}, {551, 5718}, {1213, 15492}, {3822, 5241}, {4793, 21042}

X(21251) = isotomic conjugate of isogonal conjugate of X(23645)
X(21251) = polar conjugate of isogonal conjugate of X(22429)
X(21251) = complement of X(2163)
X(21251) = complementary conjugate of X(551)

X(21252) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c)^2 (-a b^2 + b^3 - a b c + b^2 c - a c^2 + b c^2 + c^3) : :

X(21252) lies on these lines: {2, 692}, {5, 2807}, {10, 22309}, {11, 125}, {116, 926}, {124, 8677}, {141, 2876}, {942, 15116}, {1015, 15449}, {1086, 21210}, {2870, 3739}, {2875, 2886}, {2877, 2887}, {2878, 21253}, {3259, 15614}, {3763, 20468}, {4081, 4777}, {4092, 4957}, {4466, 17463}, {4858, 21045}, {6697, 16608}, {13567, 14717}, {20542, 21245}, {20544, 21237}, {20902, 21340}

X(21252) = isotomic conjugate of isogonal conjugate of X(23646)
X(21252) = polar conjugate of isogonal conjugate of X(22432)
X(21252) = complement of X(692)
X(21252) = complementary conjugate of X(650)

X(21253) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c)^2 (b + c) (-a^2 b^2 + b^4 - a^2 b c + b^3 c - a^2 c^2 + b^2 c^2 + b c^3 + c^4) : :

X(21253) lies on these lines: {2, 163}, {10, 22310}, {115, 21138}, {116, 125}, {124, 127}, {339, 21429}, {1577, 17880}, {2878, 21252}, {3454, 20540}, {7332, 8287}, {8286, 17761}, {14838, 16573}, {17886, 21046}

X(21253) = isotomic conjugate of isogonal conjugate of X(23647)
X(21253) = polar conjugate of isogonal conjugate of X(22433)
X(21253) = complement of X(163)
X(21253) = complementary conjugate of X(14838)

X(21254) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (a^4 - 2 a^2 b c + b^3 c - b^2 c^2 + b c^3) : :

X(21254) is the center of the inellipse that is the trilinear square of line X(2)X(6). The Brianchon point (perspector) of this inellipse is X(24037). (Randy Hutson, November 30, 2018)

X(21254) lies on these lines: {1, 18805}, {2, 2643}, {10, 8287}, {37, 19563}, {75, 4094}, {523, 4422}, {740, 1279}, {744, 3747}, {1109, 18151}, {3739, 20339}, {4368, 16732}, {4458, 6370}, {4585, 21756}, {16527, 17119}, {16592, 21887}, {18589, 21235}, {21231, 21238}

X(21254) = isotomic conjugate of isogonal conjugate of X(23648)
X(21254) = complement of X(2643)
X(21254) = complementary conjugate of X(24040)
X(21254) = polar conjugate of isogonal conjugate of X(22434)

X(21255) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics a b - 3 b^2 + a c + 2 b c - 3 c^2 : :

X(21255) lies on these lines: {1, 4869}, {2, 1743}, {7, 17284}, {8, 4859}, {10, 141}, {57, 20106}, {69, 3008}, {75, 4058}, {116, 21244}, {192, 3662}, {226, 1122}, {277, 6743}, {306, 17495}, {312, 4052}, {313, 18150}, {320, 17283}, {344, 17274}, {346, 4862}, {519, 4000}, {524, 17356}, {527, 7232}, {536, 4072}, {551, 4657}, {599, 3686}, {894, 4896}, {966, 20195}, {1086, 2321}, {1111, 1229}, {1125, 4349}, {1266, 4764}, {2325, 17267}, {2345, 6173}, {2835, 18589}, {2887, 11019}, {3123, 22214}, {3244, 3946}, {3454, 9843}, {3589, 4667}, {3619, 10436}, {3620, 4384}, {3625, 4361}, {3631, 17348}, {3632, 4402}, {3661, 4772}, {3707, 17337}, {3729, 4887}, {3752, 4035}, {3755, 4966}, {3763, 4675}, {3771, 10164}, {3814, 21239}, {3817, 3840}, {3821, 4356}, {3831, 3947}, {3879, 16706}, {3986, 4357}, {4021, 17304}, {4029, 17246}, {4060, 17119}, {4098, 17235}, {4101, 17674}, {4301, 12610}, {4371, 4701}, {4389, 17241}, {4395, 17372}, {4398, 17240}, {4422, 15828}, {4429, 4684}, {4431, 4821}, {4440, 17268}, {4445, 4669}, {4454, 4902}, {4464, 17386}, {4480, 17339}, {4643, 6666}, {4656, 17184}, {4741, 17338}, {4856, 5222}, {4871, 9309}, {4888, 5749}, {4967, 17228}, {5232, 16832}, {5257, 17237}, {6381, 20923}, {6646, 17266}, {6700, 18634}, {7238, 17351}, {7263, 17229}, {7321, 17285}, {15668, 16457}, {17023, 17291}, {17060, 21629}, {17133, 17309}, {17236, 17244}, {17263, 17273}, {17270, 21356}, {17301, 17311}, {17302, 17312}, {17303, 21358}, {17305, 17317}, {17341, 17347}, {17352, 17361}, {17357, 17365}, {17366, 17374}, {17367, 17375}, {17370, 17378}, {17380, 17387}, {17382, 17390}, {17383, 17391}, {17384, 17392}, {17385, 20582}

X(21255) = complement of X(1743)
X(21255) = isotomic conjugate of isogonal conjugate of X(23649)
X(21255) = polar conjugate of isogonal conjugate of X(22435)
X(21255) = complementary conjugate of complement of X(8056)

X(21256) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (-a^2 b^2 + 2 b^4 + a^2 b c - 2 b^3 c - a^2 c^2 + b^2 c^2 - 2 b c^3 + 2 c^4) : :

X(21256) lies on these lines: {2, 922}, {10, 16581}, {788, 20549}, {896, 18745}, {2887, 20305}, {20912, 21048}

X(21256) = isotomic conjugate of isogonal conjugate of X(23651)
X(21256) = polar conjugate of isogonal conjugate of X(22438)
X(21256) = complement of X(922)
X(21256) = complementary conjugate of complement of isogonal conjugate of X(922)
X(21256) = complementary conjugate of complement of isotomic conjugate of X(896)

X(21257) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (a^2 b^2 - 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21257) lies on these lines: {2, 1740}, {5, 3836}, {10, 37}, {71, 4368}, {75, 4941}, {76, 17065}, {141, 3816}, {226, 19564}, {313, 3122}, {321, 20711}, {385, 16800}, {714, 21095}, {726, 4446}, {730, 17053}, {1269, 20598}, {2228, 18137}, {3009, 21278}, {3123, 20892}, {3686, 16525}, {3741, 5224}, {3764, 18147}, {3778, 3948}, {3963, 22172}, {3971, 21035}, {4022, 18133}, {4090, 20723}, {4871, 17234}, {8287, 21235}, {12263, 17049}, {14823, 18194}, {16515, 17275}, {18589, 19563}, {21100, 22214}, {21835, 22218}, {22028, 22189}

X(21257) = isotomic conjugate of isogonal conjugate of X(23652)
X(21257) = polar conjugate of isogonal conjugate of X(22439)
X(21257) = complement of X(1740)
X(21257) = complementary conjugate of complement of X(3223)

X(21258) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :
X(21258) = 3 X(2) - X(220)

X(21258) lies on these lines: {1, 4904}, {2, 220}, {5, 116}, {7, 6554}, {10, 141}, {30, 14377}, {85, 1146}, {150, 17682}, {169, 5845}, {277, 18391}, {496, 17761}, {536, 21096}, {550, 17729}, {938, 1834}, {1062, 17045}, {1086, 3673}, {1212, 9436}, {1329, 20335}, {1446, 20905}, {2389, 2886}, {2883, 21239}, {2884, 7956}, {3123, 22219}, {3306, 7131}, {3501, 16593}, {3665, 17451}, {3742, 18214}, {3813, 20257}, {3912, 4515}, {3946, 6744}, {3991, 17243}, {4422, 8257}, {4657, 18634}, {4851, 6765}, {4882, 17296}, {5249, 13567}, {5437, 17284}, {5439, 18636}, {5690, 20328}, {5706, 15668}, {5743, 16832}, {5806, 12610}, {6703, 20266}, {8256, 21232}, {9317, 10950}, {17671, 17747}, {21346, 21931}

X(21258) = isotomic conjugate of isogonal conjugate of X(23653)
X(21258) = polar conjugate of isogonal conjugate of X(22440)
X(21258) = complement of X(220)
X(21258) = complementary conjugate of X(6554)

X(21259) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c) (a^4 b^2 - a^2 b^4 + a^4 b c - a^2 b^3 c + a^4 c^2 - a^2 b^2 c^2 + b^4 c^2 - a^2 b c^3 + 2 b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(21259) lies on these lines: {2, 810}, {10, 3907}, {333, 21761}, {521, 8062}, {788, 20549}, {1010, 22093}, {1577, 10479}, {2049, 7254}, {4369, 8678}, {5737, 21789}, {10449, 17478}, {17899, 21050}

X(21259) = isotomic conjugate of isogonal conjugate of X(23654)
X(21259) = polar conjugate of isogonal conjugate of X(22441)
X(21259) = complement of X(810)
X(21259) = complementary conjugate of X(16573)

X(21260) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c) (-a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

X(21260) lies on these lines: {2, 667}, {5, 3309}, {10, 4083}, {11, 4162}, {12, 3669}, {116, 9320}, {141, 9010}, {427, 18344}, {512, 625}, {513, 3814}, {514, 3837}, {523, 4823}, {663, 15283}, {693, 4705}, {764, 4462}, {784, 1491}, {788, 20549}, {802, 17990}, {814, 14838}, {826, 4522}, {830, 4874}, {832, 8062}, {891, 4147}, {905, 2787}, {1015, 5518}, {1329, 20317}, {1698, 4063}, {1734, 4010}, {1912, 21240}, {2530, 4391}, {2533, 14349}, {3250, 21053}, {3634, 4782}, {3716, 6004}, {3762, 3777}, {3826, 6008}, {3900, 15280}, {4041, 4728}, {4193, 6161}, {4490, 4978}, {4500, 6367}, {4775, 21302}, {4776, 4983}, {4806, 6005}, {4834, 20295}, {4885, 8678}, {4905, 7951}, {5142, 17924}, {6371, 20316}, {6373, 21191}, {17458, 21055}, {20947, 21613}, {21056, 21836}, {21099, 21348}

X(21260) = complement of X(667)
X(21260) = anticomplement of X(31288)
X(21260) = isotomic conjugate of isogonal conjugate of X(20983)
X(21260) = polar conjugate of isogonal conjugate of X(22095)
X(21260) = complementary conjugate of X(1015)

X(21261) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c) (-a b^3 + a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(21261) lies on these lines: {2, 8632}, {10, 22319}, {11, 116}, {512, 625}, {513, 21262}, {830, 4369}, {3716, 20341}, {3766, 21053}, {3768, 5224}, {4491, 17327}, {4500, 4823}, {14408, 17248}, {20949, 21055}, {20981, 21304}

X(21261) = isotomic conjugate of isogonal conjugate of X(23656)
X(21261) = polar conjugate of isogonal conjugate of X(22444)
X(21261) = complement of X(8632)
X(21261) = complementary conjugate of X(35119)

X(21262) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c) (-a b^3 - a b^2 c + b^3 c - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(21262) lies on these lines: {2, 1919}, {10, 22322}, {141, 21191}, {513, 21261}, {768, 8061}, {788, 20549}, {832, 8060}, {834, 3835}, {3261, 21053}, {3661, 17458}, {4057, 17327}, {5224, 20979}, {17308, 21389}, {20906, 21055}, {21350, 21726}

X(21262) = isotomic conjugate of isogonal conjugate of X(23657)
X(21262) = polar conjugate of isogonal conjugate of X(22449)
X(21262) = complement of X(1919)
X(21262) = complementary conjugate of X(6377)

X(21263) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (b - c) (-a^2 b^4 - a^2 b^3 c - a^2 b^2 c^2 + b^4 c^2 - a^2 b c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(21263) lies on these lines: {2, 1924}, {10, 22324}, {788, 20549}, {830, 4369}

X(21263) = isotomic conjugate of isogonal conjugate of X(23658)
X(21263) = polar conjugate of isogonal conjugate of X(22446)
X(21263) = complement of X(1924)
X(21263) = complementary conjugate of complement of X(4602)

X(21264) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics (a^2 b^2 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(21264) lies on these lines: {2, 37}, {6, 17026}, {9, 4713}, {10, 3934}, {39, 20888}, {42, 4852}, {43, 4361}, {44, 17028}, {76, 1107}, {141, 674}, {142, 3840}, {172, 17686}, {183, 4386}, {239, 21904}, {274, 4602}, {308, 16606}, {310, 16696}, {385, 20179}, {668, 16829}, {672, 17351}, {993, 3734}, {1008, 1104}, {1100, 17027}, {1376, 15271}, {1386, 17031}, {1418, 7196}, {1475, 4754}, {1573, 6381}, {1740, 15668}, {1909, 9263}, {1964, 3720}, {2140, 21240}, {2235, 4670}, {2238, 17348}, {3136, 21245}, {3510, 18170}, {3696, 3783}, {3723, 17032}, {3760, 5283}, {3761, 16975}, {3831, 17050}, {3946, 6685}, {4093, 21020}, {4363, 17754}, {4368, 15254}, {4384, 16514}, {4396, 5276}, {4399, 4685}, {4426, 7770}, {4492, 17290}, {4721, 16552}, {4851, 10453}, {4999, 7789}, {5267, 7816}, {5332, 17002}, {9055, 21101}, {10472, 17064}, {16666, 17029}, {16720, 20880}, {16819, 18140}, {17045, 17061}, {17135, 17372}, {17143, 20691}, {17149, 18144}, {17284, 19584}, {17345, 20347}, {20174, 21857}, {20545, 21246}

X(21264) = isotomic conjugate of isogonal conjugate of X(23660)
X(21264) = polar conjugate of isogonal conjugate of X(22445)
X(21264) = complement of X(2276)
X(21264) = complementary conjugate of complement of X(14621)

X(21265) = (name pending)

Barycentrics a^20 (b^2+c^2)-(b^2-c^2)^10 (b^2+c^2)-a^18 (7 b^4+10 b^2 c^2+7 c^4)+10 a^16 (2 b^6+3 b^4 c^2+3 b^2 c^4+2 c^6)+11 a^6 (b^2-c^2)^4 (5 b^8+10 b^6 c^2+12 b^4 c^4+10 b^2 c^6+5 c^8)+a^2 (b^2-c^2)^6 (8 b^8-3 b^6 c^2-11 b^4 c^4-3 b^2 c^6+8 c^8)-a^14 (27 b^8+28 b^6 c^2+26 b^4 c^4+28 b^2 c^6+27 c^8)+a^12 (7 b^10-24 b^8 c^2-34 b^6 c^4-34 b^4 c^6-24 b^2 c^8+7 c^10)-a^4 (b^2-c^2)^4 (28 b^10-b^8 c^2-28 b^6 c^4-28 b^4 c^6-b^2 c^8+28 c^10)-a^8 (b^2-c^2)^2 (63 b^10+111 b^8 c^2+128 b^6 c^4+128 b^4 c^6+111 b^2 c^8+63 c^10)+a^10 (35 b^12+55 b^10 c^2+36 b^8 c^4+36 b^6 c^6+36 b^4 c^8+55 b^2 c^10+35 c^12) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28059.

X(21265) lies on this line: {5, 51}

X(21265) = isogonal conjugate of X(5) wrt its cevian triangle

X(21266) = 7th HUNG-LOZADA-EULER POINT

Barycentrics a^2*(-16*S^3*(a+b+c)*(-a^2+b^ 2+c^2)*(a^2-b^2+c^2)*(a^2+b^2- c^2)*(a^4-b^4-4*b^2*c^2-c^4)+ a^17+(b+c)*a^16-10*(b^2+c^2)* a^15-2*(b+c)*(5*b^2-4*b*c+5*c^ 2)*a^14+2*(15*b^4+29*b^2*c^2+ 15*c^4)*a^13+2*(b+c)*(15*b^4+ 15*c^4-(16*b^2-53*b*c+16*c^2)* b*c)*a^12-2*(b^2+c^2)*(17*b^4+ 32*b^2*c^2+17*c^4)*a^11-2*(b+ c)*(17*b^6+17*c^6-(20*b^4+20* c^4-(113*b^2-92*b*c+113*c^2)* b*c)*b*c)*a^10+2*(49*b^4-6*b^ 2*c^2+49*c^4)*b^2*c^2*a^9+2*( b+c)*(73*b^4+73*c^4-2*(74*b^2- 121*b*c+74*c^2)*b*c)*b^2*c^2* a^8+2*(b^2+c^2)*(17*b^8+17*c^ 8-2*(40*b^4-67*b^2*c^2+40*c^4) *b^2*c^2)*a^7+2*(b+c)*(17*b^ 10+17*c^10-(20*b^8+20*c^8-(b^ 6+c^6+2*(28*b^4+28*c^4-(85*b^ 2-124*b*c+85*c^2)*b*c)*b*c)*b* c)*b*c)*a^6-2*(b^2-c^2)^2*(15* b^8+15*c^8-(25*b^4+108*b^2*c^ 2+25*c^4)*b^2*c^2)*a^5-2*(b^2- c^2)*(b-c)*(15*b^10+15*c^10+( 14*b^8+14*c^8+(14*b^6+14*c^6-( 26*b^4+26*c^4+(61*b^2-24*b*c+ 61*c^2)*b*c)*b*c)*b*c)*b*c)*a^ 4+2*(b^4-c^4)*(b^2-c^2)*(5*b^ 8+5*c^8-2*(3*b^4+35*b^2*c^2+3* c^4)*b^2*c^2)*a^3+2*(b^2-c^2)^ 2*(b+c)*(5*b^10+5*c^10-(4*b^8+ 4*c^8+(b^6+c^6+4*(7*b^4+7*c^4- (5*b^2-16*b*c+5*c^2)*b*c)*b*c) *b*c)*b*c)*a^2-(b^4-c^4)^2*(b^ 2-c^2)^2*(b^4+12*b^2*c^2+c^4)* a+(b^2-c^2)^4*(b+c)*(-b^8-c^8+ 2*(b^4+c^4-(4*b^2-11*b*c+4*c^ 2)*b*c)*b^2*c^2)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28060.

X(21266) lies on these lines: {2, 3}, {1295, 19406}

X(21267) = COMPLEMENT OF X(15519)

Barycentrics a^3-2 a^2 (b+c)+a (9 b^2-14 b c+9 c^2)-4 (b-c)^2 (b+c) : :
X(21267) = (4r^2+16rR-s^2) X(1) - 12r^2 X(2)

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(21267) and X(4691) are equal.

See Tran Quang Hung, Angel Montesdeoca, and César Lozada, ADGEOM 4851 and ADGEOM 4855.

X(21267) lies on these lines: {1, 2}, {165, 3021}, {1997, 4929}, {3756, 8056}, {4862, 5274}, {7963, 12625}

X(21267) = complement of X(15519)

X(21268) = MIDPOINT ON X(4) AND X(5962)

Barycentrics 2 a^16 -7 a^14 (b^2+c^2) +4 a^12 (2 b^4+5 b^2 c^2+2 c^4) -a^10 (3 b^6+17 b^4 c^2+17 b^2 c^4+3 c^6) -2 a^8 b^2 c^2 (b^4-10 b^2 c^2+c^4) +3 a^6 (b^2-c^2)^2 (b^2+c^2)^3 -4 a^4 (b^2-c^2)^2 (2 b^8-b^6 c^2+2 b^4 c^4-b^2 c^6+2 c^8) +a^2 (b^2-c^2)^4 (7 b^6+b^4 c^2+b^2 c^4+7 c^6) -2 (b^2-c^2)^6 (b^4+b^2 c^2+c^4) : :

See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4859.

X(21268) lies on these lines: {4,52}, {5,12095}, {30,131}, {381,13557}, {403,14769}, {924,13851}, {5203,9880}, {7547,14889}

X(21268) = midpoint of X(4) and X(5962)
X(21268) = reflection of X(12095) in X(5)

X(21269) = MIDPOINT ON X(265) AND X(14989)

Barycentrics 4 a^16 -9 a^14 (b^2+c^2) +a^12 (-6 b^4+44 b^2 c^2-6 c^4) +14 a^10 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6) -3 a^8 (5 b^8+15 b^6 c^2-44 b^4 c^4+15 b^2 c^6+5 c^8) -a^6 (9 b^10-71 b^8 c^2+63 b^6 c^4+63 b^4 c^6-71 b^2 c^8+9 c^10) +a^4 (b^2-c^2)^2 (4 b^8+14 b^6 c^2-63 b^4 c^4+14 b^2 c^6+4 c^8) +6 a^2 (b^2-c^2)^4 (b^6-2 b^4 c^2-2 b^2 c^4+c^6) -(b^2-c^2)^6 (3 b^4+7 b^2 c^2+3 c^4) : :

See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4867.

X(21269) lies on these lines: {30,74}, {523,3627}, {546,14934}, {10113,16340}, {12295,16168}

X(21269) = midpoint of X(265) and X(14989)
X(21269) = reflection of X(i) in X(j), for these {i, j}: {14677,12079}, {14934,546}, {16340,10113}

X(21270) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 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(21270) lies on these lines: {2, 48}, {4, 916}, {7, 80}, {8, 2893}, {10, 18650}, {12, 5736}, {40, 20291}, {56, 5740}, {69, 313}, {75, 5086}, {192, 21221}, {219, 857}, {281, 14543}, {307, 515}, {319, 3681}, {321, 5928}, {322, 5176}, {355, 1441}, {379, 16608}, {388, 5738}, {944, 17221}, {983, 7261}, {1442, 17181}, {1726, 21072}, {1826, 9028}, {1837, 17863}, {1848, 3187}, {2268, 16603}, {2784, 4336}, {2973, 18474}, {3007, 5881}, {3945, 5261}, {5224, 5260}, {5691, 18655}, {5929, 16980}, {6327, 21275}, {7253, 9253}, {7282, 7331}, {8048, 20246}, {11237, 15936}, {11442, 20242}, {12588, 15982}, {17165, 21288}, {20556, 21280}

X(21270) = complement of X(20074)
X(21270) = anticomplement of X(48)
X(21270) = isotomic conjugate of isogonal conjugate of X(23843)
X(21270) = isotomic conjugate of polar conjugate of X(17902)
X(21270) = isotomic conjugate of anticomplement of X(36033)
X(21270) = polar conjugate of isogonal conjugate of X(22130)
X(21270) = anticomplementary conjugate of X(6360)

X(21271) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 + b c^4 : :

X(21271) lies on these lines: {2, 1953}, {3, 17221}, {7, 5559}, {8, 2893}, {40, 17134}, {69, 1227}, {75, 14923}, {190, 20248}, {219, 14543}, {307, 3007}, {322, 3869}, {347, 4566}, {355, 20289}, {515, 20291}, {517, 1441}, {519, 18650}, {573, 4552}, {2099, 5736}, {2398, 12329}, {3057, 17863}, {3187, 10319}, {3212, 3672}, {3262, 20245}, {3875, 3895}, {4360, 20247}, {4872, 5564}, {5697, 17861}, {7253, 9249}, {7991, 18655}, {12702, 18661}, {16609, 17452}, {20347, 20930}, {21278, 21295}

X(21271) = anticomplement of X(1953)
X(21271) = polar conjugate of isogonal conjugate of X(23112)
X(21271) = isotomic conjugate of polar conjugate of X(18676)
X(21271) = anticomplementary conjugate of anticomplement of X(2167)

X(21272) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (a - b) (a - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21272) lies on these lines: {1, 21208}, {2, 2170}, {7, 12641}, {8, 150}, {69, 1227}, {85, 14923}, {99, 901}, {100, 658}, {101, 9086}, {145, 3212}, {190, 2415}, {322, 20245}, {346, 20248}, {514, 1018}, {517, 20347}, {644, 3732}, {668, 891}, {693, 15632}, {883, 926}, {1111, 2802}, {1145, 1565}, {1310, 9058}, {1332, 14543}, {1358, 13996}, {1909, 17164}, {3187, 8897}, {3673, 3885}, {3754, 7278}, {3869, 16284}, {3882, 4552}, {4329, 21286}, {4534, 16593}, {4561, 17780}, {4576, 7257}, {4642, 18600}, {4674, 17205}, {4872, 5176}, {4919, 9318}, {5080, 5195}, {6631, 18047}, {10914, 20880}, {16713, 21231}, {17139, 17791}, {17152, 20955}, {17220, 20930}

X(21272) = anticomplement of X(2170)
X(21272) = polar conjugate of isogonal conjugate of X(23113)
X(21272) = isotomic conjugate of isogonal conjugate of X(23845)
X(21272) = anticomplementary conjugate of anticomplement of X(4564)

X(21273) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 - a c^4 + b c^4 : :

X(21273) lies on these lines: {2, 2171}, {8, 21286}, {63, 3187}, {69, 1227}, {72, 20895}, {75, 3869}, {144, 1278}, {329, 14213}, {908, 4967}, {1441, 20347}, {1444, 17221}, {1953, 16713}, {2975, 4360}, {3596, 3952}, {3672, 20247}, {3878, 17183}, {4329, 21285}, {4552, 16574}, {5176, 5564}, {5748, 5936}, {5773, 16566}, {10436, 11682}, {11684, 17160}, {12526, 17151}, {17135, 20243}

X(21273) = isotomic conjugate of isogonal conjugate of X(23846)
X(21273) = polar conjugate of isogonal conjugate of X(23114)
X(21273) = anticomplement of X(2171)
X(21273) = anticomplementary conjugate of anticomplement of X(2185)

X(21274) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^7 + a^4 b^3 - a^3 b^4 - b^7 + a^3 b^2 c^2 - a^2 b^3 c^2 + a^4 c^3 - a^2 b^2 c^3 + b^4 c^3 - a^3 c^4 + b^3 c^4 - c^7 : :

X(21274) lies on these lines: {2, 21234}, {8, 2893}, {830, 7192}, {5189, 18657}, {17497, 21221}

X(21274) = isotomic conjugate of isogonal conjugate of X(23848)
X(21274) = polar conjugate of isogonal conjugate of X(23117)
X(21274) = anticomplement of isogonal conjugate of isotomic conjugate of X(16568)
X(21274) = anticomplement of anticomplement of X(21234)
X(21274) = anticomplementary conjugate of anticomplement of X(6)-isoconjugate of X(23)

X(21275) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - b^5 - c^5 : :

X(21275) lies on these lines: {2, 560}, {8, 17481}, {69, 9018}, {315, 17138}, {2880, 21286}, {4118, 4837}, {4329, 21295}, {4381, 16894}, {4769, 20234}, {6327, 21270}, {21221, 21299}

X(21275) = isotomic conjugate of isogonal conjugate of X(23849)
X(21275) = polar conjugate of isogonal conjugate of X(23118)
X(21275) = anticomplement of X(560)
X(21275) = anticomplementary conjugate of X(17486)

X(21276) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^3 b c + a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 - c^5 : :

X(21276) lies on these lines: {2, 2174}, {7, 18961}, {69, 313}, {75, 150}, {239, 16580}, {320, 18480}, {355, 20930}, {6646, 21221}, {7224, 17731}, {11237, 17378}, {17137, 21287}, {20289, 20347}, {20305, 20769}

X(21276) = isotomic conjugate of isogonal conjugate of X(23850)
X(21276) = polar conjugate of isogonal conjugate of X(23119)
X(21276) = anticomplement of X(2174)
X(21276) = anticomplementary conjugate of anticomplement of X(30690)

X(21277) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^3 b^2 + a^2 b^3 - b^5 + a^3 b c - a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - a b c^3 + b^2 c^3 - c^5 : :

X(21277) lies on these lines: {2, 7113}, {7, 18962}, {12, 86}, {68, 20571}, {69, 313}, {72, 319}, {75, 355}, {150, 320}, {219, 18747}, {326, 17857}, {834, 20293}, {857, 1332}, {958, 5224}, {3262, 5176}, {3879, 21077}, {4053, 6542}, {4360, 10950}, {4872, 17615}, {5080, 14616}, {5928, 20928}, {10436, 10827}, {10446, 10526}, {11236, 17378}, {11374, 17394}, {12635, 17377}, {20556, 21293}

X(21277) = isotomic conjugate of isogonal conjugate of X(1324)
X(21277) = isotomic conjugate of polar conjugate of X(37770)
X(21277) = isotomic conjugate of antigonal conjugate of X(58)
X(21277) = polar conjugate of isogonal conjugate of X(23120)
X(21277) = anticomplement of X(7113)
X(21277) = anticomplementary conjugate of anticomplement of X(18359)

X(21278) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^3 b^2 - a^2 b^3 + a^3 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(21278) lies on these lines: {2, 1964}, {6, 18082}, {8, 192}, {10, 2309}, {69, 9016}, {75, 20352}, {76, 17142}, {80, 4680}, {100, 16872}, {313, 674}, {319, 350}, {730, 3778}, {1837, 3056}, {2276, 4651}, {3009, 21257}, {3688, 3948}, {3703, 8844}, {3765, 3779}, {3770, 17165}, {4553, 18137}, {4851, 17721}, {5086, 17787}, {6327, 21270}, {9286, 17217}, {9902, 17157}, {10453, 17373}, {17049, 20913}, {17138, 20556}, {17349, 20663}, {17362, 17475}, {17792, 20891}, {21221, 21289}, {21271, 21295}

X(21278) = isotomic conjugate of isogonal conjugate of X(23851)
X(21278) = polar conjugate of isogonal conjugate of X(23121)
X(21278) = anticomplement of X(1964)
X(21278) = anticomplementary conjugate of X(21217)

X(21279) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 + a^4 b - a b^4 - b^5 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a b c^3 + 2 b^2 c^3 - a c^4 - b c^4 - c^5 : :

X(21279) lies on these lines: {2, 198}, {4, 7}, {63, 8804}, {69, 313}, {75, 1370}, {77, 515}, {149, 4452}, {150, 151}, {269, 5691}, {281, 7291}, {286, 13577}, {307, 6836}, {309, 20220}, {317, 8048}, {329, 1229}, {348, 17134}, {377, 10436}, {388, 3945}, {497, 3672}, {934, 1440}, {944, 1442}, {946, 7190}, {1014, 4293}, {1441, 17170}, {1445, 10445}, {1478, 3664}, {1479, 3663}, {1699, 4328}, {1804, 12114}, {1826, 7289}, {1848, 19785}, {2478, 4357}, {2551, 5232}, {2997, 18656}, {3583, 4862}, {3585, 4888}, {4000, 16580}, {4346, 5225}, {4902, 18514}, {5080, 21296}, {5603, 7269}, {5739, 20891}, {5905, 5928}, {6225, 6604}, {6245, 7013}, {6950, 7279}, {7318, 10785}, {8822, 14956}, {9436, 10431}, {10319, 19822}

X(21279) = isotomic conjugate of isogonal conjugate of X(22654)
X(21279) = polar conjugate of isogonal conjugate of X(23122)
X(21279) = anticomplement of X(198)
X(21279) = anticomplementary conjugate of X(20211)

X(21280) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^4 b + a b^4 - b^5 - a^4 c + b^4 c + a c^4 + b c^4 - c^5 : :

X(21280) lies on these lines: {2, 2175}, {8, 20234}, {69, 2876}, {315, 17138}, {320, 12586}, {760, 17481}, {1352, 4645}, {1899, 4388}, {2893, 3434}, {3662, 12589}, {5086, 7672}, {5207, 21281}, {6327, 11442}, {20556, 21270}

X(21280) = isotomic conjugate of isogonal conjugate of X(23852)
X(21280) = polar conjugate of isogonal conjugate of X(23123)
X(21280) = anticomplement of X(2175)
X(21280) = anticomplementary conjugate of X(21218)

X(21281) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^3 b - a b^3 + a^3 c - a^2 b c + a b^2 c - b^3 c + a b c^2 - a c^3 - b c^3 : :

X(21281) lies on these lines: {2, 1258}, {4, 1969}, {7, 8}, {76, 17753}, {86, 5710}, {145, 18600}, {150, 315}, {192, 3721}, {193, 3780}, {304, 517}, {329, 3975}, {344, 1334}, {742, 3959}, {956, 17206}, {1240, 1246}, {1403, 4447}, {1655, 4419}, {1930, 5903}, {3057, 18156}, {3208, 3912}, {3210, 6542}, {3263, 3869}, {3436, 20345}, {3662, 17752}, {3666, 17316}, {3765, 5905}, {4000, 17033}, {4050, 17296}, {4440, 20081}, {4441, 17751}, {4561, 5730}, {4713, 21025}, {4754, 15983}, {4986, 5904}, {5207, 21280}, {5228, 14829}, {5697, 14210}, {6327, 20350}, {6384, 10453}, {7019, 8817}, {9596, 14555}, {10449, 17143}, {10452, 17151}, {17026, 20257}, {17696, 17735}, {19582, 20947}, {20553, 21285}, {20719, 20923}

X(21281) = anticomplement of X(2176)
X(21281) = anticomplementary conjugate of X(21219)
X(21281) = {X(7),X(8)}-harmonic conjugate of X(1909)
X(21281) = isotomic conjugate of isogonal conjugate of X(23853)
X(21281) = polar conjugate of isogonal conjugate of X(23125)

X(21282) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 2 a^3 - a^2 b + a b^2 - 2 b^3 - a^2 c + b^2 c + a c^2 + b c^2 - 2 c^3 : :

X(21282) lies on these lines: {2, 902}, {7, 17146}, {8, 3585}, {69, 674}, {149, 4645}, {315, 20244}, {316, 512}, {320, 17145}, {516, 3006}, {518, 17491}, {519, 17953}, {528, 3936}, {752, 16704}, {908, 17780}, {1201, 17690}, {1836, 5014}, {2886, 4450}, {3058, 18139}, {3120, 17766}, {3448, 20558}, {3589, 4972}, {3722, 4892}, {3914, 17150}, {3952, 5057}, {4388, 4651}, {4432, 21026}, {4442, 5846}, {4514, 7321}, {4865, 17147}, {4914, 4980}, {5015, 17164}, {5080, 21290}, {5180, 16086}, {5300, 12699}, {5564, 17163}, {5847, 17162}, {9812, 10327}, {14360, 20351}, {16483, 17679}, {17449, 20042}

X(21282) = isotomic conjugate of isogonal conjugate of X(23854)
X(21282) = polar conjugate of isogonal conjugate of X(23126)
X(21282) = anticomplement of X(902)
X(21282) = anticomplementary conjugate of X(17487)

X(21283) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 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(21283) lies on these lines: {2, 2177}, {7, 17145}, {8, 80}, {69, 674}, {321, 4863}, {497, 4651}, {519, 4054}, {528, 1150}, {3006, 3886}, {3058, 5278}, {3242, 4442}, {3419, 3902}, {3706, 5014}, {3763, 4972}, {3873, 7321}, {3977, 4847}, {3996, 11680}, {4514, 5564}, {4673, 5178}, {4781, 5744}, {4865, 20017}, {5082, 17751}, {5233, 10707}, {5741, 11235}, {11185, 20556}, {17143, 21285}, {17163, 18697}

X(21283) = isotomic conjugate of isogonal conjugate of X(23855)
X(21283) = polar conjugate of isogonal conjugate of X(23127)
X(21283) = anticomplement of X(2177)
X(21283) = anticomplementary conjugate of X(17488)

X(21284) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(10317), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4+b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4+c^4)) : :

Let T₁ be the reflection of the orthic triangle in the orthic axis. Let T₂ be the reflection of the tangential triangle in the Euler line. T₁ and T₂ are homothetic at X(21284). (Randy Hutson, August 13, 2020)

X(21284) lies on these lines: {2, 3}, {6, 19151}, {50, 232}, {67, 19596}, {98, 9381}, {184, 15135}, {185, 2917}, {187, 8428}, {511, 19504}, {566, 10311}, {842, 933}, {935, 5938}, {1112, 15107}, {1291, 3563}, {1398, 7286}, {1495, 10117}, {1503, 13171}, {1609, 16306}, {1843, 2916}, {2079, 3291}, {2697, 20626}, {3580, 12310}, {3796, 11649}, {5012, 8537}, {5089, 15586}, {5160, 7071}, {5322, 9630}, {5966, 13863}, {6103, 11063}, {6403, 15080}, {8705, 12167}, {8907, 12164}, {9591, 11363}, {9609, 16328}, {11405, 15826}, {14908, 19330}, {15141, 18374}

X(21284) = isogonal conjugate of X(18125)
X(21284) = tangential-isogonal conjugate of X(15141)

X(21285) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^4 - a^3 b + a b^3 - b^4 - a^3 c + b^3 c + a c^3 + b c^3 - c^4 : :

X(21285) lies on these lines: {2, 41}, {4, 20347}, {7, 2475}, {8, 150}, {69, 313}, {75, 5178}, {85, 5086}, {145, 3905}, {315, 766}, {329, 2890}, {348, 17136}, {742, 4950}, {758, 4056}, {1370, 17135}, {2280, 17062}, {2389, 3434}, {3188, 9436}, {3419, 20880}, {3662, 16910}, {3868, 4911}, {3869, 4872}, {3873, 7247}, {3874, 7272}, {4329, 21273}, {4511, 17181}, {4741, 21221}, {4805, 21240}, {5176, 16284}, {9437, 21302}, {10449, 17492}, {16551, 21069}, {17143, 21283}, {20350, 20559}, {20553, 21281} X(21285) = isotomic conjugate of isogonal conjugate of X(1626)
X(21285) = polar conjugate of isogonal conjugate of X(22125)
X(21285) = complement of X(20071)
X(21285) = anticomplement of X(41)
X(21285) = anticomplementary conjugate of X(3177)

X(21286) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^3 b^2 + a^2 b^3 - b^5 + 2 a^3 b c - 2 a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 - c^5 : :

X(21286) lies on these lines: {2, 604}, {7, 5554}, {8, 21273}, {63, 17270}, {69, 313}, {75, 5176}, {86, 11681}, {144, 17343}, {150, 21296}, {319, 3869}, {329, 2893}, {908, 3879}, {1264, 3952}, {1332, 18747}, {2880, 21275}, {2975, 5224}, {4329, 21272}, {4872, 11678}, {5080, 10446}, {5905, 5933}, {6327, 11442}, {7247, 12125}, {17321, 18654}, {17373, 21221}, {20074, 20769}, {20348, 20353}, {20553, 20557}

X(21286) = isotomic conjugate of isogonal conjugate of X(2933)
X(21286) = polar conjugate of isogonal conjugate of X(23129)
X(21286) = anticomplement of X(604)
X(21286) = anticomplementary conjugate of X(3210)

X(21287) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 + a^4 b - a b^4 - b^5 + a^4 c + a^3 b c - a b^3 c - b^4 c - a b c^3 - a c^4 - b c^4 - c^5 : :

X(21287) lies on these lines: {2, 1333}, {4, 69}, {75, 2475}, {86, 5051}, {153, 322}, {297, 5317}, {313, 5080}, {319, 321}, {320, 17863}, {325, 1444}, {345, 3151}, {648, 18685}, {964, 5224}, {1231, 7282}, {1369, 18133}, {1654, 2345}, {1766, 3882}, {2997, 11604}, {4150, 5279}, {4417, 19645}, {4812, 20349}, {5046, 18147}, {5716, 17321}, {6327, 21270}, {6994, 19793}, {7270, 20336}, {17137, 21276}, {17138, 20554}, {17751, 20289}, {20355, 21221}

X(21287) = isotomic conjugate of cyclocevian conjugate of X(35058)
X(21287) = isotomic conjugate of isogonal conjugate of X(2915)
X(21287) = polar conjugate of isogonal conjugate of X(23130)
X(21287) = anticomplement of X(1333)
X(21287) = anticomplementary conjugate of X(17147)

X(21288) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^9 + a^8 b - a b^8 - b^9 + a^8 c - 2 a^4 b^4 c + b^8 c - 2 a^4 b c^4 + 2 a b^4 c^4 - a c^8 + b c^8 - c^9 : :

X(21288) lies on these lines: {2, 2156}, {5596, 17150}, {5905, 17489}, {17141, 20242}, {17165, 21270}, {17481, 17486}

X(21288) = isotomic conjugate of isogonal conjugate of X(23856)
X(21288) = polar conjugate of isogonal conjugate of X(23132)
X(21288) = anticomplement of X(2156)
X(21288) = anticomplementary conjugate of X(17481)

X(21289) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 + a^3 b^2 - a^2 b^3 - b^5 - a^2 b^2 c + a^3 c^2 - a^2 b c^2 + a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - c^5 : :

X(21289) lies on these lines: {1, 2896}, {2, 82}, {8, 17481}, {75, 1369}, {192, 744}, {1031, 4388}, {3416, 3869}, {16555, 21082}, {20538, 20932}, {21221, 21278}

X(21289) = isotomic conjugate of isogonal conjugate of X(20994)
X(21289) = polar conjugate of isogonal conjugate of X(22137)
X(21289) = complement of X(20087)
X(21289) = anticomplement of X(82)
X(21289) = anticomplementary conjugate of X(17165)

X(21290) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^4 - a^3 b + a b^3 - b^4 - a^3 c + 5 a^2 b c - 5 a b^2 c + b^3 c - 5 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 - c^4 : :

X(21290) lies on the anticomplementary circle and these lines: {1, 11814}, {2, 106}, {4, 10744}, {8, 80}, {10, 1054}, {20, 1293}, {69, 150}, {145, 10700}, {146, 2776}, {147, 2789}, {148, 1654}, {151, 2815}, {152, 2821}, {153, 2827}, {190, 1145}, {193, 10761}, {388, 1357}, {497, 6018}, {519, 13541}, {952, 3699}, {1222, 4187}, {1330, 2842}, {1387, 4997}, {1647, 9457}, {2551, 3038}, {2832, 20344}, {2841, 3436}, {2843, 14360}, {2844, 13219}, {2891, 17751}, {3030, 9534}, {3091, 5510}, {3146, 10730}, {3616, 11717}, {3622, 11731}, {3753, 7321}, {3977, 6735}, {4440, 4674}, {4723, 5176}, {4767, 12531}, {5080, 21282}, {6079, 14507}, {6224, 17780}, {6327, 21291}, {6736, 7283}, {9458, 21041}, {9527, 12384}

X(21290) = isogonal conjugate of X(34184)
X(21290) = isotomic conjugate of isogonal conjugate of X(23858)
X(21290) = complement of X(20098)
X(21290) = anticomplement of X(106)
X(21290) = anticomplementary conjugate of X(519)
X(21290) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(9059)
X(21290) = de-Longchamps-circle-inverse of X(2370)
X(21290) = polar conjugate of isogonal conjugate of X(23135)

X(21291) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 4 a^4 + 2 a^3 b - 2 a b^3 - 4 b^4 + 2 a^3 c + 5 a^2 b c - 5 a b^2 c - 2 b^3 c - 5 a b c^2 + 4 b^2 c^2 - 2 a c^3 - 2 b c^3 - 4 c^4 : :

X(21291) lies on these lines: {2, 2163}, {8, 3585}, {69, 5080}, {1654, 7229}, {3616, 5484}, {6327, 21290}

X(21291) = isotomic conjugate of isogonal conjugate of X(23859)
X(21291) = polar conjugate of isogonal conjugate of X(23136)
X(21291) = anticomplement of X(2163)
X(21291) = anticomplementary conjugate of X(3241)

X(21292) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(10485), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2*(4*a^6-5*(b^2+c^2)*a^4+(5*b^4+7*b^2*c^2+5*c^4)*a^2-(b^2+c^2)*(4*b^4-7*b^2*c^2+4*c^4)) : :

X(21292) lies on these lines: {2, 3}, {2930, 20794}, {5191, 11422}, {11842, 15019}

X(21293) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^5 - a^4 b + a b^4 - b^5 - a^4 c + a^3 b c - a b^3 c + b^4 c - a b c^3 + a c^4 + b c^4 - c^5 : :

X(21293) lies on these lines: {1, 4466}, {2, 692}, {4, 2807}, {7, 12586}, {8, 12587}, {69, 2876}, {75, 2870}, {141, 20468}, {149, 3448}, {150, 926}, {497, 1899}, {651, 5848}, {1086, 18343}, {1352, 2550}, {2875, 3434}, {2877, 6327}, {2878, 21294}, {2892, 3868}, {4000, 12589}, {4307, 5820}, {4553, 20344}, {10531, 18912}, {11433, 14717}, {11457, 12116}, {13211, 16110}, {17138, 20554}, {20556, 21277}

X(21293) = isotomic conjugate of isogonal conjugate of X(23402)
X(21293) = polar conjugate of isogonal conjugate of X(23137)
X(21293) = anticomplement of X(692)
X(21293) = anticomplementary conjugate of X(17494)

X(21294) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^7 - a^5 b^2 + a^2 b^5 - b^7 - a^5 c^2 + a^3 b^2 c^2 - a^2 b^3 c^2 + b^5 c^2 - a^2 b^2 c^3 + a^2 c^5 + b^2 c^5 - c^7 : :

X(21294) lies on these lines: {2, 163}, {150, 2774}, {1330, 20552}, {2853, 13219}, {2878, 21293}, {5906, 17492}, {13211, 17886}

X(21294) = isotomic conjugate of isogonal conjugate of X(23860)
X(21294) = polar conjugate of isogonal conjugate of X(23138)
X(21294) = complement of X(20100)
X(21294) = anticomplement of X(163)
X(21294) = anticomplementary conjugate of X(4560)

X(21295) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (a - b) (a - c) (b + c) (a^2 + a b + b^2 + a c - b c + c^2) : :

X(21295) lies on these lines: {2, 2643}, {8, 21221}, {75, 20351}, {100, 8052}, {190, 523}, {192, 4094}, {512, 3888}, {643, 4427}, {874, 3903}, {1016, 7927}, {1109, 17777}, {3057, 4459}, {3744, 4760}, {3799, 4139}, {3952, 4010}, {4132, 4553}, {4329, 21275}, {4499, 8672}, {4514, 17163}, {4552, 4613}, {7261, 18697}, {21271, 21278}

X(21295) = isotomic conjugate of isogonal conjugate of X(23861)
X(21295) = polar conjugate of isogonal conjugate of X(23139)
X(21295) = anticomplement of X(2643)
X(21295) = anticomplementary conjugate of anticomplement of X(24041)

X(21296) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Trilinears    2 sec^2(A/2) - csc^2(A/2) : :
Barycentrics    3 a^2 - 3 b^2 + 2 b c - 3 c^2 : :
Barycentrics    b c - 3 SA : :

X(21296) lies on these lines: {2, 1743}, {7, 8}, {9, 4869}, {10, 4888}, {77, 4511}, {78, 269}, {86, 5550}, {141, 4644}, {142, 391}, {144, 3161}, {145, 3663}, {150, 21286}, {189, 14206}, {193, 3662}, {200, 7271}, {239, 20080}, {253, 10538}, {306, 9965}, {314, 5556}, {326, 1443}, {329, 4358}, {344, 6172}, {346, 527}, {390, 4684}, {479, 7055}, {519, 4452}, {524, 4000}, {545, 17309}, {594, 7222}, {599, 2345}, {894, 3620}, {938, 1330}, {966, 4675}, {1086, 4402}, {1119, 5081}, {1266, 20053}, {1418, 3965}, {1992, 16706}, {2094, 17740}, {2321, 4454}, {2835, 4329}, {3146, 10442}, {3187, 19824}, {3241, 3672}, {3598, 3705}, {3616, 3945}, {3617, 4896}, {3618, 17227}, {3619, 3758}, {3623, 4021}, {3629, 17290}, {3630, 4361}, {3631, 4363}, {3632, 4902}, {3661, 7229}, {3686, 6173}, {3707, 20195}, {3729, 20059}, {3759, 11008}, {3763, 7277}, {3794, 9309}, {3872, 4328}, {3875, 4346}, {3883, 11038}, {3928, 4035}, {3936, 5744}, {3964, 4996}, {4089, 6790}, {4101, 6904}, {4310, 5847}, {4371, 7263}, {4388, 10580}, {4389, 20057}, {4417, 5435}, {4419, 4681}, {4440, 17373}, {4460, 17377}, {4461, 17294}, {4470, 17239}, {4643, 4648}, {4667, 17306}, {4671, 5905}, {4704, 6646}, {4715, 17279}, {4718, 17276}, {4741, 5308}, {4747, 5750}, {4748, 15668}, {4788, 6542}, {4795, 17385}, {4853, 7274}, {4861, 7190}, {4916, 17318}, {4966, 5698}, {5080, 21279}, {5224, 19877}, {5226, 14829}, {5232, 9780}, {5249, 14552}, {5273, 18134}, {5279, 7289}, {5739, 9776}, {6144, 17366}, {6327, 17145}, {6381, 20245}, {7179, 15589}, {9812, 10439}, {10481, 20007}, {11160, 17363}, {14210, 17170}, {15533, 17362}, {16704, 17189}, {17014, 17304}, {17146, 20290}, {17184, 19823}, {17232, 20072}, {17234, 18230}, {17236, 20090}, {17242, 20073}, {17253, 17392}, {17254, 17391}, {17255, 17390}, {17258, 17387}, {17273, 17321}, {17311, 17334}, {17312, 17333}, {17313, 17332}, {17317, 17329}, {18141, 18228}

X(21296) = isotomic conjugate of X(7319)
X(21296) = anticomplement of X(1743)
X(21296) = {X(7),X(8)}-harmonic conjugate of X(31995)
X(21296) = {X(69),X(75)}-harmonic conjugate of X(32099)
X(21296) = {X(17288),X(17364)}-harmonic conjugate of X(2)
X(21296) = polar conjugate of isogonal conjugate of X(23140)
X(21296) = anticomplementary conjugate of X(8055)

X(21297) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^2 - a b - a c + 3 b c) : :

X(21297) lies on these lines: {2, 812}, {149, 150}, {320, 350}, {321, 20950}, {514, 4120}, {664, 15343}, {668, 891}, {850, 4139}, {900, 903}, {1022, 4080}, {1639, 6009}, {2786, 6545}, {2787, 14421}, {2820, 9812}, {2832, 17777}, {3004, 17161}, {3120, 17198}, {3268, 6089}, {3667, 21183}, {3835, 4382}, {3837, 4810}, {4025, 4962}, {4155, 17163}, {4379, 4785}, {4380, 4885}, {4393, 4508}, {4608, 18004}, {4750, 21204}, {4762, 4776}, {4922, 9269}, {4958, 21115}, {4984, 14475}

X(21297) = isotomic conjugate of isogonal conjugate of X(4491)
X(21297) = polar conjugate of isogonal conjugate of X(23141)
X(21297) = anticomplement of X(1635)
X(21297) = anticomplementary conjugate of anticomplement of X(3257)

X(21298) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics 2 a^5 - a^3 b^2 + a^2 b^3 - 2 b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - 2 c^5 : :

X(21298) lies on these lines: {2, 922}, {8, 17482}, {788, 17217}, {6327, 21270}

X(21298) = isotomic conjugate of isogonal conjugate of X(23862)
X(21298) = polar conjugate of isogonal conjugate of X(23142)
X(21298) = anticomplement of X(922)
X(21298) = anticomplementary conjugate of anticomplement of isogonal conjugate of X(922)
X(21298) = anticomplementary conjugate of anticomplement of isotomic conjugate of X(896)

X(21299) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics a^3 b^2 - a^2 b^3 + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(21299) lies on these lines: {2, 1740}, {4, 4645}, {7, 871}, {8, 192}, {69, 350}, {194, 3778}, {239, 1716}, {312, 17792}, {345, 8844}, {966, 2276}, {1278, 20352}, {2550, 17787}, {3223, 20464}, {3596, 6007}, {3685, 6210}, {3781, 19582}, {5839, 17475}, {6196, 7032}, {16571, 20340}, {17135, 17343}, {17157, 20081}, {21221, 21275}

X(21299) = isotomic conjugate of isogonal conjugate of X(23863)
X(21299) = polar conjugate of isogonal conjugate of X(23143)
X(21299) = anticomplement of X(1740)
X(21299) = anticomplementary conjugate of anticomplement of X(3223)

X(21300) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (a + b) (a - b - c) (b - c) (a + c) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(21300) lies on these lines: {2, 810}, {8, 3907}, {521, 1948}, {788, 17217}, {1010, 7254}, {1577, 10449}, {3737, 4147}, {3900, 18155}, {4589, 7257}, {7192, 8678}, {9534, 14838}

X(21300) = isotomic conjugate of isogonal conjugate of X(23864)
X(21300) = polar conjugate of isogonal conjugate of X(23145)
X(21300) = anticomplement of X(810)
X(21300) = anticomplementary conjugate of anticomplement of X(811)

X(21301) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^3 - a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(21301) and X(3622) are equal.

X(21301) lies on these lines: {2, 667}, {4, 885}, {8, 4083}, {10, 4063}, {69, 9010}, {150, 9320}, {316, 512}, {388, 3669}, {497, 4162}, {513, 2517}, {514, 4088}, {522, 21124}, {649, 17072}, {659, 21051}, {663, 3835}, {693, 8678}, {764, 20060}, {788, 17217}, {812, 4041}, {814, 1491}, {830, 1577}, {832, 7253}, {1478, 4905}, {1912, 17137}, {2254, 6002}, {2530, 2787}, {2550, 6008}, {2551, 20317}, {3436, 4462}, {3667, 21186}, {3777, 21222}, {3837, 4367}, {3887, 4170}, {3900, 4106}, {4040, 4129}, {4147, 4498}, {4160, 4978}, {4504, 14413}, {4705, 17494}, {4782, 9780}, {4879, 4992}, {5046, 6161}, {6371, 20293}, {9013, 14288}, {18111, 18705}, {18135, 20350}

X(21301) = isotomic conjugate of isogonal conjugate of X(21005)
X(21301) = complement of X(31291)
X(21301) = anticomplement of X(667)
X(21301) = anticomplementary conjugate of X(9263)
X(21301) = polar conjugate of isogonal conjugate of X(22157)

X(21302) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^3 + 2 a^2 b - a b^2 + 2 a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(21302) lies on these lines: {2, 663}, {8, 514}, {10, 4040}, {69, 4406}, {145, 4449}, {316, 512}, {513, 4397}, {522, 17950}, {661, 4148}, {693, 3900}, {812, 4729}, {830, 4761}, {832, 4581}, {926, 4374}, {1577, 3887}, {1734, 4560}, {2254, 3907}, {2517, 7253}, {2533, 17751}, {3309, 4391}, {3617, 4147}, {3676, 8710}, {3716, 21052}, {3837, 4879}, {4041, 17494}, {4063, 4807}, {4162, 4885}, {4163, 4468}, {4379, 10453}, {4651, 4705}, {4775, 21260}, {4794, 9780}, {4801, 14077}, {4905, 21222}, {7192, 8678}, {9245, 20245}, {9320, 20347}, {9437, 21285}, {17135, 17166}, {17159, 21304}, {18391, 21185}

X(21302) = isotomic conjugate of isogonal conjugate of X(23865)
X(21302) = isotomic conjugate of anticomplement of X(38991)
X(21302) = polar conjugate of isogonal conjugate of X(23146)
X(21302) = anticomplement of X(663)
X(21302) = anticomplementary conjugate of X(39351)

X(21303) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (a^4 + a b^3 - 2 a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21303) lies on these lines: {2, 8632}, {149, 150}, {316, 512}, {513, 21304}, {812, 6653}, {830, 7192}, {1654, 3768}, {2254, 7261}, {4375, 21053}, {4491, 5224}

X(21303) = isotomic conjugate of isogonal conjugate of X(23866)
X(21303) = polar conjugate of isogonal conjugate of X(23147)
X(21303) = anticomplement of X(8632)
X(21303) = anticomplementary conjugate of X(39362)

X(21304) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (b - c) (-a^4 - a b^3 - a b^2 c + b^3 c - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(21304) lies on these lines: {2, 1919}, {513, 21303}, {688, 14295}, {788, 17217}, {816, 8061}, {834, 20293}, {1654, 20979}, {4057, 5224}, {6542, 17458}, {9231, 17138}, {17159, 21302}, {17300, 21191}, {20981, 21261}

X(21304) = isotomic conjugate of isogonal conjugate of X(23867)
X(21304) = polar conjugate of isogonal conjugate of X(23148)
X(21304) = anticomplement of X(1919)
X(21304) = anticomplementary conjugate of X(21224)

X(21305) = (A,B,C,X(75); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics (a + b) (b - c) (a + c) (a^3 b - a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21305) lies on these lines: {2, 1924}, {788, 17217}, {830, 7192}

X(21305) = isotomic conjugate of isogonal conjugate of X(23403)
X(21305) = polar conjugate of isogonal conjugate of X(23149)
X(21305) = anticomplement of X(1924)
X(21305) = anticomplementary conjugate of anticomplement of X(4602)

X(21306) = X(1)X(88)∩X(3)X(21229)

Barycentrics a*(a^6 - a^5*b - a^4*b^2 + 3*a^3*b^3 - 2*a*b^5 - a^5*c + 2*a^4*b*c - 4*a^3*b^2*c - 2*a^2*b^3*c + 5*a*b^4*c - a^4*c^2 - 4*a^3*b*c^2 + 15*a^2*b^2*c^2 - 9*a*b^3*c^2 + b^4*c^2 + 3*a^3*c^3 - 2*a^2*b*c^3 - 9*a*b^2*c^3 + 2*b^3*c^3 + 5*a*b*c^4 + b^2*c^4 - 2*a*c^5) : :

X(21306) lies on the cubic K1063 and these lines: {1, 88}, {3, 21229}, {8, 14513}, {765, 14260}, {953, 6079}

X(21307) = X(4)X(953)∩X(8)X(14513)

Barycentrics a*(a - b - c)*(a^8 - a^7*b - 2*a^6*b^2 + 4*a^5*b^3 + a^4*b^4 - 5*a^3*b^5 + 2*a*b^7 - a^7*c + 4*a^6*b*c - 3*a^5*b^2*c - 6*a^4*b^3*c + 11*a^3*b^4*c + 2*a^2*b^5*c - 7*a*b^6*c - 2*a^6*c^2 - 3*a^5*b*c^2 + 9*a^4*b^2*c^2 - 6*a^3*b^3*c^2 - 8*a^2*b^4*c^2 + 9*a*b^5*c^2 + b^6*c^2 + 4*a^5*c^3 - 6*a^4*b*c^3 - 6*a^3*b^2*c^3 + 12*a^2*b^3*c^3 - 4*a*b^4*c^3 + a^4*c^4 + 11*a^3*b*c^4 - 8*a^2*b^2*c^4 - 4*a*b^3*c^4 - 2*b^4*c^4 - 5*a^3*c^5 + 2*a^2*b*c^5 + 9*a*b^2*c^5 - 7*a*b*c^6 + b^2*c^6 + 2*a*c^7) : :

X(21307) lies on the cubic K1063 and these lines: {4, 953}, {8, 14513}, {21, 3737}, {40, 78}, {1745, 8666}, {4511, 13589}

X(21308) = EULER LINE INTERCEPT OF X(54)X(18874)

Barycentrics a^2*(a^8+a^4*b^2*c^2-2*(b^2+c^2)*a^6+(b^2+c^2)*(2*b^4-9*b^2*c^2+2*c^4)*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2) : :

As a point on the Euler line, X(21308) has Shinagawa coefficients (5*E-8*F, 11*E+8*F)

See César Lozada, Hyacinthos 28064.

X(21308) lies on the these lines: {2, 3}, {54, 18874}, {74, 18551}, {110, 13364}, {143, 12316}, {195, 10095}, {323, 13451}, {399, 5946}, {567, 14845}, {1614, 15047}, {3066, 18445}, {5640, 15087}, {5890, 12308}, {5898, 7693}, {5943, 10540}, {5966, 12074}, {6688, 13339}, {10564, 13570}, {10620, 15053}, {11451, 15038}, {12307, 14128}, {13321, 15068}, {13363, 14157}, {14627, 18350}, {20193, 21230}

X(21308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5899, 3), (2, 7545, 5899), (3, 18535, 15684), (5, 10096, 2), (5, 13163, 4), (5, 13595, 2070), (25, 5055, 3), (1344, 1345, 3526), (1656, 13861, 18378), (1656, 18378, 3), (3843, 6642, 3), (3851, 7506, 3), (5070, 7517, 3), (5071, 14002, 7502), (11284, 12083, 15694), (12083, 15694, 3)

X(21309) = X(3)X(6)∩X(25)X(1383)

Trilinears sin(A +ω) + 7 sin(A -ω) : :
Barycentrics (7*a^2+b^2+c^2)*a^2 : :
X(21309) = 3*S^2*X(3)-4*SW^2*X(6) = 3*X(3)-4*cot(ω)^2*X(6)

See César Lozada, Hyacinthos 28064.

X(21309) lies on the these lines: {2, 3793}, {3, 6}, {25, 1383}, {30, 1285}, {112, 1597}, {115, 14269}, {172, 7373}, {183, 12150}, {193, 8369}, {230, 5055}, {248, 3531}, {251, 5020}, {381, 7735}, {382, 5305}, {385, 11286}, {729, 3511}, {1003, 7766}, {1572, 10247}, {1598, 3172}, {1627, 16419}, {1657, 5286}, {1914, 6767}, {1992, 6390}, {2207, 10985}, {2548, 3054}, {2549, 15681}, {3295, 7031}, {3517, 8743}, {3524, 14930}, {3534, 15048}, {3619, 7767}, {3620, 7819}, {3630, 7795}, {3631, 14023}, {3731, 5266}, {3767, 3843}, {3815, 15694}, {3830, 5306}, {3851, 7745}, {3933, 11008}, {3972, 14614}, {5054, 7736}, {5073, 5254}, {5077, 14712}, {5275, 16857}, {5276, 16418}, {5277, 16863}, {5309, 15684}, {5319, 17800}, {5355, 15685}, {5359, 9909}, {5475, 19709}, {5708, 16780}, {6144, 7801}, {6660, 13192}, {6781, 11742}, {7618, 20583}, {7738, 15696}, {7739, 15689}, {7746, 18584}, {7753, 15703}, {7755, 18424}, {7774, 11288}, {7798, 15301}, {7804, 8667}, {7806, 11318}, {7813, 15534}, {7866, 20065}, {7879, 10583}, {7887, 20088}, {7897, 8366}, {8584, 11165}, {9300, 15701}, {9463, 11328}, {9755, 10788}, {10304, 14482}, {10311, 18535}, {11179, 20194}, {11284, 11580}, {11287, 16989}, {11343, 14996}, {13638, 13763}, {13644, 13758}, {14001, 20080}, {14535, 15271}, {16318, 18494}

X(21309) = midpoint of X(11485) and X(11486)
X(21309) = circumcircle-inverse of X(38010)
X(21309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 187, 15603), (6, 574, 9605), (39, 14075, 6), (187, 574, 5585), (187, 5024, 3), (187, 5585, 15655), (187, 15602, 8588), (574, 3053, 15655), (574, 5007, 6), (574, 15655, 3), (1384, 5024, 187), (2080, 5050, 3), (3053, 5585, 187), (3311, 3312, 576), (6221, 6398, 3098), (8375, 8376, 11173)

X(21310) = X(3)X(13)∩X(62)X(18350)

Barycentrics (SB+SC)*(sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(sqrt(3)*(2*S^2+SA^2)-S*SA) : :

See César Lozada, Hyacinthos 28064.

X(21310) lies on the these lines: {3, 13}, {62, 18350}, {1598, 8737}, {3457, 11486}, {1989, 21311}, {10217, 16463}, {11081, 11485}

X(21310) = {X(13), X(11142)}-harmonic conjugate of X(3)

X(21311) = X(3)X(14)∩X(61)X(18350)

Barycentrics (SB+SC)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(2*S^2+SA^2)+S*SA) : :

See César Lozada, Hyacinthos 28064.

X(21311) lies on the these lines: {3, 14}, {61, 18350}, {1598, 8738}, {3458, 11485}, {1989, 21310}, {10218, 16464}, {11086, 11486}

X(21311) = {X(14), X(11141)}-harmonic conjugate of X(3)

X(21312) = ANTICOMPLEMENT OF X(1596)

Barycentrics a^2*(a^8+16*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(21312) = 3*X(5085)-2*X(19136), 3*X(6090)-2*X(18451), 2*X(8263)-3*X(10519), 3*X(15035)-2*X(20772)

As a point on the Euler line, X(21312) has Shinagawa coefficients (E-F, -2*E+F)

See César Lozada, Hyacinthos 28064.

X(21312) lies on the these lines: {2, 3}, {64, 5562}, {74, 3565}, {98, 20187}, {99, 1294}, {110, 11820}, {112, 1033}, {155, 10575}, {159, 2935}, {185, 12160}, {394, 6000}, {511, 10602}, {691, 2693}, {841, 10420}, {999, 3100}, {1062, 1398}, {1147, 14641}, {1181, 13346}, {1204, 17834}, {1292, 1295}, {1296, 1297}, {1350, 2393}, {1351, 5890}, {1384, 10313}, {1473, 7171}, {1565, 4329}, {1578, 11473}, {1579, 11474}, {1619, 15311}, {1993, 15072}, {2691, 2694}, {2696, 2697}, {2734, 2737}, {2794, 2936}, {2979, 13445}, {3098, 14913}, {3167, 11456}, {3184, 15259}, {3295, 4296}, {3304, 9643}, {3357, 15644}, {3426, 15066}, {3527, 15043}, {3587, 7085}, {3695, 7219}, {3796, 11430}, {4293, 16541}, {5050, 15033}, {5085, 19136}, {5422, 20791}, {5891, 11472}, {5925, 9914}, {6090, 14915}, {6221, 11417}, {6241, 12164}, {6361, 12410}, {6398, 11418}, {6560, 19006}, {6561, 19005}, {7373, 9538}, {8192, 18481}, {8263, 10519}, {8717, 18475}, {8718, 9707}, {8778, 10316}, {8780, 14157}, {9729, 10982}, {9730, 9777}, {9937, 11750}, {10037, 10483}, {10574, 11432}, {10601, 16836}, {10625, 12163}, {10831, 15338}, {10832, 15326}, {10984, 11425}, {11381, 17814}, {11402, 13352}, {11406, 15941}, {11441, 12279}, {11457, 12429}, {11477, 14831}, {12017, 19121}, {12058, 13754}, {12111, 13093}, {12118, 12166}, {12256, 12303}, {12257, 12304}, {12302, 13171}, {12412, 20127}, {13142, 18916}, {13491, 16266}, {15030, 17811}, {15035, 20772}

X(21312) = isogonal conjugate of X(35512)
X(21312) = anticomplement of X(1596)
X(21312) = circumperp conjugate of X(468)
X(21312) = X(25)-of-ABC-X3-reflections-triangle
X(21312) = X(1368)-of-anti-Euler-triangle
X(21312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 6642), (3, 1593, 7395), (3, 1597, 2), (3, 5073, 7506), (3, 12083, 14070), (22, 2071, 3), (22, 7396, 5020), (22, 11413, 2071), (186, 12082, 9909), (382, 6642, 5198), (548, 7526, 3), (1593, 7484, 9818), (1995, 3543, 18535), (2041, 2042, 1906), (3146, 17928, 1598), (3528, 14865, 7509)

X(21313) = EULER LINE INTERCEPT OF X(1974)X(6391)

Barycentrics SB*SC*(SB+SC)*(4*S^2*(6*R^2-SW)-SA^2*SW) : :

See César Lozada, Hyacinthos 28064.

X(21313) lies on the these lines: {2, 3}, {1974, 6391}, {5544, 19124}, {15531, 19118}

X(21313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 1995, 1598), (25, 4232, 3517), (25, 6353, 9909), (5020, 9909, 1368)

X(21314) = X(1)X(7)∩X(57)X(1358)

Barycentrics (a^2+(b+c)*a-2*(b-c)^2)*(a+b-c)*(a-b+c) : :

See César Lozada, Hyacinthos 28065.

X(21314) lies on the these lines: {1, 7}, {57, 1358}, {85, 1698}, {241, 5219}, {738, 3338}, {948, 3911}, {1088, 1111}, {1212, 20195}, {1418, 4859}, {1565, 1699}, {1743, 7960}, {3337, 7177}, {3361, 7195}, {3632, 9312}, {3633, 6604}, {3665, 5290}, {3679, 9436}, {5526, 6180}, {10491, 18886}

X(21314) = X(1384)-of-intouch triangle
X(21314) = barycentric product X(i)*X(j) for these {i,j}: {7, 6173}, {85, 4860}, {279, 5231}
X(21314) = trilinear product X(i)*X(j) for these {i,j}: {7, 4860}, {57, 6173}, {269, 5231}
X(21314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 20121, 7), (7, 279, 1323), (7, 1323, 1), (7, 10481, 20121), (279, 10481, 1), (1323, 10481, 7), (3638, 3639, 390), (3668, 7271, 4862), (5542, 11200, 1)

X(21315) = MIDPOINT OF X(5627) AND X(14643)

Barycentrics 3*S^4+(3*R^2*(81*R^2-6*SA-34*SW)+4*SA^2-SB*SC+11*SW^2)*S^2+(27*R^2*(9*R^2-4*SW)+11*SW^2)*SB*SC : :
X(21315) = 2*X(3)+X(21269), X(1553)+2*X(20379), 2*X(3154)+X(18319), X(3258)-4*X(15088), 4*X(3628)-X(14934), X(7471)+2*X(11801), 2*X(11657)+X(18572), X(16340)-4*X(20304)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28066.

X(21315) lies on the these lines: {3, 21269}, {5, 523}, {30, 14644}, {1553, 20379}, {3154, 18319}, {3258, 15088}, {3628, 14934}, {5627, 14643}, {7471, 11801}, {11657, 18572}, {16340, 20304}

X(21315) = midpoint of X(5627) and X(14643)

X(21316) = MIDPOINT OF X(3) AND X(21269)

Barycentrics 3*S^4+(3*R^2*(27*R^2-6*SA-10*SW)+4*SA^2-SB*SC+3*SW^2)*S^2+(45*R^2*(9*R^2-4*SW)+19*SW^2)*SB*SC : :
X(21316) = 3*X(5)-X(14934), 3*X(5627)+X(7728), X(14508)-5*X(15027), 3*X(14644)-X(16340), X(14989)+3*X(15061)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28066.

X(21316) lies on the these lines: {3, 21269}, {5, 14385}, {30, 125}, {523, 546}, {1539, 6070}, {5627, 7728}, {7687, 16168}, {8901, 11563}, {14508, 15027}, {14644, 16340}, {14989, 15061}

X(21316) = midpoint of X(i) and X(j) for these {i,j}: {3, 21269}, {1539, 6070}

X(21317) = X(3)X(21269)∩X(30)X(110)

Barycentrics 3*S^4+(-3*R^2*(135*R^2+6*SA-62*SW)+4*SA^2-SB*SC-21*SW^2)*S^2+(99*R^2*(9*R^2-4*SW)+43*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28066.

X(21317) lies on the these lines: {3, 21269}, {30, 110}, {523, 15704}

X(21318) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^2 b^2 - b^4 - a^2 b c + b^3 c + a^2 c^2 + b c^3 - c^4) : :

X(21318) lies on these lines: {1, 184}, {2, 17927}, {3, 11401}, {4, 6360}, {37, 17441}, {38, 17447}, {42, 2611}, {51, 1736}, {73, 2599}, {92, 1985}, {199, 3101}, {226, 18210}, {228, 16577}, {347, 4196}, {354, 17463}, {405, 11396}, {517, 13734}, {851, 1214}, {1441, 3136}, {1829, 13724}, {1897, 7413}, {1953, 2253}, {2295, 21335}, {2969, 8226}, {3100, 16064}, {3145, 6198}, {3721, 4137}, {3914, 4516}, {5729, 9777}, {5905, 20430}, {6197, 20838}, {12135, 13442}, {17167, 18175}, {17181, 17864}, {17479, 20242}, {20305, 21028}, {20593, 21329}

X(21319) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 - 2 b^2 c^2 + b c^3) : :

X(21319) lies on these lines: {1, 51}, {2, 20256}, {3, 5905}, {5, 20242}, {7, 4191}, {25, 954}, {37, 17441}, {38, 21320}, {42, 1284}, {55, 4415}, {226, 228}, {329, 1011}, {495, 1894}, {748, 3217}, {859, 5719}, {943, 3145}, {1621, 15507}, {1836, 15624}, {3011, 20967}, {3185, 17718}, {3219, 8731}, {3295, 4186}, {3487, 13738}, {3782, 5132}, {3868, 13731}, {3925, 4557}, {3927, 16455}, {4184, 17484}, {4196, 8232}, {4210, 17483}, {4365, 4433}, {6147, 16453}, {6910, 20805}, {11043, 17016}, {13734, 18446}, {15650, 16290}, {16343, 17257}, {16373, 18228}, {16577, 18210}, {20243, 20430}, {21028, 21091}, {21327, 21341}

X(21320) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(21320) lies on these lines: {1, 2810}, {9, 20662}, {37, 17447}, {38, 21319}, {42, 3123}, {55, 1633}, {56, 6068}, {100, 4440}, {144, 20992}, {190, 8299}, {320, 4447}, {518, 1284}, {527, 2223}, {536, 4433}, {545, 4436}, {1020, 1362}, {1086, 4557}, {1423, 3779}, {1463, 1818}, {1756, 9052}, {2183, 20358}, {2330, 18162}, {2809, 4516}, {3286, 5852}, {3675, 16578}, {3750, 13097}, {3768, 9267}, {3882, 14839}, {3939, 5091}, {4053, 9020}, {4367, 17467}, {4644, 21010}, {7277, 16679}, {8053, 17334}, {8609, 9004}, {8932, 19133}, {15624, 17276}, {16684, 17332}, {17365, 20990}, {17441, 21333}

X(21321) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a - b - c) (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c - b^3 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - b c^3) : :

X(21321) lies on these lines: {1, 181}, {2, 11}, {35, 19513}, {36, 14636}, {37, 21333}, {38, 21319}, {56, 5712}, {229, 4225}, {238, 8731}, {354, 1400}, {573, 10473}, {614, 2277}, {748, 992}, {846, 15507}, {958, 14555}, {978, 3601}, {1104, 1193}, {1284, 3666}, {2269, 3720}, {2352, 17723}, {3271, 17194}, {3706, 4433}, {3846, 4199}, {4192, 17717}, {4267, 4999}, {5718, 16678}, {5745, 20967}, {7004, 17611}, {7117, 16721}, {9552, 10454}, {9554, 10478}, {10448, 13724}, {16455, 16466}, {16687, 17726}, {17056, 20470}, {17441, 17447}

X(21322) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^4 b^2 - b^6 - a^4 b c + b^5 c + a^4 c^2 - b^4 c^2 + 2 b^3 c^3 - b^2 c^4 + b c^5 - c^6) : :

X(21322) lies on these lines: {1, 206}, {37, 17441}, {1108, 17463}, {2908, 20279}, {16580, 18210}

X(21323) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^4 b^2 - b^6 - a^4 b c + b^5 c + a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 + 2 b^3 c^3 - b^2 c^4 + b c^5 - c^6) : :

X(21323) lies on these lines: {1, 18374}, {37, 17441}, {2483, 4367}, {16581, 18210}

X(21324) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (b^4 - b^3 c + b^2 c^2 - b c^3 + c^4) : :

X(21324) lies on these lines: {1, 1501}, {37, 17456}, {2240, 4463}, {2611, 21345}, {3721, 4137}, {4118, 21329}, {17441, 21341}, {18203, 20859}

X(21324) = isogonal conjugate of isotomic conjugate of X(21409)

X(21325) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^2 b^3 - b^5 + a b^3 c + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 - c^5) : :

X(21325) lies on these lines: {38, 17447}, {2611, 3666}, {3720, 17463}, {4022, 4137}

X(21326) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^2 b^3 - b^5 - a b^3 c + b^3 c^2 + a^2 c^3 - a b c^3 + b^2 c^3 - c^5) : :

X(21326) lies on these lines: {38, 17447}, {518, 2611}, {4083, 20590}, {17449, 17463}, {20593, 21339}

X(21327) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^2 b^2 - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21327) lies on these lines: {1, 3051}, {2, 1978}, {37, 42}, {38, 20356}, {39, 17147}, {76, 17486}, {292, 4418}, {310, 19565}, {321, 2229}, {893, 17763}, {1015, 17140}, {1107, 3989}, {1125, 20868}, {1215, 3121}, {2275, 17155}, {2611, 17456}, {3120, 18905}, {3720, 20363}, {3721, 4137}, {3741, 8620}, {3774, 3896}, {3995, 20688}, {4039, 16600}, {4393, 5283}, {4972, 16587}, {7226, 16975}, {15523, 18904}, {20372, 20965}, {21319, 21341}

X(21328) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^3 b^2 + a^2 b^3 - a b^4 - b^5 - a^2 b^2 c + 2 a b^3 c - b^4 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - a c^4 - b c^4 - c^5) : :

X(21328) lies on these lines: {1, 2187}, {38, 17447}, {244, 20276}, {354, 1953}, {497, 18161}, {614, 2170}, {1401, 4516}, {1458, 1824}, {1473, 4336}, {1836, 3942}, {1959, 10453}, {3720, 17441}, {7069, 8679}, {17444, 21342}

X(21329) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a b^4 - b^5 + b^4 c + a c^4 + b c^4 - c^5) : :

X(21329) lies on these lines: {1, 9447}, {38, 20589}, {3708, 3735}, {3721, 21333}, {3726, 17471}, {4118, 21324}, {20593, 21318}

X(21329) = isogonal conjugate of isotomic conjugate of X(21414)

X(21330) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a b^3 - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3) : :

X(21330) lies on these lines: {1, 1258}, {2, 3728}, {37, 38}, {75, 244}, {141, 3122}, {192, 982}, {256, 17300}, {291, 17280}, {312, 17157}, {518, 1201}, {714, 18137}, {726, 3953}, {756, 4687}, {940, 8424}, {984, 3616}, {1015, 6378}, {1739, 4709}, {2228, 17231}, {3121, 6375}, {3123, 3662}, {3661, 17065}, {3670, 3993}, {3721, 17470}, {3722, 15624}, {3747, 16574}, {3764, 4851}, {3778, 3912}, {3840, 20891}, {4118, 17463}, {4358, 21080}, {4392, 4704}, {4443, 17234}, {4446, 17233}, {4484, 17267}, {4699, 17063}, {4772, 9335}, {12782, 17242}, {17243, 21035}, {17353, 20456}, {17447, 20590}, {17452, 20356}, {18040, 21238}

X(21331) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a b^2 - 2 b^3 + b^2 c + a c^2 + b c^2 - 2 c^3) : :

X(21331) lies on these lines: {1, 2251}, {37, 517}, {38, 1107}, {44, 758}, {760, 1279}, {798, 4083}, {1575, 1739}, {1707, 16968}, {2087, 4694}, {2170, 3726}, {2809, 4864}, {3061, 16604}, {3708, 20595}, {3834, 20924}, {3959, 20691}, {8682, 17374}, {17444, 17465}, {17466, 20362}, {17760, 20255}, {18061, 20530}, {20335, 21138}

X(21332) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (2 a b^2 - b^3 + 2 b^2 c + 2 a c^2 + 2 b c^2 - c^3) : :

X(21332) lies on these lines: {1, 9351}, {37, 374}, {38, 1107}, {39, 1739}, {45, 3877}, {142, 7200}, {551, 2087}, {993, 2243}, {1212, 2295}, {2275, 17063}, {3726, 16975}, {3727, 5283}, {3753, 20331}, {3780, 4875}, {3902, 3943}

X(21333) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a - b - c) (a b^3 + b^4 - b^3 c + a c^3 - b c^3 + c^4) : :

X(21333) lies on these lines: {1, 987}, {2, 20276}, {11, 321}, {22, 55}, {37, 21321}, {38, 17447}, {192, 497}, {354, 2171}, {855, 2292}, {1214, 12721}, {1776, 2098}, {2269, 3989}, {3056, 9017}, {3721, 21329}, {4516, 4847}, {7004, 20359}, {17441, 21320}, {17463, 21342}, {20590, 20594}

X(21333) = isogonal conjugate of isotomic conjugate of X(21420)

X(21334) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a - b - c) (a^2 b^2 + a b^3 + b^3 c + a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21334) lies on these lines: {1, 3}, {4, 10372}, {11, 1211}, {38, 17447}, {69, 350}, {72, 17733}, {210, 4042}, {314, 3706}, {333, 960}, {390, 20101}, {518, 1999}, {752, 3058}, {938, 10369}, {950, 10381}, {1125, 1682}, {1355, 3318}, {1364, 2792}, {1401, 3663}, {1463, 3782}, {1738, 3819}, {1836, 10401}, {1837, 10371}, {1864, 10477}, {2262, 5275}, {2269, 3720}, {2654, 8240}, {3022, 5579}, {3030, 20103}, {3061, 7075}, {3683, 17185}, {3685, 3794}, {3688, 4847}, {3721, 18671}, {3880, 3996}, {3914, 3917}, {3959, 21001}, {4415, 8679}, {5208, 8822}, {5933, 10580}, {6051, 18180}, {7186, 15310}, {7354, 12545}, {10444, 17635}, {10478, 17605}, {10479, 17606}, {10889, 14100}, {12544, 17634}, {12721, 17625}, {15569, 18165}, {18163, 20967}, {20594, 21352}

X(21334) = isogonal conjugate of isotomic conjugate of X(21422)
X(21334) = homothetic center of Ursa-minor triangle and inverse-in-Conway-circle triangle

X(21335) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^4 - b^4 - c^4) (a^4 + b^4 - 2 b^3 c + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(21335) lies on these lines: {1, 2353}, {2295, 21318}

X(21336) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b^2 + c^2) (a^2 b^2 + b^4 + a^2 c^2 + c^4) : :

X(21336) lies on these lines: {1, 82}, {38, 4020}, {304, 9239}, {3720, 17456}, {16735, 17446}

X(21337) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a b + a c - b c) (a b^3 - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(21337) lies on these lines: {1, 727}, {8, 192}, {38, 3727}, {756, 8026}, {3721, 17470}, {17451, 20591}, {20590, 20594}

X(21338) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a - 2 b - 2 c) (2 a b^2 + 2 b^3 - b^2 c + 2 a c^2 - b c^2 + 2 c^3) : :

X(21338) lies on these lines: {37, 517}, {38, 17444}, {44, 3899}, {3721, 17465}

X(21339) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c)^2 (a b^2 - b^3 + a b c - b^2 c + a c^2 - b c^2 - c^3) : :

X(21339) lies on these lines: {38, 20589}, {649, 7004}, {824, 20901}, {2170, 2611}, {3119, 4893}, {4137, 20591}, {17435, 18210}, {20593, 21326}

X(21340) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c)^2 (b + c) (a^2 b^2 - b^4 + a^2 b c - b^3 c + a^2 c^2 - b^2 c^2 - b c^3 - c^4) : :

X(21340) lies on these lines: {1, 1576}, {692, 16599}, {2611, 18191}, {2643, 3123}, {3708, 17463}, {4016, 20589}, {4466, 18188}, {17886, 21045}, {20902, 21252}

X(21341) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^4 - 2 a^2 b c + b^3 c - b^2 c^2 + b c^3) : :

X(21341) lies on these lines: {1, 3124}, {37, 2503}, {99, 21220}, {100, 16592}, {115, 149}, {661, 3722}, {1914, 10026}, {2240, 3936}, {3218, 20666}, {3720, 20362}, {17441, 21324}, {21319, 21327}

X(21342) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a b - 3 b^2 + a c + 2 b c - 3 c^2) : :

X(21342) lies on these lines: {1, 3052}, {2, 3999}, {6, 3677}, {37, 38}, {42, 4003}, {43, 518}, {44, 614}, {45, 10582}, {55, 4864}, {57, 3242}, {63, 1279}, {72, 3953}, {192, 4891}, {210, 244}, {222, 12595}, {238, 4906}, {321, 17154}, {392, 4694}, {497, 17276}, {536, 10453}, {537, 3840}, {596, 5295}, {612, 4860}, {674, 1401}, {748, 15492}, {910, 16973}, {960, 3976}, {984, 3742}, {1086, 4847}, {1100, 17599}, {1155, 3938}, {1201, 3962}, {1214, 5083}, {1254, 9850}, {1376, 16496}, {1386, 17598}, {1427, 17625}, {1616, 12526}, {1736, 17626}, {2292, 17609}, {3218, 3744}, {3219, 3315}, {3246, 7262}, {3445, 15829}, {3452, 3756}, {3555, 3670}, {3666, 3873}, {3681, 16610}, {3703, 17231}, {3706, 4686}, {3740, 17063}, {3748, 4414}, {3750, 15570}, {3772, 4310}, {3786, 16736}, {3794, 18211}, {3870, 17595}, {3881, 3931}, {3896, 17145}, {3912, 4884}, {3957, 4689}, {3961, 18201}, {3966, 17344}, {3979, 17593}, {4358, 20068}, {4388, 17345}, {4415, 11019}, {4419, 10580}, {4423, 16814}, {4430, 4850}, {4641, 7191}, {5208, 16696}, {5220, 5272}, {5223, 5573}, {5542, 17056}, {5604, 13389}, {5605, 13388}, {5905, 17721}, {7004, 17642}, {7174, 10980}, {12675, 15852}, {15481, 17123}, {16666, 17017}, {17444, 21328}, {17448, 20358}, {17463, 21333}

X(21343) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^2 - 2 a b - 2 a c + 3 b c) : :

X(21343) lies on these lines: {1, 659}, {8, 3837}, {513, 4895}, {514, 4775}, {523, 3904}, {649, 4083}, {693, 4774}, {764, 3887}, {812, 4922}, {900, 1120}, {1482, 2826}, {1491, 14077}, {1635, 9269}, {2170, 17463}, {2787, 4810}, {2821, 7982}, {2832, 6161}, {3762, 4800}, {3768, 9267}, {3777, 3900}, {3960, 4730}, {4378, 4784}, {4498, 8656}, {4728, 9260}, {6084, 10699}, {6085, 13541}, {9032, 16496}, {9508, 14413}

X(21343) = isogonal conjugate of isotomic conjugate of X(21433)

X(21344) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^2 b^2 - 2 b^4 - a^2 b c + 2 b^3 c + a^2 c^2 - b^2 c^2 + 2 b c^3 - 2 c^4) : :

X(21344) lies on these lines: {1, 14567}, {3721, 4137}, {17458, 20598}

X(21345) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b + c) (a^2 b^2 - 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21345) lies on these lines: {1, 1197}, {2, 20363}, {37, 42}, {38, 3727}, {39, 4970}, {306, 18904}, {321, 3121}, {740, 16584}, {893, 1999}, {1107, 4393}, {1953, 3726}, {2229, 4365}, {2275, 3210}, {2611, 21324}, {3666, 17027}, {3840, 6377}, {3896, 20691}, {3914, 18905}, {3971, 20688}, {4039, 16583}, {4085, 16587}, {4359, 16604}, {4493, 9025}, {8620, 17135}, {10453, 20284}, {16703, 16742}

X(21346) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :

X(21346) lies on these lines: {1, 1170}, {2, 21039}, {6, 9502}, {7, 2310}, {37, 38}, {57, 4319}, {241, 2293}, {244, 4000}, {344, 4712}, {614, 2257}, {756, 3475}, {774, 942}, {938, 1254}, {982, 3672}, {984, 11038}, {990, 1717}, {999, 1037}, {1418, 3000}, {1458, 5728}, {1736, 5542}, {1742, 7671}, {1953, 17463}, {2191, 8557}, {2292, 11518}, {2340, 15185}, {3012, 3946}, {3242, 3304}, {3333, 4327}, {3668, 11019}, {3670, 4356}, {3779, 20275}, {3953, 4353}, {4073, 4869}, {4328, 10980}, {4334, 10394}, {4336, 5228}, {4860, 7004}, {4864, 20323}, {8257, 8271}, {10390, 16676}, {10868, 17274}, {17278, 17728}, {17447, 17451}, {17452, 20358}

X(21347) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^4 b^2 - a^2 b^4 + a^4 b c - a^2 b^3 c + a^4 c^2 - a^2 b^2 c^2 + b^4 c^2 - a^2 b c^3 + 2 b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(21347) lies on these lines: {1, 3049}, {37, 2605}, {980, 17066}, {2484, 4367}, {2489, 15411}, {3709, 5283}, {17458, 20598}

X(21348) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(21348) lies on these lines: {1, 3063}, {2, 20906}, {9, 20980}, {37, 513}, {230, 231}, {514, 3709}, {522, 665}, {657, 4449}, {693, 21225}, {798, 4083}, {824, 905}, {918, 3669}, {1415, 7012}, {1459, 3287}, {1841, 16228}, {2254, 4171}, {2484, 4367}, {2517, 4140}, {3239, 7180}, {3261, 4885}, {3667, 4526}, {4378, 5283}, {4687, 20949}, {10581, 14282}, {14324, 21104}, {16777, 21007}, {17066, 20907}, {21099, 21260}

X(21348) = isogonal conjugate of isotomic conjugate of X(21438)
X(21348) = complement of X(20906)

X(21349) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (-a b^3 + a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(21349) lies on these lines: {649, 21350}, {786, 2533}, {798, 4083}, {876, 918}, {891, 17990}, {984, 6373}, {2170, 17463}, {2483, 4367}, {3805, 21123}, {3837, 4518}, {4435, 20364}, {4490, 4802}, {20949, 21051}

X(21350) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21350) lies on these lines: {1, 1980}, {649, 21349}, {824, 3005}, {984, 20983}, {4083, 20590}, {17458, 20598}

X(21351) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^2 b^4 + a^2 b^3 c + a^2 b^2 c^2 - b^4 c^2 + a^2 b c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(21351) lies on these lines: {1, 9426}, {2483, 4367}, {17458, 20598}

X(21352) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

Barycentrics a (a^2 b^2 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(21352) lies on these lines: {1, 2}, {37, 17445}, {38, 1107}, {75, 2309}, {86, 670}, {194, 17155}, {213, 20965}, {274, 17187}, {354, 17448}, {730, 20913}, {748, 2176}, {750, 21010}, {872, 17348}, {894, 9359}, {902, 16497}, {1010, 19816}, {1045, 17117}, {1206, 1613}, {1385, 19522}, {1740, 4699}, {1918, 16684}, {1953, 17470}, {1964, 3739}, {2223, 14096}, {2234, 4688}, {2308, 16476}, {2663, 17121}, {2667, 4852}, {3051, 20963}, {3112, 18091}, {3231, 16971}, {3242, 19586}, {3248, 4670}, {3778, 17049}, {3914, 20257}, {3948, 12263}, {3989, 5283}, {4118, 17443}, {4365, 17143}, {4423, 16969}, {4772, 16571}, {7032, 10436}, {8622, 20985}, {16975, 17449}, {17142, 21080}, {17379, 18194}, {17599, 20284}, {20594, 21334}

X(21352) = isogonal conjugate of isotomic conjugate of X(21443)

X(21353) = X(35)X(37)∩X(319)X(321)

Barycentrics a*(b+c)*(a^3+(b+c)*a^2-(b^2-b*c-c^2)*a-(b+c)*(b^2-c^2))*(a^3+(b+c)*a^2+(b^2+b*c-c^2)*a+(b+c)*(b^2-c^2)) : :

See César Lozada, Hyacinthos 28071.

X(21353) lies on these lines: {35, 37}, {319, 321}, {502, 594}, {1825, 8736}, {2171, 2594}

X(21353) = barycentric product X(i)*X(j) for these {i,j}: {1, 502}, {10, 267}, {37, 1029}, {321, 3444}
X(21353) = barycentric quotient X(i)/X(j) for these (i,j): (10, 20932), (25, 2906), (31, 501), (37, 2895), (42, 191), (213, 1030), (267, 86), (502, 75), (661, 21192), (756, 21081), (1029, 274), (1402, 8614), (1824, 451)
X(21353) = trilinear product X(i)*X(j) for these {i,j}: {6, 502}, {10, 3444}, {37, 267}, {42, 1029}
X(21353) = trilinear quotient X(i)/X(j) for these (i,j): (6, 501), (10, 2895), (19, 2906), (37, 191), (42, 1030), (267, 81), (321, 20932), (502, 2), (523, 21192), (594, 21081), (1029, 86), (1400, 8614), (1826, 451)

X(21354) = X(216)X(3463)∩X(324)X(17035)

Barycentrics (S^4-SB^2*SC^2)*(3*S^2+8*R^2*(2*R^2-SB-SW)-4*SC*SA+2*SB^2+SW^2)*(3*S^2+8*R^2*(2*R^2-SC-SW)-4*SA*SB+2*SC^2+SW^2) : :
Barycentrics a^2 cos A cos(B - C)/(sec A sec(B - C) - sec B sec(C - A) - sec C sec(A - B)) : :

See César Lozada, Hyacinthos 28071.

X(21354) lies on these lines: {216, 3463}, {324, 17035}

X(21354) = isogonal conjugate of polar conjugate of X(33664)
X(21354) = barycentric product X(5)*X(3463)
X(21354) = barycentric quotient X(51)/X(3462)

X(21355) = X(32)X(15372)∩X(39)X(1915)

Barycentrics (3*S^2-4*SC*SA+2*SB^2+SW^2)*(3*S^2-4*SA*SB+2*SC^2+SW^2)*(SA^2-SW^2) : :

See César Lozada, Hyacinthos 28071.

X(21355) lies on the cubic K354 and these lines: {32, 15372}, {39, 1915}, {1031, 7779}, {1916, 8856}, {4074, 7794}, {8041, 9482}

X(21355) = isogonal conjugate of isotomic conjugate of X(33665)
X(21355) = barycentric product X(i)*X(j) for these {i,j}: {39, 1031}, {141, 14370}, {1964, 18834}
X(21355) = barycentric quotient X(i)/X(j) for these (i,j): (32, 14885), (38, 20934), (39, 2896), (1031, 308), (1401, 17083), (1964, 16556)
X(21355) = trilinear product X(i)*X(j) for these {i,j}: {38, 14370}, {1031, 1964}
X(21355) = trilinear quotient X(i)/X(j) for these (i,j): (31, 14885), (38, 2896), (39, 16556), (141, 20934), (1031, 3112), (1964, 10329)

X(21356) = ISOTOMIC CONJUGATE OF X(18842)

Barycentrics a^2-5*b^2-5*c^2 : :
X(21356) = 5*X(2)-2*X(6), 2*X(2)+X(69), X(2)-4*X(141), 7*X(2)-X(193), 7*X(2)-4*X(597), X(2)+2*X(599), 4*X(2)-X(1992), 11*X(2)-8*X(3589), 8*X(2)-5*X(3618), 4*X(2)-7*X(3619), X(2)+5*X(3620), 19*X(2)-4*X(3629), 7*X(2)+8*X(3631), 7*X(2)-10*X(3763), 31*X(2)-16*X(6329), 13*X(2)-4*X(8584), 16*X(2)-X(11008), 5*X(2)+X(11160), 7*X(2)+2*X(15533)

See César Lozada, Hyacinthos 28074.

X(21356) lies on these lines: {2, 6}, {3, 11147}, {4, 7883}, {7, 17228}, {8, 17227}, {20, 11164}, {30, 10519}, {67, 9143}, {76, 5485}, {144, 17285}, {145, 17305}, {182, 15702}, {315, 598}, {344, 17272}, {346, 17273}, {376, 1352}, {487, 6449}, {488, 6450}, {511, 3545}, {518, 4731}, {542, 3524}, {543, 7865}, {547, 1351}, {549, 6776}, {575, 3533}, {576, 5067}, {620, 8593}, {626, 7617}, {631, 7870}, {671, 19662}, {903, 4437}, {1153, 3788}, {1350, 3543}, {1353, 10124}, {1444, 16431}, {1494, 15595}, {1503, 10304}, {2345, 17288}, {2393, 7998}, {2482, 11161}, {2548, 14762}, {3090, 5476}, {3098, 11001}, {3161, 17329}, {3241, 3416}, {3448, 5648}, {3523, 15069}, {3525, 7909}, {3529, 7936}, {3564, 5054}, {3616, 17387}, {3672, 17295}, {3751, 3828}, {3785, 8369}, {3818, 15682}, {3830, 18358}, {3839, 10516}, {3849, 5207}, {3912, 16676}, {3917, 11188}, {3926, 8359}, {3934, 8176}, {4000, 17287}, {4357, 16673}, {4419, 17230}, {4644, 17292}, {4663, 19877}, {4748, 17244}, {4916, 17396}, {5050, 11539}, {5055, 14853}, {5056, 11477}, {5071, 20423}, {5085, 15708}, {5092, 15719}, {5181, 9140}, {5182, 9167}, {5206, 7795}, {5222, 17360}, {5296, 17241}, {5308, 17250}, {5343, 11304}, {5344, 11303}, {5461, 10754}, {5569, 7880}, {5642, 11061}, {5650, 8681}, {5749, 17361}, {5839, 17291}, {5921, 15692}, {6030, 11206}, {6172, 17342}, {6337, 7618}, {6393, 11287}, {6453, 11292}, {6454, 11291}, {7492, 19596}, {7615, 9466}, {7619, 7815}, {7620, 7841}, {7757, 14994}, {7767, 19661}, {7768, 16045}, {7776, 8367}, {7794, 16043}, {7811, 14039}, {7849, 14064}, {7854, 14001}, {7879, 8370}, {7937, 11054}, {8542, 11416}, {8550, 10303}, {8591, 11646}, {8703, 18440}, {10385, 12589}, {11050, 12583}, {11318, 16509}, {11812, 12017}, {11898, 15694}, {14645, 14971}, {14848, 15699}, {14912, 15709}, {15585, 20079}, {15703, 18583}, {15705, 21167}, {16475, 19883}, {17132, 17274}, {17133, 17294}, {17231, 17257}, {17236, 17314}, {17237, 17316}, {17269, 20073}, {17270, 21255}, {17289, 21296}, {17296, 17321}, {17328, 18230}

X(21356) = reflection of X(i) in X(j) for these (i,j): (3839, 10516), (5050, 11539), (5182, 9167), (14848, 15699), (16475, 19883)
X(21356) = complement of X(5032)
X(21356) = isotomic conjugate of X(18842)
X(21356) = X(3524)-of-anti-Artzt-triangle
X(21356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 1992), (2, 1271, 13637), (2, 7840, 7736), (2, 11160, 6), (69, 141, 3619), (69, 3618, 11008), (69, 3619, 3618), (141, 3631, 3763), (298, 299, 15589), (597, 3631, 15533), (597, 15533, 193), (599, 15533, 3631), (3631, 3763, 193), (3763, 15533, 597), (7840, 16986, 2), (9761, 9763, 13468)

X(21357) = X(5)X(51)∩X(54)X(632)

Barycentrics ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-5*(b^2+c^2)*a^4+7*(b^4+b^2*c^2+c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)) : :
Barycentrics (S^2+SB*SC)*(4*SA+2*SW-3*R^2) : :
X(21357) = X(5)-4*X(1209), 7*X(5)-4*X(3574), 5*X(5)-8*X(13565), 5*X(5)-2*X(20424), X(5)+2*X(21230), 7*X(1209)-X(3574), 5*X(1209)-2*X(13565), 10*X(1209)-X(20424), 2*X(1209)+X(21230), 5*X(3574)-14*X(13565), 10*X(3574)-7*X(20424), 2*X(3574)+7*X(21230), 4*X(13565)-X(20424), 4*X(13565)+5*X(21230), X(20424)+5*X(21230)

See César Lozada, Hyacinthos 28074.

X(21357) lies on these lines: {5, 51}, {54, 632}, {140, 2888}, {195, 3628}, {539, 11539}, {546, 12307}, {547, 7605}, {549, 9140}, {550, 6288}, {597, 5965}, {1216, 13368}, {1483, 12785}, {1493, 15605}, {1656, 12325}, {3090, 12316}, {3519, 8254}, {3530, 12254}, {3627, 7691}, {3858, 20584}, {6101, 6153}, {6286, 10593}, {6343, 13372}, {7356, 10592}, {8703, 18400}, {10203, 18350}, {10386, 12956}, {10610, 14869}, {12013, 15699}

X(21357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1209, 21230, 5), (13565, 20424, 5)

X(21358) = MIDPOINT OF X(3545) AND X(10519)

Barycentrics a^2+4*b^2+4*c^2 : :
X(21358) = 4*X(2)-X(6), 5*X(2)+X(69), X(2)+2*X(141), 13*X(2)-X(193), 5*X(2)-2*X(597), 2*X(2)+X(599), 7*X(2)-X(1992), 7*X(2)-4*X(3589), 11*X(2)-5*X(3618), X(2)-7*X(3619), 7*X(2)+5*X(3620), 17*X(2)-2*X(3629), 11*X(2)+4*X(3631), 2*X(2)-5*X(3763), 5*X(2)-X(5032), 23*X(2)-8*X(6329), 11*X(2)-2*X(8584), 11*X(2)+X(11160), 8*X(2)+X(15533), 10*X(2)-X(15534)

See César Lozada, Hyacinthos 28074.

X(21358) lies on these lines: {2, 6}, {3, 11178}, {30, 10516}, {45, 17237}, {67, 5642}, {76, 10302}, {125, 5646}, {182, 9703}, {381, 1350}, {487, 6429}, {488, 6430}, {511, 5055}, {518, 3921}, {542, 5054}, {543, 11287}, {547, 20423}, {549, 1352}, {551, 3416}, {576, 5070}, {598, 7770}, {626, 8176}, {631, 11180}, {635, 11306}, {636, 11305}, {671, 7937}, {1078, 8366}, {1153, 7815}, {1351, 15703}, {1503, 3524}, {1656, 5476}, {1691, 5215}, {2023, 5503}, {2076, 3849}, {2393, 5650}, {2482, 11646}, {2930, 6698}, {2979, 16776}, {3053, 7810}, {3066, 15360}, {3094, 9466}, {3096, 7841}, {3098, 3830}, {3242, 3679}, {3525, 8550}, {3526, 10168}, {3534, 3818}, {3545, 10519}, {3564, 11539}, {3642, 11295}, {3643, 11296}, {3661, 17119}, {3662, 17118}, {3734, 5077}, {3751, 19876}, {3788, 7619}, {3834, 17308}, {3912, 16672}, {3917, 9971}, {3934, 7617}, {4265, 16418}, {4357, 16675}, {4361, 17228}, {4363, 17227}, {4389, 17269}, {4445, 16706}, {4657, 17311}, {4995, 12589}, {5013, 6292}, {5017, 7818}, {5023, 7800}, {5026, 10488}, {5064, 7716}, {5071, 5480}, {5092, 15701}, {5096, 16417}, {5102, 14848}, {5116, 7622}, {5152, 9830}, {5207, 14036}, {5210, 7820}, {5254, 5485}, {5298, 12588}, {5463, 11481}, {5464, 11480}, {5569, 11288}, {5651, 18374}, {5847, 19883}, {5921, 15721}, {5969, 9166}, {6034, 14971}, {6379, 9462}, {6593, 13169}, {6697, 9924}, {6776, 15702}, {7232, 17289}, {7603, 11173}, {7618, 7795}, {7738, 11148}, {7746, 10542}, {7757, 10007}, {7761, 11159}, {7775, 7849}, {7784, 8370}, {7812, 7879}, {7814, 9731}, {7817, 7914}, {7819, 19661}, {7833, 11164}, {7845, 14535}, {7870, 11285}, {7998, 9019}, {8288, 10717}, {8360, 13881}, {8681, 15082}, {8703, 18358}, {9813, 10510}, {10150, 15514}, {10304, 21167}, {10387, 11238}, {10719, 15162}, {10720, 15163}, {11049, 12583}, {11284, 19510}, {11301, 13083}, {11302, 13084}, {11898, 15723}, {14046, 18906}, {14561, 15699}, {14810, 15681}, {15303, 16176}, {15693, 18440}, {15707, 17508}, {16041, 20112}, {16777, 17231}, {16884, 17296}, {16885, 17272}, {17132, 17281}, {17133, 17301}, {17229, 17304}, {17230, 17305}, {17233, 17323}, {17235, 17286}, {17236, 17262}, {17239, 17282}, {17240, 17324}, {17241, 17326}, {17249, 17268}, {17250, 17266}, {17252, 17341}, {17253, 17279}, {17254, 17342}, {17255, 17280}, {17270, 17356}, {17273, 17358}, {17287, 17370}, {17288, 17371}, {17294, 17382}, {17295, 17383}, {17298, 17385}, {17302, 17309}, {17303, 21255}, {17310, 17399}, {17312, 17400}

X(21358) = midpoint of X(3545) and X(10519)
X(21358) = reflection of X(i) in X(j) for these (i,j): (1691, 5215), (5102, 14848), (6034, 14971), (10304, 21167), (14561, 15699)
X(21358) = X(5054)-of-anti-Artzt-triangle
X(21358) = X(9166)-of-1st Brocard- triangle
X(21358) = X(11287)-of-Artzt-triangle
X(21358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 597), (2, 491, 13663), (2, 492, 13783), (2, 7840, 11174), (2, 11160, 3618), (69, 597, 15534), (599, 3763, 2), (3618, 3631, 6144), (3618, 11160, 8584), (3631, 8584, 11160), (7840, 16988, 2), (8252, 8253, 3054), (8584, 11160, 6144), (9761, 9763, 8667), (16644, 16645, 230), (17232, 17307, 15668)

X(21359) = X(2)X(18)∩X(13)X(298)

Barycentrics 5*S^2+sqrt(3)*(3*SA+SW)*S+3*SB*SC : :
X(21359) = X(13)+2*X(298), X(13)-4*X(623), X(298)+2*X(623), 2*X(316)+X(6779), 2*X(616)+X(19106), 2*X(618)+X(621), 4*X(620)-X(6780), X(3180)-4*X(6669), 2*X(3180)-5*X(16960), X(5473)+2*X(20428), 2*X(5978)+X(6777), 8*X(6669)-5*X(16960)

See César Lozada, Hyacinthos 28074.

X(21359) lies on these lines: {2, 18}, {13, 298}, {14, 11297}, {17, 5859}, {30, 5463}, {299, 16966}, {302, 3642}, {316, 6779}, {381, 5864}, {524, 16267}, {599, 1351}, {616, 19106}, {618, 621}, {620, 6780}, {754, 9761}, {3180, 6669}, {3411, 11289}, {3545, 7685}, {3643, 16809}, {5054, 5464}, {5858, 7759}, {5978, 6777}, {7860, 11303}, {11299, 16964}

X(21359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (298, 623, 13), (302, 3642, 16242), (3180, 6669, 16960), (9761, 11298, 16963)

X(21360) = X(2)X(17)∩X(14)X(299)

Barycentrics 5*S^2-sqrt(3)*(3*SA+SW)*S+3*SB*SC : :
X(21360) = X(14)+2*X(299), X(14)-4*X(624), X(299)+2*X(624), 2*X(316)+X(6780), 2*X(617)+X(19107), 2*X(619)+X(622), 4*X(620)-X(6779), X(3181)-4*X(6670), 2*X(3181)-5*X(16961), X(5474)+2*X(20429), 2*X(5979)+X(6778), 8*X(6670)-5*X(16961)

See César Lozada, Hyacinthos 28074.

X(21360) lies on these lines: {2, 17}, {13, 11298}, {14, 299}, {18, 5858}, {30, 5464}, {298, 16967}, {303, 3643}, {316, 6780}, {381, 5865}, {524, 16268}, {599, 1351}, {617, 19107}, {619, 622}, {620, 6779}, {754, 9763}, {3181, 6670}, {3412, 11290}, {3545, 7684}, {3642, 16808}, {5054, 5463}, {5859, 7759}, {5979, 6778}, {7860, 11304}, {11300, 16965}

X(21360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (299, 624, 14), (303, 3643, 16241), (3181, 6670, 16961), (9763, 11297, 16962)

X(21361) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(21361) lies on these lines: {1, 51}, {9, 440}, {37, 14557}, {40, 17757}, {43, 1756}, {57, 17365}, {63, 5741}, {200, 6210}, {223, 1020}, {226, 1730}, {312, 3882}, {329, 573}, {408, 1745}, {517, 22014}, {908, 1764}, {954, 17810}, {1423, 2999}, {1698, 15666}, {1736, 17441}, {2269, 4656}, {3185, 4551}, {3191, 5752}, {3198, 5927}, {3294, 5813}, {3305, 21370}, {3452, 22097}, {3588, 22000}, {4271, 4415}, {5739, 21061}, {5905, 20367}, {7291, 17292}, {9548, 12526}, {17720, 18163}, {21369, 21383}

X(21362) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a - b) (a - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21362) lies on these lines: {1, 2810}, {9, 141}, {37, 7202}, {40, 6068}, {63, 5741}, {100, 1293}, {101, 651}, {144, 573}, {169, 8545}, {190, 646}, {329, 1764}, {513, 4557}, {514, 4552}, {527, 2183}, {662, 1019}, {1023, 1332}, {1025, 4763}, {1026, 3888}, {1282, 9355}, {1423, 1743}, {1633, 3939}, {1634, 3737}, {1730, 5905}, {1756, 1757}, {1763, 21375}, {2310, 2809}, {2347, 3663}, {3216, 20805}, {3729, 3765}, {3730, 6172}, {3942, 16578}, {4069, 4553}, {4266, 4419}, {4271, 17334}, {4415, 18163}, {4416, 15983}, {4579, 6163}, {5223, 6210}, {7277, 18164}, {9945, 16528}, {10468, 17257}, {16574, 17347}, {17194, 21319}, {18206, 20072}

X(21362) = isogonal conjugate of isotomic conjugate of X(21580)
X(21362) = cevapoint of Mandart hyperbola intercepts of antiorthic axis

X(21363) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(21363) lies on these lines: {1, 181}, {2, 573}, {3, 1724}, {5, 40}, {8, 9568}, {9, 5743}, {42, 10440}, {43, 10434}, {51, 17194}, {57, 4888}, {63, 5741}, {71, 3452}, {165, 2108}, {226, 20367}, {312, 1018}, {386, 10470}, {946, 16828}, {978, 10882}, {1020, 17080}, {1125, 9569}, {1385, 5754}, {1695, 12435}, {1715, 6825}, {1746, 17277}, {1754, 19544}, {1756, 17596}, {1763, 16551}, {1766, 3305}, {2050, 19732}, {2183, 5745}, {2270, 5742}, {3430, 6986}, {3576, 5313}, {3579, 19648}, {3624, 10476}, {3687, 21061}, {3730, 18228}, {3882, 14829}, {3911, 22097}, {4220, 13329}, {4271, 18163}, {4417, 16574}, {4551, 16678}, {5278, 13478}, {5753, 11227}, {5943, 8731}, {7991, 15488}, {8703, 16528}, {9534, 10454}, {9566, 10441}, {9840, 15489}, {10888, 16832}, {13571, 18206}, {14555, 16552}, {17308, 20606}, {17825, 21483}, {18662, 22002}, {21273, 22022}

X(21364) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^7 + a^4 b^3 - a^3 b^4 - b^7 + a^4 c^3 + b^4 c^3 - a^3 c^4 + b^3 c^4 - c^7) : :

X(21364) lies on these lines: {1, 206}, {9, 440}, {19, 17861}, {35, 984}, {63, 19835}, {313, 1760}, {1716, 21381}, {1723, 16560}, {6211, 17857}, {7291, 21270}

X(21365) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^7 + a^4 b^3 - a^3 b^4 - b^7 + a^3 b^2 c^2 - a^2 b^3 c^2 + a^4 c^3 - a^2 b^2 c^3 + b^4 c^3 - a^3 c^4 + b^3 c^4 - c^7) : :

X(21365) lies on these lines: {1, 18374}, {9, 440}, {918, 1019}

X(21366) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - b^5 - c^5) : :

X(21366) lies on these lines: {1, 1501}, {6, 18203}, {9, 16555}, {57, 7217}, {1726, 1759}, {1763, 21383}, {4177, 21275}, {21381, 21387}

X(21367) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^3 b c + a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 - c^5) : :

X(21367) lies on these lines: {1, 21325}, {2, 1762}, {40, 5178}, {63, 321}, {572, 16585}, {1746, 14206}, {1782, 6734}, {1931, 3497}, {1993, 18161}, {2161, 3782}, {2182, 18607}, {2939, 16451}, {3219, 17280}, {3681, 6211}, {5773, 18662}, {7193, 21318}, {16574, 21376}, {17596, 21381}

X(21368) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - a^3 b^2 + a^2 b^3 - b^5 + a^3 b c - a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - a b c^3 + b^2 c^3 - c^5) : :

X(21368) lies on these lines: {1, 21326}, {10, 1710}, {40, 5086}, {46, 21935}, {57, 15999}, {63, 321}, {71, 1654}, {81, 2171}, {100, 2708}, {212, 20243}, {655, 16548}, {1746, 14213}, {1757, 21381}, {1760, 17787}, {1790, 16577}, {1820, 5392}, {3218, 4440}, {3955, 21318}, {4063, 4380}, {4641, 21853}, {20605, 21382}

X(21369) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^3 b^2 - a^2 b^3 + a^3 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3) : :

X(21369) lies on these lines: {1, 3051}, {2, 20372}, {9, 43}, {31, 18098}, {37, 1197}, {63, 17026}, {239, 3219}, {321, 2225}, {869, 3294}, {1726, 1759}, {2161, 2319}, {2235, 16584}, {3507, 17744}, {3508, 3961}, {3923, 5364}, {4362, 5282}, {4418, 16549}, {16555, 21381}, {16717, 18792}, {21345, 21760}, {21361, 21383}

X(21370) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 + a^4 b - a b^4 - b^5 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a b c^3 + 2 b^2 c^3 - a c^4 - b c^4 - c^5) : :

X(21370) lies on these lines: {1, 2187}, {2, 169}, {3, 3198}, {9, 20205}, {10, 7386}, {19, 57}, {22, 2000}, {33, 3220}, {40, 4847}, {58, 4206}, {63, 321}, {84, 1753}, {142, 9816}, {222, 2182}, {223, 18725}, {226, 7289}, {241, 11347}, {962, 16388}, {990, 1473}, {1210, 7713}, {1214, 15509}, {1422, 1461}, {1445, 1730}, {1448, 4185}, {1498, 5908}, {1708, 16560}, {1752, 16545}, {1760, 14829}, {1782, 5709}, {2003, 2261}, {2082, 2999}, {2270, 8808}, {3101, 5744}, {3187, 5773}, {3218, 19789}, {3305, 21361}, {3687, 20606}, {3732, 20921}, {3741, 12514}, {3928, 16548}, {4191, 15496}, {4383, 14557}, {4498, 6545}, {5437, 16547}, {5745, 10319}, {6211, 20588}, {6996, 18750}, {9799, 16389}

X(21371) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^3 b - a b^3 + a^3 c - a^2 b c + a b^2 c - b^3 c + a b c^2 - a c^3 - b c^3) : :

X(21371) lies on these lines: {1, 1258}, {2, 7}, {19, 264}, {40, 3685}, {43, 3778}, {46, 3831}, {69, 2183}, {71, 344}, {78, 10477}, {169, 17739}, {190, 20923}, {193, 2347}, {198, 20769}, {238, 16048}, {312, 4019}, {330, 16827}, {573, 3912}, {936, 3786}, {978, 1757}, {1054, 16571}, {1193, 3751}, {1405, 15988}, {1444, 2267}, {1451, 1791}, {1471, 2975}, {1716, 2239}, {1730, 11679}, {1743, 18186}, {1759, 16566}, {1760, 16560}, {1766, 17738}, {1958, 11349}, {2082, 17033}, {2245, 17279}, {2260, 3618}, {2268, 21511}, {2269, 17316}, {2282, 5271}, {2310, 12530}, {3056, 4447}, {3169, 6542}, {3208, 17242}, {3501, 17280}, {3729, 20367}, {3879, 4266}, {3882, 17296}, {3886, 17751}, {4225, 4855}, {4271, 4851}, {4384, 21061}, {4645, 6210}, {5036, 17267}, {5287, 19734}, {5783, 16412}, {6762, 20036}, {7155, 20368}, {8680, 20927}, {10475, 19861}, {16551, 20602}, {16569, 20984}, {16709, 17335}, {17022, 17185}, {18141, 22097}

X(21372) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (2 a^3 - a^2 b + a b^2 - 2 b^3 - a^2 c + b^2 c + a c^2 + b c^2 - 2 c^3) : :

X(21372) lies on these lines: {1, 2251}, {9, 484}, {44, 1739}, {63, 169}, {517, 1023}, {758, 2246}, {798, 812}, {910, 5440}, {1018, 5011}, {2082, 17736}, {3294, 3496}, {3509, 5540}, {16547, 21061}, {16548, 16561}, {16562, 20607}, {16563, 20371}, {16600, 17469}

X(21373) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (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(21373) lies on these lines: {1, 9351}, {9, 80}, {44, 3753}, {63, 169}, {672, 6205}, {993, 2246}, {1023, 3872}, {1211, 7308}, {1212, 5440}, {2082, 3294}, {2348, 16788}, {3732, 16551}, {5437, 16572}, {17736, 21384}

X(21374) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^6 - a^4 b^2 + a^2 b^4 - b^6 - a^4 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 - c^6) : :

X(21374) lies on these lines: {1, 163}, {19, 91}, {47, 17442}, {48, 2158}, {57, 20267}, {63, 1930}, {169, 21382}, {255, 18669}, {1725, 1973}, {1726, 1729}, {1755, 16545}, {1759, 19807}, {1760, 3403}, {1782, 8558}, {2172, 2312}

X(21374) = isogonal conjugate of isotomic conjugate of X(21593)

X(21375) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - a^3 b^2 + a^2 b^3 - b^5 + 2 a^3 b c - 2 a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 - c^5) : :

X(21375) lies on these lines: {1, 987}, {9, 5743}, {20, 16086}, {30, 40}, {57, 3782}, {63, 321}, {75, 1746}, {165, 6211}, {504, 556}, {517, 4641}, {545, 3928}, {573, 3219}, {846, 10434}, {894, 10478}, {1018, 3719}, {1046, 12435}, {1331, 20243}, {1708, 20367}, {1759, 19807}, {1763, 21362}, {1768, 20368}, {2267, 16579}, {2944, 10882}, {3430, 12528}, {5294, 12610}, {5816, 19822}, {6763, 10476}, {7283, 10454}, {9535, 17350}, {10860, 20601}, {17595, 19517}, {20602, 20606}

X(21376) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 + a^4 b - a b^4 - b^5 + a^4 c + a^3 b c - a b^3 c - b^4 c - a b c^3 - a c^4 - b c^4 - c^5) : :

X(21376) lies on these lines: {1, 2206}, {2, 1781}, {6, 18202}, {8, 9536}, {10, 191}, {19, 27}, {40, 5300}, {78, 2939}, {306, 2897}, {321, 16548}, {329, 14543}, {612, 846}, {1444, 16585}, {1726, 1759}, {1790, 1959}, {2185, 18714}, {2941, 4427}, {3336, 20083}, {3652, 9958}, {3916, 9895}, {4001, 7291}, {4362, 20375}, {4463, 5285}, {5278, 16547}, {5540, 19742}, {6763, 17155}, {16574, 21367}, {17147, 21378}, {20603, 21382}

X(21377) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^9 + a^8 b - a b^8 - b^9 + a^8 c - 2 a^4 b^4 c + b^8 c - 2 a^4 b c^4 + 2 a b^4 c^4 - a c^8 + b c^8 - c^9) : :

X(21377) lies on these lines: {1, 2353}, {57, 7251}, {1726, 16549}, {1730, 16544}, {2156, 16607}

X(21378) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (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(21378) lies on these lines: {1, 82}, {2, 16555}, {31, 14821}, {63, 16545}, {304, 16563}, {1930, 16546}, {2896, 16566}, {17147, 21376}

X(21379) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^4 b^2 - a^2 b^4 - 2 a^4 b c + 2 a b^4 c + a^4 c^2 + a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4) : :

X(21379) lies on these lines: {1, 727}, {63, 194}, {169, 20603}, {191, 6048}, {1759, 16566}, {20602, 20606}

X(21380) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (4 a^4 + 2 a^3 b - 2 a b^3 - 4 b^4 + 2 a^3 c + 5 a^2 b c - 5 a b^2 c - 2 b^3 c - 5 a b c^2 + 4 b^2 c^2 - 2 a c^3 - 2 b c^3 - 4 c^4) : :

X(21380) lies on these lines: {1, 21338}, {9, 484}, {63, 4659}, {846, 16676}, {1449, 1572}, {1759, 16561}, {15228, 17281}

X(21381) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5) : :

X(21381) is the perspector of the excentral triangle and the reflection of the Fuhrmann triangle in line X(5620)X(10265) (the perspectrix of excentral and Fuhrmann triangles). (Randy Hutson, August 13, 2020)

X(21381) lies on these lines: {1, 60}, {8, 191}, {9, 2503}, {11, 2607}, {43, 1726}, {46, 3464}, {55, 846}, {57, 1365}, {63, 13174}, {165, 6011}, {267, 3336}, {517, 14663}, {523, 2606}, {662, 16598}, {976, 5497}, {1046, 1710}, {1048, 3460}, {1054, 2640}, {1707, 13221}, {1716, 21364}, {1757, 21368}, {2161, 17719}, {2608, 4551}, {2629, 9355}, {2641, 21899}, {2652, 13610}, {2957, 5400}, {3218, 20369}, {3219, 21085}, {3509, 20607}, {3737, 7136}, {4039, 20602}, {4362, 20375}, {4934, 5972}, {5524, 6211}, {5540, 16562}, {6089, 12078}, {7701, 12407}, {16555, 21369}, {17596, 21367}, {21004, 21890}, {21098, 21221}, {21366, 21387}

X(21381) = isogonal conjugate of X(39137)
X(21381) = crosssum of X(523) and X(5954)
X(21381) = crossdifference of every pair of points on line X(2610)X(21341)
X(21381) = Gibert-Burek-Moses concurrent circles image of X(110)
X(21381) = trilinear product X(i)*X(j) for these {i,j}: {2, 21004}, {4, 22156}, {6, 21221}, {31, 20951}, {58, 21098}, {81, 21890}, {101, 21209}, {661, 39054}

X(21382) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^5 - a^4 b + a b^4 - b^5 - a^4 c + a^3 b c - a b^3 c + b^4 c - a b c^3 + a c^4 + b c^4 - c^5) : :

X(21382) lies on these lines: {1, 21339}, {6, 18210}, {9, 124}, {63, 15487}, {169, 21374}, {649, 1768}, {1708, 2156}, {2082, 2083}, {5540, 16562}, {17435, 20999}, {20603, 21376}, {20605, 21368}

X(21383) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a - b) (a - c) (b + c) (a^2 + a b + b^2 + a c - b c + c^2) : :

X(21383) lies on these lines: {1, 3124}, {2, 20371}, {9, 2503}, {100, 661}, {645, 3570}, {1018, 3952}, {1054, 16592}, {1211, 3512}, {1763, 21366}, {1983, 2610}, {3509, 10026}, {4557, 7239}, {4988, 6758}, {21361, 21369}

X(21383) = isogonal conjugate of isotomic conjugate of X(21604)

X(21384) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a^2 b - a b^2 + a^2 c + a b c - b^2 c - a c^2 - b c^2) : :

X(21384) lies on these lines: {1, 6}, {2, 1475}, {3, 3684}, {7, 17050}, {8, 672}, {10, 4253}, {21, 2280}, {39, 43}, {41, 2975}, {57, 85}, {63, 194}, {65, 4875}, {71, 3169}, {76, 17026}, {101, 8666}, {145, 1334}, {169, 3509}, {171, 5021}, {190, 17144}, {200, 2223}, {210, 21010}, {292, 979}, {304, 17755}, {330, 16827}, {391, 1400}, {517, 4051}, {519, 3208}, {527, 17753}, {572, 2304}, {573, 4297}, {579, 3686}, {583, 17275}, {604, 2287}, {612, 20985}, {966, 2260}, {978, 2238}, {980, 2999}, {982, 16583}, {993, 4251}, {1015, 21214}, {1018, 3632}, {1202, 5273}, {1220, 2279}, {1376, 5022}, {1423, 4416}, {1445, 7176}, {1468, 5276}, {1573, 17750}, {1575, 6048}, {1655, 17027}, {1759, 5540}, {2170, 3869}, {2241, 8616}, {2262, 15656}, {2269, 4313}, {2276, 3780}, {2285, 16824}, {2319, 20368}, {2347, 20036}, {3214, 17756}, {3218, 16816}, {3219, 4393}, {3290, 3976}, {3305, 16826}, {3306, 16815}, {3616, 17474}, {3679, 16549}, {3705, 4109}, {3729, 17143}, {3813, 17747}, {3868, 17451}, {3873, 21808}, {3880, 21872}, {3928, 5792}, {3929, 16834}, {3941, 22271}, {4095, 4737}, {4262, 5267}, {4266, 4700}, {4352, 5222}, {4378, 21390}, {4520, 5919}, {4769, 19987}, {5179, 10916}, {5279, 19851}, {5437, 5737}, {5750, 19853}, {6210, 6776}, {7119, 7719}, {7308, 16831}, {9305, 9441}, {10436, 16819}, {10453, 21071}, {15985, 17272}, {16284, 21232}, {16569, 16606}, {16818, 17306}, {17030, 17499}, {17033, 21226}, {17175, 18164}, {17736, 21373}

X(21384) = isogonal conjugate of isotomic conjugate of X(20923)
X(21384) = complement of X(36854)
X(21384) = intouch-to-excentral similarity image of X(85)

X(21385) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^2 + a b + a c - 3 b c) : :

X(21385) lies on these lines: {1, 659}, {9, 6009}, {40, 2826}, {88, 1022}, {190, 646}, {239, 514}, {484, 6550}, {513, 3245}, {764, 9508}, {900, 5541}, {905, 2516}, {1577, 18071}, {1698, 3837}, {1706, 4925}, {2254, 2832}, {2802, 13266}, {2821, 7991}, {3125, 7202}, {3226, 3227}, {4040, 4083}, {4145, 4491}, {4378, 4782}, {4382, 4791}, {4401, 4449}, {4435, 8658}, {5540, 6084}, {6264, 19916}, {8712, 14349}, {9404, 21120}

X(21385) = isogonal conjugate of isotomic conjugate of X(21606)

X(21386) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (2 a^5 - a^3 b^2 + a^2 b^3 - 2 b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - 2 c^5) : :

X(21386) lies on these lines: {1, 14567}, {802, 18197}, {1726, 1759}

X(21387) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (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(21387) lies on these lines: {1, 1197}, {9, 43}, {19, 2319}, {41, 11688}, {46, 8866}, {57, 6063}, {63, 194}, {200, 3508}, {238, 21775}, {740, 7075}, {869, 968}, {1740, 16584}, {3501, 5364}, {3507, 17742}, {3684, 20760}, {3980, 17754}, {5839, 20785}, {16569, 20372}, {17490, 20459}, {20363, 21001}, {21366, 21381}

X(21388) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a + b) (a - b - c) (b - c) (a + c) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(21388) lies on these lines: {1, 3049}, {9, 3287}, {57, 17066}, {63, 4374}, {163, 5379}, {243, 522}, {645, 4584}, {657, 4560}, {798, 6002}, {802, 18197}, {1019, 2484}, {3501, 21958}, {5214, 21061}, {5742, 8819}, {17218, 18206}

X(21389) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^3 + a b^2 + a b c - b^2 c + a c^2 - b c^2) : :

X(21389) lies on these lines: {1, 1919}, {9, 20979}, {19, 1024}, {37, 4057}, {169, 21198}, {514, 2484}, {522, 649}, {523, 2483}, {657, 4498}, {659, 3709}, {667, 21348}, {798, 812}, {802, 18197}, {824, 1019}, {1052, 5540}, {1766, 3667}, {2515, 4802}, {3063, 4083}, {3250, 3737}, {3287, 6371}, {3661, 21304}, {3875, 20370}, {4040, 4079}, {4132, 21007}, {4139, 4435}, {4369, 4509}, {4394, 6588}, {8632, 21834}, {10436, 21191}, {17308, 21262}, {18196, 18206}, {21099, 21301}, {21123, 21173}

X(21390) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^3 - 2 a^2 b + a b^2 - 2 a^2 c + a b c - b^2 c + a c^2 - b c^2) : :

X(21390) lies on these lines: {1, 3063}, {9, 513}, {37, 21007}, {514, 657}, {523, 22108}, {649, 3239}, {650, 1758}, {665, 3287}, {667, 3508}, {798, 812}, {1019, 2484}, {1021, 5214}, {1447, 21195}, {1743, 20980}, {3294, 4079}, {3305, 4776}, {3309, 4130}, {3709, 4040}, {3737, 6586}, {3887, 4171}, {4148, 4404}, {4378, 21384}, {4384, 20906}, {4827, 14077}, {5280, 22157}, {17212, 18206}, {17277, 20949}, {17797, 21061}

X(21390) = isogonal conjugate of isotomic conjugate of X(21611)

X(21391) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^4 + a b^3 - 2 a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21391) lies on these lines: {1, 21349}, {649, 2786}, {659, 17990}, {665, 3512}, {798, 812}, {846, 890}, {918, 1019}, {984, 21003}, {1757, 6373}, {5540, 6084}

X(21392) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (b - c) (a^4 + a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21392) lies on these lines: {1, 1980}, {649, 2786}, {802, 18197}, {846, 8640}, {984, 21005}, {1757, 20983}, {4063, 4380}

X(21393) = (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(1)

Barycentrics a (a + b) (b - c) (a + c) (a^3 b - a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21393) lies on these lines: {1, 9426}, {798, 18160}, {802, 18197}, {918, 1019}

X(21394) = CIRCUMCIRCLE-INVERSE OF X(19552)

Trilinears (1-2*cos(2*B))*(1-2*cos(2*C))*(cos(4*A)*cos(B-C)-cos(3*A)*cos(2*A)) : :
Barycentrics (3*S^2-SC^2)*(3*S^2-SB^2)*(7*S^4+(-8*R^2*(2*R^2+3*SA-SW)+7*SA^2-8*SB*SC-SW^2)*S^2+(4*R^2-SW)*SA^2*SW)*(SB+SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28079.

X(21394) lies on the these lines: {3, 3432}, {93, 186}, {930, 7488}

X(21394) = isogonal conjugate of antigonal conjugate of X(34418)
X(21394) = circumcircle-inverse of X(19552)

X(21395) = CIRCUMCIRCLE-INVERSE OF X(316)

Barycentrics (a^8-2*(b^2+c^2)*a^6+b^2*c^2*a^4+2*(b^6+c^6)*a^2-b^8-c^8+(b^4+b^2*c^2+c^4)*b^2*c^2)*a^2 : :
Tripolars a^2 Sqrt[2(b^2 + c^2) - a^2] : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28079.

X(21395) lies on the these lines: {3, 316}, {69, 12584}, {76, 7488}, {99, 7502}, {186, 264}, {325, 14558}, {1078, 1658}, {3455, 10512}, {5866, 7512}, {5961, 20573}, {5971, 6636}, {7492, 14360}, {7556, 11185}, {7799, 9723}, {10298, 13219}

X(21395) = 2nd Brocard circle-inverse-of X(7802)
X(21395) = circumcircle-inverse-of X(316)

X(21396) = X(4)X(13293)∩X(186)X(1093)

Trilinears cos(B)^2*cos(C)^2*((6*cos(2*A)+2*cos(4*A)+3)*cos(B-C)-10*cos(A)-6*cos(3*A)-cos(5*A)) : :
Barycentrics a^2*(a^2+b^2-c^2)^2*(a^2-b^2+c^2)^2*(a^12-4*(b^2+c^2)*a^10+5*(b^4+3*b^2*c^2+c^4)*a^8-20*b^2*c^2*(b^2+c^2)*a^6-(5*b^8+5*c^8-b^2*c^2*(10*b^4+23*b^2*c^2+10*c^4))*a^4+2*(b^2+c^2)*(2*b^8+2*c^8-b^2*c^2*(2*b^4+3*b^2*c^2+2*c^4))*a^2-(b^2-c^2)^2*(b^8+c^8+3*b^2*c^2*(b^4+b^2*c^2+c^4))) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28079.

X(21396) lies on the these lines: {4, 13293}, {186, 1093}, {2071, 18848}

X(21396) = circumcircle-inverse of X(34170)

X(21397) = CIRCUMCIRCLE-INVERSE OF X(5523)

Trilinears cos(B)*cos(C)*((4*cos(2*A)+8)*cos(B-C)+(-2*cos(A)-2*cos(3*A))*cos(2*(B-C))+7*cos(3*A)+cos(5*A)+8*cos(A))*sin(A) : :
Barycentrics a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^10-3*(b^2+c^2)*a^8+2*(b^4+6*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^4-(3*b^8+3*c^8-2*(b^2+2*c^2)*(2*b^2+c^2)*b^2*c^2)*a^2+(b^8-c^8)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28079.

X(21397) lies on the these lines: {3, 5523}, {6, 15463}, {24, 187}, {25, 5913}, {112, 6644}, {115, 378}, {186, 393}, {216, 12096}, {1560, 1995}, {8743, 17928}, {15655, 21284}

X(21397) = circumcircle-inverse-of X(5523)

X(21398) = X(1)X(6924)∩X(4)X(11009)

Barycentrics a*(a^3-(b+2*c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-2*c))*(a^3-(2*b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(2*b-c)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28080.

X(21398) lies on the Feuerbach hyperbola and these lines: {1, 6924}, {4, 11009}, {8, 3814}, {9, 17444}, {21, 5697}, {35, 2320}, {79, 2099}, {80, 1482}, {90, 7982}, {104, 5903}, {145, 11604}, {484, 11279}, {498, 1000}, {517, 15446}, {942, 15180}, {1476, 5902}, {1656, 2098}, {3057, 15175}, {3065, 11280}, {3255, 5441}, {3295, 5424}, {3296, 5425}, {3340, 7284}, {5553, 10483}, {6596, 12653}, {6598, 10912}, {6874, 12647}, {7162, 7962}, {15179, 18398}

X(21399) = X(5097)X(8541)∩X(11002)X(14906)

Barycentrics a^2*(a^4-2*(2*b^2+c^2)*a^2-4*b^2*c^2+c^4+5*b^4)*(a^4-2*(b^2+2*c^2)*a^2-4*b^2*c^2+5*c^4+b^4) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28080.

X(21399) lies on these lines: {5097, 8541}, {11002, 14906}

X(21400) = X(3)X(13851)∩X(4)X(13321)

Barycentrics (-a^2+b^2+c^2)*(2*a^4+(3*b^2-4*c^2)*a^2+2*(b^2-c^2)^2)*(2*a^4-(4*b^2-3*c^2)*a^2+2*(b^2-c^2)^2) : :
X(21400) = 3*X(381)-2*X(18504), 2*X(3627)+X(13452)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28080.

X(21400) lies on the Jerabek hyperbola and these lines: {3, 13851}, {4, 13321}, {5, 3431}, {6, 3843}, {20, 20421}, {30, 11270}, {54, 156}, {64, 3830}, {65, 18513}, {68, 18403}, {69, 18404}, {74, 382}, {185, 18550}, {546, 13472}, {568, 18434}, {1598, 18532}, {1656, 18392}, {3426, 11572}, {3519, 9927}, {3527, 14269}, {3532, 5073}, {3627, 13452}, {3851, 14528}, {3853, 11738}, {3861, 14491}, {5076, 16835}, {5449, 18561}, {5504, 12902}, {5640, 13566}, {5889, 11564}, {6145, 18376}, {6288, 13622}, {7579, 13403}, {9703, 16867}, {11744, 18381}, {14627, 18386}, {15002, 18445}, {18378, 18405}

X(21400) = {X(10113), X(18394)}-harmonic conjugate of X(382)
X(21400) = Ehrmann-mid-to-ABC similarity image of X(18504)

X(21401) = X(3)X(6)∩X(623)X(632)

Barycentrics (SB+SC)*(4*S^2+4*sqrt(3)*SA*S+SA^2-SB*SC) : :
X(21401) = X(3)+3*X(15), 5*X(3)+3*X(5611), X(3)-3*X(13350), 7*X(3)-3*X(14538), 5*X(3)-9*X(21158), 5*X(15)-X(5611), 7*X(15)+X(14538), 5*X(15)+3*X(21158), 3*X(396)-X(16001), 3*X(621)-11*X(3525), 3*X(623)-5*X(632), 7*X(3090)-3*X(20428), X(3627)-3*X(7684), 2*X(3628)-3*X(6671), 3*X(6109)-X(16002)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28080.

X(21401) lies on these lines: {3, 6}, {30, 20415}, {396, 16001}, {621, 3525}, {623, 632}, {625, 630}, {3090, 20428}, {3292, 11146}, {3627, 7684}, {3628, 6671}, {6109, 16002}, {11130, 15082}

X(21401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15, 21158, 5611), (9736, 11485, 5097)

X(21402) = X(3)X(6)∩X(624)X(632)

Barycentrics (SB+SC)*(4*S^2-4*sqrt(3)*SA*S+SA^2-SB*SC) : :
X(21402) = X(3)+3*X(16), 5*X(3)+3*X(5615), X(3)-3*X(13349), 7*X(3)-3*X(14539), 5*X(3)-9*X(21159), 5*X(16)-X(5615), 7*X(16)+X(14539), 5*X(16)+3*X(21159), 3*X(395)-X(16002), 3*X(622)-11*X(3525), 3*X(624)-5*X(632), 7*X(3090)-3*X(20429), X(3627)-3*X(7685), 2*X(3628)-3*X(6672), 3*X(6108)-X(16001)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28080.

X(21402) lies on these lines: {3, 6}, {30, 20416}, {395, 16002}, {622, 3525}, {624, 632}, {625, 629}, {3090, 20429}, {3292, 11145}, {3627, 7685}, {3628, 6672}, {6108, 16001}, {11131, 15082}

X(21402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16, 21159, 5615), (9735, 11486, 5097)

X(21403) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 - 2 b^2 c^2 + b c^3) : :

X(21403) lies on these lines: {5, 75}, {10, 17867}, {72, 349}, {321, 857}, {908, 17866}, {1441, 21077}, {1930, 21404}, {3673, 4415}, {3678, 21207}, {3963, 21067}, {21412, 21431}

X(21404) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(21404) lies on these lines: {11, 75}, {312, 16593}, {321, 20431}, {908, 20435}, {1111, 21093}, {1233, 3967}, {1930, 21403}, {2481, 17777}, {4054, 20880}, {4374, 20903}, {4728, 21433}, {5205, 10030}, {20235, 21420}

X(21405) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a+b+c) (-a^3 b-a^2 b^2-a^3 c-2 a^2 b c-a b^2 c+b^3 c-a^2 c^2-a b c^2-2 b^2 c^2+b c^3) : :

X(21405) lies on these lines: {8, 76}, {12, 75}, {226, 20880}, {321, 21420}, {1930, 21403}, {3687, 17866}, {3760, 11679}, {4357, 5051}, {10381, 20347}, {10471, 14009}, {20235, 20890}

X(21406) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics 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) : :

X(21406) lies on these lines: {19, 27}, {255, 1733}, {1930, 17865}, {1959, 17858}, {5179, 20235}, {16571, 17901}, {18648, 18671}, {18670, 18695}, {20234, 21422}, {20237, 20241}, {20890, 20891}

X(21407) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (-a^4 b^2 + b^6 + a^4 b c - b^5 c - a^4 c^2 + b^4 c^2 - 2 b^3 c^3 + b^2 c^4 - b c^5 + c^6) : :

X(21407) lies on these lines: {2, 17068}, {10, 17873}, {22, 75}, {66, 21288}, {321, 857}, {3914, 17886}, {17862, 20901}

X(21408) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (-a^4 b^2 + b^6 + a^4 b c - b^5 c - a^4 c^2 + a^2 b^2 c^2 + b^4 c^2 - 2 b^3 c^3 + b^2 c^4 - b c^5 + c^6) : :

X(21408) lies on these lines: {23, 75}, {321, 857}, {4374, 21439}, {4442, 17886}

X(21409) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (b^4 - b^3 c + b^2 c^2 - b c^3 + c^4) : :

X(21409) lies on these lines: {32, 75}, {321, 17873}, {14963, 17864}, {17867, 20912}, {17886, 21435}, {20235, 21431}, {20627, 21414}

X(21409) = isotomic conjugate of isogonal conjugate of X(21324)

X(21410) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^2 b^3 + b^5 - a b^3 c - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 + c^5) : :

X(21410) lies on these lines: {35, 75}, {1930, 17864}, {14210, 17887}, {17886, 20911}, {20888, 20901}, {21415, 21421}

X(21411) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^2 b^3 + b^5 + a b^3 c - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + c^5) : :

X(21411) lies on these lines: {36, 75}, {1930, 17864}, {3263, 17886}, {14349, 20629}, {20632, 21429}, {20893, 20901}

X(21412) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (a^2 b^2 - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21412) lies on these lines: {10, 321}, {39, 75}, {76, 21138}, {313, 22036}, {1237, 3125}, {1269, 20549}, {1930, 20433}, {3934, 20453}, {3963, 17867}, {4043, 21830}, {14963, 17864}, {17143, 17475}, {17886, 21425}, {20440, 20888}, {21403, 21431}

X(21413) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^3 b^2 - a^2 b^3 + a b^4 + b^5 + a^2 b^2 c - 2 a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 + c^5) : :

X(21413) lies on these lines: {7, 20320}, {40, 75}, {1855, 20883}, {1930, 17864}, {3673, 4858}, {3674, 17869}, {5179, 20235}, {14213, 20880}, {17170, 17860}, {17866, 21436}, {20887, 21432}

X(21414) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a b^4 + b^5 - b^4 c - a c^4 - b c^4 + c^5) : :

X(21414) lies on these lines: {41, 75}, {1930, 20628}, {17864, 20632}, {17865, 20234}, {20627, 21409}

X(21414) = isotomic conjugate of isogonal conjugate of X(21329)
X(21414) = complement of X(25247)
X(21414) = anticomplement of X(25071)

X(21415) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (a b^3 + b^3 c + a c^3 + b c^3) : :

X(21415) lies on these lines: {42, 75}, {321, 1930}, {561, 21140}, {756, 3263}, {2887, 8024}, {3266, 3846}, {3720, 18157}, {3741, 16703}, {4022, 17208}, {4121, 17047}, {18059, 20924}, {20234, 20627}, {20433, 20896}, {20435, 20889}, {20911, 21020}, {21410, 21421}

X(21416) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (a b^3 - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3) : :

X(21416) lies on these lines: {43, 75}, {76, 17889}, {141, 22206}, {305, 3944}, {321, 1930}, {561, 1111}, {740, 18138}, {3120, 8024}, {3263, 3663}, {3681, 4986}, {20234, 20438}, {20237, 20433}, {20255, 22171}, {20627, 20639}, {20629, 20890}

X(21417) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a b^2 + 2 b^3 - b^2 c - a c^2 - b c^2 + 2 c^3) : :

X(21417) lies on these lines: {44, 75}, {313, 21689}, {321, 3262}, {661, 17893}, {1930, 17760}, {3264, 4957}, {3912, 20912}, {4358, 17895}, {4858, 20432}, {17374, 20956}, {18156, 20171}, {20634, 20902}, {20887, 21427}

X(21418) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-2 a b^2 + b^3 - 2 b^2 c - 2 a c^2 - 2 b c^2 + c^3) : :

X(21418) lies on these lines: {45, 75}, {321, 3452}, {1229, 3963}, {1930, 17760}

X(21419) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(10510), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2 (a^8 - 4 a^6 b^2 + 4 a^2 b^6 - b^8 - 4 a^6 c^2 + a^4 b^2 c^2 - a^2 b^4 c^2 - a^2 b^2 c^4 + 2 b^4 c^4 + 4 a^2 c^6 - c^8) : :

X(21419) lies on these lines: {2, 3}, {187, 18374}, {574, 9971}, {1384, 19153}, {1974, 18472}, {3455, 5938}

X(21420) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a + b + c) (a b^3 + b^4 - b^3 c + a c^3 - b c^3 + c^4) : :

X(21420) lies on these lines: {8, 315}, {56, 75}, {321, 21405}, {1930, 17864}, {6358, 20880}, {17865, 20234}, {17880, 20436}, {18697, 20895}, {20235, 21404}, {20629, 20633}, {20901, 21432}

X(21420) = isotomic conjugate of isogonal conjugate of X(21333)

X(21421) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (a b^3 + b^4 + a c^3 + c^4) : :

X(21421) lies on these lines: {58, 75}, {321, 4153}, {1930, 1959}, {3262, 18697}, {4137, 17211}, {14963, 17864}, {17886, 20443}, {20630, 21429}, {21410, 21415}

X(21422) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a + b + c) (a^2 b^2 + a b^3 + b^3 c + a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21422) lies on these lines: {7, 8}, {305, 3705}, {1930, 17864}, {3210, 3673}, {3687, 17866}, {3691, 21233}, {20234, 21406}, {20633, 21443}

X(21422) = isotomic conjugate of isogonal conjugate of X(21334)

X(21423) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (-a^4 + b^4 + c^4) (a^4 + b^4 - 2 b^3 c + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(21423) lies on these lines: {66, 75}, {1441, 4972}, {2172, 17866}, {3963, 17864}

X(21424) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b^2 + c^2) (a^2 b^2 + b^4 + a^2 c^2 + c^4) : :

X(21424) lies on these lines: {31, 75}, {1930, 16747}, {9239, 18837}, {20629, 20888}

X(21425) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (b^2 + c^2) (a^2 + b^2 - b c + c^2) : :

X(21425) lies on these lines: {10, 20234}, {75, 83}, {321, 17873}, {1930, 4568}, {3008, 4359}, {3687, 19835}, {7859, 16600}, {17192, 17456}, {17886, 21412}, {20629, 20888}

X(21426) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a b - a c + b c) (-a b^3 + a b^2 c + b^3 c + a b c^2 - a c^3 + b c^3) : :

X(21426) lies on these lines: {8, 20559}, {75, 87}, {1930, 20892}, {3596, 6382}, {20234, 20438}, {20236, 20630}, {20629, 20633}

X(21427) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-2 a + b + c) (a b^2 + b^3 - 2 b^2 c + a c^2 - 2 b c^2 + c^3) : :

X(21427) lies on these lines: {75, 88}, {321, 3452}, {1109, 3626}, {1227, 4440}, {1930, 20901}, {20234, 21428}, {20887, 21417}, {20896, 20902}

X(21428) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a + 2 b + 2 c) (2 a b^2 + 2 b^3 - b^2 c + 2 a c^2 - b c^2 + 2 c^3) : :

X(21428) lies on these lines: {75, 89}, {321, 3262}, {1930, 20887}, {3625, 4647}, {20234, 21427}

X(21429) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b (b - c)^2 c (-a b^2 + b^3 - a b c + b^2 c - a c^2 + b c^2 + c^3) : :

X(21429) lies on these lines: {75, 101}, {339, 21253}, {514, 17880}, {1930, 20628}, {4361, 22144}, {4858, 17761}, {17865, 20236}, {20630, 21421}, {20632, 21411}

X(21430) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b (b - c)^2 c (b + c) (-a^2 b^2 + b^4 - a^2 b c + b^3 c - a^2 c^2 + b^2 c^2 + b c^3 + c^4) : :

X(21430) lies on these lines: {75, 110}, {17878, 20901}, {20628, 20896}

X(21431) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (a^4 - 2 a^2 b c + b^3 c - b^2 c^2 + b c^3) : :

X(21431) lies on these lines: {75, 115}, {99, 21004}, {321, 17886}, {1577, 4986}, {3008, 3948}, {3963, 4103}, {10026, 20924}, {20235, 21409}, {20439, 20888}, {21403, 21412}

X(21432) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a b + 3 b^2 - a c - 2 b c + 3 c^2) : :

X(21432) lies on these lines: {75, 145}, {85, 4696}, {321, 1930}, {1111, 3701}, {3006, 3665}, {3263, 6376}, {3673, 4358}, {4059, 17165}, {4359, 5222}, {4487, 16284}, {5014, 17170}, {7195, 10327}, {20234, 20905}, {20435, 20892}, {20887, 21413}, {20901, 21420}

X(21433) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (-a^2 + 2 a b + 2 a c - 3 b c) : :

X(21433) lies on these lines: c):: {75, 900}, {514, 4374}, {523, 4828}, {665, 4699}, {693, 4768}, {1278, 4526}, {3119, 4858}, {3261, 4777}, {4435, 17117}, {4728, 21404}, {4763, 14296}, {6009, 20950}

X(21433) = isotomic conjugate of isogonal conjugate of X(21343)

X(21434) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (-a^2 b^2 + 2 b^4 + a^2 b c - 2 b^3 c - a^2 c^2 + b^2 c^2 - 2 b c^3 + 2 c^4) : :

X(21434) lies on these lines: {75, 187}, {3250, 20637}, {14963, 17864}

X(21435) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b + c) (a^2 b^2 - 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(21435) lies on these lines: {10, 321}, {75, 194}, {76, 20440}, {313, 20491}, {1237, 21951}, {1575, 18050}, {1930, 20892}, {3721, 22028}, {3970, 20501}, {4359, 17030}, {4485, 20271}, {10009, 20891}, {14213, 20432}, {17886, 21409}, {18904, 20336}, {21830, 22016}

X(21436) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :

X(21436) lies on these lines: {75, 200}, {274, 17188}, {321, 1930}, {1111, 17889}, {3263, 4082}, {5231, 7182}, {14213, 20901}, {17866, 21413}, {20236, 20890}, {20237, 20435}

X(21437) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (-a^4 b^2 + a^2 b^4 - a^4 b c + a^2 b^3 c - a^4 c^2 + a^2 b^2 c^2 - b^4 c^2 + a^2 b c^3 - 2 b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(21437) lies on these lines: {75, 647}, {321, 4467}, {1021, 10447}, {3250, 20637}, {3596, 21719}, {4374, 6590}, {6332, 17894}

X(21438) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (-a^2 b + a b^2 - a^2 c + a b c - b^2 c + a c^2 - b c^2) : :

X(21438) lies on these lines: {8, 1938}, {75, 650}, {297, 525}, {312, 4885}, {313, 21721}, {321, 693}, {513, 20952}, {661, 17893}, {2517, 4122}, {3239, 20907}, {3261, 3700}, {3757, 8641}, {3835, 20908}, {4025, 20521}, {4088, 4397}, {4106, 20950}, {4374, 6590}, {4411, 18154}, {4468, 17894}, {4820, 20954}, {21348, 21610}

X(21438) = isotomic conjugate of isogonal conjugate of X(21348)
X(21438) = complement of X(25271)
X(21438) = anticomplement of X(25098)

X(21439) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a b^3 - a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21439) lies on these lines: {75, 659}, {313, 21722}, {514, 21440}, {661, 17893}, {3119, 4858}, {3261, 4122}, {4367, 21613}, {4374, 21408}, {4486, 20908}, {17762, 21343}

X(21440) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21440) lies on these lines: {75, 667}, {514, 21439}, {2084, 20910}, {3250, 20637}, {8045, 20907}, {14349, 20629}

X(21441) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a^2 b^4 + a^2 b^3 c + a^2 b^2 c^2 - b^4 c^2 + a^2 b c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(21441) lies on these lines: {75, 669}, {3250, 20637}, {4374, 21408}

X(21442) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (b^3 + 2 a b c + c^3) : :

X(21442) lies on these lines: {6, 75}, {8, 1352}, {321, 908}, {760, 17153}, {1227, 17344}, {1269, 16732}, {1441, 20913}, {1930, 20892}, {3006, 21804}, {3262, 3963}, {3727, 17202}, {3875, 4812}, {4359, 18698}, {4385, 5827}, {4858, 18697}, {17184, 18202}, {17288, 20955}

X(21442) = isotomic conjugate of isogonal conjugate of X(3727)

X(21443) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics b c (a^2 b^2 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(21443) lies on these lines: {10, 75}, {37, 3934}, {39, 3739}, {192, 3760}, {194, 4699}, {310, 4609}, {321, 20433}, {350, 3993}, {518, 14994}, {536, 3774}, {538, 4688}, {561, 3741}, {734, 18805}, {740, 12263}, {746, 18806}, {1920, 3840}, {1930, 17760}, {3403, 3923}, {3696, 14839}, {3783, 4441}, {3797, 4044}, {3971, 18152}, {4377, 9055}, {4495, 5263}, {4686, 21897}, {4709, 17143}, {4751, 7786}, {4772, 20081}, {5145, 10436}, {6385, 16887}, {7205, 10481}, {7697, 20430}, {20627, 20886}, {20633, 21422}, {20880, 20892}

X(21443) = isotomic conjugate of isogonal conjugate of X(21352)
X(21443) = complement of X(32453)

X(21444) = X(3)X(76)∩X(32)X(8789)

Barycentrics a^4 (a^4 b^4-a^2 b^6-a^2 b^4 c^2+a^4 c^4-a^2 b^2 c^4+2 b^4 c^4-a^2 c^6) : :

X(21444) lies on the cubic K1064 and these lines: {3,76}, {32,8789}, {111,11332}, {187,237}, {446,2794}, {682,3491}, {729,1384}, {864,3291}, {865,10418}, {1511,13210}, {1634,5026}, {3053,9431}, {3111,9155}, {3225,6179}, {3972,11328}, {5092,20775}, {5171,15429}, {5661,20975}, {13236,14691}

X(21444) = isogonal conjugate of isotomic conjugate of X(34383)
X(21444) = X(i)-Hirst inverse of X(j) for these (i,j): {237, 669}, {3231, 3288}
X(21444) = crossdifference of every pair of points on line {2, 2491}
X(21444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3511, 99), (32, 9468, 9427), (18773, 18774, 237)

X(21445) = X(3)X(194)∩X(4)X(230)

Barycentrics 3 a^8-6 a^6 b^2+5 a^4 b^4-2 a^2 b^6-6 a^6 c^2-a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2+5 a^4 c^4+2 a^2 b^2 c^4+2 b^4 c^4-2 a^2 c^6-b^2 c^6 : :
X(21445) = X[98] + 2 X[187], X[4] - 4 X[230], 2 X[3] + X[385], 2 X[325] - 5 X[631], 2 X[2080] + X[5999], X[316] - 4 X[6036], 7 X[3523] - X[7779], 4 X[549] - X[7840], 8 X[140] - 5 X[7925], 2 X[1513] + X[9862], 5 X[631] - 4 X[10256], 4 X[2030] - X[10753], X[6781] + 2 X[11623], X[9855] + 2 X[11632], 4 X[187] - X[11676], 2 X[98] + X[11676], X[5184] + 2 X[11710], X[5999] - 4 X[12042], X[2080] + 2 X[12042], 2 X[8598] + X[12243], 4 X[5914] - X[13168], 2 X[10242] - 3 X[14041], 2 X[13449] - 5 X[14061], X[6033] - 4 X[14693], X[14712] + 2 X[15980], X[6776] + 2 X[15993], X[10295] + 2 X[16315]

X(21445) lies on these lines: {2,11187}, {3,194}, {4,230}, {30,8859}, {32,262}, {98,187}, {140,2896}, {182,7771}, {186,523}, {316,6036}, {325,631}, {384,7697}, {524,3524}, {538,21166}, {549,7840}, {1078,13335}, {1384,10788}, {1513,9862}, {2030,10753}, {2080,5999}, {2782,13586}, {3329,11842}, {3398,7824}, {3399,13357}, {3523,7779}, {3526,7938}, {5023,8719}, {5171,7470}, {5184,11710}, {5206,11257}, {5914,13168}, {5965,7799}, {6033,14693}, {6179,9737}, {6308,9751}, {6776,15993}, {6781,11623}, {7417,11580}, {7694,9754}, {7757,9734}, {8598,12243}, {9855,11632}, {10242,14041}, {10295,16315}, {10359,11285}, {13449,14061}, {14712,15980}, {18993,19064}, {18994,19063}

X(21445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9755, 7709), (98, 187, 11676), (1384, 13860, 10788), (2080, 12042, 5999), (10851, 10852, 6194)
X(21445) = crosssum of X(5611) and X(5615)
X(21445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9755, 7709), (98, 187, 11676), (1384, 13860, 10788), (2080, 12042, 5999), (10851, 10852, 6194)

Tripoles of axes of perspectivities: X(21446)-X(21469)

This preamble and centers X(21446)-X(21469) were contributed by César Eliud Lozada, August 25, 2018.

The appearance of (T,i) in the following list means that the trilinear pole of the perspectrix of triangles ABC and T is X(i):

(AAOA, 6), (Andromeda, 21446), (anti-Artzt, 11163), (anti-Atik, 21447), (1st anti-Brocard, 6), (4th anti-Brocard, 21448), (anti-Conway, 8882), (2nd anti-Conway, 393), (anti-excenters-reflections, 459), (2nd anti-extouch, 24), (anti-Honsberger, 251), (anti-Hutson intouch, 394), (anti-incircle-circles, 5422), (anti-inverse-in-incircle, 76), (anti-McCay, 8860), (6th anti-mixtilinear, 193), (anti-orthocentroidal, 3), (1st anti-orthosymmedial, 1176), (1st anti-Sharygin, 21449), (anti-tangential-midarc, 57), (3rd anti-tri-squares, 486), (4th anti-tri-squares, 485), (Antlia, 21450), (Apollonius, 1), (Apus, 1), (Artzt, 6), (Atik, 346), (Ayme, 346), (BCI, 21465), (1st Brocard-reflected, 3329), (1st Brocard, 385), (2nd Brocard, 524), (3rd Brocard, 385), (4th Brocard, 468), (circummedial, 83), (circumorthic, 275), (2nd circumperp, 81), (circumsymmedial, 6), (Conway, 86), (2nd Conway, 75), (3rd Conway, 10436), (4th Conway, 75), (5th Conway, 86), (Ehrmann-side, 3580), (Ehrmann-vertex, 94), (2nd Ehrmann, 111), (2nd Euler, 6515), (5th Euler, 4), (excenters-midpoints, 145), (excenters-reflections, 8056), (excentral, 1), (1st excosine, 3), (2nd excosine, 4), (extangents, 1), (extouch, 8), (2nd extouch, 8), (3rd extouch, 7), (4th extouch, 7), (5th extouch, 7), (inner-Fermat, 395), (outer-Fermat, 396), (3rd Fermat-Dao, 8014), (4th Fermat-Dao, 8015), (7th Fermat-Dao, 21466), (8th Fermat-Dao, 21467),(11th Fermat-Dao, 11092), (12th Fermat-Dao, 11078), (13th Fermat-Dao, 21468), (14th Fermat-Dao, 21469),(15th Fermat-Dao, 18), (16th Fermat-Dao, 17), (Feuerbach, 1), (Hatzipolakis-Moses, 21451), (1st Hatzipolakis, 21452), (2nd Hatzipolakis, 1119), (hexyl, 63), (Honsberger, 21453), (Hutson extouch, 10578), (Hutson intouch, 5435), (outer-Hutson, 188), (2nd Hyacinth, 3542), (incentral, 1), (incircle-circles, 21454), (intouch, 7), (inverse-in-incircle, 279), (1st isodynamic-Dao, 17), (2nd isodynamic-Dao, 18), (1st Kenmotu diagonals, 8577), (2nd Kenmotu diagonals, 8576), (Kosnita, 1994), (Lemoine, 598), (1st Lemoine-Dao, 468), (2nd Lemoine-Dao, 468), (Lucas antipodal, 3069), (Lucas Brocard, 6), (Lucas central, 6), (Lucas inner, 6), (Lucas inner tangential, 6), (Lucas reflection, 6), (Lucas secondary central, 6), (Lucas 1st secondary tangents, 6), (Lucas 2nd secondary tangents, 6), (Lucas tangents, 6), (Lucas(-1) antipodal, 3068), (Lucas(-1) Brocard, 6), (Lucas(-1) central, 6), (Lucas(-1) inner, 6), (Lucas(-1) inner tangential, 6), (Lucas(-1) reflection, 6), (Lucas(-1) secondary central, 6), (Lucas(-1) 1st secondary tangents, 6), (Lucas(-1) 2nd secondary tangents, 6), (Lucas(-1) tangents, 6), (Macbeath, 264), (Malfatti, 21455), (Mandart-excircles, 3086), (McCay, 8859), (midarc, 18886), (2nd midarc, 21456), (midheight, 20), (mixtilinear, 57), (2nd mixtilinear, 1), (3rd mixtilinear, 57), (4th mixtilinear, 1), (6th mixtilinear, 9), (7th mixtilinear, 7), (Montesdeoca-Hung, 1), (1st Morley, 3602), (2nd Morley, 3603), (3rd Morley, 3604), (1st Morley-adjunct, 3602), (2nd Morley-adjunct, 3603), (3rd Morley-adjunct, 3604), (Moses-Hung, 21457), (inner-Napoleon, 395), (outer-Napoleon, 396), (1st Neuberg, 385), (2nd Neuberg, 3329), (orthic, 4), (orthocentroidal, 30), (1st orthosymmedial, 21458), (2nd orthosymmedial, 21459), (1st Pamfilos-Zhou, 6), (2nd Pamfilos-Zhou, 13387), (1st Parry, 1992), (2nd Parry, 9214), (3rd Parry, 21460), (Pelletier, 650), (1st Przybyłowski-Bollin, 1), (2nd Przybyłowski-Bollin, 1), (3rd Przybyłowski-Bollin, 1), (4th Przybyłowski-Bollin, 1), (reflection, 5), (Schroeter, 523), (1st Sharygin, 6), (2nd Sharygin, 6), (inner-Soddy, 7), (outer-Soddy, 7), (inner-squares, 4), (outer-squares, 4), (Steiner, 99), (symmedial, 6), (tangential, 6), (tangential-midarc, 174), (inner tri-equilateral, 21461), (outer tri-equilateral, 21462), (3rd tri-squares, 21463), (4th tri-squares, 21464), (Trinh, 323), (inner-Vecten, 3069), (outer-Vecten, 3068), (X-parabola-tangential, 115), (Yff central, 2089), (Yff contact, 190), (Yiu, 1994), (1st Zaniah, 144), (2nd Zaniah, 145)

X(21446) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND ANDROMEDA

Barycentrics a*(a-b+c)*(a^2-2*c*a+3*b^2+c^2)*(a+b-c)*(a^2-2*b*a+b^2+3*c^2) : :

Let A'B'C' be the Antlia triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(10322). The lines A'A", B'B", C'C" concur in X(21446). (Randy Hutson, August 29, 2018)

X(21446) lies on these lines: {1,1462}, {2,479}, {7,346}, {9,241}, {57,200}, {63,6605}, {142,281}, {226,15490}, {673,9311}, {1014,1445}, {1396,4183}, {3243,5228}, {4334,5223}, {5437,19605}, {5819,18725}, {7110,20195}, {7146,12560}, {15876,17284}

X(21446) = trilinear pole of the line {2254, 3669}
X(21446) = {X(1), X(10322)}-harmonic conjugate of X(21450)
X(21446) = perspector of Antlia triangle and cross-triangle of ABC and Antlia triangle

X(21447) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND ANTI-ATIK

Barycentrics b^2*c^2*(3*a^2-b^2-c^2)*(a^2-b^2+c^2)^2*(a^2+b^2-c^2)^2 : :

X(21447) lies on these lines: {2,216}, {4,6467}, {76,683}, {107,2374}, {235,6530}, {297,14615}, {648,8745}, {1093,6622}, {1593,11257}

X(21447) = polar conjugate of X(6391)
X(21447) = {X(264), X(393)}-harmonic conjugate of X(2052)

X(21448) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 4th ANTI-BROCARD

Barycentrics a^2*(a^2+b^2-5*c^2)*(a^2-5*b^2+c^2) : :

X(21448) lies on the cubics K284, K297, K657 and these lines: {2,2418}, {3,111}, {6,373}, {23,15655}, {25,187}, {37,4413}, {182,17979}, {183,3228}, {251,5020}, {263,3231}, {351,17999}, {352,1351}, {353,12017}, {381,5913}, {393,468}, {543,15304}, {599,6791}, {647,9178}, {1383,1384}, {1976,2502}, {1989,10418}, {2165,3054}, {2395,8371}, {2492,10103}, {2987,15066}, {3620,6339}, {5050,7708}, {5055,9745}, {6032,19709}, {6094,7610}, {6587,18310}, {7484,8770}, {7746,8791}, {8375,8576}, {8376,8577}, {8617,13192}, {8749,9717}, {8860,18818}, {9189,18012}, {9462,15271}, {9909,15603}, {11185,11336}, {14898,14948}

X(21448) = isogonal conjugate of X(1992)
X(21448) = isotomic conjugate of X(11059)
X(21448) = trilinear pole of the line {512, 5107}
X(21448) = X(5094)-of-4th anti-Brocard triangle
X(21448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 8585, 11284), (111, 10354, 10355), (111, 20481, 3), (1995, 11580, 1384), (3291, 8585, 6), (14262, 20481, 10354)
X(21448) = intersection, other than X(111), of hyperbola {{A,B,C,X(2),X(6)}} and line X(3)X(111)
X(21448) = perspector of ABC and unary cofactor triangle of 1st Parry triangle

X(21449) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 1st ANTI-SHARYGIN

Barycentrics SB*SC*(S^2+SA*SB)*(S^2+SA*SC)*(S^2-(4*R^2-SW)*(2*SA-SW)) : :

X(21449) lies on these lines: {2,95}, {6,16813}, {54,1075}, {264,18315}, {933,14575}, {8795,14533}, {9308,18831}

X(21450) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND ANTLIA

Barycentrics a*(a^2+(b-c)^2)*(a^2-2*b*a+b^2+3*c^2)*(a^2-2*c*a+3*b^2+c^2) : :

Let A'B'C' be the Andromeda triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10322). The lines A'A", B'B", C'C" concur in X(21450). (Randy Hutson, August 29, 2018)

X(21450) lies on these lines: {1,1462}, {2,3677}, {7,4907}, {33,354}, {37,2191}, {1024,4449}, {2285,21346}, {4000,4012}

X(21450) = trilinear pole of the line {14347, 14348}
X(21450) = {X(1), X(10322)}-harmonic conjugate of X(21446)
X(21450) = perspector of Andromeda triangle and cross-triangle of ABC and Andromeda triangle

X(21451) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND HATZIPOLAKIS-MOSES

Barycentrics (9*R^2-2*SW)*S^2+4*R^2*SB*SC : :

X(21451) lies on these lines: {2,3}, {110,10112}, {143,14643}, {195,10272}, {7666,11694}, {10095,13368}, {10539,14683}, {11692,13423}, {11800,12280}, {12359,15052}, {12897,15035}, {20193,20424}

X(21451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3089, 3146), (2, 5059, 3546), (4, 3090, 10224), (4, 18565, 3543), (5, 3518, 3153), (5, 10096, 3), (5, 13163, 381), (3091, 7505, 2), (3549, 5056, 2), (7486, 7558, 2), (7545, 10224, 4)

X(21452) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 1st HATZIPOLAKIS

Barycentrics ((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*(2*a^4-(b+c)*a^3-(b^2+c^2)*a^2+(b+c)*(b^2+c^2)*a-(b^2-c^2)^2)*(a+b-c)*(a-b+c) : :

X(21452) lies on these lines: {1118,1259}, {1813,6357}

X(21453) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND HONSBERGER

Trilinears 1/[sec^2(B/2) + sec^2(C/2)] : :
Barycentrics (a^2-(b+2*c)*a-(b-c)*c)*(a-b+c)*(a^2-(2*b+c)*a+b*(b-c))*(a+b-c) : :

Let P be any point on the Soddy line. Let U be the {X(1),X(7)}-harmonic conjugate of P. Let P' and U' be the isogonal conjugates of P and U, resp. Then X(21453) = PU'∩UP'. (Randy Hutson, August 29, 2018)

X(21453) lies on the circumhyperbola dual of Yff parabola, the hyperbola {{A,B,C,X(1),X(33)}}, the circumconic centered at X(1086), and on these lines: {1,1088}, {2,220}, {7,55}, {27,1803}, {33,273}, {64,11036}, {75,200}, {85,3870}, {86,2328}, {103,5542}, {142,1223}, {150,8226}, {226,673}, {310,1043}, {331,14004}, {354,658}, {664,3957}, {903,6606}, {963,11037}, {1434,4184}, {1440,2192}, {1996,5543}, {2400,4467}, {3748,14189}, {3945,7050}, {4229,4298}, {4845,11019}, {7056,11038}, {7072,7318}, {7073,7269}, {7218,12560}, {7580,17753}, {8817,17234}, {9441,10482}, {13727,15467}

X(21453) = isogonal conjugate of X(2293)
X(21453) = isotomic conjugate of X(4847)
X(21453) = polar conjugate of X(1855)
X(21453) = trilinear pole of the line {514, 657}
X(21453) = X(10329)-of-intouch triangle
X(21453) = cevapoint of X(1) and X(7)
X(21453) = crosspoint of X(1) and X(7) wrt the excentral triangle
X(21453) = intersection of tangents at X(1) and X(7) to rectangular hyperbola passing through X(1), X(7), and the excenters
X(21453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 2346, 10509), (7, 10578, 17093)

X(21454) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND INCIRCLE-CIRCLES

Barycentrics (3*a+b+c)*(a+b-c)*(a-b+c) : :

X(21454) = 3*X(2)-4*X(5437)

X(21454) lies on these lines: {1,3522}, {2,7}, {3,11036}, {4,5708}, {8,3339}, {10,4355}, {20,942}, {27,1119}, {40,11037}, {42,4334}, {55,11038}, {56,1621}, {65,145}, {72,10855}, {77,17011}, {79,10591}, {81,279}, {85,4359}, {89,278}, {109,9105}, {165,5542}, {171,4310}, {173,9795}, {175,13389}, {176,13388}, {189,10405}, {193,17490}, {223,17012}, {258,11891}, {269,5256}, {273,6994}, {312,4454}, {345,4869}, {346,18141}, {347,7560}, {354,390}, {376,15934}, {388,3617}, {391,19804}, {404,1260}, {497,4860}, {516,10580}, {517,10569}, {631,6147}, {664,7268}, {938,3146}, {940,3672}, {950,5059}, {962,3333}, {982,4307}, {990,9539}, {999,6909}, {1014,8025}, {1088,9533}, {1155,3475}, {1159,7967}, {1210,3832}, {1214,17092}, {1278,4032}, {1357,10473}, {1373,5393}, {1374,5405}, {1401,20012}, {1418,3666}, {1420,4323}, {1427,4850}, {1443,17013}, {1446,19788}, {1448,5262}, {1458,17018}, {1462,2221}, {1466,4188}, {1467,17576}, {1471,17127}, {1659,21169}, {1707,16020}, {1788,5261}, {1836,5274}, {1876,6995}, {1999,4452}, {2095,6916}, {2263,7191}, {2999,7271}, {3085,3336}, {3086,3337}, {3241,4315}, {3295,3296}, {3338,4295}, {3340,3623}, {3361,3616}, {3485,5265}, {3486,18221}, {3487,3523}, {3524,5719}, {3529,12433}, {3534,15935}, {3543,5722}, {3621,10106}, {3649,7288}, {3668,20061}, {3670,4340}, {3673,7406}, {3677,4344}, {3681,8581}, {3687,21296}, {3742,5698}, {3752,4644}, {3868,6904}, {3870,4321}, {3871,12631}, {3916,17558}, {3920,4327}, {3927,17582}, {3947,19877}, {3969,6604}, {4293,5902}, {4294,15228}, {4304,15933}, {4306,19767}, {4312,9812}, {4313,11518}, {4318,17024}, {4320,17016}, {4328,5287}, {4384,10521}, {4393,7176}, {4430,7672}, {4552,20092}, {4666,12560}, {4678,4848}, {4704,7201}, {5018,17017}, {5045,6361}, {5056,5714}, {5068,5704}, {5083,20095}, {5122,15692}, {5129,5439}, {5173,20075}, {5222,10481}, {5290,9780}, {5572,5918}, {5575,17364}, {5703,15717}, {5712,17595}, {5728,11220}, {5731,11529}, {5759,11227}, {5770,6843}, {5806,12246}, {5850,8580}, {6539,19825}, {6763,19855}, {6987,10202}, {7190,17019}, {7247,19822}, {7274,17022}, {7365,19785}, {7987,12563}, {7991,12577}, {8729,11889}, {8734,11890}, {9316,17126}, {9579,17578}, {10004,17093}, {10072,11552}, {10303,11374}, {10944,20054}, {13424,13459}, {13435,13437}, {17114,20036}, {17154,20020}, {17169,17185}, {17495,20043}, {18750,20905}, {18838,20067}

X(21454) = isogonal conjugate of X(34820)
X(21454) = cevapoint of X(1449) and X(3361)
X(21454) = crossdifference of every pair of points on line X(663)X(4524)
X(21454) = anticomplement of X(18228)
X(21454) = trilinear pole of the line {4778, 16533}
X(21454) = X(17810)-of-intouch triangle
X(21454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9965, 144), (2, 20059, 329), (7, 5226, 4654), (57, 226, 5435), (57, 4654, 3911), (63, 9776, 2), (142, 3928, 5273), (142, 5273, 2), (226, 5435, 2), (329, 3306, 2), (2094, 9776, 63), (3911, 5226, 2), (4114, 4654, 7), (5249, 5744, 2), (5328, 6692, 2), (5437, 18228, 2)

X(21455) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND MALFATTI

Trilinears    (cos(A/2)-cos(B/2)*cos(C/2)-cos(B/2)-cos(C/2))*(cos(B/2)+1)*(cos(C/2)+1) : :
Trilinears    1 - 2(sec(A/4) cos(B/4) cos(C/4))^2 : :
Trilinears    1 - (1 + cos B/2)(1 + cos C/2)/(1 + cos A/2) : :

X(21455) lies on these lines: {1,179}, {558,1143}

X(21455) = trilinear pole of Monge line of Malfatti circles
X(21455) = {X(1), X(1142)}-harmonic conjugate of X(179)

X(21456) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 2nd MIDARC

Trilinears sec(A/2)^3*(1-sin(A/2)) : :

X(21456) lies on these lines: {1,10491}, {7,1488}, {174,5435}, {279,555}, {4146,4452}, {9795,16664}, {10489,21314}, {11891,15495}

X(21456) = trilinear pole of antiorthic axis of intouch triangle
X(21456) = {X(279), X(555)}-harmonic conjugate of X(18886)

X(21457) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND MOSES-HUNG

Barycentrics a^9-2*(b+c)*a^8+2*(b+c)*(b^2+c^2)*a^6-(2*b^2-3*b*c+2*c^2)*(b+c)^2*a^5+(2*b-c)*(b-2*c)*(b+c)^3*a^4+2*(b^2-c^2)^2*b*c*a^3-2*(b^3+c^3)*(b^2-c^2)^2*a^2+(b^2-c^2)^2*(b-c)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c)*b*c : :

X(21457) lies on these lines: {1,29}, {3914,6738}, {11020,19642}

X(21458) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 1st ORTHOSYMMEDIAL

Barycentrics (SB+SW)*(SC+SW)*((SB+SC)*S^2-2*SB*SC*SW) : :

X(21458) lies on these lines: {2,66}, {4,10547}, {23,385}, {83,3091}, {251,393}, {253,1799}, {827,2697}, {1304,9076}, {2980,7394}, {9755,10594}, {19558,20021}

X(21458) = {X(4), X(10547)}-harmonic conjugate of X(10548)

X(21459) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 2nd ORTHOSYMMEDIAL

Barycentrics SB*SC*(SB+SW)*(SC+SW)*((6*R^2-SW)*S^2-SB*SC*SW) : :

X(21459) lies on these lines: {6,10549}, {251,393}, {308,13854}, {1249,17500}, {2489,4580}, {9308,16890}

X(21459) = {X(6), X(10549)}-harmonic conjugate of X(10550)

X(21460) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 3rd PARRY

Barycentrics (SB+SC)*(3*SB-SW)*(3*SC-SW)*((3*SA-2*SW)*S^2+SA*SW^2) : :

X(21460) lies on these lines: {2,10558}, {6,110}, {182,691}, {381,14833}, {574,12157}, {575,14246}, {597,16092}, {671,5476}, {2408,5652}, {5034,14609}, {5967,9154}, {9775,11163}, {11842,14908}, {15387,18872}

X(21460) = trilinear pole of the line {2080, 14274}
X(21460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10558, 10559), (6, 5968, 895)

X(21461) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND INNER TRI-EQUILATERAL

Barycentrics (SB+SC)*(SB+sqrt(3)*S)*(SC+sqrt(3)*S) : :

X(21461) lies on these lines: {2,17}, {3,16021}, {6,3132}, {16,2981}, {32,8565}, {37,7127}, {51,3458}, {110,2004}, {111,16806}, {184,3457}, {588,3389}, {589,3390}, {1989,8014}, {2902,16804}, {2963,8604}, {3171,6151}, {8259,15802}, {8740,8882}, {16081,16249}

X(21461) = isogonal conjugate of X(302)
X(21461) = X(61)-isoconjugate of X(75)
X(21461) = polar conjugate of the isotomic conjugate of X(32585)
X(21461) = barycentric product of vertices of outer Napoleon triangle
X(21461) = {X(32), X(15004)}-harmonic conjugate of X(21462)

X(21462) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND OUTER TRI-EQUILATERAL

Barycentrics (SB+SC)*(SB-sqrt(3)*S)*(SC-sqrt(3)*S) : :

X(21462) lies on these lines: {2,18}, {3,16022}, {6,3131}, {15,6151}, {32,8565}, {51,3457}, {110,2005}, {111,16807}, {184,3458}, {393,8740}, {588,3364}, {589,3365}, {2903,16805}, {2963,8603}, {2981,3170}, {8260,15778}, {8739,8882}, {16081,16250}

X(21462) = isogonal conjugate of X(303)
X(21462) = {X(32), X(15004)}-harmonic conjugate of X(21461)
X(21462) = X(62)-isoconjugate of X(75)
X(21462) = barycentric product of vertices of inner Napoleon triangle
X(21462) = polar conjugate of the isotomic conjugate of X(32586)

X(21463) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 3rd TRI-SQUARES

Barycentrics (SB+S)*(SC+S)*(SA-2*SB-2*SC-S) : :

X(21463) lies on the cubic K295 and these lines: {2,372}, {925,21464}, {3060,6459}, {3312,13440}, {6460,18911}

X(21464) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 4th TRI-SQUARES

Barycentrics (SB-S)*(SC-S)*(SA-2*SB-2*SC+S) : :

X(21464) lies on the cubic K295 and these lines: {2,371}, {925,21463}, {3060,6460}, {3311,13429}, {6459,18911}, {8964,19116}

X(21465) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND BCI

Trilinears cos(B/2)+cos(C/2)+2*cos(B/2)*cos(C/2)-cos(A/2) : :

X(21465) lies on these lines: {1,483}, {57,173}, {1127,1129}, {3645,8092}

X(21466) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 7th FERMAT-DAO

Barycentrics (3*SA-2*SW-sqrt(3)*S)*(sqrt(3)*SB+S)*(sqrt(3)*SC+S) : :

X(21466) lies on the cubic K295 and these lines: {2,13}, {6,11537}, {476,843}, {3090,11555}, {5334,15441}, {7739,14902}, {9117,14185}, {9214,11080}, {10654,11002}, {11581,18581}

X(21467) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 8th FERMAT-DAO

Barycentrics (3*SA-2*SW+sqrt(3)*S)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S) : :

X(21467) lies on the cubic K295 and these lines: {2,14}, {6,11549}, {476,843}, {3090,11556}, {5335,15442}, {7739,14903}, {9115,14187}, {9214,11085}, {10653,11002}, {11582,18582}

X(21468) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 13th FERMAT-DAO

Barycentrics 9*S^2-sqrt(3)*(9*R^2-2*SW)*S-9*R^2*(3*SA-2*SW)+9*SA^2-6*SW^2 : :

X(21468) lies on these lines: {2,94}, {473,648}, {1992,16771}, {11092,18776}

X(21469) = TRILINEAR POLE OF THE PERSPECTRIX OF THESE TRIANGLES: ABC AND 14th FERMAT-DAO

Barycentrics 9*S^2+sqrt(3)*(9*R^2-2*SW)*S-9*R^2*(3*SA-2*SW)+9*SA^2-6*SW^2 : :

X(21469) lies on these lines: {2,94}, {472,648}, {1992,16770}, {11078,18777}

X(21470) = (name pending)

Barycentrics a*((b+c)*a^7+(b^2+6*b*c+c^2)*a^6-(b+c)*(3*b^2+2*b*c+3*c^2)*a^5-(3*b^4+3*c^4-2*(2*b^2-21*b*c+2*c^2)*b*c)*a^4+(b+c)*(3*b^4+3*c^4+2*(6*b^2-11*b*c+6*c^2)*b*c)*a^3+(3*b^6+3*c^6-(2*b^4+2*c^4+(59*b^2+204*b*c+59*c^2)*b*c)*b*c)*a^2-(b^2-c^2)*(b-c)*(b^4+c^4+2*(6*b^2+7*b*c+6*c^2)*b*c)*a-(b^2-c^2)^2*(b+c)^2*(b^2+6*b*c+c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28085.

X(21470) lies on this line: {3295, 3646}

X(21471) = X(57)X(191)∩X(940)X(3475)

Barycentrics (3*a+b+c)*((b^2+c^2)*a+(b+c)^3)*(a-b+c)*(a+b-c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28085.

X(21471) lies on these lines: {57, 191}, {940, 3475}, {3666, 12053}, {5226, 16594}

X(21472) = (name pending)

Barycentrics a*(a+3*b+3*c)*(3*(b+c)*a^4-2*(b^2+c^2)*a^3-4*(b+c)*(b^2-3*b*c+c^2)*a^2+2*(b^4+c^4+2*(2*b^2+9*b*c+2*c^2)*b*c)*a+(b+c)^5)*(a+b-c)*(a-b+c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28085.

X(21472) lies on this line: {65, 3679}

X(21473) = 30TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2 a^12 b^4-10 a^10 b^6+20 a^8 b^8-20 a^6 b^10+10 a^4 b^12-2 a^2 b^14+2 a^12 b^2 c^2-11 a^10 b^4 c^2+12 a^8 b^6 c^2+12 a^6 b^8 c^2-28 a^4 b^10 c^2+15 a^2 b^12 c^2-2 b^14 c^2+2 a^12 c^4-11 a^10 b^2 c^4+10 a^8 b^4 c^4+5 a^6 b^6 c^4+15 a^4 b^8 c^4-33 a^2 b^10 c^4+12 b^12 c^4-10 a^10 c^6+12 a^8 b^2 c^6+5 a^6 b^4 c^6+6 a^4 b^6 c^6+20 a^2 b^8 c^6-30 b^10 c^6+20 a^8 c^8+12 a^6 b^2 c^8+15 a^4 b^4 c^8+20 a^2 b^6 c^8+40 b^8 c^8-20 a^6 c^10-28 a^4 b^2 c^10-33 a^2 b^4 c^10-30 b^6 c^10+10 a^4 c^12+15 a^2 b^2 c^12+12 b^4 c^12-2 a^2 c^14-2 b^2 c^14 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28089.

X(21473) is the intersection of the Euler lines of ABC and the polar triangle of the nine-point circle. (Randy Hutson, August 19, 2019)

X(21473) lies on this line: {2,3}

X(21473) = reflection of X(5) in X(17727)

X(21474) = (name pending)

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^2-13 a^10 b^4 c^2+19 a^8 b^6 c^2+2 a^6 b^8 c^2-20 a^4 b^10 c^2+11 a^2 b^12 c^2-b^14 c^2+a^12 c^4-13 a^10 b^2 c^4+18 a^8 b^4 c^4+18 a^6 b^6 c^4-3 a^4 b^8 c^4-27 a^2 b^10 c^4+6 b^12 c^4-5 a^10 c^6+19 a^8 b^2 c^6+18 a^6 b^4 c^6+36 a^4 b^6 c^6+17 a^2 b^8 c^6-15 b^10 c^6+10 a^8 c^8+2 a^6 b^2 c^8-3 a^4 b^4 c^8+17 a^2 b^6 c^8+20 b^8 c^8-10 a^6 c^10-20 a^4 b^2 c^10-27 a^2 b^4 c^10-15 b^6 c^10+5 a^4 c^12+11 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28089.

X(21474) lies on this line: {5,3917}

X(21475) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(15)

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 4 Sqrt[3] b c S) : :

X(21475) lies on these lines: {2, 3}, {15, 4383}, {16, 940}, {81, 11486}, {559, 1277}, {1030, 16645}, {4387, 5699}, {5124, 16644}

X(21476) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(16)

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 4 Sqrt[3] b c S) : :

X(21476) lies on these lines: {2, 3}, {15, 940}, {16, 4383}, {81, 11485}, {1030, 16644}, {1082, 1276}, {4387, 5700}, {5124, 16645}

X(21477) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(32)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 2 a^2 b c - a b^2 c + 2 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 2 b c^3) : :

X(21477) lies on these lines: {2, 3}, {6, 3882}, {32, 4383}, {36, 17284}, {39, 940}, {55, 17023}, {56, 3912}, {69, 5120}, {81, 9605}, {100, 5222}, {198, 17353}, {218, 20769}, {239, 5687}, {386, 19761}, {956, 3661}, {958, 17308}, {980, 5013}, {988, 17022}, {999, 17316}, {1014, 4869}, {1211, 7800}, {1376, 2223}, {1429, 3501}, {1444, 3619}, {2178, 17279}, {2999, 16478}, {3618, 4254}, {3752, 16974}, {3763, 5124}, {3785, 14555}, {3871, 17014}, {3913, 16834}, {3926, 18141}, {5022, 18206}, {5228, 6184}, {5253, 5308}, {5256, 5266}, {5739, 7767}, {12513, 17294}, {14997, 21309}, {17267, 21773}

X(21478) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(50)

Barycentrics a (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 + 2 a^6 b c - 3 a^5 b^2 c - 2 a^4 b^3 c + 3 a^3 b^4 c - 2 a^2 b^5 c - a b^6 c + 2 b^7 c - 3 a^6 c^2 - 3 a^5 b c^2 + 2 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + a^2 b^4 c^2 + a b^5 c^2 - 3 a^5 c^3 - 2 a^4 b c^3 + 2 a^3 b^2 c^3 + a b^4 c^3 - 2 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 + 3 a^3 c^5 - 2 a^2 b c^5 + a b^2 c^5 - 2 b^3 c^5 - a^2 c^6 - a b c^6 - a c^7 + 2 b c^7) : :

X(21478) lies on these lines: {2, 3}, {50, 4383}, {566, 940}

X(21479) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(52)

Barycentrics a (a^9 - 2 a^7 b^2 + 2 a^3 b^6 - a b^8 - 2 a^7 b c + 2 a^6 b^2 c + 6 a^5 b^3 c - 6 a^4 b^4 c - 6 a^3 b^5 c + 6 a^2 b^6 c + 2 a b^7 c - 2 b^8 c - 2 a^7 c^2 + 2 a^6 b c^2 - 4 a^5 b^2 c^2 - 2 a^4 b^3 c^2 + 10 a^3 b^4 c^2 - 2 a^2 b^5 c^2 - 4 a b^6 c^2 + 2 b^7 c^2 + 6 a^5 b c^3 - 2 a^4 b^2 c^3 - 4 a^3 b^3 c^3 - 4 a^2 b^4 c^3 - 2 a b^5 c^3 + 6 b^6 c^3 - 6 a^4 b c^4 + 10 a^3 b^2 c^4 - 4 a^2 b^3 c^4 + 10 a b^4 c^4 - 6 b^5 c^4 - 6 a^3 b c^5 - 2 a^2 b^2 c^5 - 2 a b^3 c^5 - 6 b^4 c^5 + 2 a^3 c^6 + 6 a^2 b c^6 - 4 a b^2 c^6 + 6 b^3 c^6 + 2 a b c^7 + 2 b^2 c^7 - a c^8 - 2 b c^8) : :

X(21479) lies on these lines: {2, 3}, {52, 4383}, {569, 940}, {6193, 18141}, {10320, 10832}

X(21480) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(61)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 4 Sqrt[3] b c S) : :

X(21480) lies on these lines: {2, 3}, {61, 4383}, {62, 940}, {198, 5242}, {4254, 11489}, {5120, 11488}

X(21481) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(62)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 4 Sqrt[3] b c S) : :

X(21481) lies on these lines: {2, 3}, {61, 940}, {62, 4383}, {198, 5243}, {4254, 11488}, {5120, 11489}

X(21482) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(216)

Barycentrics a (a^2 - b^2 - c^2) (a^6 + a^5 b - 2 a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + a b^5 + a^5 c - 2 a^4 b c - 2 a^3 b^2 c + a b^4 c + 2 b^5 c - 2 a^4 c^2 - 2 a^3 b c^2 - 2 a^2 b^2 c^2 - 2 a b^3 c^2 - 2 a^3 c^3 - 2 a b^2 c^3 - 4 b^3 c^3 + a^2 c^4 + a b c^4 + a c^5 + 2 b c^5) : :

X(21482) lies on these lines: {2, 3}, {6, 18603}, {7, 6349}, {63, 77}, {81, 15905}, {189, 268}, {198, 5928}, {216, 4383}, {306, 1259}, {343, 18642}, {577, 940}, {1211, 6389}, {1804, 18652}, {4254, 11433}, {5120, 11427}, {5249, 17073}, {5256, 17102}, {5739, 10432}, {5905, 6356}, {6509, 18591}, {9308, 18667}, {11020, 17011}

X(21483) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(284)

Barycentrics a (a^5 + 2 a^4 b - 2 a^2 b^3 - a b^4 + 2 a^4 c + 4 a^3 b c - 4 a^2 b^2 c - 4 a b^3 c + 2 b^4 c - 4 a^2 b c^2 - 6 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 - a c^4 + 2 b c^4) : :

X(21483) lies on these lines: {2, 3}, {37, 57}, {55, 3755}, {142, 21062}, {284, 4383}, {306, 956}, {579, 940}, {610, 7308}, {942, 5287}, {954, 7085}, {993, 20106}, {1001, 5285}, {1104, 2999}, {1212, 1763}, {1214, 1435}, {1350, 17194}, {1436, 5316}, {1467, 2213}, {2218, 5217}, {2350, 19714}, {3216, 19764}, {3345, 11212}, {3917, 5751}, {5120, 5712}, {5271, 5687}, {5646, 10857}, {5708, 17021}, {5744, 17776}, {7283, 19814}, {15934, 17019}, {17825, 21363}

X(21484) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(389)

Barycentrics a (a^9 - 2 a^7 b^2 + 2 a^3 b^6 - a b^8 - 2 a^7 b c + 2 a^6 b^2 c + 6 a^5 b^3 c - 6 a^4 b^4 c - 6 a^3 b^5 c + 6 a^2 b^6 c + 2 a b^7 c - 2 b^8 c - 2 a^7 c^2 + 2 a^6 b c^2 - 2 a^4 b^3 c^2 + 6 a^3 b^4 c^2 - 2 a^2 b^5 c^2 - 4 a b^6 c^2 + 2 b^7 c^2 + 6 a^5 b c^3 - 2 a^4 b^2 c^3 - 4 a^3 b^3 c^3 - 4 a^2 b^4 c^3 - 2 a b^5 c^3 + 6 b^6 c^3 - 6 a^4 b c^4 + 6 a^3 b^2 c^4 - 4 a^2 b^3 c^4 + 10 a b^4 c^4 - 6 b^5 c^4 - 6 a^3 b c^5 - 2 a^2 b^2 c^5 - 2 a b^3 c^5 - 6 b^4 c^5 + 2 a^3 c^6 + 6 a^2 b c^6 - 4 a b^2 c^6 + 6 b^3 c^6 + 2 a b c^7 + 2 b^2 c^7 - a c^8 - 2 b c^8) : :

X(21484) lies on these lines: {2, 3}, {81, 11426}, {389, 4383}, {578, 940}, {1040, 1872}, {1486, 7681}, {8192, 10786}, {10157, 19904}, {10321, 16541}

X(21485) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3094)

Barycentrics a (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - 2 a^4 b c - a b^4 c - 2 b^5 c - 2 a^2 b^2 c^2 - 2 a b^3 c^2 - 2 a b^2 c^3 - a^2 c^4 - a b c^4 - a c^5 - 2 b c^5) : :

X(21485) lies on these lines: {2, 3}, {183, 18144}, {940, 1691}, {1030, 7778}, {1444, 7735}, {3094, 4383}, {4254, 7774}, {5120, 16989}, {5739, 6393}

X(21486) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3095)

Barycentrics a (a^7 - a^5 b^2 + a^3 b^4 - a b^6 - 2 a^5 b c + 2 a^4 b^2 c + 2 a b^5 c - 2 b^6 c - a^5 c^2 + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 5 a b^4 c^2 + 2 b^5 c^2 + 4 a^2 b^2 c^3 + 4 a b^3 c^3 + a^3 c^4 - 5 a b^2 c^4 + 2 a b c^5 + 2 b^2 c^5 - a c^6 - 2 b c^6) : :

X(21486) lies on these lines: {2, 3}, {940, 3398}, {3095, 4383}

X(21487) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3098)

Barycentrics a (3 a^5 - 3 a b^4 - 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^2 - 10 a b^2 c^2 + 2 b^3 c^2 + 2 a b c^3 + 2 b^2 c^3 - 3 a c^4 - 2 b c^4) : :

X(21487) lies on these lines: {2, 3}, {81, 12017}, {612, 13624}, {614, 3579}, {940, 5092}, {1473, 17781}, {3098, 4383}, {7191, 12702}, {8148, 17024}, {8193, 10072}

X(21488) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3285)

Barycentrics a (a^5 + 2 a^4 b - 2 a^2 b^3 - a b^4 + 2 a^4 c + 4 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + 2 b^4 c - 2 a^2 b c^2 - 2 a b^2 c^2 - 2 a^2 c^3 - 2 a b c^3 - a c^4 + 2 b c^4) : :

X(21488) lies on these lines: {2, 3}, {306, 8666}, {394, 17191}, {940, 4286}, {3187, 3913}, {3285, 4383}, {3670, 5287}, {4255, 17012}, {16672, 17021}

X(21489) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3313)

Barycentrics a (a^8 + a^7 b + a^6 b^2 + a^5 b^3 - a^4 b^4 - a^3 b^5 - a^2 b^6 - a b^7 + a^7 c - 2 a^6 b c + a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c + 2 a^2 b^5 c - a b^6 c - 2 b^7 c + a^6 c^2 + a^5 b c^2 - 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - 3 a^2 b^4 c^2 - 3 a b^5 c^2 + a^5 c^3 + 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 4 a^2 b^3 c^3 - 3 a b^4 c^3 + 2 b^5 c^3 - a^4 c^4 - a^3 b c^4 - 3 a^2 b^2 c^4 - 3 a b^3 c^4 - a^3 c^5 + 2 a^2 b c^5 - 3 a b^2 c^5 + 2 b^3 c^5 - a^2 c^6 - a b c^6 - a c^7 - 2 b c^7) : :

X(21489) lies on these lines: {2, 3}, {940, 5157}, {3313, 4383}

X(21490) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3557)

Barycentrics (4*b*c*sqrt(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4)+a^4+(b+c)*a^3-a^2*(b-c)^2-(b+c)*(b^2+c^2)*a+2*b*c*(b^2+c^2))*a : :

X(21490) lies on these lines: {2, 3}, {940, 14631}, {3557, 4383}

X(21491) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3558)

Barycentrics (4*b*c*sqrt(a^4-(b^2+c^2)*a^2-b^2*c^2+c^4+b^4) -a^4-(b+c)*a^3+a^2*(b-c)^2+(b+c)*(b^2+c^2)*a-2*b*c*(b^2+c^2))*a : :

X(21491) lies on these lines: {2, 3}, {940, 14630}, {3558, 4383}

X(21492) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3592)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 6 b c S) : :

X(21492) lies on these lines: {2, 3}, {81, 6420}, {100, 8225}, {940, 3594}, {3083, 7982}, {3311, 14997}, {3312, 14996}, {3592, 4383}, {3746, 5405}, {4254, 13941}, {5120, 8972}, {5393, 5563}

X(21493) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3001)

Barycentrics a (a^8 + a^7 b - a^6 b^2 - a^5 b^3 + a^4 b^4 + a^3 b^5 - a^2 b^6 - a b^7 + a^7 c - 2 a^6 b c - a^5 b^2 c + 2 a^4 b^3 c + a^3 b^4 c + 2 a^2 b^5 c - a b^6 c - 2 b^7 c - a^6 c^2 - a^5 b c^2 - a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 + 2 a^4 b c^3 - a b^4 c^3 + 2 b^5 c^3 + a^4 c^4 + a^3 b c^4 - a^2 b^2 c^4 - a b^3 c^4 + a^3 c^5 + 2 a^2 b c^5 - a b^2 c^5 + 2 b^3 c^5 - a^2 c^6 - a b c^6 - a c^7 - 2 b c^7) : :

X(21493) lies on these lines: {2, 3}, {3001, 4383}

X(21494) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3003)

Barycentrics a (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 - 2 a^6 b c - 3 a^5 b^2 c + 2 a^4 b^3 c + 3 a^3 b^4 c + 2 a^2 b^5 c - a b^6 c - 2 b^7 c - 3 a^6 c^2 - 3 a^5 b c^2 + 2 a^4 b^2 c^2 + 2 a^3 b^3 c^2 + a^2 b^4 c^2 + a b^5 c^2 - 3 a^5 c^3 + 2 a^4 b c^3 + 2 a^3 b^2 c^3 - 12 a^2 b^3 c^3 + a b^4 c^3 + 2 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 + 3 a^3 c^5 + 2 a^2 b c^5 + a b^2 c^5 + 2 b^3 c^5 - a^2 c^6 - a b c^6 - a c^7 - 2 b c^7) : :

X(21494) lies on these lines: {2, 3}, {940, 5063}, {3003, 4383}

X(21495) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(3053)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 - a c^3 + b c^3) : :

X(21495) lies on these lines: {2, 3}, {35, 17023}, {36, 3912}, {39, 81}, {56, 17316}, {100, 239}, {141, 1444}, {193, 5120}, {241, 6516}, {344, 2178}, {572, 15988}, {574, 980}, {672, 20769}, {940, 5013}, {988, 5287}, {993, 17308}, {1014, 17300}, {1030, 3589}, {1332, 7113}, {1384, 14997}, {1429, 4876}, {1580, 2108}, {1621, 17397}, {2895, 7767}, {2975, 3661}, {3053, 4383}, {3785, 5739}, {3871, 4393}, {3882, 5053}, {4422, 19297}, {5024, 14996}, {5030, 18206}, {5253, 16826}, {5266, 17011}, {5303, 17292}, {6337, 18141}, {7280, 17284}, {7677, 20533}, {8299, 17798}, {8666, 17294}, {8715, 16834}, {17243, 21773}, {19719, 19769}, {19761, 19767}, {19791, 19850}

X(21496) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(7772)

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 6 a^2 b c - a b^2 c - 6 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 6 b c^3) : :

X(21496) lies on these lines: {2, 3}, {940, 5007}, {956, 17367}, {2223, 8167}, {3303, 3912}, {3304, 17023}, {3619, 4254}, {3746, 17284}, {4383, 7772}, {5241, 7795}, {5687, 17292}, {17749, 19758}

X(21497) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(8588)

Barycentrics a (9 a^4 + 9 a^3 b - 9 a^2 b^2 - 9 a b^3 + 9 a^3 c + 2 a^2 b c - 9 a b^2 c + 2 b^3 c - 9 a^2 c^2 - 9 a b c^2 - 9 a c^3 + 2 b c^3) : :

X(21497) lies on these lines: {2, 3}, {940, 8589}, {4383, 8588}, {5124, 15534}

X(21498) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(8589)

Barycentrics a (9 a^4 + 9 a^3 b - 9 a^2 b^2 - 9 a b^3 + 9 a^3 c - 2 a^2 b c - 9 a b^2 c - 2 b^3 c - 9 a^2 c^2 - 9 a b c^2 - 9 a c^3 - 2 b c^3) : :

X(21498) lies on these lines: {2, 3}, {81, 15655}, {940, 8588}, {1030, 15534}, {4383, 8589}

X(21499) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(10316)

Barycentrics a (a^2 - b^2 - c^2) (a^8 + a^7 b - a^6 b^2 - a^5 b^3 - a^4 b^4 - a^3 b^5 + a^2 b^6 + a b^7 + a^7 c + 2 a^6 b c - a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c - 2 a^2 b^5 c + a b^6 c - 2 b^7 c - a^6 c^2 - a^5 b c^2 - 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 + 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 4 a^2 b^3 c^3 - a b^4 c^3 + 2 b^5 c^3 - a^4 c^4 - a^3 b c^4 - a^2 b^2 c^4 - a b^3 c^4 - a^3 c^5 - 2 a^2 b c^5 - a b^2 c^5 + 2 b^3 c^5 + a^2 c^6 + a b c^6 + a c^7 - 2 b c^7) : :

X(21499) lies on these lines: {2, 3}, {4383, 10316}

X(21500) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(10317)

Barycentrics a (a^2 - b^2 - c^2) (a^8 + a^7 b - a^6 b^2 - a^5 b^3 - a^4 b^4 - a^3 b^5 + a^2 b^6 + a b^7 + a^7 c + 2 a^6 b c - a^5 b^2 c + 2 a^4 b^3 c - a^3 b^4 c - 2 a^2 b^5 c + a b^6 c - 2 b^7 c - a^6 c^2 - a^5 b c^2 - a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 + 2 a^4 b c^3 + 4 a^2 b^3 c^3 - a b^4 c^3 + 2 b^5 c^3 - a^4 c^4 - a^3 b c^4 - a^2 b^2 c^4 - a b^3 c^4 - a^3 c^5 - 2 a^2 b c^5 - a b^2 c^5 + 2 b^3 c^5 + a^2 c^6 + a b c^6 + a c^7 - 2 b c^7) : :

X(21500) lies on these lines: {2, 3}, {4383, 10317}

X(21501) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(10485)

Barycentrics a (3 a^6 + 3 a^5 b - 3 a^2 b^4 - 3 a b^5 + 3 a^5 c + 10 a^4 b c - 16 a^2 b^3 c - 3 a b^4 c + 10 b^5 c - 6 a^2 b^2 c^2 - 6 a b^3 c^2 - 16 a^2 b c^3 - 6 a b^2 c^3 - 16 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 - 3 a c^5 + 10 b c^5) : :

X(21501) lies on these lines: {2, 3}, {940, 8586}, {4383, 10485}

X(21502) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(10510)

Barycentrics a (a^8 + a^7 b + a^6 b^2 + a^5 b^3 - a^4 b^4 - a^3 b^5 - a^2 b^6 - a b^7 + a^7 c - 6 a^6 b c + a^5 b^2 c + 6 a^4 b^3 c - a^3 b^4 c + 6 a^2 b^5 c - a b^6 c - 6 b^7 c + a^6 c^2 + a^5 b c^2 - 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - 3 a^2 b^4 c^2 - 3 a b^5 c^2 + a^5 c^3 + 6 a^4 b c^3 - 2 a^3 b^2 c^3 - 3 a b^4 c^3 + 6 b^5 c^3 - a^4 c^4 - a^3 b c^4 - 3 a^2 b^2 c^4 - 3 a b^3 c^4 - a^3 c^5 + 6 a^2 b c^5 - 3 a b^2 c^5 + 6 b^3 c^5 - a^2 c^6 - a b c^6 - a c^7 - 6 b c^7) : :

X(21502) lies on these lines: {2, 3}, {4383, 10510}

X(21503) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(10979)

Barycentrics a (a^2 - b^2 - c^2) (3 a^6 + 3 a^5 b - 6 a^4 b^2 - 6 a^3 b^3 + 3 a^2 b^4 + 3 a b^5 + 3 a^5 c - 2 a^4 b c - 6 a^3 b^2 c + 3 a b^4 c + 2 b^5 c - 6 a^4 c^2 - 6 a^3 b c^2 - 6 a^2 b^2 c^2 - 6 a b^3 c^2 - 6 a^3 c^3 - 6 a b^2 c^3 - 4 b^3 c^3 + 3 a^2 c^4 + 3 a b c^4 + 3 a c^5 + 2 b c^5) : :

X(21503) lies on these lines: {2, 3}, {940, 22052}, {4383, 10979}

X(21504) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(12212)

Barycentrics a (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c + 2 a^4 b c + 8 a^2 b^3 c - a b^4 c + 2 b^5 c - 2 a^2 b^2 c^2 - 2 a b^3 c^2 + 8 a^2 b c^3 - 2 a b^2 c^3 + 8 b^3 c^3 - a^2 c^4 - a b c^4 - a c^5 + 2 b c^5) : :

X(21504) lies on these lines: {2, 3}, {940, 13331}, {4383, 12212}, {5120, 16990}

X(21505) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(12055)

Barycentrics a (3 a^6 + 3 a^5 b - 3 a^2 b^4 - 3 a b^5 + 3 a^5 c - 2 a^4 b c - 8 a^2 b^3 c - 3 a b^4 c - 2 b^5 c - 6 a^2 b^2 c^2 - 6 a b^3 c^2 - 8 a^2 b c^3 - 6 a b^2 c^3 - 8 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 - 3 a c^5 - 2 b c^5) : :

X(21505) lies on these lines: {2, 3}, {4383, 12055}

X(21506) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(14964)

Barycentrics a (a^6 b + 2 a^5 b^2 - 2 a^3 b^4 - a^2 b^5 + a^6 c + 2 a^5 b c - a^4 b^2 c + a^3 b^3 c - 3 a b^5 c + 2 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 - 2 b^5 c^2 + a^3 b c^3 + a^2 b^2 c^3 + 2 a b^3 c^3 + 2 b^4 c^3 - 2 a^3 c^4 - 2 a b^2 c^4 + 2 b^3 c^4 - a^2 c^5 - 3 a b c^5 - 2 b^2 c^5) : :

X(21506) lies on these lines: {2, 3}, {2176, 20367}, {4383, 14964}

X(21507) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(15513)

Barycentrics a (7 a^4 + 7 a^3 b - 7 a^2 b^2 - 7 a b^3 + 7 a^3 c + 2 a^2 b c - 7 a b^2 c + 2 b^3 c - 7 a^2 c^2 - 7 a b c^2 - 7 a c^3 + 2 b c^3) : :

X(21507) lies on these lines: {2, 3}, {940, 15515}, {4383, 15513}, {5124, 6144}, {14997, 15603}

X(21508) = (X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(15815)

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - a^2 b c - 2 a b^2 c - b^3 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 - b c^3) : :

X(21508) lies on these lines: {2, 3}, {9, 8771}, {32, 593}, {35, 17316}, {69, 1030}, {81, 3053}, {160, 20139}, {187, 980}, {193, 1444}, {573, 17209}, {574, 14997}, {940, 5023}, {988, 17011}, {2895, 3926}, {3618, 5124}, {3912, 5010}, {4262, 18206}, {4277, 16702}, {4383, 15815}, {4384, 5267}, {5206, 5337}, {5739, 6337}, {7280, 17023}

Brocard-Euler points of type 1: X(21509)-X(21543)

Suppose that X is a point in the plane of a triangle ABC and that X has barycentrics

a²(h a² + k b² + k c²) : b²(h b² + k c² + k a²) : c²(h c² + k a² + k b²),

where h and k are symmetric functions of (a,b,c) having the same degree of homogeneity. Then X is a triangle center that lies on the Brocard axis, X(3)X(6), and the point

(X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X

lies on the Euler line of ABC. This point is here named the (h,k) Brocard-Euler point of type 1. (For type 2, see the preamble just before X(21544).) The appearance of (h,k,q) in the following list means that X(q) is the (h,k) Brocard-Euler point of X(q):

(0,1,11343), (1,-5,21509), (1,-4,21510), (1,-3,21511), (1,-2,16436), (1,-1,3), (1,0,21477), (1,1,2), (1,2,21496), (1,3,21514), (1,4,21515), (1,5,21516), (2,-5,21517), (2,-3,21518), (2,-1,16431), (2,1,21519), (2,3,21520), (2,5,21521), (3,-5,21508), (3,-4,21523), (3,-2,21524), (3,-1,21495), (3,1,21526), (3,2,21527), (3,4,21528), (3,5,21529), (4,-5,21498) (4,-3,21507), (4,-1,21532), (4,1,21533), (4,3,21534), (4,5,21535), (5,-4,21497), (5,-3,21537), (5,-2,21538), (5,-1,21539), (5,1,21540), (5,2,21541), (5,3,21542), (5,4,21543)

X(21509) = (1,-5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 4 a^2 b c - 3 a b^2 c - 4 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 4 b c^3) : :

X(21509) lies on these lines: {2, 3}, {81, 21309}, {524, 4254}, {597, 5120}, {940, 1384}, {1030, 21358}, {4383, 5024}, {5217, 17284}, {6390, 14555}

X(21510) = (1,-4) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 6 a^2 b c - 5 a b^2 c - 6 b^3 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 6 b c^3) : :

X(21510) lies on these lines: {2, 3}, {4254, 11008}

X(21511) = (1,-3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a^2 b c - a b^2 c - b^3 c - a^2 c^2 - a b c^2 - a c^3 - b c^3) : :

X(21511) lies on these lines: {2, 3}, {6, 1444}, {9, 1958}, {32, 81}, {35, 3912}, {36, 17023}, {38, 18266}, {39, 1931}, {41, 63}, {55, 4447}, {77, 2199}, {100, 761}, {141, 1030}, {172, 3666}, {187, 5337}, {193, 4254}, {198, 17257}, {239, 2975}, {284, 16574}, {335, 18265}, {573, 15988}, {940, 3053}, {988, 1468}, {993, 4384}, {1014, 17379}, {1211, 7789}, {1384, 14996}, {1580, 17596}, {1621, 2223}, {1792, 19760}, {1959, 3496}, {2178, 17321}, {2220, 16696}, {2268, 21371}, {2344, 7146}, {2895, 3933}, {3008, 5267}, {3589, 5124}, {3871, 6542}, {3926, 5739}, {4251, 18206}, {4364, 19297}, {4383, 5013}, {5010, 17284}, {5024, 14997}, {5035, 16702}, {5248, 16831}, {5253, 17397}, {5266, 17019}, {5303, 17367}, {6337, 14555}, {7676, 20533}, {8666, 16834}, {8715, 17294}, {17045, 21773}, {19758, 19767}, {19834, 19850}

X(21512) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(12212), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2 (a^6 + 2 a^4 b^2 - 2 a^2 b^4 - b^6 + 2 a^4 c^2 - a^2 b^2 c^2 - 2 b^4 c^2 - 2 a^2 c^4 - 2 b^2 c^4 - c^6) : :

X(21512) lies on these lines: {2, 3}, {32, 10329}, {51, 12054}, {184, 9821}, {206, 22138}, {2076, 3117}, {3511, 7669}, {7800, 9918}, {9301, 15080}

X(21513) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(12055), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2 (a^6 - b^6 + 5 a^2 b^2 c^2 + 4 b^4 c^2 + 4 b^2 c^4 - c^6) : :

X(21513) lies on these lines: {2, 3}, {39, 20998}, {125, 6287}, {373, 3398}, {575, 3506}, {1495, 12054}, {3095, 5651}, {3284, 9822}, {5007, 5943}, {5158, 19137}, {7772, 9306}, {9301, 10545}, {9969, 22138}

X(21514) = (1,3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 4 a^2 b c - a b^2 c - 4 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 4 b c^3) : :

X(21514) lies on these lines: {2, 3}, {55, 17284}, {141, 4254}, {198, 17306}, {940, 16783}, {956, 5222}, {958, 3008}, {980, 4383}, {988, 5234}, {999, 17023}, {1384, 5337}, {2178, 17384}, {2223, 4423}, {3216, 19758}, {3295, 3912}, {3555, 5256}, {3589, 5120}, {3666, 17742}, {3933, 14555}, {4384, 9708}, {5228, 16788}, {5266, 17022}, {5743, 7795}, {6767, 17316}, {7767, 18141}, {9709, 17308}, {14997, 22246}

X(21515) = (1,4) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 10 a^2 b c - 3 a b^2 c - 10 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 10 b c^3) : :

X(21515) lies on these lines: {2, 3}, {940, 5008}, {4254, 21356}

X(21516) = (1,5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 3 a^2 b c - a b^2 c - 3 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 3 b c^3) : :

X(21516) lies on these lines: {2, 3}, {81, 5007}, {100, 17292}, {980, 7772}, {1444, 3589}, {1621, 17244}, {2223, 5284}, {2975, 17367}, {3303, 17316}, {3620, 4254}, {3746, 3912}, {3871, 17230}, {5241, 7789}, {5260, 16815}, {5266, 17021}, {5563, 17023}, {9605, 14997}

X(21517) = (2,-5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (7 a^4 + 7 a^3 b - 7 a^2 b^2 - 7 a b^3 + 7 a^3 c - 6 a^2 b c - 7 a b^2 c - 6 b^3 c - 7 a^2 c^2 - 7 a b c^2 - 7 a c^3 - 6 b c^3) : :

X(21517) lies on these lines: {2, 3}

X(21518) = (2,-3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 2 a^2 b c - 5 a b^2 c - 2 b^3 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 2 b c^3) : :

X(21518) lies on these lines: {2, 3}, {940, 5206}, {980, 5023}, {1444, 11008}, {5210, 5337}

X(21519) = (2,1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 6 a^2 b c - a b^2 c + 6 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 6 b c^3) : :

X(21519) lies on these lines: {2, 3}, {940, 7772}, {956, 17292}, {3303, 17023}, {3304, 3912}, {3619, 5120}, {4383, 5007}, {5241, 7800}, {5563, 17284}, {5687, 17367}, {17749, 19761}

X(21520) = (2,3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 10 a^2 b c - a b^2 c - 10 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 10 b c^3) : :

X(21520) lies on these lines: {2, 3}, {4383, 5041}

X(21521) = (2,5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 14 a^2 b c - 3 a b^2 c - 14 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 14 b c^3) : :

X(21521) lies on these lines: {2, 3}

X(21522) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(14964), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2 (a^5 b^2 + a^4 b^3 - a^3 b^4 - a^2 b^5 + 2 a^4 b^2 c - 2 a^2 b^4 c + a^5 c^2 + 2 a^4 b c^2 - b^5 c^2 + a^4 c^3 + b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5) : :

X(21522) lies on these lines: {2, 3}, {649, 4057}, {3011, 8618}, {5764, 21319}

X(21523) = (3,-4) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (7 a^4 + 7 a^3 b - 7 a^2 b^2 - 7 a b^3 + 7 a^3 c - 2 a^2 b c - 7 a b^2 c - 2 b^3 c - 7 a^2 c^2 - 7 a b c^2 - 7 a c^3 - 2 b c^3) : :

X(21523) lies on these lines: {2, 3}, {940, 15513}, {980, 5210}, {1030, 6144}, {4383, 15515}, {14996, 15603}

X(21524) = (3,-2) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c + 2 a^2 b c - 5 a b^2 c + 2 b^3 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 2 b c^3) : :

X(21524) lies on these lines: {2, 3}, {4383, 5206}, {5337, 15815}

X(21525) = (A,B,C,X(2); A',B',C',X(6)) COLLINEATION IMAGE OF X(14966), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics a^2 (a^2 b^2 - b^4 + a^2 c^2 - c^4) (a^8 - a^6 b^2 - a^4 b^4 + a^2 b^6 - a^6 c^2 + 3 a^4 b^2 c^2 - a^2 b^4 c^2 + b^6 c^2 - a^4 c^4 - a^2 b^2 c^4 - 2 b^4 c^4 + a^2 c^6 + b^2 c^6) : :

X(21525) lies on these lines: {2, 3}, {1177, 14910}, {2079, 3569}

X(21526) = (3,1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 4 a^2 b c - a b^2 c + 4 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 4 b c^3) : :

X(21526) lies on these lines: {2, 3}, {56, 17284}, {141, 5120}, {940, 9605}, {980, 5024}, {999, 3912}, {1376, 3008}, {2178, 17357}, {2223, 4413}, {2999, 5266}, {3216, 19761}, {3295, 17023}, {3589, 4254}, {3933, 18141}, {4383, 5337}, {4384, 9709}, {5222, 5687}, {5228, 16549}, {5743, 7800}, {7373, 17316}, {7767, 14555}, {9708, 17308}, {14996, 22246}

X(21527) = (3,2) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 10 a^2 b c - a b^2 c + 10 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 10 b c^3) : :

X(21527) lies on these lines: {2, 3}, {940, 5041}

X(21528) = (3,4) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 14 a^2 b c - a b^2 c - 14 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 14 b c^3) : :

X(21528) lies on these lines: {2, 3}, {3912, 8162}

X(21529) = (3,5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 8 a^2 b c - a b^2 c - 8 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 8 b c^3) : :

X(21529) lies on these lines: {2, 3}, {3008, 9708}, {3295, 17284}, {3763, 4254}, {3912, 6767}, {7373, 17023}

X(21530) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(1474), where A'B'C' = MEDIAL TRIANGLE

Barycentrics (b + c) (-a^2 + b^2 + c^2) (a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(21530) lies on these lines: {2, 3}, {10, 4523}, {12, 1214}, {65, 18588}, {72, 21015}, {119, 122}, {120, 127}, {125, 22076}, {169, 1213}, {339, 1228}, {495, 18643}, {668, 16100}, {942, 18635}, {975, 1060}, {1038, 5219}, {1040, 9581}, {1062, 1834}, {1211, 5044}, {1437, 11064}, {1446, 6356}, {1698, 10319}, {1714, 19728}, {1848, 9895}, {3330, 8757}, {3452, 3454}, {3564, 22136}, {3695, 20235}, {3789, 18639}, {3925, 19857}, {4466, 21671}, {5179, 14873}, {5230, 22119}, {5262, 12433}, {5706, 5803}, {5719, 18447}, {5752, 13567}, {6350, 11681}, {6376, 18749}, {9580, 18257}, {12699, 15941}, {13466, 15526}, {18651, 18732}, {18674, 21678}, {21935, 22057}

X(21530) = isotomic conjugate of cevapoint of X(2) and X(28)
X(21530) = complement of X(28)
X(21530) = complementary conjugate of X(942)

X(21531) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(9147), where A'B'C' = MEDIAL TRIANGLE

Barycentrics a^4*b^4 - a^2*b^6 - b^6*c^2 + a^4*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6 : :
Barycentrics b^3 cos(B + ω) + c^3 cos(C + ω) : :

X(21531) lies on these lines: {2, 3}, {51, 14881}, {114, 8841}, {115, 3229}, {125, 33330}, {141, 34845}, {184, 14880}, {230, 8623}, {263, 21850}, {276, 23295}, {287, 2456}, {290, 325}, {311, 20819}, {338, 3001}, {512, 625}, {524, 7668}, {623, 33490}, {624, 33491}, {626, 2387}, {1503, 36213}, {1613, 3767}, {2782, 36212}, {2979, 32521}, {3051, 5305}, {3117, 5254}, {3260, 20975}, {3564, 20021}, {3580, 18322}, {3589, 3613}, {3819, 3934}, {3933, 20023}, {5031, 39080}, {5103, 9467}, {6393, 25332}, {7838, 11225}, {9512, 22151}, {11574, 14767}, {13881, 21001}, {14570, 22087}, {16310, 18371}, {19130, 34236}, {23291, 32816}, {30737, 38368}, {36794, 37893}

X(21531) = isotomic conjugate of cevapoint of X(2) and X(237)
X(21531) = complement of X(237)
X(21531) = complementary conjugate of X(11672)
X(21531) = orthocentroidal-circle-inverse of X(11328)
X(21531) = {X(2),X(4)}-harmonic conjugate of X(11328)

X(21532) = (4,-1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c + 6 a^2 b c - 5 a b^2 c + 6 b^3 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 6 b c^3) : :

X(21532) lies on these lines: {2, 3}, {5120, 11008}

X(21533) = (4,1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c + 10 a^2 b c - 3 a b^2 c + 10 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 10 b c^3) : :

X(21533) lies on these lines: {2, 3}, {4383, 5008}, {5120, 21356}

X(21534) = (4,3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 14 a^2 b c - a b^2 c + 14 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 14 b c^3) : :

X(21534) lies on these lines: {2, 3}, {8162, 17023}

X(21535) = (4,5) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - 18 a^2 b c - a b^2 c - 18 b^3 c - a^2 c^2 - a b c^2 - a c^3 - 18 b c^3) : :

X(21535) lies on these lines: {2, 3}

X(21536) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(19559), where A'B'C' = MEDIAL TRIANGLE

Barycentrics -a^6 b^2 + b^8 - a^6 c^2 + a^2 b^4 c^2 - b^6 c^2 + a^2 b^2 c^4 - b^2 c^6 + c^8 : :

X(21536) lies on these lines: {2, 3}, {525, 4486}, {1503, 3506}

X(21536) = complement of X(6660)
X(21536) = complementary conjugate of X(19576)

X(21537) = (5,-3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c + a^2 b c - 2 a b^2 c + b^3 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 + b c^3) : :

X(21537) lies on these lines: {2, 3}, {36, 17316}, {69, 5124}, {81, 5013}, {187, 14997}, {574, 5337}, {940, 15815}, {988, 17019}, {1030, 3618}, {1444, 3620}, {2223, 4393}, {2895, 3785}, {3912, 7280}, {4383, 5023}, {5010, 17023}, {5267, 17308}

X(21538) = (5,-2) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (7 a^4 + 7 a^3 b - 7 a^2 b^2 - 7 a b^3 + 7 a^3 c + 6 a^2 b c - 7 a b^2 c + 6 b^3 c - 7 a^2 c^2 - 7 a b c^2 - 7 a c^3 + 6 b c^3) : :

X(21538) lies on these lines: {2, 3}

X(21539) = (5,-1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c + 4 a^2 b c - 3 a b^2 c + 4 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 4 b c^3) : :

X(21539) lies on these lines: {2, 3}, {524, 5120}, {597, 4254}, {940, 5024}, {1384, 4383}, {2223, 16833}, {5124, 21358}, {5204, 17284}, {5337, 9605}, {6390, 18141}

X(21540) = (5,1) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 3 a^2 b c - a b^2 c + 3 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 3 b c^3) : :

X(21540) lies on these lines: {2, 3}, {81, 7772}, {100, 17367}, {1014, 17232}, {1444, 3763}, {2223, 16815}, {2975, 17292}, {3304, 17316}, {3306, 21808}, {3620, 5120}, {3746, 17023}, {3912, 5563}, {5007, 5337}, {5253, 17244}, {5266, 17012}, {9605, 14996}

X(21541) = (5,2) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c + 14 a^2 b c - 3 a b^2 c + 14 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 14 b c^3) : :

X(21541) lies on these lines: {2, 3}

X(21542) = (5,3) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 8 a^2 b c - a b^2 c + 8 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 8 b c^3) : :

X(21542) lies on these lines: {2, 3}, {999, 17284}, {3008, 9709}, {3763, 5120}, {3912, 7373}, {5337, 21309}, {6767, 17023}

X(21543) = (5,4) BROCARD-EULER POINT OF TYPE 1

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c + 18 a^2 b c - a b^2 c + 18 b^3 c - a^2 c^2 - a b c^2 - a c^3 + 18 b c^3) : :

X(21543) lies on these lines: {2, 3}

Brocard-Euler points of type 2: X(21544)-X(21577)

Suppose that X is a point in the plane of a triangle ABC and that X has barycentrics

a(h cos A + k sin A) : b(h cos B + k sin B) : c(h cos C + k sin C)

where h and k are symmetric functions of (a,b,c) having the same degree of homogeneity. Then X is a triangle center that lies on the Brocard axis, X(3)X(6), and the point

(X(3),X(6),X(1),X(2);X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X

lies on the Euler line of ABC. This point is here named the (h,k) Brocard-Euler point of type 2. (For type 1, see the preamble just before X(21509).) The appearance of (h,k,q) in the following list means that X(q) is the (h,k) Brocard-Euler point of X(q):

(0,1,11343), (1,-5,21492), (1,-4,21545), (1,-3,21546), (1,-2,21547), (1,-1,16433), (1,0,3), (1,1,2), (1,2,21548), (1,3,21549), (1,4,21550), (1,5,21551), (2,-5,21552), (2,-3,21553), (2,-1,16641), (2,1,16440), (2,3,21554), (2,5,21555), (3,-5,21556), (3,-4,21557), (3,-2,21558), (3,-1,21559), (3,1,21560), (3,2,21561), (3,4,21562), (3,5,21563), (4,-5,21564) (4,-3,21565), (4,-1,21566), (4,1,21567), (4,3,21568), (4,5,21569), (5,-4,21570), (5,-3,21571), (5,-2,21572), (5,-1,21573), (5,1,21574), (5,2,21575), (5,3,21576), (5,4,21577)

X(21544) = (1,-5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 20 b c S) : :

X(21544) lies on these lines: {2, 3}

X(21544) = {X(2),X(3)}-harmonic conjugate of X(21551)

X(21545) = (1,-4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 16 b c S) : :

X(21545) lies on these lines: {2, 3}, {81, 6500}, {940, 6417}, {3084, 10247}, {4254, 8253}, {4383, 6418}, {5120, 8252}, {5393, 6767}, {5405, 7373}

X(21546) = (1,-3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 12 b c S) : :

X(21546) lies on these lines: {2, 3}, {81, 6427}, {940, 6419}, {3083, 15178}, {3084, 10222}, {3303, 5393}, {3304, 5405}, {4383, 6420}, {6417, 14996}, {6418, 14997}

X(21547) = (1,-2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 8 b c S) : :

X(21547) lies on these lines: {2, 3}, {81, 6417}, {590, 4254}, {615, 5120}, {940, 3311}, {999, 5405}, {1482, 3084}, {3083, 10246}, {3295, 5393}, {3312, 4383}, {5790, 6348}, {8167, 8225}

X(21548) = (1,2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 8 b c S) : :

X(21548) lies on these lines: {2, 3}, {81, 6418}, {590, 5120}, {615, 4254}, {940, 3312}, {999, 5393}, {1376, 8225}, {1482, 3083}, {3084, 10246}, {3295, 5405}, {3311, 4383}, {5790, 6347}

X(21549) = (1,3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 12 b c S) : :

X(21549) lies on these lines: {2, 3}, {81, 6428}, {940, 6420}, {3083, 10222}, {3084, 15178}, {3303, 5405}, {3304, 5393}, {4383, 6419}, {4413, 8225}, {6417, 14997}, {6418, 14996}

X(21550) = (1,4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 16 b c S) : :

X(21550) lies on these lines: {2, 3}, {81, 6501}, {940, 6418}, {3083, 10247}, {4254, 8252}, {4383, 6417}, {5120, 8253}, {5393, 7373}, {5405, 6767}

X(21551) = (1,5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 20 b c S) : :

X(21551) lies on these lines: {2, 3}

X(21551) = {X(2),X(3)}-harmonic conjugate of X(21544)

X(21552) = (2,-5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 10 b c S) : :

X(21552) lies on these lines: {2, 3}, {940, 6431}, {3084, 16200}, {4383, 6432}

X(21553) = (2,-3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + 6 b c S) : :

X(21553) lies on these lines: {2, 3}, {81, 6419}, {940, 3592}, {3084, 7982}, {3311, 14996}, {3312, 14997}, {3594, 4383}, {3746, 5393}, {4254, 8972}, {5120, 13941}, {5405, 5563}

X(21554) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1) COLLINEATION IMAGE OF X(182)

Barycentrics a*(b + c)*(a*b^3 - a*b^2*c - a*b*c^2 - b^2*c^2 + a*c^3) : :

X(21554) lies on these lines: {2, 3}, {86, 182}, {98, 17758}, {183, 17206}, {262, 17749}, {511, 17277}, {517, 16823}, {946, 9441}, {966, 10519}, {1350, 17259}, {1351, 17349}, {1352, 17234}, {1353, 20090}, {1385, 16830}, {1503, 17245}, {2271, 7736}, {3564, 17300}, {3815, 18755}, {3945, 14912}, {4648, 6776}, {5021, 7735}, {5050, 17379}, {5085, 15668}, {5205, 18908}, {5433, 17798}, {5480, 17337}, {5603, 16020}, {6626, 15819}, {7179, 11374}, {8550, 17392}, {9534, 19782}, {9751, 19862}, {10516, 17265}, {11898, 17375}, {14561, 17352}, {15069, 17313}

X(21554) = complement of X(7385)

X(21555) = (2,5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (a^4 + a^3 b - a^2 b^2 - a b^3 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 - 10 b c S) : :

X(21555) lies on these lines: {2, 3}, {940, 6432}, {3083, 16200}, {4383, 6431}

X(21556) = (3,-5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 20 b c S) : :

X(21556) lies on these lines: {2, 3}, {3084, 11278}

X(21557) = (3,-4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 16 b c S) : :

X(21557) lies on these lines: {2, 3}, {940, 6199}, {3084, 8148}, {4254, 13846}, {4383, 6395}, {5120, 13847}

X(21558) = (3,-2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 8 b c S) : :

X(21558) lies on these lines: {2, 3}, {81, 6199}, {940, 6221}, {3084, 12702}, {4383, 6398}, {6348, 18525}

X(21559) = (3,-1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 + 4 b c S) : :

X(21559) lies on these lines: {2, 3}, {81, 6221}, {940, 6200}, {1030, 13846}, {3083, 13624}, {3084, 3579}, {4254, 19054}, {4383, 6396}, {5120, 19053}, {5124, 13847}, {5204, 5405}, {5217, 5393}, {6348, 18481}, {6445, 14996}, {6446, 14997}

X(21560) = (3,1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 4 b c S) : :

X(21560) lies on these lines: {2, 3}, {81, 6398}, {940, 6396}, {1030, 13847}, {3083, 3579}, {3084, 13624}, {4254, 19053}, {4383, 6200}, {5120, 19054}, {5124, 13846}, {5204, 5393}, {5217, 5405}, {6347, 18481}, {6445, 14997}, {6446, 14996}

X(21561) = (3,2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 8 b c S) : :

X(21561) lies on these lines: {2, 3}, {81, 6395}, {940, 6398}, {3083, 12702}, {4383, 6221}, {4428, 8225}, {6347, 18525}

X(21562) = (3,4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 16 b c S) : :

X(21562) lies on these lines: {2, 3}, {940, 6395}, {3083, 8148}, {4254, 13847}, {4383, 6199}, {4421, 8225}, {5120, 13846}

X(21563) = (3,5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (3 a^4 + 3 a^3 b - 3 a^2 b^2 - 3 a b^3 + 3 a^3 c - 3 a b^2 c - 3 a^2 c^2 - 3 a b c^2 - 3 a c^3 - 20 b c S) : :

X(21563) lies on these lines: {2, 3}, {3083, 11278}

X(21564) = (4,-5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 + 10 b c S) : :

X(21564) lies on these lines: {2, 3}, {81, 6431}, {3084, 11531}

X(21565) = (4,-3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 + 6 b c S) : :

X(21565) lies on these lines: {2, 3}, {81, 3592}, {371, 14996}, {372, 14997}, {940, 6425}, {1444, 3593}, {3084, 7991}, {4383, 6426}

X(21566) = (4,-1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 + 2 b c S) : :

X(21566) lies on these lines: {2, 3}, {81, 1151}, {165, 3084}, {488, 2895}, {940, 6409}, {1030, 3068}, {1270, 1444}, {3069, 5124}, {3083, 7987}, {4297, 6348}, {4383, 6410}, {5010, 5393}, {5405, 7280}, {6200, 14996}, {6347, 10164}

X(21566) = {X(2),X(3)}-harmonic conjugate of X(21567)

X(21567) = (4,1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 - 2 b c S) : :

X(21567) lies on these lines: {2, 3}, {81, 1152}, {165, 3083}, {487, 2895}, {940, 6410}, {1030, 3069}, {1271, 1444}, {3068, 5124}, {3084, 7987}, {4297, 6347}, {4383, 6409}, {5010, 5405}, {5393, 7280}, {6200, 14997}, {6348, 10164}

X(21567) = {X(2),X(3)}-harmonic conjugate of X(21566)

X(21568) = (4,3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 - 6 b c S) : :

X(21568) lies on these lines: {2, 3}, {81, 3594}, {371, 14997}, {372, 14996}, {940, 6426}, {1444, 3595}, {3083, 7991}, {4383, 6425}

X(21569) = (4,5) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (2 a^4 + 2 a^3 b - 2 a^2 b^2 - 2 a b^3 + 2 a^3 c - 2 a b^2 c - 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 - 10 b c S) : :

X(21569) lies on these lines: {2, 3}, {81, 6432}, {3083, 11531}

X(21570) = (5,-4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 16 b c S) : :

X(21570) lies on these lines: {2, 3}

X(21571) = (5,-3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 12 b c S) : :

X(21571) lies on these lines: {2, 3}, {940, 6453}, {4383, 6454}

X(21572) = (5,-2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 8 b c S) : :

X(21572) lies on these lines: {2, 3}, {940, 6449}, {4383, 6450}, {9690, 14996}

X(21573) = (5,-1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 + 4 b c S) : :

X(21573) lies on these lines: {2, 3}, {81, 6449}, {3083, 17502}

X(21574) = (5,1) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 4 b c S) : :

X(21574) lies on these lines: {2, 3}, {81, 6450}, {3084, 17502}

X(21575) = (5,2) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 8 b c S) : :

X(21575) lies on these lines: {2, 3}, {940, 6450}, {4383, 6449}, {9690, 14997}

X(21576) = (5,3) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 12 b c S) : :

X(21576) lies on these lines: {2, 3}, {940, 6454}, {4383, 6453}

X(21577) = (5,4) BROCARD-EULER POINT OF TYPE 2

Barycentrics a (5 a^4 + 5 a^3 b - 5 a^2 b^2 - 5 a b^3 + 5 a^3 c - 5 a b^2 c - 5 a^2 c^2 - 5 a b c^2 - 5 a c^3 - 16 b c S) : :

X(21577) lies on these lines: {2, 3}

X(21578) = X(1)X(7)∩X(36)X(80)

Barycentrics 4*a^4-(b+c)*a^3-(3*b^2-4*b*c+3*c^2)*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(21578) = X(7)-3*X(18450), 3*X(36)-X(80), 3*X(36)-2*X(3911), 2*X(80)-3*X(1737), X(1317)+3*X(15326), 3*X(1319)-2*X(1387), 3*X(1737)-4*X(3911), 3*X(4511)-X(17484), 3*X(4881)-X(5080), X(5176)-3*X(13587), 2*X(6745)-3*X(15015), X(17484)+3*X(20067)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28094.

X(21578) lies on these lines: {1, 7}, {3, 5252}, {10, 4188}, {11, 5126}, {12, 13624}, {30, 1319}, {35, 10106}, {36, 80}, {46, 944}, {56, 5722}, {165, 12647}, {214, 535}, {354, 15935}, {355, 5204}, {376, 1000}, {382, 11376}, {388, 3612}, {484, 519}, {498, 5726}, {499, 5691}, {517, 1317}, {518, 10609}, {527, 4867}, {529, 5440}, {550, 3057}, {551, 2320}, {553, 5425}, {631, 10827}, {759, 18653}, {912, 12757}, {920, 10085}, {946, 10483}, {950, 5563}, {952, 1155}, {999, 18530}, {1055, 5179}, {1125, 3585}, {1385, 7354}, {1388, 12699}, {1420, 1479}, {1455, 6357}, {1478, 3576}, {1657, 12701}, {1836, 10246}, {2078, 10058}, {2093, 16236}, {2099, 3655}, {2646, 5719}, {2975, 17647}, {3245, 7972}, {3337, 6738}, {3338, 3486}, {3419, 11194}, {3474, 7967}, {3579, 10944}, {3586, 10072}, {3624, 6919}, {3636, 16118}, {3679, 5744}, {3689, 9945}, {3698, 17563}, {3811, 20076}, {3817, 18513}, {3880, 12732}, {3897, 12609}, {4031, 5902}, {4511, 17484}, {4668, 5775}, {4881, 5080}, {4996, 6735}, {5045, 10543}, {5131, 9803}, {5176, 13587}, {5183, 5844}, {5251, 6666}, {5270, 13411}, {5432, 17502}, {5433, 18480}, {5537, 10087}, {5587, 6970}, {5768, 10573}, {5770, 5881}, {5882, 5903}, {5886, 12943}, {6737, 6763}, {6745, 15015}, {6882, 18857}, {6906, 14798}, {6963, 7951}, {6979, 19925}, {7288, 10826}, {7742, 12114}, {7968, 9647}, {9615, 13905}, {9655, 11375}, {9657, 11374}, {9957, 15338}, {10074, 12119}, {11009, 13607}, {11010, 12512}, {11373, 12953}, {11709, 18968}, {15171, 20323}, {17291, 19869}

X(21578) = midpoint of X(i) and X(j) for these {i,j}: {1, 4316}, {3245, 7972}, {4511, 20067}
X(21578) = reflection of X(i) in X(j) for these (i,j): (11, 5126), (3689, 9945), (6882, 18857)
X(21578) = X(3581)-of-intouch-triangle
X(21578) = X(4316)-of-anti-Aquila-triangle
X(21578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4299, 1770), (1, 4324, 10624), (1, 4325, 4292), (1, 4330, 12575), (1, 4333, 962), (36, 80, 3911), (56, 18481, 10572), (80, 3911, 1737), (3600, 4305, 1), (4293, 5731, 1), (4294, 4308, 1), (4297, 4311, 1), (4297, 4315, 4304), (4304, 4311, 4315), (4304, 4315, 1)

X(21579) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(21579) lies on these lines: {5, 75}, {72, 18738}, {304, 21580}, {312, 857}, {349, 908}, {3820, 17867}, {4043, 22018}, {18045, 18140}, {18050, 21604}

X(21580) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a - b) (a - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21580) lies on these lines: {11, 75}, {100, 9086}, {190, 658}, {304, 21579}, {312, 8024}, {693, 3952}, {799, 3257}, {908, 20448}, {1978, 3807}, {3835, 7239}, {4043, 22032}, {4080, 16727}, {4576, 18155}, {5087, 20435}, {6063, 18142}, {16593, 18045}, {20914, 21594}

X(21580) = isotomic conjugate of isogonal conjugate of X(21362)

X(21581) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(21581) lies on these lines: {12, 75}, {69, 2478}, {76, 2051}, {304, 21579}, {311, 322}, {312, 21594}, {349, 20448}, {14829, 18140}, {16284, 20943}, {18137, 18747}, {20914, 20922}

X(21582) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^6 - a^4 b^2 + a^2 b^4 + b^6 - a^4 c^2 + 2 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 + c^6) : :

X(21582) lies on these lines: {19, 27}, {47, 4008}, {76, 5179}, {304, 18669}, {1959, 18670}, {2064, 14615}, {6374, 20451}, {17858, 18713}, {18137, 20922}, {20444, 21596}

X(21583) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^7 - a^4 b^3 + a^3 b^4 + b^7 - a^4 c^3 - b^4 c^3 + a^3 c^4 - b^3 c^4 + c^7) : :

X(21583) lies on these lines: {22, 75}, {312, 857}, {319, 3681}

X(21584) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^7 - a^4 b^3 + a^3 b^4 + b^7 - a^3 b^2 c^2 + a^2 b^3 c^2 - a^4 c^3 + a^2 b^2 c^3 - b^4 c^3 + a^3 c^4 - b^3 c^4 + c^7) : :

X(21584) lies on these lines: {23, 75}, {312, 857}, {7199, 21612}

X(21585) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + b^5 + c^5) : :

X(21585) lies on these lines: {32, 75}, {312, 20933}, {14963, 18050}, {20641, 21589}, {20914, 21604}, {20951, 21608}

X(21586) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + a^3 b^2 - a^2 b^3 + b^5 + a^3 b c - a b^3 c + a^3 c^2 - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 + c^5) : :

X(21586) lies on these lines: {35, 75}, {76, 20940}, {99, 1789}, {304, 20922}, {18138, 21595}, {20951, 20955}

X(21587) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + a^3 b^2 - a^2 b^3 + b^5 - a^3 b c + a b^3 c + a^3 c^2 - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + c^5) : :

X(21587) lies on these lines: {36, 75}, {274, 6358}, {304, 20922}, {14349, 18081}, {20646, 21602}, {20924, 20940}, {20947, 20951}

X(21588) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 - a^4 b + a b^4 + b^5 - a^4 c + 2 a^3 b c - 2 a b^3 c + b^4 c + 2 a b^2 c^2 - 2 b^3 c^2 - 2 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 + c^5) : :

X(21588) lies on these lines: {40, 75}, {76, 5179}, {85, 92}, {304, 20922}, {318, 4872}, {664, 20220}, {7210, 11109}, {20920, 21605}

X(21589) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + a^4 b - a b^4 + b^5 + a^4 c - b^4 c - a c^4 - b c^4 + c^5) : :

X(21589) lies on these lines: {41, 75}, {304, 20642}, {18055, 20927}, {20444, 21593}, {20641, 21585}, {20646, 20926}

X(21590) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^3 b + a b^3 - a^3 c + a^2 b c - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3) : :

X(21590) lies on these lines: {43, 75}, {76, 85}, {92, 18022}, {305, 908}, {322, 4087}, {2887, 6376}, {4071, 17786}, {4138, 6381}, {4485, 20930}, {4892, 18067}, {6383, 20335}, {18142, 21615}, {18157, 18743}, {18159, 20945}, {20444, 20451}, {20446, 20928}, {20641, 20940}, {20643, 20922}

X(21591) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-2 a^3 + a^2 b - a b^2 + 2 b^3 + a^2 c - b^2 c - a c^2 - b c^2 + 2 c^3) : :

X(21591) lies on these lines: {44, 75}, {304, 18137}, {312, 17791}, {661, 786}, {3912, 20956}, {4043, 4053}, {4358, 18073}, {4422, 20912}, {17789, 18151}, {18142, 20641}, {20648, 20941}, {20920, 21600}

X(21592) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-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(21592) lies on these lines: {45, 75}, {304, 18137}, {312, 3969}, {7834, 17400}, {18143, 20946}, {18152, 20922}

X(21593) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^6 + a^4 b^2 - a^2 b^4 + b^6 + a^4 c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 + c^6) : :

X(21593) lies on these lines: {48, 75}, {63, 2158}, {92, 18041}, {304, 18669}, {326, 20884}, {1760, 18595}, {1959, 18672}, {18051, 20641}, {20444, 21589}, {20927, 21602}

X(21593) = isotomic conjugate of isogonal conjugate of X(21374)

X(21594) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + a^3 b^2 - a^2 b^3 + b^5 - 2 a^3 b c + 2 a b^3 c + a^3 c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 + c^5) : :

X(21594) lies on these lines: {56, 75}, {304, 20922}, {312, 21581}, {322, 3260}, {20444, 21589}, {20449, 20454}, {20643, 20647}, {20914, 21580}, {20940, 21605}

X(21595) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 - a^4 b + a b^4 + b^5 - a^4 c - a^3 b c + a b^3 c + b^4 c + a b c^3 + a c^4 + b c^4 + c^5) : :

X(21595) lies on these lines: {58, 75}, {92, 304}, {313, 502}, {315, 18719}, {4150, 18720}, {4385, 17762}, {11683, 17206}, {14963, 18050}, {18138, 21586}, {20644, 21602}

X(21596) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^4 b - a^3 b^2 + a^2 b^3 + a b^4 - a^4 c + b^4 c - a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(21596) lies on these lines: {7, 8}, {76, 2051}, {274, 14829}, {304, 20922}, {325, 18738}, {1043, 10451}, {6374, 18157}, {20245, 22299}, {20444, 21582}, {20647, 21615}

X(21597) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^9 - a^8 b + a b^8 + b^9 - a^8 c + 2 a^4 b^4 c - b^8 c + 2 a^4 b c^4 - 2 a b^4 c^4 + a c^8 - b c^8 + c^9) : :

X(21597) lies on these lines: {66, 75}, {18040, 20926}, {18046, 20931}, {18057, 20930}

X(21598) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-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(21598) lies on these lines: {31, 75}, {76, 18744}, {304, 18715}, {18051, 20941}

X(21599) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^4 b^2 + a^2 b^4 + 2 a^4 b c - 2 a b^4 c - a^4 c^2 - a^2 b^2 c^2 + b^4 c^2 + a^2 c^4 - 2 a b c^4 + b^2 c^4) : :

X(21599) lies on these lines: {75, 87}, {304, 1921}, {20444, 20451}, {20643, 20647}, {20644, 20927}

X(21600) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^4 + a^3 b - a b^3 + b^4 + a^3 c - 5 a^2 b c + 5 a b^2 c - b^3 c + 5 a b c^2 - 4 b^2 c^2 - a c^3 - b c^3 + c^4) : :

X(21600) lies on these lines: {75, 88}, {304, 1978}, {312, 3969}, {321, 4440}, {4080, 18159}, {4671, 17762}, {20444, 21601}, {20920, 21591}, {20929, 20941}

X(21601) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-4 a^4 - 2 a^3 b + 2 a b^3 + 4 b^4 - 2 a^3 c - 5 a^2 b c + 5 a b^2 c + 2 b^3 c + 5 a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 b c^3 + 4 c^4) : :

X(21601) lies on these lines: {75, 89}, {304, 20920}, {312, 17791}, {20444, 21600}

X(21602) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^5 + a^4 b - a b^4 + b^5 + a^4 c - a^3 b c + a b^3 c - b^4 c + a b c^3 - a c^4 - b c^4 + c^5) : :

X(21602) lies on these lines: {75, 101}, {239, 22144}, {304, 20642}, {673, 17877}, {1807, 20171}, {1930, 16550}, {3732, 17880}, {4568, 20445}, {6326, 20236}, {18061, 18151}, {20644, 21595}, {20646, 21587}, {20927, 21593}, {21293, 22310}

X(21603) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a^7 + a^5 b^2 - a^2 b^5 + b^7 + a^5 c^2 - a^3 b^2 c^2 + a^2 b^3 c^2 - b^5 c^2 + a^2 b^2 c^3 - a^2 c^5 - b^2 c^5 + c^7) : :

X(21603) lies on these lines: {75, 110}, {20642, 20929}, {20940, 20941}

X(21604) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a - b) (a - c) (b + c) (a^2 + a b + b^2 + a c - b c + c^2) : :

X(21604) lies on these lines: {75, 115}, {76, 20452}, {190, 8052}, {312, 20951}, {645, 4585}, {668, 1577}, {4033, 4103}, {4043, 21090}, {4129, 4568}, {18050, 21579}, {20914, 21585}

X(21604) = isotomic conjugate of isogonal conjugate of X(21383)

X(21605) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-3 a^2 + 3 b^2 - 2 b c + 3 c^2) : :

X(21605) lies on these lines: {75, 145}, {76, 85}, {274, 16834}, {279, 345}, {309, 3260}, {322, 3264}, {3263, 10513}, {3673, 14210}, {3797, 20105}, {3926, 17078}, {3962, 21296}, {4673, 20880}, {5222, 19804}, {7270, 17170}, {9311, 18157}, {9464, 19799}, {20444, 20946}, {20448, 20923}, {20920, 21588}, {20940, 21594}

X(21605) = isotomic conjugate of isogonal conjugate of X(3928)

X(21606) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a^2 + a b + a c - 3 b c) : :

X(21606) lies on these lines: {75, 900}, {514, 1921}, {665, 4751}, {1978, 3807}, {3644, 4526}, {3762, 20568}, {4086, 18072}, {4145, 21297}, {4408, 4777}, {4479, 4800}, {4509, 18158}, {6006, 20907}, {16727, 16732}, {18151, 20940}

X(21606) = isotomic conjugate of isogonal conjugate of X(21385)

X(21607) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-2 a^5 + a^3 b^2 - a^2 b^3 + 2 b^5 + a^3 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + 2 c^5) : :

X(21607) lies on these lines: {75, 187}, {772, 3250}, {14963, 18050}

X(21608) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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(21608) lies on these lines: {75, 194}, {92, 17789}, {257, 312}, {304, 1921}, {350, 21216}, {561, 17451}, {1966, 16968}, {2998, 20363}, {3596, 17760}, {3721, 17149}, {6384, 20271}, {20081, 20440}, {20453, 20943}, {20945, 21331}, {20951, 21585}

X(21609) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (-a + b - c) (a + b - c) (a^2 - 2 a b + b^2 - 2 a c + c^2) : :

X(21609) lies on these lines: {2, 7182}, {7, 3263}, {57, 17755}, {75, 200}, {76, 85}, {92, 18031}, {344, 17093}, {883, 3873}, {1088, 4554}, {1226, 20930}, {1441, 4441}, {1446, 18135}, {1996, 1997}, {5205, 9446}, {7055, 18141}, {7112, 20921}, {7205, 20923}, {9312, 18156}, {18142, 20922}, {18152, 20567}, {18153, 20946}, {18157, 21446}, {20448, 20928}

X(21610) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a + b) (b - c) (a + c) (-a + b + c) (-a^2 b + a b^2 - a^2 c + a b c - b^2 c + a c^2 - b c^2) : :

X(21610) lies on these lines: {75, 647}, {312, 3700}, {314, 1021}, {321, 16751}, {772, 3250}, {3064, 6332}, {6590, 7199}, {21348, 21438}

X(21611) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a^3 - 2 a^2 b + a b^2 - 2 a^2 c + a b c - b^2 c + a c^2 - b c^2) : :

X(21611) lies on these lines: {75, 650}, {312, 693}, {321, 17494}, {661, 786}, {3239, 3261}, {3700, 20954}, {3766, 14321}, {3835, 20950}, {4024, 4043}, {4391, 18074}, {4411, 20923}, {4521, 20907}, {4673, 14077}, {4828, 18137}, {4885, 18743}, {4893, 20909}, {6590, 7199}

X(21611) = isotomic conjugate of isogonal conjugate of X(21390)

X(21612) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a^4 + a b^3 - 2 a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21612) lies on these lines: {75, 659}, {514, 21613}, {661, 786}, {891, 17762}, {3766, 18004}, {3837, 20947}, {4122, 20954}, {4486, 20950}, {7199, 21584}, {18151, 20940}

X(21613) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (b - c) (a^4 + a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21613) lies on these lines: {75, 667}, {514, 21612}, {772, 3250}, {2084, 20953}, {3261, 8045}, {4083, 17762}, {4367, 21439}, {5029, 20909}, {14349, 18081}, {20947, 21260}

X(21614) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b c (a + b) (b - c) (a + c) (a^3 b - a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21614) lies on these lines: {75, 669}, {772, 3250}, {7199, 21584}

X(21615) = (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics b^2 c^2 (-a^2 + a b + a c + 2 b c) : :

X(21615) lies on these lines: {10, 75}, {37, 18135}, {85, 6385}, {190, 3403}, {274, 4751}, {304, 18137}, {305, 1233}, {310, 19804}, {312, 561}, {740, 3760}, {871, 20917}, {1920, 18743}, {1965, 3769}, {1966, 4676}, {3416, 20345}, {3696, 3789}, {3701, 20435}, {3923, 4495}, {4011, 7244}, {4664, 18145}, {4687, 18044}, {18142, 21590}, {20641, 20919}, {20647, 21596}

X(21615) = isotomic conjugate of X(2279)

Wasat and anti-Wasat triangles: X(21616)-X(21663)

This preamble and centers X(21616)-X(21663) were contributed by César Eliud Lozada, August 27, 2018.

Let ABC be a triangle, A'B'C' its incentral triangle and (I) its incircle. Denote (O_a) the circle with diameter AA' and r_a the radical axis of (O_a) and (I) and cyclically (O_b), (O_c) and r_b, r_c. Let A*B*C* be the triangle bounded by r_a, r_b, r_c. Then: 1) ABC and A*B*C* share the same nine-point-center; 2) the medial triangle of ABC is the orthic triangle of A*B*C*; and 3) the medial triangle of A*B*C* is the 3rd Euler triangle of ABC. (See Antreas Hatzipolakis, Hyacinthos #28081).

The triangle A*B*C* is here named the Wasat triangle of ABC. For Wasat triangle:

The Wasat triangle is also the extraversion triangle of X(10), the complement of the excentral triangle, the reflection of the 2nd circumperp triangle in X(1125), and the mid-triangle of the Ursa-minor and Ursa-major triangles. Also, the anti-Wasat triangle is also the anticomplementary triangle of the orthic triangle, and the reflection of the circumorthic triangle in X(389). (Randy Hutson, August 29, 2018)

A* = b+c : c-a : b-a (barycentrics) |B*C*| = 2*R*cos(A/2) Area(A*B*C*) = (R*/r)*Area(ABC)/2

The appearance of (T,i) in the following list means that triangles Wasat and T are perspective with perspector X(i):
(anti-Aquila, 946), (6th anti-mixtilinear, 21616), (Ascella*, 142), (Atik*, 11019), (Ayme, 10), (1st circumperp*, 516), (2nd circumperp*, 1125), (inner-Conway*, 908), (Conway*, 5249), (2nd Conway*, 2), (3rd Conway*, 10478), (Euler, 6245), (3rd Euler*, 3817), (4th Euler*, 10), (excenters-midpoints, 142), (excenters-reflections*, 8), (excentral*, 2), (2nd extouch*, 226), (hexyl*, 4), (Honsberger*, 21617), (inner-Hutson*, 21618), (Hutson intouch*, 12053), (outer-Hutson*, 21619), (incircle-circles*, 21620), (intouch*, 226), (inverse-in-incircle*, 11019), (medial, 10), (midheight, 21621), (6th mixtilinear*, 1699), (2nd Pamfilos-Zhou*, 12610), (Schroeter, 4129), (1st Sharygin*, 4425), (tangential-midarc*, 21622), (2nd tangential-midarc*, 21623), (Ursa-major*, 11), (Ursa-minor*, 11), (Yff central*, 21624), (1st Zaniah, 946), (2nd Zaniah*, 3452)

Note: An asterisk * means that triangles Wasat and T are homothetic.
The appearance of (T,i,j) in the following list means that triangles Wasat and T are orthologic with orthologic centers X(i) = (Wasat, T) and X(j) = (T, Wasat):
(4th Euler, 10, 10), (excenters-midpoints, 21627, 1), (excenters-reflections, 10, 1), (excentral, 10, 1), (extouch, 6245, 72), (2nd extouch, 10, 72), (4th extouch, 21628, 65), (5th extouch, 21629, 65), (Fuhrmann, 21630, 8), (2nd Fuhrmann, 11263, 4), (inner-Garcia, 10265, 3869), (outer-Garcia, 946, 8), (Garcia-reflection, 21627, 8), (Gossard, 946, 12438), (inner-Grebe, 946, 3641), (outer-Grebe, 946, 3640), (hexyl, 10, 40), (Honsberger, 10, 7672), (Hutson extouch, 21631, 3555), (inner-Hutson, 10, 9805), (Hutson intouch, 10, 3057), (outer-Hutson, 10, 9806), (incentral, 5, 1), (incircle-circles, 10, 942), (intouch, 10, 65), (inverse-in-incircle, 10, 1), (Johnson, 946, 355), (inner-Johnson, 946, 355), (outer-Johnson, 946, 355), (1st Johnson-Yff, 946, 5252), (2nd Johnson-Yff, 946, 1837), (K798e, 21630, 5690), (K798i, 11263, 5), (Lucas homothetic, 946, 12440), (Lucas(-1) homothetic, 946, 12441), (Malfatti, not calculated, 1), (Mandart-excircles, 21632, 3555), (Mandart-incircle, 946, 3057), (medial, 946, 10), (midarc, 21633, 1), (2nd midarc, 21633, 1), (mixtilinear, 3452, 1), (2nd mixtilinear, 3452, 1), (5th mixtilinear, 946, 1), (6th mixtilinear, 10, 7991), (2nd Pamfilos-Zhou, 10, 9808), (1st Schiffler, 21634, 21), (2nd Schiffler, 21627, 1320), (1st Sharygin, 10, 2292), (tangential-midarc, 10, 8093), (2nd tangential-midarc, 10, 8094), (3rd tri-squares-central, 946, 8983), (4th tri-squares-central, 946, 13971), (Ursa-major, 10, 10914), (Ursa-minor, 10, 3057), (X3-ABC reflections, 946, 1482), (Yff central, 10, 12445), (inner-Yff, 946, 1), (outer-Yff, 946, 1), (inner-Yff tangents, 946, 1), (outer-Yff tangents, 946, 1), (1st Zaniah, 6245, 942), (2nd Zaniah, 10, 960)
The appearance of (T,i,j) in the following list means that triangles Wasat and T are parallelogic with parallelogic centers X(i) = (Wasat, T) and X(j) = (T, Wasat):
(Fuhrmann, 21635, 4), (K798e, 21635, 5), (1st Parry, 946, 9810), (2nd Parry, 946, 9811), (2nd Sharygin, 10, 2254)
The appearance of (T,i,j) in the following list means that triangles Wasat and T are cyclologic with cyclologic centers X(i) = (Wasat, T) and X(j) = (T, Wasat):
(2nd circumperp, 21636, 98), (K798i, 21635, 10264)
The appearance of (T,i,j) in the following list means that triangles Wasat and T are eulerologic with eulerologic centers X(i) = (Wasat, T) and X(j) = (T, Wasat):
(1st circumperp, 10164, 2), (Euler, ne, 946), (3rd Euler, 517, ne), (K798i, 11263, ne), (medial, 11, ne), (2nd Zaniah, 10, ne)

Note: "ne" means the center does not exist.

If ABC is acute then the anti-Wasat triangle A*^-B*^-C*^-, i.e. the triangle whose Wasat triangle is ABC, has the following properties:

A*^- = (S^2+SB SC) a^2 : SB (SA-SC) b^2 : SC (SA-SB) c^2 (barycentrics) |B*^-C*^-|=2 a |cos(A)| Area(A*^-B*^-C*^-) = |4-(SW/R^2)| Area(ABC).

The anti-Wasat triangle of ABC is the (3rd anti-Euler triangle)-of-(the medial triangle)-of ABC.

The appearance of (T,i) in the following list means that triangles anti-Wasat and T are perspective with perspector X(i):
(anti-Ara, 11381), (anti-Ascella*, 6), (anti-Atik*, 1899), (1st anti-circumperp*, 511), (anti-Conway*, 13366), (2nd anti-Conway*, 51), (3rd anti-Euler*, 2), (4th anti-Euler*, 4), (anti-excenters-reflections*, 11381), (2nd anti-extouch*, 184), (anti-Honsberger*, 21637), (anti-Hutson intouch*, 1204), (anti-incircle-circles*, 1181), (anti-inverse-in-incircle*, 1899), (6th anti-mixtilinear*, 3917), (1st anti-Sharygin*, 21638), (anti-tangential-midarc*, 1425), (circumorthic*, 389), (Ehrmann-side*, 13754), (Ehrmann-vertex*, 13851), (2nd Ehrmann*, 21639), (2nd Euler*, 5562), (1st excosine*, 25), (extangents*, 3611), (2nd Hyacinth, 6467), (intangents*, 3270), (1st Kenmotu diagonals*, 21640), (2nd Kenmotu diagonals*, 21641), (Kosnita*, 13367), (Lucas antipodal tangents*, 21642), (Lucas(-1) antipodal tangents*, 21643), (orthic*, 51), (1st orthosymmedial, 21644), (Pelletier, 21645), (Schroeter, 21646), (submedial*, 373), (symmedial, 217), (tangential*, 184), (inner tri-equilateral*, 21647), (outer tri-equilateral*, 21648), (Trinh*, 21663)

Note: An asterisk * means that triangles anti-Wasat and T are homothetic.
The appearance of (T,i,j) in the following list means that triangles anti-Wasat and T are orthologic with orthologic centers X(i) = (anti-Wasat, T) and X(j) = (T, anti-Wasat):
(ABC, 185, 3), (AAOA, 21649, 15136), (ABC-X3 reflections, 185, 3), (anti-Aquila, 185, 1385), (anti-Ara, 185, 4), (anti-Ascella, 5562, 12160), (anti-Atik, 5562, 6643), (5th anti-Brocard, 185, 3398), (2nd anti-circumperp-tangential, 185, 1), (1st anti-circumperp, 5562, 11412), (anti-Conway, 5562, 12161), (2nd anti-Conway, 5562, 5), (anti-Euler, 185, 20), (3rd anti-Euler, 5562, 5889), (4th anti-Euler, 5562, 11412), (anti-excenters-reflections, 5562, 12162), (2nd anti-extouch, 5562, 3), (anti-inner-Grebe, 185, 3312), (anti-outer-Grebe, 185, 3311), (anti-Honsberger, 5562, 19139), (anti-Hutson intouch, 5562, 12163), (anti-incircle-circles, 5562, 12164), (anti-inverse-in-incircle, 5562, 11411), (anti-Mandart-incircle, 185, 11248), (6th anti-mixtilinear, 5562, 1216), (anti-orthocentroidal, 21650, 3581), (1st anti-Sharygin, 5562, 19194), (anti-tangential-midarc, 5562, 7352), (anticomplementary, 185, 4), (AOA, 21649, 15123), (Aquila, 185, 40), (Ara, 185, 7387), (Aries, 21651, 7387), (1st Auriga, 185, 11252), (2nd Auriga, 185, 11253), (5th Brocard, 185, 9821), (circumorthic, 5562, 5889), (2nd circumperp tangential, 185, 11249), (Ehrmann-mid, 185, 4), (Ehrmann-side, 5562, 18436), (Ehrmann-vertex, 5562, 9927), (1st Ehrmann, 6467, 576), (2nd Ehrmann, 5562, 8548), (Euler, 185, 5), (2nd Euler, 5562, 5562), (1st excosine, 5562, 17834), (extangents, 5562, 6237), (outer-Garcia, 185, 355), (Gossard, 185, 11251), (inner-Grebe, 185, 1161), (outer-Grebe, 185, 1160), (Hatzipolakis-Moses, 21652, 9729), (1st Hyacinth, 21649, 10112), (2nd Hyacinth, 21651, 3), (intangents, 5562, 6238), (Johnson, 185, 4), (inner-Johnson, 185, 10525), (outer-Johnson, 185, 10526), (1st Johnson-Yff, 185, 1478), (2nd Johnson-Yff, 185, 1479), (1st Kenmotu diagonals, 5562, 10665), (2nd Kenmotu diagonals, 5562, 10666), (Kosnita, 5562, 1147), (Lucas antipodal, 21653, 3), (Lucas antipodal tangents, 5562, 18939), (Lucas central, 21655, 3), (Lucas homothetic, 185, 10669), (Lucas reflection, 21657, 10670), (Lucas(-1) antipodal, 21654, 3), (Lucas(-1) antipodal tangents, 5562, 18940), (Lucas(-1) central, 21656, 3), (Lucas(-1) homothetic, 185, 10673), (Lucas(-1) reflection, 21658, 10674), (Macbeath, 21659, 4), (Mandart-incircle, 185, 1), (medial, 185, 5), (midheight, 11381, 389), (5th mixtilinear, 185, 1482), (orthic, 5562, 52), (orthocentroidal, 21650, 568), (reflection, 21660, 6243), (submedial, 5562, 5462), (tangential, 5562, 155), (inner tri-equilateral, 5562, 10661), (outer tri-equilateral, 5562, 10662), (3rd tri-squares-central, 185, 8981), (4th tri-squares-central, 185, 13966), (Trinh, 5562, 7689), (X3-ABC reflections, 185, 3), (inner-Yff, 185, 55), (outer-Yff, 185, 56), (inner-Yff tangents, 185, 10679), (outer-Yff tangents, 185, 10680)
The appearance of (T,i,j) in the following list means that triangles anti-Wasat and T are parallelogic with parallelogic centers X(i) = (anti-Wasat, T) and X(j) = (T, anti-Wasat):
(AAOA, 13417, 15139), (AOA, 13417, 15126), (1st Hyacinth, 13417, 10116), (1st Parry, 185, 351), (2nd Parry, 185, 351)
The appearance of (T,i,j) in the following list means that triangles anti-Wasat and T are cyclologic with cyclologic centers X(i) = (anti-Wasat, T) and X(j) = (T, anti-Wasat):
(circumorthic, 21661, 1298)
The appearance of (T,i,j) in the following list means that triangles anti-Wasat and T are eulerologic with eulerologic centers X(i) = (anti-Wasat, T) and X(j) = (T, anti-Wasat):
(1st anti-circumperp, 3917, 2), (Ehrmann-side, 15030, 381), (orthic, 1154, ne)

Note: "ne" means the center does not exist.

X(21616) = PERSPECTOR OF THESE TRIANGLES: WASAT AND 6th ANTI-MIXTILINEAR

Barycentrics (b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2+c^2)*a-(b^2-c^2)^2 : :

X(21616) = 3*X(2)+X(11415) = 3*X(10)-2*X(8256) = 3*X(1329)-X(8256) = X(1837)-3*X(17556) = 5*X(3616)-X(20076) = 3*X(3817)-2*X(7681) = 3*X(3892)-4*X(16215) = X(5730)+3*X(17556) = 3*X(5886)-X(10680) = 4*X(6691)-5*X(19862) = 5*X(8227)-X(12704)

X(21616) lies on these lines: {1,908}, {2,46}, {3,12608}, {4,997}, {5,10}, {8,5187}, {9,6832}, {11,72}, {12,392}, {21,14526}, {40,1519}, {56,226}, {57,10200}, {58,17182}, {63,499}, {65,4187}, {78,1479}, {140,4640}, {142,3647}, {149,4420}, {165,6962}, {200,9614}, {214,2829}, {329,1728}, {355,5289}, {381,5794}, {404,1770}, {429,1828}, {442,17605}, {474,1836}, {496,518}, {497,3811}, {498,5250}, {515,6928}, {516,3149}, {519,1837}, {527,10199}, {529,551}, {535,4311}, {631,5698}, {758,1210}, {936,1699}, {938,12559}, {942,3816}, {956,11376}, {958,5886}, {962,5328}, {978,3944}, {995,13161}, {1001,11374}, {1062,16869}, {1155,13747}, {1158,6891}, {1376,12699}, {1387,11260}, {1478,19861}, {1532,14110}, {1698,6933}, {1737,3869}, {1738,17749}, {1985,3741}, {2550,6896}, {2551,5603}, {2646,11113}, {2802,6736}, {2841,11814}, {3035,3579}, {3057,10915}, {3061,5179}, {3085,5748}, {3185,19543}, {3216,3914}, {3305,19854}, {3338,5905}, {3419,10896}, {3474,17567}, {3485,5084}, {3576,6936}, {3582,6763}, {3612,6872}, {3616,13407}, {3624,5249}, {3625,5854}, {3634,5316}, {3649,5439}, {3671,5883}, {3678,4847}, {3683,7483}, {3699,5100}, {3702,5741}, {3742,6147}, {3754,8582}, {3812,17527}, {3813,7743}, {3835,5592}, {3838,8728}, {3840,14058}, {3874,11019}, {3876,11680}, {3877,10039}, {3892,16215}, {3899,18395}, {3916,5433}, {3923,12610}, {3940,9669}, {4011,21062}, {4197,10129}, {4302,4855}, {4324,15015}, {4511,5046}, {4999,11230}, {5119,5552}, {5176,5330}, {5192,19869}, {5219,10198}, {5251,5443}, {5267,10165}, {5288,16173}, {5440,6284}, {5587,15829}, {5687,12701}, {5692,6734}, {5697,6735}, {5705,7988}, {5718,6051}, {5722,12635}, {5729,5850}, {5745,6861}, {5795,13464}, {5880,16408}, {5887,6882}, {6001,6922}, {6261,6827}, {6265,12762}, {6684,6863}, {6745,8715}, {6831,12617}, {6865,12520}, {6886,18228}, {6919,18391}, {7171,16127}, {7283,17777}, {7354,17614}, {7486,18231}, {7515,18588}, {8583,9612}, {8666,12527}, {9623,11522}, {9708,18493}, {9957,12607}, {10171,18249}, {10573,11682}, {10863,12447}, {10914,21031}, {10948,17615}, {11373,12513}, {12116,17857}, {12680,13257}, {15507,19513}, {16466,17720}, {17355,21068}, {17452,21074}, {17531,20292}

X(21616) = midpoint of X(i) and X(j) for these {i,j}: {1, 3436}, {78, 1479}, {5687, 12701}, {10573, 11682}, {12116, 17857}
X(21616) = reflection of X(56) in X(1125)
X(21616) = complement of X(46)
X(21616) = X(24)-of-Wasat triangle
X(21616) = X(3436)-of-anti-Aquila triangle
X(21616) = X(7517)-of-2nd Zaniah triangle
X(21616) = X(10916)-of-2nd Johnson-Yff triangle
X(21616) = X(21077)-of-outer-Yff triangle
X(21616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 908, 21077), (2, 11415, 46), (2, 12047, 12609), (4, 997, 17647), (5, 960, 10), (8, 5187, 10826), (10, 11813, 946), (946, 3452, 10), (960, 5087, 5), (2886, 5044, 10), (3814, 3878, 10), (3820, 5836, 10), (5044, 9955, 2886), (5123, 5690, 10), (5837, 10175, 10)

X(21617) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND HONSBERGER

Barycentrics ((b+c)*a^2-2*(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :

X(21617) lies on these lines: {1,6835}, {2,7}, {4,7675}, {5,5728}, {8,11526}, {10,7672}, {11,5572}, {12,518}, {35,411}, {65,3826}, {77,948}, {78,2550}, {80,14151}, {85,17234}, {241,17245}, {273,1826}, {278,5287}, {347,5308}, {349,18137}, {390,946}, {495,1512}, {528,4870}, {651,3664}, {664,17317}, {673,2259}, {936,12560}, {938,5261}, {950,6894}, {954,3149}, {971,6831}, {1001,11344}, {1100,5723}, {1125,7677}, {1170,5526}, {1210,3947}, {1229,1441}, {1446,5179}, {1465,2197}, {1699,4326}, {1836,11495}, {2327,16054}, {2346,11218}, {2801,8068}, {2886,3059}, {2911,5228}, {3091,5809}, {3174,3434}, {3243,9578}, {3358,6833}, {3487,5720}, {3553,4000}, {3671,5692}, {3817,7671}, {3838,15587}, {3870,6601}, {3944,4335}, {3946,7269}, {4292,6986}, {4298,5251}, {4301,7673}, {4312,21153}, {4321,5290}, {4323,20007}, {4328,4859}, {4425,8238}, {4675,6180}, {5223,5705}, {5686,21075}, {5714,6865}, {5732,6836}, {5735,6962}, {5738,5816}, {5759,6988}, {5817,6855}, {5880,11509}, {6245,12669}, {6700,11263}, {6737,17097}, {6828,10394}, {6996,18650}, {7670,21633}, {8226,11018}, {8236,12053}, {8237,12610}, {8255,14100}, {8287,18635}, {8385,21618}, {8386,21619}, {8387,21622}, {8388,21623}, {8389,21624}, {8581,15844}, {9846,21625}, {10265,12755}, {10478,10889}, {10481,17092}, {10865,11019}, {10944,15570}, {11372,11919}, {12399,21626}, {12630,21627}, {12706,21628}, {12718,21629}, {12730,21630}, {12846,21631}, {12847,21632}, {12850,21634}, {14189,17084}, {14564,15492}, {16826,17086}

X(21617) = X(21637)-of-Honsberger triangle
X(21617) = X(21637)-of-Wasat triangle
X(21617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 1445), (7, 5226, 8232), (7, 8232, 8545), (7, 18230, 12848), (142, 226, 7), (226, 5219, 908), (948, 4648, 77), (1125, 12573, 7677), (5805, 11374, 954), (7672, 7679, 10)

X(21618) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND INNER-HUTSON

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) = -(b+c)*(a+b-c)*(a-b+c)
G(a,b,c) = (b-c)*(c-a+b)*(a+b-c)
H(a,b,c) = a^3+(b-c)^2*a-2*(b^2-c^2)*(b-c)

X(21618) lies on these lines: {2,363}, {4,8111}, {5,12488}, {8,11527}, {10,8380}, {11,12809}, {142,11854}, {226,5934}, {516,8107}, {908,11685}, {946,2089}, {1125,8109}, {1699,8140}, {3452,12880}, {3817,8377}, {4425,8391}, {5249,11886}, {6245,12673}, {6732,10504}, {7057,12879}, {8133,21622}, {8385,21617}, {8390,12053}, {10265,12759}, {10478,11892}, {11019,11026}, {11039,21620}, {11263,16135}, {11922,12610}, {11923,21624}, {12561,12609}, {12633,21627}, {12707,21628}, {12719,21629}, {12733,21630}, {12851,21631}, {12878,21632}, {12882,21634}, {12886,21626}, {13260,21635}

X(21618) = X(3937)-of-inner-Hutson triangle
X(21618) = X(3937)-of-Wasat triangle
X(21618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9783, 363), (8380, 9805, 10)

X(21619) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND OUTER-HUTSON

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) = (b+c)*(a+b-c)*(a-b+c)
G(a,b,c) = -(b-c)*(c-a+b)*(a+b-c)
H(a,b,c) = a^3+(b-c)^2*a-2*(b^2-c^2)*(b-c)

X(21619) lies on these lines: {2,168}, {4,8112}, {5,12489}, {8,11528}, {10,8381}, {11,17608}, {142,11855}, {177,8372}, {226,5935}, {516,8108}, {908,11686}, {946,9837}, {1125,8110}, {1699,8140}, {3452,12885}, {3817,8378}, {4425,11926}, {5249,11887}, {6245,12674}, {7707,21624}, {8135,21622}, {8138,21623}, {8386,21617}, {8392,12053}, {9836,15495}, {9849,21625}, {10265,12760}, {10478,11893}, {11019,11027}, {11040,21620}, {11263,16136}, {11925,12610}, {12562,12609}, {12708,21628}, {12720,21629}, {12734,21630}, {12881,21626}, {12883,21632}, {12884,21633}, {12887,21634}, {13261,21635}

X(21619) = X(3690)-of-outer-Hutson triangle
X(21619) = X(3690)-of-Wasat triangle
X(21619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9787, 168), (1125, 12576, 8110), (8381, 9806, 10)

X(21620) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND INCIRCLE-CIRCLES

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(21620) = X(1)-3*X(3475) = 3*X(1)-X(3486) = 3*X(1)+X(9613) = X(388)+3*X(3475) = 3*X(388)+X(3486) = 3*X(388)-X(9613) = X(1697)+3*X(4654) = 9*X(3475)-X(3486) = 9*X(3475)+X(9613) = X(4294)-3*X(10389) = X(4295)-3*X(4654) = X(9579)+3*X(10389)

X(21620) lies on these lines: {1,4}, {2,3333}, {3,4298}, {5,3947}, {7,40}, {8,4208}, {10,141}, {11,17609}, {12,354}, {20,10578}, {30,4314}, {46,553}, {55,4292}, {56,10165}, {57,3085}, {65,11362}, {118,14760}, {119,18240}, {165,4355}, {200,443}, {306,4968}, {341,17234}, {355,6738}, {377,3870}, {405,12527}, {442,3555}, {474,6745}, {484,4114}, {496,3817}, {498,3338}, {516,3295}, {517,3671}, {519,5794}, {527,12514}, {529,551}, {546,18527}, {631,3361}, {758,5837}, {908,3616}, {938,5261}, {943,15931}, {951,1065}, {958,999}, {971,12710}, {982,5530}, {1000,11531}, {1001,12572}, {1086,4646}, {1145,4004}, {1329,3742}, {1330,3883}, {1376,12436}, {1385,4315}, {1387,21635}, {1468,3011}, {1503,4349}, {1697,4295}, {1698,3296}, {1706,6173}, {1737,18398}, {1770,3746}, {1836,3303}, {1837,11237}, {1855,2294}, {2049,19868}, {2475,3957}, {2476,3889}, {2478,4666}, {2550,6765}, {2646,4311}, {2800,12709}, {2801,12564}, {3057,3649}, {3086,5219}, {3091,10580}, {3157,4667}, {3244,11263}, {3247,21068}, {3304,11375}, {3306,5552}, {3336,4031}, {3337,3584}, {3339,5657}, {3576,3600}, {3601,4293}, {3612,4317}, {3624,5316}, {3632,11041}, {3646,18228}, {3663,3931}, {3664,5711}, {3697,17529}, {3702,4054}, {3714,4966}, {3743,8680}, {3748,6284}, {3753,6736}, {3754,10915}, {3813,3838}, {3822,3881}, {3843,18530}, {3873,6734}, {3912,4385}, {3913,5880}, {3927,5850}, {3940,12447}, {3982,5119}, {4294,9579}, {4297,18990}, {4301,9957}, {4304,7354}, {4312,6361}, {4321,8726}, {4323,16200}, {4340,5269}, {4353,11042}, {4425,11043}, {4656,6051}, {4658,17197}, {4684,10449}, {4696,18139}, {4848,5902}, {5044,21060}, {5056,5558}, {5083,10265}, {5084,10582}, {5218,15803}, {5226,8227}, {5227,5750}, {5231,6856}, {5234,16845}, {5250,5905}, {5252,14563}, {5439,8582}, {5534,6826}, {5570,10954}, {5722,6744}, {5726,5818}, {5728,14872}, {5745,10198}, {5801,7174}, {5806,20330}, {5886,7373}, {5903,11551}, {6223,11372}, {6245,7680}, {6762,19843}, {6764,8000}, {6767,12575}, {6769,6916}, {6831,9850}, {6849,18528}, {6926,7091}, {7080,9776}, {7247,14828}, {7681,16215}, {7682,12915}, {8083,8088}, {8092,21624}, {8255,9943}, {8351,21623}, {8382,11033}, {8580,17582}, {9578,11518}, {9581,10590}, {9669,12571}, {10172,10588}, {10479,11021}, {10956,12736}, {11009,14526}, {11040,21619}, {11044,21622}, {11045,11046}, {11700,17724}, {12005,12616}, {12245,18421}, {12401,21626}, {12433,18480}, {12437,17647}, {12722,21629}, {12735,21630}, {12853,21631}, {12907,21632}, {12909,21634}, {15325,19862}, {15844,16193}, {15950,20323}, {16408,20103}, {17023,17681}

X(21620) = midpoint of X(i) and X(j) for these {i,j}: {1, 388}, {1697, 4295}, {4294, 9579}
X(21620) = reflection of X(i) in X(j) for these (i,j): (958, 1125), (3927, 18249)
X(21620) = X(388)-of-anti-Aquila triangle
X(21620) = X(1181)-of-incircle-circles triangle
X(21620) = X(1181)-of-Wasat triangle
X(21620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 226, 946), (1, 1478, 950), (1, 3476, 13607), (1, 3485, 13464), (1, 5270, 10572), (1, 5290, 4), (1, 9612, 497), (1, 10106, 5882), (1, 12047, 12053), (1, 13407, 226), (226, 12053, 12047), (388, 3486, 9613), (497, 9612, 18483), (1058, 5714, 1699), (12047, 12053, 946), (12608, 13464, 946)

X(21621) = PERSPECTOR OF THESE TRIANGLES: WASAT AND MIDHEIGHT

Barycentrics (b+c)*a^5+(b-c)^2*a^4+2*(b-c)^2*b*c*a^2-(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2) : :

X(21621) lies on these lines: {1,8900}, {2,169}, {4,1448}, {10,1368}, {19,20266}, {57,1848}, {63,21062}, {141,3452}, {142,6678}, {150,1999}, {222,226}, {241,19542}, {307,1764}, {496,942}, {515,1060}, {1210,1905}, {1370,2000}, {1427,1565}, {2051,8808}, {3840,14058}, {5179,18750}, {5738,10478}, {5745,18589}, {5908,6247}, {6708,21239}, {15509,17073}, {21188,21204}

X(21621) = midpoint of X(222) and X(5928)
X(21621) = complement of X(1763)
X(21621) = X(2351)-of-Wasat triangle
X(21621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57, 1848, 12610), (946, 6245, 16388)

X(21622) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND TANGENTIAL-MIDARC

Barycentrics a+(b+c)*sin(A/2)+(-b+c)*sin(B/2)+(b-c)*sin(C/2) : :

X(21622) lies on these lines: {1,21623}, {2,8078}, {4,8081}, {5,8099}, {8,11534}, {10,8087}, {11,10503}, {12,10506}, {142,8733}, {177,178}, {188,3452}, {226,2089}, {258,9807}, {515,18448}, {516,8075}, {908,11690}, {946,8091}, {1125,8077}, {1699,8089}, {1737,18399}, {3817,8085}, {4425,8249}, {5249,11888}, {6724,10504}, {8086,10967}, {8097,21630}, {8103,21635}, {8133,21618}, {8135,21619}, {8241,12053}, {8247,12610}, {8387,21617}, {9853,21625}, {10265,12771}, {10478,11894}, {10490,10493}, {11019,11032}, {11044,21620}, {11263,16146}, {12568,12609}, {12714,21628}, {12726,21629}, {12870,21631}, {12916,21626}, {13072,21632}, {13124,21634}

X(21622) = X(1425)-of-Wasat triangle
X(21622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 21633, 21623), (2, 9793, 8078), (1125, 12580, 8077), (8087, 8093, 10)

X(21623) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND 2nd TANGENTIAL-MIDARC

Barycentrics -4*sin(A/2)*a*b*c+(b+c)*(a+b-c)*(a-b+c) : :

X(21623) lies on these lines: {1,21622}, {2,258}, {4,8082}, {5,8100}, {7,173}, {8,11899}, {10,8088}, {11,10501}, {142,236}, {174,226}, {178,8422}, {515,18456}, {516,8076}, {908,8125}, {946,8092}, {1125,7588}, {1699,8090}, {1737,18409}, {3452,7028}, {3487,7590}, {3671,12445}, {3817,8086}, {3947,8382}, {4298,7587}, {4425,8250}, {5249,8126}, {5542,8083}, {6147,12491}, {6732,10504}, {7589,13405}, {8078,9807}, {8084,8085}, {8098,21630}, {8104,21635}, {8138,21619}, {8242,12053}, {8248,12610}, {8351,21620}, {8379,11019}, {8388,21617}, {9854,21625}, {10265,12772}, {10478,11895}, {10494,17761}, {11263,16147}, {11551,18408}, {12569,12609}, {12644,21627}, {12715,21628}, {12727,21629}, {12871,21631}, {13073,21632}, {13125,21634}, {13475,21626}

X(21623) = X(692)-of-intouch triangle
X(21623) = X(3270)-of-2nd tangential-midarc triangle
X(21623) = X(3270)-of-Wasat triangle
X(21623) = X(14717)-of-2nd Conway triangle
X(21623) = X(21252)-of-Ursa-minor triangle
X(21623) = X(21293)-of-inverse-in-incircle triangle
X(21623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 21633, 21622), (2, 9795, 258), (174, 226, 21624), (1125, 12581, 7588), (8088, 8094, 10), (8379, 11033, 11019)

X(21624) = HOMOTHETIC CENTER OF THESE TRIANGLES: WASAT AND YFF CENTRAL

Barycentrics 4*sin(A/2)*a*b*c+(b+c)*(a+b-c)*(a-b+c) : :

X(21624) lies on these lines: {2,173}, {4,7590}, {5,12491}, {7,258}, {8,11535}, {10,8382}, {11,10502}, {142,7028}, {174,226}, {177,178}, {236,3452}, {515,18454}, {516,7589}, {908,8126}, {1125,7587}, {1699,8423}, {1737,18408}, {3487,8082}, {3671,8094}, {3817,8379}, {3947,8088}, {4298,7588}, {4425,8425}, {5249,8125}, {5542,11033}, {6147,8100}, {6245,12685}, {7707,21619}, {8076,13405}, {8083,8086}, {8092,21620}, {8389,21617}, {10265,12774}, {10478,11896}, {10499,15997}, {11263,16151}, {11551,18409}, {11924,12053}, {11996,12610}, {12130,21625}, {12406,21626}, {12570,12609}, {12646,21627}, {12716,21628}, {12728,21629}, {12748,21630}, {12873,21631}, {13074,21632}, {13092,21633}, {13127,21634}, {13267,21635}

X(21624) = X(3611)-of-Wasat triangle
X(21624) = X(20986)-of-intouch triangle
X(21624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11891, 173), (174, 226, 21623), (1125, 12582, 7587), (8083, 8086, 11019), (8382, 12445, 10)

X(21625) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO ANDROMEDA

Barycentrics (b+c)*a^3-(b^2-14*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :

X(21625) = 3*X(2)+X(9797) = 3*X(354)+X(9848) = 3*X(1699)+X(9851) = 3*X(3817)-2*X(9842) = 3*X(5049)-X(12128) = X(9844)+5*X(17609) = X(9850)-5*X(17609)

The reciprocal orthologic center of these triangles is X(1)

Let (Oa), (Ob), (Oc) be the inverse-in-incircle of the A-, B- and C-excircles, resp. X(21625) is the radical center of (Oa), (Ob), (Oc). (Randy Hutson, August 29, 2018)

X(21625) lies on these lines: {1,2}, {4,9845}, {11,3947}, {56,4314}, {57,5493}, {65,4342}, {142,3813}, {226,9844}, {354,3671}, {390,3361}, {496,3817}, {497,4298}, {515,7373}, {516,1058}, {553,12701}, {908,12125}, {942,4301}, {946,971}, {950,3304}, {962,10980}, {999,4297}, {1482,17706}, {1699,9851}, {3295,10164}, {3303,3911}, {3338,10624}, {3339,9785}, {3452,18247}, {3555,21060}, {3663,3976}, {3742,16201}, {3812,21627}, {3874,12915}, {3881,5728}, {3892,16215}, {3913,6692}, {4304,5563}, {4313,13462}, {4345,18221}, {4353,18216}, {4355,9812}, {4425,9852}, {4848,5919}, {5249,9859}, {5265,8236}, {5274,5290}, {5603,12563}, {5836,17051}, {5837,10179}, {5882,7682}, {6245,13464}, {6684,6767}, {6762,18250}, {7288,10389}, {9589,21454}, {9710,12855}, {9846,21617}, {9849,21619}, {9853,21622}, {9854,21623}, {10478,12126}, {10520,17170}, {10569,12688}, {11036,11522}, {12129,12610}, {12130,21624}, {12432,17642}, {15178,15935}, {18240,21630}, {18481,18530}

X(21625) = midpoint of X(i) and X(j) for these {i,j}: {1, 938}, {4, 9845}, {3339, 9785}
X(21625) = complement of X(4882)
X(21625) = incircle-inverse of X(38471)
X(21625) = X(938)-of-anti-Aquila triangle
X(21625) = X(9845)-of-Euler triangle, when ABC is acute
X(21625) = X(17814)-of-incircle-circles triangle
X(21625) = X(18909)-of-Wasat triangle
X(21625) = center of inverse-in-incircle-of-excircles-radical-circle
X(21625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3086, 13405), (1, 3624, 10578), (1, 6738, 3244), (1, 10072, 13411), (1, 10580, 6744), (1, 11019, 10), (1, 14986, 1125), (2, 9797, 4882), (57, 12575, 5493), (354, 12053, 3671), (390, 3361, 12512), (496, 5049, 21620), (496, 12128, 9842), (496, 21620, 3817), (3086, 13405, 19862)

X(21626) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO ANTLIA

Barycentrics (b+c)*a^9-(3*b^2+14*b*c+3*c^2)*a^8+4*(b+c)*(b^2+6*b*c+c^2)*a^7-4*(b^4+c^4+(9*b^2+8*b*c+9*c^2)*b*c)*a^6+2*(b+c)*(b^4+c^4+24*(b^2-b*c+c^2)*b*c)*a^5+2*(b^4+c^4-2*(9*b^2-14*b*c+9*c^2)*b*c)*(b+c)^2*a^4-4*(b^2-c^2)*(b-c)*(b^4-4*b^2*c^2+c^4)*a^3+4*(b^2-c^2)^2*(b^4+c^4-3*(b-c)^2*b*c)*a^2-(b^2-c^2)*(b-c)^3*(3*b^4+3*c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a+(b^4-c^4)^2*(b-c)^2 : :

X(21626) = 3*X(2)+X(12391) = 3*X(1699)+X(12404) = 3*X(3817)-2*X(12393)

The reciprocal orthologic center of these triangles is X(1)

Let (Oa) be the inverse of the incircle in the A-excircle, and define (Ob), (Oc) cyclically. X(21626) is the radical center of (Oa), (Ob), (Oc). (Randy Hutson, August 29, 2018)

X(21626) lies on these lines: {2,12391}, {4,12398}, {8,12395}, {10,1541}, {11,17633}, {142,12385}, {226,12397}, {516,12387}, {908,12389}, {946,3836}, {1125,12388}, {1699,12404}, {3452,18248}, {3817,12393}, {4425,12405}, {5249,12390}, {10478,12392}, {11019,12386}, {12053,12400}, {12399,21617}, {12401,21620}, {12406,21624}, {12881,21619}, {12886,21618}, {12916,21622}, {13475,21623}

X(21626) = midpoint of X(i) and X(j) for these {i,j}: {4, 12398}, {8, 12395}
X(21626) = reflection of X(10) in X(12394)
X(21626) = complement of X(12396)
X(21626) = X(12398)-of-Euler triangle
X(21626) = {X(2), X(12391)}-harmonic conjugate of X(12396)

X(21627) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO EXCENTERS-MIDPOINTS

Barycentrics (-a+b+c)*((b+c)*a^2+2*(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

X(21627) = 3*X(1)-X(3189) = 3*X(2)+X(12541) = 3*X(946)-2*X(21077) = 3*X(1699)+X(11519) = 3*X(3158)-5*X(3616) = 3*X(3158)-X(12632) = 2*X(3189)-3*X(12437) = 5*X(3616)-X(12632) = 5*X(3623)-X(12536) = 4*X(3813)-X(12640) = 3*X(3817)-2*X(12607) = 3*X(3928)-X(20070) = 3*X(5603)-X(6765) = 3*X(11235)-2*X(19925) = 3*X(11236)-4*X(12571)

The reciprocal orthologic center of these triangles is X(1)

X(21627) lies on these lines: {1,142}, {2,2136}, {4,12629}, {8,3452}, {9,9785}, {10,496}, {11,3893}, {145,226}, {355,381}, {497,4853}, {516,8158}, {517,6245}, {518,4301}, {522,764}, {527,962}, {908,3621}, {952,6260}, {956,10624}, {958,12575}, {960,4342}, {1043,17197}, {1058,9623}, {1125,3913}, {1210,10914}, {1219,4659}, {1266,7185}, {1320,6598}, {1420,17784}, {1616,3008}, {1697,5745}, {1699,11519}, {1706,6692}, {2098,4863}, {2321,3061}, {2551,4915}, {2802,10265}, {3057,4847}, {3158,3616}, {3169,5257}, {3243,9797}, {3244,11263}, {3421,9614}, {3434,10106}, {3555,17653}, {3623,5249}, {3625,5854}, {3632,21075}, {3633,12047}, {3671,15733}, {3674,17158}, {3701,4939}, {3811,13464}, {3812,21625}, {3817,12607}, {3885,6734}, {3886,20258}, {3895,10527}, {3928,20070}, {4345,20007}, {4373,6553}, {4425,12642}, {4669,15862}, {4848,14923}, {4861,21634}, {5044,21631}, {5250,5325}, {5289,6743}, {5290,12127}, {5542,12128}, {5603,6765}, {5836,11019}, {5839,21068}, {5880,12577}, {6700,11373}, {6735,7705}, {6745,11376}, {6908,7966}, {7373,12436}, {8715,10165}, {9710,10179}, {9850,17668}, {10175,10915}, {10395,15558}, {11194,12512}, {11256,13271}, {12527,12701}, {12610,12638}, {12630,21617}, {12633,21618}, {12643,21622}, {12644,21623}, {12646,21624}

X(21627) = midpoint of X(i) and X(j) for these {i,j}: {4, 12629}, {8, 3680}, {145, 12625}, {3617, 4052}, {5836, 12448}, {11256, 13271}
X(21627) = reflection of X(i) in X(j) for these (i,j): (10, 3813), (3811, 13464)
X(21627) = complement of X(2136)
X(21627) = X(64)-of-Wasat triangle
X(21627) = X(6225)-of-3rd Euler triangle
X(21627) = X(6736)-of-inner-Johnson triangle
X(21627) = X(9914)-of-2nd Zaniah triangle
X(21627) = X(12250)-of-4th Euler triangle
X(21627) = X(12437)-of-5th mixtilinear triangle
X(21627) = X(12629)-of-Euler triangle
X(21627) = X(12640)-of-outer-Garcia triangle
X(21627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12541, 2136), (8, 12053, 3452), (11, 3893, 6736), (355, 946, 9842), (497, 4853, 5795), (1706, 14986, 6692), (2098, 4863, 6737), (3057, 4847, 5837), (3616, 12632, 3158), (3625, 21630, 21616)

X(21628) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 4th EXTOUCH

Barycentrics (b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(b^2-6*b*c+c^2)*a^4+4*(b^2-c^2)^2*a^3-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a^2-2*(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

X(21628) = 3*X(2)+X(9800) = X(20)-3*X(4512) = 2*X(946)+X(9949) = 3*X(1699)-X(4295) = 3*X(3817)-4*X(12558) = 3*X(3817)-2*X(12609) = 3*X(5603)-2*X(12563) = 3*X(10164)-2*X(12511)

The reciprocal orthologic center of these triangles is X(65)

X(21628) lies on these lines: {1,9799}, {2,9800}, {4,9}, {7,7992}, {8,12651}, {11,17634}, {20,4512}, {65,7965}, {84,4298}, {142,9943}, {226,12688}, {443,10860}, {495,12599}, {496,942}, {515,3295}, {758,4301}, {774,3668}, {908,12529}, {938,12560}, {962,4847}, {971,12710}, {1012,4297}, {1056,10864}, {1125,6847}, {1210,1699}, {1329,9842}, {1490,13405}, {1519,6845}, {1532,3841}, {1709,4292}, {1750,3085}, {3062,5290}, {3091,8582}, {3149,10164}, {3359,6849}, {3452,18251}, {3634,6848}, {3741,12544}, {3817,3825}, {3820,10241}, {3947,6260}, {4187,10863}, {4294,5691}, {4315,12114}, {4425,12713}, {5249,9961}, {5250,10431}, {5603,12563}, {5711,15811}, {5768,6744}, {5777,21060}, {5882,15174}, {5927,21075}, {6282,12447}, {6684,19541}, {6734,9812}, {6743,6769}, {6833,19862}, {6844,12571}, {6956,10171}, {7682,10893}, {9836,12568}, {10478,12548}, {12053,12709}, {12610,12712}, {12706,21617}, {12707,21618}, {12708,21619}, {12714,21622}, {12715,21623}, {12716,21624}

X(21628) = midpoint of X(i) and X(j) for these {i,j}: {4, 12705}, {8, 12651}, {962, 12526}, {4294, 5691}
X(21628) = reflection of X(1071) in X(12564)
X(21628) = complement of X(12565)
X(21628) = X(12705)-of-Euler triangle
X(21628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9800, 12565), (946, 6245, 11019), (946, 9948, 942), (3062, 5290, 6223), (6260, 7680, 3947), (8727, 9856, 946), (12558, 12609, 3817), (12616, 18483, 7682)

X(21629) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 5th EXTOUCH

Barycentrics (b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2-2*(b^2-c^2)^2*a+(b^4-c^4)*(b-c) : :

X(21629) = 3*X(2)+X(9801) = 3*X(1699)+X(7996) = 3*X(3817)-2*X(12610)

The reciprocal orthologic center of these triangles is X(65)

X(21629) lies on these lines: {2,1721}, {4,9}, {8,12652}, {11,17635}, {63,2961}, {141,15726}, {142,9944}, {226,12689}, {307,2310}, {314,9442}, {908,12530}, {982,3663}, {990,1125}, {1699,7996}, {1742,3912}, {2951,17284}, {3012,17075}, {3062,17272}, {3244,8271}, {3247,10186}, {3452,18252}, {3729,4847}, {3741,10444}, {3817,8228}, {3821,9843}, {4104,5927}, {4353,18216}, {4416,9355}, {4425,12725}, {4643,16112}, {5249,9962}, {5851,17345}, {7229,9812}, {9441,17353}, {10164,19544}, {10478,12549}, {11495,17279}, {12053,12721}, {12718,21617}, {12719,21618}, {12720,21619}, {12722,21620}, {12726,21622}, {12727,21623}, {12728,21624}, {17060,21255}

X(21629) = midpoint of X(i) and X(j) for these {i,j}: {4, 12717}, {8, 12652}
X(21629) = reflection of X(990) in X(1125)
X(21629) = complement of X(1721)
X(21629) = X(317)-of-Wasat triangle
X(21629) = X(12717)-of-Euler triangle
X(21629) = anticomplement, wrt 3rd Euler triangle, of X(12610)
X(21629) = {X(2), X(9801)}-harmonic conjugate of X(1721)

X(21630) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO FUHRMANN

Barycentrics (b+c)*a^3+(b^2-6*b*c+c^2)*a^2-(b+c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^2 : :

X(21630) = 3*X(1)-X(6224) = 3*X(2)+X(9802) = 3*X(10)-2*X(1145) = 3*X(10)-4*X(6702) = 5*X(10)-2*X(13996) = 3*X(11)-X(1145) = 3*X(11)-2*X(6702) = 5*X(11)-X(13996) = X(80)-3*X(10707) = 3*X(80)-X(12531) = 3*X(149)+X(6224) = 2*X(908)-3*X(11813) = 5*X(1145)-3*X(13996) = X(1320)+3*X(10707) = 3*X(1320)+X(12531) = 10*X(6702)-3*X(13996) = 9*X(10707)-X(12531)

The reciprocal orthologic center of these triangles is X(8)

X(21630) lies on these lines: {1,149}, {2,5541}, {4,6264}, {8,11524}, {10,11}, {80,519}, {100,1125}, {104,516}, {106,14028}, {119,3817}, {142,214}, {145,9897}, {153,1699}, {226,1317}, {496,6797}, {515,10738}, {517,1484}, {546,946}, {758,17638}, {950,13274}, {962,1768}, {1156,5850}, {1210,5533}, {1537,2801}, {1647,4674}, {2170,21090}, {2611,13604}, {2771,3874}, {2783,21636}, {2784,10768}, {2787,11599}, {2800,4084}, {2886,3898}, {3035,14150}, {3036,3452}, {3065,5180}, {3241,18393}, {3576,13199}, {3616,15015}, {3671,5083}, {3679,21041}, {3813,3878}, {3814,3880}, {3822,5919}, {3881,17660}, {3885,7741}, {3919,9951}, {3947,10956}, {4134,18254}, {4292,10074}, {4297,5840}, {4425,12746}, {4431,4568}, {4738,21087}, {4848,20118}, {4857,4861}, {5249,9963}, {5267,10058}, {5493,20418}, {5531,11522}, {5603,6326}, {5883,17051}, {5886,12331}, {6154,15808}, {6174,12732}, {6265,13464}, {6713,10164}, {7972,12047}, {7982,12247}, {8097,21622}, {8098,21623}, {8666,12701}, {8674,13605}, {8715,11376}, {9669,10912}, {9803,13253}, {9955,11698}, {10087,13411}, {10106,13273}, {10478,12550}, {10609,12609}, {10742,18483}, {10916,12758}, {11362,12619}, {11373,13205}, {11680,17057}, {12610,12744}, {12699,12773}, {12730,21617}, {12733,21618}, {12734,21619}, {12735,21620}, {12748,21624}, {12751,19925}, {13271,17647}, {15171,21634}, {18240,21625}

X(21630) = midpoint of X(i) and X(j) for these {i,j}: {1, 149}, {4, 6264}, {8, 12653}, {80, 1320}, {104, 14217}, {145, 9897}, {153, 7993}, {962, 1768}, {7982, 12247}, {9803, 13253}, {12699, 12773}
X(21630) = reflection of X(i) in X(j) for these (i,j): (10, 11), (100, 1125), (119, 16174), (3625, 15863), (3814, 7743), (4297, 11715), (6265, 13464), (7972, 15519), (10742, 18483), (11362, 12619), (12751, 19925)
X(21630) = complement of X(5541)
X(21630) = X(74)-of-Wasat triangle
X(21630) = X(146)-of-3rd Euler triangle
X(21630) = X(149)-of-anti-Aquila triangle
X(21630) = X(6264)-of-Euler triangle
X(21630) = X(9919)-of-2nd Zaniah triangle
X(21630) = X(11807)-of-excentral triangle
X(21630) = X(12244)-of-4th Euler triangle
X(21630) = X(13417)-of-2nd circumperp triangle
X(21630) = X(17847)-of-incircle-circles triangle
X(21630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9802, 5541), (11, 1145, 6702), (100, 16173, 1125), (119, 16174, 3817), (214, 1387, 551), (1145, 6702, 10), (1320, 10707, 80), (1699, 7993, 153), (3616, 20095, 15015), (3825, 10914, 10), (13273, 20586, 10106)

X(21631) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO HUTSON EXTOUCH

Barycentrics (b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(b^2+14*b*c+c^2)*a^4+4*(b^2+5*b*c+c^2)*(b-c)^2*a^3-(b^2-18*b*c+c^2)*(b+c)^3*a^2-2*(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

X(21631) = 3*X(2)+X(9804) = 3*X(1699)+X(8001) = 3*X(3817)-2*X(12612)

The reciprocal orthologic center of these triangles is X(3555)

X(21631) lies on these lines: {2,9804}, {4,12842}, {8,12654}, {10,6767}, {11,17639}, {40,4847}, {142,5045}, {226,12692}, {329,9874}, {516,12516}, {908,12533}, {946,10157}, {1125,12521}, {1699,8001}, {3452,18255}, {3487,8000}, {3817,12612}, {4425,12869}, {5044,21627}, {5249,12537}, {5779,12699}, {5920,12053}, {9953,11019}, {10478,12552}, {12610,12865}, {12846,21617}, {12851,21618}, {12852,21619}, {12853,21620}, {12870,21622}, {12871,21623}, {12873,21624}

X(21631) = midpoint of X(i) and X(j) for these {i,j}: {4, 12842}, {8, 12654}, {9953, 12855}
X(21631) = reflection of X(10) in X(12620)
X(21631) = complement of X(12658)
X(21631) = X(12842)-of-Euler triangle
X(21631) = {X(2), X(9804)}-harmonic conjugate of X(12658)

X(21632) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO MANDART-EXCIRCLES

Barycentrics (b+c)*a^9+(b-c)^2*a^8-2*(b+c)^3*a^7-2*(b^4+c^4+(b^2-12*b*c+c^2)*b*c)*a^6+4*(b+c)^3*b*c*a^5+2*(b^2-10*b*c+c^2)*(b+c)^2*b*c*a^4+2*(b+c)*(b^6+c^6-(2*b^4+2*c^4+(b^2-12*b*c+c^2)*b*c)*b*c)*a^3+2*(b^2-c^2)^2*(b-c)^2*(b^2+3*b*c+c^2)*a^2-(b^2-c^2)^3*(b-c)^3*a-(b^4-c^4)*(b^2-c^2)^3 : :

X(21632) = 3*X(2)+X(12542) = 3*X(1699)+X(13069) = 3*X(3817)-2*X(12613)

The reciprocal orthologic center of these triangles is X(3555)

X(21632) lies on these lines: {2,12542}, {4,12843}, {8,12655}, {10,12621}, {11,17640}, {142,12442}, {226,12693}, {516,12517}, {517,16388}, {908,12534}, {1125,12522}, {1699,13069}, {3452,18257}, {3817,12613}, {4425,13071}, {5249,12538}, {6745,10306}, {10478,12553}, {11019,12449}, {12053,12876}, {12610,13070}, {12847,21617}, {12878,21618}, {12883,21619}, {12907,21620}, {13072,21622}, {13073,21623}, {13074,21624}

X(21632) = midpoint of X(i) and X(j) for these {i,j}: {4, 12843}, {8, 12655}
X(21632) = reflection of X(10) in X(12621)
X(21632) = complement of X(12659)
X(21632) = X(12843)-of-Euler triangle
X(21632) = {X(2), X(12542)}-harmonic conjugate of X(12659)

X(21633) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO MIDARC

Barycentrics (b+c)*sin(A/2)+(-b+c)*sin(B/2)+(b-c)*sin(C/2) : :

X(21633) = 3*X(2)+X(9807) = X(167)+3*X(1699) = 3*X(3817)-2*X(12614)

The reciprocal orthologic center of these triangles is X(1)

X(21633) lies on these lines: {1,21622}, {2,164}, {4,12844}, {8,12656}, {10,12622}, {11,17641}, {142,12443}, {167,1699}, {177,226}, {178,946}, {188,5934}, {236,5935}, {516,12518}, {908,11691}, {1125,12523}, {3452,18258}, {3817,12614}, {4425,13091}, {5249,12539}, {5571,11019}, {6147,12813}, {7057,12879}, {7670,21617}, {8084,8086}, {8085,10967}, {8087,8094}, {8088,8093}, {8379,11032}, {8422,12053}, {10478,12554}, {12610,13090}, {12884,21619}, {13092,21624}

X(21633) = midpoint of X(i) and X(j) for these {i,j}: {4, 12844}, {8, 12656}, {177, 12694}, {5571, 12450}, {7057, 12879}
X(21633) = reflection of X(10) in X(12622)
X(21633) = complement of X(164)
X(21633) = X(1)-of-Wasat triangle
X(21633) = X(8)-of-3rd Euler triangle
X(21633) = X(944)-of-4th Euler triangle
X(21633) = X(9798)-of-2nd Zaniah triangle
X(21633) = X(12844)-of-Euler triangle
X(21633) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9807, 164), (21622, 21623, 1)

X(21634) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 1st SCHIFFLER

Barycentrics -(-a+b+c)*((b+c)*a^8+4*b*c*a^7-2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^6-2*(5*b^2-2*b*c+5*c^2)*b*c*a^5+(b+c)*(6*b^4+6*c^4-(16*b^2-13*b*c+16*c^2)*b*c)*a^4+2*(4*b^4+4*c^4-(3*b^2+4*b*c+3*c^2)*b*c)*b*c*a^3-(b^2-c^2)*(b-c)*(4*b^4+4*c^4-(6*b^2+5*b*c+6*c^2)*b*c)*a^2-2*(b^2-c^2)^2*(b-c)^2*b*c*a+(b^2-c^2)^3*(b-c)^3) : :

X(21634) = 3*X(2)+X(12543) = 3*X(1699)+X(13101) = 3*X(3817)-2*X(12615)

The reciprocal orthologic center of these triangles is X(21)

X(21634) lies on these lines: {2,12543}, {4,12845}, {8,12657}, {10,12267}, {11,17643}, {21,6599}, {142,12444}, {226,12695}, {516,12519}, {908,12535}, {946,12600}, {1125,12524}, {1699,13101}, {3452,18259}, {3817,12615}, {4425,13123}, {4861,21627}, {5249,12540}, {6597,10266}, {6842,10265}, {10478,12557}, {11019,12451}, {11263,15325}, {12053,12877}, {12610,13120}, {12850,21617}, {12882,21618}, {12887,21619}, {12909,21620}, {13124,21622}, {13125,21623}, {13127,21624}, {15171,21630}

X(21634) = midpoint of X(i) and X(j) for these {i,j}: {4, 12845}, {8, 12657}, {6597, 10266}
X(21634) = reflection of X(10) in X(12623)
X(21634) = complement of X(12660)
X(21634) = X(12845)-of-Euler triangle
X(21634) = {X(2), X(12543)}-harmonic conjugate of X(12660)

X(21635) = PARALLELOGIC CENTER OF THESE TRIANGLES: WASAT TO FUHRMANN

Barycentrics (b+c)*a^5-(b+c)^2*a^4-(2*b-c)*(b-2*c)*(b+c)*a^3+2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-5*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2 : :

X(21635) = 3*X(2)+X(9809) = 3*X(2)-5*X(15017) = 2*X(11)-3*X(3817) = X(20)-3*X(15015) = X(100)-3*X(5660) = X(149)-3*X(1699) = 3*X(551)-2*X(11715) = 3*X(551)-4*X(11729) = 3*X(1699)+X(5531) = X(1768)-5*X(15017) = 3*X(3817)+2*X(13257) = X(9809)+5*X(15017) = X(10698)+3*X(10711) = 3*X(10711)-X(12751) = 4*X(12611)-X(21630)

The reciprocal parallelogic center of these triangles is X(4)

X(21635) lies on these lines: {1,153}, {2,1768}, {3,16128}, {4,6326}, {5,2771}, {8,13253}, {10,119}, {11,118}, {12,17638}, {20,15015}, {79,6915}, {80,7548}, {100,516}, {104,1125}, {142,5851}, {149,1699}, {191,6960}, {214,2829}, {515,6265}, {517,11698}, {518,1538}, {519,1519}, {546,946}, {551,11715}, {758,1532}, {950,12739}, {962,5541}, {971,5087}, {1145,21075}, {1156,21617}, {1210,11570}, {1317,12053}, {1387,21620}, {1484,9955}, {1537,2802}, {1698,12767}, {1737,11571}, {2051,2783}, {2787,21636}, {2827,3716}, {2886,15064}, {2932,12679}, {3035,3452}, {3065,6888}, {3091,9803}, {3120,5400}, {3336,6979}, {3576,12248}, {3671,12736}, {3678,15908}, {3754,17654}, {3814,6001}, {3838,10157}, {3874,7681}, {3878,18242}, {4010,14680}, {4193,15071}, {4292,10090}, {4425,13265}, {5249,10171}, {5536,17484}, {5587,12247}, {5603,6264}, {5691,6224}, {5692,6932}, {5693,6941}, {5853,13271}, {5882,19907}, {5886,12773}, {5902,6945}, {6245,9946}, {6684,12515}, {6702,12609}, {6713,19862}, {6734,12532}, {6828,15096}, {6863,18233}, {6891,16127}, {6894,11604}, {6902,16132}, {6907,10176}, {6922,18243}, {6952,7701}, {6975,15016}, {7741,12528}, {7993,11522}, {8068,12617}, {8103,21622}, {8104,21623}, {9812,20095}, {9842,12019}, {9897,18393}, {10058,13411}, {10087,10624}, {10106,12740}, {10175,12619}, {10478,13244}, {10479,12551}, {10728,12119}, {10738,12738}, {10916,12665}, {10956,15558}, {12331,12699}, {12610,13262}, {12737,13464}, {12761,17647}, {13260,21618}, {13261,21619}, {13267,21624}, {13407,16173}, {14740,21060}

X(21635) = midpoint of X(i) and X(j) for these {i,j}: {1, 153}, {3, 16128}, {4, 6326}, {8, 13253}, {11, 13257}, {149, 5531}, {962, 5541}, {2932, 12679}, {5536, 17484}, {5691, 6224}, {10728, 12119}, {10738, 12738}, {12331, 12699}, {17660, 17661}
X(21635) = reflection of X(i) in X(j) for these (i,j): (10, 119), (80, 19925), (104, 1125), (1484, 9955), (5882, 19907), (10738, 18483), (12737, 13464)
X(21635) = complement of X(1768)
X(21635) = X(110)-of-Wasat-triangle
X(21635) = X(153)-of-anti-Aquila-triangle
X(21635) = X(1568)-of-Fuhrmann-triangle
X(21635) = X(3448)-of-3rd Euler-triangle
X(21635) = X(6326)-of-Euler-triangle
X(21635) = X(10265)-of-Johnson-triangle
X(21635) = X(11800)-of-excentral-triangle
X(21635) = X(12310)-of-2nd Zaniah-triangle
X(21635) = X(12383)-of-4th Euler-triangle
X(21635) = X(17838)-of-incircle-circles-triangle
X(21635) = X(21649)-of-2nd circumperp-triangle
X(21635) = X(36)-of-X(4)-Brocard-triangle
X(21635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9809, 1768), (11, 5083, 11019), (11, 12831, 5083), (1699, 5531, 149), (1768, 15017, 2), (6260, 21616, 4297), (6842, 20117, 10), (10698, 10711, 12751), (11715, 11729, 551), (12739, 12764, 950), (12740, 12763, 10106)

X(21636) = CYCLOLOGIC CENTER OF THESE TRIANGLES: WASAT TO 2nd CIRCUMPERP

Barycentrics (b+c)*a^6+2*(b-c)^2*a^5-2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^4+4*b*c*(b-c)^2*a^3+(b+c)*(2*b^2-2*b*c+c^2)*(b^2-2*b*c+2*c^2)*a^2+2*b^2*c^2*(b-c)^2*a+(b^6-c^6)*(-b+c) : :

X(21636) = 2*X(115)-3*X(3817) = X(148)-3*X(1699) = 3*X(551)-2*X(11710) = 3*X(551)-4*X(11724) = 4*X(620)-3*X(10164) = 3*X(3576)-X(9862) = 5*X(3616)-X(5984) = 3*X(3839)-X(9875) = X(4301)+2*X(14981) = 3*X(5886)-X(12188) = 4*X(6036)-5*X(19862) = 3*X(6054)+X(7970) = 3*X(6054)-X(9864) = 2*X(6055)-3*X(19883) = 2*X(6684)-3*X(15561)

The reciprocal cyclologic center of these triangles is X(1)

X(21636) lies on these lines: {1,147}, {2,9860}, {10,114}, {98,1125}, {99,516}, {115,3817}, {148,1699}, {226,3023}, {515,6033}, {519,6054}, {542,551}, {553,12351}, {620,10164}, {946,2782}, {950,12185}, {962,13174}, {993,2792}, {2783,21630}, {2787,21635}, {2789,11814}, {2794,4297}, {3027,12053}, {3576,9862}, {3616,5984}, {3839,9875}, {4292,10089}, {4301,14981}, {5886,12188}, {6036,19862}, {6055,19883}, {6321,18483}, {6684,15561}, {8227,14651}, {9812,20094}, {10053,13411}, {10086,10624}, {10106,12184}, {10165,12042}, {10171,14061}, {12182,17647}, {12512,21166}, {12571,14639}, {12699,13188}, {13178,19925}

X(21636) = midpoint of X(i) and X(j) for these {i,j}: {1, 147}, {962, 13174}, {12699, 13188}
X(21636) = reflection of X(i) in X(j) for these (i,j): (10, 114), (98, 1125), (6321, 18483), (13178, 19925)
X(21636) = complement of X(9860)
X(21636) = X(147)-of-anti-Aquila triangle
X(21636) = X(1303)-of-Wasat triangle
X(21636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6054, 7970, 9864), (11710, 11724, 551)

X(21637) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND ANTI-HONSBERGER

Barycentrics a^2*(-a^2+b^2+c^2)*(2*a^4+(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(21637) lies on these lines: {2,19122}, {4,19123}, {5,19155}, {6,25}, {39,20968}, {52,19154}, {69,3292}, {110,14913}, {125,3589}, {182,185}, {216,14575}, {217,1692}, {373,19137}, {389,19128}, {511,7488}, {570,1576}, {575,3091}, {1176,11574}, {1177,13417}, {1181,5050}, {1204,5085}, {1216,19362}, {1351,16195}, {1425,1428}, {1503,11572}, {1899,3618}, {2330,3270}, {2916,10510}, {3284,20775}, {3313,19127}, {3611,19133}, {3917,19126}, {5052,14585}, {5092,20791}, {5097,15073}, {5480,21659}, {5562,19131}, {5596,11550}, {5622,19140}, {6146,18583}, {6403,10282}, {6622,14912}, {6751,19156}, {9512,14767}, {9544,12272}, {10605,12017}, {11381,19124}, {13754,19129}, {13851,19130}, {14853,19467}, {14885,21644}, {18438,18475}, {19134,21642}, {19135,21643}, {19138,21649}, {19141,21651}, {19142,21652}, {19143,21653}, {19144,21654}, {19145,21655}, {19146,21656}, {19147,21657}, {19148,21658}, {19150,21660}, {19171,21638}

X(21637) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 154, 12167), (6, 159, 8541), (6, 184, 6467), (6, 206, 1843), (6, 1974, 51), (6, 6467, 21639), (6, 18374, 9969), (6, 19125, 184), (6, 19132, 25), (6, 19153, 1974), (206, 1843, 1495), (3618, 19119, 1899), (5622, 19140, 21650), (19126, 20806, 3917), (19131, 19139, 5562)

X(21638) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 1st ANTI-SHARYGIN

Barycentrics a^2*((b^4+c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^4+c^4)*(b^2-c^2)^2)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2)*(a^4-(2*b^2+c^2)*a^2+b^2*(b^2-c^2)) : :

X(21638) lies on these lines: {2,19167}, {4,19168}, {5,19211}, {6,16030}, {25,19180}, {51,107}, {52,19210}, {54,186}, {95,3917}, {97,511}, {184,19170}, {185,8884}, {373,19188}, {1181,19173}, {1204,19172}, {1425,19175}, {1899,19166}, {3270,19182}, {3611,19181}, {4993,5943}, {4994,10110}, {5562,19179}, {6467,19197}, {6751,19212}, {8779,8882}, {8901,13567}, {9786,16035}, {11381,19169}, {13417,19208}, {13754,19176}, {13851,19177}, {19171,21637}, {19178,21639}, {19183,21640}, {19184,21641}, {19187,21643}, {19190,21647}, {19191,21648}, {19193,21649}, {19195,21650}, {19196,21651}, {19198,21652}, {19199,21653}, {19200,21654}, {19201,21655}, {19202,21656}, {19203,21657}, {19204,21658}, {19205,21659}

X(21638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 19185, 13367), (54, 19207, 21660), (275, 9792, 51), (9792, 19209, 275), (19166, 19174, 1899), (19169, 19206, 11381), (19170, 19189, 184), (19179, 19194, 5562)

X(21639) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd EHRMANN

Barycentrics a^2*(-a^2+b^2+c^2)*(2*a^4-(b^2+c^2)*a^2-3*(b^2-c^2)^2) : :

X(21639) = 4*X(6)-X(1495) = 7*X(6)-X(12367) = 3*X(6)-X(18374) = 5*X(6)-X(19596) = 2*X(895)+X(3292) = 7*X(1495)-4*X(12367) = 3*X(1495)-4*X(18374) = 5*X(1495)-4*X(19596) = 3*X(12367)-7*X(18374) = 5*X(12367)-7*X(19596) = 5*X(18374)-3*X(19596)

X(21639) lies on these lines: {2,11443}, {4,11458}, {6,25}, {52,11255}, {125,524}, {182,15078}, {185,576}, {373,9813}, {378,10250}, {389,8537}, {401,11596}, {511,2071}, {512,1570}, {542,13851}, {575,13367}, {895,3292}, {1181,11482}, {1204,11477}, {1351,10605}, {1425,19369}, {1503,13473}, {1568,3564}, {1899,1992}, {2104,13414}, {2105,13415}, {2682,15341}, {3269,5107}, {3270,8540}, {3284,20975}, {3543,5032}, {3611,8539}, {3917,11511}, {5093,14915}, {5447,19361}, {5562,8538}, {8549,11381}, {8550,21659}, {9019,15826}, {9926,21651}, {9974,21655}, {9975,21656}, {9976,21650}, {9977,21660}, {12596,21649}, {12598,21654}, {13037,21657}, {13038,21658}, {13248,13417}, {13754,18449}, {14908,14961}, {15128,19510}, {19178,21638}, {19426,21642}

X(21639) = crossdifference of every pair of points on line X(193)X(525)
X(21639) = isogonal conjugate of isotomic conjugate of X(5159)
X(21639) = intersection of tangents to Moses-Jerabek conic at X(6) and X(1205)
X(21639) = 2nd-Lemoine-circle-inverse of X(35901)
X(21639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 6467, 21637), (6, 8541, 51), (6, 10602, 184), (6, 11216, 8541), (6, 17813, 25), (184, 10602, 6467), (575, 15073, 13367), (1992, 18919, 1899), (8538, 8548, 5562), (8549, 11470, 11381)

X(21640) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 1st KENMOTU DIAGONALS

Barycentrics a^2*(-a^2+b^2+c^2)*(2*S*a^2+(a^2+b^2-c^2)*(a^2-b^2+c^2)) : :

X(21640) lies on these lines: {2,11447}, {3,6415}, {4,11462}, {6,25}, {52,11265}, {125,590}, {185,371}, {217,5058}, {372,13367}, {373,10961}, {389,10880}, {577,6413}, {1151,1204}, {1181,3311}, {1425,2067}, {1504,19032}, {1587,19467}, {1899,3068}, {3070,21659}, {3093,11424}, {3284,8911}, {3312,19357}, {3516,19087}, {3549,19061}, {3611,5415}, {3917,11513}, {5062,14585}, {5158,6414}, {5562,10665}, {5870,19042}, {6146,7583}, {6221,10605}, {6417,19347}, {6564,13851}, {6639,13970}, {6776,7374}, {7581,18925}, {9544,11448}, {10132,15905}, {10192,13937}, {10282,10881}, {11381,11473}, {12174,19088}, {12375,21650}, {12424,21651}, {12891,21649}, {12960,21653}, {12961,21654}, {12962,21655}, {12963,21656}, {12965,21660}, {13045,21657}, {13046,21658}, {13198,19111}, {13287,13417}, {13567,13884}, {13665,18396}, {18459,18475}, {18924,19054}, {18996,19349}, {19038,19354}, {19183,21638}, {19436,21642}, {19438,21643}

X(21640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 154, 5411), (6, 184, 21641), (6, 5412, 51), (6, 10533, 5413), (6, 11241, 5412), (6, 17819, 25), (6, 19355, 184), (3068, 18923, 1899), (5413, 10533, 1495), (6467, 8779, 21641), (10665, 10897, 5562), (11473, 12964, 11381)

X(21641) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd KENMOTU DIAGONALS

Barycentrics a^2*(-a^2+b^2+c^2)*(2*S*a^2-(a^2+b^2-c^2)*(a^2-b^2+c^2)) : :

X(21641) lies on these lines: {2,11448}, {3,6416}, {4,11463}, {6,25}, {52,11266}, {125,615}, {185,372}, {216,8911}, {217,5062}, {371,13367}, {373,10963}, {389,10881}, {511,11418}, {577,6414}, {1152,1204}, {1181,3312}, {1425,6502}, {1505,19033}, {1588,19467}, {1899,3069}, {3071,21659}, {3092,11424}, {3270,5414}, {3311,19357}, {3516,19088}, {3549,19062}, {3611,5416}, {3917,11514}, {5058,14585}, {5158,6413}, {5562,10666}, {5871,19041}, {6146,7584}, {6398,10605}, {6418,19347}, {6565,13851}, {6639,13909}, {6776,7000}, {7582,18925}, {9544,11447}, {10133,15905}, {10192,13884}, {10282,10880}, {11381,11474}, {12174,19087}, {12376,21650}, {12892,21649}, {12967,21654}, {12968,21655}, {12969,21656}, {12971,21660}, {13047,21657}, {13198,19110}, {13288,13417}, {13567,13937}, {13754,18459}, {13785,18396}, {18457,18475}, {18923,19053}, {18995,19349}, {19037,19354}, {19184,21638}, {19439,21642}

X(21641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 154, 5410), (6, 184, 21640), (6, 5413, 51), (6, 10534, 5412), (6, 11242, 5413), (6, 17820, 25), (6, 19356, 184), (3069, 18924, 1899), (5158, 8908, 6413), (5412, 10534, 1495), (6467, 8779, 21640), (10666, 10898, 5562), (11474, 12970, 11381)

X(21642) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics (SB+SC)*((8*R^2+4*SA+SW)*S^2+2*S*(3*S^2+2*SA^2-SB*SC)-SB*SC*SW) : :

X(21642) lies on these lines: {2,19412}, {3,21654}, {4,19414}, {6,19404}, {25,19430}, {51,19410}, {184,8939}, {185,18980}, {373,19448}, {389,19424}, {511,19406}, {1181,19418}, {1204,13021}, {1425,19370}, {1899,18926}, {3270,19434}, {3611,19432}, {3917,19422}, {5562,18939}, {6467,12590}, {9723,21643}, {11381,19416}, {13366,19408}, {13367,19440}, {13417,19507}, {13754,18462}, {13851,18414}, {19134,21637}, {19426,21639}, {19436,21640}, {19439,21641}, {19450,21647}, {19452,21648}, {19482,21649}, {19484,21650}, {19486,21651}, {19488,21652}, {19490,21653}, {19492,21655}, {19495,21656}, {19496,21658}, {19498,21659}, {19502,21660}

X(21642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8939, 19358, 184), (18926, 19420, 1899), (18939, 19428, 5562), (19410, 19446, 51), (19416, 19500, 11381)

X(21643) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics (SB+SC)*((8*R^2+4*SA+SW)*S^2-2*S*(3*S^2+2*SA^2-SB*SC)-SB*SC*SW) : :

X(21643) lies on these lines: {2,19413}, {3,21653}, {4,19415}, {6,19405}, {51,19411}, {185,18981}, {373,19449}, {389,19425}, {511,19407}, {1425,19371}, {3917,19423}, {6467,12591}, {9723,21642}, {11381,19417}, {13366,19409}, {13754,18463}, {19135,21637}, {19187,21638}, {19438,21640}, {19491,21654}, {19494,21655}

X(21643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19411, 19447, 51), (19417, 19501, 11381)

X(21644) = PERSPECTOR OF THESE TRIANGLES: ANTI-WASAT AND 1st ORTHOSYMMEDIAL

Barycentrics a^2*(-a^2+b^2+c^2)*((b^2+c^2)*a^10-(-4*b^2*c^2+(b^2-c^2)^2)*a^8+(b^2+c^2)*b^2*c^2*a^6-2*(b^2-c^2)^2*b^2*c^2*a^4-(b^4-c^4)*(b^2-c^2)*(b^4+5*b^2*c^2+c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2) : :

X(21644) lies on these lines: {185,1907}, {14885,21637}

X(21645) = PERSPECTOR OF THESE TRIANGLES: ANTI-WASAT AND PELLETIER

Barycentrics a^2*((b+c)^2*a^5-2*(b+c)*b*c*a^4-(2*b^4+2*c^4+(b^2+c^2)*b*c)*a^3+(b+c)*(b^2+c^2)*b*c*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a+(b^2-c^2)^2*(b+c)*b*c)*(b-c) : :

X(21645) = 3*X(51)-4*X(18344)

X(21645) lies on these lines: {51,18344}, {187,237}, {3309,11381}, {11550,21301}

X(21646) = PERSPECTOR OF THESE TRIANGLES: ANTI-WASAT AND SCHROETER

Barycentrics a^2*((b^2+c^2)*a^6-2*(b^4+b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2+(b^2-c^2)^2*b^2*c^2)*(b^2-c^2) : :

X(21646) = 3*X(51)-4*X(2501) = 3*X(3917)-2*X(6563)

X(21646) lies on these lines: {2,11450}, {51,2501}, {187,237}, {1499,11381}, {1510,6130}, {3917,6563}

X(21646) = complement of X(11450)

X(21647) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND INNER TRI-EQUILATERAL

Barycentrics a^2*(2*sqrt(3)*S*a^2+(a^2+b^2-c^2)*(a^2-b^2+c^2))*(-a^2+b^2+c^2) : :

X(21647) lies on these lines: {2,11452}, {4,11466}, {6,25}, {15,185}, {16,8837}, {52,11267}, {373,10643}, {389,10632}, {397,10619}, {511,11420}, {1181,11485}, {1204,11480}, {1425,7051}, {1899,11488}, {3270,10638}, {3611,10636}, {3917,11515}, {5318,21659}, {5335,19467}, {5562,10634}, {6146,11542}, {9544,11453}, {10282,10633}, {10657,21650}, {10659,21651}, {10663,21649}, {10667,21655}, {10671,21656}, {10675,11381}, {10677,21660}, {10681,13417}, {11081,14533}, {11486,19357}, {12980,21653}, {12982,21654}, {13057,21657}, {13754,18468}, {13851,16808}, {18470,18475}, {19190,21638}, {19450,21642}

X(21647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 154, 11409), (6, 184, 21648), (6, 10641, 51), (6, 11243, 10641), (6, 17826, 25), (6, 19363, 184), (10634, 10661, 5562), (10675, 11475, 11381), (11488, 18929, 1899)

X(21648) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND OUTER TRI-EQUILATERAL

Barycentrics a^2*(-2*sqrt(3)*S*a^2+(-c^2+a^2+b^2)*(c^2+a^2-b^2))*(c^2-a^2+b^2) : :

X(21648) lies on these lines: {2,11453}, {4,11467}, {6,25}, {15,8839}, {16,185}, {373,10644}, {389,10633}, {398,10619}, {511,11421}, {1181,11486}, {1204,11481}, {1250,3270}, {1425,19373}, {1899,11489}, {3611,10637}, {3917,11516}, {5321,21659}, {5334,19467}, {5562,10635}, {6146,11543}, {9544,11452}, {10282,10632}, {10658,21650}, {10668,21655}, {10678,21660}, {11086,14533}, {11485,19357}, {13059,21657}, {13754,18470}, {13851,16809}, {18468,18475}, {19191,21638}, {19452,21642}

X(21648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 154, 11408), (6, 184, 21647), (6, 10642, 51), (6, 11244, 10642), (6, 17827, 25), (6, 19364, 184), (10635, 10662, 5562), (11489, 18930, 1899)

X(21649) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO AAOA

Barycentrics (S^2-SB*SC)*(3*R^2*(12*R^2-SA-5*SW)+2*SW^2-2*SB*SC) : :

X(21649) = 3*X(51)-2*X(113) = 3*X(51)-4*X(12236) = 2*X(110)-3*X(16223) = 3*X(125)-2*X(12358) = X(146)-3*X(3060) = 9*X(373)-8*X(12900) = 4*X(389)-3*X(16223) = X(399)-3*X(568) = 3*X(568)-2*X(11557) = 2*X(1216)-3*X(15061) = 2*X(1511)-3*X(9730) = 2*X(1986)-3*X(14831) = 3*X(3060)-2*X(11807) = 3*X(5562)-4*X(12358) = 2*X(11800)+X(12284)

The reciprocal orthologic center of these triangles is X(15136)

X(21649) lies on these lines: {2,12273}, {3,11806}, {4,11800}, {5,13358}, {6,12168}, {20,17855}, {51,113}, {52,3627}, {74,511}, {110,389}, {125,5562}, {146,3060}, {184,2931}, {185,10111}, {265,1531}, {373,12900}, {399,568}, {542,1843}, {974,6467}, {1112,1906}, {1147,17701}, {1154,10264}, {1181,12310}, {1204,12302}, {1216,15061}, {1425,19469}, {1511,9730}, {1899,12319}, {2781,15583}, {2854,19161}, {3047,10282}, {3270,12888}, {3448,5889}, {3611,12661}, {3917,6699}, {5446,7728}, {5462,14643}, {5622,11574}, {5642,9826}, {5876,11801}, {5890,12383}, {5891,20304}, {5907,14644}, {5946,10272}, {6000,10733}, {6053,12824}, {6102,11562}, {6243,10620}, {7550,10821}, {7687,12825}, {7731,12317}, {9140,12219}, {9729,15035}, {9969,14982}, {10113,12162}, {10625,12041}, {10663,21647}, {10721,13598}, {11459,15081}, {11597,15087}, {11793,15059}, {12022,19481}, {12228,13366}, {12596,21639}, {12891,21640}, {12892,21641}, {12893,13367}, {13171,17834}, {13391,14677}, {14094,16625}, {14531,16003}, {15012,15034}, {15036,17704}, {15051,16836}, {15055,15644}, {15057,15606}, {16222,16534}, {18438,19457}, {19138,21637}, {19193,21638}, {19482,21642}

X(21649) = midpoint of X(i) and X(j) for these {i,j}: {4, 12284}, {3448, 5889}, {6243, 10620}, {7731, 12317}
X(21649) = reflection of X(i) in X(j) for these (i,j): (3, 11806), (4, 11800), (5, 13358), (20, 17855), (110, 389), (146, 11807), (399, 11557), (5876, 11801), (10625, 12041), (10721, 13598)
X(21649) = complement of X(12273)
X(21649) = X(12284)-of-Euler triangle
X(21649) = X(17838)-of-anti-Ara triangle
X(21649) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113, 12236, 51), (146, 3060, 11807), (265, 19479, 13851), (399, 568, 11557), (2931, 19456, 184), (7687, 12825, 15030), (12319, 18932, 1899)

X(21650) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO ANTI-ORTHOCENTROIDAL

Barycentrics (S^2-SB*SC)*(4*S^2+3*R^2*(24*R^2-5*SA-9*SW)+4*SA^2-2*SB*SC+2*SW^2) : :

X(21650) = 3*X(4)-X(7731) = 3*X(4)-2*X(11807) = 3*X(5)-2*X(11561) = 4*X(5)-3*X(16223) = 2*X(113)-3*X(15030) = 3*X(125)-2*X(974) = 5*X(125)-4*X(16270) = 3*X(185)-4*X(974) = X(185)-4*X(15738) = 5*X(185)-8*X(16270) = X(7731)+3*X(12281) = 2*X(7731)-3*X(13417) = 2*X(11807)+3*X(12281) = 4*X(11807)-3*X(13417) = 2*X(12281)+X(13417)

The reciprocal orthologic center of these triangles is X(3581)

X(21650) lies on these lines: {2,12270}, {3,11559}, {4,7730}, {5,113}, {6,12165}, {25,17835}, {51,1986}, {52,10113}, {67,11744}, {74,186}, {110,5907}, {146,3818}, {184,399}, {206,5621}, {265,1531}, {381,11557}, {389,7722}, {511,10296}, {542,6467}, {1092,12302}, {1112,14448}, {1181,12308}, {1204,10620}, {1205,1503}, {1216,12121}, {1425,19470}, {1498,13171}, {1511,5891}, {1539,16194}, {1593,17847}, {1843,2781}, {1899,12317}, {2777,6240}, {2807,13211}, {2914,15033}, {2935,15106}, {3016,3269}, {3043,11430}, {3146,13201}, {3270,7727}, {3292,5504}, {3448,12111}, {3611,7724}, {3819,15051}, {3917,12358}, {5562,7723}, {5622,19140}, {5876,15532}, {5889,11800}, {5890,15081}, {6102,11801}, {6241,14940}, {8779,14901}, {9729,15059}, {9976,21639}, {10018,14862}, {10114,12022}, {10125,13491}, {10272,15060}, {10575,12041}, {10657,21647}, {10658,21648}, {10721,13474}, {10990,15105}, {11074,18877}, {11424,19504}, {11459,12383}, {11572,19506}, {11746,13148}, {11793,15035}, {12227,13366}, {12236,14831}, {12244,12290}, {12375,21640}, {12376,21641}, {12412,18451}, {12902,18436}, {13198,14094}, {14683,19467}, {14915,20127}, {15012,15025}, {15044,16625}, {15058,15102}, {15141,19124}, {15151,17853}, {17812,20987}, {19195,21638}, {19484,21642}

X(21650) = midpoint of X(i) and X(j) for these {i,j}: {4, 12281}, {146, 15100}, {3146, 13201}, {3448, 12111}, {12244, 12290}, {12902, 18436}
X(21650) = reflection of X(i) in X(j) for these (i,j): (52, 10113), (110, 5907), (5562, 7723), (5889, 11800), (6102, 11801), (6241, 17855), (10575, 12041), (10721, 13474)
X(21650) = complement of X(12270)
X(21650) = X(11562)-of-Johnson triangle
X(21650) = X(12281)-of-Euler triangle
X(21650) = X(17835)-of-anti-Ara triangle
X(21650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 7731, 11807), (5, 11562, 16223), (399, 19457, 184), (1986, 7687, 51), (5622, 19140, 21637), (7722, 14644, 389), (7731, 11807, 13417), (12317, 18933, 1899), (12358, 16163, 3917), (15100, 15305, 146)

X(21651) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO ARIES

Barycentrics (S^2-SB*SC)*(S^2-4*R^2*(2*R^2-SW)+SA^2+SB*SC-SW^2) : :

X(21651) = 3*X(51)-2*X(155) = 3*X(51)-4*X(12235) = 9*X(373)-8*X(9820) = 3*X(3167)-4*X(5462) = 3*X(3917)-4*X(12359) = 4*X(9927)-3*X(15030)

The reciprocal orthologic center of these triangles is X(7387)

X(21651) lies on these lines: {2,12271}, {3,6391}, {4,12282}, {6,12166}, {51,155}, {52,1843}, {68,5562}, {184,9937}, {185,12421}, {373,9820}, {382,6243}, {389,6193}, {511,11411}, {539,14831}, {569,8548}, {912,16980}, {1147,13366}, {1204,12301}, {1205,10625}, {1425,19471}, {2393,17834}, {3167,5462}, {3917,12359}, {5446,12164}, {7401,14913}, {7487,12272}, {8754,8800}, {9926,21639}, {9927,15030}, {9932,13367}, {10659,21647}, {11432,19588}, {11457,12058}, {12424,21640}, {19141,21637}, {19196,21638}, {19486,21642}

X(21651) = midpoint of X(4) and X(12282)
X(21651) = complement of X(12271)
X(21651) = X(12282)-of-Euler triangle
X(21651) = X(17836)-of-anti-Ara triangle
X(21651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (155, 12235, 51), (9937, 19458, 184)

X(21652) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO HATZIPOLAKIS-MOSES

Barycentrics (S^2-SB*SC)*((-10*R^2+2*SW)*S^2+256*R^6-8*(3*SA+17*SW)*R^4+(-10*SA^2+15*SA*SW+27*SW^2)*R^2+2*(SA^2-SA*SW-SW^2)*SW) : :

The reciprocal orthologic center of these triangles is X(9729)

X(21652) lies on these lines: {25,17837}, {184,2929}, {185,10112}, {1205,10602}, {1368,5562}, {1425,19472}, {1899,18936}, {3448,12111}, {5895,13417}, {7691,13348}, {19142,21637}, {19198,21638}, {19488,21642}

X(21652) = X(17837)-of-anti-Ara triangle
X(21652) = {X(2929), X(19460)}-harmonic conjugate of X(184)

X(21653) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS ANTIPODAL

Barycentrics (SB+SC)*(8*R^2*S^2-2*S*(S^2-4*R^2*SA+SB*SC)-(SA+SW)*SA*SW) : :

X(21653) = 3*X(51)-2*X(487) = 3*X(51)-4*X(12237) = 9*X(373)-8*X(642) = 4*X(486)-3*X(3917)

The reciprocal orthologic center of these triangles is X(3)

X(21653) lies on these lines: {2,12274}, {3,21643}, {4,12285}, {6,12169}, {51,487}, {185,21658}, {373,642}, {389,12509}, {486,3917}, {511,12221}, {1425,19473}, {3564,16655}, {5562,12601}, {11381,12296}, {12229,13366}, {12960,21640}, {12980,21647}, {19143,21637}, {19199,21638}, {19490,21642}

X(21653) = midpoint of X(4) and X(12285)
X(21653) = reflection of X(i) in X(j) for these (i,j): (5562, 12601), (11381, 12296)
X(21653) = complement of X(12274)
X(21653) = X(12285)-of-Euler triangle
X(21653) = X(17839)-of-anti-Ara triangle
X(21653) = {X(487), X(12237)}-harmonic conjugate of X(51)

X(21654) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS(-1) ANTIPODAL

Barycentrics (SB+SC)*(8*R^2*S^2+2*S*(S^2-4*R^2*SA+SB*SC)-(SA+SW)*SA*SW) : :

X(21654) = 3*X(51)-2*X(488) = 3*X(51)-4*X(12238) = 9*X(373)-8*X(641) = 4*X(485)-3*X(3917)

The reciprocal orthologic center of these triangles is X(3)

X(21654) lies on these lines: {2,12275}, {3,21642}, {4,12286}, {6,12170}, {25,17842}, {51,488}, {184,12979}, {185,21657}, {373,641}, {389,12510}, {485,3917}, {511,12222}, {1181,12312}, {1204,12304}, {1425,19474}, {1899,12321}, {3270,12911}, {3564,16655}, {3611,12663}, {5562,12602}, {11381,12297}, {12230,13366}, {12598,21639}, {12961,21640}, {12967,21641}, {12973,13367}, {12982,21647}, {13055,19358}, {19144,21637}, {19200,21638}, {19491,21643}

X(21654) = midpoint of X(4) and X(12286)
X(21654) = reflection of X(i) in X(j) for these (i,j): (5562, 12602), (11381, 12297)
X(21654) = complement of X(12275)
X(21654) = X(12286)-of-Euler triangle
X(21654) = X(17842)-of-anti-Ara triangle
X(21654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (488, 12238, 51), (12321, 18938, 1899), (12979, 19462, 184)

X(21655) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS CENTRAL

Barycentrics a^2*(-a^2+b^2+c^2)*(S*((b^2+c^2)*a^2+(b^2-c^2)^2)+(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :

X(21655) = 3*X(51)-2*X(6291) = 3*X(51)-4*X(12239) = 9*X(373)-8*X(9823) = 3*X(3917)-4*X(12360)

The reciprocal orthologic center of these triangles is X(3)

X(21655) lies on these lines: {2,12276}, {3,8910}, {4,12287}, {6,12171}, {20,185}, {25,17840}, {51,3071}, {184,1151}, {373,9823}, {389,6239}, {1181,12313}, {1204,12305}, {1425,7362}, {1899,12322}, {3270,6283}, {3611,6252}, {3917,12360}, {5023,19356}, {5562,12603}, {9974,21639}, {10667,21647}, {10668,21648}, {11381,12298}, {12231,13366}, {12962,21640}, {12968,21641}, {12974,13367}, {19145,21637}, {19201,21638}, {19492,21642}, {19494,21643}

X(21655) = midpoint of X(4) and X(12287)
X(21655) = reflection of X(i) in X(j) for these (i,j): (5562, 12603), (11381, 12298)
X(21655) = complement of X(12276)
X(21655) = X(12287)-of-Euler triangle
X(21655) = X(17840)-of-anti-Ara triangle
X(21655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (185, 6467, 21656), (1151, 19463, 184), (6291, 12239, 51), (12322, 18941, 1899)

X(21656) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS(-1) CENTRAL

Barycentrics a^2*(-a^2+b^2+c^2)*(-S*((b^2+c^2)*a^2+(b^2-c^2)^2)+(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :

X(21656) = 3*X(51)-2*X(6406) = 3*X(51)-4*X(12240) = 9*X(373)-8*X(9824) = 3*X(3917)-4*X(12361)

The reciprocal orthologic center of these triangles is X(3)

X(21656) lies on these lines: {2,12277}, {4,12288}, {6,12172}, {20,185}, {51,3070}, {184,1152}, {373,9824}, {389,6400}, {1181,12314}, {1204,12306}, {1425,7353}, {1899,12323}, {3270,6405}, {3611,6404}, {3917,12361}, {5023,19355}, {5562,12604}, {9975,21639}, {10671,21647}, {11381,12299}, {12232,13366}, {12963,21640}, {12969,21641}, {12975,13367}, {19146,21637}, {19202,21638}, {19495,21642}

X(21656) = midpoint of X(4) and X(12288)
X(21656) = reflection of X(i) in X(j) for these (i,j): (5562, 12604), (11381, 12299)
X(21656) = complement of X(12277)
X(21656) = X(12288)-of-Euler triangle
X(21656) = X(17843)-of-anti-Ara triangle
X(21656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (185, 6467, 21655), (1152, 19464, 184), (6406, 12240, 51), (12323, 18942, 1899)

X(21657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS REFLECTION

Barycentrics (S^2-SB*SC)*((10*R^2-SA-SW)*S^2+2*(R^2*(16*R^2-3*SA+3*SW)+SB*SC-SW^2)*S+2*R^2*(SA^2+SB*SC+3*SW^2)-SW^2*(SA+SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(21657) lies on these lines: {2,13015}, {4,13017}, {6,13007}, {25,17841}, {51,13013}, {184,13055}, {185,21654}, {373,13053}, {389,13035}, {511,13009}, {1181,13023}, {1204,13021}, {1425,19475}, {1899,13025}, {3270,13043}, {3611,13041}, {3917,13027}, {5562,13039}, {11381,13019}, {13011,13366}, {13037,21639}, {13045,21640}, {13047,21641}, {13049,13367}, {13057,21647}, {13059,21648}, {19147,21637}, {19203,21638}

X(21657) = midpoint of X(4) and X(13017)
X(21657) = reflection of X(i) in X(j) for these (i,j): (5562, 13039), (11381, 13019)
X(21657) = complement of X(13015)
X(21657) = X(13017)-of-Euler triangle
X(21657) = X(17841)-of-anti-Ara triangle
X(21657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13013, 13051, 51), (13025, 18943, 1899), (13055, 19465, 184)

X(21658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO LUCAS(-1) REFLECTION

Barycentrics (S^2-SB*SC)*((10*R^2-SA-SW)*S^2-2*(R^2*(16*R^2-3*SA+3*SW)+SB*SC-SW^2)*S+2*R^2*(SA^2+SB*SC+3*SW^2)-SW^2*(SA+SW)) : :

The reciprocal orthologic center of these triangles is X(3)

X(21658) lies on these lines: {2,13016}, {6,13008}, {51,13014}, {184,13056}, {185,21653}, {373,13054}, {389,13036}, {511,13010}, {1204,13022}, {1425,19476}, {3917,13028}, {5562,13040}, {13012,13366}, {13038,21639}, {13046,21640}, {13050,13367}, {19148,21637}, {19204,21638}, {19496,21642}

X(21658) = midpoint of X(4) and X(13018)
X(21658) = reflection of X(i) in X(j) for these (i,j): (5562, 13040), (11381, 13020)
X(21658) = complement of X(13016)
X(21658) = X(13018)-of-Euler triangle
X(21658) = X(17844)-of-anti-Ara triangle
X(21658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13014, 13052, 51), (13056, 19466, 184)

X(21659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO MACBEATH

Barycentrics (-a^2+b^2+c^2)*(2*a^8-2*(b^2+c^2)*a^6-(b^2-c^2)^2*a^4+(b^2-c^2)^4) : :

X(21659) = 3*X(4)-2*X(13419) = 3*X(51)-2*X(3575) = 9*X(51)-8*X(11745) = 3*X(51)-4*X(12241) = 3*X(185)-4*X(18914) = 9*X(373)-8*X(9825) = 2*X(389)-3*X(12022) = 3*X(3575)-4*X(11745) = 3*X(6146)-2*X(18914) = X(6240)-3*X(12022) = 2*X(11745)-3*X(12241) = X(12289)+2*X(13403) = 3*X(12289)+2*X(13419) = 4*X(13292)-3*X(14831) = 3*X(13403)-X(13419)

The reciprocal orthologic center of these triangles is X(4).

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(21659) is the A'B'C'-to-ABC similarity image of X(4). (Randy Hutson, August 29, 2018)

X(21659) lies on these lines: {2,12278}, {3,125}, {4,54}, {5,13367}, {6,12173}, {20,1204}, {24,18390}, {25,17845}, {30,52}, {49,5448}, {51,3575}, {113,156}, {115,14585}, {235,1495}, {287,6655}, {373,9825}, {378,18381}, {381,19357}, {382,1181}, {389,6240}, {403,10282}, {427,11572}, {511,12225}, {539,18436}, {542,12111}, {550,13470}, {576,3146}, {631,18918}, {974,11565}, {1092,12118}, {1147,1568}, {1425,7354}, {1503,1885}, {1562,7748}, {1593,11550}, {1594,11430}, {1657,10605}, {1853,3516}, {1879,14533}, {2072,12038}, {2777,6241}, {2904,11456}, {3070,21640}, {3071,21641}, {3269,7756}, {3270,6284}, {3357,11457}, {3448,11440}, {3520,20299}, {3521,15002}, {3529,18909}, {3534,17712}, {3567,18559}, {3611,6253}, {3613,15619}, {3830,19347}, {3845,15807}, {3917,12362}, {5254,8779}, {5318,21647}, {5321,21648}, {5339,8738}, {5340,8737}, {5480,21637}, {5562,12605}, {5651,6816}, {5876,15532}, {5889,10112}, {5893,13473}, {5907,14516}, {5944,10113}, {5972,11449}, {6000,18560}, {6243,10938}, {6689,14805}, {6756,16657}, {7503,8907}, {7505,11202}, {7507,11425}, {7526,18474}, {7547,18376}, {7576,10110}, {7577,18394}, {7687,11464}, {8550,21639}, {9707,12140}, {9820,10297}, {10024,18475}, {10116,18562}, {10125,11801}, {10151,16252}, {10182,14644}, {10733,13198}, {10982,18494}, {11245,13568}, {11403,19459}, {11438,18912}, {11468,20417}, {11591,15738}, {12134,15030}, {12233,13366}, {12943,19349}, {12953,19354}, {13352,18569}, {13434,18428}, {13474,16659}, {13488,16655}, {13754,18563}, {14927,18935}, {15072,19481}, {15463,19506}, {17538,18931}, {17824,19504}, {18128,18565}, {19205,21638}, {19498,21642}

X(21659) = midpoint of X(4) and X(12289)
X(21659) = reflection of X(i) in X(j) for these (i,j): (4, 13403), (382, 12897), (550, 13470), (5562, 12605), (5889, 10112)
X(21659) = complement of X(12278)
X(21659) = X(12289)-of-Euler triangle
X(21659) = X(13403)-of-anti-Euler triangle
X(21659) = X(17845)-of-anti-Ara triangle
X(21659) = X(3869)-of-orthic-triangle if ABC is acute
X(21659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 265, 5449), (4, 54, 18388), (4, 578, 3574), (4, 12254, 1614), (4, 19467, 184), (20, 1899, 1204), (20, 18945, 1899), (49, 18403, 5448), (184, 19467, 10619), (1147, 18404, 1568), (3575, 12241, 51), (6240, 12022, 389), (11430, 18383, 1594), (12118, 18531, 1092), (13367, 13851, 5)

X(21660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO REFLECTION

Barycentrics (S^2-SB*SC)*(R^2*(8*R^2-3*SA-7*SW)-2*SB*SC+2*SW^2) : :

X(21660) = 3*X(51)-2*X(6152) = 3*X(51)-4*X(12242) = 3*X(54)-2*X(389) = 3*X(54)-X(6242) = X(185)-4*X(11577) = 3*X(195)-X(6243) = 9*X(373)-8*X(9827) = 3*X(2888)-5*X(11444) = 3*X(3574)-2*X(11576) = 5*X(3574)-4*X(11743) = 3*X(3917)-4*X(12363) = X(5876)+3*X(15532) = X(6241)-3*X(12254) = 3*X(7691)-4*X(13348) = 5*X(11576)-6*X(11743)

The reciprocal orthologic center of these triangles is X(6243)

X(21660) lies on these lines: {2,12280}, {3,11806}, {4,12291}, {6,12175}, {25,17846}, {51,6152}, {52,1493}, {54,186}, {125,21230}, {184,195}, {185,550}, {235,1843}, {373,9827}, {511,12226}, {539,5562}, {1181,12316}, {1204,12307}, {1209,20303}, {1216,3519}, {1425,7356}, {1614,2914}, {1899,12325}, {2888,11444}, {2904,9707}, {3270,6286}, {3518,12380}, {3611,6255}, {3917,12359}, {5876,15532}, {5965,6467}, {6145,13622}, {6241,10628}, {6288,13851}, {7691,13348}, {8254,13368}, {9781,11808}, {9977,21639}, {10110,13433}, {10115,19357}, {10203,13198}, {10263,11803}, {10677,21647}, {10678,21648}, {11271,11412}, {11381,12300}, {12965,21640}, {12971,21641}, {13754,18442}, {14049,14448}, {19150,21637}, {19502,21642}

X(21660) = midpoint of X(i) and X(j) for these {i,j}: {4, 12291}, {11271, 11412}
X(21660) = reflection of X(i) in X(j) for these (i,j): (52, 1493), (185, 10619), (10263, 11803), (11381, 12300)
X(21660) = complement of X(12280)
X(21660) = X(12291)-of-Euler triangle
X(21660) = X(17846)-of-anti-Ara triangle
X(21660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 6242, 389), (54, 12234, 13366), (54, 19207, 21638), (195, 19468, 184), (6152, 12242, 51), (12325, 18946, 1899)

X(21661) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO CIRCUMORTHIC

Barycentrics a^2*(a^8-(b^2+3*c^2)*a^6-(b^2-2*c^2)*(2*b^2+c^2)*a^4+(b^4-c^4)*(3*b^2-c^2)*a^2-(b^6-c^6)*(b^2-c^2))*(a^8-(3*b^2+c^2)*a^6+(2*b^2-c^2)*(b^2+2*c^2)*a^4+(b^4-c^4)*(b^2-3*c^2)*a^2-(b^6-c^6)*(b^2-c^2))*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(21661) = 3*X(51)-2*X(130)

The reciprocal cyclologic center of these triangles is X(1)

X(21661) lies on these lines: {51,130}, {129,5562}, {148,5889}, {389,1298}, {511,1303}, {5667,5890}

X(21661) = X(99)-of-anti-Wasat triangle
X(21661) = X(148)-of-orthic triangle
X(21661) = X(13172)-of-2nd Euler triangle
X(21661) = X(20094)-of-2nd anti-Conway triangle

X(21662) = CENTER OF THE CIRCUMCONIC OF THESE TRIANGLES: WASAT AND ANTI-WASAT

Barycentrics a^2*(b^2-c^2)^2*(a^2-(b-c)*b)*(a^2+(b-c)*c)*(a^2-(b+c)*b)*(a^2-(b+c)*c)*((b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-c^8-b^8+b^4*c^4) : :

X(21662) lies on the nine-point circle and these lines: {2,14719}, {4,14720}, {114,5876}

X(21662) = midpoint of X(4) and X(14720)
X(21662) = complement of X(14719)
X(21662) = X(14720)-of-Euler triangle

X(21663) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND TRINH

Barycentrics a^2*(-a^2+b^2+c^2)*(2*a^6-(b^2+c^2)*a^4-4*(b^2-c^2)^2*a^2+3*(b^4-c^4)*(b^2-c^2)) : :

X(21663) = 4*X(3)-X(3292) = X(23)+5*X(15021) = 2*X(74)+X(1495) = 5*X(74)+X(12112) = 3*X(74)+X(14157) = 5*X(186)-X(12112) = 3*X(186)-X(14157) = 2*X(468)+X(10990) = 5*X(1495)-2*X(12112) = 3*X(1495)-2*X(14157) = X(1531)-4*X(6699) = X(2070)+3*X(15041) = X(2071)-3*X(15055) = 3*X(12112)-5*X(14157) = X(13445)-5*X(15021)

X(21663) lies on these lines: {2,11454}, {3,49}, {4,11270}, {6,11410}, {23,13445}, {24,3357}, {25,10606}, {30,125}, {35,1425}, {36,3270}, {51,378}, {52,11250}, {54,13382}, {64,1620}, {74,186}, {187,3269}, {230,1562}, {235,5894}, {287,13586}, {373,9818}, {376,1899}, {389,3520}, {403,2777}, {468,10990}, {511,2071}, {548,6146}, {550,13470}, {974,1154}, {1192,1593}, {1350,10602}, {1498,15750}, {1503,13399}, {1531,2072}, {1568,10257}, {1614,17506}, {1658,10575}, {2030,10766}, {2070,14915}, {2393,5621}, {2937,14641}, {2972,12096}, {3003,18877}, {3098,6467}, {3147,5878}, {3431,20421}, {3516,9777}, {3518,13474}, {3522,18913}, {3528,18909}, {3534,18396}, {3542,20427}, {3574,13568}, {3575,6696}, {3580,16386}, {3581,18859}, {3611,7688}, {5092,20791}, {5462,14130}, {5663,15646}, {5890,11430}, {5907,11440}, {5943,7527}, {6102,10226}, {6200,21640}, {6240,11572}, {6241,10282}, {6411,19355}, {6412,19356}, {6644,15030}, {6776,10304}, {7502,14855}, {7575,15738}, {7690,21655}, {7691,13348}, {7692,21656}, {7740,14264}, {7793,9289}, {8541,10249}, {9306,15078}, {9730,18570}, {9938,21651}, {10024,20191}, {10110,14865}, {10151,11598}, {10193,18388}, {10212,15806}, {10245,11820}, {10295,18400}, {10296,15057}, {10298,15072}, {10540,10620}, {10619,18914}, {10645,21647}, {10646,21648}, {11001,18918}, {11064,16976}, {11202,11456}, {11550,18533}, {11563,14677}, {12086,13598}, {12106,16194}, {12174,17821}, {12294,19136}, {12901,21649}, {12984,21653}, {12985,21654}, {13061,21657}, {13293,13417}, {13346,14531}, {13352,14831}, {13353,18364}, {13391,16270}, {13434,15012}, {13491,15331}, {14585,15513}, {15061,18403}, {15138,21284}, {15578,19161}, {15647,17856}, {18323,20397}, {19192,21638}, {19454,21642}

X(21663) = midpoint of X(i) and X(j) for these {i,j}: {23, 13445}, {74, 186}, {3580, 16386}, {3581, 18859}, {10540, 10620}, {11563, 14677}
X(21663) = reflection of X(i) in X(j) for these (i,j): (1531, 2072), (1568, 10257), (11064, 16976)
X(21663) = intersection of tangents to Moses-Jerabek conic at X(74) and X(389)
X(21663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 185, 13367), (3, 1204, 185), (3, 7689, 5562), (3, 10605, 184), (3, 12163, 1092), (4, 11270, 11468), (24, 3357, 11381), (64, 1620, 3515), (184, 1204, 10605), (184, 10605, 185), (187, 3269, 8779), (376, 18931, 1899), (378, 11438, 51), (1620, 3532, 64), (11204, 11438, 378)

X(21664) = X(118)X(133)∩X(119)X(2804)

Barycentrics    (2 a b c - a^2 (b + c) + (b - c)^2 (b + c))^2 (a^4 - (b^2 - c^2)^2) : :
Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3)^2 : :
Barycentrics    (sec A)(cos B + cos C - 1)[b(cos C + cos A - 1) + c(cos A + cos B - 1)] : :

See Angel Montesdeoca, HG210618.

X(21664) lies on the MacBeath inconic and these lines: {3,108}, {4,145}, {5,318}, {25,915}, {33,1807}, {117,522}, {118,133}, {119,2804}, {225,1830}, {342,6356}, {429,2970}, {442,2972}, {517,1361}, {1068,1398}, {1118,10306}, {1119,2095}, {1565,18026}, {1895,6922}, {5081,5844}, {5690,17555}, {5761,7524}, {5790,7046}, {5901,11109}, {7071,10679}, {7412,11849}, {7649,20619}, {10696,18340}

X(21664) = {118,133}, {119,2804}, {225,1830}, {342,6356}, {429,2970}, {442,2972}, {517,1361}, {1068,1398}, {1118,10306}, {1119,2095}, {1565,18026}, {1895,6922}, {5081,5844}, {5690,17555}, {5761,7524}, {5790,7046}, {5901,11109}, {7071,10679}, {7412,11849}, {7649,20619}, {10696,18340}

X(21664) = midpoint of X(i) and X(j) for these {i,j}: {4, 1897}, {10696, 18340}
X(21664) = reflection of X(i) in X(j) for these (i,j): {3, 15252}, {2968, 5}
X(21664) = pole wrt polar circle of line X(104)X(900)
X(21664) = MacBeath inconic antipode of X(2968)

X(21665) = X(4)X(150)∩X(25)X(917)

Barycentrics b^2c^2(-2a^3+a^2(b+c)+(b-c)^2(b+c))^2(a^4-(b^2-c^2)^2) : :
Barycentrics (csc 2A)[(csc B)/(b^2 - c^2 cos A - a^2 cos C) + (csc C)/(c^2 - a^2 cos B - b^2 cos A)] : :

See Angel Montesdeoca, HG210618.

X(21665) lies on these lines: {4,150}, {25,917}, {118,20622}, {430,2970}, {2968,8226}, {2972,3136}

X(21666) = X(4)X(151)∩X(25)X(1311)

Barycentrics b^2c^2(b-c)^2(-a+b+c)^2(a^2+b^2-c^2)(a^2-b^2+c^2) : :
Barycentrics (csc 2A)(cos B - cos C)^2 : :

See Angel Montesdeoca, HG210618.

X(21666) lies on these lines: {4,151}, {25,1311}, {124,20620}, {318,7141}, {1109,14618}, {2968,14010}, {2972,3137}, {7017,7046}, {14936,17926}

X(21666) = pole wrt polar circle of trilinear polar of X(1262) (line X(109)X(692), the tangent to the circumcircle at X(109))
X(21666) = polar conjugate of X(1262)

X(21667) = X(3)X(8754)∩X(5)X(131)

Barycentrics 2 a^14 (b^2 + c^2) - a^12 (11 b^4 + 2 b^2 c^2 + 11 c^4) + 2 a^10 (12 b^6 + b^4 c^2 + b^2 c^4 + 12 c^6) - a^8 (25 b^8 + 8 b^6 c^2 - 18 b^4 c^4 + 8 b^2 c^6 + 25 c^8) + 2 a^6 (5 b^10 + 5 b^8 c^2 - 6 b^6 c^4 - 6 b^4 c^6 + 5 b^2 c^8 + 5 c^10) + a^4 (b^2 - c^2)^2 (3 b^8 - 8 b^6 c^2 - 14 b^4 c^4 - 8 b^2 c^6 + 3 c^8) - 2 a^2 (b^2 - c^2)^4 (2 b^6 - b^4 c^2 - b^2 c^4 + 2 c^6) + (b^2 - c^2)^8 : :
X(21667) = 4 X(5) - 3 X(131) = X(131) - 2 X(136)

See Angel Montesdeoca, HG250618.

X(21667) lies on these lines: {3,8754}, {5,131}, {20,254}, {155,382}, {187,16310}, {631,925}, {9714,13558}

X(21667) = reflection of X(131) in X(136)

X(21668) = X(6)X(1511)∩X(1297)X(13398)

Barycentrics a^2 (a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-6 a^12 b^2 c^2+20 a^10 b^4 c^2-29 a^8 b^6 c^2+17 a^6 b^8 c^2-8 a^4 b^10 c^2+10 a^2 b^12 c^2-5 b^14 c^2-5 a^12 c^4+20 a^10 b^2 c^4-40 a^8 b^4 c^4+36 a^6 b^6 c^4-13 a^4 b^8 c^4-8 a^2 b^10 c^4+10 b^12 c^4+9 a^10 c^6-29 a^8 b^2 c^6+36 a^6 b^4 c^6-8 a^4 b^6 c^6+3 a^2 b^8 c^6-11 b^10 c^6-5 a^8 c^8+17 a^6 b^2 c^8-13 a^4 b^4 c^8+3 a^2 b^6 c^8+10 b^8 c^8-5 a^6 c^10-8 a^4 b^2 c^10-8 a^2 b^4 c^10-11 b^6 c^10+9 a^4 c^12+10 a^2 b^2 c^12+10 b^4 c^12-5 a^2 c^14-5 b^2 c^14+c^16) : :

See Angel Montesdeoca, HG250618.

X(21668) lies on these lines: {6,1511}, {1297,13398}

X(21669) = 31ST HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics a (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4-a b^5-a^5 c+5 a^4 b c-2 a^2 b^3 c+a b^4 c-3 b^5 c-2 a^4 c^2+2 a^2 b^2 c^2+2 a^3 c^3-2 a^2 b c^3+6 b^3 c^3+a^2 c^4+a b c^4-a c^5-3 b c^5) : :
X(21669) = 9 R X[2] - 2 (3 R - r) X[3], 2 X[3] - 3 X[21], 6 X[442] - 7 X[3090], 3 X[2475] - 5 X[3091], 4 X[3] - 3 X[3651], 5 X[3651] - 8 X[5428], 5 X[3] - 6 X[5428], 5 X[21] - 4 X[5428], 4 X[3628] - 3 X[5499], 2 X[3649] - 3 X[5603], 4 X[5] - 3 X[6175], 11 X[3525] - 12 X[6675], 5 X[3091] - 6 X[6841], 3 X[7701] + X[7982], 3 X[191] - X[7991], 4 X[6701] - 5 X[8227], 5 X[632] - 6 X[10021], 2 X[1] + X[10308], 9 X[3651] - 16 X[12104], 9 X[5428] - 10 X[12104], 9 X[21] - 8 X[12104], 3 X[3] - 4 X[12104], 4 X[12104] - 9 X[13743], 2 X[5428] - 5 X[13743], X[3651] - 4 X[13743], X[3] - 3 X[13743], 6 X[11277] - 7 X[14869], 3 X[7967] - 4 X[15174], 5 X[631] - 6 X[15670], 8 X[140] - 9 X[15671], 7 X[3523] - 9 X[15672], 3 X[3524] - 4 X[15673], 13 X[10303] - 15 X[15674], 4 X[549] - 5 X[15675], X[20] - 3 X[15677], X[3146] + 3 X[15680], 3 X[5603] - X[16116], 5 X[3651] - 4 X[16117], 5 X[3] - 3 X[16117], 5 X[21] - 2 X[16117], 5 X[13743] - X[16117], 3 X[1699] - X[16118], 3 X[1699] - 2 X[16125], 5 X[10595] - 4 X[16137], 2 X[10222] + 3 X[16138], 3 X[5426] - X[16143], 2 X[546] - 3 X[16160], 5 X[15034] - 6 X[16164], 3 X[16126] - 5 X[16189], 3 X[2] - 4 X[16617], 3 X[5657] - 4 X[18253], 16 X[5428] - 15 X[21161], 8 X[16117] - 15 X[21161], 8 X[3] - 9 X[21161], 4 X[21] - 3 X[21161], 2 X[3651] - 3 X[21161], 8 X[13743] - 3 X[21161].

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28103.

X(21669) lies on these lines: {1,10308}, {2,3}, {8,18761}, {36,18483}, {40,3647}, {65,16141}, {78,18540}, {79,104}, {84,10122}, {100,18480}, {145,18519}, {191,7991}, {515,3746}, {516,16113}, {517,3652}, {758,6762}, {943,4304}, {944,3303}, {956,3650}, {958,6361}, {962,3648}, {999,11544}, {1155,16616}, {1172,3284}, {1387,1476}, {1389,2800}, {1437,14157}, {1470,10591}, {1482,4430}, {1519,11263}, {1537,12913}, {1621,18481}, {1699,5450}, {1836,18977}, {2077,19925}, {2771,7984}, {2975,12699}, {3057,16140}, {3304,3649}, {3579,4002}, {3585,10058}, {3871,18525}, {3951,7330}, {5225,8071}, {5229,8069}, {5253,9955}, {5284,13624}, {5426,16143}, {5538,20117}, {5657,18253}, {5691,11491}, {5818,10310}, {6001,17637}, {6701,8227}, {7373,16006}, {7967,15174}, {9812,11249}, {9856,15178}, {10595,11038}, {10680,20084}, {10915,12751}, {11372,16132}, {11681,18516}, {12047,16152}, {12248,18990}, {12608,14526}, {12701,16142}, {15034,16164}, {15179,16005}, {16126,16189}

X(21669) = midpoint of X(962) and X(3648)
X(21669) = reflection of X(i) in X(j) for these {i,j}: {21, 13743}, {40, 3647}, {79, 946}, {376, 17525}, {944, 10543}, {2475, 6841}, {3651, 21}, {11684, 3652}, {15679, 381}, {16116, 3649}, {16117, 5428}, {16118, 16125}
X(21669) = anticomplement of X(37401)
X(21669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5, 17531), (3, 546, 6915), (3, 3091, 6946), (3, 3560, 16865), (3, 6912, 6920), (3, 6913, 16842), (3, 6914, 17574), (3, 16842, 631), (3, 16865, 1006), (3, 17531, 6940), (4, 1012, 6906), (4, 6833, 6941), (4, 6847, 6830), (4, 6906, 6905), (4, 6935, 6834), (4, 6942, 19541), (4, 6950, 3149), (4, 6952, 1532), (4, 6956, 6968), (4, 6977, 6848), (5, 6909, 6940), (20, 3560, 1006), (20, 6846, 6897), (20, 16865, 3), (21, 3651, 21161), (21, 15679, 404), (21, 17536, 15675), (376, 6896, 6904), (382, 6914, 411), (405, 17525, 21), (411, 17574, 3), (550, 7489, 6986), (1699, 16118, 16125), (3543, 4189, 6985), (5603, 16116, 3649), (6824, 6925, 6937), (6837, 6850, 6829), (6890, 6893, 6963), (6891, 6957, 6975), (6909, 17531, 3)

X(21670) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b^2 - a^2 b^3 + a b^4 + b^5 - a^3 c^2 - 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4 + c^5) : :

X(21670) lies on these lines: {3, 10}, {12, 7363}, {442, 21045}, {3697, 21867}, {8013, 21671}, {20653, 21031}, {20654, 21019}, {20659, 21687}, {21011, 21691}, {21054, 21674}, {21700, 21709}

X(21671) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 + b^2 + c^2) (2 a^3 + a^2 b + b^3 + a^2 c - b^2 c - b c^2 + c^3) : :

X(21671) lies on these lines: {4, 9}, {12, 201}, {65, 21717}, {80, 1794}, {210, 20653}, {212, 1837}, {306, 1265}, {440, 18673}, {3695, 3949}, {4064, 5489}, {4466, 21530}, {6047, 21933}, {8013, 21670}, {21676, 21693}, {21677, 21692}, {21681, 21711}, {21698, 21700}

X(21672) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (2 a^5 + a^4 b - 3 a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c - 3 a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(21672) lies on these lines: {5, 10}, {71, 6253}, {8013, 21670}, {18673, 20653}, {21685, 21711}

X(21673) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c)^2 (2 a^2 + a b + b^2 + a c - 2 b c + c^2) : :

X(21673) lies on these lines: {7, 10}, {210, 8013}, {4061, 6555}, {4111, 21044}, {21031, 21039}, {21033, 21713}, {21693, 21705}

X(21674) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-2 a^2 - a b + b^2 - a c - 2 b c + c^2) : :

X(21674) lies on these lines: {1, 2}, {12, 201}, {35, 11101}, {37, 21029}, {45, 10895}, {100, 409}, {115, 21816}, {227, 21686}, {377, 4414}, {407, 21811}, {442, 2292}, {846, 2475}, {896, 18253}, {966, 21076}, {986, 4197}, {1213, 20654}, {1468, 5791}, {1834, 1962}, {2308, 5717}, {2650, 17056}, {3695, 6535}, {3704, 21020}, {3750, 5178}, {3841, 4424}, {3842, 20488}, {3925, 4642}, {3932, 21713}, {3943, 21690}, {3989, 13161}, {4026, 20657}, {4205, 8040}, {5051, 6536}, {5492, 5499}, {5794, 10448}, {16589, 21044}, {21031, 21039}, {21054, 21670}, {21676, 21944}, {21700, 21725}, {21921, 21950}

X(21675) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics ((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(b+c)^2 : :

X(21675) lies on these lines: {4, 9}, {8, 21076}, {12, 594}, {37, 21029}, {48, 5794}, {115, 21810}, {201, 8736}, {210, 8013}, {442, 2294}, {1213, 21018}, {1761, 2475}, {1837, 2268}, {1901, 3958}, {1953, 2886}, {2245, 21014}, {2260, 6734}, {2265, 5830}, {2267, 5831}, {3120, 4016}, {3178, 17314}, {3294, 21065}, {3822, 22021}, {3925, 21023}, {4007, 21081}, {4062, 17299}, {4466, 17052}, {4878, 21698}, {17277, 21094}, {17647, 22054}, {20482, 21043}, {20483, 21688}, {20655, 21020}, {20658, 21706}, {21046, 21691}, {21713, 21718}, {21926, 21945}, {21942, 21961}

X(21676) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (2 a^4 - a^3 b - 2 a^2 b^2 + a b^3 - a^3 c + 2 a^2 b c + b^3 c - 2 a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21676) lies on these lines: {10, 11}, {3027, 6057}, {3925, 8286}, {4010, 21714}, {4030, 4651}, {4854, 21963}, {8013, 20656}, {17747, 20659}, {18673, 20653}, {21671, 21693}, {21674, 21944}, {21711, 21832}

X(21677) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c) (2 a^2 + a b - b^2 + a c + 2 b c - c^2) : :

X(21677) lies on these lines: {1, 5791}, {2, 11281}, {4, 12867}, {5, 5692}, {8, 21}, {9, 1837}, {10, 12}, {11, 960}, {19, 5130}, {30, 40}, {63, 5794}, {71, 594}, {78, 5432}, {79, 2093}, {80, 15910}, {144, 1654}, {145, 15674}, {307, 6046}, {329, 10895}, {341, 4126}, {377, 11246}, {380, 4034}, {392, 10916}, {443, 5221}, {495, 5904}, {498, 3940}, {517, 6841}, {518, 8261}, {519, 15670}, {528, 5178}, {740, 4918}, {908, 3614}, {938, 4423}, {944, 21161}, {950, 3683}, {952, 5258}, {956, 5427}, {997, 5433}, {1001, 12649}, {1145, 3626}, {1146, 3691}, {1213, 2294}, {1329, 3876}, {1478, 3927}, {1532, 20117}, {1697, 4863}, {1698, 11374}, {1737, 5044}, {1788, 4413}, {1802, 4390}, {1834, 2292}, {1836, 12526}, {1861, 1888}, {1867, 1869}, {1901, 3958}, {2099, 19843}, {2238, 21965}, {2264, 3686}, {2551, 3715}, {2646, 5745}, {2650, 17056}, {2771, 12665}, {2886, 3869}, {3036, 18259}, {3057, 4847}, {3058, 5250}, {3059, 4111}, {3219, 5086}, {3241, 15671}, {3419, 6284}, {3421, 18962}, {3452, 17606}, {3530, 15015}, {3621, 15676}, {3632, 5426}, {3650, 4691}, {3651, 5584}, {3681, 12607}, {3689, 6743}, {3690, 22299}, {3700, 18013}, {3710, 3714}, {3711, 7080}, {3779, 15985}, {3813, 3877}, {3820, 10523}, {3824, 11551}, {3901, 6147}, {3911, 12447}, {3916, 15326}, {3932, 17751}, {4041, 6362}, {4092, 6068}, {4187, 10176}, {4415, 21935}, {4511, 4999}, {4512, 12625}, {4534, 4875}, {4640, 15338}, {4642, 21035}, {4662, 6735}, {4668, 5441}, {4669, 17525}, {4677, 15673}, {4678, 15680}, {4679, 9581}, {4831, 20077}, {5087, 20288}, {5175, 5698}, {5204, 5744}, {5218, 18231}, {5223, 9578}, {5230, 17602}, {5231, 11376}, {5234, 5727}, {5247, 5724}, {5259, 12433}, {5267, 10609}, {5289, 10527}, {5298, 17614}, {5499, 10942}, {5587, 5812}, {5693, 6907}, {5694, 6842}, {5705, 11375}, {5730, 15950}, {5768, 8273}, {5771, 11012}, {5790, 10526}, {5818, 10894}, {5844, 10021}, {5846, 19133}, {5881, 10268}, {5883, 17529}, {5887, 15908}, {5902, 8728}, {5918, 9948}, {6174, 6684}, {6175, 11236}, {6700, 12832}, {6763, 18990}, {6986, 9803}, {7173, 21616}, {7957, 11362}, {7965, 12617}, {8013, 21693}, {8583, 17728}, {9708, 10573}, {10122, 12670}, {10306, 13743}, {10544, 18191}, {10592, 17057}, {10826, 16155}, {11277, 12738}, {11523, 17718}, {11604, 13272}, {12702, 18517}, {13911, 19026}, {13973, 19025}, {16117, 18518}, {16132, 17857}, {17747, 21029}, {18239, 18908}, {18357, 18406}, {18673, 20653}, {20718, 21926}, {21024, 21711}, {21038, 21938}, {21671, 21692}

X(21677) = complement of X(34195)
X(21677) = anticomplement of X(11281)
X(21677) = crosspoint of X(8) and X(10)
X(21677) = crosssum of X(56) and X(58)

X(21678) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 + b^2 + c^2) (a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(21678) lies on these lines: {4, 9}, {594, 21015}, {2294, 21717}, {8013, 21867}, {10327, 21076}, {18674, 21530}, {20653, 21691}, {20654, 21031}, {21033, 21687}, {21690, 21915}, {21696, 21699}

X(21679) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^5 b^2 - a^4 b^3 + a b^6 + b^7 - a^5 c^2 - a b^4 c^2 - a^4 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a c^6 + c^7) : :

X(21679) lies on these lines: {10, 22}, {8013, 21670}

X(21680) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^5 b^2 - a^4 b^3 + a b^6 + b^7 - a^5 c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - a^4 c^3 + a^2 b^2 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a c^6 + c^7) : :

X(21680) lies on these lines: {10, 23}, {523, 21722}, {858, 15523}, {8013, 21670}

X(21681) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a b^4 + b^5 + a c^4 + c^5) : :

X(21681) lies on these lines: {10, 32}, {8013, 21703}, {16886, 20655}, {20654, 21019}, {21024, 21692}, {21671, 21711}, {21709, 21716}

X(21682) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b^2 - a^2 b^3 + a b^4 + b^5 - 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^2 c^3 - a b c^3 - b^2 c^3 + a c^4 + c^5) : :

X(21682) lies on these lines: {10, 21}, {14844, 21081}, {15523, 21695}, {20488, 21023}, {20653, 21031}, {21024, 21715}, {21029, 21046}

X(21683) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b^2 - a^2 b^3 + a b^4 + b^5 + 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^2 c^3 + a b c^3 - b^2 c^3 + a c^4 + c^5) : :

X(21683) lies on these lines: {10, 36}, {20653, 21031}, {20657, 21724}, {20659, 21046}

X(21684) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^3 b + a^2 b^2 + a^3 c + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(21684) lies on these lines: {10, 38}, {594, 20659}, {1211, 3120}, {2533, 8042}, {4039, 17469}, {5741, 21085}, {17757, 20653}, {21021, 21685}, {21026, 21688}

X(21685) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^3 b^2 + a^2 b^3 + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3) : :

X(21685) lies on these lines: {10, 39}, {594, 762}, {8013, 21700}, {16886, 20659}, {20491, 21730}, {20653, 20658}, {20654, 21019}, {21021, 21684}, {21672, 21711}, {21703, 21709}

X(21686) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^4 b + 2 a^3 b^2 - 2 a b^4 - b^5 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c + 2 a^3 c^2 + 2 b^3 c^2 + 2 a b c^3 + 2 b^2 c^3 - 2 a c^4 - b c^4 - c^5) : :

X(21686) lies on these lines: {4, 9}, {12, 13853}, {201, 7140}, {227, 21674}, {1834, 21044}, {1867, 21912}, {3983, 8013}, {20653, 21031}

X(21687) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 b^3 + b^5 - a b^3 c - b^4 c - a^2 c^3 - a b c^3 - b c^4 + c^5) : :

X(21687) lies on these lines: {10, 41}, {16886, 20655}, {20482, 20653}, {20654, 21030}, {20659, 21670}, {21023, 21930}, {21029, 21692}, {21033, 21678}, {21717, 21921}

X(21688) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^2 b^2 + a b^3 - 2 a^2 b c + b^3 c + a^2 c^2 + a c^3 + b c^3) : :

X(21688) lies on these lines: {1, 2}, {321, 21713}, {1211, 21699}, {1230, 3120}, {3925, 21730}, {19835, 20590}, {20483, 21675}, {20487, 20660}, {20488, 21696}, {20494, 20655}, {20654, 21040}, {20657, 21692}, {21026, 21684}

X(21689) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 b + a b^2 + 2 b^3 - a^2 c - 2 a b c - b^2 c + a c^2 - b c^2 + 2 c^3) : :

X(21689) lies on these lines: {10, 44}, {12, 594}, {313, 21417}, {661, 20483}, {3925, 20655}, {4422, 21094}, {8013, 21708}, {17362, 21076}, {20661, 21723}, {21043, 21718}

X(21690) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-2 a^2 b - a b^2 + b^3 - 2 a^2 c - 4 a b c - 2 b^2 c - a c^2 - 2 b c^2 + c^3) : :

X(21690) lies on these lines: {10, 45}, {12, 594}, {3943, 21674}, {8013, 21043}, {21020, 21045}, {21678, 21915}

X(21691) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b^3 - a^2 b^4 + a b^5 + b^6 - a b^3 c^2 - b^4 c^2 - a^3 c^3 - a b^2 c^3 - a^2 c^4 - b^2 c^4 + a c^5 + c^6) : :

X(21691) lies on these lines: {10, 48}, {20653, 21678}, {20654, 21030}, {21011, 21670}, {21046, 21675}, {21699, 21723}

X(21692) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 b^2 + b^4 - a b^2 c - b^3 c - a^2 c^2 - a b c^2 - b c^3 + c^4) : :

X(21692) lies on these lines: {10, 55}, {594, 21015}, {3136, 21018}, {3613, 15523}, {3711, 21085}, {3925, 21023}, {4388, 21094}, {8013, 20656}, {20653, 21031}, {20657, 21688}, {21020, 21045}, {21024, 21681}, {21029, 21687}, {21671, 21677}, {21919, 21926}

X(21693) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + 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 + 2 a c^3 - b c^3 + c^4) : :

X(21693) lies on these lines: {10, 56}, {8013, 21677}, {20653, 21031}, {20654, 21030}, {20657, 20660}, {21054, 21712}, {21671, 21676}, {21673, 21705}

X(21694) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b + c)^2 (a^2 b + 2 a b^2 + b^3 + a^2 c - 2 a b c - b^2 c + 2 a c^2 - b c^2 + c^3) : :

X(21694) lies on these lines: {10, 57}, {210, 8013}, {4046, 21044}, {20487, 20488}, {20653, 21031}, {20655, 20656}

X(21695) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^3 (a^2 b^2 + 2 a b^3 + b^4 + a^2 c^2 + 2 a c^3 + c^4) : :

X(21695) lies on these lines: {10, 58}, {191, 21047}, {442, 20488}, {1089, 21043}, {15523, 21682}, {16886, 20658}, {17757, 20653}, {20654, 21019}, {20657, 21705}

X(21696) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(21696) lies on these lines: {10, 46}, {12, 7363}, {42, 2165}, {210, 8013}, {227, 21674}, {756, 7140}, {1962, 21044}, {4046, 21018}, {17757, 20653}, {20488, 21688}, {21054, 21717}, {21678, 21699}, {21708, 21943}, {21716, 21718}

X(21697) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^4 + b^4 + c^4) (2 a^5 + a^4 b + b^5 + a^4 c - b^4 c - b c^4 + c^5) : :

X(21697) lies on these lines: {10, 66}, {2980, 21011}

X(21698) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-2 a^3 - a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3) : :

X(21698) lies on these lines: {10, 69}, {42, 2165}, {2667, 21044}, {4062, 17309}, {4878, 21675}, {8013, 10026}, {21031, 21039}, {21671, 21700}

X(21699) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c)^2 (a b + a c + 2 b c) : :

X(21699) lies on these lines: {10, 75}, {42, 3711}, {291, 1268}, {594, 756}, {1211, 21688}, {1213, 3122}, {1573, 1964}, {2292, 4733}, {2643, 21810}, {2667, 4111}, {3932, 20653}, {4023, 4062}, {4039, 17277}, {5257, 22214}, {17233, 21085}, {19586, 22277}, {20482, 21043}, {20654, 21718}, {20655, 21702}, {21031, 21039}, {21041, 21725}, {21678, 21696}, {21691, 21723}

X(21700) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a^2 (b + c)^2 (a b^2 + b^2 c + a c^2 + b c^2) : :

X(21700) lies on these lines: {10, 75}, {38, 20255}, {42, 2176}, {210, 20691}, {756, 7148}, {872, 1500}, {1918, 14974}, {3122, 16589}, {3728, 21024}, {8013, 21685}, {20654, 21703}, {21670, 21709}, {21671, 21698}, {21674, 21725}

X(21701) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^3 (a^2 b + 2 a b^2 + b^3 + a^2 c + 2 a c^2 + c^3) : :

X(21701) lies on these lines: {10, 81}, {756, 21728}, {6535, 21043}, {8013, 10026}, {15523, 20531}, {17757, 20653}, {20488, 21020}, {20656, 21710}, {21707, 21723}

X(21702) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (b^2 + c^2) (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(21702) lies on these lines: {10, 82}, {594, 21037}, {20655, 21699}

X(21703) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (b^2 + c^2) (2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + a c^2 + c^3) : :

X(21703) lies on these lines: {10, 82}, {210, 21861}, {756, 16886}, {8013, 21681}, {20654, 21700}, {21685, 21709}

X(21704) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a^2 (a - b - c) (b + c)^2 (a b - b^2 + a c + 4 b c - c^2) : :

X(21704) lies on these lines: {10, 85}, {210, 20691}, {21031, 21039}

X(21705) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^3 (2 a^2 + 2 a b + b^2 + 2 a c + c^2) : :

X(21705) lies on these lines: {10, 86}, {594, 6543}, {1654, 21047}, {3932, 20653}, {4733, 20488}, {8013, 10026}, {20657, 21695}, {21673, 21693}

X(21706) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a b - a c + b c) (-a^2 b^2 - a b^3 + 2 a^2 b c + 2 a b^2 c + b^3 c - a^2 c^2 + 2 a b c^2 - a c^3 + b c^3) : :

X(21706) lies on these lines: {10, 87}, {3701, 3773}, {20654, 21040}, {20657, 20660}, {20658, 21675}

X(21707) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (2 a - b - c) (b + c)^2 (a^2 b + 2 a b^2 + b^3 + a^2 c - 4 a b c - 2 b^2 c + 2 a c^2 - 2 b c^2 + c^3) : :

X(21707) lies on these lines: {10, 88}, {8013, 21043}, {20653, 21041}, {20654, 21708}, {21701, 21723}

X(21708) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - 2 b - 2 c) (b + c)^2 (2 a^2 b + 4 a b^2 + 2 b^3 + 2 a^2 c - 2 a b c - b^2 c + 4 a c^2 - b c^2 + 2 c^3) : :

X(21708) lies on these lines: {10, 89}, {8013, 21689}, {20653, 21042}, {20654, 21707}, {21696, 21943}

X(21709) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c)^3 (-a^2 + a b + b^2 + a c + b c + c^2) : :

X(21709) lies on these lines: {10, 99}, {115, 2643}, {244, 21054}, {8013, 21711}, {20661, 21718}, {21670, 21700}, {21681, 21716}, {21685, 21703}

X(21710) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c)^3 (-a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c - a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4) : :

X(21710) lies on these lines: {10, 110}, {11, 5952}, {8901, 21046}, {20656, 21701}

X(21711) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^3 - a^2 b - a b^2 + b^3 - a^2 c - a c^2 + c^3) : :

X(21711) lies on these lines: {9, 80}, {10, 115}, {37, 16613}, {100, 21004}, {148, 190}, {210, 22280}, {1213, 3125}, {2214, 2273}, {2295, 5276}, {3690, 5509}, {3695, 4426}, {4120, 21013}, {8013, 21709}, {9293, 9396}, {10026, 21839}, {16086, 17735}, {21024, 21677}, {21671, 21681}, {21672, 21685}, {21676, 21832}

X(21712) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-2 a^2 + a b + 3 b^2 + a c - 2 b c + 3 c^2) : :

X(21712) lies on these lines: {1, 2}, {12, 6535}, {3120, 3704}, {3943, 20654}, {4535, 20488}, {21054, 21693}

X(21713) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a b^2 + b^2 c + a c^2 + b c^2) : :

X(21713) lies on these lines: {10, 192}, {42, 3974}, {313, 561}, {321, 21688}, {594, 756}, {1089, 7237}, {2643, 22196}, {3701, 3773}, {3728, 21024}, {3932, 21674}, {4024, 22260}, {4039, 17280}, {4062, 17233}, {4733, 4918}, {20654, 21043}, {21033, 21673}, {21675, 21718}

X(21714) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^2 + a b + a c - 3 b c) : :

X(21714) lies on these lines: {10, 900}, {523, 1577}, {3120, 18004}, {3700, 21720}, {4010, 21676}, {4013, 6370}, {4526, 21858}, {4651, 4800}, {14288, 14430}, {14321, 21721}

X(21715) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 b^5 - 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 c^5) : :

X(21715) lies on these lines: {10, 187}, {20654, 21019}, {21024, 21682}, {21051, 21719}, {21057, 21729}

X(21716) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^2 b^3 - a^2 b^2 c - a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3) : :

X(21716) lies on these lines: {10, 194}, {3701, 3773}, {8013, 21685}, {21681, 21709}, {21696, 21718}

X(21717) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + 4 a^2 b c + a b^2 c - 2 b^3 c - a^2 c^2 + a b c^2 + 2 b^2 c^2 - a c^3 - 2 b c^3 + c^4) : :

X(21717) lies on these lines: {1, 2}, {12, 13853}, {37, 21911}, {38, 16608}, {65, 21671}, {427, 17451}, {2171, 21015}, {2294, 21678}, {3136, 21044}, {3914, 21931}, {3925, 21023}, {4854, 21919}, {20655, 21026}, {21049, 21925}, {21054, 21696}, {21687, 21921}

X(21718) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^2 b + a b^2 + b^3 - a^2 c - 2 a b c + a c^2 + c^3) : :

X(21718) lies on these lines: {1, 2}, {756, 16886}, {3005, 4705}, {3932, 20657}, {4156, 4422}, {20654, 21699}, {20661, 21709}, {20877, 20989}, {21043, 21689}, {21675, 21713}, {21696, 21716}

X(21719) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^3 b - a b^3 + a^3 c - a^2 b c - a b^2 c + b^3 c - a b c^2 + 2 b^2 c^2 - a c^3 + b c^3) : :

X(21719) lies on these lines: {10, 647}, {523, 21721}, {594, 3700}, {661, 21958}, {1021, 3679}, {3596, 21437}, {9404, 17275}, {21051, 21715}

X(21720) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^2 b + a b^2 + a^2 c - b^2 c + a c^2 - b c^2) : :

X(21720) lies on these lines: {10, 649}, {313, 20909}, {661, 20483}, {3005, 4705}, {3700, 21714}, {4024, 4036}, {21051, 21715}

X(21721) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^3 - a b^2 + a b c + b^2 c - a c^2 + b c^2) : :

X(21721) lies on these lines: {10, 650}, {306, 4885}, {313, 21438}, {424, 2501}, {523, 21719}, {594, 4024}, {661, 20483}, {2512, 4705}, {2533, 21724}, {3661, 18154}, {8013, 21727}, {8611, 21052}, {14321, 21714}

X(21722) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^2 b^2 + a b^3 - a^2 b c - b^3 c + a^2 c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21722) lies on these lines: {10, 659}, {313, 21439}, {523, 21680}, {661, 20483}, {850, 4036}, {3120, 18004}, {3837, 15523}, {4705, 21724}, {20482, 20658}

X(21723) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c)^3 (-a^3 - a^2 b + a b^2 + b^3 - a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3) : :

X(21723) lies on these lines: {10, 501}, {11, 5952}, {115, 2643}, {20661, 21689}, {21691, 21699}, {21701, 21707}

X(21724) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c)^2 (a^2 b^2 + a b^3 + a^2 b c - b^3 c + a^2 c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21724) lies on these lines: {10, 667}, {2533, 21721}, {4705, 21722}, {16886, 21056}, {20657, 21683}, {21051, 21715}

X(21725) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b^2 - c^2)^2 (a^2 + b c) : :

X(21725) lies on these lines: {2, 7170}, {10, 274}, {148, 3571}, {244, 21054}, {388, 9413}, {1356, 6784}, {2533, 7200}, {2643, 20975}, {3122, 21043}, {3125, 4705}, {4128, 16592}, {7148, 16886}, {20494, 20653}, {21041, 21699}, {21674, 21700}

X(21725) = barycentric product X(10)*X(16592)

X(21726) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(21726) lies on these lines: {10, 669}, {523, 21680}, {21051, 21715}, {21055, 21962}, {21262, 21350}

X(21727) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b - c) (b + c)^2 (a^2 - a b - a c - b c) : :

X(21727) lies on these lines: {10, 693}, {38, 21196}, {42, 650}, {512, 661}, {756, 4024}, {984, 17161}, {1635, 7234}, {2254, 4824}, {2533, 20507}, {4036, 4804}, {4490, 22320}, {4651, 17494}, {4893, 22314}, {8013, 21721}, {21124, 21125}

X(21728) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^3 (a^2 + a b + b^2 + a c - b c + c^2) : :

X(21728) lies on these lines: {10, 894}, {210, 8013}, {594, 6543}, {756, 21701}, {3701, 3773}, {4062, 17315}, {15523, 20932}, {17332, 21047}, {20654, 21699}, {21011, 22301}, {21023, 21054}

X(21729) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (-a^3 b - a^2 b^2 + 2 a b^3 + 2 b^4 - a^3 c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + 2 a c^3 + 2 c^4) : :

X(21729) lies on these lines: {10, 896}, {313, 20904}, {661, 20483}, {3994, 21043}, {4062, 21048}, {17757, 20653}, {20488, 21026}, {21057, 21715}

X(21730) = (A,B,C,X(10); A',B',C',X(10)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c)^2 (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2) : :

X(21730) lies on these lines: {10, 75}, {12, 594}, {599, 1469}, {1211, 3120}, {2274, 5793}, {3178, 17233}, {3925, 21688}, {3932, 21674}, {4039, 5263}, {4062, 5718}, {4417, 21085}, {17300, 17751}, {20491, 21685}, {20660, 21031}

X(21731) = ISOGONAL CONJUGATE OF X(18878)

Barycentrics a^2*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(b^2-c^2) : :
X(21731) = 3*X(351)-4*X(6140), 3*X(351)-X(9409), 3*X(351)-2*X(14270), X(3005)-3*X(15451), 4*X(6140)-X(9409), 3*X(15451)-2*X(18117)

See Tran Quang Hung and César Lozada, ADGEOM 4923.

X(21731) lies on these lines: {3, 6132}, {6, 2510}, {113, 131}, {115, 2971}, {187, 237}, {399, 526}, {523, 11799}, {684, 690}, {804, 6033}, {878, 14601}, {1576, 1625}, {2780, 8552}, {2872, 9142}, {2881, 11641}, {9517, 11615}, {9934, 15478}, {11060, 14398}

X(21731) = reflection of X(3) in X(6132)
X(21731) = isogonal conjugate of X(18878)
X(21731) = anticomplement of complementary conjugate of X(39021)
X(21731) = Gibert[circumtangential conjugate of X(10420)
X(21731) = X(14687)-of-1st Parry-triangle
X(21731) = X(15919)-of-2nd Parry-triangle

X(21732) = X(3)X(523)∩X(98)X(111)

Barycentrics (3*a^10-4*(b^2+c^2)*a^8-(2*b^4-7*b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^4-(b^8+c^8-b^2*c^2*(7*b^4-16*b^2*c^2+7*c^4))*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2)*(b^2-c^2) : :
X(21732) = 3*X(3545)-2*X(18309)

See Tran Quang Hung and César Lozada, ADGEOM 4925.

X(21732) lies on these lines: {2, 9137}, {3, 523}, {4, 2489}, {6, 1499}, {30, 9178}, {98, 111}, {182, 5652}, {512, 11179}, {542, 5653}, {690, 11579}, {691, 2407}, {804, 11632}, {1352, 11182}, {2492, 19912}, {3545, 18309}, {5996, 9744}, {6088, 14666}, {7709, 8704}, {8371, 15928}, {9171, 20423}, {12106, 21006}

X(21732) = reflection of X(i) in X(j) for these (i,j): (2, 9175), (1352, 11182)
X(21732) = antipode of X(2) in circle {{X(2),X(3),X(6),X(111),X(691)}}

X(21733) = X(3)X(512)∩X(74)X(111)

Barycentrics a^2*(a^8+3*(b^2+c^2)*a^6-(11*b^4+5*b^2*c^2+11*c^4)*a^4+(b^2+c^2)*(9*b^4-2*b^2*c^2+9*c^4)*a^2-2*b^8+3*b^6*c^2-14*b^4*c^4+3*b^2*c^6-2*c^8)*(b^2-c^2) : :
X(21733) = 3*X(5054)-2*X(11183), 3*X(10516)-2*X(18309)

See Tran Quang Hung and César Lozada, ADGEOM 4925.

X(21733) lies on these lines: {2, 1499}, {3, 512}, {6, 9175}, {74, 111}, {381, 11182}, {511, 9178}, {525, 16220}, {526, 11579}, {549, 5652}, {690, 11632}, {691, 2421}, {804, 19905}, {924, 10249}, {1351, 9171}, {5054, 11183}, {5653, 5663}, {6785, 14700}, {9126, 9135}, {9137, 15066}, {10516, 18309}

X(21733) = reflection of X(i) in X(j) for these (i,j): (6, 9175), (381, 11182), (1351, 9171)
X(21733) = antipode of X(6) in circle {{X(2),X(3),X(6),X(111),X(691)}}

X(21734) = EULER LINE INTERCEPT OF X(145)X(165)

Barycentrics 7*S^2-8*SB*SC : :
X(21734) = 3*X(2)-16*X(3), 21*X(2)-8*X(4), 9*X(2)+4*X(20), 5*X(2)+8*X(376), 29*X(2)-16*X(381), 33*X(2)-20*X(3091), 15*X(2)-2*X(3146), 3*X(2)+10*X(3522), 17*X(2)-4*X(3543), 27*X(2)-14*X(3832), 25*X(2)-12*X(3839), 30*X(2)-17*X(3854), 12*X(2)+X(5059), 9*X(2)-8*X(5067), 21*X(2)-16*X(5079), 3*X(2)-8*X(10299)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28107.

X(21734) lies on these lines: {2, 3}, {6, 9543}, {8, 16192}, {40, 3623}, {144, 4855}, {145, 165}, {146, 15036}, {390, 5204}, {944, 20052}, {1587, 9680}, {1620, 11433}, {3060, 17704}, {3085, 4325}, {3086, 4330}, {3219, 9841}, {3311, 9693}, {3576, 5734}, {3600, 5217}, {3601, 21454}, {3616, 9589}, {3617, 4297}, {3621, 5731}, {3622, 4301}, {3819, 12279}, {3869, 10178}, {4309, 7280}, {4317, 5010}, {4652, 20007}, {4678, 5881}, {5013, 14930}, {5206, 5319}, {5210, 7738}, {5218, 9657}, {5253, 11495}, {5261, 15326}, {5274, 15338}, {5281, 15888}, {5286, 15513}, {5303, 17784}, {5343, 16242}, {5344, 16241}, {5642, 15023}, {5650, 11439}, {5882, 20049}, {5984, 21166}, {6194, 20105}, {6200, 9692}, {6337, 10513}, {6409, 7585}, {6411, 6460}, {6412, 6459}, {6455, 7581}, {6456, 7582}, {7288, 9670}, {7765, 8588}, {7871, 14907}, {7999, 14855}, {8596, 11623}, {9143, 15021}, {9706, 10984}, {10248, 19862}, {10574, 13348}, {11194, 12632}, {11431, 11438}, {11592, 18435}, {12307, 20585}, {14683, 15055}, {14927, 21167}, {15051, 15063}, {15057, 16163}, {15644, 20791}, {16836, 16981}

X(21734) = anticomplement of X(5068)
X(21734) = X(5079)-of-anti-Euler-triangle
X(21734) = X(10299)-of-ABC-X3-reflections-triangle
X(21734) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3146, 3854), (3, 5073, 15706), (20, 3543, 17800), (140, 3543, 15022), (382, 631, 7486), (382, 3859, 4), (382, 5070, 3859), (382, 7486, 3832), (2041, 2042, 3545), (3146, 3522, 376), (3146, 3523, 2), (3146, 15705, 3523), (3627, 15693, 3533), (5073, 14869, 5071), (5073, 15706, 14869), (11541, 15709, 3851)

X(21735) = EULER LINE INTERCEPT OF X(32)X(14482)

Barycentrics 6*S^2-7*SB*SC : :
X(21735) = 3*X(2)-14*X(3), 18*X(2)-7*X(4), 4*X(2)+7*X(376), 25*X(2)-14*X(381), 3*X(2)+8*X(548), 9*X(2)+2*X(1657), 6*X(2)-7*X(3525), 15*X(2)-4*X(3627), 21*X(2)-10*X(3843), 27*X(2)-16*X(3850), 12*X(2)-7*X(3855), 9*X(2)-7*X(5056), 15*X(2)-14*X(5070), 9*X(154)+2*X(15105), 9*X(165)+2*X(5882), 12*X(165)-X(12245), 3*X(3633)-14*X(5882), 4*X(3633)+7*X(12245), 8*X(5882)+3*X(12245)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28107.

X(21735) lies on these lines: {2, 3}, {32, 14482}, {154, 15105}, {165, 3633}, {577, 5702}, {944, 3625}, {962, 17502}, {1056, 5217}, {1058, 5204}, {1285, 5013}, {1587, 6411}, {1588, 6412}, {3098, 14912}, {3462, 15005}, {3519, 20421}, {3567, 17704}, {3576, 5493}, {3579, 7967}, {3592, 9693}, {3630, 6776}, {3819, 12290}, {4297, 4691}, {4299, 8164}, {4313, 5122}, {4316, 10588}, {4324, 10589}, {4668, 5657}, {4857, 7288}, {5082, 5303}, {5206, 5355}, {5210, 5286}, {5218, 5270}, {5254, 5585}, {5447, 15072}, {5603, 12512}, {5640, 12002}, {5731, 20053}, {5890, 13348}, {6053, 10990}, {6144, 8550}, {6200, 7581}, {6225, 11202}, {6337, 7768}, {6361, 7987}, {6396, 7582}, {6410, 9541}, {6417, 9543}, {6455, 7585}, {6456, 7586}, {7608, 18844}, {7691, 13431}, {7735, 15513}, {7736, 15515}, {7755, 8588}, {7781, 8182}, {7883, 11147}, {7917, 14907}, {7998, 10575}, {9542, 19117}, {9681, 19053}, {9778, 10595}, {10619, 12325}, {10625, 20791}, {10991, 21166}, {11204, 12324}, {11412, 13382}, {11444, 14855}, {11623, 13172}, {12244, 15051}, {12250, 17821}, {12317, 15055}, {12383, 20417}, {13199, 20418}, {13347, 15033}, {14641, 15056}, {14862, 20427}, {14927, 18553}, {18925, 21663}

X(21735) = reflection of X(2) in X(15718)
X(21735) = anticomplement of X(5072)
X(21735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 12100), (20, 3146, 15685), (20, 15682, 3529), (140, 5073, 5068), (382, 3854, 4), (382, 10303, 5071), (382, 12100, 10303), (549, 3146, 5067), (550, 5073, 20), (631, 15682, 3090), (3146, 3851, 4), (3524, 5073, 3533), (3526, 3543, 3544), (3543, 3858, 4), (5068, 5073, 4), (15684, 15718, 15723)

X(21736) = EULER LINE INTERCEPT OF X(6)X(12256)

Barycentrics S^3-SB*SC*(S- SW) : :
X(21736) = 2*S*X(3) + SW*X(4)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28107.

X(21736) lies on these lines: {2, 3}, {6, 12256}, {32, 1587}, {39, 1588}, {69, 9732}, {141, 12306}, {371, 6776}, {372, 14853}, {485, 13921}, {487, 1352}, {488, 511}, {590, 13749}, {637, 3926}, {638, 3785}, {642, 10514}, {1151, 1503}, {1152, 5480}, {3053, 3070}, {3071, 5013}, {3311, 14912}, {3424, 10851}, {3564, 6462}, {3592, 8550}, {5409, 14826}, {5870, 8721}, {5871, 9540}, {6202, 13935}, {6214, 12509}, {6278, 13710}, {6289, 6337}, {6431, 12007}, {7582, 9605}, {8213, 8420}, {8408, 9838}, {9748, 10852}, {9892, 14981}, {10132, 11206}, {10133, 11427}, {10515, 12123}, {10519, 11824}, {12975, 19130}, {13674, 13846}

X(21736) = reflection of X(13674) in X(13846)
X(21736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5, 11291), (4, 631, 6813), (20, 7374, 4), (1352, 9738, 487)

X(21737) = EULER LINE INTERCEPT OF X(76)X(487)

Barycentrics SW*S^2+SB*SC*(S-SW) : :
X(21737) = 2*SW*X(3) + S*X(4)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28107.

X(21737) lies on these lines: {2, 3}, {76, 487}, {141, 13749}, {182, 1588}, {315, 488}, {371, 5286}, {393, 11513}, {485, 11824}, {489, 12257}, {492, 12256}, {511, 1587}, {590, 12306}, {637, 6776}, {638, 10519}, {1151, 5254}, {1152, 7745}, {1161, 7583}, {1350, 3070}, {1351, 7581}, {1352, 5871}, {1578, 2207}, {3068, 9732}, {3071, 5085}, {3087, 11514}, {3564, 10783}, {5050, 7582}, {5591, 6290}, {6200, 12123}, {6202, 14561}, {6215, 18840}, {6460, 9733}, {6560, 11825}, {7584, 13949}, {9540, 9738}, {11916, 19117}, {12017, 12601}, {14242, 18440}

X(21737) = reflection of X(11916) in X(19117)
X(21737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 631, 7388), (4, 7375, 5), (3539, 19219, 5020), (14782, 14783, 11314)

X(21738) = EULER LINE INTERCEPT OF X(3070)X(5398)

Barycentrics (R+r)*S^2-(R+r-2*s)*SB*SC : :
X(21738) = (R+r)*SW*X(3) + s*X(4)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28107.

X(21738) lies on these lines: {2, 3}, {3070, 5398}, {3071, 5396}

X(21739) = ISOGONAL CONJUGATE OF X(19297)

Barycentrics (a^3+(b-c)*a^2-(b^2-b*c+c^2)*a-(b+c)*(b^2-c^2))*(a^3-(b-c)*a^2-(b^2-b*c+c^2)*a+(b+c)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28109.

X(21739) lies on these lines: {8, 191}, {29, 2906}, {92, 17483}, {190, 3578}, {312, 2895}, {323, 7359}, {333, 19302}, {3936, 4997}, {5080, 14452}, {14206, 17484}

X(21739) = isogonal conjugate of X(19297)
X(21739) = isotomic conjugate of X(17484)
X(21739) = trilinear pole of the line {522, 1125}
X(21739) = X(19)-isoconjugate of X(23071)

X(21740) = REFLECTION OF X(4) IN X(12047)

Barycentrics a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+3 a^4 b c-2 a^3 b^2 c-2 a^2 b^3 c+4 a b^4 c-b^5 c-a^4 c^2-2 a^3 b c^2+6 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+4 a^3 c^3-2 a^2 b c^3-2 a b^2 c^3+2 b^3 c^3-a^2 c^4+4 a b c^4-b^2 c^4-2 a c^5-b c^5+c^6) : :
X(21740) = 3 X[3576] - 2 X[5267] = 2 (r + R) X[1] - R X[4]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28110.

X(21740) lies on these lines: {1,4}, {3,3417}, {8,6825}, {10,6326}, {20,18664}, {21,104}, {35,2800}, {36,5884}, {40,6876}, {46,6942}, {65,6905}, {78,5657}, {84,13384}, {145,6838}, {214,18861}, {224,6916}, {355,2476}, {376,12520}, {411,517}, {519,11014}, {551,12617}, {631,997}, {758,11012}, {912,2975}, {938,18467}, {952,4861}, {960,1006}, {962,6869}, {971,8543}, {993,5693}, {1012,9960}, {1125,6852}, {1158,3612}, {1319,1858}, {1389,17097}, {1482,6985}, {1537,15171}, {1737,6949}, {1788,6880}, {1837,6941}, {1898,12740}, {2096,10884}, {2099,11500}, {2646,6001}, {3057,10698}, {3219,5694}, {3359,4855}, {3428,12635}, {3560,10246}, {3576,5267}, {3601,7971}, {3616,5768}, {3622,6837}, {3651,14110}, {3655,11114}, {3811,12245}, {3812,6946}, {3868,11249}, {3872,5534}, {3873,10680}, {3877,10267}, {3878,10902}, {3890,16202}, {3897,12528}, {3957,10222}, {4084,5535}, {4295,6934}, {4297,16116}, {4305,6938}, {4420,5690}, {5057,7491}, {5176,10942}, {5226,10599}, {5251,12691}, {5253,10202}, {5396,5797}, {5450,15071}, {5536,16126}, {5563,12005}, {5587,6874}, {5720,5818}, {5731,6868}, {5794,6937}, {5840,11015}, {5886,6828}, {5901,6841}, {5903,6796}, {6830,11375}, {6834,18391}, {6857,18443}, {6873,8227}, {6901,12609}, {6902,21616}, {6912,15178}, {6951,17647}, {6952,12616}, {6977,14647}, {7191,8229}, {7548,18480}, {7704,9669}, {9940,17614}, {10129,18525}, {10950,18242}, {11571,14792}, {12526,21165}, {12559,12704}, {12688,21669}, {13145,17654}, {14497,17098}

X(21740) = reflection of X(i) in X(j) for these {i,j}: {4, 12047}, {5086, 6842}, {6906, 2646}
X(21740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6261, 4), (1, 10571, 1870), (1, 18446, 944), (4, 7967, 3486), (944, 5603, 12116), (1071, 1385, 104), (1158, 3612, 6950), (1385, 5887, 21), (3576, 12514, 6875), (3616, 5768, 10785), (5720, 19860, 5818), (5731, 11415, 6868), (5882, 12608, 10572), (10572, 12608, 4)

X(21741) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (a + b - c) (a - b + c) (b + c) (a^2 - b^2 - b c - c^2) : :

X(21741) lies on these lines: {6, 1411}, {50, 1399}, {201, 2911}, {213, 1042}, {323, 1442}, {604, 5035}, {651, 4032}, {661, 2623}, {1055, 14597}, {1100, 20616}, {1334, 3990}, {1404, 2300}, {2149, 2150}, {2268, 22134}, {2288, 2347}, {2594, 21794}, {2908, 9247}, {4318, 15556}, {4552, 17350}, {20964, 21756}

X(21741) = isogonal conjugate of polar conjugate of X(1825)
X(21741) = isogonal conjugate of isotomic conjugate of X(16577)
X(21741) = X(92)-isoconjugate of X(1789)
X(21741) = polar conjugate of isotomic conjugate of X(22342)
X(21741) = {X(21742),X(21743)}-harmonic conjugate of X(6)

X(21742) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - a c^3 - b c^3 + c^4) : :

X(21742) lies on these lines: {6, 1411}, {41, 603}, {213, 1015}, {218, 3157}, {651, 9317}, {665, 21758}, {672, 3002}, {1409, 2347}, {2427, 17439}, {4574, 14439}, {15558, 17015}

X(21742) = isogonal conjugate of isotomic conjugate of X(16578)
X(21742) = isogonal conjugate of polar conjugate of X(1830)
X(21742) = polar conjugate of isotomic conjugate of X(22346)
X(21742) = {X(6),X(21741)}-harmonic conjugate of X(21743)

X(21743) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (a - b - c) (a^3 b + a^2 b^2 - a b^3 - b^4 + a^3 c + 2 a^2 b c + 2 a b^2 c + b^3 c + a^2 c^2 + 2 a b c^2 - a c^3 + b c^3 - c^4) : :

X(21743) lies on these lines: {6, 1411}, {41, 2220}, {213, 2347}, {1193, 1195}, {1400, 2288}, {1409, 1475}, {2280, 22134}, {2300, 21748}

X(21743) = isogonal conjugate of isotomic conjugate of X(16579)
X(21743) = isogonal conjugate of polar conjugate of X(1831)
X(21743) = polar conjugate of isotomic conjugate of X(22347)
X(21743) = {X(6),X(21741)}-harmonic conjugate of X(21742)

X(21744) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b + c) (a^4 - b^4 + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21744) lies on these lines: {6, 2172}, {42, 20969}, {71, 3954}, {72, 2245}, {213, 1042}, {2178, 22131}, {2275, 15654}, {2333, 3125}, {3721, 4456}, {8578, 20979}

X(21744) = isogonal conjugate of isotomic conjugate of X(16580)
X(21744) = polar conjugate of isotomic conjugate of X(22348)

X(21745) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b + c) (a^4 - b^4 - a^2 b c + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21745) lies on these lines: {6, 21772}, {213, 1042}, {2245, 21839}, {2503, 16611}, {5341, 11552}, {8632, 21763}

X(21745) = isogonal conjugate of isotomic conjugate of X(16581)
X(21745) = polar conjugate of isotomic conjugate of X(22349)

X(21746) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : :

X(21746) lies on these lines: {1, 256}, {6, 692}, {7, 2481}, {9, 3779}, {37, 674}, {39, 2309}, {42, 51}, {43, 5943}, {44, 22277}, {45, 7064}, {55, 181}, {56, 991}, {57, 1742}, {65, 516}, {75, 6007}, {76, 21299}, {184, 20959}, {192, 14839}, {209, 3683}, {238, 4260}, {354, 1122}, {373, 899}, {375, 4849}, {405, 10822}, {497, 10446}, {513, 13476}, {517, 11997}, {518, 3883}, {538, 17157}, {572, 17798}, {579, 20992}, {584, 4471}, {869, 21796}, {942, 15310}, {959, 4313}, {984, 9052}, {986, 12109}, {1001, 4259}, {1002, 9309}, {1015, 4116}, {1043, 9565}, {1100, 9018}, {1185, 21813}, {1196, 1197}, {1253, 1405}, {1357, 4860}, {1361, 2099}, {1362, 6180}, {1397, 10833}, {1400, 2223}, {1402, 14547}, {1425, 4332}, {1463, 5542}, {1475, 20978}, {1500, 4787}, {1573, 2388}, {1631, 2278}, {1682, 19765}, {1740, 17065}, {1843, 2354}, {1918, 4274}, {1953, 4516}, {1964, 3122}, {2092, 3764}, {2171, 2310}, {2194, 20988}, {2245, 8053}, {2269, 4343}, {2393, 18611}, {2654, 10474}, {2792, 18389}, {2808, 7201}, {2886, 18165}, {3009, 22172}, {3060, 17018}, {3125, 17872}, {3240, 5640}, {3270, 4336}, {3322, 15615}, {3340, 4907}, {3663, 20358}, {3675, 21346}, {3681, 17331}, {3707, 22312}, {3720, 3917}, {3731, 4517}, {3792, 16484}, {3873, 17364}, {3874, 17770}, {3879, 9025}, {3888, 17300}, {3912, 17792}, {3963, 21278}, {4038, 7186}, {4111, 17275}, {4268, 4497}, {4388, 5208}, {4484, 5069}, {4553, 17243}, {5052, 21760}, {5135, 20872}, {5138, 7295}, {5173, 20122}, {5248, 10974}, {5364, 21795}, {5369, 21808}, {5575, 7248}, {6467, 20963}, {6688, 16569}, {7175, 14520}, {9024, 17390}, {9047, 15569}, {9054, 17332}, {17330, 22271}, {17340, 21865}, {17369, 22279}, {17447, 18726}, {17451, 21804}, {20462, 20667}

X(21746) = isogonal conjugate of isotomic conjugate of X(2886)
X(21746) = crosssum of X(2) and X(55)
X(21746) = crosspoint of X(6) and X(7)
X(21746) = pole of Lemoine axis wrt incircle
X(21746) = polar conjugate of isotomic conjugate of X(22070)
X(21746) = X(264)-of-intouch triangle
X(21746) = pedal isotomic conjugate of X(1)
X(21746) = crossdifference of every pair of points on line X(918)X(3287)

X(21747) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Trilinears 2(1 - cos 2A) + (1 + cos A)(cos B + cos C) : :
Barycentrics a^2 (4 a + b + c) : :

X(21747) lies on these lines: {6, 31}, {58, 106}, {63, 16491}, {81, 16484}, {100, 16477}, {171, 9342}, {172, 21781}, {213, 1017}, {595, 16474}, {612, 3973}, {678, 21870}, {756, 15492}, {896, 1386}, {899, 14997}, {940, 8692}, {995, 2163}, {1193, 4257}, {1203, 4256}, {1468, 7373}, {1707, 17017}, {2054, 2502}, {2223, 5008}, {2384, 2702}, {3009, 9463}, {3230, 20985}, {3246, 17450}, {3683, 3723}, {3689, 16671}, {3720, 14996}, {3722, 4663}, {3731, 5311}, {3744, 4722}, {3745, 16814}, {3747, 16971}, {3752, 9340}, {3791, 4365}, {3868, 16498}, {3989, 7262}, {4414, 16475}, {4418, 17117}, {4641, 17469}, {7109, 21757}, {16666, 21806}, {16669, 21805}

X(21747) = isogonal conjugate of isotomic conjugate of X(551)
X(21747) = polar conjugate of isotomic conjugate of X(22357)

X(21748) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a - b - c) (2 a^2 + a b - b^2 + a c + 2 b c - c^2) : :

X(21748) lies on these lines: {6, 41}, {9, 2320}, {42, 184}, {60, 283}, {71, 2278}, {219, 1334}, {572, 672}, {663, 18000}, {909, 2259}, {1100, 2170}, {1172, 2202}, {1174, 3451}, {1200, 2280}, {1397, 10460}, {1438, 13404}, {1449, 2082}, {1818, 5135}, {2092, 7117}, {2171, 2182}, {2175, 2293}, {2194, 14547}, {2245, 22054}, {2261, 3553}, {2267, 2911}, {2273, 5114}, {2287, 3691}, {2300, 21743}, {2308, 3724}, {2330, 2340}, {2646, 21811}, {3051, 9449}, {3686, 21014}, {3713, 4390}, {5042, 14827}, {6603, 21809}, {8609, 17438}, {9310, 20818}, {10980, 16667}, {15988, 20769}, {20665, 21764}

X(21748) = isogonal conjugate of isotomic conjugate of X(5745)
X(21748) = polar conjugate of isotomic conjugate of X(22361)

X(21749) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a^4 - b^4 - c^4) (a^4 - b^4 + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21749) lies on these lines: {6, 2156}

X(21749) = isogonal conjugate of isotomic conjugate of X(16582)
X(21749) = polar conjugate of isotomic conjugate of X(22362)

X(21750) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a^2 + b^2 - 2 b c + c^2) : :

X(21750) lies on these lines: {2, 16782}, {6, 63}, {42, 213}, {228, 16584}, {306, 2238}, {614, 1184}, {1185, 2280}, {1402, 2205}, {1409, 21751}, {2176, 3870}, {2209, 5364}, {3051, 9449}, {3230, 3938}, {4666, 16781}, {5287, 16524}, {17017, 20963}

X(21750) = isogonal conjugate of isotomic conjugate of X(16583)
X(21750) = polar conjugate of isotomic conjugate of X(22363)

X(21751) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^5 (b + c) (b^2 - b c + c^2) : :

X(21751) lies on these lines: {6, 75}, {42, 21752}, {213, 6378}, {313, 2231}, {560, 9233}, {723, 825}, {1400, 21755}, {1409, 21750}, {1922, 1967}, {2210, 9288}, {2300, 3051}, {3116, 7032}, {20228, 21762}

X(21751) = isogonal conjugate of isotomic conjugate of X(16584)
X(21751) = polar conjugate of isotomic conjugate of X(22364)

X(21752) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a^2 + b c) (b^2 + c^2) : :

X(21752) lies on these lines: {6, 82}, {42, 21751}, {213, 3778}, {594, 2235}, {672, 2300}, {688, 798}, {732, 894}, {813, 1178}, {869, 9288}, {1500, 2309}, {1933, 7122}, {2231, 18082}, {2236, 16587}, {3051, 3688}, {7032, 19587}, {17049, 20965}, {20964, 21755}

X(21752) = isogonal conjugate of isotomic conjugate of X(16587)
X(21752) = polar conjugate of isotomic conjugate of X(22367)

X(21753) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a b + a c + 2 b c) : :

X(21753) lies on these lines: {2, 6}, {42, 213}, {43, 2229}, {218, 4199}, {238, 1206}, {310, 17499}, {672, 2092}, {798, 8027}, {872, 21814}, {1011, 2271}, {1171, 18268}, {1197, 1977}, {1203, 3783}, {1501, 5320}, {2176, 17018}, {2194, 14567}, {2240, 17745}, {2245, 20973}, {2276, 4272}, {2295, 4651}, {2347, 20229}, {2663, 9401}, {2667, 21820}, {3121, 3725}, {3124, 20455}, {3691, 3720}, {3780, 17135}, {3948, 17027}, {3997, 4685}, {4184, 18755}, {4191, 5021}, {4260, 20859}, {4754, 16748}, {16514, 17011}, {17034, 18152}, {20972, 21755}, {21858, 21877}

X(21753) = isogonal conjugate of isotomic conjugate of X(16589)
X(21753) = polar conjugate of isotomic conjugate of X(22369)

X(21754) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a - 2 b - 2 c) (4 a + b + c) : :

X(21754) lies on these lines: {6, 36}, {42, 1017}, {213, 902}, {8649, 16971}, {16490, 21781}

X(21754) = isogonal conjugate of isotomic conjugate of X(16590)
X(21754) = polar conjugate of isotomic conjugate of X(22372)

X(21755) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a^2 + b c) (b - c)^2 : :

X(21755) lies on these lines: {6, 662}, {213, 21756}, {798, 1084}, {1015, 9427}, {1213, 20343}, {1400, 21751}, {1977, 3124}, {2086, 16592}, {2092, 20467}, {2309, 2670}, {3051, 4274}, {3778, 21759}, {4128, 21823}, {20666, 21760}, {20964, 21752}, {20972, 21753}

X(21755) = isogonal conjugate of isotomic conjugate of X(16592)
X(21755) = polar conjugate of isotomic conjugate of X(22373)
X(21755) = crosssum of X(2) and X(799)

X(21756) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a^4 - 2 a^2 b^2 + b^4 + 2 a^2 b c - b^3 c - 2 a^2 c^2 + b^2 c^2 - b c^3 + c^4) : :

X(21756) lies on these lines: {6, 2643}, {213, 21755}, {512, 20958}, {1409, 3778}, {3013, 21043}, {3271, 5202}, {4585, 21254}, {20964, 21741}

X(21756) = isogonal conjugate of isotomic conjugate of X(16598)
X(21756) = polar conjugate of isotomic conjugate of X(22375)

X(21757) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (a b^2 - 2 a b c - b^2 c + a c^2 - b c^2) : :

X(21757) lies on these lines: {2, 20669}, {6, 43}, {31, 19587}, {42, 1977}, {81, 799}, {194, 712}, {213, 2308}, {1613, 16468}, {2235, 3791}, {3230, 17127}, {3248, 16584}, {3271, 22200}, {7109, 21747}, {7262, 16525}, {16477, 21779}, {20228, 21764}, {20229, 20959}

X(21757) = isogonal conjugate of isotomic conjugate of X(16604)
X(21757) = polar conjugate of isotomic conjugate of X(22378)

X(21758) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b - c) (a^2 - b^2 + b c - c^2) : :

X(21758) lies on these lines: {6, 1635}, {31, 8645}, {81, 812}, {649, 854}, {654, 17455}, {665, 21742}, {940, 4728}, {1015, 1977}, {1922, 3572}, {2610, 3738}, {3960, 17191}, {4435, 4984}, {4840, 4979}, {4893, 20980}, {4988, 21106}, {5040, 6373}, {8027, 8650}, {14996, 21297}

X(21758) = isogonal conjugate of X(36804)
X(21758) = crossdifference of every pair of points on line X(8)X(80)
X(21758) = crosssum of X(i) and X(j) for these {i,j}: {2, 3762}, {3239, 6735}, {4033, 24004}, {4358, 4391}
X(21758) = polar conjugate of isotomic conjugate of X(22379)
X(21758) = crosspoint of X(i) and X(j) for these {i,j}: {6, 32665}, {1415, 9456}

X(21759) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (b + c) (a b - a c - b c) (a b - a c + b c) : :

X(21759) lies on these lines: {1, 3224}, {6, 43}, {32, 2209}, {86, 4598}, {172, 7104}, {213, 6378}, {284, 18268}, {330, 20963}, {729, 932}, {894, 3225}, {981, 7155}, {1911, 9468}, {1918, 8022}, {1922, 2330}, {1974, 18262}, {3114, 14621}, {3494, 5280}, {3778, 21755}, {17349, 20669}

X(21759) = isogonal conjugate of X(31008)
X(21759) = polar conjugate of isotomic conjugate of X(22381)

X(21760) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^3 (a b^2 - b^2 c + a c^2 - b c^2) : :

X(21760) lies on these lines: {1, 6}, {2, 20549}, {31, 19587}, {32, 2209}, {42, 1197}, {43, 1613}, {51, 22200}, {87, 3501}, {172, 4279}, {511, 20861}, {519, 20501}, {667, 788}, {692, 14599}, {726, 17475}, {727, 813}, {740, 2235}, {899, 3231}, {902, 1977}, {1015, 20459}, {1016, 5383}, {1326, 18268}, {1500, 2309}, {1575, 18792}, {1692, 20958}, {1911, 2223}, {1918, 1923}, {1964, 3774}, {2162, 3550}, {2211, 2356}, {2225, 3121}, {2276, 5145}, {2308, 7109}, {2670, 20970}, {3009, 20663}, {3124, 20962}, {3240, 9463}, {3503, 7175}, {3720, 20965}, {5052, 21746}, {16569, 21001}, {17379, 17750}, {18278, 18794}, {20467, 20750}, {20666, 21755}, {21345, 21369}

X(21760) = isogonal conjugate of X(32020)
X(21760) = complement of X(20561)
X(21760) = anticomplement of X(20549)
X(21760) = crosssum of X(2) and X(350)
X(21760) = polar conjugate of isotomic conjugate of X(20777)
X(21760) = crosspoint of X(6) and X(1911)

X(21761) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 + 2 b^2 c^2 + b c^3) : :

X(21761) lies on these lines: {6, 810}, {31, 4041}, {58, 14838}, {333, 21259}, {405, 17478}, {667, 6373}, {1577, 1724}, {3907, 5247}, {8636, 20983}

X(21761) = isogonal conjugate of isotomic conjugate of X(8062)
X(21761) = polar conjugate of isotomic conjugate of X(22382)

X(21762) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^4 (b - c)^2 (a b + a c - b c) : :

X(21762) lies on these lines: {1, 9490}, {6, 190}, {11, 2086}, {330, 3224}, {890, 1977}, {1015, 9427}, {3051, 5332}, {8789, 12835}, {16666, 17459}, {18194, 19606}, {20228, 21751}

X(21762) = isogonal conjugate of isotomic conjugate of X(6377)
X(21762) = polar conjugate of isotomic conjugate of X(22386)
X(21762) = SS(a → bc) of X(11) (trilinear substitution)

X(21763) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^2 b^2 + a^2 b c + a^2 c^2 - b^2 c^2) : :

X(21763) lies on these lines: {6, 1924}, {649, 6372}, {667, 6373}, {798, 1019}, {3768, 4040}, {8630, 20868}, {8632, 21745}

X(21763) = polar conjugate of isotomic conjugate of X(22387)
X(21763) = crosssum of X(2) and X(798)
X(21763) = crosspoint of X(6) and X(799)

X(21764) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (2 a^2 + a b + b^2 + a c + c^2) : :

X(21764) lies on these lines: {1, 5282}, {6, 31}, {9, 3920}, {32, 1193}, {37, 17469}, {38, 1100}, {41, 16466}, {58, 163}, {63, 1449}, {81, 16503}, {101, 5315}, {172, 1201}, {213, 5007}, {222, 604}, {238, 5276}, {251, 1400}, {386, 7031}, {595, 1334}, {609, 995}, {748, 5275}, {896, 16666}, {899, 4386}, {1104, 17451}, {1149, 2242}, {1191, 9310}, {1203, 4251}, {1333, 17187}, {1386, 21840}, {1415, 1450}, {1453, 2082}, {1468, 16502}, {1707, 16667}, {1724, 3691}, {1743, 3961}, {1755, 2317}, {2176, 7296}, {2183, 4290}, {2251, 5008}, {2260, 4275}, {2262, 2312}, {2300, 3051}, {3496, 5262}, {3509, 7191}, {3550, 17756}, {3686, 5294}, {3744, 3930}, {3783, 16468}, {3868, 16787}, {3935, 16670}, {4262, 5313}, {4426, 10459}, {4663, 4712}, {4721, 7805}, {4766, 7792}, {5037, 5364}, {5156, 20459}, {7745, 21935}, {7787, 17033}, {8298, 16477}, {14621, 16998}, {16669, 20693}, {16782, 20985}, {16818, 17200}, {17126, 17754}, {20045, 21101}, {20228, 21757}, {20665, 21748}

X(21764) = isogonal conjugate of polar conjugate of X(1890)
X(21764) = isogonal conjugate of isotomic conjugate of X(17023)
X(21764) = polar conjugate of isotomic conjugate of X(22390)

X(21765) = X(30)X(5085)∩X(3260)X(15589)

Barycentrics (3 a^4+8 a^2 b^2+3 b^4-2 a^2 c^2-2 b^2 c^2-c^4) (3 a^4-2 a^2 b^2-b^4+8 a^2 c^2-2 b^2 c^2+3 c^4) : :

See Thanos Kalogerakis and Peter Moses, Hyacinthos 28113.

X(21765) lies on these lines: {30,5085}, {3260,15589}

X(21765) = isogonal conjugate of X(21766)

X(21766) = ISOGONAL CONJUGATE OF X(21765)

Barycentrics a^2 (a^4+2 a^2 b^2-3 b^4+2 a^2 c^2-8 b^2 c^2-3 c^4) : :
X(21766) = (27 + J^2) X[3] - 2 J^2 X[74], where (J = |OH|/R)
X(21766) = 9 R^2 X[2] + SW X[1350]

See Thanos Kalogerakis and Peter Moses, Hyacinthos 28113.

X(21766) lies on these lines: {2,1350}, {3,74}, {6,7496}, {20,16654}, {22,3819}, {141,16063}, {182,1993}, {183,4576}, {323,5085}, {376,11472}, {394,11003}, {567,7516}, {599,3448}, {631,14389}, {1351,2979}, {1995,3098}, {3060,16419}, {3124,20481}, {3292,17508}, {3580,10519}, {3763,5169}, {3830,14926}, {4563,7771}, {5013,9463}, {5054,15361}, {5189,10516}, {5359,8041}, {5447,7509}, {5544,5640}, {5645,10601}, {5891,8717}, {6030,8780}, {6636,17811}, {7592,13339}, {10170,12082}, {10299,12163}, {10303,17834}, {10605,15692}, {10691,11442}, {10989,21358}, {11064,21167}, {11258,12149}, {11284,15107}, {11422,12017}, {11477,15018}, {11820,15305}

X(21766) = isogonal conjugate of X(21765)
X(21766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6090, 15080), (3, 7998, 15066), (3, 7999, 11441), (3, 15066, 6800), (3, 15067, 11456), (1350, 5646, 3066), (2979, 7484, 5422), (3066, 5646, 2), (3098, 5650, 1995), (3819, 14810, 5651), (3917, 7485, 1993), (5651, 14810, 22), (15040, 15106, 110)

X(21767) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c + a^4 b c - a b^4 c - b^5 c - 2 a^3 c^3 + 2 b^3 c^3 - a b c^4 + a c^5 - b c^5) : :

X(21767) lies on these lines: {1, 14597}, {3, 828}, {6, 19}, {31, 10536}, {37, 6001}, {40, 3990}, {42, 11190}, {43, 10174}, {48, 1950}, {56, 21770}, {71, 7355}, {109, 577}, {154, 2352}, {198, 4559}, {216, 10571}, {219, 1761}, {222, 18161}, {281, 3330}, {573, 2818}, {1213, 20306}, {1498, 2256}, {1503, 15978}, {1609, 17966}, {1631, 21784}, {1745, 21854}, {1854, 16777}, {1945, 2164}, {2174, 20995}, {2176, 2178}, {2272, 22063}, {2286, 17849}, {4300, 6254}, {5301, 7113}, {15830, 18591}, {21775, 21776}

X(21768) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 - a b^3 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a c^5 - b c^5) : :

X(21768) lies on these lines: {6, 1411}, {37, 5887}, {184, 18612}, {1409, 8609}, {2176, 2178}, {2427, 3204}, {3230, 14597}, {7113, 16685}, {20990, 21784}

X(21768) = isogonal conjugate of isotomic conjugate of X(17479)
X(21768) = polar conjugate of isotomic conjugate of X(22457)

X(21769) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b + a b^2 + a^2 c - a b c - b^2 c + a c^2 - b c^2) : :

X(21769) lies on these lines: {1, 6}, {8, 992}, {48, 1613}, {55, 1964}, {56, 2305}, {71, 2275}, {101, 16946}, {198, 9435}, {239, 20923}, {284, 2241}, {314, 4361}, {517, 20227}, {573, 17053}, {579, 1015}, {595, 5019}, {604, 3915}, {614, 21334}, {940, 17394}, {978, 3169}, {995, 2092}, {1086, 10446}, {1149, 1400}, {1201, 2269}, {1333, 2162}, {1402, 3052}, {1500, 5105}, {1716, 9025}, {1918, 21010}, {1999, 3759}, {2178, 9259}, {2209, 3009}, {2238, 5839}, {2242, 4264}, {2262, 3290}, {2880, 16686}, {3053, 37519}, {3445, 10475}, {3741, 17275}, {3747, 7032}, {3770, 4713}, {3863, 7015}, {3924, 20594}, {4266, 21796}, {4386, 21001}, {4688, 10456}, {4974, 17733}, {5120, 14974}, {5301, 7113}, {8572, 10882}, {10027, 17786}, {10434, 21000}, {10441, 17054}, {10447, 17119}, {10449, 17362}, {10468, 17253}, {11679, 17348}, {16696, 17185}, {17314, 20037}, {21776, 21790}

X(21769) = isogonal conjugate of isotomic conjugate of X(3210)
X(21769) = polar conjugate of isotomic conjugate of X(20805)

X(21770) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a c^5 - b c^5) : :

X(21770) lies on these lines: {1, 15656}, {6, 1411}, {31, 10537}, {37, 2288}, {48, 3285}, {56, 21767}, {198, 2176}, {213, 2262}, {219, 3878}, {595, 1630}, {1108, 1409}, {1191, 3197}, {1195, 8607}, {1319, 14597}, {1625, 2150}, {2148, 2623}, {2182, 2300}, {3057, 3990}, {16679, 21784}, {18614, 20986}

X(21770) = isogonal conjugate of isotomic conjugate of X(18662)
X(21770) = polar conjugate of isotomic conjugate of X(20803)

X(21771) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 + a^4 b - a b^4 - b^5 + a^4 c + b^4 c - a c^4 + b c^4 - c^5) : :

X(21771) lies on these lines: {6, 2172}, {22, 3721}, {25, 20271}, {55, 20994}, {220, 19297}, {1191, 21773}, {2176, 2178}, {3125, 8185}, {3220, 16968}, {3959, 9798}, {7295, 16974}, {18616, 20987}

X(21771) = isogonal conjugate of isotomic conjugate of X(17481)
X(21771) = polar conjugate of isotomic conjugate of X(23068)

X(21772) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^5 + a^4 b - a b^4 - b^5 + a^4 c - a^2 b^2 c + b^4 c - a^2 b c^2 + a b^2 c^2 - a c^4 + b c^4 - c^5) : :

X(21772) lies on these lines: {6, 21745}, {2176, 2178}, {7252, 8632}, {18617, 19596}

X(21772) = isogonal conjugate of isotomic conjugate of X(17482)
X(21772) = polar conjugate of isotomic conjugate of X(23069)

X(21773) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 + a^2 b - a b^2 - b^3 + a^2 c + a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21773) lies on these lines: {1, 1030}, {3, 16777}, {6, 41}, {19, 19302}, {35, 3723}, {36, 37}, {53, 108}, {100, 17388}, {101, 583}, {404, 594}, {579, 17796}, {910, 7300}, {995, 4275}, {999, 16884}, {1014, 17365}, {1015, 2220}, {1100, 5563}, {1108, 7297}, {1191, 21771}, {1213, 2975}, {1333, 17053}, {1407, 7130}, {1415, 2965}, {1444, 4364}, {1609, 1617}, {1631, 21010}, {1743, 3196}, {1953, 2160}, {2171, 15109}, {2242, 4261}, {2305, 9259}, {2911, 5043}, {3204, 4253}, {3247, 7280}, {3290, 5322}, {3336, 21863}, {3361, 3553}, {3444, 21004}, {3554, 13462}, {3726, 5347}, {3941, 16686}, {4057, 21143}, {4188, 17314}, {4360, 19308}, {4361, 11329}, {5035, 21796}, {5120, 16885}, {5204, 16672}, {5253, 17398}, {5341, 8609}, {6186, 20988}, {6763, 21873}, {8666, 17275}, {11340, 20182}, {11349, 17366}, {14804, 16548}, {16679, 21009}, {17045, 21511}, {17243, 21495}, {17267, 21477}, {17798, 20990}

X(21773) = polar conjugate of isotomic conjugate of X(23070)
X(21773) = isogonal conjugate of isotomic conjugate of X(17483)

X(21774) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^9 b - 2 a^5 b^5 + a b^9 + a^9 c + a^8 b c - a b^8 c - b^9 c - 2 a^5 c^5 + 2 b^5 c^5 - a b c^8 + a c^9 - b c^9) : :

X(21774) lies on these lines: {6, 2156}, {19, 614}, {2352, 20993}

X(21774) = isogonal conjugate of isotomic conjugate of X(21215)
X(21774) = polar conjugate of isotomic conjugate of X(23074)

X(21775) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 b + a b^3 + a^3 c + a^2 b c - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(21775) lies on these lines: {3, 16584}, {6, 63}, {19, 1611}, {48, 1613}, {55, 869}, {57, 16782}, {213, 17594}, {238, 21387}, {345, 2238}, {354, 16781}, {561, 11339}, {940, 16524}, {982, 3496}, {1040, 16968}, {1403, 3224}, {1615, 1616}, {1716, 21876}, {2352, 3053}, {3230, 3749}, {3744, 16969}, {5299, 17591}, {16523, 20182}, {21767, 21776}

X(21775) = polar conjugate of isotomic conjugate of X(23075)
X(21775) = isogonal conjugate of isotomic conjugate of X(21216)

X(21776) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 b^3 + a^3 c^3 - b^3 c^3): :

X(21776) lies on these lines: {6, 75}, {55, 21777}, {825, 9233}, {1613, 8620}, {2162, 16679}, {2176, 20990}, {2178, 21783}, {2231, 3596}, {21767, 21775}, {21769, 21790}

X(21776) = isogonal conjugate of isotomic conjugate of X(17486)
X(21776) = polar conjugate of isotomic conjugate of X(23076)

X(21777) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (-a^3 b^3 + a^4 b c - a^3 b^2 c + a^2 b^3 c - a b^4 c - a^3 b c^2 - a b^3 c^2 - a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4) : :

X(21777) lies on these lines: {6, 82}, {55, 21776}, {1631, 2176}, {3499, 8053}, {5124, 10329}, {20990, 21783}

X(21777) = isogonal conjugate of isotomic conjugate of X(21217)
X(21777) = polar conjugate of isotomic conjugate of X(23077)

X(21778) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 - 2 a^3 c^3 + 2 b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(21778) lies on these lines: {6, 7}, {48, 1613}, {2175, 9450}, {2176, 3010}, {3197, 21779}, {16872, 20674}

X(21778) = isogonal conjugate of isotomic conjugate of X(21218)
X(21778) = polar conjugate of isotomic conjugate of X(23078)

X(21779) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b^2 + a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21779) lies on these lines: {1, 9401}, {2, 6}, {31, 1979}, {42, 21788}, {43, 7075}, {55, 869}, {100, 7109}, {171, 213}, {198, 1755}, {220, 18235}, {238, 1197}, {846, 8845}, {1030, 17735}, {1045, 21883}, {1196, 4260}, {1206, 3720}, {1627, 5371}, {1691, 2194}, {2300, 3684}, {3196, 21783}, {3197, 21778}, {3216, 3499}, {3230, 3750}, {3550, 9431}, {3666, 16514}, {3736, 21838}, {3981, 4259}, {4038, 20963}, {5021, 16420}, {5096, 10329}, {5256, 16525}, {8033, 17499}, {16477, 21757}, {16515, 20182}, {16678, 21008}, {17792, 21876}, {20468, 20998}

X(21779) = isogonal conjugate of isotomic conjugate of X(1655)
X(21779) = polar conjugate of isotomic conjugate of X(23079)

X(21780) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b^2 - 2 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - 3 b^2 c^2) : :

X(21780) lies on these lines: {6, 43}, {55, 3009}, {100, 1613}, {165, 16514}, {197, 2076}, {198, 17735}, {940, 4393}, {1030, 20855}, {1979, 4383}, {2110, 3052}, {2176, 3550}, {3959, 20359}, {4421, 21788}, {16515, 17594}, {20468, 20473}

X(21780) = isogonal conjugate of isotomic conjugate of X(21219)
X(21780) = polar conjugate of isotomic conjugate of X(23080)

X(21781) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (5 a^2 - 5 a b - b^2 - 5 a c + 7 b c - c^2) : :

X(21781) lies on these lines: {6, 101}, {45, 214}, {55, 21782}, {172, 21747}, {1055, 20672}, {1979, 2176}, {2177, 5168}, {2251, 16489}, {9324, 21885}, {16490, 21754}

X(21781) = isogonal conjugate of isotomic conjugate of X(17487)
X(21781) = polar conjugate of isotomic conjugate of X(23081)
X(21781) = crosspoint of PU(99)

X(21782) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (8 a^2 - 5 a b - 4 b^2 - 5 a c + b c - 4 c^2) : :

X(21782) lies on these lines: {6, 36}, {55, 21781}, {902, 2176}, {4262, 8649}

X(21782) = isogonal conjugate of isotomic conjugate of X(17488)
X(21782) = polar conjugate of isotomic conjugate of X(23082)

X(21783) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (-a^3 b^3 + a^4 b c + a^3 b^2 c - a^2 b^3 c - a b^4 c + a^3 b c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a b c^4) : :

X(21783) lies on these lines: {6, 662}, {741, 1084}, {1631, 21787}, {1979, 5040}, {2162, 4471}, {2176, 21784}, {2178, 21776}, {2248, 21792}, {2305, 20672}, {3196, 21779}, {5539, 21887}, {9259, 9431}, {18278, 19297}, {20675, 21788}, {20990, 21777}

X(21783) = isogonal conjugate of isotomic conjugate of X(21220)
X(21783) = polar conjugate of isotomic conjugate of X(23083)

X(21784) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a - b) (a - c) (b + c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c + 3 a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21784) lies on these lines: {6, 2643}, {59, 20188}, {512, 692}, {523, 651}, {1631, 21767}, {2176, 21783}, {2293, 4068}, {3013, 4092}, {4557, 4559}, {4585, 21295}, {5202, 16686}, {16679, 21770}, {20990, 21768}

X(21784) = isogonal conjugate of X(7372)
X(21784) = polar conjugate of isotomic conjugate of X(23084)

X(21785) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b + a b^2 + a^2 c - 3 a b c - b^2 c + a c^2 - b c^2) : :

X(21785) lies on these lines: {1, 6}, {55, 7032}, {198, 9259}, {572, 2241}, {573, 1015}, {595, 5042}, {604, 1403}, {992, 5839}, {995, 4263}, {1149, 2347}, {1575, 3169}, {1613, 2280}, {2209, 21010}, {2262, 20271}, {2269, 2275}, {3248, 20992}, {3684, 21001}, {3686, 3840}, {3831, 17275}, {3959, 20227}, {4254, 21008}, {4266, 17053}, {4268, 9456}, {4383, 17121}, {5019, 21793}, {5120, 17735}, {7155, 17475}, {11505, 18755}, {16688, 21000}, {17054, 18178}, {21214, 21892}

X(21785) = isogonal conjugate of isotomic conjugate of X(17490)
X(21785) = polar conjugate of isotomic conjugate of X(23085)

X(21786) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^3 - a b^2 + 3 a b c - b^2 c - a c^2 - b c^2) : :

X(21786) lies on these lines: {6, 1635}, {101, 109}, {105, 739}, {649, 834}, {650, 9364}, {812, 940}, {909, 2423}, {1979, 8632}, {3052, 8645}, {3125, 18191}, {4063, 7254}, {4383, 4763}, {4435, 4773}, {5040, 8027}, {17494, 18199}, {22086, 22108}

X(21786) = isogonal conjugate of isotomic conjugate of X(21222)
X(21786) = polar conjugate of isotomic conjugate of X(23087)
X(21786) = crossdifference of every pair of points on line X(10)X(11)

X(21787) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 b^3 - a^3 b^2 c + a^2 b^3 c - a^3 b c^2 - a b^3 c^2 + a^3 c^3 + a^2 b c^3 - a b^2 c^3 - b^3 c^3) : :

X(21787) lies on these lines: {6, 43}, {1613, 1918}, {1631, 21783}, {1977, 17349}, {1979, 20992}, {2176, 20990}, {3053, 21788}, {16690, 21001}

X(21787) = isogonal conjugate of isotomic conjugate of X(21223)
X(21787) = polar conjugate of isotomic conjugate of X(23088)

X(21788) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b^2 + a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - b^2 c^2) : :

X(21788) lies on these lines: {1, 6}, {42, 21779}, {55, 1613}, {81, 7109}, {86, 2295}, {100, 3231}, {101, 9264}, {172, 1918}, {190, 19565}, {667, 21007}, {692, 1691}, {729, 898}, {741, 813}, {902, 1979}, {940, 17032}, {1016, 4601}, {1018, 18792}, {1149, 20459}, {1185, 17018}, {1197, 3750}, {1332, 15994}, {1376, 21001}, {1486, 5017}, {1500, 3736}, {1611, 7074}, {1621, 3051}, {1740, 3208}, {1911, 3747}, {1914, 9454}, {2056, 20986}, {2076, 20872}, {2106, 2669}, {2110, 3009}, {2162, 3052}, {2209, 9310}, {2235, 3685}, {2242, 5156}, {2274, 2276}, {2375, 2703}, {2664, 21897}, {3053, 21787}, {3499, 8053}, {3792, 20861}, {4383, 17027}, {4421, 21780}, {4513, 17787}, {5132, 21008}, {5284, 20965}, {9259, 20470}, {17030, 17259}, {20675, 21783}, {20989, 20998}

X(21788) = isogonal conjugate of isotomic conjugate of X(17759)
X(21788) = isogonal conjugate of anticomplement of X(39028)
X(21788) = polar conjugate of isotomic conjugate of X(20796)
X(21788) = anticomplement of complementary conjugate of X(39028)

X(21789) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a + b) (a - b - c)^2 (b - c) (a + c) : :

X(21789) lies on these lines: {3, 14838}, {6, 810}, {21, 884}, {36, 238}, {55, 4041}, {58, 2424}, {99, 14727}, {110, 14733}, {162, 1624}, {333, 21300}, {405, 1577}, {512, 5060}, {514, 22160}, {520, 2605}, {523, 2074}, {650, 1946}, {652, 663}, {676, 17925}, {759, 2717}, {958, 3907}, {1021, 3900}, {1110, 4557}, {1576, 14776}, {2283, 4243}, {2299, 2432}, {3422, 4471}, {4151, 5248}, {4252, 22093}, {4426, 21901}, {4990, 7253}, {5737, 21259}, {8029, 14777}, {8636, 21005}, {8639, 16874}

X(21789) = isogonal conjugate of X(4566)
X(21789) = crossdifference of every pair of points on line X(37)X(226)
X(21789) = pole wrt circumcircle of line X(1)X(19)
X(21789) = polar conjugate of isotomic conjugate of X(23090)

X(21790) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^3 b^3 - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3) : :

X(21790) lies on these lines: {6, 190}, {890, 1979}, {2176, 4557}, {3224, 16969}, {9259, 9431}, {21769, 21776}

X(21790) = isogonal conjugate of isotomic conjugate of X(21224)
X(21790) = polar conjugate of isotomic conjugate of X(23091)

X(21791) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (b - c) (a^3 b - a^2 b^2 + a^3 c - a^2 b c - a^2 c^2 - b^2 c^2) : :

X(21791) lies on these lines: {6, 514}, {101, 20696}, {213, 4040}, {512, 1691}, {649, 18278}, {663, 2176}, {1912, 21003}, {2295, 21302}, {3063, 4083}, {3250, 7252}, {3288, 4435}, {6586, 8676}, {8631, 16695}, {18107, 20295}

X(21791) = isogonal conjugate of isotomic conjugate of X(21225)
X(21791) = polar conjugate of isotomic conjugate of X(23093)

X(21792) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (a^2 b^2 - a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21792) lies on these lines: {6, 43}, {31, 1979}, {55, 1613}, {81, 2669}, {100, 3051}, {197, 5017}, {692, 2056}, {869, 18278}, {893, 1964}, {940, 17027}, {1001, 21001}, {1185, 17126}, {1621, 3231}, {1691, 20986}, {2176, 2223}, {2235, 7081}, {2248, 21783}, {4428, 16969}, {4640, 16514}, {6043, 18268}, {8623, 21008}, {9259, 18613}, {16525, 17594}, {20988, 20998}

X(21792) = isogonal conjugate of isotomic conjugate of X(21226)
X(21792) = polar conjugate of isotomic conjugate of X(23094)

X(21793) = (A,B,C,X(1); A',B',C',X(6)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics a^2 (2 a^2 + a b + a c - b c) : :

X(21793) lies on these lines: {6, 31}, {32, 101}, {37, 8616}, {45, 5276}, {48, 20674}, {56, 17962}, {58, 2241}, {75, 4797}, {81, 16884}, {172, 3915}, {187, 995}, {190, 7766}, {213, 7031}, {238, 4386}, {385, 4713}, {609, 3230}, {739, 4588}, {901, 9111}, {985, 1001}, {1015, 4257}, {1100, 4640}, {1104, 3959}, {1191, 3053}, {1333, 2162}, {1334, 7296}, {1575, 3550}, {1613, 8620}, {1621, 16777}, {1707, 16973}, {2238, 17127}, {2305, 5329}, {2328, 16516}, {3496, 16974}, {3730, 5007}, {3747, 16515}, {3795, 16468}, {4252, 16781}, {4363, 16998}, {4426, 5255}, {4465, 17001}, {5019, 21785}, {5134, 5309}, {5306, 17747}, {5319, 17732}, {5475, 17734}, {8750, 10311}, {9259, 16483}, {12514, 16519}, {16466, 18755}, {17697, 21025}

X(21793) = isogonal conjugate of X(27494)
X(21793) = polar conjugate of isotomic conjugate of X(23095)

X(21794) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (a + b - c) (a - b + c) (b + c)^2 (a^2 - b^2 - b c - c^2) : :

X(21794) lies on these lines: {37, 21011}, {570, 17438}, {604, 2276}, {1254, 1500}, {2594, 21741}, {3969, 16577}, {13006, 17440}, {21803, 21825}

X(21795) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (a - b - c) (b + c) (a b - b^2 + a c + 2 b c - c^2) : :

X(21795) lies on these lines: {2, 2481}, {37, 226}, {39, 614}, {42, 213}, {55, 2195}, {101, 20834}, {220, 3190}, {650, 4995}, {1212, 4847}, {2276, 2999}, {2293, 8012}, {3136, 16589}, {3294, 4199}, {3687, 3693}, {3730, 16058}, {3991, 4028}, {4061, 4515}, {5364, 21746}, {8226, 16601}, {21796, 21811}, {21809, 21820}

X(21796) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21796) lies on these lines: {1, 4263}, {6, 101}, {9, 39}, {10, 37}, {19, 3199}, {32, 198}, {42, 4890}, {45, 4261}, {56, 5042}, {181, 3725}, {213, 1042}, {256, 2664}, {391, 16975}, {572, 21008}, {573, 2176}, {614, 21319}, {800, 8557}, {869, 21746}, {872, 3122}, {966, 1573}, {980, 17257}, {1100, 8610}, {1201, 2347}, {1333, 9341}, {1572, 2270}, {1574, 2345}, {1575, 6686}, {1743, 2275}, {1962, 22232}, {1964, 3271}, {2150, 5170}, {2171, 3125}, {2178, 5019}, {2183, 2300}, {2238, 21061}, {2241, 4254}, {2256, 9351}, {2269, 3230}, {2276, 3731}, {2298, 5277}, {2324, 9620}, {3161, 17756}, {3452, 3663}, {3666, 5241}, {3688, 3764}, {3690, 20966}, {3709, 8061}, {3721, 21078}, {3778, 20683}, {3930, 22171}, {3954, 21033}, {4266, 21769}, {4271, 16685}, {4277, 16777}, {4642, 21809}, {4749, 20990}, {4850, 17247}, {5007, 16470}, {5035, 21773}, {5069, 16885}, {5283, 5296}, {6210, 16970}, {7064, 21035}, {7277, 16726}, {10822, 14815}, {16696, 17332}, {21795, 21811}, {21815, 21835}, {21827, 21840}

X(21797) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - a c^3 - b c^3 + c^4) : :

X(21797) lies on these lines: {37, 21011}, {73, 1334}, {1500, 3125}, {2197, 21809}, {2276, 9259}, {13006, 17439}

X(21798) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (a - b - c) (b + c) (a^3 b + a^2 b^2 - a b^3 - b^4 + a^3 c + 2 a^2 b c + 2 a b^2 c + b^3 c + a^2 c^2 + 2 a b c^2 - a c^3 + b c^3 - c^4) : :

X(21798) lies on these lines: {37, 21011}, {1100, 22058}, {1195, 14547}, {1500, 21809}, {2092, 21811}, {2197, 21808}, {21819, 22201}

X(21799) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (a^4 - b^4 + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21799) lies on these lines: {32, 17442}, {37, 4456}, {172, 18669}, {1254, 1500}, {1953, 2241}, {2242, 18671}, {3695, 4053}, {3954, 22076}, {4079, 21134}

X(21800) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (a^4 - b^4 - a^2 b c + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21800) lies on these lines: {37, 21862}, {187, 18669}, {1254, 1500}, {3295, 5011}, {3721, 5697}, {4760, 18715}, {21832, 21836}

X(21801) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :

X(21801) lies on these lines: {1, 2267}, {6, 2098}, {9, 1389}, {19, 1802}, {37, 65}, {44, 2170}, {45, 17451}, {48, 1766}, {101, 2173}, {144, 18161}, {190, 1959}, {226, 22014}, {244, 8610}, {517, 2183}, {523, 661}, {527, 3942}, {572, 17438}, {594, 21011}, {672, 8609}, {674, 2310}, {756, 22276}, {758, 2250}, {857, 21091}, {872, 22298}, {908, 3262}, {1089, 1826}, {1213, 21012}, {1633, 18788}, {1731, 5526}, {1824, 2318}, {1901, 22073}, {2161, 17796}, {2182, 6603}, {2246, 7297}, {2265, 2323}, {2397, 17139}, {3159, 3950}, {3191, 18673}, {3247, 3333}, {3304, 16777}, {3690, 21807}, {3728, 22301}, {3781, 20430}, {3958, 21061}, {3970, 4029}, {3992, 22029}, {4266, 5697}, {4336, 12329}, {4419, 7146}, {4516, 20683}, {4552, 8680}, {4648, 7201}, {4860, 16672}, {5164, 21810}, {5289, 5782}, {5525, 18669}, {6358, 22000}, {7064, 21804}, {7113, 17439}, {8607, 22059}, {16578, 20367}, {16814, 17443}, {17277, 17868}, {17336, 18041}, {17442, 17742}, {17757, 21942}, {20703, 21829}, {21015, 21911}

X(21802) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a^2 + b^2 + c^2) : :

X(21802) lies on these lines: {1, 6}, {321, 17367}, {762, 2238}, {1500, 6155}, {1643, 4024}, {1962, 20683}, {2251, 5266}, {2295, 3125}, {3122, 20969}, {3454, 4144}, {3698, 16583}, {3721, 3997}, {3930, 20970}, {3943, 7206}, {4006, 21904}, {4079, 6161}, {4868, 21821}, {5007, 17469}, {21803, 22185}

X(21803) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (a^2 + b c) : :

X(21803) lies on these lines: {1, 3790}, {37, 3122}, {42, 2321}, {86, 4601}, {181, 756}, {594, 872}, {762, 20685}, {869, 2345}, {894, 7184}, {899, 4967}, {1215, 3963}, {1500, 6378}, {1740, 3809}, {1964, 17369}, {2234, 7227}, {2294, 20703}, {2295, 20964}, {2309, 17355}, {2329, 7122}, {2663, 6542}, {2667, 3943}, {3009, 5750}, {3589, 17445}, {3710, 4078}, {3717, 10459}, {3728, 20683}, {4111, 21805}, {5749, 7032}, {7225, 17124}, {15523, 22008}, {16589, 20706}, {17353, 21352}, {21794, 21825}, {21802, 22185}, {21818, 21823}

X(21804) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : :

X(21804) lies on these lines: {1, 17976}, {10, 7235}, {37, 4068}, {39, 17872}, {142, 3675}, {210, 21078}, {226, 21084}, {674, 17443}, {756, 21807}, {1212, 12723}, {1731, 19133}, {1781, 17798}, {1953, 3688}, {2171, 20683}, {2643, 21035}, {2886, 20236}, {3006, 21442}, {3125, 3778}, {3728, 3954}, {3925, 18698}, {3949, 4111}, {4053, 22271}, {4092, 21011}, {4890, 21808}, {7064, 21801}, {7211, 22027}, {8424, 20602}, {11997, 16601}, {12782, 17891}, {14839, 17868}, {17052, 21023}, {17245, 17463}, {17451, 21746}, {20195, 20275}, {20686, 20709}, {20703, 22167}, {20715, 21061}

X(21805) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a - b - c) : :

X(21805) is the intersection of the tangents at X(872) and X(4738) to the inellipse which is the trilinear square of the Nagel line. (Randy Hutson, October 15, 2018)

X(21805) lies on these lines: {1, 4015}, {2, 17145}, {6, 3711}, {9, 2177}, {10, 2650}, {31, 200}, {37, 42}, {38, 43}, {44, 678}, {55, 16885}, {57, 9350}, {72, 3214}, {100, 896}, {171, 4722}, {238, 3722}, {244, 518}, {321, 4090}, {375, 20961}, {512, 661}, {519, 3992}, {537, 17495}, {594, 4144}, {612, 1449}, {674, 20962}, {740, 3952}, {748, 3870}, {750, 3751}, {758, 4674}, {762, 20970}, {869, 16975}, {982, 4661}, {984, 3240}, {1215, 4651}, {1500, 21822}, {1635, 8661}, {1959, 3507}, {2229, 21893}, {2239, 4712}, {2292, 3293}, {2308, 16671}, {3121, 21830}, {3699, 17763}, {3720, 3740}, {3724, 4557}, {3743, 4547}, {3745, 16668}, {3778, 22312}, {3868, 6048}, {3873, 16569}, {3896, 3971}, {3914, 21060}, {3931, 4533}, {3932, 4062}, {3938, 4383}, {3943, 4819}, {3957, 17123}, {3958, 21858}, {3961, 17469}, {3962, 21896}, {3967, 4365}, {3979, 5284}, {3987, 4067}, {3993, 4946}, {3995, 4096}, {4005, 4646}, {4046, 6535}, {4058, 4061}, {4072, 4082}, {4111, 21803}, {4134, 4424}, {4152, 4969}, {4414, 5220}, {4420, 5247}, {4430, 17063}, {4434, 4753}, {4442, 21093}, {4457, 17163}, {4512, 17782}, {4551, 18593}, {4641, 9340}, {4649, 5297}, {4662, 10459}, {4689, 15481}, {4899, 5212}, {4974, 20045}, {4981, 6685}, {5311, 16884}, {5380, 17955}, {5531, 13329}, {5692, 17461}, {8013, 10026}, {8580, 17124}, {11075, 17796}, {16669, 21747}, {17020, 17598}, {19596, 20989}, {20685, 21833}, {20691, 21886}, {20698, 21899}, {20718, 22313}, {21075, 21935}

X(21805) = complement of X(17145)

X(21806) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (4 a + b + c) : :

X(21806) lies on these lines: {1, 88}, {2, 4732}, {31, 1449}, {37, 42}, {38, 4430}, {55, 4497}, {81, 9340}, {551, 3902}, {846, 4722}, {896, 4649}, {899, 15569}, {902, 1100}, {968, 1743}, {1213, 4819}, {1500, 3121}, {2292, 4067}, {2295, 21885}, {2308, 16668}, {2650, 3931}, {3293, 4540}, {3666, 17449}, {3683, 16671}, {3689, 3723}, {3711, 16672}, {3720, 3848}, {3743, 3988}, {3750, 17011}, {3842, 19998}, {3896, 4709}, {3938, 20182}, {3957, 17600}, {3993, 3994}, {4026, 4062}, {4085, 21026}, {4424, 4744}, {4651, 10180}, {5269, 17782}, {6155, 21808}, {7278, 16727}, {9350, 17022}, {14996, 17601}, {16484, 17012}, {16666, 21747}

X(21807) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 - b^4 + a^2 c^2 + 2 b^2 c^2 - c^4) : :

X(21807) lies on these lines: {5, 14213}, {12, 17874}, {19, 25}, {42, 4516}, {51, 1953}, {63, 20430}, {72, 1482}, {100, 5966}, {125, 21911}, {149, 3995}, {199, 16548}, {226, 18210}, {262, 321}, {516, 22002}, {692, 7073}, {756, 21804}, {851, 16577}, {1093, 6521}, {1867, 10894}, {2161, 2194}, {2171, 22168}, {2181, 3199}, {2265, 13366}, {2294, 3611}, {2643, 21936}, {3136, 6358}, {3175, 11235}, {3489, 6192}, {3490, 6191}, {3690, 21801}, {3954, 5360}, {3998, 14489}, {4463, 14495}, {4705, 15475}, {4847, 22022}, {14008, 18359}, {17616, 22067}, {21814, 21833}, {22000, 22027}

X(21808) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a b - b^2 + a c + 2 b c - c^2) : :

X(21808) lies on these lines: {1, 41}, {2, 17048}, {9, 3868}, {10, 3930}, {12, 21044}, {21, 3509}, {25, 5311}, {37, 65}, {38, 5283}, {39, 244}, {42, 16583}, {79, 5134}, {142, 1229}, {198, 1953}, {213, 2650}, {218, 15934}, {226, 857}, {228, 1962}, {321, 21071}, {335, 1655}, {354, 1212}, {405, 5282}, {442, 21029}, {514, 7278}, {518, 3691}, {672, 942}, {756, 3954}, {758, 3294}, {976, 5275}, {993, 17736}, {1018, 3754}, {1055, 2646}, {1089, 22011}, {1107, 3726}, {1111, 17758}, {1146, 15888}, {1193, 3290}, {1213, 3949}, {1467, 2285}, {1468, 16968}, {1500, 3125}, {1621, 3496}, {1697, 3247}, {1759, 5248}, {1827, 2293}, {1959, 16826}, {1961, 4220}, {2140, 7264}, {2197, 21798}, {2276, 20271}, {3061, 3616}, {3119, 17718}, {3214, 16605}, {3241, 4051}, {3293, 16611}, {3306, 21540}, {3337, 5030}, {3475, 6554}, {3649, 17747}, {3693, 3812}, {3698, 4515}, {3701, 21101}, {3720, 17456}, {3723, 17443}, {3727, 21331}, {3730, 5902}, {3746, 5011}, {3753, 3991}, {3822, 21928}, {3873, 21384}, {3874, 16552}, {3912, 20911}, {3936, 4109}, {3942, 17392}, {3986, 21078}, {3992, 21067}, {3994, 22036}, {4075, 22035}, {4253, 18398}, {4390, 19860}, {4647, 21070}, {4648, 7195}, {4695, 20691}, {4860, 5022}, {4890, 21804}, {5045, 17474}, {5179, 13407}, {5257, 21033}, {5308, 7146}, {5369, 21746}, {5883, 14439}, {6155, 21806}, {7991, 16673}, {11263, 21090}, {12609, 21073}, {15569, 20593}, {17141, 17755}, {17316, 21216}, {17319, 17868}, {17448, 21332}, {17757, 21929}, {20486, 21967}, {20686, 22172}, {21021, 21025}

X(21809) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (a - b - c) (b + c) (a b + b^2 + a c - 2 b c + c^2) : :

X(21809) lies on these lines: {9, 644}, {37, 65}, {41, 380}, {45, 1953}, {346, 19582}, {572, 17439}, {594, 21041}, {672, 12915}, {756, 21804}, {1500, 21798}, {1766, 9310}, {1959, 17261}, {2197, 21797}, {2310, 3688}, {2321, 3701}, {2340, 11997}, {2347, 3057}, {3061, 3161}, {3125, 21826}, {3294, 22197}, {3452, 20895}, {3731, 11531}, {3930, 3950}, {3942, 17334}, {3943, 3949}, {3970, 4098}, {3986, 21921}, {4006, 4072}, {4029, 22021}, {4516, 7064}, {4642, 21796}, {4695, 21892}, {5308, 7201}, {6603, 21748}, {7237, 20702}, {10868, 17792}, {10980, 16673}, {16814, 17444}, {17260, 17868}, {18698, 22019}, {20681, 22167}, {21795, 21820}

X(21810) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (a b + b^2 + a c + c^2) : :

X(21810) lies on these lines: {1, 6}, {38, 17053}, {115, 21675}, {181, 756}, {191, 2305}, {321, 3596}, {594, 762}, {966, 3735}, {992, 10176}, {1211, 18697}, {1213, 3125}, {1264, 17257}, {1500, 3949}, {1573, 1953}, {1761, 5277}, {2092, 2292}, {2294, 16589}, {2321, 3971}, {2643, 21699}, {2959, 5293}, {3686, 3727}, {3688, 5360}, {3721, 5257}, {3728, 4516}, {3930, 21820}, {3952, 14624}, {4115, 17355}, {4272, 6155}, {4424, 21857}, {4559, 10693}, {4705, 22260}, {4708, 18179}, {5164, 21801}

X(21811) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (a - b - c) (b + c) (2 a^2 + a b - b^2 + a c + 2 b c - c^2) : :

X(21811) lies on these lines: {9, 21}, {37, 65}, {45, 198}, {165, 846}, {228, 756}, {346, 18231}, {407, 21674}, {573, 17451}, {672, 8731}, {1212, 2347}, {1213, 21018}, {1214, 7147}, {1283, 2246}, {1723, 2280}, {1962, 4890}, {2092, 21798}, {2160, 16675}, {2170, 2269}, {2323, 17440}, {2646, 21748}, {3161, 3985}, {3294, 12567}, {3691, 3965}, {3692, 4390}, {3693, 18235}, {3709, 6615}, {3930, 21061}, {3958, 4053}, {3986, 4425}, {4041, 18000}, {4875, 12642}, {5051, 5257}, {5128, 16676}, {8012, 17611}, {8229, 10445}, {9840, 16601}, {10868, 15587}, {11533, 16673}, {14439, 17355}, {16814, 17454}, {18698, 22003}, {20684, 21840}, {21795, 21796}

X(21812) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c)^2 (a^4 - b^4 - c^4) (a^4 - b^4 + 2 b^3 c - 2 b^2 c^2 + 2 b c^3 - c^4) : :

X(21812) lies on these lines: {37, 21875}

X(21813) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c)^2 (a^2 + b^2 - 2 b c + c^2) : :

X(21813) lies on these lines: {31, 1196}, {37, 306}, {39, 748}, {171, 3291}, {181, 3124}, {213, 22200}, {238, 1194}, {756, 762}, {1184, 7083}, {1185, 21746}, {2176, 3981}, {2197, 21815}, {2276, 3305}, {3051, 3271}, {3290, 5249}, {3914, 16583}, {4425, 16600}, {9465, 17127}, {14580, 14975}, {21795, 21796}

X(21814) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^3 (b + c) (b^2 + c^2) : :

X(21814) lies on these lines: {2, 37}, {6, 16687}, {31, 32}, {38, 39}, {42, 3121}, {81, 292}, {100, 733}, {101, 745}, {726, 22013}, {740, 21327}, {756, 21830}, {846, 2664}, {872, 21753}, {1030, 17735}, {1107, 4981}, {1215, 2229}, {1500, 1962}, {1964, 3051}, {2092, 21815}, {2275, 3873}, {3124, 21936}, {3725, 7109}, {3736, 16717}, {3896, 20691}, {3936, 18905}, {3952, 21902}, {4093, 21035}, {4642, 20707}, {4651, 21897}, {4972, 18904}, {5741, 9284}, {10329, 21008}, {15523, 16587}, {16703, 16720}, {18059, 19565}, {21795, 21796}, {21807, 21833}

X(21815) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^4 (b + c)^2 (b^2 - b c + c^2) : :

X(21815) lies on these lines: {37, 313}, {756, 21818}, {872, 1084}, {1500, 6378}, {1964, 8265}, {2092, 21814}, {2171, 21823}, {2197, 21813}, {2275, 7189}, {2276, 17368}, {3117, 7032}, {3662, 18905}, {3778, 16584}, {4357, 8620}, {16886, 16889}, {21796, 21835}

X(21815) = isogonal conjugate of X(7307)

X(21816) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (2 a + b + c) : :

X(21816) lies on these lines: {1, 6}, {10, 4037}, {101, 15792}, {115, 21674}, {594, 4099}, {756, 762}, {846, 2135}, {1018, 9278}, {1125, 4115}, {1213, 4647}, {1930, 4364}, {1962, 20970}, {2238, 3743}, {2292, 3125}, {2295, 20708}, {4075, 21021}, {6537, 20653}, {20680, 21824}, {21821, 21833}

X(21817) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (b^2 + c^2) (2 a^2 + b^2 + c^2) : :

X(21817) lies on these lines: {37, 82}, {38, 39}, {228, 3456}, {1962, 20703}, {2092, 3930}, {5007, 17469}, {6292, 20898}, {11205, 17457}, {16587, 17456}

X(21818) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c)^2 (a^2 + b c) (b^2 + c^2) : :

X(21818) lies on these lines: {37, 18082}, {756, 21815}, {762, 3728}, {1237, 3963}, {1500, 7237}, {2092, 3930}, {21803, 21823}

X(21819) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^3 (a - b - c) (b + c) (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : :

X(21819) lies on these lines: {37, 1441}, {1334, 3774}, {21795, 21796}, {21798, 22201}

X(21820) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c)^2 (a b + a c + 2 b c) : :

X(21820) lies on these lines: {2, 37}, {190, 2668}, {292, 1255}, {756, 762}, {1655, 18059}, {1962, 3121}, {2229, 10180}, {2292, 22202}, {2667, 21753}, {3681, 21879}, {3930, 21810}, {3971, 21067}, {4079, 8034}, {4931, 21837}, {7211, 20616}, {16589, 21020}, {20702, 21833}, {21795, 21809}

X(21821) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a - b - c)^2 : :

X(21821) lies on these lines: {37, 1018}, {45, 5541}, {213, 21870}, {678, 1017}, {756, 21822}, {762, 1334}, {1500, 3121}, {2087, 14439}, {4370, 4738}, {4730, 14407}, {4868, 21802}, {21816, 21833}

X(21822) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (a - 2 b - 2 c) (b + c) (4 a + b + c) : :

X(21822) lies on these lines: {37, 758}, {45, 4752}, {756, 21821}, {1500, 21805}, {4793, 16590}

X(21823) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b^2 - c^2)^2 (a^2 + b c) : :

X(21823) lies on these lines: {37, 86}, {1015, 21824}, {1084, 2643}, {1500, 21825}, {2171, 21815}, {3121, 4516}, {3122, 4079}, {3125, 21835}, {4128, 21755}, {6378, 7237}, {20685, 21830}, {21803, 21818}

X(21824) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b^2 - c^2)^2 (a^2 - b^2 - b c - c^2) : :

X(21824) lies on these lines: {37, 101}, {115, 1365}, {1015, 21823}, {2160, 21353}, {2611, 20982}, {3125, 21833}, {4024, 21044}, {4516, 22212}, {8287, 17886}, {9447, 9636}, {20680, 21816}

X(21825) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c)^2 (a^4 - 2 a^2 b^2 + b^4 + 2 a^2 b c - b^3 c - 2 a^2 c^2 + b^2 c^2 - b c^3 + c^4) : :

X(21825) lies on these lines: {37, 21043}, {42, 3939}, {1500, 21823}, {2197, 7237}, {21794, 21803}

X(21826) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a b + b^2 + a c - 6 b c + c^2) : :

X(21826) lies on these lines: {6, 9050}, {9, 1015}, {10, 37}, {32, 1696}, {39, 3731}, {45, 5069}, {210, 22214}, {346, 1574}, {572, 8649}, {1573, 5296}, {2277, 16676}, {3122, 7064}, {3125, 21809}, {3247, 4263}, {4261, 16677}, {4277, 16674}, {8610, 16814}, {13466, 17786}, {20683, 22172}

X(21827) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a b^2 - 2 a b c - b^2 c + a c^2 - b c^2) : :

X(21827) lies on these lines: {2, 1978}, {10, 21345}, {37, 714}, {38, 1015}, {39, 17397}, {42, 21830}, {292, 846}, {321, 20688}, {756, 3121}, {893, 1961}, {1197, 3009}, {1500, 1962}, {1575, 4970}, {1908, 4682}, {2229, 3995}, {3117, 3616}, {3229, 16826}, {3720, 8620}, {3741, 20363}, {3993, 21877}, {4026, 16587}, {4096, 21893}, {4415, 16592}, {4425, 18905}, {4687, 6375}, {16524, 20760}, {16589, 22028}, {16604, 17459}, {20684, 22172}, {21796, 21840}, {22167, 22206}, {22171, 22220}, {22173, 22215}, {22174, 22201}, {22177, 22197}, {22180, 22203}, {22193, 22225}

X(21828) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b^2 - c^2) (a^2 - b^2 + b c - c^2) : :

X(21828) lies on these lines: {31, 5075}, {37, 4120}, {244, 665}, {351, 4455}, {514, 16754}, {647, 661}, {649, 834}, {650, 4802}, {654, 17455}, {905, 21124}, {3005, 7234}, {3666, 4750}, {3904, 3960}, {4526, 4958}, {4730, 22216}, {4893, 6586}, {5029, 9259}, {14404, 17990}, {16751, 21196}

X(21828) = crossdifference of every pair of points on line X(10)X(21)

X(21829) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c)^2 (a^2 - 2 b^2 + 3 b c - 2 c^2) : :

X(21829) lies on these lines: {181, 756}, {512, 20688}, {524, 17472}, {902, 968}, {2643, 4053}, {4062, 22047}, {4892, 20912}, {20703, 21801}, {21839, 22231}

X(21830) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a b^2 - b^2 c + a c^2 - b c^2) : :

X(21830) lies on these lines: {10, 37}, {39, 984}, {42, 21827}, {75, 1574}, {101, 1691}, {192, 10009}, {210, 16584}, {213, 6378}, {292, 1757}, {511, 3862}, {512, 798}, {518, 1015}, {519, 20363}, {665, 6165}, {668, 19565}, {698, 4568}, {716, 13466}, {726, 1575}, {756, 21814}, {762, 3728}, {872, 20970}, {899, 6377}, {1211, 16587}, {1334, 22205}, {2176, 4279}, {2229, 3952}, {2238, 18793}, {2664, 3229}, {3009, 20663}, {3121, 21805}, {3125, 20706}, {3912, 20549}, {3930, 22220}, {3971, 21877}, {4043, 21412}, {4090, 16606}, {4096, 21902}, {4651, 21327}, {4685, 21345}, {4687, 17030}, {16569, 20284}, {20685, 21823}, {21435, 22016}

X(21831) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b^2 - c^2) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 + 2 b^2 c^2 + b c^3) : :

X(21831) lies on these lines: {1, 7265}, {37, 3900}, {512, 20688}, {525, 17478}, {663, 4024}, {810, 3700}, {4079, 8678}, {5295, 21050}, {8062, 17899}

X(21832) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b^2 - c^2) (a^2 - b c) : :

X(21832) lies on these lines: {1, 5029}, {10, 21053}, {37, 4145}, {163, 2613}, {239, 514}, {244, 665}, {512, 661}, {522, 20979}, {523, 798}, {650, 3250}, {659, 4435}, {802, 20906}, {812, 3766}, {876, 9278}, {900, 3768}, {1018, 3952}, {1024, 3512}, {1577, 22224}, {1912, 20983}, {2610, 5213}, {3709, 4139}, {3783, 4893}, {4010, 4839}, {4024, 4039}, {4079, 4132}, {4094, 4155}, {4489, 4813}, {4526, 14408}, {4603, 4631}, {4832, 8672}, {4979, 6372}, {5540, 16562}, {6084, 20507}, {6545, 20520}, {6586, 17458}, {16755, 18196}, {17990, 22223}, {21131, 22108}, {21676, 21711}, {21800, 21836}, {21888, 21893}

X(21832) = isogonal conjugate of X(4584)
X(21832) = isotomic conjugate of X(4639)

X(21833) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b - c)^2 (b + c)^3 : :

X(21833) lies on these lines: {37, 100}, {72, 14498}, {115, 1109}, {213, 11060}, {321, 1916}, {1333, 21353}, {1648, 21944}, {1824, 2205}, {2611, 16592}, {2643, 3124}, {2796, 22033}, {3120, 4024}, {3121, 4516}, {3125, 21824}, {3930, 20708}, {3995, 6653}, {6627, 21054}, {20685, 21805}, {20690, 22206}, {20702, 21820}, {21807, 21814}, {21816, 21821}

X(21834) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b^2 - c^2) (a b + a c - b c) : :

X(21834) lies on these lines: {1, 20981}, {37, 798}, {192, 17217}, {321, 20910}, {512, 20688}, {513, 4826}, {514, 4502}, {522, 3250}, {523, 661}, {594, 21055}, {649, 21348}, {688, 22167}, {786, 20954}, {812, 21225}, {900, 21123}, {1577, 22043}, {1635, 6586}, {2084, 4170}, {2171, 4017}, {2321, 21099}, {3261, 4728}, {3287, 4879}, {3709, 4139}, {3728, 9402}, {3733, 16777}, {3835, 20906}, {4041, 4155}, {4043, 20953}, {4083, 14408}, {4369, 17159}, {4404, 21078}, {4526, 6371}, {8632, 21389}, {9400, 21349}, {20711, 22226}, {21051, 21959}, {21053, 21958}

X(21835) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^3 (b - c)^2 (b + c) (a b + a c - b c) : :

X(21835) lies on these lines: {37, 4033}, {351, 865}, {1086, 1646}, {2276, 4473}, {3123, 6377}, {3125, 21823}, {21238, 22186}, {21257, 22218}, {21796, 21815}

X(21836) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b^2 - c^2) (a^2 b^2 + a^2 b c + a^2 c^2 - b^2 c^2) : :

X(21836) lies on these lines: {37, 4083}, {512, 20688}, {514, 4079}, {784, 3250}, {2084, 4010}, {8714, 21123}, {9402, 22228}, {16589, 22224}, {21056, 21260}, {21800, 21832}

X(21837) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^3 (b^2 - c^2) (a b - b^2 + a c - b c - c^2) : :

X(21837) lies on these lines: {37, 1577}, {213, 810}, {1500, 4041}, {1734, 6586}, {2276, 14838}, {2489, 3709}, {3907, 5283}, {4822, 14991}, {4931, 21820}, {16589, 21052}

X(21838) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a^2 (b + c) (a b^2 + b^2 c + a c^2 + b c^2) : :

X(21838) lies on these lines: {2, 39}, {6, 16058}, {9, 43}, {10, 21877}, {32, 1011}, {37, 714}, {42, 213}, {83, 16957}, {99, 16956}, {115, 3136}, {187, 4184}, {210, 3774}, {232, 4213}, {291, 9401}, {292, 1961}, {511, 4476}, {574, 4191}, {672, 2670}, {740, 21883}, {756, 21814}, {851, 15487}, {1015, 3720}, {1084, 4370}, {1107, 3741}, {1197, 2309}, {1213, 1575}, {1613, 5145}, {1908, 4640}, {1962, 3121}, {2271, 13615}, {2292, 22230}, {2524, 22222}, {2548, 6818}, {2549, 6817}, {2653, 10974}, {3124, 20671}, {3199, 4207}, {3231, 17187}, {3666, 6377}, {3728, 22206}, {3736, 21779}, {3778, 22200}, {3932, 16587}, {3989, 8620}, {3993, 21345}, {3995, 20688}, {4203, 5276}, {4425, 18904}, {4642, 22173}, {4685, 20691}, {5013, 16059}, {5024, 16409}, {5206, 19346}, {5275, 11358}, {5277, 13588}, {5364, 19587}, {6651, 19579}, {6821, 7738}, {6822, 7736}, {7772, 16373}, {7804, 16955}, {7816, 16954}, {8054, 14751}, {10453, 16975}, {14936, 20967}, {16592, 16593}, {18755, 20834}, {20665, 20864}, {20666, 22080}

X(21838) = complement of X(310)
X(21838) = polar conjugate of isogonal conjugate of X(23212)

X(21839) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a^2 - b^2 - c^2) : :

X(21839) lies on these lines: {1, 6}, {10, 21057}, {100, 843}, {187, 896}, {191, 18755}, {249, 1931}, {321, 598}, {512, 661}, {519, 4037}, {524, 14210}, {758, 2238}, {762, 2295}, {1018, 20693}, {1046, 5277}, {2092, 3958}, {2205, 9515}, {2245, 21745}, {2292, 6155}, {2641, 9217}, {2650, 16589}, {2948, 9509}, {3681, 22206}, {3721, 4067}, {3780, 3878}, {3876, 17750}, {3901, 20271}, {3962, 16583}, {3997, 4134}, {4018, 16605}, {4084, 21951}, {4127, 16600}, {4424, 21904}, {4674, 9278}, {5380, 10630}, {6629, 7267}, {6763, 21008}, {10026, 21711}, {21829, 22231}

X(21840) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a^2 + a b + b^2 + a c + c^2) : :

X(21840) lies on these lines: {1, 41}, {6, 38}, {9, 5256}, {31, 16972}, {37, 42}, {81, 3509}, {213, 2292}, {306, 5257}, {672, 3666}, {910, 3745}, {968, 16970}, {976, 2271}, {1018, 4868}, {1040, 2268}, {1100, 3726}, {1213, 15523}, {1214, 1400}, {1334, 3931}, {1386, 21764}, {1449, 3873}, {1500, 6155}, {1834, 21029}, {1914, 17469}, {2092, 21814}, {2171, 18098}, {2276, 14439}, {2294, 21034}, {2295, 4642}, {2321, 3896}, {2329, 17016}, {2650, 3721}, {3210, 5749}, {3247, 3870}, {3290, 3720}, {3294, 3743}, {3684, 3920}, {3686, 4981}, {3938, 16777}, {3954, 20970}, {3985, 3995}, {3986, 4028}, {3997, 4424}, {4065, 4099}, {4071, 4972}, {4109, 5051}, {4359, 5750}, {4651, 4771}, {4850, 17754}, {4854, 17747}, {4970, 17355}, {5275, 5311}, {7191, 16503}, {10448, 16968}, {16583, 21921}, {17441, 18675}, {20684, 21811}, {21796, 21827}

X(21841) = 32ND HATZIPOLAKIS-MOSES-EULER POINT

Trilinears 2 sec A - 2 cos A - cos(B - C) : :
Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2-4 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6) : :
X(20841) = X[4] + 3 X[24], X[4] - 3 X[235], 4 X[140] - 3 X[16196], 2 X[140] - 3 X[16238]
X(20841) = 9(J^2 - 1)X[2] - (J^2 - 17)X[4] = 3(J^2 + 7)X[2] + (J^2 - 17)X[3], where J = |OH|/R

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28117.

X(21841) lies on these lines: {2,3}, {34,15325}, {53,7746}, {125,16655}, {156,13292}, {206,8550}, {230,3199}, {232,5305}, {389,16252}, {495,11399}, {496,11398}, {578,10192}, {800,1990}, {952,11363}, {1112,10272}, {1147,13142}, {1192,5878}, {1209,18358}, {1353,19118}, {1495,6146}, {1506,6748}, {1614,11245}, {1829,5901}, {1843,15074}, {1974,3564}, {2883,11438}, {3092,8981}, {3093,13966}, {3527,11427}, {5065,6749}, {5095,13431}, {5140,14693}, {5410,19116}, {5411,19117}, {5412,7584}, {5413,7583}, {5446,9820}, {5609,12828}, {5790,7718}, {5886,7713}, {5972,13598}, {6193,8780}, {6696,13474}, {6746,20193}, {6759,13567}, {6776,14530}, {7716,14561}, {8254,11576}, {9920,18946}, {9969,11808}, {10282,12241}, {10283,11396}, {10592,11392}, {10593,11393}, {10641,11543}, {10642,11542}, {11433,19347}, {11557,16534}, {11745,18388}, {11801,12140}, {12141,20253}, {12142,20252}, {12315,18913}, {13093,18931}, {13367,16657}, {13382,14862}, {13392,15463}, {13451,15806}, {15274,15594}, {16621,20299}, {18555,20771}

X(20841) = midpoint of X(i) and X(j) for these {i,j}: {24,235}, {7517,11585}
X(20841) = reflection of X(16196) in X(16238)
X(20841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1598, 1595), (3, 1596, 13488), (3, 3089, 1596), (4, 25, 7715), (4, 403, 10019), (4, 468, 140), (4, 3515, 550), (4, 4232, 3517), (4, 7715, 6756), (4, 10019, 546), (5, 25, 6756), (5, 26, 12362), (5, 7715, 4), (5, 10154, 3), (5, 13383, 6676), (5, 13490, 546), (25, 403, 13490), (25, 3542, 5), (25, 6353, 10154), (186, 1885, 548), (403, 3518, 3575), (403, 3575, 546), (427, 7505, 3628), (428, 1594, 16198), (546, 10096, 10020), (547, 16198, 1594), (1593, 3147, 549), (2072, 18378, 7553), (3089, 6353, 3), (3089, 10154, 6756), (3526, 18535, 3088), (3549, 7529, 5), (3575, 10019, 4), (3575, 13490, 6756), (3850, 18282, 140), (6240, 10151, 3853), (6622, 7487, 381), (6677, 10154, 6676), (6759, 13567, 18914), (6804, 10565, 3), (7505, 10594, 427), (7506, 15760, 9825), (10096, 13490, 6676), (10192, 15873, 578), (10201, 13861, 5), (12241, 15448, 10282), (14813, 14814, 1368)

X(21842) = ISOGONAL CONJUGATE OF X(21398)

Barycentrics a*(2*a^3-(b+c)*a^2-(2*b^2-3*b*c+2*c^2)*a+(b^2-c^2)*(b-c)) : :

X(21842) = (R-2*r)*X(1)-2*r*X(3) = R*X(8)-2*(R-2*r)*X(214)

X(21842) lies on these lines: {1,3}, {8,214}, {10,17566}, {48,1731}, {78,5288}, {79,2320}, {80,499}, {104,15071}, {140,10944}, {145,5442}, {191,5289}, {388,6902}, {497,5441}, {498,3476}, {515,6941}, {551,4311}, {579,17438}, {631,12647}, {943,15180}, {946,10483}, {952,5433}, {962,15228}, {997,5258}, {1125,3897}, {1149,4303}, {1317,5690}, {1387,6284}, {1476,15175}, {1478,3616}, {1479,5731}, {1698,17614}, {1737,5882}, {1770,13464}, {1836,4325}, {1837,3582}, {2975,5692}, {3065,11279}, {3086,6960}, {3296,5424}, {3485,4317}, {3486,10072}, {3523,5559}, {3583,11376}, {3584,3653}, {3585,5886}, {3624,10827}, {3632,5440}, {3636,4292}, {3730,17439}, {3813,10609}, {3877,5267}, {3878,11570}, {3884,4189}, {3899,3916}, {3911,13607}, {3918,17572}, {4299,5603}, {4315,13407}, {4316,12699}, {4324,12701}, {4511,5904}, {4857,11373}, {4861,4881}, {5251,19861}, {5270,11375}, {5280,9619}, {5303,5330}, {5541,10912}, {5687,15015}, {5693,6265}, {5694,17660}, {5730,6763}, {5901,7354}, {6907,10948}, {6932,10572}, {6934,11661}, {6963,10106}, {7272,17084}, {7288,7967}, {7356,12266}, {7677,18412}, {7705,20107}, {7727,11709}, {9613,17556}, {9897,18526}, {9955,18513}, {10039,10165}, {10090,11491}, {10525,12119}, {10597,11218}, {10950,15325}, {11363,17516}, {11571,19907}, {11720,19470}, {11735,18968}, {12943,18493}, {15950,18990}, {17221,17861}

X(21842) = midpoint of X(1) and X(7280)
X(21842) = isogonal conjugate of X(21398)
X(21842) = X(7280)-of-anti-Aquila triangle
X(21842) = X(16868)-of-2nd circumperp triangle
X(21842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 5903), (1, 46, 11009), (1, 484, 1482), (1, 1420, 5563), (1, 3336, 2099), (1, 5131, 11280), (1, 11280, 10247), (1, 13462, 3338), (3, 1388, 1), (36, 11009, 46), (36, 14795, 3), (46, 11009, 5903), (1319, 1385, 1), (1482, 5204, 484), (5126, 15178, 65), (13462, 15932, 56)

X(21843) = ISOGONAL CONJUGATE OF X(21399)

Barycentrics 5*a^4-4*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(21843) = 3*S^2*X(2)+(3*S^2-SW^2)*X(187)

X(21843) lies on these lines: {2,187}, {3,230}, {4,5206}, {5,5023}, {6,549}, {20,7746}, {30,5210}, {32,631}, {39,3523}, {50,18580}, {69,620}, {99,17008}, {115,376}, {140,2548}, {141,11288}, {315,7907}, {381,3054}, {439,7816}, {550,13881}, {574,3524}, {597,11173}, {616,5472}, {617,5471}, {754,1007}, {1078,7795}, {1216,15575}, {1263,2079}, {1285,7753}, {1384,3815}, {1506,3525}, {1572,10165}, {1692,10519}, {1968,3147}, {1992,7622}, {2076,14561}, {2241,7288}, {2242,5218}, {3016,8617}, {3055,15484}, {3069,9675}, {3090,7747}, {3522,7748}, {3526,7745}, {3528,7756}, {3530,5013}, {3543,18424}, {3618,15482}, {3620,7880}, {3763,8368}, {3785,3788}, {3793,9766}, {3830,15603}, {3926,7780}, {5007,14930}, {5008,15708}, {5024,5306}, {5058,13935}, {5062,9540}, {5104,20423}, {5162,9753}, {5277,6910}, {5286,15717}, {5305,15712}, {5309,8589}, {5418,12968}, {5420,12963}, {5485,15300}, {5585,8703}, {5939,9890}, {6292,14069}, {6337,7751}, {6390,8667}, {6644,8553}, {6680,16043}, {6722,16041}, {7426,20481}, {7493,10418}, {7495,9745}, {7612,11623}, {7615,8598}, {7710,10991}, {7738,7755}, {7763,7779}, {7769,7926}, {7789,15598}, {7791,7857}, {7800,7807}, {7801,15589}, {7815,14001}, {7823,16923}, {7824,7875}, {7830,14064}, {7892,16988}, {8369,15271}, {9300,15701}, {9620,10164}, {9680,12962}, {9744,21445}, {10072,10987}, {11001,18362}, {11159,15597}, {11179,15993}, {11185,13586}, {11361,17006}, {11646,12042}, {11648,19708}, {11742,15690}, {12040,15534}, {12100,15048}, {15699,18584}

X(21843) = isogonal conjugate of X(21399)
X(21843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 187, 7737), (3, 230, 2549), (115, 8588, 376), (140, 3053, 2548), (230, 2549, 3767), (574, 7735, 7739), (1078, 7835, 16990), (1078, 16925, 7795), (1384, 5054, 3815), (3524, 7735, 574), (5206, 7749, 4), (7746, 15513, 20), (7835, 16990, 7795), (15484, 15694, 3055), (16925, 16990, 7835)

X(21844) = ISOGONAL CONJUGATE OF X(21400)

Barycentrics a^2*(a^2-b^2+c^2)*(a^2+b^2-c^2)*(2*a^4-4*(b^2+c^2)*a^2+3*b^2*c^2+2*c^4+2*b^4) : :

X(21844) = 4*(4*R^2-SW)*X(3)+R^2*X(4)

As a point on the Euler line, this center has Shinagawa coefficients (-8*F, E+8*F)

X(21844) lies on these lines: {2,3}, {54,3431}, {64,12112}, {68,5963}, {74,6759}, {110,7689}, {112,5206}, {125,12289}, {185,11464}, {232,15513}, {569,15053}, {1092,3043}, {1181,1620}, {1192,7592}, {1199,9786}, {1204,1614}, {1235,7771}, {1249,8553}, {1495,12290}, {1511,7722}, {1870,7280}, {1899,12254}, {1986,6101}, {2914,12307}, {3087,15109}, {3098,11470}, {3172,15655}, {3357,14157}, {3567,11430}, {5010,6198}, {5023,8743}, {5092,6403}, {5210,8744}, {5351,8740}, {5352,8739}, {5410,6456}, {5411,6455}, {5449,12278}, {5621,15582}, {5878,12244}, {5889,12038}, {5890,13367}, {6000,11468}, {6200,10881}, {6241,10282}, {6242,10610}, {6696,16659}, {6699,11750}, {8589,10986}, {8882,14806}, {9682,13886}, {9707,10605}, {10311,15515}, {10539,11440}, {10574,18475}, {10632,10646}, {10633,10645}, {11003,13630}, {11449,13754}, {11454,12162}, {11456,17821}, {11704,13851}, {12006,14805}, {12281,17701}, {14809,18808}, {14918,15848}, {15032,19357}, {15051,15463}, {15579,19596}

X(21844) = anticomplement of X(10255)
X(21844) = isogonal conjugate of X(21400)
X(21844) = X(16868)-of-anti-Euler triangle
X(21844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,35473), (3, 26, 2071), (3, 382, 10226), (3, 7502, 3522), (4, 631, 6143), (24, 1598, 3518), (25, 14865, 4), (26, 2071, 3529), (1598, 3515, 24), (3516, 10594, 13596), (7506, 7527, 3855), (7517, 12086, 15682), (10594, 13596, 4), (11413, 12088, 11001), (12106, 14130, 3832), (15332, 18562, 20), (17714, 18859, 5059)

X(21845) = ISOGONAL CONJUGATE OF X(21401)

Barycentrics (4*S^2+4*sqrt(3)*SB*S+SB^2-SC*SA)*(4*S^2+4*sqrt(3)*SC*S+SC^2-SA*SB) : :

X(21845) lies on the Kiepert hyperbola and these lines: {2,16001}, {5340,7607}, {5478,12821}

X(21845) = isogonal conjugate of X(21401)

X(21846) = ISOGONAL CONJUGATE OF X(21402)

Barycentrics (4*S^2-4*sqrt(3)*SB*S+SB^2-SC*SA)*(4*S^2-4*sqrt(3)*SC*S+SC^2-SA*SB) : :

X(21846) lies on the Kiepert hyperbola and these lines: {2,16002}, {5339,7607}, {5479,12820}

X(21846) = isogonal conjugate of X(21402)

X(21847) = X(185)X(382)∩X(3060)X(6353)

Barycentrics a^2*(9*(b^2+c^2)*a^8-6*(3*b^4+b^2*c^2+3*c^4)*a^6+16*b^2*c^2*(b^2+c^2)*a^4+2*(9*b^8+9*c^8-b^2*c^2*(19*b^4-16*b^2*c^2+19*c^4))*a^2+(b^2-c^2)*(-9*b^4+10*b^2*c^2-9*c^4)*(b^4-c^4)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21847) lies on these lines: {185, 382}, {3060, 6353}

X(21848) = REFLECTION OF X(9957) IN X(12722)

Barycentrics a*(3*(b+c)*a^4+6*b*c*a^3+2*(b+c)*b*c*a^2-2*(b^2+c^2)*b*c*a+(b^2-c^2)*(b-c)*(-3*b^2-4*b*c-3*c^2)) : :
X(21848) = 3*X(942)-2*X(3663), 3*X(3753)-X(12530), 3*X(5902)-X(17635)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21848) lies on these lines: {517, 3056}, {518, 4133}, {942, 3663}, {2823, 9944}, {3753, 12530}, {5045, 12721}, {5902, 17635}, {7996, 11529}, {9957, 12722}

X(21848) = reflection of X(9957) in X(12722)

X(21849) = REFLECTION OF X(376) IN X(9729)

Barycentrics a^2*(3*(b^2+c^2)*a^2-3*b^4+4*b^2*c^2-3*c^4) : :
X(21849) = X(2)-3*X(51), 7*X(2)-9*X(373), 7*X(2)-3*X(2979), X(2)+3*X(3060), 4*X(2)-3*X(3819), 5*X(2)-3*X(3917), 5*X(2)-9*X(5640), 11*X(2)-9*X(5650), 2*X(2)-3*X(5943), 5*X(2)-6*X(6688), 13*X(2)-9*X(7998), 11*X(2)-12*X(10219), X(2)-9*X(11002), 11*X(2)-15*X(11451), 17*X(2)-18*X(12045), 10*X(2)-9*X(15082), 7*X(2)+9*X(16981), 7*X(51)-3*X(373), 7*X(51)-X(2979)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21849) lies on these lines: {2, 51}, {4, 14831}, {6, 9909}, {20, 15012}, {22, 575}, {23, 13366}, {25, 576}, {30, 143}, {52, 381}, {140, 16982}, {154, 5093}, {182, 9777}, {184, 5097}, {185, 3543}, {186, 13482}, {343, 19130}, {375, 9047}, {376, 3567}, {382, 13382}, {428, 542}, {524, 9969}, {539, 11808}, {541, 11807}, {547, 1216}, {549, 5462}, {568, 3830}, {578, 14070}, {597, 11574}, {599, 9822}, {970, 16370}, {1154, 5066}, {1173, 7512}, {1194, 20977}, {1196, 13330}, {1351, 9306}, {1493, 13433}, {1495, 1994}, {1503, 11225}, {1570, 1915}, {1611, 11173}, {1656, 15606}, {1843, 1992}, {2393, 8584}, {3091, 14531}, {3098, 10601}, {3167, 5102}, {3241, 16980}, {3292, 13595}, {3524, 13348}, {3527, 17834}, {3534, 9730}, {3545, 5562}, {3627, 12002}, {3628, 13421}, {3818, 6515}, {3839, 5889}, {3845, 13754}, {3981, 5052}, {5007, 6660}, {5012, 15516}, {5020, 11477}, {5032, 6467}, {5054, 10625}, {5055, 6243}, {5071, 11412}, {5092, 5422}, {5396, 19251}, {5447, 11539}, {5480, 21243}, {5663, 12101}, {5752, 16418}, {5890, 15682}, {5891, 19709}, {5892, 12100}, {5946, 8703}, {6101, 15699}, {6102, 13474}, {6353, 15010}, {6636, 15019}, {6756, 10112}, {7394, 18553}, {7667, 19924}, {7693, 15108}, {8681, 9971}, {9140, 13417}, {9967, 14848}, {10109, 10170}, {10124, 10627}, {10245, 11426}, {10304, 15043}, {10574, 15683}, {10575, 15684}, {10706, 21649}, {11402, 15520}, {11430, 18324}, {11591, 11737}, {11745, 13142}, {11812, 13363}, {12162, 14269}, {12242, 13383}, {12834, 15246}, {13292, 13419}, {13340, 15701}, {13402, 14683}, {13570, 15030}, {14128, 14892}, {14855, 15685}, {15024, 15702}, {15028, 15708}, {15045, 19708}, {15072, 15640}, {15489, 17549}, {15697, 20791}, {17530, 18180}

X(21849) = midpoint of X(i) and X(j) for these {i,j}: {4, 14831}, {52, 381}, {185, 3543}, {547, 14449}, {549, 10263}, {1843, 1992}, {3241, 16980}, {6102, 15687}, {9140, 13417}, {10575, 15684}, {10706, 21649}
X(21849) = reflection of X(i) in X(j) for these (i,j): (376, 9729), (547, 10095), (549, 5462), (599, 9822), (11591, 11737)
X(21849) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22, 15004, 575), (51, 3917, 5640), (52, 10110, 5907), (143, 5446, 389), (376, 3567, 16226), (376, 16226, 9729), (389, 5446, 13598), (2979, 3060, 16981), (3060, 11002, 51), (3917, 5640, 6688), (3917, 6688, 15082), (3917, 15082, 3819), (5640, 6688, 5943), (5650, 11451, 10219), (5943, 15082, 6688)

X(21850) = REFLECTION OF X(3) IN X(18583)

Barycentrics 5*(b^2+c^2)*a^4-4*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(21850) = 3*X(3)-5*X(3618), X(3)-3*X(14853), 3*X(4)+X(193), 5*X(4)-X(5921), 3*X(4)-X(18440), 3*X(5)-2*X(141), 3*X(5)-4*X(19130), X(141)-3*X(5480), X(193)-3*X(1351), 5*X(193)+3*X(5921), 5*X(1351)+X(5921), 3*X(1351)+X(18440), 5*X(3618)-9*X(14853), 5*X(3618)-6*X(18583), 3*X(5480)-2*X(19130), 3*X(5921)-5*X(18440), 3*X(14853)-2*X(18583)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21850) lies on these lines: {3, 3618}, {4, 193}, {5, 141}, {6, 30}, {20, 5050}, {51, 1368}, {52, 1595}, {67, 11801}, {68, 16198}, {69, 381}, {110, 10301}, {113, 1596}, {140, 1350}, {143, 19161}, {159, 7530}, {182, 550}, {235, 6403}, {265, 10752}, {325, 9993}, {376, 12017}, {382, 5093}, {427, 3060}, {428, 1993}, {495, 3056}, {496, 1469}, {517, 21629}, {524, 3818}, {542, 1539}, {546, 1352}, {547, 3763}, {548, 5085}, {549, 3098}, {567, 1176}, {575, 15704}, {576, 1353}, {597, 5092}, {599, 5066}, {611, 15171}, {613, 18990}, {648, 16264}, {858, 11002}, {895, 7728}, {1147, 7715}, {1370, 9777}, {1513, 7777}, {1533, 6467}, {1570, 7747}, {1656, 10519}, {1907, 5889}, {1974, 13352}, {1992, 3830}, {2104, 10751}, {2105, 10750}, {2456, 12110}, {2781, 10264}, {3091, 7939}, {3146, 11482}, {3167, 6995}, {3313, 13391}, {3416, 18357}, {3527, 6643}, {3545, 3620}, {3619, 5055}, {3631, 11178}, {3751, 12699}, {3820, 17792}, {3839, 20080}, {3843, 11898}, {3850, 10516}, {3853, 5102}, {3860, 15533}, {3861, 15069}, {3867, 13754}, {3933, 18906}, {5028, 7745}, {5032, 15682}, {5039, 14880}, {5052, 5254}, {5064, 6515}, {5073, 14927}, {5095, 12295}, {5097, 8550}, {5169, 16981}, {5198, 9925}, {5422, 7667}, {5448, 12002}, {5504, 11566}, {5622, 20127}, {5654, 7716}, {5847, 18480}, {6000, 15583}, {6033, 10754}, {6144, 14893}, {6243, 7403}, {6247, 16625}, {6321, 10753}, {6677, 17810}, {6756, 19139}, {6823, 9967}, {7391, 11245}, {7500, 11402}, {7576, 12383}, {7706, 11511}, {7714, 8780}, {7734, 17825}, {7897, 13862}, {8263, 9971}, {8362, 9821}, {8548, 13488}, {9024, 11698}, {9909, 11427}, {9996, 14929}, {10109, 21358}, {10128, 17811}, {10168, 17504}, {10601, 10691}, {10733, 12596}, {10738, 10759}, {10739, 10758}, {10740, 10764}, {10741, 10756}, {10742, 10755}, {10747, 10757}, {10760, 15521}, {10761, 15522}, {10766, 12918}, {10796, 13355}, {10982, 12362}, {11008, 11180}, {11061, 12902}, {11113, 15988}, {11424, 19131}, {11745, 13346}, {11818, 13562}, {12007, 15520}, {12041, 15118}, {12101, 15534}, {12161, 19149}, {12233, 13598}, {12585, 18383}, {13857, 20192}, {13860, 17008}, {14389, 15107}, {14810, 15712}, {14869, 21167}, {15033, 19121}, {15472, 19138}, {15760, 18438}, {16475, 18481}, {18533, 19118}, {18534, 19459}, {18535, 19588}, {19709, 21356}

X(21850) = midpoint of X(i) and X(j) for these {i,j}: {4, 1351}, {52, 12294}, {265, 10752}, {382, 6776}, {895, 7728}, {1992, 3830}, {2104, 10751}, {2105, 10750}, {3751, 12699}, {5073, 14927}, {5095, 12295}, {6033, 10754}, {6321, 10753}, {10738, 10759}, {10739, 10758}, {10740, 10764}, {10741, 10756}, {10742, 10755}, {10747, 10757}, {10760, 15521}, {10761, 15522}, {10766, 12918}, {11061, 12902}
X(21850) = reflection of X(i) in X(j) for these (i,j): (3, 18583), (5, 5480), (67, 11801), (69, 18358), (549, 5476), (599, 5066), (3416, 18357), (12041, 15118)
X(21850) = complement of X(33878)
X(21850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14853, 18583), (4, 193, 18440), (69, 381, 18358), (141, 5480, 19130), (141, 19130, 5), (382, 5093, 6776), (1350, 14561, 140), (1351, 18440, 193), (3098, 3589, 549), (3098, 5476, 3589)

X(21851) = REFLECTION OF X(6776) IN X(13382)

Barycentrics (SB+SC)*(4*R^2*S^2+3*(SA^2-SB*SC)*(4*R^2-SW)) : :
X(21851) = 2*X(6)-3*X(389), X(6)-3*X(19161), X(193)-3*X(14831), 7*X(3619)-6*X(11793), 5*X(3620)-3*X(5562), 3*X(5889)+X(20080), 3*X(5890)-X(6467), 3*X(5907)-4*X(18358), 3*X(10519)-2*X(15606)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21851) lies on these lines: {3, 6}, {4, 16774}, {51, 8889}, {141, 18388}, {185, 6403}, {186, 21637}, {193, 14831}, {1843, 6000}, {2781, 7687}, {3060, 7396}, {3564, 13568}, {3619, 11793}, {3620, 5562}, {5480, 6697}, {5889, 20080}, {5890, 6467}, {5907, 18358}, {6776, 13382}, {10110, 12294}, {10519, 15606}, {10605, 12167}, {11202, 19125}, {13754, 14913}

X(21851) = midpoint of X(185) and X(6403) X(21851) = reflection of X(6776) in X(13382) X(21851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3098, 11430), (9786, 16625, 389)

X(21852) = MIDPOINT OF X(5446) AND X(19161)

Barycentrics (SB+SC)*((2*R^2+SA+SW)*S^2+(SA^2-SB*SC)*(6*R^2+SA-3*SW)) : :
X(21852) = 3*X(52)+X(69), 3*X(568)+X(1843), 3*X(1216)-5*X(3763), X(3313)-3*X(5892), 5*X(3567)-X(9967), 2*X(3589)-3*X(5462), X(3818)-3*X(9969), 3*X(5946)-X(11574), 5*X(12017)-9*X(16226), 9*X(13321)-X(18438), 3*X(14831)+X(18440)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28119.

X(21852) lies on these lines: {6, 14070}, {52, 69}, {140, 143}, {568, 1843}, {1092, 1351}, {1154, 9822}, {1216, 3763}, {3060, 7386}, {3313, 5892}, {3564, 11745}, {3567, 9967}, {3818, 9969}, {5446, 14790}, {5449, 5480}, {5946, 11574}, {12017, 16226}, {13321, 18438}, {14831, 18440}

X(21852) = midpoint of X(5446) and X(19161)

X(21853) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 + a^2 b - a b^2 - b^3 + a^2 c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21853) lies on these lines: {6, 517}, {9, 5903}, {19, 2911}, {37, 65}, {40, 3553}, {44, 2262}, {46, 2178}, {53, 1872}, {72, 594}, {209, 1824}, {210, 8013}, {226, 21231}, {354, 3723}, {374, 15492}, {392, 17398}, {518, 17299}, {579, 8609}, {583, 1108}, {584, 7957}, {672, 1953}, {674, 12723}, {758, 2321}, {910, 3204}, {942, 16777}, {960, 17303}, {1018, 22021}, {1030, 3579}, {1100, 2268}, {1212, 17443}, {1213, 3753}, {1385, 5124}, {1449, 5697}, {1482, 5120}, {1500, 18591}, {1609, 11248}, {1761, 2329}, {1825, 1826}, {1880, 4559}, {2093, 2324}, {2260, 17452}, {2278, 14110}, {2289, 6603}, {2345, 3869}, {2800, 10445}, {3247, 5902}, {3554, 7982}, {3555, 17388}, {3694, 4053}, {3754, 5257}, {3868, 17314}, {3878, 5750}, {3894, 4898}, {3919, 3986}, {3943, 4018}, {3950, 4084}, {4007, 5904}, {4029, 4757}, {4058, 4067}, {4098, 4744}, {4254, 12702}, {4272, 4646}, {4641, 21368}, {4848, 21068}, {5341, 17796}, {5526, 16547}, {5836, 17275}, {5839, 14923}, {5905, 20930}, {8573, 10306}, {8804, 15556}, {9957, 16884}, {10914, 17362}, {16685, 20227}, {17747, 21933}, {20683, 21867}, {20691, 21854}, {21857, 21888}, {21874, 22300}, {21878, 21887}

X(21854) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c + a^4 b c - a b^4 c - b^5 c - 2 a^3 c^3 + 2 b^3 c^3 - a b c^4 + a c^5 - b c^5) : :

X(21854) lies on these lines: {6, 3072}, {10, 18591}, {12, 37}, {53, 18242}, {65, 21860}, {71, 3330}, {197, 1575}, {209, 3198}, {216, 515}, {577, 6796}, {1108, 1512}, {1214, 20305}, {1745, 21767}, {2182, 14737}, {2245, 21857}, {2335, 8164}, {3694, 21076}, {3990, 4551}, {4426, 10830}, {5450, 10979}, {5842, 6748}, {6047, 21896}, {6360, 18749}, {18606, 21270}, {20691, 21853}, {20713, 21891}, {21859, 21871}, {21876, 21878}

X(21855) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 - a b^3 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a c^5 - b c^5) : :

X(21855) lies on these lines: {37, 21011}, {228, 22274}, {570, 952}, {2245, 3588}, {20691, 21853}, {21865, 21891}

X(21856) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics 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) : :

X(21856) lies on these lines: {3, 9367}, {37, 226}, {57, 9444}, {71, 2238}, {516, 16588}, {1212, 2886}, {1742, 20995}, {1939, 3579}, {3177, 20935}, {3452, 6184}, {3693, 4417}, {4531, 4849}, {6181, 6244}, {7075, 9025}, {21871, 21883}

X(21857) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b + a b^2 + a^2 c - a b c - b^2 c + a c^2 - b c^2) : :

X(21857) lies on these lines: {2, 20161}, {6, 43}, {8, 2277}, {9, 6048}, {10, 37}, {39, 3686}, {44, 573}, {71, 2238}, {141, 3687}, {181, 4849}, {210, 21035}, {219, 9367}, {386, 1100}, {391, 17756}, {518, 4446}, {519, 17053}, {536, 3596}, {899, 992}, {941, 9780}, {966, 2276}, {978, 3169}, {980, 17270}, {1107, 4261}, {1400, 3214}, {1574, 4263}, {1761, 5974}, {2171, 4695}, {2245, 21854}, {2260, 3780}, {2275, 5839}, {2294, 21951}, {2300, 3216}, {2305, 5247}, {3031, 5213}, {3125, 22021}, {3666, 5224}, {3721, 3949}, {3826, 5530}, {3836, 17748}, {3965, 4429}, {3987, 21078}, {4034, 16975}, {4047, 9560}, {4270, 17750}, {4274, 16669}, {4277, 17303}, {4424, 21810}, {4642, 21033}, {4690, 16696}, {4735, 22271}, {4771, 16584}, {4850, 17238}, {6685, 17398}, {8298, 16800}, {8610, 17388}, {10822, 21866}, {14749, 17606}, {16602, 17245}, {16610, 17234}, {17362, 17448}, {20174, 21264}, {21853, 21888}, {21878, 21900}

X(21858) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b + a b^2 + a^2 c - b^2 c + a c^2 - b c^2) : :

X(21858) lies on these lines: {6, 3293}, {8, 4261}, {9, 20973}, {10, 37}, {39, 17362}, {42, 1100}, {44, 71}, {100, 1333}, {209, 3198}, {306, 3752}, {313, 536}, {319, 16696}, {583, 3780}, {872, 21865}, {980, 4445}, {1030, 5291}, {1108, 1145}, {1400, 21859}, {1574, 17398}, {1826, 5151}, {1918, 20663}, {2245, 3588}, {2262, 22313}, {2276, 4651}, {2277, 17299}, {2294, 4695}, {2295, 4272}, {2345, 4277}, {3210, 18144}, {3216, 16685}, {3666, 17239}, {3770, 17759}, {3879, 16726}, {3949, 4016}, {3958, 21805}, {3987, 22021}, {4263, 17369}, {4526, 21714}, {4850, 17228}, {5069, 5839}, {5301, 8715}, {9016, 20455}, {10950, 22071}, {16669, 20972}, {17053, 17388}, {17147, 18133}, {17495, 18143}, {20693, 21880}, {21035, 22271}, {21353, 21899}, {21753, 21877}, {21863, 21888}, {21889, 22272}

X(21859) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (a - b) (a - c) (a + b - c) (a - b + c) (b + c)^2 : :

X(21859) lies on these lines: {12, 115}, {37, 21011}, {39, 10944}, {65, 20691}, {100, 1415}, {101, 650}, {181, 1356}, {227, 4515}, {594, 2197}, {664, 3669}, {952, 11998}, {1015, 1317}, {1018, 4551}, {1319, 1575}, {1400, 21858}, {1574, 5433}, {1946, 14723}, {2276, 5252}, {2295, 2594}, {2511, 21891}, {3476, 17756}, {4033, 4552}, {4557, 22280}, {5172, 5291}, {5219, 9331}, {21854, 21871}

X(21860) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 b - 2 a^3 b^3 + a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a c^5 - b c^5) : :

X(21860) lies on these lines: {10, 5721}, {11, 233}, {37, 21011}, {42, 22063}, {44, 71}, {53, 1826}, {65, 21854}, {216, 10950}, {570, 11998}, {650, 2174}, {2197, 21933}, {4530, 21012}, {15624, 22274}, {20691, 21871}, {21891, 22279}

X(21861) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 + a^4 b - a b^4 - b^5 + a^4 c + b^4 c - a c^4 + b c^4 - c^5) : :

X(21861) lies on these lines: {37, 4456}, {210, 21703}, {1829, 4426}, {4463, 16886}, {4515, 21864}, {4646, 21863}, {9598, 20243}, {20691, 21853}

X(21862) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 + a^4 b - a b^4 - b^5 + a^4 c - a^2 b^2 c + b^4 c - a^2 b c^2 + a b^2 c^2 - a c^4 + b c^4 - c^5) : :

X(21862) lies on these lines: {37, 21800}, {284, 501}, {650, 3250}, {3125, 16784}, {20691, 21853}

X(21863) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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(21863) lies on these lines: {1, 5124}, {6, 5903}, {10, 5949}, {36, 15109}, {37, 65}, {101, 2160}, {484, 1030}, {517, 572}, {583, 1953}, {594, 758}, {672, 17443}, {942, 3723}, {1213, 3754}, {1781, 17796}, {1825, 1865}, {1845, 6748}, {1901, 15556}, {2093, 3553}, {2174, 15586}, {2260, 17444}, {2262, 2265}, {2321, 4084}, {2323, 5356}, {3057, 17440}, {3336, 21773}, {3868, 17299}, {3869, 17303}, {3874, 17388}, {3878, 17398}, {3901, 4007}, {3919, 5257}, {3943, 4757}, {3950, 4744}, {4272, 4642}, {4646, 21861}, {5697, 16884}, {5902, 16777}, {7300, 17745}, {8818, 21011}, {12432, 21933}, {20715, 21865}, {21353, 21890}, {21858, 21888}, {21889, 22277}

X(21864) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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(21864) lies on these lines: {6, 5697}, {35, 11063}, {37, 65}, {44, 517}, {45, 5903}, {101, 15586}, {210, 21943}, {484, 19297}, {502, 594}, {518, 4727}, {583, 17452}, {661, 4132}, {672, 17444}, {758, 3943}, {1018, 4053}, {1100, 5053}, {1213, 3918}, {2262, 15492}, {2321, 4067}, {2802, 4969}, {3057, 16666}, {3219, 18722}, {3697, 21019}, {3723, 5030}, {3869, 17281}, {3878, 17369}, {4029, 4084}, {4058, 4537}, {4071, 20720}, {4515, 21861}, {5526, 7297}, {5902, 16672}, {16548, 17796}, {16777, 18398}, {17484, 17791}, {20683, 21889}

X(21865) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b - a b^2 + a^2 c + b^2 c - a c^2 + b c^2) : :

X(21865) lies on these lines: {10, 15281}, {37, 3122}, {65, 21916}, {86, 3799}, {209, 14973}, {210, 8013}, {213, 22328}, {321, 22289}, {513, 17351}, {517, 19130}, {518, 3773}, {594, 4111}, {674, 17355}, {712, 6664}, {756, 4016}, {758, 4538}, {872, 21858}, {894, 4553}, {1213, 7064}, {1215, 21231}, {2238, 22327}, {2321, 22277}, {2486, 22019}, {3501, 15624}, {3589, 14839}, {3688, 17369}, {3779, 17281}, {3873, 17240}, {3912, 13476}, {4058, 22312}, {4422, 17049}, {4517, 17303}, {14624, 22301}, {16549, 20990}, {17165, 18040}, {17340, 21746}, {17357, 20358}, {20691, 21878}, {20715, 21863}, {21024, 22293}, {21855, 21891}, {21881, 21887}

X(21866) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (3 a^3 + a^2 b - 3 a b^2 - b^3 + a^2 c + 2 a b c + b^2 c - 3 a c^2 + b c^2 - c^3) : :

X(21866) lies on these lines: {6, 40}, {9, 2093}, {10, 1901}, {19, 44}, {37, 65}, {46, 219}, {48, 1155}, {55, 1100}, {57, 2256}, {209, 3198}, {210, 3958}, {227, 1409}, {284, 3579}, {517, 579}, {583, 7957}, {584, 2266}, {610, 5128}, {672, 2262}, {758, 3694}, {910, 2911}, {1030, 7688}, {1212, 15830}, {1333, 10315}, {1427, 3990}, {1743, 3987}, {1865, 1869}, {2178, 6603}, {2257, 7991}, {2260, 3057}, {2264, 5183}, {2357, 22290}, {2550, 17275}, {3101, 4641}, {3330, 21896}, {3554, 5022}, {3949, 3962}, {4018, 22021}, {4268, 16666}, {4848, 8804}, {5120, 10306}, {5124, 10902}, {5657, 5746}, {5802, 6361}, {5839, 17784}, {10822, 21857}, {11529, 16777}, {11683, 17351}, {15803, 37519}

X(21867) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 - a^2 b + a b^2 - b^3 - a^2 c + b^2 c + a c^2 + b c^2 - c^3) : :

X(21867) lies on these lines: {7, 9004}, {8, 15435}, {10, 4523}, {19, 12329}, {37, 4068}, {42, 2294}, {44, 12723}, {45, 11997}, {65, 22277}, {71, 21039}, {72, 3696}, {142, 2809}, {169, 1486}, {210, 430}, {354, 17366}, {374, 14100}, {375, 1864}, {513, 17668}, {517, 18482}, {518, 4361}, {674, 2262}, {758, 22312}, {910, 1631}, {1108, 20990}, {1212, 8053}, {1282, 18162}, {1859, 7074}, {1953, 2340}, {2171, 4878}, {2550, 3827}, {3434, 20927}, {3697, 21670}, {3740, 17293}, {3753, 22279}, {3811, 9895}, {3925, 17441}, {4058, 4538}, {4463, 4651}, {4643, 18252}, {5784, 8679}, {5813, 11677}, {8013, 21678}, {8581, 9026}, {8680, 21084}, {17334, 17635}, {20683, 21853}, {20699, 20721}, {22278, 22291}

X(21868) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a b + a c - 3 b c) : :

X(21868) lies on these lines: {8, 1575}, {10, 37}, {39, 3626}, {44, 3501}, {65, 20693}, {72, 21888}, {76, 4686}, {210, 21884}, {519, 1574}, {536, 6376}, {762, 4424}, {1015, 3625}, {1107, 3679}, {1278, 20943}, {1573, 4691}, {2176, 6048}, {2276, 3617}, {2295, 3214}, {3125, 4006}, {3632, 9336}, {3661, 3752}, {3697, 21879}, {3721, 4695}, {3730, 15492}, {3912, 16602}, {3930, 21951}, {3948, 22034}, {3954, 3987}, {4050, 16569}, {4060, 17053}, {4061, 18905}, {4103, 22036}, {4426, 5687}, {4457, 16584}, {4595, 16827}, {4651, 21877}, {4668, 16975}, {4678, 17756}, {4685, 16606}, {4882, 16973}, {7148, 22293}, {8168, 16781}, {16610, 17230}, {16666, 17750}, {17144, 20530}, {17231, 20255}, {20486, 21949}, {20697, 20719}, {21031, 21956}, {22292, 22323}

X(21869) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(16)

Barycentrics 3 a^4 - 3 a^2 b^2 - 3 a^2 c^2 - 2 Sqrt[3] S (b c + c a + a b) : :

X(21869) lies on these lines: {2, 3}, {15, 17277}, {16, 86}, {299, 17206}, {395, 18755}, {619, 6626}, {11480, 17259}, {11481, 15668}, {11485, 17349}, {11486, 17379}

X(21870) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (5 a - b - c) : :

X(21870) lies on these lines: {1, 3711}, {2, 4113}, {6, 3689}, {8, 17723}, {31, 16671}, {37, 42}, {43, 354}, {44, 2177}, {55, 1743}, {65, 3293}, {72, 4868}, {100, 4663}, {200, 1449}, {213, 21821}, {518, 3240}, {612, 16884}, {678, 21747}, {740, 4946}, {899, 17450}, {902, 16669}, {936, 2334}, {1155, 3751}, {1215, 4709}, {1386, 3935}, {1757, 4689}, {2321, 4819}, {2650, 3922}, {3175, 4090}, {3214, 3698}, {3216, 17609}, {3244, 4152}, {3246, 14997}, {3683, 16885}, {3696, 19998}, {3706, 20012}, {3740, 17018}, {3743, 4533}, {3748, 4383}, {3752, 17449}, {3753, 22295}, {3896, 3967}, {3931, 4005}, {3962, 4646}, {4046, 4058}, {4072, 6057}, {4649, 5524}, {4662, 19767}, {4685, 4732}, {4743, 21093}, {4854, 21060}, {4883, 16569}, {4906, 17020}, {5315, 22141}, {7292, 15570}, {20691, 21885}

X(21871) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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) : :

X(21871) lies on these lines: {1, 5120}, {6, 3057}, {9, 374}, {10, 21068}, {19, 220}, {37, 65}, {40, 198}, {48, 6603}, {55, 3553}, {72, 1903}, {210, 430}, {219, 1766}, {322, 329}, {346, 3869}, {354, 16777}, {391, 14923}, {392, 5750}, {518, 17314}, {644, 5279}, {672, 1108}, {674, 14100}, {758, 3950}, {910, 6602}, {942, 3247}, {960, 2345}, {966, 5836}, {1018, 3694}, {1100, 2267}, {1122, 17276}, {1155, 2178}, {1212, 1953}, {1213, 3698}, {1229, 21273}, {1436, 6282}, {1449, 9957}, {1604, 6244}, {1743, 5697}, {1778, 18178}, {1901, 22076}, {2098, 3554}, {2256, 2285}, {2264, 2911}, {2270, 7991}, {2318, 3198}, {2331, 3195}, {3230, 20227}, {3678, 4058}, {3686, 10914}, {3688, 12723}, {3723, 17609}, {3731, 5903}, {3753, 5257}, {3754, 3986}, {3779, 11997}, {3781, 5784}, {3877, 5749}, {3878, 17355}, {3880, 5839}, {3893, 17362}, {3930, 17441}, {3943, 3962}, {3949, 4515}, {3983, 21011}, {3990, 4559}, {3991, 22021}, {4005, 21873}, {4018, 4029}, {4047, 21061}, {4067, 4072}, {4084, 4098}, {4254, 5119}, {4513, 5227}, {4731, 21012}, {4877, 18180}, {5336, 14974}, {5742, 15488}, {5838, 7673}, {5902, 16673}, {7071, 12329}, {8607, 22071}, {10306, 11434}, {10440, 18236}, {10445, 12672}, {11248, 15817}, {11362, 20262}, {14872, 17732}, {17735, 20359}, {20691, 21860}, {20693, 20697}, {20694, 22301}, {20713, 20714}, {21854, 21859}, {21856, 21883}, {21888, 21892}

X(21872) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (3 a^2 - 2 a b - b^2 - 2 a c + 2 b c - c^2) : :

X(21872) lies on these lines: {1, 5022}, {2, 4520}, {3, 6603}, {6, 1697}, {9, 5836}, {10, 17747}, {37, 65}, {40, 220}, {44, 2082}, {63, 4513}, {72, 1018}, {101, 3579}, {144, 16284}, {165, 3207}, {169, 12702}, {213, 4646}, {218, 5119}, {227, 4559}, {355, 17732}, {392, 16549}, {517, 1212}, {518, 3208}, {594, 8804}, {672, 3057}, {728, 12526}, {758, 3991}, {960, 3501}, {1100, 3303}, {1104, 9620}, {1146, 11362}, {1155, 9310}, {1191, 9593}, {1213, 21068}, {1475, 5919}, {2176, 3752}, {2238, 21896}, {2267, 16666}, {2321, 4047}, {2329, 4640}, {3198, 3690}, {3294, 3753}, {3693, 3869}, {3880, 21384}, {3922, 21921}, {3930, 3962}, {3931, 3997}, {3967, 4095}, {3970, 4018}, {4253, 9957}, {4531, 4849}, {4848, 21049}, {4875, 14923}, {4919, 11260}, {5134, 18480}, {5179, 5690}, {5526, 11010}, {5537, 15855}, {5584, 7368}, {5819, 20070}, {5839, 12632}, {5903, 16601}, {6602, 7964}, {6706, 17753}, {7195, 17276}, {7957, 15853}, {10914, 16552}, {11518, 16777}, {16605, 18785}

X(21873) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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(21873) lies on these lines: {1, 6}, {10, 5949}, {71, 10693}, {191, 1030}, {210, 8013}, {319, 20538}, {321, 4886}, {502, 594}, {573, 5694}, {758, 1213}, {1761, 15586}, {2178, 3927}, {2185, 3219}, {2238, 4016}, {2245, 3958}, {2292, 4272}, {2321, 4115}, {2895, 20932}, {3204, 5282}, {3681, 4037}, {3691, 17443}, {3869, 17275}, {3876, 17303}, {3878, 17362}, {3899, 4034}, {3943, 3988}, {3950, 4537}, {3986, 4525}, {4005, 21871}, {4067, 5257}, {6763, 21773}, {10176, 17398}, {20693, 21883}, {20694, 21889}, {20716, 22289}

X(21874) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (3 a^2 - b^2 - c^2) : :

X(21874) lies on these lines: {1, 6}, {65, 2238}, {172, 1812}, {193, 18156}, {209, 21876}, {210, 2295}, {321, 5395}, {758, 16583}, {762, 4533}, {997, 5021}, {1245, 4272}, {1707, 3053}, {1824, 14248}, {2092, 4047}, {2129, 15369}, {2205, 3998}, {2271, 12514}, {2901, 4115}, {3057, 3780}, {3125, 4018}, {3198, 5360}, {3263, 20109}, {3290, 3868}, {3678, 3997}, {3721, 3962}, {3811, 14974}, {3931, 20970}, {4067, 16600}, {4084, 16611}, {4090, 4095}, {4489, 8601}, {4515, 20693}, {4531, 4849}, {4640, 18755}, {4646, 21904}, {4673, 5839}, {5044, 17750}, {5743, 20255}, {7187, 20072}, {9596, 14555}, {10822, 21857}, {21853, 22300}, {21888, 21896}, {21895, 21897}

X(21875) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^9 b - 2 a^5 b^5 + a b^9 + a^9 c + a^8 b c - a b^8 c - b^9 c - 2 a^5 c^5 + 2 b^5 c^5 - a b c^8 + a c^9 - b c^9) : :

X(21875) lies on these lines: {37, 21812}, {209, 22302}, {1826, 3914}

X(21876) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 b + a b^3 + a^3 c + a^2 b c - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(21876) lies on these lines: {37, 306}, {65, 21954}, {71, 2238}, {209, 21874}, {210, 20691}, {518, 3981}, {1107, 3966}, {1196, 5847}, {1500, 4104}, {1575, 4383}, {1613, 9025}, {1716, 21775}, {1826, 21956}, {2276, 14555}, {2348, 14737}, {2887, 4109}, {3290, 18134}, {3752, 9284}, {3925, 16605}, {17792, 21779}, {21854, 21878}

X(21877) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(21877) lies on these lines: {2, 37}, {10, 21838}, {39, 3741}, {42, 2229}, {43, 213}, {71, 2238}, {72, 695}, {100, 699}, {172, 13588}, {194, 17149}, {210, 21897}, {292, 1999}, {306, 18905}, {672, 3588}, {726, 22039}, {740, 16584}, {872, 21895}, {1011, 4426}, {1215, 3774}, {1613, 1740}, {1824, 17980}, {2275, 10453}, {3097, 22206}, {3121, 3896}, {3198, 5360}, {3294, 16569}, {3687, 9284}, {3720, 16604}, {3773, 16587}, {3840, 21070}, {3914, 18904}, {3952, 21884}, {3954, 17042}, {3971, 21830}, {3993, 21827}, {4365, 21327}, {4642, 22230}, {4651, 21868}, {4695, 22173}, {4709, 22184}, {4849, 21893}, {17135, 17448}, {20671, 22024}, {21753, 21858}, {22013, 22036}

X(21878) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 b^3 + a^3 c^3 - b^3 c^3) : :

X(21878) lies on these lines: {37, 313}, {210, 21881}, {730, 8265}, {1575, 17348}, {8620, 18133}, {16584, 21238}, {16606, 22279}, {18144, 20284}, {20691, 21865}, {21753, 21858}, {21853, 21887}, {21854, 21876}, {21857, 21900}

X(21879) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 - a b - b^2 - a c - b c - c^2) : :

X(21879) lies on these lines: {1, 6}, {8, 4037}, {10, 115}, {35, 20677}, {39, 10176}, {42, 9281}, {101, 501}, {172, 593}, {187, 3647}, {191, 5277}, {210, 20691}, {230, 18253}, {304, 4643}, {321, 3975}, {662, 2134}, {756, 2295}, {758, 16589}, {762, 1018}, {846, 18755}, {1334, 2503}, {1500, 3678}, {1573, 3878}, {1575, 5044}, {1654, 17762}, {2210, 3683}, {2238, 2292}, {2276, 3876}, {3178, 10026}, {3208, 22206}, {3681, 21820}, {3691, 3727}, {3697, 21868}, {3743, 20970}, {3931, 21904}, {3952, 21021}, {3985, 21024}, {3995, 20016}, {4095, 4096}, {4386, 12514}, {4517, 5360}, {5293, 17735}, {5969, 17760}, {17192, 17237}, {20694, 22328}, {21885, 21899}

X(21880) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^4 - a^2 b^2 - b^4 - a^2 c^2 - b^2 c^2 - c^4) : :

X(21880) lies on these lines: {37, 82}, {42, 2240}, {72, 695}, {321, 6653}, {2896, 20934}, {3496, 3954}, {4523, 21345}, {10329, 16556}, {20693, 21858}

X(21881) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (-a^3 b^3 + a^4 b c - a^3 b^2 c + a^2 b^3 c - a b^4 c - a^3 b c^2 - a b^3 c^2 - a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4) : :

X(21881) lies on these lines: {37, 18082}, {210, 21878}, {20691, 20713}, {20693, 21858}, {21865, 21887}, {21897, 22271

X(21882) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^4 b^2 - 2 a^3 b^3 + a^2 b^4 + a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 - 2 a^3 c^3 + 2 b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(21882) lies on these lines: {37, 1441}, {71, 2238}, {1818, 9367}, {4878, 20691}, {20697, 22301}

X(21883) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 + a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21883) lies on these lines: {2, 37}, {42, 2107}, {210, 20691}, {740, 21838}, {756, 21897}, {894, 2668}, {984, 22206}, {986, 22202}, {1045, 21779}, {1107, 3706}, {1215, 1500}, {1962, 2229}, {2092, 3985}, {3774, 3971}, {3970, 18208}, {3993, 16584}, {4090, 4115}, {4383, 16369}, {4854, 18904}, {6541, 16587}, {20693, 21873}, {20714, 21899}, {21856, 21871}

X(21884) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 - 2 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - 3 b^2 c^2) : :

X(21884) lies on these lines: {37, 714}, {210, 21868}, {312, 17448}, {2275, 4903}, {3175, 21893}, {3952, 21877}, {3994, 21345}, {4082, 18905}, {4090, 20691}, {4849, 20694}, {10453, 14470}, {20693, 20697}, {20714, 20721}

X(21885) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (5 a^2 - 5 a b - b^2 - 5 a c + 7 b c - c^2) : :

X(21885) lies on these lines: {37, 1018}, {44, 5541}, {210, 21886}, {2295, 21806}, {9324, 21781}, {17460, 20331}, {20691, 21870}, {21879, 21899}

X(21886) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (8 a^2 - 5 a b - 4 b^2 - 5 a c + b c - 4 c^2) : :

X(21886) lies on these lines: {37, 758}, {44, 17461}, {210, 21885}, {20691, 21805}

X(21887) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (-a^3 b^3 + a^4 b c + a^3 b^2 c - a^2 b^3 c - a b^4 c + a^3 b c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a b c^4) : :

X(21887) lies on these lines: {37, 86}, {2511, 21888}, {5539, 21783}, {16592, 21254}, {17989, 21889}, {20691, 21891}, {20698, 21897}, {20713, 21895}, {21853, 21878}, {21865, 21881}

X(21888) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 - a b - b^2 - a c + 3 b c - c^2) : :

X(21888) lies on these lines: {8, 9597}, {10, 115}, {37, 1018}, {40, 4426}, {42, 4128}, {44, 3245}, {65, 20691}, {72, 21868}, {213, 3987}, {335, 4595}, {484, 5291}, {517, 1575}, {758, 20693}, {1015, 2802}, {1054, 4919}, {1086, 21232}, {1100, 5541}, {1107, 5836}, {1334, 21951}, {1500, 3754}, {1574, 3878}, {1739, 3230}, {2170, 20331}, {2238, 4695}, {2275, 14923}, {2295, 4642}, {2511, 21887}, {2610, 21890}, {3057, 16604}, {3120, 21013}, {3208, 20271}, {3501, 3959}, {3693, 21331}, {3918, 16589}, {4083, 9267}, {4169, 22035}, {4386, 9620}, {4440, 18159}, {4730, 21889}, {5254, 8256}, {5554, 9598}, {7200, 21272}, {10914, 17448}, {11998, 17636}, {16605, 18785}, {17164, 21021}, {20718, 21897}, {21832, 21893}, {21853, 21857}, {21858, 21863}, {21871, 21892}, {21874, 21896}, {21902, 22278}

X(21889) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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(21889) lies on these lines: {10, 8287}, {37, 4068}, {42, 2643}, {149, 18151}, {190, 2805}, {210, 6535}, {518, 4716}, {594, 20495}, {1086, 2809}, {1731, 19624}, {1824, 4849}, {2161, 3939}, {2340, 17444}, {2870, 7077}, {3204, 4336}, {3696, 4523}, {4145, 22309}, {4155, 21890}, {4651, 21295}, {4730, 21888}, {5220, 15076}, {5540, 16686}, {11997, 16814}, {12723, 16669}, {13576, 16732}, {15586, 17798}, {17344, 18252}, {17441, 21949}, {17989, 21887}, {20683, 21864}, {20694, 21873}, {21035, 22287}, {21858, 22272}, {21863, 22277}, {22286, 22316}, {22289, 22304}

X(21890) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5) : :

X(21890) lies on these lines: {37, 101}, {758, 20720}, {1334, 2503}, {2079, 13204}, {2321, 4115}, {2610, 21888}, {2836, 11646}, {3678, 20700}, {4155, 21889}, {20951, 21221}, {21004, 21381}, {21353, 21863}

X(21891) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (a - b) (a - c) (b + c)^2 (a^3 + a^2 b - a b^2 - b^3 + a^2 c + 3 a b c + b^2 c - a c^2 + b c^2 - c^3) : :

X(21891) lies on these lines: {37, 21043}, {100, 22311}, {662, 8043}, {2511, 21859}, {4036, 4552}, {4557, 4705}, {20691, 21887}, {20713, 21854}, {21855, 21865}, {21860, 22279}

X(21892) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b + a b^2 + a^2 c - 3 a b c - b^2 c + a c^2 - b c^2) : :

X(21892) lies on these lines: {6, 978}, {9, 1575}, {10, 37}, {42, 22174}, {44, 579}, {142, 16602}, {198, 4426}, {210, 3778}, {391, 2275}, {518, 17065}, {966, 1107}, {992, 2183}, {995, 1100}, {1125, 4263}, {1329, 1738}, {1376, 1716}, {1400, 2238}, {1574, 17355}, {2171, 21951}, {2276, 5296}, {2664, 17792}, {3125, 21078}, {3169, 16969}, {3290, 3705}, {3662, 5233}, {3666, 17248}, {3686, 17053}, {3721, 21033}, {3752, 4357}, {4695, 21809}, {5750, 6686}, {8610, 17362}, {21214, 21785}, {21871, 21888}

X(21893) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 - 3 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21893) lies on these lines: {10, 16592}, {37, 3121}, {42, 2107}, {210, 16606}, {292, 3699}, {536, 21224}, {537, 6377}, {650, 6164}, {1054, 9360}, {1646, 17154}, {1979, 9359}, {2229, 21805}, {3175, 21884}, {3967, 21345}, {4009, 20363}, {4090, 16584}, {4096, 21827}, {4849, 21877}, {9263, 18149}, {14404, 21900}, {17989, 21887}, {20691, 21870}, {21832, 21888}

X(21894) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b - c) (b + c) (a^3 - a b^2 + 3 a b c - b^2 c - a c^2 - b c^2) : :

X(21894) lies on these lines: {11, 115}, {37, 4120}, {44, 513}, {647, 14321}, {665, 1639}, {810, 4162}, {900, 3310}, {905, 4129}, {1018, 4551}, {1577, 3669}, {2786, 3666}, {3030, 5213}, {3683, 5075}, {3700, 7180}, {3752, 4750}, {3960, 4928}, {4730, 22314}, {4944, 21348}, {6591, 16228}, {8034, 17989}, {21832, 21888}

X(21894) = midpoint of Kiepert-hyperbola-intercepts of antiorthic axis

X(21895) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 b^3 - a^3 b^2 c + a^2 b^3 c - a^3 b c^2 - a b^3 c^2 + a^3 c^3 + a^2 b c^3 - a b^2 c^3 - b^3 c^3) : :

X(21895) lies on these lines: {37, 714}, {726, 6375}, {872, 21877}, {3121, 22016}, {4043, 21345}, {4698, 16604}, {20691, 21865}, {20713, 21887}, {21874, 21897}

X(21896) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 + 2 a b + b^2 + 2 a c - 6 b c + c^2) : :

X(21896) lies on these lines: {1, 3848}, {6, 1706}, {8, 3752}, {10, 37}, {12, 21949}, {40, 44}, {42, 3698}, {43, 5836}, {72, 3987}, {145, 16610}, {210, 4642}, {341, 536}, {899, 3057}, {960, 6048}, {978, 3880}, {986, 4662}, {1104, 5687}, {1201, 3893}, {1254, 22317}, {1279, 1722}, {1427, 4848}, {1616, 2136}, {1738, 12607}, {1739, 3555}, {2238, 21872}, {2292, 3983}, {2650, 3922}, {2911, 15811}, {3017, 5956}, {3216, 10914}, {3242, 4882}, {3293, 3753}, {3330, 21866}, {3617, 3666}, {3689, 3924}, {3697, 4424}, {3701, 22034}, {3772, 7080}, {3914, 21031}, {3962, 21805}, {4018, 4674}, {4255, 9623}, {4385, 4686}, {4452, 6552}, {4675, 11024}, {4678, 4850}, {4689, 5260}, {4864, 6765}, {4875, 17756}, {5530, 9710}, {5657, 15852}, {5711, 16666}, {5815, 17276}, {6047, 21854}, {6051, 19875}, {9026, 17114}, {9957, 17749}, {12514, 15492}, {21874, 21888}

X(21897) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 + a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - b^2 c^2) : :

X(21897) lies on these lines: {10, 37}, {75, 21021}, {210, 21877}, {291, 518}, {536, 1921}, {726, 4103}, {756, 21883}, {762, 984}, {872, 2295}, {2229, 21805}, {2664, 21788}, {3121, 19998}, {4023, 9284}, {4426, 15624}, {4457, 22184}, {4651, 21814}, {4685, 16584}, {4686, 21443}, {4695, 20706}, {4698, 16819}, {4849, 16606}, {16587, 21085}, {20698, 21887}, {20718, 21888}, {21874, 21895}, {21881, 22271}

X(21898) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(15)

Barycentrics 3 a^4 - 3 a^2 b^2 - 3 a^2 c^2 + 2 Sqrt[3] S (b c + c a + a b) : :

X(21898) lies on these lines: {2, 3}, {15, 86}, {16, 17277}, {298, 17206}, {396, 18755}, {618, 6626}, {11480, 15668}, {11481, 17259}, {11485, 17379}, {11486, 17349}

X(21899) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^4 - a^2 b^2 - b^4 - a^2 c^2 + 3 b^2 c^2 - c^4) : :

X(21899) lies on these lines: {10, 6627}, {37, 100}, {148, 20939}, {321, 6653}, {1824, 17980}, {2610, 21888}, {2640, 20998}, {2641, 21381}, {3124, 9278}, {5524, 20685}, {17989, 21887}, {20693, 20720}, {20698, 21805}, {20714, 21883}, {21353, 21858}, {21879, 21885}

X(21900) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^3 b^3 - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3) : :

X(21900) lies on these lines: {10, 1084}, {37, 4033}, {1646, 18150}, {2511, 21887}, {14404, 21893}, {21857, 21878}

X(21901) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b - c) (b + c) (a^3 b - a^2 b^2 + a^3 c - a^2 b c - a^2 c^2 - b^2 c^2) : :

X(21901) lies on these lines: {37, 1577}, {523, 22229}, {650, 8632}, {798, 22320}, {804, 3709}, {1107, 3907}, {1500, 4151}, {1575, 9321}, {2276, 4560}, {4041, 20691}, {4426, 21789}, {4455, 4705}, {6586, 17072}, {14407, 22224}

X(21902) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (a^2 b^2 - a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - b^2 c^2) : :

X(21902) lies on these lines: {37, 714}, {42, 2107}, {190, 893}, {210, 21877}, {756, 2229}, {982, 22171}, {1575, 3740}, {2275, 18743}, {3121, 3995}, {3175, 21345}, {3703, 9284}, {3752, 17755}, {3774, 4090}, {3932, 18905}, {3952, 21814}, {3994, 21327}, {4096, 21830}, {4415, 18904}, {4531, 4849}, {21888, 22278}

X(21903) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(61)

Barycentrics a^4 - a^2 b^2 - a^2 c^2 + 2 Sqrt[3] S (b c + c a + a b) : :

X(21903) lies on these lines: {2, 3}, {61, 86}, {62, 17277}, {302, 17206}, {629, 6626}, {2271, 11488}, {5021, 11489}, {15668, 22236}, {17259, 22238}

X(21904) = (A,B,C,X(1); A',B',C',X(37)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics a (b + c) (2 a^2 + a b + a c - b c) : :

X(21904) lies on these lines: {2, 319}, {6, 43}, {10, 20970}, {37, 42}, {44, 751}, {65, 9278}, {71, 20697}, {200, 16972}, {213, 1018}, {239, 21264}, {291, 4663}, {350, 4852}, {386, 1107}, {594, 4685}, {672, 4271}, {899, 16666}, {1193, 3780}, {1215, 4771}, {1386, 3783}, {1449, 16569}, {1964, 20464}, {2108, 16477}, {2271, 4426}, {2294, 22173}, {2295, 3214}, {2650, 21951}, {2887, 10026}, {2999, 16973}, {3216, 16604}, {3686, 6685}, {3723, 17018}, {3741, 17362}, {3795, 16468}, {3811, 16974}, {3875, 4713}, {3896, 4037}, {3931, 21879}, {3943, 4946}, {4006, 21802}, {4424, 21839}, {4646, 21874}, {4698, 17032}, {4856, 6686}, {5247, 18755}, {5283, 5312}, {5313, 16975}, {10987, 17127}, {16671, 17756}, {17027, 20530}, {17299, 20012}, {17351, 17759}, {17366, 20335}, {21753, 21858}

X(21905) = CROSSSUM OF X(99) AND X(691)

Barycentrics a^2 (2 a^2-b^2-c^2) (b^2-c^2) (a^2 b^2+b^4+a^2 c^2-4 b^2 c^2+c^4) : :

X(21905) lies on the Jerabek hyperbola of the medial triangle and on these lines: {3,8651}, {5,1499}, {6,512}, {141,3566}, {690,5181}, {924,8542}, {1511,2780}, {1648,2682}, {3005,14824}, {11053,11183}, {11672,21731}

X(21905) = complement of the isogonal of X(11634)
X(21905) = medial isogonal conjugate of X(3143)
X(21905) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 8681}, {2489, 351}, {3565, 187}
X(21905) = X(799)-isoconjugate of X(15387)
X(21905) = crosspoint of X(i) and X(j) for these (i,j): {512, 690}, {11634, 14263}
X(21905) = crossdifference of every pair of points on line {524, 9225}
X(21905) = crosssum of X(99) and X(691)
X(21905) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3143}, {162, 8681}, {163, 6390}, {3291, 8287}, {11634, 10}, {16756, 116}, {17466, 5099}
X(21905) = barycentric product X(i)*X(j) for these {i,j}: {126, 512}, {187, 9134}, {661, 17466}, {690, 3291}, {1648, 11634}, {1649, 14263}, {5140, 14417}, {8681, 14273}
X(21905) = barycentric quotient X(i)/X(j) for these {i,j}: {126, 670}, {669, 15387}, {3291, 892}, {9134, 18023}, {17466, 799}

X(21906) = CROSSSUM OF X(99) AND X(524)

Barycentrics a^2 (2 a^2-b^2-c^2) (b^2-c^2)^2 : :

X(21906) lies on these lines: {2,14948}, {3,6}, {115,2793}, {351,865}, {524,18872}, {1648,1649}, {1916,3228}, {2854,5106}, {3291,5968}, {3455,19626}, {5166,21460}, {7801,19663}, {9227,18023}, {10567,16186}, {14824,19610}, {15525,15526}

X(21906) = X(2374)-complementary conjugate of X(21259)
X(21906) = X(i)-Ceva conjugate of X(j) for these (i,j): {111, 512}, {187, 351}, {1177, 9426}, {3266, 690}, {3455, 669}, {9227, 523}
X(21906) = X(i)-isoconjugate of X(j) for these (i,j): {662, 892}, {691, 799}, {897, 4590}, {1101, 18023}, {4610, 5380}
X(21906) = crosspoint of X(i) and X(j) for these (i,j): {6, 9178}, {111, 512}, {187, 351}, {690, 3266}
X(21906) = crossdifference of every pair of points on line {99, 523}
X(21906) = crosssum of X(i) and X(j) for these (i,j): {2, 5468}, {6, 11634}, {99, 524}, {671, 892}, {1648, 11123}
X(21906) = barycentric product X(i)*X(j) for these {i,j}: {6, 1648}, {74, 2682}, {115, 187}, {338, 14567}, {351, 523}, {468, 20975}, {512, 690}, {524, 3124}, {647, 14273}, {661, 2642}, {691, 14443}, {882, 11183}, {896, 2643}, {922, 1109}, {1084, 3266}, {1649, 9178}, {2489, 14417}, {2971, 6390}, {3122, 4062}, {3125, 21839}, {3292, 8754}, {3455, 5099}, {4079, 4750}, {4705, 14419}, {5467, 8029}, {10630, 14444}, {11053, 19610}, {14424, 18105}, {16702, 21833}
X(21906) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 18023}, {187, 4590}, {351, 99}, {512, 892}, {669, 691}, {690, 670}, {1084, 111}, {1356, 7316}, {1645, 14609}, {1648, 76}, {2642, 799}, {2682, 3260}, {2971, 17983}, {3124, 671}, {4117, 923}, {7063, 5547}, {11183, 880}, {14273, 6331}, {14419, 4623}, {14567, 249}, {15630, 9154}, {21839, 4601}
X(21906) = isogonal conjugate of isotomic conjugate of X(1648)
X(21906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 187, 9177), (5968, 14898, 3291)

X(21907) = ISOGONAL CONJUGATE OF X(17796)

Barycentrics (a^3-(b-c)*a^2-(b^2+b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3+(b-c)*a^2+(b^2-b*c-c^2)*a+(b^2-c^2)*(b-c)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28124.

X(21907) on the circumconics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(651), X(13149)}} and on these lines: {1, 149}, {2, 16732}, {23, 105}, {28, 2969}, {57, 18625}, {81, 1086}, {89, 4000}, {323, 2990}, {651, 2982}, {1170, 5723}, {1255, 17056}, {2006, 18593}, {4080, 5376}, {5249, 5483}, {17484, 17796}

X(21907) = isogonal conjugate of X(17796)
X(21907) = isotomic conjugate of X(32849)
X(21907) = trilinear pole of the line {513, 942} (the minor axis of the conic described at X(942))

X(21908) = X(5)X(141)∩X(3060)X(5140)

Barycentrics (SB+SC)*(6*S^4-(4*(3*SA-SW)*R^2-6*SB*SC+5*SW^2)*S^2-(2*SA-3*SW)*SA*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28125.

X(21908) lies on these lines: {5, 141}, {3060, 5140}

X(21909) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(371)

Barycentrics a^4 - a^2 b^2 - a^2 c^2 + 2S( b c + c a + a b) : :

See X(21992).

X(21909) lies on these lines: {2, 3}, {86, 371}, {261, 13333}, {372, 17277}, {487, 4648}, {488, 966}, {492, 17206}, {590, 18755}, {641, 6626}, {1151, 15668}, {1152, 17259}, {2271, 3068}, {3069, 5021}, {3311, 17379}, {3312, 17349}

X(21909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 21992), (4, 33035, 21992), (5, 16917, 21992), (140, 33047, 21992)

X(21910) = X(154)X(511)∩X(523)X(4855)

Barycentrics (SB+SC)*(2*S^4+((16*R^2-5*SW)*SW+2*SA^2+2*SB*SC)*S^2-(4*R^2*(6*SA+SW)-4*SA^2+4*SB*SC-3*SW^2)*SA*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28125.

X(21910) lies on these lines: {154, 511}, {523, 4885}, {7396, 9742}

X(21911) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - a c^4 - b c^4 + c^5) : :

X(21911) lies on these lines: {10, 2318}, {37, 21717}, {125, 21807}, {427, 1953}, {594, 2294}, {1861, 14547}, {2265, 11245}, {3136, 21011}, {3914, 21945}, {4466, 21318}, {7069, 13567}, {21015, 21801}, {21029, 21687}, {21936, 21944}

X(21912) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 + b^2 + c^2) (-a^3 + b^3 - b^2 c - b c^2 + c^3) : :

X(21912) lies on these lines: {2, 212}, {10, 201}, {37, 21717}, {48, 1899}, {71, 1213}, {73, 18641}, {125, 228}, {198, 1853}, {343, 1818}, {427, 2183}, {656, 22069}, {1368, 22097}, {1698, 5715}, {1861, 7069}, {1867, 21686}, {2317, 11245}, {4413, 7085}, {4466, 17441}, {4854, 21049}, {6536, 21931}, {13567, 14547}, {15523, 21914}, {17073, 20277}, {21918, 21948}, {21935, 21936}

X(21913) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^5 - 3 a^3 b^2 - a^2 b^3 + a b^4 + b^5 + a^2 b^2 c - b^4 c - 3 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 - a^2 c^3 + a c^4 - b c^4 + c^5) : :

X(21913) lies on these lines: {37, 21717}, {2486, 3925}, {4466, 21319}, {11245, 17438}, {21923, 21948}

X(21914) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^4 - 2 a^3 b - a^2 b^2 + b^4 - 2 a^3 c + 4 a^2 b c - 2 b^3 c - a^2 c^2 + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(21914) lies on these lines: {10, 4552}, {37, 21931}, {2486, 3925}, {4088, 14429}, {4413, 17325}, {4466, 4557}, {15523, 21912}, {21013, 21952}

X(21915) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 + b^2 + c^2) (-a^3 b + a^2 b^2 - a b^3 + b^4 - a^3 c - 2 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + c^4) : :

X(21915) lies on these lines: {71, 594}, {1146, 6284}, {2968, 22070}, {3925, 21930}, {4642, 21933}, {4854, 21049}, {5254, 17871}, {21015, 21020}, {21678, 21690}, {21921, 21931}

X(21916) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^5 b^2 - a^4 b^3 - a b^6 + b^7 + a^4 b^2 c - b^6 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 + b^5 c^2 - a^4 c^3 - b^4 c^3 + a b^2 c^4 - b^3 c^4 + b^2 c^5 - a c^6 - b c^6 + c^7) : :

X(21916) lies on these lines: {37, 21717}, {65, 21865}, {3949, 15523}, {4466, 21322}

X(21917) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^5 b^2 - a^4 b^3 - a b^6 + b^7 + a^4 b^2 c - b^6 c + a^5 c^2 + a^4 b c^2 - 2 a^3 b^2 c^2 + a b^4 c^2 + b^5 c^2 - a^4 c^3 - b^4 c^3 + a b^2 c^4 - b^3 c^4 + b^2 c^5 - a c^6 - b c^6 + c^7): :

X(21917) lies on these lines: {37, 21717}, {4466, 21323}, {21959, 21964}

X(21918) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b^4 + b^5 - b^4 c - a c^4 - b c^4 + c^5) : :

X(21918) lies on these lines: {37, 21939}, {17047, 21329}, {21023, 21925}, {21029, 21687}, {21912, 21948}, {21944, 21954}

X(21919) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + 2 a^2 b^2 c - a b^3 c - b^4 c + a^3 c^2 + 2 a^2 b c^2 + 4 a b^2 c^2 - a^2 c^3 - a b c^3 - a c^4 - b c^4 + c^5) : :

X(21919) lies on these lines: {2, 7073}, {10, 11553}, {594, 2294}, {3136, 21045}, {4854, 21717}, {21692, 21926}

X(21920) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + a b^3 c - b^4 c + a^3 c^2 - a^2 c^3 + a b c^3 - a c^4 - b c^4 + c^5) : :

X(21920) lies on these lines: {10, 5496}, {594, 2294}, {8611, 21052}

X(21921) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (a b - b^2 + a c + 4 b c - c^2) : :

X(21921) lies on these lines: {1, 16611}, {2, 257}, {9, 11684}, {10, 3930}, {37, 3698}, {38, 20271}, {42, 16605}, {145, 4771}, {210, 22317}, {244, 1107}, {442, 21044}, {672, 3812}, {750, 16968}, {756, 3721}, {942, 3691}, {1018, 3918}, {1125, 2170}, {1213, 2294}, {1334, 3753}, {1475, 5439}, {1500, 4695}, {1835, 7079}, {1962, 4433}, {2171, 4848}, {2238, 2650}, {2292, 3125}, {3290, 10459}, {3294, 3754}, {3496, 5047}, {3509, 5260}, {3622, 4051}, {3742, 4875}, {3842, 20706}, {3922, 21872}, {3924, 5275}, {3925, 21029}, {3985, 17164}, {3986, 21809}, {3991, 4002}, {3992, 22011}, {4515, 4731}, {4520, 10107}, {4698, 20593}, {4714, 21070}, {5011, 5259}, {5883, 16552}, {9310, 19860}, {14439, 16601}, {16583, 21840}, {16604, 21332}, {16886, 21026}, {18904, 22174}, {19318, 19557}, {21020, 21024}, {21674, 21950}, {21687, 21717}, {21915, 21931}

X(21922) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b - a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(21922) lies on these lines: {37, 21967}, {2294, 20486}, {2533, 21131}, {3589, 4124}, {3925, 21023}, {4026, 7235}, {16732, 22279}, {21021, 21022}

X(21923) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (a^2 b^2 - a b^3 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - a c^3) : :

X(21923) lies on these lines: {37, 21936}, {1194, 17445}, {1211, 20706}, {3122, 21327}, {3125, 16587}, {3703, 17451}, {3930, 4046}, {21029, 21687}, {21913, 21948}, {21939, 21944}

X(21924) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^4 b - 2 a^2 b^3 + b^5 + a^4 c + 2 a^2 b^2 c - 4 a b^3 c + b^4 c + 2 a^2 b c^2 + 8 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 4 a b c^3 - 2 b^2 c^3 + b c^4 + c^5) : :

X(21924) lies on these lines: {594, 2294}, {1086, 17871}, {1146, 1836}, {3914, 21933}, {4854, 21049}, {16608, 17860}

X(21925) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b^3 - 2 a b^4 + b^5 + 2 a b^3 c - 2 b^4 c + b^3 c^2 + a^2 c^3 + 2 a b c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4 + c^5) : :

X(21925) lies on these lines: {3136, 15523}, {3925, 21946}, {21023, 21918}, {21044, 22173}, {21049, 21717}

X(21926) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + a b^3 - 2 a b^2 c + b^3 c - a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21926) lies on these lines: {10, 7064}, {11, 126}, {12, 3696}, {37, 3914}, {75, 325}, {142, 4890}, {192, 4442}, {442, 740}, {518, 3649}, {594, 20486}, {1086, 4022}, {1213, 2486}, {1284, 2550}, {2667, 17056}, {3120, 3728}, {3136, 21020}, {3816, 4751}, {3822, 4709}, {3826, 4687}, {3841, 3993}, {3932, 4043}, {4516, 18698}, {4699, 11680}, {4732, 17757}, {4967, 20544}, {17447, 20880}, {20718, 21677}, {21018, 21045}, {21023, 21029}, {21675, 21945}, {21692, 21919}

X(21927) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b^2 + a b^3 + 2 a^2 b c - 2 a b^2 c + b^3 c - a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(21927) lies on these lines: {11, 75}, {12, 740}, {37, 3914}, {192, 2886}, {321, 20487}, {442, 3993}, {594, 2486}, {1086, 21330}, {1278, 11680}, {2321, 20486}, {3120, 22167}, {3136, 4365}, {3696, 21031}, {3728, 4415}, {3816, 4699}, {3829, 4740}, {3932, 22016}, {4043, 6057}, {4431, 20544}, {4709, 17757}, {6154, 15624}, {7064, 22031}, {20489, 21029}, {20683, 22019}, {21023, 21945}

X(21928) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b - 3 a b^2 + 2 b^3 + a^2 c + 4 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3) : :

X(21928) lies on these lines: {37, 21943}, {1737, 14439}, {3822, 21808}, {3925, 21029}, {3930, 17757}, {4171, 21052}, {11680, 17451}, {21950, 21956}

X(21929) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^2 b - 3 a b^2 + b^3 + 2 a^2 c + 8 a b c - b^2 c - 3 a c^2 - b c^2 + c^3) : :

X(21929) lies on these lines: {37, 21013}, {551, 4530}, {1647, 21332}, {3816, 17451}, {3822, 21044}, {3925, 21029}, {17757, 21808}

X(21930) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b^3 - a^2 b^4 - a b^5 + b^6 + a^2 b^3 c - b^5 c + a b^3 c^2 - b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(21930) lies on these lines: {3136, 15523}, {3142, 21016}, {3925, 21915}, {4642, 21961}, {21023, 21687}, {21044, 22197}, {21049, 21946}

X(21931) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :

X(21931) lies on these lines: {2, 4336}, {37, 21914}, {594, 2294}, {3914, 21717}, {4413, 17267}, {6536, 21912}, {20486, 21030}, {21011, 21045}, {21023, 21029}, {21049, 21955}, {21258, 21346}, {21915, 21921}

X(21932) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(62)

Barycentrics a^4 - a^2 b^2 - a^2 c^2 - 2 Sqrt[3] S(b c +c a + a b) : :

X(21932) lies on these lines: {2, 3}, {61, 17277}, {62, 86}, {303, 17206}, {630, 6626}, {2271, 11489}, {5021, 11488}, {15668, 22238}, {17259, 22236}

X(21933) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + 2 a^2 b c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + c^4) : :

X(21933) lies on these lines: {6, 281}, {8, 965}, {10, 37}, {11, 1953}, {12, 2294}, {19, 1837}, {48, 10950}, {53, 158}, {65, 1826}, {80, 1781}, {200, 17299}, {219, 10573}, {393, 3176}, {610, 5727}, {944, 37519}, {1086, 16608}, {1100, 6738}, {1108, 1210}, {1226, 17862}, {1441, 18635}, {1864, 3611}, {1865, 6354}, {2150, 3109}, {2171, 21044}, {2178, 11500}, {2197, 21860}, {2264, 8756}, {2303, 5724}, {3085, 16777}, {3723, 13405}, {3811, 17388}, {3914, 21924}, {3925, 21023}, {3930, 21030}, {3949, 21031}, {4016, 4415}, {4642, 21915}, {4848, 8804}, {6047, 21671}, {7080, 17314}, {7146, 21239}, {12432, 21863}, {17747, 21853}, {17757, 22021}, {21945, 21955}, {21954, 21956}

X(21934) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a^5-(b^4-c^4)*(b-c))*(b+c)*(a^4-b^4-c^4) : :

X(21934) lies on these lines: {3136, 21034}

X(21935) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 + b^3 - b^2 c - b c^2 + c^3) : :

X(21935) lies on these lines: {1, 2476}, {2, 10448}, {4, 31}, {5, 1193}, {6, 10895}, {10, 321}, {11, 1201}, {12, 42}, {35, 17734}, {37, 21029}, {38, 6734}, {41, 3767}, {43, 11681}, {46, 21368}, {58, 3585}, {65, 3120}, {115, 213}, {171, 2475}, {212, 10953}, {225, 1254}, {226, 2650}, {238, 5046}, {244, 1210}, {318, 17871}, {377, 750}, {381, 16466}, {386, 7951}, {387, 10590}, {388, 11269}, {407, 1400}, {443, 17124}, {496, 1149}, {595, 3583}, {601, 6923}, {602, 6928}, {603, 18961}, {614, 9581}, {672, 5254}, {748, 2478}, {774, 1785}, {899, 1329}, {902, 6284}, {950, 3011}, {976, 3419}, {978, 4193}, {995, 7741}, {1064, 6842}, {1191, 10896}, {1399, 13273}, {1453, 18492}, {1460, 4214}, {1468, 1478}, {1479, 3915}, {1496, 10629}, {1616, 11238}, {1737, 22069}, {1837, 3772}, {1861, 17872}, {2177, 3085}, {2251, 7755}, {2329, 17737}, {2886, 10459}, {2887, 17751}, {2999, 7989}, {3052, 12953}, {3214, 17757}, {3216, 3814}, {3698, 21949}, {3704, 4365}, {3714, 15523}, {3752, 17606}, {3831, 4202}, {3869, 3944}, {3961, 5178}, {4000, 20305}, {4252, 12943}, {4257, 10483}, {4300, 6907}, {4362, 5016}, {4415, 21677}, {4429, 17787}, {4766, 5025}, {4805, 7751}, {5080, 5247}, {5084, 17125}, {5141, 17717}, {5295, 20653}, {5706, 10894}, {5711, 17532}, {5721, 18242}, {5794, 17720}, {6679, 11319}, {7083, 17516}, {7270, 17763}, {7745, 21764}, {9669, 16483}, {16062, 19810}, {16583, 21016}, {16827, 17669}, {17064, 19860}, {21031, 21955}, {21075, 21805}, {21530, 22057}, {21912, 21936}

X(21936) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a^2 (b + c) (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : :

X(21936) lies on these lines: {10, 14815}, {31, 20989}, {37, 21923}, {38, 5743}, {42, 3122}, {51, 1964}, {181, 3725}, {210, 3778}, {612, 3764}, {756, 1213}, {982, 5233}, {1329, 3782}, {2643, 21807}, {3124, 21814}, {3774, 22200}, {3914, 21963}, {4046, 22167}, {4642, 4854}, {6057, 21025}, {7069, 17872}, {21029, 21939}, {21911, 21944}, {21912, 21935}

X(21937) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(187)

Barycentrics 3 a^4 - a^3 b - 3 a^2 b^2 - a b^3 - a^3 c - a^2 b c - a b^2 c - b^3 c - 3 a^2 c^2 - a b c^2 - a c^3 - b c^3 : :

X(21937) lies on these lines: {2, 3}, {86, 187}, {524, 17206}, {574, 17277}, {988, 16834}, {1384, 17379}, {1654, 6390}, {1992, 2271}, {2482, 6626}, {3793, 20090}, {5024, 17349}, {5210, 15668}, {7801, 17271}, {7810, 17297}, {7820, 17307}

X(21938) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (b^2 + c^2) (-a^3 b + a^2 b^2 - a b^3 + b^4 - a^3 c - 2 a^2 b c - b^3 c + a^2 c^2 - a c^3 - b c^3 + c^4) : :

X(21938) lies on these lines: {4642, 21023}, {4854, 21939}, {21020, 21037}, {21038, 21677}

X(21939) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (b^2 + c^2) (-2 a^3 - a b^2 + b^3 - b^2 c - a c^2 - b c^2 + c^3) : :

X(21939) lies on these lines: {37, 21918}, {1213, 20483}, {4854, 21938}, {21029, 21936}, {21923, 21944}

X(21940) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(216)

Barycentrics (a^2 - b^2 - c^2) (a^6 + a^5 b - 2 a^4 b^2 + a^2 b^4 - a b^5 + a^5 c + a^4 b c - a b^4 c - b^5 c - 2 a^4 c^2 - 2 a^2 b^2 c^2 + 2 a b^3 c^2 + 2 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 - a b c^4 - a c^5 - b c^5) : :

X(21940) lies on these lines: {2, 3}, {86, 216}, {343, 17206}, {577, 17277}, {2271, 11427}, {2968, 6542}, {5021, 11433}, {6389, 17234}, {15526, 17297}, {15905, 17349}, {16826, 17102}, {17232, 20208}

X(21941) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a b - a c + b c) (a^2 b^2 - a b^3 - 2 a^2 b c + b^3 c + a^2 c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(21941) lies on these lines: {3925, 21951}, {3932, 21031}, {15523, 21025}, {20489, 21029}

X(21942) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (2 a - b - c) (b + c) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :

X(21942) lies on these lines: {8, 16561}, {37, 21013}, {71, 1018}, {210, 6535}, {594, 21041}, {1647, 17465}, {2345, 4919}, {3925, 21945}, {4058, 21090}, {4370, 4530}, {17757, 21801}, {21029, 21943}, {21675, 21961}

X(21943) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - 2 b - 2 c) (b + c) (a^2 b - b^3 + a^2 c + a b c + b^2 c + b c^2 - c^3) : :

X(21943) lies on these lines: {37, 21928}, {210, 21864}, {594, 17757}, {11680, 17444}, {21029, 21942}, {21696, 21708}

X(21944) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2)^2 (-2 a^2 - a b + b^2 - a c + c^2) : :

X(21944) lies on these lines: {10, 3699}, {37, 21948}, {125, 2643}, {1648, 21833}, {3120, 18004}, {3125, 21046}, {8286, 21947}, {17058, 17476}, {21674, 21676}, {21911, 21936}, {21918, 21954}, {21923, 21939}, {21945, 21963}, {21946, 21950}

X(21945) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c)^2 (b + c) (a^2 - 2 a b + b^2 - 2 a c + c^2) : :

X(21945) lies on these lines: {10, 4069}, {37, 21914}, {656, 3122}, {1086, 4081}, {1109, 2632}, {2486, 21044}, {3914, 21911}, {3925, 21942}, {4466, 4516}, {4854, 21717}, {13576, 21091}, {15523, 21949}, {16560, 18343}, {21023, 21927}, {21675, 21926}, {21933, 21955}, {21944, 21963}, {21947, 21961}, {21950, 21952}

X(21946) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (a - b - c) (b - c)^2 (b + c) (a b - b^2 + a c + b c - c^2) : :

X(21946) lies on these lines: {11, 1146}, {3120, 3700}, {3925, 21925}, {21049, 21930}, {21944, 21950}

X(21947) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2)^2 (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + c^4) : :

X(21947) lies on these lines: {10, 3939}, {4036, 16732}, {4092, 15526}, {8286, 21944}, {21045, 21046}, {21945, 21961}

X(21948) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (2 a^5 - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + b^5 + 2 a^2 b^2 c - b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 - 2 a^2 c^3 + a c^4 - b c^4 + c^5) : :

X(21948) lies on these lines: {37, 21944}, {21912, 21918}, {21913, 21923}

X(21949) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 - a b + 2 b^2 - a c - 4 b c + 2 c^2) : :

X(21949) lies on these lines: {2, 4689}, {10, 3967}, {11, 16602}, {12, 21896}, {37, 3914}, {43, 3838}, {44, 1836}, {210, 3120}, {226, 4849}, {312, 3823}, {442, 4646}, {497, 17278}, {518, 17889}, {899, 17605}, {1086, 4847}, {1266, 4884}, {1279, 3434}, {1376, 17064}, {1738, 2886}, {2550, 3772}, {2887, 3696}, {3175, 4442}, {3698, 21935}, {3703, 4686}, {3706, 17231}, {3740, 3944}, {3755, 17056}, {3829, 5121}, {3834, 10453}, {3841, 3931}, {3932, 22034}, {4016, 21020}, {4042, 17344}, {4365, 21026}, {4388, 17348}, {4641, 20292}, {4685, 4892}, {4863, 4864}, {4891, 17234}, {5087, 16569}, {5272, 11235}, {9710, 13161}, {9955, 17749}, {11680, 16610}, {15523, 21945}, {16821, 17678}, {17156, 17374}, {17441, 21889}, {20486, 21868}

X(21950) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (3 a - b - c) (b - c)^2 (b + c) : :

X(21950) lies on these lines: {2, 15903}, {37, 21013}, {115, 125}, {244, 1146}, {514, 17213}, {1015, 4530}, {1647, 2170}, {3756, 4534}, {3815, 17451}, {3924, 7735}, {4103, 21041}, {4642, 21049}, {4674, 21090}, {5540, 6788}, {21029, 21951}, {21674, 21921}, {21928, 21956}, {21944, 21946}, {21945, 21952}

X(21950) = barycentric product X(10)*X(3756)

X(21951) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (b^2 - 3 b c + c^2) : :

X(21951) lies on these lines: {2, 3727}, {8, 3726}, {10, 762}, {37, 3698}, {39, 1739}, {65, 2238}, {213, 3754}, {244, 17448}, {321, 21025}, {942, 3780}, {1212, 20331}, {1237, 21435}, {1334, 21888}, {1500, 3987}, {1573, 3670}, {1575, 17451}, {1698, 3735}, {2170, 16604}, {2171, 21892}, {2243, 4426}, {2294, 21857}, {2295, 3753}, {2650, 21904}, {3290, 5836}, {3294, 4674}, {3836, 4167}, {3914, 21967}, {3918, 16600}, {3924, 4386}, {3925, 21941}, {3930, 21868}, {3992, 22036}, {4051, 17063}, {4084, 21839}, {4136, 20483}, {4424, 16589}, {4695, 20691}, {4861, 9259}, {5883, 20963}, {14923, 16969}, {16606, 22173}, {20255, 20911}, {20880, 21138}, {21029, 21950}, {21049, 21956}

X(21951) = barycentric product X(10)*X(17063)

X(21952) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b - c) (b + c) (4 a^2 - 3 a b - b^2 - 3 a c + 4 b c - c^2) : :

X(21952) lies on these lines: {10, 4088}, {1698, 14432}, {2533, 4988}, {2785, 9780}, {3617, 4458}, {3700, 21052}, {4691, 21181}, {14475, 21343}, {21013, 21914}, {21945, 21950}

X(21953) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 + a^2 b^2 c - 2 b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5) : :

X(21953) lies on these lines: {21029, 21687}, {21957, 21958}

X(21954) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b + c) (a b^3 - a b^2 c - a b c^2 - b^2 c^2 + a c^3) : :

X(21954) lies on these lines: {37, 21923}, {51, 2235}, {65, 21876}, {75, 3981}, {209, 2238}, {321, 3124}, {740, 22200}, {1211, 3721}, {1233, 21138}, {2887, 3125}, {3094, 19804}, {3122, 21345}, {3773, 22171}, {3925, 21941}, {4359, 20859}, {18134, 20271}, {21918, 21944}, {21933, 21956}

X(21955) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^3 b + 3 a^2 b^2 - 3 a b^3 + b^4 - a^3 c - 6 a^2 b c + 3 a b^2 c - 4 b^3 c + 3 a^2 c^2 + 3 a b c^2 + 6 b^2 c^2 - 3 a c^3 - 4 b c^3 + c^4) : :

X(21955) lies on these lines: {11, 4000}, {12, 3755}, {37, 3914}, {442, 4356}, {1086, 21346}, {1836, 2257}, {2886, 3672}, {3663, 6067}, {3772, 4319}, {3932, 22040}, {4081, 17861}, {4415, 21039}, {21031, 21935}, {21049, 21931}, {21933, 21945}

X(21956) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : :

X(21956) lies on these lines: {2, 20181}, {6, 3434}, {8, 5254}, {10, 3985}, {11, 1575}, {12, 20691}, {30, 5291}, {37, 3914}, {100, 230}, {115, 17757}, {120, 1738}, {141, 4441}, {209, 1824}, {313, 321}, {325, 17759}, {385, 6653}, {424, 2501}, {442, 1500}, {524, 20553}, {528, 1914}, {536, 20541}, {726, 4119}, {740, 4071}, {956, 2549}, {958, 9598}, {1086, 3726}, {1213, 4972}, {1278, 3314}, {1574, 4187}, {1714, 14974}, {1826, 21876}, {1834, 2295}, {2238, 13576}, {2242, 11112}, {2275, 3813}, {2276, 2886}, {2321, 2887}, {2550, 5275}, {3120, 3930}, {3419, 9620}, {3704, 16886}, {3767, 5687}, {3815, 11680}, {3891, 17388}, {3932, 4037}, {3943, 4442}, {4426, 6284}, {4642, 21029}, {4695, 21044}, {4863, 16973}, {5013, 10527}, {5014, 17362}, {5082, 5286}, {5231, 9574}, {5552, 13881}, {6656, 17143}, {7735, 17784}, {9597, 12513}, {9599, 11235}, {9664, 11113}, {15048, 16975}, {16583, 21073}, {16611, 21090}, {16777, 19785}, {17314, 18134}, {21031, 21868}, {21049, 21951}, {21928, 21950}, {21933, 21954}

X(21957) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b^2 - c^2) (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + 2 b^3 c^2 - a^2 c^3 + 2 b^2 c^3 + a c^4) : :

X(21957) lies on these lines: {65, 4130}, {2294, 4064}, {2489, 17478}, {2533, 4079}, {6587, 8611}, {21953, 21958}

X(21958) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(21958) lies on these lines: {10, 3709}, {424, 2501}, {512, 21099}, {523, 594}, {656, 4140}, {661, 21719}, {2276, 21347}, {2295, 3049}, {2345, 3287}, {2605, 17303}, {3501, 21388}, {3661, 4374}, {3837, 17458}, {3912, 17066}, {4079, 20491}, {4171, 21052}, {4806, 4826}, {4807, 4832}, {17072, 21348}, {17218, 17294}, {21053, 21834}, {21953, 21957}

X(21959) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a b + a c - b c) (a b - b^2 + a c - c^2) : :

X(21959) lies on these lines: {2618, 4036}, {3700, 21962}, {4120, 15523}, {4171, 21052}, {5075, 17776}, {21051, 21834}, {21917, 21964}, {21945, 21950}

X(21960) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^3 - a^2 b - a^2 c + a b c + b^2 c + b c^2) : :

X(21960) lies on these lines: {10, 22042}, {37, 2395}, {522, 5257}, {650, 7650}, {656, 3700}, {665, 4985}, {798, 4010}, {1577, 3709}, {2484, 4874}, {2533, 4079}, {3287, 8062}, {4044, 20907}, {4170, 4832}, {4171, 21052}, {4391, 21348}, {4687, 18158}, {4705, 22044}, {4885, 15413}, {8702, 17275}, {14431, 21099}

X(21961) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2)^2 (a^3 - a^2 b - a b^2 + b^3 - a^2 c - a b c - a c^2 + c^3) : :

X(21961) lies on these lines: {10, 644}, {80, 21092}, {115, 1109}, {4642, 21930}, {21044, 21046}, {21675, 21942}, {21944, 21946}, {21945, 21947}

X(21962) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^2 b^2 - a b^3 + a^2 b c - a b^2 c + b^3 c + a^2 c^2 - a b c^2 - a c^3 + b c^3) : :

X(21962) lies on these lines: {661, 15523}, {3700, 21959}, {8611, 21052}, {21055, 21726}, {21953, 21957}

X(21963) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics a (b - c)^2 (b + c) (a^2 + a b + a c - 2 b c) : :

X(21963) lies on these lines: {11, 244}, {513, 18211}, {984, 4425}, {3914, 21936}, {4854, 21676}, {21944, 21945}

X(21964) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b^2 - c^2) (a^3 b^3 - a^2 b^4 + a^3 b^2 c + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^3 c^3 - a b^2 c^3 - a^2 c^4 + b^2 c^4) : :

X(21964) lies on these lines: {21917, 21959}, {21953, 21957}

X(21965) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^2 b - 2 a b^2 + b^3 - a^2 c + 2 a b c - b^2 c - 2 a c^2 - b c^2 + c^3) : :

X(21965) lies on these lines: {1, 230}, {5, 3735}, {10, 37}, {11, 3727}, {12, 3721}, {172, 5724}, {257, 325}, {442, 3125}, {524, 17739}, {986, 5254}, {1086, 17062}, {1107, 1146}, {2238, 21677}, {2292, 21044}, {2886, 3959}, {3061, 3815}, {3496, 7745}, {3726, 15888}, {3741, 4167}, {3925, 21941}, {3954, 17757}, {4037, 4918}, {4415, 16603}, {4642, 21029}, {9597, 17595}, {16609, 17056}, {21674, 21921}

X(21966) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b - a^2 b^2 - 2 a b^3 + 2 b^4 + a^3 c + 2 a^2 b c + a b^2 c - b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 - 2 a c^3 - b c^3 + 2 c^4) : :

X(21966) lies on these lines: {3925, 21023}, {4171, 21052}, {21048, 21057}

X(21967) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(21967) lies on these lines: {37, 21922}, {1836, 5046}, {3914, 21951}, {3925, 21029}, {4124, 17023}, {4642, 4854}, {20486, 21808}, {21020, 21025}

X(21968) = X(185)X(3526)∩X(381)X(12900)

Barycentrics 21*a^6-22*(b^2+c^2)*a^4-(13*b^4-46*b^2*c^2+13*c^4)*a^2+14*(b^4-c^4)*(b^2-c^2) : :
X(21968) = 8*R^2*X(185)-7*(32*R^2-7*SW)*X(3526)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21968) lies on these lines: {185, 3526}, {381, 12900}, {1656, 12118}, {4846, 5054}, {5972, 18440}

X(21969) = X(2)X(51)∩X(30)X(52)

Barycentrics a^2*(3*(b^2+c^2)*a^2-3*b^4+2*b^2*c^2-3*c^4) : :
X(21969) = 2*X(2)-3*X(51), 8*X(2)-9*X(373), 5*X(2)-3*X(2979), X(2)-3*X(3060), 7*X(2)-6*X(3819), 4*X(2)-3*X(3917), 7*X(2)-9*X(5640), 10*X(2)-9*X(5650), 5*X(2)-6*X(5943), 11*X(2)-12*X(6688), 11*X(2)-9*X(7998), 5*X(2)-9*X(11002), 13*X(2)-15*X(11451), 19*X(2)-18*X(15082), X(2)-9*X(16981), 4*X(51)-3*X(373), 5*X(51)-2*X(2979), 7*X(51)-4*X(3819)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21969) lies on these lines: {2, 51}, {3, 15004}, {4, 14531}, {5, 13421}, {20, 16625}, {22, 576}, {25, 3292}, {30, 52}, {143, 549}, {184, 1351}, {216, 14773}, {343, 21850}, {376, 389}, {381, 5446}, {428, 524}, {468, 15010}, {519, 16980}, {539, 7540}, {541, 21649}, {542, 13417}, {547, 6101}, {568, 3534}, {573, 19346}, {575, 6636}, {597, 3313}, {599, 9969}, {970, 17549}, {1112, 5642}, {1154, 3845}, {1184, 11173}, {1194, 13330}, {1196, 20977}, {1216, 5055}, {1350, 9777}, {1495, 1993}, {1501, 1570}, {1915, 15514}, {1992, 6467}, {1994, 15107}, {2393, 15534}, {3090, 15606}, {3098, 5422}, {3124, 3787}, {3522, 15012}, {3524, 3567}, {3529, 13382}, {3543, 5889}, {3545, 10110}, {3557, 21036}, {3558, 21032}, {3796, 5093}, {3830, 13754}, {3839, 5907}, {3843, 12002}, {3860, 15060}, {5012, 5097}, {5032, 12220}, {5052, 20859}, {5054, 5462}, {5064, 12294}, {5066, 5891}, {5071, 9781}, {5102, 11402}, {5310, 19369}, {5322, 8540}, {5396, 19254}, {5447, 15694}, {5651, 17810}, {5752, 16370}, {5876, 14893}, {5890, 11001}, {5892, 13321}, {5946, 12100}, {5969, 19568}, {6000, 15682}, {6403, 7714}, {6515, 11550}, {7496, 12834}, {7667, 19161}, {8584, 9019}, {8703, 9730}, {9140, 11800}, {9729, 10304}, {9822, 21356}, {9971, 15533}, {10095, 15699}, {10109, 13451}, {10124, 15026}, {10132, 11916}, {10133, 11917}, {10245, 19357}, {10601, 21852}, {10627, 11539}, {10706, 11807}, {11160, 14913}, {11271, 13433}, {11424, 17834}, {11430, 13482}, {11433, 21851}, {11695, 15702}, {12006, 17504}, {12101, 16194}, {12162, 15687}, {13346, 15078}, {13348, 15043}, {13352, 18324}, {13363, 15713}, {13367, 14070}, {13630, 15686}, {14269, 18436}, {14855, 19710}, {15019, 15246}, {15024, 15709}, {15028, 15721}, {15045, 15698}, {16836, 19708}, {17710, 20583}, {18377, 18555}

X(21969) = midpoint of X(i) and X(j) for these {i,j}: {381, 6243}, {3543, 5889}
X(21969) = reflection of X(i) in X(j) for these (i,j): (2, 21849), (376, 389), (381, 5446), (599, 9969), (3543, 13598), (5876, 14893), (9140, 11800), (10706, 11807), (11160, 14913), (12162, 15687), (17710, 20583)
X(21969) = anticomplement of X(2) wrt orthic triangle
X(21969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3060, 21849), (2, 21849, 51), (22, 576, 13366), (51, 3917, 373), (51, 5650, 5943), (2979, 3060, 11002), (2979, 5650, 3917), (2979, 5943, 5650), (2979, 11002, 5943), (3819, 5943, 12045), (5446, 6243, 5562), (5889, 13598, 11381), (5943, 11002, 51), (10263, 14449, 52), (13421, 16982, 5)

X(21970) = X(2)X(21850)∩X(3)X(16657)

Barycentrics (3*a^2-b^2-c^2)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2): :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21970) lies on these lines: {2, 21850}, {3, 16657}, {4, 15431}, {24, 12310}, {25, 3580}, {51, 18438}, {125, 382}, {140, 5544}, {141, 5020}, {193, 3167}, {373, 3526}, {381, 1531}, {468, 1351}, {1533, 10605}, {1656, 3066}, {1899, 20850}, {3053, 6388}, {3313, 5943}, {3517, 12429}, {3527, 7542}, {3564, 4232}, {3618, 6676}, {5050, 7493}, {5055, 20192}, {5064, 7703}, {5200, 18539}, {5972, 11477}, {6515, 8780}, {7530, 9919}, {9909, 13567}, {10154, 11433}, {11432, 13383}, {12164, 21841}, {14389, 14848}, {14530, 18951}, {15066, 15360}, {17810, 19130}

X(21970) = {X(25), X(3580)}-harmonic conjugate of X(18440)

X(21971) = X(2)X(21851)∩X(51)X(6677)

Barycentrics a^2*(3*(b^2+c^2)*a^6-(3*b^4-14*b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2+(b^2-c^2)^2*(b^2-3*c^2)*(3*b^2-c^2))*(a^2-b^2-c^2) : :
X(21971) = 3*(24*R^2-5*SW)*X(373)-2*SW*X(568)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21971) lies on these lines: {2, 21851}, {51, 6677}, {69, 5943}, {373, 568}, {1092, 3527}, {1531, 4846}, {1568, 16226}

X(21972) = EULER LINE INTERCEPT OF X(5355)X(11433)

Barycentrics 3*a^8+2*(b^2+c^2)*a^6-4*(b^4-b^2*c^2+c^4)*a^4-10*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^2*(3*b^2+4*b*c+3*c^2)*(3*b^2-4*b*c+3*c^2) : :
X(21972) = (4*S^2+5*(4*R^2-SW)*SW)*X(2)-2*S^2*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21972) lies on these lines: {2, 3}, {5355, 11433}

X(21973) = MIDPOINT OF X(2) AND X(460)

Barycentrics 6*a^8+(b^2+c^2)*a^6-(11*b^4-6*b^2*c^2+11*c^4)*a^4+(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4)*a^2-(b^2-c^2)^2*(b^2-3*c^2)*(3*b^2-c^2) : :
X(21973) = (4*S^2+5*(4*R^2-SW)*SW)*X(2)-2*S^2*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21973) lies on this line: {2, 3}

X(21973) = midpoint of X(2) and X(460)

X(21974) = {X(2), X(468)}-HARMONIC CONJUGATE OF X(9909)

Barycentrics 9*a^6-6*(b^2+c^2)*a^4-(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2)*a^2+6*(b^4-c^4)*(b^2-c^2) : :
X(21974) = 2*(16*R^2-3*SW)*X(2)-SW*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28130.

X(21974) lies on this line: {2, 3}

X(21974) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 468, 9909), (2, 13361, 15703)

X(21975) = COMPLEMENT OF X(3459)

Barycentrics (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 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^8-4 a^6 c^2+5 a^4 b^2 c^2+a^2 b^4 c^2-2 b^6 c^2+6 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-2 b^2 c^6+c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28133.

Let Q be the pedal curve (a limaçon of Pascal) of the circle with center X(7728) and radius |OH|, with respect to X(265). Let A'B'C' be the triangle formed by the tangents at A,B,C to Q. Then the triangles ABC and A'B'C' are perspective, and their perspector is X(2963). Moreover, X(21975) is the only finite fixed point of the affine transformation that maps a triangle ABC onto A'B'C'. See X(21975). (Angel Montesdeoca, September 10, 2019)

X(21975) lies on the cubic K569 and these lines: {2,3459}, {3,3432}, {5,930}, {54,140}, {93,2970}, {128,6288}, {186,14111}, {195,10615}, {570,2963}, {632,11016}, {1656,18370}, {3628,20413}

X(21975) = complement X(3459)
X(21975) = reflection of X(20413) in X(3628)
X(21975) = medial isogonal conjugate of X(21230)
X(21975) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 21230}, {31, 2963}, {48, 97}, {195, 10}
X(21975) = X(2964)-isoconjugate of X(3459)
X(21975) = X(2)-Ceva conjugate of X(2963)
X(21975) = barycentric product X(195)*X(11140)
X(21975) = barycentric quotient X(i)/X(j) for these {i,j}: {195, 1994}, {2963, 3459}

X(21976) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 - a^4 b^2 + a^3 b^3 - a b^5 - a^4 b c + 2 a^3 b^2 c + 7 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 + a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + 7 a^2 b c^3 + a b^2 c^3 + 4 b^3 c^3 + 2 a b c^4 + 2 b^2 c^4 - a c^5 + 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = b c + c a + a b; P lies on X(3)X(6), and its image lies on the Euler line.

X(21976) lies on these lines: {2, 3}

X(21977) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + a^4 b c - 2 a^3 b^2 c - 7 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 - 3 a b^3 c^2 - 2 b^4 c^2 - a^3 c^3 - 7 a^2 b c^3 - 3 a b^2 c^3 - 4 b^3 c^3 - a^2 c^4 - 3 a b c^4 - 2 b^2 c^4 - 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = (a + b + c)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21977) lies on these lines: {2, 3}, {17023, 21010}

X(21978) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 + 3 a^5 b + 2 a^4 b^2 - 2 a^3 b^3 - 3 a^2 b^4 - a b^5 + 3 a^5 c - 4 a^3 b^2 c - 18 a^2 b^3 c - 7 a b^4 c - 6 b^5 c + 2 a^4 c^2 - 4 a^3 b c^2 - 10 a^2 b^2 c^2 - 8 a b^3 c^2 - 4 b^4 c^2 - 2 a^3 c^3 - 18 a^2 b c^3 - 8 a b^2 c^3 - 12 b^3 c^3 - 3 a^2 c^4 - 7 a b c^4 - 4 b^2 c^4 - a c^5 - 6 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = (b + c)^2 + (c + a)^2 + (a + b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21978) lies on these lines: {2, 3}

X(21979) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 - a^5 b - 2 a^4 b^2 + 2 a^3 b^3 + a^2 b^4 - a b^5 - a^5 c - 12 a^4 b c + 4 a^3 b^2 c - 6 a^2 b^3 c + 5 a b^4 c - 6 b^5 c - 2 a^4 c^2 + 4 a^3 b c^2 + 6 a^2 b^2 c^2 + 4 a b^3 c^2 + 4 b^4 c^2 + 2 a^3 c^3 - 6 a^2 b c^3 + 4 a b^2 c^3 - 12 b^3 c^3 + a^2 c^4 + 5 a b c^4 + 4 b^2 c^4 - a c^5 - 6 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = (b - c)^2 + (c - a)^2 + (a - b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21979) lies on these lines: {2, 3}

X(21980) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 + 2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 a^5 c + 5 a^4 b c - 2 a^3 b^2 c + a^2 b^3 c - 4 a b^4 c + 2 b^5 c + a^4 c^2 - 2 a^3 b c^2 - 6 a^2 b^2 c^2 - 5 a b^3 c^2 - 2 b^4 c^2 - a^3 c^3 + a^2 b c^3 - 5 a b^2 c^3 + 4 b^3 c^3 - 2 a^2 c^4 - 4 a b c^4 - 2 b^2 c^4 - a c^5 + 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = -(b c + c a + a b); P lies on X(3)X(6), and its image lies on the Euler line.

X(21980) lies on these lines: {2, 3}

X(21981) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 + 2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 a^5 c + 3 a^4 b c - 2 a^3 b^2 c - 3 a^2 b^3 c - 4 a b^4 c + a^4 c^2 - 2 a^3 b c^2 - 6 a^2 b^2 c^2 - 5 a b^3 c^2 - 2 b^4 c^2 - a^3 c^3 - 3 a^2 b c^3 - 5 a b^2 c^3 - 2 a^2 c^4 - 4 a b c^4 - 2 b^2 c^4 - a c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = -(a + b + c)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21981) lies on these lines: {2, 3}, {4384, 8301}, {8424, 17738}, {17023, 17798}

X(21982) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (3 a^5 + 2 a^4 b - 2 a^2 b^3 - 3 a b^4 + 2 a^4 c - 4 a^2 b^2 c - 4 a b^3 c - 2 b^4 c - 4 a^2 b c^2 - 6 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 - 3 a c^4 - 2 b c^4) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = -((b + c)^2 + (c + a)^2 + (a + b)^2); P lies on X(3)X(6), and its image lies on the Euler line.

X(21982) lies on these lines: {2, 3}, {32, 19765}, {56, 3674}, {58, 19758}, {154, 17185}, {980, 4252}, {988, 16478}, {4653, 19761}

X(21983) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (3 a^6 + a^5 b - 2 a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - 3 a b^5 + a^5 c - 8 a^4 b c + 4 a^3 b^2 c + 2 a^2 b^3 c + 3 a b^4 c - 2 b^5 c - 2 a^4 c^2 + 4 a^3 b c^2 + 2 a^2 b^2 c^2 + 4 b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 - 4 b^3 c^3 - a^2 c^4 + 3 a b c^4 + 4 b^2 c^4 - 3 a c^5 - 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = a^2 + b^2 + c^2 and k = -((b - c)^2 + (c - a)^2 + (a - b)^2); P lies on X(3)X(6), and its image lies on the Euler line.

X(21983) lies on these lines: {2, 3}, {980, 4258}

X(21984) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 - a^4 b^2 + a^3 b^3 - a b^5 - 5 a^4 b c - 2 a^3 b^2 c - 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 - 3 a b^3 c^2 - 2 b^4 c^2 + a^3 c^3 - a^2 b c^3 - 3 a b^2 c^3 - 4 b^3 c^3 - 2 a b c^4 - 2 b^2 c^4 - a c^5 - 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = b c + c a + a b and k = a^2 + b^2 + c^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21984) lies on these lines: {2, 3}, {1001, 17738}

X(21985) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + 5 a^4 b c + 2 a^3 b^2 c + a^2 b^3 c + a b^4 c + 2 b^5 c + a^4 c^2 + 2 a^3 b c^2 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 4 b^3 c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 + 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = (a + b + c)^2 and k = a^2 + b^2 + c^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21985) lies on these lines: {2, 3}, {2223, 17308}, {3912, 21010}

X(21986) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a*(a^4+(b+c)*a^3-(b^2+6*b*c+c^2)*a^2-(b+c)*(b^2+8*b*c+c^2)*a-2*b*c*(3*b^2+4*b*c+3*c^2)) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = (a + b + c)^2 and k = (b + c)^2 + (c + a)^2 + (a + b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21986) lies on these lines: {2, 3}, {956, 17397}, {1001, 17308}, {3714, 4384}, {3912, 3966}, {4383, 16589}, {8167, 16831}, {16478, 17022}

X(21987) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a*(a^6-3*(b+c)*a^5-2*(2*b^2+9*b*c+2*c^2)*a^4+4*(b^3+c^3)*a^3+3*(b^2+c^2)^2*a^2-(b+c)*(b^2+c^2)*(b^2-4*b*c+c^2)*a-6*b*c*(b^2+c^2)^2) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = (a + b + c)^2 and k = (b - c)^2 + (c - a)^2 + (a - b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21987) lies on these lines: {2, 3}

X(21988) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 + 2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 a^5 c + a^4 b c + 2 a^3 b^2 c - 7 a^2 b^3 c - 2 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 + 2 b^4 c^2 - a^3 c^3 - 7 a^2 b c^3 - a b^2 c^3 - 4 b^3 c^3 - 2 a^2 c^4 + 2 b^2 c^4 - a c^5 - 2 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = -(b c + c a + a b) and k = a^2 + b^2 + c^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21988) lies on these lines: {2, 3}

X(21989) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (a^6 + 2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 a^5 c + 3 a^4 b c + 2 a^3 b^2 c - 3 a^2 b^3 c + a^4 c^2 + 2 a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - 3 a^2 b c^3 - a b^2 c^3 - 2 a^2 c^4 + 2 b^2 c^4 - a c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = -(a + b + c)^2 and k = a^2 + b^2 + c^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21989) lies on these lines: {2, 3}, {2223, 8301}, {3912, 17798}, {5337, 18755}, {19554, 20742}

X(21990) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3558)

Barycentrics a^4 - a^3*b - a^2*b^2 - a*b^3 - a^3*c - a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - a*c^3 - b*c^3 + 2*(a*b + a*c + b*c)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :

X(21990) lies on these lines: {2, 3}, {86, 3558}, {14630, 17277}

X(21991) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3592)

Barycentrics 2 a^4 - 2 a^2 b^2 - 2 a^2 c^2 + 6 S (b c + c a + a b) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = -(a + b + c)^2 and k = -((b - c)^2 + (c - a)^2 + (a - b)^2); P lies on X(3)X(6), and its image lies on the Euler line.

X(21991) lies on these lines: {2, 3}, {86, 3592}, {2271, 8972}, {3593, 17206}, {3594, 17277}, {5021, 13941}, {6419, 17379}, {6420, 17349}, {6425, 15668}, {6426, 17259}

X(21992) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(372)

Barycentrics a^4 - a^2 b^2 - a^2 c^2 - 2S( b c + c a + a b) : :

See X(21991).

X(21992) lies on these lines: {2, 3}, {86, 372}, {261, 13332}, {371, 17277}, {487, 966}, {488, 4648}, {491, 17206}, {615, 18755}, {642, 6626}, {1151, 17259}, {1152, 15668}, {2271, 3069}, {3068, 5021}, {3311, 17349}, {3312, 17379}, {5263, 8225}

X(21992) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 21909), (4, 33035, 21909), (5, 16917, 21909), (140, 33047, 21909)

X(21993) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3094)

Barycentrics a^6 + a^5 b - a^2 b^4 + a b^5 + a^5 c + a^4 b c + a b^4 c + b^5 c - 2 a^2 b^2 c^2 - a^2 c^4 + a b c^4 + a c^5 + b c^5 : : X(21993) lies on these lines: {2, 3}, {86, 3094}, {183, 21024}, {1691, 17277}, {2271, 16989}, {3314, 17206}, {5021, 7774}, {6393, 17300}, {7792, 18755}

X(21994) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (2 a^6 + 3 a^5 b + a^4 b^2 - a^3 b^3 - 3 a^2 b^4 - 2 a b^5 + 3 a^5 c - a^4 b c - 6 a^3 b^2 c - 11 a^2 b^3 c - 9 a b^4 c - 4 b^5 c + a^4 c^2 - 6 a^3 b c^2 - 12 a^2 b^2 c^2 - 11 a b^3 c^2 - 6 b^4 c^2 - a^3 c^3 - 11 a^2 b c^3 - 11 a b^2 c^3 - 8 b^3 c^3 - 3 a^2 c^4 - 9 a b c^4 - 6 b^2 c^4 - 2 a c^5 - 4 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = b c + c a + a b and k = (b + c)^2 + (c + a)^2 + (a + b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21994) lies on these lines: {2, 3}

X(21995) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (2 a^6 - a^5 b - 3 a^4 b^2 + 3 a^3 b^3 + a^2 b^4 - 2 a b^5 - a^5 c - 13 a^4 b c + 2 a^3 b^2 c + a^2 b^3 c + 3 a b^4 c - 4 b^5 c - 3 a^4 c^2 + 2 a^3 b c^2 + 4 a^2 b^2 c^2 + a b^3 c^2 + 2 b^4 c^2 + 3 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - 8 b^3 c^3 + a^2 c^4 + 3 a b c^4 + 2 b^2 c^4 - 2 a c^5 - 4 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = b c + c a + a b and k = (b - c)^2 + (c - a)^2 + (a - b)^2; P lies on X(3)X(6), and its image lies on the Euler line.

X(21995) lies on these lines: {2, 3}

X(21996) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(386)

Barycentrics a*(-a^6-4*(b+c)*a^5-(3*b^2+7*b*c+3*c^2)*a^4+(b+c)*(3*b^2-b*c+3*c^2)*a^3+(4*b^4+4*c^4+b*c*(13*b^2+10*b*c+13*c^2))*a^2+(b+c)*(b^2+c^2)*(b^2+5*b*c+c^2)*a+2*b*c*(b^2+c^2)*(b^2+b*c+c^2)) : :

X(21996) lies on these lines: {2, 3}

X(21997) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3285)

Barycentrics (a + b) (a + c) (a^3 - a^2 b - b^3 - a^2 c - c^3) : :

X(21997) lies on these lines: {2, 3}, {58, 17367}, {86, 3285}, {239, 3670}, {284, 3662}, {320, 4273}, {333, 16722}, {1043, 17230}, {1333, 16706}, {2287, 6646}, {2303, 17302}, {3936, 18755}, {4281, 17012}, {4286, 17277}, {4653, 17244}, {4877, 17338}, {4972, 17798}, {5235, 6626}, {8822, 17350}, {16704, 17206}, {17184, 20769}

X(21998) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (2 a^6 + 5 a^5 b + 3 a^4 b^2 - 3 a^3 b^3 - 5 a^2 b^4 - 2 a b^5 + 5 a^5 c + 5 a^4 b c - 2 a^3 b^2 c - 17 a^2 b^3 c - 7 a b^4 c - 4 b^5 c + 3 a^4 c^2 - 2 a^3 b c^2 - 12 a^2 b^2 c^2 - 9 a b^3 c^2 - 2 b^4 c^2 - 3 a^3 c^3 - 17 a^2 b c^3 - 9 a b^2 c^3 - 8 b^3 c^3 - 5 a^2 c^4 - 7 a b c^4 - 2 b^2 c^4 - 2 a c^5 - 4 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = b c + c a + a b and k = -((b + c)^2 + (c + a)^2 + (a + b)^2); P lies on X(3)X(6), and its image lies on the Euler line.

X(21998) lies on these lines: {2, 3}

X(21999) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF POINT P DEFINED BELOW

Barycentrics a (2 a^6 + a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 - 2 a b^5 + a^5 c - 7 a^4 b c + 6 a^3 b^2 c - 5 a^2 b^3 c + 5 a b^4 c - 4 b^5 c - a^4 c^2 + 6 a^3 b c^2 + 4 a^2 b^2 c^2 + 3 a b^3 c^2 + 6 b^4 c^2 + a^3 c^3 - 5 a^2 b c^3 + 3 a b^2 c^3 - 8 b^3 c^3 - a^2 c^4 + 5 a b c^4 + 6 b^2 c^4 - 2 a c^5 - 4 b c^5) : :

Point P = a^2 (h a^2 + k b^2 + k c^2) : : , where h = b c + c a + a b and k = -((b - c)^2 + (c - a)^2 + (a - b)^2); P lies on X(3)X(6), and its image lies on the Euler line.

X(21999) lies on these lines: {2, 3}

X(22000) = (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics (b + c) (-a^4 b - a^3 b^2 + a^2 b^3 + a b^4 - a^4 c + b^4 c - a^3 c^2 - 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(22000) lies on these lines: {2, 2140}, {4, 3190}, {10, 3136}, {27, 101}, {37, 226}, {72, 946}, {118, 1824}, {220, 7522}, {228, 516}, {306, 1230}, {312, 22008}, {321, 908}, {329, 10478}, {430, 1867}, {464, 17732}, {469, 22018}, {894, 17182}, {1536, 3198}, {1730, 17220}, {1751, 2911}, {1826, 21072}, {1848, 22021}, {3151, 5134}, {3175, 22031}, {3219, 17167}, {3452, 5241}, {3588, 21361}, {3995, 17479}, {3998, 12610}, {4043, 4417}, {4061, 5295}, {4115, 5513}, {4641, 17197}, {5074, 18651}, {6358, 21801}, {14377, 16438}, {21079, 22023}, {21093, 22024}, {21807, 22027}

This is the end of PART 11: Centers X(20001) - X(22000)

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)