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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)


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(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 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(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) : :

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}, {237, 1503}, {338, 2871}, {343, 7467}, {420, 685}, {427, 3051}, {511, 14957}, {660, 1821}, {694, 804}, {1613, 1853}, {2548, 10014}, {2715, 9076}, {2896, 8870}, {3087, 6531}, {3404, 15523}, {3917, 4576}, {6394, 15812}, {7736, 11175}, {11328, 18440}

X(20021) = reflection of X(i) in X(j) for these {i,j}: {6, 7668}, {1634, 141}
X(20021) = complement of X(25046)
X(20021) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1581, 147}, {1910, 8782}, {15391, 6360}
X(20021) = X(2715)-Ceva conjugate of X(879)
X(20021) = X(8623)-cross conjugate of X(2)
X(20021) = crosspoint of X(98) and X(290)
X(20021) = crosssum of X(237) and X(511)
X(20021) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3405}, {82, 511}, {83, 1755}, {237, 3112}, {240, 1176}, {251, 1959}, {308, 9417}, {2491, 4593}, {3569, 4599}, {9418, 18833}, {14966, 18070}, {17209, 18098}
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) = cevapoint of X(i) and X(j) for these (i,j): {141, 732}, {804, 7668}
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) = trilinear pole of line {39, 826}
X(20021) = barycentric product X(i)*X(j) for these {i,j}: {38, 1821}, {39, 290}, {75, 3404}, {98, 141}, {248, 1235}, {287, 427}, {336, 17442}, {685, 2525}, {826, 2966}, {1910, 1930}, {1976, 8024}, {2395, 4576}, {3051, 18024}, {3917, 16081}, {3933, 6531}, {7813, 9154}
X(20021) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3405}, {38, 1959}, {39, 511}, {98, 83}, {141, 325}, {248, 1176}, {287, 1799}, {290, 308}, {427, 297}, {688, 2491}, {732, 5976}, {826, 2799}, {879, 4580}, {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}, {3933, 6393}, {4576, 2396}, {8861, 8928}, {14600, 10547}, {14617, 8840}, {17187, 17209}, {17442, 240}
X(20021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98, 287, 1976), (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) = 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(22023) = isotomic conjugate of X(263)
X(22023) = anticomplement X[3117]
X(22023) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3113, 2}, {3114, 8}, {3407, 192}, {9063, 17217}, {18898, 17486}
X(22023) = X(14994)-cross conjugate of X(183)
X(22023) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3402}, {31, 263}, {32, 2186}, {262, 560}, {327, 1917}
X(22023) = 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(22023) = 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(22023) = {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 CÚsar 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
OA = circle through A' and A'' tangent to U on the negative side of line BC; define OB and OC cyclically
Then X(20032 is the radical center of the circles OA, OB, OC.

See Kadir Altintas and Peter Moses, Hyacinthos 27800. 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 these lines: (pending)


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)

The circle externally tangent to OA, OB, OC 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 Fx.

X(20033) lies on these lines:


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, K701 and on these lines: (pending)

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 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 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) = 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) = {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(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 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(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 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) = {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) = 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(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) = 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) = 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 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) = 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 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 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 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(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(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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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, CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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(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(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(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

-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)

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(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(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(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(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(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(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) = 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) : : See X(20205).

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


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(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

-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) = 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(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(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(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(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(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(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(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(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 CÚsar 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

-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) = (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(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

-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) = -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(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

-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) = 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

-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) = 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 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 polar conjugate of X(1827)
X(20229) = polar conjugate of isotomic conjugate of X(22343)


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

-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) = 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(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

-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)

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(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(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(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

-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)

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

X(20250) lies on these lines: {75, 20162}, {4740, 20168}


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 CÚsar 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 CÚsar 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 CÚsar 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(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

-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)

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(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

-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)

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(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(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

-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) = 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 CÚsar 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

-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) = (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(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(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

-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) = (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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27938.

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 CÚsar 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 CÚsar 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

-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) = 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(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(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(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

-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) = (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

-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)

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

-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) = 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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' = a1/2, 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(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) : :

See X(20333).

X(20334) lies on these lines: {2, 365}, {3661, 20357}, {4180, 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(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(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(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(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(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(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(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(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(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) = perspector of Gemini triangle 32 and cross-triangle of Gemini triangles 32 and 34


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(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(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(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(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(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(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(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(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(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(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(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(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 CÚsar 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(20377) = X(13)X(627)∩X(17)X(671)

Barycentrics    3*S*(8*S^2+SA^2-14*SB*SC-3*SW^ 2)+sqrt(3)*(9*S^2*SA-22*S^2* SW-7*SW*SB*SC) : :
X(20377) = 3*X(13)+X(627) = X(17)-3*X(5459) = 3*X(6669)-2*X(6673)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 27952.

X(20377) lies on these lines: {13, 627}, {17, 671}, {115, 8259}, {140, 6669}, {530, 629}


X(20378) = X(14)X(628)∩X(18)X(671)

Barycentrics    -3*S*(8*S^2+SA^2-14*SB*SC-3* SW^2)+sqrt(3)*(9*S^2*SA-22*S^ 2*SW-7*SW*SB*SC) : :
X(20378) = 3*X(13)+X(627) = X(17)-3*X(5459) = 3*X(6669)-2*X(6673)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 27952.

X(20378) lies on these lines: {14, 628}, {18, 671}, {115, 8260}, {140, 6670}, {531, 630}


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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27967.

X(20402) lies on these lines: {1843, 5895}, {3051, 20232}


X(20403) = X(30)X(511) ∩X(691)X(5467)

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

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 27967.

X(20403) lies on these lines: {23, 351}, {30, 511}, {186, 17994}, {187, 2492}, {476, 805}, {477, 2698}, {669, 9137}, {691, 5467}, {842, 7418}, {1691, 14398}, {2679, 3258}, {3143, 5099}, {3569, 5104}, {3581, 19902}, {7426, 11176}, {7575, 9126}, {9138, 15107}, {9148, 10989}, {10561, 11580}, {11622, 14270}, {11631, 17414}, {16978, 16979}

X(20403) = isogonal conjugate of X(20404)


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 CÚsar 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(20405) = EULER LINE INTERCEPT OF X(15162)X(17810)

Barycentrics    (SW-4*|OH|*R)*S^2-3*SB*SC*SW : :

See Antreas Hatzipolakis and CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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 CÚsar 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) = 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 CÚsar 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) : :

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:

(1,321), (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) : :

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:

(1,312), (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(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(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) : :

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:

(1,42), (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(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(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(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(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(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(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(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(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) : :

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:

(1,55), (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(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(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) = 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) : :

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:

(1,594), (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) : :

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:

(1,2321), (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    (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(20495).

X(20503) lies on these lines: {10, 20489}, {4071, 20498}, {4153, 20501}, {20350, 20370}


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) : :

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:

(1,523), (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(20604)


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) : :

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:

(1,522), (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) : :

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:

(1,1), (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(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(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(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}


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(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(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 : :

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:

(1,2), (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(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(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(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(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(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(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) : :

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:

(1,141), (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(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(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(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(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(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(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 : :

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:

(1,69), (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(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(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(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(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(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(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) = 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) = 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) = 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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 CÚsar 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 CÚsar 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 CÚsar 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(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) : :

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:

(1,38), (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) : :

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:

(1,63), (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(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 CÚsar 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 CÚsar 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) = X(2)-Ceva conjugate of X(3064)


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 CÚsar 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 CÚsar 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 CÚsar 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(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 CÚsar 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) = X(63)-isoconjugate of X(8758)


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 CÚsar 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) = 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 CÚsar 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) : :

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:

(1,1930), (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(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) : :

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:

(1,304), (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(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) : :

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:

(1,20653), (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) : :

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:

(1,6), (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(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(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) : :

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:

(1,6), (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) : :

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), (2,1500), (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) : :

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), (2,20602), (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(20513)
X(20696) = X(i)-isoconjugate of X(j) for these (i,j): {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) : :

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:

(1,756), (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) : :

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:

(1,210), (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 CÚsar 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 CÚsar 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(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 : :

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:

(1,71), (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(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(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(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(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(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(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(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(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) = X(92)-isoconjugate of X(7087)
X(20739) = isogonal conjugate of polar conjugate of X(6327)
X(20739) = isotomic conjugate of polar conjugate of X(1631)


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 : :

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:

(1,219), (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) = X(92)-isoconjugate of X(8852)
X(20741) = isogonal conjugate of polar conjugate of X(4645)
X(20741) = isotomic conjugate of polar conjugate of X(17798)


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) = isotomic conjugate of polar conjugate of X(8301)


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(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 : :

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:

(1,3), (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(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) = X(92)-isoconjugate of X(2162)
X(20760) = crossdifference of every pair of points on line X(814)X(6591)
X(20760) = isogonal conjugate of polar conjugate of X(192)
X(20760) = isotomic conjugate of polar conjugate of X(2176)


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 : :

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:

(1,3), (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) = 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(20769) = isogonal conjugate of polar conjugate of X(350)
X(20769) = isotomic conjugate of polar conjugate of X(238)


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 CÚsar 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 CÚsar 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 CÚsar 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(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 : :

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:

(1,228), (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) = X(92)-isoconjugate of X(20332)
X(20777) = isogonal conjugate of polar conjugate of X(1575)


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(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). (CÚsar 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). (CÚsar 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). (CÚsar 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). (CÚsar 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). (CÚsar 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) = X(92)-isoconjugate of X(11051)
X(20793) = isogonal conjugate of polar conjugate of X(144)
X(20793) = isotomic conjugate of polar conjugate of X(3207)


X(20794) =  (name pending)

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

See X(20795).

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) = X(92)-isoconjugate of X(3224)
X(20794) = isogonal conjugate of polar conjugate of X(194)
X(20794) = isotomic conjugate of polar conjugate of X(1613)


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 : :

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:

(1,20760), (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(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(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(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 : :

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:

(1,394), (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) = X(92)-isoconjugate of X(3445)
X(20818) = isogonal conjugate of polar conjugate of X(145)
X(20818) = isotomic conjugate of polar conjugate of X(3052)


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 : :

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:

(1,3917), (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(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(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(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) : :

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:

(1,39), (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) : :

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:

(1,39), (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(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(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(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) : :

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(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) = 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(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(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(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(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(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(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(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(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(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(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) = 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 polar conjugate of X(1109)
X(20902) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 23964}, {4, 23357}, {19, 1101}, {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(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(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(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(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(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(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(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(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(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(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) = polar conjugate of isogonal conjugate of X(14208)


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(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(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(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(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(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 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 isotomic conjugate of X(1215)


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(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(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 isotomic conjugate of X(960)


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(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(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 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(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(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(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(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(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(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) : :

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(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(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(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(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) = 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(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(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(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(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(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(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(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, 1 6681}, {16678, 20834}, {18613, 20999}


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

X(20991) lies on these lines: {3, 2256}, {6, 41}, {9, 17614}, {19, 1319}, {36, 219}, {37, 3576}, {71, 5204}, {77, 2097}, {101, 5120}, {104, 5776}, {154, 2352}, {284, 999}, {354, 16884}, {577, 1407}, {579, 20818}, {610, 1108}, {836, 1192}, {910, 3554}, {965, 2975}, {1100, 3333}, {1191, 1333}, {1388, 1953}, {1474, 3285}, {1616, 5301}, {1617, 1630}, {1631, 10387}, {1696, 2267}, {1761, 5289}, {1839, 11376}, {1901, 4293}, {2099, 17438}, {2164, 11051}, {2257, 13462}, {2646, 7221}, {2911, 5022}, {3213, 14571}, {4252, 14597}, {4306, 15905}, {5740, 20074}, {5747, 18990}, {5781, 7677}, {7152, 14578}, {10934, 16686}

X(20991) = polar conjugate of isotomic conjugate of X(23072)


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(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(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(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(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}, {94 63, 14002}, {9924, 10836}, {15145, 17984}, {17735, 20472}


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(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) = 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) = 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) = 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) = 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) = 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) = 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) = 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) = 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) = 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) = 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) = 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) : :

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(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(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(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(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(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(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(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(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(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(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) : :

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(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(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(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(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(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(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(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) : :

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(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(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(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(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) : :

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(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(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(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(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(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(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(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(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(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(i)-Ceva conjugate of X(j) for these (i,j): {1, 25}, {17903, 3556}
crosspoint of X(1) and X(1763)
crosssum of X(1) and X(7097)
X(112)-beth conjugate of X(478)
X(i)-isoconjugate of X(j) for these (i,j): {63, 7219}, {69, 7097}, {304, 7169}
barycentric product X(i)*X(j) for these {i,j}: {4, 3556}, {6, 17903}, {19, 1763}, {25, 4329}, {1039, 8900}, {1973, 20914}
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) = {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 CÚsar Lozada, Hyacinthos 28052.

X(21152) lies on the these lines: {3, 6}, {804, 11620}

X(21152) = 1st-Lemoine circle-inverse of X(2674)







leftri  Gibert-Moses Centroids: X(21153) - X(21168)  rightri

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.

underbar

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) = 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 CÚsar 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 CÚsar 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 CÚsar 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) : :