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This is PART 17: Centers X(32001) - X(34000)

PART 1: Introduction and Centers X(1) - X(1000)
PART 2: Centers X(1001) - X(3000)
PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000)
PART 5: Centers X(7001) - X(10000)
PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000)
PART 8: Centers X(14001) - X(16000)
PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000)
PART 11: Centers X(20001) - X(22000)
PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000)
PART 14: Centers X(26001) - X(28000)
PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000)
PART 17: Centers X(32001) - X(34000)
PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000)
PART 20: Centers X(38001) - X(40000)




leftri  Points II-Caph: X(32001)-X(32007)  rightri

This preamble and centers X(32001)-X(32007) were contributed by Clark Kimberling and Peter Moses, April 15, 2019.

Suppose that P = p:q:r (barycentrics). The Point II-Caph of P is the point given by

2qr - (p+q+r)p : 2rp - (p+q+r)q : 2pq - (p+q+r)r.

The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

The appearance of {i,j} in the following list means that X(j) = Point II-Caph of X(i):

{1,3875}, {2,2}, {3,20477}, {6,7754}, {8,21296}, {9,30625}, {10,319}, {30,1494}, {37,25264}, {75,17144}, {141,7768}, {511,290}, {512,670}, {513,668}, {514,190}, {516,18025}, {517,18816}, {518,2481}, {519,903}, {520,6528}, {521,18026}, {522,664}, {523,99}, {524,671}, {525,648}, {527,1121}, {528,18821}, {532,11117}, {533,11118}, {536,3227}, {537,18822}, {538,3228}, {542,5641}, {543,18823}, {690,892}, {696,18824}, {698,3225}, {712,18825}, {714,18826}, {726,3226}, {732,14970}, {740,18827}, {758,14616}, {782,18828}, {804,18829}, {812,4562}, {824,4586}, {826,4577}, {888,886}, {891,889}, {900,4555}, {918,666}, {2574,15164}, {2575,15165}, {2799,2966}, {3413,6190}, {3414,6189}, {3900,4569}, {3910,6648}, {4083,18830}, {4777,4597}, {4977,6540}, {6362,6606}, {6368,18831}, {6550,6635}, {9033,16077}, {23870,23895}, {23871,23896}

underbar

X(32001) = POINT II-CAPH OF X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4 - 6*a^2*b^2 + 3*b^4 - 6*a^2*c^2 + 2*b^2*c^2 + 3*c^4) : :
Barycentrics    6 cot A - csc 2A : :
Barycentrics    (csc 2A) (3 cos^2 A - 1) : :
Barycentrics    tan A - cot A + cot B + cot C - cot ω : :
X(32001) lies on these lines: {2, 15905}, {4, 69}, {7, 5081}, {8, 7282}, {24, 3964}, {30, 6527}, {95, 3525}, {141, 3087}, {183, 8889}, {186, 9723}, {193, 297}, {253, 3146}, {273, 21296}, {281, 4416}, {290, 16774}, {319, 7046}, {320, 1119}, {325, 6353}, {393, 524}, {427, 15589}, {458, 3620}, {491, 3536}, {492, 3535}, {599, 6748}, {648, 11008}, {1270, 1585}, {1271, 1586}, {1597, 14929}, {1990, 6144}, {1992, 17907}, {2897, 6925}, {3089, 7776}, {3529, 20477}, {3763, 6749}, {3879, 7952}, {3933, 7487}, {3945, 17555}, {4417, 7490}, {4869, 26003}, {5232, 11109}, {6515, 6820}, {6995, 10513}, {7714, 7788}, {9308, 20080}, {10002, 11477}, {10312, 14069}, {11180, 16264}, {14362, 16096}, {18950, 19166}

X(32001) = anticomplement of X(15905)
X(32001) = polar conjugate of isotomic conjugate of X(32831)
X(32001) = {X(4),X(69)}-harmonic conjugate of X(32000)


X(32002) = POINT II-CAPH OF X(5)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4) : :
Barycentrics    3 tan A - cot A : :
Barycentrics    4 cot A - 3 csc 2A : :
Barycentrics    csc 2A sin(A + π/6) sin(A - π/6) : :

X(32002) lies on these lines: {2, 10985}, {4, 69}, {5, 95}, {53, 648}, {99, 2383}, {183, 5064}, {216, 17035}, {265, 6528}, {275, 467}, {290, 15321}, {297, 3589}, {302, 473}, {303, 472}, {325, 428}, {382, 20477}, {458, 3763}, {1007, 7714}, {1078, 15559}, {1273, 18354}, {1494, 15687}, {1907, 7750}, {1994, 14129}, {2052, 13579}, {3087, 17907}, {3518, 7769}, {3855, 8797}, {3964, 5198}, {5081, 5564}, {5976, 22480}, {6144, 9308}, {6390, 6756}, {6527, 17578}, {7282, 7321}, {7752, 9723}, {7806, 10311}, {14111, 20572}, {14516, 19169}

X(32002) = isotomic conjugate of X(3519)
X(32002) = anticomplement of X(22052)
X(32002) = polar conjugate of X(2963)


X(32003) = POINT II-CAPH OF X(7)

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

X(32003) lies on these lines: {2, 5543}, {7, 8}, {57, 29616}, {145, 3160}, {150, 962}, {279, 519}, {347, 4460}, {348, 3241}, {392, 28644}, {527, 30695}, {664, 20050}, {728, 1445}, {1323, 3633}, {1434, 4720}, {2481, 7319}, {3187, 18624}, {3188, 12536}, {3598, 4848}, {3621, 9312}, {3623, 31721}, {3632, 10481}, {3672, 6738}, {3875, 20008}, {3912, 5435}, {4323, 7179}, {4328, 5232}, {4350, 6765}, {4384, 5226}, {5175, 20244}, {5228, 29611}, {5273, 25935}, {5704, 24203}, {5734, 17181}, {6049, 17081}, {7264, 18391}, {10405, 20059}, {10914, 23839}, {12630, 14189}, {17077, 27253}, {20111, 26531}, {25719, 31145}

X(32003) = {X(7),X(8)}-harmonic conjugate of X(31994)


X(32004) = POINT II-CAPH OF X(86)

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

X(32004) lies on these lines: {1,6626}, {10,86}, {65,664}, {81,2295}, {99,3244}, {145,17103}, {274,4714}, {333,16826}, {519,1509}, {524,23905}, {648,1855}, {757,5255}, {1931,29588}, {2106,3780}, {2329,5325}, {3635,6629}, {3874,4360}, {5711,17377}, {5937,23361}, {6625,20090}, {8033,17144}, {16834,24378}, {17378,17528}, {17778,20337}


X(32005) = POINT II-CAPH OF X(192)

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

X(32005) lies on these lines: {1,87}, {2,20943}, {239,3928}, {333,16722}, {350,20105}, {385,5204}, {499,19570}, {1107,4699}, {1278,17448}, {1478,13571}, {1575,21219}, {1655,5550}, {2275,20081}, {3240,21223}, {3360,21762}, {3614,7777}, {3617,21226}, {3621,17759}, {3634,27318}, {4352,17383}, {5217,7783}, {5229,7774}, {7354,7837}, {7779,9597}, {7905,9651}, {9263,20050}, {16815,24621}, {16969,25269}, {17232,24215}, {19701,29595}, {19862,27269}, {20036,20072}, {27268,31997}


X(32006) = POINT II-CAPH OF X(69)

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

X(32006) lies on these lines: {2, 3053}, {3, 1007}, {4, 69}, {5, 3785}, {20, 325}, {30, 3926}, {32, 14064}, {83, 18841}, {99, 3529}, {115, 14023}, {127, 28696}, {148, 7946}, {183, 3091}, {193, 5254}, {290, 15077}, {350, 5225}, {376, 7763}, {381, 7767}, {382, 3933}, {385, 14063}, {388, 25303}, {491, 6459}, {492, 6460}, {524, 6392}, {543, 7916}, {546, 14929}, {626, 7737}, {631, 7752}, {754, 3767}, {1078, 3090}, {1285, 7934}, {1369, 7394}, {1370, 3266}, {1384, 8361}, {1657, 6390}, {1909, 5229}, {1975, 3146}, {1992, 5286}, {2207, 20806}, {2548, 6683}, {2549, 7759}, {2896, 16924}, {2996, 20080}, {3096, 16045}, {3314, 14035}, {3436, 20553}, {3524, 7769}, {3525, 7771}, {3528, 7814}, {3543, 7788}, {3545, 7811}, {3618, 6656}, {3619, 7770}, {3788, 3849}, {3832, 15589}, {3855, 6249}, {3972, 14069}, {4417, 7406}, {4872, 21605}, {5025, 7735}, {5177, 16992}, {5188, 7694}, {5304, 7851}, {5319, 7861}, {5475, 7800}, {5866, 11413}, {6340, 7396}, {6655, 7738}, {6658, 7897}, {6781, 7888}, {6997, 26235}, {7386, 11059}, {7388, 9540}, {7389, 13935}, {7391, 9464}, {7503, 15574}, {7736, 7785}, {7739, 7838}, {7746, 23055}, {7747, 7795}, {7748, 7758}, {7753, 7935}, {7754, 11008}, {7756, 7903}, {7775, 7830}, {7779, 20105}, {7782, 17538}, {7799, 11001}, {7803, 7812}, {7815, 31415}, {7822, 14537}, {7832, 14039}, {7833, 7941}, {7847, 7926}, {7858, 7910}, {7862, 21843}, {7866, 18907}, {7879, 8370}, {7881, 19687}, {7890, 11648}, {7893, 14041}, {7912, 14712}, {7917, 15682}, {7921, 7924}, {7929, 16044}, {7931, 14037}, {7933, 16989}, {7938, 16898}, {7939, 11361}, {7945, 19569}, {8356, 31400}, {8357, 9605}, {8362, 15484}, {8743, 22151}, {9292, 20022}, {9723, 11414}, {10159, 18844}, {10513, 17578}, {11285, 31404}, {12215, 14927}, {14062, 17129}, {14068, 17128}, {14570, 28728}, {19695, 31859}, {20556, 21281}


X(32007) = POINT II-CAPH OF X(142)

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

X(32007) lies on these lines: {1, 17078}, {7, 8}, {57, 32019}, {79, 2481}, {99, 1434}, {142, 32008}, {145, 17079}, {150, 18480}, {241, 29569}, {279, 3623}, {348, 3622}, {527, 32024}, {553, 3912}, {664, 3244}, {728, 17298}, {1125, 9436}, {2099, 7185}, {3254, 4569}, {3633, 9312}, {3649, 16823}, {4089, 11009}, {4346, 15956}, {4384, 4654}, {4675, 27253}, {4872, 12699}, {4887, 6738}, {5228, 17367}, {5845, 27000}, {5905, 30854}, {7146, 24803}, {7196, 29824}, {10025, 21258}, {11553, 17320}, {17084, 24798}, {17092, 17317}, {17181, 18493}, {17776, 21454}, {21314, 25716}






leftri  Points III-Caph: X(32008)-X(32023)  rightri

This preamble and centers X(32008)-X(32023) were contributed by Clark Kimberling and Peter Moses, April 15, 2019.

Suppose that P = p:q:r (barycentrics). The Point III-Caph of P is the point given by

(p + q)(p + r) : (q + r)(q + p) : (r + p)(r + q)

The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

The appearance of {i,j} in the following list means that X(j) = Point III-Caph of X(i):

{1,86}, {2,2}, {3,95}, {4,264}, {6,83}, {7,85}, {8,75}, {10,1268}, {13,11119}, {14,11120}, {20,69}, {22,1799}, {23,2373}, {30,1494}, {63,333}, {65,31643}, {69,76}, {75,274}, {76,308}, {81,14534}, {85,31618}, {92,31623}, {95,31617}, {99,4590}, {100,4998}, {110,18020}, {141,10159}, {144,8}, {145,7}, {146,3260}, {147,325}, {148,523}, {149,693}, {150,3261}, {153,3262}, {190,1016}, {192,1}, {193,4}, {194,6}, {239,673}, {261,31620}, {264,276}, {279,30705}, {287,9476}, {297,6330}, {308,31622}, {315,1502}, {316,18023}, {317,18027}, {320,20568}, {321,30710}, {323,2986}, {325,8781}, {329,312}, {330,87}, {346,30701}, {347,348}, {385,98}, {394,801}, {401,287}, {487,491}, {488,492}, {511,290}, {512,670}, {513,668}, {514,190}, {516,18025}, {517,18816}, {518,2481}, {519,903}, {520,6528}, {521,18026}, {522,664}, {523,99}, {524,671}, {525,648}, {527,1121}, {528,18821}, {532,11117}, {533,11118}, {536,3227}, {537,18822}, {538,3228}, {542,5641}, {543,18823}, {599,10302}, {616,298}, {617,299}, {621,300}, {622,301}, {627,302}, {628,303}, {648,23582}, {649,4598}, {650,30610}, {659,8709}, {661,27805}, {664,1275}, {668,31625}, {669,3222}, {690,892}, {693,4554}, {696,18824}, {698,3225}, {712,18825}, {714,18826}, {726,3226}, {732,14970}, {740,18827}, {758,14616}, {782,18828}, {804,18829}, {812,4562}, {824,4586}, {826,4577}, {850,6331}, {858,30786}, {888,886}, {891,889}, {894,1220}, {900,4555}, {908,4997}, {918,666}, {962,309}, {1046,18812}, {1270,5490}, {1271,5491}, {1272,7799}, {1278,330}, {1330,313}, {1352,327}, {1369,8024}, {1370,305}, {1494,31621}, {1577,15455}, {1654,10}, {1655,37}, {1764,14829}, {1909,1221}, {1944,8777}, {1992,598}, {1993,275}, {2397,5376}, {2475,1441}, {2574,15164}, {2575,15165}, {2799,2966}, {2888,311}, {2889,1232}, {2890,1233}, {2891,1269}, {2892,1236}, {2893,349}, {2895,321}, {2896,141}, {2897,1231}, {2975,261}, {2998,3224}, {3056,18299}, {3091,8797}, {3100,31637}, {3146,253}, {3151,306}, {3152,307}, {3153,328}, {3164,3}, {3177,9}, {3180,13}, {3181,14}, {3187,27}, {3210,57}, {3261,31624}, {3413,6190}, {3414,6189}, {3434,6063}, {3436,3596}, {3448,850}, {3580,16080}, {3616,30598}, {3617,5936}, {3620,18840}, {3621,4373}, {3622,28626}, {3623,30712}, {3648,319}, {3729,1222}, {3732,15742}, {3868,286}, {3869,314}, {3870,21453}, {3875,1434}, {3900,4569}, {3904,13136}, {3910,6648}, {3952,7035}, {3995,1255}, {4025,658}, {4083,18830}, {4232,10603}, {4329,304}, {4360,1509}, {4373,27818}, {4388,7018}, {4391,6335}, {4393,14621}, {4427,4600}, {4440,514}, {4452,279}, {4461,1219}, {4462,646}, {4467,4573}, {4468,3699}, {4552,4564}, {4560,662}, {4580,827}, {4608,4632}, {4645,334}, {4777,4597}, {4977,6540}, {4998,31619}, {5032,18842}, {5080,20566}, {5189,18019}, {5207,18896}, {5484,4357}, {5596,315}, {5744,30608}, {5889,8795}, {5905,92}, {5942,318}, {5984,9473}, {6193,317}, {6194,183}, {6223,322}, {6224,320}, {6225,14615}, {6327,561}, {6360,63}, {6362,6606}, {6368,18831}, {6392,393}, {6462,3068}, {6463,3069}, {6515,2052}, {6527,3926}, {6542,335}, {6550,6635}, {6563,4563}, {6646,257}, {6655,9229}, {6758,24041}, {7057,4146}, {7192,799}, {7253,811}, {7391,18018}, {7396,6340}, {7500,13575}, {7665,468}, {7674,6604}, {7766,3407}, {7774,262}, {7777,7608}, {7779,1916}, {7785,3613}, {7791,31360}, {7837,14492}, {7840,5503}, {8024,1241}, {8055,18743}, {8264,32}, {8266,1078}, {8267,251}, {8272,2896}, {8591,524}, {8782,385}, {8859,10153}, {8878,427}, {8892,8889}, {9033,16077}, {9143,9141}, {9147,9150}, {9168,9170}, {9263,513}, {9295,9267}, {9308,1105}, {9312,23618}, {9485,6082}, {9740,11172}, {9742,1007}, {9870,9084}, {9965,189}, {10025,14942}, {10336,7792}, {10340,1843}, {10453,6384}, {10529,7318}, {10754,9154}, {11004,7578}, {11054,18818}, {11061,316}, {11148,1992}, {11160,5485}, {11415,20570}, {11442,18022}, {11606,9477}, {11794,27867}, {11851,1119}, {12221,24243}, {12222,24244}, {12272,683}, {12383,340}, {12384,30737}, {12649,273}, {13174,17731}, {13219,3267}, {14360,3266}, {14361,15466}, {14570,249}, {14618,30450}, {14683,13485}, {14712,67}, {14721,6333}, {14731,3268}, {14779,4427}, {14790,20564}, {14807,22339}, {14808,22340}, {14957,18024}, {15412,18315}, {16017,556}, {16018,174}, {16019,188}, {16552,17277}, {16704,24624}, {17006,10185}, {17008,7607}, {17018,2296}, {17035,5}, {17036,11}, {17037,20}, {17134,17206}, {17135,310}, {17136,4620}, {17137,6385}, {17140,873}, {17147,81}, {17148,58}, {17149,31008}, {17154,7192}, {17161,4610}, {17165,3112}, {17166,4623}, {17217,4602}, {17257,31359}, {17300,17758}, {17316,27475}, {17350,17743}, {17479,2167}, {17480,269}, {17483,30690}, {17484,18359}, {17485,9285}, {17486,31}, {17487,519}, {17488,3679}, {17489,82}, {17490,8056}, {17493,1581}, {17494,100}, {17495,88}, {17496,651}, {17497,897}, {17498,162}, {17499,18082}, {17732,17233}, {17751,1240}, {17759,291}, {17778,226}, {17781,4102}, {17784,8817}, {17794,350}, {17896,13149}, {17950,1952}, {18105,6573}, {18133,18140}, {18287,6353}, {18301,323}, {18662,2185}, {18663,2184}, {18666,72}, {18667,1214}, {18668,2349}, {18906,3114}, {19565,292}, {19566,9468}, {19569,11058}, {19570,1989}, {19577,8791}, {19789,15474}, {20011,8049}, {20016,6650}, {20018,1246}, {20040,20028}, {20042,6548}, {20045,675}, {20055,27494}, {20059,10405}, {20061,7219}, {20064,7357}, {20065,66}, {20072,80}, {20073,1000}, {20075,13577}, {20076,8048}, {20077,8044}, {20078,2994}, {20080,2996}, {20081,2998}, {20086,1029}, {20088,1031}, {20090,6625}, {20092,8046}, {20095,8047}, {20099,13574}, {20101,7224}, {20111,6601}, {20211,329}, {20212,1034}, {20213,1032}, {20222,1444}, {20245,28660}, {20294,4561}, {20295,1978}, {20344,3263}, {20345,1921}, {20347,18031}, {20348,27424}, {20533,3912}, {20534,18297}, {20535,7155}, {20536,11599}, {20537,30545}, {20553,18895}, {20554,18891}, {21124,8052}, {21215,1760}, {21216,19}, {21217,38}, {21218,55}, {21219,192}, {21220,661}, {21221,1577}, {21222,3257}, {21223,3223}, {21224,649}, {21225,101}, {21226,256}, {21270,1969}, {21275,1928}, {21278,18833}, {21281,6383}, {21285,20567}, {21286,28659}, {21287,27801}, {21288,20641}, {21289,1930}, {21290,3264}, {21294,20948}, {21295,24037}, {21299,18832}, {21301,6386}, {21302,4572}, {22113,19712}, {22114,19713}, {22647,22468}, {23870,23895}, {23871,23896}, {24068,4360}, {24349,870}, {25046,20022}, {25054,512}, {25237,2346}, {25257,2991}, {25259,1897}, {25268,5382}, {25332,3978}, {26853,8050}, {27340,2297}, {27377,14860}, {27484,4384}, {27835,27813}, {28604,1224}, {29824,31002}, {29840,7249}, {30562,5333}, {30564,5235}, {30577,3911}, {30578,4358}, {30579,16704}, {30660,1920}, {30661,171}, {30662,30669}, {30667,238}, {30669,30663}, {30694,281}, {30695,346}, {30699,278}, {31087,1390}, {31125,10415}, {31290,3952}, {31296,110}, {31298,9295}, {31308,16826}, {31372,148}, {31527,31627}, {31670,14387}, {31859,11169}

The Point III-Caph of P is the cevapoint of X(2) and P, and the isotomic conjugate of the complement of P, and also the polar conjugate of the crosspoint of X(4) and the polar conjugate of P. (Randy Hutson, April 22, 2019)

underbar

X(32008) = POINT III-CAPH OF X(9)

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

X(32008) lies on these lines: {1, 31269}, {2, 220}, {6, 27253}, {8, 344}, {9, 85}, {10, 10482}, {29, 1861}, {92, 3305}, {189, 7367}, {190, 20880}, {218, 14828}, {257, 17260}, {312, 728}, {333, 1174}, {349, 6559}, {664, 1212}, {672, 1434}, {673, 1334}, {1121, 6606}, {1213, 27023}, {1220, 17353}, {1698, 28874}, {2338, 17095}, {2481, 3294}, {3730, 14377}, {3740, 28058}, {4422, 17947}, {4473, 17128}, {4513, 27304}, {4518, 16823}, {4997, 5316}, {5278, 29616}, {6706, 10025}, {6738, 25072}, {9312, 31169}, {14829, 29627}, {16284, 17335}, {16788, 17687}, {17338, 17743}, {18359, 26591}, {21872, 27000}, {25500, 30608}, {25878, 26059}, {27065, 30690}, {28980, 30806}

X(32008) = isogonal conjugate X(1475)
X(32008) = isotomic conjugate of X(142)
X(32008) = cevapoint of X(2) and X(9)
X(32008) = trilinear pole of line X(522)X(3935) (the isotomic conjugate of the circumconic centered at X(1212))


X(32009) = POINT III-CAPH OF X(37)

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

X(32009) lies on these lines: {1, 872}, {2, 1500}, {10, 30571}, {37, 274}, {57, 16831}, {75, 4099}, {81, 213}, {83, 5284}, {86, 3294}, {88, 29578}, {89, 29595}, {105, 8708}, {190, 17175}, {239, 1255}, {277, 17321}, {279, 26125}, {291, 1125}, {330, 5283}, {335, 21816}, {595, 985}, {668, 16589}, {959, 4323}, {961, 7677}, {1002, 3616}, {1107, 3227}, {1224, 17289}, {1258, 3230}, {1390, 16823}, {2664, 9401}, {3963, 25661}, {4384, 25430}, {4393, 27789}, {4698, 16819}, {5291, 16912}, {5550, 7786}, {16369, 28615}, {16818, 17263}, {17260, 20963}, {17320, 24790}, {18140, 27255}, {19742, 25417}

X(32009) = isogonal conjugate X(20963)
X(32009) = isotomic conjugate of X(3739)
X(32009) = cevapoint of X(2) and X(37)


X(32010) = POINT III-CAPH OF X(38)

Barycentrics    (a + b)*(a + c)*(b^2 + a*c)*(a*b + c^2) : :
Barycentrics    1/(sin(B + ω) + sin(C + ω)) : :

X(32010)lies on these lines: {2, 694}, {38, 799}, {57, 14534}, {86, 1431}, {99, 17596}, {239, 257}, {244, 873}, {256, 314}, {261, 1178}, {354, 2668}, {756, 7035}, {805, 9073}, {846, 4154}, {870, 982}, {984, 7033}, {2669, 3752}, {3027, 9791}, {3863, 27164}, {4389, 14616}, {4562, 16587}, {4594, 27922}, {4603, 6654}, {5009, 7305}, {7192, 8034}, {16739, 18021}, {16975, 18826}, {18822, 18829}

X(32010) = isogonal conjugate X(20964)
X(32010) = isotomic conjugate of X(1215)
X(32010) = polar conjugate of X(1840)
X(32010) = cevapoint of X(2) and X(38)
X(32010) = trilinear pole of line X(812)X(4560)


X(32011) = POINT III-CAPH OF X(43)

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

X(32011) lies on these lines: {2, 16969}, {42, 31002}, {43, 6384}, {75, 16569}, {86, 6685}, {310, 899}, {335, 3752}, {1575, 27447}, {17490, 27494}, {17754, 27498}

X(32011) = isogonal conjugate X(22343)
X(32011) = isotomic conjugate of X(3840)
X(32011) = cevapoint of X(2) and X(43)


X(32012) = POINT III-CAPH OF X(44)

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

X(32012) lies on these lines: {2, 1017}, {44, 20568}, {668, 17349}, {3679, 4595}, {4671, 16816}, {5219, 25529}, {5235, 17292}, {16704, 30866}, {21385, 23598}

X(32012) = isotomic conjugate of X(3834)
X(32012) = cevapoint of X(2) and X(44)


X(32013) = POINT III-CAPH OF X(45)

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

X(32013) lies on these lines: {45, 20569}, {519, 16484}, {4358, 16815}, {16704, 29569}

X(32013) = isotomic conjugate of complement of X(45)
X(32013) = cevapoint of X(2) and X(45)


X(32014) = POINT III-CAPH OF X(86)

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

X(31014) lies on these lines: {2, 1171}, {10, 86}, {76, 15668}, {83, 17398}, {98, 6578}, {99, 1125}, {226, 1434}, {274, 321}, {671, 23905}, {873, 18140}, {2368, 8701}, {3624, 17103}, {3634, 17731}, {4052, 16712}, {4080, 4632}, {4102, 29574}, {4596, 13576}, {4608, 5466}, {6539, 6542}, {6540, 17175}, {6626, 19862}, {6629, 19878}, {10159, 17245}, {11140, 26541}, {14534, 20337}, {18157, 30126}, {24051, 24058}, {25508, 28615}, {27797, 31011}

X(31014) = isogonal conjugate X(20970)
X(32014) = isotomic conjugate of X(1213)
X(32014) = polar conjugate of X(430)
X(32014) = cevapoint of X(2) and X(86)
X(32014) = barycentric product X(30593)*X(30594)


X(32015) = POINT III-CAPH OF X(142)

Barycentrics    (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) : :

X(32015) lies on these lines: {8, 3826}, {85, 10012}, {1125, 14942}, {3912, 4102}, {6706, 31640}, {17282, 31359}, {17743, 27147}

X(32015) = isotomic conjugate of X(6666)
X(32015) = cevapoint of X(2) and X(142)


X(32016) = POINT III-CAPH OF X(244)

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

X(32016) lies on these lines: {239, 6631}, {244, 7035}, {350, 1266}, {4885, 27191}, {5437, 6654}, {7033, 17063}, {10436, 27922}, {27195, 31286}

X(32016) = isotomic conjugate of X(24003)
X(32016) = cevapoint of X(2) and X(244)


X(32017) = POINT III-CAPH OF X(312)

Barycentrics    b*c*(a^2 - 2*a*b + b^2 + a*c + b*c)*(a^2 + a*b - 2*a*c + b*c + c^2) : :
Barycentrics    1/(csc^2(B/2) + csc^2(C/2)) : :

X(32017) lies on these lines: {1, 341}, {57, 312}, {75, 8056}, {76, 279}, {81, 4358}, {88, 321}, {105, 1261}, {274, 16736}, {278, 7017}, {291, 3840}, {668, 3452}, {961, 1476}, {985, 29649}, {1022, 1577}, {1224, 19864}, {1432, 3912}, {1997, 3596}, {4671, 26745}, {6685, 30571}, {14829, 29529}, {17786, 30827}, {25430, 30829}, {30710, 30818}

X(32017) = isogonal conjugate X(20228)
X(32017) = isotomic conjugate of X(3752)
X(32017) = polar conjugate of X(1828)
X(32017) = cevapoint of X(2) and X(312)
X(32017) = trilinear pole of line X(513)X(4397)


X(32018) = POINT III-CAPH OF X(319)

Barycentrics    b*c*(a + 2*b + c)*(a + b + 2*c) : :
Barycentrics    b c/(a r + S) : :

X(32018) lies on these lines: {2, 7230}, {75, 1089}, {76, 6539}, {79, 319}, {85, 4102}, {274, 321}, {286, 31902}, {304, 29602}, {334, 6538}, {668, 4647}, {767, 8701}, {870, 1126}, {1909, 31013}, {2481, 6540}, {4632, 18359}, {4717, 25303}, {4980, 18145}, {6385, 27801}, {7278, 17762}, {20568, 20911}, {20569, 30589}

X(32018) = isotomic conjugate of X(1100)
X(32018) = polar conjugate of X(2355)
X(32018) = cevapoint of X(2) and X(319)
X(32018) = trilinear pole of line X(693)X(4036)


X(32019) = POINT III-CAPH OF X(344)

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

X(32019) lies on these lines: {1, 17263}, {81, 17244}, {105, 3871}, {274, 17279}, {277, 344}, {1224, 17371}, {1255, 17367}, {2006, 30829}, {25417, 29569}

X(32019) = isotomic conjugate of X(17278)
X(32019) = cevapoint of X(2) and X(344)


X(32020) = POINT III-CAPH OF X(350)

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

X(32020) lies on these lines: {1, 668}, {2, 1978}, {28, 811}, {57, 4554}, {76, 330}, {81, 799}, {89, 30964}, {105, 8709}, {192, 20671}, {274, 4602}, {291, 350}, {670, 25534}, {727, 789}, {1002, 17794}, {1022, 20568}, {1221, 26971}, {1921, 20363}, {1929, 1966}, {3123, 18830}, {3227, 18145}, {3403, 8056}, {3840, 7018}, {9263, 18135}, {17027, 20457}, {18895, 27918}, {20372, 20610}, {21297, 31002}, {24625, 27853}, {29438, 29454}, {29750, 29811}

X(32020) = isogonal conjugate X(21760)
X(32020) = isotomic conjugate of X(1575)
X(32020) = cevapoint of X(2) and X(350)


X(32021) = POINT III-CAPH OF X(354)

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

X(32021) lies on these lines: {2481, 3742}, {4384, 5437}, {7271, 25496}

X(32021) = isotomic conjugate of X(3740)
X(32021) = cevapoint of X(2) and X(354)


X(32022) = POINT III-CAPH OF X(391)

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

X(32022) lies on these lines: {2, 2271}, {4, 17277}, {10, 344}, {69, 17758}, {76, 966}, {98, 25446}, {226, 4384}, {321, 28809}, {2478, 13576}, {3424, 7385}, {5224, 18840}, {5392, 26592}, {6625, 17349}, {7379, 14484}, {16827, 31405}, {17352, 18841}, {25006, 30701}

X(32022) = isogonal conjugate X(5021)
X(32022) = isotomic conjugate of X(4648)
X(32022) = polar conjugate of X(4196)
X(32022) = cevapoint of X(2) and X(391)


X(32023) = POINT III-CAPH OF X(497)

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

X(32023) lies on these lines: {2, 14936}, {11, 6063}, {55, 4998}, {75, 24386}, {85, 3817}, {497, 4554}, {693, 15283}, {982, 3663}, {1699, 7196}, {2481, 5274}, {3061, 3452}, {3263, 3705}, {3675, 7185}, {3794, 9309}, {6384, 27498}, {13577, 15284}, {14727, 18821}, {15149, 17917}, {20295, 26910}

X(32023) = isotomic conjugate of X(1376)
X(32023) = cevapoint of X(2) and X(497)






leftri  Points IV-Caph: X(32024)-X(32035)  rightri

This preamble and centers X(32024)-X(32035) were contributed by Clark Kimberling and Peter Moses, April 15, 2019.

Suppose that P = p:q:r (barycentrics). The Point IV-Caph of P is the point given by

qr - (p + q + r)p : rp - (p + q + r)q : pq - (p + q + r)r

The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

The appearance of {i,j} in the following list means that X(j) = Point IV-Caph of X(i):

{1,4360}, {2,2}, {4,317}, {6,7760}, {7,6604}, {8,69}, {20,6527}, {30,1494}, {69,315}, {75,17143}, {144,30695}, {145,4452}, {192,194}, {193,6392}, {194,8264}, {511,290}, {512,670}, {513,668}, {514,190}, {516,18025}, {517,18816}, {518,2481}, {519,903}, {520,6528}, {521,18026}, {522,664}, {523,99}, {524,671}, {525,648}, {527,1121}, {528,18821}, {532,11117}, {533,11118}, {536,3227}, {537,18822}, {538,3228}, {542,5641}, {543,18823}, {690,892}, {696,18824}, {698,3225}, {712,18825}, {714,18826}, {726,3226}, {732,14970}, {740,18827}, {758,14616}, {782,18828}, {804,18829}, {812,4562}, {824,4586}, {826,4577}, {888,886}, {891,889}, {900,4555}, {918,666}, {2574,15164}, {2575,15165}, {2799,2966}, {3413,6190}, {3414,6189}, {3900,4569}, {3910,6648}, {4083,18830}, {4560,14570}, {4777,4597}, {4977,6540}, {5905,6515}, {6362,6606}, {6368,18831}, {6542,6653}, {6550,6635}, {7057,7}, {7192,4576}, {9033,16077}, {16017,329}, {16018,3210}, {16019,3177}, {20346,6327}, {20534,8}, {20555,21275}, {23870,23895}, {23871,23896}

Let Gp be the centroid of ABCP (= complement of the complement of P). Let A'B'C' be the reflection in Gp of the cevian triangle of P. A'B'C' is perspective to ABC for all P, and the perspector is the isotomic conjugate of the Point IV-Caph of P. (Randy Hutson, April 20, 2019)

Let P' be the complement of P. Let A'B'C' be the anticevian triangle of P'. Let Gp' be the centroid of A'B'C'P'. Then the Point IV-Caph of P is the isotomic conjugate of the anticomplement of Gp'. (Randy Hutson, April 20, 2019)

underbar

X(32024) = POINT IV-CAPH OF X(9)

Barycentrics    -a^4 + 3*a^3*b - 3*a^2*b^2 + a*b^3 + 3*a^3*c - 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(32024) lies on these lines: {1, 31169}, {8, 190}, {9, 85}, {45, 27253}, {63, 30854}, {144, 6604}, {220, 664}, {728, 25728}, {1212, 10025}, {2481, 16552}, {3730, 3732}, {3929, 4384}, {4640, 28058}, {5282, 6559}, {6554, 31640}, {6558, 25278}, {14512, 31852}, {14828, 16601}, {16284, 17336}, {17257, 26561}, {17277, 20880}, {17350, 27288}, {17781, 25935}


X(32025) = POINT IV-CAPH OF X(10)

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

X(32025) lies on these lines: {2, 4445}, {6, 29593}, {8, 4026}, {10, 86}, {37, 29615}, {69, 3617}, {75, 537}, {141, 27191}, {190, 594}, {192, 17251}, {239, 17239}, {313, 25280}, {320, 4691}, {391, 17354}, {519, 17322}, {524, 28604}, {536, 17252}, {599, 4699}, {894, 4690}, {966, 17233}, {1100, 29610}, {1213, 4478}, {1278, 17253}, {1698, 17394}, {2321, 17256}, {2345, 17346}, {3589, 29591}, {3626, 4357}, {3631, 26806}, {3632, 17393}, {3661, 17275}, {3662, 28634}, {3664, 4745}, {3686, 17289}, {3729, 17328}, {3739, 17287}, {3759, 4034}, {3763, 16816}, {3875, 4668}, {3949, 18714}, {3995, 4102}, {4001, 19797}, {4007, 4664}, {4021, 4746}, {4361, 17238}, {4363, 17343}, {4384, 17228}, {4389, 4452}, {4393, 17327}, {4399, 17302}, {4431, 17258}, {4464, 4701}, {4470, 20080}, {4472, 20090}, {4645, 4733}, {4651, 30966}, {4657, 29617}, {4659, 17329}, {4665, 6646}, {4669, 17320}, {4686, 17254}, {4687, 17294}, {4688, 17288}, {4698, 17310}, {4708, 17319}, {4740, 17255}, {4741, 17118}, {4751, 17296}, {4772, 7232}, {4788, 24441}, {4851, 29576}, {4852, 17326}, {4889, 29580}, {5257, 17315}, {5790, 10446}, {5839, 17381}, {6540, 30594}, {7081, 30761}, {7227, 20072}, {8013, 17731}, {10436, 17360}, {10453, 31341}, {12531, 18654}, {15668, 17373}, {16777, 20055}, {16815, 17231}, {16826, 17372}, {16831, 17386}, {16832, 17241}, {16833, 17370}, {16834, 17400}, {17045, 20016}, {17116, 17344}, {17117, 17237}, {17119, 17236}, {17121, 17385}, {17143, 18133}, {17151, 17249}, {17229, 17260}, {17230, 17259}, {17248, 17299}, {17263, 29594}, {17268, 31333}, {17280, 17330}, {17281, 17331}, {17286, 17335}, {17292, 17348}, {17293, 17349}, {17303, 17363}, {17309, 27268}, {17312, 31238}, {17317, 24603}, {17337, 29587}, {17352, 29611}, {17361, 25590}, {17374, 28633}, {17772, 19856}, {17791, 18698}, {20653, 31620}, {21277, 21677}, {25498, 29584}, {26048, 26979}, {26244, 30179}, {29446, 30866}, {31025, 31143}

X(32025) = isotomic conjugate of isogonal conjugate of X(33771)


X(32026) = POINT IV-CAPH OF X(37)

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

X(32026) lies on these lines: {1, 190}, {37, 274}, {39, 27195}, {192, 5283}, {194, 4704}, {213, 17261}, {350, 25092}, {536, 16819}, {668, 1500}, {1107, 4681}, {1621, 7760}, {2276, 18140}, {3616, 7757}, {3807, 4075}, {3995, 27163}, {4043, 10471}, {4098, 24215}, {4099, 17762}, {4360, 16552}, {9331, 24524}, {16589, 17759}, {16712, 27097}, {16818, 17320}, {16997, 31451}, {17160, 29773}, {17263, 24790}, {17280, 25499}, {17319, 20963}, {18145, 27020}, {18827, 24051}, {21070, 30966}


X(32027) = POINT IV-CAPH OF X(141)

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

X(32027) lies on these lines: {2, 7826}, {39, 31168}, {69, 3096}, {76, 338}, {83, 141}, {99, 2896}, {110, 14247}, {183, 7899}, {194, 7865}, {315, 3620}, {384, 7848}, {385, 7849}, {524, 7859}, {538, 7928}, {542, 10357}, {626, 14061}, {1078, 3314}, {1975, 7936}, {3329, 7882}, {3631, 6656}, {3734, 7929}, {3763, 7878}, {3785, 7835}, {3933, 7831}, {3934, 7809}, {5008, 19694}, {5041, 16897}, {6179, 7868}, {6292, 7779}, {6392, 7790}, {6683, 7840}, {7751, 7919}, {7752, 16990}, {7754, 7937}, {7759, 16986}, {7766, 7914}, {7767, 7832}, {7770, 7850}, {7771, 7881}, {7780, 7931}, {7786, 7788}, {7793, 7869}, {7795, 7811}, {7796, 7800}, {7801, 7904}, {7805, 7948}, {7808, 7946}, {7810, 7836}, {7812, 21356}, {7814, 15271}, {7815, 7897}, {7818, 31276}, {7822, 7893}, {7824, 7895}, {7827, 22165}, {7846, 14023}, {7853, 17129}, {7855, 7876}, {7871, 11285}, {7873, 17128}, {7885, 9466}, {7898, 17130}, {7905, 8362}, {7933, 17131}, {7935, 20081}, {7941, 31239}, {7942, 8667}, {7943, 14614}, {7949, 11174}, {8370, 10302}, {19577, 21248}


X(32028) = POINT IV-CAPH OF X(190)

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

X(32028) lies on these lines: {99, 666}, {144, 30225}, {190, 514}, {660, 6372}, {668, 4462}, {898, 4367}, {4378, 9266}, {4440, 6547}, {4499, 6004}, {4562, 6540}, {6546, 6634}, {6630, 17487}, {17350, 24281}, {27191, 31647}


X(32029) = POINT IV-CAPH OF X(239)

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

X(32029) lies on these lines: {1, 17755}, {2, 1280}, {6, 190}, {8, 26561}, {69, 1086}, {75, 16973}, {81, 4576}, {141, 27191}, {145, 20533}, {193, 4440}, {239, 335}, {291, 27919}, {511, 24813}, {524, 903}, {537, 3751}, {545, 1992}, {664, 9263}, {742, 17160}, {883, 1462}, {900, 10755}, {918, 1814}, {1015, 4561}, {1100, 24358}, {1386, 6651}, {2786, 10754}, {2796, 8593}, {2876, 25048}, {3416, 29617}, {3564, 24833}, {3570, 26273}, {3589, 29586}, {3618, 4422}, {3874, 30133}, {3881, 30130}, {3961, 24631}, {4384, 16496}, {4432, 16475}, {4716, 5847}, {5032, 17487}, {5039, 24815}, {5093, 24844}, {5263, 31317}, {5846, 6653}, {6542, 9053}, {6776, 29243}, {12167, 24814}, {14839, 20455}, {14853, 24828}, {14912, 24817}, {16593, 17316}, {16830, 31306}, {16972, 17393}, {17275, 25357}, {17449, 24602}, {18440, 24827}, {19459, 24822}, {20042, 31058}, {20132, 31314}, {20154, 27484}, {20172, 24349}, {24547, 25898}


X(32030) = POINT IV-CAPH OF X(244)

Barycentrics    -(a^3*b^3) - a^2*b^4 + a^4*b*c - a^3*b^2*c + 9*a^2*b^3*c - a*b^4*c - a^3*b*c^2 - 10*a^2*b^2*c^2 - a*b^3*c^2 - a^3*c^3 + 9*a^2*b*c^3 - a*b^2*c^3 + b^3*c^3 - a^2*c^4 - a*b*c^4 : :

X(32030) lies on these lines: {244, 7035}, {668, 24191}, {903, 4106}, {3999, 17160}, {7192, 18822}


X(32031) = POINT IV-CAPH OF X(319)

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

X(32031) lies on these lines: {79, 319}, {668, 1330}, {2481, 7768}, {6327, 17143}, {7264, 17361}


X(32032) = POINT IV-CAPH OF X(320)

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

X(32032) lies on these lines: {2, 1017}, {69, 150}, {80, 320}, {316, 2481}, {3570, 10708}, {4555, 12531}, {4597, 6224}, {4738, 17360}, {17143, 21283}, {17378, 24222}


X(32033) = POINT IV-CAPH OF X(330)

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

X(32033) lies on these lines: {1, 87}, {1278, 18830}, {1575, 6384}, {2162, 4360}, {2998, 3226}, {3963, 26077}, {4361, 4598}, {4393, 21759}, {6383, 24621}, {16606, 30963}, {17302, 27341}, {21226, 26069}, {27424, 28366}


X(32034) = POINT IV-CAPH OF X(346)

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

X(32034) lies on these lines: {2, 20432}, {20, 190}, {279, 304}, {668, 30695}, {3227, 6553}, {3621, 30225}, {4461, 17143}, {7738, 17262}, {7791, 20073}


X(32035) = POINT IV-CAPH OF X(350)

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

X(32035) lies on these lines: {2, 20671}, {8, 76}, {39, 27195}, {75, 17793}, {194, 1015}, {274, 12263}, {291, 350}, {310, 4576}, {537, 4479}, {538, 3227}, {1078, 8671}, {2810, 18906}, {3097, 30963}, {4440, 18830}, {7033, 24260}, {9263, 20081}, {9902, 17144}, {12782, 18140}, {20457, 24514}, {27076, 31276}






leftri  Points Chara: X(32036)-X(32042)  rightri

This preamble and centers X(32036)-X(32042) were contributed by Clark Kimberling and Peter Moses, April 15, 2019.

Suppose that P = p:q:r (barycentrics). The Point Chara of P is the point given by

(p - q)(p - r) : (q - r)(q - p) : (r - p)(r - q).

Point Chara of P is the trilinear pole of the line PX(2), so that it lies on the Steiner circumellipse. (Randy Hutson, April 20, 2019)

underbar

X(32036) = POINT CHARA OF X(17)

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

X(32036) lies on the Steiner circumellipse and these lines: {2, 11087}, {17, 671}, {99, 16806}, {110, 930}, {338, 11130}, {530, 8172}, {531, 11600}, {648, 17402}, {1576, 14185}, {3228, 21461}, {4558, 23895}, {10409, 23872}, {11118, 11139}, {14570, 23896}

X(32036) = reflection of X(11117) in X(2)
X(32036) = Steiner circumellipse antipode of X(11117)
X(32036) = trilinear pole of line X(2)X(17)


X(32037) = POINT CHARA OF X(18)

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

X(32037) lies on the Steiner circumellipse and these lines: {2, 11082}, {18, 671}, {99, 16807}, {110, 930}, {338, 11131}, {530, 11601}, {531, 8173}, {648, 17403}, {1576, 14187}, {3228, 21462}, {4558, 23896}, {10410, 23873}, {11117, 11138}, {14570, 23895}

X(32037) = reflection of X(11118) in X(2)
X(32037) = Steiner circumellipse antipode of X(11118)
X(32037) = trilinear pole of line X(2)X(18)


X(32038) = POINT CHARA OF X(65)

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

X(32038) lies on the Steiner circumellipse and these lines: {99, 651}, {100, 6648}, {108, 648}, {190, 4551}, {646, 21859}, {664, 1020}, {668, 4552}, {670, 4554}, {903, 4654}, {941, 2481}, {959, 3227}, {1121, 31359}, {1415, 4612}, {2006, 14616}, {2092, 31643}, {4664, 9578}, {18816, 27339}

X(32038) = trilinear pole of line X(2)X(65)


X(32039) = POINT CHARA OF X(87)

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

X(32039) lies on the Steiner circumellipse and these lines: {190, 932}, {330, 3226}, {668, 4598}

X(32039) = trilinear pole of line X(2)X(87)


X(32040) = POINT CHARA OF X(165)

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

X(32040) lies on the Steiner circumellipse and these lines: {2, 18025}, {99, 26716}, {190, 2398}, {648, 4241}, {668, 30728}, {671, 2784}, {903, 1992}, {1121, 3241}, {1494, 31144}, {2481, 16834}, {3226, 14614}, {4569, 24015}

X(32040) = reflection of X(2) in X(23972)
X(32040) = reflection of X(18025) in X(2)
X(32040) = Steiner circumellipse antipode of X(18025)
X(32040) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(658)}}
X(32040) = trilinear pole of line X(2)X(165)
X(32040) = isotomic conjugate of isogonal conjugate of X(26716)


X(32041) = POINT CHARA OF X(210)

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

X(32041) lies on the Steiner circumellipse and these lines: {2, 2481}, {99, 644}, {100, 666}, {190, 1026}, {376, 2808}, {528, 14947}, {646, 670}, {648, 4238}, {651, 6606}, {664, 1018}, {668, 30730}, {671, 2795}, {903, 4664}, {1002, 3227}, {1121, 3679}, {2279, 3226}, {4552, 4569}, {4876, 18827}, {7208, 9331}, {7233, 20680}, {14727, 30610}, {17294, 18025}, {17310, 18821}

X(32041) = reflection of X(2) in X(6184)
X(32041) = reflection of X(2481) in X(2)
X(32041) = Steiner circumellipse antipode of X(2481)
X(32041) = trilinear pole of line X(2)X(210) (the line through the centroids of ABC, the intouch and extouch triangles)


X(32042) = POINT CHARA OF X(319)

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

X(32042) lies on the Steiner circumellipse and these lines: {99, 8652}, {100, 6540}, {668, 4427}, {903, 10436}, {1121, 3929}, {3227, 25417}, {3228, 28625}, {4632, 13486}, {17393, 30597}

X(32042) = isotomic conjugate of X(4802)
X(32042) = trilinear pole of line X(2)X(319)


X(32043) = X(2)X(4432)∩X(320)X(24628)

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

X(32043) lies on these lines: {2,4432}, {320,24628}, {1121,4384}, {1125,17694}, {1698,16060}, {2482,24603}, {3912,6174}, {4369,24636}, {4750,4763}, {4997,17738}, {24602,27757}


X(32044) = X(899)X(13466)∩X(1086)X(30970)

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

X(32044) lies on these lines: {899,13466}, {1086,30970}, {3248,30950}, {3768,4465}, {4033,30964}, {4379,17793}


X(32045) = X(2)X(24821)∩X(3227)X(16826)

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

X(32045) lies on these lines: {2,24821}, {3227,16826}, {4375,6544}, {4767,29584}, {5461,29610}, {24709,29615}






leftri  Circumcenters and centroids of not-equilateral central triangles: X(32046) - X(32084)  rightri

This preamble and centers X(32046)-X(32084) were contributed by CÚsar Eliud Lozada, April 16, 2019.

The appearance of (T, n) in the following list means that the circumcenter of triangle T is X(n):

(ABC, 3), (ABC-X3 reflections, 3), (anti-Aquila, 1385), (anti-Ara, 4), (anti-Artzt, 599), (anti-Ascella, 3), (anti-Atik, 18951), (1st anti-Brocard, 98), (4th anti-Brocard, 11258), (5th anti-Brocard, 3398), (6th anti-Brocard, 182), (2nd anti-circumperp-tangential, 1), (1st anti-circumperp, 3), (anti-Conway, 32046), (2nd anti-Conway, 143), (anti-Euler, 20), (3rd anti-Euler, 11412), (4th anti-Euler, 5889), (anti-excenters-reflections, 3627), (2nd anti-extouch, 12161), (anti-inner-Grebe, 3312), (anti-outer-Grebe, 3311), (anti-Honsberger, 19154), (anti-Hutson intouch, 12084), (anti-incircle-circles, 7387), (anti-inverse-in-incircle, 14790), (anti-Mandart-incircle, 11248), (anti-McCay, 671), (6th anti-mixtilinear, 140), (anti-orthocentroidal, 399), (1st anti-orthosymmedial, 19164), (1st anti-Sharygin, 19210), (anti-tangential-midarc, 32047), (anti-Ursa minor, 13371), (anti-Wasat, 52), (antiAOA, 15132), (anticomplementary, 4), (AOA, 15114), (Apollonius, 970), (Aquila, 40), (Ara, 7387), (Aries, 32048), (Artzt, 7610), (Ascella, 3), (Atik, 9856), (1st Auriga, 11252), (2nd Auriga, 11253), (Ayme, 11259), (1st Brocard-reflected, 11261), (1st Brocard, 182), (2nd Brocard, 182), (4th Brocard, 381), (5th Brocard, 9821), (6th Brocard, 11257), (circummedial, 3), (circumorthic, 3), (2nd circumperp tangential, 11249), (1st circumperp, 3), (2nd circumperp, 3), (circumsymmedial, 3), (inner-Conway, 8), (Conway, 20), (2nd Conway, 962), (3rd Conway, 1), (Ehrmann-cross, 30), (Ehrmann-mid, 4), (Ehrmann-side, 4), (Ehrmann-vertex, 18377), (1st Ehrmann, 8546), (2nd Ehrmann, 11255), (Euler, 5), (2nd Euler, 5), (3rd Euler, 5), (4th Euler, 5), (5th Euler, 5), (excenters-midpoints, 11260), (excenters-reflections, 7982), (excentral, 40), (1st excosine, 155), (extangents, 8141), (extouch, 1158), (2nd extouch, 4), (3rd extouch, 11254), (inner-Fermat, 628), (outer-Fermat, 627), (Feuerbach, 5), (Fuhrmann, 355), (2nd Fuhrmann, 16159), (inner-Garcia, 3), (outer-Garcia, 355), (Garcia-reflection, 32049), (Gossard, 11251), (inner-Grebe, 1161), (outer-Grebe, 1160), (3rd Hatzipolakis, 32050), (hexyl, 1), (Honsberger, 390), (Hutson extouch, 32051), (inner-Hutson, 9836), (Hutson intouch, 1), (outer-Hutson, 9837), (1st Hyacinth, 11264), (2nd Hyacinth, 1147), (incentral, 8143), (incircle-circles, 1), (intangents, 8144), (intouch, 1), (inverse-in-incircle, 942), (Johnson, 4), (inner-Johnson, 10525), (outer-Johnson, 10526), (1st Johnson-Yff, 1478), (2nd Johnson-Yff, 1479), (K798e, 11263), (K798i, 10), (1st Kenmotu diagonals, 11265), (2nd Kenmotu diagonals, 11266), (Kosnita, 1658), (Lemoine, 8145), (Lucas antipodal, 32052), (Lucas central, 8155), (Lucas homothetic, 10669), (Lucas inner, 6407), (Lucas tangents, 1151), (Lucas(-1) antipodal, 32053), (Lucas(-1) central, 8156), (Lucas(-1) homothetic, 10673), (Lucas(-1) inner, 6408), (Lucas(-1) tangents, 1152), (Macbeath, 8146), (Mandart-excircles, 32054), (Mandart-incircle, 1), (McCay, 7617), (medial, 5), (midarc, 1), (2nd midarc, 1), (midheight, 5893), (mixtilinear, 8147), (2nd mixtilinear, 8159), (3rd mixtilinear, 3), (4th mixtilinear, 3), (5th mixtilinear, 1482), (6th mixtilinear, 1), (1st Neuberg, 8149), (2nd Neuberg, 8150), (orthic, 5), (orthocentroidal, 381), (1st orthosymmedial, 5480), (2nd orthosymmedial, 5480), (2nd Pamfilos-Zhou, 7596), (1st Parry, 351), (2nd Parry, 351), (3rd Parry, 351), (Pelletier, 11247), (reflection, 195), (1st Schiffler, 32055), (2nd Schiffler, 11256), (Schroeter, 10279), (1st Sharygin, 9840), (2nd Sharygin, 659), (Soddy, 32056), (inner-Soddy, 176), (2nd inner-Soddy, 32057), (outer-Soddy, 175), (2nd outer-Soddy, 32058), (inner-squares, 32059), (outer-squares, 32060), (Steiner, 8151), (submedial, 3628), (symmedial, 8152), (tangential, 26), (tangential-midarc, 8091), (2nd tangential-midarc, 8092), (inner tri-equilateral, 11267), (outer tri-equilateral, 11268), (1st tri-squares-central, 13720), (2nd tri-squares-central, 13843), (3rd tri-squares-central, 8981), (4th tri-squares-central, 13966), (1st tri-squares, 13663), (2nd tri-squares, 13783), (3rd tri-squares, 13879), (4th tri-squares, 13933), (Trinh, 11250), (Ursa-major, 12672), (Ursa-minor, 65), (inner-Vecten, 642), (outer-Vecten, 641), (Wasat, 946), (X-parabola-tangential, 32061), (X3-ABC reflections, 3), (Yff central, 8351), (Yff contact, 5592), (inner-Yff, 55), (outer-Yff, 56), (inner-Yff tangents, 10679), (outer-Yff tangents, 10680), (Yiu, 8154), (1st Zaniah, 12608), (2nd Zaniah, 10)

The appearance of (T, n) in the following list means that the centroid of triangle T is X(n):

(ABC, 2), (ABC-X3 reflections, 376), (anti-Aquila, 551), (anti-Ara, 428), (anti-Artzt, 2), (anti-Ascella, 11402), (anti-Atik, 18950), (1st anti-Brocard, 2), (4th anti-Brocard, 111), (5th anti-Brocard, 12150), (6th anti-Brocard, 5182), (2nd anti-circumperp-tangential, 5434), (1st anti-circumperp, 2979), (anti-Conway, 11402), (2nd anti-Conway, 51), (anti-Euler, 376), (3rd anti-Euler, 2979), (4th anti-Euler, 5890), (anti-excenters-reflections, 32062), (2nd anti-extouch, 11402), (anti-inner-Grebe, 19053), (anti-outer-Grebe, 19054), (anti-Honsberger, 19153), (anti-Hutson intouch, 10606), (anti-incircle-circles, 32063), (anti-inverse-in-incircle, 32064), (anti-Mandart-incircle, 4421), (anti-McCay, 2), (6th anti-mixtilinear, 3819), (anti-orthocentroidal, 110), (1st anti-orthosymmedial, 9157), (1st anti-Sharygin, 19209), (anti-tangential-midarc, 32065), (3rd anti-tri-squares, 1328), (4th anti-tri-squares, 1327), (anti-Ursa minor, 23332), (anti-Wasat, 51), (antiAOA, 15131), (anticomplementary, 2), (AOA, 15113), (Apollonius, 32066), (Aquila, 3679), (Ara, 9909), (Aries, 11206), (Artzt, 2), (Ascella, 11227), (Atik, 5927), (1st Auriga, 11207), (2nd Auriga, 11208), (Ayme, 11221), (1st Brocard-reflected, 2), (1st Brocard, 2), (2nd Brocard, 10166), (3rd Brocard, 11196), (4th Brocard, 6032), (5th Brocard, 7811), (6th Brocard, 7833), (circummedial, 9829), (circumorthic, 5890), (2nd circumperp tangential, 11194), (1st circumperp, 165), (2nd circumperp, 3576), (circumsymmedial, 353), (inner-Conway, 3681), (Conway, 11220), (2nd Conway, 9812), (3rd Conway, 10439), (4th Conway, 32067), (5th Conway, 11233), (Ehrmann-cross, 18556), (Ehrmann-mid, 3845), (Ehrmann-side, 18435), (Ehrmann-vertex, 18376), (1st Ehrmann, 182), (2nd Ehrmann, 11216), (Euler, 381), (2nd Euler, 5891), (3rd Euler, 3817), (4th Euler, 10175), (5th Euler, 10162), (excenters-midpoints, 10164), (excenters-reflections, 11224), (excentral, 165), (1st excosine, 154), (extangents, 11190), (extouch, 210), (2nd extouch, 5927), (3rd extouch, 11212), (4th extouch, 11213), (5th extouch, 11214), (inner-Fermat, 2), (outer-Fermat, 2), (Feuerbach, 5947), (Fuhrmann, 5587), (2nd Fuhrmann, 1699), (inner-Garcia, 5692), (outer-Garcia, 3679), (Garcia-reflection, 1699), (Gossard, 1651), (inner-Grebe, 5861), (outer-Grebe, 5860), (3rd Hatzipolakis, 32068), (hexyl, 3576), (Honsberger, 7671), (Hutson extouch, 5918), (inner-Hutson, 11222), (Hutson intouch, 5919), (outer-Hutson, 11223), (1st Hyacinth, 11232), (2nd Hyacinth, 11245), (incentral, 1962), (incircle-circles, 5049), (intangents, 11189), (intouch, 354), (inverse-in-incircle, 354), (Johnson, 381), (inner-Johnson, 11235), (outer-Johnson, 11236), (1st Johnson-Yff, 11237), (2nd Johnson-Yff, 11238), (K798e, 11230), (K798i, 11231), (1st Kenmotu diagonals, 11241), (2nd Kenmotu diagonals, 11242), (Kosnita, 11202), (Lemoine, 32069), (Lucas antipodal tangents, 32070), (Lucas central, 11198), (Lucas homothetic, 12152), (Lucas inner, 32072), (Lucas tangents, 11199), (Lucas(-1) antipodal tangents, 32071), (Lucas(-1) central, 32077), (Lucas(-1) homothetic, 12153), (Lucas(-1) inner, 32073), (Lucas(-1) tangents, 32074), (Macbeath, 11197), (Mandart-excircles, 11246), (Mandart-incircle, 3058), (McCay, 2), (medial, 2), (midarc, 11191), (2nd midarc, 11234), (midheight, 5943), (mixtilinear, 11200), (2nd mixtilinear, 11201), (3rd mixtilinear, 32075), (4th mixtilinear, 32076), (5th mixtilinear, 3241), (6th mixtilinear, 165), (1st Neuberg, 2), (2nd Neuberg, 2), (orthic, 51), (orthocentroidal, 5640), (1st orthosymmedial, 51), (2nd orthosymmedial, 11226), (2nd Pamfilos-Zhou, 11211), (1st Parry, 9123), (2nd Parry, 9185), (3rd Parry, 11215), (Pelletier, 11193), (reflection, 3060), (1st Schiffler, 11218), (2nd Schiffler, 11219), (Schroeter, 10278), (1st Sharygin, 11203), (2nd Sharygin, 1635), (Soddy, 32079), (inner-Soddy, 32080), (2nd inner-Soddy, 32082), (outer-Soddy, 32081), (2nd outer-Soddy, 32083), (inner-squares, 11209), (outer-squares, 11210), (Steiner, 11123), (submedial, 6688), (symmedial, 11205), (tangential, 154), (tangential-midarc, 11192), (2nd tangential-midarc, 11217), (inner tri-equilateral, 11243), (outer tri-equilateral, 11244), (1st tri-squares-central, 3068), (2nd tri-squares-central, 3069), (3rd tri-squares-central, 13846), (4th tri-squares-central, 13847), (1st tri-squares, 3068), (2nd tri-squares, 3069), (3rd tri-squares, 13846), (4th tri-squares, 13847), (Trinh, 11204), (Ursa-major, 5927), (Ursa-minor, 354), (inner-Vecten, 2), (outer-Vecten, 2), (Wasat, 3817), (X-parabola-tangential, 8029), (X3-ABC reflections, 381), (Yff central, 11195), (Yff contact, 6546), (inner-Yff, 10056), (outer-Yff, 10072), (inner-Yff tangents, 11239), (outer-Yff tangents, 11240), (Yiu, 32084), (1st Zaniah, 3742), (2nd Zaniah, 3740)

underbar

X(32046) = CIRCUMCENTER OF THE ANTI-CONWAY TRIANGLE

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*a^4-(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^2-c^2)^2*b^2*c^2) : :
Barycentrics    Sin[A] (Cos[A] + 2 Cos[A - B] Cos[A - C]) : :
Trilinears    cos A + 2 cos(C - A) cos(A - B) : :
X(32046) = X(3)+3*X(11402) = 3*X(3)+X(12160) = 3*X(5050)+X(19459) = 9*X(11402)-X(12160) = 3*X(11402)-X(12161) = X(12160)-3*X(12161)

X(32046) is the center of the circle that is the locus of the crosssum of nine-point circle antipodes. (Randy Hutson, June 7, 2019)

Let A'B'C' be the antipedal triangle of X(3) (the tangential triangle). X(32046) is the radical center of the nine-point circles of triangles AA'X(3), BB'X(3), CC'X(3). (cf. Anopolis #2105, Antreas Hatzipolakis, December 1, 2014) (Randy Hutson, June 7, 2019)

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let A'B'C' be the orthic triangle. X(32046) is the radical center of the circumcircles of triangles A'AeAi, B'BeBi, C'CeCi. (Randy Hutson, June 7, 2019)

X(32046) lies on these lines: {2,49}, {3,54}, {4,567}, {5,156}, {6,26}, {17,3206}, {18,3205}, {20,31815}, {22,10263}, {24,5944}, {25,10095}, {30,578}, {35,9666}, {36,9653}, {52,7502}, {68,11264}, {110,1656}, {140,141}, {154,13364}, {155,7514}, {185,18570}, {206,18583}, {215,498}, {343,7568}, {378,13491}, {381,1614}, {382,15033}, {389,1658}, {394,7516}, {399,15058}, {499,2477}, {511,7525}, {546,6759}, {548,13346}, {549,1092}, {550,10984}, {568,1199}, {575,2393}, {576,7555}, {597,23410}, {631,9545}, {1176,1351}, {1181,5663}, {1353,19131}, {1385,31811}, {1437,6924}, {1498,31861}, {1511,17928}, {1698,9622}, {1899,13561}, {1974,7715}, {1994,6243}, {2070,3567}, {2930,15462}, {2931,13358}, {2937,3060}, {3043,15061}, {3044,15561}, {3047,14643}, {3071,9677}, {3090,9544}, {3091,10540}, {3133,20775}, {3167,7393}, {3202,11272}, {3431,16867}, {3448,11597}, {3517,12167}, {3523,13339}, {3526,9703}, {3574,11750}, {3579,31812}, {3624,9621}, {3627,11424}, {3628,9306}, {3767,9604}, {3796,13391}, {3843,14157}, {3845,26883}, {5050,6642}, {5070,9705}, {5073,8718}, {5092,5447}, {5093,19121}, {5392,25043}, {5422,7506}, {5446,17714}, {5504,14528}, {5576,14389}, {5622,15132}, {5640,26882}, {5876,7503}, {6241,14130}, {6639,18912}, {6640,18911}, {6644,12006}, {6676,13292}, {6689,10116}, {6800,7517}, {7387,11426}, {7395,14128}, {7507,11565}, {7508,13323}, {7509,15067}, {7527,18439}, {7529,18874}, {7530,10982}, {7542,11245}, {7575,16227}, {7741,9667}, {7746,9697}, {7951,9652}, {8141,11428}, {8144,11429}, {8227,9587}, {8254,10274}, {8548,9937}, {9586,31423}, {9603,31401}, {9696,31455}, {9701,26363}, {9702,26364}, {9714,9777}, {9729,12038}, {9730,13367}, {9781,18378}, {9786,18324}, {9818,19347}, {9833,11818}, {9908,19139}, {10020,13567}, {10024,12022}, {10625,22352}, {11179,18281}, {11250,11430}, {11425,12084}, {11427,14790}, {11432,14070}, {11438,15331}, {11441,15060}, {11449,15045}, {11454,18364}, {11464,15037}, {11702,15101}, {11746,20773}, {11935,15720}, {12041,13148}, {12100,13347}, {12105,15826}, {12107,16881}, {12241,15761}, {12370,15760}, {12383,15089}, {13142,16618}, {13154,17811}, {13371,23292}, {13406,18390}, {13470,18569}, {13482,15681}, {13564,15080}, {14265,14355}, {14789,23236}, {14805,15032}, {14912,18951}, {15024,15047}, {15028,15038}, {15083,31834}, {18128,20299}, {18377,18388}, {18379,18396}, {18420,18925}, {18580,26937}, {19125,19155}, {19365,32047}, {19467,30522}, {22146,26216}, {23042,23326}

X(32046) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 12161}, {5, 31804}, {20, 31815}, {550, 31802}, {1181, 7526}, {1385, 31811}, {3579, 31812}, {6101, 31810}, {6102, 31807}, {12228, 13198}
X(32046) = isogonal conjugate of X(16837)
X(32046) = X(5)-of-2nd-anti-extouch triangle
X(32046) = anti-Conway-isogonal conjugate of X(12161)
X(32046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1993, 6101), (49, 13353, 2), (11412, 11422, 195)


X(32047) = CIRCUMCENTER OF THE ANTI-TANGENTIAL-MIDARC TRIANGLE

Barycentrics    a*(a^6-(b-c)^2*a^4-(b^4+c^4+(b-c)^2*b*c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(32047) = X(7352)-3*X(32065)

X(32047) lies on these lines: {1,30}, {3,1398}, {4,18447}, {5,34}, {11,15761}, {12,13371}, {20,18455}, {26,56}, {33,3627}, {35,11250}, {36,1658}, {52,1425}, {55,12084}, {65,8141}, {140,1038}, {143,19366}, {155,221}, {156,26888}, {222,24475}, {278,6917}, {347,6868}, {382,6198}, {388,14790}, {495,23335}, {498,18281}, {499,10201}, {517,4347}, {548,1040}, {550,1062}, {614,10154}, {952,21147}, {999,7387}, {1068,6923}, {1154,7352}, {1159,17016}, {1325,2906}, {1394,24467}, {1428,19154}, {1448,24470}, {1456,5887}, {1465,6924}, {1478,18569}, {1482,4318}, {1565,7210}, {1657,3100}, {1718,24914}, {1745,1807}, {2067,11265}, {3157,16266}, {3270,10575}, {3295,12085}, {3304,9645}, {3529,9538}, {3564,19471}, {3585,18377}, {3600,31305}, {3628,19372}, {3850,9817}, {3868,23070}, {3869,22136}, {4299,9630}, {4302,9627}, {4670,30147}, {5010,10226}, {5073,9642}, {5204,18324}, {5262,5708}, {5432,23336}, {5433,10020}, {5563,17714}, {5663,6238}, {5690,8270}, {5889,19368}, {6502,11266}, {6644,11398}, {7051,11267}, {7191,9909}, {7280,15331}, {7530,11399}, {7741,13406}, {7747,9595}, {7756,9594}, {7951,10224}, {9632,23251}, {9634,13903}, {9641,17800}, {10267,15832}, {11255,19369}, {11268,19373}, {11412,19367}, {11436,13630}, {11446,12290}, {11461,12279}, {11700,26286}, {12161,19349}, {13383,15325}, {15952,20254}, {18513,18567}, {18915,18951}, {19175,19210}, {19365,32046}

X(32047) = midpoint of X(i) and X(j) for these lines: {i,j}: {6238, 7355}, {19469, 19505}
X(32047) = reflection of X(8144) in X(1)
X(32047) = anti-tangential-midarc-isogonal conjugate of X(7352)
X(32047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (34, 1060, 5), (1870, 4296, 3)


X(32048) = CIRCUMCENTER OF THE ARIES TRIANGLE

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^12-2*(b^2+c^2)*a^10-(b^2+c^2)^2*a^8+4*(b^2+c^2)*(b^4+c^4)*a^6-(b^8+6*b^4*c^4+c^8)*a^4-2*(b^4-c^4)*(b^2-c^2)^3*a^2+(b^2-c^2)^6) : :
X(32048) = 3*X(154)-X(15316) = X(9908)-3*X(9909) = 3*X(9909)+X(12309) = 3*X(10201)-2*X(20302) = 3*X(11206)-X(12420) = X(12301)-3*X(14070)

X(32048) lies on these lines: {3,125}, {22,68}, {23,6193}, {24,12118}, {25,1147}, {26,19908}, {30,9932}, {154,15316}, {155,6243}, {156,14984}, {161,1498}, {403,8907}, {539,9908}, {1069,9673}, {1598,5448}, {1658,9938}, {2937,12429}, {3157,9658}, {3167,18378}, {3564,15581}, {5480,9820}, {5654,10594}, {5899,12164}, {6642,11425}, {6644,12241}, {7506,10982}, {7530,22660}, {7689,11414}, {8185,9928}, {8548,19127}, {9591,9896}, {10117,12419}, {10201,20302}, {10625,26283}, {11206,12420}, {11411,12088}, {11477,20987}, {12083,12163}, {12166,15083}, {12225,25715}, {12301,14070}, {12319,16868}, {12328,20872}, {13371,18382}, {13383,15577}, {15761,22661}

X(32048) = midpoint of X(i) and X(j) for these lines: {i,j}: {7387, 9937}, {9908, 12309}
X(32048) = reflection of X(i) in X(j) for these (i,j): (9938, 1658), (19908, 26), (22661, 15761), (23307, 13383)
X(32048) = {X(9909), X(12309)}-harmonic conjugate of X(9908)
X(32048) = Aries-isogonal conjugate of X(12420)
X(32048) = X(1158)-of-tangential-triangle if ABC is acute


X(32049) = CIRCUMCENTER OF THE GARCIA-REFLECTION TRIANGLE

Barycentrics    a^4-(b+c)*a^3+6*b*c*a^2+(b+c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2 : :
X(32049) = X(145)-3*X(25568) = 2*X(946)-3*X(11236) = 3*X(1699)-X(3680) = 3*X(1699)-2*X(13463) = 5*X(3617)-3*X(24477) = 3*X(3679)-X(6762) = 2*X(3813)-3*X(5587) = 6*X(3829)-7*X(7989) = 2*X(4297)-3*X(4421) = 3*X(5587)-X(12629) = 3*X(5790)-2*X(10916) = 3*X(5886)-2*X(22837) = 4*X(6684)-3*X(11194) = 2*X(8666)-3*X(26446) = X(10912)-3*X(11236) = 3*X(11235)-4*X(19925) = 3*X(11235)-2*X(21627) = X(11519)-3*X(24392) = X(11531)-3*X(28609)

X(32049) lies on these lines: {1,1329}, {2,11260}, {3,10915}, {4,3880}, {7,8}, {10,999}, {11,11256}, {40,529}, {42,30960}, {46,1145}, {56,6735}, {57,8256}, {72,12647}, {78,10944}, {80,3633}, {145,1837}, {153,12679}, {354,5554}, {355,381}, {495,28628}, {515,3913}, {516,12640}, {517,6256}, {528,2136}, {612,30778}, {758,25413}, {908,2098}, {952,3811}, {956,10039}, {958,31397}, {960,3421}, {1001,5795}, {1012,8668}, {1056,3812}, {1155,20076}, {1222,3705}, {1319,5552}, {1376,6736}, {1388,27385}, {1420,3035}, {1478,10914}, {1537,5854}, {1699,3680}, {1709,27870}, {1836,14923}, {2478,5919}, {2646,10528}, {2800,12676}, {2802,10742}, {2886,4853}, {3057,3436}, {3086,5123}, {3241,26129}, {3244,5722}, {3304,24982}, {3338,3679}, {3419,3632}, {3434,3893}, {3445,5121}, {3476,7080}, {3555,5570}, {3617,24477}, {3621,4863}, {3626,5708}, {3813,5587}, {3814,11373}, {3829,7989}, {3838,5261}, {3869,25414}, {3870,10950}, {3885,5080}, {3890,4679}, {3895,6284}, {3899,5559}, {4018,10052}, {4028,15232}, {4297,4421}, {4515,24247}, {4861,11375}, {4999,31434}, {5084,10179}, {5178,31145}, {5220,5837}, {5289,21075}, {5541,10483}, {5787,12437}, {5790,10916}, {5881,6765}, {5886,22837}, {6684,11194}, {6734,18967}, {7991,17768}, {8583,9711}, {8666,26446}, {8679,31785}, {8715,18481}, {9623,25466}, {10072,17619}, {10529,17606}, {10827,24390}, {11376,11681}, {11519,24392}, {11531,28609}, {13600,26333}, {15888,19860}, {17728,25005}, {19861,21031}, {20070,28534}, {24928,26364}

X(32049) = midpoint of X(i) and X(j) for these lines: {i,j}: {2136, 5691}, {3632, 11523}, {5881, 6765}
X(32049) = reflection of X(i) in X(j) for these (i,j): (1, 12607), (3, 10915), (1482, 21077), (3680, 13463), (10912, 946), (11256, 11), (12513, 10), (12629, 3813), (18481, 8715), (21627, 19925), (24391, 3626)
X(32049) = anticomplement of X(11260)
X(32049) = Garcia-reflection-isogonal conjugate of X(3680)
X(32049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17757, 25681), (8, 388, 5836), (388, 5836, 5880)


X(32050) = CIRCUMCENTER OF THE 3rd HATZIPOLAKIS TRIANGLE

Barycentrics    (SB+SC)*((178*R^4-65*SW*R^2+6*SW^2)*S^2+(64*R^4-23*SW*R^2+2*SW^2)*(8*R^2+SA-2*SW)*SA) : :
X(32050) = X(2929)+7*X(15043) = 3*X(5946)+X(22962) = 5*X(10574)+3*X(22971) = 5*X(15026)-X(22816) = 9*X(15045)-X(22549) = X(31985)-3*X(32068)

X(32050) lies on these lines: {6,2929}, {5946,22962}, {10574,22971}, {15026,22816}, {15045,22549}, {15087,22955}, {31985,32068}

X(32050) = 3rd-Hatzipolakis-isogonal conjugate of X(31985)


X(32051) = CIRCUMCENTER OF THE HUTSON EXTOUCH TRIANGLE

Barycentrics
a*(a^9-(b+c)*a^8-4*(b^2+5*b*c+c^2)*a^7+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^6+2*(3*b^4+3*c^4+2*(15*b^2+17*b*c+15*c^2)*b*c)*a^5-6*(b+c)*(b^4+c^4-(3*b^2+4*b*c+3*c^2)*b*c)*a^4-4*(b^2+c^2)*(b^4+c^4+(15*b^2+8*b*c+15*c^2)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-(5*b^2+22*b*c+5*c^2)*b*c)*a^2+(b^4+c^4+2*(10*b^2-13*b*c+10*c^2)*b*c)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c)*(b^2-4*b*c+c^2)) : :
X(32051) = 3*X(5918)-X(31968)

X(32051) lies on these lines: {9,946}, {516,12731}, {5918,31968}, {9943,12702}, {12699,22992}, {12857,16138}

X(32051) = reflection of X(12699) in X(22992)
X(32051) = Hutson-extouch-isogonal conjugate of X(31968)


X(32052) = CIRCUMCENTER OF THE LUCAS ANTIPODAL TRIANGLE

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

X(32052) lies on these lines: {3,69}, {511,12977}, {2996,19406}, {12984,13061}, {13881,19422}, {18980,23698}

X(32052) = midpoint of X(3) and X(19490)
X(32052) = Lucas-antipodal-isogonal conjugate of X(31970)
X(32052) = {X(3), X(9723)}-harmonic conjugate of X(32053)


X(32053) = CIRCUMCENTER OF THE LUCAS(-1) ANTIPODAL TRIANGLE

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

X(32053) lies on these lines: {3,69}, {511,13068}, {2996,19407}, {12985,13062}, {13881,19423}, {18981,23698}

X(32053) = midpoint of X(3) and X(19491)
X(32053) = Lucas(-1)-antipodal-isogonal conjugate of X(31971)
X(32053) = {X(3), X(9723)}-harmonic conjugate of X(32052)


X(32054) = CIRCUMCENTER OF THE MANDART-EXCIRCLES TRIANGLE

Barycentrics
a*(a^9-(b+c)*a^8-2*(b+c)^2*a^7+2*(b+c)*(b^2+3*b*c+c^2)*a^6+2*(5*b^2-8*b*c+5*c^2)*b*c*a^5-8*(b+c)*(b^2+b*c+c^2)*b*c*a^4+2*(b^4+c^4-4*(b^2-3*b*c+c^2)*b*c)*(b+c)^2*a^3-2*(b+c)*(b^6+c^6-(b^4+c^4+(5*b^2-14*b*c+5*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(b-c)^2*(b^2+4*b*c+c^2)*a+(b^2-c^2)^4*(b+c)) : :
X(32054) = 3*X(11246)-X(31977)

X(32054) lies on these lines: {19,1598}, {65,12912}, {999,12522}, {3338,12659}, {3555,17656}, {5048,12876}, {5708,12442}, {11246,31977}, {12621,17757}, {12655,25415}, {18257,25681}, {21077,21632}

X(32054) = midpoint of X(3555) and X(17656)
X(32054) = reflection of X(12522) in X(12907)
X(32054) = Mandart-excircles-isogonal conjugate of X(31977)


X(32055) = CIRCUMCENTER OF THE 1st SCHIFFLER TRIANGLE

Barycentrics
a*(a^12-2*(b+c)*a^11-(4*b^2-9*b*c+4*c^2)*a^10+(b+c)*(10*b^2-9*b*c+10*c^2)*a^9+(5*b^4+5*c^4-(33*b^2-17*b*c+33*c^2)*b*c)*a^8-2*(b+c)*(10*b^4+10*c^4-(15*b^2-19*b*c+15*c^2)*b*c)*a^7+(48*b^4+48*c^4-(29*b^2-49*b*c+29*c^2)*b*c)*b*c*a^6+(b+c)*(20*b^6+20*c^6-(36*b^4+36*c^4-(44*b^2-29*b*c+44*c^2)*b*c)*b*c)*a^5-(5*b^8+5*c^8+(36*b^6+36*c^6-(35*b^4-6*b^2*c^2+35*c^4)*b*c)*b*c)*a^4-(b+c)*(10*b^8+10*c^8-(9*b^2-16*b*c+9*c^2)*(2*b^4+2*c^4+(2*b^2+b*c+2*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^2*(4*b^6+4*c^6+(15*b^4+15*c^4-(23*b^2-5*b*c+23*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^3*(b-c)*(2*b^4+2*c^4+(b+c)^2*b*c)*a-(b^2-c^2)^4*(b-c)^2*(b^2+5*b*c+c^2)) : :

X(32055) lies on the line {12745,31870}


X(32056) = CIRCUMCENTER OF THE SODDY TRIANGLE

Barycentrics
7*a^7-13*(b+c)*a^6+(9*b^2+2*b*c+9*c^2)*a^5-11*(b^2-c^2)*(b-c)*a^4+(9*b^2+34*b*c+9*c^2)*(b-c)^2*a^3-3*(b^2-c^2)^2*(b+c)*a^2+(7*b^2+10*b*c+7*c^2)*(b-c)^4*a-5*(b^2-c^2)^3*(b-c) : :

X(32056) lies on the line {1,4}


X(32057) = CIRCUMCENTER OF THE 2nd INNER-SODDY TRIANGLE

Barycentrics
a*((a+b+c)*(6*a^6-9*(b+c)*a^5-3*(5*b^2+8*b*c+5*c^2)*a^4+4*(b+c)*(7*b^2+2*b*c+7*c^2)*a^3-2*(b^4+c^4-2*(7*b^2-6*b*c+7*c^2)*b*c)*a^2-(b^2-c^2)*(b-c)*(11*b^2+34*b*c+11*c^2)*a+(3*b^4+3*c^4-2*(b^2+7*b*c+c^2)*b*c)*(b-c)^2)-2*S*(13*(b+c)*a^4-2*(4*b^2-3*b*c+4*c^2)*a^3-8*(b+2*c)*(2*b+c)*(b+c)*a^2+2*(4*b^4+4*c^4+(b^2-24*b*c+c^2)*b*c)*a+(b^2-c^2)*(b-c)*(3*b^2+14*b*c+3*c^2))) : :

Let A', B', C' be as at X(7133). A'B'C' is perspective to the 2nd inner Soddy triangle at X(32057). (Randy Hutson, June 7, 2019)

X(32057) lies on these lines: {1,372}, {482,11246}, {6284,31533}, {10252,16213}

X(32057) = midpoint of X(31533) and X(31568)
X(32057) = 2nd-inner-Soddy-isogonal conjugate of X(482)


X(32058) = CIRCUMCENTER OF THE 2nd OUTER-SODDY TRIANGLE

Barycentrics
a*((a+b+c)*(6*a^6-9*(b+c)*a^5-3*(5*b^2+8*b*c+5*c^2)*a^4+4*(b+c)*(7*b^2+2*b*c+7*c^2)*a^3-2*(b^4+c^4-2*(7*b^2-6*b*c+7*c^2)*b*c)*a^2-(b^2-c^2)*(b-c)*(11*b^2+34*b*c+11*c^2)*a+(3*b^4+3*c^4-2*(b^2+7*b*c+c^2)*b*c)*(b-c)^2)+2*S*(13*(b+c)*a^4-2*(4*b^2-3*b*c+4*c^2)*a^3-8*(b+2*c)*(2*b+c)*(b+c)*a^2+2*(4*b^4+4*c^4+(b^2-24*b*c+c^2)*b*c)*a+(b^2-c^2)*(b-c)*(3*b^2+14*b*c+3*c^2))) : :

X(32058) lies on these lines: {1,371}, {481,11246}, {6284,31532}, {10253,16214}

X(32058) = midpoint of X(31532) and X(31567)
X(32058) = 2nd-outer-Soddy-isogonal conjugate of X(481)


X(32059) = CIRCUMCENTER OF THE INNER-SQUARES TRIANGLE

Barycentrics    (3*SA-5*SW)*S^2-S*(2*S^2+(SA+2*SW)*(SB+SC))-4*(4*R^2-SW)*SB*SC : :

X(32059) lies on these lines: {4,371}, {11209,31987}, {11315,26958}, {31406,32060}


X(32060) = CIRCUMCENTER OF THE OUTER-SQUARES TRIANGLE

Barycentrics    (3*SA-5*SW)*S^2+S*(2*S^2+(SA+2*SW)*(SB+SC))-4*(4*R^2-SW)*SB*SC : :

X(32060) lies on these lines: {4,372}, {11210,31988}, {11316,26958}, {31406,32059}


X(32061) = CIRCUMCENTER OF THE X-PARABOLA-TANGENTIAL TRIANGLE

Barycentrics
2*a^16-6*(b^2+c^2)*a^14+3*(b^4+8*b^2*c^2+c^4)*a^12+5*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^10-(3*b^8+3*c^8+4*(2*b^4-13*b^2*c^2+2*c^4)*b^2*c^2)*a^8-2*(b^2+c^2)*(b^8+c^8-(10*b^4-21*b^2*c^2+10*c^4)*b^2*c^2)*a^6-(b^12+c^12-2*(5*b^8+5*c^8-(21*b^4-34*b^2*c^2+21*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(3*b^8+3*c^8-(13*b^4-21*b^2*c^2+13*c^4)*b^2*c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^6 : :
X(32061) = 3*X(8029)-X(31990)

X(32061) lies on these lines: {4,542}, {8029,31990}

X(32061) = X-parabola-tangential-isogonal conjugate of X(31990)


X(32062) = CENTROID OF THE ANTI-EXCENTERS-REFLECTIONS TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^6-(b^2-3*c^2)*(3*b^2-c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+10*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32062) = 4*X(4)-X(185) = 5*X(4)-2*X(389) = 11*X(4)-5*X(3567) = 3*X(4)-X(5890) = 7*X(4)-X(6241) = 13*X(4)-7*X(9781) = 7*X(4)-4*X(10110) = 2*X(4)+X(11381) = 5*X(4)+X(12290) = 13*X(4)-4*X(13382) = X(4)+2*X(13474) = 5*X(51)-4*X(389) = 11*X(51)-10*X(3567) = 3*X(51)-2*X(5890) = 7*X(51)-2*X(6241) = 13*X(51)-14*X(9781) = 7*X(51)-8*X(10110) = 3*X(3917)-4*X(5891) = 5*X(3917)-8*X(15060) = 7*X(3917)-8*X(15067) = X(3917)-4*X(16194) = 2*X(5891)-3*X(15030) = 5*X(5891)-6*X(15060) = 7*X(5891)-6*X(15067) = X(5891)-3*X(16194) = 5*X(15030)-4*X(15060) = 7*X(15030)-4*X(15067) = 7*X(15060)-5*X(15067) = 2*X(15060)-5*X(16194) = 2*X(15067)-7*X(16194)

X(32062) lies on these lines: {4,51}, {5,14855}, {6,14490}, {20,3819}, {24,11204}, {25,10606}, {30,3917}, {33,32065}, {34,11189}, {52,3853}, {53,1562}, {64,5198}, {74,13603}, {125,1596}, {154,1593}, {184,1597}, {235,23332}, {373,381}, {376,5650}, {378,1495}, {382,5562}, {428,15311}, {511,3543}, {546,10575}, {1154,3627}, {1204,1598}, {1216,5073}, {1498,11402}, {1531,31723}, {1533,15760}, {1656,14641}, {1843,2781}, {1885,16621}, {1906,6247}, {1907,2883}, {1974,10249}, {2390,12688}, {2393,12294}, {2777,7576}, {2979,3146}, {3091,6688}, {3292,18451}, {3357,10594}, {3426,10605}, {3529,11793}, {3534,10170}, {3545,16836}, {3575,16656}, {3830,13754}, {3832,9729}, {3839,5943}, {3845,9730}, {3855,11695}, {3861,13491}, {4550,12083}, {5056,17704}, {5059,13348}, {5076,5446}, {5447,17800}, {5640,13570}, {5651,21312}, {5663,14831}, {5946,14893}, {6102,12102}, {7418,10568}, {7998,15683}, {9818,22352}, {10282,14865}, {10301,10990}, {10982,12315}, {11017,15712}, {11190,11471}, {11216,11470}, {11241,11473}, {11242,11474}, {11243,11475}, {11244,11476}, {11430,13596}, {11456,13366}, {11459,15682}, {11472,18534}, {11744,15321}, {11746,17856}, {11820,22112}, {12099,17853}, {12111,13598}, {12112,15033}, {12292,13417}, {13363,23046}, {13399,13567}, {13403,16659}, {13419,18560}, {13445,13595}, {13488,16655}, {14269,16226}, {15058,15644}, {15082,15692}, {15684,23039}, {15752,31978}, {15761,18488}, {16658,18400}, {16980,31673}, {18356,18555}, {19124,19153}, {19169,19209}, {19416,32070}, {19417,32071}

X(32062) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 11455}, {51, 11381}, {382, 18435}, {2979, 3146}, {3543, 15305}, {11459, 15682}, {15684, 23039}
X(32062) = reflection of X(i) in X(j) for these (i,j): (20, 3819), (51, 4), (185, 51), (2979, 5907), (3534, 10170), (3917, 15030), (5562, 18435), (5650, 16261), (5946, 14893), (9730, 3845), (11381, 11455), (11455, 13474), (14855, 5), (15030, 16194), (15072, 5943), (16226, 14269), (17853, 12099)
X(32062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 11381, 185), (4, 12290, 389), (4, 13474, 11381)


X(32063) = CENTROID OF THE ANTI-INCIRCLE-CIRCLES TRIANGLE

Barycentrics    a^2*(3*a^8-10*(b^2+c^2)*a^6+4*(3*b^4-b^2*c^2+3*c^4)*a^4-6*(b^4-c^4)*(b^2-c^2)*a^2+(b^4+10*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32063) = 5*X(3)-2*X(64) = X(3)+2*X(1498) = 7*X(3)-4*X(3357) = X(3)-4*X(6759) = 13*X(3)-10*X(8567) = 5*X(3)-8*X(10282) = 3*X(3)-2*X(10606) = 3*X(3)-4*X(11202) = 5*X(3)-4*X(11204) = 2*X(3)+X(12315) = 4*X(3)-X(13093) = 2*X(3)-5*X(14530) = 7*X(3)-10*X(17821) = X(64)-5*X(154) = X(64)+5*X(1498) = 7*X(64)-10*X(3357) = X(64)-10*X(6759) = X(64)-4*X(10282) = 3*X(64)-5*X(10606) = 4*X(140)-X(12324)

X(32063) lies on these lines: {3,64}, {4,11402}, {5,32064}, {6,3531}, {20,25712}, {24,12174}, {25,5890}, {30,3167}, {51,1181}, {54,11403}, {110,11820}, {140,12324}, {156,12085}, {159,399}, {161,5899}, {184,1597}, {185,3517}, {206,10249}, {221,7373}, {376,6090}, {378,3426}, {381,597}, {382,2883}, {548,12250}, {550,6225}, {578,15811}, {999,10535}, {1154,7387}, {1173,3527}, {1351,2393}, {1384,1971}, {1398,9638}, {1495,10605}, {1593,1614}, {1596,6776}, {1625,15905}, {1656,14216}, {1657,5878}, {1660,22115}, {1853,5055}, {2192,6767}, {2390,10680}, {2777,15681}, {2979,11414}, {3089,18914}, {3197,22147}, {3295,11189}, {3311,11241}, {3312,11242}, {3332,15762}, {3515,6241}, {3516,9707}, {3526,6247}, {3534,15311}, {3542,26944}, {3796,15030}, {3830,18400}, {3843,18376}, {3851,14862}, {5020,5892}, {5054,10192}, {5064,16658}, {5070,20299}, {5073,17845}, {5076,5893}, {5093,17813}, {5596,15760}, {5644,14845}, {5663,14070}, {5895,17800}, {6001,10179}, {6199,17819}, {6221,10533}, {6395,17820}, {6398,10534}, {6623,19118}, {6688,11484}, {6696,15720}, {6800,15305}, {7485,15052}, {7507,16659}, {9715,12111}, {9777,15032}, {9899,31663}, {9909,13754}, {10117,12308}, {10182,15701}, {10193,15707}, {10306,11190}, {10620,15647}, {10675,11244}, {10676,11243}, {11216,11482}, {11381,19357}, {11410,11464}, {11425,13474}, {11472,18475}, {12163,16195}, {13488,18925}, {13567,15152}, {14269,18405}, {15693,23328}, {15694,23329}, {15696,20427}, {15750,26882}, {18358,20079}, {18494,31383}, {18909,21841}, {19173,19209}, {19418,32070}, {19419,32071}, {19709,23325}

X(32063) = midpoint of X(i) and X(j) for these lines: {i,j}: {154, 1498}, {5656, 11206}
X(32063) = reflection of X(i) in X(j) for these (i,j): (3, 154), (64, 11204), (154, 6759), (10249, 206), (10606, 11202), (11204, 10282), (14216, 23332), (23332, 16252), (32064, 5)
X(32063) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 12315, 13093), (1498, 6759, 3), (12315, 14530, 3)


X(32064) = CENTROID OF THE ANTI-INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics    -8*(4*R^2-SA)*S^2+16*SB*SC*SW : :
Barycentrics    3 a sec(A + ω) - b sec(B + ω) - c sec(C + ω) : :
X(32064) = 5*X(2)-4*X(10192) = 3*X(2)-4*X(23332) = 5*X(4)-2*X(5878) = 4*X(4)-X(6225) = 2*X(4)+X(12324) = X(4)+2*X(14216) = X(4)+8*X(14864) = 3*X(4)-4*X(18376) = X(4)-4*X(18381) = 5*X(4)-8*X(18383) = 7*X(4)-4*X(22802) = X(154)-3*X(1853) = 5*X(154)-6*X(10192) = 4*X(154)-3*X(11206) = 5*X(1853)-2*X(10192) = 4*X(1853)-X(11206) = 3*X(1853)-2*X(23332) = 8*X(5878)-5*X(6225) = 4*X(5878)+5*X(12324) = X(5878)+5*X(14216) = 3*X(5878)-10*X(18376) = X(5878)-10*X(18381) = X(5878)-4*X(18383) = 7*X(5878)-10*X(22802) = X(6225)+2*X(12324) = X(6225)+8*X(14216) = 3*X(6225)-16*X(18376) = X(6225)-16*X(18381) = 8*X(10192)-5*X(11206) = 3*X(10192)-5*X(23332) = 4*X(10249)-3*X(25406) = 3*X(11206)-8*X(23332)

X(32064) lies on these lines: {2,154}, {4,51}, {5,32063}, {6,7378}, {20,343}, {22,14927}, {25,23291}, {64,3146}, {66,69}, {107,459}, {125,6353}, {159,3619}, {161,6636}, {184,8889}, {193,15583}, {221,5261}, {376,11204}, {381,5644}, {382,12250}, {388,32065}, {394,5921}, {427,6776}, {428,26869}, {469,3332}, {497,11189}, {546,12315}, {631,9833}, {1154,11411}, {1352,3819}, {1368,14826}, {1498,3091}, {1585,5871}, {1586,5870}, {1593,18945}, {1619,1995}, {1892,10360}, {1992,11216}, {1993,8549}, {2192,5274}, {2390,3436}, {2550,11190}, {2777,15682}, {2781,3448}, {2883,3832}, {3068,11241}, {3069,11242}, {3088,6146}, {3089,16655}, {3090,6759}, {3147,23294}, {3357,3529}, {3410,16063}, {3434,21270}, {3522,6696}, {3524,23329}, {3525,10282}, {3535,14242}, {3536,14227}, {3541,18925}, {3542,16659}, {3543,15311}, {3545,23325}, {3546,12134}, {3556,5260}, {3566,8029}, {3575,18913}, {3580,7500}, {3618,5133}, {3620,9924}, {3627,13093}, {3628,14530}, {3681,3827}, {3818,6688}, {3839,23324}, {3917,9747}, {4232,26958}, {5032,23326}, {5056,16252}, {5059,5894}, {5064,11245}, {5159,8780}, {5225,6285}, {5229,7355}, {5284,18621}, {5422,19149}, {5480,7409}, {5891,6643}, {5892,7401}, {5895,17578}, {6145,15740}, {6193,23335}, {6619,11547}, {6622,26883}, {6756,26944}, {6815,20791}, {6995,13567}, {6997,11451}, {7394,7693}, {7408,17810}, {7492,15579}, {7493,23293}, {7494,21243}, {7496,15581}, {7667,10519}, {8972,17819}, {9143,15131}, {9306,16051}, {9544,15139}, {9920,21357}, {9934,15081}, {9968,15019}, {10182,15709}, {10193,15698}, {10250,14912}, {10299,25563}, {10303,17821}, {10304,23328}, {10535,10589}, {10588,26888}, {11180,31152}, {11225,20423}, {11243,11488}, {11244,11489}, {11392,18915}, {11393,18922}, {11432,16198}, {11548,12017}, {12174,23047}, {12278,30552}, {12279,31978}, {12359,31305}, {12964,31412}, {13364,18952}, {13941,17820}, {14683,23315}, {14855,18474}, {15246,15577}, {16419,18358}, {18435,18531}, {18533,18931}, {18917,31723}, {19174,19209}, {19420,32070}, {19421,32071}, {22967,31371}

X(32064) = reflection of X(i) in X(j) for these (i,j): (2, 1853), (20, 10606), (154, 23332), (193, 17813), (3543, 18405), (5596, 19153), (5656, 381), (9143, 15131), (9833, 11202), (9920, 21357), (10606, 6247), (11202, 20299), (11206, 2), (17813, 15583), (19153, 23300), (32063, 5)
X(32064) = isotomic conjugate of the isogonal conjugate of X(33582)
X(32064) = anticomplement of X(154)
X(32064) = anticomplementary conjugate of X(17037)
X(32064) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154, 1853, 23332), (154, 23332, 2)


X(32065) = CENTROID OF THE ANTI-TANGENTIAL-MIDARC TRIANGLE

Barycentrics    a^2*(a+b-c)*(a-b+c)*((b^2+3*b*c+c^2)*a^4-2*(b^3+c^3)*(b+c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(32065) = 4*X(1)-X(6285) = 2*X(1)+X(7355) = 2*X(3028)+X(19505) = X(6285)+2*X(7355) = X(7352)+2*X(32047)

X(32065) lies on these lines: {1,6000}, {12,23332}, {31,56}, {33,32062}, {34,51}, {35,11204}, {36,11202}, {55,10606}, {64,3303}, {65,278}, {354,5603}, {388,32064}, {497,12940}, {517,11214}, {960,6349}, {999,10535}, {1038,3819}, {1058,12950}, {1060,5891}, {1062,14855}, {1154,7352}, {1394,30493}, {1398,11402}, {1428,19153}, {1464,2352}, {1469,2393}, {1498,3304}, {1503,5434}, {1853,11237}, {1870,5890}, {2067,11241}, {2260,21767}, {2330,10249}, {2781,3028}, {2818,3576}, {2979,4296}, {3058,15311}, {3295,10076}, {3357,3746}, {3584,23329}, {3585,18376}, {3868,18633}, {4309,20427}, {4317,9833}, {4857,22802}, {4995,23328}, {5270,18381}, {5298,10192}, {5563,6759}, {5895,9670}, {6198,11455}, {6247,15888}, {6502,11242}, {6688,19372}, {6767,10060}, {7051,11243}, {10571,22341}, {11216,19369}, {11244,19373}, {18435,18447}, {18915,18950}, {19175,19209}, {19370,32070}, {19371,32071}, {24308,24806}

X(32065) = midpoint of X(7355) and X(11189)
X(32065) = reflection of X(i) in X(j) for these (i,j): (6285, 11189), (11189, 1)
X(32065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7355, 6285), (56, 221, 26888)


X(32066) = CENTROID OF THE APOLLONIUS TRIANGLE

Barycentrics
a^2*(3*(b+c)^2*a^8+4*(b+c)*(2*b^2+3*b*c+2*c^2)*a^7+(b^4+c^4+2*(5*b^2+11*b*c+5*c^2)*b*c)*a^6-10*(b^3+c^3)*(b+c)^2*a^5-(10*b^4+10*c^4+(17*b^2+6*b*c+17*c^2)*b*c)*b*c*a^4+2*(b^2+c^2)*(b+c)*(5*b^4-9*b^2*c^2+5*c^4)*a^3-(b^4+c^4+2*(5*b^2+8*b*c+5*c^2)*b*c)*(b^4+c^4)*a^2-2*(b+c)*(4*b^8+4*c^8+(7*b^6+7*c^6-(b^4+c^4+(5*b^2-2*b*c+5*c^2)*b*c)*b*c)*b*c)*a-(3*b^8+3*c^8+(2*b^6+2*c^6-(5*b^4+5*c^4+2*(b^2+c^2)*b*c)*b*c)*b*c)*(b+c)^2) : :

X(32066) lies on the line {970,5529}


X(32067) = CENTROID OF THE 4th CONWAY TRIANGLE

Barycentrics
3*b*c*a^7-(b+c)*(4*b^2+3*b*c+4*c^2)*a^6-2*(4*b^4+4*c^4+(10*b^2+7*b*c+10*c^2)*b*c)*a^5-6*(b+c)*(b^2+3*b*c+c^2)*b*c*a^4+(8*b^6+8*c^6+(11*b^4+11*c^4-2*(4*b^2+7*b*c+4*c^2)*b*c)*b*c)*a^3+(b+c)*(4*b^6+4*c^6+(9*b^2+22*b*c+9*c^2)*(b-c)^2*b*c)*a^2+6*(b+c)*(b^2-c^2)*(b^3-c^3)*b*c*a+2*(b^2-c^2)^2*(b+c)*b^2*c^2 : :

X(32067) lies on these lines: {1,11233}, {2,10439}, {3984,11679}


X(32068) = CENTROID OF THE 3rd HATZIPOLAKIS TRIANGLE

Barycentrics    2*a^6-4*(b^2+c^2)*a^4+(3*b^4-8*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(32068) = 5*X(389)+X(12605) = 5*X(6688)-4*X(13361) = 2*X(9826)+X(19481) = 2*X(10095)+X(18128) = X(10116)+5*X(15026) = 2*X(11695)+X(13292) = X(12022)+3*X(16226) = X(12241)+2*X(15012) = X(13142)+2*X(17704) = 5*X(13403)+X(18565) = X(31985)+2*X(32050)

X(32068) lies on these lines: {2,5965}, {5,15130}, {6,30771}, {51,29012}, {182,7494}, {389,12605}, {511,10691}, {539,13363}, {542,5943}, {575,13567}, {1209,15047}, {1368,5097}, {1853,5476}, {1899,19130}, {3060,19924}, {3448,12834}, {3564,6688}, {5422,21243}, {5644,10516}, {5946,18400}, {5972,13366}, {6388,14153}, {6677,12007}, {6723,15516}, {9826,19481}, {10095,18128}, {10116,15026}, {10601,24206}, {11427,22234}, {11695,13292}, {12022,16226}, {12241,15012}, {13142,17704}, {13403,18565}, {14561,18950}, {15004,18911}, {21849,29317}, {31985,32050}

X(32068) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 11225}, {5943, 11245}
X(32068) = {X(5422), X(21243)}-harmonic conjugate of X(25555)


X(32069) = CENTROID OF THE LEMOINE TRIANGLE

Barycentrics    (4*a^4+11*(b^2+c^2)*a^2-2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2))*(4*a^2+b^2+c^2) : :
X(32069) = 2*X(8145)+X(31969)

X(32069) lies on these lines: {2,10183}, {8029,12073}, {8145,31969}

X(32069) = reflection of X(2) in X(10183)


X(32070) = CENTROID OF THE LUCAS ANTIPODAL TANGENTIAL TRIANGLE

Barycentrics    (SB+SC)*((12*R^2+6*SA+SW)*S^2-2*SB*SC*SW+S*(4*S^2+2*(SA+2*SW)*SA)) : :
X(32070) = 2*X(18980)+X(19500)

X(32070) lies on these lines: {51,19410}, {154,8939}, {1151,19442}, {1154,18939}, {2393,12590}, {2781,19507}, {2979,19406}, {3819,19422}, {5890,19414}, {5891,19428}, {6000,18980}, {6688,19448}, {8681,11198}, {9723,32071}, {10238,17842}, {10606,13021}, {11189,19434}, {11190,19432}, {11202,19440}, {11204,19454}, {11216,19426}, {11241,19436}, {11242,19439}, {11243,19450}, {11244,19452}, {11402,19358}, {18376,18414}, {18435,18462}, {18926,18950}, {19134,19153}, {19186,19209}, {19370,32065}, {19416,32062}, {19418,32063}, {19420,32064}, {23298,23332}

X(32070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19358, 19404, 19408), (19446, 21642, 19410)


X(32071) = CENTROID OF THE LUCAS(-1) ANTIPODAL TANGENTIAL TRIANGLE

Barycentrics    (SB+SC)*((12*R^2+6*SA+SW)*S^2-2*SB*SC*SW-S*(4*S^2+2*(SA+2*SW)*SA)) : :
X(32071) = 2*X(18981)+X(19501)

X(32071) lies on these lines: {51,19411}, {154,8943}, {1152,19443}, {1154,18940}, {2393,12591}, {2781,19508}, {2979,19407}, {3819,19423}, {5890,19415}, {5891,19429}, {6000,18981}, {6688,19449}, {8681,32077}, {9723,32070}, {10240,17839}, {10606,13022}, {11189,19435}, {11190,19433}, {11202,19441}, {11204,19455}, {11216,19427}, {11241,19438}, {11242,19437}, {11243,19451}, {11244,19453}, {11402,19359}, {18376,18415}, {18435,18463}, {18927,18950}, {19135,19153}, {19187,19209}, {19371,32065}, {19417,32062}, {19419,32063}, {19421,32064}, {23299,23332}

X(32071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19359, 19405, 19409), (19447, 21643, 19411)


X(32072) = CENTROID OF THE LUCAS INNER TRIANGLE

Barycentrics    (SB+SC)*(260*(3*SA+2*SW)*S^2+260*SA*SW^2+S*(1351*S^2+901*SA^2-900*SB*SC+150*SW^2)) : :

X(32072) lies on the Parry circle and the line {356,8496}


X(32073) = CENTROID OF THE LUCAS(-1) INNER TRIANGLE

Barycentrics    (SB+SC)*(260*(3*SA+2*SW)*S^2+260*SA*SW^2-S*(1351*S^2+901*SA^2-900*SB*SC+150*SW^2)) : :

X(32073) lies on the Parry circle and these lines: {}


X(32074) = CENTROID OF THE LUCAS(-1) TANGENTS TRIANGLE

Barycentrics    a^2*(-2*(6*a^4-6*(b^2+c^2)*a^2-3*c^4-3*b^4-14*b^2*c^2)*S+10*(b^2+c^2)*a^4-5*(3*b^4+5*b^2*c^2+3*c^4)*a^2+5*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)) : :
X(32074) = 2*X(1152)+X(31975)

X(32074) lies on the Parry circle and these lines: {23,6454}, {352,6481}, {1152,8156}, {11171,11199}


X(32075) = CENTROID OF THE 3rd MIXTILINEAR TRIANGLE

Barycentrics
a*(12*a^6-20*(b+c)*a^5-(24*b^2-97*b*c+24*c^2)*a^4+2*(b+c)*(19*b^2-45*b*c+19*c^2)*a^3+2*(5*b^4+5*c^4-4*(13*b^2-24*b*c+13*c^2)*b*c)*a^2-6*(b^2-c^2)*(b-c)*(3*b^2-7*b*c+3*c^2)*a+(b^2-c^2)^2*(2*b-c)*(b-2*c)) : :
X(32075) = 2*X(3)+X(31979)

X(32075) lies on these lines: {3,31979}, {1420,3924}


X(32076) = CENTROID OF THE 4th MIXTILINEAR TRIANGLE

Barycentrics
a*(12*a^8-28*(b+c)*a^7-(20*b^2-73*b*c+20*c^2)*a^6+6*(b+c)*(17*b^2-22*b*c+17*c^2)*a^5-(70*b^4+70*c^4+(91*b^2-162*b*c+91*c^2)*b*c)*a^4-8*(b^2-c^2)*(b-c)*(4*b^2-13*b*c+4*c^2)*a^3+3*(16*b^4+16*c^4+9*(b-c)^2*b*c)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)^3*(5*b^2+14*b*c+5*c^2)*a-(b+2*c)*(2*b+c)*(b-c)^6) : :
X(32076) = 2*X(3)+X(31980)

X(32076) lies on these lines: {3,31980}, {165,5011}


X(32077) = CENTROID OF THE LUCAS(-1) CENTRAL TRIANGLE

Barycentrics    a^2*(3*a^6-17*(b^2+c^2)*a^4+(21*b^4+26*b^2*c^2+21*c^4)*a^2+(12*a^4-8*(b^2+c^2)*a^2-4*b^4-40*b^2*c^2-4*c^4)*S-(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)) : :
X(32077) = 2*X(3)+X(10673) = 2*X(8156)+X(31973)

X(32077) lies on these lines: {3,494}, {5409,18940}, {8156,31973}, {8373,8400}, {8681,32071}, {8780,10133}

X(32077) = {X(3), X(11950)}-harmonic conjugate of X(26293)


X(32078) = CENTROID OF THE CEVIAN TRIANGLE OF X(3)

Barycentrics    a^2*(-a^2+b^2+c^2)*(2*a^4-3*(b^2+c^2)*a^2+(b^2-c^2)^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(32078) = 4*X(3)-X(31388), 4*X(140)-X(14978), 4*X(10184)-3*X(11197), X(10184)-3*X(12012), X(11197)-4*X(12012)

X(32078) lies on these lines: {2,10184}, {3,54}, {4,14635}, {5,14129}, {51,216}, {110,31626}, {140,14978}, {154,157}, {184,10979}, {233,3078}, {852,6688}, {2972,3819}, {2974,6676}, {3131,10632}, {3132,10633}, {3135,23635}, {6000,23719}, {6638,11451}, {7494,17008}, {7499,30737}, {7503,23709}, {10003,30506}, {13366,22052}, {18376,18416}, {18435,18464}, {20975,23195}, {23332,26905}, {26870,32064}, {26874,30258}

X(32078) = reflection of X(i) in X(j) for these (i,j): (2, 12012), (4, 14635), (11197, 2)
X(32078) = anticomplement of X(10184)
X(32078) = isogonal conjugate of the polar conjugate of X(233)
X(32078) = X(14635)-of-anti-Euler triangle
X(32078) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 233}, {5, 22052}, {97, 3078}, {140, 216}, {217, 1232}, {343, 13366}, {577, 14978}
X(32078) = barycentric quotient X(i)/X(j) for these (i,j): (3, 31617), (140, 276), (184, 288), (217, 1173), (233, 264), (418, 31626)
X(32078) = trilinear product X(i)*X(j) for these lines: {i,j}: {48, 233}, {216, 17438}, {217, 20879}
X(32078) = trilinear quotient X(i)/X(j) for these (i,j): (48, 288), (63, 31617), (233, 92)
X(32078) = {X(216), X(26907)}-harmonic conjugate of X(418)


X(32079) = CENTROID OF THE SODDY TRIANGLE

Barycentrics    (a+b-c)*(a-b+c)*(7*a^4-4*(b+c)*a^3-14*(b-c)^2*a^2+12*(b^2-c^2)*(b-c)*a-(b^2+14*b*c+c^2)*(b-c)^2) : :
X(32079) = X(7)+2*X(15913) = X(7)-4*X(17113) = X(15913)+2*X(17113)

X(32079) lies on these lines: {7,1699}, {479,17728}, {658,9778}, {3160,10164}, {10004,10171}

X(32079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9533, 15511, 10136), (15913, 17113, 7)


X(32080) = CENTROID OF THE INNER-SODDY TRIANGLE

Barycentrics
(a+b-c)*(a-b+c)*(-2*(6*a^4-32*(b+c)*a^3+(43*b^2+24*b*c+43*c^2)*a^2-14*(b^2-c^2)*(b-c)*a-3*(b^2+4*b*c+c^2)*(b-c)^2)*S+5*(-a+b+c)*(2*a^5+(b+c)*a^4-2*(4*b^2+3*b*c+4*c^2)*a^3+4*(b^2-c^2)*(b-c)*a^2+2*(b^2+5*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3)) : :

X(32080) lies on these lines: {}


X(32081) = CENTROID OF THE OUTER-SODDY TRIANGLE

Barycentrics
(a+b-c)*(a-b+c)*(2*(6*a^4-32*(b+c)*a^3+(43*b^2+24*b*c+43*c^2)*a^2-14*(b^2-c^2)*(b-c)*a-3*(b^2+4*b*c+c^2)*(b-c)^2)*S+5*(-a+b+c)*(2*a^5+(b+c)*a^4-2*(4*b^2+3*b*c+4*c^2)*a^3+4*(b^2-c^2)*(b-c)*a^2+2*(b^2+5*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3)) : :

X(32081) lies on the line {175,16214}


X(32082) = CENTROID OF THE 2nd INNER-SODDY TRIANGLE

Barycentrics    a^3-6*S*a-3*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c) : :
X(32082) = 2*X(1)+X(31533) = X(482)+2*X(32057)

X(32082) lies on these lines: {1,4}, {55,31538}, {176,9778}, {482,11246}, {1371,3474}, {5393,17728}, {10164,13389}, {13388,31570}, {16213,31535}

X(32082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3475, 32083), (176, 16663, 10134)


X(32083) = CENTROID OF THE 2nd OUTER-SODDY TRIANGLE

Barycentrics    a^3+6*S*a-3*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c) : :
X(32083) = 2*X(1)+X(31532) = X(481)+2*X(32058)

X(32083) lies on these lines: {1,4}, {55,31539}, {175,9778}, {481,11246}, {1372,3474}, {5405,17728}, {10164,13388}, {13389,31569}, {16214,31534}

X(32083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3475, 32082), (175, 16662, 10135)


X(32084) = CENTROID OF THE YIU TRIANGLE

Barycentrics
a^16-5*(b^2+c^2)*a^14+3*(3*b^4+4*b^2*c^2+3*c^4)*a^12-2*(b^2+c^2)*(3*b^4+b^2*c^2+3*c^4)*a^10+(2*b^4-3*b^2*c^2+2*c^4)*b^2*c^2*a^8-(b^2+c^2)*(b^8+c^8-(4*b^4-3*b^2*c^2+4*c^4)*b^2*c^2)*a^6+(b^2-c^2)^2*(5*b^8+5*c^8-2*(3*b^4+b^2*c^2+3*c^4)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)^3*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4+c^4)*(b^2-c^2)^6 : :
X(32084) = 2*X(8154)+X(31991)

X(32084) lies on these lines: {5,30529}, {381,9143}, {1989,14643}, {5663,30685}, {8154,31991}


X(32085) = ISOGONAL CONJUGATE OF X(3917)

Barycentrics    (a^2 + b^2) (a^2 + b^2 - c^2) (a^2 + c^2) (a^2 - b^2 + c^2) : :
Barycentrics    (4 R^2 SB + 4 R^2 SC + SB SC + 4 R^2 SW - SB SW - SC SW - SW^2)S^2 + 2 SB SC SW^2 : :
Barycentrics    SB SC (SB + SW) (SC + SW) : : (Peter Moses, April 14, 2019)
Barycentrics    (tan A)/(b^2 + c^2) : : (Randy Hutson, June 7, 2019)
Trilinears    tan A csc(A + ω) : : (Randy Hutson, June 7, 2019)
Trilinears    1/(2 b c - a b cos B - a c cos C) : : (Randy Hutson, June 7, 2019)

See Kadir Altintas and Ercole Suppa, Hyacinthos 28982.

Let (PA) be the pedal circle of the A-vertex of the anticomplementary triangle (and of the A-vertex of the tangential triangle), and define (PB) and (PC) cyclically. Let RA be the radical center of (O), (PB), (PC) (where (O) is the circumcircle). Define RB and RC cyclically. The lines ARA, BRB, CRC concur in X(32085). (Randy Hutson, June 7, 2019)

Generalization: Let X = x : y : z and X' = 1/x : 1/y : 1/z (trilinears) be two isogonal conjugate points. Let (PA) be the pedal circle of the A-vertex of the anticevian triangle of X (and of the A-vertex of the anticevian triangle of X'), and define (PB) and (PC) cyclically. Let RA be the radical center of (O), (PB), (PC). Define RB and RC cyclically. The lines ARA, BRB, CRC concur in the polar conjugate of the isotomic conjugate of the cevapoint of X and X', with trilinears 1/(x (y^2 + z^2) (a^2 - b^2 - c^2)) : :. (Randy Hutson, June 7, 2019)

X(32085) lies on the circumhyperbola {{A,B,C,X(4),X(93)}} (with center X(136)) and on these lines {2,8801}, {4,83}, {20,26224}, {25,183}, {53,6531}, {82,225}, {95,237}, {107,9076}, {235,14860}, {242,4222}, {251,393}, {297,16890}, {317,20022}, {458,18092}, {648,1843}, {689,2374}, {827,1300}, {847,10594}, {1093,1598}, {1105,3575}, {1217,7487}, {1596,16264}, {1826,2201}, {2052,16277}, {3089,18855}, {4194,27067}, {4200,27005}, {4232,10130}, {6530,6756}, {7716,9308}, {8793,11547}, {9755,14486}, {10002,10548}, {10301,17983}, {12110,14575}, {12122,22078}, {12173,18848}, {14018,29568}, {15424,26863}, {16230,18010}, {16263,18494}, {18105,23290}

X(32085) = isogonal conjugate of X(3917)
X(32085) = isotomic conjugate of X(3933)
X(32085) = cevapoint of X(i) and X(j) for these (i,j): {2, 7754}, {4, 25}, {6620, 9755}, {18101, 18108}
X(32085) = crosssum of X(3) and X(22138)
X(32085) = trilinear pole of line {419, 2501}
X(32085) = polar conjugate of X(141)
X(32085) = isogonal conjugate of the anticomplement of X(5943)
X(32085) = isogonal conjugate of the complement of X(3060)
X(32085) = isotomic conjugate of the anticomplement of X(5305)
X(32085) = isotomic conjugate of the complement of X(7754)
X(32085) = polar conjugate of the isotomic of X(83)
X(32085) = polar conjugate of the isogonal of X(251)
X(32085) = X(i)-cross conjugate of X(j) for these (i,j): {251, 83}, {428, 4}, {2489, 648}, {5305, 2}, {14618, 107}, {17922, 1897}
X(32085) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3917}, {2, 4020}, {3, 38}, {31, 3933}, {39, 63}, {48, 141}, {69, 1964}, {71, 16696}, {72, 17187}, {75, 20775}, {77, 3688}, {78, 1401}, {163, 2525}, {184, 1930}, {212, 3665}, {228, 16887}, {255, 427}, {304, 3051}, {305, 1923}, {326, 1843}, {394, 17442}, {577, 20883}, {603, 3703}, {656, 1634}, {810, 4576}, {826, 4575}, {906, 16892}, {1331, 2530}, {1332, 21123}, {1437, 15523}, {1444, 21035}, {1459, 4553}, {1790, 3954}, {2084, 4563}, {2200, 16703}, {3005, 4592}, {3990, 17171}, {4055, 16747}, {4558, 8061}, {4568, 22383}, {6507, 27376}, {7116, 16720}, {8024, 9247}, {17206, 21814}, {18604, 21016}, {19215, 26347}, {22367, 32010}
X(32085) = barycentric product of X(i)*X(j) for these lines: {i,j}: {4,83}, {19,3112}, {25,308}, {27,18082}, {29,18097}, {82,92}, {107,4580}, {162,18070}, {251,264}, {275,17500}, {286,18098}, {393,1799}, {419,14970}, {689,2489}, {733,17984}, {827,14618}, {1093,28724}, {1176,2052}, {1897,10566}, {1973,18833}, {2373,21459}, {2501,4577}, {4599,24006}, {6330,21458}, {6331,18105}, {6335,18108}, {6531,20022}, {10547,18027}, {10548,14860}, {16277,17907}
X(32085) = barycentric quotient of X(i) and X(j) for these lines: {i,j}: {2,3933}, {4,141}, {6,3917}, {19,38}, {25,39}, {27,16887}, {28,16696}, {31,4020}, {32,20775}, {82,63}, {83,69}, {92,1930}, {112,1634}, {141,4175}, {158,20883}, {251,3}, {264,8024}, {278,3665}, {281,3703}, {286,16703}, {308,305}, {393,427}, {419,732}, {427,7794}, {428,6292}, {458,14994}, {468,7813}, {523,2525}, {607,3688}, {608,1401}, {648,4576}, {827,4558}, {1096,17442}, {1176,394}, {1194,22424}, {1474,17187}, {1783,4553}, {1799,3926}, {1824,3954}, {1826,15523}, {1843,8041}, {1897,4568}, {1973,1964}, {1974,3051}, {2052,1235}, {2207,1843}, {2333,21035}, {2489,3005}, {2501,826}, {3112,304}, {4577,4563}, {4580,3265}, {4599,4592}, {4628,1331}, {5007,22078}, {6524,27376}, {6531,20021}, {6591,2530}, {6995,8362}, {7009,16720}, {7649,16892}, {8743,3313}, {8744,9019}, {8747,17171}, {8793,23115}, {8882,16030}, {8948,26347}, {10311,14096}, {10547,577}, {10549,21243}, {10551,22416}, {10566,4025}, {14273,14424}, {14569,27371}, {14618,23285}, {14885,22138}, {16277,14376}, {16890,4121}, {17409,23208}, {17500,343}, {18070,14208}, {18082,306}, {18097,307}, {18098,72}, {18099,4019}, {18101,26932}, {18105,647}, {18108,905}, {19118,3787}, {20022,6393}, {21458,441}, {21459,858}, {22105,14417}, {28724,3964}
X(32085) = trilinear product of X(i)*X(j) for these lines: {i,j}: {4,82}, {19,83}, {19,83}, {25,3112}, {25,3112}, {27,18098}, {28,18082}, {28,18082}, {92,251}, {112,18070}, {158,1176}, {308,1973}, {308,1973}, {811,18105}, {827,24006}, {1096,1799}, {1096,1799}, {1172,18097}, {1172,18097}, {1783,10566}, {1897,18108}, {1974,18833}, {1974,18833}, {2190,17500}, {2190,17500}, {2216,10550}, {2489,4593}, {2489,4593}, {2501,4599}, {2501,4599}, {3405,6531}, {4580,24019}, {4628,17924}, {4628,17924}, {6520,28724}, {6520,28724}, {7012,18101}, {7012,18101}, {8767,21458}, {8767,21458}
X(32085) = trilinear quotient of X(i) and X(j) for these lines: {i,j}: {1,3917}, {6,4020}, {19,39}, {25,1964}, {28,17187}, {31,20775}, {33,3688}, {34,1401}, {75,3933}, {158,427}, {162,1634}, {273,3665}, {318,3703}, {393,17442}, {419,2236}, {428,17457}, {811,4576}, {827,4575}, {862,4093}, {1096,1843}, {1824,21035}, {1897,4553}, {1930,4175}, {1969,8024}, {1973,3051}, {2052,20883}, {2190,16030}, {2333,21814}, {2501,8061}, {4577,4592}, {6520,27376}, {6591,21123}, {16600,22077}, {17446,22424}, {17469,22078}, {20964,22367}
X(32085) = {X(i),X(j)}-harmonic conjugate of X(k) for these lines: {i,j,k}: {4,17500,10550}, {1176,17500,83}






leftri  Points V-Caph: X(32086)-X(32097)  rightri

This preamble and centers X(32086)-X(32097) were contributed by Clark Kimberling and Peter Moses, April 19, 2019.

Suppose that P = p:q:r (barycentrics). The Point V-Caph of P is the point given by

qr + 2(p + q + r)p : rp + 2(p + q + r)q : pq + 2(p + q + r)r

The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

underbar

X(32086) = POINT V-CAPH OF X(7)

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

X(32086) lies on these lines: {2, 10481}, {7, 8}, {12, 24797}, {226, 8055}, {279, 3616}, {348, 5550}, {479, 17169}, {664, 5543}, {728, 4454}, {938, 1111}, {1119, 5342}, {1125, 21314}, {1323, 3622}, {1358, 3485}, {1698, 20121}, {2481, 5558}, {3160, 17079}, {3241, 9312}, {3598, 16823}, {3665, 5226}, {3673, 10580}, {4308, 7223}, {4323, 27818}, {4384, 10521}, {4452, 7274}, {4872, 10248}, {4888, 6738}, {6172, 32008}, {7196, 26103}, {7288, 24805}, {7960, 24774}, {9436, 9780}, {9812, 17170}, {10588, 24798}, {18230, 32024}, {20053, 25719}

X(32086) = {X(7),X(8)}-harmonic conjugate of X(32098)
X(32086) = {X(85),X(6604)}-harmonic conjugate of X(31994)


X(32087) = POINT V-CAPH OF X(8)

Barycentrics    -a^2 + b^2 + 6*b*c + c^2 : :
Trilinears    sec^2(A/2) + 2 csc^2(A/2) : :

X(32087) lies on these lines {1, 4460}, {2, 2321}, {6, 4371}, {7, 8}, {9, 4461}, {10, 3672}, {40, 28638}, {63, 9536}, {77, 4853}, {78, 7269}, {86, 3241}, {142, 4007}, {144, 3686}, {145, 10436}, {192, 5296}, {193, 17116}, {200, 7190}, {239, 5749}, {269, 4915}, {273, 7046}, {314, 7320}, {321, 18228}, {326, 4861}, {329, 14213}, {346, 4384}, {347, 4847}, {350, 26038}, {391, 3729}, {519, 3945}, {524, 7222}, {527, 4034}, {536, 966}, {594, 3763}, {938, 5295}, {962, 4647}, {1014, 12513}, {1058, 28644}, {1100, 4470}, {1213, 28635}, {1219, 7176}, {1266, 4678}, {1278, 17257}, {1442, 3872}, {1447, 7172}, {1654, 4740}, {1698, 4021}, {2345, 3589}, {3161, 17277}, {3187, 19825}, {3243, 4923}, {3475, 4046}, {3596, 4441}, {3616, 4360}, {3617, 4357}, {3620, 29615}, {3621, 3879}, {3622, 28626}, {3623, 4464}, {3626, 4346}, {3629, 4363}, {3632, 3664}, {3663, 3679}, {3687, 5226}, {3694, 24554}, {3706, 10580}, {3739, 5308}, {3883, 30332}, {3886, 8236}, {3932, 9780}, {3950, 16832}, {3966, 9812}, {3977, 5271}, {3986, 28313}, {4058, 17284}, {4060, 17296}, {4072, 31211}, {4328, 4882}, {4359, 34255}, {4373, 17274}, {4395, 17293}, {4398, 32025}, {4405, 7227}, {4416, 4454}, {4419, 4686}, {4445, 7263}, {4478, 7232}, {4546, 30181}, {4643, 4726}, {4644, 6144}, {4648, 4688}, {4651, 30946}, {4668, 4862}, {4677, 4888}, {4699, 17316}, {4739, 4851}, {4748, 17246}, {4764, 17256}, {4772, 6542}, {4816, 4896}, {4821, 6646}, {4859, 29594}, {4869, 17294}, {4873, 6666}, {4916, 17392}, {4971, 15668}, {4980, 5739}, {4986, 21290}, {5082, 21279}, {5435, 11679}, {5543, 20007}, {5550, 28653}, {5750, 17014}, {7319, 30479}, {11024, 28612}, {14552, 28610}, {16816, 26685}, {16833, 17355}, {17143, 30022}, {17163, 18697}, {17184, 19826}, {17233, 29627}, {17322, 19877}, {17331, 20073}, {17353, 24599}, {17377, 20053}, {17394, 20057}, {20054, 30712}, {20055, 26806}, {26626, 28604}, {27147, 29583}, {27304, 27544}

X(32087) = isotomic conjugate of X(5558)
X(32087) = anticomplement of X(3247)
X(32087) = {X(7),X(8)}-harmonic conjugate of X(32099)
X(32087) = {X(69),X(75)}-harmonic conjugate of X(31995)


X(32088) = POINT V-CAPH OF X(9)

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

X(32088) lies on these lines: {2, 32007}, {8, 4702}, {9, 85}, {44, 27253}, {220, 31269}, {6173, 32015}, {6604, 18230}, {17336, 20880}, {27065, 30854}

X(32088) = {X(9),X(85)}-harmonic conjugate of X(32100)


X(32089) = POINT V-CAPH OF X(10)

Barycentrics    a^2 + 5*a*b + 4*b^2 + 5*a*c + 9*b*c + 4*c^2 : :

X(32089) lies on these lines: {2, 4399}, {10, 86}, {75, 3992}, {594, 31248}, {1698, 4360}, {3632, 30598}, {3634, 5564}, {3828, 17322}, {3932, 9780}, {4389, 5936}, {4460, 19877}, {4464, 31253}, {6535, 30594}, {17271, 21296}, {17279, 29576}, {17285, 24603}, {17311, 29593}, {17332, 28604}, {17359, 31311}, {17380, 28635}, {17384, 28633}

X(32089) = {X(10),X(86)}-harmonic conjugate of X(32101)


X(32090) = POINT V-CAPH OF X(37)

Barycentrics    4*a^2*b^2 + 9*a^2*b*c + 5*a*b^2*c + 4*a^2*c^2 + 5*a*b*c^2 + b^2*c^2 : :

X(32090) lies on these lines: {1, 4753}, {37, 274}, {4687, 17143}, {4755, 16819}, {17264, 25512}, {18145, 27255}, {21830, 27268}

X(32090) = {X(37),X(274)}-harmonic conjugate of X(32102)


X(32091) = POINT V-CAPH OF X(44)

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

X(32091) lies on these lines: {44, 20568}


X(32092) = POINT V-CAPH OF X(75)

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

X(32092) lies on these lines: {1, 75}, {2, 3760}, {7, 9534}, {10, 3761}, {19, 16747}, {35, 16992}, {63, 169}, {76, 1698}, {85, 3339}, {192, 31996}, {194, 4699}, {213, 4363}, {239, 20893}, {310, 17210}, {321, 16831}, {330, 16829}, {350, 3624}, {609, 16915}, {980, 31993}, {1089, 30758}, {1107, 4688}, {1111, 28611}, {1125, 4441}, {1218, 2663}, {1268, 30596}, {1449, 20174}, {1909, 3679}, {1975, 5251}, {2176, 17118}, {3208, 29383}, {3294, 3729}, {3633, 25303}, {3634, 18135}, {3665, 7243}, {3673, 16832}, {3739, 5283}, {3925, 3933}, {3926, 19854}, {4000, 16818}, {4063, 23807}, {4253, 24592}, {4352, 19853}, {4358, 25585}, {4361, 20963}, {4410, 17251}, {4664, 32009}, {4668, 24524}, {4671, 29578}, {4687, 32026}, {4691, 25278}, {4726, 25130}, {4739, 16975}, {4980, 29597}, {5088, 16824}, {5271, 18206}, {5280, 11321}, {5299, 20172}, {6376, 19875}, {6381, 9780}, {6390, 24953}, {6533, 7264}, {7031, 16998}, {7176, 16821}, {11679, 21422}, {16502, 20181}, {16705, 19858}, {16748, 16887}, {16815, 30563}, {16826, 28605}, {16827, 17116}, {16830, 31130}, {16834, 21432}, {17135, 17169}, {17211, 17889}, {17303, 25499}, {17308, 20913}, {17321, 25512}, {17753, 20245}, {17754, 29433}, {17759, 27255}, {18044, 29757}, {18145, 19876}, {20911, 28612}, {21384, 29773}, {24199, 29960}, {25457, 25661}, {26035, 30107}, {26959, 27318}, {30035, 30038}

X(32092) = {X(1),X(75)}-harmonic conjugate of X(32104)


X(32093) = POINT V-CAPH OF X(145)

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

X(32093) lies on these lines: {2, 1743}, {7, 145}, {8, 4888}, {69, 3617}, {75, 31145}, {141, 4747}, {144, 17300}, {193, 24599}, {320, 1279}, {346, 17311}, {391, 4675}, {464, 9965}, {1122, 3873}, {3241, 4862}, {3244, 4902}, {3621, 4896}, {3623, 3663}, {3631, 4470}, {3672, 17378}, {3870, 7271}, {4454, 4851}, {4488, 29573}, {4644, 4869}, {4648, 17332}, {4678, 25590}, {4699, 11160}, {5059, 10442}, {5232, 17361}, {6646, 29624}, {7222, 17374}, {17014, 20090}, {17151, 20014}, {17316, 20059}, {17375, 29616}, {17778, 21454}, {20043, 26842}, {20080, 26806}, {29583, 31300}


X(32094) = POINT V-CAPH OF X(190)

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

X(32094) lies on these lines: {10, 24428}, {100, 2748}, {190, 514}, {519, 1757}, {644, 4568}, {660, 6005}, {1018, 4063}, {3218, 3912}, {3570, 4169}, {3693, 9321}, {3732, 30720}, {4422, 6547}, {4440, 31647}, {4553, 6004}, {4562, 32042}, {4760, 21821}, {6632, 10196}, {17262, 24281}, {21272, 30732}, {23891, 30731}, {24196, 24419}


X(32095) = POINT V-CAPH OF X(192)

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

X(32095) lies on these lines: {1, 87}, {2, 17144}, {37, 20168}, {42, 14823}, {75, 25130}, {220, 20180}, {239, 7308}, {274, 4740}, {385, 3303}, {519, 27269}, {551, 27318}, {1107, 4704}, {1278, 31997}, {1500, 7786}, {1655, 3241}, {2176, 4393}, {3058, 7823}, {3210, 29570}, {3304, 7783}, {3622, 17759}, {3623, 21226}, {3746, 7793}, {3913, 16999}, {3934, 30998}, {4309, 14712}, {4360, 16969}, {4479, 24656}, {4664, 14897}, {4699, 17143}, {7762, 15170}, {7787, 16785}, {9263, 20057}, {9331, 26959}, {16777, 20170}, {16826, 17490}, {17024, 31088}, {17242, 30038}, {17244, 20257}, {17383, 27248}, {19738, 31036}, {20081, 25303}, {21830, 27268}, {24620, 29595}, {24621, 29580}, {25107, 30963}


X(32096) = POINT V-CAPH OF X(239)

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

X(32096) lies on these lines: {2, 3722}, {190, 4686}, {239, 335}, {1086, 3629}, {1100, 27191}, {1279, 29607}, {4366, 4384}, {4402, 4440}, {4437, 29617}, {4974, 24715}, {6650, 28558}, {6653, 29590}, {16593, 29628}, {16816, 17755}, {16833, 17738}, {17367, 26582}, {17380, 25357}


X(32097) = POINT V-CAPH OF X(320)

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

X(32097) lies on these lines: {7, 21290}, {69, 1111}, {80, 320}, {668, 7321}, {4555, 26726}, {4986, 31995}, {17274, 24222}






leftri  Points VI-Caph: X(32098)-X(32109)  rightri

This preamble and centers X(32098)-X(32109) were contributed by Clark Kimberling and Peter Moses, April 19, 2019.

Suppose that P = p:q:r (barycentrics). The Point VI-Caph of P is the point given by

qr - 2(p + q + r)p : rp - 2(p + q + r)q : pq - 2(p + q + r)r

The points in a Caph family of a point P all lie on the line PP*, where P* is the isotomic conjugate of P.

underbar

X(32098) = POINT VI-CAPH OF X(7)

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

X(32098) lies on these lines: {7, 8}, {57, 29627}, {145, 10481}, {279, 3241}, {348, 5543}, {728, 4869}, {938, 7264}, {1323, 3623}, {1565, 5734}, {2099, 24797}, {2481, 5556}, {3160, 20057}, {3244, 21314}, {3616, 9436}, {3633, 20121}, {3665, 4323}, {3912, 21454}, {4373, 20008}, {4862, 6738}, {9312, 20050}, {9812, 17753}, {9965, 25935}, {17079, 25718}, {20059, 30625}

X(32098) = {X(7),X(8)}-harmonic conjugate of X(32086)


X(32099) = POINT VI-CAPH OF X(8)

Barycentrics    3*a^2 - 3*b^2 - 2*b*c - 3*c^2 : :
Barycentrics    b c + 3 SA : :
Trilinears    sec^2(A/2) - 2 csc^2(A/2) : :

X(32099) lies on these lines: {1, 5232}, {2, 1449}, {6, 29611}, {7, 8}, {9, 29616}, {10, 3945}, {77, 200}, {78, 1442}, {86, 9780}, {141, 5222}, {142, 4034}, {144, 2321}, {145, 4357}, {150, 4738}, {189, 26872}, {193, 3661}, {239, 3620}, {269, 4882}, {279, 6743}, {306, 5273}, {307, 3160}, {314, 7319}, {318, 32001}, {326, 4420}, {329, 2893}, {344, 17295}, {346, 4416}, {347, 6737}, {355, 31780}, {391, 3912}, {519, 3672}, {524, 2345}, {527, 4007}, {594, 4644}, {599, 4000}, {651, 3713}, {894, 20080}, {914, 5744}, {938, 5814}, {966, 4690}, {1014, 1376}, {1086, 4371}, {1264, 5423}, {1266, 20052}, {1654, 5296}, {1743, 29594}, {1943, 18624}, {1992, 17289}, {3161, 17233}, {3187, 19823}, {3241, 17271}, {3421, 21279}, {3474, 4046}, {3578, 17776}, {3596, 25278}, {3616, 4966}, {3617, 5936}, {3618, 17228}, {3619, 3759}, {3621, 3875}, {3624, 4909}, {3625, 4346}, {3626, 25590}, {3629, 17293}, {3630, 4363}, {3631, 4361}, {3632, 3663}, {3633, 4021}, {3662, 4402}, {3664, 3679}, {3687, 5435}, {3694, 24635}, {3705, 15589}, {3706, 9812}, {3751, 5772}, {3758, 11008}, {3763, 4969}, {3872, 7269}, {3883, 8236}, {3886, 30332}, {3949, 18161}, {3964, 7279}, {3966, 10580}, {3974, 4872}, {3986, 29602}, {4001, 28610}, {4005, 10978}, {4042, 14828}, {4060, 4659}, {4328, 4915}, {4344, 5847}, {4358, 5739}, {4360, 20050}, {4384, 4869}, {4389, 20053}, {4399, 7232}, {4419, 4718}, {4431, 4454}, {4452, 17274}, {4464, 20014}, {4488, 17347}, {4643, 4681}, {4648, 17275}, {4657, 4725}, {4665, 7222}, {4668, 4888}, {4677, 4862}, {4678, 4967}, {4704, 6542}, {4720, 8822}, {4748, 4916}, {4788, 6646}, {4816, 4887}, {4853, 7190}, {4856, 29598}, {4886, 18141}, {4971, 17255}, {5044, 28644}, {5081, 32000}, {5226, 11679}, {5227, 7291}, {5257, 29624}, {5550, 17394}, {5815, 21270}, {5932, 31527}, {6144, 17369}, {6381, 10449}, {7046, 7282}, {7172, 7179}, {7320, 30479}, {11160, 17364}, {14555, 30829}, {15533, 17365}, {16667, 29604}, {16706, 21356}, {16786, 17349}, {16833, 21255}, {17014, 17306}, {17135, 30946}, {17184, 19824}, {17230, 26685}, {17236, 20016}, {17238, 26626}, {17248, 29585}, {17250, 20057}, {17251, 17390}, {17252, 17389}, {17253, 17388}, {17256, 17386}, {17260, 29583}, {17277, 29627}, {17282, 24599}, {17283, 31189}, {17288, 29617}, {17309, 17332}, {17310, 17331}, {17311, 17330}, {17315, 17328}, {17376, 28634}, {17378, 32025}, {18623, 26942}, {20090, 29593}, {26038, 30962}, {26041, 27097}, {28780, 28795}

X(32099) = isotomic conjugate of X(5556)
X(32099) = anticomplement of X(1449)
X(32099) = {X(7),X(8)}-harmonic conjugate of X(32087)
X(32099) = {X(69),X(75)}-harmonic conjugate of X(21296)


X(32100) = POINT VI-CAPH OF X(9)

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

X(32100) lies on these lines: {8, 15481}, {9, 85}, {144, 32007}, {220, 31169}, {3219, 30854}, {6172, 6604}, {10025, 31269}, {16814, 27253}, {17335, 20880}

X(32100) = {X(9),X(85)}-harmonic conjugate of X(32088)


X(32101) = POINT VI-CAPH OF X(10)

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

X(32101) lies on these lines: {2, 4478}, {7, 11237}, {10, 86}, {594, 31144}, {1654, 7227}, {1698, 30598}, {3617, 3672}, {3626, 4464}, {3679, 4360}, {4021, 4691}, {4357, 4745}, {4668, 17393}, {4887, 4967}, {4888, 17270}, {5278, 30590}, {6542, 31248}, {6666, 17285}, {7277, 28604}, {9780, 17377}, {17228, 20195}, {17239, 17291}, {17268, 31311}, {17275, 17368}, {17277, 17293}, {17287, 28633}, {17295, 29576}, {17305, 28634}, {17307, 17366}, {17394, 19875}, {26842, 27790}

X(32101) = {X(10),X(86)}-harmonic conjugate of X(32089)


X(32102) = POINT VI-CAPH OF X(37)

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

X(32101) lies on these lines: {1, 17336}, {37, 274}, {4664, 17143}, {4681, 16819}, {4704, 5283}

X(32102) = {X(37),X(274)}-harmonic conjugate of X(32090)


X(32103) = POINT VI-CAPH OF X(44)

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

X(32103) lies on these lines: {44, 20568}


X(32104) = POINT VI-CAPH OF X(75)

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

X(32194) lies on these lines: {1, 75}, {8, 3761}, {9, 20174}, {10, 3760}, {76, 3679}, {85, 18421}, {192, 16819}, {194, 4740}, {213, 4361}, {239, 26223}, {312, 16832}, {321, 3294}, {350, 1698}, {536, 5283}, {668, 4668}, {1107, 4686}, {1269, 17270}, {1278, 25264}, {1909, 3632}, {1975, 5258}, {2176, 17119}, {2321, 17050}, {2345, 16818}, {2481, 4385}, {3501, 29433}, {3617, 6381}, {3644, 32026}, {3672, 19853}, {3729, 16552}, {3730, 24592}, {3746, 16992}, {3770, 4034}, {4050, 29699}, {4359, 16831}, {4363, 20963}, {4431, 29960}, {4461, 27304}, {4479, 18140}, {4494, 4665}, {4659, 21384}, {4669, 25278}, {4671, 16815}, {4677, 24524}, {4699, 31996}, {4714, 7264}, {4717, 24331}, {4726, 16975}, {4732, 21615}, {4751, 32009}, {4980, 16834}, {5280, 20172}, {7243, 30617}, {7773, 31159}, {7776, 31140}, {9331, 27255}, {11321, 16785}, {11520, 20880}, {11679, 20367}, {16549, 17026}, {16823, 31130}, {16828, 17321}, {17030, 17759}, {17294, 20913}, {17301, 25499}, {17320, 19871}, {17754, 29742}, {17866, 20895}, {21070, 29966}, {24170, 30942}, {24190, 31027}, {24790, 27248}, {26234, 28612}

X(32104) = {X(1),X(75)}-harmonic conjugate of X(32092)


X(32105) = POINT VI-CAPH OF X(145)

Barycentrics    -5*a^2 - 6*a*b - b^2 - 6*a*c + 14*b*c - c^2 : :

X(32105) lies on these lines: {2, 3950}, {7, 145}, {69, 20054}, {75, 3622}, {192, 24599}, {319, 31145}, {346, 17366}, {1278, 17014}, {3617, 3672}, {3621, 3663}, {3623, 30712}, {4000, 30833}, {4346, 17361}, {4402, 6666}, {4454, 4852}, {4461, 17368}, {4862, 20050}, {9797, 21314}, {17272, 20052}, {17484, 20043}, {20014, 21296}


X(32106) = POINT VI-CAPH OF X(190)

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

X(32106) lies on these lines: {2, 31647}, {190, 514}, {545, 6547}, {3912, 17484}, {4473, 6549}, {17496, 25267}


X(32107) = POINT VI-CAPH OF X(192)

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

X(32107) lies on these lines: {1, 87}, {39, 30998}, {239, 3929}, {274, 27268}, {385, 5217}, {498, 19570}, {1107, 1278}, {1479, 13571}, {1655, 9780}, {1909, 20105}, {2176, 25269}, {2276, 20081}, {3177, 11684}, {3210, 5278}, {3617, 17759}, {3621, 21226}, {3634, 27269}, {3644, 17448}, {4352, 17280}, {4699, 5283}, {4704, 31997}, {4788, 17144}, {5204, 7783}, {5225, 7774}, {5333, 29595}, {6284, 7837}, {7173, 7777}, {7779, 9598}, {7905, 9664}, {16666, 20168}, {16815, 17490}, {17129, 31448}, {17232, 24214}, {17242, 24215}, {17495, 30563}, {19862, 27318}, {20018, 20072}, {20036, 20073}, {20691, 21219}, {24621, 29578}


X(32108) = POINT VI-CAPH OF X(239)

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

X(32108) lies on these lines: {190, 4718}, {239, 335}, {1086, 3630}, {3759, 9055}, {4366, 16468}, {4393, 17755}, {4422, 17393}, {4437, 17367}, {4864, 17266}, {12630, 20533}, {15570, 29575}, {16593, 17389}, {17287, 27191}, {24715, 28498}, {26582, 29617}


X(32109) = POINT VI-CAPH OF X(320)

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

X(32109) lies on these lines: {69, 4738}, {80, 320}, {150, 21286}






leftri  Points on Walsmith rectangular hyperbola: X(32110)-X(32114) and X(32119)-X(32127)  rightri

Centers X(32110)-X(32114) and X(32119)-X(32127) were contributed by Peter Moses, April 20, 2019.

The centers X(32110)-X(32114) and X(32119)-X(32127) lie on the hyperbola denoted by H1 at K1091 (Walsmith Focal Cubic), here named the Walsmith rectangular hyperbola.

The center of the Walsmith rectangular hyperbola is X(468), and the hyperbola passes through these points: X(6), X(74), X(110), X(113), X(125), X(1495), X(2574), X(2575), X(2931), X(3569), X(3580), X(5000), X(5001), X(7699), X(7703), X(10117), X(11472), X(15904)

underbar

X(32110) = MIDPOINT OF X(23) AND X(74)

Barycentrics    a^2*(2*a^8 - 3*a^6*b^2 - 3*a^4*b^4 + 7*a^2*b^6 - 3*b^8 - 3*a^6*c^2 + 6*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 2*b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 + 2*b^4*c^4 + 7*a^2*c^6 + 2*b^2*c^6 - 3*c^8) : :
X(32110) = {X[110] - 3 X[186],X[323] - 3 X[15035],3 X[2070] + X[10620],3 X[2072] - 4 X[6723],X[3292] - 4 X[18571],X[3830] - 3 X[15362],X[7464] - 3 X[15055],X[7574] - 3 X[15061],X[10296] - 3 X[14644],2 X[10297] - 3 X[23515],3 X[10540] - X[12308],X[10990] + 2 X[16619],4 X[11801] - 3 X[13851],2 X[12041] - 3 X[21663],5 X[15040] - 3 X[22115],5 X[15051] - X[23061],3 X[15055] + X[15107],3 X[16111] - 2 X[20725]

X(32110) lies on these lines: {2, 4549}, {3, 6}, {4, 7703}, {5, 1531}, {22, 8717}, {23, 74}, {24, 7689}, {25, 11472}, {26, 1204}, {30, 125}, {51, 18570}, {64, 9714}, {110, 186}, {113, 468}, {155, 15750}, {184, 18324}, {185, 1658}, {323, 15035}, {376, 15360}, {382, 23325}, {512, 19902}, {517, 10149}, {541, 7426}, {549, 13857}, {550, 18555}, {858, 6699}, {1154, 14708}, {1209, 31833}, {1216, 22467}, {1495, 5663}, {1503, 8262}, {1511, 3292}, {1533, 10990}, {1594, 20191}, {1995, 4550}, {2070, 6000}, {2071, 11692}, {2072, 6723}, {2393, 11579}, {2777, 11799}, {2931, 21284}, {3066, 9818}, {3146, 11468}, {3357, 7517}, {3431, 11422}, {3515, 10539}, {3518, 11440}, {3520, 5446}, {3530, 8254}, {3564, 30714}, {3569, 14696}, {3580, 10295}, {3830, 15362}, {4846, 7493}, {5447, 7691}, {5448, 10018}, {5449, 6240}, {5462, 14118}, {5642, 18579}, {5651, 5891}, {5889, 12038}, {5890, 10298}, {5892, 15053}, {6102, 13367}, {6696, 7553}, {6756, 18488}, {7464, 15055}, {7514, 22112}, {7542, 13568}, {7556, 15072}, {7574, 15061}, {7687, 18323}, {9909, 11820}, {10110, 14130}, {10226, 10263}, {10272, 22249}, {10296, 14644}, {10297, 23515}, {10540, 12308}, {10605, 14070}, {10606, 18534}, {10619, 12899}, {11202, 18445}, {11572, 13561}, {11645, 20126}, {12088, 14641}, {12106, 15030}, {12107, 13491}, {12112, 15054}, {12307, 15606}, {12367, 16010}, {13474, 18378}, {14002, 16261}, {14927, 18931}, {15040, 22115}, {15051, 23061}, {15139, 17835}, {15712, 31802}, {18325, 20127}, {18474, 18533}, {18559, 23293}, {18572, 20304}, {19506, 31726}, {20300, 23328}, {20417, 29012}, {23329, 31723}

X(32110) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 3581}, {23, 74}, {376, 15360}, {1533, 10990}, {3580, 10295}, {7464, 15107}, {12112, 15054}, {12367, 16010}, {15139, 17835}, {18325, 20127}
X(32110) = reflection of X(i) in X(j) for these lines: {i,j}: {113, 468}, {858, 6699}, {1495, 7575}, {1511, 18571}, {1531, 5}, {1533, 16619}, {3292, 1511}, {5642, 18579}, {10272, 22249}, {13857, 549}, {18323, 7687}, {18572, 20304}, {25641, 11657}
X(32110) = reflection of X(25641) in the Orthic axis
X(32110) = circumcircle-inverse of X(14687)
X(32110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 568, 11430}, {3, 9786, 569}, {3, 11438, 9730}, {24, 7689, 12162}, {26, 1204, 10575}, {182, 1350, 9967}, {1379, 1380, 14687}, {1620, 17834, 3}, {3515, 12163, 10539}, {5889, 21844, 12038}, {5890, 10298, 18475}, {6102, 15331, 13367}, {13349, 13350, 22463}, {15055, 15107, 7464}


X(32111) = MIDPOINT OF X(23) AND X(146)

Barycentrics    3*a^8*b^2 - 8*a^6*b^4 + 6*a^4*b^6 - b^10 + 3*a^8*c^2 + 10*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 10*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^6*c^4 - 6*a^4*b^2*c^4 + 20*a^2*b^4*c^4 - 2*b^6*c^4 + 6*a^4*c^6 - 10*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :
X(32111) = 2 X[125] - 3 X[403],5 X[125] - 3 X[13399],3 X[186] - X[12244],3 X[186] - 4 X[15448],3 X[376] - 2 X[20725],5 X[403] - 2 X[13399],2 X[1514] + X[12112],3 X[10540] - X[12121],X[10721] + 3 X[14157],X[12244] - 4 X[15448],X[12902] - 3 X[31726],X[13619] - 4 X[15152],3 X[14643] - 2 X[15122],3 X[14912] - 4 X[15471]

X(32111) lies on these lines: {2, 11472}, {4, 6}, {5, 7703}, {22, 4549}, {23, 146}, {24, 5878}, {30, 110}, {54, 13488}, {64, 7505}, {74, 468}, {113, 858}, {125, 403}, {186, 10117}, {235, 6241}, {376, 20725}, {381, 18911}, {399, 18325}, {427, 7699}, {511, 1533}, {541, 7426}, {550, 26882}, {1495, 2777}, {1499, 1513}, {1531, 29012}, {1539, 18323}, {1555, 2794}, {1594, 11381}, {1596, 5890}, {1614, 1885}, {1906, 3567}, {1995, 4846}, {3146, 22660}, {3147, 12250}, {3292, 6053}, {3357, 10018}, {3426, 5094}, {3520, 16252}, {3521, 31830}, {3542, 6225}, {3564, 14094}, {3580, 5663}, {3581, 16619}, {3628, 13623}, {4550, 7495}, {5133, 16194}, {5894, 21844}, {6001, 15904}, {6240, 22802}, {6247, 16868}, {6696, 14940}, {6759, 18560}, {6823, 15058}, {7464, 11064}, {7542, 15062}, {8718, 12362}, {9820, 12086}, {9919, 21284}, {10151, 25739}, {10257, 13445}, {10294, 13202}, {10564, 16534}, {10733, 12419}, {11440, 13383}, {11457, 12315}, {11585, 12279}, {11744, 15139}, {11745, 26863}, {11820, 31152}, {12174, 18912}, {12902, 31726}, {13367, 14862}, {13474, 15559}, {13596, 23292}, {13619, 15152}, {14389, 31861}, {14643, 15122}, {14677, 18571}, {15030, 24206}, {15305, 15760}, {15531, 21850}, {15761, 18439}, {18388, 32062}

X(32111) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 12112}, {23, 146}, {399, 18325}, {1533, 15063}, {11744, 15139}
X(32111) = reflection of X(i) in X(j) for these lines: {i,j}: {4, 1514}, {74, 468}, {858, 113}, {3292, 6053}, {3580, 11799}, {3581, 16619}, {7464, 11064}, {10295, 1495}, {10564, 16534}, {13445, 10257}, {14677, 18571}, {18323, 1539}, {25739, 10151}
X(32111) = reflection of X(477) in the Orthic axis
X(32111) = crossdifference of every pair of points on line {520, 5651}
X(32111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5656, 11456}, {4, 11456, 12022}, {4, 15032, 16657}, {235, 6241, 26879}, {1515, 6530, 4}, {5893, 16655, 4}, {22802, 26883, 6240}


X(32112) = CROSSDIFFERENCE OF X(30) AND X(2420)

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

The trilinear polar of X(32112) passes through X(3569) and the complement of X(877). (Randy Hutson, June 7, 2019)

X(32112) lies on the curve Q101 and these lines: {6, 647}, {74, 512}, {110, 250}, {113, 525}, {125, 523}, {232, 3569}, {262, 2394}, {264, 850}, {325, 6333}, {511, 684}, {526, 1495}, {924, 3447}, {2931, 8673}, {3425, 8723}, {3580, 9033}, {6530, 16230}, {8552, 10564}, {9139, 9213}, {11472, 30209}

X(32112) = reflection of X(10564) in X(8552)
X(32112) = X(i)-isoconjugate of X(j) for these (i,j): {248, 24001}, {293, 4240}, {336, 23347}, {1821, 2420}, {1910, 2407}, {2173, 2966}, {2715, 14206}
X(32112) = crossdifference of every pair of points on line {30, 2420}
X(32112) = intersection of Simson line of X(74) and trilinear polar of X(74)
X(32112) = barycentric product X(i)*X(j) for these lines: {i,j}: {74, 2799}, {297, 14380}, {325, 2433}, {511, 2394}, {684, 16080}, {1494, 3569}, {2421, 12079}, {6333, 8749}, {14919, 16230}
X(32112) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {74, 2966}, {232, 4240}, {237, 2420}, {240, 24001}, {511, 2407}, {684, 11064}, {2211, 23347}, {2394, 290}, {2433, 98}, {2491, 1495}, {2799, 3260}, {3569, 30}, {8430, 9214}, {8749, 685}, {14380, 287}, {14919, 17932}, {16080, 22456}, {17994, 1990}, {18808, 16081}


X(32113) = MIDPOINT OF X(23) AND X(69)

Barycentrics    3*a^6*b^2 - a^4*b^4 - 3*a^2*b^6 + b^8 + 3*a^6*c^2 - 4*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - a^4*c^4 + 3*a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 + c^8 : :
X(32113) = 3 X[186] - X[6776],3 X[403] - 2 X[5480],3 X[2070] + X[11898],3 X[2072] - 4 X[24206],7 X[3619] - 5 X[30745],5 X[3620] - X[5189],5 X[3763] - 4 X[5159],X[7464] - 3 X[10519],2 X[10297] - 3 X[10516],X[10989] - 3 X[21356],4 X[15448] - 3 X[18374]}

X(32113) lies on these lines: {6, 468}, {23, 69}, {30, 599}, {67, 74}, {110, 524}, {113, 511}, {125, 2393}, {141, 858}, {159, 21284}, {186, 6776}, {373, 16511}, {403, 5480}, {523, 3569}, {1316, 16321}, {1594, 12061}, {2070, 11898}, {2072, 9967}, {2452, 16324}, {2854, 3580}, {2930, 2931}, {3589, 15826}, {3619, 30745}, {3620, 5189}, {3763, 5159}, {3818, 18323}, {5654, 11477}, {7464, 10519}, {8547, 18911}, {8550, 11464}, {10117, 19596}, {10297, 10516}, {10510, 11064}, {10989, 21356}, {11179, 18579}, {11645, 16111}, {11663, 13371}, {15068, 16619}, {16776, 20113}, {17710, 26156}, {18358, 18572}, {31959, 31983}

X(32113) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 69}, {67, 12367}
X(32113) = reflection of X(i) in X(j) for these lines: {i,j}: {6, 468}, {858, 141}, {1316, 16321}, {2452, 16324}, {2453, 16334}, {3580, 8262}, {10510, 11064}, {11179, 18579}, {15826, 3589}, {18323, 3818}, {18572, 18358}
X(32113) = reflection of X(2453) in the Orthic axis
X(32113) = {X(8263),X(16789)}-harmonic conjugate of X(599)
X(32113) = crossdifference of every pair of points on line {182, 30209}


X(32114) = REFLECTION OF X(125) IN X(5181)

Barycentrics    4*a^6*b^2 - a^4*b^4 - 4*a^2*b^6 + b^8 + 4*a^6*c^2 - 14*a^4*b^2*c^2 + 8*a^2*b^4*c^2 - a^4*c^4 + 8*a^2*b^2*c^4 - 2*b^4*c^4 - 4*a^2*c^6 + c^8 : :
X(32114) = X[6] - 3 X[5642],X[6] - 3 X[5648],5 X[69] - 3 X[13169],3 X[110] - X[193],5 X[110] - 3 X[25321],3 X[113] - 2 X[21850],3 X[125] - 4 X[141],7 X[125] - 8 X[6698],5 X[125] - 4 X[25328],2 X[141] - 3 X[5181],7 X[141] - 6 X[6698],5 X[141] - 3 X[25328],2 X[193] - 3 X[5095],5 X[193] - 9 X[25321],3 X[895] - 5 X[3618],5 X[2930] - X[25336],5 X[3618] - 6 X[5972],5 X[3620] - 3 X[9140],2 X[3629] - 3 X[15303],5 X[5095] - 6 X[25321],7 X[5181] - 4 X[6698],5 X[5181] - 2 X[25328],10 X[6698] - 7 X[25328],4 X[6723] - 3 X[25320],3 X[9143] + X[20080],3 X[10519] - 2 X[20417],3 X[14912] - 5 X[15034],5 X[24981] - 2 X[25336]

X(32114) lies on these lines: {6, 5642}, {69, 74}, {110, 193}, {113, 1596}, {125, 126}, {159, 2930}, {511, 1533}, {524, 1495}, {526, 6131}, {895, 3618}, {1350, 10990}, {1351, 16534}, {2931, 19588}, {3448, 15812}, {3564, 30714}, {3569, 9033}, {3580, 8681}, {3620, 9140}, {3629, 15303}, {5486, 5651}, {5965, 25714}, {6053, 10752}, {6723, 25320}, {7699, 11188}, {7703, 12272}, {9143, 20080}, {10519, 20417}, {11472, 17702}, {11898, 23236}, {13202, 14982}, {14912, 15034}
X(32114) = midpoint of X(11898) and X(23236)
X(32114) = reflection of X(i) in X(j) for these lines: {i,j}: {125, 5181}, {895, 5972}, {1351, 16534}, {5095, 110}, {5642, 5648}, {10752, 6053}, {10990, 1350}, {13202, 14982}, {24981, 2930}
X(32114) = crossdifference of every pair of points on line {2780, 5622}






leftri  Incenters of not-equilateral central triangles: X(32115) - X(32118)  rightri

This preamble and centers X(32115)-X(32118) were contributed by CÚsar Eliud Lozada, April 19, 2019.

The appearance of (T, n) in the following list means that the incenter of triangle T-of-ABC is X(n), for any ABC:

(ABC-X3 reflections, 40), (anti-Aquila, 1), (anti-Ara, 1829), (1st anti-Brocard, 1281), (5th anti-Brocard, 12194), (6th anti-Brocard, 32115), (2nd anti-circumperp-tangential, 65), (anti-Euler, 944), (anti-inner-Grebe, 18992), (anti-outer-Grebe, 18991), (anti-Mandart-incircle, 3), (anti-orthocentroidal, 6126), (1st anti-orthosymmedial, 32116), (anticomplementary, 8), (Aquila, 1), (Ara, 9798), (Ascella, 12443), (Atik, 12450), (1st Auriga, 55), (2nd Auriga, 55), (1st Brocard, 3923), (5th Brocard, 9941), (6th Brocard, 32117), (2nd circumperp tangential, 3), (1st circumperp, 12518), (2nd circumperp, 12523), (inner-Conway, 11691), (Conway, 12539), (2nd Conway, 9807), (3rd Conway, 12554), (Ehrmann-mid, 18480), (Euler, 946), (3rd Euler, 12614), (4th Euler, 12622), (excenters-reflections, 12656), (excentral, 164), (2nd extouch, 12694), (Feuerbach, 5960), (inner-Garcia, 8), (outer-Garcia, 8), (Gossard, 12438), (inner-Grebe, 3641), (outer-Grebe, 3640), (hexyl, 12844), (Honsberger, 7670), (inner-Hutson, 12879), (Hutson intouch, 8422), (outer-Hutson, 12884), (incircle-circles, 12908), (intouch, 177), (inverse-in-incircle, 5571), (Johnson, 355), (inner-Johnson, 355), (outer-Johnson, 355), (1st Johnson-Yff, 5252), (2nd Johnson-Yff, 1837), (Lucas central, 1151), (Lucas homothetic, 12440), (Lucas inner tangential, 6407), (Lucas(-1) homothetic, 12441), (Lucas(-1) inner tangential, 6408), (Malfatti, 483), (Mandart-incircle, 3057), (medial, 10), (5th mixtilinear, 1), (6th mixtilinear, 167), (orthocentroidal, 5902), (1st orthosymmedial, 32118), (2nd Pamfilos-Zhou, 13090), (1st Parry, 9810), (2nd Parry, 9811), (1st Sharygin, 13091), (2nd Sharygin, 13301), (tangential-midarc, 1), (2nd tangential-midarc, 1), (3rd tri-squares-central, 8983), (4th tri-squares-central, 13971), (Ursa-major, 17657), (Ursa-minor, 17641), (Wasat, 21633), (X3-ABC reflections, 1482), (Yff central, 13092), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1), (2nd Zaniah, 18258)

For some triangles T-of-ABC, the incenter of T is a triangle center when ABC is acute but it is not when ABC is obtuse, as can be seen in the following table:

Triangle T-of-ABC Incenter of T-of-ABC when ABC is acute Incenter of T-of-ABC when ABC is obtuse and angles A ≤ B ≤ C (Barycentric coordinates)
anti-Ascella X(11414) -SB*SC*(SB+SC) : -SA*SC*(SC+SA) : (2*S^2+2*SC^2-SA*SB)*(SA+SB)
anti-Atik X(1593) 4*R^2-SA : 4*R^2-SB : -SC
1st anti-circumperp X(20) 1 : 1 : -1
anti-Conway X(578) SA*(SB+SC)^2 : SB*(SC+SA)^2 : (2*S^2-SA*SB+SC^2)*(SA+SB)
2nd anti-Conway X(389) (S^2+SB*SC)*(SB+SC) : (S^2+SA*SC)*(SC+SA) : -(SC^2-SA*SB)*(SA+SB)
3rd anti-Euler X(12111) (S^2-SA^2+2*SB*SC)*(SB+SC) : (S^2-SB^2+2*SA*SC)*(SC+SA) : -(3*S^2-2*SA*SB+5*SC^2)*(SA+SB)
4th anti-Euler X(6241) (SB+SC)*(2*S^4+SB*SC*(S^2-SA^2)) :
(SC+SA)*(2*S^4+SA*SC*(S^2-SB^2)) :
-(SA+SB)*(2*S^4-(SA*SB-4*SC^2)*S^2+SA*SB*SC^2)
anti-excenters-reflections X(4) SB*SC : SA*SC : SB*SA-S^2/2
2nd anti-extouch X(1181) S^2*(SB+SC) : S^2*(SC+SA) : (SA-SC)*S^2-(SB+SC)*SA^2
anti-Honsberger X(182) SB*(SB+SC)^2 : SA*(SC+SA)^2 : -(SA*SB+SW^2)*(SA+SB)
anti-Hutson intouch X(3) SB*(S^2+2*SA^2)*(SB+SC) : SA*(S^2+2*SB^2)*(SC+SA) : -S^2*(8*R^2-SW)*(SA+SB)
anti-incircle-circles X(3) SB*(2*S^2+SA^2)*(SB+SC) : SA*(2*S^2+SB^2)*(SC+SA) : -S^2*(4*R^2+SW)*(SA+SB)
anti-inverse-in-incircle X(4) SB*SW : SA*SW : -SW^2+(SB+SC)*(SA+SC)
6th anti-mixtilinear X(3) 1 : 1 : 0
1st anti-Sharygin X(8884) SB/(S^2+SB*SC) : SA/(S^2+SA*SC) : -S^2/(SC*S^2+SA*SB*SC)
anti-tangential-midarc X(1) (a^2-b^2+c^2)*(c+a-b)*(b-c+a)*a : (b^2+c^2-a^2)*(c-a+b)*(b-c+a)*b : -(c^2+a^2+2*a*b+b^2)*(c-a+b)*(c+a-b)*c
anti-Ursa minor X(5) (SA+SW)*SB : (SB+SW)*SA : S^2+SA*SB+SC^2-SW^2
anti-Wasat X(185) (S^2+SB*SC)*(SB+SC) : (S^2+SA*SC)*(SC+SA) : -SC*(SW+SC)*(SA+SB)
circumorthic X(4) SB*SC : SA*SC : SB*SA-2*S^2
Ehrmann-side X(3) S^2+3*SB*SC : S^2+3*SA*SC : -3*S^2+3*SA*SB
Ehrmann-vertex X(4) 4*SB*(3*R^2-SW)*S^2 : 4*SA*(3*R^2-SW)*S^2 : -S^4-(SC^2-SW^2)*S^2-3*SA^2*SB^2
2nd Ehrmann X(576) SB*(3*SA+SW)*(SB+SC) : SA*(3*SB+SW)*(SA+SC) : (3*SB*SA-SW^2)*(SA+SB)
2nd Euler X(3) S^2+SB*SC : S^2+SA*SC : -S^2+SA*SB
1st excosine X(1498) S^2*(SB+SC) : S^2*(SC+SA) : -(S^2+2*SC^2)*(SA+SB)
extangents X(40) (a^2-b^2+c^2)*(b+c+a)*a : (b^2+c^2-a^2)*(b+c+a)*b : -(c^3+(a+b)*c^2+(a+b)^2*c+(a^2-b^2)*(a-b))*c
1st Hyacinth X(12370) 4*SA*(S^4-(3*SB^2+4*SB*SC+7*SC^2)*S^2+SB^2*SC^2) :
4*SA*(S^4-(3*SC^2-4*SB*SC+7*SB^2)*S^2+SB^2*SC^2) :
-4*(SA+5*SB-3*SC)*S^4-4*(4*SA*SC-3*SB^2)*SA*S^2-4*(SB+SC)*SA^3*SB
intangents X(1) a*(c-a+b)*(a^2-b^2+c^2) : -(b-c-a)*(b^2+c^2-a^2)*b : (c-a-b)*(c^2+a^2-2*a*b+b^2)*c
1st Kenmotu diagonals X(371) SB*(S-SA)*(SB+SC) : SA*(S-SB)*(SC+SA) : -(S*SW+SA*SB)*(SA+SB)
2nd Kenmotu diagonals X(372) SB*(S+SA)*(SB+SC) : SA*(S+SB)*(SC+SA) : -(S*SW-SA*SB)*(SA+SB)
Kosnita X(3) SB*(S^2-SA^2)*(SB+SC) : SA*(S^2-SB^2)*(SC+SA) : 2*S^2*(2*R^2-SW)*(SA+SB)
orthic X(4) 0 : 0 : 1
submedial X(5) 1 : 1 : 2
tangential X(3) SB*(SB+SC) : SA*(SA+SC) : -SW*(SA+SB)
inner tri-equilateral X(15) (SA-sqrt(3)*S)*(SB+SC)*SB : (SB-sqrt(3)*S)*(SA+SC)*SA : (SA*SB+sqrt(3)*S*SW)*(SA+SB)
outer tri-equilateral X(16) (SA+sqrt(3)*S)*(SB+SC)*SB : (SB+sqrt(3)*S)*(SA+SC)*SA : (SA*SB-sqrt(3)*S*SW)*(SA+SB)

underbar

X(32115) = INCENTER OF THE 6th ANTI-BROCARD TRIANGLE

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

X(32115) lies on these lines: {10,98}, {58,99}, {83,11599}, {182,2783}, {239,1281}, {291,4586}, {519,8298}, {740,1691}, {2456,29057}, {2792,12177}, {2796,5182}, {3573,4368}, {4154,19561}, {4672,5038}, {5977,16825}, {5988,10352}, {5992,10353}, {10131,32117}, {10789,16834}, {12151,28558}, {13196,17768}, {13355,24728}, {27838,27950}

X(32115) = inverse of X(5150) in the 1st Lemoine circle


X(32116) = INCENTER OF THE 1st ANTI-ORTHOSYMMEDIAL TRIANGLE

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

X(32116) lies on these lines: {28,105}, {109,1297}, {251,32118}, {2794,15971}, {2831,3744}, {2844,9157}, {5078,10016}


X(32117) = INCENTER OF THE 6th BROCARD TRIANGLE

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

X(32117) lies on these lines: {1,7187}, {3,1281}, {8,3177}, {20,24280}, {55,17789}, {72,15310}, {75,20992}, {76,2795}, {194,740}, {304,3685}, {384,3923}, {516,17760}, {744,24696}, {956,29010}, {1975,5695}, {2783,11257}, {2792,9863}, {2796,7833}, {2896,4655}, {3705,25083}, {3821,7876}, {4019,21299}, {4201,9791}, {4451,25252}, {4672,7787}, {5025,5988}, {5992,6655}, {7081,7580}, {7750,17768}, {7791,24248}, {7893,17770}, {8680,24351}, {9873,29097}, {9939,28558}, {10131,32115}, {16895,24295}, {17135,25257}, {17688,24342}, {20065,24695}, {20432,23407}, {27549,32034}


X(32118) = INCENTER OF THE 1st ORTHOSYMMEDIAL TRIANGLE

Barycentrics    a*((b+c)*a^4+2*b*c*a^3+(b+c)*b*c*a^2-(b^2-c^2)*(b^3-c^3)) : :
X(32118) = 3*X(354)-2*X(4353) = 3*X(354)-X(12721) = X(3663)+2*X(21848) = 2*X(3821)-3*X(5883) = 3*X(5902)-X(24248) = 3*X(10176)-4*X(24295)

X(32118) lies on these lines: {1,572}, {3,25065}, {4,15314}, {21,16566}, {31,1726}, {33,57}, {46,24309}, {51,2844}, {58,3497}, {65,516}, {72,17355}, {226,12618}, {251,32116}, {354,4353}, {405,25081}, {517,12722}, {518,2321}, {527,24476}, {726,2901}, {758,3923}, {761,29044}, {942,3663}, {971,24471}, {984,3730}, {991,7146}, {1009,25078}, {1011,25080}, {1210,12610}, {1400,1736}, {1451,1825}, {1469,2801}, {1721,2955}, {1746,26000}, {1753,23052}, {1764,11031}, {1770,29050}, {1781,4223}, {1876,3668}, {2352,16577}, {2690,2711}, {2809,3751}, {2825,12294}, {2831,5480}, {2838,18907}, {3671,21629}, {3729,3868}, {3754,4660}, {3812,18252}, {3821,5883}, {4329,5807}, {4356,11997}, {4463,5294}, {5256,20243}, {5884,29057}, {5902,24248}, {8094,12728}, {9021,17351}, {9532,19161}, {10158,19744}, {10176,24295}, {10319,10382}, {10572,29024}, {11221,19732}, {11529,12717}, {12445,12727}, {12652,18421}, {17132,24473}, {18389,29069}

X(32118) = midpoint of X(i) and X(j) for these lines: {i,j}: {65, 12723}, {942, 21848}, {3729, 3868}
X(32118) = reflection of X(i) in X(j) for these (i,j): (72, 17355), (3663, 942), (4660, 3754), (12721, 4353), (18252, 3812)
X(32118) = {X(354), X(12721)}-harmonic conjugate of X(4353)


X(32119) = X(6)X(1637)∩X(74)X(98)

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

X(32119) lies lies on these lines: {6, 1637}, {74, 98}, {107, 110}, {113, 132}, {125, 526}, {520, 3580}, {523, 1495}, {542, 31953}, {684, 5972}, {2492, 6793}, {2777, 9409}, {2797, 16163}, {3569, 6103}, {7669, 10117}

X(32119) = reflection of X(i) in X(j) for these lines: {i,j}: {125, 6130}, {684, 5972}
X(32119) = crossdifference of every pair of points on line {1625, 3269}


X(32120) = X(6)X(523)∩X(74)X(1499)

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

X(32120) lies on these lines: {6, 523}, {74, 1499}, {110, 525}, {113, 30209}, {125, 512}, {468, 3569}, {520, 32114}, {526, 32112}, {690, 1495}, {858, 24284}, {2780, 32110}, {3566, 10117}, {3580, 9517}, {9033, 32113}, {14417, 18860}

X(32120) = reflection of X(i) in X(j) for these lines: {i,j}: {858, 24284}, {3569, 468}


X(32121) = X(6)X(14273)∩X(74)X(2374)

Barycentrics    (b^2 - c^2)*(2*a^12 - 4*a^10*b^2 + a^8*b^4 + 3*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 4*a^10*c^2 + 14*a^8*b^2*c^2 - 13*a^6*b^4*c^2 + 13*a^4*b^6*c^2 - 7*a^2*b^8*c^2 + b^10*c^2 + a^8*c^4 - 13*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 4*a^2*b^6*c^4 + 3*a^6*c^6 + 13*a^4*b^2*c^6 + 4*a^2*b^4*c^6 - 2*b^6*c^6 - 3*a^4*c^8 - 7*a^2*b^2*c^8 + a^2*c^10 + b^2*c^10) : :

X(32121) lies on these lines: {6, 14273}, {74, 2374}, {99, 110}, {113, 126}, {125, 5139}, {512, 3580}, {523, 32113}, {526, 6131}, {924, 32112}, {1495, 3566}, {1499, 32109}, {3569, 10418}, {7665, 9138}, {16316, 32111}, {20186, 32110}

X(32121) = reflection of X(32114) and X(6131)
X(32121) = crossdifference of every pair of points on line {3124, 14984}


X(32122) = X(6)X(9033)∩X(74)X(2373)

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

X(32121) lies on these lines: {6, 9033}, {74, 2373}, {110, 1289}, {125, 127}, {520, 32112}, {525, 1495}, {690, 10117}, {3580, 8673}, {8057, 32113}, {30209, 32110}


X(32123) = X(5)X(6)∩X(74)X(858)

Barycentrics    (a^2 - b^2 - c^2)*(a^12*b^2 - 2*a^10*b^4 - a^8*b^6 + 4*a^6*b^8 - a^4*b^10 - 2*a^2*b^12 + b^14 + a^12*c^2 - 2*a^10*b^2*c^2 + 7*a^8*b^4*c^2 - 14*a^6*b^6*c^2 + 5*a^4*b^8*c^2 + 8*a^2*b^10*c^2 - 5*b^12*c^2 - 2*a^10*c^4 + 7*a^8*b^2*c^4 + 12*a^6*b^4*c^4 - 4*a^4*b^6*c^4 - 14*a^2*b^8*c^4 + 9*b^10*c^4 - a^8*c^6 - 14*a^6*b^2*c^6 - 4*a^4*b^4*c^6 + 16*a^2*b^6*c^6 - 5*b^8*c^6 + 4*a^6*c^8 + 5*a^4*b^2*c^8 - 14*a^2*b^4*c^8 - 5*b^6*c^8 - a^4*c^10 + 8*a^2*b^2*c^10 + 9*b^4*c^10 - 2*a^2*c^12 - 5*b^2*c^12 + c^14) : :
X(32123) = 5 X[5] - 4 X[15090],2 X[15123] - 3 X[23515]

X(32123) lies on these lines: {4, 9938}, {5, 6}, {23, 12319}, {24, 22661}, {30, 10117}, {74, 858}, {110, 403}, {125, 1568}, {235, 12293}, {427, 11472}, {468, 2931}, {912, 15904}, {1147, 10024}, {1368, 4549}, {1495, 11799}, {1503, 15133}, {1594, 7703}, {3167, 10254}, {5133, 7699}, {5448, 5576}, {6193, 16868}, {6622, 12318}, {7505, 9932}, {9820, 13160}, {10255, 12164}, {10564, 15115}, {11064, 15136}, {11459, 12359}, {11585, 12163}, {12118, 15761}, {14984, 32112}, {15123, 23515}, {18323, 19479}, {18404, 19908}, {20427, 23335}

X(32123) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 12319}, {68, 12364}
X(32123) = reflection of X(i) in X(j) for these lines: {i,j}: {858, 23306}, {2931, 468}, {10564, 15115}, {15136, 11064}, {18323, 19479}
X(32123) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {68, 5654, 15068}, {235, 23307, 12293}


X(32124) = X(6)X(23)∩X(74)X(7575)

Barycentrics    a^2*(8*a^10 - 10*a^8*b^2 - 12*a^6*b^4 + 16*a^4*b^6 + 4*a^2*b^8 - 6*b^10 - 10*a^8*c^2 + 24*a^6*b^2*c^2 - 11*a^4*b^4*c^2 - 14*a^2*b^6*c^2 + 11*b^8*c^2 - 12*a^6*c^4 - 11*a^4*b^2*c^4 + 22*a^2*b^4*c^4 - 5*b^6*c^4 + 16*a^4*c^6 - 14*a^2*b^2*c^6 - 5*b^4*c^6 + 4*a^2*c^8 + 11*b^2*c^8 - 6*c^10) : :
X(32124) = 3 X[403] - 2 X[15432]

X(32124) lies on these lines: {6, 23}, {30, 7699}, {74, 7575}, {113, 10295}, {125, 7426}, {186, 11472}, {403, 15432}, {468, 7703}, {10546, 18571}

X(32124) = midpoint of X(23) and X(7712)
X(32124) = reflection of X(7703) and X(468)


X(32125) = X(6)X(66)∩X(74)X(403)

Barycentrics    a^10*b^2 - a^8*b^4 - 2*a^6*b^6 + 2*a^4*b^8 + a^2*b^10 - b^12 + a^10*c^2 + 2*a^6*b^4*c^2 + 2*a^4*b^6*c^2 - 7*a^2*b^8*c^2 + 2*b^10*c^2 - a^8*c^4 + 2*a^6*b^2*c^4 - 8*a^4*b^4*c^4 + 6*a^2*b^6*c^4 + b^8*c^4 - 2*a^6*c^6 + 2*a^4*b^2*c^6 + 6*a^2*b^4*c^6 - 4*b^6*c^6 + 2*a^4*c^8 - 7*a^2*b^2*c^8 + b^4*c^8 + a^2*c^10 + 2*b^2*c^10 - c^12 : :
X(32125) = 2 X[1514] + X[12379], 2 X[11064] - 3 X[15131]

X(32125) lies on these lines: {5, 64}, {6, 66}, {23, 13203}, {30, 2931}, {74, 403}, {110, 858}, {113, 2072}, {125, 15126}, {154, 1368}, {159, 31152}, {161, 1370}, {235, 1192}, {468, 10117}, {1498, 11585}, {1594, 6241}, {1596, 31860}, {1619, 30771}, {2393, 32114}, {2450, 3566}, {2777, 11799}, {2781, 3580}, {2883, 15072}, {3357, 10024}, {3564, 17847}, {5133, 7703}, {5169, 20300}, {5576, 9730}, {5654, 13371}, {5894, 11454}, {6696, 13160}, {6823, 8567}, {7396, 28419}, {7495, 15578}, {9932, 14790}, {10192, 15080}, {10255, 13093}, {10564, 18400}, {10606, 15760}, {11206, 31101}, {12118, 23335}, {12250, 16868}, {13352, 18381}, {15138, 15142}, {15472, 25739}, {15559, 16657}, {15577, 16063}, {15761, 20427}, {17824, 18914}, {18323, 19506}, {18382, 31133}, {31074, 32064}

X(32125) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 13203}, {11744, 12379}
X(32125) = reflection of X(i) in X(j) for these lines: {i,j}: {125, 15126}, {858, 23315}, {10117, 468}, {11744, 1514}, {18323, 19506}
X(32125) = crosspoint of X(16080) and X(18018)
X(32125) = crosssum of X(206) and X(3284)
X(32125) = crossdifference of every pair of points on line {8673, 9306}


X(32126) = X(6)X(5089)∩X(74)X(2766)

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

X(32125) lies on these lines: {6, 5089}, {40, 5692}, {72, 20989}, {74, 2766}, {110, 518}, {113, 517}, {125, 3827}, {468, 15904}, {650, 3569}, {910, 21873}, {912, 2931}, {2771, 32109}, {2778, 32110}, {2836, 3580}, {3740, 7703}, {7686, 7699}, {9004, 32114}

X(32125) = reflection of X(15904) and X(468)


X(32127) = REFLECTION OF X(3292) IN X(6)

Barycentrics    a^2*(2*a^6 - 3*a^4*b^2 - 2*a^2*b^4 + 3*b^6 - 3*a^4*c^2 + 16*a^2*b^2*c^2 - 7*b^4*c^2 - 2*a^2*c^4 - 7*b^2*c^4 + 3*c^6) : :
X(32127) = 2 X[1350] - 3 X[21663]

X(32127) lies on these lines: {6, 373}, {74, 511}, {110, 8681}, {113, 3564}, {125, 524}, {193, 7703}, {520, 2451}, {538, 2452}, {576, 15030}, {1350, 10602}, {1351, 1597}, {1352, 1992}, {1495, 2854}, {1974, 6391}, {2030, 10765}, {2080, 22143}, {2393, 10117}, {2987, 9190}, {3003, 9145}, {6467, 8547}, {8548, 13352}, {9004, 15904}, {11003, 15531}, {18860, 20975}

X(32127) = reflection of X(3292) in X(6)
X(32127) = {X(6),X(8542)}-harmonic conjugate of X(373)
X(32127) = crossdifference of every pair of points on line {1499, 6776}


X(32128) = X(3)X(8)∩X(4)X(93)

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

See J.L. Ayme and Ercole Suppa, Hyacinthos 28984.

X(32128) lies on these lines: {3,8}, {4,93}, {498,2594}, {6928,10449}, {10441,10526}


X(32129) = CENTROID OF GEMINI TRIANGLE 104

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

X(32129) lies on these lines: {1, 32044}, {2, 3761}, {1978, 4664}, {4364, 30629}, {17237, 25378}, {26738, 30990}, {30967, 30991}


X(32130) = (name pending)

Barycentrics    (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (a^4-16 a^2 b^2+7 b^4+14 a^2 c^2-16 b^2 c^2+c^4) (a^4+14 a^2 b^2+b^4-16 a^2 c^2-16 b^2 c^2+7 c^4) : :
Barycentrics    (108 R^2 S^2+18 S^2 SB-21 S^2 SW-4 SW^3) (108 R^2 S^2+18 S^2 SC-21 S^2 SW-4 SW^3) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28987.

X(32130) lies on Kiepert hyperbola and this line: {4,17952}

X(32130) = isogonal conjugate of X(32131)
X(32130) = barycentric quotient of X(i) and X(j) for these lines: {i,j}: {5485, 9741}


X(32131) = (name pending)

Barycentrics    a^2 (5 a^2-b^2-c^2) (7 a^4-16 a^2 b^2+b^4-16 a^2 c^2+14 b^2 c^2+c^4) : :
Barycentrics    18 S^4 + (108 R^2 SB+108 R^2 SC-18 SB SC-21 SB SW-21 SC SW)S^2 -4 SB SW^3-4 SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28987.

X(32131) lies on this line: {3,6}

X(32131) = isogonal conjugate of X(32130)
X(32131) = barycentric product of X(i) and X(j) for these lines: {i,j}: {1384, 9741}


X(32132) = X(2)X(254)∩X(3)X(2165)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 28987.

Let A'B'C' be the Schroeter triangle. Let P be a point on the nine-point circle. Let A" be the intersection, other than P, of the nine-point circle and line A'P; define B" and C" cyclically. Triangle A"B"C" is perspective to ABC for all P. If P = X(135), the perspector is X(32132). (Randy Hutson, June 7, 2019)

X(32132) lies on these lines: {2,254}, {3,2165}, {24,16172}, {68,394}, {96,97}, {925,3147}, {3546,5392}, {3926,20563}, {6642,14593}

X(32132) = barycentric product of X(i) and X(j) for these lines: {i,j}: {68,6504}, {5392,15316)
X(32132) = barycentric quotient of X(i) and X(j) for these lines: {i,j}: {68,6515}, {115,135}, {254,11547}, {921,1748}, {1820,920}, {2165,3542}, {2351,1609}, {6504,317}, {8800,467}, {15316,1993}, {16391,6503}
X(32132) = trilinear product of X(i) and X(j) for these lines: {i,j}: {68, 921}, {68, 921}, {91, 15316}, {1820, 6504}, {1820, 6504}
X(32132) = trilinear quotient of X(i) and X(j) for these lines: {i,j}: {68,920}, {91,3542}


X(32133) = X(2)X(12505)∩X(67)X(10354)

Barycentrics    (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (a^4-4 a^2 b^2+b^4-c^4) (a^4-b^4-4 a^2 c^2+c^4) : :
Barycentrics    (648 R^4-162 R^2 SW+9 SW^2)S^4 + (-54 R^2 SB SC SW+9 SB SC SW^2-24 R^2 SW^3-3 SB SW^3-3 SC SW^3+6 SW^4)S^2 + 2 SB SC SW^4 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28988.

X(32133) lies on these lines: {2,12505}, {67,10354}, {599,5486}, {858,14262}, {1296,16063}, {5094,21448}, {5485,16051}

X(32133) = barycentric product of X(i) and X(j) for these lines: {i,j}: {338,15406}, {5485,5486}
X(32133) = barycentric quotient of X(i) and X(j) for these lines: {i,j}: {115,5512}, {5485,11185}, {5486,1992}, {15406,249}, {21448,1995}
X(32133) = trilinear product of X(i) and X(j) for these lines: {i,j}: {1109,15406}, {1109,15406}
X(32133) = trilinear quotient of X(i) and X(j) for these lines: {i,j}: {1109, 5512}






leftri  X(5)-of-central (not equilateral) triangles: X(32134) - X(32215)  rightri

This preamble and centers X(32134)-X(32215) were contributed by CÚsar Eliud Lozada, April 23, 2019.

The appearance of (T, n) in the following list means that X(5)-of-triangle-T is X(n):

(ABC, 5), (ABC-X3 reflections, 550), (anti-Aquila, 5901), (anti-Ara, 6756), (anti-Artzt, 597), (anti-Ascella, 12161), (anti-Atik, 18952), (1st anti-Brocard, 114), (4th anti-Brocard, 14650), (5th anti-Brocard, 32134), (6th anti-Brocard, 32135), (2nd anti-circumperp-tangential, 18990), (1st anti-circumperp, 6101), (anti-Conway, 32136), (2nd anti-Conway, 10095), (anti-Euler, 3), (3rd anti-Euler, 3), (4th anti-Euler, 3), (anti-excenters-reflections, 32137), (2nd anti-extouch, 32046), (anti-inner-Grebe, 19116), (anti-outer-Grebe, 19117), (anti-Honsberger, 19155), (anti-Hutson intouch, 32138), (anti-incircle-circles, 32139), (anti-inverse-in-incircle, 32140), (anti-Mandart-incircle, 32141), (anti-McCay, 2482), (6th anti-mixtilinear, 32142), (anti-orthocentroidal, 1511), (1st anti-orthosymmedial, 19165), (1st anti-Sharygin, 19211), (anti-tangential-midarc, 32143), (anti-Ursa minor, 13561), (anti-Wasat, 5), (antiAOA, 13371), (anticomplementary, 3), (AOA, 32144), (Aquila, 355), (Ara, 26), (Aries, 32145), (Artzt, 9771), (Ascella, 9940), (Atik, 9947), (1st Auriga, 32146), (2nd Auriga, 32147), (Ayme, 32148), (1st Brocard-reflected, 32149), (1st Brocard, 24206), (2nd Brocard, 32150), (4th Brocard, 31840), (5th Brocard, 32151), (6th Brocard, 32152), (circummedial, 31744), (circumorthic, 6102), (2nd circumperp tangential, 32153), (1st circumperp, 3579), (2nd circumperp, 1385), (circumsymmedial, 31727), (inner-Conway, 72), (Conway, 1071), (2nd Conway, 4), (3rd Conway, 10441), (Ehrmann-cross, 30), (Ehrmann-mid, 546), (Ehrmann-side, 5876), (Ehrmann-vertex, 18379), (1st Ehrmann, 32154), (2nd Ehrmann, 32155), (Euler, 546), (2nd Euler, 11591), (3rd Euler, 9955), (4th Euler, 9956), (5th Euler, 32156), (excenters-midpoints, 32157), (excenters-reflections, 1482), (excentral, 3), (1st excosine, 26), (extangents, 32158), (extouch, 32159), (2nd extouch, 5777), (3rd extouch, 32160), (inner-Fermat, 18), (outer-Fermat, 17), (Feuerbach, 32161), (Fuhrmann, 5), (2nd Fuhrmann, 16160), (inner-Garcia, 5694), (outer-Garcia, 5690), (Garcia-reflection, 13463), (Gossard, 32162), (inner-Grebe, 5875), (outer-Grebe, 5874), (3rd Hatzipolakis, 32163), (hexyl, 3), (Honsberger, 5728), (Hutson extouch, 32164), (inner-Hutson, 12488), (Hutson intouch, 9957), (outer-Hutson, 12489), (1st Hyacinth, 32165), (2nd Hyacinth, 32166), (incentral, 32167), (incircle-circles, 5045), (intangents, 32168), (intouch, 942), (inverse-in-incircle, 5045), (Johnson, 5), (inner-Johnson, 10943), (outer-Johnson, 10942), (1st Johnson-Yff, 495), (2nd Johnson-Yff, 496), (K798e, 10021), (K798i, 140), (1st Kenmotu diagonals, 32169), (2nd Kenmotu diagonals, 32170), (Kosnita, 32171), (Lemoine, 32172), (Lucas antipodal, 32173), (Lucas central, 32175), (Lucas homothetic, 32177), (Lucas tangents, 32179), (Lucas(-1) antipodal, 32174), (Lucas(-1) central, 32176), (Lucas(-1) homothetic, 32178), (Lucas(-1) tangents, 32180), (Macbeath, 32181), (Mandart-excircles, 32182), (Mandart-incircle, 15171), (McCay, 7619), (medial, 140), (midarc, 12908), (2nd midarc, 32183), (midheight, 32184), (mixtilinear, 32185), (2nd mixtilinear, 32186), (3rd mixtilinear, 32187), (4th mixtilinear, 32188), (5th mixtilinear, 1483), (6th mixtilinear, 40), (1st Neuberg, 32189), (2nd Neuberg, 32190), (orthic, 143), (orthocentroidal, 5946), (1st orthosymmedial, 32191), (2nd orthosymmedial, 32192), (2nd Pamfilos-Zhou, 12490), (1st Parry, 14610), (2nd Parry, 32193), (3rd Parry, 32194), (Pelletier, 32195), (reflection, 32196), (1st Schiffler, 32197), (2nd Schiffler, 32198), (Schroeter, 10280), (1st Sharygin, 9959), (2nd Sharygin, 9508), (Soddy, 32199), (2nd inner-Soddy, 32200), (2nd outer-Soddy, 32201), (inner-squares, 32202), (outer-squares, 32203), (Steiner, 32204), (submedial, 32205), (symmedial, 32206), (tangential, 156), (tangential-midarc, 8099), (2nd tangential-midarc, 8100), (inner tri-equilateral, 32207), (outer tri-equilateral, 32208), (1st tri-squares-central, 13721), (2nd tri-squares-central, 13844), (3rd tri-squares-central, 13925), (4th tri-squares-central, 13993), (1st tri-squares, 13664), (2nd tri-squares, 13784), (3rd tri-squares, 13924), (4th tri-squares, 32209), (Trinh, 32210), (Ursa-major, 355), (Ursa-minor, 1), (inner-Vecten, 6119), (outer-Vecten, 6118), (Wasat, 5), (X-parabola-tangential, 32211), (X3-ABC reflections, 4), (Yff central, 12491), (Yff contact, 32212), (inner-Yff, 495), (outer-Yff, 496), (inner-Yff tangents, 32213), (outer-Yff tangents, 32214), (Yiu, 32215), (1st Zaniah, 18260), (2nd Zaniah, 5044)

underbar

X(32134) = X(5)-OF-5th ANTI-BROCARD TRIANGLE

Barycentrics    2*a^8-4*(b^2+c^2)*a^6+(-4*b^2*c^2+(b^2-c^2)^2)*a^4+(b^2-c^2)^2*b^2*c^2+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2 : :
X(32134) = X(384)+3*X(22521) = X(3398)-3*X(12150) = X(12110)+3*X(12150)

X(32134) lies on these lines: {3,3329}, {4,11842}, {5,32}, {26,10790}, {30,3398}, {83,140}, {98,546}, {182,550}, {187,11272}, {355,10789}, {384,22521}, {495,10797}, {496,10798}, {548,12054}, {549,5171}, {632,7808}, {952,12194}, {1078,3628}, {1353,5039}, {1483,10800}, {1506,10631}, {1656,7793}, {1691,18583}, {2782,5007}, {3095,3972}, {3564,12193}, {3627,14880}, {4027,19687}, {5008,6248}, {5097,7816}, {5188,15870}, {5690,10791}, {5844,12195}, {5874,10793}, {5875,10792}, {5901,11364}, {5999,13111}, {6179,7697}, {6249,10991}, {6661,10333}, {6756,11380}, {7766,13108}, {7781,15520}, {7819,10350}, {7830,25555}, {7839,13188}, {7858,15561}, {7878,11171}, {8369,10349}, {8722,15712}, {9301,10346}, {9821,10348}, {10794,10943}, {10795,10942}, {10799,15171}, {10803,32213}, {10804,32214}, {11490,32141}, {11837,32146}, {11838,32147}, {11839,32162}, {11840,32177}, {11841,32178}, {12176,12206}, {12197,28174}, {12201,12208}, {12204,22523}, {12205,22522}, {12835,18990}, {13335,14881}, {13885,13925}, {13938,13993}, {18993,19116}, {18994,19117}, {22330,32135}, {22520,32153}

X(32134) = midpoint of X(i) and X(j) for these lines: {i,j}: {3398, 12110}, {12201, 13193}
X(32134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7787, 10788, 3), (11842, 18501, 4), (12210, 12211, 12212)


X(32135) = X(5)-OF-6th ANTI-BROCARD TRIANGLE

Barycentrics    2*a^10-6*(b^2+c^2)*a^8+6*(b^4+c^4)*a^6-(b^2+c^2)*(3*b^4-8*b^2*c^2+3*c^4)*a^4+(b^4+c^4)*(b^4-5*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(32135) = 3*X(6)+X(13188) = X(98)-3*X(182) = X(98)-9*X(5182) = X(98)+3*X(12177) = X(114)+3*X(18800) = 5*X(114)-3*X(25562) = X(147)+3*X(11179) = X(182)-3*X(5182) = X(2080)+3*X(12151) = 3*X(5055)+X(10488) = 3*X(5182)+X(12177) = 3*X(5476)-X(6321) = 2*X(6036)-3*X(10168) = 4*X(6721)-3*X(24206) = X(10754)-3*X(15520) = X(11676)+3*X(22525) = X(13172)+3*X(20423) = 5*X(18800)+X(25562) = 5*X(22234)+X(23235)

X(32135) lies on these lines: {2,98}, {6,1569}, {99,576}, {115,5038}, {262,8289}, {511,5026}, {543,10796}, {575,2782}, {690,25556}, {1691,5477}, {2030,12829}, {2080,2482}, {2456,29012}, {3098,10753}, {3398,14981}, {5055,10488}, {5095,14590}, {5097,5969}, {5476,6321}, {7697,10485}, {7787,22234}, {7808,20398}, {8590,8787}, {8724,11842}, {10104,20399}, {10131,32152}, {10351,11257}, {10754,15520}, {10991,12054}, {10992,12110}, {11645,22505}, {11676,22525}, {12042,20190}, {13172,20423}, {14159,25561}, {19120,19130}, {22330,32134}

X(32135) = midpoint of X(i) and X(j) for these lines: {i,j}: {99, 576}, {182, 12177}, {3098, 10753}, {8593, 11178}
X(32135) = reflection of X(i) in X(j) for these (i,j): (115, 25555), (12042, 20190)
X(32135) = {X(5182), X(12177)}-harmonic conjugate of X(182)


X(32136) = X(5)-OF-ANTI-CONWAY TRIANGLE

Barycentrics    a^2*(2*a^8-7*(b^2+c^2)*a^6+3*(3*b^4+2*b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(5*b^4-9*b^2*c^2+5*c^4)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32136) = X(3)-9*X(11402) = 5*X(3)+3*X(12160) = X(3)+3*X(12161) = X(3)-3*X(32046) = X(3529)+3*X(31815) = X(3627)+3*X(31804) = 15*X(11402)+X(12160) = 3*X(11402)+X(12161) = 3*X(11402)-X(32046) = 5*X(11482)+3*X(19459) = X(12160)-5*X(12161) = X(12160)+5*X(32046) = X(15704)+3*X(31802) = X(24680)+3*X(31811)

X(32136) lies on these lines: {3,54}, {5,11264}, {6,156}, {22,13421}, {26,17809}, {49,1199}, {110,15026}, {143,184}, {155,14128}, {182,32142}, {265,3091}, {389,32171}, {546,12241}, {567,5876}, {568,5944}, {569,11591}, {575,3564}, {576,11536}, {578,5663}, {632,3292}, {1147,12006}, {1173,7545}, {1511,9545}, {1614,14627}, {1994,10263}, {2393,22330}, {2914,15101}, {2916,11477}, {3448,11702}, {3525,9716}, {3529,31815}, {3567,9704}, {3627,21659}, {5893,12102}, {6243,11003}, {6593,22234}, {8254,21243}, {8550,13371}, {9306,32205}, {9703,15043}, {9826,13358}, {10116,12242}, {10226,13382}, {10282,16881}, {10539,13364}, {11424,32137}, {11425,32138}, {11426,32139}, {11427,32140}, {11428,32158}, {11429,32168}, {11430,32210}, {11482,19459}, {11565,18569}, {11750,20424}, {12086,13491}, {12107,16625}, {12233,30522}, {13321,26882}, {13353,15067}, {13434,15060}, {13561,23292}, {14912,18952}, {15704,31802}, {17710,19150}, {18912,20304}, {19365,32143}, {24680,31811}

X(32136) = midpoint of X(12161) and X(32046)
X(32136) = X(140)-of-2nd anti-extouch triangle
X(32136) = X(3628)-of-anti-Ascella triangle
X(32136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11422, 1493), (195, 5012, 6101), (11422, 11423, 3)


X(32137) = X(5)-OF-ANTI-EXCENTERS-REFLECTIONS TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^6-3*(b^2-c^2)^2*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4+7*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32137) = 3*X(2)-4*X(11017) = X(3)-5*X(11439) = X(3)+3*X(11455) = 7*X(4)-3*X(568) = 3*X(4)-X(6102) = 17*X(4)-9*X(11002) = 3*X(4)+X(18439) = 3*X(5)-X(10575) = X(5)-3*X(16194) = X(20)-3*X(15060) = 7*X(143)-6*X(568) = 3*X(143)-2*X(6102) = 17*X(143)-18*X(11002) = 3*X(143)+2*X(18439) = 9*X(568)-7*X(6102) = 9*X(568)+7*X(18439) = X(10575)+3*X(11381) = X(10575)-9*X(16194) = X(11381)+3*X(16194) = 5*X(11439)+3*X(11455) = 3*X(15060)-2*X(32142)

X(32137) lies on these lines: {2,11017}, {3,11439}, {4,94}, {5,10575}, {20,15060}, {24,32210}, {25,32138}, {26,11472}, {30,1216}, {33,32143}, {34,32168}, {49,13596}, {52,15687}, {64,13861}, {140,14915}, {156,1593}, {185,3845}, {195,14094}, {235,13561}, {376,11592}, {378,32171}, {381,10574}, {382,5876}, {389,3861}, {546,5462}, {550,14128}, {632,14855}, {1154,3627}, {1209,1533}, {1498,31861}, {1597,32139}, {1656,12279}, {1657,15058}, {1885,30522}, {2070,15062}, {3091,32205}, {3146,6101}, {3357,12106}, {3426,6642}, {3530,14641}, {3534,15056}, {3543,18436}, {3545,12046}, {3567,14269}, {3830,10263}, {3832,15026}, {3843,5946}, {3850,13363}, {3851,11465}, {3853,13754}, {3856,5943}, {3858,9730}, {4550,7525}, {4846,16620}, {5066,9729}, {5073,11459}, {5076,5889}, {5079,20791}, {5446,12102}, {5576,32111}, {5878,11818}, {5891,15704}, {5892,12811}, {5944,14130}, {6146,15807}, {6243,17578}, {7527,10610}, {7999,15681}, {10096,20191}, {10110,14893}, {10540,14865}, {11250,20773}, {11403,12161}, {11424,32136}, {11440,18378}, {11444,17800}, {11470,32155}, {11471,32158}, {11473,32169}, {11474,32170}, {11475,32207}, {11476,32208}, {11793,12103}, {11819,16654}, {12086,18350}, {12324,18952}, {12358,18565}, {12363,16658}, {13368,22584}, {13421,13598}, {13565,15760}, {15061,21451}, {15311,31830}, {17814,22334}, {18570,26883}, {19124,19155}, {19169,19211}, {20299,20304}, {20396,23294}, {28154,31752}, {28168,31751}

X(32137) = midpoint of X(i) and X(j) for these lines: {i,j}: {5, 11381}, {382, 5876}, {1539, 12292}, {3146, 6101}, {3627, 12162}, {6102, 18439}, {10263, 12111}, {12290, 13491}
X(32137) = reflection of X(i) in X(j) for these (i,j): (20, 32142), (143, 4), (185, 10095), (389, 3861), (550, 14128), (5446, 12102), (5889, 16982), (6146, 15807), (10627, 5907), (12103, 11793), (13421, 13598), (13491, 12006), (13630, 546), (14641, 3530)
X(32137) = anticomplement of the anticomplement of X(11017)
X(32137) = X(13491)-of-Ehrmann-mid triangle
X(32137) = X(32138)-of-anti-Ara triangle
X(32137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18439, 6102), (11439, 11455, 3)


X(32138) = X(5)-OF-ANTI-HUTSON INTOUCH TRIANGLE

Barycentrics    a^2*(a^8-(b^2+c^2)*a^6-3*(b^2-c^2)^2*a^4+(b^2+c^2)*(5*b^4-9*b^2*c^2+5*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :
X(32138) = 3*X(3)-X(32139) = 3*X(3)-2*X(32171) = X(155)-5*X(8567) = 3*X(156)-2*X(32139) = 3*X(156)-4*X(32171) = X(156)-4*X(32210) = X(1147)-3*X(11204) = X(1498)-3*X(18324) = 3*X(5085)-2*X(19155) = X(5878)-3*X(10201) = X(5925)+3*X(14852) = 2*X(10224)-3*X(23329) = 2*X(10226)-3*X(11204) = 3*X(10606)-X(12084) = 3*X(10606)+X(12163) = X(13093)+3*X(14070) = 2*X(18567)-3*X(23325) = X(22660)-3*X(23328) = 3*X(23328)-2*X(23336) = X(32139)-6*X(32210) = X(32171)-3*X(32210)

X(32138) lies on these lines: {3,74}, {4,13561}, {5,1204}, {20,30522}, {25,32137}, {26,64}, {30,3357}, {55,32143}, {56,32168}, {143,1593}, {146,14940}, {155,8567}, {185,18570}, {186,18439}, {378,6102}, {381,11704}, {382,18379}, {511,15579}, {541,20191}, {546,11438}, {568,14865}, {1147,10226}, {1151,32169}, {1152,32170}, {1154,10606}, {1192,11472}, {1498,18324}, {1657,13445}, {1658,6000}, {2070,12290}, {2071,18436}, {2777,5449}, {2883,10020}, {2904,13148}, {2937,12279}, {3146,3581}, {3516,12161}, {3532,17814}, {3534,7691}, {3628,4550}, {3851,15053}, {5020,11017}, {5085,19155}, {5448,25563}, {5584,32158}, {5878,10201}, {5890,14130}, {5895,15114}, {5925,14852}, {6101,11413}, {6243,12086}, {6696,13371}, {6759,15331}, {7502,10575}, {7526,10605}, {7575,26883}, {7728,16868}, {9544,23040}, {9786,10095}, {9818,12006}, {9833,15138}, {10224,23329}, {10539,15646}, {10540,21844}, {11250,13754}, {11264,18917}, {11270,18350}, {11381,32110}, {11412,18859}, {11425,32136}, {11455,18378}, {11477,32155}, {11479,32205}, {11480,32207}, {11481,32208}, {11750,13399}, {12085,13391}, {12162,21663}, {13093,14070}, {13406,22802}, {14915,17714}, {15060,17928}, {15311,15761}, {15704,20725}, {16003,21659}, {17702,18356}, {18377,20299}, {18403,23294}, {18435,22467}, {18562,25739}, {18567,23325}, {18913,18952}, {19172,19211}, {22660,23328}

X(32138) = midpoint of X(i) and X(j) for these lines: {i,j}: {20, 32140}, {26, 64}, {3357, 7689}, {5894, 12359}, {12084, 12163}, {15054, 15132}
X(32138) = reflection of X(i) in X(j) for these (i,j): (3, 32210), (4, 13561), (156, 3), (382, 18379), (1147, 10226), (2883, 10020), (5448, 25563), (6759, 15331), (11477, 32155), (13371, 6696), (18377, 20299), (22660, 23336), (22802, 13406), (32139, 32171)
X(32138) = X(156)-of-ABC-X3 reflections triangle
X(32138) = X(13561)-of-anti-Euler triangle
X(32138) = X(32137)-of-Ara triangle
X(32138) = X(32143)-of-anti-Mandart-incircle triangle
X(32138) = X(32168)-of-2nd circumperp tangential triangle
X(32138) = X(32210)-of-X3-ABC reflections triangle
X(32138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11456, 5944), (11468, 12111, 3), (32139, 32171, 156)


X(32139) = X(5)-OF-ANTI-INCIRCLE-CIRCLES TRIANGLE

Barycentrics    a^2*(a^8-4*(b^2+c^2)*a^6+6*(b^4+c^4)*a^4-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2) : :
Trilinears    3 cos A - 2 cos B cos C - 4 sin B sin C cos^2 A : :
X(32139) = 3*X(3)-2*X(32138) = 3*X(3)-4*X(32171) = 5*X(3)-4*X(32210) = 3*X(154)-2*X(1658) = 3*X(154)-X(12163) = 3*X(156)-X(32138) = 3*X(156)-2*X(32171) = 5*X(156)-2*X(32210) = 2*X(1498)+X(16266) = 5*X(1656)-4*X(13561) = 3*X(1853)-4*X(10224) = 3*X(3167)-X(12085) = 3*X(3167)+X(12315) = 5*X(3843)-4*X(18379) = 3*X(5050)-4*X(19155) = X(7387)-3*X(32063) = X(9925)+2*X(9968) = X(12164)+3*X(32063) = 5*X(32138)-6*X(32210) = 5*X(32171)-3*X(32210)

Let A'B'C' be the Kosnita triangle. Let A" be the circumcenter of triangle AB'C', and define B" and C" cyclically. Then X(32139) = X(20)-of-A"B"C". (Randy Hutson, June 7, 2019)

X(32139) lies on these lines: {3,74}, {4,1994}, {5,1181}, {6,546}, {22,18436}, {24,10540}, {25,6102}, {26,6759}, {30,155}, {49,378}, {52,7530}, {54,15305}, {64,11250}, {68,15761}, {140,17814}, {143,1598}, {154,1658}, {184,7526}, {185,6644}, {195,3830}, {323,3529}, {381,7592}, {382,1993}, {389,13861}, {394,550}, {403,25738}, {511,9925}, {542,8548}, {547,15805}, {568,10594}, {569,15030}, {578,31861}, {999,32143}, {1069,32047}, {1092,10575}, {1147,6000}, {1154,7387}, {1199,3832}, {1493,11403}, {1503,18569}, {1539,19504}, {1596,13292}, {1597,32137}, {1620,18571}, {1656,13561}, {1853,10224}, {2072,11457}, {2979,8718}, {3089,18951}, {3090,15052}, {3091,15032}, {3146,12112}, {3157,8144}, {3167,12085}, {3295,32168}, {3311,32169}, {3312,32170}, {3357,12038}, {3448,16868}, {3520,9544}, {3530,17811}, {3542,18917}, {3544,15018}, {3564,19149}, {3843,15087}, {3845,10982}, {3851,5422}, {3853,15811}, {5012,15058}, {5020,12006}, {5050,19155}, {5072,15037}, {5448,18381}, {5462,15011}, {5654,13371}, {5656,6193}, {5889,7517}, {5890,7506}, {5891,7516}, {5907,7514}, {5925,17847}, {5946,7529}, {6101,11414}, {6238,26888}, {6247,9820}, {6642,13630}, {6823,31831}, {7352,10535}, {7393,14128}, {7395,15060}, {7503,18435}, {7525,31834}, {7553,31815}, {7564,18388}, {7689,10282}, {7728,12419}, {8717,13348}, {8743,22146}, {9140,11704}, {9545,14865}, {9704,14130}, {9786,12106}, {9815,23410}, {9818,19347}, {10024,11442}, {10095,11432}, {10113,19456}, {10116,18390}, {10170,13154}, {10201,12359}, {10226,10606}, {10263,12160}, {10306,32158}, {10323,23039}, {10661,10676}, {10662,10675}, {10665,12970}, {10666,12964}, {11002,26863}, {11381,13352}, {11412,12083}, {11413,22115}, {11423,16261}, {11424,16194}, {11425,11472}, {11426,32136}, {11439,15033}, {11477,11663}, {11482,32155}, {11484,32205}, {11485,32207}, {11486,32208}, {11818,12233}, {11819,31383}, {12293,17824}, {12317,14940}, {13346,14915}, {13406,14852}, {13509,22120}, {14070,14530}, {14644,18504}, {15047,19709}, {15063,21659}, {15331,17821}, {15690,16936}, {16659,31723}, {16881,17810}, {17702,22802}, {17714,17834}, {17928,18350}, {18405,18567}, {18570,19357}, {19173,19211}, {20299,31283}, {31725,32111}

X(32139) = midpoint of X(i) and X(j) for these lines: {i,j}: {155, 1498}, {5878, 12118}, {7387, 12164}, {7728, 12419}, {12085, 12315}, {12308, 12412}
X(32139) = reflection of X(i) in X(j) for these (i,j): (3, 156), (26, 6759), (64, 11250), (68, 15761), (3357, 12038), (6247, 9820), (7689, 10282), (12084, 1147), (12163, 1658), (12359, 16252), (14216, 13371), (15132, 5609), (16266, 155), (17834, 17714), (18356, 13406), (18381, 5448), (18569, 22660), (32138, 32171), (32140, 5)
X(32139) = X(156)-of-X3-ABC reflections triangle
X(32139) = X(6102)-of-Ara triangle
X(32139) = X(32140)-of-Johnson triangle
X(32139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (399, 11456, 15068), (1614, 12111, 3), (11441, 11456, 3)


X(32140) = X(5)-OF-ANTI-INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics    a^10-3*(b^2+c^2)*a^8+4*(b^4+c^4)*a^6-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*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(32140) = 3*X(2)-4*X(13561) = 3*X(4)-4*X(18379) = 3*X(154)-4*X(10020) = X(155)-3*X(1853) = 3*X(376)-4*X(32210) = 5*X(631)-4*X(32171) = 2*X(1147)-3*X(18281) = X(1498)-3*X(14852) = 3*X(1853)-2*X(13371) = 3*X(1992)-4*X(32155) = 5*X(3618)-4*X(19155) = 2*X(5448)-3*X(23325) = 4*X(5449)-3*X(10201) = 3*X(5654)-4*X(10224) = 2*X(6759)-3*X(10201) = X(11411)+3*X(32064) = X(14216)+2*X(18356) = X(14790)-3*X(32064) = 3*X(14852)-2*X(15761) = 3*X(18281)-4*X(20299)

X(32140) lies on these lines: {2,156}, {3,70}, {4,94}, {5,1181}, {20,30522}, {26,1503}, {30,64}, {52,11550}, {66,3564}, {69,10627}, {74,12278}, {110,6640}, {125,10539}, {140,1352}, {154,10020}, {155,1853}, {182,18128}, {185,18474}, {376,2888}, {381,18912}, {388,32143}, {389,11818}, {399,10255}, {427,12161}, {497,32168}, {511,14864}, {539,13346}, {542,1147}, {578,10116}, {631,3410}, {1154,11411}, {1199,5169}, {1204,16003}, {1209,10984}, {1368,31831}, {1370,6101}, {1498,14852}, {1593,12370}, {1594,18445}, {1595,13292}, {1614,6639}, {1656,18911}, {1658,9833}, {1992,32155}, {2072,11441}, {2550,32158}, {3068,32169}, {3069,32170}, {3088,11264}, {3098,17712}, {3357,17702}, {3519,13340}, {3538,11592}, {3546,5921}, {3580,7517}, {3581,31304}, {3618,19155}, {3796,7568}, {3818,5462}, {5448,23325}, {5449,6759}, {5562,14791}, {5576,7592}, {5596,19154}, {5609,15114}, {5654,10224}, {5876,18531}, {5889,31723}, {5946,7528}, {6000,9927}, {6143,9544}, {6146,7526}, {6243,7391}, {6247,12084}, {6515,10263}, {6642,18440}, {6643,11591}, {6644,12134}, {6776,32046}, {6816,15060}, {6997,15026}, {7386,32142}, {7392,32205}, {7401,12006}, {7403,11245}, {7505,10540}, {7506,26879}, {7529,26869}, {7530,16655}, {7564,12233}, {7689,18400}, {7706,13382}, {8144,10071}, {9140,26917}, {9786,31830}, {9820,23332}, {10024,11456}, {10055,32047}, {10095,11433}, {10264,26937}, {10516,15805}, {10620,18565}, {11202,20191}, {11250,12118}, {11424,18488}, {11427,32136}, {11440,12289}, {11488,32207}, {11489,32208}, {11585,15068}, {11695,18553}, {12038,23329}, {12085,12429}, {12106,16010}, {12111,18404}, {12241,31861}, {12290,31725}, {12419,15061}, {13364,18950}, {13367,18580}, {13567,13861}, {13630,18420}, {13754,18381}, {14449,31670}, {16172,20422}, {18383,18568}, {18570,19467}, {18918,22533}, {19139,23300}, {19174,19211}

X(32140) = midpoint of X(i) and X(j) for these lines: {i,j}: {64, 12293}, {68, 14216}, {11411, 14790}, {12085, 12429}
X(32140) = reflection of X(i) in X(j) for these (i,j): (20, 32138), (26, 12359), (68, 18356), (155, 13371), (156, 13561), (1147, 20299), (1498, 15761), (5596, 19154), (5609, 15114), (6759, 5449), (9833, 1658), (12084, 6247), (12118, 11250), (16266, 23335), (18569, 18381), (19139, 23300), (32139, 5)
X(32140) = anticomplement of X(156)
X(32140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (156, 13561, 2), (11442, 11457, 3)


X(32141) = X(5)-OF-ANTI-MANDART-INCIRCLE TRIANGLE

Barycentrics    a*(a^6-(b+c)*a^5-2*(b^2+c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3+(b^4+c^4-(b+c)^2*b*c)*a^2-(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*b*c) : :
X(32141) = 3*X(165)+X(5534) = 3*X(165)-X(24467) = 3*X(3158)+X(5709) = 3*X(4421)-X(11248) = 3*X(4421)+X(11500) = X(12338)+3*X(22556)

X(32141) lies on these lines: {1,6924}, {3,8}, {4,11849}, {5,55}, {21,5790}, {26,197}, {30,4421}, {35,355}, {40,14988}, {46,24475}, {56,1483}, {119,6284}, {140,1376}, {145,6942}, {149,6949}, {155,7074}, {156,692}, {165,5534}, {200,26921}, {227,32047}, {390,6944}, {404,10246}, {411,12702}, {474,16202}, {495,11501}, {496,11502}, {497,6959}, {499,1484}, {515,26285}, {517,6796}, {519,26286}, {546,11496}, {547,4428}, {550,10310}, {571,21855}, {632,4413}, {912,3579}, {958,7508}, {1001,3628}, {1012,18518}, {1058,6970}, {1385,5836}, {1388,10090}, {1478,14882}, {1482,3871}, {1486,13861}, {1621,1656}, {1771,5399}, {2077,18481}, {2098,10087}, {2886,31659}, {3085,6917}, {3149,10679}, {3158,5709}, {3198,8141}, {3293,5398}, {3295,5901}, {3303,10283}, {3434,6863}, {3560,18357}, {3564,12328}, {3617,6875}, {3740,18233}, {3746,5886}, {3753,24299}, {3913,5844}, {4188,7967}, {4294,6929}, {4302,11698}, {5010,5881}, {5070,5284}, {5082,6954}, {5172,10573}, {5217,22758}, {5218,6862}, {5248,9956}, {5264,5396}, {5281,6824}, {5432,26470}, {5450,26086}, {5541,11014}, {5552,6928}, {5560,5587}, {5693,12738}, {5697,6265}, {5762,6600}, {5763,6985}, {5779,7676}, {5818,7489}, {5840,18242}, {5843,11495}, {5874,11498}, {5875,11497}, {6154,15908}, {6326,11010}, {6756,11383}, {6825,17784}, {6834,20075}, {6868,7080}, {6883,9709}, {6906,18525}, {6915,18493}, {6921,10806}, {6923,10786}, {6934,10528}, {6941,10738}, {6950,26321}, {6958,12116}, {6960,20095}, {6971,27529}, {7491,17757}, {8273,15712}, {9778,16117}, {9897,15446}, {10525,13205}, {10902,26446}, {10944,14793}, {10993,11826}, {11490,32134}, {11492,32146}, {11493,32147}, {11494,32151}, {11503,32177}, {11504,32178}, {11509,18990}, {11510,15325}, {11517,31789}, {11729,26358}, {11848,32162}, {12114,28224}, {12178,12339}, {12255,12535}, {12334,12341}, {12336,22558}, {12337,22557}, {12338,22556}, {12514,31835}, {12515,15071}, {12737,21842}, {12775,18542}, {13887,13925}, {13940,13993}, {14795,18395}, {14798,24914}, {15624,29010}, {16203,16371}, {17522,24808}, {17714,20872}, {18999,19116}, {19000,19117}, {20986,32046}, {21740,25413}, {22457,23981}, {22935,30144}, {24680,25439}

X(32141) = midpoint of X(i) and X(j) for these lines: {i,j}: {3913, 11249}, {5534, 24467}, {6796, 8715}, {6985, 10306}, {11248, 11500}, {12334, 13204}
X(32141) = reflection of X(i) in X(j) for these (i,j): (5450, 26086), (10943, 140), (32153, 3)
X(32141) = X(156)-of-1st circumperp triangle
X(32141) = X(1483)-of-2nd circumperp tangential triangle
X(32141) = X(13561)-of-excentral triangle
X(32141) = X(26286)-of-inner-Garcia triangle
X(32141) = X(32138)-of-2nd circumperp triangle
X(32141) = X(32153)-of-ABC-X3 reflections triangle
X(32141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5687, 5690), (3, 18526, 104), (100, 11491, 3)


X(32142) = X(5)-OF-6th ANTI-MIXTILINEAR TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^6-3*(b^2+c^2)^2*a^4+3*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32142) = 3*X(2)+X(6101) = 9*X(2)-X(6243) = 9*X(2)-5*X(15026) = 3*X(3)+X(5876) = 9*X(3)-X(6241) = X(3)-9*X(7998) = X(3)+7*X(7999) = 3*X(3)+5*X(11444) = 5*X(3)+3*X(11459) = 7*X(3)+X(12111) = 5*X(3)-X(13491) = X(3)+3*X(15067) = 11*X(3)-3*X(15072) = 3*X(143)-X(6243) = 3*X(143)-5*X(15026) = 3*X(5876)+X(6241) = X(5876)-5*X(11444) = 5*X(5876)-9*X(11459) = X(5876)-3*X(11591) = X(5876)+6*X(11592) = 7*X(5876)-3*X(12111) = 5*X(5876)+3*X(13491) = X(5876)-9*X(15067) = 3*X(6101)+X(6243) = 3*X(6101)+5*X(15026) = X(6101)+2*X(32205) = X(6243)-5*X(15026) = X(6243)-6*X(32205) = 5*X(15026)-6*X(32205)

X(32142) lies on these lines: {2,143}, {3,74}, {4,11017}, {5,3917}, {6,13154}, {20,15060}, {26,17811}, {30,5447}, {51,13421}, {52,632}, {69,18952}, {95,19211}, {125,21230}, {140,389}, {141,12061}, {182,32136}, {185,15712}, {186,12300}, {323,1493}, {338,25043}, {394,7516}, {511,3628}, {546,10170}, {547,5446}, {548,5907}, {549,5562}, {550,5891}, {568,3525}, {599,15074}, {631,6102}, {1038,32143}, {1040,32168}, {1112,14940}, {1350,13861}, {1368,13561}, {1594,11576}, {1656,2979}, {1657,15056}, {3025,7144}, {3060,5070}, {3091,13340}, {3098,17714}, {3519,15108}, {3522,18435}, {3523,18436}, {3526,5946}, {3528,18439}, {3530,13754}, {3534,15058}, {3548,31807}, {5054,5889}, {5066,13598}, {5449,6698}, {5462,15606}, {5690,31849}, {5888,15037}, {5890,15720}, {5943,14449}, {5965,32165}, {6143,6242}, {6292,11675}, {6636,18350}, {7386,32140}, {7393,11426}, {7484,12161}, {7514,11425}, {7525,9306}, {7568,11064}, {7574,22804}, {8703,12162}, {9729,12108}, {9730,14869}, {10124,11695}, {10203,11702}, {10319,32158}, {10610,22115}, {10691,31831}, {11227,31836}, {11230,31737}, {11231,31738}, {11439,15681}, {11465,13321}, {11511,32155}, {11513,32169}, {11514,32170}, {11515,32207}, {11516,32208}, {11561,14448}, {11573,31835}, {11583,31376}, {12010,12900}, {12100,31834}, {12279,15688}, {12362,30522}, {13347,15083}, {13624,31752}, {14627,23061}, {14831,15713}, {15030,15704}, {15043,15694}, {15088,15465}, {15305,15696}, {15443,24470}, {18379,18531}, {19126,19155}, {26910,26915}, {26911,26914}, {31663,31751}

X(32142) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 11591}, {5, 10627}, {20, 32137}, {140, 1216}, {143, 6101}, {546, 15644}, {548, 5907}, {2979, 13364}, {5447, 11793}, {5462, 15606}, {5562, 13630}, {11573, 31835}, {13624, 31752}, {31663, 31751}
X(32142) = reflection of X(i) in X(j) for these (i,j): (3, 11592), (4, 11017), (143, 32205), (5446, 18874), (5462, 16239), (9729, 12108), (10095, 3628), (12006, 140), (14128, 11793), (16881, 11695), (16982, 10095)
X(32142) = complementary conjugate of X(31376)
X(32142) = anticomplement of X(32205)
X(32142) = complement of X(143)
X(32142) = X(11017)-of-anti-Euler triangle
X(32142) = X(11592)-of-X3-ABC reflections triangle
X(32142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 143, 32205), (2, 6101, 143), (6101, 15026, 6243)


X(32143) = X(5)-OF-ANTI-TANGENTIAL-MIDARC TRIANGLE

Barycentrics    a^2*((b+c)^2*a^6-(3*b^2-4*b*c+3*c^2)*(b+c)^2*a^4+(3*b^6+3*c^6-2*(b^4+c^4+(b-c)^2*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^2-b*c+c^2)^2) : :
X(32143) = X(7352)+3*X(32065) = X(32047)-3*X(32065)

X(32143) lies on these lines: {1,3024}, {3,19367}, {5,1425}, {12,13561}, {26,221}, {33,32137}, {34,143}, {35,32210}, {36,32171}, {55,32138}, {56,156}, {65,32158}, {388,32140}, {1038,32142}, {1060,11591}, {1154,7352}, {1398,12161}, {1428,19155}, {1870,6102}, {2067,32169}, {3585,18379}, {4296,6101}, {5876,18447}, {6502,32170}, {7051,32207}, {7354,30522}, {7355,8144}, {9641,12279}, {9642,12290}, {9817,11017}, {10095,19366}, {13491,18455}, {18915,18952}, {19175,19211}, {19349,32046}, {19365,32136}, {19369,32155}, {19372,32205}, {19373,32208}

X(32143) = midpoint of X(i) and X(j) for these lines: {i,j}: {7352, 32047}, {7355, 8144}
X(32143) = reflection of X(32168) in X(1)
X(32143) = X(156)-of-2nd anti-circumperp-tangential triangle
X(32143) = X(32138)-of-Mandart-incircle triangle
X(32143) = X(32168)-of-5th mixtilinear triangle
X(32143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7352, 32065, 32047), (19367, 19368, 3)


X(32144) = X(5)-OF-AOA TRIANGLE

Barycentrics    2*a^10-7*(b^2+c^2)*a^8+2*(3*b^4+8*b^2*c^2+3*c^4)*a^6+4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4-8*(b^6-c^6)*(b^2-c^2)*a^2+3*(b^4-c^4)*(b^2-c^2)^3 : :
X(32144) = 9*X(2)-X(7387) = X(1147)+3*X(23332) = 3*X(15113)-X(15114) = 3*X(15113)+X(15115) = X(22660)+3*X(23329)

As a point on the Euler line, X(32144) has Shinagawa coefficients (E-10*F, -3*E-2*F).

X(32144) lies on these lines: {2,3}, {125,13292}, {1147,23332}, {3564,6697}, {5448,6696}, {5663,22967}, {6699,13568}, {6723,10110}, {9820,20299}, {11064,31831}, {11264,15129}, {11801,25487}, {12161,19360}, {13754,32184}, {15113,15114}, {15120,32165}, {15132,15142}, {22660,23329}

X(32144) = midpoint of X(i) and X(j) for these lines: {i,j}: {140, 13371}, {546, 11250}, {548, 18377}, {5448, 6696}, {9820, 20299}, {10224, 23336}, {11801, 25487}, {13383, 23335}, {15114, 15115}
X(32144) = reflection of X(i) in X(j) for these (i,j): (3530, 5498), (10020, 16239), (15330, 11540), (15331, 12108)
X(32144) = complement of X(13383)
X(32144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 1593, 546), (5, 3548, 140), (140, 546, 6644)


X(32145) = X(5)-OF-ARIES TRIANGLE

Barycentrics    SA*((4*R^2-2*SW)*S^2+(SB+SC)*(3*SW^2+2*R^4+(5*SA-13*SW)*R^2)) : :
X(32145) = 3*X(11206)+X(12420) = 3*X(11206)-X(32048)

X(32145) lies on these lines: {25,32166}, {3564,15580}, {5448,31804}, {6193,12087}, {11206,12420}

X(32145) = midpoint of X(12420) and X(32048)
X(32145) = X(32166)-of-Ara triangle
X(32145) = {X(11206), X(12420)}-harmonic conjugate of X(32048)


X(32146) = X(5)-OF-1st AURIGA TRIANGLE

Barycentrics    (2*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*D+8*S^2*(-a+b+c)*a^2 : : , where D=4*S*sqrt(R*(4*R+r))

X(32146) lies on these lines: {3,5601}, {4,11875}, {5,5597}, {26,8190}, {30,9834}, {55,952}, {140,5599}, {145,11876}, {355,8186}, {495,11869}, {496,11871}, {546,8196}, {550,11822}, {1483,5598}, {3564,12415}, {5602,12645}, {5690,8197}, {5844,11253}, {5874,8199}, {5875,8198}, {5901,11366}, {6756,11384}, {8201,32177}, {8202,32178}, {9835,28224}, {10942,11867}, {10943,11865}, {11492,32141}, {11493,32153}, {11837,32134}, {11844,18526}, {11861,32151}, {11863,32162}, {11873,15171}, {11881,32213}, {11883,32214}, {12179,12476}, {12458,28174}, {12474,22668}, {13890,13925}, {13944,13993}, {18955,18990}, {19007,19116}, {19008,19117}

X(32146) = midpoint of X(11253) and X(12454)
X(32146) = reflection of X(32147) in X(55)
X(32146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5597, 8200, 5), (5601, 11843, 3)


X(32147) = X(5)-OF-2nd AURIGA TRIANGLE

Barycentrics    -(2*a^4-(2*(b+c))*a^3-(b^2-4*b*c+c^2)*a^2+(2*(b^2-c^2))*(b-c)*a-(b^2-c^2)^2)*D+8*S^2*(-a+b+c)*a^2 : : , where D=4*S*sqrt(R*(4*R+r))

X(32147) lies on these lines: {3,5602}, {4,11876}, {5,5598}, {26,8191}, {30,9835}, {55,952}, {140,5600}, {145,11875}, {355,8187}, {495,11870}, {496,11872}, {546,8203}, {550,11823}, {1483,5597}, {5601,12645}, {5690,8204}, {5844,11252}, {5874,8206}, {5875,8205}, {5901,11367}, {6756,11385}, {8208,32177}, {8209,32178}, {9834,28224}, {10942,11868}, {10943,11866}, {11492,32153}, {11493,32141}, {11838,32134}, {11843,18526}, {11862,32151}, {11874,15171}, {11882,32213}, {11884,32214}, {12180,12477}, {12459,28174}, {12475,22672}, {13891,13925}, {13945,13993}, {18956,18990}, {19009,19116}, {19010,19117}

X(32147) = midpoint of X(11252) and X(12455)
X(32147) = reflection of X(32146) in X(55)
X(32147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5598, 8207, 5), (5602, 11844, 3)


X(32148) = X(5)-OF-AYME TRIANGLE

Barycentrics
a*((b+c)*a^8-2*b*c*a^7-(b+c)*(2*b^2-b*c+2*c^2)*a^6+(5*b^2+8*b*c+5*c^2)*b*c*a^5-2*(b+c)*(3*b^2-b*c+3*c^2)*b*c*a^4-10*(b^2+c^2)*(b+c)^2*b*c*a^3+(b+c)*(2*b^6+2*c^6+(3*b^4+3*c^4-2*(4*b^2+7*b*c+4*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^2*(7*b^2+12*b*c+7*c^2)*b*c*a-(b^2-c^2)^2*(b+c)^3*(b^2-4*b*c+c^2)) : :
X(32148) = X(9958)+3*X(11221) = 3*X(11221)-X(11259)

X(32148) lies on these lines: {5,10}, {612,32167}, {9958,11221}, {10618,30142}

X(32148) = midpoint of X(9958) and X(11259)
X(32148) = {X(9958), X(11221)}-harmonic conjugate of X(11259)


X(32149) = X(5)-OF-1st BROCARD-REFLECTED TRIANGLE

Barycentrics    3*(b^2+c^2)*a^8-4*(b^4+b^2*c^2+c^4)*a^6-15*(b^2+c^2)*b^2*c^2*a^4+(b^4+4*b^2*c^2+c^4)*(b^4-5*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(32149) = 3*X(2)+X(31958) = X(575)+2*X(3934) = X(6194)+3*X(14561) = X(6248)+2*X(20190) = X(14994)+2*X(15516)

X(32149) lies on these lines: {2,51}, {182,7697}, {538,7606}, {575,3934}, {620,24256}, {2782,10168}, {5969,7619}, {6248,20190}, {6683,8179}, {6704,25555}, {10347,22521}, {14994,15516}, {22681,29012}

X(32149) = midpoint of X(i) and X(j) for these lines: {i,j}: {182, 7697}, {5476, 22712}, {11261, 31958}
X(32149) = complement of X(11261)
X(32149) = {X(2), X(31958)}-harmonic conjugate of X(11261)


X(32150) = X(5)-OF-2nd BROCARD TRIANGLE

Barycentrics    (27*R^2*(SA+2*SW)-25*SW^2)*S^4+(SB*SC*(27*R^2+7*SW)+SW*(2*SA^2-3*SW^2))*SW*S^2-SB*SC*SW^4 : :
X(32150) = X(182)-3*X(10166) = 3*X(353)+X(1352) = 3*X(10166)+X(31959)

X(32150) lies on these lines: {141,31727}, {182,9169}, {353,1352}, {511,31608}, {542,31742}, {3098,14867}, {3818,31731}, {6719,10160}, {7619,9830}, {8705,18583}

X(32150) = midpoint of X(i) and X(j) for these lines: {i,j}: {141, 31727}, {182, 31959}, {3098, 14867}, {3818, 31731}
X(32150) = X(16776)-of-McCay triangle
X(32150) = X(31744)-of-1st Brocard triangle
X(32150) = X(32150)-of-circumsymmedial triangle
X(32150) = {X(10166), X(31959)}-harmonic conjugate of X(182)


X(32151) = X(5)-OF-5th BROCARD TRIANGLE

Barycentrics    2*a^8-3*(b^2+c^2)*a^6+2*(b^4+c^4)*a^4+3*(b^2+c^2)*b^2*c^2*a^2-(b^6-c^6)*(b^2-c^2) : :
X(32151) = 3*X(3)-5*X(7904) = 3*X(5)-2*X(7745) = 3*X(381)-X(7823) = 3*X(3095)-X(7877) = X(7756)-3*X(32152) = 3*X(7811)-X(9821) = 3*X(7811)+X(9873) = 5*X(7904)+3*X(9863) = X(9983)+3*X(22678)

X(32151) lies on these lines: {3,147}, {4,9301}, {5,32}, {26,10828}, {30,76}, {98,7911}, {140,3096}, {355,3099}, {381,7823}, {495,10038}, {496,10047}, {542,7830}, {546,9993}, {549,3788}, {550,1503}, {626,12042}, {632,7914}, {754,14881}, {952,9941}, {1078,6033}, {1352,2076}, {1483,9997}, {1656,10583}, {2782,7756}, {3094,3564}, {3095,7877}, {3104,9981}, {3398,7859}, {3406,10345}, {3628,7846}, {5690,9857}, {5844,12495}, {5874,9995}, {5875,9994}, {5901,11368}, {6287,12110}, {6655,12188}, {6756,11386}, {7502,15270}, {7575,16335}, {7761,14880}, {7801,8703}, {7831,12054}, {7870,12100}, {7883,14830}, {7915,13335}, {7916,9737}, {8362,10349}, {8782,13108}, {9983,22678}, {9985,12501}, {10346,11842}, {10871,10943}, {10872,10942}, {10875,32177}, {10876,32178}, {10877,15171}, {10878,32213}, {10879,32214}, {11494,32141}, {11861,32146}, {11862,32147}, {11885,32162}, {12497,28174}, {13892,13925}, {13946,13993}, {18957,18990}, {19011,19116}, {19012,19117}, {19780,22907}, {19781,22861}, {22744,32153}

X(32151) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 9863}, {9821, 9873}, {12501, 13210}
X(32151) = X(7823)-of-Ehrmann-mid triangle
X(32151) = X(19130)-of-6th Brocard triangle
X(32151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2896, 9862, 3), (9301, 18503, 4), (9986, 9987, 3094)


X(32152) = X(5)-OF-6th BROCARD TRIANGLE

Barycentrics    2*a^8-4*(b^2+c^2)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*a^4+2*(b^2+c^2)*b^2*c^2*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(32152) = 3*X(262)-X(7823) = 3*X(7753)-4*X(11272) = X(7756)+2*X(32151) = 3*X(7811)-X(12251) = 3*X(7833)+X(9863) = 3*X(7833)-X(11257) = 5*X(7904)-3*X(22712) = 3*X(8356)-2*X(13334)

X(32152) lies on these lines: {3,114}, {4,1078}, {5,187}, {20,1352}, {30,5188}, {76,23698}, {98,6655}, {115,10104}, {140,7853}, {182,7791}, {194,5965}, {262,7823}, {315,9737}, {376,7883}, {384,24206}, {401,21243}, {491,9738}, {492,9739}, {511,7750}, {542,7833}, {543,13108}, {576,20065}, {631,7899}, {754,3095}, {1353,9607}, {1975,10992}, {2782,7756}, {3398,4045}, {5025,6036}, {6055,7841}, {6200,6290}, {6249,8150}, {6289,6396}, {6656,13335}, {6721,7907}, {6781,9996}, {7616,17578}, {7737,10358}, {7746,23514}, {7753,11272}, {7763,9734}, {7787,25555}, {7811,12251}, {7828,21445}, {7829,11842}, {7836,21166}, {7842,15980}, {7873,18860}, {7904,22712}, {8356,13334}, {10131,32135}, {10991,14880}, {11293,12975}, {11294,12974}, {12110,14712}

X(32152) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 7802}, {20, 9873}, {9863, 11257}
X(32152) = reflection of X(i) in X(j) for these (i,j): (3, 7830), (7747, 5)
X(32152) = inverse of X(19165) in the 2nd Brocard circle
X(32152) = X(7747)-of-Johnson triangle
X(32152) = X(7802)-of-Euler triangle
X(32152) = X(7830)-of-X3-ABC reflections triangle
X(32152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 14907, 5171), (7833, 9863, 11257)


X(32153) = X(5)-OF-2nd CIRCUMPERP TANGENTIAL TRIANGLE

Barycentrics    a*(a^6-(b+c)*a^5-2*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)*a^3+(b^4+c^4-3*(b-c)^2*b*c)*a^2-(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*b*c) : :
X(32153) = X(944)+3*X(5770) = X(5534)-5*X(7987) = 3*X(11194)-X(11249) = 3*X(11194)+X(12114) = 3*X(22680)+X(22779)

X(32153) lies on these lines: {1,1399}, {3,8}, {4,20067}, {5,56}, {21,10246}, {26,22654}, {30,10525}, {36,355}, {55,1483}, {119,5433}, {140,958}, {145,6950}, {153,6949}, {388,6862}, {392,24927}, {404,5790}, {405,16203}, {495,22759}, {496,22760}, {515,26286}, {517,5450}, {519,26285}, {546,18761}, {550,3428}, {912,960}, {997,31835}, {999,3485}, {1001,5843}, {1012,10680}, {1056,6892}, {1319,5887}, {1329,6713}, {1388,10074}, {1420,7330}, {1455,32047}, {1479,1484}, {1482,6906}, {1656,5253}, {1768,11014}, {2077,5288}, {2098,10058}, {3086,6929}, {3149,18519}, {3304,10283}, {3421,6961}, {3436,6958}, {3526,5260}, {3564,22595}, {3600,6824}, {3616,7489}, {3628,25524}, {3655,10902}, {3813,5840}, {3878,11715}, {4189,7967}, {4293,6917}, {4861,25413}, {5080,6971}, {5126,5777}, {5204,11499}, {5248,15178}, {5251,24954}, {5258,26446}, {5265,6944}, {5267,5882}, {5445,12751}, {5534,7987}, {5563,5886}, {5584,8703}, {5603,13743}, {5693,6265}, {5697,12737}, {5714,6913}, {5779,7677}, {5816,21773}, {5844,11248}, {5874,22757}, {5875,22756}, {5881,7280}, {5885,30147}, {6256,22775}, {6264,11010}, {6756,22479}, {6796,28204}, {6833,20076}, {6863,12115}, {6905,18525}, {6909,12702}, {6910,10805}, {6911,18357}, {6912,18493}, {6923,10527}, {6928,10785}, {6930,14986}, {6938,10529}, {6941,10742}, {6942,18524}, {6952,20060}, {6959,7288}, {6974,10597}, {6976,10586}, {6985,28186}, {7354,26470}, {7508,10267}, {8715,26086}, {10950,14793}, {10966,15171}, {11012,18481}, {11279,15175}, {11492,32147}, {11493,32146}, {11500,28224}, {11928,22938}, {12001,28444}, {13925,22763}, {13993,22764}, {14806,21855}, {16150,21669}, {16202,16370}, {19013,19116}, {19014,19117}, {19478,22586}, {21031,21154}, {22504,22780}, {22520,32134}, {22680,22779}, {22744,32151}, {22755,32162}, {22761,32177}, {22762,32178}, {22768,32213}, {22770,28174}, {22771,22773}, {22772,22774}, {22836,26287}, {23961,25440}

X(32153) = midpoint of X(i) and X(j) for these lines: {i,j}: {1, 24467}, {5450, 8666}, {11248, 12513}, {11249, 12114}, {19478, 22586}
X(32153) = reflection of X(i) in X(j) for these (i,j): (8715, 26086), (10942, 140), (22836, 26287), (32141, 3)
X(32153) = X(156)-of-2nd circumperp triangle
X(32153) = X(1483)-of-anti-Mandart-incircle triangle
X(32153) = X(13561)-of-hexyl triangle
X(32153) = X(24467)-of-anti-Aquila triangle
X(32153) = X(26285)-of-inner-Garcia triangle
X(32153) = X(32138)-of-1st circumperp triangle
X(32153) = X(32141)-of-ABC-X3 reflections triangle
X(32153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 12773, 944), (3, 18526, 11491), (104, 2975, 3)


X(32154) = X(5)-OF-1st EHRMANN TRIANGLE

Barycentrics    a^2*(2*a^6-(b^2+c^2)*a^4-2*(b^2+c^2)^2*a^2+(b^2+c^2)*((b^2+c^2)^2-9*b^2*c^2)) : :
X(32154) = 3*X(182)+X(8542) = 3*X(182)-X(8546) = X(8547)-5*X(12017) = 3*X(10168)-X(16511)

X(32154) lies on these lines: {2,67}, {3,9019}, {6,5888}, {23,16776}, {30,25488}, {140,524}, {141,3292}, {182,1511}, {187,30489}, {576,6101}, {597,13857}, {599,11422}, {1995,19127}, {2393,20190}, {2930,5012}, {3589,5159}, {5092,8705}, {5609,11178}, {7492,9971}, {7495,8262}, {7496,10510}, {7509,11477}, {7708,28662}, {7804,11594}, {8547,12017}, {9716,21356}, {10272,24206}, {10541,17928}, {11179,23236}, {12041,25489}, {13366,22165}, {16042,18374}

X(32154) = midpoint of X(i) and X(j) for these lines: {i,j}: {5092, 12039}, {8542, 8546}, {12041, 25489}
X(32154) = reflection of X(20113) in X(3589)
X(32154) = {X(182), X(8542)}-harmonic conjugate of X(8546)


X(32155) = X(5)-OF-2nd EHRMANN TRIANGLE

Barycentrics    (SB+SC)*((48*R^2-13*SW)*S^2+(2*R^2-3*SA+2*SW)*SA*SW) : :
X(32155) = 3*X(6)-X(156) = X(26)+3*X(17813) = 2*X(156)-3*X(19155) = 3*X(895)+X(15132) = 3*X(1992)+X(32140) = X(8548)+3*X(11216) = 3*X(10250)-X(11250) = 3*X(11216)-X(11255) = 5*X(11482)-X(32139) = X(13371)-3*X(23326) = X(18377)-3*X(23048)

X(32155) lies on these lines: {3,11443}, {5,21639}, {6,156}, {26,17813}, {143,8541}, {511,32210}, {524,13561}, {542,18379}, {575,32171}, {576,5663}, {895,15132}, {1154,8548}, {1353,11264}, {1493,15531}, {1992,32140}, {5876,18449}, {6101,11416}, {6102,8537}, {8538,11591}, {8539,32158}, {8540,32168}, {8550,30522}, {9813,32205}, {10250,11250}, {10602,32046}, {11405,12161}, {11470,32137}, {11477,32138}, {11482,32139}, {11511,32142}, {13371,23326}, {18377,23048}, {18919,18952}, {19178,19211}, {19369,32143}

X(32155) = midpoint of X(i) and X(j) for these lines: {i,j}: {8548, 11255}, {11477, 32138}
X(32155) = reflection of X(i) in X(j) for these (i,j): (19155, 6), (32171, 575)
X(32155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8548, 11216, 11255), (11443, 11458, 3)


X(32156) = X(5)-OF-5th EULER TRIANGLE

Barycentrics    6*(9*R^2-4*SW)*S^4+(27*R^2*SA*(SB+SC)+SW^2*(3*SA-4*SW))*S^2+SB*SC*SW^3 : :
X(32156) = X(3)+3*X(6032) = X(5)-3*X(10162) = 3*X(5)-X(14866) = 3*X(381)-X(31824) = 3*X(549)-X(31729) = 5*X(632)-3*X(10163) = 5*X(1656)-X(12505) = 11*X(3525)-3*X(6031) = 7*X(3526)-3*X(9829) = 2*X(3628)-3*X(10173) = 3*X(5055)+X(31961) = 3*X(5946)-X(31745) = 3*X(10162)+X(12506) = 9*X(10162)-X(14866) = 3*X(10173)-X(31606) = 3*X(11230)-X(31747) = 3*X(11231)-X(31746) = 3*X(12506)+X(14866) = 3*X(15067)-X(31736) = X(31744)+3*X(31840)

X(32156) lies on these lines: {2,31744}, {3,6032}, {5,9172}, {30,31749}, {140,3849}, {381,31824}, {517,31755}, {549,31729}, {575,20379}, {632,10163}, {1154,31753}, {1656,12505}, {3525,6031}, {3526,9829}, {3628,10173}, {5055,31961}, {5946,31745}, {8704,11272}, {11230,31747}, {11231,31746}, {15060,31743}, {15067,31736}, {30516,31727}

X(32156) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 31840}, {5, 12506}, {15060, 31743}, {31749, 31762}, {31753, 31763}, {31755, 31758}
X(32156) = reflection of X(31606) in X(3628)
X(32156) = complement of X(31744)
X(32156) = X(31824)-of-Ehrmann-mid triangle
X(32156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10162, 12506, 5), (10173, 31606, 3628)


X(32157) = X(5)-OF-EXCENTERS-MIDPOINTS TRIANGLE

Barycentrics    2*a^4+2*(b+c)*a^3-(3*b^2+8*b*c+3*c^2)*a^2-2*(b+c)*(b^2-3*b*c+c^2)*a+(b^2-c^2)^2 : :
X(32157) = X(8)+3*X(4421) = 3*X(165)+X(32049) = 3*X(549)-X(22837) = 5*X(631)-X(10912) = 5*X(1698)-3*X(3829) = X(2136)+7*X(9588) = 3*X(3654)+X(3811) = X(3813)-3*X(26446) = X(3913)+3*X(5657) = X(6361)+3*X(11236) = 7*X(9780)-3*X(11235) = 3*X(10164)-X(11260) = 3*X(10164)+X(12640)

X(32157) lies on these lines: {2,13463}, {8,4421}, {10,528}, {21,13205}, {35,1145}, {40,12607}, {55,5554}, {100,32198}, {140,2802}, {165,32049}, {518,31787}, {519,12100}, {529,3579}, {549,22837}, {631,10912}, {952,26086}, {1329,5119}, {1376,27870}, {1385,5854}, {1697,3816}, {1698,3829}, {1706,3826}, {2136,9588}, {3035,3057}, {3036,10572}, {3058,25005}, {3654,3811}, {3679,5441}, {3813,26446}, {3880,6684}, {3885,5433}, {3895,24914}, {3913,5657}, {4640,6736}, {4642,17061}, {4861,13996}, {4999,10914}, {5086,6154}, {5123,10624}, {5176,15338}, {5204,12648}, {5221,11239}, {5250,9711}, {5428,5690}, {5541,24390}, {5559,15015}, {5836,6690}, {5855,11362}, {6361,11236}, {6667,12053}, {6691,9957}, {6940,22560}, {9780,11235}, {10107,13405}, {10164,11260}, {11010,17757}, {15813,26357}, {30337,31190}

X(32157) = midpoint of X(i) and X(j) for these lines: {i,j}: {40, 12607}, {100, 32198}, {3579, 10915}, {5690, 8715}, {11260, 12640}
X(32157) = complement of X(13463)
X(32157) = X(6247)-of-K798i triangle
X(32157) = X(32184)-of-excentral triangle
X(32157) = {X(10164), X(12640)}-harmonic conjugate of X(11260)


X(32158) = X(5)-OF-EXTANGENTS TRIANGLE

Barycentrics
a^2*((b+c)^2*a^9-(b^2-c^2)*(b-c)*a^8-2*(2*b^2-b*c+2*c^2)*(b+c)^2*a^7+2*(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a^6+(6*b^6+6*c^6+(6*b^4+6*c^4+(3*b^2+4*b*c+3*c^2)*b*c)*b*c)*a^5-(b+c)*(6*b^6+6*c^6-(6*b^4+6*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*a^4-2*(2*b^6+2*c^6-3*(b^4+c^4-(b^2-b*c+c^2)*b*c)*b*c)*(b+c)^2*a^3+2*(b^2-c^2)*(b-c)*(2*b^6+2*c^6+3*(b^4+c^4+(b^2+b*c+c^2)*b*c)*b*c)*a^2+(b^2-c^2)^2*(b+c)^2*(b^2-b*c+c^2)^2*a-(b^2-c^2)^3*(b-c)*(b^2+b*c+c^2)^2) : :
X(32158) = X(6237)+3*X(11190) = X(8141)-3*X(11190)

X(32158) lies on these lines: {3,11445}, {5,3611}, {19,143}, {26,3197}, {40,2940}, {55,156}, {65,32143}, {1154,6237}, {2550,32140}, {3101,6101}, {3925,13561}, {5415,32169}, {5416,32170}, {5584,32138}, {5876,18453}, {6102,6197}, {6243,9536}, {6253,30522}, {7688,32210}, {8251,11591}, {8539,32155}, {9537,18436}, {9816,32205}, {10095,11435}, {10306,32139}, {10319,32142}, {10636,32207}, {10637,32208}, {10902,32171}, {11406,12161}, {11428,32136}, {11471,32137}, {18379,18406}, {18921,18952}, {19133,19155}, {19181,19211}, {19350,32046}

X(32158) = reflection of X(32168) in X(156)
X(32158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6237, 11190, 8141), (11445, 11460, 3)


X(32159) = X(5)-OF-EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^8-2*(b^2+b*c+c^2)*a^7-2*(b^2-c^2)*(b-c)*a^6+2*(b^2+b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a^5-4*(b+c)*(2*b^2-b*c+2*c^2)*b*c*a^4-2*(b^2+c^2)*(3*b^2+5*b*c+3*c^2)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4+4*(b^2+b*c+c^2)*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4-(b+c)^2*b*c)*a-(b^2-c^2)^4*(b+c)) : :
X(32159) = 3*X(210)-X(1158) = 3*X(210)+X(18239) = 5*X(3697)-X(17649) = 3*X(3740)-X(18238) = 3*X(5657)+X(12666) = X(6245)-3*X(15064)

X(32159) lies on these lines: {2,18260}, {40,17615}, {72,6256}, {210,1158}, {515,960}, {518,12608}, {912,18242}, {956,6261}, {971,6796}, {2829,31837}, {3697,17649}, {3740,18238}, {3820,12616}, {3916,17661}, {4662,5690}, {5044,5450}, {5570,6941}, {5657,12666}, {6245,15064}, {12675,15325}

X(32159) = midpoint of X(i) and X(j) for these lines: {i,j}: {72, 6256}, {1158, 18239}, {6261, 14872}
X(32159) = reflection of X(5450) in X(5044)
X(32159) = anticomplement of X(18260)
X(32159) = {X(210), X(18239)}-harmonic conjugate of X(1158)


X(32160) = X(5)-OF-3rd EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^14+2*(2*b^2+b*c+2*c^2)*a^13+(b^2-c^2)*(b-c)*a^12-8*(2*b^4+2*c^4+(b+c)^2*b*c)*a^11-(b+c)*(19*b^4+19*c^4-2*(4*b^2+b*c+4*c^2)*b*c)*a^10+2*(b^2+c^2)*(10*b^4+10*c^4+(5*b^2+2*b*c+5*c^2)*b*c)*a^9+(b+c)*(45*b^6+45*c^6-(10*b^4+10*c^4+(29*b^2-4*b*c+29*c^2)*b*c)*b*c)*a^8-16*(2*b^2-3*b*c+2*c^2)*(b+c)^2*b^2*c^2*a^7-3*(b^2-c^2)*(b-c)*(15*b^6+15*c^6+(30*b^4+30*c^4+(25*b^2+36*b*c+25*c^2)*b*c)*b*c)*a^6-2*(b^2-c^2)^2*(10*b^6+10*c^6+(5*b^4+5*c^4-6*(b^2-b*c+c^2)*b*c)*b*c)*a^5+(b^2-c^2)*(b-c)*(19*b^8+19*c^8+2*(24*b^6+24*c^6+(18*b^4+18*c^4+(32*b^2+25*b*c+32*c^2)*b*c)*b*c)*b*c)*a^4+8*(b^2-c^2)^2*(b-c)^2*(2*b^6+2*c^6+(5*b^4+5*c^4+2*(3*b^2+4*b*c+3*c^2)*b*c)*b*c)*a^3-(b^2-c^2)^3*(b-c)*(b^6+c^6+(10*b^4+10*c^4+(19*b^2+36*b*c+19*c^2)*b*c)*b*c)*a^2-2*(b^4-c^4)*(b^2-c^2)^3*(b+c)^2*(2*b^2-3*b*c+2*c^2)*a-(b^2-c^2)^7*(b-c)) : :

X(32160) lies on these lines: {5909,6684}, {11212,11254}

X(32160) = midpoint of X(11254) and X(31965)
X(32160) = {X(11212), X(31965)}-harmonic conjugate of X(11254)


X(32161) = X(5)-OF-FEUERBACH TRIANGLE

Barycentrics
2*(b^2+b*c+c^2)*a^11+2*(b+c)*(b^2+b*c+c^2)*a^10-(10*b^4+10*c^4+(9*b^2+8*b*c+9*c^2)*b*c)*a^9-2*(b+c)*(5*b^4+5*c^4+(5*b^2+6*b*c+5*c^2)*b*c)*a^8+2*(10*b^6+10*c^6+(7*b^4+7*c^4-(b^2+3*b*c+c^2)*b*c)*b*c)*a^7+2*(b+c)*(2*b^2+b*c+2*c^2)*(5*b^4+5*c^4+(2*b^2-3*b*c+2*c^2)*b*c)*a^6-(20*b^8+20*c^8+(8*b^6+8*c^6-(34*b^4+34*c^4+(37*b^2+24*b*c+37*c^2)*b*c)*b*c)*b*c)*a^5-2*(b^2-c^2)*(b-c)*(10*b^6+10*c^6+(27*b^4+27*c^4+(36*b^2+37*b*c+36*c^2)*b*c)*b*c)*a^4+(b^2-c^2)^2*(10*b^6+10*c^6-(20*b^2+19*b*c+20*c^2)*b^2*c^2)*a^3+2*(b^2-c^2)^4*(b+c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^4*(b+c)^2*(2*b-c)*(b-2*c)*a-2*(b^2-c^2)^6*(b+c) : :

X(32161) lies on these lines: {5,5947}, {11,10281}, {30,31750}, {517,31756}, {952,10277}, {1154,31754}, {5499,10209}, {7951,10208}

X(32161) = midpoint of X(i) and X(j) for these lines: {i,j}: {5, 5948}, {5499, 10209}, {31750, 31764}, {31754, 31765}, {31756, 31759}
X(32161) = complement of X(5) wrt Feuerbach triangle
X(32161) = {X(5947), X(5948)}-harmonic conjugate of X(5)


X(32162) = X(5)-OF-GOSSARD TRIANGLE

Barycentrics    (S^2-3*SB*SC)*(5*S^2-2*R^2*(36*R^2+12*SA-19*SW)+6*SA^2-4*SB*SC-5*SW^2) : :
X(32162) = X(355)-3*X(11852) = X(1482)-3*X(16212) = 2*X(5901)-3*X(11831) = X(12794)+3*X(22698)

As a point on the Euler line, X(32162) has Shinagawa coefficients (11EF-34F2-2S2,3EF+30F2-2S2).

X(32162) lies on these lines: {2,3}, {355,11852}, {495,11905}, {496,11906}, {523,7740}, {952,12438}, {1482,16212}, {1483,11910}, {1511,9033}, {3564,12418}, {5690,11900}, {5844,12626}, {5874,11902}, {5875,11901}, {5901,11831}, {10942,11904}, {10943,11903}, {11839,32134}, {11848,32141}, {11885,32151}, {11907,32177}, {11908,32178}, {11914,32213}, {11915,32214}, {12181,12795}, {12696,28174}, {12790,12797}, {12792,22897}, {12793,22852}, {12794,22698}, {13894,13925}, {13948,13993}, {18958,18990}, {19017,19116}, {19018,19117}, {22755,32153}

X(32162) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 20128}, {3, 4240}, {4, 18508}, {402, 15774}, {11251, 12113}, {12790, 13212}
X(32162) = reflection of X(i) in X(j) for these (i,j): (5, 402), (1650, 140), (18507, 546)
X(32162) = X(18508)-of-Euler triangle
X(32162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1651, 12113, 11251), (11897, 18507, 546), (11911, 18508, 4)


X(32163) = X(5)-OF-3rd HATZIPOLAKIS TRIANGLE

Barycentrics    (512*R^6-R^4*(223*SA+79*SW)+12*R^2*SW*(-2*SW+7*SA)-4*SW^2*(-SW+2*SA))*S^2+4*(R^4*(64*R^2-53*SW)+SW^2*(13*R^2-SW))*SB*SC : :
X(32163) = X(31985)+3*X(32068)

X(32163) lies on the line {31985,32050}

X(32163) = midpoint of X(31985) and X(32050)
X(32163) = {X(31985), X(32068)}-harmonic conjugate of X(32050)


X(32164) = X(5)-OF-HUTSON EXTOUCH TRIANGLE

Barycentrics
a*((b+c)*a^11-(3*b^2-4*b*c+3*c^2)*a^10-(b+c)*(b^2+24*b*c+c^2)*a^9+(11*b^4+11*c^4+2*(8*b^2-23*b*c+8*c^2)*b*c)*a^8-2*(b+c)*(3*b^4+3*c^4-4*(8*b^2+9*b*c+8*c^2)*b*c)*a^7-2*(7*b^6+7*c^6+(36*b^4+36*c^4-(43*b^2+24*b*c+43*c^2)*b*c)*b*c)*a^6+2*(b+c)*(7*b^6+7*c^6-(24*b^4+24*c^4+(73*b^2-4*b*c+73*c^2)*b*c)*b*c)*a^5+2*(3*b^8+3*c^8+2*(20*b^6+20*c^6-(4*b^4+4*c^4+(16*b^2-29*b*c+16*c^2)*b*c)*b*c)*b*c)*a^4-(b^2-c^2)*(b-c)*(11*b^6+11*c^6+(22*b^4+22*c^4-(47*b^2+84*b*c+47*c^2)*b*c)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)*(b^4+c^4-2*(14*b^2+13*b*c+14*c^2)*b*c)*a^2+(b^2-c^2)^3*(b-c)*(3*b^4+3*c^4+2*(7*b^2+13*b*c+7*c^2)*b*c)*a-(b^2-c^2)^6) : :
X(32164) = 3*X(5918)+X(31968) = 3*X(5918)-X(32051) = 3*X(10167)-X(12731)

X(32164) lies on these lines: {5918,31968}, {10167,12731}

X(32164) = midpoint of X(31968) and X(32051)
X(32164) = {X(5918), X(31968)}-harmonic conjugate of X(32051)


X(32165) = X(5)-OF-1st HYACINTH TRIANGLE

Barycentrics    (5*R^2-4*SA+2*SW)*S^2+(R^2-2*SW)*SB*SC : :
X(32165) = X(140)-3*X(11245) = X(143)-3*X(11225) = 3*X(143)-X(13419) = X(389)+3*X(11232) = 3*X(6102)+X(21659) = X(10116)+3*X(11225) = 3*X(10116)+X(13419) = 9*X(11225)-X(13419) = 3*X(11232)-X(11264) = 2*X(11745)-3*X(16881) = X(13142)-5*X(13292) = 3*X(13142)+5*X(18914) = 3*X(13292)+X(18914) = 3*X(32068)-2*X(32205)

X(32165) lies on these lines: {5,1199}, {6,18356}, {30,52}, {54,140}, {125,1493}, {143,10116}, {184,18282}, {265,546}, {389,6153}, {397,11600}, {539,12006}, {542,10095}, {575,3564}, {1154,27552}, {1353,13371}, {1594,2914}, {1614,25338}, {1658,18925}, {1885,7722}, {1994,15002}, {2888,15037}, {3448,14627}, {3850,18388}, {3861,18379}, {5663,19481}, {5965,32142}, {6515,7525}, {6748,15557}, {6776,17714}, {8254,13366}, {10112,13630}, {10224,12161}, {11250,18913}, {11536,25328}, {11745,16881}, {11801,25402}, {11898,13154}, {12107,31804}, {12134,13163}, {12899,13367}, {13391,18128}, {13392,16238}, {15114,15130}, {15120,32144}, {15330,19357}, {16982,29012}, {20424,25739}, {22249,32171}, {32068,32205}

X(32165) = midpoint of X(i) and X(j) for these lines: {i,j}: {143, 10116}, {389, 11264}, {10112, 13630}, {10114, 13358}
X(32165) = reflection of X(12134) in X(13163)
X(32165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10116, 11225, 143), (13353, 21230, 140)


X(32166) = X(5)-OF-2nd HYACINTH TRIANGLE

Barycentrics    SA*(2*(6*R^2-SW)*S^2+(SB+SC)*(2*R^4-(3*SA+5*SW)*R^2+SW^2)) : :
X(32166) = X(1147)-3*X(11245) = 3*X(11245)+X(12421)

X(32166) lies on these lines: {25,32145}, {68,7592}, {575,3564}, {1147,11245}, {6776,32048}, {10116,12235}, {11264,13630}, {11433,12420}, {12241,13292}, {17702,18914}, {18951,19908}

X(32166) = midpoint of X(i) and X(j) for these lines: {i,j}: {1147, 12421}, {10116, 12235}
X(32166) = X(32145)-of-anti-Ara triangle
X(32166) = {X(11245), X(12421)}-harmonic conjugate of X(1147)


X(32167) = X(5)-OF-INCENTRAL TRIANGLE

Barycentrics    a*(3*(b+c)*a^5+(b^2+4*b*c+c^2)*a^4-(b+c)*(6*b^2-b*c+6*c^2)*a^3-2*(b^4+c^4+3*(b^2+b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(3*b^2+5*b*c+3*c^2)*a+(b^2-c^2)^2*(b+c)^2) : :
X(32167) = X(500)+3*X(1962) = 3*X(1962)-X(8143)

Let LA be the trilinear polar of A wrt BCX(1), and define LB and LC cyclically. Let LI be the trilinear polar of X(1) wrt ABC (the antiorthic axis). Then X(32167) = QL-P11 (Nine-point Center QL-Diagonal Triangle) of quadrilateral LALBLCLI. (Randy Hutson, June 7, 2019)

X(32167) lies on these lines: {1,3}, {81,22937}, {500,1962}, {612,32148}, {2771,3743}, {9955,15569}, {11230,24936}, {11231,24883}

X(32167) = midpoint of X(i) and X(j) for these lines: {i,j}: {500, 8143}, {3743, 5453}
X(32167) = reflection of X(1) in X(10618)
X(32167) = X(10618)-of-Aquila triangle
X(32167) = X(15557)-of-excentral triangle
X(32167) = {X(500), X(1962)}-harmonic conjugate of X(8143)
X(32167) = centroid of X(11)X(214)X(500)X(4065)
X(32167) = centroid of X(500) and the vertices of the incentral triangle
X(32167) = QA-P13 (Nine-point Center of the QA-Diagonal Triangle) of quadrangle ABCX(1)


X(32168) = X(5)-OF-INTANGENTS TRIANGLE

Barycentrics    a^2*((b-c)^2*a^6-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^4+(3*b^6+3*c^6+2*(b^4+c^4-(b+c)^2*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^2+b*c+c^2)^2) : :
X(32168) = X(6238)+3*X(11189) = X(8144)-3*X(11189)

X(32168) lies on these lines: {1,3024}, {3,11446}, {5,3270}, {11,13561}, {26,2192}, {33,143}, {34,32137}, {35,32171}, {36,32210}, {55,156}, {56,32138}, {497,32140}, {1040,32142}, {1062,11591}, {1154,6238}, {1250,32208}, {2066,32169}, {2330,19155}, {3100,6101}, {3295,32139}, {3583,18379}, {5414,32170}, {5876,18455}, {5889,9642}, {5944,9638}, {6102,6198}, {6243,9539}, {6284,30522}, {6285,32047}, {7071,12161}, {8540,32155}, {9538,18436}, {9641,11412}, {9817,32205}, {10095,11436}, {10638,32207}, {11017,19372}, {11429,32136}, {13491,18447}, {18922,18952}, {19182,19211}

X(32168) = midpoint of X(i) and X(j) for these lines: {i,j}: {6238, 8144}, {6285, 32047}
X(32168) = reflection of X(32158) in X(156)
X(32168) = X(156)-of-Mandart-incircle triangle
X(32168) = X(32138)-of-2nd anti-circumperp-tangential triangle
X(32168) = X(32143)-of-5th mixtilinear triangle
X(32168) = reflection of X(32143) in X(1)
X(32168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6238, 11189, 8144), (11446, 11461, 3)


X(32169) = X(5)-OF-1st KENMOTU DIAGONALS TRIANGLE

Barycentrics    (SB+SC)*(S^2+4*(4*R^2-SW)*S-(2*R^2-3*SA+2*SW)*SA) : :
X(32169) = X(10665)+3*X(11241) = 3*X(11241)-X(11265)

X(32169) lies on these lines: {3,11447}, {5,21640}, {6,156}, {26,17819}, {143,5412}, {371,5663}, {372,32171}, {590,13561}, {1151,32138}, {1154,10665}, {2066,32168}, {2067,32143}, {3068,32140}, {3070,30522}, {3311,32139}, {5410,12161}, {5415,32158}, {5876,18457}, {5944,18459}, {6101,11417}, {6102,10880}, {6200,32210}, {6564,18379}, {10201,19061}, {10533,11266}, {10897,11591}, {10961,32205}, {11473,32137}, {11513,32142}, {13909,18356}, {18923,18952}, {19183,19211}, {19355,32046}

X(32169) = midpoint of X(10665) and X(11265)
X(32169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 156, 32170), (11447, 11462, 3)


X(32170) = X(5)-OF-2nd KENMOTU DIAGONALS TRIANGLE

Barycentrics    (SB+SC)*(S^2-4*(4*R^2-SW)*S-(2*R^2-3*SA+2*SW)*SA) : :
X(32170) = X(10666)+3*X(11242) = 3*X(11242)-X(11266)

X(32170) lies on these lines: {3,11448}, {5,21641}, {6,156}, {26,17820}, {143,5413}, {371,32171}, {372,5663}, {615,13561}, {1152,32138}, {1154,10666}, {3069,32140}, {3071,30522}, {3312,32139}, {5411,12161}, {5414,32168}, {5416,32158}, {5876,18459}, {5944,18457}, {6101,11418}, {6102,10881}, {6396,32210}, {6502,32143}, {6565,18379}, {10201,19062}, {10534,11265}, {10898,11591}, {10963,32205}, {11474,32137}, {11514,32142}, {13970,18356}, {18924,18952}, {19184,19211}, {19356,32046}

X(32170) = midpoint of X(10666) and X(11266)
X(32170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 156, 32169), (11448, 11463, 3)


X(32171) = X(5)-OF-KOSNITA TRIANGLE

Barycentrics    a^2*(2*a^8-5*(b^2+c^2)*a^6+3*(b^2+c^2)^2*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(32171) = 3*X(3)-X(32138) = 3*X(3)+X(32139) = X(26)-5*X(17821) = 3*X(154)+X(12084) = X(155)+3*X(18324) = 3*X(156)+X(32138) = 3*X(156)-X(32139) = 2*X(156)+X(32210) = 5*X(631)-X(32140) = X(1147)+3*X(11202) = X(1658)-3*X(11202) = X(5449)-3*X(10182) = X(9833)+3*X(18281) = 2*X(10125)-3*X(10182) = 3*X(10192)-X(15761) = 3*X(10193)-4*X(10212) = 3*X(10201)+X(12118) = 5*X(15034)-X(15132) = 2*X(32138)-3*X(32210) = 2*X(32139)+3*X(32210)

Let NANBNC be the reflection triangle of X(5). Let OA be the circumcenter of ANBNC, and define OB and OC cyclically; then X(32171) is the circumcenter of OAOBOC. (Randy Hutson, June 7, 2019)

X(32171) lies on these lines: {2,6288}, {3,74}, {4,16665}, {5,13367}, {15,32208}, {16,32207}, {24,143}, {26,13391}, {30,5448}, {35,32168}, {36,32143}, {49,186}, {52,7575}, {54,5946}, {140,13561}, {154,12084}, {155,18324}, {184,13630}, {185,15646}, {265,14940}, {371,32170}, {372,32169}, {376,18442}, {378,32137}, {382,26882}, {389,32136}, {468,12370}, {511,12107}, {539,15330}, {542,20191}, {546,11430}, {567,15026}, {568,1493}, {569,9827}, {575,32155}, {578,10095}, {631,3410}, {1092,7502}, {1147,1154}, {1495,3627}, {1503,23336}, {1657,26881}, {2070,10263}, {3090,14805}, {3091,3431}, {3515,12161}, {3520,10540}, {5020,12046}, {5072,10546}, {5092,12108}, {5449,10125}, {5498,20299}, {5889,9703}, {5890,9704}, {6000,10226}, {6101,7488}, {6642,32205}, {6644,12006}, {6759,11250}, {7506,13364}, {7555,15644}, {7712,17538}, {9306,14128}, {9544,21844}, {9706,15087}, {9818,11017}, {9833,18281}, {10113,16868}, {10192,15761}, {10193,10212}, {10201,12118}, {10224,18400}, {10254,12278}, {10255,12289}, {10264,17701}, {10298,18436}, {10539,18570}, {10564,15704}, {10902,32158}, {11425,13861}, {13163,19130}, {13346,17714}, {13406,17702}, {13754,15331}, {14070,16266}, {15060,18350}, {15074,15462}, {15088,18396}, {15647,22802}, {15712,22352}, {15806,18388}, {16223,21660}, {18925,18952}, {19154,23041}, {19185,19211}, {19479,20773}, {22249,32165}, {23292,31830}

X(32171) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 156}, {1147, 1658}, {6759, 11250}, {10282, 12038}, {13346, 17714}, {15647, 25487}, {32138, 32139}
X(32171) = reflection of X(i) in X(j) for these (i,j): (5449, 10125), (13561, 140), (18379, 5), (20299, 5498), (32155, 575), (32210, 3)
X(32171) = X(18379)-of-Johnson triangle
X(32171) = X(32210)-of-ABC-X3 reflections triangle
X(32171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 399, 11440), (3, 1614, 13491), (3, 32139, 32138)


X(32172) = X(5)-OF-LEMOINE TRIANGLE

Barycentrics    9*(6*R^2-3*SA+7*SW)*S^4+3*(9*R^2*SA*(SB+SC)+SW*(21*SA^2-30*SA*SW+14*SW^2))*S^2+SB*SC*SW^3 : :
X(32172) = X(8145)-3*X(32069) = X(31969)+3*X(32069)

X(32172) lies on these lines: {5397,8523}, {8145,31969}

X(32172) = midpoint of X(8145) and X(31969)
X(32172) = {X(31969), X(32069)}-harmonic conjugate of X(8145)


X(32173) = X(5)-OF-LUCAS ANTIPODAL TRIANGLE

Barycentrics    4*(4*R^2-2*SA+SW)*S^4-(8*R^2*SA*(SB+SC)+SW*(2*SA^2-6*SA*SW+3*SW^2))*S^2+SB*SC*SW^3-S*(8*S^4+4*(2*(SB+SC)*R^2+SA*(5*SA-4*SW))*S^2-4*SB*SC*SW^2) : :

X(32173) lies on the line {31970,32052}

X(32173) = midpoint of X(31970) and X(32052)


X(32174) = X(5)-OF-LUCAS(-1) ANTIPODAL TRIANGLE

Barycentrics    4*(4*R^2-2*SA+SW)*S^4-(8*R^2*SA*(SB+SC)+SW*(2*SA^2-6*SA*SW+3*SW^2))*S^2+SB*SC*SW^3+S*(8*S^4+4*(2*(SB+SC)*R^2+SA*(5*SA-4*SW))*S^2-4*SB*SC*SW^2) : :

X(32174) lies on the line {31971,32053}

X(32174) = midpoint of X(31971) and X(32053)


X(32175) = X(5)-OF-LUCAS CENTRAL TRIANGLE

Barycentrics    (SB+SC)*(30*S^4-(4*R^2*SA-23*SA^2+17*SB*SC-3*SW^2)*S^2+SA*SW^3-S*((2*R^2-16*SA-13*SW)*S^2-3*(SA+2*SW)*SA*SW)) : :
X(32175) = X(8155)-3*X(11198)

X(32175) lies on these lines: {1151,7598}, {8155,11198}

X(32175) = midpoint of X(8155) and X(31972)
X(32175) = {X(11198), X(31972)}-harmonic conjugate of X(8155)


X(32176) = X(5)-OF-LUCAS(-1) CENTRAL TRIANGLE

Barycentrics    (SB+SC)*(30*S^4-(4*R^2*SA-23*SA^2+17*SB*SC-3*SW^2)*S^2+SA*SW^3+S*((2*R^2-16*SA-13*SW)*S^2-3*(SA+2*SW)*SA*SW)) : :
X(32176) = X(8156)-3*X(32077) = X(31973)+3*X(32077)

X(32176) lies on these lines: {1152,7599}, {8156,31973}

X(32176) = midpoint of X(8156) and X(31973)
X(32176) = {X(31973), X(32077)}-harmonic conjugate of X(8156)


X(32177) = X(5)-OF-LUCAS HOMOTHETIC TRIANGLE

Barycentrics    (4*SA^2-3*SW^2)*S^2+SB*SC*SW^2-4*S*((2*R^2+SB+SC)*S^2-2*R^2*SB*SC) : :
X(32177) = X(9838)+3*X(12152) = X(10669)-3*X(12152) = X(12992)+3*X(22709)

X(32177) lies on these lines: {3,6462}, {4,11949}, {5,493}, {26,8194}, {30,9838}, {140,8222}, {355,8188}, {495,11930}, {496,11932}, {546,8212}, {550,11828}, {952,12440}, {1483,8210}, {3564,12426}, {5690,8214}, {5844,12636}, {5874,8218}, {5875,8216}, {5901,11377}, {6461,32178}, {6756,11394}, {8201,32146}, {8208,32147}, {10875,32151}, {10942,10951}, {10943,10945}, {11503,32141}, {11840,32134}, {11907,32162}, {11947,15171}, {11955,32213}, {11957,32214}, {12186,12994}, {12894,12998}, {12988,22908}, {12990,22863}, {12992,22709}, {13899,13925}, {13956,13993}, {18963,18990}, {19031,19116}, {19032,19117}, {22761,32153}, {22841,28174}

X(32177) = midpoint of X(i) and X(j) for these lines: {i,j}: {9838, 10669}, {12894, 13215}
X(32177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (493, 8220, 5), (6462, 11846, 3)


X(32178) = X(5)-OF-LUCAS(-1) HOMOTHETIC TRIANGLE

Barycentrics    (4*SA^2-3*SW^2)*S^2+SB*SC*SW^2+4*S*((2*R^2+SB+SC)*S^2-2*R^2*SB*SC) : :
X(32178) = X(9839)+3*X(12153) = X(10673)-3*X(12153) = X(12993)+3*X(22710)

X(32178) lies on these lines: {3,6463}, {4,11950}, {5,494}, {26,8195}, {30,9839}, {140,8223}, {355,8189}, {495,11931}, {496,11933}, {546,8213}, {550,11829}, {952,12441}, {1483,8211}, {3564,12427}, {5690,8215}, {5844,12637}, {5874,8219}, {5875,8217}, {5901,11378}, {6461,32177}, {6756,11395}, {8202,32146}, {8209,32147}, {10876,32151}, {10942,10952}, {10943,10946}, {11504,32141}, {11841,32134}, {11908,32162}, {11948,15171}, {11956,32213}, {11958,32214}, {12187,12995}, {12895,12999}, {12989,22909}, {12991,22864}, {12993,22710}, {13900,13925}, {13957,13993}, {18964,18990}, {19033,19116}, {19034,19117}, {22762,32153}, {22842,28174}

X(32178) = midpoint of X(i) and X(j) for these lines: {i,j}: {9839, 10673}, {12895, 13216}
X(32178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (494, 8221, 5), (6463, 11847, 3)


X(32179) = X(5)-OF-LUCAS TANGENTS TRIANGLE

Barycentrics    (SB+SC)*(44*S^4+(4*R^2*SA+31*SA^2-33*SB*SC+11*SW^2)*S^2+3*SA*SW^3+S*((2*R^2+20*SA+37*SW)*S^2+(17*SA^2-18*SB*SC+2*SW^2)*SW)) : :
X(32179) = X(1151)-3*X(11199) = 3*X(11199)+X(31974)

X(32179) lies on the line {1151,8155}

X(32179) = midpoint of X(1151) and X(31974)
X(32179) = {X(11199), X(31974)}-harmonic conjugate of X(1151)


X(32180) = X(5)-OF-LUCAS(-1) TANGENTS TRIANGLE

Barycentrics    (SB+SC)*(44*S^4+(4*R^2*SA+31*SA^2-33*SB*SC+11*SW^2)*S^2+3*SA*SW^3-S*((2*R^2+20*SA+37*SW)*S^2+(17*SA^2-18*SB*SC+2*SW^2)*SW)) : :
X(32180) = X(1152)-3*X(32074) = X(31975)+3*X(32074)

X(32180) lies on these lines: {1152,8156}, {6398,31973}

X(32180) = midpoint of X(1152) and X(31975)
X(32180) = {X(31975), X(32074)}-harmonic conjugate of X(1152)


X(32181) = X(5)-OF-MACBEATH TRIANGLE

Barycentrics    3*S^4-(R^2*(7*SA-3*SW)-2*SA^2+SB*SC+SW^2)*S^2+(2*R^2-SW)*(16*R^2-5*SW)*SB*SC : :
X(32181) = X(8146)-3*X(11197) = 3*X(11197)+X(31976)

X(32181) lies on these lines: {5,10274}, {5576,14674}, {8146,11197}

X(32181) = midpoint of X(8146) and X(31976)
X(32181) = {X(11197), X(31976)}-harmonic conjugate of X(8146)


X(32182) = X(5)-OF-MANDART-EXCIRCLES TRIANGLE

Barycentrics
2*a^13-(7*b^2+4*b*c+7*c^2)*a^11-(b+c)*(b^2-8*b*c+c^2)*a^10+(7*b^2-2*b*c+7*c^2)*(b+c)^2*a^9+(b+c)*(3*b^4+3*c^4-2*(11*b^2-b*c+11*c^2)*b*c)*a^8+2*(b^6+c^6-(8*b^4+8*c^4-(b^2+8*b*c+c^2)*b*c)*b*c)*a^7-2*(b+c)*(b^2-8*b*c+c^2)*(b^2+c^2)^2*a^6-8*(b^6+c^6-(3*b^4+3*c^4-4*(b^2+c^2)*b*c)*b*c)*(b^2+b*c+c^2)*a^5-2*(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a^4+(b^2-c^2)^2*(b-c)^2*(5*b^4+5*c^4-2*(b^2-7*b*c+c^2)*b*c)*a^3+(b^2-c^2)^3*(b-c)^3*(3*b^2+4*b*c+3*c^2)*a^2-(b^2-c^2)^4*(b-c)^4*a-(b^2-c^2)^5*(b-c)*(b^2+c^2) : :
X(32182) = 3*X(11246)+X(31977) = 3*X(11246)-X(32054)

X(32182) lies on these lines: {4338,12659}, {11246,31977}

X(32182) = midpoint of X(31977) and X(32054)
X(32182) = {X(11246), X(31977)}-harmonic conjugate of X(32054)


X(32183) = X(5)-OF-2nd MIDARC TRIANGLE

Barycentrics    a*(6*(-a+b+c)*sin(A/2)*a*b*c-(a+b-c)*(a^2-4*c*a+c^2-b^2)*sin(C/2)*b-(a-b+c)*(a^2-4*b*a+b^2-c^2)*sin(B/2)*c) : :
X(32183) = 3*X(1)-X(177) = 5*X(1)-3*X(11191) = X(1)-3*X(11234) = X(177)+3*X(8422) = 3*X(3241)+X(11691) = 3*X(5919)+X(17641) = X(31767)+2*X(31796)

X(32183) lies on these lines: {1,167}, {3,31800}, {30,31734}, {164,31393}, {495,12614}, {496,12622}, {517,5571}, {519,18258}, {999,12518}, {1317,10501}, {1385,31790}, {3241,11691}, {3488,12694}, {5045,31768}, {5882,8100}, {5919,17641}, {8094,10968}, {11040,11528}, {12443,24929}, {15172,31769}, {31766,31792}

X(32183) = midpoint of X(i) and X(j) for these lines: {i,j}: {1, 8422}, {5571, 31767}, {31734, 31770}
X(32183) = reflection of X(i) in X(j) for these (i,j): (5571, 31796), (12908, 1), (31766, 31792), (31768, 5045), (31769, 15172), (31790, 1385), (31800, 3)
X(32183) = inverse of X(30423) in the incircle
X(32183) = X(40)-of-incircle-circles triangle
X(32183) = X(942)-of-2nd tangential-midarc triangle
X(32183) = X(1385)-of-Hutson intouch triangle
X(32183) = X(1902)-of-2nd circumperp triangle
X(32183) = X(8422)-of-anti-Aquila triangle
X(32183) = X(9955)-of-Ursa-minor triangle
X(32183) = X(12699)-of-inverse-in-incircle triangle
X(32183) = X(12908)-of-5th mixtilinear triangle
X(32183) = X(31800)-of-ABC-X3 reflections triangle
X(32183) = {X(8422), X(11234)}-harmonic conjugate of X(1)


X(32184) = X(5)-OF-MIDHEIGHT TRIANGLE

Barycentrics    (SB+SC)*((10*R^2-SA-3*SW)*S^2+(4*R^2*(16*R^2-5*SW)-SA^2+SB*SC+2*SW^2)*SA) : :
X(32184) = 3*X(51)+X(5894) = X(52)+3*X(23328) = X(64)+7*X(15043) = X(185)+3*X(23332) = X(1498)-9*X(15045) = 3*X(1853)+5*X(10574) = 3*X(3060)+5*X(8567) = 5*X(3091)+3*X(7729) = X(3357)+3*X(5946) = 5*X(3567)+3*X(10606) = 9*X(5640)-X(5895) = 3*X(5892)-X(16252) = X(5893)-3*X(5943) = X(5925)+7*X(9781) = 3*X(5943)+X(31978) = X(6102)+3*X(23329) = X(6247)+3*X(9730) = 3*X(10193)-X(10627) = X(10263)+3*X(11204) = X(10575)+3*X(23324)

X(32184) lies on these lines: {3,9972}, {5,22800}, {51,5894}, {52,23328}, {64,15043}, {185,23332}, {389,2781}, {974,1594}, {1154,25563}, {1192,10249}, {1498,15045}, {1503,9729}, {1620,6403}, {1853,10574}, {1885,11746}, {2393,17704}, {2777,10095}, {3060,8567}, {3091,7729}, {3357,5946}, {3567,10606}, {3574,23315}, {3850,6000}, {5462,15311}, {5640,5895}, {5890,12300}, {5892,16252}, {5893,5943}, {5925,9781}, {6102,23329}, {6247,7403}, {6746,21663}, {8991,12240}, {9786,32191}, {10193,10627}, {10263,11204}, {10575,23324}, {10628,30531}, {11598,14865}, {12239,13980}, {13358,25564}, {13491,23325}, {13630,20299}, {13754,32144}, {14644,22948}, {15026,22802}, {17845,20791}, {18381,22804}

X(32184) = midpoint of X(i) and X(j) for these lines: {i,j}: {389, 6696}, {5893, 31978}, {13358, 25564}, {13630, 20299}
X(32184) = {X(5943), X(31978)}-harmonic conjugate of X(5893)


X(32185) = X(5)-OF-MIXTILINEAR TRIANGLE

Barycentrics
4*(b+c)*a^7-(9*b^2+8*b*c+9*c^2)*a^6+24*(b+c)*b*c*a^5+(11*b^4+11*c^4-14*(2*b^2+b*c+2*c^2)*b*c)*a^4-4*(b^2-c^2)^2*(b+c)*a^3-(3*b^4+3*c^4-2*(b^2+9*b*c+c^2)*b*c)*(b-c)^2*a^2+8*(b^2-c^2)*(b-c)^3*b*c*a+(b^2-c^2)^2*(b-c)^4 : :
X(32185) = X(7961)+3*X(11200) = X(8147)-3*X(11200)

X(32185) lies on these lines: {1,32186}, {7961,8147}

X(32185) = midpoint of X(7961) and X(8147)
X(32185) = {X(7961), X(11200)}-harmonic conjugate of X(8147)


X(32186) = X(5)-OF-2nd MIXTILINEAR TRIANGLE

Barycentrics
4*(b+c)*a^9-(17*b^2+8*b*c+17*c^2)*a^8+22*(b+c)*(b^2+c^2)*a^7+2*(b^4+c^4-(41*b^2-24*b*c+41*c^2)*b*c)*a^6-2*(b+c)*(13*b^4+13*c^4-6*(12*b^2-17*b*c+12*c^2)*b*c)*a^5+2*(8*b^4+8*c^4-(5*b^2+54*b*c+5*c^2)*b*c)*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)*(b^2-10*b*c+c^2)*(b^2-4*b*c+c^2)*a^3-2*(b^4+c^4-(b^2+8*b*c+c^2)*b*c)*(b-c)^4*a^2-2*(b^2-c^2)*(b-c)^7*a+(b^2-c^2)^2*(b-c)^6 : :
X(32186) = X(7960)+3*X(11201) = X(8159)-3*X(11201)

X(32186) lies on these lines: {1,32185}, {7960,8159}

X(32186) = midpoint of X(7960) and X(8159)
X(32186) = {X(7960), X(11201)}-harmonic conjugate of X(8159)


X(32187) = X(5)-OF-3rd MIXTILINEAR TRIANGLE

Barycentrics
a*(8*a^9-22*(b+c)*a^8-8*(b^2-14*b*c+c^2)*a^7+4*(b+c)*(16*b^2-47*b*c+16*c^2)*a^6-(24*b^4+24*c^4+(143*b^2-448*b*c+143*c^2)*b*c)*a^5-(b+c)*(60*b^4+60*c^4-(371*b^2-668*b*c+371*c^2)*b*c)*a^4+2*(20*b^6+20*c^6-(23*b^4+23*c^4+(173*b^2-380*b*c+173*c^2)*b*c)*b*c)*a^3+2*(b+c)*(8*b^6+8*c^6-(87*b^4+87*c^4-11*(27*b^2-40*b*c+27*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(16*b^4+16*c^4-7*(11*b^2-18*b*c+11*c^2)*b*c)*a+(b^2-c^2)^3*(b-c)*(2*b-c)*(b-2*c)) : :
X(32187) = X(3)-3*X(32075) = X(31979)+3*X(32075)

X(32187) lies on the line {3,31979}

X(32187) = midpoint of X(3) and X(31979)
X(32187) = {X(31979), X(32075)}-harmonic conjugate of X(3)


X(32188) = X(5)-OF-4th MIXTILINEAR TRIANGLE

Barycentrics
a*(8*a^9-10*(b+c)*a^8-8*(4*b^2-b*c+4*c^2)*a^7+4*(b+c)*(12*b^2-7*b*c+12*c^2)*a^6+(32*b^4+32*c^4-(7*b^2-48*b*c+7*c^2)*b*c)*a^5-(b+c)*(68*b^4+68*c^4-(31*b^2-36*b*c+31*c^2)*b*c)*a^4+2*(19*b^2+39*b*c+19*c^2)*(b-c)^2*b*c*a^3+2*(b^2-c^2)*(b-c)*(16*b^4+16*c^4+(11*b^2-15*b*c+11*c^2)*b*c)*a^2-(b^2-c^2)^2*(b-c)^2*(8*b^2+23*b*c+8*c^2)*a-(b^2-c^2)*(b-c)^5*(b+2*c)*(2*b+c)) : :
X(32188) = X(3)-3*X(32076) = X(31980)+3*X(32076)

X(32188) lies on the line {3,31980}

X(32188) = midpoint of X(3) and X(31980)
X(32188) = X(31744)-of-1st circumperp triangle
X(32188) = X(32156)-of-excentral triangle
X(32188) = {X(31980), X(32076)}-harmonic conjugate of X(3)


X(32189) = X(5)-OF-1st NEUBERG TRIANGLE

Barycentrics    (b^2+c^2)*a^6-(b^2+c^2)^2*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-2*b^4*c^4 : :
X(32189) = 3*X(2)+X(31981) = X(76)+3*X(13085) = 3*X(262)-X(7759) = X(6309)-5*X(7786) = X(7781)-3*X(11171) = X(18768)-5*X(31276)

X(32189) lies on these lines: {2,39}, {3,8150}, {5,736}, {140,698}, {230,18806}, {262,7759}, {511,7780}, {626,2023}, {732,7764}, {754,14881}, {1078,1916}, {2021,7816}, {3094,7815}, {3095,7751}, {3202,3506}, {5976,7749}, {6680,24256}, {7752,9983}, {7775,10356}, {7781,11171}, {7804,13357}, {7895,14994}, {9462,18573}, {9862,18769}, {10357,13086}

X(32189) = midpoint of X(i) and X(j) for these lines: {i,j}: {3095, 7751}, {8149, 31981}
X(32189) = reflection of X(7764) in X(11272)
X(32189) = complement of X(8149)
X(32189) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31981, 8149), (76, 7746, 3934), (76, 7786, 7836)


X(32190) = X(5)-OF-2nd NEUBERG TRIANGLE

Barycentrics    (3*b^4+8*b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(b^4-8*b^2*c^2+c^4)*a^2-c^8-(b^2-c^2)^2*b^2*c^2-b^8 : :
X(32190) = 3*X(2)+X(31982) = X(6308)+3*X(9765) = X(6308)-5*X(31268) = 3*X(9765)+5*X(31268)

X(32190) lies on these lines: {2,32}, {39,9478}, {114,6287}, {732,7764}, {3788,24273}, {5031,6683}, {7697,8149}, {7769,8290}, {7771,9990}, {8177,25555}

X(32190) = midpoint of X(i) and X(j) for these lines: {i,j}: {2896, 18548}, {7775, 13086}, {8150, 31982}
X(32190) = complement of X(8150)
X(32190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31982, 8150), (1506, 6292, 83), (9765, 31268, 6308)


X(32191) = X(5)-OF-1st ORTHOSYMMEDIAL TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^4+c^4)*a^6-5*(b^2+c^2)*b^2*c^2*a^4+2*(b^2-c^2)^2*(b^4+3*b^2*c^2+c^4)*a^2-(b^6+c^6)*(b^2-c^2)^2) : :
X(32191) = X(6)-5*X(3567) = 3*X(6)+X(6403) = 5*X(6)-X(15073) = 3*X(51)-X(5480) = 9*X(51)-X(12294) = 3*X(51)+X(19161) = X(182)-3*X(5946) = 3*X(568)+X(1352) = 3*X(597)-X(9967) = X(1350)+3*X(3060) = X(1351)-9*X(13321) = X(1352)-3*X(16776) = 15*X(3567)+X(6403) = X(3589)+2*X(21852) = 5*X(3763)-X(11412) = 3*X(5050)-X(17710) = 3*X(5085)-7*X(15043) = 3*X(5480)-X(12294) = 5*X(6403)+3*X(15073) = X(12294)+3*X(19161)

X(32191) lies on these lines: {4,15321}, {6,24}, {51,125}, {52,141}, {140,143}, {159,11432}, {182,5946}, {251,19165}, {389,1503}, {518,31760}, {524,10127}, {542,13358}, {568,1352}, {575,12107}, {576,22828}, {597,9967}, {1154,24206}, {1350,3060}, {1351,13321}, {1353,2854}, {1843,6746}, {2393,12007}, {3098,10263}, {3564,16881}, {3763,11412}, {3818,6102}, {5050,17710}, {5085,15043}, {5092,12006}, {5097,14984}, {5446,29181}, {5449,10095}, {5889,10516}, {5900,22336}, {5943,11548}, {6467,12061}, {6593,16222}, {6776,9971}, {7592,20987}, {7716,15581}, {9786,32184}, {9822,16625}, {9973,14912}, {10110,16198}, {10625,21167}, {10691,21849}, {11002,16063}, {11438,15578}, {11649,15516}, {13391,14810}, {13630,29012}, {16224,28343}, {17810,19149}, {20391,21851}

X(32191) = midpoint of X(i) and X(j) for these lines: {i,j}: {52, 141}, {389, 9969}, {568, 16776}, {1843, 8550}, {3098, 10263}, {3818, 6102}, {5462, 21852}, {5480, 19161}, {6467, 12061}, {9822, 16625}
X(32191) = reflection of X(i) in X(j) for these (i,j): (3589, 5462), (5092, 12006), (19130, 10095)
X(32191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 19161, 5480), (5480, 13567, 20300)


X(32192) = X(5)-OF-2nd ORTHOSYMMEDIAL TRIANGLE

Barycentrics    (SB+SC)*(3*(6*R^2-7*SW)*(3*R^2-SW)*S^4-(9*R^2*SA*(12*R^2+3*SA-8*SW)+14*R^2*SW^2-SW*(9*SA^2-12*SA*SW+7*SW^2))*SW*S^2+(4*R^2+3*SA-4*SW)*SA*SW^4) : :
X(32192) = X(5480)-3*X(11226) = 3*X(11226)+X(31983)

X(32192) lies on these lines: {182,7575}, {2781,31771}, {5480,11226}

X(32192) = midpoint of X(5480) and X(31983)
X(32192) = X(31744)-of-1st orthosymmedial triangle
X(32192) = {X(11226), X(31983)}-harmonic conjugate of X(5480)


X(32193) = X(5)-OF-2nd PARRY TRIANGLE

Barycentrics    (3*a^4-(b^2+c^2)*a^2-b^4+b^2*c^2-c^4)*(b^2-c^2) : :
X(32193) = 5*X(351)-3*X(9123) = 3*X(351)-X(9131) = X(351)-3*X(9185) = 3*X(1637)-X(9134) = 2*X(5113)+X(14316) = 4*X(6587)-X(23301) = 9*X(9123)-5*X(9131) = X(9123)-5*X(9185) = 3*X(9123)+5*X(9979) = 6*X(9123)-5*X(14610) = X(9131)-9*X(9185) = X(9131)+3*X(9979) = 2*X(9131)-3*X(14610) = 2*X(9134)-3*X(10278) = 3*X(9185)+X(9979) = 6*X(9185)-X(14610) = 3*X(9189)-X(14417) = 2*X(9979)+X(14610) = X(13307)+3*X(22734) = 3*X(14420)+X(14424)

X(32193) lies on these lines: {2,9479}, {111,9076}, {351,523}, {525,9208}, {526,14697}, {690,5461}, {804,1637}, {826,4142}, {900,9811}, {2450,3566}, {2799,10190}, {3124,7668}, {3800,5027}, {4977,9810}, {8029,9147}, {8704,32194}, {9007,13303}, {9138,13291}, {9148,10189}, {9189,14417}, {9200,22889}, {9201,22934}, {9512,20998}, {10276,14698}, {13250,28221}, {13251,28217}, {13307,22734}, {14272,14977}

X(32193) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 14420}, {351, 9979}, {8029, 9147}, {9138, 13291}, {14272, 14977}
X(32193) = reflection of X(i) in X(j) for these (i,j): (9148, 10189), (10190, 11176), (10278, 1637), (14610, 351)
X(32193) = complement of X(14424)
X(32193) = X(550)-of-1st Parry triangle
X(32193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9185, 9979, 351), (13319, 13320, 3569)


X(32194) = X(5)-OF-3rd PARRY TRIANGLE

Barycentrics    a^2*(10*a^8-29*(b^2+c^2)*a^6+3*(2*b^4+7*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(31*b^4-82*b^2*c^2+31*c^4)*a^2-14*b^8+(34*b^4+15*b^2*c^2+34*c^4)*b^2*c^2-14*c^8)*(b^2-c^2) : :
X(32194) = X(351)-3*X(11215) = X(31986)+3*X(11215)

X(32194) lies on these lines: {351,31986}, {523,31772}, {8704,32193}

X(32194) = midpoint of X(i) and X(j) for these lines: {i,j}: {351, 31986}, {31772, 31773}
X(32194) = {X(31986), X(11215)}-harmonic conjugate of X(351)


X(32195) = X(5)-OF-PELLETIER TRIANGLE

Barycentrics    a*(a^5-(b+c)*a^4-5*b*c*a^3+6*(b+c)*b*c*a^2-(b^2+c^2)*(b^2+b*c+c^2)*a+(b^2-c^2)^2*(b+c))*(b-c) : :
X(32195) = X(1)+3*X(11193) = 11*X(5550)-3*X(30613) = 3*X(11193)-X(11247)

X(32195) lies on these lines: {1,11193}, {513,5045}, {521,3813}, {650,3746}, {676,2775}, {885,1058}, {946,3309}, {2646,11934}, {3900,5044}, {5550,30613}, {6198,18344}, {8760,15178}, {9957,14077}

X(32195) = midpoint of X(1) and X(11247)
X(32195) = X(8151)-of-incircle-circles triangle
X(32195) = X(10280)-of-Ursa-minor triangle
X(32195) = X(11247)-of-anti-Aquila triangle
X(32195) = {X(1), X(11193)}-harmonic conjugate of X(11247)


X(32196) = X(5)-OF-REFLECTION TRIANGLE

Barycentrics    (SB+SC)*((17*R^2-2*SA-6*SW)*S^2+(2*SB*SC+4*SW^2-2*SA^2+2*R^4+(-10*SW+3*SA)*R^2)*SA) : :
X(32196) = 3*X(2)-4*X(13365) = X(3)-3*X(7730) = 3*X(51)-2*X(8254) = X(195)-3*X(3060) = 3*X(568)-X(12254) = 4*X(973)-3*X(5946) = 3*X(3060)+X(13423) = X(3519)+2*X(13421) = X(5876)-4*X(11576) = 3*X(5891)-4*X(20584) = 3*X(5946)-2*X(10610) = 2*X(6152)+X(10263) = 4*X(6689)-5*X(15026) = 4*X(10095)-X(12226) = 4*X(13565)-3*X(15067) = X(15801)-4*X(16982)

X(32196) lies on these lines: {2,13365}, {3,7730}, {4,93}, {5,11808}, {51,8254}, {54,143}, {156,195}, {206,576}, {511,6153}, {550,11802}, {568,12254}, {569,973}, {1112,11702}, {1209,6101}, {3518,11597}, {3574,13406}, {3627,10628}, {5446,11563}, {5609,14668}, {5891,20584}, {6102,10116}, {6145,15101}, {6639,12363}, {6689,15026}, {7530,17824}, {7691,13391}, {9019,9977}, {9707,12291}, {9920,12161}, {10095,12226}, {10203,15107}, {10254,12606}, {11441,12280}, {11805,22660}, {12134,14449}, {13565,15067}, {14652,15787}, {21660,22051}

X(32196) = midpoint of X(i) and X(j) for these lines: {i,j}: {195, 13423}, {2888, 6243}, {10263, 13368}, {12280, 12316}
X(32196) = reflection of X(i) in X(j) for these (i,j): (5, 11808), (54, 143), (550, 11802), (5876, 22804), (6101, 1209), (10610, 973), (11702, 1112), (13368, 6152), (15532, 1493), (20424, 5446), (21230, 6153), (21660, 22051), (22804, 11576)
X(32196) = anticomplement of the anticomplement of X(13365)
X(32196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (973, 10610, 5946), (3060, 13423, 195)


X(32197) = X(5)-OF-1st SCHIFFLER TRIANGLE

Barycentrics
2*(b+c)*a^12-5*(b^2+c^2)*a^11-(b+c)*(5*b^2+4*b*c+5*c^2)*a^10+(21*b^4+21*c^4-(5*b^2-34*b*c+5*c^2)*b*c)*a^9-(b+c)*(b^4+c^4-2*(11*b^2-3*b*c+11*c^2)*b*c)*a^8-(34*b^6+34*c^6-(18*b^4+18*c^4-(43*b^2-16*b*c+43*c^2)*b*c)*b*c)*a^7+(b+c)*(14*b^6+14*c^6-(46*b^4+46*c^4-(21*b^2-32*b*c+21*c^2)*b*c)*b*c)*a^6+(26*b^8+26*c^8-(24*b^6+24*c^6-(4*b^2+3*b*c+4*c^2)*(b^2-4*b*c+c^2)*b*c)*b*c)*a^5-2*(b+c)*(8*b^8+8*c^8-(23*b^6+23*c^6-(3*b^4+3*c^4+(14*b^2-3*b*c+14*c^2)*b*c)*b*c)*b*c)*a^4-(9*b^6+9*c^6-(14*b^4+14*c^4-(8*b^2-21*b*c+8*c^2)*b*c)*b*c)*(b^2-c^2)^2*a^3+(b^2-c^2)^3*(b-c)*(7*b^4+7*c^4-(8*b^2+11*b*c+8*c^2)*b*c)*a^2+(b^2-c^2)^3*(b-c)^3*(b^3+c^3)*a-(b^2-c^2)^5*(b-c)^3 : :
X(32197) = 3*X(11218)-X(32055)

X(32197) lies on the line {11218,32055}


X(32198) = X(5)-OF-2nd SCHIFFLER TRIANGLE

Barycentrics
2*(b+c)*a^6-(3*b^2+8*b*c+3*c^2)*a^5-(b+c)*(3*b^2-16*b*c+3*c^2)*a^4+(2*b-c)*(b-2*c)*(3*b^2+8*b*c+3*c^2)*a^3-2*(b+c)*(7*b^2-15*b*c+7*c^2)*b*c*a^2-(b^2-c^2)^2*(3*b^2-7*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c) : :
X(32198) = 3*X(3829)-4*X(6702) = 3*X(4421)-X(6224) = 3*X(5657)-X(22560) = 3*X(11219)-X(11256) = 3*X(11219)+X(12641)

X(32198) lies on these lines: {8,13205}, {9,80}, {11,8256}, {100,32157}, {529,12515}, {952,5450}, {1329,12758}, {1387,10199}, {1484,2802}, {1737,17652}, {1768,32049}, {2771,10915}, {2800,12607}, {2932,12647}, {3035,12740}, {3816,15558}, {3829,6702}, {3880,10265}, {3913,12247}, {4421,6224}, {5657,22560}, {5854,12737}, {6735,17638}, {7701,12751}, {10944,17100}, {11219,11256}, {19914,25438}

X(32198) = midpoint of X(i) and X(j) for these lines: {i,j}: {8, 13205}, {1768, 32049}, {3913, 12247}, {11256, 12641}, {19914, 25438}
X(32198) = reflection of X(i) in X(j) for these (i,j): (100, 32157), (3813, 12619), (13463, 11)
X(32198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8256, 27870, 13463), (11219, 12641, 11256)


X(32199) = X(5)-OF-SODDY TRIANGLE

Barycentrics
7*a^7-5*(b+c)*a^6-2*(4*b^2-9*b*c+4*c^2)*a^5-8*(b^2-c^2)*(b-c)*a^4+(19*b^2+12*b*c+19*c^2)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(b^2+12*b*c+c^2)*a^2-2*(b^2+c^2)*(b-c)^4*a-2*(b^3+c^3)*(b-c)^4 : :
X(32199) = X(32056)-3*X(32079)

X(32199) lies on the line {32056,32079}


X(32200) = X(5)-OF-2nd INNER-SODDY TRIANGLE

Barycentrics
(a+b+c)*(22*(b+c)*a^8-(47*b^2+42*b*c+47*c^2)*a^7-(b+c)*(27*b^2+61*b*c+27*c^2)*a^6+(117*b^4+117*c^4+(123*b^2+136*b*c+123*c^2)*b*c)*a^5-(b+c)*(43*b^4+43*c^4-152*(b^2-b*c+c^2)*b*c)*a^4-(57*b^4+57*c^4+4*(55*b^2+57*b*c+55*c^2)*b*c)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(35*b^4+35*c^4-(25*b^2+74*b*c+25*c^2)*b*c)*a^2+(3*b^4+3*c^4+(45*b^2+86*b*c+45*c^2)*b*c)*(b-c)^4*a-(b^2-c^2)*(b-c)^3*(b^2-3*c^2)*(3*b^2-c^2))+2*S*(6*a^8-6*(b+c)*a^7-6*(7*b^2+13*b*c+7*c^2)*a^6+5*(b+c)*(10*b^2-11*b*c+10*c^2)*a^5+2*(21*b^4+21*c^4+(109*b^2+82*b*c+109*c^2)*b*c)*a^4-2*(b+c)*(29*b^4+29*c^4-(15*b^2+61*b*c+15*c^2)*b*c)*a^3-2*(3*b^4+3*c^4+2*(29*b^2+61*b*c+29*c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(14*b^4+14*c^4+(9*b^2-68*b*c+9*c^2)*b*c)*a+8*(b^2-c^2)^2*(b-c)^2*b*c) : :

X(32200) lies on these lines: {176,31545}, {942,31541}


X(32201) = X(5)-OF-2nd OUTER-SODDY TRIANGLE

Barycentrics
(a+b+c)*(22*(b+c)*a^8-(47*b^2+42*b*c+47*c^2)*a^7-(b+c)*(27*b^2+61*b*c+27*c^2)*a^6+(117*b^4+117*c^4+(123*b^2+136*b*c+123*c^2)*b*c)*a^5-(b+c)*(43*b^4+43*c^4-152*(b^2-b*c+c^2)*b*c)*a^4-(57*b^4+57*c^4+4*(55*b^2+57*b*c+55*c^2)*b*c)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(35*b^4+35*c^4-(25*b^2+74*b*c+25*c^2)*b*c)*a^2+(3*b^4+3*c^4+(45*b^2+86*b*c+45*c^2)*b*c)*(b-c)^4*a-(b^2-c^2)*(b-c)^3*(b^2-3*c^2)*(3*b^2-c^2))-2*S*(6*a^8-6*(b+c)*a^7-6*(7*b^2+13*b*c+7*c^2)*a^6+5*(b+c)*(10*b^2-11*b*c+10*c^2)*a^5+2*(21*b^4+21*c^4+(109*b^2+82*b*c+109*c^2)*b*c)*a^4-2*(b+c)*(29*b^4+29*c^4-(15*b^2+61*b*c+15*c^2)*b*c)*a^3-2*(3*b^4+3*c^4+2*(29*b^2+61*b*c+29*c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(14*b^4+14*c^4+(9*b^2-68*b*c+9*c^2)*b*c)*a+8*(b^2-c^2)^2*(b-c)^2*b*c) : :

X(32201) lies on these lines: {175,31544}, {13390,31555}


X(32202) = X(5)-OF-INNER-SQUARES TRIANGLE

Barycentrics    (16*R^2-5*SA+5*SW)*S^2-2*(8*R^2-3*SW)*SB*SC+S*(6*S^2-(SB+SC)*(SA-4*SW)) : :
X(32202) = 3*X(11209)+X(31987)

X(32202) lies on the line {11209,31987}

X(32202) = midpoint of X(31987) and X(32059)
X(32202) = {X(11209), X(31987)}-harmonic conjugate of X(32059)


X(32203) = X(5)-OF-OUTER-SQUARES TRIANGLE

Barycentrics    (16*R^2-5*SA+5*SW)*S^2-2*(8*R^2-3*SW)*SB*SC-S*(6*S^2-(SB+SC)*(SA-4*SW)) : :
X(32203) = 3*X(11210)+X(31988)

X(32203) lies on the line {11210,31988}

X(32203) = midpoint of X(31988) and X(32060)
X(32203) = {X(11210), X(31988)}-harmonic conjugate of X(32060)


X(32204) = X(5)-OF-STEINER TRIANGLE

Barycentrics    (3*a^8-8*(b^2+c^2)*a^6+(8*b^4+7*b^2*c^2+8*c^4)*a^4-(b^2+c^2)*(4*b^4-3*b^2*c^2+4*c^4)*a^2+(b^4+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :
X(32204) = X(3)+3*X(11123) = X(4)-9*X(9168) = X(5)-3*X(10190) = 5*X(632)-3*X(10278) = 9*X(1649)-5*X(1656) = 7*X(3523)-3*X(16220) = 7*X(3526)-3*X(8029) = 17*X(3533)-9*X(5466) = X(8151)-3*X(11123) = 3*X(10189)-4*X(16239)

X(32204) lies on these lines: {2,10280}, {3,8151}, {4,9168}, {5,10190}, {140,523}, {195,30219}, {512,5447}, {525,1147}, {550,1499}, {632,10278}, {669,13564}, {1649,1656}, {2501,10018}, {3523,16220}, {3526,8029}, {3533,5466}, {5926,6563}, {10189,16239}

X(32204) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 8151}, {5926, 6563}
X(32204) = reflection of X(10279) in X(140)
X(32204) = anticomplement of X(10280)
X(32204) = {X(3), X(11123)}-harmonic conjugate of X(8151)


X(32205) = X(5)-OF-SUBMEDIAL TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-13*b^2*c^2+3*c^4)*a^2-(b^4-5*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(32205) = 3*X(2)+X(143) = 9*X(2)-X(6101) = 15*X(2)+X(6243) = 3*X(2)+5*X(15026) = X(3)+15*X(11451) = X(3)-17*X(11465) = X(3)+3*X(13364) = 7*X(5)+X(185) = X(5)-9*X(373) = 5*X(5)+3*X(9730) = 9*X(5)-X(12162) = X(5)+3*X(13363) = 3*X(5)+X(13630) = 11*X(5)-3*X(15030) = 3*X(143)+X(6101) = 5*X(143)-X(6243) = X(143)-5*X(15026) = 2*X(185)+7*X(11017) = X(185)-7*X(12006) = X(185)+14*X(12046) = 9*X(185)+7*X(12162) = 3*X(185)-7*X(13630) = 15*X(373)+X(9730) = 18*X(373)-X(11017) = 9*X(373)+X(12006) = 9*X(373)-2*X(12046) = 3*X(373)+X(13363) = 5*X(6101)+3*X(6243) = X(6101)+15*X(15026) = X(6101)-3*X(32142) = X(6243)+5*X(32142) = 5*X(11451)-X(13364) = 17*X(11465)+3*X(13364) = 5*X(15026)+X(32142)

X(32205) lies on these lines: {2,143}, {3,11451}, {5,113}, {26,17825}, {30,11695}, {49,15018}, {51,632}, {54,5643}, {110,15047}, {140,5446}, {156,5020}, {381,12279}, {389,547}, {511,16239}, {546,5892}, {568,5067}, {1112,6143}, {1154,3628}, {1511,13434}, {1614,15038}, {1656,5889}, {3090,6102}, {3091,32137}, {3411,11624}, {3412,11626}, {3526,5640}, {3530,10110}, {3544,18439}, {3567,5070}, {3589,10020}, {3627,14845}, {3819,14449}, {3845,27355}, {3850,9729}, {3851,11439}, {3853,16836}, {3856,14915}, {3857,10575}, {3859,13474}, {3917,13421}, {5054,9781}, {5055,5876}, {5056,15060}, {5072,10574}, {5079,5890}, {5447,10124}, {5544,7393}, {5562,15699}, {5907,12812}, {5972,8254}, {6000,12811}, {6241,19709}, {6642,32171}, {6689,13365}, {6746,14940}, {7392,32140}, {7486,18436}, {7999,13321}, {9306,32136}, {9813,32155}, {9817,32168}, {9818,32210}, {9825,30522}, {10545,18378}, {10601,32046}, {10625,11539}, {10643,32207}, {10644,32208}, {10961,32169}, {10963,32170}, {11284,12161}, {11444,15703}, {11479,32138}, {11484,32139}, {11812,13348}, {12045,15606}, {12100,13598}, {12834,14627}, {13403,15465}, {13565,14788}, {13861,15805}, {14641,14893}, {14984,25555}, {15022,18435}, {16042,18350}, {16105,30507}, {18379,18420}, {18928,18952}, {19137,19155}, {19188,19211}, {19372,32143}, {32068,32165}

X(32205) = midpoint of X(i) and X(j) for these lines: {i,j}: {5, 12006}, {140, 10095}, {143, 32142}, {389, 14128}, {3530, 10110}, {3628, 5462}, {3850, 9729}, {6689, 13365}, {10627, 16982}, {11793, 16881}
X(32205) = reflection of X(i) in X(j) for these (i,j): (5, 12046), (11017, 5), (11592, 140)
X(32205) = complement of X(32142)
X(32205) = X(11017)-of-Johnson triangle
X(32205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 143, 32142), (2, 15026, 143)


X(32206) = X(5)-OF-SYMMEDIAL TRIANGLE

Barycentrics    a^2*(3*(b^2+c^2)*a^6-6*(b^4+b^2*c^2+c^4)*a^4+(b^2+c^2)*(2*b^4-15*b^2*c^2+2*c^4)*a^2+(b^4+c^4)*(b^4-b^2*c^2+c^4)) : :
X(32206) = X(8152)-3*X(11205) = 3*X(11205)+X(31989)

X(32206) lies on these lines: {511,14133}, {1199,8718}, {8152,11205}

X(32206) = midpoint of X(8152) and X(31989)
X(32206) = {X(11205), X(31989)}-harmonic conjugate of X(8152)


X(32207) = X(5)-OF-INNER TRI-EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(4*SA*SB*SC+sqrt(3)*S*(S^2+2*R^2*SA-SA^2-2*SB*SC)) : :
X(32207) = X(10661)+3*X(11243) = 3*X(11243)-X(11267)

X(32207) lies on these lines: {3,11452}, {5,21647}, {6,156}, {15,5663}, {16,32171}, {26,17826}, {143,10641}, {1154,10661}, {5318,30522}, {5876,18468}, {5944,18470}, {6101,11420}, {6102,10632}, {7051,32143}, {10634,11591}, {10636,32158}, {10638,32168}, {10643,32205}, {10645,32210}, {11268,30402}, {11408,12161}, {11475,32137}, {11480,32138}, {11485,32139}, {11488,32140}, {11515,32142}, {13561,23302}, {16808,18379}, {18929,18952}, {19190,19211}, {19363,32046}

X(32207) = midpoint of X(10661) and X(11267)
X(32207) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 156, 32208), (11452, 11466, 3)


X(32208) = X(5)-OF-OUTER TRI-EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(4*SA*SB*SC-sqrt(3)*S*(S^2+2*R^2*SA-SA^2-2*SB*SC)) : :
X(32208) = X(10662)+3*X(11244) = 3*X(11244)-X(11268)

X(32208) lies on these lines: {3,11453}, {5,21648}, {6,156}, {15,32171}, {16,5663}, {26,17827}, {143,10642}, {1154,10662}, {1250,32168}, {5321,30522}, {5876,18470}, {5944,18468}, {6101,11421}, {6102,10633}, {10635,11591}, {10637,32158}, {10644,32205}, {10646,32210}, {11267,30403}, {11409,12161}, {11476,32137}, {11481,32138}, {11486,32139}, {11489,32140}, {11516,32142}, {13561,23303}, {16809,18379}, {18930,18952}, {19191,19211}, {19364,32046}, {19373,32143}

X(32208) = midpoint of X(10662) and X(11268)
X(32208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 156, 32207), (11453, 11467, 3)


X(32209) = X(5)-OF-4th TRI-SQUARES TRIANGLE

Barycentrics    2*a^4-15*(b^2+c^2)*a^2+(20*a^2+6*b^2+6*c^2)*S+5*(b^2-c^2)^2 : :
X(32209) = 3*X(13847)-X(13933) = 3*X(13847)+X(13934)

X(32209) lies on these lines: {20,486}, {615,640}, {642,3069}, {6251,13951}, {6290,13961}, {8184,13972}, {8252,13921}, {12123,13939}, {12787,13942}, {12963,13821}

X(32209) = midpoint of X(13933) and X(13934)
X(32209) = X(642)-of-4th tri-squares-central triangle
X(32209) = {X(13847), X(13934)}-harmonic conjugate of X(13933)


X(32210) = X(5)-OF-TRINH TRIANGLE

Barycentrics    a^2*(2*a^8-3*(b^2+c^2)*a^6-(b^2-3*c^2)*(3*b^2-c^2)*a^4+(b^2+c^2)*(7*b^4-13*b^2*c^2+7*c^4)*a^2-(3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :
X(32210) = 3*X(3)-X(156) = 5*X(3)-X(32139) = X(26)+3*X(10606) = X(64)+3*X(18324) = 3*X(74)+X(15132) = X(156)+3*X(32138) = 5*X(156)-3*X(32139) = 2*X(156)-3*X(32171) = 3*X(376)+X(32140) = X(5448)-3*X(10193) = 2*X(5498)-3*X(10193) = X(7689)+3*X(11204) = 3*X(7689)+X(13346) = 5*X(8567)-X(12084) = 15*X(8567)+X(17834) = 3*X(11204)-X(11250) = 9*X(11204)-X(13346) = 3*X(11250)-X(13346) = 3*X(12084)+X(17834) = 5*X(32138)+X(32139) = 2*X(32138)+X(32171) = 2*X(32139)-5*X(32171)

X(32210) lies on these lines: {2,3521}, {3,74}, {5,21663}, {24,32137}, {26,10606}, {30,5449}, {35,32143}, {36,32168}, {64,18324}, {143,378}, {343,550}, {376,2888}, {511,32155}, {569,1204}, {1154,7689}, {1192,31861}, {1539,16868}, {1620,11472}, {1658,3357}, {1994,3520}, {2071,6101}, {2777,13406}, {2937,13445}, {3581,12086}, {3627,32110}, {3631,14810}, {5012,18364}, {5092,19155}, {5448,5498}, {5621,15074}, {5894,15761}, {5946,14130}, {6000,15331}, {6200,32169}, {6396,32170}, {6642,11017}, {7526,10601}, {7527,15026}, {7555,14641}, {7575,11381}, {7688,32158}, {7728,14940}, {8567,12084}, {9818,32205}, {10020,15311}, {10095,11438}, {10201,20427}, {10224,25563}, {10226,13754}, {10249,11255}, {10264,21659}, {10540,17506}, {10605,32046}, {10645,32207}, {10646,32208}, {11410,12161}, {11430,32136}, {11479,12046}, {11598,19506}, {12107,14915}, {12162,15646}, {12226,13340}, {13371,23328}, {13567,15807}, {15060,22467}, {15332,18400}, {18377,23329}, {18439,21844}, {18562,23294}, {18565,23293}, {18931,18952}, {19192,19211}

X(32210) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 32138}, {1658, 3357}, {5894, 15761}, {7689, 11250}
X(32210) = reflection of X(i) in X(j) for these (i,j): (5448, 5498), (10224, 25563), (13406, 20191), (18379, 13561), (19155, 5092), (32171, 3)
X(32210) = X(32171)-of-ABC-X3 reflections triangle
X(32210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 74, 13491), (3, 11468, 12041), (3, 12111, 1511)


X(32211) = X(5)-OF-X-PARABOLA-TANGENTIAL TRIANGLE

Barycentrics
2*a^16-6*(b^2+c^2)*a^14+3*(b^4+8*b^2*c^2+c^4)*a^12+5*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^10+(b^8+c^8-4*(6*b^4-19*b^2*c^2+6*c^4)*b^2*c^2)*a^8-2*(b^2+c^2)*(7*b^8+7*c^8-(34*b^4-57*b^2*c^2+34*c^4)*b^2*c^2)*a^6+(11*b^12+11*c^12-2*(13*b^8+13*c^8+(3*b^4-22*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(b^8+c^8+(5*b^4-13*b^2*c^2+5*c^4)*b^2*c^2)*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^6 : :
X(32211) = 3*X(8029)+X(31990) = 3*X(8029)-X(32061)

X(32211) lies on the line {8029,31990}

X(32211) = midpoint of X(31990) and X(32061)
X(32211) = {X(8029), X(31990)}-harmonic conjugate of X(32061)


X(32212) = X(5)-OF-YFF CONTACT TRIANGLE

Barycentrics    (a^3+2*(b+c)*a^2-(2*b^2+7*b*c+2*c^2)*a+(b+c)*(b^2+b*c+c^2))*(b-c) : :
X(32212) = X(1)-3*X(10196) = X(8)+3*X(6546) = X(145)-9*X(31992) = 5*X(1698)-3*X(21204) = 5*X(3616)-9*X(6544) = X(5592)-3*X(6546) = 3*X(6545)-7*X(9780) = 9*X(14475)-13*X(19877)

X(32212) lies on these lines: {1,10196}, {8,5592}, {10,514}, {40,3667}, {145,31992}, {513,4662}, {1698,21204}, {3239,19582}, {3616,6544}, {4528,28521}, {6545,9780}, {14077,31792}, {14475,19877}, {19963,21211}, {20001,21143}, {21198,28155}, {21201,23757}

X(32212) = midpoint of X(8) and X(5592)
X(32212) = {X(8), X(6546)}-harmonic conjugate of X(5592)


X(32213) = X(5)-OF-INNER-YFF TANGENTS TRIANGLE

Barycentrics    a^4*(b^2+8*b*c+c^2)*(a-b-c)-2*(b^4+c^4+2*(b^2-4*b*c+c^2)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(32213) = X(119)-3*X(10956) = 4*X(5719)-3*X(10283) = 2*X(5901)-3*X(17718) = 3*X(10056)-X(22758) = X(10679)-3*X(11239) = 3*X(11239)+X(12115) = X(13109)+3*X(22731)

X(32213) lies on these lines: {1,5}, {3,10528}, {4,12000}, {10,13373}, {26,10834}, {30,10679}, {56,15867}, {140,956}, {145,6842}, {381,10596}, {442,12645}, {518,5690}, {546,10531}, {549,10269}, {550,11248}, {632,26364}, {912,31397}, {1056,6911}, {1329,15178}, {1385,12607}, {1532,10247}, {1656,10586}, {2077,8703}, {3421,6883}, {3564,12430}, {3621,6937}, {3623,6941}, {3627,6256}, {3845,26333}, {3871,13199}, {5554,8728}, {5840,25439}, {5844,6907}, {5874,10930}, {5875,10929}, {5885,8256}, {6735,10202}, {6756,11400}, {6767,6929}, {6831,18526}, {6834,12001}, {6863,10530}, {6882,7967}, {6959,7373}, {7491,20060}, {7680,28204}, {7715,26378}, {8666,31659}, {8727,28224}, {10056,22758}, {10246,17757}, {10386,26358}, {10532,18518}, {10599,18544}, {10680,10786}, {10803,32134}, {10965,15171}, {11112,12331}, {11509,18990}, {11881,32146}, {11882,32147}, {11914,32162}, {11929,12116}, {11955,32177}, {11956,32178}, {12189,13112}, {12513,26487}, {12608,22791}, {12703,28174}, {12905,13121}, {13104,22931}, {13109,22731}, {13906,13925}, {13964,13993}, {18242,24680}, {19047,19116}, {19048,19117}, {21075,31838}, {22768,32153}, {24927,27385}

X(32213) = midpoint of X(i) and X(j) for these lines: {i,j}: {10679, 12115}, {12905, 13217}
X(32213) = reflection of X(i) in X(j) for these (i,j): (5, 495), (956, 140)
X(32213) = X(10942)-of-inner-Yff triangle
X(32213) = X(10943)-of-1st Johnson-Yff triangle
X(32213) = outer-Yff-to-inner-Yff similarity image of X(5)
X(32213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10942, 5), (1, 10958, 496), (5, 1483, 32214)


X(32214) = X(5)-OF-OUTER-YFF TANGENTS TRIANGLE

Barycentrics    a^4*(b^2-8*b*c+c^2)*(a-b-c)-2*(b^4+c^4-4*(b-c)^2*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
X(32214) = 3*X(5886)-X(17857) = 3*X(10072)-X(11499) = X(10680)-3*X(11240) = 3*X(10959)-X(26470) = 3*X(11240)+X(12116) = X(13110)+3*X(22732)

X(32214) lies on these lines: {1,5}, {3,10529}, {4,12001}, {26,10835}, {30,10680}, {55,15868}, {140,5687}, {145,6882}, {381,10597}, {546,10532}, {549,4421}, {550,11249}, {632,26363}, {912,12053}, {946,3892}, {1058,3560}, {1385,3813}, {1532,18526}, {1656,10587}, {1898,30384}, {2886,15178}, {3434,16203}, {3564,12431}, {3621,6963}, {3622,6881}, {3623,6830}, {3655,15908}, {3845,26332}, {3880,5690}, {3913,26492}, {4187,12645}, {4847,31838}, {5770,9785}, {5844,6922}, {5874,10932}, {5875,10931}, {5884,21630}, {6001,22791}, {6713,8715}, {6756,11401}, {6767,6862}, {6831,10247}, {6833,12000}, {6841,10595}, {6842,7967}, {6911,14986}, {6914,15172}, {6917,7373}, {6958,10530}, {6968,18545}, {7681,28204}, {7715,26377}, {8703,11012}, {10072,11499}, {10085,12699}, {10246,24390}, {10386,26357}, {10531,18519}, {10598,18542}, {10679,10785}, {10804,32134}, {10879,32151}, {10902,15712}, {10966,15171}, {11510,15325}, {11883,32146}, {11884,32147}, {11915,32162}, {11928,12115}, {11957,32177}, {11958,32178}, {12190,13113}, {12331,13747}, {12558,13464}, {12704,28174}, {12906,13122}, {13106,22932}, {13107,22887}, {13110,22732}, {13607,24387}, {13907,13925}, {13965,13993}, {16216,21625}, {18967,18990}, {19049,19116}, {19050,19117}, {20418,26285}

X(32214) = midpoint of X(i) and X(j) for these lines: {i,j}: {10073, 12737}, {10085, 12699}, {10680, 12116}, {12906, 13218}
X(32214) = reflection of X(i) in X(j) for these (i,j): (5, 496), (5687, 140)
X(32214) = X(10942)-of-2nd Johnson-Yff triangle
X(32214) = X(10943)-of-outer-Yff triangle
X(32214) = inner-Yff-to-outer-Yff similarity image of X(5)
X(32214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 1483, 32213), (11, 10942, 5), (1483, 1484, 5)


X(32215) = X(5)-OF-YIU TRIANGLE

Barycentrics    (R^2*(21*R^2-2*SA-12*SW)-2*SB*SC+2*SW^2)*(S^2+SB*SC) : :
X(32215) = X(8154)-3*X(32084) = X(31991)+3*X(32084)

X(32215) lies on these lines: {5,15869}, {30,30685}, {6750,20424}, {8154,31991}

X(32215) = midpoint of X(8154) and X(31991)
X(32215) = {X(31991), X(32084)}-harmonic conjugate of X(8154)


X(32216) = COMPLEMENT OF X(26255)

Barycentrics    a^6-4 a^4 b^2-a^2 b^4+4 b^6-4 a^4 c^2+18 a^2 b^2 c^2-4 b^4 c^2-a^2 c^4-4 b^2 c^4+4 c^6 : :
Barycentrics    (36 R^2-5 SW)S^2 -3 SB SC SW : :

As a point on the Euler line, X(32216) has Shinagawa coefficients [4 E - 5 F, -3 E -3 F].

See Kadir Altintas and Ercole Suppa, Hyacinthos 28993.

X(32216) lies on these lines: {2,3}, {6,13857}, {115,21448}, {125,599}, {126,3014}, {524,26869}, {542,6090}, {2393,5650}, {2549,24855}, {2790,23234}, {5024,9745}, {5642,14982}, {6054,9717}, {7998,14984}, {9140,15066}, {9155,9759}, {11064,11179}, {15059,21766}, {15080,20772}, {15131,19153}, {16187,25561}, {19136,22112}

X(32216) = complement of X(26255)
X(32216) = {X(i),X(j)}-harmonic conjugate of X(k) for these lines: {i,j,k}: {2,376,468}, {2,381,11284}, {2,858,381}, {2,10989,1995}, {2,16063,7426}, {381,3534,18325}, {427,468,6623}, {549,5159,2}, {1368,5159,18531}, {1995,10989,3830}, {7426,16063,3534}, {11284,21312,25}






leftri  Points associated with the Walsmith triangle: X(32217)-X(32229)  rightri

This preamble and centers X(32217)-X(32229) were contributed by Peter Moses, April 27, 2019.

For a definition of the Walsmith triangle, see K1091 (Walsmith Focal Cubic). See also the preamble just before X(32110).

The appearance of {i,j} in the following list means that X(j) = X(i)-of-Walsmith-triangle: {2,7426}, {4,32113}, {30,524}, {370,7426}, {523,1499}, {1144,7426}, {3413,542}, {3414,690}

The circumcircle of the Walsmith triangle passes through X(i) for i = 6, 23, 1316, 32221, 32222, 32224. This circle meets the Walsmith hyperbola and the Brocard circle in X(6). See X(32443).

underbar

X(32217) = X(3)-OF-WALSMITH-TRIANGLE

Barycentrics    a^2*(2*a^6 - 2*a^2*b^4 + 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4) : :
X(32217) = 2 X[23] + X[15826],X[110] - 3 X[18374],3 X[186] - X[1350],X[576] + 2 X[12105],X[691] - 3 X[1691],X[1351] + 3 X[2070],3 X[1495] + X[32127],5 X[3618] - X[5189],3 X[5085] - X[7464],3 X[7426] - X[32113],X[7574] - 3 X[14561],X[8550] + 2 X[16619],2 X[14810] - 3 X[15646]

X(32217) is the common radical trace of the harmonic circles of pairs of the Ehrmann circles. (Randy Hutson, June 7, 2019)

X(32217) lies on these lines: {6, 23}, {30, 182}, {110, 524}, {141, 468}, {184, 8584}, {186, 1350}, {206, 3629}, {237, 5467}, {265, 1177}, {511, 1511}, {523, 5027}, {576, 12105}, {691, 729}, {858, 3589}, {895, 12367}, {1112, 9019}, {1176, 6329}, {1316, 6322}, {1351, 2070}, {1428, 7286}, {1495, 2854}, {1513, 24975}, {1576, 2080}, {1976, 6094}, {2030, 28662}, {2330, 5160}, {2393, 11800}, {2781, 32110}, {3098, 18571}, {3564, 25338}, {3580, 27085}, {3581, 9970}, {3618, 5189}, {5085, 7464}, {5097, 11649}, {5159, 19126}, {5201, 9407}, {7574, 14561}, {8550, 16619}, {9306, 22165}, {9544, 15534}, {9973, 11443}, {10113, 11645}, {10295, 15472}, {10297, 19131}, {10510, 15107}, {11216, 20850}, {11692, 15516}, {12039, 16776}, {12112, 16010}, {14810, 15646}, {15118, 29012}, {16316, 19558}, {18323, 19129}, {18572, 19130}, {19118, 21284}, {23327, 31726}

X(32217) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 23}, {895, 12367}, {3581, 9970}, {10510, 15107}, {12112, 16010}
X(32217) = reflection of X(i) in X(j) for these lines: {i,j}: {141, 468}, {858, 3589}, {3098, 18571}, {15826, 6}, {18572, 19130}
X(32217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 6800, 8546}, {19127, 19136, 597}
X(32217) = isogonal of the isotomic of X(26276)
X(32217) = crossdifference of every pair of points on line {3094, 3906}
X(32217) = barycentric product X(6)*X(26276)
X(32217) = barycentric quotient X(26276) / X(76)


X(32218) = X(5)-OF-WALSMITH-TRIANGLE

Barycentrics    2*a^8 + 3*a^6*b^2 - 3*a^4*b^4 - 3*a^2*b^6 + b^8 + 3*a^6*c^2 + 2*a^2*b^4*c^2 - 3*a^4*c^4 + 2*a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 + c^8 : :
X(32218) = X[1352] + 3 X[2070],X[3448] + 3 X[19596],7 X[3619] + X[20063],5 X[3763] - X[5189],3 X[7426] + X[32113],X[7464] - 3 X[21167],3 X[10096] - X[18583],3 X[18374] - X[25329]

X(32218) lies on these lines: {23, 141}, {30, 14810}, {110, 524}, {373, 468}, {511, 10272}, {523, 5113}, {755, 1287}, {858, 5888}, {1352, 2070}, {1495, 8262}, {1503, 7575}, {3448, 19596}, {3619, 20063}, {3631, 13562}, {3763, 5189}, {5092, 22249}, {5972, 9019}, {6329, 15826}, {6698, 29012}, {7464, 21167}, {10096, 11649}, {11594, 16321}, {11799, 29181}, {12367, 25328}, {15067, 16619}

X(32218) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 141}, {1495, 8262}, {12367, 25328}
X(32218) = reflection of X(i) in X(j) for these lines: {i,j}: {3589, 468}, {5092, 22249}, {15826, 6329}


X(32219) = X(6)-OF-WALSMITH-TRIANGLE

Barycentrics    a^2*(2*a^8 - 6*a^6*b^2 - 4*a^4*b^4 + 12*a^2*b^6 - 4*b^8 - 6*a^6*c^2 + 8*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 3*b^6*c^2 - 4*a^4*c^4 - 7*a^2*b^2*c^4 + 2*b^4*c^4 + 12*a^2*c^6 + 3*b^2*c^6 - 4*c^8) : :

X(32219) lies on these lines: {1495, 5663}, {8588, 19156}


X(32220) = X(20)-OF-WALSMITH-TRIANGLE

Barycentrics    4*a^8 - 3*a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 - 3*a^6*c^2 + 12*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 + 2*b^4*c^4 + 3*a^2*c^6 - c^8 : :
X(32220) = X[323] - 3 X[25321],3 X[403] - 2 X[1352],3 X[2072] - 4 X[18583],5 X[3618] - 4 X[5159],3 X[5032] - X[10989],3 X[5050] - 2 X[15122],3 X[5093] - X[7574],3 X[7426] - 2 X[32113],X[7464] - 3 X[14912],2 X[10297] - 3 X[14853]

X(32220) lies on these lines: {6, 858}, {23, 159}, {30, 1351}, {69, 468}, {110, 524}, {323, 25321}, {399, 3564}, {403, 1352}, {511, 1986}, {542, 32111}, {576, 12022}, {895, 1503}, {1513, 2407}, {2072, 18583}, {2366, 10423}, {3618, 5159}, {3629, 8705}, {5032, 10989}, {5050, 15122}, {5093, 7574}, {5486, 6800}, {7391, 11216}, {7464, 14912}, {8550, 15072}, {8681, 24981}, {9035, 32120}, {10169, 31074}, {10297, 14853}, {11064, 15471}, {18323, 21850}, {19504, 22151}

X(32220) = midpoint of X(23) and X(193)
X(32220) = reflection of X(i) in X(j) for these lines: {i,j}: {69, 468}, {858, 6}, {11064, 15471}, {18323, 21850}
X(32220) = crossdifference of every pair of points on line {5028, 7652}


X(32221) = X(99)-OF-WALSMITH-TRIANGLE

Barycentrics    2*a^14 - 4*a^12*b^2 + 5*a^10*b^4 - 10*a^8*b^6 + 11*a^6*b^8 - 4*a^4*b^10 - 4*a^12*c^2 + 10*a^8*b^4*c^2 - 6*a^6*b^6*c^2 - 3*a^4*b^8*c^2 + 7*a^2*b^10*c^2 - 2*b^12*c^2 + 5*a^10*c^4 + 10*a^8*b^2*c^4 - 20*a^6*b^4*c^4 + 11*a^4*b^6*c^4 - 23*a^2*b^8*c^4 + 7*b^10*c^4 - 10*a^8*c^6 - 6*a^6*b^2*c^6 + 11*a^4*b^4*c^6 + 30*a^2*b^6*c^6 - 5*b^8*c^6 + 11*a^6*c^8 - 3*a^4*b^2*c^8 - 23*a^2*b^4*c^8 - 5*b^6*c^8 - 4*a^4*c^10 + 7*a^2*b^2*c^10 + 7*b^4*c^10 - 2*b^2*c^12 : :

X(32221) lies on these lines: {6, 598}, {110, 1316}, {542, 2682}, {543, 14999}, {9144, 14559}


X(32222) = X(110)-OF-WALSMITH-TRIANGLE

Barycentrics    a^2*(2*a^16 - 8*a^14*b^2 - a^12*b^4 + 21*a^10*b^6 - 6*a^8*b^8 - 18*a^6*b^10 + 7*a^4*b^12 + 5*a^2*b^14 - 2*b^16 - 8*a^14*c^2 + 44*a^12*b^2*c^2 - 41*a^10*b^4*c^2 - 58*a^8*b^6*c^2 + 93*a^6*b^8*c^2 + a^4*b^10*c^2 - 44*a^2*b^12*c^2 + 13*b^14*c^2 - a^12*c^4 - 41*a^10*b^2*c^4 + 118*a^8*b^4*c^4 - 51*a^6*b^6*c^4 - 140*a^4*b^8*c^4 + 130*a^2*b^10*c^4 - 23*b^12*c^4 + 21*a^10*c^6 - 58*a^8*b^2*c^6 - 51*a^6*b^4*c^6 + 230*a^4*b^6*c^6 - 87*a^2*b^8*c^6 - 13*b^10*c^6 - 6*a^8*c^8 + 93*a^6*b^2*c^8 - 140*a^4*b^4*c^8 - 87*a^2*b^6*c^8 + 50*b^8*c^8 - 18*a^6*c^10 + a^4*b^2*c^10 + 130*a^2*b^4*c^10 - 13*b^6*c^10 + 7*a^4*c^12 - 44*a^2*b^2*c^12 - 23*b^4*c^12 + 5*a^2*c^14 + 13*b^2*c^14 - 2*c^16) : :

X(32222) lies on these lines: {6, 110}, {23, 2780}, {543, 7471}, {14688, 16165}, {15329, 15566}


X(32223) = MIDPOINT OF X(23) AND X(125)

Barycentrics    2*a^6 - 3*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6 : :
X(32223) = 3 X[2] + X[15107],3 X[186] - X[16163],X[265] + 3 X[2070],X[323] - 3 X[5642],X[323] + 3 X[15360],3 X[403] - X[1531],3 X[468] - X[11064],X[1495] - 3 X[7426],X[1539] - 3 X[11563],X[3580] + 3 X[7426],X[5189] - 5 X[15059],3 X[5972] - 2 X[11064],X[7574] - 3 X[23515],3 X[10096] - X[10272],X[12317] + 3 X[14157],2 X[16619] + X[20417]

X(32223) lies on these lines: {2, 3098}, {6, 21970}, {23, 125}, {25, 3818}, {30, 6699}, {51, 14389}, {69, 1974}, {74, 1533}, {110, 5965}, {113, 3581}, {140, 13598}, {143, 12242}, {182, 7493}, {184, 11225}, {186, 16163}, {265, 2070}, {316, 419}, {323, 5642}, {373, 7495}, {381, 31860}, {382, 25563}, {389, 13383}, {403, 1531}, {436, 32002}, {468, 511}, {476, 14595}, {541, 15361}, {542, 1495}, {575, 13394}, {826, 4142}, {858, 6723}, {1154, 10096}, {1209, 13621}, {1352, 4232}, {1539, 11563}, {1648, 8627}, {1658, 13403}, {1691, 6388}, {1853, 20850}, {1995, 24206}, {2777, 11799}, {2917, 22550}, {3147, 13346}, {3167, 6144}, {3231, 10418}, {3564, 15448}, {3589, 5943}, {3627, 20191}, {3629, 10192}, {3763, 5020}, {3819, 6677}, {4240, 14918}, {4846, 11438}, {5012, 32068}, {5112, 23698}, {5159, 29181}, {5189, 15059}, {5446, 10020}, {5449, 13419}, {5640, 25555}, {5663, 25338}, {5892, 25337}, {5907, 21841}, {6053, 13754}, {6292, 21513}, {6688, 7499}, {6689, 10095}, {6719, 22103}, {6759, 18917}, {7517, 20299}, {7542, 10110}, {7574, 23515}, {7575, 17702}, {7691, 21451}, {7820, 11328}, {8705, 15118}, {9714, 18381}, {9909, 26958}, {10112, 10282}, {10114, 20773}, {10154, 13567}, {10182, 13352}, {10201, 18388}, {11178, 26255}, {12317, 14157}, {12897, 15331}, {13163, 13565}, {13391, 14156}, {14070, 18390}, {14810, 30739}, {14915, 16619}, {15644, 16238}, {16111, 18325}, {16618, 16836}, {18534, 23329}, {21230, 30551}, {21849, 23292}

X(32223) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 125}, {74, 1533}, {113, 3581}, {1495, 3580}, {5642, 15360}, {11799, 32110}, {16111, 18325}
X(32223) = reflection of X(i) in X(j) for these lines: {i,j}: {858, 6723}, {5972, 468}
X(32223) = reflection of X(22104) in the orthic axis
X(32223) = X(i)-complementary conjugate of X(j) for these (i,j): {74, 21249}, {82, 113}, {2159, 6292}, {2349, 21248}
X(32223) = crosspoint of X(476) and X(18020)
X(32223) = crosssum of X(526) and X(20975)
X(32223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 20193, 18874}, {3580, 7426, 1495}


X(32224) = X(6)X(30)∩X(23)X(385)

Barycentrics    a^8 + a^6*b^2 - 2*a^4*b^4 + a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6 : :
X(32224) = 3 X[7426] - 2 X[16321]

X(32223) lies on these lines: {6, 30}, {23, 385}, {183, 7426}, {186, 8266}, {468, 15271}, {858, 11174}, {1316, 6322}, {2452, 8705}, {3329, 10989}, {5467, 11676}, {8177, 9462}, {8667, 16312}, {19221, 20897}

X(32224) = reflection of X(858) and X(16324)
X(32224) = crossdifference of every pair of points on line {39, 8675}


X(32225) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WALSMITH AND 4th BROCARD

Barycentrics    (2*a^2 - b^2 - c^2)*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) : :
X(32224) = X[381] - 3 X[15362],4 X[468] - X[3292],X[1495] + 2 X[3580],X[1531] + 2 X[3581],X[1531] - 6 X[15362],X[3581] + 3 X[15362],X[13857] + 2 X[15360],4 X[15448] - X[24981],X[16003] + 2 X[16619],2 X[19510] - 3 X[21358],2 X[32113] + X[32127]

X(32225) lies on these lines: {2, 51}, {5, 20192}, {6, 30516}, {23, 9140}, {30, 125}, {187, 1648}, {230, 6791}, {376, 18390}, {381, 1531}, {389, 7552}, {468, 524}, {541, 11799}, {542, 1495}, {543, 5112}, {549, 10564}, {597, 16789}, {599, 5651}, {858, 19924}, {1209, 23410}, {1316, 3849}, {1352, 26255}, {1995, 11178}, {2030, 6792}, {3066, 5055}, {3906, 9208}, {4232, 11180}, {5054, 22112}, {5449, 7540}, {5648, 9027}, {5655, 13754}, {6090, 15533}, {7493, 11179}, {7495, 10168}, {7505, 14531}, {9019, 12099}, {9169, 9181}, {10201, 14831}, {10418, 15993}, {10989, 15107}, {11284, 19510}, {11317, 13378}, {12824, 25566}, {13567, 22352}, {14002, 18553}, {14915, 20126}, {15004, 18449}, {15331, 18555}, {15448, 24981}, {15682, 18376}, {16003, 16619}, {26958, 31152}, {31174, 31953}, {32113, 32127}

X(32225) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 15360}, {23, 9140}, {381, 3581}, {3580, 7426}, {10989, 15107}
X(32225) = reflection of X(i) in X(j) for these lines: {i,j}: {1495, 7426}, {1531, 381}, {3292, 5642}, {5642, 468}, {10564, 549}, {13857, 2}, {32110, 15361}
X(32225) = {X(3581),X(15362)}-harmonic conjugate of X(381),X(897)-isoconjugate of X(3431)
X(32225) = crossdifference of every pair of points on line {353, 3288}
X(32225) = X(2)-of-pedal-triangle-of-X(23)
X(32225) = barycentric product X(381) and X(524)
X(32225) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {187, 3431}, {381, 671}, {5158, 895}, {18487, 9214}


X(32226) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WALSMITH AND REFLECTION TRIANGLE

Barycentrics    a^2*(2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 + 10*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 3*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 7*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 4*a^4*c^6 + 3*a^2*b^2*c^6 - 4*a^2*c^8 - b^2*c^8 + c^10) : :
X(32226) = 5 X[54] - X[74],X[74] + 5 X[2914],5 X[1493] + X[5609],X[5609] - 5 X[11702],5 X[12307] - 13 X[15042],X[12308] + 5 X[15089],7 X[15020] + 5 X[15801]

X(32226) lies on these lines: {49, 11557}, {51, 110}, {54, 74}, {113, 137}, {125, 11245}, {184, 10117}, {195, 568}, {323, 27866}, {373, 9977}, {389, 3043}, {399, 11424}, {539, 32123}, {575, 15059}, {578, 21650}, {826, 32120}, {1112, 1495}, {1154, 14708}, {1594, 10114}, {1614, 11807}, {1986, 13367}, {2904, 9707}, {3292, 3580}, {3448, 7703}, {3569, 14397}, {5095, 21637}, {5562, 12228}, {5663, 15739}, {6467, 13248}, {6593, 32114}, {7699, 18390}, {10264, 32136}, {10610, 25487}, {11003, 13201}, {11381, 15472}, {11402, 17847}, {11472, 12308}, {11806, 15087}, {12161, 21649}, {12174, 17824}, {12254, 22802}, {12307, 15042}, {12893, 14831}, {13171, 17809}, {13472, 18912}, {14531, 22109}, {15020, 15801}, {18400, 32111}

X(32226) = midpoint of X(i) and X(j) for these lines: {i,j}: {54, 2914}, {195, 11597}, {1493, 11702}, {3574, 14049}
X(32226) = crossdifference of every pair of points on line {3448, 13527}
X(32226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 1994, 11800}, {184, 19504, 13417}, {389, 3043, 17701}, {12227, 15463, 185}


X(32227) = X(2)X(265)∩X(6)X(5642)

Barycentrics    3*a^12 - 5*a^10*b^2 - 3*a^8*b^4 + 8*a^6*b^6 - a^4*b^8 - 3*a^2*b^10 + b^12 - 5*a^10*c^2 + 9*a^8*b^2*c^2 - 2*a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 11*a^2*b^8*c^2 - 2*b^10*c^2 - 3*a^8*c^4 - 2*a^6*b^2*c^4 + 16*a^4*b^4*c^4 - 8*a^2*b^6*c^4 - b^8*c^4 + 8*a^6*c^6 - 11*a^4*b^2*c^6 - 8*a^2*b^4*c^6 + 4*b^6*c^6 - a^4*c^8 + 11*a^2*b^2*c^8 - b^4*c^8 - 3*a^2*c^10 - 2*b^2*c^10 + c^12 : :

X(32227) lies on these lines: {2, 265}, {6, 5642}, {22, 20127}, {23, 7728}, {25, 113}, {110, 468}, {125, 5085}, {427, 10733}, {542, 26864}, {858, 12121}, {1495, 14982}, {1986, 6353}, {1995, 14643}, {3581, 5655}, {3618, 12099}, {5094, 17702}, {5181, 6090}, {5523, 15144}, {5663, 7493}, {5972, 11284}, {6676, 10264}, {6677, 12228}, {7484, 19457}, {7495, 15061}, {11579, 13394}, {12022, 15034}, {12827, 18440}, {12828, 19118}, {12893, 21284}, {14852, 30714}, {15035, 30739}, {16042, 25714}, {16163, 31152}


X(32228) = X(30)X(511)∩X(74)X(6325)

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

X(32228) lies on these lines: {30,511}, {74,6325}, {110,6236}, {1649,9126}, {3569,6032}, {9184,11568}, {9191,16235}, {9829,19902}, {10097,11564}, {10163,24284}, {10294,11215}, {13241,20404}

X(32228) = isogonal conjugate of X(32229)
X(32228) = crossdifference of every pair of points on line {6, 16510}


X(32229) = ISOGONAL CONJUGATE OF X(32228)

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

X(32229) lies on the circumcircle and these lines: {30,6325}, {74,8705}, {111,7575}, {381,2770}, {523,6236}, {2373,10296}, {7464,14388}, {10102,14002}

X(32229) = reflection of X(6236) in the Euler line
X(32229) = isogonal conjugate of X(32228)
X(32229) = trilinear pole of line {6, 16510}


X(32230) = X(30)X(250)∩X(107)X(523)

Barycentrics    (a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2*(a^2 + b^2 - c^2)^3*(a^2 - b^2 + c^2)^3 : :
Barycentrics    (tan^3 A)/(b^2 - c^2)^2 : :
X(32230) = 2X[107]+X[15459]

Line X(107)X(112) (the trilinear polar of X(32230)) is the locus of the trilinear pole of the tangent at P to hyperbola {{A,B,C,X(4),P}}, as P moves on the van Aubel line. Line X(107)X(112) is also the tangent to conic {{A,B,C,X(107),X(648)}} (the circumconic centered at X(1249)) at X(107). (Randy Hutson, June 7, 2019)

X(32230) lies on K1097 and these lines:{30,250}, {107,523}, {450,3260}, {648,8057}, {685,15352}, {1093,14254}, {1503,1559}, {1971,1990}, {2404,2409}, {2489,20031}, {6524,9214}, {16813,23286}, {23692,24000}

X(32230) = isogonal conjugate of X(2972)
X(32230) = polar conjugate of X(15526)
X(32230) = isotomic of the polar conjugate of X(23590)
X(32230) = polar conjugate of the isotomic of X(23582)
X(32230) = polar conjugate of the isogonal of X(23964)
X(32230) = X(i)-cross conjugate of X(j) for these (i,j): {{4, 107}, {6, 16813}, {133, 4240}, {184, 112}, {393, 15352}, {1249, 648}, {1614, 933}, {1990, 15459}, {6524, 6529}, {6759, 110}, {8779, 2966}, {9833, 925}, {11206, 99}, {14157, 1304}, {15262, 687}, {18400, 476}, {18685, 6335}, {23964, 23582}, {26883, 1301}, {31383, 1289}}
X(32230) = X(i)-isoconjugate of X(j) for these (i,j): {{1, 2972}, {3, 2632}, {25, 24020}, {48, 15526}, {63, 3269}, {115, 6507}, {122, 19614}, {125, 255}, {163, 23616}, {184, 17879}, {201, 1364}, {212, 1367}, {228, 17216}, {244, 4158}, {273, 7065}, {318, 1363}, {326, 20975}, {338, 4100}, {394, 3708}, {520, 656}, {525, 822}, {577, 20902}, {603, 7068}, {647, 24018}, {756, 7215}, {798, 4143}, {810, 3265}, {1092, 1109}, {1102, 3124}, {1425, 24031}, {1826, 16730}, {1973, 23974}, {2638, 6356}, {2643, 3964}, {2968, 7138}, {3682, 18210}, {3990, 4466}, {4064, 23224}, {4575, 5489}, {7004, 7066}, {18604, 21046}, {23606, 23994}}
X(32230) = cevapoint of X(i) and X(j) for these (i,j): {{4, 107}, {20, 110}, {112, 184}, {648, 17907}, {6524, 6529}}
X(32230) = trilinear pole of line {107, 112}
X(32230) = barycentric product X(i)*X(j) for these lines: {i,j}: {{4, 23582}, {19, 23999}, {63, 24021}, {69, 23590}, {92, 24000}, {99, 6529}, {107, 648}, {110, 15352}, {112, 6528}, {162, 823}, {249, 1093}, {250, 2052}, {264, 23964}, {304, 24022}, {305, 23975}, {393, 18020}, {811, 24019}, {877, 20031}, {1101, 6521}, {2326, 24032}, {4240, 15459}, {4590, 6524}, {6520, 24041}, {15384, 15466}}
X(32230) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {{4, 15526}, {6, 2972}, {19, 2632}, {25, 3269}, {27, 17216}, {63, 24020}, {69, 23974}, {92, 17879}, {99, 4143}, {107, 525}, {112, 520}, {158, 20902}, {162, 24018}, {249, 3964}, {250, 394}, {278, 1367}, {281, 7068}, {393, 125}, {523, 23616}, {593, 7215}, {648, 3265}, {823, 14208}, {850, 23107}, {1093, 338}, {1096, 3708}, {1101, 6507}, {1249, 122}, {1252, 4158}, {1437, 16730}, {1990, 1650}, {2052, 339}, {2189, 1364}, {2207, 20975}, {2326, 24031}, {2501, 5489}, {4590, 4176}, {5317, 18210}, {5379, 3998}, {6520, 1109}, {6521, 23994}, {6524, 115}, {6525, 1562}, {6528, 3267}, {6529, 523}, {7115, 7066}, {8747, 4466}, {10002, 12037}, {15352, 850}, {15384, 1073}, {18020, 3926}, {18831, 15414}, {20031, 879}, {23347, 1636}, {23357, 1092}, {23582, 69}, {23590, 4}, {23963, 23606}, {23964, 3}, {23975, 25}, {23984, 6356}, {23985, 1425}, {23995, 4100}, {23999, 304}, {24000, 63}, {24019, 656}, {24021, 92}, {24022, 19}, {24041, 1102}}


X(32231) = MIDPOINT OF X(3) AND X(17414)

Barycentrics    a^2*(b^2 - c^2)*(4*a^6 - 10*a^4*b^2 + 8*a^2*b^4 - 2*b^6 - 10*a^4*c^2 + 15*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 - 2*c^6) : :

X(32231) lies on these lines: {3, 17414}, {182, 9044}, {187, 30491}, {351, 1499}, {512, 5926}, {523, 549}, {647, 2780}, {3288, 21733}, {8552, 8675}, {8704, 11621}, {11615, 30230}, {16235, 23878}

X(32231) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 17414}, {3288, 21733}
X(32231) = reflection of X(5926) in X(9126)


X(32232) = X(523)X(5054)∩X(549)X(17414)

Barycentrics    (b^2 - c^2)*(5*a^8 - 13*a^6*b^2 + 11*a^4*b^4 - 3*a^2*b^6 - 13*a^6*c^2 + 18*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + b^6*c^2 + 11*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 + b^2*c^6) : :
X(32232) = 2 X[549] + X[17414],X[5996] + 2 X[9126]

X(32232) lies on these lines: {523, 5054}, {549, 17414}, {5215, 8704}, {5996, 9126}


X(32233) = MIDPOINT OF X(20) AND X(11061)

Barycentrics    3*a^12 - 5*a^10*b^2 - a^8*b^4 + 4*a^6*b^6 - a^4*b^8 + a^2*b^10 - b^12 - 5*a^10*c^2 + 5*a^8*b^2*c^2 + 2*a^6*b^4*c^2 - a^4*b^6*c^2 - 3*a^2*b^8*c^2 + 2*b^10*c^2 - a^8*c^4 + 2*a^6*b^2*c^4 - 4*a^4*b^4*c^4 + 2*a^2*b^6*c^4 + b^8*c^4 + 4*a^6*c^6 - a^4*b^2*c^6 + 2*a^2*b^4*c^6 - 4*b^6*c^6 - a^4*c^8 - 3*a^2*b^2*c^8 + b^4*c^8 + a^2*c^10 + 2*b^2*c^10 - c^12 : :
X(32233) = 2 X[5] - 3 X[15462],2 X[125] - 3 X[5085],2 X[141] - 3 X[15035],3 X[182] - 2 X[20301],3 X[186] - 2 X[8262],3 X[265] - 4 X[20301],5 X[631] - 4 X[6698],X[3448] - 3 X[25406],4 X[3589] - 3 X[14644],2 X[3818] - 3 X[14643],3 X[5050] - X[12902],4 X[5092] - 3 X[15061],3 X[5621] - 2 X[16003],3 X[5621] - X[25335],3 X[5622] - 2 X[25328],3 X[5648] - 4 X[12584],4 X[5972] - 3 X[10516],2 X[9969] - 3 X[16223],2 X[10113] - 3 X[14561],X[10296] - 3 X[22151],2 X[11799] - 3 X[18374],5 X[15027] - 8 X[20190],7 X[15036] - 6 X[21167],2 X[16111] + X[25336]

See Q152, the Walsmith circular quartic.

X(32233) lies on these lines: {3, 67}, {4, 6593}, {5, 15133}, {6, 7706}, {20, 2781}, {30, 9970}, {110, 858}, {125, 5085}, {141, 15035}, {146, 14927}, {182, 265}, {186, 8262}, {382, 15140}, {511, 11562}, {524, 10295}, {611, 18968}, {613, 12896}, {631, 6698}, {895, 8550}, {974, 2854}, {1071, 2836}, {1350, 16163}, {1352, 1511}, {1428, 12904}, {1498, 15063}, {1853, 15142}, {2330, 12903}, {2777, 11820}, {3313, 10628}, {3448, 25406}, {3589, 14644}, {3818, 7579}, {4549, 5663}, {5026, 11005}, {5050, 12902}, {5092, 15061}, {5094, 5642}, {5095, 11477}, {5480, 10733}, {5609, 14791}, {5622, 6146}, {5655, 7574}, {5972, 10516}, {6102, 12118}, {7493, 16165}, {7495, 9140}, {7519, 12824}, {7728, 19140}, {9143, 16063}, {9969, 16223}, {10113, 14561}, {10249, 12827}, {10296, 22151}, {10575, 18442}, {10752, 25329}, {11744, 19149}, {11799, 18374}, {15027, 20190}, {15036, 21167}, {15106, 24981}, {16111, 25336}, {19374, 19510}

X(32233) = midpoint of X(i) and X(j) for these lines: {i,j}: {20, 11061}, {146, 14927}, {6776, 12383}
X(32233) = reflection of X(i) in X(j) for these lines: {i,j}: {4, 6593}, {67, 3}, {265, 182}, {895, 8550}, {1350, 16163}, {1352, 1511}, {2930, 30714}, {7728, 19140}, {10733, 5480}, {10752, 25329}, {11005, 5026}, {11477, 5095}, {11744, 19149}, {14982, 110}, {15069, 5181}, {25335, 16003}
X(32233) = {X(5621),X(25335)}-harmonic conjugate of X(16003)


X(32234) = MIDPOINT OF X(7731) AND X(12283)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^8 - 12*a^6*b^2 + 11*a^4*b^4 - 6*a^2*b^6 + 2*b^8 - 12*a^6*c^2 + 11*a^4*b^2*c^2 - a^2*b^4*c^2 + 11*a^4*c^4 - a^2*b^2*c^4 - 4*b^4*c^4 - 6*a^2*c^6 + 2*c^8) : :
X(32234) = 4 X[6] - 3 X[14644],2 X[67] - 3 X[5622],2 X[69] - 3 X[15035],2 X[113] - 3 X[25321],2 X[125] - 3 X[14912],3 X[403] - 4 X[15471],8 X[468] - 9 X[19128],6 X[5050] - 5 X[15059],3 X[5093] - 2 X[10113],4 X[5181] - 5 X[15034],3 X[5622] - 4 X[8550],X[5921] - 3 X[25321],2 X[7574] - 3 X[11416],4 X[9970] - 3 X[10706],6 X[10519] - 7 X[15036],10 X[11482] - 7 X[15044],2 X[14913] - 3 X[16223],3 X[18449] - 2 X[18572]

See Q152, the Walsmith circular quartic.

X(32234) lies on these lines: {4, 542}, {6, 7699}, {24, 2930}, {54, 67}, {68, 16534}, {69, 15035}, {74, 5486}, {110, 468}, {113, 5921}, {125, 14912}, {182, 3043}, {186, 12584}, {193, 17702}, {265, 1353}, {378, 16010}, {403, 15471}, {511, 7722}, {524, 10295}, {1177, 1614}, {1351, 10733}, {1503, 10721}, {1511, 11898}, {1594, 25328}, {1986, 2854}, {2501, 14932}, {2781, 5925}, {2892, 11457}, {2914, 9976}, {4232, 9143}, {4235, 8593}, {5050, 15059}, {5093, 10113}, {5094, 9140}, {5181, 15034}, {5477, 6103}, {5663, 15531}, {5889, 13148}, {6193, 30714}, {6467, 10628}, {6593, 15069}, {7574, 11416}, {7577, 20301}, {7731, 12283}, {8548, 11441}, {9744, 9769}, {9967, 12219}, {10519, 15036}, {10990, 19467}, {11179, 13169}, {11180, 15303}, {11443, 18440}, {11482, 15044}, {11579, 15463}, {12292, 25336}, {14357, 14649}, {14913, 16223}, {14982, 25329}, {15116, 23294}, {15118, 18912}, {18449, 18572}, {18909, 20417}, {18947, 24981}

X(32234) = midpoint of X(7731) and X(12283)
X(32234) = reflection of X(i) in X(j) for these lines: {i,j}: {4, 5095}, {67, 8550}, {74, 6776}, {265, 1353}, {5921, 113}, {6403, 1986}, {10721, 10752}, {10733, 1351}, {11005, 5477}, {11180, 15303}, {11898, 1511}, {12219, 9967}, {13169, 11179}, {14094, 11061}, {14982, 25329}, {15069, 6593}
X(32234) = anticomplement of X(32275)
X(32234) = polar-circle-inverse of X(9880)
X(32234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {67, 8550, 5622}, {5921, 25321, 113}, {17983, 20774, 4}


X(32235) = X(2)X(98)∩X(6)X(16510)

Barycentrics    a^2*(2*a^10 - 4*a^8*b^2 + 4*a^4*b^6 - 2*a^2*b^8 - 4*a^8*c^2 + 6*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 5*b^8*c^2 - 2*a^4*b^2*c^4 + 10*a^2*b^4*c^4 - 5*b^6*c^4 + 4*a^4*c^6 - 5*a^2*b^2*c^6 - 5*b^4*c^6 - 2*a^2*c^8 + 5*b^2*c^8) : :

See Q152, the Walsmith circular quartic.

X(32235) lies on these lines: {2, 98}, {6, 16510}, {265, 7579}, {373, 6593}, {1350, 15106}, {1495, 5663}, {1511, 5650}, {1995, 19140}, {2088, 2502}, {2854, 3292}, {2930, 6090}, {5609, 9730}, {5640, 25556}, {5655, 7706}, {7574, 15136}, {10113, 15432}, {10264, 13394}, {10620, 12893}, {11438, 14094}, {11562, 20772}, {11801, 13413}, {12041, 14855}, {12584, 15066}, {13339, 20397}, {13352, 15132}, {14389, 20301}, {16010, 26864}

X(32235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 5651, 5642}, {110, 9140, 11003}, {110, 11579, 184}
X(32235) = crossdifference of every pair of points on line {3569, 6032}


X(32236) = (name pending)

Barycentrics    4 a^12-21 a^10 b^2-87 a^8 b^4-94 a^6 b^6-30 a^4 b^8+3 a^2 b^10+b^12-21 a^10 c^2+324 a^8 b^2 c^2-9 a^6 b^4 c^2+546 a^4 b^6 c^2-72 a^2 b^8 c^2-87 a^8 c^4-9 a^6 b^2 c^4-954 a^4 b^4 c^4+93 a^2 b^6 c^4-9 b^8 c^4-94 a^6 c^6+546 a^4 b^2 c^6+93 a^2 b^4 c^6-16 b^6 c^6-30 a^4 c^8-72 a^2 b^2 c^8-9 b^4 c^8+3 a^2 c^10+c^12 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28997.

X(32236) lies on this line: {2,6}






leftri  More centers related to Walsmith triangle: X(32237) - X(32317)  rightri

This preamble and centers X(32237)-X(32317) were contributed by CÚsar Eliud Lozada, May 1, 2019.

This section continues the lists of centers related to Walsmith triangle given in X(32217)-X(32229).

The appearance of (T,n) in the following list means that triangles Walsmith and T are perspective with perspector X(n):

(ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (2nd Brocard, 6), (circumsymmedial, 6), (Ehrmann-mid, 7703), (2nd Ehrmann, 6), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6)

The appearance of (T,i,j) in the following list means that triangles Walsmith and T are orthologic with orthologic centers X(i) and X(j):

(ABC, 125, 67), (ABC-X3 reflections, 125, 32233), (anti-Aquila, 125, 32238), (anti-Ara, 125, 32239), (anti-Ascella, 110, 32240), (anti-Atik, 110, 32241), (5th anti-Brocard, 125, 32242), (2nd anti-circumperp-tangential, 125, 32243), (1st anti-circumperp, 110, 32244), (anti-Conway, 110, 32245), (2nd anti-Conway, 110, 32246), (anti-Euler, 125, 32247), (3rd anti-Euler, 110, 32248), (4th anti-Euler, 110, 32249), (anti-excenters-reflections, 110, 32250), (2nd anti-extouch, 110, 32251), (anti-inner-Grebe, 125, 32252), (anti-outer-Grebe, 125, 32253), (anti-Honsberger, 110, 110), (anti-Hutson intouch, 110, 16010), (anti-incircle-circles, 110, 32254), (anti-inverse-in-incircle, 110, 32255), (anti-Mandart-incircle, 125, 32256), (6th anti-mixtilinear, 110, 32257), (anti-orthocentroidal, 1495, 6), (1st anti-Sharygin, 110, 32258), (anti-tangential-midarc, 110, 32259), (anti-Ursa minor, 110, 25328), (anti-Wasat, 110, 32260), (antiAOA, 3580, 67), (anticomplementary, 125, 11061), (AOA, 3580, 15118), (Aquila, 125, 32261), (Ara, 125, 32262), (Aries, 32263, 32264), (1st Auriga, 125, 32265), (2nd Auriga, 125, 32266), (4th Brocard, 32225, 6), (5th Brocard, 125, 32268), (circummedial, 1316, 110), (circumorthic, 110, 32234), (2nd circumperp tangential, 125, 32270), (Ehrmann-mid, 125, 32271), (Ehrmann-side, 110, 32272), (Ehrmann-vertex, 110, 32273), (1st Ehrmann, 6, 16510), (2nd Ehrmann, 110, 895), (Euler, 125, 32274), (2nd Euler, 110, 32275), (5th Euler, 1316, 125), (1st excosine, 110, 32276), (extangents, 110, 32277), (outer-Garcia, 125, 32278), (Gossard, 125, 32279), (inner-Grebe, 125, 32280), (outer-Grebe, 125, 32281), (Hatzipolakis-Moses, 32316, 32317), (3rd Hatzipolakis, 32282, 32283), (1st Hyacinth, 3580, 32284), (2nd Hyacinth, 32263, 32285), (intangents, 110, 32286), (Johnson, 125, 9970), (inner-Johnson, 125, 32287), (outer-Johnson, 125, 32288), (1st Johnson-Yff, 125, 32289), (2nd Johnson-Yff, 125, 32290), (1st Kenmotu diagonals, 110, 32291), (2nd Kenmotu diagonals, 110, 32292), (Kosnita, 110, 12584), (Lucas antipodal tangents, 110, 32293), (Lucas homothetic, 125, 32295), (Lucas(-1) antipodal tangents, 110, 32294), (Lucas(-1) homothetic, 125, 32296), (Mandart-incircle, 125, 32297), (medial, 125, 6593), (midheight, 10117, 15118), (5th mixtilinear, 125, 32298), (orthic, 110, 5095), (orthocentroidal, 1495, 6), (reflection, 32226, 16176), (submedial, 110, 32300), (tangential, 110, 2930), (inner tri-equilateral, 110, 32301), (outer tri-equilateral, 110, 32302), (3rd tri-squares-central, 125, 32303), (4th tri-squares-central, 125, 32304), (Trinh, 110, 32305), (X3-ABC reflections, 125, 32306), (inner-Yff, 125, 32307), (outer-Yff, 125, 32308), (inner-Yff tangents, 125, 32309), (outer-Yff tangents, 125, 32310)

The appearance of (T,i,j) in the following list means that triangles Walsmith and T are parallelogic with parallelogic centers X(i) and X(j):

(circummedial, 32224, 74), (1st Ehrmann, 23, 32305), (5th Euler, 32224, 113), (1st Parry, 125, 32312), (2nd Parry, 125, 32313)

The Walsmith triangle is directly similar to the 2nd Brocard and the 2nd orthosymmedial triangles. In both cases, the center of direct similitude is X(1316).

The Walsmith triangle is inversely similar to the circummedial and the 1st Ehrmann triangles with centers of inverse similitude X(32314) and X(32315), respectively. It is also inversely similar to the 5th Euler triangle, with a non interesting center on inverse similitude.

underbar

X(32237) = CENTER OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: WALSMITH TO ABC-X3 REFLECTIONS

Barycentrics    a^2*(4*a^4-(b^2+c^2)*a^2-3*b^4+4*b^2*c^2-3*c^4) : :
X(32237) = 3*X(23)+X(110) = 7*X(23)+X(323) = 5*X(23)+X(3292) = 5*X(23)-X(15107) = 11*X(23)+X(23061) = 7*X(110)-3*X(323) = X(110)-3*X(1495) = 5*X(110)-3*X(3292) = 5*X(110)+3*X(15107) = 11*X(110)-3*X(23061) = X(323)-7*X(1495) = 5*X(323)-7*X(3292) = 5*X(323)+7*X(15107) = 11*X(323)-7*X(23061) = 5*X(1495)-X(3292) = 5*X(1495)+X(15107) = 11*X(1495)-X(23061) = 11*X(3292)-5*X(23061) = X(5972)-3*X(32267) = 2*X(15448)-3*X(32267)

See preamble just before X(31353).

X(32237) lies on these lines: {3,5646}, {22,3819}, {23,110}, {25,182}, {26,5907}, {30,5972}, {51,11003}, {125,7426}, {154,1351}, {184,5097}, {187,20998}, {373,14002}, {468,6723}, {542,32269}, {567,10110}, {575,6800}, {576,26864}, {858,29323}, {1112,11649}, {1350,9306}, {1503,32223}, {1533,10295}, {1614,16625}, {1658,13474}, {1995,5092}, {2030,3124}, {2070,6000}, {2080,14673}, {2393,11800}, {2937,11793}, {3129,13350}, {3130,13349}, {3291,8627}, {3448,32225}, {3518,9729}, {3581,12308}, {3818,7493}, {5050,31860}, {5085,5544}, {5640,5645}, {5650,7492}, {5663,12105}, {6030,10219}, {6353,14927}, {6644,8717}, {6660,18860}, {6688,13595}, {6759,9714}, {7464,15036}, {7517,10282}, {7530,11430}, {7555,10170}, {7556,15030}, {7575,12041}, {8547,19136}, {8589,21292}, {8681,12310}, {9544,21969}, {10154,21243}, {10244,17814}, {10254,15432}, {10298,32062}, {10301,13394}, {10330,26276}, {10564,15040}, {11002,11663}, {11064,29317}, {11202,18534}, {11284,17508}, {11381,11454}, {11472,14070}, {11695,13339}, {11799,12295}, {11801,25338}, {11807,20773}, {12039,19127}, {12045,16042}, {12088,13348}, {12106,16836}, {13335,20897}, {13383,13419}, {14169,21402}, {14170,21401}, {14683,15360}, {14913,20987}, {15606,18350}, {15644,17714}, {16619,17702}

X(32237) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 1495}, {1533, 10295}, {3292, 15107}
X(32237) = reflection of X(5972) in X(15448)
X(32237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22, 5651, 14810), (5651, 14810, 3819)


X(32238) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO WALSMITH

Barycentrics
(b+c)*a^8-(b^2+c^2)*a^7-(b+c)*(b^2+c^2)*a^6+2*(b^4+c^4)*a^5+(b+c)*b^2*c^2*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^3+(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-2*(b^4-c^4)^2*a-(b^4-c^4)^2*(b+c) : :
X(32238) = 3*X(1)+X(32261) = 3*X(1)-X(32298) = 3*X(67)-X(32261) = 3*X(67)+X(32298) = 5*X(3616)-X(11061) = 3*X(5603)+X(32247) = 3*X(5886)-X(9970) = 3*X(10246)+X(32306) = 3*X(10516)-X(12368) = 3*X(11831)-X(32279)

The reciprocal orthologic center of these triangles is X(125).

X(32238) lies on these lines: {1,67}, {2,32278}, {10,6698}, {125,518}, {511,12261}, {515,32274}, {542,1385}, {942,15116}, {946,2781}, {960,2836}, {999,32270}, {1125,6593}, {1319,32243}, {1386,11735}, {1503,11709}, {2646,32297}, {2854,13605}, {3242,13211}, {3295,32256}, {3416,7984}, {3616,11061}, {4663,15118}, {5603,32247}, {5886,9970}, {9955,32271}, {10246,32306}, {10516,12368}, {10693,24476}, {11363,32239}, {11364,32242}, {11365,32262}, {11366,32265}, {11367,32266}, {11368,32268}, {11370,32280}, {11371,32281}, {11373,32287}, {11374,32288}, {11375,32289}, {11376,32290}, {11377,32295}, {11378,32296}, {11831,32279}, {12259,14984}, {13883,32303}, {13936,32304}, {18991,32252}, {18992,32253}

X(32238) = midpoint of X(i) and X(j) for these lines: {i,j}: {1, 67}, {3242, 13211}, {3416, 7984}, {10693, 24476}, {32261, 32298}
X(32238) = reflection of X(i) in X(j) for these (i,j): (10, 6698), (1386, 11735), (4663, 15118), (6593, 1125), (32271, 9955)
X(32238) = complement of X(32278)
X(32238) = X(67)-of-anti-Aquila-triangle
X(32238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32261, 32298), (67, 32298, 32261)


X(32239) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO WALSMITH

Barycentrics    SC*SB*(R^2*(13*SW^2-6*SA*SW+9*SA^2)-SW*(3*SW^2-2*SA*SW+3*SA^2)) : :

The reciprocal orthologic center of these triangles is X(125).

X(32239) lies on these lines: {4,542}, {23,32257}, {25,67}, {33,32297}, {34,32243}, {66,125}, {235,32274}, {427,6593}, {468,6698}, {511,12140}, {1205,11381}, {1503,12133}, {1593,32233}, {1598,32306}, {1843,13417}, {1853,15128}, {2781,3575}, {3541,15462}, {3867,25329}, {5090,32278}, {5169,32300}, {5410,32253}, {5411,32252}, {5596,13198}, {5622,14216}, {7487,32247}, {7519,32244}, {7530,32275}, {7713,32261}, {11363,32238}, {11380,32242}, {11383,32256}, {11384,32265}, {11385,32266}, {11386,32268}, {11388,32280}, {11389,32281}, {11390,32287}, {11391,32288}, {11392,32289}, {11393,32290}, {11394,32295}, {11395,32296}, {11396,32298}, {11398,32307}, {11399,32308}, {11400,32309}, {11401,32310}, {11746,26926}, {11832,32279}, {12134,14984}, {12294,17702}, {13884,32303}, {13937,32304}, {22479,32270}, {32246,32285}, {32251,32264}

X(32239) = reflection of X(i) in X(j) for these (i,j): (26926, 11746), (32285, 32246)
X(32239) = X(67)-of-anti-Ara-triangle


X(32240) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO WALSMITH

Barycentrics    a^2*(a^8-(2*b^4+15*b^2*c^2+2*c^4)*a^4+17*(b^2+c^2)*b^2*c^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(32240) = 3*X(11402)-4*X(32245) = 3*X(11402)-2*X(32251)

The reciprocal orthologic center of these triangles is X(110)

X(32240) lies on these lines: {3,32244}, {4,32254}, {6,32260}, {24,9925}, {25,2930}, {110,19118}, {184,32276}, {193,12310}, {195,8537}, {427,32255}, {511,12165}, {524,21284}, {542,1593}, {895,11405}, {1177,26864}, {1398,32259}, {1598,5609}, {1993,32248}, {2781,12174}, {2836,11396}, {2854,12167}, {3043,5050}, {3515,12584}, {3516,16010}, {3564,12168}, {5094,25328}, {5410,32291}, {5411,32292}, {6467,17847}, {7071,32286}, {7395,32275}, {7484,32257}, {7592,32249}, {9777,32246}, {9818,32272}, {9924,13417}, {10602,15141}, {11160,21844}, {11216,15140}, {11245,32241}, {11284,32300}, {11402,32245}, {11403,32250}, {11406,32277}, {11408,32301}, {11409,32302}, {11410,32305}, {12160,14984}, {13171,19459}, {16030,32258}, {16176,32262}, {18386,32273}, {19404,32293}, {19405,32294}

X(32240) = reflection of X(i) in X(j) for these (i,j): (12167, 19504), (13171, 19459), (32251, 32245)
X(32240) = X(1156)-of-anti-Ascella-triangle if ABC is acute
X(32240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2930, 5095, 25), (32245, 32251, 11402)


X(32241) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO WALSMITH

Barycentrics    SA*((3*R^2*(12*R^2-7*SW)+2*SW^2)*S^2+(3*SA+SW)*(SB+SC)*R^2*SW) : :

The reciprocal orthologic center of these triangles is X(110).

X(32241) lies on these lines: {2,32251}, {4,67}, {69,125}, {110,19119}, {511,18933}, {542,18909}, {631,5181}, {1177,6353}, {1843,13203}, {1899,32255}, {2854,18935}, {2930,6776}, {3043,14912}, {3548,23296}, {3564,18932}, {5095,11433}, {6515,32244}, {6643,14984}, {8263,22467}, {8889,15116}, {10519,19457}, {11061,32285}, {11245,32240}, {11411,32275}, {12324,32250}, {12584,18925}, {13567,32276}, {14683,26926}, {16010,18913}, {18911,32248}, {18912,32249}, {18914,32254}, {18915,32259}, {18916,32234}, {18917,32272}, {18918,32273}, {18921,32277}, {18922,32286}, {18923,32291}, {18924,32292}, {18926,32293}, {18927,32294}, {18928,32300}, {18929,32301}, {18930,32302}, {18931,32305}, {19166,32258}, {23291,25328}

X(32241) = X(1156)-of-anti-Atik-triangle if ABC is acute
X(32241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (67, 32246, 2892), (2892, 32246, 4)


X(32242) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32242) lies on these lines: {32,67}, {83,6593}, {98,32274}, {125,1691}, {182,265}, {511,12201}, {542,3398}, {1078,6698}, {1503,12192}, {2781,12110}, {5038,15118}, {7787,11061}, {9970,10796}, {10788,32247}, {10789,32261}, {10790,32262}, {10791,32278}, {10792,32280}, {10793,32281}, {10795,32288}, {10797,32289}, {10798,32290}, {10800,32298}, {10801,32307}, {10802,32308}, {10803,32309}, {10804,32310}, {11364,32238}, {11380,32239}, {11490,32256}, {11837,32265}, {11838,32266}, {11839,32279}, {11840,32295}, {11841,32296}, {11842,32306}, {12193,14984}, {12835,32243}, {13885,32303}, {13938,32304}, {18502,32271}, {18993,32252}, {18994,32253}, {22520,32270}

X(32242) = X(67)-of-5th-anti-Brocard-triangle


X(32243) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32243) lies on these lines: {1,542}, {3,32307}, {4,32290}, {6,12903}, {11,32274}, {12,6593}, {34,32239}, {55,32233}, {56,67}, {57,32261}, {110,12588}, {125,1428}, {265,613}, {388,11061}, {498,15462}, {511,18968}, {999,32306}, {1319,32238}, {1352,10091}, {1503,3024}, {2099,32298}, {2781,7354}, {3056,17702}, {3585,32271}, {4293,32247}, {5095,19369}, {5252,32278}, {5433,6698}, {5621,9672}, {11509,32256}, {12835,32242}, {14984,18970}, {18954,32262}, {18955,32265}, {18956,32266}, {18957,32268}, {18958,32279}, {18959,32280}, {18960,32281}, {18961,32287}, {18962,32288}, {18963,32295}, {18964,32296}, {18965,32303}, {18966,32304}, {18967,32310}, {18995,32252}, {18996,32253}

X(32243) = reflection of X(32297) in X(1)
X(32243) = X(67)-of-2nd-anti-circumperp-tangential-triangle
X(32243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 11061, 32289), (999, 32306, 32308)


X(32244) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO WALSMITH

Barycentrics    3*a^8-4*(b^2+c^2)*a^6-(b^4-9*b^2*c^2+c^4)*a^4+(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^2-2*(b^4-c^4)^2 : :
X(32244) = 3*X(2)-4*X(32257) = 9*X(2)-8*X(32300) = 3*X(6)-4*X(6698) = 4*X(6)-5*X(15059) = 4*X(67)-3*X(9140) = 2*X(67)-3*X(13169) = 3*X(67)-2*X(25328) = 3*X(69)-2*X(5181) = 3*X(69)-X(11061) = 3*X(110)-4*X(5181) = 3*X(110)-2*X(11061) = 4*X(858)-3*X(11416) = 2*X(895)-3*X(9140) = X(895)-3*X(13169) = 3*X(895)-4*X(25328) = X(2930)-3*X(15533) = 3*X(5095)-4*X(32300) = 16*X(6698)-15*X(15059) = 9*X(9140)-8*X(25328) = 9*X(13169)-4*X(25328) = 3*X(32257)-2*X(32300)

The reciprocal orthologic center of these triangles is X(110).

X(32244) lies on these lines: {2,5095}, {3,32240}, {4,32275}, {6,6698}, {20,542}, {22,2930}, {30,32272}, {67,524}, {69,110}, {74,3564}, {97,32258}, {125,193}, {325,9769}, {394,32276}, {511,10296}, {576,15025}, {599,6593}, {690,16093}, {1205,8681}, {1297,20404}, {1351,14644}, {1352,10752}, {1353,15061}, {1370,32255}, {1494,4226}, {1992,15118}, {1993,32251}, {2071,32305}, {2407,30789}, {2777,5921}, {2781,5895}, {2836,3962}, {2854,12220}, {2892,11442}, {2979,32248}, {3043,19131}, {3060,32246}, {3100,32286}, {3101,32277}, {3146,32250}, {3153,32273}, {3448,20080}, {3620,5972}, {3631,25329}, {4296,32259}, {5012,32245}, {5093,20304}, {5505,18125}, {5622,15057}, {5663,11898}, {5847,7984}, {5965,11579}, {6515,32241}, {6699,14912}, {6776,15055}, {7488,12584}, {7519,32239}, {7691,32233}, {8541,31857}, {10519,15051}, {10721,18440}, {11008,25320}, {11411,16003}, {11412,12429}, {11413,16010}, {11414,32254}, {11417,32291}, {11418,32292}, {11420,32301}, {11421,32302}, {11477,15044}, {12272,13201}, {13171,19588}, {13417,14913}, {14683,32114}, {15116,23293}, {15303,21356}, {15534,30744}, {16266,20379}, {19406,32293}, {19407,32294}, {19510,22151}

X(32244) = midpoint of X(i) and X(j) for these lines: {i,j}: {3448, 20080}, {11412, 32249}, {12272, 13201}
X(32244) = reflection of X(i) in X(j) for these (i,j): (4, 32275), (110, 69), (193, 125), (895, 67), (3146, 32250), (5095, 32257), (9140, 13169), (10721, 18440), (10752, 1352), (11061, 5181), (11477, 32274), (13417, 14913), (14683, 32114), (15054, 32247), (16176, 6593), (25329, 3631), (32234, 3)
X(32244) = anticomplement of X(5095)
X(32244) = X(1156)-of-1st-anti-circumperp-triangle if ABC is acute
X(32244) = anticomplementary conjugate of the anticomplement of X(15398)
X(32244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (67, 895, 9140), (895, 13169, 67), (5095, 32257, 2)


X(32245) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO WALSMITH

Barycentrics    (SB+SC)*((6*R^2-SW)^2*S^2-(4*R^2-SW)*(3*SA-2*SW)*SA*SW) : :
X(32245) = 3*X(11402)+X(32240) = 3*X(11402)-X(32251)

The reciprocal orthologic center of these triangles is X(110).

X(32245) lies on these lines: {6,110}, {54,67}, {159,1112}, {182,32257}, {184,1177}, {389,12584}, {511,12227}, {542,578}, {567,32272}, {569,32275}, {1147,5181}, {1181,2781}, {1353,19138}, {1503,15472}, {1614,2904}, {1899,15116}, {2892,6776}, {3043,14912}, {3047,25321}, {3564,12228}, {5012,32244}, {6467,13248}, {8537,9971}, {9306,32300}, {11402,32240}, {11423,32249}, {11424,32250}, {11425,16010}, {11426,32254}, {11427,32255}, {11428,32277}, {11429,32286}, {11430,32305}, {11536,32284}, {12161,14984}, {13366,32260}, {15471,18374}, {17809,32276}, {18388,32273}, {19128,32113}, {19365,32259}, {19408,32293}, {19409,32294}, {19459,19504}, {23292,25328}

X(32245) = midpoint of X(i) and X(j) for these lines: {i,j}: {19459, 19504}, {32240, 32251}
X(32245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2930, 32246), (54, 32234, 5622)
X(32245) = X(1156)-of-anti-Conway-triangle if ABC is acute


X(32246) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO WALSMITH

Barycentrics    (SB+SC)*((6*R^2*(6*R^2-5*SW)+5*SW^2)*S^2-(4*R^2-SW)*(3*SA-2*SW)*SA*SW) : :
X(32246) = 3*X(51)-X(5095) = 3*X(51)+X(32260) = X(67)+3*X(9971) = 3*X(125)-X(1205) = 3*X(568)+X(32272) = X(895)+3*X(11188) = X(1205)+3*X(1843) = 3*X(3060)+X(32244) = 5*X(3567)-X(32234) = X(5181)-3*X(29959) = 9*X(5640)-X(32248) = 3*X(5943)-2*X(32300) = X(6403)+3*X(14644) = X(6593)-3*X(16776) = 7*X(9781)+X(32249) = X(9967)-3*X(23515) = X(11061)-3*X(12824) = 3*X(12099)-2*X(15118) = X(12220)-5*X(15059)

The reciprocal orthologic center of these triangles is X(110).

X(32246) lies on these lines: {4,67}, {5,5181}, {6,110}, {24,5622}, {25,1177}, {51,5095}, {52,32275}, {125,1205}, {159,13198}, {185,32250}, {389,542}, {427,15116}, {468,2393}, {511,7687}, {568,32272}, {578,12584}, {599,11444}, {858,6698}, {974,1503}, {1112,9969}, {1192,5621}, {3060,32244}, {3313,13416}, {3564,12236}, {3567,32234}, {3575,31978}, {3867,23315}, {5663,19161}, {5943,14763}, {5972,9822}, {6217,32281}, {6218,32280}, {6403,14644}, {6642,15462}, {6723,11574}, {7716,10117}, {9777,32240}, {9781,32249}, {9786,16010}, {9792,32258}, {9967,23515}, {10821,12367}, {11061,12824}, {11433,32255}, {11435,32277}, {11436,32286}, {11438,32305}, {11579,12412}, {11585,16789}, {11800,14913}, {12061,20303}, {12167,13248}, {12220,15059}, {13567,25328}, {15128,23327}, {15647,20987}, {17810,32276}, {18390,32273}, {19039,32252}, {19040,32253}, {19366,32259}, {19410,32293}, {19411,32294}, {22968,23047}, {23292,25488}, {32239,32285}

X(32246) = midpoint of X(i) and X(j) for these lines: {i,j}: {52, 32275}, {125, 1843}, {185, 32250}, {5095, 32260}, {11800, 14913}, {32239, 32285}
X(32246) = reflection of X(i) in X(j) for these (i,j): (6, 11746), (1112, 9969), (3313, 13416), (5972, 9822), (11574, 6723), (15738, 32274)
X(32246) = polar-circle-inverse of X(34163)
X(32246) = X(1156)-of-2nd-anti-Conway-triangle if ABC is acute
X(32246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 32241, 2892), (2892, 32241, 67)


X(32247) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO WALSMITH

Barycentrics    3*S^4-3*(R^2*(3*SA+SW)-SA^2+SB*SC)*S^2+(9*R^2-SW)*SB*SC*SW : :
X(32247) = 3*X(4)-4*X(32274) = 3*X(67)-2*X(32274) = 3*X(74)-X(32234) = 2*X(110)-3*X(10519) = 4*X(125)-3*X(14853) = 4*X(182)-3*X(25321) = 3*X(376)-2*X(32233) = 5*X(631)-4*X(6593) = 2*X(1351)-3*X(25320) = 7*X(3090)-8*X(6698) = 5*X(3091)-4*X(32271) = 2*X(5095)-3*X(5622) = 3*X(5621)-2*X(8550) = 3*X(5621)-X(16176) = 3*X(5622)-4*X(20417) = 3*X(6776)-2*X(32234) = 4*X(10264)-3*X(25320) = 2*X(10752)-3*X(14853) = 3*X(11180)-4*X(32275) = 3*X(13169)-2*X(32275)

The reciprocal orthologic center of these triangles is X(125).

X(32247) lies on these lines: {2,9970}, {3,11061}, {4,67}, {20,542}, {24,32262}, {30,32306}, {66,11564}, {69,5663}, {74,5486}, {104,32270}, {110,10519}, {125,10752}, {146,1352}, {182,25321}, {193,11579}, {376,32233}, {388,32307}, {468,15106}, {497,32308}, {511,3448}, {515,32261}, {524,7464}, {541,11180}, {578,5095}, {631,6593}, {842,13574}, {895,16003}, {1205,15073}, {1350,12383}, {1351,10264}, {1503,12244}, {1992,20126}, {2393,32249}, {2777,32250}, {2854,12317}, {3085,32289}, {3086,32290}, {3090,6698}, {3091,32271}, {3520,5621}, {3523,15462}, {3541,11431}, {3547,16534}, {3564,10620}, {3618,15061}, {3619,14643}, {4293,32243}, {4294,32297}, {5085,25329}, {5181,14094}, {5480,15081}, {5603,32238}, {5655,21356}, {5657,32278}, {5900,22336}, {6103,10766}, {7487,32239}, {7492,9143}, {7581,32253}, {7582,32252}, {7731,19161}, {7967,32298}, {9140,31099}, {9862,32268}, {10753,15357}, {10783,32280}, {10784,32281}, {10785,32287}, {10786,32288}, {10788,32242}, {10805,32309}, {10806,32310}, {11411,14984}, {11477,25328}, {11491,32256}, {11843,32265}, {11844,32266}, {11845,32279}, {11846,32295}, {11847,32296}, {12041,25406}, {12112,32113}, {12164,23296}, {13886,32303}, {13939,32304}, {14927,20127}, {15063,32257}, {15122,22151}, {20301,20423}, {25336,31884}

X(32247) = midpoint of X(15054) and X(32244)
X(32247) = reflection of X(i) in X(j) for these (i,j): (4, 67), (146, 1352), (193, 11579), (895, 16003), (1351, 10264), (1992, 20126), (5095, 20417), (6776, 74), (7731, 19161), (10752, 125), (10753, 15357), (11061, 3), (11180, 13169), (11477, 25328), (12112, 32113), (12164, 23296), (12383, 1350), (14094, 5181), (14927, 20127), (15063, 32257), (15073, 1205), (16176, 8550)
X(32247) = anticomplement of X(9970)
X(32247) = X(67)-of-anti-Euler-triangle
X(32247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 10752, 14853), (5095, 20417, 5622)


X(32248) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO WALSMITH

Barycentrics    (SB+SC)*((3*R^2*(48*R^2-25*SW)+10*SW^2)*S^2-(7*R^2-2*SW)*(3*SA-2*SW)*SA*SW) : :
X(32248) = 2*X(895)-3*X(15531) = 2*X(1843)-3*X(25321) = 3*X(2979)-2*X(32244) = 3*X(3060)-4*X(5095) = 9*X(5640)-8*X(32246) = 4*X(6593)-3*X(11188) = 9*X(7998)-8*X(32257) = 5*X(11439)-4*X(32250) = 5*X(11444)-4*X(32275) = 15*X(11451)-16*X(32300) = 3*X(11459)-2*X(32272)

The reciprocal orthologic center of these triangles is X(110).

X(32248) lies on these lines: {2,32260}, {3,32249}, {6,110}, {22,32276}, {511,12270}, {542,12111}, {1177,26881}, {1843,25321}, {1993,32240}, {2393,11061}, {2781,12279}, {2979,32244}, {3060,5095}, {3448,6467}, {3564,12273}, {5012,32251}, {5486,11442}, {5663,12283}, {5889,13148}, {7998,32257}, {9019,16176}, {9973,25329}, {11416,15141}, {11439,32250}, {11440,16010}, {11441,32254}, {11444,32275}, {11445,32277}, {11446,32286}, {11449,12584}, {11451,32300}, {11454,32305}, {11459,32272}, {12220,13201}, {12281,18438}, {15074,32306}, {18392,32273}, {18911,32241}, {19167,32258}, {19367,32259}, {19412,32293}, {19413,32294}, {23293,25328}

X(32248) = reflection of X(i) in X(j) for these (i,j): (3448, 6467), (5889, 32234), (9973, 25329), (12272, 110), (12281, 18438), (13201, 12220), (32249, 3), (32306, 15074)
X(32248) = anticomplement of X(32260)
X(32248) = X(1156)-of-3rd-anti-Euler-triangle if ABC is acute
X(32248) = {X(12272), X(15531)}-harmonic conjugate of X(11443)


X(32249) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO WALSMITH

Barycentrics    (SB+SC)*((3*R^2*(48*R^2-25*SW)+2*(3*SA+5*SW)*SW)*S^2-(R^2*(36*R^2+21*SA-22*SW)-6*SA^2+6*SB*SC+4*SW^2)*SA*SW) : :
X(32249) = 5*X(3567)-4*X(5095) = 3*X(5890)-2*X(32234) = 7*X(7999)-8*X(32257) = 7*X(9781)-8*X(32246) = 2*X(9970)-3*X(11188) = 3*X(11455)-4*X(32250) = 3*X(11459)-4*X(32275) = 17*X(11465)-16*X(32300)

The reciprocal orthologic center of these triangles is X(110).

X(32249) lies on these lines: {3,32248}, {4,32260}, {24,32276}, {54,32251}, {67,15073}, {74,1296}, {110,19123}, {511,12281}, {542,6241}, {895,11458}, {1177,26882}, {1205,11457}, {1614,2930}, {2393,32247}, {2781,12290}, {3564,12284}, {3567,5095}, {5663,12272}, {5890,32234}, {6403,7731}, {7592,32240}, {7999,32257}, {8537,15141}, {9781,32246}, {9970,11188}, {11412,12429}, {11423,32245}, {11455,32250}, {11456,32254}, {11459,32275}, {11460,32277}, {11461,32286}, {11462,32291}, {11463,32292}, {11464,12584}, {11465,32300}, {11466,32301}, {11467,32302}, {11468,32305}, {11898,12273}, {12111,32272}, {15106,23061}, {18394,32273}, {18912,32241}, {19168,32258}, {19368,32259}, {19414,32293}, {19415,32294}, {23294,25328}

X(32249) = reflection of X(i) in X(j) for these (i,j): (4, 32260), (7731, 6403), (11412, 32244), (12111, 32272), (12273, 11898), (12283, 74), (15073, 67), (32248, 3)
X(32249) = X(1156)-of-4th-anti-Euler-triangle if ABC is acute


X(32250) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO WALSMITH

Barycentrics    (8*a^8-12*(b^2+c^2)*a^6+(5*b^4+14*b^2*c^2+5*c^4)*a^4-2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+5*(b^4-c^4)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(32250) = 3*X(4)-X(32234) = 3*X(125)-4*X(32274) = 5*X(3091)-4*X(32300) = 3*X(5095)-2*X(32234) = 3*X(5642)-2*X(32233) = 4*X(6723)-3*X(25406) = 3*X(10151)-2*X(15471) = 5*X(11439)-X(32248) = 3*X(11455)+X(32249) = 4*X(18440)-X(32114)

The reciprocal orthologic center of these triangles is X(110).

X(32250) lies on these lines: {4,542}, {20,32257}, {24,32305}, {25,16010}, {30,32275}, {33,32259}, {34,32286}, {64,67}, {110,15431}, {113,15432}, {125,468}, {185,32246}, {235,25328}, {378,12584}, {382,32272}, {403,20301}, {511,12292}, {541,32306}, {1177,26883}, {1205,15738}, {1352,16163}, {1498,32251}, {1593,2930}, {1597,32254}, {1843,5663}, {1902,2836}, {1974,11579}, {2777,32247}, {2781,11381}, {2854,12133}, {3091,32300}, {3146,32244}, {3515,5621}, {3564,12295}, {4232,9140}, {5094,5642}, {5622,6759}, {5921,10733}, {6103,11646}, {6723,25406}, {6776,7687}, {10151,15471}, {10295,11645}, {10301,12828}, {11403,32240}, {11424,32245}, {11439,32248}, {11455,32249}, {11471,32277}, {11472,17702}, {11473,32291}, {11474,32292}, {11475,32301}, {11476,32302}, {12134,18488}, {12162,14984}, {12324,32241}, {14216,20417}, {14982,24981}, {15118,19153}, {15133,16534}, {15811,32276}, {19169,32258}, {19416,32293}, {19417,32294}

X(32250) = midpoint of X(i) and X(j) for these lines: {i,j}: {382, 32272}, {3146, 32244}, {5921, 10733}, {11381, 32260}
X(32250) = reflection of X(i) in X(j) for these (i,j): (20, 32257), (185, 32246), (1205, 15738), (5095, 4), (6776, 7687), (10990, 67), (12294, 12133), (16163, 1352), (24981, 14982)
X(32250) = X(1156)-of-anti-excenters-reflections-triangle if ABC is acute


X(32251) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH TO WALSMITH

Barycentrics
a^2*(-a^2+b^2+c^2)*(a^10-(b^2+c^2)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4)) : :
X(32251) = 3*X(5050)-2*X(12228) = 3*X(11402)-X(32240) = 3*X(11402)-2*X(32245)

The reciprocal orthologic center of these triangles is X(110).

X(32251) lies on these lines: {2,32241}, {3,895}, {5,9512}, {6,67}, {25,1177}, {54,32249}, {69,23296}, {110,19125}, {155,32275}, {184,2930}, {185,16010}, {371,19385}, {372,19384}, {394,32257}, {427,2892}, {511,19457}, {542,1181}, {974,11579}, {1204,1205}, {1498,32250}, {1593,2781}, {1843,10117}, {1899,25328}, {1992,18281}, {1993,32244}, {2393,21284}, {2854,13198}, {2931,19131}, {2935,19124}, {3448,26926}, {3564,19456}, {3594,19399}, {5012,32248}, {5050,12228}, {5159,22151}, {6593,11284}, {6776,32255}, {7592,32234}, {9970,15738}, {10510,21639}, {10601,32300}, {10605,32305}, {11402,32240}, {12167,13171}, {12584,19357}, {14683,19119}, {14853,18933}, {15059,26206}, {15122,19348}, {16063,18919}, {18396,32273}, {18445,32272}, {19170,32258}, {19347,32254}, {19349,32259}, {19350,32277}, {19354,32286}, {19355,32291}, {19356,32292}, {19358,32293}, {19359,32294}, {19363,32301}, {19364,32302}, {32239,32264}

X(32251) = midpoint of X(12167) and X(13171)
X(32251) = reflection of X(i) in X(j) for these (i,j): (19459, 13198), (19504, 6), (32240, 32245)
X(32251) = X(1156)-of-2nd-anti-extouch-triangle if ABC is acute
X(32251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 67, 15141), (6, 32276, 5095), (125, 15118, 15128)


X(32252) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32252) lies on these lines: {2,32304}, {6,67}, {141,19111}, {372,32233}, {511,19051}, {542,3312}, {1503,19059}, {1587,32274}, {1588,2781}, {3068,6698}, {3069,6593}, {3299,32307}, {3301,32308}, {3763,8998}, {5085,13969}, {5411,32239}, {5648,32291}, {6418,32306}, {7582,32247}, {7584,9970}, {7585,32303}, {7586,11061}, {7968,32298}, {9971,19043}, {12367,19001}, {13785,32271}, {13910,15059}, {13936,32278}, {13966,15462}, {13979,14561}, {14984,19061}, {18991,32238}, {18993,32242}, {18995,32243}, {18999,32256}, {19003,32261}, {19005,32262}, {19007,32265}, {19009,32266}, {19011,32268}, {19013,32270}, {19017,32279}, {19023,32287}, {19025,32288}, {19027,32289}, {19029,32290}, {19031,32295}, {19033,32296}, {19037,32297}, {19039,32246}, {19047,32309}, {19049,32310}

X(32252) = X(67)-of-anti-inner-Grebe-triangle
X(32252) = {X(6), X(67)}-harmonic conjugate of X(32253)


X(32253) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32253) lies on these lines: {2,32303}, {6,67}, {141,19110}, {371,32233}, {511,19052}, {542,3311}, {1503,19060}, {1587,2781}, {1588,32274}, {3068,6593}, {3069,6698}, {3299,32308}, {3301,32307}, {3763,13990}, {5085,8994}, {5410,32239}, {5648,32292}, {6417,32306}, {7581,32247}, {7583,9970}, {7585,11061}, {7586,32304}, {7969,32298}, {8981,15462}, {9971,19044}, {12367,19002}, {13665,32271}, {13883,32278}, {13915,14561}, {13972,15059}, {14984,19062}, {18992,32238}, {18994,32242}, {18996,32243}, {19000,32256}, {19004,32261}, {19006,32262}, {19008,32265}, {19010,32266}, {19012,32268}, {19014,32270}, {19018,32279}, {19024,32287}, {19026,32288}, {19028,32289}, {19030,32290}, {19032,32295}, {19034,32296}, {19038,32297}, {19040,32246}, {19048,32309}, {19050,32310}

X(32253) = X(67)-of-anti-outer-Grebe-triangle
X(32253) = {X(6), X(67)}-harmonic conjugate of X(32252)


X(32254) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO WALSMITH

Barycentrics    (SB+SC)*((18*R^2-6*SW)*S^2+(9*R^2-6*SA+2*SW)*SA*SW) : :
X(32254) = 7*X(3)-6*X(5621) = 3*X(3)-4*X(12584) = 3*X(3)-2*X(16010) = 5*X(3)-4*X(32305) = 4*X(110)-3*X(5050) = 3*X(399)-2*X(9970) = 4*X(895)-5*X(11482) = 4*X(1177)-5*X(14530) = 3*X(1351)-4*X(9970) = 7*X(2930)-3*X(5621) = 3*X(2930)-2*X(12584) = 3*X(2930)-X(16010) = 5*X(2930)-2*X(32305) = 8*X(5609)-5*X(11482) = 9*X(5621)-14*X(12584) = 9*X(5621)-7*X(16010) = 15*X(5621)-14*X(32305) = 5*X(12584)-3*X(32305) = 5*X(16010)-6*X(32305) = 4*X(32275)-3*X(32306)

The reciprocal orthologic center of these triangles is X(110).

X(32254) lies on these lines: {3,67}, {4,32240}, {5,32255}, {23,3564}, {25,32234}, {110,5050}, {399,1351}, {511,12308}, {575,18350}, {895,3527}, {999,32259}, {1177,14530}, {1181,32260}, {1352,8546}, {1482,2836}, {1597,32250}, {1598,5095}, {1656,25328}, {1995,9143}, {2781,12315}, {3295,32286}, {3311,32291}, {3312,32292}, {3843,32273}, {5055,20301}, {5093,19140}, {5655,16510}, {5663,11820}, {5921,12168}, {5965,25336}, {6403,12165}, {6759,32276}, {7530,11061}, {9920,15581}, {9924,10628}, {10272,25320}, {10306,32277}, {10510,19377}, {11414,32244}, {11426,32245}, {11432,32246}, {11441,32248}, {11456,32249}, {11477,12316}, {11484,32300}, {11485,32301}, {11486,32302}, {11579,12017}, {12106,18951}, {12164,12271}, {12902,14982}, {15039,15462}, {17702,19588}, {18440,31861}, {18583,20125}, {18914,32241}, {19173,32258}, {19347,32251}, {19418,32293}, {19419,32294}, {24206,25330}

X(32254) = reflection of X(i) in X(j) for these (i,j): (3, 2930), (895, 5609), (1351, 399), (12902, 14982), (16010, 12584), (32255, 5), (32262, 15581), (32276, 6759)
X(32254) = inverse of X(8724) in the Stammler circle
X(32254) = X(1156)-of-anti-incircle-circles-triangle if ABC is acute
X(32254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2930, 16010, 12584), (12584, 16010, 3)


X(32255) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO WALSMITH

Barycentrics    a^8+2*(b^2+c^2)*a^6-11*b^2*c^2*a^4-(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-(b^4-c^4)^2 : :
X(32255) = 3*X(2)-4*X(25328) = 3*X(4)-4*X(32273) = 4*X(67)-3*X(69) = 2*X(67)-3*X(3448) = 4*X(110)-5*X(3618) = 3*X(110)-4*X(15118) = 2*X(110)-3*X(25320) = 8*X(125)-7*X(3619) = 2*X(141)-3*X(25330) = 3*X(193)-2*X(16176) = 3*X(376)-4*X(32305) = 2*X(399)-3*X(14853) = 4*X(895)-3*X(1992) = 3*X(895)-2*X(5095) = 9*X(1992)-8*X(5095) = 3*X(1992)-2*X(11061) = 15*X(3618)-16*X(15118) = 5*X(3618)-6*X(25320) = 4*X(5095)-3*X(11061) = 8*X(15118)-9*X(25320)

The reciprocal orthologic center of these triangles is X(110).

X(32255) lies on these lines: {2,2930}, {4,542}, {5,32254}, {6,7533}, {8,2836}, {20,16010}, {67,69}, {110,3618}, {125,3619}, {141,25330}, {193,16176}, {265,18489}, {376,32305}, {388,32259}, {399,14853}, {427,32240}, {497,32286}, {511,12317}, {524,5189}, {631,12584}, {1177,11206}, {1205,11457}, {1370,32244}, {1503,17812}, {1843,18947}, {1899,32241}, {2407,5984}, {2550,32277}, {2781,12324}, {2892,32064}, {3068,32291}, {3069,32292}, {3090,20301}, {3522,5621}, {3564,7574}, {3629,25336}, {4558,10991}, {5181,9140}, {5486,11442}, {5622,13336}, {5648,6698}, {5987,7735}, {6593,9143}, {6643,32275}, {6776,32251}, {7386,32257}, {7392,32300}, {8550,14516}, {9976,14912}, {10264,10519}, {11008,13203}, {11411,14984}, {11427,32245}, {11433,32246}, {11488,32301}, {11489,32302}, {11579,12383}, {12308,21850}, {14561,20125}, {14791,32306}, {14927,17702}, {15583,17847}, {18531,32272}, {19174,32258}, {19420,32293}, {19421,32294}

X(32255) = reflection of X(i) in X(j) for these (i,j): (20, 16010), (69, 3448), (2930, 25328), (11061, 895), (12308, 21850), (12383, 11579), (14683, 6), (17847, 15583), (25336, 3629), (32254, 5)
X(32255) = anticomplementary conjugate of the anticomplement of X(13574)
X(32255) = anticomplement of X(2930)
X(32255) = inverse of X(671) in the anticomplementary circle
X(32255) = X(1156)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(32255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (895, 11061, 1992), (2930, 25328, 2)


X(32256) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32256) lies on these lines: {3,32270}, {35,32261}, {55,67}, {56,32298}, {100,11061}, {197,32262}, {511,12334}, {542,11248}, {1001,6698}, {1376,6593}, {1503,12327}, {2781,11500}, {2836,3811}, {2930,4062}, {3295,32238}, {5687,32278}, {5846,22586}, {9970,11499}, {10310,32233}, {11383,32239}, {11490,32242}, {11491,32247}, {11492,32265}, {11493,32266}, {11494,32268}, {11496,32274}, {11497,32280}, {11498,32281}, {11501,32289}, {11502,32290}, {11503,32295}, {11504,32296}, {11507,32307}, {11508,32308}, {11509,32243}, {11510,32310}, {11848,32279}, {11849,32306}, {12328,14984}, {13887,32303}, {13940,32304}, {15622,16010}, {18491,32271}, {18999,32252}, {19000,32253}

X(32256) = reflection of X(i) in X(j) for these (i,j): (32270, 3), (32287, 6593)
X(32256) = X(67)-of-anti-Mandart-incircle-triangle


X(32257) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO WALSMITH

Barycentrics    (-a^2+b^2+c^2)*(2*a^6-2*(b^2+c^2)*a^4-(b^4-4*b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)) : :
X(32257) = 3*X(2)+X(32244) = 3*X(3)+X(32272) = X(67)+3*X(599) = 3*X(67)+X(2930) = 5*X(67)+3*X(5648) = 5*X(67)-X(25335) = 3*X(69)+X(895) = 5*X(69)+3*X(25320) = 3*X(125)-X(895) = 5*X(125)-3*X(25320) = 9*X(599)-X(2930) = 3*X(599)-X(5181) = 5*X(599)-X(5648) = 15*X(599)+X(25335) = 5*X(895)-9*X(25320) = X(2930)-3*X(5181) = 5*X(2930)-9*X(5648) = 5*X(2930)+3*X(25335) = 5*X(5181)-3*X(5648) = 5*X(5181)+X(25335) = 3*X(5648)+X(25335) = 3*X(20417)-2*X(32305) = X(32244)+2*X(32300) = X(32272)-3*X(32275)

The reciprocal orthologic center of these triangles is X(110).

X(32257) lies on these lines: {2,5095}, {3,67}, {6,6723}, {20,32250}, {23,32239}, {69,125}, {95,32258}, {110,3620}, {141,5972}, {182,32245}, {193,15059}, {394,32251}, {511,7687}, {524,5159}, {631,32234}, {1038,32259}, {1040,32286}, {1092,5622}, {1112,9822}, {1177,9306}, {1216,14984}, {1270,13774}, {1271,13654}, {1351,23515}, {1352,2777}, {1368,22165}, {2781,5893}, {2854,3631}, {3284,15993}, {3292,32285}, {3448,15812}, {3564,6699}, {3917,32260}, {5642,11061}, {5847,11735}, {5894,8263}, {5921,15055}, {7386,32255}, {7484,32240}, {7998,32248}, {7999,32249}, {8542,15116}, {9970,11487}, {9972,14076}, {10319,32277}, {10510,19378}, {10519,16163}, {11178,32271}, {11284,12828}, {11513,32291}, {11514,32292}, {11515,32301}, {11516,32302}, {11585,20301}, {11850,15027}, {11898,15061}, {12359,15115}, {12363,15605}, {14645,15359}, {15063,32247}, {15303,16176}, {15533,30771}, {16111,18440}, {17811,32276}, {18323,19924}, {18531,32273}, {19422,32293}, {19423,32294}, {29012,32113}

X(32257) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 32275}, {20, 32250}, {67, 5181}, {69, 125}, {3448, 32114}, {5095, 32244}, {5642, 13169}, {15063, 32247}, {16111, 18440}, {30714, 32306}
X(32257) = reflection of X(i) in X(j) for these (i,j): (6, 6723), (1112, 9822), (5095, 32300), (5972, 141), (11574, 13416), (15118, 6698)
X(32257) = anticomplement of X(32300)
X(32257) = complement of X(5095)
X(32257) = inverse of X(9876) in the circumcircle
X(32257) = X(1156)-of-6th-anti-mixtilinear-triangle if ABC is acute
X(32257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5095, 32300), (2, 32244, 5095)


X(32258) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN TO WALSMITH

Barycentrics    SB*SC*(S^2+SA*SB)*(S^2+SA*SC)*(3*S^4-(12*(3*SA+SW)*R^2-12*SA^2+3*SB*SC-3*SW^2)*S^2+SB*SC*SW^2) : :

The reciprocal orthologic center of these triangles is X(110).

X(32258) lies on these lines: {54,67}, {95,32257}, {97,32244}, {110,19171}, {275,5095}, {511,19195}, {542,8884}, {895,8795}, {1177,26887}, {2781,19206}, {2854,19197}, {2930,19189}, {3564,19193}, {9792,32246}, {12584,19185}, {14984,19194}, {16010,19172}, {16030,32240}, {19166,32241}, {19167,32248}, {19168,32249}, {19169,32250}, {19170,32251}, {19173,32254}, {19174,32255}, {19175,32259}, {19176,32272}, {19177,32273}, {19179,32275}, {19180,32276}, {19181,32277}, {19182,32286}, {19183,32291}, {19184,32292}, {19186,32293}, {19187,32294}, {19188,32300}, {19190,32301}, {19191,32302}, {19192,32305}, {21638,32260}, {23295,25328}

X(32258) = X(1156)-of-1st-anti-Sharygin-triangle if ABC is acute


X(32259) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32259) lies on these lines: {1,542}, {12,25328}, {33,32250}, {34,5095}, {35,32305}, {36,12584}, {55,16010}, {56,2930}, {65,651}, {67,73}, {110,1428}, {221,32276}, {388,32255}, {399,613}, {511,19470}, {524,7286}, {999,32254}, {1038,32257}, {1060,32275}, {1177,26888}, {1398,32240}, {1425,32260}, {1431,2842}, {1442,12739}, {1469,2854}, {1503,5160}, {1870,32234}, {2067,32291}, {2330,10088}, {2781,7355}, {3056,5663}, {3448,12588}, {3564,19469}, {3585,32273}, {4296,32244}, {4551,5018}, {5217,5621}, {5297,9140}, {6126,9976}, {6502,32292}, {7051,32301}, {7352,14984}, {7951,20301}, {8540,9970}, {12904,14982}, {14094,32290}, {18447,32272}, {18915,32241}, {19175,32258}, {19349,32251}, {19365,32245}, {19366,32246}, {19367,32248}, {19368,32249}, {19370,32293}, {19371,32294}, {19372,32300}, {19373,32302}

X(32259) = reflection of X(i) in X(j) for these (i,j): (1469, 3028), (32286, 1)
X(32259) = X(1156)-of-anti-tangential-midarc-triangle if ABC is acute
X(32259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (895, 32289, 19369), (10088, 11579, 2330)


X(32260) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO WALSMITH

Barycentrics    a^2*(-a^2+b^2+c^2)*((b^2+c^2)*a^8-4*b^2*c^2*a^6-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+3*(b^2-c^2)^2*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4)) : :
X(32260) = 3*X(51)-2*X(5095) = 3*X(51)-4*X(32246) = 9*X(373)-8*X(32300) = 3*X(3917)-4*X(32257) = 2*X(6593)-3*X(29959) = 3*X(9971)-X(16176) = X(11061)-3*X(11188)

The reciprocal orthologic center of these triangles is X(110).

X(32260) lies on these lines: {2,32248}, {4,32249}, {6,32240}, {25,32276}, {51,5095}, {67,1205}, {69,18125}, {110,14913}, {125,126}, {182,17701}, {184,2930}, {185,542}, {193,11800}, {373,32300}, {389,32234}, {511,10296}, {895,3292}, {1147,19361}, {1177,1495}, {1181,32254}, {1204,16010}, {1216,15133}, {1425,32259}, {1843,13417}, {1899,32241}, {2781,11381}, {2888,11444}, {2892,11550}, {3270,32286}, {3448,12272}, {3564,21649}, {3611,32277}, {3917,32257}, {5562,14984}, {5622,12584}, {6403,10628}, {6593,29959}, {7722,21851}, {8541,15141}, {9924,13171}, {9971,16176}, {11061,11188}, {12167,17847}, {13366,32245}, {13754,32272}, {13851,32273}, {16270,30714}, {21638,32258}, {21640,32291}, {21641,32292}, {21642,32293}, {21643,32294}, {21647,32301}, {21648,32302}, {21663,32305}

X(32260) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 32249}, {3448, 12272}
X(32260) = reflection of X(i) in X(j) for these (i,j): (110, 14913), (193, 11800), (1205, 67), (5095, 32246), (5562, 32275), (6467, 125), (7722, 21851), (11381, 32250), (13417, 1843), (32234, 389)
X(32260) = complement of X(32248)
X(32260) = X(1156)-of-anti-Wasat-triangle if ABC is acute
X(32260) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2930, 32251, 184), (5095, 32246, 51)


X(32261) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO WALSMITH

Barycentrics
3*a^9+2*(b+c)*a^8-2*(b^2+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6-(2*b^4-3*b^2*c^2+2*c^4)*a^5+2*(b+c)*b^2*c^2*a^4+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^3+2*(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-(b^4-c^4)^2*a-2*(b^4-c^4)^2*(b+c) : :
X(32261) = 3*X(1)-4*X(32238) = 3*X(1)-2*X(32298) = 3*X(67)-2*X(32238) = 3*X(67)-X(32298) = 4*X(125)-3*X(16475) = 3*X(165)-2*X(32233) = 5*X(1698)-4*X(6593) = 3*X(1699)-4*X(32274) = 7*X(3624)-8*X(6698) = 3*X(3679)-2*X(32278) = 3*X(5587)-2*X(9970) = 3*X(11852)-2*X(32279) = 6*X(15462)-7*X(31423) = 5*X(18492)-4*X(32271)

The reciprocal orthologic center of these triangles is X(125).

X(32261) lies on these lines: {1,67}, {10,11061}, {35,32256}, {36,32270}, {40,542}, {57,32243}, {125,16475}, {165,32233}, {511,12407}, {515,32247}, {517,32306}, {1503,9904}, {1697,32297}, {1698,6593}, {1699,32274}, {2781,5691}, {2836,5904}, {2948,3416}, {3099,32268}, {3448,5847}, {3624,6698}, {3679,32278}, {3751,13211}, {4663,16176}, {5587,9970}, {5588,32281}, {5589,32280}, {7713,32239}, {8185,32262}, {8186,32265}, {8187,32266}, {8188,32295}, {8189,32296}, {9578,32289}, {9581,32290}, {9896,14984}, {10789,32242}, {10826,32287}, {10827,32288}, {11852,32279}, {13888,32303}, {13942,32304}, {15462,31423}, {18492,32271}, {19003,32252}, {19004,32253}

X(32261) = reflection of X(i) in X(j) for these (i,j): (1, 67), (2948, 3416), (3751, 13211), (11061, 10), (16176, 4663), (32298, 32238)
X(32261) = X(67)-of-Aquila-triangle
X(32261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (67, 32298, 32238), (32238, 32298, 1)


X(32262) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO WALSMITH

Barycentrics    (SB+SC)*(3*(6*R^2-SW)*R^2*S^2+(2*SW^2+(3*SA-11*SW)*R^2)*SA*SW) : :
X(32262) = 3*X(159)-2*X(2930) = 3*X(1177)-2*X(6593) = 4*X(1177)-3*X(19153) = X(2930)-3*X(10117) = 4*X(6593)-3*X(15141) = 8*X(6593)-9*X(19153) = 2*X(15141)-3*X(19153)

The reciprocal orthologic center of these triangles is X(125).

X(32262) lies on these lines: {3,1177}, {6,1205}, {22,11061}, {24,32247}, {25,67}, {159,2930}, {197,32256}, {206,17847}, {382,25328}, {511,12412}, {542,7387}, {597,14130}, {1204,19136}, {1503,9919}, {1593,5621}, {1598,32274}, {1995,2892}, {2393,32276}, {2777,23049}, {2916,25331}, {2937,15582}, {3556,5904}, {5020,6698}, {5594,32281}, {5595,32280}, {6293,9968}, {7517,32306}, {8185,32261}, {8190,32265}, {8191,32266}, {8192,32298}, {8193,32278}, {8194,32295}, {8195,32296}, {9818,32271}, {9908,14984}, {9914,16010}, {9920,15581}, {10037,32307}, {10046,32308}, {10628,19149}, {10790,32242}, {10828,32268}, {10829,32287}, {10830,32288}, {10831,32289}, {10832,32290}, {10833,32297}, {10834,32309}, {10835,32310}, {11284,15116}, {11365,32238}, {11414,32233}, {11853,32279}, {13201,20806}, {13203,23300}, {13889,32303}, {13943,32304}, {15579,20427}, {16176,32240}, {18954,32243}, {19005,32252}, {19006,32253}, {22654,32270}

X(32262) = reflection of X(i) in X(j) for these (i,j): (159, 10117), (13203, 23300), (15141, 1177), (17847, 206), (32254, 15581)
X(32262) = X(67)-of-Ara-triangle
X(32262) = {X(1177), X(15141)}-harmonic conjugate of X(19153)


X(32263) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WALSMITH TO ARIES

Barycentrics    SA*((28*R^2-SA-3*SW)*S^2-(SB+SC)*(2*(6*SA-SW)*R^2-SA^2+SB*SC+SW^2)) : :
X(32263) = 5*X(113)-4*X(22660) = X(12295)+2*X(15085)

The reciprocal orthologic center of these triangles is X(32264).

X(32263) lies on these lines: {6,5181}, {20,68}, {52,113}, {110,6353}, {125,343}, {161,542}, {511,32125}, {569,5504}, {1209,10170}, {1495,3564}, {1511,13292}, {2777,17834}, {2931,3515}, {3569,14391}, {5448,11002}, {5449,7999}, {5642,32226}, {6053,12164}, {6146,16163}, {8057,32112}, {8673,32120}, {8681,32113}, {9517,32121}, {11472,12295}, {12893,15078}, {13198,19131}, {13399,20725}, {13754,32111}

X(32263) = reflection of X(i) in X(j) for these (i,j): (12164, 6053), (30714, 2931)


X(32264) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARIES TO WALSMITH

Barycentrics    (24*(3*SA-2*SW)*R^4-(39*SA-29*SW)*R^2*SW+(5*SA-4*SW)*SW^2)*S^2-(14*R^2-3*SW)*SB*SC*SW^2 : :
X(32264) = 3*X(154)-2*X(5181) = 3*X(1853)-4*X(15118) = 2*X(2892)-3*X(15131) = 3*X(5622)-2*X(6247) = 4*X(5972)-5*X(19132) = 4*X(6593)-3*X(15131) = 2*X(11598)-3*X(25406) = X(13203)-3*X(25321) = 2*X(15116)-3*X(19153) = X(20079)-3*X(25320)

The reciprocal orthologic center of these triangles is X(32263).

X(32264) lies on these lines: {20,2781}, {25,32285}, {67,468}, {69,15647}, {125,19118}, {154,5181}, {159,2930}, {382,17813}, {542,1498}, {895,1503}, {1853,15118}, {2854,5596}, {2892,6593}, {3564,9934}, {5095,5895}, {5622,6247}, {5648,31166}, {5972,19132}, {9833,14984}, {11598,25406}, {13203,25321}, {14982,19149}, {15116,19153}, {15126,15128}, {15141,31152}, {15462,16196}, {20079,25320}, {32239,32251}

X(32264) = reflection of X(i) in X(j) for these (i,j): (67, 1177), (69, 15647), (2892, 6593), (5648, 31166), (14982, 19149)
X(32264) = {X(2892), X(6593)}-harmonic conjugate of X(15131)


X(32265) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO WALSMITH

Barycentrics
(2*a^9+(b+c)*a^8-(b^2+c^2)*a^7-(b+c)*(b^2+c^2)*a^6-2*(b^4-b^2*c^2+c^4)*a^5+(b+c)*b^2*c^2*a^4+(b^6+c^6)*a^3+(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-(b^4-c^4)^2*(b+c))*D+a^2*(a+b+c)*(a-b-c)*(a^2+b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : : , where D = 4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(125).

X(32265) lies on these lines: {55,32266}, {67,5597}, {542,11252}, {2781,9834}, {5598,32298}, {5599,6593}, {5601,11061}, {5846,13209}, {8186,32261}, {8190,32262}, {8196,32274}, {8197,32278}, {8198,32280}, {8199,32281}, {8200,9970}, {11366,32238}, {11384,32239}, {11492,32256}, {11493,32270}, {11822,32233}, {11837,32242}, {11843,32247}, {11861,32268}, {11865,32287}, {11867,32288}, {11869,32289}, {11871,32290}, {11873,32297}, {11875,32306}, {11877,32307}, {11879,32308}, {11881,32309}, {11883,32310}, {13890,32303}, {13944,32304}, {18495,32271}, {18955,32243}, {19007,32252}, {19008,32253}

X(32265) = reflection of X(32266) in X(55)
X(32265) = X(67)-of-1st-Auriga-triangle


X(32266) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO WALSMITH

Barycentrics
-(2*a^9+(b+c)*a^8-(b^2+c^2)*a^7-(b+c)*(b^2+c^2)*a^6-2*(b^4-b^2*c^2+c^4)*a^5+(b+c)*b^2*c^2*a^4+(b^6+c^6)*a^3+(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-(b^4-c^4)^2*(b+c))*D+a^2*(a+b+c)*(a-b-c)*(a^2+b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : : , where D = 4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(125).

X(32266) lies on these lines: {55,32265}, {67,5598}, {542,11253}, {2781,9835}, {5597,32298}, {5600,6593}, {5602,11061}, {5846,13208}, {8187,32261}, {8191,32262}, {8203,32274}, {8204,32278}, {8205,32280}, {8206,32281}, {8207,9970}, {11367,32238}, {11385,32239}, {11492,32270}, {11493,32256}, {11823,32233}, {11838,32242}, {11844,32247}, {11862,32268}, {11866,32287}, {11868,32288}, {11870,32289}, {11872,32290}, {11874,32297}, {11876,32306}, {11878,32307}, {11880,32308}, {11882,32309}, {11884,32310}, {13891,32303}, {13945,32304}, {18497,32271}, {18956,32243}, {19009,32252}, {19010,32253}

X(32266) = reflection of X(32265) in X(55)
X(32266) = X(67)-of-2nd-Auriga-triangle


X(32267) = CENTER OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: WALSMITH TO 4th BROCARD

Barycentrics    10*a^6-4*(b^2+c^2)*a^4-(7*b^4-12*b^2*c^2+7*c^4)*a^2+(b^4-c^4)*(b^2-c^2) : :
X(32267) = 5*X(1495)+X(3580) = 2*X(1495)+X(32223) = 3*X(1495)+X(32225) = 3*X(2070)+X(5655) = X(3580)-5*X(7426) = 2*X(3580)-5*X(32223) = 3*X(3580)-5*X(32225) = X(5972)-4*X(15448) = X(5972)+2*X(32237) = 3*X(7426)-X(32225) = 2*X(12105)+X(16534) = X(15303)-3*X(18374) = 2*X(15448)+X(32237) = 3*X(32223)-2*X(32225)

See preamble just before X(31353).

X(32267) lies on these lines: {2,6030}, {23,5642}, {25,5476}, {30,5972}, {154,11225}, {182,26255}, {376,1533}, {468,11645}, {541,7575}, {542,1495}, {575,20192}, {1995,10168}, {2070,5655}, {3524,16187}, {4232,11179}, {5646,15706}, {5965,15360}, {7493,11178}, {12100,15082}, {12105,16534}, {13857,29317}, {14002,25555}, {14848,31860}, {14915,18579}, {15303,18374}

X(32267) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 5642}, {376, 1533}, {1495, 7426}
X(32267) = reflection of X(32223) in X(7426)
X(32267) = {X(15448), X(32237)}-harmonic conjugate of X(5972)


X(32268) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32268) lies on these lines: {32,67}, {511,12501}, {542,7826}, {1503,9984}, {2781,9873}, {2896,11061}, {3096,6593}, {3098,32233}, {3099,32261}, {6698,7846}, {9301,32306}, {9857,32278}, {9862,32247}, {9923,14984}, {9970,9996}, {9993,32274}, {9994,32280}, {9995,32281}, {9997,32298}, {10038,32307}, {10047,32308}, {10828,32262}, {10871,32287}, {10872,32288}, {10873,32289}, {10874,32290}, {10875,32295}, {10876,32296}, {10877,32297}, {10878,32309}, {10879,32310}, {11368,32238}, {11386,32239}, {11494,32256}, {11861,32265}, {11862,32266}, {11885,32279}, {13892,32303}, {13946,32304}, {18500,32271}, {18957,32243}, {19011,32252}, {19012,32253}, {22744,32270}

X(32268) = X(67)-of-5th-Brocard-triangle


X(32269) = CENTER OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: WALSMITH TO CIRCUMORTHIC

Barycentrics    2*a^6+(b^2+c^2)*a^4-4*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2) : :
X(32269) = 3*X(23)+X(3448) = X(110)-3*X(7426) = X(110)+3*X(15360) = X(125)-3*X(32225) = 3*X(468)-2*X(5972) = 3*X(858)-5*X(15059) = 3*X(1495)-X(24981) = X(1514)+2*X(3581) = 3*X(1514)-2*X(7728) = X(3448)-3*X(3580) = 3*X(3581)+X(7728) = 4*X(5972)-3*X(11064) = X(5972)-3*X(32223) = 3*X(7426)-2*X(15448) = X(7728)-3*X(11799) = X(11064)-4*X(32223) = X(12041)-3*X(15361) = 5*X(15059)+3*X(15107) = 3*X(15360)+2*X(15448) = 4*X(16111)-3*X(20725) = X(16111)-3*X(32110) = X(20725)-4*X(32110) = X(25329)-3*X(32217)

See preamble just before X(31353).

X(32269) lies on these lines: {2,1350}, {3,16657}, {6,7493}, {22,13567}, {23,1503}, {25,343}, {26,6146}, {30,125}, {51,6676}, {52,13383}, {68,9714}, {69,4232}, {107,297}, {110,524}, {140,373}, {141,1995}, {154,6515}, {184,1353}, {187,6388}, {193,15585}, {230,3124}, {237,23181}, {381,4549}, {394,6353}, {428,21243}, {460,13449}, {462,20429}, {463,20428}, {468,511}, {542,32237}, {549,22112}, {599,26255}, {858,15059}, {1211,4228}, {1213,7474}, {1330,4248}, {1370,26958}, {1495,3564}, {1514,3581}, {1620,30552}, {1624,5201}, {1648,4226}, {1853,7500}, {1899,9909}, {1993,10192}, {2070,12310}, {3060,23292}, {3098,30739}, {3146,6696}, {3542,17834}, {3589,5640}, {3631,10546}, {3658,15447}, {3796,10565}, {3818,10301}, {3917,6677}, {4239,26543}, {5054,5544}, {5094,31670}, {5104,24855}, {5446,7542}, {5449,7553}, {5562,21841}, {5663,16619}, {5889,16252}, {5943,7499}, {5946,25337}, {6243,9820}, {6329,15019}, {6723,19924}, {6800,8550}, {7391,23332}, {7465,25964}, {7488,12241}, {7494,10601}, {7503,15873}, {7517,12359}, {7530,16654}, {7556,12022}, {7568,10095}, {7570,7693}, {7706,15760}, {9019,11746}, {9730,16618}, {9970,32227}, {10020,10263}, {10516,31860}, {10625,16238}, {11002,14389}, {11003,12007}, {11594,32311}, {11649,11800}, {11745,13160}, {12088,26879}, {12107,12370}, {12828,16165}, {13142,13367}, {14449,18282}, {14693,15366}, {15993,20998}, {16195,19467}, {16789,19136}, {17522,26579}, {20850,31383}

X(32269) = midpoint of X(i) and X(j) for these lines: {i,j}: {23, 3580}, {858, 15107}, {3581, 11799}, {7426, 15360}
X(32269) = reflection of X(i) in X(j) for these (i,j): (110, 15448), (468, 32223), (1514, 11799), (11064, 468)
X(32269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7493, 13394), (5640, 7495, 3589)


X(32270) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO WALSMITH

Barycentrics
a*(a^11-(b^2-4*b*c+c^2)*a^9+2*(b+c)*b*c*a^8-(2*b^4+2*c^4+(2*b^2-b*c+2*c^2)*b*c)*a^7-2*(b+c)*(b^2+c^2)*b*c*a^6+2*(b^6+c^6-(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c)*a^5+2*(b+c)*b^3*c^3*a^4+(b^3-c^3)*(b-c)*(b^2+c^2)*(b^2+3*b*c+c^2)*a^3+2*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c*a^2-(b^4-c^4)^2*(b^2+c^2)*a-2*(b^4-c^4)^2*(b+c)*b*c) : :

The reciprocal orthologic center of these triangles is X(125).

X(32270) lies on these lines: {1,15141}, {3,32256}, {36,32261}, {55,32298}, {56,67}, {104,32247}, {511,19478}, {542,11249}, {956,32278}, {958,6593}, {999,32238}, {1503,22583}, {2781,12114}, {2930,23361}, {2975,11061}, {3428,32233}, {5846,13204}, {6698,25524}, {9970,22758}, {10966,32297}, {11492,32266}, {11493,32265}, {14984,22659}, {18761,32271}, {19013,32252}, {19014,32253}, {22479,32239}, {22520,32242}, {22654,32262}, {22744,32268}, {22753,32274}, {22755,32279}, {22756,32280}, {22757,32281}, {22759,32289}, {22760,32290}, {22761,32295}, {22762,32296}, {22763,32303}, {22764,32304}, {22765,32306}, {22766,32307}, {22767,32308}, {22768,32309}

X(32270) = reflection of X(i) in X(j) for these (i,j): (32256, 3), (32288, 6593)
X(32270) = X(67)-of-2nd-circumperp-tangential-triangle


X(32271) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO WALSMITH

Barycentrics
4*(b^2+c^2)*a^10-(7*b^4-2*b^2*c^2+7*c^4)*a^8-2*(b^4-c^4)*(b^2-c^2)*a^6+2*(4*b^8+4*c^8-(5*b^4-4*b^2*c^2+5*c^4)*b^2*c^2)*a^4-2*(b^6-c^6)*(b^4-c^4)*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(32271) = 3*X(4)+X(11061) = 3*X(5)-2*X(6698) = X(20)-3*X(15462) = X(67)-3*X(381) = X(74)-3*X(14561) = 3*X(113)-X(5181) = X(146)+3*X(14853) = 3*X(182)-4*X(32300) = 3*X(265)-X(25335) = X(895)+3*X(10706) = X(895)-3*X(20423) = X(1350)-3*X(14643) = 3*X(1352)-X(32244) = X(2930)-3*X(5655) = 5*X(3091)-X(32247) = 3*X(5476)-2*X(15118) = 3*X(5476)-X(32305) = 3*X(9970)-X(11061) = 3*X(10752)+X(32244) = X(11579)-3*X(14853)

The reciprocal orthologic center of these triangles is X(125).

X(32271) lies on these lines: {4,542}, {5,2781}, {6,7728}, {20,15462}, {23,5642}, {30,6593}, {67,381}, {74,14561}, {110,7519}, {113,511}, {125,5169}, {146,11579}, {182,1177}, {265,25335}, {382,15140}, {389,16003}, {541,5476}, {546,32274}, {611,12374}, {613,12373}, {858,12824}, {1350,14643}, {1351,14982}, {1352,10752}, {1478,32290}, {1479,32289}, {1503,1539}, {1511,29181}, {2854,21850}, {2930,5655}, {3091,32247}, {3098,5972}, {3583,32297}, {3585,32243}, {3589,12041}, {3618,12244}, {3843,32306}, {5085,20127}, {5092,16111}, {5480,5663}, {7527,10990}, {7530,12584}, {7540,30714}, {9818,32262}, {9955,32238}, {10296,13202}, {10510,18325}, {10605,15128}, {10895,32307}, {10896,32308}, {11178,32257}, {11444,18504}, {11645,15303}, {12061,14984}, {12699,32278}, {13665,32253}, {13785,32252}, {14483,18125}, {16163,29317}, {16176,32272}, {17702,19139}, {18491,32256}, {18492,32261}, {18495,32265}, {18497,32266}, {18500,32268}, {18502,32242}, {18507,32279}, {18509,32280}, {18511,32281}, {18516,32287}, {18517,32288}, {18520,32295}, {18522,32296}, {18525,32298}, {18538,32303}, {18542,32309}, {18544,32310}, {18553,32275}, {18761,32270}, {18762,32304}

X(32271) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 9970}, {6, 7728}, {110, 31670}, {146, 11579}, {382, 32233}, {1351, 14982}, {1352, 10752}, {10510, 18325}, {10706, 20423}, {12699, 32278}, {16176, 32272}, {18507, 32279}, {18525, 32298}
X(32271) = reflection of X(i) in X(j) for these (i,j): (125, 19130), (3098, 5972), (5642, 25566), (12041, 3589), (12584, 16534), (16003, 20301), (16111, 5092), (32238, 9955), (32274, 546), (32275, 18553), (32305, 15118)
X(32271) = X(67)-of-Ehrmann-mid-triangle
X(32271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (146, 14853, 11579), (5476, 32305, 15118)


X(32272) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO WALSMITH

Barycentrics    SA*(12*(3*R^2-SW)*S^2-(SB+SC)*(27*R^2-3*SA-7*SW)*SW) : :
X(32272) = 3*X(3)-4*X(32257) = 4*X(67)-3*X(20126) = 3*X(67)-2*X(32305) = 3*X(265)-2*X(895) = 3*X(381)-2*X(5095) = 3*X(568)-4*X(32246) = 3*X(1352)-2*X(6593) = 4*X(1352)-3*X(14643) = 2*X(1353)-3*X(14644) = X(2930)-3*X(15069) = 4*X(2930)-3*X(23236) = 9*X(5055)-8*X(32300) = 3*X(5093)-4*X(7687) = 3*X(5655)-2*X(11061) = 8*X(6593)-9*X(14643) = 4*X(10297)-3*X(18449) = X(11061)-3*X(11180) = 4*X(15069)-X(23236) = 9*X(20126)-8*X(32305) = 2*X(32257)-3*X(32275)

The reciprocal orthologic center of these triangles is X(110).

X(32272) lies on these lines: {3,67}, {5,32234}, {30,32244}, {49,5622}, {69,12121}, {110,19129}, {193,10113}, {265,895}, {381,5095}, {382,32250}, {511,22584}, {524,18323}, {567,32245}, {568,32246}, {1177,10540}, {1352,6593}, {1353,14644}, {1503,20127}, {2072,25328}, {2781,18439}, {2854,7723}, {3448,16051}, {3545,10294}, {5055,32300}, {5093,7687}, {5159,9140}, {5655,11061}, {5663,5921}, {6288,9970}, {6698,11179}, {6776,15061}, {7728,18440}, {9818,32240}, {10255,20301}, {10510,18441}, {11459,32248}, {11562,14913}, {11898,17702}, {12111,32249}, {12272,12281}, {13754,32260}, {14516,15054}, {14912,20304}, {14984,18436}, {15027,15128}, {16176,32271}, {18403,32273}, {18445,32251}, {18447,32259}, {18451,32276}, {18453,32277}, {18455,32286}, {18457,32291}, {18459,32292}, {18462,32293}, {18463,32294}, {18468,32301}, {18470,32302}, {18531,32255}, {18917,32241}, {19176,32258}, {20397,26944}

X(32272) = midpoint of X(i) and X(j) for these lines: {i,j}: {12111, 32249}, {12272, 12281}
X(32272) = reflection of X(i) in X(j) for these (i,j): (3, 32275), (193, 10113), (382, 32250), (5655, 11180), (7728, 18440), (11562, 14913), (12121, 69), (16176, 32271), (18438, 7723), (32234, 5)
X(32272) = X(1156)-of-Ehrmann-side-triangle if ABC is acute
X(32272) = Stammler-circle-inverse of X(9876)


X(32273) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO WALSMITH

Barycentrics    3*(3*SA-2*SW)*R^2*S^2-(27*R^2-8*SW)*SB*SC*SW : :
X(32273) = 3*X(4)+X(32255) = X(67)-3*X(265) = 3*X(182)-4*X(15118) = 3*X(381)-X(2930) = 3*X(576)-2*X(5095) = 3*X(1351)-X(16176) = X(1657)-3*X(5621) = 5*X(3843)-X(32254) = 2*X(5181)-3*X(11178) = 3*X(5476)-2*X(6593) = X(10620)-3*X(25330) = X(11061)-3*X(20423) = X(12383)-3*X(14561) = 3*X(14644)-2*X(24206) = 2*X(14810)-3*X(15061) = 3*X(14853)-2*X(25556) = 2*X(15116)-3*X(23325) = X(15141)-3*X(23049) = 3*X(15462)-4*X(25555) = 2*X(16163)-3*X(17508)

The reciprocal orthologic center of these triangles is X(110).

X(32273) lies on these lines: {3,20301}, {4,542}, {5,12584}, {6,12902}, {30,25328}, {67,265}, {74,29317}, {110,7533}, {125,3098}, {141,11801}, {182,15118}, {381,2930}, {382,16010}, {524,18572}, {575,32233}, {578,30714}, {1177,18400}, {1351,16176}, {1352,18387}, {1503,9976}, {1657,5621}, {2781,10263}, {2836,18480}, {2854,3818}, {3153,32244}, {3448,16981}, {3564,19479}, {3583,32286}, {3585,32259}, {3843,32254}, {3845,16510}, {5092,12121}, {5181,11178}, {5189,9140}, {5476,6593}, {5480,19140}, {5622,21659}, {5648,25561}, {5987,9993}, {6288,18553}, {6564,32291}, {6565,32292}, {9143,25566}, {9927,11591}, {10264,29181}, {10620,25330}, {10733,11579}, {11477,32306}, {11564,18125}, {11645,18325}, {12370,18428}, {12383,14561}, {13293,23300}, {13434,15462}, {13851,32260}, {14644,24206}, {14810,15061}, {14853,25556}, {15116,23325}, {15141,23049}, {16163,17508}, {16808,32301}, {16809,32302}, {18382,19506}, {18386,32240}, {18388,32245}, {18390,32246}, {18392,32248}, {18394,32249}, {18396,32251}, {18403,32272}, {18404,32275}, {18405,32276}, {18406,32277}, {18414,32293}, {18415,32294}, {18420,32300}, {18531,32257}, {18918,32241}, {19177,32258}

X(32273) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 12902}, {382, 16010}, {3448, 31670}, {10733, 11579}, {11477, 32306}
X(32273) = reflection of X(i) in X(j) for these (i,j): (3, 20301), (110, 19130), (141, 11801), (3098, 125), (3818, 10113), (5648, 25561), (9143, 25566), (12121, 5092), (12584, 5), (13293, 23300), (19140, 5480), (19506, 18382), (32233, 575), (32305, 25328)
X(32273) = X(1156)-of-Ehrmann-vertex-triangle if ABC is acute


X(32274) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO WALSMITH

Barycentrics    3*S^4-3*(R^2*(SW+3*SA)-SA^2+SB*SC)*S^2-(18*R^2-5*SW)*SB*SC*SW : :
X(32274) = 3*X(4)+X(32247) = X(6)-3*X(14644) = 3*X(6)-X(32234) = 3*X(67)-X(32247) = X(110)-3*X(10516) = 3*X(125)+X(32250) = 3*X(381)-X(9970) = 3*X(381)+X(32306) = 5*X(1656)-3*X(15462) = 3*X(1699)+X(32261) = 5*X(3091)-X(11061) = 2*X(3589)-3*X(23515) = 5*X(3763)-3*X(15035) = 3*X(5085)-5*X(15059) = 3*X(5587)-X(32278) = 3*X(5603)-X(32298) = X(5921)+3*X(25320) = X(6776)-5*X(15081) = 3*X(9140)-X(16010) = 9*X(14644)-X(32234)

The reciprocal orthologic center of these triangles is X(125).

X(32274) lies on these lines: {2,32233}, {3,6698}, {4,67}, {5,542}, {6,7699}, {11,32243}, {12,32297}, {24,5621}, {30,8262}, {66,10293}, {98,32242}, {110,10516}, {113,25488}, {125,468}, {141,17702}, {182,20304}, {235,32239}, {265,1352}, {371,32303}, {372,32304}, {381,9970}, {511,10113}, {515,32238}, {524,10297}, {546,32271}, {895,15069}, {1350,10733}, {1478,32308}, {1479,32307}, {1511,24206}, {1587,32252}, {1588,32253}, {1598,32262}, {1614,5622}, {1656,15462}, {1699,32261}, {1986,32191}, {1995,9140}, {2493,11005}, {2836,5777}, {3091,11061}, {3448,14982}, {3564,11801}, {3574,5095}, {3575,10990}, {3589,23515}, {3763,15035}, {3818,5663}, {5085,15059}, {5480,7687}, {5587,32278}, {5603,32298}, {5921,25320}, {6103,28343}, {6201,32281}, {6202,32280}, {6247,13419}, {6776,15081}, {7495,16165}, {7533,12824}, {7547,16176}, {7574,9019}, {7575,11645}, {8196,32265}, {8203,32266}, {8212,32295}, {8213,32296}, {9517,18312}, {9815,25711}, {9927,11591}, {9969,10628}, {9993,32268}, {10531,32309}, {10532,32310}, {10893,32287}, {10894,32288}, {10895,32289}, {10896,32290}, {11178,12584}, {11477,15044}, {11496,32256}, {11579,18440}, {11897,32279}, {12041,29012}, {12061,18383}, {12106,20379}, {12295,29181}, {12588,12904}, {12589,12903}, {14094,25335}, {14787,23236}, {15577,19457}, {19161,21650}, {22753,32270}

X(32274) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 67}, {265, 1352}, {895, 15069}, {1350, 10733}, {3448, 14982}, {9970, 32306}, {11005, 11646}, {11477, 32244}, {11579, 18440}, {14094, 25335}, {15738, 32246}, {19161, 21650}
X(32274) = reflection of X(i) in X(j) for these (i,j): (3, 6698), (182, 20304), (1511, 24206), (1986, 32191), (5480, 7687), (6593, 5), (8550, 15118), (32271, 546), (32305, 20379)
X(32274) = complement of X(32233)
X(32274) = X(67)-of-Euler-triangle
X(32274) = {X(381), X(32306)}-harmonic conjugate of X(9970)


X(32275) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO WALSMITH

Barycentrics    SA*(12*(3*R^2-SW)*S^2-(SB+SC)*(18*R^2-3*SA-5*SW)*SW) : :
X(32275) = 2*X(6)-3*X(23515) = 3*X(67)-X(16010) = 3*X(113)-2*X(9970) = X(193)-3*X(14644) = 3*X(599)-X(32233) = 3*X(1352)-X(9970) = 5*X(1656)-4*X(32300) = 5*X(3620)-3*X(15035) = 3*X(5050)-4*X(6723) = 3*X(5181)-2*X(12584) = 3*X(5622)-4*X(20397) = 4*X(9822)-3*X(16222) = 3*X(11180)+X(32247) = 4*X(12584)-3*X(30714) = 3*X(13169)-X(32247) = 2*X(15069)+X(16003) = 3*X(15069)+X(16010) = 3*X(16003)-2*X(16010) = X(32254)+3*X(32306) = 2*X(32257)+X(32272)

The reciprocal orthologic center of these triangles is X(110).

X(32275) lies on these lines: {2,32234}, {3,67}, {4,32244}, {5,5095}, {6,23515}, {30,32250}, {52,32246}, {68,895}, {69,4549}, {74,5921}, {110,19131}, {113,1352}, {125,3292}, {155,32251}, {193,14644}, {265,11898}, {511,7723}, {524,10297}, {541,11180}, {569,32245}, {575,24572}, {1060,32259}, {1062,32286}, {1147,5622}, {1177,10539}, {1209,6593}, {1351,7687}, {1353,20304}, {1503,16111}, {1656,32300}, {1995,12828}, {2072,20301}, {2777,18440}, {2781,12162}, {2854,9967}, {3620,15035}, {5050,6723}, {5562,14984}, {5609,31831}, {5965,18449}, {6334,9003}, {6403,12219}, {6643,32255}, {6698,8550}, {6699,6776}, {7395,32240}, {7530,32239}, {8251,32277}, {9140,16051}, {9822,16222}, {10510,19380}, {10628,14913}, {10634,32301}, {10635,32302}, {10897,32291}, {10898,32292}, {11061,16534}, {11178,15303}, {11411,32241}, {11444,32248}, {11459,32249}, {11585,25328}, {14912,15059}, {17814,32276}, {18404,32273}, {18553,32271}, {19179,32258}, {19428,32293}, {19429,32294}

X(32275) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 32272}, {4, 32244}, {67, 15069}, {74, 5921}, {265, 11898}, {5562, 32260}, {6403, 12219}, {11180, 13169}
X(32275) = reflection of X(i) in X(j) for these (i,j): (3, 32257), (52, 32246), (113, 1352), (1351, 7687), (1353, 20304), (5095, 5), (6776, 6699), (8550, 6698), (9967, 12358), (11061, 16534), (15303, 11178), (16003, 67), (30714, 5181), (32271, 18553)
X(32275) = complement of X(32234)
X(32275) = X(1156)-of-2nd-Euler-triangle if ABC is acute


X(32276) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EXCOSINE TO WALSMITH

Barycentrics    (SB+SC)*((3*R^2*(24*R^2-11*SW)+2*(SA+2*SW)*SW)*S^2-2*((3*SA+SW)*R^2-SA^2+SB*SC)*SA*SW) : :
X(32276) = 3*X(6)-2*X(15141) = 2*X(66)-3*X(25330) = 4*X(110)-5*X(19132) = 3*X(154)-4*X(1177) = 3*X(154)-2*X(2930) = 4*X(895)-3*X(17813) = 3*X(1853)-2*X(2892) = 3*X(1853)-4*X(25328) = 6*X(5621)-5*X(8567) = 3*X(10606)-4*X(32305) = 4*X(12584)-5*X(17821) = 4*X(15118)-3*X(15131) = 4*X(15141)-3*X(17847) = 3*X(18405)-4*X(32273) = 2*X(23315)-3*X(25320)

The reciprocal orthologic center of these triangles is X(110).

X(32276) lies on these lines: {6,67}, {22,32248}, {24,32249}, {25,32260}, {64,895}, {66,25330}, {110,19132}, {154,1177}, {184,32240}, {221,32259}, {394,32244}, {511,15138}, {542,1498}, {1181,32234}, {1205,10602}, {1351,10628}, {1503,17812}, {1853,2892}, {1992,17823}, {2192,32286}, {2393,32262}, {2854,9924}, {2888,11061}, {2935,11579}, {3197,32277}, {3564,17838}, {5505,31860}, {5621,8567}, {5622,11425}, {6467,13171}, {6759,32254}, {10606,32305}, {10982,15738}, {12167,13417}, {12584,17821}, {13203,15583}, {13567,32241}, {14984,17834}, {15811,32250}, {16003,17822}, {17809,32245}, {17810,32246}, {17811,32257}, {17814,32275}, {17819,32291}, {17820,32292}, {17825,32300}, {17826,32301}, {17827,32302}, {18405,32273}, {18451,32272}, {19180,32258}, {19430,32293}, {19431,32294}, {23315,25320}

X(32276) = reflection of X(i) in X(j) for these (i,j): (64, 16010), (2892, 25328), (2930, 1177), (2935, 11579), (9924, 10117), (13203, 15583), (17847, 6), (32254, 6759)
X(32276) = X(1156)-of-1st-excosine-triangle if ABC is acute
X(32276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1177, 2930, 154), (5095, 32251, 6)


X(32277) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32277) lies on these lines: {19,5095}, {40,542}, {55,2930}, {65,651}, {67,71}, {110,1963}, {511,7724}, {1177,10536}, {2550,32255}, {2781,6254}, {2854,3779}, {3101,32244}, {3197,32276}, {3564,12661}, {3611,32260}, {3925,25328}, {5415,32291}, {5416,32292}, {5584,16010}, {6197,32234}, {6237,14984}, {7067,16563}, {7688,32305}, {8251,32275}, {9816,32300}, {10306,32254}, {10319,32257}, {10636,32301}, {10637,32302}, {10902,12584}, {11406,32240}, {11428,32245}, {11435,32246}, {11445,32248}, {11460,32249}, {11471,32250}, {18406,32273}, {18453,32272}, {18921,32241}, {19181,32258}, {19350,32251}, {19432,32293}, {19433,32294}

X(32277) = reflection of X(32286) in X(2930)
X(32277) = X(1156)-of-extangents-triangle if ABC is acute


X(32278) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO WALSMITH

Barycentrics    a*(a^8+2*(b+c)*a^7-(b^2+c^2)*a^6-(b+c)*(b^2+c^2)*a^5+b^2*c^2*a^4-2*(b+c)*(b^4-b^2*c^2+c^4)*a^3+(b^4-c^4)*(b^2-c^2)*a^2+(b^6+c^6)*(b+c)*a-(b^4-c^4)^2) : :
X(32278) = 2*X(1385)-3*X(15462) = 5*X(1698)-4*X(6698) = 3*X(3679)-X(32261) = 3*X(5085)-2*X(11709) = 3*X(5587)-2*X(32274) = 3*X(5657)-X(32247) = 3*X(5790)-X(32306) = 2*X(12261)-3*X(14561)

The reciprocal orthologic center of these triangles is X(125).

X(32278) lies on these lines: {1,6593}, {2,32238}, {8,11061}, {10,67}, {40,2781}, {65,651}, {72,692}, {110,518}, {355,542}, {511,12778}, {515,32233}, {517,9970}, {519,32298}, {956,32270}, {1385,15462}, {1386,7984}, {1503,12368}, {1698,6698}, {1737,32308}, {1837,32297}, {2771,11579}, {2854,2948}, {3057,32290}, {3242,11720}, {3679,32261}, {3827,13248}, {5085,11709}, {5090,32239}, {5252,32243}, {5587,32274}, {5657,32247}, {5687,32256}, {5688,32281}, {5689,32280}, {5790,32306}, {5846,25329}, {8193,32262}, {8197,32265}, {8204,32266}, {8214,32295}, {8215,32296}, {9857,32268}, {9928,14984}, {10039,32307}, {10791,32242}, {10914,32287}, {10915,32309}, {10916,32310}, {11900,32279}, {12261,14561}, {12699,32271}, {13883,32253}, {13893,32303}, {13936,32252}, {13947,32304}

X(32278) = midpoint of X(i) and X(j) for these lines: {i,j}: {8, 11061}, {2948, 3751}
X(32278) = reflection of X(i) in X(j) for these (i,j): (1, 6593), (67, 10), (895, 4663), (3242, 11720), (7984, 1386), (12699, 32271)
X(32278) = anticomplement of X(32238)
X(32278) = X(67)-of-outer-Garcia-triangle


X(32279) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO WALSMITH

Barycentrics    (S^2-3*SB*SC)*(3*S^4+(9*R^2*(12*R^2-SA-5*SW)+3*SA^2-3*SB*SC+4*SW^2)*S^2-54*R^4*(2*SW^2+3*SB*SC)+R^2*SW*(47*SW^2+12*SA^2+69*SB*SC)-SW^2*(3*SA^2+7*SB*SC+5*SW^2)) : :
X(32279) = 4*X(6698)-5*X(15183) = 3*X(11831)-2*X(32238) = 3*X(11845)-X(32247) = 3*X(11852)-X(32261) = 3*X(11897)-2*X(32274) = 3*X(11911)-X(32306)

The reciprocal orthologic center of these triangles is X(125).

X(32279) lies on these lines: {30,9970}, {67,402}, {511,12790}, {542,11251}, {1503,12369}, {1650,6593}, {2781,12113}, {4240,11061}, {6698,15183}, {11831,32238}, {11832,32239}, {11839,32242}, {11845,32247}, {11848,32256}, {11852,32261}, {11853,32262}, {11885,32268}, {11897,32274}, {11900,32278}, {11901,32280}, {11902,32281}, {11903,32287}, {11904,32288}, {11905,32289}, {11906,32290}, {11907,32295}, {11908,32296}, {11909,32297}, {11910,32298}, {11911,32306}, {11912,32307}, {11913,32308}, {11914,32309}, {11915,32310}, {12418,14984}, {13894,32303}, {13948,32304}, {18507,32271}, {18958,32243}, {19017,32252}, {19018,32253}, {22755,32270}

X(32279) = midpoint of X(4240) and X(11061)
X(32279) = reflection of X(i) in X(j) for these (i,j): (67, 402), (1650, 6593), (18507, 32271)
X(32279) = X(67)-of-Gossard-triangle


X(32280) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32280) lies on these lines: {6,67}, {511,12803}, {542,1161}, {1271,11061}, {1503,7725}, {2781,5871}, {2930,8903}, {5589,32261}, {5591,6593}, {5595,32262}, {5605,32298}, {5689,32278}, {6202,32274}, {6215,9970}, {6218,32246}, {7720,9971}, {8198,32265}, {8205,32266}, {8216,32295}, {8217,32296}, {8974,32303}, {9929,14984}, {9994,32268}, {10040,32307}, {10048,32308}, {10783,32247}, {10792,32242}, {10814,12367}, {10919,32287}, {10921,32288}, {10923,32289}, {10925,32290}, {10927,32297}, {10929,32309}, {10931,32310}, {11370,32238}, {11388,32239}, {11497,32256}, {11824,32233}, {11901,32279}, {11916,32306}, {13949,32304}, {18509,32271}, {18959,32243}, {22756,32270}

X(32280) = reflection of X(32281) in X(67)
X(32280) = X(67)-of-inner-Grebe-triangle


X(32281) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32281) lies on these lines: {6,67}, {511,12804}, {542,1160}, {1270,11061}, {1503,7726}, {2781,5870}, {2930,8904}, {5588,32261}, {5590,6593}, {5594,32262}, {5604,32298}, {5688,32278}, {6201,32274}, {6214,9970}, {6217,32246}, {7721,9971}, {8199,32265}, {8206,32266}, {8218,32295}, {8219,32296}, {8975,32303}, {9930,14984}, {9995,32268}, {10041,32307}, {10049,32308}, {10784,32247}, {10793,32242}, {10815,12367}, {10920,32287}, {10922,32288}, {10924,32289}, {10926,32290}, {10928,32297}, {10930,32309}, {10932,32310}, {11371,32238}, {11389,32239}, {11498,32256}, {11825,32233}, {11902,32279}, {11917,32306}, {13950,32304}, {18511,32271}, {18960,32243}, {22757,32270}

X(32281) = reflection of X(32280) in X(67)
X(32281) = X(67)-of-outer-Grebe-triangle


X(32282) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WALSMITH TO 3rd HATZIPOLAKIS

Barycentrics    (336*R^6-R^4*(49*SA+183*SW)+R^2*SW*(33*SW+20*SA)-2*SW^2*(SW+SA))*S^2-(4*R^2-SW)*R^2*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(32283).

X(32282) lies on these lines: {74,5894}, {110,13567}, {113,389}, {370,18604}, {1112,32125}, {1899,10117}, {2929,2931}, {11444,23308}, {11472,11801}, {15059,22830}, {15739,22530}


X(32283) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO WALSMITH

Barycentrics    (6*(21*SA+SW)*R^4-3*(25*SA-6*SW)*R^2*SW+2*(5*SA-2*SW)*SW^2)*S^2-(11*R^2-2*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(32282).

X(32283) lies on these lines: {6,67}, {542,13630}, {5012,5181}

X(32283) = midpoint of X(15118) and X(32285)


X(32284) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO WALSMITH

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^4+5*b^2*c^2+c^4)*a^6+16*(b^2+c^2)*b^2*c^2*a^4+2*(b^4+c^4+(b-c)^2*b*c)*(b^4+c^4-(b+c)^2*b*c)*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(32284) = X(4)+3*X(15531) = 3*X(6)-2*X(5462) = 3*X(51)-5*X(11482) = X(52)-3*X(1992) = 3*X(193)+X(11412) = 4*X(575)-3*X(5892) = 3*X(1353)-X(6102) = X(1843)-3*X(5093) = 3*X(1992)+X(15073) = 5*X(3567)-9*X(5032) = 3*X(6776)-X(10575) = 7*X(7999)-3*X(11160) = 3*X(9967)-X(11412) = 5*X(10574)-9*X(14912) = 2*X(10627)-3*X(11574) = 3*X(12294)-X(18439)

The reciprocal orthologic center of these triangles is X(3580).

X(32284) lies on these lines: {4,15531}, {5,8681}, {6,1147}, {51,8780}, {52,1992}, {67,15134}, {143,15471}, {182,22829}, {193,9967}, {195,18449}, {389,14984}, {511,550}, {524,1216}, {542,12370}, {567,32127}, {575,5892}, {576,2393}, {578,8548}, {1181,1351}, {1493,6153}, {1503,12897}, {1843,5093}, {1993,8538}, {1994,8537}, {2854,16534}, {2883,21850}, {3564,5907}, {3567,5032}, {5097,9969}, {5447,18951}, {5965,12899}, {6403,21852}, {6776,10575}, {7999,11160}, {9027,10170}, {9306,9925}, {10574,14912}, {10627,11574}, {11426,21651}, {11470,18445}, {11511,16266}, {11536,32245}, {12007,22952}, {12242,14913}, {12294,18439}, {15116,15129}, {15118,15120}

X(32284) = midpoint of X(i) and X(j) for these lines: {i,j}: {52, 15073}, {193, 9967}, {1351, 6467}
X(32284) = reflection of X(i) in X(j) for these (i,j): (182, 22829), (5446, 576), (6403, 21852), (9969, 5097), (12585, 13292), (14913, 18583)
X(32284) = {X(1992), X(15073)}-harmonic conjugate of X(52)


X(32285) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd HYACINTH TO WALSMITH

Barycentrics    SA*((36*R^2*(2*R^2-SW)-(SA-5*SW)*SW)*S^2-(SB+SC)*((3*SA-5*SW)*R^2-SA^2+SB*SC+SW^2)*SW) : :

The reciprocal orthologic center of these triangles is X(32263).

X(32285) lies on these lines: {6,67}, {25,32264}, {69,13198}, {184,5181}, {185,542}, {858,21639}, {895,1899}, {974,3564}, {1885,2781}, {2854,26926}, {3091,9970}, {3292,32257}, {5622,15057}, {5972,21637}, {6146,14984}, {6699,15136}, {9825,25711}, {10938,17702}, {11061,32241}, {15063,32306}, {17818,32233}, {32239,32246}

X(32285) = reflection of X(i) in X(j) for these (i,j): (15118, 32283), (32239, 32246)
X(32285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 67, 15116), (67, 32251, 125)


X(32286) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32286) lies on these lines: {1,542}, {11,25328}, {33,5095}, {34,32250}, {35,12584}, {36,32305}, {55,2930}, {56,16010}, {110,2330}, {399,611}, {497,32255}, {511,7727}, {524,5160}, {895,8540}, {1040,32257}, {1062,32275}, {1177,10535}, {1250,32302}, {1428,10091}, {1469,5663}, {1503,7286}, {2066,32291}, {2192,32276}, {2781,6285}, {2836,3057}, {2854,3024}, {3100,32244}, {3270,32260}, {3295,32254}, {3448,12589}, {3564,12888}, {3583,32273}, {5204,5621}, {5414,32292}, {6198,32234}, {6238,14984}, {7071,32240}, {7281,7312}, {7292,9140}, {7343,9976}, {7741,20301}, {9817,32300}, {9970,19369}, {10638,32301}, {11429,32245}, {11436,32246}, {11446,32248}, {11461,32249}, {11670,24471}, {12903,14982}, {14094,32289}, {18455,32272}, {18922,32241}, {19182,32258}, {19354,32251}, {19434,32293}, {19435,32294}

X(32286) = reflection of X(i) in X(j) for these (i,j): (11670, 24471), (32259, 1), (32277, 2930)
X(32286) = X(1156)-of-intangents-triangle if ABC is acute
X(32286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (895, 32290, 8540), (10091, 11579, 1428)


X(32287) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32287) lies on these lines: {11,67}, {12,32309}, {355,9970}, {511,12889}, {542,10525}, {1376,6593}, {1503,12371}, {2781,12114}, {3434,11061}, {10523,32307}, {10785,32247}, {10794,32242}, {10826,32261}, {10829,32262}, {10871,32268}, {10893,32274}, {10914,32278}, {10919,32280}, {10920,32281}, {10944,32289}, {10945,32295}, {10946,32296}, {10947,32297}, {10948,32308}, {10949,32310}, {11373,32238}, {11390,32239}, {11826,32233}, {11865,32265}, {11866,32266}, {11903,32279}, {11928,32306}, {12422,14984}, {13895,32303}, {13952,32304}, {18516,32271}, {18961,32243}, {19023,32252}, {19024,32253}

X(32287) = reflection of X(i) in X(j) for these (i,j): (32256, 6593), (32288, 9970)
X(32287) = X(67)-of-inner-Johnson-triangle
X(32287) = X(32309)-of-outer-Johnson-triangle


X(32288) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO WALSMITH

Barycentrics
a*(a^11-4*(b^2+b*c+c^2)*a^9-2*(b+c)*b*c*a^8+(4*b^4+4*c^4+(2*b^2+b*c+2*c^2)*b*c)*a^7+2*(b+c)*(b^2+c^2)*b*c*a^6+(2*b^6+2*c^6+(4*b^4+4*c^4-(b^2+4*b*c+c^2)*b*c)*b*c)*a^5-2*(b+c)*b^3*c^3*a^4-(b^3+c^3)*(b+c)*(5*b^4+5*c^4-(3*b^2+2*b*c+3*c^2)*b*c)*a^3-2*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4+b^2*c^2+2*c^4)*a+2*(b^4-c^4)^2*(b+c)*b*c) : :

The reciprocal orthologic center of these triangles is X(125).

X(32288) lies on these lines: {11,32310}, {12,67}, {72,692}, {355,9970}, {511,12890}, {542,10526}, {895,15232}, {958,6593}, {1503,12372}, {2781,11500}, {3436,11061}, {10523,32308}, {10786,32247}, {10795,32242}, {10827,32261}, {10830,32262}, {10872,32268}, {10894,32274}, {10921,32280}, {10922,32281}, {10950,32290}, {10951,32295}, {10952,32296}, {10953,32297}, {10954,32307}, {10955,32309}, {11374,32238}, {11391,32239}, {11827,32233}, {11867,32265}, {11868,32266}, {11904,32279}, {11929,32306}, {12423,14984}, {13896,32303}, {13953,32304}, {18517,32271}, {18962,32243}, {19025,32252}, {19026,32253}

X(32288) = reflection of X(i) in X(j) for these (i,j): (32270, 6593), (32287, 9970)
X(32288) = X(67)-of-outer-Johnson-triangle
X(32288) = X(32310)-of-inner-Johnson-triangle


X(32289) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32289) lies on these lines: {1,9970}, {4,32297}, {5,32308}, {6,3028}, {12,67}, {36,15462}, {55,2781}, {56,6593}, {65,651}, {73,1177}, {74,2330}, {110,1469}, {113,12589}, {182,10081}, {388,11061}, {495,32307}, {511,10088}, {542,1478}, {1479,32271}, {1503,12373}, {1870,8540}, {3056,10752}, {3085,32247}, {3157,14984}, {5480,12904}, {7286,10510}, {7354,32233}, {9578,32261}, {9654,32306}, {10091,19140}, {10797,32242}, {10831,32262}, {10873,32268}, {10895,32274}, {10923,32280}, {10924,32281}, {10944,32287}, {10956,32309}, {10957,32310}, {11375,32238}, {11392,32239}, {11501,32256}, {11579,19470}, {11869,32265}, {11870,32266}, {11905,32279}, {11930,32295}, {11931,32296}, {12896,31670}, {13897,32303}, {13954,32304}, {14094,32286}, {15128,26955}, {19027,32252}, {19028,32253}, {22759,32270}

X(32289) = reflection of X(32307) in X(495)
X(32289) = X(67)-of-1st-Johnson-Yff-triangle
X(32289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9970, 32290), (19369, 32259, 895)


X(32290) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32290) lies on these lines: {1,9970}, {4,32243}, {5,32307}, {6,3024}, {11,67}, {35,15462}, {55,6593}, {56,2781}, {74,1428}, {110,3056}, {113,12588}, {182,10065}, {496,32308}, {497,11061}, {511,10091}, {542,1479}, {613,5663}, {895,8540}, {1069,14984}, {1469,10752}, {1478,32271}, {1503,12374}, {3057,32278}, {3086,32247}, {5160,10510}, {5480,12903}, {6198,19369}, {6284,32233}, {7727,11579}, {9581,32261}, {9669,32306}, {10088,19140}, {10798,32242}, {10832,32262}, {10874,32268}, {10896,32274}, {10925,32280}, {10926,32281}, {10950,32288}, {10958,32309}, {10959,32310}, {11376,32238}, {11393,32239}, {11502,32256}, {11871,32265}, {11872,32266}, {11906,32279}, {11932,32295}, {11933,32296}, {13898,32303}, {13955,32304}, {14094,32259}, {15128,26956}, {18968,31670}, {19029,32252}, {19030,32253}, {22760,32270}

X(32290) = reflection of X(32308) in X(496)
X(32290) = X(67)-of-2nd-Johnson-Yff-triangle
X(32290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9970, 32289), (497, 11061, 32297)


X(32291) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32291) lies on these lines: {6,110}, {67,6413}, {159,13288}, {371,542}, {372,12584}, {511,12375}, {590,25328}, {1151,16010}, {1177,10533}, {2066,32286}, {2067,32259}, {2781,12964}, {2836,7969}, {3068,32255}, {3311,32254}, {3564,12891}, {5095,5412}, {5181,10960}, {5410,32240}, {5415,32277}, {5621,6409}, {5648,32252}, {6200,32305}, {6564,32273}, {8253,13774}, {10576,20301}, {10665,14984}, {10819,11579}, {10880,32234}, {10897,32275}, {10961,32300}, {11417,32244}, {11462,32249}, {11473,32250}, {11513,32257}, {17819,32276}, {18457,32272}, {18923,32241}, {19183,32258}, {19355,32251}, {19436,32293}, {19438,32294}, {21640,32260}

X(32291) = X(1156)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(32291) = {X(6), X(2930)}-harmonic conjugate of X(32292)


X(32292) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32292) lies on these lines: {6,110}, {67,6414}, {159,13287}, {371,12584}, {372,542}, {511,12376}, {615,25328}, {1152,16010}, {1177,10534}, {2781,12970}, {2836,7968}, {3069,32255}, {3312,32254}, {3564,12892}, {5095,5413}, {5181,10962}, {5411,32240}, {5414,32286}, {5416,32277}, {5621,6410}, {5648,32253}, {6396,32305}, {6502,32259}, {6565,32273}, {8252,13654}, {10577,20301}, {10666,14984}, {10820,11579}, {10881,32234}, {10898,32275}, {10963,32300}, {11418,32244}, {11463,32249}, {11474,32250}, {11514,32257}, {17820,32276}, {18459,32272}, {18924,32241}, {19184,32258}, {19356,32251}, {19437,32294}, {19439,32293}, {21641,32260}

X(32292) = {X(6), X(2930)}-harmonic conjugate of X(32291)
X(32292) = X(1156)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute


X(32293) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO WALSMITH

Barycentrics
(SB+SC)*(4*SW*S^4-2*(12*(3*R^2-SW)*R^2+(SA+SW)*SW)*S^3-((72*R^2+18*SA-29*SW)*R^2-4*SA^2+4*SB*SC+3*SW^2)*SW*S^2+2*SA*SW*(2*SW^2-SA^2+SB*SC+(-8*SW+3*SA)*R^2)*S+(4*R^2-SW)*(3*SA-2*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(110).

X(32293) lies on these lines: {110,19134}, {511,19484}, {542,18980}, {895,19426}, {2781,19500}, {2854,12590}, {2930,8939}, {3564,19482}, {5095,19446}, {9723,32294}, {12584,19440}, {13021,16010}, {14984,18939}, {18414,32273}, {18462,32272}, {18926,32241}, {19186,32258}, {19358,32251}, {19370,32259}, {19404,32240}, {19406,32244}, {19408,32245}, {19410,32246}, {19412,32248}, {19414,32249}, {19416,32250}, {19418,32254}, {19420,32255}, {19422,32257}, {19424,32234}, {19428,32275}, {19430,32276}, {19432,32277}, {19434,32286}, {19436,32291}, {19439,32292}, {19448,32300}, {19450,32301}, {19452,32302}, {19454,32305}, {21642,32260}, {23298,25328}

X(32293) = X(1156)-of-Lucas-antipodal-tangents-triangle if ABC is acute


X(32294) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO WALSMITH

Barycentrics
(SB+SC)*(4*SW*S^4+2*(12*(3*R^2-SW)*R^2+(SA+SW)*SW)*S^3-((72*R^2+18*SA-29*SW)*R^2-4*SA^2+4*SB*SC+3*SW^2)*SW*S^2-2*SA*SW*(2*SW^2-SA^2+SB*SC+(-8*SW+3*SA)*R^2)*S+(4*R^2-SW)*(3*SA-2*SW)*SA*SW^2) : :

The reciprocal orthologic center of these triangles is X(110).

X(32294) lies on these lines: {110,19135}, {511,19485}, {542,18981}, {895,19427}, {2781,19501}, {2854,12591}, {2930,8943}, {3564,19483}, {5095,19447}, {9723,32293}, {12584,19441}, {13022,16010}, {14984,18940}, {18415,32273}, {18463,32272}, {18927,32241}, {19187,32258}, {19359,32251}, {19371,32259}, {19405,32240}, {19407,32244}, {19409,32245}, {19411,32246}, {19413,32248}, {19415,32249}, {19417,32250}, {19419,32254}, {19421,32255}, {19423,32257}, {19425,32234}, {19429,32275}, {19431,32276}, {19433,32277}, {19435,32286}, {19437,32292}, {19438,32291}, {19449,32300}, {19451,32301}, {19453,32302}, {19455,32305}, {21643,32260}, {23299,25328}

X(32294) = X(1156)-of-Lucas(-1)-antipodal-tangents-triangle if ABC is acute


X(32295) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO WALSMITH

Barycentrics    2*((12*(3*SA-2*SW)*R^4-(3*SA+SW)*SW*R^2+2*(SB+SC)*SW^2)*S^2-4*R^2*SB*SC*SW^2)*S+((3*(6*SA^2+3*SA*SW-5*SW^2)*R^2-(4*SA^2+3*SA*SW-4*SW^2)*SW)*S^2-SB*SC*SW^3)*SW : :

The reciprocal orthologic center of these triangles is X(125).

X(32295) lies on these lines: {67,493}, {511,12894}, {542,10669}, {1503,12377}, {2781,9838}, {6461,32296}, {6462,11061}, {6593,8222}, {8188,32261}, {8194,32262}, {8210,32298}, {8212,32274}, {8214,32278}, {8218,32281}, {8220,9970}, {8408,19398}, {10875,32268}, {10945,32287}, {10951,32288}, {11377,32238}, {11394,32239}, {11503,32256}, {11828,32233}, {11840,32242}, {11846,32247}, {11907,32279}, {11930,32289}, {11932,32290}, {11947,32297}, {11949,32306}, {11951,32307}, {11953,32308}, {11955,32309}, {11957,32310}, {12426,14984}, {13899,32303}, {13956,32304}, {18520,32271}, {18963,32243}, {19031,32252}, {19032,32253}, {22761,32270}

X(32295) = X(67)-of-Lucas-homothetic-triangle


X(32296) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO WALSMITH

Barycentrics    -2*((12*(3*SA-2*SW)*R^4-(3*SA+SW)*SW*R^2+2*(SB+SC)*SW^2)*S^2-4*R^2*SB*SC*SW^2)*S+((3*(6*SA^2+3*SA*SW-5*SW^2)*R^2-(4*SA^2+3*SA*SW-4*SW^2)*SW)*S^2-SB*SC*SW^3)*SW : :

The reciprocal orthologic center of these triangles is X(125).

X(32296) lies on these lines: {67,494}, {511,12895}, {542,10673}, {1503,12378}, {2781,9839}, {6461,32295}, {6463,11061}, {6593,8223}, {8189,32261}, {8195,32262}, {8211,32298}, {8213,32274}, {8215,32278}, {8217,32280}, {8219,32281}, {8221,9970}, {8420,19399}, {10876,32268}, {10946,32287}, {10952,32288}, {11378,32238}, {11395,32239}, {11504,32256}, {11829,32233}, {11841,32242}, {11847,32247}, {11908,32279}, {11931,32289}, {11933,32290}, {11948,32297}, {11950,32306}, {11952,32307}, {11954,32308}, {11956,32309}, {11958,32310}, {12427,14984}, {13900,32303}, {13957,32304}, {18522,32271}, {18964,32243}, {19033,32252}, {19034,32253}, {22762,32270}

X(32296) = X(67)-of-Lucas(-1)-homothetic-triangle


X(32297) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32297) lies on these lines: {1,542}, {3,32308}, {4,32289}, {6,12904}, {11,6593}, {12,32274}, {33,32239}, {55,67}, {56,32233}, {110,12589}, {125,2330}, {265,611}, {497,11061}, {499,15462}, {511,12896}, {1352,10088}, {1479,9970}, {1503,3028}, {1697,32261}, {1837,32278}, {1858,2836}, {2098,32298}, {2646,32238}, {2781,6284}, {3295,32306}, {3583,32271}, {4294,32247}, {5095,8540}, {5432,6698}, {5621,9659}, {10833,32262}, {10877,32268}, {10927,32280}, {10928,32281}, {10947,32287}, {10953,32288}, {10965,32309}, {10966,32270}, {11873,32265}, {11874,32266}, {11909,32279}, {11947,32295}, {11948,32296}, {12428,14984}, {13901,32303}, {13958,32304}, {19037,32252}, {19038,32253}

X(32297) = reflection of X(32243) in X(1)
X(32297) = X(67)-of-Mandart-incircle-triangle
X(32297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 11061, 32290), (3295, 32306, 32307)


X(32298) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO WALSMITH

Barycentrics
3*a^9+(b+c)*a^8-(b^2+c^2)*a^7-(b^2+c^2)*(b+c)*a^6-(4*b^4-3*b^2*c^2+4*c^4)*a^5+(b+c)*b^2*c^2*a^4+(b^4+c^4)*(b^2+c^2)*a^3+(b^4-c^4)*(b^2-c^2)*(b+c)*a^2+(b^4-c^4)^2*a-(b^4-c^4)^2*(b+c) : :
X(32298) = 3*X(1)-2*X(32238) = 3*X(1)-X(32261) = 3*X(67)-4*X(32238) = 3*X(67)-2*X(32261) = 3*X(5603)-2*X(32274) = 3*X(10516)-4*X(11723)

The reciprocal orthologic center of these triangles is X(125).

X(32298) lies on these lines: {1,67}, {8,6593}, {55,32270}, {56,32256}, {110,5846}, {145,11061}, {511,12898}, {517,32233}, {519,32278}, {542,1482}, {944,2781}, {952,9970}, {1386,13211}, {1503,7978}, {2098,32297}, {2099,32243}, {2836,3555}, {3416,11720}, {5603,32274}, {5648,28538}, {8192,32262}, {8211,32296}, {9053,25329}, {9933,14984}, {10516,11723}, {10944,32287}, {11396,32239}, {11910,32279}, {13959,32304}, {18525,32271}

X(32298) = midpoint of X(145) and X(11061)
X(32298) = reflection of X(i) in X(j) for these (i,j): (8, 6593), (67, 1), (3416, 11720), (13211, 1386), (18525, 32271), (32261, 32238)
X(32298) = X(67)-of-5th-mixtilinear-triangle
X(32298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32261, 32238), (32238, 32261, 67), (32309, 32310, 67)


X(32299) = CENTER OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: REFLECTION TO WALSMITH

Barycentrics    (SB+SC)*((9*R^2*(10*R^2-7*SW)+10*SW^2)*S^2-(7*R^2-2*SW)*(3*SA-2*SW)*SA*SW) : :
X(32299) = 3*X(6)-X(32248) = X(2930)-3*X(11188) = 3*X(3060)-X(16176) = 2*X(6593)-3*X(16776) = 3*X(9971)-X(11061) = 3*X(16776)-4*X(32246)

See preamble just before X(31353).

X(32299) lies on these lines: {6,110}, {67,5189}, {125,17710}, {382,2781}, {524,32260}, {542,6102}, {1205,8705}, {2393,8262}, {3060,16176}, {3448,9973}, {3629,11800}, {9927,11591}, {9969,25329}, {9971,11061}, {10095,11536}, {11412,15738}, {11477,32249}, {15074,20301}, {19127,32251}

X(32299) = midpoint of X(i) and X(j) for these lines: {i,j}: {3448, 9973}, {11477, 32249}
X(32299) = reflection of X(i) in X(j) for these (i,j): (3629, 11800), (6593, 32246), (15074, 20301), (17710, 125), (25329, 9969)
X(32299) = {X(6593), X(32246)}-harmonic conjugate of X(16776)


X(32300) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO WALSMITH

Barycentrics    6*a^8-4*(b^2+c^2)*a^6-(7*b^4-12*b^2*c^2+7*c^4)*a^4+2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)^2 : :
X(32300) = 3*X(2)+X(5095) = 9*X(2)-X(32244) = 3*X(6)+X(5181) = X(67)+3*X(15303) = X(113)+3*X(5050) = X(125)-5*X(3618) = 3*X(125)+X(11061) = 3*X(182)+X(32271) = 9*X(373)-X(32260) = 3*X(597)+X(6593) = 3*X(597)-X(15118) = 9*X(597)-X(25328) = X(89)+3*X(5642) = X(1205)+3*X(12824) = 15*X(3618)+X(11061) = 3*X(5095)+X(32244) = X(5181)-3*X(5972) = 3*X(6593)+X(25328) = 3*X(15118)-X(25328) = X(32244)-3*X(32257)

The reciprocal orthologic center of these triangles is X(110).

X(32300) lies on these lines: {2,5095}, {5,542}, {6,5181}, {67,15303}, {110,19137}, {113,5050}, {125,3618}, {182,1177}, {373,32260}, {511,9826}, {578,9815}, {895,5642}, {1112,11574}, {1205,12824}, {1656,32275}, {1974,7519}, {2781,9729}, {2854,6329}, {2930,5020}, {3090,32234}, {3091,32250}, {3564,12900}, {3589,6698}, {5055,32272}, {5159,15471}, {5169,32239}, {5182,16278}, {5462,14984}, {5622,15063}, {5943,14763}, {6053,11579}, {6642,12584}, {6721,24975}, {7392,32255}, {7493,11511}, {7687,14561}, {9306,32245}, {9816,32277}, {9817,32286}, {9818,32305}, {9967,16222}, {9970,20417}, {10168,18580}, {10601,32251}, {10643,32301}, {10644,32302}, {10961,32291}, {10963,32292}, {11284,32240}, {11451,32248}, {11465,32249}, {11479,16010}, {11484,32254}, {12017,16111}, {13202,25406}, {14853,16163}, {15059,25321}, {15119,19510}, {15141,15805}, {15473,19128}, {16238,22330}, {17702,18583}, {17825,32276}, {18420,32273}, {18928,32241}, {19188,32258}, {19372,32259}, {19448,32293}, {19449,32294}, {22973,32163}, {24981,25320}

X(32300) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 5972}, {1112, 11574}, {5095, 32257}, {5159, 15471}, {6053, 11579}, {6593, 15118}, {9970, 20417}
X(32300) = reflection of X(6723) in X(3589)
X(32300) = complement of X(32257)
X(32300) = X(1156)-of-submedial-triangle if ABC is acute
X(32300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5095, 32257), (597, 6593, 15118)


X(32301) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32301) lies on these lines: {6,110}, {15,542}, {16,12584}, {159,10682}, {187,13859}, {511,10657}, {599,11130}, {1177,30402}, {2781,10675}, {3564,10663}, {5095,10641}, {5181,10639}, {5191,14173}, {5648,11081}, {7051,32259}, {9142,14179}, {10632,32234}, {10634,32275}, {10636,32277}, {10638,32286}, {10643,32300}, {10645,32305}, {10661,14984}, {11408,32240}, {11420,32244}, {11466,32249}, {11475,32250}, {11480,16010}, {11488,32255}, {11515,32257}, {16808,32273}, {16966,20301}, {17826,32276}, {18468,32272}, {18929,32241}, {19190,32258}, {19363,32251}, {19450,32293}, {19451,32294}, {21647,32260}, {23302,25328}

X(32301) = X(1156)-of-inner-tri-equilateral-triangle if ABC is acute
X(32301) = {X(6), X(2930)}-harmonic conjugate of X(32302)


X(32302) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(110).

X(32302) lies on these lines: {6,110}, {15,12584}, {16,542}, {159,10681}, {187,13858}, {511,10658}, {574,13859}, {599,11131}, {1177,30403}, {1250,32286}, {2781,10676}, {3564,10664}, {5095,10642}, {5181,10640}, {5191,14179}, {5648,11086}, {9142,14173}, {10633,32234}, {10635,32275}, {10637,32277}, {10644,32300}, {10646,32305}, {10662,14984}, {11409,32240}, {11421,32244}, {11467,32249}, {11476,32250}, {11481,16010}, {11486,32254}, {11489,32255}, {11516,32257}, {16809,32273}, {16967,20301}, {17827,32276}, {18470,32272}, {18930,32241}, {19191,32258}, {19364,32251}, {19373,32259}, {19452,32293}, {19453,32294}, {21648,32260}, {23303,25328}

X(32302) = X(1156)-of-outer-tri-equilateral-triangle if ABC is acute
X(32302) = {X(6), X(2930)}-harmonic conjugate of X(32301)


X(32303) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO WALSMITH

Barycentrics    3*S^4-(3*R^2*(3*SA+7*SW)-3*SA^2+3*SB*SC-4*SW^2)*S^2-2*(9*R^2-2*SW)*(SB+SC)*SW*S+SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(125).

X(32303) lies on these lines: {2,32253}, {6,6698}, {67,3068}, {371,32274}, {485,2781}, {511,13915}, {542,8981}, {590,6593}, {1503,8994}, {3763,19110}, {6723,13972}, {7585,32252}, {8972,11061}, {8974,32280}, {8975,32281}, {8976,9970}, {9540,32233}, {10516,19060}, {13883,32238}, {13884,32239}, {13886,32247}, {13887,32256}, {13888,32261}, {13889,32262}, {13890,32265}, {13891,32266}, {13892,32268}, {13893,32278}, {13895,32287}, {13896,32288}, {13897,32289}, {13898,32290}, {13899,32295}, {13900,32296}, {13901,32297}, {13902,32298}, {13903,32306}, {13904,32307}, {13905,32308}, {13906,32309}, {13907,32310}, {13909,14984}, {18538,32271}, {18965,32243}, {22763,32270}

X(32303) = X(67)-of-3rd-tri-squares-central-triangle
X(32303) = {X(6), X(6698)}-harmonic conjugate of X(32304)


X(32304) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO WALSMITH

Barycentrics    3*S^4-(3*R^2*(3*SA+7*SW)-3*SA^2+3*SB*SC-4*SW^2)*S^2+2*(9*R^2-2*SW)*(SB+SC)*SW*S+SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(125).

X(32304) lies on these lines: {2,32252}, {6,6698}, {67,3069}, {372,32274}, {486,2781}, {511,13979}, {542,13966}, {615,6593}, {1503,13969}, {3763,19111}, {6723,13910}, {7586,32253}, {9970,13951}, {10516,19059}, {11061,13941}, {13935,32233}, {13936,32238}, {13937,32239}, {13938,32242}, {13939,32247}, {13940,32256}, {13942,32261}, {13943,32262}, {13944,32265}, {13945,32266}, {13946,32268}, {13947,32278}, {13948,32279}, {13949,32280}, {13950,32281}, {13952,32287}, {13953,32288}, {13954,32289}, {13955,32290}, {13956,32295}, {13957,32296}, {13958,32297}, {13959,32298}, {13961,32306}, {13962,32307}, {13963,32308}, {13964,32309}, {13965,32310}, {13970,14984}, {18762,32271}, {18966,32243}, {22764,32270}

X(32304) = X(67)-of-4th-tri-squares-central-triangle
X(32304) = {X(6), X(6698)}-harmonic conjugate of X(32303)


X(32305) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO WALSMITH

Barycentrics    (SB+SC)*((9*R^2-3*SW)*S^2+(27*R^2-3*SA-4*SW)*SA*SW) : :
X(32305) = 3*X(3)-X(2930) = X(3)-3*X(5621) = 5*X(3)-X(32254) = X(67)-3*X(20126) = 3*X(67)-X(32272) = 3*X(74)+X(895) = 2*X(74)+X(9976) = 2*X(895)-3*X(9976) = X(895)-3*X(11579) = X(2930)-9*X(5621) = 2*X(2930)-3*X(12584) = X(2930)+3*X(16010) = 5*X(2930)-3*X(32254) = 6*X(5621)-X(12584) = 3*X(5621)+X(16010) = 15*X(5621)-X(32254) = X(12584)+2*X(16010) = 5*X(12584)-2*X(32254) = 5*X(16010)+X(32254) = 9*X(20126)-X(32272) = 3*X(20417)-X(32257)

The reciprocal orthologic center of these triangles is X(110).

X(32305) lies on these lines: {3,67}, {4,20301}, {6,10620}, {23,9140}, {24,32250}, {30,25328}, {35,32259}, {36,32286}, {74,511}, {98,20404}, {110,5092}, {125,1995}, {146,14561}, {182,4550}, {185,575}, {265,29012}, {376,32255}, {378,5095}, {399,5085}, {541,5476}, {576,2781}, {690,13233}, {1177,6000}, {1350,15041}, {1428,7727}, {1503,7575}, {1511,17508}, {1691,14901}, {1974,12292}, {1986,19124}, {2071,32244}, {2330,19470}, {2393,16219}, {2777,23049}, {2836,3579}, {2854,3098}, {3448,7492}, {3520,32234}, {3564,12901}, {5050,25556}, {5097,10752}, {5655,10168}, {5907,7550}, {6200,32291}, {6396,32292}, {6698,11178}, {6800,32315}, {7530,18381}, {7545,15027}, {7555,12359}, {7556,11457}, {7688,32277}, {7689,14984}, {7691,15021}, {7728,19130}, {8546,16510}, {9517,13247}, {9818,32300}, {10249,15141}, {10510,19402}, {10605,32251}, {10606,32276}, {10628,19150}, {10645,32301}, {10646,32302}, {10733,29323}, {11061,11179}, {11410,32240}, {11430,32245}, {11438,32246}, {11454,32248}, {11468,32249}, {11572,15044}, {11643,19905}, {12017,12308}, {12106,20379}, {12244,25320}, {12317,25406}, {12902,25330}, {14677,29181}, {14810,15055}, {14865,22330}, {14982,15061}, {15116,23329}, {16042,25561}, {18931,32241}, {19192,32258}, {19454,32293}, {19455,32294}, {19506,23300}, {20127,29317}, {21663,32260}

X(32305) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 16010}, {6, 10620}, {74, 11579}, {9970, 15054}, {12244, 31670}
X(32305) = reflection of X(i) in X(j) for these (i,j): (4, 20301), (110, 5092), (3098, 12041), (3818, 125), (5655, 10168), (7728, 19130), (9970, 575), (9976, 11579), (10752, 5097), (12584, 3), (14982, 24206), (16510, 8546), (19140, 182), (19506, 23300), (32271, 15118), (32273, 25328), (32274, 20379)
X(32305) = circumperp conjugate of X(8724)
X(32305) = circumtangential isogonal conjugate of X(23)
X(32305) = inverse of X(14830) in the circumcircle
X(32305) = X(1156)-of-Trinh-triangle if ABC is acute
X(32305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5621, 16010, 3), (5622, 9970, 575), (5622, 15054, 9970)


X(32306) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO WALSMITH

Barycentrics    (3*(6*SA-SW)*R^2-2*(3*SA-SW)*SW)*S^2+(9*R^2-4*SW)*SB*SC*SW : :
X(32306) = 3*X(3)-2*X(32233) = 3*X(6)-4*X(20301) = 3*X(67)-X(32233) = 4*X(125)-3*X(5050) = 3*X(381)-2*X(9970) = 3*X(381)-4*X(32274) = 3*X(599)-2*X(12584) = 2*X(1353)-3*X(25320) = 5*X(1656)-4*X(6593) = 3*X(2070)-4*X(8262) = 7*X(3526)-8*X(6698) = 7*X(3526)-6*X(15462) = 5*X(3843)-4*X(32271) = 4*X(5095)-5*X(11482) = 3*X(5622)-4*X(20379) = 3*X(5790)-2*X(32278) = 4*X(6698)-3*X(15462) = 3*X(9140)-X(32234) = 2*X(9976)-3*X(25330) = X(32254)-4*X(32275)

The reciprocal orthologic center of these triangles is X(125).

X(32306) lies on these lines: {3,67}, {5,9512}, {6,7579}, {30,32247}, {125,5050}, {182,11597}, {195,15141}, {265,1351}, {323,858}, {381,9970}, {382,2781}, {399,1352}, {511,12902}, {517,32261}, {524,7574}, {541,32250}, {576,16176}, {895,11255}, {999,32243}, {1353,25320}, {1503,10620}, {1598,32239}, {1656,6593}, {2070,8262}, {2854,11898}, {3295,32297}, {3526,6698}, {3843,32271}, {5094,9140}, {5095,11482}, {5169,15135}, {5622,15132}, {5663,11188}, {5790,32278}, {5921,12317}, {6193,23296}, {6403,15100}, {6417,32253}, {6418,32252}, {6776,10264}, {7495,9143}, {7517,32262}, {9301,32268}, {9654,32289}, {9669,32290}, {9976,25330}, {10113,10752}, {10246,32238}, {10247,32298}, {10516,19140}, {11412,12429}, {11477,32273}, {11579,11935}, {11801,14853}, {11842,32242}, {11849,32256}, {11875,32265}, {11876,32266}, {11911,32279}, {11916,32280}, {11917,32281}, {11928,32287}, {11929,32288}, {11949,32295}, {11950,32296}, {12000,32309}, {12001,32310}, {12017,15061}, {12308,14982}, {13903,32303}, {13961,32304}, {14561,25329}, {14677,14927}, {14791,32255}, {15027,15118}, {15063,32285}, {15074,32248}, {15081,18583}, {19510,22115}, {22765,32270}, {25331,25556}

X(32306) = midpoint of X(i) and X(j) for these lines: {i,j}: {5921, 12317}, {6403, 15100}, {15069, 25335}
X(32306) = reflection of X(i) in X(j) for these (i,j): (3, 67), (399, 1352), (1351, 265), (6776, 10264), (9970, 32274), (10752, 10113), (11061, 5), (11477, 32273), (12308, 14982), (14927, 14677), (16176, 576), (23236, 5181), (25336, 19140), (30714, 32257), (32248, 15074)
X(32306) = X(67)-of-X3-ABC-reflections-triangle
X(32306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9970, 32274, 381), (32243, 32308, 999)


X(32307) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32307) lies on these lines: {1,67}, {3,32243}, {5,32290}, {12,9970}, {35,32233}, {55,542}, {125,613}, {141,10091}, {265,3056}, {388,32247}, {495,32289}, {498,6593}, {499,6698}, {511,12903}, {1350,18968}, {1352,3024}, {1428,15061}, {1478,2781}, {1479,32274}, {1503,10065}, {3085,11061}, {3295,32297}, {3299,32252}, {3301,32253}, {3818,12374}, {5432,15462}, {5622,26956}, {5663,12588}, {7727,14982}, {10037,32262}, {10038,32268}, {10039,32278}, {10041,32281}, {10055,14984}, {10387,12896}, {10523,32287}, {10801,32242}, {10895,32271}, {10954,32288}, {11398,32239}, {11507,32256}, {11877,32265}, {11878,32266}, {11912,32279}, {11951,32295}, {11952,32296}, {13904,32303}, {13962,32304}, {22766,32270}

X(32307) = midpoint of X(67) and X(32309)
X(32307) = reflection of X(32289) in X(495)
X(32307) = X(67)-of-inner-Yff-triangle
X(32307) = X(32309)-of-outer-Yff-triangle
X(32307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 67, 32308), (3295, 32306, 32297)


X(32308) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32308) lies on these lines: {1,67}, {3,32297}, {5,32289}, {11,9970}, {36,32233}, {56,542}, {125,611}, {141,10088}, {265,1469}, {496,32290}, {497,32247}, {498,6698}, {499,6593}, {511,12904}, {999,32243}, {1350,12896}, {1352,3028}, {1478,32274}, {1479,2781}, {1503,10081}, {1737,32278}, {2330,15061}, {3086,11061}, {3299,32253}, {3301,32252}, {3818,12373}, {5433,15462}, {5622,26955}, {5663,12589}, {10046,32262}, {10047,32268}, {10048,32280}, {10049,32281}, {10071,14984}, {10523,32288}, {10802,32242}, {10896,32271}, {11399,32239}, {11508,32256}, {11879,32265}, {11880,32266}, {11913,32279}, {11953,32295}, {11954,32296}, {13905,32303}, {13963,32304}, {14982,19470}, {22767,32270}

X(32308) = midpoint of X(67) and X(32310)
X(32308) = reflection of X(32290) in X(496)
X(32308) = X(67)-of-outer-Yff-triangle
X(32308) = X(32310)-of-inner-Yff-triangle
X(32308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 67, 32307), (999, 32306, 32243)


X(32309) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32309) lies on these lines: {1,67}, {12,32287}, {511,12905}, {542,10679}, {1503,12381}, {2781,12115}, {5552,6593}, {9970,10942}, {10528,11061}, {10531,32274}, {10803,32242}, {10805,32247}, {10834,32262}, {10878,32268}, {10915,32278}, {10929,32280}, {10930,32281}, {10955,32288}, {10956,32289}, {10958,32290}, {10965,32297}, {11248,32233}, {11400,32239}, {11509,32243}, {11881,32265}, {11882,32266}, {11914,32279}, {11955,32295}, {11956,32296}, {12430,14984}, {13906,32303}, {13964,32304}, {18542,32271}, {19047,32252}, {19048,32253}, {22768,32270}

X(32309) = reflection of X(67) in X(32307)
X(32309) = X(67)-of-inner-Yff-tangents-triangle
X(32309) = {X(67), X(32298)}-harmonic conjugate of X(32310)


X(32310) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO WALSMITH

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

The reciprocal orthologic center of these triangles is X(125).

X(32310) lies on these lines: {1,67}, {11,32288}, {511,12906}, {542,10680}, {1503,12382}, {2781,12116}, {6593,10527}, {9970,10943}, {10529,11061}, {10532,32274}, {10804,32242}, {10806,32247}, {10835,32262}, {10879,32268}, {10916,32278}, {10931,32280}, {10932,32281}, {10949,32287}, {10957,32289}, {10959,32290}, {10966,32270}, {11249,32233}, {11401,32239}, {11510,32256}, {11883,32265}, {11884,32266}, {11915,32279}, {11957,32295}, {11958,32296}, {12001,32306}, {12431,14984}, {13907,32303}, {13965,32304}, {18544,32271}, {18967,32243}, {19049,32252}, {19050,32253}

X(32310) = reflection of X(67) in X(32308)
X(32310) = X(67)-of-outer-Yff-tangents-triangle
X(32310) = {X(67), X(32298)}-harmonic conjugate of X(32309)


X(32311) = CENTER OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: REFLECTION TO WALSMITH

Barycentrics
4*a^12-(b^2+c^2)*a^10-14*(b^4-b^2*c^2+c^4)*a^8+(b^2+c^2)*(12*b^4-17*b^2*c^2+12*c^4)*a^6+2*(4*b^8+4*c^8-(8*b^4-9*b^2*c^2+8*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(11*b^8+11*c^8-5*(5*b^4-6*b^2*c^2+5*c^4)*b^2*c^2)*a^2+2*(b^4-4*b^2*c^2+c^4)*(b^4-c^4)^2 : :
X(32311) = X(110)-3*X(9829) = X(3448)+3*X(6031) = 2*X(5972)-3*X(10163) = 3*X(6032)-5*X(15059) = 4*X(6723)-3*X(10162) = 3*X(23515)-2*X(31749)

See preamble just before X(31353).

X(32311) lies on these lines: {74,12505}, {110,599}, {113,31606}, {125,3849}, {2777,14866}, {3448,6031}, {5663,31744}, {5972,10163}, {6032,15059}, {6699,12506}, {6723,10162}, {11594,32269}, {17702,31729}, {23515,31749}

X(32311) = midpoint of X(74) and X(12505)
X(32311) = reflection of X(i) in X(j) for these (i,j): (113, 31606), (12506, 6699)


X(32312) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO WALSMITH

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

The reciprocal parallelogic center of these triangles is X(125).

X(32312) lies on these lines: {351,690}, {512,13290}, {525,9138}, {3566,15139}, {9131,9517}, {9208,13291}

X(32312) = reflection of X(i) in X(j) for these (i,j): (13291, 9208), (32313, 351)
X(32312) = X(67)-of-1st-Parry-triangle
X(32312) = X(32233)-of-2nd-Parry-triangle


X(32313) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO WALSMITH

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

The reciprocal parallelogic center of these triangles is X(125).

X(32313) lies on these lines: {67,18310}, {110,525}, {113,1560}, {351,690}, {512,13291}, {542,1640}, {1499,1551}, {1648,14114}, {2780,14698}, {3288,19140}, {3566,5653}, {5027,13290}, {5095,9033}, {6593,18311}, {8429,18332}, {9209,9759}, {9517,9979}, {11061,14977}, {22105,22255}

X(32313) = midpoint of X(11061) and X(14977)
X(32313) = reflection of X(i) in X(j) for these (i,j): (67, 18310), (3569, 14697), (13290, 5027), (18311, 6593), (32312, 351)
X(32313) = X(67)-of-2nd-Parry-triangle
X(32313) = X(32233)-of-1st-Parry-triangle


X(32314) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: CIRCUMMEDIAL AND WALSMITH

Barycentrics    2*(9*R^2-2*SW)^2*S^4+(216*R^4*SW+3*R^2*(9*SA^2-12*SA*SW-35*SW^2)-2*SW*(3*SA^2-3*SA*SW-7*SW^2))*SW*S^2+6*(3*R^2-SW)*SB*SC*SW^3 : :

X(32314) lies on these lines: {1316,14682}, {5642,9463}, {9140,15080}, {9745,15484}


X(32315) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 1st EHRMANN AND WALSMITH

Barycentrics    (SB+SC)*((54*R^2*(3*R^2-SW)+3*SW^2)*S^2+(SW^2*(4*SW+3*SA)+9*R^2*(9*(3*SA-SW)*R^2-SW*(9*SA-SW)))*SA) : :

X(32315) lies on these lines: {1511,5651}, {1995,5476}, {6800,32305}, {9125,14270}, {16510,32225}


X(32316) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WALSMITH TO HATZIPOLAKIS-MOSES

Barycentrics    (5*R^4*(24*R^2-5*SA)-R^2*SW*(97*R^2-14*SA-25*SW)-2*SW^2*(SW+SA))*S^2+(2*R^2-SW)*R^2*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(32317).

X(32316) lies on these lines: {68,1658}, {74,6145}, {110,343}, {113,1209}, {161,3448}, {973,11746}, {1154,32123}, {6293,12281}, {7703,23315}, {7730,26917}, {10117,31383}, {15068,17824}, {18400,32110}


X(32317) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO WALSMITH

Barycentrics    (6*(15*SA-SW)*R^4-9*(7*SA-2*SW)*SW*R^2+2*(5*SA-2*SW)*SW^2)*S^2-(7*R^2-2*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(32316).

X(32317) lies on these lines: {6,67}, {542,6102}, {1511,5965}, {2781,13403}, {3564,11561}, {9925,9932}, {11061,19136}, {16511,32244}, {32154,32257}

X(32317) = reflection of X(67) in X(32283)
X(32317) = {X(125), X(5095)}-harmonic conjugate of X(15140)


X(32318) = X(1)X(389)∩X(3)X(2262)

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

See Tran Quang Hung and CÚsar Lozada, Hyacinthos 28999.

X(32318) lies on these lines: {1, 389}, {3, 2262}


X(32319) = X(4)X(19209)∩X(20)X(2979)

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

See Angel Montesdeoca, Hyacinthos 29000 and HG020519

X(32319) lies on these lines: {4,19209}, {20,2979}, {51,3087}, {97,6759}, {154,160}, {217,3172}, {512,23613}, {3198,11190}

X(32319) = isogonal conjugate of X(20477)
X(32319) = isogonal of the anticomplement of X(53)
X(32319) = isogonal of the isotomic of X(15318)
X(32319) = polar conjugate of the isotomic of X(18890)
X(32319) = X(i)-Ceva conjugate of X(j) for these (i,j): {14371, 6}, {15318, 18890}
X(32319) = X(3199)-cross conjugate of X(6)
X(32319) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20477}, {75, 6759}, {799, 30442}
X(32319) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 18890}, {6, 15318}, {53, 14371}, {512, 30441}
X(32319) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {6, 20477}, {32, 6759}, {669, 30442}, {3199, 14363}, {15318, 76}, {18890, 69}, {30441, 670}


X(32320) = REFLECTION OF X(17434) IN X(647)

Barycentrics    a^4 (a^2 - b^2 - c^2)^3 (b^2 - c^2) : :
Barycentrics    (sin^2 2A)(sin 2B - sin 2C) : :
X(32320) = 2X[647]-3X[1636] = 3X[1636]-X[17434] = 4X[2501]-3X[14391] = 4X[3265]-3X[23616] = 8X[6587]-9X[14401]

See Angel Montesdeoca, Hyacinthos 29000 and HG020519

X(32320) lies on these lines: {6,2430}, {112,6080}, {323,401}, {394,3265}, {416,23090}, {418,23613}, {450,2451}, {520,647}, {651,2639}, {1640,23128}, {3569,6753}, {9033,12077}

X(32320) = reflection of X(17434) and X(647)
X(32320) = isogonal conjugate of X(15352)
X(32320) = isogonal of the polar conjugate of X(520)
X(32320) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 418}, {394, 2972}, {648, 3}, {651, 408}, {1625, 22052}, {2430, 1636}, {2986, 1650}, {4558, 417}, {4563, 426}, {15316, 20975}
X(32320) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15352}, {4, 823}, {19, 6528}, {75, 6529}, {92, 107}, {99, 6520}, {110, 6521}, {158, 648}, {162, 2052}, {264, 24019}, {393, 811}, {525, 24021}, {653, 1896}, {662, 1093}, {799, 6524}, {1096, 6331}, {1577, 32230}, {1784, 15459}, {2501, 23999}, {2617, 8794}, {3267, 24022}, {6330, 24024}, {6335, 8747}, {8748, 18026}, {14208, 23590}, {14618, 24000}, {17926, 24032}, {23582, 24006}
X(32320) = crosspoint of X(i) and X(j) for these (i,j): {3, 648}, {97, 110}
X(32320) = crosssum of X(i) and X(j) for these (i,j): {4, 647}, {53, 523}, {107, 6529}, {133, 1637}, {235, 2501}, {1896, 17926}, {2052, 14618}, {6750, 12077}
X(32320) = crosssum of Jerabek-hyperbola-intercepts of orthic axis
X(32320) = crossdifference of every pair of points on line {4, 51}
X(32320) = pole of Euler line wrt MacBeath circumconic
X(32320) = perspector of hyperbola {{A,B,C,X(3),X(95)}}
X(32320) = barycentric product X(i) X(j) for these lines: {i,j}: {3, 520}, {32, 4143}, {48, 24018}, {63, 822}, {71, 4091}, {72, 23224}, {97, 17434}, {110, 2972}, {184, 3265}, {217, 15414}, {228, 4131}, {255, 656}, {326, 810}, {394, 647}, {512, 3964}, {521, 22341}, {523, 1092}, {525, 577}, {661, 6507}, {669, 4176}, {684, 17974}, {798, 1102}, {850, 23606}, {905, 3990}, {924, 16391}, {1364, 23067}, {1459, 3682}, {1577, 4100}, {1636, 14919}, {1783, 16730}, {2200, 30805}, {2584, 2585}, {2632, 4575}, {3049, 3926}, {3267, 14585}, {3269, 4558}, {3733, 4158}, {3998, 22383}, {4025, 4055}, {4557, 7215}, {5562, 23286}, {6056, 17094}, {6368, 19210}, {7066, 23189}, {7125, 8611}, {8057, 14379}, {14642, 20580}, {23103, 32230}, {23107, 23963}, {23357, 23616}
X(32320) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {3, 6528}, {6, 15352}, {32, 6529}, {48, 823}, {184, 107}, {255, 811}, {394, 6331}, {512, 1093}, {520, 264}, {525, 18027}, {577, 648}, {647, 2052}, {661, 6521}, {669, 6524}, {798, 6520}, {810, 158}, {822, 92}, {1092, 99}, {1102, 4602}, {1363, 17094}, {1576, 32230}, {1946, 1896}, {2623, 8794}, {2972, 850}, {3049, 393}, {3265, 18022}, {3269, 14618}, {3964, 670}, {3990, 6335}, {4055, 1897}, {4100, 662}, {4143, 1502}, {4158, 27808}, {4176, 4609}, {4575, 23999}, {6507, 799}, {9247, 24019}, {14533, 16813}, {14585, 112}, {14600, 20031}, {15451, 13450}, {16730, 15413}, {17434, 324}, {17974, 22456}, {18877, 15459}, {19210, 18831}, {22341, 18026}, {23224, 286}, {23286, 8795}, {23606, 110}, {23613, 2972}, {23616, 23962}, {24018, 1969}, {30451, 11547}
X(32320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 6587, 30442}, {647, 30451, 3049}, {1636, 17434, 647}


X(32321) = MIDPOINT OF X(9914) AND X(12085)

Barycentrics    a^2*(a^14 - 3*a^12*b^2 + a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - a^4*b^10 + 3*a^2*b^12 - b^14 - 3*a^12*c^2 + 10*a^10*b^2*c^2 - 13*a^8*b^4*c^2 + 8*a^6*b^6*c^2 - a^4*b^8*c^2 - 2*a^2*b^10*c^2 + b^12*c^2 + a^10*c^4 - 13*a^8*b^2*c^4 + 2*a^6*b^4*c^4 + 2*a^4*b^6*c^4 + 5*a^2*b^8*c^4 + 3*b^10*c^4 + 5*a^8*c^6 + 8*a^6*b^2*c^6 + 2*a^4*b^4*c^6 - 12*a^2*b^6*c^6 - 3*b^8*c^6 - 5*a^6*c^8 - a^4*b^2*c^8 + 5*a^2*b^4*c^8 - 3*b^6*c^8 - a^4*c^10 - 2*a^2*b^2*c^10 + 3*b^4*c^10 + 3*a^2*c^12 + b^2*c^12 - c^14) : :
X(32321) = (1 + J^2) X[3] - 2 X[64], (7 - J^2) X[3] - 6 X[154], (7 - J^2) X[64] - 3 (1 + J^2) X[154]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29002

X(32321) lies on these lines: {3, 64}, {4, 18532}, {22, 9833}, {24, 14216}, {25, 13419}, {26, 1503}, {30, 9938}, {159, 2918}, {161, 2937}, {186, 12324}, {265, 7517}, {378, 5878}, {511, 9908}, {542, 9937}, {1216, 1660}, {1593, 18388}, {1598, 18383}, {1853, 7506}, {2071, 12250}, {2777, 9914}, {2781, 16266}, {2883, 7526}, {3515, 13171}, {3517, 14864}, {3518, 32064}, {3520, 6225}, {3818, 6642}, {5073, 9919}, {5198, 18376}, {5656, 14118}, {5893, 31861}, {6053, 13293}, {6247, 6644}, {6293, 18445}, {6640, 32125}, {7387, 12293}, {7512, 11206}, {7514, 16252}, {7516, 10192}, {7525, 15577}, {7529, 23325}, {7555, 15581}, {9934, 12292}, {10249, 15805}, {10628, 12164}, {11413, 20427}, {12083, 17845}, {12084, 15311}, {22658, 22978}

X(32321) = midpoint of X(9914) and X(12085)
X(32321) = crosspoint of X(1288) and X(15384)
X(32321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1619, 6759}, {64, 1498, 18439}, {3357, 6759, 5907}






leftri  Centers related to the Hatzipolakis-Moses triangle: X(32322) - X(32413)  rightri

This preamble and centers X(32322)-X(32413) were contributed by CÚsar Eliud Lozada, May 4, 2019.

Hatzipolakis-Moses triangle is defined in X(6145).

The appearance of (T,i) in the following list means that triangles Hatizpolakis-Moses and T are perspective with perspector X(i) (an asterisk * means they are homothetic):

(ABC, 54), (2nd anti-extouch, 32322), (anti-inner-Grebe, 32323), (anti-outer-Grebe, 32324), (anti-orthocentroidal, 32325), (1st excosine, 32326), (inner-Grebe, 32327), (outer-Grebe, 32328), (3rd Hatzipolakis, 54), (2nd Hyacinth, 6146), (Kosnita, 54), (midheight, 389), (orthic, 1986), (orthocentroidal, 32329), (reflection, 6241)

The appearance of (T,i,j) in the following list means that triangles Hatizpolakis-Moses and T are orthologic with orthologic centers X(i) and X(j):

(ABC, 6146, 6145), (ABC-X3 reflections, 6146, 32330), (anti-Aquila, 6146, 32331), (anti-Ara, 6146, 32332), (anti-Ascella, 389, 32333), (anti-Atik, 389, 32334), (5th anti-Brocard, 6146, 32335), (2nd anti-circumperp-tangential, 6146, 32336), (1st anti-circumperp, 389, 7691), (anti-Conway, 389, 54), (2nd anti-Conway, 389, 973), (anti-Euler, 6146, 32337), (3rd anti-Euler, 389, 32338), (4th anti-Euler, 389, 32339), (anti-excenters-reflections, 389, 32340), (2nd anti-extouch, 389, 32341), (anti-inner-Grebe, 6146, 32342), (anti-outer-Grebe, 6146, 32343), (anti-Honsberger, 389, 32344), (anti-Hutson intouch, 389, 32345), (anti-incircle-circles, 389, 9920), (anti-inverse-in-incircle, 389, 32346), (anti-Mandart-incircle, 6146, 32347), (6th anti-mixtilinear, 389, 32348), (anti-orthocentroidal, 1986, 32349), (1st anti-Sharygin, 389, 54), (anti-tangential-midarc, 389, 32350), (anti-Ursa minor, 389, 32351), (anti-Wasat, 389, 32352), (antiAOA, 10114, 32353), (anticomplementary, 6146, 32354), (AOA, 10114, 32355), (Aquila, 6146, 32356), (Ara, 6146, 32357), (Aries, 32358, 32359), (1st Auriga, 6146, 32360), (2nd Auriga, 6146, 32361), (5th Brocard, 6146, 32362), (circumorthic, 389, 54), (2nd circumperp tangential, 6146, 32363), (Ehrmann-mid, 6146, 32364), (Ehrmann-side, 389, 6288), (Ehrmann-vertex, 389, 32365), (1st Ehrmann, 32366, 32367), (2nd Ehrmann, 389, 32368), (Euler, 6146, 32369), (2nd Euler, 389, 1209), (1st excosine, 389, 17824), (extangents, 389, 32370), (outer-Garcia, 6146, 32371), (Gossard, 6146, 32372), (inner-Grebe, 6146, 32373), (outer-Grebe, 6146, 32374), (3rd Hatzipolakis, 32375, 32376), (1st Hyacinth, 10114, 10115), (2nd Hyacinth, 32358, 32377), (intangents, 389, 32378), (Johnson, 6146, 32379), (inner-Johnson, 6146, 32380), (outer-Johnson, 6146, 32381), (1st Johnson-Yff, 6146, 32382), (2nd Johnson-Yff, 6146, 32383), (1st Kenmotu diagonals, 389, 32384), (2nd Kenmotu diagonals, 389, 32385), (Kosnita, 389, 23358), (Lucas antipodal tangents, 389, 32386), (Lucas homothetic, 6146, 32388), (Lucas(-1) antipodal tangents, 389, 32387), (Lucas(-1) homothetic, 6146, 32389), (Mandart-incircle, 6146, 32390), (medial, 6146, 32391), (midheight, 32392, 32393), (5th mixtilinear, 6146, 32394), (orthic, 389, 3574), (orthocentroidal, 1986, 32395), (reflection, 54, 17824), (submedial, 389, 32396), (tangential, 389, 2917), (inner tri-equilateral, 389, 32397), (outer tri-equilateral, 389, 32398), (3rd tri-squares-central, 6146, 32399), (4th tri-squares-central, 6146, 32400), (Trinh, 389, 32401), (Walsmith, 32317, 32316), (X3-ABC reflections, 6146, 32402), (inner-Yff, 6146, 32403), (outer-Yff, 6146, 32404), (inner-Yff tangents, 6146, 32405), (outer-Yff tangents, 6146, 32406)

The appearance of (T,i,j) in the following list means that triangles Hatizpolakis-Moses and T are parallelogic with parallelogic centers X(i) and X(j): (1st Parry, 6146, 32407), (2nd Parry, 6146, 32408).

The appearance of (T,i,j) in the following list means that triangles Hatizpolakis-Moses and T are cyclologic with cyclologic centers X(i) and X(j): (2nd anti-Conway, 389, 32409), (orthic, 32410, 1986).

The appearance of (T,i,j) in the following list means that triangles Hatizpolakis-Moses and T are eulerologic with eulorologic centers X(i) and X(j): (2nd anti-Conway, 32411, x), (Lucas(-1) reflection, 0, x), where x means "not-existing-eulerologic center" and 0 means "not calculated center".

underbar

X(32322) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND 2nd ANTI-EXTOUCH

Barycentrics    (S^2-SB*SC)*((4*R^2-SW)*S^2-72*R^6-6*(SA-10*SW)*R^4+(4*SA+9*SW)*(SA-2*SW)*R^2-(SA^2-2*SW^2)*SW) : :

X(32322) lies on these lines: {26,1181}, {182,32251}, {184,2931}, {389,10274}, {567,19361}, {568,19362}, {1533,18534}, {1593,13403}, {1899,10264}, {6146,32345}, {8479,21331}, {9932,19357}, {10620,10938}, {12022,22533}, {12168,21243}, {14805,19348}, {19468,32358}


X(32323) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND ANTI-INNER-GREBE

Barycentrics    (SB+SC)*(2*(18*R^2-5*SW)*S^2+(3*S^2+4*R^2*(40*R^2-SA-26*SW)+SA^2-2*SB*SC+16*SW^2)*S-2*(4*R^2-SW)*(SA-2*SW)*SA) : :

X(32323) lies on these lines: {389,19095}, {1588,13403}, {3312,6102}, {6146,32342}, {6241,19087}


X(32324) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND ANTI-OUTER-GREBE

Barycentrics    (SB+SC)*(2*(18*R^2-5*SW)*S^2-(3*S^2+4*R^2*(40*R^2-SA-26*SW)+SA^2-2*SB*SC+16*SW^2)*S-2*(4*R^2-SW)*(SA-2*SW)*SA) : :

X(32324) lies on these lines: {389,19096}, {1587,13403}, {3311,6102}, {6146,32343}, {6241,19088}


X(32325) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND ANTI-ORTHOCENTROIDAL

Barycentrics    (SB+SC)*((431*R^4-2*(21*SA+92*SW)*R^2+4*(3*SA+5*SW)*SW)*S^2+3*(7*R^2-2*SW)*(84*R^4+(5*SA-34*SW)*R^2-2*SA^2+4*SW^2+2*SB*SC)*SA) : :

X(32325) lies on these lines: {74,13403}, {389,2914}, {399,6102}, {1986,32349}, {6241,17812}, {10114,12380}


X(32326) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND 1st EXCOSINE

Barycentrics    (SB+SC)*((72*R^4-3*(2*SA+11*SW)*R^2+2*(SA+2*SW)*SW)*S^2+2*(3*R^2-SW)*(48*R^4-16*R^2*SW-SA^2+SB*SC+2*SW^2)*SA) : :

X(32326) lies on these lines: {3,32276}, {64,13403}, {389,15139}, {1498,6102}, {6759,17837}, {9932,17821}, {14677,17835}, {17846,32358}


X(32327) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND INNER-GREBE

Barycentrics    (SB+SC)*(3*S^4+(4*R^2*(40*R^2-SA-4*SW)+SA^2-2*SB*SC)*S^2-(2*(18*R^2-SA-3*SW)*S^2-2*(4*R^2*SA+SA^2-SB*SC-2*SW^2)*SA)*S-16*R^2*SB*SC*SW) : :

X(32327) lies on these lines: {389,6277}, {1161,6102}, {5871,13403}, {6146,32373}, {6241,6267}


X(32328) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND OUTER-GREBE

Barycentrics    (SB+SC)*(3*S^4+(4*R^2*(40*R^2-SA-4*SW)+SA^2-2*SB*SC)*S^2+(2*(18*R^2-SA-3*SW)*S^2-2*(4*R^2*SA+SA^2-SB*SC-2*SW^2)*SA)*S-16*R^2*SB*SC*SW) : :

X(32328) lies on these lines: {389,6276}, {1160,6102}, {5870,13403}, {6146,32374}, {6241,6266}


X(32329) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND ORTHOCENTROIDAL

Barycentrics    (SB+SC)*((43*R^2-10*SW)*S^2+(7*R^2-2*SW)*(28*R^2-SA-6*SW)*SA) : :

X(32329) lies on these lines: {74,19361}, {195,1511}, {381,3567}, {389,6143}, {1986,32395}, {2777,22949}, {5890,13403}, {7730,10114}, {11561,12280}, {15033,32210}, {25739,32392}


X(32330) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO HATZIPOLAKIS-MOSES

Barycentrics    (R^2*(32*R^2+3*SA-23*SW)-(SA-4*SW)*SW)*S^2-(8*R^2*(7*R^2-5*SW)+7*SW^2)*SB*SC : :
X(32330) = 3*X(3)-X(32402) = 2*X(20)+X(32359) = 3*X(165)-X(32356) = 3*X(376)-X(32337) = 3*X(3576)-2*X(32331) = 4*X(6689)-3*X(32395) = 5*X(17821)-2*X(32340) = 3*X(18405)-4*X(32393)

The reciprocal orthologic center of these triangles is X(6146).

X(32330) lies on these lines: {2,32369}, {3,161}, {4,32391}, {20,32354}, {30,32372}, {35,32403}, {36,32404}, {54,6240}, {55,32336}, {56,32390}, {110,12225}, {154,5448}, {165,32356}, {182,32335}, {371,32343}, {372,32342}, {376,32337}, {382,32364}, {515,32371}, {517,32394}, {973,18533}, {1154,12118}, {1593,32332}, {3098,32362}, {3428,32363}, {3574,11425}, {3576,32331}, {4549,5876}, {5925,11820}, {6000,18442}, {6242,6776}, {6284,32383}, {6689,32395}, {7354,32382}, {7487,11743}, {7507,17821}, {7691,15138}, {9540,32399}, {9630,32350}, {10274,13352}, {10282,31724}, {10310,32347}, {10619,32341}, {10625,10628}, {11248,32405}, {11249,32406}, {11414,32357}, {11824,32373}, {11825,32374}, {11826,32380}, {11827,32381}, {11828,32388}, {11829,32389}, {13160,32351}, {13935,32400}, {17818,17846}, {18405,32393}, {18569,32171}

X(32330) = midpoint of X(i) and X(j) for these lines: {i,j}: {20, 32354}, {17845, 32345}
X(32330) = reflection of X(i) in X(j) for these (i,j): (4, 32391), (382, 32364), (6145, 3), (6288, 23358), (32359, 32354)
X(32330) = anticomplement of X(32369)
X(32330) = X(6145)-of-ABC-X3-reflections-triangle


X(32331) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO HATZIPOLAKIS-MOSES

Barycentrics
(b+c)*a^15-2*b*c*a^14-(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^12+(b+c)*(b^4+c^4-(4*b^2-7*b*c+4*c^2)*b*c)*a^11-2*(2*b^6+2*c^6-(b^4+c^4+(b^2-3*b*c+c^2)*b*c)*b*c)*a^10+(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+3*(b-c)^2*b*c)*b*c)*a^9+(5*b^6+5*c^6+(2*b^4+2*c^4+(b+c)^2*b*c)*b*c)*(b-c)^2*a^8-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*b*c)*a^7+2*(b^2-c^2)^2*(b^4+c^4-(b^2-3*b*c+c^2)*b*c)*b*c*a^6-(b^2-c^2)^2*(b+c)*(b^6+c^6+(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*b*c)*a^5-(5*b^8+5*c^8-2*(2*b^2-b*c+2*c^2)*(b^4+c^4-(b^2+b*c+c^2)*b*c)*b*c)*(b^2-c^2)^2*a^4+(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*(3*b^2+5*b*c+3*c^2)*a^3+2*(b^4-c^4)*(b^2-c^2)^3*(2*b^4+2*c^4-(b-c)^2*b*c)*a^2-(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6 : :
X(32331) = 3*X(3576)-X(32330) = 5*X(3616)-X(32354) = 3*X(5603)+X(32337) = 3*X(5886)-X(32379) = 7*X(9624)-X(32359) = 3*X(10246)+X(32402)

The reciprocal orthologic center of these triangles is X(6146).

X(32331) lies on these lines: {1,6145}, {2,32371}, {515,32369}, {960,1209}, {999,32363}, {1125,32391}, {1154,12259}, {1319,32336}, {1385,18400}, {2646,32390}, {3295,32347}, {3576,32330}, {3616,32354}, {5603,32337}, {5886,32379}, {9624,32359}, {9955,32364}, {10246,32402}, {10628,12261}, {11363,32332}, {11364,32335}, {11365,32357}, {11368,32362}, {11370,32373}, {11371,32374}, {11373,32380}, {11374,32381}, {11375,32382}, {11376,32383}, {11377,32388}, {11378,32389}, {11831,32372}, {13883,32399}, {13936,32400}, {18991,32342}, {18992,32343}

X(32331) = midpoint of X(i) and X(j) for these lines: {i,j}: {1, 6145}, {32356, 32394}
X(32331) = reflection of X(i) in X(j) for these (i,j): (32364, 9955), (32391, 1125)
X(32331) = complement of X(32371)
X(32331) = X(6145)-of-anti-Aquila-triangle
X(32331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32356, 32394), (6145, 32394, 32356)


X(32332) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO HATZIPOLAKIS-MOSES

Barycentrics    S^4-(R^2*(12*R^2+3*SA-10*SW)-SA^2+SB*SC+2*SW^2)*S^2+(R^2*(16*R^2-19*SW)+5*SW^2)*SB*SC : :
X(32332) = 3*X(428)-2*X(11743) = 4*X(6756)-X(32377)

The reciprocal orthologic center of these triangles is X(6146).

X(32332) lies on these lines: {4,54}, {25,6145}, {26,1209}, {33,32390}, {34,32336}, {66,1204}, {159,32333}, {235,32369}, {427,32391}, {428,11743}, {539,7540}, {973,6756}, {1154,11819}, {1352,7691}, {1593,32330}, {1598,32402}, {1843,32352}, {5090,32371}, {5410,32343}, {5411,32342}, {5576,6689}, {5965,12272}, {6288,7517}, {7487,32337}, {7488,32348}, {7564,11750}, {7713,32356}, {10628,12140}, {11363,32331}, {11380,32335}, {11383,32347}, {11386,32362}, {11388,32373}, {11389,32374}, {11390,32380}, {11391,32381}, {11392,32382}, {11393,32383}, {11394,32388}, {11395,32389}, {11396,32394}, {11398,32403}, {11399,32404}, {11400,32405}, {11401,32406}, {11572,32393}, {11832,32372}, {12133,16621}, {13884,32399}, {13937,32400}, {15761,22804}, {18563,32062}, {22479,32363}, {32341,32359}

X(32332) = reflection of X(i) in X(j) for these (i,j): (973, 6756), (32377, 973)
X(32332) = X(6145)-of-anti-Ara-triangle
X(32332) = {X(4), X(32379)}-harmonic conjugate of X(3574)


X(32333) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(R^2*S^2+(2*SW^2+R^2*(16*R^2-SA-11*SW))*SA) : :
X(32333) = 4*X(54)-3*X(11402) = 4*X(578)-X(12175) = X(1593)+2*X(19468) = 3*X(11402)-2*X(32341)

The reciprocal orthologic center of these triangles is X(389).

X(32333) lies on these lines: {3,54}, {4,9920}, {6,32352}, {25,2917}, {26,20424}, {159,32332}, {184,6293}, {378,12254}, {427,32346}, {539,12166}, {578,12175}, {973,9777}, {1181,10628}, {1209,7395}, {1398,32350}, {1593,18400}, {1658,22051}, {2888,7503}, {2937,31815}, {3515,12242}, {3516,10619}, {3520,31804}, {5094,32351}, {5410,32384}, {5411,32385}, {6242,11426}, {6288,9818}, {6644,8254}, {7071,32378}, {7484,32348}, {7488,31802}, {7509,26869}, {7514,21230}, {7526,12168}, {9704,12412}, {10274,19357}, {10831,13079}, {10832,18984}, {10982,11808}, {11245,32334}, {11284,32396}, {11403,32340}, {11405,32368}, {11406,32370}, {11408,32397}, {11409,32398}, {11410,32401}, {11424,17846}, {11425,21660}, {12233,21284}, {13423,15033}, {14528,32226}, {15739,26864}, {18386,32365}, {19118,32344}, {19404,32386}, {19405,32387}

X(32333) = reflection of X(i) in X(j) for these (i,j): (12160, 195), (32341, 54)
X(32333) = X(21)-of-anti-Ascella-triangle if ABC is acute
X(32333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 32339, 7592), (54, 32341, 11402)


X(32334) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO HATZIPOLAKIS-MOSES

Barycentrics    2*S^4+2*(R^2*(8*R^2-3*SA-3*SW)+SA^2-SB*SC)*S^2-(8*R^2-3*SW)*R^2*SB*SC : :

The reciprocal orthologic center of these triangles is X(389).

X(32334) lies on these lines: {2,32341}, {4,973}, {54,69}, {68,11802}, {195,3548}, {539,18934}, {1154,6643}, {1199,6143}, {1209,11411}, {1658,27552}, {1899,32346}, {2888,6815}, {2917,6776}, {3541,32251}, {3574,11433}, {6288,18917}, {6353,32379}, {6515,7691}, {7730,11457}, {9920,18914}, {10628,18933}, {11245,32333}, {11808,14216}, {12254,18910}, {12324,32340}, {13418,14528}, {13567,17824}, {15801,16051}, {18400,18909}, {18911,32338}, {18912,32339}, {18913,32345}, {18915,32350}, {18918,32365}, {18919,32368}, {18921,32370}, {18922,32378}, {18923,32384}, {18924,32385}, {18925,23358}, {18926,32386}, {18927,32387}, {18928,32396}, {18929,32397}, {18930,32398}, {18931,32401}, {19119,32344}, {19468,21844}, {22051,31283}, {23291,32351}, {32354,32377}

X(32334) = X(21)-of-anti-Atik-triangle if ABC is acute
X(32334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (973, 32337, 4), (1899, 32352, 32346)


X(32335) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO HATZIPOLAKIS-MOSES

Barycentrics
((72*R^2+3*SA-46*SW)*R^2-(SA-7*SW)*SW)*S^4+(8*(5*SA^2-12*SB*SC-6*SW^2)*R^4-(23*SA^2-63*SB*SC-29*SW^2)*SW*R^2+(3*SA^2-10*SB*SC-4*SW^2)*SW^2)*S^2-(2*R^2-SW)*(8*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32335) lies on these lines: {32,6145}, {83,32391}, {98,32369}, {182,32330}, {1154,12193}, {3398,18400}, {7787,32354}, {10628,12201}, {10788,32337}, {10789,32356}, {10790,32357}, {10791,32371}, {10792,32373}, {10793,32374}, {10794,32380}, {10795,32381}, {10796,32379}, {10797,32382}, {10798,32383}, {10799,32390}, {10800,32394}, {10801,32403}, {10802,32404}, {10803,32405}, {10804,32406}, {11364,32331}, {11380,32332}, {11490,32347}, {11839,32372}, {11840,32388}, {11841,32389}, {11842,32402}, {12835,32336}, {13885,32399}, {13938,32400}, {18502,32364}, {18993,32342}, {18994,32343}, {22520,32363}

X(32335) = X(6145)-of-5th-anti-Brocard-triangle


X(32336) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO HATZIPOLAKIS-MOSES

Barycentrics    (3*R^2-SW)*(8*R^2-SA-3*SW)*S^2+(SB+SC)*((5*R^2-2*SW)*(4*R^2-SW)*b*c-(12*R^2-5*SW)*(3*R^2-SW)*SA) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32336) lies on these lines: {1,18400}, {3,32403}, {4,32383}, {11,32369}, {12,32391}, {34,32332}, {55,32330}, {56,6145}, {57,32356}, {388,32354}, {999,32402}, {1154,18970}, {1319,32331}, {1478,32379}, {2099,32394}, {2917,9659}, {3574,19365}, {3585,32364}, {4293,32337}, {5252,32371}, {9657,32359}, {9672,32345}, {10628,18968}, {11509,32347}, {12835,32335}, {18954,32357}, {18957,32362}, {18958,32372}, {18959,32373}, {18960,32374}, {18961,32380}, {18962,32381}, {18963,32388}, {18964,32389}, {18965,32399}, {18966,32400}, {18967,32406}, {18995,32342}, {18996,32343}

X(32336) = reflection of X(32390) in X(1)
X(32336) = X(6145)-of-2nd-anti-circumperp-tangential-triangle
X(32336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 32354, 32382), (999, 32402, 32404)


X(32337) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO HATZIPOLAKIS-MOSES

Barycentrics    S^4+(R^2*(8*R^2-3*SA-3*SW)+SA^2-SB*SC)*S^2-(R^2*(SW+4*R^2)-SW^2)*SB*SC : :
X(32337) = 3*X(4)-4*X(32369) = 3*X(376)-2*X(32330) = 5*X(631)-2*X(32359) = 5*X(631)-4*X(32391) = 3*X(1853)-X(17824) = 3*X(1853)-2*X(32351) = X(2888)+2*X(14216) = 5*X(3091)-4*X(32364) = 3*X(5603)-4*X(32331) = 3*X(5657)-2*X(32371) = X(6225)-4*X(22804) = 4*X(6247)-X(12254) = 2*X(6288)+X(12324) = 3*X(7967)-2*X(32394) = 3*X(11845)-2*X(32372) = 3*X(32064)-X(32346)

The reciprocal orthologic center of these triangles is X(6146).

X(32337) lies on these lines: {2,32379}, {3,32354}, {4,973}, {20,2888}, {24,32357}, {30,32402}, {54,66}, {64,12244}, {70,18533}, {104,32363}, {376,32330}, {388,32403}, {427,32341}, {497,32404}, {515,32356}, {631,32359}, {1154,11411}, {1209,3547}, {1352,32348}, {1503,2916}, {1853,7592}, {1899,3574}, {2892,8549}, {3085,32382}, {3086,32383}, {3088,32377}, {3091,32364}, {3410,9833}, {3448,5889}, {3520,6247}, {4293,32336}, {4294,32390}, {5012,10274}, {5189,14864}, {5596,32344}, {5603,32331}, {5657,32371}, {6143,15139}, {6146,30100}, {6225,22804}, {6288,12324}, {6759,14076}, {7487,32332}, {7581,32343}, {7582,32342}, {7967,32394}, {9862,32362}, {9920,21230}, {10783,32373}, {10784,32374}, {10785,32380}, {10786,32381}, {10788,32335}, {10805,32405}, {10806,32406}, {11431,18912}, {11491,32347}, {11550,32352}, {11845,32372}, {11846,32388}, {11847,32389}, {12300,18910}, {13423,32247}, {13886,32399}, {13939,32400}, {15033,32412}, {15801,31099}, {22533,25739}

X(32337) = reflection of X(i) in X(j) for these (i,j): (4, 6145), (5596, 32344), (6759, 14076), (9833, 23358), (9920, 21230), (10274, 20299), (12254, 32345), (17824, 32351), (32345, 6247), (32354, 3), (32359, 32391)
X(32337) = anticomplement of X(32379)
X(32337) = X(6145)-of-anti-Euler-triangle
X(32337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 32334, 973), (7691, 11442, 2888)


X(32338) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((7*R^2-2*SB-2*SC)*S^2+((32*R^2-7*SA-22*SW)*R^2+2*SA^2-2*SB*SC+4*SW^2)*SA) : :
X(32338) = 4*X(5)-3*X(7730) = 3*X(381)-2*X(32196) = 3*X(568)-4*X(8254) = 5*X(631)-4*X(11802) = 8*X(973)-9*X(5640) = 4*X(1209)-5*X(11444) = 3*X(2979)-2*X(7691) = 3*X(3060)-4*X(3574) = 5*X(3091)-4*X(11808) = 9*X(5055)-8*X(13365) = 4*X(5562)-X(12280) = X(5889)-4*X(12606) = 3*X(5890)-4*X(10610) = 3*X(5891)-2*X(6153) = 4*X(6152)-7*X(15056) = 2*X(6242)-5*X(11444) = 2*X(6288)-3*X(11459) = 5*X(10574)-8*X(12363) = X(11412)+2*X(22815) = 3*X(11459)-X(13423)

The reciprocal orthologic center of these triangles is X(389).

X(32338) lies on these lines: {2,32352}, {3,54}, {5,7730}, {20,10628}, {22,17824}, {30,15103}, {68,12319}, {110,2917}, {381,32196}, {539,12271}, {568,8254}, {631,11802}, {973,5640}, {1209,6242}, {1216,6143}, {1658,11597}, {2888,3153}, {2937,7731}, {3060,3574}, {3091,11808}, {3519,16000}, {5055,13365}, {5891,6153}, {5925,12279}, {5965,15073}, {6152,7547}, {6243,20424}, {6288,11459}, {6689,15043}, {7488,10274}, {7998,32348}, {9920,11441}, {10224,21230}, {10625,13619}, {11439,32340}, {11440,32345}, {11442,32346}, {11443,32368}, {11445,32370}, {11446,32378}, {11447,32384}, {11448,32385}, {11449,23358}, {11451,32396}, {11452,32397}, {11453,32398}, {11454,32401}, {11464,15091}, {11576,18386}, {11591,13368}, {11750,15100}, {12111,12226}, {12254,13754}, {12272,15069}, {12273,12281}, {12380,15068}, {13340,15332}, {15058,22804}, {18392,32365}, {18911,32334}, {19122,32344}, {19367,32350}, {19412,32386}, {19413,32387}, {19467,21660}, {23293,32351}, {26881,32379}

X(32338) = reflection of X(i) in X(j) for these (i,j): (54, 12606), (2888, 5562), (5889, 54), (6242, 1209), (6243, 20424), (12280, 2888), (12307, 6101), (13368, 11591), (13423, 6288), (32339, 3), (32341, 31807)
X(32338) = anticomplement of X(32352)
X(32338) = X(21)-of-3rd-anti-Euler-triangle if ABC is acute
X(32338) = {X(11459), X(13423)}-harmonic conjugate of X(6288)


X(32339) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((7*R^2-2*SB-2*SC)*S^2+((52*R^2-7*SA-30*SW)*R^2+2*SA^2-2*SB*SC+4*SW^2)*SA) : :
X(32339) = 3*X(2)-4*X(11802) = 2*X(4)-3*X(7730) = 3*X(4)-4*X(11808) = 2*X(54)-3*X(5890) = 4*X(185)-X(12291) = 3*X(568)-2*X(20424) = 8*X(973)-7*X(9781) = 4*X(1209)-3*X(11459) = 5*X(3567)-4*X(3574) = 7*X(3851)-8*X(13365) = 4*X(6152)-X(12290) = X(6241)+2*X(6242) = 8*X(6689)-9*X(15045) = 9*X(7730)-8*X(11808) = 3*X(7730)-4*X(32352) = 7*X(7999)-8*X(32348) = 7*X(9781)-4*X(12300) = 5*X(10574)-4*X(10610) = 5*X(10574)-2*X(22815) = 2*X(11808)-3*X(32352)

The reciprocal orthologic center of these triangles is X(389).

X(32339) lies on these lines: {2,11802}, {3,54}, {4,7730}, {24,17824}, {52,3153}, {74,6799}, {143,14644}, {185,12254}, {186,10274}, {382,32196}, {389,6143}, {539,12282}, {568,10224}, {973,7547}, {1204,16013}, {1209,5448}, {1614,2917}, {1986,2929}, {2888,13754}, {2914,11438}, {3060,18377}, {3432,14979}, {3567,3574}, {3851,13365}, {5498,22051}, {5663,13368}, {5895,6152}, {6145,16000}, {6153,12162}, {6241,6242}, {6288,12111}, {6689,15045}, {7999,32348}, {9905,31728}, {9920,11456}, {11271,22647}, {11455,32340}, {11457,32346}, {11458,32368}, {11460,32370}, {11461,32378}, {11462,32384}, {11463,32385}, {11464,23358}, {11465,32396}, {11466,32397}, {11467,32398}, {11468,32401}, {11557,21451}, {12270,12278}, {13565,15056}, {15305,22804}, {17855,22949}, {18394,32365}, {18436,21230}, {18912,32334}, {19123,32344}, {19368,32350}, {19414,32386}, {19415,32387}, {21357,31834}, {23294,32351}, {26882,32379}

X(32339) = midpoint of X(6241) and X(13423)
X(32339) = reflection of X(i) in X(j) for these (i,j): (4, 32352), (195, 6102), (382, 32196), (9905, 31728), (11412, 7691), (12111, 6288), (12162, 6153), (12254, 185), (12291, 12254), (12300, 973), (13423, 6242), (18436, 21230), (22815, 10610), (32338, 3)
X(32339) = anticomplement of the anticomplement of X(11802)
X(32339) = X(21)-of-4th-anti-Euler-triangle if ABC is acute
X(32339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 32352, 7730), (7592, 32333, 54)


X(32340) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO HATZIPOLAKIS-MOSES

Barycentrics    4*a^10-6*(b^2+c^2)*a^8-3*(b^2-c^2)^2*a^6+5*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :
X(32340) = 3*X(4)-X(54) = 4*X(4)-X(10619) = 5*X(4)-2*X(12242) = 5*X(4)-X(12254) = 3*X(51)-4*X(11743) = 2*X(54)-3*X(3574) = 4*X(54)-3*X(10619) = 5*X(54)-6*X(12242) = 5*X(54)-3*X(12254) = X(195)-5*X(5076) = 3*X(381)-2*X(6689) = 3*X(382)+X(12307) = X(1493)-4*X(12102) = X(2888)+3*X(3543) = 5*X(3574)-4*X(12242) = 5*X(3574)-2*X(12254) = 9*X(3830)-X(12316) = 3*X(6288)-X(12307) = 5*X(10619)-8*X(12242) = 5*X(10619)-4*X(12254)

The reciprocal orthologic center of these triangles is X(389).

X(32340) lies on these lines: {4,54}, {5,15432}, {20,15431}, {24,32401}, {25,32345}, {30,1209}, {33,32350}, {34,32378}, {51,11743}, {64,6145}, {125,3575}, {185,973}, {195,5076}, {235,32351}, {378,23358}, {381,6689}, {382,6288}, {468,20376}, {539,3830}, {546,10610}, {548,20584}, {550,13565}, {1112,11262}, {1154,3627}, {1493,12102}, {1495,23047}, {1498,32341}, {1539,3853}, {1568,18350}, {1593,2917}, {1597,9920}, {2888,3543}, {3091,32396}, {3146,7691}, {3527,18396}, {3861,8254}, {5921,5965}, {6000,11808}, {6152,10628}, {6564,8995}, {6565,13986}, {6756,13851}, {7507,17821}, {7576,18383}, {7730,12290}, {10594,18376}, {11381,11576}, {11403,32333}, {11439,32338}, {11455,32339}, {11470,32368}, {11471,32370}, {11473,32384}, {11474,32385}, {11475,32397}, {11476,32398}, {11802,14915}, {12060,16337}, {12063,13423}, {12101,22051}, {12266,18483}, {12315,32377}, {12324,32334}, {12606,18323}, {12943,12956}, {12946,12953}, {13473,16656}, {15004,18945}, {15687,20424}, {15739,21660}, {15811,17824}, {18559,20299}, {19095,23259}, {19096,23249}, {19124,32344}, {19416,32386}, {19417,32387}

X(32340) = midpoint of X(i) and X(j) for these lines: {i,j}: {382, 6288}, {3146, 7691}, {11381, 32352}
X(32340) = reflection of X(i) in X(j) for these (i,j): (20, 32348), (185, 973), (548, 20584), (550, 13565), (1209, 22804), (3574, 4), (8254, 3861), (10610, 546), (10619, 3574), (12254, 12242), (12266, 18483), (21660, 15739), (32345, 32393), (32352, 11576)
X(32340) = X(21)-of-anti-excenters-reflections-triangle if ABC is acute
X(32340) = {X(3575), X(11572)}-harmonic conjugate of X(125)


X(32341) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((3*R^2-SB-SC)*S^2+((8*R^2-3*SA-3*SW)*R^2+SA^2-SB*SC)*SA) : :
X(32341) = 2*X(54)-3*X(11402) = X(1181)+2*X(12234) = 3*X(11402)-X(32333)

The reciprocal orthologic center of these triangles is X(389).

X(32341) lies on these lines: {2,32334}, {3,54}, {4,19362}, {6,3574}, {25,973}, {155,1209}, {184,2917}, {185,32345}, {389,10274}, {394,32348}, {427,32337}, {539,19458}, {578,10628}, {1147,11802}, {1181,12173}, {1199,1594}, {1498,32340}, {1614,7730}, {1899,32351}, {1994,12225}, {2888,3564}, {2904,11426}, {2914,15106}, {3515,32391}, {3575,32354}, {5094,12242}, {5198,11743}, {6152,9920}, {6240,12254}, {6276,19352}, {6277,19351}, {6288,18445}, {6759,11808}, {6776,32346}, {7558,12325}, {8254,18916}, {9905,31811}, {10224,27552}, {10601,32396}, {10602,32368}, {10605,32401}, {10619,32330}, {10982,18376}, {11576,12167}, {11702,15132}, {11803,14791}, {12022,14627}, {12175,19459}, {13371,22051}, {13565,15068}, {15037,26879}, {15047,26917}, {18386,32369}, {18396,32365}, {18569,20424}, {18997,19095}, {18998,19096}, {19125,32344}, {19349,32350}, {19350,32370}, {19354,32378}, {19355,32384}, {19356,32385}, {19357,23358}, {19358,32386}, {19359,32387}, {19363,32397}, {19364,32398}, {19468,21284}, {20376,26937}, {22804,32139}, {32332,32359}

X(32341) = reflection of X(i) in X(j) for these (i,j): (195, 12161), (9905, 31811), (12254, 31804), (32333, 54), (32338, 31807)
X(32341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7592, 12161, 11402), (11402, 32333, 54)
X(32341) = X(21)-of-2nd-anti-extouch-triangle if ABC is acute


X(32342) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO HATZIPOLAKIS-MOSES

Barycentrics    ((8*R^2-3*SA-3*SW)*R^2+SA*SW)*S^2-2*(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146).

X(32342) lies on these lines: {2,32400}, {6,3574}, {372,32330}, {973,19039}, {1154,19061}, {1587,32369}, {3069,32391}, {3299,32403}, {3301,32404}, {3312,18400}, {5411,32332}, {6146,32323}, {6293,19045}, {6418,32402}, {7582,32337}, {7584,32379}, {7585,32399}, {7586,32354}, {7968,32394}, {10628,19051}, {13785,32364}, {13936,32371}, {18991,32331}, {18993,32335}, {18995,32336}, {18997,32345}, {18999,32347}, {19003,32356}, {19005,32357}, {19011,32362}, {19013,32363}, {19017,32372}, {19023,32380}, {19025,32381}, {19027,32382}, {19029,32383}, {19031,32388}, {19033,32389}, {19037,32390}, {19047,32405}, {19049,32406}, {19096,32351}

X(32342) = X(6145)-of-anti-inner-Grebe-triangle
X(32342) = {X(6), X(6145)}-harmonic conjugate of X(32343)


X(32343) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO HATZIPOLAKIS-MOSES

Barycentrics    ((8*R^2-3*SA-3*SW)*R^2+SA*SW)*S^2+2*(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146).

X(32343) lies on these lines: {2,32399}, {6,3574}, {371,32330}, {973,19040}, {1154,19062}, {1588,32369}, {3068,32391}, {3299,32404}, {3301,32403}, {3311,18400}, {5410,32332}, {6146,32324}, {6293,19046}, {6417,32402}, {7581,32337}, {7583,32379}, {7585,32354}, {7586,32400}, {7969,32394}, {10628,19052}, {13665,32364}, {13883,32371}, {18992,32331}, {18994,32335}, {18996,32336}, {18998,32345}, {19000,32347}, {19004,32356}, {19006,32357}, {19012,32362}, {19014,32363}, {19018,32372}, {19024,32380}, {19026,32381}, {19028,32382}, {19030,32383}, {19032,32388}, {19034,32389}, {19038,32390}, {19048,32405}, {19050,32406}, {19095,32351}

X(32343) = X(6145)-of-anti-outer-Grebe-triangle
X(32343) = {X(6), X(6145)}-harmonic conjugate of X(32342)


X(32344) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((10*R^2*(2*R^2-SW)+SW^2)*S^2+(2*R^2-SW)*(4*R^2+SA-2*SW)*SA*SW) : :
X(32344) = 2*X(2917)+X(32368) = 5*X(3618)-X(32346) = 3*X(5050)+X(9920) = 3*X(5085)-X(32345) = X(9972)+2*X(32367) = X(9977)+2*X(10282) = X(10274)-3*X(23042) = X(17824)-5*X(19132) = 3*X(23041)-2*X(32391)

The reciprocal orthologic center of these triangles is X(389).

X(32344) lies on these lines: {6,24}, {110,343}, {159,9827}, {161,5012}, {182,18400}, {206,1209}, {511,23358}, {539,19141}, {569,9815}, {1154,19139}, {1176,1503}, {1177,10628}, {1428,32350}, {1974,3574}, {2330,32378}, {3589,32351}, {3618,32346}, {5050,9920}, {5085,32345}, {5092,32401}, {5596,32337}, {5965,10274}, {6288,19129}, {7691,19121}, {9967,15462}, {9972,32367}, {9977,10282}, {17824,19132}, {19118,32333}, {19119,32334}, {19122,32338}, {19123,32339}, {19124,32340}, {19125,32341}, {19126,32348}, {19127,19149}, {19130,32365}, {19133,32370}, {19134,32386}, {19135,32387}, {19137,32396}, {21637,32352}

X(32344) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 2917}, {5596, 32337}
X(32344) = reflection of X(i) in X(j) for these (i,j): (32351, 3589), (32365, 19130), (32368, 6), (32379, 206), (32401, 5092)
X(32344) = X(21)-of-anti-Honsberger-triangle if ABC is acute


X(32345) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((3*R^2-SA-SW)*S^2-(2*(16*R^2-SA-9*SW)*R^2+SA^2-SB*SC+2*SW^2)*SA) : :
X(32345) = 3*X(2)-4*X(20376) = 3*X(3)-X(9920) = 3*X(3)-2*X(23358) = 2*X(4)-3*X(32395) = 2*X(54)+X(64) = 3*X(154)-4*X(32391) = X(195)+2*X(3357) = X(1498)-4*X(10610) = X(2888)-4*X(6696) = 3*X(2917)-2*X(9920) = 3*X(2917)-4*X(23358) = X(2917)-4*X(32401) = 3*X(3060)-4*X(11262) = 4*X(3574)-X(5895) = 3*X(5085)-2*X(32344) = X(5878)-4*X(8254) = 2*X(6247)+X(12254) = 2*X(7691)-5*X(8567) = X(9914)-3*X(32357) = 4*X(32351)-3*X(32395)

The reciprocal orthologic center of these triangles is X(389).

X(32345) lies on these lines: {2,20376}, {3,161}, {4,10117}, {20,32346}, {24,11704}, {25,32340}, {26,18379}, {40,12341}, {54,64}, {55,32350}, {56,32378}, {74,6799}, {125,2929}, {154,7503}, {185,32341}, {195,2935}, {221,32382}, {382,32365}, {389,19361}, {539,12301}, {578,6293}, {973,9786}, {1147,19376}, {1151,32384}, {1152,32385}, {1154,10606}, {1192,11576}, {1204,1205}, {1350,7691}, {1498,7526}, {1503,32354}, {1593,3574}, {2070,18383}, {2071,2888}, {2192,32383}, {2883,7527}, {3060,11262}, {3516,10619}, {3518,23324}, {3520,6247}, {3522,15578}, {5085,32344}, {5448,12412}, {5584,32370}, {5646,7509}, {5878,8254}, {5894,12086}, {5898,25564}, {6000,10274}, {6102,17835}, {6146,32322}, {6644,22804}, {6689,9818}, {7514,17821}, {7730,17823}, {7973,12266}, {9659,32390}, {9672,32336}, {9905,12262}, {9934,32137}, {10282,14926}, {10625,12307}, {11250,12118}, {11438,11808}, {11440,32338}, {11456,32349}, {11468,13423}, {11477,32368}, {11479,32396}, {11480,32397}, {11481,32398}, {11597,18439}, {11743,17810}, {12041,13368}, {13021,32386}, {13022,32387}, {13366,32392}, {13367,15139}, {14216,18570}, {14864,18364}, {14865,15311}, {15100,17847}, {15579,16010}, {15622,32347}, {17822,17846}, {18376,18378}, {18382,31304}, {18431,20424}, {18913,32334}, {18997,32342}, {18998,32343}, {19351,32373}, {19352,32374}, {19460,31978}, {22467,23332}, {23292,30100}, {26937,32316}

X(32345) = midpoint of X(i) and X(j) for these lines: {i,j}: {20, 32346}, {64, 17824}, {12254, 32337}
X(32345) = reflection of X(i) in X(j) for these (i,j): (3, 32401), (4, 32351), (382, 32365), (1498, 32379), (2917, 3), (9920, 23358), (11477, 32368), (17824, 54), (17845, 32330), (32337, 6247), (32340, 32393), (32379, 10610), (32402, 18381)
X(32345) = anticomplement of the anticomplement of X(20376)
X(32345) = X(21)-of-anti-Hutson-intouch-triangle if ABC is acute
X(32345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9920, 23358), (9920, 23358, 2917)


X(32346) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO HATZIPOLAKIS-MOSES

Barycentrics    (4*R^2+SA-2*SW)*R^2*S^2+2*(2*R^2-SW)^2*SB*SC : :
X(32346) = 3*X(2)-4*X(32351) = 3*X(4)-4*X(32365) = 3*X(376)-4*X(32401) = 5*X(631)-4*X(23358) = 3*X(1992)-4*X(32368) = 5*X(3091)-6*X(32395) = 7*X(3523)-8*X(20376) = 5*X(3618)-4*X(32344) = 3*X(11206)-4*X(32379) = X(12325)-4*X(18381) = 3*X(32064)-2*X(32337)

The reciprocal orthologic center of these triangles is X(389).

X(32346) lies on these lines: {2,2917}, {4,54}, {5,9920}, {20,32345}, {69,1225}, {159,7566}, {195,31723}, {376,32401}, {388,32350}, {427,32333}, {497,32378}, {539,12318}, {631,23358}, {973,11433}, {1154,11411}, {1209,6643}, {1370,7691}, {1503,17824}, {1899,32334}, {1992,32368}, {2550,32370}, {3068,32384}, {3069,32385}, {3091,15582}, {3153,14516}, {3523,20376}, {3618,32344}, {5189,6247}, {6193,12319}, {6288,11487}, {6689,7401}, {6776,32341}, {6816,15435}, {7386,32348}, {7391,12324}, {7392,32396}, {7544,32391}, {7730,18912}, {8254,11818}, {10610,18420}, {10628,12284}, {11382,11576}, {11412,12325}, {11442,32338}, {11457,32339}, {11488,32397}, {11489,32398}, {13423,25739}, {14791,21230}, {18489,22804}, {19420,32386}, {19421,32387}

X(32346) = reflection of X(i) in X(j) for these (i,j): (20, 32345), (2888, 6145), (2917, 32351), (9833, 10274), (9920, 5), (32354, 54)
X(32346) = anticomplement of X(2917)
X(32346) = X(21)-of-anti-inverse-in-incircle-triangle if ABC is acute
X(32346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 12254, 18925), (2917, 32351, 2)


X(32347) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO HATZIPOLAKIS-MOSES

Barycentrics    a*(((5*R^2-2*SW)*(4*R^2-SW)*(b+c)*a-((8*R^2-3*SA-3*SW)*R^2+SA*SW)*b*c-(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC))*S^2-(2*R^2-SW)*(8*R^2-3*SW)*SB*SC*b*c) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32347) lies on these lines: {3,32363}, {35,32356}, {55,6145}, {56,32394}, {100,32354}, {197,32357}, {1154,12328}, {1376,32380}, {3295,32331}, {5687,32371}, {10310,32330}, {10628,12334}, {11248,18400}, {11383,32332}, {11490,32335}, {11491,32337}, {11494,32362}, {11496,32369}, {11497,32373}, {11498,32374}, {11499,32379}, {11500,32381}, {11501,32382}, {11502,32383}, {11503,32388}, {11504,32389}, {11507,32403}, {11508,32404}, {11509,32336}, {11510,32406}, {11848,32372}, {11849,32402}, {13887,32399}, {13940,32400}, {15622,32345}, {18491,32364}, {18999,32342}, {19000,32343}

X(32347) = reflection of X(i) in X(j) for these (i,j): (32363, 3), (32380, 32391)
X(32347) = X(6145)-of-anti-Mandart-incircle-triangle


X(32348) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO HATZIPOLAKIS-MOSES

Barycentrics    2*a^10-4*(b^2+c^2)*a^8-(b^4+c^4)*a^6+(b^2+c^2)*(7*b^4-2*b^2*c^2+7*c^4)*a^4-(b^2-c^2)^2*(5*b^4+8*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(32348) = 3*X(2)+X(7691) = 3*X(3)+X(6288) = X(54)-5*X(631) = 5*X(54)-X(11271) = 3*X(54)+X(12325) = 3*X(140)-X(8254) = 7*X(140)-X(11803) = 4*X(140)-X(12242) = 5*X(140)-X(22051) = 15*X(631)+X(12325) = 3*X(1209)-X(6288) = 3*X(5892)-X(10115) = 3*X(6689)-2*X(8254) = 7*X(6689)-2*X(11803) = 5*X(6689)-2*X(22051) = X(7691)+2*X(32396) = 7*X(8254)-3*X(11803) = 4*X(8254)-3*X(12242) = 5*X(8254)-3*X(22051) = 3*X(11271)+5*X(12325) = 4*X(11803)-7*X(12242) = 5*X(11803)-7*X(22051) = 5*X(12242)-4*X(22051) = 3*X(13565)-2*X(20584) = X(18381)+3*X(23358)

The reciprocal orthologic center of these triangles is X(389).

X(32348) lies on these lines: {2,3574}, {3,161}, {5,32223}, {20,15431}, {24,24206}, {30,13565}, {54,69}, {140,389}, {141,32391}, {185,7495}, {195,5054}, {343,10112}, {394,32341}, {511,973}, {539,549}, {550,22804}, {569,11225}, {632,20424}, {1038,32350}, {1040,32378}, {1352,32337}, {1368,32351}, {1493,14869}, {1899,2888}, {3098,3541}, {3153,18428}, {3519,14528}, {3524,12254}, {3526,5646}, {3530,6699}, {3547,5878}, {3549,32364}, {3567,25555}, {3576,12785}, {3628,13566}, {3818,9715}, {3917,32352}, {5204,12946}, {5217,12956}, {5418,8995}, {5420,13986}, {5432,18984}, {5433,13079}, {5657,7979}, {5894,6823}, {5907,6676}, {5972,7542}, {6153,23336}, {6745,9940}, {7386,32346}, {7484,32333}, {7488,32332}, {7494,12324}, {7499,9729}, {7502,13419}, {7512,29012}, {7516,9932}, {7568,13754}, {7998,32338}, {7999,32339}, {9306,11487}, {9540,19096}, {9827,11808}, {10018,12300}, {10020,10170}, {10165,12266}, {10257,12363}, {10303,15801}, {10319,32370}, {10516,16195}, {11017,25338}, {11438,14542}, {11511,32368}, {11513,32384}, {11514,32385}, {11515,32397}, {11516,32398}, {11548,11745}, {11574,16196}, {11576,13348}, {11743,13598}, {12108,20585}, {12362,32393}, {13347,26937}, {13365,13391}, {13367,32377}, {13474,16618}, {13564,18488}, {13935,19095}, {14389,14531}, {14862,18435}, {15559,29317}, {15712,21357}, {16976,22581}, {17811,17824}, {17821,32359}, {18531,32365}, {19126,32344}, {19422,32386}, {19423,32387}, {21167,23300}

X(32348) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 1209}, {20, 32340}, {550, 22804}, {1216, 11802}, {2888, 10619}, {3574, 7691}, {10610, 21230}, {11808, 15644}, {13564, 18488}
X(32348) = reflection of X(i) in X(j) for these (i,j): (3574, 32396), (6689, 140), (12242, 6689), (13598, 11743), (15605, 21230)
X(32348) = anticomplement of X(32396)
X(32348) = complement of X(3574)
X(32348) = X(21)-of-6th-anti-Mixtilinear-triangle if ABC is acute
X(32348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3574, 32396), (2, 7691, 3574)


X(32349) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((R^2*(11*R^2-18*SA-10*SW)+2*(3*SA+SW)*SW)*S^2-6*(3*R^2-SW)*((8*R^2-3*SA-3*SW)*R^2+SA^2-SB*SC)*SA) : :

The reciprocal orthologic center of these triangles is X(1986).

X(32349) lies on these lines: {6,3574}, {64,32401}, {195,18405}, {399,15091}, {1986,32325}, {2917,15068}, {3581,10628}, {6000,11597}, {10274,14805}, {11441,12226}, {11456,32345}, {11750,32359}, {15032,32351}


X(32350) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC TO HATZIPOLAKIS-MOSES

Barycentrics    a*((12*R^2-5*SW)*SB*SC*b*c-((4*R^2+SA-2*SW)*b*c-4*(5*R^2-2*SW)*(4*R^2-SW))*S^2) : :
X(32350) = X(7355)+2*X(13079)

The reciprocal orthologic center of these triangles is X(389).

X(32350) lies on these lines: {1,18400}, {12,32351}, {33,32340}, {34,3574}, {35,32401}, {36,23358}, {54,65}, {55,32345}, {56,2917}, {73,6145}, {221,17824}, {388,32346}, {539,19471}, {973,19366}, {999,9920}, {1038,32348}, {1060,1209}, {1154,7352}, {1398,32333}, {1425,32352}, {1428,32344}, {1469,18984}, {2067,32384}, {3468,4551}, {3585,32365}, {4296,7691}, {5432,20376}, {6284,10118}, {6286,10628}, {6288,18447}, {6502,32385}, {7051,32397}, {7355,13079}, {9630,32330}, {10082,26888}, {10535,32383}, {10895,32395}, {18915,32334}, {18970,19469}, {19349,32341}, {19367,32338}, {19368,32339}, {19369,32368}, {19370,32386}, {19371,32387}, {19372,32396}, {19373,32398}

X(32350) = reflection of X(32378) in X(1)
X(32350) = X(21)-of-anti-tangential-midarc-triangle if ABC is acute


X(32351) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA MINOR TO HATZIPOLAKIS-MOSES

Barycentrics    (R^2*(24*R^2+SA-15*SW)+2*SW^2)*S^2+2*(2*R^2-SW)^2*SB*SC : :
X(32351) = 3*X(2)+X(32346) = X(4)-3*X(32395) = 5*X(1656)-X(9920) = 3*X(1853)+X(17824) = 3*X(1853)-X(32337) = 2*X(3574)+X(6247) = 2*X(8254)+X(18381) = 2*X(11804)+X(23315) = 2*X(20299)+X(20424) = 3*X(23324)-2*X(32369) = 3*X(23324)-4*X(32393) = X(32345)+3*X(32395)

The reciprocal orthologic center of these triangles is X(389).

X(32351) lies on these lines: {2,2917}, {3,20376}, {4,10117}, {5,5944}, {11,32378}, {12,32350}, {30,32365}, {52,11262}, {54,1594}, {125,6746}, {140,23358}, {141,1209}, {161,7569}, {185,427}, {195,15141}, {235,32340}, {394,2888}, {524,32368}, {539,23307}, {590,32384}, {615,32385}, {858,7691}, {973,13567}, {1147,10224}, {1154,12359}, {1216,14076}, {1368,32348}, {1498,5169}, {1503,5576}, {1656,9920}, {1853,7592}, {1899,32341}, {2072,6288}, {2883,32137}, {3549,18382}, {3589,32344}, {3925,32370}, {5094,32333}, {5133,16252}, {5965,23326}, {6102,10115}, {7577,12254}, {7730,26917}, {8254,10274}, {8550,12242}, {10066,32404}, {10082,32403}, {11743,15873}, {13160,32330}, {13406,18376}, {15032,32349}, {15311,15559}, {19095,32343}, {19096,32342}, {19176,32181}, {21243,31807}, {23291,32334}, {23293,32338}, {23294,32339}, {23298,32386}, {23299,32387}, {23302,32397}, {23303,32398}, {23328,23335}

X(32351) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 32345}, {54, 6145}, {2917, 32346}, {10274, 18381}, {17824, 32337}, {32365, 32401}
X(32351) = reflection of X(i) in X(j) for these (i,j): (3, 20376), (52, 11262), (2883, 32364), (10274, 8254), (21230, 14076), (23358, 140), (32344, 3589), (32369, 32393), (32391, 6689)
X(32351) = complement of X(2917)
X(32351) = X(21)-of-anti-Ursa-minor-triangle if ABC is acute
X(32351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 32346, 2917), (32345, 32395, 4)


X(32352) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(S^2+SB*SC)*((3*R^2-SW)*S^2-(4*R^2-SW)*SA^2) : :
X(32352) = X(4)-3*X(7730) = 3*X(5)-4*X(13365) = 3*X(51)-4*X(973) = 3*X(51)-2*X(3574) = 9*X(51)-4*X(15739) = X(185)+2*X(6152) = X(195)-3*X(568) = 9*X(373)-8*X(32396) = 2*X(389)+X(6242) = 4*X(389)-X(21660) = 3*X(568)-2*X(10115) = 3*X(973)-X(15739) = 3*X(3574)-2*X(15739) = 3*X(5891)-4*X(13565) = 2*X(6242)+X(21660) = 3*X(7730)-2*X(11808) = 3*X(7730)+X(32339) = 2*X(11597)-3*X(16223) = 2*X(11808)+X(32339) = X(15801)-4*X(16625)

The reciprocal orthologic center of these triangles is X(389).

Let HAHBHC be as described in Anapolis #2471 (March 19, 2015, Antreas Hatzipolakis). X(32352) is the intersection of the Euler lines of the orthic triangle and HAHBHC. (Randy Hutson, June 7, 2019)

X(32352) lies on these lines: {2,32338}, {3,11802}, {4,7730}, {5,51}, {6,32333}, {24,10274}, {25,17824}, {30,32196}, {54,186}, {125,6746}, {184,2917}, {185,6152}, {195,568}, {373,32396}, {511,7691}, {539,14831}, {1181,9920}, {1204,1205}, {1425,32350}, {1495,32379}, {1594,14076}, {1843,32332}, {1899,32334}, {2888,5889}, {3270,32378}, {3292,15801}, {3567,14940}, {3611,32370}, {3917,32348}, {5446,18403}, {5462,22815}, {5890,12254}, {5946,8254}, {6101,15426}, {6102,11562}, {6145,11572}, {6153,6288}, {6243,11424}, {6403,32354}, {6467,8550}, {6689,12606}, {9729,12226}, {9730,10610}, {9786,12175}, {10018,12242}, {10296,13598}, {11262,13567}, {11381,11576}, {11550,32337}, {11561,14049}, {12162,22804}, {12280,32248}, {12380,15032}, {12897,18563}, {13382,13433}, {13451,25402}, {13568,21652}, {13851,32365}, {14449,16657}, {15060,20584}, {16227,22249}, {16881,22051}, {21637,32344}, {21639,32368}, {21640,32384}, {21641,32385}, {21642,32386}, {21643,32387}, {21647,32397}, {21648,32398}, {21663,32401}

X(32352) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 32339}, {54, 6242}, {2888, 5889}, {6102, 13368}, {6243, 12307}, {12254, 13423}
X(32352) = reflection of X(i) in X(j) for these (i,j): (3, 11802), (4, 11808), (54, 389), (195, 10115), (3574, 973), (5562, 1209), (6288, 6153), (11381, 32340), (12162, 22804), (12606, 6689), (20424, 143), (21660, 54), (22051, 16881), (32340, 11576)
X(32352) = complement of X(32338)
X(32352) = X(21)-of-anti-Wasat-triangle if ABC is acute
X(32352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 7730, 11808), (7730, 32339, 4)


X(32353) = ORTHOLOGIC CENTER OF THESE TRIANGLES: antiAOA TO HATZIPOLAKIS-MOSES

Barycentrics    (96*R^6-21*R^4*(SA+3*SW)+R^2*SW*(10*SW+13*SA)-2*SA*SW^2)*S^2+(R^2*(72*R^4-104*R^2*SW+45*SW^2)-6*SW^3)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10114)

X(32353) lies on these lines: {3,161}, {265,19402}, {1154,15133}, {5094,32355}, {6143,32391}, {7574,10628}, {10115,15134}, {10224,32379}, {11202,18432}, {15131,19381}

X(32353) = X(11604)-of-AAOA-triangle if ABC is acute


X(32354) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO HATZIPOLAKIS-MOSES

Barycentrics    S^4-(R^2*(12*R^2+3*SA-10*SW)-SA^2+SB*SC+2*SW^2)*S^2+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC : :
X(32354) = 3*X(2)-4*X(32391) = 3*X(4)-4*X(32364) = X(20)+2*X(32359) = 3*X(1853)-4*X(20376) = 5*X(3091)-4*X(32369) = 5*X(3616)-4*X(32331) = 2*X(9833)+X(12254) = 3*X(11202)-2*X(14076) = 2*X(32364)-3*X(32379)

The reciprocal orthologic center of these triangles is X(6146).

X(32354) lies on these lines: {2,6145}, {3,32337}, {4,54}, {5,32402}, {8,32371}, {10,32356}, {20,32330}, {22,32357}, {26,9920}, {69,7691}, {70,21844}, {100,32347}, {145,32394}, {146,1498}, {193,31304}, {195,31802}, {343,1601}, {388,32336}, {497,32383}, {973,7487}, {1154,6193}, {1270,32374}, {1271,32373}, {1503,32345}, {1853,20376}, {1993,17824}, {2892,11413}, {2896,32362}, {2975,32363}, {3085,32403}, {3086,32404}, {3091,32369}, {3434,32380}, {3436,32381}, {3549,6288}, {3575,32341}, {3616,32331}, {4240,32372}, {6403,32352}, {6462,32388}, {6463,32389}, {6995,11743}, {7564,8254}, {7585,32343}, {7586,32342}, {7787,32335}, {8972,32399}, {10528,32405}, {10529,32406}, {10628,11412}, {11202,14076}, {11271,14531}, {12363,18439}, {13941,32400}, {14216,32401}, {14389,32395}, {32334,32377}

X(32354) = midpoint of X(i) and X(j) for these lines: {i,j}: {17824, 17845}, {32330, 32359}
X(32354) = reflection of X(i) in X(j) for these (i,j): (4, 32379), (8, 32371), (20, 32330), (145, 32394), (2888, 2917), (4240, 32372), (6145, 32391), (14216, 32401), (32337, 3), (32346, 54), (32356, 10), (32402, 5)
X(32354) = anticomplementary conjugate of the anticomplement of X(7488)
X(32354) = anticomplement of X(6145)
X(32354) = X(6145)-of-anticomplementary-triangle
X(32354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 3574, 11427), (6145, 32391, 2)


X(32355) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AOA TO HATZIPOLAKIS-MOSES

Barycentrics    (528*R^6+6*(7*SA-79*SW)*R^4-(19*SA-134*SW)*SW*R^2+2*(SA-6*SW)*SW^2)*S^2-(144*R^6-31*R^2*SW*(4*R^2-SW)-2*SW^3)*SB*SC : :

The reciprocal orthologic center of these triangles is X(10114).

X(32355) lies on these lines: {5,5944}, {1154,15115}, {5094,32353}, {5965,23296}, {10115,15120}, {10274,18281}, {10628,15122}, {11597,15116}

X(32355) = X(11604)-of-AOA-triangle if ABC is acute


X(32356) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO HATZIPOLAKIS-MOSES

Barycentrics
3*a^16-(b+c)*a^15-(9*b^2-2*b*c+9*c^2)*a^14+(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(5*b^4+5*c^4-(4*b-c)*(b-4*c)*b*c)*a^12-(b+c)*(b^4+c^4-(4*b^2-7*b*c+4*c^2)*b*c)*a^11+(7*b^6+7*c^6-(2*b^4+2*c^4+(5*b^2-6*b*c+5*c^2)*b*c)*b*c)*a^10-(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+3*(b-c)^2*b*c)*b*c)*a^9-(5*b^6+5*c^6+(2*b^4+2*c^4+7*(b+c)^2*b*c)*b*c)*(b-c)^2*a^8+(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*b*c)*a^7-(3*b^6+3*c^6+(2*b^2-b*c+2*c^2)*(b-c)^2*b*c)*(b^2-c^2)^2*a^6+(b^2-c^2)^2*(b+c)*(b^6+c^6+(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*b*c)*a^5-(b^8+c^8+(4*b^6+4*c^6+(3*b^4+3*c^4+2*(b^2+6*b*c+c^2)*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^4-(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*(3*b^2+5*b*c+3*c^2)*a^3+(b^4-c^4)*(b^2-c^2)^3*(5*b^4+5*c^4+2*(b^2+b*c+c^2)*b*c)*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-2*(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32356) lies on these lines: {1,6145}, {10,32354}, {35,32347}, {36,32363}, {40,12785}, {57,32336}, {165,32330}, {515,32337}, {517,32402}, {1154,9896}, {1697,32390}, {1698,32391}, {1699,32369}, {3099,32362}, {3679,32371}, {5587,32379}, {5588,32374}, {5589,32373}, {7713,32332}, {8185,32357}, {8188,32388}, {8189,32389}, {9578,32382}, {9581,32383}, {10628,12407}, {10789,32335}, {10826,32380}, {10827,32381}, {11852,32372}, {13888,32399}, {13942,32400}, {16473,17824}, {18492,32364}, {19003,32342}, {19004,32343}

X(32356) = reflection of X(i) in X(j) for these (i,j): (1, 6145), (32354, 10), (32394, 32331)
X(32356) = X(6145)-of-Aquilla-triangle
X(32356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6145, 32394, 32331), (32331, 32394, 1)


X(32357) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((2*R^2-SW)*R^2*S^2+(8*(4*R^2+SA-6*SW)*R^4-SW*((3*SA-19*SW)*R^2+2*SW^2))*SA) : :
X(32357) = X(9914)+2*X(32345)

The reciprocal orthologic center of these triangles is X(6146).

X(32357) lies on these lines: {3,32379}, {22,32354}, {24,32337}, {25,6145}, {64,13171}, {155,10628}, {159,2917}, {184,6293}, {197,32347}, {382,9919}, {1154,9908}, {1593,3574}, {1598,32369}, {2937,9833}, {3556,5693}, {5562,22658}, {5594,32374}, {5595,32373}, {7387,12293}, {7517,32402}, {8185,32356}, {8192,32394}, {8193,32371}, {8194,32388}, {8195,32389}, {9818,32364}, {10037,32403}, {10046,32404}, {10790,32335}, {10828,32362}, {10829,32380}, {10830,32381}, {10831,32382}, {10832,32383}, {10833,32390}, {10834,32405}, {10835,32406}, {11365,32331}, {11414,32330}, {11477,32262}, {11853,32372}, {13889,32399}, {13943,32400}, {14130,15311}, {18954,32336}, {19005,32342}, {19006,32343}, {22654,32363}

X(32357) = X(6145)-of-Ara-triangle


X(32358) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO ARIES

Barycentrics    (R^2-2*SA+SW)*S^2+(R^2-SW)*SB*SC : :
X(32358) = 3*X(5)-2*X(31831) = 2*X(140)-3*X(11245) = 4*X(143)-3*X(13490) = 3*X(568)-X(14516) = 3*X(568)-2*X(31830) = X(1216)-3*X(11232) = 2*X(5462)-3*X(11225) = 3*X(5889)+X(12289) = 3*X(6102)-2*X(13568) = 5*X(6146)-4*X(11565) = 3*X(6146)-2*X(13470) = 3*X(10112)-X(13403) = 3*X(11245)-4*X(32165) = 5*X(11264)-2*X(11565) = 3*X(11264)-X(13470) = 6*X(11565)-5*X(13470) = 3*X(12022)-X(18436) = 2*X(12134)-3*X(13490) = 3*X(12370)-2*X(13403) = 3*X(13292)-X(31831)

The reciprocal orthologic center of these triangles is X(32359).

X(32358) lies on these lines: {5,6}, {26,6515}, {30,5889}, {49,3580}, {52,11819}, {54,140}, {69,7516}, {70,1993}, {143,12134}, {193,14790}, {195,1594}, {323,11271}, {343,7568}, {389,539}, {394,18952}, {427,18356}, {511,10116}, {524,6101}, {542,5446}, {550,10605}, {568,14516}, {1154,6146}, {1199,2888}, {1209,13366}, {1216,5965}, {1493,23292}, {1503,10263}, {1899,16266}, {1986,3575}, {1994,5576}, {3627,13142}, {3628,11422}, {3853,7728}, {5133,14627}, {5462,11225}, {5876,12241}, {6102,13568}, {6193,6644}, {6759,16619}, {7393,11898}, {7502,31804}, {7526,11411}, {7553,14449}, {7574,12316}, {8537,16198}, {8681,12585}, {9703,10018}, {10112,12370}, {10255,25740}, {10602,14791}, {11449,16531}, {11750,14531}, {12022,18436}, {12084,18917}, {12160,18569}, {12359,32166}, {13160,15087}, {13413,22051}, {15073,26926}, {15083,18390}, {15644,18128}, {15761,18445}, {15801,25739}, {17846,32326}, {18400,32392}, {19468,32322}, {22660,23323}

X(32358) = midpoint of X(11750) and X(14531)
X(32358) = reflection of X(i) in X(j) for these (i,j): (5, 13292), (68, 22663), (140, 32165), (550, 18914), (3627, 13142), (5876, 12241), (6146, 11264), (7553, 14449), (11819, 52), (12134, 143), (12359, 32166), (12370, 10112), (14516, 31830), (15644, 18128)
X(32358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49, 3580, 10020), (68, 12161, 5)


X(32359) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARIES TO HATZIPOLAKIS-MOSES

Barycentrics    3*S^4-(R^2*(16*R^2+9*SA-17*SW)-3*SA^2+3*SB*SC+4*SW^2)*S^2+(8*R^2*(R^2-2*SW)+5*SW^2)*SB*SC : :
X(32359) = 2*X(5)-3*X(32379) = 2*X(20)-3*X(32330) = X(20)-3*X(32354) = 3*X(154)-2*X(1209) = 5*X(631)-3*X(32337) = 5*X(631)-6*X(32391) = 3*X(1853)-4*X(6689) = X(2888)-3*X(11206) = 7*X(3832)-6*X(32369) = 5*X(3843)-6*X(32364) = 5*X(3843)-3*X(32402) = 7*X(9624)-6*X(32331) = 2*X(11362)-3*X(32371) = 2*X(15774)-3*X(32372) = 5*X(17821)-4*X(32348)

The reciprocal orthologic center of these triangles is X(32358).

X(32359) lies on these lines: {5,6145}, {20,32330}, {25,32377}, {54,1503}, {154,1209}, {159,2917}, {195,382}, {539,7387}, {548,15138}, {631,32337}, {973,9971}, {1154,6293}, {1593,10619}, {1619,9920}, {1853,6689}, {2888,11206}, {2937,3519}, {3574,11402}, {3832,32369}, {3843,32364}, {5596,11271}, {5965,9924}, {6152,9973}, {6278,32374}, {6281,32373}, {6288,6759}, {9624,32331}, {9657,32336}, {9670,32390}, {10535,12956}, {10610,14216}, {10628,23236}, {11362,32371}, {11576,31383}, {11750,32349}, {12234,13419}, {12242,17809}, {12254,15739}, {12946,26888}, {15774,32372}, {15888,32382}, {17821,32348}, {32332,32341}

X(32359) = reflection of X(i) in X(j) for these (i,j): (6145, 32379), (6288, 6759), (14216, 10610), (32330, 32354), (32337, 32391), (32402, 32364)


X(32360) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO HATZIPOLAKIS-MOSES

Barycentrics    (2*a^16-(b+c)*a^15-2*(3*b^2-b*c+3*c^2)*a^14+(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(3*b^4+3*c^4-4*(b^2-3*b*c+c^2)*b*c)*a^12-(b+c)*(b^4+c^4-(4*b^2-7*b*c+4*c^2)*b*c)*a^11+2*(3*b^6+3*c^6-(b^4+c^4+b*c*(2*b^2-3*b*c+2*c^2))*b*c)*a^10-(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+3*b*c*(b-c)^2)*b*c)*a^9-(5*b^6+5*c^6+(2*b^4+2*c^4+5*b*c*(b+c)^2)*b*c)*(b-c)^2*a^8+(b^2-c^2)*(b-c)*(5*b^6+5*c^6+b*c*(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2))*a^7-2*(b^2-c^2)^2*(b^6+c^6+b*c*(b^2-b*c+c^2)^2)*a^6+(b^2-c^2)^2*(b+c)*(b^6+c^6+b*c*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2))*a^5+(b^8+c^8-2*(b^2-b*c+c^2)*(2*b^4+2*c^4+b*c*(2*b^2+b*c+2*c^2))*b*c)*(b^2-c^2)^2*a^4-(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*(3*b^2+5*b*c+3*c^2)*a^3+2*(b^3+c^3)*(b+c)*(b^2-c^2)^3*(b^4-c^4)*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6)*D-4*S^2*(-a+b+c)*a^2*(-a^2+b^2+c^2)*(a^2+c^2-b^2)*(a^2-c^2+b^2)*(a^6-(b^2+c^2)*a^4-(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(6146).

X(32360) lies on these lines: {}

X(32360) = X(6145)-of-1st-Auriga-triangle


X(32361) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO HATZIPOLAKIS-MOSES

Barycentrics    (-2*a^16+(b+c)*a^15+2*(3*b^2-b*c+3*c^2)*a^14-(b+c)*(3*b^2-2*b*c+3*c^2)*a^13-(3*b^4+3*c^4-4*(b^2-3*b*c+c^2)*b*c)*a^12+(b+c)*(b^4+c^4-(4*b^2-7*b*c+4*c^2)*b*c)*a^11-2*(3*b^6+3*c^6-(b^4+c^4+b*c*(2*b^2-3*b*c+2*c^2))*b*c)*a^10+(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+3*b*c*(b-c)^2)*b*c)*a^9+(5*b^6+5*c^6+(2*b^4+2*c^4+5*b*c*(b+c)^2)*b*c)*(b-c)^2*a^8-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+b*c*(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2))*a^7+2*(b^2-c^2)^2*(b^6+c^6+b*c*(b^2-b*c+c^2)^2)*a^6-(b^2-c^2)^2*(b+c)*(b^6+c^6+b*c*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2))*a^5-(b^8+c^8-2*(b^2-b*c+c^2)*(2*b^4+2*c^4+b*c*(2*b^2+b*c+2*c^2))*b*c)*(b^2-c^2)^2*a^4+(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*(3*b^2+5*b*c+3*c^2)*a^3-2*(b^3+c^3)*(b+c)*(b^2-c^2)^3*(b^4-c^4)*a^2-(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a+(b^2+c^2)^2*(b^2-c^2)^6)*D-4*S^2*(-a+b+c)*a^2*(-a^2+b^2+c^2)*(a^2+c^2-b^2)*(a^2-c^2+b^2)*(a^6-(b^2+c^2)*a^4-(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(6146)

X(32361) lies on these lines: {}

X(32361) = X(6145)-of-2nd-Auriga-triangle


X(32362) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO HATZIPOLAKIS-MOSES

Barycentrics    (3*R^2-SW)*(8*R^2-SA-3*SW)*S^4+(8*(7*SA^2-12*SA*SW+2*SW^2)*R^4-(40*SA^2-75*SA*SW+17*SW^2)*SW*R^2+(7*SA^2-14*SA*SW+4*SW^2)*SW^2)*S^2-3*(2*R^2-SW)*(8*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32362) lies on these lines: {32,6145}, {1154,9923}, {2896,32354}, {3096,32391}, {3098,32330}, {3099,32356}, {9301,32402}, {9821,18400}, {9857,32371}, {9862,32337}, {9993,32369}, {9994,32373}, {9995,32374}, {9996,32379}, {9997,32394}, {10038,32403}, {10047,32404}, {10628,12501}, {10828,32357}, {10871,32380}, {10872,32381}, {10873,32382}, {10874,32383}, {10875,32388}, {10876,32389}, {10877,32390}, {10878,32405}, {10879,32406}, {11368,32331}, {11386,32332}, {11494,32347}, {11885,32372}, {13892,32399}, {13946,32400}, {18500,32364}, {18957,32336}, {19011,32342}, {19012,32343}, {22744,32363}

X(32362) = X(6145)-of-5th-Brocard-triangle


X(32363) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO HATZIPOLAKIS-MOSES

Barycentrics    a*(a^18-(b+c)*a^17-2*(2*b^2-3*b*c+2*c^2)*a^16+4*(b^3+c^3)*a^15+(4*b^4+4*c^4-(18*b^2-17*b*c+18*c^2)*b*c)*a^14-(b+c)*(2*b^2-3*b*c+2*c^2)^2*a^13+(4*b^6+4*c^6+(8*b^4+8*c^4-19*(b-c)^2*b*c)*b*c)*a^12-(b+c)*(4*b^6+4*c^6+(4*b^4+4*c^4-(19*b^2-28*b*c+19*c^2)*b*c)*b*c)*a^11-2*(5*b^6+5*c^6-(b^4+c^4+2*(b-c)^2*b*c)*b*c)*(b-c)^2*a^10+2*(b^2-c^2)*(b-c)*(5*b^6+6*b^3*c^3+5*c^6)*a^9+2*(2*b^8+2*c^8-(6*b^6+6*c^6-(b^4+c^4-2*(b^2+5*b*c+c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^8-2*(b^2-c^2)*(b-c)*(2*b^8+2*c^8-(6*b^6+6*c^6-(-4*b^2*c^2+(b^2-c^2)^2)*b*c)*b*c)*a^7+(b^2-c^2)^2*(b-c)^2*(4*b^6+4*c^6+(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*b*c)*a^6-(b^2-c^2)^3*(b-c)*(4*b^6+4*c^6+(4*b^4+4*c^4+(b^2+10*b*c+c^2)*b*c)*b*c)*a^5-(b^2-c^2)^2*(4*b^10+4*c^10-(8*b^8+8*c^8-(15*b^6+15*c^6-(6*b^4+6*c^4-(5*b^2+12*b*c+5*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+(b^4-c^4)*(b^2-c^2)^2*(b-c)^3*(4*b^4+4*c^4+(4*b^2+3*b*c+4*c^2)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b^6+c^6+(2*b^4+2*c^4+(7*b^2-4*b*c+7*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^5*(b-c)^3*(b^2+c^2)^2*a-2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32363) lies on these lines: {3,32347}, {36,32356}, {55,32394}, {56,6145}, {104,32337}, {956,32371}, {958,32381}, {999,32331}, {1154,22659}, {2917,23361}, {2975,32354}, {3428,32330}, {10628,19478}, {10966,32390}, {11249,18400}, {12114,32380}, {18761,32364}, {19013,32342}, {19014,32343}, {22479,32332}, {22520,32335}, {22654,32357}, {22744,32362}, {22753,32369}, {22755,32372}, {22756,32373}, {22757,32374}, {22758,32379}, {22759,32382}, {22760,32383}, {22761,32388}, {22762,32389}, {22763,32399}, {22764,32400}, {22765,32402}, {22766,32403}, {22767,32404}, {22768,32405}

X(32363) = reflection of X(i) in X(j) for these (i,j): (32347, 3), (32381, 32391)
X(32363) = X(6145)-of-2nd-circumperp-tangential-triangle


X(32364) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO HATZIPOLAKIS-MOSES

Barycentrics    S^4-(R^2*(12*R^2+3*SA-10*SW)-SA^2+SB*SC+2*SW^2)*S^2-(R^2*(44*R^2-25*SW)+3*SW^2)*SB*SC : :
X(32364) = 3*X(4)+X(32354) = X(1498)+3*X(32395) = 5*X(3091)-X(32337) = X(3357)-4*X(32396) = 5*X(3843)+X(32359) = 5*X(3843)-X(32402) = 2*X(5893)+X(10610) = 2*X(6689)+X(22802) = 5*X(18492)-X(32356) = X(32354)-3*X(32379)

The reciprocal orthologic center of these triangles is X(6146).

X(32364) lies on these lines: {4,54}, {26,5448}, {30,32391}, {113,1209}, {235,973}, {381,6145}, {382,32330}, {546,32369}, {1154,15761}, {1478,32383}, {1479,32382}, {1498,32395}, {1539,5893}, {1568,7488}, {1596,11743}, {2883,32137}, {2917,7517}, {3091,32337}, {3357,4846}, {3549,32348}, {3583,32390}, {3585,32336}, {3818,9977}, {3843,32359}, {5576,6000}, {6288,17824}, {6689,7526}, {7564,18128}, {7703,10574}, {9818,32357}, {9955,32331}, {10895,32403}, {10896,32404}, {11802,22800}, {11819,16252}, {12699,32371}, {13665,32343}, {13785,32342}, {14076,18504}, {15311,20376}, {18491,32347}, {18492,32356}, {18500,32362}, {18502,32335}, {18507,32372}, {18509,32373}, {18511,32374}, {18516,32380}, {18517,32381}, {18520,32388}, {18522,32389}, {18525,32394}, {18538,32399}, {18542,32405}, {18544,32406}, {18761,32363}, {18762,32400}, {20424,21850}

X(32364) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 32379}, {382, 32330}, {2883, 32351}, {6288, 17824}, {6759, 32365}, {12699, 32371}, {18507, 32372}, {18525, 32394}, {22802, 32401}, {32359, 32402}
X(32364) = reflection of X(i) in X(j) for these (i,j): (18381, 32393), (32331, 9955), (32369, 546), (32401, 6689)
X(32364) = X(6145)-of-Ehrmann-mid-triangle


X(32365) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO HATZIPOLAKIS-MOSES

Barycentrics    (4*R^2+SA-2*SW)*R^2*S^2+(68*R^4-47*R^2*SW+8*SW^2)*SB*SC : :
X(32365) = X(3)-3*X(32395) = 3*X(4)+X(32346) = 3*X(381)-X(2917) = 5*X(3843)-X(9920) = X(17824)+3*X(18405) = 3*X(18376)-2*X(32369) = 3*X(18405)-X(32402) = 3*X(23325)-4*X(32393)

The reciprocal orthologic center of these triangles is X(389).

X(32365) lies on these lines: {3,32395}, {4,54}, {5,18428}, {30,32351}, {52,265}, {381,2917}, {382,32345}, {539,18568}, {542,32368}, {550,20376}, {576,18382}, {973,18390}, {1154,9927}, {1209,18404}, {2888,18387}, {3153,7691}, {3357,31723}, {3583,32378}, {3585,32350}, {3627,11804}, {3818,22804}, {3843,9920}, {5449,18569}, {5907,6288}, {6102,11262}, {6564,32384}, {6565,32385}, {10113,32196}, {11204,14790}, {11576,18385}, {12370,20424}, {12893,13371}, {13851,32352}, {16808,32397}, {16809,32398}, {17824,18405}, {18386,32333}, {18392,32338}, {18394,32339}, {18396,32341}, {18406,32370}, {18414,32386}, {18415,32387}, {18420,32396}, {18531,32348}, {18567,19479}, {18572,21230}, {18918,32334}, {19130,32344}

X(32365) = midpoint of X(i) and X(j) for these lines: {i,j}: {382, 32345}, {17824, 32402}
X(32365) = reflection of X(i) in X(j) for these (i,j): (550, 20376), (6102, 11262), (6145, 18383), (6759, 32364), (7691, 14076), (10274, 3574), (23358, 5), (32344, 19130), (32401, 32351)
X(32365) = X(21)-of-Ehrmann-vertex-triangle if ABC is acute
X(32365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3574, 21659, 54), (18383, 31724, 19506)


X(32366) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO 1st EHRMANN

Barycentrics    a^2*((b^2+c^2)*a^4-6*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)) : :
X(32366) = 5*X(6)-3*X(51) = 3*X(6)-X(1843) = 7*X(6)-3*X(9971) = 5*X(6)-X(9973) = 9*X(51)-5*X(1843) = 3*X(51)+5*X(6467) = 6*X(51)-5*X(9969) = 7*X(51)-5*X(9971) = 3*X(51)-X(9973) = 3*X(51)-10*X(22829) = 5*X(69)-9*X(7998) = X(69)+3*X(15531) = X(1843)+3*X(6467) = 2*X(1843)-3*X(9969) = 7*X(1843)-9*X(9971) = 5*X(1843)-3*X(9973) = X(1843)-6*X(22829) = 2*X(6467)+X(9969) = 7*X(6467)+3*X(9971) = 3*X(7998)+5*X(15531)

The reciprocal orthologic center of these triangles is X(32367).

X(32366) lies on these lines: {6,25}, {66,17040}, {69,3266}, {141,8681}, {157,5065}, {160,800}, {182,8548}, {193,3313}, {389,12007}, {511,550}, {524,6665}, {570,20975}, {597,9822}, {1216,5965}, {1503,13403}, {1511,5892}, {1992,12220}, {2386,15048}, {2854,3589}, {2929,13367}, {2979,11008}, {3003,20775}, {3564,11264}, {3618,12272}, {3631,3819}, {5097,5446}, {5140,5254}, {5181,26156}, {5421,23635}, {5943,6329}, {6146,15739}, {6241,6776}, {7668,14767}, {10574,27082}, {12235,32046}, {12283,14853}, {14778,23642}, {14912,15073}, {15126,23300}, {15140,32226}, {18583,18874}, {20583,21849}, {25320,32248}

X(32366) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 6467}, {193, 3313}, {1353, 15074}, {3629, 17710}, {15073, 19161}
X(32366) = reflection of X(i) in X(j) for these (i,j): (6, 22829), (389, 12007), (5446, 5097), (9969, 6), (14913, 3589), (21849, 20583)
X(32366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 159, 19136), (6, 19459, 206), (6467, 22829, 9969)


X(32367) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(3*(2*R^2*(5*R^2-3*SW)+(SA+SW)*SW)*S^2-((16*R^2+9*SA-17*SW)*R^2-3*SA^2+3*SB*SC+4*SW^2)*SA*SW) : :
X(32367) = X(576)-3*X(10274) = X(9972)-3*X(32344) = 2*X(15579)-3*X(32401)

The reciprocal orthologic center of these triangles is X(32366).

X(32367) lies on these lines: {54,2393}, {159,3574}, {206,576}, {5965,9925}, {8542,32391}, {9970,32379}, {9972,32344}, {12584,23358}, {15579,32401}, {15581,18400}


X(32368) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((6*R^2*(10*R^2-7*SW)+7*SW^2)*S^2-(2*R^2-SW)*(4*R^2+SA-2*SW)*SA*SW) : :
X(32368) = 3*X(6)-X(2917) = 3*X(1992)+X(32346) = 2*X(2917)-3*X(32344) = X(9920)-5*X(11482) = X(15069)-3*X(32395) = 3*X(17813)+X(17824) = 3*X(23049)-2*X(32369)

The reciprocal orthologic center of these triangles is X(389).

X(32368) lies on these lines: {6,24}, {68,5965}, {161,11422}, {511,32401}, {524,32351}, {539,9926}, {542,32365}, {575,23358}, {576,18400}, {895,6145}, {1154,8548}, {1209,8538}, {1992,32346}, {2393,32379}, {3574,8541}, {5486,12242}, {6288,18449}, {7691,11416}, {8539,32370}, {8540,32378}, {9813,32396}, {9920,11482}, {9976,10628}, {10602,32341}, {11405,32333}, {11443,32338}, {11458,32339}, {11470,32340}, {11477,32345}, {11511,32348}, {15069,32395}, {17813,17824}, {18919,32334}, {19369,32350}, {19426,32386}, {19427,32387}, {21639,32352}, {23049,32369}

X(32368) = midpoint of X(11477) and X(32345)
X(32368) = reflection of X(i) in X(j) for these (i,j): (23358, 575), (32344, 6)
X(32368) = X(21)-of-2nd-Ehrmann-triangle if ABC is acute


X(32369) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO HATZIPOLAKIS-MOSES

Barycentrics    S^4+(8*R^4-3*R^2*SA-3*R^2*SW+SA^2-SB*SC)*S^2+(56*R^4-40*R^2*SW+7*SW^2)*SB*SC : :
X(32369) = 3*X(4)+X(32337) = X(54)-3*X(32395) = 3*X(381)-X(32379) = 3*X(381)+X(32402) = 3*X(1699)+X(32356) = X(2917)+3*X(18405) = 5*X(3091)-X(32354) = 7*X(3832)-X(32359) = 3*X(5587)-X(32371) = 3*X(5603)-X(32394) = 3*X(11897)-X(32372) = 3*X(18376)-X(32365) = 2*X(18383)+X(22804) = 3*X(23049)-X(32368) = 3*X(23324)-X(32351) = 3*X(23324)-2*X(32393) = 3*X(23325)-X(32401)

The reciprocal orthologic center of these triangles is X(6146).

X(32369) lies on these lines: {2,32330}, {4,973}, {5,5944}, {11,32336}, {12,32390}, {24,11704}, {54,7547}, {98,32335}, {125,3575}, {235,32332}, {265,11262}, {371,32399}, {372,32400}, {381,32379}, {515,32331}, {546,32364}, {1154,9927}, {1209,12605}, {1352,6288}, {1478,32404}, {1479,32403}, {1587,32342}, {1588,32343}, {1598,32357}, {1658,23325}, {1699,32356}, {2917,7503}, {3091,32354}, {3574,12241}, {3832,32359}, {3861,27552}, {5446,10113}, {5587,32371}, {5603,32394}, {6152,15738}, {6201,32374}, {6202,32373}, {7542,20376}, {7569,17845}, {7691,18392}, {7706,13630}, {8212,32388}, {8213,32389}, {9993,32362}, {10531,32405}, {10532,32406}, {10893,32380}, {10894,32381}, {10895,32382}, {10896,32383}, {10982,17824}, {11496,32347}, {11897,32372}, {13565,23358}, {18386,32341}, {20191,23332}, {22753,32363}, {23049,32368}

X(32369) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 6145}, {32379, 32402}
X(32369) = reflection of X(i) in X(j) for these (i,j): (23358, 13565), (32351, 32393), (32364, 546), (32391, 5)
X(32369) = complement of X(32330)
X(32369) = X(6145)-of-Euler-triangle
X(32369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 32402, 32379), (23324, 32351, 32393)


X(32370) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO HATZIPOLAKIS-MOSES

Barycentrics
a*(a^15+(b+c)*a^14-3*(b^2+c^2)*a^13-3*(b+c)*(b^2+c^2)*a^12+(b^2-b*c+c^2)^2*a^11+(b+c)*(b^2+b*c+c^2)^2*a^10+(5*b^4+5*c^4-3*(b-c)^2*b*c)*(b+c)^2*a^9+(b^2-c^2)*(b-c)*(5*b^4+5*c^4+3*(b+c)^2*b*c)*a^8-(b^2+c^2)*(5*b^4+5*c^4-2*(b^2+c^2)*b*c)*(b+c)^2*a^7-(b^4-c^4)*(b-c)*(5*b^4+5*c^4+2*(b^2+c^2)*b*c)*a^6-(b^4-c^4)*(b^2-c^2)*(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^5-(b^4-c^4)*(b^2-c^2)*(b+c)*(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^4+(b^2-c^2)^2*(b+c)^2*(3*b^6+3*c^6-2*(2*b^4+2*c^4-3*(b^2-b*c+c^2)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(3*b^6+3*c^6+2*(2*b^4+2*c^4+3*(b^2+b*c+c^2)*b*c)*b*c)*a^2-(b^3+c^3)*(b+c)*(b^2-c^2)^3*(b^4-c^4)*a-(b^3-c^3)*(b^2-c^2)^5*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(389).

X(32370) lies on these lines: {19,3574}, {40,12785}, {54,65}, {55,2917}, {71,6145}, {539,12417}, {973,11435}, {1154,6237}, {1209,8251}, {2550,32346}, {2888,9537}, {3101,7691}, {3197,17824}, {3611,32352}, {3925,32351}, {5415,32384}, {5416,32385}, {5584,32345}, {6288,18453}, {7688,32401}, {7724,10119}, {8274,9572}, {8539,32368}, {9816,32396}, {9920,10306}, {10319,32348}, {10536,32379}, {10636,32397}, {10637,32398}, {10901,12946}, {10902,23358}, {11406,32333}, {11445,32338}, {11460,32339}, {11471,32340}, {18406,32365}, {18921,32334}, {19133,32344}, {19350,32341}, {19432,32386}, {19433,32387}

X(32370) = reflection of X(32378) in X(2917)
X(32370) = X(21)-of-extangents-triangle if ABC is acute


X(32371) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(2*R^2-3*SA-SW)*S^2+4*(5*R^2-2*SW)*(4*R^2-SW)*R*a*S-(SB+SC)*(2*R^2*(4*R^2-4*SA+SW)+3*SA^2-3*SB*SC-2*SW^2)*SA-2*(5*R^2-2*SW)*(4*R^2-SW)*(SB*b+SC*c)*a : :
X(32371) = 3*X(5587)-2*X(32369) = 3*X(5657)-X(32337) = 3*X(5790)-X(32402) = 2*X(11362)+X(32359)

The reciprocal orthologic center of these triangles is X(6146).

X(32371) lies on these lines: {1,32391}, {2,32331}, {8,32354}, {10,6145}, {54,65}, {72,32381}, {355,18400}, {515,32330}, {517,32379}, {519,32394}, {956,32363}, {1154,9928}, {1737,32404}, {1837,32390}, {3057,32383}, {3679,32356}, {3751,9905}, {5090,32332}, {5252,32336}, {5587,32369}, {5657,32337}, {5687,32347}, {5688,32374}, {5689,32373}, {5790,32402}, {7713,11743}, {8193,32357}, {8214,32388}, {8215,32389}, {9857,32362}, {10039,32403}, {10628,12778}, {10791,32335}, {10914,32380}, {10915,32405}, {10916,32406}, {11362,32359}, {11900,32372}, {12699,32364}, {13883,32343}, {13893,32399}, {13936,32342}, {13947,32400}

X(32371) = midpoint of X(8) and X(32354)
X(32371) = reflection of X(i) in X(j) for these (i,j): (1, 32391), (6145, 10), (12699, 32364)
X(32371) = anticomplement of X(32331)
X(32371) = X(6145)-of-outer-Garcia-triangle


X(32372) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO HATZIPOLAKIS-MOSES

Barycentrics
(S^2-3*SB*SC)*(S^4-(R^2*(10*R^2+3*SA-6*SW)-SA^2+SB*SC+SW^2)*S^2-16*(72*R^2+15*SA-98*SW)*R^6-2*(17*SA^2-107*SA*SW+378*SW^2)*R^4+(17*SA^2-59*SA*SW+153*SW^2)*SW*R^2-(2*SA^2-5*SA*SW+11*SW^2)*SW^2) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32372) lies on these lines: {30,32330}, {402,6145}, {1154,12418}, {1650,32391}, {4240,32354}, {10628,12790}, {11251,18400}, {11831,32331}, {11832,32332}, {11839,32335}, {11845,32337}, {11848,32347}, {11852,32356}, {11853,32357}, {11885,32362}, {11897,32369}, {11900,32371}, {11901,32373}, {11902,32374}, {11903,32380}, {11904,32381}, {11905,32382}, {11906,32383}, {11907,32388}, {11908,32389}, {11909,32390}, {11910,32394}, {11911,32402}, {11912,32403}, {11913,32404}, {11914,32405}, {11915,32406}, {13894,32399}, {13948,32400}, {15774,32359}, {18507,32364}, {18958,32336}, {19017,32342}, {19018,32343}, {22755,32363}

X(32372) = midpoint of X(4240) and X(32354)
X(32372) = reflection of X(i) in X(j) for these (i,j): (1650, 32391), (6145, 402), (18507, 32364)
X(32372) = X(6145)-of-Gossard-triangle


X(32373) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO HATZIPOLAKIS-MOSES

Barycentrics    (((8*R^2-3*SA-3*SW)*R^2+SA*SW)*S^2+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC)*S+2*(2*R^2-SW)*(3*S^4-(2*R^2*(5*SA-3*SW)-3*SA^2+3*SB*SC+2*SW^2)*S^2-(8*R^2-3*SW)*SB*SC*SW) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32373) lies on these lines: {6,3574}, {973,6218}, {1154,9929}, {1161,18400}, {1271,32354}, {2917,8903}, {5589,32356}, {5591,32391}, {5595,32357}, {5605,32394}, {5689,32371}, {6146,32327}, {6202,32369}, {6215,32379}, {6220,6293}, {6281,32359}, {8216,32388}, {8217,32389}, {8974,32399}, {9994,32362}, {10040,32403}, {10048,32404}, {10628,12803}, {10783,32337}, {10792,32335}, {10919,32380}, {10921,32381}, {10923,32382}, {10925,32383}, {10927,32390}, {10929,32405}, {10931,32406}, {11370,32331}, {11388,32332}, {11497,32347}, {11824,32330}, {11901,32372}, {11916,32402}, {13949,32400}, {18509,32364}, {18959,32336}, {19351,32345}, {22756,32363}

X(32373) = X(6145)-of-inner-Grebe-triangle


X(32374) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO HATZIPOLAKIS-MOSES

Barycentrics    -(((8*R^2-3*SA-3*SW)*R^2+SA*SW)*S^2+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC)*S+2*(2*R^2-SW)*(3*S^4-(2*R^2*(5*SA-3*SW)-3*SA^2+3*SB*SC+2*SW^2)*S^2-(8*R^2-3*SW)*SB*SC*SW) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32374) lies on these lines: {6,3574}, {973,6217}, {1154,9930}, {1160,18400}, {1270,32354}, {2917,8904}, {5588,32356}, {5590,32391}, {5594,32357}, {5604,32394}, {5688,32371}, {6146,32328}, {6201,32369}, {6214,32379}, {6219,6293}, {6278,32359}, {8218,32388}, {8219,32389}, {8975,32399}, {9995,32362}, {10041,32403}, {10049,32404}, {10628,12804}, {10784,32337}, {10793,32335}, {10920,32380}, {10922,32381}, {10924,32382}, {10926,32383}, {10928,32390}, {10930,32405}, {10932,32406}, {11371,32331}, {11389,32332}, {11498,32347}, {11825,32330}, {11902,32372}, {11917,32402}, {13950,32400}, {18511,32364}, {18960,32336}, {19352,32345}, {22757,32363}

X(32374) = X(6145)-of-outer-Grebe-triangle


X(32375) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO 3rd HATZIPOLAKIS

Barycentrics    (2*R^4*(32*R^2-73*SA+39*SW)+(59*SA-38*SW)*R^2*SW-2*(3*SA-2*SW)*SW^2)*S^2+(16*R^2-3*SW)*(R^2*(16*R^2-13*SW)+2*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(32376).

X(32375) lies on these lines: {6,17837}, {54,22533}, {3521,22979}, {9932,22962}, {19511,22834}

X(32375) = reflection of X(22466) in X(31985)


X(32376) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO HATZIPOLAKIS-MOSES

Barycentrics    (64*R^6-2*(37*SA-3*SW)*R^4+(43*SA-22*SW)*SW*R^2-2*(3*SA-2*SW)*SW^2)*S^2+(8*R^2-3*SW)*(16*R^4-13*R^2*SW+2*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(32375).

X(32376) lies on these lines: {6,3574}, {54,22533}, {368,20917}, {539,14076}, {13630,18128}

X(32376) = midpoint of X(i) and X(j) for these lines: {i,j}: {6145, 32412}, {32377, 32393}


X(32377) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd HYACINTH TO HATZIPOLAKIS-MOSES

Barycentrics    SA*((12*R^2-SA-3*SW)*S^2+(SB+SC)*((8*R^2-3*SA-11*SW)*R^2+SA^2-SB*SC+3*SW^2)) : :
X(32377) = 3*X(973)-2*X(6756) = 4*X(6756)-3*X(32332) = 3*X(12022)-X(12300)

The reciprocal orthologic center of these triangles is X(32358).

X(32377) lies on these lines: {3,539}, {6,3574}, {25,32359}, {54,70}, {125,6689}, {184,1209}, {185,6152}, {542,13160}, {973,6756}, {974,18914}, {1154,6146}, {1181,6288}, {1493,13371}, {1503,11576}, {1594,12242}, {2888,6776}, {3088,32337}, {3542,32379}, {5965,6467}, {6243,10938}, {7691,19467}, {9827,12134}, {10112,12225}, {11271,18946}, {11416,15801}, {12022,12300}, {12234,18381}, {12254,18909}, {12315,32340}, {12946,19349}, {12956,19354}, {13367,32348}, {13431,14791}, {17818,17846}, {21230,31804}, {32334,32354}

X(32377) = reflection of X(i) in X(j) for these (i,j): (32332, 973), (32393, 32376)
X(32377) = {X(6145), X(32341)}-harmonic conjugate of X(3574)


X(32378) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO HATZIPOLAKIS-MOSES

Barycentrics    a*((4*R^2*(20*R^2-13*SW)+b*c*(4*R^2+SA-2*SW)+8*SW^2)*S^2-(12*R^2-5*SW)*SB*SC*b*c) : :
X(32378) = X(6285)+2*X(18984)

The reciprocal orthologic center of these triangles is X(389).

X(32378) lies on these lines: {1,18400}, {11,32351}, {33,3574}, {34,32340}, {35,23358}, {36,32401}, {54,6198}, {55,2917}, {56,32345}, {195,9642}, {497,32346}, {539,9931}, {973,11436}, {1040,32348}, {1062,1209}, {1154,6238}, {1250,32398}, {2066,32384}, {2192,17824}, {2330,32344}, {2888,9538}, {3056,13079}, {3057,7979}, {3100,7691}, {3270,32352}, {3295,9920}, {3583,32365}, {5414,32385}, {5433,20376}, {6145,9630}, {6285,18984}, {6288,18455}, {7071,32333}, {7354,19505}, {7356,7727}, {8540,32368}, {9577,9905}, {9627,12946}, {9641,12307}, {9817,32396}, {10066,10535}, {10149,22954}, {10638,32397}, {10896,32395}, {11446,32338}, {11461,32339}, {12428,12888}, {18922,32334}, {19354,32341}, {19434,32386}, {19435,32387}, {26888,32382}

X(32378) = reflection of X(i) in X(j) for these (i,j): (32350, 1), (32370, 2917)
X(32378) = X(21)-of-intangents-triangle if ABC is acute


X(32379) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((2*R^2-3*SA-SW)*S^2-(2*(4*R^2-4*SA+SW)*R^2+3*SA^2-3*SB*SC-2*SW^2)*SA) : :
X(32379) = 2*X(5)+X(32359) = X(54)+2*X(6759) = 3*X(154)-X(2917) = 3*X(154)+X(17824) = 2*X(195)+X(9935) = X(195)+5*X(14530) = 3*X(381)-2*X(32369) = 3*X(381)-X(32402) = X(1498)+2*X(10610) = 2*X(3574)+X(9833) = 3*X(5587)-X(32356) = 3*X(5886)-2*X(32331) = X(6288)-4*X(16252) = 4*X(6689)-X(14216) = X(7691)-4*X(10282) = X(9920)-5*X(14530) = X(9934)+2*X(11597) = X(9935)-10*X(14530) = 3*X(11206)+X(32346) = X(32354)+2*X(32364)

The reciprocal orthologic center of these triangles is X(6146).

X(32379) lies on these lines: {1,32382}, {2,32337}, {3,32357}, {4,54}, {5,6145}, {11,32404}, {12,32403}, {23,15801}, {25,973}, {26,154}, {30,32330}, {110,5562}, {140,15139}, {159,195}, {161,12161}, {182,32396}, {206,1209}, {355,32380}, {381,32369}, {517,32371}, {569,32393}, {952,32394}, {1177,3519}, {1478,32336}, {1479,32390}, {1493,7530}, {1495,32352}, {1498,7526}, {1503,5576}, {1598,11743}, {1843,11808}, {2393,32368}, {2777,8718}, {2781,13564}, {2883,7728}, {2918,10117}, {3542,32377}, {5012,18381}, {5587,32356}, {5886,32331}, {5887,24301}, {6000,15062}, {6214,32374}, {6215,32373}, {6247,20376}, {6288,10024}, {6293,7502}, {6353,32334}, {6689,14216}, {7529,19153}, {7552,10203}, {7564,32046}, {7583,32343}, {7584,32342}, {8220,32388}, {8221,32389}, {8976,32399}, {9306,11487}, {9544,31304}, {9652,13079}, {9667,18984}, {9707,12300}, {9970,32367}, {9996,32362}, {10066,10535}, {10082,26888}, {10192,18350}, {10224,32353}, {10533,32384}, {10534,32385}, {10536,32370}, {10627,17847}, {10796,32335}, {10942,32405}, {10943,32406}, {11262,15087}, {11499,32347}, {11585,15462}, {11819,20424}, {12363,19149}, {13434,18383}, {13951,32400}, {15132,15331}, {15647,22815}, {15739,26864}, {18374,21841}, {21230,31831}, {22758,32363}, {26881,32338}, {26882,32339}, {30402,32397}, {30403,32398}

X(32379) = midpoint of X(i) and X(j) for these lines: {i,j}: {4, 32354}, {195, 9920}, {1498, 32345}, {2917, 17824}, {6145, 32359}, {6759, 10274}, {32380, 32381}
X(32379) = reflection of X(i) in X(j) for these (i,j): (3, 32391), (4, 32364), (54, 10274), (6145, 5), (6247, 20376), (7691, 23358), (9935, 9920), (23358, 10282), (32344, 206), (32345, 10610), (32402, 32369)
X(32379) = complement of X(32337)
X(32379) = X(6145)-of-Johnson-triangle
X(32379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (184, 3574, 54), (184, 6759, 9833), (32382, 32383, 1)


X(32380) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO HATZIPOLAKIS-MOSES

Barycentrics
a*(a^18-(b+c)*a^17-(5*b^2-6*b*c+5*c^2)*a^16+(b+c)*(5*b^2-4*b*c+5*c^2)*a^15+(9*b^4+9*c^4-(20*b^2-19*b*c+20*c^2)*b*c)*a^14-(b+c)*(9*b^4+9*c^4-(14*b^2-19*b*c+14*c^2)*b*c)*a^13-(5*b^6+5*c^6-(16*b^4+16*c^4-(21*b^2-38*b*c+21*c^2)*b*c)*b*c)*a^12+(b^2-c^2)*(b-c)*(5*b^4+5*c^4-2*(b^2-6*b*c+c^2)*b*c)*a^11-(5*b^6+5*c^6-2*(b^4+c^4+3*(b^2+c^2)*b*c)*b*c)*(b-c)^2*a^10+(b^2-c^2)*(b-c)*(5*b^6+5*c^6-2*(b-c)^2*b^2*c^2)*a^9+(b^2+c^2)*(9*b^6+9*c^6-(2*b^4+2*c^4+5*(b^2+c^2)*b*c)*b*c)*(b-c)^2*a^8-(b^2-c^2)*(b-c)*(9*b^8+9*c^8-2*(b^2-b*c+c^2)*(b^4+c^4-(b^2+3*b*c+c^2)*b*c)*b*c)*a^7-(b^2-c^2)^2*(5*b^8+5*c^8-(4*b^6+4*c^6-(13*b^4+13*c^4-4*(4*b^2-5*b*c+4*c^2)*b*c)*b*c)*b*c)*a^6+(b^3+c^3)*(b^2-c^2)^2*(5*b^6+5*c^6-(b^4+c^4-(7*b^2-6*b*c+7*c^2)*b*c)*b*c)*a^5+(b^2-c^2)^2*(b+c)^2*(b^8+c^8-2*(b^2+3*b*c+c^2)*(b^2-b*c+c^2)^2*b*c)*a^4-(b^4-c^4)*(b^2-c^2)^2*(b-c)*(b^6+c^6+(6*b^4+6*c^4+(5*b^2+4*b*c+5*c^2)*b*c)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(4*b^4+4*c^4+(3*b^2+2*b*c+3*c^2)*b*c)*b*c*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^2+b*c+2*c^2)*b*c*a-2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32380) lies on these lines: {11,6145}, {12,32405}, {355,32379}, {1154,12422}, {1376,32347}, {3434,32354}, {10523,32403}, {10525,18400}, {10628,12889}, {10785,32337}, {10794,32335}, {10826,32356}, {10829,32357}, {10871,32362}, {10893,32369}, {10914,32371}, {10919,32373}, {10920,32374}, {10944,32382}, {10945,32388}, {10946,32389}, {10947,32390}, {10948,32404}, {10949,32406}, {11373,32331}, {11390,32332}, {11826,32330}, {11903,32372}, {11928,32402}, {12114,32363}, {13895,32399}, {13952,32400}, {18516,32364}, {18961,32336}, {19023,32342}, {19024,32343}

X(32380) = reflection of X(i) in X(j) for these (i,j): (32347, 32391), (32381, 32379)
X(32380) = X(6145)-of-inner-Johnson-triangle


X(32381) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO HATZIPOLAKIS-MOSES

Barycentrics
a*(a^18-(b+c)*a^17-(5*b^2+2*b*c+5*c^2)*a^16+5*(b+c)*(b^2+c^2)*a^15+(9*b^4+9*c^4+(4*b^2+11*b*c+4*c^2)*b*c)*a^14-(b^3+c^3)*(9*b^2+7*b*c+9*c^2)*a^13-(5*b^6+5*c^6-(4*b^4+4*c^4-5*(b+c)^2*b*c)*b*c)*a^12+(b+c)*(5*b^2-8*b*c+5*c^2)*(b^4+c^4)*a^11-(5*b^6+5*c^6+2*(b^4+c^4-(7*b^2-10*b*c+7*c^2)*b*c)*b*c)*(b+c)^2*a^10+(5*b^6+5*c^6-2*(5*b^2-8*b*c+5*c^2)*b^2*c^2)*(b+c)^3*a^9+3*(b^2-c^2)^2*(3*b^6+3*c^6+(b^2+4*b*c+c^2)*b^2*c^2)*a^8-(b^2-c^2)^2*(b+c)*(9*b^6+9*c^6+(3*b^2+8*b*c+3*c^2)*b^2*c^2)*a^7-(b^2-c^2)^2*(5*b^8+5*c^8-(4*b^2-7*b*c+4*c^2)*(3*b^4+3*c^4+4*(b^2+b*c+c^2)*b*c)*b*c)*a^6+(b^2-c^2)^2*(b+c)*(5*b^8+5*c^8-(2*b^2-b*c+2*c^2)*(5*b^4-4*b^2*c^2+5*c^4)*b*c)*a^5+(b^10+c^10-(b^2-b*c+c^2)*(4*b^6+4*c^6-(7*b^4+7*c^4+3*(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)^2*(b-c)*(b^6+c^6-3*(2*b^4+2*c^4+(b^2+c^2)*b*c)*b*c)*a^3-(b^4-c^4)*(b^2-c^2)^3*(4*b^4+4*c^4+(5*b^2-2*b*c+5*c^2)*b*c)*b*c*a^2-(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^2-b*c+2*c^2)*b*c*a+2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32381) lies on these lines: {11,32406}, {12,6145}, {54,15232}, {72,32371}, {355,32379}, {958,32363}, {1154,12423}, {3436,32354}, {10523,32404}, {10526,18400}, {10628,12890}, {10786,32337}, {10795,32335}, {10827,32356}, {10830,32357}, {10872,32362}, {10894,32369}, {10921,32373}, {10922,32374}, {10950,32383}, {10951,32388}, {10952,32389}, {10953,32390}, {10954,32403}, {10955,32405}, {11374,32331}, {11391,32332}, {11500,32347}, {11827,32330}, {11904,32372}, {11929,32402}, {13896,32399}, {13953,32400}, {18517,32364}, {18962,32336}, {19025,32342}, {19026,32343}

X(32381) = reflection of X(i) in X(j) for these (i,j): (32363, 32391), (32380, 32379)
X(32381) = X(6145)-of-outer-Johnson-triangle


X(32382) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(-2*(5*R^2-2*SW)*(4*R^2-SW)*b*c+2*(4*R^2*SA-S^2-4*SA^2+SA*SW)*R^2+SW*(S^2+3*SA^2-2*SA*SW)) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32382) lies on these lines: {1,32379}, {4,32390}, {5,32404}, {12,6145}, {54,65}, {56,32391}, {221,32345}, {388,32336}, {495,32403}, {973,11398}, {1154,3157}, {1478,10066}, {1479,32364}, {2917,18984}, {3085,32337}, {3574,11393}, {7354,32330}, {9578,32356}, {9654,32402}, {10082,10274}, {10088,10628}, {10797,32335}, {10831,32357}, {10873,32362}, {10895,32369}, {10923,32373}, {10924,32374}, {10944,32380}, {10956,32405}, {10957,32406}, {11375,32331}, {11392,32332}, {11501,32347}, {11905,32372}, {11930,32388}, {11931,32389}, {13079,17824}, {13897,32399}, {13954,32400}, {15888,32359}, {19027,32342}, {19028,32343}, {22759,32363}, {26888,32378}

X(32382) = reflection of X(32403) in X(495)
X(32382) = X(6145)-of-1st-Johnson-Yff-triangle
X(32382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32379, 32383), (388, 32354, 32336)


X(32383) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(2*(5*R^2-2*SW)*(4*R^2-SW)*b*c+2*(4*R^2*SA-S^2-4*SA^2+SA*SW)*R^2+SW*(S^2+3*SA^2-2*SA*SW)) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32383) lies on these lines: {1,32379}, {4,32336}, {5,32403}, {11,6145}, {54,6198}, {55,32391}, {496,32404}, {497,32354}, {973,11399}, {1069,1154}, {1478,32364}, {1479,10082}, {2192,32345}, {2917,13079}, {3057,32371}, {3086,32337}, {3574,11392}, {6284,32330}, {9581,32356}, {9669,32402}, {10066,10274}, {10091,10628}, {10535,32350}, {10798,32335}, {10832,32357}, {10874,32362}, {10896,32369}, {10925,32373}, {10926,32374}, {10950,32381}, {10958,32405}, {10959,32406}, {11376,32331}, {11393,32332}, {11502,32347}, {11906,32372}, {11932,32388}, {11933,32389}, {13898,32399}, {13955,32400}, {17824,18984}, {19029,32342}, {19030,32343}, {22760,32363}

X(32383) = reflection of X(32404) in X(496)
X(32383) = X(6145)-of-2nd-Johnson-Yff-triangle
X(32383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32379, 32382), (497, 32354, 32390)


X(32384) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((3*R^2-SW)*S^2+(5*R^2-2*SW)*(4*R^2-SW)*S+(2*R^2-SW)*(4*R^2+SA-2*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(389).

X(32384) lies on these lines: {6,24}, {371,18400}, {372,23358}, {539,12424}, {590,32351}, {1151,32345}, {1154,10665}, {1209,10897}, {2066,32378}, {2067,32350}, {3068,32346}, {3311,9920}, {3574,5412}, {5410,32333}, {5415,32370}, {6145,6413}, {6200,32401}, {6288,18457}, {6564,32365}, {7691,11417}, {10274,12971}, {10533,32379}, {10628,12375}, {10961,32396}, {11447,32338}, {11462,32339}, {11473,32340}, {11513,32348}, {17819,17824}, {18923,32334}, {19355,32341}, {19436,32386}, {19438,32387}, {21640,32352}

X(32384) = X(21)-of-1st-Kenmotu-diagonals-triangle if ABC is acute
X(32384) = {X(6), X(2917)}-harmonic conjugate of X(32385)


X(32385) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((3*R^2-SW)*S^2-(5*R^2-2*SW)*(4*R^2-SW)*S+(2*R^2-SW)*(4*R^2+SA-2*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(389).

X(32385) lies on these lines: {6,24}, {371,23358}, {372,18400}, {539,12425}, {615,32351}, {1152,32345}, {1154,10666}, {1209,10898}, {3069,32346}, {3312,9920}, {3574,5413}, {5411,32333}, {5414,32378}, {5416,32370}, {6145,6414}, {6288,18459}, {6396,32401}, {6502,32350}, {6565,32365}, {7691,11418}, {10274,12965}, {10534,32379}, {10628,12376}, {10963,32396}, {11448,32338}, {11463,32339}, {11474,32340}, {11514,32348}, {17820,17824}, {18924,32334}, {19356,32341}, {19437,32387}, {19439,32386}, {21641,32352}

X(32385) = X(21)-of-2nd-Kenmotu-diagonals-triangle if ABC is acute
X(32385) = {X(6), X(2917)}-harmonic conjugate of X(32384)


X(32386) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO HATZIPOLAKIS-MOSES

Barycentrics
(SB+SC)*(4*S^4-(R^2*(20*R^2+10*SA-11*SW)-4*SA^2+4*SB*SC+SW^2)*S^2-(2*(2*R^2-SA-SW)*S^2-2*(R^2*(8*R^2-3*SA-8*SW)+SA^2-SB*SC+2*SW^2)*SA)*S+(2*R^2-SW)*(4*R^2+SA-2*SW)*SA*SW) : :

The reciprocal orthologic center of these triangles is X(389).

X(32386) lies on these lines: {54,19186}, {539,19486}, {973,19410}, {1154,18939}, {1209,19428}, {2917,8939}, {3574,19446}, {6288,18462}, {7691,19406}, {9723,32387}, {9920,19418}, {10628,19484}, {12590,12998}, {13021,32345}, {17824,19430}, {18400,18980}, {18414,32365}, {18926,32334}, {19134,32344}, {19358,32341}, {19370,32350}, {19404,32333}, {19412,32338}, {19414,32339}, {19416,32340}, {19420,32346}, {19422,32348}, {19426,32368}, {19432,32370}, {19434,32378}, {19436,32384}, {19439,32385}, {19440,23358}, {19448,32396}, {19450,32397}, {19452,32398}, {19454,32401}, {21642,32352}, {23298,32351}

X(32386) = X(21)-of-Lucas-antipodal-tangents-triangle if ABC is acute


X(32387) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO HATZIPOLAKIS-MOSES

Barycentrics
(SB+SC)*(4*S^4-(R^2*(20*R^2+10*SA-11*SW)-4*SA^2+4*SB*SC+SW^2)*S^2+(2*(2*R^2-SA-SW)*S^2-2*(R^2*(8*R^2-3*SA-8*SW)+SA^2-SB*SC+2*SW^2)*SA)*S+(2*R^2-SW)*(4*R^2+SA-2*SW)*SA*SW) : :

The reciprocal orthologic center of these triangles is X(389).

X(32387) lies on these lines: {54,19187}, {539,19487}, {973,19411}, {1154,18940}, {1209,19429}, {2917,8943}, {3574,19447}, {6288,18463}, {7691,19407}, {9723,32386}, {9920,19419}, {10628,19485}, {12591,12999}, {13022,32345}, {17824,19431}, {18400,18981}, {18415,32365}, {18927,32334}, {19135,32344}, {19359,32341}, {19371,32350}, {19405,32333}, {19413,32338}, {19415,32339}, {19417,32340}, {19421,32346}, {19423,32348}, {19427,32368}, {19433,32370}, {19435,32378}, {19437,32385}, {19438,32384}, {19441,23358}, {19449,32396}, {19451,32397}, {19453,32398}, {19455,32401}, {21643,32352}, {23299,32351}

X(32387) = X(21)-of-Lucas(-1)-antipodal-tangents-triangle if ABC is acute


X(32388) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO HATZIPOLAKIS-MOSES

Barycentrics
(8*(5*SA^2-4*SW^2)*R^4-(SA+SW)*(26*SA-23*SW)*R^2*SW+(4*SA^2+SA*SW-4*SW^2)*SW^2)*S^2+(-2*(48*R^6-4*(2*SA+5*SW)*R^4+(9*SA-5*SW)*SW*R^2-2*(SA-SW)*SW^2)*S^2+8*(2*R^2-SW)*(8*R^2-3*SW)*R^2*SB*SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32388) lies on these lines: {493,6145}, {1154,12426}, {6461,32389}, {6462,32354}, {8188,32356}, {8194,32357}, {8210,32394}, {8212,32369}, {8214,32371}, {8216,32373}, {8218,32374}, {8220,32379}, {8222,32391}, {10628,12894}, {10669,18400}, {10875,32362}, {10945,32380}, {10951,32381}, {11377,32331}, {11394,32332}, {11503,32347}, {11828,32330}, {11840,32335}, {11846,32337}, {11907,32372}, {11930,32382}, {11932,32383}, {11947,32390}, {11949,32402}, {11951,32403}, {11953,32404}, {11955,32405}, {11957,32406}, {13899,32399}, {13956,32400}, {18520,32364}, {18963,32336}, {19031,32342}, {19032,32343}, {22761,32363}

X(32388) = X(6145)-of-Lucas-homothetic-triangle


X(32389) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO HATZIPOLAKIS-MOSES

Barycentrics
(8*(5*SA^2-4*SW^2)*R^4-(SA+SW)*(26*SA-23*SW)*R^2*SW+(4*SA^2+SA*SW-4*SW^2)*SW^2)*S^2-(-2*(48*R^6-4*(2*SA+5*SW)*R^4+(9*SA-5*SW)*SW*R^2-2*(SA-SW)*SW^2)*S^2+8*(2*R^2-SW)*(8*R^2-3*SW)*R^2*SB*SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32389) lies on these lines: {494,6145}, {1154,12427}, {6461,32388}, {6463,32354}, {8189,32356}, {8195,32357}, {8211,32394}, {8213,32369}, {8215,32371}, {8217,32373}, {8219,32374}, {8221,32379}, {8223,32391}, {10628,12895}, {10673,18400}, {10876,32362}, {10946,32380}, {10952,32381}, {11378,32331}, {11395,32332}, {11504,32347}, {11829,32330}, {11841,32335}, {11847,32337}, {11908,32372}, {11931,32382}, {11933,32383}, {11948,32390}, {11950,32402}, {11952,32403}, {11954,32404}, {11956,32405}, {11958,32406}, {13900,32399}, {13957,32400}, {18522,32364}, {18964,32336}, {19033,32342}, {19034,32343}, {22762,32363}

X(32389) = X(6145)-of-Lucas(-1)-homothetic-triangle


X(32390) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO HATZIPOLAKIS-MOSES

Barycentrics    (3*R^2-SW)*(8*R^2-SA-3*SW)*S^2-(SB+SC)*((5*R^2-2*SW)*(4*R^2-SW)*b*c+(12*R^2-5*SW)*(3*R^2-SW)*SA) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32390) lies on these lines: {1,18400}, {3,32404}, {4,32382}, {11,32391}, {12,32369}, {33,32332}, {55,6145}, {56,32330}, {497,32354}, {1154,12428}, {1479,32379}, {1697,32356}, {1837,32371}, {1858,6152}, {2098,32394}, {2646,32331}, {2917,9672}, {3295,32402}, {3574,11429}, {3583,32364}, {4294,32337}, {9659,32345}, {9670,32359}, {10628,12896}, {10799,32335}, {10833,32357}, {10877,32362}, {10927,32373}, {10928,32374}, {10947,32380}, {10953,32381}, {10965,32405}, {10966,32363}, {11909,32372}, {11947,32388}, {11948,32389}, {13901,32399}, {13958,32400}, {19037,32342}, {19038,32343}

X(32390) = reflection of X(32336) in X(1)
X(32390) = X(6145)-of-Mandart-incircle-triangle
X(32390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 32354, 32383), (3295, 32402, 32403)


X(32391) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO HATZIPOLAKIS-MOSES

Barycentrics    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))*(2*a^6-3*(b^2+c^2)*a^4+(b^4-c^4)*(b^2-c^2)) : :
X(32391) = 3*X(2)+X(32354) = X(54)+5*X(17821) = 5*X(54)+X(17846) = 3*X(154)+X(32345) = 5*X(631)-X(32337) = 5*X(631)+X(32359) = 5*X(1656)-X(32402) = 5*X(1698)-X(32356) = X(2917)-5*X(17821) = 5*X(2917)-X(17846) = 3*X(10182)-X(14076) = X(10274)+3*X(11202) = 2*X(10282)+X(10610) = 3*X(11202)-X(23358) = X(17845)+3*X(32395) = 3*X(23041)-X(32344)

The reciprocal orthologic center of these triangles is X(6146).

X(32391) lies on these lines: {1,32371}, {2,6145}, {3,32357}, {4,32330}, {5,5944}, {6,24}, {8,32394}, {11,32390}, {12,32336}, {25,11743}, {30,32364}, {55,32383}, {56,32382}, {64,15080}, {83,32335}, {113,12605}, {141,32348}, {154,7503}, {389,15872}, {394,7691}, {427,32332}, {468,10619}, {498,32403}, {499,32404}, {590,32399}, {615,32400}, {631,32337}, {942,12266}, {958,32363}, {1125,32331}, {1147,1154}, {1209,7542}, {1216,1511}, {1376,32347}, {1493,7575}, {1495,23047}, {1503,20376}, {1650,32372}, {1656,32402}, {1698,32356}, {2781,7512}, {3068,32343}, {3069,32342}, {3096,32362}, {3515,32341}, {3574,3575}, {4550,6759}, {5552,32405}, {5590,32374}, {5591,32373}, {6143,32353}, {6288,6639}, {6696,22352}, {6750,8146}, {7505,12254}, {7506,9920}, {7544,32346}, {7547,26882}, {8222,32388}, {8223,32389}, {8254,31830}, {8542,32367}, {10182,14076}, {10527,32406}, {11597,12606}, {15116,16196}, {16195,19153}, {17845,32395}, {22249,27552}

X(32391) = midpoint of X(i) and X(j) for these lines: {i,j}: {1, 32371}, {3, 32379}, {4, 32330}, {8, 32394}, {54, 2917}, {1650, 32372}, {6145, 32354}, {6759, 32401}, {7691, 17824}, {10274, 23358}, {32337, 32359}, {32347, 32380}, {32363, 32381}
X(32391) = reflection of X(i) in X(j) for these (i,j): (32331, 1125), (32351, 6689), (32369, 5), (32393, 32396)
X(32391) = complement of X(6145)
X(32391) = complementary conjugate of X(1594)
X(32391) = X(6145)-of-medial-triangle
X(32391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 32354, 6145), (6689, 18475, 10610)


X(32392) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO MIDHEIGHT

Barycentrics    (SB+SC)*(2*R^2*S^2+(8*R^2-3*SW)*(8*R^2-SA-SW)*SA) : :
X(32392) = X(64)-3*X(185) = X(64)+3*X(6293) = 5*X(64)-9*X(7729) = 2*X(64)-3*X(31978) = 5*X(185)-3*X(7729) = 3*X(185)-2*X(22967) = 3*X(5562)-5*X(17821) = 5*X(6293)+3*X(7729) = 3*X(6293)+2*X(22967) = 2*X(6293)+X(31978) = 9*X(7729)-10*X(22967) = 6*X(7729)-5*X(31978) = 4*X(15012)-3*X(23332) = 4*X(22967)-3*X(31978)

The reciprocal orthologic center of these triangles is X(32393).

X(32392) lies on these lines: {6,64}, {26,6759}, {369,11761}, {389,32393}, {1216,1511}, {1660,12164}, {2393,5889}, {2929,15139}, {3627,5446}, {5562,17821}, {5878,18917}, {5892,32396}, {5895,16879}, {6053,16252}, {6144,14531}, {6241,18945}, {6247,13382}, {12315,18534}, {13366,32345}, {13367,17824}, {13403,15311}, {15012,23332}, {18400,32358}, {22802,32140}, {25739,32329}

X(32392) = midpoint of X(i) and X(j) for these lines: {i,j}: {185, 6293}, {14531, 17845}
X(32392) = reflection of X(i) in X(j) for these (i,j): (64, 22967), (6247, 13382), (31978, 185)
X(32392) = X(6145)-of-5th-mixtilinear-triangle
X(32392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64, 185, 22967), (64, 22967, 31978)


X(32393) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MIDHEIGHT TO HATZIPOLAKIS-MOSES

Barycentrics    S^4+(2*R^2*(16*R^2-SA-9*SW)+SA^2-SB*SC+2*SW^2)*S^2+(8*R^2-3*SW)^2*SB*SC : :
X(32393) = 3*X(3574)-X(17824) = X(3574)-3*X(32395) = X(6689)+2*X(18383) = X(17824)-9*X(32395) = 3*X(18376)+X(32401) = 3*X(18405)+X(32330) = 3*X(23324)+X(32351) = 3*X(23324)-X(32369) = 3*X(23325)+X(32365)

The reciprocal orthologic center of these triangles is X(32392).

X(32393) lies on these lines: {4,11747}, {5,5944}, {6,3574}, {25,32340}, {26,18376}, {54,18945}, {206,3091}, {389,32392}, {569,32379}, {1154,12235}, {1209,18531}, {2917,7395}, {5449,18569}, {6676,20376}, {6756,7687}, {6816,15435}, {7514,23358}, {7564,18128}, {9969,11743}, {10628,12236}, {11431,18912}, {11572,32332}, {11576,32246}, {12362,32348}, {15077,15801}, {18405,32330}

X(32393) = midpoint of X(i) and X(j) for these lines: {i,j}: {3574, 6145}, {18381, 32364}, {32340, 32345}, {32351, 32369}
X(32393) = reflection of X(i) in X(j) for these (i,j): (32377, 32376), (32391, 32396)
X(32393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6145, 32395, 3574), (23324, 32351, 32369)


X(32394) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO HATZIPOLAKIS-MOSES

Barycentrics
3*a^16-2*(b+c)*a^15-(9*b^2-4*b*c+9*c^2)*a^14+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(4*b^4+4*c^4-(8*b^2-19*b*c+8*c^2)*b*c)*a^12-2*(b+c)*(b^4+c^4-(4*b^2-7*b*c+4*c^2)*b*c)*a^11+(11*b^6+11*c^6-(4*b^4+4*c^4+(7*b^2-12*b*c+7*c^2)*b*c)*b*c)*a^10-2*(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+3*(b-c)^2*b*c)*b*c)*a^9-2*(b^2-b*c+c^2)*(5*b^4+5*c^4+(7*b^2+6*b*c+7*c^2)*b*c)*(b-c)^2*a^8+2*(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*b*c)*a^7-(3*b^6+3*c^6+(4*b^4+4*c^4-(7*b^2-12*b*c+7*c^2)*b*c)*b*c)*(b^2-c^2)^2*a^6+2*(b^2-c^2)^2*(b+c)*(b^6+c^6+(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*b*c)*a^5+(b^2-c^2)^2*(4*b^8+4*c^8-(8*b^6+8*c^6-(3*b^4+3*c^4-2*(2*b^2+3*b*c+2*c^2)*b*c)*b*c)*b*c)*a^4-2*(b^2-c^2)*(b-c)^2*(b^3+c^3)*(b^4-c^4)*(3*b^2+5*b*c+3*c^2)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2+2*(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32394) lies on these lines: {1,6145}, {8,32391}, {55,32363}, {56,32347}, {145,32354}, {517,32330}, {519,32371}, {952,32379}, {1154,9933}, {1482,18400}, {2098,32390}, {2099,32336}, {5603,32369}, {5604,32374}, {5605,32373}, {7718,11743}, {7967,32337}, {7968,32342}, {7969,32343}, {8192,32357}, {8210,32388}, {8211,32389}, {9997,32362}, {10247,32402}, {10628,12898}, {10800,32335}, {10944,32380}, {10950,32381}, {11396,32332}, {11910,32372}, {13902,32399}, {13959,32400}, {18525,32364}

X(32394) = midpoint of X(145) and X(32354)
X(32394) = reflection of X(i) in X(j) for these (i,j): (8, 32391), (6145, 1), (18525, 32364), (32356, 32331)
X(32394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 32356, 32331), (32331, 32356, 6145), (32405, 32406, 6145)


X(32395) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHOCENTROIDAL TO HATZIPOLAKIS-MOSES

Barycentrics    (R^2*(24*R^2+SA-15*SW)+2*SW^2)*S^2+2*(8*R^2-3*SW)*(3*R^2-SW)*SB*SC : :
X(32395) = X(3)+2*X(32365) = 2*X(4)+X(32345) = X(4)+2*X(32351) = 4*X(5)-X(2917) = X(20)-4*X(20376) = X(54)+2*X(32369) = X(382)+2*X(32401) = X(1498)-4*X(32364) = 5*X(1656)-2*X(23358) = 5*X(3091)+X(32346) = 4*X(3574)-X(17824) = X(3574)+2*X(32393) = 7*X(3851)-X(9920) = X(5889)-4*X(11262) = 4*X(6689)-X(32330) = X(10274)+2*X(18383) = 2*X(10274)+X(32402) = X(17824)+8*X(32393) = 4*X(18383)-X(32402) = X(32345)-4*X(32351)

The reciprocal orthologic center of these triangles is X(1986).

X(32395) lies on these lines: {3,32365}, {4,10117}, {5,2917}, {6,3574}, {20,20376}, {54,7547}, {154,381}, {195,265}, {382,32401}, {567,10274}, {568,10628}, {1154,14852}, {1498,32364}, {1503,7565}, {1656,23358}, {1853,5890}, {1986,32329}, {2931,10224}, {3091,15582}, {3851,9920}, {5889,11262}, {6288,17814}, {6689,32330}, {7564,32046}, {7730,14644}, {10516,11188}, {10539,11597}, {10895,32350}, {10896,32378}, {12022,23324}, {12307,14076}, {14389,32354}, {15069,32368}, {17835,20299}, {17845,32391}

X(32395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 32351, 32345), (3574, 6145, 17824), (3574, 32393, 6145)


X(32396) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO HATZIPOLAKIS-MOSES

Barycentrics    (24*R^2+SA-10*SW)*S^2+(8*R^2-3*SW)*SB*SC : :
X(32396) = 3*X(2)+X(3574) = 9*X(2)-X(7691) = 3*X(5)+X(10610) = 5*X(5)-X(22804) = X(195)+3*X(1209) = X(195)+15*X(1656) = 5*X(195)+3*X(3519) = X(195)-3*X(12242) = 7*X(195)-3*X(13431) = 19*X(195)-3*X(13432) = X(1209)-5*X(1656) = 5*X(1209)-X(3519) = 7*X(1209)+X(13431) = 19*X(1209)+X(13432) = 3*X(1209)-X(15605) = 5*X(1656)+X(12242) = 15*X(1656)-X(15605) = X(3519)+5*X(12242) = 7*X(3519)+5*X(13431) = 3*X(3519)-5*X(15605) = 3*X(3574)+X(7691) = 3*X(6689)-X(10610) = 5*X(6689)+X(22804) = X(7691)-3*X(32348) = 5*X(10610)+3*X(22804) = 7*X(12242)-X(13431) = 19*X(12242)-X(13432)

The reciprocal orthologic center of these triangles is X(389).

X(32396) lies on these lines: {2,3574}, {5,5944}, {6,17}, {54,3090}, {182,32379}, {373,32352}, {485,13986}, {486,8995}, {539,547}, {973,5943}, {1154,3628}, {1493,16254}, {2888,7486}, {2917,5020}, {3091,32340}, {3357,4846}, {3549,19130}, {3818,14530}, {5055,6288}, {5056,10619}, {5071,12254}, {5449,32068}, {5576,29012}, {5892,32392}, {5893,20376}, {5972,14788}, {6642,23358}, {6723,9826}, {7392,32346}, {7569,7592}, {8889,13347}, {9813,32368}, {9816,32370}, {9817,32378}, {9818,32401}, {9920,11484}, {10109,20584}, {10112,14389}, {10115,10170}, {10175,12266}, {10601,32341}, {10643,32397}, {10644,32398}, {10961,32384}, {10963,32385}, {11284,32333}, {11451,32338}, {11465,32339}, {11479,32345}, {11597,23515}, {11808,12363}, {12046,15350}, {12900,15088}, {13469,25339}, {15699,21230}, {17824,17825}, {18420,32365}, {18928,32334}, {19137,32344}, {19372,32350}, {19448,32386}, {19449,32387}

X(32396) = midpoint of X(i) and X(j) for these lines: {i,j}: {5, 6689}, {195, 15605}, {1209, 12242}, {3574, 32348}, {8254, 13565}, {11808, 12363}, {32391, 32393}
X(32396) = complement of X(32348)
X(32396) = X(21)-of-submedial-triangle if ABC is acute
X(32396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3574, 32348), (195, 1209, 15605), (12242, 15605, 195)


X(32397) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(sqrt(3)*(5*R^2-2*SW)*(4*R^2-SW)*S+3*(3*R^2-SW)*S^2+3*SA*(2*R^2-SW)*(4*R^2+SA-2*SW)) : :

The reciprocal orthologic center of these triangles is X(389).

X(32397) lies on these lines: {6,24}, {15,18400}, {16,23358}, {539,10659}, {1154,10661}, {1209,8837}, {3166,10635}, {3574,10641}, {6288,18468}, {7051,32350}, {7691,11420}, {9920,11485}, {10274,10678}, {10628,10657}, {10636,32370}, {10638,32378}, {10643,32396}, {10645,32401}, {11408,32333}, {11452,32338}, {11466,32339}, {11475,32340}, {11480,32345}, {11488,32346}, {11515,32348}, {16808,32365}, {17824,17826}, {18929,32334}, {19363,32341}, {19450,32386}, {19451,32387}, {21647,32352}, {23302,32351}, {30402,32379}

X(32397) = X(21)-of-inner-tri-equilateral-triangle if ABC is acute
X(32397) = {X(6), X(2917)}-harmonic conjugate of X(32398)


X(32398) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(-sqrt(3)*(5*R^2-2*SW)*(4*R^2-SW)*S+3*(3*R^2-SW)*S^2+3*SA*(2*R^2-SW)*(4*R^2+SA-2*SW)) : :

The reciprocal orthologic center of these triangles is X(389).

X(32398) lies on these lines: {6,24}, {15,23358}, {16,18400}, {539,10660}, {1154,10662}, {1209,8839}, {1250,32378}, {3165,10634}, {3574,10642}, {6288,18470}, {7691,11421}, {9920,11486}, {10274,10677}, {10628,10658}, {10637,32370}, {10644,32396}, {10646,32401}, {11409,32333}, {11453,32338}, {11467,32339}, {11476,32340}, {11481,32345}, {11489,32346}, {11516,32348}, {16809,32365}, {17824,17827}, {18930,32334}, {19364,32341}, {19373,32350}, {19452,32386}, {19453,32387}, {21648,32352}, {23303,32351}, {30403,32379}

X(32398) = X(21)-of-outer-tri-equilateral-triangle if ABC is acute
X(32398) = {X(6), X(2917)}-harmonic conjugate of X(32397)


X(32399) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO HATZIPOLAKIS-MOSES

Barycentrics    S^4+(R^2*(48*R^2-3*SA-29*SW)+SA^2-SB*SC+4*SW^2)*S^2+2*(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146).

X(32399) lies on these lines: {2,32343}, {6,32400}, {371,32369}, {590,32391}, {1154,13909}, {3068,6145}, {7585,32342}, {8972,32354}, {8974,32373}, {8976,32379}, {8981,18400}, {9540,32330}, {10628,13915}, {13883,32331}, {13884,32332}, {13885,32335}, {13886,32337}, {13887,32347}, {13888,32356}, {13889,32357}, {13893,32371}, {13894,32372}, {13895,32380}, {13897,32382}, {13898,32383}, {13899,32388}, {13900,32389}, {13901,32390}, {13902,32394}, {13903,32402}, {13904,32403}, {13905,32404}, {13906,32405}, {13907,32406}, {18538,32364}, {18965,32336}, {22763,32363}

X(32399) = X(6145)-of-3rd-tri-squares-central-triangle


X(32400) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO HATZIPOLAKIS-MOSES

Barycentrics    S^4+(R^2*(48*R^2-3*SA-29*SW)+SA^2-SB*SC+4*SW^2)*S^2-2*(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S+(2*R^2-SW)*(8*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146).

X(32400) lies on these lines: {2,32342}, {6,32399}, {372,32369}, {615,32391}, {1154,13970}, {3069,6145}, {7586,32343}, {10628,13979}, {13935,32330}, {13936,32331}, {13937,32332}, {13938,32335}, {13939,32337}, {13940,32347}, {13941,32354}, {13942,32356}, {13943,32357}, {13946,32362}, {13947,32371}, {13948,32372}, {13949,32373}, {13950,32374}, {13951,32379}, {13952,32380}, {13953,32381}, {13954,32382}, {13955,32383}, {13956,32388}, {13957,32389}, {13958,32390}, {13959,32394}, {13961,32402}, {13962,32403}, {13963,32404}, {13964,32405}, {13965,32406}, {13966,18400}, {18762,32364}, {18966,32336}, {22764,32363}

X(32400) = X(6145)-of-4th-tri-squares-central-triangle


X(32401) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*((3*R^2-SA-SW)*S^2-((52*R^2-2*SA-31*SW)*R^2+SA^2-SB*SC+4*SW^2)*SA) : :
X(32401) = 3*X(3)-X(2917) = 5*X(3)-X(9920) = X(195)+5*X(8567) = 3*X(376)+X(32346) = X(382)-3*X(32395) = 3*X(1853)-X(32402) = 5*X(2917)-3*X(9920) = 2*X(2917)-3*X(23358) = X(2917)+3*X(32345) = X(3357)+2*X(10610) = X(5894)+2*X(8254) = 4*X(6689)-X(22802) = 3*X(10606)+X(17824) = 2*X(15579)+X(32367) = 3*X(18376)-4*X(32393) = 3*X(23325)-2*X(32369) = 4*X(32184)-X(32196)

The reciprocal orthologic center of these triangles is X(389).

X(32401) lies on these lines: {3,161}, {5,13289}, {24,32340}, {26,18376}, {30,32351}, {35,32350}, {36,32378}, {49,19402}, {54,74}, {64,32349}, {186,11572}, {195,8567}, {376,32346}, {378,3574}, {382,32395}, {511,32368}, {539,9938}, {973,11438}, {1154,7689}, {1204,12234}, {1658,23325}, {2071,7691}, {2777,3521}, {3098,15578}, {3357,10274}, {3516,12242}, {4550,6759}, {5092,32344}, {5894,8254}, {6000,15062}, {6200,32384}, {6293,14805}, {6396,32385}, {6689,7526}, {6699,22962}, {7488,18383}, {7688,32370}, {9818,32396}, {9977,11802}, {10226,12901}, {10263,11262}, {10605,32341}, {10606,17824}, {10645,32397}, {10646,32398}, {11410,32333}, {11454,32338}, {11457,12254}, {11468,32339}, {11793,22955}, {12038,15132}, {12107,23324}, {12893,13561}, {13403,19457}, {14216,32354}, {15331,23332}, {15579,32367}, {16003,25564}, {18128,32412}, {18931,32334}, {19454,32386}, {19455,32387}, {21230,22978}, {21663,32352}, {32184,32196}

X(32401) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 32345}, {3357, 10274}, {14216, 32354}
X(32401) = reflection of X(i) in X(j) for these (i,j): (5, 20376), (6145, 20299), (6288, 14076), (6759, 32391), (10263, 11262), (10274, 10610), (14076, 25563), (22802, 32364), (23358, 3), (32344, 5092), (32364, 6689), (32365, 32351), (32412, 18128)
X(32401) = circumtangential-isogonal conjugate of X(7488)
X(32401) = X(21)-of-Trinh-triangle if ABC is acute
X(32401) = {X(11750), X(20299)}-harmonic conjugate of X(18381)


X(32402) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO HATZIPOLAKIS-MOSES

Barycentrics    (R^2*(4*R^2+6*SA-7*SW)+2*(SB+SC)*SW)*S^2-(R^2*(52*R^2-41*SW)+8*SW^2)*SB*SC : :
X(32402) = 3*X(3)-2*X(32330) = 3*X(381)-4*X(32369) = 3*X(381)-2*X(32379) = 5*X(1656)-4*X(32391) = 3*X(1853)-2*X(32401) = 5*X(3843)-2*X(32359) = 5*X(3843)-4*X(32364) = 3*X(5790)-2*X(32371) = 3*X(10246)-4*X(32331) = 3*X(10247)-2*X(32394) = 2*X(10274)-3*X(32395) = 3*X(11911)-2*X(32372) = X(12315)-4*X(32340) = X(17824)-3*X(18405) = 4*X(18383)-3*X(32395) = 3*X(18405)-2*X(32365)

The reciprocal orthologic center of these triangles is X(6146).

X(32402) lies on these lines: {3,161}, {4,19362}, {5,32354}, {30,32337}, {54,7507}, {195,31724}, {381,32369}, {517,32356}, {567,10274}, {973,18494}, {999,32336}, {1154,12429}, {1594,12254}, {1598,32332}, {1656,32391}, {2888,12225}, {3295,32390}, {3574,11426}, {3843,32359}, {5790,32371}, {6193,12319}, {6241,7730}, {6243,10628}, {6417,32343}, {6418,32342}, {7517,32357}, {9301,32362}, {9654,32382}, {9669,32383}, {10246,32331}, {10247,32394}, {10620,14216}, {11842,32335}, {11849,32347}, {11911,32372}, {11916,32373}, {11917,32374}, {11928,32380}, {11929,32381}, {11949,32388}, {11950,32389}, {12000,32405}, {12001,32406}, {12315,32340}, {13903,32399}, {13961,32400}, {17824,18405}, {22765,32363}

X(32402) = reflection of X(i) in X(j) for these (i,j): (3, 6145), (9920, 6288), (10274, 18383), (17824, 32365), (17845, 23358), (32345, 18381), (32354, 5), (32359, 32364), (32379, 32369)
X(32402) = X(6145)-of-X3-ABC-reflections-triangle
X(32402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10274, 18383, 32395), (32369, 32379, 381)


X(32403) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO HATZIPOLAKIS-MOSES

Barycentrics    (3*R^2-SW)*(8*R^2-SA-3*SW)*S^2+(SB+SC)*((5*R^2-2*SW)*(4*R^2-SW)*b*c-(2*R^2-SW)*(8*R^2-3*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32403) lies on these lines: {1,6145}, {3,32336}, {5,32383}, {12,32379}, {35,32330}, {55,12946}, {388,32337}, {495,32382}, {498,32391}, {973,11392}, {1154,10055}, {1479,32369}, {3295,32390}, {3299,32342}, {3301,32343}, {6288,18455}, {10037,32357}, {10038,32362}, {10039,32371}, {10040,32373}, {10041,32374}, {10082,32351}, {10523,32380}, {10628,12903}, {10801,32335}, {10895,32364}, {10954,32381}, {11398,32332}, {11507,32347}, {11912,32372}, {11951,32388}, {11952,32389}, {13904,32399}, {13962,32400}, {22766,32363}

X(32403) = midpoint of X(6145) and X(32405)
X(32403) = reflection of X(32382) in X(495)
X(32403) = X(6145)-of-inner-Yff-triangle
X(32403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6145, 32404), (3295, 32402, 32390)


X(32404) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO HATZIPOLAKIS-MOSES

Barycentrics    (3*R^2-SW)*(8*R^2-SA-3*SW)*S^2+(SB+SC)*(-(5*R^2-2*SW)*(4*R^2-SW)*b*c-(2*R^2-SW)*(8*R^2-3*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(6146).

X(32404) lies on these lines: {1,6145}, {3,32390}, {5,32382}, {11,32379}, {36,32330}, {56,12956}, {496,32383}, {497,32337}, {499,32391}, {973,11393}, {999,32336}, {1154,10071}, {1478,32369}, {1737,32371}, {3086,32354}, {3299,32343}, {3301,32342}, {6288,18447}, {10046,32357}, {10047,32362}, {10048,32373}, {10049,32374}, {10066,32351}, {10523,32381}, {10628,12904}, {10802,32335}, {10896,32364}, {10948,32380}, {11399,32332}, {11508,32347}, {11913,32372}, {11953,32388}, {11954,32389}, {13905,32399}, {13963,32400}, {22767,32363}

X(32404) = midpoint of X(6145) and X(32406)
X(32404) = reflection of X(32383) in X(496)
X(32404) = X(6145)-of-outer-Yff-triangle
X(32404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6145, 32403), (999, 32402, 32336)


X(32405) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO HATZIPOLAKIS-MOSES

Barycentrics
a^19-(b+c)*a^18-4*(b-c)^2*a^17+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^16+(5*b^4+5*c^4-(24*b^2-19*b*c+24*c^2)*b*c)*a^15-(b+c)*(5*b^2-3*b*c+5*c^2)*(b^2-3*b*c+c^2)*a^14-(b^6+c^6-12*(b^4+c^4-2*(b-c)^2*b*c)*b*c)*a^13+(b+c)*(b^6+c^6-2*(4*b^4+4*c^4-(12*b^2-19*b*c+12*c^2)*b*c)*b*c)*a^12-(b^8+c^8-2*(12*b^6+12*c^6-(5*b^4+5*c^4+(8*b^2-15*b*c+8*c^2)*b*c)*b*c)*b*c)*a^11+(b+c)*(b^8+c^8-2*(11*b^6+11*c^6-(5*b^4+5*c^4+(7*b^2-15*b*c+7*c^2)*b*c)*b*c)*b*c)*a^10-(b^8+c^8+2*(11*b^6+11*c^6+(3*b^4+3*c^4+(7*b^2+17*b*c+7*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^9+(b^2-c^2)*(b-c)*(b^8+c^8+2*(11*b^6+11*c^6+(3*b^4+3*c^4+(5*b^2+13*b*c+5*c^2)*b*c)*b*c)*b*c)*a^8-(b^8+c^8+(8*b^6+8*c^6+(11*b^4+11*c^4-16*(b-c)^2*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^7+(b^2-c^2)^2*(b+c)*(b^8+c^8+(6*b^6+6*c^6+(11*b^4+11*c^4-2*(7*b^2-16*b*c+7*c^2)*b*c)*b*c)*b*c)*a^6+(b^2-c^2)^2*(b+c)*(b^3+c^3)*(5*b^6+5*c^6-(b^4+c^4+(11*b^2-6*b*c+11*c^2)*b*c)*b*c)*a^5-(b^2-c^2)^3*(b-c)*(5*b^8+5*c^8+(18*b^6+18*c^6+(19*b^4+19*c^4+2*(13*b^2+16*b*c+13*c^2)*b*c)*b*c)*b*c)*a^4-4*(b^4-c^4)*(b^2-c^2)^3*(b^6+c^6-(2*b^4+2*c^4+(b^2+c^2)*b*c)*b*c)*a^3+2*(b^2-c^2)^6*(b+c)*(b^2+c^2)*(2*b^2-b*c+2*c^2)*a^2+(b^2-4*b*c+c^2)*(b^2+c^2)^2*(b^2-c^2)^6*a-(b^2-c^2)^7*(b-c)*(b^2+c^2)^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32405) lies on these lines: {1,6145}, {12,32380}, {1154,12430}, {5552,32391}, {10528,32354}, {10531,32369}, {10628,12905}, {10679,18400}, {10803,32335}, {10805,32337}, {10834,32357}, {10878,32362}, {10915,32371}, {10929,32373}, {10930,32374}, {10942,32379}, {10955,32381}, {10956,32382}, {10958,32383}, {10965,32390}, {11248,32330}, {11400,32332}, {11509,32336}, {11914,32372}, {11955,32388}, {11956,32389}, {12000,32402}, {13906,32399}, {13964,32400}, {18542,32364}, {19047,32342}, {19048,32343}, {22768,32363}

X(32405) = reflection of X(6145) in X(32403)
X(32405) = X(6145)-of-inner-Yff-tangents-triangle
X(32405) = {X(6145), X(32394)}-harmonic conjugate of X(32406)


X(32406) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO HATZIPOLAKIS-MOSES

Barycentrics
a^19-(b+c)*a^18-4*(b^2+b*c+c^2)*a^17+2*(b+c)*(2*b^2+b*c+2*c^2)*a^16+(5*b^4+5*c^4+3*(4*b^2+b*c+4*c^2)*b*c)*a^15-(b+c)*(5*b^4+5*c^4+3*(2*b^2+b*c+2*c^2)*b*c)*a^14-(b^6+c^6+4*(b^4+c^4-(2*b^2-7*b*c+2*c^2)*b*c)*b*c)*a^13+(b+c)*(b^6+c^6-2*(4*b^2-9*b*c+4*c^2)*b^2*c^2)*a^12-(b^8+c^8+2*(10*b^6+10*c^6-3*(b^4+c^4+(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*b*c)*a^11+(b+c)*(b^8+c^8+2*(9*b^6+9*c^6-(3*b^4+3*c^4+(5*b^2-9*b*c+5*c^2)*b*c)*b*c)*b*c)*a^10-(b^8+c^8-2*(9*b^6+9*c^6+(5*b^4+5*c^4+3*(3*b^2+5*b*c+3*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^9+(b^4-c^4)*(b-c)*(b^6+c^6-(18*b^4+18*c^4+(11*b^2-4*b*c+11*c^2)*b*c)*b*c)*a^8-(b^8+c^8-(4*b^6+4*c^6+(5*b^4+5*c^4-4*(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^7+(b^2-c^2)^2*(b+c)*(b^8+c^8-(2*b^2-3*b*c+2*c^2)*(b^4+c^4+4*(b^2+c^2)*b*c)*b*c)*a^6+(b^4-c^4)*(b^2-c^2)*(5*b^8+5*c^8-3*(4*b^6+4*c^6-5*(b^4+c^4)*b*c)*b*c)*a^5-(b^2-c^2)^3*(b-c)*(5*b^8+5*c^8-3*(2*b^2-b*c+2*c^2)*(b^4+c^4)*b*c)*a^4-4*(b-c)*(b^3+c^3)*(b^4-c^4)^3*a^3+2*(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^4+2*c^4-(b-c)^2*b*c)*a^2+(b^2+c^2)^3*(b^2-c^2)^6*a-(b^2-c^2)^7*(b-c)*(b^2+c^2)^2 : :

The reciprocal orthologic center of these triangles is X(6146).

X(32406) lies on these lines: {1,6145}, {11,32381}, {1154,12431}, {10527,32391}, {10529,32354}, {10532,32369}, {10628,12906}, {10680,18400}, {10804,32335}, {10806,32337}, {10835,32357}, {10879,32362}, {10916,32371}, {10931,32373}, {10932,32374}, {10943,32379}, {10949,32380}, {10957,32382}, {10959,32383}, {10966,32363}, {11249,32330}, {11401,32332}, {11510,32347}, {11915,32372}, {11957,32388}, {11958,32389}, {12001,32402}, {13907,32399}, {13965,32400}, {18544,32364}, {18967,32336}, {19049,32342}, {19050,32343}

X(32406) = reflection of X(6145) in X(32404)
X(32406) = X(6145)-of-outer-Yff-tangents-triangle
X(32406) = {X(6145), X(32394)}-harmonic conjugate of X(32405)


X(32407) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO HATZIPOLAKIS-MOSES

Barycentrics    SA*(SB-SC)*((18*R^2-6*SW)*S^4+(20*(3*SA-4*SW)*R^4+(9*SA^2-48*SA*SW+49*SW^2)*R^2-(3*SA^2-9*SA*SW+7*SW^2)*SW)*S^2-(8*R^2-3*SW)*SB*SC*SW^2) : :

The reciprocal parallelogic center of these triangles is X(6146).

X(32407) lies on these lines: {351,6368}, {1510,13223}

X(32407) = reflection of X(32408) in X(351)
X(32407) = X(6145)-of-1st-Parry-triangle


X(32408) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO HATZIPOLAKIS-MOSES

Barycentrics    SA*(SB-SC)*((-18*R^2+6*SW)*S^4+(20*(3*SA-4*SW)*R^4-(9*SA^2+30*SA*SW-55*SW^2)*R^2+3*(SA^2+SA*SW-3*SW^2)*SW)*S^2-(2*R^2-SW)*SB*SC*SW^2) : :

The reciprocal parallelogic center of these triangles is X(6146).

X(32408) lies on these lines: {110,16039}, {351,6368}, {1510,13224}

X(32408) = reflection of X(32407) in X(351)
X(32408) = X(6145)-of-2nd-Parry-triangle


X(32409) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO HATZIPOLAKIS-MOSES

Barycentrics    (SB+SC)*(S^2+SB*SC)*(3*S^4-(R^2*(16*R^2-13*SW)+SA^2+3*SW^2)*S^2+(3*R^2-SW)*(4*R^2-SW)*SA^2) : :
X(32409) = 3*X(51)-X(16337) = 3*X(568)+X(19552) = X(1157)-5*X(3567) = 3*X(5946)-X(6150)

The reciprocal cyclologic center of these triangles is X(389).

X(32409) lies on these lines: {5,51}, {54,3432}, {568,19552}, {1157,3567}, {5462,10615}, {5946,6150}, {11077,15401}, {18400,32410}

X(32409) = midpoint of X(52) and X(16336)
X(32409) = reflection of X(10615) in X(5462)
X(32409) = complement of X(16337) wrt orthic triangle
X(32409) = X(1325)-of-2nd-anti-Conway-triangle if ABC is acute


X(32410) = CYCLOLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO ORTHIC

Barycentrics    (4*S^2-R^2*(24*R^2+5*SA-19*SW)+2*SA^2-2*SB*SC-4*SW^2)*((2*R^2-SA)*S^2+(3*R^2-SW)*SB*SC) : :

The reciprocal cyclologic center of these triangles is X(1986).

X(32410) lies on these lines: {5,49}, {137,12241}, {12370,25150}, {15959,19467}, {18400,32409}

X(32410) = reflection of X(137) in X(12241)


X(32411) = EULEROLOGIC CENTER OF THESE TRIANGLES: HATZIPOLAKIS-MOSES TO 2nd ANTI-CONWAY

Barycentrics    (SB+SC)*((13*R^2-3*SW)*S^2+(4*R^2-SW)*(3*R^2+SA-2*SW)*SA) : :
X(32411) = 3*X(51)-X(403) = X(186)-5*X(3567) = 3*X(568)+X(18403) = X(2070)-9*X(13321) = X(2071)+3*X(3060) = 3*X(5946)-X(15646) = X(10297)+2*X(16625)

X(32411) lies on these lines: {6,2070}, {30,143}, {51,403}, {52,2072}, {186,578}, {468,973}, {511,10257}, {568,18390}, {1112,6000}, {1154,11746}, {1986,13851}, {2071,3060}, {5899,11432}, {5946,11430}, {6746,10110}, {7687,12236}, {7722,23043}, {9786,18859}, {10114,11557}, {10297,13376}, {10540,12227}, {11262,11745}, {11431,20063}, {11563,12233}, {11807,15311}, {15473,18400}, {15644,16976}

X(32411) = midpoint of X(i) and X(j) for these lines: {i,j}: {52, 2072}, {1986, 13851}, {13376, 16625}
X(32411) = reflection of X(i) in X(j) for these (i,j): (10151, 10110), (10297, 13376), (15644, 16976)
X(32411) = {X(12241), X(16881)}-harmonic conjugate of X(389)


X(32412) = ORTHOCENTER OF THE HATZIPOLAKIS-MOSES TRIANGLE

Barycentrics    (32*R^6-2*(31*SA-9*SW)*R^4+(39*SA-22*SW)*SW*R^2-2*(3*SA-2*SW)*SW^2)*S^2+(8*R^2-3*SW)*(R^2*(8*R^2-9*SW)+2*SW^2)*SB*SC : :
X(32412) = 3*X(11225)-2*X(11262) = 3*X(11225)-4*X(32413)

X(32412) lies on these lines: {6,3574}, {539,9932}, {1511,21230}, {5449,10274}, {6102,10116}, {6689,14076}, {11225,11262}, {15033,32337}, {18128,32401}

X(32412) = reflection of X(i) in X(j) for these (i,j): (6145, 32376), (11262, 32413), (32401, 18128)
X(32412) = {X(11262), X(32413)}-harmonic conjugate of X(11225)


X(32413) = X(5)-OF-HATZIPOLAKIS-MOSES TRIANGLE

Barycentrics    (96*R^6-(81*SA+23*SW)*R^4+4*(13*SA-4*SW)*SW*R^2-4*(2*SA-SW)*SW^2)*S^2+2*(R^2*(16*R^4-29*R^2*SW+14*SW^2)-2*SW^3)*SB*SC : :
X(32413) = 3*X(11225)-X(11262) = 3*X(11225)+X(32412) = 3*X(11245)-X(20376)

X(32413) lies on these lines: {11225,11262}, {11245,20376}, {12242,15118}, {18400,32165}

X(32413) = midpoint of X(11262) and X(32412)
X(32413) = {X(11225), X(32412)}-harmonic conjugate of X(11262)


X(32414) = X(2)X(1495)∩X(5)X(26614)

Barycentrics    24*S^4+(3*SA-4*SW)*SW*S^2+9*SB*SC*SW^2 : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29004.

X(32414) lies on these lines: {2, 1495}, {5, 26614}, {30, 1153}, {381, 5215}, {511, 7610}, {542, 9771}, {575, 6055}, {3830, 8588}, {6054, 10486}, {7603, 14830}, {13355, 21358}, {15597, 19924}, {23053, 31670}


X(32415) = X(2)X(32401)∩X(140)X(6000)

Barycentrics    (R^2*(188*R^2+SA-96*SW)+12*SW^2)*S^2-(R^2*(52*R^2-31*SW)+4*SW^2)*SB*SC : :
X(32415) = 15*X(15694)+X(32321)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29004

X(32415) lies on these lines: {2, 32401}, {140, 6000}, {2777, 5498}, {6143, 13289}, {6288, 11202}, {10018, 18383}, {10125, 18400}, {10182, 14076}, {13363, 15426}, {15694, 32321}, {16532, 32351}


X(32416) = X(511)X(25488)∩X(3589)X(5066)

Barycentrics    4*S^4-(3*R^2*(3*SA-4*SW)-4*SA^2+4*SB*SC+12*SW^2)*S^2-27*R^2*SB*SC*SW : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29004.

X(32416) lies on these lines: {511, 25488}, {3589, 5066}, {3818, 5012}, {6676, 10219}, {7605, 32305}, {14389, 25561}


X(32417) = X(3)X(31378)∩X(4)X(5627)

Barycentrics    5*S^4+3*(3*R^2*(12*R^2-3*SA-4*SW)+2*SA^2-SB*SC+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SB*SC : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29004.

X(32417) lies on these lines: {3, 31378}, {4, 5627}, {20, 14480}, {30, 511}, {74, 3258}, {107, 18809}, {113, 22104}, {133, 1552}, {146, 476}, {381, 18279}, {477, 1138}, {550, 18285}, {1304, 5667}, {3154, 20417}, {3233, 6053}, {5502, 20128}, {7471, 15063}, {7687, 12079}, {7728, 14993}, {10096, 16244}, {10620, 20957}, {10745, 16177}, {12041, 31379}, {13382, 14895}, {14508, 14731}, {14611, 16163}, {14934, 16111}, {15054, 17511}, {16978, 21649}

X(32417) = isogonal conjugate of X(32418}


X(32418) = X(107)X(12121)∩X(476)X(16163)

Barycentrics    (SB+SC)*(5*S^4+3*(3*R^2*(12*R^2-3*SB-4*SW)+2*SB^2-SA*SC+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SC*SA)*(5*S^4+3*(3*R^2*(12*R^2-3*SC-4*SW)+2*SC^2-SA*SB+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SA*SB) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29004.

X(32418) lies on the circumcircle and these lines: {107, 12121}, {476, 16163}, {1304, 1511}, {2132, 15035}

X(32418) = isogonal conjugate of X(32417}


X(32419) = INTERSECTION OF THE EULER LINES OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES, 4th TRI-SQUARES, AND INNER-VECTEN

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

Centers X(32419) to X(32441) were contributed by César Lozada, May 5, 2019.

X(32419) lies on these lines: {2,371}, {3,591}, {4,5861}, {6,13644}, {13,22607}, {14,22608}, {20,26288}, {30,511}, {69,6561}, {187,13989}, {193,6560}, {298,22601}, {299,22603}, {323,12376}, {372,489}, {376,5860}, {381,1991}, {395,6307}, {396,6306}, {428,12147}, {485,12322}, {488,13712}, {491,6565}, {492,6200}, {551,12268}, {615,9675}, {640,3071}, {641,1151}, {1161,13748}, {1270,9541}, {1271,23259}, {1327,2996}, {1328,5491}, {1504,5309}, {1651,12799}, {2462,5182}, {2858,32422}, {3058,13081}, {3068,13650}, {3069,13770}, {3102,8356}, {3241,7980}, {3543,12296}, {3592,11313}, {3679,9906}, {3830,22809}, {3845,22596}, {4421,12343}, {5434,18989}, {5591,23273}, {5875,14230}, {6118,8981}, {6231,9892}, {6278,13710}, {6289,12313}, {6337,13701}, {6419,7389}, {6420,11293}, {6425,11315}, {6459,26619}, {6463,13798}, {7811,9986}, {7833,9991}, {7837,22613}, {7840,22501}, {9732,9766}, {9742,13681}, {9758,9770}, {9909,9921}, {10008,13684}, {10056,10067}, {10072,10083}, {10653,22612}, {10654,22611}, {11194,22595}, {11207,12484}, {11208,12485}, {11235,12928}, {11236,12938}, {11237,12948}, {11238,12958}, {11239,13132}, {11240,13133}, {12150,12210}, {12152,13002}, {12153,13003}, {12222,22644}, {12237,21849}, {12323,22615}, {12962,13846}, {12963,13821}, {12972,14070}, {13926,13927}, {16267,22602}, {16268,22604}, {19053,19104}, {19054,19105}, {19493,32060}, {21653,21969}, {23267,26339}

X(32419) = isogonal conjugate of X(32420)


X(32420) = ISOGONAL CONJUGATE OF X(32419)

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

X(32420) lies on the circumcircle and these lines: {99,491}, {110,372}, {112,5412}, {485,925}, {691,2460}, {1307,6396}, {2858,32421}, {2872,9142}, {3565,6200}

X(32420) = isogonal conjugate of X(32419)
X(32420) = reflection of X(98) in the line X(3)X(9892)


X(32421) = INTERSECTION OF THE EULER LINES OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES, 3rd TRI-SQUARES, AND OUTER-VECTEN

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

X(32421) lies on these lines: {2,372}, {3,1991}, {4,5860}, {6,13763}, {13,22636}, {14,22637}, {20,26289}, {30,511}, {69,6560}, {187,8997}, {193,6561}, {298,22630}, {299,22632}, {323,12375}, {371,490}, {376,5861}, {381,591}, {395,6303}, {396,6302}, {428,12148}, {486,12323}, {487,13835}, {491,6396}, {492,6564}, {551,12269}, {639,3070}, {642,1152}, {1160,13749}, {1270,23249}, {1327,5490}, {1328,2996}, {1505,5309}, {1651,12800}, {2461,5182}, {2858,32420}, {3058,13082}, {3068,13651}, {3069,13771}, {3103,8356}, {3241,7981}, {3543,12297}, {3594,11314}, {3679,9907}, {3830,22810}, {3845,22625}, {4421,12344}, {5434,18988}, {5590,23267}, {5874,14233}, {6119,13966}, {6230,9894}, {6281,13830}, {6290,12314}, {6337,13821}, {6419,11294}, {6420,7388}, {6426,11316}, {6460,26620}, {6462,13678}, {7811,9987}, {7833,9992}, {7837,22642}, {7840,22502}, {8989,12973}, {8996,9909}, {9733,9766}, {9742,13801}, {9757,9770}, {10008,13804}, {10056,10068}, {10072,10084}, {10653,22641}, {10654,22640}, {11194,22624}, {11207,12486}, {11208,12487}, {11235,12929}, {11236,12939}, {11237,12949}, {11238,12959}, {11239,13134}, {11240,13135}, {12150,12211}, {12152,13004}, {12153,13005}, {12221,22615}, {12238,21849}, {12322,22644}, {12968,13701}, {12969,13847}, {13873,13874}, {16267,22631}, {16268,22633}, {19053,19102}, {19054,19103}, {19492,32059}, {21654,21969}, {23273,26340}

X(32421) = isogonal conjugate of X(32422)


X(32422) = ISOGONAL CONJUGATE OF X(32421)

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

X(32422) lies on the circumcircle and these lines: {99,492}, {110,371}, {112,5413}, {486,925}, {691,2459}, {1306,6200}, {2858,32419}, {2872,9142}, {3565,6396}

X(32422) = isogonal conjugate of X(32421)
X(32422) = reflection of X(98) in the line X(3)X(9894)


X(32423) = INTERSECTION OF THE EULER LINES OF THESE TRIANGLES: AOA, AAOA, AND 1st HYACINTH

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

X(32423) lies on the cubic K466, the curve Q110 and these lines: {2,11694}, {3,2888}, {4,195}, {5,49}, {20,10620}, {26,9920}, {30,511}, {52,32196}, {68,1658}, {69,32306}, {74,550}, {99,15545}, {113,137}, {125,128}, {141,12584}, {143,10112}, {146,382}, {155,18377}, {156,9927}, {182,25328}, {252,19268}, {323,7574}, {343,22109}, {355,2948}, {376,15041}, {381,9143}, {389,6153}, {427,15463}, {495,10066}, {496,10082}, {547,5642}, {548,12041}, {549,9140}, {568,7730}, {576,25329}, {631,15040}, {632,15027}, {895,1353}, {946,11699}, {973,12236}, {974,18914}, {1112,6756}, {1147,10224}, {1216,13470}, {1350,25335}, {1351,11061}, {1352,2930}, {1385,13605}, {1483,7979}, {1484,10778}, {1495,25338}, {1513,5987}, {1539,3853}, {1594,3043}, {1595,15472}, {1656,15081}, {1657,12244}, {1770,11670}, {1885,12292}, {1986,3575}, {2935,14216}, {3028,3327}, {3047,10024}, {3070,12375}, {3071,12376}, {3090,15039}, {3091,20125}, {3530,6699}, {3545,15046}, {3580,7575}, {3581,12380}, {3585,14101}, {3589,20301}, {3627,7728}, {3628,5972}, {3845,5655}, {3850,7687}, {3857,15044}, {3861,6053}, {4857,7343}, {5050,25320}, {5066,23516}, {5073,13432}, {5085,25330}, {5093,25321}, {5102,25331}, {5254,14901}, {5270,6126}, {5318,10657}, {5321,10658}, {5448,18379}, {5449,10125}, {5462,13365}, {5480,19140}, {5498,12038}, {5501,25340}, {5504,6145}, {5654,32395}, {5690,12778}, {5874,6276}, {5875,6277}, {5876,15532}, {5889,13423}, {5901,11720}, {5946,16223}, {6101,11750}, {6102,11562}, {6193,12319}, {6240,6242}, {6243,7731}, {6247,13293}, {6253,6255}, {6284,6286}, {6321,15342}, {6593,18583}, {6644,26869}, {6696,25564}, {6723,16239}, {6776,32251}, {6794,22146}, {7354,7356}, {7487,18947}, {7526,12168}, {7577,9703}, {7579,11935}, {7689,15332}, {7723,12605}, {8550,9976}, {8703,15055}, {8981,10819}, {8995,8998}, {9544,10254}, {9825,9826}, {9833,10117}, {9934,32359}, {9970,21850}, {9985,12501}, {10020,32391}, {10021,16164}, {10095,13163}, {10111,14708}, {10116,11802}, {10126,18016}, {10127,12099}, {10205,30484}, {10226,12901}, {10263,13417}, {10282,18282}, {10299,15042}, {10540,11563}, {10663,32397}, {10664,32398}, {10706,15687}, {10721,16659}, {10820,13966}, {10942,12890}, {10943,12889}, {11064,32235}, {11202,15330}, {11250,12118}, {11412,15100}, {11442,18570}, {11477,25336}, {11536,18428}, {11559,13418}, {11565,32142}, {11577,15738}, {11579,32233}, {11807,13419}, {11818,14853}, {11819,14449}, {12079,15468}, {12084,32337}, {12103,16111}, {12105,32269}, {12107,13289}, {12108,20397}, {12112,18325}, {12133,13488}, {12165,12173}, {12201,12208}, {12219,12225}, {12227,12233}, {12270,12278}, {12273,12281}, {12293,17824}, {12334,12341}, {12358,12362}, {12359,12893}, {12364,32349}, {12428,12888}, {12596,32368}, {12661,32370}, {12790,12797}, {12824,13451}, {12891,32384}, {12892,32385}, {12894,12998}, {12895,12999}, {12897,32137}, {12900,15088}, {12905,13121}, {12906,13122}, {13142,32332}, {13188,18331}, {13198,31804}, {13202,13431}, {13353,27866}, {13383,20773}, {13393,15605}, {13413,23292}, {13979,13986}, {14480,20957}, {14791,15106}, {14869,15020}, {14912,18420}, {14934,16340}, {15036,15057}, {15051,15712}, {15054,15704}, {15068,18396}, {15085,17835}, {15091,18572}, {15093,22265}, {15113,15114}, {15122,15124}, {15123,32355}, {15136,32353}, {15345,27868}, {15465,23411}, {15761,32379}, {16165,25337}, {16266,17847}, {16337,24385}, {18358,32274}, {18381,23315}, {18440,31861}, {18563,22584}, {18567,19479}, {18874,23409}, {18932,32334}, {18933,18945}, {18970,19469}, {19051,19095}, {19052,19096}, {19138,32344}, {19139,23049}, {19195,19205}, {19456,32341}, {19457,19467}, {19478,22586}, {19482,32386}, {19483,32387}, {19484,19498}, {19485,19499}, {19552,24144}, {20030,20414}, {22115,25739}, {32114,32275}

X(32423) = isogonal conjugate of X(14979)


X(32424) = INTERSECTION OF THE EULER LINES OF THESE TRIANGLES: 4th BROCARD, CIRCUMMEDIAL, AND 5th EULER

Barycentrics    3*(9*R^2-SW)*S^4+(3*SW*SA^2-3*(27*R^2-5*SW)*SB*SC-2*SW^3)*S^2+3*SB*SC*SW^3 : :

X(32424) lies on these lines: {2,10748}, {3,9829}, {4,31824}, {5,9172}, {30,511}, {110,11159}, {111,381}, {113,13994}, {125,11569}, {126,549}, {140,31606}, {143,31763}, {376,6031}, {546,31749}, {547,6719}, {550,31729}, {1296,3534}, {1316,32222}, {1385,31747}, {3543,20099}, {3579,31746}, {3830,10734}, {3845,5512}, {5077,9140}, {5476,28662}, {5694,31823}, {5946,31743}, {5987,9855}, {6101,31736}, {6102,31745}, {6232,9129}, {9940,31818}, {9955,31755}, {9956,31758}, {11591,31753}, {12041,32311}, {12161,31809}, {13643,13644}, {13762,13763}, {14657,18324}, {15303,18907}, {31727,31748}, {31761,32161}, {32219,32225}

X(32424) = isogonal conjugate of X(32425)


X(32425) = ISOGONAL CONJUGATE OF X(32424)

Barycentrics    (SB+SC)*(3*(9*R^2-SW)*S^4+(3*SB^2*SW-3*(27*R^2-5*SW)*SC*SA-2*SW^3)*S^2+3*SA*SC*SW^3)*(3*(9*R^2-SW)*S^4+(3*SC^2*SW-3*(27*R^2-5*SW)*SA*SB-2*SW^3)*S^2+3*SA*SB*SW^3) : :

X(32425) lies on the circumcircle and these lines: {476,11159}, {523,11568}, {524,6236}, {1296,8705}, {1499,6325}

X(32425) = isogonal conjugate of X(32424)
X(32425) = (circumsymmedial)-isogonal conjugate of-X(2780)
X(32425) = X(2696) of circumsymmedial triangle
X(32425) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (74, 3, 352), (111, 3, 11663), (477, 3, 11628), (1296, 3, 17414), (11568, 2, 3)


X(32426) = INTERSECTION OF THE EULER LINES OF THESE TRIANGLES: EXCENTERS-MIDPOINTS, GARCIA-REFLECTION, AND 2nd SCHIFFLER

Barycentrics    2*a^4-4*(b+c)*a^3-(b^2-20*b*c+c^2)*a^2+2*(b+c)*(2*b^2-7*b*c+2*c^2)*a-(b^2-c^2)^2 : :

X(32426) lies on these lines: {8,1997}, {30,511}, {145,1376}, {497,3621}, {1699,3680}, {2886,12648}, {3338,3633}, {3476,5221}, {3632,15829}, {3813,5790}, {3913,7967}, {4969,16561}, {5506,13606}, {5603,10912}, {6667,6735}, {8256,17728}, {10164,11260}, {10915,11230}, {11219,11256}, {11246,14923}, {12645,26333}, {17784,20014}

X(32426) = isogonal conjugate of X(32427)


X(32427) = ISOGONAL CONJUGATE OF X(32426)

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

X(32427) lies on the circumcircle and these lines: {100,20323}, {30236,31798}

X(32427) = isogonal conjugate of X(32426)
X(32427) = X(110)-of-3rd mixtilinear triangle


X(32428) = POINT AT INFINITY OF THE BROCARD AXIS OF THE CIRCUMORTHIC TRIANGLE

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

X(32428) lies on these lines: {2,10184}, {3,95}, {4,3164}, {5,53}, {20,31388}, {26,157}, {30,511}, {52,6751}, {68,8612}, {110,32439}, {140,6709}, {143,15912}, {155,17849}, {324,418}, {381,14635}, {1209,15780}, {1316,19128}, {1513,12131}, {2052,6638}, {2070,30716}, {2980,17714}, {5872,8175}, {5873,8174}, {6662,10627}, {6663,10095}, {9747,9909}, {10125,13467}, {13371,23333}, {13599,17039}, {17041,31505}, {18377,18380}, {18951,18953}, {19154,19156}, {19210,19212}, {27352,27361}

X(32428) = complementary conjugate of X(129)
X(32428) = isogonal conjugate of X(1298)
X(32428) = point at infinity of Brocard axis of any triangle homothetic to circumorthic triangle


X(32429) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: 6th ANTI-BROCARD, ANTI-EULER, AND 1st NEUBERG

Barycentrics    (SA+SW)*S^4+(5*SA^2+2*SB*SC-4*SW^2)*SW*S^2+3*SB*SC*SW^3 : :
X(32429) = 3*X(182)-2*X(24256) = X(1352)-3*X(7709) = 2*X(1352)-3*X(11261) = 3*X(5085)-X(13108) = 4*X(10007)-3*X(11178) = 3*X(11171)-2*X(24206) = 3*X(11179)-X(18906) = 3*X(13331)-2*X(19130)

X(32429) lies on these lines: {6,18501}, {20,185}, {39,3818}, {76,5092}, {182,2782}, {542,1569}, {732,3098}, {1352,7709}, {3095,29012}, {5085,13108}, {5116,12188}, {5476,7739}, {6248,7803}, {7757,11645}, {7765,13331}, {7795,13334}, {8149,14880}, {10007,11178}, {11171,14981}, {11179,18906}

X(32429) = reflection of X(i) in X(j) for these (i,j): (76, 5092), (3818, 39), (11261, 7709), (32430, 14712)
X(32429) = (1st Neuberg)-isogonal conjugate of-X(6194)
X(32429) = (1st Ehrmann)-anticomplement of-X(24256)
X(32429) = X(9821)-of-1st Brocard triangle


X(32430) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: ANTI-EULER, 1st BROCARD-REFLECTED, AND 2nd NEUBERG

Barycentrics    3*(SA+14*SW)*S^6+(30*SA^2-57*SB*SC+19*SW^2)*SW*S^4+(11*SA^2-57*SB*SC-4*SW^2)*SW^3*S^2-3*SB*SC*SW^5 : :

X(32430) lies on these lines: {20,185}, {8150,14881}, {10796,11261}

X(32430) = reflection of X(32429) in X(14712)


X(32431) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL, EXCENTRAL, AND FEUERBACH

Barycentrics    a^5+b*c*a^3+(b^3+c^3)*a^2-(b^2-c^2)^2*a-(b^2-c^2)^2*(b+c) : :
X(32431) = 6*X(381)-X(3019)

X(32431) lies on these lines: {2,17190}, {3,31872}, {4,9}, {5,572}, {6,13}, {7,24224}, {8,24048}, {30,1213}, {37,18480}, {80,2171}, {191,21014}, {219,24045}, {284,6841}, {355,4053}, {391,3839}, {403,1474}, {429,15946}, {430,2328}, {517,21873}, {594,18357}, {604,7741}, {941,5560}, {1030,13743}, {1100,9955}, {1400,3585}, {1568,22133}, {1765,6839}, {1781,21044}, {1999,10478}, {2160,22798}, {2173,7110}, {2178,18761}, {2268,7951}, {2269,3583}, {2285,10826}, {3686,18483}, {3723,28204}, {3845,17330}, {4034,31162}, {4253,6849}, {4909,5712}, {5080,21061}, {5086,21078}, {5257,31673}, {5443,17440}, {5747,6866}, {6894,18865}, {6996,17307}, {7377,17352}, {7384,17300}, {8287,24884}, {9355,13610}, {9665,21785}, {10446,17343}, {10883,19642}, {12699,17275}, {13478,14534}, {15971,26079}, {16777,18525}, {16884,18493}, {17205,28091}, {17258,29069}, {17362,22791}, {17748,27871}, {20546,24267}, {21773,26321}, {24271,27688}

X(32431) = (2nd Fuhrmann)-isotomic conjugate of-X(22791)
X(32431) = intersection of the Brocard axes of these triangles: anti-orthocentroidal, 2nd extouch and Feuerbach
X(32431) = intersection of the Brocard axes of these triangles: anti-orthocentroidal, Feuerbach and 2nd Zaniah
X(32431) = inverse-in-Kiepert-hyperbola of X(3017)
X(32431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5816, 573), (13, 14, 3017)


X(32432) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES, EULER, AND OUTER-VECTEN

Barycentrics    ((b^2+c^2)*a^2-b^4-c^4)*a^2+(2*(b^2+c^2)*a^2-4*b^4+4*b^2*c^2-4*c^4)*S : :
X(32432) = X(6566)-5*X(31275) = X(6567)+3*X(31173)

X(32432) lies on these lines: {2,2459}, {5,141}, {316,2460}, {371,7773}, {372,7887}, {487,23259}, {641,3070}, {642,13449}, {1691,11314}, {2456,6290}, {3102,7752}, {3103,5025}, {6567,31173}, {7825,9738}, {7862,9739}, {13665,22623}

X(32432) = midpoint of X(316) and X(2460)
X(32432) = reflection of X(32435) in X(625)
X(32432) = complement of X(2459)
X(32432) = (outer-Vecten)-isogonal conjugate of-X(13926)
X(32432) = (1st Brocard)-complement of-X(2459)
X(32432) = inverse of X(639) in the nine-point circle
X(32432) = X(6567)-of-outer-Vecten triangle
X(32432) = intersection of the Brocard axes of these triangles: 3rd anti-tri-squares, medial and outer-Vecten
X(32432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5031, 32435), (2039, 2040, 639)


X(32433) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES, 4th TRI-SQUARES, AND INNER-VECTEN

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

X(32433) lies on these lines: {30,511}, {371,7851}, {486,489}, {487,23259}, {642,3071}, {2459,12221}, {3103,7921}, {6119,18762}, {6316,12601}

X(32433) = isogonal conjugate of X(32434)


X(32434) = ISOGONAL CONJUGATE OF X(32433)

Barycentrics    a^2*(-5*(a^4-b^2*a^2-(b^2-c^2)*c^2)*b^2+(4*a^4+2*(b^2-2*c^2)*a^2-8*b^4+2*b^2*c^2+4*c^4)*S)*(-5*(a^4-c^2*a^2+(b^2-c^2)*b^2)*c^2+(4*a^4-2*(2*b^2-c^2)*a^2+4*b^4+2*b^2*c^2-8*c^4)*S) : :

X(32434) lies on the circumcircle and the line {110,32074}

X(32434) = isogonal conjugate of X(32433)


X(32435) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES, EULER, AND INNER-VECTEN

Barycentrics    -2*((b^2+c^2)*a^2-2*b^4+2*b^2*c^2-2*c^4)*S+((b^2+c^2)*a^2-b^4-c^4)*a^2 : :
X(32435) = X(6566)+3*X(31173) = X(6567)-5*X(31275)

X(32435) lies on these lines: {2,2460}, {5,141}, {316,2459}, {371,7887}, {372,7773}, {488,23249}, {641,13449}, {642,3071}, {1691,11313}, {2456,6289}, {3102,5025}, {3103,7752}, {6566,31173}, {7825,9739}, {7862,9738}, {13785,22594}

X(32435) = midpoint of X(316) and X(2459)
X(32435) = reflection of X(32432) in X(625)
X(32435) = (inner-Vecten)-isogonal conjugate of-X(13873)
X(32435) = (1st Brocard)-complement of-X(2460)
X(32435) = complement of X(2460)
X(32435) = inverse of X(640) in the nine-point circle
X(32435) = intersection of the Brocard axes of these triangles: 4th anti-tri-squares, medial and inner-Vecten
X(32435) = X(6567)-of-inner-Vecten triangle
X(32435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5031, 32432), (2039, 2040, 640)


X(32436) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES, 3rd TRI-SQUARES, AND OUTER-VECTEN

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

X(32436) lies on these lines: {30,511}, {187,13879}, {372,7851}, {485,490}, {488,23249}, {641,3070}, {2460,12222}, {3102,7921}, {6118,18538}, {6312,12602}

X(32436) = isogonal conjugate of X(32437)


X(32437) = ISOGONAL CONJUGATE OF X(32436)

Barycentrics    a^2*(5*(a^4-b^2*a^2-(b^2-c^2)*c^2)*b^2+(4*a^4+2*(b^2-2*c^2)*a^2-8*b^4+2*b^2*c^2+4*c^4)*S)*(5*(a^4-c^2*a^2+(b^2-c^2)*b^2)*c^2+(4*a^4-2*(2*b^2-c^2)*a^2+4*b^4+2*b^2*c^2-8*c^4)*S) : :

X(32437) lies on the circumcircle and the line {110,11199}

X(32437) = isogonal conjugate of X(32436)


X(32438) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: AAOA, AOA, AND 1st HYACINTH

Barycentrics    (SB+SC)*(3*S^4-(6*R^2*(8*R^2+3*SA-4*SW)-5*SA^2+2*SB*SC+3*SW^2)*S^2+(4*R^2-SW)*(6*R^2+SA-2*SW)*SA*SW) : :

X(32438) lies on these lines: {30,511}, {74,1303}, {97,110}, {113,130}, {125,129}, {265,6528}, {1511,22052}, {5504,8612}, {12310,22551}, {17838,22552}, {21649,21661}

X(32438) = isogonal conjugate of X(32439)


X(32439) = ISOGONAL CONJUGATE OF X(32438)

Barycentrics    (3*S^4-(6*R^2*(8*R^2+3*SB-4*SW)-5*SB^2+2*SA*SC+3*SW^2)*S^2+(4*R^2-SW)*(6*R^2+SB-2*SW)*SB*SW)*(3*S^4-(6*R^2*(8*R^2+3*SC-4*SW)-5*SC^2+2*SA*SB+3*SW^2)*S^2+(4*R^2-SW)*(6*R^2+SC-2*SW)*SC*SW) : :

X(32439) lies on the circumcircle and these lines: {30,1303}, {418,476}, {436,1304}, {523,1298}, {2713,12022}, {8613,10420}

X(32439) = isogonal conjugate of X(32438)


X(32440) = INTERSECTION OF THE BROCARD AXES OF THESE TRIANGLES: EXCENTERS-MIDPOINTS, GARCIA-REFLECTION AND 2nd SCHIFFLER

Barycentrics    a*(2*(b+c)*a^7-(7*b^2+8*b*c+7*c^2)*a^6+(b+c)*(3*b^2+26*b*c+3*c^2)*a^5+2*(6*b^4+6*c^4-(20*b^2+19*b*c+20*c^2)*b*c)*a^4-4*(b+c)*(b^2-b*c-3*c^2)*(3*b^2+b*c-c^2)*a^3-(3*b^6+3*c^6-(44*b^4+44*c^4-53*(b^2+c^2)*b*c)*b*c)*a^2+(b^2-c^2)*(b-c)*(7*b^4+7*c^4-2*(10*b^2-3*b*c+10*c^2)*b*c)*a-2*(b^2-c^2)^2*(b^2-b*c+c^2)^2) : :

X(32440) lies on these lines: {8,18811}, {30,511}

X(32440) = isogonal conjugate of X(32441)


X(32441) = ISOGONAL CONJUGATE OF X(32440)

Barycentrics    a*(2*a^8-(7*b+4*c)*a^7+(3*b^2+27*b*c+2*c^2)*a^6+(12*b^3+4*c^3-(44*b+19*c)*b*c)*a^5-(12*b^4+8*c^4-(4*b^2+53*b*c-c^2)*b*c)*a^4-(3*b^5-4*c^5-(40*b^3-52*b^2*c-c^3)*b*c)*a^3+(b-c)*(7*b^5-2*c^5-(22*b^3-17*c^3-4*(4*b-9*c)*b*c)*b*c)*a^2-(b^2-c^2)*(b-c)*(2*b^4+4*c^4-(6*b^2-25*b*c+23*c^2)*b*c)*a-(b^2-c^2)^2*(b-c)*(b-2*c)*(2*b-c)*c)*(2*a^8-(4*b+7*c)*a^7+(2*b^2+27*b*c+3*c^2)*a^6+(4*b^3+12*c^3-(19*b+44*c)*b*c)*a^5-(8*b^4+12*c^4+(b^2-53*b*c-4*c^2)*b*c)*a^4+(4*b^5-3*c^5-(b^3+52*b*c^2-40*c^3)*b*c)*a^3+(b-c)*(2*b^5-7*c^5-(17*b^3-22*c^3-4*(9*b-4*c)*b*c)*b*c)*a^2-(b^2-c^2)*(b-c)*(4*b^4+2*c^4-(23*b^2-25*b*c+6*c^2)*b*c)*a+(b^2-c^2)^2*(b-c)*(b-2*c)*(2*b-c)*b) : :

X(32441) lies on the circumcircle and these lines: {}

X(32441) = isogonal conjugate of X(32440)
X(32441) = X(99)-of-3rd mixtilinear triangle


X(32442) = REFLECTION OF X(6786) IN X(27088)

Barycentrics    a^2*(a^4*b^4 - a^2*b^6 - 8*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + b^6*c^2 + a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + b^2*c^6) : :
X(32442) = 4 X[187] - X[5167],X[6787] - 3 X[26613]

X(32442) lies on the cubic K1099 and these lines: {30, 6784}, {187, 237}, {376, 511}, {1691, 19136}, {2080, 8717}, {3111, 3849}, {6179, 14135}, {6786, 27088}, {6787, 26613}, {14830, 14915}

X(32442) = reflection of X(6786) in X(27088)


X(32443) = WALSMITH-LEMOINE POINT

Barycentrics    Sqrt[2*a^8 - 3*a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 + 2*b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 3*b^6*c^2 + 2*a^4*c^4 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - 3*a^2*c^6 - 3*b^2*c^6 + 2*c^8 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(a^6 - a^4*b^2 - 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 - b^2*c^4 + c^6)]*((-5*a^4 + 4*a^2*b^2 + b^4 + 4*a^2*c^2 - 2*b^2*c^2 + c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] + 4*(2*a^2 - b^2 - c^2)*S^2) + (b^2 - c^2)*(13*a^8 - 21*a^6*b^2 + a^4*b^4 + 9*a^2*b^6 - 2*b^8 - 21*a^6*c^2 + 43*a^4*b^2*c^2 - 17*a^2*b^4*c^2 + 3*b^6*c^2 + a^4*c^4 - 17*a^2*b^2*c^4 - 2*b^4*c^4 + 9*a^2*c^6 + 3*b^2*c^6 - 2*c^8 - 8*(2*a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*S^2) : :

See the preamble just before X(32215). The connection with the Lemoine point (X(6), the symmedian point) is that X(32443)-of-Walsmith-Lemoine-triangle = X(6); see the preamble just before X(32217). Contributed by Peter Moses, May 6, 2019.

X(32443) lies on the circumcircle and on no lines X(i)X(j) for 0 < i < j < 32444.


X(32444) = X(2)X(3)∩X(39)X(6000)

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

As a point on the Euler line, X(32444) has Shinagawa coefficients (EF+F2+S2,2E2+EF-F2 -S2).

X(32444) lies on these lines: {2, 3}, {39, 6000}, {217, 9605}, {3095, 13754}, {3331, 3426}, {4550, 5167}, {9155, 16261}, {11171, 14915}, {12294, 30258}, {18440, 20794}

X(32444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 381, 11328}, {3, 12083, 21512}, {376, 14096, 3}, {378, 3148, 3}


X(32445) = X(3)X(1625)∩X(4)X(6)

Barycentrics    a^2*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6) : :
X(32445) = 3 X[154] - 2 X[15270]

X(32445) lies on the cubics K622 and K1100 and also on these lines: {3, 1625}, {4, 6}, {20, 3289}, {32, 1971}, {39, 6000}, {64, 5013}, {112, 1614}, {154, 237}, {159, 3499}, {172, 10535}, {184, 1968}, {185, 232}, {187, 10282}, {206, 1691}, {216, 5907}, {221, 16781}, {230, 16252}, {389, 3199}, {394, 7750}, {512, 3202}, {574, 3357}, {1384, 14530}, {1506, 20299}, {1660, 2056}, {1914, 26888}, {1993, 7823}, {2052, 27359}, {2076, 11674}, {2174, 8761}, {2176, 10537}, {2275, 7355}, {2276, 6285}, {2393, 13330}, {2548, 14216}, {2549, 5878}, {2777, 7756}, {3016, 7746}, {3051, 11206}, {3094, 12502}, {3224, 14601}, {3269, 6241}, {3566, 10685}, {3815, 6247}, {5023, 17821}, {5024, 13093}, {5033, 23042}, {5038, 12202}, {5206, 11202}, {5301, 7113}, {5475, 18381}, {6225, 7738}, {7736, 12324}, {7737, 9833}, {7747, 18400}, {7748, 22802}, {7755, 14862}, {7904, 15066}, {8571, 13406}, {8778, 9412}, {9291, 9308}, {9418, 11325}, {9419, 11257}, {9574, 9899}, {9597, 12940}, {9598, 12950}, {9605, 12315}, {9863, 11441}, {9985, 13236}, {10060, 31448}, {10192, 21001}, {10311, 26883}, {10312, 14157}, {10533, 12963}, {10534, 12968}, {10574, 15355}, {10575, 14961}, {10606, 15815}, {10790, 27374}, {11204, 15515}, {12111, 22240}, {14264, 18573}, {15068, 32151}, {15305, 26216}, {18388, 27371}, {18621, 21788}, {19780, 30403}, {19781, 30402}, {20775, 31382}, {20965, 32064}, {23128, 32139}, {23329, 31455}, {30435, 32063}

X(32445) = isogonal conjugate of the isotomic conjugate of X(3164)
X(32445) = isogonal conjugate of the polar conjugate of X(3168)
X(32445) = polar conjugate of the isotomic conjugate of X(6638)
X(32445) = X(i)-Ceva conjugate of X(j) for these (i,j): {184, 6}, {1968, 3053}, {3164, 6638}
X(32445) = X(3168)-cross conjugate of X(6)
X(32445) = X(75)-isoconjugate of X(1988)
X(32445) = crosspoint of X(3164) and X(3168)
X(32445) = crossdifference of every pair of points on line {520, 6130}
X(32445) = tangential-isotomic conjugate of X(159)
X(32445) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 3168}, {4, 6638}, {5, 26887}, {6, 3164}
X(32445) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {32, 1988}, {3164, 76}, {3168, 264}, {6638, 69}, {26887, 95}
X(32445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 217, 6}, {32, 6759, 1971}, {112, 1614, 14585}, {184, 1968, 1970}, {217, 3331, 4}, {1181, 2207, 6}, {2211, 6776, 6}, {12111, 22240, 22416}, {12964, 12970, 19149}


X(32446) = X(2)X(2124)∩X(10)X(971)

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

See Angel Montesdeoca, Hyacinthos 29006.

X(32446) lies on these lines: {2,2124}, {10,971}, {142,10004}, {279,19605}, {2391,3452}, {6706,20205}

X(32446) = complement of X(2124)
X(32446) = X(i)-complementary conjugate of X(j) for these (i,j): {55, 17113}, {2125, 10}, {8917, 142}


X(32447) = MIDPOINT OF X(262) AND X(7757)

Barycentrics    a^2*(a^4*b^2 - 3*a^2*b^4 + 2*b^6 + a^4*c^2 - 7*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + 2*c^6) : :
Trilinears    cos A - 2 cos(A + 2ω) : :
X(32447) = X[3] - 4 X[39],X[3] + 2 X[3095],7 X[3] - 4 X[5188],5 X[3] - 2 X[9821],2 X[5] + X[194],4 X[5] - X[13108],2 X[39] + X[3095],7 X[39] - X[5188],10 X[39] - X[9821],2 X[76] - 5 X[1656],X[76] - 4 X[11272],4 X[140] - X[12251],2 X[194] + X[13108],5 X[262] - 2 X[22681],X[381] + 2 X[7757],5 X[381] - 4 X[22681],X[382] + 2 X[11257],X[382] - 4 X[14881],X[1351] + 2 X[3094],X[1482] + 2 X[12782],2 X[1569] + X[6321],5 X[1656] - 8 X[11272],2 X[1916] + X[13188],7 X[3090] - X[20081],7 X[3095] + 2 X[5188],5 X[3095] + X[9821],7 X[3526] - 10 X[7786],8 X[3628] - 5 X[31276],7 X[3851] - 4 X[6248],8 X[3934] - 11 X[5070],3 X[5054] - 2 X[22712],3 X[5055] - 2 X[7697],11 X[5056] + X[20105],10 X[5188] - 7 X[9821],2 X[7709] + X[22728],5 X[7757] + 2 X[22681],2 X[7838] + X[32152],2 X[7976] + X[12645],4 X[9466] - 7 X[15703],X[9902] - 4 X[9956],2 X[11152] + X[12355],X[11257] + 2 X[14881],3 X[14269] - 4 X[22682],3 X[15688] - 2 X[22676],5 X[15694] - 4 X[15819],4 X[18583] - X[18906].

Let X be a point on the 2nd Brocard circle. As X varies, the locus of X(381) of triangle XPU(1) is a circle with center X(32447). (Randy Hutson, June 7, 2019)

Let (O1) and (O2) be the circles that are the loci of the 1st and 2nd Isogonic Centers, respectively, in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC). The center of (O1) is X(3107). The center of (O2) is X(3106). The insimilicenter of (O1) and (O2) is X(6), and the exsimilicenter is X(32447). (Randy Hutson, June 7, 2019)

X(32447) lies on these lines: {2, 8179}, {3, 6}, {5, 194}, {30, 7709}, {76, 1656}, {99, 10796}, {140, 7806}, {237, 11002}, {262, 381}, {382, 11257}, {517, 3097}, {538, 5055}, {549, 6194}, {597, 31958}, {599, 11261}, {698, 14561}, {726, 5886}, {730, 5790}, {732, 13085}, {736, 9766}, {999, 12837}, {1482, 12782}, {1569, 5475}, {1916, 11170}, {3090, 20081}, {3202, 9704}, {3295, 12836}, {3399, 7754}, {3526, 7786}, {3534, 12156}, {3552, 32134}, {3628, 31276}, {3851, 6248}, {3934, 5070}, {5054, 22712}, {5056, 20105}, {5191, 11422}, {5355, 6036}, {5476, 8724}, {5640, 9155}, {5663, 32444}, {5969, 11165}, {6234, 18872}, {6390, 18583}, {6660, 6800}, {7753, 23698}, {7760, 10104}, {7781, 10358}, {7783, 18501}, {7813, 24206}, {7820, 25555}, {7838, 32152}, {7976, 12645}, {8704, 9178}, {9466, 15703}, {9654, 18982}, {9669, 13077}, {9902, 9956}, {10063, 31479}, {10247, 14839}, {10567, 30209}, {11152, 11317}, {11188, 20794}, {12150, 21166}, {12188, 13860}, {13586, 22521}, {14269, 22682}, {14981, 19130}, {15048, 15980}, {15072, 31952}, {15531, 23635}, {15688, 22676}, {15694, 15819}, {22650, 28204}

X(32447) = midpoint of X(262) and X(7757)
X(32447) = reflection of X(i) in X(j) for these lines: {i,j}: {3, 11171}, {381, 262}, {599, 11261}, {6194, 549}, {31958, 597}
X(32447) = midpoint of centers of circles {{X(1340),X(3557),PU(1)}} and {{X(1341),X(3558),PU(1)}}
X(32447) = harmonic center of circumcircle and circle (X(6), |OK|/2)
X(32447) = Brocard-circle-inverse of X(11842)
X(32447) = 2nd-Brocard-circle-inverse of X(575)
X(32447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 11842}, {3, 1351, 9301}, {5, 194, 13108}, {39, 3094, 5024}, {39, 3095, 3}, {76, 11272, 1656}, {574, 576, 2080}, {574, 2080, 3}, {575, 18860, 26316}, {576, 8586, 1351}, {1351, 5024, 3}, {1670, 1671, 575}, {1689, 1690, 187}, {2021, 13330, 1384}, {3102, 3103, 13330}, {3106, 3107, 6}, {3398, 9737, 3}, {5640, 9155, 11328}, {7772, 9737, 3398}, {9605, 10983, 3}, {9734, 15520, 32}, {9821, 13334, 3}, {11257, 14881, 382}, {11482, 12313, 3311}, {11482, 12314, 3312}, {12054, 30270, 3}, {13343, 13344, 2076}, {14961, 30258, 3}, {18860, 26316, 3}


X(32448) = MIDPOINT OF X(3) AND X(194)

Barycentrics    2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6 + 2*a^6*c^2 - 6*a^4*b^2*c^2 - b^6*c^2 - 3*a^4*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6 : :
X(32448) = 5 X[3] - 3 X[6194],X[3] - 3 X[7709],3 X[3] - X[12251],X[4] - 3 X[32447],3 X[5] - 2 X[6248],3 X[5] - 4 X[11272],3 X[39] - X[6248],3 X[39] - 2 X[11272],X[76] - 3 X[11171],2 X[140] - 3 X[11171],5 X[194] + 3 X[6194],X[194] + 3 X[7709],3 X[194] + X[12251],3 X[262] - 2 X[546],X[355] - 3 X[3097],3 X[549] - 4 X[13334],5 X[631] - X[20081],5 X[632] - 4 X[3934],X[3095] - 3 X[7757],X[3146] - 3 X[22728],7 X[3523] + X[20105],7 X[3526] - 5 X[31276],4 X[3530] - 3 X[22712],4 X[3628] - 3 X[7697],4 X[3628] - 5 X[7786],7 X[3857] - 6 X[22681],3 X[5050] - X[18906],2 X[5188] - 3 X[8703],X[6194] - 5 X[7709],9 X[6194] - 5 X[12251],3 X[7697] - 5 X[7786],9 X[7709] - X[12251],3 X[7757] + X[11257],2 X[9466] - 3 X[11539],X[9902] - 3 X[26446],X[11055] + 2 X[12100],3 X[13331] - 2 X[18583],2 X[14711] - 5 X[15713],7 X[14869] - 6 X[15819],5 X[15712] - 6 X[21163]

X(32448) lies on these lines: {2, 13108}, {3, 194}, {4, 32447}, {5, 39}, {6, 32134}, {30, 3095}, {54, 15093}, {76, 140}, {83, 23235}, {99, 3398}, {182, 698}, {262, 546}, {355, 3097}, {384, 10353}, {397, 3107}, {398, 3106}, {495, 18982}, {496, 13077}, {511, 550}, {538, 549}, {548, 9821}, {574, 10104}, {575, 5026}, {631, 20081}, {632, 3934}, {726, 1385}, {730, 5690}, {952, 12782}, {1483, 14839}, {1503, 32429}, {1595, 12143}, {2080, 7760}, {3094, 3564}, {3146, 22728}, {3523, 20105}, {3526, 31276}, {3530, 22712}, {3552, 11842}, {3627, 14881}, {3628, 7697}, {3857, 22681}, {5050, 18906}, {5171, 7798}, {5188, 8703}, {5844, 7976}, {5965, 7830}, {6287, 14692}, {7755, 14693}, {7772, 10796}, {7827, 8724}, {7828, 15561}, {7839, 11676}, {8370, 11152}, {9466, 11539}, {9737, 14880}, {9902, 26446}, {10079, 15325}, {11055, 12100}, {12836, 15171}, {12837, 18990}, {13331, 18583}, {14711, 15713}, {14869, 15819}, {15712, 21163}, {19522, 31036}

X(32448) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 194}, {3095, 11257}
X(32448) = reflection of X(i) in X(j) for these lines: {i,j}: {5, 39}, {76, 140}, {3627, 14881}, {6248, 11272}, {9821, 548}
X(32448) = complement of X(13108)
X(32448) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(115)
X(32448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 6248, 11272}, {76, 11171, 140}, {194, 7709, 3}, {3102, 3103, 115}, {6248, 11272, 5}, {7697, 7786, 3628}, {7757, 11257, 3095}, {32470, 32471, 6}


X(32449) = MIDPOINT OF X(6) AND X(194)

Barycentrics    (b^2 + c^2)*(2*a^4 + a^2*b^2 + a^2*c^2 - b^2*c^2) : :
X(32449) = 3 X[6] - X[18906],3 X[39] - 2 X[10007],3 X[39] - X[14994],X[76] - 3 X[13331],3 X[141] - 4 X[10007],3 X[141] - 2 X[14994],3 X[194] + X[18906],3 X[597] - 2 X[24256],X[1350] - 3 X[7709],X[1352] - 3 X[32447],X[3094] - 3 X[7757],3 X[3097] - X[3416],2 X[3589] - 3 X[13331],5 X[3618] - X[20081],2 X[5052] - 3 X[8584],3 X[5085] - X[12251],X[13108] - 3 X[14561],4 X[13334] - 3 X[21167]}

X(32449) lies on these lines: {6, 194}, {30, 32429}, {39, 141}, {76, 3589}, {99, 12212}, {182, 7798}, {385, 5116}, {427, 17949}, {511, 550}, {524, 3094}, {538, 597}, {726, 1386}, {736, 15048}, {1078, 12055}, {1100, 17760}, {1350, 7709}, {1352, 32447}, {1503, 3095}, {1569, 5052}, {1691, 7760}, {1916, 12830}, {2076, 7783}, {2782, 5480}, {3051, 4576}, {3097, 3416}, {3106, 23000}, {3107, 23009}, {3231, 31088}, {3329, 24273}, {3618, 20081}, {3867, 12143}, {4074, 19568}, {5017, 31859}, {5031, 7764}, {5039, 7781}, {5085, 12251}, {5092, 7805}, {5103, 5254}, {5207, 13571}, {5305, 8149}, {5306, 5976}, {5846, 12782}, {6309, 7789}, {7754, 8177}, {7766, 10335}, {7792, 9865}, {7838, 29012}, {7976, 9053}, {8024, 11205}, {8267, 20965}, {8891, 10191}, {9605, 31981}, {11257, 29181}, {13108, 14561}, {13334, 21167}, {20583, 22486}, {31406, 32189}

X(32449) = midpoint of X(6) and X(194)
X(32449) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 3589}, {141, 39}, {14994, 10007}, {22486, 20583}
X(32449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 14994, 10007}, {76, 13331, 3589}, {7839, 12215, 6}, {10007, 14994, 141}
X(32449) = X(14617)-Ceva conjugate of X(141)
X(32449) = crossdifference of every pair of points on line {3221, 18105}
X(32449) = barycentric product X(i)*X(j) for these lines: {i,j}: {141, 7766}, {3051, 10010}, {4576, 25423}, {10335, 14617}
X(32449) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {1634, 25424}, {7766, 83}


X(32450) = MIDPOINT OF X(39) AND X(194)

Barycentrics    3*a^2*b^2 + 3*a^2*c^2 - 2*b^2*c^2 : :
X(32450) = 3 X[2] - 5 X[39],9 X[2] - 5 X[76],3 X[2] + 5 X[194],6 X[2] - 5 X[3934],9 X[2] - 10 X[6683],X[2] - 5 X[7757],21 X[2] - 25 X[7786],7 X[2] - 5 X[9466],7 X[2] + 5 X[11055],11 X[2] - 5 X[14711],21 X[2] - 5 X[20081],27 X[2] + 5 X[20105],27 X[2] - 25 X[31239],33 X[2] - 25 X[31276],3 X[39] - X[76],3 X[39] - 2 X[6683],X[39] - 3 X[7757],7 X[39] - 5 X[7786],7 X[39] - 3 X[9466],7 X[39] + 3 X[11055],11 X[39] - 3 X[14711],7 X[39] - X[20081],9 X[39] + X[20105],9 X[39] - 5 X[31239],11 X[39] - 5 X[31276],X[76] + 3 X[194]

X(32450) lies on these lines: {2, 39}, {3, 7798}, {6, 7781}, {30, 7838}, {32, 31859}, {99, 5007}, {148, 7858}, {187, 7760}, {262, 3855}, {316, 13571}, {325, 7765}, {382, 3095}, {384, 5041}, {511, 550}, {524, 7830}, {543, 7745}, {546, 2782}, {574, 7754}, {620, 5305}, {625, 5254}, {626, 15048}, {698, 6329}, {726, 3636}, {730, 3626}, {732, 3631}, {736, 8357}, {1015, 25264}, {1078, 31652}, {1500, 25303}, {1569, 19687}, {1916, 14042}, {1975, 7772}, {2482, 5368}, {2549, 7759}, {2996, 31415}, {3244, 14839}, {3528, 5188}, {3529, 11257}, {3530, 13334}, {3552, 5008}, {3632, 12782}, {3734, 9605}, {3849, 7756}, {3851, 6248}, {3933, 4045}, {5013, 7751}, {5024, 7815}, {5079, 13108}, {5206, 14614}, {5319, 6337}, {5346, 16925}, {5355, 7807}, {5969, 20583}, {6179, 15513}, {6390, 6680}, {6459, 26340}, {6460, 26339}, {6655, 7845}, {6656, 7813}, {7738, 7758}, {7748, 7774}, {7750, 7890}, {7766, 7782}, {7773, 11648}, {7776, 7872}, {7778, 7902}, {7779, 7847}, {7784, 7916}, {7788, 7935}, {7789, 7829}, {7790, 7821}, {7791, 7848}, {7792, 7863}, {7793, 8589}, {7796, 7853}, {7802, 7837}, {7825, 9766}, {7826, 8356}, {7833, 7877}, {7835, 7920}, {7840, 7911}, {7841, 7903}, {7851, 7888}, {7856, 7891}, {7866, 7908}, {7871, 7933}, {7881, 7913}, {7896, 11287}, {7897, 7918}, {7898, 7949}, {7909, 7923}, {7910, 7946}, {7917, 7924}, {7919, 7947}, {7921, 14537}, {7941, 31173}, {7976, 20050}, {8716, 30435}, {9821, 15688}, {10299, 12251}, {11171, 15720}, {11174, 17130}, {11285, 17131}, {12263, 15808}, {14482, 18841}, {14881, 15687}, {15482, 22332}, {22330, 32134}

X(32450) = midpoint of X(i) and X(j) for these lines: {i,j}: {39, 194}, {7750, 7890}, {7756, 7762}, {9466, 11055}
X(32450) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 6683}, {3934, 39}
X(32450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7798, 7805}, {6, 7781, 7816}, {39, 76, 6683}, {39, 9466, 7786}, {76, 6683, 3934}, {99, 7839, 5007}, {194, 7757, 39}, {194, 20081, 11055}, {325, 7765, 7861}, {574, 7754, 7780}, {1975, 7772, 7804}, {2549, 7759, 7842}, {3552, 7894, 5008}, {3788, 5286, 7817}, {3926, 7739, 7834}, {3926, 7834, 7880}, {3933, 4045, 7849}, {3933, 9607, 4045}, {5013, 22253, 7751}, {5254, 7764, 625}, {5309, 7763, 7886}, {6655, 7905, 7845}, {6656, 7813, 7895}, {7738, 7758, 7761}, {7748, 7774, 7843}, {7758, 7761, 7882}, {7760, 7783, 187}, {7779, 7847, 7873}, {7781, 7816, 15301}, {7786, 11055, 20081}, {7786, 20081, 9466}, {7790, 7906, 7821}, {7791, 7855, 7848}, {7796, 7864, 7853}, {7797, 7799, 7874}, {7801, 7803, 7915}, {7827, 7836, 7852}, {7829, 14148, 7789}


X(32451) = MIDPOINT OF X(193) AND X(194)

Barycentrics    2*a^4*b^2 + 2*a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 : :
X(32451) = 3X[6] - 2 X[24256],3 X[76] - 4 X[24256],4 X[141] - 5 X[7786],2 X[141] - 3 X[13331],4 X[182] - 3 X[22712],3 X[262] - 2 X[1352],3 X[599] - 4 X[10007],3 X[1992] - 2 X[5052],3 X[1992] - X[18906],2 X[3094] - 3 X[7757],5 X[3618] - 4 X[3934],7 X[3619] - 8 X[6683],4 X[5052] - 3 X[22486],3 X[5093] - X[13108],3 X[5182] - 2 X[5976],2 X[5188] - 3 X[25406],2 X[6248] - 3 X[14853],3 X[7697] - 4 X[18583],3 X[7757] - 4 X[32449],5 X[7786] - 6 X[13331],3 X[10519] - 4 X[13334],X[11008] + 4 X[32450],X[11055] + 2 X[15534],3 X[11257] - 4 X[32429],X[11898] - 3 X[32447],X[12251] - 3 X[14912],2 X[12263] - 3 X[16475],2 X[13354] - 3 X[14912],2 X[18906] - 3 X[22486]

X(32451) lies on these lines: {2, 11175}, {6, 76}, {20, 185}, {32, 6309}, {39, 69}, {99, 5017}, {141, 7786}, {182, 385}, {262, 1352}, {287, 1993}, {305, 3051}, {384, 5039}, {518, 7976}, {524, 3094}, {538, 1992}, {542, 1916}, {599, 10007}, {698, 3629}, {730, 3751}, {736, 5028}, {1350, 31859}, {1351, 2782}, {1503, 7762}, {1569, 14645}, {1691, 6179}, {1692, 7805}, {2023, 9766}, {2024, 7763}, {2025, 3767}, {2076, 7782}, {3056, 25264}, {3060, 8267}, {3095, 3564}, {3098, 7783}, {3104, 23000}, {3105, 23009}, {3231, 11059}, {3266, 9463}, {3589, 7856}, {3618, 3934}, {3619, 6683}, {3763, 7909}, {3818, 7785}, {3926, 13357}, {3972, 4048}, {5031, 7814}, {5034, 7751}, {5038, 8177}, {5092, 7793}, {5093, 13108}, {5116, 7771}, {5182, 5976}, {5188, 25406}, {5207, 7759}, {5476, 19570}, {5847, 12782}, {5969, 8593}, {6248, 6392}, {7697, 18583}, {7766, 9865}, {7777, 24206}, {7823, 29012}, {7839, 9983}, {7926, 11646}, {7998, 31088}, {8878, 11550}, {9917, 19459}, {10519, 13334}, {11008, 32450}, {11898, 32447}, {12143, 12167}, {12251, 13354}, {12263, 16475}, {13877, 19103}, {13930, 19104}, {14467, 30739}, {14881, 18440}

X(32451) = midpoint of X(193) and X(194)
X(32451) = reflection of X(i) in X(j) for these lines: {i,j}: {69, 39}, {76, 6}, {3094, 32449}, {12251, 13354}, {13330, 3629}, {18440, 14881}, {18906, 5052}, {22486, 1992}
X(32451) = anticomplement of X(14994)
X(32451) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {263, 21289}, {2186, 1369}, {3402, 2896}
X(32451) = crosssum of X(1084) and X(3288)
X(32451) = crossdifference of every pair of points on line {688, 2451}
X(32451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 13331, 7786}, {1992, 18906, 5052}, {3094, 32449, 7757}, {4048, 12212, 3972}, {5052, 18906, 22486}, {6272, 19090, 76}, {6273, 19089, 76}, {12251, 14912, 13354}


X(32452) = MIDPOINT OF X(194) and X(315)

Barycentrics   a^2*(b^6 - a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 + c^6) : :
Trilinears    cos(A + 2ω) (cos(A - 2ω) sin A + cos(B - 2ω) sin B + cos(C - 2ω) sin C) + (sin^2 A + sin^2 B + sin^2 C) sin A : :
Trilinears    2 sin(A + 2ω) (sin(A - 2ω) sin A + sin(B - 2ω) sin B + sin(C - 2ω) sin C) + (sin 2A + sin 2B + sin 2C) cos A : :
X(32452) = 4 X[3934] - 5 X[7867],4 X[6680] - 5 X[7786]

X(32452) lies on these lines: {2, 18806}, {3, 6}, {69, 31981}, {76, 115}, {99, 10350}, {183, 32189}, {194, 315}, {262, 1506}, {298, 6294}, {299, 6581}, {325, 8149}, {538, 7788}, {698, 3933}, {730, 4769}, {732, 7855}, {754, 7757}, {760, 12782}, {1015, 12836}, {1500, 12837}, {1569, 2794}, {1571, 3097}, {2023, 7746}, {2782, 7748}, {2979, 8623}, {3117, 20859}, {3202, 9696}, {3229, 3981}, {3735, 3865}, {3767, 12251}, {3788, 5976}, {3934, 7867}, {5149, 10349}, {5475, 14881}, {5969, 7801}, {6309, 7813}, {6680, 7786}, {7749, 22712}, {7768, 9983}, {7782, 10351}, {7795, 18906}, {7796, 9865}, {7810, 13085}, {7822, 24256}, {7836, 8782}, {7906, 10335}, {9466, 11318}, {9651, 18982}, {9664, 13077}, {9753, 31401}, {10000, 10333}, {11272, 31455}

X(32452) = midpoint of X(194) and X(315)
X(32452) = reflection of X(i) in X(j) for these lines: {i,j}: {32, 39}, {76, 626}
X(32452) = reflection of X(32) in line PU(1)
X(32452) = anticomplement of X(18806)
X(32452) = 2nd-Brocard-Circle-inverse of X(1691)
X(32452) = X(30254)-Ceva conjugate of X(512)
X(32452) = circle-{{X(3102),X(3103),PU(1)}}-inverse of X(182)
X(32452) = X(163)-isoconjugate of X(30492)
X(32452) = barycentric quotient X(523)/X(30492)
X(32452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 30270, 5162}, {39, 5041, 13331}, {39, 5188, 2021}, {1569, 7756, 11257}, {1670, 1671, 1691}, {1689, 1690, 3398}, {2021, 5188, 5206}, {2458, 5028, 6}, {2562, 2563, 11171}, {3094, 3095, 39}, {3102, 3103, 182}, {5013, 32447, 39}, {5028, 30270, 32}, {13325, 13326, 3094}


X(32453) = MIDPOINT OF X(192) and X(194)

Barycentrics   a^3*b^3 + a^3*b^2*c + a^3*b*c^2 + a^3*c^3 - b^3*c^3 : :
X(32453) = X[3644] + 4 X[32450],4 X[3739] - 5 X[7786],4 X[3934] - 5 X[4687],5 X[4704] - X[20081],7 X[4751] - 8 X[6683],7 X[27268] - 5 X[31276]

X(32453) lies on these lines: {1, 87}, {2, 21443}, {37, 76}, {39, 75}, {42, 17486}, {511, 30273}, {518, 7976}, {536, 7757}, {538, 4664}, {561, 21814}, {579, 17034}, {700, 24696}, {716, 3774}, {730, 984}, {740, 12782}, {742, 3094}, {1045, 19566}, {1278, 26801}, {1575, 10009}, {1655, 9902}, {1921, 27091}, {2350, 17027}, {2782, 20430}, {3095, 29010}, {3097, 17759}, {3210, 17026}, {3550, 30661}, {3644, 32450}, {3739, 7786}, {3875, 24727}, {3934, 4687}, {3995, 17032}, {4660, 17493}, {4704, 20081}, {4751, 6683}, {6376, 21830}, {6382, 21877}, {8620, 30964}, {9055, 32449}, {16468, 30667}, {17028, 17495}, {17033, 17489}, {17262, 21788}, {24621, 27478}, {27268, 31276}, {27481, 31036}

X(32453) = midpoint of X(192) and X(194)
X(32453) = reflection of X(i) in X(j) for these lines: {i,j}: {75, 39}, {76, 37}
X(32453) = anticomplement of X(21443)
X(32453) = {X(192),X(19565)}-harmonic conjugate of X(1)


X(32454) = MIDPOINT OF X(149) and X(194)

Barycentrics    a*(a*b^2 - b^3 - a*b*c + a*c^2)*(a*b^2 - a*b*c + a*c^2 - c^3) : :
X(32454) = 2 X[119] - 3 X[262],4 X[3035] - 5 X[7786],3 X[3097] - X[5541],4 X[3934] - 5 X[31272],4 X[6713] - 3 X[22712],3 X[7709] - X[13199],2 X[12263] - 3 X[16173],X[12331] - 3 X[32447]

X(32454) lies on these lines: {3, 12199}, {6, 13194}, {11, 76}, {39, 100}, {80, 730}, {104, 511}, {106, 24516}, {119, 262}, {149, 194}, {528, 7757}, {538, 10707}, {726, 21630}, {952, 3095}, {1317, 12837}, {1320, 14839}, {1916, 2787}, {2530, 31129}, {2782, 10738}, {2802, 12782}, {3035, 7786}, {3045, 3202}, {3094, 9024}, {3097, 5541}, {3934, 31272}, {5533, 10079}, {5840, 11257}, {6713, 22712}, {7709, 13199}, {8068, 10063}, {9263, 23646}, {10742, 14881}, {12263, 16173}, {12331, 32447}, {13077, 13274}, {13273, 18982}

X(32454) = midpoint of X(149) and X(194)
X(32454) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 11}, {100, 39}, {10742, 14881}
X(32454) = {X(12836),X(12923)}-harmonic conjugate of X(76)


X(32455) = MIDPOINT OF X(6) and X(3629)

Barycentrics    -6*a^2 + b^2 + c^2 : :
X(32455) = 3 X[2] - 7 X[6],15 X[2] - 7 X[69],9 X[2] - 7 X[141],9 X[2] + 7 X[193],5 X[2] - 7 X[597],11 X[2] - 7 X[599],X[2] + 7 X[1992],6 X[2] - 7 X[3589],27 X[2] - 35 X[3618],57 X[2] - 49 X[3619],51 X[2] - 35 X[3620],3 X[2] + 7 X[3629],12 X[2] - 7 X[3631],39 X[2] - 35 X[3763],5 X[2] - 21 X[5032],3 X[2] + X[6144],9 X[2] - 14 X[6329],X[2] - 7 X[8584],33 X[2] + 7 X[11008],23 X[2] - 7 X[11160],19 X[2] - 7 X[15533],5 X[2] + 7 X[15534],39 X[2] - 7 X[20080],8 X[2] - 7 X[20582],2 X[2] - 7 X[20583],29 X[2] - 21 X[21356],25 X[2] - 21 X[21358],13 X[2] - 7 X[22165],X[5] - 3 X[15520],5 X[6] - X[69],3 X[6] - X[141],3 X[6] + X[193],5 X[6] - 3 X[597],11 X[6] - 3 X[599],X[6] + 3 X[1992],9 X[6] - 5 X[3618],19 X[6] - 7 X[3619],17 X[6] - 5 X[3620],7 X[6] - X[3630],4 X[6] - X[3631],13 X[6] - 5 X[3763],5 X[6] - 9 X[5032],7 X[6] + X[6144],3 X[6] - 2 X[6329],X[6] - 3 X[8584],11 X[6] + X[11008],23 X[6] - 3 X[11160],19 X[6] - 3 X[15533],5 X[6] + 3 X[15534],13 X[6] - X[20080],8 X[6] - 3 X[20582],2 X[6] - 3 X[20583],29 X[6] - 9 X[21356],25 X[6] - 9 X[21358],13 X[6] - 3 X[22165],X[66] - 3 X[23326],3 X[69] - 5 X[141],3 X[69] + 5 X[193],X[69] - 3 X[597],11 X[69] - 15 X[599],X[69] + 15 X[1992],2 X[69] - 5 X[3589],9 X[69] - 25 X[3618],19 X[69] - 35 X[3619],17 X[69] - 25 X[3620],X[69] + 5 X[3629],7 X[69] - 5 X[3630],4 X[69] - 5 X[3631],13 X[69] - 25 X[3763],X[69] - 9 X[5032],7 X[69] + 5 X[6144],3 X[69] - 10 X[6329],X[69] - 15 X[8584],11 X[69] + 5 X[11008],23 X[69] - 15 X[11160],19 X[69] - 15 X[15533],X[69] + 3 X[15534],13 X[69] - 5 X[20080],8 X[69] - 15 X[20582],2 X[69] - 15 X[20583],29 X[69] - 45 X[21356],5 X[69] - 9 X[21358],13 X[69] - 15 X[22165],5 X[141] - 9 X[597],11 X[141] - 9 X[599],X[141] + 9 X[1992],2 X[141] - 3 X[3589],3 X[141] - 5 X[3618],19 X[141] - 21 X[3619],17 X[141] - 15 X[3620],X[141] + 3 X[3629],7 X[141] - 3 X[3630],4 X[141] - 3 X[3631],13 X[141] - 15 X[3763],5 X[141] - 27 X[5032],7 X[141] + 3 X[6144],X[141] - 9 X[8584],11 X[141] + 3 X[11008],23 X[141] - 9 X[11160],19 X[141] - 9 X[15533],5 X[141] + 9 X[15534],13 X[141] - 3 X[20080],8 X[141] - 9 X[20582],2 X[141] - 9 X[20583],29 X[141] - 27 X[21356]

X(32455) lies on these lines: {2, 6}, {5, 15520}, {39, 3793}, {44, 17390}, {66, 23326}, {140, 15516}, {182, 15712}, {239, 7277}, {487, 6432}, {488, 6431}, {511, 548}, {518, 3635}, {536, 4856}, {542, 14893}, {545, 4852}, {575, 12108}, {576, 1353}, {594, 17120}, {625, 5305}, {648, 6748}, {698, 5052}, {742, 4726}, {894, 4399}, {895, 25329}, {1086, 17121}, {1100, 17332}, {1285, 8716}, {1350, 21735}, {1351, 1657}, {1352, 5072}, {1449, 4364}, {1743, 17243}, {1974, 15471}, {1990, 27377}, {2321, 28337}, {2325, 4889}, {2393, 21847}, {2854, 9969}, {3124, 10552}, {3564, 3850}, {3625, 4663}, {3633, 3751}, {3663, 28333}, {3686, 4472}, {3707, 28639}, {3729, 28309}, {3739, 4700}, {3758, 7227}, {3759, 4395}, {3818, 23046}, {3843, 5093}, {3875, 28297}, {3879, 4422}, {3912, 16671}, {3946, 4715}, {4357, 16668}, {4361, 7222}, {4363, 4371}, {4370, 17315}, {4393, 17334}, {4402, 4644}, {4405, 17118}, {4416, 16666}, {4478, 17363}, {4643, 16667}, {4665, 5839}, {4667, 17348}, {4691, 5847}, {4718, 9055}, {4725, 17355}, {4755, 4909}, {4851, 16670}, {4946, 9024}, {4966, 16477}, {4971, 17351}, {4991, 17771}, {5007, 7789}, {5008, 6390}, {5041, 7767}, {5092, 14891}, {5102, 6776}, {5111, 19695}, {5368, 8361}, {5596, 17813}, {5965, 12812}, {6467, 8705}, {6593, 32114}, {6664, 31506}, {6749, 9308}, {7238, 17364}, {7745, 7760}, {7762, 7790}, {7798, 18907}, {7801, 14075}, {7819, 7890}, {7839, 14712}, {7877, 7937}, {7882, 8364}, {7949, 8363}, {8787, 14928}, {9019, 32366}, {9027, 9822}, {9607, 20065}, {9973, 15531}, {10022, 28634}, {10115, 32284}, {10168, 14890}, {10192, 21970}, {11179, 15689}, {11216, 15583}, {11422, 32269}, {11477, 14912}, {12017, 15706}, {12161, 16252}, {12212, 13196}, {13330, 32449}, {14032, 18906}, {14892, 18358}, {15492, 29574}, {15585, 32218}, {15684, 31670}, {15826, 32220}, {16176, 25320}, {17014, 17255}, {17246, 20072}, {17340, 17377}, {17347, 17395}, {17350, 17388}, {21639, 26926}, {25331, 32255}, {25488, 32127}

X(32455) = midpoint of X(i) and X(j) for these lines: {i,j}: {6, 3629}, {141, 193}, {576, 1353}, {597, 15534}, {895, 25329}, {1351, 8550}, {1992, 8584}, {3630, 6144}, {7798, 18907}, {13330, 32449}, {15826, 32220}
X(32455) = reflection of X(i) in X(j) for these lines: {i,j}: {140, 15516}, {141, 6329}, {3589, 6}, {3631, 3589}, {18583, 22330}, {20583, 8584}
X(32455) = complement of X(3630)
X(32455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6144, 3630}, {6, 69, 597}, {6, 141, 6329}, {6, 193, 141}, {6, 1992, 3629}, {6, 6144, 2}, {6, 15534, 69}, {69, 5032, 6}, {141, 3629, 193}, {141, 6329, 3589}, {597, 8584, 5032}, {894, 4969, 4399}, {1992, 5032, 15534}, {3589, 3631, 20582}, {3589, 20583, 6}, {3629, 8584, 6}, {3629, 20583, 3631}, {3758, 17362, 7227}, {3759, 17365, 4395}, {3763, 20080, 22165}, {3879, 16669, 4422}, {4416, 16666, 17045}, {5032, 15534, 597}, {7585, 26340, 591}, {7586, 26339, 1991}, {17330, 17379, 6707}, {17363, 17369, 4478}, {17364, 17366, 7238}
X(32455) = crosssum of X(6) and X(15513)
X(32455) = crossdifference of every pair of points on line {512, 15514}


X(32456) = MIDPOINT OF X(99) and X(187)

Barycentrics    4*a^4 - 3*a^2*b^2 - 3*a^2*c^2 + 2*b^2*c^2 : :
X(32456) = 3 X[99] + X[385],X[99] + 3 X[13586],3 X[187] - X[385],X[187] - 3 X[13586],2 X[187] + X[15301],X[230] - 3 X[27088],X[325] - 3 X[2482],X[325] + 3 X[8598],X[385] - 9 X[13586],2 X[385] + 3 X[15301],X[671] - 3 X[5215],3 X[1692] - X[10754],3 X[2482] + X[6781],5 X[5026] - 3 X[13196],X[5107] - 3 X[5182],X[6781] - 3 X[8598],3 X[7799] - X[7845],3 X[7799] + X[14712],5 X[7925] + 3 X[9855],5 X[7925] - 3 X[31173],X[8352] - 3 X[9167],X[8591] + 3 X[26613],3 X[10150] - 4 X[22247],X[11676] + 3 X[21166],X[13449] - 3 X[15561],6 X[13586] + X[15301],3 X[14041] - 5 X[31275],3 X[14568] + X[20094],X[18860] - 3 X[21166],3 X[21445] + X[23235]

X(32456) lies on these lines: {2, 8589}, {3, 3734}, {20, 3788}, {30, 620}, {32, 31859}, {39, 3552}, {76, 15513}, {83, 31652}, {99, 187}, {126, 7426}, {141, 8703}, {183, 8588}, {230, 543}, {232, 4235}, {325, 2482}, {376, 7761}, {382, 7862}, {384, 6683}, {439, 3767}, {511, 5026}, {524, 14148}, {548, 7789}, {550, 626}, {574, 1003}, {671, 5215}, {754, 6390}, {1153, 11164}, {1384, 7798}, {1495, 5108}, {1506, 19687}, {1657, 7825}, {1692, 10754}, {1975, 5206}, {2030, 5969}, {2549, 7817}, {3053, 7781}, {3363, 7619}, {3522, 7795}, {3528, 7800}, {3534, 7778}, {3926, 7882}, {4027, 10631}, {4045, 8369}, {4048, 14810}, {4226, 11052}, {5007, 7783}, {5008, 7757}, {5023, 7751}, {5031, 29323}, {5107, 5182}, {5118, 9418}, {5148, 15452}, {5149, 11676}, {6337, 7759}, {6636, 15822}, {6655, 7874}, {6658, 7769}, {6704, 19697}, {7492, 30749}, {7603, 11361}, {7618, 7736}, {7622, 11159}, {7748, 7886}, {7750, 7863}, {7756, 7807}, {7763, 7843}, {7770, 15515}, {7771, 9466}, {7777, 14537}, {7784, 15696}, {7791, 7915}, {7799, 7845}, {7801, 7848}, {7802, 7821}, {7808, 15815}, {7820, 8356}, {7833, 7835}, {7836, 7873}, {7844, 11288}, {7847, 7852}, {7865, 15688}, {7870, 7898}, {7897, 11057}, {7910, 7945}, {7925, 9855}, {8176, 11147}, {8182, 15589}, {8352, 9167}, {8591, 26613}, {8667, 15655}, {9341, 25264}, {10150, 22247}, {11286, 15482}, {13349, 22689}, {13350, 22687}, {13449, 15561}, {14035, 31455}, {14041, 31275}, {14568, 20094}, {14762, 15491}, {14767, 18570}, {14928, 15993}, {15031, 16923}, {15300, 22329}, {15810, 16986}, {16589, 17693}, {21445, 23235}, {21937, 24275}, {26276, 31128}

X(32456) = midpoint of X(i) and X(j) for these lines: {i,j}: {99, 187}, {325, 6781}, {2482, 8598}, {7845, 14712}, {9855, 31173}, {11676, 18860}, {14928, 15993}, {15300, 22329}
X(32456) = reflection of X(i) in X(j) for these lines: {i,j}: {625, 620}, {15301, 99}
X(32456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7816, 3934}, {20, 3788, 7842}, {99, 13586, 187}, {325, 8598, 6781}, {548, 7789, 7830}, {574, 1003, 7804}, {1384, 8716, 7798}, {1975, 5206, 7780}, {2482, 6781, 325}, {3053, 7781, 7805}, {3552, 7782, 39}, {7748, 16925, 7886}, {7750, 7863, 7895}, {7756, 7807, 7861}, {7789, 7830, 7849}, {7799, 14712, 7845}, {7801, 14907, 7848}, {7802, 7891, 7821}, {7833, 7835, 7853}, {11676, 21166, 18860}


X(32457) = MIDPOINT OF X(148) and X(187)

Barycentrics   a^2*b^2 + 2*b^4 + a^2*c^2 - 6*b^2*c^2 + 2*c^4 : :
X(32457) = 3 X[115] - X[325],5 X[115] - X[7813],X[148] + 3 X[14568],X[187] - 3 X[14568],5 X[230] - 3 X[27088],X[316] + 3 X[19570],2 X[325] - 3 X[625],5 X[325] - 3 X[7813],X[385] + 3 X[671],5 X[385] + 3 X[8597],5 X[625] - 2 X[7813],5 X[671] - X[8597],3 X[5215] - X[8591],4 X[5461] - 3 X[10150],3 X[5461] - X[14148],X[6781] - 3 X[22329],X[7779] + 3 X[11054],X[7779] - 3 X[31173],3 X[7799] - 5 X[31275],X[7845] - 3 X[14041],5 X[7925] - 9 X[9166],X[8596] + 3 X[26613],9 X[10150] - 4 X[14148],3 X[14651] - X[18860]}

X(32457) lies on these lines: {4, 7805}, {5, 32450}, {6, 18546}, {39, 16921}, {76, 7849}, {115, 325}, {148, 187}, {183, 11648}, {230, 543}, {316, 19570}, {381, 7798}, {385, 671}, {546, 7838}, {620, 15301}, {754, 15480}, {1975, 7886}, {2996, 3767}, {3314, 14711}, {3629, 3845}, {3734, 7817}, {3934, 4045}, {5008, 11361}, {5025, 7895}, {5041, 16044}, {5215, 8591}, {5304, 7620}, {5309, 7804}, {5346, 14035}, {5355, 8370}, {5461, 10150}, {6390, 6722}, {6392, 7759}, {6683, 7765}, {6781, 22329}, {7603, 7757}, {7615, 7736}, {7617, 31489}, {7748, 7780}, {7751, 7842}, {7754, 7843}, {7761, 15589}, {7766, 14537}, {7774, 18424}, {7775, 22253}, {7779, 11054}, {7781, 13881}, {7790, 9466}, {7799, 31275}, {7814, 20105}, {7821, 20081}, {7825, 7882}, {7839, 15031}, {7841, 7848}, {7844, 7880}, {7845, 14041}, {7851, 7915}, {7852, 17128}, {7855, 14063}, {7864, 31239}, {7873, 17129}, {7877, 14062}, {7908, 11318}, {7917, 14045}, {7925, 9166}, {8589, 17004}, {8596, 26613}, {11174, 14762}, {14651, 18860}, {15048, 15491}

X(32457) = midpoint of X(i) and X(j) for these lines: {i,j}: {148, 187}, {11054, 31173}
X(32457) = reflection of X(i) in X(j) for these lines: {i,j}: {625, 115}, {6390, 6722}, {15301, 620}
X(32457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 7861, 7849}, {148, 14568, 187}, {5309, 11185, 7804}, {7841, 17131, 7848}, {7851, 17130, 7915}


X(32458) = MIDPOINT OF X(99) and X(315)

Barycentrics   (a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :
X(32458) = X[5989] + 3 X[7788],4 X[6680] - 5 X[31274],4 X[6722] - 5 X[7867]

X(32458) lies on these lines: {2, 2987}, {20, 99}, {32, 620}, {69, 98}, {76, 115}, {114, 325}, {132, 877}, {141, 2023}, {183, 6036}, {305, 1972}, {439, 20065}, {524, 12829}, {542, 5989}, {543, 7818}, {670, 15526}, {736, 1569}, {754, 2482}, {1007, 9753}, {1506, 18806}, {1975, 23698}, {2458, 4027}, {2782, 3933}, {2967, 23098}, {3266, 23965}, {3329, 6680}, {3425, 9723}, {4558, 11610}, {5017, 9766}, {5026, 12830}, {5149, 7759}, {5152, 7768}, {5181, 23342}, {5503, 21356}, {5984, 10513}, {6033, 7776}, {6337, 21166}, {6722, 7867}, {7767, 12042}, {7836, 10350}, {7896, 8178}, {7897, 8782}, {8024, 23962}, {9146, 30789}, {10722, 32006}

X(32458) = midpoint of X(99) and X(315)
X(32458) = reflection of X(i) in X(j) for these lines: {i,j}: {32, 620}, {115, 626}
X(32458) = X(i)-isoconjugate of X(j) for these (i,j): {1821, 14601}, {1910, 1976}
X(32458) = crosspoint of X(2396) and X(4590)
X(32458) = crosssum of X(i) and X(j) for these (i,j): {1501, 14601}, {2422, 3124}
X(32458) = crossdifference of every pair of points on line {2422, 14601}
X(32458) = barycentric product X(i)*X(j)for these lines: {i,j}: {297, 6393}, {305, 2967}, {325, 325}, {561, 23996}, {850, 15631}, {877, 6333}, {1502, 11672}, {2396, 2799}, {16725, 27801}, {18024, 23098}
X(32458) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {237, 14601}, {297, 6531}, {325, 98}, {511, 1976}, {684, 878}, {877, 685}, {1355, 1397}, {1959, 1910}, {2396, 2966}, {2421, 2715}, {2799, 2395}, {2967, 25}, {3289, 14600}, {3569, 2422}, {6333, 879}, {6393, 287}, {7062, 2175}, {9419, 1501}, {11672, 32}, {15631, 110}, {16725, 1333}, {18829, 18858}, {23098, 237}, {23611, 9418}, {23996, 31}
X(32458) = X(325),X(5976)}-harmonic conjugate of X(114),is on K778,X(i)-Ceva conjugate of X(j) for these (i,j): {670, 6333}, {4590, 2396}


X(32459) = MIDPOINT OF X(99) and X(230)

Barycentrics   (2*a^2 - b^2 - c^2)*(3*a^2 - b^2 - c^2) : :
X(32459) = 5 X[99] + 3 X[14568],X[187] + 3 X[2482],3 X[187] - X[3793],3 X[187] + X[7813],X[187] - 3 X[27088],5 X[230] - 3 X[14568],X[316] + 3 X[8598],X[316] - 3 X[22110],X[325] + 3 X[13586],3 X[325] + X[14712],3 X[620] - X[625],X[1513] + 3 X[21166],9 X[2482] + X[3793],3 X[2482] - X[6390],9 X[2482] - X[7813],X[3793] + 3 X[6390],X[3793] - 9 X[27088],3 X[5215] + X[15300],3 X[6390] - X[7813],X[6390] + 3 X[27088],X[7813] + 9 X[27088],9 X[9167] - 5 X[31275],3 X[10256] - X[15980],9 X[13586] - X[14712]

X(32459) lies on these lines: {2, 11147}, {3, 66}, {5, 9734}, {30, 620}, {39, 6329}, {69, 5210}, {99, 230}, {126, 468}, {140, 7816}, {187, 524}, {193, 439}, {316, 8598}, {325, 13586}, {376, 7778}, {548, 626}, {549, 3734}, {550, 3788}, {574, 3589}, {597, 5024}, {599, 5585}, {698, 2021}, {1003, 3815}, {1213, 21937}, {1384, 3629}, {1513, 21166}, {1975, 17008}, {2080, 13196}, {2549, 11288}, {3054, 11185}, {3055, 8370}, {3522, 7784}, {3524, 15271}, {3530, 3934}, {3552, 7745}, {3566, 3798}, {3580, 5866}, {3618, 5013}, {3627, 7862}, {3630, 15655}, {3631, 7801}, {3712, 7267}, {3926, 5023}, {3933, 5206}, {3972, 9300}, {4045, 8368}, {4760, 7181}, {5215, 15300}, {5241, 21511}, {5254, 16925}, {5306, 31859}, {5467, 15471}, {5743, 16436}, {6683, 19697}, {7472, 16320}, {7664, 24855}, {7735, 8716}, {7750, 7891}, {7756, 8361}, {7761, 8703}, {7767, 7863}, {7769, 19687}, {7782, 7790}, {7815, 15712}, {7820, 8359}, {7825, 15704}, {7835, 7937}, {7842, 12103}, {7853, 8354}, {7874, 8357}, {7899, 19695}, {7908, 14929}, {7934, 8353}, {8182, 22165}, {8355, 22247}, {8362, 15515}, {8584, 11165}, {9167, 31275}, {9737, 21850}, {9771, 11159}, {10011, 23698}, {10256, 15980}, {11149, 11151}, {11286, 15491}, {13468, 21843}, {14001, 15815}, {14033, 31489}, {17005, 19686}, {18860, 29181}, {19661, 20583}, {21163, 24256}

X(32459) = midpoint of X(i) and X(j) for these lines: {i,j}: {99, 230}, {187, 6390}, {2482, 27088}, {3793, 7813}, {7472, 16320}, {8598, 22110}
X(32459) = reflection of X(8355) in X(22247)
X(32459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 2482, 6390}, {187, 7813, 3793}, {439, 6337, 3053}, {574, 8369, 3589}, {3793, 6390, 7813}, {6390, 27088, 187}, {7820, 8589, 8359}, {7863, 15513, 7767}
X(32459) = X(468)-Ceva conjugate of X(524)
X(32459) = X(i)-isoconjugate of X(j) for these (i,j): {111, 8769}, {897, 8770}, {923, 2996}, {3565, 23894}
X(32459) = crossdifference of every pair of points on line {2485, 8770}
X(32459) = barycentric product X(i)*X(j)for these lines: {i,j}: {193, 524}, {468, 6337}, {896, 18156}, {1707, 14210}, {3053, 3266}, {3566, 5468}, {3712, 17081}, {4028, 6629}, {6353, 6390}, {16741, 21874}
X(32459) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {187, 8770}, {193, 671}, {524, 2996}, {896, 8769}, {1707, 897}, {3053, 111}, {3167, 895}, {3292, 6391}, {3566, 5466}, {5095, 5203}, {5467, 3565}, {6091, 15398}, {6337, 30786}, {6353, 17983}, {6390, 6340}, {8651, 9178}, {19118, 8753}


X(32460) = EULER LINE INTERCEPT OF TRILINEAR POLAR OF X(14)

Barycentrics    -2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)*S+3*(a^2+b^2-c^2)*(2*a^2-b^2-c^2)*(a^2+c^2-b^2) : :
Barycentrics    9*(4*R^2-SW)*S^2+3*SB*SC*SW+S*sqrt(3)*(S^2-3*SB*SC) : :

As a point on the Euler line, X(32460) has Shinagawa coefficients (-3*F+sqrt(3)*S/3, E+F-sqrt(3)*S).

See Konstantin Knop, Vladimir Dubrovsky and CÚsar Lozada, Hyacinthos 29009.

X(32460) lies on these lines: {2, 3}, {13, 11657}, {14, 3166}, {15, 16319}, {110, 11092}, {396, 523}, {398, 8015}, {476, 8838}, {530, 32225}, {531, 5642}, {533, 3292}, {627, 15794}, {635, 15785}, {1525, 2777}, {2453, 16644}, {3163, 23713}, {3284, 23715}, {3642, 5651}, {5318, 30468}, {6105, 6107}, {8836, 20253}, {14704, 22510}, {18487, 23712}

X(32460) = midpoint of X(i) and X(j) for these lines: {i,j}: {110, 11092}, {16179, 16181}
X(32460) = reflection of X(32461) in X(468)
X(32460) = isogonal conjugate of the antigonal conjugate of X(2993)
X(32460) = circumcircle-inverse-of X(3130)
X(32460) = inner-Napoleon circle-inverse-of X(4)
X(32460) = orthoptic circle of Steiner inellipse-inverse-of X(383)
X(32460) = polar circle-inverse-of X(471)
X(32460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1316, 32461), (3, 4, 23722), (1113, 1114, 3130)


X(32461) = EULER LINE INTERCEPT OF TRILINEAR POLAR OF X(13)

Barycentrics    2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)*S+3*(a^2+b^2-c^2)*(2*a^2-b^2-c^2)*(a^2+c^2-b^2) : :
Barycentrics    9*(4*R^2-SW)*S^2+3*SB*SC*SW-S*sqrt(3)*(S^2-3*SB*SC) : :

As a point on the Euler line, X(32461) has Shinagawa coefficients (-3*F-sqrt(3)*S/3, E+F+sqrt(3)*S).

See Konstantin Knop, Vladimir Dubrovsky and CÚsar Lozada, Hyacinthos 29009.

X(32461) lies on these lines: {2, 3}, {13, 3165}, {14, 11657}, {16, 16319}, {62, 8919}, {110, 11078}, {395, 523}, {397, 8014}, {476, 8836}, {530, 5642}, {531, 32225}, {532, 3292}, {628, 15793}, {636, 15784}, {1524, 2777}, {2453, 16645}, {3163, 23712}, {3284, 23714}, {3643, 5651}, {5321, 30465}, {6104, 6106}, {8838, 20252}, {14705, 22511}, {18487, 23713}

X(32461) = midpoint of X(i) and X(j) for these lines: {i,j}: {110, 11078}, {16180, 16182}
X(32461) = reflection of X(32460) in X(468)
X(32461) = isogonal conjugate of the antigonal conjugate of X(2992)
X(32461) = circumcircle-inverse-of X(3129)
X(32461) = outer-Napoleon circle-inverse-of X(4)
X(32461) = orthoptic circle of Steiner inellipse-inverse-of X(1080)
X(32461) = polar circle-inverse-of X(470)
X(32461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1316, 32460), (3, 4, 23721), (1113, 1114, 3129)






leftri  Points associated with the inscribed triply bilogic triangle: X(32462)-X(32480)  rightri

Centers X(32462)-X(32480) were contributed by Peter Moses, May 11, 2019.

The appearance of (i,j) in the following list means that X(j) = ITB-isogonal conjugate of X(i), where the ITB triangle (inscribed triply bilogic triangle) is described at K003 and K1098.

(1,32462), (2,32463), (5,32464), (15,32465, (16,32466), (25,32514), (30,32515), (39,32467), (40,32468), (110,32592), (140,32516), (186,32517), (187,32469), (237,32518), (371,32470), (372,32471), (381,32519), (382,32520), (399,32595), (550,32521), (616,32596), (617,32597), (631,32522), (632,32523), (729,32472), (755,32473), (1350,32474), (2370,32475), (3398,32476), (5171,32477), (5966,32478), (8600,32479), (11842,32480)

underbar

X(32462) = X(1)X(85)∩X(3)X(238)

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

X(32462) lies on these lines: {1, 85}, {3, 238}, {10, 26068}, {43, 170}, {78, 32117}, {497, 20731}, {954, 4335}, {1045, 1047}, {1053, 2082}, {2808, 12782}, {3010, 3212}, {15726, 24696}, {19545, 20368}, {21778, 21882}

X(32462) = excentral-isogonal conjugate of X(1759)
X(32462) = excentral-isotomic conjugate of X(1766)
X(32462) = X(41)-Ceva conjugate of X(1)
X(32462) = X(21882)-cross conjugate of X(21218)
X(32462) = crosspoint of X(21218) and X(31604)
X(32462) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 21218}, {9, 31604}, {75, 21778}, {86, 21882}, {92, 23078}
X(32462) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {21218, 75}, {21778, 1}, {21882, 10}, {23078, 63}, {31604, 85}


X(32463) = X(3)X(3231)∩X(30)X(1351)

Barycentrics    a^2*(a^6*b^2 - 4*a^4*b^4 + 3*a^2*b^6 + a^6*c^2 + 5*a^4*b^2*c^2 - 9*a^2*b^4*c^2 - 3*b^6*c^2 - 4*a^4*c^4 - 9*a^2*b^2*c^4 + 10*b^4*c^4 + 3*a^2*c^6 - 3*b^2*c^6) : :
X(32463) = 2 X[11472] - 3 X[32444]

X(32463) lies on these lines: {3, 3231}, {30, 1351}, {182, 30229}, {2080, 8717}, {3095, 14915}, {3426, 10983}, {6321, 7706}, {11472, 32444}


X(32464) = X(576)X(3146)∩X(1176)X(30496)

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

X(32464) lies on these lines: {576, 3146}, {1176, 30496}, {5206, 6030}, {7802, 23061}


X(32465) = REFLECTION OF X(13) IN X(3107)

Barycentrics    3*a^6*b^2 - 5*a^4*b^4 + 2*a^2*b^6 + 3*a^6*c^2 - 11*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - 2*b^2*c^6 - 2*Sqrt[3]*a^2*(a^2*b^2 + a^2*c^2 + b^2*c^2)*S : :
X(32465) = 3 X[5469] - 4 X[22692],3 X[16530] - 2 X[22702]

X(32465) lies on these lines: {6, 1569}, {13, 2782}, {14, 32447}, {16, 7709}, {17, 13108}, {18, 39}, {61, 194}, {62, 32448}, {262, 11602}, {511, 5473}, {538, 16962}, {3095, 16964}, {3105, 11257}, {3106, 5463}, {5238, 12251}, {5469, 9762}, {6194, 10645}, {6777, 22695}, {7697, 16966}, {8594, 9114}, {9112, 22701}, {11171, 16242}, {19107, 22728}

X(32465) = reflection of X(13) in X(3107)
X(32465) = reflection of X(32466) in X(1569)


X(32466) = REFLECTION OF X(14) IN X(3106)

Barycentrics    3*a^6*b^2 - 5*a^4*b^4 + 2*a^2*b^6 + 3*a^6*c^2 - 11*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - 2*b^2*c^6 + 2*Sqrt[3]*a^2*(a^2*b^2 + a^2*c^2 + b^2*c^2)*S : :
X(32466) = 3 X[5470] - 4 X[22691],3 X[16529] - 2 X[22701]

X(32466) lies on these lines: {6, 1569}, {13, 32447}, {14, 2782}, {15, 7709}, {17, 39}, {18, 13108}, {61, 32448}, {62, 194}, {262, 11603}, {511, 5474}, {538, 16963}, {3095, 16965}, {3104, 11257}, {3107, 5464}, {5237, 12251}, {5470, 9760}, {6194, 10646}, {6778, 22696}, {7697, 16967}, {8595, 9116}, {9113, 22702}, {11171, 16241}, {19106, 22728}

X(32466) = reflection of X(14) in X(3106)
X(32466) = reflection of X(32465) in X(1569)


X(32467) = REFLECTION OF X(3399) IN X(39)

Barycentrics    a^8 + a^6*b^2 - 3*a^4*b^4 + a^2*b^6 + a^6*c^2 - 9*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6 : :
X(32467) = 2 X[4] - 3 X[14492],14 X[3526] - 9 X[10302],4 X[7839] + X[12122],3 X[9302] - 4 X[11623],3 X[9751] - 4 X[12054]

X(32467) lies on these lines: {3, 6179}, {4, 7739}, {5, 6054}, {6, 11257}, {20, 576}, {30, 12156}, {32, 7709}, {39, 98}, {54, 826}, {83, 2782}, {99, 3398}, {114, 7797}, {140, 7799}, {182, 194}, {262, 9605}, {384, 575}, {385, 13334}, {401, 13366}, {511, 7839}, {572, 12197}, {631, 7622}, {1078, 11171}, {1353, 7750}, {1506, 14651}, {1569, 12176}, {1656, 7884}, {1975, 5050}, {2896, 5965}, {3095, 12203}, {3329, 6248}, {3413, 14630}, {3414, 14631}, {3526, 7607}, {3734, 10359}, {5007, 11676}, {5013, 9755}, {5028, 7738}, {5111, 12007}, {5171, 7766}, {5254, 14639}, {5286, 9744}, {6776, 9873}, {7608, 9302}, {7754, 22712}, {7783, 13335}, {7798, 12251}, {7805, 21163}, {7817, 23234}, {7829, 14981}, {7858, 15980}, {8550, 9607}, {8721, 9993}, {9751, 12054}, {10722, 15048}, {11272, 12188}, {14880, 32447}, {15819, 17129}

X(32467) = reflection of X(3399) in X(39)
X(32467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 11257, 12110}, {3398, 32448, 99}, {7783, 13335, 21166}


X(32468) = X(31)-CEVA CONJUGATE OF X(9)

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

X(32468) lies on these lines: {6, 979}, {8, 23640}, {9, 341}, {40, 740}, {55, 2053}, {200, 4531}, {1018, 7031}, {1043, 4050}, {2136, 24578}, {3503, 3875}, {3729, 24579}, {3880, 17448}, {5710, 17754}, {14829, 17144}, {17752, 19591}

X(32468) = X(31)-Ceva conjugate of X(9)
X(32468) = barycentric product X(318)*X(23159)
X(32468) = barycentric quotient X(23159)/X(77)
X(32468) = {X(2319),X(3208)}-harmonic conjugate of X(2329)


X(32469) = REFLECTION OF X(98) IN X(7709)

Barycentrics    3*a^10*b^2 - 11*a^8*b^4 + 13*a^6*b^6 - 7*a^4*b^8 + 2*a^2*b^10 + 3*a^10*c^2 - 23*a^8*b^2*c^2 + 23*a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 4*a^2*b^8*c^2 - 2*b^10*c^2 - 11*a^8*c^4 + 23*a^6*b^2*c^4 + 3*a^4*b^4*c^4 - 6*a^2*b^6*c^4 + 9*b^8*c^4 + 13*a^6*c^6 - 11*a^4*b^2*c^6 - 6*a^2*b^4*c^6 - 14*b^6*c^6 - 7*a^4*c^8 + 4*a^2*b^2*c^8 + 9*b^4*c^8 + 2*a^2*c^10 - 2*b^2*c^10 : :
X(32469) = X[98] - 4 X[1569],2 X[194] + X[23235],4 X[262] - 3 X[14639],3 X[5182] - 2 X[31958],2 X[6194] - 3 X[21166],4 X[11152] - X[12117]

X(32469) lies on these lines: {39, 7608}, {98, 574}, {99, 2080}, {194, 576}, {262, 381}, {511, 8593}, {538, 19911}, {5182, 31958}, {6194, 21166}, {7617, 23234}, {7697, 8179}, {8586, 10753}, {8724, 11054}, {9166, 14159}, {10723, 22728}, {12191, 15520}

X(32469) = reflection of X(i) in X(j) for these lines: {i,j}: {98, 7709}, {671, 32447}, {7709, 1569}, {10723, 22728}


X(32470) = X(6)X(32134)∩X(39)X(486)

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

X(32470) lies on these lines: {6, 32134}, {39, 486}, {76, 5418}, {194, 371}, {372, 7709}, {485, 2782}, {511, 12124}, {590, 13108}, {698, 19145}, {1504, 1569}, {3071, 32447}, {3095, 6561}, {3102, 7757}, {3103, 6560}, {5420, 11171}, {6200, 12251}, {9540, 20081}, {9683, 9917}, {11152, 13669}, {14881, 22615}

X(32470) = {X(6),X(32448)}-harmonic conjugate of X(32471)
X(32470) = {X(3103),X(11257)}-harmonic conjugate of X(6560)


X(32471) = X(6)X(32134)∩X(39)X(485)

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

X(32471) lies on these lines: {6, 32134}, {39, 485}, {76, 5420}, {194, 372}, {371, 7709}, {486, 2782}, {511, 12123}, {615, 13108}, {698, 19146}, {1505, 1569}, {3070, 32447}, {3095, 6560}, {3102, 6561}, {3103, 7757}, {5418, 11171}, {11152, 13789}, {13935, 20081}, {14881, 22644}

X(32471) = {X(6),X(32448)}-harmonic conjugate of X(32470)
X(32471) = {X(3103),X(11257)}-harmonic conjugate of X(6561)


X(32472) = X(2)X(8644)∩X(3)X(9489)

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

X(32472) lies on these lines: {2, 8644}, {3, 9489}, {4, 2489}, {30, 511}, {98, 5970}, {114, 9152}, {182, 9426}, {351, 31176}, {381, 9175}, {647, 5996}, {669, 9148}, {850, 3804}, {887, 11616}, {2485, 11620}, {3818, 11182}, {3830, 9178}, {3835, 8653}, {4108, 31174}, {4140, 4761}, {4170, 7212}, {5466, 14458}, {5926, 16235}, {8430, 10722}, {8651, 11176}, {9123, 13306}, {9137, 10989}, {9185, 13307}, {24782, 26148}, {25084, 25299}

> X(32472) = Thomson-isogonal conjugate of X(729)
X(32472) = Lucas-isogonal conjugate of X(729)
X(32472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {850, 31299, 3804}, {5996, 9147, 647}


X(32473) = X(3)X(18105)∩X(30)X(511)

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

X(32473) lies on these lines: {3, 18105}, {30, 511}, {1637, 5996}, {1649, 11633}, {3005, 14420}, {3098, 14318}, {3804, 6563}, {4108, 14417}, {5466, 14492}, {9123, 13308}, {9125, 15724}, {9134, 31176}, {9180, 9302}, {9185, 13309}, {11616, 21006}, {21732, 32467}

X(32473) = Lucas-isogonal conjugate of X(755)
X(32473) = Thomson-isogonal conjugate of X(755)


X(32474) = REFLECTION OF X(4) IN X(9764)

Barycentrics    3*a^6*b^2 - 8*a^4*b^4 - 3*a^2*b^6 + 3*a^6*c^2 - 7*a^4*b^2*c^2 - a^2*b^4*c^2 + 3*b^6*c^2 - 8*a^4*c^4 - a^2*b^2*c^4 + 10*b^4*c^4 - 3*a^2*c^6 + 3*b^2*c^6 : :
X(32474) = 3 X[5485] - 4 X[9466],3 X[7709] - 4 X[8716],2 X[7757] - 3 X[9741]

X(32474) lies on these lines: {4, 9764}, {39, 18841}, {194, 1003}, {376, 538}, {511, 15428}, {698, 5085}, {2782, 3424}, {5485, 9466}, {7757, 9741}, {8782, 11148}

X(32474) = reflection of X(4) in X(9764)


X(32475) = X(3)X(1459)∩X(4)X(20293)

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

X(32475) lies on these lines: {3, 1459}, {4, 20293}, {5, 20316}, {30, 511}, {40, 21173}, {573, 6586}, {1293, 15403}, {3261, 10446}, {4791, 15488}

X(32475) = isogonal conjugate of X(32704)
X(32475) = Lucas-isogonal conjugate of X(2370)
X(32475) = Thomson-isogonal conjugate of X(2370)
X(32475) = crossdifference of every pair of points on line {6, 8756}


X(32476) = MIDPOINT OF X(194) AND X(2896)

Barycentrics    2*a^4*b^4 + a^2*b^6 + 3*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 2*a^4*c^4 + 2*a^2*b^2*c^4 - b^4*c^4 + a^2*c^6 : :
X(32476) = 3 X[262] - 2 X[6249],3 X[3097] - X[9903],4 X[3934] - 5 X[31268],4 X[6704] - 5 X[7786],3 X[7709] - X[12252],3 X[7833] - X[9990],3 X[9751] - 4 X[13334],X[13111] - 3 X[32447]

X(194) and X(2896) are the centers of the 1st and 2nd Neuberg radical circles, resp.

X(32476) lies on these lines: {5, 9772}, {6, 10131}, {20, 3095}, {39, 83}, {69, 194}, {76, 4045}, {115, 32190}, {262, 6249}, {385, 6308}, {511, 7839}, {538, 31168}, {574, 8150}, {730, 12783}, {736, 7847}, {754, 7757}, {1975, 24273}, {2549, 31982}, {2782, 3399}, {3097, 9903}, {3314, 6309}, {3663, 17760}, {3934, 7923}, {4048, 10345}, {5254, 9478}, {5976, 7797}, {6296, 6775}, {6297, 6772}, {6337, 16898}, {6655, 9866}, {6656, 9865}, {6683, 16896}, {6704, 7786}, {7750, 32449}, {7766, 9821}, {7787, 13331}, {7790, 8149}, {7827, 18806}, {7977, 14839}, {9751, 13334}, {9765, 14041}, {9855, 12156}, {9873, 32429}, {10329, 16985}, {10333, 12215}, {10350, 12216}, {11257, 29012}, {12836, 13078}, {12837, 18983}, {12944, 18982}, {12954, 13077}, {13111, 32447}, {13357, 13586}

X(32476) = midpoint of X(194) and X(2896)
X(32476) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 6292}, {83, 39}
X(32476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {194, 7791, 9983}, {7864, 10335, 76}


X(32477) = REFLECTION OF X(76) IN X(6337)

Barycentrics    6*a^6*b^2 - 8*a^4*b^4 + 2*a^2*b^6 + 6*a^6*c^2 - 15*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 3*b^6*c^2 - 8*a^4*c^4 + 4*a^2*b^2*c^4 + 10*b^4*c^4 + 2*a^2*c^6 - 3*b^2*c^6 : :
X(32477) = 3 X[7612] - 4 X[13334],5 X[7786] - 4 X[13881]

X(32477) lies on these lines: {39, 2996}, {76, 631}, {99, 3053}, {194, 5052}, {1975, 5050}, {2782, 8781}, {3564, 7750}, {5033, 10131}, {5395, 32450}, {7757, 8370}, {7786, 13881}, {11152, 14645}

X(32477) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 6337}, {2996, 39}


X(32478) = X(2)X(18308)∩X(3)X(18336)

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

X(32478) lies on these lines: {2, 18308}, {3, 18336}, {4, 18335}, {30, 511}, {381, 1116}, {669, 14417}, {879, 16835}, {1637, 31176}, {2519, 7651}, {2525, 8664}, {3569, 22889}, {3830, 15475}, {3845, 15543}, {4807, 18004}, {5027, 13315}, {5466, 33698}, {5652, 6030}, {8718, 8723}, {9135, 22888}, {9208, 13318}, {9218, 15342}, {9494, 14824}, {9979, 12075}, {10097, 15321}, {11550, 14582}, {12077, 13412}, {18310, 18313}

X(32478) = Lucas-isogonal conjugate of X(5966)
X(32478) = Thomson-isogonal conjugate of X(5966)
X(32478) = crossdifference of every pair of points on line {6, 11451}


X(32479) = X(2)X(8589)∩X(3)X(1153)

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

X(32479) lies on these lines: {2, 8589}, {3, 1153}, {4, 7618}, {5, 7619}, {20, 7620}, {21, 7621}, {30, 511}, {39, 598}, {99, 8597}, {115, 8598}, {187, 671}, {316, 8591}, {325, 15300}, {376, 5569}, {381, 7622}, {382, 7775}, {549, 20112}, {550, 16509}, {574, 11317}, {625, 2482}, {1657, 7780}, {2080, 12355}, {3146, 7843}, {3529, 5485}, {3534, 7610}, {3734, 5077}, {3830, 11184}, {3845, 9771}, {3934, 7833}, {5007, 19696}, {5008, 33687}, {5032, 7737}, {5073, 7781}, {5092, 7606}, {5107, 8593}, {5140, 12132}, {5148, 12354}, {5184, 9875}, {5194, 18969}, {5215, 9166}, {5254, 19661}, {5309, 33193}, {5461, 27088}, {5999, 9877}, {6658, 7827}, {6683, 7756}, {6781, 22329}, {7746, 33208}, {7748, 7817}, {7751, 17800}, {7758, 11541}, {7759, 33703}, {7761, 21356}, {7801, 7842}, {7804, 11159}, {7810, 32819}, {7812, 32450}, {7816, 7841}, {7845, 20094}, {7848, 32815}, {7849, 19695}, {7854, 33271}, {7861, 8369}, {7870, 33019}, {7872, 33237}, {7873, 19691}, {7880, 33017}, {7883, 33256}, {7886, 32985}, {7915, 33190}, {8355, 22247}, {8586, 10488}, {8588, 8860}, {8596, 11054}, {8667, 15685}, {8703, 15597}, {8716, 15684}, {8724, 13449}, {9167, 10150}, {9466, 33264}, {9770, 15682}, {10302, 17128}, {11055, 19569}, {11057, 14711}, {11147, 32984}, {12040, 15687}, {12117, 18860}, {13330, 33683}, {13468, 19710}, {14042, 31652}, {26613, 33265}, {31709, 33476}, {31710, 33477}

X(32479) = Lucas-isogonal conjugate of X(8600)
X(32479) = Thomson-isogonal conjugate of X(8600)
X(32479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7617, 1153}, {4, 7618, 8176}, {20, 7620, 8182}, {99, 8597, 31173}, {376, 7615, 5569}, {671, 9855, 187}, {2482, 8352, 625}, {8596, 14712, 11054}, {9166, 13586, 5215}


X(32480) = REFLECTION OF X(76) IN X(15810)

Barycentrics    4*a^4 - 7*a^2*b^2 - 2*b^4 - 7*a^2*c^2 + 5*b^2*c^2 - 2*c^4 : :
X(32480) = 5 X[194] + 4 X[7750],X[194] + 2 X[7833],7 X[194] + 2 X[7893],2 X[194] + X[9939],2 X[7750] - 5 X[7833],14 X[7750] - 5 X[7893],8 X[7750] - 5 X[9939],2 X[7756] + X[7812],5 X[7786] - 4 X[14762],4 X[7810] - X[20081],8 X[7830] + X[20105],7 X[7833] - X[7893],4 X[7833] - X[9939],X[7837] + 2 X[8353],2 X[7837] + X[14976],4 X[7893] - 7 X[9939],4 X[8353] - X[14976],8 X[8359] - 5 X[31276],X[9878] + 2 X[11152]

X(32480) lies on these lines: {2, 99}, {3, 8859}, {4, 10484}, {6, 9855}, {20, 576}, {30, 7709}, {39, 598}, {76, 15810}, {182, 12117}, {194, 524}, {376, 2080}, {384, 11164}, {1975, 21358}, {1992, 8586}, {2896, 11148}, {3091, 7608}, {3094, 9830}, {3329, 11159}, {3543, 9744}, {3552, 7827}, {3849, 7757}, {5024, 11317}, {5038, 7738}, {5077, 7840}, {5116, 7606}, {5309, 26613}, {5569, 14568}, {7748, 8176}, {7756, 7812}, {7777, 8352}, {7781, 7883}, {7782, 7817}, {7783, 7841}, {7785, 23334}, {7786, 14762}, {7791, 21356}, {7793, 8182}, {7801, 7847}, {7806, 27088}, {7810, 20081}, {7830, 20105}, {7837, 8353}, {7864, 8369}, {7870, 7933}, {7891, 8360}, {7923, 8366}, {7924, 8716}, {7937, 15301}, {8359, 31276}, {8597, 11163}, {8598, 15048}, {9606, 14066}, {10168, 10992}, {11147, 16925}, {11178, 23235}, {11184, 14041}, {11257, 11645}, {14042, 22332}

X(32480) = reflection of X(i) in X(j) for these lines: {i,j}: {76, 15810}, {598, 39}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8596, 11185}, {194, 7833, 9939}, {574, 671, 2}, {2482, 7790, 2}, {5077, 7840, 7898}, {5077, 31859, 7840}, {7622, 9166, 2}, {7622, 11648, 9166}, {7837, 8353, 14976}


X(32481) = X(3)X(76)∩X(114)X(2567)

Barycentrics    a^6*b^2 - a^4*b^4 + a^6*c^2 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6 - a^2*(a^2 - b^2 - c^2)*Sqrt[(a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)] : :
X(32481) = (Sqrt[Csc[w]^2 - 4] Sin[w] - 1) X[3] + X[76]

X(32481) lies on the the 2nd Brocard circle, the cubics K019 and K1103, and on these lines: {3, 76}, {114, 2567}, {115, 2566}, {1663, 12215}, {1664, 6776}, {1665, 18906}, {1670, 3414}, {1671, 3413}, {2564, 3023}, {2565, 3027}, {2572, 9860}, {2573, 13174}

X(32481) = reflection of X(32482) in X(3)
X(32481) = antipode in 2nd Brocard circle of X(32482)
X(32481) = {X(76),X(99)}-harmonic conjugate of X(32482)
X(32481) = {X(98),X(11257)}-harmonic conjugate of X(32482)
X(32481) = X(1663)-of-6th-Brocard-triangle
X(32481) = X(1665)-of-1st-anti-Brocard-triangle


X(32482) = X(3)X(76)∩X(114)X(2566)

Barycentrics    a^6*b^2 - a^4*b^4 + a^6*c^2 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6 + a^2*(a^2 - b^2 - c^2)*Sqrt[(a^2*b^2 + a^2*c^2 + b^2*c^2)*(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)] : :
X(32482) = (Sqrt[Csc[w]^2 - 4] Sin[w] + 1) X[3] - X[76]

X(32482) lies on the the 2nd Brocard circle, the cubics K019 and K1103, and on these lines: {3, 76}, {114, 2566}, {115, 2567}, {1662, 12215}, {1664, 18906}, {1665, 6776}, {1670, 3413}, {1671, 3414}, {2564, 3027}, {2565, 3023}, {2572, 13174}, {2573, 9860}

X(32482) = reflection of X(32481) in X(3)
X(32482) = antipode in 2nd Brocard circle of X(32481)
X(32482) = {X(76),X(99)}-harmonic conjugate of X(32481)
X(32482) = {X(98),X(11257)}-harmonic conjugate of X(32481)
X(32482) = X(1662)-of-6th-Brocard-triangle
X(32482) = X(1664)-of-1st-anti-Brocard-triangle


X(32483) = X(98)X(385)∩X(880)X(2782)

Barycentrics    a^14*b^6 - a^12*b^8 + a^10*b^10 - a^8*b^12 - a^16*b^2*c^2 + 3*a^14*b^4*c^2 - 8*a^12*b^6*c^2 + 9*a^10*b^8*c^2 - 8*a^8*b^10*c^2 + 4*a^6*b^12*c^2 - a^4*b^14*c^2 + 3*a^14*b^2*c^4 - 7*a^12*b^4*c^4 + 10*a^10*b^6*c^4 - a^8*b^8*c^4 + a^6*b^10*c^4 + 3*a^4*b^12*c^4 + a^2*b^14*c^4 + a^14*c^6 - 8*a^12*b^2*c^6 + 10*a^10*b^4*c^6 - 15*a^8*b^6*c^6 + 3*a^6*b^8*c^6 - 8*a^4*b^10*c^6 - 3*a^2*b^12*c^6 - a^12*c^8 + 9*a^10*b^2*c^8 - a^8*b^4*c^8 + 3*a^6*b^6*c^8 + 9*a^4*b^8*c^8 + 2*a^2*b^10*c^8 - b^12*c^8 + a^10*c^10 - 8*a^8*b^2*c^10 + a^6*b^4*c^10 - 8*a^4*b^6*c^10 + 2*a^2*b^8*c^10 + 2*b^10*c^10 - a^8*c^12 + 4*a^6*b^2*c^12 + 3*a^4*b^4*c^12 - 3*a^2*b^6*c^12 - b^8*c^12 - a^4*b^2*c^14 + a^2*b^4*c^14 : :

X(32483) lies on the cubic K1103 and these lines: {98, 385}, {880, 2782}


X(32484) = X(39)X(512)∩X(99)X(511)

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

X(32484) lies on the cubic K1103 and these lines: {39, 512}, {99, 511}, {1916, 14509}, {2023, 6071}, {2679, 2782}, {8870, 13137}

> X(32484) = midpoint of X(1916) and X(14509)
X(32484) = reflection of X(i) in X(j) for these lines: {i,j}: {6071, 2023}, {16068, 39}
X(32484) = reflection of X(16068) in the Brocard axis
X(32484) = orthogonal projection of X(99) on line PU(1)


X(32485) = (name pending)

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

X(32485) lies on this line: {3,6}


X(32486) = MIDPOINT OF X(1) AND X(5400)

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

See Angel Montesdeoca, Hyacinthos 29011 and HG130519

X(32486) lies on these lines: {1,5}, {3,23404}, {40,1054}, {43,16200}, {56,1777}, {57,1361}, {102,105}, {104,106}, {117,614}, {244,2800}, {386,10595}, {392,25939}, {484,23153}, {515,1149}, {517,1739}, {551,1064}, {581,3622}, {759,953}, {899,28234}, {912,4694}, {946,1201}, {962,28370}, {978,7982}, {995,4000}, {999,6180}, {1086,1537}, {1125,26095}, {1193,13464}, {1318,14511}, {1331,13279}, {1482,3216}, {1616,3149}, {1647,10265}, {1724,10680}, {1742,3576}, {1772,12758}, {2718,28219}, {3073,5563}, {3293,10222}, {3953,5887}, {3976,5693}, {3987,23340}, {4202,19861}, {4657,17044}, {5844,31855}, {6261,28011}, {6684,28352}, {6788,12247}, {8583,25882}, {8686,28233}, {10090,23703}, {11362,27627}, {12245,17749}, {12608,23675}, {12616,28018}, {14217,24715}, {15558,24028}, {16483,22753}, {17154,30196}

X(32486) = midpoint of X(i) and X(j), for these lines: {i, j}: {1, 5400}, {17154, 30196}


X(32487) = X(2292)X(3754)∩X(4535)X(18697)

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

See Angel Montesdeoca, Hyacinthos 29011 and HG130519

X(32487) lies on these lines: {2292,3754}, {4535,18697}






leftri  2nd and 3rd Vecten triangles: X(32488) - X(32513)  rightri

This preamble and centers X(32488)-X(32513) were contributed by CÚsar Eliud Lozada, May 14, 2019.

Given a triangle ABC, build the square σ'a = AA'bA'aA'c, with the same orientation as ABC, and such that B and C lie on the lines A'aA'b and A'aA'c, respectively. Define σ'b and σ'c cyclically.

Build now the square σ"a = AA"bA"aA"c, with opposite orientation of ABC, and such that B and C lie on the lines A"aA"b and A"aA"c, respectively, and construct σ"b and σ"c cyclically.

A'a and A"a have barycentric coordinates:

   A'a = 1/SA : 1/(SB-S) : 1/(SC-S).

   A"a = 1/SA : 1/(SB+S) : 1/(SC+S).

The lines AA'a, BB`b and CC'a concur at X(486) and similarly the lines AA"a, BB"b and CC"a concur at X(485). Moreover, the triangle bounded by the lines A'bA'c, B'aB'c and C'aC'b is the inner-Vecten triangle of ABC and the triangle enclosed by the lines A"bA"c, B"aB"c and C"aC"b is the outer-Vecten triangle of ABC. For the latter reasons, the triangle A'aB'bC'c are here named here the 2nd inner-Vecten triangle of ABC and the triangle A"aB"bC"c are here named the 2nd outer-Vecten triangle of ABC.

Let A', A" be the centers of σ'a and σ"a, respectively, and define (B', B"), (C', C") cyclically. The triangles A'B'C' and A"B"C" are here named the 3rd inner-Vecten triangle of ABC and 3rd outer-Vecten triangle of ABC, respectively. The have first vertices barycentrics coordinates:

   A' = (3*S^2+SB*SC-2*S*SW)/SA : SC-S : SB-S

   A" = (3*S^2+SB*SC+2*S*SW)/SA : SC+S : SB+S

Perspective triangles and perspectors:

   2nd inner-Vecten triangle: (ABC, 486), (anticomplementary, 32488), (outer-squares, 486), (inner-Vecten, 486), (3rd inner-Vecten, 486).
   2nd outer-Vecten triangle: (ABC, 485), (anticomplementary, 32489), (inner-squares, 485), (outer-Vecten, 485), (3rd outer-Vecten, 485).
   3rd inner-Vecten triangle: (ABC, 486), (medial, 32490), (outer-squares, 486), (inner-Vecten, 486), (2nd inner-Vecten, 486).
   3rd outer-Vecten triangle: (ABC, 485), (medial, 32491), (inner-squares, 485), (outer-Vecten, 485), (2nd outer-Vecten, 485).

Orthologic triangles and orthologic centers:

   2nd inner-Vecten triangle: (3rd anti-tri-squares, 486, 32492), (Lucas(-1) reflection, 3071, 32493), (4th tri-squares, 486, 32494), (inner-Vecten, 486, 3071).
   2nd outer-Vecten triangle: (4th anti-tri-squares, 485, 32495), (Lucas reflection, 3070, 32496), (3rd tri-squares, 485, 32497), (outer-Vecten, 485, 3070).
   3rd inner-Vecten triangle: (3rd anti-tri-squares, 486, 32498), (4th tri-squares, 486, 13933), (inner-Vecten, 486, 642).
   3rd outer-Vecten triangle: (4th anti-tri-squares, 485, 32499), (3rd tri-squares, 485, 13879), (outer-Vecten, 485, 641).

The following pairs of triangles are directly similar: {3rd inner-Vecten, Lucas(-1) central}, {3rd outer-Vecten, Lucas(+1) central}

underbar

X(32488) = PERSPECTOR OF THESE TRIANGLES: 2nd INNER-VECTEN AND ANTICOMPLEMENTARY

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

As a point on the Euler line, X(32488) has Shinagawa coefficients (E+F-S, -2*S).

X(32488) lies on these lines: {2,3}, {193,9975}, {486,638}, {487,23259}, {489,23261}, {490,615}, {491,3071}, {637,640}, {1132,1271}, {1587,7797}, {1588,7785}, {3069,12323}, {3070,7851}, {5591,12322}, {6776,26469}, {13758,13881}

X(32488) = anticomplement of the anticomplement of X(32490)
X(32488) = inverse of X(11293) in the orthocentroidal circle
X(32488) = X(12963)-of-3rd anti-tri-squares triangle, when ABC is obtuse
X(32488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11293), (4, 5025, 32489), (4, 11292, 14035)


X(32489) = PERSPECTOR OF THESE TRIANGLES: 2nd OUTER-VECTEN AND ANTICOMPLEMENTARY

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

As a point on the Euler line, X(32489) has Shinagawa coefficients (E+F+S, 2*S).

X(32489) lies on these lines: {2,3}, {69,31412}, {193,9974}, {485,637}, {488,23249}, {489,590}, {490,23251}, {492,3070}, {638,639}, {1131,1270}, {1587,7785}, {1588,7797}, {3068,12322}, {3071,7851}, {3594,13757}, {5590,12323}, {5860,31414}, {6118,6200}, {6250,11825}, {6425,13663}, {6776,26468}, {7581,7921}, {7582,7920}, {7585,12221}, {7745,13758}, {8376,13941}, {8960,32419}, {13638,13881}

X(32489) = anticomplement of the anticomplement of X(32491)
X(32489) = inverse of X(11294) in the orthocentroidal circle
X(32489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11294), (4, 5025, 32488), (4, 11291, 14035)


X(32490) = PERSPECTOR OF THESE TRIANGLES: 3rd INNER-VECTEN AND MEDIAL

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

As a point on the Euler line, X(32490) has Shinagawa coefficients (2*E+2*F-3*S, -S).

X(32490) lies on these lines: {2,3}, {69,13951}, {115,32494}, {141,10577}, {372,23312}, {487,13785}, {491,7584}, {590,1506}, {597,8960}, {615,640}, {637,18762}, {638,13966}, {639,7867}, {642,3071}, {1270,3317}, {1271,13939}, {1588,26362}, {3589,10576}, {3595,7582}, {3618,8976}, {6318,22725}, {6395,12222}, {6398,12323}, {10195,13663}, {31455,32497}

X(32490) = complement of the complement of X(32488)
X(32490) = inverse of X(11316) in the orthocentroidal circle
X(32490) = X(371)-of-inner-Vecten triangle, when ABC is obtuse
X(32490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11316), (5, 8361, 32491), (3090, 6811, 5)


X(32491) = PERSPECTOR OF THESE TRIANGLES: 3rd OUTER-VECTEN AND MEDIAL

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

As a point on the Euler line, X(32491) has Shinagawa coefficients (2*E+2*F+3*S, S).

X(32491) lies on these lines: {2,3}, {69,8976}, {115,32497}, {141,10576}, {371,23311}, {488,13665}, {492,7583}, {524,8960}, {590,639}, {615,1506}, {637,8981}, {638,18538}, {640,7867}, {641,3070}, {1270,13886}, {1271,3316}, {1587,26361}, {3589,10577}, {3593,7581}, {3618,13951}, {6199,12221}, {6221,12322}, {6314,22724}, {10194,13783}, {26288,31414}, {31454,32419}, {31455,32494}

X(32491) = complement of the complement of X(32489)
X(32491) = inverse of X(11315) in the orthocentroidal circle
X(32491) = X(371)-of-outer-Vecten triangle
X(32491) = X(6213)-of-3rd outer-Vecten triangle
X(32491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11315), (5, 8361, 32490), (3090, 6813, 5)


X(32492) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 2nd INNER-VECTEN

Barycentrics    a^6-7*(b^2+c^2)*a^4+(5*b^4-6*b^2*c^2+5*c^4)*a^2+2*(5*a^4-3*(b^2-c^2)^2)*S+(b^4-c^4)*(b^2-c^2) : :
X(32492) = 3*X(486)-2*X(32494) = 3*X(3071)-X(32494)

The reciprocal orthologic center of these triangles is X(486).

X(32492) lies on these lines: {3,486}, {30,12969}, {315,32419}, {371,6251}, {487,23259}, {489,642}, {616,22601}, {617,22603}, {1322,13036}, {1503,9975}, {1588,12296}, {2782,22501}, {6290,23261}, {12221,20081}, {12509,23275}, {20423,32495}, {22484,22591}, {32506,32508}

X(32492) = reflection of X(i) in X(j) for these (i,j): (486, 3071), (489, 642), (32506, 32508)
X(32492) = (3rd anti-tri-squares)-isogonal conjugate of-X(486)
X(32492) = X(372)-of-3rd anti-tri-squares triangle, when ABC is obtuse
X(32492) = X(8982)-of-inner-Vecten triangle, when ABC is obtuse
X(32492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6561, 22617, 486), (13785, 13934, 486)


X(32493) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) REFLECTION TO 2nd INNER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3071).

X(32493) lies on these lines: {32,6402}, {184,3071}

X(32493) = X(1)-of-Lucas(-1) reflection triangle
X(32493) = (Lucas(-1) reflection)-isogonal conjugate of-X(32493)


X(32494) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 2nd INNER-VECTEN

Barycentrics    (a^2-3*b^2-3*c^2)*a^2+2*(a^2+b^2+c^2)*S : :
X(32494) = 3*X(486)-X(32492) = 3*X(3071)-2*X(32492)

The reciprocal orthologic center of these triangles is X(486).

X(32494) lies on these lines: {2,1975}, {3,486}, {4,19493}, {6,487}, {32,32419}, {37,31535}, {39,590}, {69,6463}, {115,32490}, {372,3564}, {489,3053}, {494,11091}, {524,12969}, {574,6119}, {591,3785}, {597,12962}, {1152,1503}, {1322,13020}, {1352,15884}, {1384,13770}, {1588,12509}, {2548,9767}, {2549,11314}, {2782,13926}, {3070,6290}, {3767,11316}, {3815,7389}, {3926,5491}, {5023,13798}, {5024,13711}, {5206,32209}, {5210,13941}, {6414,8420}, {6776,8406}, {7375,31400}, {7745,11293}, {8252,11292}, {8253,22332}, {8576,13046}, {8855,8964}, {9605,19105}, {9906,13973}, {10991,13989}, {11313,31401}, {12169,19005}, {12296,23261}, {12963,13972}, {13036,13429}, {13880,13968}, {18937,19411}, {19104,30435}, {31455,32491}

X(32494) = midpoint of X(489) and X(12221)
X(32494) = reflection of X(3071) in X(486)
X(32494) = (4th tri-squares)-isogonal conjugate of-X(486)
X(32494) = X(371)-of-4th tri-squares triangle, when ABC is acute
X(32494) = X(372)-of-4th tri-squares triangle, when ABC is obtuse
X(32494) = X(489)-of-4th tri-squares-central triangle
X(32494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5013, 32497), (5254, 7789, 32497), (6337, 13881, 32497)


X(32495) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 2nd OUTER-VECTEN

Barycentrics    a^6-7*(b^2+c^2)*a^4+(5*b^4-6*b^2*c^2+5*c^4)*a^2-2*(5*a^4-3*(b^2-c^2)^2)*S+(b^4-c^4)*(b^2-c^2) : :
X(32495) = 3*X(485)-2*X(32497) = 3*X(3070)-X(32497)

The reciprocal orthologic center of these triangles is X(485).

X(32495) lies on these lines: {3,485}, {30,12962}, {315,32421}, {372,6250}, {488,23249}, {490,641}, {616,22630}, {617,22632}, {1321,13035}, {1503,9974}, {1587,12297}, {2782,22502}, {5254,19102}, {6118,6396}, {6200,13879}, {6228,7389}, {6289,23251}, {6561,19103}, {8352,12159}, {8376,13834}, {12222,20081}, {12510,23269}, {20423,32492}, {22485,22592}, {32507,32509}

X(32495) = reflection of X(i) in X(j) for these (i,j): (485, 3070), (490, 641), (32507, 32509)
X(32495) = (4th anti-tri-squares)-isogonal conjugate of-X(485)
X(32495) = X(8982)-of-outer-Vecten triangle
X(32495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6560, 22646, 485), (13665, 13882, 485)


X(32496) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS REFLECTION TO 2nd OUTER-VECTEN

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

The reciprocal orthologic center of these triangles is X(3070).

X(32496) lies on these lines: {32,6401}, {184,3070}

X(32496) = (Lucas reflection)-isogonal conjugate of-X(32496)
X(32496) = X(1)-of-Lucas reflection triangle


X(32497) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 2nd OUTER-VECTEN

Barycentrics    (a^2-3*b^2-3*c^2)*a^2-2*(a^2+b^2+c^2)*S : :
X(32497) = 3*X(485)-X(32495) = 3*X(3070)-2*X(32495)

The reciprocal orthologic center of these triangles is X(485).

X(32497) lies on these lines: {2,1975}, {3,485}, {4,19492}, {6,488}, {32,32421}, {39,615}, {69,6462}, {115,32491}, {187,13879}, {371,3564}, {490,3053}, {493,11090}, {524,12962}, {574,6118}, {597,12969}, {1144,6529}, {1151,1503}, {1321,13019}, {1352,15883}, {1384,13651}, {1587,12510}, {1991,3785}, {2548,9768}, {2549,11313}, {2782,13873}, {3071,6289}, {3767,11315}, {3815,7388}, {3926,5490}, {5023,13678}, {5024,13834}, {5206,13924}, {5210,8972}, {6413,8408}, {6776,8414}, {7376,31400}, {7745,11294}, {8252,22332}, {8253,11291}, {8577,13047}, {8997,10991}, {9605,19102}, {9907,13911}, {11314,31401}, {12170,19006}, {12297,23251}, {12968,13910}, {13035,13440}, {13908,13921}, {18938,19410}, {19103,30435}, {31455,32490}

X(32497) = midpoint of X(490) and X(12222)
X(32497) = reflection of X(3070) in X(485)
X(32497) = (3rd tri-squares)-isogonal conjugate of-X(485)
X(32497) = X(372)-of-3rd tri-squares triangle
X(32497) = X(490)-of-3rd tri-squares-central triangle
X(32497) = X(31559)-of-3rd outer-Vecten triangle
X(32497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5013, 32494), (5254, 7789, 32494), (6337, 13881, 32494)


X(32498) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 3rd INNER-VECTEN

Barycentrics    -(3*a^4+2*(b^2+c^2)*a^2-3*(b^2-c^2)^2)*S+2*(b^2+c^2)*a^4-(b^4-4*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(32498) = 3*X(486)-2*X(13933) = 3*X(1328)-X(22617)

The reciprocal orthologic center of these triangles is X(486).

X(32498) lies on these lines: {4,372}, {6,22596}, {30,13934}, {487,23259}, {642,6561}, {1161,12601}, {1328,5491}, {3071,6290}, {3564,9975}, {6119,11291}, {6229,12322}, {6564,19105}, {7000,13926}, {7900,12221}, {8184,11293}, {12296,23263}, {12306,23261}, {13881,22625}, {19130,32499}

X(32498) = (3rd anti-tri-squares)-isogonal conjugate of-X(22617)
X(32498) = (3rd anti-tri-squares)-complement of-X(22617)
X(32498) = X(3)-of-3rd anti-tri-squares triangle


X(32499) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 3rd OUTER-VECTEN

Barycentrics    (3*a^4+2*(b^2+c^2)*a^2-3*(b^2-c^2)^2)*S+2*(b^2+c^2)*a^4-(b^4-4*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(32499) = 3*X(485)-2*X(13879) = 3*X(1327)-X(22646)

The reciprocal orthologic center of these triangles is X(485).

X(32499) lies on these lines: {4,371}, {6,22625}, {30,13882}, {488,23249}, {641,6560}, {1160,12602}, {1327,5490}, {3070,6289}, {3564,9974}, {6118,11292}, {6228,12323}, {6565,19102}, {7374,13873}, {7900,12222}, {8180,11294}, {12297,23253}, {12305,23251}, {13881,22596}, {19130,32498}

X(32499) = (4th anti-tri-squares)-isogonal conjugate of-X(22646)
X(32499) = (4th anti-tri-squares)-complement of-X(22646)
X(32499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (488, 23249, 32495), (12257, 31412, 485)


X(32500) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO 3rd INNER-VECTEN

Barycentrics    3*S^4-3*(SA^2+SB*SC)*S^2-SB*SC*SW^2-S*(2*(12*R^2-SA)*S^2-(SA^2+4*SB*SC+SW^2)*SW) : :

There is not reciprocal eulerologic center for these triangles

X(32500) lies on these lines: {2,7599}, {136,641}, {369,20799}

X(32500) = complement of X(13520)
X(32500) = X(125)-of-inner-Vecten triangle
X(32500) = orthoptic-circle-of-Steiner-inellipse-inverse of X(33340)


X(32501) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO 3rd INNER-VECTEN

Barycentrics    3*S^4-3*(SA^2+SB*SC)*S^2-SB*SC*SW^2+S*(2*(12*R^2-SA)*S^2-(SA^2+4*SB*SC+SW^2)*SW) : :

There is not reciprocal eulerologic center for these triangles

X(32501) lies on these lines: {2,7598}, {136,642}

X(32501) = complement of X(13521)
X(32501) = X(100)-of-3rd outer-Vecten triangle
X(32501) = X(125)-of-outer-Vecten triangle
X(32501) = orthoptic-circle-of-Steiner-inellipse-inverse of X(33341)


X(32502) = CENTROID OF THE 2nd INNER-VECTEN TRIANGLE

Barycentrics    28*S^4+4*(11*SA^2-29*SB*SC-12*SW^2)*S^2-20*SB*SC*SW^2-S*(8*(2*R^2+5*SA-17*SW)*S^2-4*(19*SA-3*SW)*(SA-SW)*SW) : :
X(32502) = X(486)+2*X(32508) = X(642)-4*X(32512) = 2*X(3071)+X(32506) = X(3071)+2*X(32510) = X(32506)-4*X(32510)

X(32502) lies on these lines: {2,32433}, {486,32508}, {642,32512}, {3071,32506}

X(32502) = reflection of X(2) in X(32504)
X(32502) = {X(3071), X(32510)}-harmonic conjugate of X(32506)
X(32502) = X(3060)-of-inner-Vecten triangle


X(32503) = CENTROID OF THE 2nd OUTER-VECTEN TRIANGLE

Barycentrics    28*S^4+4*(11*SA^2-29*SB*SC-12*SW^2)*S^2-20*SB*SC*SW^2+S*(8*(2*R^2+5*SA-17*SW)*S^2-4*(19*SA-3*SW)*(SA-SW)*SW) : :
X(32503) = X(485)+2*X(32509) = X(641)-4*X(32513) = 2*X(3070)+X(32507) = X(3070)+2*X(32511) = X(32507)-4*X(32511)

X(32503) lies on these lines: {2,32436}, {485,32509}, {641,32513}, {3070,32507}

X(32503) = reflection of X(2) in X(32505)
X(32503) = X(1699)-of-3rd outer-Vecten triangle
X(32503) = X(3060)-of-outer-Vecten triangle
X(32503) = {X(3070), X(32511)}-harmonic conjugate of X(32507)


X(32504) = CENTROID OF THE 3rd INNER-VECTEN TRIANGLE

Barycentrics    3*S^4-(11*SA^2-29*SB*SC-22*SW^2)*S^2+5*SB*SC*SW^2+S*(2*(4*R^2+5*SA-15*SW)*S^2+(3*SA^2-22*SB*SC-5*SW^2)*SW) : :

X(32504) lies on these lines: {2,32433}, {6119,32508}

X(32504) = midpoint of X(2) and X(32502)
X(32504) = X(51)-of-inner-Vecten triangle
X(32504) = {X(6119), X(32512)}-harmonic conjugate of X(32508)


X(32505) = CENTROID OF THE 3rd OUTER-VECTEN TRIANGLE

Barycentrics    3*S^4-(11*SA^2-29*SB*SC-22*SW^2)*S^2+5*SB*SC*SW^2-S*(2*(4*R^2+5*SA-15*SW)*S^2+(3*SA^2-22*SB*SC-5*SW^2)*SW) : :
X(32505) = 2*X(6118)+X(32509)

X(32505) lies on these lines: {2,32436}, {6118,32509}

X(32505) = midpoint of X(2) and X(32503)
X(32505) = X(51)-of-outer-Vecten triangle
X(32505) = {X(6118), X(32513)}-harmonic conjugate of X(32509)


X(32506) = ORTHOCENTER OF THE 2nd INNER-VECTEN TRIANGLE

Barycentrics    2*(2*R^2+5*SW)*S^4+(8*(SA-SW)*SA*R^2+(SA^2-5*SA*SW+5*SW^2)*SW)*S^2+SB*SC*SW^3-S*(S^4-(SA*(4*R^2+3*SA)-SB*SC-13*SW^2)*S^2+(4*SA-SW)*(SA-SW)*SW^2) : :
X(32506) = 2*X(3071)-3*X(32502)

X(32506) lies on these lines: {486,489}, {3071,32502}, {32492,32508}

X(32506) = reflection of X(i) in X(j) for these (i,j): (3071, 32510), (32492, 32508)
X(32506) = {X(3071), X(32510)}-harmonic conjugate of X(32502)


X(32507) = ORTHOCENTER OF THE 2nd OUTER-VECTEN TRIANGLE

Barycentrics    2*(2*R^2+5*SW)*S^4+(8*(SA-SW)*SA*R^2+(SA^2-5*SA*SW+5*SW^2)*SW)*S^2+SB*SC*SW^3+S*(S^4-(SA*(4*R^2+3*SA)-SB*SC-13*SW^2)*S^2+(4*SA-SW)*(SA-SW)*SW^2) : :
X(32507) = 2*X(3070)-3*X(32503)

X(32507) lies on these lines: {485,490}, {3070,32503}, {32495,32509}

X(32507) = reflection of X(i) in X(j) for these (i,j): (3070, 32511), (32495, 32509)
X(32507) = {X(3070), X(32511)}-harmonic conjugate of X(32503)


X(32508) = ORTHOCENTER OF THE 3rd INNER-VECTEN TRIANGLE

Barycentrics    (4*R^2+15*SA-67*SW)*S^4-(SA-SW)*(2*R^2*SA+SW*(47*SA-12*SW))*S^2-4*SB*SC*SW^3-S*(20*S^4+(2*(SA+SW)*R^2+23*SA^2-54*SB*SC-49*SW^2)*S^2-2*(12*SA-SW)*(SA-SW)*SW^2) : :

X(32508) lies on these lines: {486,32502}, {642,3071}, {6119,32504}, {32492,32506}

X(32508) = midpoint of X(32492) and X(32506)
X(32508) = reflection of X(6119) in X(32512)
X(32508) = X(52)-of-inner-Vecten triangle
X(32508) = {X(6119), X(32512)}-harmonic conjugate of X(32504)


X(32509) = ORTHOCENTER OF THE 3rd OUTER-VECTEN TRIANGLE

Barycentrics    (4*R^2+15*SA-67*SW)*S^4-(SA-SW)*(2*R^2*SA+SW*(47*SA-12*SW))*S^2-4*SB*SC*SW^3+S*(20*S^4+(2*(SA+SW)*R^2+23*SA^2-54*SB*SC-49*SW^2)*S^2-2*(12*SA-SW)*(SA-SW)*SW^2) : :

X(32509) lies on these lines: {485,32503}, {641,3070}, {6118,32505}, {32495,32507}

X(32509) = midpoint of X(32495) and X(32507)
X(32509) = reflection of X(6118) in X(32513)
X(32509) = X(52)-of-outer-Vecten triangle
X(32509) = {X(6118), X(32513)}-harmonic conjugate of X(32505)


X(32510) = X(5)-OF-2nd INNER-VECTEN TRIANGLE

Barycentrics    (5*SA-33*SW)*S^4-(4*(SA-SW)*SA*R^2+(19*SA^2-28*SA*SW+10*SW^2)*SW)*S^2-3*SB*SC*SW^3-S*(8*S^4+2*((SA+SW)*R^2+6*SA^2-10*SB*SC-16*SW^2)*S^2-2*(7*SA-SW)*(SA-SW)*SW^2) : :

X(32510) lies on these lines: {3071,32502}, {6119,18762}

X(32510) = midpoint of X(3071) and X(32506)
X(32510) = {X(32502), X(32506)}-harmonic conjugate of X(3071)


X(32511) = X(5)-OF-2nd OUTER-VECTEN TRIANGLE

Barycentrics    (5*SA-33*SW)*S^4-(4*(SA-SW)*SA*R^2+(19*SA^2-28*SA*SW+10*SW^2)*SW)*S^2-3*SB*SC*SW^3+S*(8*S^4+2*((SA+SW)*R^2+6*SA^2-10*SB*SC-16*SW^2)*S^2-2*(7*SA-SW)*(SA-SW)*SW^2) : :

X(32511) lies on these lines: {3070,32503}, {6118,18538}

X(32511) = midpoint of X(3070) and X(32507)
X(32511) = {X(32503), X(32507)}-harmonic conjugate of X(3070)


X(32512) = X(5)-OF-3rd INNER-VECTEN TRIANGLE

Barycentrics
(20*R^2+35*SA-203*SW)*S^4-(2*(SA-SW)*SA*R^2+(103*SA^2-44*(3*SA-SW)*SW)*SW)*S^2-9*SB*SC*SW^3-S*(24*S^4+(2*(SA+5*SW)*R^2+55*SA^2-122*SB*SC-97*SW^2)*S^2+(5*SA^2-58*SB*SC-7*SW^2)*SW^2) : :
X(32512) = X(642)+3*X(32502)

X(32512) lies on these lines: {642,32502}, {6119,32504}

X(32512) = midpoint of X(6119) and X(32508)
X(32512) = X(143)-of-inner-Vecten triangle
X(32512) = {X(32504), X(32508)}-harmonic conjugate of X(6119)


X(32513) = X(5)-OF-3rd OUTER-VECTEN TRIANGLE

Barycentrics
(20*R^2+35*SA-203*SW)*S^4-(2*(SA-SW)*SA*R^2+(103*SA^2-44*(3*SA-SW)*SW)*SW)*S^2-9*SB*SC*SW^3+S*(24*S^4+(2*(SA+5*SW)*R^2+55*SA^2-122*SB*SC-97*SW^2)*S^2+(5*SA^2-58*SB*SC-7*SW^2)*SW^2) : :
X(32513) = X(641)+3*X(32503)

X(32513) lies on these lines: {641,32503}, {6118,32505}

X(32513) = midpoint of X(6118) and X(32509)
X(32513) = X(143)-of-outer-Vecten triangle
X(32513) = {X(32505), X(32509)}-harmonic conjugate of X(6118)


X(32514) = X(3)X(194)∩X(699)X(1384)

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

X(32514) = X(25)-of-inscribed-triply-bilogic-triangle; see the preamble just before X(32562)

X(32514) lies on these lines: {3, 194}, {699, 1384}, {2076, 32442}, {3552, 19597}, {5017, 9431}


X(32515) = X(3)X(194)∩X(5)X(76)

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

X(32515) lies on these lines: {2, 8179}, {3, 194}, {4, 7779}, {5, 76}, {6, 22525}, {15, 32465}, {16, 32466}, {20, 20105}, {26, 9917}, {30, 511}, {39, 140}, {98, 10290}, {99, 2080}, {114, 7813}, {141, 11261}, {182, 7798}, {187, 1569}, {298, 20426}, {299, 20425}, {316, 6321}, {323, 1316}, {355, 9902}, {381, 7840}, {384, 22521}, {395, 3106}, {396, 3107}, {397, 3105}, {398, 3104}, {495, 10063}, {496, 10079}, {546, 6248}, {547, 9466}, {548, 5188}, {549, 7757}, {550, 9821}, {576, 3734}, {620, 14693}, {632, 7786}, {1151, 32470}, {1152, 32471}, {1351, 18906}, {1353, 32451}, {1483, 7976}, {1484, 32454}, {1513, 9772}, {1656, 7881}, {1916, 15980}, {3094, 15048}, {3097, 26446}, {3102, 7584}, {3103, 7583}, {3186, 22152}, {3398, 7760}, {3530, 13334}, {3580, 11007}, {3589, 32149}, {3628, 3934}, {3926, 9752}, {4385, 15973}, {5054, 8859}, {5066, 14711}, {5093, 11286}, {5097, 7804}, {5149, 13196}, {5171, 7781}, {5254, 32452}, {5690, 12782}, {5874, 6272}, {5875, 6273}, {5901, 12263}, {5976, 6390}, {5999, 12188}, {6234, 18829}, {6683, 16239}, {6756, 12143}, {6781, 10992}, {6795, 32599}, {7467, 8267}, {7617, 14159}, {7694, 7758}, {7751, 9737}, {7763, 9754}, {7766, 11842}, {7789, 18806}, {7799, 15561}, {7805, 13335}, {7809, 14639}, {7826, 32152}, {7845, 13449}, {8597, 12355}, {8598, 11152}, {8703, 11055}, {8782, 9301}, {8992, 13925}, {9743, 9764}, {9748, 32830}, {9751, 12054}, {9753, 32833}, {9756, 31981}, {9983, 22678}, {10263, 16983}, {10358, 17130}, {10942, 12933}, {10943, 12923}, {11054, 11632}, {12042, 18860}, {12100, 21163}, {12338, 22556}, {12474, 22668}, {12475, 22672}, {12794, 22698}, {12992, 22709}, {12993, 22710}, {13077, 15171}, {13109, 22731}, {13110, 22732}, {13172, 14712}, {13330, 18907}, {13983, 13993}, {14651, 19570}, {14853, 32836}, {14880, 30270}, {15122, 16315}, {16266, 23128}, {16316, 16619}, {16320, 25338}, {18583, 24256}, {18971, 18982}, {19063, 19089}, {19064, 19090}, {22680, 22779}, {22735, 33873}

X(32515) = Lucas-isogonal conjugate of X(25424)
X(32515) = Thomson-isogonal conjugate of X(25424)
X(32515) = X(30)-of-inscribed-triply-bilogic-triangle

X(32515) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 194, 32448}, {4, 20081, 13108}, {76, 262, 7697}, {76, 3095, 5}, {194, 6194, 7709}, {194, 12251, 3}, {262, 7697, 5}, {576, 3734, 10796}, {3095, 7697, 262}, {3934, 11272, 3628}, {6194, 7709, 3}, {6248, 14881, 546}, {6248, 22682, 22681}, {7709, 12251, 6194}, {7751, 9737, 10104}, {7757, 22712, 11171}, {9301, 13188, 11676}, {9821, 11257, 550}, {10063, 12837, 495}, {10063, 22729, 22705}, {10079, 12836, 496}, {10079, 22730, 22706}, {11152, 22564, 8598}, {11171, 22712, 549}, {12836, 22706, 22730}, {12837, 22705, 22729}, {14881, 22681, 22682}, {22681, 22682, 546}, {22705, 22729, 495}, {22706, 22730, 496}


X(32516) = X(3)X(194)∩X(30)X(39)

Barycentrics    4*a^6*b^2 - 5*a^4*b^4 + a^2*b^6 + 4*a^6*c^2 - 8*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 5*a^4*c^4 - 2*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6 : :
X(32516) = 3 X[3] + X[194],7 X[3] - 3 X[6194],X[3] + 3 X[7709],5 X[3] - X[12251],3 X[5] - 5 X[7786],X[5] - 3 X[11171],X[20] + 3 X[32447],3 X[39] - X[14881],X[76] - 3 X[549],3 X[140] - 2 X[3934],7 X[194] + 9 X[6194],X[194] - 9 X[7709],5 X[194] + 3 X[12251],X[194] - 3 X[32448],3 X[262] - X[3627],3 X[547] - 4 X[6683],5 X[631] - X[13108],5 X[632] - 3 X[7697],3 X[3097] + X[18481],9 X[3524] - X[20081],X[3529] + 3 X[22728],2 X[3530] - 3 X[21163],X[3934] - 3 X[13334],9 X[5054] - 5 X[31276],X[6194] + 7 X[7709],15 X[6194] - 7 X[12251],3 X[6194] + 7 X[32448],9 X[7618] - X[18768],15 X[7709] + X[12251],3 X[7709] - X[32448],3 X[7757] + X[9821],5 X[7786] - 9 X[11171],5 X[7786] + 3 X[11257],3 X[8356] - X[32151],3 X[8703] - X[9821],3 X[8716] + X[31981],6 X[10124] - 5 X[31239],7 X[10541] - 3 X[31958],X[11055] + 5 X[15711],3 X[11171] + X[11257],5 X[12017] - X[18906],2 X[12102] - 3 X[22682],4 X[12108] - 3 X[15819],X[12251] + 5 X[32448],4 X[12811] - 3 X[22681],3 X[13331] - X[21850],15 X[15692] + X[20105],5 X[15712] - 3 X[22712]

X(32516) lies on these lines: {3, 194}, {5, 7786}, {20, 32447}, {30, 39}, {76, 549}, {99, 12054}, {140, 620}, {141, 32429}, {262, 3627}, {511, 548}, {538, 12100}, {546, 11272}, {547, 6683}, {550, 3095}, {574, 14880}, {631, 13108}, {632, 7697}, {698, 5092}, {726, 13624}, {1151, 32471}, {1152, 32470}, {1569, 12042}, {2021, 5305}, {2080, 32467}, {3097, 18481}, {3098, 32449}, {3524, 20081}, {3529, 22728}, {3530, 21163}, {3628, 6248}, {5054, 31276}, {5237, 32465}, {5238, 32466}, {6033, 7847}, {7467, 31088}, {7618, 18768}, {7757, 8703}, {7782, 26316}, {7824, 12188}, {7832, 8724}, {7839, 9301}, {8356, 32151}, {8716, 31981}, {9466, 11812}, {10007, 18358}, {10124, 31239}, {10541, 31958}, {11055, 15711}, {11676, 32134}, {12017, 18906}, {12102, 22682}, {12108, 15819}, {12811, 22681}, {13077, 15325}, {13331, 21850}, {15692, 20105}, {15712, 22712}, {18806, 32459}

X(32516) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 32448}, {5, 11257}, {141, 32429}, {550, 3095}, {1569, 12042}, {3098, 32449}, {7757, 8703}
X(32516) = reflection of X(i) in X(j) for these lines: {i,j}: {140, 13334}, {546, 11272}, {6248, 3628}, {9466, 11812}, {18358, 10007}
X(32516) = X(140)-of-inscribed-triply-bilogic-triangle
X(32516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7709, 32448}, {11171, 11257, 5}


X(32517) = X(3)X(194)∩X(187)X(699)

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

X(32517) lies on these lines: {3, 194}, {187, 699}, {9491, 13586}

X(32517) = circumcircle-inverse of X(194)
X(32517) = X(186)-of-inscribed-triply-bilogic-triangle


X(32518) = X(3)X(194)∩X(32)X(9490)

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

X(32518) lies on these lines: {3, 194}, {32, 9490}, {187, 237}, {805, 2080}, {865, 5913}, {1613, 23221}, {3053, 3224}, {3360, 5023}, {3511, 13586}, {7771, 14096}, {11332, 11580}, {16514, 22386}

X(32518) = {X(187),X(21444)}-harmonic conjugate of X(237)
X(32518) = X(237)-of-inscribed-triply-bilogic-triangle


X(32519) = X(3)X(194)∩X(6)X(15690)

Barycentrics    3*a^6*b^2 - 5*a^4*b^4 + 2*a^2*b^6 + 3*a^6*c^2 - 11*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - 2*b^2*c^6 : :
X(32519) = X[3] + 2 X[194],3 X[3] - 2 X[6194],5 X[3] - 2 X[12251],X[3] - 4 X[32448],8 X[39] - 5 X[1656],4 X[39] - X[13108],4 X[76] - 7 X[3526],4 X[140] - X[20081],3 X[194] + X[6194],5 X[194] + X[12251],X[194] + 2 X[32448],4 X[262] - 3 X[381],X[262] - 3 X[7757],3 X[262] - 2 X[22681],2 X[262] - 3 X[32447],X[381] - 4 X[7757],9 X[381] - 8 X[22681],X[382] - 4 X[3095],X[382] - 16 X[32450],5 X[631] + X[20105],X[1351] - 4 X[32449],4 X[1569] - X[13188]

X(32519) lies on these lines: {3, 194}, {6, 1569}, {39, 1656}, {76, 3526}, {99, 11842}, {140, 20081}, {262, 381}, {382, 3095}, {511, 3534}, {538, 1153}, {631, 20105}, {698, 5050}, {726, 10246}, {732, 22677}, {1351, 32449}, {1499, 32463}, {1657, 11257}, {2070, 22655}, {2080, 7798}, {3094, 11898}, {3097, 5790}, {3311, 32470}, {3312, 32471}, {3398, 7781}, {3818, 14692}, {3906, 8723}, {5072, 6248}, {5076, 14881}, {5079, 11272}, {5309, 15561}, {7748, 10242}, {7839, 22521}, {9466, 15723}, {9756, 12188}, {9917, 13564}, {10256, 18768}, {10796, 23235}, {11055, 15693}, {11152, 11159}, {12645, 12782}, {13077, 22730}, {15696, 22676}, {15700, 21163}, {18525, 22650}, {18982, 22729}

X(32519) = midpoint of X(i) and X(j) for these lines: {i,j}: {194, 7709}, {11055, 22712}
X(32519) = reflection of X(i) in X(j) for these lines: {i,j}: {3, 7709}, {381, 32447}, {382, 22728}, {5790, 3097}, {7697, 39}, {7709, 32448}, {13108, 7697}, {18525, 22650}, {22728, 3095}, {32447, 7757}
X(32519) = X(381)-of-inscribed-triply-bilogic-triangle
X(32519) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 13108, 1656}, {194, 32448, 3}, {32465, 32466, 6}


X(32520) = X(3)X(194)∩X(4)X(20105)

Barycentrics    a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6 + a^6*c^2 - 9*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^4*c^4 + 3*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - 2*b^2*c^6 : :
X(32520) = 7 X[3] - 6 X[6194],5 X[3] - 6 X[7709],3 X[3] - 2 X[12251],3 X[3] - 4 X[32448],8 X[39] - 7 X[3526],4 X[76] - 5 X[1656],3 X[76] - 4 X[11272],2 X[76] - 3 X[32447],7 X[194] - 3 X[6194],5 X[194] - 3 X[7709],3 X[194] - X[12251],3 X[194] - 2 X[32448],12 X[262] - 11 X[5072],3 X[381] - 4 X[3095],9 X[381] - 8 X[6248],3 X[381] - 2 X[13108],15 X[1656] - 16 X[11272],5 X[1656] - 6 X[32447],3 X[3095] - 2 X[6248],X[3534] - 4 X[11055],3 X[3534] - 4 X[11257]

X(32520) lies on these lines: {3, 194}, {4, 20105}, {5, 7906}, {39, 3526}, {76, 1656}, {262, 5072}, {381, 538}, {382, 2782}, {511, 1657}, {576, 18501}, {698, 1351}, {726, 1482}, {730, 12645}, {732, 11898}, {1569, 3053}, {2080, 7781}, {2937, 9917}, {3102, 18510}, {3103, 18512}, {3398, 7798}, {3534, 11055}, {5050, 32449}, {5054, 7757}, {5070, 31276}, {5076, 22728}, {5079, 7697}, {5093, 18906}, {5188, 15688}, {5790, 9902}, {6221, 32470}, {6321, 7759}, {6398, 32471}, {7760, 11842}, {7890, 23698}, {9821, 15696}, {10983, 31981}, {11171, 15720}, {13334, 15693}, {15069, 18503}, {22236, 32465}, {22238, 32466}

X(32520) = midpoint of X(4) and X(20105)
X(32520) = reflection of X(i) in X(j) for these lines: {i,j}: {3, 194}, {12251, 32448}, {13108, 3095}, {20081, 5}
X(32520) = X(382)-of-inscribed-triply-bilogic-triangle
X(32520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 32447, 1656}, {194, 12251, 32448}, {3095, 13108, 381}, {12251, 32448, 3}


X(32521) = X(3)X(194)∩X(5)X(141)

Barycentrics    2*a^6*b^2 - a^4*b^4 - a^2*b^6 + 2*a^6*c^2 + 2*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 - a^4*c^4 - 4*a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + b^2*c^6 : :
X(32521) = 3 X[3] - X[194],X[3] - 3 X[6194],5 X[3] - 3 X[7709],3 X[5] - 4 X[3934],3 X[5] - 2 X[14881],2 X[39] - 3 X[549],6 X[140] - 5 X[7786],2 X[140] - 3 X[22712],X[194] - 9 X[6194],5 X[194] - 9 X[7709],X[194] + 3 X[12251],2 X[194] - 3 X[32448],3 X[262] - 4 X[3628],3 X[376] + X[20081],3 X[381] - 5 X[31276],2 X[546] - 3 X[7697],5 X[631] - 3 X[32447],5 X[632] - 4 X[11272],5 X[632] - 6 X[15819],5 X[3091] - 3 X[22728],3 X[3095] - 5 X[7786],X[3095] - 3 X[22712]

X(32521) lies on these lines: {3, 194}, {4, 7929}, {5, 141}, {20, 13108}, {30, 76}, {39, 549}, {140, 3095}, {230, 32452}, {262, 3628}, {343, 21536}, {376, 20081}, {381, 7879}, {384, 9301}, {395, 3104}, {396, 3105}, {524, 8149}, {538, 8703}, {546, 7697}, {548, 11257}, {550, 2782}, {631, 32447}, {632, 11272}, {698, 3098}, {726, 3579}, {1353, 13354}, {1569, 15513}, {2979, 21531}, {3091, 22728}, {3094, 5305}, {3102, 13966}, {3103, 8981}, {3106, 16773}, {3107, 16772}, {3530, 11171}, {3627, 6248}, {3845, 9466}, {3857, 22682}, {3933, 5976}, {5092, 7805}, {5351, 32466}, {5352, 32465}, {6033, 7768}, {6179, 26316}, {6409, 32470}, {6410, 32471}, {6644, 9917}, {6683, 11539}, {7470, 8782}, {7757, 12100}, {7760, 12054}, {7780, 12042}, {7797, 10357}, {7842, 22515}, {7854, 9996}, {8367, 22486}, {8370, 22564}, {8667, 31981}, {9902, 18481}, {10063, 18990}, {10079, 15171}, {10104, 30270}, {10304, 20105}, {11055, 15759}, {11477, 31958}, {12103, 22676}, {12263, 22791}, {12836, 15325}, {13334, 15712}, {13468, 32189}, {14711, 19710}, {15699, 31239}, {17504, 32450}

X(32521) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 12251}, {20, 13108}, {76, 9821}, {8782, 12188}, {9902, 18481}
X(32521) = reflection of X(i) in X(j) for these lines: {i,j}: {550, 5188}, {1353, 13354}, {3095, 140}, {3627, 6248}, {3845, 9466}, {7757, 12100}, {11257, 548}, {14881, 3934}, {21850, 24256}, {22791, 12263}, {32151, 7767}, {32448, 3}, {32449, 5092}
X(32521) = X(550)-of-inscribed-triply-bilogic-triangle
X(32521) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3095, 22712, 140}, {3098, 7751, 14880}, {3934, 14881, 5}, {6194, 12251, 3}, {7470, 17129, 12188}, {11272, 15819, 632}


X(32522) = X(3)X(194)∩X(20)X(39)

Barycentrics    5*a^6*b^2 - 6*a^4*b^4 + a^2*b^6 + 5*a^6*c^2 - 9*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 - 6*a^4*c^4 - 3*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6 : :
X(32522) = 9 X[2] - 4 X[6248],3 X[2] + 2 X[11257],3 X[2] - 8 X[13334],4 X[3] + X[194],8 X[3] - 3 X[6194],2 X[3] + 3 X[7709],6 X[3] - X[12251],3 X[3] + 2 X[32448],X[4] - 6 X[11171],3 X[4] - 8 X[11272],X[20] + 4 X[39],2 X[76] - 7 X[3523],X[76] - 6 X[21163],2 X[194] + 3 X[6194],X[194] - 6 X[7709],3 X[194] + 2 X[12251],3 X[194] - 8 X[32448]

X(32522) lies on these lines: {2, 6248}, {3, 194}, {4, 11171}, {20, 39}, {76, 3523}, {114, 7933}, {147, 7791}, {182, 3552}, {262, 3146}, {376, 3095}, {511, 3522}, {538, 15692}, {549, 13108}, {550, 32447}, {574, 12203}, {631, 2782}, {726, 7987}, {1513, 7864}, {1569, 15515}, {2021, 5286}, {3091, 7786}, {3094, 25406}, {3096, 14981}, {3097, 4297}, {3102, 9541}, {3525, 7697}, {3528, 9821}, {3529, 14881}, {3564, 7904}, {3934, 10303}, {5013, 5999}, {5056, 6683}, {5085, 18906}, {5171, 7766}, {5188, 7757}, {5218, 18982}, {5351, 32465}, {5352, 32466}, {5731, 12782}, {6200, 32471}, {6396, 32470}, {6655, 9744}, {7288, 13077}, {7760, 8722}, {7787, 11676}, {7815, 15483}, {8356, 9863}, {8782, 21166}, {9466, 15708}, {9902, 10164}, {15704, 22728}, {15717, 20081}, {20190, 31958}, {31884, 32449}

X(32522) = reflection of X(i) in X(j) for these lines: {i,j}: {3091, 7786}, {31276, 631}
X(32522) = X(631)-of-inscribed-triply-bilogic-triangle
X(32522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 194, 6194}, {3, 7709, 194}, {3, 32448, 12251}, {76, 21163, 3523}, {5171, 32467, 7766}, {7709, 12251, 32448}, {11257, 13334, 2}, {12251, 32448, 194}, {15717, 20081, 22712}


X(32523) = X(3)X(194)∩X(39)X(3627)

Barycentrics    10*a^6*b^2 - 13*a^4*b^4 + 3*a^2*b^6 + 10*a^6*c^2 - 22*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 3*b^6*c^2 - 13*a^4*c^4 - 4*a^2*b^2*c^4 + 6*b^4*c^4 + 3*a^2*c^6 - 3*b^2*c^6 : :
X(32523) = 7 X[3] + 3 X[194],19 X[3] - 9 X[6194],X[3] + 9 X[7709],13 X[3] - 3 X[12251],2 X[3] + 3 X[32448],6 X[39] - X[3627],3 X[76] - 8 X[12108],19 X[194] + 21 X[6194],X[194] - 21 X[7709],13 X[194] + 7 X[12251],2 X[194] - 7 X[32448]

X(32523) lies on these lines: {3, 194}, {39, 3627}, {76, 12108}, {262, 12102}, {538, 15712}, {546, 11257}, {548, 7757}, {549, 14711}, {632, 2782}, {726, 31666}, {3095, 12103}, {3529, 32447}, {3628, 11171}, {3857, 11272}, {6425, 32471}, {6426, 32470}, {7786, 12812}, {10303, 13108}, {11055, 14891}, {13334, 14869}

X(32523) = X(632)-of-inscribed-triply-bilogic-triangle


X(32524) = X(3)X(194)∩X(39)X(3224)

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

X(32524) lies on these s {3, 194}, {39, 3224}, {99, 11328}, {574, 3229}, {1975, 14937}, {3225, 7757}, {3360, 5013}, {3511, 8716}, {8841, 9737}, {9490, 9605}

X(32524) = Thomson-isogonal conjugate of X(6194)


X(32525) = MIDPOINT OF X(2) AND X(9169)

Barycentrics    a^4*b^2 + 3*a^2*b^4 - b^6 + a^4*c^2 - 10*a^2*b^2*c^2 + 2*b^4*c^2 + 3*a^2*c^4 + 2*b^2*c^4 - c^6 : :
X(32525) = 3 X[2] + X[6792],5 X[2] - X[14916],7 X[3090] - X[15098],9 X[5055] - X[18346],X[5108] + 3 X[9169],5 X[5108] - 3 X[14916],X[5971] + 3 X[11647],X[6792] - 3 X[9169],5 X[6792] + 3 X[14916],5 X[9169] + X[14916]

X(32525) lies on the cubic K249 and these s {2, 6}, {5, 1499}, {126, 5969}, {373, 2679}, {542, 6719}, {3090, 15098}, {4045, 11007}, {5026, 10418}, {5055, 18346}, {6034, 30786}, {6082, 14658}, {7603, 9151}, {7804, 22104}, {9152, 31654}, {9172, 9830}, {10160, 10168}, {10278, 22260}, {20382, 23297}

X(32525) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 9169}, {126, 6791}, {5108, 6792}
X(32525) = complement of X(5108)
X(32525) = nine-point-circle-inverse of X(23301)
X(32525) = orthoptic-circle-of-Steiner-inellipe-inverse of X(385)
X(32525) = circumcircle-of-inner-Napoleon-triangle-inverse of X(3180)
X(32525) = circumcircle-of-outer-Napoleon-triangle inverse of X(3181)
X(32525) = complement of the isogonal conjugate of X(14948)
X(32525) = psi-transform of X(148)
X(32525) = X(14948)-complementary conjugate of X(10)
X(32525) = crossdifference of every pair of points lies on these lines: {512, 9225}
X(32525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 11053}, {2, 1648, 141}, {2, 6792, 5108}, {5108, 9169, 6792}


X(32526) = X(2)X(39)∩X(23)X(5106)

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

X(32526) lies on the Parry circle, the cubic K794, and on these lines: {2, 39}, {23, 5106}, {110, 1691}, {187, 9999}, {237, 9157}, {353, 5092}, {511, 9998}, {647, 5996}, {694, 20977}, {698, 13518}, {1084, 7840}, {3292, 9463}, {3796, 21001}, {7711, 11580}, {7931, 8265}, {8620, 9978}, {9138, 9210}, {9156, 17414}, {9208, 9212}, {14684, 20481}

X(32526) = reflection of X(9998) in the Lemoine axis
X(32526) = orthoptic-circle-of-Steiner-inellipse-inverse of X(39)
X(32526) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(194)
X(32526) = psi-transform of X(3094)
X(32526) = crossdifference of every pair of points lies on these lines: {669, 11182}
X(32526) = X(5970)-of-2nd-Parry-triangle


X(32527) = X(4)X(147)∩X(25)X(5152)

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

X(32527) lies on these lines: {4, 147}, {25, 5152}, {804, 2489}

X(32527) = polar circle inverse of X(194)
X(32527) = {X(5186),X(12131)}-harmonic conjugate of X(12143)


X(32528) = X(2)X(4159)∩X(4)X(147)

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

X(32528) lies on these lines: {2, 4159}, {4, 147}, {83, 115}, {99, 626}, {315, 8782}, {316, 9865}, {542, 19570}, {543, 7809}, {620, 7944}, {671, 7753}, {804, 2514}, {2794, 14712}, {2896, 5976}, {3926, 20094}, {5025, 5989}, {5103, 12215}, {5939, 7932}, {5987, 19577}, {6034, 9830}, {6656, 8290}, {7752, 8178}, {7790, 10334}, {7791, 7945}, {7793, 9862}, {7803, 10353}, {7839, 12830}, {7889, 14061}, {9880, 14492}, {11646, 24256}, {12177, 14639}

X(32528) = anticomplement of X(5152)
X(32528) = polar circle inverse of X(12143)
X(32528) = circumcircle-of-anticomplementary-triangle-inverse of X(194)
X(32528) = X(30530)-anticomplementary conjugate of X(17217)
X(32528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 4027, 7797}, {147, 148, 194}, {148, 7785, 1916}, {1916, 6033, 7785}, {6033, 6321, 14881}, {11606, 16044, 115}


X(32529) = X(20)X(185)∩X(22)X(1613)

Barycentrics    a^2*(a^6*b^4 - a^2*b^8 - a^6*b^2*c^2 - a^4*b^4*c^2 + 2*a^2*b^6*c^2 + a^6*c^4 - a^4*b^2*c^4 + a^2*b^4*c^4 - b^6*c^4 + 2*a^2*b^2*c^6 - b^4*c^6 - a^2*c^8) : :
X(32529) = 4 X[2076] - 3 X[11673]

X(32529) lies on these lines: {20, 185}, {22, 1613}, {401, 19585}, {512, 16084}, {805, 1297}, {858, 5103}, {1691, 19121}, {1843, 6658}, {2456, 2698}, {2882, 12215}, {5104, 32464}, {5111, 11416}, {11634, 14965}

X(32529) = crosssum of X(3229) and X(6467)
X(32529) = de Longchamps-circle-inverse of X(194)
X(32529) = anticomplement of the isogonal conjugate of X(8858)
X(32529) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {699, 21216}, {3225, 5905}, {8858, 8}


X(32530) = X(2)X(39)∩X(8651)X(11176)

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

X(32530) lies on these lines: {2, 39}, {8651, 11176}

X(32530) = orthoptic-circle-of-Steiner-inellipse-inverse of X(194)
X(32530) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(20105)


X(32531) = X(2)X(669)∩X(3)X(194)

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

X(32531) lies on these lines: {2, 669}, {3, 194}, {182, 805}, {353, 3231}, {574, 5970}, {3407, 14870}, {5092, 15920}, {5108, 7496}, {8289, 21444}

X(32531) = circumcircle-inverse of X(385)
X(32531) = Brocard-circle-inverse of X(805)
X(32531) = orthoptic-circle-of-Steiner-inellipse-inverse of X(23301)
X(32531) = 2nd-Brocard-circle-inverse of X(194)
X(32531) = psi-transform of X(5027)


X(32532) = ISOGONAL CONJUGATE OF X(15655)

Barycentrics    (7*a^2-11*b^2+7*c^2)*(7*a^2+7*b^2-11*c^2) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32532) lies on the Kiepert hyperbola and these lines: {4, 8584}, {98, 15682}, {115, 10153}, {376, 7607}, {631, 10185}, {1992, 17503}, {2996, 8352}, {3424, 3830}, {3545, 7608}, {3845, 14484}, {5254, 18843}, {5286, 18844}, {5395, 11317}, {5485, 15533}, {7612, 11001}, {11668, 15719}

X(32532) = reflection of X(10153) in X(115)
X(32532) = isogonal conjugate of X(15655)
X(32532) = antigonal conjugate of X(10153)
X(32532) = antitomic conjugate of X(10153)
X(32532) = isotomic conjugate of the anticomplement of X(15534)
X(32532) = antipode of X(10153) in the Kiepert hyperbola


X(32533) = X(5)X(14528)∩X(6)X(546)

Barycentrics    (-a^2+b^2+c^2)*(3*a^4-2*(3*b^2-2*c^2)*a^2+3*(b^2-c^2)^2)*(3*a^4+2*(2*b^2-3*c^2)*a^2+3*(b^2-c^2)^2) : :
Barycentrics    SA*(S^2-5*SB^2)*(S^2-5*SC^2) : :
Barycentrics    (Cos[A] Sin[A])/(2+3 Cos[2 A]) : :
X(32533) = 7*X(3090)-3*X(25712)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32533) lies on the Jerabek hyperbola and these lines: {5, 14528}, {6, 546}, {30, 3532}, {54, 3091}, {64, 3627}, {66, 18383}, {67, 18569}, {68, 13851}, {69, 9927}, {70, 18394}, {74, 3146}, {895, 15083}, {1352, 13622}, {1899, 3521}, {3090, 3431}, {3426, 5076}, {3519, 18404}, {3529, 11270}, {3839, 10116}, {5486, 18553}, {10113, 11744}, {10293, 31725}, {10297, 15316}, {10594, 18532}, {12102, 22334}, {13754, 15077}, {14542, 18390}, {15740, 18918}, {16625, 18376}, {17538, 20421}, {21400, 25738}

X(32533) = isogonal conjugate of X(32534)
X(32533) = anticomplement of X(33556)


X(32534) = ISOGONAL CONJUGATE OF X(32533)

Barycentrics    a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(3*a^4-6*(b^2+c^2)*a^2+3*b^4+4*b^2*c^2+3*c^4) : :
Barycentrics    SB*SC*(SB+SC)*(S^2-5*SA^2) : :
Barycentrics    (2+3 Cos[2 A]) Tan[A] : :
Trilinears    6 cos A - sec A : :

As a point on the Euler line, X(32534) has Shinagawa coefficients (-6*F, E+6*F).

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32534) lies on these lines: {2, 3}, {54, 9786}, {64, 11468}, {74, 1498}, {110, 12163}, {112, 5023}, {154, 6241}, {155, 11449}, {184, 13382}, {185, 9707}, {187, 8743}, {232, 5206}, {340, 9723}, {1151, 10881}, {1152, 10880}, {1181, 11464}, {1192, 5890}, {1199, 3431}, {1204, 10282}, {1350, 19128}, {1495, 3357}, {1511, 13148}, {1614, 1620}, {1829, 17502}, {1843, 17508}, {1870, 5204}, {1968, 15513}, {1974, 14810}, {1986, 11412}, {1990, 8553}, {1993, 12038}, {2207, 5210}, {2351, 5963}, {2904, 16879}, {2931, 9938}, {3043, 15040}, {3092, 6411}, {3093, 6412}, {3567, 11425}, {3580, 12118}, {3581, 16266}, {5013, 10312}, {5085, 6403}, {5217, 6198}, {5237, 8740}, {5238, 8739}, {5410, 6450}, {5411, 6449}, {5621, 15581}, {5702, 8573}, {6030, 15086}, {6759, 21663}, {7592, 11438}, {7689, 11441}, {8541, 20190}, {8744, 8778}, {8907, 12893}, {8960, 9682}, {10193, 13419}, {10606, 12290}, {10619, 23358}, {10632, 11481}, {10633, 11480}, {10990, 13289}, {11204, 11381}, {11270, 12315}, {11363, 31663}, {11440, 18451}, {11550, 25563}, {12112, 13093}, {12278, 14852}, {12279, 12292}, {12383, 12429}, {13481, 19189}, {13558, 16080}, {15036, 15472}, {15578, 20987}, {16655, 23328}, {17845, 25739}, {18396, 26917}, {18445, 32171}, {18474, 20191}, {19467, 26879}, {20427, 32111}

X(32534) = reflection of X(3) in X(23323)
X(32534) = isogonal conjugate of X(32533)
X(32534) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7387, 2071), (4, 3515, 24), (14865, 23040, 11410)


X(32535) = X(3)X(1487)∩X(4)X(1493)

Barycentrics    (7*R^2-3*SC-SW)*(7*R^2-3*SB-SW)*(S^2+SB*SC) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32535) lies on these lines: {3, 1487}, {4, 1493}, {20, 3459}, {3853, 15619}, {17507, 20424}


X(32536) = X(4)X(137)∩X(5)X(27684)

Barycentrics    3*S^4-(R^2*(12*R^2+5*SA-12*SW)-2*SA^2+13*SB*SC+3*SW^2)*S^2-(R^2*(28*R^2-17*SW)+SW^2)*SB*SC : :
X(32536) = 3*X(381)-X(30484)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32536) lies on these lines: {4, 137}, {5, 27684}, {30, 13856}, {128, 24573}, {381, 30484}, {539, 23337}, {546, 20413}, {3853, 5893}, {8146, 28341}, {15307, 31376}, {15619, 17507}, {18807, 28237}, {20120, 25150}

X(32536) = reflection of X(i) in X(j) for these (i,j): (13856, 31879), (18016, 23338)


X(32537) = X(5)X(519)∩X(7)X(8)

Barycentrics    2*a^4-3*(b+c)*a^3+10*a^2*b*c+(b+c)*(3*b^2-8*b*c+3*c^2)*a-2*(b^2-c^2)^2 : :
X(32537) = X(3621)+3*X(25568), 3*X(3679)-X(12513), 5*X(4668)-X(6762), 3*X(4669)-X(24391), 3*X(4677)+X(11523), 7*X(4678)-3*X(24477), 3*X(5587)-X(10912), X(7982)-3*X(11236)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32537) lies on these lines: {1, 5123}, {5, 519}, {7, 8}, {10, 6691}, {145, 11376}, {355, 3880}, {474, 3679}, {517, 32159}, {528, 12640}, {529, 11362}, {946, 5854}, {952, 10915}, {960, 12647}, {1012, 3913}, {1210, 3036}, {1317, 27385}, {1319, 17566}, {1387, 3244}, {1482, 22835}, {1837, 12648}, {2098, 5087}, {2802, 18480}, {3057, 5046}, {3241, 6931}, {3337, 4668}, {3621, 25568}, {3625, 5855}, {3632, 10827}, {3633, 23708}, {3742, 5554}, {3811, 12645}, {3838, 9578}, {3893, 5086}, {4134, 15862}, {4297, 32157}, {4669, 24391}, {4677, 11523}, {4678, 24477}, {4861, 7504}, {5048, 11681}, {5258, 19525}, {5587, 10912}, {5795, 15254}, {5837, 15481}, {5844, 21077}, {6735, 10944}, {6738, 15570}, {6871, 31145}, {7483, 10039}, {7982, 11236}, {7991, 28534}, {8256, 10106}, {8715, 28204}, {9037, 31785}, {9956, 22837}, {11567, 22836}, {12641, 13271}, {12672, 12751}, {13463, 19925}, {15888, 25962}, {20323, 25005}, {21627, 32426}

X(32537) = midpoint of X(i) and X(j) for these lines: {i,j}: {8, 32049}, {3632, 12635}, {3811, 12645}, {3913, 5881}, {12641, 13271}
X(32537) = reflection of X(i) in X(j) for these (i,j): (4297, 32157), (11260, 10), (13463, 19925), (22837, 9956)
X(32537) = {X(8), X(5252)}-harmonic conjugate of X(5836)


X(32538) = X(381)X(7620)∩X(3734)X(16509)

Barycentrics    126*S^4+3*(3*SA+SW)*(15*SA-14*SW)*S^2-4*(3*SA+SW)*(3*SA-2*SW)*SW^2 : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32538) lies on these lines: {381, 7620}, {3734, 16509}, {5485, 18907}


X(32539) = X(4)X(52)∩X(125)X(1147)

Barycentrics    SA*((16*R^2-2*SA-2*SW)*S^2+(SB+SC)*(10*R^4-(7*SA+11*SW)*R^2+2*SA^2-2*SB*SC+2*SW^2)) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29016.

X(32539) lies on these lines: {4, 52}, {5, 32166}, {125, 1147}, {155, 10255}, {539, 18281}, {1899, 12038}, {3357, 17702}, {3448, 12118}, {3564, 10224}, {5449, 9306}, {5504, 23294}, {6696, 11250}, {12163, 18565}, {13383, 32145}, {15316, 19477}

X(32539) = reflection of X(32145) in X(13383)
X(32539) = {X(68), X(11442)}-harmonic conjugate of X(9927)


X(32540) = X(3)X(76)∩X(6)X(14251)

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

X(32540) lies on the cubic K1104 and these on lines {3, 76}, {6, 14251}, {32, 512}, {39, 15630}, {248, 30496}, {446, 1503}, {1634, 19120}, {1910, 2053}, {3095, 13137}, {3224, 14601}, {5025, 10342}, {17932, 19597}

X(32540) = X(i)-isoconjugate of X(j) for these (i,j): {240, 8858}, {1959, 3225}
X(32540) = crosssum of X(i) and X(j) for these (i,j): {511, 5976}, {2679, 2799}
X(32540) = crossdifference of every pair of points on line {325, 2491}
X(32540) = barycentric product X(i)*X(j) for these lines: {i,j}: {98, 3229}, {698, 1976}, {1910, 2227}
X(32540) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {248, 8858}, {1976, 3225}, {3229, 325}, {9429, 3569}, {14601, 699}
X(32540) = {X(98),X(8870)}-harmonic conjugate of X(14382)


X(32541) = X(3)X(3224)∩X(3360)X(18829)

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

X(32541) lies on the cubic K1104 and these lines: {3, 3224}, {3360, 18829}, {8789, 10131}

X(32541) = barycentric product X(9468)*X(9493)
X(32541) = barycentric quotient X(9493) / X(14603)


X(32542) = X(3)X(1625)∩X(24)X(2698)

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

X(32542) lies on the cubics K389 and K1104, and on these lines: {3, 1625}, {24, 2698}, {3148, 14251}, {3406, 8743}

X(32542) = isogonal conjugate of the anticomplement of X(14382)
X(32542) = X(i)-isoconjugate of X(j) for these (i,j): {401, 1581}, {1916, 1955}, {1934, 1971}
X(32542) = barycentric product X(i)*X(j) for these lines: {i,j}: {385, 1987}, {419, 14941}, {1580, 1956}, {1691, 1972}
X(32542) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {419, 16089}, {1691, 401}, {1933, 1955}, {1956, 1934}, {1972, 18896}, {1987, 1916}, {5027, 6130}, {14602, 1971}


X(32543) = (name pending)

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

X(32543) lies on the cubic K1104 and this line: {3360, 32514}

X(32543) = isogonal conjugate of X(32548)


X(32544) = X(32)X(99)∩X(880)X(1691)

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

X(32544) lies on the cubics K444 and K1013, and on these lines: {32, 99}, {880, 1691}, {14602, 17941}

X(32544) = X(1691)-cross conjugate of X(699)
X(32544) = X(i)-isoconjugate of X(j) for these (i,j): {694, 2227}, {698, 1967}, {1581, 3229}
X(32544) = cevapoint of X(1691) and X(4027)
X(32544) = barycentric product X(i)*X(j) for these lines: {i,j}: {385, 3225}, {419, 8858}, {699, 3978}
X(32544) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {385, 698}, {699, 694}, {1580, 2227}, {1691, 3229}, {3225, 1916}


X(32545) = X(3)X(2966)∩X(4)X(32)

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

X(32545) lies on the cubics K444, K629, K790, and on these lines: {3, 2966}, {4, 32}, {20, 9218}, {54, 826}, {194, 17974}, {1316, 14265}, {1975, 17932}, {2065, 9307}, {2715, 11257}, {3398, 14382}, {14251, 18858}

X(32545) = X(i)-isoconjugate of X(j) for these (i,j): {240, 14941}, {511, 1956}, {1755, 1972}, {1959, 1987}
X(32545) = trilinear pole of line {1971, 6130}
X(32545) = barycentric product X(i)*X(j) for these lines: {i,j}: {98, 401}, {248, 16089}, {290, 1971}, {1821, 1955}, {2966, 6130}
X(32545) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {98, 1972}, {248, 14941}, {401, 325}, {1910, 1956}, {1955, 1959}, {1971, 511}, {1976, 1987}, {6130, 2799}


X(32546) = (name pending)

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

X(32546) lies on the cubic K444 and on no line X(i)X(j) for 0 < i < j < 32546.


X(32547) = X(2)X(9292)∩X(69)X(32529)

Barycentrics    a^2*(a^4*b^4 - a^2*b^6 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 + b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 - b^4*c^4 - a^2*c^6 + b^2*c^6) : :
X(32547) = 3 X[2979] - 4 X[7750],3 X[3060] - 2 X[7823],2 X[4173] - 3 X[7833],9 X[5640] - 8 X[7745],10 X[7904] - 9 X[7998],3 X[11459] - 4 X[32151]

X(32547) lies on these lines: {2, 9292}, {69, 32529}, {76, 512}, {110, 32445}, {699, 3224}, {1297, 11440}, {1503, 12272}, {1968, 2001}, {2387, 7802}, {2698, 10104}, {2979, 7750}, {3060, 7823}, {3111, 7940}, {3491, 3552}, {3852, 12220}, {4173, 7833}, {5167, 7793}, {5640, 7745}, {6787, 7746}, {7860, 14962}, {7904, 7998}, {9863, 12111}, {11459, 32151}, {15270, 26881}, {19597, 32464}

X(32547) = reflection of X(12111) in X(9863)
X(32547) = crosspoint of X(3224) and X(30496)
X(32547) = crosssum of X(194) and X(3552)


X(32548) = X(2)X(3224)∩X(69)X(698)

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

X(32548) lies on these lines: {2, 3224}, {69, 698}, {76, 9493}, {315, 1899}, {5207, 32529}, {6337, 22089}, {6374, 9492}, {7785, 13518}, {10453, 17137}, {17138, 21281}

X(32548) = isogonal conjugate of X(32543)
X(32548) = anticomplement of X(3224)
X(32548) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 20081}, {2, 21299}, {6, 21223}, {31, 2998}, {81, 17157}, {88, 23633}, {100, 20979}, {162, 2451}, {190, 20983}, {194, 8}, {651, 23655}, {660, 23656}, {662, 669}, {673, 20863}, {799, 3221}, {897, 20977}, {1424, 145}, {1613, 192}, {1740, 2}, {3186, 5905}, {3221, 21220}, {4598, 23464}, {6374, 6327}, {7075, 144}, {11325, 21216}, {17082, 7}, {17149, 69}, {18837, 315}, {20332, 20464}, {20794, 6360}, {20910, 3448}, {21080, 2895}, {21191, 149}, {21877, 1654}, {22028, 1330}, {23301, 21221}, {23503, 25054}, {23572, 9263}, {23807, 150}, {27644, 3223}
X(32548) = X(6374)-Ceva conjugate of X(2)


X(32549) = X(5)X(2883)∩X(546)X(14374)

Barycentrics    SA*(a*b*c*(8*S^2+(SB+SC)*(6*R^2-7*SA-3*SW))+2*(SB+SC)*(10*R^2-SA-SW)*OH*S) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29027.

X(32549) lies on these lines: {5, 2883}, {546, 14374}, {1113, 32110}, {1312, 13754}, {1313, 14915}, {1346, 9730}, {1347, 16194}, {1531, 10751}, {2575, 6699}


X(32550) = X(5)X(2883)∩X(546)X(14375)

Barycentrics    SA*(a*b*c*(8*S^2+(SB+SC)*(6*R^2-7*SA-3*SW))-2*(SB+SC)*(10*R^2-SA-SW)*OH*S) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29027.

X(32550) lies on these lines: {5, 2883}, {546, 14375}, {1114, 32110}, {1312, 14915}, {1313, 13754}, {1346, 16194}, {1347, 9730}, {1531, 10750}, {2574, 6699}


X(32551) = COMPLEMENT OF X(15345)

Barycentrics    (4*S^2+R^2*(R^2-5*SA-SW)+2*SA^2-2*SB*SC)*(S^2+SB*SC) : :
X(32551) = 3*X(2)+X(25043)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29027.

X(32551) lies on these lines: {2, 3459}, {5, 51}, {140, 6592}, {252, 6150}, {1493, 19553}, {1510, 20327}, {3628, 13856}, {5501, 20414}, {8254, 24385}, {10285, 14143}, {14051, 24306}, {14142, 18400}, {15957, 31879}, {18282, 32428}

X(32551) = midpoint of X(i) and X(j) for these lines: {i,j}: {10285, 14143}, {15345, 25043}
X(32551) = reflection of X(i) in X(j) for these (i,j): (13856, 3628), (18016, 140), (20414, 5501), (31879, 15957)
X(32551) = complement of X(15345)
X(32551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 25043, 15345), (5, 21230, 16336)


X(32552) = COMPLEMENT OF X(6777)

Barycentrics    (2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)+2*S*(2*a^4-2*(b^2+c^2)*a^2+b^4+c^4) : :
Barycentrics    2*sqrt(3)*(3*SB*SC*SW+(3*SA-2*SW)*S^2)-2*S*(4*S^2-SA^2-2*SB*SC-SW^2) : :
X(32552) = 3*X(13)-X(148), 3*X(14)-5*X(14061), X(99)-3*X(5464), 5*X(99)-3*X(9116), 2*X(115)-3*X(5459), 5*X(115)-3*X(31696), X(148)+3*X(617), 3*X(618)-4*X(620), 3*X(619)-2*X(620), 5*X(5459)-2*X(31696), 3*X(5460)-4*X(6722), 3*X(5464)+X(6778), 5*X(5464)-X(9116), 3*X(5478)-2*X(22515), 3*X(5613)-X(6033), 6*X(6669)-5*X(14061), X(6773)-3*X(21156), 5*X(6778)+3*X(9116), 3*X(9114)-X(20094), 3*X(15561)-X(22507)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29029.

X(32552) lies onthese lines: {2, 6777}, {3, 22509}, {6, 6299}, {13, 148}, {14, 6669}, {69, 6582}, {98, 635}, {99, 299}, {114, 630}, {115, 396}, {140, 25560}, {141, 542}, {385, 533}, {532, 7813}, {621, 25236}, {623, 6771}, {624, 5613}, {633, 5982}, {2782, 25559}, {3412, 20394}, {3642, 12188}, {3643, 9862}, {5460, 6722}, {5478, 22515}, {5859, 22570}, {5981, 6115}, {6114, 7792}, {6295, 11646}, {6672, 6782}, {6770, 22687}, {6773, 21156}, {6780, 31710}, {8594, 8597}, {9113, 9760}, {9114, 20094}, {10645, 14904}, {11602, 20377}, {14136, 16529}, {15561, 22507}

X(32552) = reflection of X(i) in X(j) for these (i,j): (14, 6669), (618, 619), (6782, 6672), (11602, 20377), (25560, 140)
X(32552) = complement of X(6777)
X(32552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5464, 6778, 99), (5613, 22689, 624)


X(32553) = COMPLEMENT OF X(6778)

Barycentrics    (2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)-2*S*(2*a^4-2*(b^2+c^2)*a^2+b^4+c^4) : :
Barycentrics    2*sqrt(3)*(3*SB*SC*SW+(3*SA-2*SW)*S^2)+2*S*(4*S^2-SA^2-2*SB*SC-SW^2) : :
X(32553) = 3*X(13)-X(148), 3*X(14)-5*X(14061), X(99)-3*X(5464), 5*X(99)-3*X(9116), 2*X(115)-3*X(5459), 5*X(115)-3*X(31696), X(148)+3*X(617), 3*X(618)-4*X(620), 3*X(619)-2*X(620), 5*X(5459)-2*X(31696), 3*X(5460)-4*X(6722), 3*X(5464)+X(6778), 5*X(5464)-X(9116), 3*X(5478)-2*X(22515), 3*X(5613)-X(6033), 6*X(6669)-5*X(14061), X(6773)-3*X(21156), 5*X(6778)+3*X(9116), 3*X(9114)-X(20094), 3*X(15561)-X(22507)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29029.

X(32553) lies these lines: {2, 6778}, {3, 22507}, {6, 6298}, {13, 6670}, {14, 148}, {69, 6295}, {98, 636}, {99, 298}, {114, 629}, {115, 395}, {140, 25559}, {141, 542}, {385, 532}, {533, 7813}, {622, 25235}, {623, 5617}, {624, 6774}, {634, 5983}, {2782, 25560}, {3411, 20395}, {3642, 9862}, {3643, 12188}, {5459, 6722}, {5479, 22515}, {5858, 22568}, {5980, 6114}, {6115, 7792}, {6582, 11646}, {6671, 6783}, {6770, 21157}, {6773, 22689}, {6779, 31709}, {8595, 8597}, {9112, 9762}, {9116, 20094}, {10646, 14905}, {11603, 20378}, {14137, 16530}, {15561, 22509}

X(32553) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 22507}, {14, 616}, {99, 6777}, {622, 25235}, {5858, 22568}, {5979, 22998}
X(32553) = reflection of X(i) in X(j) for these (i,j): (13, 6670), (619, 618), (6783, 6671), (11603, 20378), (25559, 140)
X(32553) = complement of X(6778)
X(32553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5463, 6777, 99), (5617, 22687, 623)


X(32554) = MIDPOINT OF X(104) AND X(3436)

Barycentrics    (b-c)^2*a^8-2*(b+c)*(b^2+c^2)*a^7-2*(b^4+c^4-7*(b^2+c^2)*b*c)*a^6+2*(b+c)*(b^2+b*c+c^2)*(3*b^2-8*b*c+3*c^2)*a^5-4*(5*b^4+5*c^4-4*(b^2+b*c+c^2)*b*c)*b*c*a^4-2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4-4*(b+c)^2*b*c)*a^3+2*(b^2-c^2)^2*(b-c)^2*(b^2+5*b*c+c^2)*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :
X(32554) = 2*X(7681)-3*X(23513)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29029.

X(32554) lies these lines: {3, 119}, {11, 517}, {12, 21154}, {46, 8068}, {56, 6713}, {78, 952}, {100, 6827}, {104, 3436}, {153, 6926}, {1385, 10956}, {1537, 4187}, {2478, 12775}, {2800, 21616}, {3826, 6980}, {4413, 6923}, {5533, 30323}, {5690, 18802}, {5840, 6928}, {5854, 19914}, {6174, 28459}, {6667, 7680}, {6842, 11231}, {6863, 31246}, {6948, 10728}, {6961, 11681}, {6963, 10698}, {6971, 7681}, {6978, 31272}, {7491, 24466}, {8256, 26470}, {10265, 24391}, {10955, 26287}, {10958, 26285}, {12611, 31788}, {12616, 18254}, {15017, 30503}, {15528, 21077}, {22799, 31775}

X(32554) = midpoint of X(i) and X(j) for these lines: {i,j}: {104, 3436}, {10310, 12764}
X(32554) = reflection of X(i) in X(j) for these (i,j): (56, 6713), (119, 1329), (18802, 5690)


X(32555) = X(1)X(372)∩X(3)X(9)

Barycentrics    a*( a*(-a^2+b^2+c^2)+(-a+b+c)*S) : :
Trilinears    1 + (sec A - tan A) - (sec B - tan B) - (sec C - tan C) : :

See Vu Thanh Tung and CÚsar Lozada, Hyacinthos 29031.

In the configuration at X(6213), the lines IAJA, IBJB, ICJC concur in X(32555), which is also the perspector of A'B'C' and the excentral triangle. (Randy Hutson, June 7, 2019)

X(32555) lies on the cubic K414 and these lines: {1, 372}, {2, 31540}, {3, 9}, {7, 31541}, {20, 30412}, {37, 1152}, {40, 30556}, {41, 30400}, {43, 1685}, {44, 1151}, {45, 6410}, {48, 19067}, {57, 16432}, {63, 16440}, {101, 31564}, {165, 6212}, {169, 31563}, {223, 13388}, {371, 1743}, {487, 4416}, {488, 3912}, {515, 7090}, {573, 31438}, {1100, 3594}, {1123, 31533}, {1124, 7074}, {1449, 3312}, {1587, 5393}, {2183, 19068}, {2951, 31545}, {3218, 21568}, {3219, 21567}, {3247, 6398}, {3305, 16441}, {3306, 21492}, {3311, 16670}, {3523, 30413}, {3576, 30557}, {3592, 16669}, {3723, 6438}, {3731, 6396}, {3911, 8957}, {3928, 21561}, {3929, 21560}, {3973, 6200}, {4292, 30324}, {4297, 31595}, {4357, 11291}, {5405, 13935}, {5437, 21548}, {6204, 15803}, {6351, 6460}, {6409, 16885}, {6411, 15492}, {6412, 16814}, {6420, 16667}, {6426, 16777}, {6430, 16672}, {6431, 16671}, {6432, 16666}, {6434, 16677}, {6450, 16676}, {6454, 16673}, {6469, 16674}, {6684, 14121}, {7308, 16433}, {9583, 13332}, {10164, 31594}, {11292, 17353}, {13411, 30325}, {21566, 27065}, {21569, 27003}

X(32555) = (excentral)-isogonal conjugate of-X(6213)
X(32555) = X(485)-of-excentral triangle
X(32555) = X(488)-of-1st circumperp triangle
X(32555) = X(641)-of-6th mixtilinear triangle
X(32555) = X(6289)-of-hexyl triangle
X(32555) = X(12257)-of-2nd circumperp triangle
X(32555) = X(40)-Ceva conjugate of X(32556)
X(32555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9, 32556), (20, 30412, 31562), (84, 198, 32556), (610, 1490, 32556), (6502, 7133, 1)


X(32556) = X(1)X(371)∩X(3)X(9)

Barycentrics    a*(a*(-a^2 + b^2 + c^2) - (-a + b + c)*S) : :
Trilinears    1 + (sec A + tan A) - (sec B + tan B) - (sec C + tan C) : :
X(32556) = 2 s X[3] - (r + 4 R) X[9], s^2 X[1] - (SW + S) X[371]

X(32556) lies on the cubic K414, the Kiepert circumhyperbola of the excentral triangle, and on these lines: {1, 371}, {2, 31541}, {3, 9}, {7, 31540}, {20, 30413}, {37, 1151}, {40, 30557}, {41, 30401}, {43, 1686}, {44, 1152}, {45, 6409}, {48, 19068}, {57, 16433}, {63, 16441}, {101, 31563}, {165, 6213}, {169, 31564}, {223, 13389}, {372, 1743}, {487, 3912}, {488, 4416}, {505, 3645}, {515, 14121}, {572, 31438}, {846, 9582}, {950, 8957}, {1100, 3592}, {1335, 7074}, {1336, 31532}, {1449, 3311}, {1588, 5405}, {2183, 19067}, {2951, 31544}, {3218, 21565}, {3219, 21566}, {3247, 6221}, {3305, 16440}, {3306, 21553}, {3312, 16670}, {3523, 30412}, {3576, 30556}, {3594, 16669}, {3723, 6437}, {3731, 6200}, {3928, 21558}, {3929, 21559}, {3973, 6396}, {4292, 30325}, {4297, 31594}, {4357, 11292}, {5393, 9540}, {5437, 21547}, {6203, 15803}, {6352, 6459}, {6410, 16885}, {6411, 16814}, {6412, 15492}, {6419, 16667}, {6425, 16777}, {6429, 16672}, {6431, 16666}, {6432, 16671}, {6433, 16677}, {6449, 16676}, {6453, 16673}, {6468, 16674}, {6684, 7090}, {7308, 16432}, {10164, 31595}, {10436, 21909}, {11291, 17353}, {13411, 30324}, {21564, 27003}, {21567, 27065}

X(32556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9, 32555}, {20, 30413, 31561}, {84, 198, 32555}, {610, 1490, 32555}
X(32556) = excentral-isogonal conjugate of X(6212)
X(32556) = X(485)-of-excentral-triangle
X(32556) = X(487)-of-1st-circumperp-triangle
X(32556) = X(12257)-of-2nd-circumperp-triangle
X(32556) = X(642)-of-6th-mixtilinear-triangle
X(32556) = X(6290)-of-hexyl-triangle
X(32556) = X(i)-Ceva conjugate of X(j) for these (i,j): {40, 32555}, {30557, 1}


X(32557) = X(1)X(6702)∩X(2)X(2802)

Barycentrics    2*a^4 - a^3*b - 4*a^2*b^2 + a*b^3 + 2*b^4 - a^3*c + 6*a^2*b*c - 4*a^2*c^2 - 4*b^2*c^2 + a*c^3 + 2*c^4 : :
X(32557) = X[1] + 2 X[6702],5 X[1] + X[12531],2 X[1] + X[15863],X[1] + 5 X[31272],X[3] + 2 X[16174],2 X[5] + X[11715],X[10] + 2 X[1387],X[10] - 4 X[6667],2 X[11] + X[214],X[11] + 2 X[1125],5 X[11] + X[10609],7 X[11] - X[12690],X[80] + 5 X[3616],X[100] - 7 X[3624],X[104] + 5 X[8227],X[124] + 2 X[29008],X[149] + 11 X[5550],X[214] - 4 X[1125],5 X[214] - 2 X[10609],7 X[214] + 2 X[12690],4 X[551] - X[11274],5 X[631] + X[14217],X[946] + 2 X[6713],10 X[1125] - X[10609],14 X[1125] + X[12690],X[1145] - 4 X[3634],X[1317] - 4 X[3636],X[1320] + 5 X[1698],2 X[1385] + X[6246],X[1387] + 2 X[6667],5 X[1656] + X[12737],2 X[3035] - 5 X[19862],2 X[3035] + X[21630],2 X[3036] + X[3244],7 X[3090] - X[12751],5 X[3617] + X[26726],7 X[3622] - X[7972],2 X[3626] + X[25416],5 X[3698] + X[17652],2 X[3754] + X[12758],X[3874] - 4 X[18240],X[3874] + 2 X[18254],X[3878] + 2 X[12736],X[4973] + 2 X[11813],X[4973] - 4 X[15325],5 X[5439] + X[17638],2 X[5901] + X[12619],4 X[5901] - X[25485],2 X[6681] + X[30384],10 X[6702] - X[12531],4 X[6702] - X[15863],2 X[6702] - 5 X[31272],5 X[7987] + X[10724],7 X[9624] - X[10698],X[10265] + 2 X[11729],7 X[10609] + 5 X[12690],X[11813] + 2 X[15325],2 X[12019] + 7 X[15808],X[12515] + 5 X[18493],2 X[12531] - 5 X[15863],X[12531] - 25 X[31272],X[12532] + 5 X[18398],2 X[12619] + X[25485],2 X[13624] + X[22938],X[13996] - 10 X[31253],2 X[15528] + X[31803],X[15863] - 10 X[31272],2 X[18240] + X[18254],2 X[18857] + X[24042],5 X[19862] + X[21630],8 X[19878] - 5 X[31235],2 X[20418] + X[21635]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29037.

X(32557) lies these lines: {1, 6702}, {2, 2802}, {3, 16174}, {5, 11715}, {10, 1387}, {11, 214}, {80, 3616}, {100, 3624}, {104, 8227}, {124, 29008}, {149, 5550}, {499, 3878}, {515, 23513}, {516, 21154}, {528, 19883}, {547, 551}, {631, 14217}, {758, 3582}, {946, 6713}, {1145, 3634}, {1317, 3636}, {1320, 1698}, {1385, 6246}, {1656, 12737}, {2771, 3742}, {2800, 5883}, {2801, 5817}, {2829, 3817}, {3035, 14150}, {3036, 3244}, {3057, 20107}, {3086, 3874}, {3090, 12751}, {3617, 26726}, {3622, 7972}, {3626, 25416}, {3698, 17652}, {3754, 12758}, {3825, 8068}, {3892, 10072}, {4973, 11813}, {4996, 5259}, {5083, 11375}, {5248, 10090}, {5439, 17638}, {5443, 11570}, {5840, 10165}, {5901, 12619}, {6265, 30143}, {6681, 30384}, {7968, 8988}, {7969, 13976}, {7987, 10724}, {9624, 10698}, {10057, 10584}, {10058, 23708}, {10265, 11729}, {10527, 14740}, {10707, 15015}, {11108, 22560}, {11263, 12611}, {11376, 15558}, {11717, 24222}, {12019, 15808}, {12515, 18493}, {12532, 18398}, {12740, 30147}, {13205, 16408}, {13624, 22938}, {13902, 19077}, {13959, 19078}, {13996, 31253}, {15528, 31803}, {15950, 20118}, {17636, 20104}, {17724, 23869}, {18857, 24042}, {19878, 31235}, {20418, 21635}

X(32557) = midpoint of X(i) and X(j) for these lines: {i,j}: {2, 16173}, {10707, 15015}
X(32557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6702, 15863}, {1, 31272, 6702}, {11, 1125, 214}, {1387, 6667, 10}, {5886, 10199, 5883}, {5901, 12619, 25485}, {11813, 15325, 4973}, {18240, 18254, 3874}, {19862, 21630, 3035}


X(32558) = X(2)X(2802)∩X(8)X(1387)

Barycentrics    3*a^4 - 2*a^3*b - 6*a^2*b^2 + 2*a*b^3 + 3*b^4 - 2*a^3*c + 11*a^2*b*c - a*b^2*c - 6*a^2*c^2 - a*b*c^2 - 6*b^2*c^2 + 2*a*c^3 + 3*c^4 : :
X(32558) = X[2] + 2 X[16173],X[8] + 8 X[1387],5 X[8] + 4 X[25416],X[8] - 10 X[31272],4 X[11] + 5 X[3616],8 X[11] + X[6224],X[20] + 8 X[16174],2 X[80] + 7 X[3622],2 X[100] - 11 X[5550],X[145] + 8 X[6702],X[149] + 8 X[1125],X[149] + 2 X[15015],X[153] - 10 X[8227],X[962] + 8 X[6713],4 X[1125] - X[15015],4 X[1145] - 13 X[19877],X[1320] + 8 X[6667],2 X[1320] + 7 X[9780],10 X[1387] - X[25416],4 X[1387] + 5 X[31272],8 X[3035] + X[9802],8 X[3036] + X[20050],7 X[3090] + 2 X[12737],5 X[3091] + 4 X[11715],7 X[3523] + 2 X[14217],10 X[3616] - X[6224],5 X[3623] + 4 X[15863],7 X[3624] + 2 X[21630],8 X[3634] + X[12653],8 X[3636] + X[9897],5 X[3890] + 4 X[6797],7 X[4678] + 2 X[26726],11 X[5056] - 2 X[12751],X[5180] + 8 X[15325],X[5541] - 10 X[19862],8 X[5901] + X[12247],16 X[6667] - 7 X[9780],7 X[9624] + 2 X[10265],X[9778] - 4 X[21154],X[9803] + 8 X[11729],X[9809] + 8 X[20418],8 X[9955] + X[12248],5 X[10595] + 4 X[12619],2 X[12531] + 7 X[20057],X[12532] + 8 X[18240],2 X[25416] + 25 X[31272]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29037.

X(32558) lies these lines: {2, 2802}, {8, 1387}, {11, 2476}, {20, 16174}, {80, 3622}, {100, 5550}, {145, 6702}, {149, 1125}, {153, 8227}, {214, 31418}, {952, 5055}, {962, 6713}, {1145, 19877}, {1320, 6667}, {2771, 5886}, {2829, 9779}, {3035, 9802}, {3036, 20050}, {3090, 12737}, {3091, 11715}, {3523, 14217}, {3623, 15863}, {3624, 21630}, {3634, 12653}, {3636, 9897}, {3890, 6797}, {4323, 12832}, {4678, 26726}, {5047, 22560}, {5056, 12751}, {5180, 15325}, {5541, 19862}, {5901, 12247}, {9624, 10265}, {9778, 21154}, {9803, 11729}, {9809, 20418}, {9955, 12248}, {10588, 20586}, {10595, 12619}, {11373, 27529}, {11376, 25414}, {12531, 20057}, {12532, 18240}, {13205, 17531}, {15558, 18220}

X(32558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 3616, 6224}, {1320, 6667, 9780}, {1387, 31272, 8}






leftri  Mutual polar conics: X(32559) - X(32588)  rightri

This preamble and centers X(32559)-X(32588) were contributed by CÚsar Eliud Lozada, May 22, 2019.

Two triangles are mutually polar with respect to a conic if each edge of one is the polar of a vertex of the other.

Two triangles are perspective if and only if they are mutually polar with respect to a conic.. (Lord, Eric (2013). Symmetry and Pattern in Projective Geometry, (pp 88). London: Springer).

The conic with respect to which two perspective triangles T1 and T2 are mutually polar are here named the mutual polar conic of T1 and T2. A list of these conics involving ABC and some triangles can be seen in Wolfram's Polar Triangle.

If T' = A'B'C' is a central triangle perspective to ABC with perspector P = u:v:w (trilinears) then there exists a function f(a,b,c) such that AA'=f(a,b,c)*AP, BB'=f(b,c,a)*BP and CC'=f(c,a,b)*CP (in particular, if T' and ABC are homothetic then f(a,b,c) is a constant). The center O of the mutual polar conic of ABC and T' has trilinears O = u/f(a,b,c) : v/f(b,c,a) : w/f(c,a,b) (O=P if ABC and T' are homothetic). The perspector of this conic is P.

The appearance of (T, n) in the following list means that the center of the mutual polar conic of ABC and T is X(n) (ABC and T non-homothetic):

(Andromeda, 32559), (anti-Artzt, 2), (anti-Atik, 4), (1st anti-Brocard, 2), (4th anti-Brocard, 21448), (anti-Conway, 54), (2nd anti-Conway, 4), (anti-excenters-reflections, 64), (2nd anti-extouch, 6), (anti-Honsberger, 1176), (anti-Hutson intouch, 3), (anti-incircle-circles, 3), (anti-inverse-in-incircle, 69), (anti-McCay, 2), (6th anti-mixtilinear, 3), (anti-orthocentroidal, 6), (1st anti-orthosymmedial, 251), (1st anti-Sharygin, 54), (anti-tangential-midarc, 73), (3rd anti-tri-squares, 486), (4th anti-tri-squares, 485), (AAOA, 3), (Antlia, 32560), (Apollonius, 9560), (Apus, 32561), (Artzt, 2), (Atik, 8), (Ayme, 10), (BCI, 1489), (1st Brocard-reflected, 2), (1st Brocard, 2), (2nd Brocard, 574), (3rd Brocard, 3117), (4th Brocard, 5094), (circummedial, 10130), (circumorthic, 54), (2nd circumperp, 21), (circumsymmedial, 574), (Conway, 21), (2nd Conway, 8), (3rd Conway, 1), (4th Conway, 11679), (5th Conway, 17185), (Ehrmann-side, 3), (Ehrmann-vertex, 265), (2nd Ehrmann, 895), (2nd Euler, 3), (5th Euler, 5094), (excenters-midpoints, 9), (excenters-reflections, 3680), (excentral, 9), (1st excosine, 6), (2nd excosine, 1033), (extangents, 71), (extouch, 9), (2nd extouch, 9), (3rd extouch, 223), (4th extouch, 1038), (5th extouch, 1038), (inner-Fermat, 2), (outer-Fermat, 2), (3rd Fermat-Dao, 13), (4th Fermat-Dao, 14), (5th Fermat-Dao, 11080), (6th Fermat-Dao, 11085), (7th Fermat-Dao, 13), (8th Fermat-Dao, 14), (9th Fermat-Dao, 11080), (10th Fermat-Dao, 11085), (11th Fermat-Dao, 14), (12th Fermat-Dao, 13), (13th Fermat-Dao, 14), (14th Fermat-Dao, 13), (15th Fermat-Dao, 18), (16th Fermat-Dao, 17), (Feuerbach, 5949), (Hatzipolakis-Moses, 6), (1st Hatzipolakis, 17054), (2nd Hatzipolakis, 17054), (3rd Hatzipolakis, 6), (hexyl, 1), (Honsberger, 2346), (Hutson extouch, 9), (Hutson intouch, 1), (outer-Hutson, 7707), (2nd Hyacinth, 6), (incentral, 37), (incircle-circles, 1), (intouch, 1), (inverse-in-incircle, 7), (1st isodynamic-Dao, 17), (2nd isodynamic-Dao, 18), (3rd isodynamic-Dao, 4), (4th isodynamic-Dao, 4), (1st Kenmotu diagonals, 6413), (2nd Kenmotu diagonals, 6414), (Kosnita, 3), (Lemoine, 597), (1st Lemoine-Dao, 4), (2nd Lemoine-Dao, 4), (Lucas antipodal, 10132), (Lucas Brocard, 32562), (Lucas central, 10132), (Lucas inner, 32563), (Lucas inner tangential, 32564), (Lucas reflection, 184), (Lucas secondary central, 32565), (Lucas 1st secondary tangents, 32566), (Lucas 2nd secondary tangents, 32567), (Lucas tangents, 32568), (Lucas(-1) antipodal, 10133), (Lucas(-1) Brocard, 32569), (Lucas(-1) central, 10133), (Lucas(-1) inner, 32570), (Lucas(-1) inner tangential, 32571), (Lucas(-1) reflection, 184), (Lucas(-1) secondary central, 32572), (Lucas(-1) 1st secondary tangents, 32573), (Lucas(-1) 2nd secondary tangents, 32574), (Lucas(-1) tangents, 32575), (Macbeath, 5), (Malfatti, 32576), (Mandart-excircles, 6), (McCay, 2), (midarc, 2089), (2nd midarc, 1488), (midheight, 6), (mixtilinear, 55), (2nd mixtilinear, 220), (3rd mixtilinear, 32577), (4th mixtilinear, 32578), (6th mixtilinear, 1), (7th mixtilinear, 2124), (Montesdeoca-Hung, 32579), (1st Morley, 3604), (2nd Morley, 3602), (3rd Morley, 3603), (1st Morley-adjunct, 16839), (2nd Morley-adjunct, 16840), (3rd Morley-adjunct, 16841), (Moses-Hung, 32580), (inner-Napoleon, 2), (outer-Napoleon, 2), (1st Neuberg, 2), (2nd Neuberg, 2), (orthic, 6), (orthocentroidal, 6), (1st orthosymmedial, 251), (2nd orthosymmedial, 32581), (1st Pamfilos-Zhou, 32582), (2nd Pamfilos-Zhou, 7133), (1st Parry, 110), (2nd Parry, 111), (3rd Parry, 32583), (Pelletier, 11), (reflection, 6), (1st Sharygin, 21), (2nd Sharygin, 100), (Soddy, 3160), (inner-Soddy, 10134), (2nd inner-Soddy, 13389), (outer-Soddy, 10135), (2nd outer-Soddy, 13388), (inner-squares, 3068), (outer-squares, 3069), (Steiner, 523), (symmedial, 39), (tangential, 3), (tangential-midarc, 15997), (inner tri-equilateral, 32585), (outer tri-equilateral, 32586), (3rd tri-squares, 485), (4th tri-squares, 486), (Trinh, 3), (inner-Vecten, 2), (2nd inner-Vecten, 32587), (3er inner-Vecten, 32587), (outer-Vecten, 2), (2nd outer-Vecten, 32588), (3er outer-Vecten, 32588), (Walsmith, 74), (X-parabola-tangential, 523), (Yff central, 7707), (Yff contact, 514), (Yiu, 1154), (1st Zaniah, 1), (2nd Zaniah, 9).
underbar

X(32559) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND ANDROMEDA

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

X(32559) lies on these lines: {1,738}, {2,3677}, {37,32560}, {3666,21446}


X(32560) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND ANTLIA

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

X(32560) lies on these lines: {1,728}, {2,479}, {37,32559}, {10582,21450}


X(32561) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND APUS

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

X(32561) lies on these lines: {3,169}, {6,255}, {9,35}, {21,6554}, {32,16588}, {37,8573}, {41,55}, {56,17451}, {71,11434}, {101,10267}, {198,23843}, {218,11507}, {219,584}, {384,26068}, {411,5819}, {672,11509}, {958,1146}, {1376,26036}, {1466,5022}, {2082,26357}, {2170,10966}, {2178,7742}, {2183,15592}, {2225,7085}, {2246,5217}, {2478,6506}, {2911,15830}, {3208,8668}, {3295,6603}, {3560,5179}, {3730,11248}, {5251,23058}, {7367,8606}, {8069,16601}, {9310,11510}, {10306,21872}, {11496,17747}, {12329,23851}, {16367,30854}, {16865,27541}

X(32561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15288, 1212), (1802, 8012, 220)


X(32562) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS BROCARD

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

X(32562) lies on these lines: {574,8373}, {1151,1584}


X(32563) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS INNER

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

X(32563) lies on these lines: {2,6568}, {110,32072}, {574,32570}, {6221,8854}, {6407,32564}


X(32564) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS INNER TANGENTIAL

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

X(32564) lies on the line {6407,32563}


X(32565) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS SECONDARY CENTRAL

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

X(32565) lies on these lines: {1152,32568}, {8407,10133}, {22112,32572}


X(32566) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS 1st SECONDARY TANGENTS

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

X(32566) lies on the line {1151,10133}


X(32567) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS 2nd SECONDARY TANGENTS

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

X(32567) lies on these lines: {22,6480}, {372,1600}, {3156,6431}


X(32568) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS TANGENTS

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

X(32568) lies on these lines: {2,7598}, {3,493}, {6,6216}, {22,6200}, {110,11199}, {154,1151}, {371,1599}, {574,20859}, {1152,32565}, {1501,9675}, {1583,15883}, {3167,6221}, {6413,8939}, {6560,11209}, {7484,9600}, {8216,21642}, {8414,10133}, {13882,32588}, {19358,26347}

X(32568) = isogonal conjugate of X(18819)
X(32568) = {X(3), X(493)}-harmonic conjugate of X(8577)


X(32569) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) BROCARD

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

X(32569) lies on these lines: {574,8374}, {1152,1583}


X(32570) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) INNER

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

X(32570) lies on these lines: {2,6569}, {110,32073}, {574,32563}, {6398,8855}, {6408,32571}


X(32571) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) INNER TANGENTIAL

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

X(32571) lies on the line {6408,32570}


X(32572) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) SECONDARY CENTRAL

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

X(32572) lies on these lines: {1151,32575}, {8400,10132}, {22112,32565}


X(32573) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) 1st SECONDARY TANGENTS

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

X(32573) lies on the line {1152,10132}


X(32574) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) 2nd SECONDARY TANGENTS

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

X(32574) lies on these lines: {22,6481}, {371,1599}, {3155,6432}


X(32575) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND LUCAS(-1) TANGENTS

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

X(32575) lies on these lines: {2,7599}, {3,494}, {6,6397}, {22,6396}, {110,32074}, {154,1152}, {372,1600}, {574,20859}, {1151,32572}, {1584,15884}, {3167,6398}, {6414,8943}, {6561,11210}, {8219,21643}, {8406,10132}, {13934,32587}

X(32575) = isogonal conjugate of X(18820)
X(32575) = {X(3), X(494)}-harmonic conjugate of X(8576)


X(32576) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND MALFATTI

Barycentrics    a*(2*a*(cos(B/2)*cos(C/2)+cos(B/2)+cos(C/2))-cos(A/2)*(a-b-c)+a+b+c)*(cos(B/2)+1)*(cos(C/2)+1) : :
Barycentrics    Sin[A] (Sec[A / 4]^2) (1 + Sin[A / 2]) : : (Peter Moses, May 21, 2019)

X(32576) lies on this line: {258,483}

X(32576) = X(289)-isoconjugate of X(557)
X(32576) = barycentric product X(236)*X(483)
X(32576) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {7010, 21456}, {7014, 1488}


X(32577) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 3rd MIXTILINEAR

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

X(32577) lies on these lines: {1,88}, {3,1149}, {8,9350}, {25,8278}, {31,56}, {36,3915}, {38,19861}, {41,1015}, {42,3304}, {57,15839}, {200,7963}, {220,23649}, {227,1319}, {238,28370}, {355,28096}, {515,28018}, {551,16393}, {602,22765}, {604,17053}, {614,1420}, {748,2975}, {756,8583}, {899,12513}, {944,28074}, {956,27627}, {958,8688}, {976,17614}, {995,1203}, {999,1066}, {1064,16203}, {1125,5192}, {1253,10966}, {1357,16945}, {1385,19548}, {1388,17054}, {1450,26437}, {1458,15287}, {1477,9439}, {1496,22767}, {1647,1837}, {1739,22837}, {2209,20470}, {2267,8610}, {2275,9259}, {2650,3333}, {3086,21935}, {3120,11376}, {3303,17782}, {3576,28011}, {3616,4195}, {3720,16405}, {3752,20323}, {3756,10950}, {3890,17596}, {3953,30144}, {3976,4511}, {4251,9336}, {4298,24725}, {4310,24558}, {4694,22836}, {4861,24174}, {5193,10571}, {5288,17749}, {5303,8616}, {5450,32486}, {5484,25960}, {5731,28016}, {6762,21805}, {9619,21808}, {9708,28257}, {10165,28027}, {10459,17124}, {11260,16610}, {11682,18193}, {12635,17449}, {14986,26050}, {16466,23070}, {21842,30117}, {22437,23057}, {26561,30816}

X(32577) = isogonal conjugate of the isotomic conjugate of X(4862)
X(32577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1054, 14923), (1, 4855, 3722), (1, 5253, 750)


X(32578) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 4th MIXTILINEAR

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

X(32578) lies on these lines: {9,100}, {31,15263}, {41,55}, {57,17451}, {165,169}, {200,15838}, {672,1405}, {899,6181}, {1155,1212}, {1253,14936}, {2078,9310}, {2280,2323}, {3730,5537}, {5745,24596}

X(32578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 8012, 6602), (1212, 15855, 1155)


X(32579) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND MONTESDEOCA-HUNG

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

X(32579) lies on these lines: {}


X(32580) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND MOSES-HUNG

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

X(32580) lies on these lines: {4,6}, {37,115}, {3553,8818}, {6354,8736}


X(32581) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 2nd ORTHOSYMMEDIAL

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

X(32581) lies on these lines: {4,83}, {5,26224}, {25,14535}, {66,8801}, {112,251}, {406,27005}, {475,27067}, {1594,28724}, {1799,8889}, {5094,10130}, {5523,10549}, {7507,10547}, {7770,12220}

X(32581) = polar conjugate of X(23297)


X(32582) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 1st PAMFILOS-ZHOU

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

X(32582) lies on these lines: {2,31546}, {165,16441}, {1621,5314}


X(32583) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 3rd PARRY

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

X(32583) lies on these lines: {2,10558}, {110,249}, {111,352}, {323,895}, {647,9216}, {892,5466}, {1296,15406}, {2421,9186}, {2434,2444}, {3292,15899}, {3906,9146}, {5968,15066}, {6137,9202}, {6138,9203}, {9124,9178}, {9145,17414}, {9872,10510}, {10989,14833}, {11215,13241}, {15398,23061}

X(32583) = midpoint of X(9872) and X(10510)
X(32583) = isogonal conjugate of X(23287)
X(32583) = trilinear pole of the line {574, 8542}
X(32583) = homothetic center of ABC and 3rd Parry of 3rd Parry triangle
X(32583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10559, 10558), (2, 10560, 21460), (10560, 21460, 10558)


X(32584) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND SCHROETER

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

X(32584) lies on the line {115,523}


X(32585) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND INNER TRI-EQUILATERAL

Barycentrics    a^2*(-a^2+b^2+c^2)*(2*sqrt(3)*S+a^2-b^2+c^2)*(2*sqrt(3)*S+a^2-c^2+b^2) : :
Barycentrics    Sin[A] Csc[A+Pi/6] Sin[2 A] : : (Peter Moses, May 21, 2019)

X(32585) lies on the Jerabek hyperbola and these lines: {2,10639}, {3,10661}, {4,15}, {6,3132}, {16,54}, {61,1173}, {62,13472}, {64,10675}, {65,7051}, {67,32301}, {68,10634}, {69,11515}, {74,3166}, {110,10640}, {184,10979}, {216,21648}, {290,32036}, {1176,11516}, {2963,3458}, {2992,19779}, {3131,30402}, {3431,10646}, {3527,11485}, {5238,16835}, {5352,13452}, {6145,32397}, {8795,19190}, {11081,11139}, {11144,11420}, {11421,14170}, {11481,14528}, {21647,22052}, {22466,22974}

X(32585) = isogonal conjugate of X(473)
X(32585) = isotomic conjugate of the polar conjugate of X(21461)
X(32585) = anticomplement of the complementary conjugate of X(465)
X(32585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15, 8837, 10632), (184, 10979, 32586)


X(32586) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(-a^2+b^2+c^2)*(-2*sqrt(3)*S+a^2-b^2+c^2)*(-2*sqrt(3)*S+a^2-c^2+b^2) : :
Barycentrics    Sin[A] Csc[A-Pi/6] Sin[2 A] : : (Peter Moses, May 21, 2019)

X(32586) lies on the Jerabek hyperbola and these lines: {2,10640}, {3,10662}, {4,16}, {6,3131}, {15,54}, {61,13472}, {62,1173}, {64,10676}, {65,10637}, {67,32302}, {68,10635}, {69,11516}, {74,3165}, {110,10639}, {184,10979}, {216,21647}, {290,32037}, {1176,11515}, {2963,3457}, {2992,11131}, {3132,30403}, {3431,10645}, {3527,11486}, {5237,16835}, {5351,13452}, {6145,32398}, {8795,19191}, {11143,11421}, {11420,14169}, {11480,14528}, {21648,22052}, {22466,22975}

X(32586) = isogonal conjugate of X(472)
X(32586) = isotomic conjugate of the polar conjugate of X(21462)
X(32586) = anticomplement of the complementary conjugate of X(466)
X(32586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16, 8839, 10633), (184, 10979, 32585)


X(32587) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 2nd INNER-VECTEN

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)*(b^2+c^2-2*S) : :
X(32587) = (4*R^2-SW)*X(3)+(2*R^2+S-SW)*X(4)

As a point on the Euler line, X(32587) has Shinagawa coefficients (F, E+F-2*S).

X(32587) lies on these lines: {2,3}, {51,12299}, {69,24243}, {136,5408}, {184,13749}, {275,14238}, {486,494}, {488,11395}, {490,12172}, {591,8754}, {615,8564}, {640,8946}, {1249,19041}, {1322,1899}, {1351,13439}, {1588,11245}, {1853,23261}, {2052,14231}, {2969,13387}, {2979,6400}, {3070,12232}, {3917,6406}, {5409,6290}, {5591,26376}, {6414,13960}, {6561,8940}, {6747,11474}, {7140,13386}, {11090,14593}, {11550,13748}, {12601,13428}, {13429,13951}, {13934,32575}, {14230,23292}

X(32587) = polar conjugate of the isogonal conjugate of X(1505)
X(32587) = anticomplement of X(8964)
X(32587) = X(22553)-of-4th tri-squares triangle, when ABC is obtuse
X(32587) = X(22554)-of-4th tri-squares triangle, when ABC is acute
X(32587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 427, 32588), (4, 1586, 25), (4, 3128, 427), (4, 5200, 428)


X(32588) = CENTER OF THE MUTUAL POLAR CONIC OF THESE TRIANGLES: ABC AND 2nd OUTER-VECTEN

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)*(b^2+c^2+2*S) : :
X(32588) = (4*R^2-SW)*X(3)+(2*R^2-S-SW)*X(4)

As a point on the Euler line, X(32587) has Shinagawa coefficients (F, E+F+2*S).

X(32588) lies on these lines: {2,3}, {51,12298}, {69,24244}, {136,5409}, {184,13748}, {275,14234}, {485,493}, {487,11394}, {489,12171}, {590,8563}, {639,8948}, {1249,19042}, {1321,1899}, {1351,13428}, {1587,11245}, {1853,23251}, {1991,8754}, {2052,14245}, {2969,13386}, {2979,6239}, {3071,12231}, {3917,6291}, {5408,6289}, {5590,26375}, {6413,8966}, {6560,8944}, {6564,8956}, {6747,11473}, {7140,13387}, {8976,13440}, {11091,14593}, {11550,13749}, {12602,13439}, {13882,32568}, {14233,23292}

X(32588) = polar conjugate of the isogonal conjugate of X(1504)
X(32588) = X(15891)-of-3rd outer-Vecten triangle
X(32588) = X(22553)-of-3rd tri-squares triangle
X(32588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 427, 32587), (4, 1585, 25), (4, 3535, 5200)


X(32589) = SS( S → -S) of X(8954)

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

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32589) lies on these lines: {3, 6}, {24, 6414}, {155, 10960}, {485, 6810}, {486, 1586}, {590, 8955}, {1075, 3069}, {1600, 26922}, {5562, 8963}, {6413, 7592}, {6565, 8887}, {6642, 10962}, {8855, 15199}, {10880, 26920}

X(32589) = SS( S → -S) of X(8954)
X(32589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (216, 389, 8954), (3371, 3372, 577), (3385, 3386, 578)


X(32590) = X(1)X(372)∩X(4)X(48)

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

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32590) lies on these lines: {1, 372}, {4, 48}, {41, 1588}, {101, 31562}, {572, 31561}, {604, 1587}, {610, 6212}, {637, 20769}, {1745, 2067}, {2066, 3362}, {2174, 3071}, {2286, 3093}, {3070, 7113}, {3092, 7124}, {5709, 19215}, {7330, 19216}

X(32590) = isogonal conjugate of X(32591)
X(32590) = {X(4),X(48)}-harmonic conjugate of X(32592)


X(32591) = ISOGONAL CONJUGATE OF X(32590)

Barycentrics    a*(-(a^2-b^2+c^2)*(a^2-b^2-c^2)*a*b+2*(a^2+b^2-c^2)*S*c^2)*(-(a^2-b^2-c^2)*(a^2+b^2-c^2)*a*c+2*(a^2-b^2+c^2)*S*b^2) : :
Barycentrics    a*(S^3+(a*b-S)*SA*SB)*(S^3+SA*(-S+c*a)*SC) : :
Trilinears    sin(A)*(2*cos(A)*cos(C)+sin(2*B))*(2*cos(A)*cos(B)+sin(2*C)) : :

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32591) lies on these lines: {1, 32589}, {1745, 6212}

X(32591) = isogonal conjugate of X(32590)


X(32592) = X(1)X(371)∩X(4)X(48)

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

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32592) lies on these lines: {1, 371}, {4, 48}, {41, 1587}, {101, 31561}, {572, 31562}, {604, 1588}, {610, 6213}, {638, 20769}, {1745, 6502}, {2174, 3070}, {2286, 3092}, {3071, 7113}, {3093, 7124}, {3362, 5414}, {5709, 19216}, {7330, 19215}

X(32592) = isogonal conjugate of X(32593)
X(32592) = {X(4),X(48)}-harmonic conjugate of X(32590)


X(32593) = ISOGONAL CONJUGATE OF X(32592)

Barycentrics    a*(-(a^2-b^2+c^2)*(a^2-b^2-c^2)*a*b-2*(a^2+b^2-c^2)*S*c^2)*(-(a^2-b^2-c^2)*(a^2+b^2-c^2)*a*c-2*(a^2-b^2+c^2)*S*b^2) : :
Barycentrics    a*(S^3+(a*b+S)*SA*SB)*(S^3+SA*(a*c+S)*SC) : :
Trilinears    sin(A)*(2*cos(A)*cos(C)-sin(2*B))*(2*cos(A)*cos(B)-sin(2*C)) : :

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32593) lies on these lines: {1, 8954}, {1745, 6213}

X(32593) = isogonal conjugate of X(32592)


X(32594) = X(3)X(8287)∩X(5)X(6)

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

See Vu Thanh Tung, Vu Quoc My and CÚsar Lozada, Hyacinthos 29049.

X(32594) lies on these lines: {3, 8287}, {5, 6}, {368, 18453}, {1736, 1893}, {2478, 25000}, {5221, 5729}, {5706, 14873}, {5740, 6835}


X(32595) = X(3)X(25424)∩X(6)X(1569)

Barycentrics    a^2 (a^12 b^4-4 a^10 b^6+6 a^8 b^8-4 a^6 b^10+a^4 b^12-2 a^12 b^2 c^2+4 a^10 b^4 c^2-11 a^8 b^6 c^2+13 a^6 b^8 c^2-2 a^4 b^10 c^2-2 a^2 b^12 c^2+a^12 c^4+4 a^10 b^2 c^4+a^8 b^4 c^4-9 a^6 b^6 c^4+25 a^4 b^8 c^4+4 a^2 b^10 c^4+b^12 c^4-4 a^10 c^6-11 a^8 b^2 c^6-9 a^6 b^4 c^6-21 a^4 b^6 c^6-11 a^2 b^8 c^6+2 b^10 c^6+6 a^8 c^8+13 a^6 b^2 c^8+25 a^4 b^4 c^8-11 a^2 b^6 c^8-6 b^8 c^8-4 a^6 c^10-2 a^4 b^2 c^10+4 a^2 b^4 c^10+2 b^6 c^10+a^4 c^12-2 a^2 b^2 c^12+b^4 c^12) : :
X(32595) = 3 X[3] - 2 X[25424]

X(32595) lies on the Stammler circle, the cubic K1105, and on these lines: {3,25424}, {6,1569}


X(32596) = X(4)X(3104)∩X(16)X(7709)

Barycentrics    Sqrt[3] (a^8 b^2+3 a^6 b^4-5 a^4 b^6+a^2 b^8+a^8 c^2+5 a^6 b^2 c^2-6 a^4 b^4 c^2-3 a^2 b^6 c^2-b^8 c^2+3 a^6 c^4-6 a^4 b^2 c^4+4 a^2 b^4 c^4+b^6 c^4-5 a^4 c^6-3 a^2 b^2 c^6+b^4 c^6+a^2 c^8-b^2 c^8)+2 (3 a^6 b^2-2 a^4 b^4-a^2 b^6+3 a^6 c^2+a^4 b^2 c^2-5 a^2 b^4 c^2+b^6 c^2-2 a^4 c^4-5 a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6) S : :
X(32596) = 3 X[7709] - 2 X[32465]

X(32596) lies on the cubic K1105 and on these lines: {4, 3104}, {16, 7709}, {98, 3098}, {511, 3180}, {616, 2782}, {3094, 16941}, {5980, 18906}, {9821, 22532}, {11257, 22531}, {12251, 14540}, {16940, 22707}, {31683, 31701}


X(32597) = X(4)X(3105)∩X(15)X(7709)

Barycentrics    Sqrt[3] (a^8 b^2+3 a^6 b^4-5 a^4 b^6+a^2 b^8+a^8 c^2+5 a^6 b^2 c^2-6 a^4 b^4 c^2-3 a^2 b^6 c^2-b^8 c^2+3 a^6 c^4-6 a^4 b^2 c^4+4 a^2 b^4 c^4+b^6 c^4-5 a^4 c^6-3 a^2 b^2 c^6+b^4 c^6+a^2 c^8-b^2 c^8)-2 (3 a^6 b^2-2 a^4 b^4-a^2 b^6+3 a^6 c^2+a^4 b^2 c^2-5 a^2 b^4 c^2+b^6 c^2-2 a^4 c^4-5 a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6) S : :
X(32597) = 3 X[7709] - 2 X[32466]

X(32597) lies on the cubic K1105 and these lines: {4, 3105}, {15, 7709}, {98, 3098}, {511, 3181}, {617, 2782}, {3094, 16940}, {5981, 18906}, {9821, 22531}, {11257, 22532}, {12251, 14541}, {16941, 22708}, {31684, 31702}


X(32598) = X(54)X(511)∩X(2916)X(2931)

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

X(32598) lies on the cubic K1107 and these lines: {54, 511}, {2916, 2931}, {7488, 29012}, {9019, 19129}, {13564, 32351}\


X(32599) = X(3)X(524)∩X(23)X(542)

Barycentrics    a^2*(a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 7*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 11*a^2*b^6*c^2 - 4*b^8*c^2 - 2*a^6*c^4 - 13*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 5*b^6*c^4 + 2*a^4*c^6 + 11*a^2*b^2*c^6 + 5*b^4*c^6 + a^2*c^8 - 4*b^2*c^8 - c^10) : :
X(32599) = 3 X[186] - 2 X[12584],4 X[575] - 3 X[22151],3 X[2070] - X[32254],2 X[3292] - 3 X[15462],2 X[5609] - 3 X[18374],3 X[5622] - X[23061],4 X[12105] - 3 X[19596]

X(32599) lies on the cubic K1107 and these lines: {3, 524}, {23, 542}, {30, 16010}, {54, 20190}, {68, 7530}, {74, 511}, {110, 32315}, {182, 323}, {186, 12584}, {399, 32217}, {575, 1199}, {576, 5889}, {1154, 10510}, {1204, 15073}, {1352, 1995}, {1503, 9919}, {2070, 32254}, {2854, 3581}, {2930, 2931}, {3098, 15531}, {3292, 15462}, {5609, 18374}, {5622, 23061}, {5648, 15361}, {5965, 14049}, {6776, 7492}, {6795, 32515}, {7574, 25328}, {8537, 14865}, {8548, 11477}, {9027, 32110}, {9970, 13754}, {9972, 32352}, {10296, 32273}, {10990, 19924}, {11178, 16042}, {11180, 14002}, {12105, 19596}, {12106, 15069}, {15534, 18570}, {20423, 31861}

X(32599) = reflection of X(i) in X(j) for these lines: {i,j}: {323, 182}, {399, 32217}, {1352, 3580}, {2930, 7575}, {5648, 15361}, {7464, 32305}, {7574, 25328}, {10296, 32273}


X(32600) = X(3)X(2916)∩X(74)X(827)

Barycentrics    a^2*(a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 - a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 2*b^8*c^2 - 2*a^6*c^4 + 2*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 3*b^6*c^4 + 2*a^4*c^6 + 2*a^2*b^2*c^6 + 3*b^4*c^6 + a^2*c^8 - 2*b^2*c^8 - c^10) : :
X(32600) = 3 X[3] - X[2916]

X(32600) lies on the cubic K1107 and on these lines: {3, 2916}, {74, 827}, {141, 12901}, {182, 7689}, {206, 3357}, {511, 7691}, {549, 13289}, {1352, 16013}, {1495, 5888}, {1503, 32401}, {1531, 7550}, {1843, 3520}, {3098, 9019}, {5085, 11999}, {7512, 29323}, {8718, 15030}, {10226, 21167}, {14130, 29317}, {20190, 21637}

X(32600) = reflection of X(1176) in X(5092)


X(32601) = X(4)X(64)∩X(20)X(12164)

Barycentrics    7*a^10 - 3*a^8*b^2 - 30*a^6*b^4 + 38*a^4*b^6 - 9*a^2*b^8 - 3*b^10 - 3*a^8*c^2 + 52*a^6*b^2*c^2 - 38*a^4*b^4*c^2 - 20*a^2*b^6*c^2 + 9*b^8*c^2 - 30*a^6*c^4 - 38*a^4*b^2*c^4 + 58*a^2*b^4*c^4 - 6*b^6*c^4 + 38*a^4*c^6 - 20*a^2*b^2*c^6 - 6*b^4*c^6 - 9*a^2*c^8 + 9*b^2*c^8 - 3*c^10 : :
X(32601) - 3 X[4] - 4 X[9786],7 X[4] - 8 X[15873],7 X[9786] - 6 X[15873]

X(32601) lies on the cubic K1108 and on these lines: {4, 64}, {20, 12164}, {74, 3090}, {376, 1092}, {511, 3529}, {546, 15431}, {578, 20427}, {631, 11468}, {1204, 6622}, {1620, 2883}, {2777, 18909}, {3357, 8889}, {5878, 6353}, {5925, 6776}, {6225, 22750}, {6240, 20079}, {9825, 11469}, {11456, 17538}, {11457, 15682}, {12244, 15463}, {13347, 15740}, {18931, 22802}, {25712, 32139}

X(32601) = {X(5895),X(18913)}-harmonic conjugate of X(4)


X(32602) = X(20)X(154)∩X(25)X(64)

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^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 - 4*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 28*a^2*b^6*c^2 - 19*b^8*c^2 + 2*a^6*c^4 - 2*a^4*b^2*c^4 - 50*a^2*b^4*c^4 + 18*b^6*c^4 + 2*a^4*c^6 + 28*a^2*b^2*c^6 + 18*b^4*c^6 - 3*a^2*c^8 - 19*b^2*c^8 + c^10) : :

X(32602) lies on the cubic K1108 and these lines: {20, 154}, {25, 64}, {511, 1498}, {1181, 16105}, {1853, 16656}, {5085, 5893}, {8567, 10117}, {11744, 14528}, {13598, 17813}, {17812, 17818}, {17822, 18535}

X(32602) = reflection of X(64) in X(1192)
X(32602) = {X(9914),X(15811)}-harmonic conjugate of X(64)


X(32603) = X(3)X(3620)∩X(125)X(6353)

Barycentrics    15*a^12 - 20*a^10*b^2 - 17*a^8*b^4 + 24*a^6*b^6 + 5*a^4*b^8 - 4*a^2*b^10 - 3*b^12 - 20*a^10*c^2 + 78*a^8*b^2*c^2 - 32*a^6*b^4*c^2 + 12*a^4*b^6*c^2 - 44*a^2*b^8*c^2 + 6*b^10*c^2 - 17*a^8*c^4 - 32*a^6*b^2*c^4 - 34*a^4*b^4*c^4 + 48*a^2*b^6*c^4 + 3*b^8*c^4 + 24*a^6*c^6 + 12*a^4*b^2*c^6 + 48*a^2*b^4*c^6 - 12*b^6*c^6 + 5*a^4*c^8 - 44*a^2*b^2*c^8 + 3*b^4*c^8 - 4*a^2*c^10 + 6*b^2*c^10 - 3*c^12 : :

X(32603) lies on the cubic K1108 and on these lines: {3, 3620}, {125, 6353}, {185, 14912}, {376, 12283}, {5907, 25406}, {6776, 10606}


X(32604) = X(185)X(14914)∩X(511)X(6391)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(3*a^12 - 10*a^10*b^2 + 9*a^8*b^4 + 4*a^6*b^6 - 11*a^4*b^8 + 6*a^2*b^10 - b^12 - 10*a^10*c^2 + 74*a^8*b^2*c^2 - 68*a^6*b^4*c^2 - 12*a^4*b^6*c^2 - 18*a^2*b^8*c^2 + 34*b^10*c^2 + 9*a^8*c^4 - 68*a^6*b^2*c^4 + 110*a^4*b^4*c^4 + 12*a^2*b^6*c^4 - 127*b^8*c^4 + 4*a^6*c^6 - 12*a^4*b^2*c^6 + 12*a^2*b^4*c^6 + 188*b^6*c^6 - 11*a^4*c^8 - 18*a^2*b^2*c^8 - 127*b^4*c^8 + 6*a^2*c^10 + 34*b^2*c^10 - c^12) : :

X(32604) lies on the cubic K1108) and on these lines: {185, 14914}, {511, 6391}, {11820, 19458}


X(32605) = X(2)X(185)∩X(4)X(3167)

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 - 5*a^4*b^2 + 7*a^2*b^4 - 3*b^6 - 5*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b^4*c^2 + 7*a^2*c^4 + 3*b^2*c^4 - 3*c^6) : :
X(32605) = 5 X[3091] - 2 X[15077],7 X[3523] - 4 X[3532]

X(32605) lies on the cubics K182, K329, K1108, the Thomson-Gibert-Moses hyperbola, the Feuerbach circumhyperbola of the orthic triangle, and on these lines: {2, 185}, {4, 3167}, {5, 5644}, {6, 3091}, {20, 154}, {25, 15741}, {30, 25712}, {52, 3089}, {110, 3146}, {113, 6193}, {155, 6623}, {193, 235}, {354, 1858}, {648, 6526}, {1498, 7396}, {1986, 3542}, {3090, 5544}, {3522, 6030}, {3523, 3532}, {3541, 22948}, {3546, 13491}, {3574, 3832}, {3839, 14516}, {5056, 18928}, {5068, 11442}, {5643, 15022}, {5646, 10303}, {5888, 11440}, {5889, 16879}, {6053, 14216}, {6225, 11064}, {6247, 30769}, {6622, 12164}, {6816, 19125}, {7398, 12233}, {7400, 7999}, {7487, 22660}, {7712, 11449}, {10151, 15751}, {10565, 16252}, {12174, 16051}, {12250, 15063}\

X(32605) = reflection of X(i) in X(j) for these lines: {i,j}: {20, 27082}, {5889, 16879}
X(32605) = {X(3091),X(11441)}-harmonic conjugate of X(5921)
X(32605) = Thomson-isogonal conjugate of X(9909)
X(32605) = orthic-isogonal conjugate of X(20)
X(32605) = X(4)-Ceva conjugate of X(20)
X(32605) = crosspoint of X(4) and X(6622)
X(32605) = barycentric product X(12164)*X(15466)
X(32605) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {6622, 459}, {12164, 1073}


X(32606) = (name pending)

Barycentrics    a*(2*a^9 - 5*a^8*b - a^7*b^2 + 12*a^6*b^3 - 9*a^5*b^4 - 6*a^4*b^5 + 13*a^3*b^6 - 4*a^2*b^7 - 5*a*b^8 + 3*b^9 - 5*a^8*c + 20*a^7*b*c - 19*a^6*b^2*c - 19*a^5*b^3*c + 46*a^4*b^4*c - 22*a^3*b^5*c - 15*a^2*b^6*c + 21*a*b^7*c - 7*b^8*c - a^7*c^2 - 19*a^6*b*c^2 + 56*a^5*b^2*c^2 - 36*a^4*b^3*c^2 - 34*a^3*b^4*c^2 + 57*a^2*b^5*c^2 - 21*a*b^6*c^2 - 2*b^7*c^2 + 12*a^6*c^3 - 19*a^5*b*c^3 - 36*a^4*b^2*c^3 + 84*a^3*b^3*c^3 - 38*a^2*b^4*c^3 - 21*a*b^5*c^3 + 18*b^6*c^3 - 9*a^5*c^4 + 46*a^4*b*c^4 - 34*a^3*b^2*c^4 - 38*a^2*b^3*c^4 + 52*a*b^4*c^4 - 12*b^5*c^4 - 6*a^4*c^5 - 22*a^3*b*c^5 + 57*a^2*b^2*c^5 - 21*a*b^3*c^5 - 12*b^4*c^5 + 13*a^3*c^6 - 15*a^2*b*c^6 - 21*a*b^2*c^6 + 18*b^3*c^6 - 4*a^2*c^7 + 21*a*b*c^7 - 2*b^2*c^7 - 5*a*c^8 - 7*b*c^8 + 3*c^9) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29052.

X(32606) lies on this line: {1, 23279}


X(32607) = X(3)X(125)∩X(4)X(13289)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^12 - 2*a^10*b^2 - a^8*b^4 + 4*a^6*b^6 - a^4*b^8 - 2*a^2*b^10 + b^12 - 2*a^10*c^2 + 7*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - b^10*c^2 - a^8*c^4 - 5*a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 3*a^2*b^6*c^4 - 5*b^8*c^4 + 4*a^6*c^6 - 4*a^4*b^2*c^6 - 3*a^2*b^4*c^6 + 10*b^6*c^6 - a^4*c^8 + 5*a^2*b^2*c^8 - 5*b^4*c^8 - 2*a^2*c^10 - b^2*c^10 + c^12) : :
X(32607) = X[74] + 2 X[11430],2 X[12041] + X[13352]

X(32607) lies on the cubic K1107 and these lines: {3, 125}, {4, 13289}, {24, 7687}, {26, 12295}, {49, 22584}, {54, 74}, {64, 17856}, {110, 5907}, {113, 7526}, {159, 32250}, {184, 5663}, {186, 13851}, {378, 2777}, {399, 19357}, {511, 2071}, {550, 11804}, {569, 14708}, {578, 1986}, {631, 18933}, {974, 1204}, {1092, 12358}, {1112, 11424}, {1147, 7723}, {1177, 12294}, {1181, 10620}, {1350, 32251}, {1425, 10081}, {1511, 15738}, {1593, 10117}, {1594, 19506}, {1658, 10113}, {2935, 3516}, {3043, 12281}, {3047, 12111}, {3270, 10065}, {3357, 17854}, {3448, 19467}, {5504, 5562}, {5621, 11410}, {5650, 15035}, {5890, 16219}, {5972, 7503}, {6146, 10264}, {6240, 32365}, {6467, 11579}, {6644, 23515}, {6723, 17928}, {6759, 12292}, {6776, 32305}, {7488, 10733}, {7728, 14130}, {9934, 11381}, {9970, 21637}, {9976, 15073}, {10114, 20417}, {10605, 15041}, {10606, 17853}, {10619, 18364}, {10681, 11476}, {10682, 11475}, {10721, 14865}, {11473, 13288}, {11474, 13287}, {11562, 12228}, {11801, 15331}, {12084, 16111}, {12133, 15647}, {12163, 19456}, {12168, 24981}, {12317, 18925}, {12412, 15063}, {12605, 23306}, {13358, 16270}, {13417, 15472}, {14049, 32333}, {14448, 17835}, {14585, 14901}, {15059, 22467}, {15081, 21844}, {15089, 22815}, {16010, 19459}, {18379, 20304}, {18563, 19479}

X(32607) = midpoint of X(74) and X(15463)
X(32607) = reflection of X(i) in X(j) for these lines: {i,j}: {15463, 11430}, {22109, 3}}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{3, 265, 12893}, {3, 12302, 16163}, {3, 19457, 125}, {54, 74, 7722}, {54, 7722, 12227}, {74, 3520, 13293}, {74, 13198, 185}, {125, 21659, 265}, {974, 12041, 1204}, {1593, 10117, 13202}, {2935, 13171, 10990}, {3516, 13171, 2935}, {11425, 17835, 19504}, {12133, 15647, 26883}, {13367, 21650, 110}, {17835, 19504, 14448}


X(32608) = X(3)X(54)∩X(30)X(3448)

Barycentrics    a^2*(a^8 - 6*a^4*b^4 + 8*a^2*b^6 - 3*b^8 + a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 - 3*a^2*b^2*c^4 + 2*b^4*c^4 + 8*a^2*c^6 + 2*b^2*c^6 - 3*c^8) : :
X(32608) = 5 X[3] - 2 X[23061],5 X[399] - 8 X[1495],X[399] - 4 X[3581],3 X[399] - 4 X[10540],4 X[1495] - 5 X[2070],2 X[1495] - 5 X[3581],6 X[1495] - 5 X[10540],4 X[1568] - 5 X[1656],3 X[2070] - 2 X[10540],3 X[3581] - X[10540],9 X[5054] - 8 X[14156],5 X[15040] - 4 X[22115],5 X[15040] - 8 X[32110],3 X[15041] - 2 X[18859]

X(32608) lies on the cubic K1107 and these lines: {3, 54}, {30, 3448}, {52, 14130}, {74, 13391}, {185, 13564}, {323, 15646}, {381, 17810}, {382, 9927}, {389, 15047}, {399, 1495}, {511, 5621}, {539, 15085}, {567, 14831}, {568, 15004}, {1350, 15688}, {1568, 1656}, {1657, 11750}, {1658, 9544}, {3526, 9786}, {3534, 10605}, {3580, 18403}, {3830, 18392}, {5054, 14156}, {5055, 7699}, {5663, 5899}, {5876, 13621}, {5907, 18369}, {6241, 15086}, {6243, 7689}, {7545, 18435}, {7666, 32534}, {9306, 18436}, {9703, 18324}, {9818, 13321}, {10263, 11440}, {11438, 23039}, {12111, 18378}, {12308, 14157}, {12359, 31724}, {13403, 18442}, {13414, 28447}, {13415, 28448}, {14118, 14627}, {14128, 22462}, {14449, 14865}, {15040, 22115}, {15053, 15067}, {15060, 21308}, {15707, 21766}, {17835, 18400}

X(32608) = reflection of X(i) in X(j) for these lines: {i,j}: {323, 15646}, {399, 2070}, {2070, 3581}, {12308, 14157}, {18403, 3580}, {22115, 32110}
X(32608) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5889, 195}, {7691, 13630, 3}


X(32609) = X(2)X(11694)∩X(3)X(74)

Barycentrics    - a^2*(3*a^8 - 8*a^6*b^2 + 6*a^4*b^4 - b^8 - 8*a^6*c^2 + 11*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 - 5*a^2*b^2*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8) : :
X(32609) = X[2] - 4 X[11694],5 X[3] - 2 X[74],X[3] + 2 X[110],2 X[3] + X[399],X[3] - 4 X[1511],5 X[3] + 4 X[5609],4 X[3] - X[10620],7 X[3] - 4 X[12041],5 X[3] + X[12308],7 X[3] + 2 X[14094],5 X[3] - 14 X[15020],19 X[3] - 10 X[15021],29 X[3] - 38 X[15023],X[3] - 10 X[15034],11 X[3] - 14 X[15036],2 X[3] + 7 X[15039],2 X[3] - 5 X[15040],10 X[3] - 13 X[15042],7 X[3] - 10 X[15051],11 X[3] - 2 X[15054],3 X[3] - 2 X[15055],X[4] - 4 X[10272],2 X[5] + X[12383],4 X[5] - X[12902],X[5] - 4 X[13392],X[6] + 2 X[12584],X[20] + 5 X[20125],X[40] + 2 X[11699],2 X[54] + X[5898],X[64] - 4 X[25564],X[74] + 5 X[110],4 X[74] + 5 X[399],X[74] - 10 X[1511],X[74] + 2 X[5609],8 X[74] - 5 X[10620],7 X[74] - 10 X[12041],2 X[74] + X[12308],7 X[74] + 5 X[14094],X[74] - 7 X[15020],19 X[74] - 25 X[15021],29 X[74] - 95 X[15023],X[74] - 25 X[15034],X[74] - 5 X[15035],11 X[74] - 35 X[15036],4 X[74] + 35 X[15039],4 X[74] - 25 X[15040],4 X[74] - 5 X[15041],4 X[74] - 13 X[15042],7 X[74] - 25 X[15051],11 X[74] - 5 X[15054],3 X[74] - 5 X[15055],4 X[110] - X[399],X[110] + 2 X[1511],5 X[110] - 2 X[5609],8 X[110] + X[10620],7 X[110] + 2 X[12041],10 X[110] - X[12308],7 X[110] - X[14094],5 X[110] + 7 X[15020],19 X[110] + 5 X[15021],29 X[110] + 19 X[15023],X[110] + 5 X[15034],11 X[110] + 7 X[15036],4 X[110] - 7 X[15039],4 X[110] + 5 X[15040],4 X[110] + X[15041],20 X[110] + 13 X[15042],7 X[110] + 5 X[15051],11 X[110] + X[15054],3 X[110] + X[15055]

X(32609) lies on the cubic K1107 and these lines: {2, 11694}, {3, 74}, {4, 7666}, {5, 12383}, {6, 11935}, {20, 20125}, {24, 3043}, {25, 15463}, {40, 11699}, {49, 9730}, {54, 5898}, {64, 25564}, {113, 382}, {125, 3526}, {140, 3448}, {141, 32306}, {146, 550}, {154, 2777}, {155, 12893}, {182, 2930}, {184, 17701}, {195, 568}, {265, 1656}, {323, 7575}, {373, 567}, {381, 5642}, {394, 22109}, {511, 2070}, {541, 15688}, {542, 5054}, {548, 12244}, {549, 9143}, {569, 15089}, {620, 15545}, {631, 10264}, {974, 19347}, {999, 10088}, {1092, 2937}, {1112, 3517}, {1151, 12376}, {1152, 12375}, {1350, 19140}, {1351, 6593}, {1385, 2948}, {1482, 11720}, {1498, 13293}, {1539, 5073}, {1598, 15472}, {1657, 7728}, {1658, 12307}, {1986, 3515}, {2079, 9696}, {2574, 28447}, {2575, 28448}, {2771, 3576}, {2778, 28450}, {2842, 22586}, {2854, 5050}, {2888, 10125}, {2935, 6759}, {3090, 11801}, {3200, 30439}, {3201, 30440}, {3233, 14934}, {3292, 3581}, {3295, 10091}, {3311, 10819}, {3312, 10820}, {3431, 14926}, {3516, 12292}, {3522, 14677}, {3523, 12317}, {3527, 5504}, {3628, 15081}, {3796, 15693}, {3843, 10733}, {3851, 10113}, {5013, 14901}, {5055, 14644}, {5070, 20304}, {5072, 7687}, {5079, 12900}, {5092, 16010}, {5093, 13321}, {5181, 11898}, {5204, 19470}, {5217, 7727}, {5465, 12355}, {5621, 17508}, {5640, 15038}, {5650, 18475}, {5651, 14805}, {5889, 11561}, {5907, 18364}, {6053, 16111}, {6101, 7731}, {6102, 12273}, {6243, 11557}, {6417, 19110}, {6418, 19111}, {6642, 12228}, {6644, 9703}, {6699, 15720}, {6723, 15027}, {7387, 20773}, {7517, 20771}, {7545, 13352}, {7574, 11064}, {7722, 32534}, {8909, 12892}, {8998, 19052}, {9129, 11258}, {9140, 15694}, {9545, 13472}, {9654, 18968}, {9669, 12896}, {9704, 17928}, {9705, 12284}, {9706, 12006}, {9786, 12227}, {9833, 23315}, {9904, 31663}, {9919, 15647}, {9934, 14530}, {9956, 12407}, {10117, 10282}, {10170, 13367}, {10540, 14915}, {10627, 13201}, {10628, 11202}, {10657, 11481}, {10658, 11480}, {10706, 15681}, {10721, 17800}, {11002, 12106}, {11426, 11746}, {11477, 25556}, {11562, 18436}, {11579, 12017}, {11598, 13093}, {11702, 12316}, {11999, 15748}, {12038, 12302}, {12083, 16165}, {12085, 25487}, {12165, 15750}, {12645, 12898}, {12903, 31479}, {13188, 18332}, {13289, 17821}, {13358, 15043}, {13434, 22462}, {13743, 16164}, {13990, 19051}, {14695, 32204}, {15033, 21308}, {15063, 15696}, {15141, 15577}, {17845, 19506}, {18440, 32233}, {18534, 20772}, {19145, 32292}, {19146, 32291}, {27866, 32046}

X(32609) = midpoint of X(i) and X(j) for these lines: {i,j}: {110, 15035}, {399, 15041}, {23515, 30714}
X(32609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 110, 399}, {3, 399, 10620}, {3, 1511, 15040}, {3, 12308, 74}, {5, 12383, 12902}, {24, 3043, 19504}, {74, 110, 5609}, {74, 5609, 12308}, {110, 1511, 3}, {110, 15020, 74}, {110, 15034, 1511}, {110, 15040, 10620}, {110, 15051, 14094}, {113, 12121, 382}, {265, 5972, 1656}, {381, 14643, 15046}, {399, 15039, 110}, {399, 15040, 3}, {399, 15042, 74}, {631, 14683, 10264}, {1511, 15039, 10620}, {5609, 12308, 399}, {5609, 15020, 3}, {5609, 15042, 10620}, {5972, 30714, 265}, {6644, 9703, 15087}, {7728, 16163, 1657}, {11720, 12778, 1482}, {12017, 32254, 11579}, {12038, 18350, 14130}, {12041, 15051, 3}, {12308, 15020, 15042}, {14094, 15051, 12041}, {15039, 15040, 399}, {16163, 16534, 7728}, {17821, 17847, 13289}

X(32609) = circumcircle- inverse of X(5609)
X(32609) = crosssum of X(523) and X(6070)
X(32609) = pole of line X(526)X(5607) wrt circumcircle


X(32610) = X(35)X(79)∩X(5010)X(20277)

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

See Ercole Suppa, Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29057.

X(32610) lies on these lines: {35, 79}, {5010, 20277}

X(32610) = isogonal conjugate of X(32611)
X(32610) = {X(35), X(8606)}-harmonic conjugate of X(226)


X(32611) = ISOGONAL CONJUGATE OF X(32610)

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

See Ercole Suppa, Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29057.

X(32611) lies on this line: {500, 9642}

X(32611) = isogonal conjugate of X(32610)


X(32612) = MIDPOINT OF X(3) AND X(56)

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-4*b*c*(b^2-b*c+c^2))*a-(b^3+c^3)*(b-c)^2) : :
X(32612) = 3*X(3)-X(10310), 3*X(3)+X(10680), X(46)+3*X(3576), 3*X(56)+X(10310), 3*X(56)-X(10680), 5*X(631)-X(3436), 7*X(3523)+X(20076), 7*X(3526)-5*X(31246), 3*X(5054)-X(31141), 3*X(10165)-X(21616), X(11499)-3*X(16371), 3*X(21154)-X(32554)

See Ercole Suppa, Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29059 and Hyacinthos 29062.

X(32612) lies these lines: {1, 3}, {4, 10584}, {5, 2829}, {10, 32153}, {12, 21154}, {24, 1828}, {30, 7681}, {84, 31828}, {104, 355}, {119, 13747}, {140, 993}, {182, 8679}, {214, 5884}, {388, 6961}, {474, 9956}, {496, 5840}, {499, 6923}, {515, 6924}, {528, 32214}, {529, 549}, {631, 3436}, {944, 4188}, {952, 8256}, {958, 11231}, {971, 15297}, {997, 5694}, {1006, 5303}, {1012, 9955}, {1125, 6914}, {1158, 22775}, {1478, 6958}, {1483, 5854}, {1656, 18515}, {1766, 21773}, {1837, 10090}, {2390, 11202}, {2975, 6940}, {3035, 10942}, {3086, 6948}, {3149, 28160}, {3523, 20076}, {3526, 5251}, {3560, 3824}, {3585, 6971}, {3616, 6950}, {3624, 7489}, {3653, 17549}, {3655, 11491}, {4190, 10785}, {4293, 6891}, {4299, 6928}, {4511, 26877}, {4640, 31838}, {4881, 21740}, {4973, 31806}, {5054, 31141}, {5229, 6978}, {5253, 5886}, {5267, 10165}, {5298, 15908}, {5322, 16434}, {5428, 17768}, {5433, 6842}, {5587, 26321}, {5690, 8666}, {5731, 6942}, {5818, 17572}, {5881, 12773}, {5882, 32141}, {5887, 17614}, {6256, 6959}, {6265, 18861}, {6850, 7288}, {6875, 11415}, {6882, 7354}, {6885, 18517}, {6905, 18481}, {6909, 12699}, {6911, 12114}, {6918, 18761}, {6921, 12115}, {6929, 10200}, {6944, 18516}, {6955, 10527}, {6966, 10532}, {6970, 12667}, {6980, 15446}, {7491, 15326}, {7741, 12764}, {8227, 13743}, {8583, 22936}, {8703, 12511}, {9657, 11929}, {10058, 11376}, {10943, 20418}, {11112, 26470}, {11263, 17009}, {11499, 16371}, {12737, 14923}, {13731, 27657}, {14988, 30144}, {15325, 15866}, {18446, 26201}, {19525, 19861}, {19548, 28096}, {22753, 22793}, {22836, 24475}, {28459, 30264}

X(32612) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 56}, {4299, 6928}, {10310, 10680}
X(32612) = reflection of X(i) in X(j) for these (i,j): (5, 6691), (1329, 140)
X(32612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1,3,26285}, {3,36,26286}, {3,55,26086}, {3, 1385, 32613}, {3,999,11248}, {3,1482,2077}, {3,3428,31663}, {3,10246,35}, {3,10269,1385}, {3,10680,10310}, {3,11249,3579}, {3,16202,5217}, {3,16203,55}, {3,22765,40}, {36,14800,1}, {55,16203,15178}, {56,8069,24928}, {56,10310,10680}, {104,404,355}, {474,22758,9956}, {997,24467,5694}, {999,1466,31794}, {999,11248,10222}, {1319,25414,1}, {1385,23961,3}, {1470,22766,942}, {2077,5563,1482}, {2975,6940,26446}, {3086,6948,10525}, {3560,25524,11230}, {3576,7280,3}, {4293,6891,10526}, {5253,6906,5886}, {5885,13624,26287}, {5885,26287,1}, {6911,12114,18480}, {8071,22768,24929}, {9940,13624,1385}, {13528,20323,23340}, {15178,26086,55}, {31663,31797,3579}


X(32613) = MIDPOINT OF X(3) AND X(55)

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+(b+c)*(2*b^2-b*c+2*c^2)*a^2+(b^4+c^4)*a-(b^2-c^2)*(b^3-c^3)) : :
X(32613) = 3*X(3)-X(3428), 3*X(3)+X(10679), 3*X(55)+X(3428), 3*X(55)-X(10679), 5*X(631)-X(3434), 5*X(1656)-X(18499), X(3419)-3*X(26446), 7*X(3523)+X(20075), 7*X(3526)-5*X(31245), X(3870)+3*X(21165), 3*X(5054)-X(31140), 4*X(6690)-X(18407), 3*X(16370)-X(22758)

See Ercole Suppa, Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 29059 and Hyacinthos 29062.

X(32613) lies these lines: {1, 3}, {4, 10585}, {5, 5248}, {8, 6875}, {10, 32141}, {11, 21155}, {12, 7491}, {21, 355}, {24, 1824}, {30, 7680}, {31, 5396}, {42, 5398}, {47, 2594}, {100, 1006}, {104, 3655}, {119, 11113}, {140, 2886}, {153, 15677}, {182, 674}, {255, 5399}, {405, 9956}, {411, 12699}, {497, 6954}, {498, 6928}, {500, 601}, {515, 6914}, {528, 549}, {529, 32213}, {631, 3434}, {902, 1064}, {912, 4640}, {943, 5812}, {944, 4189}, {952, 993}, {962, 6876}, {971, 15296}, {991, 19624}, {997, 22935}, {1001, 6911}, {1012, 28160}, {1030, 2079}, {1125, 6924}, {1283, 14663}, {1324, 3185}, {1376, 6883}, {1479, 6863}, {1483, 5855}, {1490, 31828}, {1621, 5886}, {1656, 5259}, {1871, 14017}, {2771, 18446}, {2875, 18475}, {3085, 6868}, {3145, 9959}, {3149, 9955}, {3523, 20075}, {3526, 31245}, {3560, 11500}, {3583, 6980}, {3584, 15175}, {3616, 6942}, {3651, 16159}, {3652, 12528}, {3653, 13587}, {3654, 21161}, {3679, 12331}, {3811, 22937}, {3870, 21165}, {3877, 6265}, {4192, 29678}, {4220, 29665}, {4294, 6825}, {4302, 6923}, {4420, 26878}, {4421, 28466}, {4428, 22753}, {4512, 5720}, {4995, 28459}, {4996, 12737}, {4999, 10943}, {5054, 31140}, {5070, 25542}, {5218, 6827}, {5251, 5790}, {5258, 12645}, {5267, 5882}, {5281, 6987}, {5284, 6946}, {5310, 19544}, {5428, 5690}, {5432, 6882}, {5534, 31424}, {5552, 6936}, {5587, 7489}, {5691, 13743}, {5694, 12514}, {5731, 6950}, {5762, 8255}, {5818, 16865}, {5840, 6907}, {5844, 25439}, {5887, 20846}, {6097, 13754}, {6253, 6841}, {6284, 6842}, {6824, 18517}, {6872, 10786}, {6902, 27529}, {6906, 18481}, {6910, 12116}, {6913, 18491}, {6917, 10198}, {6930, 18516}, {6962, 10531}, {6970, 26105}, {6985, 11496}, {7301, 7609}, {7330, 22936}, {7483, 26470}, {7488, 20243}, {7580, 28146}, {8053, 14723}, {9670, 11928}, {10282, 10537}, {11604, 13199}, {13323, 22276}, {15623, 17524}, {15733, 31658}, {15865, 31789}, {15888, 30264}, {16139, 31660}, {16370, 22758}, {19524, 19861}, {19649, 29680}, {20999, 23206}, {24309, 29243}, {24466, 28458}, {28208, 28444}

X(32613) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 55}, {3428, 10679}, {4302, 6923}
X(32613) = reflection of X(i) in X(j) for these (i,j): (5, 6690), (993, 7508), (2886, 140), (5173, 13373), (10537, 10282), (18407, 5)
X(32613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1,3,26286}, {1,165,5535}, {1,1454,942} ,{3,35,26285}, {3, 1385, 32612}, {3,1482,11012}, {3,3295,11249}, {3,5217,26086}, {3,10246,36}, {3,10267,1385}, {3,10269,23961}, {3,10310,31663}, {3,10679,3428}, {3,11248,3579}, {3,11849,40}, {3,16202,56}, {3,16203,5204}, {21,11491,355}, {35,10902,3}, {35,14795,1}, {35,14799,5010}, {35,15931,2077}, {55,3428,10679}, {55,5172,1}, {55,8069,24929}, {56,16202,15178}, {100,1006,26446}, {405,11499,9956}, {1001,6911,11230}, {1376,6883,11231}, {1385,17502,18857}, {1385,23961,10269}, {1621,6905,5886}, {2077,10902,15931}, {2077,15931,3}, {3085,6868,10526}, {3295,11249,10222}, {3560,11500,18480}, {3576,5010,3}, {3746,11012,1482}, {3746,14804,1}, {4294,6825,10525}, {5248,6796,5}, {5267,5882,32153}, {6985,11496,22793}, {7489,18524,5587}, {8071,11510,24928}, {11508,26357,9957}, {13624,26086,3}, {26398,26422,1}

X(32613) = center of circle that is inverse-in-circumcircle of antiorthic axis


X(32614) = X(4)X(13414)∩X(20)X(13415)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2) - (5*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*J*R^2 : :
X(32614) = J X[110] - X[3146]

X(32614) lies on the cubic K1108 and these lines: {4, 13414}, {20, 13415}, {64, 2574}, {110, 3146}, {185, 24651}, {511, 15156}, {1114, 21663}, {1495, 15157}, {2575, 12164}, {5907, 25407}, {13598, 24650}, {15155, 32608}

X(32614) = reflection of X(32615) in X(13346)
X(32614) = {X(110),X(3146)}-harmonic conjugate of X(32615)


X(32615) = X(4)X(13415)∩X(20)X(13414)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2) + (5*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*J*R^2 : :
X(32615) = J X[110] + X[3146]

X(32615) lies on the cubic K1108 and these lines: {4, 13415}, {20, 13414}, {64, 2575}, {110, 3146}, {185, 24650}, {511, 15157}, {1113, 21663}, {1495, 15156}, {2574, 12164}, {5907, 25408}, {13598, 24651}, {15154, 32608}

X(32615) = reflection of X(32614) in X(13346)
X(32615) = {X(110),X(3146)}-harmonic conjugate of X(32614)


X(32616) = X(3)X(74)∩X(4)X(2575)

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

X(32616) lies on the cubic K1108 and these lines: {3, 74}, {4, 2575}, {20, 2574}, {64, 15461}, {185, 13415}, {511, 15156}, {1113, 6000}, {1114, 13754}, {1154, 15155}, {1344, 15305}, {1345, 5890}, {1347, 18388}, {5562, 25407}, {5889, 24651}, {5907, 13414}, {10751, 25739}, {12162, 14709}, {14915, 15157}, {18439, 20478}

X(32616) = reflection of X(i) in X(j) for these lines: {i,j}: {24650, 31955}, {31954, 13415}
X(32616) = reflection of X(32617) in X(3)
X(32616) = {X(74),X(6241)}-harmonic conjugate of X(32617)
X(32616) = {X(110),X(12111)}-harmonic conjugate of X(32617)


X(32617) = X(3)X(74)∩X(4)X(2574)

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

X(32617) lies on the cubic K1108 and these lines: {3, 74}, {4, 2574}, {20, 2575}, {64, 15460}, {185, 13414}, {511, 15157}, {1113, 13754}, {1114, 6000}, {1154, 15154}, {1344, 5890}, {1345, 15305}, {1346, 18388}, {5562, 25408}, {5889, 24650}, {5907, 13415}, {10750, 25739}, {12162, 14710}, {14915, 15156}, {18439, 20479}

X(32617) = reflection of X(i) in X(j) for these lines: {i,j}: {24651, 31954}, {31955, 13414}
X(32617) = reflection of X(32616) in X(3)
X(32617) = {X(74),X(6241)}-harmonic conjugate of X(32616)
X(32617) = {X(110),X(12111)}-harmonic conjugate of X(32616)


X(32618) = ISOGONAL CONJUGATE OF X(5000)

Barycentrics    1/(S*SB*SC + SA*Sqrt[SA*SB*SC*SW]) : :

X(32618) lies on the following curves: Jerabek circumhyperbola, K018, K270, K305, K336, K337, K1096, Q021, Q030, Q044, Q094, and on these lines: {2, 98}, {10264, 31665}

X(32618) = reflection of X(32619) in X(125)
X(32618) = isogonal conjugate of X(5000)
X(32618) = isogonal conjugate of the complement of X(5002)
X(32618) = Brocard-circle inverse of X(32619)
X(32618) = antigonal image of X(32619)
X(32618) = psi-transform of X(5001)
X(32618) = X(511)-cross conjugate of X(32619)
X(32618) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5000}, {240,32619}
X(32618) = barycentric product X(287)*X(5001)
X(32618) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {6, 5000}, {248, 32619}, {5001, 297}
X(32618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6776, 32619}, {98, 287, 32619}, {110, 5622, 32619}, {182, 184, 32619}, {1352, 1899, 32619}, {1976, 17974, 32619}, {5921, 23291, 32619}, {13198, 15462, 32619}, {13414, 13415, 32619}


X(32619) = ISOGONAL CONJUGATE OF X(5001)

Barycentrics    1/(S*SB*SC - SA*Sqrt[SA*SB*SC*SW]) : :

X(32619) lies on the following curves: Jerabek circumhyperbola, K018, K270, K305, K336, K337, K1096, Q021, Q030, Q044, Q094, and on these lines: {2, 98}, {10264, 31664}

X(32619) = reflection of X(32618) in X(125)
X(32619) = Brocard-circle inverse of X(32618)
X(32619) = isogonal conjugate of X(5001)
X(32619) = isogonal conjugate of the complement of X(5003)
X(32619) = antigonal image of X(32618)
X(32619) = psi-transform of X(5000)
X(32619) = X(511)-cross conjugate of X(32618)
X(32619) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5001}, {240, 32618}
X(32619) = barycentric product X(287)*X(5000)
X(32619) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {6, 5001}, {248, 32618}, {5000, 297}
X(32619 = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6776, 32618}, {98, 287, 32618}, {110, 5622, 32618}, {182, 184, 32618}, {1352, 1899, 32618}, {1976, 17974, 32618}, {5921, 23291, 32618}, {13198, 15462, 32618}, {13414, 13415, 32618}


X(32620) = X(3)X(1495)∩X(5)X(4549)

Barycentrics    a^2*(3*a^8 - 4*a^6*b^2 - 6*a^4*b^4 + 12*a^2*b^6 - 5*b^8 - 4*a^6*c^2 + 4*a^4*b^2*c^2 + 8*a^2*b^4*c^2 - 8*b^6*c^2 - 6*a^4*c^4 + 8*a^2*b^2*c^4 + 26*b^4*c^4 + 12*a^2*c^6 - 8*b^2*c^6 - 5*c^8) : :
X(32620) = 5 X[3] + X[3426],X[3] + 2 X[4550],5 X[3] - 2 X[8717],2 X[3] + X[11472],7 X[3] - X[11820],2 X[5] + X[4549],4 X[140] - X[4846],X[1350] + 2 X[31861],5 X[1656] - 2 X[7706],X[3426] - 10 X[4550],X[3426] + 2 X[8717],2 X[3426] - 5 X[11472],7 X[3426] + 5 X[11820],5 X[4550] + X[8717],4 X[4550] - X[11472],14 X[4550] + X[11820],4 X[8717] + 5 X[11472],14 X[8717] - 5 X[11820],7 X[11472] + 2 X[11820]

X(32620) lies on the cubic K1107 and on these lines: {3, 1495}, {5, 4549}, {6, 32599}, {22, 16261}, {30, 10516}, {54, 155}, {64, 7516}, {140, 3532}, {154, 15060}, {378, 7998}, {511, 9813}, {541, 5054}, {549, 10606}, {1154, 5102}, {1192, 3628}, {1350, 31861}, {1656, 7706}, {3066, 3581}, {3336, 7986}, {3796, 18435}, {5050, 13754}, {5085, 5621}, {5646, 12041}, {5891, 6090}, {6000, 17508}, {6800, 18451}, {7393, 16836}, {7395, 9730}, {7464, 21766}, {7509, 15072}, {7525, 15811}, {7526, 15067}, {9786, 13363}, {9827, 17834}, {10938, 19357}, {10982, 32352}, {11248, 15625}, {11284, 32110}, {11387, 11817}, {13154, 32138}, {16072, 23515}, {17811, 18570}, {18377, 20584}, {32607, 32609}

X(32620) = reflection of X(5085) in X(7514)
X(32620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3426, 8717}, {3, 4550, 11472}, {7395, 12163, 15805}


X(32621) = X(3)X(524)∩X(6)X(25)

Barycentrics    a^2*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 10*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6) : :
X(32621) = X[3] - 4 X[8546],X[1351] + 2 X[8547],X[5486] + 2 X[8550],X[15069] - 4 X[16511],X[16010] + 2 X[16510]

Let A'B'C' be the reflection of ABC in X(6). Let AB = BC∩C'A', AC = BC∩A'B', and define BC, BA, CA, CB cyclically. AB, AC, BC, BA, CA, CB lie on the 2nd Lemoine circle. Let A" be the intersection of the tangents to the 2nd Lemoine circle at BA and CA, and define B" and C" cyclically. X(32621) = X(4)-of-A"B"C". (see also X(5050)) (Randy Hutson, June 7, 2019)

X(32621) lies on the cubic K1107 and on these lines: {3, 524}, {6, 25}, {22, 1992}, {23, 5032}, {24, 11431}, {67, 32240}, {69, 7485}, {140, 9925}, {141, 16419}, {160, 8573}, {182, 8681}, {186, 14912}, {193, 6636}, {378, 6776}, {542, 9818}, {575, 6642}, {576, 7387}, {578, 8549}, {597, 5020}, {599, 7484}, {1181, 9914}, {1351, 8547}, {1353, 7502}, {1449, 1486}, {1503, 1597}, {1598, 15581}, {1609, 20775}, {2854, 5050}, {2930, 5642}, {2936, 18800}, {3158, 12329}, {3455, 5477}, {3517, 15582}, {3564, 7514}, {3576, 9004}, {4663, 9798}, {5012, 15531}, {5024, 9142}, {5085, 9027}, {5093, 5899}, {5095, 32262}, {5422, 11188}, {5480, 18535}, {5621, 11410}, {6515, 16789}, {7395, 15069}, {7517, 11482}, {7592, 15073}, {7666, 32154}, {8537, 11423}, {8540, 10833}, {8548, 9937}, {8584, 9909}, {8787, 9876}, {8889, 18935}, {10169, 18919}, {10249, 11430}, {10541, 14528}, {10601, 29959}, {11160, 15246}, {11255, 32136}, {11414, 11477}, {11416, 11422}, {12007, 15577}, {12161, 15074}, {15141, 32245}, {16010, 16510}, {18388, 23049}, {18534, 20423}, {18954, 19369}, {20583, 20850}, {23292, 23327}, {32254, 32274}

X(32621) = reflection of X(9813) in X(575)
X(32621) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 154, 19136}, {6, 184, 19153}, {6, 9924, 9969}, {6, 9971, 9777}, {6, 10602, 11216}, {6, 19459, 159}, {206, 22829, 6}, {6467, 13366, 8541}, {8541, 13366, 6}, {10602, 11402, 6}


X(32622) = 1st MOSES-STEVANOVIC POINT

Barycentrics    a^2*((a^2 - b^2 - c^2)*Sqrt[2*(b*c + a*b + a*c) - (a^2 + b^2 + c^2)] + 2*(a - b - c)*S) : :
Tripolars    Sqrt[b c] : :

"Tripolars" abbreviates "homogeneous tripolar coordinates"; i.e., to say that x : y : z are tripolars for a point X means that x, y, z, are respectively proportional to |AX|, |BX|, |CX|.

X(32622) is one of two points (X(32623) is the other) whose pedal antipodal perspector is X(7). Also, X(32622) is the perspector of the excentral triangles of ABC and the circumcevian triangle of X(3513). (Randy Hutson, June 5, 2019 )

X(32622) lies on the Stevanovic circle, the curves K352 and Q037, and on this line: {1, 3}

X(32622) = reflection of X(32623) in X(1155)
X(32622) = X(513)-vertex conjugate of X(32623)
X(32622) = X(972)-Ceva conjugate of X(32623)
X(32622) = circumcircle-inverse of X(32623)
X(32622) = Bevan-circle-inverse of X(32623)
X(32622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 32623}, {3, 55, 32623}, {35, 15931, 32623}, {36, 5537, 32623}, {40, 57, 32623}, {55, 3514, 1}, {56, 6244, 32623}, {354, 3579, 32623}, {484, 5536, 32623}, {942, 7964, 32623}, {1155, 3514, 36}, {1381, 1382, 32623}, {1617, 10310, 32623}, {2077, 2078, 32623}, {2448, 2449, 32623}, {3256, 11012, 32623}, {3660, 13528, 32623}, {3748, 31663, 32623}, {7987, 31508, 32623}, {7994, 15803, 32623}, {8186, 8187, 32623}, {10268, 10383, 32623}, {10270, 10388, 32623}


X(32623) = 2nd MOSES-STEVANOVIC POINT

Barycentrics    a^2*((a^2 - b^2 - c^2)*Sqrt[2*(b*c + a*b + a*c) - (a^2 + b^2 + c^2)] - 2*(a - b - c)*S) : :
Tripolars    Sqrt[b c] : :

X(32623) is one of two points (X(32622) is the other) whose pedal antipodal perspector is X(7). Also, X(32623) is the perspector of the excentral triangles of ABC and the circumcevian triangle of X(3514). (Randy Hutson, June 5, 2019 )

p> X(32623) lies on the Stevanovic circle, the curves K352 and Q037, and on this line: {1, 3}

X(32623) = reflection of X(32622) in X(1155)
X(32623) = X(513)-vertex conjugate of X(32622)
X(32623) = X(972)-Ceva conjugate of X(32622)
X(32623) = circumcircle-inverse of X(32622)
X(32623) = Bevan-circle-inverse of X(32622)
X(32623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 32622}, {3, 55, 32622}, {35, 15931, 32622}, {36, 5537, 32622}, {40, 57, 32622}, {55, 3513, 1}, {56, 6244, 32622}, {354, 3579, 32622}, {484, 5536, 32622}, {942, 7964, 32622}, {1155, 3513, 36}, {1381, 1382, 32622}, {1617, 10310, 32622}, {2077, 2078, 32622}, {2448, 2449, 32622}, {3256, 11012, 32622}, {3660, 13528, 32622}, {3748, 31663, 32622}, {7987, 31508, 32622}, {7994, 15803, 32622}, {8186, 8187, 32622}, {10268, 10383, 32622}, {10270, 10388, 32622}


X(32624) = CIRCUMCIRCLE-INVERSE OF X(7)

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

X(32624) lies on these lines: {3,7}, {1358,5172}, {4367,8638}, {5543,26357}, {7742,14878}, {9436,27086}, {23839,32613}

X(32624) = circumcircle-inverse of X(7)


X(32625) = CIRCUMCIRCLE-INVERSE OF X(9)

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

X(32625) lies on these lines: {3, 9}, {35, 3207}, {36, 910}, {48, 4262}, {101, 2077}, {104, 8074}, {404, 8568}, {667, 3900}, {934, 2391}, {1055, 2078}, {1108, 7297}, {1615, 15931}, {2170, 5193}, {2178, 18594}, {2183, 5030}, {2317, 4251}, {3911, 11349}, {4394, 4730}, {5537, 6603}, {7987, 32561}, {12114, 23058}

X(32625) = circumcircle-inverse of X(9)
X(32625) = crosspoint of X(2291) and X(3451)
X(32625) = crosssum of X(i) and X(j) for these (i,j): {527, 3452}, {5514, 6366}
X(32625) = crossdifference of every pair of points on line {3752, 6129}
X(32625) = {X(1436),X(1604)}-harmonic conjugate of X(9)


X(32626) = CIRCUMCIRCLE-INVERSE OF X(12)

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

X(32626) lies on these lines: {3, 12}, {2915, 14882}

X(32626) = circumcircle-inverse of X(12)


X(32627) = CIRCUMCIRCLE-INVERSE OF X(17)

Barycentrics    a^2*(a^2 + b^2 - c^2 + 2*Sqrt[3]*S)*(a^2 - b^2 + c^2 + 2*Sqrt[3]*S)*(Sqrt[3]*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4) - 2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*S) : :

X(32627) lies on the cubic K937 and these lines: {3, 13}, {16, 16806}, {61, 1337}, {3130, 16530}, {3439, 11602}, {3442, 15444}, {5012, 21461}, {10677, 16257}, {11087, 22511}

X(32627) = isogonal conjugate of X(34219)
X(32627) = circumcircle-inverse of X(17)
X(32627) = {X(6104),X(31939)}-harmonic conjugate of X(3)


X(32628) = CIRCUMCIRCLE-INVERSE OF X(18)

Barycentrics    a^2*(a^2 + b^2 - c^2 - 2*Sqrt[3]*S)*(a^2 - b^2 + c^2 - 2*Sqrt[3]*S)*(Sqrt[3]*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4) + 2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*S) : :

X(32628) lies on the cubic K937 and these lines: {3, 14}, {15, 16807}, {62, 1338}, {3129, 16529}, {3438, 11603}, {3443, 15445}, {5012, 21462}, {10678, 16258}, {11082, 22510}

X(32628) = isogonal conjugate of X(34220)
X(32628) = circumcircle-inverse of X(18)
X(32628) = {X(6105),X(31940)}-harmonic conjugate of X(3)


X(32629) = X(13624)X(28185)∩X(15178)X(28173)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29066.

X(32629) lies on the circumcircle and these lines: {13624, 28185}, {15178, 28173}


X(32630) = X(1)X(28535)∩X(101)X(14422)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29068.

X(32630) lies on the circumcircle and these lines: {1,28535}, {101,14422}, {103,10246}, {244,28317}, {2291,4860}, {3576,28159}, {5425,28471}, {14413,2899}


X(32631) = ISOGONAL CONJUGATE OF X(19654)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29068.

X(32631) lies on these lines: {2,1155}, {4671,27756}, {4870,27761}, {4945,27741}, {4956,27753}, {5219,27737}, {5235,27740}, {5603,28537}, {9093,28899}

X(32631) = isogonal conjugate of X(19654)
X(32631) = barycentric quotient of X(i) and X(j) for these lines: {i,j}: {6,19654}, {45,5220}, {3679,17294}, {4777,28898}, {28899,4588}
X(32631) = trilinear product of X(i) and X(j) for these lines: {i,j}: {4791,28899}
X(32631) = trilinear quotient of X(i) and X(j) for these lines: {i,j}: {1,19654}, {3679,5220}, {4671,17294}


X(32632) = X(1)X(4127)∩X(9)X(19872)

Barycentrics    a*(9*a^3 + 9*a^2*b - 9*a*b^2 - 9*b^3 + 9*a^2*c - 13*a*b*c - 21*b^2*c - 9*a*c^2 - 21*b*c^2 - 9*c^3) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29070.

X(32632) lies on the Jerabek circumhyperbola of the excentral triangle and these lines: {1, 4127}, {9, 19872}, {40, 3627}, {191, 9780}, {2136, 4816}, {3219, 3337}, {3626, 5541}, {3646, 6763}, {3647, 13146}, {6326, 13624}

X(32632) = excentral isogonal conjugate of X(31663)
X(32632) = X(3634)-Ceva conjugate of X(1)


X(32633) = X(1)X(4757)∩X(3)X(5260)

Barycentrics    a*(9*a^3 - 9*a*b^2 - a*b*c - 3*b^2*c - 9*a*c^2 - 3*b*c^2) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29070.

X(32633) lies on these lines: {1, 4757}, {3, 5260}, {8, 19704}, {21, 19862}, {100, 3626}, {404, 19872}, {1001, 4189}, {1621, 19535}, {2975, 3621}, {3634, 13587}, {3736, 16477}, {4816, 5010}, {5251, 22266}, {5284, 7280}, {5550, 16370}, {6265, 13624}, {7173, 15680}, {19705, 19877}, {21161, 22936}, {22355, 27026}

X(32633) = {X(7280),X(17574)}-harmonic conjugate of X(5284)


X(32634) = X(1)X(3968)∩X(8)X(381)

Barycentrics    a*(a^3 - 4*a^2*b - a*b^2 + 4*b^3 - 4*a^2*c + 23*a*b*c - 19*b^2*c - a*c^2 - 19*b*c^2 + 4*c^3) : :
X(32634) = 2 X[8148] - 5 X[16615]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29070.

X(32634) lies on these lines: {1, 3968}, {8, 381}, {10, 13606}, {100, 13624}, {2550, 3621}, {3617, 3813}, {3626, 21630}, {4420, 10222}, {4853, 5303}, {5178, 12751}, {5223, 14923}, {10914, 11684}, {11024, 20050}

X(32634) = reflection of X(13606) in X(10)


X(32635) = X(1)X(748)∩X(2)X(3296)

Barycentrics    a*(a - b - c)*(a + 2*b + c)*(a + b + 2*c) : :
X(32635) = X[1] - 3 X[5506],8 X[3634] - 3 X[5557],7 X[9780] - 3 X[9782]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29070.

X(32635) lies on Feuerbach circumhyperbola and these lines: {1, 748}, {2, 3296}, {4, 3617}, {7, 12}, {8, 3058}, {9, 3871}, {10, 79}, {21, 210}, {44, 2298}, {45, 941}, {80, 3626}, {84, 1796}, {100, 3065}, {104, 6986}, {191, 3956}, {256, 3214}, {314, 3701}, {354, 17546}, {404, 7284}, {484, 4540}, {517, 16615}, {518, 17536}, {519, 13606}, {943, 3935}, {958, 2320}, {960, 1320}, {1000, 3621}, {1171, 5297}, {1389, 11278}, {1392, 5330}, {2346, 15254}, {2478, 5686}, {2481, 6540}, {2891, 25479}, {3062, 7997}, {3219, 3579}, {3254, 6734}, {3475, 5550}, {3625, 5559}, {3634, 5557}, {3679, 5560}, {3680, 3877}, {3691, 4876}, {3740, 17531}, {3811, 16858}, {3869, 17098}, {3873, 17534}, {3885, 4900}, {3889, 7308}, {3983, 15481}, {4096, 27368}, {4430, 16842}, {4532, 16126}, {4537, 5425}, {4647, 4756}, {4661, 11108}, {4866, 5250}, {4943, 23836}, {5044, 15179}, {5251, 5424}, {5551, 5708}, {5665, 15556}, {5719, 17552}, {6595, 18259}, {6684, 13243}, {6763, 9342}, {7320, 20050}, {10266, 21031}, {11544, 17484}, {11604, 13272}, {11698, 16006}, {12019, 24298}, {17420, 23838}, {18483, 25006}, {21077, 31254}, {26792, 31419}, {27549, 30479}

X(32635) = isogonal conjugate of X(32636)
X(32635) = X(1268)-Ceva conjugate of X(1255)
X(32635) = X(i)-cross conjugate of X(j) for these (i,j): {3737, 3699}, {4041, 644}, {4662, 8}
X(32635) = X(i)-isoconjugate of X(j) for these (i,j): {6, 553}, {7, 2308}, {34, 3916}, {56, 1125}, {57, 1100}, {58, 3649}, {73, 31900}, {77, 2355}, {101, 30724}, {106, 5298}, {109, 4977}, {181, 30593}, {222, 1839}, {269, 3683}, {273, 23201}, {278, 22054}, {604, 4359}, {608, 4001}, {651, 4979}, {1014, 1962}, {1106, 3702}, {1213, 1412}, {1230, 16947}, {1269, 1397}, {1396, 3958}, {1400, 8025}, {1402, 16709}, {1407, 3686}, {1408, 4647}, {1411, 4973}, {1414, 4983}, {1415, 4978}, {1416, 4966}, {1417, 4975}, {1431, 4697}, {1434, 20970}, {1461, 4976}, {2163, 4870}, {2171, 30581}, {4565, 4988}, {4990, 6614}, {7341, 8013}
X(32635) = cevapoint of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
X(32635) = crosspoint of X(1268) and X(4102)
X(32635) = trilinear pole of line {650, 4501}
X(32635) = barycentric product X(i) X(j) for these lines: {i,j}: {1, 4102}, {8, 1255}, {9, 1268}, {21, 6539}, {55, 32018}, {210, 32014}, {312, 1126}, {318, 1796}, {643, 31010}, {644, 4608}, {650, 6540}, {1171, 3701}, {1320, 31011}, {2185, 6538}, {3596, 28615}, {3700, 4596}, {4041, 4632}, {4086, 4629}, {4391, 8701}
X(32635) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {1, 553}, {8, 4359}, {9, 1125}, {21, 8025}, {33, 1839}, {37, 3649}, {41, 2308}, {44, 5298}, {45, 4870}, {55, 1100}, {60, 30581}, {78, 4001}, {200, 3686}, {210, 1213}, {212, 22054}, {219, 3916}, {220, 3683}, {312, 1269}, {333, 16709}, {346, 3702}, {513, 30724}, {522, 4978}, {607, 2355}, {644, 4427}, {650, 4977}, {663, 4979}, {1126, 57}, {1171, 1014}, {1172, 31900}, {1255, 7}, {1268, 85}, {1334, 1962}, {1796, 77}, {2185, 30593}, {2318, 3958}, {2321, 4647}, {2323, 4973}, {2325, 4975}, {2329, 4697}, {3158, 4856}, {3208, 4970}, {3239, 4985}, {3684, 4974}, {3686, 6533}, {3689, 4969}, {3693, 4966}, {3700, 30591}, {3701, 1230}, {3709, 4983}, {3900, 4976}, {4041, 4988}, {4069, 4115}, {4102, 75}, {4130, 4990}, {4420, 3578}, {4515, 4046}, {4526, 30592}, {4578, 30729}, {4596, 4573}, {4608, 24002}, {4629, 1414}, {4632, 4625}, {4873, 4717}, {4895, 4984}, {6538, 6358}, {6539, 1441}, {6540, 4554}, {7064, 21816}, {8701, 651}, {28615, 56}, {31010, 4077}, {32018, 6063}
X(32635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {210, 5302, 4420}, {4420, 5302, 21}


X(32636) = MIDPOINT OF X(3336) AND X(5563)

Barycentrics    a*(a + b - c)*(a - b + c)*(2*a + b + c) : :
X(32636) = (r+3 R) X[1] - 3 r X[3], X[1] + 3 X[3336], X[1] - 3 X[5563], 2 X[1] - 3 X[20323], 3 X[404] - X[4420], 2 X[3336] + X[20323]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29070.

X(32636) lies on these lines: {1, 3}, {2, 5302}, {4, 17728}, {7, 5550}, {10, 5434}, {11, 1354}, {12, 3634}, {21, 3742}, {34, 17523}, {44, 583}, {45, 2285}, {58, 14158}, {60, 757}, {63, 25524}, {79, 3582}, {81, 4719}, {88, 961}, {89, 959}, {104, 7686}, {108, 1887}, {140, 13407}, {142, 24953}, {145, 9352}, {210, 474}, {226, 5433}, {229, 4228}, {244, 1104}, {329, 24954}, {355, 4317}, {388, 5435}, {404, 518}, {496, 1770}, {499, 17605}, {501, 1412}, {529, 24982}, {553, 1125}, {584, 604}, {595, 18360}, {603, 1418}, {614, 4252}, {631, 17718}, {749, 1469}, {758, 17614}, {896, 28352}, {908, 6691}, {910, 1475}, {936, 4005}, {946, 11246}, {950, 15326}, {956, 3698}, {958, 3306}, {960, 3218}, {976, 21342}, {978, 4641}, {993, 5439}, {997, 3962}, {1001, 4652}, {1042, 1450}, {1100, 17454}, {1106, 1427}, {1193, 1464}, {1203, 8614}, {1210, 7354}, {1357, 2842}, {1358, 10521}, {1386, 17092}, {1393, 1455}, {1398, 1452}, {1406, 16466}, {1434, 1447}, {1445, 5220}, {1458, 2594}, {1468, 3752}, {1478, 17606}, {1698, 11237}, {1708, 15650}, {1737, 18357}, {1768, 9856}, {1788, 3600}, {1836, 3086}, {1837, 4293}, {2096, 12679}, {2173, 2260}, {2306, 19373}, {2348, 4253}, {2975, 3812}, {3052, 28011}, {3058, 31730}, {3149, 12680}, {3296, 3524}, {3474, 12701}, {3475, 3523}, {3476, 3621}, {3485, 5265}, {3555, 3689}, {3556, 26866}, {3584, 5442}, {3585, 31776}, {3598, 24796}, {3616, 4640}, {3624, 4654}, {3625, 4315}, {3626, 10106}, {3669, 4782}, {3671, 4031}, {3674, 7181}, {3681, 17572}, {3720, 11553}, {3740, 17531}, {3753, 8666}, {3811, 16371}, {3848, 5047}, {3869, 23958}, {3873, 4188}, {3874, 5440}, {3925, 12436}, {3928, 8583}, {3983, 4413}, {4018, 30144}, {4295, 11376}, {4299, 5722}, {4301, 17613}, {4308, 20050}, {4311, 10950}, {4324, 31795}, {4325, 28160}, {4333, 9668}, {4340, 17723}, {4355, 5219}, {4383, 11512}, {4423, 31424}, {4650, 21214}, {4676, 26093}, {4857, 28146}, {4861, 10107}, {4999, 5249}, {5030, 16601}, {5044, 6763}, {5057, 20084}, {5083, 12432}, {5123, 20060}, {5218, 11037}, {5225, 10431}, {5229, 5704}, {5247, 16610}, {5267, 5427}, {5270, 9956}, {5290, 19872}, {5294, 25914}, {5393, 10910}, {5405, 10911}, {5432, 21620}, {5542, 15837}, {5586, 14150}, {5587, 9657}, {5714, 6832}, {5880, 10527}, {5905, 25681}, {6001, 26877}, {6284, 11019}, {6684, 15888}, {6692, 12527}, {6848, 12678}, {6904, 24477}, {6905, 12675}, {6906, 13374}, {6911, 14872}, {7173, 8226}, {7175, 16477}, {7198, 9436}, {7248, 27655}, {7989, 9656}, {8686, 8698}, {9579, 10896}, {9581, 12943}, {9655, 10826}, {9776, 30478}, {9965, 28647}, {10072, 12699}, {10074, 17636}, {10085, 19541}, {10090, 12738}, {10091, 11670}, {10308, 16141}, {10916, 11112}, {11038, 15717}, {11194, 19860}, {11544, 12047}, {12019, 20118}, {12688, 22753}, {12832, 18976}, {13369, 17637}, {13747, 21077}, {14027, 31524}, {15299, 31391}, {16736, 27660}, {17351, 25591}, {17736, 25066}, {17751, 24593}, {17863, 18661}, {18253, 24564}, {19861, 31165}, {20470, 22345}, {22344, 23383}, {22836, 24473}, {24628, 30038}, {31190, 31246}

X(32636) = isogonal conjugate of X(32635)
X(32636) = midpoint of X(3336) and X(5563)
X(32636) = reflection of X(20323) in X(5563)
X(32636) = X(i)-Ceva conjugate of X(j) for these (i,j): {553, 1100}, {1414, 3669}, {26700, 513}
X(32636) = X(2308)-cross conjugate of X(1100)
X(32636) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4102}, {8, 1126}, {9, 1255}, {41, 32018}, {55, 1268}, {60, 6538}, {281, 1796}, {284, 6539}, {312, 28615}, {522, 8701}, {663, 6540}, {1171, 2321}, {1334, 32014}, {2316, 31011}, {3700, 4629}, {3709, 4632}, {3939, 4608}, {4041, 4596}, {5546, 31010}, {5547, 31013}
X(32636) = crosspoint of X(i) and X(j) for these (i,j): {1, 10308}, {57, 1014}
X(32636) = crosssum of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
X(32636) = crossdifference of every pair of points on line {650, 4501}
X(32636) = barycentric product X(i) X(j) for these lines: {i,j}: {1, 553}, {7, 1100}, {12, 30581}, {34, 4001}, {56, 4359}, {57, 1125}, {65, 8025}, {77, 1839}, {81, 3649}, {85, 2308}, {88, 5298}, {89, 4870}, {100, 30724}, {109, 4978}, {269, 3686}, {273, 22054}, {278, 3916}, {279, 3683}, {331, 23201}, {348, 2355}, {552, 21816}, {604, 1269}, {651, 4977}, {664, 4979}, {934, 4976}, {1014, 1213}, {1214, 31900}, {1230, 1408}, {1400, 16709}, {1407, 3702}, {1412, 4647}, {1414, 4988}, {1432, 4697}, {1434, 1962}, {1461, 4985}, {1462, 4966}, {2006, 4973}, {2171, 30593}, {3669, 4427}, {4115, 7203}, {4565, 30591}, {4573, 4983}, {4617, 4990}, {4856, 19604}, {4970, 7153}, {4989, 21446}
X(32636) = barycentric quotient X(i) / X(j) for these lines: {i,j}: {1, 4102}, {7, 32018}, {56, 1255}, {57, 1268}, {65, 6539}, {553, 75}, {603, 1796}, {604, 1126}, {651, 6540}, {1014, 32014}, {1100, 8}, {1125, 312}, {1213, 3701}, {1269, 28659}, {1319, 31011}, {1397, 28615}, {1408, 1171}, {1414, 4632}, {1415, 8701}, {1839, 318}, {1962, 2321}, {2171, 6538}, {2308, 9}, {2355, 281}, {3649, 321}, {3669, 4608}, {3683, 346}, {3686, 341}, {3916, 345}, {3958, 3710}, {4001, 3718}, {4017, 31010}, {4359, 3596}, {4427, 646}, {4565, 4596}, {4647, 30713}, {4697, 17787}, {4870, 4671}, {4969, 4723}, {4970, 4110}, {4974, 3975}, {4976, 4397}, {4977, 4391}, {4979, 522}, {4983, 3700}, {4984, 4768}, {4988, 4086}, {4989, 30854}, {5298, 4358}, {8025, 314}, {16709, 28660}, {17454, 4420}, {20970, 210}, {21816, 6057}, {22054, 78}, {22080, 3694}, {23201, 219}, {30581, 261}, {30724, 693}, {31900, 31623}
X(32636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 13624}, {1, 46, 12702}, {1, 57, 5221}, {1, 5221, 65}, {1, 5903, 11278}, {1, 11278, 5048}, {1, 12702, 3057}, {1, 13624, 2646}, {3, 3338, 354}, {3, 12704, 7957}, {7, 7288, 11375}, {35, 5045, 3748}, {36, 942, 2646}, {36, 3337, 942}, {40, 3304, 5919}, {46, 999, 3057}, {46, 3057, 5183}, {55, 3333, 17609}, {56, 57, 65}, {56, 65, 1319}, {56, 1388, 13462}, {56, 2099, 1420}, {56, 5221, 1}, {56, 11509, 1617}, {57, 1420, 3339}, {57, 3361, 56}, {63, 25524, 25917}, {65, 1319, 11011}, {79, 3582, 9955}, {388, 5435, 24914}, {553, 1125, 3649}, {553, 5298, 4870}, {942, 13624, 1}, {999, 12702, 1}, {1125, 3649, 4870}, {1125, 3916, 3683}, {1125, 4973, 3916}, {1418, 1471, 1456}, {1420, 3339, 2099}, {1434, 1447, 4059}, {1447, 4059, 24805}, {1788, 3600, 5252}, {2099, 3339, 65}, {2975, 27003, 3812}, {3218, 5253, 960}, {3333, 15803, 55}, {3340, 13462, 1388}, {3361, 15932, 13370}, {3474, 14986, 12701}, {3513, 3514, 3748}, {3555, 25440, 3689}, {3649, 5298, 1125}, {3746, 5131, 31663}, {3911, 4298, 12}, {4315, 4848, 10944}, {4860, 5204, 1}, {5045, 5122, 35}, {5049, 31663, 3746}, {5126, 31794, 1}, {5265, 21454, 3485}, {5903, 24928, 5048}, {7280, 18398, 24929}, {11278, 24928, 1}, {12009, 31794, 942}, {13388, 13389, 20182}, {15325, 24470, 12047}


X(32637) = ISOGONAL CONJUGATE OF X(27357)

Barycentrics    (a^4-2 a^2 b^2+b^4-a^2 c^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) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)^2 : :
Barycentrics    (17 R^4-12 SB SC-4 R^2 SW-4 SW^2)S^4 + (-132 R^8-125 R^6 SB-125 R^6 SC-67 R^4 SB SC+293 R^6 SW+150 R^4 SB SW+150 R^4 SC SW+44 R^2 SB SC SW-225 R^4 SW^2-60 R^2 SB SW^2-60 R^2 SC SW^2-4 SB SC SW^2+72 R^2 SW^3+8 SB SW^3+8 SC SW^3-8 SW^4)S^2 + 12 S^6-18 R^8 SB SC+27 R^6 SB SC SW-9 R^4 SB SC SW^2 : :

See Antreas Hatzipolakis, Ercole Suppa and Peter Moses, Hyacinthos 29077 and Hyacinthos 29079.

X(32637) lies on these lines: {3,252}, {137,1487}, {1157,6592}, {3519,14072}, {19268,19552}, {27868,31392}

X(32637) = isogonal conjugate of X(27357)
X(32637) = barycentric quotient X(6)/X(27357)
X(32637) = trilinear quotient X(1)/X(27357)
X(32637) = {X(252),X(930)}-harmonic conjugate of X(24144)


X(32638) = X(5)X(49)∩X(1263)X(15038)

Barycentrics    (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 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) (2 a^16-11 a^14 b^2+25 a^12 b^4-31 a^10 b^6+25 a^8 b^8-17 a^6 b^10+11 a^4 b^12-5 a^2 b^14+b^16-11 a^14 c^2+34 a^12 b^2 c^2-31 a^10 b^4 c^2-4 a^8 b^6 c^2+31 a^6 b^8 c^2-38 a^4 b^10 c^2+27 a^2 b^12 c^2-8 b^14 c^2+25 a^12 c^4-31 a^10 b^2 c^4-5 a^6 b^6 c^4+34 a^4 b^8 c^4-51 a^2 b^10 c^4+28 b^12 c^4-31 a^10 c^6-4 a^8 b^2 c^6-5 a^6 b^4 c^6-14 a^4 b^6 c^6+29 a^2 b^8 c^6-56 b^10 c^6+25 a^8 c^8+31 a^6 b^2 c^8+34 a^4 b^4 c^8+29 a^2 b^6 c^8+70 b^8 c^8-17 a^6 c^10-38 a^4 b^2 c^10-51 a^2 b^4 c^10-56 b^6 c^10+11 a^4 c^12+27 a^2 b^2 c^12+28 b^4 c^12-5 a^2 c^14-8 b^2 c^14+c^16) : :
Barycentrics    (4 R^2+8 SB+8 SC)S^4 + (-93 R^6+14 R^4 SB+14 R^4 SC+52 R^2 SB SC+76 R^4 SW-8 R^2 SB SW-8 R^2 SC SW-16 SB SC SW-16 R^2 SW^2) S^2 -9 R^6 SB SC : :

See Antreas Hatzipolakis, Ercole Suppa and Peter Moses, Hyacinthos 29077 and Hyacinthos 29079.

X(32638) lies on these lines: {5,49}, {1263,15038}, {6343,13621}, {6592,11584}, {10096,14071}, {14627,31674}, {14857,24306}

X(32638) = barycentric quotient X(24385)/X(13582)
X(32638) = trilinear product of X(i) and X(j) for these lines: {i,j}: {1749,24385}, {1749,24385}


X(32639) = X(30)X(1141)∩X(140)X(10277)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29079.

X(32639) lies on these lines: {30, 1141}, {140, 10277}

X(32639) = barycentric product X(8254)*X(13582)






leftri  Centers associated with barycentric products of circumcircle-P-antipodes: X(32640) - X(32741)  rightri

Let P = p : q : r (barycentrics). The locus of the barycentric product of circumcircle-P-antipodes is the circumconic with perspector X(6)*P = a2p : b2q : c2r.

Let L be a line. The barycentric product of the (real or nonreal) circumcircle intercepts of L is the trilinear pole of the isogonal conjugate of the isotomic conjugate of L (or equivalently, X(6)*L). These intercepts are also circumcircle-P-antipodes for all P on L.

Contributed by Randy Hutson, June 14, 2019.

underbar



X(32640) = BARYCENTRIC PRODUCT X(74)*X(110)

Barycentrics    a^4/((b^2 - c^2) (2 a^4 - a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Trilinears    (sin^2 A) (cos A)/((tan B - tan C) (2 tan A - tan B - tan C)) : :

As the barycentric product of circumcircle antipodes, X(32640) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32640) lies on these lines: {32, 14385}, {74, 187}, {110, 32681}, {112, 647}, {115, 14989}, {186, 3003}, {237, 14908}, {249, 4558}, {1384, 9717}, {1576, 32715}, {23357, 32320} et al

X(32640) = isogonal conjugate of polar conjugate of X(1304)
X(32640) = isotomic conjugate of polar conjugate of X(32715)
X(32640) = trilinear pole of line X(184)X(1576)
X(32640) = center of bianticevian conic of X(15) and X(16)
X(32640) = barycentric product X(i)*X(j) for these lines: {i,j}: {69, 32715}, {74, 110}, {112, 14919}, {163, 2349}, {249, 2433}, {476, 14385}, {691, 9717}, {4558, 8749}, {15066, 32681}
X(32640) = barycentric quotient X(i)/X(j) for these (i,j): (32, 1637), (74, 850), (110, 3260), (163, 14206), (2159, 32671), (2349, 20948), (2433, 338), (8749, 14618), (14385, 3268), (14919, 3267), (32715, 4)


X(32641) = BARYCENTRIC PRODUCT X(100)*X(104)

Barycentrics    a^2/((b - c) (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)) : :
Barycentrics    a^2/((b - c) (cos B + cos C - 1)) : :

As the barycentric product of circumcircle antipodes, X(32641) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32641) lies on these lines: {36, 909}, {44, 14578}, {100, 32685}, {101, 652}, {104, 294}, {112, 1309}, {248, 5291}, {644, 906}, {645, 4558}, {650, 1415}, {651, 905}, {654, 32675}, {666, 2401}, {692, 32723}, {1110, 3939}, {1576, 14776}, {1795, 2338}, {1983, 23703}, {2250, 2341}, {2323, 15629}, {2342, 4845}, {5546, 32661}, {8609, 15500}, {15627, 18877}, {23987, 32647} et al

X(32641) = isogonal conjugate of X(10015)
X(32641) = trilinear pole of line X(55)X(184)
X(32641) = crossdifference of every pair of points on line X(3259)X(3326)
X(32641) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 1309}, {6, 13136}, {69, 14776}, {100, 104}, {108, 1809}, {190, 909}, {662, 2250}, {664, 2342}, {692, 18816}, {906, 16082}, {1252, 2401}, {1795, 1897}, {6335, 14578}
X(32641) = barycentric quotient X(i)/X(j) for these (i,j): (6, 10015), (100, 3262), (101, 908), (104, 693), (109, 22464), (692, 517), (909, 514), (1252, 2397), (1309, 264), (1415, 1465), (1576, 859), (1795, 4025), (1983, 16586), (2250, 1577), (2342, 522), (2401, 23989), (3939, 6735), (13136, 76), (14578, 905), (14776, 4)


X(32642) = BARYCENTRIC PRODUCT X(101)*X(103)

Barycentrics    a^4/((b - c) (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c))) : :
Barycentrics    a^4/((b - c) ((a - b) cot C + (a - c) cot B)) : :

As the barycentric product of circumcircle antipodes, X(32642) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32642) lies on these lines: {101, 32684}, {103, 32682}, {112, 21789}, {248, 17735}, {677, 4558}, {692, 24027}, {906, 1110}, {911, 2223}, {1415, 1946}, {6586, 8750}, {8608, 32699}, {23990, 32656}, {32659, 32719}, {32721, 32739} et al

X(32642) = isogonal conjugate of isotomic conjugate of X(677)
X(32642) = trilinear pole of line X(184)X(14827)
X(32642) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 677}, {100, 911}, {101, 103}, {109, 2338}, {18025, 32739}
X(32642) = barycentric quotient X(i)/X(j) for these (i,j): (32, 676), (103, 3261), (677, 76), (911, 693), (24027, 24015), (32656, 26006), (32739, 516)


X(32643) = BARYCENTRIC PRODUCT X(102)*X(109)

Barycentrics    a^4/((2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2) (b - c) (b + c - a)) : :
Barycentrics    (sin^3 A)/((sin B (sec A - sec B) + sin C (sec A - sec C)) (cos B - cos C)) : :

As the barycentric product of circumcircle antipodes, X(32643) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32643) lies on these lines: {102, 32689}, {109, 32683}, {248, 17966}, {906, 2149}, {1415, 32667}, {1946, 32652}, {2425, 2432}, {5053, 15629}, {6589, 32653}, {7113, 14578}, {8607, 32707}, {23979, 32660}

X(32643) = barycentric product X(i)*X(j) for these lines: {i,j}: {63, 32667}, {102, 109}, {1262, 2432}, {1461, 15629}, {2399, 23979}
X(32643) = barycentric quotient X(i)/X(j) for these (i,j): (2432, 23978), (23979, 2406), (32667, 92)


X(32644) = BARYCENTRIC PRODUCT X(105)*X(1292)

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

As the barycentric product of circumcircle antipodes, X(32644) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32644) lies on these lines: {666, 2414}, {906, 919}, {2428, 2440}, {3290, 8751}, {32656, 32666}

X(32644) = barycentric product X(i)*X(j) for these lines: {i,j}: {105, 1292}, {277, 919}, {2428, 6185}
X(32644) = barycentric quotient X(i)/X(j) for these (i,j): (919, 344), (1292, 3263), (2428, 4437)


X(32645) = BARYCENTRIC PRODUCT X(106)*X(1293)

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

As the barycentric product of circumcircle antipodes, X(32645) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32645) lies on these lines: {1293, 32686}, {2429, 2441}, {4558, 4591}, {8610, 8752}, {32656, 32719}

X(32645) = barycentric product X(i)*X(j) for these lines: {i,j}: {106, 1293}, {2226, 2429}, {4373, 32719}
X(32645) = barycentric quotient X(i)/X(j) for these (i,j): (1293, 3264), (32719, 145)


X(32646) = BARYCENTRIC PRODUCT X(107)*X(1294)

Barycentrics    1/((b^2 - c^2) (a^2 - b^2 - c^2)^2 (a^6 (b^2 + c^2) - a^4 (3 b^4 - 4 b^2 c^2 + 3 c^4) + 3 a^2 (b^2 - c^2)^2 (b^2 + c^2) - (b^2 - c^2)^2 (b^4 + 4 b^2 c^2 + c^4))) : :

As the barycentric product of circumcircle antipodes, X(32646) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32646) lies on these lines: {112, 23590}, {1990, 15311}, {2404, 16077}, {2430, 2442}, {4558, 23582}, {6529, 6587}, {23964, 32661}

X(32646) = trilinear pole of line X(184)X(6525)
X(32646) = barycentric product X(i)*X(j) for these lines: {i,j}: {107, 1294}, {2416, 23590}
X(32646) = barycentric quotient X(i)/X(j) for these (i,j): (1294, 3265), (2416, 23974), (23590, 2404)


X(32647) = BARYCENTRIC PRODUCT X(108)*X(1295)

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

As the barycentric product of circumcircle antipodes, X(32647) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32647) lies on these lines: {906, 7115}, {1295, 32688}, {1415, 23985}, {1455, 14571}, {1457, 32667}, {2431, 2443}, {23987, 32641}

X(32647) = trilinear pole of line X(184)X(3195)
X(32647) = barycentric product X(i)*X(j) for these lines: {i,j}: {108, 1295}, {2417, 23985}, {2431, 23984}
X(32647) = barycentric quotient X(23985)/X(2405)


X(32648) = BARYCENTRIC PRODUCT X(111)*X(1296)

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

As the barycentric product of circumcircle antipodes, X(32648) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32648) lies on these lines: {691, 1296}, {892, 2418}, {2434, 2444}, {3291, 8753}, {10422, 15899}, {32661, 32729}

X(32648) = barycentric product X(i)*X(j) for these lines: {i,j}: {111, 1296}, {691, 21448}, {2434, 10630}, {5485, 32729}
X(32648) = barycentric quotient X(i)/X(j) for these (i,j): (691, 11059), (1296, 3266), (32729, 1992)


X(32649) = BARYCENTRIC PRODUCT X(112)*X(1297)

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

As the barycentric product of circumcircle antipodes, X(32649) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32649) lies on these lines: {112, 32687}, {186, 1297}, {232, 248}, {250, 4558}, {2409, 2966}, {2435, 2445}, {2485, 32713}, {18374, 18877}

X(32649) = trilinear pole of line X(184)X(17409)
X(32649) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 32687}, {112, 1297}, {2435, 23964}
X(32649) = barycentric quotient X(i)/X(j) for these (i,j): (112, 30737), (1297, 3267), (32687, 264)


X(32650) = BARYCENTRIC PRODUCT X(476)*X(477)

Barycentrics    1/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) (a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4))) : :
Barycentrics    (sin^2 A)/((1 + 2 cos 2A) sin(B - C) (4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C))) : :

As the barycentric product of circumcircle antipodes, X(32650) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32650) lies on these lines: {476, 32690}, {1637, 32640}, {1989, 18320}, {2436, 2437}, {3018, 32711}, {4558, 30528}, {14401, 32661}, {14560, 32733}, {23588, 32662}

X(32650) = trilinear pole of line X(184)X(14560)
X(32650) = barycentric product X(i)*X(j) for these lines: {i,j}: {476, 477}, {1989, 30528}
X(32650) = barycentric quotient X(i)/X(j) for these (i,j): (477, 3268), (30528, 7799)


X(32651) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(7)

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

As the barycentric product of circumcircle antipodes, X(32651) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32651) lies on these lines: {109, 15439}, {651, 906}, {943, 9372}, {1415, 32714}, {1461, 32660}, {1462, 32658}, {2259, 32657}, {2982, 14578}, {4558, 4573}, {4565, 32661} et al

X(32651) = trilinear pole of line X(56)X(184)
X(32651) = barycentric product X(i)*X(j) for these lines: {i,j}: {7, 15439}, {651, 2982}, {934, 943}
X(32651) = barycentric quotient X(i)/X(j) for these (i,j): (109, 6734), (943, 4397), (1576, 8021), (2982, 4391), (15439, 8)


X(32652) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(9)

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

As the barycentric product of circumcircle antipodes, X(32652) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32652) lies on these lines: {31, 2188}, {32, 7367}, {109, 8059}, {248, 1903}, {643, 4558}, {692, 32660}, {906, 3939}, {1415, 8750}, {1436, 2195}, {1946, 32643}, {2192, 2342}, {2208, 2352}, {6129, 32714}, {7151, 32655}, {21059, 32657} et al

X(32652) = isogonal conjugate of X(17896)
X(32652) = trilinear pole of line X(41)X(184)
X(32652) = crossdifference of every pair of points on line X(5514)X(16596)
X(32652) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 13138}, {9, 8059}, {84, 101}, {100, 1436}, {108, 268}, {109, 282}, {110, 1903}, {189, 692}, {190, 2208}, {280, 1415}, {644, 1413}, {651, 2192}, {653, 2188}, {934, 7367}, {1422, 3939}, {1332, 7151}, {7020, 32660}
X(32652) = barycentric quotient X(i)/X(j) for these (i,j): (6, 17896), (31, 14837), (32, 6129), (84, 3261), (101, 322), (692, 329), (1413, 24002), (1415, 347), (1436, 693), (1903, 850), (2188, 6332), (2192, 4391), (2208, 514), (7151, 17924), (7367, 4397), (8059, 85), (13138, 76), (32660, 7013)


X(32653) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(10)

Barycentrics    a^2/(b^2 (a - b) (a + b - c) - c^2 (a - c) (a - b + c)) : :
Barycentrics    a^2/((b - c) (b^3 + c^3 - a^2 b - a^2 c + a b c)) : :
Barycentrics    a^4/((cos C - cos A) (csc^2 B) + (cos A - cos B) (csc^2 C)) : :

As the barycentric product of circumcircle antipodes, X(32653) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32653) lies on these lines: {32, 1146}, {101, 32661}, {112, 26704}, {190, 4558}, {248, 15232}, {577, 23986}, {906, 1018}, {1576, 8750}, {2178, 32659}, {2217, 16968}, {2250, 14578}, {2425, 4559}, {4557, 32656}, {5301, 32655}, {6589, 32643} et al

X(32653) = trilinear pole of line X(42)X(184)
X(32653) = crossdifference of every pair of points on line X(124)X(2968)
X(32653) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 26704}, {100, 2217}, {101, 13478}, {109, 10570}, {110, 15232}, {692, 2995}, {4559, 19607}, {8687, 19608}
X(32653) = barycentric quotient X(i)/X(j) for these (i,j): (32, 6589), (101, 4417), (692, 3869), (2217, 693), (13478, 3261), (15232, 850), (26704, 264)


X(32654) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(512)

Barycentrics    a^4/(2 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) : :
Trilinears    (sin A)/(cos A cos(A + ω) - cos B cos(B + ω) - cos C cos(C + ω)) : :

As the barycentric product of circumcircle antipodes, X(32654) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184), and as the barycentric product of circumcircle-X(512)-antipodes, X(32654) also lies on conic {{A,B,C,X(6),X(32),X(83),X(213),X(729)}} with perspector X(669).

Let A'B'C' be the circumcevian triangle of X(511). Let A" be the barycentric product B'*C', and define B" and C" cyclically. A", B", C" are collinear on line X(2491)X(3289). The lines AA", BB", CC" concur in X(32654). (Randy Hutson, August 19, 2019)

X(32654) lies on these lines: {6, 2987}, {24, 112}, {32, 1147}, {50, 32740}, {83, 3815}, {187, 5961}, {213, 906}, {230, 297}, {237, 14601}, {248, 2422}, {571, 1576}, {577, 1084}, {729, 10425}, {1691, 2065}, {1918, 32656} et al

X(32654) = isogonal conjugate of isotomic conjugate of X(2987)
X(32654) = trilinear pole of line X(184)X(669)
X(32654) = crossdifference of every pair of points on line X(114)X(2974)
X(32654) = X(92)-isoconjugate of X(3564)
X(32654) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 3563}, {6, 2987}, {31, 8773}, {32, 8781}
X(32654) = barycentric quotient X(i)/X(j) for these (i,j): (31, 1733), (32, 230), (184, 3564), (237, 114), (1576, 4226), (2987, 76), (3563, 264), (8773, 561), (8781, 1502)


X(32655) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(513)

Barycentrics    a^3/(a^3 (b + c) - a^2 (b^2 + c^2) - a (b - c)^2 (b + c) + (b^2 - c^2)^2) : :
Barycentrics    (sin^3 A)/(sin B (cos A + cos B - 1) + sin C (cos A + cos C - 1)) : :

As the barycentric product of circumcircle antipodes, X(32655) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

Let A'B'C' be the circumcevian triangle of X(517). Let A" be the barycentric product B'*C', and define B" and C" cyclically. A", B", C" are collinear with X(3310). The lines AA", BB", CC" concur in X(32655). (Randy Hutson, August 19, 2019)

X(32655) lies on these lines: {6, 906}, {31, 32656}, {32, 23980}, {81, 2990}, {112, 915}, {248, 3657}, {563, 604}, {571, 608}, {577, 1015}, {654, 2423}, {739, 6099}, {1333, 32661}, {1576, 2203}, {5301, 32653}, {7151, 32652}, {8609, 15500} et al

X(32655) = isogonal conjugate of polar conjugate of X(915)
X(32655) = trilinear pole of line X(184)X(667)
X(32655) = X(92)-isoconjugate of X(912)
X(32655) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 915}, {6, 2990}, {110, 3657}, {905, 32698}
X(32655) = barycentric quotient X(i)/X(j) for these (i,j): (31, 1737), (32, 8609), (184, 912), (915, 264), (2990, 76), (3657, 850), (32698, 6335)


X(32656) = BARYCENTRIC PRODUCT X(3)*X(101)

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

X(32656) is the barycentric product of the circumcircle intercepts of line X(3)X(71). As the barycentric product of circumcircle antipodes, X(32656) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32656) lies on these lines: {31, 32655}, {42, 14910}, {48, 23203}, {71, 248}, {100, 15440}, {101, 112}, {109, 15439}, {110, 28624}, {184, 32659}, {190, 2966}, {212, 2638}, {560, 21059}, {577, 20776}, {692, 1415}, {1110, 3939}, {1331, 4558}, {1459, 1813}, {1576, 32739}, {1897, 16813}, {1918, 32654}, {4557, 32653}, {7649, 32699}, {23990, 32642}, {32644, 32666}, {32645, 32719} et al

X(32656) = isogonal conjugate of polar conjugate of X(101)
X(32656) = isotomic conjugate of polar conjugate of X(32739)
X(32656) = X(92)-isoconjugate of X(514)
X(32656) = trilinear pole of line X(184)X(2200)
X(32656) = crossdifference of every pair of points on X(116)X(2973)
X(32656) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 906}, {3, 101}, {6, 1331}, {8, 32660}, {10, 32661}, {31, 1332}, {32, 4561}, {41, 6516}, {42, 4558}, {48, 100}, {55, 1813}, {56, 4587}, {58, 4574}, {63, 692}, {69, 32739}, {71, 110}, {72, 163}, {108, 2289}, {109, 219}, {112, 3682}, {162, 3990}, {184, 190}, {212, 651}, {577, 1897}, {905, 1110}, {1252, 1459}, {1790, 4557}, {1918, 4563}, {3977, 32719}, {17780, 32659}, {25083, 32666}, {26006, 32642}
X(32656) = barycentric quotient X(i)/X(j) for these (i,j): (3, 3261), (31, 17924), (32, 7649), (42, 14618), (48, 693), (71, 850), (72, 20948), (100, 1969), (101, 264), (109, 331), (163, 286), (184, 514), (190, 18022), (212, 4391), (560, 6591), (577, 4025), (649, 2973), (692, 92), (906, 75), (1110, 6335), (1331, 76), (1332, 561), (1459, 23989), (1813, 6063), (1897, 18027), (1918, 2501), (3682, 3267), (3990, 14208), (4558, 310), (4561, 1502), (4574, 313), (4587, 3596), (6516, 20567), (32659, 6548), (32660, 7), (32661, 86), (32719, 6336), (32739, 4)


X(32657) = BARYCENTRIC PRODUCT X(3)*X(103)

Barycentrics    a^4 (a^2 - b^2 - c^2)/(2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :
Trilinears    (sin 2A)/(a^2 - b^2 cos C - c^2 cos B) : :

X(32657) is the barycentric product of the circumcircle intercepts of line X(3)X(4091). As the barycentric product of circumcircle antipodes, X(32657) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32657) lies on these lines: {41, 603}, {58, 103}, {255, 906}, {577, 20776}, {1795, 2338}, {1815, 2327}, {1886, 32701}, {2259, 32651}, {7335, 32660}, {21059, 32652} et al

X(32657) = isogonal conjugate of polar conjugate of X(103)
X(32657) = X(92)-isoconjugate of X(516)
X(32657) = crossdifference of every pair of points on line X(118)X(20622)
X(32657) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 103}, {6, 1815}, {63, 911}, {184, 18025}, {222, 2338}, {677, 1459}, {916, 15380}
X(32657) = barycentric quotient X(i)/X(j) for these (i,j): (32, 1886), (103, 264), (184, 516), (577, 26006), (911, 92), (1815, 76), (2338, 7017), (18025, 18022)


X(32658) = BARYCENTRIC PRODUCT X(3)*X(105)

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

X(32658) is the barycentric product of the circumcircle intercepts of line X(3)X(905). As the barycentric product of circumcircle antipodes, X(32658) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32658) lies on these lines: {3, 906}, {6, 3433}, {28, 105}, {32, 56}, {48, 23203}, {104, 294}, {248, 10099}, {448, 2481}, {603, 9247}, {665, 2440}, {911, 2223}, {1333, 1576}, {1384, 3420}, {1436, 2195}, {1437, 32661}, {1444, 1814}, {1462, 32651}, {2217, 16968}, {3053, 3435}, {5089, 32703} et al

X(32658) = isogonal conjugate of polar conjugate of X(105)
X(32658) = X(92)-isoconjugate of X(518)
X(32658) = crossdifference of every pair of points on line X(120)X(20621)
X(32658) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 105}, {6, 1814}, {31, 31637}, {48, 673}, {63, 1438}, {77, 2195}, {78, 1416}, {109, 23696}, {110, 10099}, {184, 2481}, {219, 1462}, {222, 294}, {603, 14942}, {905, 919}, {1437, 13576}, {6185, 20752}, {9247, 18031}
X(32658) = barycentric quotient X(i)/X(j) for these (i,j): (3, 3263), (31, 1861), (32, 5089), (48, 3912), (105, 264), (184, 518), (222, 27818), (294, 7017), (577, 25083), (603, 9436), (604, 5236), (673, 1969), (919, 6335), (1333, 15149), (1416, 273), (1437, 30941), (1438, 92), (1462, 331), (1576, 4238), (1814, 76), (2195, 318), (2481, 18022), (9247, 672), (10099, 850), (20752, 4437), (31637, 561)


X(32659) = BARYCENTRIC PRODUCT X(3)*X(106)

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

X(32659) is the barycentric product of the circumcircle intercepts of line X(3)X(1459). As the barycentric product of circumcircle antipodes, X(32659) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32659) lies on these lines: {36, 909}, {48, 906}, {106, 112}, {184, 32656}, {604, 1415}, {903, 2966}, {1576, 2206}, {1790, 1797}, {2178, 32653}, {2208, 2352}, {2359, 22054}, {8756, 32705}, {32642, 32719} et al

X(32659) = isogonal conjugate of polar conjugate of X(106)
X(32659) = X(92)-isoconjugate of X(519)
X(32659) = crossdifference of every pair of points on line X(121)X(4768)
X(32659) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 106}, {6, 1797}, {48, 88}, {63, 9456}, {78, 1417}, {184, 903}, {222, 2316}, {577, 6336}, {603, 1320}, {901, 1459}, {905, 32665}, {906, 1022}, {4025, 32719}, {4049, 32661}, {6548, 32656}, {9247, 20568}
X(32659) = barycentric quotient X(i)/X(j) for these (i,j): (3, 3264), (32, 8756), (48, 4358), (88, 1969), (106, 264), (184, 519), (577, 3977), (903, 18022), (906, 24004), (1417, 273), (1797, 76), (2316, 7017), (6336, 18027), (9247, 44), (9456, 92), (32656, 17780), (32665, 6335), (32719, 1897)


X(32660) = BARYCENTRIC PRODUCT X(3)*X(109)

Barycentrics    a^4 (a^2 - b^2 - c^2)/((b - c) (a - b - c)) : :
Trilinears    a^2 (cos A)/(cos B - cos C) : :

X(32660) is the barycentric product of the circumcircle intercepts of line X(3)X(73). As the barycentric product of circumcircle antipodes, X(32660) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

The trilinear polar of X(32660) passes through X(184).

X(32660) lies on these lines: {48, 14578}, {73, 248}, {101, 652}, {109, 112}, {563, 604}, {603, 9247}, {653, 16813}, {664, 2966}, {692, 32652}, {1400, 14910}, {1404, 2252}, {1415, 32739}, {1461, 32651}, {1813, 4558}, {2425, 4559}, {3064, 32707}, {7335, 32657}, {23979, 32643} et al

X(32660) = isogonal conjugate of polar conjugate of X(109)
X(32660) = X(92)-isoconjugate of X(522)
X(32660) = crossdifference of every pair of points on line X(124)X(20620)
X(32660) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 109}, {6, 1813}, {7, 32656}, {31, 6516}, {48, 651}, {63, 1415}, {65, 4575}, {71, 4565}, {73, 110}, {77, 692}, {100, 603}, {101, 222}, {108, 255}, {162, 22341}, {163, 1214}, {184, 664}, {212, 4397}, {219, 1461}, {226, 32661}, {228, 1414}, {307, 1576}, {348, 32739}, {577, 653}, {604, 1332}, {652, 1262}, {1400, 4558}, {1790, 4559}, {1897, 7335}, {4554, 9247}, {6332, 23979}, {7013, 32652}, {14578, 24029}
X(32660) = barycentric quotient X(i)/X(j) for these (i,j): (32, 3064), (48, 4391), (73, 850), (101, 7017), (109, 264), (163, 31623), (184, 522), (212, 934), (222, 3261), (228, 4086), (577, 6332), (603, 693), (604, 17924), (651, 1969), (652, 23978), (653, 18027), (664, 18022), (692, 318), (1214, 20948), (1332, 28659), (1400, 14618), (1415, 92), (1459, 59), (1461, 331), (1576, 29), (1813, 76), (4558, 28660), (4575, 314), (6516, 561), (7335, 4025), (9247, 650), (22341, 14208), (23979, 653), (32656, 8), (32661, 333), (32739, 281), (32652, 7020)


X(32661) = BARYCENTRIC PRODUCT X(3)*X(110)

Barycentrics    a^4 (a^2 - b^2 - c^2)/(b^2 - c^2) : :
Trilinears    sin 2A csc(B - C) : :

X(32661) is the barycentric product of the circumcircle intercepts of line X(3)X(49). As the barycentric product of circumcircle antipodes, X(32661) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32661) lies on these lines: {3, 248}, {5, 1970}, {6, 1511}, {30, 1971}, {32, 1147}, {39, 12038}, {49, 217}, {50, 15136}, {99, 249}, {101, 32653}, {110, 112}, {115, 17702}, {155, 3053}, {163, 1415}, {184, 14908}, {187, 13754}, {206, 2882}, {216, 5961}, {394, 18876}, {520, 9145}, {577, 18877}, {647, 32662}, {648, 16813}, {686, 23181}, {827, 26714}, {906, 4575}, {912, 5006}, {916, 1326}, {925, 32692}, {933, 6570}, {1092, 10316}, {1333, 32655}, {1384, 1501}, {1437, 32658}, {1562, 16163}, {1576, 14270}, {1691, 3564}, {1915, 18907}, {1968, 10539}, {2030, 8681}, {2071, 13509}, {2193, 14578}, {2421, 4611}, {2501, 32708}, {3060, 10986}, {3124, 12310}, {3202, 15257}, {3284, 32663}, {3289, 10317}, {3331, 10540}, {3767, 12118}, {4558, 8552}, {4565, 32651}, {5023, 12163}, {5206, 7689}, {5448, 7747}, {5449, 7749}, {5546, 32641}, {14401, 32650}, {23357, 32640}, {23964, 32646}, {32648, 32729} et al

X(32661) = isogonal conjugate of X(14618)
X(32661) = isotomic conjugate of polar conjugate of X(1576)
X(32661) = X(i)-isoconjugate of X(j) for these lines: {i,j}: {1, 14618}, {4, 1577}, {92, 523}, {107, 20902}
X(32661) = trilinear pole of line X(184)X(418) (the tangent at X(577) to the inellipse that is the barycentric square of the Brocard axis)
X(32661) = crossdifference of every pair of points on line X(125)X(136)
X(32661) = crosspoint of [X(371)-isoconjugate of X(1577)] and [X(372)-isoconjugate of X(1577)]
X(32661) = crosssum of polar circle intercepts of Brocard axis
X(32661) = trilinear product of vertices of circumanticevian triangle of X(3)
X(32661) = barycentric product of MacBeath circumconic intercepts of Brocard axis
X(32661) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 4575}, {3, 110}, {5, 15958}, {6, 4558}, {31, 4592}, {32, 4563}, {48, 662}, {49, 930}, {54, 23181}, {58, 1331}, {63, 163}, {69, 1576}, {81, 906}, {86, 32656}, {99, 184}, {100, 1437}, {101, 1790}, {107, 1092}, {109, 283}, {112, 394}, {155, 13398}, {162, 255}, {216, 18315}, {219, 4565}, {222, 5546}, {248, 2421}, {249, 647}, {250, 520}, {284, 1813}, {287, 14966}, {323, 32662}, {476, 22115}, {577, 648}, {651, 2193}, {686, 18879}, {691, 3292}, {692, 1444}, {827, 3917}, {907, 3796}, {925, 1147}, {933, 5562}, {1332, 1333}, {1415, 1812}, {1511, 2425}, {2396, 14600}, {2407, 18877}, {2966, 3289}, {3167, 3565}, {3267, 23963}, {4230, 17974), {5468, 14908}, {6390, 32729}, {10317, 17708}, {10420, 13754}, {11064, 32640}, {15136, 16167}
X(32661) = barycentric quotient X(i)/X(j) for these (i,j): (3, 850), (6, 14618), (31, 24006), (32, 2501), (48, 1577), (63, 20948), (99, 18022), (110, 264), (112, 2052), (163, 92), (184, 523), (216, 18314), (217, 12077), (249, 6331), (250, 6528), (255, 14208), (394, 3267), (476, 18817), (520, 339), (577, 525), (647, 338), (648, 18027), (662, 1969), (906, 321), (930, 20572), (933, 8795), (1092, 3265), (1147, 6563), (1331, 313), (1332, 27801), (1333, 17924), (1437, 693), (1501, 2489), (1576, 4), (1790, 3261), (1813, 349), (2193, 4391), (3289, 2799), (3917, 23285), (4558, 76), (4563, 1502), (4565, 331), (4575, 75), (4592, 561), (5546, 7017), (10317, 9979), (14600, 2395), (14908, 5466), (14966, 297), (15958, 95), (18315, 276), (18877, 2394), (20975, 23105), (22115, 3268), (23181, 311), (23963, 112), (23964, 15352), (32640, 16080), (32656, 10), (32662, 94), (32729, 17983)


X(32662) = BARYCENTRIC PRODUCT X(3)*X(476)

Barycentrics    a^2 (a^2 - b^2 - c^2)/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :
Trilinears    (sin 2A)/((1 + 2 cos 2A) sin(B - C)) : :

X(32662) is the barycentric product of the circumcircle intercepts of line X(3)X(125). As the barycentric product of circumcircle antipodes, X(32662) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32662) lies on these lines: {30, 50}, {32, 23967}, {94, 401}, {112, 476}, {187, 5961}, {248, 265}, {297, 18883}, {328, 15013}, {441, 18876}, {512, 1576}, {525, 4558}, {577, 32663}, {647, 32661}, {691, 23969}, {1415, 32678}, {1625, 2623}, {2420, 2433}, {2966, 14592}, {3284, 13754}, {14590, 16077}, {14591, 30510}, {14966, 23968}, {16813, 23582}, {23588, 32650}, {32708, 32711} et al

X(32662) = isogonal conjugate of polar conjugate of X(476)
X(32662) = isotomic conjugate of polar conjugate of X(14560)
X(32662) = X(92)-isoconjugate of X(526)
X(32662) = trilinear pole of line X(184)X(5158)
X(32662) = crossdifference of every pair of points on line X(3258)X(16186)
X(32662) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 476}, {63, 32678}, {69, 14560}, {94, 32661}, {110, 265}, {249, 14582}, {328, 1576}, {925, 5961}, {1141, 23181}, {1989, 4558}, {2166, 4575}, {2410, 32663}, {8552, 23588}, {14592, 23357}
X(32662) = barycentric quotient X(i)/X(j) for these (i,j): (3, 3268), (48, 32679), (110, 340), (112, 14165), (184, 526), (265, 850), (476, 264), (577, 8552), (1576, 186), (1625, 14918), (1989, 14618), (2420, 14920), (3049, 2088), (3284, 5664), (4558, 7799), (5961, 6563), (8552, 23965), (14560, 4), (14582, 338), (14592, 23962), (14908, 9213), (23181, 1273), (23357, 14590), (32661, 323), (32663, 2411), (32678, 92)


X(32663) = BARYCENTRIC PRODUCT X(3)*X(477)

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

X(32663) is the barycentric product of the circumcircle intercepts of line X(3)X(9033). As the barycentric product of circumcircle antipodes, X(32663) lies on the conic {{A,B,C,X(112),X(248)}} with center X(22391) and perspector X(184).

X(32663) lies on these lines: {6, 32640}, {50, 112}, {248, 14220}, {577, 32662}, {647, 18877}, {841, 32690}, {1495, 1576}, {2966, 30528}, {3284, 32661}, {4558, 11064}, {6587, 14910}, {14586, 18365}

X(32663) = isogonal conjugate of polar conjugate of X(477)
X(32663) = X(92)-isoconjugate of X(5663)
X(32663) = trilinear pole of line X(184)X(9409)
X(32663) = crossdifference of every pair of points on line X(11251)X(18809)
X(32663) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 477}, {110, 14220}, {647, 30528}, {2411, 32662}
X(32663) = barycentric quotient X(i)/X(j) for these (i,j): (477, 264), (14220, 850), (30528, 6331), (32662, 2410)


X(32664) = CENTER OF LOCUS OF BARYCENTRIC PRODUCT OF CIRCUMCIRCLE-X(1)-ANTIPODES

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

The locus of the barycentric product of circumcircle-X(1)-antipodes is a circumconic that is also the locus of the trilinear product of circumcircle-X(6)-antipodes, and the locus of trilinear poles of lines passing through X(31). The conic is the isogonal conjugate of line X(514)X(661), and passes through X(101), X(163), X(662), X(693), X(909), X(913), X(923), X(1415), X(1438), X(1461), X(1910), X(2159), X(2576), X(2577), X(4586), X(4593), X(8632), X(9456), X(18268), X(24019), X(32665), X(32666), X(32667), X(32668), X(32669), X(32670), X(32671), X(32672), X(32673), X(32674), X(32675), X(32676), X(32677), X(32678). The perspector of this conic is X(31).

X(32664) lies on these lines: {2, 7357}, {31, 1501}, {41, 6186}, {48, 354}, {101, 3681}, {184, 2225}, {379, 2140}, {561, 4586}, {572, 11220}, {604, 1401}, {748, 2112}, {1495, 16588}, {3185, 32739} et al

X(32664) = complement of X(7357)
X(32664) = perspector of circumconic centered at X(31)
X(32664) = X(2)-Ceva conjugate of X(31)


X(32665) = BARYCENTRIC PRODUCT X(100)*X(106)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32665) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32665) lies on these lines: {36, 909}, {41, 16944}, {88, 2224}, {100, 32686}, {101, 649}, {106, 919}, {662, 1019}, {667, 692}, {813, 6551}, {859, 3285}, {913, 8752}, {1262, 1461}, {1320, 5011}, {1415, 2149}, {1783, 32705}, {7115, 32674}, {23981, 32675} et al

X(32665) = isogonal conjugate of X(3762)
X(32665) = trilinear pole of line X(31)X(692)
X(32665) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 901}, {6, 3257}, {31, 4555}, {75, 32719}, {88, 101}, {100, 106}, {109, 1320}, {110, 4674}, {163, 4080}, {244, 6551}, {649, 5376}, {692, 903}, {1332, 8752}, {1415, 4997}, {6335, 32659}
X(32665) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3762), (31, 900), (32, 1635), (41, 1639), (88, 3261), (100, 3264), (101, 4358), (106, 693), (163, 16704), (667, 1647), (692, 519), (901, 75), (1415, 3911), (3257, 76), (4080, 20948), (4555, 561), (4674, 850), (5376, 1978), (6551, 7035), (8752, 17924), (16944, 4453), (32659, 905), (32719, 1)


X(32666) = BARYCENTRIC PRODUCT X(101)*X(105)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32666) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32666) lies on these lines: {101, 663}, {105, 2224}, {662, 3737}, {666, 4586}, {667, 1415}, {692, 3063}, {884, 32675}, {909, 2195}, {911, 2223}, {1416, 1457}, {1438, 2210}, {1461, 24027}, {8750, 32703}, {8751, 32699}, {9456, 32719}, {32644, 32656}, {32724, 32739}

X(32666) = trilinear pole of line X(31)X(9447)
X(32666) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 919}, {31, 666}, {100, 1438}, {101, 105}, {109, 294}, {110, 18785}, {163, 13576}, {644, 1416}, {649, 5377}, {651, 2195}, {673, 692}, {884, 4564}, {1415, 14942}, {1461, 28071}, {2481, 32739}, {24027, 28132}
X(32666) = barycentric quotient X(i)/X(j) for these (i,j): (31, 918), (101, 3263), (105, 3261), (109, 27818), (110, 18157), (163, 30941), (663, 693), (666, 561), (692, 3912), (884, 4858), (919, 75), (1415, 9436), (1416, 24002), (2195, 4391), (5377, 1978), (13576, 20948), (18785, 850), (32656, 25083), (32739, 518)


X(32667) = BARYCENTRIC PRODUCT X(102)*X(108)

Barycentrics    a^3/((b - c) (a - b - c) (a^2 - b^2 - c^2) (2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2)) : :
Barycentrics    a^3/((sec B - sec C) (sin B (sec A - sec B) + sin C (sec A - sec C))) : :

As the barycentric product of circumcircle-X(1)-antipodes, X(32667) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32667) lies on these lines: {101, 7115}, {102, 32688}, {108, 32683}, {909, 32702}, {1415, 32643}, {1457, 32647}, {2432, 2443}, {23985, 32674}

X(32667) = barycentric product X(i)*X(j) for these lines: {i,j}: {92, 32643}, {102, 108}, {2432, 7128}
X(32667) = barycentric quotient X(i)/X(j) for these (i,j): (32643, 63)


X(32668) = BARYCENTRIC PRODUCT X(103)*X(934)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32668) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32668) lies on these lines: {101, 1262}, {692, 24027}, {934, 32684}, {1019, 24019}, {1461, 23971}, {6614, 32674)}, {32701, 32714}

X(32668) = trilinear pole of line X(31)X(7099)
X(32668) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 24016}, {103, 934}, {1815, 32714}, {2400, 24027}
X(32668) = barycentric quotient X(i)/X(j) for these (i,j): (103, 4397), (1815, 15416), (23971, 24015), (24016, 75), (24027, 2398)


X(32669) = BARYCENTRIC PRODUCT X(104)*X(109)

Barycentrics    a^3/((b - c) (a - b - c) (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)) : :
Barycentrics    a^3/((cos B - cos C) (1 - cos B - cos C)) : :

As the barycentric product of circumcircle-X(1)-antipodes, X(32669) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32669) lies on these lines: {101, 652}, {109, 32685}, {649, 32674}, {692, 1946}, {909, 1404}, {1415, 23979}, {2423, 2425}, {7113, 14578}

X(32669) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 2720}, {63, 32702}, {104, 109}, {108, 1795}, {651, 909}, {653, 14578}, {2423, 4564}
X(32669) = barycentric quotient X(i)/X(j) for these (i,j): (109, 3262), (692, 6735), (909, 4391), (1415, 908), (2423, 4858), (2720, 75), (14578, 6332), (23979, 24029), (32702, 92)


X(32670) = BARYCENTRIC PRODUCT X(107)*X(26701)

Barycentrics    a^2/((b + c) (b^2 - c^2) (a^2 - b^2 - c^2)^2 (a^6 - a^4 (2 b^2 - 3 b c + 2 c^2) + a^2 (b - c)^2 (b^2 + c^2) - b c (b^2 - c^2)^2)) : :
Barycentrics    (a tan A)/((tan B - tan C) ((b + c) tan A - b tan B - c tan C)) : :

As the barycentric product of circumcircle-X(1)-antipodes, X(32670) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32670) lies on these lines: {163, 23964}, {662, 23582}, {2159, 32695}, {23590, 24019}, {26701, 32687}

X(32670) = barycentric product X(i)*X(j) for these lines: {i,j}: {107, 26701}
X(32670) = barycentric quotient X(i)/X(j) for these (i,j): (26701, 3265)


X(32671) = BARYCENTRIC PRODUCT X(110)*X(759)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32671) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32671) lies on these lines: {163, 23357}, {249, 662}, {759, 1910}, {923, 32729}, {2159, 32640}, {9274, 99500}, {23964,24019}

X(32671) = trilinear pole of line X(31)X(1576)
X(32671) = barycentric product X(i)*X(j) for these lines: {i,j}: {110, 759}, {163, 24624}, {1576, 14616}
X(32671) = barycentric quotient X(i)/X(j) for these (i,j): (163, 3936), (759, 850), (1576, 758), (23357, 4585), (24624, 20948)


X(32672) = BARYCENTRIC PRODUCT X(111)*X(8691)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32672) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32672) lies on these lines: {163, 32729}, {662, 691}


X(32673) = BARYCENTRIC PRODUCT X(112)*X(26702)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32673) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32673) lies on these lines: {250, 662}, {1910, 32696}, {2159, 32715}, {10423, 26702}


X(32674) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(4)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2)) : :
Barycentrics    a^3/(sec B - sec C) : :

As the barycentric product of circumcircle-X(1)-antipodes, X(32674) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the barycentric product of circumcircle-X(4)-antipodes, X(32674) also lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32674) lies on these lines: {1, 20613}, {6, 3209}, {19, 604}, {34, 1438}, {41, 208}, {48, 2331}, {56, 607}, {65, 2332}, {101, 108}, {109, 112}, {196, 2301}, {204, 2187}, {207, 7114}, {221, 3172}, {225, 1910}, {232, 17966}, {278, 2224}, {281, 3476}, {478, 21148}, {608, 8752}, {648, 653}, {649, 32669}, {692, 2498}, {906, 23981}, {923, 8753}, {1033, 3197}, {1249, 1630}, {1319, 5089}, {1400, 2159}, {1402, 1945}, {1411, 1474}, {1415, 2425}, {1420, 7719}, {1429, 5236}, {1461, 32714}, {1767, 7125}, {1826, 7120}, {1877, 2201}, {1886, 2202}, {2172, 21147}, {3192, 17408}, {6589, 32643}, {6591, 32675}, {6614, 32668}, {7115, 32665}, {8687, 32691}, {23985, 32667} et al

X(32674) = isogonal conjugate of X(6332)
X(32674) = polar conjugate of isotomic conjugate of X(109)
X(32674) = crossdifference of every pair of points on line X(2968)X(4082)
X(32674) = trilinear pole of line X(25)X(31) (the isogonal conjugate of the isotomic conjugate of line X(1)X(4))
X(32674) = barycentric product of PU(100)
X(32674) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 108}, {4, 109}, {6, 653}, {7, 8750}, {9, 32714}, {19, 651}, {25, 664}, {27, 4559}, {28, 4551}, {31, 18026}, {33, 934}, {34, 100}, {41, 13149}, {56, 1897}, {57, 1783}, {65, 162}, {73, 107}, {92, 1415}, {101, 278}, {102, 23987}, {104, 23706}, {110, 225}, {112, 226}, {190, 608}, {208, 13138}, {273, 692}, {281, 1461}, {388, 32691}, {514, 7115}, {521, 24033}, {604, 6335}, {607, 658}, {648, 1400}, {811, 1402}, {901, 1877}, {919, 5236}, {1411, 4242}, {1474, 4552}, {1826, 4565}, {1945, 1981}, {1973, 4554}, {3476, 9088}, {4564, 6591}, {6332, 23985}, {6614, 7046}, {17923, 32675}, {23979, 32707}
X(32674) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6332), (19, 4391), (25, 522), (28, 18155), (31, 521), (33, 4397), (34, 693), (56, 4025), (57, 15413), (65, 14208), (73, 3265), (100, 3718), (101, 345), (108, 75), (109, 69), (110, 332), (112, 333), (162, 314), (163, 1812), (208, 17896), (225, 850), (226, 3267), (278, 3261), (604, 905), (607, 3239), (608, 514), (648, 28660), (649, 26932), (651, 304), (653, 76), (664, 305), (692, 78), (906, 3719), (934, 7182), (1400, 525), (1402, 656), (1415, 63), (1461, 348), (1474, 4560), (1783, 312), (1897, 3596), (1973, 650), (3172, 14331), (3209, 14837), (4551, 20336), (4559, 306), (6335, 28659), (6591, 4858), (6614, 7056), (7115, 190), (7120, 17899), (8750, 8), (13149, 20567), (18026, 561), (23706, 3262), (23979, 1813), (23985, 653), (24027, 6516), (24033, 18026), (32676, 21), (32691, 30479), (32714, 85)


X(32675) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(5)

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

As the barycentric product of circumcircle-X(1)-antipodes, X(32675) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the barycentric product of circumcircle-X(5)-antipodes, X(32675) also lies on conic {{A,B,C,X(112),X(1987)}} with perspector X(51).

X(32675) lies on these lines: {101, 650}, {163, 1625}, {649, 1415}, {654, 32641}, {655, 662}, {663, 692}, {884, 32666}, {909, 1319}, {911, 1055}, {913, 14571}, {1404, 1457}, {1411, 1438}, {1429, 2006}, {1461, 3669}, {2159, 19297}, {2183, 32677}, {2423, 2425}, {17923, 32674}, {23981, 32665} et al

X(32675) = isogonal conjugate of X(3904)
X(32675) = trilinear pole of line X(31)X(51) (the isogonal conjugate of the isotomic conjugate of line X(1)X(5))
X(32675) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 2222}, {6, 655}, {80, 109}, {100, 1411}, {101, 2006}, {108, 1807}, {654, 23592}, {692, 18815}, {1415, 18359}, {14628, 32665}
X(32675) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3904), (109, 320), (655, 76), (692, 4511), (1411, 693), (1415, 3218), (2006, 3261), (2222, 75), (32674, 17923)


X(32676) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(19)

Barycentrics    a^3/((b^2 - c^2) (a^2 - b^2 - c^2)) : :
Trilinears    a^2/(sin 2B - sin 2C) : :

As the barycentric product of circumcircle-X(1)-antipodes, X(32676) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the barycentric product of circumcircle-X(19)-antipodes, X(32676) also lies on conic {{A,B,C,X(162),X(8750)}} with perspector X(1973).

X(32676) lies on these lines: {19, 560}, {28, 2224}, {31, 2159}, {101, 112}, {107, 15440}, {110, 32691}, {162, 662}, {648, 4586}, {692, 2873}, {811, 4593}, {823, 24000}, {909, 2206}, {911, 1333}, {923, 1973}, {1086, 14119}, {1415, 1576}, {1438, 1474}, {1582, 20883}, {1974, 18268}, {2194, 2288}, {2203, 9456} et al

X(32676) = isogonal conjugate of X(14208)
X(32676) = polar conjugate of X(20948)
X(32676) = X(i)-isoconjugate of X(j) for these lines: {i,j}: {1, 14208}, {2, 525}, {4, 3265}, {6, 3267}, {48, 20948}, {690, 30786}, {3566, 6340}
X(32676) = trilinear pole of line X(31)X(1932)
X(32676) = crossdifference of every pair of points on line X(3708)X(4466)
X(32676) = barycentric product of PU(108)
X(32676) = trilinear product of circumcircle intercepts of Moses radical circle
X(32676) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 112}, {3, 24019}, {4, 163}, {6, 162}, {19, 110}, {21, 32674}, {25, 662}, {27, 692}, {28, 101}, {29, 1415}, {31, 648}, {32, 811}, {33, 4565}, {34, 5546}, {48, 107}, {58, 1783}, {63, 32713}, {92, 1576}, {99, 1973}, {100, 1474}, {108, 284}, {109, 1172}, {184, 823}, {190, 2203}, {240, 2715}, {286, 32739}, {560, 6331}, {647, 24000}, {649, 5379}, {651, 2299}, {653, 2194}, {799, 1974}, {909, 4246}, {911, 4241}, {923, 4235}, {1333, 1897}, {1461, 4183}, {1969, 14574}, {2159, 4240}, {2206, 6335}, {2224, 4249}, {2303, 32691}, {4593, 27369}, {4630, 20883}
X(32676) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3267), (4, 20948), (6, 14208), (19, 850), (25, 1577), (28, 3261), (31, 525), (32, 656), (48, 3265), (58, 15413), (101, 20336), (107, 1969), (108, 349), (109, 1231), (110, 304), (112, 75), (162, 76), (163, 69), (184, 24018), (560, 647), (648, 561), (662, 305), (669, 3708), (692, 306), (811, 1502), (823, 18022), (923, 14977), (1333, 4025), (1415, 307), (1474, 693), (1576, 63), (1783, 313), (1897, 27801), (1973, 523), (1974, 661), (2194, 6332), (2203, 514), (2206, 905), (2299, 4391), (2715, 336), (4565, 7182), (5379, 1978), (5546, 3718), (6331, 1928), (8750, 321), (9780, 4466), (14574, 48), (24000, 6331), (24019, 264), (27369, 8061), (32674, 1441), (32713, 92), (32739, 72)


X(32677) = BARYCENTRIC PRODUCT X(1)*X(102)

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

X(32677) is the barycentric product of the circumcircle intercepts of line X(1)X(521). As the barycentric product of circumcircle-X(1)-antipodes, X(32677) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32677) lies on these lines: {6, 3209}, {25, 2192}, {48, 1415}, {101, 102}, {163, 2193}, {197, 2875}, {212, 692}, {222, 1461}, {654, 909}, {662, 1812}, {1172, 1905}, {2182, 6001}, {2183, 32675}, {2194, 2288}, {2431, 8677}, {7113, 14578}

X(32677) = X(2)-isoconjugate of X(515)
X(32677) = trilinear pole of line X(31)X(1946)
X(32677) = trilinear product X(6)*X(102)
X(32677) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 102}, {57, 15629}, {651, 2432}, {1415, 2399}
X(32677) = barycentric quotient X(i)/X(j) for these (i,j): (31, 515), (102, 75), (1415, 2406), (2432, 4391), (15629, 312)


X(32678) = BARYCENTRIC PRODUCT X(1)*X(476)

Barycentrics    a/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :
Trilinears    a/((1 + 2 cos 2A) sin(B - C)) : :

X(32678) is the barycentric product of the circumcircle intercepts of line X(1)X(564). As the barycentric product of circumcircle-X(1)-antipodes, X(32678) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(32678) lies on these lines: {101, 476}, {163, 661}, {662, 1577}, {923, 8772}, {1415, 32662}, {1725, 2159}, {1910, 2166}, {9455, 11076}, {11060, 18268}

X(32678) = isogonal conjugate of X(32679)
X(32678) = trilinear pole of line X(31)X(2153)
X(32678) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 476}, {6, 32680}, {75, 14560}, {92, 32662}, {94, 163}, {110, 2166}, {162, 265}, {328, 32676}, {662, 1989}, {799, 11060}
X(32678) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3268), (6, 32679), (48, 8552), (94, 20948), (162, 340), (163, 323), (265, 14208), (476, 75), (662, 7799), (923, 9213), (1989, 1577), (2166, 850), (11060, 661), (14560, 1), (32680, 76), (32662, 63), (32676, 186)
X(32787) = {X(2),X(6)}-harmonic conjugate of X(32788)


X(32679) = ISOGONAL CONJUGATE OF X(32678)

Barycentrics    a (b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) : :
Barycentrics    (1 + 2 cos 2A) sin(B - C) : :
Trilinears    directed distance from A to Fermat axis : :

X(32679) lies on these lines: {163, 662}, {323, 14838}, {442, 6366}, {514, 661}, {2349, 2631}, {2573, 19178}, {2610, 3960}, {4707, 5664}, {7202, 8287}, {8773, 23894}, {10015, 17056}, {16559, 16563}

X(32679) = isogonal conjugate of X(32678)
X(32679) = isotomic conjugate of X(32680)
X(32679) = crossdifference of every pair of points on line X(31)X(2153)
X(32679) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 3268}, {50, 20948}, {75, 526}, {92, 8552}, {186, 14208}, {323, 1577}, {340, 656}, {561, 14270}, {661, 7799}, {799, 2088}, {850, 6149}, {2349, 5664}, {3219, 4707}, {3936, 14838}, {3960, 3969}, {4585, 8287}, {14590, 20902}
X(32679) = barycentric quotient X(i)/X(j) for these (i,j): (1, 476), (6, 32678), (31, 14560), (48, 32662), (50, 163), (186, 162), (323, 662), (340, 811), (526, 1), (656, 265), (661, 1989), (1577, 94), (2088, 661), (2610, 8818), (3268, 75), (3936, 15455), (4707, 30690), (5664, 14206), (6149, 110), (7799, 799), (8552, 63), (14208, 328), (14270, 31), (14838, 24624), (20902, 14592), (20948, 20573), (21828, 2160)


X(32680) = TRILINEAR PRODUCT X(2)*X(476)

Barycentrics    b c/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :
Barycentrics    csc(B - C)/(1 + 2 cos 2A) : :

X(32680) is the trilinear product of the circumcircle intercepts of line X(2)X(94) (the isotomic conjugate of the isogonal conjugate of the Fermat axis). As the trilinear product of circumcircle-X(2)-antipodes, X(32680) lies on conic {{A,B,C,X(100),X(162)}} with center X(9) and perspector X(1). X(32680) is also the barycentric product of the circumcircle intercepts of line X(75)X(2166).

Let H be the rectangular hyperbola passing through X(1), X(5), X(30) and the excenters (centered at X(476)); then X(32680) is the perspector of ABC and the tangential triangle, wrt excentral triangle, of H.

X(32680) lies on these lines: {94, 24624}, {100, 476}, {162, 24006}, {662, 1577}, {799, 20948}, {897, 1733}, {1101, 2618}, {1492, 14560}, {1821, 16568}, {2349, 14206}, {5802, 10991}

X(32680) = isogonal conjugate of X(2624)
X(32680) = isotomic conjugate of X(32679)
X(32680) = X(6)-isoconjugate of X(526)
X(32680) = trilinear pole of line X(1)X(564)
X(32680) = trilinear product X(i)*X(j) for these lines: {i,j}: {2, 476}, {75, 32678}, {76, 14560}, {94, 110}, {99, 1989}, {112, 328}, {250, 14592}, {662, 2166}, {759, 15455}, {1576, 20573}, {3268, 23588}, {6742, 24624}
X(32680) = trilinear quotient X(i)/X(j) for these (i,j): (2, 526), (6, 14270), (75, 32679), (76, 3268), (94, 523), (99, 323), (110, 50), (250, 14591), (329, 3267), (476, 6), (526, 18334), (662, 6149), (1989, 512), (2166, 661), (6742, 2245), (14560, 32), (14592, 125), (15455, 758), (20573, 850), (23588, 14560), (24624, 2605), (32678, 31)
X(32680) = barycentric product X(i)*X(j) for these lines: {i,j}: {75, 476}, {76, 32678}, {94, 662}, {99, 2166}, {162, 328}, {163, 20573}, {561, 14560}, {799, 1989}, {6742, 14616}, {15455, 24624}
X(32680) = barycentric quotient X(i)/X(j) for these (i,j): (1, 526), (31, 14270), (63, 8552), (75, 3268), (94, 1577), (110, 6149), (162, 186), (328, 14208), (476, 1), (662, 323), (759, 2605), (799, 7799), (897, 9213), (1989, 661), (2166, 523), (3268, 32679), (6742, 758), (14206, 5664), (14560, 31), (14616, 4467), (15455, 3936), (20573, 20948), (24624, 14838), (32678, 6)


X(32681) = BARYCENTRIC PRODUCT X(74)*X(1302)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32681) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(30) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32681).

X(32681) lies on the circumcircle and these lines: {6, 841}, {74, 3003}, {110, 32640}, {1300, 1990}, {1302, 9209}, {2420, 10420}, {2433, 9060} et al

X(32681) = Λ(PU(168))
X(32681) = Λ(X(1637), X(5664))
X(32681) = Λ(X(8552), X(14566))
X(32681) = Ψ(X(2), X(74))
X(32681) = barycentric product X(74)*X(1302)
X(32681) = barycentric quotient X(i)/X(j) for these (i,j): (74, 30474), (1302, 3260), (32640, 15066)


X(32682) = BARYCENTRIC PRODUCT X(101)*X(675)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32682) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(31) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32682).

X(32682) lies on the circumcircle and these lines: {98, 17734}, {99, 4570}, {100, 1110}, {101, 6586}, {102, 13329}, {103, 32642}, {105, 2224}, {106, 32719}, {675, 3011}, {759, 5009}, {789, 5384}, {813, 1983}, {840, 4257}, {917, 32699}, {934, 24027}, {953, 995}, {1311, 1737} et al

X(32682) = isogonal conjugate of ideal point of 1st and 2nd isobarycs of X(101)
X(32682) = Ψ(X(2), X(101))
X(32682) = barycentric product X(i)*X(j) for these lines: {i,j}: {100, 2224}, {101, 675}
X(32682) = barycentric quotient X(i)/X(j) for these (i,j): (101, 3006), (2224, 693), (675, 3261)


X(32683) = BARYCENTRIC PRODUCT X(102)*X(9056)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32683) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(652) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32683).

X(32683) lies on the circumcircle and these lines: {102, 8607}, {108, 32667}, {109, 32643}, {8755, 32706}, {26704, 32700}, {26715, 32720}

X(32683) = Ψ(X(2), X(102))


X(32684) = BARYCENTRIC PRODUCT X(103)*X(9057)

Barycentrics    a^2/((b - c) (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) (a^3 (b + c) - a^2 (b^2 + b c + c^2) - a (b^3 + b^2 c + b c^2 + c^3) + b^4 + b^3 c + 4 b^2 c^2 + b c^3 + c^4)) : :
Barycentrics    a^3/(((c - a) cot B - (a - b) cot C) ((b^3 - c^3) cos A + a^2 (b cos B - c cos C) + b c (b - c))) : :

As the barycentric product of circumcircle-X(2)-antipodes, X(32684) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(657) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32684).

X(32684) lies on the circumcircle and these lines: {101, 32642}, {103, 8608}, {917, 1886}, {934, 32668}, {26705, 32701}, {26716, 32721}

X(32684) = Ψ(X(2), X(103))


X(32685) = BARYCENTRIC PRODUCT X(104)*X(9058)

Barycentrics    a^2/((b - c) (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c) (a^3- a^2 (b + c) - a (b - c)^2 + (b + c) (b^2 + c^2))) : :
Barycentrics    a^2/((cos B + cos C - 1) (2 (sin B - sin C) (cos A - 1) + sin 2B + sin 2C)) : :

As the barycentric product of circumcircle-X(2)-antipodes, X(32685) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(650) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32685).

X(32685) lies on the circumcircle and these lines: {100, 32641}, {102, 5053}, {104, 8609}, {108, 32702}, {109, 32669}, {915, 14571}, {998, 2717}, {1983, 2743}, {2427, 6099}, {32722, 32723}

X(32685) = Ψ(X(2), X(104))


X(32686) = BARYCENTRIC PRODUCT X(106)*X(9059)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32686) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(649) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32686).

X(32686) lies on the circumcircle and these lines: {99, 4582}, {100, 32665}, {101, 32719}, {106, 5053}, {110, 5548}, {996, 2726}, {1293, 32645}, {3257, 13396}, {32704, 32705}

X(32686) = Ψ(X(2), X(106))
X(32686) = barycentric product X(i)*X(j) for these lines: {i,j}: {106, 9059}, {901, 996}
X(32686) = barycentric quotient X(i)/X(j) for these (i,j): (901, 4389), (9059, 3264), (32719, 995)


X(32687) = BARYCENTRIC PRODUCT X(107)*X(1297)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32687) lies on the circumcircle.

Let A', B', C' be the intersections of the van Aubel line and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32687).

X(32687) lies on the circumcircle and these lines: {74, 8744}, {98, 6530}, {99, 20580}, {107, 6587}, {110, 23964}, {112, 32649}, {232, 1297}, {393, 2697}, {647, 1301}, {1289, 6529}, {26701, 32670}, {26717, 32725} et al

X(32687) = Λ(X(i), X(j)) for these lines: {i,j}: {2, 14345}, {287, 2419}, {648, 2404}, {3269, 15526}
X(32687) = Ψ(X(2), X(107))
X(32687) = Moses-radical-circle-inverse of X(1301)
X(32687) = polar-circle-inverse of X(33504)
X(32687) = barycentric product X(i)*X(j) for these lines: {i,j}: {107, 1297}, {264, 32649}
X(32687) = barycentric quotient X(i)/X(j) for these (i,j): (1297, 3265), (32649, 3)


X(32688) = BARYCENTRIC PRODUCT X(108)*X(26703)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2) (a^4 (b + c) - 2 a^3 b c - (b - c)^2 (b + c) (b^2 + c^2))) : :
Barycentrics    a^2/((sec B - sec C) (b^2 sec B + c^2 sec C - (b^2 + c^2) sec A)) : :

As the barycentric product of circumcircle-X(2)-antipodes, X(32688) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(19) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32688).

X(32688) lies on the circumcircle and these lines: {100, 7115}, {102, 32667}, {104, 32702}, {108, 6588}, {1295, 32647}, {1415, 26706}, {5089, 26703}, {32726, 32727}

X(32688) = Ψ(X(2), X(108))


X(32689) = BARYCENTRIC PRODUCT X(109)*X(1311)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32689) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(41) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32689).

X(32689) lies on the circumcircle and these lines: {100, 2149}, {102, 32643}, {104, 32669}, {109, 6589}, {2291, 32728}, {3064, 26704}, {8558, 26703}, {32706, 32707}

X(32689) = Ψ(X(2), X(109))


X(32690) = BARYCENTRIC PRODUCT X(477)*X(9060)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32690) lies on the circumcircle.

Let A', B', C' be the intersections of line X(6)X(1637) and lines BC, CA, AB, resp. The circumcircles of triangles AB'C', BC'A', CA'B' concur in X(32690).

X(32690) lies on the circumcircle and these lines: {99, 30528}, {476, 32650}, {477, 3018}, {841, 32663}, {1304, 32712}, {32732, 32733}

X(32690) = Ψ(X(2), X(477))


X(32691) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(19)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32691) lies on the circumcircle, and as the barycentric product of circumcircle-X(19)-antipodes, X(32691) also lies on conic {{A,B,C,X(162),X(8750)}} with perspector X(1973).

Let P be a point of the line X(2)X(19), other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (i.e., A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(32691).

X(32691) lies on the circumcircle and these lines: {4, 5515}, {25, 28476}, {74, 1245}, {99, 162}, {100, 8750}, {102, 1036}, {103, 2221}, {104, 1039}, {105, 5338}, {110, 32676}, {595, 11383}, {689, 32691}, {759, 17520}, {833, 4242}, {1897, 8707}, {2249, 2281}, {2339, 26703}, {2370, 11319}, {8687, 32674}.

X(32691) = isogonal conjugate of X(23874)
X(32691) = trilinear pole of line X(6)X(1245)
X(32691) = inverse-in-polar-circle of X(5515)
X(32691) = cevapoint of X(8646) and X(21750)
X(32691) = Ψ(X(i), X(j)) for these (i,j): (1, 25), (2, 19), (3, 31), (6, 1245), (63, 6), (69, 1), (75, 4), (1010, 2)
X(32691) = X(834)-cross conjugate of X(4)
X(32691) = trilinear product of circumcircle intercepts of line X(1)X(25)
X(32691) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23874}, {2, 2522}, {3, 2517}, {63, 6590}, {69, 8678}, {100, 26933}, {304, 2484}, {305, 8646}, {388, 521}, {514, 5227}, {522, 1038}, {525, 2303}, {612, 4025}, {649, 19799}, {656, 1010}, {693, 7085}, {905, 2345}, {1019, 3610}, {1459, 4385}, {2285, 6332}, {2286, 4391}, {3265, 4206}, {4131, 7102}, {7004, 14594}, {8898, 15411}
X(32691) = barycentric product X(i)*X(j) for these lines: {i,j}: {19, 1310}, {108, 2339}, {648, 1245}, {651, 1039}, {653, 1036}, {811, 2281}, {1897, 2221}, {30479, 32674}
X(32691) = barycentric quotient X(i)/X(j) for these (i,j): (19, 2517), (25, 6590), (31, 2522), (100, 19799), (112, 1010), (1036, 6332), (1039, 4391), (1245, 525), (1310, 304), (1415, 1038), (2221, 4025), (2281, 656), (8750, 2345), (32674, 388), (32676, 2303)


X(32692) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(54)

Barycentrics    a^2/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2)) : :
Barycentrics    a^2/(tan 2B - tan 2C) : :
Trilinears    (sin A)/(sin 4B - sin 4C) : :
Trilinears    sin A sec 2A csc(2B - 2C) : :

As the barycentric product of circumcircle-X(2)-antipodes, X(32692) lies on the circumcircle.

Let P be a point of the line X(2)X(54), other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (i.e., A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(32692).

X(32692) lies on the circumcircle and these lines: {4, 139}, {6, 2383}, {54, 3563}, {74, 15664}, {96, 98}, {99, 18315}, {110, 14586}, {112, 32734}, {759, 2168}, {847, 14585}, {925, 32661}, {1141, 2165}, {1297, 10625}, {1299, 8882}, {1300, 14533}, {1971, 5962}, {2623, 10420} et al

X(32692) = trilinear pole of line X(6)X(2351) (the tangent to hyperbola {A,B,C,X(5),X(6)} at X(6))
X(32692) = inverse-in-polar-circle of X(139)
X(32692) = Ψ(X(i), X(j)) for these (i,j): (2, 54), (5, 6), (6, 2351), (51, 31), (76, 95), (317, 2)
X(32692) = Λ(trilinear polar of isotomic conjugate of X(96))
X(32692) = X(52)-isoconjugate of X(1577)
X(32692) = barycentric product X(i)*X(j) for these lines: {i,j}: {54, 925}, {68, 933}, {95, 32692}, {96, 110}, {662, 2168}, {847, 15958}, {2165, 18315}, {2351, 18831}, {5392, 14586}, {14533, 30450}
X(32692) = barycentric quotient X(i)/X(j) for these (i,j): (54, 6563), (96, 850), (925, 311), (933, 317), (1576, 52), (2165, 18314), (2168, 1577), (2351, 6368), (5392, 15415), (14586, 1993), (15958, 9723), (18315, 7763), (32692, 5)


X(32693) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(65)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 + a b + a c + 2 b c)) : :
Trilinears    a/(b (cos C + cos A) - c (cos A + cos B)) : :

As the barycentric product of circumcircle-X(2)-antipodes, X(32693) lies on the circumcircle, and as the barycentric product of circumcircle-X(65)-antipodes, X(32693) also lies on conic {{A,B,C,X(651),X(1415)}} with perspector X(1402).

Let P be a point of the line X(2)X(65), other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (i.e., A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(32693).

X(32693) lies on the circumcircle and these lines: {99, 651}, {100, 4559}, {104, 941}, {105, 959}, {110, 1415}, {644, 8707}, {692, 8687}, {1311, 31359}, {1783, 26704}, {2258, 2291}, {2284, 6574} et al

X(32693) = trilinear pole of line X(6)X(1402)
X(32693) = Ψ(X(i), X(j)) for these (i,j): (2, 65), (6, 1402), (21, 6), (314, 2)
X(32693) = Λ(X(i), X(j)) for these lines: {i,j}: {650, 3975}, {693, 3669}, {905, 1577}
X(32693) = Λ(polar of X(10435) wrt Conway circle)
X(32693) = barycentric product X(i)*X(j) for these lines: {i,j}: {65, 931}, {100, 959}, {109, 31359}, {651, 941}, {664, 2258}
X(32693) = barycentric quotient X(i)/X(j) for these (i,j): (101, 11679), (109, 10436), (931, 314), (941, 4391), (959, 693), (1415, 940), (2258, 522)


X(32694) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(353)

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

As the barycentric product of circumcircle-X(2)-antipodes, X(32694) lies on the circumcircle.

X(32694) lies on the circumcircle and these lines: {6, 9831}, {32, 843}, {74, 2080}, {98, 8859}, {99, 3906}, {110, 17414}, {111, 1691}, {182, 842}, {187, 6323}, {249, 12074}, {511, 14388}, {512, 11636}, {805, 5467}, {1296, 9218}, {1297, 11416}, {2698, 11842}, {2710, 8722}, {2770, 9169} et al

X(32694) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(9208)
X(32694) = inverse-in-Schoute-circle of X(6323)
X(32694) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(111)
X(32694) = Ψ(X(2), X(353))
X(32694) = Λ(X(i), X(j)) for these lines: {i,j}: {76, 9180}, {98, 14388}, {99, 5467}, {115, 17416}, {262, 14223}
X(32694) = barycentric quotient X(i)/X(j) for these (i,j): (32, 9208), (110, 7840)


X(32695) = BARYCENTRIC PRODUCT X(74)*X(107)

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2)^2 (2 a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2)) : :
Trilinears    (tan A)/((tan B - tan C) (cos A - 2 cos B cos C)) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32695) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32695) lies on these lines: {74, 8744}, {112, 647}, {512, 32713}, {525, 648}, {878, 32696}, {1552, 16318}, {1990, 15311}, {2159, 32670}, {2433, 2442}, {2501, 6529} et al

X(32695) = polar conjugate of isotomic conjugate of X(1304)
X(32695) = trilinear pole of line X(25)X(8749) (the isogonal conjugate of the isotomic conjugate of line X(4)X(74))
X(32695) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 1304}, {6, 15459}, {25, 16077}, {74, 107}, {112, 16080}, {264, 32715}, {648, 8749}, {823, 2159}, {1494, 32713}, {2433, 23582}, {6529, 14919}
X(32695) = barycentric quotient X(i)/X(j) for these (i,j): (25, 9033), (32, 1636), (74, 3265), (107, 3260), (112, 11064), (512, 1650), (1304, 69), (2159, 24018), (2433, 15526), (8749, 525), (14919, 4143), (15459, 76), (16077, 305), (16080, 3267), (32713, 30), (32715, 3)


X(32696) = BARYCENTRIC PRODUCT X(98)*X(112)

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2)) : :
Barycentrics    sin^2 A tan A sec(A + ω) csc(B - C) : :
Barycentrics    (sin^3 A)/(sec B sin^3 C - sec C sin^3 B) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32696) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32696) lies on these lines: {98, 403}, {112, 512}, {232, 248}, {250, 523}, {351, 1304}, {378, 14355}, {460, 6531}, {669, 23964}, {827, 22456}, {878, 32695}, {933, 6037}, {1783, 4705}, {1910, 32673}, {1976, 8749}, {2386, 11610}, {2422, 2445}, {2489, 32713}, {4079, 8750}, {4230, 32697}, {6529, 20031} et al

X(32696) = isogonal conjugate of X(6333)
X(32696) = polar conjugate of isotomic conjugate of X(2715)
X(32696) = trilinear pole of line X(25)X(1501) (the isogonal conjugate of the isotomic conjugate of line X(4)X(32))
X(32696) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 20031}, {4, 2715}, {6, 685}, {25, 2966}, {32, 22456}, {98, 112}, {107, 248}, {110, 6531}, {162, 1910}, {250, 2395}, {287, 32713}, {648, 1976}, {878, 23582}, {879, 23964}, {1289, 11610}, {2207, 17932}, {2422, 18020}, {2445, 9476}, {6037, 10311}, {6331, 14601}, {6528, 14600}, {6529, 17974}
X(32696) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6333), (25, 2799), (32, 684), (98, 3267), (112, 325), (248, 3265), (250, 2396), (685, 76), (878, 15526), (1910, 14208), (1974, 3569), (1976, 525), (2207, 16230), (2395, 339), (2422, 125), (2445, 15595), (2489, 868), (2715, 69), (2966, 305), (6531, 850), (14600, 520), (14601, 647), (17974, 4143), (20031, 264), (22456, 1502), (23964, 877), (32713, 297)


X(32697) = BARYCENTRIC PRODUCT X(99)*X(3563)

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2) (2 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2)) : :
Barycentrics    sec A csc(B - C)/(b^2 cos^2 C + c^2 cos^2 B - b c cos A) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32697) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32697) lies on these lines: {112, 249}, {186, 691}, {230, 297}, {250, 7468}, {648, 4590}, {1783, 4567}, {2489, 4558}, {4230, 32696}, {18879, 32708} et al

X(32697) = polar conjugate of isotomic conjugate of X(10425)
X(32697) = trilinear pole of line X(25)X(110) (the isogonal conjugate of the isotomic conjugate of line X(4)X(99))
X(32697) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 10425}, {99, 3563}, {112, 8781}, {162, 8773}, {648, 2987}
X(32697) = barycentric quotient X(i)/X(j) for these (i,j): (110, 3564), (112, 230), (162, 1733), (250, 4226), (2987, 525), (3563, 523), (4230, 114), (8773, 14208), (8781, 3267), (10425, 69)


X(32698) = BARYCENTRIC PRODUCT X(100)*X(915)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32698) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32698) lies on these lines: {112, 6099}, {648, 4567}, {906, 6591}, {913, 8752}, {915, 919}, {1110, 8750}, {1252, 1783}, {1262, 32714}, {2149, 32674}, {8609, 15500} et al

X(32698) = polar conjugate of isotomic conjugate of X(6099)
X(32698) = trilinear pole of line X(25)X(692) (the isogonal conjugate of the isotomic conjugate of line X(4)X(100))
X(32698) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 6099}, {100, 915}, {190, 913}, {1783, 2990}, {6335, 32655}
X(32698) = barycentric quotient X(i)/X(j) for these (i,j): (101, 914), (913, 514), (915, 693), (2990, 15413), (6099, 69), (8750, 1737), (32655, 905)


X(32699) = BARYCENTRIC PRODUCT X(101)*X(917)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32699) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32699) lies on these lines: {648, 4570}, {917, 32682}, {1110, 1783}, {7649, 32656}, {8608, 32642}, {8750, 23990}, {8751, 32666}, {8752, 32719}, {24027, 32714}

X(32699) = barycentric product X(i)*X(j) for these lines: {i,j}: {101, 917}, {2989, 8750}
X(32699) = barycentric quotient X(i)/X(j) for these (i,j): (917, 3261)


X(32700) = BARYCENTRIC PRODUCT X(102)*X(26704)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32700) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32700) lies on these lines: {6589, 32643}, {26704, 32683}


X(32701) = BARYCENTRIC PRODUCT X(103)*X(26705)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32701) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32701) lies on these lines: {1734, 1783}, {1886, 32657}, {6586, 8750}, {6589, 32643}, {26705, 32684}, {32668, 32714}


X(32702) = BARYCENTRIC PRODUCT X(104)*X(108)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32702) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32702) lies on these lines: {104, 32688}, {108, 32685}, {112, 2720}, {648, 4560}, {649, 32669}, {650, 1415}, {663, 8750}, {909, 32667}, {1309, 8687}, {1455, 14571}, {2423, 2443}, {3669, 32714}, {6591, 23985}

X(32702) = polar conjugate of isotomic conjugate of X(2720)
X(32702) = trilinear pole of line X(25)X(1397)
X(32702) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 2720}, {7, 14776}, {56, 1309}, {92, 32669}, {104, 108}, {653, 909}, {1415, 16082}
X(32702) = barycentric quotient X(i)/X(j) for these (i,j): (108, 3262), (909, 6332), (1309, 3596), (2720, 69), (8750, 6735), (14776, 8), (32669, 63)


X(32703) = BARYCENTRIC PRODUCT X(105)*X(26706)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32703) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32703) lies on these lines: {919, 1783}, {5089, 32658}, {8750, 32666}


X(32704) = CIRCUMCIRCLE-X(4)-ANTIPODE OF X(106)

Barycentrics    1/((b - c) (a^2 - b^2 - c^2) (a^2 (b + c) - 3 a b c - (b + c) (b^2 - 3 b c + c^2))) : :

X(32704) lies on the circumcircle and these lines: {3, 2370}, {4, 106}, {25, 9083}, {186, 2758}, {901, 1897}, {972, 30273}, {1311, 15626}, {32686, 32705}

X(32704) = reflection of X(i) in X(j) for these (i,j): (4, 20619), (2370, 3)
X(32704) = isogonal conjugate of X(32475)
X(32704) = circumcircle antipode of X(2370)
X(32704) = trilinear pole of line X(6)X(8756)
X(32704) = inverse-in-polar-circle of X(5510)
X(32704) = Λ(X(3), X(1459))
X(32704) = Ψ(X(i), X(j)) for these (i,j): (3, 519), (6, 8756), (106, 4), (1797, 2)
X(32704) = barycentric product X(3264)*X(32705)
X(32704) = 2nd-circumperp-isogonal conjugate of X(2390)
X(32704) = circumorthic-isogonal conjugate of X(2390) if ABC is acute


X(32705) = BARYCENTRIC PRODUCT X(106)*X(32704)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32705) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32705) lies on these lines: {648, 4591}, {1783, 32665}, {8756, 32659}, {32686, 32704}

X(32705) = barycentric quotient X(32704)/X(3264)


X(32706) = CIRCUMCIRCLE-X(4)-ANTIPODE OF X(109)

Barycentrics    1/((a^2 - b^2 - c^2) (a^4 (b^2 + c^2) - a^3 b c (b + c) - a^2 (b - c)^2 (2 b^2 + 3 b c + 2 c^2) + a b c (b - c)^2 (b + c) + (b^2 - c^2)^2 (b^2 - b c + c^2))) : :
Trilinears    (tan A)/(cos B csc C (cos A - cos B) + cos C csc B (cos A - cos C)) : :

X(32706) lies on the circumcircle and these lines: {3, 21666}, {4, 109}, {24, 26704}, {25, 9056}, {29, 110}, {99, 7436}, {100, 318}, {101, 281}, {108, 158}, {112, 8748}, {186, 2689}, {242, 2728}, {273, 934}, {411, 13397}, {476, 2075}, {925, 4225}, {944, 7040}, {1294, 7454}, {1295, 7456}, {1297, 7441}, {1305, 7420}, {1766, 29014}, {2370, 7457}, {2373, 7439}, {3518, 26709}, {3565, 7415}, {5176, 6099}, {7424, 10420}, {7443, 26703}, {8750, 32719}, {8755, 32683}, {32689, 32707}

X(32706) = reflection of X(4) in X(20620)
X(32706) = trilinear pole of line X(6)X(3064)
X(32706) = inverse-in-polar-circle of X(117)
X(32706) = X(63)-isoconjugate of X(8607)
X(32706) = Λ(X(3), X(73))
X(32706) = Ψ(X(i), X(j)) for these (i,j): (3, 522), (6, 3064), (109, 4), (1813, 2)


X(32707) = BARYCENTRIC PRODUCT X(109)*X(32706)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32707) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32707) lies on these lines: {1783, 2149}, {3064, 32660}, {8607, 32643}, {23979, 32674}, {32689, 32706}


X(32708) = BARYCENTRIC PRODUCT X(110)*X(1300)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32708) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32708) lies on these lines: {112, 6753}, {186, 3003}, {249, 648}, {297, 2986}, {1300, 2715}, {2501, 32661}, {4240, 14591}, {6529, 23964}, {8753, 32729}, {18879, 32697}, {32662, 32711}

X(32708) = isogonal conjugate of X(6334)
X(32708) = polar conjugate of isotomic conjugate of X(10420)
X(32708) = trilinear pole of line X(25)X(1576)
X(32708) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 10420}, {6, 687}, {25, 18878}, {107, 5504}, {110, 1300}, {112, 2986}, {250, 15328}, {648, 14910}, {2501, 18879}, {4240, 10419}, {15421, 23964}
X(32708) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6334), (32, 686), (112, 3580), (687, 76), (1300, 850), (2986, 3267), (5504, 3265), (10420, 69), (14910, 525), (15328, 339), (18878, 305), (18879, 4563), (23964, 16237), (32713, 403)


X(32709) = BARYCENTRIC PRODUCT X(111)*X(30247)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32709) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32709) lies on these lines: {112, 32729}, {468, 10422}, {648, 691}, {8753, 14580}


X(32710) = CIRCUMCIRCLE-X(4)-ANTIPODE OF X(476)

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

X(32710) lies on the circumcircle and these lines: {2, 16167}, {3, 10420}, {4, 476}, {24, 1304}, {25, 9060}, {30, 925}, {74, 924}, {99, 340}, {107, 403}, {110, 186}, {112, 3003}, {250, 1986}, {378, 691}, {468, 1302}, {523, 1300}, {562, 930}, {935, 18533}, {1287, 7576}, {1290, 7414}, {1291, 3520}, {1825, 2222}, {1835, 26700}, {2071, 13398}, {23969, 32711} et al

X(32710) = reflection of X(i) in X(j) for these (i,j): (4, 16221), (10420, 3)
X(32710) = isogonal conjugate of X(17702)
X(32710) = inverse-in-polar-circle of X(25641)
X(32710) = circumcircle antipode of X(10420)
X(32710) = X(63)-isoconjugate of X(3018)
X(32710) = trilinear pole of line X(6)X(686)
X(32710) = Λ(X(3), X(125))
X(32710) = Λ(X(4), X(110))
X(32710) = Ψ(X(i), X(j)) for these (i,j): (3, 526), (6, 686), (476, 4), (687, 2)
X(32710) = barycentric quotient X(i)/X(j) for these (i,j): (25, 3018), (112, 7471)


X(32711) = BARYCENTRIC PRODUCT X(476)*X(32710)

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

As the barycentric product of circumcircle-X(4)-antipodes, X(32711) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32711) lies on these lines: {3018, 32650}, {23969, 32710}, {32662, 32708}

X(32711) = trilinear pole of line X(25)X(14560)


X(32712) = BARYCENTRIC PRODUCT X(477)*X(1304)

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2) (2 a^4 - b^4 - c^4 - a^2 b^2 - a^2 c^2 + 2 b^2 c^2) (a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4))) : :
Trilinears    (sin^2 A)/((tan B - tan C) (2 tan A - tan B - tan C) (4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C))) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32712) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25).

X(32712) lies on these lines: {648, 5664}, {1304, 32690}, {1637, 32640}, {2088, 8749}, {32715, 32733}


X(32713) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF VAN AUBEL LINE

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2)^2) : :
Trilinears    (sin A) (tan A)/(sin 2B - sin 2C) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32713) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25), and as the barycentric product of circumcircle-X(6)-antipodes, X(32713) also lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32713) lies on these lines: {4, 1177}, {25, 8749}, {107, 110}, {112, 1576}, {132, 23583}, {133, 9934}, {159, 15259}, {162, 1633}, {182, 10002}, {184, 6525}, {206, 1249}, {250, 7468}, {393, 1974}, {512, 32695}, {520, 15384}, {685, 15352}, {692, 1783}, {823, 1492}, {1304, 9064}, {1632, 2409}, {1660, 3079}, {1990, 18374}, {2207, 8753}, {2393, 15262}, {2445, 14398}, {2485, 32649}, {2489, 32696}, {4230, 4558}, {4577, 6528}, {5317, 8751}, {6523, 6759}, {6526, 26883}, {6529, 23977}, {8743, 20410}, {8745, 19118}, {10561, 23964}, {14569, 18384}, {20031, 32716}, {23590, 32725}

X(32713) = isogonal conjugate of X(3265)
X(32713) = polar conjugate of X(3267)
X(32713) = pole wrt polar circle of trilinear polar of X(3267) (line X(125)X(339))
X(32713) = trilinear pole of line X(25)X(32) (the isogonal conjugate of the isotomic conjugate of the van Aubel line)
X(32713) = center of bianticevian conic of X(6) and X(25)
X(32713) = X(i)-isoconjugate of X(j) for these lines: {i,j}: {1, 3265}, {2, 24018}, {3, 14208}, {48, 3267}
X(32713) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 24019}, {3, 6529}, {4, 112}, {6, 107}, {19, 162}, {25, 648}, {27, 8750}, {28, 1783}, {30, 32695}, {31, 823}, {32, 6528}, {51, 16813}, {53, 933}, {92, 32676}, {99, 2207}, {100, 5317}, {101, 8747}, {108, 1172}, {109, 8748}, {110, 393}, {158, 163}, {184, 15352}, {232, 685}, {250, 2501}, {297, 32696}, {403, 32708}, {511, 20031}, {512, 23582}, {520, 23590}, {523, 23964}, {662, 1096}, {811, 1973}, {925, 8745}, {1249, 1301}, {1289, 8743}, {1297, 23977}, {1304, 1990}, {1474, 1897}, {1576, 2052}, {1974, 6331}, {2445, 6330}, {2489, 18020}, {4230, 6531}, {4235, 8753}, {4238, 8751}, {4240, 8749}, {4558, 6524}, {6587, 15384}, {14569, 18315}, {14590, 18384}, {15262, 22239}
X(32713) = barycentric quotient X(i)/X(j) for these (i,j): (3, 4143), (4, 3267), (6, 3265), (19, 14208), (25, 525), (28, 15413), (31, 24018), (32, 520), (107, 76), (108, 1231), (110, 3926), (112, 69), (158, 20948), (162, 304), (163, 326), (232, 6333), (250, 4563), (393, 850), (512, 15526), (520, 23974), (648, 305), (692, 3998), (823, 561), (1096, 1577), (1474, 4025), (1576, 394), (1783, 20336), (1973, 656), (1974, 647), (2207, 523), (2445, 441), (2489, 125), (2501, 339), (4230, 6393), (4558, 4176), (5317, 693), (6524, 14618), (6528, 1502), (6529, 264), (8745, 6563), (8747, 3261), (8750, 306), (14398, 1650), (14569, 18314), (15352, 18022), (18384, 14592), (20031, 290), (23582, 670), (23590, 6528), (23964, 99), (23977, 30737), (24019, 75), (32676, 63), (32695, 1494), (32696, 287)


X(32714) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(4)X(7)

Barycentrics    a/((a - b - c)^2 (b - c) (a^2 - b^2 - c^2)) : :
Barycentrics    a^2 (tan A/2)/(sec B - sec C) : :
Barycentrics    a^2 (tan A)/((1 + cos A) (cos B - cos C)) : :
Barycentrics    a^2 tan A sec^2(A/2)/(cos B - cos C) : :

As the barycentric product of circumcircle-X(4)-antipodes, X(32713) lies on conic {{A,B,C,X(112),X(648)}} with center X(3162) and perspector X(25), and as the barycentric product of circumcircle-X(7)-antipodes, X(32713) also lies on conic {{A,B,C,X(109),X(651)}} with center X(478) and perspector X(56).

X(32714) lies on these lines: {4, 9372}, {19, 269}, {28, 1042}, {108, 109}, {112, 934}, {196, 222}, {207, 1394}, {223, 1767}, {240, 5018}, {278, 1086}, {281, 6180}, {513, 24033}, {608, 1119}, {648, 4569}, {651, 653}, {1118, 1406}, {1172, 3668}, {1262, 32698}, {1415, 32651}, {1419, 2331}, {1426, 7316}, {1427, 8749}, {1435, 8752}, {1461, 32674}, {1712, 2956}, {3669, 32702}, {6129, 32652}, {24027, 32699}, {32668, 32701}

X(32714) = isogonal conjugate of complement of X(17896)
X(32714) = isotomic conjugate of X(15416)
X(32714) = trilinear pole of line X(25)X(34)
X(32714) = barycentric product X(i)*X(j) for these lines: {i,j}: {4, 934}, {6, 13149}, {7, 108}, {19, 658}, {25, 4569}, {28, 4566}, {56, 18026}, {57, 653}, {85, 32674}, {92, 1461}, {99, 1426}, {100, 1119}, {101, 1847}, {107, 1439}, {109, 273}, {112, 1446}, {162, 3668}, {190, 1435}, {269, 1897}, {278, 651}, {281, 4617}, {331, 1415}, {608, 4554}, {648, 1427}, {811, 1042}, {1118, 6516}, {1262, 17924}, {4025, 24033}
X(32714) = barycentric quotient X(i)/X(j) for these (i,j): (4, 4397), (19, 3239), (25, 3900), (28, 7253), (56, 521), (57, 6332), (100, 1265), (101, 3692), (108, 8), (109, 78), (110, 1792), (112, 2287), (162, 1043), (269, 4025), (278, 4391), (513, 2968), (608, 650), (651, 345), (653, 312), (658, 304), (934, 69), (1042, 656), (1119, 693), (1262, 1332), (1415, 219), (1426, 523), (1427, 525), (1435, 514), (1439, 3265), (1446, 3267), (1461, 63), (1847, 3261), (1897, 341), (3668, 14208), (3669, 26932), (4566, 20336), (4569, 305), (4617, 348), (6516, 1264), (8059, 271), (13149, 76), (17924, 23978), (18026, 3596), (24027, 1331), (24033, 1897), (32668, 1815), (32674, 9)


X(32715) = BARYCENTRIC PRODUCT X(74)*X(112)

Barycentrics    a^4/((b^2 - c^2) (a^2 - b^2 - c^2) (2 a^4 - a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Barycentrics    a^4/(SA (SB - SC) (SA SB + SA SC - 2 SB SC)) : :
Barycentrics    a^4/((tan B - tan C) (2 tan A - tan B - tan C)) : :
Barycentrics    a^4/((sin 2B - sin 2C) (2 sin 2A - sin 2B - sin 2C)) : :
Barycentrics    a^3/((sin 2B - sin 2C) (cos A - 2 cos B cos C)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32715) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

The trilinear polar of X(32715) passes through X(32) and the barycentric square of X(647).

X(32715) lies on these lines: {74, 1177}, {110, 250}, {112, 32738}, {512, 32695}, {685, 879}, {687, 4240}, {1576, 32640}, {1976, 8749}, {2159, 32673}, {2211, 32740}, {2393, 15291}, {2433, 2445}, {18374, 18877}, {32712, 32733} et al

X(32715) = polar conjugate of isotomic conjugate of X(32640)
X(32715) = barycentric product X(i)*X(j) for these lines: {i,j}: {3, 32695}, {4, 32640}, {6, 1304}, {32, 16077}, {74, 112}, {110, 8749}, {162, 2159}, {250, 2433}, {1301, 15291}, {1576, 16080}
X(32715) = barycentric quotient X(i)/X(j) for these (i,j): (32, 9033), (74, 3267), (112, 3260), (1304, 76), (1576, 11064), (2159, 14208), (2433, 339), (8749, 850), (16077, 1502), (32695, 264), (32640, 69)


X(32716) = BARYCENTRIC PRODUCT X(98)*X(26714)

Barycentrics    a^2/((b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2) (a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2)) : :
Barycentrics    a^3 sec(A + ω)/((b^2 - c^2) sin 2A + b^2 sin 2C - c^2 sin 2B) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32716) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32716) lies on these lines: {110, 2966}, {262, 14355}, {1177, 11653}, {1576, 2715}, {1976, 9418}, {4577, 17932}, {20031, 32713}

X(32716) = trilinear pole of line X(32)X(263)
X(32716) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 6037}, {98, 26714}, {262, 2715}, {263, 2966}
X(32716) = barycentric quotient X(i)/X(j) for these (i,j): (263, 2799), (2715, 183), (2966, 20023), (6037, 76), (26714, 325)


X(32717) = BARYCENTRIC PRODUCT X(99)*X(729)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32717) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32717) lies on these lines: {110, 4590}, {249, 1576}, {691, 729}, {692, 4567}, {886, 4577}, {892, 5027}, {1976, 2966}, {5467, 17938} et al

X(32717) = isogonal conjugate of X(9148)
X(32717) = trilinear pole of line X(32)X(110)
X(32717) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 9150}, {32, 886}, {99, 729}, {110, 3228}, {691, 14608}
X(32717) = barycentric quotient X(i)/X(j) for these (i,j): (6, 9148), (32, 888), (99, 30736), (110, 538), (729, 523), (886, 1502), (1576, 3231), (3228, 850), (9150, 76)


X(32718) = BARYCENTRIC PRODUCT X(100)*X(739)

Barycentrics    a^3/((b - c) (a b + a c - 2 b c)) : :
Barycentrics    a^4/((csc B - csc C) (2 csc A - csc B - csc C)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32718) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32718) lies on these lines: {110, 898}, {692, 1252}, {739, 919}, {889, 4577}, {1110, 32739}, {1492, 4607}, {2210, 32665}

X(32718) = trilinear pole of line X(32)X(692)
X(32718) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 898}, {31, 4607}, {32, 889}, {100, 739}, {692, 3227}, {31002, 32739}
X(32718) = barycentric quotient X(i)/X(j) for these (i,j): (31, 4728), (32, 891), (101, 6381), (692, 536), (739, 693), (889, 1502), (898, 76), (4607, 561), (32739, 899)


X(32719) = BARYCENTRIC PRODUCT X(101)*X(106)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32719) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32719) lies on these lines: {59, 4057}, {101, 32686}, {106, 32682}, {110, 901}, {667, 692}, {898, 6551}, {900, 13136}, {1492, 3257}, {1919, 23990}, {2426, 2441}, {5548, 32736}, {8750, 32705}, {8752, 32699}, {9456, 32666}, {32642, 32659}, {32645, 32656} et al

X(32719) = trilinear pole of line X(32)X(1977)
X(32719) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 32665}, {6, 901}, {31, 3257}, {32, 4555}, {88, 692}, {100, 9456}, {101, 106}, {145, 32645}, {649, 9268}, {667, 5376}, {902, 4638}, {903, 32739}, {995, 32686}, {1015, 6551}, {1331, 8752}, {1797, 8750}, {1897, 32659}, {6336, 32656}, {6635, 23990}
X(32719) = barycentric quotient X(i)/X(j) for these (i,j): (31, 3762), (32, 900), (101, 3264), (106, 3261), (692, 4358), (901, 76), (1919, 1647), (3257, 561), (4555, 1502), (5376, 6386), (6551, 31625), (9268, 1978), (9456, 693), (23990, 6550), (32645, 4373), (32656, 3977), (32659, 4025), (32665, 75), (32739, 519)


X(32720) = BARYCENTRIC PRODUCT X(102)*X(26715)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32720) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32720) lies on the line {26715, 32683}


X(32721) = BARYCENTRIC PRODUCT X(103)*X(26716)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32721) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32721) lies on these lines: {26716, 32684}, {32642, 32739}


X(32722) = CIRCUMCIRCLE-X(6)-ANTIPODE OF X(104)

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

X(32722) lies on the circumcircle and these lines: {6, 104}, {100, 2427}, {103, 4256}, {105, 957}, {187, 2699}, {644, 9059}, {739, 1384}, {1309, 1783}, {1415, 2720}, {2726, 3230}, {2751, 5526}, {32685, 32723} et al

X(32722) = Λ(X(2), X(905))
X(32722) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(650)
X(32722) = circumcircle intercept, other than X(104), of circle {{X(15),X(16),X(104)}} (or V(X(104))
X(32722) = inverse-in-Schoute-circle of X(2699)
X(32722) = barycentric product X(100)*X(957)
X(32722) = barycentric quotient X(957)/X(693)


X(32723) = BARYCENTRIC PRODUCT X(104)*X(32722)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32723) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32723) lies on these lines: {692, 32641}, {32685, 32722}


X(32724) = BARYCENTRIC PRODUCT X(105)*X(8693)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32724) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32724) lies on these lines: {692, 919}, {32666, 32739}

X(32724) = barycentric product X(i)*X(j) for these lines: {i,j}: {105, 8693}, {919, 1002}
X(32724) = barycentric quotient X(i)/X(j) for these (i,j): (919, 4441), (8693, 3263)


X(32725) = BARYCENTRIC PRODUCT X(107)*X(26717)

Barycentrics    a^2/((b^2 - c^2) (a^2 - b^2 - c^2)^2 (a^6 (b^2 + c^2) - 2 a^4 (b^4 - b^2 c^2 + c^4) + a^2 (b^2 - c^2)^2 (b^2 + c^2) - 2 b^2 c^2 (b^2 - c^2)^2)) : :
Barycentrics    a^4/((csc 2B - csc 2C) (2 csc 2A - csc 2B - csc 2C)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32725) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32725) lies on these lines: {110, 23582}, {1576, 23964}, {1976, 20031}, {23590, 32713}, {26717, 32687}

X(32725) = trilinear pole of line X(32)X(32713)
X(32725) = barycentric product X(i)*X(j) for these lines: {i,j}: {107, 26717}
X(32725) = barycentric quotient X(i)/X(j) for these (i,j): (26717, 3265)


X(32726) = CIRCUMCIRCLE-X(6)-ANTIPODE OF X(108)

Barycentrics    a^2/(a^4 (b + c) - a^3 (b^2 + c^2) - a^2 (b - c)^2 (b + c) + a (b^2 - c^2)^2 - 2 b c (b - c)^2 (b + c)) : :
Trilinears    (sin A)/(2 sec A - sec B - sec C) : :

X(32726) lies on the circumcircle and these lines: {6, 108}, {48, 109}, {99, 1812}, {100, 219}, {101, 212}, {107, 1172}, {110, 2193}, {112, 2194}, {187, 2714}, {222, 934}, {927, 1814}, {1055, 1949}, {1295, 2431}, {1951, 14733}, {2720, 14578}, {2722, 13509}, {2730, 5526}, {32688, 32727} et al

X(32726) = trilinear pole of line X(6)X(1946)
X(32726) = Ψ(X(i), X(j)) for these (i,j): (4, 650), (6, 1946), (651, 3), (18026, 2)
X(32726) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(19)
X(32726) = circumcircle intercept, other than X(108), of circle {{X(15),X(16),X(108)}} (or V(X(108))
X(32726) = inverse-in-Schoute-circle of X(2714)
X(32726) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 23707}
X(32726) = barycentric quotient X(i)/X(j) for these (i,j): (23707, 75), (32727, 108)


X(32727) = BARYCENTRIC PRODUCT X(108)*X(32726)

Barycentrics    a^3/((b - c) (a - b - c) (a^2 - b^2 - c^2) (a^4 (b + c) - a^3 (b^2 + c^2) - a^2 (b - c)^2 (b + c) + a (b^2 - c^2)^2 - 2 b c (b - c)^2 (b + c))) : :
Barycentrics    a^4/((sec B - sec C) (2 sec A - sec B - sec C)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32727) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32727) lies on these lines: {692, 7115}, {32688, 32726}


X(32728) = BARYCENTRIC PRODUCT X(109)*X(2291)

Barycentrics    a^4/((b - c) (a - b - c) (a (2 a - b - c) - (b - c)^2)) : :
Barycentrics    a^4/((cos B - cos C) (2 cos A - cos B - cos C)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32728) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32728) lies on these lines: {110, 14733}, {692, 2149}, {884, 32735}, {2291, 32689}

X(32728) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 14733}, {109, 2291}
X(32728) = barycentric quotient X(i)/X(j) for these (i,j): (14733, 76)


X(32729) = BARYCENTRIC PRODUCT X(110)*X(111)

Barycentrics    a^4/((b^2 - c^2) (2 a^2 - b^2 - c^2)) : :
Barycentrics    a^4/((cot B - cot C) (2 cot A - cot B - cot C)) : :
Barycentrics    (SB + SC)^2/((SB - SC) (2 SA - SB - SC)) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32729) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

Let A1B1C1 and A2B2C2 be the 1st and 2nd Parry triangles. Let A' be the barycentric product A1*A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X(32729).

X(32729) lies on these lines: {6, 32741}, {23, 895}, {110, 249}, {111, 1495}, {112, 32709}, {163, 32672}, {184, 10558}, {187, 9215}, {237, 14908}, {669, 1576}, {685, 4240}, {690, 17708}, {881, 17938}, {892, 4577}, {923, 32671}, {2420, 2444}, {2966, 9147}, {5968, 6800}, {8753, 32708}, {9178, 9206}, {9544, 10559}, {10561, 23964}, {11003, 21460}, {14567, 18374}, {14602, 19626}, {32648, 32661}

X(32729) = trilinear pole of line X(32)X(1084)
X(32729) = barycentric product of PU(62)
X(32729) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 691}, {15, 9206}, {16, 9207}, {32, 892}, {110, 111}, {112, 895}, {163, 897}, {249, 9178}, {250, 10097}, {648, 14908}, {662, 923}, {670, 19626}, {671, 1576}, {1992, 32648}, {2420, 9139}, {2715, 5968}, {4558, 8753}, {11634, 15387}, {14574, 18023}, {17983, 32661}
X(32729) = barycentric quotient X(i)/X(j) for these (i,j): (32, 690), (110, 3266), (111, 850), (163, 14210), (184, 14417), (691, 76), (892, 1502), (895, 3267), (897, 20948), (923, 1577), (1576, 524), (8753, 14618), (9178, 338), (9206, 300), (9207, 301), (10097, 339), (14567, 1649), (14574, 187), (14602, 11183), (14908, 525), (19626, 512), (32648, 5485), (32661, 6390)


X(32730) = CIRCUMCIRCLE-X(6)-ANTIPODE OF X(476)

Barycentrics    a^2/(a^6 (b^2 + c^2) - 2 a^4 (b^4 + c^4) + a^2 (b^6 + c^6) - 2 b^2 c^2 (b^2 - c^2)^2) : :
Trilinears    a^2/(csc B sin(A - B) (1 + 2 cos 2C) + csc C sin(A - C) (1 + 2 cos 2B)) : :

X(32730) lies on the circumcircle and these lines: {6, 476}, {50, 110}, {98, 5996}, {99, 323}, {187, 9160}, {325, 9066}, {477, 2436}, {691, 10560}, {805, 15080}, {1141, 2623}, {1384, 32732}, {2222, 21741}, {2715, 9463}, {23969, 32731}

X(32730) = trilinear pole of line X(6)X(14270)
X(32730) = Ψ(X(6), X(14270))
X(32730) = Λ(X(2), X(94))
X(32730) = trilinear pole, wrt circumsymmedial triangle, of Fermat axis
X(32730) = circumcircle intercept, other than X(476), of circle {X(15),X(16),X(476)} (or V(X(476))
X(32730) = barycentric product X(3268)*X(32731)
X(32730) = barycentric quotient X(32731)/X(476)


X(32731) = BARYCENTRIC PRODUCT X(476)*X(32730)

Barycentrics    a^2/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) (a^6 (b^2 + c^2) - 2 a^4 (b^4 + c^4) + a^2 (b^6 + c^6) - 2 b^2 c^2 (b^2 - c^2)^2)) : :
Barycentrics    a^4/(sin(B - C) (1 + 2 cos 2A) (csc B sin(A - B) (1 + 2 cos 2C) + csc C sin(A - C) (1 + 2 cos 2B))) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32731) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32731) lies on these lines: {14560, 23588}, {23969, 32730}

X(32731) = trilinear pole of line X(32)X(14560)
X(32731) = barycentric quotient X(32730)/X(3268)


X(32732) = CIRCUMCIRCLE-X(6)-ANTIPODE OF X(477)

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

X(32732) lies on the circumcircle and these lines: {6, 477}, {98, 12112}, {187, 9161}, {476, 2437}, {1297, 3581}, {1384, 32730}, {32690, 32733}

X(32732) = Λ(X(2), X(2411))
X(32732) = trilinear pole, wrt circumsymmedial triangle, of line X(6)X(1637)
X(32732) = circumcircle intercept, other than X(477), of circle {X(15),X(16),X(477)} (or V(X(477))


X(32733) = BARYCENTRIC PRODUCT X(477)*X(32732)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32733) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32733) lies on these lines: {110, 30528}, {14560, 32650}, {32690, 32732}, {32712, 32715}


X(32734) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(5)X(6)

Barycentrics    a^2/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)) : :
Barycentrics    a^4/(sec 2B - sec 2C) : :
Trilinears    a^2/(b cos(A - B) - c cos(A - C)) : :

As the barycentric product of circumcircle-X(5)-antipodes, X(32734) lies on conic {{A,B,C,X(112),X(1987)}} with perspector X(51), and as the barycentric product of circumcircle-X(6)-antipodes, X(32734) also lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32734) lies on these lines: {68, 542}, {110, 925}, {112, 32692}, {154, 2351}, {159, 2871}, {184, 8754}, {206, 1976}, {485, 30427}, {486, 30428}, {685, 30450}, {847, 1614}, {924, 4558}, {1498, 16391}, {1576, 1625} et al

X(32734) = isogonal conjugate of X(6563)
X(32734) = trilinear pole of line X(32)X(51) (the isogonal conjugate of the isotomic conjugate of line X(5)X(6))
X(32734) = barycentric product X(i)*X(j) for these lines: {i,j}: {5, 32692}, {6, 925}, {68, 112}, {96, 1625}, {184, 30450}, {648, 2351}, {847, 32661}, {1576, 5392}, {4558, 14593}, {6529, 16391}
X(32734) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6563), (68, 3267), (112, 317), (847, 32661), (925, 76), (1576, 1993), (2351, 525), (14593, 14618), (16391, 4143), (30450, 18022), (32692, 95), (32661, 9723)


X(32735) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(7)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32735) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(7)-antipodes, X(32735) also lies on conic {{A,B,C,X(109),X(651)}} with center X(478) and perspector X(56).

X(32735) lies on these lines: {59, 513}, {100, 30626}, {105, 2720}, {109, 649}, {110, 927}, {294, 2182}, {666, 4581}, {667, 1262}, {677, 926}, {883, 3573}, {884, 32728}, {885, 14776}, {971, 28071}, {1404, 1438}, {1428, 1456}, {1458, 2195}, {1461, 24027}, {1576, 3733}, {1814, 3827}, {1876, 15382}, {1960, 14733}, {4617, 7339}, {7316, 32740} et al

X(32735) = trilinear pole of line X(32)X(56) (the isogonal conjugate of the isotomic conjugate of line X(6)X(7))
X(32735) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 927}, {7, 919}, {56, 666}, {100, 1462}, {105, 651}, {108, 1814}, {109, 673}, {294, 934}, {658, 2195}, {664, 1438}, {884, 1275}, {885, 1262}, {1461, 14942}, {4617, 28071}, {7339, 28132}
X(32735) = barycentric quotient X(i)/X(j) for these (i,j): (56, 918), (105, 4391), (109, 3912), (294, 4397), (651, 3263), (666, 3596), (667, 17435), (884, 1146), (885, 23978), (919, 8), (927, 76), (934, 27818), (1262, 883), (1438, 522), (1461, 9436), (1462, 693), (2195, 3239)


X(32736) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(8)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32736) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(8)-antipodes, X(32736) also lies on conic {{A,B,C,X(101),X(651)}} with center X(5452) and perspector X(55).

X(32736) lies on these lines: {101, 8687}, {110, 645}, {294, 2264}, {644, 692}, {666, 4581}, {1169, 2311}, {1576, 4557}, {1976, 14624}, {2308, 2316}, {2338, 2359}, {3939, 32739}, {5547, 32740}, {5548, 32719} et al

X(32736) = isogonal conjugate of X(3004)
X(32736) = trilinear pole of line X(32)X(55) (the isogonal conjugate of the isotomic conjugate of line X(6)X(8))
X(32736) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 8707}, {8, 8687}, {55, 6648}, {100, 2298}, {101, 1220}, {110, 14624}, {644, 961}, {692, 30710}, {1169, 3952}, {1240, 32739}, {1252, 4581}, {1897, 2359}, {4557, 14534}
X(32736) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3004), (55, 3910), (100, 20911), (101, 4357), (110, 16705), (692, 3666), (961, 24002), (1169, 7192), (1220, 3261), (2298, 693), (2359, 4025), (3952, 1228), (4557, 1211), (4581, 23989), (6648, 6063), (8687, 7), (8707, 76), (14624, 850), (32739, 1193)


X(32737) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF NAPOLEON AXIS

Barycentrics    a^2/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2)) : :
Barycentrics    a^3/((3 sin^2 A - cos^2 A)(cos B sin C - sin B cos C)) : :
Barycentrics    a^2/(sin^2 A sin(2B - 2C) - sin^2 B sin(2C - 2A) - sin^2 C sin(2A - 2B)) : :
Trilinears    csc(B - C)/(3 - cot^2 A) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32737) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32).

X(32737) lies on these lines: {110, 930}, {156, 25043}, {184, 11082}, {252, 1614}, {1177, 3519}, {1510, 18315}, {1576, 6140}, {1976, 2963} et al

X(32737) = isogonal conjugate of isotomic conjugate of X(930)
X(32737) = trilinear pole of line X(32)X(8565)
X(32737) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 930}, {17, 16807}, {18, 16806}, {110, 2963}, {112, 3519}, {163, 2962}, {252, 1625}, {1576, 11140}, {14586, 25043}
X(32737) = barycentric quotient X(i)/X(j) for these (i,j): (110, 7769), (930, 76), (1576, 1994), (2962, 20948), (2963, 850), (3519, 3267), (16806, 303), (16807, 302), (25043, 15415)


X(32738) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(30)

Barycentrics    a^2/((b^2 - c^2) (a^4 - 2 a^2 (b^2 + c^2) + b^4 + 4 b^2 c^2 + c^4)) : :
Trilinears    a^2/((cos A) (sin 2B - sin 2C) + cos B sin C - sin B cos C) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32738) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(30)-antipodes, X(32738) also lies on conic {{A,B,C,X(6),X(112)}} with perspector X(1495).

X(32738) lies on these lines: {110, 1302}, {112, 32715}, {182, 1177}, {1576, 2420}, {1974, 16240}

X(32738) = isogonal conjugate of X(30474)
X(32738) = trilinear pole of line X(32)X(1495) (the isogonal conjugate of the isotomic conjugate of line X(6)X(30))
X(32738) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 1302}, {112, 4846}
X(32738) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30474), (1302, 76), (1576, 15066), (4846, 3267)


X(32739) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(31)

Barycentrics    a^4/(b - c) : :

As the barycentric product of circumcircle-X(6)-antipodes, X(32739) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(31)-antipodes, X(32739) also lies on conic {{A,B,C,X(163),X(825)}} with perspector X(560).

X(32739) lies on these lines: {31, 9447}, {32, 1977}, {41, 9247}, {42, 1976}, {71, 1177}, {99, 769}, {100, 825}, {101, 110}, {109, 649}, {112, 28624}, {184, 14827}, {190, 4577}, {514, 15378}, {657, 14776}, {685, 1897}, {692, 2874}, {1110, 32718}, {1415, 32660}, {1576, 32656}, {1918, 32740}, {1919, 23990}, {1922, 2205}, {2225, 2361}, {3185, 32664}, {3234, 9057}, {3939, 32736}, {23989, 24279}, {32666, 32724}, {32642, 32721} et al

X(32739) = isogonal conjugate of X(3261)
X(32739) = polar conjugate of isotomic conjugate of X(32656)
X(32739) = trilinear pole of line X(32)X(560) (the isogonal conjugate of the isotomic conjugate of line X(6)X(31))
X(32739) = crossdifference of every pair of points on line X(1111)X(3120)
X(32739) = barycentric product of vertices of 1st circumperp triangle
X(32739) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 692}, {4, 32656}, {6, 101}, {9, 1415}, {10, 1576}, {19, 906}, {31, 100}, {32, 190}, {37, 163}, {41, 651}, {42, 110}, {48, 1783}, {55, 109}, {56, 3939}, {71, 112}, {72, 32676}, {99, 1918}, {103, 2426}, {106, 23344}, {107, 4055}, {108, 212}, {162, 228}, {184, 1897}, {281, 32660}, {513, 1110}, {514, 23990}, {516, 32642}, {518, 32666}, {519, 32719}, {648, 2200}, {649, 1252}, {657, 1262}, {658, 14827}, {799, 2205}, {825, 2276}, {899, 32718}, {1011, 28624}, {1016, 1919}, {1193, 32736}, {1292, 21059}, {1501, 1978}, {1922, 3570}, {1977, 6632}, {2222, 2361}, {4554, 9447}, {6335, 9247}, {6586, 15378}, {14776, 22350}
X(32739) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3261), (31, 693), (32, 514), (37, 20948), (41, 4391), (42, 850), (48, 15413), (71, 3267), (100, 561), (101, 76), (109, 6063), (110, 310), (163, 274), (184, 4025), (190, 1502), (228, 14208), (649, 23989), (651, 20567), (657, 23978), (692, 75), (906, 304), (1110, 668), (1252, 1978), (1415, 85), (1501, 649), (1576, 86), (1783, 1969), (1897, 18022), (1918, 523), (1919, 1086), (1922, 4444), (1977, 6545), (2200, 525), (2205, 661), (3939, 3596), (4055, 3265), (9247, 905), (9447, 650), (14827, 3239), (15378, 31624), (23344, 3264), (23990, 190), (32642, 18025), (32656, 69), (32660, 348), (32666, 2481), (32718, 31002), (32719, 903), (32736, 1240), (32676, 286)


X(32740) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(512)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32740) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(512)-antipodes, X(32740) also lies on conic {{A,B,C,X(6),X(32),X(83),X(213),X(729)}} with perspector X(669).

X(32740) lies on these lines: {2, 9516}, {3, 15268}, {6, 110}, {23, 5166}, {32, 1084}, {50, 32654}, {67, 1648}, {83, 597}, {115, 15118}, {182, 14609}, {187, 13493}, {213, 692}, {512, 32741}, {524, 9225}, {575, 9966}, {685, 1990}, {691, 729}, {892, 3225}, {897, 1492}, {1177, 2492}, {1613, 10559}, {1692, 11060}, {1918, 32739}, {1976, 2422}, {2030, 9192}, {2056, 8584}, {2207, 8753}, {2211, 32715}, {3051, 10558}, {3114, 18023}, {3228, 17941}, {3589, 30786}, {5023, 6091}, {5095, 6791}, {5104, 11643}, {5106, 9145}, {5181, 10418}, {5380, 27810}, {5547, 32736}, {5618, 10036}, {6792, 11061}, {7316, 32735}, {9084, 15638}, {9407, 14601}, {9463, 10560}, {9468, 17938}, {11646, 25328}, {14567, 18374}, {16092, 16306}

X(32740) = isogonal conjugate of X(3266)
X(32740) = trilinear pole of line X(32)X(669) (the isogonal conjugate of the isotomic conjugate of line X(6)X(512))
X(32740) = crossdifference of every pair of points on line X(690)X(5181)
X(32740) = barycentric product of circumcircle-intercepts of Schoute circle
X(32740) = barycentric product X(i)*X(j) for these lines: {i,j}: {1, 923}, {3, 8753}, {4, 14908}, {6, 111}, {25, 895}, {31, 897}, {32, 671}, {55, 7316}, {56, 5547}, {76, 19626}, {110, 9178}, {163, 23894}, {184, 17983}, {187, 10630}, {237, 9154}, {669, 892}, {729, 14609}, {1501, 18023}, {1974, 30786}, {1976, 5968}, {3455, 14246}, {5380, 9780}, {6091, 14248}, {10415, 18374}
X(32740) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3266), (31, 14210), (32, 524), (111, 76), (163, 24039), (184, 6390), (669, 690), (671, 1502), (892, 4609), (895, 305), (897, 561), (923, 75), (1084, 1648), (1501, 187), (1974, 468), (5380, 28626), (5547, 3596), (7316, 6063), (8753, 264), (9154, 18024), (9178, 850), (9407, 5642), (9426, 351), (10630, 18023), (14567, 2482), (14601, 5967), (14609, 30736), (14908, 69), (17983, 18022), (18374, 7664), (19626, 6), (23894, 20948)


X(32741) = BARYCENTRIC PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(690)

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

As the barycentric product of circumcircle-X(6)-antipodes, X(32741) lies on conic {{A,B,C,X(110),X(685)}} with center X(206) and perspector X(32), and as the barycentric product of circumcircle-X(690)-antipodes, X(32741) also lies on hyperbola {{A,B,C,X(6),X(187),X(249),X(512),X(524),X(598),X(843)}} with perspector X(351).

X(32741) lies on these lines: {6, 32729}, {110, 524}, {182, 9192}, {187, 1576}, {249, 6593}, {512, 32740}, {692, 21839}, {1177, 3566}, {10630, 28662}, {10765, 12367}, {17938, 18872}

X(32741) = trilinear pole of line X(32)X(351) (the isogonal conjugate of the isotomic conjugate of line X(6)X(690))
X(32741) = barycentric product X(6)*X(2770)
X(32741) = barycentric quotient X(i)/X(j) for these (i,j): (32, 2854), (2770, 76)


X(32742) = (name pending)

Barycentrics    12 a^12 b^2-46 a^10 b^4+64 a^8 b^6-54 a^6 b^8+32 a^4 b^10-8 a^2 b^12+12 a^12 c^2-40 a^10 b^2 c^2+8 a^8 b^4 c^2+65 a^6 b^6 c^2-88 a^4 b^8 c^2+47 a^2 b^10 c^2-4 b^12 c^2-46 a^10 c^4+8 a^8 b^2 c^4-10 a^6 b^4 c^4+32 a^4 b^6 c^4-82 a^2 b^8 c^4+14 b^10 c^4+64 a^8 c^6+65 a^6 b^2 c^6+32 a^4 b^4 c^6+86 a^2 b^6 c^6-10 b^8 c^6-54 a^6 c^8-88 a^4 b^2 c^8-82 a^2 b^4 c^8-10 b^6 c^8+32 a^4 c^10+47 a^2 b^2 c^10+14 b^4 c^10-8 a^2 c^12-4 b^2 c^12 : :
Barycentrics    (54 R^2 SB SC SW^2-18 SB SC SW^3+SB SW^4+SC SW^4)S^2 + (-54 R^2 SB SW-54 R^2 SC SW-18 R^2 SW^2+12 SB SW^2+12 SC SW^2-2 SW^3)S^4 + (108 R^2 + 27 SB + 27 SC - 18 SW)S^6 -2 SB SC SW^5 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29086.

X(32742) lies on this line: {597,6055}


X(32743) = COMPLEMENT OF X(13289)

Barycentrics    a^14 b^2-3 a^12 b^4+a^10 b^6+5 a^8 b^8-5 a^6 b^10-a^4 b^12+3 a^2 b^14-b^16+a^14 c^2-4 a^12 b^2 c^2+6 a^10 b^4 c^2-5 a^8 b^6 c^2+9 a^4 b^10 c^2-11 a^2 b^12 c^2+4 b^14 c^2-3 a^12 c^4+6 a^10 b^2 c^4-6 a^8 b^4 c^4+5 a^6 b^6 c^4-13 a^4 b^8 c^4+15 a^2 b^10 c^4-4 b^12 c^4+a^10 c^6-5 a^8 b^2 c^6+5 a^6 b^4 c^6+10 a^4 b^6 c^6-7 a^2 b^8 c^6-4 b^10 c^6+5 a^8 c^8-13 a^4 b^4 c^8-7 a^2 b^6 c^8+10 b^8 c^8-5 a^6 c^10+9 a^4 b^2 c^10+15 a^2 b^4 c^10-4 b^6 c^10-a^4 c^12-11 a^2 b^2 c^12-4 b^4 c^12+3 a^2 c^14+4 b^2 c^14-c^16 : :
Barycentrics    (84 R^4 + R^2 SB + R^2 SC - 37 R^2 SW + 4 SW^2)S^2 + 36 R^4 SB SC-27 R^2 SB SC SW+4 SB SC SW^2 : :
X(32743) = 3*X[2]-X[13289], X[3]+X[19506], X[4]+X[13293], X[66]+X[19140], X[110]+X[18381], 3*X[381]+X[2935], X[399]+3*X[1853], 5*X[1656]-X[10117], X[2892]+3*X[14561], 7*X[3090]+X[13203], X[3357]+X[7728], 9*X[5055]-X[9919], 2*X[5972]-X[10282], X[6759]-3*X[14643], X[9976]-3*X[23327], X[10264]-3*X[23332], X[10606]+X[23043], X[10733]-3*X[18376], X[12893]+X[18569], X[12901]-3*X[18281], X[14864]+2*X[16534], 5*X[15040]-X[17845], 5*X[20125]+3*X[32064]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29086.

X(32743) lies on these lines: {2,13289}, {3,19506}, {4,13293}, {5,1539}, {30,15090}, {66,19140}, {74,7577}, {110,18381}, {113,2072}, {125,389}, {265,13352}, {381,2935}, {399,1853}, {403,13202}, {427,7687}, {511,15116}, {542,23300}, {858,16163}, {974,18388}, {1503,10272}, {1511,14156}, {1568,12825}, {1656,10117}, {1899,12227}, {2778,9955}, {2781,6697}, {2892,14561}, {3043,25739}, {3090,13203}, {3357,7728}, {5055,9919}, {5092,12900}, {5094,19457}, {5097,20300}, {5448,5663}, {5576,23515}, {5972,10282}, {6759,14643}, {7579,17835}, {7722,23294}, {7741,10118}, {7951,19505}, {8889,18933}, {9976,23327}, {10024,16111}, {10113,12897}, {10254,11204}, {10264,23332}, {10576,13287}, {10577,13288}, {10606,23043}, {10721,16868}, {10733,18376}, {11572,17701}, {11597,15139}, {12219,23293}, {12358,21243}, {12893,18569}, {12901,18281}, {13371,17702}, {14864,16534}, {15040,17845}, {15051,31101}, {15472,18390}, {20125,32064}

X(32743) = complement of X(13289)
X(32743) = midpoint of X(i) and X(j) for these lines: {i,j}: {3,19506}, {4,13293}, {5,23315}, {66,19140}, {110,18381}, {3357,7728}, {12893,18569}, {15131,23325}
X(32743) = reflection of X(i) in X(j) for these lines: {i,j}: {10282,5972}, {12041,25563}, {20301,20300}


X(32744) = X(3)X(14140)∩X(5)X(252)

Barycentrics    2 a^16-11 a^14 b^2+25 a^12 b^4-29 a^10 b^6+15 a^8 b^8+3 a^6 b^10-9 a^4 b^12+5 a^2 b^14-b^16-11 a^14 c^2+34 a^12 b^2 c^2-33 a^10 b^4 c^2+4 a^8 b^6 c^2+5 a^6 b^8 c^2+12 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2+25 a^12 c^4-33 a^10 b^2 c^4+4 a^8 b^4 c^4+a^6 b^6 c^4-2 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4-29 a^10 c^6+4 a^8 b^2 c^6+a^6 b^4 c^6-2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6+15 a^8 c^8+5 a^6 b^2 c^8-2 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+3 a^6 c^10+12 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-9 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : :
Barycentrics    2 S^4 + (R^4+10 R^2 SB+10 R^2 SC-6 SB SC-6 R^2 SW-4 SB SW-4 SC SW+2 SW^2)S^2 -3 R^4 SB SC-2 R^2 SB SC SW+2 SB SC SW^2 : :
X(32744) = X[3]-X[14140], X[5]-X[252], X[12]-X[14102], X[30]-X[511], X[137]-X[24147], X[140]-X[6150], X[546]-X[16337], X[550]-X[14141], X[930]-X[6345], X[1263]-X[19553], X[3628]-X[10615], X[5501]-X[20414], X[10096]-X[10227], X[10126]-X[13856], X[10205]-X[14095], X[13372]-X[27090], X[22051]-X[27246], X[25043]-X[28237], X[27868]-X[30484]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29089.

X(32744) lies on these lines: {3, 14140}, {5, 252}, {12, 14102}, {30, 511}, {137, 24147}, {140, 6150}, {546, 16337}, {550, 14141}, {930, 6345}, {1263, 19553}, {3628, 10615}, {5501, 20414}, {10096, 10227}, {10126, 13856}, {10205, 14095}, {13372, 27090}, {16881, 32409}, {20030, 32536}, {22051, 27246}, {25043, 28237}, {27868, 30484}, {30480, 32638}

X(32744) = isogonal conjugate of X(32749)


X(32745) = (name pending)

Barycentrics    4 a^16-5 a^14 b^2-23 a^12 b^4+45 a^10 b^6-3 a^8 b^8-39 a^6 b^10+23 a^4 b^12-a^2 b^14-b^16-5 a^14 c^2+12 a^12 b^2 c^2-30 a^10 b^4 c^2-21 a^8 b^6 c^2+84 a^6 b^8 c^2-33 a^4 b^10 c^2-13 a^2 b^12 c^2+6 b^14 c^2-23 a^12 c^4-30 a^10 b^2 c^4+94 a^8 b^4 c^4-55 a^6 b^6 c^4-51 a^4 b^8 c^4+49 a^2 b^10 c^4-8 b^12 c^4+45 a^10 c^6-21 a^8 b^2 c^6-55 a^6 b^4 c^6+106 a^4 b^6 c^6-35 a^2 b^8 c^6-6 b^10 c^6-3 a^8 c^8+84 a^6 b^2 c^8-51 a^4 b^4 c^8-35 a^2 b^6 c^8+18 b^8 c^8-39 a^6 c^10-33 a^4 b^2 c^10+49 a^2 b^4 c^10-6 b^6 c^10+23 a^4 c^12-13 a^2 b^2 c^12-8 b^4 c^12-a^2 c^14+6 b^2 c^14-c^16 : :
Barycentrics    (162 R^4+27 R^2 SB+27 R^2 SC-99 R^2 SW-9 SB SW-9 SC SW+15 SW^2)S^4 + (-486 R^4 SB SC+243 R^2 SB SC SW+18 R^2 SB SW^2+18 R^2 SC SW^2-27 SB SC SW^2+6 R^2 SW^3-5 SB SW^3-5 SC SW^3-SW^4)S^2 + 18 R^2 SB SC SW^3-3 SB SC SW^4 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29086.

X(32745) lies on this line: {541,597}


X(32746) = X(2)-CEVA CONJUGATE OF X(194)

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

X(32746) lies on these lines: {69, 25319}, {126, 7897}, {141, 5025}, {160, 9491}, {194, 6374}, {670, 2998}, {1368, 3314}, {3662, 3741}, {3739, 17448}, {6383, 24621}, {6389, 7836}

X(32746) = complement of the isogonal conjugate of X(21001)
X(32746) = complement of the isotomic conjugate of X(20081)
X(32746) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 194}, {21001, 20081}
X(32746) = crosspoint of X(2) and X(20081)
X(32746) = barycentric product X(i)*X(j) for these lines: {i,j}: {194, 20081}, {1740, 20945}, {6374, 21001}, {16571, 17149}
X(32746) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {16571, 3223}, {20081, 2998}, {20945, 18832}, {21001, 3224}, {22152, 3504}
X(32746) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 194}, {16571, 141}, {17091, 17046}, {20081, 2887}, {20945, 626}, {21001, 10}, {21095, 21245}, {21206, 21252}, {22152, 18589}


X(32747) = X(194)-CEVA CONJUGATE OF X(2)

Barycentrics    a^4*b^4 - 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + a^4*c^4 + 2*a^2*b^2*c^4 - 3*b^4*c^4 : :
X(32747) = 3 X[2] - 4 X[6374],9 X[2] - 8 X[6375],3 X[2998] - 4 X[6375],3 X[6374] - 2 X[6375]

X(32747) lies on these lines: {2, 2998}, {69, 698}, {75, 330}, {192, 17149}, {193, 25332}, {194, 15968}, {670, 8264}, {1278, 17135}, {1370, 7779}, {3186, 8782}, {3797, 18750}, {7774, 19583}, {9865, 14615}, {20245, 20535}, {21080, 21219}

X(32747) = anticomplement of X(2998)
X(32747) = reflection of X(2998) and X(6374)
X(32747) = anticomplement of the isogonal conjugate of X(1613)
X(32747) = anticomplement of the isotomic conjugate of X(194)
X(32747) = isotomic conjugate of the isogonal conjugate of X(3360)
X(32747) = X(194)-Ceva conjugate of X(2)
X(32747) = crosspoint of X(3360) and X(15965)
X(32747) = {X(2998),X(6374)}-harmonic conjugate of X(2)
X(32747) = trilinear pole, wrt anticomplementary triangle, of Lemoine axis
X(32747) = barycentric product X(i)*X(j) for these lines: {i,j}: {76, 3360}, {6374, 15965}
X(32747) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {194, 15968}, {3360, 6}, {15965, 3224}
X(32747) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 21299}, {31, 20081}, {32, 21223}, {101, 20983}, {163, 669}, {194, 6327}, {560, 2998}, {662, 3221}, {692, 20979}, {923, 20977}, {1333, 17157}, {1415, 23655}, {1424, 3434}, {1438, 20863}, {1613, 8}, {1740, 69}, {3186, 21270}, {3221, 21221}, {6374, 21275}, {7075, 3436}, {9456, 23633}, {9491, 21220}, {11325, 5905}, {17082, 21285}, {17149, 315}, {20794, 4329}, {21080, 21287}, {21191, 21293}, {21877, 1330}, {23301, 21294}, {23503, 148}, {23572, 149}, {24037, 4609}


X(32748) = X(1)X(6)∩X(31)X(7104)

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

X(32748) lies on the cubic K789 and these lines: {2, 6}, {31, 7104}, {32, 9490}, {187, 9427}, {237, 9468}, {669, 688}, {699, 805}, {1186, 7772}, {1197, 23576}, {1501, 15389}, {1576, 18902}, {1692, 9419}, {1914, 21762}, {1922, 19554}, {3224, 3552}, {3499, 7839}, {4173, 14820}, {6380, 9150}, {8023, 23200}, {9431, 13586}

X(32748) = isogonal conjugate of the isotomic conjugate of X(3229)
X(32748) = crosspoint of X(6) and X(9468)
X(32748) = crosssum of X(i) and X(j) for these (i,j): {2, 3978}, {3225, 8858}, {3766, 6377}, {3948, 6382}
X(32748) = crossdifference of every pair of points on line {76, 512}
X(32748) = X(i)-Ceva conjugate of X(j) for these (i,j): {805, 669}, {1691, 237}, {8789, 3051}
X(32748) = X(i)-isoconjugate of X(j) for these (i,j): {75, 3225}, {92, 8858}, {561, 699}, {1934, 32544}
X(32748) = barycentric product X(i)*X(j) for these lines: {i,j}: {6, 3229}, {31, 2227}, {32, 698}, {99, 9429}, {511, 32540}
X(32748) = barycentric quotient X(i)/X(j) for these lines: {i,j}: {32, 3225}, {184, 8858}, {698, 1502}, {1501, 699}, {2227, 561}, {3229, 76}, {9429, 523}, {14602, 32544}, {32540, 290}
X(32748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1613, 7766}, {6, 3329, 20965}, {6, 18899, 3051}


X(32749) = ISOGONAL CONJUGATE OF X(32744)

Barycentrics    a^2 (a^16-6 a^14 b^2+16 a^12 b^4-26 a^10 b^6+30 a^8 b^8-26 a^6 b^10+16 a^4 b^12-6 a^2 b^14+b^16-5 a^14 c^2+17 a^12 b^2 c^2-21 a^10 b^4 c^2+9 a^8 b^6 c^2+9 a^6 b^8 c^2-21 a^4 b^10 c^2+17 a^2 b^12 c^2-5 b^14 c^2+9 a^12 c^4-12 a^10 b^2 c^4+2 a^8 b^4 c^4+2 a^6 b^6 c^4+2 a^4 b^8 c^4-12 a^2 b^10 c^4+9 b^12 c^4-3 a^10 c^6-5 a^8 b^2 c^6-a^6 b^4 c^6-a^4 b^6 c^6-5 a^2 b^8 c^6-3 b^10 c^6-15 a^8 c^8-4 a^6 b^2 c^8-4 a^4 b^4 c^8-4 a^2 b^6 c^8-15 b^8 c^8+29 a^6 c^10+33 a^4 b^2 c^10+33 a^2 b^4 c^10+29 b^6 c^10-25 a^4 c^12-34 a^2 b^2 c^12-25 b^4 c^12+11 a^2 c^14+11 b^2 c^14-2 c^16) (a^16-5 a^14 b^2+9 a^12 b^4-3 a^10 b^6-15 a^8 b^8+29 a^6 b^10-25 a^4 b^12+11 a^2 b^14-2 b^16-6 a^14 c^2+17 a^12 b^2 c^2-12 a^10 b^4 c^2-5 a^8 b^6 c^2-4 a^6 b^8 c^2+33 a^4 b^10 c^2-34 a^2 b^12 c^2+11 b^14 c^2+16 a^12 c^4-21 a^10 b^2 c^4+2 a^8 b^4 c^4-a^6 b^6 c^4-4 a^4 b^8 c^4+33 a^2 b^10 c^4-25 b^12 c^4-26 a^10 c^6+9 a^8 b^2 c^6+2 a^6 b^4 c^6-a^4 b^6 c^6-4 a^2 b^8 c^6+29 b^10 c^6+30 a^8 c^8+9 a^6 b^2 c^8+2 a^4 b^4 c^8-5 a^2 b^6 c^8-15 b^8 c^8-26 a^6 c^10-21 a^4 b^2 c^10-12 a^2 b^4 c^10-3 b^6 c^10+16 a^4 c^12+17 a^2 b^2 c^12+9 b^4 c^12-6 a^2 c^14-5 b^2 c^14+c^16) : :
Barycentrics    (68 R^2+4 SB+4 SC-20 SW)S^6 + (-142 R^6-116 R^4 SB-116 R^4 SC-92 R^2 SB SC+296 R^4 SW+64 R^2 SB SW+64 R^2 SC SW+20 SB SC SW-160 R^2 SW^2-8 SB SW^2-8 SC SW^2+24 SW^3)S^4 +(12 R^10-59 R^8 SB-59 R^8 SC-82 R^6 SB SC+53 R^8 SW-8 R^6 SB SW-8 R^6 SC SW-72 R^4 SB SC SW-14 R^6 SW^2+68 R^4 SB SW^2+68 R^4 SC SW^2+112 R^2 SB SC SW^2-48 R^4 SW^3-32 R^2 SB SW^3-32 R^2 SC SW^3-24 SB SC SW^3+28 R^2 SW^4+4 SB SW^4+4 SC SW^4-4 SW^5)S^2 -18 R^10 SB SC -21 R^8 SB SC SW+30 R^6 SB SC SW^2+16 R^4 SB SC SW^3-20 R^2 SB SC SW^4+4 SB SC SW^5 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29089.

X(32749) lies on the circumcircle and these lines: {110,15345}, {930,10205}, {1510,6345}

X(32749) = isogonal conjugate of X(32744)


X(32750) = X(2)X(15351)∩X(14401)X(15183)

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)^2*(a^8-(b^2+c^2)*a^6-((2*b^2-2*c^2)^2-b^2*c^2)*a^4+7*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^2)^2)*(a^8-(b^2+c^2)*a^6-(2*b^2-c^2)*(b^2-2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
Barycentrics    (4*S^2+3*(SB+SC)*(12*R^2-SA-3*SW))*(5*S^2-4*(6*SA-SW)*R^2+6*SA^2-4*SB*SC-SW^2)*(9*S^2-4*(12*SA+SW)*R^2+12*SA^2-8*SB*SC+SW^2) : :

See Tran Quang Hung and CÚsar Lozada, Hyacinthos 29090.

X(32750) lies on these lines: {2, 15351}, {14401, 15183}

X(32750) = medial-isotomic conjugate of-X(15184)
X(32750) = complement of the isotomic conjugate of X(402)
X(32750) = barycentric product X(402)*X(15184)






leftri  Points Selected for Their Tripolar Coordinates: X(32751)-X(32766)  rightri

This preamble and centers X(32751)-X(32766) were contributed by Peter Moses and Clark Kimberling, June 18, 2019.

Homogeneous tripolar coordinates (henceforth simply tripolars) for a point X are any triple x : y : z of numbers (or functions of a,b,c) that are respectively proportional to the distances |AX|, |BX|, |CX|. A point X and its circumcircle-inverse have identical tripolars. The appearance of i, j; f(a,b,c) in the following table means that X(i) and X(j) are such a pair, with tripolars f(a,b,c,) : f(b,c,a) : f(c,a,b).

1 36 Sqrt[b c(b + c - a)]
2 23 Sqrt[2 b^2 + 2 c^2 - a^2]
3 (none) 1
4 186 |cos A|
5 2070 Sqrt[a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6]
6 187 b c Sqrt[(-a^2+2 b^2+2 c^2)]
7 32624 (b+c-a) Sqrt[a (a-b-c) (a^2+a b-2 b^2+a c+4 b c-2 c^2)]
8 17100 Sqrt[a (a^2-a b-2 b^2-a c+4 b c-2 c^2)]
9 32625 Sqrt[b c (a^4-2 a^2 b^2+b^4-4 b^3 c-2 a^2 c^2+6 b^2 c^2-4 b c^3+c^4)]
10 1324 Sqrt[a^3-2 a b^2-b^3+a b c-2 a c^2-c^3]
11 14667 (See X(11).)
12 332626 (See X(12).)
13 6104 (See X(13).)
14 6105 (See X(14).)
15 16 b c
17 32627 (See X(17).)
18 32628 (See X(18).)
19 32756 (See X(19).)
20 2071 (See X(20).)
34 32757 (See X(34).)
37 32758 (See X(37).)
42 32759 (See X(42).)
46 32760 (See X(46).)
50 32761 (See X(50).)
52 32762 (See X(52).)
265 5961 a^2((a^2 - b^2 - c^2)^2 - b^2 c^2)
1687 1688 b c Sqrt[b^2 + b^2]
3513 3514 b + c - a
5000 5001 Sqrt[SA]
5002 5003 a
5004 5005 Sqrt[b^2 + c^2]
5980 5981 a Sqrt[b^4 + b^2 c^2 + c^4]
32622 32623 Sqrt[b c]
32753 32754 Sqrt[b + c - a]
32757 32770 (pending)
32763 32764 Sqrt[a]
32765 32766 Sqrt[a^2 + b c]

If X has tripolars x: y : z, then barycentrics for X are

a^2 (D SA + T S) : : and a^2( D SA - T S ) : : , where

SA = b c cos A, S = 2 area(ABC), T = b^2 y^2 + c^2 z^2 - a^2 x^2, and

D = Sqrt[(a x + b y + c z)(- a x + b y + c z)(a x - b y + c z)(a x + b y - c z)];

For details, see X(5002). Further developments are given at

Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines

Albrecht Hess, Transforming Tripolar into Barycentric Coordinates

underbar

X(32751) = X(3)X(76)∩X(83)X(1688)

Barycentrics    a^2 (-a^2 + b^2 + c^2) Sqrt[b^2 c^2 + c^2 a^2 + a^2 b^2] - (a^4 - b^2 c^2 - c^2 a^2 - a^2 b^2) S : :
Tripolars    sin(A + ω) : :
X(32751) = 4 (1 + Sin[w]) X[3] + Csc[w] X[76]

X(32751) lies on these lines: {3,76}, {83,1688}, {1690,7760}

X(32751) = circumcircle-inverse of X(32752)
X(32751) = {X(i),X(j)}-harmonic conjugate of X(k) for these lines: {i,j,k}: {3, 1078, 32752}, {76, 12203, 32752}, {98, 99, 32752}


X(32752) = X(3)X(76)∩X(83)X(1687)

Barycentrics    a^2 (-a^2 + b^2 + c^2) Sqrt[b^2 c^2 + c^2 a^2 + a^2 b^2] + (a^4 - b^2 c^2 - c^2 a^2 - a^2 b^2) S : :
Tripolars    sin(A + ω) : :
X(32752) = 4 (1 - Sin[w]) X[3] - Csc[w] X[76]

X(32752) lies on these lines: {3,76}, {83,1687}, {1689,7760}

X(32752) = circumcircle-inverse of X(32751)
X(32752) = {X(i),X(j)}-harmonic conjugate of X(k) for these lines: {i,j,k}: {3, 1078, 32751}, {76, 12203, 32751}, {98, 99, 32751}


X(32753) = (name pending)

Barycentrics    a^2 ((-a^2+b^2+c^2) Sqrt[-a^6+2 a^5 b-3 a^4 b^2+4 a^3 b^3-3 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-3 a^4 c^2+6 a^2 b^2 c^2-3 b^4 c^2+4 a^3 c^3+4 b^3 c^3-3 a^2 c^4-2 a b c^4-3 b^2 c^4+2 a c^5+2 b c^5-c^6]+2 (a^3-a^2 b+a b^2-b^3-a^2 c+b^2 c+a c^2+b c^2-c^3) S) : :
Tripolars    Sqrt[b+c-a] : :

X(32753) lies on the Stevanovic circle and this line: {3,142}

X(32753) = reflection of X(32754) in the Gergonne line
X(32753) = circumcircle-inverse of X(32754)
X(32753) = {X(3),X(1486)}-harmonic conjugate of X(32754)


X(32754) = (name pending)

Barycentrics    a^2 ((-a^2+b^2+c^2) Sqrt[-a^6+2 a^5 b-3 a^4 b^2+4 a^3 b^3-3 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-3 a^4 c^2+6 a^2 b^2 c^2-3 b^4 c^2+4 a^3 c^3+4 b^3 c^3-3 a^2 c^4-2 a b c^4-3 b^2 c^4+2 a c^5+2 b c^5-c^6]-2 (a^3-a^2 b+a b^2-b^3-a^2 c+b^2 c+a c^2+b c^2-c^3) S) : :
Tripolars    Sqrt[b+c-a] : :

X(32754) lies on the Stevanovic circle and this line: {3,142}

X(32754) = reflection of X(32753) in the Gergonne line
X(32754) = circumcircle-inverse of X(32753)
X(32754) = {X(3),X(1486)}-harmonic conjugate of X(32753)


X(32755) = (name pending)

Barycentrics    a^2*((-a^2 + b^2 + c^2)*Sqrt[-a^8 + 2*a^4*b^4 - b^8 - 4*a^6*b*c + 4*a^5*b^2*c + 4*a^2*b^5*c - 4*a*b^6*c + 4*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 + 4*a*b^5*c^2 + 8*a^3*b^2*c^3 + 8*a^2*b^3*c^3 + 2*a^4*c^4 - 4*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*b*c^5 + 4*a*b^2*c^5 - 4*a*b*c^6 - c^8] + 2*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - c^4)*S) : :
Tripolars    (pending) : :

X(32755) lies on this line: {3, 8}

X(32755) = {X(100),X(104)}-harmonic conjugate of X(32770)
X(32755) = circumcircle-inverse of X(32770)


X(32756) = CIRCUMCIRCLE-INVERSE OF X(19)

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

X(32756) lies on these lines: {3, 19}, {650, 1946}, {5089, 5172}

X(32756) = circumcircle-inverse of X(19)


X(32757) = CIRCUMCIRCLE-INVERSE OF X(34)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 2*a^7*c + 7*a^6*b*c - 13*a^4*b^3*c + 6*a^3*b^4*c + 5*a^2*b^5*c - 4*a*b^6*c + b^7*c - 2*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^3*b^3*c^2 - 2*a*b^5*c^2 + 6*a^5*c^3 - 13*a^4*b*c^3 + 2*a^3*b^2*c^3 + 10*a^2*b^3*c^3 - 4*a*b^4*c^3 - b^5*c^3 + 6*a^3*b*c^4 - 4*a*b^3*c^4 + 2*b^4*c^4 - 6*a^3*c^5 + 5*a^2*b*c^5 - 2*a*b^2*c^5 - b^3*c^5 + 2*a^2*c^6 - 4*a*b*c^6 + 2*a*c^7 + b*c^7 - c^8) : :
Tripolars   

X(32757) lies on these lines: {3, 34}, {1785, 10058}, {1870, 18861}, {1946, 6129}

X(32757) = circumcircle-inverse of X(34)


X(32758) = CIRCUMCIRCLE-INVERSE OF X(37)

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

X(32758) lies on these 3, 37}, {36, 3290}, {172, 2915}, {187, 20872}, {502, 3444}, {650, 667}, {1325, 17737}, {3230, 5078}, {5251, 19309}, {18447, 21861}

X(32758) = circumcircle-inverse of X(37)
X(32758) = Stevanovic circle inverse of X(20989)


X(32759) = CIRCUMCIRCLE-INVERSE OF X(42)

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

X(32759) lies on these lines: {3, 42}, {649, 4079}, {3011, 17729}, {5196, 17734}, {5264, 17019}

X(32759) = circumcircle-inverse of X(42)


X(32760) = CIRCUMCIRCLE-INVERSE OF X(46)

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 + 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 + 2*a^2*c^3 - a*b*c^3 + b^2*c^3 + a*c^4 - c^5) : :
Tripolars   
X(32760) = (R - r) R X[1] - 2 r^2 X[3], X[36] + 2 X[55], 2 X[1319] + X[5119], X[3434] - 4 X[6681], 8 X[6690] - 5 X[31263], X[10087] + 2 X[17010], X[10679] + 2 X[23961], 4 X[25405] - X[25415]

X(32760) lies on these lines: {1, 3}, {21, 5176}, {90, 17857}, {100, 1737}, {186, 15500}, {405, 5123}, {497, 6880}, {498, 2478}, {499, 3434}, {515, 10057}, {519, 10087}, {528, 3582}, {535, 10056}, {611, 9037}, {855, 1283}, {902, 22350}, {906, 8608}, {912, 1727}, {915, 1785}, {920, 3811}, {954, 28534}, {993, 12647}, {1399, 5399}, {1478, 6938}, {1479, 6796}, {1532, 3583}, {1698, 25875}, {1725, 5497}, {1728, 2900}, {1772, 30117}, {1837, 32141}, {1878, 11398}, {1914, 13006}, {2886, 13747}, {3085, 5080}, {3086, 20075}, {3149, 22835}, {3419, 18395}, {3476, 6950}, {3488, 10051}, {3560, 10827}, {3584, 11113}, {3585, 7680}, {3651, 16153}, {4187, 5259}, {4294, 6838}, {4302, 6925}, {4309, 6962}, {4881, 17100}, {5180, 5703}, {5218, 6947}, {5252, 6914}, {5281, 6992}, {5312, 16472}, {5433, 10948}, {6149, 19624}, {6284, 10523}, {6905, 30384}, {6909, 21578}, {6911, 23708}, {6924, 11376}, {6929, 7951}, {6936, 31452}, {6959, 7741}, {8715, 10573}, {10090, 21630}, {10572, 11491}, {10826, 11499}, {10896, 18499}, {11813, 13411}, {12739, 14988}, {13405, 16152}

X(32760) = midpoint of X(55) and X(5172)
X(32760) = reflection of X(36) and X(5172)
X(32760) = circumcircle-inverse of X(46)
X(32760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5010, 14793}, {3, 55, 5119}, {3, 1319, 36}, {3, 3295, 10966}, {3, 11508, 1}, {35, 36, 2077}, {35, 14795, 10902}, {35, 14798, 3}, {35, 15931, 5010}, {36, 5537, 484}, {55, 56, 10679}, {55, 8069, 1}, {55, 24929, 3746}, {55, 32613, 35}, {56, 10679, 25415}, {56, 23961, 36}, {1381, 1382, 46}, {2077, 2078, 36}, {3295, 22765, 5048}, {3295, 22766, 1}, {3560, 11501, 10827}, {11249, 26358, 30323}, {14796, 14797, 5697}, {23961, 25405, 56}


X(32761) = CIRCUMCIRCLE-INVERSE OF X(50)

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

X(32761) lies on these lines: {2, 249}, {3, 6}, {4, 14591}, {30, 19627}, {49, 18321}, {110, 3016}, {112, 9161}, {115, 17702}, {184, 512}, {246, 20976}, {477, 2549}, {691, 10560}, {842, 3431}, {868, 5475}, {1147, 31848}, {1511, 2493}, {1625, 9696}, {1915, 14537}, {2871, 3455}, {3269, 32607}, {5012, 9218}, {5477, 15357}, {6785, 15033}, {6792, 21843}, {7748, 14585}, {7749, 10413}, {14355, 18333}, {18334, 32662}

X(32761) = circumcircle-inverse of X(50)
X(32761) = Brocard-circle-inverse of X(2088)
X(32761) = second-Lemoine-circle-inverse of X(18449)
X(32761) = Moses-circle-inverse of X(3284)
X(32761) = X(1577)-isoconjugate of X(9160)
X(32761) = crosssum of X(31862) and X(31863)
X(32761) = crossdifference of every pair of points on line {523, 3580}
X(32761) = barycentric quotient X(1576)/X(9160)
X(32761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 2088}, {6, 2420, 32}, {371, 372, 18114}, {1379, 1380, 50}, {1666, 1667, 18449}, {2028, 2029, 3284}


X(32762) = CIRCUMCIRCLE-INVERSE OF X(52)

Barycentrics    a^2*(a^10 - 4*a^8*b^2 + 7*a^6*b^4 - 7*a^4*b^6 + 4*a^2*b^8 - b^10 - 4*a^8*c^2 + 7*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 2*b^8*c^2 + 7*a^6*c^4 - 2*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - b^6*c^4 - 7*a^4*c^6 - 3*a^2*b^2*c^6 - b^4*c^6 + 4*a^2*c^8 + 2*b^2*c^8 - c^10) : :
Barycentrics    Sin[A] (Cos[A + w] - (2 Cos[2 w] - 1) Csc[w] Sin[2 A] Sin[B] Sin[C]) : :
Tripolars   

X(32762) lies on these lines: {3, 6}, {5, 2934}, {24, 5963}, {26, 14693}, {249, 14587}, {316, 7509}, {625, 7393}, {691, 2383}, {7395, 13449}, {16188, 21284}

X(32762) = circumcircle-inverse of X(52)
X(32762) = {X(1379),X(1380)}-harmonic conjugate of X(52)


X(32763) = (name pending)

Barycentrics    a^2*(a^6 - 2*a^3*b^3 + b^6 - 2*a^3*c^3 - 2*b^3*c^3 + c^6)*((a^2 - b^2 - c^2)*Sqrt[-a^6 + 2*a^3*b^3 - b^6 + 2*a^3*c^3 + 2*b^3*c^3 - c^6] + 2*(a^3 - b^3 - c^3)*S) : :
Tripolars    Sqrt[a] : :

The barycentrics are real-valued only when ABC is acute.

X(32763) lies on this line: {3, 142}

X(32763) = circumcircle-inverse of X(32764)


X(32764) = (name pending)

Barycentrics    a^2*(a^6 - 2*a^3*b^3 + b^6 - 2*a^3*c^3 - 2*b^3*c^3 + c^6)*((a^2 - b^2 - c^2)*Sqrt[-a^6 + 2*a^3*b^3 - b^6 + 2*a^3*c^3 + 2*b^3*c^3 - c^6] - 2*(a^3 - b^3 - c^3)*S) : :
Tripolars    Sqrt[a] : :

The barycentrics are real-valued only when ABC is acute.

X(32764) lies on this line: {3, 142}

X(32764) = circumcircle-inverse of X(32763)


X(32765) = (name pending)

Barycentrics    a^2*((a^2 - b^2 - c^2)*Sqrt[-a^8 + 2*a^4*b^4 - b^8 - 2*a^6*b*c + 2*a^5*b^2*c + 2*a^2*b^5*c - 2*a*b^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 + 2*a*b^5*c^2 + 2*a^3*b^2*c^3 + 2*a^2*b^3*c^3 + 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + 2*a^2*b*c^5 + 2*a*b^2*c^5 - 2*a*b*c^6 - c^8] + (2*a^4 - 2*b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - 2*c^4)*S) : :
Tripolars    Sqrt[a^2 + b c] : :

The barycentrics are real-valued only when ABC is acute.

X(32765) lies on this line: {3, 10}

X(32765) = circumcircle-inverse of X(32766)


X(32766) = (name pending)

Barycentrics    a^2*((a^2 - b^2 - c^2)*Sqrt[-a^8 + 2*a^4*b^4 - b^8 - 2*a^6*b*c + 2*a^5*b^2*c + 2*a^2*b^5*c - 2*a*b^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 + 2*a*b^5*c^2 + 2*a^3*b^2*c^3 + 2*a^2*b^3*c^3 + 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + 2*a^2*b*c^5 + 2*a*b^2*c^5 - 2*a*b*c^6 - c^8] - (2*a^4 - 2*b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - 2*c^4)*S) : :
Tripolars    Sqrt[a^2 + b c] : :

The barycentrics are real-valued only when ABC is acute.

X(32766) lies on this line: {3, 10}

X(32766) = circumcircle-inverse of X(32765)


X(32767) = COMPLEMENT OF X(10282)

Barycentrics    a^8*b^2 - a^6*b^4 - 3*a^4*b^6 + 5*a^2*b^8 - 2*b^10 + a^8*c^2 - 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 6*b^8*c^2 - a^6*c^4 + 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 4*b^6*c^4 - 3*a^4*c^6 - 6*a^2*b^2*c^6 - 4*b^4*c^6 + 5*a^2*c^8 + 6*b^2*c^8 - 2*c^10 : :
X(32767) = 9 X[2] - X[9833],3 X[2] + X[18381],5 X[3] + 3 X[18405],X[3] + 3 X[23325],X[4] + 3 X[23329],5 X[5] - X[2883],3 X[5] + X[6247],X[5] + 3 X[23332],X[20] + 3 X[18376],X[64] + 7 X[3851],3 X[154] - 11 X[5070],3 X[381] + X[3357],9 X[381] - X[5895],X[382] + 3 X[11204],3 X[547] - X[16252],X[550] - 3 X[10193],X[550] + 3 X[23324],5 X[632] - 3 X[10182],X[1147] - 5 X[31283],X[1498] - 9 X[5055],5 X[1656] + 3 X[1853],5 X[1656] - X[6759],5 X[1656] + X[14864],3 X[1853] + X[6759],3 X[1853] - X[14864],3 X[2883] + 5 X[6247],X[2883] + 5 X[20299],X[2883] + 15 X[23332],7 X[3090] + X[14216],15 X[3091] + X[12250],5 X[3091] - X[22802],3 X[3357] + X[5895],7 X[3526] - 3 X[11202],9 X[3545] - X[5878],X[3627] + 3 X[23328],3 X[3830] + 5 X[8567],7 X[3832] + X[20427],5 X[3843] + 3 X[10606],3 X[3845] + X[5894],9 X[5054] - X[17845],3 X[5066] - X[5893],13 X[5067] + 3 X[32064],15 X[5071] + X[12324],3 X[5656] - 19 X[15022],X[6247] - 3 X[20299],X[6247] - 9 X[23332],X[8549] + 3 X[11178],X[9833] - 3 X[10282],X[9833] + 3 X[18381],X[9927] + 3 X[18281],3 X[10250] + X[15069],X[12250] + 3 X[22802],X[13093] + 15 X[19709],X[13289] - 5 X[15059],X[13293] + 3 X[14644],X[13346] + 3 X[14852],3 X[14076] - X[21230],5 X[14530] - 21 X[15703],5 X[15027] + 3 X[15131],7 X[15057] - 3 X[16219],3 X[15061] + X[19506],5 X[18383] - 3 X[18405],X[18383] - 3 X[23325],X[18405] - 5 X[23325],X[20299] - 3 X[23332],X[21230] + 3 X[32351]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29094.

X(32767) lies on these lines: {2, 9833}, {3, 18383}, {4, 11270}, {5, 2883}, {20, 18376}, {30, 20191}, {51, 26917}, {64, 3851}, {125, 389}, {140, 13470}, {154, 5070}, {185, 7577}, {186, 11572}, {343, 15606}, {381, 3357}, {382, 11204}, {403, 13474}, {427, 10110}, {468, 13419}, {511, 5449}, {542, 9820}, {546, 2777}, {547, 16252}, {550, 10193}, {575, 18952}, {578, 5094}, {632, 10182}, {858, 15644}, {1092, 30744}, {1147, 31283}, {1204, 7547}, {1209, 3819}, {1216, 14076}, {1493, 11232}, {1495, 14940}, {1498, 5055}, {1503, 3628}, {1531, 11440}, {1656, 1853}, {1885, 7687}, {2072, 5907}, {2781, 10095}, {3090, 14216}, {3091, 7703}, {3520, 13851}, {3526, 11202}, {3541, 18390}, {3545, 5878}, {3581, 32395}, {3627, 23328}, {3830, 8567}, {3832, 20427}, {3843, 10606}, {3845, 5894}, {3850, 15311}, {5020, 32321}, {5054, 17845}, {5066, 5893}, {5067, 32064}, {5071, 12324}, {5498, 30522}, {5562, 23293}, {5576, 5943}, {5651, 31282}, {5656, 15022}, {5663, 32184}, {5972, 12134}, {6143, 12254}, {6640, 18474}, {6723, 16238}, {7505, 11550}, {7507, 11438}, {7553, 32223}, {7574, 32365}, {7603, 32445}, {8549, 11178}, {8681, 23307}, {9927, 15123}, {10113, 25564}, {10117, 18369}, {10224, 13561}, {10226, 18379}, {10250, 15069}, {10254, 10575}, {10255, 12162}, {10274, 13353}, {11245, 12242}, {11264, 32376}, {11381, 16868}, {11585, 11793}, {12006, 13413}, {12370, 15113}, {13093, 19709}, {13154, 15577}, {13160, 16836}, {13289, 15059}, {13293, 14644}, {13346, 14852}, {13382, 18388}, {13383, 29012}, {13406, 14915}, {14530, 15703}, {14627, 17847}, {14810, 18382}, {15027, 15131}, {15037, 17824}, {15057, 16219}, {15061, 19506}, {15116, 20301}, {15578, 17714}, {17702, 23336}, {17712, 25337}, {18488, 23515}, {18567, 32210}, {23300, 24206}, {31724, 32110}

X(32767) = complement of X(10282)
X(32767) = complement of the isogonal of X(15319)
X(32767) = X(15319)-complementary conjugate of X(10)
X(32767) = midpoint of X(i) and X(j) for these lines: {i,j}: {3, 18383}, {5, 20299}, {125, 32743}, {546, 6696}, {5449, 13371}, {6697, 20300}, {6759, 14864}, {10113, 25564}, {10193, 23324}, {10224, 13561}, {10226, 18379}, {10282, 18381}, {14076, 32351}, {14810, 18382}, {15116, 20301}, {18567, 32210}, {23300, 24206}
X(32767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18381, 10282}, {3, 23325, 18383}, {5, 23332, 20299}, {125, 1594, 389}, {125, 3574, 26879}, {1594, 26879, 3574}, {1656, 1853, 6759}, {1853, 6759, 14864}, {3574, 26879, 389}, {6143, 25739, 13367}, {7577, 23294, 185}, {11585, 21243, 11793}, {13353, 15139, 10274}


X(32768) = COMPLEMENT OF X(32769)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29097.

X(32768) lies on this line: {2, 32769}

X(32768) = complement of X(32769)


X(32769) = ANTICOMPLEMENT OF X(32768)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29097.

X(32769) lies on these lines: {2, 32768}, {10625, 15331}

X(32769) = anticomplement of X(32768)


X(32770) = (name pending)

Barycentrics    a^2*((-a^2 + b^2 + c^2)*Sqrt[-a^8 + 2*a^4*b^4 - b^8 - 4*a^6*b*c + 4*a^5*b^2*c + 4*a^2*b^5*c - 4*a*b^6*c + 4*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 + 4*a*b^5*c^2 + 8*a^3*b^2*c^3 + 8*a^2*b^3*c^3 + 2*a^4*c^4 - 4*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*b*c^5 + 4*a*b^2*c^5 - 4*a*b*c^6 - c^8] - 2*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - c^4)*S) : :

X(32770) lies on this line: {3, 8}

X(32770) = {X(100),X(104)}-harmonic conjugate of X(32755)
X(32770) = circumcircle inverse of X(32755)






leftri  Centers of degree 3 with coefficients in {-1,0,1}: X(32771)-X(32784)  rightri

Centers X(32771)-X(32784) were contributed by Peter Moses, June 20, 2019:

Each center in this section has barycentrics of the form p(a,b,c) : : , where p(a,b,c) is a polynomial homogeneous of degree 3 in a,b,c, with coefficents in the set {-1, 0 , 1}. For more of these, see also X(32842)-X(32866), X(32911)-X(32950), and X(33064)-X(33175).

underbar

X(32771) = X(1)X(321)∩X(2)X(38)

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

X(32771) lies on these lines: {1, 321}, {2, 38}, {7, 26034}, {8, 2650}, {10, 3681}, {31, 894}, {37, 17157}, {42, 75}, {43, 4359}, {51, 17049}, {55, 4363}, {56, 7211}, {57, 29828}, {72, 31339}, {81, 4362}, {86, 5311}, {100, 3980}, {142, 25961}, {171, 26227}, {192, 1962}, {200, 25590}, {210, 3739}, {226, 25760}, {238, 26223}, {274, 869}, {312, 3720}, {354, 30942}, {518, 30969}, {551, 4135}, {561, 1909}, {611, 26000}, {612, 3263}, {726, 28606}, {740, 17018}, {748, 16823}, {750, 7081}, {870, 3112}, {873, 7033}, {899, 19804}, {908, 25960}, {940, 17763}, {968, 3729}, {976, 1010}, {977, 1220}, {986, 26115}, {1125, 3971}, {1278, 21806}, {1621, 3923}, {1757, 5278}, {1911, 18895}, {2177, 17116}, {2308, 3758}, {2345, 3475}, {2887, 29667}, {3009, 31997}, {3120, 29685}, {3159, 27785}, {3175, 15569}, {3187, 4649}, {3190, 18698}, {3293, 28612}, {3315, 29668}, {3474, 7222}, {3589, 29852}, {3616, 3994}, {3666, 17155}, {3677, 29826}, {3683, 17351}, {3703, 17056}, {3705, 31120}, {3715, 17259}, {3736, 30599}, {3741, 3873}, {3742, 30818}, {3745, 4670}, {3751, 5271}, {3772, 29631}, {3782, 4026}, {3797, 17032}, {3836, 27186}, {3846, 31053}, {3869, 30076}, {3874, 10479}, {3896, 4980}, {3932, 29854}, {3938, 5263}, {3961, 24342}, {3974, 4648}, {4011, 5284}, {4037, 16777}, {4046, 4665}, {4054, 24210}, {4082, 29571}, {4088, 24353}, {4358, 26102}, {4388, 24725}, {4446, 20966}, {4519, 4891}, {4655, 17483}, {4660, 20292}, {4661, 27798}, {4671, 29814}, {4672, 17127}, {4683, 5905}, {4688, 4849}, {4692, 30116}, {4697, 17126}, {4699, 21805}, {4703, 17484}, {4739, 21870}, {4850, 6685}, {4892, 25958}, {4972, 17889}, {5205, 17124}, {5220, 19732}, {5293, 16454}, {5550, 25079}, {5718, 29849}, {5750, 21101}, {5835, 15888}, {6057, 17243}, {6382, 18059}, {6533, 17749}, {6535, 17233}, {6679, 29681}, {6703, 17602}, {7191, 25496}, {7229, 10578}, {10434, 29069}, {11680, 25385}, {12263, 17397}, {13476, 14973}, {13740, 28082}, {15523, 18134}, {16569, 24589}, {16706, 29663}, {16720, 21010}, {16915, 18266}, {17030, 20456}, {17061, 29636}, {17122, 26627}, {17135, 31025}, {17147, 17592}, {17150, 19717}, {17163, 20011}, {17234, 29687}, {17279, 29851}, {17289, 24943}, {17369, 29638}, {17449, 31241}, {17591, 29825}, {17716, 20045}, {17718, 29846}, {17720, 29845}, {17724, 29848}, {17874, 20237}, {18139, 29674}, {18743, 30950}, {19784, 24159}, {19786, 29647}, {19869, 26728}, {20711, 26107}, {20909, 23655}, {21350, 24674}, {24174, 26030}, {24295, 29672}, {24321, 24333}, {24330, 24357}, {24456, 26976}, {24709, 26105}, {24789, 29850}, {24924, 30584}, {25525, 29857}, {25526, 30142}, {25959, 28595}, {27475, 30821}

X(32771) =


X(32772) = X(1)X(321)∩X(2)X(31)

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

X(32772) lies on these lines: {1, 321}, {2, 31}, {6, 4042}, {10, 1203}, {11, 6703}, {21, 36}, {38, 894}, {42, 3996}, {55, 16405}, {57, 29826}, {58, 19863}, {75, 17017}, {81, 3741}, {86, 310}, {100, 6685}, {142, 4224}, {190, 3989}, {239, 21020}, {244, 1281}, {274, 21415}, {312, 5311}, {333, 2308}, {354, 4670}, {474, 27639}, {497, 2268}, {561, 870}, {614, 10436}, {740, 17011}, {756, 16830}, {940, 30942}, {975, 25591}, {978, 16454}, {984, 26223}, {1001, 1011}, {1010, 1193}, {1100, 3706}, {1215, 3920}, {1220, 10459}, {1386, 31993}, {1449, 17156}, {1471, 27339}, {1621, 4203}, {1699, 19645}, {1724, 19858}, {1757, 4981}, {1836, 4657}, {1848, 4206}, {1909, 18064}, {1961, 4358}, {1962, 3685}, {2049, 16466}, {2298, 4071}, {2344, 32664}, {2886, 29631}, {2895, 3775}, {2975, 30076}, {3112, 18091}, {3120, 14012}, {3218, 4697}, {3219, 4672}, {3550, 29825}, {3589, 3925}, {3616, 4195}, {3624, 16342}, {3666, 4418}, {3703, 17369}, {3745, 17763}, {3750, 29822}, {3757, 17469}, {3772, 29636}, {3821, 20292}, {3842, 27065}, {3873, 29652}, {3914, 17023}, {3923, 28606}, {3966, 17303}, {3980, 4850}, {4038, 8025}, {4046, 14459}, {4357, 4683}, {4359, 24342}, {4360, 4365}, {4363, 17155}, {4383, 26037}, {4387, 16777}, {4423, 15668}, {4429, 29663}, {4432, 10180}, {4438, 29664}, {4514, 29685}, {4649, 17135}, {4682, 30818}, {4693, 27804}, {4716, 17163}, {4722, 17120}, {4854, 17045}, {4865, 29667}, {4886, 8013}, {4972, 29633}, {4974, 27798}, {5014, 29659}, {5248, 16452}, {5255, 26115}, {5269, 29828}, {5271, 16475}, {5278, 16468}, {5695, 20182}, {5718, 29846}, {6536, 17322}, {6686, 9342}, {7081, 31264}, {7083, 16352}, {7191, 24325}, {8053, 19339}, {8692, 19749}, {9313, 30909}, {9347, 29649}, {10453, 17379}, {10582, 24179}, {10886, 13478}, {11680, 29635}, {13733, 28628}, {14006, 17923}, {14829, 31241}, {15523, 17289}, {15988, 24997}, {16445, 25498}, {16477, 19742}, {16480, 29576}, {16484, 19740}, {16706, 29684}, {16827, 16926}, {17056, 29632}, {17061, 29834}, {17063, 26627}, {17140, 17598}, {17147, 17600}, {17150, 31025}, {17234, 29677}, {17279, 29854}, {17368, 26061}, {17381, 29647}, {17686, 23682}, {17716, 26227}, {17718, 29848}, {17720, 29847}, {17723, 29849}, {17889, 19271}, {18134, 24943}, {18139, 29637}, {19738, 28650}, {20132, 31027}, {20148, 27035}, {24295, 29653}, {24725, 27184}, {24789, 29852}, {25385, 29645}, {25441, 25639}, {25524, 28348}, {25525, 29855}, {26102, 31005}, {26105, 30943}, {26128, 29648}, {27186, 29666}, {29860, 30588}

X(32772) = complement of X(33083)


X(32773) = X(1)X(977)∩X(2)X(11)

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

X(32773) lies on these lines: {1, 977}, {2, 11}, {4, 1220}, {6, 4388}, {8, 1211}, {10, 312}, {31, 29631}, {33, 5174}, {37, 29641}, {38, 1227}, {42, 4417}, {43, 3846}, {56, 4201}, {63, 24723}, {75, 305}, {76, 18057}, {81, 6327}, {141, 10453}, {171, 4660}, {192, 3703}, {209, 3869}, {238, 25453}, {239, 3966}, {319, 17156}, {321, 29667}, {350, 5224}, {354, 3662}, {495, 16052}, {518, 27184}, {595, 20083}, {614, 16706}, {748, 17352}, {750, 29845}, {846, 4438}, {894, 1836}, {899, 25960}, {902, 29863}, {940, 4645}, {958, 26117}, {962, 5799}, {968, 29857}, {982, 3821}, {984, 4425}, {999, 11359}, {1213, 23918}, {1215, 3944}, {1233, 3673}, {1478, 17677}, {1479, 13740}, {1486, 25494}, {1654, 4042}, {1738, 19804}, {1757, 4703}, {1818, 24550}, {1848, 1869}, {1962, 29643}, {1999, 3416}, {2049, 19865}, {2177, 29846}, {2345, 21956}, {2476, 26115}, {2975, 17676}, {3006, 28606}, {3120, 29685}, {3136, 27042}, {3175, 3790}, {3240, 5741}, {3314, 17302}, {3475, 26132}, {3616, 4202}, {3617, 3974}, {3661, 3706}, {3666, 3705}, {3677, 17304}, {3681, 26580}, {3687, 3755}, {3717, 4656}, {3720, 17234}, {3722, 29848}, {3739, 21949}, {3743, 30172}, {3744, 29634}, {3745, 29841}, {3750, 3771}, {3757, 3772}, {3782, 24349}, {3836, 26102}, {3840, 24217}, {3873, 17184}, {3886, 19815}, {3920, 5014}, {3936, 17018}, {3959, 21954}, {3980, 24715}, {3995, 31079}, {3996, 30832}, {4030, 17602}, {4073, 4357}, {4193, 26030}, {4195, 6284}, {4205, 19853}, {4234, 4302}, {4358, 29679}, {4387, 17280}, {4398, 17155}, {4442, 28605}, {4450, 17126}, {4657, 20541}, {4676, 5294}, {4683, 17347}, {4872, 10436}, {4884, 17246}, {4891, 17231}, {4999, 19278}, {5016, 17016}, {5057, 26223}, {5241, 26038}, {5259, 19846}, {5264, 25441}, {5484, 12513}, {5550, 17674}, {5698, 26065}, {5749, 9812}, {6679, 8616}, {6682, 17774}, {6685, 17717}, {7069, 25024}, {7081, 17720}, {8543, 28776}, {9664, 24275}, {9668, 11354}, {10582, 17282}, {11109, 11393}, {11110, 19854}, {11269, 14829}, {13725, 19843}, {13728, 24390}, {15171, 17698}, {15523, 17233}, {15985, 21299}, {16060, 27309}, {16484, 29642}, {16823, 24789}, {17017, 17380}, {17063, 24169}, {17127, 29868}, {17235, 21342}, {17306, 24392}, {17354, 26061}, {17378, 31134}, {17381, 29647}, {17469, 29636}, {17522, 20988}, {17550, 26965}, {17592, 29671}, {17598, 29844}, {17671, 30107}, {17715, 29656}, {17716, 17766}, {17719, 29670}, {17722, 29650}, {17792, 21334}, {17889, 24325}, {18139, 25959}, {19270, 26363}, {19273, 31493}, {19823, 19993}, {20011, 31037}, {20173, 25006}, {21020, 23928}, {21026, 29854}, {22267, 26686}, {23682, 31997}, {24357, 25345}, {24551, 24678}, {24703, 27064}, {24752, 25978}, {25496, 29633}, {25914, 26093}, {25961, 30950}, {27047, 27313}, {28595, 29674}, {29632, 31237}, {29839, 30811}

X(32773) =


X(32774) = X(1)X(4202)∩X(2)X(37)

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

X(32774) lies on these lines: {1, 4202}, {2, 37}, {6, 17184}, {10, 3891}, {31, 3821}, {38, 25453}, {42, 26128}, {55, 26230}, {57, 21376}, {63, 17304}, {81, 3662}, {86, 17173}, {100, 29634}, {141, 3187}, {142, 20268}, {171, 29636}, {226, 7225}, {238, 29852}, {244, 29635}, {306, 3946}, {333, 17305}, {354, 29829}, {379, 2140}, {443, 1058}, {693, 25603}, {726, 26061}, {740, 24943}, {748, 4425}, {750, 24169}, {940, 17290}, {942, 4463}, {964, 23537}, {968, 24542}, {975, 17674}, {982, 29631}, {984, 29850}, {1104, 17676}, {1125, 3914}, {1211, 17366}, {1215, 29663}, {1255, 17244}, {1386, 6327}, {1824, 5439}, {1961, 25961}, {1962, 29642}, {1999, 17291}, {2177, 29656}, {2275, 16717}, {2308, 4655}, {2887, 17017}, {2895, 3759}, {2999, 5741}, {3006, 17599}, {3120, 25496}, {3219, 4389}, {3315, 29843}, {3416, 17150}, {3578, 17272}, {3589, 3782}, {3618, 5905}, {3624, 4442}, {3663, 5294}, {3670, 20083}, {3702, 19836}, {3744, 29831}, {3750, 29638}, {3758, 17483}, {3765, 26965}, {3836, 5311}, {3875, 3969}, {3920, 4429}, {3936, 5256}, {3938, 4085}, {3948, 30107}, {4357, 5278}, {4383, 26580}, {4392, 29868}, {4414, 6679}, {4417, 17012}, {4438, 29867}, {4514, 17024}, {4641, 17235}, {4643, 19742}, {4656, 31191}, {4660, 17469}, {4675, 8025}, {4683, 16468}, {4852, 20017}, {4865, 29819}, {4968, 19784}, {5016, 5262}, {5222, 5739}, {5233, 17020}, {5263, 29648}, {5268, 24988}, {5271, 17306}, {5287, 17282}, {5333, 17397}, {5745, 31229}, {5826, 14557}, {6682, 24892}, {6703, 26627}, {7490, 9776}, {10453, 26150}, {12610, 19645}, {16484, 29853}, {17011, 17380}, {17019, 17234}, {17025, 25958}, {17027, 30965}, {17061, 26227}, {17063, 29845}, {17064, 29826}, {17122, 29847}, {17127, 24723}, {17253, 19723}, {17325, 19732}, {17352, 27065}, {17361, 20086}, {17367, 27184}, {17372, 20046}, {17394, 20553}, {17591, 29856}, {17592, 29632}, {17594, 29855}, {17596, 29859}, {17597, 29835}, {17600, 29643}, {17715, 29836}, {17716, 29834}, {17750, 18098}, {17861, 20886}, {17889, 19271}, {18044, 28654}, {23292, 26651}, {23681, 29598}, {24046, 25441}, {24175, 24594}, {24325, 29647}, {25581, 25598}, {25760, 29821}, {27064, 29630}, {29671, 31237}

X(32774) =


X(32775) = X(1)X(3454)∩X(2)X(38)

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

X(32775) lies on these lines: {1, 3454}, {2, 38}, {6, 29636}, {9, 29855}, {31, 4683}, {37, 29632}, {42, 19786}, {55, 29848}, {81, 29645}, {100, 3821}, {141, 17602}, {171, 17184}, {210, 29850}, {238, 26230}, {312, 24943}, {321, 23689}, {354, 29845}, {518, 29631}, {612, 25527}, {614, 25960}, {750, 3662}, {896, 6646}, {899, 16706}, {902, 24723}, {908, 1125}, {940, 29847}, {976, 16062}, {1001, 20999}, {1150, 29658}, {1155, 17235}, {1211, 17061}, {1279, 29836}, {1283, 1621}, {1386, 29834}, {1757, 29859}, {1961, 18139}, {1962, 29839}, {2239, 31004}, {2887, 3920}, {2895, 3791}, {3011, 4357}, {3120, 5263}, {3219, 6679}, {3616, 5484}, {3666, 29846}, {3681, 25453}, {3712, 17246}, {3717, 30768}, {3771, 28606}, {3772, 31330}, {3782, 4418}, {3836, 5297}, {3846, 7191}, {3873, 29635}, {3874, 25441}, {3935, 4085}, {3944, 24552}, {3961, 4972}, {3989, 29865}, {3994, 17280}, {4009, 17357}, {4023, 17366}, {4026, 17724}, {4062, 4360}, {4104, 26723}, {4202, 5293}, {4204, 30961}, {4358, 29637}, {4364, 9318}, {4383, 29852}, {4388, 17469}, {4389, 4414}, {4413, 17290}, {4417, 17017}, {4423, 29853}, {4430, 29864}, {4649, 29833}, {4655, 17126}, {4657, 17718}, {4661, 29868}, {4672, 17484}, {4696, 19879}, {4697, 17483}, {4703, 17127}, {4766, 16826}, {4852, 14459}, {4865, 25958}, {5205, 17291}, {5219, 29826}, {5268, 25961}, {5284, 29672}, {5311, 18134}, {5363, 31019}, {5741, 29821}, {5904, 20083}, {6327, 17716}, {6536, 29689}, {6693, 6763}, {7018, 18064}, {7174, 29857}, {11375, 28386}, {11680, 29652}, {14829, 29683}, {17150, 31037}, {17305, 25374}, {17306, 29828}, {17599, 29849}, {17720, 30942}, {17722, 29823}, {17725, 26227}, {18743, 29677}, {19812, 29647}, {19836, 25591}, {19863, 24160}, {21093, 24295}, {24412, 24450}, {24542, 29860}, {24709, 29660}, {24789, 26037}, {25466, 27687}, {25496, 29648}, {27065, 29871}, {27131, 29666}, {29641, 31237}, {29643, 30811}, {29657, 30834}, {29671, 30831}

X(32775) =


X(32776) = X(1)X(6327)∩X(2)X(846)

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

X(32776) lies on these lines: {1, 6327}, {2, 846}, {6, 4683}, {9, 29850}, {10, 28605}, {31, 19786}, {37, 25957}, {38, 1227}, {42, 27184}, {43, 26580}, {55, 29848}, {57, 29845}, {63, 29631}, {81, 4655}, {141, 4854}, {149, 29652}, {171, 29847}, {190, 26061}, {191, 20083}, {192, 15523}, {238, 29852}, {354, 17235}, {614, 17304}, {726, 29667}, {748, 16706}, {756, 4429}, {894, 29647}, {902, 29634}, {964, 24851}, {968, 25527}, {984, 4972}, {986, 5051}, {1001, 16064}, {1125, 1770}, {1621, 26128}, {1699, 29826}, {1738, 26037}, {1836, 4657}, {1962, 18134}, {2292, 16062}, {2887, 28606}, {3175, 3844}, {3218, 29635}, {3219, 25453}, {3661, 4365}, {3662, 3720}, {3663, 17155}, {3666, 25760}, {3681, 4085}, {3685, 24943}, {3703, 17246}, {3706, 17237}, {3735, 20859}, {3752, 25960}, {3763, 4387}, {3782, 4026}, {3816, 25378}, {3846, 4850}, {3914, 4357}, {3920, 4660}, {3924, 26117}, {3925, 4364}, {3936, 17592}, {3966, 17301}, {3971, 29679}, {3989, 17247}, {3995, 29674}, {4042, 17253}, {4388, 17017}, {4392, 29655}, {4423, 17290}, {4450, 17716}, {4512, 29855}, {4645, 5311}, {4722, 17347}, {5057, 25496}, {5224, 21020}, {5249, 16778}, {5278, 24697}, {6682, 11680}, {6685, 31053}, {6703, 11246}, {7018, 30964}, {7226, 29673}, {8616, 26230}, {10453, 17236}, {12514, 27661}, {12579, 16865}, {16342, 24161}, {16705, 16891}, {17126, 29645}, {17127, 29654}, {17156, 17272}, {17165, 29659}, {17187, 17202}, {17291, 29677}, {17320, 31134}, {17323, 17599}, {17383, 29684}, {17449, 29843}, {17491, 19717}, {17594, 29846}, {17763, 26034}, {17766, 29815}, {21241, 29664}, {23537, 31339}, {24210, 30942}, {24349, 29685}, {24551, 26573}, {24627, 29662}, {25058, 30984}, {25958, 29671}, {25959, 29653}, {26223, 29633}, {26840, 29837}, {27064, 29663}, {27804, 31017}, {30632, 31008}

X(32776) =


X(32777) = X(1)X(3695)∩X(2)X(37)

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

X(32777) lies on these lines: {1, 3695}, {2, 37}, {3, 23847}, {6, 306}, {8, 1104}, {9, 440}, {10, 55}, {25, 1376}, {31, 3416}, {38, 24943}, {42, 26061}, {44, 5739}, {57, 7198}, {63, 141}, {69, 4641}, {81, 4851}, {100, 9078}, {171, 29674}, {190, 27184}, {209, 10477}, {226, 17355}, {238, 3966}, {239, 16974}, {244, 29677}, {318, 18679}, {333, 3661}, {355, 1746}, {406, 9371}, {429, 1329}, {474, 27802}, {553, 21255}, {594, 5271}, {599, 4001}, {612, 3932}, {726, 26128}, {740, 25453}, {750, 29687}, {894, 18134}, {936, 1040}, {940, 3912}, {958, 8192}, {968, 4026}, {982, 29637}, {1009, 2352}, {1125, 17599}, {1214, 14376}, {1215, 3771}, {1403, 3831}, {1427, 28739}, {1441, 28776}, {1621, 29667}, {1698, 3712}, {1708, 26942}, {1714, 5295}, {1722, 25992}, {1724, 5814}, {1766, 19542}, {1836, 2887}, {1914, 5278}, {1962, 29647}, {1999, 17233}, {2161, 2339}, {2325, 4656}, {2886, 29857}, {2901, 20083}, {2915, 25440}, {3006, 24552}, {3120, 31237}, {3187, 3969}, {3219, 4376}, {3452, 21062}, {3501, 24310}, {3589, 5256}, {3679, 3749}, {3686, 19723}, {3687, 3713}, {3705, 17721}, {3714, 5230}, {3715, 4104}, {3719, 26543}, {3729, 3782}, {3741, 4438}, {3750, 29659}, {3758, 17778}, {3763, 3977}, {3773, 4362}, {3790, 29634}, {3836, 3980}, {3840, 17728}, {3844, 4640}, {3846, 4011}, {3891, 26230}, {3914, 5695}, {3929, 17272}, {3931, 19784}, {3936, 26223}, {3961, 30615}, {3965, 23600}, {4024, 25684}, {4195, 7270}, {4204, 26037}, {4361, 26723}, {4363, 5249}, {4365, 29867}, {4387, 24210}, {4388, 4676}, {4415, 17340}, {4417, 17354}, {4418, 5880}, {4647, 19846}, {4659, 23681}, {4660, 28595}, {4675, 18139}, {4689, 9780}, {4760, 21221}, {4798, 5333}, {4863, 29673}, {4886, 17349}, {5014, 31079}, {5016, 11319}, {5057, 25958}, {5090, 17520}, {5226, 30823}, {5241, 7308}, {5247, 10371}, {5263, 29641}, {5273, 14021}, {5287, 6703}, {5328, 28656}, {5332, 19742}, {5336, 11679}, {5712, 5749}, {5737, 17293}, {5745, 21483}, {5750, 19701}, {5772, 10578}, {5791, 10479}, {5793, 24987}, {5794, 13733}, {5835, 19860}, {5905, 17351}, {6057, 17602}, {6541, 29645}, {6690, 29828}, {7283, 16062}, {7580, 12618}, {7795, 25083}, {8299, 31330}, {11374, 25645}, {11680, 29872}, {14555, 26685}, {14829, 17285}, {15985, 22370}, {16577, 25078}, {16594, 20196}, {16844, 19857}, {17023, 20182}, {17056, 17369}, {17061, 29855}, {17080, 28780}, {17184, 17276}, {17227, 26840}, {17229, 24597}, {17242, 29841}, {17334, 25734}, {17339, 30832}, {17345, 20078}, {17592, 29633}, {17595, 29596}, {17598, 29660}, {17600, 29646}, {17723, 24295}, {18141, 29579}, {18235, 26066}, {19744, 31477}, {20292, 25959}, {21245, 27052}, {21376, 24335}, {24325, 29642}, {24330, 30985}, {24691, 30965}, {24703, 25760}, {24954, 25079}, {25019, 26958}, {25591, 25681}, {27272, 31997}, {29678, 31264}, {30831, 31053}

X(32777) =


X(32778) = X(1)X(2)∩X(57)X(12588)

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

X(32778) lies on these lines: {1, 2}, {57, 12588}, {75, 2887}, {141, 982}, {171, 3416}, {190, 4703}, {192, 4425}, {238, 3966}, {305, 1930}, {312, 3773}, {321, 3944}, {333, 4438}, {345, 846}, {573, 5282}, {584, 3684}, {594, 2886}, {726, 27184}, {984, 1211}, {1215, 4417}, {1376, 5096}, {1757, 5739}, {2321, 24210}, {2345, 4071}, {3120, 25958}, {3210, 3821}, {3509, 5227}, {3596, 6382}, {3662, 24165}, {3686, 5037}, {3690, 5692}, {3717, 4104}, {3742, 17231}, {3744, 4914}, {3752, 3844}, {3790, 3971}, {3836, 19804}, {3883, 8616}, {3923, 4388}, {3932, 5743}, {3980, 4645}, {4011, 17280}, {4026, 17592}, {4038, 4851}, {4358, 25960}, {4359, 25957}, {4418, 6327}, {4429, 28595}, {4657, 17600}, {4671, 6535}, {4680, 14012}, {4865, 5263}, {5247, 5814}, {5278, 8300}, {5294, 16468}, {5300, 19271}, {5515, 20551}, {5846, 17716}, {5904, 10974}, {7281, 27542}, {8167, 17267}, {10009, 30631}, {16478, 17698}, {17123, 17279}, {17140, 31017}, {17155, 17184}, {17165, 31037}, {17229, 24217}, {17281, 24703}, {17289, 25496}, {17293, 17722}, {17490, 24169}, {17596, 17740}, {18134, 24325}, {18203, 20911}, {18788, 26118}, {19822, 24342}, {20292, 31134}, {21024, 23918}, {21241, 21829}, {21726, 31003}, {21920, 31245}, {21962, 25666}, {24589, 25961}, {26590, 27474}, {30599, 30984}, {31327, 31419}

X(32778) =


X(32779) = X(2)X(37)∩X(8)X(5266)

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

X(32779) lies on these lines: {2, 37}, {8, 5266}, {9, 21368}, {10, 21}, {22, 1376}, {43, 26061}, {55, 29667}, {63, 17272}, {81, 306}, {88, 24170}, {141, 3218}, {171, 15523}, {190, 26580}, {226, 30831}, {244, 29637}, {319, 16704}, {320, 31017}, {333, 32025}, {464, 5273}, {740, 29631}, {750, 29674}, {894, 3936}, {908, 17355}, {940, 17311}, {982, 24943}, {1150, 3661}, {1155, 3844}, {1211, 3219}, {1214, 18632}, {1215, 29846}, {1332, 26637}, {1465, 28780}, {1698, 21935}, {1738, 30768}, {1836, 25958}, {1861, 7466}, {1999, 3969}, {2064, 20896}, {2177, 29659}, {2886, 29872}, {2887, 4418}, {2895, 4641}, {2901, 25441}, {3006, 5263}, {3306, 5525}, {3416, 17126}, {3589, 17012}, {3687, 5294}, {3703, 3920}, {3704, 17016}, {3705, 24552}, {3712, 4026}, {3750, 29685}, {3758, 31034}, {3763, 17595}, {3773, 17763}, {3891, 29634}, {3912, 16785}, {3923, 5057}, {3925, 29873}, {3932, 5297}, {3966, 17127}, {3977, 4357}, {3980, 25957}, {4011, 25960}, {4062, 4649}, {4195, 5016}, {4256, 19867}, {4360, 29833}, {4363, 30811}, {4365, 29863}, {4370, 27776}, {4416, 31143}, {4417, 26223}, {4422, 5241}, {4427, 24723}, {4438, 31330}, {4642, 19879}, {4851, 14996}, {4886, 19742}, {4933, 21806}, {5051, 7283}, {5233, 17354}, {5247, 20653}, {5249, 20106}, {5262, 17698}, {5295, 24883}, {5333, 17189}, {5718, 17369}, {5739, 26065}, {5741, 27064}, {5743, 27065}, {5744, 29611}, {5791, 13726}, {5880, 25959}, {5955, 9780}, {6292, 17292}, {6350, 24635}, {6535, 29683}, {6703, 17019}, {7270, 11115}, {9709, 13730}, {16729, 25280}, {16823, 24542}, {17021, 17243}, {17061, 29874}, {17063, 29677}, {17122, 29687}, {17155, 26128}, {17234, 26627}, {17256, 27081}, {17293, 21488}, {17351, 17484}, {17360, 31303}, {17367, 24277}, {17521, 19842}, {17557, 19857}, {17592, 29647}, {17598, 29686}, {17599, 29648}, {17889, 31237}, {19262, 26446}, {19846, 28612}, {21857, 27320}, {24160, 25669}, {24325, 29632}, {24342, 29862}, {24593, 29587}, {24594, 29629}, {25496, 29849}, {26064, 31445}, {26070, 29591}, {26685, 30479}, {26738, 30828}, {30566, 30867}

X(32779) =


X(32780) = X(1)X(3695)∩X(2)X(38)

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

X(32780) lies on these lines: {1, 3695}, {2, 38}, {8, 3791}, {10, 58}, {12, 57}, {31, 29667}, {36, 19867}, {55, 29659}, {65, 19879}, {75, 25453}, {81, 15523}, {86, 29653}, {190, 4425}, {238, 5294}, {292, 16587}, {306, 4649}, {312, 29635}, {321, 29631}, {334, 8033}, {345, 17592}, {354, 29637}, {409, 5260}, {583, 17303}, {726, 19786}, {750, 29679}, {846, 4026}, {894, 2887}, {940, 29674}, {958, 5329}, {985, 5278}, {986, 19784}, {1089, 25441}, {1125, 17598}, {1126, 21081}, {1211, 1757}, {1213, 3509}, {1376, 4265}, {1460, 9708}, {1621, 29685}, {1723, 5831}, {1762, 3925}, {1961, 3932}, {1999, 3773}, {2161, 2886}, {2895, 4722}, {3219, 24697}, {3329, 3705}, {3339, 19880}, {3589, 29821}, {3624, 3677}, {3634, 18201}, {3666, 29633}, {3678, 24931}, {3679, 5269}, {3741, 17289}, {3742, 17357}, {3757, 6679}, {3772, 29856}, {3790, 29841}, {3846, 27064}, {3873, 24943}, {3891, 29636}, {3912, 4038}, {3953, 19881}, {3966, 16468}, {3976, 19836}, {3980, 4429}, {4011, 17354}, {4284, 5750}, {4358, 29845}, {4359, 29850}, {4363, 17889}, {4385, 19806}, {4388, 4672}, {4418, 4972}, {4645, 4697}, {4650, 26034}, {4671, 29864}, {4703, 17350}, {4850, 29663}, {5143, 18235}, {5249, 30768}, {5251, 5285}, {5263, 29673}, {5749, 26098}, {5955, 6048}, {7186, 17792}, {7262, 26065}, {16706, 24165}, {17023, 17600}, {17056, 29862}, {17061, 29859}, {17123, 17353}, {17263, 25501}, {17279, 26102}, {17280, 29837}, {17355, 24210}, {17381, 29644}, {17597, 29660}, {17599, 29646}, {18165, 22279}, {19684, 29643}, {19792, 20083}, {20368, 26446}, {24295, 29655}, {24586, 29576}, {24725, 25958}, {24746, 28653}, {25760, 26223}, {25961, 26627}, {26040, 26929}, {27798, 28604}, {28605, 29868}, {28606, 29647}, {31019, 31237}

X(32780) =


X(32781) = X(2)X(31)∩X(6)X(29663)

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

X(32781) lies on these lines: {2, 31}, {6, 29663}, {10, 38}, {37, 29687}, {42, 141}, {55, 3763}, {63, 1698}, {81, 29633}, {210, 17237}, {226, 31264}, {321, 3821}, {333, 29850}, {354, 29685}, {536, 6535}, {595, 19881}, {612, 17306}, {672, 1213}, {756, 3263}, {758, 19867}, {896, 3634}, {899, 1211}, {940, 29647}, {968, 17284}, {982, 29667}, {984, 29679}, {1001, 29677}, {1125, 17469}, {1150, 25453}, {1215, 17184}, {1386, 29684}, {1468, 19784}, {1621, 29637}, {1737, 11031}, {1738, 21020}, {1962, 3912}, {2177, 3619}, {2223, 6292}, {2228, 20966}, {2308, 3589}, {2886, 31241}, {2975, 19879}, {3006, 6682}, {3416, 17017}, {3666, 3844}, {3683, 17357}, {3715, 17253}, {3720, 4026}, {3741, 4972}, {3744, 29686}, {3745, 17384}, {3757, 17291}, {3769, 29636}, {3773, 17147}, {3775, 4651}, {3812, 27714}, {3831, 5051}, {3873, 29659}, {3915, 19836}, {3925, 30970}, {3932, 3989}, {3936, 6685}, {3969, 4970}, {4001, 4722}, {4085, 17135}, {4358, 4425}, {4413, 7085}, {4418, 17289}, {4429, 31330}, {4655, 26223}, {4657, 5311}, {4660, 24552}, {4683, 27064}, {4698, 8040}, {4987, 5750}, {5014, 29652}, {5224, 26037}, {5251, 7293}, {5282, 17303}, {5314, 24931}, {5372, 29868}, {5745, 30768}, {5846, 29819}, {6057, 17246}, {6690, 29865}, {8013, 17239}, {8620, 16587}, {8728, 28274}, {9342, 31247}, {9345, 18141}, {11246, 17369}, {11680, 29827}, {14829, 29631}, {16062, 19810}, {16570, 19876}, {16707, 16887}, {16818, 20985}, {17283, 29851}, {17370, 29852}, {17398, 21764}, {17674, 28269}, {17716, 25539}, {17763, 19786}, {19812, 29847}, {19877, 26065}, {24697, 27065}, {25351, 27798}, {25527, 29828}, {25914, 28352}, {26128, 26227}, {28599, 29823}, {28606, 29674}, {29678, 30811}

X(32781) =


X(32782) = X(2)X(6)∩X(7)X(19822)

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

X(32782) lies on these lines: {2, 6}, {7, 19822}, {8, 3891}, {9, 21367}, {10, 3681}, {63, 17272}, {75, 17184}, {76, 321}, {100, 26034}, {192, 3969}, {210, 3844}, {238, 24943}, {305, 16703}, {306, 4357}, {308, 18096}, {312, 17228}, {319, 3187}, {320, 19808}, {329, 29611}, {405, 26064}, {464, 18642}, {518, 29667}, {594, 3782}, {748, 29637}, {756, 29674}, {956, 19258}, {961, 10372}, {964, 1330}, {980, 7794}, {984, 15523}, {986, 20653}, {997, 6505}, {1008, 4388}, {1043, 17676}, {1255, 17316}, {1257, 15474}, {1352, 4220}, {1386, 29648}, {1757, 26061}, {1999, 17287}, {2049, 26131}, {2345, 5905}, {2476, 3454}, {2886, 25958}, {2887, 3775}, {3175, 17229}, {3210, 17236}, {3219, 4376}, {3305, 17284}, {3416, 3920}, {3662, 4359}, {3666, 17237}, {3679, 23681}, {3686, 26723}, {3687, 4850}, {3703, 7226}, {3739, 27186}, {3741, 11680}, {3791, 29636}, {3821, 21085}, {3836, 26037}, {3840, 25960}, {3842, 29854}, {3846, 30942}, {3923, 4683}, {3925, 25959}, {3948, 17550}, {3966, 7191}, {3995, 17233}, {4026, 17018}, {4062, 17592}, {4193, 17182}, {4202, 9534}, {4360, 20017}, {4363, 17483}, {4384, 26724}, {4389, 17147}, {4415, 4671}, {4416, 5294}, {4418, 4655}, {4429, 4651}, {4469, 8024}, {4641, 17344}, {4649, 29647}, {4656, 29594}, {4657, 17011}, {4851, 17019}, {4886, 16706}, {4966, 29814}, {4974, 29852}, {4981, 29641}, {5044, 18732}, {5051, 10449}, {5256, 17306}, {5262, 5814}, {5263, 6327}, {5271, 17270}, {5273, 26932}, {5287, 17296}, {5337, 7854}, {5564, 19796}, {5793, 20060}, {5846, 29815}, {6682, 29849}, {6989, 12359}, {7232, 26842}, {7321, 19797}, {7465, 11442}, {7761, 24271}, {7789, 21508}, {7795, 21511}, {7800, 21495}, {10371, 17016}, {10519, 26118}, {13728, 19767}, {14907, 16046}, {16342, 25650}, {16585, 25078}, {16887, 18601}, {17123, 29677}, {17227, 19804}, {17239, 31019}, {17257, 17776}, {17275, 24789}, {17279, 27065}, {17289, 26223}, {17292, 27064}, {17293, 17484}, {17325, 20182}, {17355, 17781}, {17360, 19812}, {17361, 19827}, {17717, 31241}, {17889, 21020}, {18135, 26601}, {18200, 24921}, {18229, 31266}, {18747, 31018}, {19819, 32087}, {20173, 26593}, {20208, 21482}, {20929, 20955}, {21024, 31060}, {21240, 29576}, {24988, 26038}, {25539, 29684}, {25914, 27625}, {26747, 27641}, {27038, 27096}, {27047, 27263}, {27131, 30818}, {27476, 27495}, {28654, 30473}

X(32782) =


X(32783) = X(1)X(2)∩X(75)X(26128)

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

X(32783) lies on these lines: {1, 2}, {75, 26128}, {141, 171}, {238, 1211}, {242, 1848}, {261, 7305}, {319, 3791}, {320, 4697}, {321, 23689}, {333, 3775}, {350, 18697}, {594, 17061}, {740, 19786}, {846, 4357}, {1011, 23850}, {1213, 3684}, {1215, 17289}, {1283, 4199}, {1376, 3763}, {1707, 17272}, {1757, 5294}, {1891, 4212}, {2238, 16470}, {2308, 2895}, {2309, 27270}, {2345, 21101}, {2887, 5263}, {3416, 17716}, {3550, 26034}, {3662, 3980}, {3681, 26061}, {3683, 24697}, {3685, 4425}, {3740, 17357}, {3750, 4026}, {3769, 17228}, {3846, 30832}, {3923, 27184}, {3967, 17359}, {3971, 17280}, {3996, 4085}, {4038, 4966}, {4046, 4716}, {4104, 17353}, {4204, 5259}, {4279, 28242}, {4417, 25496}, {4418, 17184}, {4640, 17237}, {4643, 4797}, {4647, 19787}, {4657, 17592}, {4676, 4703}, {4682, 17231}, {4693, 4854}, {4734, 17383}, {4886, 4974}, {4970, 17302}, {5227, 17754}, {5247, 17698}, {5249, 24342}, {5284, 31247}, {5285, 16056}, {5329, 11358}, {5739, 16468}, {5743, 17123}, {5814, 16478}, {5955, 24174}, {7110, 14844}, {8300, 24542}, {10180, 17322}, {16800, 17277}, {17291, 24169}, {17293, 17725}, {17306, 17594}, {17358, 27538}, {17490, 26150}, {17591, 17740}, {17889, 25527}, {18491, 19540}, {19808, 24325}, {24259, 31004}, {24295, 27064}, {24552, 25760}

X(32783) =


X(32784) = X(1)X(141)∩X(2)X(31)

Barycentrics    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(32784) lies on these lines: {1, 141}, {2, 31}, {3, 29020}, {5, 6210}, {6, 29633}, {8, 3775}, {9, 46}, {10, 75}, {11, 29827}, {12, 1423}, {37, 3844}, {38, 29667}, {43, 1211}, {69, 4649}, {81, 29647}, {165, 19542}, {192, 3773}, {242, 5117}, {256, 5051}, {306, 17592}, {312, 4425}, {333, 25453}, {405, 7295}, {451, 2212}, {516, 7377}, {518, 17237}, {519, 17399}, {524, 28650}, {545, 19875}, {740, 3661}, {756, 29679}, {894, 4655}, {958, 19879}, {960, 25144}, {982, 24163}, {1001, 3763}, {1125, 3883}, {1150, 29631}, {1215, 27184}, {1279, 29660}, {1386, 17384}, {1429, 12588}, {1621, 24943}, {1716, 4205}, {1757, 4643}, {1861, 24341}, {2209, 27270}, {2214, 16470}, {2345, 24248}, {2354, 5142}, {3097, 25349}, {3219, 26061}, {3220, 5251}, {3454, 27042}, {3589, 16468}, {3616, 17232}, {3618, 16477}, {3619, 16484}, {3624, 7290}, {3634, 17353}, {3662, 24325}, {3670, 4446}, {3679, 7174}, {3685, 17292}, {3696, 17239}, {3705, 6682}, {3751, 17272}, {3755, 4923}, {3757, 26128}, {3758, 17770}, {3769, 19812}, {3790, 17247}, {3823, 4708}, {3831, 21257}, {3842, 4019}, {3873, 29685}, {3915, 21289}, {3923, 17289}, {3925, 24310}, {3932, 4364}, {3966, 29821}, {3980, 19808}, {3993, 17233}, {4073, 6734}, {4197, 28287}, {4202, 31339}, {4356, 29594}, {4362, 19786}, {4393, 17772}, {4417, 6685}, {4432, 29613}, {4488, 7229}, {4514, 29652}, {4660, 5263}, {4663, 17344}, {4664, 6541}, {4672, 17368}, {4676, 17371}, {4683, 26223}, {4687, 25354}, {4693, 29611}, {4703, 27064}, {4710, 30473}, {4741, 17771}, {4972, 31330}, {4974, 17367}, {5047, 7301}, {5220, 17253}, {5234, 19880}, {5247, 19784}, {5271, 19834}, {5278, 29850}, {5284, 29677}, {5294, 7262}, {5361, 29868}, {5372, 29864}, {5625, 17391}, {5692, 19867}, {5695, 17293}, {5718, 29825}, {5743, 16569}, {5749, 24695}, {5847, 17023}, {5904, 22277}, {6211, 26446}, {6536, 29687}, {7083, 11108}, {8728, 27626}, {9791, 17280}, {11680, 31241}, {12587, 18162}, {14829, 29635}, {15254, 17357}, {15523, 28606}, {15569, 17231}, {15601, 19872}, {16475, 29598}, {16476, 16818}, {16706, 16825}, {16739, 18057}, {16823, 17291}, {16830, 17326}, {16927, 30174}, {17064, 18229}, {17236, 24349}, {17308, 17738}, {17327, 19856}, {17350, 26083}, {17469, 29648}, {17593, 17740}, {17719, 29828}, {17722, 29826}, {17725, 26227}, {17755, 25351}, {17889, 31993}, {19717, 20290}, {19804, 24169}, {19855, 27509}, {19877, 26685}, {20994, 24273}, {21912, 26040}, {24206, 31394}, {24217, 30942}, {24231, 31178}, {24693, 29610}, {24931, 25440}, {25005, 25024}, {26030, 26772}, {26064, 27320}, {26251, 26580}, {28595, 29641}, {29640, 30811}, {29678, 30831}, {29822, 31017}, {30940, 30966}, {31053, 31264}

X(32784) =






leftri  Points with first barycentric of type h f(a,b,c) + k S: X(32785)-X(32841)  rightri

Centers X(32785)-X(32841) were contributed by Clark Kimberling, June 20, 2019:

Centers X(32785) - X(32790) . . . . . . . of form h a^2 + k S : :
Centers X(32791) - X(32804) . . . . . . . of form h b c + k S : :
Centers X(32804) - X(32841) . . . . . . . of form h SA^2 + k S : :

underbar

X(32785) = (name pending)

Barycentrics    a^2 + 3 S : :

X(32785) lies on these lines: {2, 6}, {3, 18538}, {4, 5418}, {5, 6221}, {10, 13902}, {20, 6411}, {30, 6451}, {125, 18923}, {140, 1587}, {187, 11292}, {239, 32795}, {371, 3090}, {372, 3525}, {376, 6564}, {381, 6445}, {485, 631}, {486, 5067}, {488, 8376}, {546, 6449}, {547, 13785}, {549, 6452}, {550, 23253}, {574, 11291}, {605, 17123}, {606, 17122}, {632, 3312}, {638, 33364}, {641, 13651}, {642, 7376}, {894, 32796}, {1030, 21565}, {1125, 13893}, {1131, 15717}, {1151, 3091}, {1267, 17117}, {1327, 19708}, {1588, 1656}, {1589, 22052}, {1590, 10979}, {1600, 8939}, {1698, 8983}, {2045, 5335}, {2046, 5334}, {2066, 10589}, {2067, 10588}, {3035, 31413}, {3070, 3523}, {3071, 5056}, {3085, 9661}, {3086, 9646}, {3146, 6409}, {3247, 5393}, {3311, 3628}, {3317, 6435}, {3366, 18581}, {3391, 18582}, {3522, 23251}, {3524, 6560}, {3526, 6395}, {3533, 5420}, {3545, 6480}, {3590, 6434}, {3616, 13911}, {3624, 13883}, {3627, 6455}, {3634, 18991}, {3731, 6351}, {3817, 9616}, {3832, 6433}, {3851, 9690}, {3855, 9680}, {4254, 21546}, {4294, 31499}, {4361, 32799}, {4363, 32800}, {4413, 13887}, {5054, 6446}, {5068, 6468}, {5070, 7584}, {5071, 6565}, {5072, 6407}, {5094, 13884}, {5120, 21549}, {5124, 21568}, {5326, 19037}, {5391, 17116}, {5412, 8889}, {5432, 13898}, {5433, 13897}, {5550, 7968}, {6118, 13834}, {6119, 13770}, {6419, 13939}, {6424, 31404}, {6425, 15022}, {6427, 13993}, {6439, 9543}, {6447, 12812}, {6450, 14869}, {6456, 12108}, {6481, 15702}, {6496, 15704}, {6519, 12811}, {6622, 11473}, {6667, 19113}, {6683, 19090}, {6721, 19056}, {6722, 19109}, {6723, 19111}, {7288, 31472}, {7294, 18995}, {7389, 9600}, {7484, 13889}, {7486, 31454}, {7509, 8276}, {7582, 10577}, {7749, 31411}, {7786, 8992}, {7808, 13885}, {7914, 13892}, {7969, 9780}, {8167, 18999}, {8227, 13912}, {8993, 31268}, {8997, 14061}, {8998, 15059}, {9582, 18483}, {9583, 10175}, {9602, 18584}, {9615, 19925}, {9675, 31415}, {9679, 31418}, {10299, 23269}, {10533, 32064}, {10819, 15081}, {11444, 12239}, {11513, 16051}, {12240, 15028}, {12257, 26951}, {12900, 19060}, {13888, 13936}, {13894, 15184}, {13896, 24953}, {13906, 26364}, {13907, 26363}, {13922, 31272}, {13924, 19102}, {13973, 19877}, {14241, 15719}, {15492, 30412}, {15694, 18512}, {15703, 18510}, {16239, 19117}, {16419, 19006}, {16814, 30413}, {17538, 22644}, {17819, 23332}, {18755, 21991}, {18992, 19862}, {19004, 19872}, {19058, 22247}, {19089, 31239}, {19108, 31274}, {19112, 31235}, {19116, 31487}, {19355, 23291}, {24842, 27191}}

X(32785) = {X(2),X(6)}-harmonic conjugate of X(32786)


X(32786) = (name pending)

Barycentrics    a^2 - 3 S : :

X(32786) lies on these lines: {2, 6}, {3, 18762}, {4, 5420}, {5, 6398}, {10, 13959}, {20, 6412}, {30, 6452}, {125, 18924}, {140, 1588}, {187, 11291}, {239, 32796}, {371, 3525}, {372, 3090}, {376, 6565}, {381, 6446}, {485, 5067}, {486, 631}, {546, 6450}, {547, 13665}, {549, 6451}, {550, 23263}, {574, 11292}, {605, 17122}, {606, 17123}, {632, 3311}, {637, 9675}, {641, 7375}, {642, 13770}, {894, 32795}, {1030, 21568}, {1125, 13947}, {1132, 15717}, {1151, 10303}, {1152, 3091}, {1267, 17116}, {1328, 19708}, {1587, 1656}, {1589, 10979}, {1590, 22052}, {1599, 8943}, {1698, 13971}, {2045, 5334}, {2046, 5335}, {3070, 5056}, {3071, 3523}, {3146, 6410}, {3247, 5405}, {3312, 3628}, {3316, 6436}, {3367, 18581}, {3392, 18582}, {3522, 23261}, {3524, 6561}, {3526, 6199}, {3533, 5418}, {3545, 6481}, {3591, 6433}, {3616, 13973}, {3624, 13902}, {3627, 6456}, {3634, 18992}, {3731, 6352}, {3832, 6434}, {3851, 23253}, {4254, 21549}, {4361, 32800}, {4363, 32799}, {4413, 13940}, {5054, 6445}, {5068, 6469}, {5070, 7583}, {5071, 6564}, {5072, 6408}, {5094, 13937}, {5120, 21546}, {5124, 21565}, {5326, 19038}, {5391, 17117}, {5413, 8889}, {5414, 10589}, {5432, 13955}, {5433, 13954}, {5550, 7969}, {6118, 13651}, {6119, 13711}, {6420, 13886}, {6423, 31404}, {6426, 15022}, {6428, 13925}, {6448, 12812}, {6449, 14869}, {6455, 12108}, {6480, 15702}, {6490, 9692}, {6497, 15704}, {6502, 10588}, {6522, 12811}, {6622, 11474}, {6667, 19112}, {6683, 19089}, {6721, 19055}, {6722, 19108}, {6723, 19110}, {7294, 18996}, {7388, 13934}, {7484, 13943}, {7486, 31414}, {7509, 8277}, {7581, 10576}, {7786, 13983}, {7808, 13938}, {7914, 13946}, {7968, 9780}, {8167, 19000}, {8227, 13975}, {9602, 12221}, {10299, 23275}, {10534, 32064}, {10820, 15081}, {11444, 12240}, {11514, 16051}, {12239, 15028}, {12900, 19059}, {13883, 13942}, {13911, 19877}, {13948, 15184}, {13953, 24953}, {13964, 26364}, {13965, 26363}, {13989, 14061}, {13991, 31272}, {14226, 15719}, {15492, 30413}, {15694, 18510}, {15703, 18512}, {16239, 19116}, {16419, 19005}, {16814, 30412}, {17538, 22615}, {17820, 23332}, {17851, 19709}, {18966, 31408}, {18991, 19862}, {19003, 19872}, {19057, 22247}, {19090, 31239}, {19105, 32209}, {19109, 31274}, {19113, 31235}, {19356, 23291}, {24843, 27191}}

X(32786) = {X(2),X(6)}-harmonic conjugate of X(32785)


X(32787) = (name pending)

Barycentrics    3 a^2 + 2 S : :

X(32787) lies on these lines: {2, 6}, {3, 9680}, {4, 3592}, {5, 6419}, {16, 15764}, {20, 6425}, {30, 371}, {32, 32421}, {39, 13821}, {44, 5393}, {61, 18585}, {62, 15765}, {99, 19058}, {115, 13703}, {140, 6420}, {239, 32801}, {372, 549}, {376, 1151}, {381, 485}, {397, 2043}, {398, 2044}, {428, 5412}, {486, 5055}, {519, 7969}, {547, 7584}, {550, 6453}, {551, 7968}, {588, 1989}, {589, 30537}, {631, 3594}, {671, 19109}, {894, 32802}, {903, 24819}, {1124, 10072}, {3524, 7581}, {1327, 3830}, {1335, 10056}, {1384, 13712}, {1503, 11241}, {1504, 5309}, {1506, 13921}, {1585, 1990}, {1586, 6749}, {1588, 3545}, {1656, 6427}, {1657, 6447}, {1702, 31162}, {2066, 3058}, {2067, 5434}, {2482, 8997}, {2549, 13644}, {2965, 26912}, {3003, 8962}, {3094, 22722}, {3146, 31414}, {3163, 8968}, {3167, 11209}, {3241, 19066}, {3299, 3582}, {3301, 3584}, {3304, 31475}, {3312, 5054}, {3365, 16962}, {3390, 16963}, {3523, 6426}, {3526, 6428}, {3530, 6454}, {3534, 6221}, {3543, 6459}, {3590, 15022}, {3679, 13911}, {3828, 13936}, {3839, 6470}, {3845, 6564}, {4254, 21558}, {4361, 32797}, {4363, 32798}, {4370, 24842}, {4428, 13887}, {4995, 5414}, {5058, 7753}, {5064, 5410}, {5066, 6565}, {5070, 10195}, {5071, 7582}, {5120, 21561}, {5254, 12962}, {5286, 26289}, {5298, 6502}, {5319, 11313}, {5405, 16666}, {5413, 13884}, {5415, 19024}, {5420, 6418}, {5421, 8963}, {5459, 13917}, {5460, 13916}, {5463, 19074}, {5464, 19076}, {5642, 8998}, {6054, 19056}, {6055, 8980}, {6174, 13922}, {6200, 8703}, {6351, 16885}, {6352, 16884}, {6395, 15701}, {6396, 12100}, {6398, 15693}, {6407, 15689}, {6408, 15718}, {6409, 6460}, {6410, 15692}, {6411, 19708}, {6412, 15698}, {6415, 10262}, {6422, 7739}, {6424, 31411}, {6432, 13935}, {6433, 9542}, {6435, 10109}, {6437, 9541}, {6438, 15719}, {6441, 23273}, {6445, 15695}, {6449, 15688}, {6450, 15700}, {6452, 15716}, {6455, 14093}, {6456, 15706}, {6468, 15697}, {6471, 15721}, {6480, 15690}, {6481, 19711}, {6500, 13951}, {6519, 15696}, {6772, 13646}, {6775, 13645}, {6811, 8550}, {7392, 20197}, {7576, 10880}, {7615, 12158}, {7737, 8375}, {7755, 32491}, {7757, 19090}, {7772, 31483}, {7865, 19012}, {7991, 31440}, {8376, 21843}, {8573, 8939}, {8909, 19062}, {8992, 9466}, {9140, 19111}, {9166, 19057}, {9167, 13989}, {9605, 19105}, {9607, 11293}, {9692, 10147}, {10128, 10961}, {10192, 11242}, {10577, 15699}, {10691, 11513}, {10706, 19060}, {10707, 19113}, {10711, 19082}, {10718, 19115}, {10819, 19052}, {11157, 12159}, {11179, 19145}, {11237, 18996}, {11238, 19038}, {11291, 31465}, {11539, 13966}, {11646, 13640}, {12239, 14831}, {12240, 16226}, {13651, 22485}, {13674, 13749}, {13701, 13720}, {13711, 13932}, {13785, 19709}, {13832, 33457}, {13833, 13835}, {13882, 19103}, {13888, 18992}, {13893, 13973}, {13947, 19876}, {13961, 15723}, {13968, 14971}, {15682, 23249}, {15684, 22644}, {16777, 30413}, {17120, 32792}, {17121, 32791}, {19064, 22712}, {19073, 22489}, {19075, 22490}, {19092, 31168}, {28198, 31439}}

X(32787) = complement of X(32808)


X(32788) = (name pending)

Barycentrics    3 a^2 - 2 S : :

X(32788) lies on these lines: {2, 6}, {3, 9681}, {4, 3594}, {5, 6420}, {15, 15764}, {20, 6426}, {30, 372}, {32, 32419}, {39, 13701}, {44, 5405}, {61, 15765}, {62, 18585}, {99, 19057}, {115, 13823}, {140, 6419}, {239, 32802}, {371, 549}, {376, 1152}, {381, 486}, {397, 2044}, {398, 2043}, {428, 5413}, {485, 5055}, {519, 7968}, {547, 7583}, {550, 6454}, {551, 7969}, {588, 30537}, {589, 1989}, {631, 3592}, {671, 19108}, {894, 32801}, {903, 24818}, {1124, 10056}, {1151, 3524}, {1328, 3830}, {1335, 10072}, {1384, 13835}, {1503, 11242}, {1505, 5309}, {1506, 13880}, {1585, 6749}, {1586, 1990}, {1587, 3545}, {1656, 6428}, {1657, 6448}, {1703, 31162}, {2066, 4995}, {2067, 5298}, {2482, 13989}, {2549, 13763}, {3058, 5414}, {3094, 22723}, {3167, 11210}, {3241, 19065}, {3299, 3584}, {3301, 3582}, {3311, 5054}, {3364, 16962}, {3389, 16963}, {3523, 6425}, {3526, 6427}, {3530, 6453}, {3534, 6398}, {3543, 6460}, {3591, 15022}, {3628, 8960}, {3679, 13973}, {3828, 13883}, {3839, 6471}, {3845, 6565}, {4254, 21561}, {4361, 32798}, {4363, 32797}, {4370, 24843}, {4428, 13940}, {5062, 7753}, {5064, 5411}, {5066, 6564}, {5068, 31414}, {5070, 10194}, {5071, 7581}, {5120, 21558}, {5254, 12969}, {5286, 26288}, {5319, 11314}, {5393, 16666}, {5412, 13937}, {5416, 19023}, {5418, 6417}, {5434, 6502}, {5459, 13982}, {5460, 13981}, {5463, 19073}, {5464, 19075}, {5642, 13990}, {6054, 19055}, {6055, 13967}, {6174, 13991}, {6199, 15701}, {6200, 12100}, {6221, 15693}, {6351, 16884}, {6352, 16885}, {6396, 8703}, {6407, 15718}, {6408, 15689}, {6409, 15692}, {6410, 6459}, {6411, 15698}, {6412, 9541}, {6416, 10261}, {6421, 7739}, {6431, 9540}, {6436, 10109}, {6437, 15719}, {6438, 11001}, {6442, 23267}, {6446, 15695}, {6449, 15700}, {6450, 15688}, {6451, 15716}, {6455, 15706}, {6456, 14093}, {6469, 15697}, {6470, 15721}, {6480, 19711}, {6481, 15690}, {6488, 9693}, {6501, 8976}, {6522, 15696}, {6772, 13765}, {6775, 13764}, {6813, 8550}, {7494, 20197}, {7576, 10881}, {7615, 12159}, {7737, 8376}, {7755, 32490}, {7757, 19089}, {7865, 19011}, {8375, 21843}, {8573, 8943}, {8981, 11539}, {8983, 19883}, {8997, 9167}, {9140, 19110}, {9166, 19058}, {9466, 13983}, {9605, 19102}, {9607, 11294}, {9680, 15720}, {10128, 10963}, {10192, 11241}, {10576, 15699}, {10691, 11514}, {10706, 19059}, {10707, 19112}, {10711, 19081}, {10718, 19114}, {11158, 12158}, {11179, 19146}, {11237, 18995}, {11238, 19037}, {11646, 13760}, {12239, 16226}, {12240, 14831}, {13665, 19709}, {13712, 13769}, {13748, 13794}, {13821, 13843}, {13831, 33456}, {13834, 13850}, {13893, 19876}, {13903, 15723}, {13908, 14971}, {13911, 13947}, {13942, 18991}, {13954, 31472}, {15682, 23259}, {15684, 22615}, {16777, 30412}, {17120, 32791}, {17121, 32792}, {19063, 22712}, {19074, 22489}, {19076, 22490}, {19091, 31168}

X(32788) = complement of X(32809)
X(32788) = {X(2),X(6)}-harmonic conjugate of X(32787)


X(32789) = (name pending)

Barycentrics    a^2 + 6 S : :

X(32789) lies on these lines: {2, 6}, {3, 22644}, {4, 6411}, {5, 6200}, {140, 3070}, {187, 32432}, {371, 3628}, {372, 632}, {381, 6451}, {485, 3526}, {486, 5070}, {547, 6480}, {549, 6564}, {574, 6118}, {631, 6412}, {641, 11614}, {642, 32491}, {1030, 21553}, {1151, 3090}, {3525, 23267}, {1327, 15701}, {1587, 3533}, {1656, 3071}, {2045, 5318}, {2046, 5321}, {3091, 6409}, {3316, 13935}, {3366, 33416}, {3391, 33417}, {3523, 23251}, {3594, 13886}, {3624, 13911}, {3634, 7969}, {3723, 5393}, {4361, 32795}, {4363, 32796}, {5024, 13711}, {5054, 6452}, {5055, 6445}, {5056, 6433}, {5067, 6437}, {5071, 9541}, {5072, 6455}, {5079, 6449}, {5124, 21492}, {5159, 11513}, {5210, 11292}, {5326, 5414}, {5420, 6395}, {6351, 16677}, {6352, 16674}, {6410, 10303}, {6420, 13925}, {6431, 13939}, {6434, 6460}, {6436, 19117}, {6440, 31414}, {6441, 7582}, {6446, 13665}, {6459, 6468}, {6481, 11539}, {6502, 7294}, {7583, 16239}, {7741, 31499}, {7968, 19862}, {8960, 13966}, {8981, 10577}, {9600, 11313}, {9675, 32490}, {9680, 9690}, {10299, 23253}, {10533, 23332}, {10645, 18585}, {10646, 15765}, {11314, 31415}, {13785, 15703}, {13834, 13882}, {13883, 19878}, {13902, 19877}, {13924, 31483}, {13936, 31253}, {14813, 16967}, {14814, 16966}, {15723, 18512}, {17116, 32792}, {17117, 32791}, {17118, 32800}, {17119, 32799}, {18991, 19872}

X(32789) = complement of X(32807)
X(32789) = {X(2),X(6)}-harmonic conjugate of X(32790)


X(32790) = (name pending)

Barycentrics    a^2 - 6 S : :

X(32790) lies on these

X(32790) lies on these lines: {2, 6}, {3, 22615}, {4, 6412}, {5, 6396}, {140, 3071}, {187, 32435}, {371, 632}, {372, 3628}, {381, 6452}, {485, 5070}, {486, 3526}, {547, 6481}, {549, 6565}, {574, 6119}, {631, 6411}, {639, 9675}, {641, 32490}, {642, 11614}, {1030, 21492}, {1151, 3525}, {3090, 23249}, {1328, 15701}, {1588, 3533}, {1656, 3070}, {2045, 5321}, {2046, 5318}, {2066, 5326}, {2067, 7294}, {3091, 6410}, {3317, 9540}, {3367, 33416}, {3392, 33417}, {3523, 23261}, {3592, 13939}, {3624, 13973}, {3634, 7968}, {3723, 5405}, {4361, 32796}, {4363, 32795}, {5024, 13834}, {5054, 6451}, {5055, 6446}, {5056, 6434}, {5067, 6438}, {5072, 6456}, {5079, 6450}, {5124, 21553}, {5159, 11514}, {5210, 11291}, {5418, 6199}, {6351, 16674}, {6352, 16677}, {6409, 10303}, {6419, 13993}, {6426, 31412}, {6432, 13886}, {6433, 6459}, {6435, 19116}, {6436, 8960}, {6442, 7581}, {6445, 13785}, {6460, 6469}, {6480, 11539}, {7584, 16239}, {7969, 19862}, {9541, 15702}, {9600, 11315}, {10299, 23263}, {10534, 23332}, {10576, 13966}, {10645, 15765}, {10646, 18585}, {11313, 31415}, {13665, 15703}, {13883, 31253}, {13936, 19878}, {13959, 19877}, {14813, 16966}, {14814, 16967}, {15723, 18510}, {17116, 32791}, {17117, 32792}, {17118, 32799}, {17119, 32800}, {18992, 19872}

X(32790) = complement of isotomic conjugate of X(10194)
X(32790) = {X(2),X(6)}-harmonic conjugate of X(32789)


X(32791) = (name pending)

Barycentrics    b c + 2 S : :

X(32791) lies on these

X(32791) = lines: {2, 37}, {7, 32805}, {8, 32806}, {86, 5405}, {239, 590}, {274, 3302}, {319, 491}, {320, 492}, {591, 17364}, {615, 894}, {1270, 17361}, {1271, 17360}, {1991, 17363}, {3068, 3759}, {3069, 3758}, {4360, 5393}, {4361, 8253}, {4363, 8252}, {5590, 17227}, {5591, 17228}, {7321, 32807}, {10436, 16513}, {17116, 32790}, {17117, 32789}, {17120, 32788}, {17121, 32787}, {17335, 30413}, {17336, 30412}, {21296, 32810}, {32099, 32811}

X(32791) = {X(2),X(75)}-harmonic conjugate of X(32792)


X(32792) = (name pending)

Barycentrics    b c - 2 S : :

X(32792) lies on these

X(32792) = lines: {2, 37}, {7, 32806}, {8, 32805}, {86, 5393}, {239, 615}, {274, 3300}, {319, 492}, {320, 491}, {590, 894}, {591, 17363}, {1270, 17360}, {1271, 17361}, {1991, 17364}, {3068, 3758}, {3069, 3759}, {4360, 5405}, {4361, 8252}, {4363, 8253}, {5564, 32807}, {5590, 17228}, {5591, 17227}, {10436, 16512}, {17116, 32789}, {17117, 32790}, {17120, 32787}, {17121, 32788}, {17335, 30412}, {17336, 30413}, {21296, 32811}, {32099, 32810}

X(32792) = {X(2),X(75)}-harmonic conjugate of X(32791)


X(32793) = (name pending)

Barycentrics    2 b c + S : :

X(32793) lies on these

X(32793) = lines: {2, 37}, {7, 1270}, {8, 176}, {239, 7585}, {320, 32814}, {590, 17119}, {594, 5591}, {615, 17118}, {894, 7586}, {1086, 5590}, {1991, 4399}, {3068, 4361}, {3069, 4363}, {3593, 31995}, {3595, 32087}, {3729, 30412}, {4384, 30413}, {4969, 26339}, {5393, 17151}, {5405, 25590}, {5860, 17365}, {5861, 17362}, {7277, 26340}, {8972, 17117}, {13941, 17116}

X(32793) = anticomplement of X(6351)
X(32793) = {X(2),X(75)}-harmonic conjugate of X(32794)


X(32794) = (name pending)

Barycentrics    2 b c - S : :

X(32794) lies on these lines: {2, 37}, {7, 1271}, {8, 175}, {239, 7586}, {319, 32814}, {590, 17118}, {591, 4399}, {594, 5590}, {615, 17119}, {894, 7585}, {1086, 5591}, {3068, 4363}, {3069, 4361}, {3593, 32087}, {3595, 31995}, {3729, 30413}, {4384, 30412}, {4969, 26340}, {5393, 25590}, {5405, 17151}, {5860, 17362}, {5861, 17365}, {7277, 26339}, {8972, 17116}, {13941, 17117}

X(32794) = anticomplement of X(6352)
X(32794) = {X(2),X(75)}-harmonic conjugate of X(32793)


X(32795) = (name pending)

Barycentrics    b c + 3 S : :

X(32795) lies on these lines: {2, 37}, {7, 32807}, {239, 32785}, {319, 3595}, {320, 3593}, {491, 32099}, {492, 21296}, {894, 32786}, {3068, 17121}, {3069, 17120}, {3661, 26362}, {3662, 26361}, {3758, 13941}, {3759, 8972}, {4361, 32789}, {4363, 32790}

X(32795) =


X(32796) = (name pending)

Barycentrics    b c - 3 S : :

X(32796) lies on these

X(32796) = lines: {2, 37}, {8, 32807}, {239, 32786}, {319, 3593}, {320, 3595}, {491, 21296}, {492, 32099}, {894, 32785}, {3068, 17120}, {3069, 17121}, {3661, 26361}, {3662, 26362}, {3758, 8972}, {3759, 13941}, {4361, 32790}, {4363, 32789}


X(