leftri rightri

This is PART 21: Centers X(40001) - X(42000)

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

X(40001) = X(2)X(37)∩X(76)X(18744)

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

X(40001) lies on these lines: {2, 37}, {76, 18744}, {82, 17763}, {304, 17241}, {313, 18151}, {1089, 13741}, {1230, 20916}, {1930, 17283}, {3702, 32850}, {3718, 17335}, {3912, 18714}, {4044, 20445}, {4673, 5100}, {10159, 33944}, {17285, 18697}, {17788, 17791}, {17789, 18143}, {20444, 20917}, {20915, 20927}, {32930, 33760}

X(40002) = X(2)X(3108)∩X(69)X(7394)

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

X(40002) lies on these lines: {2, 3108}, {69, 7394}, {75, 33091}, {76, 1369}, {1272, 6636}, {7768, 37349}, {8024, 14360}, {9464, 19583}, {14023, 16952}, {20934, 39728}, {33090, 33944}

X(40002) = anticomplement of X(3108)

X(40003) = X(2)X(32)∩X(69)X(14247)

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

X(40003) lies on these lines: {2, 32}, {69, 14247}, {76, 38946}, {827, 7767}, {7826, 14885}, {10159, 40000}

X(40004) = X(2)X(39735)∩X(37)X(16727)

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

X(40004) lies on these lines: {2, 39735}, {37, 16727}, {75, 3873}, {76, 17234}, {85, 4751}, {86, 2481}, {274, 20448}, {310, 4043}, {870, 16709}, {1218, 2350}

X(40005) = X(2)X(40008)∩X(76)X4043)

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

X(40005) lies on these lines: {2, 40008}, {76, 4043}, {310, 2388}, {561, 33932}, {1500, 23989}, {6063, 33298}, {17911, 17913}, {18031, 18140}, {29433, 39797}

X(40005) = isotomic conjugate of X(8053)

X(40006) = X(1)X(2)∩X(69)X(3730)

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

X(40006) lies on these lines: {1, 2}, {69, 3730}, {75, 17758}, {76, 4043}, {100, 29473}, {101, 34065}, {141, 1500}, {304, 4568}, {334, 29757}, {1018, 17137}, {1043, 29775}, {1330, 20533}, {1930, 3930}, {1978, 33778}, {2140, 17143}, {2141, 17295}, {2276, 16887}, {2321, 20888}, {3263, 4006}, {3454, 26590}, {3501, 17296}, {3695, 4437}, {3726, 24166}, {3797, 24068}, {3881, 24631}, {3933, 6184}, {3969, 20913}, {3970, 20911}, {3987, 26562}, {3996, 17682}, {4103, 33932}, {4153, 24057}, {4551, 28777}, {4851, 17750}, {6376, 17240}, {6381, 21071}, {7270, 29743}, {8715, 24586}, {12782, 33087}, {14210, 33299}, {16060, 33771}, {16549, 30941}, {16589, 17243}, {17144, 17761}, {17206, 24047}, {17231, 20691}, {17232, 24190}, {17280, 17499}, {18047, 29792}, {18152, 40008}, {18720, 20914}, {20932, 20933}, {21067, 33931}, {22011, 33935}, {24170, 30945}, {29789, 35978}, {30949, 32104}

X(40006) = anticomplement of X(2350)

X(40007) = X(2)X(2350)∩X(75)X(3681)

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

X(40007) lies on these lines: {2, 2350}, {75, 3681}, {76, 17135}, {1330, 1369}, {3952, 18138}, {16684, 37632}, {17018, 30946}, {17149, 29824}, {18133, 30941}, {20012, 32105}, {24215, 27635}, {29814, 36854}

X(40008) = X(2)X(40005)∩X(76)X(17135)

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

X(40008) lies on these lines: {2, 40005}, {76, 17135}, {310, 29767}, {561, 18137}, {6063, 17077}, {18031, 18064}, {18152, 40006}

leftri

Points on cubics: X(40009)-X(40046)

rightri

This preamble is contributed by Clark Kimberling, October 13, 2020; modified January 31, 2021.

In column 1 of the following table, the appearance of u(a,b,c) in a row means that the points indicated in column 2 lie on the cubic

u(a,b,c)(by-cz)(ax+by)(ax+cz) + u(b,c,a)(cz-ax)(by+cz)(by+cz) + u(c,a,b)(ax-by)(cz+ax)(cz+by) = 0.

The appearance of {i,j} in column 2 means that {X(i),X(j)} are a pair of X(560)-isoconjugates that lie on the cubic.

b+c {2998,6374}
(b+c)^2 {18133,40010}, {18140,40013}
a^2 (b+c)^2 {17758,18152}, {18137,39735}
(b+c)(b+c-a) {18135,40012}
(b+c)^2 cos^2 A {18134,40011}
(b+c)(2a-b-c) {4358,20568}, {18145,39994}, {39995,40039}, {39996,40040}, {39997,40041}
(b+c)(3a+b+c) {18135,40012}
(b+c)(-3a+b+c) {18134,40013}, {20934,40026}
(b+c)(4a+b+c) {4671,205690}
(b+c)(2a+3b+3c) {4359,32018}
(b+c)(-a+2b+2c) {18146,40021}, {30829,40029}
(b+c)(a^2+b^2+c^2+a(a+b+c)) {870,33931}
(b+c)(a+2b+2c)) {19804,40023}
(b+c)(bc+ca+ab+a(a+b+c)) {20913,40024}
(b+c)(bc+ca+ab-a(a+b+c)) {3948,40017}, {18032,20947}
(b+c)(bc+ca+ab-2a(a+b+c)) {31060,40031}
(b+c)(2(bc+ca+ab)-a(a+b+c)) {30758,40028}, {30830,40030}
(b+c)(abc+a(a^2+b^2+c^2)) {10159,40020}, {33944,40033}
(b+c)(2abc+a(a^2+b^2+c^2)) {32000,40032}
(b+c)(abc+a(bc+ca+ab)) {10,310}, {4043,40004}
(b+c)(abc-a(bc+ca+ab)) {17758,18152}
(b+c)(2abc-a(bc+ca+ab)) {20923,40025}
(b+c)(b^2c^2+c^2a^2+a^2b^2+a^2bc) {3934,31630}
(b+c)(b^2c^2+c^2a^2+a^2b^2-a^2bc) {39,40016}
(b+c)(2a+b+c) {274,321}, {35058,40034}
(b+c)(abc-a(a^2+b^2+c^2)) {83,8024}, {1031,40035}, {1369,40036}, {20933,40037}, {33938,40038}

X(40009) = X(560)-ISOCONJUGATE OF X(1370)

Barycentrics    b^2*c^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(-a^6 - a^4*b^2 + a^2*b^4 + b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 - b^2*c^4 - c^6) : :
Barycentrics    (csc^2 A) (a^2 tan A - b^2 tan B + c^2 tan C) (a^2 tan A + b^2 tan B - c^2 tan C) : :

X(40009) lies on these lines: {76, 17907}, {83, 26209}, {305, 315}, {2207, 36793}, {4150, 20914}, {28695, 31636}

X(40009) = isotomic conjugate of X(159)
X(40009) = polar conjugate of X(3162)
X(40009) = X(4)-cross conjugate of X(76)
X(40009) = trilinear pole of line X(3267)X(33294)

X(40010) = X(560)-ISOCONJUGATE OF X(18133)

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

X(40010) lies on these lines: {2, 18040}, {7, 18147}, {27, 18742}, {75, 24046}, {86, 13741}, {272, 18738}, {312, 39700}, {335, 18137}, {350, 8049}, {673, 29437}, {2296, 30963}, {4286, 29712}, {4360, 29454}, {6376, 30598}, {6384, 30596}, {14621, 18046}, {18143, 27145}, {18149, 20932}, {18739, 37633}, {20028, 30939}, {29486, 29772}

X(40010) = isotomic conjugate of X(3216)

X(40011) = X(560)-ISOCONJUGATE OF X(18134)

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

X(40011) lies on these lines: {2, 349}, {8, 313}, {29, 264}, {76, 333}, {85, 18726}, {92, 20926}, {272, 1234}, {312, 27801}, {1235, 19768}, {1259, 34387}, {1305, 1311}, {1502, 28660}, {19607, 28917}

X(40011) = isotomic conjugate of X(579)
X(40011) = polar conjugate of isogonal conjugate of isotomic conjugate of X(5125)

X(40012) = X(560)-ISOCONJUGATE OF X(18135)

Barycentrics    b*c*(a*b + b^2 - 2*a*c + b*c)*(-2*a*b + a*c + b*c + c^2) : :
Barycentrics    1/(a^2 - 4 R r) : :

X(40012) lies on these lines: {2, 34283}, {4, 18141}, {10, 982}, {83, 940}, {98, 8690}, {141, 34258}, {226, 17234}, {312, 4052}, {321, 3662}, {345, 30866}, {801, 25934}, {1751, 14829}, {2051, 18134}, {3963, 6539}, {4035, 37865}, {4417, 14554}, {6557, 36805}, {8033, 32014}, {10453, 13576}, {14534, 37674}, {17678, 19792}, {17758, 18136}, {17786, 24177}

X(40012) = isogonal conjugate of X(16946)
X(40012) = isotomic conjugate of X(4383)
X(40012) = polar conjugate of X(4186)
X(40012) = cevapoint of X(1086) and X(4391)
X(40012) = trilinear pole of line X(523)X(3777)

X(40013) = X(560)-ISOCONJUGATE OF X(18140)

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

X(40013) lies on the Kiepert hyperbola and these lines: {2, 3770}, {4, 37482}, {10, 38}, {75, 6539}, {76, 16703}, {81, 83}, {98, 19649}, {141, 321}, {226, 4358}, {239, 29757}, {312, 4080}, {873, 18140}, {1086, 28654}, {1150, 1751}, {1577, 8042}, {2051, 3936}, {3416, 4863}, {3720, 30982}, {3765, 26978}, {3912, 22010}, {3948, 17758}, {4049, 4391}, {5192, 18169}, {5741, 14554}, {7248, 10404}, {8024, 16727}, {8025, 18046}, {14534, 37633}, {14829, 24624}, {17147, 18040}, {17307, 30599}, {17790, 26842}, {18059, 24731}, {19742, 29484}, {20917, 28606}, {26540, 37874}, {27797, 28605}, {30807, 36907}, {32782, 34258}

X(40013) = isogonal conjugate of X(2220)
X(40013) = isotomic conjugate of X(32911)
X(40013) = polar conjugate of X(4222)
X(40013) = trilinear pole of line X(523)X(2530)
X(40013) = cevapoint of X(1086) and X(1577)

X(40014) = X(560)-ISOCONJUGATE OF X(18743)

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

X(40014) lies on these lines: {75, 3617}, {76, 4052}, {85, 5226}, {274, 8056}, {279, 37758}, {286, 30939}, {304, 20568}, {334, 20943}, {341, 1111}, {767, 1293}, {870, 3445}, {1088, 27829}, {2481, 3680}, {6383, 21615}, {7319, 21296}, {10563, 32104}, {19604, 31643}, {20569, 27820}, {20942, 21605}, {24796, 36926}, {27834, 37130}, {32018, 33934}

X(40014) = isotomic conjugate of X(1743)

X(40015) = X(560)-ISOCONJUGATE OF X(20914)

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

X(40015) lies on these lines: {75, 17903}, {305, 20914}, {312, 18629}, {341, 1370}, {2064, 14615}

X(40015) = isotomic conjugate of X(1763)
X(40015) = polar conjugate of X(36103)
X(40015) = X(4)-cross conjugate of X(75)

X(40016) = X(560)-ISOCONJUGATE OF X(39)

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

X(40016) lies on the Kiepert hyperbola and these lines: {2, 308}, {4, 18022}, {10, 18833}, {76, 19562}, {83, 1207}, {98, 689}, {99, 34452}, {141, 31630}, {251, 3407}, {262, 305}, {561, 18066}, {690, 18008}, {804, 17995}, {1916, 4609}, {3051, 9230}, {11606, 23962}, {16890, 18901}, {16893, 18896}, {24624, 37204}, {34087, 34294}

X(40016) = isogonal conjugate of X(41331)
X(40016) = isotomic conjugate of X(3051)
X(40016) = polar conjugate of X(27369)
X(40016) = cevapoint of X(i) and X(j) for these {i,j}: {75, 18050}, {76, 1502}
X(40016) = trilinear pole of line X(523)X(14603)
X(40016) = trilinear product X(i)*X(j) for these {i,j}: {2, 18833}, {75, 308}, {76, 3112}, {82, 1502}, {83, 561}, {251, 1928}, {523, 37204}, {670, 18070}, {689, 1577}, {850, 4593}, {1799, 1969}, {1926, 14970}, {3115, 20627}, {3405, 18024}, {4577, 20948}, {6385, 18082}, {6386, 10566}, {18022, 34055}, {18090, 38812}, {18097, 40072}, {20889, 31622}, {30505, 33778}, {32085, 40364}, {37221, 40074}

X(40017) = X(560)-ISOCONJUGATE OF X(3948)

Barycentrics    b*(a + b)*c*(a + c)*(b^2 - a*c)*(a*b - c^2) : :
Barycentrics    (sec A)/(cot C sin 2B sin(C - A) + cot B sin 2C sin(B - A)) : :

X(40017) lies on these lines: {2, 799}, {4, 811}, {10, 274}, {76, 4602}, {83, 1509}, {86, 741}, {98, 36036}, {99, 8299}, {142, 34021}, {226, 4554}, {292, 31996}, {310, 321}, {350, 9505}, {670, 1086}, {1581, 18298}, {2051, 34020}, {2311, 17206}, {2394, 33805}, {2669, 3783}, {3834, 30938}, {4049, 20568}, {4080, 4639}, {4440, 36860}, {4444, 31001}, {4583, 18157}, {4589, 4645}, {4623, 25536}, {6384, 30953}, {6539, 16748}, {7245, 16712}, {7304, 37676}, {11611, 20924}, {16705, 30669}, {17234, 34022}, {17758, 18140}, {30588, 30990}, {30964, 31006}, {30992, 30993}, {30997, 32020}, {39786, 39925}

X(40017) = isogonal conjugate of X(41333)
X(40017) = isotomic conjugate of X(2238)
X(40017) = polar conjugate of X(862)
X(40017) = cevapoint of X(i) and X(j) for these {i,j}: {2, 30941}, {6, 16876}, {75, 3948}, {334, 335}, {514, 23822}, {1086, 3766}, {18827, 36800}
X(40017) = trilinear pole of line X(75)X(523)
X(40017) = trilinear product X(i)*X(j) for these {i,j}: {2, 18827}, {7, 36800}, {27, 337}, {58, 18895}, {75, 37128}, {76, 741}, {81, 334}, {86, 335}, {99, 4444}, {274, 291}, {292, 310}, {331, 1808}, {333, 7233}, {513, 4639}, {514, 4589}, {561, 18268}, {660, 7199}, {670, 3572}, {693, 4584}, {799, 876}, {875, 4602}, {982, 40834}, {1019, 4583}, {1434, 4518}, {1577, 36066}, {1581, 8033}, {1911, 6385}, {1916, 17103}, {1930, 39276}, {2311, 6063}, {4017, 36806}, {4369, 18829}, {4374, 37134}, {4481, 41072}, {4562, 7192}, {4610, 35352}, {5378, 16727}, {17096, 36801}, {30663, 30940}, {30669, 32010}, {33295, 40098}, {39747, 40093}, {39950, 40094}

X(40018) = X(560)-ISOCONJUGATE OF X(4398)

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

X(40018) lies on these lines: {7, 3264}, {76, 39710}, {313, 4373}, {673, 29541}, {903, 3596}, {1269, 36588}, {6548, 35519}

X(40019) = X(560)-ISOCONJUGATE OF X(8032)

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

X(40019) lies on these lines: {}

X(40020) = X(560)-ISOCONJUGATE OF X(40007)

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

X(40020) lies on these lines: on lines {18137, 40006}, {18152, 40007}

X(40021) = X(560)-ISOCONJUGATE OF X(18146)

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

X(40021) lies on these lines: {2, 39960}, {10, 4392}, {83, 14996}, {321, 17227}, {1029, 18141}, {1751, 5372}, {4080, 17232}, {30588, 30829}

X(40021) = isotomic conjugate of X(14997)

X(40022) = X(560)-ISOCONJUGATE OF X(18840)

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

X(40022) lies on these lines: {2, 39}, {3, 16276}, {4, 16275}, {22, 1078}, {25, 183}, {32, 16950}, {51, 69}, {83, 5359}, {99, 7485}, {111, 1239}, {115, 8890}, {141, 3981}, {251, 6179}, {262, 31630}, {311, 7494}, {315, 6997}, {316, 7394}, {325, 37439}, {338, 8556}, {350, 612}, {428, 7750}, {614, 1909}, {1007, 1232}, {1184, 7770}, {1235, 6353}, {1241, 8770}, {1269, 30758}, {1369, 7533}, {1370, 11185}, {1611, 11324}, {1613, 24256}, {1627, 3972}, {1915, 8177}, {1975, 7484}, {1995, 33651}, {2052, 37187}, {2979, 33798}, {3596, 26234}, {3734, 16951}, {3760, 5268}, {3761, 5272}, {3785, 6995}, {3846, 18067}, {3917, 18906}, {3963, 26274}, {4074, 21001}, {4417, 18052}, {4563, 17811}, {5025, 21248}, {5249, 21590}, {5475, 8878}, {5651, 37894}, {5943, 14994}, {6374, 16986}, {6394, 6641}, {6636, 7771}, {7398, 14615}, {7467, 22712}, {7499, 37688}, {7500, 14907}, {7667, 32819}, {7752, 37990}, {7782, 15246}, {7802, 34603}, {7878, 34482}, {9230, 16990}, {10327, 17143}, {10565, 30737}, {11174, 33769}, {13595, 26233}, {13881, 30785}, {15004, 39099}, {15437, 32983}, {15466, 18022}, {17234, 18138}, {18142, 18143}, {18835, 29634}, {19188, 34384}, {20965, 32451}, {21415, 25961}, {21609, 21617}, {26257, 34481}, {30786, 31255}, {33854, 34283}, {34816, 37876}

X(40022) = isotomic conjugate of X(39951)
X(40022) = polar conjugate of isogonal conjugate of X(3785)

X(40023) = X(560)-ISOCONJUGATE OF X(19840)

Barycentrics    b*c*(a + 3*b + c)*(a + b + 3*c) : :

X(40023) lies on these lines: on lines {75, 3701}, {76, 30713}, {85, 321}, {274, 312}, {286, 318}, {304, 20569}, {767, 8694}, {870, 2334}, {2481, 4385}, {4606, 37130}, {5556, 32099}, {6385, 28659}, {20568, 33935}, {31643, 39126}

X(40023) = isotomic conjugate of X(1449)
X(40023) = polar conjugate of X(5338)

X(40024) = X(560)-ISOCONJUGATE OF X(20913)

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

X(40024) lies on these lines: {2, 30940}, {10, 350}, {83, 2238}, {226, 10030}, {274, 17758}, {308, 594}, {321, 1921}, {3112, 4651}, {3783, 5263}, {4444, 7199}, {4665, 31625}, {11599, 39028}, {14534, 37676}, {21443, 34475}, {21897, 25368}, {29792, 34016}

X(40024) = isotomic conjugate of X(24512)

X(40025) = X(560)-ISOCONJUGATE OF X(20923)

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

X(40025) lies on these lines: {9, 274}, {33, 286}, {37, 85}, {75, 210}, {76, 2321}, {312, 6385}, {331, 1826}, {870, 34445}, {2481, 3875}, {4664, 39735}, {6383, 20335}, {18032, 20930}, {18159, 39467}

X(40025) = isotomic conjugate of X(21384)

X(40026) = X(560)-ISOCONJUGATE OF X(20942)

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

X(40026) lies on these lines: {75, 4678}, {85, 36621}, {274, 36603}, {767, 8699}, {20568, 33780}, {20925, 32018}, {20942, 21605}

X(40026) = isotomic conjugate of X(3973)

X(40027) = X(560)-ISOCONJUGATE OF X(20943)

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

X(40027) lies on these lines: {1, 32011}, {2, 17448}, {7, 24495}, {75, 3840}, {86, 18192}, {244, 8026}, {310, 30957}, {312, 27494}, {335, 18743}, {350, 4373}, {673, 36630}, {675, 29227}, {903, 34020}, {2296, 30950}, {4106, 38238}, {4479, 36588}, {4871, 6384}, {14621, 36614}, {17149, 31002}, {17234, 20528}, {20335, 27498}, {30947, 39741}

X(40027) = isotomic conjugate of X(16569)

X(40028) = X(560)-ISOCONJUGATE OF X(30758)

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

X(40028) lies on these lines: {75, 966}, {76, 4087}, {85, 350}, {274, 988}, {312, 334}, {767, 28847}, {4385, 32018}, {4479, 18032}, {20930, 39735}, {30758, 30830}

X(40028) = isotomic conjugate of X(3751)

X(40029) = X(560)-ISOCONJUGATE OF X(30829)

Barycentrics    b*(a + b - 5*c)*(-a + 5*b - c)*c : :

X(40029) lies on these lines: {75, 4723}, {85, 4358}, {274, 39963}, {312, 20568}, {767, 6014}, {2481, 4900}, {18146, 20569}

X(40029) = isotomic conjugate of X(16670)

X(40030) = X(560)-ISOCONJUGATE OF X(30830)

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

X(40030) lies on the Kiepert hyperbola and these lines: {4, 30941}, {10, 3761}, {69, 13576}, {310, 34258}, {17758, 18135}

X(40030) = isotomic conjugate of X(37657)
X(40030) = trilinear pole of the line X(523)X(4411)

X(40031) = X(560)-ISOCONJUGATE OF X(31060)

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

X(40031) lies on these lines: {2, 39952}, {10, 24524}, {321, 31028}, {4080, 30964}

X(40031) = isotomic conjugate of X(37673)

X(40032) = X(560)-ISOCONJUGATE OF X(32000)

Barycentrics    b^2*c^2*(-a^2 + b^2 + c^2)*(a^4 + 6*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 + 6*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    2*Cot[A] / (3 + Cos[2*A]) : :

X(40032) lies on these lines: {2, 800}, {69, 185}, {76, 253}, {95, 16035}, {235, 264}, {311, 36889}, {3260, 8797}

X(40032) = isotomic conjugate of X(1593)

X(40033) = X(560)-ISOCONJUGATE OF X(33944)

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

X(40033) lies on these lines: {2, 21021}, {7, 7211}, {12, 7249}, {27, 1840}, {75, 24169}, {86, 1215}, {310, 1237}, {312, 14621}, {321, 6650}, {4385, 6384}, {17725, 30598}, {17762, 20715}, {17763, 18099}, {20934, 39728}

X(40033) = isotomic conjugate of X(29821)

X(40034) = X(560)-ISOCONJUGATE OF X(35058)

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

X(40034) lies on these lines: {10, 35538}, {75, 596}, {76, 321}, {141, 21412}, {274, 27163}, {670, 1509}, {1921, 18050}, {1930, 35544}, {1978, 18140}, {3159, 18133}, {3262, 14615}, {3739, 6374}, {4075, 6376}, {4087, 33940}, {4485, 33945}, {17495, 29765}, {18040, 22011}, {18135, 26774}, {18146, 35652}, {18152, 33775}, {20924, 21596}, {21240, 21435}, {21595, 21598}, {33764, 34016}

X(40034) = isotomic conjugate of isogonal conjugate of X(17147)

X(40035) = X(560)-ISOCONJUGATE OF X(1031)

Barycentrics    b^2*c^2*(-a^4 + a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 + c^4) : :
Barycentrics    A-power of symmedial circle : :

X(40035) lies on these lines: {2, 39999}, {6, 76}, {75, 29673}, {304, 17786}, {305, 3314}, {570, 7799}, {626, 24733}, {1031, 7779}, {1502, 17949}, {2896, 28677}, {3313, 7768}, {7788, 14615}, {18835, 24732}, {20934, 21083}, {32452, 39468}

X(40035) = isotomic conjugate of X(14370)

X(40036) = X(560)-ISOCONJUGATE OF X(1369)

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

X(40036) lies on these lines: {76, 38946}, {1369, 5189}, {7878, 18018}, {20933, 21064}

X(40036) = isogonal conjugate of X(39)-Ceva conjugate of X(32)
X(40036) = isotomic conjugate of X(2916)
X(40036) = polar conjugate of X(8792)

X(40037) = X(560)-ISOCONJUGATE OF X(20933)

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

X(40037) lies on these lines: {1369, 20934}, {8024, 20933}, {17087, 28780}

X(40037) = isotomic conjugate of X(16555)

X(40038) = X(560)-ISOCONJUGATE OF X(33938)

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

X(40038) lies on these lines: {2, 16720}, {75, 29673}, {86, 7194}, {335, 3782}, {673, 24631}, {1369, 39723}, {3665, 7249}, {3673, 6384}, {3961, 20955}, {8024, 33938}, {20924, 29656}, {24326, 39745}, {27918, 39746}, {29655, 33940}

X(40038) = isotomic conjugate of X(3961)

X(40039) = X(560)-ISOCONJUGATE OF X(39995)

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

Let P and U be the circumcircle intercepts of the Nagel line. Then X(40039) = isotomic conjugate of {P,U}-harmonic conjugate of X(1). (Randy Hutson, October 29, 2020)

X(40039) lies on these lines: {2, 4033}, {7, 18133}, {27, 6335}, {75, 21208}, {86, 668}, {310, 6386}, {313, 17205}, {903, 18145}, {6376, 39704}, {6548, 21606}, {6650, 17790}, {18149, 20937}

X(40040) = X(560)-ISOCONJUGATE OF X(39996)

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

X(40040) lies on these lines: {4358, 39699}, {18145, 30939}

X(40041) = X(560)-ISOCONJUGATE OF X(39997)

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

X(40041) lies on these lines: {519, 39995}, {18145, 39997}, {20568, 39699}

X(40042) = X(560)-ISOCONJUGATE OF X(39999)

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

X(40042) lies on these lines: {3589, 7839}, {39998, 39999}, {40000, 40002}

X(40043) = X(560)-ISOCONJUGATE OF X(40000)

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

X(40043) lies on these lines: {2, 39999}, {76, 1031}, {141, 28677}, {15523, 28676}, {39998, 40000}

X(40044) = X(560)-ISOCONJUGATE OF X(40001)

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

X(40044) lies on these lines: {75, 33091}, {85, 17371}, {21598, 39999}, {33944, 40003}, {39998, 40001}

X(40045) = X(560)-ISOCONJUGATE OF X(40002)

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

X(40045) lies on these lines: {39998, 40002}, {39999, 40003}

X(40046) = X(560)-ISOCONJUGATE OF X(40003)

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

X(40046) lies on these lines: {76, 1369}, {39998, 40003}

X(40047) = X(74)X(15478)∩X(131)X(15329)

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

X(40047) lies on the cubic K039 and these lines: {74, 15478}, {131, 15329}, {186, 925}, {13496, 13754}

X(40047) = circumcircle inverse of X(40048)

X(40048) = X(68)X(526)∩X(265)X(924)

Barycentrics    1/((a^2 - b^2)*(a^2 - c^2)*(a^10*b^2 - 3*a^8*b^4 + 2*a^6*b^6 + 2*a^4*b^8 - 3*a^2*b^10 + b^12 + a^10*c^2 - 2*a^8*b^2*c^2 + 3*a^6*b^4*c^2 - 7*a^4*b^6*c^2 + 8*a^2*b^8*c^2 - 3*b^10*c^2 - 3*a^8*c^4 + 3*a^6*b^2*c^4 + 6*a^4*b^4*c^4 - 5*a^2*b^6*c^4 + 3*b^8*c^4 + 2*a^6*c^6 - 7*a^4*b^2*c^6 - 5*a^2*b^4*c^6 - 2*b^6*c^6 + 2*a^4*c^8 + 8*a^2*b^2*c^8 + 3*b^4*c^8 - 3*a^2*c^10 - 3*b^2*c^10 + c^12)) : :

X(40048) lies on the Jerabek circumhyperbola and these lines: {68, 526}, {265, 924}, {523, 5504}, {690, 34801}, {2970, 15328}, {3566, 34802}, {6391, 9003}, {9033, 15316}, {11559, 20184}

X(40048) = isogonal conjugate of X(40049)
X(40048) = circumcircle-inverse of X(40047)

X(40049) = X(2)X(3)∩X(99)X(16167)

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

X(40049) lies on these lines: {2, 3}, {99, 16167}, {110, 924}, {476, 925}, {523, 23181}, {1147, 39371}, {1304, 13398}, {1624, 3233}, {3258, 23217}, {3565, 9060}, {14480, 36829}, {16166, 20185}, {17702, 39986}

X(40049) = isogonal conjugate of X(40048)
X(40049) = circumcircle-inverse of X(30512)
X(40049) = X(523)-vertex conjugate of X(30512)
X(40049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1113, 1114, 30512}, {3658, 7477, 37964}, {4226, 7468, 7482}, {4230, 7472, 7468}, {7468, 7480, 7471}, {7471, 15329, 7480}

X(40050) = X(560)-ISOCONJUGATE OF X(25)

Barycentrics    b^4*c^4*(-a^2 + b^2 + c^2) : :
Barycentrics    Cot[A]*Csc[A]^4 : :

X(40050) lies on these lines: {2, 1241}, {4, 683}, {69, 4173}, {75, 23664}, {76, 141}, {83, 9516}, {99, 15270}, {194, 35540}, {304, 20727}, {305, 1368}, {315, 670}, {706, 33786}, {1235, 5117}, {1613, 3978}, {1975, 16084}, {3266, 7906}, {3673, 18891}, {3926, 28438}, {4176, 20023}, {5025, 35524}, {7760, 16285}, {7770, 9230}, {11059, 31406}, {14376, 34254}, {18840, 40016}, {39129, 40009}

X(40050) = isotomic conjugate of X(1974)
X(40050) = isogonal conjugate of isotomic conjugate of X(40360)
X(40050) = polar conjugate of X(36417)

leftri

Dao-perspeconics: X(40051)-X(40070)

rightri

This preamble and centers X(40051)-X(40070) were contributed by César Eliud Lozada, October 14, 2020.

Let ABC, A'B'C' be two perspective triangles, neither inscribed in the other. Let T1 be the triangle bounded by the lines BC', CA', AB' and let T2 be the triangle bounded by the lines BA', CB', AC'. Then the vertices of T1 and T2 lie all on a conic (Dao Thanh Oai, October 13, 2020). This conic will be named here the Dao-perspeconic of ABC and A'B'C'.

The appearance of (T, n) in the following partial list means that the center of the Dao-perspeconic of triangles ABC and T is X(n):

(ABC-X3 reflections, 3), (anti-Aquila, 40051), (anti-Ara, 40052), (anti-Conway, 15648), (2nd anti-Conway, 15649), (anti-excenters-reflections, 40053), (anti-Honsberger, 40054), (anti-inverse-in-incircle, 7800), (anti-tangential-midarc, 40055), (Apus, 40056), (Aquila, 15650), (Ara, 15651), (4th Brocard, 40057), (9th Brocard, 2996), (circummedial, 15652), (circumorthic, 15653), (2nd circumperp, 15654), (circumsymmedial, 15655), (Conway, 15656), (2nd Conway, 966), (5th Euler, 40058), (excenters-reflections, 40059), (extangents, 40060), (outer-Garcia, 10), (Gossard, 402), (Honsberger, 15657), (infinite-altitude, 3), (inverse-in-Conway, 40061), (inverse-in-incircle, 15658), (Johnson, 5), (2nd Johnson-Yff, 40062), (1st Kenmotu diagonals, 15659), (2nd Kenmotu diagonals, 15660), (Mandart-incircle, 40062), (midheight, 15661), (2nd mixtilinear, 15662), (3rd mixtilinear, 15663), (4th mixtilinear, 40063), (5th mixtilinear, 1), (6th mixtilinear, 40064), (orthic axes, 40065), (reflection, 15664), (1st Sharygin, 40066), (inner-squares, 40067), (outer-squares, 40068), (2nd inner-Vecten, 1132), (2nd outer-Vecten, 1131), (1st Zaniah, 40069), (2nd Zaniah, 40070)

Definitions of all triangles above mentioned can be found in the index of triangles.

X(40051) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ANTI-AQUILA

Barycentrics    a*(2*a^5+7*(b+c)*a^4+5*(b^2+b*c+c^2)*a^3-(b+c)*(5*b^2+29*b*c+5*c^2)*a^2-(7*b^4+7*c^4+(47*b^2+87*b*c+47*c^2)*b*c)*a-(b+2*c)*(2*b+c)*(b+c)*(b^2+4*b*c+c^2))*(2*a+b+c) : :

X(40051) lies on these lines: {1,15650}, {3647,11281}

X(40051) = midpoint of X(1) and X(15650)

X(40052) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ANTI-ARA

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

X(40052) lies on these lines: {25,15651}, {1843,19595}, {3867,9969}

X(40053) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ANTI-EXCENTERS-REFLECTIONS

Barycentrics    (SB+SC)*((SA+24*R^2)*S^2+4*(SW^2+4*(2*SA-SW)*R^2-32*R^4)*SA) : :

X(40053) lies on these lines: {3,13474}, {4,33580}, {11414,18840}

X(40054) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ANTI-HONSBERGER

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

X(40054) lies on these lines: {141,206}, {1974,17409}

X(40055) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ANTI-TANGENTIAL-MIDARC

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

X(40055) lies on these lines: {2,10571}, {1042,1402}

X(40056) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND APUS

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

X(40056) lies on these lines: {2271,9406}, {15817,32664}

X(40057) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 4th BROCARD

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

X(40057) lies on these lines: {111,251}, {15820,15899}

X(40058) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 5th EULER

Barycentrics    (3*a^10-32*(b^2+c^2)*a^8-2*(27*b^4+46*b^2*c^2+27*c^4)*a^6-100*(b^2+c^2)*b^2*c^2*a^4+(19*b^8+19*c^8-2*(10*b^4+31*b^2*c^2+10*c^4)*b^2*c^2)*a^2+20*(b^4-c^4)*(b^2-c^2)*b^2*c^2)*(b^2+c^2) : :

X(40058) lies on these lines: {39,15880}, {141,3787}

X(40059) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND EXCENTERS-REFLECTIONS

Barycentrics    a*(3*a^5-7*(b+c)*a^4-2*(5*b^2-2*b*c+5*c^2)*a^3+2*(b+c)*(5*b^2-8*b*c+5*c^2)*a^2+(7*b^2-18*b*c+7*c^2)*(b+c)^2*a-(b^2-c^2)*(b-c)*(b-3*c)*(3*b-c))*(a-3*b-3*c) : :

X(40059) lies on these lines: {10,3090}, {3340,10563}

X(40060) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND EXTANGENTS

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

X(40060) lies on these lines: {228,1334}, {15830,38015}

X(40061) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND INVERSE-IN-CONWAY

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

X(40061) lies on these lines: {10,3781}, {10471,10473}

X(40062) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND MANDART-INCIRCLE

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

X(40062) lies on the line {15280,15845}

X(40063) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 4th MIXTILINEAR

Barycentrics    a^2*(a^7-3*(b+c)*a^6+(b^2+12*b*c+c^2)*a^5+(b+c)*(5*b^2-8*b*c+5*c^2)*a^4-(5*b^4+5*c^4+2*(16*b^2-29*b*c+16*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(b^2-38*b*c+c^2)*a^2+3*(b^4+c^4-2*(b^2+7*b*c+c^2)*b*c)*(b-c)^2*a-(b^2-c^2)*(b-c)^3*(b^2+4*b*c+c^2)) : :

X(40063) lies on these lines: {3,15855}, {41,37541}, {55,32625}, {101,1615}, {165,198}, {284,11051}, {2267,15288}

X(40063) = center of the cross-perspeconic of these triangles: ABC and 4th mixtilinear

X(40064) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 6th MIXTILINEAR

Barycentrics    a*(a^8+8*(b+c)*a^7-4*(b^2-4*b*c+c^2)*a^6-8*(b+c)*(9*b^2-16*b*c+9*c^2)*a^5+2*(67*b^2+86*b*c+67*c^2)*(b-c)^2*a^4-72*(b^2-c^2)^2*(b+c)*a^3-4*(b^4+c^4-6*(3*b^2+5*b*c+3*c^2)*b*c)*(b-c)^2*a^2+8*(b^2-c^2)*(b-c)^3*(b^2+4*b*c+c^2)*a+(b^2+6*b*c+c^2)*(b-c)^6) : :

X(40064) lies on these lines: {55,1419}, {223,15856}, {14522,14547}

X(40065) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND ORTHIC AXES

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

X(40065) lies on these lines: {2,15905}, {4,6}, {5,38292}, {9,34231}, {25,37665}, {30,15851}, {51,6618}, {69,36794}, {193,458}, {216,376}, {232,7714}, {264,1992}, {275,459}, {281,1743}, {284,37417}, {317,3618}, {340,3619}, {378,1285}, {386,37379}, {389,3183}, {391,11109}, {427,5304}, {469,37666}, {562,2963}, {566,35503}, {572,37410}, {577,631}, {579,37028}, {1033,1597}, {1073,14362}, {1119,4644}, {1449,7952}, {1585,7586}, {1586,7585}, {1609,3520}, {1656,33636}, {1724,7498}, {1745,22063}, {1785,16667}, {1870,3553}, {1885,33893}, {1968,13342}, {1993,6819}, {2193,6988}, {2548,6622}, {3068,3536}, {3069,3535}, {3079,17810}, {3088,30435}, {3090,3284}, {3091,36413}, {3147,5063}, {3329,37187}, {3524,36748}, {3528,36751}, {3529,5158}, {3554,6198}, {3815,38282}, {3945,26003}, {4254,37305}, {4383,37276}, {5024,37460}, {5065,6353}, {5081,5749}, {5094,37689}, {5120,7412}, {5200,5410}, {5222,7282}, {5413,19219}, {5422,6820}, {5839,7046}, {6524,15004}, {6620,12167}, {6621,15873}, {6623,15484}, {6803,23115}, {6995,14930}, {7378,16318}, {7401,22120}, {7487,9605}, {7494,10313}, {7577,9722}, {7718,9575}, {7735,8889}, {8553,35473}, {8779,11427}, {9748,37074}, {10299,22052}, {10979,21735}, {11348,20208}, {11513,36701}, {11514,36703}, {13567,19039}, {14860,15077}, {16328,35489}, {22124,34048}, {22240,34608}, {34545,37192}, {36743,37441}, {37448,37681}

X(40065) = polar conjugate of the isogonal conjugate of X(17809)
X(40065) = polar conjugate of the isotomic conjugate of X(3523)
X(40065) = barycentric product X(i)*X(j) for these {i, j}: {4, 3523}, {264, 17809}, {631, 11282}
X(40065) = trilinear product X(i)*X(j) for these {i, j}: {19, 3523}, {92, 17809}
X(40065) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(3523)}} and {{A, B, C, X(6), X(17809)}}
X(40065) = crosssum of X(3) and X(15851)
X(40065) = orthosymmedial-circle-inverse of X(1249)
X(40065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 27377, 32001), (4, 6, 1249), (4, 33630, 53), (6, 1249, 5702), (6, 3087, 4), (6, 6748, 393), (6, 6749, 3087), (193, 458, 32000), (393, 3087, 6748), (393, 6748, 4), (1587, 1588, 12233), (7736, 10311, 6353), (19039, 19040, 13567), (19041, 19042, 427)

X(40066) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 1st SHARYGIN

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

X(40066) lies on these lines: {6651,15864}, {8424,16372}, {8845,8932}

X(40067) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND INNER-SQUARES

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

X(40067) lies on these lines: {6,641}, {25,371}, {32,8939}, {6680,40068}

X(40068) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND OUTER-SQUARES

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

X(40068) lies on these lines: {6,642}, {25,372}, {32,8943}, {6680,40067}

X(40068) = {X(372), X(8855)}-harmonic conjugate of X(8996)

X(40069) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 1st ZANIAH

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

X(40069) lies on these lines: {1,11051}, {281,1886}

X(40070) = CENTER OF THE DAO-PERSPECONIC OF THESE TRIANGLES: ABC AND 2nd ZANIAH

Barycentrics    a*(a^5-9*(b+c)*a^4+2*(11*b^2+6*b*c+11*c^2)*a^3-2*(b+c)*(11*b^2-14*b*c+11*c^2)*a^2+3*(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3)*(-a+b+c)*(a+c-3*b)*(a+b-3*c) : :

X(40070) lies on these lines: {277,5745}, {3680,15853}

X(40071) = X(560)-ISOCONJUGATE OF X(27)

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

X(40071) lies on these lines: {10, 75}, {42, 1241}, {69, 11573}, {99, 19842}, {183, 19844}, {190, 4456}, {274, 19808}, {304, 305}, {321, 1228}, {325, 19839}, {336, 3682}, {346, 27250}, {349, 6358}, {350, 37042}, {668, 7270}, {683, 2333}, {714, 23664}, {1010, 1909}, {1078, 19841}, {1231, 26942}, {1969, 18022}, {1975, 19845}, {1978, 33805}, {2064, 21595}, {3695, 20235}, {3719, 19807}, {3765, 19281}, {3948, 16583}, {3975, 37086}, {4087, 18835}, {4150, 20914}, {4384, 19792}, {4386, 33731}, {7283, 16085}, {7763, 19795}, {8024, 19835}, {18135, 19785}, {18140, 19786}, {18145, 19796}, {18152, 19787}, {18153, 19790}, {19794, 32832}, {19798, 40025}, {19806, 30022}, {19810, 28660}, {19811, 34384}, {19822, 34284}, {20917, 37097}, {21063, 21094}

X(40071) = isotomic conjugate of X(1474)
X(40071) = polar conjugate of isogonal conjugate of isotomic conjugate of X(8747)

X(40072) = X(560)-ISOCONJUGATE OF X(65)

Barycentrics    b^3*(a + b)*c^3*(a + c)*(-a + b + c) : :
Barycentrics    (csc^3 A) / (cos B + cos C) : :

X(40072) lies on these lines: {38, 75}, {76, 1211}, {86, 4485}, {264, 305}, {274, 1920}, {312, 28659}, {314, 3706}, {321, 4469}, {333, 20665}, {668, 22275}, {670, 18816}, {869, 7033}, {1812, 4631}, {1921, 3666}, {3596, 3703}, {3665, 6063}, {6386, 20566}, {7018, 18891}, {8024, 31089}, {13588, 14195}, {16703, 35543}, {18138, 21596}

X(40072) = isotomic conjugate of X(1402)
X(40072) = polar conjugate of isogonal conjugate of isotomic conjugate of X(1880)

X(40073) = X(560)-ISOCONJUGATE OF X(66)

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

X(40073) lies on these lines: {6, 76}, {264, 305}, {311, 20023}, {315, 3313}, {393, 3926}, {570, 7763}, {670, 14615}, {1975, 12143}, {3596, 35551}, {4150, 20641}, {6374, 35542}, {6382, 35547}, {7774, 8024}, {7792, 40025}, {8264, 9865}, {16276, 32085}, {16989, 39998}, {17907, 34254}, {18024, 20563}, {19562, 35530}, {20806, 31636}, {20968, 38842}, {23642, 33734}, {39129, 40009}

X(40073) = isogonal conjugate of X(40146)
X(40073) = isotomic conjugate of X(2353)
X(40073) = polar conjugate of isogonal conjugate of X(34254)

X(40074) = X(560)-ISOCONJUGATE OF X(67)

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

X(40074) lies on these lines: {6, 33301}, {76, 524}, {251, 308}, {264, 305}, {311, 33769}, {316, 9019}, {328, 18024}, {338, 3978}, {523, 14603}, {670, 3260}, {702, 9865}, {892, 1236}, {1235, 38294}, {2393, 39266}, {3266, 18023}, {4590, 15014}, {7840, 8024}, {8859, 26235}, {14295, 33919}, {18896, 35542}, {20944, 21094}, {22329, 40025}, {37765, 37804}

X(40074) = isotomic conjugate of X(3455)
X(40074) = polar conjugate of isogonal conjugate of X(37804)

X(40075) = X(560)-ISOCONJUGATE OF X(80)

Barycentrics    b^2*c^2*(-a^2 + b^2 - b*c + c^2) : :
Barycentrics    (1 - 2 Cos[A])*Csc[A]^3 : :

X(40075) lies on these lines: {75, 33120}, {76, 4358}, {86, 310}, {274, 33129}, {305, 561}, {693, 784}, {1233, 18152}, {1909, 32927}, {1920, 8024}, {1921, 3266}, {3836, 18066}, {3936, 20924}, {4766, 17789}, {7018, 21415}, {7112, 20944}, {17234, 18054}, {18037, 24602}, {18835, 20880}, {18895, 35545}, {20893, 21241}, {20947, 29854}, {25760, 33930}, {33108, 33933}, {34284, 37759}

X(40075) = isotomic conjugate of X(6187)

X(40076) = ISOGONAL CONJUGATE OF X(5134)

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

X(40076) lies on the cubic K039 and these lines: {3, 2772}, {27, 116}, {58, 22084}, {63, 35342}, {101, 1796}, {103, 186}, {222, 36075}, {2392, 17972}, {4466, 14377}, {5196, 39993}

X(40076) = isogonal conjugate of X(5134)
X(40076) = isogonal conjugate of the anticomplement of X(17729)
X(40076) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5134}, {37, 5196}
X(40076) = trilinear pole of line {1459, 2308}
X(40076) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5134}, {58, 5196}

X(40077) = ISOGONAL CONJUGATE OF X(38947)

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

X(40077) lies on the cubic K039 and these lines: {3, 2421}, {186, 2698}, {419, 2679}, {511, 2966}, {805, 15391}

X(40077) = reflection of X(2966) in the Lemoine axis
X(40077) = circumcircle-inverse of X(9513)
X(40077) = isogonal conjugate of X(38947)
X(40077) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38947}, {1316, 1581}
X(40077) = barycentric product X(385)*X(9513)
X(40077) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 38947}, {1691, 1316}, {9513, 1916}

X(40078) = ISOGONAL CONJUGATE OF X(34169)

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

X(40078) lies on the cubic K039 and these lines: {3, 351}, {126, 4235}, {186, 1296}, {187, 4558}, {3455, 6091}, {7472, 34171}, {10717, 13586}, {14417, 34161}

X(40078) = isogonal conjugate of X(34169)
X(40078) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34169}, {897, 10418}, {7472, 23894}
X(40078) = cevapoint of X(187) and X(9177)
X(40078) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34169}, {187, 10418}, {5467, 7472}, {9177, 31655}

X(40079) = X(3)X(525)∩X(74)X(187)

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

X(40079) lies on the cubic K039 and these lines: {3, 525}, {74, 187}, {98, 186}, {237, 1503}, {378, 35906}, {2071, 2966}, {3520, 32545}, {5621, 34369}, {5866, 6394}, {6091, 11589}, {7418, 34366}, {13754, 17974}, {14355, 15032}, {15407, 36212}

p> X(40079) = circumcircle-inverse of X(879)
X(40079) = X(240)-isoconjugate of X(2697)
X(40079) = barycentric product X(287)*X(2781)
X(40079) = barycentric quotient X(i)/X(j) for these {i,j}: {248, 2697}, {2781, 297}

X(40080) = X(3)X(684)∩X(74)X(3455)

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

X(40080) lies on the cubic K039 and these lines: {3, 684}, {74, 3455}, {112, 186}, {132, 2409}, {248, 10766}, {2781, 5191}, {2794, 7422}, {2881, 23350}, {5649, 38699}, {5866, 11589}, {5961, 18876}, {8429, 14998}, {9475, 28343}, {19165, 21525}

X(40080) = circumcircle-inverse of X(35909)
X(40080) = X(i)-isoconjugate of X(j) for these (i,j): {542, 8767}, {2247, 6330}, {18312, 36046}
X(40080) = barycentric product X(i)*X(j) for these {i,j}: {441, 842}, {2409, 35911}, {5641, 8779}, {34211, 35909}
X(40080) = barycentric quotient X(i)/X(j) for these {i,j}: {842, 6330}, {2445, 35907}, {8779, 542}, {35911, 2419}

X(40081) = X(29)X(124)∩X(102)X(186)

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

X(40081) lies on the conic {{A,B,C,X(1), X(3)}}, the cubic K039, and these lines: {3, 2779}, {29, 124}, {102, 186}, {283, 4996}, {758, 1807}, {7100, 11700}, {7424, 39992}

X(40081) =isogonal conjugate of X(38945)
X(40081) =X(i)-isoconjugate of X(j) for these (i,j): {1, 38945}, {65, 7424}
X(40081) =trilinear pole of line {652, 21748}
X(40081) =barycentric quotient X(i)/X(j) for these {i,j}: {6, 38945}, {284, 7424}

X(40082) = ISOGONAL CONJUGATE OF X(34170)

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

X(40082) lies on the cubic K039 and these lines: {3, 113}, {24, 5879}, {25, 38956}, {107, 1105}, {133, 6644}, {186, 1294}, {1204, 2972}, {1609, 28783}, {1658, 38621}, {2071, 10152}, {3515, 34426}, {5866, 6394}, {6716, 17928}, {6760, 13997}, {7488, 38714}, {7526, 36520}, {9530, 15078}, {10714, 37941}, {12096, 13754}, {15469, 15478}

X(40082) = isogonal conjugate of X(34170)
X(40082) = isogonal conjugate of the anticomplement of X(12096)
X(40082) = circumcircle-inverse of X(11744)
X(40082) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34170}, {92, 15262}, {158, 2071}, {1784, 38937}
X(40082) = crosspoint of X(5504) and X(5897)
X(40082) = crosssum of X(403) and X(15311)
X(40082) = barycentric product X(394)*X(11744)
X(40082) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34170}, {184, 15262}, {577, 2071}, {11744, 2052}, {18877, 38937}, {22239, 15352}
X(40082) = {X(3),X(14703)}-harmonic conjugate of X(3184)

X(40083) = ISOGONAL CONJUGATE OF X(34175)

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

X(40083) lies on the cubic K039 and these lines: {3, 690}, {74, 6091}, {99, 186}, {114, 4230}, {187, 13754}, {684, 34157}, {2931, 2936}, {3455, 5961}, {7468, 34174}, {12177, 32599}, {15478, 18876}

X(40083) = isogonal conjugate of X(34175)
X(40083) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34175}, {1821, 2493}, {14984, 36120}
X(40083) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34175}, {237, 2493}, {2421, 14221}, {3289, 14984}, {14966, 7468}

X(40084) = ISOGONAL CONJUGATE OF X(34173)

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

X(40084) lies on the cubic K039 and these lines: {3, 2775}, {120, 4238}, {186, 1292}, {187, 906}, {5172, 5866}, {6091, 34442}

X(40084) = isogonal conjugate of X(34173)

X(40085) = X(81)-ISOCONJUGATE OF X(595)

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

The cubic pK(X(594), X(3995)) is given by

(b+c) (b+c) (y/(c+a)-z/(a+b)) (x/(b+c)+y/(c+a)) (x/(b+c)+z/(a+b))+(c+a) (c+a) (z/(a+b)-x/(b+c)) (y/(c+a)+z/(a+b)) (y/(c+a)+x/(b+c))+(a+b) (a+b) (x/(b+c)-y/(c+a)) (z/(a+b)+x/(b+c)) (z/(a+b)+y/(c+a)) = 0.

Let La be the line tangent to this cubic at A, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(40085). (Peter Moses, October 16, 2020)

The cubic passes through the following 14 points: A, B, C, the vertices of the cevian tiangle of X(3995), the vertices of the anticevian triangle of X(10), and X(i) for i = 10, 37 ,321 ,3159, 3995}. (Peter Moses, October 16, 2020)

X(40085) lies on these lines: {6, 3891}, {37, 39}, {79, 32846}, {86, 1255}, {141, 321}, {335, 4043}, {536, 22012}, {594, 3954}, {674, 1824}, {756, 1213}, {1575, 22013}, {2161, 21061}, {2171, 3649}, {2321, 22035}, {3175, 17392}, {3982, 22034}, {4024, 21143}, {4365, 15320}, {17307, 31025}, {21067, 21858}, {21257, 34475}

X(40085) = X(39747)-Ceva conjugate of X(10)
X(40085) = X(i)-cross conjugate of X(j) for these (i,j): {3122, 523}, {6535, 10}
X(40085) = X(i)-isoconjugate of X(j) for these (i,j): {58, 32911}, {81, 595}, {86, 2220}, {110, 4063}, {162, 22154}, {163, 20295}, {593, 3293}, {662, 4057}, {849, 3995}, {1333, 4360}, {1412, 3871}, {1576, 20949}, {1790, 4222}, {2206, 18140}, {4132, 4556}, {4567, 8054}, {4575, 17922}
X(40085) = cevapoint of X(i) and X(j) for these (i,j): {826, 3120}, {3124, 6367}, {3125, 4024}
X(40085) = crosspoint of X(596) and X(40013)
X(40085) = crosssum of X(595) and X(2220)
X(40085) = trilinear pole of line {3005, 4705}
X(40085) = crossdifference of every pair of points on line {4057, 22154}
X(40085) = barycentric product X(i)*X(j) for these {i,j}: {10, 596}, {37, 40013}, {321, 39798}, {523, 8050}, {594, 39747}, {1089, 39949}, {3701, 20615}, {4024, 37205}, {4036, 34594}
X(40085) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 4360}, {37, 32911}, {42, 595}, {210, 3871}, {213, 2220}, {321, 18140}, {512, 4057}, {523, 20295}, {594, 3995}, {596, 86}, {647, 22154}, {661, 4063}, {756, 3293}, {1577, 20949}, {1824, 4222}, {2501, 17922}, {3120, 21208}, {3122, 8054}, {4024, 4129}, {4705, 4132}, {6535, 4075}, {8013, 4065}, {8050, 99}, {20615, 1014}, {37205, 4610}, {39747, 1509}, {39798, 81}, {39949, 757}, {40013, 274}

X(40086) = X(100)-ISOCONJUGATE OF X(595)

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

The cubic pK(X(1086), X(20295)) is given by

(b+c) (b+c) (y/(c-a)-z/(a-b)) (x/(b-c)+y/(c-a)) (x/(b-c)+z/(a-b))+(c+a) (c+a) (z/(a-b)-x/(b-c)) (y/(c-a)+z/(a-b)) (y/(c-a)+x/(b-c))+(a+b) (a+b) (x/(b-c)-y/(c-a)) (z/(a-b)+x/(b-c)) (z/(a-b)+y/(c-a)) = 0.

Let La be the line tangent to this cubic at A, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(40086). (Peter Moses, October 16, 2020)

The cubic passes through the following 14 points: A, B, C, the vertices of the cevian tiangle of X(20295), the vertices of the anticevian triangle of X(514), and X(i) for i = 513, 514, 693, 14078, 20295. (Peter Moses, October 16, 2020)

X(40086) lies on these lines: {513, 11813}, {522, 596}, {523, 2530}, {659, 27167}, {661, 1639}, {764, 4036}, {834, 24720}, {900, 4017}, {1290, 34594}, {3261, 35367}, {3699, 8050}, {3733, 18108}, {19947, 31947}, {21051, 28213}, {21173, 24161}, {21260, 28195}, {29362, 39798}, {35353, 40013}, {37135, 37205}

X(40086) = midpoint of X(764) and X(4036)
X(40086) = reflection of X(i) in X(j) for these {i,j}: {31946, 3837}, {31947, 19947}
X(40086) = X(8050)-Ceva conjugate of X(596)
X(40086) = X(i)-cross conjugate of X(j) for these (i,j): {4024, 514}, {21143, 1086}, {30591, 523}
X(40086) = X(i)-isoconjugate of X(j) for these (i,j): {100, 595}, {101, 32911}, {109, 3871}, {110, 3293}, {163, 3995}, {190, 2220}, {692, 4360}, {765, 4057}, {1110, 20295}, {1252, 4063}, {1331, 4222}, {4132, 4570}, {18140, 32739}, {20949, 23990}
X(40086) = cevapoint of X(764) and X(3120)
X(40086) = crosspoint of X(596) and X(8050)
X(40086) = crosssum of X(595) and X(4057)
X(40086) = trilinear pole of line {3125, 4530}
X(40086) = crossdifference of every pair of points on line {595, 2220}
X(40086) = barycentric product X(i)*X(j) for these {i,j}: {513, 40013}, {514, 596}, {523, 39747}, {693, 39798}, {1086, 8050}, {1577, 39949}, {3120, 37205}, {4391, 20615}, {16732, 34594}
X(40086) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 4063}, {513, 32911}, {514, 4360}, {523, 3995}, {596, 190}, {649, 595}, {650, 3871}, {661, 3293}, {667, 2220}, {693, 18140}, {1015, 4057}, {1086, 20295}, {1111, 20949}, {2969, 17922}, {3120, 4129}, {3125, 4132}, {3837, 27044}, {3937, 22154}, {4024, 4075}, {4988, 4065}, {6545, 21208}, {6591, 4222}, {8050, 1016}, {20615, 651}, {21143, 8054}, {34594, 4567}, {37205, 4600}, {39747, 99}, {39798, 100}, {39949, 662}, {40013, 668}

X(40087) = X(2)X(1978)∩X(38)X(75)

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

X(40087) lies on these lines: {2, 1978}, {10, 18833}, {37, 27035}, {38, 75}, {76, 6539}, {274, 27163}, {305, 31130}, {321, 1921}, {333, 33764}, {350, 4365}, {668, 17165}, {670, 873}, {799, 32939}, {874, 1621}, {1920, 4359}, {1965, 4418}, {1966, 32914}, {2170, 30074}, {3112, 5263}, {3210, 30964}, {3403, 5271}, {3617, 20023}, {3995, 18140}, {4033, 18052}, {4087, 26234}, {4110, 18054}, {4572, 6063}, {4699, 6374}, {4772, 6383}, {6376, 32925}, {9230, 28604}, {15523, 30631}, {17063, 31002}, {17141, 25294}, {17143, 17163}, {17147, 31008}, {17151, 18078}, {17495, 34020}, {18059, 24325}, {18064, 32922}, {20440, 31004}, {23538, 24343}, {27798, 35532}, {28660, 33935}, {30632, 32778}, {32025, 33769}, {32930, 39044}

X(40087) = anticomplement of X(21827)
X(40087) = isotomic conjugate of X(40148)
X(40087) = isotomic conjugate of the isogonal conjugate of X(4360)
X(40087) = X(i)-cross conjugate of X(j) for these (i,j): {20295, 1978}, {21208, 20949}
X(40087) = X(i)-isoconjugate of X(j) for these (i,j): {32, 39798}, {560, 596}, {669, 34594}, {1501, 40013}, {1918, 39949}, {1924, 37205}, {1980, 8050}, {2175, 20615}, {2205, 39747}
X(40087) = cevapoint of X(i) and X(j) for these (i,j): {75, 40034}, {20949, 21208}
X(40087) = trilinear pole of line {4129, 20949}
X(40087) = barycentric product X(i)*X(j) for these {i,j}: {75, 18140}, {76, 4360}, {310, 3995}, {561, 32911}, {595, 1502}, {668, 20949}, {670, 4129}, {1928, 2220}, {1978, 20295}, {3293, 6385}, {3871, 20567}, {4063, 6386}, {4132, 4602}, {21208, 31625}
X(40087) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 39798}, {76, 596}, {85, 20615}, {274, 39949}, {310, 39747}, {561, 40013}, {595, 32}, {670, 37205}, {799, 34594}, {1978, 8050}, {2220, 560}, {3293, 213}, {3871, 41}, {3995, 42}, {4057, 1919}, {4063, 667}, {4065, 20970}, {4075, 1500}, {4129, 512}, {4132, 798}, {4222, 1973}, {4360, 6}, {8054, 1977}, {18140, 1}, {20295, 649}, {20949, 513}, {21208, 1015}, {27044, 3009}, {32911, 31}
X(40087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6382, 1978}, {75, 561, 310}, {75, 17149, 17155}, {321, 1921, 18152}, {4359, 35543, 1920}, {6382, 10009, 2}, {20889, 21020, 75}

X(40088) = X(37)X(308)∩X(38)X(75)

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

X(40088) lies on these lines: {37, 308}, {38, 75}, {76, 594}, {313, 18891}, {668, 33769}, {871, 1268}, {1218, 14624}, {1278, 30638}, {1502, 3596}, {1920, 3739}, {1978, 18137}, {4043, 18152}, {4772, 30637}, {6376, 28593}, {9230, 17790}, {20891, 35543}, {21238, 24732}, {21615, 28659}

X(40088) = isotomic conjugate of the isogonal conjugate of X(17143)
X(40088) = X(31625)-Ceva conjugate of X(6386)
X(40088) = X(i)-isoconjugate of X(j) for these (i,j): {32, 2350}, {560, 13476}, {1501, 17758}, {2205, 39950}
X(40088) = barycentric product X(i)*X(j) for these {i,j}: {75, 18152}, {76, 17143}, {310, 4043}, {561, 17277}, {1502, 1621}, {1928, 4251}, {1978, 20954}, {3996, 20567}, {4151, 4602}, {4651, 6385}, {6386, 17494}
X(40088) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 2350}, {76, 13476}, {310, 39950}, {561, 17758}, {1621, 32}, {2486, 3121}, {3294, 1918}, {3996, 41}, {4040, 1919}, {4043, 42}, {4151, 798}, {4251, 560}, {4651, 213}, {6385, 39734}, {14004, 1973}, {17143, 6}, {17277, 31}, {17494, 667}, {17761, 3248}, {18152, 1}, {20954, 649}, {21007, 1980}, {29447, 16679}, {33765, 1106}
X(40088) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 561, 6385}, {1502, 3596, 6386}, {3728, 20889, 75}

X(40089) = X(2)X(36645)∩X(38)X(75)

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

X(40088) lies on these lines: {2, 36645}, {38, 75}, {76, 27797}, {850, 4608}, {1920, 24589}, {1921, 1978}, {3264, 40075}, {4671, 6382}, {6381, 18891}, {17786, 18054}, {18075, 32922}, {28660, 40034}

X(40089) = isotomic conjugate of the isogonal conjugate of X(17160)
X(40089) = X(i)-isoconjugate of X(j) for these (i,j): {32, 39981}, {560, 39697}, {1501, 39994}
X(40089) = barycentric product X(i)*X(j) for these {i,j}: {75, 18145}, {76, 17160}, {561, 37680}, {668, 21606}, {1928, 33882}, {1978, 21297}, {4145, 4602}, {6385, 31855}, {6386, 21385}
X(40089) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 39981}, {76, 39697}, {561, 39994}, {4145, 798}, {4491, 1919}, {17160, 6}, {18145, 1}, {21297, 649}, {21385, 667}, {21606, 513}, {21714, 4079}, {31855, 213}, {33882, 560}, {37680, 31}
X(40089) = {X(1921),X(35543)}-harmonic conjugate of X(1978)

X(40090) = X(10)X(75)∩X(850)X(4608)

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

X(40090) lies on these lines: {10,75}, {850,4608}, {14210,35544}

X(40091) = X(1)X(21)∩X(36)X(106)

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

X(40091) = {2 X[36] + X[39148]}

X(40091) lies on these lines: {1, 21}, {2, 37610}, {3, 1616}, {6, 6767}, {10, 4514}, {11, 17734}, {32, 16969}, {35, 1201}, {36, 106}, {40, 5573}, {42, 5315}, {43, 25439}, {46, 28011}, {55, 995}, {71, 16488}, {101, 1914}, {109, 1319}, {145, 1724}, {171, 551}, {172, 9327}, {187, 9259}, {212, 7962}, {238, 519}, {244, 484}, {284, 16685}, {386, 1191}, {392, 3744}, {405, 37542}, {512, 1326}, {517, 1279}, {572, 21769}, {580, 1482}, {581, 16202}, {582, 8148}, {602, 7982}, {614, 5119}, {672, 16784}, {674, 16796}, {727, 898}, {748, 3679}, {750, 25055}, {859, 23404}, {946, 24160}, {962, 24159}, {978, 8715}, {986, 30148}, {997, 3749}, {999, 3052}, {1001, 4279}, {1015, 5030}, {1018, 33854}, {1058, 5292}, {1064, 34486}, {1086, 28174}, {1104, 9957}, {1125, 5255}, {1126, 1203}, {1193, 3746}, {1253, 9819}, {1293, 12029}, {1331, 10700}, {1334, 5299}, {1450, 3256}, {1453, 37556}, {1457, 2078}, {1471, 18421}, {1739, 7292}, {1770, 23675}, {1834, 15172}, {1870, 8750}, {1918, 16484}, {2176, 2241}, {2177, 5313}, {2209, 15485}, {2242, 21793}, {2275, 24047}, {2308, 16474}, {2361, 5048}, {3011, 30384}, {3017, 15170}, {3072, 13464}, {3073, 5882}, {3216, 3871}, {3218, 4694}, {3241, 17127}, {3244, 5247}, {3246, 3880}, {3290, 5011}, {3303, 16466}, {3616, 5264}, {3622, 37522}, {3636, 37607}, {3730, 16502}, {3822, 33106}, {3924, 5697}, {3938, 5692}, {3961, 10176}, {3997, 16503}, {4252, 7373}, {4253, 14974}, {4259, 16794}, {4264, 16777}, {4290, 16672}, {4301, 37570}, {4306, 34040}, {4322, 34043}, {4364, 25432}, {4424, 7191}, {4482, 10027}, {4642, 37563}, {4692, 32930}, {4695, 5541}, {4803, 16690}, {4857, 21935}, {4868, 29821}, {4880, 17449}, {4881, 35281}, {4975, 17763}, {5053, 9456}, {5080, 24222}, {5180, 33148}, {5259, 10459}, {5301, 33628}, {5398, 10247}, {5445, 28096}, {5493, 24171}, {5687, 17749}, {5710, 16302}, {5883, 29820}, {5903, 28082}, {6051, 20715}, {6905, 32486}, {7031, 9310}, {7280, 32577}, {7290, 31393}, {7299, 37738}, {7322, 31435}, {7798, 17262}, {8624, 38865}, {9316, 13462}, {9441, 28228}, {10197, 17717}, {10246, 37469}, {10571, 11510}, {10595, 37530}, {10624, 23537}, {11010, 24443}, {11529, 21059}, {11813, 17719}, {12000, 36754}, {12047, 28027}, {12702, 17054}, {16490, 21747}, {16497, 18900}, {16501, 23344}, {16785, 21764}, {17053, 37508}, {17125, 19875}, {17126, 38314}, {17152, 33953}, {17541, 29381}, {17686, 29383}, {18393, 33127}, {20040, 27660}, {20703, 27785}, {21214, 25440}, {24231, 28026}, {24390, 24880}, {24864, 38455}, {26687, 29697}, {26725, 29689}, {30108, 35274}, {31855, 37680}

X(40091) = reflection of X(30117) in X(1279)
X(40091) = isogonal conjugate of X(39697)
X(40091) = isogonal conjugate of the isotomic conjugate of X(17160)
X(40091) = X(i)-Ceva conjugate of X(j) for these (i,j): {5376, 101}, {30576, 5053}
X(40091) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39697}, {2, 39981}, {6, 39994}
X(40091) = crosspoint of X(i) and X(j) for these (i,j): {106, 1126}, {110, 9268}, {765, 901}
X(40091) = crosssum of X(i) and X(j) for these (i,j): {244, 900}, {519, 1125}, {523, 1647}, {1086, 21115}
X(40091) = crossdifference of every pair of points on line {661, 1213}
X(40091) = barycentric product X(i)*X(j) for these {i,j}: {1, 37680}, {6, 17160}, {31, 18145}, {75, 33882}, {81, 31855}, {100, 21385}, {101, 21297}, {190, 4491}, {662, 4145}, {692, 21606}, {1897, 23141}, {4556, 21714}, {5376, 38979}
X(40091) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 39994}, {6, 39697}, {31, 39981}, {4145, 1577}, {4491, 514}, {17160, 76}, {18145, 561}, {21297, 3261}, {21385, 693}, {23141, 4025}, {31855, 321}, {33882, 1}, {37680, 75}
X(40091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 595, 58}, {1, 1046, 3881}, {1, 1621, 4653}, {1, 3915, 595}, {1, 8616, 993}, {1, 32913, 3892}, {36, 1149, 106}, {36, 16489, 1149}, {55, 995, 4256}, {55, 16483, 995}, {392, 3744, 30115}, {902, 1149, 36}, {902, 16489, 106}, {999, 3052, 4257}, {1015, 17735, 5030}, {1104, 9957, 15955}, {1191, 3295, 386}, {1193, 3746, 33771}, {1914, 3230, 101}, {2176, 2241, 4251}, {3052, 16486, 999}, {4653, 4658, 10458}, {4653, 38832, 58}, {14974, 16781, 4253}

X(40092) = X(2)X(38)∩X(1268)X(35352)

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

X(40092) lies on this line: {2,38}, {1268,35352}

X(40093) = X(2)X(39717)∩X(10)X(274)

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

X(40093) lies on these lines: {2,39717}, {10,274}, {75,308}, {83,16549}, {86,4553}, {239,292}, {335,1268}, {3864,29633}, {4075,18140}, {4444,21385}, {4876,17023}, {19973,20345}, {29767,37128}, {32010,32780}

X(40093) = barycentric product X(i)*X(j) for these {i, j}: {291, 18140}, {292, 40087}, {334, 32911}, {335, 4360}, {595, 18895}, {660, 20949}
X(40093) = barycentric quotient X(i)/X(j) for these (i, j): (291, 39798), (334, 40013), (335, 596), (595, 1914), (2220, 2210)
X(40093) = trilinear product X(i)*X(j) for these {i, j}: {291, 4360}, {292, 18140}, {334, 595}, {335, 32911}, {337, 4222}, {660, 20295}
X(40093) = trilinear quotient X(i)/X(j) for these (i, j): (334, 596), (335, 39798), (595, 2210), (2220, 14599)
X(40093) = trilinear pole of the line {3995, 20295}
X(40093) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(3293)}} and {{A, B, C, X(75), X(16887)}}
X(40093) = cevapoint of X(1575) and X(22279)
X(40093) = X(i)-isoconjugate-of-X(j) for these {i,j}: {596, 2210}, {1914, 39798}
X(40093) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (291, 39798), (334, 40013), (335, 596), (595, 1914)
X(40093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (291, 334, 18827), (18827, 40095, 334)

X(40094) = X(2)X(3112)∩X(10)X(274)

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

X(40094) lies on these lines: {2,3112}, {10,274}, {75,40024}, {292,16819}, {335,4359}, {350,6541}, {874,13576}, {1921,3263}, {3783,32922}, {3978,20486}, {4562,30109}, {4876,29960}, {17143,17761}, {26752,26978}, {30997,32035}

X(40094) = barycentric product X(i)*X(j) for these {i, j}: {291, 18152}, {292, 40088}, {334, 17277}, {335, 17143}, {1621, 18895}
X(40094) = barycentric quotient X(i)/X(j) for these (i, j): (291, 2350), (334, 17758), (335, 13476), (1621, 1914), (2486, 39786)
X(40094) = trilinear product X(i)*X(j) for these {i, j}: {291, 17143}, {292, 18152}, {334, 1621}, {335, 17277}, {337, 14004}, {660, 20954}
X(40094) = trilinear quotient X(i)/X(j) for these (i, j): (334, 13476), (335, 2350), (1621, 2210)
X(40094) = trilinear pole of the line {4043, 20954}
X(40094) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(16887)}} and {{A, B, C, X(10), X(4651)}}
X(40094) = X(1914)-isoconjugate-of-X(2350)
X(40094) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (291, 2350), (334, 17758), (335, 13476), (1621, 1914)
X(40094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (291, 334, 40017), (4518, 18895, 4583)

X(40095) = X(2)X(4562)∩X(10)X(274)

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

X(40095) lies on these lines: {2,4562}, {10,274}, {75,31625}, {292,16815}, {335,4688}, {1268,35352}, {3252,4751}, {3263,19955}, {4389,36801}, {4876,29596}, {9780,22116}, {17292,21264}, {19950,33931}, {19951,30758}

X(40095) = barycentric product X(i)*X(j) for these {i, j}: {291, 18145}, {292, 40089}, {334, 37680}, {335, 17160}, {660, 21606}
X(40095) = barycentric quotient X(i)/X(j) for these (i, j): (291, 39982), (334, 39994), (335, 39697)
X(40095) = trilinear product X(i)*X(j) for these {i, j}: {291, 17160}, {292, 18145}, {334, 40091}, {335, 37680}, {660, 21297}, {813, 21606}
X(40095) = trilinear quotient X(i)/X(j) for these (i, j): (334, 39697), (335, 39982)
X(40095) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(21297)}} and {{A, B, C, X(10), X(31855)}}
X(40095) = X(1914)-isoconjugate-of-X(39982)
X(40095) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (291, 39982), (334, 39994), (335, 39697)
X(40095) = {X(334), X(40093)}-harmonic conjugate of X(18827)

X(40096) = X(1)X(16702)∩X(6)X(31)

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

X(40096) lies on these lines: {1,16702}, {3,34814}, {6,31}, {187,21009}, {512,1326}, {922,3285}, {1333,4068}, {16777,21829}, {20675,33704}, {20999,23366}

X(40097) = X(4)X(123)∩X(24)X(104)

Barycentrics    a*(a - b)*(a - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*b*c + 2*a*b^2*c - 2*a*b*c^2 - c^4)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a^2*c^2 + 2*a*b*c^2 + c^4) : :
Trilinears    (sec A)/(b^2 (sec A - sec B) + c^2 (sec C - sec A)) : :

X(40097) lies on the circumcircle and these lines: {4, 123}, {24, 104}, {25, 2968}, {28, 39435}, {74, 31384}, {99, 4244}, {105, 6353}, {110, 7435}, {186, 2694}, {242, 2723}, {403, 2687}, {650, 32688}, {691, 37965}, {759, 30733}, {915, 3542}, {925, 4246}, {1294, 7414}, {1297, 4231}, {1300, 31385}, {1305, 4250}, {1783, 8687}, {1897, 9058}, {2370, 4222}, {2373, 7438}, {2693, 37979}, {2720, 15385}, {2752, 37777}, {3565, 4238}, {3651, 5897}, {3658, 13398}, {4220, 34168}, {4242, 13397}, {10420, 37966}, {26253, 39436}, {30267, 39434}

X(40097) = Stevanovic-circle-inverse of X(32688)
X(40097) = polar-circle-inverse of X(123)
X(40097) = Collings transform of X(i) for these i: {431, 39167}
X(40097) = X(i)-cross conjugate of X(j) for these (i,j): {521, 4}, {650, 34277}, {3435, 15385}, {14312, 104}
X(40097) = cevapoint of X(i) and X(j) for these (i,j): {25, 650}, {431, 523}, {521, 39167}
X(40097) = trilinear pole of line {6, 1854}
X(40097) = Ψ(X(6), X(1854))
X(40097) = X(i)-isoconjugate of X(j) for these (i,j): {3, 21186}, {63, 6588}, {109, 123}, {197, 4025}, {205, 15413}, {478, 6332}, {514, 22132}, {521, 21147}, {656, 16049}, {905, 1766}, {1459, 3436}, {7254, 21074}, {20928, 22383}
X(40097) = barycentric product X(i)*X(j) for these {i,j}: {108, 34277}, {1783, 8048}, {3435, 6335}, {4391, 15385}
X(40097) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 21186}, {25, 6588}, {112, 16049}, {650, 123}, {692, 22132}, {1783, 3436}, {1897, 20928}, {3435, 905}, {8048, 15413}, {8750, 1766}, {15385, 651}, {32674, 21147}, {34277, 35518}

X(40098) = ISOTOMIC CONJUGATE OF X(4366)

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

X(40098) lies on the cubic K768 and these lines: {2, 19897}, {239, 291}, {334, 3263}, {335, 726}, {660, 20683}, {894, 24479}, {1015, 35172}, {1916, 17789}, {3507, 18787}, {3948, 4583}, {4562, 6542}, {7233, 9436}, {15149, 17927}, {17798, 30664}, {18891, 18895}, {19308, 34067}

X(40098) = isotomic conjugate of X(4366)
X(40098) = isotomic conjugate of the anticomplement of X(26582)
X(40098) = isotomic conjugate of the complement of X(6653)
X(40098) = X(i)-cross conjugate of X(j) for these (i,j): {75, 1916}, {141, 40017}, {523, 4583}, {1086, 4444}, {26582, 2}
X(40098) = X(i)-isoconjugate of X(j) for these (i,j): {6, 8300}, {9, 12835}, {31, 4366}, {32, 39044}, {238, 1914}, {239, 2210}, {350, 14599}, {593, 4094}, {692, 4375}, {849, 35068}, {1110, 35119}, {1333, 4368}, {1428, 3684}, {1691, 18786}, {1911, 6652}, {1921, 18892}, {1933, 17493}, {2150, 3027}, {2201, 7193}, {2238, 5009}, {3573, 8632}, {18264, 27916}, {18891, 18894}, {27855, 32739}
X(40098) = cevapoint of X(i) and X(j) for these (i,j): {2, 6653}, {1086, 4444}
X(40098) = trilinear pole of line {918, 3837}
X(40098) = barycentric product X(i)*X(j) for these {i,j}: {75, 30663}, {256, 30642}, {291, 334}, {292, 18895}, {335, 335}, {876, 4583}, {1916, 30669}, {1928, 18267}, {1934, 18787}, {4444, 4562}, {4518, 7233}, {4589, 35352}, {23596, 37207}
X(40098) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8300}, {2, 4366}, {10, 4368}, {12, 3027}, {56, 12835}, {75, 39044}, {239, 6652}, {291, 238}, {292, 1914}, {295, 7193}, {334, 350}, {335, 239}, {514, 4375}, {594, 35068}, {660, 3573}, {693, 27855}, {741, 5009}, {756, 4094}, {876, 659}, {984, 3802}, {1086, 35119}, {1215, 4154}, {1581, 18786}, {1911, 2210}, {1916, 17493}, {1922, 14599}, {3572, 8632}, {3862, 16514}, {3864, 3783}, {3912, 27919}, {4444, 812}, {4518, 3685}, {4562, 3570}, {4583, 874}, {4876, 3684}, {6542, 27926}, {6645, 4027}, {7233, 1447}, {7245, 4396}, {14598, 18892}, {18267, 560}, {18787, 1580}, {18827, 33295}, {18895, 1921}, {18897, 18894}, {22116, 8299}, {23596, 4486}, {30642, 1909}, {30657, 172}, {30663, 1}, {30669, 385}, {35352, 4010}, {40017, 30940}

X(40099) = ISOTOMIC CONJUGATE OF X(6645)

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

X(40099) lies on these lines: {75, 1916}, {239, 256}, {257, 4357}, {1967, 24575}, {3687, 3797}, {3688, 3903}, {3865, 27447}, {4027, 8424}, {7018, 20891}, {17280, 27805}, {17302, 32010}, {18891, 21442}, {23868, 30670}

X(40099) = isotomic conjugate of X(6645)
X(40099) = isotomic conjugate of the anticomplement of X(26558)
X(40099) = X(26558)-cross conjugate of X(2)
X(40099) = X(i)-isoconjugate of X(j) for these (i,j): {31, 6645}, {57, 10799}, {171, 172}, {894, 7122}, {904, 7369}, {1252, 7207}, {1691, 18787}, {1911, 27982}, {1933, 30669}, {2149, 3023}, {2330, 7175}, {3955, 7119}, {4579, 20981}
X(40099) = trilinear pole of line {3910, 4486}
X(40099) = barycentric product X(i)*X(j) for these {i,j}: {256, 7018}, {257, 257}, {291, 30643}, {1916, 17493}, {1934, 18786}, {4451, 7249}, {18895, 30658}
X(40099) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6645}, {11, 3023}, {55, 10799}, {239, 27982}, {244, 7207}, {256, 171}, {257, 894}, {893, 172}, {894, 7369}, {904, 7122}, {982, 7188}, {1432, 7175}, {1581, 18787}, {1916, 30669}, {3865, 7184}, {3903, 4579}, {4366, 4027}, {4451, 7081}, {4496, 4400}, {7015, 3955}, {7018, 1909}, {7249, 7176}, {17493, 385}, {18786, 1580}, {27805, 18047}, {30643, 350}, {30658, 1914}, {32010, 17103}

X(40100) = ANTICOMPLEMENT OF X(38614)

Barycentrics    a^10 - 2*a^9*b - 2*a^8*b^2 + 4*a^7*b^3 + a^6*b^4 - a^4*b^6 - 4*a^3*b^7 + 2*a^2*b^8 + 2*a*b^9 - b^10 - 2*a^9*c + 8*a^8*b*c - 2*a^7*b^2*c - 10*a^6*b^3*c + 2*a^5*b^4*c - 2*a^4*b^5*c + 10*a^3*b^6*c + 2*a^2*b^7*c - 8*a*b^8*c + 2*b^9*c - 2*a^8*c^2 - 2*a^7*b*c^2 + 7*a^6*b^2*c^2 + 2*a^5*b^3*c^2 + 4*a^4*b^4*c^2 - 2*a^3*b^5*c^2 - 12*a^2*b^6*c^2 + 2*a*b^7*c^2 + 3*b^8*c^2 + 4*a^7*c^3 - 10*a^6*b*c^3 + 2*a^5*b^2*c^3 - 4*a^4*b^3*c^3 - 4*a^3*b^4*c^3 - 2*a^2*b^5*c^3 + 22*a*b^6*c^3 - 8*b^7*c^3 + a^6*c^4 + 2*a^5*b*c^4 + 4*a^4*b^2*c^4 - 4*a^3*b^3*c^4 + 20*a^2*b^4*c^4 - 18*a*b^5*c^4 - 2*b^6*c^4 - 2*a^4*b*c^5 - 2*a^3*b^2*c^5 - 2*a^2*b^3*c^5 - 18*a*b^4*c^5 + 12*b^5*c^5 - a^4*c^6 + 10*a^3*b*c^6 - 12*a^2*b^2*c^6 + 22*a*b^3*c^6 - 2*b^4*c^6 - 4*a^3*c^7 + 2*a^2*b*c^7 + 2*a*b^2*c^7 - 8*b^3*c^7 + 2*a^2*c^8 - 8*a*b*c^8 + 3*b^2*c^8 + 2*a*c^9 + 2*b*c^9 - c^10 : :
X(40100) = 4 X[140] - 3 X[38705], 3 X[381] - 2 X[31841], 3 X[381] - X[38584], 2 X[550] - 3 X[38707], 5 X[1656] - 4 X[22102], 2 X[3627] + X[38682], 2 X[6073] - 3 X[38755]

X(40100) lies on the Johnson circle and these lines: {2, 38614}, {3, 3259}, {4, 38954}, {5, 901}, {20, 38617}, {30, 953}, {140, 38705}, {381, 31841}, {382, 38586}, {513, 10738}, {517, 10742}, {550, 38707}, {952, 31512}, {1478, 13756}, {1479, 3025}, {1656, 22102}, {2070, 39479}, {3585, 23153}, {3627, 38682}, {5722, 33645}, {6073, 38755}, {7517, 10016}, {12645, 18326}, {18342, 38385}

X(40100) = midpoint of X(382) and X(38586)
X(40100) = reflection of X(i) in X(j) for these {i,j}: {3, 3259}, {20, 38617}, {901, 5}, {18342, 38385}, {38584, 31841}, {38954, 4}
X(40100) = anticomplement of X(38614)
X(40100) = X(901)-of-Johnson-triangle
X(40100) = {X(381),X(38584)}-harmonic conjugate of X(31841)

X(40101) = X(519)-CROSS CONJUGATE OF X(4)

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

X(40100) lies on the circumcircle and these lines: {4, 121}, {9, 29014}, {19, 29149}, {24, 32704}, {25, 9059}, {28, 34594}, {99, 4247}, {100, 4222}, {101, 17314}, {109, 1724}, {110, 4248}, {186, 2692}, {242, 1308}, {404, 13397}, {925, 7419}, {1294, 7444}, {1295, 7447}, {1297, 7434}, {1305, 4245}, {2373, 7448}, {3518, 26713}, {3565, 4234}, {6353, 9088}, {7459, 26703}, {7478, 10420}, {8074, 35182}, {8756, 32686}, {15383, 35186}

X(40101) = polar-circle-inverse of X(121)
X(40101) = X(519)-cross conjugate of X(4)
X(40101) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1739}, {63, 8610}, {71, 16753}, {88, 22428}, {121, 36058}, {1797, 17465}, {5440, 39264}, {21427, 32659}
X(40101) = cevapoint of X(25) and X(8756)
X(40101) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 1739}, {25, 8610}, {28, 16753}, {902, 22428}, {8752, 39264}, {8756, 121}, {15383, 1797}, {38462, 21427}

X(40102) = ISOGONAL CONJUGATE OF X(11188)

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

Let ABC be a triangle, P a point and A'B'C' the circumcevian triangle of P. Let Ab, Ac be the orthogonal projections of P in A'C and A'B, respectively, and build Bc, Ba and Ca, Cb cyclically. Let A"B"C" be the triangle bounded by the lines AbAc, BcBa and CaCb. Then A"B"C" and ABC are perspective. (Abdilkadir Altintas, problem 1540).

If P=x:y:z (barycentrics) then the given perspector Q(P) is the isogonal conjugate of (2*(b^2*z+c^2*y)*a^2*y*z*cos(A)-(a^2*y*z+b^2*x*z+c^2*x*y)*b*c*x-2*(b^3*z^2*cos(B)+c^3*y^2*cos(C))*a*x)*a : :.

The appearance of (i, j) in the following partial list means that Q(X(i))=X(j): (1, 15446), (2, 40102), (3, 4), (4, 22261), (6, 40103), (13, 40104), (14, 40105), (15, 15), (16, 16), (23, 468), (36, 1), (54, 3459), (59, 650), (186, 523), (187, 10630), (249, 523), (250, 647), (501, 1), (1157, 4), (2065, 230), (2070, 3459) (César Lozada, October 20, 2020).

X(40102) lies on these lines: {2,14908}, {3,3266}, {25,37778}, {32,468}, {184,524}, {2200,4062}, {5181,9516}, {5967,14600}, {7493,10547}\

X(40102) = isogonal conjugate of X(11188)
X(40102) = trilinear pole of the line {690, 3049}
X(40102) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(468)}} and {{A, B, C, X(3), X(25)}}

X(40103) = ISOGONAL CONJUGATE OF X(15534)

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

See X(40102).

X(40103) lies on these lines: {2,32457}, {6,9716}, {25,11580}, {69,34898}, {111,7492}, {694,8617}, {1383,3291}, {6094,17008}, {8585,39389}, {8589,39576}, {8770,15246}, {34288,37689}

X(40103) = isogonal conjugate of X(15534)
X(40103) = isotomic conjugate of the anticomplement of X(39576)
X(40103) = anticomplement of the complementary conjugate of X(22165)
X(40103) = barycentric product X(523)*X(33638)
X(40103) = trilinear product X(661)*X(33638)
X(40103) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(16042)}}
X(40103) = cevapoint of X(6) and X(15655)
X(40103) = X(25)-vertex conjugate of-X(1383)

X(40104) = ISOGONAL CONJUGATE OF X(36979)

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

See X(40102).

X(40104) lies on the Kiepert hyperbola and these lines: {2,3200}, {13,11136}, {94,6105}, {11140,37848}

X(40104) = isogonal conjugate of X(36979)
X(40104) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(15), X(1141)}}

X(40105) = ISOGONAL CONJUGATE OF X(36981)

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

See X(40102).

X(40105) lies on the Kiepert hyperbola and these lines: {2,3201}, {14,11135}, {94,6104}, {11140,37850}

X(40105) = isogonal conjugate of X(36981)
X(40105) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(16), X(1141)}}

X(40106) = (name pending)

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

X(40106) lies on this line:: {8,192}

X(40107) = COMPLEMENT OF X(576)

Barycentrics    2*a^4*b^2 - 3*a^2*b^4 + b^6 + 2*a^4*c^2 - 4*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6 : :
X(40107) = X[3] + 3 X[599], 3 X[3] + X[15069], X[4] - 3 X[11178], X[4] - 9 X[21356], X[5] - 3 X[141], 5 X[5] - 3 X[5480], 4 X[5] - 3 X[19130], 7 X[5] - 3 X[21850], 2 X[5] - 3 X[24206], 13 X[5] - 9 X[38136], 3 X[6] - 7 X[3526], X[20] + 3 X[1352], X[20] - 3 X[3098], X[20] + 15 X[3620], X[20] - 9 X[10519], 3 X[67] + X[23236], 3 X[69] + 5 X[631], 5 X[69] + 3 X[14912], 3 X[69] + 2 X[33749], X[76] + 3 X[22677], 4 X[140] - 3 X[10168], 5 X[141] - X[5480], 4 X[141] - X[19130], 7 X[141] - X[21850], 13 X[141] - 3 X[38136], 3 X[182] - 5 X[631], 5 X[182] - 3 X[14912], 3 X[182] - 2 X[33749], X[193] - 3 X[39561], X[382] + 3 X[1350], X[382] - 3 X[3818], 2 X[546] - 3 X[25561], 2 X[548] - 3 X[14810], 3 X[549] - X[8550], 3 X[549] - 2 X[20190], 2 X[575] - 3 X[10168], 3 X[597] - 5 X[632], 3 X[597] - 2 X[22330], 9 X[599] - X[15069], 3 X[599] - X[34507], 25 X[631] - 9 X[14912], 5 X[631] - 2 X[33749], 5 X[632] - 2 X[22330], X[1351] - 5 X[3763], 3 X[1351] - 11 X[5070], X[1351] - 3 X[38317], X[1352] - 5 X[3620], X[1352] + 3 X[10519], 5 X[1656] - 3 X[5476], 5 X[1656] - X[11477], 5 X[1656] - 9 X[21358], 3 X[1992] - 11 X[3525], 3 X[1992] - 5 X[22234], 7 X[3090] - 3 X[20423]

X(40107) is the radical trace of the Ehrmann cirles of the 1st and 2nd Ehrmann inscribed triangles. (Randy Hutson, December 18, 2020)

X(40107) lies on these lines: {2, 576}, {3, 67}, {4, 7883}, {5, 141}, {6, 3411}, {20, 1352}, {30, 18553}, {39, 15993}, {54, 69}, {76, 22677}, {114, 3314}, {125, 7998}, {126, 16938}, {140, 524}, {183, 6036}, {193, 39561}, {262, 16986}, {298, 6774}, {299, 6771}, {325, 15819}, {340, 37124}, {343, 3819}, {382, 1350}, {394, 15135}, {487, 12974}, {488, 12975}, {518, 5885}, {546, 25561}, {548, 1503}, {549, 8550}, {550, 11645}, {597, 632}, {620, 32135}, {858, 3917}, {1176, 9705}, {1351, 3763}, {1353, 3630}, {1469, 37719}, {1506, 13330}, {1656, 5476}, {1843, 15559}, {1972, 15595}, {1992, 3525}, {2080, 7820}, {2393, 5447}, {2781, 11591}, {2854, 20379}, {2979, 5169}, {3056, 37720}, {3090, 20423}, {3094, 7765}, {3095, 6292}, {3096, 12251}, {3292, 7495}, {3398, 7826}, {3399, 10292}, {3416, 37727}, {3522, 11180}, {3523, 11179}, {3528, 33751}, {3530, 3564}, {3548, 11511}, {3580, 5650}, {3589, 5097}, {3618, 15520}, {3619, 5067}, {3628, 20582}, {3629, 15516}, {3642, 22737}, {3643, 22736}, {3832, 31670}, {3843, 10516}, {3853, 18358}, {3933, 13334}, {4045, 32515}, {4309, 12589}, {4317, 12588}, {4663, 11231}, {5012, 15108}, {5054, 15533}, {5079, 38072}, {5085, 11898}, {5104, 7747}, {5171, 7795}, {5182, 33259}, {5449, 6698}, {5477, 39560}, {5651, 32223}, {5891, 11799}, {5921, 21734}, {5972, 15066}, {5980, 25559}, {5981, 25560}, {6143, 8537}, {6228, 6229}, {6393, 14994}, {6515, 32068}, {6640, 8538}, {6697, 14076}, {6721, 7778}, {6723, 37638}, {6776, 15717}, {6791, 39576}, {6937, 10477}, {6998, 17297}, {7486, 14853}, {7493, 9306}, {7499, 34986}, {7505, 11470}, {7509, 10112}, {7525, 15582}, {7552, 9970}, {7752, 31958}, {7756, 11646}, {7758, 13086}, {7767, 13335}, {7769, 39099}, {7771, 38748}, {7782, 14928}, {7800, 9737}, {7811, 35925}, {7813, 11171}, {7818, 37348}, {7833, 10992}, {7836, 12177}, {7841, 19662}, {7865, 37242}, {7869, 37466}, {7870, 38751}, {7877, 10359}, {7880, 37459}, {7909, 10753}, {7915, 20576}, {7922, 37446}, {7931, 38227}, {7934, 23514}, {7999, 11704}, {8252, 9975}, {8253, 9974}, {8263, 13348}, {8541, 37119}, {8542, 18281}, {8548, 15115}, {8549, 23329}, {8584, 11539}, {8681, 12359}, {9003, 23108}, {9019, 10627}, {9466, 15980}, {9714, 37485}, {9967, 24572}, {9968, 14862}, {9971, 37484}, {9972, 11416}, {9976, 15061}, {9977, 15137}, {10303, 11160}, {10304, 13399}, {10350, 16898}, {10357, 12203}, {10541, 15720}, {10625, 29959}, {11303, 16001}, {11304, 16002}, {11411, 13347}, {11412, 14789}, {11444, 18504}, {11459, 15063}, {11579, 15057}, {13083, 33385}, {13084, 33384}, {13169, 15034}, {13564, 19596}, {13862, 33706}, {14485, 18840}, {14499, 25407}, {14500, 25408}, {15074, 15532}, {15080, 24981}, {15082, 37648}, {15178, 28538}, {15360, 16042}, {15462, 32244}, {15534, 15694}, {15605, 25738}, {15644, 16789}, {15696, 18440}, {16241, 16530}, {16242, 16529}, {16921, 22486}, {17004, 36859}, {17271, 21554}, {17529, 26543}, {17702, 33533}, {17800, 36990}, {18114, 23098}, {18381, 34787}, {18388, 23039}, {18400, 34118}, {18583, 34573}, {18800, 33274}, {19905, 23235}, {20191, 32283}, {20415, 34509}, {20416, 34508}, {21849, 37439}, {21969, 37990}, {22112, 37644}, {22493, 32909}, {22494, 32907}, {22866, 33418}, {22911, 33419}, {23327, 34788}, {24309, 29255}, {29323, 39884}, {31394, 33087}, {31848, 36165}, {31857, 33884}, {32317, 34114}, {32782, 37521}, {32863, 37527}, {33081, 37619}, {33217, 35431}, {33245, 35377}, {33362, 33363}, {34885, 35375}, {37450, 37671}

X(40107) = midpoint of X(i) and X(j) for these {i,j}: {3, 34507}, {67, 12584}, {69, 182}, {549, 22165}, {1350, 3818}, {1352, 3098}, {1353, 3630}, {18381, 34787}
X(40107) = reflection of X(i) in X(j) for these {i,j}: {575, 140}, {576, 25555}, {3629, 15516}, {5097, 3589}, {8550, 20190}, {9968, 14862}, {18583, 34573}, {19130, 24206}, {20301, 6698}, {20423, 25565}, {24206, 141}, {25556, 5972}, {32135, 620}
X(40107) = anticomplement of X(25555)
X(40107) = complement of X(576)
X(40107) = complement of the isogonal conjugate of X(7607)
X(40107) = medial-isogonal conjugate of X(15850)
X(40107) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 15850}, {661, 35132}, {7607, 10}, {35178, 4369}
X(40107) = crossdifference of every pair of points on line {2492, 3050}
X(40107) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 576, 25555}, {3, 599, 34507}, {140, 575, 10168}, {549, 8550, 20190}, {627, 628, 1078}, {635, 636, 626}, {639, 640, 625}, {1351, 3763, 38317}, {1352, 10519, 3098}, {1656, 11477, 5476}, {3314, 22712, 114}, {3620, 10519, 1352}, {3629, 38110, 15516}, {3917, 37636, 21243}, {6228, 6229, 7761}, {11477, 21358, 1656}

X(40108) = COMPLEMENT OF X(7697)

Barycentrics    3*a^6*b^2 - 5*a^4*b^4 + 2*a^2*b^6 + 3*a^6*c^2 - 8*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + b^6*c^2 - 5*a^4*c^4 - 5*a^2*b^2*c^4 - 2*b^4*c^4 + 2*a^2*c^6 + b^2*c^6 : :
X(40108) = 3 X[2] + X[7709], X[3] + 5 X[7786], X[3] + 2 X[11272], 2 X[3] + X[14881], 3 X[3] - X[22676], 3 X[3] + X[22728], X[5] - 4 X[6683], X[5] + 2 X[13334], X[39] + 2 X[140], X[76] - 7 X[3526], X[182] + 2 X[10007], X[194] + 11 X[3525], X[262] - 5 X[7786], 3 X[262] + X[22676], 3 X[262] - X[22728], 5 X[631] + X[3095], 5 X[631] - X[6194], 5 X[632] - 2 X[3934], 5 X[632] + X[32448], 5 X[1656] + X[11257], X[1916] + 5 X[38750], 2 X[2023] + X[33813], 7 X[3090] + 5 X[32522], 7 X[3523] - X[9821], 7 X[3526] + X[32519], 4 X[3530] - X[5188], 17 X[3533] - 5 X[31276], 3 X[3576] + X[22650], 4 X[3628] - X[6248], 2 X[3628] + X[32516], 2 X[3934] + X[32448], 3 X[5054] - X[22712], 3 X[5054] + X[32447], X[6248] + 2 X[32516], 2 X[6683] + X[13334], 8 X[6683] - X[22681], X[7697] + 3 X[11171], X[7709] - 3 X[11171], X[7757] + 5 X[15694], 5 X[7786] - 2 X[11272], 10 X[7786] - X[14881], 15 X[7786] + X[22676], 15 X[7786] - X[22728], 4 X[8359] - X[34510], X[9466] - 4 X[10124], X[9772] - 3 X[15561], 3 X[10246] - X[22713], 13 X[10303] - X[12251], 4 X[11272] - X[14881], 6 X[11272] + X[22676], 6 X[11272] - X[22728], 4 X[13334] + X[22681], 7 X[14869] - X[32521], 3 X[14881] + 2 X[22676], 3 X[14881] - 2 X[22728], 5 X[15026] - 2 X[27375], 7 X[15701] - X[33706], 8 X[16239] - 5 X[31239], 3 X[21163] + X[22682], X[22697] - 3 X[26446], X[22698] - 3 X[26451], X[32454] + 5 X[38762]

Let BA, CA be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define CB, AB, AC, BC cyclically. Triangles ABACA, ABBCB, ACBCC are similar to each other and inversely similar to ABC. Let SA be the similitude center of triangles ABBCB and ACBCC. Define SB and SC cyclically. X(40108) is the circumcenter of triangle SASBSC. (Randy Hutson, October 29, 2020)

X(40108) lies on these lines: {2, 2782}, {3, 83}, {5, 4045}, {6, 22677}, {24, 22480}, {30, 21163}, {35, 22711}, {36, 18971}, {39, 140}, {55, 22730}, {56, 22729}, {76, 3526}, {182, 10007}, {194, 3525}, {498, 22705}, {499, 22706}, {511, 549}, {517, 22475}, {538, 7619}, {574, 2023}, {620, 24256}, {631, 3095}, {632, 3934}, {730, 11231}, {1078, 39093}, {1656, 7919}, {1916, 38750}, {2021, 3815}, {2080, 3329}, {3090, 32522}, {3094, 31958}, {3102, 22727}, {3103, 22726}, {3104, 22686}, {3105, 22684}, {3106, 16241}, {3107, 16242}, {3311, 19063}, {3312, 19064}, {3398, 7824}, {3523, 9821}, {3530, 5188}, {3533, 31276}, {3576, 22650}, {3589, 37459}, {3628, 6248}, {5026, 39498}, {5038, 22525}, {5054, 22712}, {5969, 7606}, {6200, 35839}, {6396, 35838}, {6642, 22655}, {6655, 10242}, {6771, 33479}, {6774, 33478}, {7583, 22720}, {7584, 22721}, {7612, 32978}, {7694, 22505}, {7757, 8860}, {7771, 11842}, {7787, 22679}, {7792, 14693}, {7803, 9754}, {8359, 34510}, {8719, 35930}, {9466, 10124}, {9743, 37071}, {9744, 9996}, {9755, 10104}, {9756, 14880}, {10160, 11176}, {10246, 22713}, {10267, 22556}, {10269, 22680}, {10303, 12251}, {10359, 33004}, {12054, 37334}, {12143, 37119}, {13330, 21843}, {13357, 31406}, {14839, 38028}, {14869, 32521}, {15026, 27375}, {15122, 16324}, {15701, 33706}, {16202, 22732}, {16203, 22731}, {16239, 31239}, {22515, 37348}, {22678, 26316}, {22697, 26446}, {22698, 26451}, {22699, 26341}, {22700, 26348}, {22703, 26492}, {22704, 26487}, {22724, 32497}, {22725, 32494}, {32454, 38762}, {32465, 33416}, {32466, 33417}, {33273, 38225}, {35002, 37455}, {36177, 38613}, {37647, 39266}

X(40108) = midpoint of X(i) and X(j) for these {i,j}: {2, 11171}, {3, 262}, {6, 22677}, {39, 15819}, {76, 32519}, {182, 11261}, {2080, 22503}, {3094, 31958}, {3095, 6194}, {3102, 22727}, {3103, 22726}, {3104, 22686}, {3105, 22684}, {3106, 22715}, {3107, 22714}, {7697, 7709}, {22676, 22728}, {22712, 32447}
X(40108) = reflection of X(i) in X(j) for these {i,j}: {262, 11272}, {11261, 10007}, {14881, 262}, {15819, 140}, {22681, 5}, {24256, 32149}
X(40108) = complement of X(7697)
X(40108) = X(22681)-of-Johnson-triangle
X(40108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7709, 7697}, {3, 7786, 11272}, {3, 11174, 10796}, {3, 11272, 14881}, {3, 22728, 22676}, {262, 22676, 22728}, {632, 32448, 3934}, {3628, 32516, 6248}, {5054, 32447, 22712}, {6683, 13334, 5}, {7697, 11171, 7709}

X(40109) = X(2)X(36)∩X(44)X(513)

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

X(40109) lies on the curve Q158 and these lines: {2, 36}, {42, 517}, {43, 484}, {44, 513}, {100, 752}, {730, 17763}, {750, 2267}, {908, 3724}, {1011, 5172}, {1319, 3720}, {1403, 33098}, {1468, 19543}, {2077, 37400}, {3035, 15447}, {3240, 3245}, {3783, 14513}, {4203, 29846}, {5057, 5143}, {5078, 16405}, {5122, 16056}, {5126, 30950}, {5131, 16569}, {5176, 31330}, {9037, 37676}, {11269, 19647}, {16058, 29661}, {16778, 30852}, {19540, 22765}, {24405, 32856}, {28845, 36002}, {29632, 35992}

X(40109) = isogonal conjugate of X(40110)
X(40109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5080, 30981}, {36, 5080, 28377}

X(40110) = X(190)X(5692)∩X(662)X(4276)

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

X(40110) lies on the circumconic having center X(9), and on the curve Q157, and on these lines: {190, 5692}, {662, 4276}, {1492, 2278}

X(40110) = isogonal conjugate of X(40109)

X(40111) = X(2)X(9703)∩X(30)X(110)

Barycentrics    a^2*(2*a^8 - 6*a^6*b^2 + 6*a^4*b^4 - 2*a^2*b^6 - 6*a^6*c^2 + 8*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 - 2*a^2*c^6 + b^2*c^6) : :
X(40111) = X[23] - 7 X[15039], 3 X[110] - X[10540], 5 X[110] - X[14157], 5 X[110] + X[37477], 3 X[186] - X[32608], X[186] - 3 X[32609], 2 X[3292] + X[7575], X[3580] - 4 X[13392], 2 X[5609] + X[37950], X[5899] - 3 X[35265], 5 X[10540] - 3 X[14157], X[10540] + 3 X[22115], 5 X[10540] + 3 X[37477], X[10620] - 3 X[37948], 2 X[12105] + X[23061], X[14157] + 5 X[22115], 5 X[15034] - 2 X[18571], 3 X[15035] - 2 X[37968], 5 X[15040] - 3 X[37941], X[18572] + 2 X[30714], 5 X[22115] - X[37477], X[32608] - 9 X[32609]

X(40111) lies on these lines: {2, 9703}, {3, 9544}, {5, 578}, {23, 15039}, {30, 110}, {49, 140}, {54, 3628}, {155, 1192}, {156, 550}, {182, 11539}, {184, 549}, {186, 32608}, {195, 16881}, {215, 15325}, {230, 9696}, {323, 2070}, {394, 7502}, {395, 3201}, {396, 3200}, {399, 2071}, {403, 3043}, {511, 37936}, {539, 5972}, {542, 14156}, {546, 18350}, {547, 567}, {548, 1614}, {631, 9704}, {632, 32046}, {1154, 3292}, {1216, 5944}, {1353, 8263}, {1437, 5428}, {1493, 5462}, {1495, 13391}, {1511, 13754}, {1568, 18572}, {1656, 9545}, {1993, 12106}, {2072, 32423}, {2979, 7555}, {3167, 6644}, {3202, 32521}, {3205, 16772}, {3206, 16773}, {3289, 35324}, {3518, 14449}, {3530, 9705}, {3564, 15462}, {3580, 11597}, {3627, 10539}, {3845, 13352}, {3850, 37472}, {3853, 37495}, {3857, 11424}, {3860, 13482}, {5054, 11003}, {5055, 11935}, {5066, 15033}, {5305, 9603}, {5453, 17104}, {5504, 23323}, {5562, 32171}, {5609, 6000}, {5651, 15699}, {5663, 34152}, {5876, 12038}, {5886, 9586}, {5899, 35265}, {5946, 34986}, {6090, 7514}, {6101, 10282}, {6150, 33526}, {6640, 18356}, {6759, 15704}, {7525, 9707}, {7530, 8780}, {7542, 21230}, {8254, 14788}, {9621, 26446}, {9653, 10592}, {9666, 10593}, {9706, 13353}, {10151, 15463}, {10224, 14516}, {10226, 12111}, {10257, 10264}, {10610, 11793}, {10620, 37948}, {11004, 13321}, {11064, 37938}, {11250, 11441}, {11412, 12107}, {11430, 15060}, {11449, 15331}, {11464, 23039}, {11591, 13367}, {11695, 36153}, {11812, 13339}, {11818, 37645}, {12105, 23061}, {12112, 35452}, {12278, 18567}, {12383, 18403}, {13160, 15806}, {13292, 22955}, {13340, 26881}, {13363, 13366}, {13434, 35018}, {13451, 13595}, {14791, 37669}, {15034, 18571}, {15035, 37968}, {15040, 37941}, {15067, 18475}, {15068, 18570}, {15122, 15132}, {15686, 37480}, {16238, 32358}, {16266, 33586}, {19504, 37951}, {20424, 31830}, {23236, 25739}, {23293, 34331}, {26882, 37484}, {32139, 35602}, {34397, 37935}, {35259, 39522}, {37496, 37925}

X(40111) = midpoint of X(i) and X(j) for these {i,j}: {110, 22115}, {323, 2070}, {399, 2071}, {1568, 30714}, {12112, 35452}, {12383, 18403}, {14157, 37477}, {23236, 25739}, {37496, 37925}
X(40111) = reflection of X(i) in X(j) for these {i,j}: {403, 10272}, {10264, 10257}, {15646, 1511}, {18572, 1568}, {37938, 11064}, {37947, 1495}
X(40111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {156, 1092, 550}, {11449, 18436, 15331}, {18350, 34148, 546}

X(40112) = X(2)X(6)∩X(30)X(110)

Barycentrics    4*a^6 - 7*a^4*b^2 + 2*a^2*b^4 + b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4 + c^6 : :
X(40112) = X[323] + X[3580], X[323] + 2 X[11064], 2 X[468] + X[23061], X[858] + 2 X[3292], X[1992] - 3 X[22151], X[3580] - 4 X[11064], 7 X[15020] - 4 X[37934], 4 X[15448] - 3 X[37909], X[32111] + 2 X[37477], 2 X[32269] - 3 X[37907], X[32599] - 3 X[38064], 3 X[35265] - X[37901]

X(40112) lies on these lines: {2, 6}, {23, 35266}, {30, 110}, {287, 37858}, {297, 9141}, {376, 6800}, {381, 6090}, {401, 8591}, {441, 14919}, {468, 15360}, {511, 5642}, {525, 1636}, {541, 10564}, {542, 858}, {549, 5890}, {671, 2986}, {1092, 38323}, {1154, 15361}, {1495, 19924}, {1499, 9137}, {1503, 9143}, {1995, 20423}, {2434, 38951}, {2450, 22566}, {2482, 18334}, {3167, 31152}, {3431, 35254}, {3524, 21766}, {3534, 26864}, {3564, 9140}, {3581, 18579}, {4563, 7799}, {5107, 10418}, {5133, 25561}, {5476, 5651}, {5477, 39602}, {5648, 10510}, {5650, 10168}, {5972, 32225}, {6034, 9225}, {6390, 9146}, {7552, 9820}, {7575, 11694}, {8550, 9716}, {8703, 15080}, {9155, 37461}, {10294, 10295}, {10539, 34613}, {10546, 21850}, {11002, 20192}, {11130, 35303}, {11131, 35304}, {11284, 14848}, {11412, 34351}, {11422, 30739}, {13394, 33884}, {13623, 15759}, {15020, 37934}, {15107, 37904}, {15122, 20126}, {15303, 32220}, {15448, 37909}, {16092, 32583}, {18911, 32216}, {25565, 37990}, {29181, 35265}, {30685, 31173}, {32269, 37907}, {32515, 34094}, {32599, 38064}, {33879, 38110}, {34148, 34664}

X(40112) = midpoint of X(i) and X(j) for these {i,j}: {2, 323}, {3292, 13857}, {5648, 10510}, {5655, 37477}, {9143, 10989}, {15360, 23061}
X(40112) = reflection of X(i) in X(j) for these {i,j}: {2, 11064}, {23, 35266}, {858, 13857}, {3580, 2}, {3581, 18579}, {7426, 5642}, {7575, 11694}, {15107, 37904}, {15360, 468}, {20126, 15122}, {32111, 5655}, {32220, 15303}, {32225, 5972}
X(40112) = reflection of X(9158) in the Orthic axis
X(40112) = isotomic conjugate of the polar conjugate of X(10295)
X(40112) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34802}, {661, 9060}
X(40112) = crossdifference of every pair of points on line {512, 34417}
X(40112) = anticomplement of polar conjugate of X(37984)
X(40112) = barycentric product X(i)*X(j) for these {i,j}: {69, 10295}, {99, 9003}
X(40112) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34802}, {110, 9060}, {9003, 523}, {10295, 4}
X(40112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {323, 11064, 3580}, {323, 22151, 1993}, {8115, 8116, 15066}, {15066, 37645, 14389}, {22151, 37669, 11064}

X(40113) = X(6)X(17)∩X(30)X(110)

Barycentrics    (a^2 - b^2 - c^2)*(3*a^8 - 9*a^6*b^2 + 8*a^4*b^4 - a^2*b^6 - b^8 - 9*a^6*c^2 - a^4*b^2*c^2 + a^2*b^4*c^2 + 4*b^6*c^2 + 8*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(40113) = X[265] + 4 X[3292], 2 X[323] + 3 X[14643], 6 X[1568] - X[12902], X[3519] + 4 X[15091], 9 X[5655] - 4 X[32111], 3 X[5655] + 2 X[37477], 4 X[6053] + X[35001], 4 X[10272] + X[23061], 8 X[11064] - 3 X[15061], X[12121] - 6 X[22115], 4 X[16534] + X[37496], 2 X[32111] + 3 X[37477]

X(40113) lies on these lines: {3, 13623}, {6, 17}, {30, 110}, {265, 3292}, {323, 14643}, {631, 13630}, {1092, 3521}, {1511, 10294}, {1568, 12902}, {3564, 15027}, {6053, 35001}, {10272, 23061}, {11064, 15061}, {12293, 17505}, {13392, 22248}, {13754, 38728}, {16534, 37496}

X(40113) = reflection of X(22248) in X(13392)

X(40114) = X(6)X(25)∩X(30)X(110)

Barycentrics    a^4*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 11*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 3*b^6*c^2 - 6*a^2*b^2*c^4 + 8*b^4*c^4 + 2*a^2*c^6 - 3*b^2*c^6 - c^8) : :
X(40114) = 3 X[18374] - 2 X[19136], 2 X[20772] - 3 X[35265], 3 X[35265] - X[37980]

X(40114) lies on these lines: {6, 25}, {23, 14984}, {30, 110}, {49, 7530}, {156, 31815}, {237, 14908}, {468, 5622}, {1596, 15033}, {1614, 37458}, {5651, 32216}, {6000, 15106}, {6090, 14915}, {6644, 6800}, {6759, 37196}, {9306, 11645}, {9703, 18534}, {10293, 10295}, {11003, 26255}, {12099, 37962}, {12106, 15043}, {12824, 18449}, {13171, 21663}, {13198, 15448}, {14791, 18350}, {15066, 18435}, {15139, 36201}, {15462, 35266}, {20772, 35265}

X(40114) = reflection of X(i) in X(j) for these {i,j}: {25, 1495}, {37980, 20772}
X(40114) = isogonal conjugate of the isotomic conjugate of X(7464)
X(40114) = X(75)-isoconjugate of X(10293)
X(40114) = crossdifference of every pair of points on line {525, 37648}
X(40114) = barycentric product X(6)*X(7464)
X(40114) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 10293}, {7464, 76}
X(40114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {154, 19596, 1495}, {184, 1495, 18374}, {184, 18374, 34397}, {10540, 37477, 5655}, {35265, 37980, 20772}

X(40115) = X(3)X(6)∩X(30)X(111)

Barycentrics    a^2*(a^8 - 3*a^6*b^2 + a^4*b^4 + 3*a^2*b^6 - 2*b^8 - 3*a^6*c^2 + 25*a^4*b^2*c^2 - 14*a^2*b^4*c^2 + a^4*c^4 - 14*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - 2*c^8) : :

X(40115) lies on these lines: {3, 6}, {30, 111}, {352, 5663}, {381, 8585}, {542, 9872}, {647, 30230}, {2393, 34106}, {3291, 35001}, {5655, 14653}, {7464, 11580}, {11799, 24855}, {15685, 34481}, {15759, 38862}, {15993, 20126}, {20481, 31861}, {22115, 39689}

X(40115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 37811, 182}

X(40116) = ISOGONAL CONJUGATE OF X(39470)

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

Let A', B', C' be the intersections of line X(4)X(9) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(40116). (Randy Hutson, October 29, 2020)

X(40116) lies on the circumcircle and these lines: {4, 1566}, {9, 2739}, {10, 2741}, {19, 2717}, {71, 2738}, {98, 17927}, {99, 15411}, {102, 2338}, {103, 2272}, {104, 911}, {105, 5089}, {107, 17926}, {108, 650}, {109, 652}, {110, 677}, {112, 21789}, {281, 2723}, {905, 934}, {910, 972}, {1826, 2688}, {1897, 9057}, {2333, 2700}, {2432, 36067}, {2725, 7719}, {8750, 26716}, {10535, 32726}, {14776, 22108}

X(40116) = isogonal conjugate of X(39470)
X(40116) = Stevanovic-circle-inverse of X(108)
X(40116) = polar-circle-inverse of X(1566)
X(40116) = polar conjugate of the isotomic conjugate of X(677)
X(40116) = polar conjugate of the isogonal conjugate of X(32642)
X(40116) = X(i)-cross conjugate of X(j) for these (i,j): {926, 4}, {8608, 1252}, {32642, 677}
X(40116) = cevapoint of X(i) and X(j) for these (i,j): {647, 39690}, {650, 5089}
X(40116) = trilinear pole of line {6, 3270}
X(40116) = Ψ(X(3), X(101))
X(40116) = Ψ(X(6), X(3270))
X(40116) = Λ(X(651), X(653))
X(40116) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39470}, {63, 676}, {513, 26006}, {516, 905}, {656, 14953}, {910, 4025}, {1456, 6332}, {1459, 30807}, {1886, 4131}, {2398, 3942}, {3270, 24015}, {22383, 35517}, {23696, 39063}, {23973, 34591}
X(40116) = barycentric product X(i)*X(j) for these {i,j}: {4, 677}, {92, 36039}, {100, 36122}, {103, 1897}, {264, 32642}, {653, 2338}, {911, 6335}, {1783, 36101}, {2424, 15742}, {3681, 36109}, {7046, 24016}, {7101, 32668}, {8750, 18025}, {17233, 32701}
X(40116) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39470}, {25, 676}, {101, 26006}, {103, 4025}, {112, 14953}, {677, 69}, {911, 905}, {1783, 30807}, {1815, 30805}, {1897, 35517}, {2338, 6332}, {2424, 1565}, {7128, 24015}, {8750, 516}, {24016, 7056}, {32642, 3}, {32657, 4091}, {32668, 7177}, {32701, 14377}, {36039, 63}, {36056, 4131}, {36101, 15413}, {36122, 693}

X(40117) = X(4)X(972)∩X(19)X(102)

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

X(40117) lies on the circumcircle and these lines: {4, 972}, {19, 102}, {28, 26702}, {55, 31893}, {74, 1903}, {84, 103}, {104, 1436}, {105, 7154}, {106, 7129}, {109, 1783}, {110, 13138}, {112, 32652}, {189, 37378}, {242, 2724}, {268, 281}, {271, 2365}, {280, 26703}, {607, 1433}, {650, 36067}, {653, 934}, {739, 7151}, {1172, 26701}, {1301, 24019}, {1311, 7020}, {2173, 2732}, {2192, 32726}, {2249, 2357}, {2291, 7008}, {2333, 29056}, {2716, 8756}, {2739, 8074}, {3176, 8886}, {3341, 7156}, {7046, 38902}, {8059, 32674}, {13395, 14543}, {32714, 36079}

X(40117) = X(37141)-Ceva conjugate of X(108)
X(40117) = X(i)-cross conjugate of X(j) for these (i,j): {650, 282}, {652, 1172}, {1946, 1433}, {3900, 4}, {32652, 13138}, {32674, 1783}
X(40117) = Stevanovic-circle-inverse of X(36067)
X(40117) = polar-circle-inverse of X(5514)
X(40117) = polar conjugate of X(17896)
X(40117) = polar conjugate of the isotomic conjugate of X(13138)
X(40117) = polar conjugate of the isogonal conjugate of X(32652)
X(40117) = Collings transform of X(7367)
X(40117) = X(i)-isoconjugate of X(j) for these (i,j): {3, 14837}, {7, 10397}, {40, 905}, {48, 17896}, {63, 6129}, {77, 14298}, {109, 16596}, {198, 4025}, {221, 6332}, {222, 8058}, {223, 521}, {322, 22383}, {329, 1459}, {342, 36054}, {347, 652}, {514, 7078}, {522, 7011}, {525, 2360}, {647, 8822}, {650, 7013}, {656, 1817}, {1461, 7358}, {1813, 38357}, {1819, 7178}, {2187, 15413}, {2199, 35518}, {2331, 4131}, {3194, 24018}, {3195, 30805}, {4091, 7952}, {4391, 7114}, {4587, 38374}, {7254, 21075}
X(40117) = cevapoint of X(i) and X(j) for these (i,j): {19, 650}, {607, 1946}, {3900, 7367}
X(40117) = trilinear pole of line {6, 33}
X(40117) = Ψ(X(3), X(9))
X(40117) = Ψ(X(6), X(33))
X(40117) = barycentric product X(i)*X(j) for these {i,j}: {4, 13138}, {84, 1897}, {92, 36049}, {108, 280}, {109, 7020}, {162, 39130}, {189, 1783}, {190, 7129}, {264, 32652}, {271, 36127}, {281, 37141}, {282, 653}, {309, 8750}, {318, 8059}, {645, 2358}, {648, 1903}, {651, 7003}, {664, 7008}, {668, 7151}, {811, 2357}, {1436, 6335}, {2192, 18026}, {4554, 7154}, {7367, 13149}, {32674, 34404}
X(40117) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 17896}, {19, 14837}, {25, 6129}, {33, 8058}, {41, 10397}, {84, 4025}, {108, 347}, {109, 7013}, {112, 1817}, {162, 8822}, {189, 15413}, {280, 35518}, {282, 6332}, {607, 14298}, {650, 16596}, {692, 7078}, {1415, 7011}, {1433, 4131}, {1436, 905}, {1783, 329}, {1897, 322}, {1903, 525}, {2192, 521}, {2208, 1459}, {2357, 656}, {2358, 7178}, {3900, 7358}, {7003, 4391}, {7008, 522}, {7020, 35519}, {7118, 652}, {7129, 514}, {7151, 513}, {7154, 650}, {8059, 77}, {8750, 40}, {13138, 69}, {14776, 15501}, {18344, 38357}, {32652, 3}, {32674, 223}, {32676, 2360}, {32713, 3194}, {32714, 14256}, {36049, 63}, {36127, 342}, {37141, 348}, {39130, 14208}

X(40118) = ISOGONAL CONJUGATE OF X(14984)

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

X(40118) lies on the circumcircle and these lines: {2, 10420}, {4, 691}, {5, 11635}, {23, 925}, {24, 935}, {25, 476}, {30, 3565}, {69, 10425}, {74, 3566}, {99, 186}, {107, 37777}, {110, 468}, {111, 2501}, {112, 230}, {183, 2855}, {378, 2696}, {427, 1291}, {523, 3563}, {542, 35191}, {827, 37943}, {858, 13398}, {930, 21284}, {1287, 3518}, {1289, 37951}, {1290, 4231}, {1292, 37979}, {1296, 10295}, {1297, 36166}, {1300, 14618}, {1302, 37962}, {1304, 6353}, {1995, 16167}, {2691, 7414}, {2693, 7422}, {2694, 7425}, {2697, 7418}, {2715, 36472}, {3542, 10423}, {4232, 9060}, {5189, 20185}, {6103, 23969}, {6792, 35188}, {7464, 20187}, {10098, 18533}, {10101, 31384}, {12131, 14734}, {13397, 37959}, {20189, 37920}, {39193, 39828}

X(40118) = reflection of X(3563) in the Euler line
X(40118) = isogonal conjugate of X(14984)
X(40118) = polar-circle-inverse of X(16188)
X(40118) = orthoptic-circle-of-Steiner-inellipe-inverse of X(16221)
X(40118) = Collings transform of X(39021)
X(40118) = X(542)-cross conjugate of X(4)
X(40118) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14984}, {63, 2493}, {656, 7468}, {810, 14221}
X(40118) = cevapoint of X(25) and X(6103)
X(40118) = trilinear pole of line {6, 14273}
X(40118) = barycentric product X(16081)*X(40083)
X(40118) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14984}, {25, 2493}, {112, 7468}, {648, 14221}, {6103, 16188}, {6531, 34175}, {40083, 36212}

X(40119) = X(4)X(2696)∩X(23)X(3565)

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

X(40119) lies on the circumcircle and these lines: {4, 2696}, {23, 3565}, {24, 10098}, {25, 691}, {30, 20187}, {74, 20186}, {99, 468}, {107, 16315}, {110, 8681}, {111, 2489}, {112, 3291}, {186, 1296}, {403, 30247}, {476, 4232}, {523, 2374}, {925, 7426}, {935, 6353}, {1290, 7438}, {1294, 36166}, {1995, 10420}, {2373, 36168}, {2691, 4231}, {2693, 7418}, {2694, 7423}, {2697, 7417}, {2971, 15398}, {6090, 10425}, {9084, 36898}, {10295, 30256}, {11635, 13595}, {13398, 37980}, {16167, 26255}, {33638, 37969}, {37951, 39382}

X(40119) = reflection of X(2374) in the Euler line
X(40119) = polar-circle-inverse of X(31655)
X(40119) = X(2854)-cross conjugate of X(4)
X(40119) = Ψ(X(3), X(351))
X(40119) = X(i)-isoconjugate of X(j) for these (i,j): {63, 10418}, {656, 7472}
X(40119) = barycentric product X(17983)*X(40078)
X(40119) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 10418}, {112, 7472}, {8753, 34169}, {40078, 6390}

X(40120) = X(2)X(135)∩X(24)X(99)

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

X(40120) lies on the circumcircle and the lines: {2, 135}, {4, 3565}, {24, 99}, {25, 925}, {110, 6353}, {112, 3542}, {378, 20187}, {403, 691}, {427, 20185}, {468, 10420}, {476, 37777}, {487, 1306}, {488, 1307}, {847, 39416}, {935, 37951}, {1292, 31384}, {1296, 18533}, {4231, 13397}, {5897, 7422}, {5966, 36898}, {7418, 34168}, {11635, 37943}, {16167, 37962}, {26706, 31385}, {33638, 35480}

X(40120) = polar-circle-inverse of X(31842)
X(40120) = orthoptic-circle-of-Steiner-inellipe-inverse of X(135)
X(40120) = isogonal conjugate of X(34382)
X(40120) = X(3564)-cross conjugate of X(4)
X(40120) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34382}, {31842, 36051}
X(40120) = cevapoint of X(i) and X(j) for these (i,j): {25, 230}, {193, 35296}
X(40120) = trilinear pole of line {6, 38359}
X(40120) = Λ(X(3), X(6467))
X(40120) = Λ(X(68), X(69))
X(40120) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34382}, {230, 31842}

X(40121) = X(3)X(19164)∩X(25)X(111)

Barycentrics    a^2*(2*a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - b^8 - a^6*c^2 + b^6*c^2 - a^4*c^4 + a^2*c^6 + b^2*c^6 - c^8) : :
X(40121) = X[112] + 3 X[9157], 3 X[9157] - X[11641]

X(40121) lies on these lines: {3, 19164}, {5, 2794}, {25, 111}, {26, 19165}, {127, 6676}, {132, 6756}, {206, 1511}, {1297, 9715}, {1576, 39857}, {2799, 22105}, {2871, 14574}, {2881, 14270}, {2909, 6102}, {3202, 5944}, {3542, 13200}, {3549, 10749}, {5027, 9517}, {5938, 10313}, {5946, 19156}, {6031, 7493}, {7395, 38699}, {7507, 10735}, {7514, 14649}, {9714, 13310}, {9969, 28343}, {10547, 14885}, {10766, 19125}, {11819, 19160}, {12362, 14689}, {13236, 14691}, {14676, 15562}, {14900, 21841}, {15818, 18876}

X(40121) = midpoint of X(i) and X(j) for these {i,j}: {3, 19164}, {112, 11641}, {5938, 10313}, {14676, 15562}
X(40121) = reflection of X(38624) in X(34217)
X(40121) = barycentric product X(25)*X(28726)
X(40121) = barycentric quotient X(28726)/X(305)
X(40121) = {X(112),X(9157)}-harmonic conjugate of X(11641)

X(40122) = X(6)X(538)∩X(729)X(8667)

Barycentrics    a^2*(8*a^4*b^4 - 11*a^4*b^2*c^2 + a^2*b^4*c^2 + 8*a^4*c^4 + a^2*b^2*c^4 - 16*b^4*c^4) : :

X(40122) lies on the cubic K1161 and these lines: {6, 538}, {729, 8667}, {3053, 3231}, {3288, 33979}

X(40122) = isotomic conjugate of anticomplement of X(40125)

X(40123) = X(2)X(6)∩X(4)X(8024)

Barycentrics    a^6 + a^4*b^2 - 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 : :

X(40123) lies on these lines: {2, 6}, {4, 8024}, {22, 3926}, {25, 3933}, {76, 6997}, {251, 14001}, {304, 10327}, {305, 315}, {427, 7776}, {1194, 7758}, {1196, 7855}, {1369, 16063}, {1799, 7763}, {1975, 7500}, {2548, 8891}, {2549, 19568}, {2979, 4176}, {3266, 7386}, {3785, 7485}, {4872, 19799}, {5133, 32816}, {6337, 6636}, {6340, 31101}, {6995, 32830}, {7391, 9464}, {7392, 39998}, {7484, 7767}, {7493, 7796}, {7494, 26233}, {7495, 32825}, {7762, 11324}, {8362, 39951}, {10691, 14929}, {16276, 32833}, {16951, 20065}, {18018, 28706}, {18916, 37450}, {25053, 32458}, {32064, 33796}, {32815, 34603}, {32817, 34608}, {32824, 37900}, {32828, 37990}, {39978, 40022}

X(40123) = anticomplement of X(1184)
X(40123) = isotomic conjugate of the isogonal conjugate of X(37485)
X(40123) = barycentric product X(76)*X(37485)
X(40123) = barycentric quotient X(37485)/X(6)
X(40123) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 193, 5359}, {305, 315, 1370}, {7796, 33651, 34254}, {10327, 39732, 304}, {33651, 34254, 7493}

X(40124) = X(2)X(64)∩X(25)X(17808)

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

X(40124) lies on these lines: {2, 64}, {25, 17808}, {612, 10375}, {3162, 39951}, {7484, 33581}

X(40124) = X(1496)-complementary conjugate of X(15259)

X(40125) = X(2)X(159)∩X(25)X(39)

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

X(40125) lies on the Kiepert circumhyperbola of the medial triangle and on these lines: {2, 159}, {22, 6337}, {25, 39}, {160, 6503}, {1125, 15497}, {1184, 19459}, {1486, 3666}, {2482, 39857}, {5359, 32621}, {6292, 7484}, {9909, 11165}

X(40125) = complement of the isogonal conjugate of X(37485)
X(40125) = complement of isotomic conjugate of X(40122)
X(40125) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1184}, {37485, 10}
X(40125) = X(2)-Ceva conjugate of X(1184)
X(40125) = barycentric product X(5286)*X(37485)

X(40126) = X(2)X(3933)∩X(25)X(32)

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

X(40126) lies on these lines: {2, 3933}, {3, 9465}, {6, 373}, {23, 1384}, {25, 32}, {111, 21309}, {115, 15433}, {468, 2452}, {612, 4515}, {1180, 16419}, {1194, 1611}, {1351, 9463}, {1627, 9909}, {1995, 5354}, {3051, 9777}, {3066, 5039}, {3266, 22253}, {3767, 5094}, {5007, 30734}, {5020, 5359}, {5024, 11580}, {5093, 39024}, {5254, 31152}, {5286, 30739}, {5304, 16317}, {5309, 32216}, {11173, 20977}, {11324, 17128}, {14567, 26864}, {15302, 20481}, {31404, 37439}, {31885, 34417}

X(40127) = X(1)X(8074)∩X(2)X(7)

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

X(40127) lies on these lines: {1, 8074}, {2, 7}, {6, 3756}, {8, 9310}, {11, 5819}, {25, 1604}, {37, 5218}, {41, 938}, {56, 6554}, {101, 18391}, {108, 281}, {169, 3086}, {198, 33849}, {220, 1788}, {346, 5205}, {497, 910}, {612, 4336}, {614, 5304}, {631, 16601}, {919, 2726}, {956, 19309}, {1055, 5731}, {1108, 3290}, {1146, 3476}, {1212, 7288}, {1436, 4224}, {1696, 2345}, {1743, 5121}, {2082, 14986}, {2246, 5838}, {2256, 5275}, {2280, 10580}, {2291, 9057}, {2348, 17728}, {3011, 37689}, {3161, 14439}, {3177, 17081}, {3207, 3486}, {3212, 26658}, {3241, 17439}, {3474, 17747}, {3501, 26062}, {3600, 27541}, {3616, 17451}, {3684, 36845}, {3689, 17314}, {4000, 26007}, {4223, 38902}, {4293, 5179}, {4315, 5199}, {4339, 37055}, {5011, 30305}, {5089, 6353}, {5222, 9502}, {5540, 10072}, {5657, 6998}, {5703, 21808}, {6921, 25082}, {7176, 30694}, {7228, 30754}, {10106, 23058}, {16502, 28016}, {16780, 28080}, {16845, 25086}, {16997, 17316}, {17567, 25066}, {20752, 37657}, {24477, 37658}, {32625, 37254}

X(40128) = X(1)X(26258)∩X(2)X(1743)

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

X(40128) lies on these lines: {1, 26258}, {2, 1743}, {6, 17728}, {609, 5179}, {612, 2324}, {614, 5304}, {910, 3914}, {2323, 5276}, {2348, 37646}, {2911, 5275}, {3008, 26229}, {3011, 7735}, {3290, 5306}, {3912, 17001}, {4644, 30742}, {4896, 31071}, {5299, 28018}, {5305, 23536}, {5750, 17124}, {17023, 26279}, {20072, 30798}, {26265, 39595}

X(40129) = X(2)X(6)∩X(9)X(11031)

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

X(40129) lies on these lines: {2, 6}, {9, 11031}, {32, 411}, {39, 6986}, {57, 2312}, {100, 10315}, {232, 4233}, {284, 4220}, {579, 19649}, {614, 2257}, {938, 16502}, {949, 5222}, {961, 7119}, {1108, 7191}, {1172, 16318}, {1210, 5299}, {1249, 37394}, {1333, 10313}, {1901, 37456}, {2548, 6991}, {3149, 30435}, {3767, 6828}, {5007, 6915}, {5254, 6895}, {5280, 13411}, {5286, 6836}, {5305, 6831}, {5319, 6943}, {5324, 39690}, {5746, 26118}, {6894, 7745}, {7466, 10311}, {8557, 26242}, {15048, 37428}, {22240, 36018}

X(40130) = X(2)X(11175)∩X(6)X(373)

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

X(40130) lies on these lines: {2, 11175}, {6, 373}, {32, 1495}, {39, 3231}, {51, 1196}, {110, 5354}, {184, 1184}, {194, 35275}, {230, 30516}, {353, 38010}, {511, 9463}, {732, 30749}, {1180, 3819}, {1194, 1613}, {1843, 14580}, {1995, 5039}, {2021, 3117}, {2030, 11003}, {2225, 17053}, {2502, 5008}, {3053, 35268}, {3787, 20859}, {3981, 21969}, {5097, 39024}, {5305, 11064}, {5306, 5642}, {5309, 13857}, {5359, 9306}, {6656, 14467}, {7882, 14463}, {8617, 15082}, {8627, 35007}, {12212, 20998}, {12294, 35325}, {13366, 39764}, {15820, 39691}, {30435, 35259}

X(40131) = X(1)X(41)∩X(2)X(7)

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

X(40131) lies on these lines: {1, 41}, {2, 7}, {3, 16601}, {6, 354}, {10, 17742}, {12, 208}, {19, 25}, {31, 16970}, {38, 16517}, {40, 1334}, {44, 4860}, {45, 1155}, {46, 3730}, {56, 1212}, {65, 220}, {72, 19309}, {78, 19310}, {85, 6559}, {120, 5880}, {165, 846}, {172, 16968}, {190, 30758}, {200, 3930}, {218, 942}, {219, 5173}, {239, 26274}, {284, 4228}, {346, 32932}, {388, 6554}, {404, 25082}, {474, 25066}, {497, 5819}, {518, 37658}, {609, 37817}, {610, 2268}, {728, 1706}, {936, 33299}, {965, 16352}, {966, 5227}, {975, 18596}, {1002, 3751}, {1055, 3576}, {1146, 5252}, {1174, 39943}, {1190, 2264}, {1194, 2277}, {1196, 21796}, {1201, 9575}, {1202, 2257}, {1281, 3501}, {1319, 34522}, {1376, 3693}, {1434, 32024}, {1449, 7191}, {1475, 3333}, {1478, 5179}, {1541, 11372}, {1572, 3230}, {1617, 15288}, {1642, 5091}, {1697, 39587}, {1698, 17744}, {1731, 26228}, {1743, 5272}, {1759, 2198}, {1760, 4687}, {1770, 17732}, {1836, 17747}, {1837, 21049}, {1929, 3097}, {2099, 6603}, {2171, 2324}, {2178, 5322}, {2182, 17603}, {2183, 29639}, {2256, 2262}, {2266, 2294}, {2267, 22099}, {2269, 2270}, {2287, 5208}, {2291, 9058}, {2303, 5324}, {2316, 39393}, {2321, 10327}, {2329, 19860}, {2345, 26040}, {2646, 3207}, {3008, 10520}, {3011, 7735}, {3061, 19861}, {3085, 7719}, {3099, 15485}, {3125, 9620}, {3145, 5277}, {3177, 7176}, {3247, 3920}, {3263, 3729}, {3329, 36406}, {3338, 4253}, {3423, 24320}, {3496, 5250}, {3601, 37254}, {3616, 33950}, {3660, 22163}, {3673, 17682}, {3679, 5525}, {3684, 3870}, {3691, 39581}, {3692, 29641}, {3726, 16973}, {3748, 16777}, {3811, 3970}, {3812, 30618}, {3838, 30755}, {3916, 19313}, {3927, 19321}, {3951, 19316}, {3980, 17355}, {3984, 19318}, {3991, 5687}, {4007, 33091}, {4034, 33090}, {4051, 36846}, {4258, 37080}, {4363, 30748}, {4384, 26234}, {4390, 9623}, {4513, 5836}, {4648, 7289}, {4652, 19314}, {4659, 31130}, {4666, 16503}, {4875, 12513}, {4911, 17671}, {5011, 5119}, {5020, 20760}, {5022, 32636}, {5044, 16852}, {5253, 26690}, {5261, 27541}, {5283, 13738}, {5286, 23536}, {5297, 16548}, {5308, 7291}, {5320, 16972}, {5341, 16675}, {5359, 16470}, {5440, 19322}, {5526, 5902}, {5587, 21044}, {5781, 10391}, {5838, 10580}, {6167, 9312}, {6180, 34855}, {6714, 25557}, {6734, 26036}, {7081, 21387}, {7084, 8751}, {7146, 25930}, {7290, 21764}, {7292, 16670}, {7297, 16672}, {7410, 26878}, {7412, 17916}, {7484, 22060}, {7736, 20785}, {7964, 37499}, {7994, 21809}, {8074, 31397}, {8583, 39244}, {8609, 33925}, {8804, 26052}, {9578, 23058}, {9593, 24443}, {10129, 30787}, {10388, 17452}, {10857, 19649}, {11018, 25514}, {11108, 25086}, {11227, 16434}, {11688, 38869}, {14828, 27475}, {16059, 25074}, {16193, 22153}, {16408, 25068}, {16409, 25075}, {16412, 25083}, {16502, 28011}, {16552, 17736}, {16567, 21382}, {16580, 31261}, {16589, 37225}, {16667, 30350}, {16779, 29820}, {16780, 28082}, {16823, 21384}, {16831, 20602}, {16849, 31445}, {16998, 39252}, {17001, 26247}, {17007, 17270}, {17022, 21370}, {17056, 39690}, {17064, 17737}, {17107, 24796}, {17683, 20880}, {17745, 18398}, {18615, 23203}, {19297, 34879}, {21258, 30617}, {21872, 37567}, {22108, 26275}, {24005, 26063}, {24471, 25878}, {24512, 36404}, {24590, 37555}, {26244, 29828}, {27129, 33867}, {28043, 37580}, {30385, 30556}, {30386, 30557}, {30400, 32556}, {30401, 32555}, {32561, 37579}, {37272, 37597}

X(40131) = {X(6203),X(6204)}-harmonic conjugate of X(7)

X(40132) = X(2)X(3)∩X(51)X(37669)

Barycentrics    3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - a^4*c^2 + 14*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6 : :

X(40132) lies on these lines: {2, 3}, {51, 37669}, {69, 5651}, {110, 14912}, {125, 35904}, {182, 35260}, {184, 18928}, {193, 6090}, {264, 10603}, {373, 3618}, {895, 5642}, {1007, 37804}, {1196, 5319}, {1249, 14580}, {1285, 5913}, {1352, 37643}, {1495, 25406}, {1992, 3292}, {2892, 6698}, {2986, 14494}, {3066, 11064}, {3068, 10963}, {3069, 10961}, {3260, 34229}, {3266, 32817}, {3291, 7735}, {3589, 8547}, {3819, 33522}, {4319, 5218}, {4320, 7288}, {5085, 15448}, {5268, 31452}, {5297, 37696}, {5304, 16317}, {5544, 38110}, {5640, 37645}, {5646, 21167}, {5656, 37475}, {5921, 26869}, {5943, 11427}, {5972, 14561}, {6337, 11059}, {6688, 34750}, {6699, 18489}, {6719, 35282}, {6776, 35259}, {7292, 37697}, {7612, 34289}, {7665, 9155}, {7736, 10314}, {7765, 34481}, {8549, 10192}, {9172, 23583}, {9214, 15398}, {9306, 11225}, {9826, 20125}, {9936, 18934}, {10519, 32269}, {10546, 18911}, {10643, 11489}, {10644, 11488}, {11061, 32241}, {11185, 37803}, {11469, 34469}, {12828, 32244}, {13567, 14826}, {15030, 18931}, {16187, 32223}, {16276, 19583}, {18289, 35812}, {18290, 35813}, {18852, 34334}, {21356, 32225}, {21448, 37689}, {29181, 31860}, {35283, 37638}

X(40133) = X(1)X(6)∩X(2)X(4875)

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

X(40133) lies on these lines: {1, 6}, {2, 4875}, {19, 1398}, {39, 4646}, {40, 5022}, {41, 1319}, {42, 22317}, {56, 910}, {57, 7955}, {58, 34862}, {65, 1475}, {69, 25887}, {75, 27340}, {81, 16699}, {101, 24928}, {145, 3693}, {169, 999}, {172, 294}, {210, 39244}, {241, 2275}, {269, 2124}, {279, 1418}, {322, 25971}, {354, 1202}, {387, 15852}, {496, 5179}, {517, 4253}, {519, 4515}, {536, 17158}, {579, 31793}, {583, 7957}, {604, 2264}, {650, 21105}, {672, 3057}, {673, 7176}, {948, 3772}, {1015, 16583}, {1030, 35202}, {1086, 10481}, {1146, 1210}, {1170, 18889}, {1200, 22088}, {1201, 9502}, {1249, 1841}, {1323, 17366}, {1334, 5919}, {1385, 4251}, {1420, 3207}, {1427, 23653}, {1434, 27000}, {1451, 7118}, {1572, 5021}, {1575, 21896}, {1766, 8158}, {2260, 2262}, {2266, 37080}, {2271, 9619}, {2280, 2646}, {2310, 10939}, {2340, 3780}, {2348, 9310}, {2391, 10521}, {3008, 6692}, {3241, 25082}, {3244, 3991}, {3290, 5304}, {3501, 3880}, {3576, 4258}, {3579, 5030}, {3600, 5819}, {3666, 17014}, {3679, 25068}, {3686, 12447}, {3691, 25917}, {3721, 21342}, {3730, 9957}, {3739, 27304}, {3815, 25614}, {3890, 4520}, {4051, 5836}, {4255, 9592}, {4262, 13624}, {4308, 5838}, {4350, 6610}, {4383, 25930}, {4513, 36846}, {4642, 23649}, {4856, 25078}, {5011, 37582}, {5065, 5301}, {5305, 15251}, {5540, 5563}, {5584, 36743}, {5839, 20007}, {6184, 12640}, {6554, 14986}, {6736, 8568}, {6743, 17362}, {6764, 17299}, {7208, 24790}, {7735, 16020}, {7743, 24045}, {8273, 36744}, {9312, 24600}, {9441, 33863}, {9797, 17314}, {10460, 37593}, {10914, 16549}, {11019, 21049}, {11997, 36635}, {11998, 21764}, {12053, 17747}, {13370, 32625}, {14100, 20978}, {16679, 21867}, {16716, 16726}, {16728, 33296}, {16834, 25083}, {17609, 21808}, {17721, 28052}, {18663, 19790}, {19861, 37658}, {20905, 26818}, {24352, 30625}, {24597, 25939}, {25055, 25086}, {25067, 37681}, {26563, 26964}, {27253, 31269}, {29571, 37662}, {30271, 37507}, {34497, 34855}, {35092, 35116}, {37500, 37551}, {37665, 39587}

X(40133) = complement of X(16284)
X(40133) = crossdifference of every pair of points on line X(513)X(5537) (the de Longchamps line of the excentral triangle, and the radical axis of any pair of {1st, 2nd and 3rd antipedal circles of X(1)})

X(40134) = X(2)X(905)∩X(230)X(231)

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

X(40134) lies on these lines: {2, 905}, {25, 1946}, {111, 2687}, {230, 231}, {513, 21786}, {612, 3900}, {649, 6615}, {1639, 2509}, {1734, 5268}, {2522, 3239}, {3803, 26249}, {4468, 27400}, {4521, 16612}, {4893, 14413}, {5020, 22160}, {7484, 22091}, {9058, 32685}, {14298, 22383}, {16757, 31209}, {21894, 31946}, {24562, 25084}, {25009, 26146}

X(40134) = complement of isotomic conjugate of X(9058)
X(40134) = crosspoint of X(2) and X(9058)
X(40134) = crosssum of X(6) and X(9001)

X(40135) = X(3)X(6)∩X(115)X(1990)

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

X(40135) lies on these lines: {3, 6}, {115, 1990}, {230, 37911}, {232, 15262}, {237, 21639}, {419, 11596}, {647, 657}, {1843, 34416}, {3163, 16310}, {5702, 7735}, {6128, 18487}, {6749, 7747}, {8721, 18919}, {8749, 14581}, {8779, 14567}, {9407, 20975}, {10991, 15471}, {11060, 11079}, {11443, 37465}, {12167, 33578}, {14537, 34288}, {14836, 39593}, {15525, 23967}, {23976, 23992}, {34570, 37941}, {36212, 37784}, {37665, 39602}

X(40136) = X(6)X(17)∩X(32)X(393)

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

X(40136) lies on these lines: on lines {6, 17}, {32, 393}, {115, 6748}, {216, 14836}, {570, 36422}, {571, 6781}, {577, 7765}, {800, 11062}, {3051, 31883}, {3163, 39018}, {3767, 40065}, {5065, 5355}, {5309, 15905}, {6709, 10220}, {7735, 38282}, {8573, 9699}

X(40136) = barycentric product X(397)*X(398)

X(40137) = X(44)X(513)∩X(521)X(4521)

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

X(40137) lies on these lines: {44, 513}, {521, 4521}, {2490, 9001}, {3239, 3900}, {3887, 14350}, {3910, 20318}, {4131, 31209}, {5375, 15632}, {14303, 15313}

X(40138) = X(2)X(648)∩X(4)X(6)

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

X(40138) lies on these lines: {2, 648}, {4, 6}, {19, 1475}, {20, 3284}, {32, 37460}, {44, 7952}, {69, 11331}, {112, 5063}, {184, 6525}, {193, 340}, {216, 3523}, {232, 4232}, {281, 1100}, {297, 1992}, {376, 36427}, {378, 14836}, {468, 2452}, {470, 37640}, {471, 37641}, {550, 15905}, {577, 3522}, {685, 35906}, {1033, 3516}, {1119, 17366}, {1217, 36752}, {1384, 37934}, {1585, 19054}, {1586, 19053}, {1609, 32534}, {1640, 18808}, {1656, 15851}, {1657, 38292}, {1785, 16670}, {1968, 33871}, {2331, 3554}, {3003, 35486}, {3088, 7772}, {3089, 5319}, {3091, 15860}, {3172, 7738}, {3515, 8573}, {3535, 32787}, {3536, 32788}, {3543, 18487}, {3553, 7129}, {3589, 32000}, {3618, 9308}, {3629, 32001}, {3854, 36412}, {4846, 18850}, {5007, 7487}, {5032, 37174}, {5094, 7736}, {5306, 6353}, {5309, 6623}, {5667, 9408}, {5967, 6531}, {6032, 37665}, {6110, 10653}, {6111, 10654}, {6524, 11402}, {6618, 17809}, {7046, 17369}, {7412, 37503}, {7739, 14581}, {7748, 34569}, {7757, 35940}, {8014, 8737}, {8015, 8738}, {8553, 17506}, {8557, 23710}, {8882, 38808}, {8889, 9300}, {9722, 35487}, {10295, 16303}, {10299, 36751}, {10312, 33872}, {11063, 21844}, {12174, 35711}, {14361, 23292}, {14614, 37187}, {16666, 34231}, {17555, 37654}, {18533, 34288}, {21735, 36748}, {30435, 37458}, {31400, 37118}, {35481, 39176}, {35484, 39662}, {36744, 37289}

leftri

Hodpieces: X(40139)-X(40173)

rightri

This preamble is based on notes received from Radosław Żak (October 29, 2020) and Peter Moses (October 29-30, 2020).

In the plane of a triangle ABC, let P be a point, not on a sideline of ABC, and let DEF be the cevian triangle of P. The isogonal conjugate of line EF is a conic. Let A' be the center of of the conic, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in a point here named the hodpiece of P, denoted by H(P). The name hodpiece is taken from James Joyce's book, Finnegans Wake. The unique point P such that H(P) = P, indexed below as X(40139), is named the Bloom point after Leopold Bloom, the main character in Joyce's Ulysses. The point H(X(5)) = X(40140) is the Dedalus point, and the H(X(7)) = X(40141), the Zana point.

An article (in Polish) about hodpieces by Żak won a gold medal in a competition for high school students organized by the Polish Mathematical Society. For an English translation, see Isogonal conjugate and a few properties of the point X(25).

If P = p : q : r (barycentrics), then H(P) = a2/(p*(-a2/p + b2/q + c2/r) : : .

Let P* = P-Ceva conjugate of X(6). Then H(P) = isogonal conjugate of P*-cross conjugate of P.

The appearance of (i,j) in the following list means that H(X(i)) = X(j):

(1,57), (2,25), (3,459), (4,394), (5,40140), (6,2), (7,40141), (9,1422), (10,40142), (13,40156), (14,40157), (15, 40158), (16,40159), (19,6513), (21,40160), (25,6384), (28,40161), (31,6384), (32,40162), (37,40143), (39,40163), (41,40164), (48,40165), (54,324), (55,36620), (56,6557), (57,200), (58,321), (59, 40166), (61,40167), (62,40168), (63,40169), (64,40170), (69,40144), (75,40145), (76,40146), (81,42), (83,3051), (86,40147), (87,40171), (88,40172), (162,37755), (163,6358), (190,40148), (249,8029), (251,8024), (259,16664), (266,7028), (275,418), (284,40149), (288,3078), (493,8038), (512,37880), (514,40150), (588,8035), (589,8036), (644,40151), (648,184), (651,55), (662,756), (765,8042), (1016,8027), (1073,3079), (1126,8025), (1170,8012), (1171,8013), (1172,401523), (1252,6545), (1262,23615), (1461,7046), (1783,222), (2226,8028), (2298,40153), (2981,8014), (2982,8021), (3939,40154), (4558,14593), (4577,8041), (4629,8040), (4638,8028), (5381,8027), (6151,8015), (6185,23612), (7121,8026), (8115,25), (8116, 25), (9268,6545), (10630,8030), (13138,6611), (18018,36414), (20332,40155), (23964,23616), (23984,23614), (345071,6382), (34536,23611), (34537,23610), (34538,23613), (34568,3081), (34574,8030), (36049,196), (38810,8022), (38826,8039), (38828,6555), (38830,8023)

Note that H(X(2)) = H(X(8115)) = H(X(8116)) = X(25).

For Vu Thanh Tung's generalization to U-hodpieces, see the preamble just before X(40212).

X(40139) = BLOOM POINT

Barycentrics    a*(a + sqrt(t + a^2) : : , where t is the positive solution of t+a^2+b^2+c^2 = a*sqrt(t+a^2)+b*sqrt(t+b^2)+c*sqrt(t+c^2)
Trilinears    a+sqrt(t+a^2) : : , where t is as just above

X(40139) is the fixed point of the hodpiece transform.

X(40139) lies on the cubic K102 and these lines: (pending)

X(40140) = DEDALUS POINT

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

X(40140) lies on these lines: {24,3432}, {18883,45832}

X(40140) = hodpiece of X(5)
X(40140) = X(54)-reciprocal conjugate of-X(2888)
X(40140) = intersection, other than A, B, C, of circumconics {A, B, C, X(2), X(24)} and {A, B, C, X(97), X(25044)}
X(40140) = barycentric product X(95)*X(3432)
X(40140) = barycentric quotient X(54)/X(2888)
X(40140) = trilinear quotient X(i)/X(j) for these (i, j): (2167, 2888), (2169, 45800)

X(40141) = ZANA POINT

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

X(40141) lies on these lines: {2, 1814}, {6, 5089}, {48, 672}, {212, 2340}, {218, 222}, {219, 3693}, {650, 11502}, {2194, 37908}, {5063, 14578}, {19350, 32677}, {26706, 32726}

X(40141) = isogonal conjugate of X(37800)
X(40141) = hodpiece of X(7)
X(40141) = crossdifference of every pair of points on line X(11934)X(21185)

X(40142) = HODPIECE OF X(10)

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

X(40142) lies on these lines: {2, 8044}, {48, 28606}, {184, 386}, {2359, 38822}

X(40142) = isogonal conjugate of X(21076
X(40140) = hodpiece of X(10)

X(40143) = HODPIECE OF X(37)

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

X(40143) lies on these lines: {1, 229}, {2, 1029}, {502, 1224}, {1255, 21353}, {3733, 8029}, {19623, 35058}

X(40143) = isogonal conjugate of X(21873)
X(40143) = hodpiece of X(37)

X(40144) = HODPIECE OF X(69)

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

X(40144) lies on these lines: {2, 2138}, {6, 17409}, {37, 21148}, {111, 39417}, {378, 3108}, {428, 34288}, {455, 2386}, {1241, 40009}, {2165, 16318}, {2207, 13854}, {3172, 39951}, {6339, 37784}, {6753, 34212}, {8770, 14580}, {14910, 21213}, {15262, 34608}, {23115, 36414}

X(40144) = isogonal conjugate of X(28419)
X(40144) = polar conjugate of isotomic conjugate of X(34207)

X(40145) = HODPIECE OF X(75)

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

X(40145) lies on these lines: lines {2, 7357}, {748, 19559}, {2174, 2276}, {7296, 26892}

X(40145) = isogonal conjugate of X(20444)
X(40145) = hodpiece of X(75)

X(40146) = HODPIECE OF X(76)

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

X(40146) lies on these lines: {2, 66}, {32, 39466}, {39, 184}, {1501, 27369}, {1976, 13854}, {2001, 18018}, {3051, 14575}, {3852, 36414}, {9306, 34138}, {15389, 19558}, {19156, 37649}

X(40146) = isogonal conjugate of X(40073)
X(40146) = hodpiece of X(76)

X(40147) = HODPIECE OF X(86)

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

X(40147) lies on these lines: {2, 2140}, {6, 34444}, {111, 6577}, {213, 2350}, {672, 39798}, {941, 3588}, {995, 39965}, {1218, 39735}, {2183, 39974}, {2205, 38346}

X(40147) = isogonal conjugate of X(29767)
X(40148) = hodpiece of X(86)

X(40148) = HODPIECE OF X(190)

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

X(40148) lies on these lines: {1, 596}, {2, 8050}, {31, 16679}, {39, 14751}, {42, 1100}, {86, 3112}, {87, 32925}, {213, 2308}, {593, 595}, {741, 1621}, {756, 3248}, {899, 3791}, {902, 1402}, {1015, 8041}, {1042, 1319}, {1201, 1245}, {1459, 40086}, {1977, 21827}, {2296, 17394}, {2309, 40085}, {3223, 24661}, {3231, 23533}, {3720, 30982}, {3730, 30651}, {3920, 31111}, {4075, 39748}, {5311, 7032}, {17150, 18792}, {17193, 39712}, {18194, 26037}, {37132, 37205}

X(40148) = isogonal conjugate of X(4360)
X(40148) = isotomic conjugate of X(40087)
X(40148) = hodpiece of X(190)

X(40149) = HODPIECE OF X(284)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :
Barycentrics    (sec A)/(cos B + cos C) : :

X(40149) lies on the Kiepert hyperbola and these lines: {2, 92}, {4, 65}, {7, 37181}, {8, 37189}, {10, 201}, {19, 1708}, {27, 653}, {28, 1940}, {34, 5136}, {57, 5307}, {76, 331}, {98, 108}, {226, 1826}, {243, 4219}, {264, 34258}, {275, 1409}, {286, 1396}, {321, 8736}, {393, 17903}, {485, 1659}, {486, 13390}, {651, 2986}, {664, 31631}, {671, 18026}, {801, 1944}, {857, 18588}, {1029, 7282}, {1068, 3085}, {1172, 2982}, {1426, 1867}, {1427, 16732}, {1446, 20618}, {1737, 1838}, {1785, 4424}, {1788, 14018}, {1812, 1943}, {1824, 1893}, {1825, 1873}, {1848, 2051}, {1860, 2181}, {1865, 6354}, {1868, 12709}, {1869, 4848}, {1870, 5397}, {1891, 3429}, {1897, 3896}, {1947, 17950}, {2052, 6521}, {2333, 16609}, {3668, 8808}, {3696, 7046}, {3772, 17902}, {3931, 7952}, {4213, 17985}, {4296, 11103}, {4331, 17871}, {4554, 8781}, {4605, 22000}, {4654, 38461}, {5174, 13583}, {5236, 17758}, {5905, 6504}, {6198, 7073}, {6830, 13599}, {6844, 31363}, {6905, 22341}, {7098, 14016}, {7178, 14618}, {7554, 8762}, {13405, 23710}, {14213, 22464}, {14571, 18676}, {16603, 21016}, {16608, 17862}, {17863, 37543}, {17869, 21924}, {17918, 19786}, {20928, 23600}, {28950, 30807}, {33133, 37770}, {37263, 38860}, {37788, 40012}

X(40149) = isogonal conjugate of X(2193)
X(40149) = isotomic conjugate of X(1812)
X(40149) = hodpiece of X(284)
X(40149) = polar conjugate of X(21)
X(40149) = antigonal conjugate of polar conjugate of X(425)
X(40149) = cevapoint of X(i) and X(j) for these {i,j}: {1, 1744}, {65, 1880}, {225, 1826}
X(40149) = crosspoint of X(92) and X(2052)
X(40149) = crosssum of X(48) and X(577)
X(40149) = crossdifference of every pair of points on line X(1946)X(36054)
X(40149) = Danneels point of X(92)
X(40149) = trilinear pole of line X(523)X(24006)
X(40149) = perspector of ABC and orthoanticevian triangle of X(1441)
X(40149) = pole wrt polar circle of trilinear polar of X(21) (line X(521)X(650))
X(40149) = trilinear product X(i)*X(j) for these {i,j}: {62, 95}, {108, 1577}
X(40149) = barycentric product X(108)*X(850)

X(40150) = HODPIECE OF X(514)

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

X(40150) lies on these lines: {2, 39026}, {31624, 31634}

X(40150) = isogonal conjugate of X(21202)
X(40150) = hodpiece of X(514)

X(40151) = HODPIECE OF X(644)

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

X(40151) lies on these lines: {1, 2137}, {2, 27825}, {6, 9050}, {7, 1997}, {55, 1293}, {56, 1149}, {57, 1122}, {63, 27819}, {65, 3680}, {222, 38828}, {388, 6556}, {553, 4052}, {951, 1466}, {1376, 31343}, {1407, 38266}, {1412, 33628}, {1434, 16711}, {1458, 38289}, {2415, 32933}, {3161, 8051}, {3218, 27827}, {3304, 14261}, {3339, 10563}, {4373, 21454}, {5228, 7153}, {5437, 24151}, {17743, 27830}, {26627, 27823}

X(40151) = isogonal conjugate of X(3161)
X(40151) = hodpiece of X(644)
X(40151) = cevapoint of X(649) and X(1357)

X(40152) = HODPIECE OF X(1172)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - c^2)^2 : :
Barycentrics    (cos A) (sec B + sec C) : :

X(40152) lies on these lines: {1, 1779}, {2, 7}, {3, 73}, {28, 1935}, {40, 3182}, {42, 22069}, {48, 3173}, {65, 10901}, {71, 1214}, {72, 856}, {97, 22128}, {109, 1297}, {219, 1073}, {223, 573}, {225, 1217}, {276, 349}, {278, 24310}, {283, 951}, {284, 2003}, {394, 1804}, {651, 1817}, {916, 23171}, {1020, 36908}, {1102, 3719}, {1334, 16577}, {1396, 4269}, {1407, 37500}, {1410, 22076}, {1422, 6282}, {1427, 2245}, {1465, 5755}, {1764, 34050}, {1812, 1949}, {1813, 14919}, {1936, 4219}, {2183, 11347}, {2252, 26934}, {2260, 37543}, {2318, 23067}, {3074, 37275}, {3075, 7549}, {3682, 7066}, {4055, 22057}, {4466, 18588}, {5751, 14547}, {6360, 34287}, {7175, 23602}, {7177, 8813}, {8021, 20122}, {8804, 8808}, {13726, 37523}, {15934, 18477}, {16609, 20235}, {18876, 32660}, {22270, 37612}, {23207, 39796}, {26931, 37872}, {37264, 37694}

X(40152) = isogonal conjugate of X(8748)
X(40152) = isotomic conjugate of polar conjugate of X(73)
X(40152) = hodpiece of X(11172)
X(40152) = crossdifference of every pair of points on line X(663)X(3064)
X(40152) = X(i)-isoconjugate of X(j) for these {i,j}: {19, 29}, {92, 2299}
X(40152) = trilinear product X(i)*X(j) for these {i,j}: {48, 307}, {63, 73}
X(40152) = barycentric product X(63)*X(1214)

X(40153) = HODPIECE OF X(2298)

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

X(40153) lies on these lines:s {1, 19259}, {2, 6}, {21, 1191}, {31, 2274}, {42, 18185}, {55, 3736}, {56, 58}, {57, 16700}, {63, 16696}, {110, 28479}, {171, 18792}, {213, 4641}, {221, 5323}, {238, 18169}, {292, 28643}, {386, 16374}, {553, 17205}, {595, 17524}, {614, 18165}, {958, 27660}, {1001, 10458}, {1010, 5710}, {1014, 1407}, {1043, 20037}, {1171, 4629}, {1193, 4267}, {1201, 10457}, {1203, 20744}, {1333, 1790}, {1453, 37523}, {1724, 19243}, {1964, 16687}, {1999, 30939}, {2185, 7303}, {2193, 22119}, {2300, 3666}, {2999, 18163}, {3052, 4184}, {3218, 18601}, {3306, 16736}, {3733, 8027}, {3772, 17167}, {3782, 17139}, {4216, 4252}, {4257, 19254}, {4363, 30599}, {4393, 16722}, {4481, 7252}, {4653, 16483}, {4658, 30116}, {5208, 17597}, {5315, 19247}, {5711, 25526}, {7290, 17194}, {7304, 18021}, {9022, 19835}, {9575, 16699}, {10455, 31993}, {13588, 37540}, {16468, 18192}, {16470, 18724}, {16685, 28606}, {16717, 30647}, {16753, 27003}, {17173, 33129}, {17174, 33133}, {17182, 17720}, {17189, 37543}, {17202, 19786}, {17599, 35623}, {19262, 36746}, {19550, 36754}, {20182, 21769}, {32939, 34063}

X(40153) = isogonal conjugate of X(14624)
X(40153) = hodpiece of X(2298)
X(40153) = cevapoint of X(1193) and X(2300)
X(40153) = crosspoint of X(i) and X(j) for these {i,j}: {58, 1178}, {81, 593}, {1014, 1509}
X(40153) = crosssum of X(i) and X(j) for these {i,j}: {10, 1215}, {37, 594}, {210, 1500}
X(40153) = crossdifference of every pair of points on line X(512)X(3700)
X(40153) = trilinear product X(i)*X(j) for these {i,j}: {57, 4267}, {58, 3666}, {81, 1193}, {86, 2300}, {163, 3004}, {593, 2292}, {662, 6371}, {757, 2092}, {849, 1211}, {960, 1412}, {1014, 2269}, {1333, 4357}, {1408, 3687}, {1437, 1848}, {1444, 2354}, {1509, 3725}, {1576, 4509}, {1790, 1829}, {2194, 3674}, {3733, 3882}

X(40154) = HODPIECE OF X(3939)

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

X(40154) lies on these lines: {2, 37206}, {7, 3434}, {55, 1292}, {57, 169}, {85, 8817}, {222, 1462}, {226, 15490}, {269, 2191}, {279, 1617}, {354, 14268}, {479, 3660}, {1014, 3598}, {1119, 34855}, {3664, 19604}, {8814, 24471}

X(40154) = isogonal conjugate of X(6600)
X(40154) = hodpiece of X(3939)

X(40155) = HODPIECE OF X(20332)

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

X(40155) lies on these lines: {2, 38}, {6, 2109}, {31, 813}, {42, 649}, {43, 660}, {55, 1911}, {192, 39918}, {292, 16515}, {2276, 3252}, {3097, 30663}, {4562, 36817}, {4583, 32925}, {6376, 24421}, {12782, 40098}, {17596, 18787}, {17756, 36906}, {19567, 27853}, {20358, 20456}, {22116, 37596}, {24420, 30963}, {24426, 37678}

X(40155) = isogonal conjugate of X(3253)
X(40155) = hodpiece of X(20332)

X(40156) = HODPIECE OF X(13)

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

X(40156) lies on these lines: {2, 2992}, {16, 184}, {62, 8919}, {186, 34394}, {3480, 5616}, {5063, 40157}, {8739, 14165}, {15412, 35443}

X(40156) = hodpiece of X(13)

X(40157) = HODPIECE OF X(14)

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

X(40157) lies on these lines: {2, 2993}, {15, 184}, {61, 8918}, {186, 34395}, {3479, 5612}, {5063, 40156}, {8740, 14165}, {15412, 35444}

X(40157) = hodpiece of X(14)

X(40158) = HODPIECE OF X(15)

Barycentrics    (3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S))/(2*S^2 + (a^2 - b^2 - c^2)*(3*a^2 + Sqrt[3]*S)) : :

X(40158) lies on these lines: {2, 19776}, {4, 8014}, {13, 34296}, {14, 3440}, {17, 8919}, {18, 32461}, {2394, 20578}, {8737, 16080}, {11078, 11121}, {11550, 12816}

X(40158) = hodpiece of X(15)

X(40159) = HODPIECE OF X(16)

Barycentrics    (3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S)/(2*S^2 + (a^2 - b^2 - c^2)*(3*a^2 - Sqrt[3]*S)) : :

X(40159) lies on these lines: {2, 19777}, {4, 8015}, {13, 3441}, {14, 34295}, {17, 32460}, {18, 8918}, {2394, 20579}, {8738, 16080}, {11092, 11122}, {11550, 12817}

X(40159) = hodpiece of X(16)

X(40160) = HODPIECE OF X(21)

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

X(40160) lies on these lines: {2, 2995}, {12, 73}, {56, 225}, {65, 15267}, {86, 19607}, {222, 226}, {348, 349}, {1214, 6358}, {1400, 37646}, {5930, 10570}, {19701, 37695}, {25525, 37523}

X(40160) = hodpiece of X(21)

X(40161) = HODPIECE OF X(28)

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

X(40161) lies on these lines: {2, 2335}, {10, 55}, {37, 6358}, {71, 440}, {219, 306}, {272, 32779}, {281, 17776}, {345, 40071}, {2318, 3695}, {3682, 7515}, {25515, 33116}

X(40161) = hodpiece of X(28)
X(40161) = isotomic conjugate of polar conjugate of X(41506)

X(40162) = HODPIECE OF X(32)

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

X(40162) lies on these lines: {2, 2998}, {4, 3978}, {10, 6382}, {76, 3981}, {83, 3224}, {98, 3222}, {226, 18275}, {264, 37892}, {305, 1916}, {670, 21001}, {850, 23610}, {2996, 20023}, {3407, 24733}, {19606, 20965}

X(40162) = hodpiece of X(32)

X(40163) = HODPIECE OF X(39)

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

X(40163) lies on these lines: {2, 1031}, {76, 14370}, {251, 11606}, {262, 8928}, {1916, 8856}, {10159, 33665}, {32085, 37892}

X(40163) = hodpiece of X(39)

X(40164) = HODPIECE OF X(41)

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

X(40164) lies on these lines: {2, 1931}, {75, 8033}, {81, 6650}, {86, 4425}, {333, 27483}, {335, 20362}, {1089, 40033}, {1268, 32780}, {1434, 7249}, {2296, 18757}, {3120, 6628}, {5196, 8049}, {7192, 8029}, {24041, 31632}, {25496, 30598}

X(40164) = hodpiece of X(41)

X(40165) = HODPIECE OF X(48)

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

X(40165) lies on these lines: {2, 1947}, {8, 7049}, {29, 3362}, {333, 18751}, {1948, 2994}, {6521, 34591}, {34234, 37279}

X(40165) = hodpiece of X(48)

X(40166) = HODPIECE OF X(59)

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

X(40166) lies on these lines: {2, 650}, {11, 15914}, {226, 514}, {278, 2401}, {312, 4391}, {497, 885}, {498, 35100}, {513, 1836}, {522, 4847}, {523, 17874}, {652, 4382}, {654, 812}, {666, 31633}, {850, 20896}, {905, 24789}, {918, 36038}, {1088, 24002}, {1211, 1577}, {1479, 11247}, {1734, 32865}, {3126, 3925}, {3900, 4863}, {3914, 23811}, {4036, 15523}, {4077, 21104}, {4106, 5928}, {4554, 31619}, {4791, 29594}, {4823, 21198}, {5432, 11124}, {6545, 23760}, {6923, 8760}, {10947, 11927}, {11393, 18344}, {12647, 14077}, {14298, 23813}, {16732, 21141}, {17420, 30591}, {21132, 23615}, {30613, 33110}, {30713, 35519}

X(40166) = isotomic conjugate of X(31615)
X(40166) = Danneels point of X(693)
X(40166) = hodpiece of X(59)

X(40167) = HODPIECE OF X(61)

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

X(40167) lies on these lines: {2, 19712}, {4, 36304}, {13, 8174}, {14, 39134}, {18, 3489}, {275, 8741}, {3131, 32627}, {5487, 19779}, {11122, 11144}

X(40167) = hodpiece of X(61)

X(40168) = HODPIECE OF X(62)

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

X(40168) lies on these lines: {2, 19713}, {4, 36305}, {13, 39135}, {14, 8175}, {17, 3490}, {275, 8742}, {3132, 32628}, {5488, 19778}, {11121, 11143}

X(40168) = hodpiece of X(62)

X(40169) = HODPIECE OF X(63)

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

X(40169) lies on these lines: {2, 7219}, {4, 1448}, {19, 15259}, {33, 2285}, {204, 612}, {1249, 2345}, {2000, 40015}, {2303, 4183}

X(40169) = hodpiece of X(63)

X(40170) = HODPIECE OF X(64)

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

X(40170) lies on these lines: {2, 15851}, {20, 33893}, {69, 37878}, {122, 23608}, {1032, 11064}, {1105, 3091}, {3523, 3532}, {5273, 9533}, {7396, 34168}, {32831, 34403}

X(40170) = hodpiece of X(64)

X(40171) = HODPIECE OF X(87)

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

X(40171) lies on these lines: {2, 1334}, {87, 8616}, {256, 13097}, {1221, 40025}, {3208, 31008}, {23415, 37677}

X(40171) = hodpiece of X(87)

X(40172) = HODPIECE OF X(88)

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

X(40172) lies on these lines: {1, 1168}, {2, 80}, {42, 663}, {45, 55}, {57, 840}, {200, 6065}, {484, 13589}, {759, 28210}, {1319, 1647}, {1644, 5440}, {1807, 3478}, {2099, 34431}, {3689, 4908}, {3992, 17780}, {5727, 14629}, {15343, 24402}, {21805, 23344}

X(40172) = hodpiece of X(88)

X(40173) = HODPIECE OF X(523)

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

X(40173) lies on these lines: {2, 9514}, {22, 3447}, {32, 14164}, {1993, 35910}, {5012, 9513}, {6328, 36163}

X(40173) = hodpiece of X(523)

X(40174) = X(2)X(64)∩X(25)X(1249)

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

X(40174) lies on the cubic K1162 and these lines: {2, 64}, {25, 1249}, {154, 36413}, {614, 1042}, {1297, 10565}, {3079, 3172}, {3424, 7398}, {7386, 17808}, {7392, 20207}, {15246, 15874}, {15589, 40032}

X(40175) = X(2)X(269)∩X(25)X(7079)

Barycentrics    a*(a - b - c)^2*(a^2 + b^2 - 2*b*c + c^2)*(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(40175) lies on the cubic K1162 and these lines: {2, 269}, {25, 7079}, {614, 6554}, {7083, 28070}, {15487, 40124}

X(40176) = X(2)X(6359)∩X(204)X(612)

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

X(40176) lies on the cubic K1162 and these lines: {2, 6359}, {204, 612}, {6554, 7386}

X(40177) = X(2)X(2139)∩X(25)X(800)

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

X(40177) lies on the cubic K1162 and these lines: {2, 2139}, {25, 800}, {1184, 40124}

X(40178) = X(2)X(159)∩X(4)X(3162)

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

X(40178) lies on the cubic K1162 and these lines: {2, 159}, {4, 3162}, {10, 15487}, {76, 1370}, {83, 6997}, {321, 11677}, {459, 33584}, {614, 36907}, {671, 39842}, {2996, 7391}, {5395, 7394}, {7386, 18840}, {7392, 18841}, {7735, 16277}, {18845, 37349}

X(40179) = X(2)X(3933)∩X(4)X(3162)

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

X(40179) lies on the cubic K1162 and these lines: {2, 3933}, {4, 3162}, {6, 7392}, {25, 1249}, {32, 34608}, {115, 15437}, {216, 1194}, {233, 7736}, {251, 7714}, {376, 1627}, {612, 2345}, {614, 6554}, {631, 1180}, {1184, 5286}, {1196, 5319}, {1285, 7500}, {1370, 5354}, {1560, 8889}, {3618, 40022}, {4000, 15487}, {5368, 34481}, {6353, 9465}, {6392, 11324}, {6995, 30435}, {6997, 34482}, {7408, 18907}, {7499, 37689}, {7766, 37895}, {8267, 32817}, {8877, 36877}, {8878, 16041}, {8879, 14091}, {10299, 38862}, {11433, 15595}, {16045, 39998}, {16949, 32822}, {16989, 37891}, {37439, 37665}, {37669, 40130}

X(40180) = X(2)X(269)∩X(388)X(612)

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

X(40180) lies on the cubic K1162 and these lines: {2, 269}, {388, 612}, {478, 1122}, {614, 1042}, {1427, 15487}, {1435, 3162}

X(40181) = X(2)X(17742)∩X(9)X(33163)

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

X(40181) lies on the cubic K1162 and these lines: {2, 17742}, {9, 33163}, {10, 15487}, {25, 7079}, {37, 614}, {169, 29667}, {612, 1184}, {1213, 40131}, {1766, 37456}, {2345, 7386}, {3162, 23050}, {3305, 17755}, {4204, 21838}, {16593, 32777}

X(40182) = X(2)X(14259)∩X(612)X(23051)

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

X(40182) lies on the cubic K1162 and these lines: {2, 14259}, {612, 23051}, {907, 7485}, {1184, 7772}, {3763, 17810}, {7386, 18840}

X(40183) = X(2)X(6359)∩X(25)X(34)

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

X(40183) lies on the cubic K1162 and these lines: {2, 6359}, {25, 34}, {612, 10375}, {15487, 36908}

X(40184) = X(2)X(169)∩X(19)X(614)

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

X(40184) lies on the cubic K1162 and these lines: {2, 169}, {19, 614}, {612, 40125}, {2298, 4224}, {2345, 7386}, {4359, 24605}, {5272, 16545}

X(40185) = X(2)X(2138)∩X(7386)X(40125)

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

X(40185) lies on the cubic K1162 and these lines: {2, 2138}, {7386, 40125}, {7494, 34207}, {15487, 18589}, {28419, 34254}

X(40186) = X(2)X(2139)∩X(1249)X(3162)

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

X(40186) lies on the cubic K1162 and these lines: {2, 2139}, {1249, 3162}, {3344, 5304}, {13854, 35968}, {15487, 36908}, {36417, 39020}

X(40187) = X(2)X(3933)∩X(3)X(907)

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

X(40187) lies on the cubics K169 and K657, and also on these lines: {2, 3933}, {3, 907}, {64, 159}, {269, 17742}, {1907, 8801}

X(40187) = barycentric product X(i)*X(j) for these {i,j}: {17811, 18840}, {32000, 34817}, {32830, 39951}
X(40187) = barycentric quotient X(i)/X(j) for these {i,j}: {1593, 6995}, {5065, 30435}, {17811, 3618}, {18840, 37874}, {32830, 40022}, {34817, 15740}
X(40187) = {X(907),X(14259)}-harmonic conjugate of X(3)

X(40188) = X(1)X(159)∩X(2)X(169)

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

X(40188) lies on the conic {{A,B,C,X(1),X(2)}}, the cubics K169 and K1062, and on these lines: {1, 159}, {2, 169}, {20, 1219}, {40, 39959}, {46, 291}, {57, 5299}, {69, 17742}, {77, 2172}, {269, 2138}, {278, 7195}, {330, 5088}, {961, 1448}, {985, 3338}, {1280, 3868}, {1973, 3942}, {2224, 16780}, {3333, 39958}, {4222, 36122}, {14953, 39747}

X(40188) = isogonal conjugate of X(17742)
X(40188) = X(i)-cross conjugate of X(j) for these (i,j): {25, 269}, {1565, 1019}, {16502, 1}
X(40188) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17742}, {2, 12329}, {6, 10327}, {9, 8270}, {55, 28739}, {63, 23050}, {78, 20613}, {100, 2509}, {1801, 1826}, {4557, 17498}, {7123, 11677}
X(40188) = cevapoint of X(649) and X(3942)
X(40188) = barycentric product X(i)*X(j) for these {i,j}: {1, 39732}, {81, 36907}
X(40188) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 10327}, {6, 17742}, {25, 23050}, {31, 12329}, {56, 8270}, {57, 28739}, {608, 20613}, {614, 11677}, {649, 2509}, {1019, 17498}, {1437, 1801}, {7289, 28409}, {16502, 15487}, {36907, 321}, {39732, 75}
X(40188) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 18725, 18727}, {7291, 17170, 18596}

X(40189) = X(2)X(14259)∩X(6)X(3917)

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

X(40189) lies on the cubic K169 and these lines: {2, 14259}, {6, 3917}, {20, 13575}, {22, 907}, {1350, 4175}, {7392, 18840}

X(40189) = isogonal conjugate of X(40222)
X(40189) = barycentric product X(i)*X(j) for these {i,j}: {18840, 37485}, {39951, 40123}
X(40189) = barycentric quotient X(i)/X(j) for these {i,j}: {37485, 3618}, {40123, 40022}

X(40190) = X(6)X(20)∩X(1285)X(27082)

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

X(40190) lies on the cubics K041, K055, and K169, and on lines {6, 20}, {1285, 27082}, {18841, 37874}

X(40190) = X(33580)-cross conjugate of X(6995)
X(40190) = X(i)-isoconjugate of X(j) for these (i,j): {1496, 18840}, {17811, 23051}
X(40190) = barycentric product X(i)*X(j) for these {i,j}: {6995, 15740}, {30435, 37874}
X(40190) = barycentric quotient X(i)/X(j) for these {i,j}: {3618, 32830}, {6995, 32000}, {30435, 17811}, {33580, 33537}

X(40191) = X(2)X(14259)∩X(40125)X(40179)

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

X(40191) lies on the cubic K1162 and these lines: {2, 14259}, {40125, 40179}

X(40191) = X(1184)-cross conjugate of X(40179)
X(40191) = barycentric product X(39978)*X(40179)
X(40191) = barycentric quotient X(1184)/X(40182)

X(40192) = X(2)X(40190)∩X(612)X(40175)

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

X(40192) lies on the cubic K1162 and these lines: {2, 40190}, {612, 40175}, {1184, 40174}, {7386, 37665}, {40177, 40178}

X(40192) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 40174}, {2155, 18840}
X(40192) = X(2)-Ceva conjugate of X(40174)

X(40193) = X(2)X(17742)∩X(612)X(23051)

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

X(40193) lies on the cubic K1162 and these lines: {2, 17742}, {612, 23051}, {614, 40125}, {1435, 3162}, {2191, 37538}, {4000, 15487}

X(40193) = X(1184)-cross conjugate of X(614)
X(40193) = barycentric quotient X(1184)/X(40181)

X(40194) = X(612)X(39951)∩X(1184)X(40175)

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

X(40194) lies on the cubic K1162 and these lines: {612, 39951}, {1184, 40175}, {40174, 40184}, {40178, 40183}

X(40194) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 40175}, {1106, 23051}
X(40194) = X(2)-Ceva conjugate of X(40175)

X(40195) = X(2)X(40190)∩X(25)X(40182)

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

X(40195) lies on the cubic K1162 and these lines: {2, 40190}, {25, 40182}, {40124, 40125}, {40180, 40181}

X(40195) = X(1184)-cross conjugate of X(40124)

X(40196) = X(20)X(154)∩X(30)X(1351)

Barycentrics    (3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*(3*a^6 - a^4*b^2 - 7*a^2*b^4 + 5*b^6 - a^4*c^2 + 22*a^2*b^2*c^2 - 5*b^4*c^2 - 7*a^2*c^4 - 5*b^2*c^4 + 5*c^6)
3X(40196) = 5 X[3619] - 32 X[4550], 8 X[11820] - 5 X[14927]

X(40196) lies on these lines: {20, 154}, {30, 1351}, {69, 15311}, {1885, 15740}, {1993, 5059}, {2777, 35513}, {3146, 11433}, {3543, 17810}, {3619, 4550}, {5890, 13598}, {7691, 33522}, {10304, 20725}, {10574, 15887}, {12233, 31371}, {16251, 20477}, {16386, 35260}

leftri

Points associated with Vu parallels conics: X(40197)-X(40211)

rightri

This preamble is based on notes received from Vu Thanh Tung, October 31, 2020.

In the plane of a triangle ABC, let P = p:q:r and U = u:v:w (barycentrics) be points. Let A1 be the point on BC such that PA1 is parallel to AU, and define B1 and C1 cyclically. Let A2 be the point on BC such that PA2 is parallel to AP, and define B2 and C2 cyclically. The six points A1, A2, B1, B2, C1, C2 lies on a conic, here named the Vu parallels conic of P and U. See Vu Parallels Conic.

Peter Moses (October 31, 2020) found that V(P,U) = (p*u + r*u + p*v)*(p*u + q*u + p*w)*(p^3*r^2*u^3*v^2 - p^2*q*r^2*u^3*v^2 - 4*p*q^2*r^2*u^3*v^2 - 2*q^3*r^2*u^3*v^2 - 2*p*q*r^3*u^3*v^2 - 2*q^2*r^3*u^3*v^2 - 6*p^2*q*r^2*u^2*v^3 - 10*p*q^2*r^2*u^2*v^3 - 4*q^3*r^2*u^2*v^3 - 5*p^2*r^3*u^2*v^3 - 11*p*q*r^3*u^2*v^3 - 6*q^2*r^3*u^2*v^3 - 2*p*r^4*u^2*v^3 - 2*q*r^4*u^2*v^3 - 2*p^2*q*r^2*u*v^4 - 3*p*q^2*r^2*u*v^4 - q^3*r^2*u*v^4 - 2*p^2*r^3*u*v^4 - 3*p*q*r^3*u*v^4 - q^2*r^3*u*v^4 - 2*p^3*q*r*u^3*v*w - 3*p^2*q^2*r*u^3*v*w - 2*p*q^3*r*u^3*v*w - 3*p^2*q*r^2*u^3*v*w - 8*p*q^2*r^2*u^3*v*w - 4*q^3*r^2*u^3*v*w - 2*p*q*r^3*u^3*v*w - 4*q^2*r^3*u^3*v*w - 3*p^3*q*r*u^2*v^2*w - 8*p^2*q^2*r*u^2*v^2*w - 10*p*q^3*r*u^2*v^2*w - 2*q^4*r*u^2*v^2*w - p^3*r^2*u^2*v^2*w - 15*p^2*q*r^2*u^2*v^2*w - 32*p*q^2*r^2*u^2*v^2*w - 14*q^3*r^2*u^2*v^2*w - 6*p^2*r^3*u^2*v^2*w - 21*p*q*r^3*u^2*v^2*w - 16*q^2*r^3*u^2*v^2*w - 2*p*r^4*u^2*v^2*w - 4*q*r^4*u^2*v^2*w - 2*p^3*q*r*u*v^3*w - 10*p^2*q^2*r*u*v^3*w - 14*p*q^3*r*u*v^3*w - 3*q^4*r*u*v^3*w - 2*p^3*r^2*u*v^3*w - 21*p^2*q*r^2*u*v^3*w - 37*p*q^2*r^2*u*v^3*w - 13*q^3*r^2*u*v^3*w - 11*p^2*r^3*u*v^3*w - 26*p*q*r^3*u*v^3*w - 13*q^2*r^3*u*v^3*w - 3*p*r^4*u*v^3*w - 3*q*r^4*u*v^3*w - 2*p^2*q^2*r*v^4*w - 3*p*q^3*r*v^4*w - 4*p^2*q*r^2*v^4*w - 6*p*q^2*r^2*v^4*w - 2*p^2*r^3*v^4*w - 3*p*q*r^3*v^4*w + p^3*q^2*u^3*w^2 - p^2*q^2*r*u^3*w^2 - 2*p*q^3*r*u^3*w^2 - 4*p*q^2*r^2*u^3*w^2 - 2*q^3*r^2*u^3*w^2 - 2*q^2*r^3*u^3*w^2 - p^3*q^2*u^2*v*w^2 - 6*p^2*q^3*u^2*v*w^2 - 2*p*q^4*u^2*v*w^2 - 3*p^3*q*r*u^2*v*w^2 - 15*p^2*q^2*r*u^2*v*w^2 - 21*p*q^3*r*u^2*v*w^2 - 4*q^4*r*u^2*v*w^2 - 8*p^2*q*r^2*u^2*v*w^2 - 32*p*q^2*r^2*u^2*v*w^2 - 16*q^3*r^2*u^2*v*w^2 - 10*p*q*r^3*u^2*v*w^2 - 14*q^2*r^3*u^2*v*w^2 - 2*q*r^4*u^2*v*w^2 - 4*p^3*q^2*u*v^2*w^2 - 10*p^2*q^3*u*v^2*w^2 - 3*p*q^4*u*v^2*w^2 - 8*p^3*q*r*u*v^2*w^2 - 32*p^2*q^2*r*u*v^2*w^2 - 37*p*q^3*r*u*v^2*w^2 - 6*q^4*r*u*v^2*w^2 - 4*p^3*r^2*u*v^2*w^2 - 32*p^2*q*r^2*u*v^2*w^2 - 68*p*q^2*r^2*u*v^2*w^2 - 24*q^3*r^2*u*v^2*w^2 - 10*p^2*r^3*u*v^2*w^2 - 37*p*q*r^3*u*v^2*w^2 - 24*q^2*r^3*u*v^2*w^2 - 3*p*r^4*u*v^2*w^2 - 6*q*r^4*u*v^2*w^2 - 2*p^3*q^2*v^3*w^2 - 4*p^2*q^3*v^3*w^2 - p*q^4*v^3*w^2 - 4*p^3*q*r*v^3*w^2 - 14*p^2*q^2*r*v^3*w^2 - 13*p*q^3*r*v^3*w^2 - 2*p^3*r^2*v^3*w^2 - 16*p^2*q*r^2*v^3*w^2 - 24*p*q^2*r^2*v^3*w^2 - 6*p^2*r^3*v^3*w^2 - 13*p*q*r^3*v^3*w^2 - p*r^4*v^3*w^2 - 5*p^2*q^3*u^2*w^3 - 2*p*q^4*u^2*w^3 - 6*p^2*q^2*r*u^2*w^3 - 11*p*q^3*r*u^2*w^3 - 2*q^4*r*u^2*w^3 - 10*p*q^2*r^2*u^2*w^3 - 6*q^3*r^2*u^2*w^3 - 4*q^2*r^3*u^2*w^3 - 2*p^3*q^2*u*v*w^3 - 11*p^2*q^3*u*v*w^3 - 3*p*q^4*u*v*w^3 - 2*p^3*q*r*u*v*w^3 - 21*p^2*q^2*r*u*v*w^3 - 26*p*q^3*r*u*v*w^3 - 3*q^4*r*u*v*w^3 - 10*p^2*q*r^2*u*v*w^3 - 37*p*q^2*r^2*u*v*w^3 - 13*q^3*r^2*u*v*w^3 - 14*p*q*r^3*u*v*w^3 - 13*q^2*r^3*u*v*w^3 - 3*q*r^4*u*v*w^3 - 2*p^3*q^2*v^2*w^3 - 6*p^2*q^3*v^2*w^3 - p*q^4*v^2*w^3 - 4*p^3*q*r*v^2*w^3 - 16*p^2*q^2*r*v^2*w^3 - 13*p*q^3*r*v^2*w^3 - 2*p^3*r^2*v^2*w^3 - 14*p^2*q*r^2*v^2*w^3 - 24*p*q^2*r^2*v^2*w^3 - 4*p^2*r^3*v^2*w^3 - 13*p*q*r^3*v^2*w^3 - p*r^4*v^2*w^3 - 2*p^2*q^3*u*w^4 - 2*p^2*q^2*r*u*w^4 - 3*p*q^3*r*u*w^4 - 3*p*q^2*r^2*u*w^4 - q^3*r^2*u*w^4 - q^2*r^3*u*w^4 - 2*p^2*q^3*v*w^4 - 4*p^2*q^2*r*v*w^4 - 3*p*q^3*r*v*w^4 - 2*p^2*q*r^2*v*w^4 - 6*p*q^2*r^2*v*w^4 - 3*p*q*r^3*v*w^4) : :

and T(P,U) = (p*u + r*u + p*v)*(p*u + q*u + p*w)*(3*p^2*q*r*u^3*v + 5*p*q^2*r*u^3*v + 2*q^3*r*u^3*v + 2*p*q*r^2*u^3*v + 2*q^2*r^2*u^3*v + p^3*r*u^2*v^2 + 10*p^2*q*r*u^2*v^2 + 14*p*q^2*r*u^2*v^2 + 5*q^3*r*u^2*v^2 + 4*p^2*r^2*u^2*v^2 + 11*p*q*r^2*u^2*v^2 + 7*q^2*r^2*u^2*v^2 + 2*p*r^3*u^2*v^2 + 2*q*r^3*u^2*v^2 + p^3*r*u*v^3 + 8*p^2*q*r*u*v^3 + 10*p*q^2*r*u*v^3 + 3*q^3*r*u*v^3 + 5*p^2*r^2*u*v^3 + 10*p*q*r^2*u*v^3 + 5*q^2*r^2*u*v^3 + 2*p*r^3*u*v^3 + 2*q*r^3*u*v^3 + p^2*q*r*v^4 + p*q^2*r*v^4 + p^2*r^2*v^4 + p*q*r^2*v^4 + p^2*q^2*u^3*w + p*q^3*u^3*w + 4*p*q^2*r*u^3*w + 2*q^3*r*u^3*w + 2*q^2*r^2*u^3*w + 3*p^3*q*u^2*v*w + 10*p^2*q^2*u^2*v*w + 8*p*q^3*u^2*v*w + q^4*u^2*v*w + 12*p^2*q*r*u^2*v*w + 28*p*q^2*r*u^2*v*w + 10*q^3*r*u^2*v*w + 10*p*q*r^2*u^2*v*w + 11*q^2*r^2*u^2*v*w + 2*q*r^3*u^2*v*w + 5*p^3*q*u*v^2*w + 14*p^2*q^2*u*v^2*w + 10*p*q^3*u*v^2*w + q^4*u*v^2*w + 4*p^3*r*u*v^2*w + 28*p^2*q*r*u*v^2*w + 40*p*q^2*r*u*v^2*w + 10*q^3*r*u*v^2*w + 10*p^2*r^2*u*v^2*w + 28*p*q*r^2*u*v^2*w + 14*q^2*r^2*u*v^2*w + 4*p*r^3*u*v^2*w + 5*q*r^3*u*v^2*w + 2*p^3*q*v^3*w + 5*p^2*q^2*v^3*w + 3*p*q^3*v^3*w + 2*p^3*r*v^3*w + 10*p^2*q*r*v^3*w + 10*p*q^2*r*v^3*w + 5*p^2*r^2*v^3*w + 8*p*q*r^2*v^3*w + p*r^3*v^3*w + 4*p^2*q^2*u^2*w^2 + 5*p*q^3*u^2*w^2 + q^4*u^2*w^2 + 10*p*q^2*r*u^2*w^2 + 5*q^3*r*u^2*w^2 + 4*q^2*r^2*u^2*w^2 + 2*p^3*q*u*v*w^2 + 11*p^2*q^2*u*v*w^2 + 10*p*q^3*u*v*w^2 + q^4*u*v*w^2 + 10*p^2*q*r*u*v*w^2 + 28*p*q^2*r*u*v*w^2 + 8*q^3*r*u*v*w^2 + 12*p*q*r^2*u*v*w^2 + 10*q^2*r^2*u*v*w^2 + 3*q*r^3*u*v*w^2 + 2*p^3*q*v^2*w^2 + 7*p^2*q^2*v^2*w^2 + 5*p*q^3*v^2*w^2 + 2*p^3*r*v^2*w^2 + 11*p^2*q*r*v^2*w^2 + 14*p*q^2*r*v^2*w^2 + 4*p^2*r^2*v^2*w^2 + 10*p*q*r^2*v^2*w^2 + p*r^3*v^2*w^2 + 2*p^2*q^2*u*w^3 + 2*p*q^3*u*w^3 + 4*p*q^2*r*u*w^3 + q^3*r*u*w^3 + q^2*r^2*u*w^3 + 2*p^2*q^2*v*w^3 + 2*p*q^3*v*w^3 + 2*p^2*q*r*v*w^3 + 5*p*q^2*r*v*w^3 + 3*p*q*r^2*v*w^3)*(p^2*r^2*u^3*v + 4*p*q*r^2*u^3*v + 2*q^2*r^2*u^3*v + p*r^3*u^3*v + 2*q*r^3*u^3*v + 4*p^2*r^2*u^2*v^2 + 10*p*q*r^2*u^2*v^2 + 4*q^2*r^2*u^2*v^2 + 5*p*r^3*u^2*v^2 + 5*q*r^3*u^2*v^2 + r^4*u^2*v^2 + 2*p^2*r^2*u*v^3 + 4*p*q*r^2*u*v^3 + q^2*r^2*u*v^3 + 2*p*r^3*u*v^3 + q*r^3*u*v^3 + 3*p^2*q*r*u^3*w + 2*p*q^2*r*u^3*w + 5*p*q*r^2*u^3*w + 2*q^2*r^2*u^3*w + 2*q*r^3*u^3*w + 3*p^3*r*u^2*v*w + 12*p^2*q*r*u^2*v*w + 10*p*q^2*r*u^2*v*w + 2*q^3*r*u^2*v*w + 10*p^2*r^2*u^2*v*w + 28*p*q*r^2*u^2*v*w + 11*q^2*r^2*u^2*v*w + 8*p*r^3*u^2*v*w + 10*q*r^3*u^2*v*w + r^4*u^2*v*w + 2*p^3*r*u*v^2*w + 10*p^2*q*r*u*v^2*w + 12*p*q^2*r*u*v^2*w + 3*q^3*r*u*v^2*w + 11*p^2*r^2*u*v^2*w + 28*p*q*r^2*u*v^2*w + 10*q^2*r^2*u*v^2*w + 10*p*r^3*u*v^2*w + 8*q*r^3*u*v^2*w + r^4*u*v^2*w + 2*p^2*q*r*v^3*w + 3*p*q^2*r*v^3*w + 2*p^2*r^2*v^3*w + 5*p*q*r^2*v^3*w + 2*p*r^3*v^3*w + p^3*q*u^2*w^2 + 4*p^2*q^2*u^2*w^2 + 2*p*q^3*u^2*w^2 + 10*p^2*q*r*u^2*w^2 + 11*p*q^2*r*u^2*w^2 + 2*q^3*r*u^2*w^2 + 14*p*q*r^2*u^2*w^2 + 7*q^2*r^2*u^2*w^2 + 5*q*r^3*u^2*w^2 + 4*p^3*q*u*v*w^2 + 10*p^2*q^2*u*v*w^2 + 4*p*q^3*u*v*w^2 + 5*p^3*r*u*v*w^2 + 28*p^2*q*r*u*v*w^2 + 28*p*q^2*r*u*v*w^2 + 5*q^3*r*u*v*w^2 + 14*p^2*r^2*u*v*w^2 + 40*p*q*r^2*u*v*w^2 + 14*q^2*r^2*u*v*w^2 + 10*p*r^3*u*v*w^2 + 10*q*r^3*u*v*w^2 + r^4*u*v*w^2 + 2*p^3*q*v^2*w^2 + 4*p^2*q^2*v^2*w^2 + p*q^3*v^2*w^2 + 2*p^3*r*v^2*w^2 + 11*p^2*q*r*v^2*w^2 + 10*p*q^2*r*v^2*w^2 + 7*p^2*r^2*v^2*w^2 + 14*p*q*r^2*v^2*w^2 + 5*p*r^3*v^2*w^2 + p^3*q*u*w^3 + 5*p^2*q^2*u*w^3 + 2*p*q^3*u*w^3 + 8*p^2*q*r*u*w^3 + 10*p*q^2*r*u*w^3 + 2*q^3*r*u*w^3 + 10*p*q*r^2*u*w^3 + 5*q^2*r^2*u*w^3 + 3*q*r^3*u*w^3 + 2*p^3*q*v*w^3 + 5*p^2*q^2*v*w^3 + p*q^3*v*w^3 + 2*p^3*r*v*w^3 + 10*p^2*q*r*v*w^3 + 8*p*q^2*r*v*w^3 + 5*p^2*r^2*v*w^3 + 10*p*q*r^2*v*w^3 + 3*p*r^3*v*w^3 + p^2*q^2*w^4 + p^2*q*r*w^4 + p*q^2*r*w^4 + p*q*r^2*w^4) : :

Let V(P,U) denote the center, and T(P,U) the perspector, of the Vu parallels conic of P and U. The appearance of (i,j,k) in the following list means that V(X(i),X(j)) = X(k): (1,2,40197), (1,6,40199), (2,3,40201), (2,4,40203), (2,6,40305), (2,6,40205), (3,4,14767), (3,5,6709), (3,6,40209)

The appearance of (i,j,k) in the following list means that T(X(i),X(j)) = X(k): (1,2,40198), (1,6,40200), (2,3,40202), (2,4,40204), (2,6,40206), (3,4,40207), (3,5,40208), (3,6,40210)

X(40197) = CENTER OF VU PARALLELS CONIC OF X(1) AND (2)

Barycentrics    (2*a + b)*(2*a + c)*(5*a^3*b^2 + 23*a^2*b^3 + 6*a*b^4 + 14*a^3*b*c + 67*a^2*b^2*c + 79*a*b^3*c + 10*b^4*c + 5*a^3*c^2 + 67*a^2*b*c^2 + 154*a*b^2*c^2 + 50*b^3*c^2 + 23*a^2*c^3 + 79*a*b*c^3 + 50*b^2*c^3 + 6*a*c^4 + 10*b*c^4) : :

X(40198) = PERSPECTOR OF VU PARALLELS CONIC OF X(1) AND X(2)

Barycentrics    (2*a + b)*(2*a + c)*(14*a^3*b + 56*a^2*b^2 + 46*a*b^3 + 4*b^4 + 10*a^3*c + 95*a^2*b*c + 173*a*b^2*c + 46*b^3*c + 29*a^2*c^2 + 95*a*b*c^2 + 56*b^2*c^2 + 10*a*c^3 + 14*b*c^3)*(10*a^3*b + 29*a^2*b^2 + 10*a*b^3 + 14*a^3*c + 95*a^2*b*c + 95*a*b^2*c + 14*b^3*c + 56*a^2*c^2 + 173*a*b*c^2 + 56*b^2*c^2 + 46*a*c^3 + 46*b*c^3 + 4*c^4) : :

X(40199) = CENTER OF VU PARALLELS CONIC OF X(1) AND X(6)

Barycentrics    a*(a^2 + b^2 + a*c)*(a^2 + a*b + c^2)*(a^8*b^2 - a^7*b^3 - 4*a^6*b^4 - 8*a^5*b^5 - 10*a^4*b^6 - 6*a^3*b^7 - 3*a^2*b^8 - a*b^9 - 2*a^8*b*c - 3*a^7*b^2*c - 7*a^6*b^3*c - 15*a^5*b^4*c - 23*a^4*b^5*c - 20*a^3*b^6*c - 17*a^2*b^7*c - 6*a*b^8*c - 3*b^9*c + a^8*c^2 - 3*a^7*b*c^2 - 10*a^6*b^2*c^2 - 25*a^5*b^3*c^2 - 42*a^4*b^4*c^2 - 47*a^3*b^5*c^2 - 42*a^2*b^6*c^2 - 21*a*b^7*c^2 - 7*b^8*c^2 - a^7*c^3 - 7*a^6*b*c^3 - 25*a^5*b^2*c^3 - 50*a^4*b^3*c^3 - 63*a^3*b^4*c^3 - 67*a^2*b^5*c^3 - 35*a*b^6*c^3 - 16*b^7*c^3 - 4*a^6*c^4 - 15*a^5*b*c^4 - 42*a^4*b^2*c^4 - 63*a^3*b^3*c^4 - 78*a^2*b^4*c^4 - 49*a*b^5*c^4 - 25*b^6*c^4 - 8*a^5*c^5 - 23*a^4*b*c^5 - 47*a^3*b^2*c^5 - 67*a^2*b^3*c^5 - 49*a*b^4*c^5 - 26*b^5*c^5 - 10*a^4*c^6 - 20*a^3*b*c^6 - 42*a^2*b^2*c^6 - 35*a*b^3*c^6 - 25*b^4*c^6 - 6*a^3*c^7 - 17*a^2*b*c^7 - 21*a*b^2*c^7 - 16*b^3*c^7 - 3*a^2*c^8 - 6*a*b*c^8 - 7*b^2*c^8 - a*c^9 - 3*b*c^9) : :

X(40200) = PERSPECTOR OF VU PARALLELS CONIC OF X(1) AND X(6)

Barycentrics    a*(a^2 + b^2 + a*c)*(a^2 + a*b + c^2)*(3*a^7*b + 6*a^6*b^2 + 12*a^5*b^3 + 15*a^4*b^4 + 13*a^3*b^5 + 10*a^2*b^6 + 4*a*b^7 + b^8 + a^7*c + 6*a^6*b*c + 16*a^5*b^2*c + 24*a^4*b^3*c + 27*a^3*b^4*c + 22*a^2*b^5*c + 12*a*b^6*c + 4*b^7*c + 4*a^6*c^2 + 14*a^5*b*c^2 + 34*a^4*b^2*c^2 + 40*a^3*b^3*c^2 + 44*a^2*b^4*c^2 + 22*a*b^5*c^2 + 10*b^6*c^2 + 6*a^5*c^3 + 17*a^4*b*c^3 + 33*a^3*b^2*c^3 + 40*a^2*b^3*c^3 + 27*a*b^4*c^3 + 13*b^5*c^3 + 10*a^4*c^4 + 17*a^3*b*c^4 + 34*a^2*b^2*c^4 + 24*a*b^3*c^4 + 15*b^4*c^4 + 6*a^3*c^5 + 14*a^2*b*c^5 + 16*a*b^2*c^5 + 12*b^3*c^5 + 4*a^2*c^6 + 6*a*b*c^6 + 6*b^2*c^6 + a*c^7 + 3*b*c^7)*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 10*a^4*b^4 + 6*a^3*b^5 + 4*a^2*b^6 + a*b^7 + 3*a^7*c + 6*a^6*b*c + 14*a^5*b^2*c + 17*a^4*b^3*c + 17*a^3*b^4*c + 14*a^2*b^5*c + 6*a*b^6*c + 3*b^7*c + 6*a^6*c^2 + 16*a^5*b*c^2 + 34*a^4*b^2*c^2 + 33*a^3*b^3*c^2 + 34*a^2*b^4*c^2 + 16*a*b^5*c^2 + 6*b^6*c^2 + 12*a^5*c^3 + 24*a^4*b*c^3 + 40*a^3*b^2*c^3 + 40*a^2*b^3*c^3 + 24*a*b^4*c^3 + 12*b^5*c^3 + 15*a^4*c^4 + 27*a^3*b*c^4 + 44*a^2*b^2*c^4 + 27*a*b^3*c^4 + 15*b^4*c^4 + 13*a^3*c^5 + 22*a^2*b*c^5 + 22*a*b^2*c^5 + 13*b^3*c^5 + 10*a^2*c^6 + 12*a*b*c^6 + 10*b^2*c^6 + 4*a*c^7 + 4*b*c^7 + c^8) : :

X(40201) = CENTER OF VU PARALLELS CONIC OF X(2) AND X(3)

Barycentrics    (2*a^4 - 3*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2)*(-2*a^4 + 2*a^2*b^2 + 3*a^2*c^2 + b^2*c^2 - c^4)*(-5*a^16*b^4 + 48*a^14*b^6 - 171*a^12*b^8 + 310*a^10*b^10 - 315*a^8*b^12 + 180*a^6*b^14 - 53*a^4*b^16 + 6*a^2*b^18 - 14*a^16*b^2*c^2 + 114*a^14*b^4*c^2 - 305*a^12*b^6*c^2 + 273*a^10*b^8*c^2 + 178*a^8*b^10*c^2 - 548*a^6*b^12*c^2 + 419*a^4*b^14*c^2 - 127*a^2*b^16*c^2 + 10*b^18*c^2 - 5*a^16*c^4 + 114*a^14*b^2*c^4 - 380*a^12*b^4*c^4 + 257*a^10*b^6*c^4 + 171*a^8*b^8*c^4 + 240*a^6*b^10*c^4 - 870*a^4*b^12*c^4 + 573*a^2*b^14*c^4 - 100*b^16*c^4 + 48*a^14*c^6 - 305*a^12*b^2*c^6 + 257*a^10*b^4*c^6 - 68*a^8*b^6*c^6 + 128*a^6*b^8*c^6 + 557*a^4*b^10*c^6 - 1017*a^2*b^12*c^6 + 400*b^14*c^6 - 171*a^12*c^8 + 273*a^10*b^2*c^8 + 171*a^8*b^4*c^8 + 128*a^6*b^6*c^8 - 106*a^4*b^8*c^8 + 565*a^2*b^10*c^8 - 860*b^12*c^8 + 310*a^10*c^10 + 178*a^8*b^2*c^10 + 240*a^6*b^4*c^10 + 557*a^4*b^6*c^10 + 565*a^2*b^8*c^10 + 1100*b^10*c^10 - 315*a^8*c^12 - 548*a^6*b^2*c^12 - 870*a^4*b^4*c^12 - 1017*a^2*b^6*c^12 - 860*b^8*c^12 + 180*a^6*c^14 + 419*a^4*b^2*c^14 + 573*a^2*b^4*c^14 + 400*b^6*c^14 - 53*a^4*c^16 - 127*a^2*b^2*c^16 - 100*b^4*c^16 + 6*a^2*c^18 + 10*b^2*c^18) : :

X(40202) = PERSPECTOR OF VU PARALLELS CONIC OF X(2) AND X(3)

Barycentrics    (2*a^4-(3*b^2+2*c^2)*a^2+(b^2-c^2)*b^2)*(2*a^4-(2*b^2+3*c^2)*a^2-(b^2-c^2)*c^2)*(2*(7*b^2+5*c^2)*a^14-(112*b^4+143*b^2*c^2+69*c^4)*a^12+3*(118*b^6+62*c^6+(149*b^2+115*c^2)*b^2*c^2)*a^10-2*(290*b^8+127*c^8+(182*b^4+133*b^2*c^2+108*c^4)*b^2*c^2)*a^8+2*(b^2-c^2)*(265*b^8-93*c^8+(79*b^4+84*b^2*c^2+15*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(264*b^8+69*c^8-(237*b^4+217*b^2*c^2+207*c^4)*b^2*c^2)*a^4+(62*b^8-10*c^8-(219*b^4+78*b^2*c^2-113*c^4)*b^2*c^2)*(b^2-c^2)^3*a^2-2*(b^2-c^2)^5*(2*b^4-21*b^2*c^2+7*c^4)*b^2)*(2*(5*b^2+7*c^2)*a^14-(69*b^4+143*b^2*c^2+112*c^4)*a^12+3*(62*b^6+118*c^6+(115*b^2+149*c^2)*b^2*c^2)*a^10-2*(127*b^8+290*c^8+(108*b^4+133*b^2*c^2+182*c^4)*b^2*c^2)*a^8+2*(b^2-c^2)*(93*b^8-265*c^8-(15*b^4+84*b^2*c^2+79*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(69*b^8+264*c^8-(207*b^4+217*b^2*c^2+237*c^4)*b^2*c^2)*a^4+(10*b^8-62*c^8-(113*b^4-78*b^2*c^2-219*c^4)*b^2*c^2)*(b^2-c^2)^3*a^2+2*(b^2-c^2)^5*(7*b^4-21*b^2*c^2+2*c^4)*c^2) : :

X(40203) = CENTER OF VU PARALLELS CONIC OF X(2) AND X(4)

Barycentrics    (a^2 + 3*b^2 - c^2)*(a^2 - b^2 + 3*c^2)* (12*a^12 - 27*a^10*b^2 - 27*a^8*b^4 + 74*a^6*b^6 - 18*a^4*b^8 - 15*a^2*b^10 + b^12 - 27*a^10*c^2 + 18*a^8*b^2*c^2 + 46*a^6*b^4*c^2 - 44*a^4*b^6*c^2 + 5*a^2*b^8*c^2 + 2*b^10*c^2 - 27*a^8*c^4 + 46*a^6*b^2*c^4 + 124*a^4*b^4*c^4 + 10*a^2*b^6*c^4 - 17*b^8*c^4 + 74*a^6*c^6 - 44*a^4*b^2*c^6 + 10*a^2*b^4*c^6 + 28*b^6*c^6 - 18*a^4*c^8 + 5*a^2*b^2*c^8 - 17*b^4*c^8 - 15*a^2*c^10 + 2*b^2*c^10 + c^12) : :

X(40204) = PERSPECTOR OF VU PARALLELS CONIC OF X(2) AND X(4)

Barycentrics    (a^2 + 3*b^2 - c^2)*(a^2 - b^2 + 3*c^2)*(12*a^12 - 29*a^10*b^2 - 28*a^8*b^4 + 90*a^6*b^6 - 28*a^4*b^8 - 29*a^2*b^10 + 12*b^12 - 25*a^10*c^2 + 5*a^8*b^2*c^2 + 20*a^6*b^4*c^2 + 20*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 25*b^10*c^2 - 13*a^8*c^4 + 68*a^6*b^2*c^4 + 86*a^4*b^4*c^4 + 68*a^2*b^6*c^4 - 13*b^8*c^4 + 62*a^6*c^6 - 40*a^4*b^2*c^6 - 40*a^2*b^4*c^6 + 62*b^6*c^6 - 38*a^4*c^8 + a^2*b^2*c^8 - 38*b^4*c^8 - 5*a^2*c^10 - 5*b^2*c^10 + 7*c^12)*(12*a^12 - 25*a^10*b^2 - 13*a^8*b^4 + 62*a^6*b^6 - 38*a^4*b^8 - 5*a^2*b^10 + 7*b^12 - 29*a^10*c^2 + 5*a^8*b^2*c^2 + 68*a^6*b^4*c^2 - 40*a^4*b^6*c^2 + a^2*b^8*c^2 - 5*b^10*c^2 - 28*a^8*c^4 + 20*a^6*b^2*c^4 + 86*a^4*b^4*c^4 - 40*a^2*b^6*c^4 - 38*b^8*c^4 + 90*a^6*c^6 + 20*a^4*b^2*c^6 + 68*a^2*b^4*c^6 + 62*b^6*c^6 - 28*a^4*c^8 + 5*a^2*b^2*c^8 - 13*b^4*c^8 - 29*a^2*c^10 - 25*b^2*c^10 + 12*c^12) : :

X(40205) = CENTER OF VU PARALLELS CONIC OF X(2) AND X(6)

Barycentrics    (2*a^2 + b^2)*(2*a^2 + c^2)*(5*a^6*b^4 + 23*a^4*b^6 + 6*a^2*b^8 + 14*a^6*b^2*c^2 + 67*a^4*b^4*c^2 + 79*a^2*b^6*c^2 + 10*b^8*c^2 + 5*a^6*c^4 + 67*a^4*b^2*c^4 + 154*a^2*b^4*c^4 + 50*b^6*c^4 + 23*a^4*c^6 + 79*a^2*b^2*c^6 + 50*b^4*c^6 + 6*a^2*c^8 + 10*b^2*c^8) : :

X(40206) = PERSPECTOR OF VU PARALLELS CONIC OF X(2) AND X(6)

Barycentrics    (2*a^2 + b^2)*(2*a^2 + c^2)*(14*a^6*b^2 + 56*a^4*b^4 + 46*a^2*b^6 + 4*b^8 + 10*a^6*c^2 + 95*a^4*b^2*c^2 + 173*a^2*b^4*c^2 + 46*b^6*c^2 + 29*a^4*c^4 + 95*a^2*b^2*c^4 + 56*b^4*c^4 + 10*a^2*c^6 + 14*b^2*c^6)*(10*a^6*b^2 + 29*a^4*b^4 + 10*a^2*b^6 + 14*a^6*c^2 + 95*a^4*b^2*c^2 + 95*a^2*b^4*c^2 + 14*b^6*c^2 + 56*a^4*c^4 + 173*a^2*b^2*c^4 + 56*b^4*c^4 + 46*a^2*c^6 + 46*b^2*c^6 + 4*c^8) : :

X(40207) = PERSPECTOR OF VU PARALLELS CONIC OF X(3) AND X(4)

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

The center of the Vu parallels conic of X(3) and X(5) is X(14767).

X(40208) = PERSPECTOR OF VU PARALLELS CONIC OF X(3) AND X(5)

Barycentrics    (4*a^8 - 17*a^6*b^2 + 26*a^4*b^4 - 17*a^2*b^6 + 4*b^8 - 15*a^6*c^2 + 15*a^4*b^2*c^2 + 15*a^2*b^4*c^2 - 15*b^6*c^2 + 19*a^4*c^4 + 11*a^2*b^2*c^4 + 19*b^4*c^4 - 9*a^2*c^6 - 9*b^2*c^6 + c^8)*(4*a^8 - 15*a^6*b^2 + 19*a^4*b^4 - 9*a^2*b^6 + b^8 - 17*a^6*c^2 + 15*a^4*b^2*c^2 + 11*a^2*b^4*c^2 - 9*b^6*c^2 + 26*a^4*c^4 + 15*a^2*b^2*c^4 + 19*b^4*c^4 - 17*a^2*c^6 - 15*b^2*c^6 + 4*c^8) : :

The center of the Vu parallels conic of X(3) and X(5) is X(6709).

X(40209) = CENTER OF VU PARALLELS CONIC OF X(3) AND X(6)

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

X(40210) = PERSPECTOR OF VU PARALLELS CONIC OF X(3) AND X(6)

Barycentrics    a^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)*(3*a^14*b^2 - 9*a^12*b^4 + 5*a^10*b^6 + 9*a^8*b^8 - 11*a^6*b^10 + a^4*b^12 + 3*a^2*b^14 - b^16 + 3*a^14*c^2 - 9*a^12*b^2*c^2 - 11*a^10*b^4*c^2 + 33*a^8*b^6*c^2 + a^6*b^8*c^2 - 27*a^4*b^10*c^2 + 7*a^2*b^12*c^2 + 3*b^14*c^2 - 6*a^12*c^4 - 23*a^10*b^2*c^4 + 29*a^8*b^4*c^4 + 34*a^6*b^6*c^4 - 8*a^4*b^8*c^4 - 27*a^2*b^10*c^4 + b^12*c^4 - 7*a^10*c^6 + 29*a^8*b^2*c^6 + 50*a^6*b^4*c^6 + 34*a^4*b^6*c^6 + a^2*b^8*c^6 - 11*b^10*c^6 + 20*a^8*c^8 + 29*a^6*b^2*c^8 + 29*a^4*b^4*c^8 + 33*a^2*b^6*c^8 + 9*b^8*c^8 - 7*a^6*c^10 - 23*a^4*b^2*c^10 - 11*a^2*b^4*c^10 + 5*b^6*c^10 - 6*a^4*c^12 - 9*a^2*b^2*c^12 - 9*b^4*c^12 + 3*a^2*c^14 + 3*b^2*c^14)*(3*a^14*b^2 - 6*a^12*b^4 - 7*a^10*b^6 + 20*a^8*b^8 - 7*a^6*b^10 - 6*a^4*b^12 + 3*a^2*b^14 + 3*a^14*c^2 - 9*a^12*b^2*c^2 - 23*a^10*b^4*c^2 + 29*a^8*b^6*c^2 + 29*a^6*b^8*c^2 - 23*a^4*b^10*c^2 - 9*a^2*b^12*c^2 + 3*b^14*c^2 - 9*a^12*c^4 - 11*a^10*b^2*c^4 + 29*a^8*b^4*c^4 + 50*a^6*b^6*c^4 + 29*a^4*b^8*c^4 - 11*a^2*b^10*c^4 - 9*b^12*c^4 + 5*a^10*c^6 + 33*a^8*b^2*c^6 + 34*a^6*b^4*c^6 + 34*a^4*b^6*c^6 + 33*a^2*b^8*c^6 + 5*b^10*c^6 + 9*a^8*c^8 + a^6*b^2*c^8 - 8*a^4*b^4*c^8 + a^2*b^6*c^8 + 9*b^8*c^8 - 11*a^6*c^10 - 27*a^4*b^2*c^10 - 27*a^2*b^4*c^10 - 11*b^6*c^10 + a^4*c^12 + 7*a^2*b^2*c^12 + b^4*c^12 + 3*a^2*c^14 + 3*b^2*c^14 - c^16) : :

X(40211) = CENTER OF VU PARALLELS CONIC OF X(4) AND X(6)

Barycentrics    (3*a^2 + b^2 - c^2)*(3*a^2 - b^2 + c^2)*(3*a^12*b^2 - a^10*b^4 - 18*a^8*b^6 + 30*a^6*b^8 - 17*a^4*b^10 + 3*a^2*b^12 + 3*a^12*c^2 - 4*a^10*b^2*c^2 - 36*a^8*b^4*c^2 + 10*a^6*b^6*c^2 + 77*a^4*b^8*c^2 - 38*a^2*b^10*c^2 - 12*b^12*c^2 - a^10*c^4 - 36*a^8*b^2*c^4 - 32*a^6*b^4*c^4 + 132*a^4*b^6*c^4 - 19*a^2*b^8*c^4 - 44*b^10*c^4 - 18*a^8*c^6 + 10*a^6*b^2*c^6 + 132*a^4*b^4*c^6 + 108*a^2*b^6*c^6 + 56*b^8*c^6 + 30*a^6*c^8 + 77*a^4*b^2*c^8 - 19*a^2*b^4*c^8 + 56*b^6*c^8 - 17*a^4*c^10 - 38*a^2*b^2*c^10 - 44*b^4*c^10 + 3*a^2*c^12 - 12*b^2*c^12) : :

leftri

U-Hodpieces: X(40212)-X(40218)

rightri

This preamble is based on notes received from Vu Thanh Tung, November 1, 2020.

The definition of hodpiece in the preamble just before X(40137) generalizes as follows. Let P be a point, not on a sideline of ABC, and let DEF be the cevian triangle of P. Let U = u:v:w be a point. The P-reciprocal conjugate of U (defined as u/p : v/q : w/r in the Glossary of ETC), of the line EF is a conic. Let A' be the center of the conic, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in the point

u / (p*(-u/p + v/q + w/r)) : v / (q*(u/p - v/q + w/r)) : w / (r*(u/p + v/q - w/r)),

here named the U-hodpiece of P, so that the hodpiece of P is the X(6)-hodpiece of P.

X(40212) = X(8)-HODPIECE OF X(40)

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

X(40212) lies on these lines: {1, 9786}, {2, 7}, {40, 196}, {84, 5928}, {108, 7070}, {198, 223}, {278, 2270}, {610, 34032}, {934, 34499}, {1020, 1763}, {1419, 7125}, {1422, 34371}, {1435, 2183}, {1490, 3182}, {1706, 6358}, {1766, 1767}, {3074, 3361}, {3342, 7078}, {5909, 38290}, {7183, 33066}, {9121, 15498}, {19366, 37993}, {22464, 39592}

X(40212) = X(i)-Ceva conjugate of X(j) for these (i,j): {329, 223}, {7013, 40}
X(40212) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1256}, {84, 282}, {189, 2192}, {271, 7129}, {280, 1436}, {285, 1903}, {309, 7118}, {1433, 7003}, {1440, 7367}, {2208, 34404}, {7054, 7157}
X(40212) = barycentric product X(i)*X(j) for these {i,j}: {7, 1103}, {40, 347}, {221, 322}, {223, 329}, {227, 8822}, {342, 7078}, {2324, 14256}, {3318, 7045}, {7013, 7952}
X(40212) = barycentric quotient X(i)/X(j) for these {i,j}: {40, 280}, {56, 1256}, {198, 282}, {221, 84}, {223, 189}, {227, 39130}, {329, 34404}, {347, 309}, {1103, 8}, {1254, 7157}, {2187, 2192}, {2199, 1436}, {2331, 7003}, {2360, 285}, {3195, 7008}, {3209, 7129}, {3318, 24026}, {6611, 1422}, {7078, 271}, {7114, 1433}, {7952, 7020}

X(40213) = X(1)-HODPIECE OF X(11)

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

X(40213) lies on these lines: {2, 1577}, {27, 1019}, {333, 1021}, {661, 2051}, {693, 5249}, {3687, 4391}, {3703, 4086}, {3737, 17188}, {4467, 20879}, {6545, 23100}

X(40213) = X(4560)-Ceva conjugate of X(4858)
X(40213) = crosspoint of X(4560) and X(26856)
X(40213) = X(i)-isoconjugate of X(j) for these (i,j): {42, 4619}, {59, 4559}, {1018, 24027}, {1020, 1110}, {1262, 4557}, {1402, 31615}, {2149, 4551}, {3952, 23979}, {4566, 23990}, {7115, 23067}
X(40213) = barycentric product X(i)*X(j) for these {i,j}: {11, 18155}, {99, 1090}, {314, 21132}, {333, 40166}, {1014, 23104}, {1019, 23978}, {1021, 23989}, {1111, 7253}, {1146, 7199}, {1577, 26856}, {2287, 23100}, {3239, 16727}, {3737, 34387}, {4391, 17197}, {4397, 17205}, {4560, 4858}, {4625, 5532}, {7192, 24026}, {7257, 7336}, {18191, 35519}
X(40213) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 4551}, {81, 4619}, {333, 31615}, {764, 1042}, {1019, 1262}, {1021, 1252}, {1086, 1020}, {1090, 523}, {1111, 4566}, {1146, 1018}, {2170, 4559}, {2310, 4557}, {3733, 24027}, {3737, 59}, {4081, 4069}, {4560, 4564}, {4858, 4552}, {5532, 4041}, {6545, 1427}, {7004, 23067}, {7192, 7045}, {7199, 1275}, {7203, 7339}, {7252, 2149}, {7253, 765}, {7336, 4017}, {8042, 1407}, {16726, 1461}, {16727, 658}, {16732, 4605}, {17197, 651}, {17205, 934}, {17219, 6516}, {17925, 7128}, {18155, 4998}, {18191, 109}, {21044, 21859}, {21132, 65}, {21789, 1110}, {23100, 1446}, {23104, 3701}, {23105, 1091}, {23615, 210}, {23978, 4033}, {24026, 3952}, {26856, 662}, {34591, 4574}, {40166, 226}

X(40214) = X(100)-HODPIECE OF X(35)

Barycentrics    a^2*(a + b)*(a + c)*(a^2 - b^2 - b*c - c^2) : :
Trilinears    cot A' : :, where A'B'C' is the incentral triangle

X(40214) lies on the cubic K577 and these lines: {1, 229}, {2, 662}, {3, 60}, {6, 593}, {21, 90}, {31, 1326}, {35, 17104}, {36, 9275}, {41, 1931}, {46, 37294}, {48, 28606}, {55, 110}, {57, 77}, {58, 5313}, {63, 37783}, {65, 37405}, {86, 17173}, {99, 32933}, {100, 7095}, {101, 33761}, {162, 37441}, {163, 4262}, {186, 500}, {222, 4565}, {226, 18653}, {249, 9273}, {261, 5278}, {270, 7501}, {321, 27958}, {323, 17454}, {386, 849}, {394, 7054}, {445, 14165}, {572, 21363}, {584, 757}, {759, 37525}, {842, 36069}, {960, 37032}, {991, 4575}, {1029, 8818}, {1098, 4189}, {1150, 7058}, {1214, 18609}, {1444, 4280}, {1474, 14014}, {1479, 3615}, {1812, 27174}, {1836, 5196}, {1837, 37158}, {1993, 36744}, {1994, 4271}, {2003, 35192}, {2167, 4560}, {2174, 3219}, {2193, 18605}, {2194, 4184}, {2206, 3736}, {2287, 6514}, {2326, 24553}, {2605, 9213}, {2646, 11101}, {3187, 19623}, {3240, 6043}, {3285, 40153}, {3295, 33669}, {3450, 4267}, {3578, 7799}, {4210, 5135}, {4251, 30581}, {4258, 38811}, {4287, 4383}, {4511, 17512}, {4558, 6511}, {4591, 40215}, {4610, 8033}, {4637, 9533}, {4641, 16702}, {5009, 17187}, {5010, 5127}, {5012, 5132}, {5794, 37152}, {6061, 20835}, {6507, 24635}, {7096, 40145}, {7113, 17011}, {10572, 13746}, {11375, 37369}, {11507, 23059}, {11680, 19642}, {14355, 22094}, {14570, 17479}, {14829, 30606}, {16579, 34544}, {17139, 26830}, {17147, 18042}, {17168, 17197}, {18048, 40013}, {18165, 33325}, {22130, 32661}, {24041, 24504}, {27644, 27661}, {31393, 33903}, {32950, 35916}, {37571, 37816}

X(40214) = isogonal conjugate of X(8818)
X(40214) = isogonal conjugate of the isotomic conjugate of X(34016)
X(40214) = X(662)-Ceva conjugate of X(14838)
X(40214) = X(i)-cross conjugate of X(j) for these (i,j): {500, 1442}, {2174, 17104}, {9404, 110}, {17454, 35}, {20982, 2605}, {22094, 4467}, {35192, 35193}
X(40214) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8818}, {6, 6757}, {10, 2160}, {37, 79}, {42, 30690}, {65, 7110}, {94, 3724}, {213, 20565}, {226, 7073}, {321, 6186}, {476, 2610}, {512, 15455}, {661, 6742}, {758, 1989}, {1789, 8736}, {1826, 7100}, {1983, 10412}, {2166, 2245}, {2171, 3615}, {3700, 26700}, {4024, 13486}, {4041, 38340}, {4092, 35049}, {4242, 14582}, {4585, 15475}, {6370, 32678}, {8606, 40149}, {11060, 35550}, {21873, 30602}
X(40214) = cevapoint of X(i) and X(j) for these (i,j): {35, 2174}, {284, 501}, {2605, 20982}, {3024, 9404}, {17104, 35192}
X(40214) = crosspoint of X(249) and X(662)
X(40214) = crosssum of X(i) and X(j) for these (i,j): {37, 21863}, {79, 14844}, {115, 661}, {4988, 21044}
X(40214) = trilinear pole of line {526, 2605}
X(40214) = crossdifference of every pair of points on line {4041, 4838}
X(40214) = barycentric product X(i)*X(j) for these {i,j}: {6, 34016}, {7, 35193}, {21, 1442}, {35, 86}, {58, 319}, {75, 17104}, {77, 11107}, {81, 3219}, {85, 35192}, {99, 2605}, {101, 16755}, {110, 4467}, {163, 18160}, {249, 8287}, {261, 2594}, {274, 2174}, {283, 7282}, {284, 17095}, {314, 1399}, {323, 24624}, {333, 2003}, {593, 3969}, {662, 14838}, {757, 3678}, {811, 23226}, {1014, 4420}, {1101, 17886}, {1154, 39277}, {1171, 3578}, {1255, 17190}, {1333, 33939}, {1414, 35057}, {1444, 6198}, {2185, 16577}, {2611, 24041}, {3268, 36069}, {3615, 7279}, {4556, 7265}, {4567, 7202}, {4573, 9404}, {4590, 20982}, {6149, 14616}, {7799, 34079}, {17454, 32014}, {18020, 22094}, {32679, 37140}
X(40214) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6757}, {6, 8818}, {35, 10}, {50, 2245}, {58, 79}, {60, 3615}, {81, 30690}, {86, 20565}, {110, 6742}, {186, 860}, {284, 7110}, {319, 313}, {323, 3936}, {500, 442}, {526, 6370}, {662, 15455}, {759, 2166}, {1333, 2160}, {1399, 65}, {1437, 7100}, {1442, 1441}, {1511, 6739}, {2003, 226}, {2174, 37}, {2194, 7073}, {2206, 6186}, {2477, 2594}, {2594, 12}, {2605, 523}, {2611, 1109}, {2624, 2610}, {3024, 6741}, {3219, 321}, {3578, 1230}, {3647, 4647}, {3678, 1089}, {3969, 28654}, {4420, 3701}, {4467, 850}, {4565, 38340}, {6149, 758}, {7186, 2887}, {7202, 16732}, {7266, 17886}, {8287, 338}, {9273, 39295}, {9404, 3700}, {11107, 318}, {14838, 1577}, {14975, 1824}, {16577, 6358}, {16718, 20886}, {16755, 3261}, {17095, 349}, {17104, 1}, {17190, 4359}, {17454, 1213}, {17886, 23994}, {18160, 20948}, {20982, 115}, {21741, 2171}, {22094, 125}, {22342, 201}, {23226, 656}, {24624, 94}, {32671, 32678}, {33939, 27801}, {34016, 76}, {34079, 1989}, {35057, 4086}, {35192, 9}, {35193, 8}, {35195, 27529}, {36069, 476}, {37140, 32680}
X(40214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {35, 17104, 35193}, {284, 1790, 81}, {501, 15792, 1}, {662, 2185, 2}

X(40215) = X(100)-HODPIECE OF X(36)

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

X(40215) lies on the cubic K577 and these lines: {1, 39148}, {2, 3257}, {6, 2226}, {31, 106}, {36, 16944}, {55, 840}, {57, 88}, {81, 1019}, {89, 679}, {354, 14190}, {593, 4556}, {999, 1318}, {1168, 5902}, {1262, 1407}, {1320, 3873}, {1417, 7248}, {1478, 36590}, {2094, 36887}, {3218, 4585}, {3418, 36058}, {4080, 17483}, {4582, 32933}, {4591, 40214}, {4615, 8033}, {4674, 32913}, {4945, 31164}, {4997, 31053}, {5332, 9456}, {8034, 23345}, {9352, 14193}, {11246, 19636}, {34583, 39154}, {36814, 37604}

X(40215) = X(i)-Ceva conjugate of X(j) for these (i,j): {679, 106}, {3257, 3960}
X(40215) = X(i)-cross conjugate of X(j) for these (i,j): {654, 901}, {7113, 106}, {17455, 36}, {34586, 1443}
X(40215) = X(i)-isoconjugate of X(j) for these (i,j): {2, 40172}, {9, 14584}, {44, 80}, {55, 14628}, {519, 2161}, {655, 4895}, {759, 3943}, {902, 18359}, {1168, 4370}, {1319, 36910}, {1411, 2325}, {1639, 2222}, {1807, 8756}, {1960, 36804}, {2006, 3689}, {2251, 20566}, {3285, 15065}, {3992, 34079}, {4358, 6187}, {4768, 32675}, {16704, 34857}, {21805, 24624}
X(40215) = cevapoint of X(i) and X(j) for these (i,j): {6, 14260}, {36, 17455}, {654, 3025}, {2316, 39148}
X(40215) = crosssum of X(i) and X(j) for these (i,j): {1635, 35092}, {3943, 4370}, {4530, 6544}
X(40215) = trilinear pole of line {36, 39478}
X(40215) = crossdifference of every pair of points on line {4895, 21805}
X(40215) = barycentric product X(i)*X(j) for these {i,j}: {36, 903}, {75, 16944}, {88, 3218}, {106, 320}, {214, 679}, {901, 4453}, {1022, 4585}, {1320, 1443}, {1797, 17923}, {2316, 17078}, {3257, 3960}, {4089, 9268}, {4591, 4707}, {4615, 21828}, {6336, 22128}, {7113, 20568}, {9456, 20924}, {17191, 30575}
X(40215) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 40172}, {36, 519}, {56, 14584}, {57, 14628}, {88, 18359}, {106, 80}, {214, 4738}, {320, 3264}, {654, 1639}, {758, 3992}, {903, 20566}, {1318, 36590}, {1417, 1411}, {1870, 38462}, {1983, 1023}, {2245, 3943}, {2316, 36910}, {2323, 2325}, {2361, 3689}, {3218, 4358}, {3257, 36804}, {3724, 21805}, {3738, 4768}, {3792, 4439}, {3960, 3762}, {4511, 4723}, {4585, 24004}, {4674, 15065}, {4881, 4487}, {4973, 4975}, {7113, 44}, {8648, 4895}, {9456, 2161}, {16944, 1}, {17191, 16729}, {17455, 4370}, {21758, 1635}, {21828, 4120}, {22128, 3977}, {34586, 1145}, {36058, 1807}, {39148, 36909}
X(40215) = {X(999),X(14260)}-harmonic conjugate of X(1318)

X(40216) = X(100)-HODPIECE OF X(2)

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

X(40216) lies on these lines: {38, 16727}, {75, 3873}, {85, 3681}, {92, 15149}, {274, 3112}, {313, 1233}, {321, 1930}, {561, 33933}, {693, 2886}, {756, 1111}, {1086, 8041}, {1441, 4967}, {2350, 4359}, {2481, 5284}, {2550, 13577}, {2995, 25590}, {2997, 10436}, {3925, 23989}, {4651, 20448}, {4972, 39712}, {6063, 33108}, {6358, 20901}, {8049, 20718}, {11680, 32023}, {14549, 17863}, {20632, 24199}, {20892, 30047}, {30473, 30636}, {32092, 39950}

X(40216) = isotomic conjugate of X(1621)
X(40216) = isotomic conjugate of the anticomplement of X(3925)
X(40216) = isotomic conjugate of the complement of X(33110)
X(40216) = isotomic conjugate of the isogonal conjugate of X(13476)
X(40216) = X(39734)-anticomplementary conjugate of X(2890)
X(40216) = X(40004)-Ceva conjugate of X(17758)
X(40216) = X(i)-cross conjugate of X(j) for these (i,j): {594, 76}, {2294, 1446}, {3925, 2}, {15523, 40013}, {21020, 321}, {21026, 39994}, {21924, 2052}, {23989, 693}
X(40216) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4251}, {31, 1621}, {32, 17277}, {59, 38365}, {101, 21007}, {184, 14004}, {560, 17143}, {692, 4040}, {1252, 38346}, {1253, 38859}, {1333, 3294}, {1397, 3996}, {1501, 18152}, {1576, 4151}, {1917, 40088}, {2149, 38347}, {2150, 20616}, {2206, 4651}, {8750, 22160}, {14827, 33765}, {17494, 32739}, {17761, 23990}, {18892, 40094}
X(40216) = cevapoint of X(i) and X(j) for these (i,j): {2, 33110}, {75, 33943}, {523, 1111}, {1086, 2530}, {4858, 6362}
X(40216) = crosspoint of X(75) and X(40005)
X(40216) = trilinear pole of line {918, 1577}
X(40216) = barycentric product X(i)*X(j) for these {i,j}: {10, 40004}, {75, 17758}, {76, 13476}, {313, 39950}, {321, 39734}, {561, 2350}
X(40216) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4251}, {2, 1621}, {10, 3294}, {11, 38347}, {12, 20616}, {75, 17277}, {76, 17143}, {92, 14004}, {244, 38346}, {279, 38859}, {312, 3996}, {313, 4043}, {321, 4651}, {513, 21007}, {514, 4040}, {561, 18152}, {693, 17494}, {905, 22160}, {1088, 33765}, {1111, 17761}, {1502, 40088}, {1577, 4151}, {2170, 38365}, {2350, 31}, {3261, 20954}, {4024, 21727}, {13476, 6}, {16732, 2486}, {17758, 1}, {18895, 40094}, {20888, 29773}, {23807, 27168}, {39734, 81}, {39950, 58}, {40004, 86}
X(40216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 16708, 17140}, {75, 40004, 13476}

X(40217) = X(105)-HODPIECE OF X(1)

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

X(40217) lies on these lines: {2, 38}, {57, 4998}, {63, 813}, {312, 4583}, {321, 693}, {337, 4876}, {518, 27919}, {660, 3681}, {1911, 3938}, {3252, 3930}, {3509, 3570}, {3661, 40098}, {3675, 3912}, {3961, 18787}, {4441, 18034}, {4562, 17294}, {6063, 6358}, {6654, 9451}, {8047, 32863}, {16708, 40017}, {17780, 24628}, {21101, 24318}, {24326, 39712}, {33676, 39959}, {36483, 36800}

X(40217) = anticomplement of X(27942)
X(40217) = isotomic conjugate of X(6654)
X(40217) = isotomic conjugate of the isogonal conjugate of X(3252)
X(40217) = X(335)-Ceva conjugate of X(3912)
X(40217) = X(4437)-cross conjugate of X(3912)
X(40217) = X(i)-isoconjugate of X(j) for these (i,j): {31, 6654}, {105, 1914}, {238, 1438}, {242, 32658}, {294, 1428}, {659, 919}, {673, 2210}, {812, 32666}, {1416, 3684}, {1429, 2195}, {2201, 36057}, {2481, 14599}, {4435, 32735}, {5009, 18785}, {7193, 8751}, {8632, 36086}, {18031, 18892}
X(40217) = cevapoint of X(3930) and X(4712)
X(40217) = crosspoint of X(335) and X(40098)
X(40217) = trilinear pole of line {918, 3932}
X(40217) = crossdifference of every pair of points on line {2210, 8632}
X(40217) = barycentric product X(i)*X(j) for these {i,j}: {75, 22116}, {76, 3252}, {291, 3263}, {334, 518}, {335, 3912}, {337, 1861}, {672, 18895}, {918, 4562}, {1934, 4447}, {2254, 4583}, {3717, 7233}, {3930, 40017}, {3932, 18827}, {4088, 4589}, {4518, 9436}, {4639, 24290}, {17755, 40098}
X(40217) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6654}, {241, 1429}, {291, 105}, {292, 1438}, {295, 36057}, {334, 2481}, {335, 673}, {337, 31637}, {518, 238}, {660, 36086}, {665, 8632}, {672, 1914}, {813, 919}, {876, 1027}, {918, 812}, {1026, 3573}, {1458, 1428}, {1818, 7193}, {1861, 242}, {2196, 32658}, {2223, 2210}, {2254, 659}, {3252, 6}, {3263, 350}, {3286, 5009}, {3675, 27846}, {3693, 3684}, {3717, 3685}, {3912, 239}, {3930, 2238}, {3932, 740}, {4088, 4010}, {4437, 17755}, {4447, 1580}, {4518, 14942}, {4562, 666}, {4712, 8299}, {4876, 294}, {4966, 4974}, {5089, 2201}, {5378, 5377}, {7077, 2195}, {8299, 8300}, {9436, 1447}, {9454, 14599}, {9455, 18892}, {15149, 31905}, {17755, 4366}, {18157, 30940}, {18895, 18031}, {20683, 3747}, {22116, 1}, {24290, 21832}, {25083, 20769}, {27919, 6652}, {30671, 29956}, {30941, 33295}, {34067, 32666}, {36801, 36802}

X(40218) = X(9)-HODPIECE OF X(44)

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

X(40218) lies on the cubic K577 and these lines: {2, 222}, {55, 104}, {57, 514}, {196, 40151}, {200, 36819}, {202, 14359}, {203, 14358}, {345, 1016}, {519, 23703}, {603, 37043}, {996, 38955}, {1397, 2720}, {1997, 13136}, {3086, 28347}, {3476, 10428}, {5435, 37136}, {6630, 37683}, {14266, 14584}, {15635, 17625}, {23615, 39771}, {34523, 36795}, {36037, 36845}

X(40218) = X(34234)-Ceva conjugate of X(3911)
X(40218) = X(i)-cross conjugate of X(j) for these (i,j): {44, 104}, {14425, 1309}, {14584, 7}
X(40218) = X(i)-isoconjugate of X(j) for these (i,j): {9, 14260}, {517, 2316}, {1320, 2183}, {1769, 5548}, {2427, 23838}, {2804, 32665}, {6735, 9456}
X(40218) = cevapoint of X(i) and X(j) for these (i,j): {44, 1317}, {4530, 39771}
X(40218) = crosssum of X(2183) and X(23980)
X(40218) = trilinear pole of line {900, 1319}
X(40218) = barycentric product X(i)*X(j) for these {i,j}: {7, 36944}, {1319, 18816}, {3762, 37136}, {3911, 34234}, {4358, 34051}, {13136, 30725}
X(40218) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 14260}, {104, 1320}, {519, 6735}, {900, 2804}, {909, 2316}, {1317, 1145}, {1319, 517}, {1404, 2183}, {1647, 35015}, {1846, 21664}, {1877, 1785}, {2720, 901}, {3259, 3326}, {3911, 908}, {10428, 1318}, {12832, 119}, {13136, 4582}, {14027, 3259}, {30725, 10015}, {32641, 5548}, {32669, 32665}, {34051, 88}, {34234, 4997}, {36944, 8}, {37136, 3257}, {39771, 23757}

X(40219) = X(2)X(64)∩X(25)X(40190)

Barycentrics    (a^4 + 6*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 + 6*a^2*c^2 - 2*b^2*c^2 + c^4)*(5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 22*a^4*b^6 - 7*a^2*b^8 - b^10 - 5*a^8*c^2 + 60*a^6*b^2*c^2 - 38*a^4*b^4*c^2 - 20*a^2*b^6*c^2 + 3*b^8*c^2 - 14*a^6*c^4 - 38*a^4*b^2*c^4 + 54*a^2*b^4*c^4 - 2*b^6*c^4 + 22*a^4*c^6 - 20*a^2*b^2*c^6 - 2*b^4*c^6 - 7*a^2*c^8 + 3*b^2*c^8 - c^10)::

X(40219) lies on the cubic K169 and these lines: {2, 64}, {25, 40190}, {2139, 40189}

X(40219) = X(69)-Ceva conjugate of X(40190)

X(40220) = X(1)X(40190)∩X(2)X(269)

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

X(40220) lies on the cubic K169 and these lines: {1, 40190}, {2, 269}, {64, 17742}, {3692, 6574}, {7097, 40189}

X(40220) = barycentric product X(1219)*X(12565)
X(40220) = barycentric quotient X(12565)/X(3672)

X(40221) = X(2)X(2139)∩X(25)X(64)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*(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 + 18*a^10*b^2*c^2 - 13*a^8*b^4*c^2 - 36*a^6*b^6*c^2 + 51*a^4*b^8*c^2 - 14*a^2*b^10*c^2 - 3*b^12*c^2 + a^10*c^4 - 13*a^8*b^2*c^4 + 82*a^6*b^4*c^4 - 50*a^4*b^6*c^4 - 35*a^2*b^8*c^4 + 15*b^10*c^4 + 5*a^8*c^6 - 36*a^6*b^2*c^6 - 50*a^4*b^4*c^6 + 92*a^2*b^6*c^6 - 11*b^8*c^6 - 5*a^6*c^8 + 51*a^4*b^2*c^8 - 35*a^2*b^4*c^8 - 11*b^6*c^8 - a^4*c^10 - 14*a^2*b^2*c^10 + 15*b^4*c^10 + 3*a^2*c^12 - 3*b^2*c^12 - c^14)::

X(40221) lies on the cubic K169 and these lines: {2, 2139}, {25, 64}, {269, 2184}, {1073, 13567}, {1301, 1619}, {3343, 14390}, {13575, 40190}, {14457, 37072}

X(40221) = X(69)-Ceva conjugate of X(64)
X(40221) = barycentric product X(17807)*X(34403)
X(40221) = barycentric quotient X(17807)/X(1249)
X(40221) = perspector of pedal triangle of X(20) and anticevian triangle of X(64)
X(40221) = {X(3343),X(17811)}-harmonic conjugate of X(14390)

X(40222) = X(2)X(159)∩X(6)X(39978)

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

X(40222) lies on the cubic K169 and these lines: {2, 159}, {6, 39978}, {69, 40189}, {2138, 40190}

X(40222) = isogonal conjugate of X(40189)
X(40222) = X(25)-cross conjugate of X(40190)
X(40222) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40189}, {23051, 37485}
X(40222) = barycentric product X(3618)*X(40178)
X(40222) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40189}, {3618, 40123}, {30435, 37485}, {40178, 18840}

X(40223) = X(1)X(64)∩X(2)X(6359)

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

X(40223) lies on the cubic K169 and these lines: {1, 64}, {2, 6359}, {25, 269}, {1422, 14524}, {2139, 17742}, {5738, 7177}, {40188, 40190}
X(40223) = X(69)-Ceva conjugate of X(269)
X(40223) = barycentric product X(i)*X(j) for these {i,j}: {279, 16389}, {348, 8899}
X(40223) = barycentric quotient X(i)/X(j) for these {i,j}: {8899, 281}, {16389, 346}

X(40224) = X(2)X(14259)∩X(159)X(40190)

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

X(40224) lies on the cubic K169 and these lines: {2, 14259}, {159, 40190}

X(40225) = X(1)X(40189)∩X(2)X(17742)

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

X(40225) lies on the cubic K169 and these lines: {1, 40189}, {2, 17742}, {159, 269}, {1763, 40190}

X(40226) = X(1)X(40187)∩X(2)X(40194)

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

X(40226) lies on the cubic K169 and these lines: {1, 40187}, {2, 40194}, {6, 200}, {20, 1219}

X(40227) = X(2)X(40190)∩X(64)X(40189)

Barycentrics    a^2*(a^4 + 10*a^2*b^2 + 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 10*a^2*c^2 + 10*b^2*c^2 + 5*c^4)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 - 44*a^4*b^2*c^2 + 36*a^2*b^4*c^2 + 12*b^6*c^2 + 6*a^4*c^4 + 36*a^2*b^2*c^4 - 26*b^4*c^4 - 4*a^2*c^6 + 12*b^2*c^6 + c^8)::

X(40227) lies on the cubic K169 and these lines: {2, 40190}, {64, 40189}

X(40228) = X(6)X(110)∩X(542)X(17854)

Barycentrics    a^2*(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 - 4*a^8*b^2*c^2 + 11*a^6*b^4*c^2 + a^4*b^6*c^2 - 12*a^2*b^8*c^2 + 3*b^10*c^2 - a^8*c^4 + 11*a^6*b^2*c^4 - 32*a^4*b^4*c^4 + 17*a^2*b^6*c^4 + b^8*c^4 - 2*a^6*c^6 + a^4*b^2*c^6 + 17*a^2*b^4*c^6 - 6*b^6*c^6 + 2*a^4*c^8 - 12*a^2*b^2*c^8 + b^4*c^8 + a^2*c^10 + 3*b^2*c^10 - c^12) : :
X(40228) = 5 X[110] - 3 X[15531], 3 X[895] - 4 X[11746]

X(40228) lies on the cubic K1163 and these lines: {6, 110}, {542, 17854}, {7728, 14984}, {8681, 24981}

X(40229) = (name pending)

Barycentrics    (a^2 - 2*a*b + b^2 + c^2)*(a^2 + 2*a*b + b^2 + c^2)*(a^2 + b^2 - 2*a*c + c^2)*(a^2 + b^2 + 2*a*c + c^2)*(5*a^10*b^2 + 3*a^8*b^4 - 6*a^6*b^6 - 2*a^4*b^8 + a^2*b^10 - b^12 + 5*a^10*c^2 - 12*a^8*b^2*c^2 + 12*a^6*b^4*c^2 + 18*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 2*b^10*c^2 + 3*a^8*c^4 + 12*a^6*b^2*c^4 - 48*a^4*b^4*c^4 + 8*a^2*b^6*c^4 + b^8*c^4 - 6*a^6*c^6 + 18*a^4*b^2*c^6 + 8*a^2*b^4*c^6 - 4*b^6*c^6 - 2*a^4*c^8 - 9*a^2*b^2*c^8 + b^4*c^8 + a^2*c^10 + 2*b^2*c^10 - c^12) : :

X(40229) lies on the cubic K1163 and this line: {25, 69}

X(40230) = X(25)X(111)∩X(69)X(146)

Barycentrics    a^2*(a^12*b^2 - 3*a^8*b^6 + 3*a^4*b^10 - b^14 + a^12*c^2 - 2*a^10*b^2*c^2 + 6*a^8*b^4*c^2 + 8*a^6*b^6*c^2 - 11*a^4*b^8*c^2 - 6*a^2*b^10*c^2 + 4*b^12*c^2 + 6*a^8*b^2*c^4 - 30*a^6*b^4*c^4 + 12*a^4*b^6*c^4 + 18*a^2*b^8*c^4 - 6*b^10*c^4 - 3*a^8*c^6 + 8*a^6*b^2*c^6 + 12*a^4*b^4*c^6 - 24*a^2*b^6*c^6 + 3*b^8*c^6 - 11*a^4*b^2*c^8 + 18*a^2*b^4*c^8 + 3*b^6*c^8 + 3*a^4*c^10 - 6*a^2*b^2*c^10 - 6*b^4*c^10 + 4*b^2*c^12 - c^14) : :

X(40230) lies on the cubic K1163 and these lines: {25, 111}, {69, 146}, {2794, 38323}, {9517, 39904}

X(40231) = (name pending)

Barycentrics    (a^4 + 2*a^2*b^2 + b^4 - 4*a^2*c^2 + 2*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4)*(6*a^10*b^2 - a^8*b^4 - 16*a^6*b^6 - 6*a^4*b^8 + 2*a^2*b^10 - b^12 + 6*a^10*c^2 - 22*a^8*b^2*c^2 + 34*a^6*b^4*c^2 + 42*a^4*b^6*c^2 - 16*a^2*b^8*c^2 + 4*b^10*c^2 - a^8*c^4 + 34*a^6*b^2*c^4 - 120*a^4*b^4*c^4 + 26*a^2*b^6*c^4 + b^8*c^4 - 16*a^6*c^6 + 42*a^4*b^2*c^6 + 26*a^2*b^4*c^6 - 8*b^6*c^6 - 6*a^4*c^8 - 16*a^2*b^2*c^8 + b^4*c^8 + 2*a^2*c^10 + 4*b^2*c^10 - c^12) : :

X(40231) lies on the cubic K1163 and this line: {69, 111}

X(40232) = X(23)X(2353)∩X(66)X(69)

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

X(40232) lies on the cubic K1163 and these lines: {23, 2353}, {66, 69}, {111, 1289}, {5485, 16277}, {14376, 16051}

X(40233) = (name pending)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*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)*(a^12 - 4*a^10*b^2 - a^8*b^4 + 8*a^6*b^6 - a^4*b^8 - 4*a^2*b^10 + b^12 - 4*a^10*c^2 + 21*a^8*b^2*c^2 - 31*a^6*b^4*c^2 - 25*a^4*b^6*c^2 + 27*a^2*b^8*c^2 - 4*b^10*c^2 - a^8*c^4 - 31*a^6*b^2*c^4 + 114*a^4*b^4*c^4 - 37*a^2*b^6*c^4 - b^8*c^4 + 8*a^6*c^6 - 25*a^4*b^2*c^6 - 37*a^2*b^4*c^6 + 8*b^6*c^6 - a^4*c^8 + 27*a^2*b^2*c^8 - b^4*c^8 - 4*a^2*c^10 - 4*b^2*c^10 + c^12) : :

X(40233) lies on the cubic K1163 and this line: {111, 2393}

X(40234) = X(6)X(1562)∩X(20)X(112)

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

X(40234) lies on the Moses-Parry circle, the cubic K1163, and these lines: {6, 1562}, {20, 112}, {111, 1289}, {115, 235}, {187, 16318}, {1368, 1560}, {2079, 3515}, {3569, 14391}, {5913, 14580}, {8428, 9909}, {15078, 21397}

X(40235) = X(3)X(126)∩X(235)X(1560)

Barycentrics    3*a^12*b^4 - 4*a^10*b^6 - 5*a^8*b^8 + 8*a^6*b^10 + a^4*b^12 - 4*a^2*b^14 + b^16 - 14*a^12*b^2*c^2 + 28*a^10*b^4*c^2 + 38*a^8*b^6*c^2 - 56*a^6*b^8*c^2 - 18*a^4*b^10*c^2 + 28*a^2*b^12*c^2 - 6*b^14*c^2 + 3*a^12*c^4 + 28*a^10*b^2*c^4 - 138*a^8*b^4*c^4 + 64*a^6*b^6*c^4 + 95*a^4*b^8*c^4 - 60*a^2*b^10*c^4 + 8*b^12*c^4 - 4*a^10*c^6 + 38*a^8*b^2*c^6 + 64*a^6*b^4*c^6 - 156*a^4*b^6*c^6 + 36*a^2*b^8*c^6 + 6*b^10*c^6 - 5*a^8*c^8 - 56*a^6*b^2*c^8 + 95*a^4*b^4*c^8 + 36*a^2*b^6*c^8 - 18*b^8*c^8 + 8*a^6*c^10 - 18*a^4*b^2*c^10 - 60*a^2*b^4*c^10 + 6*b^6*c^10 + a^4*c^12 + 28*a^2*b^2*c^12 + 8*b^4*c^12 - 4*a^2*c^14 - 6*b^2*c^14 + c^16 : :

X(40235) is the singular focus of the cubic K1163.

X(40235) lies these lines: {3, 126}, {235, 1560}, {2373, 37201}, {3542, 30247}, {5656, 9968}

leftri

Tetrahedral projections: X(40236)-X(40296)

rightri

This preamble and centers X(40236)-X(40296) were contributed by César Eliud Lozada, November 4, 2020.

Let ABC be a triangle on a plane XY. Consider three segments AA', BB', CC' with lengths U, V, W, respectively, and each having one fixed extreme in A, B and C, respectively, and the other extremes free to move outside the plane XY. Suppose that these segments are rotated around their fixed extremes in such a way that their free extremes coincide at a point D, forming, together with the sides of ABC, the edges of a tetrahedron ABCD. Let D* be the orthogonal projection of D on the plane of ABC. The point D* is here named the tetrahedral projection of ABC by (U, V, W) or the tetrahedral projection of ABC to A'B'C'.

The point D* has barycentric coordinates:

    D* = a2 (SA - U2) + SB W2 + SC V2 : b2 (SB - V2) + SC U2 + SA W2 : c2 (SC - W2) + SA V2 + SB U2     (1)

Z(D), the Z-coordinate of D , i.e., the height of the point D measured from D* and orthogonally to the plane of ABC, is given by:

    Z(D) = ±sqrt(∑ [2 (a2 U2 + V2 W2) SA - a2 U4] - (a b c)2)/(2 S)                     (2)

Equation (2) shows that D is real or imaginary according to the sign of the quantity under the square root. If this quantity is zero then D and D* coincide on the plane of ABC. Moreover, the ± sign indicates that there are two possible points D and D', each in different sides with respect to the plane of ABC.

Equation (1) shows that if U, V, W are real numbers then D* is always real and also that, if U, V, W are cyclic values, i.e., if there exists a degree-1 function ƒ(a,b,c) such that U=ƒ(a,b,c), V=ƒ(b,c,a) and W=ƒ(c,a,b), then D* is a triangle center.

Some calculated values:

Definitions of all triangles above mentioned can be found in the index of triangles.

Preamble edited on June 28, 2022.

X(40236) = TETRAHEDRAL PROJECTION OF ABC TO 1st ANTI-BROCARD TRIANGLE

Barycentrics    a^8+3*(b^2+c^2)*a^6-(2*b^4-b^2*c^2+2*c^4)*a^4-(b^2+c^2)*(b^4+c^4)*a^2-(b^6-c^6)*(b^2-c^2) : :
X(40236) = 3*X(2)-4*X(1513) = 15*X(2)-16*X(10011) = 2*X(20)-3*X(33265) = 5*X(1513)-4*X(10011)

X(40236) lies on these lines: {2, 3}, {98, 8784}, {114, 29317}, {147, 511}, {182, 9993}, {183, 36990}, {194, 8721}, {325, 29181}, {385, 1503}, {516, 1281}, {1350, 3314}, {1352, 6194}, {2080, 9862}, {2456, 10353}, {2794, 14712}, {2896, 5188}, {3095, 40278}, {3329, 5480}, {3398, 12252}, {3424, 37667}, {3818, 22712}, {5085, 7875}, {5171, 9873}, {5207, 5976}, {5476, 9774}, {5986, 18400}, {5987, 17702}, {5992, 29057}, {6033, 9772}, {6054, 19924}, {6310, 11381}, {6776, 7766}, {7710, 7774}, {7735, 14927}, {7759, 9764}, {7761, 22676}, {7797, 12203}, {7802, 36997}, {7809, 38745}, {7836, 30270}, {7837, 11477}, {7868, 31884}, {8844, 20539}, {9474, 36899}, {9744, 31670}, {9753, 39750}, {9756, 17004}, {9821, 40253}, {10334, 13355}, {10516, 16986}, {11177, 11645}, {12830, 15514}, {14931, 23698}, {15072, 40254}, {16989, 25406}, {17984, 30737}, {19570, 38664}, {29323, 38227}, {33706, 34507}, {35389, 39882}

X(40236) = reflection of X(i) in X(j) for these (i,j): (20, 11676), (5189, 36173), (5984, 385), (5999, 1513), (7779, 147), (9862, 2080), (15683, 9855), (40246, 3543)
X(40236) = anticomplement of X(5999)
X(40236) = intersection, other than A,B,C, of conics {{A, B, C, X(25), X(34214)}} and {{A, B, C, X(98), X(420)}}
X(40236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13862, 2), (4, 37182, 2), (5, 37455, 2), (382, 40279, 4), (1513, 5999, 2), (5059, 33244, 20), (6039, 6040, 5), (17578, 33018, 4), (20854, 21536, 420)

X(40237) = TETRAHEDRAL PROJECTION OF ABC TO 4th ANTI-BROCARD TRIANGLE

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+(b^2+7*b*c+c^2)*(b^2-7*b*c+c^2)*a^4+(b^2+c^2)*(3*b^4+19*b^2*c^2+3*c^4)*a^2-2*(b^4-16*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(40237) = 4*X(111)-3*X(40115)

X(40237) lies on these lines: {30, 111}, {574, 3830}, {3534, 8585}, {9872, 19924}

X(40238) = TETRAHEDRAL PROJECTION OF ABC TO 5th ANTI-BROCARD TRIANGLE

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

X(40238) lies on these lines: {3, 6}, {98, 7944}, {4027, 6656}, {10345, 40250}, {12203, 37243}, {12252, 40239}

X(40238) = {X(1342), X(1343)}-harmonic conjugate of X(35422)

X(40239) = TETRAHEDRAL PROJECTION OF ABC TO 6th ANTI-BROCARD TRIANGLE

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

X(40239) lies on these lines: {3, 10333}, {30, 2456}, {83, 546}, {182, 40250}, {550, 10350}, {575, 10796}, {2782, 35377}, {5171, 7908}, {12110, 32448}, {12177, 32515}, {12252, 40238}, {32139, 33786}

X(40240) = TETRAHEDRAL PROJECTION OF ABC TO 2nd ANTI-CONWAY TRIANGLE

Barycentrics    2*a^10-6*(b^2+c^2)*a^8+(7*b^4+12*b^2*c^2+7*c^4)*a^6-5*(b^4-c^4)*(b^2-c^2)*a^4+(3*b^4-8*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(40240) = 9*X(51)-X(6240) = 3*X(51)+X(13403) = 3*X(381)+X(10112) = 3*X(389)+X(1885) = X(389)+3*X(16657) = 3*X(546)+X(11264) = X(1885)-9*X(16657) = 3*X(3845)+X(10116) = 3*X(5946)+X(12897) = X(6240)+3*X(13403) = X(6756)-3*X(10110) = X(6756)+3*X(12241) = 7*X(9781)+X(21659) = 3*X(11225)+X(12162) = 3*X(11245)+X(13474) = 3*X(12022)+X(13419) = X(12605)+3*X(21849)

X(40240) lies on these lines: {4, 1173}, {30, 12002}, {51, 6240}, {113, 14627}, {125, 35482}, {235, 37505}, {381, 10112}, {389, 974}, {397, 35715}, {398, 35714}, {403, 12242}, {524, 40247}, {539, 3850}, {542, 546}, {578, 3542}, {1493, 16534}, {1596, 14862}, {2914, 3574}, {3088, 20299}, {3357, 11433}, {3845, 10116}, {3853, 18128}, {5097, 22660}, {5480, 18383}, {5946, 12897}, {5972, 37472}, {6756, 10110}, {7507, 10982}, {9781, 21659}, {9927, 19130}, {10095, 17702}, {10182, 11425}, {10282, 15873}, {10619, 34484}, {11225, 12162}, {11245, 13474}, {11424, 37119}, {11432, 22802}, {11793, 13142}, {12022, 13419}, {12605, 21849}, {13382, 13488}, {13567, 25563}, {13851, 32377}, {14865, 20417}, {14940, 15033}, {15807, 16881}, {17810, 34785}, {18369, 30714}, {18388, 35488}, {18555, 37347}, {29317, 32191}, {33332, 36253}

X(40240) = midpoint of X(i) and X(j) for these {i,j}: {3853, 18128}, {10110, 12241}, {11793, 13142}, {13382, 13488}, {15807, 16881}
X(40240) = crosssum of X(3) and X(12006)

X(40241) = TETRAHEDRAL PROJECTION OF ABC TO 3rd ANTI-EULER TRIANGLE

Barycentrics    4*a^10-9*(b^2+c^2)*a^8+(5*b^4+b^2*c^2+5*c^4)*a^6-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4+(3*b^4+5*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(40241) = 9*X(3060)-8*X(7553) = 15*X(3060)-16*X(13292) = 3*X(3060)-4*X(34224) = 9*X(5640)-8*X(13419) = 5*X(6241)-4*X(34798) = 5*X(7553)-6*X(13292) = 2*X(7553)-3*X(34224) = 9*X(7998)-8*X(12134) = 5*X(11439)-4*X(16659) = 4*X(11750)-3*X(15305) = 3*X(12111)-4*X(12225) = 4*X(13292)-5*X(34224)

X(40241) lies on these lines: {1503, 12111}, {3060, 7553}, {3146, 11645}, {5012, 7566}, {5640, 13419}, {6241, 34798}, {7558, 15080}, {7998, 12134}, {9705, 31181}, {9833, 11449}, {10298, 14864}, {11439, 16659}, {11440, 34780}, {11454, 14216}, {11750, 15305}, {12279, 12280}, {12283, 15084}, {13163, 15024}, {13445, 17845}, {18381, 26881}, {29012, 34799}, {38397, 38435}

X(40242) = TETRAHEDRAL PROJECTION OF ABC TO 4th ANTI-EULER TRIANGLE

Barycentrics    4*a^10-7*(b^2+c^2)*a^8-(b^4-13*b^2*c^2+c^4)*a^6+5*(b^4-c^4)*(b^2-c^2)*a^4+(b^4-5*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(40242) = 5*X(3567)-4*X(6240) = 15*X(3567)-16*X(12241) = 2*X(5876)-3*X(18561) = 3*X(5890)-4*X(21659) = 3*X(5890)-2*X(34797) = 3*X(6240)-4*X(12241) = 3*X(6241)-4*X(34224) = 7*X(7999)-8*X(12605) = 7*X(9781)-8*X(13403) = 7*X(9781)-6*X(18559) = 4*X(10116)-3*X(34796) = 5*X(11444)-6*X(18564) = 9*X(11455)-8*X(16655) = 3*X(11455)-4*X(18560) = 3*X(11459)-2*X(12278) = 3*X(11459)-4*X(18563) = 17*X(11465)-16*X(31833) = 3*X(12289)-2*X(34224) = 4*X(13403)-3*X(18559) = 2*X(16655)-3*X(18560)

X(40242) lies on these lines: {3, 18379}, {4, 1495}, {20, 9927}, {26, 10733}, {30, 5889}, {54, 35480}, {74, 1657}, {195, 5073}, {382, 1614}, {550, 23294}, {1147, 10296}, {1498, 10721}, {1593, 9920}, {2931, 11413}, {3043, 40276}, {3520, 34786}, {3529, 11457}, {3567, 6240}, {3830, 9707}, {5876, 18561}, {5890, 21659}, {6143, 18376}, {7488, 16013}, {7999, 12605}, {9781, 13403}, {10116, 34796}, {10295, 26917}, {10483, 19368}, {11412, 12219}, {11444, 18564}, {11449, 18403}, {11455, 16655}, {11459, 12278}, {11462, 35820}, {11463, 35821}, {11465, 31833}, {11466, 19106}, {11467, 19107}, {11572, 35475}, {11704, 13851}, {11750, 22949}, {12111, 18562}, {12173, 15033}, {12283, 29012}, {12290, 12291}, {14157, 17845}, {14644, 32534}, {15072, 18565}, {15685, 34469}, {17800, 32608}, {18383, 35473}, {18405, 35477}, {23040, 32767}, {29323, 39874}

X(40242) = reflection of X(i) in X(j) for these (i,j): (6241, 12289), (12111, 18562), (12278, 18563), (34797, 21659)
X(40242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 34785, 26882), (20, 25739, 11468), (12278, 18563, 11459), (13403, 18559, 9781), (13851, 21844, 11704), (17845, 35490, 14157), (21659, 34797, 5890)

X(40243) = TETRAHEDRAL PROJECTION OF ABC TO ANTI-INNER-GREBE TRIANGLE

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

X(40243) lies on these lines: {3, 6}, {3069, 36711}, {3843, 39661}, {6460, 36712}, {14242, 36714}, {14269, 35830}, {14930, 36702}

X(40243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39658, 3), (372, 8416, 6395), (1152, 9605, 3), (6395, 11917, 3312), (6410, 7772, 39649), (6410, 39649, 3), (21309, 40268, 40244)

X(40244) = TETRAHEDRAL PROJECTION OF ABC TO ANTI-OUTER-GREBE TRIANGLE

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

X(40244) lies on these lines: {3, 6}, {3068, 36712}, {3843, 39660}, {6459, 36711}, {11292, 32814}, {14227, 36709}, {14269, 35831}, {14930, 36717}

X(40244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39649, 3), (371, 8396, 6199), (1151, 9605, 3), (6199, 11916, 3311), (6409, 7772, 39658), (6409, 39658, 3), (21309, 40268, 40243)

X(40245) = TETRAHEDRAL PROJECTION OF ABC TO ANTI-MANDART-INCIRCLE TRIANGLE

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

X(40245) lies on these lines: {1, 3}, {100, 6934}, {149, 6890}, {1259, 10526}, {1376, 6917}, {1399, 36747}, {3560, 26066}, {4421, 34696}, {5763, 33814}, {5812, 11517}, {5840, 12332}, {5841, 11500}, {5844, 8668}, {6796, 21077}, {6831, 10525}, {6833, 11680}, {6862, 11496}, {6911, 25681}, {6928, 11502}, {11231, 37224}, {13346, 38607}, {26446, 37228}

X(40245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1482, 22766), (3, 10679, 2646), (3, 35448, 14110), (3, 37541, 34339), (46, 2077, 3), (10306, 35000, 11248), (10310, 11509, 3), (11248, 11249, 55), (11248, 35238, 26285)

X(40246) = TETRAHEDRAL PROJECTION OF ABC TO ANTI-MCCAY TRIANGLE

Barycentrics    11*a^4-7*b^4+13*b^2*c^2-7*c^4-5*(b^2+c^2)*a^2 : :
X(40246) = 3*X(2)-4*X(8352) = 15*X(2)-16*X(8355) = 5*X(2)-4*X(8598) = 7*X(2)-6*X(13586) = 5*X(2)-6*X(14041) = 9*X(2)-8*X(27088) = 11*X(2)-12*X(33228) = 4*X(2)-3*X(33265) = 13*X(2)-12*X(35297) = 7*X(2)-8*X(37350)

X(40246) lies on these lines: {2, 3}, {148, 3849}, {316, 8591}, {524, 8596}, {530, 25166}, {531, 25156}, {543, 7779}, {671, 14712}, {1992, 33683}, {6781, 9166}, {7748, 34604}, {7809, 15300}, {7823, 15534}, {7840, 20094}, {8584, 20088}, {8593, 29012}, {9889, 11606}, {11161, 19924}, {22165, 32819}

X(40246) = reflection of X(i) in X(j) for these (i,j): (2, 8597), (8591, 316), (9855, 8352), (14712, 671), (15683, 5999), (20094, 7840), (37901, 36174), (40236, 3543)
X(40246) = anticomplement of X(9855)
X(40246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5077, 11361, 2), (7833, 11317, 2), (8352, 9855, 2), (8598, 14041, 2), (13586, 37350, 2)

X(40247) = TETRAHEDRAL PROJECTION OF ABC TO 6th ANTI-MIXTILINEAR TRIANGLE

Barycentrics    (3*(b^2+c^2)*a^6-(9*b^4+4*b^2*c^2+9*c^4)*a^4+(b^2+c^2)*(9*b^4-2*b^2*c^2+9*c^4)*a^2-3*(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*a^2 : :
X(40247) = 5*X(3)-9*X(3819) = X(3)-9*X(5891) = X(3)+3*X(5907) = 11*X(3)-3*X(10575) = X(3)-3*X(11793) = 5*X(3)+3*X(12162) = 17*X(3)-9*X(14855) = 7*X(3)+9*X(18435) = 13*X(3)+3*X(18439) = X(3819)-5*X(5891) = 3*X(3819)+5*X(5907) = 3*X(3819)-5*X(11793) = 3*X(3819)+X(12162) = 17*X(3819)-5*X(14855) = 7*X(3819)+5*X(18435) = 3*X(5891)+X(5907) = 3*X(5891)-X(11793) = 15*X(5891)+X(12162) = 17*X(5891)-X(14855) = 7*X(5891)+X(18435)

X(40247) lies on these lines: {2, 13382}, {3, 64}, {4, 15606}, {5, 16254}, {51, 3544}, {52, 5072}, {185, 3525}, {373, 389}, {511, 546}, {524, 40240}, {575, 15083}, {632, 5876}, {1154, 12811}, {1216, 3627}, {1352, 18383}, {3060, 3091}, {3146, 11444}, {3292, 35500}, {3529, 3917}, {3545, 14531}, {3628, 10219}, {3850, 16982}, {3851, 21849}, {3855, 21969}, {3856, 12002}, {3857, 5446}, {3859, 13421}, {5056, 14831}, {5076, 10625}, {5079, 5943}, {5092, 32139}, {5447, 12103}, {5462, 12812}, {5650, 6241}, {5663, 12108}, {5889, 15022}, {6102, 6688}, {6643, 14864}, {7486, 16226}, {7568, 16534}, {7723, 38795}, {7999, 11381}, {9730, 40284}, {10263, 13570}, {10303, 12111}, {11439, 36987}, {11541, 32062}, {12109, 31836}, {12219, 15029}, {12358, 38791}, {12825, 38729}, {13348, 15067}, {13598, 23039}, {13857, 35482}, {14826, 34785}, {14862, 16197}, {14915, 32142}, {15025, 21649}, {15034, 21650}, {18553, 18569}, {22660, 24206}

X(40247) = midpoint of X(i) and X(j) for these {i,j}: {4, 15606}, {5462, 31834}, {5562, 10110}, {5876, 9729}, {5907, 11793}, {12109, 31836}
X(40247) = reflection of X(i) in X(j) for these (i,j): (12002, 3856), (15012, 3628)
X(40247) = complement of X(13382)
X(40247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3628, 15012, 11695), (3819, 5907, 12162), (3917, 15058, 13474), (5876, 10170, 9729), (5891, 5907, 11793), (11444, 15030, 15644)

X(40248) = TETRAHEDRAL PROJECTION OF ABC TO ARTZT TRIANGLE

Barycentrics    a^8-15*(b^2+c^2)*a^6+(19*b^4+10*b^2*c^2+19*c^4)*a^4-(b^2+c^2)*(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2)*a^2+4*(b^4-b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(40248) = 5*X(3)+4*X(40279)

X(40248) lies on these lines: {2, 3}, {98, 8860}, {114, 599}, {183, 6054}, {230, 11179}, {511, 11184}, {542, 7610}, {598, 39656}, {1351, 11163}, {1352, 11168}, {1503, 15597}, {2782, 9743}, {2794, 5569}, {3055, 31670}, {3815, 20423}, {4846, 24855}, {5050, 38227}, {5663, 9759}, {6055, 11646}, {6776, 23055}, {7694, 8182}, {8722, 31173}, {8859, 9755}, {9744, 22329}, {9753, 14848}, {9756, 11645}, {11177, 17004}, {11178, 15271}, {11180, 34229}, {15819, 21358}, {17008, 39899}, {18440, 37688}, {22712, 23234}

X(40248) = midpoint of X(7694) and X(8182)
X(40248) = anti-Artzt-to-Artzt similarity image of X(3)
X(40248) = X(7610)-of-Artzt-triangle
X(40248) = X(3)-of-Artzt-of-Artzt-triangle
X(40248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1513, 381), (3, 381, 5077), (549, 10011, 2)

X(40249) = TETRAHEDRAL PROJECTION OF ABC TO ASCELLA TRIANGLE

Barycentrics    a*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*(a^5-(b+c)*a^4-(2*b^2-b*c+2*c^2)*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-b*c*(b^2-4*b*c+c^2))*a-(b^3+c^3)*(b-c)^2) : :
X(40249) = 3*X(1071)+X(18239) = 5*X(5439)-X(12664) = X(6245)-3*X(10202) = 3*X(6260)-X(18239)

X(40249) lies on these lines: {3, 214}, {4, 30274}, {57, 5884}, {84, 6912}, {142, 12616}, {499, 11219}, {515, 942}, {546, 971}, {758, 37623}, {944, 5083}, {946, 12711}, {1071, 1210}, {1125, 6001}, {1158, 4512}, {2095, 3874}, {2801, 18242}, {3149, 18389}, {3244, 24474}, {3678, 5771}, {3754, 37281}, {3811, 5709}, {3817, 6245}, {5439, 12664}, {5450, 18443}, {5693, 5744}, {5745, 20117}, {5768, 6246}, {6905, 15556}, {6927, 18397}, {6949, 12691}, {6960, 9964}, {7971, 19861}, {9776, 15016}, {10122, 12671}, {10571, 12016}, {11018, 13464}, {12114, 30143}, {12436, 34339}, {12564, 13374}, {12672, 17603}, {12687, 19860}, {13369, 37290}, {31649, 34862}, {31671, 40265}

X(40249) = midpoint of X(i) and X(j) for these {i,j}: {942, 9942}, {1071, 6260}, {3874, 11500}, {5884, 6261}
X(40249) = reflection of X(6705) in X(9940)
X(40249) = trilinear product X(1210)*X(11012)

X(40250) = TETRAHEDRAL PROJECTION OF ABC TO 1st BROCARD-REFLECTED TRIANGLE

Barycentrics    a^8+(b^4+4*b^2*c^2+c^4)*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(40250) = 5*X(4)+3*X(32986) = 5*X(381)-X(11159) = 3*X(381)-X(35930) = X(382)+3*X(11287)

X(40250) lies on these lines: {2, 3}, {76, 6287}, {147, 32447}, {148, 10335}, {182, 40239}, {262, 6033}, {511, 7848}, {538, 18553}, {1352, 32515}, {2023, 5475}, {2080, 9993}, {2782, 3818}, {2794, 10796}, {3095, 7905}, {3398, 9873}, {3734, 5031}, {3972, 38741}, {4045, 29012}, {5480, 35431}, {5663, 40254}, {6249, 7747}, {7748, 22803}, {7761, 24256}, {7777, 32528}, {7823, 13111}, {8721, 32516}, {9862, 11842}, {9863, 18503}, {10033, 11632}, {10345, 40238}, {10356, 30270}, {10358, 36997}, {13334, 40278}, {13449, 22682}, {15048, 39884}, {18907, 38136}, {20428, 22693}, {20429, 22694}, {22512, 36759}, {22513, 36760}, {22515, 22681}, {32134, 36998}, {34615, 34734}

X(40250) = midpoint of X(i) and X(j) for these {i,j}: {4, 37242}, {15048, 39884}
X(40250) = reflection of X(10796) in X(19130)
X(40250) = tetrahedral projection of ABC to 1st Brocard triangle
X(40250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6655, 382), (4, 37336, 3), (381, 13860, 5), (3851, 7887, 5)

X(40251) = TETRAHEDRAL PROJECTION OF ABC TO 2nd BROCARD TRIANGLE

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

X(40251) lies on these lines: {2, 1495}, {353, 511}, {575, 1383}, {2782, 11655}

X(40252) = TETRAHEDRAL PROJECTION OF ABC TO 5th BROCARD TRIANGLE

Barycentrics    ((b^2+c^2)*a^8-(b^4+c^4)*a^6+(b^2+c^2)*(b^4+c^4)*a^4+(4*b^4+3*b^2*c^2+4*c^4)*b^2*c^2*a^2-(b^6+c^6)*(b^4+b^2*c^2+c^4))*a^2 : :
X(40252) = X(9983)-3*X(22678) = 3*X(22678)-2*X(32151)

X(40252) lies on these lines: {3, 6}, {4, 8782}, {76, 9996}, {194, 9862}, {262, 7940}, {2782, 9873}, {2896, 12251}, {3399, 10345}, {5976, 7752}, {6033, 8149}, {6194, 7932}, {6248, 18500}, {6656, 32521}, {7697, 10356}, {7811, 34510}, {7846, 11272}, {7884, 33706}, {7942, 22712}, {9983, 22678}, {10038, 12837}, {10047, 12836}, {10063, 10873}, {10079, 10874}, {10263, 39684}, {10346, 35925}, {13108, 18503}, {15821, 40107}, {31670, 31982}, {35700, 38733}

X(40252) = reflection of X(9983) in X(32151)
X(40252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3094, 9821, 3), (3095, 9821, 32), (9821, 35248, 5188), (9983, 22678, 32151), (34870, 35002, 3)

X(40253) = TETRAHEDRAL PROJECTION OF ABC TO 6th BROCARD TRIANGLE

Barycentrics    3*(b^2+c^2)*a^10-2*(b^4+c^4)*a^8+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^6+(5*b^4+3*b^2*c^2+5*c^4)*b^2*c^2*a^4-(b^2+c^2)*(2*b^8+2*c^8-b^2*c^2*(b^4+c^4))*a^2-(b^6-c^6)*(b^2-c^2)*b^2*c^2 : :
X(40253) = 2*X(3)-3*X(22678)

X(40253) lies on these lines: {3, 10333}, {76, 36997}, {384, 35387}, {511, 7893}, {3095, 9862}, {3146, 12251}, {3314, 5188}, {6776, 32476}, {7876, 13354}, {9821, 40236}, {13862, 35430}

X(40253) = reflection of X(9983) in X(9863)

X(40254) = TETRAHEDRAL PROJECTION OF ABC TO 7th BROCARD TRIANGLE

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

X(40254) lies on these lines: {3, 695}, {4, 51}, {511, 2549}, {1181, 11325}, {1204, 35476}, {5167, 9744}, {5309, 31850}, {5562, 7791}, {5663, 40250}, {5889, 6655}, {6310, 9729}, {6759, 27369}, {7709, 11674}, {9730, 37348}, {11444, 33021}, {11695, 32968}, {11793, 16043}, {12111, 37336}, {13630, 40279}, {13754, 37242}, {14831, 33017}, {15028, 33020}, {15043, 16044}, {15072, 40236}, {16226, 33016}, {34783, 37243}

X(40254) = crosssum of X(3) and X(35930)

X(40255) = TETRAHEDRAL PROJECTION OF ABC TO 2nd CIRCUMPERP TANGENTIAL TRIANGLE

Barycentrics    a^2*(a^8-2*(b+c)*a^7-2*(b^2-3*b*c+c^2)*a^6+2*(b+c)*(3*b^2-5*b*c+3*c^2)*a^5-4*(2*b-c)*(b-2*c)*b*c*a^4-2*(b+c)*(3*b^4+3*c^4-2*b*c*(5*b^2-9*b*c+5*c^2))*a^3+2*(b^6+c^6-(b^4+c^4+b*c*(7*b^2-18*b*c+7*c^2))*b*c)*a^2+2*(b^2-c^2)*(b-c)*(b^4+c^4-b*c*(3*b^2-8*b*c+3*c^2))*a-(b^2-c^2)^2*(b-c)^4) : :

X(40255) lies on these lines: {1, 3}, {945, 39173}, {958, 6929}, {1145, 11499}, {1532, 10526}, {1872, 22479}, {2975, 6938}, {3913, 22775}, {5840, 12114}, {5886, 25875}, {6265, 11517}, {6834, 11681}, {6838, 20060}, {6911, 37828}, {6923, 22759}, {6959, 22753}, {10043, 10090}, {10525, 22758}, {10785, 13279}, {11194, 34708}

X(40255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1482, 11508), (3, 10680, 1319), (3, 35460, 10310), (3428, 10966, 3), (5119, 11012, 3), (11248, 11249, 56), (11249, 35239, 26286), (13528, 34880, 3)

X(40256) = TETRAHEDRAL PROJECTION OF ABC TO 1st CIRCUMPERP TRIANGLE

Barycentrics    a*(a^6-3*(b^2+c^2)*a^4+3*(b+c)*b*c*a^3+(3*b^2+5*b*c+3*c^2)*(b-c)^2*a^2-3*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(40256) = 3*X(40)+X(84) = 7*X(40)+X(10864) = X(84)-3*X(1158) = 7*X(84)-3*X(10864) = 3*X(165)-X(6261) = 7*X(1158)-X(10864) = 3*X(3655)-2*X(32905) = 3*X(5657)-X(6256) = X(6361)+3*X(14647) = X(7971)-5*X(35242) = 4*X(12616)-X(40265) = X(12667)+3*X(14646)

X(40256) lies on these lines: {1, 6950}, {3, 214}, {4, 484}, {8, 20}, {10, 6923}, {46, 499}, {55, 5884}, {57, 13464}, {72, 13528}, {100, 5693}, {104, 5697}, {165, 6261}, {191, 2950}, {355, 40264}, {516, 10525}, {517, 5450}, {519, 24467}, {551, 37612}, {601, 4424}, {758, 11248}, {912, 8715}, {920, 4848}, {944, 1768}, {950, 10051}, {962, 5535}, {993, 37562}, {1012, 37567}, {1071, 37568}, {1376, 5780}, {1388, 25485}, {1389, 5903}, {1479, 10265}, {1490, 16558}, {1537, 5433}, {1621, 15016}, {1697, 13607}, {1709, 31673}, {1727, 6938}, {1788, 26333}, {2077, 3869}, {2093, 7098}, {2098, 11715}, {2771, 32141}, {2829, 5690}, {3218, 7982}, {3295, 12005}, {3336, 5603}, {3337, 10595}, {3357, 3579}, {3359, 3452}, {3474, 26332}, {3560, 3754}, {3576, 17548}, {3652, 5790}, {3655, 32905}, {3874, 10679}, {3877, 37561}, {3881, 37622}, {3884, 10269}, {3892, 12000}, {3898, 16203}, {4084, 37533}, {4301, 37532}, {4640, 31788}, {4868, 36742}, {4973, 10680}, {5010, 21740}, {5119, 5882}, {5128, 6844}, {5180, 6972}, {5248, 34339}, {5250, 10165}, {5330, 38693}, {5445, 6941}, {5493, 6245}, {5553, 7162}, {5687, 14740}, {5709, 6705}, {5734, 23958}, {5842, 33899}, {5887, 25440}, {6211, 29497}, {6223, 12849}, {6361, 14647}, {6763, 12245}, {6871, 10175}, {6905, 37572}, {6914, 30147}, {6924, 10225}, {6952, 18393}, {6958, 11813}, {7967, 37563}, {7971, 35242}, {8227, 31224}, {9588, 26878}, {9624, 27003}, {9803, 20066}, {10310, 31806}, {10698, 21842}, {10826, 24042}, {11491, 15071}, {11496, 31870}, {11499, 31803}, {11500, 13465}, {12114, 12702}, {12647, 37002}, {12667, 14646}, {14110, 17613}, {14988, 22836}, {15528, 26358}, {17102, 20324}, {18389, 37287}, {18491, 31871}, {19919, 38112}, {20070, 24468}, {31663, 37837}, {36866, 38755}, {37469, 37598}, {37822, 37828}

X(40256) = midpoint of X(i) and X(j) for these {i,j}: {40, 1158}, {5493, 6245}, {12114, 12702}
X(40256) = reflection of X(i) in X(j) for these (i,j): (6796, 3579), (12608, 6684), (22836, 26285), (22837, 32153), (37837, 31663), (40257, 3), (40264, 355)
X(40256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 63, 11362), (1768, 11010, 944), (3359, 12514, 6684), (5445, 34789, 6941), (6914, 35004, 30147), (11496, 36279, 31870)

X(40257) = TETRAHEDRAL PROJECTION OF ABC TO 2nd CIRCUMPERP TRIANGLE

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+(b+c)*(4*b^2-7*b*c+4*c^2)*a^3-(b^2+5*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(2*b^2-3*b*c+2*c^2)*a+(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(40257) = 3*X(1)+X(1490) = 5*X(1)-X(12650) = 3*X(551)-X(6245) = X(1158)-3*X(3576) = 3*X(1385)-X(34862) = X(1490)-3*X(6261) = 5*X(1490)+3*X(12650) = 3*X(3576)+X(7971) = 3*X(3655)+X(6259) = 3*X(5450)-2*X(34862) = 3*X(5790)-4*X(40260) = X(5812)-3*X(34647) = 5*X(6261)+X(12650) = 3*X(6796)-4*X(40262) = 3*X(7967)+X(12667) = X(7992)-9*X(30392) = X(9799)-9*X(38314) = 3*X(10246)-X(12114) = 9*X(10246)-X(12684) = 4*X(12608)-X(40264) = 3*X(37837)-2*X(40262)

In the plane of a triangle ABC, let
O = circumcenter;
A' = reflection of A in O, and define B'; and C' cyclically;
H = orthocenter;
A''B''C'' = circumcevian triangle of H wrt A''B''C'';
Ab = AC∩B'A'', and define Bc and Ca cyclically;
Ac = AB∩C'A'', and define Ba and Cb cyclically;
Ao = circumcenter of AAbAc, and define Bo and Cocyclically;
σ = the affine transformation that maps ABC onto AoBoCocyclically;
I = incenter;
Then σ(I) = X(40257). Moreover, if X = x : y : z, then
σ(X) = (-a+b+c)^2 (a^4-2 a^2 (b-c)^2+(b-c)^2 (b^2-b c+c^2)) x-a^2 (a-c) c (a-b+c)^2 y-a^2 (a-b) b (a+b-c)^2 z : : . See X(40257). (Angel Montesdeoca, June 28, 2022)

X(40257) lies on these lines: {1, 4}, {3, 214}, {5, 30147}, {8, 6326}, {10, 6863}, {20, 5180}, {40, 4511}, {56, 5884}, {78, 6962}, {80, 6941}, {84, 2320}, {104, 15071}, {221, 11700}, {355, 6980}, {484, 6942}, {499, 10265}, {517, 6796}, {519, 37700}, {551, 6245}, {758, 11249}, {912, 8666}, {952, 3813}, {958, 20117}, {962, 20066}, {971, 15178}, {993, 5887}, {997, 5837}, {999, 12005}, {1012, 34471}, {1071, 1319}, {1125, 6862}, {1158, 3576}, {1385, 5248}, {1388, 11715}, {1482, 11500}, {1532, 10950}, {1537, 6284}, {1727, 37618}, {2098, 12739}, {2099, 3149}, {2360, 17515}, {2646, 12672}, {2771, 32153}, {2829, 19907}, {2975, 5693}, {3057, 33597}, {3304, 10122}, {3428, 5730}, {3616, 6888}, {3655, 6259}, {3656, 34745}, {3754, 6911}, {3811, 12640}, {3869, 11012}, {3872, 17857}, {3874, 10680}, {3877, 10902}, {3884, 10267}, {3890, 34486}, {3892, 12001}, {3895, 7982}, {3898, 16202}, {4084, 37532}, {4301, 37533}, {4861, 5881}, {5141, 5587}, {5253, 15016}, {5443, 6830}, {5538, 6361}, {5697, 10087}, {5731, 15680}, {5790, 40260}, {5812, 34647}, {5842, 22791}, {5886, 30143}, {5903, 6905}, {6003, 35050}, {6224, 37437}, {6264, 20085}, {6705, 6892}, {6831, 15950}, {6834, 10573}, {6906, 37525}, {6910, 10165}, {6924, 35004}, {6928, 11813}, {6933, 10175}, {6949, 12247}, {6950, 37616}, {6974, 9948}, {7680, 37737}, {7681, 37730}, {7992, 30392}, {9669, 16174}, {9799, 38314}, {9942, 24929}, {10246, 12114}, {10283, 16160}, {10609, 11826}, {10786, 12647}, {11010, 13253}, {11260, 32159}, {11372, 30284}, {11567, 28160}, {11928, 12737}, {12246, 12255}, {12520, 37611}, {12635, 22770}, {12645, 12738}, {12675, 24928}, {12699, 40265}, {12705, 13384}, {14986, 18467}, {14988, 26286}, {15955, 37699}, {18389, 26437}, {18480, 33281}, {18493, 40259}, {18761, 31871}, {21635, 37821}, {22753, 31870}, {22758, 31803}, {23340, 25439}, {25440, 37562}, {26087, 28204}, {28082, 32486}, {28194, 37531}, {32905, 37727}, {33899, 38028}

X(40257) = midpoint of X(i) and X(j) for these {i,j}: {1, 6261}, {944, 6256}, {1158, 7971}, {1482, 11500}, {5882, 6260}, {12635, 22770}
X(40257) = reflection of X(i) in X(j) for these (i,j): (5450, 1385), (6796, 37837), (12616, 1125), (37727, 32905), (40256, 3), (40265, 12699)
X(40257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18446, 5882), (3, 6265, 30144), (946, 5882, 950), (3428, 5730, 31806), (3576, 7971, 1158), (6326, 11014, 8), (6949, 12247, 18395), (10698, 11491, 5697), (13464, 13607, 40270), (15071, 21842, 104)

X(40258) = TETRAHEDRAL PROJECTION OF ABC TO 1st EHRMANN TRIANGLE

Barycentrics    (a^10-4*(b^2+c^2)*a^8+2*(2*b^4-7*b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(2*b^4+21*b^2*c^2+2*c^4)*a^4-(b^4+3*b^2*c^2+c^4)*(5*b^4-8*b^2*c^2+5*c^4)*a^2+2*(b^4-c^4)^2*(b^2+c^2))*a^2 : :

X(40258) lies on these lines: {6, 12308}, {382, 576}, {511, 8547}, {575, 5907}, {3818, 15019}, {5092, 21766}, {5476, 14094}, {5890, 12584}, {7516, 20190}, {9970, 37077}, {11422, 32305}, {11935, 17508}, {15032, 32599}, {15087, 16010}

X(40259) = TETRAHEDRAL PROJECTION OF ABC TO 3rd EULER TRIANGLE

Barycentrics    (b+c)*a^6+(b^2-6*b*c+c^2)*a^5-(b+c)*(4*b^2-7*b*c+4*c^2)*a^4-(2*b^2-3*b*c+2*c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(5*b^2-b*c+5*c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-2*(b^2-c^2)^3*(b-c) : :
X(40259) = 3*X(3)+X(40265) = 3*X(946)+X(12616) = 3*X(1699)+X(5450) = 5*X(3843)-X(40264) = X(6256)-9*X(9779) = X(6261)-9*X(38021) = X(6796)-5*X(8227) = X(12650)+15*X(30308) = 5*X(18493)-X(40257)

X(40259) lies on these lines: {3, 40265}, {4, 21842}, {5, 3884}, {11, 65}, {515, 546}, {1389, 37718}, {1484, 3881}, {1621, 6796}, {1699, 5450}, {3585, 11715}, {3843, 40264}, {5126, 18483}, {5330, 5587}, {5603, 37702}, {5790, 15862}, {5882, 17605}, {5884, 18393}, {5886, 35016}, {6256, 9779}, {6261, 38021}, {6701, 38028}, {7504, 38134}, {7743, 13464}, {8070, 30384}, {11680, 31806}, {11813, 20117}, {12005, 12047}, {12611, 31871}, {12650, 30308}, {16160, 33592}, {18480, 19907}, {18493, 40257}, {37621, 38062}, {37722, 38039}

X(40259) = reflection of X(40260) in X(5)
X(40259) = X(40260)-of-Johnson-triangle
X(40259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 34352, 38183), (11, 946, 31870), (11813, 26470, 20117)

X(40260) = TETRAHEDRAL PROJECTION OF ABC TO 4th EULER TRIANGLE

Barycentrics    (b+c)*a^6-3*(b+c)^2*a^5+7*(b+c)*b*c*a^4+(6*b^2+11*b*c+6*c^2)*(b-c)^2*a^3-3*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2-(b^2-c^2)^2*(3*b^2-7*b*c+3*c^2)*a+2*(b^2-c^2)^3*(b-c) : :
X(40260) = 3*X(3)+X(40264) = 3*X(10)+X(12608) = X(1158)-9*X(19875) = 5*X(1698)-X(5450) = 5*X(3843)-X(40265) = 3*X(5587)+X(6796) = 3*X(5790)+X(40257) = X(6256)+7*X(9780) = X(12616)-5*X(31399) = X(12650)-17*X(30315) = 3*X(18242)+X(33899) = X(18242)+3*X(38042) = X(33899)-9*X(38042)

X(40260) lies on these lines: {3, 40264}, {4, 11661}, {5, 3884}, {10, 119}, {12, 31870}, {21, 5587}, {35, 6246}, {140, 515}, {631, 38411}, {946, 3614}, {1158, 19875}, {1210, 10955}, {1385, 6702}, {1698, 5450}, {1737, 12005}, {3057, 16174}, {3576, 7705}, {3647, 26446}, {3652, 38755}, {3843, 40265}, {4187, 10175}, {5123, 6684}, {5499, 18242}, {5559, 5603}, {5790, 40257}, {5818, 6853}, {5882, 17606}, {5884, 18395}, {5953, 31759}, {6256, 9780}, {6949, 37710}, {7967, 15079}, {10165, 17619}, {10609, 32910}, {11681, 31806}, {12616, 12671}, {12650, 30315}, {19843, 34918}, {19925, 37290}, {24042, 37568}, {37230, 38162}, {37561, 38133}

X(40260) = midpoint of X(10609) and X(32910)
X(40260) = reflection of X(40259) in X(5)
X(40260) = X(40259)-of-Johnson-triangle
X(40260) = {X(10), X(119)}-harmonic conjugate of X(20117)

X(40261) = TETRAHEDRAL PROJECTION OF ABC TO 5th EULER TRIANGLE

Barycentrics    4*a^10-21*(b^2+c^2)*a^8+(13*b^4+16*b^2*c^2+13*c^4)*a^6+5*(b^2+c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^4-(17*b^8+17*c^8-2*(9*b^4+11*b^2*c^2+9*c^4)*b^2*c^2)*a^2+6*(b^4-c^4)*(b^2-c^2)^3 : :

X(40261) lies on these lines: {2, 1495}, {30, 10173}, {3055, 11574}

X(40262) = TETRAHEDRAL PROJECTION OF ABC TO EXCENTERS-MIDPOINTS TRIANGLE

Barycentrics    a*(4*a^6-5*(b+c)*a^5-(7*b^2-4*b*c+7*c^2)*a^4+10*(b^3+c^3)*a^3+2*(b^2-b*c+c^2)*(b-c)^2*a^2-5*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b+c)^2) : :
X(40262) = 5*X(3)-X(84) = 3*X(3)+X(1490) = 9*X(3)-X(12684) = 3*X(3)-X(34862) = 3*X(84)+5*X(1490) = 9*X(84)-5*X(12684) = 3*X(84)-5*X(34862) = 3*X(376)+X(6259) = 3*X(549)-X(6245) = 5*X(631)-X(5787) = 3*X(1490)+X(12684) = 3*X(3158)+X(8158) = 5*X(3522)+3*X(5658) = 9*X(3524)-X(9799) = 7*X(3528)+X(6223) = 3*X(6796)+X(40257) = 3*X(10164)-X(33899) = 9*X(10304)-X(12246) = X(12684)-3*X(34862) = 3*X(37837)-X(40257)

X(40262) lies on these lines: {3, 9}, {20, 22792}, {21, 10157}, {35, 9856}, {100, 31798}, {140, 515}, {355, 6954}, {376, 6259}, {404, 11227}, {411, 5440}, {474, 10156}, {517, 6796}, {549, 6245}, {550, 6260}, {631, 5787}, {916, 15489}, {942, 6905}, {944, 5126}, {946, 10386}, {993, 9947}, {1071, 5122}, {1329, 4297}, {1385, 6911}, {1538, 6284}, {3149, 5806}, {3158, 8158}, {3419, 6962}, {3520, 12136}, {3522, 5658}, {3524, 9799}, {3528, 6223}, {3530, 6705}, {3576, 16408}, {3579, 6261}, {3601, 19541}, {3824, 37281}, {4188, 10167}, {4189, 5927}, {4255, 9620}, {4314, 7956}, {4640, 31821}, {4855, 7580}, {5010, 12688}, {5691, 37600}, {5703, 5805}, {5722, 6927}, {5731, 17567}, {5842, 9955}, {6001, 31663}, {6449, 19068}, {6450, 19067}, {6668, 19925}, {6745, 31799}, {6862, 18480}, {6891, 18481}, {6924, 9940}, {7161, 14794}, {7280, 12680}, {7681, 31795}, {8726, 16417}, {9942, 31837}, {9957, 11491}, {10164, 33899}, {10304, 12246}, {10884, 16371}, {11012, 34790}, {11220, 37307}, {12114, 17502}, {12608, 28146}, {13411, 20420}, {16845, 38318}, {17558, 38108}, {17573, 37526}, {17580, 38122}, {17616, 37293}, {18242, 28160}, {18446, 37582}, {24299, 37251}, {25440, 31787}, {31231, 37605}, {31937, 33862}, {36999, 37692}, {37623, 37700}

X(40262) = midpoint of X(i) and X(j) for these {i,j}: {20, 22792}, {550, 6260}, {1385, 11500}, {1490, 34862}, {3579, 6261}, {6796, 37837}, {9942, 31837}, {37623, 37700}
X(40262) = reflection of X(6705) in X(3530)
X(40262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 936, 31658), (3, 1490, 34862), (3, 5720, 31445), (411, 5440, 31793), (1071, 6942, 5122), (3149, 24929, 5806), (6905, 33597, 942)

X(40263) = TETRAHEDRAL PROJECTION OF ABC TO EXTOUCH TRIANGLE

Barycentrics    a*((b+c)*a^5-(b-c)^2*a^4-2*(b^3+c^3)*a^3+2*(b^4+c^4)*a^2+(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*(b+c)^2) : :
X(40263) = 3*X(3)-4*X(5044) = 5*X(3)-4*X(31805) = 11*X(3)-12*X(33575) = 3*X(4)-X(3868) = 6*X(5)-5*X(5439) = 2*X(5)-3*X(5927) = 4*X(5)-3*X(10202) = 3*X(1071)-5*X(5439) = X(1071)-3*X(5927) = 2*X(1071)-3*X(10202) = X(3868)+3*X(12528) = 2*X(3868)-3*X(24474) = 2*X(5044)-3*X(5777) = 5*X(5044)-3*X(31805) = 11*X(5044)-9*X(33575) = 5*X(5439)-9*X(5927) = 10*X(5439)-9*X(10202) = 5*X(5777)-2*X(31805) = 11*X(5777)-6*X(33575) = 2*X(12528)+X(24474) = 11*X(31805)-15*X(33575)

X(40263) lies on the cubic K680 and these lines: {1, 1898}, {2, 13369}, {3, 9}, {4, 912}, {5, 1071}, {7, 6849}, {10, 37401}, {19, 37489}, {20, 31837}, {30, 72}, {33, 3157}, {35, 7701}, {37, 500}, {40, 18518}, {46, 18491}, {52, 916}, {58, 2341}, {63, 6985}, {65, 79}, {90, 37579}, {119, 12616}, {140, 10167}, {191, 210}, {200, 35448}, {222, 37696}, {226, 6841}, {329, 6851}, {354, 9955}, {355, 5836}, {376, 3876}, {381, 942}, {382, 517}, {389, 2808}, {392, 34773}, {405, 13151}, {474, 17616}, {495, 12711}, {496, 17625}, {511, 22036}, {515, 3878}, {518, 12699}, {546, 24475}, {550, 31835}, {568, 2262}, {631, 11220}, {651, 6198}, {758, 31673}, {908, 37356}, {943, 29007}, {944, 3890}, {946, 2801}, {952, 12672}, {960, 18481}, {990, 36754}, {1012, 33596}, {1062, 34048}, {1066, 2310}, {1158, 11499}, {1214, 35194}, {1385, 5259}, {1467, 38271}, {1478, 1858}, {1482, 9856}, {1519, 10943}, {1656, 9940}, {1657, 31793}, {1698, 40296}, {1699, 18544}, {1709, 11248}, {1745, 24430}, {1750, 5709}, {1824, 13754}, {1836, 18517}, {1837, 18516}, {1872, 12162}, {1902, 12293}, {1935, 3465}, {2096, 6885}, {2261, 37506}, {2772, 31728}, {2886, 18243}, {3057, 28204}, {3062, 6769}, {3073, 9355}, {3149, 24467}, {3219, 3651}, {3339, 18529}, {3359, 7992}, {3421, 12529}, {3487, 10394}, {3526, 11227}, {3555, 22791}, {3560, 18446}, {3601, 28444}, {3652, 4640}, {3654, 4662}, {3656, 34791}, {3678, 31730}, {3680, 8148}, {3681, 6361}, {3698, 13145}, {3753, 17653}, {3817, 12005}, {3818, 24476}, {3827, 34775}, {3843, 5806}, {3845, 24473}, {3874, 18483}, {3927, 37411}, {3931, 5492}, {4005, 16113}, {4084, 34648}, {4292, 28452}, {4297, 20117}, {4303, 7069}, {4420, 10308}, {4523, 29040}, {5045, 8581}, {5076, 31822}, {5251, 16132}, {5533, 12611}, {5534, 10679}, {5570, 10896}, {5587, 15071}, {5657, 9961}, {5658, 6825}, {5687, 17615}, {5690, 18908}, {5694, 14110}, {5696, 36973}, {5714, 6866}, {5728, 6147}, {5731, 31838}, {5758, 36991}, {5761, 37434}, {5768, 6893}, {5770, 6848}, {5787, 6928}, {5790, 9947}, {5811, 6827}, {5817, 6887}, {5840, 12665}, {5884, 19925}, {5885, 38140}, {5886, 12675}, {5891, 11573}, {5902, 18492}, {5918, 31663}, {5928, 18531}, {6000, 29958}, {6223, 6850}, {6245, 6882}, {6260, 6842}, {6261, 22758}, {6264, 10222}, {6684, 15064}, {6734, 37406}, {6831, 13257}, {6833, 37713}, {6845, 31053}, {6863, 9942}, {6864, 36996}, {6883, 10884}, {6899, 31018}, {6913, 37615}, {6914, 33597}, {6915, 13243}, {6918, 37612}, {6920, 18444}, {6948, 12246}, {6958, 18238}, {6990, 31019}, {7082, 7742}, {7411, 26878}, {7580, 26921}, {7688, 16143}, {7957, 28146}, {7989, 15016}, {8143, 37593}, {8227, 13373}, {8726, 30326}, {9578, 18545}, {9579, 18397}, {9614, 18543}, {9708, 18251}, {9844, 12433}, {9848, 31792}, {9928, 37194}, {9943, 26446}, {9957, 18526}, {10085, 10269}, {10391, 11374}, {10525, 12679}, {10728, 12532}, {10826, 18838}, {10855, 16863}, {10861, 17582}, {10864, 37611}, {10895, 13750}, {10914, 37705}, {10950, 34697}, {11230, 26201}, {11412, 31836}, {11496, 16112}, {11500, 32159}, {11529, 30290}, {12047, 26475}, {12259, 37368}, {12526, 12702}, {12572, 28459}, {12608, 26470}, {12709, 37730}, {12738, 16138}, {12773, 24928}, {13624, 18515}, {13743, 24929}, {14923, 34627}, {15030, 23154}, {15528, 23513}, {15800, 22793}, {16116, 20292}, {16465, 37447}, {17484, 37433}, {17637, 22798}, {17649, 33899}, {17745, 37509}, {18254, 38761}, {18440, 34381}, {18534, 37547}, {18541, 37544}, {19541, 37532}, {21669, 34772}, {23156, 31751}, {28164, 31806}, {28208, 31165}, {30304, 37534}, {31786, 31821}, {35631, 38485}, {36865, 37837}, {37251, 37582}

X(40263) = midpoint of X(i) and X(j) for these {i,j}: {4, 12528}, {5691, 5693}, {10728, 12532}, {12664, 18239}, {12688, 14872}, {18525, 40266}
X(40263) = reflection of X(i) in X(j) for these (i,j): (1, 31937), (3, 5777), (20, 31837), (65, 18480), (550, 31835), (946, 31871), (1071, 5), (1482, 9856), (1657, 31793), (3555, 22791), (3874, 18483), (4297, 20117), (5884, 19925), (5887, 31803), (10202, 5927), (10914, 37705), (11412, 31836), (11500, 32159), (12680, 1385), (12688, 31828), (12702, 34790), (14110, 5694), (15071, 34339), (17637, 22798), (17649, 33899), (17660, 12611), (18481, 960), (18526, 9957), (23156, 31751), (23340, 12672), (24473, 3845), (24474, 4), (24475, 546), (24476, 3818), (31730, 3678), (31786, 31821), (31788, 9947), (37562, 355), (37585, 72), (38761, 18254)
X(40263) = anticomplement of X(13369)
X(40263) = circumcenter of triangle A*B*C* as described at X(5905)
X(40263) = X(1071)-of-Johnson-triangle
X(40263) = intersection, other than A,B,C, of conics {{A, B, C, X(265), X(268)}} and {{A, B, C, X(282), X(2166)}}
X(40263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18540, 37234), (5, 1071, 10202), (40, 18528, 18518), (84, 5720, 3), (355, 6259, 6923), (936, 7171, 3), (1012, 37700, 33596), (1071, 5927, 5), (1490, 7330, 3), (1709, 17857, 11248), (1745, 24430, 37565), (3560, 18446, 24299), (3927, 37411, 37584), (4654, 10399, 942), (5534, 12705, 10679), (5587, 15071, 34339), (5787, 37822, 6928), (9940, 10157, 1656), (9947, 31788, 5790), (17781, 31938, 72)

X(40264) = TETRAHEDRAL PROJECTION OF ABC TO FUHRMANN TRIANGLE

Barycentrics    3*a^7-4*(b+c)*a^6-3*(b^2-4*b*c+c^2)*a^5+(3*b-2*c)*(2*b-3*c)*(b+c)*a^4-(3*b^2+11*b*c+3*c^2)*(b-c)^2*a^3+9*(b^2-c^2)*(b-c)*b*c*a^2+(3*b^2-7*b*c+3*c^2)*(b^2-c^2)^2*a-2*(b^2-c^2)^3*(b-c) : :
X(40264) = 3*X(3)-4*X(40260) = 5*X(3843)-4*X(40259) = 3*X(5658)-5*X(6256) = 4*X(12608)-3*X(40257)

X(40264) lies on these lines: {1, 4}, {3, 40260}, {56, 6246}, {355, 40256}, {382, 40265}, {2800, 18525}, {2829, 33899}, {3576, 5154}, {3843, 40259}, {4188, 5587}, {4297, 6958}, {5450, 6924}, {6796, 26086}, {6921, 10175}, {6931, 10165}, {6941, 36975}, {6959, 19925}, {6971, 18481}, {7354, 12832}, {9655, 31870}, {10265, 37002}, {10896, 11715}, {11681, 12119}, {12114, 37251}, {18242, 28186}, {19535, 38134}, {22793, 23960}, {28208, 37837}

X(40264) = reflection of X(i) in X(j) for these (i,j): (5450, 18480), (40256, 355), (40265, 382)

X(40265) = TETRAHEDRAL PROJECTION OF ABC TO 2nd FUHRMANN TRIANGLE

Barycentrics    3*a^7-2*(b+c)*a^6-5*(b^2+c^2)*a^5+(b+c)*(2*b^2+b*c+2*c^2)*a^4+(b^2+3*b*c+c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-2*(b^2-c^2)^3*(b-c) : :
X(40265) = 3*X(3)-4*X(40259) = 5*X(3843)-4*X(40260) = 4*X(12616)-3*X(40256)

X(40265) lies on these lines: {3, 40259}, {4, 7161}, {145, 515}, {382, 40264}, {516, 10525}, {946, 3612}, {3843, 40260}, {4305, 9580}, {5450, 28146}, {6796, 22793}, {6848, 18483}, {6890, 31730}, {7967, 16118}, {9579, 13607}, {9668, 31870}, {12699, 40257}, {18499, 31803}, {31671, 40249}

X(40265) = reflection of X(i) in X(j) for these (i,j): (6796, 22793), (40257, 12699), (40264, 382)

X(40266) = TETRAHEDRAL PROJECTION OF ABC TO INNER-GARCIA TRIANGLE

Barycentrics    a*((b+c)*a^5-(b^2+b*c+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+(2*b^4+2*c^4+b*c*(b^2-4*b*c+c^2))*a^2+(b^2-c^2)*(b-c)^3*a-(b^4-c^4)*(b^2-c^2)) : :
X(40266) = 3*X(3)-4*X(960) = 5*X(3)-4*X(9943) = 13*X(3)-12*X(10178) = 2*X(65)-3*X(381) = 3*X(381)-4*X(31937) = 3*X(392)-2*X(13369) = 4*X(942)-5*X(18493) = 2*X(960)-3*X(5887) = 5*X(960)-3*X(9943) = 13*X(960)-9*X(10178) = 3*X(1482)-2*X(3555) = X(3555)-3*X(12672) = 2*X(3893)-3*X(12645) = X(3893)-3*X(14872) = 3*X(5693)-X(5904) = 5*X(5887)-2*X(9943) = 13*X(5887)-6*X(10178) = 13*X(9943)-15*X(10178) = 3*X(13743)-2*X(17637) = X(25413)-4*X(31803)

X(40266) lies on these lines: {1, 399}, {3, 960}, {4, 14988}, {5, 10129}, {30, 3869}, {40, 5694}, {46, 37251}, {63, 13465}, {65, 381}, {72, 3426}, {78, 35000}, {221, 18447}, {355, 2800}, {382, 517}, {392, 13369}, {518, 8148}, {550, 9961}, {758, 12699}, {912, 1482}, {942, 18493}, {952, 3885}, {993, 3652}, {1071, 10246}, {1385, 15071}, {1537, 10943}, {1656, 34339}, {1657, 14110}, {1698, 13145}, {1768, 32612}, {1836, 37230}, {1854, 18455}, {1858, 37234}, {2099, 18761}, {2390, 18435}, {2778, 38790}, {2801, 37727}, {2818, 12162}, {3057, 18526}, {3340, 18540}, {3534, 31165}, {3579, 5692}, {3654, 3678}, {3655, 3884}, {3656, 3874}, {3697, 38066}, {3812, 5055}, {3827, 18440}, {3843, 7686}, {3868, 22791}, {3877, 34773}, {3878, 18481}, {3901, 31162}, {3940, 35448}, {4067, 28194}, {4084, 18483}, {5054, 25917}, {5248, 33858}, {5250, 37292}, {5450, 6265}, {5587, 35004}, {5603, 24475}, {5657, 31835}, {5687, 35460}, {5697, 28204}, {5730, 35459}, {5777, 5790}, {5884, 5886}, {5885, 8227}, {5902, 9955}, {5903, 18480}, {6326, 26285}, {6583, 11522}, {6825, 18231}, {6841, 39542}, {6882, 33899}, {6914, 21740}, {6958, 14647}, {6971, 12616}, {6980, 12608}, {7171, 15829}, {7330, 7971}, {7741, 11571}, {7986, 16466}, {7991, 18528}, {7992, 37611}, {7995, 37531}, {8715, 12738}, {9856, 18544}, {10106, 34698}, {10167, 31838}, {10540, 14529}, {10573, 18516}, {10620, 10693}, {10624, 34745}, {10680, 37252}, {10942, 13257}, {11230, 15016}, {11376, 11570}, {11682, 35457}, {11849, 37700}, {12331, 17857}, {12515, 25440}, {12526, 37584}, {12532, 14923}, {12675, 37624}, {12705, 37533}, {12709, 15934}, {12758, 37738}, {14269, 16616}, {15726, 17800}, {16200, 26200}, {18254, 37828}, {18446, 37621}, {18491, 37567}, {20117, 26446}, {22765, 24467}, {22793, 37625}, {24806, 35194}, {25414, 37708}, {31019, 33668}, {31788, 31821}, {34718, 34790}, {34880, 35451}

X(40266) = reflection of X(i) in X(j) for these (i,j): (3, 5887), (40, 5694), (65, 31937), (355, 31803), (382, 12688), (1482, 12672), (1657, 14110), (3534, 31165), (3868, 22791), (4084, 18483), (5691, 31828), (5903, 18480), (9961, 550), (10620, 10693), (11571, 12611), (12645, 14872), (12702, 72), (12773, 17638), (14923, 37705), (15071, 1385), (18481, 3878), (18525, 40263), (18526, 3057), (24474, 9856), (25413, 355), (31788, 31821), (37562, 5777), (37625, 22793)
X(40266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 31937, 381), (5777, 37562, 5790), (25917, 40296, 5054)

X(40267) = TETRAHEDRAL PROJECTION OF ABC TO GARCIA-REFLECTION TRIANGLE

Barycentrics    3*a^7-3*(b+c)*a^6-2*(2*b^2-7*b*c+2*c^2)*a^5+2*(2*b-c)*(b-2*c)*(b+c)*a^4-(b-c)^2*(b^2+8*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*(b^2+8*b*c+c^2)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(40267) = 3*X(3)-4*X(18242) = 3*X(381)-2*X(12114) = 2*X(1158)-3*X(5790) = 5*X(1656)-4*X(5450) = 3*X(3534)-4*X(6796) = 5*X(3617)-3*X(14646) = 3*X(5587)-2*X(34862) = 3*X(6256)-2*X(18242) = 3*X(10246)-4*X(12608) = 4*X(11249)-3*X(34740) = 3*X(14647)-4*X(18357)

X(40267) lies on these lines: {1, 22792}, {3, 119}, {4, 496}, {20, 17757}, {30, 10306}, {65, 971}, {84, 18480}, {153, 5687}, {221, 18340}, {381, 10199}, {382, 515}, {452, 38031}, {516, 12640}, {517, 18239}, {529, 8158}, {944, 1537}, {956, 37437}, {1012, 9654}, {1158, 5790}, {1317, 40272}, {1420, 1538}, {1479, 30283}, {1490, 28160}, {1532, 37002}, {1656, 5450}, {1657, 11500}, {1699, 9657}, {2098, 34789}, {2800, 12645}, {3295, 12115}, {3338, 10864}, {3421, 31777}, {3436, 6244}, {3534, 6796}, {3585, 22766}, {3617, 14646}, {3830, 12001}, {4297, 25681}, {5048, 12953}, {5073, 5842}, {5080, 37022}, {5229, 8727}, {5570, 12680}, {5587, 34862}, {5708, 5787}, {5779, 5794}, {5841, 37411}, {6001, 18525}, {6260, 18481}, {6850, 9708}, {6906, 31479}, {6918, 18516}, {6935, 10592}, {6941, 12248}, {7354, 19541}, {7373, 26333}, {7686, 18541}, {7952, 10731}, {7971, 28204}, {9613, 9856}, {9709, 31775}, {10246, 12608}, {10525, 40290}, {10572, 12678}, {10724, 25416}, {11249, 34740}, {11849, 18545}, {11928, 12761}, {12246, 33899}, {12330, 18518}, {12650, 22793}, {12666, 14988}, {12763, 26358}, {14647, 18357}, {16127, 18499}, {18237, 18519}, {21077, 28164}, {22758, 31493}, {25415, 36999}, {31822, 33697}

X(40267) = reflection of X(i) in X(j) for these (i,j): (1, 22792), (3, 6256), (84, 18480), (1657, 11500), (5787, 31673), (12246, 33899), (12650, 22793), (12773, 12761), (18481, 6260)
X(40267) = {X(4), X(3600)}-harmonic conjugate of X(7956)

X(40268) = TETRAHEDRAL PROJECTION OF ABC TO INNER-GREBE TRIANGLE

Barycentrics    a^2*(a^6+5*(b^2+c^2)*a^4+3*(b^4+6*b^2*c^2+c^4)*a^2-(b^2+c^2)*(9*b^4-2*b^2*c^2+9*c^4)) : :
X(40268) = 9*X(3)-8*X(5171) = 7*X(3)-8*X(9737) = 7*X(5171)-9*X(9737)

X(40268) lies on these lines: {3, 6}, {4, 10513}, {5, 14484}, {194, 14532}, {1270, 36709}, {1271, 36714}, {3640, 12697}, {3641, 12698}, {5590, 6202}, {5591, 6201}, {6214, 36711}, {6215, 36712}, {6226, 6319}, {6227, 6320}, {7725, 7733}, {7726, 7732}, {7758, 29181}, {7855, 36990}, {9748, 33185}, {10927, 18960}, {10928, 18959}, {12753, 13270}, {12754, 13269}, {12805, 13283}, {12806, 13282}, {15312, 34938}, {21542, 37521}

X(40268) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1160, 1161, 6), (1350, 9605, 3), (3095, 33878, 3), (5024, 5188, 3), (5864, 5865, 575), (9821, 10983, 3), (12305, 39649, 3), (12306, 39658, 3), (23115, 34815, 3), (30270, 30435, 3)

X(40269) = TETRAHEDRAL PROJECTION OF ABC TO HONSBERGER TRIANGLE

Barycentrics    a*(-a+b+c)*(2*(b+c)*a^3-(2*b^2-3*b*c+2*c^2)*a^2-2*(b^2-c^2)*(b-c)*a+(2*b^2+b*c+2*c^2)*(b-c)^2) : :
X(40269) = 5*X(7)-6*X(5902) = 3*X(7)-4*X(30329) = 5*X(390)-4*X(3057) = 3*X(390)-4*X(14100) = 2*X(3057)-5*X(10394) = 3*X(3057)-5*X(14100) = 5*X(3059)-6*X(4711) = 4*X(5045)-5*X(5728) = 16*X(5045)-15*X(11038) = 6*X(5728)-5*X(11025) = 4*X(5728)-3*X(11038) = 3*X(5902)-5*X(18412) = 9*X(5902)-10*X(30329) = 4*X(8581)-5*X(30340) = 3*X(10394)-2*X(14100) = 10*X(11025)-9*X(11038) = 8*X(15587)-9*X(38092) = 3*X(18412)-2*X(30329)

X(40269) lies on these lines: {1, 29007}, {7, 80}, {8, 3255}, {9, 2320}, {55, 4661}, {144, 145}, {497, 4430}, {971, 7672}, {1156, 10698}, {1445, 18450}, {1864, 3873}, {2099, 16112}, {2646, 15481}, {2771, 11041}, {3059, 4711}, {3100, 3751}, {3240, 7004}, {3241, 12532}, {3487, 5045}, {3616, 5825}, {3681, 5281}, {3811, 5223}, {3889, 18220}, {3957, 30223}, {4313, 5904}, {4323, 31803}, {4860, 27778}, {5086, 5832}, {5261, 14872}, {5265, 12675}, {5686, 7080}, {5704, 12005}, {5729, 7677}, {5731, 18397}, {5759, 35250}, {5779, 8543}, {7069, 29814}, {7226, 14547}, {7678, 20330}, {8581, 30340}, {10399, 11037}, {10950, 17768}, {11372, 11526}, {11502, 23958}, {12669, 12671}, {13243, 37541}, {14151, 19907}, {15587, 38092}, {17018, 24430}, {17620, 39779}, {17636, 20085}, {30312, 31657}

X(40269) = reflection of X(i) in X(j) for these (i,j): (7, 18412), (390, 10394)
X(40269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1864, 3873, 5274), (3681, 10391, 5281)

X(40270) = TETRAHEDRAL PROJECTION OF ABC TO INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics    2*a^4-(b+c)*a^3-(b^2+14*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(40270) = 3*X(1)+X(950) = 5*X(1)-X(10106) = 7*X(1)+X(10572) = 3*X(354)+X(10624) = X(942)+3*X(15170) = 5*X(950)+3*X(10106) = 7*X(950)-3*X(10572) = 3*X(3058)+X(4292) = 3*X(3058)+5*X(17609) = 5*X(3698)+3*X(34699) = X(4292)-5*X(17609) = X(4298)-3*X(5049) = 3*X(5045)-X(24470) = 3*X(5049)+X(15171) = 7*X(10106)+5*X(10572) = X(12575)-3*X(15170) = 3*X(15172)+X(24470)

X(40270) lies on these lines: {1, 4}, {3, 21625}, {8, 3646}, {10, 6767}, {30, 12577}, {40, 10580}, {354, 10624}, {355, 18530}, {390, 3333}, {496, 13405}, {514, 32195}, {516, 5045}, {517, 6744}, {519, 4015}, {527, 3881}, {528, 12436}, {596, 28557}, {938, 11362}, {942, 12575}, {999, 4314}, {1125, 3813}, {1210, 3303}, {1837, 8162}, {2177, 28018}, {3058, 4292}, {3085, 10172}, {3086, 10389}, {3244, 5289}, {3295, 6684}, {3296, 4312}, {3304, 4304}, {3698, 34699}, {3746, 3911}, {3748, 13411}, {3913, 9843}, {3946, 30148}, {3947, 9669}, {4114, 4338}, {4297, 7373}, {4298, 5049}, {4301, 15934}, {4460, 28644}, {4882, 17559}, {5082, 10582}, {5129, 9797}, {5436, 34625}, {5493, 5708}, {5542, 12699}, {5572, 18241}, {5703, 37704}, {5763, 10222}, {5850, 15008}, {6361, 10980}, {6601, 12864}, {6692, 8715}, {6738, 9957}, {6765, 26105}, {6766, 37423}, {7982, 14563}, {8227, 10578}, {8236, 10165}, {9589, 30350}, {9785, 11529}, {10122, 18839}, {10198, 24386}, {10385, 15803}, {10386, 12512}, {11036, 31162}, {11518, 30305}, {12005, 12710}, {12245, 30337}, {12563, 22791}, {12572, 34791}, {13374, 16201}, {15174, 25405}, {18391, 37556}, {18527, 19925}, {18990, 28172}, {19843, 38316}, {20008, 36922}, {28158, 31776}, {28164, 31795}, {28228, 31794}, {29655, 39559}, {31435, 36845}

X(40270) = midpoint of X(i) and X(j) for these {i,j}: {942, 12575}, {3244, 5795}, {4298, 15171}, {5045, 15172}, {5542, 15006}, {6738, 9957}, {12433, 31792}, {12572, 34791}
X(40270) = reflection of X(17706) in X(6744)
X(40270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 497, 21620), (1, 1058, 946), (1, 3488, 5882), (1, 9614, 3475), (390, 3333, 31730), (497, 21620, 18483), (938, 31393, 11362), (942, 15170, 12575), (946, 5882, 1490), (3058, 17609, 4292), (3295, 11019, 6684), (5049, 15171, 4298), (12710, 12915, 12005), (13464, 13607, 40257), (21625, 30331, 3)

X(40271) = TETRAHEDRAL PROJECTION OF ABC TO 1st JOHNSON-YFF TRIANGLE

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

X(40271) lies on these lines: {3, 12}, {4, 26437}, {226, 9657}, {515, 1836}, {535, 17532}, {912, 4338}, {993, 10895}, {2476, 5229}, {3585, 22758}, {3626, 37567}, {3822, 5204}, {4293, 6830}, {6224, 12831}, {18962, 37468}, {22791, 40272}, {31266, 37605}

X(40272) = TETRAHEDRAL PROJECTION OF ABC TO 2nd JOHNSON-YFF TRIANGLE

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

X(40272) lies on these lines: {3, 11}, {4, 26358}, {515, 2098}, {1317, 40267}, {1388, 9670}, {3419, 3626}, {3583, 11499}, {3825, 5217}, {4193, 5225}, {4294, 6941}, {5046, 8165}, {5687, 12764}, {5697, 5881}, {10087, 18542}, {10896, 25440}, {22791, 40271}

X(40272) = {X(1479), X(10090)}-harmonic conjugate of X(9669)

X(40273) = TETRAHEDRAL PROJECTION OF ABC TO K798E 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-3*(b^2-c^2)^2 : :
X(40273) = 7*X(3)-11*X(5550) = X(3)+3*X(9812) = X(3)-3*X(38034) = 7*X(4)+X(145) = 3*X(4)+X(1482) = 5*X(4)-X(18525) = 3*X(5)-X(40) = 7*X(5)-5*X(1698) = X(5)-3*X(1699) = 13*X(5)-7*X(9588) = 5*X(5)+X(9589) = 5*X(5)-3*X(26446) = 7*X(40)-15*X(1698) = X(40)-9*X(1699) = 5*X(40)+3*X(9589) = X(40)+3*X(12699) = 5*X(40)-9*X(26446) = 3*X(145)-7*X(1482) = 5*X(145)+7*X(18525) = X(145)-7*X(22791) = 5*X(1482)+3*X(18525) = X(1482)-3*X(22791) = X(18525)+5*X(22791)

X(40273) lies on these lines: {1, 3627}, {3, 5284}, {4, 145}, {5, 40}, {8, 3843}, {10, 3850}, {11, 3336}, {12, 37563}, {20, 18493}, {30, 551}, {46, 10593}, {79, 37722}, {140, 516}, {143, 2807}, {165, 632}, {226, 15172}, {354, 11544}, {355, 3845}, {381, 962}, {382, 5603}, {484, 7173}, {495, 12701}, {496, 1836}, {497, 6147}, {515, 3853}, {517, 546}, {519, 14893}, {547, 6684}, {548, 1125}, {549, 8227}, {550, 5886}, {944, 3830}, {1387, 7354}, {1479, 12433}, {1483, 3656}, {1484, 16159}, {1519, 20420}, {1656, 6361}, {1657, 3616}, {1770, 15325}, {2951, 38111}, {3091, 12702}, {3146, 10246}, {3241, 38335}, {3485, 9668}, {3526, 9778}, {3530, 11230}, {3534, 38022}, {3543, 10595}, {3545, 20070}, {3576, 15704}, {3579, 3628}, {3583, 37730}, {3614, 11010}, {3622, 33703}, {3624, 15712}, {3634, 12812}, {3636, 28172}, {3649, 4857}, {3652, 5536}, {3653, 19710}, {3654, 7989}, {3655, 33699}, {3671, 18527}, {3679, 23046}, {3828, 14892}, {3832, 5790}, {3839, 12245}, {3851, 5657}, {3856, 11362}, {3857, 7991}, {3858, 5587}, {3859, 28228}, {3861, 4301}, {3874, 31828}, {4292, 7743}, {4295, 9669}, {4338, 17728}, {5057, 24390}, {5066, 9956}, {5072, 9780}, {5073, 5731}, {5076, 10247}, {5119, 10592}, {5274, 5708}, {5443, 15338}, {5482, 29349}, {5493, 11231}, {5698, 31493}, {5714, 6767}, {5719, 12047}, {5734, 18526}, {5771, 6841}, {5805, 37534}, {5882, 33697}, {5903, 12019}, {6265, 13146}, {6284, 18393}, {6824, 31671}, {6915, 35000}, {6960, 38114}, {6972, 34126}, {7514, 9911}, {7956, 37356}, {7967, 17578}, {7982, 37705}, {8144, 34036}, {8226, 26878}, {8703, 38021}, {8727, 37532}, {9579, 11373}, {9580, 10386}, {9612, 37556}, {9626, 37947}, {9654, 30305}, {9856, 14988}, {10164, 16239}, {10165, 33923}, {10175, 12811}, {10222, 12102}, {10283, 11522}, {10591, 36279}, {10707, 14450}, {10944, 18513}, {10950, 18514}, {11012, 31649}, {11036, 18530}, {11246, 37720}, {11365, 12084}, {11372, 24467}, {11698, 14217}, {11735, 34584}, {12100, 12512}, {12101, 28204}, {12103, 13624}, {12108, 19862}, {12645, 14269}, {12679, 32214}, {12688, 24475}, {13373, 15726}, {13407, 15170}, {13451, 31760}, {13464, 28160}, {13607, 28208}, {13925, 31439}, {14869, 35242}, {14891, 19883}, {14986, 18541}, {15178, 28164}, {15326, 37735}, {15684, 38314}, {15686, 25055}, {15699, 30308}, {15759, 34638}, {15808, 31666}, {16118, 16173}, {16417, 26129}, {16881, 31728}, {17768, 24387}, {18492, 38138}, {18990, 20323}, {19541, 32141}, {19709, 34632}, {21669, 22765}, {21677, 31159}, {24703, 31419}, {26725, 31651}, {29309, 34466}, {33668, 37433}, {33814, 37251}, {34123, 37256}, {35272, 37435}, {36002, 37621}, {38038, 38602}, {38044, 38761}

X(40273) = midpoint of X(i) and X(j) for these {i,j}: {1, 3627}, {4, 22791}, {5, 12699}, {382, 34773}, {946, 22793}, {962, 5690}, {1483, 5691}, {1484, 34789}, {1537, 22938}, {3655, 33699}, {3656, 15687}, {3845, 31162}, {3874, 31828}, {4301, 18480}, {5882, 33697}, {7982, 37705}, {9812, 38034}, {10222, 31673}, {11698, 14217}, {12679, 32214}, {12688, 24475}, {33668, 37433}
X(40273) = reflection of X(i) in X(j) for these (i,j): (10, 3850), (140, 9955), (546, 18483), (548, 1125), (3579, 3628), (5901, 946), (9956, 12571), (12103, 13624), (18357, 546), (18480, 3861), (31673, 12102), (31728, 16881), (31730, 3530), (34638, 15759)
X(40273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 18493, 38028), (381, 962, 5690), (382, 5603, 34773), (496, 1836, 24470), (944, 10248, 3830), (1479, 39542, 12433), (1483, 15687, 5691), (1699, 12699, 5), (3091, 12702, 38042), (3579, 3817, 3628), (3656, 5691, 1483), (6284, 18393, 37737), (6361, 9779, 1656), (7965, 26470, 16160), (9580, 11374, 10386), (9956, 12571, 5066), (11230, 31730, 3530), (11522, 18481, 10283), (12047, 15171, 5719), (12699, 26446, 9589)

X(40274) = TETRAHEDRAL PROJECTION OF ABC TO 1st KENMOTU-FREE-VERTICES TRIANGLE

Barycentrics    a^2*(a^4+(b^2+c^2)*a^2-2*b^4-2*c^4-2*b^2*c^2-2*S*(b^2+c^2)) : :
X(40274) = 3*X(371)-2*X(1504)

X(40274) lies on these lines: {3, 6}, {385, 6312}, {488, 9541}, {637, 6561}, {639, 6565}, {1078, 6316}, {6813, 10576}, {8956, 33586}, {13828, 33273}, {32419, 32808}

X(40274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3102, 372), (39, 3098, 40275), (371, 6396, 32), (371, 11825, 372), (371, 35840, 6419), (1160, 39649, 19145), (6221, 12962, 371), (12305, 39648, 2459)

X(40275) = TETRAHEDRAL PROJECTION OF ABC TO 2nd KENMOTU-FREE-VERTICES TRIANGLE

Barycentrics    a^2*(a^4+(b^2+c^2)*a^2-2*b^4-2*c^4-2*b^2*c^2+2*S*(b^2+c^2)) : :
X(40275) = 3*X(372)-2*X(1505)

X(40275) lies on these lines: {3, 6}, {385, 6316}, {638, 6560}, {640, 6564}, {1078, 6312}, {6811, 10577}, {7484, 8956}, {13708, 33273}, {32421, 32809}

X(40275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3103, 371), (39, 3098, 40274), (372, 6200, 32), (372, 11824, 371), (372, 35841, 6420), (1161, 39658, 19146), (6398, 12969, 372), (12306, 39679, 2460)

X(40276) = TETRAHEDRAL PROJECTION OF ABC TO KOSNITA TRIANGLE

Barycentrics    (a^14-4*(b^2+c^2)*a^12+(5*b^4+8*b^2*c^2+5*c^4)*a^10-5*(b^2+c^2)*b^2*c^2*a^8-(5*b^8+5*c^8-(5*b^4-2*b^2*c^2+5*c^4)*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^4-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)*(b^4-b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2)*a^2 : :
X(40276) = 3*X(154)+X(40285) = 5*X(14530)-X(32321)

X(40276) lies on these lines: {3, 64}, {26, 10628}, {49, 34786}, {54, 18376}, {110, 34785}, {156, 5448}, {184, 7547}, {206, 18553}, {381, 10274}, {567, 18434}, {578, 18386}, {1503, 10224}, {1614, 7577}, {3043, 40242}, {3153, 9833}, {5878, 13619}, {6143, 14216}, {9704, 18405}, {10594, 11808}, {11459, 23358}, {12279, 13293}, {12281, 13289}, {14157, 22802}, {15060, 32391}, {15311, 15332}, {16000, 16868}, {17824, 18378}, {26883, 35480}

X(40276) = {X(6759), X(10539)}-harmonic conjugate of X(10282)

X(40277) = TETRAHEDRAL PROJECTION OF ABC TO MCCAY TRIANGLE

Barycentrics    5*a^8-6*(b^2+c^2)*a^6-(7*b^4+4*b^2*c^2+7*c^4)*a^4+(b^2+c^2)*(15*b^4-31*b^2*c^2+15*c^4)*a^2-(7*b^4-13*b^2*c^2+7*c^4)*(b^2-c^2)^2 : :
X(40277) = 4*X(5)-3*X(14161) = 8*X(546)+X(40279)

X(40277) lies on these lines: {2, 3}, {5476, 8787}, {7777, 12355}

X(40277) = {X(381), X(35930)}-harmonic conjugate of X(5066)

X(40278) = TETRAHEDRAL PROJECTION OF ABC TO MOSES-STEINER OSCULATORY TRIANGLE

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

X(40278) lies on these lines: {3, 3096}, {4, 11171}, {5, 7913}, {20, 6033}, {30, 7775}, {140, 3818}, {147, 9821}, {315, 35705}, {316, 1657}, {548, 7761}, {631, 6287}, {1153, 11645}, {1503, 10104}, {1513, 14880}, {2782, 8721}, {3095, 40236}, {3627, 5475}, {5167, 10575}, {7764, 29317}, {7802, 38744}, {7858, 22728}, {7898, 17538}, {8722, 32151}, {9744, 14881}, {12054, 13862}, {12252, 26316}, {13334, 40250}, {14692, 20081}, {16924, 22681}, {29012, 32190}, {33014, 38742}

X(40279) = TETRAHEDRAL PROJECTION OF ABC TO 1st NEUBERG TRIANGLE

Barycentrics    a^8-(b^4+c^4)*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(40279) = 5*X(3)-9*X(40248) = 8*X(546)-9*X(40277)

X(40279) lies on these lines: {2, 3}, {76, 6033}, {114, 7863}, {115, 14880}, {147, 13108}, {182, 7902}, {183, 32151}, {265, 38520}, {315, 32521}, {316, 9821}, {511, 7843}, {1078, 10722}, {2023, 7748}, {2549, 32516}, {2794, 10104}, {3095, 7858}, {3098, 5031}, {3398, 7856}, {3818, 24256}, {3934, 9996}, {4846, 30496}, {5092, 7861}, {5188, 13449}, {5475, 14881}, {6310, 13754}, {6321, 11257}, {7694, 7758}, {7728, 38523}, {7746, 12042}, {7749, 38749}, {7752, 35002}, {7756, 39809}, {7782, 38730}, {7790, 12054}, {7801, 22566}, {7823, 9301}, {7828, 26316}, {7854, 22505}, {7860, 33706}, {7936, 22712}, {7946, 12251}, {9744, 32448}, {9753, 32134}, {9863, 38744}, {9993, 18502}, {10242, 22676}, {10741, 38522}, {10742, 38521}, {10749, 38529}, {12188, 32528}, {12203, 14639}, {12918, 38525}, {13630, 40254}, {19127, 34981}, {22338, 38524}, {25157, 37824}, {25167, 37825}, {32152, 39838}, {38526, 38953}

X(40279) = orthocentroidal circle-inverse of-X(37243)
X(40279) = tetrahedral projection of ABC to 2nd Neuberg triangle
X(40279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 37243), (3, 381, 5025), (3, 33233, 549), (4, 16044, 381), (4, 37348, 5), (4, 40236, 382), (5, 33185, 547), (376, 33259, 3), (381, 7770, 5)

X(40280) = TETRAHEDRAL PROJECTION OF ABC TO ORTHOCENTROIDAL TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4-5*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-11*b^2*c^2+3*c^4)*a^2-(b^4+c^4)*(b^2-c^2)^2) : :
Barycentrics    (SB + SC) (S^2 + 9 R^2 SA - SA SW) : :
X(40280) = 5*X(2)-2*X(15060) = 4*X(2)-X(18435) = 7*X(3)+2*X(52) = 5*X(3)+4*X(389) = 2*X(3)+X(568) = 8*X(3)+X(6243) = X(3)+8*X(9729) = X(3)+2*X(9730) = 11*X(3)-2*X(10625) = 4*X(3)-X(13340) = 17*X(3)-8*X(13348) = 11*X(3)+16*X(15012) = 13*X(3)-4*X(15644) = 19*X(3)+8*X(16625) = X(3)-4*X(16836) = 7*X(3)-16*X(17704) = 4*X(3)+5*X(37481) = 10*X(3)-X(37484) = 5*X(52)-14*X(389) = 4*X(52)-7*X(568) = 16*X(52)-7*X(6243) = X(52)-7*X(9730) = 11*X(52)+7*X(10625) = 8*X(52)+7*X(13340) = 13*X(52)+14*X(15644) = X(52)+14*X(16836) = X(52)+8*X(17704) = 20*X(52)+7*X(37484) = 8*X(15060)-5*X(18435)

X(40280) lies on these lines: {2, 5655}, {3, 6}, {4, 7693}, {5, 7703}, {20, 12006}, {30, 5640}, {51, 3534}, {125, 10938}, {140, 10574}, {143, 3522}, {184, 17701}, {185, 3526}, {186, 34513}, {265, 18911}, {373, 381}, {376, 5946}, {399, 5651}, {546, 15028}, {547, 15305}, {548, 3567}, {549, 5890}, {550, 15043}, {631, 13630}, {632, 12111}, {974, 38794}, {1112, 35485}, {1154, 3524}, {1204, 34864}, {1511, 11003}, {1656, 15030}, {1657, 5462}, {2070, 35268}, {2781, 38064}, {2854, 11179}, {2979, 12100}, {3060, 8703}, {3066, 35237}, {3090, 13491}, {3146, 15026}, {3426, 5544}, {3523, 6102}, {3525, 5876}, {3528, 10263}, {3529, 10095}, {3530, 5889}, {3533, 14128}, {3543, 13364}, {3627, 15024}, {3628, 6241}, {3819, 15701}, {3830, 5943}, {3832, 32205}, {3845, 11451}, {3850, 11465}, {3851, 10575}, {3917, 15693}, {4550, 10620}, {4846, 7728}, {5012, 15035}, {5054, 5650}, {5055, 6000}, {5066, 11455}, {5068, 32137}, {5070, 12162}, {5072, 11381}, {5076, 14641}, {5446, 15696}, {5562, 15720}, {5891, 15082}, {5904, 15229}, {5913, 30515}, {6090, 18445}, {6101, 15717}, {6288, 6815}, {6293, 25563}, {6403, 37934}, {6644, 6800}, {6688, 16194}, {6776, 16270}, {7464, 15018}, {7496, 33533}, {7502, 15053}, {7574, 7706}, {7575, 15080}, {7722, 10294}, {7999, 12108}, {8717, 34417}, {9744, 12093}, {9781, 15704}, {9826, 20127}, {9833, 32184}, {10110, 17800}, {10254, 23515}, {10299, 10627}, {10303, 11591}, {10304, 13391}, {10540, 35259}, {10605, 32620}, {10653, 11624}, {10654, 11626}, {11204, 38633}, {11412, 15712}, {11424, 15047}, {11444, 14869}, {11799, 37648}, {11806, 15040}, {12045, 15703}, {12112, 16042}, {12308, 16187}, {13321, 15688}, {13451, 15686}, {13570, 35403}, {14093, 36987}, {14269, 14845}, {14389, 15122}, {14708, 18580}, {14831, 15700}, {15041, 16223}, {15055, 18570}, {15056, 16239}, {15695, 21849}, {16003, 24206}, {16222, 38788}, {16658, 23410}

X(40280) = midpoint of X(i) and X(j) for these {i,j}: {376, 11002}, {5890, 7998}, {13321, 15688}, {15045, 20791}, {15072, 16261}
X(40280) = reflection of X(i) in X(j) for these (i,j): (373, 5892), (381, 373), (5891, 15082), (7998, 549), (11002, 5946), (13321, 16226), (14269, 14845), (16261, 5), (23039, 7998)
X(40280) = Brocard circle-inverse of-X(37477)
X(40280) = X(16261)-of-Johnson-triangle
X(40280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 37477), (3, 182, 14805), (3, 389, 37484), (3, 568, 13340), (3, 9730, 568), (3, 15037, 13352), (3, 36752, 37472), (3, 36753, 37495), (3, 37481, 6243), (3, 37514, 13353), (52, 17704, 3), (140, 10574, 34783), (182, 37470, 3), (568, 9730, 37481), (568, 13340, 6243), (5092, 32110, 3), (5943, 14855, 3830), (9729, 16836, 9730), (9730, 16836, 3), (10575, 11695, 3851), (11465, 12279, 3850), (13340, 37481, 568), (30260, 30261, 566)

X(40281) = TETRAHEDRAL PROJECTION OF ABC TO 1st ORTHOSYMMEDIAL TRIANGLE

Barycentrics    ((b^2+c^2)*a^10-(b^2-c^2)^2*a^8-(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^6+(2*b^8+2*c^8-(7*b^4+22*b^2*c^2+7*c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(b^4+5*b^2*c^2+c^4)*a^2-(b^2-c^2)^2*(b^8+c^8+(b^4+4*b^2*c^2+c^4)*b^2*c^2))*a^2 : :
X(40281) = 3*X(3060)+X(14532) = X(5889)+3*X(11287) = 3*X(5946)-X(35930) = 3*X(7739)+X(37473) = 3*X(11286)-7*X(15043)

X(40281) lies on these lines: {3, 1180}, {30, 143}, {3060, 14532}, {5663, 40250}, {5889, 11287}, {5946, 35930}, {6102, 37242}, {7739, 37473}, {11286, 15043}, {15048, 19161}

X(40281) = midpoint of X(i) and X(j) for these {i,j}: {6102, 37242}, {15048, 19161}

X(40282) = TETRAHEDRAL PROJECTION OF ABC TO 1st PARRY TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^8+(5*b^4-18*b^2*c^2+5*c^4)*a^6-3*(b^2+c^2)*(3*b^4-8*b^2*c^2+3*c^4)*a^4+(4*b^8+4*c^8+b^2*c^2*(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2))*a^2-(b^2+c^2)*(b^8+c^8-(b^4+b^2*c^2+c^4)*b^2*c^2)) : :

X(40282) lies on these lines: {3, 74}, {351, 2854}, {511, 8644}, {542, 9125}, {2782, 9123}, {9129, 9142}, {9130, 9145}, {9156, 33962}, {35357, 39689}

X(40282) = reflection of X(40283) in X(351)
X(40282) = crosspoint of X(843) and X(14948)
X(40282) = crosssum of X(543) and X(5108)
X(40282) = X(351)-of-1st-Parry-triangle
X(40282) = {X(110), X(9215)}-harmonic conjugate of X(3)

X(40283) = TETRAHEDRAL PROJECTION OF ABC TO 2nd PARRY TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^8-(7*b^4-6*b^2*c^2+7*c^4)*a^6+3*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^4-(2*b^8+2*c^8+b^2*c^2*(11*b^4-18*b^2*c^2+11*c^4))*a^2-(b^2+c^2)*(b^8+c^8-b^2*c^2*(7*b^4-11*b^2*c^2+7*c^4))) : :

X(40283) lies on these lines: {3, 111}, {351, 2854}, {511, 647}, {543, 9189}, {1649, 5969}, {2502, 5467}, {2782, 9185}, {5106, 9177}, {5663, 9138}, {7664, 15000}, {9129, 9145}, {9130, 9142}

X(40283) = reflection of X(40282) in X(351)
X(40283) = crossdifference of every pair of points on line {X(1316), X(9125)}
X(40283) = X(351)-of-2nd-Parry-triangle
X(40283) = {X(111), X(9216)}-harmonic conjugate of X(3)

X(40284) = TETRAHEDRAL PROJECTION OF ABC TO SUBMEDIAL TRIANGLE

Barycentrics    a^2*(5*(b^2+c^2)*a^6-5*(3*b^4-4*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(15*b^4-62*b^2*c^2+15*c^4)*a^2-(5*b^4-12*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :
X(40284) = X(5)+15*X(5892) = 7*X(5)-15*X(6688) = 3*X(5)+5*X(9729) = X(5)-5*X(11695) = 3*X(376)+5*X(10110) = 5*X(389)+11*X(3525) = 7*X(389)+9*X(5650) = 9*X(389)+7*X(7999) = X(1657)+15*X(5943) = 9*X(3060)+7*X(15644) = X(3146)+15*X(16836) = 7*X(5892)+X(6688) = 9*X(5892)-X(9729) = 3*X(5892)+X(11695) = 9*X(6688)+7*X(9729) = 3*X(6688)-7*X(11695) = X(9729)+3*X(11695) = 9*X(9729)-X(13491) = 3*X(10110)-11*X(15024) = 5*X(11592)-9*X(12108)

X(40284) lies on these lines: {5, 2883}, {52, 5054}, {376, 10110}, {389, 3525}, {511, 11592}, {1657, 5943}, {3060, 3523}, {3146, 15028}, {5462, 12100}, {5876, 12045}, {5907, 15703}, {9730, 40247}, {10095, 12002}, {10124, 12006}, {10219, 13630}, {11465, 13474}, {11591, 15012}, {11793, 37481}, {12103, 13363}, {13382, 15045}, {15043, 15606}, {15718, 21849}

X(40285) = TETRAHEDRAL PROJECTION OF ABC TO TANGENTIAL TRIANGLE

Barycentrics    a^2*(a^14-5*(b^2+c^2)*a^12+3*(3*b^4+2*b^2*c^2+3*c^4)*a^10-(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^8-(5*b^8+5*c^8-2*b^2*c^2*(4*b^4-7*b^2*c^2+4*c^4))*a^6+3*(b^4-c^4)*(b^2-c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^4-(b^2-c^2)^2*(5*b^8+5*c^8+2*b^2*c^2*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2))*a^2+(b^4-c^4)*(b^2-c^2)^3*(b^4+6*b^2*c^2+c^4)) : :
X(40285) = 3*X(154)-4*X(40276)

X(40285) lies on these lines: {3, 64}, {6, 18383}, {155, 18400}, {161, 18436}, {381, 6145}, {394, 34785}, {546, 34117}, {1181, 7507}, {1503, 18569}, {1594, 11456}, {2393, 15083}, {3818, 19149}, {5878, 6240}, {6225, 12112}, {6293, 7517}, {7503, 32379}, {9833, 11441}, {10274, 37506}, {10628, 12310}, {10982, 18376}, {11591, 15577}, {12173, 13419}, {12324, 37119}, {15068, 34782}, {17824, 18405}, {18358, 34118}, {18445, 31724}, {23325, 36752}, {32767, 37514}, {34781, 37444}, {34786, 36747}

X(40285) = reflection of X(32321) in X(6759)
X(40285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1498, 18451, 6759), (17824, 18405, 36749)

X(40286) = TETRAHEDRAL PROJECTION OF ABC TO 1st TRI-SQUARES TRIANGLE

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

X(40286) lies on these lines: {30, 3068}, {187, 13846}, {385, 13657}, {597, 40287}, {1160, 7583}, {3070, 38425}, {7374, 13886}, {8960, 8980}, {8975, 18511}, {8976, 13638}, {8981, 35945}, {12602, 13879}, {13910, 18440}

X(40287) = TETRAHEDRAL PROJECTION OF ABC TO 2nd TRI-SQUARES TRIANGLE

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

X(40287) lies on these lines: {30, 3069}, {187, 13847}, {385, 13777}, {597, 40286}, {1161, 7584}, {3071, 38426}, {7000, 13939}, {10991, 13967}, {12601, 13933}, {13758, 13951}, {13949, 18509}, {13966, 35944}, {13972, 18440}

X(40288) = TETRAHEDRAL PROJECTION OF ABC TO 3rd TRI-SQUARES TRIANGLE

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

X(40288) lies on these lines: {371, 3629}, {1151, 1503}, {3529, 9540}, {12974, 13910}, {15815, 40289}

X(40289) = TETRAHEDRAL PROJECTION OF ABC TO 4th TRI-SQUARES TRIANGLE

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

X(40289) lies on these lines: {372, 3629}, {1152, 1503}, {3529, 13935}, {12975, 13972}, {15815, 40288}

X(40290) = TETRAHEDRAL PROJECTION OF ABC TO URSA MAJOR TRIANGLE

Barycentrics    2*a^10-4*(b+c)*a^9-(3*b^2-26*b*c+3*c^2)*a^8+2*(b+c)*(5*b^2-18*b*c+5*c^2)*a^7-2*(b^4+c^4+2*b*c*(7*b^2-23*b*c+7*c^2))*a^6-2*(b+c)*(3*b^4+3*c^4-2*b*c*(14*b^2-27*b*c+14*c^2))*a^5+4*(b^4+c^4-3*b*c*(b^2+6*b*c+c^2))*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*(b^2+8*b*c+c^2)*a^3+4*(b^2-c^2)^2*(5*b^2-12*b*c+5*c^2)*b*c*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :
X(40290) = 2*X(12114)-3*X(34697)

X(40290) lies on these lines: {2, 12114}, {8, 2829}, {11, 6256}, {84, 355}, {515, 3057}, {1476, 7681}, {1490, 34773}, {1532, 15866}, {7995, 37708}, {10525, 40267}, {10893, 14986}, {10947, 37001}, {11827, 17615}, {12629, 12700}, {12761, 38669}, {26492, 38319}

X(40291) = TETRAHEDRAL PROJECTION OF ABC TO WALSMITH TRIANGLE

Barycentrics    a^2*(4*a^10-6*(b^2+c^2)*a^8-2*(2*b^4-7*b^2*c^2+2*c^4)*a^6+4*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-(13*b^4-20*b^2*c^2+13*c^4)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*(-2*c^4+7*b^2*c^2-2*b^4)) : :

X(40291) lies on these lines: {5, 1511}, {23, 32235}, {25, 19140}, {74, 7556}, {110, 576}, {125, 15080}, {399, 37489}, {542, 1495}, {3629, 20772}, {3818, 32227}, {5092, 16165}, {5642, 10546}, {5663, 12105}, {7712, 9140}, {9976, 26864}, {10117, 12315}, {11060, 20998}, {11800, 32284}, {12584, 35259}, {13394, 20301}, {15035, 16187}, {15448, 32423}

X(40291) = midpoint of X(23) and X(32235)
X(40291) = {X(110), X(34417)}-harmonic conjugate of X(25556)

X(40292) = TETRAHEDRAL PROJECTION OF ABC TO INNER-YFF TRIANGLE

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2+b*c+c^2)*a^3+2*(b^3+c^3)*a^2+(b^4+c^4+2*b*c*(b^2+3*b*c+c^2))*a-(b^4-c^4)*(b-c)) : :
X(40292) = 3*X(3)+X(40294)

X(40292) lies on these lines: {1, 3}, {8, 20846}, {10, 11344}, {11, 6883}, {12, 6985}, {21, 3434}, {80, 9708}, {90, 31445}, {197, 11334}, {212, 1064}, {219, 2174}, {222, 4337}, {255, 4300}, {278, 378}, {347, 2071}, {387, 16452}, {388, 3651}, {390, 37106}, {405, 1479}, {411, 3085}, {474, 6690}, {497, 1006}, {498, 3149}, {528, 10058}, {601, 22361}, {674, 36740}, {859, 1486}, {920, 12711}, {943, 3485}, {954, 38454}, {956, 37286}, {958, 3419}, {960, 11517}, {984, 3465}, {993, 4304}, {1001, 30384}, {1011, 33137}, {1012, 4302}, {1030, 2256}, {1036, 1794}, {1125, 37282}, {1212, 1752}, {1253, 22350}, {1259, 12514}, {1260, 5692}, {1478, 7580}, {1496, 4303}, {1593, 1838}, {1621, 30305}, {1698, 16293}, {1714, 16287}, {1780, 4267}, {1858, 26921}, {2328, 4276}, {2550, 37306}, {2975, 4305}, {3058, 28466}, {3086, 6986}, {3145, 9798}, {3173, 13754}, {3560, 6284}, {3583, 6913}, {3585, 37411}, {3586, 5251}, {3600, 37105}, {3616, 37301}, {3624, 16410}, {4189, 20075}, {4292, 12511}, {4293, 7411}, {4299, 37426}, {4423, 23708}, {4996, 9802}, {5047, 10591}, {5218, 6905}, {5225, 6920}, {5248, 10624}, {5259, 9614}, {5428, 10386}, {5432, 6911}, {5441, 37292}, {5540, 15288}, {5687, 32157}, {6737, 8715}, {6825, 10523}, {6842, 10953}, {6906, 37000}, {6908, 10629}, {6909, 7676}, {6988, 10321}, {7489, 9668}, {7514, 15253}, {7741, 11108}, {7951, 19541}, {8192, 23850}, {9655, 16117}, {9818, 37695}, {10039, 11500}, {10198, 37229}, {10385, 21161}, {10590, 36002}, {11365, 13738}, {11496, 37302}, {11502, 26446}, {12114, 37287}, {12953, 18407}, {13730, 23361}, {13743, 18499}, {15175, 15909}, {16058, 33138}, {16173, 38031}, {16346, 19858}, {16418, 31140}, {17549, 34625}, {18481, 22759}, {18961, 37401}, {20834, 30366}, {22097, 30269}, {34868, 37246}, {35193, 38850}

X(40292) = intersection, other than A,B,C, of conics {{A, B, C, X(21), X(7742)}} and {{A, B, C, X(57), X(3422)}}
X(40292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 7742), (1, 5010, 15931), (3, 999, 37578), (3, 3295, 37579), (3, 26357, 8071), (35, 36, 30282), (35, 5119, 55), (55, 56, 24929), (55, 2099, 3295), (55, 3428, 1), (55, 5217, 32613), (1697, 10902, 11508), (3601, 11012, 22766), (5010, 14793, 3), (5010, 31508, 35), (5173, 24929, 1), (5217, 37564, 3), (11492, 11493, 34879), (14801, 14802, 35202), (26357, 37601, 3)

X(40293) = TETRAHEDRAL PROJECTION OF ABC TO OUTER-YFF TRIANGLE

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2-3*b*c+c^2)*a^3+2*(b^3+c^3)*a^2+(b^4+c^4-6*b*c*(b^2-b*c+c^2))*a-(b^4-c^4)*(b-c)) : :
X(40293) = 3*X(3)+X(40295)

X(40293) lies on these lines: {1, 3}, {25, 5121}, {47, 2122}, {90, 34862}, {100, 36977}, {104, 1788}, {279, 14878}, {378, 7744}, {382, 12764}, {388, 6940}, {404, 3436}, {405, 6691}, {474, 1329}, {497, 37403}, {499, 1012}, {529, 16371}, {601, 1450}, {920, 17649}, {956, 8256}, {993, 8582}, {1106, 22350}, {1398, 1845}, {1406, 34586}, {1413, 36052}, {1436, 1723}, {1479, 37022}, {1604, 1743}, {1737, 12114}, {1768, 18237}, {1770, 22753}, {1838, 37245}, {2829, 3149}, {2932, 5854}, {3086, 6909}, {3435, 36058}, {3560, 5433}, {3585, 6918}, {3824, 37692}, {3911, 5450}, {4188, 7080}, {4292, 21616}, {4295, 5253}, {4311, 6736}, {4316, 37411}, {4413, 10827}, {4996, 27383}, {5229, 6946}, {5267, 9843}, {5445, 9708}, {5687, 38455}, {6700, 37282}, {6745, 37309}, {6882, 18961}, {6891, 10523}, {6905, 12667}, {6906, 7288}, {6911, 7354}, {6926, 10629}, {6985, 15326}, {7951, 16408}, {8668, 17648}, {8679, 36741}, {9612, 37244}, {9709, 37710}, {10483, 19541}, {10526, 32554}, {10589, 21669}, {10590, 17531}, {11415, 34758}, {11500, 21578}, {11502, 18481}, {11570, 12635}, {12047, 25524}, {13587, 34619}, {13738, 27657}, {15654, 37257}, {15817, 36743}, {16417, 31141}, {16572, 32625}, {17606, 18761}, {17768, 37308}, {19537, 35023}, {20842, 22654}, {20849, 30362}, {22758, 24914}, {22759, 26446}, {28348, 28393}, {30283, 37706}, {36972, 37707}

X(40293) = X(1)-Gimel conjugate of-X(56)
X(40293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 36, 7742), (3, 56, 8069), (3, 1470, 8071), (36, 46, 56), (46, 30323, 2093), (55, 56, 24928), (56, 2098, 999), (56, 5204, 32612), (56, 10310, 1), (56, 37567, 10680), (57, 37561, 22766), (999, 35448, 2098), (1155, 34880, 11249), (1385, 13601, 1), (1420, 2077, 11508), (5126, 26285, 11510), (7280, 14793, 3), (10680, 35448, 8148), (32612, 37582, 56), (36279, 37535, 26437)

X(40294) = TETRAHEDRAL PROJECTION OF ABC TO INNER-YFF TANGENTS TRIANGLE

Barycentrics    a^2*(a^8-2*(b+c)*a^7-2*(b^2-b*c+c^2)*a^6+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^5-2*(b^2-4*b*c+c^2)*b*c*a^4-2*(b+c)*(3*b^4+3*c^4-2*b*c*(2*b^2-7*b*c+2*c^2))*a^3+2*(b^6+c^6-(b^4+c^4+3*b*c*(b^2-6*b*c+c^2))*b*c)*a^2+2*(b^2-c^2)*(b-c)*(b^4+10*b^2*c^2+c^4)*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2) : :
X(40294) = 3*X(3)-4*X(40292)

X(40294) lies on these lines: {1, 3}, {1597, 2969}, {3434, 37234}, {6883, 10596}, {6985, 10528}, {7580, 32213}, {10530, 37356}, {10915, 18518}, {18545, 37411}

X(40294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7688, 10269), (55, 3428, 13624), (10679, 35251, 55), (11248, 12703, 10679), (37533, 37584, 37544)

X(40295) = TETRAHEDRAL PROJECTION OF ABC TO OUTER-YFF TANGENTS TRIANGLE

Barycentrics    a^2*(a^8-2*(b+c)*a^7-2*(b^2-5*b*c+c^2)*a^6+6*(b^2-c^2)*(b-c)*a^5-6*(3*b^2-4*b*c+3*c^2)*b*c*a^4-2*(b+c)*(3*b^4+3*c^4-2*b*c*(6*b^2-11*b*c+6*c^2))*a^3+2*(b^6+c^6+(3*b^4+3*c^4-b*c*(11*b^2-26*b*c+11*c^2))*b*c)*a^2+2*(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(2*b-c)*(b-2*c))*a-(b^4-c^4)*(b^2-c^2)*(b-c)^2) : :
X(40295) = 3*X(3)-4*X(40293)

X(40295) lies on these lines: {1, 3}, {6985, 20076}, {10527, 37234}, {10530, 37406}, {10916, 18519}, {32214, 37022}

X(40295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 10310, 13624), (10680, 35252, 56), (10680, 35448, 1), (11249, 12704, 10680)

X(40296) = TETRAHEDRAL PROJECTION OF ABC TO 1st ZANIAH TRIANGLE

Barycentrics    a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+2*(b^2+b*c+c^2)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*(b-c)^2) : :
X(40296) = 3*X(3)+X(65) = 5*X(3)-X(14110) = X(40)+3*X(10202) = 3*X(40)+5*X(18398) = 5*X(65)+3*X(14110) = X(65)-3*X(34339) = 3*X(165)+5*X(15016) = 3*X(165)+X(24474) = 3*X(354)+X(12702) = 3*X(1385)-X(9957) = 5*X(1385)-X(10284) = X(1385)-3*X(11227) = 9*X(3576)-X(5697) = 3*X(3576)+X(37562) = X(3660)-3*X(18856) = X(5045)-3*X(9940) = 2*X(5045)-3*X(13373) = X(5045)+3*X(31787) = 5*X(5045)+3*X(31797) = 3*X(5049)-X(11278)

X(40296) lies on these lines: {1, 3}, {2, 31937}, {5, 9943}, {10, 13369}, {30, 3812}, {140, 6001}, {355, 4002}, {377, 18480}, {382, 5918}, {442, 22798}, {515, 3918}, {546, 15726}, {549, 960}, {550, 7686}, {631, 5887}, {912, 3678}, {971, 3826}, {975, 7986}, {1071, 3697}, {1158, 6883}, {1538, 6831}, {1656, 12688}, {1698, 40263}, {1737, 37401}, {1770, 28459}, {1827, 37414}, {2355, 37117}, {2478, 10940}, {2771, 3035}, {2800, 31838}, {2801, 4540}, {2818, 17704}, {3090, 9961}, {3524, 3869}, {3530, 14988}, {3555, 3654}, {3655, 10914}, {3698, 18525}, {3742, 22791}, {3753, 4190}, {3816, 9955}, {3827, 5092}, {3833, 18483}, {4067, 5884}, {5054, 25917}, {5439, 6899}, {5440, 33858}, {5690, 12675}, {5777, 6889}, {5790, 12680}, {5806, 28146}, {5818, 11220}, {5836, 34773}, {5880, 18482}, {5883, 31730}, {5886, 6890}, {6833, 9856}, {6836, 22793}, {6876, 9352}, {6903, 20292}, {6907, 10395}, {6911, 12520}, {6977, 12672}, {6989, 14647}, {7171, 18761}, {9942, 33899}, {10157, 31828}, {10172, 31871}, {10572, 28458}, {10693, 38728}, {10884, 11499}, {11112, 28208}, {12512, 31870}, {13374, 28174}, {15071, 31423}, {15693, 31165}, {17647, 28204}, {28154, 31822}, {28160, 31805}, {28168, 37468}, {28202, 37428}

X(40296) = midpoint of X(i) and X(j) for these {i,j}: {3, 34339}, {5, 9943}, {10, 13369}, {550, 7686}, {942, 3579}, {1385, 31788}, {5690, 12675}, {5836, 34773}, {5884, 31837}, {5885, 31663}, {9940, 31787}, {9942, 33899}, {10222, 31798}, {12512, 31870}, {13145, 13624}, {31786, 35004}
X(40296) = reflection of X(13373) in X(9940)
X(40296) = complement of X(31937)
X(40296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 46, 3579), (65, 3612, 9957), (165, 15016, 24474), (1385, 3579, 55), (1385, 18856, 9940), (3359, 8726, 10267), (3576, 16209, 3), (3579, 9940, 16216), (3660, 9957, 5045), (5054, 40266, 25917), (5884, 10164, 31837), (5902, 35242, 37585), (7686, 10178, 550), (11227, 31788, 1385), (12609, 37356, 9955), (12616, 37438, 9956), (17502, 35004, 31786), (18443, 37560, 11248), (30503, 37534, 11249)

leftri

Points associated with the power curve: X(40297)-X(40305)

rightri

This preamble is based on notes contributed by Suren, November 4, 2020.

In the plane of a triangle ABC, the locus of a point at : bt : ct (barycentrics [or trilinears]) as t varies through the real numbers is the power curve, PC(ABC), of ABC. (The term is introduced in Clark Kimberling, "Major Centers of Triangles," American Math. Monthly 104 (1997), 431-438.) Note that PC(ABC) passes through X(i) for i = 1,2,6,31,75,76, and that eliminating t shows that PC(ABC) is given by the equations

(log x)/(log a) = (log y)/(log b) = (log z)/(log c).

(Here, "log" signifies the natural logarithm, but equivalent equations result under change of base for "log".) Centers X(40297)-X(40305) involve the line tangent to PC at X(1), X(2), and X(6).

In general, the line tangent to the power curve at a point at : bt : ct has the direction (i.e., a point on the infinity line) given by

(a*c)t log(a/c) + (a*b)t log(a/b) : : ,

and the trilinear pole of that point is the point at log(c/b) : bt log(a/c) : ct log(b/a).

X(40297) = INFINITE POINT ON THE LINE TANGENT TO THE POWER CURVE AT X(1)

Barycentrics    a c log(a/c) + a b log(a/b) : :

X(40297) lies on this line: (30,511)

X(40297) = isogonal conjugate of X(40303)

X(40298) = INFINITE POINT ON THE LINE TANGENT TO THE POWER CURVE AT X(2)

Barycentrics    log(a^2/(b c)) : log(b^2/(c a)) : log(c^2/(a b))

X(40298) lies on this line: (30,511)

X(40298) = isogonal conjugate of X(40304)

X(40299) = INFINITE POINT ON THE LINE TANGENT TO THE POWER CURVE AT X(6)

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

X(40299) = isogonal conjugate of X(40304)

X(40300) = TRILINEAR POLE OF THE LINE TANGENT TO THE POWER CURVE AT X(1)

Barycentrics    a/log(c/b) : b/log(a/c): c/log(b/a)

X(40301) = TRILINEAR POLE OF THE LINE TANGENT TO THE POWER CURVE AT X(2)

Barycentrics    1/log(c/b) : 1/log(a/c): 1/log(b/a)

X(40301) lies on the Steiner circumellipse and these lines: {99, 40302}, {190, 40300}

X(40301) = isotomic conjugate of X(40327)
X(40301) = isotomic conjugate of the isogonal conjugate of X(40302)
X(40301) = X(40327)-cross conjugate of X(2)
X(40301) = X(31)-isoconjugate of X(40327)
X(40301) = cevapoint of X(2) and X(40327)
X(40301) = trilinear pole of line {2, 40298}
X(40301) = barycentric product X(i)*X(j) for these {i,j}: {75, 40300}, {76, 40302}
X(40301) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40327}, {40300, 1}, {40302, 6}

X(40302) = TRILINEAR POLE OF THE LINE TANGENT TO THE POWER CURVE AT X(6)

Barycentrics    a^2/log(c/b) : b^2/log(a/c): c^2/log(b/a)

X(40302) lies on the circumcircle and these lines: {}

X(40303) = ISOGONAL CONJUGATE OF X(40297)

Barycentrics    a/(c log(a/c) + b log(a/b)) : :

X(40303) lies on the circumcircle and these lines: {}

X(40303) = isogonal conjugate of X(40297)

X(40304) = ISOGONAL CONJUGATE OF X(40298)

Barycentrics    a^2/(log(a^2/(b c))) : :

X(40304) lies on the circumcircle and these lines: {}

X(40304) = isogonal conjugate of X(40298)

X(40305) = ISOGONAL CONJUGATE OF X(40299)

Barycentrics    1/(c^2 log(a/c) + b^2 log(a/b)) : :

X(40305) lies on the circumcircle and these lines: {}

X(40305) = isogonal conjugate of X(40299)

leftri

Points on Vu orthogonal conics: X(40306)-X(40315)

rightri

This preamble is based on notes contributed by Vu Thanh Tung, November 5, 2020.

In the plane of a triangle ABC, let P and U be points. Let L be the line through P perpendicular to line AU, and let A1 = L∩BC. Define B1 and C1 cyclically. Let L' be the line through U perpendicular to line AP. and A2 = L'∩BC. Define B2 and C2 cyclically. The six points A1, B1, C1, A2, B2, C2 lie on a conic, here named the Vu orthogonal conic of P and U, denoted by VOC(P,U).

Let V(P,U) denote the center, and T(P,U) the perspector, of VOC(P,U). Note that VOC(U,P) = VOC(P,U), V(U,P) = V(P,U), and T(U,P) = T(P,U).

See Vu Orthogonal Conic.

X(40306) = CENTER OF THE CONIC VOC(X(1),X(2))

Barycentrics    (2*a - b)*(2*a - c)*(b - c)*(9*a^4 - 15*a^3*b - 5*a^2*b^2 + a*b^3 + 6*b^4 - 15*a^3*c + 44*a^2*b*c - 3*a*b^2*c - 20*b^3*c - 5*a^2*c^2 - 3*a*b*c^2 + 20*b^2*c^2 + a*c^3 - 20*b*c^3 + 6*c^4) : :

X(40307) = PERSPECTOR OF THE CONIC VOC(X(1),X(2))

Barycentrics    (2*a - b)*(2*a - c)*(b - c)*(6*a^4 - 3*a^3*b - 2*a^2*b^2 - 3*a*b^3 + 6*b^4 - 18*a^3*c + 8*a^2*b*c + 8*a*b^2*c - 18*b^3*c + 10*a^2*c^2 + 13*a*b*c^2 + 10*b^2*c^2 - 10*a*c^3 - 10*b*c^3 + 4*c^4)*(6*a^4 - 18*a^3*b + 10*a^2*b^2 - 10*a*b^3 + 4*b^4 - 3*a^3*c + 8*a^2*b*c + 13*a*b^2*c - 10*b^3*c - 2*a^2*c^2 + 8*a*b*c^2 + 10*b^2*c^2 - 3*a*c^3 - 18*b*c^3 + 6*c^4) : :

X(40308) = CENTER OF THE CONIC VOC(X(1),X(3))

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

X(40309) = PERSPECTOR OF THE CONIC VOC(X(1),X(3))

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

X(40310) = CENTER OF THE CONIC VOC(X(1),X(6))

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

X(40311) = PERSPECTOR OF THE CONIC VOC(X(1),X(6))

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

X(40312) = CENTER OF THE CONIC VOC(X(2),X(3))

Barycentrics    (2*a^2 - b^2)*(b - c)*(b + c)*(2*a^2 - c^2)*(9*a^4 - 11*a^2*b^2 + 6*b^4 - 11*a^2*c^2 + 6*c^4) : :

X(40313) = PERSPECTOR OF THE CONIC VOC(X(2),X(3))

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

X(40314) = CENTER OF THE CONIC VOC(X(2),X(6))

Barycentrics    (b - c)*(b + c)*(-2*a^4 - 3*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2)*(2*a^4 - 2*a^2*b^2 + 3*a^2*c^2 + b^2*c^2 - c^4)*(-9*a^10 + 32*a^8*b^2 - 12*a^6*b^4 - 14*a^4*b^6 - 3*a^2*b^8 + 6*b^10 + 32*a^8*c^2 - 140*a^6*b^2*c^2 + 122*a^4*b^4*c^2 + 24*a^2*b^6*c^2 - 22*b^8*c^2 - 12*a^6*c^4 + 122*a^4*b^2*c^4 - 58*a^2*b^4*c^4 - 16*b^6*c^4 - 14*a^4*c^6 + 24*a^2*b^2*c^6 - 16*b^4*c^6 - 3*a^2*c^8 - 22*b^2*c^8 + 6*c^10) : :

X(40315) = PERSPECTOR OF THE CONIC VOC(X(2),X(6))

Barycentrics    (b - c)*(b + c)*(-2*a^4 - 3*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2)*(2*a^4 - 2*a^2*b^2 + 3*a^2*c^2 + b^2*c^2 - c^4)*(-6*a^12 + 26*a^10*b^2 + 8*a^8*b^4 - 12*a^6*b^6 - 6*a^4*b^8 - 14*a^2*b^10 + 4*b^12 + 17*a^10*c^2 + 6*a^8*b^2*c^2 - 414*a^6*b^4*c^2 + 40*a^4*b^6*c^2 + 45*a^2*b^8*c^2 - 14*b^10*c^2 - 30*a^8*c^4 - 32*a^6*b^2*c^4 + 828*a^4*b^4*c^4 + 40*a^2*b^6*c^4 - 6*b^8*c^4 + 38*a^6*c^6 - 32*a^4*b^2*c^6 - 414*a^2*b^4*c^6 - 12*b^6*c^6 - 30*a^4*c^8 + 6*a^2*b^2*c^8 + 8*b^4*c^8 + 17*a^2*c^10 + 26*b^2*c^10 - 6*c^12)*(6*a^12 - 17*a^10*b^2 + 30*a^8*b^4 - 38*a^6*b^6 + 30*a^4*b^8 - 17*a^2*b^10 + 6*b^12 - 26*a^10*c^2 - 6*a^8*b^2*c^2 + 32*a^6*b^4*c^2 + 32*a^4*b^6*c^2 - 6*a^2*b^8*c^2 - 26*b^10*c^2 - 8*a^8*c^4 + 414*a^6*b^2*c^4 - 828*a^4*b^4*c^4 + 414*a^2*b^6*c^4 - 8*b^8*c^4 + 12*a^6*c^6 - 40*a^4*b^2*c^6 - 40*a^2*b^4*c^6 + 12*b^6*c^6 + 6*a^4*c^8 - 45*a^2*b^2*c^8 + 6*b^4*c^8 + 14*a^2*c^10 + 14*b^2*c^10 - 4*c^12) : :

X(40316) = X(2)X(6)∩X(206)X(32220)

Barycentrics    2*a^8 - 3*a^6*b^2 - 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 - a^4*c^4 - 5*a^2*b^2*c^4 + 2*b^4*c^4 + 3*a^2*c^6 - c^8 : :
X(40316) = 3 X[15531] - 2 X[26926] = 3*(1 + J^2)*X[2] - (5 + 3*J^2)*X[6]

X(40316) lies on theser lines: {2, 6}, {206, 32220}, {895, 23300}, {1351, 37197}, {1353, 34148}, {1885, 3564}, {5889, 31829}, {6391, 11442}, {6467, 12058}, {6776, 30552}, {7507, 14914}, {14516, 34382}, {15116, 39125}, {15531, 26926}, {34622, 39899}

X(40316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 69, 26156}, {193, 20080, 6515}, {1992, 28408, 6}, {6515, 15066, 3580}

X(40317) = X(66)X(69)∩X(110)X(193)

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

X(40317 lies on these lines: on lines {25, 40316}, {66, 69}, {110, 193}, {524, 20987}, {1992, 19122}, {3580, 6391}, {3618, 5486}, {3620, 23293}, {5059, 5921}, {5181, 28408}, {6338, 9146}, {6467, 18911}, {8263, 26206}, {10602, 26156}, {11416, 28419}, {11441, 34380}, {11449, 14912}, {11898, 14516}, {14913, 27365}, {15107, 20080}

X(40317) = reflection of X(193) in X(1974)
X(40317) = {X(69),X(12272)}-harmonic conjugate of X(11442)

X(40318) = X(2)X(6)∩X(22)X(6467)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 8*a^2*b^2*c^2 - 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :
X(40318) = 4 X[1974] - 3 X[35264] = 6*X[2] - (7 + J^2)*X[6]

X(40318) lies on these lines: {2, 6}, {22, 6467}, {23, 9924}, {24, 34382}, {25, 6391}, {74, 38263}, {110, 19118}, {155, 6622}, {235, 3564}, {439, 6461}, {511, 1204}, {648, 8745}, {895, 32262}, {1176, 32621}, {1351, 1593}, {1353, 6823}, {1609, 4558}, {1843, 32127}, {1974, 8681}, {1995, 14913}, {2207, 6392}, {2393, 35219}, {2854, 20987}, {2916, 8547}, {3003, 9723}, {3053, 5866}, {3060, 12167}, {3089, 17836}, {3092, 12222}, {3093, 12221}, {3167, 19122}, {3448, 32276}, {5013, 34883}, {5050, 34148}, {5093, 11459}, {5157, 22829}, {6776, 37201}, {6800, 15531}, {7387, 12283}, {7391, 15583}, {7494, 17040}, {7754, 8743}, {9544, 19132}, {9707, 19154}, {10602, 12220}, {10607, 35296}, {10996, 14912}, {11456, 39899}, {11477, 12086}, {11482, 15801}, {12310, 32248}, {15073, 37488}, {15262, 32001}, {16196, 34380}, {18931, 37498}, {19131, 32284}, {22241, 30435}, {33748, 37514}

X(40318) = reflection of X(20806) in X(6)
X(40318) = X(63)-isoconjugate of X(15591)
X(40318) = crosssum of X(647) and X(6388)
X(40318) = polar conjugate of isogonal conjugate of X(41619)
X(40318) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 15591}, {15261, 6391}
X(40318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 69, 26206}, {6, 193, 1993}, {25, 6391, 12272}, {69, 193, 40316}, {69, 26206, 15066}, {193, 37784, 6}, {1351, 8548, 39588}, {1993, 3580, 15066}, {3580, 40316, 69}, {10602, 37491, 12220}, {15531, 19121, 19459}, {19118, 19588, 110}, {19121, 19459, 6800}

X(40319) = X(3)X(6391)∩X(20)X(98)

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

X(40319) lies on the cubic K1164 and these lines: {3, 6391}, {20, 98}, {25, 1611}, {32, 11326}, {184, 682}, {187, 2353}, {237, 33581}, {1402, 38252}, {1799, 6340}, {1885, 5203}, {3425, 13335}, {3455, 5206}, {5023, 9924}, {5139, 15591}, {8408, 21642}, {8420, 21643}, {8884, 34208}, {9292, 9306}, {10316, 14908}, {23099, 39201}, {27364, 34449}, {30739, 40102}

X(40319) = isogonal conjugate of the anticomplement of X(22401)
X(40319) = isogonal conjugate of the isotomic conjugate of X(6391)
X(40319) = isogonal conjugate of the polar conjugate of X(8770)
X(40319) = X(i)-cross conjugate of X(j) for these (i,j): {577, 184}, {1084, 647}
X(40319) = X(i)-isoconjugate of X(j) for these (i,j): {4, 18156}, {63, 21447}, {75, 6353}, {92, 193}, {158, 6337}, {264, 1707}, {286, 4028}, {318, 17081}, {561, 19118}, {811, 3566}, {1969, 3053}, {3798, 6335}, {5139, 24037}, {6521, 10607}, {17876, 18020}
X(40319) = crosspoint of X(6391) and X(8770)
X(40319) = crosssum of X(i) and X(j) for these (i,j): {4, 6392}, {193, 6353}
X(40319) = barycentric product X(i)*X(j) for these {i,j}: {3, 8770}, {6, 6391}, {32, 6340}, {48, 8769}, {63, 38252}, {184, 2996}, {394, 14248}, {577, 34208}, {647, 3565}, {3049, 35136}, {14533, 27364}
X(40319) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 21447}, {32, 6353}, {48, 18156}, {184, 193}, {577, 6337}, {1084, 5139}, {1501, 19118}, {2200, 4028}, {2996, 18022}, {3049, 3566}, {3565, 6331}, {6340, 1502}, {6391, 76}, {8769, 1969}, {8770, 264}, {9247, 1707}, {14248, 2052}, {14575, 3053}, {14585, 3167}, {23200, 32459}, {23606, 10607}, {34208, 18027}, {38252, 92}

X(40320) = X(3)X(6)∩X(112)X(3542)

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

X(40320) lies on the cubic K1164 and these lines: {3, 6}, {112, 3542}, {115, 31725}, {206, 682}, {230, 235}, {237, 1660}, {439, 4558}, {468, 17409}, {1611, 36417}, {1627, 7493}, {1656, 18373}, {1885, 10311}, {7735, 37201}, {7755, 19220}, {8770, 21313}, {10313, 30552}, {11326, 19136}, {14908, 20960}, {14910, 37460}, {34481, 37973}

X(40320) = X(6353)-Ceva conjugate of X(1974)
X(40320) = X(304)-isoconjugate of X(15591)
X(40320) = polar conjugate of isotomic conjugate of X(41619)
X(40320) = barycentric product X(6353)*X(15261)
X(40320) = barycentric quotient X(i)/X(j) for these {i,j}: {1974, 15591}, {15261, 6340}
X(40320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 187, 10316}, {32, 3053, 571}, {571, 3003, 5063}

X(40321) = X(3)X(69)∩X(25)X(15591)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^6 - 5*a^4*b^2 - 5*a^2*b^4 + b^6 - 5*a^4*c^2 + 18*a^2*b^2*c^2 - b^4*c^2 - 5*a^2*c^4 - b^2*c^4 + c^6) : :

X(40321) lies on the cubic K1164 and these lines: {3, 69}, {25, 15591}, {154, 5023}, {187, 20993}, {237, 1661}, {1593, 9756}, {1974, 3053}, {2996, 33974}, {3515, 14900}, {10602, 22401}

X(40321) = X(6353)-Ceva conjugate of X(6)
X(40321) = crosssum of X(525) and X(5139)
X(40321) = crossdifference of every pair of points on line {2489, 14341}
X(40321) = {X(3),X(682)}-harmonic conjugate of X(19459)

X(40322) = X(6)X(6337)∩X(32)X(3167)

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

X(40322) lies on the cubics K1047 and K1164, and also on these lines: {6, 6337}, {32, 3167}, {1498, 9431}, {1611, 2207}, {1974, 3053}, {2129, 21775}, {3224, 17811}, {5013, 39238}, {5023, 6091}

X(40322) = isogonal conjugate of X(6392)
X(40322) = isogonal conjugate of the anticomplement of X(3926)
X(40322) = isogonal conjugate of the isotomic conjugate of X(6339)
X(40322) = isotomic conjugate of the polar conjugate of X(15369)
X(40322) = X(30558)-Ceva conjugate of X(3)
X(40322) = X(394)-cross conjugate of X(6)
X(40322) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6392}, {2, 33781}, {4, 2128}, {6, 33787}, {19, 19583}, {75, 1611}, {92, 19588}, {158, 6461}, {811, 2519}, {1096, 6338}
X(40322) = cevapoint of X(9427) and X(39201)
X(40322) = crosspoint of X(8770) and X(39128)
X(40322) = crosssum of X(i) and X(j) for these (i,j): {193, 18287}, {1611, 19588}, {6462, 6463}
X(40322) = barycentric product X(i)*X(j) for these {i,j}: {6, 6339}, {63, 2129}, {69, 15369}, {8770, 30558}
X(40322) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33787}, {3, 19583}, {6, 6392}, {31, 33781}, {32, 1611}, {48, 2128}, {184, 19588}, {394, 6338}, {577, 6461}, {2129, 92}, {3049, 2519}, {6339, 76}, {15369, 4}

X(40323) = X(6)X(6387)∩X(25)X(15591)

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

X(40323) lies on the conic {{A,B,C,X(2)X(6)}}, the cubic K1164, and on these lines: {6, 6387}, {25, 15591}, {30, 36616}, {111, 1370}, {468, 40144}, {1368, 8770}, {1660, 1976}, {2987, 37669}, {3291, 13854}, {8749, 38282}, {18928, 30535}, {21448, 31255}

X(40323) = isotomic conjugate of the polar conjugate of X(15591)
X(40323) = X(i)-cross conjugate of X(j) for these (i,j): {5139, 523}, {22401, 6}
X(40323) = cevapoint of X(647) and X(6388)
X(40323) = barycentric product X(69)*X(15591)
X(40323) = barycentric quotient X(15591)/X(4)

X(40324) = X(25)X(19583)∩X(69)X(15369)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 5*a^4*c^2 + 18*a^2*b^2*c^2 - 5*b^4*c^2 - 5*a^2*c^4 - 5*b^2*c^4 + c^6)*(a^6 - 5*a^4*b^2 - 5*a^2*b^4 + b^6 - a^4*c^2 + 18*a^2*b^2*c^2 - 5*b^4*c^2 - a^2*c^4 - 5*b^2*c^4 + c^6) : :

X(40324) lies on the cubic K1164 and these lines: {25, 19583}, {69, 15369}, {1611, 36417}, {1974, 8780}, {5020, 15261}

X(40325) = isogonal conjugate of the anticomplement of X(6391)
X(40325) = trilinear pole of line {2519, 20186}

X(40325) = X(4)X(69)∩X(25)X(1611)

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

X(40325) lies on the cubic K1165 and these lines: {4, 69}, {25, 1611}, {51, 460}, {132, 235}, {232, 11325}, {427, 30749}, {682, 1196}, {1974, 2207}, {2386, 3767}, {2489, 23099}, {2971, 3199}, {2996, 12272}, {3089, 9752}, {3853, 16983}, {3917, 7784}, {5254, 6467}, {5395, 5640}, {6392, 8681}, {6525, 6620}, {8754, 27376}, {9822, 32971}, {10151, 11397}, {11574, 32974}, {12220, 32982}

X(40325) = isotomic conjugate of the isogonal conjugate of X(3080)
X(40325) = polar conjugate of the isotomic conjugate of X(1196)
X(40325) = orthic isogonal conjugate of X(5254)
X(40325) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 5254}, {107, 2489}
X(40325) = X(255)-isoconjugate of X(683)
X(40325) = crosspoint of X(4) and X(2207)
X(40325) = crosssum of X(3) and X(3926)
X(40325) = crossdifference of every pair of points on line {3049, 4143}
X(40325) = barycentric product X(i)*X(j) for these {i,j}: {4, 1196}, {19, 17872}, {25, 5254}, {76, 3080}, {112, 12075}, {393, 6467}, {682, 2052}, {1096, 18671}, {1368, 2207}, {1824, 16716}, {6524, 22401}
X(40325) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 683}, {682, 394}, {1196, 69}, {3080, 6}, {5254, 305}, {6467, 3926}, {12075, 3267}, {17872, 304}, {22401, 4176}
X(40325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2971, 27369, 3199}, {6291, 6406, 12294}

X(40326) = X(2)X(6)∩X(4)X(8770)

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

X(40326) lies on the cubic K1165 and these lines: {2, 6}, {4, 8770}, {25, 15591}, {30, 34481}, {32, 6677}, {111, 34603}, {126, 19568}, {187, 10154}, {427, 3291}, {428, 16317}, {468, 17409}, {574, 7734}, {1194, 9607}, {1196, 1368}, {1513, 2883}, {1691, 10192}, {2056, 8550}, {3053, 6353}, {3767, 30771}, {3787, 6388}, {5020, 7745}, {5023, 10565}, {5475, 10128}, {6340, 6392}, {6393, 35294}, {6791, 21969}, {8889, 13881}, {9729, 37451}, {31255, 40126}, {37990, 39576}

X(40326) = orthic-isogonal conjugate of X(6467)
X(40326) = X(4)-Ceva conjugate of X(6467)
X(40326) = crosspoint of X(4) and X(21447)
X(40326) = barycentric product X(i)*X(j) for these {i,j}: {193, 5254}, {1368, 6353}, {17872, 18156}, {21447, 22401}
X(40326) = barycentric quotient X(i)/X(j) for these {i,j}: {1196, 8770}, {1368, 6340}, {5254, 2996}, {6467, 6391}, {17872, 8769}
X(40326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1611, 230}, {1196, 1368, 5254}

X(40327) = SS(a → log(b/c)) OF X(1)

Barycentrics    log(b/c) : log(c/a) : log(a/b)
Barycentrics    log(c/b) : log(a/c) : log(b/a)
Barycentrics    log b - log c : log c - log a : log a - log b
Trilinears    b c log(b/c) : c a log(c/a) : a b log(a/b)

See the preamble just before X(40297). For symbolic substitutions SS(# → #), see X(3221).

X(40327) lies on this line: X(30)X(511)

X(40327) = isotomic conjugate of X(40301)

leftri

Osiris points: X(40328)-X(40336)

rightri

This preamble and centers X(40328)-X(40336) were contributed by César Eliud Lozada, November 9, 2020.

Let ABC be a triangle, P a point and Q the isotomic conjugate of P. Denote by A' the centroid of the quadrangle BCPQ and define B' and C'. cyclically. Then AA', BB', CC' concur at a point O(P)=O(Q), here named the Osiris point of P.

For P=x:y:z (barycentrics), O(P) = (y^2+5*y*z+z^2)*x+2*(y+z)*(x^2+y*z) : :

The appearance of (i, j) in the following list means that the Osiris point of X(i) is X(j):
(1, 40328), (2, 2), (3, 40329), (4, 40330), (5, 40331), (6, 40332), (7, 40333), (8, 40333), (13, 40334), (14, 40335), (69, 40330), (75, 40328), (76, 40332), (95, 40331), (98, 40336), (99, 523), (190, 514), (264, 40329), (290, 511), (298, 40334), (299, 40335), (325, 40336), (648, 525), (664, 522), (666, 918), (668, 513), (670, 512), (671, 524), (886, 888), (889, 891), (892, 690), (903, 519), (1121, 527), (1494, 30), (2481, 518), (2966, 2799), (3225, 698), (3226, 726), (3227, 536), (3228, 538), (4555, 900), (4562, 812), (4569, 3900), (4577, 826), (4586, 824), (4597, 4777), (5641, 542), (6189, 3414), (6190, 3413), (6540, 4977), (6606, 6362), (6635, 6550), (6648, 3910), (11117, 532), (11118, 533), (14616, 758), (14728, 33906), (14970, 732), (15164, 2574), (15165, 2575), (16077, 9033), (18025, 516), (18026, 521), (18816, 517), (18821, 528), (18822, 537), (18823, 543), (18824, 696), (18826, 714), (18827, 740), (18828, 782), (18829, 804), (18830, 4083), (18831, 6368), (23895, 23870), (23896, 23871), (32036, 23872), (32037, 23873), (32038, 23880), (32039, 23886), (32041, 4762), (32042, 4802), (34393, 515), (35136, 3566), (35137, 7927), (35138, 3906), (35139, 526), (35140, 1503), (35141, 17768), (35142, 3564), (35143, 35101), (35144, 35102), (35145, 8680), (35146, 5969), (35147, 2787), (35148, 2786), (35149, 2792), (35150, 2784), (35151, 2783), (35152, 2795), (35153, 2796), (35154, 2785), (35155, 35103), (35156, 8674), (35157, 6366), (35158, 5845), (35159, 35104), (35160, 5853), (35162, 17770), (35164, 2801), (35168, 545), (35170, 4715), (35171, 3887), (35172, 9055), (35174, 3738), (35175, 2802), (35179, 1499), (35181, 4160), (39626, 39624)

If P or Q lie on the cubic K953, then its Osiris point lies on the Euler line of ABC.

The mapping O takes certain cubics onto lines: O(K296) = X(1)X(2), O(K185) = X(2)X(6), O(K953) = X(2)X(3). (Peter Moses, November 10, 2020)

X(40328) = OSIRIS POINT OF X(1)

Barycentrics    2*(b+c)*a^2+(b^2+5*b*c+c^2)*a+2*(b+c)*b*c : :
X(40328) = 3*X(1)+2*X(3696) = X(1)+4*X(3739) = 6*X(2)-X(984) = 9*X(2)-4*X(3842) = 3*X(2)+2*X(24325) = 9*X(2)+X(24349) = 4*X(2)+X(31178) = 3*X(984)-8*X(3842) = X(984)+4*X(24325) = 3*X(984)+2*X(24349) = 2*X(984)+3*X(31178) = 7*X(984)-2*X(31302) = X(3696)-6*X(3739) = 2*X(3842)+3*X(24325) = 4*X(3842)+X(24349) = 16*X(3842)+9*X(31178) = 6*X(24325)-X(24349) = 8*X(24325)-3*X(31178) = 14*X(24325)+X(31302) = 4*X(24349)-9*X(31178) = 7*X(24349)+3*X(31302)

X(40328) lies on these lines: {1, 3696}, {2, 38}, {7, 24697}, {10, 4684}, {36, 19287}, {37, 3624}, {75, 1125}, {86, 16825}, {142, 32784}, {145, 4732}, {192, 5550}, {238, 10436}, {312, 25501}, {496, 21926}, {518, 1698}, {631, 29054}, {726, 4687}, {740, 3616}, {742, 4798}, {872, 17749}, {1001, 4436}, {1213, 25557}, {1699, 30271}, {1757, 17259}, {1921, 31997}, {3210, 10180}, {3242, 36531}, {3636, 4709}, {3751, 16832}, {3773, 17244}, {3775, 29576}, {3790, 29581}, {3797, 29612}, {3826, 29659}, {3836, 27147}, {3976, 19858}, {4026, 34824}, {4032, 7288}, {4038, 5271}, {4169, 19963}, {4357, 39580}, {4359, 17592}, {4384, 4649}, {4389, 25354}, {4393, 5625}, {4441, 30571}, {4472, 24357}, {4648, 32846}, {4655, 26806}, {4664, 19883}, {4670, 16468}, {4675, 33082}, {4688, 15569}, {4698, 34595}, {4704, 28516}, {4758, 4989}, {4968, 29982}, {4974, 17379}, {5251, 27471}, {5257, 24231}, {5263, 24331}, {5333, 17017}, {5904, 13476}, {6536, 33146}, {7201, 11375}, {10165, 30273}, {10453, 27798}, {11230, 20430}, {14005, 28082}, {16408, 34247}, {16476, 17175}, {16478, 25526}, {16610, 29825}, {16826, 27474}, {17245, 29674}, {17260, 32935}, {17278, 29633}, {17303, 29637}, {17331, 17771}, {17368, 31289}, {17391, 17772}, {17398, 29646}, {18157, 33945}, {19701, 29821}, {19732, 32913}, {19808, 29642}, {19822, 33158}, {21020, 29814}, {24174, 24778}, {24199, 33149}, {24369, 24440}, {25507, 29644}, {26102, 31993}, {26115, 27311}, {26363, 28755}, {26627, 32917}, {26724, 29647}, {27475, 29604}, {29981, 31339}, {31336, 31347}, {33076, 39581}

X(40328) = midpoint of X(3616) and X(4699)
X(40328) = reflection of X(i) in X(j) for these (i, j): (1698, 31238), (4687, 19862)
X(40328) = intersection, other than A,B,C, of conics {{A, B, C, X(291), X(10013)}} and {{A, B, C, X(335), X(39711)}}
X(40328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24325, 984), (2, 24349, 3842), (984, 24325, 31178), (1698, 20195, 31252), (3842, 24325, 24349), (3842, 24349, 984)

X(40329) = OSIRIS POINT OF X(3)

Barycentrics    2*(b^2+c^2)*a^10-(7*b^4+9*b^2*c^2+7*c^4)*a^8+(b^2+c^2)*(9*b^4-b^2*c^2+9*c^4)*a^6-5*(b^6-c^6)*(b^2-c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(b^2-3*b*c+c^2)*(b^2+3*b*c+c^2)*a^2+2*(b^2-c^2)^4*b^2*c^2 : :
X(40329) = 9*X(2)-4*X(10003) = 6*X(2)-X(30258) = X(3)+4*X(14767) = 3*X(3)+2*X(39530) = 4*X(140)+X(264) = 2*X(216)-7*X(3526) = X(3164)-11*X(3525) = 8*X(10003)-3*X(30258) = 6*X(14767)-X(39530)

X(40329) lies on these lines: {2, 1972}, {3, 14767}, {140, 264}, {216, 3526}, {233, 40107}, {511, 1656}, {631, 32428}, {3164, 3525}

X(40330) = OSIRIS POINT OF X(4)

Barycentrics    a^6-(b^2+c^2)*a^4+(3*b^2+c^2)*(b^2+3*c^2)*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(40330) = 9*X(2)-4*X(182) = 3*X(2)+2*X(1352) = 9*X(2)+X(5921) = 6*X(2)-X(6776) = 13*X(2)-8*X(10168) = X(2)+4*X(11178) = 7*X(2)-2*X(11179) = 4*X(2)+X(11180) = 3*X(2)-8*X(24206) = 11*X(2)-6*X(38064) = 2*X(182)+3*X(1352) = 4*X(182)+X(5921) = 8*X(182)-3*X(6776) = 13*X(182)-18*X(10168) = X(182)+9*X(11178) = 14*X(182)-9*X(11179) = 16*X(182)+9*X(11180) = X(182)-6*X(24206) = 6*X(1352)-X(5921) = 4*X(1352)+X(6776) = 13*X(1352)+12*X(10168) = X(1352)-6*X(11178) = 7*X(1352)+3*X(11179) = 8*X(1352)-3*X(11180) = X(1352)+4*X(24206) = 11*X(1352)+9*X(38064)

X(40330) lies on these lines: {2, 98}, {3, 3619}, {4, 141}, {5, 69}, {6, 3090}, {10, 39898}, {20, 3818}, {66, 7383}, {113, 32247}, {140, 18440}, {159, 7509}, {183, 9752}, {193, 5056}, {235, 11382}, {262, 14994}, {343, 3066}, {376, 21358}, {381, 21356}, {428, 33522}, {487, 37343}, {488, 37342}, {511, 3091}, {518, 5818}, {524, 5071}, {546, 33878}, {547, 1353}, {567, 6193}, {576, 15022}, {590, 39876}, {599, 3545}, {611, 10588}, {613, 10589}, {615, 39875}, {631, 1503}, {632, 12017}, {639, 10514}, {640, 10515}, {895, 23515}, {1125, 39885}, {1176, 10539}, {1370, 21766}, {1469, 10590}, {1614, 5157}, {1656, 3564}, {1843, 11793}, {1853, 5646}, {1992, 5055}, {1995, 37488}, {2080, 3785}, {2854, 15081}, {3056, 10591}, {3085, 12589}, {3086, 12588}, {3088, 37480}, {3098, 3146}, {3167, 11548}, {3313, 7999}, {3316, 13910}, {3317, 13972}, {3416, 5603}, {3522, 29012}, {3523, 18553}, {3524, 20582}, {3525, 5085}, {3528, 21167}, {3529, 31884}, {3544, 3631}, {3589, 5067}, {3624, 39870}, {3628, 5050}, {3630, 5102}, {3751, 10175}, {3819, 7396}, {3832, 31670}, {3839, 25561}, {3844, 5657}, {3851, 21850}, {3917, 7378}, {4259, 6984}, {4413, 39877}, {4648, 7380}, {4869, 7407}, {5052, 31415}, {5068, 19130}, {5070, 38110}, {5072, 38136}, {5079, 5093}, {5084, 26543}, {5092, 10303}, {5094, 39871}, {5181, 14644}, {5432, 39892}, {5433, 39891}, {5476, 11160}, {5544, 18928}, {5550, 38029}, {5562, 9822}, {5590, 6813}, {5591, 6811}, {5596, 7558}, {5714, 24471}, {5800, 6854}, {5820, 6983}, {5846, 10595}, {5847, 8227}, {6090, 37454}, {6214, 11314}, {6215, 11313}, {6248, 32974}, {6292, 8721}, {6403, 29959}, {6515, 37990}, {6530, 32000}, {6623, 12294}, {6698, 14982}, {6759, 20079}, {6803, 18913}, {6815, 26156}, {6843, 10477}, {6920, 36740}, {6933, 15988}, {6946, 36741}, {6997, 37636}, {7379, 17232}, {7385, 17238}, {7395, 18945}, {7400, 15812}, {7401, 37489}, {7404, 13352}, {7405, 11411}, {7410, 17327}, {7484, 32064}, {7486, 15516}, {7499, 11206}, {7512, 20987}, {7528, 37494}, {7539, 11427}, {7697, 10008}, {7709, 10007}, {7795, 18860}, {7800, 8722}, {7808, 39872}, {7914, 39882}, {7998, 31099}, {8263, 16072}, {8889, 17811}, {9753, 15589}, {9863, 16898}, {9967, 10170}, {9969, 11412}, {10109, 14848}, {10272, 32306}, {10446, 36671}, {10565, 32237}, {10594, 37485}, {10752, 32257}, {10753, 36519}, {10754, 23514}, {10755, 23513}, {10762, 36520}, {11061, 14643}, {11257, 33202}, {11284, 37643}, {11387, 15644}, {11433, 37439}, {11459, 19161}, {11482, 12812}, {11645, 15692}, {12140, 35485}, {12215, 32829}, {12319, 14926}, {12383, 32274}, {12900, 32275}, {13862, 16990}, {14001, 39647}, {14913, 15073}, {15058, 34146}, {15184, 39886}, {15577, 35921}, {16986, 37182}, {17578, 29317}, {17792, 31418}, {17825, 18950}, {18141, 37360}, {19145, 32785}, {19146, 32786}, {20304, 25320}, {21851, 40247}, {22165, 38072}, {24220, 36673}, {24273, 35925}, {24953, 39890}, {26118, 33172}, {26363, 39903}, {26364, 39902}, {26468, 32488}, {26469, 32489}, {31742, 34512}, {32152, 32981}, {32217, 37943}, {32255, 38724}, {32815, 37242}, {32956, 39646}, {33198, 36998}, {34229, 37071}, {35283, 37638}, {37174, 39530}, {38118, 39878}

X(40330) = midpoint of X(3091) and X(3620)
X(40330) = reflection of X(i) in X(j) for these (i, j): (631, 3763), (3618, 1656), (12017, 632)
X(40300) = isogonal conjugate of X(40338)
X(40300) = isotomic conjugate of X(40339)
X(40330) = X(6)-isoconjugate of X(40327)
X(40330) = barycentric quotient X(6)/X(40338)
X(40330) = trilinear product X(i)*X(j) for these {i,j}: {2, 40302}, {6, 40301}
X(40330) = trilinear quotient X(i)/X(j) for these (i,j): (1, 40338), (40301, 2), (40302, 6)
X(40330) = intersection, other than A,B,C, of conics {{A, B, C, X(98), X(8797)}} and {{A, B, C, X(287), X(18840)}}
X(40330) = X(3091)-of-1st Brocard triangle
X(40330) = X(3620)-of-McCay triangle
X(40330) = X(6) of cross-triangle of Euler and anti-Euler triangles
X(40330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1352, 6776), (2, 5921, 182), (3, 39884, 14927), (4, 141, 10519), (5, 69, 14853), (140, 18440, 25406), (141, 10516, 4), (182, 1352, 5921), (182, 5921, 6776), (193, 5056, 14561), (1352, 6776, 11180), (1352, 24206, 2), (3525, 39874, 5085), (3589, 15069, 14912), (5055, 11898, 18583), (5067, 14912, 3589), (5085, 34573, 3525), (11178, 24206, 1352), (11898, 18583, 1992), (14561, 34507, 193), (32257, 36518, 10752)

X(40331) = OSIRIS POINT OF X(5)

Barycentrics    4*a^12-25*(b^2+c^2)*a^10+16*(4*b^4+5*b^2*c^2+4*c^4)*a^8-(b^2+c^2)*(85*b^4-29*b^2*c^2+85*c^4)*a^6+(61*b^4+71*b^2*c^2+61*c^4)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*(22*b^4-53*b^2*c^2+22*c^4)*a^2+(3*b^4-11*b^2*c^2+3*c^4)*(b^2-c^2)^4 : :
X(40331) = X(5)+4*X(6709) = X(95)+4*X(3628) = 11*X(5070)-X(17035)

X(40331) lies on these lines: {2, 35311}, {5, 6709}, {95, 3628}, {233, 6749}, {632, 32428}, {5070, 17035}

X(40332) = OSIRIS POINT OF X(6)

Barycentrics    2*(b^2+c^2)*a^4+(b^4+5*b^2*c^2+c^4)*a^2+2*(b^2+c^2)*b^2*c^2 : :
X(40332) = 6*X(2)-X(3094) = 9*X(2)-4*X(10007) = 9*X(2)+X(18906) = 3*X(2)+2*X(24256) = X(6)+4*X(3934) = 3*X(6)+2*X(14994) = X(76)+4*X(3589) = 2*X(76)+3*X(13331) = 3*X(76)+2*X(32449) = 3*X(3094)-8*X(10007) = 3*X(3094)+2*X(18906) = X(3094)+4*X(24256) = 8*X(3589)-3*X(13331) = 6*X(3589)-X(32449) = 6*X(3934)-X(14994) = 2*X(5976)+3*X(6034) = 4*X(10007)+X(18906) = 2*X(10007)+3*X(24256) = 9*X(13331)-4*X(32449) = X(18906)-6*X(24256)

X(40332) lies on these lines: {2, 694}, {6, 3934}, {76, 3589}, {83, 8177}, {141, 7752}, {182, 7697}, {183, 12212}, {511, 1656}, {597, 32451}, {599, 5052}, {698, 7786}, {732, 3618}, {1350, 15819}, {1691, 7770}, {1975, 12055}, {2021, 33237}, {2076, 7815}, {3095, 7822}, {3096, 5103}, {3619, 32999}, {3734, 5116}, {5017, 15271}, {5031, 16921}, {5085, 6248}, {5207, 33020}, {5480, 22712}, {6292, 9821}, {6704, 8149}, {7820, 11171}, {10168, 32429}, {10485, 39141}, {10516, 13354}, {12263, 38047}, {13910, 19089}, {13972, 19090}, {20582, 22486}, {24206, 36519}, {33249, 34573}

X(40332) = midpoint of X(3618) and X(31276)
X(40332) = reflection of X(3763) in X(31239)
X(40332) = X(7786)-of-1st Brocard triangle
X(40332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24256, 3094), (76, 3589, 13331), (10007, 18906, 3094), (10007, 24256, 18906)

X(40333) = OSIRIS POINT OF X(7)

Barycentrics    a^3-(b+c)*a^2+(b+3*c)*(3*b+c)*a-3*(b^2-c^2)*(b-c) : :
X(40333) = 6*X(1)-X(12630) = X(1)-6*X(38204) = 6*X(2)-X(390) = 9*X(2)-4*X(1001) = 3*X(2)+2*X(2550) = 3*X(2)-8*X(3826) = 11*X(2)-6*X(38025) = 2*X(2)+3*X(38092) = 3*X(390)-8*X(1001) = X(390)+4*X(2550) = X(390)-16*X(3826) = X(390)+9*X(38092) = 2*X(1001)+3*X(2550) = X(1001)-6*X(3826) = X(2550)+4*X(3826) = 11*X(2550)+9*X(38025) = 4*X(2550)-9*X(38092) = 4*X(3035)+X(20119) = 2*X(3035)+3*X(38202) = 16*X(3826)+9*X(38092)
X(40333) = 2*X(7)+X(8)+2*X(9)

X(40333) lies on these lines: {1, 12630}, {2, 11}, {3, 38149}, {5, 35514}, {7, 10}, {8, 142}, {9, 5128}, {20, 5251}, {75, 39570}, {140, 38170}, {141, 38185}, {144, 5880}, {145, 15570}, {355, 21151}, {391, 4645}, {392, 7673}, {405, 7676}, {442, 7679}, {443, 956}, {474, 7677}, {480, 27525}, {516, 1698}, {518, 3617}, {942, 34784}, {944, 38122}, {954, 9709}, {962, 38150}, {971, 5818}, {984, 4346}, {1125, 8236}, {1156, 34122}, {1320, 38205}, {1482, 38171}, {1699, 36835}, {1738, 3672}, {1890, 7378}, {2345, 3823}, {2346, 5687}, {2551, 37161}, {2951, 19925}, {3008, 4344}, {3059, 3812}, {3062, 38158}, {3146, 11495}, {3241, 38093}, {3523, 19854}, {3525, 38031}, {3616, 5853}, {3624, 30331}, {3626, 38054}, {3634, 30332}, {3654, 38073}, {3679, 5542}, {3696, 27475}, {3717, 31995}, {3753, 7672}, {3755, 5308}, {3832, 34501}, {3886, 29627}, {3932, 4461}, {3945, 4649}, {4000, 39587}, {4187, 7678}, {4197, 7080}, {4294, 17554}, {4307, 16468}, {4312, 6172}, {4323, 12447}, {4454, 27549}, {4470, 5845}, {4566, 10004}, {4669, 38024}, {4678, 25557}, {4731, 8581}, {4745, 38094}, {4999, 38203}, {5056, 38037}, {5070, 38043}, {5082, 17529}, {5187, 15254}, {5220, 20059}, {5226, 8580}, {5260, 37435}, {5265, 17580}, {5316, 9779}, {5435, 12573}, {5439, 11025}, {5550, 38316}, {5587, 36991}, {5657, 5805}, {5690, 38107}, {5698, 6871}, {5750, 5838}, {5759, 6843}, {5779, 38042}, {5784, 40269}, {5790, 31657}, {5817, 9956}, {5819, 17303}, {6042, 16819}, {6067, 9710}, {6173, 24393}, {6361, 18482}, {6601, 12632}, {6666, 6919}, {6844, 31658}, {6856, 8543}, {6904, 26060}, {6908, 18491}, {6984, 21168}, {7308, 9812}, {8583, 18220}, {9776, 25006}, {10005, 24349}, {10175, 11372}, {10394, 15587}, {10591, 38059}, {11362, 38036}, {12245, 20330}, {12730, 34123}, {14543, 24342}, {14986, 17582}, {15717, 24953}, {19860, 30284}, {20007, 28629}, {21931, 29674}, {22754, 37462}, {24599, 38186}

X(40333) = reflection of X(i) in X(j) for these (i, j): (3616, 20195), (11025, 5439), (18230, 1698)
X(40333) = intersection, other than A,B,C, of conics {{A, B, C, X(55), X(14626)}} and {{A, B, C, X(105), X(10390)}}
X(40333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 38121, 35514), (7, 10, 5686), (8, 142, 11038), (10, 4208, 5261), (10, 25590, 5772), (10, 38052, 7), (142, 38200, 8), (984, 7613, 4346), (5177, 9780, 8165), (5880, 38057, 144), (17580, 19843, 5265), (17582, 31419, 14986)

X(40334) = OSIRIS POINT OF X(13)

Barycentrics    -2*(a^2+2*b^2+2*c^2)*S+(a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*sqrt(3) : :
X(40334) = 6*X(2)-X(15) = 9*X(2)+X(621) = 3*X(2)+2*X(623) = 9*X(2)-4*X(6671) = 4*X(3)+X(36992) = 4*X(5)+X(14538) = 3*X(15)+2*X(621) = X(15)+4*X(623) = 3*X(15)-8*X(6671) = X(16)+4*X(625) = 4*X(140)+X(20428) = 8*X(140)-3*X(21158) = X(298)+4*X(6669) = 2*X(298)+3*X(16267) = X(621)-6*X(623) = X(621)+4*X(6671) = 3*X(623)+2*X(6671) = 8*X(3589)-3*X(36757) = 3*X(5464)+2*X(33518) = X(5978)+4*X(6670) = 2*X(20428)+3*X(21158)

X(40334) lies on these lines: {2, 14}, {3, 36992}, {5, 14538}, {16, 625}, {17, 69}, {18, 3589}, {20, 33387}, {30, 36770}, {61, 7886}, {62, 302}, {140, 20428}, {141, 16966}, {298, 6669}, {303, 635}, {316, 6672}, {325, 22511}, {381, 36755}, {396, 21359}, {511, 1656}, {524, 16960}, {616, 33560}, {618, 36969}, {620, 23004}, {628, 6673}, {629, 5237}, {1975, 16630}, {3090, 7684}, {3104, 7887}, {3105, 7862}, {3106, 7844}, {3525, 36993}, {3526, 13350}, {3618, 16961}, {3624, 11707}, {5070, 5611}, {5238, 11309}, {5318, 5463}, {5395, 10187}, {6722, 22510}, {7778, 25167}, {7808, 11311}, {10646, 11303}, {10653, 31705}, {11133, 22846}, {11301, 36970}, {11308, 30560}, {11310, 35229}, {11542, 22489}, {13449, 21159}, {16239, 33386}, {16242, 37352}, {16809, 37340}, {19106, 31693}, {19107, 37172}, {22691, 23000}, {22693, 37071}, {23005, 33228}, {23006, 38412}, {33416, 37341}, {34509, 34540}, {36968, 37170}

X(40334) = reflection of X(40335) in X(31275)
X(40334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 621, 6671), (2, 623, 15), (140, 20428, 21158), (298, 6669, 16267), (316, 6672, 39554), (621, 6671, 15), (623, 6671, 621), (1656, 3763, 40335)

X(40335) = OSIRIS POINT OF X(14)

Barycentrics    2*(a^2+2*b^2+2*c^2)*S+(a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*sqrt(3) : :
X(40335) = 6*X(2)-X(16) = 9*X(2)+X(622) = 3*X(2)+2*X(624) = 9*X(2)-4*X(6672) = 4*X(3)+X(36994) = 4*X(5)+X(14539) = X(15)+4*X(625) = 3*X(16)+2*X(622) = X(16)+4*X(624) = 3*X(16)-8*X(6672) = 4*X(140)+X(20429) = 8*X(140)-3*X(21159) = X(299)+4*X(6670) = 2*X(299)+3*X(16268) = X(622)-6*X(624) = X(622)+4*X(6672) = 3*X(624)+2*X(6672) = 8*X(3589)-3*X(36758) = 3*X(5463)+2*X(33517) = X(5979)+4*X(6669) = 2*X(20429)+3*X(21159)

X(40335) lies on these lines: {2, 13}, {3, 36994}, {5, 14539}, {15, 625}, {17, 3589}, {18, 69}, {20, 33386}, {61, 303}, {62, 7886}, {140, 20429}, {141, 16967}, {299, 6670}, {302, 636}, {316, 6671}, {325, 22510}, {381, 36756}, {395, 21360}, {511, 1656}, {524, 16961}, {617, 33561}, {619, 36970}, {620, 23005}, {627, 6674}, {630, 5238}, {1975, 16631}, {3090, 7685}, {3104, 7862}, {3105, 7887}, {3107, 7844}, {3525, 36995}, {3526, 13349}, {3618, 16960}, {3624, 11708}, {5070, 5615}, {5237, 11310}, {5321, 5464}, {5395, 10188}, {6722, 22511}, {7778, 25157}, {7808, 11312}, {10645, 11304}, {10654, 31706}, {11132, 22891}, {11302, 36969}, {11307, 30559}, {11309, 35230}, {11543, 22490}, {13449, 21158}, {16239, 33387}, {16241, 37351}, {16808, 37341}, {19106, 37173}, {19107, 31694}, {22692, 23009}, {22694, 37071}, {23004, 33228}, {33417, 37340}, {34508, 34541}, {36967, 37171}

X(40335) = reflection of X(40334) in X(31275)
X(40335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 622, 6672), (2, 624, 16), (140, 20429, 21159), (299, 6670, 16268), (316, 6671, 39555), (622, 6672, 16), (624, 6672, 622), (1656, 3763, 40334)

X(40336) = OSIRIS POINT OF X(98)

Barycentrics    4*a^8-9*(b^2+c^2)*a^6+(13*b^4+4*b^2*c^2+13*c^4)*a^4-(b^2+c^2)*(11*b^4-14*b^2*c^2+11*c^4)*a^2+(3*b^4-4*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(40336) = 6*X(2)-X(1513) = 9*X(2)+X(5999) = 9*X(2)-4*X(10011) = 2*X(3)+3*X(33228) = 4*X(140)+X(15980) = 8*X(140)-3*X(35297) = X(376)+4*X(8355) = 4*X(549)+X(8352) = 3*X(1513)+2*X(5999) = 3*X(1513)-8*X(10011) = 7*X(1513)-2*X(40236)

X(40336) lies on these lines: {2, 3}, {98, 13196}, {99, 10256}, {182, 37647}, {230, 5111}, {325, 5965}, {538, 38740}, {625, 38737}, {1007, 9755}, {1350, 9754}, {2794, 31275}, {3054, 22712}, {3564, 7925}, {6390, 14651}, {6722, 18860}, {7607, 13468}, {7612, 37668}, {16984, 18583}

X(40336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3523, 33253, 3), (5999, 10011, 1513), (6039, 6040, 3529)

X(40337) = X(4)X(12271)∩X(6)X(1196)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 8*a^2*b^2*c^2 - 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :
X(40337) = 4 X[10110] - 3 X[14914]

X(40337) lies on Feuerbach circumhyperbola of the orthic triangle, the cubic K1165, and on these lines: {4, 12271}, {6, 1196}, {185, 3564}, {511, 5895}, {520, 38359}, {1368, 6467}, {1858, 34381}, {2854, 15583}, {3574, 38136}, {6193, 9730}, {6403, 15741}, {9825, 21651}, {10110, 14914}, {12166, 13352}, {13202, 14984}, {13754, 20080}, {14091, 15143}, {14961, 19597}

X(40337) = reflection of X(6391) in X(14913)
X(40337) = orthic-isogonal conjugate of X(1368)
X(40337) = X(4)-Ceva conjugate of X(1368)
X(40337) = barycentric product X(1368)*X(40318)
X(40337) = barycentric quotient X(i)/X(j) for these {i,j}: {1196, 15591}, {6467, 40323}

X(40338) = BARYCENTRIC PRODUCT X(1)*X(40327)

Barycentrics    a log(b/c) : b log(c/a) : c log(a/b)
Trilinears    log(b/c) : log(c/a) : log(a/b)

X(40338) lies on the line {44, 513}

X(40338) = isogonal conjugate of X(40300)
X(40338) = crossdifference of every pair of points on line {X(1), X(40297)}
X(40338) = crosspoint of X(1) and X(40300)
X(40338) = X(i)-isoconjugate-of-X(j) for these {i,j}: {1, 40300}, {2, 40302}, {6, 40301}
X(40338) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 40301), (31, 40302)
X(40338) = X(i)-Zayin conjugate of-X(j) for these (i,j): (9, 40302), (43, 40301)
X(40338) = trilinear SS(a → log(b/c))
X(40338) = barycentric product X(i)*X(j) for these {i, j}: {1, 40327}, {6, 40339}
X(40338) = barycentric quotient X(i)/X(j) for these (i, j): (1, 40301), (31, 40302)
X(40338) = trilinear product X(i)*X(j) for these {i j}: {6, 40327}, {31, 40339}
X(40338) = trilinear quotient X(i)/X(j) for these (i, j): (1, 40300), (2, 40301), (6, 40302), (6, 40300)

X(40339) = BARYCENTRIC PRODUCT X(75)*X(40327)

Barycentrics    b c/log(b/c) : c a/log(c/a) : a b/log(a/b)
Barycentrics    b c /(log b - log c) : :
Trilinears    b^2 c^2 /log(b/c) : :

X(40339) lies on the line {514, 693}

X(40339) = isotomic conjugate of X(40300)
X(40339) = barycentric product X(i)*X(j) for these {i, j}: {75, 40327}, {76, 40338}
X(40339) = barycentric quotient X(i)/X(j) for these (i, j): (1, 40302), (75, 40301)
X(40339) = trilinear product X(i)*X(j) for these {i, j}: {2, 40327}, {75, 40338}
X(40339) = trilinear quotient X(i)/X(j) for these (i, j): (2, 40302), (75, 40300), (76, 40301), (40338, 31)
X(40339) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 40302}, {31, 40300}, {32, 40301}
X(40339) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 40302), (75, 40301)

X(40340) = MIDPOINT OF X(5) AND X(126)

Barycentrics    3*a^8*b^2 - 7*a^6*b^4 - a^4*b^6 + 7*a^2*b^8 - 2*b^10 + 3*a^8*c^2 - 22*a^6*b^2*c^2 + 28*a^4*b^4*c^2 - 39*a^2*b^6*c^2 + 10*b^8*c^2 - 7*a^6*c^4 + 28*a^4*b^2*c^4 + 40*a^2*b^4*c^4 - 8*b^6*c^4 - a^4*c^6 - 39*a^2*b^2*c^6 - 8*b^4*c^6 + 7*a^2*c^8 + 10*b^2*c^8 - 2*c^10 : :
X(40340) = 3 X[2] + X[10748], 9 X[2] - X[14654], 5 X[2] - X[14666], 9 X[2] - 5 X[38806], 3 X[3] + X[10734], 3 X[4] + X[38797], 3 X[5] - X[5512], X[111] - 5 X[1656], 3 X[126] + X[5512], 3 X[381] + X[1296], 9 X[381] - X[38800], X[382] + 3 X[38716], 5 X[632] - 3 X[38804], 3 X[1296] + X[38800], 7 X[3090] + X[14360], 7 X[3090] - 3 X[38796], 5 X[3091] - X[22338], X[3146] + 3 X[38798], 7 X[3526] - 3 X[38698], 7 X[3851] + X[38593], 3 X[5055] + X[10717], 9 X[5055] - X[11258], 11 X[5072] + X[38688], 11 X[5072] - 3 X[38799], 13 X[5079] - X[38675], 3 X[5790] + X[10704], 17 X[7486] - X[20099], X[9172] - 3 X[15699], 3 X[10717] + X[11258], 3 X[10748] + X[14654], 5 X[10748] + 3 X[14666], 3 X[10748] + 5 X[38806], X[10779] + 3 X[38752], 3 X[11230] - X[11721], 4 X[12811] - X[38801], X[14360] + 3 X[38796], 3 X[14561] + X[36883], 3 X[14650] - X[14654], 5 X[14650] - 3 X[14666], 3 X[14650] - 5 X[38806], 5 X[14654] - 9 X[14666], X[14654] - 5 X[38806], 9 X[14666] - 25 X[38806], X[28662] - 3 X[38317], 3 X[38623] - X[38797], X[38688] + 3 X[38799]

X(40340) lies on these lines: {2, 10748}, {3, 10734}, {4, 38623}, {5, 126}, {30, 38803}, {111, 1656}, {140, 23699}, {381, 1296}, {382, 38716}, {543, 547}, {632, 38804}, {2854, 16511}, {3048, 18350}, {3090, 14360}, {3091, 22338}, {3146, 38798}, {3325, 7951}, {3526, 38698}, {3627, 38805}, {3628, 6719}, {3818, 14688}, {3851, 38593}, {5055, 10717}, {5072, 38688}, {5079, 38675}, {5790, 10704}, {6019, 7741}, {6593, 36832}, {7486, 20099}, {7514, 14657}, {9172, 15699}, {10779, 38752}, {11230, 11721}, {11835, 23261}, {11836, 23251}, {12811, 38801}, {14561, 36883}, {28662, 38317}

X(40340) = midpoint of X(i) and X(j) for these {i,j}: {4, 38623}, {5, 126}, {3627, 38805}, {3818, 14688}, {6593, 36832}, {10748, 14650}
X(40340) = reflection of X(6719) in X(3628)
X(40340) = complement of X(14650)
X(40340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 10748, 14650}, {2, 14654, 38806}, {3090, 14360, 38796}, {10748, 38806, 14654}, {14654, 38806, 14650}

X(40341) = MIDPOINT OF X(69) AND X(20080)

Barycentrics    3*a^2 - 2*b^2 - 2*c^2 : :
Barycentrics    A'-power of A-Moses-Steiner osculating circle : : , where A'B'C' is the anticomplementary triangle
X(40341) = 6 X[2] - 5 X[6], 3 X[2] - 5 X[69], 9 X[2] - 10 X[141], 9 X[2] - 5 X[193], 11 X[2] - 10 X[597], 4 X[2] - 5 X[599], 7 X[2] - 5 X[1992], 21 X[2] - 20 X[3589], 27 X[2] - 25 X[3618], 33 X[2] - 35 X[3619], 21 X[2] - 25 X[3620], 3 X[2] - 10 X[3630], 3 X[2] - 4 X[3631], 24 X[2] - 25 X[3763], 19 X[2] - 15 X[5032], 12 X[2] - 5 X[6144], 9 X[2] - 8 X[6329], 13 X[2] - 10 X[8584], X[2] - 5 X[11160], 2 X[2] - 5 X[15533], 8 X[2] - 5 X[15534], 3 X[2] + 5 X[20080], 19 X[2] - 20 X[20582], 5 X[2] - 4 X[20583], 13 X[2] - 15 X[21356], 14 X[2] - 15 X[21358], 7 X[2] - 10 X[22165], 4 X[5] - 3 X[5102], 3 X[6] - 4 X[141], 3 X[6] - 2 X[193], 11 X[6] - 12 X[597], 2 X[6] - 3 X[599], 7 X[6] - 6 X[1992], 7 X[6] - 8 X[3589], 9 X[6] - 10 X[3618], 11 X[6] - 14 X[3619], 7 X[6] - 10 X[3620], 5 X[6] - 4 X[3629], X[6] - 4 X[3630], 5 X[6] - 8 X[3631], 4 X[6] - 5 X[3763], 19 X[6] - 18 X[5032], 15 X[6] - 16 X[6329], 13 X[6] - 12 X[8584], 5 X[6] - 2 X[11008], X[6] - 6 X[11160], X[6] - 3 X[15533], 4 X[6] - 3 X[15534], X[6] + 2 X[20080], 19 X[6] - 24 X[20582], 25 X[6] - 24 X[20583], 13 X[6] - 18 X[21356], 7 X[6] - 9 X[21358], 7 X[6] - 12 X[22165], 4 X[67] - 3 X[25330], 3 X[69] - 2 X[141], 3 X[69] - X[193], 11 X[69] - 6 X[597], 4 X[69] - 3 X[599], 7 X[69] - 3 X[1992], 7 X[69] - 4 X[3589], 9 X[69] - 5 X[3618], 11 X[69] - 7 X[3619], 7 X[69] - 5 X[3620], 5 X[69] - 2 X[3629], 5 X[69] - 4 X[3631], 8 X[69] - 5 X[3763], 19 X[69] - 9 X[5032], 4 X[69] - X[6144], 15 X[69] - 8 X[6329], 13 X[69] - 6 X[8584], 5 X[69] - X[11008], X[69] - 3 X[11160], 2 X[69] - 3 X[15533], 8 X[69] - 3 X[15534], 19 X[69] - 12 X[20582], 25 X[69] - 12 X[20583], 13 X[69] - 9 X[21356], 14 X[69] - 9 X[21358], 7 X[69] - 6 X[22165], 4 X[110] - 3 X[25331], 11 X[141] - 9 X[597], 8 X[141] - 9 X[599], 14 X[141] - 9 X[1992], 7 X[141] - 6 X[3589], 6 X[141] - 5 X[3618], 22 X[141] - 21 X[3619], 14 X[141] - 15 X[3620], 5 X[141] - 3 X[3629], X[141] - 3 X[3630], 5 X[141] - 6 X[3631], 16 X[141] - 15 X[3763], 38 X[141] - 27 X[5032], 8 X[141] - 3 X[6144], 5 X[141] - 4 X[6329], 13 X[141] - 9 X[8584], 10 X[141] - 3 X[11008], 2 X[141] - 9 X[11160], 4 X[141] - 9 X[15533], 16 X[141] - 9 X[15534], 2 X[141] + 3 X[20080], 19 X[141] - 18 X[20582]

Let LA be the reflection of line BC in A, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. A' is also the reflection of A in the A-vertex of the anticomplementary triangle. A'B'C' is homothetic to, and 5 times the size, of ABC. X(40341) = X(6)-of-A'B'C'. (Randy Hutson, December 18, 2020)

X(40341) lies on these lines: {1, 17253}, {2, 6}, {3, 5965}, {5, 5102}, {7, 4371}, {8, 7222}, {9, 17311}, {20, 16775}, {32, 33242}, {37, 29602}, {45, 4416}, {53, 32001}, {67, 6391}, {110, 16176}, {144, 3943}, {145, 17246}, {159, 2930}, {182, 15720}, {190, 17309}, {239, 7232}, {315, 33229}, {316, 34505}, {319, 4363}, {320, 4361}, {338, 14615}, {340, 9308}, {381, 7845}, {382, 511}, {487, 6410}, {488, 6409}, {518, 3632}, {519, 17276}, {527, 17299}, {542, 15681}, {546, 1352}, {550, 1350}, {576, 5079}, {594, 4644}, {623, 33465}, {624, 33464}, {625, 5111}, {631, 12007}, {633, 5340}, {634, 5339}, {637, 23251}, {638, 23261}, {698, 33256}, {732, 33234}, {742, 3644}, {894, 4445}, {1030, 21518}, {1086, 4402}, {1100, 17272}, {1351, 3851}, {1353, 14869}, {1384, 7801}, {1449, 17237}, {1503, 3529}, {1634, 22152}, {1656, 5097}, {1743, 17231}, {1853, 34777}, {1975, 7893}, {2076, 33235}, {2345, 7277}, {2482, 15655}, {2525, 30511}, {2549, 14929}, {2854, 12220}, {2916, 19588}, {2979, 17710}, {3053, 3793}, {3094, 32450}, {3098, 15688}, {3242, 3244}, {3284, 20208}, {3313, 8681}, {3416, 3626}, {3526, 39561}, {3528, 6776}, {3530, 5085}, {3544, 14853}, {3636, 38315}, {3664, 17275}, {3686, 4675}, {3729, 4715}, {3758, 17287}, {3759, 17288}, {3770, 34282}, {3779, 9038}, {3785, 15815}, {3818, 14269}, {3855, 5480}, {3875, 4725}, {3879, 4643}, {3882, 5036}, {3886, 28570}, {3912, 16885}, {3917, 32366}, {3926, 5023}, {3964, 8553}, {4000, 4969}, {4034, 4688}, {4042, 32949}, {4053, 18161}, {4265, 19535}, {4357, 16884}, {4360, 4741}, {4384, 17376}, {4393, 17273}, {4398, 20016}, {4414, 4938}, {4419, 17388}, {4667, 17303}, {4670, 17270}, {4681, 29605}, {4690, 10436}, {4700, 21255}, {4852, 17274}, {4859, 31138}, {5008, 33237}, {5013, 7758}, {5024, 7810}, {5028, 33241}, {5033, 12151}, {5050, 40107}, {5092, 15700}, {5093, 24206}, {5096, 19537}, {5124, 21524}, {5207, 14062}, {5210, 6390}, {5220, 32846}, {5286, 33232}, {5486, 34817}, {5621, 12901}, {5695, 17770}, {5846, 20050}, {5848, 6154}, {5921, 29181}, {6179, 7881}, {6467, 9027}, {6542, 17262}, {6646, 17318}, {6697, 11216}, {6748, 32000}, {7321, 29617}, {7703, 23061}, {7716, 10301}, {7750, 33253}, {7754, 7768}, {7760, 7879}, {7761, 22253}, {7770, 7877}, {7773, 7946}, {7775, 18584}, {7780, 7916}, {7789, 22331}, {7793, 13196}, {7794, 30435}, {7796, 35006}, {7798, 7848}, {7805, 7866}, {7811, 31859}, {7820, 21309}, {7841, 7850}, {7851, 7939}, {7854, 7890}, {7871, 33233}, {7887, 7917}, {7894, 32027}, {7895, 32954}, {7905, 11285}, {7908, 11288}, {8266, 20794}, {8550, 10299}, {8588, 39785}, {8589, 11165}, {8716, 14907}, {9019, 12272}, {9035, 39232}, {9053, 20054}, {9054, 25304}, {9306, 21970}, {9466, 15484}, {9939, 11742}, {9971, 14913}, {10300, 15812}, {10387, 39873}, {10452, 21769}, {10488, 14928}, {10541, 14912}, {11179, 17504}, {11225, 16419}, {11482, 38317}, {11646, 14645}, {11737, 38072}, {12017, 15707}, {12088, 15580}, {12215, 33276}, {12383, 17835}, {13142, 33537}, {13330, 14994}, {14042, 18906}, {14561, 35018}, {15526, 15905}, {15687, 31670}, {16496, 28538}, {16666, 17306}, {16667, 17384}, {16668, 29598}, {16669, 17284}, {16670, 17357}, {16672, 17257}, {16674, 29574}, {16675, 17316}, {16808, 22493}, {16809, 22494}, {16814, 29573}, {16826, 17328}, {16834, 17235}, {16866, 37492}, {17120, 17228}, {17121, 17227}, {17233, 20072}, {17249, 29584}, {17252, 17394}, {17254, 17393}, {17256, 17391}, {17258, 17389}, {17260, 17387}, {17261, 17386}, {17267, 17296}, {17269, 17295}, {17294, 17351}, {17298, 17348}, {17310, 17336}, {17312, 17335}, {17314, 17334}, {17315, 17333}, {17317, 17331}, {17319, 17329}, {17340, 29616}, {17571, 36740}, {17573, 36741}, {17813, 23300}, {18358, 20423}, {20477, 39352}, {20850, 20987}, {21241, 32853}, {21242, 32946}, {22034, 34371}, {22892, 38412}, {32113, 37897}, {32234, 33851}, {35482, 39588}, {37900, 40317}, {39710, 39720}

X(40341) = midpoint of X(69) and X(20080)
X(40341) = reflection of X(i) in X(j) for these {i,j}: {6, 69}, {69, 3630}, {193, 141}, {599, 15533}, {1351, 34507}, {1992, 22165}, {2549, 14929}, {3629, 3631}, {3729, 17372}, {3875, 17345}, {6144, 6}, {7798, 7848}, {11008, 3629}, {11477, 1352}, {13330, 14994}, {15069, 11898}, {15533, 11160}, {15534, 599}, {16176, 110}, {22253, 7761}, {25336, 2930}, {32234, 33851}, {36990, 15069}, {39899, 3098}
X(40341) = complement of X(11008)
X(40341) = anticomplement of X(3629)
X(40341) = isotomic conjugate of the isogonal conjugate of X(5206)
X(40341) = isotomic conjugate of the polar conjugate of X(37453)
X(40341) = X(5206)-cross conjugate of X(37453)
X(40341) = barycentric product X(i)*X(j) for these {i,j}: {69, 37453}, {76, 5206}
X(40341) = barycentric quotient X(i)/X(j) for these {i,j}: {5206, 6}, {37453, 4}
X(40341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17344, 17253}, {2, 69, 3631}, {2, 3629, 6}, {2, 11008, 3629}, {6, 69, 599}, {6, 599, 3763}, {6, 6144, 15534}, {6, 15533, 69}, {6, 21358, 3589}, {7, 17362, 17119}, {8, 17365, 17118}, {9, 17374, 17311}, {44, 17296, 17267}, {69, 193, 141}, {69, 1992, 3620}, {69, 3620, 22165}, {69, 3630, 15533}, {69, 6144, 3763}, {69, 11008, 2}, {69, 11160, 3630}, {86, 17343, 17251}, {110, 16176, 25331}, {141, 193, 6}, {141, 3629, 6329}, {141, 6329, 2}, {141, 32455, 3618}, {183, 7779, 9766}, {183, 9766, 31489}, {190, 17373, 17309}, {193, 3618, 32455}, {239, 17361, 7232}, {298, 5859, 16644}, {299, 5858, 16645}, {319, 17364, 4363}, {320, 17363, 4361}, {325, 8667, 37637}, {385, 7788, 7778}, {491, 492, 37647}, {491, 591, 8252}, {492, 1991, 8253}, {599, 6144, 6}, {894, 17360, 4445}, {1100, 17272, 17325}, {1270, 5861, 590}, {1271, 5860, 615}, {1351, 34507, 10516}, {1654, 17378, 15668}, {1992, 3589, 6}, {1992, 3620, 3589}, {1992, 22165, 21358}, {3589, 3620, 21358}, {3589, 22165, 3620}, {3618, 32455, 6}, {3629, 3631, 2}, {3630, 20080, 6}, {3631, 6329, 141}, {3631, 11008, 6}, {3758, 17287, 17293}, {3759, 17288, 17290}, {3763, 15534, 6}, {3815, 15589, 8556}, {3879, 4643, 16777}, {3933, 14023, 3053}, {4034, 4888, 4688}, {4360, 4741, 17255}, {4393, 17273, 17323}, {4416, 4851, 45}, {4644, 32099, 594}, {4869, 37654, 17337}, {5839, 21296, 1086}, {6189, 6190, 7925}, {6542, 17347, 17262}, {6646, 17377, 17318}, {7751, 7776, 13881}, {7751, 7882, 7776}, {7754, 7768, 7784}, {7758, 7767, 5013}, {7774, 37671, 15271}, {7798, 7848, 11287}, {7805, 7896, 7866}, {7826, 7855, 3}, {7845, 17131, 381}, {7854, 7890, 9605}, {7946, 17129, 7773}, {8177, 39099, 6}, {11160, 20080, 69}, {15533, 20080, 6144}, {17257, 17390, 16672}, {17271, 17379, 17327}, {17277, 17375, 17313}, {17295, 17350, 17269}, {17297, 17349, 17265}, {17300, 17346, 17259}, {17316, 17332, 16675}, {17319, 17329, 24441}, {21358, 22165, 599}, {22844, 22845, 3}

X(40342) = REFLECTION OF X(6698) IN X(6593)

Barycentrics    8*a^8 - 5*a^6*b^2 - 6*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - 5*a^6*c^2 + 8*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 6*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 5*a^2*c^6 - 2*c^8 : :
X(40342) = 9 X[2] - 5 X[67], 3 X[2] - 5 X[6593], 6 X[2] - 5 X[6698], 3 X[2] + 5 X[11061], X[2] - 5 X[34319], X[67] - 3 X[6593], 2 X[67] - 3 X[6698], X[67] + 3 X[11061], X[67] - 9 X[34319], 3 X[110] + X[16176], X[110] + 3 X[25331], X[382] - 5 X[9970], X[3529] - 5 X[32233], 3 X[3629] - 5 X[5095], X[3629] - 5 X[25329], X[3632] - 5 X[32278], 7 X[3851] - 5 X[32274], 13 X[5079] - 5 X[32306], 5 X[5095] + 3 X[24981], X[5095] - 3 X[25329], 6 X[6329] - 5 X[15118], X[6593] - 3 X[34319], X[6698] + 2 X[11061], X[6698] - 6 X[34319], 13 X[10299] - 5 X[32247], X[11061] + 3 X[34319], 3 X[12824] - X[32299], 3 X[15303] - X[25328], 15 X[15462] - 11 X[15720], 3 X[15687] - 5 X[32271], 7 X[15808] - 5 X[32238], X[16176] - 9 X[25331], X[20050] - 5 X[32298], X[24981] + 5 X[25329], 9 X[25321] - X[32255]

X(40342) lies on these lines: {2, 67}, {110, 16176}, {382, 9970}, {524, 32267}, {542, 546}, {550, 2781}, {1112, 1843}, {3529, 32233}, {3632, 32278}, {3851, 32274}, {5079, 32306}, {5965, 25338}, {6053, 37984}, {6329, 15118}, {8550, 12162}, {9019, 37900}, {10299, 32247}, {12824, 32299}, {15303, 25328}, {15462, 15720}, {15687, 32271}, {15808, 32238}, {16534, 31831}, {20050, 32298}, {25321, 32255}

X(40342) = midpoint of X(i) and X(j) for these {i,j}: {3629, 24981}, {6593, 11061}
X(40342) = reflection of X(6698) in X(6593)
X(40342) = {X(11061),X(34319)}-harmonic conjugate of X(6593)

X(40343) = X(67)X(524)∩X(111)X(5189)

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

X(40343) lies on these lines: {67, 524}, {111, 5189}, {126, 13574}, {625, 691}, {8877, 31101}, {10989, 15398}

X(40343) ={X(858),X(34320)}-harmonic conjugate of X(15899)

X(40344) = X(2)X(187)∩X(3)X(7849)

Barycentrics    4*a^4 - 5*a^2*b^2 - 2*b^4 - 5*a^2*c^2 - 2*b^2*c^2 - 2*c^4 : :
X(40344) = 7 X[2] - 3 X[598], 3 X[2] + X[11057], 4 X[2] - 3 X[14762], 7 X[2] + X[14976], X[2] - 3 X[15810], 9 X[2] - X[19569], 3 X[39] - X[7837], 7 X[39] - X[7877], 5 X[39] + X[7893], X[39] + 5 X[7904], 3 X[376] + X[14458], 9 X[598] + 7 X[11057], 9 X[598] - 7 X[14537], 4 X[598] - 7 X[14762], 3 X[598] + X[14976], X[598] - 7 X[15810], 27 X[598] - 7 X[19569], X[3934] + 2 X[7830], 2 X[6683] + X[7750], 2 X[7767] + X[32450], X[7802] + 5 X[31239], 3 X[7810] - X[37671], 3 X[7811] + X[7837], 7 X[7811] + X[7877], 5 X[7811] - X[7893], X[7811] - 5 X[7904], 7 X[7837] - 3 X[7877], 5 X[7837] + 3 X[7893], X[7837] + 15 X[7904], 5 X[7877] + 7 X[7893], X[7877] + 35 X[7904], X[7893] - 25 X[7904], 3 X[8356] + X[37671], 3 X[8359] - X[9300], 3 X[9774] - 7 X[15698], 3 X[10033] + X[11001], 4 X[11057] + 9 X[14762], 7 X[11057] - 3 X[14976], X[11057] + 9 X[15810], 3 X[11057] + X[19569], 4 X[14537] - 9 X[14762], 7 X[14537] + 3 X[14976], X[14537] - 9 X[15810], 3 X[14537] - X[19569], 21 X[14762] + 4 X[14976], X[14762] - 4 X[15810], 27 X[14762] - 4 X[19569], X[14976] + 21 X[15810], 9 X[14976] + 7 X[19569], 27 X[15810] - X[19569]

X(40344) lies on these lines: on lines {2, 187}, {3, 7849}, {30, 3934}, {39, 7811}, {141, 8703}, {183, 11648}, {376, 7800}, {381, 7815}, {385, 39593}, {512, 3819}, {524, 8358}, {538, 7810}, {543, 8354}, {549, 626}, {574, 7788}, {620, 12100}, {754, 8359}, {1078, 7861}, {2896, 7799}, {3096, 15513}, {3314, 8589}, {3524, 3788}, {3534, 3734}, {3631, 14148}, {3642, 36755}, {3643, 36756}, {3785, 7739}, {3830, 15271}, {4045, 5306}, {5007, 33021}, {5013, 7882}, {5023, 7914}, {5054, 7784}, {5055, 7825}, {5077, 8556}, {5206, 7915}, {5309, 7780}, {6292, 6661}, {6655, 39563}, {6683, 7750}, {7746, 33251}, {7748, 33263}, {7759, 32990}, {7767, 32450}, {7768, 31652}, {7778, 15693}, {7789, 34200}, {7793, 7884}, {7795, 10304}, {7801, 33008}, {7802, 31239}, {7809, 7824}, {7817, 11287}, {7821, 7936}, {7822, 33255}, {7833, 9466}, {7841, 18362}, {7843, 11285}, {7847, 19570}, {7854, 32833}, {7862, 15694}, {7868, 8588}, {7874, 7928}, {7876, 35007}, {7879, 15515}, {7883, 33273}, {7886, 7935}, {7896, 15815}, {7910, 39565}, {7911, 38223}, {7922, 33022}, {8353, 32479}, {9774, 15698}, {9830, 36521}, {10033, 11001}, {10130, 10989}, {13586, 31168}, {15759, 32459}, {32828, 38259}, {32832, 33278}, {33184, 34506}

X(40344) = midpoint of X(i) and X(j) for these {i,j}: {39, 7811}, {549, 34510}, {7750, 7753}, {7810, 8356}, {7833, 9466}, {11057, 14537}
X(40344) = reflection of X(7753) in X(6683)
X(40344) = complement of X(14537)
X(40344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11057, 14537}, {2, 14976, 598}, {3, 7865, 7880}, {2896, 37512, 7895}, {5077, 8556, 18546}, {7865, 7880, 7849}, {7936, 33004, 7821}

leftri

Points on the tangential power curve: X(40345)-X(40346)

rightri

This preamble is contributed by Clark Kimberling, November 11, 2020.

Let PC(ABC) by the power curve; i.e., the locus of the point at : bt : ct (barycentrics) as t varies through the real numbers. If P = at : bt : ct for fixed t, then the isotomic conjugate of P is the point P' = a-t : b-t : c-t. Equations for the lines tangent to PC(ABC) at P and P' are found using Suren's points on the line at infinity; see the preamble just before X(40296). If P is not the centroid of ABC, then the tangent lines are distinct, and they meet in the point

P'' = at(b2t - c2t)/(log b - log c) : bt(c2t - a2t)/(log c - log a) : ct(a2t - b2t)/(log a - log b)

The locus of P'' as t varies through the positive real numbers is here named the (barycentric) tangential power curve.

The corresponding normal lines at P and P' meet in a point whose locus is the normal power curve.

The trilinear tangential and normal power curves are defined in the same manner using trilinear coordindates throughout, using isogonal conjugates instead of isotomic.

X(40345) = TANGENTIAL POWER POINT OF X(1)

Barycentrics    a(b2 - c2)/(log b - log c) : b(c2 - a2)/(log c - log a) : c(a2 - b2)/(log a - log b)

X(40345) lies on these lines: {1, 40297}, {759, 40302}, {897, 40300}, {18827, 40301}

X(40345) = barycentric product X(i)*X(j) for these {i, j}: {83, 40346}, {523, 40300}, {661, 40301}, {1577, 40302}
X(40345) = barycentric quotient X(i)/X(j) for these (i, j): (512, 40338), (523, 40339), (661, 40327)
X(40345) = trilinear product X(i)*X(j) for these {i, j}: {82, 40346}, {512, 40301}, {523, 40302}, {661, 40300}
X(40345) = trilinear quotient X(i)/X(j) for these (i, j): (523, 40327), (661, 40338), (1577, 40339)
X(40345) = X(i)-isoconjugate-of-X(j) for these {i,j}: {110, 40327}, {163, 40339}, {662, 40338}
X(40345) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (512, 40338), (523, 40339), (661, 40327)
X(40345) = X(661)-Zayin conjugate of-X(40338)

X(40346) = TANGENTIAL POWER POINT OF X(6)

Barycentrics    a2(b4 - c4)/(log b - log c) : :

X(40346) lies on these lines: {6, 40299}, {755, 40302}, {14970, 40301}

X(40346) = barycentric product X(i)*X(j) for these {i, j}: {38, 40345}, {826, 40302}
X(40346) = barycentric quotient X(2084)/X(40338)
X(40346) = trilinear product X(i)*X(j) for these {i, j}: {39, 40345}, {2084, 40301}
X(40346) = trilinear quotient X(826)/X(40339)
X(40346) = X(827)-isoconjugate-of-X(40339)

X(40347) = ISOGONAL CONJUGATE OF X(37784)

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

X(40347) lies on the conic {{A,B,C,X(2),X(6)}, the cubics K478 and K1166, and these lines: {6, 5181}, {25, 1560}, {111, 858}, {112, 40326}, {115, 8770}, {230, 8749}, {393, 2493}, {1611, 40144}, {1976, 10836}, {2987, 11064}, {3291, 8791}, {6339, 28419}, {6587, 14998}, {9606, 39389}, {13881, 21448}, {14772, 14948}, {30535, 37648}, {34609, 36616}

X(40347) = isogonal conjugate of X(37784)
X(40347) = isotomic conjugate of X(37803)
X(40347) = X(i)-cross conjugate of X(j) for these (i,j): {14908, 67}, {14961, 6}, {21639, 4}
X(40347) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37784}, {19, 5866}, {31, 37803}, {63, 37777}, {2349, 20772}
X(40347) = cevapoint of X(647) and X(1648)
X(40347) = trilinear pole of line {512, 6467}
X(40347) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 37803}, {3, 5866}, {6, 37784}, {25, 37777}, {1495, 20772}, {14908, 39169}

X(40348) = X(20)X(68)∩X(25)X(53)

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

X(40348) lies on the cubic K1166 and these lines: {20, 68}, {25, 53}, {5392, 34608}, {7386, 37802}, {11181, 30739}, {12362, 34853}

X(40348) = X(47)-isoconjugate of X(36889)
X(40348) = barycentric product X(i)*X(j) for these {i,j}: {68, 40138}, {376, 2165}, {925, 9209}, {5392, 26864}
X(40348) = barycentric quotient X(i)/X(j) for these {i,j}: {376, 7763}, {2165, 36889}, {9209, 6563}, {26864, 1993}, {40138, 317}

X(40349) = X(3)X(6)∩X(112)X(37948)

Barycentrics    a^2*(a^2 - b^2 - c^2)^2*(2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4) : :
Barycentrics    Cos[A]*Csc[B]*Csc[C]*(5*Cot[w] + Cot[B]*Cot[C]*Cot[w] - 6*Csc[A]*Csc[B]*Csc[C]) : :

X(40349) lies on these lines: {3, 6}, {112, 37948}, {115, 10257}, {230, 16976}, {232, 2071}, {441, 32459}, {625, 35923}, {647, 22089}, {3199, 12084}, {3289, 21663}, {3548, 7748}, {5866, 36212}, {6390, 15526}, {6640, 39565}, {6644, 33843}, {7386, 39602}, {7746, 15075}, {7756, 11585}, {7816, 28407}, {9155, 34147}, {10311, 15078}, {10313, 37941}, {11598, 11672}, {13509, 15035}, {14581, 34152}, {15013, 32456}, {23967, 39020}, {35067, 39008}

X(40349) = isogonal conjugate of the polar conjugate of X(5159)
X(40349) = isotomic conjugate of the polar conjugate of X(21639)
X(40349) = X(40347)-complementary conjugate of X(20305)
X(40349) = X(5159)-Ceva conjugate of X(21639)
X(40349) = crosssum of X(6) and X(37777)
X(40349) = crossdifference of every pair of points on line {523, 6353}
X(40349) = barycentric product X(i)*X(j) for these {i,j}: {3, 5159}, {69, 21639}
X(40349) = barycentric quotient X(i)/X(j) for these {i,j}: {5159, 264}, {21639, 4}
X(40349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 574, 216}, {3, 10316, 15513}, {3, 14961, 187}, {3, 15905, 5210}, {3, 23115, 5206}, {187, 14961, 3284}, {574, 5065, 5013}, {15166, 15167, 22401}

X(40350) = X(23)X(111)∩X(25)X(32)

Barycentrics    a^2*(2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4) : :
X(40350) = X[9225] - 3 X[20998]

X(40350) is the intersection of the tangents to the Moses-Lemoine conic at X(111) and X(1194). (Randy Hutson, December 18, 2020)

X(40350) lies on these lines: {2, 7748}, {4, 15820}, {22, 15513}, {23, 111}, {25, 32}, {39, 1995}, {50, 37972}, {110, 1570}, {115, 468}, {148, 37803}, {230, 37897}, {232, 15262}, {511, 9225}, {574, 11284}, {625, 7665}, {858, 10418}, {1194, 5041}, {1495, 1692}, {1503, 6388}, {1611, 20850}, {1691, 32237}, {2056, 21849}, {2079, 37928}, {2393, 32740}, {2489, 8651}, {2493, 3284}, {2502, 3292}, {2549, 40132}, {3066, 5034}, {3117, 34098}, {3767, 4232}, {3832, 15880}, {3934, 26257}, {4239, 16589}, {5007, 9465}, {5013, 5020}, {5023, 8770}, {5028, 35259}, {5038, 5943}, {5052, 34417}, {5106, 18860}, {5189, 39602}, {5309, 26255}, {5913, 37900}, {6688, 10329}, {6781, 16317}, {7398, 31404}, {7453, 21838}, {7492, 39576}, {7493, 7746}, {7664, 31275}, {7747, 10301}, {7756, 30739}, {8585, 8589}, {8588, 21448}, {8854, 12962}, {8855, 12969}, {13192, 23061}, {15822, 26235}, {16042, 31652}, {16055, 30749}, {16306, 18487}, {16320, 23991}, {17129, 33651}, {19577, 32457}, {26864, 39764}, {35265, 39024}

X(40350) = polar conjugate of the isotomic conjugate of X(21639)
X(40350) = crosspoint of X(25) and X(111)
X(40350) = crosssum of X(i) and X(j) for these (i,j): {69, 524}, {394, 5866}, {1648, 3566}
X(40350) = crossdifference of every pair of points on line {1649, 3265}
X(40350) = barycentric product X(i)*X(j) for these {i,j}: {4, 21639}, {25, 5159}
X(40350) = barycentric quotient X(i)/X(j) for these {i,j}: {5159, 305}, {21639, 69}
X(40350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {23, 111, 3291}, {23, 3291, 187}, {25, 34481, 1196}, {1495, 3124, 1692}, {2502, 20977, 3292}, {3292, 20977, 5107}

X(40351) = X(251)X(8749)∩X(699)X(1304)

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

X(40351) lies on these lines: {251, 8749}, {699, 1304}, {1084, 36417}, {3407, 16080}

X(40351) = X(i)-isoconjugate of X(j) for these (i,j): {304, 3260}, {305, 14206}, {561, 11064}, {1928, 3284}, {2173, 40050}, {2631, 4609}, {4602, 9033}
X(40351) = barycentric product X(i)*X(j) for these {i,j}: {32, 8749}, {74, 1974}, {512, 32715}, {560, 36119}, {669, 1304}, {798, 36131}, {1501, 16080}, {1973, 2159}, {2207, 18877}, {2489, 32640}, {3049, 32695}, {9426, 16077}, {14574, 18808}, {14601, 35908}, {14919, 36417}, {22455, 34416}
X(40351) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 40050}, {1304, 4609}, {1501, 11064}, {1974, 3260}, {8749, 1502}, {9233, 3284}, {9426, 9033}, {23216, 1650}, {32715, 670}, {36119, 1928}, {36131, 4602}

X(40352) = X(3)X(74)∩X(6)X(32738)

Barycentrics    a^4*(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) : :

X(40352) lies on the cubics K594 and K1171, and on these lines: {3, 74}, {6, 32738}, {25, 8749}, {30, 2986}, {98, 468}, {154, 2351}, {184, 1576}, {228, 692}, {235, 8884}, {237, 14908}, {265, 34104}, {351, 878}, {974, 14703}, {1112, 14673}, {1177, 14380}, {1399, 1410}, {1402, 14975}, {1492, 2349}, {1494, 1799}, {1495, 3003}, {1624, 13198}, {1632, 36178}, {1660, 3135}, {1885, 10152}, {2200, 32739}, {2394, 9147}, {2491, 32740}, {3425, 35908}, {3542, 34449}, {4630, 10547}, {6795, 36789}, {7493, 36875}, {8644, 32741}, {8651, 39840}, {9140, 30510}, {9407, 32715}, {11402, 15291}, {11799, 34150}, {13558, 15647}, {14177, 36311}, {14181, 36308}, {14567, 14600}, {14989, 18325}, {15270, 40319}, {15627, 32736}, {17938, 17970}, {32695, 32725}

X(40352) = isogonal conjugate of X(3260)
X(40352) = isogonal conjugate of the anticomplement of X(3003)
X(40352) = isogonal conjugate of the isotomic conjugate of X(74)
X(40352) = isogonal conjugate of the polar conjugate of X(8749)
X(40352) = polar conjugate of the isotomic conjugate of X(18877)
X(40352) = X(i)-Ceva conjugate of X(j) for these (i,j): {74, 18877}, {1304, 2433}, {10419, 6}
X(40352) = X(i)-cross conjugate of X(j) for these (i,j): {9407, 32}, {14270, 1576}
X(40352) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3260}, {2, 14206}, {30, 75}, {69, 1784}, {76, 2173}, {85, 7359}, {92, 11064}, {99, 36035}, {304, 1990}, {312, 6357}, {321, 18653}, {328, 35201}, {525, 24001}, {561, 1495}, {668, 11125}, {799, 1637}, {811, 9033}, {1099, 1494}, {1502, 9406}, {1577, 2407}, {1650, 23999}, {1733, 36891}, {1928, 9407}, {1969, 3284}, {1978, 14399}, {2166, 6148}, {2349, 36789}, {2420, 20948}, {2631, 6331}, {3163, 33805}, {4240, 14208}, {4554, 14400}, {4602, 14398}, {5664, 32680}, {6739, 14616}, {9214, 14210}
X(40352) = cevapoint of X(i) and X(j) for these (i,j): {32, 9407}, {14575, 19627}, {20975, 21731}, {34394, 34395}
X(40352) = crosspoint of X(i) and X(j) for these (i,j): {6, 34178}, {74, 8749}, {1989, 11744}
X(40352) = crosssum of X(i) and X(j) for these (i,j): {2, 146}, {30, 11064}, {69, 1272}, {323, 2071}, {23097, 36789}
X(40352) = trilinear pole of line {32, 3049}
X(40352) = crossdifference of every pair of points on line {1637, 5664}
X(40352) = barycentric product of circumcircle intercepts of line X(6)X(647)
X(40352) = barycentric product X(i)*X(j) for these {i,j}: {1, 2159}, {3, 8749}, {4, 18877}, {6, 74}, {19, 35200}, {25, 14919}, {31, 2349}, {32, 1494}, {48, 36119}, {50, 5627}, {56, 15627}, {64, 15291}, {110, 2433}, {111, 9717}, {112, 14380}, {184, 16080}, {186, 11079}, {187, 9139}, {248, 35908}, {520, 32695}, {523, 32640}, {525, 32715}, {560, 33805}, {647, 1304}, {656, 36131}, {661, 36034}, {1576, 2394}, {1976, 35910}, {1989, 14385}, {2088, 15395}, {2623, 36831}, {2715, 32112}, {3003, 10419}, {3049, 16077}, {3470, 14579}, {5158, 22455}, {8675, 32681}, {8739, 39377}, {8740, 39378}, {9404, 36064}, {9407, 31621}, {9409, 34568}, {10152, 14642}, {12079, 23357}, {14264, 14910}, {14270, 39290}, {15459, 39201}, {18808, 32661}, {32654, 36875}, {32740, 36890}, {34178, 36896}, {34394, 36308}, {34395, 36311}
X(40352) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3260}, {31, 14206}, {32, 30}, {50, 6148}, {74, 76}, {184, 11064}, {217, 1568}, {560, 2173}, {669, 1637}, {798, 36035}, {1304, 6331}, {1397, 6357}, {1494, 1502}, {1495, 36789}, {1501, 1495}, {1576, 2407}, {1917, 9406}, {1919, 11125}, {1973, 1784}, {1974, 1990}, {1980, 14399}, {2159, 75}, {2175, 7359}, {2206, 18653}, {2349, 561}, {2433, 850}, {3049, 9033}, {5627, 20573}, {8749, 264}, {9139, 18023}, {9233, 9407}, {9406, 1099}, {9407, 3163}, {9408, 23097}, {9426, 14398}, {9717, 3266}, {11060, 14254}, {11079, 328}, {12079, 23962}, {14270, 5664}, {14380, 3267}, {14385, 7799}, {14567, 5642}, {14574, 2420}, {14575, 3284}, {14581, 34334}, {14600, 35912}, {14601, 35906}, {14919, 305}, {15291, 14615}, {15627, 3596}, {16080, 18022}, {18877, 69}, {19627, 1511}, {32640, 99}, {32654, 36891}, {32676, 24001}, {32695, 6528}, {32715, 648}, {32740, 9214}, {33805, 1928}, {34397, 14920}, {34416, 18487}, {35200, 304}, {36034, 799}, {36119, 1969}, {36131, 811}

X(40353) = X(6)X(11074)∩X(50)X(18877)

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

X(40353) lies on the cubic K1171 and these lines: {6, 11074}, {50, 18877}, {74, 3003}, {323, 3284}, {2433, 2436}, {5063, 14385}, {5627, 6128}, {8749, 14581}, {9407, 32715}, {34568, 35906}

X(40353) = isogonal conjugate of X(36789)
X(40353) = X(3049)-cross conjugate of X(32715)
X(40353) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36789}, {2, 1099}, {30, 14206}, {63, 34334}, {75, 3163}, {85, 6062}, {92, 16163}, {304, 16240}, {312, 1354}, {561, 9408}, {811, 14401}, {1553, 36102}, {1577, 3233}, {1784, 11064}, {2173, 3260}, {2349, 23097}, {2407, 36035}, {3081, 33805}, {9033, 24001}, {18750, 38956}, {23999, 39008}
X(40353) = crosssum of X(3163) and X(16163)
X(40353) = crossdifference of every pair of points on line {1553, 23097}
X(40353) = barycentric product X(i)*X(j) for these {i,j}: {32, 31621}, {74, 74}, {647, 34568}, {1304, 14380}, {2159, 2349}, {2394, 32640}, {5627, 14385}, {8749, 14919}, {9139, 9717}, {10419, 14264}, {16080, 18877}, {32715, 34767}, {35200, 36119}
X(40353) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36789}, {25, 34334}, {31, 1099}, {32, 3163}, {74, 3260}, {184, 16163}, {1397, 1354}, {1495, 23097}, {1501, 9408}, {1576, 3233}, {1974, 16240}, {2159, 14206}, {2175, 6062}, {3049, 14401}, {9407, 3081}, {14385, 6148}, {18877, 11064}, {31621, 1502}, {32640, 2407}, {32715, 4240}, {33581, 38956}, {34568, 6331}, {36131, 24001}
X(40353) = {X(74),X(36896)}-harmonic conjugate of X(3003)

X(40354) = X(6)X(74)∩X(83)X(16080)

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

X(40354) lies on the cubic K1171 and these lines: {6, 74}, {83, 16080}, {113, 6103}, {729, 1304}, {1300, 1990}, {1494, 6661}, {2159, 2281}, {2207, 3124}, {2211, 32715}, {2420, 5504}, {3225, 16077}, {6531, 20031}, {11060, 14581}, {18268, 36131}, {32640, 32654}

X(40354) = isogonal conjugate of the isotomic conjugate of X(8749)
X(40354) = X(i)-isoconjugate of X(j) for these (i,j): {30, 304}, {63, 3260}, {69, 14206}, {75, 11064}, {305, 2173}, {561, 3284}, {670, 2631}, {799, 9033}, {1784, 3926}, {2407, 14208}, {3265, 24001}, {3718, 6357}, {4563, 36035}, {4572, 14395}, {4602, 9409}, {7182, 7359}, {9406, 40050}, {16163, 33805}, {18653, 20336}
X(40354) = trilinear pole of line {669, 1974}
X(40354) = barycentric product X(i)*X(j) for these {i,j}: {6, 8749}, {19, 2159}, {25, 74}, {31, 36119}, {32, 16080}, {112, 2433}, {393, 18877}, {512, 1304}, {523, 32715}, {608, 15627}, {647, 32695}, {661, 36131}, {669, 16077}, {1096, 35200}, {1494, 1974}, {1576, 18808}, {1973, 2349}, {1976, 35908}, {2207, 14919}, {2501, 32640}, {3049, 15459}, {5627, 34397}, {8753, 9717}, {10152, 33581}, {14380, 32713}, {14385, 18384}, {14398, 34568}, {22455, 34417}, {32112, 32696}
X(40354) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 3260}, {32, 11064}, {74, 305}, {669, 9033}, {1304, 670}, {1494, 40050}, {1501, 3284}, {1924, 2631}, {1973, 14206}, {1974, 30}, {2159, 304}, {2433, 3267}, {8749, 76}, {9407, 16163}, {9426, 9409}, {14581, 36789}, {14601, 35912}, {16077, 4609}, {16080, 1502}, {18877, 3926}, {32640, 4563}, {32695, 6331}, {32715, 99}, {34397, 6148}, {34416, 1531}, {36119, 561}, {36131, 799}, {36417, 1990}

X(40355) = X(30)X(74)∩X(462)X(14372)

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

X(40355) lies on the cubicx K497 and K1171, and on these lines: {30, 74}, {462, 14372}, {463, 14373}, {1495, 3003}, {1511, 10419}, {4550, 39170}, {8749, 34397}, {9139, 15395}, {11060, 14581}, {11074, 11080}, {14254, 34289}

X(40355) = reflection of X(14560) in X(15295)
X(40355) = isogonal conjugate of X(6148)
X(40355) = isogonal conjugate of the anticomplement of X(6128)
X(40355) = isogonal conjugate of the isotomic conjugate of X(5627)
X(40355) = polar conjugate of the isotomic conjugate of X(11079)
X(40355) = X(i)-Ceva conjugate of X(j) for these (i,j): {74, 11074}, {5627, 11079}
X(40355) = X(i)-cross conjugate of X(j) for these (i,j): {32, 8749}, {512, 14560}, {3124, 2433}, {20975, 15475}
X(40355) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6148}, {63, 14920}, {69, 35201}, {75, 1511}, {304, 39176}, {323, 14206}, {662, 5664}, {2173, 7799}, {2407, 32679}, {3258, 24041}, {3260, 6149}, {8552, 24001}, {10411, 36035}
X(40355) = cevapoint of X(3457) and X(3458)
X(40355) = crosssum of X(3258) and X(5664)
X(40355) = trilinear pole of line {11060, 14398}
X(40355) = barycentric product X(i)*X(j) for these {i,j}: {4, 11079}, {6, 5627}, {74, 1989}, {115, 15395}, {265, 8749}, {476, 2433}, {512, 39290}, {1138, 11074}, {1304, 14582}, {1494, 11060}, {2159, 2166}, {2394, 14560}, {3457, 36311}, {3458, 36308}, {3470, 11071}, {6344, 18877}, {8737, 39378}, {8738, 39377}, {10412, 32640}, {14592, 32715}, {14919, 18384}, {18808, 32662}
X(40355) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6148}, {25, 14920}, {32, 1511}, {74, 7799}, {512, 5664}, {1973, 35201}, {1974, 39176}, {1989, 3260}, {2433, 3268}, {3124, 3258}, {5627, 76}, {8749, 340}, {11060, 30}, {11074, 1272}, {11079, 69}, {14560, 2407}, {14583, 36789}, {15395, 4590}, {32640, 10411}, {32715, 14590}, {39290, 670}

X(40356) = X(1511)X(3163)∩X(14270)X(14398)

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

X(40355) lies on the cubic K1171 and these lines: {1511, 3163}, {14270, 14398}, {14581, 34397}

X(40356) = isogonal conjugate of the isotomic conjugate of X(11070)
X(40356) = X(32)-cross conjugate of X(14581)
X(40356) = X(i)-isoconjugate of X(j) for these (i,j): {399, 33805}, {1272, 2349}
X(40356) = barycentric product X(i)*X(j) for these {i,j}: {6, 11070}, {25, 20123}, {1138, 1495}
X(40356) = barycentric quotient X(i)/X(j) for these {i,j}: {1495, 1272}, {9407, 399}, {11070, 76}, {14398, 14566}, {20123, 305}

X(40357) = X(2)X(19615)∩X(4)X(251)

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

X(40357) lies on the cubic K644 and these lines: {2, 19615}, {4, 251}, {6, 18018}, {83, 26209}, {8793, 17407}

X(40357) = X(83)-Ceva conjugate of X(40404)
X(40357) = X(3162)-cross conjugate of X(8793)
X(40357) = X(i)-isoconjugate of X(j) for these (i,j): {2172, 39129}, {20883, 39172}, {23208, 39733}
X(40357) = cevapoint of X(3162) and X(17407)
X(40357) = barycentric product X(i)*X(j) for these {i,j}: {1370, 16277}, {1799, 17407}, {8793, 18018}
X(40357) = barycentric quotient X(i)/X(j) for these {i,j}: {66, 39129}, {159, 3313}, {8793, 22}, {10547, 39172}, {16277, 13575}, {17407, 427}

X(40358) = X(2)X(2138)∩X(22)X(39172)

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

X(40358) lies on the cubics K555 and K644, and on these lines: {2, 2138}, {22, 39172}, {25, 39417}, {83, 26209}, {1176, 19153}, {20806, 36414}

X(40358) = polar conjugate of the isogonal conjugate of X(39172)
X(40358) = X(40009)-Ceva conjugate of X(34207)
X(40358) = X(i)-cross conjugate of X(j) for these (i,j): {6, 8743}, {2485, 39417}, {17409, 22}
X(40358) = X(i)-isoconjugate of X(j) for these (i,j): {63, 17407}, {66, 18596}, {1370, 2156}, {2353, 21582}
X(40358) = barycentric product X(i)*X(j) for these {i,j}: {22, 13575}, {206, 40009}, {264, 39172}, {315, 34207}, {2172, 39733}, {34254, 40144}
X(40358) = barycentric quotient X(i)/X(j) for these {i,j}: {22, 1370}, {25, 17407}, {206, 159}, {1760, 21582}, {2172, 18596}, {10316, 23115}, {13575, 18018}, {17409, 3162}, {20806, 28419}, {34207, 66}, {39172, 3}, {39417, 1289}, {40144, 13854}

leftri

Ceva-conjugates associated with the power curve: X(40359)-X(40375)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, November 17, 2020.

Let P(t) = at : bt : ct, on the power curve, as in the preambles just before X(40297) and X(40345). The P(t)-Ceva conjugate of P(u), denoted by P(t)©P(u) is given by

au(-au-t + bu-t + cu-t) : bu(au-t - bu-t + cu-t) : cu(au-t + bu-t - cu-t),

P(t)©P(u) is the perspector of the cevian triangle of P(t) and the anticevian triangle of P(u).

The appearance of (i,j,k) in the following list means that P(t)©P(u) = X(k):

(-9,-1,33807), (-8,-8,40359), (-8,-6,40360), (-8,-2,40361), (-8,0,33797), (-7,-1,33806), (-7,1,33791), (-6,-6,40362), (-6,-4,40050), (-6,-2,40073), (-6,-1,21585), (-6,0,33796), (-6,2,33802), (-5,-5,1928), (-5,-4,40363), (-5,-3,40364), (-5,-2,40365), (-5,-1,20641), (-5,0,21275), (-5,1,33790), (-4,-4,1502), (-4,-3,28659), (-4,-2,305), (-4,-1,20444), (-4,0,315), (-4,1,21366), (-4,2,33801), (-4,4,40366), (-3,-3,561), (-3,-2,3596), (-3,-1,304), (-3,0,6327), (3,1,1760), (-3,2,23849), (-2,-3,40367), (-2,-2,76), (-2,-1,312), (-2,0,69), (-2,1,1759), (-2,2,22), (-2,4,18796), (-1,-3,18837), (-1,-2,6382), (-1,1,75), (-1,0,8), (-1,1,63), (-1,2,1631), (-1,3,2172), (0,-2,6374), (0,-1,6376), (0,0,2), (0,1,9), (0,2,3), (0,3,32664), (0,4,206), (0,6,,40368), (0,8,,40369), (1,1,17149), (1,0,192), (1,1,1), (1,2,55), (1,3,48), (1,4,,40370), (1,5,17), (2,-2,19562), (2,0,194), (2,1,43), (2,2,6), (2,3,41), (2,4,184), (2,5,40371), (2,6,20968), (3,-1,33788), (3,0,17486), (3,1,1740), (3,2,2176), (3,3,31), (3,4,2175), (3,5,9247), (4,0,8264), (4,2,1613), (4,3,2209), (4,4,32), (4,5,9447), (4,6,14575), (4,8,2,(5,1,33782), (5,2,21776), (5,5,560), (5,6,9448), (6,2,33786), (6,6,1501), (6,8,40373), (7,1,33783), (7,7,1917), (8,8,9233)

A few more: (0,1/2 40374), (1/2,1/2,366), (1/2,1,364), (1/2,3/2,4166), (1/2,2,20469), (1,1/2,40375)

For fixed t = t0 and variable u, the locus of P(t0)©P(u) is here named the P(t0)©P(u)-Ceva power curve. For fixed variable t and fixed u = u0, the locus of P(t)©P(u0) is here named the P(t)©P(u0)-Ceva power curve.

X(40359) = P(-8)©P(-8)

Barycentrics    b^8*c^8 : :

X(40359) lies on these lines: {6, 38812}, {76, 14820}, {561, 23626}, {626, 1502}, {4609, 7796}, {8149, 14603}, {9065, 23849}

X(40359) = isotomic conjugate of X(9233)
X(40359) = polar conjugate of isogonal conjugate of X(40360)
X(40359) = complement of X(40381)
X(40359) = anticomplement of X(40376)
X(40359) = barycentric square of X(1502)

X(40360) = P(-8)©P(-6)

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

X(40360) lies on these lines: {194, 14603}, {305, 20819}, {670, 12220}, {1502, 3314}, {4609, 40073}, {6374, 23642}, {8264, 35528}

X(40360) = isogonal conjugate of polar conjugate of X(40359)
X(40360) = isotomic conjugate of isogonal conjugate of X(40050)
X(40360) = barycentric quotient X(69)/X(1501)

X(40361) = P(-8)©P(-2)

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

X(40361) lies on these lines: {32, 76}, {305, 7871}, {3001, 40073}, {4174, 33806}, {7752, 40074}, {7855, 8024}

X(40361) = barycentric product X(76)*X(33796)

X(40362) = P(-6)©P(-6)

Barycentrics    b^6 c^6 : :

X(40362) lies on these lines: {1, 35529}, {2, 14603}, {6, 35530}, {22, 689}, {75, 35527}, {76, 19562}, {305, 4609}, {308, 1239}, {561, 2887}, {670, 2979}, {1235, 5117}, {1502, 3314}, {1928, 35523}, {3124, 40162}, {6374, 8041}, {6386, 32862}, {8039, 23962}, {10010, 39998}, {18018, 40073}, {20023, 20024}, {33802, 38842}

X(40362) = isogonal conjugate of X(9233)
X(40362) = isotomic conjugate of X(1501)
X(40362) = complement of X(40382)
X(40362) = anticomplement of X(40377)
X(40362) = barycentric square of X(561)

X(40363) = P(-5)©P(-4)

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

X(40363) lies on these lines: {76, 3782}, {305, 6386}, {312, 20684}, {313, 561}, {704, 33782}, {871, 40033}, {1211, 6382}, {1928, 40050}, {1978, 4417}, {3596, 3703}, {5224, 40087}, {6376, 7034}, {17149, 35539}, {28659, 30713}

X(40363) = isogonal conjugate of X(41280)
X(40363) = isotomic conjugate of X(1397)
X(40363) = cevapoint of X(75) and X(21594)
X(40363) = trilinear product X(i)*X(j) for these {i, j}: {2, 28659}, {8, 561}, {9, 1502}, {10, 40072}
X(40363) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 1917}, {31, 1397}, {32, 604}, {34, 14575}, {56, 560}, {57, 1501}
X(40363) = barycentric product X(8)*X(1502)

X(40364) = P(-5)©P(-3)

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

X(40364) lies on these lines: {38, 75}, {192, 35545}, {304, 18671}, {305, 20336}, {326, 336}, {349, 1502}, {720, 21776}, {799, 1760}, {1102, 3403}, {1920, 2345}, {1921, 4000}, {1928, 1969}, {2643, 18069}, {4602, 33805}, {9239, 18837}, {18068, 33781}, {18833, 23051}, {20641, 21582}, {20915, 20944}, {40050, 40071}

X(40364) = isogonal conjugate of polar conjugate of X(1928)
X(40364) = isotomic conjugate of X(1973)
X(40364) = polar conjugate of trilinear square of X(25)
X(40364) = barycentric product X(69)*X(561)

X(40365) = P(-5)©P(-2)

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

X(40365) lies on these lines: {1, 76}, {305, 3006}, {310, 29829}, {1492, 18796}, {1932, 4172}, {3596, 6757}, {3616, 30893}, {4153, 20444}, {6374, 35541}, {6382, 35546}, {8024, 29832}, {17143, 36500}, {18152, 29830}, {19562, 35529}, {26230, 40022}, {29831, 39998}, {32773, 33940}, {33108, 33933}, {35517, 40071}, {35552, 40073}

X(40365) = isotomic conjugate of X(7087)
X(40365) = barycentric product X(76)*X(6327)

X(40366) = P(-4)©P(4)

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

X(40366) lies on these lines: {2, 66}, {110, 34254}, {184, 1180}, {315, 39466}, {2001, 18796}, {2909, 5012}, {4630, 36414}, {28710, 37183}

X(40366) = barycentric product X(32)*X(33797)

X(40367) = P(-2)©P(-3)

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

X(40367) lies on these lines: {76, 21138}, {561, 21140}, {700, 33788}, {1502, 1928}, {1921, 10010}, {1925, 4485}, {3596, 14603}, {7034, 33938}, {18837, 35538}

X(40367) = isotomic conjugate of isogonal conjugate of X(6382)
X(40367) = barycentric product X(192)*X(1502)

X(40368) = P(0)©P(6)

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

X(40368) lies on these lines: {31, 40145}, {51, 5007}, {1501, 19556}, {1576, 2979}, {5133, 7792}, {6327, 34069}

X(40368) = centroid of X(31) and its extraversions
X(40368) = barycentric product X(1501)*X(33796)

X(40369) = P(0)©P(8)

Barycentrics    a^8*(a^8 - b^8 - c^8) : :

X(40369) lies on these lines: {32, 39466}, {315, 4630}, {6680, 6697}, {10316, 14574}

X(40369) = barycentric product X(9233)*X(33797)

X(40370) = P(1)©P(4)

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

X(40370) lies on these lines: {1, 7096}, {32, 1917}, {76, 1492}, {110, 34016}, {184, 1475}, {206, 942}, {215, 3202}, {692, 3730}, {766, 2172}, {1631, 20739}, {1932, 4116}, {1974, 2333}, {2242, 18759}, {8618, 9247}, {14963, 23849}, {22164, 35327}

X(40370) = barycentric product X(32)*X(6327)

X(40371) = P(2)©P(5)

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

X(40371) lies on these lines: {6, 7087}, {75, 825}, {560, 9233}, {2260, 4275}, {9247, 22363}, {9407, 9449}, {15624, 32739}, {20444, 38840}

X(40371) = isogonal conjugate of isotomic conjugate of X(32664)
X(40371) = barycentric product X(560)*X(6327)

X(40372) = P(2)©P(5)

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

X(40372) lies on these lines: {2, 4630}, {32, 39466}, {184, 14574}, {206, 36414}, {8023, 9233}, {19556, 33728}, {20968, 22075}

X(40372) = barycentric product X(315)*X(9233)

X(40373) = P(6)©P(8)

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

X(40373) lies on these lines: {32, 2909}, {184, 4173}, {1974, 14573}, {1976, 10312}, {3202, 19627}, {3492, 35924}, {9247, 22364}, {9418, 20968}, {9967, 21637}, {14575, 14585}, {19558, 20960}

X(40373) = isogonal conjugate of polar conjugate of X(1501)
X(40373) = isotomic conjugate of polar conjugate of X(9233)
X(40373) = barycentric product X(69)*X(9233)

X(40374) = P(0)©P(1/2)

Barycentrics    Sqrt[a]*(Sqrt[a] - Sqrt[b] - Sqrt[c]) : :

X(40374) lies on these lines: {1, 366}, {2, 4182}, {7, 20527}

X(40374) = barycentric product X(366)*X(20534)

X(40375) = P(1)©P(1/2)

Barycentrics    a*(Sqrt[b*c] - Sqrt[c*a] - Sqrt[a*b]) : :

X(40375) lies on these lines: {1, 366}, {43, 365}, {87, 20664}, {238, 20673}

leftri

Complements and anticomplements associated with the power curve: X(40376)-X(40383)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, November 19, 2020.

Suppose that P(t) = at : bt : ct (barycentrics) is a point on the power curve. The complement of P(t) is the point bt + ct : ct + at : at + bt. The anticomplement of P(t) is the point -at + bt + ct : at - bt + ct : at + bt - ct.

The appearance of (i,j) in the following list means that the (complement of X(i)) = X(j):

(-8,40376), (-6,40377), (-4,8265), (-3,16584), (-2,39), (-1,37), (-1/2,40378), (0,2), (1/2,20527), (1,10), (3/2,20334), (2,141), (5/2,20543), (3,2887), (4,626), (5,21235), (6,40379), (8,40380)

The appearance of (i,j) in the following list means that the (anticomplement of X(i)) = X(j):

(-8,40381), (-6,40382), (-4,8264), (-3,17486), (-2,194), (-1,192), (-1/2,40383), (0,2), (1/2,20534), (1,8), (3/2,20346), (2,69), (5/2,20555), (3,6327), (4,315), (5,21275), (6,33796), (8,33797)

X(40376) = COMPLEMENT OF POWER POINT P(-8)

Barycentrics    a^8 (b^8 + c^8) : :

X(40376) lies on these lines: {2, 40359}, {32, 14946}, {6680, 8265}, {9233, 40369}

X(40376) = complement of X(40359)
X(40376) = barycentric product X(9233)*X(40380)

X(40377) = COMPLEMENT OF POWER POINT P(-6)

Barycentrics    a^6 (b^6 + c^6) : :

X(40377) lies on these lines: {2, 14603}, {6, 23173}, {31, 14945}, {39, 4074}, {51, 1084}, {427, 35971}, {1194, 7792}, {1501, 19556}, {1915, 9468}, {6679, 16584}

X(40377) = complement of X(40362)
X(40377) = barycentric product X(1501)*X(40379)

X(40378) = COMPLEMENT OF POWER POINT P(-1/2)

Barycentrics    1/Sqrt[b] + 1/Sqrt[c] : :

X(40378) lies on these lines: {1, 366}, {2, 18297}, {6, 20743}, {28, 20779}, {57, 364}, {81, 2069}, {20357, 20682}

X(40378) = complement of X(18297)
X(40378) = barycentric product X(366)*X(20527)

X(40379) = COMPLEMENT OF POWER POINT P(6)

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

X(40379) lies on these lines: {2, 1501}, {51, 5103}, {116, 35972}, {141, 427}, {184, 30747}, {316, 33301}, {458, 7784}, {625, 5943}, {626, 2387}, {698, 4121}, {746, 4177}, {779, 37845}, {1853, 7778}, {2076, 16275}, {3096, 11338}, {3763, 11324}, {3981, 5025}, {4048, 11550}, {5133, 24256}, {8024, 16893}, {8041, 31107}, {8878, 12212}, {16584, 30877}

X(40379) = isogonal conjugate of X(38829)
X(40379) = complement of X(1501)
X(40379) = barycentric product X(141)*X(5025)

X(40380) = COMPLEMENT OF POWER POINT P(8)

Barycentrics    b^8 + c^8 : :

X(40380) lies on these lines: {2, 9233}, {141, 21536}, {626, 3852}, {1502, 15449}

X(40380) = complement of X(9233)

X(40381) = ANTICOMPLEMENT OF POWER POINT P(-8)

Barycentrics    a^8*b^8 + a^8*c^8 - b^8*c^8 : :

X(40381) lies on these lines: {32, 8264}, {194, 1186}, {16985, 40366}, {19566, 31981}

X(40381) = anticomplement of X(40359)

X(40382) = ANTICOMPLEMENT OF POWER POINT P(-6)

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

X(40382) lies on these lines: {2, 14603}, {31, 17486}, {2998, 33798}, {7766, 8264}, {20064, 39347}

X(40382) = anticomplement of X(40362)

X(40383) = ANTICOMPLEMENT OF POWER POINT P(-1/2)

Barycentrics    Sqrt[a]*Sqrt[b] + Sqrt[a]*Sqrt[c] - Sqrt[b]*Sqrt[c] : :

X(40383) lies on these lines: {2, 18297}, {192, 366}, {239, 364}, {330, 367}, {2068, 17350}, {2069, 4393}

X(40383) = anticomplement of X(18297)

X(40384) = CEVAPOINT OF X(6) AND X(74)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)^2*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)^2 : :
Barycentrics    1/(cos A - 2 cos B cos C)^2 : :

X(40384) lies on the cubic K1172 and these lines: {2, 39290}, {6, 34568}, {15, 39378}, {16, 39377}, {74, 186}, {323, 3284}, {842, 40355}, {1138, 3258}, {1494, 3580}, {1990, 14165}, {2349, 18593}, {2394, 2411}, {3431, 14385}, {3470, 15032}, {7799, 11064}, {8431, 15404}, {9139, 9213}, {9717, 14355}, {11430, 38933}, {14264, 14685}, {36210, 36311}, {36211, 36308}

X(40384) = isogonal conjugate of X(3163)
X(40384) = isotomic conjugate of X(36789)
X(40384) = polar conjugate of X(34334)
X(40384) = isogonal conjugate of the complement of X(1494)
X(40384) = isotomic conjugate of the isogonal conjugate of X(40353)
X(40384) = isogonal conjugate of the isotomic conjugate of X(31621)
X(40384) = X(i)-cross conjugate of X(j) for these (i,j): {6, 74}, {647, 1304}, {974, 69}, {3269, 14380}, {11079, 10419}, {14264, 1494}
X(40384) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3163}, {6, 1099}, {9, 1354}, {19, 16163}, {30, 2173}, {31, 36789}, {48, 34334}, {57, 6062}, {63, 16240}, {75, 9408}, {162, 14401}, {610, 38956}, {661, 3233}, {1495, 14206}, {1553, 36151}, {1784, 3284}, {2159, 23097}, {2349, 3081}, {2420, 36035}, {2631, 4240}, {3260, 9406}, {9409, 24001}, {24000, 39008}
X(40384) = cevapoint of X(i) and X(j) for these (i,j): {6, 74}, {3269, 14380}, {14385, 18877}
X(40384) = X(40384) = crosssum of X(i) and X(j) for these (i,j): {30, 34582}, {3081, 36435}, {14401, 39008}
X(40384) = trilinear pole of line {74, 526} (the tangent to the circumcircle at X(74))
X(40384) = crossdifference of every pair of points on line {3081, 14401}
X(40384) = barycentric square of X(2349)
X(40384) = barycentric product X(i)*X(j) for these {i,j}: {6, 31621}, {74, 1494}, {76, 40353}, {525, 34568}, {1304, 34767}, {2159, 33805}, {2349, 2349}, {9139, 36890}, {14380, 16077}, {14919, 16080}
X(40384) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1099}, {2, 36789}, {3, 16163}, {4, 34334}, {6, 3163}, {25, 16240}, {30, 23097}, {32, 9408}, {55, 6062}, {56, 1354}, {64, 38956}, {74, 30}, {110, 3233}, {647, 14401}, {1304, 4240}, {1494, 3260}, {1495, 3081}, {2159, 2173}, {2349, 14206}, {2433, 1637}, {3269, 39008}, {3470, 10272}, {5627, 14254}, {5663, 1553}, {8749, 1990}, {9139, 9214}, {9408, 36435}, {9717, 5642}, {10419, 15454}, {14264, 113}, {14380, 9033}, {14385, 1511}, {14919, 11064}, {15627, 7359}, {18877, 3284}, {31621, 76}, {32640, 2420}, {32715, 23347}, {34568, 648}, {36119, 1784}, {40352, 1495}, {40353, 6}, {40354, 14581}, {40355, 14583}

X(40385) = X(4)X(6128)∩X(1302)X(3163)

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

X(40385) lies on the cubic K1172 and these lines: {4, 6128}, {1302, 3163}, {1494, 3580}, {18877, 32681}

X(40385) = barycentric product X(74)*X(39263)
X(40385) = barycentric quotient X(i)/X(j) for these {i,j}: {26864, 10564}, {39263, 3260}

X(40386) = X(4)X(3426)∩X(1495)X(9064)

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

X(40386) lies on the cubic K1172 and these lines: {4, 3426}, {1495, 9064}, {3580, 11070}, {9140, 18554}

X(40385) = barycentric product X(12112)*X(36889)
X(40385) = barycentric quotient X(i)/X(j) for these {i,j}: {3426, 18317}, {12112, 376}

X(40387) = X(2)X(74)∩X(1300)X(1990)

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

X(40387) lies on the cubic K1172 and these lines: {2, 74}, {1300, 1990}, {15472, 32738}, {36789, 39263}

X(40388) = X(6)X(38936)∩X(186)X(3003)

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

X(40388) lies on the cubic K1172 and these lines: {6, 38936}, {186, 3003}, {340, 687}, {1300, 1990}, {2501, 14222}, {3284, 10420}, {5962, 18877}, {11079, 32710}, {15454, 40138}

X(40388) = isogonal conjugate of the complement of X(2986)
X(40388) = polar conjugate of the isotomic conjugate of X(10419)
X(40388) = X(i)-cross conjugate of X(j) for these (i,j): {6, 8749}, {25, 1300}, {512, 10420}, {2501, 32695}
X(40388) = X(i)-isoconjugate of X(j) for these (i,j): {63, 113}, {1725, 11064}, {2315, 3260}, {13754, 14206}
X(40388) = https://en.wikipedia.org/wiki/Aslackby_and_Laughton#/media/File:St.James'_church,_Aslackby,_Lincs._-_geograph.org.uk_-_90690.jpg of X(i) and X(j) for these (i,j): {6, 14910}, {25, 40354}
X(40388) = trilinear pole of line {21731, 40352}
X(40388) = barycentric product X(i)*X(j) for these {i,j}: {4, 10419}, {74, 1300}, {403, 39379}, {687, 2433}, {1304, 15328}, {2394, 32708}, {2986, 8749}, {5627, 38936}, {10420, 18808}, {10421, 35373}, {14910, 16080}, {15421, 32695}, {36053, 36119}
X(40388) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 113}, {1300, 3260}, {2433, 6334}, {8749, 3580}, {10419, 69}, {14910, 11064}, {32695, 16237}, {32708, 2407}, {32715, 15329}, {38936, 6148}, {40352, 13754}, {40354, 3003}, {40355, 39170}

X(40389) = X(2)X(39290)∩X(6)X(5627)

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

X(40389) lies on the cubic K1172 and these lines: {2, 39290}, {6, 5627}, {74, 1989}, {3470, 11079}, {14582, 18808}

X(40389) = X(16080)-Ceva conjugate of X(5627)
X(40389) = crosspoint of X(10421) and X(16080)
X(40389) = barycentric product X(i)*X(j) for these {i,j}: {265, 10421}, {5627, 12383}
X(40389) = barycentric quotient X(i)/X(j) for these {i,j}: {10421, 340}, {12383, 6148}, {40355, 35372}

X(40390) = X(4)X(18781)∩X(74)X(35373)

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

X(40390) lies on the cubic K1172 and these lines: {4, 18781}, {74, 35373}, {186, 35372}, {1990, 3580}, {34834, 39176}

X(40390) = X(6)-cross conjugate of X(186)
X(40390) = X(32678)-isoconjugate of X(38401)
X(40390) = cevapoint of X(6) and X(35372)
X(40390) = barycentric product X(340)*X(35372)
X(40390) = barycentric quotient X(i)/X(j) for these {i,j}: {186, 12383}, {526, 38401}, {35372, 265}, {35373, 12028}

X(40391) = X(4)X(5627)∩X(6)X(34568)

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

X(40391) lies on the cubic K1172 and these lines: {4, 5627}, {6, 34568}, {11070, 16080}

X(40391) = X(2173)-isoconjugate of X(20123)
X(40391) = barycentric product X(14566)*X(34568)
X(40391) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 20123}, {399, 16163}, {8749, 11070}, {40354, 40356}

X(40392) = X(2)X(5627)∩X(6)X(38936)

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

X(40392) lies on the cubic K1172 and these lines: {2, 5627}, {6, 38936}, {74, 35373}

X(40393) = CEVAPOINT OF X(5) AND X(6)

Barycentrics    (a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 - 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :
Barycentrics    (sec A)/(sec A + 2 cos(B - C)) : :
Trilinears    1/(sin A + sin 2A cos(B - C)) : :

X(40393) lies on the Kiepert circumhyperbola and these lines: {2, 571}, {4, 569}, {5, 96}, {6, 5392}, {10, 2216}, {22, 262}, {76, 1993}, {94, 34545}, {98, 5133}, {275, 467}, {311, 1994}, {648, 9381}, {1176, 30505}, {2052, 5422}, {2986, 23292}, {6504, 11427}, {7494, 14494}, {7495, 7608}, {7500, 14484}, {7503, 13599}, {9221, 35921}, {10601, 34289}, {14492, 34603}, {37765, 39284}

X(40393) = isogonal conjugate of X(570)
X(40393) = isotomic conjugate of X(37636)
X(40393) = polar conjugate of X(1594)
X(40393) = isogonal conjugate of the complement of X(311)
X(40393) = isotomic conjugate of the anticomplement of X(37649)
X(40393) = isotomic conjugate of the complement of X(1994)
X(40393) = isotomic conjugate of the polar conjugate of X(1179)
X(40393) = X(i)-cross conjugate of X(j) for these (i,j): {6, 1166}, {1510, 99}, {2623, 110}, {5576, 264}, {13353, 95}, {16040, 107}, {18314, 648}, {37649, 2}
X(40393) = X(i)-isoconjugate of X(j) for these (i,j): {1, 570}, {19, 1216}, {31, 37636}, {42, 16698}, {48, 1594}, {92, 23195}, {1209, 2148}, {1238, 1973}, {4020, 10550}
X(40393) = cevapoint of X(i) and X(j) for these (i,j): {2, 1994}, {5, 6}, {216, 34951}
X(40393) = trilinear pole of line {523, 2070} (the polar of X(5) wrt the circumcircle)
X(40393) = barycentric product X(i)*X(j) for these {i,j}: {69, 1179}, {75, 2216}, {311, 1166}
X(40393) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 37636}, {3, 1216}, {4, 1594}, {5, 1209}, {6, 570}, {69, 1238}, {81, 16698}, {184, 23195}, {311, 1225}, {1166, 54}, {1179, 4}, {2216, 1}, {3518, 6152}, {13621, 6153}, {32085, 10550}

X(40394) = CEVAPOINT OF X(6) AND X(10)

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

X(40394) lies on these lines: {6, 28654}, {8, 595}, {10, 2206}, {92, 26223}, {257, 3219}, {312, 3187}, {333, 32025}, {835, 20966}, {894, 30690}, {1220, 5176}, {2994, 26065}, {3920, 4518}, {4997, 29833}, {5260, 31359}, {18359, 27064}

X(40394) = isotomic conjugate of X(17184)
X(40394) = isogonal conjugate of the complement of X(313)
X(40394) = isotomic conjugate of the anticomplement of X(5294)
X(40394) = X(i)-cross conjugate of X(j) for these (i,j): {6, 3453}, {3050, 101}, {4129, 190}, {5294, 2}, {7252, 100}, {24083, 4632}
X(40394) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3670}, {19, 11573}, {28, 22073}, {31, 17184}, {42, 18601}, {58, 4016}, {81, 20966}, {92, 23197}, {163, 21121}, {649, 3909}, {849, 20654}, {1333, 3454}, {2206, 20896}
X(40394) = cevapoint of X(i) and X(j) for these (i,j): {6, 10}, {9, 3293}, {220, 4097}
X(40394) = trilinear pole of line {522, 1324} (the polar of X(10) wrt the circumcircle)
X(40394) = barycentric product X(313)*X(3453)
X(40394) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3670}, {2, 17184}, {3, 11573}, {10, 3454}, {37, 4016}, {42, 20966}, {71, 22073}, {81, 18601}, {100, 3909}, {184, 23197}, {321, 20896}, {523, 21121}, {594, 20654}, {3453, 58}

X(40395) = CEVAPOINT OF X(6) AND X(28)

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

X(40395) lies on the Kiepert circumhyperbola and these lines: {2, 7054}, {4, 1175}, {6, 36419}, {10, 29}, {27, 226}, {28, 228}, {76, 7058}, {81, 1446}, {98, 37362}, {107, 1859}, {270, 580}, {321, 2287}, {447, 5294}, {469, 1751}, {1172, 2982}, {2052, 36421}, {2326, 37279}, {3149, 13599}, {5397, 37381}, {5466, 14775}, {5736, 40214}, {17758, 37389}, {26023, 32014}

X(40395) = isogonal conjugate of X(18591)
X(40395) = isogonal conjugate of the complement of X(286)
X(40395) = isotomic conjugate of isogonal conjugate of X(40570)
X(40395) = isotomic conjugate of complement of X(40571)
X(40395) = polar conjugate of X(442)
X(40395) = polar conjugate of the isogonal conjugate of X(1175)
X(40395) = X(i)-cross conjugate of X(j) for these (i,j): {6, 943}, {650, 107}, {15313, 99}, {17796, 39439}, {17924, 648}, {21007, 112}
X(40395) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18591}, {3, 2294}, {9, 39791}, {10, 14597}, {37, 4303}, {42, 18607}, {48, 442}, {71, 942}, {72, 2260}, {226, 23207}, {228, 5249}, {255, 1865}, {906, 23752}, {1214, 14547}, {1234, 9247}, {1409, 6734}, {1437, 21675}, {1838, 3990}, {1841, 3682}, {1859, 40152}, {8021, 37755}
X(40395) = cevapoint of X(i) and X(j) for these (i,j): {4, 1172}, {6, 28}, {284, 580}
X(40395) = trilinear pole of line {523, 2074} (the polar of X(28) wrt the circumcircle)
X(40395) = barycentric product X(i)*X(j) for these {i,j}: {99, 14775}, {264, 1175}, {286, 943}, {2982, 31623}
X(40395) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 442}, {6, 18591}, {19, 2294}, {27, 5249}, {28, 942}, {29, 6734}, {56, 39791}, {58, 4303}, {81, 18607}, {264, 1234}, {393, 1865}, {943, 72}, {1175, 3}, {1333, 14597}, {1474, 2260}, {1794, 3682}, {1826, 21675}, {2194, 23207}, {2259, 71}, {2299, 14547}, {2982, 1214}, {5317, 1841}, {7649, 23752}, {8747, 1838}, {11107, 31938}, {13739, 39772}, {14775, 523}, {15439, 23067}, {30733, 14054}, {31902, 3824}

X(40396) = CEVAPOINT OF X(6) AND X(33)

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

X(40396) lies on the Feuerbach circumhyperbola and these lines: {1, 947}, {4, 221}, {6, 7003}, {7, 412}, {8, 7078}, {9, 17916}, {33, 84}, {34, 3577}, {65, 36121}, {79, 1785}, {104, 6198}, {108, 3075}, {318, 3562}, {1172, 2182}, {1389, 1870}, {1476, 15500}, {1771, 7412}, {1838, 15909}, {1896, 3194}, {2000, 3561}, {2956, 3062}, {5706, 7149}, {18283, 34046}, {23710, 34485}

X(40396) = isogonal conjugate of X(17102)
X(40396) = isogonal conjugate of the complement of X(318)
X(40396) = X(i)-cross conjugate of X(j) for these (i,j): {1459, 108}, {1887, 4}
X(40396) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17102}, {2, 22063}, {3, 946}, {63, 2262}, {222, 20262}, {603, 23528}, {1804, 1856}
X(40396) = cevapoint of X(i) and X(j) for these (i,j): {1, 1771}, {6, 33}, {19, 3195}
X(40396) = trilinear pole of line {650, 39199} (the polar of X(33) wrt the circumcircle)
X(40396) = barycentric product X(92)*X(947)
X(40396) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17102}, {19, 946}, {25, 2262}, {31, 22063}, {33, 20262}, {281, 23528}, {947, 63}
X(40396) = {X(33),X(603)}-harmonic conjugate of X(38870)

X(40397) = CEVAPOINT OF X(6) AND X(34)

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

X(40397) lies on these lines: {4, 2192}, {6, 196}, {34, 40}, {48, 223}, {208, 937}, {219, 278}, {222, 14256}, {1396, 1465}, {1427, 14578}, {1875, 2194}

X(40397) = isogonal conjugate of the complement of X(273)
X(40397) = X(i)-cross conjugate of X(j) for these (i,j): {6, 1167}, {649, 108}
X(40397) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1071}, {63, 1864}, {78, 1108}, {212, 17862}, {219, 1210}, {268, 6260}, {283, 21933}, {312, 23204}, {333, 3611}, {3692, 37566}
X(40397) = cevapoint of X(i) and X(j) for these (i,j): {6, 34}, {208, 608}
X(40397) = trilinear pole of line {1946, 6129} (the polar of X(34) wrt the circumcircle)
X(40397) = barycentric product X(273)*X(1167)
X(40397) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1864}, {34, 1210}, {56, 1071}, {208, 6260}, {273, 1226}, {278, 17862}, {608, 1108}, {1167, 78}, {1397, 23204}, {1398, 37566}, {1402, 3611}, {1875, 1532}, {1880, 21933}

X(40398) = CEVAPOINT OF X(6) AND X(38)

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

X(40398) lies on these lines: {6, 7794}, {58, 518}, {81, 3912}, {241, 1412}, {593, 18206}, {741, 22116}, {831, 20969}, {1396, 5236}, {1509, 18157}, {4251, 39957}, {5276, 17758}

X(40398) = isogonal conjugate of X(16600)
X(40398) = isogonal conjugate of the complement of X(1930)
X(40398) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16600}, {6, 4972}, {10, 5299}, {37, 7191}, {42, 16706}, {65, 33950}, {82, 17456}, {83, 20969}, {92, 23203}, {213, 33940}, {251, 21249}, {512, 33951}, {692, 27712}, {1400, 4514}, {1500, 33955}, {1826, 7293}, {4628, 21125}, {18098, 18183}, {22077, 32085}
X(40398) = cevapoint of X(6) and X(38)
X(40398) = trilinear pole of line {2254, 3733} (the polar of X(38) wrt the circumcircle)
X(40398) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4972}, {6, 16600}, {21, 4514}, {38, 21249}, {39, 17456}, {58, 7191}, {81, 16706}, {86, 33940}, {141, 21425}, {184, 23203}, {284, 33950}, {514, 27712}, {662, 33951}, {757, 33955}, {1333, 5299}, {1437, 7293}, {1964, 20969}, {2530, 21125}, {3954, 21037}, {4020, 22077}, {16696, 17192}, {17187, 18183}

X(40399) = CEVAPOINT OF X(6) AND X(40)

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

X(40399) lies on the circumconic {{A,B,C,X(1),X(2}} and these lines: {1, 1167}, {2, 2256}, {28, 517}, {40, 2208}, {57, 2289}, {63, 1422}, {105, 17642}, {219, 278}, {279, 394}, {291, 25941}, {321, 16082}, {525, 2401}, {957, 4245}, {1123, 1377}, {1214, 34051}, {1336, 1378}, {2006, 3452}, {2192, 17784}, {2810, 16100}, {3219, 34056}, {15474, 33146}, {17658, 36122}, {25243, 35058}, {26591, 30710}, {26637, 39747}, {35057, 35348}

X(40399) = isogonal conjugate of X(1108)
X(40399) = isotomic conjugate of X(17862)
X(40399) = isogonal conjugate of the complement of X(322)
X(40399) = isotomic conjugate of the anticomplement of X(25091)
X(40399) = X(i)-cross conjugate of X(j) for these (i,j): {652, 100}, {14837, 651}, {25091, 2}
X(40399) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1108}, {6, 1210}, {9, 37566}, {19, 1071}, {27, 3611}, {31, 17862}, {32, 1226}, {57, 1864}, {58, 21933}, {92, 23204}, {909, 1532}, {1436, 6260}, {8602, 18239}
X(40399) = cevapoint of X(i) and X(j) for these (i,j): {1, 219}, {6, 40}, {9, 5687}
X(40399) = trilinear pole of line {513, 2077} (the polar of X(40) wrt the circumcircle)
X(40399) = barycentric product X(75)*X(1167)
X(40399) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1210}, {2, 17862}, {3, 1071}, {6, 1108}, {37, 21933}, {40, 6260}, {55, 1864}, {56, 37566}, {75, 1226}, {184, 23204}, {228, 3611}, {517, 1532}, {1167, 1}, {10310, 18239}, {11012, 40249}

X(40400) = CEVAPOINT OF X(6) AND X(44)

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

X(40400) lies on these lines: {6, 644}, {9, 38266}, {31, 678}, {44, 5548}, {81, 645}, {100, 20972}, {101, 604}, {294, 23836}, {608, 1783}, {651, 1407}, {666, 1462}, {739, 6079}, {1333, 1811}, {1635, 2316}, {5549, 28607}, {16671, 28615}

X(40400) = isogonal conjugate of X(16610)
X(40400) = isogonal conjugate of the complement of X(4358)
X(40400) = isogonal conjugate of the isotomic conjugate of X(36805)
X(40400) = polar conjugate of the isotomic conjugate of X(1811)
X(40400) = X(36805)-Ceva conjugate of X(1811)
X(40400) = X(i)-cross conjugate of X(j) for these (i,j): {1960, 100}, {3689, 1}, {21786, 101}
X(40400) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16610}, {2, 1149}, {6, 1266}, {42, 16711}, {57, 3880}, {63, 1878}, {81, 4695}, {88, 17460}, {92, 23205}, {101, 4927}, {106, 16594}, {190, 6085}, {514, 23832}, {901, 21129}, {903, 20972}, {1797, 5151}, {1978, 8660}, {3669, 23705}, {4358, 17109}, {6336, 22082}, {9456, 20900}
X(40400) = cevapoint of X(i) and X(j) for these (i,j): {6, 44}, {650, 2087}
X(40400) = trilinear pole of line {55, 667} (the polar of X(44) wrt the circumcircle)
X(40400) = crossdifference of every pair of points on line {6018, 6085}
X(40400) = barycentric product of circumcircle intercepts of line X(8)X(513)
X(40400) = barycentric product X(i)*X(j) for these {i,j}: {1, 1120}, {4, 1811}, {6, 36805}, {8, 8686}, {100, 23836}, {513, 6079}, {3699, 37627}
X(40400) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1266}, {6, 16610}, {25, 1878}, {31, 1149}, {42, 4695}, {44, 16594}, {55, 3880}, {81, 16711}, {184, 23205}, {513, 4927}, {519, 20900}, {667, 6085}, {692, 23832}, {902, 17460}, {1120, 75}, {1635, 21129}, {1811, 69}, {1980, 8660}, {2251, 20972}, {3939, 23705}, {6079, 668}, {8686, 7}, {21805, 21041}, {23202, 22082}, {23836, 693}, {36805, 76}, {37627, 3676}

X(40401) = CEVAPOINT OF X(6) AND X(45)

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

X(40401) lies on these lines: {1, 9456}, {6, 519}, {9, 609}, {31, 44}, {33, 2203}, {37, 604}, {45, 993}, {81, 312}, {100, 751}, {226, 1407}, {513, 750}, {608, 1826}, {739, 5276}, {940, 4795}, {1100, 38266}, {1743, 28615}, {2177, 39974}, {2221, 4383}, {2276, 17961}, {2295, 14584}, {2718, 32686}, {4945, 37633}, {14621, 17790}, {16885, 34819}

X(40401) = isogonal conjugate of X(4850)
X(40401) = isotomic conjugate of X(33934)
X(40401) = isogonal conjugate of the anticomplement of X(30818)
X(40401) = isogonal conjugate of the complement of X(4671)
X(40401) = X(4775)-cross conjugate of X(100)
X(40401) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4850}, {2, 995}, {6, 4389}, {7, 4266}, {31, 33934}, {42, 16712}, {56, 5233}, {57, 3877}, {58, 26580}, {81, 4424}, {89, 17461}, {92, 23206}, {190, 9002}, {306, 4247}, {901, 23888}, {4588, 21130}, {17196, 28658}, {20973, 39704}
X(40401) = cevapoint of X(6) and X(45)
X(40401) = crosssum of X(995) and X(4266)
X(40401) = trilinear pole of line {667, 1635} (the polar of X(45) wrt the circumcircle)
X(40401) = barycentric product X(i)*X(j) for these {i,j}: {1, 996}, {513, 9059}, {900, 36091}, {3762, 32686}
X(40401) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4389}, {2, 33934}, {6, 4850}, {9, 5233}, {31, 995}, {37, 26580}, {41, 4266}, {42, 4424}, {55, 3877}, {81, 16712}, {184, 23206}, {667, 9002}, {996, 75}, {1635, 23888}, {2177, 17461}, {2203, 4247}, {4653, 17196}, {4893, 21130}, {9059, 668}, {32686, 3257}, {36091, 4555}

X(40402) = CEVAPOINT OF X(6) AND X(53)

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

X(40402) lies on these lines: {4, 577}, {6, 1093}, {53, 1970}, {184, 393}, {216, 436}, {264, 394}, {1217, 13346}, {1352, 18855}, {1826, 4055}, {3087, 6526}, {6748, 18877}

X(40402) = isogonal conjugate of the complement of X(324)
X(40402) = X(i)-cross conjugate of X(j) for these (i,j): {2623, 112}, {15451, 107}
X(40402) = X(i)-isoconjugate of X(j) for these (i,j): {63, 389}, {2169, 34836}
X(40402) = cevapoint of X(i) and X(j) for these (i,j): {4, 436}, {6, 53}, {25, 217}
X(40402) = trilinear pole of line {2501, 39201} (the polar of X(53) wrt the circumcircle)
X(40402) = polar conjugate of isotomic conjugate of X(40448)
X(40402) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 389}, {53, 34836}, {8882, 19170}, {14569, 6750}

X(40403) = CEVAPOINT OF X(6) AND X(63)

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

X(40403) lies on these lines: {6, 3926}, {58, 1792}, {63, 1973}, {81, 7123}, {284, 30676}, {333, 1396}, {1098, 30733}, {1172, 30688}, {1310, 23620}, {1412, 1708}, {1509, 2303}, {2287, 17206}

X(40403) = isogonal conjugate of X(16583)
X(40403) = isogonal conjugate of the complement of X(304)
X(40403) = X(i)-cross conjugate of X(j) for these (i,j): {6586, 100}, {21789, 99}
X(40403) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16583}, {4, 23620}, {6, 3914}, {10, 16502}, {19, 17441}, {25, 18589}, {37, 614}, {42, 4000}, {65, 2082}, {69, 8020}, {71, 1851}, {75, 21750}, {86, 21813}, {92, 22363}, {210, 28017}, {213, 3673}, {225, 7124}, {226, 7083}, {393, 22057}, {497, 1400}, {512, 3732}, {661, 1633}, {872, 16750}, {1020, 17115}, {1040, 1880}, {1042, 6554}, {1245, 5286}, {1334, 7195}, {1427, 4319}, {1473, 1826}, {1474, 21015}, {1824, 7289}, {1843, 18084}, {1973, 20235}, {2171, 5324}, {2333, 17170}, {3668, 30706}, {3949, 4211}, {8750, 21107}
X(40403) = cevapoint of X(i) and X(j) for these (i,j): {6, 63}, {81, 2287}
X(40403) = crosssum of X(21750) and X(22363)
X(40403) = trilinear pole of line {3733, 8646} (the polar of X(63) wrt the circumcircle)
X(40403) = barycentric product X(i)*X(j) for these {i,j}: {21, 8817}, {81, 30701}, {274, 7123}, {310, 7084}, {314, 1037}, {332, 1041}, {333, 7131}, {2287, 30705}, {7253, 8269}
X(40403) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3914}, {3, 17441}, {6, 16583}, {21, 497}, {28, 1851}, {32, 21750}, {48, 23620}, {58, 614}, {60, 5324}, {63, 18589}, {69, 20235}, {72, 21015}, {81, 4000}, {86, 3673}, {110, 1633}, {184, 22363}, {213, 21813}, {255, 22057}, {283, 1040}, {284, 2082}, {662, 3732}, {905, 21107}, {1014, 7195}, {1037, 65}, {1041, 225}, {1333, 16502}, {1412, 28017}, {1437, 1473}, {1444, 17170}, {1509, 16750}, {1790, 7289}, {1812, 27509}, {1973, 8020}, {2193, 7124}, {2194, 7083}, {2287, 6554}, {2303, 5286}, {2328, 4319}, {4183, 1863}, {7084, 42}, {7123, 37}, {7131, 226}, {8269, 4566}, {8817, 1441}, {14935, 4516}, {16728, 17060}, {21789, 17115}, {30701, 321}, {30705, 1446}, {34055, 18084}

X(40404) = CEVAPOINT OF X(6) AND X(66)

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

X(40404) lies on the cubic K644 and these lines: {2, 66}, {6, 18018}, {69, 10316}, {83, 264}, {95, 7832}, {251, 13575}, {253, 10548}, {305, 20806}, {1289, 1843}, {2353, 31360}, {2419, 4580}, {3589, 37801}, {3618, 13854}, {6330, 32085}, {6340, 28708}, {9229, 10333}, {10547, 26926}, {18024, 31636}, {20563, 28695}, {27372, 28723}

X(40404) = isogonal conjugate of X(40938)
X(40404) = isogonal conjugate of the complement of X(18018)
X(40404) = isotomic conjugate of the polar conjugate of X(16277)
X(40404) = polar conjugate of X(41375)
X(40404) = X(i)-cross conjugate of X(j) for these (i,j): {6, 1176}, {647, 1289}, {10547, 1799}, {26926, 69}
X(40404) = cevapoint of X(6) and X(66)
X(40404) = trilinear pole of the polar of X(66) wrt the circumcircle
X(40404) = X(i)-isoconjugate of X(j) for these (i,j): {19, 3313}, {22, 17442}, {38, 8743}, {42, 16715}, {63, 27373}, {92, 23208}, {206, 20883}, {427, 2172}, {1235, 17453}, {1760, 1843}, {1930, 17409}, {1964, 17907}, {16747, 21034}, {19595, 19616}, {20641, 27369}, {23881, 32676}
X(40404) = barycentric product X(i)*X(j) for these {i,j}: {66, 1799}, {69, 16277}, {83, 14376}, {1176, 18018}
X(40404) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3313}, {25, 27373}, {66, 427}, {81, 16715}, {83, 17907}, {184, 23208}, {251, 8743}, {525, 23881}, {1176, 22}, {1799, 315}, {2156, 17442}, {2353, 1843}, {4580, 33294}, {9076, 11605}, {10547, 206}, {13854, 27376}, {14376, 141}, {16277, 4}, {18018, 1235}, {28724, 20806}, {34055, 1760}, {40146, 27369}

X(40405) = CEVAPOINT OF X(6) AND X(69)

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

X(40405) lies on these lines: {32, 193}, {69, 1974}, {76, 683}, {99, 6467}, {141, 32740}, {305, 40318}, {1918, 4028}, {1975, 6391}, {3618, 39238}, {6337, 17040}, {6394, 14601}, {6531, 9230}, {12272, 16276}, {22468, 35140}

X(40405) = isogonal conjugate of X(1196)
X(40405) = isotomic conjugate of X(5254)
X(40405) = isogonal conjugate of the complement of X(305)
X(40405) = isotomic conjugate of the anticomplement of X(7789)
X(40405) = isotomic conjugate of the complement of X(1975)
X(40405) = isogonal conjugate of the polar conjugate of X(683)
X(40405) = X(i)-cross conjugate of X(j) for these (i,j): {647, 99}, {7789, 2}
X(40405) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1196}, {6, 17872}, {19, 6467}, {25, 18671}, {31, 5254}, {42, 16716}, {63, 40325}, {92, 682}, {163, 12075}, {304, 3080}, {1096, 22401}, {1368, 1973}, {1974, 21406}, {38252, 40326}
X(40405) = cevapoint of X(i) and X(j) for these (i,j): {2, 1975}, {6, 69}, {394, 6337}
X(40405) = trilinear pole of line {669, 3265} (the polar of X(69) wrt the circumcircle)
X(40405) = barycentric product X(3)*X(683)
X(40405) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17872}, {2, 5254}, {3, 6467}, {6, 1196}, {25, 40325}, {63, 18671}, {69, 1368}, {81, 16716}, {184, 682}, {193, 40326}, {304, 21406}, {394, 22401}, {523, 12075}, {683, 264}, {1974, 3080}, {17206, 18648}

X(40406) = CEVAPOINT OF X(6) AND X(72)

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

X(40406) lies on these lines: {6, 17776}, {31, 3811}, {72, 2203}, {321, 5317}, {604, 1708}, {608, 5739}, {1333, 3998}, {1462, 4359}, {3693, 28615}, {4976, 24115}, {9456, 25091}

X(40406) = isogonal conjugate of X(40941)
X(40406) = isogonal conjugate of the complement of X(20336)
X(40406) = X(647)-cross conjugate of X(100)
X(40406) = X(i)-isoconjugate of X(j) for these (i,j): {6, 23537}, {19, 18732}, {25, 18651}, {28, 18674}, {1474, 21530}
X(40406) = cevapoint of X(i) and X(j) for these (i,j): {6, 72}, {37, 5687}, {213, 12329}, {3990, 11517}
X(40406) = trilinear pole of line {667, 15313} (the polar of X(72) wrt the circumcircle)
X(40406) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23537}, {3, 18732}, {63, 18651}, {71, 18674}, {72, 21530}, {3949, 21678}

X(40407) = CEVAPOINT OF X(6) AND X(73)

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

X(40407) lies on these lines: {1, 7008}, {6, 7011}, {9, 16577}, {19, 223}, {55, 581}, {73, 2299}, {222, 1436}, {226, 8748}, {284, 2003}, {333, 17095}, {1427, 2160}

X(40407) = isogonal conjugate of X(40942)
X(40407) = isogonal conjugate of the complement of X(307)
X(40407) = X(647)-cross conjugate of X(109)
X(40407) = X(i)-isoconjugate of X(j) for these (i,j): {6, 23661}, {9, 4292}, {21, 1901}, {29, 18675}, {33, 18652}, {650, 14544}, {1172, 18641}
X(40407) = cevapoint of X(i) and X(j) for these (i,j): {6, 73}, {48, 1399}, {221, 1400}
X(40407) = crosssum of X(1901) and X(18675)
X(40407) = trilinear pole of line {663, 39199} (the polar of X(73) wrt the circumcircle)
X(40407) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23661}, {56, 4292}, {73, 18641}, {109, 14544}, {222, 18652}, {1400, 1901}, {1409, 18675}

X(40408) = CEVAPOINT OF X(6) AND X(81)

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

X(40408) lies on these lines: {6, 1509}, {32, 593}, {58, 1918}, {81, 213}, {99, 20963}, {741, 8708}, {757, 4251}, {981, 7760}, {1396, 31919}, {1974, 17562}, {2207, 36419}, {2238, 32014}, {3997, 32004}, {20970, 37128}

X(40408) = isogonal conjugate of X(16589)
X(40408) = isogonal conjugate of the anticomplement of X(36812)
X(40408) = isogonal conjugate of the complement of X(274)
X(40408) = X(i)-cross conjugate of X(j) for these (i,j): {667, 99}, {21007, 110}, {21788, 741}
X(40408) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16589}, {2, 2667}, {6, 21020}, {9, 39793}, {10, 20963}, {37, 3720}, {42, 3739}, {57, 4111}, {65, 3691}, {75, 21753}, {81, 21699}, {86, 21820}, {92, 22369}, {213, 20888}, {661, 4436}, {756, 18166}, {872, 16748}, {1018, 6372}, {1334, 4059}, {1400, 3706}, {1500, 17175}, {1826, 22060}, {18089, 21035}
X(40408) = cevapoint of X(i) and X(j) for these (i,j): {6, 81}, {58, 4251}
X(40408) = crosssum of X(i) and X(j) for these (i,j): {21699, 21820}, {21753, 22369}
X(40408) = trilinear pole of line {669, 2106} (the polar of X(81) wrt the circumcircle)
X(40408) = barycentric product X(i)*X(j) for these {i,j}: {81, 32009}, {7192, 8708}
X(40408) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21020}, {6, 16589}, {21, 3706}, {31, 2667}, {32, 21753}, {42, 21699}, {55, 4111}, {56, 39793}, {58, 3720}, {81, 3739}, {86, 20888}, {110, 4436}, {184, 22369}, {213, 21820}, {284, 3691}, {593, 18166}, {757, 17175}, {1014, 4059}, {1333, 20963}, {1437, 22060}, {1509, 16748}, {3733, 6372}, {8708, 3952}, {16948, 4891}, {32009, 321}
X(40408) = {X(81),X(213)}-harmonic conjugate of X(33770)

X(40409) = CEVAPOINT OF X(6) AND X(86)

Barycentrics    (a + b)*(a + 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(40409) lies on these lines: {6, 7304}, {32, 1509}, {81, 21759}, {86, 171}, {87, 1178}, {99, 2309}, {213, 274}, {1434, 7175}, {2663, 18787}, {9468, 37128}, {28369, 40017}

X(40409) = isogonal conjugate of X(21838)
X(40409) = isotomic conjugate of X(21024)
X(40409) = isogonal conjugate of the complement of X(310)
X(40409) = isotomic conjugate of the complement of X(33296)
X(40409) = X(i)-cross conjugate of X(j) for these (i,j): {649, 99}, {16737, 4573}, {18278, 741}, {21791, 110}
X(40409) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21838}, {6, 3728}, {9, 39780}, {10, 1197}, {31, 21024}, {37, 2309}, {42, 1107}, {58, 22206}, {81, 21700}, {92, 23212}, {213, 3741}, {872, 16738}, {893, 27880}, {1333, 21713}, {1500, 18169}, {1824, 22065}, {1826, 22389}, {1918, 20891}, {18091, 21814}
X(40409) = cevapoint of X(i) and X(j) for these (i,j): {2, 33296}, {6, 86}, {81, 17103}, {274, 34020}
X(40409) = trilinear pole of line {669, 4367} (the polar of X(86) wrt the circumcircle)
X(40409) = barycentric product X(i)*X(j) for these {i,j}: {81, 1221}, {274, 1258}
X(40409) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3728}, {2, 21024}, {6, 21838}, {10, 21713}, {37, 22206}, {42, 21700}, {56, 39780}, {58, 2309}, {81, 1107}, {86, 3741}, {171, 27880}, {184, 23212}, {274, 20891}, {757, 18169}, {1221, 321}, {1258, 37}, {1333, 1197}, {1434, 30097}, {1437, 22389}, {1509, 16738}, {1790, 22065}

X(40410) = CEVAPOINT OF X(2) AND X(5)

Barycentrics    (a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :
Barycentrics    (csc A)/(cos A + 2 sin B sin C) : :

X(40410) lies on these lines: {2, 10979}, {4, 36948}, {5, 95}, {69, 576}, {233, 648}, {253, 7486}, {261, 39280}, {264, 1656}, {287, 3589}, {288, 14389}, {305, 7539}, {307, 7321}, {311, 1487}, {316, 14788}, {317, 5056}, {340, 35018}, {547, 1494}, {1232, 26862}, {1441, 7504}, {1799, 37439}, {1972, 14767}, {5067, 8797}, {5070, 20477}, {7569, 20563}, {7570, 18019}, {7571, 18018}, {7887, 31360}, {9229, 32967}, {11090, 32807}, {14977, 39183}, {30786, 37454}, {32223, 38833}

X(40410) = isogonal conjugate of X(13366)
X(40410) = isotomic conjugate of X(140)
X(40410) = polar conjugate of X(6748)
X(40410) = isotomic conjugate of the anticomplement of X(3628)
X(40410) = isotomic conjugate of the complement of X(5)
X(40410) = isotomic conjugate of the isogonal conjugate of X(1173)
X(40410) = isotomic conjugate of the polar conjugate of X(39284)
X(40410) = polar conjugate of the isogonal conjugate of X(31626)
X(40410) = X(i)-Ceva conjugate of X(j) for these (i,j): {31617, 31626}, {39289, 1173}
X(40410) = X(i)-cross conjugate of X(j) for these (i,j): {2, 31617}, {5, 31610}, {1173, 39284}, {3628, 2}, {6368, 648}, {23061, 671}
X(40410) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13366}, {6, 17438}, {19, 22052}, {31, 140}, {32, 20879}, {48, 6748}, {213, 17168}, {233, 2148}, {560, 1232}, {661, 35324}, {692, 21103}, {810, 35311}, {1333, 21012}, {2190, 32078}
X(40410) = cevapoint of X(i) and X(j) for these (i,j): {2, 5}, {3, 1994}, {302, 303}, {1173, 31626}
X(40410) = trilinear pole of line {525, 15340}
X(40410) = barycentric product X(i)*X(j) for these {i,j}: {5, 31617}, {69, 39284}, {76, 1173}, {95, 31610}, {99, 39183}, {141, 39289}, {264, 31626}, {288, 311}, {305, 33631}, {343, 39286}, {525, 33513}, {1487, 7769}, {6331, 39180}
X(40410) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17438}, {2, 140}, {3, 22052}, {4, 6748}, {5, 233}, {6, 13366}, {10, 21012}, {75, 20879}, {76, 1232}, {86, 17168}, {110, 35324}, {140, 36422}, {216, 32078}, {288, 54}, {324, 14978}, {514, 21103}, {648, 35311}, {1173, 6}, {1487, 2963}, {1994, 1493}, {6368, 35441}, {20574, 14533}, {31610, 5}, {31617, 95}, {31626, 3}, {33513, 648}, {33631, 25}, {34545, 36153}, {35360, 35318}, {36412, 3078}, {39180, 647}, {39181, 23286}, {39183, 523}, {39284, 4}, {39286, 275}, {39289, 83}
X(40410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31610, 31626}, {31610, 31626, 39284}

X(40411) = CEVAPOINT OF X(2) AND X(19)

Barycentrics    (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) : :

X(40411) lies on these lines: {2, 2207}, {19, 304}, {27, 19799}, {28, 1043}, {232, 33828}, {264, 17682}, {274, 2322}, {333, 1396}, {475, 17277}, {1968, 33821}, {7058, 14013}, {7131, 16054}, {14829, 37382}, {17680, 27376}, {17907, 33833}, {27109, 35974}

X(40411) = isogonal conjugate of X(23620)
X(40411) = isotomic conjugate of X(18589)
X(40411) = polar conjugate of X(3914)
X(40411) = isotomic conjugate of the complement of X(19)
X(40411) = X(i)-cross conjugate of X(j) for these (i,j): {4228, 86}, {7192, 648}, {21300, 6528}, {21302, 18026}, {26153, 76}
X(40411) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23620}, {2, 22363}, {3, 16583}, {6, 17441}, {19, 22057}, {31, 18589}, {32, 20235}, {37, 1473}, {42, 7289}, {48, 3914}, {65, 7124}, {69, 21750}, {71, 614}, {72, 16502}, {73, 2082}, {213, 17170}, {228, 4000}, {326, 8020}, {497, 1409}, {647, 1633}, {692, 21107}, {810, 3732}, {1040, 1400}, {1214, 7083}, {1333, 21015}, {1402, 27509}, {1410, 6554}, {1439, 30706}, {1444, 21813}, {1851, 3990}, {1964, 18084}, {2197, 5324}, {2200, 3673}, {2281, 7386}, {2318, 28017}
X(40411) = cevapoint of X(i) and X(j) for these (i,j): {2, 19}, {27, 2322}, {3730, 3811}
X(40411) = trilinear pole of line {7253, 14954}
X(40411) = barycentric product X(i)*X(j) for these {i,j}: {27, 30701}, {29, 8817}, {314, 1041}, {2322, 30705}, {7131, 31623}
X(40411) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17441}, {2, 18589}, {3, 22057}, {4, 3914}, {6, 23620}, {10, 21015}, {19, 16583}, {21, 1040}, {27, 4000}, {28, 614}, {29, 497}, {31, 22363}, {58, 1473}, {75, 20235}, {81, 7289}, {83, 18084}, {86, 17170}, {162, 1633}, {270, 5324}, {284, 7124}, {286, 3673}, {333, 27509}, {514, 21107}, {648, 3732}, {1010, 7386}, {1037, 73}, {1041, 65}, {1172, 2082}, {1396, 28017}, {1474, 16502}, {1973, 21750}, {2207, 8020}, {2299, 7083}, {2322, 6554}, {2332, 30706}, {2333, 21813}, {4183, 4319}, {7084, 228}, {7123, 71}, {7131, 1214}, {8747, 1851}, {8817, 307}, {30701, 306}

X(40412) = CEVAPOINT OF X(2) AND X(21)

Barycentrics    (a + b)*(a + c)*(a^3 - a^2*b - a*b^2 + b^3 - 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c - 2*a*b*c - b^2*c - a*c^2 + c^3) : :
Barycentrics    1/((cos B + cos C) (b (cos A + cos B) + c (cos A + cos C))) : :

X(40412) lies on these lines: {2, 7054}, {21, 286}, {69, 261}, {81, 3990}, {85, 1789}, {86, 283}, {95, 7483}, {253, 17558}, {264, 405}, {287, 25536}, {305, 16992}, {306, 319}, {314, 943}, {757, 14828}, {1494, 15670}, {1793, 14616}, {1799, 37664}, {2982, 37870}, {5084, 8797}, {9229, 33047}, {17561, 36889}, {20291, 37369}

X(40412) = isogonal conjugate of X(40952)
X(40412) = isotomic conjugate of X(442)
X(40412) = polar conjugate of X(1865)
X(40412) = isotomic conjugate of the anticomplement of X(6675)
X(40412) = isotomic conjugate of the complement of X(21)
X(40412) = isotomic conjugate of the isogonal conjugate of X(1175)
X(40412) = X(i)-cross conjugate of X(j) for these (i,j): {521, 648}, {693, 99}, {6675, 2}, {22160, 110}
X(40412) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2294}, {19, 18591}, {31, 442}, {33, 39791}, {37, 2260}, {42, 942}, {48, 1865}, {65, 14547}, {71, 1841}, {73, 1859}, {213, 5249}, {225, 23207}, {228, 1838}, {560, 1234}, {692, 23752}, {1020, 33525}, {1254, 8021}, {1333, 21675}, {1402, 6734}, {1824, 4303}, {1826, 14597}, {2333, 18607}
X(40412) = cevapoint of X(i) and X(j) for these (i,j): {2, 21}, {3, 81}, {2328, 4251}
X(40412) = trilinear pole of line {448, 525}
X(40412) = barycentric product X(i)*X(j) for these {i,j}: {76, 1175}, {274, 943}, {310, 2259}, {314, 2982}, {4563, 14775}
X(40412) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2294}, {2, 442}, {3, 18591}, {4, 1865}, {10, 21675}, {27, 1838}, {28, 1841}, {58, 2260}, {76, 1234}, {81, 942}, {86, 5249}, {222, 39791}, {284, 14547}, {333, 6734}, {514, 23752}, {943, 37}, {1172, 1859}, {1175, 6}, {1437, 14597}, {1444, 18607}, {1790, 4303}, {1794, 71}, {2193, 23207}, {2259, 42}, {2982, 65}, {5333, 3824}, {7054, 8021}, {14775, 2501}, {15439, 4559}, {21789, 33525}, {35320, 35307}, {36048, 1020}, {40214, 500}

X(40413) = CEVAPOINT OF X(2) AND X(25)

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

X(40413) lies on these lines: {2, 1968}, {4, 6340}, {25, 305}, {69, 1974}, {95, 6676}, {183, 40032}, {232, 9229}, {264, 5020}, {287, 1915}, {427, 30786}, {468, 1799}, {648, 1196}, {1078, 38282}, {1995, 18018}, {8770, 9308}, {13595, 18019}, {37962, 39998}

X(40413) = isogonal conjugate of X(6467)
X(40413) = isotomic conjugate of X(1368)
X(40413) = polar conjugate of X(5254)
X(40413) = isogonal conjugate of the anticomplement of X(14913)
X(40413) = isogonal conjugate of the complement of X(12272)
X(40413) = isotomic conjugate of the anticomplement of X(6677)
X(40413) = isotomic conjugate of the complement of X(25)
X(40413) = isogonal conjugate of the isotomic conjugate of X(683)
X(40413) = X(i)-cross conjugate of X(j) for these (i,j): {512, 648}, {6677, 2}, {11326, 6}, {26156, 76}, {32529, 3225}
X(40413) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6467}, {3, 17872}, {6, 18671}, {19, 22401}, {31, 1368}, {32, 21406}, {48, 5254}, {63, 1196}, {71, 16716}, {75, 682}, {213, 18648}, {326, 40325}, {4575, 12075}
X(40413) = cevapoint of X(i) and X(j) for these (i,j): {2, 25}, {3, 193}, {4, 9308}
X(40413) = trilinear pole of line {525, 2451}
X(40413) = barycentric product X(6)*X(683)
X(40413) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18671}, {2, 1368}, {3, 22401}, {4, 5254}, {6, 6467}, {19, 17872}, {25, 1196}, {28, 16716}, {32, 682}, {75, 21406}, {86, 18648}, {683, 76}, {2207, 40325}, {2501, 12075}, {6353, 40326}, {36417, 3080}, {40318, 40337}

X(40414) = CEVAPOINT OF X(2) AND X(27)

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

X(40414) lies on these lines: {2, 36419}, {27, 306}, {69, 7058}, {95, 7536}, {264, 7522}, {286, 2064}, {307, 333}, {447, 20106}, {1441, 31623}, {29163, 39438}

X(40414) = isotomic conjugate of X(440)
X(40414) = polar conjugate of X(1834)
X(40414) = isotomic conjugate of the anticomplement of X(6678)
X(40414) = isotomic conjugate of the complement of X(27)
X(40414) = X(i)-cross conjugate of X(j) for these (i,j): {514, 648}, {6678, 2}, {13442, 7}, {20293, 6528}, {20294, 99}, {25015, 75}, {26167, 76}, {37113, 86}
X(40414) = X(i)-isoconjugate of X(j) for these (i,j): {6, 18673}, {31, 440}, {48, 1834}, {71, 1104}, {73, 2264}, {213, 18650}, {810, 14543}, {950, 1409}, {1333, 21671}, {1842, 3990}, {2200, 17863}
X(40414) = cevapoint of X(i) and X(j) for these (i,j): {2, 27}, {4, 2322}, {333, 18134}
X(40414) = trilinear pole of line {447, 525}
X(40414) = barycentric product X(286)*X(1257)
X(40414) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18673}, {2, 440}, {4, 1834}, {10, 21671}, {28, 1104}, {29, 950}, {86, 18650}, {286, 17863}, {648, 14543}, {951, 73}, {1172, 2264}, {1257, 72}, {2983, 71}, {8747, 1842}, {17925, 29162}, {29163, 4574}

X(40415) = CEVAPOINT OF X(2) AND X(31)

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

X(40415) lies on these lines: {2, 1501}, {21, 32010}, {31, 561}, {81, 4621}, {86, 7179}, {171, 334}, {238, 7018}, {261, 7305}, {286, 14006}, {314, 983}, {333, 3661}, {701, 9063}, {3736, 7303}, {4586, 16584}, {7132, 37870}, {7307, 18021}, {7369, 30657}, {17126, 30636}, {17127, 30635}

X(40415) = isogonal conjugate of X(3778)
X(40415) = isotomic conjugate of X(2887)
X(40415) = isotomic conjugate of the anticomplement of X(6679)
X(40415) = isotomic conjugate of the complement of X(31)
X(40415) = isotomic conjugate of the isogonal conjugate of X(38813)
X(40415) = X(i)-cross conjugate of X(j) for these (i,j): {788, 4586}, {6679, 2}, {17217, 99}, {17743, 38810}, {20561, 3226}, {21298, 671}, {21300, 648}, {21301, 190}, {21304, 668}, {21305, 670}, {24995, 75}, {26176, 76}
X(40415) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3778}, {2, 16584}, {6, 3721}, {7, 4531}, {10, 7032}, {19, 20727}, {31, 2887}, {32, 20234}, {37, 2275}, {41, 16888}, {42, 982}, {57, 20684}, {58, 7237}, {65, 3056}, {76, 21751}, {181, 3794}, {210, 7248}, {213, 3662}, {225, 20753}, {226, 20665}, {264, 22364}, {274, 21815}, {292, 18904}, {512, 3888}, {561, 8022}, {604, 4136}, {649, 7239}, {692, 3801}, {722, 14945}, {789, 17415}, {798, 33946}, {872, 33947}, {893, 18905}, {1042, 4073}, {1333, 16886}, {1400, 3061}, {1402, 3705}, {1824, 3784}, {1918, 33930}, {1964, 16889}, {2295, 3863}, {3777, 4557}, {3865, 20964}, {16606, 20284}, {21759, 33890}
X(40415) = cevapoint of X(i) and X(j) for these (i,j): {2, 31}, {21, 27644}, {81, 13588}, {983, 17743}
X(40415) = crosssum of X(4531) and X(16584)
X(40415) = trilinear pole of line {824, 4560}
X(40415) = barycentric product X(i)*X(j) for these {i,j}: {1, 38810}, {42, 7307}, {76, 38813}, {81, 7033}, {86, 17743}, {190, 7255}, {274, 983}, {314, 7132}, {321, 7305}, {824, 33514}, {1333, 7034}, {3114, 3736}, {3407, 30966}, {4621, 7192}, {7096, 38840}, {14124, 16584}
X(40415) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3721}, {2, 2887}, {3, 20727}, {6, 3778}, {7, 16888}, {8, 4136}, {10, 16886}, {21, 3061}, {31, 16584}, {37, 7237}, {41, 4531}, {55, 20684}, {58, 2275}, {75, 20234}, {81, 982}, {83, 16889}, {86, 3662}, {99, 33946}, {100, 7239}, {171, 18905}, {238, 18904}, {274, 33930}, {284, 3056}, {333, 3705}, {514, 3801}, {560, 21751}, {662, 3888}, {983, 37}, {1019, 3777}, {1178, 3863}, {1333, 7032}, {1412, 7248}, {1434, 7185}, {1501, 8022}, {1509, 33947}, {1790, 3784}, {1918, 21815}, {2185, 3794}, {2193, 20753}, {2194, 20665}, {2287, 4073}, {3736, 3094}, {4273, 4787}, {4560, 3810}, {4621, 3952}, {7033, 321}, {7034, 27801}, {7132, 65}, {7192, 3776}, {7255, 514}, {7305, 81}, {7307, 310}, {8685, 4559}, {9247, 22364}, {17103, 7187}, {17743, 10}, {30966, 3314}, {31909, 5117}, {33295, 33891}, {33296, 33890}, {33514, 4586}, {38810, 75}, {38813, 6}, {38832, 20284}, {38837, 21776}, {38840, 20444}, {40214, 7186}
X(40415) = {X(31),X(561)}-harmonic conjugate of X(33767)

X(40416) = CEVAPOINT OF X(2) AND X(32)

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

X(40416) lies on these lines: {2, 9233}, {32, 710}, {83, 3613}, {141, 1078}, {385, 1627}, {427, 7792}, {1031, 10583}, {1084, 36432}, {4577, 8265}, {18105, 35222}, {30167, 38847}

X(40416) = isogonal conjugate of X(20859)
X(40416) = isotomic conjugate of X(626)
X(40416) = isogonal conjugate of the anticomplement of X(4074)
X(40416) = isotomic conjugate of the anticomplement of X(6680)
X(40416) = isotomic conjugate of the complement of X(32)
X(40416) = isotomic conjugate of the isogonal conjugate of X(38826)
X(40416) = isogonal conjugate of the isotomic conjugate of X(38830)
X(40416) = X(3115)-Ceva conjugate of X(38830)
X(40416) = X(i)-cross conjugate of X(j) for these (i,j): {688, 4577}, {6680, 2}, {14295, 2966}, {21304, 190}, {28759, 4554}
X(40416) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20859}, {2, 2085}, {6, 4118}, {10, 16717}, {19, 20819}, {31, 626}, {32, 20627}, {41, 7217}, {42, 18167}, {75, 8265}, {92, 4173}, {213, 16891}, {604, 4178}, {692, 21110}, {1333, 16894}, {1917, 8039}, {1928, 8023}, {1964, 16890}, {1969, 23209}, {1973, 4121}, {3112, 3118}
X(40416) = cevapoint of X(i) and X(j) for these (i,j): {2, 32}, {4027, 8623}
X(40416) = crosssum of X(4173) and X(8265)
X(40416) = trilinear pole of line {826, 5027}
X(40416) = barycentric product X(i)*X(j) for these {i,j}: {1, 38847}, {6, 38830}, {39, 3115}, {76, 38826}, {826, 33515}, {2353, 38842}
X(40416) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4118}, {2, 626}, {3, 20819}, {6, 20859}, {7, 7217}, {8, 4178}, {10, 16894}, {31, 2085}, {32, 8265}, {69, 4121}, {75, 20627}, {81, 18167}, {83, 16890}, {86, 16891}, {141, 16893}, {184, 4173}, {514, 21110}, {1333, 16717}, {1502, 8039}, {3051, 3118}, {3115, 308}, {9233, 8023}, {14575, 23209}, {16985, 710}, {33515, 4577}, {38826, 6}, {38830, 76}, {38838, 33786}, {38842, 40073}, {38847, 75}
X(40416) = {X(32),X(1502)}-harmonic conjugate of X(33768)

X(40417) = CEVAPOINT OF X(2) AND X(40)

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

X(40417) lies on these lines: {8, 1804}, {40, 309}, {63, 7101}, {69, 7080}, {75, 7013}, {307, 34393}, {322, 7182}, {332, 947}, {345, 5744}, {3718, 33932}, {8822, 35516}

X(40417) = isotomic conjugate of X(946)
X(40417) = isotomic conjugate of the anticomplement of X(6684)
X(40417) = isotomic conjugate of the complement of X(40)
X(40417) = isotomic conjugate of the isogonal conjugate of X(947)
X(40417) = X(i)-cross conjugate of X(j) for these (i,j): {4131, 664}, {4397, 190}, {6684, 2}
X(40417) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2262}, {19, 22063}, {25, 17102}, {31, 946}, {603, 1856}, {604, 20262}, {1397, 23528}
X(40417) = cevapoint of X(i) and X(j) for these (i,j): {2, 40}, {8, 63}, {200, 3730}, {37558, 40152}
X(40417) = trilinear pole of line {6332, 17496}
X(40417) = barycentric product X(76)*X(947)
X(40417) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2262}, {2, 946}, {3, 22063}, {8, 20262}, {63, 17102}, {281, 1856}, {312, 23528}, {947, 6}

X(40418) = CEVAPOINT OF X(2) AND X(42)

Barycentrics    (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(40418) lies on the circumconic {{A,B,C,X(2),X(7)}} and these lines: {1, 6384}, {2, 1258}, {7, 1403}, {27, 7119}, {42, 310}, {43, 75}, {65, 7249}, {86, 171}, {190, 21838}, {192, 39967}, {335, 3666}, {350, 1240}, {727, 33682}, {893, 894}, {1911, 2668}, {2162, 17379}, {3210, 27494}, {3502, 40038}, {3720, 31002}, {5936, 26038}, {9315, 27498}, {16712, 24215}, {17234, 27264}, {18170, 23460}, {24512, 39746}, {26102, 40027}, {27483, 31993}, {29822, 33947}, {35916, 40164}

X(40418) = isogonal conjugate of X(2309)
X(40418) = isotomic conjugate of X(3741)
X(40418) = isotomic conjugate of the anticomplement of X(6685)
X(40418) = isotomic conjugate of the complement of X(42)
X(40418) = X(i)-cross conjugate of X(j) for these (i,j): {512, 190}, {4374, 664}, {6685, 2}, {17159, 99}, {17217, 668}, {24533, 4598}, {24782, 658}, {28758, 4554}, {29487, 799}
X(40418) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2309}, {2, 1197}, {4, 22389}, {6, 1107}, {19, 22065}, {21, 39780}, {31, 3741}, {32, 20891}, {41, 30097}, {42, 18169}, {58, 3728}, {81, 21838}, {213, 16738}, {286, 23212}, {593, 22206}, {757, 21700}, {849, 21713}, {983, 23473}, {1178, 27880}, {1333, 21024}, {1964, 18091}
X(40418) = cevapoint of X(i) and X(j) for these (i,j): {1, 894}, {2, 42}, {10, 192}
X(40418) = trilinear pole of line {514, 19565}
X(40418) = barycentric product X(i)*X(j) for these {i,j}: {1, 1221}, {75, 1258}
X(40418) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1107}, {2, 3741}, {3, 22065}, {6, 2309}, {7, 30097}, {10, 21024}, {31, 1197}, {37, 3728}, {42, 21838}, {48, 22389}, {75, 20891}, {81, 18169}, {83, 18091}, {86, 16738}, {594, 21713}, {756, 22206}, {1221, 75}, {1258, 1}, {1400, 39780}, {1500, 21700}, {2200, 23212}, {2275, 23473}, {2295, 27880}

X(40419) = CEVAPOINT OF X(2) AND X(55)

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

X(40419) lies on these lines: {2, 14827}, {55, 6063}, {100, 40216}, {171, 3664}, {666, 16588}, {693, 1621}, {1001, 32023}, {2223, 7176}, {2329, 3912}, {3263, 7081}, {3449, 29839}, {4219, 7009}, {4998, 5432}, {5218, 8817}, {5253, 32021}, {7196, 15931}, {31637, 39712}

X(40419) = isogonal conjugate of X(21746)
X(40419) = isotomic conjugate of X(2886)
X(40419) = isotomic conjugate of the anticomplement of X(6690)
X(40419) = isotomic conjugate of the complement of X(55)
X(40419) = isotomic conjugate of the isogonal conjugate of X(3449)
X(40419) = X(i)-cross conjugate of X(j) for these (i,j): {926, 666}, {4374, 99}, {6690, 2}, {21302, 190}
X(40419) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21746}, {6, 17451}, {19, 22070}, {31, 2886}, {32, 20236}, {42, 18165}, {57, 16588}, {58, 21804}, {85, 9449}, {273, 22368}, {692, 21118}, {1333, 21029}, {1400, 16699}, {1434, 21819}, {1964, 18088}
X(40419) = cevapoint of X(i) and X(j) for these (i,j): {2, 55}, {11, 17494}, {385, 8299}
X(40419) = trilinear pole of line {918, 3287}
X(40419) = barycentric product X(76)*X(3449)
X(40419) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17451}, {2, 2886}, {3, 22070}, {6, 21746}, {10, 21029}, {21, 16699}, {37, 21804}, {55, 16588}, {75, 20236}, {81, 18165}, {83, 18088}, {514, 21118}, {2175, 9449}, {3449, 6}

X(40420) = CEVAPOINT OF X(2) AND X(57)

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

X(40420) lies on these lines: {1, 26720}, {2, 1407}, {7, 1997}, {8, 56}, {29, 1877}, {57, 312}, {85, 738}, {92, 1435}, {171, 1416}, {189, 6612}, {226, 4997}, {241, 257}, {333, 1412}, {345, 8828}, {664, 3752}, {1121, 6613}, {1150, 30711}, {1220, 8582}, {1427, 27002}, {1434, 28660}, {1477, 8706}, {3699, 17625}, {4518, 8581}, {5226, 38255}, {5745, 32008}, {7020, 37278}, {7153, 27424}, {7196, 18031}, {8056, 9312}, {8583, 31225}, {9364, 32942}, {17283, 28774}, {17862, 18359}, {20205, 31640}, {26125, 37682}, {30608, 31231}

X(40420) = isogonal conjugate of X(2347)
X(40420) = isotomic conjugate of X(3452)
X(40420) = isotomic conjugate of the anticomplement of X(6692)
X(40420) = isotomic conjugate of the complement of X(57)
X(40420) = isotomic conjugate of the isogonal conjugate of X(3451)
X(40420) = X(i)-cross conjugate of X(j) for these (i,j): {2, 32017}, {513, 664}, {4462, 190}, {5176, 903}, {5253, 86}, {6692, 2}, {10106, 7}, {20293, 18026}, {21302, 4569}, {23617, 1222}, {24982, 75}, {32850, 35160}
X(40420) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2347}, {6, 3057}, {8, 20228}, {9, 1201}, {19, 22072}, {21, 21796}, {31, 3452}, {32, 20895}, {41, 3663}, {42, 18163}, {55, 3752}, {58, 21809}, {101, 6615}, {213, 17183}, {219, 1828}, {220, 1122}, {281, 22344}, {284, 4642}, {604, 6736}, {644, 6363}, {650, 23845}, {663, 21362}, {667, 25268}, {692, 21120}, {1333, 21031}, {1946, 17906}, {1964, 18086}, {2175, 26563}, {2194, 4415}, {3063, 21272}, {12640, 38266}, {14284, 34080}, {18344, 23113}
X(40420) = cevapoint of X(i) and X(j) for these (i,j): {2, 57}, {7, 9312}, {9, 145}, {1400, 37558}, {1476, 23617}
X(40420) = trilinear pole of line {522, 4318}
X(40420) = barycentric product X(i)*X(j) for these {i,j}: {7, 1222}, {57, 32017}, {75, 1476}, {76, 3451}, {85, 23617}, {522, 6613}, {1088, 1261}, {3676, 8706}
X(40420) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3057}, {2, 3452}, {3, 22072}, {6, 2347}, {7, 3663}, {8, 6736}, {10, 21031}, {34, 1828}, {37, 21809}, {56, 1201}, {57, 3752}, {65, 4642}, {75, 20895}, {81, 18163}, {83, 18086}, {85, 26563}, {86, 17183}, {109, 23845}, {145, 12640}, {190, 25268}, {226, 4415}, {269, 1122}, {513, 6615}, {514, 21120}, {603, 22344}, {604, 20228}, {651, 21362}, {653, 17906}, {664, 21272}, {1222, 8}, {1261, 200}, {1400, 21796}, {1434, 18600}, {1476, 1}, {1813, 23113}, {3451, 6}, {3667, 14284}, {4369, 28006}, {4554, 21580}, {6613, 664}, {7153, 27499}, {8706, 3699}, {23617, 9}, {32017, 312}
X(40420) = {X(57),X(30567)}-harmonic conjugate of X(39126)

X(40421) = CEVAPOINT OF X(2) AND X(66)

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

X(40421) lies on these lines: {3, 16097}, {66, 315}, {76, 5523}, {264, 40009}, {305, 858}, {683, 11185}, {1236, 40050}, {1241, 13854}, {2353, 38907}, {7763, 14376}, {11056, 37801}, {16277, 33651}, {21017, 40071}

X(40421) = isogonal conjugate of X(20968)
X(40421) = isotomic conjugate of X(206)
X(40421) = polar conjugate of X(17409)
X(40421) = isotomic conjugate of the anticomplement of X(6697)
X(40421) = isotomic conjugate of the complement of X(66)
X(40421) = isotomic conjugate of the isogonal conjugate of X(18018)
X(40421) = X(i)-cross conjugate of X(j) for these (i,j): {2, 1502}, {1235, 76}, {6697, 2}, {21407, 75}
X(40421) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20968}, {6, 17453}, {19, 22075}, {22, 560}, {31, 206}, {32, 2172}, {41, 7251}, {48, 17409}, {75, 40372}, {213, 17186}, {315, 1917}, {604, 4548}, {692, 21122}, {1333, 21034}, {1501, 1760}, {1924, 4611}, {1973, 10316}, {7210, 9448}, {8743, 9247}, {9233, 20641}, {9417, 11610}
X(40421) = cevapoint of X(i) and X(j) for these (i,j): {2, 66}, {75, 21583}, {23285, 36793}
X(40421) = trilinear pole of line {3267, 23881}
X(40421) = barycentric product X(i)*X(j) for these {i,j}: {66, 1502}, {76, 18018}, {1928, 2156}, {2353, 40362}, {13854, 40050}, {14376, 18022}, {18024, 34138}, {40146, 40359}
X(40421) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17453}, {2, 206}, {3, 22075}, {4, 17409}, {6, 20968}, {7, 7251}, {8, 4548}, {10, 21034}, {32, 40372}, {66, 32}, {69, 10316}, {75, 2172}, {76, 22}, {86, 17186}, {141, 23208}, {264, 8743}, {290, 11610}, {305, 20806}, {313, 4456}, {315, 36414}, {339, 38356}, {514, 21122}, {561, 1760}, {670, 4611}, {850, 2485}, {1502, 315}, {1928, 20641}, {2156, 560}, {2353, 1501}, {3267, 8673}, {8024, 3313}, {13854, 1974}, {14376, 184}, {18018, 6}, {18022, 17907}, {18024, 31636}, {20567, 7210}, {27801, 4463}, {28659, 4123}, {34138, 237}, {37801, 18374}, {40050, 34254}, {40146, 9233}, {40362, 40073}

X(40422) = CEVAPOINT OF X(2) AND X(72)

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

X(40422) lies on these lines: {8, 264}, {69, 6063}, {72, 286}, {75, 78}, {271, 309}, {312, 3305}, {314, 943}, {319, 349}, {321, 2287}, {668, 1234}, {1265, 3596}, {1809, 18816}, {2893, 21403}, {2982, 30710}, {2997, 3876}, {5564, 20566}, {31643, 39765}

X(40422) = isotomic conjugate of X(942)
X(40422) = polar conjugate of X(1841)
X(40422) = isotomic conjugate of the anticomplement of X(5044)
X(40422) = isotomic conjugate of the complement of X(72)
X(40422) = isotomic conjugate of the isogonal conjugate of X(943)
X(40422) = X(i)-cross conjugate of X(j) for these (i,j): {850, 668}, {5044, 2}, {7253, 190}, {23683, 18026}
X(40422) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2260}, {19, 14597}, {25, 4303}, {31, 942}, {32, 5249}, {34, 23207}, {48, 1841}, {56, 14547}, {184, 1838}, {442, 2206}, {500, 6186}, {603, 1859}, {1042, 8021}, {1333, 2294}, {1397, 6734}, {1461, 33525}, {1474, 18591}, {1576, 23752}, {1973, 18607}, {2299, 39791}
X(40422) = cevapoint of X(i) and X(j) for these (i,j): {2, 72}, {8, 321}, {75, 319}, {200, 3294}
X(40422) = trilinear pole of line {4391, 17494}
X(40422) = barycentric product X(i)*X(j) for these {i,j}: {76, 943}, {561, 2259}, {1175, 27801}, {1794, 1969}, {2982, 3596}
X(40422) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2260}, {2, 942}, {3, 14597}, {4, 1841}, {9, 14547}, {10, 2294}, {63, 4303}, {69, 18607}, {72, 18591}, {75, 5249}, {92, 1838}, {219, 23207}, {281, 1859}, {312, 6734}, {319, 16585}, {321, 442}, {943, 6}, {1089, 21675}, {1175, 1333}, {1214, 39791}, {1577, 23752}, {1794, 48}, {2259, 31}, {2287, 8021}, {2982, 56}, {3219, 500}, {3900, 33525}, {14775, 6591}, {15439, 1415}, {17776, 14054}, {27801, 1234}, {28605, 3824}, {33116, 39772}, {36048, 1461}

X(40423) = CEVAPOINT OF X(2) AND X(74)

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

X(40423) lies on these lines: {2, 40353}, {69, 39379}, {74, 3260}, {264, 38937}, {298, 36311}, {299, 36308}, {340, 687}, {1494, 7799}, {5627, 6148}, {12028, 31621}

X(40423) = isotomic conjugate of X(113)
X(40423) = isotomic conjugate of the anticomplement of X(6699)
X(40423) = isotomic conjugate of the complement of X(74)
X(40423) = isotomic conjugate of the isogonal conjugate of X(10419)
X(40423) = X(i)-cross conjugate of X(j) for these (i,j): {69, 1494}, {850, 16077}, {6699, 2}, {15454, 2986}
X(40423) = X(i)-isoconjugate of X(j) for these (i,j): {31, 113}, {1495, 1725}, {1990, 2315}, {2173, 3003}, {3580, 9406}
X(40423) = cevapoint of X(i) and X(j) for these (i,j): {2, 74}, {525, 12079}, {2986, 15454}, {16080, 38937}
X(40423) = trilinear pole of line {2394, 2986}
X(40423) = barycentric product X(i)*X(j) for these {i,j}: {76, 10419}, {305, 40388}, {687, 34767}, {1494, 2986}, {2394, 18878}, {15421, 16077}, {15454, 31621}, {33805, 36053}
X(40423) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 113}, {74, 3003}, {687, 4240}, {1300, 1990}, {1494, 3580}, {2349, 1725}, {2433, 21731}, {2986, 30}, {3580, 34104}, {5504, 3284}, {10419, 6}, {10420, 2420}, {14380, 686}, {14910, 1495}, {14919, 13754}, {15328, 1637}, {15421, 9033}, {15454, 3163}, {16077, 16237}, {16080, 403}, {18878, 2407}, {32708, 23347}, {34767, 6334}, {35200, 2315}, {36053, 2173}, {38936, 39176}, {39379, 14910}, {40384, 14264}, {40388, 25}

X(40424) = CEVAPOINT OF X(2) AND X(78)

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

X(40424) lies on the circumconic {{A,B,C,X(2),X(7)}} and these lines: {2, 2256}, {7, 404}, {27, 908}, {69, 1440}, {75, 936}, {78, 273}, {86, 1167}, {326, 1088}, {329, 1436}, {965, 27282}, {5736, 30712}, {18815, 20895}

X(40424) = isogonal conjugate of X(40958)
X(40424) = isotomic conjugate of X(1210)
X(40424) = isotomic conjugate of the anticomplement of X(6700)
X(40424) = isotomic conjugate of the complement of X(78)
X(40424) = isotomic conjugate of the isogonal conjugate of X(1167)
X(40424) = trilinear pole of line X(514)X(40863)
X(40424) = X(i)-cross conjugate of X(j) for these (i,j): {521, 190}, {6700, 2}, {17896, 664}
X(40424) = X(i)-isoconjugate of X(j) for these (i,j): {4, 23204}, {6, 1108}, {25, 1071}, {28, 3611}, {31, 1210}, {32, 17862}, {55, 37566}, {56, 1864}, {560, 1226}, {1333, 21933}, {1532, 34858}, {2208, 6260}
X(40424) = cevapoint of X(i) and X(j) for these (i,j): {1, 329}, {2, 78}
X(40424) = barycentric product X(76)*X(1167)
X(40424) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1108}, {2, 1210}, {9, 1864}, {10, 21933}, {48, 23204}, {57, 37566}, {63, 1071}, {71, 3611}, {75, 17862}, {76, 1226}, {329, 6260}, {908, 1532}, {1167, 6}

X(40425) = CEVAPOINT OF X(2) AND X(83)

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

X(40425) lies on these lines: {83, 141}, {251, 16988}, {308, 3108}, {427, 32085}, {1502, 7808}, {3589, 4577}, {7859, 38946}, {7953, 39427}, {14970, 18092}, {15523, 17285}

X(40425) = isogonal conjugate of X(11205)
X(40425) = isotomic conjugate of X(6292)
X(40425) = isotomic conjugate of the anticomplement of X(6704)
X(40425) = isotomic conjugate of the complement of X(83)
X(40425) = X(i)-cross conjugate of X(j) for these (i,j): {2, 10159}, {523, 4577}, {6704, 2}, {7779, 14970}
X(40425) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11205}, {6, 17457}, {19, 22078}, {31, 6292}, {32, 20898}, {38, 5007}, {39, 17469}, {58, 21817}, {213, 17193}, {428, 4020}, {688, 18062}, {692, 21126}, {1333, 21038}, {1923, 39998}, {1964, 3589}, {2084, 10330}, {17187, 21802}, {17200, 21814}, {17442, 22352}
X(40425) = cevapoint of X(i) and X(j) for these (i,j): {2, 83}, {251, 14247}, {3108, 10159}
X(40425) = trilinear pole of line {826, 14318}
X(40425) = barycentric product X(i)*X(j) for these {i,j}: {83, 10159}, {308, 3108}, {4577, 31065}
X(40425) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17457}, {2, 6292}, {3, 22078}, {6, 11205}, {10, 21038}, {37, 21817}, {75, 20898}, {82, 17469}, {83, 3589}, {86, 17193}, {251, 5007}, {308, 39998}, {427, 28666}, {514, 21126}, {1176, 22352}, {1799, 7767}, {3108, 39}, {4577, 10330}, {4593, 18062}, {7953, 1634}, {10159, 141}, {18098, 21802}, {18105, 8664}, {31065, 826}, {31067, 2528}, {31068, 7813}, {32085, 428}, {35137, 4576}, {39668, 39784}
X(40425) = {X(3589),X(40000)}-harmonic conjugate of X(4577)

X(40426) = CEVAPOINT OF X(2) AND X(89)

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

X(40426) lies on these lines: {89, 4671}, {996, 1150}, {3306, 23598}, {3758, 30607}, {4597, 4850}, {4945, 37633}, {5219, 30588}, {5235, 30608}, {9059, 39428}, {29908, 30818}

X(40426) = isogonal conjugate of X(20973)
X(40426) = isotomic conjugate of the complement of X(89)
X(40426) = X(513)-cross conjugate of X(4597)
X(40426) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20973}, {6, 17461}, {19, 22083}, {45, 995}, {213, 17196}, {692, 21130}, {1333, 21042}, {1405, 3877}, {2099, 4266}, {2177, 4850}, {4273, 4424}, {4752, 9002}
X(40426) = cevapoint of X(2) and X(89)
X(40426) = trilinear pole of line {4777, 29908}
X(40426) = barycentric product X(996)*X(39704)
X(40426) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17461}, {3, 22083}, {6, 20973}, {10, 21042}, {86, 17196}, {89, 4850}, {514, 21130}, {996, 3679}, {2163, 995}, {2320, 3877}, {2364, 4266}, {9059, 4767}, {20569, 33934}, {30588, 26580}, {30608, 5233}, {39704, 4389}

X(40427) = CEVAPOINT OF X(2) AND X(94)

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

X(40427) lies on these lines: {94, 323}, {186, 476}, {264, 39290}, {2411, 15421}, {3431, 15454}, {3580, 35139}, {7799, 20573}, {10420, 39430}, {14165, 18883}, {14254, 34289}

X(40427) = isotomic conjugate of X(34834)
X(40427) = polar conjugate of X(1986)
X(40427) = isotomic conjugate of the complement of X(94)
X(40427) = polar conjugate of the isogonal conjugate of X(12028)
X(40427) = X(i)-cross conjugate of X(j) for these (i,j): {2, 2986}, {523, 35139}
X(40427) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34834}, {48, 1986}, {50, 1725}, {186, 2315}, {2624, 15329}, {3003, 6149}
X(40427) = cevapoint of X(i) and X(j) for these (i,j): {2, 94}, {338, 14592}, {1989, 14254}
X(40427) = trilinear pole of line {265, 526}
X(40427) = barycentric product X(i)*X(j) for these {i,j}: {94, 2986}, {264, 12028}, {328, 1300}, {687, 14592}, {1494, 39375}, {5504, 18817}, {10412, 18878}, {14910, 20573}, {15328, 35139}
X(40427) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34834}, {4, 1986}, {94, 3580}, {265, 13754}, {476, 15329}, {687, 14590}, {1300, 186}, {1989, 3003}, {2166, 1725}, {2970, 16221}, {2986, 323}, {5504, 22115}, {5627, 14264}, {6344, 403}, {10419, 14385}, {12028, 3}, {14254, 113}, {14582, 686}, {14592, 6334}, {14910, 50}, {15328, 526}, {15421, 8552}, {15454, 1511}, {15475, 21731}, {18878, 10411}, {32708, 14591}, {35361, 2081}, {36053, 6149}, {38936, 3043}, {39170, 34333}, {39375, 30}

X(40428) = CEVAPOINT OF X(2) AND X(98)

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

X(40428) lies on thje cubic K776 and these lines: {98, 325}, {183, 36897}, {230, 297}, {290, 19599}, {327, 14382}, {385, 2987}, {3563, 22456}, {5641, 6055}, {5967, 34803}, {9154, 34229}, {14253, 14265}

X(40428) = isotomic conjugate of X(114)
X(40428) = isotomic conjugate of the anticomplement of X(6036)
X(40428) = isotomic conjugate of the complement of X(98)
X(40428) = isotomic conjugate of the isogonal conjugate of X(2065)
X(40428) = X(i)-cross conjugate of X(j) for these (i,j): {2, 8781}, {69, 290}, {523, 2966}, {6036, 2}, {34157, 2987}
X(40428) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17462}, {31, 114}, {230, 1755}, {237, 1733}, {511, 8772}, {1692, 1959}
X(40428) = cevapoint of X(i) and X(j) for these (i,j): {2, 98}, {647, 15630}, {2987, 34157}
X(40428) = trilinear pole of line {287, 2395}
X(40428) = barycentric product X(i)*X(j) for these {i,j}: {76, 2065}, {98, 8781}, {287, 35142}, {290, 2987}, {1821, 8773}, {18024, 32654}
X(40428) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17462}, {2, 114}, {98, 230}, {287, 3564}, {1821, 1733}, {1910, 8772}, {1976, 1692}, {2065, 6}, {2966, 4226}, {2987, 511}, {3563, 232}, {5967, 5477}, {6531, 460}, {8773, 1959}, {8781, 325}, {10425, 2421}, {32654, 237}, {32697, 4230}, {34157, 11672}, {34536, 14265}, {35142, 297}, {35364, 3569}, {36051, 1755}

X(40429) = CEVAPOINT OF X(2) AND X(115)

Barycentrics    (a^4 - 2*a^2*b^2 + 2*b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + 2*c^4) : :
Barycentrics    1/(4 SA a^2 - b^4 - c^4) : :
X(40429) = 6 X[2] - X[33799], X[99] + 4 X[31644], 3 X[99] - 8 X[36953], 4 X[115] + X[4590], 7 X[671] + 8 X[9164], 3 X[671] + 2 X[14588], X[892] + 4 X[23991], 3 X[892] + 2 X[35511], 16 X[5461] - X[18823], 12 X[9164] - 7 X[14588], 9 X[9166] + X[31998], 6 X[23991] - X[35511], X[31372] - 6 X[35087], 3 X[31644] + 2 X[36953]

X(40429) lies on these lines: {2, 33799}, {99, 31644}, {115, 4590}, {468, 30716}, {523, 14061}, {524, 5103}, {671, 9164}, {892, 23991}, {3266, 7925}, {3618, 5967}, {5461, 14728}, {31372, 35087}

X(40429) = isogonal conjugate of X(20976)
X(40429) = isotomic conjugate of X(620)
X(40429) = isotomic conjugate of the anticomplement of X(6722)
X(40429) = isotomic conjugate of the complement of X(115)
X(40429) = X(i)-cross conjugate of X(j) for these (i,j): {5468, 671}, {6722, 2}, {33919, 892}
X(40429) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20976}, {6, 17467}, {19, 22085}, {31, 620}, {32, 20903}, {163, 11123}, {213, 17199}, {692, 21135}, {798, 14588}, {1101, 23991}, {1333, 21047}, {33906, 36142}
X(40429) = cevapoint of X(i) and X(j) for these (i,j): {2, 115}, {523, 31644}
X(40429) = trilinear pole of line {148, 690}
X(40429) = barycentric product X(690)*X(14728)
X(40429) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17467}, {2, 620}, {3, 22085}, {6, 20976}, {10, 21047}, {75, 20903}, {86, 17199}, {99, 14588}, {115, 23991}, {514, 21135}, {523, 11123}, {690, 33906}, {14728, 892}

X(40430) = CEVAPOINT OF X(1) AND X(21)

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

X(40430) lies on these lines: {1, 1098}, {10, 1043}, {19, 2326}, {21, 65}, {29, 225}, {37, 2287}, {75, 10448}, {81, 31503}, {86, 3668}, {158, 1982}, {267, 5426}, {409, 662}, {757, 969}, {759, 35016}, {994, 5248}, {1010, 23604}, {1125, 5620}, {1621, 34434}, {2185, 2217}, {2975, 13476}, {3612, 11116}, {3615, 3616}, {6740, 26095}, {7259, 16601}, {10543, 19642}, {11115, 25536}, {35991, 37600}

X(40430) = isogonal conjugate of X(2650)
X(40430) = isotomic conjugate of X(18698)
X(40430) = isotomic conjugate of the anticomplement of X(25081)
X(40430) = isotomic conjugate of the complement of X(25255)
X(40430) = X(i)-cross conjugate of X(j) for these (i,j): {1, 17097}, {650, 662}, {1758, 37142}, {21189, 162}, {21390, 799}, {25081, 2}
X(40430) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2650}, {3, 407}, {6, 17056}, {31, 18698}, {42, 3664}, {56, 21677}, {57, 21811}, {58, 21674}, {65, 2646}, {101, 23755}, {225, 22361}, {226, 21748}, {512, 17136}, {649, 22003}, {1042, 6737}, {1400, 5745}, {4588, 30604}, {15232, 37836}
X(40430) = cevapoint of X(i) and X(j) for these (i,j): {1, 21}, {2, 25255}
X(40430) = trilinear pole of line {661, 1021}
X(40430) = barycentric product X(333)*X(17097)
X(40430) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17056}, {2, 18698}, {6, 2650}, {9, 21677}, {19, 407}, {21, 5745}, {37, 21674}, {55, 21811}, {81, 3664}, {100, 22003}, {284, 2646}, {513, 23755}, {662, 17136}, {2193, 22361}, {2194, 21748}, {2287, 6737}, {4893, 30604}, {17097, 226}
X(40430) = {X(409),X(2646)}-harmonic conjugate of X(662)

X(40431) = CEVAPOINT OF X(1) AND X(28)

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

X(40431) lies on these lines: {1, 2326}, {21, 1214}, {27, 306}, {28, 72}, {29, 226}, {63, 1098}, {162, 1104}, {270, 3868}, {1426, 4183}, {5436, 11107}

X(40431) = isogonal conjugate of X(18673)
X(40431) = X(i)-cross conjugate of X(j) for these (i,j): {1, 1257}, {513, 162}
X(40431) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18673}, {3, 1834}, {6, 440}, {42, 18650}, {58, 21671}, {72, 1104}, {73, 950}, {228, 17863}, {647, 14543}, {1214, 2264}, {1842, 3682}, {4574, 29162}
X(40431) = cevapoint of X(i) and X(j) for these (i,j): {1, 28}, {19, 4183}, {21, 3868}
X(40431) = trilinear pole of line {656, 1021}
X(40431) = barycentric product X(i)*X(j) for these {i,j}: {1, 40414}, {27, 1257}, {286, 2983}, {951, 31623}
X(40431) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 440}, {6, 18673}, {19, 1834}, {27, 17863}, {37, 21671}, {81, 18650}, {162, 14543}, {951, 1214}, {1172, 950}, {1257, 306}, {1474, 1104}, {2299, 2264}, {2983, 72}, {5317, 1842}, {40414, 75}

X(40432) = CEVAPOINT OF X(1) AND X(39)

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

X(40432) lies on these lines: {1, 1581}, {6, 24519}, {21, 238}, {27, 7249}, {37, 27954}, {39, 83}, {56, 2363}, {58, 3865}, {81, 1429}, {82, 16689}, {86, 16744}, {239, 257}, {261, 40099}, {274, 33891}, {330, 8033}, {404, 27665}, {662, 21008}, {694, 39971}, {799, 21226}, {805, 3110}, {882, 24286}, {964, 27642}, {1015, 1509}, {1016, 1500}, {1201, 2106}, {1244, 36214}, {1431, 5331}, {2185, 7303}, {2275, 14621}, {2276, 17743}, {3571, 9424}, {3752, 24378}, {3905, 28606}, {4850, 24595}, {5209, 26959}, {6625, 16592}, {7018, 19786}, {7257, 26752}, {7260, 16722}, {16591, 17084}, {16705, 16738}, {16975, 34016}, {17448, 17731}, {18140, 25530}, {18600, 26802}, {18829, 35172}, {19281, 24598}, {24555, 37228}, {24617, 37233}, {25520, 29983}, {26978, 27189}, {27368, 32921}

X(40432) = isogonal conjugate of X(2295)
X(40432) = isotomic conjugate of X(3963)
X(40432) = isogonal conjugate of the complement of X(17152)
X(40432) = X(7303)-Ceva conjugate of X(1178)
X(40432) = X(i)-cross conjugate of X(j) for these (i,j): {256, 32010}, {893, 1178}, {3271, 7192}, {6377, 3733}, {18169, 86}, {29545, 190}, {29821, 757}, {33295, 37128}
X(40432) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2295}, {2, 20964}, {3, 1840}, {4, 22061}, {6, 1215}, {10, 172}, {25, 4019}, {31, 3963}, {32, 1237}, {37, 171}, {39, 18099}, {42, 894}, {55, 4032}, {56, 4095}, {58, 21021}, {65, 2329}, {71, 7009}, {72, 7119}, {81, 21803}, {82, 16587}, {101, 2533}, {109, 4140}, {181, 27958}, {190, 7234}, {210, 7175}, {213, 1909}, {226, 2330}, {284, 7211}, {292, 4039}, {321, 7122}, {512, 18047}, {661, 4579}, {765, 16592}, {804, 813}, {872, 8033}, {983, 18905}, {1016, 4128}, {1018, 4367}, {1020, 4477}, {1258, 27880}, {1334, 7176}, {1400, 7081}, {1402, 17787}, {1500, 17103}, {1826, 3955}, {1918, 1920}, {2197, 14006}, {2238, 18787}, {3112, 21752}, {3287, 4551}, {3709, 6649}, {3747, 30669}, {3907, 4559}, {3952, 20981}, {4368, 30657}, {4369, 4557}, {4447, 18785}, {4562, 5027}, {4567, 21725}, {4600, 21823}, {7035, 21755}, {17752, 23493}
X(40432) = cevapoint of X(i) and X(j) for these (i,j): {1, 39}, {256, 893}, {1015, 1019}
X(40432) = crosssum of X(i) and X(j) for these (i,j): {7234, 21755}, {16587, 21752}, {20691, 21879}
X(40432) = trilinear pole of line {659, 3737}
X(40432) = barycentric product X(i)*X(j) for these {i,j}: {1, 32010}, {10, 7303}, {21, 7249}, {28, 7019}, {58, 7018}, {75, 1178}, {81, 257}, {86, 256}, {274, 893}, {286, 7015}, {310, 904}, {314, 1431}, {333, 1432}, {513, 4594}, {514, 4603}, {649, 7260}, {659, 18829}, {694, 30940}, {805, 3766}, {812, 37134}, {1014, 4451}, {1019, 27805}, {1581, 33295}, {1934, 5009}, {3863, 38810}, {3865, 40415}, {3903, 7192}, {4560, 37137}, {6385, 7104}, {17493, 37128}, {18155, 29055}, {18786, 18827}, {27447, 27644}, {39292, 39786}
X(40432) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1215}, {2, 3963}, {6, 2295}, {9, 4095}, {19, 1840}, {21, 7081}, {28, 7009}, {31, 20964}, {37, 21021}, {39, 16587}, {42, 21803}, {48, 22061}, {57, 4032}, {58, 171}, {63, 4019}, {65, 7211}, {75, 1237}, {81, 894}, {82, 18099}, {86, 1909}, {110, 4579}, {238, 4039}, {256, 10}, {257, 321}, {270, 14006}, {274, 1920}, {284, 2329}, {333, 17787}, {513, 2533}, {650, 4140}, {659, 804}, {662, 18047}, {667, 7234}, {741, 18787}, {757, 17103}, {805, 660}, {893, 37}, {904, 42}, {1014, 7176}, {1015, 16592}, {1019, 4369}, {1021, 4529}, {1178, 1}, {1333, 172}, {1412, 7175}, {1414, 6649}, {1431, 65}, {1432, 226}, {1434, 7196}, {1437, 3955}, {1474, 7119}, {1509, 8033}, {1977, 21755}, {2185, 27958}, {2194, 2330}, {2206, 7122}, {2275, 18905}, {2309, 27880}, {3051, 21752}, {3121, 21823}, {3122, 21725}, {3248, 4128}, {3286, 4447}, {3666, 27697}, {3733, 4367}, {3737, 3907}, {3766, 14295}, {3863, 3721}, {3865, 2887}, {3903, 3952}, {4267, 18235}, {4451, 3701}, {4594, 668}, {4603, 190}, {4833, 4774}, {4960, 4842}, {5009, 1580}, {7015, 72}, {7018, 313}, {7019, 20336}, {7104, 213}, {7116, 71}, {7192, 4374}, {7249, 1441}, {7252, 3287}, {7260, 1978}, {7303, 86}, {8300, 4154}, {16695, 24533}, {16696, 16720}, {16702, 7267}, {16726, 7200}, {17302, 27966}, {17493, 3948}, {17938, 34067}, {18166, 4754}, {18191, 4459}, {18786, 740}, {18829, 4583}, {20775, 22367}, {21789, 4477}, {21814, 21818}, {22096, 22373}, {27644, 17752}, {27805, 4033}, {29055, 4551}, {30670, 4613}, {30940, 3978}, {32010, 75}, {33295, 1966}, {37128, 30669}, {37134, 4562}, {37137, 4552}, {38814, 27954}, {39179, 18111}, {39915, 27890}, {40153, 28369}
X(40432) = {X(2275),X(17103)}-harmonic conjugate of X(37128)

X(40433) = CEVAPOINT OF X(1) AND X(42)

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

X(40433) lies on the conic {{A,B,C,X(1),X(6)}} and these lines: {1, 872}, {2, 10013}, {6, 1621}, {37, 25426}, {42, 86}, {56, 5132}, {58, 1918}, {100, 18166}, {106, 8708}, {190, 2663}, {238, 1126}, {269, 33765}, {292, 1100}, {870, 4360}, {1001, 2334}, {1411, 17015}, {1449, 2279}, {1911, 30593}, {2191, 5256}, {2309, 37129}, {3240, 15668}, {3736, 39949}, {3979, 39977}, {4393, 20140}, {4651, 25508}, {17259, 29814}, {27164, 29822}

X(40433) = isogonal conjugate of X(3720)
X(40433) = isotomic conjugate of X(20888)
X(40433) = isogonal conjugate of the complement of X(4651)
X(40433) = isotomic conjugate of the anticomplement of X(25092)
X(40433) = isotomic conjugate of the complement of X(25264)
X(40433) = X(i)-cross conjugate of X(j) for these (i,j): {798, 190}, {1019, 100}, {21763, 4598}, {25092, 2}, {28840, 37138}
X(40433) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3720}, {2, 20963}, {4, 22060}, {6, 3739}, {21, 39793}, {31, 20888}, {37, 18166}, {39, 18089}, {42, 17175}, {55, 4059}, {56, 3706}, {57, 3691}, {58, 21020}, {81, 16589}, {86, 2667}, {100, 6372}, {213, 16748}, {274, 21753}, {286, 22369}, {513, 4436}, {757, 21699}, {893, 4754}, {1014, 4111}, {1509, 21820}, {2350, 29773}, {3445, 4891}
X(40433) = cevapoint of X(i) and X(j) for these (i,j): {1, 42}, {2, 25264}, {9, 4097}, {10, 32925}
X(40433) = crosssum of X(2667) and X(16589)
X(40433) = trilinear pole of line {649, 2664}
X(40433) = barycentric product X(i)*X(j) for these {i,j}: {1, 32009}, {10, 40408}, {514, 8708}
X(40433) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3739}, {2, 20888}, {6, 3720}, {9, 3706}, {31, 20963}, {37, 21020}, {42, 16589}, {48, 22060}, {55, 3691}, {57, 4059}, {58, 18166}, {81, 17175}, {82, 18089}, {86, 16748}, {101, 4436}, {171, 4754}, {213, 2667}, {649, 6372}, {872, 21820}, {1334, 4111}, {1400, 39793}, {1500, 21699}, {1621, 29773}, {1743, 4891}, {1918, 21753}, {2200, 22369}, {8708, 190}, {32009, 75}, {40408, 86}
X(40433) = {X(2663),X(2667)}-harmonic conjugate of X(190)

X(40434) = CEVAPOINT OF X(1) AND X(45)

Barycentrics    a*(a + 4*b + c)*(a + b + 4*c) : :

X(40434) lies on the conic {{A,B,C,X(1),X(2)} and these lines: {1, 4015}, {2, 3943}, {28, 1900}, {37, 88}, {44, 81}, {45, 89}, {57, 16676}, {105, 5297}, {274, 4358}, {291, 30950}, {330, 29595}, {519, 24857}, {551, 4767}, {661, 1022}, {899, 30571}, {1150, 26071}, {1224, 19862}, {1255, 17012}, {1390, 7292}, {3227, 16826}, {3666, 39962}, {3912, 34914}, {4789, 31992}, {4850, 39963}, {4945, 30588}, {5287, 39948}, {5333, 39747}, {8056, 28606}, {11010, 27784}, {16666, 35595}, {16815, 32009}, {16816, 39738}, {16831, 36871}, {17013, 27789}, {17022, 39980}, {17023, 34892}, {17595, 26745}, {21907, 37691}, {25417, 32911}, {29007, 34051}, {29571, 34578}, {31035, 39706}, {33761, 37520}

X(40434) = isogonal conjugate of X(16666)
X(40434) = isotomic conjugate of X(24589)
X(40434) = isogonal conjugate of the complement of X(17360)
X(40434) = isotomic conjugate of the complement of X(31035)
X(40434) = X(i)-cross conjugate of X(j) for these (i,j): {4893, 100}, {5049, 7}
X(40434) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16666}, {2, 21747}, {4, 22357}, {6, 551}, {31, 24589}, {42, 26860}, {55, 4031}, {56, 3707}, {81, 21806}, {101, 28209}, {604, 3902}, {649, 4781}, {901, 14435}, {1333, 4714}, {2163, 16590}, {2364, 39782}, {3939, 30722}, {4793, 28607}, {21754, 39704}
X(40434) = cevapoint of X(i) and X(j) for these (i,j): {1, 45}, {2, 31035}, {6, 5010}
X(40434) = trilinear pole of line {513, 3245}
X(40434) = barycentric product X(i)*X(j) for these {i,j}: {81, 27797}, {693, 28210}
X(40434) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 551}, {2, 24589}, {6, 16666}, {8, 3902}, {9, 3707}, {10, 4714}, {31, 21747}, {42, 21806}, {45, 16590}, {48, 22357}, {57, 4031}, {81, 26860}, {100, 4781}, {513, 28209}, {644, 30727}, {984, 4407}, {1635, 14435}, {2099, 39782}, {3669, 30722}, {3679, 4793}, {27797, 321}, {28210, 100}

X(40435) = CEVAPOINT OF X(1) AND X(71)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 - 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c - 2*a*b*c - b^2*c - a*c^2 + c^3) : :
Barycentrics    1/(1 + cos B + cos C) : :

X(40435) lies on these lines: {2, 219}, {8, 405}, {9, 92}, {10, 29}, {27, 71}, {48, 7573}, {63, 85}, {100, 8021}, {189, 268}, {220, 27287}, {306, 319}, {312, 3305}, {469, 26063}, {664, 16585}, {756, 26000}, {1073, 6349}, {1175, 33078}, {1220, 5294}, {1311, 15439}, {1441, 3219}, {1762, 21231}, {1796, 18653}, {1815, 18652}, {1952, 25091}, {2983, 17923}, {3757, 4518}, {4102, 17264}, {4552, 39770}, {5235, 25515}, {5657, 7497}, {5745, 34234}, {6998, 26885}, {14829, 30608}, {14942, 25006}, {18359, 27065}, {19607, 32779}, {19810, 28660}, {19860, 31359}, {20305, 30841}, {25255, 33761}, {25935, 32008}, {28980, 33066}

X(40435) = isogonal conjugate of X(2260)
X(40435) = isotomic conjugate of X(5249)
X(40435) = polar conjugate of X(1838)
X(40435) = isotomic conjugate of the complement of X(3219)
X(40435) = isotomic conjugate of the isogonal conjugate of X(2259)
X(40435) = polar conjugate of the isogonal conjugate of X(1794)
X(40435) = X(40412)-Ceva conjugate of X(943)
X(40435) = X(i)-cross conjugate of X(j) for these (i,j): {1021, 100}, {1577, 190}, {5259, 86}, {8611, 1897}, {24084, 4632}, {26017, 37206}, {35057, 664}
X(40435) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2260}, {3, 1841}, {4, 14597}, {6, 942}, {19, 4303}, {25, 18607}, {28, 18591}, {31, 5249}, {48, 1838}, {57, 14547}, {58, 2294}, {163, 23752}, {222, 1859}, {278, 23207}, {442, 1333}, {500, 2160}, {604, 6734}, {849, 21675}, {934, 33525}, {1172, 39791}, {1427, 8021}, {1437, 1865}, {2982, 37993}, {3824, 34819}, {6186, 16585}, {23595, 32656}
X(40435) = cevapoint of X(i) and X(j) for these (i,j): {1, 71}, {2, 3219}, {9, 10}, {1794, 2259}
X(40435) = trilinear pole of line {522, 3465}
X(40435) = barycentric product X(i)*X(j) for these {i,j}: {10, 40412}, {75, 943}, {76, 2259}, {264, 1794}, {306, 40395}, {312, 2982}, {313, 1175}, {4397, 36048}, {4561, 14775}, {15439, 35519}
X(40435) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 942}, {2, 5249}, {3, 4303}, {4, 1838}, {6, 2260}, {8, 6734}, {10, 442}, {19, 1841}, {33, 1859}, {35, 500}, {37, 2294}, {48, 14597}, {55, 14547}, {63, 18607}, {71, 18591}, {73, 39791}, {212, 23207}, {313, 1234}, {523, 23752}, {594, 21675}, {657, 33525}, {943, 1}, {1175, 58}, {1698, 3824}, {1794, 3}, {1826, 1865}, {2259, 6}, {2328, 8021}, {2982, 57}, {3219, 16585}, {3811, 14054}, {4420, 31938}, {6198, 1844}, {14547, 37993}, {14775, 7649}, {15439, 109}, {17924, 23595}, {32651, 1461}, {34772, 39772}, {36048, 934}, {40395, 27}, {40412, 86}

X(40436) = CEVAPOINT OF X(1) AND X(78)

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

X(40436) lies on the conic {{A,B,C,X(1),X(6)}} and on these lines: {1, 4438}, {6, 26690}, {8, 1411}, {34, 78}, {56, 1259}, {58, 22836}, {106, 3976}, {269, 320}, {447, 1043}, {976, 1220}, {977, 1193}, {998, 3811}, {1027, 3810}, {1098, 5692}, {1222, 3938}, {1431, 4259}, {1438, 3061}, {1474, 2327}, {2191, 19861}, {3445, 17597}, {7253, 25253}, {7259, 25087}

X(40436) = isogonal conjugate of X(3924)
X(40436) =isotomic conjugate of X(17861)
X(40436) =isotomic conjugate of the anticomplement of X(25078)
X(40436) =isotomic conjugate of the complement of X(25252)
X(40436) =X(i)-cross conjugate of X(j) for these (i,j): {652, 190}, {16612, 662}, {21189, 100}, {25078, 2}
X(40436) =X(i)-isoconjugate of X(j) for these (i,j): {1, 3924}, {6, 3772}, {9, 36570}, {19, 26934}, {31, 17861}, {42, 17189}, {56, 1837}, {58, 21935}, {213, 16749}
X(40436) =cevapoint of X(i) and X(j) for these (i,j): {1, 78}, {2, 25252}, {42, 21078}
X(40436) =trilinear pole of line {649, 6003}
X(40436) =barycentric product X(i)*X(j) for these {i,j}: {9, 34399}, {63, 34406}
X(40436) =barycentric quotient X(i)/X(j) for these {i,j}: {1, 3772}, {2, 17861}, {3, 26934}, {6, 3924}, {9, 1837}, {37, 21935}, {56, 36570}, {81, 17189}, {86, 16749}, {34399, 85}, {34406, 92}

X(40437) = CEVAPOINT OF X(1) AND X(80)

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

X(40437) lies on these lines: {1, 14628}, {2, 36590}, {11, 953}, {36, 80}, {54, 10950}, {59, 952}, {60, 3109}, {514, 1168}, {517, 655}, {519, 2323}, {859, 24624}, {860, 1309}, {1317, 1391}, {1318, 1387}, {1319, 14204}, {1411, 1870}, {1443, 17895}, {1807, 4358}, {1837, 3417}, {6740, 16704}, {6830, 39270}, {6882, 38954}, {10428, 37222}, {14266, 14584}, {18359, 38460}, {32899, 37718}, {34079, 37168}, {36909, 36915}

X(40437) = isogonal conjugate of X(34586)
X(40437) = isogonal conjugate of the complement of X(38955)
X(40437) = X(i)-cross conjugate of X(j) for these (i,j): {1, 104}, {6, 24624}, {523, 1309}, {650, 655}, {2605, 2720}, {14584, 80}
X(40437) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34586}, {3, 1845}, {6, 16586}, {36, 517}, {214, 14260}, {654, 24029}, {758, 859}, {908, 7113}, {1145, 16944}, {1457, 4511}, {1465, 2323}, {1870, 22350}, {1983, 10015}, {2183, 3218}, {2361, 22464}, {2397, 21758}, {2427, 3960}, {3310, 4585}, {3724, 17139}, {3738, 23981}, {4242, 8677}, {7128, 38353}, {11570, 39173}, {14571, 22128}, {15906, 39166}
X(40437) = cevapoint of X(i) and X(j) for these (i,j): {1, 80}, {36944, 38955}
X(40437) = trilinear pole of line {654, 900}
X(40437) = barycentric product X(i)*X(j) for these {i,j}: {80, 34234}, {104, 18359}, {909, 20566}, {1411, 36795}, {1807, 16082}, {2161, 18816}, {2250, 14616}, {24624, 38955}, {36590, 40218}
X(40437) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16586}, {6, 34586}, {19, 1845}, {80, 908}, {104, 3218}, {909, 36}, {1411, 1465}, {1795, 22128}, {2006, 22464}, {2161, 517}, {2222, 24029}, {2250, 758}, {2342, 2323}, {2401, 4453}, {3270, 38353}, {6187, 2183}, {10428, 40215}, {18359, 3262}, {18816, 20924}, {24624, 17139}, {32675, 23981}, {34051, 1443}, {34079, 859}, {34234, 320}, {34857, 21801}, {34858, 7113}, {36037, 4585}, {36123, 17923}, {36910, 6735}, {36921, 27757}, {38955, 3936}

X(40438) = CEVAPOINT OF X(1) AND X(81)

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

X(40438) lies on these lines: {1, 757}, {6, 24944}, {10, 86}, {31, 39737}, {37, 81}, {65, 1014}, {75, 873}, {99, 30593}, {171, 38836}, {225, 7282}, {314, 32018}, {320, 34920}, {596, 1509}, {662, 1100}, {741, 2667}, {759, 6578}, {940, 24530}, {1414, 7269}, {1444, 31503}, {1931, 3723}, {2166, 14616}, {2185, 2214}, {2363, 20360}, {3664, 5620}, {3875, 39711}, {4038, 17322}, {4596, 4674}, {4629, 18785}, {4663, 32635}, {6539, 8025}, {10436, 39708}, {17103, 17393}, {17394, 31359}, {18166, 37128}, {20090, 31064}, {25417, 40214}, {26860, 31011}

X(40438) = reflection of X(662) in X(39042)
X(40438) = isogonal conjugate of X(1962)
X(40438) = isotomic conjugate of X(4647)
X(40438) = isogonal conjugate of the anticomplement of X(27798)
X(40438) = isogonal conjugate of the complement of X(17163)
X(40438) = isotomic conjugate of the anticomplement of X(3743)
X(40438) = X(i)-cross conjugate of X(j) for these (i,j): {1, 1255}, {484, 24624}, {513, 662}, {1126, 1171}, {1255, 32014}, {1734, 162}, {1757, 37128}, {3743, 2}, {4063, 799}
X(40438) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1962}, {2, 20970}, {3, 430}, {4, 22080}, {6, 1213}, {10, 2308}, {19, 3958}, {31, 4647}, {32, 1230}, {37, 1100}, {42, 1125}, {55, 3649}, {56, 4046}, {58, 8013}, {65, 3683}, {71, 1839}, {72, 2355}, {81, 21816}, {99, 8663}, {100, 4983}, {101, 4988}, {110, 6367}, {210, 32636}, {213, 4359}, {512, 4427}, {523, 35327}, {553, 1334}, {649, 4115}, {661, 35342}, {692, 30591}, {762, 30581}, {872, 16709}, {1018, 4979}, {1126, 8040}, {1269, 1918}, {1400, 3686}, {1402, 3702}, {1500, 8025}, {1824, 3916}, {1826, 22054}, {2333, 4001}, {3690, 31900}, {3700, 36075}, {4065, 40148}, {4557, 4977}, {4559, 4976}, {4822, 35339}, {4970, 23493}, {4973, 34857}, {7180, 30729}, {8818, 17454}
X(40438) = cevapoint of X(i) and X(j) for these (i,j): {1, 81}, {6, 4068}, {58, 40214}, {63, 14868}, {86, 4360}, {1126, 1255}
X(40438) = trilinear pole of line {661, 1019}
X(40438) = barycentric product X(i)*X(j) for these {i,j}: {1, 32014}, {58, 32018}, {75, 1171}, {81, 1268}, {86, 1255}, {274, 1126}, {286, 1796}, {310, 28615}, {513, 4632}, {514, 4596}, {662, 4608}, {693, 4629}, {757, 6539}, {763, 6538}, {1014, 4102}, {1019, 6540}, {1434, 32635}, {1577, 6578}, {7192, 37212}, {7199, 8701}, {30581, 30594}, {30582, 30593}
X(40438) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1213}, {2, 4647}, {3, 3958}, {6, 1962}, {9, 4046}, {19, 430}, {21, 3686}, {28, 1839}, {31, 20970}, {37, 8013}, {42, 21816}, {48, 22080}, {57, 3649}, {58, 1100}, {75, 1230}, {81, 1125}, {86, 4359}, {100, 4115}, {110, 35342}, {163, 35327}, {274, 1269}, {284, 3683}, {333, 3702}, {513, 4988}, {514, 30591}, {643, 30729}, {649, 4983}, {661, 6367}, {662, 4427}, {757, 8025}, {763, 30593}, {798, 8663}, {1014, 553}, {1019, 4977}, {1021, 4990}, {1100, 8040}, {1126, 37}, {1171, 1}, {1255, 10}, {1268, 321}, {1333, 2308}, {1412, 32636}, {1437, 22054}, {1444, 4001}, {1474, 2355}, {1509, 16709}, {1790, 3916}, {1796, 72}, {3733, 4979}, {3737, 4976}, {4102, 3701}, {4184, 17746}, {4560, 4985}, {4596, 190}, {4608, 1577}, {4627, 35339}, {4629, 100}, {4632, 668}, {5235, 4717}, {6539, 1089}, {6540, 4033}, {6578, 662}, {7192, 4978}, {7203, 30724}, {8025, 6533}, {8701, 1018}, {16704, 4975}, {16948, 4856}, {17104, 17454}, {18197, 4992}, {18206, 4966}, {27644, 4970}, {28615, 42}, {30582, 6538}, {31010, 4036}, {31011, 3992}, {32014, 75}, {32018, 313}, {32635, 2321}, {32911, 4065}, {33635, 210}, {37212, 3952}, {38836, 21879}, {40214, 3647}
X(40438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 81, 33766}, {81, 1255, 1171}, {86, 1268, 32014}, {86, 32004, 319}, {1100, 1963, 662}

X(40439) = CEVAPOINT OF X(1) AND X(86)

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

X(40439) lies on these lines: {1, 873}, {31, 757}, {42, 86}, {81, 213}, {274, 3896}, {741, 5625}, {799, 2107}, {1014, 1402}, {1206, 2106}, {1509, 1621}, {1962, 18827}, {1973, 31904}, {8025, 37128}, {8033, 29814}, {10013, 34022}, {10458, 23493}, {17450, 32010}, {32004, 32864}

X(40439) = isogonal conjugate of X(2667)
X(40439) = isotomic conjugate of X(21020)
X(40439) = isotomic conjugate of the anticomplement of X(10180)
X(40439) = isotomic conjugate of the complement of X(27804)
X(40439) = X(i)-cross conjugate of X(j) for these (i,j): {649, 799}, {2664, 37128}, {4040, 662}, {10180, 2}
X(40439) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2667}, {2, 21753}, {4, 22369}, {6, 16589}, {31, 21020}, {37, 20963}, {42, 3720}, {55, 39793}, {56, 4111}, {58, 21699}, {81, 21820}, {213, 3739}, {512, 4436}, {872, 17175}, {1400, 3691}, {1402, 3706}, {1500, 18166}, {1824, 22060}, {1918, 20888}, {4557, 6372}, {7109, 16748}, {18089, 21814}
X(40439) = cevapoint of X(i) and X(j) for these (i,j): {1, 86}, {2, 27804}, {81, 1621}, {274, 34022}
X(40439) = trilinear pole of line {798, 1019}
X(40439) = barycentric product X(i)*X(j) for these {i,j}: {75, 40408}, {86, 32009}, {7199, 8708}
X(40439) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16589}, {2, 21020}, {6, 2667}, {9, 4111}, {21, 3691}, {31, 21753}, {37, 21699}, {42, 21820}, {48, 22369}, {57, 39793}, {58, 20963}, {81, 3720}, {86, 3739}, {274, 20888}, {333, 3706}, {662, 4436}, {757, 18166}, {873, 16748}, {1019, 6372}, {1434, 4059}, {1509, 17175}, {1790, 22060}, {8708, 1018}, {17103, 4754}, {32009, 10}, {40408, 1}
X(40439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {42, 86, 33779}, {2668, 3720, 799}

X(40440) = CEVAPOINT OF X(1) AND X(92)

Barycentrics    b*c*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + 2*a^2*c^2 + b^2*c^2 - c^4) : : Barycentrics    csc 2A sec(B - C) : :
Trilinears    1/(csc 2B + csc 2C) : :

X(40440) lies on these lines: {48, 92}, {73, 8795}, {75, 255}, {95, 404}, {275, 321}, {276, 313}, {326, 561}, {823, 1953}, {1821, 2148}, {1955, 2169}, {1964, 36120}, {1969, 20571}, {14616, 18831}, {17858, 35200}

X(40440) = polar conjugate of X(1953)
X(40440) = isotomic conjugate of the isogonal conjugate of X(2190)
X(40440) = polar conjugate of the isogonal conjugate of X(2167)
X(40440) = X(i)-cross conjugate of X(j) for these (i,j): {1, 2167}, {656, 823}, {1955, 1821}, {17859, 75}, {21173, 653}
X(40440) = X(i)-isoconjugate of X(j) for these (i,j): {2, 217}, {3, 51}, {4, 418}, {5, 184}, {6, 216}, {22, 27372}, {25, 5562}, {32, 343}, {48, 1953}, {52, 2351}, {53, 577}, {55, 30493}, {63, 2179}, {110, 15451}, {112, 17434}, {154, 8798}, {212, 1393}, {213, 16697}, {228, 18180}, {255, 2181}, {311, 14575}, {324, 14585}, {394, 3199}, {512, 23181}, {560, 18695}, {603, 7069}, {647, 1625}, {810, 2617}, {933, 34983}, {1092, 14569}, {1173, 32078}, {1437, 21807}, {1501, 28706}, {1568, 40352}, {1576, 6368}, {1799, 27374}, {1820, 2180}, {2081, 32662}, {2200, 17167}, {3049, 14570}, {3078, 20574}, {3527, 26907}, {9247, 14213}, {9409, 36831}, {12077, 32661}, {13450, 23606}, {14391, 32640}, {14533, 36412}, {14587, 24862}, {17500, 20775}, {17810, 31504}, {21102, 32656}, {35360, 39201}
X(40440) = cevapoint of X(i) and X(j) for these (i,j): {1, 92}, {4, 18676}, {2167, 2190}
X(40440) = trilinear pole of line {822, 1577}
X(40440) = barycentric product X(i)*X(j) for these {i,j}: {1, 276}, {19, 34384}, {54, 1969}, {63, 8795}, {75, 275}, {76, 2190}, {92, 95}, {158, 34386}, {264, 2167}, {304, 8884}, {326, 8794}, {561, 8882}, {811, 15412}, {933, 20948}, {1577, 18831}, {1748, 34385}, {2148, 18022}, {2169, 18027}, {2616, 6331}, {14208, 16813}, {15414, 36126}, {20879, 39286}, {20883, 39287}
X(40440) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 216}, {4, 1953}, {19, 51}, {24, 2180}, {25, 2179}, {27, 18180}, {31, 217}, {48, 418}, {54, 48}, {57, 30493}, {63, 5562}, {75, 343}, {76, 18695}, {86, 16697}, {92, 5}, {95, 63}, {96, 1820}, {97, 255}, {158, 53}, {162, 1625}, {186, 2290}, {264, 14213}, {275, 1}, {276, 75}, {278, 1393}, {281, 7069}, {286, 17167}, {324, 1087}, {393, 2181}, {561, 28706}, {648, 2617}, {656, 17434}, {661, 15451}, {662, 23181}, {811, 14570}, {823, 35360}, {933, 163}, {1096, 3199}, {1309, 35321}, {1577, 6368}, {1748, 52}, {1826, 21807}, {1969, 311}, {2083, 6751}, {2148, 184}, {2156, 27372}, {2167, 3}, {2168, 2351}, {2169, 577}, {2184, 8798}, {2190, 6}, {2616, 647}, {2623, 810}, {4993, 18477}, {6520, 14569}, {6521, 13450}, {8794, 158}, {8795, 92}, {8882, 31}, {8884, 19}, {8901, 3708}, {14206, 1568}, {14618, 2618}, {15412, 656}, {16030, 4020}, {16813, 162}, {17438, 32078}, {17924, 21102}, {18315, 4575}, {18831, 662}, {19166, 6508}, {19174, 17442}, {19180, 820}, {19189, 1755}, {19210, 4100}, {20902, 35442}, {21449, 1954}, {23286, 822}, {24006, 12077}, {34384, 304}, {34386, 326}, {35196, 2193}, {36035, 14391}, {36134, 32661}, {38808, 610}, {39177, 23189}, {39287, 34055}
X(40440) = {X(1953),X(9252)}-harmonic conjugate of X(823)

X(40441) = CEVAPOINT OF X(3) AND X(49)

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

X(40441) lies on the Jerabek circumhyperbola and these lines: {2, 70}, {3, 19362}, {4, 569}, {5, 6145}, {6, 26}, {23, 1173}, {49, 343}, {52, 54}, {64, 7526}, {65, 2216}, {66, 182}, {67, 140}, {68, 184}, {69, 1147}, {72, 24301}, {74, 11562}, {110, 1209}, {265, 6146}, {578, 14542}, {895, 12235}, {973, 2070}, {1176, 9967}, {1177, 34155}, {1181, 34801}, {1614, 16000}, {2917, 5944}, {3431, 34148}, {3521, 18563}, {3527, 7517}, {3532, 32210}, {4846, 10984}, {5462, 19128}, {5504, 13367}, {5576, 13353}, {5622, 18125}, {5900, 6699}, {6293, 18570}, {6391, 9937}, {6413, 10898}, {6414, 10897}, {6759, 38443}, {6776, 18124}, {7525, 19151}, {7527, 16835}, {7556, 13472}, {8795, 37127}, {9706, 11271}, {9908, 19125}, {9970, 34437}, {9977, 10282}, {10610, 12228}, {10634, 32586}, {10635, 32585}, {12359, 19129}, {14528, 17834}, {15316, 19357}, {15761, 22466}, {18400, 18428}, {18532, 35603}, {19506, 32364}, {22115, 34483}, {22334, 31861}, {34114, 34438}, {34117, 34207}

X(40441) = midpoint of X(3) and X(19362)
X(40441) = isogonal conjugate of X(1594)
X(40441) = isogonal conjugate of the anticomplement of X(7542)
X(40441) = isogonal conjugate of the complement of X(7488)
X(40441) = isogonal conjugate of the polar conjugate of X(40393)
X(40441) = X(6368)-cross conjugate of X(110)
X(40441) = X(2949)-of-orthic-triangle if ABC is acute
X(40441) = cevapoint of X(i) and X(j) for these (i,j): {3, 49}, {52, 34116}, {184, 216}
X(40441) = trilinear pole of line {647, 9380}
X(40441) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1594}, {19, 37636}, {38, 10550}, {92, 570}, {158, 1216}, {1096, 1238}, {1209, 2190}, {1826, 16698}, {2962, 6152}
X(40441) = barycentric product X(i)*X(j) for these {i,j}: {3, 40393}, {63, 2216}, {343, 1166}, {394, 1179}
X(40441) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 37636}, {6, 1594}, {184, 570}, {216, 1209}, {251, 10550}, {343, 1225}, {394, 1238}, {577, 1216}, {1166, 275}, {1179, 2052}, {1437, 16698}, {2216, 92}, {2965, 6152}, {14585, 23195}, {40393, 264}

X(40442) = CEVAPOINT OF X(3) AND X(73)

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

X(40442) lies on the conic {{A,B,C,X(1),X(3)}} and these lines: {1, 411}, {2, 10570}, {3, 1425}, {29, 225}, {73, 283}, {78, 201}, {102, 10902}, {109, 40081}, {219, 2197}, {284, 1400}, {307, 332}, {945, 10267}, {947, 11012}, {1036, 37579}, {1037, 26357}, {1758, 9398}, {1794, 22350}, {1795, 4303}, {1807, 33597}, {1813, 17973}, {3422, 36152}, {3478, 11510}, {10571, 20846}, {35979, 37558}

X(40442) = isogonal conjugate of X(40950)
X(40442) = X(i)-cross conjugate of X(j) for these (i,j): {647, 1813}, {23090, 651}
X(40442) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2646}, {19, 5745}, {21, 407}, {27, 21811}, {28, 21677}, {29, 2650}, {33, 3664}, {34, 6737}, {92, 21748}, {158, 22361}, {270, 21674}, {1172, 17056}, {2299, 18698}, {17136, 18344}
X(40442) = cevapoint of X(3) and X(73)
X(40442) = crosssum of X(407) and X(2650)
X(40442) = trilinear pole of line {652, 17975}
X(40442) = barycentric product X(63)*X(17097)
X(40442) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 5745}, {48, 2646}, {71, 21677}, {73, 17056}, {184, 21748}, {219, 6737}, {222, 3664}, {228, 21811}, {577, 22361}, {1214, 18698}, {1400, 407}, {1409, 2650}, {1813, 17136}, {2197, 21674}, {17097, 92}, {23067, 22003}

X(40443) = CEVAPOINT OF X(3) AND X(77)

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

X(40443) lies on these lines: {3, 7056}, {7, 55}, {63, 1802}, {69, 1260}, {77, 212}, {81, 241}, {85, 1621}, {189, 7367}, {286, 4183}, {1174, 1708}, {1214, 1814}, {3219, 6605}, {6606, 18816}, {7084, 10482}

X(40443) = isogonal conjugate of X(1827)
X(40443) = isotomic conjugate of the polar conjugate of X(1170)
X(40443) = isogonal conjugate of the polar conjugate of X(31618)
X(40443) = X(31618)-Ceva conjugate of X(1170)
X(40443) = X(22160)-cross conjugate of X(1813)
X(40443) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1827}, {4, 2293}, {6, 1855}, {19, 1212}, {25, 4847}, {27, 21795}, {28, 21039}, {33, 354}, {34, 3059}, {92, 20229}, {108, 6608}, {142, 607}, {158, 22079}, {278, 8012}, {281, 1475}, {653, 10581}, {1172, 21808}, {1229, 1973}, {1418, 7079}, {1783, 21127}, {1824, 17194}, {1847, 8551}, {1857, 22053}, {1897, 2488}, {2212, 20880}, {2299, 3925}, {2333, 16713}, {3064, 35326}, {6362, 8750}, {6591, 35341}, {6607, 36118}, {7071, 10481}, {18344, 35338}
X(40443) = cevapoint of X(3) and X(77)
X(40443) = trilinear pole of line {905, 23146}
X(40443) = barycentric product X(i)*X(j) for these {i,j}: {3, 31618}, {63, 21453}, {69, 1170}, {75, 1803}, {77, 32008}, {78, 10509}, {348, 2346}, {905, 6606}, {1174, 7182}, {6605, 7056}
X(40443) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1855}, {3, 1212}, {6, 1827}, {48, 2293}, {63, 4847}, {69, 1229}, {71, 21039}, {73, 21808}, {77, 142}, {184, 20229}, {212, 8012}, {219, 3059}, {222, 354}, {228, 21795}, {348, 20880}, {577, 22079}, {603, 1475}, {652, 6608}, {905, 6362}, {1170, 4}, {1174, 33}, {1214, 3925}, {1331, 35341}, {1444, 16713}, {1459, 21127}, {1790, 17194}, {1803, 1}, {1813, 35338}, {1946, 10581}, {2346, 281}, {6605, 7046}, {6606, 6335}, {7053, 1418}, {7125, 22053}, {7177, 10481}, {7182, 1233}, {10482, 7079}, {10509, 273}, {21453, 92}, {22383, 2488}, {23067, 35310}, {23144, 15185}, {31618, 264}, {32008, 318}, {36059, 35326}

X(40444) = CEVAPOINT OF X(4) AND X(9)

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

X(40444) lies on these lines: {2, 268}, {4, 1260}, {27, 908}, {63, 1847}, {92, 3692}, {219, 278}, {342, 1767}, {1167, 1785}, {1864, 1897}, {7151, 7952}, {9612, 37278}, {11109, 27287}

X(40444) = isogonal conjugate of isotomic conjugate of polar conjugate of X(40958)
X(40444) = polar conjugate of X(1210)
X(40444) = polar conjugate of the isogonal conjugate of X(1167)
X(40444) = X(650)-cross conjugate of X(1897)
X(40444) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23204}, {3, 1108}, {6, 1071}, {48, 1210}, {81, 3611}, {184, 17862}, {219, 37566}, {222, 1864}, {1226, 9247}, {1437, 21933}, {1532, 14578}
X(40444) = cevapoint of X(i) and X(j) for these (i,j): {4, 9}, {19, 7952}, {71, 3191}, {278, 1767}
X(40444) = trilinear pole of line {7649, 8058}
X(40444) = barycentric product X(i)*X(j) for these {i,j}: {92, 40399}, {264, 1167}, {312, 40397}
X(40444) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1071}, {4, 1210}, {19, 1108}, {31, 23204}, {33, 1864}, {34, 37566}, {42, 3611}, {92, 17862}, {264, 1226}, {1167, 3}, {1785, 1532}, {1826, 21933}, {7952, 6260}, {40397, 57}, {40399, 63}

X(40445) = CEVAPOINT OF X(4) AND X(10)

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

X(40445) lies on these lines: {4, 346}, {8, 278}, {10, 2322}, {27, 306}, {29, 40161}, {75, 1847}, {92, 341}, {280, 377}, {318, 6358}, {917, 29163}, {951, 10106}, {1834, 1897}, {3692, 11471}, {5016, 37279}, {6559, 36124}

X(40445) = isotomic conjugate of X(18650)
X(40445) = polar conjugate of X(40940)
X(40445) = polar conjugate of the isogonal conjugate of X(2983)
X(40445) = X(i)-cross conjugate of X(j) for these (i,j): {523, 1897}, {17926, 6335}
X(40445) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1104}, {31, 18650}, {58, 18673}, {184, 17863}, {222, 2264}, {255, 1842}, {440, 1333}, {603, 950}, {849, 21671}, {906, 29162}, {1437, 1834}, {14543, 22383}
X(40445) = cevapoint of X(i) and X(j) for these (i,j): {1, 1782}, {4, 10}, {71, 3190}, {1826, 7046}
X(40445) = trilinear pole of line {3239, 4064}
X(40445) = barycentric product X(i)*X(j) for these {i,j}: {10, 40414}, {92, 1257}, {264, 2983}, {951, 7017}
X(40445) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18650}, {10, 440}, {19, 1104}, {33, 2264}, {37, 18673}, {92, 17863}, {281, 950}, {393, 1842}, {594, 21671}, {951, 222}, {1257, 63}, {1826, 1834}, {1897, 14543}, {2983, 3}, {7649, 29162}, {29163, 1331}, {40414, 86}

X(40446) = CEVAPOINT OF X(4) AND X(34)

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

X(40446) lies on these lines: {4, 496}, {29, 1877}, {34, 318}, {281, 608}, {653, 1828}, {1261, 4200}, {2316, 4848}, {3451, 8748}, {4318, 4696}

X(40446) = isogonal conjugate of X(22072)
X(40446) = polar conjugate of X(3452)
X(40446) = polar conjugate of the isotomic conjugate of X(40420)
X(40446) = polar conjugate of the isogonal conjugate of X(3451)
X(40446) = X(i)-cross conjugate of X(j) for these (i,j): {3451, 40420}, {6591, 653}
X(40446) = X(i)-isoconjugate of X(j) for these (i,j): {1, 22072}, {3, 3057}, {8, 22344}, {48, 3452}, {63, 2347}, {71, 18163}, {78, 1201}, {184, 20895}, {212, 3663}, {219, 3752}, {228, 17183}, {283, 4642}, {345, 20228}, {521, 23845}, {603, 6736}, {650, 23113}, {652, 21362}, {906, 21120}, {1122, 1260}, {1259, 1828}, {1331, 6615}, {1437, 21031}, {1790, 21809}, {1812, 21796}, {1946, 21272}, {2193, 4415}, {4020, 18086}, {4571, 6363}, {17906, 36054}, {22383, 25268}
X(40446) = cevapoint of X(i) and X(j) for these (i,j): {1, 1788}, {4, 34}
X(40446) = barycentric product X(i)*X(j) for these {i,j}: {4, 40420}, {34, 32017}, {92, 1476}, {264, 3451}, {273, 23617}, {278, 1222}, {1261, 1847}, {3064, 6613}
X(40446) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3452}, {6, 22072}, {19, 3057}, {25, 2347}, {27, 17183}, {28, 18163}, {34, 3752}, {92, 20895}, {108, 21362}, {109, 23113}, {225, 4415}, {273, 26563}, {278, 3663}, {281, 6736}, {604, 22344}, {608, 1201}, {653, 21272}, {1222, 345}, {1261, 3692}, {1395, 20228}, {1435, 1122}, {1476, 63}, {1824, 21809}, {1826, 21031}, {1880, 4642}, {1897, 25268}, {3451, 3}, {6591, 6615}, {7649, 21120}, {18026, 21580}, {23617, 78}, {32017, 3718}, {32085, 18086}, {32674, 23845}, {36127, 17906}, {40420, 69}

X(40447) = CEVAPOINT OF X(4) AND X(37)

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

X(40447) lies on these lines: {2, 331}, {4, 3690}, {9, 92}, {29, 3191}, {200, 318}, {264, 17776}, {281, 2052}, {286, 3219}, {321, 2287}, {346, 7017}, {445, 18026}, {943, 1896}, {2982, 16082}, {15439, 39429}

X(40447) = isogonal conjugate of X(14597)
X(40447) = isotomic conjugate of X(18607)
X(40447) = polar conjugate of X(942)
X(40447) = polar conjugate of the isogonal conjugate of X(943)
X(40447) = X(i)-cross conjugate of X(j) for these (i,j): {14618, 6335}, {17926, 1897}
X(40447) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14597}, {3, 2260}, {6, 4303}, {31, 18607}, {48, 942}, {57, 23207}, {58, 18591}, {184, 5249}, {222, 14547}, {255, 1841}, {284, 39791}, {577, 1838}, {1437, 2294}, {1859, 7125}, {23752, 32661}
X(40447) = cevapoint of X(i) and X(j) for these (i,j): {4, 37}, {9, 3191}, {321, 17776}
X(40447) = trilinear pole of line {3900, 4036}
X(40447) = barycentric product X(i)*X(j) for these {i,j}: {264, 943}, {321, 40395}, {668, 14775}, {1969, 2259}, {2982, 7017}
X(40447) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4303}, {2, 18607}, {4, 942}, {6, 14597}, {19, 2260}, {33, 14547}, {37, 18591}, {55, 23207}, {65, 39791}, {92, 5249}, {158, 1838}, {318, 6734}, {393, 1841}, {943, 3}, {1175, 1437}, {1794, 255}, {1826, 2294}, {1857, 1859}, {1859, 37993}, {2259, 48}, {2982, 222}, {4183, 8021}, {5174, 39772}, {6198, 500}, {14775, 513}, {15439, 36059}, {24006, 23752}, {40395, 81}, {40412, 1444}

X(40448) = CEVAPOINT OF X(3) AND X(5)

Barycentrics    (a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 3*a^6*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - b^6*c^2 + 2*a^4*c^4 + 3*a^2*b^2*c^4 + 3*b^4*c^4 - 2*a^2*c^6 - 3*b^2*c^6 + c^8) : :
Trilinears    1/(cos A - cos 2A cos(B - C)) : :
Trilinears    csc(A + T) : :, T as at X(389)

Let A'B'C' be the orthic triangle. Let BA and CA be the orthogonal projections of B' and C' on line BC, resp. Let (OA) be the circle with segment BACA as diameter. Define (OB), (OC) cyclically. X(40448) is the radical center of circles (OA), (OB), (OA). (Randy Hutson, December 18, 2020)

X(40448) lies on the Kiepert circumhyperbola and these lines: {2, 578}, {3, 2052}, {4, 577}, {5, 275}, {6, 13599}, {20, 8796}, {30, 39284}, {76, 3964}, {83, 7399}, {94, 14118}, {95, 9291}, {96, 12022}, {98, 6146}, {140, 16080}, {226, 3075}, {262, 10982}, {321, 7549}, {418, 8884}, {459, 631}, {485, 6809}, {486, 6810}, {671, 34664}, {1093, 6641}, {1181, 13380}, {2394, 38933}, {3525, 38253}, {5392, 7503}, {5562, 9290}, {6504, 6816}, {6831, 40395}, {6905, 22341}, {8613, 8887}, {9381, 34864}, {11414, 20792}, {11538, 34007}, {13160, 40393}, {13322, 19169}, {17928, 34289}, {23239, 40082}, {37334, 37892}

X(40448) = midpoint of X(4) and X(17401)
X(40448) = isogonal conjugate of X(389)
X(40448) = isogonal conjugate of the anticomplement of X(11793)
X(40448) = isogonal conjugate of the complement of X(5562)
X(40448) = isotomic conjugate of the polar conjugate of X(40402)
X(40448) = X(i)-cross conjugate of X(j) for these (i,j): {12241, 4}, {17434, 648}, {23286, 110}, {23290, 925}, {26897, 3}, {34965, 264}
X(40448) = X(i)-isoconjugate of X(j) for these (i,j): {1, 389}, {1953, 19170}, {2148, 34836}, {2169, 6750}
X(40448) = cevapoint of X(i) and X(j) for these (i,j): {3, 5}, {6, 418}, {216, 34985}
X(40448) = trilinear pole of line {523, 32320}
X(40448) = barycentric product X(69)*X(40402)
X(40448) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 34836}, {6, 389}, {53, 6750}, {54, 19170}, {40402, 4}
X(40448) = {X(5),X(2055)}-harmonic conjugate of X(275)

X(40449) = CEVAPOINT OF X(5) AND X(143)

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

X(40449) lies on the conic {{A,B,C,X(4),X(5)}} and these lines: {2, 16837}, {3, 3613}, {4, 569}, {32, 2165}, {52, 311}, {143, 25043}, {315, 327}, {1141, 1166}, {1487, 31610}, {1625, 7745}, {3843, 17703}, {7401, 8797}, {10412, 20188}, {11816, 15226}, {11818, 34449}, {13450, 30506}, {31724, 38305}

X(40449) = X(512)-cross conjugate of X(1625)
X(40449) = X(i)-isoconjugate of X(j) for these (i,j): {570, 2167}, {1216, 2190}, {1594, 2169}, {2148, 37636}
X(40449) = cevapoint of X(i) and X(j) for these (i,j): {5, 143}, {51, 36412}
X(40449) = crosssum of X(570) and X(23195)
X(40449) = barycentric product X(i)*X(j) for these {i,j}: {5, 40393}, {343, 1179}, {2216, 14213}
X(40449) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 37636}, {51, 570}, {53, 1594}, {216, 1216}, {217, 23195}, {343, 1238}, {1179, 275}, {2216, 2167}, {14577, 6152}, {18180, 16698}, {36412, 1209}, {40393, 95}

X(40450) = CEVAPOINT OF X(1) AND X(11)

Barycentrics    (a^4 - a^3*b - a*b^3 + b^4 - a^3*c + 2*a^2*b*c + 2*a*b^2*c - b^3*c - a^2*c^2 - 2*a*b*c^2 - b^2*c^2 + a*c^3 + b*c^3)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 2*a^2*b*c - 2*a*b^2*c + b^3*c + 2*a*b*c^2 - b^2*c^2 - a*c^3 - b*c^3 + c^4) : :
Trilinears    1/(2 - cos(A - B) - cos(A - C)) : :

X(40450) lies on these lines: {1, 1090}, {11, 59}, {36, 516}, {54, 496}, {60, 37722}, {952, 1391}, {953, 1387}, {2323, 4700}, {3417, 11373}, {3582, 13329}, {4511, 4742}, {24002, 24203}

X(40450) = X(1983)-cross conjugate of X(24624)
X(40450) = X(i)-isoconjugate of X(j) for these (i,j): {2, 21742}, {3, 1830}, {4, 22346}, {6, 16578}, {56, 14740}, {81, 21797}
X(40450) = cevapoint of X(1) and X(11)
X(40450) = trilinear pole of line {654, 1768}
X(40450) = crosspoint of X(1) and X(11) wrt the excentral triangle
X(40450) = intersection of tangents at X(1) and X(11) to the rectangular hyperbola passing through X(1), X(11), and the excenters
X(40450) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16578}, {9, 14740}, {19, 1830}, {31, 21742}, {42, 21797}, {48, 22346}

X(40451) = CEVAPOINT OF X(11) AND X(244)

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

X(40450) lies on these lines: {10, 106}, {11, 1357}, {80, 1210}, {171, 1416}, {244, 17888}, {291, 3840}, {1015, 1146}, {1261, 4847}, {3663, 4554}, {3976, 39697}, {6736, 31343}, {18191, 34590}, {23617, 32944}

X(40451) = X(i)-isoconjugate of X(j) for these (i,j): {59, 3057}, {100, 23845}, {101, 21362}, {692, 21272}, {765, 1201}, {906, 17906}, {1016, 20228}, {1110, 3663}, {1122, 6065}, {1252, 3752}, {1415, 25268}, {1783, 23113}, {2149, 3452}, {2347, 4564}, {4567, 21796}, {4570, 4642}, {6736, 24027}, {7012, 22072}, {15742, 22344}, {21580, 32739}, {23990, 26563}
X(40451) = cevapoint of X(i) and X(j) for these (i,j): {11, 244}, {1086, 21139}, {1647, 34590}, {2310, 4534}
X(40451) = crosssum of X(1201) and X(23845)
X(40451) = trilinear pole of line {21143, 23764}
X(40451) = barycentric product X(i)*X(j) for these {i,j}: {11, 40420}, {244, 32017}, {1086, 1222}, {1111, 23617}, {1476, 4858}, {3451, 34387}, {6545, 8706}
X(40451) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 3452}, {244, 3752}, {513, 21362}, {514, 21272}, {522, 25268}, {649, 23845}, {693, 21580}, {1015, 1201}, {1086, 3663}, {1111, 26563}, {1146, 6736}, {1222, 1016}, {1459, 23113}, {1476, 4564}, {2170, 3057}, {3120, 4415}, {3122, 21796}, {3125, 4642}, {3248, 20228}, {3271, 2347}, {3451, 59}, {4516, 21809}, {4534, 12640}, {4858, 20895}, {7117, 22072}, {7649, 17906}, {8706, 6632}, {17197, 17183}, {17205, 18600}, {18101, 18086}, {18191, 18163}, {21044, 21031}, {21132, 21120}, {21143, 6363}, {23617, 765}, {32017, 7035}, {40420, 4998}

X(40452) = X(1)X(849)∩X(21)X(961)

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

X(40452) lies on the cubic K1173 and these lines: {1, 849}, {21, 961}, {28, 1791}, {261, 2975}, {314, 16049}, {1043, 5285}, {1169, 17521}, {1220, 5251}, {1610, 7058}, {37265, 37583}

X(40452) = X(i)-isoconjugate of X(j) for these (i,j): {1193, 15232}, {1400, 19608}, {2092, 13478}, {2217, 2292}, {2269, 40160}, {2995, 3725}, {21124, 32653}
X(40452) = cevapoint of X(i) and X(j) for these (i,j): {21, 1610}, {2975, 16049}, {3869, 4225}
X(40452) = barycentric product X(i)*X(j) for these {i,j}: {2363, 4417}, {3869, 14534}, {4225, 30710}, {8707, 16754}
X(40452) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 19608}, {573, 2292}, {961, 40160}, {1169, 2217}, {2298, 15232}, {2363, 13478}, {3185, 2092}, {3869, 1211}, {4225, 3666}, {4417, 18697}, {14534, 2995}, {16754, 3004}, {21078, 20653}, {21189, 21124}, {22134, 22076}, {22276, 21810}
X(40452) = {X(21),X(961)}-harmonic conjugate of X(14534)

X(40453) = X(21)X(1220)∩X(58)X(961)

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

X(40453) lies on the cubic K1173 and these lines: {21, 1220}, {58, 961}, {60, 1610}, {261, 2975}, {284, 2298}, {2051, 37399}, {16049, 20028}

X(40453) = X(31)-cross conjugate of X(2298)
X(40453) = X(i)-isoconjugate of X(j) for these (i,j): {572, 1211}, {960, 37558}, {1193, 17751}, {2092, 14829}, {2292, 2975}, {3666, 21061}, {11109, 22076}, {17074, 21033}, {18697, 20986}
X(40453) = barycentric product X(i)*X(j) for these {i,j}: {2051, 2363}, {2298, 20028}, {14534, 34434}
X(40453) = barycentric quotient X(i)/X(j) for these {i,j}: {1169, 2975}, {2051, 18697}, {2298, 17751}, {2363, 14829}, {20028, 20911}, {34434, 1211}

X(40454) = X(4)X(961)∩X(8)X(197)

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

X(40454) lies on the Feuerbach circumhyperbola, the cubic K1173, and these lines: {4, 961}, {8, 197}, {9, 205}, {21, 1798}, {314, 16049}, {1169, 1172}, {1220, 30513}, {2975, 8048}

X(40454) = isogonal conjugate of X(41600)
X(40454) = X(25)-cross conjugate of X(1169)
X(40454) = X(i)-isoconjugate of X(j) for these (i,j): {197, 4357}, {205, 20911}, {478, 3687}, {960, 21147}, {1193, 3436}, {1766, 3666}, {1848, 22132}, {2292, 16049}, {2300, 20928}, {3882, 6588}, {21074, 40153}
X(40454) = barycentric product X(i)*X(j) for these {i,j}: {961, 34277}, {2298, 8048}, {3435, 30710}, {15420, 40097}
X(40454) = barycentric quotient X(i)/X(j) for these {i,j}: {1169, 16049}, {1220, 20928}, {2298, 3436}, {3435, 3666}, {8048, 20911}

X(40455) = X(1)X(572)∩X(21)X(1220)

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

X(40455) lies on the cubic K1173 and these lines: {1, 572}, {21, 1220}, {958, 14624}, {1791, 2217}, {2975, 30710}, {4216, 5552}, {4224, 29828}, {16049, 17751}

X(40455) = cevapoint of X(22299) and X(23361)
barycentric product X(i)*X(j) for these {i,j}: {1220, 1764}, {2298, 20245}, {2363, 22020}, {14534, 22299}, {23361, 30710}, {23799, 36147}
barycentric quotient X(i)/X(j) for these {i,j}: {1764, 4357}, {3588, 2292}, {20245, 20911}, {22020, 18697}, {22299, 1211}, {23361, 3666}, {23799, 4509}

X(40456) = X(1)X(1437)∩X(21)X(572)

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

X(40456) lies on the cubic K1173 and these lines: {1, 1437}, {21, 572}, {58, 961}, {859, 1724}, {1764, 16049}, {1791, 21061}, {4225, 21363}

X(40457) = X(2)X(14257)∩X(4)X(34277)

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

X(40457) lies on the curves K1173 and Q066, and on these lines: {2, 14257}, {4, 34277}, {63, 12089}, {78, 1766}, {345, 3436}, {1610, 1812}, {17408, 39167}, {34188, 39990}, {34259, 34263}

X(40457) = isotomic conjugate of the anticomplement of X(1880)
X(40457) = X(i)-cross conjugate of X(j) for these (i,j): {1880, 2}, {2217, 1}
X(40457) = X(i)-isoconjugate of X(j) for these (i,j): {21, 12089}, {1400, 28944}
X(40457) = cevapoint of X(i) and X(j) for these (i,j): {123, 523}, {512, 35072}, {513, 34588}
X(40457) = trilinear pole of line {521, 6588}
X(40457) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 28944}, {1400, 12089}

X(40458) = MIDPOINT OF THE 1ST AND 2ND MOSES POINTS

Barycentrics    a*(-2*a^6*b^4 + 6*a^5*b^5 - 6*a^4*b^6 + 2*a^3*b^7 + a^7*b^2*c - 3*a^6*b^3*c + 5*a^4*b^5*c - 4*a^3*b^6*c + 2*a^2*b^7*c - a*b^8*c + a^7*b*c^2 + 2*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 4*a^3*b^5*c^2 - 4*a^2*b^6*c^2 + 3*a*b^7*c^2 - 3*a^6*b*c^3 + 6*a^4*b^3*c^3 - a^3*b^4*c^3 + 5*a^2*b^5*c^3 - 4*a*b^6*c^3 + b^7*c^3 - 2*a^6*c^4 - 6*a^4*b^2*c^4 - a^3*b^3*c^4 - 6*a^2*b^4*c^4 + 2*a*b^5*c^4 - 4*b^6*c^4 + 6*a^5*c^5 + 5*a^4*b*c^5 + 4*a^3*b^2*c^5 + 5*a^2*b^3*c^5 + 2*a*b^4*c^5 + 6*b^5*c^5 - 6*a^4*c^6 - 4*a^3*b*c^6 - 4*a^2*b^2*c^6 - 4*a*b^3*c^6 - 4*b^4*c^6 + 2*a^3*c^7 + 2*a^2*b*c^7 + 3*a*b^2*c^7 + b^3*c^7 - a*b*c^8) : :
X(40458) = 3 X[354] + X[15615]

The 1st and 2nd Moses points are the incircle-inverses of the 1st and 2nd Brocard points. See P(195) in Bicentric Pairs.

X(40458) lies on these lines: {1, 813}, {354, 15615}, {3333, 12032}, {9320, 14760}, {18240, 37998}

X(40458) = midpoint of PU(195)

X(40459) = IDEAL POINT OF THE 1ST AND 2ND MOSES POINTS

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

X(40459) lies on these lines: {1, 4444}, {10, 27929}, {30, 511}, {98, 12032}, {99, 813}, {115, 4129}, {148, 39362}, {239, 661}, {350, 1577}, {671, 18822}, {1575, 9321}, {2482, 35123}, {3008, 25666}, {3023, 15615}, {3912, 4369}, {4107, 24290}, {4367, 8299}, {4375, 6161}, {4560, 17759}, {4761, 32847}, {5216, 38481}, {6542, 7192}, {13178, 13576}, {17266, 24924}, {17310, 31148}, {20016, 31290}, {23596, 24286}, {24281, 24289}, {27321, 27527}, {30225, 36216}, {34342, 34343}, {34362, 34363}, {35352, 38348}, {36230, 36232}, {36234, 36235}

X(40459) = crossdifference of every pair of points on line {6, 25817}
X(40459) = ideal point of PU(195)
X(40459) = barycentric quotient X(2823)/X(21285)

X(40460) = BICENTRIC SUM OF THE 1ST AND 2ND MOSES POINTS

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

X(40460) lies on these lines: {1, 39}, {528, 5572}, {938, 13576}, {3022, 24203}, {3271, 3732}, {3673, 39789}, {3753, 14523}, {3887, 14760}, {5728, 18413}, {11019, 17761}

X(40460) = crosssum of X(55) and X(21320)
X(40460) = bicentric sum of PU(195)

X(40461) = CROSSDIFFERENCE OF THE 1ST AND 2ND MOSES POINTS

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

X(40461) lies on these lines: {6, 25817}, {220, 3570}, {1901, 2238}

X(40461) = crossdifference of PU(195)

leftri

Centers and perspectors of cevapoint conics: X(40462)-X(40529)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, November 30, 2020.

In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point u : v : w, this being the isotomic conjugate of the trilinear pole of L. Let P = p : q : r be a point. The (U,P)-cevapoint conic, introduced here as the locus of X such that the cevapoint of P and X is on the line LU is given by

u (q x + p y)(r x + p z) + v (r y + q z)(p y + q x) + w (p z + r x)(q z + r y) = 0.

The center of the conic is the point

p*(p^2*(p - q - r) u^2 + q^2 (p - q + r) v^2 + r^2 (p + q - r) w^2 - 2 p q r v w + 2 p r (p - r) w u + 2 p q (p - q) u v) : : ,

and the perspector, by

p/(-p u + q r + r w) : q/(p u - q v + r w) : r/(p u + q v - r w).

For every point U, the (U,P)-cevapoint conic passes through the vertices of the anticevian triangle of P.

The appearance of (i,j,k) in the following list means that the center of the (X(i),X(j))-cevapoint conic is X(k):

(1,1,15487), (1,2,7), (1,37,10), (1,514,693), (1,661,523), (1,1577,850), (1,3239,4397), (2,2,2), (6,6,14713), (2,1,40), (2,6,159), (2,37,22271), (2,512,512), (2,513,513), (2,514,514), (2,522,522), (2,523,523), (2,900,900), (75,1,3973), (75,6,15494), (75,513,649), (75,649,667), (75,661,512), (75,798,669), (75,822,39201), (115,523,6722), (264,3,38292), (76,6,5023), (274,37,21868), (190,514,1086), (6,2,4), (6,514,522), (6,523,850), (10,1,36808), (10,2,75), (10,514,7192), (10,649,3733), (514,1,100), (514,2,190), (514,6,14723), (514,8,3699), (514,10,3952), (514,42,4557), (514,200,4578), (514,519,17780), (514,1125,4427), (514,1647,900)

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(1),X(i))-cevapoint conic passes through the points X(j1), X(j2),... :

{1, {2640,5540,16550,16559,16560,16561,16562,16563}}
{2, {149,4440,20355,20533,21220,21221,30578,37781}}
{6, {2932,20871,20999,21004,23402,23860}}
{10, {21090,21100,22029,22031,22035}}
{37, {20716,21888,22306,22308,22313,22321}}
{513, {650,905,6588,14079}}
{514, {514,522,14078,14837,20516,21192,21198,21199}}
{522, {514,3239,4521,14331}}
{523, {661,1577,3700,14086,21051}}
{650, {513,521,11934,14298}}
{661, {523,656,661,6587,13636,13722,14086,17431,17432,23301,31946}}
{1577, {523,525,1577,14086,14566,17898,18310,20910}}
{3239, {522,3239,8058,14302}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(2),X(i))-cevapoint conic passes through the points X(j1), X(j2),... :

{1, {1054,1282,1768,2100,2101,2448,2449,2948,3464,5539,5540,5541,9860,9904,12408,13174,13221,13513,20114,20375,21381,34196,34464,39156}}
{2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}
{6,{2930,7669,10117,15588,16686,20468,20998,20999,23858}}
{37, {20694,21889,21893,22313,22323}}
{512, {512,647,661,2519,14090}}
{513, {513,650,6129,6728,14079,17427,31947,33646}}
{514, {514,3835,7658,14078,21196,25381}}
{522, {522,4521,6728,6730}}
{523, {523,656,661,6587,13636,13722,14086,17431,17432,23301,31946}}
{661, {512,523,4041,21834,22226}}
{900, {900,1647,6544,23757}}
{1647, {900,6550,14442,24131}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(75),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {1054,2629,2636,2640,9324,9355,9359,39335,39336,39337,39338,39339,39340,39341,39342,39343,39344}}
{6, {3196,9259,9509,16686,20672,21004,21783}}
{512, {661,798,3709,14090}}
{513, {513,4083,9269,14079}}
{514, {3835,14078,21191,21195}}
{523, {9276,14086,21051,31946}}
{649, {649,663,6729,14088}}
{661, {512,647,661,2519,14090}}
{798, {512,798,810,3221,14090}}
{822, {647,810,822,2524}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(6),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {16560,16565,20601,21381,21382,39335}}
{2, {146,147,148,149,150,151,152,153,3448,11671,12384,13219,13510,14360,14731,14732,14807,14808,20344,21290,33650,34186,34188,34193,34547,34548,34549,34550}}
{6, {2936,7669,16873,23402,39857}}
{10, {20496,21091,21093,22031,22032}}
{37, {21889,22308,22309,22310}}
{75, {18151,18159,20937,20951}}
{514, {522,4025,14078,20518,21186,21187,21196,21197}}
{523, {523,525,1577,14086,14566,17898,18310,20910}}
{525, {523,3265,8057,38401}}
{690, {1649,14417,18311,21906}}
{693, {693,4391,14080,17896}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(10),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {5540,9359,16554,24578}}
{2, {4440,17154,21224,30579,33888}}
{6, {8301,9259,20999,23392,23404}}
{513, {514,649,650,4083,6589,14079}}
{514, {513,514,905,14078,14079,21172,21191,21194}}
{522, {650,4521,14837,20317}}
{649, {513,649,1459,14079,14088}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(514),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {1,9,40,188,191,366,1045,1050,1490,2136,2949,2950,2951,3174,3307,3308,3646,5506,5528,5541,6326,12658,12660,13144,13146,16009,16550,16558,18598,24578,25427,32632,38004,39131}}
{2, {2,144,192,366,1654,3151,4182,17487,17488,20533,24313,24314,27484,31308,33888,37881}}
{3, {6,3157,7078,22133}}
{6, {3,55,197,199,8301,11505,11506,12335,18755,20871,20996,23858,23859,36943}}
{8, {8,188,3161,6731,8834,19582,30412,30413,39800}}
{9, {1,200,3158,7070}}
{10, {10,37,72,3159,8804,20722,21080,21083,22271,22299,22306,22307,39131}}
{11, {522,523,650,17420}}
{37, {10,42,210,20691,20700,22276,28600}}
{42, {37,42,71,3198,3588,21858,21877,21880}}
{44, {214,678,1960,3689}}
{55, {6,219,5452,7074}}
{115, {523,6367,12069,12071}}
{200, {9,200,2324,4182,6731,24771}}
{512, {1015,1084,3122,14090,16613}}
{513, {244,1015,3756,14079}}
{514, {1086,4904,14078,17761,24185}}
{518, {1575,2254,3693,6184,8299}}
{519, {519,900,1145,4370,34587,36945}}
{521, {2968,7004,34588,35072}}
{522, {11,1146,3036,34589}}
{523, {11,115,3120,8286,14086,23938}}
{650, {11,2310,3271,38375}}
{740, {2238,4010,10026,17793,20723,35068}}
{900, {519,1647,34590,35092}}
{1125, {1125,1213,3650,4065}}
{1279, {659,2348,2976,39048}}
{1647, {900,1647,6544,23757}}

The appearance of {i, {j(1),j(2),...}} in the following list means that the (X(i),X(i))-cevapoint conic passes through the points X(j1), X(j2),... :

{1, {2640,5540,16550,16559,16560,16561,16562,16563}}
{2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}
{6, {2936,7669,16873,23402,39857}}
{514, {1086,4904,14078,17761,24185}}
{523, {115,5461,6128,7668,14086,39022,39023}}
{525, {127,2454,2455,15526}}

Let X*(i) denote the isotomic conjugate of X(i). The appearance of {i, {j(1),j(2),...}} in the following list means that the (X*(i),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {1054,2629,2636,2640,9324,9355,9359,39335,39336,39337,39338,39339,39340,39341,39342,39343,39344}}
{2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}
{3, {20795,22143,22148,22158,23081,23180}}
{6, {1979,9259,9412,9431,20998,21781}}
{37, {21885,21888,21893,21899}}
{514, {14078,21200,21204,21211}}

Let X^2(i) denote the barycentric square of X(i). The appearance of {i, {j(1),j(2),...}} in the following list means that the (X^2(i),X(i))-cevapoint conic passes through the points X(j1), X(j2),...:

{1, {16560,16565,20601,21381,21382,39335}}
{2, {148,4440,8591,9263,17487,25054,39345,39346,39347,39348,39349,39350,39351,39352,39353,39354,39355,39356,39357,39358,39359,39360,39361,39362,39363,39364,39365,39366,39367,39368}}
{30, {2,402,23583,24975}}
{512, {2,3589,4698,6375,6387,6677,6685,6719,14090,15895,15896,34236}}
{513, {2,1125,6692,6703,6714,14079,16604,28600,36812}}
{514, {2,142,3739,4859,6678,6707,14078,15497,27478,31312,31351,31380}}
{520, {2,140,3788,20203,34841}}
{521, {2,5745,6675,6700}}
{522, {2,10,6706,6708,20205,21198,23058}}
{523, {2,5,2023,3413,3414,3634,3934,5461,6036,6118,6119,6669,6670,6673,6674,6704,9478,9756,13881,14086,14566,14762,16509,22847,22893,36597,37691,39143}}
{525, {2,141,6709,14767,18310,20106,20208}}
{526, {2,6671,6672,11064,16760}}
{690, {2,523,524,16511,37911}}
{812, {2,3008,4369,20530,27800}}
{900, {2,514,519,34024,35466}}
{924, {2,6689,16238,23292}}
{1510, {2,6694,6695,37649}}

X(40462) = CENTER OF THE (X(1),X(6))-CEVAPOINT CONIC

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

X(40462) lies on these lines: {1, 11334}, {21, 1626}, {100, 1610}, {1001, 23850}, {1324, 3913}, {3736, 7087}

X(40463) = CENTER OF THE (X(1),X(10))-CEVAPOINT CONIC

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

X(40463) lies on these lines: {37, 42}, {244, 21877}, {321, 1930}, {2205, 3722}, {3294, 3681}, {3936, 22009}, {3954, 21820}, {3994, 22039}, {3995, 40007}, {4043, 18138}, {21020, 21808}, {22000, 24071}, {22021, 24067}, {30821, 31993}, {35892, 36808}

X(40464) = CENTER OF THE (X(1),X(513))-CEVAPOINT CONIC

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

X(40464) lies on these lines: {514, 16604}, {650, 29226}, {663, 1575}, {812, 14838}, {978, 21791}, {4147, 17448}, {25127, 31286}

X(40465) = CENTER OF THE (X(1),X(522))-CEVAPOINT CONIC

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

X(40465) lies on these lines: {9, 3900}, {1212, 1734}

X(40466) = CENTER OF THE (X(1),X(523))-CEVAPOINT CONIC

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

X(40466) lies on these lines: {3907, 23905}, {4129, 28840}, {4151, 6537}, {21052, 23897}

X(40467) = CENTER OF THE (X(1),X(650))-CEVAPOINT CONIC

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

X(40467) lies on these lines: {55, 21173}, {65, 32475}, {513, 4162}, {521, 4086}, {522, 3057}, {1459, 2646}, {1837, 20293}, {17606, 20316}

X(40468) = CENTER OF THE (X(514),X(514))-CEVAPOINT CONIC

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

X(40468) lies on these lines: {2, 32094}, {1086, 6545}, {3452, 24198}, {6546, 6547}, {24232, 33117}

X(40469) = CENTER OF THE (X(523),X(523))-CEVAPOINT CONIC

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

X(40469) lies on these lines: {2, 14588}, {115, 8029}, {1648, 10189}, {11123, 23991}

X(40470) = CENTER OF THE (X(525),X(525))-CEVAPOINT CONIC

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

X(40470) lies on these lines: {15526, 23616}

X(40471) = CENTER OF THE (X(2),X(661))-CEVAPOINT CONIC

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

X(40471) lies on these lines: {38, 7192}, {244, 4369}, {512, 4895}, {513, 21350}, {523, 2254}, {661, 756}, {984, 31290}, {4132, 4784}, {4151, 4467}, {4642, 4761}, {4988, 21727}, {24325, 26822}, {28758, 31264}

X(40472) = CENTER OF THE (X(2),X(1647))-CEVAPOINT CONIC

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

X(40472) lies on these lines: {514, 20042}, {764, 1647}, {900, 4088}, {6546, 17780}

X(40473) = CENTER OF THE (X(75),X(512))-CEVAPOINT CONIC

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

X(40473) lies on these lines: {14838, 14991}

X(40474) = CENTER OF THE (X(75),X(514))-CEVAPOINT CONIC

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

X(40474) lies on these lines: {2, 21390}, {142, 513}, {649, 25604}, {812, 14838}, {2254, 3667}, {3008, 3063}, {3664, 20980}, {3912, 20906}, {4049, 17758}, {4129, 21188}, {4667, 39521}, {4775, 17050}, {4776, 5249}, {17234, 20949}, {17278, 21007}, {20516, 21200}, {21099, 27485}, {21348, 29571}, {21617, 24002}, {23696, 24220}, {23828, 27147}, {30835, 33864}

X(40475) = CENTER OF THE (X(75),X(523))-CEVAPOINT CONIC

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

X(40475) lies on these lines: {661, 5949}, {2610, 14321}

X(40476) = CENTER OF THE (X(6),X(1))-CEVAPOINT CONIC

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

X(40476) lies on these lines: {2, 1726}, {20, 10005}, {35, 984}, {75, 16551}, {1782, 2550}, {4859, 16560}, {21368, 25728}

X(40477) = CENTER OF THE (X(3163),X(30))-CEVAPOINT CONIC

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

X(40477) lies on these lines: {2, 648}, {542, 547}, {549, 6720}, {2799, 22247}

X(40478) = CENTER OF THE (X(1084),X(512))-CEVAPOINT CONIC

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

X(40478) lies on these lines: {2, 670}, {804, 6722}, {2882, 6680}, {3589, 34383}, {5969, 6683}, {22110, 32530}

X(40479) = CENTER OF THE (X(1015),X(513))-CEVAPOINT CONIC

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

X(40479) lies on these lines: {2, 668}, {10, 24739}, {39, 30963}, {291, 3624}, {537, 4698}, {538, 20530}, {1001, 8671}, {1125, 6683}, {2787, 6667}, {2810, 3589}, {3816, 4045}, {3825, 7861}, {3934, 16604}, {6680, 6691}, {7786, 20671}, {7808, 25524}, {7834, 10200}, {17290, 24497}, {17793, 19862}, {22247, 35103}, {24508, 27191}, {31239, 31997}, {32020, 39736}

X(40480) = CENTER OF THE (X(1086),X(514))-CEVAPOINT CONIC

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

X(40480) lies on these lines: {2, 45}, {10, 9041}, {44, 7238}, {75, 29629}, {140, 29243}, {141, 4384}, {142, 3589}, {239, 28337}, {320, 29607}, {335, 4751}, {524, 3008}, {527, 6687}, {528, 1125}, {536, 17067}, {537, 3634}, {549, 24827}, {583, 29749}, {594, 17283}, {597, 4675}, {673, 15668}, {900, 6667}, {1213, 17291}, {1656, 24828}, {1738, 4702}, {2325, 28297}, {2786, 6722}, {2796, 19878}, {3090, 24813}, {3526, 24833}, {3533, 24817}, {3619, 32029}, {3624, 24715}, {3629, 17298}, {3631, 17348}, {3662, 17329}, {3664, 6329}, {3739, 9055}, {3763, 4437}, {3823, 9053}, {3826, 36480}, {3836, 5846}, {3912, 4395}, {3932, 31252}, {3943, 17266}, {3946, 29606}, {4000, 17243}, {4014, 16482}, {4360, 29589}, {4361, 29616}, {4393, 17234}, {4399, 17231}, {4402, 17309}, {4405, 17294}, {4432, 19862}, {4643, 31183}, {4644, 31189}, {4657, 16593}, {4659, 4859}, {4665, 17284}, {4688, 29596}, {4708, 31211}, {4748, 17259}, {4759, 17768}, {4767, 33148}, {4781, 24542}, {4871, 17070}, {4969, 17297}, {5222, 17313}, {5437, 16560}, {6547, 6633}, {6666, 17235}, {6707, 17384}, {7227, 17357}, {7228, 17353}, {7232, 37650}, {8252, 24843}, {8253, 24842}, {9780, 24841}, {16419, 24822}, {16706, 16826}, {17119, 29579}, {17227, 17330}, {17232, 17362}, {17241, 17388}, {17244, 17395}, {17246, 17263}, {17255, 18230}, {17293, 36807}, {17320, 29626}, {17334, 17338}, {17340, 17341}, {17352, 17365}, {17367, 17392}, {17370, 17398}, {17376, 32455}, {17382, 29571}, {17394, 32096}, {17399, 29581}, {17724, 17780}, {17755, 31238}, {19872, 24821}, {24593, 35466}, {24690, 31199}, {24691, 31200}, {24818, 32786}, {24819, 32785}, {26982, 27159}, {29598, 36834}, {29853, 34612}

X(40480) = complement of X(4422)

X(40481) = CENTER OF THE (X(35071),X(520))-CEVAPOINT CONIC

Barycentrics    (3*a^12*b^4 - 12*a^10*b^6 + 18*a^8*b^8 - 12*a^6*b^10 + 3*a^4*b^12 - 4*a^12*b^2*c^2 + 10*a^10*b^4*c^2 - 4*a^8*b^6*c^2 - 8*a^6*b^8*c^2 + 8*a^4*b^10*c^2 - 2*a^2*b^12*c^2 + 3*a^12*c^4 + 10*a^10*b^2*c^4 - 26*a^8*b^4*c^4 + 20*a^6*b^6*c^4 - 15*a^4*b^8*c^4 + 6*a^2*b^10*c^4 + 2*b^12*c^4 - 12*a^10*c^6 - 4*a^8*b^2*c^6 + 20*a^6*b^4*c^6 + 8*a^4*b^6*c^6 - 4*a^2*b^8*c^6 - 8*b^10*c^6 + 18*a^8*c^8 - 8*a^6*b^2*c^8 - 15*a^4*b^4*c^8 - 4*a^2*b^6*c^8 + 12*b^8*c^8 - 12*a^6*c^10 + 8*a^4*b^2*c^10 + 6*a^2*b^4*c^10 - 8*b^6*c^10 + 3*a^4*c^12 - 2*a^2*b^2*c^12 + 2*b^4*c^12) : :

X(40481) lies on these lines: {2, 6528}, {2797, 6722}, {3526, 14941}

X(40482) = CENTER OF THE (X(35072),X(521))-CEVAPOINT CONIC

Barycentrics    (3*a^8*b^2 - 6*a^7*b^3 - 3*a^6*b^4 + 12*a^5*b^5 - 3*a^4*b^6 - 6*a^3*b^7 + 3*a^2*b^8 - 4*a^8*b*c + 4*a^7*b^2*c + 10*a^6*b^3*c - 10*a^5*b^4*c - 8*a^4*b^5*c + 8*a^3*b^6*c + 2*a^2*b^7*c - 2*a*b^8*c + 3*a^8*c^2 + 4*a^7*b*c^2 - 12*a^6*b^2*c^2 - 2*a^5*b^3*c^2 + 13*a^4*b^4*c^2 - 4*a^3*b^5*c^2 - 6*a^2*b^6*c^2 + 2*a*b^7*c^2 + 2*b^8*c^2 - 6*a^7*c^3 + 10*a^6*b*c^3 - 2*a^5*b^2*c^3 - 4*a^4*b^3*c^3 + 2*a^3*b^4*c^3 - 2*a^2*b^5*c^3 + 6*a*b^6*c^3 - 4*b^7*c^3 - 3*a^6*c^4 - 10*a^5*b*c^4 + 13*a^4*b^2*c^4 + 2*a^3*b^3*c^4 + 6*a^2*b^4*c^4 - 6*a*b^5*c^4 - 2*b^6*c^4 + 12*a^5*c^5 - 8*a^4*b*c^5 - 4*a^3*b^2*c^5 - 2*a^2*b^3*c^5 - 6*a*b^4*c^5 + 8*b^5*c^5 - 3*a^4*c^6 + 8*a^3*b*c^6 - 6*a^2*b^2*c^6 + 6*a*b^3*c^6 - 2*b^4*c^6 - 6*a^3*c^7 + 2*a^2*b*c^7 + 2*a*b^2*c^7 - 4*b^3*c^7 + 3*a^2*c^8 - 2*a*b*c^8 + 2*b^2*c^8) : :

X(40482) lies on these lines: {2, 18026}, {140, 2808}, {2798, 6722}

X(40483) = CENTER OF THE (X(1146),X(522))-CEVAPOINT CONIC

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

X(40483) lies on these lines: {2, 664}, {116, 5845}, {528, 3828}, {952, 6710}, {1213, 23674}, {1565, 31273}, {2785, 6722}, {3015, 6707}, {3039, 24318}, {3634, 28850}, {5834, 30808}, {6366, 6667}, {9317, 31192}, {9780, 14942}, {21044, 26007}

X(40484) = CENTER OF THE (X(15526),X(525))-CEVAPOINT CONIC

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

X(40484) lies on these lines: {2, 648}, {5, 9530}, {140, 542}, {287, 3619}, {2799, 6722}, {3763, 15595}, {6723, 9033}

X(40485) = CENTER OF THE (X(18334),X(522))-CEVAPOINT CONIC

Barycentrics    (3*a^12*b^4 - 12*a^10*b^6 + 18*a^8*b^8 - 12*a^6*b^10 + 3*a^4*b^12 - 4*a^12*b^2*c^2 + 8*a^10*b^4*c^2 - 4*a^8*b^6*c^2 - 2*a^6*b^8*c^2 + 4*a^4*b^10*c^2 - 2*a^2*b^12*c^2 + 3*a^12*c^4 + 8*a^10*b^2*c^4 - 16*a^8*b^4*c^4 + 10*a^6*b^6*c^4 - 8*a^4*b^8*c^4 + 4*a^2*b^10*c^4 + 2*b^12*c^4 - 12*a^10*c^6 - 4*a^8*b^2*c^6 + 10*a^6*b^4*c^6 + 4*a^4*b^6*c^6 - 2*a^2*b^8*c^6 - 8*b^10*c^6 + 18*a^8*c^8 - 2*a^6*b^2*c^8 - 8*a^4*b^4*c^8 - 2*a^2*b^6*c^8 + 12*b^8*c^8 - 12*a^6*c^10 + 4*a^4*b^2*c^10 + 4*a^2*b^4*c^10 - 8*b^6*c^10 + 3*a^4*c^12 - 2*a^2*b^2*c^12 + 2*b^4*c^12) : :

X(40485) lies on this line: {2, 18334}

X(40486) = CENTER OF THE (X(23992),X(690))-CEVAPOINT CONIC

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

X(40486) lies on these lines: {2, 892}, {99, 23991}, {523, 6722}, {524, 22244}, {7804, 18122}, {9182, 31274}

X(40487) = CENTER OF THE (X(35119),X(812))-CEVAPOINT CONIC

Barycentrics    (3*a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 - 4*a^4*b*c - 2*a^3*b^2*c - 4*a^2*b^3*c - 2*a*b^4*c + 3*a^4*c^2 - 2*a^3*b*c^2 + 16*a^2*b^2*c^2 - 2*a*b^3*c^2 + 3*b^4*c^2 - 2*a^3*c^3 - 4*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + 2*a^2*c^4 - 2*a*b*c^4 + 3*b^2*c^4) : :

X(40487) lies on these lines: {2, 4562}, {350, 29607}, {3008, 20530}, {17278, 36232}

X(40488) = CENTER OF THE (X(35092),X(900))-CEVAPOINT CONIC

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

X(40488) lies on these lines: {2, 4555}, {190, 6547}, {519, 6687}, {3008, 22247}, {4384, 36230}, {4409, 32106}, {6722, 25666}, {25031, 31285}, {29629, 35957}

X(40489) = CENTER OF THE (X(39013),X(924))-CEVAPOINT CONIC

Barycentrics    (3*a^12*b^4 - 12*a^10*b^6 + 18*a^8*b^8 - 12*a^6*b^10 + 3*a^4*b^12 - 4*a^12*b^2*c^2 + 6*a^10*b^4*c^2 - 4*a^8*b^6*c^2 + 4*a^6*b^8*c^2 - 2*a^2*b^12*c^2 + 3*a^12*c^4 + 6*a^10*b^2*c^4 - 2*a^8*b^4*c^4 - 4*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 - 12*a^10*c^6 - 4*a^8*b^2*c^6 - 4*a^6*b^4*c^6 - 8*a^4*b^6*c^6 - 8*b^10*c^6 + 18*a^8*c^8 + 4*a^6*b^2*c^8 + 5*a^4*b^4*c^8 + 12*b^8*c^8 - 12*a^6*c^10 + 2*a^2*b^4*c^10 - 8*b^6*c^10 + 3*a^4*c^12 - 2*a^2*b^2*c^12 + 2*b^4*c^12) : :

X(40489) lies on this line: {2, 39013}

X(40490) = CENTER OF THE (X(39018),X(1510))-CEVAPOINT CONIC

Barycentrics    (3*a^12*b^4 - 12*a^10*b^6 + 18*a^8*b^8 - 12*a^6*b^10 + 3*a^4*b^12 - 4*a^12*b^2*c^2 + 4*a^10*b^4*c^2 - 4*a^8*b^6*c^2 + 10*a^6*b^8*c^2 - 4*a^4*b^10*c^2 - 2*a^2*b^12*c^2 + 3*a^12*c^4 + 4*a^10*b^2*c^4 + 16*a^8*b^4*c^4 - 22*a^6*b^6*c^4 + 24*a^4*b^8*c^4 + 2*b^12*c^4 - 12*a^10*c^6 - 4*a^8*b^2*c^6 - 22*a^6*b^4*c^6 - 28*a^4*b^6*c^6 + 2*a^2*b^8*c^6 - 8*b^10*c^6 + 18*a^8*c^8 + 10*a^6*b^2*c^8 + 24*a^4*b^4*c^8 + 2*a^2*b^6*c^8 + 12*b^8*c^8 - 12*a^6*c^10 - 4*a^4*b^2*c^10 - 8*b^6*c^10 + 3*a^4*c^12 - 2*a^2*b^2*c^12 + 2*b^4*c^12) : :

X(40490) lies on this line: {2, 39018}

X(40491) = CENTER OF THE (X(6),X(10))-CEVAPOINT CONIC

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

X(40491) lies on these lines: {10, 15281}, {313, 22008}, {321, 908}, {3947, 15282}, {20245, 21061}

X(40492) = CENTER OF THE (X(6),X(37))-CEVAPOINT CONIC

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

X(40492) lies on these lines: {321, 22271}, {1233, 22275}, {21867, 31993}, {21883, 21889}

X(40493) = CENTER OF THE (X(6),X(75))-CEVAPOINT CONIC

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

X(40493) lies on these lines: {75, 354}, {76, 85}, {210, 16284}, {305, 17786}, {322, 325}, {345, 7196}, {561, 20923}, {1088, 3693}, {1214, 31627}, {1909, 3974}, {4847, 4967}, {6374, 30048}, {10030, 18141}, {17026, 19804}, {17241, 18045}, {18142, 20942}, {18157, 40072}, {18743, 20448}, {20646, 20946}, {20930, 20945}, {24524, 30615}, {27538, 30806}, {30988, 33116}

X(40494) = CENTER OF THE (X(6),X(525))-CEVAPOINT CONIC

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

X(40494) lies on these lines: {3, 520}, {523, 2072}, {2972, 12079}, {34333, 36169}

X(40495) = CENTER OF THE (X(6),X(693))-CEVAPOINT CONIC

Barycentrics    b^3*(b - c)*c^3 : :

X(40495) lies on these lines: {75, 1734}, {76, 4391}, {99, 2864}, {274, 905}, {667, 7255}, {670, 35156}, {693, 784}, {772, 2084}, {824, 1577}, {826, 850}, {3122, 24238}, {3126, 33933}, {3900, 17143}, {4077, 23877}, {4142, 20518}, {4705, 20906}, {8714, 20888}, {15413, 17924}, {16992, 22160}, {17072, 20907}, {17496, 34284}, {20909, 35560}, {21056, 35554}, {21438, 24290}, {23100, 23596}, {23685, 33935}

X(40495) = isotomic conjugate of X(692)
X(40495) = polar conjugate of isogonal conjugate of X(15413)
X(40495) = polar conjugate of barycentric product of circumcircle intercepts of Stevanovic circle
X(40495) = complement of polar conjugate of isogonal conjugate of X(23191)
X(40495) = anticomplement of polar conjugate of isogonal conjugate of X(23228)
X(40495) = crossdifference of every pair of points on line X(560)X(1501)
X(40495) = trilinear pole of line X(16732)X(17878)

X(40496) = CENTER OF THE (X(10),X(6))-CEVAPOINT CONIC

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

X(40496) lies on these lines: {1, 7428}, {55, 14753}, {56, 34281}, {58, 23383}, {333, 1610}, {978, 20470}, {3679, 39578}, {8053, 37296}, {11194, 15654}

X(40497) = CENTER OF THE (X(10),X(522))-CEVAPOINT CONIC

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

X(40497) lies on this line: {14838, 23058}

X(40498) = CENTER OF THE (X(514),X(3))-CEVAPOINT CONIC

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

X(40498) lies on this line: {1331, 22154}

X(40499) = CENTER OF THE (X(514),X(9))-CEVAPOINT CONIC

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

X(40499) lies on these lines: {1, 17048}, {8, 27010}, {100, 3903}, {101, 692}, {512, 8671}, {522, 4568}, {644, 663}, {660, 932}, {668, 3907}, {831, 29067}, {874, 4561}, {997, 32941}, {1026, 4595}, {1293, 28528}, {1310, 29052}, {2605, 3908}, {2705, 28486}, {3009, 19589}, {3810, 33946}, {4069, 30728}, {4073, 20753}, {8691, 29187}, {14714, 28071}, {28552, 28564}

X(40500) = CENTER OF THE (X(514),X(9))-CEVAPOINT CONIC

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

X(40500) lies on these lines: {522, 4546}, {523, 1459}, {650, 28161}, {900, 21119}, {1638, 14353}, {17420, 24457}, {21106, 28179}

X(40501) = CENTER OF THE (X(514),X(37))-CEVAPOINT CONIC

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

X(40501) lies on these lines: {10, 20529}, {100, 17943}, {210, 21890}, {1018, 4705}, {2533, 21272}, {3293, 8298}, {3699, 3799}, {3939, 21891}, {4103, 4155}, {4553, 17934}, {21295, 21604}, {21383, 23861}, {21725, 21888}

X(40502) = CENTER OF THE (X(514),X(115))-CEVAPOINT CONIC

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

X(40502) lies on these lines: {523, 2487}, {690, 31290}, {4024, 4705}

X(40503) = CENTER OF THE (X(514),X(900))-CEVAPOINT CONIC

Barycentrics    (2*a - b - c)*(b - c)^2*(a^4 - 2*a^3*b - 4*a^2*b^2 + 5*a*b^3 - 2*b^4 - 2*a^3*c + 14*a^2*b*c - 7*a*b^2*c + 3*b^3*c - 4*a^2*c^2 - 7*a*b*c^2 - b^2*c^2 + 5*a*c^3 + 3*b*c^3 - 2*c^4) : :

X(40503) lies on these lines: {519, 1279}, {1086, 6550}

X(40504) = PERSPECTOR OF THE (X(1),X(37))-CEVAPOINT CONIC

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

X(40504) lies on these lines: {1, 5132}, {10, 15281}, {37, 40147}, {42, 13476}, {75, 3681}, {517, 4698}, {518, 596}, {759, 6577}, {969, 22282}, {2214, 2280}, {2218, 21059}, {3739, 22325}, {4850, 39739}, {9278, 21863}, {19874, 22299}, {22279, 39712}, {22293, 40005}

X(40505) = PERSPECTOR OF THE (X(1),X(650))-CEVAPOINT CONIC

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

X(40505) lies on these lines: {354, 1122}, {497, 17183}, {1201, 1279}, {1827, 1828}, {2347, 2348}, {3057, 3059}, {3271, 6067}, {7083, 26357}, {8605, 11246}, {9309, 24477}

X(40506) = PERSPECTOR OF THE (X(3163),X(30))-CEVAPOINT CONIC

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

X(40506) lies on this line: {5055, 35912}

X(40507) = PERSPECTOR OF THE (X(1084),X(512))-CEVAPOINT CONIC

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

X(40507) lies on this line: {2, 31646}

X(40508) = PERSPECTOR OF THE (X(1015),X(513))-CEVAPOINT CONIC

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

X(40508) lies on these lines: {2, 31645}, {536, 20688}, {1015, 36957}, {27195, 31625}

X(40509) = PERSPECTOR OF THE (X(1086),X(514))-CEVAPOINT CONIC

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

X(40509) lies on these lines: {2, 31647}, {519, 3836}, {1000, 25031}, {1016, 27191}, {1086, 36954}, {3008, 16704}, {3912, 31011}, {4358, 20432}, {17305, 32013}

X(40510) = PERSPECTOR OF THE (X(1086),X(514))-CEVAPOINT CONIC

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

X(40510) lies on these lines: {2, 31648}, {1146, 36956}, {1275, 31640}

X(40511) = PERSPECTOR OF THE (X(115),X(523))-CEVAPOINT CONIC

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

X(40511) lies on these lines: {2, 31644}, {115, 36953}, {325, 31068}, {523, 6722}, {524, 1570}, {3589, 5967}, {4045, 15464}, {4590, 14061}, {5461, 9164}

X(40512) = PERSPECTOR OF THE (X(15526),X(525))-CEVAPOINT CONIC

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

X(40512) lies on this line: {140, 35912}

X(40513) = PERSPECTOR OF THE (X(23992),X(690))-CEVAPOINT CONIC

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

X(40513) lies on these lines: {543, 23991}, {1641, 22247}, {6722, 8371}

X(40514) = PERSPECTOR OF THE (X(35092),X(900))-CEVAPOINT CONIC

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

X(40514) lies on this line: {545, 6547}

X(40515) = PERSPECTOR OF THE (X(6),X(10))-CEVAPOINT CONIC

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

X(40515) lies on these lines: {2, 2140}, {10, 15281}, {37, 17758}, {76, 4043}, {98, 6577}, {101, 29775}, {213, 17761}, {218, 1751}, {226, 20616}, {321, 4006}, {1001, 22006}, {3293, 13576}, {3912, 22010}, {4049, 22046}, {4079, 23100}, {5134, 6625}, {17152, 29773}, {22018, 24072}, {22020, 34258}

X(40516) = PERSPECTOR OF THE (X(6),X(37))-CEVAPOINT CONIC

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

X(40516) lies on these lines: {8, 22271}, {55, 5283}, {76, 22275}, {13576, 22300}

X(40517) = PERSPECTOR OF THE (X(6),X(690))-CEVAPOINT CONIC

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

X(40517) lies on these lines: {3, 36880}, {39, 597}, {83, 31128}, {126, 1506}, {574, 34161}, {1649, 3005}, {6680, 7664}, {7794, 23992}, {7813, 21906}, {7820, 14357}

X(40518) = PERSPECTOR OF THE (X(514),X(3))-CEVAPOINT CONIC

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

X(40518) lies on these lines: {651, 1625}, {1813, 22154}

X(40519) = PERSPECTOR OF THE (X(514),X(6))-CEVAPOINT CONIC

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

X(40519) lies on these lines: {3, 596}, {35, 39949}, {55, 40148}, {100, 1634}, {190, 4057}, {692, 8671}, {3733, 4553}, {4436, 4613}, {4557, 21003}, {8053, 37586}, {8683, 23703}, {23344, 36075}

X(40520) = PERSPECTOR OF THE (X(514),X(6))-CEVAPOINT CONIC

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

X(40520) lies on these lines: {}

X(40521) = PERSPECTOR OF THE (X(514),X(37))-CEVAPOINT CONIC

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

X(40521) lies on these lines: {10, 2486}, {37, 3122}, {72, 7206}, {100, 8701}, {190, 513}, {210, 6535}, {518, 4439}, {594, 4092}, {644, 692}, {651, 3908}, {674, 2325}, {756, 2643}, {765, 8702}, {872, 4094}, {1018, 4069}, {1023, 35327}, {1026, 4436}, {1084, 1500}, {1215, 21254}, {2321, 22271}, {2511, 21859}, {3271, 4370}, {3294, 22328}, {3678, 4535}, {3688, 17340}, {3690, 6057}, {3710, 22299}, {3882, 23343}, {3900, 3939}, {3932, 20718}, {3943, 20683}, {3950, 22277}, {3952, 4010}, {3967, 4377}, {3985, 20723}, {4009, 38472}, {4015, 6538}, {4043, 22289}, {4072, 22312}, {4082, 22276}, {4103, 4155}, {4422, 14839}, {4473, 16482}, {4505, 33948}, {4517, 17281}, {4605, 6370}, {4712, 17463}, {6386, 36863}, {13476, 17243}, {17142, 29396}, {20691, 21900}, {20713, 20714}, {20715, 21864}, {21070, 22292}, {21071, 22293}, {21096, 22317}, {22280, 30730}, {35309, 35310}

X(40522) = PERSPECTOR OF THE (X(514),X(44))-CEVAPOINT CONIC

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

X(40522) lies on these lines: {42, 678}, {214, 39697}, {662, 28210}, {667, 4557}

X(40523) = PERSPECTOR OF THE (X(514),X(55))-CEVAPOINT CONIC

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

X(40523) lies on these lines: {}

X(40524) = PERSPECTOR OF THE (X(514),X(115))-CEVAPOINT CONIC

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

X(40524) lies on this line: {190, 12078}

X(40525) = PERSPECTOR OF THE (X(514),X(512))-CEVAPOINT CONIC

Barycentrics    a^2*(-b + c)^2*(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(40525) lies on these lines: {190, 21838}, {1015, 9427}, {1084, 1086}, {1258, 17946}, {3125, 21823}, {9468, 37128}

X(40526) = PERSPECTOR OF THE (X(514),X(518))-CEVAPOINT CONIC

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

X(40526) lies on these lines: {37, 142}, {513, 4557}, {4552, 24002}

X(40527) = PERSPECTOR OF THE (X(514),X(521))-CEVAPOINT CONIC

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

X(40527) lies on these lines: {1086, 16596}, {1167, 17102}, {1214, 34051}, {3942, 39006}, {18191, 35014}, {36100, 40397}

X(40528) = PERSPECTOR OF THE (X(514),X(650))-CEVAPOINT CONIC

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

X(40528) lies on these lines: {11, 1357}, {190, 1222}, {210, 1261}, {1156, 1476}, {2310, 3248}, {2330, 14100}, {3119, 38991}, {3271, 4081}, {17604, 40420}, {20359, 32017}

X(40529) = PERSPECTOR OF THE (X(514),X(740))-CEVAPOINT CONIC

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

X(40529) lies on these lines: {1086, 1213}

leftri

Centers and perspectors of 1st Ceva conics: X(40530)-X(40564)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, November 30, 2020.

In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point u : v : w, this being the isotomic conjugate of the trilinear pole of L. Let P = p : q : r be a point. The 1st (U,P)-Ceva conic, introduced here as the locus of X such that the P-Ceva conjugate of X is on the line L, is given by

u p (-p/x + q/y + r/z) + v q (p/x - q/y + r/z) + w r (p/x + q/y - r/z) = 0.

The center of the conic is the point

p (-p q^2 u v - 3 p q r u v + p q^2 v^2 - q^2 r v^2 - 3 p q r u w - p r^2 u w - 2 p q r v w - 3 q^2 r v w - 3 q r^2 v w + p r^2 w^2 - q r^2 w^2) : :

If U = X(2), then the center of the 1st (U,P)-Ceva conic is the complement of the complement of P, which is also the centroid of {A,B,C,P}, and also the center of the bicevian conic of X(2) and P. (Randy Hutson, December 18, 2020)

The appearance of (i,j,k) in the following list means that the center of the 1st (X(i),X(j))-Ceva conic is X(k):

(2,1,1125), (2,2,2), (2,3,140), (2,4,5), (2,5,3628), (2,6,3589), (2,7,142), (2,8,10), (2,10,3634), (2,25,6677), (2,54,6689), (2,56,6691), (2,57,6692), (2,58,6693), (2,63,5745), (2,69,141), (2,74,6699), (2,75,3739), (2,76,3934), (2,81,6703), (2,85,6706), (2,86,6707), (2,92,6708), (2,98,6036), (2,99,620), (2,100,3035), (2,101,6710), (2,105,6714), (2,107,6716), (2,108,6717), (2,109,6718), (2,110,5972), (2,111,6719), (2,112,6720), (2,190,4422), (2,264,14767), (2,476,22104), (2,511,511), (2,512,512), (2,513,513), (2,514,514), (2,517,517), (2,518,518), (2,522,522), (2,523,523), (2,525,525), (2,648,23583), (2,651,36949), (2,664,17044), (2,668,27076), (2,670,36950), (2,690,690), (2,693,4885), (2,805,22103), (2,842,16760), (2,850,30476), (2,901,22102), (2,925,34844), (2,930,13372), (2,1303,34839), (2,1897,15252), (1,2,3739), (1,75,10), (1,85,40216), (75,1,37), (75,2,1125), (75,6,14751), (75,57,1), (75,99,14750), (75,190,14752), (75,513,4979), (75,1029,8143), (7,7,5452), (264,3,216), (69,4,6), (76,6,39), (8,7,1), (7,8,9), (298,13,396), (299,14,395), (274,37,16589), (34387,59,13006), (4,69,3), (6,76,141), (523,99,523), (693,100,650), (850,110,647), (514,190,514), (338,249,34990), (3,264,5), (37,274,3739), (99,523,115), (599,598,597), (525,648,525), (513,668,513), (1111,765,24036), (1086,1016,4422), (23994,1101,23993), (23989,1252,23988), (1146,1275,17044), (32,1502,626), (594,1509,17045), (10,2,6707), (514,1,14752), (514,2,4422), (514,4,14774), (514,6,14780), (514,7,14759), (514,8,14740), (514,264,14771), (514,664,17494), (514,668,31290), (514,903,39349), (514,1897,522),

The appearance of {i, {j(1),j(2),...}} in the following list means that the 1st (X(2),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{1, {11,214,244,1015,8054,8299,10494,14714,17417,17419,17421,17761,17793,34586,34587,34588,34589,34590,34591,34592,34593,38978,38979,38980,38981,38982,38983,38984,38985,38986,39046}
{2, {115,1015,1084,1086,1146,2454,2455,2482,3163,4370,5997,6184,11672,13466,15166,15167,15449,15525,15526,15527,17416,17429,18334,20532,23967,23972,23976,23980,23986,23992,35066,35067,35068,35069,35070,35071,35072,35073,35074,35075,35076,35077,35078,35079,35080,35081,35082,35083,35084,35085,35086,35087,35088,35089,35090,35091,35092,35093,35094,35095,35110,35111,35112,35113,35114,35115,35116,35117,35118,35119,35120,35121,35122,35123,35124,35125,35126,35127,35128,35129,35130,35131,35132,35133,35134,35135,35508,35509,39008,39009,39010,39011,39012,39013,39014,39015,39016,39017,39018,39019,39020,39021,39022,39023,39206,39207,39208,39209}
{3, {125,1511,2972,17423,34467,35071,38983,38999,39000,39001,39002,39003,39004,39005,39006,39007,39071}
{4, {11,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,1312,1313,1560,1566,2039,2040,2679,3258,3259,5099,5139,5190,5509,5510,5511,5512,5513,5514,5515,5516,5517,5518,5519,5520,5521,5522,5950,5952,5993,6092,9151,9152,9193,10017,11569,11792,12494,12624,13141,13234,13249,13499,13517,13612,13613,13870,13871,13872,13994,13999,14103,14672,15169,15241,15607,15608,15609,15610,15611,15612,15613,15614,16177,16178,16188,16221,16938,18402,18809,20033,20389,20551,20619,20620,20621,20622,20623,20625,21662,22474,25640,25641,25642,31653,31654,31655,31841,31842,31843,31844,31845,33330,33331,33333,33504,34111,34113,35579,35580,35581,35582,35583,35584,35585,35586,35587,35588,35589,35590,35591,35592,35593,35594,35967,35968,35969,35970,35971,35972,36471,36472,38957,38958,38959,38960,38961,38962,38963,38964,38965,38966,38967,38968,38969,38970,38971,38972,38973,38974,38975,38976,38977,39535,40357,40358}
{5, {137,2972,6592,8902,17433,35442,39019}
{6, {125,1084,3124,6593,7668,8054,15450,17413,36213,38987,38988,38989,38990,38991,38992,38993,38994,38995,38996,38997,38998,39067,39068,39075,39079,39080}
{7, {11,1086,8287,10427,13609,16591,16592,16593,16594,16595,16596,16597,20343,21623,26932,34846,38989,39007,39063}
{8, {11,1145,1146,2968,3756,4904,6739,6741,7358,8286,16613,38992,39004,39050}
{10, {115,244,3120,6741,21709,24185}
{19, {244,5521,14936,17463,38991,39069,39070}
{25, {1084,5139,17423,20975,39025,39072}
{54, {125,8901,11597,17433,38984,39013,39027,39045,39233}
{56, {3259,8054,20982,34467,39015,39025}
{57, {1015,2170,5514,19593,24237,38991,39006,39048}
{58, {124,8054,18191,39006,39016,39029}
{59, {15608,38984,38989,39004,39017,39026}
{63, {6506,26932,31653,34591,35072,39006}
{69, {125,5181,6388,7358,15526,15595,17421,26932}
{74, {3,125,2088,3134,39174,39987}
{75, {244,1086,2968,4858,5515,6377,16586,17755,38995,39040}
{76, {115,339,3124,5976,7664,21208,36901,39000}
{81, {1015,3125,5517,17197,19557,26932}
{85, {1111,1146,3119,36905,38959,38980}
{86, {1086,3120,6627,6651,8054,16726,38960}
{92, {1146,4858,5190,16596,34591,39039}
{98, {3,115,868,17423,34156,34810,38997,39078}
{99, {2,3,39,114,618,619,629,630,641,642,1125,1649,2482,3413,3414,3666,5664,5745,5976,6292,6337,6503,6509,6626,7710,8290,8299,8786,10291,10335,11147,11165,13701,13821,13882,13934,14713,15349,15810,15814,15819,15850,22848,22892,27929,30471,30472,33364,33365,33614,33615,33616,33617,33618,33619,33620,33621,34452,34834,34835,38998,39090,39091,39094,39096,39098,39100,39102,40125}
{100, {1,3,9,10,119,142,214,442,600,1145,2092,3126,3307,3308,3647,5507,6184,6260,6594,6600,10427,10472,11517,11530,12631,12639,12640,12864,13089,15346,15347,15348,17057,17060,18258,18642,19557,19584,22754,34261,35204,39041,39048}
{101, {3,118,354,2140,3136,3789,5452,20970,32664,39029,39046}
{105, {3,1015,3140,3675,5511,34160,39025}
{107, {3,4,133,800,1249,3184,6523,14363,15259,16253,20208,23976,33549,33580}
{108, {3,56,429,12610,25640,36103}
{109, {3,65,117,478,3142,3454,24220,34281,36033,39037,39070}
{110, {3,5,6,113,141,206,942,960,1147,1209,1493,1511,2574,2575,2883,4550,5181,6593,8542,10639,10640,10960,10962,11597,11598,11672,15116,15748,16254,17713,19576,19602,21905,22333,22966,32391,33537,33556,34116,34472,34586,34830,34831,37836,37890,39072,39083,39084}
{111, {3,1084,3143,5512,21906,34158}
{112, {3,32,132,427,3162,21248,22391,39045,39071,39086}
{162, {9,1104,36033,36103,39038,39039}
{190, {2,9,37,440,1213,3161,4370,5513,6544,6651,15487,16590,16593,17755,21838,24771,27481,31336,36911,38015,39056,39059,40181}
{264, {136,338,2972,14920,15526,34834,36901,38987}
{291, {10,1015,5518,22116,27846,38995}
{476, {3,30,523,3003,6663,14993,15295,23967,25641,31378,39170}
{511, {511,2679,38987,39000,39009,39073}
{512, {512,2679,3005,21905,23301,38978,38988,39001,39010}
{513, {513,661,3259,3835,6615,14434,17115,27854,31946,38979,38989,39002,39011}
{514, {514,650,661,1566,4521,4988,6544,27929,35092,38980,38990,39003}
{517, {517,3259,33646,35014,38981,39004}
{518, {518,5519,17435,38980,38989,39012,39077}
{522, {522,650,656,6129,6608,6615,7658,10017,23757,35091,38981}
{523, {523,647,1649,3005,3258,4988,8562,13636,13722,17433,17436,21196,23992,31945,31947,35443,35444,38982,38987,39005}
{525, {525,647,6587,14401,17434,33504,35441,39000,39008}
{644, {1,220,1040,4847,5452,24152,24153,24181,24771}
{648, {2,6,216,233,1196,1249,1560,3162,3163,8105,8106,8968,14091,14401,15595,18311,32750,37891,37895,39034,39078,39081,40179}
{651, {6,9,223,226,478,1211,5452,13388,13389,18591,20262,20623,23980,39032,39049,39050,39055,39063}
{653, {9,57,281,1108,1214,1249,1901,23986,39033}
{655, {9,216,650,2245,3911,8609}
{658, {7,9,3160,17113,23058,23972,40133}
{660, {9,141,513,518,2238,3789,9470,20335,36906}
{662, {9,1100,5249,5949,6505,32664,34544,35069,39040,39042,39043,39069}
{664, {1,2,223,1212,1214,2582,2583,3160,3752,6505,16585,16586,17056,18641,31534,31535,35110,36905,39035,39046,39047,39066}
{666, {2,650,2238,3008,3290,5375,5452,16588,27942,35113}
{668, {2,10,120,1211,3452,3789,6376,6552,6554,13466,14434,16589,16594,17793,21530,28651,36912,39028}
{670, {2,126,141,1368,3739,3741,6338,6374,6389,10472,20339,21246,21248,27854,32746,34021,35073,39080}
{677, {6,518,521,6600,16608,39026}
{685, {206,232,523,1249,1503,6389,7710,36899,39085}
{687, {577,1249,2501,3003,14918,36830}
{690, {690,1648,1649,11053,21905,35582}
{691, {3,187,858,3005,15477,15899,16188,36830,39169}
{693, {514,522,905,1577,3126,15612,35094}
{789, {3,3821,6376,6651,26601,37596}
{799, {9,1107,4357,6376,6626,34021,35068,39044,39057}
{805, {3,511,512,626,8623,9467,21531,33330,39088,39092}
{835, {3,37,958,4205,4657,37592}
{842, {3,5099,18334,36189,38987,39233}
{850, {523,525,2485,5664,18311,18314,21187,23285,35088,38971}
{874, {740,812,1966,6651,8299,26582,39028}
{876, {244,661,665,3005,3837,4369,9508,24003}
{879, {125,647,5972,6130,24284,30476}
{885, {11,650,676,3035,3716,4885,17115}
{889, {2,513,536,2229,4871,9296}
{892, {2,523,524,3291,5159,8542,9165,15899,23991,31655,31998,35087,39061}
{901, {3,513,517,1329,2245,31841,39026}
{925, {3,131,216,6389,10600,24245,24246,31377,33553,34833,34851,34853,35067,37565,37864}
{927, {3,241,514,516,857,3160,6554,33331,35093}
{930, {3,128,140,570,6592,8562,15345,17707,21975,23702,34828,39171}
{934, {3,57,223,946,3452,6609,20205,36908}
{1296, {3,126,574,10354,30739,39027}
{1303, {3,129,389,3819,21243,34850}
{1304, {3,403,12096,14385,18809,36896,40135}
{1309, {3,515,522,860,7952,36944,39535}
{1576, {51,206,22391,34452,34845,40368}
{1633, {614,1486,15487,15497,18589,40125}
{1783, {10,614,3162,5452,7079,20621,36103,40181}
{1897, {1,37,1249,1834,2588,2589,4000,7952,17102,18643,20619,23050,23757,23972,36103}
{1978, {37,75,3061,6374,6376,16604,18277,19584,20532,21024}

The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X(1),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{2, {244,1086,2968,4858,5515,6377,16586,17755,38995,39040}
{75, {244,1099,1109,1111,4712,4736,4738,10504,17879,23996,24010,24014,24023,24026,24028,24031,24034,24038}
{76, {1086,1111,1227,3123,21208,34387}
{85, {1111,4858,17880,17886,20443,20900}
{92, {1109,4858,20431,20639,20901,20902,21427}
{99, {8,1125,1631,4736,4996,23928}
{100, {63,142,1631,1962,22271,27474}
{190, {2,8,63,321,1281,2292,3413,3414,3578,4712,17741,20880,21129,33890}
{304, {17875,17876,17877,17878,17879,17880,17881,20902}
{514, {514,523,20504,20906,21120,21124,21129,21130}
{651, {63,226,2650,20896,21147,22130}
{662, {1,63,1930,2172,2582,2583,3687,5249,14213,16586,17746,17866,19572,19600,23996,38822}
{664, {1,8,347,1441,17797,18697,20504,21147,23528,23555,23668,23669,24028}
{668, {8,10,75,3728,4647,4738,4793,11677,22271,28616}
{789, {75,1281,1631,3778,3821,17797,26234}
{799, {38,63,75,1227,4357,4359,17755,20898,24038}
{811, {1,75,1099,1895,2588,2589,6734,17863,17872,17875,23537,23661,23665}
{927, {347,516,523,1631,5002,5003,11677}
{1577, {656,1577,8061,21124,21192,30591}
{1978, {75,321,6382,18697,20431,20895,20900,21020,27474}

The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X(75),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{1, {244,678,2310,2632,2638,2643,3248,4094,4117,10501,23063,24012}
{2, {11,214,244,1015,8054,8299,10494,14714,17417,17419,17421,17761,17793,34586,34587,34588,34589,34590,34591,34592,34593,38978,38979,38980,38981,38982,38983,38984,38985,38986,39046}
{6, {1015,2170,3248,3270,11998,17455,17475,21762}
{19, {2170,2643,3708,17462,17463,17464,17465,17466,20600,38345}
{57, {244,2170,2446,2447,2611,4128,7004,17460,20366,35065}
{92, {1109,2632,34589,35201,37754,38350}
{99, {55,192,2309,3666,4094,4366,11997,23928,38814,39915}
{100, {1,42,55,678,1962,3158,3251,3795,8298,8299,18673,27787,38349}
{101, {6,37,48,55,354,2294,2590,2591,3725,5638,5639,17454,17455,19561,19586,20284}
{108, {33,55,56,73,204,207}
{109, {31,55,65,221,2067,2292,6502}
{110, {55,202,203,215,501,942,2308,3157,40370}
{163, {31,38,48,563,1953,2260,2269,2578,2579,17453,19578,19603}
{190, {1,37,192,2292,2667,3057,3159,4065,4319,5497,8393,8394,17460,17461,17464,17475,18674,19582,23757,34587,39916}
{513, {512,513,663,3251,4162,4983,38348}
{514, {513,523,650,1459,14284,21104,23752,23757,23758,34590}
{651, {1,6,73,221,500,1201,1419,1480,2293,2574,2575,2650,3157,6126,9502,18675,28369,34586,35197}
{653, {1,65,207,1108,1148,2294,2331,2658,3924,4336}
{658, {1,354,614,20277,31526,40133}
{662, {1,48,214,501,820,1100,1193,1964,2584,2585,2646,17457,17462,38348,38814}
{664, {192,1419,3158,3752,7032,14100,23668,31526}
{666, {6,192,385,518,523,3290,6163,33674}
{668, {2,42,192,3056,3728,17149,19581,19586,28369}
{799, {1,38,1107,3720,17149,17466,17793,18671,20362,21334,21336,39915}
{813, {38,55,649,672,3747,20358,40155}
{901, {15,16,55,512,517,902,1480,17461,39148}
{927, {55,241,6654,9358,21104,31526}
{1018, {37,42,836,1100,2269,3720,4094,17441,17456}
{1020, {37,65,73,1104,2260,2599,2646,2654}
{1029, {11,115,3024,10036,14101,23063}
{1783, {6,42,204,614,2331,8105,8106,21148,22063}

The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X(10),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{2, {1086,3120,6627,6651,8054,16726,38960}
{99, {1,21,86,1125,2309,8053,18650,28627}
{100, {2,42,142,8021,8053,18166,22279}
{190, {1,2,3294,3995,4368,4375,8025,17170,17175,17185,17192,30568}
{662, {2,21,81,1193,1790,5249,17169,17190,17191}
{799, {2,86,3720,4357,17183,17195,17196,18133,18651,31008}
{1414, {7,21,58,4303,4357,10571,12047}

The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X(75),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{1, {1,37,192,2292,2667,3057,3159,4065,4319,5497,8393,8394,17460,17461,17464,17475,18674,19582,23757,34587,39916}
{2, {2,9,37,440,1213,3161,4370,5513,6544,6651,15487,16590,16593,17755,21838,24771,27481,31336,36911,38015,39056,39059,40181}
{4, {46,121,193,1213,2899,2901,3057}
{6, {1,43,194,213,19579,19587,20665,20671,21757,21838,22024,23552,23553,33688,39929}
{7, {2,57,145,3021,3057,3175,8055,16594}
{8, {8,9,40,72,144,1145,3057,3059,3307,3308,3588,3650,6068,12665,12670,12682,16008,18239,21677,31938,36922}
{9, {165,3057,3177,4712,4936,20665,22027,27538}
{10, {10,71,191,1213,1654,2292,21035,21038,21677}
{37, {846,1334,1655,2292,3971,4368,21838}
{42, {42,1045,2667,20681,21035,21080,21838}
{69, {20,63,72,329,440,1763,2582,2583,17170,17742,22001,31547,31548}
{75, {2,8,63,321,1281,2292,3413,3414,3578,4712,17741,20880,21129,33890}
{80, {10,484,513,519,2183,3057,16590,20072,36926}
{85, {7,9,169,3970,17170,17464,20880}
{86, {1,2,3294,3995,4368,4375,8025,17170,17175,17185,17192,30568}
{92, {9,19,5905,18674,20431,22021}
{100, {513,522,649,4057,8640,17494,38349}
{101, {4040,4057,4064,4079,21225,38367}
{141, {2896,3954,16555,17192,17744,22026}
{190, {514,649,3234,3239,4024,4375,6544,24979,31182}
{238, {659,672,2108,4368,17475,30667}
{239, {239,17475,17755,21832,24578,33888}
{264, {4,321,440,1726,3730,17776,22000,34335}
{274, {37,75,16552,17175,20880,22011,25082}
{291, {37,513,672,726,894,1757,9334,9339,17759,21035}
{304, {346,4329,17170,18596,18674,20336,22005}
{306, {71,306,440,3151,8804,18598,18674}
{312, {9,321,329,346,1766,21078}
{314, {312,321,1764,3057,3869,17185,22022}
{319, {2895,3219,3578,3648,17781,31938}
{333, {9,63,573,17185,20665,21061}
{334, {10,3912,4391,4645,17192,20602,20880,31647}
{335, {2,3509,4024,4037,6542,40217}
{350, {726,812,3685,4368,17738,17755,17794}
{514, {4370,4440,5540,14442,17464,22035}
{518, {672,1282,4088,4712,10025,17464,17794,39350}
{519, {519,900,4370,5541,17460,17487}
{664, {514,522,6332,7265,30719,31605,38371}
{668, {513,514,4063,4391,20954,20979}
{673, {2,239,364,649,672,20665}
{740, {4037,4088,4368,13174,17759,39367}
{765, {100,644,1331,3952,4712,17460}
{870, {1,4384,17755,20880,22048,24349}
{903, {2,514,519,3218,17461,24428,31647,32094,36591}
{1016, {190,644,1018,4115,4370,17475,30720}
{1441, {226,440,1762,2475,3219,21677}
{1897, {522,4024,4064,23757,25259,38360}

The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X(1),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{2, {115,1015,1084,1086,1146,2454,2455,2482,3163,4370,5997,6184,11672,13466,15166,15167,15449,15525,15526,15527,17416,17429,18334,20532,23967,23972,23976,23980,23986,23992,35066,35067,35068,35069,35070,35071,35072,35073,35074,35075,35076,35077,35078,35079,35080,35081,35082,35083,35084,35085,35086,35087,35088,35089,35090,35091,35092,35093,35094,35095,35110,35111,35112,35113,35114,35115,35116,35117,35118,35119,35120,35121,35122,35123,35124,35125,35126,35127,35128,35129,35130,35131,35132,35133,35134,35135,35508,35509,39008,39009,39010,39011,39012,39013,39014,39015,39016,39017,39018,39019,39020,39021,39022,39023,39206,39207,39208,39209}
{7, {3022,3271,4904,26932}
{99, {148,2482,7669,12076,14443}
{100, {16560,17060,22308,23402}
{190, {4370,4440,5540,14442,17464,22035}
{513, {9263,13466,14441,22323}

Let X*(i) denote the isotomic conjugate of X(i). The appearance of {i, {j(1),j(2),...}
in the following list means that the 1st (X*(1),X(i))-Ceva conic passes through the points X(j1), X(j2),... :

{1, {244,678,2310,2632,2638,2643,3248,4094,4117,10501,23063,24012}
{2, {115,1015,1084,1086,1146,2454,2455,2482,3163,4370,5997,6184,11672,13466,15166,15167,15449,15525,15526,15527,17416,17429,18334,20532,23967,23972,23976,23980,23986,23992,35066,35067,35068,35069,35070,35071,35072,35073,35074,35075,35076,35077,35078,35079,35080,35081,35082,35083,35084,35085,35086,35087,35088,35089,35090,35091,35092,35093,35094,35095,35110,35111,35112,35113,35114,35115,35116,35117,35118,35119,35120,35121,35122,35123,35124,35125,35126,35127,35128,35129,35130,35131,35132,35133,35134,35135,35508,35509,39008,39009,39010,39011,39012,39013,39014,39015,39016,39017,39018,39019,39020,39021,39022,39023,39206,39207,39208,39209}
{3, {2972,3270,3937,20759,20776,20975,22096,22371,23216}
{4, {125,2969,3270,5095,6754,8754,16240,24862,34980}
{6, {1015,1017,1977,2028,2029,3124,3269,9408,9419,9427,14936,20671,24973,35505,35506,39686,39687,39689}
{7, {11,1314,1315,1317,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,2446,2447,3020,3021,3022,3023,3024,3025,3026,3027,3028,3318,3319,3320,3321,3322,3323,3324,3325,3326,3327,3328,5577,5578,5579,5580,5581,5582,5997,6018,6019,6020,6021,6022,6023,6024,6025,6026,6027,6028,6029,7158,7159,7333,7334,10491,10501,10504,10505,10506,12809,13756,14027,15615,15616,16184,16185,22106,22107,31522,31524,31889,31890,31891,31892,31893,33964,33965,33966,34194,34228,35504}
{8, {11,3271,4081,4092,4152,4542,6062,6068,7062,7063,7065,7067,7068}
{13, {30452,30454,30459,30460,30461,30465,30466,30467}
{14, {30453,30455,30462,30463,30464,30468,30469,30470}
{37, {3121,3125,20690,21821,21833,36197}
{59, {55,56,181,202,203,215,1124,1335,1362,1397,1672,1673,1682,2007,2008,3235,3236,3237,3238,6056,7005,7006,7066,10799,12835,12836,12837,12838,12839,12840,12841,37993,39641,39642}
{69, {125,1565,2968,3937,16163,38554}
{75, {244,1099,1109,1111,4712,4736,4738,10504,17879,23996,24010,24014,24023,24026,24028,24031,24034,24038}
{76, {338,1086,3124,4437,23970,23978,23983,23989,26611,36789,36790,36791,36792,36793}
{99, {523,669,1649,2528,3233,3265,3733,7192,7253,24974,30508,30509}
{100, {513,667,3126,3251,3900,4705,4825}
{110, {512,520,3733,9426,21789,34983}
{190, {514,649,3234,3239,4024,4375,6544,24979,31182}
{249, {6,394,593,1501,1599,1600,7054,8041,11130,11131,35069,36790,39689}
{264, {339,1312,1313,2967,2968,2969,2970,2971,2972,2973,2974,21664,21665,21666,24977,34332,34333,34334,34335,34336,34337,34338,35012,38552}
{274, {3121,16725,16726,16727,16728,16729,16730,16731,16732,16733}
{523, {1649,5489,8029,8034,14443,23099}
{598, {8288,20380,20381,20382,20383,20384,20385,20386,35507}
{648, {525,2501,14401,15639,17925,17926,23090,32320}
{668, {513,693,4036,4397,14434,15632,25142,27855}
{765, {1,31,200,678,756,4712,8300}
{1016, {2,6,346,594,4366,4370,4437,7109,13425,13458}
{1101, {31,255,849,1094,1095,1917,23996}
{1252, {6,32,220,1017,1500,6184}
{1275, {2,220,279,394,1407,6354,6645,13436,13453,26611,35110,39686}
{1502, {115,23962,23965,23974,23989,32458}
{1509, {1086,1977,4366,26844,26846,26856}

X(40530) = CENTER OF 1ST (X(2),X(19))-CEVA CONIC

Barycentrics    2*a^5 - a^4*b - 2*a*b^4 + b^5 - a^4*c + 2*a^2*b^2*c - b^4*c + 2*a^2*b*c^2 + 4*a*b^2*c^2 - 2*a*c^4 - b*c^4 + c^5 : :
X(40530) =3 X(2) + X(19)

X(40530) lines on these lines: {2, 19}, {4, 21160}, {5, 516}, {10, 7535}, {48, 25935}, {142, 24315}, {226, 34176}, {379, 1826}, {515, 15943}, {610, 26130}, {631, 30265}, {857, 1839}, {1125, 9895}, {1376, 1486}, {1441, 8756}, {1731, 24884}, {1788, 2263}, {1838, 25443}, {1842, 25015}, {1861, 4223}, {1953, 26006}, {2173, 18650}, {2182, 25964}, {2264, 18635}, {2822, 38601}, {2876, 9822}, {3589, 3812}, {3668, 3911}, {3739, 5745}, {4319, 5218}, {4698, 6677}, {5338, 26052}, {6642, 6796}, {6692, 14743}, {6711, 20202}, {7392, 11677}, {7522, 8804}, {9028, 16608}, {12047, 25651}, {14213, 28705}, {17073, 31184}, {17259, 26066}, {18594, 24683}, {18634, 24316}, {25514, 34822}, {28258, 34851}

X(40530) = complement of X(18589)
X(40530) = centroid of {A,B,C,X(19)}
X(40530) = center of bicevian conic of X(2) and X(19)

X(40531) = CENTER OF 1ST (X(2),X(59))-CEVA CONIC

Barycentrics    2*a^8 - 4*a^7*b + 4*a^5*b^3 - 3*a^4*b^4 + 2*a^3*b^5 - 2*a*b^7 + b^8 - 4*a^7*c + 12*a^6*b*c - 8*a^5*b^2*c - 2*a^4*b^3*c + 2*a^3*b^4*c - 4*a^2*b^5*c + 6*a*b^6*c - 2*b^7*c - 8*a^5*b*c^2 + 12*a^4*b^2*c^2 - 4*a^3*b^3*c^2 + 4*a^2*b^4*c^2 - 4*a*b^5*c^2 + 4*a^5*c^3 - 2*a^4*b*c^3 - 4*a^3*b^2*c^3 + 2*b^5*c^3 - 3*a^4*c^4 + 2*a^3*b*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 + 2*a^3*c^5 - 4*a^2*b*c^5 - 4*a*b^2*c^5 + 2*b^3*c^5 + 6*a*b*c^6 - 2*a*c^7 - 2*b*c^7 + c^8 : :
X(40531) = 3 X(2) + X(59)

X(40531) lies on these lines: {2, 59}, {513, 36949}, {518, 15325}, {521, 3035}, {6718, 22102}, {21189, 23593}

X(40531) = complement of complement of X(59)
X(40531) = centroid of {A,B,C,X(59)}
X(40531) = center of bicevian conic of X(2) and X(59)

X(40532) = CENTER OF 1ST (X(2),X(162))-CEVA CONIC

Barycentrics    2*a^9 - 2*a^7*b^2 + a^6*b^3 - 2*a^5*b^4 - a^4*b^5 + 2*a^3*b^6 - a^2*b^7 + b^9 - a^6*b^2*c + 2*a^4*b^4*c - a^2*b^6*c - 2*a^7*c^2 - a^6*b*c^2 + 6*a^5*b^2*c^2 - a^4*b^3*c^2 - 2*a^3*b^4*c^2 + 3*a^2*b^5*c^2 - 2*a*b^6*c^2 - b^7*c^2 + a^6*c^3 - a^4*b^2*c^3 - a^2*b^4*c^3 + b^6*c^3 - 2*a^5*c^4 + 2*a^4*b*c^4 - 2*a^3*b^2*c^4 - a^2*b^3*c^4 + 4*a*b^4*c^4 - b^5*c^4 - a^4*c^5 + 3*a^2*b^2*c^5 - b^4*c^5 + 2*a^3*c^6 - a^2*b*c^6 - 2*a*b^2*c^6 + b^3*c^6 - a^2*c^7 - b^2*c^7 + c^9 : :
X(40532) = 3 X(2) + X(162)

X(40532) lies on these lines: {2, 162}, {5, 25448}, {2806, 3035}, {2846, 6716}, {2850, 5972}, {4422, 15252}, {6679, 6708}, {8062, 24030}

X(40532) = complement of X(34846)
X(40532) = centroid of {A,B,C,X(162)}
X(40532) = center of bicevian conic of X(2) and X(162)

X(40533) = CENTER OF 1ST (X(2),X(291))-CEVA CONIC

Barycentrics    a^3*b^2 + 2*a^2*b^3 - 2*a^3*b*c - a^2*b^2*c - a*b^3*c + a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 + 2*a^2*c^3 - a*b*c^3 + b^2*c^3 : :
X(40533) = 3 X(2) + X(291)

X(40533) lies on these lines: {1, 4595}, {2, 38}, {10, 1015}, {37, 20671}, {75, 32020}, {325, 3836}, {350, 28516}, {668, 1698}, {726, 20530}, {740, 1575}, {812, 3837}, {1086, 36217}, {1125, 6683}, {2023, 5750}, {2108, 4432}, {2787, 6702}, {2810, 6686}, {3035, 6685}, {3097, 30963}, {3227, 19875}, {3248, 18793}, {3634, 25109}, {3756, 3773}, {3821, 25350}, {3828, 33908}, {4368, 20331}, {4472, 25347}, {4672, 17754}, {4974, 37686}, {5248, 8671}, {6714, 31289}, {9263, 9780}, {14829, 16569}, {20457, 24512}, {21238, 39798}, {21337, 24443}, {24508, 24715}, {28850, 34460}

X(40533) = complement of X(17793)
X(40533) = centroid of {A,B,C,X(291)}
X(40533) = center of bicevian conic of X(2) and X(291)

X(40534) = CENTER OF 1ST (X(2),X(644))-CEVA CONIC

Barycentrics    2*a^4 - 4*a^3*b + 3*a^2*b^2 - 2*a*b^3 + b^4 - 4*a^3*c + 4*a^2*b*c - 2*b^3*c + 3*a^2*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4 : :
X(40534) = 3 X(2) + X(644)

X(40534) lies on these lines: {2, 644}, {9, 1565}, {101, 16593}, {120, 1083}, {218, 28740}, {344, 4561}, {514, 3039}, {518, 11730}, {525, 25095}, {551, 3589}, {918, 3960}, {997, 1807}, {1018, 26007}, {1086, 25600}, {1280, 3616}, {1292, 38386}, {1387, 6666}, {2006, 16594}, {2802, 33970}, {3035, 3887}, {5432, 36639}, {6714, 14839}, {15252, 24003}, {15903, 25072}, {20328, 24333}, {24398, 24795}, {25430, 34892}, {26074, 30857}, {27132, 28961}, {30618, 34847}, {30728, 36807}, {34625, 37650}

X(40534) = complement of X(4904)
X(40534) = centroid of {A,B,C,X(644)}
X(40534) = center of bicevian conic of X(2) and X(644)

X(40535) = CENTER OF 1ST (X(2),X(653))-CEVA CONIC

Barycentrics    2*a^8 - 2*a^7*b - 3*a^6*b^2 + 2*a^5*b^3 + a^4*b^4 + 2*a^3*b^5 - a^2*b^6 - 2*a*b^7 + b^8 - 2*a^7*c + 8*a^6*b*c - 2*a^5*b^2*c - 4*a^4*b^3*c - 2*a^3*b^4*c - 4*a^2*b^5*c + 6*a*b^6*c - 3*a^6*c^2 - 2*a^5*b*c^2 + 6*a^4*b^2*c^2 + a^2*b^4*c^2 + 2*a*b^5*c^2 - 4*b^6*c^2 + 2*a^5*c^3 - 4*a^4*b*c^3 + 8*a^2*b^3*c^3 - 6*a*b^4*c^3 + a^4*c^4 - 2*a^3*b*c^4 + a^2*b^2*c^4 - 6*a*b^3*c^4 + 6*b^4*c^4 + 2*a^3*c^5 - 4*a^2*b*c^5 + 2*a*b^2*c^5 - a^2*c^6 + 6*a*b*c^6 - 4*b^2*c^6 - 2*a*c^7 + c^8 : :
X(40535) = 3 X(2) + X(653)

X(40535) lies on these lines: {2, 196}, {5, 1158}, {676, 2804}, {1375, 37805}, {2846, 6716}, {5437, 20197}, {6692, 6708}, {14837, 23982}, {17044, 23583}, {17073, 20204}, {31190, 37695}, {36949, 39470}

X(40535) = complement of X(16596)
X(40535) = centroid of {A,B,C,X(653)}
X(40535) = center of bicevian conic of X(2) and X(653)

X(40536) = CENTER OF 1ST (X(2),X(655))-CEVA CONIC

Barycentrics    2*a^8 - 4*a^7*b + 5*a^5*b^3 - 5*a^4*b^4 + 2*a^3*b^5 + 2*a^2*b^6 - 3*a*b^7 + b^8 - 4*a^7*c + 12*a^6*b*c - 9*a^5*b^2*c - 2*a^4*b^3*c + 7*a^3*b^4*c - 9*a^2*b^5*c + 6*a*b^6*c - b^7*c - 9*a^5*b*c^2 + 16*a^4*b^2*c^2 - 9*a^3*b^3*c^2 + 2*a^2*b^4*c^2 + 2*a*b^5*c^2 - 2*b^6*c^2 + 5*a^5*c^3 - 2*a^4*b*c^3 - 9*a^3*b^2*c^3 + 10*a^2*b^3*c^3 - 5*a*b^4*c^3 + b^5*c^3 - 5*a^4*c^4 + 7*a^3*b*c^4 + 2*a^2*b^2*c^4 - 5*a*b^3*c^4 + 2*b^4*c^4 + 2*a^3*c^5 - 9*a^2*b*c^5 + 2*a*b^2*c^5 + b^3*c^5 + 2*a^2*c^6 + 6*a*b*c^6 - 2*b^2*c^6 - 3*a*c^7 - b*c^7 + c^8 : :
X(40536) = 3 X(2) + X(655)

X(40536) lies on these lines: {2, 655}, {514, 36949}, {516, 6702}, {522, 3035}, {908, 7359}, {3911, 26011}, {10015, 23593}, {14838, 16578}

X(40536) = complement of complement of X(655)
X(40536) = centroid of {A,B,C,X(655)}
X(40536) = center of bicevian conic of X(2) and X(655)

X(40537) = CENTER OF 1ST (X(2),X(658))-CEVA CONIC

Barycentrics    2*a^6 - 2*a^5*b - 7*a^4*b^2 + 12*a^3*b^3 - 4*a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c + 16*a^4*b*c - 12*a^3*b^2*c - 16*a^2*b^3*c + 14*a*b^4*c - 7*a^4*c^2 - 12*a^3*b*c^2 + 40*a^2*b^2*c^2 - 12*a*b^3*c^2 - 9*b^4*c^2 + 12*a^3*c^3 - 16*a^2*b*c^3 - 12*a*b^2*c^3 + 16*b^3*c^3 - 4*a^2*c^4 + 14*a*b*c^4 - 9*b^2*c^4 - 2*a*c^5 + c^6 : :
X(40537) = 3 X(2) + X(658)

X(40537) lies on these lines: {2, 658}, {142, 5851}, {3035, 6366}, {7658, 15252}

X(40537) = complement of X(13609)
X(40537) = centroid of {A,B,C,X(658)}
X(40537) = center of bicevian conic of X(2) and X(658)

X(40538) = CENTER OF 1ST (X(2),X(660))-CEVA CONIC

Barycentrics    a*(-(a^3*b^3) + a^2*b^4 + 2*a^4*b*c - 3*a^3*b^2*c + 4*a^2*b^3*c - 3*a*b^4*c + b^5*c - 3*a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - 2*b^4*c^2 - a^3*c^3 + 4*a^2*b*c^3 - a*b^2*c^3 + 4*b^3*c^3 + a^2*c^4 - 3*a*b*c^4 - 2*b^2*c^4 + b*c^5) : :
X(40538) = 3 X(2) + X(660)

X(40538) lies on these lines: {2, 660}, {9, 39344}, {238, 4447}, {513, 4422}, {518, 3008}, {1083, 36086}, {3035, 31286}, {4369, 24003}, {6005, 36954}, {17338, 36294}, {34807, 36280}

X(40538) = complement of X(38989)
X(40538) = centroid of {A,B,C,X(660)}
X(40538) = center of bicevian conic of X(2) and X(660)

X(40539) = CENTER OF 1ST (X(2),X(662))-CEVA CONIC

Barycentrics    2*a^5 - 2*a^3*b^2 - a^2*b^3 + b^5 + a^2*b^2*c - 2*a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 + c^5 : :
X(40539) = 3 X(2) + X(662)

X(40539) lies on these lines: {2, 662}, {141, 31186}, {190, 24636}, {620, 2786}, {645, 25469}, {3035, 5972}, {8286, 25533}, {9034, 36949}, {14838, 16578}, {17044, 23583}, {17359, 24384}, {24617, 24957}, {31201, 35466}

X(40539) = complement of X(8287)
X(40539) = centroid of {A,B,C,X(662)}
X(40539) = center of bicevian conic of X(2) and X(662)

X(40540) = CENTER OF 1ST (X(2),X(666))-CEVA CONIC

Barycentrics    2*a^6 - 4*a^5*b + 4*a^4*b^2 - 4*a^3*b^3 + 3*a^2*b^4 - 2*a*b^5 + b^6 - 4*a^5*c + 4*a^4*b*c + 2*a*b^4*c - 2*b^5*c + 4*a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 - 4*a^3*c^3 + 3*a^2*c^4 + 2*a*b*c^4 + b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6 : :
X(40540) = 3 X(2) + X(666)

X(40540) lies on these lines: {2, 666}, {514, 6710}, {522, 4422}, {620, 14838}, {997, 36230}, {1566, 34906}, {1944, 23593}, {3008, 34852}, {3814, 5461}, {4763, 22102}, {6547, 24203}, {6554, 35093}, {6714, 15325}, {10025, 29607}, {26685, 34361}

X(40540) = complement of X(35094)
X(40540) = centroid of {A,B,C,X(666)}
X(40540) = center of bicevian conic of X(2) and X(666)

X(40541) = CENTER OF 1ST (X(2),X(677))-CEVA CONIC

Barycentrics    2*a^10 - 4*a^9*b + 2*a^8*b^2 - 2*a^7*b^3 + a^6*b^4 + 6*a^5*b^5 - 7*a^4*b^6 + 2*a^3*b^7 + a^2*b^8 - 2*a*b^9 + b^10 - 4*a^9*c + 8*a^8*b*c - 2*a^7*b^2*c - 4*a^5*b^4*c - 2*a^4*b^5*c + 6*a^3*b^6*c - 4*a^2*b^7*c + 4*a*b^8*c - 2*b^9*c + 2*a^8*c^2 - 2*a^7*b*c^2 - 2*a^5*b^3*c^2 + 3*a^4*b^4*c^2 - 2*a^3*b^5*c^2 + 2*a^2*b^6*c^2 - 2*a*b^7*c^2 + b^8*c^2 - 2*a^7*c^3 - 2*a^5*b^2*c^3 + 12*a^4*b^3*c^3 - 6*a^3*b^4*c^3 - 4*a^2*b^5*c^3 + 2*a*b^6*c^3 + a^6*c^4 - 4*a^5*b*c^4 + 3*a^4*b^2*c^4 - 6*a^3*b^3*c^4 + 10*a^2*b^4*c^4 - 2*a*b^5*c^4 - 2*b^6*c^4 + 6*a^5*c^5 - 2*a^4*b*c^5 - 2*a^3*b^2*c^5 - 4*a^2*b^3*c^5 - 2*a*b^4*c^5 + 4*b^5*c^5 - 7*a^4*c^6 + 6*a^3*b*c^6 + 2*a^2*b^2*c^6 + 2*a*b^3*c^6 - 2*b^4*c^6 + 2*a^3*c^7 - 4*a^2*b*c^7 - 2*a*b^2*c^7 + a^2*c^8 + 4*a*b*c^8 + b^2*c^8 - 2*a*c^9 - 2*b*c^9 + c^10 : :
X(40541) = 3 X(2) + X(677)

X(40541) lies on these lines: {2, 677}, {521, 36949}, {3239, 15252}, {6712, 22102}, {8062, 23583}

X(40541) = complement of complement of X(677)
X(40541) = centroid of {A,B,C,X(677)}
X(40541) = center of bicevian conic of X(2) and X(677)

X(40542) = CENTER OF 1ST (X(2),X(685))-CEVA CONIC

Barycentrics    2*a^16 - 4*a^14*b^2 + 2*a^12*b^4 - 2*a^10*b^6 + 3*a^8*b^8 - 2*a^2*b^14 + b^16 - 4*a^14*c^2 + 8*a^12*b^2*c^2 - 2*a^10*b^4*c^2 - 6*a^6*b^8*c^2 + 2*a^4*b^10*c^2 + 4*a^2*b^12*c^2 - 2*b^14*c^2 + 2*a^12*c^4 - 2*a^10*b^2*c^4 - 4*a^8*b^4*c^4 + 6*a^6*b^6*c^4 - 4*a^4*b^8*c^4 + 2*b^12*c^4 - 2*a^10*c^6 + 6*a^6*b^4*c^6 + 4*a^4*b^6*c^6 - 2*a^2*b^8*c^6 - 6*b^10*c^6 + 3*a^8*c^8 - 6*a^6*b^2*c^8 - 4*a^4*b^4*c^8 - 2*a^2*b^6*c^8 + 10*b^8*c^8 + 2*a^4*b^2*c^10 - 6*b^6*c^10 + 4*a^2*b^2*c^12 + 2*b^4*c^12 - 2*a^2*c^14 - 2*b^2*c^14 + c^16 : :
X(40542) = 3 X(2) + X(685)

X(40542) lies on these lines: {2, 685}, {523, 23583}, {5972, 11595}, {6036, 37911}, {6716, 14341}

X(40542) = complement of complement of X(685)
X(40542) = centroid of {A,B,C,X(685)}
X(40542) = center of bicevian conic of X(2) and X(685)

X(40543) = CENTER OF 1ST (X(2),X(687))-CEVA CONIC

Barycentrics    2*a^20 - 8*a^18*b^2 + 8*a^16*b^4 + 10*a^14*b^6 - 31*a^12*b^8 + 32*a^10*b^10 - 17*a^8*b^12 + 2*a^6*b^14 + 5*a^4*b^16 - 4*a^2*b^18 + b^20 - 8*a^18*c^2 + 36*a^16*b^2*c^2 - 54*a^14*b^4*c^2 + 20*a^12*b^6*c^2 + 22*a^10*b^8*c^2 - 24*a^8*b^10*c^2 + 22*a^6*b^12*c^2 - 28*a^4*b^14*c^2 + 18*a^2*b^16*c^2 - 4*b^18*c^2 + 8*a^16*c^4 - 54*a^14*b^2*c^4 + 104*a^12*b^4*c^4 - 74*a^10*b^6*c^4 + 11*a^8*b^8*c^4 - 18*a^6*b^10*c^4 + 50*a^4*b^12*c^4 - 30*a^2*b^14*c^4 + 3*b^16*c^4 + 10*a^14*c^6 + 20*a^12*b^2*c^6 - 74*a^10*b^4*c^6 + 68*a^8*b^6*c^6 - 6*a^6*b^8*c^6 - 52*a^4*b^10*c^6 + 22*a^2*b^12*c^6 + 12*b^14*c^6 - 31*a^12*c^8 + 22*a^10*b^2*c^8 + 11*a^8*b^4*c^8 - 6*a^6*b^6*c^8 + 50*a^4*b^8*c^8 - 6*a^2*b^10*c^8 - 36*b^12*c^8 + 32*a^10*c^10 - 24*a^8*b^2*c^10 - 18*a^6*b^4*c^10 - 52*a^4*b^6*c^10 - 6*a^2*b^8*c^10 + 48*b^10*c^10 - 17*a^8*c^12 + 22*a^6*b^2*c^12 + 50*a^4*b^4*c^12 + 22*a^2*b^6*c^12 - 36*b^8*c^12 + 2*a^6*c^14 - 28*a^4*b^2*c^14 - 30*a^2*b^4*c^14 + 12*b^6*c^14 + 5*a^4*c^16 + 18*a^2*b^2*c^16 + 3*b^4*c^16 - 4*a^2*c^18 - 4*b^2*c^18 + c^20 : :
X(40543) = 3 X(2) + X(687)

X(40543) lies on these lines: {2, 687}, {6716, 12068}

X(40543) = complement of complement of X(687)
X(40543) = centroid of {A,B,C,X(687)}
X(40543) = center of bicevian conic of X(2) and X(687)

X(40544) = CENTER OF 1ST (X(2),X(691))-CEVA CONIC

Barycentrics    2*a^10 - 4*a^8*b^2 + 3*a^4*b^6 - 2*a^2*b^8 + b^10 - 4*a^8*c^2 + 12*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 2*b^8*c^2 - 7*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 + 2*a^2*b^2*c^6 + b^4*c^6 - 2*a^2*c^8 - 2*b^2*c^8 + c^10 : :
X(40544) = 3 X(2) + X(691)

X(40544) lies on these lines: {2, 691}, {3, 16188}, {4, 38702}, {5, 38611}, {30, 5461}, {115, 7472}, {125, 9181}, {140, 16760}, {187, 858}, {249, 3448}, {316, 30745}, {468, 5140}, {511, 6699}, {512, 5972}, {523, 620}, {538, 16315}, {549, 31379}, {625, 5159}, {631, 842}, {1692, 32220}, {2072, 13449}, {2453, 11288}, {2482, 16092}, {2794, 36170}, {3523, 38704}, {3525, 38679}, {3526, 38582}, {3767, 14659}, {5054, 38583}, {5206, 36187}, {5215, 7426}, {5432, 6027}, {5433, 6023}, {5569, 36194}, {6680, 36157}, {6722, 14120}, {7464, 38227}, {7574, 38225}, {7575, 34837}, {7665, 15398}, {7749, 36165}, {7857, 36182}, {7907, 38526}, {9218, 15059}, {10257, 34841}, {10277, 38230}, {10303, 38680}, {10415, 14360}, {10989, 26613}, {14061, 36174}, {14971, 36196}, {14999, 15357}, {18911, 32761}, {21843, 36163}, {22104, 36597}, {34473, 36173}, {36166, 38737}

X(40544) = complement of X(5099)
X(40544) = centroid of {A,B,C,X(691)}
X(40544) = center of bicevian conic of X(2) and X(691)

X(40545) = CENTER OF 1ST (X(2),X(789))-CEVA CONIC

Barycentrics    a^4*b^4 + 2*a^6*b*c - a^4*b^3*c - 2*a^3*b^4*c + a*b^6*c - a^4*b*c^3 - a*b^4*c^3 + a^4*c^4 - 2*a^3*b*c^4 - a*b^3*c^4 + 2*b^4*c^4 + a*b*c^6 : :
X(40545) = 3 X(2) + X(789)

X(40545) lies on these lines: {2, 743}, {6710, 27076}

X(40545) = complement of complement of X(789)
X(40545) = centroid of {A,B,C,X(789)}
X(40545) = center of bicevian conic of X(2) and X(789)

X(40546) = CENTER OF 1ST (X(2),X(799))-CEVA CONIC

Barycentrics    a^3*b^3 + 2*a^4*b*c - a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c - a^3*b*c^2 - a*b^3*c^2 + a^3*c^3 - 2*a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 + a*b*c^4 : :
X(40546) = 3 X(2) + X(799)

X(40546) lies on these lines: {2, 799}, {100, 30996}, {141, 25652}, {620, 2787}, {4369, 24003}, {4422, 36950}, {16613, 25472}, {24384, 25107}, {24505, 27805}, {27008, 27306}

X(40546) = complement of X(16592)
X(40546) = centroid of {A,B,C,X(799)}
X(40546) = center of bicevian conic of X(2) and X(799)

X(40547) = CENTER OF 1ST (X(2),X(835))-CEVA CONIC

Barycentrics    2*a^6 + 2*a^5*b - a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + 2*a^5*c + 2*a^4*b*c - 3*a^3*b^2*c - 3*a^2*b^3*c + a*b^4*c + b^5*c - 3*a^3*b*c^2 - 2*a*b^3*c^2 + b^4*c^2 - a^3*c^3 - 3*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 + b*c^5 + c^6 : :
X(40547) = 3 X(2) + X(835)

X(40547) lies on these lines: {2, 835}, {4422, 5972}, {4687, 37842}, {6710, 24003}, {6711, 13731}, {6715, 11814}, {6718, 16578}, {6720, 15252}

X(40547) = complement of X(5515)
X(40547) = centroid of {A,B,C,X(835)}
X(40547) = center of bicevian conic of X(2) and X(835)

X(40548) = CENTER OF 1ST (X(2),X(874))-CEVA CONIC

Barycentrics    (a^2 - b*c)*(-(a*b^3) + 2*a^2*b*c - a*b^2*c - a*b*c^2 + 2*b^2*c^2 - a*c^3) : :
X(40548) = 3 X(2) + X(874)

X(40548) lies on these lines: {2, 874}, {620, 804}, {740, 1125}, {812, 4422}, {1966, 17289}, {4155, 21254}, {4432, 20333}, {17357, 18904}, {24254, 24327}, {27838, 38989}, {28604, 30940}

X(40548) = complement of X(39786)
X(40548) = centroid of {A,B,C,X(874)}
X(40548) = center of bicevian conic of X(2) and X(874)

X(40549) = CENTER OF 1ST (X(2),X(876))-CEVA CONIC

Barycentrics    (b - c)*(a^4*b - 2*a^2*b^3 + a^4*c + 2*a^3*b*c - 2*a^2*b^2*c - 2*a^2*b*c^2 + 2*a*b^2*c^2 + b^3*c^2 - 2*a^2*c^3 + b^2*c^3) : :
X(40549) = 3 X(2) + X(876)

X(40549) lies on these lines: {2, 876}, {512, 1125}, {513, 4698}, {514, 3634}, {523, 3739}, {665, 3837}, {3005, 27167}, {3766, 30795}, {3812, 4083}, {4151, 6532}, {4367, 16830}, {4784, 29578}, {6372, 23814}, {7180, 25126}, {16826, 38348}, {18004, 23829}, {19948, 19949}, {25380, 30665}

X(40549) = complement of complement of X(876)
X(40549) = centroid of {A,B,C,X(876)}
X(40549) = center of bicevian conic of X(2) and X(876)

X(40550) = CENTER OF 1ST (X(2),X(879))-CEVA CONIC

Barycentrics    (b - c)*(b + c)*(a^10 - 2*a^8*b^2 + a^6*b^4 - 2*a^8*c^2 - a^6*b^2*c^2 + 3*a^4*b^4*c^2 - a^2*b^6*c^2 + b^8*c^2 + a^6*c^4 + 3*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - b^6*c^4 - a^2*b^2*c^6 - b^4*c^6 + b^2*c^8) : :
X(40550) = 3 X(2) + X(879)

X(40550) lies on these lines: {2, 879}, {5, 512}, {140, 525}, {141, 520}, {182, 18312}, {523, 3589}, {526, 6698}, {690, 6036}, {804, 14271}, {826, 6689}, {924, 6697}, {3566, 6696}, {3800, 10280}, {3906, 40108}, {5449, 8673}, {6130, 24284}, {8675, 16511}, {8723, 23105}, {9030, 32154}, {9033, 15118}, {9517, 20304}, {10168, 23878}, {11182, 35364}, {14096, 38354}, {14618, 37124}, {15059, 35909}, {33752, 38317}

X(40550) = complement of X(41167)
X(40550) = complement of complement of X(879)
X(40550) = centroid of {A,B,C,X(879)}
X(40550) = center of bicevian conic of X(2) and X(879)

X(40551) = CENTER OF 1ST (X(2),X(885))-CEVA CONIC

Barycentrics    (b - c)*(a^5 - 2*a^4*b + a^3*b^2 - 2*a^4*c - a^3*b*c + 3*a^2*b^2*c - a*b^3*c + b^4*c + a^3*c^2 + 3*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - a*b*c^3 - b^2*c^3 + b*c^4) : :
X(40551) = 3 X(2) + X(885)

X(40551) lies on these lines: {2, 885}, {5, 3309}, {10, 3900}, {142, 513}, {514, 1125}, {522, 6666}, {523, 25081}, {667, 4223}, {676, 20516}, {1387, 6366}, {2488, 26017}, {2826, 6713}, {3716, 24285}, {3887, 6702}, {3925, 11193}, {4391, 16823}, {4423, 40166}, {4806, 6701}, {6362, 6675}, {8641, 25009}, {8728, 11247}, {15584, 31287}, {31419, 32195}, {35355, 36848}

X(40551) = complement of X(3126)
X(40551) = centroid of {A,B,C,X(885)}
X(40551) = center of bicevian conic of X(2) and X(885)

X(40552) = CENTER OF 1ST (X(2),X(889))-CEVA CONIC

Barycentrics    a^4*b^4 - 4*a^4*b^3*c + 8*a^4*b^2*c^2 - 4*a^3*b^3*c^2 + 2*a^2*b^4*c^2 - 4*a^4*b*c^3 - 4*a^3*b^2*c^3 + 8*a^2*b^3*c^3 - 4*a*b^4*c^3 + a^4*c^4 + 2*a^2*b^2*c^4 - 4*a*b^3*c^4 + 2*b^4*c^4 : :
X(40552) = 3 X(2) + X(889)

X(40552) lies on these lines: {2, 889}, {513, 27076}, {4369, 36950}, {4422, 31286}, {9263, 31625}, {9296, 27195}, {21264, 25382}

X(40552) = complement of X(39011)
X(40552) = centroid of {A,B,C,X(889)}
X(40552) = center of bicevian conic of X(2) and X(889)

X(40553) = CENTER OF 1ST (X(2),X(892))-CEVA CONIC

Barycentrics    2*a^8 - 4*a^6*b^2 + 2*a^4*b^4 + b^8 - 4*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 4*b^6*c^2 + 2*a^4*c^4 - 4*a^2*b^2*c^4 + 8*b^4*c^4 - 4*b^2*c^6 + c^8 : :
X(40553) = 3 X(2) + X(892)

X(40553) lies on these lines: {2, 892}, {115, 9182}, {126, 16092}, {148, 4590}, {230, 6719}, {385, 23589}, {523, 620}, {524, 625}, {888, 22103}, {2482, 17948}, {7778, 36207}, {9164, 36521}, {9183, 33915}, {11053, 33921}, {14061, 23991}, {14341, 23583}, {18310, 24975}, {30476, 36950}

X(40553) = complement of X(23992)
X(40553) = centroid of {A,B,C,X(892)}
X(40553) = center of bicevian conic of X(2) and X(892)
X(40553) = crosssum of PU(62)

X(40554) = CENTER OF 1ST (X(2),X(927))-CEVA CONIC

Barycentrics    2*a^8 - 4*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 + 3*a^4*b^4 - 2*a*b^7 + b^8 - 4*a^7*c + 8*a^6*b*c - 2*a^5*b^2*c - 6*a^3*b^4*c + 2*a^2*b^5*c + 4*a*b^6*c - 2*b^7*c + 2*a^6*c^2 - 2*a^5*b*c^2 - 4*a^4*b^2*c^2 + 6*a^3*b^3*c^2 - 4*a^2*b^4*c^2 + 2*b^6*c^2 - 2*a^5*c^3 + 6*a^3*b^2*c^3 + 4*a^2*b^3*c^3 - 2*a*b^4*c^3 - 6*b^5*c^3 + 3*a^4*c^4 - 6*a^3*b*c^4 - 4*a^2*b^2*c^4 - 2*a*b^3*c^4 + 10*b^4*c^4 + 2*a^2*b*c^5 - 6*b^3*c^5 + 4*a*b*c^6 + 2*b^2*c^6 - 2*a*c^7 - 2*b*c^7 + c^8 : :
X(40554) = 3 X(2) + X(927)

X(40554) lies on these lines: {2, 927}, {3, 33331}, {101, 14505}, {103, 6074}, {514, 6710}, {516, 6712}, {631, 2724}, {1565, 34805}, {3035, 4885}, {3234, 5845}, {3323, 9318}, {4369, 5972}, {4778, 36956}, {5074, 6699}, {5532, 9317}, {34906, 35094}

X(40554) = complement of X(1566)
X(40554) = centroid of {A,B,C,X(927)}
X(40554) = center of bicevian conic of X(2) and X(927)

X(40555) = CENTER OF 1ST (X(2),X(934))-CEVA CONIC

Barycentrics    2*a^7 - 2*a^6*b - 5*a^5*b^2 + 5*a^4*b^3 + 4*a^3*b^4 - 4*a^2*b^5 - a*b^6 + b^7 - 2*a^6*c + 12*a^5*b*c - 5*a^4*b^2*c - 12*a^3*b^3*c + 8*a*b^5*c - b^6*c - 5*a^5*c^2 - 5*a^4*b*c^2 + 16*a^3*b^2*c^2 + 4*a^2*b^3*c^2 - 7*a*b^4*c^2 - 3*b^5*c^2 + 5*a^4*c^3 - 12*a^3*b*c^3 + 4*a^2*b^2*c^3 + 3*b^4*c^3 + 4*a^3*c^4 - 7*a*b^2*c^4 + 3*b^3*c^4 - 4*a^2*c^5 + 8*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7 : :
X(40555) = 3 X(2) + X(934)

X(40555) lies on these lines: {2, 934}, {9, 28344}, {116, 20418}, {142, 6713}, {631, 972}, {1125, 6712}, {1360, 5433}, {3035, 6366}, {6691, 6714}, {6710, 36949}, {7483, 15725}

X(40555) = complement of X(5514)
X(40555) = centroid of {A,B,C,X(934)}
X(40555) = center of bicevian conic of X(2) and X(934)

X(40556) = CENTER OF 1ST (X(2),X(1296))-CEVA CONIC

Barycentrics    2*a^10 - 10*a^8*b^2 + 3*a^6*b^4 + 9*a^4*b^6 - 5*a^2*b^8 + b^10 - 10*a^8*c^2 + 60*a^6*b^2*c^2 - 49*a^4*b^4*c^2 + 20*a^2*b^6*c^2 - 5*b^8*c^2 + 3*a^6*c^4 - 49*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 4*b^6*c^4 + 9*a^4*c^6 + 20*a^2*b^2*c^6 + 4*b^4*c^6 - 5*a^2*c^8 - 5*b^2*c^8 + c^10 : :
X(40556) = 3 X(2) + X(1296)

X(40556) lies on these lines: {2, 1296}, {3, 126}, {4, 38716}, {5, 38623}, {30, 38803}, {111, 631}, {140, 6719}, {141, 14688}, {376, 10734}, {381, 38797}, {382, 38798}, {485, 11836}, {486, 11835}, {543, 549}, {620, 2793}, {1656, 22338}, {2780, 5972}, {2805, 6713}, {2813, 6712}, {2819, 6718}, {2824, 6710}, {2830, 3035}, {2847, 34842}, {2852, 6711}, {2854, 6699}, {3325, 5432}, {3523, 14360}, {3524, 10717}, {3525, 38688}, {3526, 38593}, {3628, 38801}, {5054, 9172}, {5055, 38800}, {5070, 38799}, {5085, 36883}, {5433, 6019}, {5657, 10704}, {6714, 9522}, {6715, 9526}, {6716, 9529}, {6717, 9531}, {9129, 38793}, {10165, 11721}, {10303, 38675}, {10519, 10765}, {10779, 34474}, {12100, 32424}, {14643, 35447}, {14666, 15693}, {15122, 16760}, {16239, 38802}, {17566, 38518}, {18580, 34840}, {37450, 38651}

X(40556) = complement of X(5512)
X(40556) = centroid of {A,B,C,X(1296)}
X(40556) = center of bicevian conic of X(2) and X(1296)

X(40557) = CENTER OF 1ST (X(2),X(1304))-CEVA CONIC

Barycentrics    2*a^18 - 4*a^16*b^2 - 8*a^14*b^4 + 27*a^12*b^6 - 16*a^10*b^8 - 17*a^8*b^10 + 24*a^6*b^12 - 7*a^4*b^14 - 2*a^2*b^16 + b^18 - 4*a^16*c^2 + 28*a^14*b^2*c^2 - 31*a^12*b^4*c^2 - 54*a^10*b^6*c^2 + 116*a^8*b^8*c^2 - 48*a^6*b^10*c^2 - 23*a^4*b^12*c^2 + 18*a^2*b^14*c^2 - 2*b^16*c^2 - 8*a^14*c^4 - 31*a^12*b^2*c^4 + 142*a^10*b^4*c^4 - 99*a^8*b^6*c^4 - 96*a^6*b^8*c^4 + 119*a^4*b^10*c^4 - 22*a^2*b^12*c^4 - 5*b^14*c^4 + 27*a^12*c^6 - 54*a^10*b^2*c^6 - 99*a^8*b^4*c^6 + 240*a^6*b^6*c^6 - 89*a^4*b^8*c^6 - 42*a^2*b^10*c^6 + 17*b^12*c^6 - 16*a^10*c^8 + 116*a^8*b^2*c^8 - 96*a^6*b^4*c^8 - 89*a^4*b^6*c^8 + 96*a^2*b^8*c^8 - 11*b^10*c^8 - 17*a^8*c^10 - 48*a^6*b^2*c^10 + 119*a^4*b^4*c^10 - 42*a^2*b^6*c^10 - 11*b^8*c^10 + 24*a^6*c^12 - 23*a^4*b^2*c^12 - 22*a^2*b^4*c^12 + 17*b^6*c^12 - 7*a^4*c^14 + 18*a^2*b^2*c^14 - 5*b^4*c^14 - 2*a^2*c^16 - 2*b^2*c^16 + c^18 : :
X(40557) = 3 X(2) + X(1304)

X(40557) lies on these lines: {2, 1304}, {3, 18809}, {4, 38719}, {5, 31379}, {30, 34842}, {122, 31510}, {402, 22104}, {403, 12096}, {468, 12145}, {520, 5972}, {523, 6716}, {631, 2693}, {1552, 3184}, {3526, 38595}, {6000, 6699}, {6036, 37911}, {6677, 16760}, {12068, 34844}, {23583, 31945}, {32417, 38605}

X(40557) = complement of X(16177)
X(40557) = centroid of {A,B,C,X(1304)}
X(40557) = center of bicevian conic of X(2) and X(1304)

X(40558) = CENTER OF 1ST (X(2),X(1309))-CEVA CONIC

Barycentrics    2*a^12 - 4*a^11*b - 2*a^10*b^2 + 10*a^9*b^3 - 7*a^8*b^4 - 4*a^7*b^5 + 12*a^6*b^6 - 8*a^5*b^7 - 4*a^4*b^8 + 8*a^3*b^9 - 2*a^2*b^10 - 2*a*b^11 + b^12 - 4*a^11*c + 16*a^10*b*c - 14*a^9*b^2*c - 16*a^8*b^3*c + 38*a^7*b^4*c - 26*a^6*b^5*c - 10*a^5*b^6*c + 34*a^4*b^7*c - 18*a^3*b^8*c - 6*a^2*b^9*c + 8*a*b^10*c - 2*b^11*c - 2*a^10*c^2 - 14*a^9*b*c^2 + 48*a^8*b^2*c^2 - 34*a^7*b^3*c^2 - 36*a^6*b^4*c^2 + 82*a^5*b^5*c^2 - 48*a^4*b^6*c^2 - 22*a^3*b^7*c^2 + 38*a^2*b^8*c^2 - 12*a*b^9*c^2 + 10*a^9*c^3 - 16*a^8*b*c^3 - 34*a^7*b^2*c^3 + 100*a^6*b^3*c^3 - 64*a^5*b^4*c^3 - 50*a^4*b^5*c^3 + 94*a^3*b^6*c^3 - 40*a^2*b^7*c^3 - 6*a*b^8*c^3 + 6*b^9*c^3 - 7*a^8*c^4 + 38*a^7*b*c^4 - 36*a^6*b^2*c^4 - 64*a^5*b^3*c^4 + 136*a^4*b^4*c^4 - 62*a^3*b^5*c^4 - 36*a^2*b^6*c^4 + 40*a*b^7*c^4 - 9*b^8*c^4 - 4*a^7*c^5 - 26*a^6*b*c^5 + 82*a^5*b^2*c^5 - 50*a^4*b^3*c^5 - 62*a^3*b^4*c^5 + 92*a^2*b^5*c^5 - 28*a*b^6*c^5 - 4*b^7*c^5 + 12*a^6*c^6 - 10*a^5*b*c^6 - 48*a^4*b^2*c^6 + 94*a^3*b^3*c^6 - 36*a^2*b^4*c^6 - 28*a*b^5*c^6 + 16*b^6*c^6 - 8*a^5*c^7 + 34*a^4*b*c^7 - 22*a^3*b^2*c^7 - 40*a^2*b^3*c^7 + 40*a*b^4*c^7 - 4*b^5*c^7 - 4*a^4*c^8 - 18*a^3*b*c^8 + 38*a^2*b^2*c^8 - 6*a*b^3*c^8 - 9*b^4*c^8 + 8*a^3*c^9 - 6*a^2*b*c^9 - 12*a*b^2*c^9 + 6*b^3*c^9 - 2*a^2*c^10 + 8*a*b*c^10 - 2*a*c^11 - 2*b*c^11 + c^12 : :
X(40558) = 3 X(2) + X(1309)

X(40558) lies on these lines: {2, 1309}, {3, 39535}, {5, 38617}, {515, 6711}, {522, 6718}, {631, 2734}, {3326, 24410}, {5972, 8062}, {22102, 35013}

X(40558) = complement of X(10017)
X(40558) = centroid of {A,B,C,X(1309)}
X(40558) = center of bicevian conic of X(2) and X(1309)

X(40559) = CENTER OF 1ST (X(2),X(1576))-CEVA CONIC

Barycentrics    2*a^8 - 2*a^6*b^2 - a^2*b^6 + b^8 - 2*a^6*c^2 + 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^2*b^2*c^4 - a^2*c^6 - b^2*c^6 + c^8 : :
X(40559) = 3 X(2) + X(1576)

X(40559) lies on these lines: {2, 1576}, {140, 2781}, {338, 15000}, {420, 39231}, {523, 23583}, {526, 5972}, {620, 9479}, {1316, 34981}, {2871, 3589}, {2881, 6720}, {2882, 6680}

X(40559) = complement of complement of X(1576)
X(40559) = centroid of {A,B,C,X(1576)}
X(40559) = center of bicevian conic of X(2) and X(1576)

X(40560) = CENTER OF 1ST (X(2),X(1633))-CEVA CONIC

Barycentrics    2*a^5 - 2*a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 2*a^4*c + a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - a^2*c^3 + 2*a*b*c^3 - a*c^4 - b*c^4 + c^5 : :
X(40560) = 3 X(2) + X(1633)

X(40560) lies on these lines: {2, 1633}, {11, 16686}, {45, 5432}, {140, 12608}, {190, 26231}, {468, 1155}, {513, 36949}, {523, 16599}, {692, 5848}, {900, 3035}, {1086, 24346}, {1125, 2835}, {2820, 6710}, {2823, 6684}, {2849, 6717}, {3246, 15325}, {3579, 16618}, {3683, 7499}, {3826, 36477}, {4364, 6690}, {4640, 6676}, {5849, 7193}, {6713, 38607}, {12329, 27509}, {21293, 35280}, {23305, 24309}, {23845, 25968}

X(40560) = complement of complement of X(1633)
X(40560) = centroid of {A,B,C,X(1633)}
X(40560) = center of bicevian conic of X(2) and X(1633)

X(40561) = CENTER OF 1ST (X(2),X(1783))-CEVA CONIC

Barycentrics    2*a^7 - 2*a^6*b + a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 - a*b^6 + b^7 - 2*a^6*c + a^4*b^2*c + 2*a^3*b^3*c + 2*a^2*b^4*c - 2*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 - 4*a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*a*b^3*c^3 - b^4*c^3 - 2*a^3*c^4 + 2*a^2*b*c^4 + a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 - 2*a*b*c^5 + b^2*c^5 - a*c^6 - b*c^6 + c^7 : :
X(40561) = 3 X(2) + X(1783)

X(40561) lies on these lines: {2, 1783}, {2806, 3035}, {2812, 6710}, {3820, 20204}, {15252, 24003}, {23583, 27076}, {36949, 39470}

X(40561) = complement of complement of X(1783)
X(40561) = centroid of {A,B,C,X(1783)}
X(40561) = center of bicevian conic of X(2) and X(1783)

X(40562) = CENTER OF 1ST (X(2),X(1978))-CEVA CONIC

Barycentrics    a^3*b^3 - a^3*b^2*c - a^2*b^3*c - a^3*b*c^2 + 4*a^2*b^2*c^2 - 2*a*b^3*c^2 + a^3*c^3 - a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 : :
X(40562) = 3 X(2) + X(1978)

X(40562) lies on these lines: {2, 1978}, {891, 4928}, {1015, 18149}, {3835, 36951}, {3934, 17760}, {4422, 36950}, {4598, 24502}, {11052, 13006}, {11814, 20333}, {21893, 33908}

X(40562) = complement of X(6377)
X(40562) = centroid of {A,B,C,X(1978)}
X(40562) = center of bicevian conic of X(2) and X(1978)

X(40563) = CENTER OF 1ST (X(1),X(76))-CEVA CONIC

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

X(40563) lies on these lines: {141, 40216}, {693, 3741}, {1269, 35544}, {4357, 4972}, {4980, 20888}, {14751, 27800}, {17184, 26582}, {20292, 20347}

X(40564) = CENTER OF 1ST (X(1),X(92))-CEVA CONIC

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

X(40564) lies on these lines: {37, 4858}, {72, 3696}, {75, 16574}, {321, 908}, {1089, 15281}, {1214, 6358}, {1229, 21070}, {1441, 4605}, {1577, 20305}, {1838, 1861}, {3262, 22008}, {3588, 29069}, {3613, 30171}, {4404, 20910}, {22018, 26165}

X(40565) = ISOGONAL CONJUGATE OF X(3513)

Barycentrics    a*(a^2 - 2*a*b + b^2 - c^2 - 4*r*Sqrt[r*(r + 4*R)])*(a^2 - b^2 - 2*a*c + c^2 - 4*r*Sqrt[r*(r + 4*R)]) : :

X(40565) lies on the Feuerbach circumhyperbola, the curve Q044, and these lines: {2, 11}, {7, 3514}, {104, 38014}, {516, 32622}, {2801, 39145}, {3513, 7677}

X(40565) = reflection of X(40566) in X(11)
X(40565) = isogonal conjugate of X(3513)
X(40565) = cevapoint of X(1) and X(32622)
X(40565) = barycentric product X(3514)*X(36796)
X(40565) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3513}, {3514, 241}
X(40565) = {X(2),X(390)}-harmonic conjugate of X(40566)

X(40566) = ISOGONAL CONJUGATE OF X(3514)

Barycentrics    a*(a^2 - 2*a*b + b^2 - c^2 + 4*r*Sqrt[r*(r + 4*R)])*(a^2 - b^2 - 2*a*c + c^2 + 4*r*Sqrt[r*(r + 4*R)]) : :

X(40566) lies on the Feuerbach circumhyperbola, the curve Q044, and these lines: {2, 11}, {7, 3513}, {104, 38013}, {516, 32623}, {2801, 39144}, {3514, 7677}

X(40566) = reflection of X(40565) in X(11)
X(40566) = isogonal conjugate of X(3514)
X(40566) = cevapoint of X(1) and X(32623)
X(40566) = barycentric product X(3513)*X(36796)
X(40566) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3514}, {3513, 241}
X(40566) = {X(2),X(390)}-harmonic conjugate of X(40565)

X(40567) = X(105)X(516)∩X(518)X(677)

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

X(40567) lies on the cubic K1175 and these lines: {105, 516}, {518, 677}

X(40568) = X(3)X(348)∩X(105)X(175)

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

X(40568) lies on the cubic K1175 and these lines: {3, 348}, {105, 175}, {1814, 13388}

X(40568) = circumcircle-inverse of X(40569)
X(40568) = X(5089)-isoconjugate of X(7348)
X(40568) = barycentric product X(6203)*X(31637)
X(40568) = barycentric quotient X(i)/X(j) for these {i,j}: {6203, 1861}, {36057, 7348}

X(40569) = X(3)X(348)∩X(105)X(176)

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

X(40569) lies on the cubic K1175 and these lines: {3, 348}, {105, 176}, {1814, 13389}

X(40569) = circumcircle-inverse of X(40568)
X(40569) = X(5089)-isoconjugate of X(7347)
X(40569) = barycentric product X(6204)*X(31637)
X(40569) = barycentric quotient X(i)/X(j) for these {i,j}: {6204, 1861}, {36057, 7347}

X(40570) = X(2)X(7054)∩X(6)X(1175)

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

X(40570) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1174, and on these lines:" {2, 7054}, {6, 1175}, {32, 36420}, {37, 943}, {42, 2259}, {112, 1841}, {1333, 1396}, {1400, 1474}, {1880, 2204}, {2395, 14775}

X(40570) = polar conjugate of X(1234)
X(40570) = isogonal conjugate of the isotomic conjugate of X(40395)
X(40570) = polar conjugate of the isotomic conjugate of X(1175)
X(40570) = X(40395)-Ceva conjugate of X(1175)
X(40570) = X(i)-cross conjugate of X(j) for these (i,j): {3063, 32713}, {6591, 112}
X(40570) = X(i)-isoconjugate of X(j) for these (i,j): {10, 18607}, {48, 1234}, {63, 442}, {69, 2294}, {72, 5249}, {75, 18591}, {306, 942}, {312, 39791}, {313, 14597}, {321, 4303}, {326, 1865}, {349, 23207}, {1214, 6734}, {1231, 14547}, {1332, 23752}, {1444, 21675}, {1838, 3998}, {2260, 20336}
X(40570) = cevapoint of X(i) and X(j) for these (i,j): {25, 2204}, {32, 2203}
X(40570) = barycentric product X(i)*X(j) for these {i,j}: {4, 1175}, {6, 40395}, {25, 40412}, {27, 2259}, {28, 943}, {110, 14775}, {1172, 2982}, {1794, 8747}, {17926, 32651}
X(40570) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1234}, {25, 442}, {32, 18591}, {943, 20336}, {1175, 69}, {1333, 18607}, {1397, 39791}, {1474, 5249}, {1973, 2294}, {2203, 942}, {2206, 4303}, {2207, 1865}, {2259, 306}, {2299, 6734}, {2333, 21675}, {2982, 1231}, {14775, 850}, {40395, 76}, {40412, 305}

X(40571) = X(2)X(6)∩X(21)X(72)

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

X(40571) lies on the cubics K610 and K1174 and on these lines: {2, 6}, {21, 72}, {27, 5905}, {28, 3868}, {29, 1069}, {58, 78}, {63, 284}, {92, 1172}, {100, 209}, {110, 2203}, {184, 36018}, {218, 16050}, {239, 17866}, {286, 40395}, {307, 2003}, {411, 5752}, {518, 2194}, {579, 37312}, {593, 40403}, {648, 2990}, {651, 1396}, {758, 21376}, {894, 37095}, {912, 4227}, {1029, 13583}, {1043, 20013}, {1170, 39747}, {1210, 27412}, {1333, 3998}, {1412, 1445}, {1441, 2982}, {1444, 4280}, {1778, 27396}, {1780, 3811}, {1790, 18206}, {1817, 3218}, {2206, 32912}, {2328, 3870}, {2651, 26893}, {2893, 27052}, {2911, 17776}, {3060, 7466}, {3149, 12160}, {3152, 20077}, {3194, 5081}, {3434, 5327}, {3564, 37362}, {3666, 4273}, {3759, 19788}, {4001, 24632}, {4184, 7085}, {4215, 20760}, {4228, 5208}, {4558, 18605}, {5320, 10477}, {5358, 20602}, {5810, 6828}, {6915, 15801}, {6986, 34148}, {7538, 20018}, {7754, 11341}, {8822, 20078}, {9965, 14953}, {10974, 35979}, {11115, 20007}, {14054, 30733}, {14868, 37301}, {17188, 26015}, {17189, 26723}, {17498, 23090}, {20043, 20212}, {21997, 26840}

X(40571) = anticomplement of the isotomic conjugate of X(40395)
X(40571) = isotomic conjugate of the polar conjugate of X(30733)
X(40571) = isotomic conjugate of isogonal conjugate of X(41332)
X(40571) = polar conjugate of isogonal conjugate of X(41608)
X(40571) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1175, 4329}, {1474, 2894}, {14775, 21294}, {40395, 6327}
X(40571) = X(i)-Ceva conjugate of X(j) for these (i,j): {286, 21}, {40395, 2}
X(40571) = X(2911)-cross conjugate of X(1780)
X(40571) = X(i)-isoconjugate of X(j) for these (i,j): {6, 23604}, {19, 28787}, {42, 15474}, {65, 39943}, {71, 39267}, {661, 13397}
X(40571) = cevapoint of X(i) and X(j) for these (i,j): {1708, 3173}, {2911, 3811}
X(40571) = crosspoint of X(648) and X(4567)
X(40571) = crosssum of X(647) and X(3125)
X(40571) = barycentric product X(i)*X(j) for these {i,j}: {69, 30733}, {75, 1780}, {81, 17776}, {86, 3811}, {99, 15313}, {274, 2911}, {286, 11517}, {314, 37579}, {333, 1708}, {1043, 4341}, {3173, 31623}, {4570, 17877}, {14054, 40412}
X(40571) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23604}, {3, 28787}, {28, 39267}, {81, 15474}, {110, 13397}, {284, 39943}, {1708, 226}, {1780, 1}, {2911, 37}, {3173, 1214}, {3215, 73}, {3811, 10}, {4341, 3668}, {11517, 72}, {14054, 442}, {15313, 523}, {17776, 321}, {17877, 21207}, {26217, 33294}, {30733, 4}, {37579, 65}
X(40571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 284, 27174}, {81, 1812, 26637}, {81, 2287, 2}, {81, 37783, 1812}, {965, 19716, 2}, {5278, 5736, 2}, {5320, 10477, 37325}, {20078, 26830, 8822}

X(40572) = X(2)X(219)∩X(4)X(2911)

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

X(40572) lies on the cubic K1174 and these lines: {2, 219}, {4, 2911}, {6, 943}, {35, 71}, {2174, 7430}, {2323, 2983}, {9085, 15439}

X(40572) = X(40395)-Ceva conjugate of X(943)
X(40572) = X(i)-isoconjugate of X(j) for these (i,j): {272, 2294}, {942, 1751}, {2218, 5249}, {2260, 2997}
X(40572) = barycentric product X(i)*X(j) for these {i,j}: {209, 40412}, {943, 3868}, {1794, 5125}, {2259, 18134}, {2982, 27396}, {15439, 20294}
X(40572) = barycentric quotient X(i)/X(j) for these {i,j}: {209, 442}, {579, 5249}, {943, 2997}, {1175, 272}, {2198, 2294}, {2259, 1751}, {2352, 942}, {3190, 6734}, {15439, 1305}

X(40573) = X(4)X(12)∩X(9)X(92)

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

X(40573) lies on the cubic K1174 and these lines: {4, 12}, {6, 278}, {9, 92}, {27, 226}, {57, 1847}, {225, 2299}, {273, 1708}, {333, 349}, {917, 15439}, {1175, 14016}, {1214, 36019}, {1436, 7490}, {1794, 1838}, {1860, 2195}, {2164, 7363}, {2316, 6336}, {2339, 37086}, {6994, 8232}, {14775, 23351}

X(40573) = polar conjugate of X(6734)
X(40573) = X(40395)-Ceva conjugate of X(2982)
X(40573) = X(661)-cross conjugate of X(36127)
X(40573) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23207}, {8, 14597}, {9, 4303}, {21, 18591}, {48, 6734}, {55, 18607}, {63, 14547}, {78, 2260}, {212, 5249}, {219, 942}, {283, 2294}, {394, 1859}, {442, 2193}, {1214, 8021}, {1259, 1841}, {1838, 2289}, {2287, 39791}, {6516, 33525}, {8606, 16585}
X(40573) = cevapoint of X(i) and X(j) for these (i,j): {19, 225}, {34, 1400}, {226, 1708}
X(40573) = trilinear pole of line {663, 7649}
X(40573) = barycentric product X(i)*X(j) for these {i,j}: {92, 2982}, {225, 40412}, {226, 40395}, {273, 943}, {331, 2259}, {664, 14775}
X(40573) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6734}, {25, 14547}, {31, 23207}, {34, 942}, {56, 4303}, {57, 18607}, {225, 442}, {278, 5249}, {604, 14597}, {608, 2260}, {943, 78}, {1042, 39791}, {1096, 1859}, {1118, 1838}, {1175, 283}, {1400, 18591}, {1794, 1259}, {1880, 2294}, {2259, 219}, {2299, 8021}, {2982, 63}, {6198, 31938}, {8736, 21675}, {14775, 522}, {15439, 1331}, {32651, 1813}, {36048, 6516}, {40395, 333}, {40412, 332}

X(40574) = X(6)-CROSS CONJUGATE OF X(28)

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

X(40574) lies on the cubic K1174 and these lines: {4, 580}, {27, 272}, {28, 1612}, {29, 40161}, {92, 1172}, {278, 1474}, {1214, 1305}, {1396, 1847}, {5137, 5146}

X(40574) = isogonal conjugate of the complement of X(2997)
X(40574) = polar conjugate of the isotomic conjugate of X(272)
X(40574) = X(i)-cross conjugate of X(j) for these (i,j): {6, 28}, {513, 1305}
X(40574) = X(i)-isoconjugate of X(j) for these (i,j): {3, 22021}, {63, 209}, {69, 2198}, {71, 3868}, {72, 579}, {73, 27396}, {228, 18134}, {306, 2352}, {1214, 3190}, {3694, 4306}, {3990, 5125}, {4574, 23800}
X(40574) = cevapoint of X(6) and X(2218)
X(40574) = barycentric product X(i)*X(j) for these {i,j}: {4, 272}, {27, 1751}, {28, 2997}, {286, 2218}, {1474, 40011}, {2299, 15467}, {36419, 40161}
X(40574) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 22021}, {25, 209}, {27, 18134}, {28, 3868}, {272, 69}, {1172, 27396}, {1474, 579}, {1751, 306}, {1973, 2198}, {2203, 2352}, {2218, 72}, {2299, 3190}, {2997, 20336}, {8747, 5125}, {40011, 40071}

X(40575) = X(4)X(18687)∩X(63)X(284)

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

X(40575) lies on the cubic K1174 and these lines: {4, 18687}, {63, 284}, {226, 13395}

X(40576) = X(7)X(1037)∩X(8)X(8283)

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

X(40576) lies on these lines: {7, 1037}, {8, 8283}, {56, 528}, {59, 513}, {77, 24309}, {100, 108}, {105, 37771}, {109, 13397}, {934, 1292}, {1376, 4081}, {1486, 37800}, {1804, 11495}, {1813, 35338}, {2222, 9058}, {2961, 4319}, {3939, 24029}, {4236, 4565}, {4331, 37576}, {5723, 16686}, {19850, 37579}, {21147, 35998}

X(40576) = X(4569)-Ceva conjugate of X(651)
X(40576) = X(i)-cross conjugate of X(j) for these (i,j): {11934, 169}, {21185, 4228}
X(40576) = X(i)-isoconjugate of X(j) for these (i,j): {55, 26721}, {514, 40141}, {522, 3433}, {663, 13577}, {7004, 26706}
X(40576) = cevapoint of X(169) and X(11934)
X(40576) = trilinear pole of line {169, 5452}
X(40576) = crossdifference of every pair of points on line {7117, 17435}
X(40576) = barycentric product X(i)*X(j) for these {i,j}: {59, 26546}, {100, 37800}, {108, 28420}, {109, 20927}, {169, 664}, {190, 34036}, {651, 3434}, {1275, 11934}, {1414, 21073}, {1486, 4554}, {4228, 4552}, {4564, 21185}, {4569, 5452}, {4573, 21867}, {6516, 17905}, {18026, 22131}
X(40576) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 26721}, {169, 522}, {651, 13577}, {692, 40141}, {1415, 3433}, {1486, 650}, {3434, 4391}, {4228, 4560}, {5452, 3900}, {7115, 26706}, {11934, 1146}, {20927, 35519}, {21073, 4086}, {21185, 4858}, {21867, 3700}, {22131, 521}, {26546, 34387}, {28420, 35518}, {34036, 514}, {37800, 693}

X(40577) = X(7)X(3446)∩X(36)X(516)

Barycentrics    a*(a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2 - c^3) : :
X(40577) = 3 X[165] - X[29374], 2 X[1618] - 3 X[35280]

X(40577) lies on the cubic K578 and these lines: {7, 3446}, {36, 516}, {59, 513}, {100, 522}, {101, 21127}, {105, 3322}, {108, 901}, {109, 1290}, {165, 29374}, {404, 24410}, {484, 1725}, {517, 3100}, {656, 36031}, {927, 24002}, {934, 1308}, {1086, 38863}, {1155, 9358}, {1284, 5172}, {1292, 14733}, {1305, 2722}, {1319, 4318}, {1769, 39026}, {2077, 16869}, {2078, 22464}, {2310, 2957}, {2720, 13397}, {3315, 24201}, {4236, 17942}, {5091, 21746}, {13589, 23981}, {16686, 37771}, {24025, 34464}, {38682, 39756}

X(40577) = reflection of X(i) in X(j) for these {i,j}: {651, 59}, {4318, 1319}
X(40577) = reflection of X(59) in the OI line
X(40577) = X(7)-Ceva conjugate of X(651)
X(40577) = X(11193)-cross conjugate of X(5540)
X(40577) = X(i)-isoconjugate of X(j) for these (i,j): {109, 34896}, {522, 3446}, {663, 8047}
X(40577) = cevapoint of X(i) and X(j) for these (i,j): {513, 38863}, {5540, 11193}
X(40577) = crosspoint of X(7) and X(37771)
X(40577) = trilinear pole of line {1421, 5540}
X(40577) = isogonal conjugate of X(11) wrt the anticevian triangle of X(11)
X(40577) = barycentric product X(i)*X(j) for these {i,j}: {7, 5375}, {100, 37771}, {109, 18151}, {149, 651}, {190, 1421}, {513, 31633}, {664, 5540}, {1275, 11193}, {1414, 21090}, {3669, 11607}, {4554, 16686}, {4564, 21201}, {4573, 21889}, {18026, 22144}
X(40577) = barycentric quotient X(i)/X(j) for these {i,j}: {149, 4391}, {650, 34896}, {651, 8047}, {1415, 3446}, {1421, 514}, {5375, 8}, {5540, 522}, {11193, 1146}, {11607, 646}, {16686, 650}, {18151, 35519}, {21090, 4086}, {21201, 4858}, {21889, 3700}, {22144, 521}, {31633, 668}, {37771, 693}

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Centers of 2nd Ceva conics: X(40578)-X(40629)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 6, 2020.

In the plane of a triangle ABC, let L be the line u x + v y + w z = 0, and let U be the point u : v : w, this being the trilinear pole of L. Let P = p : q : r be a point. The 2nd (U,P)-Ceva conic is introduced here as the locus of X such that the X-Ceva conjugate of P is on the line L. This conic circumscribes ABC and is given by

p(-u p + v q + w r)/x + q(u p - v q + w r)/y + r(u p + v q - w r)/z = 0.

The center of the conic is the point

p*((p+q+r) p^2 u^2 + (p+q-r) q^2 v^2 + (p-q+r) r^2 w^2 + 2 p q r v w - 2 p r (p+r) w u - 2 p q (p+q) u v) : :

If U = X(2), then the center of the 2nd (U,P)-Ceva conic is the X(2)-Ceva conjugate of P, and the perspector of the 2nd (U,P)-Ceva conic is P. (Randy Hutson, December 18, 2020)

X(40578) = CENTER OF 2ND (X(2),X(13))-CEVA CONIC

Barycentrics    (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S)*(5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4 - 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

See also X(41889).

X(40578) lies on these lines: {2, 19776}, {3, 5623}, {13, 15}, {14, 39170}, {298, 1494}, {470, 8737}, {476, 36298}, {2153, 37772}, {2992, 3180}, {8014, 37640}, {16645, 18777}

X(40578) = complement of X(19776)
X(40578) = complementary conjugate of complement of X(15)-Ceva conjugate of X(6)
X(40578) = X(2)-Ceva conjugate of X(13)
X(40578) = perspector of circumconic centered at X(13)

X(40579) = CENTER OF 2nd (X(2),X(14))-CEVA CONIC

Barycentrics    (-2*S+(a^2+b^2-c^2)*sqrt(3))*(2*sqrt(3)*(-a^2+b^2+c^2)*S+5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2)*(-2*S+(a^2-b^2+c^2)*sqrt(3)) : :
X(40579) = X(3181)+3*X(19773) = 2*X(10218)+X(45779)

X(40579) lies on the cubics K277, K341b, K472 and these lines: {2,19777}, {3,5624}, {13,39170}, {14,16}, {299,1494}, {471,8738}, {476,36299}, {2154,37772}, {2993,3181}, {3642,40580}, {8015,37641}, {11556,42989}, {14583,51270}, {16241,40695}, {16644,18776}, {25641,46465}, {36840,46825}, {37640,52040}

X(40579) = midpoint of X(i) and X(j) for these {i, j}: {14, 45779}, {23896, 36311}
X(40579) = reflection of X(14) in X(10218)
X(40579) = complement of X(19777)
X(40579) = perspector of the circumconic {A, B, C, X(36840), X(39133)}
X(40579) = center of the circumconic {A, B, C, X(476), X(23896)}
X(40579) = intersection, other than A, B, C, of circumconics {A, B, C, X(13), X(15442)} and {A, B, C, X(14), X(617)}
X(40579) = barycentric product X(i)*X(j) for these {i, j}: {14, 617}, {533, 39133}, {622, 46059}
X(40579) = barycentric quotient X(i)/X(j) for these (i, j): (14, 19777), (617, 299)
X(40579) = trilinear product X(i)*X(j) for these {i, j}: {14, 19299}, {617, 2154}
X(40579) = trilinear quotient X(2154)/X(3441)
X(40579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 36298, 51277), (14, 46074, 15442), (16, 36210, 14), (395, 11085, 14), (395, 11549, 11085), (11549, 47142, 36210), (11582, 16268, 14), (23715, 47482, 51269)

X(40580) = CENTER OF 2ND (X(2),X(15))-CEVA CONIC

Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) - 2*S)*(Sqrt[3]*(a^4 - (b^2 - c^2)^2) + 2*(-a^2 + b^2 + c^2)*S) : :

X(40580) lies on these lines: {2, 2992}, {3, 3165}, {5, 13}, {6, 3170}, {15, 1511}, {16, 4550}, {61, 1147}, {110, 36296}, {113, 5668}, {202, 942}, {216, 10639}, {298, 340}, {300, 23896}, {577, 10640}, {2005, 11088}, {5158, 9306}, {5237, 34328}, {5238, 33556}, {10217, 10272}, {11126, 17035}, {15748, 36836}, {22238, 33537}, {30383, 34830}, {32586, 34425}

X(40580) = complement of X(2992)
X(40580) = complementary conjugate of complement of X(3129)
X(40580) = X(2)-Ceva conjugate of X(15)
X(40580) = perspector of circumconic centered at X(15)
X(40580) = crosssum of circumcircle intercepts of inner Napoleon circle
X(40580) = {X(5158),X(9306)}-harmonic conjugate of X(40581)

X(40581) = CENTER OF 2ND (X(2),X(16))-CEVA CONIC

Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) + 2*S)*(Sqrt[3]*(a^4 - (b^2 - c^2)^2) - 2*(-a^2 + b^2 + c^2)*S) : :

X(40581) lies on these lines: {2, 2993}, {3, 3166}, {5, 14}, {6, 3171}, {15, 4550}, {16, 1511}, {62, 1147}, {110, 36297}, {113, 5669}, {203, 942}, {216, 10640}, {299, 340}, {301, 23895}, {577, 10639}, {2004, 11083}, {5158, 9306}, {5237, 33556}, {5238, 34327}, {10218, 10272}, {11127, 17035}, {15748, 36843}, {22236, 33537}, {30382, 34830}, {32585, 34424}

X(40581) = complement of X(2993)
X(40581) = complementary conjugate of complement of X(3130)
X(40581) = X(2)-Ceva conjugate of X(16)
X(40581) = perspector of circumconic centered at X(16)
X(40581) = crosssum of circumcircle intercepts of outer Napoleon circle
X(40581) = {X(5158),X(9306)}-harmonic conjugate of X(40580)

X(40582) = CENTER OF 2ND (X(2),X(21))-CEVA CONIC

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

X(40582) lies on the hyperbola {{X(2),X(6),X(216),X(233),X(1249)}} and these lines: {6, 2476}, {9, 35193}, {19, 1325}, {21, 270}, {37, 5546}, {60, 2264}, {81, 3664}, {216, 404}, {229, 1781}, {233, 7504}, {281, 13746}, {284, 2170}, {377, 1249}, {403, 15947}, {442, 2906}, {648, 1441}, {662, 1442}, {857, 2905}, {1100, 2303}, {1196, 2670}, {1560, 30770}, {1731, 2150}, {2182, 23059}, {2287, 2323}, {2322, 11103}, {2907, 5051}, {3163, 6175}, {4560, 18311}, {5778, 11441}, {14401, 23090}, {15595, 15988}, {17686, 37891}, {33841, 37895}

X(40582) = X(2)-Ceva conjugate of X(21)
X(40582) = perspector of circumconic centered at X(21)

X(40583) = CENTER OF 2ND (X(2),X(23))-CEVA CONIC

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

X(40583) lies on the hyperbola {{X(2),X(6),X(216),X(233),X(1249)}} and these lines: {2, 8792}, {6, 3448}, {23, 8744}, {111, 251}, {112, 6636}, {115, 34482}, {216, 7496}, {233, 6103}, {323, 15595}, {648, 18019}, {1249, 16063}, {1560, 30745}, {3162, 31101}, {3163, 10989}, {5189, 22121}, {7394, 40179}, {7711, 13114}, {18311, 31128}, {31107, 37895}

X(40583) = complement of isogonal conjugate of X(19596)
X(40583) = complementary conjugate of complement of X(19596)
X(40583) = X(2)-Ceva conjugate of X(23)
X(40583) = perspector of circumconic centered at X(23)

X(40584) = CENTER OF 2ND (X(2),X(36))-CEVA CONIC

Barycentrics    a^2*(a^2 - b^2 + b*c - c^2)*(a^4 - b^4 + a^2*b*c - a*b^2*c - a*b*c^2 + 2*b^2*c^2 - c^4) : :
Trilinears    (1 - 2 cos A) (a (1 - 2 cos A) - b (1 - 2 cos B) - c (1 - 2 cos C)) : :

X(40584) is the antipode of X(1211) in the hyperbola described at X(36949). (Randy Hutson, December 18, 2020)

X(40584) lies on these lines: {6, 1718}, {9, 1060}, {81, 226}, {758, 1870}, {1015, 16470}, {1211, 36949}, {1415, 2193}, {1983, 26744}, {2245, 6149}, {2610, 3738}, {3013, 8068}, {3284, 23980}, {5299, 9456}, {5526, 16307}, {6703, 26932}, {7110, 20262}, {13006, 18591}, {16548, 22123}, {34544, 34586}, {35069, 35204}

X(40584) = reflection of X(1211) in X(36949)
X(40584) = complementary conjugate of complement of X(20989)
X(40584) = X(2)-Ceva conjugate of X(36)
X(40584) = perspector of circumconic centered at X(36)

X(40585) = CENTER OF 2ND (X(2),X(38))-CEVA CONIC

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

X(40585) lies on these lines: {2, 7239}, {38, 8041}, {42, 1100}, {321, 1930}, {3112, 4562}, {3661, 17177}, {4876, 33123}, {5949, 29687}, {19584, 32771}, {22013, 40013}, {24484, 33166}, {25440, 32664}, {31161, 35123}

X(40585) = complement of isogonal conjugate of X(20990)
X(40585) = complement of isotomic conjugate of X(17165)
X(40585) = complement of X(19)-isoconjugate of X(22164)
X(40585) = complementary conjugate of complement of X(20990)
X(40585) = X(2)-Ceva conjugate of X(38)
X(40585) = perspector of circumconic centered at X(38)

X(40586) = CENTER OF 2ND (X(2),X(42))-CEVA CONIC

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

X(50586) lies on these lines: {2, 2140}, {9, 3588}, {37, 38}, {42, 213}, {71, 1213}, {101, 4184}, {190, 310}, {220, 1011}, {228, 8012}, {321, 17755}, {430, 2333}, {518, 40463}, {573, 9812}, {649, 4640}, {756, 39258}, {899, 21877}, {902, 2205}, {1001, 36808}, {1018, 4651}, {1386, 14751}, {1400, 39793}, {2183, 16590}, {2225, 3683}, {2238, 21858}, {2328, 32739}, {3006, 22009}, {3159, 17031}, {3161, 10453}, {3207, 19346}, {3219, 6651}, {3230, 23632}, {3501, 26037}, {3736, 38853}, {3995, 17027}, {4024, 22027}, {4115, 22013}, {4210, 24047}, {4253, 29814}, {4370, 31136}, {5513, 30751}, {12514, 15487}, {14752, 21345}, {15830, 38015}, {16552, 17135}, {16593, 30821}, {17149, 17336}, {17208, 27097}, {20966, 21813}, {21879, 21880}

X(40586) = complement of X(8049)
X(40586) = complementary conjugate of complement of X(8053)
X(40586) = X(2)-Ceva conjugate of X(42)
X(40586) = perspector of circumconic centered at X(42)

X(40587) = CENTER OF 2ND (X(2),X(45))-CEVA CONIC

Barycentrics    a*(a - 2*b - 2*c)*(a^2 - b^2 + 4*b*c - c^2) : :
X(40587) = 3 X[1] + X[4900], X[4900] - 3 X[11525], X[4900] + 6 X[64732], X[11525] + 2 X[64732], X[9] - 3 X[9623], 2 X[9] - 3 X[9708], 3 X[5790] - 2 X[64335], 3 X[15934] - 2 X[42871], 3 X[10246] - 2 X[64735], 3 X[3753] - X[39779], 3 X[1056] - 5 X[62778], 5 X[1698] - X[8275], 3 X[3679] + X[16236], 3 X[3679] - X[36922], 3 X[6767] - 4 X[42819], 2 X[42819] - 3 X[54318], 3 X[26446] - 2 X[64315], 3 X[59388] - X[64313]

X(40587) lies on the Feuerbach circumhyperbola of the medial triangle and these lines: {1, 3689}, {2, 1000}, {3, 5836}, {5, 64322}, {8, 442}, {9, 374}, {10, 1482}, {40, 51576}, {45, 4752}, {55, 5426}, {80, 31140}, {100, 2320}, {119, 2886}, {142, 519}, {145, 37462}, {200, 50194}, {210, 25415}, {214, 1376}, {355, 6260}, {392, 51378}, {405, 14923}, {474, 4861}, {514, 996}, {515, 43182}, {518, 1159}, {529, 18541}, {936, 10222}, {940, 49494}, {942, 4853}, {952, 2550}, {956, 3218}, {958, 3647}, {960, 8148}, {993, 51570}, {997, 10247}, {999, 3306}, {1001, 2802}, {1056, 62778}, {1100, 34261}, {1125, 10912}, {1319, 16417}, {1329, 18493}, {1385, 1706}, {1480, 55432}, {1538, 5587}, {1698, 2098}, {2092, 16777}, {2099, 3679}, {2551, 22791}, {2800, 5779}, {3057, 11108}, {3059, 4915}, {3126, 4825}, {3245, 16558}, {3246, 41453}, {3295, 3895}, {3338, 3922}, {3340, 34790}, {3421, 39542}, {3434, 12690}, {3452, 3656}, {3488, 60959}, {3526, 37828}, {3616, 47742}, {3617, 5730}, {3621, 50237}, {3626, 12635}, {3632, 41862}, {3654, 5745}, {3680, 31792}, {3697, 11682}, {3715, 3899}, {3754, 5708}, {3812, 7373}, {3820, 5328}, {3826, 5854}, {3877, 35595}, {3878, 51573}, {3880, 6600}, {3890, 16842}, {3898, 8167}, {3900, 57178}, {3913, 30147}, {3918, 22837}, {3925, 12647}, {3927, 5903}, {4002, 19861}, {4004, 62874}, {4018, 63135}, {4304, 34707}, {4555, 20569}, {4669, 62822}, {4674, 16499}, {4677, 5425}, {4678, 62830}, {4701, 62860}, {4731, 5048}, {5044, 7982}, {5045, 12629}, {5055, 5123}, {5082, 37730}, {5119, 16418}, {5126, 64112}, {5176, 17532}, {5221, 5288}, {5252, 17528}, {5258, 37567}, {5273, 50810}, {5330, 46933}, {5436, 64202}, {5437, 51788}, {5438, 15178}, {5439, 36846}, {5554, 24390}, {5690, 6862}, {5698, 28212}, {5719, 34619}, {5774, 16821}, {5791, 11362}, {5794, 12645}, {5795, 12699}, {5904, 64963}, {5919, 64203}, {6001, 60884}, {6184, 34522}, {6547, 24864}, {6700, 61276}, {6735, 31479}, {6736, 11374}, {6762, 31794}, {6832, 12245}, {6850, 52683}, {6857, 63133}, {6907, 64317}, {6923, 33898}, {6955, 64191}, {6974, 59417}, {7080, 37737}, {7171, 31788}, {7686, 8158}, {7971, 9947}, {7991, 31445}, {8256, 26363}, {8580, 16200}, {8582, 11373}, {8666, 37545}, {9460, 52553}, {9654, 64087}, {9819, 15837}, {9843, 64205}, {9957, 64673}, {10107, 62858}, {10915, 28628}, {11278, 15829}, {11499, 64286}, {12433, 64068}, {12515, 22758}, {12609, 32049}, {12619, 64330}, {12631, 64675}, {12649, 64200}, {12650, 31787}, {12773, 64745}, {13253, 58659}, {14151, 38092}, {14260, 40594}, {15015, 61152}, {15347, 16863}, {15668, 59711}, {16126, 58658}, {16370, 63136}, {16494, 48805}, {16853, 58679}, {17054, 50637}, {17060, 36479}, {17571, 37568}, {17573, 37618}, {17647, 18526}, {18357, 31418}, {18393, 31141}, {18421, 60937}, {18525, 47032}, {19541, 64332}, {19875, 63210}, {20085, 33110}, {21888, 31449}, {24870, 31139}, {24929, 63137}, {25055, 41702}, {25466, 49169}, {25917, 30323}, {26446, 64315}, {26492, 45080}, {26727, 29676}, {28444, 35460}, {30478, 61524}, {30827, 51709}, {31145, 63159}, {31246, 37735}, {31420, 61249}, {31485, 35775}, {34918, 45631}, {35457, 38066}, {37075, 40863}, {37562, 49171}, {37624, 51577}, {37727, 57284}, {37739, 63146}, {38028, 59572}, {41541, 51767}, {44663, 60973}, {48661, 57288}, {48907, 49734}, {49712, 53115}, {50193, 57279}, {50726, 64744}, {50798, 51569}, {53114, 64165}, {53620, 62826}, {56938, 62582}, {58641, 64896}, {59388, 64313}, {61146, 64328}, {61158, 64137}, {63130, 64951}

X(40587) = midpoint of X(i) and X(j) for these {i,j}: {1, 11525}, {8, 11041}, {100, 24297}, {4752, 4792}, {4915, 11529}, {16236, 36922}
X(40587) = reflection of X(i) in X(j) for these {i,j}: {1, 64732}, {3, 64733}, {1000, 64109}, {1482, 64731}, {6767, 54318}, {7966, 1385}, {9708, 9623}, {36867, 14563}, {37727, 64323}, {64322, 5}, {64326, 64318}, {64330, 12619}, {64734, 10}
X(40587) = complement of X(1000)
X(40587) = anticomplement of X(64109)
X(40587) = complement of the isogonal conjugate of X(999)
X(40587) = complement of the isotomic conjugate of X(42697)
X(40587) = medial-isogonal conjugate of X(3820)
X(40587) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3820}, {6, 5316}, {31, 45}, {48, 6350}, {56, 31397}, {106, 51362}, {999, 10}, {2163, 64109}, {3306, 141}, {3753, 3454}, {3872, 1329}, {4054, 21245}, {7113, 36913}, {17079, 17046}, {17519, 34831}, {20925, 626}, {21183, 21252}, {22129, 18589}, {28808, 21244}, {35281, 513}, {40587, 21251}, {42697, 2887}, {52428, 9}, {55432, 3452}, {56426, 3814}
X(40587) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 45}, {1320, 4867}
X(40587) = X(i)-isoconjugate of X(j) for these (i,j): {1000, 2163}, {28607, 65029}, {34446, 39704}
X(40587) = X(i)-Dao conjugate of X(j) for these (i,j): {45, 2}, {36911, 65029}, {40587, 1000}, {51362, 1145}, {52148, 2320}
X(40587) = crosspoint of X(2) and X(42697)
X(40587) = crosssum of X(6) and X(34446)
X(40587) = barycentric product X(i)*X(j) for these {i,j}: {9, 36595}, {45, 42697}, {88, 36919}, {999, 4671}, {1320, 36914}, {2099, 28808}, {2177, 20925}, {3306, 3679}, {3711, 17079}, {3753, 5235}, {3872, 5219}, {4054, 4653}, {4752, 21183}, {4791, 35281}, {4792, 62621}, {27757, 56426}
X(40587) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 1000}, {999, 89}, {3306, 39704}, {3679, 65029}, {3711, 36916}, {3753, 30588}, {3872, 30608}, {3940, 30680}, {4671, 58029}, {4752, 51564}, {35281, 4604}, {36595, 85}, {36919, 4358}, {42697, 20569}, {52428, 2364}, {55432, 2320}, {62621, 63226}
X(40587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3698, 16408}, {1, 4413, 35272}, {2, 1000, 64109}, {2099, 3679, 3940}, {2099, 3711, 4867}, {2099, 3940, 4930}, {3679, 4867, 3711}, {3679, 5219, 51362}, {3679, 16236, 36922}, {3711, 4867, 3940}, {3753, 3872, 999}, {3754, 12513, 5708}, {3918, 22837, 25524}, {4674, 16499, 17595}, {4677, 5425, 41711}, {10914, 19860, 3295}, {37624, 59691, 51577}

X(40588) = CENTER OF 2ND (X(2),X(51))-CEVA CONIC

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

X(40588) is the center of the conic that is the locus of the barycentric product of circumcircle-X(5)-antipodes. (Randy Hutson, December 18, 2020)

X(40588) lies on these lines: {32, 184}, {53, 232}, {206, 22391}, {343, 35319}, {647, 10192}, {1180, 9993}, {2211, 23195}, {3917, 11672}, {5480, 14773}, {6636, 13236}, {34990, 40379}

X(40588) = isogonal conjugate of isotomic conjugate of X(41480)
X(40588) = complement of isogonal conjugate of X(160)
X(40588) = complement of isotomic conjugate of X(2979)
X(40588) = complement of polar conjugate of X(39575)
X(40588) = complementary conjugate of X(34845)
X(40588) = X(2)-Ceva conjugate of X(51)
X(40588) = perspector of circumconic centered at X(51)

X(40589) = CENTER OF 2ND (X(2),X(58))-CEVA CONIC

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

Let (OA) be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(ra2 + s2), where ra is the A-exradius). Define (OB) and (OC) cyclically. Let AB be the intersection, other than B, of line BC and (OB). Define BC and CA cyclically. Let AC be the intersection, other than C, of line BC and (OC). Define BA and CB cyclically. AB, AC, BC, BA, CA, CB lie on a common conic, here named the Conway conic, with center X(40589). (Randy Hutson, December 18, 2020)

X(40589) lies on these lines: {2, 8044}, {3, 34440}, {5, 572}, {6, 2248}, {27, 86}, {36, 2150}, {48, 3868}, {60, 2260}, {71, 110}, {101, 21873}, {141, 7536}, {199, 22133}, {229, 2294}, {284, 501}, {319, 662}, {573, 1147}, {579, 17104}, {849, 16470}, {960, 2360}, {1201, 38858}, {1325, 1953}, {1326, 5301}, {1412, 24471}, {1437, 4269}, {1511, 35069}, {1798, 28266}, {1963, 40214}, {2189, 4303}, {2193, 34586}, {2252, 23059}, {4280, 16519}, {4288, 37836}, {5006, 17053}, {7054, 22054}, {18417, 34544}, {21011, 37158}

X(40589) = complement of X(8044)
X(40589) = complementary conjugate of X(34119)
X(40589) = X(2)-Ceva conjugate of X(58)
X(40589) = perspector of circumconic centered at X(58)

X(40590) = CENTER OF 2ND (X(2),X(65))-CEVA CONIC

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

X(40590) lies on these lines: {2, 2995}, {3, 478}, {6, 41}, {9, 37694}, {12, 37}, {58, 38864}, {65, 2092}, {71, 4559}, {109, 37508}, {201, 21033}, {216, 2182}, {221, 37499}, {226, 1465}, {313, 4552}, {314, 32038}, {573, 10571}, {608, 36744}, {651, 1444}, {800, 2264}, {828, 18591}, {941, 3485}, {946, 14749}, {980, 10401}, {1030, 1950}, {1211, 1214}, {1319, 17053}, {1399, 2305}, {1409, 2245}, {1415, 2193}, {1441, 27042}, {1457, 2269}, {2285, 4261}, {2321, 21859}, {3185, 3192}, {4417, 17080}, {4551, 21061}, {5257, 5930}, {5433, 28244}, {5723, 28366}, {16578, 21244}, {16584, 21796}, {17321, 37800}, {20623, 38977}, {32431, 38945}, {34042, 40152}, {34528, 35069}, {34586, 37620}

X(40590) = isogonal conjugate of X(19607)
X(40590) = complement of X(2995)
X(40590) = complementary conjugate of complement of X(3185)
X(40590) = crosssum of X(6) and X(2217)
X(40590) = X(2)-Ceva conjugate of X(65)
X(40590) = perspector of circumconic centered at X(65)
X(40590) = trilinear product X(i)*X(j) for these {i,j}: {37, 10571}, {42, 17080}, {65, 573}, {226, 3185}, {1214, 3192}, {1400, 3869}, {1402, 4417}, {1409, 17555}, {2171, 4225}, {4551, 6589}

X(40591) = CENTER OF 2ND (X(2),X(71))-CEVA CONIC

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

X(40591) lies on these lines: {12, 42}, {58, 32656}, {72, 17102}, {73, 228}, {201, 2658}, {354, 1193}, {386, 2140}, {2198, 8776}, {3159, 4064}, {3191, 3192}, {3682, 18643}, {3811, 23050}, {3990, 22063}, {20970, 21796}, {22072, 22400}

X(40591) = complement of isogonal conjugate of X(23383)
X(40591) = complement of isotomic conjugate of X(17220)
X(40591) = complementary conjugate of complement of X(23383)
X(40591) = X(2)-Ceva conjugate of X(71)
X(40591) = perspector of circumconic centered at X(71)

X(40592) = CENTER OF 2ND (X(2),X(81))-CEVA CONIC

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

X(40592) lies on these lines: {2, 1029}, {3, 32782}, {9, 19620}, {10, 37294}, {21, 36}, {38, 1326}, {39, 1931}, {56, 37032}, {60, 3916}, {63, 37783}, {81, 593}, {86, 26842}, {99, 321}, {100, 8935}, {110, 4640}, {114, 4220}, {191, 501}, {261, 4359}, {333, 19302}, {553, 1014}, {641, 16441}, {642, 16440}, {662, 3219}, {958, 37405}, {993, 1325}, {1030, 2895}, {1214, 4565}, {1386, 33774}, {1649, 3733}, {1790, 34544}, {1817, 5235}, {2352, 5867}, {2482, 31143}, {2886, 5196}, {2975, 15349}, {3434, 35915}, {3616, 37029}, {4184, 8299}, {4560, 5664}, {4972, 35916}, {4999, 37369}, {5260, 35991}, {5739, 6337}, {5976, 26243}, {6292, 21495}, {6505, 35193}, {6509, 37659}, {6626, 19308}, {6763, 15792}, {7279, 26942}, {7354, 37152}, {8290, 31089}, {11102, 16752}, {11104, 24552}, {11165, 16436}, {15819, 19649}, {16948, 37599}, {17103, 19684}, {17147, 19623}, {17512, 19785}, {21566, 33364}, {21567, 33365}, {24617, 26724}, {27958, 32933}, {34834, 35069}

X(40592) = isogonal conjugate of X(21353)
X(40592) = complement of X(1029)
X(40592) = complementary conjugate of complement of X(1030)
X(40592) = X(2)-Ceva conjugate of X(81)
X(40592) = perspector of circumconic centered at X(81)
X(40592) = X{i}-isoconjugate of X(j) for these {i,j}: {1, 21353}, {6, 502}, {10, 3444}, {37, 267}, {42, 1029}

X(40593) = CENTER OF 2ND (X(2),X(85))-CEVA CONIC

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

X(40593) lies on these lines: {7, 34019}, {9, 4569}, {75, 4081}, {85, 142}, {87, 7209}, {348, 2275}, {658, 30988}, {927, 24309}, {982, 3663}, {3729, 4554}, {4572, 17786}, {4859, 34018}, {6063, 24199}, {17073, 17095}, {20195, 31618}, {20206, 34863}, {20935, 31526}, {21348, 24002}, {30854, 39063}

X(40593) = isotomic conjugate of X(2)-cross conjugate of X(9)
X(40593) = complement of isogonal conjugate of X(20995)
X(40593) = complement of isotomic conjugate of X(3177)
X(40593) = complement of X(19)-isoconjugate of X(20793)
X(40593) = complementary conjugate of complement of X(20995)
X(40593) = X(2)-Ceva conjugate of X(85)
X(40593) = perspector of circumconic centered at X(85)

X(40594) = CENTER OF 2ND (X(2),X(88))-CEVA CONIC

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

X(40594) lies on these lines: {2, 8046}, {9, 19618}, {44, 88}, {80, 519}, {903, 17484}, {3306, 40215}, {3689, 14190}, {3911, 36592}, {3936, 4997}, {4358, 4555}, {4792, 21805}, {5541, 39148}, {6631, 30566}, {31171, 35121}

X(40594) = complement of X(8046)
X(40594) = complementary conjugate of complement of X(3196)
X(40594) = X(2)-Ceva conjugate of X(88)
X(40594) = perspector of circumconic centered at X(88)
X(40594) = center of conic {{A,B,C,PU(50)}}

X(40595) = CENTER OF 2ND (X(2),X(106))-CEVA CONIC

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

X(40595) lies on these lines: {44, 517}, {101, 35129}, {106, 5053}, {320, 908}, {901, 2183}, {1320, 21801}, {2245, 17969}, {2265, 5375}, {5548, 22356}, {8752, 8756}, {9326, 40215}

X(40595) = complement of isogonal conjugate of X(23858)
X(40595) = complement of isotomic conjugate of X(21290)
X(40595) = complement of X(19)-isoconjugate of X(23135)
X(40595) = complementary conjugate of complement of X(23858)
X(40595) = X(2)-Ceva conjugate of X(106)
X(40595) = perspector of circumconic centered at X(106)

X(40596) = CENTER OF 2ND (X(2),X(112))-CEVA CONIC

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

X(40596) lies on these lines: {2, 13573}, {24, 3447}, {110, 8057}, {112, 2485}, {186, 12096}, {232, 15262}, {250, 4558}, {403, 1503}, {523, 32713}, {924, 32715}, {935, 39417}, {1301, 1304}, {1974, 36191}, {2409, 16237}, {9934, 18809}

X(40596) = complement of X(13573)
X(40596) = complementary conjugate of X(23315)
X(40596) = X(2)-Ceva conjugate of X(112)
X(40596) = perspector of circumconic centered at X(112)

X(40597) = CENTER OF 2ND (X(2),X(171))-CEVA CONIC

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

X(40597) lies on these lines: {2, 7224}, {3, 8866}, {6, 982}, {83, 226}, {171, 1691}, {238, 9285}, {385, 39928}, {419, 1215}, {1915, 9284}, {1920, 19574}, {2174, 10026}, {2344, 26098}, {2887, 19557}, {3496, 23150}, {3684, 32861}, {3955, 36213}, {4426, 14823}, {4586, 7018}, {5247, 23447}, {7234, 23865}, {8290, 8857}, {20995, 23143}, {25760, 32664}, {27967, 27976}

X(40597) = complement of X(7224)
X(40597) = complementary conjugate of complement of X(23868)
X(40597) = X(2)-Ceva conjugate of X(171)
X(40597) = perspector of circumconic centered at X(171)

X(40598) = CENTER OF 2ND (X(2),X(192))-CEVA CONIC

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

X(40598) lies on these lines: {2, 17448}, {8, 17793}, {10, 3662}, {76, 4740}, {120, 3314}, {192, 4110}, {330, 668}, {1211, 23897}, {1278, 20943}, {1575, 21219}, {3177, 4462}, {3452, 3661}, {3617, 3789}, {3679, 31276}, {3730, 20979}, {4661, 22293}, {5233, 16594}, {6554, 17280}, {7885, 31141}, {7904, 34606}, {14434, 25142}, {16589, 26772}, {17294, 30863}, {17349, 27430}, {17350, 24343}, {18140, 32095}, {25120, 25311}, {25277, 25625}, {29572, 37663}, {30713, 31060}

X(40598) = complement of X(38247)
X(40598) = complementary conjugate of complement of X(16969)
X(40598) = X(2)-Ceva conjugate of X(192)
X(40598) = perspector of circumconic centered at X(192)

X(40599) = CENTER OF 2ND (X(2),X(210))-CEVA CONIC

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

X(40599) lies on these lines: {37, 3914}, {41, 55}, {142, 14746}, {209, 39258}, {210, 21795}, {226, 35310}, {345, 28797}, {354, 6184}, {1214, 3991}, {1500, 16584}, {2276, 3720}, {2321, 3693}, {3666, 24175}, {3689, 16588}, {3700, 3971}, {3744, 3997}, {4046, 4515}, {5452, 6600}, {9049, 36808}

X(40599) = complement of isogonal conjugate of X(3941)
X(40599) = complement of isotomic conjugate of X(3873)
X(40599) = complementary conjugate of complement of X(3941)
X(40599) = X(2)-Ceva conjugate of X(210)
X(40599) = perspector of circumconic centered at X(210)

X(40600) = CENTER OF 2ND (X(2),X(213))-CEVA CONIC

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

X(40600) lies on these lines: {1, 5132}, {3, 35628}, {9, 3185}, {10, 13731}, {42, 181}, {55, 14749}, {71, 6184}, {100, 314}, {142, 16056}, {171, 18724}, {214, 37620}, {442, 4026}, {572, 16872}, {573, 22301}, {1045, 5143}, {1376, 10472}, {1918, 1964}, {2175, 33718}, {4097, 12640}, {4191, 10473}, {4210, 35614}, {4557, 21061}, {6600, 23853}, {10470, 23361}, {10477, 11517}, {16574, 16678}, {22286, 35552}, {22299, 23846}

X(40600) = complement of isogonal conjugate of X(16678)
X(40600) = complement of isotomic conjugate of X(17137)
X(40600) = complement of polar conjugate of X(17913)
X(40600) = complementary conjugate of complement of X(16678)
X(40600) = complementary conjugate of nine-point-circle pole of antiorthic axis
X(40600) = X(2)-Ceva conjugate of X(213)
X(40600) = perspector of circumconic centered at X(213)

X(40601) = CENTER OF 2ND (X(2),X(237))-CEVA CONIC

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

X(40601) lies on these lines: {2, 25053}, {6, 3613}, {51, 1196}, {125, 20965}, {216, 9475}, {237, 2211}, {648, 18024}, {1249, 37190}, {1625, 25046}, {2967, 7467}, {3569, 36213}, {5007, 38997}, {8623, 38987}, {14957, 14965}, {37891, 39931}

X(40601) = complement of isotomic conjugate of X(14957)
X(40601) = X(2)-Ceva conjugate of X(237)
X(40601) = perspector of circumconic centered at X(237)

X(40602) = CENTER OF 2ND (X(2),X(284))-CEVA CONIC

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

X(40602) lies on these lines: {5, 580}, {6, 2653}, {21, 7004}, {29, 270}, {58, 942}, {60, 14547}, {73, 110}, {141, 7515}, {162, 1935}, {212, 3876}, {238, 1780}, {411, 23692}, {501, 1511}, {581, 1147}, {960, 2328}, {1064, 38850}, {1451, 37791}, {2360, 37836}, {11107, 24430}, {13739, 37591}

X(40602) = complement of isogonal conjugate of X(3145)
X(40602) = complement of isotomic conjugate of X(2893)
X(40602) = complement of polar conjugate of X(18679)
X(40602) = complementary conjugate of complement of X(3145)
X(40602) = X(2)-Ceva conjugate of X(284)
X(40602) = perspector of circumconic centered at X(284)

X(40603) = CENTER OF 2ND (X(2),X(321))-CEVA CONIC

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

X(40603) lies on these lines: {2, 18040}, {10, 38}, {37, 27041}, {42, 17793}, {63, 29511}, {75, 30603}, {81, 668}, {306, 3452}, {313, 321}, {2895, 17790}, {3219, 29508}, {3264, 17184}, {3596, 32782}, {3765, 26035}, {3780, 25298}, {3789, 4651}, {3936, 22020}, {3948, 3969}, {3975, 33157}, {3995, 4033}, {4036, 14434}, {6376, 28606}, {6554, 17776}, {10371, 17751}, {16589, 21827}, {16594, 37662}, {17147, 18133}, {17495, 18136}, {17757, 21530}, {18147, 20017}, {18601, 27102}, {19804, 28651}, {19810, 25280}, {21443, 40563}, {26563, 40071}, {27792, 31025}, {27793, 31993}, {30710, 31247}

X(40603) = complement of X(35058)
X(40603) = complementary conjugate of complement of X(16685)
X(40603) = X(2)-Ceva conjugate of X(321)
X(40603) = perspector of circumconic centered at X(321)

X(40604) = CENTER OF 2ND (X(2),X(323))-CEVA CONIC

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

X(40604) lies on these lines: {2, 13582}, {3, 2888}, {23, 114}, {39, 2981}, {50, 323}, {94, 99}, {97, 6509}, {618, 6105}, {619, 6104}, {1125, 4996}, {1994, 4558}, {2482, 35296}, {3518, 14111}, {3628, 10276}, {5664, 24978}, {6337, 37644}, {7492, 7710}, {7496, 15819}, {10272, 14354}, {11063, 37779}, {15850, 16042}, {16023, 22892}, {16024, 22848}, {34545, 34990}

X(40604) = isogonal conjugate of X(11071)
X(40604) = complement of X(13582)
X(40604) = complementary conjugate of complement of X(11063)
X(40604) = X(2)-Ceva conjugate of X(323)
X(40604) = perspector of circumconic centered at X(323)
X(40604) = trilinear product X(i)*X(j) for these {i,j}: {63,2914}, {323,1749}, {662,8562}

X(40605) = CENTER OF 2ND (X(2),X(333))-CEVA CONIC

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

X(40605) lies on these lines: {1, 15349}, {2, 5110}, {3, 18134}, {21, 21321}, {86, 3666}, {99, 226}, {114, 7413}, {171, 643}, {261, 284}, {314, 20882}, {345, 27958}, {662, 33116}, {811, 1947}, {1010, 1125}, {1043, 2646}, {1211, 6626}, {1326, 29671}, {1649, 7253}, {1944, 7106}, {2305, 17778}, {2887, 35916}, {3736, 17477}, {3752, 25536}, {5333, 17302}, {5712, 6337}, {5976, 39915}, {8299, 13588}, {11104, 26098}, {15604, 17770}, {16050, 40432}, {17190, 27757}, {18155, 27929}, {18755, 27319}, {19270, 24931}, {19803, 25523}, {24378, 24789}, {27398, 27399}, {33113, 40214}

X(40605) = complement of isogonal conjugate of X(2305)
X(40605) = complement of isotomic conjugate of X(17778)
X(40605) = complement of polar conjugate of X(3144)
X(40605) = complementary conjugate of complement of X(2305)
X(40605) = X(2)-Ceva conjugate of X(333)
X(40605) = perspector of circumconic centered at X(333)

X(40606) = CENTER OF 2ND (X(2),X(354))-CEVA CONIC

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

X(40606) lies on these lines: {3, 5452}, {6, 31}, {11, 21856}, {210, 6184}, {226, 241}, {354, 21795}, {650, 10164}, {651, 40443}, {1155, 16588}, {1211, 25066}, {1212, 1855}, {1214, 39063}, {2887, 24036}, {3748, 23653}, {4640, 23988}, {4847, 35310}, {5542, 14746}, {5718, 25074}, {8012, 22053}, {14827, 34879}, {18134, 25082}, {20331, 21954}, {23636, 39258}, {25075, 37663}, {26690, 32773}

X(40606) = complement of isogonal conjugate of X(15624)
X(40606) = complement of isotomic conjugate of X(3681)
X(40606) = complement of polar conjugate of X(17916)
X(40606) = complementary conjugate of complement of X(15624)
X(40606) = X(2)-Ceva conjugate of X(354)
X(40606) = perspector of circumconic centered at X(354)

X(40607) = CENTER OF 2ND (X(1),X(181))-CEVA CONIC

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

The 2nd (X(1),X(181))-Ceva conic is also the 2nd (X(2),X(1500))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40607) lies on these lines: {2, 13476}, {9, 4557}, {10, 15281}, {37, 42}, {44, 20964}, {72, 16828}, {75, 3952}, {141, 25323}, {513, 17332}, {518, 1125}, {536, 4096}, {594, 4092}, {740, 4015}, {984, 3216}, {1089, 3696}, {1211, 21249}, {1213, 20683}, {1215, 3739}, {1654, 4553}, {2664, 16696}, {3681, 4687}, {3688, 17330}, {3715, 34247}, {3789, 17279}, {3799, 32025}, {3943, 4111}, {3948, 22289}, {3956, 4732}, {3971, 22316}, {3986, 22312}, {4043, 4651}, {4104, 18589}, {4517, 17275}, {4662, 28581}, {4735, 21892}, {5257, 22277}, {10176, 34587}, {14992, 21830}, {15569, 34790}, {16552, 20990}, {16589, 22292}, {17328, 25279}, {20715, 21873}, {21068, 22276}, {21699, 21803}, {21879, 21897}, {39735, 40216}

X(40607) = complement of X(13476)
X(40607) = complementary conjugate of X(3925)
X(40607) = X(2)-Ceva conjugate of X(1500)
X(40607) = perspector of circumconic centered at X(1500)

X(40608) = CENTER OF 2ND (X(1),X(512))-CEVA CONIC

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

The 2nd (X(1),X(512))-Ceva conic is also the 2nd (X(2),X(3709))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40608) lies on these lines: {2, 3903}, {8, 7257}, {10, 4531}, {11, 7063}, {115, 512}, {1111, 4761}, {1146, 3271}, {1966, 32850}, {2170, 4041}, {2642, 2643}, {3023, 3907}, {3056, 23902}, {3753, 4085}, {4111, 4711}, {4128, 16592}, {4433, 19589}, {4705, 20982}, {4807, 17761}, {5976, 14839}, {6741, 18191}, {7234, 22373}, {15864, 37568}, {20359, 23922}

X(40608) = complement of X(3903)
X(40608) = complementary conjugate of X(21051)
X(40608) = excentral-to-ABC barycentric image of X(99)
X(40608) = X(99)com(extouch triangle)
X(40608) = X(2)-Ceva conjugate of X(3709)
X(40608) = perspector of circumconic centered at X(3709)

X(40609) = CENTER OF 2ND (X(1),X(518))-CEVA CONIC

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

The 2nd (X(1),X(518))-Ceva conic is also the 2nd (X(2),X(3693))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40609) lies on these lines: {1, 40534}, {2, 1280}, {8, 220}, {10, 4904}, {11, 210}, {120, 518}, {200, 1040}, {341, 27390}, {668, 34018}, {918, 2254}, {1145, 3887}, {1211, 8286}, {1639, 38376}, {2348, 3021}, {3243, 30813}, {3693, 3717}, {3870, 4952}, {4046, 6741}, {4152, 6745}, {4383, 36845}, {4422, 4578}, {4543, 14430}, {4738, 24014}, {4899, 9436}, {4953, 25268}, {5845, 20344}, {8580, 33169}, {9451, 26007}, {10025, 32850}, {10580, 25531}, {11019, 24003}, {16589, 16613}, {16594, 26015}, {21530, 34790}

X(40609) = complement of X(1280)
X(40609) = complementary conjugate of X(3823)
X(40609) = X(2)-Ceva conjugate of X(3693)
X(40609) = perspector of circumconic centered at X(3693)

X(40610) = CENTER OF 2ND (X(75),X(513))-CEVA CONIC

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

The 2nd (X(75),X(513))-Ceva conic is also the 2nd (X(2),X(4083))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40610) lies on the Steiner inellipse and these lines: {2, 18830}, {37, 20532}, {39, 14823}, {115, 5518}, {190, 20671}, {192, 23643}, {244, 22227}, {256, 7168}, {1015, 1960}, {1146, 39786}, {1500, 4033}, {2092, 35068}, {3123, 6377}, {3124, 40525}, {3666, 35070}, {4364, 21250}, {6184, 21796}, {9294, 35119}, {17321, 27289}, {20979, 21762}, {21830, 35126}

X(40610) = complement of X(18830)
X(40610) = complementary conjugate of complement of X(8640)
X(40610) = crosspoint of X(2) and X(4083)
X(40610) = crosssum of X(6) and X(932)
X(40610) = barycentric square of X(4083)
X(40610) = X(2)-Ceva conjugate of X(4083)
X(40610) = perspector of circumparabola centered at X(4083)

X(40611) = CENTER OF 2ND (X(10),X(65))-CEVA CONIC

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

The 2nd (X(10),X(65))-Ceva conic is also the 2nd (X(2),X(1400))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40611) lies on these lines: {1, 9551}, {2, 10571}, {12, 73}, {42, 10474}, {56, 34281}, {65, 1193}, {109, 4225}, {213, 1042}, {221, 13738}, {664, 28660}, {959, 39797}, {1064, 6831}, {1201, 1402}, {1212, 30456}, {1214, 12089}, {1458, 28350}, {1880, 17442}, {4296, 39035}, {4300, 13734}, {4551, 17751}, {6505, 21147}, {14529, 36033}, {19513, 34586}, {24806, 31339}

X(40611) = complement of isogonal conjugate of X(23361)
X(40611) = complement of isotomic conjugate of X(20245)
X(40611) = complement of X(19)-isoconjugate of X(23131)
X(40611) = complementary conjugate of complement of X(23361)
X(40611) = X(2)-Ceva conjugate of X(1400)
X(40611) = perspector of circumconic centered at X(1400)

X(40612) = CENTER OF 2ND (X(10),X(320))-CEVA CONIC

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

The 2nd (X(10),X(320))-Ceva conic is also the 2nd (X(2),X(3218))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40612) lies on these lines: {1, 149}, {2, 21739}, {81, 88}, {223, 27131}, {226, 13582}, {323, 1443}, {651, 1214}, {664, 18359}, {1086, 17011}, {1212, 35595}, {1442, 2006}, {2610, 3960}, {3160, 31018}, {3580, 18644}, {5249, 24145}, {5483, 17021}, {6505, 18625}, {11078, 36933}, {11092, 36932}, {11126, 37773}, {11127, 37772}, {14918, 17923}, {16578, 16585}, {26611, 35110}, {27186, 37771}, {30144, 30991}

X(40612) = isogonal conjugate of X(11075)
X(40612) = complement of X(21739)
X(40612) = complementary conjugate of complement of X(19297)
X(40612) = X(2)-Ceva conjugate of X(3218)
X(40612) = perspector of circumconic centered at X(3218)
X(40612) = trilinear product X(i)*X(j) for these {i,j}: {2,6126}, {484,3218}

X(40613) = CENTER OF 2ND (X(10),X(517))-CEVA CONIC

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

The 2nd (X(10),X(517))-Ceva conic is also the 2nd (X(2),X(2183))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40613) lies on these lines: {1, 26095}, {11, 1193}, {37, 22063}, {65, 244}, {73, 38985}, {392, 17102}, {995, 4000}, {997, 23050}, {1015, 2260}, {1104, 8054}, {1149, 34590}, {1191, 23404}, {1361, 1457}, {1459, 11700}, {1769, 14299}, {2646, 38983}, {8299, 14414}, {17757, 22350}, {23757, 34587}

X(40613) = X(2)-Ceva conjugate of X(2183)
X(40613) = perspector of circumconic centered at X(2183)

X(40614) = CENTER OF 2ND (X(10),X(536))-CEVA CONIC

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

The 2nd (X(10),X(536))-Ceva conic is also the 2nd (X(2),X(899))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40614) lies on these lines: {2, 1018}, {9, 30942}, {11, 1213}, {37, 244}, {42, 38986}, {190, 31002}, {214, 5029}, {649, 4432}, {672, 4370}, {836, 38985}, {899, 3230}, {1015, 3720}, {1100, 8054}, {1635, 8299}, {2238, 38979}, {3161, 30947}, {3218, 6651}, {3768, 4465}, {4094, 38978}, {4358, 17755}, {5163, 9283}, {6377, 14752}, {6544, 21832}, {17441, 34591}, {17754, 36911}, {21580, 30964}, {21894, 39046}, {27481, 31035}

X(40614) = complement of isotomic conjugate of X(29824)
X(40614) = X(2)-Ceva conjugate of X(899)
X(40614) = perspector of circumconic centered at X(899)

X(40615) = CENTER OF 2ND (X(514),X(7))-CEVA CONIC

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

The 2nd (X(514),X(7))-Ceva conic is also the 2nd (X(2),X(3676))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40615) lies on these lines: {2, 37206}, {7, 100}, {11, 1111}, {57, 26007}, {226, 16593}, {1086, 14936}, {1357, 3323}, {1617, 17093}, {3119, 26932}, {3665, 5219}, {3676, 3756}, {3699, 35160}, {4106, 38384}, {4468, 5519}, {4904, 38375}, {5173, 36905}, {5435, 31226}, {17107, 20269}, {17272, 19604}, {17718, 24796}, {20343, 36482}, {21208, 38374}

X(40615) = complement of X(37206)
X(40615) = X(2)-Ceva conjugate of X(3676)
X(40615) = perspector of circumconic centered at X(3676)

X(40616) = CENTER OF 2ND (X(514),X(20))-CEVA CONIC

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

The 2nd (X(514),X(20))-Ceva conic is also the 2nd (X(2),X(21172))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40616) lies on these lines: {2, 36118}, {3, 6554}, {101, 38554}, {123, 5514}, {189, 1073}, {268, 281}, {441, 37774}, {1146, 2968}, {1565, 3942}, {2822, 3184}, {4534, 35014}, {13609, 13611}, {15905, 27382}

X(40616) = complement of X(36118)
X(40616) = X(2)-Ceva conjugate of X(21172)
X(40616) = perspector of circumconic centered at X(21172)

X(40617) = CENTER OF 2ND (X(514),X(57))-CEVA CONIC

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

The 2nd (X(514),X(57))-Ceva conic is also the 2nd (X(2),X(3669))-Ceva conic.

X(40617) lies on these lines: {2, 27825}, {7, 190}, {11, 1357}, {12, 2885}, {56, 1633}, {57, 21362}, {65, 10427}, {226, 16594}, {553, 36913}, {1086, 1358}, {1122, 14524}, {1477, 3021}, {2976, 3756}, {3649, 16597}, {3937, 38351}, {4675, 17107}, {4859, 19604}, {5435, 31227}, {6173, 24796}, {6557, 8051}, {8287, 10933}, {16603, 20343}, {21454, 30577}

X(40617) = complement of X(27834)
X(40617) = complementary conjugate of X(4816)
X(40617) = X(2)-Ceva conjugate of X(3669)
X(40617) = perspector of circumconic centered at X(3669)

X(40618) = CENTER OF 2ND (X(514),X(69))-CEVA CONIC

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

The 2nd (X(514),X(69))-Ceva conic is also the 2nd (X(2),X(4025))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40618) lies on these lines: {2, 1815}, {69, 1331}, {77, 4551}, {116, 17198}, {124, 693}, {125, 1565}, {141, 23988}, {1111, 3120}, {1897, 18025}, {2968, 4025}, {5181, 18650}, {17170, 29579}, {17219, 17421}, {25006, 36905}

X(40618) = isotomic conjugate of isogonal conjugate of X(22084)
X(40618) = isotomic conjugate of polar conjugate of X(116)
X(40618) = isotomic conjugate of X(63)-isoconjugate of X(20974)
X(40618) = complement of isogonal conjugate of X(6586)
X(40618) = complement of isotomic conjugate of X(25259)
X(40618) = complement of trilinear pole of line X(3)X(142)
X(40618) = complementary conjugate of complement of X(6586)
X(40618) = X(2)-Ceva conjugate of X(4025)
X(40618) = perspector of circumconic centered at X(4025)

X(40619) = CENTER OF 2ND (X(514),X(75))-CEVA CONIC

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

The 2nd (X(514),X(75))-Ceva conic is also the 2nd (X(2),X(693))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40619) lies on these lines: {2, 40216}, {11, 693}, {75, 3952}, {76, 33089}, {85, 35312}, {100, 2481}, {244, 1111}, {321, 20433}, {339, 2968}, {650, 27009}, {1086, 3124}, {1211, 40563}, {2973, 5521}, {3119, 4858}, {3121, 14296}, {3673, 4850}, {4358, 20435}, {4359, 17755}, {4554, 31272}, {4957, 20906}, {5057, 10030}, {5701, 27190}, {13476, 39735}, {14936, 31150}, {16586, 17866}, {17165, 18142}, {17494, 26846}, {18152, 40094}, {20295, 38390}, {20448, 29824}, {20880, 24589}, {21404, 30566}, {23822, 38995}, {25009, 26565}, {27072, 35310}

X(40619) = complement of isogonal conjugate of X(21007)
X(40619) = complement of isotomic conjugate of X(17494)
X(40619) = complement of trilinear pole of line X(10)X(141)
X(40619) = complement of X(19)-isoconjugate of X(22160)
X(40619) = complementary conjugate of complement of X(21007)
X(40619) = X(2)-Ceva conjugate of X(693)
X(40619) = perspector of circumconic centered at X(693)

X(40620) = CENTER OF 2ND (X(514),X(86))-CEVA CONIC

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

The 2nd (X(514),X(86))-Ceva conic is also the 2nd (X(2),X(7192))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40620) lies on these lines: {86, 4427}, {244, 7192}, {274, 27812}, {2669, 17145}, {3120, 17198}, {3121, 16726}, {3952, 18827}, {8025, 17199}, {16700, 39734}, {16887, 27081}, {17169, 37635}, {17175, 29578}

X(40620) = complement of isotomic conjugate of X(31290)
X(40620) = X(2)-Ceva conjugate of X(7192)
X(40620) = perspector of circumconic centered at X(7192)

X(40621) = CENTER OF 2ND (X(514),X(145))-CEVA CONIC

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

The 2nd (X(514),X(145))-Ceva conic is also the 2nd (X(2),X(3667))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40621) lies on the Steiner inellipse and these lines: {1, 3039}, {5, 3244}, {6, 644}, {44, 13540}, {115, 2789}, {145, 30720}, {230, 35085}, {346, 24150}, {1086, 1358}, {1107, 35095}, {1108, 23980}, {1146, 2087}, {2885, 3815}, {3554, 23986}, {3756, 4534}, {4521, 5516}, {6184, 12640}, {7735, 23972}, {8609, 35129}, {13466, 29600}, {14759, 16593}, {15637, 31182}, {15993, 35117}, {24918, 35110}

X(40621) = complement of isogonal conjugate of X(8643)
X(40621) = complement of isotomic conjugate of X(3667)
X(40621) = complementary conjugate of complement of X(8643)
X(40621) = crosspoint of X(2) and X(3667)
X(40621) = crosssum of X(6) and X(1293)
X(40621) = barycentric square of X(3667)
X(40621) = X(2)-Ceva conjugate of X(3667)
X(40621) = perspector of circumparabola centered at X(3667)

X(40622) = CENTER OF 2ND (X(514),X(226))-CEVA CONIC

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

The 2nd (X(514),X(226))-Ceva conic is also the 2nd (X(2),X(7178))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40622) lies on these lines: {7, 662}, {11, 1365}, {226, 22003}, {1086, 7117}, {1358, 1367}, {3649, 10427}, {4466, 8287}, {7178, 17058}, {11375, 16597}, {16593, 21617}, {16888, 27691}, {20662, 39063}

X(40622) = X(2)-Ceva conjugate of X(7178)
X(40622) = perspector of circumconic centered at X(7178)

X(40623) = CENTER OF 2ND (X(514),X(238))-CEVA CONIC

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

The 2nd (X(514),X(238))-Ceva conic is also the 2nd (X(2),X(659))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40623) lies on these lines: {11, 31003}, {244, 649}, {661, 8054}, {663, 38986}, {1015, 4367}, {1279, 1575}, {2238, 4974}, {2295, 12264}, {3230, 17475}, {4366, 17495}, {4375, 27918}, {4583, 35172}, {32922, 33854}

X(40623) = complement of isogonal conjugate of X(21003)
X(40623) = complement of isotomic conjugate of anticomplement of X(659)
X(40623) = complement of X(19)-isoconjugate of X(22155)
X(40623) = complementary conjugate of complement of X(21003)
X(40623) = X(2)-Ceva conjugate of X(659)
X(40623) = perspector of circumconic centered at X(659)

X(40624) = CENTER OF 2ND (X(514),X(312))-CEVA CONIC

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

The 2nd (X(514),X(312))-Ceva conic is also the 2nd (X(2),X(4391))-Ceva conic and the isotomic conjugate of line X(1)X(3). (Randy Hutson, December 18, 2020)

X(40624) lies on these lines: {2, 56188}, {8, 4553}, {75, 4552}, {116, 40626}, {244, 17888}, {312, 25268}, {321, 20879}, {338, 1086}, {651, 18816}, {2170, 3904}, {2517, 21252}, {2968, 2972}, {4359, 16586}, {4391, 23978}, {14920, 29833}, {17023, 18690}, {17755, 20891}, {17790, 28813}, {20892, 20895}, {21422, 52882}, {24220, 54121}, {24224, 40619}

X(40624) = complement of X(56188)
X(40624) = complement of the isotomic conjugate of X(17496)
X(40624) = isotomic conjugate of the isogonal conjugate of X(11998)
X(40624) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 4391}, {572, 513}, {604, 21189}, {649, 12}, {667, 37662}, {1919, 21796}, {2975, 3835}, {11998, 124}, {14829, 21260}, {17074, 17072}, {17496, 2887}, {20986, 514}, {21061, 31946}, {21173, 141}, {22118, 20315}, {23187, 18589}, {24237, 21252}, {38344, 123}, {51662, 17052}, {52139, 4129}, {53566, 21253}, {57091, 21244}, {57125, 21246}, {57129, 1193}, {57244, 626}
X(40624) = X(2)-Ceva conjugate of X(4391)
X(40624) = X(i)-isoconjugate of X(j) for these (i,j): {1415, 56194}, {2149, 34434}
X(40624) = X(i)-Dao conjugate of X(j) for these (i,j): {650, 34434}, {1146, 56194}, {1577, 2051}, {4391, 2}, {21189, 573}, {34589, 4559}, {40624, 56188}
X(40624) = barycentric product X(i)*X(j) for these {i,j}: {75, 34589}, {76, 11998}, {312, 24237}, {314, 53566}, {522, 57244}, {693, 57091}, {850, 57125}, {1969, 38344}, {2975, 34387}, {4391, 17496}, {4858, 14829}, {11109, 17880}, {17074, 23978}, {21173, 35519}
X(40624) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 34434}, {522, 56194}, {572, 2149}, {2975, 59}, {4391, 56188}, {4858, 2051}, {11109, 7012}, {11998, 6}, {14829, 4564}, {17074, 1262}, {17197, 53083}, {17496, 651}, {18191, 52150}, {21173, 109}, {23187, 36059}, {24237, 57}, {34387, 54121}, {34589, 1}, {35519, 56252}, {38344, 48}, {43728, 53702}, {51662, 53321}, {52322, 35307}, {53566, 65}, {57091, 100}, {57125, 110}, {57244, 664}
X(40624) = {X(23978),X(26932)}-harmonic conjugate of X(4391)
X(40624) = pole of line {40467, 50518} with respect to the Feuerbach circumhyperbola
X(40624) = pole of line {4391, 21189} with respect to {{A,B,C,X(2),X(7)}}
X(40624) = pole of line {4391, 55187} with respect to the {{A,B,C,X(1),X(2)}}
X(40624) = pole of line {24237, 34589} with respect to the Steiner inellipse
X(40624) = pole of line {37, 1953} with respect to the Mandart parabola

X(40625) = CENTER OF 2ND (X(514),X(333))-CEVA CONIC

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

The 2nd (X(514),X(333))-Ceva conic is also the 2nd (X(2),X(4560))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40625) lies on these lines: {21, 4436}, {1654, 17183}, {3120, 23821}, {4552, 14616}, {4560, 4858}, {6740, 25536}, {7192, 24237}, {7200, 16726}, {17182, 31037}, {17185, 27065}, {17197, 21044}, {25268, 28828}

X(40625) = complement of isotomic conjugate of anticomplement of X(4560)
X(40625) = complement of trilinear pole of line X(758)X(942)
X(40625) = X(2)-Ceva conjugate of X(4560)
X(40625) = perspector of circumconic centered at X(4560)

X(40626) = CENTER OF 2ND (X(514),X(345))-CEVA CONIC

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

The 2nd (X(514),X(345))-Ceva conic is also the 2nd (X(2),X(6332))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40626) lies on these lines: {63, 5741}, {69, 1813}, {125, 2968}, {141, 23585}, {244, 24031}, {653, 34393}, {3687, 16586}, {4384, 25000}, {4466, 17880}, {4858, 21044}, {5181, 37836}, {6332, 16596}, {6377, 6388}, {7117, 16731}, {27108, 27509}

X(40626) = isotomic conjugate of polar conjugate of X(124)
X(40626) = isotomic conjugate of X(2)-cross conjugate of X(653)
X(40626) = isotomic conjugate of trilinear pole of line X(109)X(23987)
X(40626) = complement of isogonal conjugate of X(6589)
X(40626) = complement of isotomic conjugate of polar conjugate of X(26704)
X(40626) = complement of isotomic conjugate of anticomplement of X(6332)
X(40626) = complement of isotomic conjugate of trilinear pole of line X(124)X(2968)
X(40626) = complement of isotomic conjugate of crossdifference of X(42) and X(184)
X(40626) = complement of isotomic conjugate of Steiner-circumellipse pole of line X(1)X(4)
X(40626) = complement of trilinear pole of line X(3)X(10)
X(40626) = complementary conjugate of complement of X(6589)
X(40626) = X(2)-Ceva conjugate of X(6332)
X(40626) = perspector of circumconic centered at X(6332)

X(40627) = CENTER OF 2ND (X(514),X(512))-CEVA CONIC

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

The 2nd (X(514),X(512))-Ceva conic is also the 2nd (X(2),X(3122))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40627) lies on these lines: {512, 798}, {514, 1921}, {649, 16695}, {650, 3250}, {661, 2533}, {772, 20906}, {2084, 4705}, {4151, 21836}, {6372, 21143}, {14825, 27929}, {27469, 30096}

X(40627) = complement of isotomic conjugate of anticomplement of X(3122)
X(40627) = X(2)-Ceva conjugate of X(3122)
X(40627) = perspector of circumconic centered at X(3122)

X(40628) = CENTER OF 2ND (X(514),X(512))-CEVA CONIC

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

The 2nd (X(514),X(512))-Ceva conic is also the 2nd (X(2),X(7004))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40628) lies on these lines: {284, 1021}, {514, 3064}, {521, 652}, {647, 18675}, {650, 1459}, {661, 8819}, {1566, 38375}, {4521, 14331}, {4988, 21127}, {20297, 24562}, {20314, 25925}

X(40628) = X(2)-Ceva conjugate of X(7004)
X(40628) = perspector of circumconic centered at X(7004)

X(40629) = CENTER OF 2ND (X(514),X(527))-CEVA CONIC

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

The 2nd (X(514),X(527))-Ceva conic is also the 2nd (X(2),X(1638))-Ceva conic. (Randy Hutson, December 18, 2020)

X(40629) lies on these lines: {2, 37131}, {7, 37139}, {11, 514}, {527, 1155}, {650, 1086}, {661, 3942}, {812, 38385}, {908, 16593}, {1639, 35094}, {3218, 24582}, {3259, 6084}, {3321, 3323}, {4465, 16597}, {4521, 26932}, {4643, 9458}, {4988, 16732}, {9318, 17718}, {16594, 30823}, {17484, 31020}, {24499, 33151}, {27929, 38989}, {28534, 39308}, {37757, 37787}

X(40629) = complement of X(37143)
X(40629) = complementary conjugate of complement of X(22108)
X(40629) = X(2)-Ceva conjugate of X(1638)
X(40629) = perspector of circumconic centered at X(1638)

X(40630) = MIDPOINT OF X(74) AND X(5627)

Barycentrics    (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)*(2*a^8 - 2*a^6*b^2 - 3*a^4*b^4 + 4*a^2*b^6 - b^8 - 2*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 4*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(40630) j= X[74] + 2 X[12079], 5 X[74] + X[14989], 2 X[74] + X[34150], 2 X[3233] + X[12317], 5 X[5627] - X[14989], X[6070] + 2 X[20417], 2 X[6070] + X[36164], 4 X[6699] - X[14611], X[7471] + 2 X[16003], 4 X[12068] - X[14094], 10 X[12079] - X[14989], 4 X[12079] - X[34150], X[14480] - 7 X[15057], 2 X[14989] - 5 X[34150], X[14993] + 3 X[20126], X[15054] + 2 X[36169], 4 X[20379] - X[36184], 4 X[20417] - X[36164]

X(40630) lies on the curve X161 and these lines: {2, 14264}, {30, 74}, {125, 32417}, {140, 3470}, {186, 17986}, {403, 16080}, {523, 1138}, {542, 15468}, {549, 14385}, {1494, 7799}, {2394, 15543}, {3233, 12317}, {3524, 36875}, {5054, 9717}, {6070, 20417}, {6699, 14611}, {7471, 16003}, {10257, 14919}, {10295, 10421}, {11539, 39239}, {12068, 14094}, {14480, 15057}, {14568, 38894}, {15054, 36169}, {16319, 40384}, {20379, 36184}, {26879, 38933}, {32836, 36890}

X(40630) = midpoint of X(74) and X(5627)
X(40630) = reflection of X(i) in X(j) for these {i,j}: {5627, 12079}, {14611, 31378}, {31378, 6699}, {34150, 5627}
X(40630) = X(99)-Ceva conjugate of X(2394)
X(40630) = barycentric product X(i)*X(j) for these {i,j}: {1494, 6128}, {2394, 14611}, {6699, 16080}
X(40630) = barycentric quotient X(i)/X(j) for these {i,j}: {6128, 30}, {6699, 11064}, {14611, 2407}
X(40630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 12079, 34150}, {6070, 20417, 36164}, {36308, 36311, 3580}

X(40631) = MIDPOINT OF X(3) AND X(19553)

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

X(40631) lies on the curve Q161 and these lines: {3, 19553}, {4, 35724}, {5, 23338}, {30, 1141}, {54, 140}, {95, 1238}, {96, 275}, {128, 539}, {186, 523}, {230, 14586}, {403, 933}, {1154, 24147}, {1166, 14788}, {1493, 13856}, {3479, 3480}, {3530, 25042}, {3575, 8883}, {6150, 25150}, {7604, 35018}, {8901, 37938}, {12026, 32744}, {12060, 18400}, {12242, 32904}, {16336, 34837}, {16768, 36966}, {19210, 37452}, {23337, 24573}

X(40631) = midpoint of X(i) and X(j) for these {i,j}: {3, 19553}, {1141, 1157}, {1263, 35729}, {24147, 38618}
X(40631) = reflection of X(i) in X(j) for these {i,j}: {128, 10615}, {16336, 34837}
X(40631) = barycentric product X(i)*X(j) for these {i,j}: {95, 231}, {275, 539}, {13582, 27423}
X(40631) = barycentric quotient X(i)/X(j) for these {i,j}: {231, 5}, {539, 343}, {8882, 2383}, {27423, 37779}
X(40631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 25044, 36842}, {54, 252, 140}

leftri

Points associated with bicevian triangles: X(40632)-X(40661)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 8, 2020.

Let P = p : q : r and U = u : v : w be points in the plane of a triangle ABC. Let A'B'C' be the cevian triangle of P and A"B"C" the cevian triangle. Let A* be the midpoint of A' and A", and define B* and C* cyclically. The triangle A*B*C* is here named the (P,U)-bicevian triangle:

A* = 0 : 2 q v + q w + r v : 2 r w + q w + r v
B* = 2 p u + r u + p w : 0 : 2 r w + r u + p w
C* = 2 p u + p v + q u : 2 q v + p v + q u : 0

For example, let DEF be the (X(2),X(4))-bicevian triangle, so that D = 0 : 2 a^2 + b^2 - c^2 : a^2 - b^2 + c^2. The vertices D, E, F lie on the cubics K054 and K124 and on the Moses-Steiner ellipse (see X(6070).

These points lie on the Euler line of DEF:
X(5943) = X(2)-of-DEF
X(8254) = X(3)-of-DEF
X(6153) = X(4)-of-DEF
X(13365) = X(5)-of-DEF
X(40632) = X(20)-of-DEF

The following triangles are perspective to DEF, all with perspector X(5): 3rd and 4th Euler triangles, submedial, infinite altitude, Ehrmann mid-triangle, Gemini 110, 1st and 2nd half-diamonds equilateral triangles, and 1st and 2nd half-diamonds triangles (X(33338).

The circumcircle (M), of DEF, passes through X(i) for i = 125, 137, 11702, 14071, 30480 and has squared radius

(2*(a^2 + b^2 - c^2)^2 - a^2*b^2*(-2 + J^2))*(2*(a^2 - b^2 + c^2)^2 - a^2*c^2*(-2 + J^2))*(2*(-a^2 + b^2 + c^2)^2 - b^2*c^2*(-2 + J^2))/(64*a^2*b^2*c^2*(-2 + J)^2*(2 + J)^2*S^2)

Note that (M) meets the nine-point circle in the points X(125) and X(137).

DEF is the reflection triangle of the medial-of-medial triangle (which is also Gemini triangle 110, the X(2)-midcevian triangle, and the excentral triangle of the submedial triangle, if ABC is acute), and DEF is homothetic to the reflection triangle at X(2). (Randy Hutson, December 18, 2020)

Let D' be the point, other than D, where (M) meets the line BC, and define E' and F' cyclically. Then

Let D' = 0 : (a^2 + 2*b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2) : (a^2 - b^2 + 2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4).

The triangle D'E'F', here named the Maia triangle, is perspective to the following triangles, with perspectors as shown:

1st orthosymmedial (see X(6792); perspector X(40633)
infinite altitude; perspector X(54)
orthic axes triangle (see X(25010)); perspector X(275)
Yiu tangents triangle (see X(7495); perspector X(40634)

The Maia triangle is homothetic to the polar triangle of the nine-point circle at X(8901). (Randy Hutson, December 18, 2020)

X(40632) = X(20)-OF-(X(2),X(4))-BICEVIAN TRIANGLE

Barycentrics    a^2*(a^12*b^2 - 4*a^10*b^4 + 5*a^8*b^6 - 5*a^4*b^10 + 4*a^2*b^12 - b^14 + a^12*c^2 - 10*a^10*b^2*c^2 + 22*a^8*b^4*c^2 - 19*a^6*b^6*c^2 + 10*a^4*b^8*c^2 - 7*a^2*b^10*c^2 + 3*b^12*c^2 - 4*a^10*c^4 + 22*a^8*b^2*c^4 - 16*a^6*b^4*c^4 - 5*a^4*b^6*c^4 + 6*a^2*b^8*c^4 - 3*b^10*c^4 + 5*a^8*c^6 - 19*a^6*b^2*c^6 - 5*a^4*b^4*c^6 - 6*a^2*b^6*c^6 + b^8*c^6 + 10*a^4*b^2*c^8 + 6*a^2*b^4*c^8 + b^6*c^8 - 5*a^4*c^10 - 7*a^2*b^2*c^10 - 3*b^4*c^10 + 4*a^2*c^12 + 3*b^2*c^12 - c^14) : :
X(40632) = 3 X[51] - X[13423], 5 X[54] - X[6242], 3 X[54] - X[32352], 5 X[389] - 2 X[6242], X[389] + 2 X[21660], 3 X[389] - 2 X[32352], 3 X[3819] - 2 X[21230], 3 X[3917] - X[12325], 3 X[5943] - 2 X[6153], 3 X[5943] - 4 X[8254], 9 X[5943] - 8 X[13365], 9 X[5943] - 8 X[40632], 3 X[6153] - 4 X[13365], 3 X[6153] - 4 X[40632], X[6242] + 5 X[21660], 3 X[6242] - 5 X[32352], 3 X[8254] - 2 X[13365], 3 X[8254] - 2 X[40632], 2 X[10110] + X[12291], 4 X[10610] - 3 X[16836], X[11271] + 2 X[15606], 8 X[11577] + X[13474], 3 X[11577] + X[15739], 4 X[11695] - X[12280], 4 X[12242] - X[13433], 3 X[13474] - 8 X[15739], 3 X[21660] + X[32352], 3 X[21849] - 2 X[32196]

X(40632) lies on these lines: {5, 15532}, {51, 13423}, {54, 186}, {195, 511}, {548, 1154}, {1209, 5181}, {1216, 11264}, {1885, 11577}, {2888, 11793}, {3519, 13622}, {3574, 11817}, {3819, 21230}, {3917, 12325}, {5446, 22051}, {5462, 13368}, {5907, 32423}, {5943, 6153}, {5965, 11574}, {6000, 12254}, {6467, 18946}, {9920, 19596}, {9969, 11808}, {10110, 12291}, {10203, 22352}, {10274, 19468}, {10610, 16836}, {10619, 10628}, {10625, 12316}, {11271, 15606}, {11430, 32333}, {11692, 15806}, {11695, 12280}, {12307, 13348}, {13382, 32339}, {13598, 20424}, {13754, 36966}, {15073, 23048}, {15801, 16661}, {18368, 33565}, {21849, 32196}

X(40632) = midpoint of X(i) and X(j) for these {i,j}: {5, 15532}, {54, 21660}, {10625, 12316}
X(40632) = reflection of X(i) in X(j) for these {i,j}: {389, 54}, {2888, 11793}, {5446, 22051}, {6153, 8254}, {11808, 12242}, {12307, 13348}, {13368, 5462}, {13433, 11808}, {13598, 20424}, {32339, 13382}
X(40632) = crosspoint of X(54) and X(13418)
X(40632) = crosssum of X(5) and X(13621)
X(40632) = {X(6153),X(8254)}-harmonic conjugate of X(5943)

X(40633) = PERSPECTOR OF THESE TRIANGLES: MAIA AND 1ST ORTHOSYMMEDIAL

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

X(40633) lies on the cubic K055 and these lines: {6, 24}, {32, 14586}, {1141, 3767}, {1157, 3053}, {1992, 18315}, {5013, 25042}, {7604, 31415}, {11815, 35906}, {18907, 36842}

X(40633) = X(13622)-isoconjugate of X(14213)
X(40633) = barycentric product X(54)*X(13595)
X(40633) = barycentric quotient X(13595)/X(311)
X(40633) = {X(32),X(14586)}-harmonic conjugate of X(25044)

X(40634) = PERSPECTOR OF THESE TRIANGLES: MAIA AND YIU TANGENTS

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

X(40634) lies on these lines: {2, 38429}, {5, 23338}, {54, 550}, {98, 275}, {549, 1157}, {1141, 3845}, {1166, 11585}, {1493, 35720}, {2623, 3051}, {3520, 16035}, {3574, 35728}, {6636, 16030}, {8254, 35888}, {10619, 35721}, {12242, 30484}, {14073, 27423}, {14586, 18907}, {33992, 35885}

X(40634) = barycentric quotient X(39454)/X(15031)
X(40634) = pedal homothetic center of X(54) (see X(3066))
X(40634) = {X(25044),X(36842)}-harmonic conjugate of X(5)

X(40635) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(4))-BICEVIAN AND AYME

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

X(40635) lies on these lines: {5, 10}, {12, 1829}, {19, 25}, {26, 32613}, {42, 1953}, {48, 5311}, {65, 3772}, {200, 21867}, {206, 10537}, {210, 2262}, {226, 3827}, {312, 3434}, {518, 4362}, {528, 35652}, {674, 9969}, {756, 2183}, {942, 17061}, {1402, 8609}, {1828, 10895}, {1871, 11500}, {1872, 11496}, {1900, 6284}, {2182, 3745}, {2217, 37539}, {2265, 2308}, {2270, 7322}, {2643, 3725}, {2875, 14717}, {2900, 19589}, {3052, 12723}, {3419, 3714}, {3428, 7395}, {3742, 29645}, {3752, 20276}, {4463, 26227}, {4523, 29670}, {5173, 15253}, {5842, 6756}, {5903, 17064}, {6051, 23846}, {6676, 6690}, {7493, 20243}, {7528, 37820}, {7529, 10679}, {7539, 31245}, {8758, 21318}, {9958, 13754}, {10831, 26377}, {11365, 37696}, {11818, 18407}, {13407, 18732}, {15940, 16072}, {17441, 17718}, {20961, 34857}, {21370, 22769}, {24476, 33144}, {24929, 30142}, {25466, 37613}, {37000, 37122}

X(40636) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(7))-BICEVIAN AND INCIRCLE-INVERSE OF ABC

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

X(40636) lies on these lines: {1, 3286}, {354, 1418}, {516, 5045}, {674, 11018}, {916, 16216}, {5572, 34830}

X(40637) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(9))-BICEVIAN AND ANTICOMPLEMENTARY

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

X(40637) lies on these lines: {2, 40216}, {37, 39735}, {55, 17494}, {190, 33798}, {192, 4661}, {239, 3219}, {1655, 21217}, {3177, 4430}, {3957, 10025}, {3995, 40007}, {8267, 16684}, {16588, 27009}, {17495, 25249}, {21795, 23989}

X(40638) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(9))-BICEVIAN AND TANGENTIAL

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

X(40638) lies on these lines: {1, 5132}, {31, 32}, {55, 5277}, {100, 1078}, {595, 38887}, {976, 37575}, {1030, 20994}, {1621, 32009}, {3185, 36014}, {3294, 8053}, {3688, 22369}, {4251, 16693}, {4557, 16552}, {5283, 34247}, {23851, 35342}

X(40639) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(9))-BICEVIAN AND ANTICEVIAN OF X(523)

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

X(40639) lies on these lines: {5, 13576}, {11, 116}, {442, 8299}, {528, 3584}, {1018, 2886}, {1111, 6362}, {2170, 28473}, {3140, 18101}, {5511, 38959}, {14839, 24390}

X(40640) = PERSPECTOR OF THESE TRIANGLES: (X(3),X(4))-BICEVIAN AND ANTI-ORTHOCENTROIDAL

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

X(40640) lies on these lines: {5, 399}, {23, 11557}, {74, 11562}, {110, 389}, {113, 25739}, {125, 32396}, {182, 15100}, {186, 323}, {265, 34545}, {569, 15102}, {1181, 18933}, {1199, 32423}, {1994, 12383}, {3523, 17847}, {5622, 19140}, {5663, 13353}, {6126, 11570}, {6593, 12825}, {6636, 7731}, {7512, 38898}, {7527, 12270}, {7574, 11805}, {7592, 14683}, {7728, 13470}, {9143, 19456}, {10296, 13202}, {10628, 27866}, {10733, 34155}, {11003, 12412}, {11807, 37945}, {12112, 18403}, {12219, 15462}, {12364, 32325}, {13392, 16532}, {14643, 34826}, {14940, 15068}, {15051, 38448}, {15101, 37471}, {18445, 18932}, {33565, 37347}

X(40641) = PERSPECTOR OF THESE TRIANGLES: (X(3),X(4))-BICEVIAN AND MEDIAL-OF-ORTHIC

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

X(40641) lies on these lines: {30, 5462}, {51, 216}, {5640, 15466}, {14249, 15043}

X(40642) = PERSPECTOR OF THESE TRIANGLES: (X(3),X(6))-BICEVIAN AND ANTICOMPLEMENTARY

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

X(40642) lies on these lines: {2, 11794}, {184, 31296}, {385, 6636}, {401, 1994}, {14570, 33798}

X(40643) = PERSPECTOR OF THESE TRIANGLES: (X(3),X(6))-BICEVIAN AND TANGENTIAL

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

X(40643) lies on these lines: {3, 3202}, {5, 182}, {6, 27375}, {25, 27370}, {32, 184}, {39, 9418}, {49, 2080}, {54, 12110}, {83, 5012}, {98, 1614}, {110, 1078}, {156, 10104}, {567, 18502}, {569, 10358}, {626, 36213}, {1092, 8722}, {1147, 5171}, {1207, 1915}, {1627, 38854}, {1691, 11360}, {1974, 5034}, {3044, 39652}, {3506, 8150}, {3796, 20993}, {4045, 14133}, {5038, 18374}, {5118, 7782}, {7787, 11003}, {7793, 9544}, {7815, 9306}, {10274, 14676}, {10790, 11402}, {10796, 32046}, {10984, 37479}, {11380, 34397}, {12177, 39840}, {14574, 15257}

X(40644) = PERSPECTOR OF THESE TRIANGLES: (X(3),X(7))-BICEVIAN AND ASCELLA

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

X(40644) lies on these lines: {3, 102}, {4, 34956}, {57, 1745}, {142, 14058}, {222, 578}, {389, 1465}, {515, 942}, {970, 7352}, {1214, 11793}, {1364, 3468}, {1425, 6905}, {3075, 39791}, {5562, 17080}, {5907, 37565}, {6000, 17102}, {6942, 19368}, {8726, 21228}, {10110, 20122}, {21484, 34032}

X(40645) = PERSPECTOR OF THESE TRIANGLES: (X(4),X(6))-BICEVIAN AND MEDIAL-OF-ORTHIC

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

X(40645) lies on these lines: {6, 27375}, {51, 217}, {1503, 10110}, {5943, 34850}

X(40646) = PERSPECTOR OF THESE TRIANGLES: (X(4),X(7))-BICEVIAN AND INCIRCLE-INVERSE OF ABC

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

X(40646) lies on these lines: {354, 1827}, {971, 12005}, {4860, 7004}, {8679, 9969}, {11028, 13476}

X(40647) = PERSPECTOR OF THESE TRIANGLES: (X(4),X(20))-BICEVIAN AND 3RD HATZIPOLAKIS

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

X(40647) lies on these lines: {2, 6241}, {3, 49}, {4, 4846}, {5, 2883}, {6, 12085}, {20, 52}, {23, 8718}, {26, 11438}, {30, 143}, {51, 382}, {54, 2071}, {64, 9818}, {68, 15740}, {74, 11562}, {113, 17854}, {125, 10024}, {140, 5663}, {182, 3357}, {195, 37477}, {217, 14961}, {265, 14861}, {373, 3851}, {376, 5889}, {378, 569}, {381, 11381}, {511, 550}, {541, 25711}, {546, 5943}, {548, 1154}, {549, 5876}, {568, 1657}, {578, 12084}, {631, 5891}, {632, 15060}, {916, 31837}, {974, 6146}, {1038, 6238}, {1040, 7352}, {1192, 14070}, {1199, 7464}, {1209, 12827}, {1368, 22660}, {1385, 2807}, {1425, 18455}, {1498, 6642}, {1503, 31833}, {1511, 22966}, {1568, 37452}, {1593, 36752}, {1614, 22467}, {1656, 15030}, {1899, 9927}, {1986, 16111}, {2393, 34785}, {2772, 20117}, {2777, 11557}, {2979, 3528}, {3060, 3529}, {3090, 15305}, {3091, 12290}, {3146, 3567}, {3270, 18447}, {3516, 37506}, {3518, 15053}, {3519, 13623}, {3520, 5012}, {3521, 18403}, {3522, 11412}, {3523, 11459}, {3524, 11444}, {3525, 15056}, {3526, 18435}, {3530, 3819}, {3534, 6243}, {3543, 9781}, {3545, 11439}, {3546, 5654}, {3547, 18913}, {3549, 26937}, {3581, 13564}, {3627, 5946}, {3767, 15575}, {3830, 16226}, {3832, 11455}, {3843, 32062}, {3845, 15026}, {3850, 13363}, {3853, 10095}, {3855, 11451}, {3861, 13364}, {4297, 31728}, {4550, 7395}, {5020, 12315}, {5050, 12294}, {5066, 32205}, {5068, 11465}, {5073, 12002}, {5133, 18488}, {5422, 35502}, {5448, 11585}, {5449, 15760}, {5650, 15720}, {5878, 34944}, {5893, 9826}, {5944, 15646}, {6101, 8703}, {6153, 11802}, {6225, 18537}, {6240, 11750}, {6285, 37696}, {6293, 10606}, {6467, 10937}, {6560, 12239}, {6561, 12240}, {6583, 15229}, {6593, 20190}, {6644, 6759}, {6689, 6696}, {6776, 12118}, {6800, 32534}, {6823, 12359}, {6842, 34462}, {7355, 37697}, {7387, 9786}, {7401, 12324}, {7488, 32110}, {7503, 13336}, {7506, 26883}, {7514, 37515}, {7516, 13347}, {7542, 20191}, {7550, 15054}, {7592, 11413}, {7722, 15055}, {7723, 38727}, {7728, 16223}, {7998, 10299}, {7999, 15717}, {8717, 11414}, {9306, 32139}, {9707, 15078}, {9820, 14156}, {9822, 39884}, {9937, 17818}, {9967, 25406}, {9969, 11819}, {10019, 12133}, {10112, 11232}, {10115, 17712}, {10116, 18914}, {10263, 15704}, {10264, 34826}, {10282, 37814}, {10323, 37478}, {10539, 11456}, {10564, 15032}, {10610, 10628}, {10619, 16163}, {10620, 13339}, {10627, 33923}, {10938, 14852}, {10996, 11411}, {11017, 12812}, {11250, 11430}, {11402, 12058}, {11424, 36753}, {11440, 35921}, {11457, 18474}, {11472, 11479}, {11645, 38322}, {11807, 34584}, {12083, 37490}, {12086, 15033}, {12100, 31834}, {12103, 13391}, {12121, 21649}, {12160, 37483}, {12161, 13346}, {12174, 18451}, {12228, 13293}, {12233, 23335}, {12254, 12280}, {12292, 23515}, {12370, 22952}, {12825, 38793}, {12901, 13198}, {12918, 16225}, {13202, 16222}, {13353, 14130}, {13366, 18859}, {13371, 18388}, {13399, 37347}, {13417, 20127}, {13419, 31830}, {13434, 13445}, {14216, 18420}, {14531, 15696}, {14627, 35452}, {14677, 38898}, {14864, 34514}, {14869, 40247}, {15041, 18364}, {15061, 21650}, {15062, 35500}, {15063, 17853}, {15067, 15712}, {15087, 37495}, {15123, 15129}, {15138, 32345}, {15681, 21969}, {15738, 20397}, {16238, 16252}, {16266, 37480}, {16270, 36253}, {16868, 26913}, {17834, 35243}, {17856, 36518}, {18390, 18952}, {18570, 32392}, {18916, 37201}, {18925, 27082}, {21312, 36747}, {22584, 38728}, {23128, 39913}, {25739, 34007}, {28150, 31757}, {28164, 31760}, {31730, 31732}, {32068, 40240}, {34224, 38323}, {35237, 39568}, {35477, 39242}, {37198, 37486}, {37374, 39271}, {38730, 39817}, {38741, 39846}, {39805, 39860}, {39831, 39834}

X(40647) = midpoint of X(3) and X(185)
X(40647) = reflection of X(1216) in X(3)
X(40647) = complement of X(12162)
X(40647) = X(20) of X(5)-Brocard triangle

X(40648) = PERSPECTOR OF THESE TRIANGLES: (X(5),X(7))-BICEVIAN AND ASCELLA

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

X(40648) lies on these lines: {3, 37806}, {57, 3460}, {5122, 22835}, {9940, 16870}

X(40649) = PERSPECTOR OF THESE TRIANGLES: (X(6),X(7))-BICEVIAN AND ASCELLA

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

X(40649) lies on these lines: {3, 595}, {43, 57}, {142, 3840}, {511, 3752}, {519, 942}, {982, 9052}, {1015, 21792}, {2999, 3784}, {3216, 29958}, {3666, 3819}, {3688, 17591}, {3742, 39543}, {3779, 18193}, {3917, 4850}, {3937, 32911}, {4000, 37521}, {4014, 33096}, {4253, 20995}, {5650, 28606}, {5745, 6686}, {5943, 16610}, {6688, 16602}, {6904, 20037}, {9776, 10453}, {10219, 31197}, {11227, 29353}, {12109, 24046}, {17063, 21746}, {33150, 33852}

X(40650) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(8))-BICEVIAN AND 1ST VIJAY-PAASCHE-HUTSON

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

X(40650) lies on these lines: {2, 586}, {326, 1267}, {3086, 17869}, {38487, 38491}, {38488, 39312}

X(40650) = {X(2),X(40651)}-harmonic conjugate of X(40652)

X(40651) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(8))-BICEVIAN AND 4TH VIJAY-PAASCHE-HUTSON

Barycentrics    a*(b*c + S)*(2*a*b*c + b*S + c*S) : :
Barycentrics    1/(1 + sin B) + 1/(1 + sin C) : :
Barycentrics    1/(b + 2 R) + 1/(c + 2 R) : :

X(40651) lies on these lines: {2, 586}, {9, 13389}, {63, 10252}, {394, 1124}, {440, 31591}, {1125, 6509}, {1214, 31535}, {1267, 38488}, {37861, 38015}, {38487, 38489}, {39314, 39609}

X(40651) = complement of isogonal conjugate of X(605)
X(40651) = complement of isotomic conjugate of X(3083)
X(40651) = complement of polar conjugate of X(6212)
X(40651) = complement of complement of X(37881)
X(40651) = {X(40650),X(40652)}-harmonic conjugate of X(2)

X(40652) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(8))-BICEVIAN AND 8TH VIJAY-PAASCHE-HUTSON

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

X(40652) lies on these lines: {2, 586}, {3083, 38488}, {38487, 38495}, {39314, 39610}

X(40652) = {X(2),X(40651)}-harmonic conjugate of X(40650)

X(40653) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(8))-BICEVIAN AND 9TH VIJAY-PAASCHE-HUTSON

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

X(40653) lies on these lines: {1, 1123}, {2, 30416}, {3, 31595}, {142, 13360}, {6600, 18234}, {38487, 38493}, {39313, 39616}

X(40654) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(11))-BICEVIAN AND ASCELLA

Barycentrics    2*a^8 - 4*a^7*b + 2*a^6*b^2 + 2*a^5*b^3 - 7*a^4*b^4 + 6*a^3*b^5 + 2*a^2*b^6 - 4*a*b^7 + b^8 - 4*a^7*c + 8*a^6*b*c - 6*a^5*b^2*c + 10*a^4*b^3*c - 6*a^3*b^4*c - 14*a^2*b^5*c + 16*a*b^6*c - 4*b^7*c + 2*a^6*c^2 - 6*a^5*b*c^2 - 4*a^4*b^2*c^2 + 28*a^2*b^4*c^2 - 24*a*b^5*c^2 + 4*b^6*c^2 + 2*a^5*c^3 + 10*a^4*b*c^3 - 32*a^2*b^3*c^3 + 12*a*b^4*c^3 + 4*b^5*c^3 - 7*a^4*c^4 - 6*a^3*b*c^4 + 28*a^2*b^2*c^4 + 12*a*b^3*c^4 - 10*b^4*c^4 + 6*a^3*c^5 - 14*a^2*b*c^5 - 24*a*b^2*c^5 + 4*b^3*c^5 + 2*a^2*c^6 + 16*a*b*c^6 + 4*b^2*c^6 - 4*a*c^7 - 4*b*c^7 + c^8 : :

X(40654) lies on these lines: {3, 37815}, {57, 2957}, {142, 40531}, {516, 5122}, {3667, 13226}

X(40655) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(12))-BICEVIAN AND ASCELLA

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

X(40655) lies on these lines: {3, 37816}, {942, 35063}, {3008, 6678}, {5087, 5122}, {5972, 12047}, {6723, 14873}, {28239, 28258}

X(40656) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(19))-BICEVIAN AND ASCELLA

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

X(40656) lies on these lines: {3, 1104}, {57, 238}, {142, 3846}, {518, 3771}, {579, 20227}, {614, 37581}, {758, 942}, {1108, 20254}, {1427, 37507}, {2886, 12722}, {3306, 16352}, {3666, 8731}, {3812, 32916}, {3848, 25498}, {3911, 6676}, {4260, 11018}, {4463, 31229}, {5437, 16852}, {5439, 16343}, {8727, 9944}, {11227, 29353}, {11997, 33135}, {12723, 17064}, {16056, 16610}, {17441, 24597}, {19788, 30943}, {28389, 37566}

X(40657) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(20))-BICEVIAN AND ASCELLA

Barycentrics    4*a^10 + 3*a^9*b - 5*a^8*b^2 - 8*a^7*b^3 - 8*a^6*b^4 + 6*a^5*b^5 + 14*a^4*b^6 - 4*a^2*b^8 - a*b^9 - b^10 + 3*a^9*c + 6*a^8*b*c + 4*a^7*b^2*c - 4*a^6*b^3*c - 14*a^5*b^4*c - 8*a^4*b^5*c + 4*a^3*b^6*c + 4*a^2*b^7*c + 3*a*b^8*c + 2*b^9*c - 5*a^8*c^2 + 4*a^7*b*c^2 + 24*a^6*b^2*c^2 + 8*a^5*b^3*c^2 - 14*a^4*b^4*c^2 - 12*a^3*b^5*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^7*c^3 - 4*a^6*b*c^3 + 8*a^5*b^2*c^3 + 16*a^4*b^3*c^3 + 8*a^3*b^4*c^3 - 4*a^2*b^5*c^3 - 8*a*b^6*c^3 - 8*b^7*c^3 - 8*a^6*c^4 - 14*a^5*b*c^4 - 14*a^4*b^2*c^4 + 8*a^3*b^3*c^4 + 24*a^2*b^4*c^4 + 6*a*b^5*c^4 - 2*b^6*c^4 + 6*a^5*c^5 - 8*a^4*b*c^5 - 12*a^3*b^2*c^5 - 4*a^2*b^3*c^5 + 6*a*b^4*c^5 + 12*b^5*c^5 + 14*a^4*c^6 + 4*a^3*b*c^6 - 8*a^2*b^2*c^6 - 8*a*b^3*c^6 - 2*b^4*c^6 + 4*a^2*b*c^7 - 8*b^3*c^7 - 4*a^2*c^8 + 3*a*b*c^8 + 3*b^2*c^8 - a*c^9 + 2*b*c^9 - c^10 : :

X(40657) lies on these lines: {3, 36908}, {4, 57}, {77, 3345}, {12436, 20205}, {12572, 20206}, {15803, 34050}, {24604, 26723}

X(40658) = PERSPECTOR OF THESE TRIANGLES: (X(7),X(20))-BICEVIAN AND INFINITE ALTITUDE

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

X(40658) lies on these lines: {1, 84}, {3, 12335}, {10, 16252}, {40, 154}, {64, 3576}, {65, 10535}, {108, 15498}, {110, 1295}, {159, 9911}, {165, 17821}, {184, 1902}, {185, 11363}, {515, 2883}, {516, 34782}, {517, 6759}, {518, 19149}, {651, 18239}, {912, 32139}, {944, 5656}, {946, 1386}, {962, 11206}, {971, 8144}, {1062, 9942}, {1103, 7070}, {1108, 3073}, {1125, 6247}, {1201, 28381}, {1319, 7355}, {1385, 6000}, {1482, 32063}, {1702, 17819}, {1703, 17820}, {1829, 26883}, {1853, 8227}, {2393, 31812}, {2646, 6285}, {2777, 11699}, {2781, 31738}, {2829, 5930}, {2917, 9591}, {3057, 26888}, {3100, 12671}, {3197, 6769}, {3357, 13624}, {3428, 37250}, {3556, 22770}, {3579, 10282}, {3612, 10060}, {3616, 12324}, {4297, 15311}, {4663, 34117}, {5603, 34781}, {5706, 7686}, {5731, 6225}, {5878, 18481}, {5886, 14216}, {5893, 31673}, {6198, 12664}, {6254, 11189}, {6684, 10192}, {6696, 10165}, {6700, 20307}, {7957, 10536}, {7968, 12964}, {7969, 12970}, {7987, 9899}, {9538, 9960}, {9583, 19088}, {9626, 10117}, {9833, 12699}, {9955, 18381}, {10076, 37618}, {10246, 12315}, {11202, 31663}, {11230, 20299}, {12162, 24301}, {12330, 38288}, {12571, 23324}, {12688, 38336}, {12702, 14530}, {13374, 37543}, {14529, 31786}, {14925, 31788}, {18400, 22793}, {18493, 34780}, {20323, 32065}, {22802, 28160}, {28146, 34785}, {32380, 40263}, {36851, 38035}

X(40659) = PERSPECTOR OF THESE TRIANGLES: (X(8),X(9))-BICEVIAN AND 2ND ZANIAH

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

X(40659) lies on these lines: {2, 11025}, {7, 3681}, {8, 1229}, {9, 55}, {10, 141}, {46, 5223}, {65, 38200}, {72, 2550}, {144, 17615}, {219, 28043}, {281, 1827}, {354, 20195}, {390, 3876}, {516, 3678}, {517, 18482}, {527, 9954}, {528, 18254}, {756, 4343}, {960, 5853}, {971, 1158}, {1001, 3811}, {2340, 21039}, {2346, 3935}, {3036, 4711}, {3085, 3697}, {3219, 7676}, {3452, 24389}, {3555, 19855}, {3617, 7672}, {3626, 7686}, {3634, 20116}, {3740, 5572}, {3868, 40333}, {3890, 12630}, {4067, 38201}, {4092, 4111}, {4533, 5698}, {5173, 21617}, {5221, 8581}, {5686, 7080}, {5732, 14872}, {5779, 35448}, {5856, 14740}, {5904, 38052}, {6172, 25722}, {7064, 11997}, {7678, 27131}, {8271, 25878}, {8732, 17625}, {10176, 30331}, {10177, 18230}, {10198, 16216}, {14523, 37650}, {15254, 18233}, {15481, 18232}, {15570, 30143}, {17620, 37787}, {18412, 31434}, {25917, 38316}

X(40660) = PERSPECTOR OF THESE TRIANGLES: (X(8),X(20))-BICEVIAN AND INFINITE ALTITUDE

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

X(40660) lies on these lines: {1, 154}, {3, 960}, {4, 2182}, {6, 7713}, {8, 11206}, {9, 37320}, {10, 1503}, {19, 5706}, {20, 12779}, {26, 912}, {28, 65}, {40, 219}, {46, 11347}, {48, 37528}, {57, 221}, {63, 37250}, {64, 165}, {66, 3844}, {72, 2915}, {159, 518}, {161, 8185}, {184, 1829}, {206, 942}, {208, 34032}, {284, 3931}, {355, 9833}, {387, 2264}, {406, 5928}, {515, 34782}, {516, 2883}, {517, 6759}, {611, 1773}, {946, 16252}, {1071, 3220}, {1125, 10192}, {1155, 7355}, {1177, 2836}, {1214, 1782}, {1385, 10282}, {1439, 34043}, {1452, 19349}, {1482, 14530}, {1486, 12710}, {1495, 11363}, {1610, 14110}, {1619, 8193}, {1633, 30267}, {1697, 2192}, {1698, 1853}, {1708, 13737}, {1709, 37046}, {1763, 7078}, {1836, 14018}, {1842, 5721}, {1854, 3601}, {1858, 14017}, {1864, 4222}, {1902, 26883}, {1944, 37088}, {2390, 37582}, {2393, 4663}, {2771, 13289}, {2778, 9934}, {2781, 31737}, {2818, 37623}, {2917, 9626}, {2939, 3198}, {2948, 9591}, {2956, 3182}, {3057, 10535}, {3079, 3176}, {3157, 34371}, {3194, 30456}, {3211, 19149}, {3295, 18621}, {3357, 31663}, {3416, 5596}, {3562, 7291}, {3576, 17821}, {3579, 6000}, {3616, 35260}, {3634, 23332}, {3683, 13726}, {3694, 38868}, {3743, 24929}, {3751, 9924}, {3811, 39600}, {3812, 7535}, {3869, 7520}, {4219, 12688}, {4295, 7490}, {4401, 8676}, {5090, 31383}, {5656, 6361}, {5657, 34781}, {5691, 17845}, {5709, 15509}, {5745, 20306}, {5786, 39585}, {5847, 34774}, {5894, 12512}, {6197, 38860}, {6225, 9778}, {6244, 12335}, {6247, 6684}, {6285, 11190}, {6678, 12609}, {6696, 10164}, {7387, 9928}, {7412, 12664}, {7497, 7686}, {7523, 25917}, {7959, 37551}, {7968, 10534}, {7969, 10533}, {7973, 7991}, {8282, 20224}, {8567, 16192}, {9306, 37613}, {9616, 19088}, {9712, 14454}, {9780, 32064}, {9956, 18381}, {10391, 13730}, {10606, 35242}, {11202, 13624}, {11231, 20299}, {11396, 26864}, {12259, 13383}, {12675, 22654}, {12702, 32063}, {12785, 32359}, {14216, 26446}, {14925, 31786}, {15254, 16290}, {15311, 31730}, {15324, 40117}, {15726, 15951}, {15823, 19262}, {16475, 19132}, {16980, 34750}, {17594, 19764}, {17819, 18991}, {17820, 18992}, {18383, 38140}, {18400, 18480}, {18405, 18492}, {22802, 28146}, {24474, 32379}, {28160, 34785}, {28538, 31166}, {32065, 32636}, {32278, 38885}, {36851, 38047}

X(40661) = PERSPECTOR OF THESE TRIANGLES: (X(8),X(20))-BICEVIAN AND 2ND ZANIAH

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

X(40661) lies on these lines: {1, 12867}, {2, 39772}, {4, 5692}, {8, 6598}, {9, 21}, {10, 12}, {30, 5777}, {63, 35979}, {165, 191}, {329, 2475}, {405, 10176}, {498, 18397}, {517, 15911}, {518, 11281}, {936, 1708}, {943, 15910}, {950, 960}, {1125, 14054}, {1762, 3430}, {1794, 3465}, {1858, 18249}, {1864, 10543}, {1901, 21873}, {2771, 20417}, {2900, 5250}, {2949, 6905}, {3036, 4662}, {3419, 3878}, {3452, 10395}, {3487, 5904}, {3647, 5217}, {3650, 17653}, {3679, 5715}, {3681, 11523}, {3682, 16577}, {3715, 33857}, {3740, 8261}, {3868, 25525}, {3877, 12625}, {3929, 12528}, {4420, 31660}, {4866, 16126}, {5044, 6675}, {5128, 11684}, {5220, 12059}, {5552, 18231}, {5693, 6908}, {5694, 6907}, {5728, 25917}, {5791, 18389}, {5812, 37230}, {5884, 6889}, {6175, 28609}, {6745, 14454}, {6829, 31870}, {6843, 37625}, {7580, 31803}, {10123, 15587}, {10399, 16845}, {11499, 16139}, {12053, 24389}, {12572, 18254}, {12635, 37224}, {12691, 26878}

X(40662) = MIDPOINT OF X(1138) AND X(14451)

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

X(406652) lies on the curve Q161 and these lines: {30, 146}, {14354, 31378}

midpoint of X(1138) and X(14451)
on Q161

X(40663) = MIDPOINT OF X(80) AND X(484)

Barycentrics    (2*a - b - c)*(a + b - c)*(a - b + c)*(b + c) : :
X(40663) = 5 X[11] - 4 X[7743], 3 X[11] - 2 X[30384], 3 X[80] + X[15228], X[484] + 2 X[11545], 3 X[484] - X[15228], X[1317] - 4 X[3911], X[1317] - 3 X[5298], X[1317] + 2 X[36920], 2 X[1319] - 3 X[5298], 2 X[1387] - 3 X[3582], 5 X[1737] - 2 X[7743], 3 X[1737] - X[30384], 2 X[3036] + X[3218], X[3245] + 2 X[12019], 2 X[3814] - 3 X[34122], 4 X[3911] - 3 X[5298], 2 X[3911] + X[36920], 3 X[5131] + X[9897], 3 X[5131] - X[36975], X[5180] - 3 X[37375], 3 X[5298] + 2 X[36920], 2 X[5440] - 3 X[6174], X[6224] - 3 X[13587], 4 X[6681] - 3 X[34123], 4 X[6702] - 3 X[17533], 6 X[7743] - 5 X[30384], 6 X[11545] + X[15228], 2 X[11813] - 3 X[17533], X[13996] + 2 X[26015], X[20085] + 3 X[36004]

X(40663) lies on the curve Q161 and these lines: {1, 140}, {2, 2099}, {3, 10573}, {4, 37567}, {5, 5903}, {7, 11237}, {8, 56}, {10, 12}, {11, 517}, {21, 14882}, {30, 80}, {35, 5428}, {36, 952}, {40, 1728}, {43, 24806}, {44, 1877}, {46, 355}, {55, 1006}, {57, 3679}, {63, 34606}, {71, 21933}, {73, 3214}, {78, 37828}, {100, 5172}, {109, 2758}, {119, 13141}, {145, 1388}, {165, 5727}, {171, 5724}, {190, 36926}, {201, 1834}, {214, 519}, {227, 21896}, {298, 36929}, {299, 36928}, {329, 31141}, {354, 31397}, {377, 18962}, {388, 3617}, {395, 7052}, {396, 33655}, {429, 1825}, {474, 26437}, {485, 38235}, {495, 5902}, {496, 5697}, {499, 1482}, {515, 1155}, {516, 5183}, {518, 6735}, {523, 656}, {524, 24324}, {527, 38099}, {528, 37787}, {529, 3036}, {549, 37525}, {550, 37572}, {553, 4745}, {594, 1400}, {604, 17362}, {611, 38116}, {631, 34471}, {664, 7181}, {672, 1146}, {855, 23845}, {899, 1457}, {908, 5123}, {912, 37725}, {920, 11826}, {938, 3303}, {942, 10039}, {944, 5204}, {946, 7173}, {950, 37568}, {956, 1470}, {958, 5554}, {960, 24982}, {962, 10896}, {999, 12647}, {1056, 4860}, {1125, 7294}, {1149, 3756}, {1159, 31479}, {1210, 3057}, {1213, 2171}, {1227, 3264}, {1284, 3932}, {1320, 32198}, {1329, 3869}, {1334, 21049}, {1358, 9436}, {1385, 21155}, {1387, 3582}, {1389, 6952}, {1399, 5247}, {1402, 4046}, {1403, 3703}, {1404, 4969}, {1405, 17369}, {1406, 9370}, {1411, 26727}, {1415, 5291}, {1420, 3632}, {1423, 33165}, {1428, 5846}, {1429, 32847}, {1452, 5090}, {1454, 5794}, {1460, 5774}, {1464, 4551}, {1466, 22759}, {1467, 4882}, {1478, 5790}, {1479, 12702}, {1483, 21842}, {1512, 6001}, {1532, 2800}, {1616, 28074}, {1698, 3340}, {1706, 37550}, {1708, 3419}, {1770, 18480}, {1826, 21866}, {1836, 2093}, {1846, 38462}, {1852, 6197}, {1858, 31788}, {1861, 1875}, {1866, 1883}, {1869, 1882}, {1901, 21011}, {1935, 18360}, {1940, 5174}, {2098, 3086}, {2197, 21858}, {2238, 4559}, {2245, 2250}, {2285, 17275}, {2294, 21012}, {2295, 21965}, {2348, 8074}, {2362, 13911}, {2475, 12745}, {2550, 12848}, {2594, 3293}, {2646, 6684}, {2802, 20118}, {2975, 37293}, {3017, 24912}, {3035, 4511}, {3058, 3654}, {3109, 5127}, {3212, 3665}, {3244, 17663}, {3245, 3583}, {3256, 5251}, {3336, 18990}, {3339, 9578}, {3361, 4668}, {3416, 39897}, {3428, 11502}, {3436, 18961}, {3452, 31165}, {3474, 12943}, {3485, 9780}, {3486, 5217}, {3530, 37616}, {3555, 10915}, {3576, 37740}, {3579, 10572}, {3584, 5425}, {3585, 18357}, {3600, 4678}, {3601, 9588}, {3612, 37739}, {3614, 9956}, {3621, 5265}, {3626, 10106}, {3628, 5443}, {3656, 23708}, {3704, 3969}, {3746, 12433}, {3782, 37716}, {3812, 24987}, {3813, 14923}, {3814, 34122}, {3816, 3877}, {3820, 5692}, {3826, 7672}, {3828, 4870}, {3868, 12607}, {3871, 32157}, {3876, 9711}, {3878, 4187}, {3880, 13996}, {3881, 13751}, {3895, 34699}, {3913, 11510}, {3930, 21013}, {3943, 21942}, {4031, 38098}, {4032, 4732}, {4295, 5818}, {4298, 4691}, {4299, 18525}, {4315, 4669}, {4316, 28186}, {4317, 37545}, {4323, 19877}, {4424, 4854}, {4654, 5726}, {4661, 18419}, {4677, 13462}, {4714, 6358}, {4863, 34720}, {4868, 16577}, {4880, 24465}, {4973, 15863}, {4995, 24929}, {5044, 13601}, {5048, 28234}, {5080, 13273}, {5086, 7098}, {5122, 21578}, {5128, 5691}, {5131, 9897}, {5180, 37375}, {5219, 18421}, {5222, 31230}, {5225, 20070}, {5228, 36487}, {5260, 18253}, {5270, 24470}, {5326, 11231}, {5552, 12635}, {5563, 34753}, {5599, 18956}, {5600, 18955}, {5687, 37579}, {5730, 26364}, {5740, 21271}, {5836, 6734}, {5837, 8582}, {5854, 38460}, {5881, 15803}, {5882, 37605}, {5886, 25415}, {5901, 11009}, {5904, 26482}, {5905, 11236}, {5919, 11019}, {6048, 37694}, {6049, 20053}, {6147, 37719}, {6224, 13587}, {6361, 12953}, {6681, 34123}, {6702, 11813}, {6736, 24391}, {6738, 37080}, {6788, 40091}, {6842, 35004}, {6905, 12247}, {6907, 18397}, {6922, 26475}, {7146, 29659}, {7211, 21020}, {7280, 34773}, {7483, 30147}, {7741, 22791}, {7807, 30136}, {7951, 38042}, {7968, 18966}, {7969, 18965}, {7982, 11376}, {7991, 9581}, {8162, 10580}, {8362, 30124}, {8581, 24393}, {8666, 34880}, {9579, 37714}, {9583, 9663}, {9661, 35641}, {9708, 37541}, {9710, 15844}, {10056, 15934}, {10058, 35000}, {10072, 34718}, {10090, 19914}, {10164, 37600}, {10175, 17605}, {10265, 17636}, {10310, 22760}, {10401, 17270}, {10529, 10912}, {10593, 15079}, {10826, 12699}, {10914, 10916}, {10954, 37438}, {10955, 34339}, {10958, 15908}, {11010, 15171}, {11238, 30305}, {11280, 37735}, {11373, 30323}, {11529, 17718}, {11571, 11698}, {11682, 25681}, {11822, 11872}, {11823, 11871}, {13145, 37401}, {13407, 31794}, {13724, 23844}, {13747, 30144}, {13973, 16232}, {14026, 23832}, {14584, 23703}, {15172, 37563}, {15298, 38126}, {15500, 23711}, {15829, 24954}, {15867, 26487}, {16137, 37731}, {16210, 18958}, {16236, 25055}, {16589, 20616}, {17023, 31221}, {17051, 31188}, {17619, 21616}, {17747, 21044}, {17950, 24836}, {18481, 37711}, {18591, 21860}, {18635, 21231}, {18967, 25524}, {18995, 19066}, {18996, 19065}, {19029, 35775}, {19030, 35774}, {19636, 36590}, {19860, 24953}, {20085, 36004}, {21273, 24986}, {21672, 21674}, {21857, 40590}, {21871, 24005}, {21888, 21956}, {23846, 28238}, {23958, 34605}, {24223, 26742}, {24440, 37591}, {24541, 31260}, {24633, 26575}, {25466, 27186}, {25557, 30312}, {26481, 31419}, {26752, 28771}, {28212, 37718}, {30852, 34647}, {32141, 36152}, {36574, 37542}, {37524, 37705}, {37618, 37727}

X(40663) = midpoint of X(i) and X(j) for these {i,j}: {80, 484}, {1319, 36920}, {3218, 5176}, {3245, 3583}, {4316, 37006}, {4973, 15863}, {6905, 12247}, {9897, 36975}, {19914, 22765}
X(40663) = reflection of X(i) in X(j) for these {i,j}: {1, 15325}, {11, 1737}, {80, 11545}, {908, 5123}, {1317, 1319}, {1319, 3911}, {3583, 12019}, {4511, 3035}, {5176, 3036}, {6882, 12619}, {11813, 6702}, {15326, 1155}, {17757, 10}, {21578, 5122}
X(40663) = X(i)-Ceva conjugate of X(j) for these (i,j): {14584, 1317}, {23703, 900}
X(40663) = X(21805)-cross conjugate of X(3943)
X(40663) = X(i)-isoconjugate of X(j) for these (i,j): {21, 106}, {29, 36058}, {58, 1320}, {60, 4674}, {81, 2316}, {88, 284}, {110, 23838}, {283, 36125}, {333, 9456}, {643, 23345}, {650, 4591}, {663, 4622}, {901, 3737}, {903, 2194}, {1019, 5548}, {1022, 5546}, {1043, 1417}, {1172, 1797}, {1333, 4997}, {1812, 8752}, {2150, 4080}, {2193, 6336}, {2341, 40215}, {3063, 4615}, {3257, 7252}, {4560, 32665}, {6740, 16944}, {9268, 18191}, {18155, 32719}, {31623, 32659}
X(40663) = crosspoint of X(i) and X(j) for these (i,j): {10, 38955}, {519, 38462}, {655, 4998}
X(40663) = crosssum of X(i) and X(j) for these (i,j): {58, 859}, {106, 36058}, {654, 3271}, {2194, 4282}
X(40663) = trilinear pole of line {4120, 30572}
X(40663) = crossdifference of every pair of points on line {284, 7252}
X(40663) = barycentric product X(i)*X(j) for these {i,j}: {7, 3943}, {10, 3911}, {12, 16704}, {44, 1441}, {57, 3992}, {65, 4358}, {72, 37790}, {85, 21805}, {190, 30572}, {225, 3977}, {226, 519}, {306, 1877}, {307, 8756}, {313, 1404}, {321, 1319}, {349, 902}, {653, 14429}, {664, 4120}, {758, 14628}, {900, 4552}, {1020, 4768}, {1023, 4077}, {1214, 38462}, {1317, 4080}, {1400, 3264}, {1427, 4723}, {1446, 3689}, {1577, 23703}, {1639, 4566}, {2171, 30939}, {2325, 3668}, {3285, 34388}, {3649, 31011}, {3676, 4169}, {3762, 4551}, {3936, 14584}, {3952, 30725}, {4017, 24004}, {4554, 4730}, {4572, 14407}, {5298, 6539}, {5440, 40149}, {7178, 17780}, {17757, 40218}, {24816, 27809}, {26942, 37168}, {30588, 36920}
X(40663) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 4997}, {12, 4080}, {37, 1320}, {42, 2316}, {44, 21}, {65, 88}, {73, 1797}, {109, 4591}, {225, 6336}, {226, 903}, {519, 333}, {651, 4622}, {661, 23838}, {664, 4615}, {900, 4560}, {902, 284}, {1023, 643}, {1317, 16704}, {1319, 81}, {1400, 106}, {1402, 9456}, {1404, 58}, {1409, 36058}, {1441, 20568}, {1464, 40215}, {1635, 3737}, {1639, 7253}, {1647, 17197}, {1877, 27}, {1880, 36125}, {1960, 7252}, {2087, 18191}, {2171, 4674}, {2251, 2194}, {2325, 1043}, {3264, 28660}, {3285, 60}, {3689, 2287}, {3762, 18155}, {3911, 86}, {3943, 8}, {3952, 4582}, {3977, 332}, {3992, 312}, {4017, 1022}, {4120, 522}, {4169, 3699}, {4358, 314}, {4434, 27958}, {4551, 3257}, {4552, 4555}, {4554, 4634}, {4557, 5548}, {4559, 901}, {4730, 650}, {4783, 3975}, {4819, 391}, {4848, 31227}, {4895, 1021}, {4908, 4720}, {5298, 8025}, {5440, 1812}, {7178, 6548}, {7180, 23345}, {8756, 29}, {14407, 663}, {14429, 6332}, {14584, 24624}, {14628, 14616}, {16609, 27922}, {16704, 261}, {17780, 645}, {21805, 9}, {21821, 3689}, {21942, 6735}, {22086, 23189}, {22356, 283}, {23202, 2193}, {23344, 5546}, {23703, 662}, {24004, 7257}, {30572, 514}, {30725, 7192}, {30731, 7256}, {36920, 5235}, {37790, 286}, {38462, 31623}, {40172, 2341}
X(40663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5445, 140}, {1, 24914, 5433}, {1, 26446, 5432}, {2, 2099, 15950}, {3, 10573, 10950}, {8, 56, 10944}, {8, 1788, 56}, {8, 5435, 3476}, {10, 65, 12}, {10, 72, 21031}, {10, 3753, 3925}, {10, 3754, 442}, {10, 3822, 38058}, {10, 3919, 3822}, {10, 4848, 65}, {12, 65, 3649}, {35, 37730, 10543}, {40, 1837, 6284}, {46, 355, 7354}, {57, 3679, 5252}, {57, 5252, 5434}, {145, 7288, 1388}, {165, 30286, 5727}, {484, 1727, 12515}, {549, 37728, 37525}, {942, 10039, 15888}, {946, 17606, 7173}, {1145, 12832, 1317}, {1210, 3057, 37722}, {1210, 11362, 3057}, {1317, 5298, 1319}, {1319, 3911, 5298}, {1420, 3632, 37738}, {1478, 36279, 11246}, {1698, 3340, 11375}, {1788, 3476, 5435}, {2093, 5587, 1836}, {2362, 13911, 19028}, {3086, 12245, 2098}, {3212, 33298, 3665}, {3293, 37558, 2594}, {3336, 37710, 18990}, {3339, 9578, 10404}, {3361, 4668, 37709}, {3476, 5435, 56}, {3579, 10572, 15338}, {3584, 5425, 5719}, {3654, 5722, 5119}, {3754, 15556, 65}, {3869, 25005, 1329}, {3911, 36920, 1317}, {4295, 5818, 10895}, {4424, 37715, 4854}, {5119, 5722, 3058}, {5131, 9897, 36975}, {5657, 18391, 55}, {5790, 36279, 1478}, {5837, 8582, 25917}, {5903, 18395, 5}, {6702, 11813, 17533}, {6735, 18838, 10956}, {7280, 37706, 34773}, {7991, 9581, 12701}, {9956, 12047, 3614}, {10056, 15934, 37703}, {11010, 37702, 15171}, {11529, 31434, 17718}, {13973, 16232, 19027}, {18421, 19875, 5219}, {19860, 26066, 24953}, {24633, 26575, 30847}, {38042, 39542, 7951}

X(40664) = MIDPOINT OF X(5667) AND X(6761)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12 - 2*a^10*b^2 - 2*a^8*b^4 + 8*a^6*b^6 - 7*a^4*b^8 + 2*a^2*b^10 - 2*a^10*c^2 + 7*a^8*b^2*c^2 - 8*a^6*b^4*c^2 + 2*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - b^10*c^2 - 2*a^8*c^4 - 8*a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 4*a^2*b^6*c^4 + 4*b^8*c^4 + 8*a^6*c^6 + 2*a^4*b^2*c^6 - 4*a^2*b^4*c^6 - 6*b^6*c^6 - 7*a^4*c^8 + 2*a^2*b^2*c^8 + 4*b^4*c^8 + 2*a^2*c^10 - b^2*c^10) : :
X(40664) = 2 X[12096] - 3 X[23239], 2 X[34109] + X[38672]

X(40664) lies on the curve Q161 and these lines: {3, 1075}, {4, 64}, {30, 5667}, {55, 1148}, {56, 7049}, {107, 6000}, {140, 3462}, {186, 523}, {275, 389}, {324, 15053}, {376, 14361}, {378, 3168}, {395, 36303}, {396, 36302}, {436, 5890}, {450, 13754}, {1093, 1204}, {1249, 3524}, {1294, 11589}, {2052, 11438}, {2071, 35360}, {2322, 21162}, {2777, 34170}, {3357, 14249}, {3484, 16813}, {6524, 18931}, {6760, 38605}, {12096, 23239}, {14379, 15318}, {15045, 37124}, {15312, 39221}, {16080, 34329}, {16226, 36794}, {34109, 38672}, {37127, 37481}

X(40664) = midpoint of X(5667) and X(6761)
X(40664) = reflection of X(i) in X(j) for these {i,j}: {1294, 11589}, {6760, 38605}
X(40664) = X(1294)-Ceva conjugate of X(4)
X(40664) = cevapoint of X(1075) and X(5667)
X(40664) = barycentric product X(i)*X(j) for these {i,j}: {2052, 6760}, {16080, 38605}
X(40664) = barycentric quotient X(i)/X(j) for these {i,j}: {6760, 394}, {38605, 11064}
X(40664) = {X(6523),X(12250)}-harmonic conjugate of X(4)

X(40665) = X(4)X(6)∩X(17)X(6113)

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

X(40665) lies on the curve Q161 and these lines: {4, 6}, {17, 6113}, {30, 5668}, {125, 23714}, {140, 8837}, {298, 19772}, {395, 3130}, {396, 36296}, {470, 11243}, {471, 2993}, {523, 14446}, {1495, 23715}, {6000, 6110}, {6111, 18400}, {14634, 35469}

X(40665) = midpoint of X(5668) and X(38943)
X(40665) = reflection of X(40666) in X(1990)
X(40665) = crosspoint of X(13) and X(19775)
X(40665) = crosssum of X(15) and X(11244)
X(40665) = crossdifference of every pair of points on line {61, 520}

X(40666) = X(4)X(6)∩X(18)X(6112)

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

X(40666) lies on the curve Q161 and these lines: {4, 6}, {18, 6112}, {30, 5669}, {125, 23715}, {140, 8839}, {299, 19773}, {395, 36297}, {396, 3129}, {470, 2992}, {471, 11244}, {523, 14447}, {1495, 23714}, {6000, 6111}, {6110, 18400}, {14634, 35470}

X(40666) = midpoint of X(5669) and X(38944)
X(40666) = reflection of X(40665) in X(1990)
X(40666) = crosspoint of X(14) and X(19774)
X(40666) = crosssum of X(16) and X(11243)
X(40666) = crossdifference of every pair of points on line {62, 520}

X(40667) = X(16)X(17)∩X(30)X(8172)

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

X(40667) lies on the curve Q161 and these lines: {16, 17}, {30, 8172}, {299, 11145}, {395, 11087}, {396, 15802}, {523, 14446}, {524, 32036}, {532, 18803}, {930, 40668}, {11078, 11119}, {19294, 23714}, {23303, 36300}

X(40667) = midpoint of X(8172) and X(11600)
X(40667) = X(2981)-isoconjugate of X(3376)
X(40667) = barycentric product X(i)*X(j) for these {i,j}: {17, 532}, {299, 36304}, {396, 19779}, {11139, 14922}, {14446, 32036}
X(40667) = barycentric quotient X(i)/X(j) for these {i,j}: {17, 11117}, {396, 16771}, {532, 302}, {14446, 23872}, {19294, 11146}, {21461, 2380}, {23714, 473}, {30462, 6671}, {36304, 14}

X(40668) = X(15)X(18)∩X(30)X(8173)

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

X(40668) lies on the curve Q161 and these lines: {15, 18}, {30, 8173}, {298, 11146}, {395, 15778}, {396, 11082}, {523, 14447}, {524, 32037}, {533, 18804}, {930, 40667}, {11092, 11120}, {19295, 23715}, {23302, 36301}

X(40668) = midpoint of X(8173) and X(11601)
X(40668) = X(3383)-isoconjugate of X(6151)
X(40668) = barycentric product X(i)*X(j) for these {i,j}: {18, 533}, {298, 36305}, {395, 19778}, {11138, 14921}, {14447, 32037}
X(40668) = barycentric quotient X(i)/X(j) for these {i,j}: {18, 11118}, {395, 16770}, {533, 303}, {14447, 23873}, {19295, 11145}, {21462, 2381}, {23715, 472}, {30459, 6672}, {36305, 13}

X(40669) = X(4)X(5221)∩X(30)X(3464)

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

X(40669) lies on the curve Q161 and these lines: {4, 5221}, {30, 3464}, {140, 3468}, {226, 14873}, {523, 656}, {3649, 27555}

X(40669) = midpoint of X(3464) and X(34301)

leftri

Points associated with the pedal triangle of the centroid: X(40670)-X(40673)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 9, 2020.

Let T denote the pedal triangle of X(2); T is perspective to these triangles:

orthocentroidal, with perspector X(1992)
1st Ehrmann, with perspector X(1995)
Artzt, with perspector X(2)
infinite altitude, with persector X(2)
anti-Artzt, with perspector X(2)
Gemini 105 triangle, with perspector X(145)
Gemini 107 triangle, with perspector X(1992)

X(2)-of-T = X(373)
X(3)-of-T = X(597)
X(4)-of-T = X(29959)
X(5)-of-T = X(40670)
X(6)-of-T = X(3363)
X(15)-of-T = X(40671)
X(16)-of-T = X(40672)
X(20)-of-T = X(40673)
X(30)-of-T = X(3854)

X(40670) = NINE-POINT CENTER OF PEDAL TRIANGLE OF X(2)

Barycentrics    a^2*(a^4*b^2 - b^6 + a^4*c^2 + 4*a^2*b^2*c^2 + 4*b^4*c^2 + 4*b^2*c^4 - c^6) : :
X(40670) = 3 X[2] + X[9971], X[6] - 5 X[11451], 3 X[373] - X[597], 3 X[373] + X[29959], X[575] - 4 X[32205], X[599] + 3 X[5640], X[2979] - 5 X[3763], X[3060] + 3 X[21358], X[3589] + 2 X[9822], 11 X[5056] + X[37473], X[5480] - 3 X[14845], X[5890] + 3 X[10516], 2 X[6329] + X[14913], X[9969] + 2 X[34573], X[9971] - 3 X[16776], 2 X[10095] + X[40107], 2 X[12006] + X[18553], 11 X[15024] + X[15069], 5 X[15026] + X[34507], 3 X[20791] + X[36990], 3 X[21167] - X[36987], 2 X[24206] + X[32191]

X(40670) lies on these lines: {2, 9019}, {5, 2781}, {6, 11451}, {51, 141}, {83, 16175}, {182, 15580}, {373, 597}, {511, 547}, {524, 5943}, {542, 13363}, {575, 32205}, {599, 5640}, {1154, 24206}, {1503, 5892}, {1576, 21513}, {1995, 19127}, {2393, 3589}, {2871, 34236}, {2930, 5643}, {2979, 3763}, {3060, 21358}, {3819, 9969}, {5020, 19153}, {5056, 37473}, {5480, 14845}, {5544, 8547}, {5663, 25561}, {5890, 10516}, {5946, 11178}, {6329, 14913}, {6593, 16042}, {8254, 25555}, {9027, 20583}, {10095, 40107}, {11746, 20113}, {12006, 18553}, {15024, 15069}, {15026, 34507}, {15435, 18950}, {15581, 15805}, {20791, 36990}, {21167, 36987}, {23048, 38317}, {25488, 35370}, {34990, 37338}

X(40670) = midpoint of X(i) and X(j) for these {i,j}: {2, 16776}, {51, 141}, {597, 29959}, {3819, 9969}, {5946, 11178}, {6688, 9822}
X(40670) = reflection of X(i) in X(j) for these {i,j}: {3589, 6688}, {3819, 34573}
X(40670) = {X(373),X(29959)}-harmonic conjugate of X(597)

X(40671) = 1ST ISODYNAMIC POINT OF PEDAL TRIANGLE OF X(2)

Barycentrics    4*a^4 + 5*a^2*b^2 - 5*b^4 + 5*a^2*c^2 + 14*b^2*c^2 - 5*c^4 + 2*Sqrt[3]*(4*a^2 + b^2 + c^2)*S : :

X(40671) lies on these lines: {2, 13}, {6, 22574}, {14, 8593}, {17, 9116}, {18, 33398}, {115, 597}, {395, 33477}, {396, 543}, {524, 5472}, {542, 31693}, {671, 12154}, {1992, 9112}, {2482, 33475}, {5077, 22513}, {5182, 5470}, {5471, 8787}, {6771, 35303}, {6772, 11159}, {6778, 11161}, {7603, 9115}, {8370, 36251}, {8599, 27551}, {9885, 16644}, {10654, 22576}, {11054, 37786}, {11121, 22487}, {11153, 36764}, {11295, 25154}, {11303, 38664}, {11317, 31710}, {14711, 25183}, {16001, 37340}, {20415, 37341}, {32907, 37352}

X(40671) = reflection of X(6115) in X(5459)
X(40671) = circumcircle-of-inner-Napoleon-triangle-inverse of X(22492)
X(40671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8595, 618}, {13, 5463, 22492}, {115, 18800, 40672}, {597, 3363, 40672}

X(40672) = 2ND ISODYNAMIC POINT OF PEDAL TRIANGLE OF X(2)

Barycentrics    4*a^4 + 5*a^2*b^2 - 5*b^4 + 5*a^2*c^2 + 14*b^2*c^2 - 5*c^4 - 2*Sqrt[3]*(4*a^2 + b^2 + c^2)*S : :

X(40672) lies on these lines: {2, 14}, {6, 22573}, {13, 8593}, {17, 33399}, {18, 9114}, {115, 597}, {395, 543}, {396, 33476}, {524, 5471}, {542, 31694}, {671, 12155}, {1992, 9113}, {2482, 33474}, {5077, 22512}, {5182, 5469}, {5472, 8787}, {6774, 35304}, {6775, 11159}, {6777, 11161}, {7603, 9117}, {8370, 36252}, {8599, 27550}, {9886, 16645}, {10653, 22575}, {11054, 37785}, {11122, 22488}, {11296, 25164}, {11304, 38664}, {11317, 31709}, {14711, 25187}, {16002, 37341}, {20416, 37340}, {32909, 37351}

X(40672) = reflection of X(6114) in X(5460)
X(40672) = {circumcircle-of-outer-Napoleon-triangle-inverse of X(22491)}
X(40672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8594, 619}, {14, 5464, 22491}, {115, 18800, 40671}, {597, 3363, 40671}

X(40673) = DE LONGCHAMPS POINT OF PEDAL TRIANGLE OF X(2)

Barycentrics    a^2*(a^4*b^2 - b^6 + a^4*c^2 - 8*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :
X(40673) = X[3] + 2 X[32284], 4 X[6] - X[1843], 2 X[6] + X[6467], 5 X[6] - 2 X[9969], 3 X[6] - X[9971], 7 X[6] - X[9973], X[6] - 4 X[22829], X[6] + 2 X[32366], 5 X[51] - 4 X[9969], 3 X[51] - 2 X[9971], 7 X[51] - 2 X[9973], X[51] - 8 X[22829], X[51] + 4 X[32366], X[52] + 2 X[15074], X[185] - 4 X[8550], X[193] + 2 X[11574], 3 X[373] - 4 X[597], 3 X[373] - 2 X[29959], 2 X[389] + X[15073], 2 X[599] - 3 X[5650], X[1205] + 2 X[5095], 2 X[1353] + X[9967], X[1843] + 2 X[6467], 5 X[1843] - 8 X[9969], 3 X[1843] - 4 X[9971], 7 X[1843] - 4 X[9973], X[1843] - 16 X[22829], X[1843] + 8 X[32366], X[3060] - 3 X[5032], X[3313] + 2 X[3629], 5 X[3618] - 4 X[6688], 5 X[3618] - 2 X[14913], 4 X[4663] - X[16980], 3 X[5050] - 2 X[5892], 2 X[5446] - 5 X[11482], X[5890] - 3 X[14912], 5 X[6467] + 4 X[9969], 3 X[6467] + 2 X[9971], 7 X[6467] + 2 X[9973], X[6467] + 8 X[22829], X[6467] - 4 X[32366], 2 X[6776] + X[12294], 3 X[7998] - X[11160], 4 X[8548] - X[21651], 4 X[8584] - X[21969], 4 X[9822] - 5 X[11451], 4 X[9822] - X[12272], 6 X[9969] - 5 X[9971], 14 X[9969] - 5 X[9973], X[9969] - 10 X[22829], X[9969] + 5 X[32366], 7 X[9971] - 3 X[9973], X[9971] - 12 X[22829], X[9971] + 6 X[32366], X[9973] - 28 X[22829], X[9973] + 14 X[32366], 5 X[11451] - X[12272], 4 X[12007] - X[19161], 3 X[14845] - 4 X[18583], 4 X[15118] - X[32260], X[17710] + 2 X[32455], 2 X[22829] + X[32366]

X(40673) lies on these lines: {2, 8681}, {3, 32284}, {6, 25}, {39, 682}, {52, 15074}, {54, 575}, {69, 3819}, {182, 32127}, {185, 1205}, {193, 2979}, {373, 597}, {376, 511}, {389, 15073}, {524, 3917}, {542, 12022}, {567, 39562}, {569, 8548}, {576, 7592}, {578, 10250}, {599, 5650}, {800, 20775}, {1154, 1353}, {1181, 11470}, {1199, 8537}, {1351, 8717}, {1503, 32062}, {1587, 6291}, {1588, 6406}, {1993, 11511}, {1994, 11416}, {2386, 7739}, {3060, 5032}, {3148, 33871}, {3284, 34396}, {3313, 3629}, {3398, 22143}, {3564, 5891}, {3618, 6688}, {4558, 13335}, {4663, 16980}, {5012, 37784}, {5013, 40321}, {5050, 5892}, {5097, 37925}, {5254, 8754}, {5286, 40325}, {5422, 9813}, {5446, 11482}, {5486, 35371}, {5622, 11430}, {5943, 11188}, {6000, 6776}, {6248, 25051}, {7827, 16175}, {7998, 11160}, {8263, 32114}, {8538, 12161}, {8546, 19127}, {8549, 11424}, {8584, 9019}, {8705, 20583}, {9730, 14984}, {9822, 11451}, {10510, 15135}, {10765, 30534}, {11423, 22330}, {11427, 18919}, {11443, 27365}, {11455, 39874}, {11596, 32467}, {12007, 19161}, {14644, 25561}, {14845, 18583}, {14853, 23048}, {15087, 18449}, {15118, 32260}, {15121, 19510}, {17710, 32455}, {18435, 39899}, {18553, 32255}, {18912, 34507}, {18935, 32064}, {19126, 40318}, {22112, 38396}, {22151, 34986}, {26879, 40107}, {32245, 34470}, {34854, 40138}

X(40673) = midpoint of X(i) and X(j) for these {i,j}: {2, 15531}, {51, 6467}, {193, 2979}, {11455, 39874}, {18435, 39899}
X(40673) = reflection of X(i) in X(j) for these {i,j}: {51, 6}, {69, 3819}, {1843, 51}, {2979, 11574}, {11188, 5943}, {14913, 6688}, {29959, 597}
X(40673) = isogonal conjugate of the isotomic conjugate of X(30739)
X(40673) = X(30247)-Ceva conjugate of X(647)
X(40673) = crosspoint of X(i) and X(j) for these (i,j): {4, 21448}, {6, 5486}, {25, 36878}
X(40673) = crosssum of X(i) and X(j) for these (i,j): {2, 1995}, {3, 1992}
X(40673) = barycentric product X(6)*X(30739)
X(40673) = barycentric quotient X(30739)/X(76)
X(40673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 6467, 1843}, {6, 10602, 8541}, {6, 19459, 1974}, {6, 32366, 6467}, {6, 32621, 184}, {597, 29959, 373}, {13366, 21639, 6}, {22829, 32366, 6}

leftri

Points associated with midcevian triangles: X(40674)-X(40696)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 9, 2020.

Let U = u : v : w be a point in the plane of a triangle ABC. Let A'B'C' be the cevian triangle of U, and let A'' be the midpoint of the segment AA'. Define B'' and C'' cyclically, so that

A'' = v + w : v : w
B'' = u : w + u : w
C'' = u : v : u + v.

The triangle A''B''C'' is here named the U-midcevian triangle. Examples include

X(1)-midcevian triangle = Gemini triangle 15
X(2)-midcevian triangle = Gemini triangle 110
X(4)-midcevian triangle = half-altitude triangle
X(7)-midcevian triangle = 1st Zaniah triangle
X(8)-midcevian triangle = 2nd Zaniah triangle
X(69)-midcevian triangle = orthic-of-medial triangle = anti-6th-mixtilinear = anticomplement of submedial triangle; see X(11363)
X(75)-midcevian triangle = Gemini triangle 16 = complement of incentral triangle = n(Incentral)*n(Medial) (ETC preamble before X(3739))
X(523)-midcevian triangle = anticevian triangle of X(523) = Schroeter triangle (ETC X(8286) = diagonal triangle of Feuerbach quadrangle of ABC (ETC X(10276)

In general, the U-midcevian triangle is perspective to the following triangles:

ABC, with perspector U
medial triangle, with perspector u v + u w : :
Wasat triangle (see X(21616)), with perspector a u (b + c) - (b v - c w)(b - c) : :
Gemini 7 triangle, with perspector a u (a - b - c) - (b - c)((a - b + c) v + (a + b - c) w) : :

Let T(U) denote the midcevian triangle of U, and let C(U) denote the cevian triangle of U.

The locus of a point X such that T(U) is perspective to C(X) is the cubic pK(X(2),U*), where U* = isotomic conjugate of U.

The locus of X such that T(U) is perspective to the anticevian triangle of X is the cubic pK(u*(v + w) : : , -u + v + w : :). For example, if U = X(3), then the cubic is K044.

The locus of X such that T(X(1)) is perspective to C(X) is the cubic K034.
The locus of X such that T(X(3)) is perspective to C(X) is the cubic K045.
The locus of X such that T(X(4)) is perspective to C(X) is the cubic K007.
The locus of X such that T(X(6)) is perspective to C(X) is the cubic K141.
The locus of X such that T(X(7)) is perspective to C(X) is the cubic K200.
The locus of X such that T(X(8)) is perspective to C(X) is the cubic K1078.
The locus of X such that T(X13)) is perspective to C(X) is the cubic K264a.
The locus of X such that T(X(14)) is perspective to C(X) is the cubic K264b.

X(40674) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND CIRCUM-MEDIAL

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

X(40674) lies on these lines: {3, 251}, {25, 32078}, {112, 7485}, {216, 1194}, {1368, 6032}, {2548, 7386}, {7499, 22240}, {8879, 39575}, {10691, 15302}, {14570, 40022}, {17409, 37126}

X(40675) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND 2ND EXCOSINE

Barycentrics    a^2*(a^2 - b^2 - c^2)*(3*a^12 - 6*a^10*b^2 + a^8*b^4 - 4*a^6*b^6 + 21*a^4*b^8 - 22*a^2*b^10 + 7*b^12 - 6*a^10*c^2 + 14*a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 36*a^4*b^6*c^2 + 34*a^2*b^8*c^2 - 10*b^10*c^2 + a^8*c^4 + 4*a^6*b^2*c^4 + 30*a^4*b^4*c^4 - 12*a^2*b^6*c^4 - 23*b^8*c^4 - 4*a^6*c^6 - 36*a^4*b^2*c^6 - 12*a^2*b^4*c^6 + 52*b^6*c^6 + 21*a^4*c^8 + 34*a^2*b^2*c^8 - 23*b^4*c^8 - 22*a^2*c^10 - 10*b^2*c^10 + 7*c^12) : :

X(40675) lies on these lines: {2, 3183}, {3, 64}, {4, 20208}, {5, 6525}, {30, 35711}, {590, 22838}, {615, 22839}, {1033, 7395}, {1368, 17830}, {2130, 39268}, {2972, 3516}, {3348, 15394}, {3851, 10745}, {5020, 18288}, {5562, 15905}, {10319, 17831}, {12164, 23163}, {17928, 34993}, {18017, 18405}, {20410, 38689}, {26937, 37072}

X(40676) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND 1ST ANTI-ORTHOSYMMEDIAL

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^22 - 2*a^20*b^2 - a^18*b^4 + 2*a^16*b^6 + 2*a^14*b^8 + 4*a^12*b^10 - 10*a^10*b^12 - 4*a^8*b^14 + 13*a^6*b^16 - 2*a^4*b^18 - 5*a^2*b^20 + 2*b^22 - 2*a^20*c^2 - 4*a^18*b^2*c^2 + 14*a^16*b^4*c^2 - 5*a^14*b^6*c^2 - 5*a^12*b^8*c^2 + 19*a^10*b^10*c^2 - 29*a^8*b^12*c^2 - 7*a^6*b^14*c^2 + 27*a^4*b^16*c^2 - 3*a^2*b^18*c^2 - 5*b^20*c^2 - a^18*c^4 + 14*a^16*b^2*c^4 + 3*a^14*b^4*c^4 - 25*a^12*b^6*c^4 + 5*a^10*b^8*c^4 + 23*a^8*b^10*c^4 - 31*a^6*b^12*c^4 - 11*a^4*b^14*c^4 + 24*a^2*b^16*c^4 - b^18*c^4 + 2*a^16*c^6 - 5*a^14*b^2*c^6 - 25*a^12*b^4*c^6 + 4*a^10*b^6*c^6 + 10*a^8*b^8*c^6 + 27*a^6*b^10*c^6 - 13*a^4*b^12*c^6 - 10*a^2*b^14*c^6 + 10*b^16*c^6 + 2*a^14*c^8 - 5*a^12*b^2*c^8 + 5*a^10*b^4*c^8 + 10*a^8*b^6*c^8 - 4*a^6*b^8*c^8 - a^4*b^10*c^8 - 3*a^2*b^12*c^8 - 4*b^14*c^8 + 4*a^12*c^10 + 19*a^10*b^2*c^10 + 23*a^8*b^4*c^10 + 27*a^6*b^6*c^10 - a^4*b^8*c^10 - 6*a^2*b^10*c^10 - 2*b^12*c^10 - 10*a^10*c^12 - 29*a^8*b^2*c^12 - 31*a^6*b^4*c^12 - 13*a^4*b^6*c^12 - 3*a^2*b^8*c^12 - 2*b^10*c^12 - 4*a^8*c^14 - 7*a^6*b^2*c^14 - 11*a^4*b^4*c^14 - 10*a^2*b^6*c^14 - 4*b^8*c^14 + 13*a^6*c^16 + 27*a^4*b^2*c^16 + 24*a^2*b^4*c^16 + 10*b^6*c^16 - 2*a^4*c^18 - 3*a^2*b^2*c^18 - b^4*c^18 - 5*a^2*c^20 - 5*b^2*c^20 + 2*c^22) : :

X(40676) lies on these lines: {112, 37126}, {1297, 7499}, {10749, 12362}, {12145, 21284}

X(40677) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND WASAT

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

X(40677) lies on these lines: {2, 1726}, {10, 21243}, {48, 20268}, {142, 6678}, {212, 29307}, {226, 1465}, {321, 21429}, {379, 2140}, {908, 33113}, {946, 4314}, {1751, 24789}, {1848, 24220}, {2339, 25527}, {3452, 4422}, {6260, 16388}, {13478, 37695}, {14213, 21072}, {21375, 28776}, {24618, 26724}

X(40678) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND CEVIAN TRIANGLE OF X(254)

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

X(40678) lies on these lines: {2, 254}, {3, 34428}, {5, 8800}, {6, 1147}, {24, 12095}, {136, 11585}, {1594, 16172}, {2383, 7488}, {3133, 14576}, {6504, 7401}, {9818, 15827}, {40674, 40678}

X(40678) = complement of X(40698)

X(40679) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND GEMINI 80

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

X(40679) lies on these lines: {2, 1068}, {3, 31}, {3144, 17080}

X(40680) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND GEMINI 82

Barycentrics    (a^2 - b^2 - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - 6*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) : :
Barycentrics    cot A - tan B - tan C : :

X(40680) lies on these lines: {2, 216}, {3, 69}, {4, 20477}, {6, 34828}, {20, 317}, {76, 7400}, {83, 28717}, {95, 253}, {141, 36751}, {157, 5596}, {183, 7494}, {193, 577}, {286, 6847}, {309, 17095}, {311, 3547}, {322, 6350}, {325, 7386}, {339, 1272}, {340, 3522}, {376, 32001}, {401, 3087}, {441, 3618}, {464, 4417}, {491, 1590}, {492, 1589}, {524, 36748}, {590, 19439}, {615, 19438}, {631, 32000}, {1007, 1368}, {1232, 32836}, {1235, 7383}, {1494, 15692}, {1656, 8797}, {1975, 10996}, {1976, 6394}, {1992, 15905}, {3146, 32002}, {3260, 3546}, {3537, 32817}, {3538, 32818}, {3548, 32839}, {3549, 32838}, {3619, 20208}, {3620, 10979}, {4648, 21940}, {5054, 36889}, {5224, 25876}, {6146, 10608}, {6349, 19804}, {6617, 18928}, {6639, 32883}, {6640, 32884}, {6641, 14826}, {6643, 32816}, {6676, 34229}, {7763, 14615}, {7782, 22468}, {7803, 28425}, {8573, 11433}, {10565, 26880}, {11206, 33582}, {12362, 32006}, {14555, 21482}, {14853, 30258}, {15705, 35510}, {17102, 17321}, {17234, 25932}, {18531, 32827}, {18589, 29965}, {20080, 22052}, {20563, 34853}, {27377, 35941}, {30771, 34803}

X(40680) = isogonal conjugate of polar conjugate of trilinear product of vertices of anti-Atik triangle
X(40680) = isotomic conjugate of X(1217)

X(40681) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND GEMINI 84

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

X(40681) lies on these lines: {2, 216}, {3, 1176}, {566, 34828}, {570, 37188}, {3618, 28696}, {6643, 26870}, {7485, 35211}, {11174, 28701}, {26216, 36794}, {34990, 36751}

X(40682) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND 3RD ISODYNAMIC-DAO EQUILATERAL

Barycentrics    (a^2 - b^2 - c^2)*((a^4 - b^4 - 2*a^2*b*c + 2*b^2*c^2 - c^4)*(a^4 - b^4 + 2*a^2*b*c + 2*b^2*c^2 - c^4) + 2*Sqrt[3]*a^4*(a^2 - b^2 - c^2)*S) : :

X(40682) lies on these lines: {3, 13}, {5, 31687}, {30, 35714}, {115, 577}, {216, 5472}, {465, 5459}, {466, 530}, {590, 31689}, {615, 31692}, {1368, 6108}, {2058, 18403}, {3129, 12142}, {3165, 5972}, {6115, 6676}, {12362, 36251}

X(40682) = {X(577),X(18531)}-harmonic conjugate of X(40683)

X(40683) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(3) AND 4TH ISODYNAMIC-DAO EQUILATERAL

Barycentrics    (a^2 - b^2 - c^2)*((a^4 - b^4 - 2*a^2*b*c + 2*b^2*c^2 - c^4)*(a^4 - b^4 + 2*a^2*b*c + 2*b^2*c^2 - c^4) - 2*Sqrt[3]*a^4*(a^2 - b^2 - c^2)*S) : :

X(40683) lies on these lines: {3, 14}, {5, 31688}, {30, 35715}, {115, 577}, {216, 5471}, {465, 531}, {466, 5460}, {590, 31691}, {615, 31690}, {1368, 6109}, {2059, 18403}, {3130, 12141}, {3166, 5972}, {6114, 6676}, {12362, 36252}

X(40683) = {X(577),X(18531)}-harmonic conjugate of X(40682)

X(40684) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(5) AND MACBEATH

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    1 + 2 sec A sin B sin C : :

X(40684) lies on these lines: {2, 216}, {4, 1216}, {5, 6662}, {92, 27131}, {97, 276}, {140, 14978}, {141, 467}, {275, 323}, {297, 14129}, {311, 394}, {338, 14920}, {340, 15108}, {343, 3260}, {458, 1235}, {511, 30506}, {648, 34545}, {850, 38240}, {1075, 15028}, {1093, 5056}, {1232, 6748}, {1629, 6636}, {1656, 13450}, {1896, 37162}, {1947, 27003}, {1948, 27065}, {1994, 36794}, {2972, 11197}, {3108, 16081}, {3168, 11451}, {3266, 18022}, {3917, 39530}, {5012, 37124}, {5066, 34334}, {5068, 14249}, {5392, 37645}, {5422, 9308}, {5943, 35360}, {6194, 6995}, {6515, 6819}, {6530, 37990}, {6531, 34945}, {6747, 24206}, {7485, 33971}, {8884, 37126}, {13366, 35311}, {14566, 14618}, {14768, 35325}, {14918, 37636}, {15526, 34836}, {18026, 26842}, {20477, 37068}, {32142, 35719}, {34289, 38253}

X(40684) = isotomic conjugate of X(31626)
X(40684) = polar conjugate of X(1173)
X(40684) = crosspoint of [polar conjugate of X(61)] and [polar conjugate of X(62)]

X(40685) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(5) AND ANTI-ORTHOCENTROIDAL

Barycentrics    2*a^10 - 5*a^8*b^2 + 10*a^4*b^6 - 10*a^2*b^8 + 3*b^10 - 5*a^8*c^2 + 14*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 13*a^2*b^6*c^2 - 9*b^8*c^2 - 13*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 6*b^6*c^4 + 10*a^4*c^6 + 13*a^2*b^2*c^6 + 6*b^4*c^6 - 10*a^2*c^8 - 9*b^2*c^8 + 3*c^10 : :

X(40685) lies on these lines: {2, 399}, {3, 11801}, {5, 74}, {30, 6699}, {67, 38110}, {110, 632}, {113, 547}, {125, 128}, {141, 9976}, {146, 5055}, {265, 549}, {323, 10821}, {381, 14677}, {389, 30531}, {427, 11566}, {541, 10109}, {542, 10124}, {546, 12041}, {548, 10113}, {550, 14644}, {631, 34153}, {1216, 13358}, {1539, 5066}, {2771, 3634}, {2777, 3850}, {2914, 6143}, {2929, 11250}, {3054, 14901}, {3090, 10620}, {3091, 15041}, {3448, 3526}, {3523, 12902}, {3525, 32609}, {3530, 17702}, {3533, 14683}, {3545, 38790}, {3564, 6698}, {3581, 37938}, {3589, 25556}, {3620, 39562}, {3627, 15055}, {3628, 5663}, {3845, 20127}, {3851, 12244}, {3853, 16111}, {3857, 15021}, {3858, 10721}, {3861, 34584}, {5054, 12383}, {5056, 38789}, {5159, 12358}, {5498, 11430}, {5844, 11735}, {5892, 11561}, {5972, 16239}, {6000, 15350}, {7486, 15046}, {7984, 38112}, {8703, 10733}, {8994, 13979}, {9140, 11539}, {10065, 10593}, {10081, 10592}, {10114, 25401}, {10224, 11438}, {10303, 15040}, {10627, 11800}, {10628, 12006}, {11231, 13605}, {11487, 19348}, {11557, 13363}, {11591, 11806}, {11699, 19862}, {11709, 18357}, {11749, 14993}, {11807, 13364}, {12100, 16163}, {12103, 12295}, {12108, 36253}, {12121, 15712}, {12133, 37942}, {12140, 37935}, {12227, 34331}, {12270, 40280}, {12375, 32789}, {12376, 32790}, {12812, 36518}, {13211, 38028}, {13289, 23332}, {13413, 32743}, {13417, 15026}, {13418, 21230}, {13915, 13969}, {14805, 26913}, {14869, 15027}, {15025, 15704}, {15032, 15806}, {15067, 21649}, {15101, 16223}, {15118, 34380}, {15357, 34127}, {15699, 20126}, {16881, 32144}, {18281, 37643}, {18377, 37487}, {20417, 35018}, {21167, 32273}, {21315, 36164}, {22251, 23236}, {23302, 36208}, {23303, 36209}, {23306, 33533}, {32226, 36153}

X(40685) = complement of X(10272)

X(40686) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(5) AND 1ST EXCOSINE

Barycentrics    a^10 - 2*a^8*b^2 - 2*a^6*b^4 + 8*a^4*b^6 - 7*a^2*b^8 + 2*b^10 - 2*a^8*c^2 + 12*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 6*b^8*c^2 - 2*a^6*c^4 - 8*a^4*b^2*c^4 + 6*a^2*b^4*c^4 + 4*b^6*c^4 + 8*a^4*c^6 + 4*a^2*b^2*c^6 + 4*b^4*c^6 - 7*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

X(40686) lies on these lines: {2, 1498}, {3, 161}, {4, 1192}, {5, 64}, {6, 3541}, {20, 18405}, {24, 35217}, {30, 8567}, {66, 5085}, {68, 37497}, {74, 7547}, {125, 1593}, {140, 154}, {155, 18281}, {185, 5094}, {221, 498}, {378, 23294}, {381, 3357}, {382, 23325}, {427, 9786}, {485, 19087}, {486, 19088}, {499, 2192}, {546, 20427}, {549, 9833}, {578, 26944}, {599, 8549}, {631, 1503}, {1075, 15274}, {1092, 15069}, {1181, 37119}, {1204, 7507}, {1350, 23300}, {1352, 16196}, {1587, 13980}, {1588, 8991}, {1594, 10605}, {1595, 17810}, {1620, 18533}, {1656, 6000}, {1657, 11204}, {1698, 6001}, {1737, 1854}, {1899, 11425}, {2777, 3843}, {2781, 3567}, {2883, 3090}, {3088, 13567}, {3091, 15311}, {3146, 23324}, {3147, 16655}, {3515, 11550}, {3520, 18396}, {3523, 32064}, {3525, 10192}, {3526, 6759}, {3534, 34786}, {3538, 36851}, {3542, 15811}, {3545, 5893}, {3546, 11487}, {3548, 17814}, {3575, 37487}, {3614, 12940}, {3624, 40658}, {3851, 22802}, {5054, 10282}, {5055, 13093}, {5056, 6225}, {5067, 5656}, {5068, 15105}, {5070, 12315}, {5073, 18376}, {5418, 17819}, {5420, 17820}, {5449, 12085}, {5587, 12262}, {5654, 32144}, {5663, 31283}, {5886, 7973}, {5890, 12300}, {5907, 30771}, {6143, 11456}, {6293, 9730}, {6353, 16621}, {6624, 35711}, {6640, 18451}, {6697, 10516}, {7173, 12950}, {7378, 11745}, {7401, 34944}, {7404, 17825}, {7506, 10117}, {7512, 15578}, {7529, 18488}, {7552, 20391}, {7566, 15053}, {7729, 12162}, {7741, 10060}, {7951, 10076}, {7988, 9899}, {8252, 12970}, {8253, 12964}, {8254, 17824}, {8889, 12233}, {9934, 34128}, {10175, 12779}, {10249, 11457}, {10264, 12161}, {10303, 11206}, {10519, 15583}, {10574, 31236}, {10620, 32743}, {10625, 34751}, {10982, 26879}, {11202, 14864}, {11250, 12293}, {11410, 21659}, {11411, 37672}, {11413, 23293}, {11424, 26869}, {11439, 15059}, {11442, 35602}, {11468, 35480}, {11477, 23327}, {11572, 37196}, {11598, 14644}, {11744, 23515}, {12084, 13561}, {12111, 30744}, {12163, 13371}, {12173, 21663}, {12174, 13399}, {12241, 23291}, {12325, 40341}, {12359, 37498}, {12902, 25564}, {13293, 38724}, {13568, 18931}, {14070, 20191}, {14528, 31804}, {14530, 15694}, {15030, 31978}, {15041, 19506}, {15058, 31282}, {15063, 15113}, {15116, 16010}, {15126, 15138}, {15131, 16003}, {15238, 20208}, {15559, 20300}, {15873, 37643}, {16195, 29012}, {16266, 17823}, {17809, 18914}, {17834, 23335}, {17835, 23315}, {18909, 23292}, {19843, 20307}, {20376, 32337}, {23336, 32140}, {25739, 35477}, {26883, 37453}, {26917, 35502}, {31074, 32351}, {31423, 40660}, {31489, 32445}, {36201, 38729}

X(40687) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(5) AND WASAT

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

X(40687) lies on these lines: {2, 21361}, {10, 3819}, {57, 24179}, {81, 24618}, {142, 6678}, {226, 4896}, {946, 15325}, {1746, 17074}, {1764, 17077}, {2051, 3911}, {2140, 9776}, {3218, 22000}, {3452, 17332}, {3752, 17197}, {4416, 30006}, {5435, 10478}, {11019, 39543}, {14829, 22020}, {16551, 28951}, {17167, 27003}, {17182, 24627}, {17761, 24177}, {22019, 32939}, {28748, 29529}, {30035, 30567}, {30097, 39595}

X(40688) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(5) AND GEMINI 7

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

X(40688) lies on these lines: {1, 34612}, {2, 45}, {7, 4383}, {11, 17063}, {12, 24174}, {38, 3826}, {42, 25557}, {57, 1723}, {63, 17278}, {65, 24178}, {81, 17366}, {141, 4359}, {142, 3666}, {210, 24231}, {222, 5723}, {226, 16610}, {238, 11246}, {241, 24181}, {244, 2886}, {277, 2982}, {306, 3834}, {321, 7263}, {354, 1738}, {377, 17054}, {442, 24046}, {443, 37549}, {474, 24159}, {495, 1739}, {497, 7613}, {537, 4126}, {553, 3008}, {594, 33172}, {614, 5880}, {748, 17768}, {750, 17061}, {908, 16602}, {940, 4000}, {978, 3649}, {982, 3925}, {1054, 5432}, {1211, 3662}, {1266, 3175}, {1376, 17724}, {1407, 37800}, {1427, 30379}, {1647, 3829}, {1714, 5708}, {1722, 10404}, {1724, 24470}, {1836, 5272}, {1999, 37756}, {2185, 24617}, {2550, 17597}, {2999, 6173}, {3035, 33127}, {3058, 24715}, {3120, 3816}, {3187, 4395}, {3210, 17234}, {3216, 6147}, {3218, 26724}, {3219, 17337}, {3305, 17276}, {3306, 3772}, {3315, 33110}, {3474, 16020}, {3616, 19336}, {3670, 8728}, {3677, 38052}, {3703, 3836}, {3712, 29642}, {3742, 3914}, {3752, 5249}, {3755, 4883}, {3756, 9335}, {3763, 19822}, {3812, 23536}, {3822, 24168}, {3841, 24167}, {3929, 31183}, {3932, 17155}, {3946, 37595}, {3953, 31419}, {3999, 4847}, {4001, 17348}, {4023, 33064}, {4026, 33125}, {4028, 4706}, {4046, 33087}, {4310, 26040}, {4413, 33144}, {4417, 24620}, {4423, 24248}, {4648, 20182}, {4654, 23511}, {4675, 5256}, {4850, 17056}, {4854, 26102}, {4860, 33137}, {4862, 7308}, {4886, 17288}, {4966, 32860}, {4995, 29675}, {5121, 17605}, {5219, 8056}, {5241, 27184}, {5284, 33102}, {5287, 17301}, {5294, 17356}, {5433, 24161}, {5435, 6354}, {5437, 17720}, {5439, 23537}, {5440, 26728}, {5573, 17721}, {5721, 10202}, {5739, 7232}, {5743, 17184}, {5836, 23675}, {5905, 37679}, {6154, 17715}, {6703, 26627}, {7228, 26223}, {7238, 32859}, {7292, 20292}, {7321, 27064}, {7336, 34583}, {9342, 33153}, {9352, 29681}, {9965, 37650}, {10589, 38357}, {11112, 30117}, {11375, 11512}, {12436, 37539}, {13747, 24160}, {15888, 24440}, {16569, 33103}, {16736, 17167}, {16752, 40153}, {16753, 17173}, {16823, 33068}, {16885, 20078}, {17011, 17392}, {17019, 17395}, {17050, 37596}, {17064, 17728}, {17067, 37520}, {17070, 29662}, {17074, 37771}, {17122, 17602}, {17123, 32857}, {17124, 33143}, {17125, 33098}, {17147, 17243}, {17165, 24988}, {17245, 28606}, {17265, 17776}, {17277, 26840}, {17282, 32777}, {17291, 19808}, {17292, 19797}, {17293, 19825}, {17334, 27065}, {17362, 32863}, {17365, 26842}, {17483, 37680}, {17484, 37687}, {17490, 18134}, {17495, 18139}, {17775, 31019}, {18201, 33138}, {18635, 19788}, {19512, 21375}, {19785, 37674}, {20255, 20913}, {21342, 25006}, {21949, 26015}, {24169, 24325}, {24199, 31993}, {24200, 29653}, {24443, 25466}, {24693, 29652}, {24779, 37543}, {24911, 25448}, {25351, 29673}, {25502, 33154}, {26007, 36538}, {27003, 33129}, {28244, 30007}, {29851, 32845}, {30950, 33145}, {31151, 32866}, {31252, 33164}, {33150, 37633}

X(40689) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(6) AND EXCENTRAL OF TANGENTIAL

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

X(40689) lies on these lines: {3, 3589}, {25, 39}, {32, 39653}, {159, 9605}, {1486, 25066}, {1576, 30435}, {1598, 8721}, {1995, 3926}, {5020, 7795}, {7506, 10983}, {7772, 19459}, {7800, 37491}, {7822, 11284}, {8362, 37485}, {9914, 40053}, {9969, 23115}, {11414, 37479}, {27802, 37592}

X(40689) = eigencenter of Ara triangle

X(40690) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(6) AND WASAT

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

X(40690) lies on these lines: {1, 5074}, {2, 1759}, {10, 626}, {31, 20267}, {41, 4056}, {46, 30742}, {65, 116}, {101, 4911}, {142, 3647}, {226, 241}, {315, 30108}, {519, 4950}, {758, 17046}, {908, 29596}, {946, 15251}, {1125, 25497}, {1155, 24784}, {1770, 17729}, {1836, 14377}, {1930, 4153}, {2051, 36907}, {2140, 12047}, {3120, 24790}, {3585, 9317}, {3673, 24045}, {3674, 5179}, {3730, 7179}, {3741, 30954}, {3825, 17048}, {3835, 21201}, {3878, 17062}, {3997, 24211}, {4129, 34959}, {4251, 4872}, {4253, 17181}, {4797, 6680}, {4920, 16600}, {5011, 33867}, {5030, 17095}, {5757, 24220}, {7272, 9310}, {12609, 39580}, {16549, 33864}, {17044, 18990}, {17198, 39950}, {17266, 31053}, {17605, 24774}, {17671, 33949}, {17736, 28734}, {17745, 24712}, {18589, 37565}, {21258, 39542}, {29578, 31019}

X(40690) = complement of X(1759)

X(40691) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(6) AND GEMINI 82

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

X(40691) lies on these lines: {6, 3926}, {39, 6389}, {4558, 32973}, {5489, 34291}, {7736, 26226}, {26218, 37665}

X(40692) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(6) AND GEMINI 84

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

X(40692) lies on these lines: {6, 28724}, {39, 28696}, {7789, 28710}

X(40693) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(13) AND OUTER NAPOLEON

Barycentrics    a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 + 2*Sqrt[3]*a^2*S : :

X(40693) lies on these lines: {2, 17}, {3, 396}, {4, 13}, {5, 6}, {14, 3091}, {15, 20}, {16, 631}, {18, 3090}, {30, 5340}, {32, 5472}, {69, 635}, {115, 22509}, {140, 16644}, {141, 11311}, {156, 11137}, {193, 623}, {194, 628}, {202, 3086}, {203, 388}, {299, 11289}, {303, 7763}, {371, 2041}, {372, 2042}, {376, 5238}, {381, 398}, {382, 5318}, {395, 1656}, {497, 7005}, {498, 7127}, {524, 11305}, {530, 37172}, {533, 37170}, {546, 5339}, {548, 11480}, {549, 36843}, {550, 36836}, {568, 11624}, {576, 20415}, {597, 11306}, {617, 16529}, {618, 36763}, {622, 7787}, {624, 3618}, {633, 3180}, {1075, 36302}, {1249, 6117}, {1250, 31452}, {1478, 2307}, {1587, 3365}, {1588, 3364}, {1992, 5459}, {2043, 35822}, {2044, 35823}, {2045, 8960}, {2912, 3457}, {3068, 3389}, {3069, 3390}, {3085, 7006}, {3087, 6116}, {3104, 22691}, {3105, 12251}, {3146, 5344}, {3181, 22511}, {3201, 9545}, {3205, 9544}, {3411, 5067}, {3448, 36208}, {3522, 5352}, {3523, 5237}, {3524, 5351}, {3525, 16242}, {3526, 11486}, {3528, 10645}, {3529, 36967}, {3530, 11481}, {3542, 8739}, {3543, 5366}, {3589, 11312}, {3594, 35738}, {3628, 16645}, {3643, 6694}, {3830, 5350}, {3832, 5334}, {3839, 5343}, {3843, 5321}, {3855, 16809}, {4197, 5362}, {4309, 10638}, {4317, 7051}, {5007, 37825}, {5056, 37835}, {5070, 23303}, {5071, 16268}, {5286, 6783}, {5309, 37824}, {5353, 37719}, {5357, 37720}, {5613, 7772}, {5617, 7755}, {5859, 37352}, {5862, 21359}, {5984, 6778}, {6107, 18912}, {6114, 37665}, {6115, 7735}, {6243, 36978}, {6515, 33529}, {6772, 16001}, {6773, 22510}, {6776, 7684}, {6782, 36771}, {7486, 16967}, {7753, 16627}, {7765, 22907}, {8259, 16629}, {8742, 36612}, {8838, 37644}, {8930, 21467}, {9763, 34511}, {9833, 11243}, {10573, 33655}, {10611, 16626}, {10641, 37122}, {10646, 15717}, {10677, 11271}, {11080, 11555}, {11134, 32046}, {11298, 33458}, {11303, 37786}, {11304, 22492}, {12155, 32985}, {14138, 21158}, {16628, 16634}, {16630, 22491}, {17578, 19107}, {18586, 32788}, {18587, 32787}, {19106, 33703}, {20416, 22234}, {22114, 22846}, {22237, 33607}, {30328, 39153}, {33417, 34755}, {36995, 39555}

X(40693) = {X(5),X(6)}-harmonic conjugate of X(40694)

X(40694) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(14) AND INNER NAPOLEON

Barycentrics    a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 - 2*Sqrt[3]*a^2*S : :

X(40694) lies on these lines: {2, 18}, {3, 395}, {4, 14}, {5, 6}, {13, 3091}, {15, 631}, {16, 20}, {17, 3090}, {30, 5339}, {32, 5471}, {69, 636}, {115, 22507}, {140, 16645}, {141, 11312}, {156, 11134}, {193, 624}, {194, 627}, {202, 388}, {203, 3086}, {298, 11290}, {302, 7763}, {371, 2042}, {372, 2041}, {376, 5237}, {381, 397}, {382, 5321}, {396, 1656}, {497, 7006}, {499, 2307}, {524, 11306}, {531, 37173}, {532, 37171}, {546, 5340}, {548, 11481}, {549, 36836}, {550, 36843}, {568, 11626}, {576, 20416}, {597, 11305}, {616, 16530}, {621, 7787}, {623, 3618}, {634, 3181}, {1075, 36303}, {1249, 6116}, {1250, 4309}, {1479, 7127}, {1587, 3390}, {1588, 3389}, {1992, 5460}, {2043, 35823}, {2044, 35822}, {2046, 8960}, {2913, 3458}, {3068, 3364}, {3069, 3365}, {3085, 7005}, {3087, 6117}, {3104, 12251}, {3105, 22692}, {3146, 5343}, {3180, 22510}, {3200, 9545}, {3206, 9544}, {3412, 5067}, {3448, 36209}, {3522, 5351}, {3523, 5238}, {3524, 5352}, {3525, 16241}, {3526, 11485}, {3528, 10646}, {3529, 36968}, {3530, 11480}, {3542, 8740}, {3543, 5365}, {3589, 11311}, {3592, 35738}, {3628, 16644}, {3642, 6695}, {3830, 5349}, {3832, 5335}, {3839, 5344}, {3843, 5318}, {3855, 16808}, {4197, 5367}, {4317, 19373}, {5007, 37824}, {5056, 37832}, {5070, 23302}, {5071, 16267}, {5286, 6782}, {5309, 37825}, {5353, 37720}, {5357, 37719}, {5613, 7755}, {5617, 7772}, {5858, 37351}, {5863, 21360}, {5984, 6777}, {6106, 18912}, {6114, 7735}, {6115, 37665}, {6243, 36980}, {6515, 33530}, {6770, 22511}, {6775, 16002}, {6776, 7685}, {7052, 10573}, {7486, 16966}, {7753, 16626}, {7765, 22861}, {8260, 16628}, {8741, 36612}, {8836, 37644}, {8929, 21466}, {9761, 34511}, {9833, 11244}, {10612, 16627}, {10638, 31452}, {10642, 37122}, {10645, 15717}, {10678, 11271}, {11085, 11556}, {11137, 32046}, {11297, 33459}, {11303, 22491}, {11304, 37785}, {12154, 32985}, {14136, 36765}, {14139, 21159}, {16629, 16635}, {16631, 22492}, {17578, 19106}, {18586, 32787}, {18587, 32788}, {19107, 33703}, {20415, 22234}, {22113, 22891}, {22235, 33606}, {30327, 39152}, {33416, 34754}, {36993, 39554}

X(40694) = {X(5),X(6)}-harmonic conjugate of X(40693)

X(40695) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(15) AND MEDIAL

Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) - 2*S)*(Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) + 2*(b^2 + c^2)*S) : :

X(40695) lies on these lines: {2, 94}, {6, 2981}, {15, 1511}, {16, 11083}, {39, 395}, {50, 11146}, {216, 23302}, {396, 3003}, {470, 11062}, {570, 23303}, {1576, 3131}, {2058, 37848}, {3104, 36980}, {3106, 11626}, {5663, 30260}, {6593, 38431}, {8562, 23284}, {11142, 37776}, {11489, 13351}, {13337, 37641}, {16644, 18573}

X(40695) = isogonal conjugate of X(41907)
X(40695) = complement of X(300)
X(40695) = crosspoint of X(2) and X(15)
X(40695) = crosssum of X(6) and X(13)
X(40695) = X(2)-Ceva conjugate of X(623)
X(40695) = perspector of circumconic centered at X(623)
X(40695) = {X(2),X(566)}-harmonic conjugate of X(40696)

X(40696) = PERSPECTOR OF THESE TRIANGLES: MIDCEVIAN OF X(16) AND MEDIAL

Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) + 2*S)*(Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) - 2*(b^2 + c^2)*S) : :

X(40696) lies on these lines: {2, 94}, {6, 6151}, {15, 11088}, {16, 1511}, {39, 396}, {50, 11145}, {216, 23303}, {395, 3003}, {471, 11062}, {570, 23302}, {1576, 3132}, {2059, 37850}, {3105, 36978}, {3107, 11624}, {5663, 30261}, {6593, 38432}, {8562, 23283}, {11141, 37775}, {11488, 13351}, {13337, 37640}, {16242, 40578}, {16645, 18573}

X(40696) = isogonal conjugate of X(41908)
X(40696) = complement of X(301)
X(40696) = crosspoint of X(2) and X(16)
X(40696) = crosssum of X(6) and X(14)
X(40696) = X(2)-Ceva conjugate of X(624)
X(40696) = perspector of circumconic centered at X(624)
X(40696) = {X(2),X(566)}-harmonic conjugate of X(40695)

X(40697) = ISOTOMIC CONJUGATE OF X(254)

Barycentrics    (a^2 - b^2 - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) : :
Barycentrics    (cot A) (cos^2 B + cos^2 C - cos^2 A) : :
Barycentrics    sin^2 A sec 2A - sin^2 B sec 2B - sin^2 C sec 2C : :
Barycentrics    A-power of Dou circle : :

X(40697) lies on the cubic K045 and these lines: {2, 311}, {3, 69}, {4, 8905}, {20, 1273}, {68, 15827}, {75, 7318}, {76, 7383}, {99, 317}, {193, 571}, {253, 35520}, {254, 264}, {325, 1370}, {343, 36751}, {393, 14570}, {394, 34828}, {427, 1007}, {441, 28708}, {524, 10607}, {1225, 7558}, {1232, 32830}, {1272, 32837}, {1369, 37668}, {1609, 6503}, {1975, 6815}, {3260, 6527}, {3265, 34291}, {3547, 28706}, {3620, 14806}, {5596, 37183}, {6340, 8797}, {6389, 28419}, {7499, 34229}, {7799, 14615}, {8220, 19463}, {8221, 19464}, {13512, 31723}, {14360, 31099}, {14790, 32816}, {15574, 40123}, {15589, 40002}, {17135, 17221}, {18354, 18420}, {18750, 32851}, {20806, 37188}, {28406, 28710}, {30698, 32841}, {36181, 39193}

X(40697) = isogonal conjugate of X(39109)
X(40697) = isotomic conjugate of X(254)
X(40697) = anticomplement of X(2165)
X(40697) = anticomplement of the isogonal conjugate of X(1993)
X(40697) = anticomplement of the isotomic conjugate of X(7763)
X(40697) = isotomic conjugate of the anticomplement of X(34853)
X(40697) = isotomic conjugate of the isogonal conjugate of X(155)
X(40697) = isotomic conjugate of the polar conjugate of X(6515)
X(40697) = polar conjugate of the isogonal conjugate of X(6503)
X(40697) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3, 18664}, {24, 5905}, {47, 2}, {63, 37444}, {92, 68}, {162, 14618}, {317, 21270}, {563, 3164}, {571, 192}, {662, 924}, {924, 21221}, {1101, 4558}, {1147, 6360}, {1444, 18658}, {1748, 4}, {1993, 8}, {2167, 11412}, {2180, 17035}, {2190, 5392}, {2349, 25739}, {6563, 21294}, {7763, 6327}, {9723, 4329}, {11547, 5906}, {18605, 1}, {34948, 4440}, {34952, 21220}, {36034, 6334}
X(40697) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 69}, {7763, 2}
X(40697) = X(i)-cross conjugate of X(j) for these (i,j): {155, 6515}, {34853, 2}
X(40697) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39109}, {25, 921}, {31, 254}, {1096, 15316}, {1973, 6504}
X(40697) = cevapoint of X(155) and X(6503)
X(40697) = crosssum of X(2971) and X(3049)
X(40697) = crossdifference of every pair of points on line {2489, 34952}
X(40697) = barycentric product X(i)*X(j) for these {i,j}: {63, 33808}, {69, 6515}, {76, 155}, {264, 6503}, {304, 920}, {305, 1609}, {3542, 3926}, {7763, 34853}, {8883, 28706}, {9723, 39116}
X(40697) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 254}, {6, 39109}, {63, 921}, {69, 6504}, {155, 6}, {343, 8800}, {394, 15316}, {454, 1609}, {920, 19}, {925, 39416}, {1609, 25}, {1993, 34756}, {3542, 393}, {3580, 16172}, {4558, 13398}, {6503, 3}, {6515, 4}, {8883, 8882}, {15478, 14910}, {27087, 16310}, {33808, 92}, {34853, 2165}, {35603, 8745}, {39113, 39114}, {39116, 847}
X(40697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 6337, 9723}, {487, 488, 6193}, {3926, 40680, 69}, {6389, 36212, 28419}, {13430, 13441, 2}

X(40698) = ISOGONAL CONJUGATE OF X(39110)

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

X(40698) lies on the cubic K045 and these lines: {2, 254}, {3, 96}, {4, 8906}, {24, 925}, {68, 69}, {1093, 30450}, {2165, 3547}, {3548, 37802}, {5962, 37444}, {5963, 7488}, {6193, 39111}, {7401, 14593}

X(40698) = anticomplement of X(40678)
X(40698) = isogonal conjugate of X(39110)
X(40698) = isotomic conjugate of the isogonal conjugate of X(39111)
X(40698) = X(2190)-anticomplementary conjugate of X(254)
X(40698) = X(264)-Ceva conjugate of X(5392)
X(40698) = X(8905)-cross conjugate of X(6193)
X(40698) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39110}, {47, 34428}
X(40698) = cevapoint of X(8906) and X(34853)
X(40698) = crosssum of X(6754) and X(30451)
X(40698) = barycentric product X(i)*X(j) for these {i,j}: {76, 39111}, {5392, 6193}
X(40698) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39110}, {2165, 34428}, {6193, 1993}, {39111, 6}, {39116, 39115}, {39117, 39114}
X(40698) = {X(32132),X(34853)}-harmonic conjugate of X(2)

X(40699) = ISOTOMIC CONJUGATE OF X(175)

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

X(40699) lies on the cubic K200 and these lines: {8, 175}, {20, 30303}, {144, 13387}, {176, 280}, {346, 15891}, {347, 4847}, {1043, 30336}, {5815, 31551}

X(40699) = isotomic conjugate of X(175)
X(40699) = isotomic conjugate of the anticomplement of X(14121)
X(40699) = isotomic conjugate of the isogonal conjugate of X(30336)
X(40699) = X(30336)-anticomplementary conjugate of X(13386)
X(40699) = X(14121)-cross conjugate of X(2)
X(40699) = X(i)-isoconjugate of X(j) for these (i,j): {31, 175}, {41, 16662}, {604, 30413}, {30335, 34033}
X(40699) = cevapoint of X(15891) and X(34911)
X(40699) = barycentric product X(i)*X(j) for these {i,j}: {75, 15891}, {76, 30336}, {85, 34911}
X(40699) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 175}, {7, 16662}, {8, 30413}, {15891, 1}, {30336, 6}, {30412, 9778}, {34911, 9}
X(40699) = {X(4847),X(32087)}-harmonic conjugate of X(40700)

X(40700) = ISOTOMIC CONJUGATE OF X(176)

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

X(40700) lies on the cubic K200 and these lines: {8, 176}, {20, 30302}, {144, 13386}, {175, 280}, {346, 15892}, {347, 4847}, {1043, 30335}, {5815, 31552}

X(40700) = isotomic conjugate of X(176)
X(40700) = isotomic conjugate of the anticomplement of X(7090)
X(40700) = isotomic conjugate of the isogonal conjugate of X(30335)
X(40700) = X(30335)-anticomplementary conjugate of X(13387)
X(40700) = X(7090)-cross conjugate of X(2)
X(40700) = X(i)-isoconjugate of X(j) for these (i,j): {31, 176}, {41, 16663}, {604, 30412}, {30336, 34033}
X(40700) = cevapoint of X(15892) and X(34912)
X(40700) = barycentric product X(i)*X(j) for these {i,j}: {75, 15892}, {76, 30335}, {85, 34912}, {556, 5451}
X(40700) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 176}, {7, 16663}, {8, 30412}, {5451, 174}, {15892, 1}, {30335, 6}, {30413, 9778}, {34912, 9}
X(40700) = {X(4847),X(32087)}-harmonic conjugate of X(40699)

X(40701) = ISOTOMIC CONJUGATE OF X(268)

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

X(40701) lies on the cubic K1069 and these lines: {8, 18026}, {85, 92}, {264, 1441}, {322, 342}, {348, 13149}, {349, 7017}, {1969, 6063}

X(40701) = isotomic conjugate of X(268)
X(40701) = polar conjugate of X(2192)
X(40701) = isotomic conjugate of the isogonal conjugate of X(196)
X(40701) = polar conjugate of the isogonal conjugate of X(347)
X(40701) = X(i)-Ceva conjugate of X(j) for these (i,j): {1969, 331}, {6063, 264}
X(40701) = X(16596)-cross conjugate of X(17896)
X(40701) = X(i)-isoconjugate of X(j) for these (i,j): {3, 7118}, {6, 2188}, {31, 268}, {32, 271}, {41, 1433}, {48, 2192}, {184, 282}, {212, 1436}, {219, 2208}, {255, 7154}, {280, 9247}, {285, 2200}, {577, 7008}, {603, 7367}, {652, 32652}, {1413, 1802}, {1946, 36049}, {2193, 2357}, {2289, 7151}, {6056, 7129}, {7020, 14585}, {14575, 34404}
X(40701) = cevapoint of X(i) and X(j) for these (i,j): {196, 347}, {16596, 17896}
X(40701) = barycentric product X(i)*X(j) for these {i,j}: {75, 342}, {76, 196}, {208, 561}, {221, 18022}, {223, 1969}, {264, 347}, {273, 322}, {329, 331}, {1502, 3209}, {2331, 20567}, {6063, 7952}, {7011, 18027}, {7017, 14256}, {17896, 18026}
X(40701) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2188}, {2, 268}, {4, 2192}, {7, 1433}, {19, 7118}, {34, 2208}, {40, 212}, {75, 271}, {92, 282}, {108, 32652}, {158, 7008}, {196, 6}, {208, 31}, {221, 184}, {223, 48}, {225, 2357}, {227, 228}, {264, 280}, {273, 84}, {278, 1436}, {281, 7367}, {286, 285}, {322, 78}, {329, 219}, {331, 189}, {342, 1}, {347, 3}, {393, 7154}, {653, 36049}, {1118, 7151}, {1119, 1413}, {1817, 2193}, {1847, 1422}, {1969, 34404}, {2052, 7003}, {2199, 9247}, {2324, 1802}, {2331, 41}, {3194, 2194}, {3195, 2175}, {3209, 32}, {6129, 1946}, {7011, 577}, {7013, 255}, {7078, 6056}, {7080, 1260}, {7952, 55}, {8822, 283}, {13149, 37141}, {14256, 222}, {14837, 652}, {16596, 35072}, {17896, 521}, {18026, 13138}, {21075, 2318}, {27398, 2327}, {36118, 8059}, {38357, 3270}, {38362, 3271}, {38374, 3937}, {40149, 1903}

X(40702) = ISOTOMIC CONJUGATE OF X(282)

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

X(40702) lies on the cubic K184 and these lines: {2, 85}, {7, 2478}, {29, 38298}, {57, 17048}, {63, 1847}, {69, 1034}, {75, 225}, {76, 7182}, {78, 664}, {208, 342}, {223, 27398}, {227, 322}, {304, 4554}, {312, 1231}, {329, 10402}, {404, 3188}, {411, 6516}, {658, 7183}, {936, 9312}, {938, 6604}, {1210, 3673}, {1323, 6700}, {1441, 9780}, {1445, 1760}, {1447, 16048}, {1565, 6922}, {1895, 18026}, {3149, 5088}, {3160, 27383}, {3668, 8582}, {3732, 29464}, {3869, 4566}, {4346, 17863}, {4572, 28659}, {4872, 6836}, {5226, 30845}, {5704, 38468}, {6831, 17181}, {6865, 17170}, {7179, 15844}, {9843, 10481}, {12649, 17158}, {18135, 20946}, {18140, 21609}, {18635, 40593}, {18721, 28742}, {18739, 18751}, {18743, 30843}, {18747, 21617}, {20895, 36640}, {21579, 26611}, {26229, 38859}, {27832, 40014}, {28736, 28739}, {31526, 36854}, {31638, 34018}, {34497, 35102}

X(40702) = isogonal conjugate of X(7118)
X(40702) = isotomic conjugate of X(282)
X(40702) = polar conjugate of X(7008)
X(40702) = isotomic conjugate of the anticomplement of X(20206)
X(40702) = isotomic conjugate of the complement of X(5932)
X(40702) = isotomic conjugate of the isogonal conjugate of X(223)
X(40702) = isotomic conjugate of the polar conjugate of X(342)
X(40702) = polar conjugate of the isogonal conjugate of X(7013)
X(40702) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 85}, {7182, 75}
X(40702) = X(i)-cross conjugate of X(j) for these (i,j): {223, 342}, {329, 322}, {14256, 85}, {20206, 2}
X(40702) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7118}, {3, 7154}, {6, 2192}, {9, 2208}, {19, 2188}, {25, 268}, {31, 282}, {32, 280}, {41, 84}, {48, 7008}, {55, 1436}, {56, 7367}, {184, 7003}, {189, 2175}, {212, 7129}, {213, 285}, {219, 7151}, {220, 1413}, {271, 1973}, {284, 2357}, {309, 9447}, {480, 6612}, {560, 34404}, {607, 1433}, {650, 32652}, {657, 8059}, {663, 36049}, {1253, 1422}, {1440, 14827}, {1903, 2194}, {1946, 40117}, {3063, 13138}, {7020, 9247}, {8641, 37141}
X(40702) = cevapoint of X(i) and X(j) for these (i,j): {2, 5932}, {223, 7013}, {329, 347}
X(40702) = trilinear pole of line {8058, 17896}
X(40702) = barycentric product X(i)*X(j) for these {i,j}: {7, 322}, {40, 6063}, {69, 342}, {75, 347}, {76, 223}, {85, 329}, {196, 304}, {198, 20567}, {208, 305}, {221, 561}, {227, 310}, {264, 7013}, {312, 14256}, {349, 1817}, {664, 17896}, {1088, 7080}, {1441, 8822}, {1446, 27398}, {1502, 2199}, {1969, 7011}, {3209, 40364}, {4554, 14837}, {4569, 8058}, {4572, 6129}, {6611, 28659}, {7035, 38374}, {7114, 18022}, {7182, 7952}
X(40702) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2192}, {2, 282}, {3, 2188}, {4, 7008}, {6, 7118}, {7, 84}, {9, 7367}, {19, 7154}, {34, 7151}, {40, 55}, {56, 2208}, {57, 1436}, {63, 268}, {65, 2357}, {69, 271}, {75, 280}, {76, 34404}, {77, 1433}, {85, 189}, {86, 285}, {92, 7003}, {109, 32652}, {196, 19}, {198, 41}, {208, 25}, {221, 31}, {223, 6}, {226, 1903}, {227, 42}, {264, 7020}, {269, 1413}, {278, 7129}, {279, 1422}, {322, 8}, {329, 9}, {342, 4}, {347, 1}, {651, 36049}, {653, 40117}, {658, 37141}, {664, 13138}, {738, 6612}, {934, 8059}, {1088, 1440}, {1103, 7074}, {1440, 1256}, {1441, 39130}, {1446, 8808}, {1817, 284}, {2187, 2175}, {2199, 32}, {2324, 220}, {2331, 607}, {2360, 2194}, {3182, 28784}, {3194, 2299}, {3195, 2212}, {3209, 1973}, {3342, 7037}, {5514, 3119}, {5932, 3341}, {6063, 309}, {6129, 663}, {6260, 1864}, {6611, 604}, {7011, 48}, {7013, 3}, {7074, 1253}, {7078, 212}, {7080, 200}, {7114, 184}, {7368, 6602}, {7952, 33}, {8058, 3900}, {8822, 21}, {14256, 57}, {14298, 657}, {14837, 650}, {15501, 2342}, {16596, 34591}, {17896, 522}, {21075, 210}, {21871, 1334}, {27398, 2287}, {37421, 10382}, {38357, 2310}, {38374, 244}, {40212, 198}
X(40702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1446, 85}, {75, 18749, 33672}, {78, 34059, 664}, {85, 31627, 348}, {273, 307, 75}, {279, 26563, 85}, {307, 6734, 33298}, {26563, 37780, 279}

X(40703) = ISOTOMIC CONJUGATE OF X(293)

Barycentrics    b*c*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 + c^4) : :
Barycentrics    (csc 2A) (a^2 cos B cos C - b c cos^2 A) : :

X(40703) is the perspector of the circumconic through the polar conjugates of PU(23). (Randy Hutson, December 18, 2020)

X(40703) lies on the cubic K995 and these lines: {4, 24282}, {19, 27}, {158, 304}, {240, 23996}, {242, 5991}, {264, 7018}, {326, 1096}, {331, 33930}, {561, 18695}, {811, 1784}, {1581, 17901}, {1582, 1954}, {1725, 17881}, {1733, 36036}, {1895, 18156}, {1930, 1969}, {1959, 17875}, {2181, 20627}, {3112, 40440}, {6335, 20947}, {7017, 33931}, {8747, 21595}, {14208, 18076}, {14571, 35551}, {18026, 35149}, {18056, 33808}, {18694, 23994}, {20944, 23999}

X(40703) = isotomic conjugate of X(293)
X(40703) = polar conjugate of X(1910)
X(40703) = isotomic conjugate of the isogonal conjugate of X(240)
X(40703) = polar conjugate of the isogonal conjugate of X(1959)
X(40703) = X(17875)-cross conjugate of X(75)
X(40703) = X(i)-isoconjugate of X(j) for these (i,j): {2, 14600}, {3, 1976}, {6, 248}, {25, 17974}, {31, 293}, {32, 287}, {48, 1910}, {69, 14601}, {98, 184}, {110, 878}, {290, 14575}, {336, 560}, {520, 32696}, {577, 6531}, {647, 2715}, {669, 17932}, {685, 39201}, {810, 36084}, {822, 36104}, {879, 1576}, {1691, 15391}, {1821, 9247}, {1974, 6394}, {2395, 32661}, {2422, 4558}, {2966, 3049}, {5967, 14908}, {9154, 23200}, {10547, 20021}, {14585, 16081}, {18024, 40373}, {18877, 35906}, {20031, 32320}, {35912, 40352}
X(40703) = cevapoint of X(240) and X(1959)
X(40703) = crossdifference of every pair of points on line {810, 9247}
X(40703) = barycentric product X(i)*X(j) for these {i,j}: {75, 297}, {76, 240}, {92, 325}, {158, 6393}, {232, 561}, {264, 1959}, {304, 6530}, {336, 36426}, {511, 1969}, {799, 16230}, {811, 2799}, {823, 6333}, {877, 1577}, {1235, 3405}, {1755, 18022}, {1928, 2211}, {1934, 39931}, {2396, 24006}, {4230, 20948}, {4602, 17994}, {6330, 17875}, {20022, 20883}, {32458, 36120}, {34854, 40364}
X(40703) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 248}, {2, 293}, {4, 1910}, {19, 1976}, {31, 14600}, {63, 17974}, {75, 287}, {76, 336}, {92, 98}, {107, 36104}, {132, 2312}, {158, 6531}, {162, 2715}, {232, 31}, {237, 9247}, {240, 6}, {264, 1821}, {297, 1}, {304, 6394}, {318, 15628}, {325, 63}, {427, 3404}, {511, 48}, {648, 36084}, {661, 878}, {684, 822}, {799, 17932}, {811, 2966}, {823, 685}, {868, 3708}, {877, 662}, {1577, 879}, {1581, 15391}, {1755, 184}, {1784, 35906}, {1959, 3}, {1969, 290}, {1973, 14601}, {2052, 36120}, {2211, 560}, {2396, 4592}, {2421, 4575}, {2450, 2083}, {2799, 656}, {2967, 1755}, {3405, 1176}, {3569, 810}, {4230, 163}, {5360, 2200}, {5968, 36060}, {6331, 36036}, {6333, 24018}, {6393, 326}, {6530, 19}, {9417, 14575}, {14206, 35912}, {15595, 8766}, {16230, 661}, {17209, 1437}, {17875, 441}, {17994, 798}, {19189, 2148}, {20022, 34055}, {20883, 20021}, {23996, 3289}, {23997, 32661}, {24006, 2395}, {24019, 32696}, {34854, 1973}, {35908, 2159}, {35910, 35200}, {36126, 20031}, {36212, 255}, {36426, 240}, {39569, 1953}, {39931, 1580}
X(40703) = {X(1784),X(14210)}-harmonic conjugate of X(811)

X(40704) = ISOTOMIC CONJUGATE OF X(294)

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

X(40704) lies on the cubic K994 and these lines: {2, 7182}, {7, 8}, {57, 24602}, {76, 1229}, {226, 4766}, {239, 1462}, {241, 16728}, {274, 1170}, {279, 304}, {312, 1088}, {321, 6063}, {335, 18033}, {344, 348}, {345, 17093}, {655, 37214}, {664, 4318}, {894, 25001}, {918, 3261}, {1016, 1275}, {1111, 23690}, {1280, 34018}, {1323, 14210}, {1427, 18138}, {1458, 39775}, {1876, 30941}, {1921, 4572}, {1930, 10481}, {1975, 3188}, {1996, 28808}, {2171, 3674}, {2263, 3886}, {2284, 28961}, {3100, 31637}, {3160, 18156}, {3263, 3717}, {3662, 17435}, {3673, 4310}, {3685, 14189}, {4327, 26234}, {4358, 4554}, {4569, 35158}, {4573, 16741}, {4847, 21436}, {4869, 10004}, {5328, 30796}, {5807, 14548}, {7056, 18141}, {7081, 9446}, {7112, 30807}, {7205, 20891}, {8817, 10327}, {17078, 17264}, {17087, 19815}, {17095, 17263}, {18135, 20946}, {18743, 31627}, {23839, 24282}, {25585, 26083}, {26167, 26168}, {29824, 35312}, {30062, 30097}, {32851, 37757}, {32939, 33765}, {40030, 40149}

X(40704) = isotomic conjugate of X(294)
X(40704) = isotomic conjugate of the anticomplement of X(17060)
X(40704) = isotomic conjugate of the isogonal conjugate of X(241)
X(40704) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1037, 20533}, {7131, 20344}, {8817, 20552}
X(40704) = X(i)-cross conjugate of X(j) for these (i,j): {918, 883}, {3912, 3263}, {6184, 40216}, {17060, 2}, {35094, 693}
X(40704) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2195}, {31, 294}, {32, 14942}, {33, 32658}, {41, 105}, {55, 1438}, {101, 884}, {212, 8751}, {220, 1416}, {560, 36796}, {604, 28071}, {607, 36057}, {650, 32666}, {657, 32735}, {663, 919}, {673, 2175}, {692, 1024}, {885, 32739}, {1253, 1462}, {1397, 6559}, {1814, 2212}, {1919, 36802}, {2194, 18785}, {2481, 9447}, {3063, 36086}, {6654, 18265}, {8641, 36146}, {9448, 18031}, {14599, 33676}
X(40704) = cevapoint of X(3912) and X(9436)
X(40704) = crosspoint of X(2481) and X(32023)
X(40704) = crossdifference of every pair of points on line {2175, 3063}
X(40704) = barycentric product X(i)*X(j) for these {i,j}: {7, 3263}, {75, 9436}, {76, 241}, {85, 3912}, {226, 18157}, {304, 5236}, {305, 1876}, {331, 25083}, {334, 39775}, {349, 18206}, {518, 6063}, {561, 1458}, {672, 20567}, {693, 883}, {918, 4554}, {1025, 3261}, {1088, 3717}, {1231, 15149}, {1441, 30941}, {1861, 7182}, {2254, 4572}, {2283, 40495}, {3596, 34855}, {4088, 4625}, {4437, 34018}, {10029, 18743}, {10030, 40217}, {18033, 22116}, {18895, 34253}
X(40704) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2195}, {2, 294}, {7, 105}, {8, 28071}, {57, 1438}, {75, 14942}, {76, 36796}, {77, 36057}, {85, 673}, {109, 32666}, {222, 32658}, {226, 18785}, {241, 6}, {269, 1416}, {273, 36124}, {278, 8751}, {279, 1462}, {312, 6559}, {334, 33676}, {348, 1814}, {513, 884}, {514, 1024}, {518, 55}, {651, 919}, {658, 36146}, {664, 36086}, {665, 3063}, {668, 36802}, {672, 41}, {693, 885}, {883, 100}, {918, 650}, {926, 8641}, {934, 32735}, {1025, 101}, {1026, 3939}, {1362, 2223}, {1441, 13576}, {1458, 31}, {1818, 212}, {1861, 33}, {1876, 25}, {2223, 2175}, {2254, 663}, {2283, 692}, {2340, 1253}, {2356, 2212}, {3126, 926}, {3263, 8}, {3286, 2194}, {3323, 3675}, {3675, 3271}, {3676, 1027}, {3693, 220}, {3717, 200}, {3912, 9}, {3930, 1334}, {3932, 210}, {4025, 23696}, {4088, 4041}, {4391, 28132}, {4437, 3693}, {4447, 2330}, {4554, 666}, {4569, 927}, {4684, 4512}, {4712, 2340}, {4899, 3158}, {4925, 4162}, {4966, 3683}, {4998, 5377}, {5089, 607}, {5236, 19}, {6063, 2481}, {6168, 9310}, {7182, 31637}, {9311, 6169}, {9436, 1}, {9454, 9447}, {9455, 9448}, {10029, 8056}, {10030, 6654}, {15149, 1172}, {16593, 2348}, {17094, 10099}, {17435, 14936}, {17755, 3684}, {18157, 333}, {18206, 284}, {20567, 18031}, {21609, 31638}, {22116, 7077}, {23829, 3737}, {24290, 3709}, {25083, 219}, {27509, 23601}, {30941, 21}, {34018, 6185}, {34253, 1914}, {34855, 56}, {35094, 17435}, {36819, 2342}, {36905, 9441}, {39063, 910}, {39775, 238}, {40217, 4876}
X(40704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {85, 1231, 20911}, {4358, 37780, 4554}, {7182, 21609, 2}

X(40705) = ISOTOMIC CONJUGATE OF X(399)

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

X(40705) lies on the cubic K276 and these lines: {340, 37779}, {1138, 1272}, {40423, 40427}

X(40705) = isotomic conjugate of X(399)
X(40705) = isotomic conjugate of the anticomplement of X(10264)
X(40705) = isotomic conjugate of the complement of X(12317)
X(40705) = isotomic conjugate of the isogonal conjugate of X(1138)
X(40705) = X(i)-cross conjugate of X(j) for these (i,j): {94, 76}, {1494, 264}, {10264, 2}, {39235, 2986}
X(40705) = X(i)-isoconjugate of X(j) for these (i,j): {6, 19303}, {31, 399}, {560, 1272}
X(40705) = cevapoint of X(2) and X(12317)
X(40705) = trilinear pole of line {3268, 14566}
X(40705) = barycentric product X(76)*X(1138)
X(40705) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19303}, {2, 399}, {76, 1272}, {94, 14993}, {850, 14566}, {1138, 6}, {5627, 11074}, {11070, 1495}, {14451, 11063}, {18781, 3003}, {20123, 3284}, {37779, 15766}, {40356, 9407}

X(40706) = ISOTOMIC CONJUGATE OF X(395)

Barycentrics    (Sqrt[3]*b^2 - 2*S)*(Sqrt[3]*c^2 - 2*S) : :
Barycentrics    (csc A)/(cos(B - C) + 2 cos(A + π/3)) : :

X(40706) = 3 X[18] - 4 X[6670], X[617] - 3 X[628], 5 X[14061] - 4 X[22847], 3 X[16627] - 2 X[22797], 3 X[21360] - X[22850]

X(40706) lies on the Kiepert circumhyperbola, the cubic K867a, and these lines: {2, 6151}, {4, 617}, {13, 99}, {14, 299}, {17, 630}, {18, 298}, {83, 396}, {98, 5979}, {115, 11121}, {141, 6034}, {531, 12817}, {598, 9763}, {634, 6774}, {671, 14905}, {1078, 3643}, {1327, 33443}, {1328, 33442}, {3180, 7838}, {3642, 31703}, {5460, 33606}, {5464, 12816}, {5978, 14492}, {5981, 14458}, {6114, 22665}, {6674, 10187}, {6771, 14145}, {6775, 11122}, {7811, 22861}, {10612, 22114}, {13582, 14922}, {16644, 33220}, {22487, 40672}, {22846, 25187}, {33603, 33624}, {33605, 33627}

X(40706) = reflection of X(i) in X(j) for these {i,j}: {99, 30471}, {11121, 115}, {22114, 10612}, {36368, 5460}
X(40706) = isotomic conjugate of X(395)
X(40706) = anticomplement of X(22848)
X(40706) = polar conjugate of X(462)
X(40706) = antigonal image of X(11121)
X(40706) = antitomic conjugate of X(11121)
X(40706) = symgonal image of X(30471)
X(40706) = isotomic conjugate of the complement of X(299)
X(40706) = isotomic conjugate of the isogonal conjugate of X(6151)
X(40706) = isotomic conjugate of the polar conjugate of X(38427)
X(40706) = X(i)-cross conjugate of X(j) for these (i,j): {2, 11120}, {6151, 38427}, {11092, 1494}, {23871, 99}
X(40706) = X(i)-isoconjugate of X(j) for these (i,j): {31, 395}, {48, 462}, {661, 35330}, {798, 35315}, {923, 9117}, {2152, 8015}
X(40706) = cevapoint of X(i) and X(j) for these (i,j): {2, 299}, {6, 34009}
X(40706) = trilinear pole of line {298, 523}
X(40706) = barycentric product X(i)*X(j) for these {i,j}: {69, 38427}, {76, 6151}, {298, 11118}, {299, 11120}, {301, 38404}, {850, 10410}
X(40706) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 395}, {4, 462}, {14, 8015}, {18, 36305}, {99, 35315}, {110, 35330}, {298, 533}, {299, 619}, {303, 6672}, {323, 19295}, {470, 23715}, {524, 9117}, {533, 30459}, {617, 15769}, {628, 15778}, {1338, 3129}, {2381, 3457}, {2992, 3480}, {2993, 38932}, {4427, 35344}, {6151, 6}, {7799, 14921}, {10410, 110}, {11089, 11060}, {11118, 13}, {11120, 14}, {16460, 3458}, {19713, 39135}, {19777, 34295}, {19778, 40668}, {23870, 14447}, {23871, 35444}, {27551, 13305}, {34322, 21462}, {35314, 35345}, {38404, 16}, {38427, 4}, {39261, 3130}
X(40706) = {X(299),X(624)}-harmonic conjugate of X(7809)

X(40707) = ISOTOMIC CONJUGATE OF X(396)

Barycentrics    (Sqrt[3]*b^2 + 2*S)*(Sqrt[3]*c^2 + 2*S) : :
Barycentrics    (csc A)/(cos(B - C) + 2 cos(A - π/3)) : :
X(40707) = 3 X[17] - 4 X[6669], X[616] - 3 X[627], 6 X[629] - 5 X[36770], 5 X[14061] - 4 X[22893], 3 X[16626] - 2 X[22796], 3 X[21359] - X[22894], 5 X[36770] - 3 X[36782]

X(40707) lies on the Kiepert circumhyperbola, the cubic K867b, and these lines: {2, 2981}, {4, 616}, {13, 298}, {14, 99}, {17, 299}, {18, 629}, {83, 395}, {98, 5978}, {115, 11122}, {141, 6034}, {530, 12816}, {598, 9761}, {633, 6771}, {671, 14904}, {1078, 3642}, {1327, 33441}, {1328, 33440}, {3181, 7838}, {3643, 31704}, {5459, 33607}, {5463, 12817}, {5979, 14492}, {5980, 14458}, {6115, 22666}, {6673, 10188}, {6772, 11121}, {6774, 14144}, {7811, 22907}, {10611, 22113}, {13582, 14921}, {16645, 33220}, {22488, 40671}, {22891, 25183}, {33602, 33622}, {33604, 33626}

X(40707) = reflection of X(i) in X(j) for these {i,j}: {99, 30472}, {11122, 115}, {22113, 10611}, {36366, 5459}, {36782, 629}
X(40707) = isotomic conjugate of X(396)
X(40707) = polar conjugate of X(463)
X(40707) = anticomplement of X(22892)
X(40707) = antigonal image of X(11122)
X(40707) = antitomic conjugate of X(11122)
X(40707) = symgonal image of X(30472)
X(40707) = isotomic conjugate of the complement of X(298)
X(40707) = isotomic conjugate of the isogonal conjugate of X(2981)
X(40707) = isotomic conjugate of the polar conjugate of X(38428)
X(40707) = X(i)-cross conjugate of X(j) for these (i,j): {2, 11119}, {2981, 38428}, {11078, 1494}, {23870, 99}
X(40707) = X(i)-isoconjugate of X(j) for these (i,j): {31, 396}, {48, 463}, {661, 35329}, {798, 35314}, {923, 9115}, {2151, 8014}
X(40707) = cevapoint of X(i) and X(j) for these (i,j): {2, 298}, {6, 34008}
X(40707) = trilinear pole of line {299, 523}
X(40707) = barycentric product X(i)*X(j) for these {i,j}: {69, 38428}, {76, 2981}, {298, 11119}, {299, 11117}, {300, 38403}, {850, 10409}
X(40707) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 396}, {4, 463}, {13, 8014}, {17, 36304}, {99, 35314}, {110, 35329}, {298, 618}, {299, 532}, {302, 6671}, {323, 19294}, {471, 23714}, {524, 9115}, {532, 30462}, {616, 15768}, {627, 15802}, {1337, 3130}, {2380, 3458}, {2981, 6}, {2992, 38931}, {2993, 3479}, {4427, 35343}, {7799, 14922}, {10409, 110}, {11084, 11060}, {11117, 14}, {11119, 13}, {16459, 3457}, {19712, 39134}, {19776, 34296}, {19779, 40667}, {23870, 35443}, {23871, 14446}, {27550, 13304}, {34321, 21461}, {35315, 35345}, {38403, 15}, {38428, 4}, {39262, 3129}
X(40707) = {X(298),X(623)}-harmonic conjugate of X(7809)

X(40708) = ISOTOMIC CONJUGATE OF X(419)

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

X(40708) lies on the cubic K1023 and these lines: {2, 694}, {69, 20819}, {99, 19571}, {125, 305}, {141, 9229}, {264, 5117}, {287, 12215}, {306, 7019}, {307, 337}, {334, 1441}, {805, 2373}, {1494, 18829}, {1799, 3917}, {1976, 39292}, {2076, 16985}, {2396, 20021}, {3506, 5152}, {12294, 40413}, {14603, 18024}, {17938, 37183}, {37134, 37202}

X(40708) = isotomic conjugate of X(419)
X(40708) = isotomic conjugate of the isogonal conjugate of X(36214)
X(40708) = isotomic conjugate of the polar conjugate of X(1916)
X(40708) = isogonal conjugate of the polar conjugate of X(18896)
X(40708) = X(18896)-Ceva conjugate of X(1916)
X(40708) = X(i)-cross conjugate of X(j) for these (i,j): {6393, 69}, {36214, 1916}
X(40708) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1933}, {19, 1691}, {25, 1580}, {31, 419}, {92, 14602}, {162, 5027}, {172, 2201}, {242, 7122}, {385, 1973}, {560, 17984}, {804, 32676}, {1914, 7119}, {1966, 1974}, {1969, 18902}, {2203, 4039}, {2210, 7009}, {4164, 8750}
X(40708) = cevapoint of X(i) and X(j) for these (i,j): {125, 6333}, {3917, 36212}
X(40708) = trilinear pole of line {525, 3933}
X(40708) = barycentric product X(i)*X(j) for these {i,j}: {3, 18896}, {63, 1934}, {69, 1916}, {76, 36214}, {125, 39292}, {257, 337}, {304, 1581}, {305, 694}, {335, 7019}, {525, 18829}, {805, 3267}, {1502, 17970}, {1967, 40364}, {3933, 14970}, {6333, 39291}, {6393, 36897}, {7015, 18895}, {8789, 40360}, {9468, 40050}, {14208, 37134}
X(40708) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 419}, {3, 1691}, {48, 1933}, {63, 1580}, {69, 385}, {76, 17984}, {184, 14602}, {256, 2201}, {257, 242}, {291, 7119}, {295, 172}, {304, 1966}, {305, 3978}, {306, 4039}, {325, 39931}, {335, 7009}, {337, 894}, {525, 804}, {647, 5027}, {694, 25}, {805, 112}, {882, 2489}, {905, 4164}, {1581, 19}, {1916, 4}, {1934, 92}, {1967, 1973}, {2196, 7122}, {3265, 24284}, {3267, 14295}, {3564, 12829}, {3917, 8623}, {3926, 12215}, {3933, 732}, {4025, 4107}, {4563, 17941}, {6390, 5026}, {6393, 5976}, {7015, 1914}, {7019, 239}, {7116, 2210}, {8552, 39495}, {8842, 458}, {8858, 32544}, {9468, 1974}, {12215, 4027}, {14251, 2211}, {14417, 11183}, {14575, 18902}, {14941, 32542}, {14970, 32085}, {15391, 1976}, {15413, 14296}, {17970, 32}, {17980, 2207}, {18829, 648}, {18896, 264}, {20975, 2086}, {32010, 31905}, {34897, 36820}, {36212, 36213}, {36214, 6}, {36800, 14006}, {36897, 6531}, {37134, 162}, {37894, 16985}, {39291, 685}, {39292, 18020}, {40050, 14603}, {40360, 18901}, {40364, 1926}
X(40708) = {X(694),X(8842)}-harmonic conjugate of X(20027)

X(40709) = ISOTOMIC CONJUGATE OF X(470)

Barycentrics    SA*(S + Sqrt[3]*SB)*(S + Sqrt[3]*SC) : :

X(40709) lies on the cubic K3421 and these lines: {2, 13}, {3, 125}, {15, 3580}, {18, 94}, {61, 37644}, {62, 14389}, {69, 36296}, {95, 303}, {141, 11081}, {264, 300}, {287, 38414}, {298, 1494}, {343, 465}, {395, 14836}, {476, 36185}, {621, 19772}, {623, 15441}, {627, 8919}, {858, 14538}, {1989, 16645}, {2373, 5995}, {3129, 32223}, {3130, 3818}, {3132, 21243}, {3170, 22998}, {3448, 14170}, {3589, 11083}, {3763, 11142}, {5473, 36186}, {5617, 32461}, {6330, 36306}, {9205, 20578}, {9761, 18777}, {10217, 11064}, {11092, 36967}, {11127, 37779}, {11581, 40334}, {15066, 36208}, {16771, 16964}, {20428, 37974}, {34417, 37333}, {36299, 36889}, {37172, 37643}, {37340, 37648}

X(40709) = isogonal conjugate of X(8739)
X(40709) = isotomic conjugate of X(470)
X(40709) = isotomic conjugate of the complement of X(19772)
X(40709) = isotomic conjugate of the isogonal conjugate of X(36296)
X(40709) = isotomic conjugate of the polar conjugate of X(13)
X(40709) = isogonal conjugate of the polar conjugate of X(300)
X(40709) = X(300)-Ceva conjugate of X(13)
X(40709) = X(36296)-cross conjugate of X(13)
X(40709) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8739}, {4, 2151}, {15, 19}, {31, 470}, {92, 34394}, {162, 6137}, {186, 2154}, {298, 1973}, {1094, 8737}, {2148, 6117}, {2159, 6110}, {2624, 36309}, {3384, 10641}, {6149, 8738}, {8742, 35198}, {23870, 32676}
X(40709) = cevapoint of X(2) and X(19772)
X(40709) = barycentric product X(i)*X(j) for these {i,j}: {3, 300}, {13, 69}, {16, 328}, {76, 36296}, {265, 299}, {298, 10217}, {304, 2153}, {305, 3457}, {525, 23895}, {850, 38414}, {3260, 39377}, {3265, 36306}, {3267, 5995}, {3926, 8737}, {4563, 20578}, {6390, 36307}, {11064, 36308}, {14592, 17403}
X(40709) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 470}, {3, 15}, {5, 6117}, {6, 8739}, {13, 4}, {16, 186}, {30, 6110}, {48, 2151}, {62, 10633}, {69, 298}, {125, 30465}, {184, 34394}, {265, 14}, {299, 340}, {300, 264}, {328, 301}, {343, 33529}, {395, 23715}, {471, 14165}, {476, 36309}, {525, 23870}, {622, 11094}, {647, 6137}, {1989, 8738}, {2153, 19}, {3457, 25}, {3564, 6782}, {4558, 17402}, {5612, 2914}, {5995, 112}, {6104, 10632}, {7100, 39152}, {8014, 463}, {8737, 393}, {8838, 473}, {8919, 36302}, {10217, 13}, {10218, 36210}, {10661, 2902}, {11078, 471}, {11080, 8737}, {11081, 8740}, {11082, 8742}, {11083, 10642}, {11118, 38427}, {11119, 38428}, {11139, 8741}, {11142, 10641}, {11537, 23712}, {11542, 31687}, {14417, 9204}, {14582, 20579}, {16770, 472}, {17403, 14590}, {18777, 23713}, {20578, 2501}, {23895, 648}, {30452, 8754}, {30454, 5095}, {30468, 35235}, {32585, 8603}, {32662, 5994}, {33530, 14918}, {34395, 34397}, {36296, 6}, {36297, 11086}, {36299, 1990}, {36306, 107}, {36307, 17983}, {36308, 16080}, {36839, 36306}, {38414, 110}, {38943, 36303}, {39153, 1870}, {39377, 74}
X(40709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11078, 13}, {2, 16770, 8838}, {3, 37638, 40710}, {8838, 11078, 16770}, {8838, 16770, 13}

X(40710) = ISOTOMIC CONJUGATE OF X(471)

Barycentrics    SA*(S - Sqrt[3]*SB)*(S - Sqrt[3]*SC) : :

X(40710) lies on the cubic K342b and these lines: {2, 14}, {3, 125}, {16, 3580}, {17, 94}, {61, 14389}, {62, 37644}, {69, 36297}, {95, 302}, {141, 11086}, {264, 301}, {287, 38413}, {299, 1494}, {343, 466}, {396, 14836}, {476, 36186}, {622, 19773}, {624, 15442}, {628, 8918}, {858, 14539}, {1989, 16644}, {2373, 5994}, {3129, 3818}, {3130, 32223}, {3131, 21243}, {3171, 22997}, {3448, 14169}, {3589, 11088}, {3763, 11141}, {5474, 36185}, {5613, 32460}, {6330, 36309}, {9204, 20579}, {9763, 18776}, {10218, 11064}, {11078, 36968}, {11126, 37779}, {11582, 40335}, {15066, 36209}, {16770, 16965}, {20429, 37975}, {34417, 37332}, {36298, 36889}, {37173, 37643}, {37341, 37648}

X(40710) = isogonal conjugate of X(8740)
X(40710) = isotomic conjugate of X(471)
X(40710) = isotomic conjugate of the complement of X(19773)
X(40710) = isotomic conjugate of the isogonal conjugate of X(36297)
X(40710) = isotomic conjugate of the polar conjugate of X(14)
X(40710) = isogonal conjugate of the polar conjugate of X(301)
X(40710) = X(301)-Ceva conjugate of X(14)
X(40710) = X(36297)-cross conjugate of X(14)
X(40710) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8740}, {4, 2152}, {16, 19}, {31, 471}, {92, 34395}, {162, 6138}, {186, 2153}, {299, 1973}, {1095, 8738}, {2148, 6116}, {2159, 6111}, {2624, 36306}, {3375, 10642}, {6149, 8737}, {8741, 35199}, {23871, 32676}
X(40710) = cevapoint of X(2) and X(19773)
X(40710) = barycentric product X(i)*X(j) for these {i,j}: {3, 301}, {14, 69}, {15, 328}, {76, 36297}, {265, 298}, {299, 10218}, {304, 2154}, {305, 3458}, {525, 23896}, {850, 38413}, {3260, 39378}, {3265, 36309}, {3267, 5994}, {3926, 8738}, {4563, 20579}, {6390, 36310}, {11064, 36311}, {14592, 17402}
X(40710) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 471}, {3, 16}, {5, 6116}, {6, 8740}, {14, 4}, {15, 186}, {30, 6111}, {48, 2152}, {61, 10632}, {69, 299}, {125, 30468}, {184, 34395}, {265, 13}, {298, 340}, {301, 264}, {328, 300}, {343, 33530}, {396, 23714}, {470, 14165}, {476, 36306}, {525, 23871}, {621, 11093}, {647, 6138}, {1989, 8737}, {2154, 19}, {3458, 25}, {3564, 6783}, {4558, 17403}, {5616, 2914}, {5994, 112}, {6105, 10633}, {7100, 39153}, {8015, 462}, {8738, 393}, {8836, 472}, {8918, 36303}, {10217, 36211}, {10218, 14}, {10662, 2903}, {11085, 8738}, {11086, 8739}, {11087, 8741}, {11088, 10641}, {11092, 470}, {11117, 38428}, {11120, 38427}, {11138, 8742}, {11141, 10642}, {11543, 31688}, {11549, 23713}, {14417, 9205}, {14582, 20578}, {16771, 473}, {17402, 14590}, {18776, 23712}, {20579, 2501}, {23896, 648}, {30453, 8754}, {30455, 5095}, {30465, 35235}, {32586, 8604}, {32662, 5995}, {33529, 14918}, {34394, 34397}, {36296, 11081}, {36297, 6}, {36298, 1990}, {36309, 107}, {36310, 17983}, {36311, 16080}, {36840, 36309}, {38413, 110}, {38944, 36302}, {39152, 1870}, {39378, 74}
X(40710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11092, 14}, {2, 16771, 8836}, {3, 37638, 40709}, {8836, 11092, 16771}, {8836, 16771, 14}

X(40711) = ISOTOMIC CONJUGATE OF X(472)

Barycentrics    SA*(Sqrt[3]*S - SB)*(Sqrt[3]*S - SC) : :

X(40711) lies on the cubic K867a and these lines: {2, 18}, {3, 539}, {13, 11140}, {15, 37636}, {69, 11516}, {95, 298}, {264, 299}, {303, 40410}, {343, 465}, {472, 33530}, {524, 8604}, {617, 8175}, {636, 15445}, {930, 5473}, {1494, 32037}, {1993, 10678}, {2373, 16807}, {2963, 16644}, {3131, 34507}, {5064, 5865}, {5464, 37850}, {8797, 36301}, {11082, 33458}, {11131, 15108}, {11442, 14539}, {21969, 37332}

X(40711) = isogonal conjugate of X(10641)
X(40711) = isotomic conjugate of X(472)
X(40711) = isotomic conjugate of the anticomplement of X(466)
X(40711) = isotomic conjugate of the isogonal conjugate of X(32586)
X(40711) = isotomic conjugate of the polar conjugate of X(18)
X(40711) = isogonal conjugate of the polar conjugate of X(34390)
X(40711) = X(34390)-Ceva conjugate of X(18)
X(40711) = X(i)-cross conjugate of X(j) for these (i,j): {466, 2}, {32586, 18}
X(40711) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10641}, {19, 62}, {31, 472}, {303, 1973}, {2153, 10633}, {2964, 8741}, {3383, 8739}, {8737, 35198}, {23873, 32676}
X(40711) = barycentric product X(i)*X(j) for these {i,j}: {3, 34390}, {18, 69}, {76, 32586}, {302, 3519}, {305, 21462}, {525, 32037}, {3267, 16807}, {3926, 8742}, {34386, 36301}
X(40711) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 472}, {3, 62}, {6, 10641}, {15, 10633}, {18, 4}, {61, 3518}, {69, 303}, {302, 32002}, {525, 23873}, {2963, 8741}, {3519, 17}, {8175, 36302}, {8604, 8740}, {8742, 393}, {10218, 11582}, {10678, 10632}, {11082, 8737}, {11138, 8738}, {11143, 473}, {16807, 112}, {19778, 470}, {21462, 25}, {32037, 648}, {32586, 6}, {34390, 264}, {36296, 11142}, {36297, 11088}, {36301, 53}, {36305, 462}, {40668, 23715}
X(40711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11143, 18}, {2, 19778, 11143}

X(40712) = ISOTOMIC CONJUGATE OF X(473)

Barycentrics    SA*(Sqrt[3]*S + SB)*(Sqrt[3]*S + SC) : :

X(40712) lies on the cubic K867b and these lines: {2, 17}, {3, 539}, {14, 11140}, {16, 37636}, {69, 11515}, {95, 299}, {264, 298}, {302, 40410}, {343, 466}, {473, 33529}, {524, 8603}, {616, 8174}, {635, 15444}, {930, 5474}, {1494, 32036}, {1993, 10677}, {2373, 16806}, {2963, 16645}, {3132, 34507}, {5064, 5864}, {5463, 37848}, {8797, 36300}, {11087, 33459}, {11130, 15108}, {11442, 14538}, {21969, 37333}

X(40712) = isogonal conjugate of X(10642)
X(40712) = isotomic conjugate of X(473)
X(40712) = isotomic conjugate of the anticomplement of X(465)
X(40712) = isotomic conjugate of the isogonal conjugate of X(32585)
X(40712) = isotomic conjugate of the polar conjugate of X(17)
X(40712) = isogonal conjugate of the polar conjugate of X(34389)
X(40712) = X(34389)-Ceva conjugate of X(17)
X(40712) = X(i)-cross conjugate of X(j) for these (i,j): {465, 2}, {32585, 17}
X(40712) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10642}, {19, 61}, {31, 473}, {302, 1973}, {2154, 10632}, {2964, 8742}, {3376, 8740}, {8738, 35199}, {23872, 32676}
X(40712) = barycentric product X(i)*X(j) for these {i,j}: {3, 34389}, {17, 69}, {76, 32585}, {303, 3519}, {305, 21461}, {525, 32036}, {3267, 16806}, {3926, 8741}, {34386, 36300}
X(40712) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 473}, {3, 61}, {6, 10642}, {16, 10632}, {17, 4}, {62, 3518}, {69, 302}, {303, 32002}, {525, 23872}, {2963, 8742}, {3519, 18}, {8174, 36303}, {8603, 8739}, {8741, 393}, {10217, 11581}, {10677, 10633}, {11087, 8738}, {11139, 8737}, {11144, 472}, {16806, 112}, {19779, 471}, {21461, 25}, {32036, 648}, {32585, 6}, {34389, 264}, {36296, 11083}, {36297, 11141}, {36300, 53}, {36304, 463}, {40667, 23714}
X(40712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11144, 17}, {2, 19779, 11144}

X(40713) = ISOTOMIC CONJUGATE OF X(1081)

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

X(40713) lies on the cubic K867a and these lines: {1, 2}, {9, 7089}, {13, 321}, {57, 36928}, {63, 617}, {75, 299}, {100, 12781}, {226, 36929}, {298, 319}, {303, 5564}, {312, 7043}, {318, 473}, {333, 7026}, {395, 17362}, {396, 594}, {466, 2968}, {472, 5081}, {533, 3578}, {894, 3180}, {956, 21475}, {2345, 37640}, {3181, 17363}, {3452, 5246}, {4060, 5243}, {4363, 5859}, {4385, 11303}, {4644, 5863}, {4665, 33458}, {4886, 5240}, {5015, 11304}, {5245, 5745}, {5295, 37144}, {5687, 21476}, {5814, 37145}, {5839, 37641}, {14829, 36668}, {17117, 34541}

X(40713) = reflection of X(40714) in X(3687)
X(40713) = isotomic conjugate of X(1081)
X(40713) = isotomic conjugate of the isogonal conjugate of X(1250)
X(40713) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2306}, {31, 1081}, {56, 1251}, {513, 36072}, {559, 6186}, {2153, 19373}, {3457, 37772}, {7051, 11072}
X(40713) = barycentric product X(i)*X(j) for these {i,j}: {76, 1250}, {298, 7026}, {312, 1082}, {2307, 3596}, {33653, 33939}
X(40713) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2306}, {2, 1081}, {9, 1251}, {15, 19373}, {101, 36072}, {1082, 57}, {1250, 6}, {2307, 56}, {3219, 559}, {5239, 39153}, {5240, 3179}, {5353, 7051}, {7006, 2307}, {7026, 13}, {7126, 11072}, {19551, 2153}, {33653, 2160}
X(40713) = {X(2),X(8)}-harmonic conjugate of X(40714)
X(40713) = {X(200),X(17294)}-harmonic conjugate of X(40714)

X(40714) = ISOTOMIC CONJUGATE OF X(554)

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

X(40714) lies on the cubic K867band these lines: {1, 2}, {9, 7088}, {14, 321}, {57, 36929}, {63, 616}, {75, 298}, {100, 12780}, {226, 36928}, {299, 319}, {302, 5564}, {312, 7026}, {318, 472}, {333, 7043}, {395, 594}, {396, 17362}, {465, 2968}, {473, 5081}, {532, 3578}, {894, 3181}, {956, 21476}, {2345, 37641}, {3180, 17363}, {3219, 7150}, {3452, 5245}, {4060, 5242}, {4363, 5858}, {4385, 11304}, {4644, 5862}, {4665, 33459}, {4886, 5239}, {5015, 11303}, {5246, 5745}, {5295, 37145}, {5687, 21475}, {5814, 37144}, {5839, 37640}, {14829, 36669}, {17117, 34540}
X(40714) = reflection of X(40713) in X(3687)
X(40714) = isotomic conjugate of X(554)
X(40714) = isotomic conjugate of the isogonal conjugate of X(10638)
X(40714) = X(i)-isoconjugate of X(j) for these (i,j): {6, 33654}, {31, 554}, {56, 33653}, {513, 36073}, {1082, 6186}, {2154, 7051}, {2160, 2307}, {3458, 37773}, {11073, 19373}
X(40714) = barycentric product X(i)*X(j) for these {i,j}: {76, 10638}, {299, 7043}, {312, 559}, {1251, 33939}
X(40714) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33654}, {2, 554}, {9, 33653}, {16, 7051}, {35, 2307}, {101, 36073}, {559 , 57}, {1251, 2160}, {3219, 1082}, {5240, 39152}, {5357, 19373}, {7043, 14}, {7126, 2154}, {7150, 7052}, {10638, 6}, {19551, 11073}
X(40714) = {X(2),X(8)}-harmonic conjugate of X(40713)
X(40714) = {X(200),X(17294)}-harmonic conjugate of X(40713)

X(40715) = ISOTOMIC CONJUGATE OF X(447)

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

X(40715) lies on the cubic K296 and these lines: {2, 1762}, {69, 17216}, {264, 17861}, {287, 9028}, {306, 15526}, {307, 1367}, {519, 1494}, {648, 40414}, {3187, 39352}, {4357, 40412}, {14429, 34767}, {17879, 20336}

X(40715) = midpoint of X(3187) and X(39352)
X(40715) = reflection of X(306) in X(15526)
X(40715) = isotomic conjugate of X(447)
X(40715) = antitomic conjugate of X(306)
X(40715) = X(i)-isoconjugate of X(j) for these (i,j): {31, 447}, {2203, 16086}
X(40715) = trilinear pole of line {440, 525}
X(40715) = barycentric product X(i)*X(j) for these {i,j}: {306, 16099}, {525, 35169}
X(40715) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 447}, {306, 16086}, {4466, 867}, {16099, 27}, {35169, 648}

X(40716) = ISOTOMIC CONJUGATE OF X(484)

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

X(40716) lies on the cubic K276 and these lines: {75, 1272}, {312, 2895}, {314, 1227}, {3260, 17791}, {3596, 20932}, {17361, 20570}, {20565, 34387}

X(40716) = isotomic conjugate of X(484)
X(40716) = isotomic conjugate of the anticomplement of X(11813)
X(40716) = isotomic conjugate of the complement of X(5180)
X(40716) = isotomic conjugate of the isogonal conjugate of X(3065)
X(40716) = X(i)-cross conjugate of X(j) for these (i,j): {320, 75}, {11813, 2}
X(40716) = X(i)-isoconjugate of X(j) for these (i,j): {6, 19297}, {25, 23071}, {31, 484}, {32, 17484}, {560, 17791}, {1333, 21864}, {2174, 11076}, {6126, 6187}
X(40716) = cevapoint of X(i) and X(j) for these (i,j): {2, 5180}, {3904, 24026}, {3936, 4647}
X(40716) = trilinear pole of line {4359, 4391}
X(40716) = barycentric product X(i)*X(j) for these {i,j}: {75, 21739}, {76, 3065}, {561, 19302}, {11075, 40075}
X(40716) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19297}, {2, 484}, {10, 21864}, {63, 23071}, {75, 17484}, {76, 17791}, {79, 11076}, {320, 40612}, {3065, 6}, {3218, 6126}, {4511, 26744}, {4560, 35055}, {7343, 2174}, {11075, 6187}, {14452, 11069}, {19302, 31}, {21739, 1}, {26743, 1411}, {34921, 1415}

X(40717) = ISOTOMIC CONJUGATE OF X(295)

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

X(40717) lies on these lines: {4, 1969}, {75, 24430}, {92, 264}, {242, 1921}, {273, 6384}, {286, 6385}, {317, 21276}, {318, 33938}, {331, 40028}, {350, 1874}, {561, 1851}, {811, 1870}, {1875, 18026}, {3261, 4025}, {3583, 23994}, {3975, 35544}, {5089, 6335}, {23978, 30737}

X(40717) = isotomic conjugate of X(295)
X(40717) = polar conjugate of X(292)
X(40717) = isotomic conjugate of the isogonal conjugate of X(242)
X(40717) = polar conjugate of the isotomic conjugate of X(1921)
X(40717) = polar conjugate of the isogonal conjugate of X(239)
X(40717) = X(239)-cross conjugate of X(1921)
X(40717) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1911}, {6, 2196}, {31, 295}, {48, 292}, {63, 1922}, {69, 14598}, {71, 18268}, {77, 18265}, {171, 17970}, {184, 291}, {228, 741}, {304, 18897}, {305, 18893}, {334, 14575}, {335, 9247}, {337, 560}, {603, 7077}, {813, 22383}, {875, 1331}, {876, 32656}, {906, 3572}, {1402, 1808}, {1409, 2311}, {1459, 34067}, {1967, 3955}, {2200, 37128}, {3049, 4584}, {3252, 32658}, {5378, 22096}, {7122, 36214}
X(40717) = cevapoint of X(239) and X(242)
X(40717) = crosssum of X(i) and X(j) for these (i,j): {3, 23186}, {228, 20777}, {22096, 23225}
X(40717) = pole wrt polar circle of trilinear polar of X(292) (line X(42)X(649), or PU(8))
X(40717) = perspector of circumconic through the polar conjugates of PU(8)
X(40717) = barycentric product X(i)*X(j) for these {i,j}: {4, 1921}, {19, 18891}, {27, 35544}, {76, 242}, {92, 350}, {238, 1969}, {239, 264}, {257, 17984}, {273, 3975}, {278, 4087}, {281, 18033}, {286, 3948}, {313, 31905}, {318, 10030}, {331, 3685}, {349, 14024}, {561, 2201}, {862, 6385}, {874, 17924}, {1447, 7017}, {1874, 28660}, {1914, 18022}, {3766, 6335}, {4010, 6331}, {6528, 24459}, {7193, 18027}, {7649, 27853}, {17982, 18035}, {34856, 40071}
X(40717) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2196}, {2, 295}, {4, 292}, {19, 1911}, {25, 1922}, {27, 741}, {28, 18268}, {29, 2311}, {76, 337}, {92, 291}, {238, 48}, {239, 3}, {242, 6}, {257, 36214}, {264, 335}, {281, 7077}, {286, 37128}, {318, 4876}, {331, 7233}, {333, 1808}, {350, 63}, {385, 3955}, {419, 172}, {607, 18265}, {659, 22383}, {740, 71}, {811, 4584}, {812, 1459}, {862, 213}, {874, 1332}, {893, 17970}, {1281, 20741}, {1284, 1409}, {1429, 603}, {1447, 222}, {1783, 34067}, {1861, 3252}, {1874, 1400}, {1897, 813}, {1914, 184}, {1921, 69}, {1969, 334}, {1973, 14598}, {1974, 18897}, {2201, 31}, {2210, 9247}, {2238, 228}, {3570, 1331}, {3573, 906}, {3684, 212}, {3685, 219}, {3716, 652}, {3747, 2200}, {3766, 905}, {3797, 3781}, {3948, 72}, {3975, 78}, {3985, 2318}, {4010, 647}, {4037, 3690}, {4039, 22061}, {4087, 345}, {4107, 22093}, {4124, 7117}, {4366, 7193}, {4375, 22384}, {4432, 22356}, {4435, 1946}, {4448, 22086}, {4455, 3049}, {4760, 3292}, {4974, 22054}, {6331, 4589}, {6335, 660}, {6591, 875}, {6651, 17976}, {6654, 36057}, {7017, 4518}, {7193, 577}, {7235, 2197}, {7649, 3572}, {8299, 20752}, {10030, 77}, {14024, 284}, {14599, 14575}, {14618, 35352}, {16609, 73}, {17031, 22099}, {17475, 20777}, {17493, 7015}, {17755, 1818}, {17793, 20785}, {17924, 876}, {17982, 9506}, {17984, 894}, {18022, 18895}, {18033, 348}, {18786, 7116}, {18891, 304}, {18894, 40373}, {19579, 23186}, {20457, 23223}, {20769, 255}, {21832, 810}, {24459, 520}, {27853, 4561}, {27912, 22148}, {27918, 3937}, {27919, 20778}, {27920, 20797}, {27922, 1797}, {27942, 20786}, {27945, 20761}, {27947, 20804}, {30940, 1444}, {31905, 58}, {33295, 1790}, {33891, 3784}, {34252, 15373}, {34856, 1474}, {35544, 306}, {39044, 20769}, {39914, 23086}, {39916, 20796}

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Points associated with CCC cubics: X(40718)-X(40725)

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This preamble is contributed by Clark Kimberling and Peter Moses, December 16, 2020.

Let P = p : q : r and U = u : v : w be points in the plane of a triangle ABC, and let
A'B'C' = cevian triangle of P, D'E'F' = cevian triangle of U
A"B"C" = anticevian triangle of P, D"E"F" = anticevian triangle of U
A* = A'D" ∩ A"E', and define B* and C* cyclically, so that

A* = - p u : q u + p v : r u + p w
B* = p v + q u : - q v : r v + q w
C* = p w + r u : q w + r v : - r w

The triangle A*B*C* is here named the (P,U)-cevian-cross triangle (not to be confused with the cross-cevian triangle in TCCT, p. 201)..

The locus of a point X = x : y : z such that the (P,U)-cevian-cross triangle is perspective to the cevian triangle of X is the (P,U)-CCC cubic, given by

(r u + p w)(q^2 u^2 + p q u v + p^2 v^2) y z^2 - (q u + p v) (r^2 u^2 + p r u w + p^2 w^2) y^2 z + (cyclic) = 0

The (P,U)-CCC cubic is the cubic pK(P*,U*), where

P* = q^2 u^2 + p q u v + p^2 v^2)(r^2 u^2 + p r u w + p^2w^2) : :
U* = (q^2 u^2 + p q u v + p^2*v^2)(r^2 u^2 + p r u w + p^2 w^2)(r v + q w) : :

Examples:
(X(15), X(16))-CCC cubic = pK(X(6), X(30)) = K001
(X(2), X(6))-CCC cubic = pK(X(3407), X(14617)) = K421

The locus of a point X = x : y : z such that the (P,U)-cevian-cross triangle is perspective to the anticevian triangle of X is the (P,U)-CCA cubic, given by

2 p u (r^2 u v + 2 q r u w + 2 p r v w + p q w^2) y^2 z - 2 p u (2 q r u v + p r v^2 + q^2 u w + 2 p q v w) y z^2 + (cyclic) = 0.

The (P,U)-CCA cubic is the cubic pK(P*,U*), where

P* = p u : : , and U* = q r u^2 + p^2 v w + 2 p u (r v + q w)) : :

Examples:
((X(2), X(4))_CCA cubic = pK(X(4), X(458)) = K677
((X(2), X(6))_CCA cubic = pK(X(6), X(3329)) = K423
((X(2), X(13))_CCA cubic = pK(X(13), X(8838)) = K420b
((X(2), X(14))_CCA cubic = pK(X(13), X(8836)) = K420a
((X(2), X(30))_CCA cubic = pK(X(30), X(2)) = K472
((X(6), X(98))_CCA cubic = pK(X(1976), X(98)) = K380
((X(13), X(14))_CCA cubic = pK(X(1989), X(265)) = K060
((X(15), X(16))_CCA cubic = pK(X(50), X(3)) = K073

X(40718) = X(1)X(76)∩X(2)X(31)

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

X(40718) lies on the Kiepert circumhyperbola, the (X(1),X(2))-CCC cubic, and these lines: {1, 76}, {2, 31}, {4, 1973}, {10, 213}, {37, 4368}, {40, 3597}, {42, 321}, {43, 2258}, {83, 16889}, {86, 741}, {98, 825}, {226, 1284}, {242, 37892}, {256, 291}, {262, 3402}, {513, 875}, {516, 2051}, {518, 25368}, {671, 923}, {672, 4672}, {726, 24330}, {871, 2296}, {984, 24514}, {1042, 1446}, {1064, 28850}, {1096, 2052}, {1125, 16850}, {1386, 17031}, {1492, 14009}, {1716, 10436}, {1751, 25453}, {1874, 40149}, {1918, 27042}, {2276, 3923}, {2475, 13584}, {2996, 38252}, {3112, 40016}, {3223, 40162}, {3399, 7594}, {3666, 24259}, {3696, 21904}, {3720, 30982}, {3741, 5847}, {3783, 5263}, {3789, 36480}, {3840, 4349}, {3993, 21101}, {4052, 4356}, {4080, 4613}, {4085, 13576}, {4272, 22316}, {4424, 11611}, {4441, 32921}, {4651, 6539}, {4865, 31330}, {4892, 30588}, {4974, 24592}, {5018, 7196}, {5057, 5143}, {6625, 18757}, {7248, 36538}, {8300, 20179}, {8781, 36051}, {10159, 29637}, {10290, 33946}, {10791, 16788}, {12609, 36907}, {13478, 29046}, {16475, 17026}, {16476, 17030}, {16826, 30571}, {17018, 32920}, {17135, 17772}, {17379, 21299}, {17469, 26237}, {17768, 25349}, {17770, 24690}, {19998, 27797}, {23660, 24512}, {24260, 29650}, {24325, 26234}, {25526, 32014}, {26102, 40012}, {26128, 30985}, {28498, 31241}, {28639, 34585}, {29207, 37365}, {30116, 36873}, {30950, 39994}, {30965, 32949}, {30966, 33082}, {31027, 32846}, {34087, 37132}

X(40718) = isogonal conjugate of X(3736)
X(40718) = isotomic conjugate of X(30966)
X(40718) = polar conjugate of X(31909)
X(40718) = X(789)-Ceva conjugate of X(4817)
X(40718) = X(i)-cross conjugate of X(j) for these (i,j): {37, 25425}, {4026, 10}, {4806, 3952}
X(40718) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3736}, {21, 1469}, {28, 3781}, {31, 30966}, {48, 31909}, {56, 3786}, {58, 984}, {81, 2276}, {86, 869}, {99, 788}, {101, 4481}, {110, 1491}, {163, 824}, {284, 7146}, {295, 17569}, {310, 18900}, {662, 3250}, {670, 8630}, {741, 3783}, {759, 3792}, {849, 3773}, {985, 4476}, {1014, 4517}, {1333, 3661}, {1408, 3790}, {1509, 3774}, {2150, 16603}, {2194, 7179}, {2206, 33931}, {2328, 7204}, {3116, 40415}, {3117, 38810}, {3733, 3799}, {3797, 18268}, {3864, 5009}, {4475, 4570}, {4615, 14436}, {14574, 30870}, {16514, 37128}, {17938, 30639}, {30654, 37134}
X(40718) = cevapoint of X(i) and X(j) for these (i,j): {1, 24342}, {6, 16372}, {10, 3993}, {894, 16826}
X(40718) = crosspoint of X(870) and X(14621)
X(40718) = crosssum of X(869) and X(2276)
X(40718) = trilinear pole of line {523, 798}
X(40718) = crossdifference of every pair of points on line {3250, 16514}
X(40718) = barycentric product X(i)*X(j) for these {i,j}: {10, 14621}, {37, 870}, {213, 871}, {321, 985}, {512, 37133}, {514, 4613}, {523, 4586}, {661, 789}, {825, 850}, {1441, 2344}, {1492, 1577}, {2887, 3407}, {3113, 3721}, {3114, 3778}, {3122, 5388}, {3952, 4817}, {4010, 37207}, {5384, 16732}, {14617, 16889}, {20948, 34069}, {24349, 25425}
X(40718) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30966}, {4, 31909}, {6, 3736}, {9, 3786}, {10, 3661}, {12, 16603}, {37, 984}, {42, 2276}, {65, 7146}, {71, 3781}, {213, 869}, {226, 7179}, {321, 33931}, {512, 3250}, {513, 4481}, {523, 824}, {594, 3773}, {661, 1491}, {740, 3797}, {789, 799}, {798, 788}, {825, 110}, {870, 274}, {871, 6385}, {872, 3774}, {984, 4469}, {985, 81}, {1018, 3799}, {1213, 3775}, {1334, 4517}, {1400, 1469}, {1427, 7204}, {1492, 662}, {1924, 8630}, {2201, 17569}, {2205, 18900}, {2238, 3783}, {2245, 3792}, {2276, 4476}, {2321, 3790}, {2344, 21}, {2887, 3314}, {3113, 38810}, {3125, 4475}, {3407, 40415}, {3696, 27474}, {3700, 4522}, {3747, 16514}, {3778, 3094}, {3842, 27495}, {3943, 4439}, {3952, 3807}, {3993, 27481}, {3997, 3809}, {4010, 4486}, {4024, 4122}, {4033, 4505}, {4586, 99}, {4613, 190}, {4817, 7192}, {4841, 4818}, {4931, 4951}, {5027, 30654}, {5384, 4567}, {8022, 18899}, {14621, 86}, {16584, 3116}, {18898, 38813}, {20948, 30870}, {21010, 25429}, {21832, 30665}, {21904, 3795}, {30664, 4584}, {30670, 4603}, {34069, 163}, {35352, 23596}, {37133, 670}, {37207, 4589}
X(40718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 26098, 30953}, {2, 33112, 30969}, {1386, 21264, 17031}, {5263, 37678, 3783}

X(40719) = X(1)X(85)∩X(2)X(7)

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

X(40719) lies on the (X(1),X(2))-CCC cubic and these lines: {1, 85}, {2, 7}, {6, 24600}, {10, 6604}, {12, 30617}, {21, 38859}, {29, 1847}, {37, 24352}, {40, 17753}, {56, 4059}, {65, 24805}, {69, 4847}, {75, 200}, {77, 4666}, {78, 20880}, {86, 269}, {145, 5543}, {150, 5587}, {169, 2140}, {218, 24774}, {220, 6706}, {241, 16831}, {273, 14004}, {279, 3616}, {320, 5231}, {331, 39585}, {348, 1125}, {349, 3760}, {350, 6063}, {461, 1119}, {481, 13453}, {482, 13436}, {551, 1323}, {946, 17170}, {948, 17023}, {988, 24214}, {997, 38468}, {1210, 36660}, {1212, 30625}, {1215, 7274}, {1231, 33945}, {1319, 7223}, {1358, 11730}, {1418, 15668}, {1419, 17379}, {1420, 7176}, {1434, 3361}, {1441, 3870}, {1442, 29817}, {1446, 4350}, {1462, 5276}, {1536, 5805}, {1565, 5886}, {1698, 32007}, {1699, 4872}, {1758, 4389}, {2887, 17272}, {3160, 3622}, {3212, 3340}, {3338, 7183}, {3485, 3674}, {3576, 5088}, {3617, 32003}, {3623, 25718}, {3624, 17095}, {3663, 13405}, {3664, 11019}, {3665, 11375}, {3668, 17093}, {3671, 10521}, {3672, 10578}, {3693, 3729}, {3742, 34855}, {3789, 39792}, {3872, 30806}, {3877, 23839}, {3879, 36845}, {3886, 4441}, {3945, 10580}, {3957, 7269}, {4071, 17296}, {4292, 36706}, {4298, 13725}, {4327, 26234}, {4384, 5228}, {4512, 33765}, {4554, 30963}, {4659, 21101}, {4853, 16284}, {4862, 17596}, {4911, 9612}, {4955, 5221}, {5136, 38461}, {5195, 31162}, {5263, 12560}, {5290, 7247}, {5542, 10520}, {5714, 36682}, {5722, 36722}, {6700, 25583}, {6762, 36854}, {7131, 17758}, {7177, 17169}, {7185, 17084}, {7196, 26102}, {7198, 10404}, {7228, 25355}, {7271, 25496}, {7289, 34830}, {7411, 18655}, {7580, 10444}, {8227, 17181}, {8551, 25878}, {9778, 15506}, {9780, 32098}, {10431, 18650}, {10473, 24471}, {11520, 20247}, {14189, 38316}, {17018, 25721}, {17078, 21314}, {17151, 32920}, {17234, 30813}, {17270, 25006}, {17300, 31038}, {17378, 31146}, {18443, 36027}, {19604, 27829}, {19860, 26563}, {21258, 40483}, {21446, 27475}, {23058, 26531}, {24349, 39959}, {24411, 35157}, {26134, 26959}, {27086, 34865}, {30545, 30982}, {30712, 36620}, {31269, 32024}, {31643, 40025}

X(40719) = isotomic conjugate of the isogonal conjugate of X(1471)
X(40719) = X(21446)-Ceva conjugate of X(9312)
X(40719) = X(1001)-cross conjugate of X(4384)
X(40719) = X(i)-isoconjugate of X(j) for these (i,j): {9, 2279}, {41, 27475}, {55, 1002}, {650, 8693}, {663, 37138}, {3063, 32041}
X(40719) = cevapoint of X(i) and X(j) for these (i,j): {1001, 5228}, {24349, 29627}
X(40719) = barycentric product X(i)*X(j) for these {i,j}: {7, 4384}, {56, 21615}, {57, 4441}, {75, 5228}, {76, 1471}, {85, 1001}, {269, 28809}, {273, 23151}, {279, 3886}, {307, 31926}, {664, 4762}, {1014, 4044}, {1088, 37658}, {1434, 3696}, {1893, 17206}, {2280, 6063}, {4554, 4724}, {4573, 4804}
X(40719) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 27475}, {56, 2279}, {57, 1002}, {109, 8693}, {651, 37138}, {664, 32041}, {1001, 9}, {1471, 6}, {1893, 1826}, {2280, 55}, {3696, 2321}, {3886, 346}, {4044, 3701}, {4384, 8}, {4441, 312}, {4702, 2325}, {4724, 650}, {4762, 522}, {4804, 3700}, {5228, 1}, {21615, 3596}, {23151, 78}, {27474, 3790}, {28044, 7079}, {28809, 341}, {31926, 29}, {32735, 36138}, {37658, 200}
X(40719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 85, 9312}, {1, 9312, 25716}, {2, 7, 9436}, {2, 10025, 9}, {7, 1447, 57}, {145, 31994, 25719}, {226, 36538, 57}, {1125, 10481, 348}, {1441, 7190, 3875}, {3160, 3622, 25723}, {3485, 7195, 3674}, {3616, 32086, 279}, {3664, 11019, 14548}, {5543, 31994, 145}, {7274, 25590, 39126}, {9318, 30949, 40131}, {11375, 24796, 3665}, {13405, 24283, 17594}, {20335, 24333, 9}, {20335, 25521, 25527}

X(40720) = X(1)X(87)∩X(2)X(1977)

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

X(40720) lies on the (X(1),X(2))-CCC cubic and these lines: {1, 87}, {2, 1977}, {42, 33784}, {86, 26143}, {238, 7121}, {932, 1001}, {1740, 20971}, {2295, 37677}, {3226, 24343}, {3618, 34249}, {8843, 9791}, {15485, 17105}, {17232, 26986}, {17349, 20669}, {27455, 28395}, {27633, 27672}

X(40720) = X(i)-cross conjugate of X(j) for these (i,j): {25376, 10009}, {30963, 4393}
X(40720) = X(2209)-isoconjugate of X(27494)
X(40720) = barycentric product X(i)*X(j) for these {i,j}: {87, 30963}, {330, 4393}, {2162, 10009}, {4598, 4785}, {4782, 18830}, {6383, 21793}, {6384, 16468}
X(40720) = barycentric quotient X(i)/X(j) for these {i,j}: {330, 27494}, {3993, 3971}, {4393, 192}, {4782, 4083}, {4785, 3835}, {4806, 21051}, {4991, 4970}, {10009, 6382}, {16468, 43}, {21793, 2176}, {21904, 20691}, {23095, 20760}, {25376, 21250}, {30963, 6376}, {34476, 38832}
X(40720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 33681, 192}, {87, 39914, 330}

X(40721) = X(1)X(1655)∩X(2)X(6)

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

X(40721) lies on the (X(1),X(2))-CCC cubic and these lines: {1, 1655}, {2, 6}, {9, 17032}, {42, 894}, {144, 11688}, {192, 714}, {194, 19767}, {257, 2650}, {274, 20970}, {335, 21840}, {350, 1100}, {672, 17120}, {870, 4393}, {1008, 7754}, {1218, 39961}, {1449, 17027}, {1509, 5277}, {2234, 3240}, {2271, 16915}, {2276, 3758}, {2280, 14621}, {2295, 4595}, {2475, 6625}, {3616, 16476}, {3720, 23532}, {3783, 33682}, {3879, 31027}, {3882, 17754}, {4360, 24330}, {4651, 28604}, {4670, 21904}, {4713, 16884}, {4754, 33296}, {5021, 17684}, {5625, 30571}, {7277, 25349}, {8040, 17248}, {16497, 38314}, {16666, 17029}, {16667, 17026}, {16826, 25427}, {17023, 31004}, {17103, 17693}, {17121, 24592}, {17126, 18900}, {17302, 20347}, {17312, 30821}, {17363, 31330}, {17367, 30949}, {17750, 26752}, {20072, 29822}, {20963, 26801}, {26626, 30946}, {26815, 26971}, {29841, 30961}

X(40721) = anticomplement of X(30966)
X(40721) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {798, 39345}, {825, 7192}, {870, 17138}, {985, 17135}, {1492, 512}, {2344, 20245}, {3407, 561}, {4586, 17217}, {4613, 21301}, {9426, 39347}, {14621, 17137}, {34069, 523}, {37133, 21305}
X(40721) = X(4649)-Ceva conjugate of X(4393)
X(40721) = cevapoint of X(1045) and X(25427)
X(40721) = crosspoint of X(4586) and X(4590)
X(40721) = crosssum of X(3124) and X(3250)
X(40721) = crossdifference of every pair of points on line {512, 21763}
X(40721) = barycentric quotient X(30571)/X(30570)
X(40721) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17499, 1655}, {2, 20090, 30941}, {6, 20140, 17349}, {6, 37632, 2}, {42, 894, 17759}, {86, 2238, 2}, {1654, 20536, 2895}, {2663, 3510, 42}, {2665, 39916, 30667}, {17103, 18755, 17693}, {24512, 37678, 2}

X(40722) = X(1)X(257)∩X(2)X(2112)

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

X(40722) lies on the (X(1),X(2))-CCC cubic and these lines: {1, 257}, {2, 2112}, {86, 1333}, {870, 17962}, {894, 8424}, {985, 3616}, {3661, 26244}, {4586, 35162}, {6625, 18757}, {17084, 17689}, {17762, 18755}, {22267, 33867}

X(40722) = X(i)-isoconjugate of X(j) for these (i,j): {869, 6625}, {984, 2248}, {2276, 13610}, {3661, 18757}, {3774, 40164}
X(40722) = barycentric product X(i)*X(j) for these {i,j}: {846, 870}, {985, 17762}, {1654, 14621}, {4586, 21196}
X(40722) = barycentric quotient X(i)/X(j) for these {i,j}: {846, 984}, {985, 13610}, {1654, 3661}, {2905, 31909}, {6626, 30966}, {14621, 6625}, {17084, 7179}, {17762, 33931}, {18755, 2276}, {21085, 3773}, {21196, 824}, {22139, 3781}, {27691, 16603}

X(40723) = X(1)X(18299)∩X(2)X(2114)

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

X(40723) lies on the cubic VT(X(1),X(2)) and these lines: {1, 18299}, {2, 2114}, {57, 7249}, {85, 25496}, {86, 269}, {223, 9312}, {226, 6625}, {348, 26098}, {664, 1215}, {870, 6063}, {2887, 17095}, {4865, 33298}, {5018, 7196}, {10030, 29821}, {15903, 33144}, {17739, 27963}, {24333, 31526}, {32942, 39775}

X(40723) = X(7196)-Ceva conjugate of X(57)
X(40723) = X(8424)-cross conjugate of X(17739)
X(40723) = X(i)-isoconjugate of X(j) for these (i,j): {9, 18784}, {2175, 18760}, {7077, 16366}
X(40723) = barycentric product X(i)*X(j) for these {i,j}: {7, 17739}, {57, 30660}, {85, 8424}, {7249, 27963}, {18759, 20567}
X(40723) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 18784}, {85, 18760}, {1429, 16366}, {8424, 9}, {17739, 8}, {18759, 41}, {27963, 7081}, {30660, 312}

X(40724) = X(1)X(85)∩X(2)X(2115)

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

X(40724) lies on the (X(1),X(2))-CCC cubic and these lines: {1, 85}, {2, 2115}, {3, 34179}, {10, 666}, {79, 5377}, {105, 5253}, {673, 1492}, {894, 33676}, {927, 2700}, {1027, 1220}, {2475, 6625}, {5263, 33674}, {6185, 17023}, {6646, 9501}, {6654, 14267}, {9441, 24980}, {11109, 36124}, {24234, 37607}, {24723, 36086}

X(40724) = cevapoint of X(1281) and X(4645)
X(40724) = trilinear pole of line {3509, 4458}
X(40724) = X(i)-isoconjugate of X(j) for these (i,j): {518, 8852}, {672, 3512}, {1458, 7281}, {2223, 7261}, {8299, 30648}, {9455, 18036}
X(40724) = barycentric product X(i)*X(j) for these {i,j}: {105, 17789}, {666, 4458}, {673, 4645}, {2481, 3509}, {5018, 36796}, {17798, 18031}
X(40724) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 3512}, {294, 7281}, {673, 7261}, {1281, 17755}, {1438, 8852}, {3509, 518}, {4071, 3932}, {4458, 918}, {4645, 3912}, {4987, 4966}, {5018, 241}, {17789, 3263}, {17798, 672}, {18031, 18036}, {18262, 9454}, {19554, 2223}, {19557, 8299}, {20715, 3930}, {20741, 1818}

X(40725) = X(2)X(846)∩X(81)X(17930)

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

X(40725) lies on the (X(1),X(2))-CCC cubic and these lines: {2, 846}, {81, 17930}, {239, 27916}, {321, 6634}, {350, 27912}, {870, 17962}, {873, 16727}, {894, 9506}, {2702, 9073}, {4375, 6545}, {4760, 27922}, {6652, 27918}, {9278, 25368}, {18822, 29584}, {19936, 29609}

X(40725) = X(6650)-Ceva conjugate of X(239)
X(40725) = X(i)-cross conjugate of X(j) for these (i,j): {659, 17930}, {4366, 239}
X(40725) = X(i)-isoconjugate of X(j) for these (i,j): {291, 17735}, {292, 1757}, {335, 18266}, {660, 5029}, {741, 20693}, {813, 9508}, {1911, 6542}, {1922, 20947}, {2196, 17927}, {2786, 34067}, {4584, 17990}, {6541, 18268}, {18035, 18267}
X(40725) = cevapoint of X(4375) and X(27918)
X(40725) = trilinear pole of line {812, 4974}
X(40725) = barycentric product X(i)*X(j) for these {i,j}: {238, 18032}, {239, 6650}, {350, 1929}, {812, 35148}, {1921, 17962}, {3766, 37135}, {4010, 17930}, {9278, 30940}, {9505, 39044}, {11599, 33295}, {17972, 40717}
X(40725) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 1757}, {239, 6542}, {242, 17927}, {350, 20947}, {659, 9508}, {740, 6541}, {812, 2786}, {1914, 17735}, {1929, 291}, {2210, 18266}, {2238, 20693}, {2702, 813}, {4010, 18004}, {4366, 6651}, {4375, 27929}, {4448, 28602}, {4455, 17990}, {5009, 1326}, {6650, 335}, {6652, 27926}, {7193, 17976}, {8300, 8298}, {8632, 5029}, {9505, 30663}, {17930, 4589}, {17962, 292}, {17972, 295}, {18014, 35352}, {18032, 334}, {31905, 423}, {33295, 17731}, {35148, 4562}, {37135, 660}

X(40726) = X(1)X(4004)∩X(2)X(12)

Barycentrics    a*(3*a^3 - 3*a*b^2 + 8*a*b*c + 2*b^2*c - 3*a*c^2 + 2*b*c^2) : :
X(40726) = 2 X[57] + X[5289], 2 X[999] + X[1376], 8 X[999] + X[8168], 4 X[1125] - X[24703], 4 X[1376] - X[8168], X[3474] + 5 X[3616], 2 X[3816] + X[4293], X[4315] + 2 X[6692], X[8168] - 8 X[16417], X[8169] - 4 X[25524], 2 X[10269] + X[22753]

X(40726) lies on these lines: {1, 4004}, {2, 12}, {3, 551}, {8, 36006}, {30, 7956}, {35, 19705}, {36, 1001}, {55, 4345}, {57, 5289}, {104, 3545}, {115, 22565}, {165, 10179}, {214, 15934}, {354, 35262}, {381, 10199}, {404, 3241}, {474, 3679}, {499, 17530}, {519, 999}, {547, 32153}, {549, 11249}, {553, 34647}, {597, 22769}, {758, 35272}, {954, 38024}, {956, 19875}, {960, 3361}, {993, 8167}, {1012, 38021}, {1056, 3035}, {1125, 16418}, {1149, 37540}, {1191, 37608}, {1201, 8688}, {1319, 3306}, {1420, 3812}, {1478, 17533}, {1616, 37603}, {2066, 9689}, {2099, 27003}, {2482, 22514}, {3058, 22768}, {3086, 3829}, {3303, 4188}, {3337, 5730}, {3338, 12635}, {3428, 3524}, {3445, 5255}, {3474, 3616}, {3488, 17051}, {3550, 16486}, {3576, 3742}, {3582, 17532}, {3622, 5217}, {3624, 17542}, {3654, 10680}, {3655, 11500}, {3720, 16395}, {3746, 19537}, {3813, 6904}, {3816, 4293}, {3825, 9655}, {3828, 8666}, {3847, 5229}, {3878, 37545}, {3919, 10247}, {3929, 8583}, {4187, 4317}, {4190, 37722}, {4193, 9657}, {4252, 21214}, {4298, 25681}, {4315, 6692}, {4370, 24826}, {4423, 16858}, {4511, 4860}, {4666, 37600}, {4669, 9709}, {4795, 24328}, {4855, 17609}, {4870, 34880}, {4930, 5708}, {4995, 10966}, {5054, 10197}, {5055, 22758}, {5066, 18761}, {5123, 31190}, {5154, 9656}, {5251, 19536}, {5258, 16862}, {5437, 13462}, {5438, 34791}, {5439, 37618}, {5459, 22773}, {5460, 22774}, {5550, 16861}, {5584, 15692}, {5642, 22586}, {5710, 32577}, {5883, 10246}, {5886, 28444}, {6055, 22504}, {6174, 11239}, {6284, 10586}, {6667, 10590}, {6681, 31479}, {6826, 20418}, {6911, 28204}, {6914, 38022}, {6921, 15888}, {6946, 38074}, {7223, 26229}, {7280, 19704}, {7373, 25440}, {7963, 39980}, {8301, 35110}, {8715, 17573}, {9466, 22779}, {9670, 37256}, {9710, 17580}, {10056, 22767}, {10058, 38026}, {10072, 10948}, {10181, 10606}, {10200, 18990}, {10304, 11495}, {10596, 24466}, {10894, 26492}, {10912, 20323}, {11113, 34620}, {11179, 39883}, {11238, 17579}, {11240, 34612}, {11263, 28453}, {11274, 12331}, {11281, 21161}, {11346, 19769}, {11357, 19762}, {11358, 31137}, {11496, 32612}, {11813, 18541}, {12100, 35239}, {12607, 17567}, {13370, 28609}, {13738, 19722}, {16431, 29597}, {17556, 34739}, {17572, 31145}, {17577, 22760}, {17798, 19325}, {18967, 34743}, {19013, 19054}, {19014, 19053}, {19709, 26321}, {19796, 37091}, {19861, 31165}, {22763, 32787}, {22764, 32788}, {25893, 31164}, {26286, 28466}, {31146, 37270}, {31162, 37561}, {34123, 38056}, {36740, 38023}, {37617, 37674}, {38031, 38054}, {38053, 38454}

X(40726) = midpoint of X(999) and X(16417)
X(40726) = reflection of X(1376) in X(16417)
X(40726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16371, 4421}, {2, 56, 11194}, {2, 5434, 11236}, {2, 11194, 958}, {2, 34605, 31141}, {3, 551, 4428}, {36, 25055, 16370}, {56, 5253, 25524}, {56, 25524, 958}, {404, 3304, 3913}, {474, 5563, 12513}, {993, 19883, 16857}, {3338, 17614, 12635}, {10072, 11112, 11235}, {11194, 25524, 2}, {11238, 17579, 34706}, {13587, 38314, 55}, {16370, 25055, 1001}, {16857, 19883, 8167}

X(40727) = X(2)X(2418)∩X(3)X(543)

Barycentrics    a^4 - a^2*b^2 + 4*b^4 - a^2*c^2 - 16*b^2*c^2 + 4*c^4 : :
X(50727) = 5 X[2] - X[11148], 3 X[3] - 4 X[5569], 7 X[3] - 4 X[34504], X[3] + 2 X[34505], 5 X[3] - 8 X[34506], 3 X[381] - 4 X[20112], 8 X[1153] - 7 X[15701], 5 X[1656] - 4 X[9771], 5 X[1656] - 2 X[34511], X[3534] - 4 X[13468], X[3830] + 2 X[8667], X[3830] - 4 X[18546], 5 X[3843] + 4 X[7751], 7 X[3851] - 4 X[7775], 3 X[5054] - 2 X[7618], 3 X[5054] - 4 X[15597], 3 X[5055] - 4 X[7617], 3 X[5055] - 2 X[11184], 11 X[5072] - 2 X[7758], X[5073] + 8 X[7780], 3 X[5485] + X[9741], 5 X[5485] + X[11148], 2 X[5485] + X[11165], 3 X[5485] + 2 X[12040], X[5485] + 2 X[16509], X[5503] - 3 X[9166], 2 X[5569] - 3 X[7610], 7 X[5569] - 3 X[34504], 2 X[5569] + 3 X[34505], 5 X[5569] - 6 X[34506], 7 X[7610] - 2 X[34504], 5 X[7610] - 4 X[34506], 3 X[7615] - 2 X[20112], 4 X[7622] - 5 X[15694], 4 X[8176] - 5 X[19709], X[8667] + 2 X[18546], 2 X[8716] - 5 X[15694], 5 X[9741] - 3 X[11148], 2 X[9741] - 3 X[11165], X[9741] - 6 X[16509], 2 X[9766] - 5 X[19709], 2 X[11148] - 5 X[11165], 3 X[11148] - 10 X[12040], X[11148] - 10 X[16509], 3 X[11165] - 4 X[12040], X[11165] - 4 X[16509], X[12040] - 3 X[16509], 2 X[13085] + X[13108], 2 X[34504] + 7 X[34505], 5 X[34504] - 14 X[34506], 5 X[34505] + 4 X[34506]

Let A'B'C' be the antipedal triangle of X(2) wrt the medial triangle. Then X(40727) = X(4)-of-A'B'C'. (Randy Hutson, December 18, 2020)

X(40727) lies on the Kiepert circumhyperbola of the Brocard triangle and these lines: {2, 2418}, {3, 543}, {4, 9740}, {5, 9770}, {30, 7620}, {69, 37350}, {76, 5503}, {99, 8860}, {115, 599}, {148, 35955}, {183, 671}, {381, 524}, {385, 10807}, {525, 8371}, {538, 5055}, {597, 14535}, {598, 12156}, {754, 14269}, {1003, 8859}, {1153, 15701}, {1384, 11159}, {1656, 9771}, {1992, 3363}, {2482, 37637}, {2549, 11168}, {2782, 9743}, {2896, 7841}, {2996, 33215}, {3094, 9466}, {3534, 8182}, {3642, 5459}, {3643, 5460}, {3767, 33237}, {3821, 3828}, {3830, 3849}, {3843, 7751}, {3845, 23334}, {3851, 7775}, {3933, 32984}, {5032, 32983}, {5054, 7618}, {5072, 7758}, {5073, 7780}, {5286, 8367}, {5309, 24273}, {5461, 7778}, {5475, 15534}, {5969, 22677}, {7622, 8716}, {7746, 9167}, {7754, 33013}, {7761, 36523}, {7776, 33006}, {7801, 13881}, {8176, 9766}, {8359, 32828}, {8366, 17128}, {8370, 30435}, {8556, 11648}, {8586, 15533}, {8591, 17004}, {8598, 15655}, {9178, 9462}, {9486, 11162}, {9737, 32414}, {9774, 39646}, {9830, 12188}, {10000, 11286}, {10008, 21356}, {10717, 20481}, {11054, 11163}, {11164, 26613}, {12505, 14262}, {12525, 34383}, {13085, 13108}, {13188, 19911}, {13191, 33962}, {14033, 19661}, {14711, 18362}, {15271, 32457}, {15681, 32479}, {16644, 36775}, {18424, 40341}, {23055, 27088}, {32538, 37689}, {32834, 33190}, {32874, 33285}, {33896, 37690}

X(40727) = midpoint of X(i) and X(j) for these {i,j}: {2, 5485}, {4, 9740}, {7610, 34505}
X(40727) = reflection of X(i) in X(j) for these {i,j}: {2, 16509}, {3, 7610}, {381, 7615}, {3534, 8182}, {7618, 15597}, {8182, 13468}, {8716, 7622}, {9737, 32414}, {9741, 12040}, {9766, 8176}, {9770, 5}, {11165, 2}, {11184, 7617}, {13188, 19911}, {23334, 3845}, {34511, 9771}
X(40727) = isotomic conjugate of the isogonal conjugate of X(22111)
X(40727) = anticomplement of X(12040)
X(40727) = complement of X(9741)
X(40727) = complement of the isogonal conjugate of X(39236)
X(40727) = X(39236)-complementary conjugate of X(10)
X(40727) = X(9770)-of-Johnson-triangle
X(40727) = barycentric product X(76)*X(22111)
X(40727) = barycentric quotient X(22111)/X(6)
X(40727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9741, 12040}, {183, 671, 5077}, {1992, 3363, 15484}, {5485, 16509, 11165}, {7617, 11184, 5055}, {7618, 15597, 5054}, {8667, 18546, 3830}, {9741, 12040, 11165}, {11054, 11163, 22253}, {11159, 22329, 1384}, {11185, 22329, 11159}, {17131, 31173, 15533}, {23055, 32815, 27088}, {42035, 42036, 599}

X(40728) = X(1)X(6)∩X(31)X(1911)

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

X(40728) lies on the cubic K1019 and these lines: {1, 6}, {31, 1911}, {32, 560}, {39, 2274}, {41, 904}, {42, 1185}, {43, 7075}, {51, 21813}, {55, 1197}, {81, 17032}, {86, 17750}, {100, 717}, {101, 731}, {171, 1613}, {172, 5156}, {181, 1196}, {190, 32453}, {748, 20965}, {750, 3231}, {869, 3774}, {1201, 20459}, {1206, 17018}, {1334, 2309}, {1500, 3688}, {1740, 3499}, {1964, 39258}, {2053, 21759}, {2211, 2212}, {2225, 30647}, {2235, 3923}, {2271, 23863}, {2276, 3736}, {3094, 3792}, {3271, 5052}, {3550, 21792}, {3730, 5145}, {3997, 33682}, {4259, 20861}, {4383, 17026}, {5017, 7295}, {5364, 16584}, {9463, 17126}, {14974, 20992}, {14997, 17029}, {16549, 18792}, {17027, 32911}, {17028, 37680}, {17030, 17277}, {17033, 17743}, {17034, 18147}, {17122, 21001}, {17349, 26801}, {17350, 19565}, {17379, 26082}, {17475, 32921}, {18899, 19587}, {23632, 36808}
X(40728) = isogonal conjugate of the isotomic conjugate of X(2276)
X(40728) = X(i)-Ceva conjugate of X(j) for these (i,j): {5388, 100}, {8693, 667}
X(40728) = X(i)-isoconjugate of X(j) for these (i,j): {2, 870}, {6, 871}, {75, 14621}, {76, 985}, {244, 5388}, {513, 37133}, {514, 789}, {668, 4817}, {693, 4586}, {825, 40495}, {982, 3114}, {1492, 3261}, {2344, 6063}, {3113, 3662}, {3407, 33930}, {3766, 37207}, {4583, 23597}, {4613, 7199}, {5384, 23989}
X(40728) = crosspoint of X(100) and X(5388)
X(40728) = crosssum of X(i) and X(j) for these (i,j): {2, 4441}, {75, 20917}, {76, 10009}, {28959, 34387}
X(40728) = crossdifference of every pair of points on line {513, 3261}
X(40728) = barycentric product X(i)*X(j) for these {i,j}: {1, 869}, {6, 2276}, {25, 3781}, {31, 984}, {32, 3661}, {41, 7146}, {42, 3736}, {55, 1469}, {56, 4517}, {75, 18900}, {81, 3774}, {100, 788}, {101, 3250}, {292, 16514}, {560, 33931}, {667, 3799}, {668, 8630}, {692, 1491}, {824, 32739}, {983, 3116}, {1110, 4475}, {1253, 7204}, {1397, 3790}, {1402, 3786}, {1576, 4122}, {1911, 3783}, {1914, 3862}, {1918, 30966}, {1919, 3807}, {1922, 3797}, {1980, 4505}, {2175, 7179}, {2200, 31909}, {2206, 3773}, {2210, 3864}, {2284, 29956}, {3117, 17743}, {3257, 14436}, {3792, 6187}, {5386, 14402}, {30665, 34067}
X(40728) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 871}, {31, 870}, {32, 14621}, {101, 37133}, {560, 985}, {692, 789}, {788, 693}, {869, 75}, {984, 561}, {1252, 5388}, {1469, 6063}, {1491, 40495}, {1919, 4817}, {2276, 76}, {3116, 33930}, {3117, 3662}, {3250, 3261}, {3661, 1502}, {3736, 310}, {3774, 321}, {3781, 305}, {3783, 18891}, {3786, 40072}, {3790, 40363}, {3792, 40075}, {3799, 6386}, {3862, 18895}, {4517, 3596}, {7146, 20567}, {8630, 513}, {9447, 2344}, {14436, 3762}, {16514, 1921}, {17415, 3801}, {18899, 2275}, {18900, 1}, {19587, 20917}, {32739, 4586}, {33931, 1928}
X(40728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 24727}, {6, 1001, 23660}, {6, 2176, 238}, {6, 21769, 16503}, {6, 21788, 1}, {32, 2175, 14599}, {213, 21760, 6}, {1918, 9454, 32}, {3051, 7109, 31}, {3230, 23660, 1001}

X(40729) = X(1)X(2670)∩X(9)X(43)

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

X(40729) lies on the cubics K220 and K1019, and on these lines: {1, 2670}, {6, 694}, {7, 16592}, {9, 43}, {37, 4039}, {41, 904}, {42, 22201}, {115, 19637}, {257, 17033}, {798, 1964}, {874, 17280}, {1334, 3774}, {1400, 16584}, {1431, 2279}, {1432, 39970}, {1469, 18784}, {1967, 19587}, {2229, 27447}, {2245, 3863}, {2295, 6378}, {3124, 30647}, {3229, 28369}, {4116, 9427}, {7032, 21755}, {17349, 40432}, {18785, 21796}, {20964, 21815}, {29055, 35106}

X(40729) = isogonal conjugate of X(8033)
X(40729) = X(37137)-Ceva conjugate of X(512)
X(40729) = X(i)-cross conjugate of X(j) for these (i,j): {3725, 213}, {7063, 512}
X(40729) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8033}, {2, 17103}, {7, 27958}, {21, 7196}, {58, 1920}, {81, 1909}, {86, 894}, {99, 4369}, {100, 16737}, {171, 274}, {172, 310}, {190, 17212}, {261, 4032}, {284, 7205}, {314, 7175}, {333, 7176}, {348, 14006}, {385, 18827}, {552, 4095}, {593, 1237}, {662, 4374}, {668, 18200}, {670, 20981}, {741, 3978}, {757, 3963}, {799, 4367}, {873, 2295}, {880, 3572}, {1014, 17787}, {1215, 1509}, {1434, 7081}, {1580, 40017}, {1926, 18268}, {1966, 37128}, {2162, 27891}, {2533, 4610}, {3287, 4625}, {3907, 4573}, {4107, 4589}, {4128, 34537}, {4164, 4639}, {4444, 17941}, {4459, 4620}, {4477, 4635}, {4529, 4616}, {4560, 6649}, {4576, 18111}, {4579, 7199}, {4584, 14296}, {4600, 7200}, {4615, 4922}, {4697, 32014}, {4754, 40439}, {6331, 22093}, {6385, 7122}, {6628, 21021}, {6645, 32010}, {7009, 17206}, {7184, 38810}, {7187, 40415}, {7192, 18047}, {16592, 24037}, {18787, 30940}, {27954, 40164}, {30669, 33295}
X(40729) = cevapoint of X(798) and X(1084)
X(40729) = crosspoint of X(893) and X(904)
X(40729) = crosssum of X(i) and X(j) for these (i,j): {7, 34061}, {333, 39915}, {894, 1909}, {3023, 3287}, {4367, 21755}, {17103, 27958}
X(40729) = crossdifference of every pair of points on line {804, 1966}
X(40729) = barycentric product X(i)*X(j) for these {i,j}: {10, 904}, {37, 893}, {42, 256}, {210, 1431}, {213, 257}, {321, 7104}, {512, 3903}, {694, 2238}, {740, 1967}, {756, 1178}, {798, 27805}, {805, 4155}, {862, 36214}, {872, 32010}, {874, 881}, {882, 3573}, {1334, 1432}, {1402, 4451}, {1500, 40432}, {1581, 3747}, {1824, 7015}, {1826, 7116}, {1918, 7018}, {1927, 35544}, {3709, 37137}, {3948, 9468}, {4041, 29055}, {4079, 4603}
X(40729) = trilinear product X(i)*X(j) for these {i,j}: {10, 7104}, {37, 904}, {42, 893}, {213, 256}, {257, 1918}, {694, 3747}, {733, 4093}, {798, 3903}, {881, 3570}, {1178, 1500}, {1334, 1431}, {1824, 7116}, {1927, 3948}, {1967, 2238}, {2205, 7018}, {2333, 7015}
X(40729) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8033}, {31, 17103}, {37, 1920}, {41, 27958}, {42, 1909}, {43, 27891}, {65, 7205}, {213, 894}, {256, 310}, {257, 6385}, {512, 4374}, {649, 16737}, {667, 17212}, {669, 4367}, {694, 40017}, {740, 1926}, {756, 1237}, {798, 4369}, {862, 17984}, {872, 1215}, {881, 876}, {893, 274}, {904, 86}, {1084, 16592}, {1178, 873}, {1334, 17787}, {1400, 7196}, {1402, 7176}, {1500, 3963}, {1918, 171}, {1919, 18200}, {1924, 20981}, {1927, 741}, {1967, 18827}, {2205, 172}, {2212, 14006}, {2238, 3978}, {3121, 7200}, {3573, 880}, {3747, 1966}, {3903, 670}, {3948, 14603}, {4117, 4128}, {4155, 14295}, {4451, 40072}, {4455, 14296}, {4826, 4842}, {7063, 40608}, {7104, 81}, {7109, 2295}, {7116, 17206}, {8789, 18268}, {9427, 21755}, {9468, 37128}, {16584, 7187}, {17938, 36066}, {21753, 4754}, {21814, 16720}, {21815, 18905}, {23216, 22373}, {27805, 4602}, {29055, 4625}

X(40730) = X(1)X(2111)∩X(2)X(660)

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

X(40730) lies on the cubics K577 and K1019, and on these lines: {1, 2111}, {2, 660}, {31, 1911}, {38, 25813}, {42, 649}, {43, 57}, {55, 813}, {292, 2279}, {335, 3873}, {518, 27919}, {672, 3252}, {1026, 4447}, {1397, 2149}, {2196, 18265}, {2276, 18783}, {4583, 32937}, {4589, 8033}, {8041, 35505}, {17754, 36906}, {24514, 39918}

X(40730) = isogonal conjugate of the isotomic conjugate of X(22116)
X(40730) = X(i)-Ceva conjugate of X(j) for these (i,j): {660, 665}, {1911, 2223}
X(40730) = X(i)-isoconjugate of X(j) for these (i,j): {2, 6654}, {105, 350}, {238, 2481}, {239, 673}, {242, 31637}, {294, 10030}, {666, 812}, {874, 1027}, {927, 3716}, {1416, 4087}, {1429, 36796}, {1438, 1921}, {1447, 14942}, {1462, 3975}, {1914, 18031}, {2195, 18033}, {3684, 34018}, {3766, 36086}, {4124, 39293}, {4435, 34085}, {6185, 17755}, {8632, 36803}, {13576, 33295}, {18785, 30940}, {36057, 40717}
X(40730) = crosssum of X(i) and X(j) for these (i,j): {239, 8299}, {350, 39044}, {659, 35119}, {673, 33674}
X(40730) = crossdifference of every pair of points on line {239, 3766}
X(40730) = barycentric product X(i)*X(j) for these {i,j}: {1, 3252}, {6, 22116}, {31, 40217}, {241, 7077}, {291, 672}, {292, 518}, {295, 5089}, {334, 9454}, {335, 2223}, {660, 665}, {694, 4447}, {741, 3930}, {813, 2254}, {876, 2284}, {918, 34067}, {1026, 3572}, {1458, 4876}, {1861, 2196}, {1911, 3912}, {1922, 3263}, {3932, 18268}, {9455, 18895}, {18265, 40704}, {18827, 39258}, {20683, 37128}
X(40730) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 6654}, {241, 18033}, {291, 18031}, {292, 2481}, {518, 1921}, {660, 36803}, {665, 3766}, {672, 350}, {1026, 27853}, {1458, 10030}, {1911, 673}, {1922, 105}, {2196, 31637}, {2223, 239}, {2284, 874}, {2340, 3975}, {3252, 75}, {3286, 30940}, {3693, 4087}, {3912, 18891}, {3930, 35544}, {4447, 3978}, {5089, 40717}, {7077, 36796}, {8638, 4435}, {9454, 238}, {9455, 1914}, {14598, 1438}, {18265, 294}, {20683, 3948}, {22116, 76}, {34067, 666}, {39258, 740}, {39686, 8299}, {40217, 561}

X(40731) = X(1)X(21)∩X(6)X(694)

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

X(40731) lies on the cubic K1019 and these lines: {1, 21}, {6, 694}, {41, 18784}, {42, 17209}, {171, 1909}, {172, 3955}, {741, 21010}, {757, 983}, {1469, 3736}, {2209, 38814}, {4279, 21511}, {4579, 18787}, {5021, 16058}, {5145, 33718}, {5255, 11104}, {16887, 33064}, {18266, 40214}

X(40731) = X(523)-isoconjugate of X(30670)
X(40731) = crossdifference of every pair of points on line {661, 804}
X(40731) = barycentric product X(i)*X(j) for these {i,j}: {172, 30966}, {662, 3805}, {869, 8033}, {894, 3736}, {1469, 27958}, {2276, 17103}, {3786, 7175}, {3799, 18200}, {3955, 31909}, {4481, 4579}, {4589, 30654}, {17941, 30671}
X(40731) = barycentric quotient X(i)/X(j) for these {i,j}: {163, 30670}, {3736, 257}, {3805, 1577}, {8033, 871}, {30654, 4010}

X(40732) = X(1)X(2110)∩X(6)X(2223)

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

X(40732) lies on the cubic K1019 and these lines: {1, 2110}, {6, 2223}, {31, 21753}, {39, 20455}, {42, 20459}, {43, 55}, {100, 14621}, {869, 3774}, {1001, 3696}, {1469, 3736}, {2092, 3764}, {2309, 2347}, {3169, 6600}, {4254, 20992}, {4255, 18758}, {4433, 32941}, {4649, 21010}, {5132, 37586}, {20142, 23407}, {20967, 30706}

X(40732) = isogonal conjugate of the isotomic conjugate of X(3789)
X(40732) = X(i)-Ceva conjugate of X(j) for these (i,j): {105, 16514}, {2346, 984}
X(40732) = X(i)-isoconjugate of X(j) for these (i,j): {870, 1002}, {4817, 32041}, {14621, 27475}
X(40732) = crosspoint of X(6) and X(1001)
X(40732) = crosssum of X(2) and X(1002)
X(40732) = barycentric product X(i)*X(j) for these {i,j}: {6, 3789}, {31, 27474}, {869, 4384}, {984, 2280}, {1001, 2276}, {1469, 37658}, {4517, 5228}, {18900, 21615}
X(40732) = barycentric quotient X(i)/X(j) for these {i,j}: {869, 27475}, {2280, 870}, {3789, 76}, {4384, 871}, {18900, 2279}, {27474, 561}

X(40733) = X(1)X(1573)∩X(6)X(3009)

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

X(40733) lies on the cubic K1019 and these lines: {1, 1573}, {6, 3009}, {41, 1914}, {42, 16515}, {43, 17475}, {213, 7296}, {292, 2279}, {649, 38367}, {869, 2276}, {872, 16525}, {1017, 5008}, {1100, 4687}, {1107, 3876}, {2277, 23548}, {3230, 21754}, {3747, 10987}, {4393, 21904}, {16369, 23407}, {17275, 26772}, {20284, 21779}, {21352, 37673}

X(40733) = isogonal conjugate of the isotomic conjugate of X(27481)
X(40733) = X(6)-Ceva conjugate of X(2276)
X(40733) = X(985)-isoconjugate of X(27494)
X(40733) = crosspoint of X(6) and X(21793)
X(40733) = crosssum of X(2) and X(27494)
X(40733) = crossdifference of every pair of points on line {4784, 4785}
X(40733) = barycentric product X(i)*X(j) for these {i,j}: {1, 3795}, {6, 27481}, {869, 30963}, {984, 16468}, {2276, 4393}, {3661, 21793}, {3736, 3993}, {3773, 34476}, {3799, 4782}
X(40733) = barycentric quotient X(i)/X(j) for these {i,j}: {2276, 27494}, {3795, 75}, {16468, 870}, {21793, 14621}, {27481, 76}, {30963, 871}
X(40733) = {X(869),X(16514)}-harmonic conjugate of X(2276)

X(40734) = X(1)X(2106)∩X(6)X(741)

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

X(40734) lies on the cubic K1019 and these lines: {1, 2106}, {6, 741}, {31, 1326}, {41, 58}, {43, 81}, {662, 985}, {688, 875}, {1185, 5145}, {2276, 3736}, {4658, 17175}, {9455, 17104}

X(40734) = barycentric product X(i)*X(j) for these {i,j}: {58, 27495}, {3736, 16826}, {3781, 31904}
X(40734) = barycentric quotient X(i)/X(j) for these {i,j}: {3736, 27483}, {27495, 313}

X(40735) = X(1)X(20332)∩X(6)X(3009)

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

X(40735) lies on these lines: {1, 20332}, {6, 3009}, {31, 19587}, {42, 2162}, {43, 81}, {55, 36614}, {604, 18266}, {739, 2177}, {1333, 2209}, {1918, 34819}, {4393, 4649}, {17379, 25311}

X(40735) = isogonal conjugate of X(30963)
X(40735) = X(869)-cross conjugate of X(31)
X(40735) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30963}, {2, 4393}, {6, 10009}, {75, 16468}, {76, 21793}, {86, 3993}, {99, 4806}, {190, 4785}, {264, 23095}, {274, 21904}, {306, 31912}, {313, 34476}, {668, 4782}, {870, 3795}, {903, 4759}, {1268, 4991}, {14621, 27481}
X(40735) = cevapoint of X(788) and X(3248)
X(40735) = barycentric product X(i)*X(j) for these {i,j}: {31, 27494}, {1333, 34475}
X(40735) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 10009}, {6, 30963}, {31, 4393}, {32, 16468}, {213, 3993}, {560, 21793}, {667, 4785}, {798, 4806}, {869, 27481}, {1918, 21904}, {1919, 4782}, {2203, 31912}, {2251, 4759}, {9247, 23095}, {27494, 561}, {34475, 27801}

X(40736) = X(6)X(43)∩X(31)X(7104)

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

X(40736) lies on the cubic K1019 and these lines: {6, 43}, {31, 7104}, {41, 1922}, {81, 38810}, {213, 2053}, {717, 932}, {4598, 37678}, {18899, 19587}

X(40736) = X(i)-isoconjugate of X(j) for these (i,j): {192, 870}, {789, 3835}, {871, 2176}, {985, 6382}, {3113, 33890}, {3123, 5388}, {4083, 37133}, {4586, 20906}, {4817, 36863}, {6376, 14621}
X(40736) = barycentric product X(i)*X(j) for these {i,j}: {87, 869}, {788, 932}, {984, 7121}, {1469, 2053}, {2162, 2276}, {3250, 34071}, {3736, 23493}, {6384, 18900}, {8630, 18830}
X(40736) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 871}, {788, 20906}, {869, 6376}, {2276, 6382}, {3117, 33890}, {3661, 40367}, {7121, 870}, {8630, 4083}, {18899, 20284}, {18900, 43}, {34071, 37133}
X(40736) = {X(6),X(2162)}-harmonic conjugate of X(34252)

X(40737) = ISOGONAL CONJUGATE OF X(1045)

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

X(40737) lies on the cubics K131, K1026, , and K1176, and on these lines: {1, 2668}, {2, 2107}, {31, 2106}, {42, 894}, {171, 213}, {846, 16362}, {873, 4117}, {1402, 7175}, {1740, 2258}, {1967, 37128}, {1973, 15148}, {3223, 10436}, {4128, 32010}, {5539, 17596}, {13610, 18786}, {16826, 23493}, {18793, 24342}, {25528, 38275}, {33779, 37132}

X(40737) = isogonal conjugate of X(1045)
X(40737) = isogonal conjugate of the isotomic conjugate of X(18298)
X(40737) = X(i)-cross conjugate of X(j) for these (i,j): {86, 1}, {893, 57}
X(40737) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1045}, {2, 21779}, {4, 23079}, {6, 1655}, {42, 39915}, {75, 18756}, {81, 21883}, {99, 9402}, {213, 34021}, {904, 27890}
X(40737) = cevapoint of X(i) and X(j) for these (i,j): {513, 4128}, {649, 4117}, {659, 38978}
X(40737) = trilinear pole of line {798, 4367}
X(40737) = barycentric product X(i)*X(j) for these {i,j}: {6, 18298}, {37128, 39926}
X(40737) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1655}, {6, 1045}, {31, 21779}, {32, 18756}, {42, 21883}, {48, 23079}, {81, 39915}, {86, 34021}, {798, 9402}, {894, 27890}, {18298, 76}, {39926, 3948}

X(40738) = X(1)X(257)∩X(2)X(893)

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

X(40738) lies on the conic {{A,B,C,X(1),X(2)}}, the cubic K1176, and these lines: {1, 257}, {2, 893}, {57, 7249}, {81, 32010}, {86, 3863}, {105, 30670}, {256, 291}, {274, 33891}, {330, 870}, {733, 789}, {1002, 40721}, {1255, 26243}, {1431, 17379}, {1432, 1447}, {2224, 30111}, {2344, 3407}, {4817, 17212}, {10436, 27447}, {16826, 25425}

X(40738) = X(i)-isoconjugate of X(j) for these (i,j): {37, 40731}, {101, 3805}, {171, 2276}, {172, 984}, {660, 30654}, {788, 18047}, {869, 894}, {1469, 2329}, {1580, 3862}, {1691, 3864}, {1909, 40728}, {1920, 18900}, {2295, 3736}, {2330, 7146}, {3250, 4579}, {3661, 7122}, {3774, 17103}, {3781, 7119}, {3799, 20981}, {3802, 30657}, {4517, 7175}, {16514, 18787}
X(40738) = trilinear pole of line {513, 4107}
X(40738) = barycentric product X(i)*X(j) for these {i,j}: {256, 870}, {257, 14621}, {693, 30670}, {871, 904}, {985, 7018}, {3113, 3865}, {3114, 3863}, {4817, 27805}, {32010, 40718}
X(40738) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 40731}, {256, 984}, {257, 3661}, {513, 3805}, {694, 3862}, {870, 1909}, {893, 2276}, {904, 869}, {985, 171}, {1178, 3736}, {1431, 1469}, {1432, 7146}, {1492, 4579}, {1581, 3864}, {2344, 2329}, {3766, 30639}, {3863, 3094}, {3903, 3799}, {4451, 3790}, {4586, 18047}, {4817, 4369}, {7015, 3781}, {7018, 33931}, {7104, 40728}, {7249, 7179}, {8632, 30654}, {14438, 30656}, {14621, 894}, {17493, 3797}, {18786, 3783}, {23597, 4107}, {27805, 3807}, {30670, 100}, {32010, 30966}, {33891, 9865}, {40718, 1215}, {40722, 27954}, {40729, 3774}

X(40739) = X(1)X(39923)∩X(2)X(2116)

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

X(40739) lies on the cubic K1176 and these lines: {1, 39923}, {2, 2116}, {1001, 14621}, {1002, 40721}, {3923, 32041}, {4334, 40718}, {5263, 37138}, {8926, 24342}

X(40739) = X(i)-isoconjugate of X(j) for these (i,j): {7, 40732}, {56, 3789}, {604, 27474}, {869, 40719}, {984, 1471}, {1001, 1469}, {2276, 5228}, {2280, 7146}
X(40739) = cevapoint of X(1) and X(9746)
X(40739) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 27474}, {9, 3789}, {41, 40732}, {985, 5228}, {1002, 7146}, {2279, 1469}, {2344, 1001}, {14621, 40719}, {27475, 7179}

X(40740) = X(1)X(335)∩X(2)X(2113)

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

X(40740) lies on the cubic K1176 and these lines: {1, 335}, {2, 2113}, {660, 27495}, {894, 24479}, {985, 40217}, {1757, 27926}, {9505, 40718}, {24358, 24510}

X(40740) = X(i)-isoconjugate of X(j) for these (i,j): {869, 40725}, {1929, 16514}, {2702, 30665}, {3783, 17962}, {3802, 9506}
X(40740) = barycentric product X(2786)*X(37207)
X(40740) = barycentric quotient X(i)/X(j) for these {i,j}: {1757, 3783}, {2786, 4486}, {6542, 3797}, {8298, 3802}, {9508, 30665}, {14621, 40725}, {17735, 16514}, {30664, 37135}, {37207, 35148}

X(40741) = X(2)X(2053)∩X(86)X(26143)

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

X(40741) lies on the cubic K1176 and these lines: {2, 2053}, {86, 26143}, {330, 870}, {932, 6645}, {2295, 4598}, {7121, 14621}, {7153, 40719}, {16826, 23493}, {26801, 34252}

X(40741) = X(40718)-Ceva conjugate of X(40720)
X(40741) = barycentric product X(i)*X(j) for these {i,j}: {330, 30661}, {6384, 18754}
X(40741) = barycentric quotient X(i)/X(j) for these {i,j}: {18754, 43}, {30661, 192}

X(40742) = X(1)X(18795)∩X(2)X(292)

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

X(40742) lies on the cubic K1176 and these lines: {1, 18795}, {2, 292}, {894, 24576}, {2111, 14621}

X(40742) = X(2665)-isoconjugate of X(16514)
X(40742) = barycentric quotient X(i)/X(j) for these {i,j}: {2664, 3783}, {15148, 17569}, {17759, 3797}, {21788, 16514}

X(40743) = X(1)X(7168)∩X(2)X(893)

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

X(40743) lies on the cubic K1176 and these lines: {1, 7168}, {2, 893}, {86, 870}, {894, 19567}, {1045, 27890}, {9401, 26102}

X(40743) = X(40718)-Ceva conjugate of X(870)
X(40743) = X(18298)-isoconjugate of X(18900)
X(40743) = barycentric product X(i)*X(j) for these {i,j}: {870, 1655}, {871, 21779}, {34021, 40718}
X(40743) = barycentric quotient X(i)/X(j) for these {i,j}: {1045, 2276}, {1655, 984}, {18756, 40728}, {21779, 869}, {34021, 30966}

X(40744) = X(1)X(2210)∩X(2)X(41)

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

X(40744) lies on these lines: {1, 2210}, {2, 41}, {21, 72}, {42, 1580}, {48, 17379}, {75, 584}, {81, 172}, {86, 2174}, {101, 16826}, {171, 18266}, {218, 16367}, {239, 4251}, {251, 17011}, {284, 894}, {379, 31019}, {572, 17120}, {604, 37677}, {662, 4670}, {1100, 18042}, {1468, 37617}, {1931, 2185}, {1993, 23150}, {2112, 16503}, {2268, 17350}, {2278, 3758}, {2280, 2344}, {2289, 26059}, {2329, 6542}, {2663, 7122}, {3204, 4687}, {3218, 21511}, {3661, 16788}, {3868, 13723}, {4289, 4363}, {4390, 20055}, {5371, 16519}, {5813, 26626}, {8300, 21352}, {9310, 29570}, {9454, 20132}, {11320, 24514}, {11349, 27003}, {11364, 20985}, {16704, 31039}, {16783, 17397}, {17023, 27950}, {17103, 40214}, {17302, 18162}

X(40744) = crosspoint of X(4567) and X(4586)
X(40744) = crosssum of X(3125) and X(3250)

X(40745) = X(7)X(604)∩X(81)X(4586)

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

X(40745) lies on these lines: {7, 604}, {81, 4586}, {171, 7369}, {172, 385}, {192, 15370}, {330, 985}, {789, 35105}, {870, 16998}, {894, 1580}, {961, 40738}, {1446, 3407}, {6625, 18757}, {7187, 17103}, {16915, 20911}, {30664, 40742}

X(40745) = X(i)-isoconjugate of X(j) for these (i,j): {256, 2276}, {257, 869}, {694, 3783}, {788, 27805}, {893, 984}, {904, 3661}, {1432, 4517}, {1581, 16514}, {1967, 3797}, {3250, 3903}, {3774, 32010}, {3862, 18786}, {7018, 40728}, {7104, 33931}, {30966, 40729}
X(40745) = trilinear pole of line {4164, 4369}
X(40745) = barycentric product X(i)*X(j) for these {i,j}: {171, 870}, {789, 4367}, {871, 7122}, {894, 14621}, {985, 1909}, {1492, 4374}, {2344, 7196}, {3113, 7184}, {3407, 7187}, {4107, 37207}, {4369, 4586}, {4613, 17212}, {4817, 18047}, {6645, 40738}, {14296, 30664}, {17103, 40718}, {20981, 37133}
X(40745) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 984}, {172, 2276}, {385, 3797}, {870, 7018}, {894, 3661}, {985, 256}, {1215, 3773}, {1492, 3903}, {1580, 3783}, {1691, 16514}, {1909, 33931}, {2330, 4517}, {2533, 4122}, {3907, 4522}, {3955, 3781}, {4032, 16603}, {4107, 4486}, {4164, 30665}, {4367, 1491}, {4369, 824}, {4434, 4439}, {4579, 3799}, {4586, 27805}, {4697, 3775}, {4774, 4951}, {7081, 3790}, {7122, 869}, {7175, 7146}, {7176, 7179}, {7187, 3314}, {14621, 257}, {17103, 30966}, {18047, 3807}, {18200, 4481}, {18787, 3864}, {20981, 3250}, {40731, 4476}, {40738, 40099}

X(40746) = X(1)X(32)∩X(6)X(560)

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

X(40746) lies on the conic {{A,B,C,X(1),X(6)}} and these lines: {1, 32}, {6, 560}, {9, 39977}, {31, 292}, {56, 21771}, {58, 2275}, {86, 1333}, {87, 1716}, {101, 40733}, {106, 825}, {604, 1431}, {713, 789}, {870, 16998}, {996, 5291}, {1126, 4251}, {1220, 4426}, {1252, 5378}, {1400, 18898}, {1438, 5332}, {1449, 9277}, {1492, 5035}, {2220, 40433}, {2276, 37586}, {2280, 25426}, {3226, 4586}, {3661, 4386}, {4372, 33935}, {5042, 36598}, {5156, 14599}, {5337, 30945}, {9111, 30664}, {16946, 39972}, {21010, 21793}, {30670, 35105}

X(40746) = isogonal conjugate of X(3661)
X(40746) = isogonal conjugate of the anticomplement of X(17023)
X(40746) = isogonal conjugate of the complement of X(4393)
X(40746) = isogonal conjugate of the isotomic conjugate of X(14621)
X(40746) = X(21764)-cross conjugate of X(6)
X(40746) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3661}, {2, 984}, {6, 33931}, {8, 7146}, {9, 7179}, {21, 16603}, {37, 30966}, {57, 3790}, {72, 31909}, {75, 2276}, {76, 869}, {81, 3773}, {85, 4517}, {88, 4439}, {92, 3781}, {100, 824}, {190, 1491}, {226, 3786}, {239, 3864}, {291, 3797}, {310, 3774}, {312, 1469}, {321, 3736}, {334, 16514}, {335, 3783}, {346, 7204}, {350, 3862}, {513, 3807}, {514, 3799}, {561, 40728}, {649, 4505}, {651, 4522}, {660, 4486}, {662, 4122}, {668, 3250}, {788, 1978}, {874, 30671}, {983, 3314}, {1002, 27474}, {1016, 4475}, {1255, 3775}, {1502, 18900}, {3094, 7033}, {3117, 7034}, {3573, 23596}, {3789, 27475}, {3792, 18359}, {3795, 27494}, {3802, 40098}, {3805, 27805}, {3952, 4481}, {4407, 40434}, {4469, 40718}, {4562, 30665}, {4604, 4951}, {4606, 4818}, {5386, 33904}, {27495, 30571}, {30870, 32739}
X(40746) = cevapoint of X(6) and X(21793)
X(40746) = crosssum of X(2276) and X(3781)
X(40746) = trilinear pole of line {649, 1980}
X(40746) = crossdifference of every pair of points on line {824, 1491}
X(40746) = barycentric product X(i)*X(j) for these {i,j}: {1, 985}, {6, 14621}, {31, 870}, {57, 2344}, {58, 40718}, {101, 4817}, {172, 40738}, {244, 5384}, {513, 1492}, {514, 825}, {560, 871}, {649, 4586}, {659, 30664}, {667, 789}, {693, 34069}, {813, 23597}, {893, 40745}, {1471, 40739}, {1919, 37133}, {1977, 5388}, {2248, 40722}, {2275, 3407}, {3113, 7032}, {3662, 18898}, {3733, 4613}, {4367, 30670}, {8632, 37207}
X(40746) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33931}, {6, 3661}, {31, 984}, {32, 2276}, {42, 3773}, {55, 3790}, {56, 7179}, {58, 30966}, {100, 4505}, {101, 3807}, {184, 3781}, {512, 4122}, {560, 869}, {604, 7146}, {649, 824}, {663, 4522}, {667, 1491}, {692, 3799}, {693, 30870}, {789, 6386}, {825, 190}, {870, 561}, {871, 1928}, {902, 4439}, {985, 75}, {1106, 7204}, {1397, 1469}, {1400, 16603}, {1474, 31909}, {1492, 668}, {1501, 40728}, {1911, 3864}, {1914, 3797}, {1917, 18900}, {1919, 3250}, {1922, 3862}, {1980, 788}, {2175, 4517}, {2194, 3786}, {2205, 3774}, {2206, 3736}, {2210, 3783}, {2275, 3314}, {2280, 27474}, {2308, 3775}, {2344, 312}, {3113, 7034}, {3248, 4475}, {3572, 23596}, {4164, 30639}, {4586, 1978}, {4613, 27808}, {4775, 4951}, {4817, 3261}, {5384, 7035}, {8632, 4486}, {14599, 16514}, {14621, 76}, {18898, 17743}, {21747, 4407}, {21793, 27481}, {30664, 4583}, {34069, 100}, {40718, 313}, {40745, 1920}

X(40747) = X(1)X(32)∩X(6)X(75)

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

X(40747) lies on the cubic K1177 and these lines: {1, 32}, {6, 75}, {9, 17038}, {10, 213}, {19, 1974}, {37, 1918}, {65, 21861}, {81, 4586}, {83, 18833}, {158, 2207}, {171, 292}, {596, 20963}, {649, 876}, {729, 789}, {759, 825}, {869, 4386}, {897, 1492}, {910, 34434}, {940, 16524}, {994, 5011}, {1100, 13476}, {1258, 40738}, {1449, 39742}, {1910, 14601}, {1922, 32115}, {2166, 11060}, {2176, 5275}, {2186, 21010}, {2218, 16974}, {3224, 18832}, {4649, 25426}, {5276, 16514}, {8773, 32654}, {16777, 39737}, {16782, 17023}, {16826, 40722}, {16884, 39739}, {16971, 39697}, {16973, 23051}, {17475, 39714}, {17750, 29633}, {17754, 17795}, {21352, 21764}, {21793, 23407}, {21904, 22327}, {29610, 37673}, {36119, 40354}, {37207, 40742}

X(40747) = isogonal conjugate of X(40773)
X(40747) = X(14621)-Ceva conjugate of X(40718)
X(40747) = X(i)-cross conjugate of X(j) for these (i,j): {3288, 4559}, {21840, 37}
X(40747) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3736}, {3, 31909}, {6, 30966}, {21, 7146}, {27, 3781}, {57, 3786}, {58, 3661}, {60, 16603}, {81, 984}, {86, 2276}, {99, 3250}, {100, 4481}, {110, 824}, {257, 40731}, {274, 869}, {284, 7179}, {310, 40728}, {333, 1469}, {593, 3773}, {662, 1491}, {741, 3797}, {788, 799}, {873, 3774}, {985, 4469}, {1019, 3799}, {1171, 3775}, {1333, 33931}, {1412, 3790}, {1434, 4517}, {2287, 7204}, {3094, 40415}, {3116, 38810}, {3314, 38813}, {3733, 3807}, {3783, 37128}, {3792, 24624}, {3805, 4603}, {3862, 33295}, {4122, 4556}, {4475, 4567}, {4476, 14621}, {4522, 4565}, {4584, 30665}, {4602, 8630}, {4627, 4818}, {4634, 14436}, {6385, 18900}, {16514, 18827}, {18829, 30654}, {27483, 40734}
X(40747) = cevapoint of X(i) and X(j) for these (i,j): {37, 21904}, {171, 4649}
X(40747) = crosspoint of X(985) and X(14621)
X(40747) = crosssum of X(984) and X(2276)
X(40747) = trilinear pole of line {661, 669}
X(40747) = crossdifference of every pair of points on line {788, 1491}
X(40747) = trilinear product X(i)*X(j) for these {i,j}: {6, 40718}, {10, 40746}, {37, 985}, {42, 14621}, {65, 2344}, {213, 870}, {512, 4586}, {523, 825}, {649, 4613}, {661, 1492}, {669, 37133}, {789, 798}, {871, 2205}, {1577, 34069}, {2295, 40763}, {2887, 18898}, {3113, 16584}, {3125, 5384}, {3407, 3778}, {4455, 37207}, {4557, 4817}, {20964, 40738}, {21010, 25425}, {21832, 30664}
X(40747) = barycentric product X(i)*X(j) for these {i,j}: {1, 40718}, {10, 985}, {37, 14621}, {42, 870}, {226, 2344}, {512, 789}, {513, 4613}, {523, 1492}, {661, 4586}, {798, 37133}, {825, 1577}, {850, 34069}, {871, 1918}, {1018, 4817}, {2295, 40738}, {2533, 30670}, {3113, 3778}, {3114, 16584}, {3120, 5384}, {3121, 5388}, {3407, 3721}, {4010, 30664}, {17754, 25425}, {18898, 20234}, {21832, 37207}
X(40747) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 30966}, {10, 33931}, {19, 31909}, {31, 3736}, {37, 3661}, {42, 984}, {55, 3786}, {65, 7179}, {210, 3790}, {213, 2276}, {228, 3781}, {512, 1491}, {649, 4481}, {661, 824}, {669, 788}, {756, 3773}, {789, 670}, {798, 3250}, {804, 30639}, {825, 662}, {850, 30870}, {869, 4476}, {870, 310}, {985, 86}, {1018, 3807}, {1042, 7204}, {1400, 7146}, {1402, 1469}, {1492, 99}, {1918, 869}, {1962, 3775}, {2171, 16603}, {2205, 40728}, {2238, 3797}, {2276, 4469}, {2344, 333}, {3122, 4475}, {3407, 38810}, {3721, 3314}, {3724, 3792}, {3747, 3783}, {3952, 4505}, {4041, 4522}, {4455, 30665}, {4557, 3799}, {4586, 799}, {4613, 668}, {4705, 4122}, {4770, 4951}, {4817, 7199}, {4822, 4818}, {5384, 4600}, {7109, 3774}, {7122, 40731}, {7234, 3805}, {9426, 8630}, {14621, 274}, {16584, 3094}, {21751, 3117}, {21805, 4439}, {21806, 4407}, {21832, 4486}, {21904, 27481}, {30664, 4589}, {30670, 4594}, {34069, 110}, {37133, 4602}, {37207, 4639}, {40718, 75}, {40745, 8033}

X(40748) = X(1)X(18789)∩X(985)X(1001)

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

X(40748) lies on the cubic K1177 and these lines: {1, 18789}, {985, 1001}, {4384, 14621}, {4649, 25426}

X(40748) = isogonal conjugate of X(40774)
X(40748) = X(3720)-cross conjugate of X(870)
X(40748) = X(i)-isoconjugate of X(j) for these (i,j): {6, 27495}, {10, 40734}, {984, 4649}, {2276, 16826}, {3736, 3842}, {3799, 4784}, {3862, 20142}
X(40748) = trilinear product X(i)*X(j) for these {i,j}: {985, 30571}, {4817, 28841}, {14621, 25426}, {27483, 40746}
X(40748) = barycentric product X(i)*X(j) for these {i,j}: {870, 25426}, {985, 27483}, {14621, 30571}
X(40748) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 27495}, {985, 16826}, {1333, 40734}, {25426, 984}, {27483, 33931}, {28841, 3799}, {30571, 3661}

X(40749) = X(1)X(21)∩X(6)X(1045)

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

X(40749) lies on the cubic K1177 and these lines: {1, 21}, {6, 1045}, {43, 2229}, {44, 24450}, {171, 213}, {239, 2308}, {274, 4697}, {869, 17126}, {980, 4650}, {1761, 16972}, {1781, 13610}, {1918, 2663}, {1961, 3294}, {2111, 5091}, {3791, 17143}, {4039, 17499}, {4384, 14621}, {4393, 4427}, {4443, 5165}, {5283, 7262}, {13174, 32115}, {16369, 25427}, {16475, 37555}, {16831, 37604}, {17034, 24259}, {20367, 29821}, {21010, 23194}, {21352, 21747}

X(40749) = isogonal conjugate of X(40775)
X(40749) = X(4649)-isoconjugate of X(30570)
X(40749) = crosspoint of X(1492) and X(24041)
X(40749) = crosssum of X(1491) and X(2643)
X(40749) = trilinear product X(6)*X(40721)
X(40749) = barycentric product X(1)*X(40721)
X(40749) = barycentric quotient X(i)/X(j) for these {i,j}: {25426, 30570}, {40721, 75}
X(40749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {81, 3747, 1}, {171, 213, 2664}

X(40750) = X(1)X(1929)∩X(2)X(6)

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

X(40750) lies on the cubic K1177 and these lines: {1, 1929}, {2, 6}, {9, 37604}, {37, 171}, {45, 896}, {55, 199}, {58, 16589}, {187, 4653}, {191, 21816}, {213, 37559}, {220, 18253}, {573, 19516}, {644, 2295}, {750, 2276}, {894, 20947}, {942, 16519}, {980, 21989}, {985, 1001}, {1010, 21024}, {1046, 21879}, {1100, 3684}, {1107, 37607}, {1220, 21025}, {1509, 17499}, {1575, 17122}, {1655, 17103}, {1914, 3720}, {2176, 5711}, {2242, 30116}, {2243, 17021}, {2280, 9345}, {2475, 23903}, {3053, 13723}, {3145, 21808}, {3247, 3550}, {3290, 3745}, {3723, 3750}, {3726, 3920}, {3770, 8033}, {3985, 4697}, {3996, 17388}, {4037, 4418}, {4362, 25124}, {4363, 33931}, {4425, 4987}, {4649, 21904}, {4657, 24586}, {4658, 20970}, {5228, 16518}, {5283, 33863}, {5337, 21981}, {5710, 16969}, {6543, 6625}, {8258, 17750}, {9347, 20998}, {11358, 39967}, {14621, 30963}, {16369, 25427}, {16372, 21010}, {16502, 29646}, {16516, 37543}, {16521, 37520}, {16583, 37594}, {16672, 37540}, {16678, 21773}, {16826, 40722}, {16884, 17017}, {16917, 33296}, {16968, 37554}, {17016, 21951}, {17019, 35216}, {17275, 32853}, {21764, 30950}, {21769, 29644}, {21771, 27802}, {21785, 29650}, {23897, 26051}, {23905, 26117}, {25499, 29473}, {25809, 25817}, {29649, 34261}, {29671, 38408}, {36659, 36746}

X(40750) = isogonal conjugate of X(40776)
X(40750) = X(i)-Ceva conjugate of X(j) for these (i,j): {16826, 1001}, {40722, 8424}
X(40750) = crosspoint of X(789) and X(34537)
X(40750) = crosssum of X(i) and X(j) for these (i,j): {788, 1084}, {824, 8287}
X(40750) = crossdifference of every pair of points on line {512, 9508}
X(40750) = trilinear product X(i)*X(j) for these {i,j}: {6, 24342}, {662, 9279}, {1001, 18791}
X(40750) = barycentric product X(i)*X(j) for these {i,j}: {1, 24342}, {99, 9279}, {4384, 18791}
X(40750) = barycentric quotient X(i)/X(j) for these {i,j}: {9279, 523}, {18791, 27475}, {24342, 75}
X(40750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5277, 18755}, {37, 171, 17735}, {81, 2238, 6}, {81, 37675, 2238}, {940, 5275, 6}, {1030, 20472, 199}, {1929, 8298, 8301}, {1961, 3509, 37}, {3684, 4038, 1100}, {5276, 24512, 6}, {5276, 37633, 24512}, {5283, 37522, 33863}, {16999, 20132, 37678}

X(40751) = X(1)X(3506)∩X(6)X(256)

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

X(40751) lies on the cubic K1177 and these lines: {1, 3506}, {6, 256}, {81, 982}, {171, 19554}, {1449, 9277}, {1654, 40722}, {4649, 40744}, {6650, 14621}, {29840, 32853}

X(40751) = isogonal conjugate of X(40777)
X(40751) = X(i)-isoconjugate of X(j) for these (i,j): {984, 13610}, {2248, 3661}, {2276, 6625}, {15377, 31909}, {18757, 33931}
X(40751) = trilinear product X(i)*X(j) for these {i,j}: {6, 40722}, {825, 21196}, {846, 985}, {1654, 40746}, {14621, 18755}, {38814, 40747}
X(40751) = barycentric product X(i)*X(j) for these {i,j}: {1, 40722}, {846, 14621}, {870, 18755}, {985, 1654}, {1492, 21196}, {2344, 17084}, {38814, 40718}
X(40751) = barycentric quotient X(i)/X(j) for these {i,j}: {846, 3661}, {985, 6625}, {1654, 33931}, {18755, 984}, {21879, 3773}, {38814, 30966}, {40722, 75}

X(40752) = X(1)X(257)∩X(6)X(19579)

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

X(40752) lies on the cubic K1177 and these lines: {1, 257}, {6, 19579}, {81, 310}, {171, 19565}, {985, 39925}, {1655, 18756}, {4649, 40718}, {17032, 26243}, {21779, 40743}

X(40752) = isogonal conjugate of X(40778)
X(40752) = X(i)-isoconjugate of X(j) for these (i,j): {2276, 40737}, {18298, 40728}
X(40752) = cevapoint of X(30661) and X(40721)
X(40752) = trilinear product X(i)*X(j) for these {i,j}: {6, 40743}, {870, 21779}, {985, 1655}, {1045, 14621}, {39915, 40747}
X(40752) = barycentric product X(i)*X(j) for these {i,j}: {1, 40743}, {870, 1045}, {871, 18756}, {1655, 14621}, {39915, 40718}
X(40752) = barycentric quotient X(i)/X(j) for these {i,j}: {870, 18298}, {985, 40737}, {1045, 984}, {1655, 3661}, {18756, 869}, {21779, 2276}, {23079, 3781}, {39915, 30966}, {40743, 75}

X(40753) = X(1)X(727)∩X(6)X(43)

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

X(40753) lies on the cubic K1177 and these lines: {1, 727}, {6, 43}, {2280, 34071}, {16468, 40720}

X(40753) = isogonal conjugate of X(40780)
X(40753) = X(4393)-cross conjugate of X(16468)
X(40753) = X(i)-isoconjugate of X(j) for these (i,j): {2176, 27494}, {6376, 40735}, {34475, 38832}
X(40753) = cevapoint of X(4393) and X(40720)
X(40753) = barycentric product X(i)*X(j) for these {i,j}: {1, 40720}, {87, 4393}, {330, 16468}, {932, 4785}, {2162, 30963}, {4598, 4782}, {6384, 21793}, {7121, 10009}
X(40753) = trilinear product X(i)*X(j) for these {i,j}: {6, 40720}, {87, 16468}, {330, 21793}, {932, 4782}, {2162, 4393}, {4785, 34071}, {7121, 30963}
X(40753) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 27494}, {4393, 6376}, {4782, 3835}, {4785, 20906}, {16468, 192}, {16606, 34475}, {21793, 43}, {21904, 3971}, {23095, 22370}, {25376, 21426}, {30963, 6382}, {34476, 27644}, {40720, 75}
X(40753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7121, 17105}, {2162, 34252, 87}

X(40754) = X(1)X(9453)∩X(6)X(7)

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

X(40754) lies on the cubic K1177 and these lines: {1, 9453}, {6, 7}, {37, 36086}, {105, 825}, {666, 894}, {885, 2298}, {984, 9501}, {1438, 7194}, {1781, 13610}, {4336, 28071}, {4645, 40724}, {4649, 9505}, {5018, 19554}, {5276, 9318}, {17300, 31637}

X(40754) = isogonal conjugate of X(40781)
X(40754) = cevapoint of X(3509) and X(19557)
X(40754) = trilinear product X(i)*X(j) for the X(40754) = X(i)-isoconjugate of X(j) for these (i,j): {241, 7281}, {518, 3512}, {672, 7261}, {3912, 8852}, {8299, 24479}, {9454, 18036}, {17755, 30648}
X(40754) = barycentric product X(i)*X(j) for these {i,j}: {1, 40724}, {105, 4645}, {673, 3509}, {1438, 17789}, {2481, 17798}, {4458, 36086}, {5018, 14942}, {18031, 19554}
X(40754) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 7261}, {1438, 3512}, {2195, 7281}, {2481, 18036}, {3509, 3912}, {4645, 3263}, {5018, 9436}, {17798, 518}, {18262, 2223}, {19554, 672}, {19557, 17755}, {19561, 8299}, {20715, 3932}, {20741, 25083}, {40724, 75}

X(40755) = X(1)X(727)∩X(213)X(8709)

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

X(40755) lies on the cubic K1177 and these lines: {1, 727}, {213, 8709}, {14621, 20332}

X(40755) = isogonal conjugate of X(40782)
X(40755) = cevapoint of X(18278) and X(19580)
X(40755) = X(i)-isoconjugate of X(j) for these (i,j): {1575, 7168}, {17475, 24576}
X(40755) = trilinear product X(i)*X(j) for these {i,j}: {727, 19565}, {3226, 18278}, {3510, 20332}, {19567, 34077}
X(40755) = barycentric product X(i)*X(j) for these {i,j}: {727, 19567}, {3226, 3510}, {18275, 34077}, {18278, 32020}, {19565, 20332}
X(40755) = barycentric quotient X(i)/X(j) for these {i,j}: {727, 7168}, {3510, 726}, {18274, 17475}, {18278, 1575}, {19567, 35538}, {19580, 17793}, {30634, 20663}

X(40756) = ISOGONAL CONJUGATE OF X(40783)

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

X(40756) lies on the cubic K1177 and these lines: {4393, 4649}

X(40756) = isogonal conjugate of X(40783)
X(40756) = cevapoint of X(43) and X(40780)
X(40756) = X(i)-isoconjugate of X(j) for these (i,j): {87, 3795}, {330, 40733}, {2162, 27481}, {2276, 40720}, {10009, 40736}
X(40756) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 27481}, {985, 40720}, {2176, 3795}, {2209, 40733}

X(40757) = X(985)X(1002)∩X(4386)X(37138)

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

X(40757) lies on the cubic K1177 and these lines: {985, 1002}, {4386, 37138}, {5276, 8693}, {37658, 40739}

X(40757) = isogonal conjugate of X(40784)
X(40757) = X(i)-isoconjugate of X(j) for these (i,j): {56, 27474}, {57, 3789}, {85, 40732}, {984, 5228}, {1001, 7146}, {1469, 4384}, {1471, 3661}, {2276, 40719}, {2280, 7179}, {7204, 37658}
X(40757) = trilinear product X(i)*X(j) for these {i,j}: {6, 40739}, {1002, 2344}
X(40757) = barycentric product X(i)*X(j) for these {i,j}: {1, 40739}, {2344, 27475}
X(40757) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 27474}, {55, 3789}, {985, 40719}, {1002, 7179}, {2175, 40732}, {2279, 7146}, {2344, 4384}, {40739, 75}

X(40758) = X(1)X(20361)∩X(6)X(3212)

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

X(40758) lies on the cubic K1177 and these lines: {1, 20361}, {6, 3212}, {171, 8932}

X(40758) = isogonal conjugate of X(40785)
X(40758) = X(7220)-isoconjugate of X(28391)
X(40758) = barycentric product X(i)*X(j) for these {i,j}: {4334, 39924}, {18299, 21010}
X(40758) = barycentric quotient X(i)/X(j) for these {i,j}: {17754, 17760}, {21010, 17792}

X(40759) = X(1)X(1326)∩X(6)X(2669)

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

X(40759) lies on the cubic K1177 and these lines: {1, 1326}, {6, 2669}, {81, 4610}

X(40759) = isogonal conjugate of X(40786)

X(40760) = ISOGONAL CONJUGATE OF X(40787)

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

X(40760) lies on the cubic K1177 and these lines: {5228, 16518}

X(40760) = isogonal conjugate of X(40787)
X(40760) = barycentric quotient X(i)/X(j) for these {i,j}: {17754, 27478}, {21010, 28600}

X(40761) = X(1)X(41)∩X(6)X(6654)

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

X(40761) lies on the cubic K1177 and these lines: {1, 41}, {6, 6654}, {673, 24512}

X(40761) = isogonal conjugate of X(40788)
X(40761) = trilinear product X(105)*X(39252)
X(40761) = barycentric product X(673)*X(39252)
X(40761) = barycentric quotient X(39252)/X(3912)

X(40762) = X(6)X(190)∩X(727)X(20985)

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

X(40762) lies on the cubic K1177 and these lines: {6, 190}, {727, 20985}, {8709, 16826}

X(40762) = isogonal conjugate of X(40789)

X(40763) = X(1)X(257)∩X(6)X(256)

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

X(40763) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K1177, and these lines: {1, 257}, {6, 256}, {56, 985}, {58, 3865}, {86, 1178}, {87, 14621}, {106, 30670}, {171, 292}, {1220, 40718}, {1429, 1431}, {1916, 32115}, {3329, 17795}, {4451, 39977}, {4586, 14970}, {4835, 9277}, {7018, 25496}, {7249, 29821}

X(40763) = isogonal conjugate of X(40790)
X(40763) = cevapoint of X(1) and X(17795)
X(40763) = trilinear pole of line {649, 4164}
X(40763) = X(i)-isoconjugate of X(j) for these (i,j): {10, 40731}, {100, 3805}, {171, 984}, {172, 3661}, {385, 3862}, {869, 1909}, {894, 2276}, {1215, 3736}, {1469, 7081}, {1491, 4579}, {1580, 3864}, {1920, 40728}, {2329, 7146}, {2330, 7179}, {3250, 18047}, {3774, 8033}, {3781, 7009}, {3783, 18787}, {3799, 4367}, {3807, 20981}, {4517, 7176}, {4562, 30654}, {5386, 30656}, {7122, 33931}, {16514, 30669}, {20964, 30966}, {22061, 31909}, {30639, 34067}
X(40763) = trilinear product X(i)*X(j) for these {i,j}: {6, 40738}, {256, 985}, {257, 40746}, {513, 30670}, {870, 904}, {893, 14621}, {1178, 40718}, {1432, 2344}, {3407, 3863}, {40432, 40747}
X(40763) = barycentric product X(i)*X(j) for these {i,j}: {1, 40738}, {256, 14621}, {257, 985}, {514, 30670}, {870, 893}, {871, 7104}, {2344, 7249}, {3113, 3863}, {3407, 3865}, {3903, 4817}, {40432, 40718}
X(40763) = barycentric quotient X(i)/X(j) for these {i,j}: {256, 3661}, {257, 33931}, {649, 3805}, {694, 3864}, {812, 30639}, {825, 4579}, {870, 1920}, {893, 984}, {904, 2276}, {985, 894}, {1333, 40731}, {1431, 7146}, {1432, 7179}, {1492, 18047}, {1967, 3862}, {2344, 7081}, {3865, 3314}, {3903, 3807}, {4817, 4374}, {7104, 869}, {7116, 3781}, {14621, 1909}, {18786, 3797}, {23597, 14296}, {27805, 4505}, {30670, 190}, {40432, 30966}, {40718, 3963}, {40738, 75}

X(40764) = X(1)X(3506)∩X(85)X(14621)

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

X(40764) lies on the cubic K1177 and these lines: {1, 3506}, {85, 14621}, {171, 30648}, {514, 20513}, {985, 9499}

X(40764) = isogonal conjugate of X(40791)
X(40764) = X(2276)-isoconjugate of X(40724)
X(40764) = barycentric quotient X(985)/X(40724)

X(40765) = X(6)X(3212)∩X(56)X(985)

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

X(40765) lies on the cubic K1177 and these lines: {6, 3212}, {56, 985}, {65, 4649}, {81, 279}, {85, 14621}, {171, 28391}, {221, 388}, {651, 2295}, {664, 6645}, {1442, 3721}, {6604, 20090}, {7779, 33298}, {17739, 27963}

X(40765) = isogonal conjugate of X(40792)
X(40765) = trilinear product X(i)*X(j) for these {i,j}: {6, 40723}, {56, 17739}, {57, 8424}, {85, 18759}, {604, 30660}, {1431, 27963}
X(40765) = X(i)-Ceva conjugate of X(j) for these (i,j): {7176, 56}, {40723, 8424}
X(40765) = X(i)-isoconjugate of X(j) for these (i,j): {8, 18784}, {41, 18760}, {4876, 16366}
X(40765) = barycentric product X(i)*X(j) for these {i,j}: {1, 40723}, {7, 8424}, {56, 30660}, {57, 17739}, {1432, 27963}, {5018, 39920}, {6063, 18759}
X(40765) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 18760}, {604, 18784}, {1428, 16366}, {8424, 8}, {17739, 312}, {18759, 55}, {27963, 17787}, {30660, 3596}, {40723, 75}

X(40766) = X(1)X(18783)∩X(6)X(291)

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

X(40766) lies on the cubic K1177 and these lines: {1, 18783}, {6, 291}, {171, 30648}, {1757, 27926}, {2712, 30664}, {37207, 40718}

X(40766) = isogonal conjugate of X(40793)
X(40766) = trilinear product X(i)*X(j) for these {i,j}: {6, 40740}, {5029, 37207}, {9508, 30664}
X(40766) = X(i)-isoconjugate of X(j) for these (i,j): {1929, 3783}, {2276, 40725}, {2702, 4486}, {3797, 17962}, {3802, 9505}, {6650, 16514}, {30665, 37135}
X(40766) = barycentric product X(i)*X(j) for these {i,j}: {1, 40740}, {2786, 30664}, {9508, 37207}
X(40766) = barycentric quotient X(i)/X(j) for these {i,j}: {985, 40725}, {1757, 3797}, {5029, 30665}, {9508, 4486}, {17735, 3783}, {18266, 16514}, {30664, 35148}, {40740, 75}

X(40767) = X(1)X(1929)∩X(10)X(1016)

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

X(40767) lies on the cubic K1177 and these lines: {1, 1929}, {10, 1016}, {83, 11599}, {238, 39786}, {239, 27916}, {764, 1019}, {1509, 17205}, {2054, 39971}, {2111, 5091}, {2702, 12194}, {3500, 17972}, {3673, 18032}, {4649, 9505}, {6650, 14621}, {16477, 37135}, {17023, 19936}, {35148, 35172}

X(40767) = isogonal conjugate of X(40794)
X(40767) = X(1929)-Ceva conjugate of X(238)
X(40767) = X(8300)-cross conjugate of X(238)
X(40767) = crossdifference of every pair of points on line {9508, 20693}
X(40767) = X(i)-isoconjugate of X(j) for these (i,j): {291, 1757}, {292, 6542}, {295, 17927}, {334, 18266}, {335, 17735}, {660, 9508}, {741, 6541}, {813, 2786}, {1911, 20947}, {2276, 40740}, {4562, 5029}, {4589, 17990}, {8298, 30663}, {17943, 35352}, {20693, 37128}
X(40767) = trilinear product X(i)*X(j) for these {i,j}: {6, 40725}, {238, 1929}, {239, 17962}, {242, 17972}, {659, 37135}, {812, 2702}, {1914, 6650}, {2054, 33295}, {2210, 18032}, {4010, 17940}, {4366, 9506}, {4455, 17930}, {5009, 11599}, {7193, 17982}, {8300, 9505}, {8632, 35148}
X(40767) = barycentric product X(i)*X(j) for these {i,j}: {1, 40725}, {238, 6650}, {239, 1929}, {350, 17962}, {659, 35148}, {812, 37135}, {1914, 18032}, {2054, 30940}, {2702, 3766}, {4366, 9505}, {9278, 33295}, {9506, 39044}, {17930, 21832}, {17982, 20769}
X(40767) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 6542}, {239, 20947}, {659, 2786}, {985, 40740}, {1914, 1757}, {1929, 335}, {2201, 17927}, {2210, 17735}, {2238, 6541}, {2702, 660}, {3747, 20693}, {5009, 1931}, {6650, 334}, {8300, 6651}, {8632, 9508}, {9505, 40098}, {9506, 30663}, {14599, 18266}, {17930, 4639}, {17940, 4584}, {17962, 291}, {18032, 18895}, {21832, 18004}, {35148, 4583}, {37135, 4562}, {39044, 18035}, {40725, 75}

X(40768) = X(1)X(20361)∩X(87)X(14621)

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

X(40768) lies on the cubic K1177 and these lines: {1, 20361}, {87, 14621}, {932, 20964}, {4649, 21759}, {15966, 17105}, {18754, 40741}

X(40768) = isogonal conjugate of X(40795)
X(40768) = X(30661)-cross conjugate of X(18754)
X(40768) = cevapoint of X(30661) and X(40741)
X(40768) = trilinear product X(i)*X(j) for these {i,j}: {6, 40741}, {87, 18754}, {2162, 30661}, {16362, 34252}
X(40768) = barycentric product X(i)*X(j) for these {i,j}: {1, 40741}, {87, 30661}, {330, 18754}, {16362, 39914}
X(40768) = barycentric quotient X(i)/X(j) for these {i,j}: {18754, 192}, {30661, 6376}, {40741, 75}

X(40769) = X(6)X(1045)∩X(765)X(1918)

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

X(40769) lies on the cubic K1177 and these lines: {6, 1045}, {765, 1918}, {985, 39925}, {2107, 8298}, {2382, 16484}, {3733, 18166}

X(40769) = isogonal conjugate of X(40796)
X(40769) = X(4366)-cross conjugate of X(238)
X(40769) = trilinear product X(i)*X(j) for these {i,j}: {238, 2665}, {1914, 39925}, {2107, 33295}
X(40769) = X(i)-isoconjugate of X(j) for these (i,j): {291, 2664}, {292, 17759}, {335, 21788}, {2276, 40742}, {21897, 37128}
X(40769) = barycentric product X(i)*X(j) for these {i,j}: {238, 39925}, {239, 2665}, {2107, 30940}
X(40769) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 17759}, {985, 40742}, {1914, 2664}, {2210, 21788}, {2665, 335}, {3747, 21897}, {4366, 39028}, {5009, 2106}, {8300, 39916}, {39925, 334}

X(40770) = X(1)X(9431)∩X(6)X(2669)

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

X(40770) lies on the cubics K10067 and K1177 and these lines: {1, 9431}, {6, 2669}, {171, 213}, {172, 1918}, {729, 33770}, {741, 9468}, {940, 3224}, {1509, 9427}, {2086, 6625}, {4649, 21759}, {21008, 21783}, {21755, 40432}

X(40770) = isogonal conjugate of X(1655)
X(40770) = isogonal conjugate of the anticomplement of X(274)
X(40770) = X(i)-cross conjugate of X(j) for these (i,j): {81, 6}, {904, 56}
X(40770) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1655}, {2, 1045}, {37, 39915}, {42, 34021}, {75, 21779}, {76, 18756}, {86, 21883}, {92, 23079}, {799, 9402}, {893, 27890}, {2276, 40743}
X(40770) = cevapoint of X(i) and X(j) for these (i,j): {649, 21755}, {667, 9427}
X(40770) = crosssum of X(21779) and X(23079)
X(40770) = trilinear pole of line {669, 20981}
X(40770) = barycentric product X(i)*X(j) for these {i,j}: {1, 40737}, {31, 18298}, {741, 39926}
X(40770) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1655}, {31, 1045}, {32, 21779}, {58, 39915}, {81, 34021}, {171, 27890}, {184, 23079}, {213, 21883}, {560, 18756}, {669, 9402}, {985, 40743}, {18298, 561}, {39926, 35544}, {40737, 75}

X(40771) = X(1)X(18784)∩X(6)X(7061)

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

X(40771) lies on the cubic K1177 and these lines: {1, 18784}, {6, 7061}, {171, 19554}

X(40771) = isogonal conjugate of X(40797)
X(40771) = X(i)-isoconjugate of X(j) for these (i,j): {1469, 17739}, {2276, 40723}, {7146, 8424}
X(40771) = barycentric quotient X(i)/X(j) for these {i,j}: {985, 40723}, {2344, 17739}, {18784, 7146}

X(40772) = X(1)X(335)∩X(2664)X(40742)

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

X(40722) lies on the cubic K1177 and these lines: {1, 335}, {2664, 40742}

X(40772) = isogonal conjugate of X(40798)
X(40772) = trilinear product X(6)*X(40742)
X(40772) = X(i)-isoconjugate of X(j) for these (i,j): {2665, 3783}, {16514, 39925}
X(40772) = barycentric product X(1)*X(40742)
X(40772) = barycentric quotient X(i)/X(j) for these {i,j}: {2664, 3797}, {21788, 3783}, {40742, 75}

X(40773) = ISOGONAL CONJUGATE OF X(40747)

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

X(40773) lies on the cubic K1178 and these lines: {1, 21}, {2, 39}, {9, 27644}, {37, 86}, {55, 16876}, {75, 27164}, {99, 743}, {110, 761}, {192, 314}, {193, 941}, {213, 3219}, {239, 257}, {241, 1434}, {284, 3512}, {321, 10471}, {325, 26601}, {330, 37870}, {612, 13588}, {662, 16521}, {756, 2664}, {759, 29145}, {869, 984}, {940, 16367}, {988, 37442}, {1010, 16735}, {1014, 2285}, {1015, 29586}, {1045, 3728}, {1213, 24530}, {1214, 7176}, {1255, 39950}, {1409, 1442}, {1412, 16577}, {1444, 2303}, {1500, 6542}, {1575, 29610}, {1654, 2092}, {1778, 16972}, {1790, 3497}, {1975, 19281}, {2176, 40153}, {2185, 7305}, {2223, 3920}, {2234, 24450}, {2256, 23125}, {2275, 17397}, {2276, 3661}, {2277, 17248}, {2287, 16517}, {2667, 24437}, {2669, 31323}, {3009, 3989}, {3247, 18164}, {3286, 21010}, {3294, 33761}, {3672, 16713}, {3729, 10455}, {3752, 16815}, {3770, 27042}, {3774, 27495}, {3799, 3864}, {3802, 4475}, {3809, 4517}, {3888, 14945}, {3912, 16887}, {3995, 27163}, {4016, 18714}, {4225, 37575}, {4261, 5224}, {4277, 17346}, {4278, 30142}, {4359, 16819}, {4360, 29767}, {4374, 21347}, {4384, 4850}, {4393, 16704}, {4414, 5184}, {4419, 17139}, {4649, 20166}, {4656, 17182}, {4664, 30939}, {4687, 16709}, {4704, 17178}, {4921, 16834}, {5030, 37633}, {5069, 17381}, {5088, 24606}, {5089, 14013}, {5249, 24214}, {5256, 21384}, {5266, 37296}, {5275, 11329}, {5276, 21511}, {5277, 19308}, {5297, 35983}, {5308, 17169}, {5333, 16831}, {6385, 34022}, {6586, 16755}, {6707, 24944}, {7096, 40145}, {7146, 25429}, {7179, 31909}, {7291, 22345}, {8025, 18171}, {9331, 29605}, {11110, 16823}, {14005, 39586}, {14008, 29680}, {14009, 29639}, {14552, 20018}, {16047, 33955}, {16053, 16601}, {16054, 24617}, {16349, 16992}, {16366, 17611}, {16476, 17017}, {16552, 32911}, {16571, 17038}, {16604, 29609}, {16672, 18198}, {16673, 17207}, {16687, 23370}, {16700, 25507}, {16710, 27268}, {16742, 29630}, {16744, 29614}, {16777, 18166}, {17000, 19224}, {17011, 20963}, {17023, 24625}, {17143, 17147}, {17183, 20348}, {17196, 24441}, {17202, 17247}, {17205, 29571}, {17210, 17308}, {17212, 21348}, {17244, 33947}, {17316, 30941}, {17324, 28358}, {17326, 27633}, {17448, 29584}, {17458, 18196}, {17524, 37590}, {17776, 27248}, {18165, 20358}, {18172, 29580}, {18602, 31631}, {19310, 19758}, {19822, 19853}, {20691, 29615}, {21840, 39252}, {21858, 32025}, {22172, 25421}, {24239, 37373}, {24464, 30116}, {24557, 26669}, {24790, 26724}, {25255, 40625}, {25512, 26747}, {25660, 26979}, {25946, 37675}, {27065, 27643}, {27784, 28620}, {27785, 28619}, {28618, 31318}, {29641, 33730}, {29643, 30984}, {29766, 34064}, {32009, 39747}, {37096, 37664}, {39957, 39971}

X(40773) = isogonal conjugate of X(40747)
X(40773) = X(i)-Ceva conjugate of X(j) for these (i,j): {30966, 3786}, {40438, 40734}
X(40773) = X(i)-cross conjugate of X(j) for these (i,j): {984, 30966}, {2276, 3736}, {3250, 3799}, {7146, 31909}
X(40773) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40747}, {6, 40718}, {10, 40746}, {37, 985}, {42, 14621}, {65, 2344}, {213, 870}, {512, 4586}, {523, 825}, {649, 4613}, {661, 1492}, {669, 37133}, {789, 798}, {871, 2205}, {1577, 34069}, {2295, 40763}, {2887, 18898}, {3113, 16584}, {3125, 5384}, {3407, 3778}, {4455, 37207}, {4557, 4817}, {20964, 40738}, {21010, 25425}, {21832, 30664}
X(40773) = cevapoint of X(984) and X(2276)
X(40773) = crosspoint of X(256) and X(30571)
X(40773) = crosssum of X(i) and X(j) for these (i,j): {1, 40749}, {6, 40750}, {37, 21904}, {81, 40759}, {171, 4649}, {985, 40751}, {5228, 40765}, {14621, 40752}, {40753, 40768}, {40754, 40761}, {40755, 40762}, {40766, 40772}
X(40773) = trilinear pole of line {788, 1491}
X(40773) = crossdifference of every pair of points on line {661, 669}
X(40773) = trilinear product X(i)*X(j) for these {i,j}: {2, 3736}, {3, 31909}, {6, 30966}, {21, 7146}, {27, 3781}, {57, 3786}, {58, 3661}, {60, 16603}, {81, 984}, {86, 2276}, {99, 3250}, {100, 4481}, {110, 824}, {257, 40731}, {274, 869}, {284, 7179}, {310, 40728}, {333, 1469}, {593, 3773}, {662, 1491}, {741, 3797}, {788, 799}, {873, 3774}, {985, 4469}, {1019, 3799}, {1171, 3775}, {1333, 33931}, {1412, 3790}, {1434, 4517}, {2287, 7204}, {3094, 40415}, {3116, 38810}, {3314, 38813}, {3733, 3807}, {3783, 37128}, {3792, 24624}, {3805, 4603}, {3862, 33295}, {4122, 4556}, {4475, 4567}, {4476, 14621}, {4522, 4565}, {4584, 30665}, {4602, 8630}, {4627, 4818}, {4634, 14436}, {6385, 18900}, {16514, 18827}, {18829, 30654}, {27483, 40734}
X(40773) = barycentric product X(i)*X(j) for these {i,j}: {1, 30966}, {7, 3786}, {21, 7179}, {58, 33931}, {63, 31909}, {75, 3736}, {81, 3661}, {86, 984}, {99, 1491}, {190, 4481}, {274, 2276}, {286, 3781}, {310, 869}, {314, 1469}, {333, 7146}, {337, 17569}, {662, 824}, {670, 788}, {757, 3773}, {799, 3250}, {805, 30639}, {870, 4476}, {1014, 3790}, {1019, 3807}, {1043, 7204}, {1414, 4522}, {1576, 30870}, {2185, 16603}, {3094, 38810}, {3733, 4505}, {3775, 40438}, {3783, 18827}, {3792, 14616}, {3797, 37128}, {3799, 7192}, {3805, 4594}, {3862, 30940}, {3864, 33295}, {4469, 14621}, {4475, 4600}, {4486, 4584}, {4589, 30665}, {4609, 8630}, {4614, 4818}, {6385, 40728}, {7018, 40731}, {16514, 40017}
X(40773) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40718}, {6, 40747}, {58, 985}, {81, 14621}, {86, 870}, {99, 789}, {100, 4613}, {110, 1492}, {163, 825}, {284, 2344}, {310, 871}, {662, 4586}, {788, 512}, {799, 37133}, {824, 1577}, {869, 42}, {984, 10}, {1019, 4817}, {1178, 40763}, {1333, 40746}, {1469, 65}, {1491, 523}, {1576, 34069}, {2276, 37}, {3094, 3721}, {3116, 3778}, {3117, 16584}, {3250, 661}, {3314, 20234}, {3661, 321}, {3736, 1}, {3773, 1089}, {3774, 1500}, {3775, 4647}, {3781, 72}, {3783, 740}, {3786, 8}, {3789, 3696}, {3790, 3701}, {3792, 758}, {3795, 3993}, {3797, 3948}, {3799, 3952}, {3802, 4368}, {3805, 2533}, {3807, 4033}, {4122, 4036}, {4407, 4714}, {4439, 3992}, {4469, 3661}, {4475, 3120}, {4476, 984}, {4481, 514}, {4505, 27808}, {4517, 210}, {4522, 4086}, {4570, 5384}, {4584, 37207}, {4601, 5388}, {4818, 4815}, {7146, 226}, {7179, 1441}, {7204, 3668}, {8630, 669}, {14436, 14407}, {16514, 2238}, {16603, 6358}, {17569, 242}, {18899, 21751}, {18900, 1918}, {19584, 21101}, {25429, 17754}, {27474, 4044}, {30639, 14295}, {30665, 4010}, {30966, 75}, {31909, 92}, {33931, 313}, {38810, 3114}, {38814, 40722}, {39915, 40743}, {40415, 3113}, {40432, 40738}, {40728, 213}, {40731, 171}, {40733, 21904}, {40734, 4649}, {40736, 21759}
X(40773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 846, 3747}, {1, 18206, 81}, {2, 1655, 3948}, {2, 18600, 16752}, {37, 16696, 86}, {37, 37596, 16826}, {38, 10458, 5208}, {81, 1931, 58}, {81, 28606, 25058}, {192, 16738, 314}, {333, 3666, 25059}, {333, 33296, 239}, {980, 5283, 2}, {984, 3736, 3786}, {1107, 3666, 239}, {3912, 16887, 30965}, {4687, 16709, 25508}, {16831, 17175, 5333}, {18169, 35623, 3794}

X(40774) = ISOGONAL CONJUGATE OF X(40748)

Barycentrics    a*(a^2 + 2*a*b + 2*a*c + b*c)*(b^2 + b*c + c^2) : :
X(40774) = X[1] - 4 X[25092], 2 X[10] + X[25264], 5 X[1698] - 2 X[20888]

X(40774) lies on the cubic K1178 and these lines: {1, 672}, {2, 726}, {10, 1655}, {37, 291}, {42, 846}, {190, 40718}, {813, 40766}, {984, 2276}, {1698, 18135}, {2308, 8616}, {3773, 30966}, {3778, 25421}, {3799, 3864}, {3932, 25349}, {4368, 17261}, {5283, 12782}, {6541, 31027}, {15481, 21904}, {24690, 32846}, {32935, 37632}

X(40774) = isogonal conjugate of X(40748)
X(40774) = X(i)-Ceva conjugate of X(j) for these (i,j): {291, 3783}, {1268, 3661}, {40433, 869}
X(40774) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40748}, {985, 30571}, {4817, 28841}, {14621, 25426}, {27483, 40746}
X(40774) = trilinear product X(i)*X(j) for these {i,j}: {6, 27495}, {10, 40734}, {984, 4649}, {2276, 16826}, {3736, 3842}, {3799, 4784}, {3862, 20142}
X(40774) = barycentric product X(i)*X(j) for these {i,j}: {1, 27495}, {321, 40734}, {984, 16826}, {3661, 4649}, {3799, 28840}, {3807, 4784}, {3864, 20142}
X(40774) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40748}, {869, 25426}, {984, 27483}, {2276, 30571}, {4649, 14621}, {4784, 4817}, {16826, 870}, {27495, 75}, {40734, 81}
X(40774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 28600, 30571}, {291, 30571, 28600}, {984, 2276, 3783}, {984, 3795, 3789}, {2276, 3789, 3795}, {3789, 3795, 3783}

X(40775) = ISOGONAL CONJUGATE OF X(40749)

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

X(40775) lies on the cubic K1178 and these lines: {1, 2106}, {10, 1655}, {19, 15148}, {37, 1045}, {75, 34021}, {274, 18298}, {846, 18785}, {1573, 35040}, {2276, 30570}, {2665, 40737}, {9278, 24578}, {13476, 24437}, {13610, 18206}, {24342, 40742}, {25347, 25457}, {39252, 40747}

X(40775) = isogonal conjugate of X(40749)
X(40775) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40749}, {6, 40721}
X(40775) = cevapoint of X(1491) and X(2643)
X(40775) = barycentric product X(16826)*X(30570)
X(40775) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40721}, {6, 40749}, {30570, 27483}

X(40776) = ISOGONAL CONJUGATE OF X(40750)

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

X(40776) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1178, and these lines: {1, 2054}, {6, 1931}, {37, 319}, {42, 846}, {81, 2248}, {1400, 1442}, {1655, 27809}, {1880, 7282}, {1989, 14616}, {4649, 40744}, {5224, 39982}, {6625, 6650}, {9281, 28606}, {24530, 39798}, {40721, 40740}

X(40776) = isogonal conjugate of X(40750)
X(40776) = X(25426)-cross conjugate of X(1002)
X(40776) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40750}, {6, 24342}, {662, 9279}, {1001, 18791}
X(40776) = cevapoint of X(i) and X(j) for these (i,j): {788, 1084}, {824, 8287}
X(40776) = trilinear pole of line {512, 9508}
X(40776) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24342}, {6, 40750}, {512, 9279}, {2279, 18791}

X(40777) = ISOGONAL CONJUGATE OF X(40751)

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

X(40777) lies on the cubic K1178 and these lines: {37, 171}, {313, 1920}, {983, 5311}, {1655, 6625}

X(40777) = isogonal conjugate of X(40751)
X(40777) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40751}, {6, 40722}, {825, 21196}, {846, 985}, {1654, 40746}, {14621, 18755}, {38814, 40747}
X(40777) = trilinear product X(i)*X(j) for these {i,j}: {984, 13610}, {2248, 3661}, {2276, 6625}, {15377, 31909}, {18757, 33931}
X(40777) = barycentric product X(i)*X(j) for these {i,j}: {984, 6625}, {2248, 33931}, {3661, 13610}
X(40777) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40722}, {6, 40751}, {869, 18755}, {984, 1654}, {1491, 21196}, {2248, 985}, {2276, 846}, {3661, 17762}, {3736, 38814}, {3773, 27569}, {6625, 870}, {7146, 17084}, {13610, 14621}, {18757, 40746}

X(40778) = ISOGONAL CONJUGATE OF X(40752)

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

X(40778) lies on the cubic K1178 and these lines: {1, 2670}, {37, 1655}, {172, 1918}, {846, 16362}, {40728, 40731}

X(40778) = isogonal conjugate of X(40752)
X(40778) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40752}, {6, 40743}, {870, 21779}, {985, 1655}, {1045, 14621}, {39915, 40747}
X(40778) = crosssum of X(30661) and X(40721)
X(40778) = trilinear product X(i)*X(j) for these {i,j}: {2276, 40737}, {18298, 40728}
X(40778) = barycentric product X(i)*X(j) for these {i,j}: {869, 18298}, {984, 40737}, {3661, 40770}
X(40778) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40743}, {6, 40752}, {869, 1045}, {2276, 1655}, {3736, 39915}, {3774, 21883}, {18298, 871}, {18900, 18756}, {40728, 21779}, {40737, 870}, {40770, 14621}

X(40779) = ISOGONAL CONJUGATE OF X(5228)

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

X(40779) lies on the Feuerbach circumhyperbola, the cubic K1178, and these lines: {1, 672}, {2, 2481}, {4, 5089}, {6, 2346}, {7, 37}, {8, 1212}, {9, 2293}, {21, 220}, {45, 1156}, {55, 294}, {79, 17732}, {104, 8693}, {218, 943}, {256, 21811}, {279, 27253}, {314, 346}, {650, 885}, {941, 3779}, {1172, 7071}, {1252, 5377}, {1320, 34522}, {1621, 7123}, {1642, 14947}, {1655, 18299}, {1742, 3062}, {2276, 5222}, {2295, 17097}, {2298, 20992}, {2320, 6603}, {2344, 40757}, {2345, 2997}, {3000, 16676}, {3008, 17756}, {3161, 7155}, {3208, 3680}, {3247, 10390}, {3475, 40606}, {3691, 4866}, {4050, 31509}, {4814, 23893}, {4876, 24498}, {5134, 5561}, {5226, 21856}, {5281, 16588}, {5283, 39587}, {5435, 9444}, {5526, 15175}, {7160, 16572}, {7320, 40133}, {9330, 36197}, {9442, 9502}, {10481, 30494}, {17316, 24635}, {18166, 34820}, {24036, 36479}, {25066, 39581}, {25242, 34284}, {26242, 37597}, {27109, 27304}, {27396, 30479}, {29571, 30949}

X(40779) = isogonal conjugate of X(5228)
X(40779) = X(27475)-Ceva conjugate of X(1002)
X(40779) = X(i)-cross conjugate of X(j) for these (i,j): {4517, 8}, {6182, 100}
X(40779) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5228}, {2, 1471}, {6, 40719}, {7, 2280}, {34, 23151}, {56, 4384}, {57, 1001}, {73, 31926}, {109, 4762}, {269, 37658}, {604, 4441}, {651, 4724}, {1106, 28809}, {1397, 21615}, {1407, 3886}, {1408, 4044}, {1412, 3696}, {1790, 1893}, {4565, 4804}, {7177, 28044}
X(40779) = crosssum of X(i) and X(j) for these (i,j): {1471, 2280}, {3243, 17754}
X(40779) = trilinear pole of line {650, 926} (the line through X(650) parallel to its trilinear polar)
X(40779) = barycentric product X(i)*X(j) for these {i,j}: {8, 1002}, {9, 27475}, {312, 2279}, {522, 37138}, {650, 32041}, {984, 40739}, {3661, 40757}, {4391, 8693}
X(40779) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40719}, {6, 5228}, {8, 4441}, {9, 4384}, {31, 1471}, {41, 2280}, {55, 1001}, {200, 3886}, {210, 3696}, {219, 23151}, {220, 37658}, {312, 21615}, {346, 28809}, {650, 4762}, {663, 4724}, {1002, 7}, {1172, 31926}, {1824, 1893}, {2279, 57}, {2321, 4044}, {3689, 4702}, {4041, 4804}, {4517, 3789}, {7071, 28044}, {8693, 651}, {27475, 85}, {32041, 4554}, {32724, 32735}, {36138, 36146}, {37138, 664}, {40739, 870}, {40757, 14621}

X(40780) = ISOGONAL CONJUGATE OF X(40753)

Barycentrics    a*(a*b + a*c - b*c)*(a*b + 2*b^2 - a*c + b*c)*(a*b - a*c - b*c - 2*c^2) : :
X(40780 = 4 X[37] - X[87], 2 X[75] - 5 X[31270], X[192] + 2 X[34832], 2 X[4664] + X[31170]

X(40780) lies on the cubic K1178 and these lines: {1, 20332}, {2, 726}, {37, 87}, {43, 17459}, {75, 31270}, {192, 34832}, {4664, 31170}, {4704, 25284}, {8026, 31008}

X(40780) = isogonal conjugate of X(40753)
X(40780) = X(40756)-Ceva conjugate of X(43)
X(40780) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40753}, {6, 40720}, {87, 16468}, {330, 21793}, {932, 4782}, {2162, 4393}, {4785, 34071}, {7121, 30963}
X(40780) = crosssum of X(4393) and X(40720)
X(40780) = trilinear product X(i)*X(j) for these {i,j}: {2176, 27494}, {6376, 40735}, {34475, 38832}
X(40780) = barycentric product X(i)*X(j) for these {i,j}: {43, 27494}, {3661, 40756}, {6382, 40735}, {27644, 34475}
X(40780) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40720}, {6, 40753}, {43, 4393}, {192, 30963}, {2176, 16468}, {2209, 21793}, {4083, 4785}, {6376, 10009}, {20691, 3993}, {20979, 4782}, {21337, 25376}, {21834, 4806}, {27494, 6384}, {40735, 2162}, {40756, 14621}

X(40781) = ISOGONAL CONJUGATE OF X(40754)

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

X(40781) lies on the cubic K1178 and these lines: {1, 2115}, {2, 20940}, {55, 846}, {518, 40764}, {650, 824}, {1252, 3219}

X(40781) = isogonal conjugate of X(40754)
X(40781) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40754}, {6, 40724}, {105, 3509}, {294, 5018}, {673, 17798}, {919, 4458}, {1438, 4645}, {2481, 19554}, {18031, 18262}, {20741, 36124}
X(40781) = crosspoint of X(3512) and X(24479)
X(40781) = crosssum of X(3509) and X(19557)
X(40781) = trilinear product X(i)*X(j) for these {i,j}: {241, 7281}, {518, 3512}, {672, 7261}, {3912, 8852}, {8299, 24479}, {9454, 18036}, {17755, 30648}
X(40781) = barycentric product X(i)*X(j) for these {i,j}: {518, 7261}, {2223, 18036}, {3263, 8852}, {3512, 3912}, {3661, 40764}, {7281, 9436}, {17755, 24479}
X(40781) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40724}, {6, 40754}, {518, 4645}, {672, 3509}, {1458, 5018}, {2223, 17798}, {2254, 4458}, {3512, 673}, {3912, 17789}, {3930, 4071}, {7261, 2481}, {7281, 14942}, {8299, 1281}, {8852, 105}, {9454, 19554}, {9455, 18262}, {17755, 18037}, {20683, 20715}, {20752, 20741}, {40764, 14621}

X(40782) = ISOGONAL CONJUGATE OF X(40755)

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

X(40782) lies on the cubic K1178 and these lines: {1, 40736}, {37, 33680}, {1107, 4083}, {1655, 2276}

X(40782) = isogonal conjugate of X(40755)
X(40782) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40755}, {727, 19565}, {3226, 18278}, {3510, 20332}, {19567, 34077}
X(40782) = crosssum of X(18278) and X(19580)
X(40782) = trilinear product X(i)*X(j) for these {i,j}: {1575, 7168}, {17475, 24576}
X(40782) = barycentric product X(i)*X(j) for these {i,j}: {726, 7168}, {17793, 24576}, {20663, 30633}
X(40782) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40755}, {726, 19567}, {1575, 19565}, {3009, 3510}, {7168, 3226}, {17475, 19579}, {17793, 19581}, {20663, 19580}, {20777, 23186}, {21760, 18278}

X(40783) = ISOGONAL CONJUGATE OF X(40756)

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

X(40783) lies on the cubic K1178 and these lines: {1, 2053}, {37, 87}, {330, 1655}, {846, 2162}, {1107, 14823}, {2276, 3117}, {2319, 17592}, {4704, 7155}, {16514, 40736}, {16525, 21759}, {21793, 40753}, {27458, 32776}

X(40783) = isogonal conjugate of X(40756)
X(40783) = X(27481)-cross conjugate of X(3795)
X(40783) = crosspoint of X(87) and X(40753)
X(40783) = crosssum of X(43) and X(40780)
X(40783) = trilinear product X(i)*X(j) for these {i,j}: {87, 3795}, {330, 40733}, {2162, 27481}, {2276, 40720}, {10009, 40736}
X(40783) = barycentric product X(i)*X(j) for these {i,j}: {87, 27481}, {330, 3795}, {984, 40720}, {3661, 40753}, {6384, 40733}
X(40783) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40756}, {3795, 192}, {27481, 6376}, {40720, 870}, {40733, 43}, {40736, 40735}, {40753, 14621}

X(40784) = ISOGONAL CONJUGATE OF X(40757)

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

X(40784) lies on the cubic K1178 and these lines: {1, 2114}, {2, 10030}, {7, 37}, {57, 846}, {85, 1655}, {344, 26125}, {497, 3666}, {980, 3663}, {984, 1469}, {1001, 1471}, {1423, 3731}, {1429, 16484}, {1462, 8543}, {2275, 17084}, {2276, 7179}, {3674, 5283}, {4310, 37596}, {4335, 4907}, {4657, 17077}, {5701, 38186}, {5805, 24248}, {16591, 33149}, {17257, 25099}, {20367, 31394}, {20616, 30617}, {21615, 28809}, {25065, 33869}, {28091, 28093}, {37632, 39930}

X(40784) = isogonal conjugate of X(40757)
X(40784) = X(7)-Ceva conjugate of X(1469)
X(40784) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40757}, {6, 40739}, {1002, 2344}
X(40784) = trilinear product X(i)*X(j) for these {i,j}: {56, 27474}, {57, 3789}, {85, 40732}, {984, 5228}, {1001, 7146}, {1469, 4384}, {1471, 3661}, {2276, 40719}, {2280, 7179}, {7204, 37658}
X(40784) = barycentric product X(i)*X(j) for these {i,j}: {7, 3789}, {57, 27474}, {984, 40719}, {1001, 7179}, {1469, 4441}, {1471, 33931}, {3661, 5228}, {3886, 7204}, {4384, 7146}, {6063, 40732}
X(40784) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40739}, {6, 40757}, {1469, 1002}, {1471, 985}, {2280, 2344}, {3789, 8}, {5228, 14621}, {7146, 27475}, {27474, 312}, {40719, 870}, {40732, 55}

X(40785) = ISOGONAL CONJUGATE OF X(40758)

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

X(40785) lies on the cubic K1178 and these lines: {1, 2053}, {2, 10030}, {984, 7220}

X(40785) = isogonal conjugate of X(40758)
X(40785) = trilinear product X(7220)*X(28391)
X(40785) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40758}, {7220, 39924}, {17760, 20917}, {17792, 24349}, {18758, 21010}

X(40786) = ISOGONAL CONJUGATE OF X(40759)

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

X(40786) lies on the cubic K1178 and these lines: {846, 8845}, {1655, 4037}, {21085, 21883}

X(40786) = isogonal conjugate of X(40759)
X(40786) = barycentric quotient X(6)/X(40759)

X(40787) = ISOGONAL CONJUGATE OF X(40760)

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

X(40787) lies on the cubic K1178 and these lines: {2276, 5222}, {4817, 25425}

X(40787) = isogonal conjugate of X(40760)
X(40787) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40760}, {27478, 20917}, {28600, 24349}

X(40788) = ISOGONAL CONJUGATE OF X(40761)

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

X(40788) lies on the cubic K1178 and these lines: {1, 2110}, {85, 1655}, {514, 27854}, {846, 9499}, {2276, 27475}

X(40788) = isogonal conjugate of X(40761)
X(40788) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40761}, {105, 39252}
X(40788) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40761}, {672, 39252}

X(40789) = ISOGONAL CONJUGATE OF X(40762)

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

X(40789) lies on the cubic K1178 and these lines: {1, 2109}, {649, 38348}, {846, 2162}

X(40789) = isogonal conjugate of X(40762)
X(40789) = barycentric quotient X(6)/X(40762)

X(40790) = ISOGONAL CONJUGATE OF X(40763)

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

X(40790) lies on the cubic K1178 and these lines: {1, 2}, {12, 21531}, {35, 237}, {36, 14096}, {37, 256}, {38, 18208}, {55, 11328}, {86, 38810}, {87, 5749}, {171, 172}, {291, 37596}, {319, 872}, {420, 6198}, {756, 1959}, {846, 1334}, {894, 7184}, {980, 12782}, {984, 1469}, {1045, 2321}, {1215, 1237}, {1376, 21008}, {1429, 17122}, {1442, 39977}, {1478, 37190}, {1500, 3229}, {1655, 3971}, {1740, 2345}, {1964, 17289}, {2276, 3117}, {2309, 17280}, {2330, 36213}, {2344, 4386}, {2663, 3879}, {2667, 17315}, {3051, 5280}, {3061, 21332}, {3208, 17594}, {3219, 17799}, {3231, 16785}, {3250, 29955}, {3585, 14957}, {3589, 18170}, {3618, 18194}, {3736, 3773}, {3746, 37338}, {3750, 19589}, {3761, 20023}, {3765, 32931}, {3799, 3864}, {3997, 40749}, {5010, 37184}, {5299, 20965}, {5337, 11364}, {6358, 17901}, {7032, 17368}, {7229, 25570}, {7951, 37988}, {12197, 37527}, {16696, 21865}, {16706, 17445}, {17137, 33085}, {17263, 24757}, {17281, 24696}, {17592, 20284}, {18169, 33164}, {20556, 33106}, {21278, 27261}, {22277, 24437}, {25144, 28358}, {26244, 40763}, {26978, 33147}

X(40790) = isogonal conjugate of X(40763)
X(40790) = X(3862)-Ceva conjugate of X(3783)
X(40790) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40763}, {6, 40738}, {256, 985}, {257, 40746}, {513, 30670}, {870, 904}, {893, 14621}, {1178, 40718}, {1432, 2344}, {3407, 3863}, {40432, 40747}
X(40790) = crosssum of X(1) and X(17795)
X(40790) = crossdifference of every pair of points on line {649, 4164}
X(40790) = trilinear product X(i)*X(j) for these {i,j}: {10, 40731}, {100, 3805}, {171, 984}, {172, 3661}, {385, 3862}, {869, 1909}, {894, 2276}, {1215, 3736}, {1469, 7081}, {1491, 4579}, {1580, 3864}, {1920, 40728}, {2329, 7146}, {2330, 7179}, {3250, 18047}, {3774, 8033}, {3781, 7009}, {3783, 18787}, {3799, 4367}, {3807, 20981}, {4517, 7176}, {4562, 30654}, {5386, 30656}, {7122, 33931}, {16514, 30669}, {20964, 30966}, {22061, 31909}, {30639, 34067}
X(40790) = barycentric product X(i)*X(j) for these {i,j}: {171, 3661}, {172, 33931}, {190, 3805}, {321, 40731}, {385, 3864}, {813, 30639}, {824, 4579}, {869, 1920}, {894, 984}, {1469, 17787}, {1491, 18047}, {1909, 2276}, {1966, 3862}, {2295, 30966}, {2329, 7179}, {3736, 3963}, {3783, 30669}, {3786, 4032}, {3790, 7175}, {3797, 18787}, {3799, 4369}, {3807, 4367}, {4505, 20981}, {4517, 7196}, {4583, 30654}, {7081, 7146}
X(40790) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40738}, {6, 40763}, {101, 30670}, {171, 14621}, {172, 985}, {869, 893}, {894, 870}, {984, 257}, {1469, 1432}, {1920, 871}, {2276, 256}, {2295, 40718}, {2330, 2344}, {3094, 3865}, {3116, 3863}, {3661, 7018}, {3736, 40432}, {3783, 17493}, {3799, 27805}, {3805, 514}, {3862, 1581}, {3864, 1916}, {4164, 23597}, {4367, 4817}, {4579, 4586}, {7122, 40746}, {7146, 7249}, {16514, 18786}, {18047, 789}, {18900, 7104}, {20964, 40747}, {30654, 659}, {30656, 4809}, {40728, 904}, {40731, 81}
X(40790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3507, 42}, {8, 43, 4489}, {37, 17792, 256}, {171, 2329, 1580}, {869, 3661, 3783}, {894, 7184, 7240}, {2295, 4447, 171}, {3661, 3809, 869}, {7081, 17752, 4039}

X(40791) = ISOGONAL CONJUGATE OF X(40764)

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

X(40791) lies on the cubic K1178 and these lines: {1, 41}, {1281, 40724}, {1655, 18760}, {2276, 7179}, {3509, 40754}

X(40791) = isogonal conjugate of X(40764)
X(40791) = trilinear product X(2276)*X(40724)
X(40791) = barycentric product X(i)*X(j) for these {i,j}: {984, 40724}, {3661, 40754}
X(40791) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40764}, {40724, 870}, {40754, 14621}

X(40792) = ISOGONAL CONJUGATE OF X(40765)

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

X(40792) lies on the cubic K1178 and these lines: {37, 17084}, {846, 1334}, {1655, 18760}, {2329, 40771}

X(40792) = isogonal conjugate of X(40765)
X(40792) = X(17611)-cross conjugate of X(9)
X(40792) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40765}, {6, 40723}, {56, 17739}, {57, 8424}, {85, 18759}, {604, 30660}, {1431, 27963}
X(40792) = trilinear product X(i)*X(j) for these {i,j}: {8, 18784}, {41, 18760}, {4876, 16366}
X(40792) = barycentric product X(i)*X(j) for these {i,j}: {55, 18760}, {312, 18784}, {3661, 40771}, {4518, 16366}
X(40792) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40723}, {6, 40765}, {8, 30660}, {9, 17739}, {55, 8424}, {2175, 18759}, {2329, 27963}, {7281, 39920}, {16366, 1447}, {18760, 6063}, {18784, 57}, {40771, 14621}

X(40793) = ISOGONAL CONJUGATE OF X(40766)

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

X(40793) lies on the cubic K1178 and these lines: {1, 2113}, {2, 846}, {37, 9505}, {238, 39786}, {1001, 17962}

X(40793) = isogonal conjugate of X(40766)
X(40793) = X(3802)-cross conjugate of X(3783)
X(40793) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40766}, {6, 40740}, {5029, 37207}, {9508, 30664}
X(40793) = trilinear product X(i)*X(j) for these {i,j}: {1929, 3783}, {2276, 40725}, {2702, 4486}, {3797, 17962}, {3802, 9505}, {6650, 16514}, {30665, 37135}
X(40793) = barycentric product X(i)*X(j) for these {i,j}: {984, 40725}, {1929, 3797}, {3661, 40767}, {3783, 6650}, {4486, 37135}, {16514, 18032}, {30665, 35148}
X(40793) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40740}, {6, 40766}, {2702, 30664}, {3783, 6542}, {3797, 20947}, {3802, 6651}, {16514, 1757}, {17569, 423}, {30665, 2786}, {37135, 37207}, {40725, 870}, {40767, 14621}

X(40794) = ISOGONAL CONJUGATE OF X(40767)

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

X(40794) lies on the cubic K1178 and these lines: {1, 39}, {8, 6630}, {10, 4562}, {35, 34067}, {37, 9505}, {58, 4567}, {334, 18140}, {335, 29569}, {756, 9510}, {813, 1334}, {846, 8933}, {876, 6372}, {984, 2113}, {1655, 6625}, {1909, 4583}, {3634, 40095}, {3842, 24505}, {4447, 17799}, {4517, 40730}, {4589, 6626}, {6651, 40740}, {16826, 40098}, {17316, 40217}, {17735, 40766}, {18895, 33943}, {24518, 32931}

X(40794) = isogonal conjugate of X(40767)
X(40794) = X(30663)-Ceva conjugate of X(291)
X(40794) = X(1757)-cross conjugate of X(291)
X(40794) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40767}, {6, 40725}, {238, 1929}, {239, 17962}, {242, 17972}, {659, 37135}, {812, 2702}, {1914, 6650}, {2054, 33295}, {2210, 18032}, {4010, 17940}, {4366, 9506}, {4455, 17930}, {5009, 11599}, {7193, 17982}, {8300, 9505}, {8632, 35148}
X(40794) = trilinear pole of line {9508, 20693}
X(40794) = trilinear product X(i)*X(j) for these {i,j}: {291, 1757}, {292, 6542}, {295, 17927}, {334, 18266}, {335, 17735}, {660, 9508}, {741, 6541}, {813, 2786}, {1911, 20947}, {2276, 40740}, {4562, 5029}, {4589, 17990}, {8298, 30663}, {17943, 35352}, {20693, 37128}
X(40794) = barycentric product X(i)*X(j) for these {i,j}: {291, 6542}, {292, 20947}, {334, 17735}, {335, 1757}, {660, 2786}, {984, 40740}, {3661, 40766}, {4562, 9508}, {4583, 5029}, {4584, 18004}, {4639, 17990}, {6541, 37128}, {6651, 30663}, {8298, 40098}, {18266, 18895}, {18827, 20693}
X(40794) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40725}, {6, 40767}, {291, 6650}, {292, 1929}, {335, 18032}, {660, 35148}, {813, 37135}, {1757, 239}, {1911, 17962}, {1931, 33295}, {2196, 17972}, {2786, 3766}, {4584, 17930}, {5029, 659}, {6541, 3948}, {6542, 350}, {6651, 39044}, {8298, 4366}, {9508, 812}, {17731, 30940}, {17735, 238}, {17976, 20769}, {17990, 21832}, {18266, 1914}, {18267, 18263}, {20693, 740}, {20947, 1921}, {27929, 27855}, {34067, 2702}, {38348, 4375}, {40740, 870}, {40766, 14621}
X(40794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 22116, 291}, {292, 3864, 291}

X(40795) = ISOGONAL CONJUGATE OF X(40768)

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

X(40795) lies on the cubic K1178 and these lines: {846, 16360}, {1655, 3971}

X(40795) = isogonal conjugate of X(40768)
X(40795) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40768}, {6, 40741}, {87, 18754}, {2162, 30661}, {16362, 34252}
X(40795) = crosssum of X(30661) and X(40741)
X(40795) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40741}, {6, 40768}, {43, 30661}, {2176, 18754}

X(40796) = ISOGONAL CONJUGATE OF X(40769)

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

X(40796) lies on the cubic K1178 and these lines: {1, 2111}, {2, 38}, {42, 660}, {43, 9361}, {171, 813}, {846, 16362}, {1500, 35040}, {1621, 1911}, {2276, 18795}, {3572, 4979}, {3971, 4583}, {4589, 39915}, {21788, 40772}, {24169, 40094}, {39916, 40742}

X(40796) = isogonal conjugate of X(40769)
X(40796) = X(17759)-cross conjugate of X(291)
X(40796) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40769}, {238, 2665}, {1914, 39925}, {2107, 33295}
X(40796) = trilinear product X(i)*X(j) for these {i,j}: {291, 2664}, {292, 17759}, {335, 21788}, {2276, 40742}, {21897, 37128}
X(40796) = barycentric product X(i)*X(j) for these {i,j}: {291, 17759}, {334, 21788}, {335, 2664}, {984, 40742}, {3661, 40772}, {18827, 21897}, {30663, 39916}
X(40796) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40769}, {291, 39925}, {292, 2665}, {2106, 33295}, {2664, 239}, {2669, 30940}, {15148, 31905}, {17759, 350}, {20796, 20769}, {21788, 238}, {21897, 740}, {27854, 27855}, {39916, 39044}, {40742, 870}, {40772, 14621}
X(40796) = {X(2),X(40155)}-harmonic conjugate of X(291)

X(40797) = ISOGONAL CONJUGATE OF X(40771)

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

X(40797) lies on the cubic K1178 and these lines: {1, 256}, {2, 20940}, {37, 17084}, {8424, 40765}

X(40797) = isogonal conjugate of X(40771)
X(40797) = trilinear product X(i)*X(j) for these {i,j}: {1469, 17739}, {2276, 40723}, {7146, 8424}
X(40797) = barycentric product X(i)*X(j) for these {i,j}: {984, 40723}, {1469, 30660}, {3661, 40765}, {7146, 17739}, {7179, 8424}
X(40797) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40771}, {7179, 18760}, {40723, 870}, {40765, 14621}

X(40798) = ISOGONAL CONJUGATE OF X(40772)

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

X(40798) lies on the cubic K1178 and these lines: {1, 1655}, {2, 18795}, {846, 8934}, {1914, 40769}, {18786, 27919}

X(40798) = isogonal conjugate of X(40772)
X(40798) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40772}, {6, 40742}
X(40798) = trilinear product X(i)*X(j) for these {i,j}: {2665, 3783}, {16514, 39925}
X(40798) = barycentric product X(i)*X(j) for these {i,j}: {2665, 3797}, {3661, 40769}, {3783, 39925}
X(40798) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40742}, {6, 40772}, {3783, 17759}, {3802, 39916}, {16514, 2664}, {40769, 14621}

X(40799) = X(2)X(6394)∩X(3)X(232)

Barycentrics    a^4*(a^4 + 3*b^4 - 2*a^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 3*c^4) : :
Barycentrics    a^2/(a^4 - SB*SC) : :
Trilinears    (cos 2A)/(sin A tan ω - cos B cos C) : :

X(40799) lies on the cubic K1179 and these lines: {2, 6394}, {3, 232}, {6, 3964}, {32, 1092}, {39, 16391}, {83, 7736}, {95, 17907}, {184, 11672}, {237, 577}, {248, 9306}, {574, 11060}, {647, 2422}, {729, 35575}, {3224, 34870}, {5063, 32740}, {10311, 11328}, {10313, 37465}, {10314, 37338}, {10316, 19210}, {15355, 37183}, {33871, 39238}

X(40799) = isogonal conjugate of X(40814)
X(40799) = isotomic conjugate of X(40822)
X(40799) = X(33569)-cross conjugate of X(14966)
X(40799) = cevapoint of X(i) and X(j) for these (i,j): {3, 11328}, {182, 9306}
X(40799) = crosssum of X(6776) and X(7735)
X(40799) = trilinear pole of line {669, 32320}
X(40799) = crossdifference of every pair of points on line {1513, 30735}
X(40799) = X(i)-isoconjugate of X(j) for these (i,j): {2, 4008}, {75, 7735}, {92, 6776}, {158, 37188}, {304, 6620}, {662, 30735}, {1513, 1821}, {1577, 35278}
X(40799) = trilinear product X(798)*X(35575)
X(40799) = barycentric product X(512)*X(35575)
X(40799) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 4008}, {32, 7735}, {184, 6776}, {237, 1513}, {512, 30735}, {577, 37188}, {1576, 35278}, {1974, 6620}, {34396, 9755}, {35575, 670}

X(40800) = ISOGONAL CONJUGATE OF X(3168)

Barycentrics    a^2*(a^2 - b^2 - c^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)*(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(40800) lies on the cubic K1179 and these lines: {3, 3164}, {5, 13855}, {182, 14379}, {577, 1971}, {5020, 28783}, {6374, 6394}, {22341, 37694}, {36608, 38283}

X(40800) = isogonal conjugate of X(3168)
X(40800) = isotomic conjugate of the polar conjugate of X(1988)
X(40800) = X(2)-cross conjugate of X(3)
X(40800) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3168}, {19, 3164}, {92, 32445}, {158, 6638}
X(40800) = cevapoint of X(i) and X(j) for these (i,j): {3, 38283}, {6, 31382}
X(40800) = trilinear pole of line {22089, 32320}
X(40800) = barycentric product X(69)*X(1988)
X(40800) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3164}, {6, 3168}, {184, 32445}, {577, 6638}, {1988, 4}, {14533, 26887}

X(40801) = ISOGONAL CONJUGATE OF X(6776)

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

X(40801) lies on the hyperbolas {{A,B,C,X(2),X(3)}} and {{A,B,C,X(4),X(25)}}, the cubic K1179, and these lines: {2, 6524}, {3, 232}, {4, 325}, {5, 14376}, {6, 2967}, {22, 97}, {24, 28724}, {25, 394}, {98, 9307}, {114, 37074}, {132, 37071}, {250, 37930}, {262, 458}, {264, 13860}, {378, 5968}, {381, 34897}, {427, 14593}, {523, 9756}, {648, 9755}, {842, 36176}, {1073, 5020}, {1214, 19544}, {1297, 15355}, {1351, 10311}, {1485, 19165}, {1593, 9737}, {1824, 3998}, {1843, 14486}, {1885, 15591}, {1995, 14919}, {2333, 3682}, {3092, 9732}, {3093, 9733}, {3172, 13335}, {3199, 30270}, {3515, 5171}, {3563, 35575}, {5094, 14356}, {5191, 11181}, {5481, 22240}, {5999, 33971}, {6644, 18876}, {6677, 15312}, {7390, 8813}, {7485, 31626}, {7866, 39604}, {8430, 14687}, {9734, 11410}, {10519, 37187}, {10607, 39803}, {11174, 37124}, {11472, 30209}, {14576, 37485}, {17907, 37450}, {23350, 35911}, {34129, 34841}, {34854, 37344}, {37581, 40152}

X(40801) = isogonal conjugate of X(6776)
X(40801) = isogonal conjugate of the anticomplement of X(1352)
X(40801) = isogonal conjugate of the complement of X(5921)
X(40801) = X(i)-cross conjugate of X(j) for these (i,j): {3148, 6}, {3288, 648}, {12294, 4}
X(40801) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6776}, {3, 4008}, {19, 37188}, {63, 7735}, {293, 1513}, {326, 6620}, {656, 35278}, {4575, 30735}
X(40801) = cevapoint of X(i) and X(j) for these (i,j): {3, 1351}, {458, 9308}, {1843, 14096}
X(40801) = trilinear pole of line {520, 2451}
X(40801) = polar conjugate of X(40814)
X(40801) = pole wrt polar circle of trilinear polar of X(40814) (line X(1513)X(30735))
X(40801) = barycentric product X(2501)*X(35575)
X(40801) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 37188}, {6, 6776}, {19, 4008}, {25, 7735}, {112, 35278}, {232, 1513}, {2207, 6620}, {2501, 30735}, {10311, 9755}, {35575, 4563}

X(40802) = ISOGONAL CONJUGATE OF X(7735)

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

X(40802) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic 1179, and these lines: {2, 4176}, {3, 1976}, {6, 3964}, {25, 394}, {42, 611}, {69, 297}, {76, 16081}, {111, 15066}, {141, 2165}, {251, 1993}, {263, 1351}, {287, 1975}, {323, 1383}, {343, 13854}, {458, 18906}, {524, 34288}, {525, 2395}, {599, 1989}, {941, 15988}, {1583, 8576}, {1584, 8577}, {1691, 35302}, {1915, 37672}, {1994, 39955}, {2963, 3763}, {3108, 5422}, {3981, 8770}, {5017, 18898}, {5024, 11166}, {6660, 33878}, {8675, 9178}, {8749, 35910}, {8791, 37638}, {8794, 34384}, {8882, 20806}, {9605, 11175}, {10601, 39951}, {10602, 16098}, {15595, 39645}

X(40802) = isogonal conjugate of X(7735)
X(40802) = isotomic conjugate of X(40814)
X(40802) = isogonal conjugate of the anticomplement of X(7778)
X(40802) = isogonal conjugate of the complement of X(37668)
X(40802) = X(5028)-cross conjugate of X(6)
X(40802) = cevapoint of X(i) and X(j) for these (i,j): {6, 1350}, {69, 18906}, {183, 1975}
X(40802) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7735}, {6, 4008}, {19, 6776}, {63, 6620}, {163, 30735}, {661, 35278}, {1096, 37188}, {1513, 1910}, {2186, 9755}
X(40802) = trilinear pole of line {512, 684}
X(40802) = barycentric product X(523)*X(35575)
X(40802) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4008}, {3, 6776}, {6, 7735}, {25, 6620}, {110, 35278}, {182, 9755}, {394, 37188}, {511, 1513}, {523, 30735}, {1350, 7710}, {1351, 9752}, {5921, 9747}, {35575, 99}

X(40803) = ISOGONAL CONJUGATE OF X(9755)

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

X(40803) lies on the cubic K1179 and these lines: {262, 1007}, {263, 1351}, {327, 40330}, {1352, 23878}, {13354, 14252}, {14927, 39682}, {26714, 35387}

X(40803) = isogonal conjugate of X(9755)
X(40803) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9755}, {182, 4008}
X(40803) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9755}, {263, 7735}, {2186, 4008}, {26714, 35278}

X(40804) = ISOGONAL CONJUGATE OF X(32545)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(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(40804) lies on the cubics K12, K630, K1179, and these lines:" {3, 1625}, {5, 525}, {76, 39604}, {114, 9289}, {182, 15407}, {249, 1092}, {827, 1298}, {1972, 15595}, {6663, 36952}, {9306, 34157}, {23098, 36212}

X(40804) = isogonal conjugate of X(32545)
X(40804) = X(14941)-Ceva conjugate of X(511)
X(40804) = X(2967)-cross conjugate of X(511)
X(40804) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32545}, {98, 1955}, {401, 1910}, {1821, 1971}, {6130, 36084}
X(40804) = crossdifference of every pair of points on line {1971, 6130}
X(40804) = trilinear product X(i)*X(j) for these {i,j}: {240, 14941}, {511, 1956}, {1755, 1972}, {1959, 1987}
X(40804) = barycentric product X(i)*X(j) for these {i,j}: {297, 14941}, {325, 1987}, {511, 1972}, {1956, 1959}
X(40804) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 32545}, {237, 1971}, {297, 16089}, {511, 401}, {1755, 1955}, {1956, 1821}, {1972, 290}, {1987, 98}, {3569, 6130}, {14941, 287}
X(40804) = {X(1987),X(14941)}-harmonic conjugate of X(39683)

X(40805) = X(2)X(6)∩X(3)X(1625)

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

X(40805) lies on the cubic K1179 and these lines: {2, 6}, {3, 1625}, {20, 38297}, {39, 11793}, {154, 160}, {216, 3819}, {217, 631}, {232, 3917}, {327, 458}, {376, 3331}, {571, 1915}, {577, 1971}, {1092, 1970}, {2211, 10519}, {2979, 15355}, {3199, 15644}, {3224, 32654}, {3269, 11459}, {5063, 9225}, {5651, 10311}, {5891, 14961}, {5907, 22401}, {6090, 6786}, {7998, 22240}, {7999, 39575}, {9308, 16089}, {9418, 20885}, {9419, 22712}, {11444, 22416}, {15068, 39849}, {15905, 38283}, {28407, 34850}, {33786, 34870}

X(40805) = isogonal conjugate of X(40815)
X(40805) = crosspoint of X(34537) and X(35575)
X(40805) = crossdifference of every pair of points on line {512, 6130}
X(40805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3289, 6}, {6, 21001, 230}, {577, 9306, 1971}, {3051, 7736, 6}

X(40806) = X(2)X(36897)∩X(3)X(3224)

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

X(40806) lies on the cubic K1179 and these lines: {2, 36897}, {3, 3224}

X(40806) = isogonal conjugate of X(40816)

X(40807) = X(2)X(6331)∩X(3)X(3164)

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

X(40807) lies on the cubic K1179 and these lines: {2, 6331}, {3, 3164}, {6, 194}, {877, 3552}, {5999, 33971}, {6776, 39355}, {11003, 11794}

X(40807) = isogonal conjugate of X(40817)
X(40807) = anticomplement of X(40822)
X(40807) = X(35575)-anticomplementary conjugate of X(21305)

X(40808) = X(2)X(34208)∩X(182)X(3224)

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

X(40808) lies on the cubic K1179 and these lines: {2, 34208}, {182, 3224}

X(40808) = isogonal conjugate of X(40818)

X(40809) = X(2)X(34208)∩X(3)X(2971)

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

X(40809) lies on the cubics X297 and K1179, and on these lines: {2, 34208}, {3, 2971}, {5, 2996}, {6, 1196}, {183, 35136}, {381, 5203}, {6340, 8797}, {11479, 14489}

X(40809) = isogonal conjugate of X(40819)
X(40809) = trilinear product X(i)*X(j) for these {i,j}: {1007, 38252}, {1351, 8769}
X(40809) = X(1707)-isoconjugate of X(7612)
X(40809) = barycentric product X(i)*X(j) for these {i,j}: {1007, 8770}, {1351, 2996}, {6391, 37174}, {10008, 14248}
X(40809) = barycentric quotient X(i)/X(j) for these {i,j}: {1351, 193}, {8770, 7612}

X(40810) = X(2)X(36897)∩X(3)X(3493)

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

Let AB, AC, BC, BA, CA, CB be as in the construction of the conic described in ADGEOM #4589 (Tran Quang Hung, Randy Hutson, 5/26/2018) for P,Q = PU(1). Let A' be the intersection of the tangents to the conic at AB and AC. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(40810). (Randy Hutson, January 22, 2021)

X(40810) lies on the cubics X1012 and K1179, and on these lines: {2, 36897}, {3, 3493}, {6, 694}, {69, 18829}, {114, 262}, {141, 523}, {160, 17938}, {182, 2065}, {250, 1974}, {264, 5117}, {446, 511}, {805, 842}, {1581, 7146}, {3425, 17970}, {3613, 6665}, {3818, 38947}, {5968, 6786}, {9307, 15595}, {9513, 39291}, {14970, 17500}

X(40810) = isogonal conjugate of X(40820)
X(40810) = isotomic conjugate of X(14382)
X(40810) = isotomic conjugate of the isogonal conjugate of X(14251)
X(40810) = X(699)-complementary conjugate of X(16609)
X(40810) = X(694)-Ceva conjugate of X(511)
X(40810) = X(i)-cross conjugate of X(j) for these (i,j): {684, 18829}, {2679, 3569}, {36790, 511}
X(40810) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14382}, {98, 1580}, {290, 1933}, {293, 419}, {385, 1910}, {804, 36084}, {1691, 1821}, {1926, 14601}, {1966, 1976}, {5027, 36036}, {24284, 36104}
X(40810) = cevapoint of X(2679) and X(3569)
X(40810) = crosssum of X(i) and X(j) for these (i,j): {6, 32540}, {385, 4027}
X(40810) = crossdifference of every pair of points on line {804, 1691}
X(40810) = trilinear product X(i)*X(j) for these {i,j}: {75, 14251}, {237, 1934}, {240, 36214}, {325, 1967}, {511, 1581}, {694, 1959}, {1755, 1916}, {3569, 37134}, {9417, 18896}, {17970, 40703}, {23996, 36897}
X(40810) = barycentric product X(i)*X(j) for these {i,j}: {76, 14251}, {232, 40708}, {237, 18896}, {297, 36214}, {325, 694}, {511, 1916}, {805, 2799}, {882, 2396}, {1581, 1959}, {1755, 1934}, {3569, 18829}, {6393, 17980}, {32458, 34238}, {36790, 36897}
X(40810) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14382}, {232, 419}, {237, 1691}, {297, 17984}, {325, 3978}, {511, 385}, {684, 24284}, {694, 98}, {805, 2966}, {881, 2422}, {882, 2395}, {1581, 1821}, {1755, 1580}, {1916, 290}, {1959, 1966}, {1967, 1910}, {2396, 880}, {2421, 17941}, {2491, 5027}, {2679, 35078}, {2799, 14295}, {2967, 39931}, {3569, 804}, {8789, 14601}, {9155, 5026}, {9417, 1933}, {9418, 14602}, {9468, 1976}, {11672, 36213}, {14251, 6}, {17938, 2715}, {17970, 248}, {17980, 6531}, {18872, 5967}, {18896, 18024}, {36212, 12215}, {36213, 4027}, {36214, 287}, {36790, 5976}, {36897, 34536}, {37134, 36036}, {39092, 39941}

X(40811) = X(2)X(4176)∩X(3)X(9292)

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

X(40811) lies on the cubic K1179 and these lines: {2, 4176}, {3, 9292}, {695, 5013}, {699, 35575}, {3224, 34870}

X(40811) = isogonal conjugate of X(40821)
X(40811) = X(i)-isoconjugate of X(j) for these (i,j): {3223, 7735}, {3224, 4008}
X(40811) = barycentric product X(23301)*X(35575)
X(40811) = barycentric quotient X(i)/X(j) for these {i,j}: {1613, 7735}, {1740, 4008}, {11325, 6620}, {20794, 6776}, {23301, 30735}, {35575, 3222}

X(40812) = X(6)X(2987)∩X(3148)X(3563)

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

X(40812) lies on the cubic K1179 and these lines: {6, 2987}, {3148, 3563}, {9306, 34157}, {14253, 36212}

X(40813) = X(2)X(34403)∩X(3)X(64)

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

X(40813) lies on the cubic K1179 and these lines: {2, 34403}, {3, 64}, {6, 15394}, {141, 253}, {459, 18840}, {1352, 5922}, {3343, 17825}, {31942, 33537}

X(40813) = X(610)-isoconjugate of X(3424)
X(40813) = barycentric product X(i)*X(j) for these {i,j}: {64, 37668}, {253, 1350}, {10002, 15394}, {19611, 23052}
X(40813) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 3424}, {1350, 20}, {10002, 14249}, {23052, 1895}, {37668, 14615}

X(40814) = X(2)X(39)∩X(4)X(51)

Barycentrics    b^2*c^2*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) : :
Barycentrics    a^4 - SB*SC : :
Barycentrics    (sec A) (sin A tan ω - cos B cos C) : :

Let A'B'C' be the Artzt triangle. Let A" be the perspector of conic {{A,B,C,B',C'}}, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(40814). (Randy Hutson, January 22, 2021)

X(40814) lies on the cubic K790 and these lines: {2, 39}, {4, 51}, {6, 264}, {22, 12203}, {25, 39646}, {32, 401}, {83, 5392}, {94, 598}, {98, 3148}, {125, 5117}, {184, 419}, {193, 14615}, {237, 11257}, {262, 37988}, {297, 3981}, {311, 3618}, {315, 6515}, {316, 37644}, {327, 3815}, {343, 6656}, {394, 7754}, {441, 5305}, {460, 11245}, {511, 37190}, {578, 37124}, {671, 34289}, {800, 3164}, {1232, 3619}, {1235, 11427}, {1236, 37645}, {1249, 21447}, {1316, 14265}, {1975, 37344}, {1992, 3260}, {1993, 7760}, {1994, 7894}, {1995, 38664}, {2782, 11328}, {2996, 37874}, {3053, 35941}, {3060, 14957}, {3095, 21531}, {3096, 37636}, {3186, 6467}, {3580, 7790}, {3596, 26665}, {3673, 26001}, {3710, 4385}, {3917, 12251}, {4027, 33336}, {4054, 25935}, {5013, 37067}, {5222, 34387}, {5304, 30737}, {5749, 34388}, {5943, 6248}, {6376, 25007}, {6620, 6776}, {6660, 14880}, {7388, 11090}, {7389, 11091}, {7735, 37188}, {7770, 10601}, {7782, 35296}, {7850, 37779}, {7878, 34545}, {8573, 20477}, {9747, 9755}, {9786, 37200}, {10063, 40790}, {10349, 33301}, {11188, 25051}, {11331, 26958}, {11333, 35275}, {11438, 35474}, {13335, 35926}, {14096, 22712}, {15988, 34283}, {17862, 26531}, {18033, 26016}, {19768, 25875}, {26913, 33314}, {37778, 40138}

X(40814) = isogonal conjugate of X(40799)
X(40814) = isotomic conjugate of X(40802)
X(40814) = polar conjugate of X(40801)
X(40814) = pole wrt polar circle of trilinear polar of X(40801) (line X(520)X(2451))
X(40814) = isotomic conjugate of the isogonal conjugate of X(7735)
X(40814) = polar conjugate of the isogonal conjugate of X(6776)
X(40814) = X(264)-Ceva conjugate of X(9747)
X(40814) = X(798)-isoconjugate of X(35575)
X(40814) = cevapoint of X(6776) and X(7735)
X(40814) = crosspoint of X(i) and X(j) for these (i,j): {4, 19222}, {262, 9307}
X(40814) = crosssum of X(i) and X(j) for these (i,j): {3, 11328}, {182, 9306}
X(40814) = trilinear pole of line {1513, 30735}
X(40814) = crossdifference of every pair of points on line {669, 32320}
X(40814) = trilinear product X(i)*X(j) for these {i,j}: {2, 4008}, {75, 7735}, {92, 6776}, {158, 37188}, {304, 6620}, {662, 30735}, {1513, 1821}, {1577, 35278}
X(40814) = barycentric product X(i)*X(j) for these {i,j}: {75, 4008}, {76, 7735}, {99, 30735}, {264, 6776}, {290, 1513}, {305, 6620}, {327, 9755}, {850, 35278}, {2052, 37188}
X(40814) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 35575}, {1513, 511}, {4008, 1}, {6620, 25}, {6776, 3}, {7710, 1350}, {7735, 6}, {9747, 5921}, {9752, 1351}, {9755, 182}, {30735, 523}, {35278, 110}, {37188, 394}
X(40814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 194, 36212}, {4, 3168, 34854}, {76, 3978, 305}, {237, 39906, 11257}, {5254, 13567, 297}, {6392, 26164, 76}

X(40815) = X(6)X(401)∩X(25)X(3168)

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

X(40815) lies on the conic {{A,B,C,X(2)X(6)}}, the cubic K790, and these lines: {6, 401}, {25, 3168}, {194, 2987}, {263, 6776}, {1976, 32545}

X(40815) = isogonal conjugate of X(40805)
X(40815) = trilinear pole of line {512, 6130}

X(40816) = ISOGONAL CONJUGATE OF X(40806)

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

X(40816) lies on the cubic K790 and this line: {194, 36213}

X(40816) = isogonal conjugate of X(40806)

X(40817) = X(194)X(3289)∩X(1613)X(6638)

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

X(40817) lies on the cubic K790 and these lines: {194, 3289}, {1613, 6638}, {2211, 3168}, {9418, 11325}

X(40817) = isogonal conjugate of X(40807)

X(40818) = ISOGONAL CONJUGATE OF X(40818)

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

X(40818) lies on the cubic K790 and this line: {194, 3167}

X(40818) = isogonal conjugate of X(40808)

X(40819) = X(2)X(3167)∩X(6)X(34208)

Barycentrics    (3*a^2 - b^2 - c^2)*(3*a^4 - 2*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + c^4)*(3*a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 4*b^2*c^2 + 3*c^4) : :
X(40819) 4 X[6] - X[34208]

X(40819) lies on the cubics K295 and K790, and on these lines: {2, 3167}, {6, 34208}

X(40819) = isogonal conjugate of X(40809)
X(40819) = X(i)-isoconjugate of X(j) for these (i,j): {1007, 38252}, {1351, 8769}
X(40819) = trilinear product X(1707)*X(7612)
X(40819) = barycentric product X(193)*X(7612)
X(40819) = barycentric quotient X(i)/X(j) for these {i,j}: {193, 1007}, {3053, 1351}, {6337, 10008}, {6353, 37174}, {7612, 2996}

X(40820) = X(2)X(98)∩X(6)X(36897)

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

X(40820) lies on the cubics K693, K79, and K1013, and on these lines: {2, 98}, {6, 36897}, {25, 685}, {32, 8870}, {51, 13137}, {248, 19222}, {251, 2395}, {262, 2065}, {290, 3114}, {305, 31614}, {419, 14602}, {1215, 39043}, {1316, 14265}, {1403, 36065}, {1501, 2715}, {1580, 16609}, {1910, 2344}, {2966, 14614}, {3117, 15391}, {3167, 17932}, {3407, 34238}, {3978, 14382}, {5286, 8861}, {5306, 34369}, {5943, 15630}, {7735, 36899}, {8623, 20026}, {11328, 32540}, {35906, 36874}

X(40820) = isogonal conjugate of X(40810)
X(40820) = isogonal conjugate of the isotomic conjugate of X(14382)
X(40820) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 32545}, {685, 5027}
X(40820) = X(i)-cross conjugate of X(j) for these (i,j): {6, 32544}, {385, 98}, {12829, 385}
X(40820) = X(i)-isoconjugate of X(j) for these (i,j): {75, 14251}, {237, 1934}, {240, 36214}, {325, 1967}, {511, 1581}, {694, 1959}, {1755, 1916}, {3569, 37134}, {9417, 18896}, {17970, 40703}, {23996, 36897}
X(40820) = cevapoint of X(i) and X(j) for these (i,j): {6, 32540}, {385, 4027}
X(40820) = crosspoint of X(6) and X(32542)
X(40820) = crosssum of X(2679) and X(3569)
X(40820) = trilinear pole of line {804, 1691}
X(40820) = trilinear product X(i)*X(j) for these {i,j}: {31, 14382}, {98, 1580}, {290, 1933}, {293, 419}, {385, 1910}, {804, 36084}, {1691, 1821}, {1926, 14601}, {1966, 1976}, {5027, 36036}, {24284, 36104}
X(40820) = barycentric product X(i)*X(j) for these {i,j}: {6, 14382}, {98, 385}, {248, 17984}, {287, 419}, {290, 1691}, {685, 24284}, {804, 2966}, {880, 2422}, {1580, 1821}, {1910, 1966}, {1976, 3978}, {2395, 17941}, {2715, 14295}, {4027, 36897}, {5026, 9154}, {6531, 12215}, {12829, 40428}, {14601, 14603}, {14602, 18024}, {34536, 36213}
X(40820) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 14251}, {98, 1916}, {248, 36214}, {287, 40708}, {290, 18896}, {385, 325}, {419, 297}, {804, 2799}, {1580, 1959}, {1691, 511}, {1821, 1934}, {1910, 1581}, {1933, 1755}, {1976, 694}, {2422, 882}, {2715, 805}, {2966, 18829}, {4027, 5976}, {5027, 3569}, {5976, 32458}, {12215, 6393}, {12829, 114}, {14382, 76}, {14600, 17970}, {14601, 9468}, {14602, 237}, {17941, 2396}, {18902, 9418}, {24284, 6333}, {36084, 37134}, {36213, 36790}
X(40820) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5967, 34761, 14355}, {34396, 34536, 32545}
X(40820) =

X(40821) = X(6)X(194)∩X(3504)X(5020)

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

X(40821) lies on the cubic K790 and these lines: {6, 194}, {3504, 5020}

X(40821) = isogonal conjugate of X(40811)
X(40821) = trilinear product X(i)*X(j) for these {i,j}: {3223, 7735}, {3224, 4008}
X(40821) = barycentric product X(i)*X(j) for these {i,j}: {2998, 7735}, {3223, 4008}
X(40821) = barycentric quotient X(i)/X(j) for these {i,j}: {4008, 17149}, {6620, 3186}, {7735, 194}

X(40822) = X(6)X(6331)∩X(3)X(17984)

Barycentrics    b^4*c^4*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) : :
Barycentrics    b^2 c^2 (a^4 - SB*SC) : :
Barycentrics    (csc A) (csc 2A) (sin A tan ω - cos B cos C) : :
Barycentrics    A'-power of circumcircle : :, where A'B'C' is the 7th Brocard triangle

X(40822) lies on these lines: {2, 6331}, {3, 17984}, {5, 264}, {76, 141}, {182, 14382}, {276, 14376}, {290, 1352}, {308, 2165}, {327, 24206}, {7876, 26166}, {13862, 30737}, {16089, 33971}

X(40822) = isogonal conjugate of X(40823)
X(40822) = isotomic conjugate of X(40799)
X(40822) = complement of X(40807)
X(40822) = X(1924)-isoconjugate of X(35575)
X(40822) = barycentric product X(i)*X(j) for these {i,j}: {561, 4008}, {670, 30735}, {1502, 7735}, {1513, 18024}, {6620, 40050}, {6776, 18022}, {18027, 37188}
X(40822) = barycentric quotient X(i)/X(j) for these {i,j}: {670, 35575}, {1513, 237}, {4008, 31}, {6620, 1974}, {6776, 184}, {7735, 32}, {9755, 34396}, {30735, 512}, {35278, 1576}, {37188, 577}
X(40822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 6374, 6393}, {1502, 14603, 40050}

X(40823) = X(6)X(2967)∩X(32)X(1092)

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

X(40823) lies on these lines: {6, 2967}, {32, 1092}, {54, 8743}, {184, 2211}, {251, 1993}, {699, 35575}, {1501, 23606}, {3407, 7774}, {9418, 14585}, {9419, 14575}

X(40823) = isogonal conjugate of X(40822)
X(40823) = trilinear product X(1924)*X(35575)
X(40823) = X(i)-isoconjugate of X(j) for these (i,j): {76, 4008}, {561, 7735}, {799, 30735}, {1969, 6776}, {6620, 40364}, {20948, 35278}
X(40823) = barycentric product X(669)*X(35575)
X(40823) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 4008}, {669, 30735}, {1501, 7735}, {9418, 1513}, {14574, 35278}, {14575, 6776}, {14585, 37188}, {35575, 4609}

X(40824) = ISOTOMIC CONJUGATE OF X(7735)

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

X(40824) lies on the Kiepert circumhyperbola and these lines: {2, 4176}, {4, 325}, {69, 98}, {76, 14064}, {83, 7736}, {94, 9464}, {99, 7710}, {183, 7612}, {262, 1007}, {275, 34254}, {305, 2052}, {384, 5395}, {598, 7799}, {599, 11172}, {631, 3406}, {671, 16041}, {2996, 5025}, {3090, 3399}, {3266, 34289}, {3407, 7774}, {3424, 5921}, {5207, 14458}, {5392, 8024}, {5466, 30474}, {5485, 32836}, {5989, 39874}, {6337, 9744}, {7607, 34229}, {7608, 34803}, {7868, 18840}, {7892, 32835}, {7897, 11606}, {7901, 32834}, {8781, 37690}, {8889, 37892}, {10153, 23055}, {10155, 37647}, {10159, 32832}, {11163, 14039}, {11167, 21356}, {11174, 14069}, {13862, 14484}, {14035, 18845}, {14046, 32869}, {14047, 32870}, {14063, 32840}, {14067, 32871}, {16277, 40123}, {18844, 32876}, {32532, 32896}, {32838, 32953}, {32839, 32952}, {32879, 32996}, {32880, 33287}, {32970, 39095}

X(40824) = isogonal conjugate of X(40825)
X(40824) = isotomic conjugate of X(7735)
X(40824) = polar conjugate of X(6620)
X(40824) = isotomic conjugate of the anticomplement of X(7778)
X(40824) = isotomic conjugate of the complement of X(37668)
X(40824) = X(7778)-cross conjugate of X(2)
X(40824) = cevapoint of X(i) and X(j) for these (i,j): {2, 37668}, {3926, 10008}
X(40824) = trilinear pole of line {523, 4143}
X(40824) = X(i)-isoconjugate of X(j) for these (i,j): {31, 7735}, {32, 4008}, {48, 6620}, {798, 35278}, {1973, 6776}, {3402, 9755}
X(40824) = trilinear product X(1577)*X(35575)
X(40824) = barycentric product X(850)*X(35575)
X(40824) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7735}, {4, 6620}, {69, 6776}, {75, 4008}, {99, 35278}, {183, 9755}, {325, 1513}, {850, 30735}, {1007, 9752}, {3926, 37188}, {35575, 110}, {37668, 7710}

X(40825) = MIDPOINT OF X(6) AND X(3053)

Barycentrics    a^2*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) : :
X(40825) = 5 X[3618] - X[32006], X[6776] + 3 X[9752]

X(40825) lies on these lines: {3, 6}, {25, 1501}, {69, 7807}, {81, 21485}, {115, 36990}, {141, 32954}, {154, 1196}, {172, 611}, {184, 1184}, {193, 6393}, {217, 19125}, {230, 1352}, {251, 5422}, {385, 39141}, {524, 11288}, {597, 11287}, {613, 1914}, {1003, 18906}, {1194, 3796}, {1285, 22521}, {1353, 37459}, {1386, 1572}, {1428, 16502}, {1503, 3767}, {1513, 6776}, {1569, 36784}, {1611, 9306}, {1613, 3167}, {1627, 1993}, {1899, 22135}, {1915, 5020}, {1971, 34809}, {1974, 2207}, {1992, 35297}, {2211, 3172}, {2548, 3589}, {2715, 5941}, {2916, 9700}, {3051, 11402}, {3231, 6090}, {3291, 35259}, {3506, 8780}, {3564, 37466}, {3618, 6656}, {3763, 7749}, {3787, 37672}, {3818, 13881}, {3830, 6034}, {3981, 9909}, {5012, 5359}, {5026, 22253}, {5182, 5976}, {5207, 7887}, {5286, 25406}, {5304, 37182}, {5306, 11179}, {5354, 11003}, {5480, 7737}, {5622, 38641}, {5921, 37689}, {6531, 33971}, {6800, 9465}, {6811, 39875}, {6813, 39876}, {7083, 14599}, {7485, 34945}, {7736, 37450}, {7745, 14561}, {7746, 10516}, {7754, 12215}, {8363, 31404}, {8667, 14994}, {8743, 19128}, {9300, 38064}, {10312, 39588}, {11286, 24256}, {11360, 16285}, {11898, 15993}, {12177, 12829}, {13860, 39095}, {13910, 31411}, {14537, 38072}, {14567, 26864}, {14585, 19459}, {14605, 15303}, {14901, 16010}, {17349, 21993}, {18583, 18907}, {19153, 21177}, {24206, 37637}, {32738, 32740}, {35302, 36790}, {36696, 38651}, {38642, 39656}

X(40825) = midpoint of X(6) and X(3053)
X(40825) = isogonal conjugate of X(40824)
X(40825) = isogonal conjugate of the isotomic conjugate of X(7735)
X(40825) = isogonal conjugate of the polar conjugate of X(6620)
X(40825) = X(1577)-isoconjugate of X(35575)
X(40825) = crosspoint of X(6620) and X(7735)
X(40825) = crosssum of X(i) and X(j) for these (i,j): {2, 37668}, {3926, 10008}
X(40825) = crossdifference of every pair of points on line {523, 4143}
X(40825) = trilinear product X(i)*X(j) for these {i,j}: {31, 7735}, {32, 4008}, {48, 6620}, {798, 35278}, {1973, 6776}, {3402, 9755}
X(40825) = barycentric product X(i)*X(j) for these {i,j}: {3, 6620}, {6, 7735}, {25, 6776}, {31, 4008}, {263, 9755}, {512, 35278}, {1513, 1976}, {1576, 30735}, {2207, 37188}
X(40825) = barycentric quotient X(i)/X(j) for these {i,j}: {1576, 35575}, {4008, 561}, {6620, 264}, {6776, 305}, {7735, 76}, {9755, 20023}, {35278, 670}
X(40825) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1350, 5028}, {6, 1384, 11173}, {6, 1691, 3}, {6, 5017, 1351}, {6, 5085, 39}, {6, 11477, 1570}, {6, 13330, 5093}, {6, 31884, 10542}, {32, 1692, 6}, {32, 39764, 5052}, {39, 5033, 5085}, {187, 5028, 1350}, {193, 16925, 6393}, {575, 5039, 6}, {1342, 1343, 13356}, {1351, 1384, 5017}, {1351, 5017, 11173}, {1687, 1688, 13355}, {1692, 5052, 39764}, {2021, 5052, 3094}, {2024, 35432, 3095}, {3172, 19118, 2211}, {5007, 5034, 6}, {5050, 30435, 6}, {5052, 39764, 6}, {5058, 5062, 7772}, {6423, 6424, 3}, {12050, 12051, 32}, {19145, 19146, 3}

X(40826) = ISOTOMIC CONJUGATE OF X(574)

Barycentrics    b^2*c^2*(-2*a^2 + b^2 - 2*c^2)*(2*a^2 + 2*b^2 - c^2) : :
Barycentrics    A'-power of circumcircle : : , where A'B'C' = 2nd Brocard triangle

X(40826) lies on these lines: {2, 18023}, {76, 524}, {264, 468}, {290, 5967}, {308, 1383}, {313, 4062}, {327, 3260}, {599, 8785}, {892, 8542}, {1502, 3266}, {2367, 11636}, {3734, 4590}, {5486, 11185}, {7771, 11594}, {7835, 36953}, {14295, 34763}, {18027, 37778}, {20573, 40822}

X(40826) = isotomic conjugate of X(574)
X(40826) = polar conjugate of X(8541)
X(40826) = isotomic conjugate of the complement of X(11185)
X(40826) = isotomic conjugate of the isogonal conjugate of X(598)
X(40826) = anticomplement of crosspoint of X(2) and X(574)
X(40826) = anticomplement of crosssum of X(6) and X(598)
X(40826) = X(i)-cross conjugate of X(j) for these (i,j): {23297, 598}, {26235, 76}
X(40826) = X(i)-isoconjugate of X(j) for these (i,j): {31, 574}, {32, 36263}, {48, 8541}, {163, 17414}, {560, 599}, {798, 9145}, {1917, 9464}, {1919, 3908}, {1923, 10130}, {1924, 9146}, {5094, 9247}, {8288, 23995}
X(40826) = cevapoint of X(i) and X(j) for these (i,j): {2, 11185}, {76, 11059}
X(40826) = trilinear pole of line {690, 850}
X(40826) = barycentric product X(i)*X(j) for these {i,j}: {76, 598}, {308, 23297}, {316, 10512}, {670, 8599}, {850, 35138}, {1383, 1502}, {3266, 18818}, {10511, 40074}, {30489, 40016}
X(40826) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 574}, {4, 8541}, {75, 36263}, {76, 599}, {99, 9145}, {264, 5094}, {308, 10130}, {316, 10510}, {338, 8288}, {523, 17414}, {598, 6}, {668, 3908}, {670, 9146}, {850, 3906}, {892, 32583}, {1236, 19510}, {1383, 32}, {1502, 9464}, {3260, 13857}, {3264, 4141}, {3266, 39785}, {8599, 512}, {8785, 8586}, {10511, 3455}, {10512, 67}, {11054, 9872}, {11059, 11165}, {11185, 8542}, {11636, 1576}, {18818, 111}, {20380, 39689}, {23287, 351}, {23297, 39}, {26235, 15810}, {30489, 3051}, {30491, 3049}, {35138, 110}

X(40827) = ISOTOMIC CONJUGATE OF X(2092)

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

X(40827) lies on these lines: {65, 314}, {76, 940}, {86, 313}, {264, 4185}, {274, 1920}, {290, 1798}, {308, 1169}, {310, 349}, {670, 20911}, {1502, 34284}, {2368, 8707}, {5209, 37607}, {18896, 40017}, {19701, 30022}, {30940, 40409}

X(40827) = isotomic conjugate of X(2092)
X(40827) = isotomic conjugate of the complement of X(314)
X(40827) = isotomic conjugate of the isogonal conjugate of X(14534)
X(40827) = X(i)-cross conjugate of X(j) for these (i,j): {2, 31643}, {693, 670}, {17496, 99}, {37759, 14616}
X(40827) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3725}, {31, 2092}, {32, 2292}, {42, 2300}, {213, 1193}, {228, 2354}, {429, 9247}, {560, 1211}, {669, 3882}, {872, 40153}, {1228, 1917}, {1397, 21033}, {1400, 20967}, {1402, 2269}, {1501, 18697}, {1829, 2200}, {1918, 3666}, {1923, 27067}, {1973, 22076}, {2205, 4357}, {2206, 21810}, {2333, 22345}
X(40827) = cevapoint of X(i) and X(j) for these (i,j): {2, 314}, {75, 27792}, {76, 274}, {86, 14829}, {1240, 30710}
X(40827) = trilinear pole of line {850, 4374}
X(40827) = barycentric product X(i)*X(j) for these {i,j}: {76, 14534}, {86, 1240}, {274, 30710}, {310, 1220}, {314, 31643}, {561, 2363}, {670, 4581}, {961, 40072}, {1169, 1502}, {1798, 18022}, {2298, 6385}, {6331, 15420}
X(40827) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3725}, {2, 2092}, {21, 20967}, {27, 2354}, {69, 22076}, {75, 2292}, {76, 1211}, {81, 2300}, {86, 1193}, {261, 4267}, {264, 429}, {274, 3666}, {286, 1829}, {308, 27067}, {310, 4357}, {312, 21033}, {313, 20653}, {314, 960}, {321, 21810}, {333, 2269}, {561, 18697}, {799, 3882}, {961, 1402}, {1169, 32}, {1220, 42}, {1240, 10}, {1444, 22345}, {1502, 1228}, {1509, 40153}, {1791, 228}, {1798, 184}, {1812, 22074}, {1920, 27697}, {2298, 213}, {2359, 2200}, {2363, 31}, {3261, 21124}, {3596, 3704}, {4581, 512}, {6385, 20911}, {6648, 4559}, {7192, 6371}, {8033, 28369}, {8707, 4557}, {14534, 6}, {14624, 1500}, {15420, 647}, {17206, 22097}, {18155, 17420}, {18697, 6042}, {28660, 3687}, {30710, 37}, {31643, 65}, {40452, 3185}

X(40828) = ISOTOMIC CONJUGATE OF X(5019)

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

X(40828) lies on these lines: {12, 3596}, {76, 1211}, {264, 429}, {290, 34259}, {308, 941}, {313, 20653}, {349, 561}, {931, 2367}, {1228, 1502}, {4417, 28660}, {5224, 34265}, {5331, 37678}, {5718, 30022}, {6376, 31359}, {18140, 37870}, {18152, 40011}, {27801, 40363}

X(40828) = isotomic conjugate of X(5019)
X(40828) = isotomic conjugate of the isogonal conjugate of X(34258)
X(40828) = X(i)-isoconjugate of X(j) for these (i,j): {31, 5019}, {32, 1468}, {163, 8639}, {560, 940}, {1397, 2268}, {1501, 10436}, {1917, 34284}, {4185, 9247}, {5307, 14575}
X(40828) = cevapoint of X(4417) and X(5224)
X(40828) = barycentric product X(i)*X(j) for these {i,j}: {76, 34258}, {561, 31359}, {941, 1502}, {959, 40363}, {1928, 2258}, {18022, 34259}, {27801, 37870}
X(40828) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5019}, {75, 1468}, {76, 940}, {264, 4185}, {312, 2268}, {523, 8639}, {561, 10436}, {850, 8672}, {931, 1576}, {941, 32}, {959, 1397}, {1502, 34284}, {1969, 5307}, {2258, 560}, {3596, 958}, {5224, 34281}, {5331, 2206}, {27801, 31993}, {28659, 11679}, {31359, 31}, {32038, 1415}, {34258, 6}, {34259, 184}, {35519, 17418}, {37870, 1333}

X(40829) = ISOTOMIC CONJUGATE OF X(14537)

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

X(40829) lies on these lines: on lines {5094, 7788}, {11057, 11058}, {19601, 35138}

X(40829) = isotomic conjugate of X(14537)
X(40829) = isotomic conjugate of the anticomplement of X(40344)
X(40829) = isotomic conjugate of the complement of X(11057)
X(40829) = X(40344)-cross conjugate of X(2)
X(40829) = X(31)-isoconjugate of X(14537)
X(40829) = cevapoint of X(2) and X(11057)
X(40829) = trilinear pole of line {3906, 7799}
X(40829) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14537}, {11057, 19601}

X(40830) = ISOTOMIC CONJUGATE OF X(800)

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

X(40830) lies on these lines: {64, 14615}, {76, 801}, {264, 1105}, {308, 26166}, {1502, 32830}, {3926, 18027}

X(40830) = isotomic conjugate of X(800)
X(40830) = isotomic conjugate of the complement of X(14615)
X(40830) = isotomic conjugate of the isogonal conjugate of X(801)
X(40830) = X(3265)-cross conjugate of X(670)
X(40830) = X(i)-isoconjugate of X(j) for these (i,j): {31, 800}, {32, 774}, {185, 1973}, {235, 9247}, {560, 13567}, {798, 1624}, {820, 2207}, {1501, 17858}, {1918, 18603}, {1974, 6508}, {2179, 16035}
X(40830) = cevapoint of X(i) and X(j) for these (i,j): {2, 14615}, {76, 3926}
X(40830) = barycentric product X(i)*X(j) for these {i,j}: {76, 801}, {305, 1105}, {561, 775}
X(40830) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 800}, {69, 185}, {75, 774}, {76, 13567}, {95, 16035}, {99, 1624}, {264, 235}, {274, 18603}, {304, 6508}, {326, 820}, {561, 17858}, {775, 31}, {801, 6}, {821, 1096}, {1105, 25}, {3926, 6509}, {3964, 417}, {14615, 2883}, {34384, 19166}, {34386, 19180}

X(40831) = ISOTOMIC CONJUGATE OF X(1184)

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

X(40831) lies on the Kiepert circumhyperbola and these lines: {4, 305}, {83, 11324}, {99, 40125}, {1502, 2052}, {2996, 8024}, {3266, 5395}, {9464, 38259}, {11059, 18841}, {18840, 40022}, {40123, 40178}

X(40831) = isotomic conjugate of X(1184)
X(40831) = isotomic conjugate of the complement of X(40123)
X(40831) = X(14064)-cross conjugate of X(264)
X(40831) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1184}, {560, 5286}, {1973, 19459}, {2474, 34072}
X(40831) = cevapoint of X(i) and X(j) for these (i,j): {2, 40123}, {305, 40022}
X(40831) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1184}, {69, 19459}, {76, 5286}, {305, 7386}, {826, 2474}, {8817, 1460}, {30479, 7083}, {40022, 40179}, {40123, 40125}

X(40832) = ISOTOMIC CONJUGATE OF X(3003)

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

X(40832 lies on these lines: {74, 3260}, {76, 2986}, {99, 264}, {290, 1236}, {308, 14910}, {313, 4561}, {327, 11185}, {1502, 32833}, {2367, 10420}, {3926, 23962}, {6331, 7763}, {7799, 20573}, {14387, 15454}

X(40832) = isotomic conjugate of X(3003)
X(40832) = isotomic conjugate of the complement of X(3260)
X(40832) = isotomic conjugate of the isogonal conjugate of X(2986)
X(40832) = X(i)-cross conjugate of X(j) for these (i,j): {2, 40423}, {2394, 35139}, {3268, 670}, {15421, 18878}
X(40832) = cevapoint of X(i) and X(j) for these (i,j): {2, 3260}, {76, 7799}
X(40832) = trilinear pole of line {69, 850}
X(40832) = X(i)-isoconjugate of X(j) for these (i,j): {25, 2315}, {31, 3003}, {32, 1725}, {163, 21731}, {403, 9247}, {560, 3580}, {686, 32676}, {798, 15329}, {1918, 18609}, {1973, 13754}, {9406, 14264}
X(40832) = barycentric product X(i)*X(j) for these {i,j}: {76, 2986}, {305, 1300}, {561, 36053}, {670, 15328}, {687, 3267}, {850, 18878}, {1502, 14910}, {3260, 40423}, {5504, 18022}, {6331, 15421}, {7799, 40427}, {18879, 23962}
X(40832) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3003}, {63, 2315}, {69, 13754}, {75, 1725}, {76, 3580}, {99, 15329}, {264, 403}, {274, 18609}, {316, 12824}, {328, 39170}, {340, 1986}, {523, 21731}, {525, 686}, {687, 112}, {1236, 12827}, {1300, 25}, {1494, 14264}, {2986, 6}, {3260, 113}, {3267, 6334}, {5504, 184}, {6331, 16237}, {7799, 34834}, {10419, 40352}, {10420, 1576}, {14910, 32}, {15328, 512}, {15421, 647}, {15454, 1495}, {15470, 14270}, {18878, 110}, {18879, 23357}, {36053, 31}, {36114, 32676}, {38936, 34397}, {39375, 14583}, {39988, 5663}, {40388, 40354}, {40423, 74}, {40427, 1989}, {40705, 18781}

X(40833) = ISOTOMIC CONJUGATE OF X(4908)

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

X(40833) lies on the conidc {{A,B,C,X(1),X(2)}} and these lines: {1, 903}, {2, 4403}, {274, 16723}, {668, 36912}, {1002, 36887}, {2006, 17078}, {4555, 36593}, {4945, 30588}, {17320, 24857}, {18145, 36805}, {30608, 39963}

X(40833) = isotomic conjugate of X(4908)
X(40833) = X(4850)-cross conjugate of X(679)
X(40833) = cevapoint of X(17078) and X(17079)
X(40833) = trilinear pole of line {513, 6548}
X(40833) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4908}, {41, 36920}, {44, 2177}, {45, 902}, {1017, 4792}, {1023, 4775}, {1404, 3711}, {1405, 3689}, {1960, 4752}, {2251, 3679}, {4273, 21805}, {4671, 9459}, {4893, 23344}
X(40833) = barycentric product X(i)*X(j) for these {i,j}: {88, 20569}, {89, 20568}, {903, 39704}, {4597, 6548}
barycentric quotient X(i)/X(j) for these {i,j}: {2, 4908}, {7, 36920}, {88, 45}, {89, 44}, {106, 2177}, {320, 36923}, {679, 4792}, {903, 3679}, {1022, 4893}, {1320, 3711}, {2163, 902}, {2320, 3689}, {3257, 4752}, {4049, 4931}, {4555, 4767}, {4588, 23344}, {4597, 17780}, {4604, 1023}, {4997, 4873}, {6548, 4777}, {14421, 14410}, {14422, 14409}, {17078, 36913}, {17079, 36914}, {20568, 4671}, {20569, 4358}, {23345, 4775}, {23352, 4825}, {23838, 4814}, {27922, 4693}, {28607, 2251}, {30588, 3943}, {30589, 4727}, {30608, 2325}, {39704, 519}

X(40834) = ISOTOMIC CONJUGATE OF X(18904)

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

X(40834) lies on these lines: {2, 3114}, {86, 38810}, {171, 334}, {292, 1920}, {1434, 40017}, {1509, 17743}, {1581, 1966}, {2368, 8684}, {7146, 36800}

X(40834) = isotomic conjugate of X(18904)
X(40834) = isotomic conjugate of the complement of X(1966)
X(40834) = cevapoint of X(i) and X(j) for these (i,j): {2, 1966}, {292, 334}
X(40834) = trilinear pole of line {3805, 7033}
X(40834) = X(i)-isoconjugate of X(j) for these (i,j): {31, 18904}, {238, 16584}, {350, 21751}, {1284, 20665}, {1428, 20684}, {1429, 4531}, {1914, 3778}, {1918, 33891}, {1921, 8022}, {2210, 3721}, {2238, 7032}, {2275, 3747}, {2887, 14599}, {9468, 19563}, {18892, 20234}, {21815, 33295}
X(40834) = barycentric product X(i)*X(j) for these {i,j}: {334, 40415}, {335, 38810}, {741, 7034}, {4583, 7255}, {7033, 18827}, {17743, 40017}
X(40834) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18904}, {274, 33891}, {291, 3778}, {292, 16584}, {334, 2887}, {335, 3721}, {741, 7032}, {983, 3747}, {1808, 20753}, {1922, 21751}, {1966, 19563}, {2311, 20665}, {4562, 7239}, {4589, 3888}, {4639, 33946}, {4876, 20684}, {7033, 740}, {7034, 35544}, {7077, 4531}, {7192, 3808}, {7255, 659}, {7305, 5009}, {7307, 30940}, {8684, 4557}, {14598, 8022}, {17743, 2238}, {18827, 982}, {18895, 20234}, {30669, 18905}, {36800, 3061}, {37128, 2275}, {38810, 239}, {38813, 2210}, {40017, 3662}, {40415, 238}

X(40835) = ISOTOMIC CONJUGATE OF X(18905)

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

X(40835) lies on these lines: {2, 3114}, {238, 7018}, {333, 38810}, {893, 1921}, {984, 7033}, {1043, 3783}, {1581, 1965}, {31008, 40432}

X(40835) = isotomic conjugate of X(18905)
X(40835) = isotomic conjugate of the complement of X(1965)
X(40835) = cevapoint of X(i) and X(j) for these (i,j): {2, 1965}, {893, 7018}, {17493, 39044}
X(40835) = trilinear pole of line {7253, 30665}
X(40835) = X(i)-isoconjugate of X(j) for these (i,j): {31, 18905}, {171, 16584}, {172, 3778}, {213, 7184}, {1909, 21751}, {1918, 7187}, {1920, 8022}, {2275, 20964}, {2295, 7032}, {3721, 7122}, {4531, 7175}, {7188, 40729}, {17103, 21815}
X(40835) = barycentric product X(i)*X(j) for these {i,j}: {257, 38810}, {1178, 7034}, {7018, 40415}, {7033, 32010}
X(40835) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18905}, {86, 7184}, {256, 3778}, {257, 3721}, {274, 7187}, {893, 16584}, {983, 20964}, {1178, 7032}, {1965, 19564}, {4594, 3888}, {7018, 2887}, {7033, 1215}, {7034, 1237}, {7104, 21751}, {7255, 4367}, {7260, 33946}, {7307, 8033}, {17103, 7188}, {17493, 18904}, {17743, 2295}, {27805, 7239}, {32010, 982}, {38810, 894}, {38813, 7122}, {39044, 19563}, {40415, 171}, {40432, 2275}, {40729, 21815}

X(40836) = ISOGONAL CONJUGATE OF X(7078)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3) : :
Barycentrics    (tan A)/(cos A - cos B - cos C + 1) : :

Let A' be the center of the polar-circle-inverse of the A-excircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(40836). (Randy Hutson, January 22, 2021)

X(40836) lies on the conic {{A,B,C,G(1),X(2)}}, the cubic K879, and these lines: {1, 281}, {2, 280}, {4, 57}, {7, 6355}, {8, 40399}, {11, 1118}, {28, 1436}, {29, 81}, {34, 1256}, {56, 1857}, {78, 15500}, {88, 5125}, {89, 7518}, {92, 14986}, {105, 7154}, {108, 3149}, {158, 278}, {196, 946}, {235, 7103}, {243, 7288}, {268, 405}, {271, 6734}, {273, 279}, {277, 475}, {412, 5435}, {459, 7149}, {497, 1940}, {499, 1784}, {653, 962}, {936, 7046}, {950, 37028}, {959, 2358}, {1034, 7358}, {1068, 2006}, {1097, 37774}, {1148, 5603}, {1170, 7513}, {1255, 5703}, {1413, 34051}, {1496, 7076}, {1498, 10365}, {1593, 1863}, {1715, 1788}, {1724, 36049}, {1728, 1741}, {1737, 39947}, {1767, 12705}, {1785, 8056}, {1870, 7040}, {1897, 27383}, {1903, 5746}, {2188, 3488}, {2192, 2982}, {2282, 2357}, {2345, 23052}, {2990, 12649}, {3195, 40396}, {3333, 39574}, {3487, 7551}, {3911, 37417}, {4298, 39531}, {6361, 8762}, {6621, 34050}, {6831, 14257}, {7078, 27382}, {7132, 7718}, {8059, 32706}, {8894, 20263}, {9376, 37305}, {11019, 39585}, {13411, 25430}, {13853, 37372}, {17920, 39954}, {37141, 37258}

X(40836) = isogonal conjugate of X(7078)
X(40836) = polar conjugate of X(329)
X(40836) = isotomic conjugate of the isogonal conjugate of X(7151)
X(40836) = polar conjugate of the isotomic conjugate of X(189)
X(40836) = polar conjugate of the isogonal conjugate of X(1436)
X(40836) = X(i)-complementary conjugate of X(j) for these (i,j): {2208, 3351}, {3347, 1329}, {34167, 6260}
X(40836) = X(i)-Ceva conjugate of X(j) for these (i,j): {1440, 278}, {7020, 4}
X(40836) = X(i)-cross conjugate of X(j) for these (i,j): {34, 4}, {393, 278}, {1146, 17924}, {1436, 189}, {7008, 7003}, {17054, 39267}
X(40836) = cevapoint of X(i) and X(j) for these (i,j): {1, 1728}, {11, 7649}, {1436, 7151}, {7008, 7129}, {7117, 18344}
X(40836) = trilinear pole of line {513, 3064}
X(40836) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7078}, {3, 40}, {8, 7114}, {9, 7011}, {48, 329}, {55, 7013}, {63, 198}, {65, 1819}, {69, 2187}, {71, 1817}, {72, 2360}, {77, 7074}, {78, 221}, {184, 322}, {196, 2289}, {208, 1259}, {212, 347}, {219, 223}, {222, 2324}, {227, 283}, {228, 8822}, {255, 7952}, {268, 40212}, {326, 3195}, {342, 6056}, {345, 2199}, {394, 2331}, {603, 7080}, {651, 10397}, {906, 14837}, {1103, 1433}, {1331, 6129}, {1409, 27398}, {1437, 21075}, {1790, 21871}, {1802, 14256}, {1813, 14298}, {2149, 16596}, {3194, 3682}, {3209, 3719}, {3692, 6611}, {7177, 7368}, {7358, 24027}, {8058, 36059}, {15501, 22350}, {17896, 32656}
X(40836) = barycentric product X(i)*X(j) for these {i,j}: {4, 189}, {7, 7003}, {19, 309}, {27, 39130}, {29, 8808}, {34, 34404}, {57, 7020}, {75, 7129}, {76, 7151}, {84, 92}, {85, 7008}, {264, 1436}, {273, 282}, {278, 280}, {281, 1440}, {285, 40149}, {286, 1903}, {314, 2358}, {318, 1422}, {331, 2192}, {693, 40117}, {1413, 7017}, {1433, 2052}, {1857, 34400}, {1969, 2208}, {6063, 7154}, {6355, 36421}, {13138, 17924}
X(40836) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 329}, {6, 7078}, {11, 16596}, {19, 40}, {25, 198}, {27, 8822}, {28, 1817}, {29, 27398}, {33, 2324}, {34, 223}, {56, 7011}, {57, 7013}, {84, 63}, {92, 322}, {189, 69}, {208, 40212}, {268, 1259}, {271, 3719}, {273, 40702}, {278, 347}, {280, 345}, {281, 7080}, {282, 78}, {284, 1819}, {285, 1812}, {309, 304}, {393, 7952}, {604, 7114}, {607, 7074}, {608, 221}, {663, 10397}, {1096, 2331}, {1118, 196}, {1119, 14256}, {1146, 7358}, {1395, 2199}, {1398, 6611}, {1413, 222}, {1422, 77}, {1433, 394}, {1436, 3}, {1440, 348}, {1474, 2360}, {1824, 21871}, {1826, 21075}, {1880, 227}, {1903, 72}, {1973, 2187}, {2188, 2289}, {2192, 219}, {2207, 3195}, {2208, 48}, {2331, 1103}, {2357, 71}, {2358, 65}, {3064, 8058}, {5317, 3194}, {6591, 6129}, {6612, 7053}, {7003, 8}, {7008, 9}, {7020, 312}, {7071, 7368}, {7118, 212}, {7129, 1}, {7151, 6}, {7154, 55}, {7337, 3209}, {7367, 1260}, {7649, 14837}, {8059, 1813}, {8735, 38357}, {8808, 307}, {8886, 6617}, {13138, 1332}, {13853, 6356}, {17924, 17896}, {18344, 14298}, {32652, 906}, {34400, 7055}, {34404, 3718}, {36049, 1331}, {37141, 6516}, {39130, 306}, {40117, 100}
X(40836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1895, 7952}, {29, 938, 34231}, {158, 3086, 278}

X(40837) = POLAR CONJUGATE OF X(1034)

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

X(40837) lies on the cubics K879 and K1060, and on these lines: {2, 342}, {4, 57}, {9, 1767}, {108, 7011}, {196, 226}, {207, 1490}, {223, 1249}, {278, 393}, {329, 653}, {345, 18026}, {442, 14257}, {948, 18679}, {1035, 8885}, {1214, 3346}, {1895, 37421}, {3183, 9118}, {5435, 37279}, {7103, 37376}, {7149, 20264}, {8804, 40212}, {11051, 40573}, {18623, 36118}, {23986, 32714}, {34231, 37543}

X(40837) = complement of X(41514)
X(40837) = complement of the isogonal conjugate of X(3197)
X(40837) = polar conjugate of X(1034)
X(40837) = polar conjugate of the isotomic conjugate of X(5932)
X(40837) = polar conjugate of the isogonal conjugate of X(1035)
X(40837) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 6245}, {31, 278}, {41, 282}, {198, 20209}, {207, 16608}, {1035, 142}, {1490, 141}, {3176, 20305}, {3197, 10}, {3341, 21239}, {5932, 17046}, {8885, 34830}, {13614, 21246}, {32652, 8063}, {33672, 626}
X(40837) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 278}, {342, 4}, {653, 8063}
X(40837) = X(1035)-cross conjugate of X(5932)
X(40837) = crosssum of X(6) and X(7152)
X(40837) = X(i)-isoconjugate of X(j) for these (i,j): {48, 1034}, {63, 7037}, {78, 7152}, {219, 3345}, {268, 3342}, {394, 7007}, {2193, 8806}, {2289, 7149}, {2327, 8811}
X(40837) = barycentric product X(i)*X(j) for these {i,j}: {4, 5932}, {7, 3176}, {34, 33672}, {75, 207}, {264, 1035}, {273, 1490}, {331, 3197}, {342, 3341}, {1441, 8885}, {14302, 36118}
X(40837) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1034}, {25, 7037}, {34, 3345}, {207, 1}, {208, 3342}, {225, 8806}, {608, 7152}, {1035, 3}, {1096, 7007}, {1118, 7149}, {1426, 8811}, {1490, 78}, {3176, 8}, {3197, 219}, {3341, 271}, {5932, 69}, {8885, 21}, {13614, 1792}, {33672, 3718}
X(40837) = {X(393),X(1427)}-harmonic conjugate of X(278)

X(40838) = POLAR CONJUGATE OF X(5932)

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

X(40838) lies on the cubic K879 and these lines: {2, 342}, {4, 282}, {9, 1249}, {200, 7007}, {278, 2184}, {281, 6523}, {1034, 2287}, {4183, 7037}, {5776, 7152}

X(40838) = polar conjugate of X(5932)
X(40838) = polar conjugate of the isotomic conjugate of X(1034)
X(40838) = polar conjugate of the isogonal conjugate of X(7037)
X(40838) = X(3354)-complementary conjugate of X(20307)
X(40838) = X(i)-cross conjugate of X(j) for these (i,j): {393, 281}, {7007, 7149}, {7008, 4}, {7037, 1034}
X(40838) = cevapoint of X(3064) and X(5514)
X(40838) = X(i)-isoconjugate of X(j) for these (i,j): {48, 5932}, {63, 1035}, {77, 3197}, {207, 394}, {222, 1490}, {3176, 7125}, {3341, 7011}, {8885, 40152}
X(40838) = barycentric product X(i)*X(j) for these {i,j}: {4, 1034}, {8, 7149}, {29, 8806}, {75, 7007}, {264, 7037}, {318, 3345}, {3342, 7020}, {7017, 7152}
X(40838) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5932}, {25, 1035}, {33, 1490}, {318, 33672}, {607, 3197}, {1034, 69}, {1096, 207}, {1857, 3176}, {3342, 7013}, {3345, 77}, {4183, 13614}, {7007, 1}, {7008, 3341}, {7037, 3}, {7149, 7}, {7152, 222}, {8806, 307}, {8811, 1439}

X(40839) = POLAR CONJUGATE OF X(14365)

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 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*(5*a^12 - 10*a^10*b^2 - 9*a^8*b^4 + 36*a^6*b^6 - 29*a^4*b^8 + 6*a^2*b^10 + b^12 - 10*a^10*c^2 + 34*a^8*b^2*c^2 - 36*a^6*b^4*c^2 + 4*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - 6*b^10*c^2 - 9*a^8*c^4 - 36*a^6*b^2*c^4 + 50*a^4*b^4*c^4 - 20*a^2*b^6*c^4 + 15*b^8*c^4 + 36*a^6*c^6 + 4*a^4*b^2*c^6 - 20*a^2*b^4*c^6 - 20*b^6*c^6 - 29*a^4*c^8 + 14*a^2*b^2*c^8 + 15*b^4*c^8 + 6*a^2*c^10 - 6*b^2*c^10 + c^12) : :

X(40839) lies on the cubic K879 and these lines: {4, 1073}, {278, 2184}, {393, 459}, {1249, 3343}, {1301, 3079}, {3183, 28785}, {6509, 6621}, {33537, 39268}

X(40839) = polar conjugate of X(14365)
X(40839) = polar conjugate of the isotomic conjugate of X(14362)
X(40839) = polar conjugate of the isogonal conjugate of X(28785)
X(40839) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 459}, {48, 33546}, {3183, 20305}
X(40839) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 459}, {14362, 3183}
X(40839) = X(28785)-cross conjugate of X(14362)
X(40839) = crosssum of X(6) and X(31956)
X(40839) = X(i)-isoconjugate of X(j) for these (i,j): {48, 14365}, {63, 28781}, {610, 3348}
X(40839) = barycentric product X(i)*X(j) for these {i,j}: {4, 14362}, {253, 3183}, {264, 28785}
X(40839) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14365}, {25, 28781}, {64, 3348}, {3183, 20}, {14362, 69}, {28785, 3}

leftri

Points associated with (X(1),U)-cevian-cross triangles: X(40840)-X(40842)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 31, 2020.

Let U = u : v : w be a point in the plane of a triangle ABC. The (X(1),U)-cevian-cross triangle A'B'C' is introduced in the preamble just before X(40718). The vertices are given by.

A'' = u : u + v : u + w
B'' = v + u : - v : v + w
C'' = w + u : w + v : - w.

Triangles perspective to A'B'C' and the perspectors include the following:

ABC, (u+v)*(u+w) : :
medial, u*(u - v - w) : :
anticomplementary, u*v + u*w - v*w : :
Gemini 107, (u - v)*(u - w) : :
Gemini 109, 3*u^2 - u*u*v - 7*u*w - 5*v*w : :

X(6)-of-A'B'C' = 2*(a^2 - 2*b^2 - 2*c^2)*u^3 + (2*a^2 - b^2 - 7*c^2)*u^2*v + (2*a^2 - 7*b^2 - c^2)*u^2*w + (7*a^2 - 5*b^2 - 5*c^2)*u*v*w - (2*a^2 - b^2 + 2*c^2)*v^2*w - (2*a^2 + 2*b^2 - c^2)*v*w^2 : :

Following are four points on the Euler line of A'B'C':

X(2)-of-A'B'C' = X(2)

X(3)-of-A'B'C' = (b - c)*(b + c)*(a^2 + b^2 - 5*c^2)*u^2*v - 2*(b - c)*(b + c)*(2*a^2 - b^2 + 2*c^2)*u*v^2 - (b - c)*(b + c)*(a^2 - 5*b^2 + c^2)*u^2*w + 3*(a - b - c)*(a + b - c)*(a - b + c)*(a + b + c)*u*v*w - (3*a^2 + b^2 - c^2)*(2*a^2 - b^2 + 2*c^2)*v^2*w + 2*(b - c)*(b + c)*(2*a^2 + 2*b^2 - c^2)*u*w^2 - (2*a^2 + 2*b^2 - c^2)*(3*a^2 - b^2 + c^2)*v*w^2 : :

X(4)-of-A'B'C' = (3*a^4 - 8*a^2*b^2 + b^4 - 4*a^2*c^2 + 6*b^2*c^2 - 7*c^4)*u^2*v + (a^2 + b^2 - 5*c^2)*(3*a^2 - b^2 + c^2)*u*v^2 + (3*a^4 - 4*a^2*b^2 - 7*b^4 - 8*a^2*c^2 + 6*b^2*c^2 + c^4)*u^2*w + 3*(a - b - c)*(a + b - c)*(a - b + c)*(a + b + c)*u*v*w + (5*a^2 - b^2 - c^2)*(3*a^2 - b^2 + c^2)*v^2*w + (3*a^2 + b^2 - c^2)*(a^2 - 5*b^2 + c^2)*u*w^2 + (5*a^2 - b^2 - c^2)*(3*a^2 + b^2 - c^2)*v*w^2 : :

X(5)-of-A'B'C' = (a^2 - 2*b^2 - 2*c^2)*(3*a^2 - b^2 + c^2)*u^2*v + (3*a^4 - 2*a^2*b^2 + b^4 - 10*a^2*c^2 - c^4)*u*v^2 + (a^2 - 2*b^2 - 2*c^2)*(3*a^2 + b^2 - c^2)*u^2*w + 6*(a - b - c)*(a + b - c)*(a - b + c)*(a + b + c)*u*v*w + (9*a^4 - 7*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*v^2*w + (3*a^4 - 10*a^2*b^2 - b^4 - 2*a^2*c^2 + c^4)*u*w^2 + (9*a^4 - 2*a^2*b^2 + b^4 - 7*a^2*c^2 - 3*b^2*c^2 + 2*c^4)*v*w^2 : :

Further examples of triangle centers of A'B'C':

For U = X(1), X(6)-of-A'B'C' = X(40841)

For U = X(2), X(3)-of-A'B'C' = X(381)
For U = X(2), X(4)-of-A'B'C' = X(376)
For U = X(2), X(5)-of-A'B'C' = X(549)
For U = X(2), X(6)-of-A'B'C' = X(599)

For U = X(6), X(3)-of-A'B'C' = X(11261)
For U = X(6), X(4)-of-A'B'C' = X(31958)
For U = X(6), X(5)-of-A'B'C' = X(32149)
For U = X(6), X(6)-of-A'B'C' = X(40842)

For U = X(75), X(3)-of-A'B'C' = X(40840)

For U = X(523), X(3)-of-A'B'C' = X(2793)
For U = X(523), X(4)-of-A'B'C' = X(2793)
For U = X(523), X(5)-of-A'B'C' = X(2793)

X(40840) = X(3)-OF-(X(1),X(75))-CEVIAN-CROSS TRIANGLE

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

X(40840) lies on these lines: {497,3666}, {1111,4008}

X(40841) = X(6)-OF-(X(1),X(1))-CEVIAN-CROSS TRIANGLE

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

X(40841) lies on these lines: {896,3758}, {993,4432}

X(40842) = X(6)-OF-(X(1),X(6))-CEVIAN-CROSS TRIANGLE

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

X(40842) lies on these lines: {2,187}, {182,2782}, {6055,26316}, {7816,32522}, {10796,11261}, {21163,32456}

leftri

Points associated with paratriangles: X(40843)-X(40850)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, December 31, 2020.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The point P' = p : r : q is here named the parapoint of P. To contruct P', let L be the line through P parallel to BC. Let Q = L∩AG , where G denotes the centroid of ABC. Then P' is the reflection of P in Q.

The paratriangle of P is the triangle A'B'C' with vertices given by

A' = p : r : q
B' = r : q : p
C' = q : p : r

If P is a triangle center other than X(1), then A'B'C' is a noncentral triangle, but its perspector with other triangles can be a triangle center.

(More generally, the paratriangle of a triangle UVW is here defined as the triangle U'V'W'.)

For fixed P with paratriangle A'B'C', the locus of X such that the cevian triangle of X is perspective to A'B'C' is the cevian-associated cubic given by

q (p q - r^2) y z^2 - r (p r - q^2) y^2 z + (cyclic) = 0,

which is the cubic pK(P1,P2), where P1 = (p r - q^2)(p q - r^2) and P2 = P*P1.

The appearance of (i,K___) in the following list means that if P = X(i), then the K___ is the cevian-associated cubic;

(1, K768)
(4, K776)
(6, K322)
(13, K859b)
(14, K859a)
(30, K860)
(69, K777)
(75, K769)
(76, K354)
(98, K718)
(239, K766)
(287, K778)
(290, K357)
(291, K770)
(297, K779)
(298, K419b)
(299, K419a)
(325, K780)
(335, K767)
(350, K323)
(385, K738)
(511, K355)
(694, K739)
(903, K1148)
(1494, K472)
(1916, K356)
(3912, K623)
(3978, K128)
(11078, K867a)
(11092, K867b)

The following list shows, as an example, that if P = X(7), then the points listed lie on the cevian-associated cubic:

{7,{2,8,9,239,673,2319,2481,4373,5853,9312,9436,14942,33676}}
{8,{1,2,7,516,673,2481,3729,3912,6185,10025,10405,14942,39914}}
{10,{2,86,350,1125,6542,6625,6650,9505,11599,17770,20536,28604,39921}}
{83,{2,76,141,732,6292,9477,11606,17949,24733,40000}}
{86,{2,10,75,740,1213,6650,9278,11599,17731,20016,24731}}
{92,{2,63,1214,1947,1948,1952,2994,17950,35145}}
{141,{2,83,1031,3589,3978,7779,11606,17949,39938}}
{253,{2,20,297,1249,6330,14944,16096,35140,35510}}
{256,{1,2,385,1909,2669,7168,7346,39929,39933}}
{264,{2,3,69,216,511,1972,1987,9291,14941,16089}}
{330,{2,192,350,3226,6376,30545,32020,33680,36799,38247}}
{518,{2,7,8,239,335,518,2481,10025,17794,39350}}
{673,{2,7,85,145,335,518,3912,9436,10029,14942,16593,30545,35160,39775}}
{698,{2,194,694,698,2998,3225,3978,8782,9493}}
{726,{2,192,291,330,350,726,3226,33888,39354}}
{732,{2,6,76,732,7779,11606,14970,15588,39939}}
{740,{1,2,75,740,6542,6650,13174,17759,18827,39367,39718,39719,39925}}
{894,{2,76,257,350,3509,3512,4357,18760,39920}}
{1502,{2,32,66,694,695,3505,3852,8265,14946,36214}}
{1503,{2,20,147,253,287,297,1503,9473,35140}}
{1909,{2,6,239,256,1107,3508,7168,16364,19567,24576,39915,39933}}
{1972,{2,4,264,276,290,401,14941,16089,32428}}
{2319,{2,192,335,1916,3912,4876,9311,17752,17760,17786,20528,30545,39914}}
{2481,{1,2,9,291,518,2481,3912,6184,9436,14943,17755,33675,33700,33701,36905,36906}}
{3226,{2,75,335,726,1575,3226,6376,17793,20532,33678}}
{3505,{2,384,1916,5989,6660,9229,16101,16985,36432}}
{4876,{2,239,330,385,673,2319,3500,3729,7167,10030,27447,39914}}
{6330,{2,253,287,441,1503,3146,14944,16096,34403}}
{6542,{2,10,86,239,1509,6650,13174,13610,17731,17770,18827}}
{6650,{1,2,86,6542,6651,11599,17731,18827,32014}}
{7168,{2,75,256,1916,19567,19579,21226,24732,39933}}
{7779,{2,83,141,385,9483,11606,14370,14970,15588}}
{9436,{2,7,8,346,350,3912,4876,5853,14942,14943,18025,20533,33677,36807}}
{10030,{1,2,6,2319,4876,7155,14942,27424,30545,36799,39914}}
{11599,{2,10,291,1268,1654,6542,6650,17731,35162}}
{14941,{2,3,95,98,401,1972,3164,9290,16089,32428}}
{14942,{2,8,75,144,518,673,3912,4437,9311,9436,18025,20935}}
{14946,{2,32,384,385,710,8264,16985,19585,22252,39082,39927,40416}}
{14970,{2,39,141,385,732,3978,8290,14970,17949}}
{16089,{2,3,264,287,401,511,577,1988,5374,14941}}
{16101,{2,32,384,694,3492,3505,6660,19571,21355}}
{30545,{2,8,291,694,1575,2319,3501,3507,3551,4876,10030,17787,17792}}
{35140,{2,4,98,441,1249,1503,15595,16096,23976,35140,36899}}
{35142,{2,69,114,230,287,3564,6337,35067,35142}}
{35145,{2,226,1214,1944,1948,8680,11608,35075,35145,39035,39036}}
{35160,{2,8,3008,3161,5853,14942,16593,35111,35160}}
{36214,{2,98,385,1975,2998,3504,16985,17984,39927}}
{39914,{2,75,76,1423,2319,3212,9436,10030,30545}}
{39933,{2,76,335,1221,1909,3510,7168,19567,19581}}

The next list shows the points P1 and P*P1 for each cubic in the list just above. For example, if P = X(7), then the cevian-associated cubic is pK(X(14942), X(67312)).

{7,{14942,673}}
{8,{673,14942}}
{10,{6650,11599}}
{83,{17949,11606}}
{86,{11599,6650}}
{92,{40843,1952}}
{141,{11606,17949}}
{253,{14944,6330}}
{256,{39933,7168}}
{264,{14941,1972}}
{330,{40844,32020}}
{518,{2,518}}
{673,{9436,7}}
{698,{2,698}}
{726,{2,726}}
{732,{2,732}}
{740,{2,740}}
{894,{40845,40846}}
{1502,{14946,40847}}
{1503,{2,1503}}
{1909,{7168,39933}}
{1972,{16089,264}}
{2319,{40848,4876}}
{2481,{518,2}}
{3226,{726,2}}
{3505,{384,6660}}
{4876,{39914,2319}}
{6330,{16096,253}}
{6542,{86,17731}}
{6650,{17731,86}}
{7168,{40849,256}}
{7779,{83,40850}}
{9436,{8,3912}}
{10030,{2319,39914}}
{11599,{6542,10}}
{14941,{401,3}}
{14942,{3912,8}}
{14946,{16985,32}}
{14970,{732,2}}
{16089,{3,401}}
{16101,{6660,384}}
{17731,{10,6542}}
{17984,{3504,39927}}
{18827,{740,2}}
{19567,{256,40849}}
{30545,{4876,40848}}
{35140,{1503,2}}
{35142,{3564,2}}
{35145,{8680,2}}
{35160,{5853,2}}
{36214,{39927,3504}}
{39914,{30545,10030}}
{39933,{19567,1909}}

X(40843) = X(2)X(1952)∩X(21)X(73)

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

X(40843) lies on these lines: {2, 1952}, {21, 73}, {63, 6509}, {78, 296}, {226, 1947}, {1812, 1949}, {1945, 2339}, {2249, 13395}, {2808, 18446}, {3912, 36795}, {5905, 7361}

X(40844) = X(2)X(1978)∩X(43)X(8026)

Barycentrics    b*c*(-(a*b) - a*c + 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(40844) lies on these lines: {2, 1978}, {43, 8026}, {75, 87}, {192, 23643}, {239, 20332}, {274, 20899}, {668, 35958}, {740, 7168}, {1281, 8851}, {3223, 18793}, {6376, 40780}, {8709, 9082}, {17459, 31008}, {17755, 36799}, {27644, 36860}

X(40845) = X(2)X(20940)∩X(8)X(7261)

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

X(40845) lies on these lines: {2, 20940}, {8, 7261}, {29, 33296}, {75, 1281}, {92, 30545}, {257, 1146}, {304, 27424}, {312, 18035}, {325, 3932}, {333, 3512}, {664, 1220}, {740, 7281}, {918, 21612}, {1228, 21604}, {1502, 17788}, {3978, 17789}, {8926, 28850}, {17743, 20926}, {18760, 40777}, {20934, 25332}, {23989, 30690}, {29821, 39920}

X(40846) = X(1)X(18760)∩X(2)X(20940)

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

X(40846) lies on these lines: {1, 18760}, {2, 20940}, {69, 7155}, {76, 3496}, {350, 1934}, {1821, 10030}, {1909, 19571}, {3912, 24479}, {5976, 7061}

X(40846) = isogonal conjugate of X(41882)
X(40846) = isotomic conjugate of X(40873)
X(40846) = cevapoint of X(3512) and X(39920)
X(40846) = trilinear product X(i)*X(j) for these {i,j}: {2, 7061}, {75, 41534}, {171, 40845}, {172, 18036}, {894, 7261}, {1909, 3512}, {1920, 8852}, {1966, 24479}, {3978, 30648}, {7196, 7281}

X(40847) = X(2)X(14946)∩X(66)X(8264)

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

X(40847) lies on these lines: {2, 14946}, {66, 8264}, {141, 9229}, {384, 694}, {385, 3505}, {427, 1916}, {698, 17949}, {702, 18828}, {783, 9076}, {1031, 14970}, {1502, 15449}, {3852, 16985}, {4577, 8265}

X(40848) = X(2)X(4876)∩X(8)X(291)

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

X(40848) lies on these lines: {2, 4876}, {8, 291}, {10, 257}, {75, 141}, {76, 7034}, {100, 18265}, {192, 3123}, {239, 292}, {244, 25746}, {384, 3500}, {385, 3507}, {694, 17787}, {894, 3888}, {1740, 1911}, {1978, 18277}, {2227, 17759}, {2998, 4019}, {3116, 12782}, {3551, 3729}, {3799, 39916}, {3912, 33891}, {4595, 16742}, {4904, 17670}, {6376, 21138}, {6377, 36288}, {6653, 7261}, {7233, 9436}, {7245, 18827}, {8256, 9311}, {15310, 33681}, {16571, 18787}, {17033, 27634}, {17292, 21264}, {17319, 28358}, {18895, 20911}, {19973, 20333}, {22319, 35352}

X(40849) = X(1)X(39917)∩X(2)X(256)

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

X(40849) lies on these lines: {1, 39917}, {2, 256}, {76, 257}, {141, 3863}, {350, 694}, {384, 904}, {893, 37678}, {1575, 27805}, {1581, 3912}, {1916, 33891}, {2106, 4594}, {3903, 10027}, {17738, 18786}, {21264, 32010}, {24514, 40729}, {37137, 39930}

X(40850) = X(2)X(32)∩X(76)X(14370)

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

X(40850) lies on these lines: {2, 32}, {76, 14370}, {141, 40000}, {184, 7877}, {237, 39557}, {385, 9477}, {511, 8928}, {524, 4577}, {689, 3978}, {732, 15588}, {733, 3229}, {826, 14318}, {1915, 7768}, {3186, 32085}, {3225, 18828}, {3405, 3508}, {3510, 7166}, {3511, 20854}, {4447, 36081}, {7760, 13511}, {8782, 8856}, {9483, 17949}, {11328, 38908}, {14096, 34888}, {34573, 40425}

X(40851) = X(3)X(39162)∩X(4)X(3413)

Barycentrics    (S^2-3*SB*SC)*e+2*(3*SA-SW)*S^2-3*(SA^2-SB*SC)*(SB+SC) + 2*SB*SC*sqrt(2*(9R^2-2*SW)*e-3*S^2+5*SW^2-18*R^2*SW) : : , where e = sqrt(SW^2-3*S^2)
X(40851) = 2*X(3)-3*X(39162) = 4*X(5)-3*X(39163) = X(20)-3*X(39158) = 5*X(3091)-3*X(39159)

Contributed by César Lozada, December 31, 2020. See X(39158).

X(40851) lies on the cubics K004 (Darboux cubic), K800, and these lines: {3, 39162}, {4, 3413}, {5, 39163}, {20, 39158}, {3091, 39159}

X(40851) = reflection of X(40852) in X(4)
X(40851) = isogonal conjugate of X(40993)
X(40851) = Miquel point of X(39158)

X(40852) = X(3)X(39163)∩X(4)X(3413)

Barycentrics    (S^2-3*SB*SC)*e+2*(3*SA-SW)*S^2-3*(SA^2-SB*SC)*(SB+SC) - 2*SB*SC*sqrt(2*(9R^2-2*SW)*e-3*S^2+5*SW^2-18*R^2*SW) : : , where e = sqrt(SW^2-3*S^2)
X(40852) = 2*X(3)-3*X(39163) = 4*X(5)-3*X(39162) = X(20)-3*X(39159) = 5*X(3091)-3*X(39158)

Contributed by César Lozada, December 31, 2020. See X(39159).

X(40852) lies on the cubics K004 (Darboux cubic), K800 and these lines: {3, 39163}, {4, 3413}, {5, 39162}, {20, 39159}, {3091, 39158}

X(40852) = reflection of X(40851) in X(4)
X(40852) = isogonal conjugate of X(40994)
X(40852) = Miquel point of X(39159)

leftri

Points associated with paratriangles: X(40853)-X(40893)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, January 1, 2021.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. As in the preamble just before X(40843), the paratriangle of P is the triangle A'B'C' with vertices given by

A' = p : r : q
B' = r : q : p
C' = q : p : r

For fixed P with paratriangle A'B'C', the locus of X such that the anticevian triangle of X is perspective to A'B'C' is the anticevian-associated cubic given by

p (p r - q^2) y z^2 - p (p q - r^2) y^2 z + (cyclic) = 0,

which is the cubic pK(P,P'), where P' = p^2 - q r : q^2 - r p : r^2 - p q is the X(2)-Hirst inverse of P.

The appearance of (i,K___) in the following list means that if P = X(i), then the K___ is the anticevian-associated cubic;

(1, K323)
(4, K780)
(6, K128)
(7, K623)
(13, K419a)
(14, K419b)
(30, K472)
(69, K779)
(75, K766)
(76, K738)
(98, K777)
(239, K770)
(287, K776)
(291, K769)
(297, K718)
(298, K867a)
(299, K867b)
(325, K778)
(335, K768)
(350, K767)
(385, K739)
(511, K357)
(519, K1148)
(694, K354)
(1916, K322)
(3978, K356)
(11078, K859a)
(11092, K859b)

The appearance of (i,j) in the following list means that the X(2)-Hirst inverse of X(i) is X(j):

(1,239), (3,401), (4,297), (5,40853), (6,385), (7,9436), (8,3912), (9,10025), (10,6542), (13,11078), (14,11092), (15,40854), (16,40855), (20,441), (21,448), (22,15013), (23,40856), (25,15014), (27,447), (30,2), (32,16985), (37,17759), (38,40857), (39,40858), (42,40859), (43,10027), (44,40860), (55,40861), (57,40862), (63,1944), (69,325), (75,350), (76,3978), (78,40863), (81,19623), (85,40864), (86,17731), (89,29908), (92,1948), (98,287), (99,99), (100,40865), (110,40866), (114,40869), (115,148), (141,7779), (142,40868), (144,40869), (145,3008), (147,15595), (148,115), (182,40870), (183,39099), (187,40871), (190,190), (192,1575), (193,230), (194,3229), (200,40872), (257,40873), (274,40874), (312,40875), (315,40876), (316,40877), (320,40878), (323,40879), (329,40880), (330,40881), (333,40882), (346,40883), (376,40884), (381,40885), (386,40886), (393,408878), (394,40888), (427,40889), (468,40890), (551,408891), (553,40892), (556,40893)

Note that X(99) and X(190) are self-X(2)-Hirst inverse. In general, a point P is self-X-Hirst inverse if and only if X lies on the circumconic with perspector X; thus, X is self-X(2)-Hirst inverse if and only if X is on the Steiner circumellipse.

X(40853) = X(2)-HIRST INVERSE OF X(5)

Barycentrics    a^8 - 3*a^6*b^2 + 2*a^4*b^4 + a^2*b^6 - b^8 - 3*a^6*c^2 + a^4*b^2*c^2 - a^2*b^4*c^2 + 3*b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 - 4*b^4*c^4 + a^2*c^6 + 3*b^2*c^6 - c^8 : :
Barycentrics    A'-power of circumcircle : : , where A'B'C' = anti-Wasat triangle

X(40853) lies on these lines: {2, 3}, {95, 36412}, {97, 14129}, {147, 39682}, {216, 17035}, {217, 1994}, {287, 29012}, {316, 36212}, {317, 3164}, {323, 3331}, {324, 9291}, {340, 39352}, {343, 32819}, {394, 38297}, {525, 15340}, {1972, 32428}, {1993, 7823}, {3284, 37765}, {5207, 36790}, {5422, 7864}, {6709, 40410}, {7737, 14602}, {7748, 40814}, {7779, 39355}, {11427, 14567}, {11433, 20977}, {14918, 39062}, {15595, 29317}, {17128, 37636}, {23357, 23582}, {30227, 37644}, {30529, 32662}, {31610, 31617}, {34981, 39231}

X(40853) = anticomplement of X(401)
X(40853) = Steiner-circumellipse-inverse of X(5)
X(40853) = {X(2479),X(2480)}-harmonic conjugate of X(5)

X(40854) = X(2)-HIRST INVERSE OF X(15)

Barycentrics    a^4*(Sqrt[3]*(-a^2 + b^2 + c^2) + 2*S)^2 - (b^2*c^2*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) + 2*S)) : :

X(40854) lies on these lines: {2, 14}, {3, 14185}, {6, 23895}, {98, 14170}, {99, 11131}, {110, 1316}, {115, 8838}, {542, 11078}, {1995, 25234}, {3580, 32461}, {4226, 14169}, {5191, 14181}, {6777, 40709}, {6778, 16770}, {11126, 23871}, {12177, 25233}, {12188, 14177}, {18776, 21469}, {25155, 37776}, {32036, 34990}

X(40855) = X(2)-HIRST INVERSE OF X(16)

Barycentrics    a^4*(Sqrt[3]*(-a^2 + b^2 + c^2) - 2*S)^2 - (b^2*c^2*(Sqrt[3]*(a^2 + b^2 - c^2) - 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) - 2*S)) : :

X(40855) lies on these lines: {2, 13}, {3, 14187}, {6, 23896}, {98, 14169}, {99, 11130}, {110, 1316}, {115, 8836}, {542, 11092}, {1995, 25233}, {3580, 32460}, {4226, 14170}, {5191, 14177}, {6777, 16771}, {6778, 40710}, {11127, 23870}, {12177, 25234}, {12188, 14181}, {18777, 21468}, {25165, 37775}, {32037, 34990}

X(40856) = X(2)-HIRST INVERSE OF X(23)

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

X(40856) lies on these lines: {2, 3}, {6, 525}, {98, 38608}, {112, 339}, {115, 6720}, {264, 23582}, {287, 5663}, {543, 23583}, {598, 5641}, {1289, 13166}, {2966, 3972}, {3618, 30227}, {3734, 14966}, {5475, 35088}, {5661, 7804}, {10722, 19163}, {15595, 17702}, {18019, 36415}, {23699, 35282}, {36794, 39062}

X(40856) = complement of X(35923)
X(40856) = Steiner-circumellipse-inverse of X(23)
X(40856) = {X(2479),X(2480)}-harmonic conjugate of X(23)

X(40857) = X(2)-HIRST INVERSE OF X(38)

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

X(40857) lies on these lines: {2, 38}, {39, 190}, {86, 1015}, {141, 668}, {239, 3792}, {274, 1086}, {812, 4481}, {3112, 8041}, {3227, 17392}, {3736, 4366}, {4260, 32029}, {5969, 35119}, {6652, 24436}, {9055, 17790}, {9263, 17300}, {14839, 32922}, {17297, 33908}, {17307, 27076}, {17398, 27195}, {21217, 40585}, {24358, 24508}, {24484, 30667}, {24505, 39044}, {35103, 37756}

X(40857) = Steiner-circumellipse-inverse of X(38)

X(40858) = X(2)-HIRST INVERSE OF X(39)

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

X(40858) lies on these lines: {2, 39}, {38, 257}, {69, 706}, {99, 8623}, {110, 16985}, {148, 14957}, {193, 263}, {237, 385}, {335, 1959}, {384, 3051}, {512, 14712}, {524, 25054}, {670, 702}, {694, 732}, {698, 25332}, {1186, 7787}, {1613, 1975}, {1916, 22735}, {1962, 39926}, {2076, 15588}, {2227, 17759}, {2896, 3118}, {3009, 39916}, {3314, 33734}, {7754, 11328}, {7779, 14970}, {7783, 14096}, {7793, 37184}, {7839, 20965}, {8041, 32476}, {8265, 33769}, {9463, 34341}, {10342, 33786}, {20016, 21224}, {20021, 40847}, {20080, 36648}, {25264, 40790}, {28419, 33785}, {31613, 31622}, {32480, 34344}

X(40858) = reflection of X(2) in X(41143)
X(40858) = isotomic conjugate of X(39939)
X(40858) = anticomplement of X(3978)
X(40858) = polar conjugate of isogonal conjugate of X(23174)
X(40858) = Steiner-circumellipse-inverse of X(39)

X(40859) = X(2)-HIRST INVERSE OF X(42)

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

X(40859) lies on these lines: {1, 2}, {39, 34063}, {58, 6645}, {75, 24254}, {76, 2176}, {99, 17735}, {101, 385}, {148, 5134}, {190, 538}, {194, 3730}, {213, 1909}, {220, 7754}, {238, 730}, {257, 16600}, {274, 2295}, {310, 7109}, {312, 35274}, {330, 4253}, {335, 758}, {350, 3230}, {384, 595}, {514, 19565}, {668, 2238}, {742, 20924}, {894, 3997}, {1003, 3052}, {1015, 37686}, {1016, 4601}, {1018, 17759}, {1078, 21008}, {1191, 7770}, {1334, 25264}, {1500, 33296}, {1573, 17277}, {1655, 3294}, {1975, 14974}, {2106, 5209}, {2235, 24502}, {3228, 4555}, {3596, 27623}, {3701, 26689}, {3761, 24514}, {3963, 27644}, {3972, 21793}, {4366, 40091}, {4465, 18145}, {4481, 40459}, {4482, 20142}, {4562, 18826}, {4568, 33889}, {5264, 16915}, {5291, 18047}, {5710, 11321}, {6767, 20162}, {7783, 24047}, {8682, 36226}, {8750, 15014}, {9708, 20154}, {12338, 23863}, {14839, 32922}, {16552, 21226}, {16704, 31061}, {16712, 25349}, {16788, 16998}, {17152, 26978}, {17497, 21272}, {17750, 31997}, {20963, 25303}, {21281, 24190}, {24272, 33760}, {27643, 28654}

X(40859) = anticomplement of X(30109)
X(40859) = Steiner-circumellipse-inverse of X(42)

X(40860) = X(2)-HIRST INVERSE OF X(44)

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

X(40860) lies on these lines: {2, 44}, {6, 27922}, {37, 35959}, {101, 4510}, {192, 4777}, {239, 9318}, {894, 9458}, {1017, 20568}, {3241, 39349}, {3570, 4363}, {4393, 18822}, {4781, 17487}, {16495, 17379}, {17316, 39363}, {17780, 24344}, {24408, 29584}, {24502, 24514}, {24505, 40721}

X(40860) = Steiner-circumellipse-inverse of X(44)

X(40861) = X(2)-HIRST INVERSE OF X(55)

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

X(40861) lies on these lines: {2, 11}, {6, 664}, {86, 17044}, {101, 24279}, {190, 220}, {238, 28850}, {384, 4676}, {918, 3287}, {1086, 21008}, {1146, 17277}, {1952, 16560}, {4422, 17947}, {5452, 21218}, {5845, 17950}, {6063, 14827}, {17259, 31640}, {17349, 39351}

X(40861) = Steiner-circumellipse-inverse of X(55)

X(40862) = X(2)-HIRST INVERSE OF X(57)

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

X(40862) lies on these lines: {2, 7}, {6, 39126}, {45, 31225}, {56, 4676}, {75, 6180}, {77, 192}, {85, 4363}, {190, 241}, {222, 1999}, {223, 3210}, {239, 651}, {269, 3729}, {279, 4454}, {312, 1407}, {348, 4419}, {522, 4318}, {536, 664}, {545, 17078}, {666, 1462}, {726, 5018}, {1016, 1275}, {1323, 17132}, {1362, 6007}, {1414, 19623}, {1418, 17351}, {1419, 3875}, {1422, 39694}, {1427, 32939}, {1441, 17116}, {1442, 17319}, {1443, 4552}, {1456, 32922}, {1458, 3685}, {1467, 17697}, {2263, 3872}, {3227, 34056}, {3663, 17086}, {3758, 5228}, {3923, 4334}, {4306, 7283}, {4307, 28849}, {4341, 25252}, {4350, 25242}, {4364, 17095}, {4396, 6649}, {4440, 22464}, {4643, 33298}, {4644, 6604}, {4645, 6735}, {4659, 9312}, {5088, 29069}, {5205, 9364}, {5263, 8581}, {5723, 37756}, {7081, 9316}, {7190, 17379}, {7196, 24330}, {14942, 15726}, {18623, 30699}, {18663, 20223}, {24352, 31627}, {25268, 28982}, {34050, 37759}, {35154, 35176}

X(40862) = anticomplement of X(40880)
X(40862) = Steiner-circumellipse-inverse of X(57)

X(40863) = X(2)-HIRST INVERSE OF X(78)

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

X(40863) lies on these lines: {1, 2}, {9, 29001}, {75, 28965}, {219, 322}, {345, 1943}, {346, 28966}, {379, 14923}, {448, 643}, {644, 30807}, {664, 25083}, {857, 5176}, {1016, 37790}, {1145, 1375}, {1231, 20924}, {1332, 1944}, {1429, 21232}, {1442, 27514}, {3057, 37086}, {3208, 24268}, {3476, 37280}, {3560, 29331}, {3685, 29016}, {3692, 25252}, {5252, 37445}, {5279, 21271}, {5836, 16054}, {7291, 21272}, {16284, 23151}, {20895, 37659}, {35935, 37568}, {37075, 40587}

X(40863) = Steiner-circumellipse-inverse of X(78)

X(40864) = X(2)-HIRST INVERSE OF X(85)

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

X(40864) lies on these lines: {2, 85}, {7, 34019}, {8, 2898}, {497, 6604}, {527, 4569}, {664, 2340}, {673, 10030}, {693, 3900}, {2481, 9436}, {3008, 34018}, {3779, 31604}, {3912, 4554}, {4384, 6063}, {5057, 32007}, {6666, 31618}, {8012, 32024}, {14189, 28058}, {14727, 35160}

X(40864) = isotomic conjugate of X(14943)
X(40864) = Steiner-circumellipse-inverse of X(85)

X(40865) = X(2)-HIRST INVERSE OF X(100)

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

X(40865) is the intersection of the perspectrices of every pair of {ABC, 1st Montesdeoca bisector triangle, 2nd Montesdeoca bisector triangle}. (Randy Hutson, January 22, 2021)

X(40865) lies on these lines: {2, 11}, {101, 812}, {190, 644}, {384, 6162}, {545, 35110}, {666, 4762}, {692, 4164}, {693, 1252}, {1086, 9259}, {1146, 4422}, {1633, 21003}, {3732, 6084}, {4432, 24294}, {4473, 39351}, {4554, 31615}, {4776, 14513}, {4885, 4998}, {5375, 17494}, {10944, 28995}, {17197, 25536}, {30610, 31628}

X(40865) = Steiner-circumellipse-inverse of X(100)

X(40866) = X(2)-HIRST INVERSE OF X(110)

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

X(40866) lies on these lines: {2, 98}, {99, 112}, {115, 23583}, {384, 33695}, {458, 15928}, {543, 3163}, {620, 15526}, {804, 1576}, {850, 9514}, {2793, 35278}, {2794, 35923}, {2966, 23878}, {4108, 32729}, {7422, 10753}, {9530, 38738}, {11159, 23348}, {11794, 40173}, {14971, 40477}, {18020, 30476}, {31274, 40484}, {31296, 36830}

X(40866) = Steiner-circumellipse-inverse of X(110)
X(40866) = perspector of conic {{A,B,C,PU(145)}}

X(40867) = X(2)-HIRST INVERSE OF X(114)

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

X(40867) lies on these lines: {2, 98}, {6, 17035}, {20, 38873}, {69, 1972}, {148, 30226}, {193, 317}, {297, 3564}, {384, 14516}, {401, 1503}, {458, 18440}, {524, 39358}, {694, 40815}, {1993, 8878}, {2502, 37643}, {3124, 11433}, {3163, 5032}, {3580, 7665}, {3620, 10979}, {4235, 12383}, {5025, 11441}, {5191, 18437}, {5663, 35923}, {6330, 15258}, {6646, 37781}, {6655, 12111}, {6658, 12278}, {7779, 9473}, {9033, 39905}, {10553, 11064}, {11427, 20976}, {11440, 33260}, {11449, 33259}, {12168, 35936}, {12317, 38551}, {13567, 20998}, {15988, 26081}, {17037, 20080}, {18914, 26155}, {26205, 31804}, {28407, 32140}, {28723, 34224}, {32605, 32980}, {32762, 35296}

X(40867) = anticomplement of X(287)
X(40867) = Steiner-circumellipse-inverse of X(114)

X(40868) = X(2)-HIRST INVERSE OF X(142)

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

X(40868) lies on these lines: {2, 7}, {8, 20089}, {145, 28849}, {279, 20111}, {320, 3693}, {522, 26824}, {666, 29590}, {902, 10578}, {1212, 32007}, {1252, 1275}, {1536, 5843}, {1742, 3870}, {1914, 4644}, {2293, 3957}, {3000, 3935}, {3177, 6604}, {3726, 4419}, {4645, 4712}, {4666, 17247}, {4862, 24600}, {5088, 20096}, {6542, 25257}, {6603, 17078}, {9801, 36845}, {10580, 17449}, {11019, 33099}, {14828, 17365}, {14942, 17768}, {17116, 25006}, {17170, 26790}, {17347, 37658}, {17753, 26839}, {20015, 20080}, {21258, 32024}, {25728, 30813}, {26531, 30625}, {29569, 40779}

X(40868) = anticomplement of X(10025)
X(40868) = Steiner-circumellipse-inverse of X(142)

X(40869) = X(2)-HIRST INVERSE OF X(144)

Barycentrics    (a - b - c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) : :
Barycentrics    b sec^2(B/2) (cos A - cos B) + c sec^2(C/2) (cos A - cos C) : :

Let LA, LB, LC be the lines through A, B, C, resp. parallel to the Gergonne line. Let MA, MB, MC be the reflections of BC, CA, AB in LA, LB, LC, resp. Let A' = MB∩MC, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Gergonne line. The triangle A"B"C" is homothetic to ABC, and the center of homothety is X(40869). (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, January 22, 2021)

X(40869) lies on these lines: {1, 6554}, {2, 7}, {6, 11019}, {8, 23058}, {10, 220}, {11, 2348}, {19, 21068}, {31, 40128}, {33, 200}, {37, 800}, {39, 9367}, {41, 950}, {101, 515}, {118, 516}, {169, 946}, {198, 7580}, {218, 1210}, {219, 3686}, {346, 19605}, {348, 30625}, {350, 36796}, {406, 7079}, {517, 8074}, {518, 3041}, {519, 1146}, {522, 650}, {551, 34522}, {644, 6735}, {666, 2338}, {728, 7080}, {758, 31896}, {760, 960}, {997, 34526}, {1001, 15288}, {1012, 38902}, {1125, 1212}, {1323, 17044}, {1329, 30618}, {1449, 10580}, {1642, 24980}, {1699, 5819}, {1737, 5526}, {1802, 1855}, {1886, 24014}, {2082, 12053}, {2124, 34060}, {2182, 37374}, {2323, 4700}, {2329, 5795}, {2345, 8580}, {2391, 3732}, {2911, 24005}, {3008, 34852}, {3039, 11730}, {3086, 16572}, {3089, 7719}, {3119, 3930}, {3160, 30695}, {3207, 4297}, {3247, 10578}, {3663, 24352}, {3684, 5853}, {3692, 27522}, {3707, 5231}, {3726, 17435}, {3730, 6684}, {3811, 21096}, {3814, 6506}, {3879, 31038}, {3935, 31648}, {4119, 40609}, {4205, 18250}, {4258, 4314}, {4370, 35091}, {4384, 28827}, {4386, 16283}, {4513, 6736}, {4534, 5048}, {4667, 14548}, {5011, 28194}, {5134, 28150}, {5248, 32561}, {5274, 5838}, {5304, 7290}, {5514, 17757}, {5540, 30384}, {5574, 25568}, {5687, 7368}, {6181, 31477}, {6605, 7110}, {6700, 25066}, {6738, 21049}, {6909, 32625}, {7735, 16970}, {8756, 21801}, {9310, 10106}, {9312, 26658}, {9502, 26006}, {10405, 25718}, {10445, 19541}, {13411, 16601}, {14004, 21065}, {15629, 15633}, {15817, 20835}, {16600, 34937}, {17023, 30854}, {17095, 32024}, {17266, 39353}, {17351, 25355}, {17355, 20103}, {17732, 31730}, {18483, 24045}, {24771, 38015}, {24987, 27068}, {25082, 27385}, {25728, 30228}, {26036, 36660}, {26068, 27255}, {30457, 40161}, {34361, 34524}, {35094, 35111}, {36956, 37780}

X(40869) = isotomic conjugate of trilinear pole of line X(7)X(522)
X(40869) = complement of X(9436)
X(40869) = anticomplement of Steiner-inellipse-inverse of X(7)
X(40869) = Steiner-circumellipse-inverse of X(144)
X(40869) = Steiner-inellipse-inverse of X(9)

X(40870) = X(2)-HIRST INVERSE OF X(182)

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

X(40870) lies on these lines: {2, 98}, {22, 1972}, {248, 39355}, {275, 11606}, {323, 14931}, {385, 1971}, {394, 5989}, {401, 2782}, {458, 12188}, {648, 7766}, {1599, 8316}, {1600, 8317}, {1629, 36426}, {1915, 12829}, {1916, 1993}, {2023, 14153}, {2799, 31296}, {5152, 36212}, {8289, 15066}, {9478, 37649}, {12176, 22735}, {13188, 35941}

X(40870) = Steiner-circumellipse-inverse of X(182)

X(40871) = X(2)-HIRST INVERSE OF X(187)

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

X(40871) lies on these lines: {2, 187}, {111, 12191}, {194, 3906}, {353, 32480}, {384, 5108}, {385, 1316}, {669, 36182}, {1992, 39356}, {3734, 17941}, {5468, 34364}, {6792, 34604}, {7766, 35146}, {7823, 15000}, {8591, 35356}, {9878, 9999}, {11152, 39689}

X(40871) = anticomplement of X(40877)
X(40871) = Steiner-circumellipse-inverse of X(187)

X(40872) = X(2)-HIRST INVERSE OF X(200)

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

X(40872) lies on these lines: {1, 2}, {77, 27544}, {85, 4513}, {220, 16284}, {322, 27420}, {515, 20533}, {644, 10025}, {664, 3693}, {673, 3880}, {728, 9312}, {1016, 1275}, {1018, 5088}, {1447, 21232}, {1697, 17691}, {3436, 27129}, {3501, 7176}, {3685, 28850}, {4645, 28849}, {4875, 32008}, {4881, 31020}, {4919, 20335}, {4936, 30625}, {6913, 29331}, {9957, 17681}, {10914, 17682}, {14923, 27000}, {16593, 38455}, {27487, 34024}

X(40872) = Steiner-circumellipse-inverse of X(200)

X(40873) = X(2)-HIRST INVERSE OF X(257)

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

X(40873) lies on these lines: {1, 2653}, {2, 257}, {6, 256}, {39, 893}, {239, 1916}, {661, 3907}, {673, 2669}, {894, 40099}, {1575, 1581}, {1931, 40432}, {1967, 20456}, {2277, 40729}, {2886, 23897}, {3912, 27805}, {4039, 4876}, {8301, 8936}, {9230, 17788}, {17023, 32010}

X(40873) = isogonal conjugate of X(41534)
X(40873) = isotomic conjugate of X(40846)
X(40873) = cevapoint of X(256) and X(8936)
X(40873) = trilinear product X(i)*X(j) for these {i,j}: {256, 3509}, {257, 17798}, {694, 1281}, {893, 4645}, {904, 17789}, {1178, 4071}, {1581, 19557}, {1916, 19561}, {1934, 18038}, {1967, 18037}, {7018, 19554}, {20715, 40432}
X(40873) = Steiner-circumellipse-inverse of X(257)

X(40874) = X(2)-HIRST INVERSE OF X(274)

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

X(40874) lies on these lines: {1, 2668}, {2, 39}, {38, 17143}, {75, 34021}, {99, 2223}, {213, 7304}, {239, 799}, {335, 4639}, {350, 18827}, {512, 7192}, {536, 670}, {726, 24576}, {873, 16826}, {886, 3227}, {1920, 21883}, {2106, 39916}, {2111, 17738}, {2481, 18829}, {2664, 2669}, {3912, 40017}, {4623, 19623}, {18021, 37596}, {21820, 32026}, {24287, 25667}, {29578, 33779}

X(40874) = isotomic conjugate of isogonal conjugate of X(2106)
X(40874) = Steiner-circumellipse-inverse of X(274)

X(40875) = X(2)-HIRST INVERSE OF X(312)

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

X(40875) lies on these lines: {2, 37}, {7, 17786}, {8, 9025}, {10, 24463, {69, 4110}, {76, 4494}, {190, 2183}, {314, 4431}, {320, 4033}, {513, 4397}, {527, 668}, {646, 3912}, {889, 1121}, {1149, 32922}, {1909, 4363}, {1966, 10027}, {2064, 20237}, {2550, 36222}, {3006, 36238}, {3208, 30092}, {3262, 17789}, {3596, 3729}, {3963, 17116}, {4007, 34282}, {4398, 18044}, {4419, 6376}, {4429, 24338}, {4470, 31997}, {4562, 18816}, {4643, 25280}, {4644, 24524}, {4670, 25303}, {5263, 25382}, {6381, 17132}, {6692, 32017}, {7321, 18040}, {8680, 35544}, {10030, 31625}, {17276, 30473}, {17788, 20895}, {18145, 28301}, {20072, 25298}, {20902, 35550}, {21608, 25242}, {30693, 39126}, {30713, 32939}

X(40875) = Steiner-circumellipse-inverse of X(312)

X(40876) = X(2)-HIRST INVERSE OF X(315)

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

X(40876) lies on these lines: {2, 32}, {76, 5117}, {125, 3978}, {141, 37891}, {206, 33797}, {297, 39266}, {316, 419}, {420, 5207}, {702, 15449}, {826, 850}, {1502, 6697}, {5976, 21536}, {7747, 33336}, {7768, 37894}, {7799, 19599}, {8265, 40380}, {8920, 24206}, {18896, 19573}, {19571, 29012}, {20023, 23293}, {40359, 40421}

X(40876) = complement of X(16985)
X(40876) = Steiner-circumellipse-inverse of X(315)

X(40877) = X(2)-HIRST INVERSE OF X(316)

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

X(40877) lies on these lines: {2, 187}, {69, 22254}, {76, 850}, {99, 36194}, {183, 5641}, {325, 11007}, {599, 892}, {671, 8288}, {1648, 7790}, {3005, 38526}, {3314, 35077}, {5108, 7883}, {5468, 7850}, {6656, 32525}, {6698, 18023}, {6792, 7827}, {7664, 11628}, {7768, 38940}, {7802, 15000}, {11331, 18020}, {23301, 36165}

X(40877) = complement of X(40871)
X(40877) = Steiner-circumellipse-inverse of X(316)

X(40878) = X(2)-HIRST INVERSE OF X(320)

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

X(40878) lies on these lines: {2, 44}, {75, 693}, {214, 32032}, {1647, 4389}, {3262, 20935}, {3661, 35123}, {3679, 4555}, {3912, 24318}, {4357, 25377}, {4384, 18821}, {6702, 20568}, {9458, 17271}, {17057, 20569}, {17274, 27922}, {17360, 17780}, {23816, 25031}

X(40878) = isotomic conjugate of antitomic conjugate of X(80)
X(40878) = Steiner-circumellipse-inverse of X(320)

X(40879) = X(2)-HIRST INVERSE OF X(323)

Barycentrics    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 : :

X(40879) lies on these lines: {2, 6}, {3, 523}, {30, 15919}, {50, 3260}, {76, 4590}, {98, 2854}, {237, 32224}, {264, 14590}, {338, 4558}, {340, 34827}, {526, 18332}, {648, 34990}, {892, 7771}, {924, 31848}, {1078, 31998}, {1316, 5467}, {1350, 7422}, {1510, 18321}, {1634, 9512}, {1975, 14588}, {1989, 6148}, {2453, 4226}, {2782, 9145}, {2847, 38747}, {2966, 30528}, {3734, 14966}, {5463, 11659}, {5464, 11658}, {5914, 21448}, {5968, 9832}, {5999, 8705}, {7493, 16320}, {7668, 22143}, {7746, 23991}, {7761, 40553}, {7782, 33799}, {9123, 9130}, {9129, 9185}, {9142, 12042}, {9308, 16237}, {9769, 9775}, {11074, 39290}, {11079, 40423}, {11130, 23895}, {11131, 23896}, {11594, 11676}, {14995, 36194}, {16315, 30739}, {17948, 35955}, {18301, 40604}, {18311, 39078}, {24345, 37522}, {31859, 35936}, {35473, 38294}, {37283, 37455}

X(40879) = isotomic conjugate of isogonal conjugate of X(32761)
X(40879) = isotomic conjugate of antigonal conjugate of X(94)
X(40879) = complement of isotomic conjugate of antigonal conjugate of X(2986)
X(40879) = complement of crossdifference of X(512) and X(3284)
X(40879) = anticomplement of X(18122)
X(40879) = Steiner-circumellipse-inverse of X(323)
X(40879) = {X(6189),X(6190)}-harmonic conjugate of X(323)

X(40880) = X(2)-HIRST INVERSE OF X(329)

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

X(40880) lies on these lines: {2, 7}, {10, 24341}, {69, 2324}, {75, 20262}, {101, 6518}, {190, 37774}, {219, 4416}, {220, 4643}, {281, 3729}, {312, 20205}, {522, 3717}, {524, 6603}, {536, 1146}, {651, 26006}, {666, 15629}, {984, 31397}, {1012, 24320}, {1086, 34852}, {1212, 4364}, {1266, 4858}, {1420, 25905}, {1696, 10401}, {1785, 1948}, {3000, 21914}, {3041, 6007}, {3220, 6909}, {3718, 34404}, {3872, 3883}, {3912, 26932}, {4389, 30854}, {4419, 6554}, {4454, 27541}, {4480, 7359}, {4659, 23058}, {5179, 29069}, {5199, 17132}, {5795, 26117}, {6610, 17044}, {6916, 26939}, {7360, 16870}, {9623, 28849}, {10446, 21068}, {10468, 15479}, {12053, 26116}, {13466, 35091}, {17279, 34524}, {22464, 30807}, {25364, 30618}, {25681, 34807}, {35086, 35130}

X(40880) = complement of X(40862)
X(40880) = Steiner-circumellipse-inverse of X(329)

X(40881) = X(2)-HIRST INVERSE OF X(330)

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

X(40881) lies on these lines: {1, 2162}, {2, 330}, {37, 87}, {75, 32033}, {194, 8026}, {239, 4598}, {335, 20363}, {536, 18830}, {649, 4083}, {932, 9111}, {982, 2319}, {1100, 21759}, {1575, 20467}, {2109, 3509}, {4657, 27341}, {7113, 34071}, {7146, 7153}, {15966, 18194}, {20868, 31061}, {23493, 40148}, {27436, 28358}, {27444, 28366}, {27465, 28395}, {29570, 40720}

X(40881) = Steiner-circumellipse-inverse of X(330)

X(40882) = X(2)-HIRST INVERSE OF X(333)

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

X(40882) lies on these lines: {2, 6}, {44, 25536}, {63, 40605}, {92, 2905}, {99, 527}, {190, 4053}, {226, 7058}, {261, 4416}, {314, 17788}, {319, 21076}, {320, 662}, {523, 4833}, {645, 3912}, {648, 8755}, {811, 1948}, {892, 1121}, {1001, 36223}, {1043, 5327}, {1086, 24378}, {1326, 17770}, {1509, 4667}, {1931, 20072}, {1944, 35145}, {2185, 33066}, {2966, 34393}, {4590, 4620}, {4643, 6626}, {4644, 17103}, {4655, 35916}, {4715, 16702}, {4758, 32014}, {5196, 17491}, {5263, 24345}, {5529, 18792}, {6646, 38814}, {7175, 7364}, {11104, 24695}, {11608, 17931}, {14616, 35148}, {17950, 17966}, {18151, 30939}, {20337, 26081}, {20349, 31297}, {20654, 32025}, {29824, 36239}, {32859, 40214}

X(40882) = isotomic conjugate of X(11608)
X(40882) = Steiner-circumellipse-inverse of X(333)
X(40882) = {X(6189),X(6190)}-harmonic conjugate of X(333)

X(40883) = X(2)-HIRST INVERSE OF X(346)

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

X(40883) lies on these lines: {2, 37}, {8, 9311}, {144, 6555}, {190, 910}, {304, 4515}, {337, 4562}, {341, 25242}, {513, 4468}, {517, 4568}, {519, 14759}, {1376, 3729}, {3699, 10025}, {3975, 39350}, {3991, 33942}, {4019, 14973}, {4110, 20935}, {4119, 24318}, {4386, 17351}, {4437, 9436}, {4561, 6603}, {4696, 25244}, {4885, 20907}, {5014, 31080}, {5836, 17760}, {6552, 30695}, {6554, 32034}, {6556, 10405}, {6706, 33933}, {21216, 21896}, {21432, 27096}, {25066, 33937}, {25263, 37548}, {30730, 30806}

X(40883) = Steiner-circumellipse-inverse of X(346)

X(40884) = X(2)-HIRST INVERSE OF X(376)

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

X(40884) lies on these lines: {2, 3}, {99, 11064}, {287, 524}, {323, 22146}, {343, 7811}, {394, 32833}, {525, 1636}, {1272, 22151}, {1503, 35278}, {3972, 37648}, {5306, 40814}, {5641, 22110}, {6389, 27377}, {11645, 15595}, {14907, 37638}, {14919, 16076}, {18487, 23583}, {22329, 23967}, {31859, 37645}, {34828, 36794}, {36427, 36889}

X(40884) = anticomplement of X(40885)
X(40884) = Steiner-circumellipse-inverse of X(376)
X(40884) = {X(2479),X(2480)}-harmonic conjugate of X(376)

X(40885) = X(2)-HIRST INVERSE OF X(381)

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

X(40885) lies on these lines: {2, 3}, {148, 3580}, {287, 11645}, {323, 1625}, {524, 39358}, {525, 14391}, {648, 18487}, {1494, 1972}, {1495, 10722}, {2966, 8859}, {3163, 36426}, {5641, 7840}, {7809, 36212}, {7898, 15066}, {11648, 40814}, {13114, 36900}, {14918, 39019}, {15595, 19924}, {32223, 39809}

X(40885) = anticomplement of X(40884)
X(40885) = Steiner-circumellipse-inverse of X(381)
X(40885) = {X(2479),X(2480)}-harmonic conjugate of X(381)

X(40886) = X(2)-HIRST INVERSE OF X(386)

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

X(40886) lies on these lines: {1, 2}, {71, 17148}, {81, 25303}, {313, 16685}, {514, 31296}, {730, 3747}, {740, 28375}, {742, 28369}, {1334, 31036}, {2176, 3765}, {2238, 25298}, {2300, 3963}, {3057, 19791}, {3219, 21226}, {3230, 3948}, {3915, 11320}, {5847, 24809}, {6360, 36858}, {7976, 23407}, {9022, 35550}, {9049, 32029}, {9840, 29331}, {17152, 17184}, {20110, 20348}, {20892, 28350}, {27469, 40459}

X(40886) = anticomplement of X(30059)
X(40886) = Steiner-circumellipse-inverse of X(386)
X(40886) = {X(2479),X(2480)}-harmonic conjugate of X(386)

X(40887) = X(2)-HIRST INVERSE OF X(393)

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

X(40887) lies on these lines: {2, 216}, {4, 9289}, {107, 385}, {114, 6530}, {148, 34170}, {193, 6525}, {194, 14249}, {325, 36426}, {520, 16230}, {538, 6528}, {2996, 6526}, {3926, 36434}, {6392, 6523}, {6529, 15014}, {9306, 9308}, {9418, 37070}, {21849, 27377}, {34854, 39931}

X(40887) = isotomic conjugate of trilinear pole of line X(520)X(6389)
X(40887) = polar conjugate of isogonal conjugate of X(15143)
X(40887) = Steiner-circumellipse-inverse of X(393)

X(40888) = X(2)-HIRST INVERSE OF X(394)

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

X(40888) lies on these lines: {2, 6}, {63, 7108}, {98, 8681}, {262, 9813}, {378, 36207}, {401, 3260}, {523, 2071}, {577, 14615}, {648, 36212}, {1943, 6359}, {2393, 5999}, {2452, 22087}, {3164, 9723}, {3284, 36841}, {3964, 9308}, {4590, 15014}, {6360, 6511}, {6512, 7361}, {7783, 19221}, {8062, 21761}, {8743, 28441}, {9818, 32515}, {10607, 20477}, {21531, 22143}

X(40888) = complement of isotomic conjugate of antigonal conjugate of X(801)
X(40888) = Steiner-circumellipse-inverse of X(394)
X(40888) = {X(6189),X(6190)}-harmonic conjugate of X(394)

X(40889) = X(2)-HIRST INVERSE OF X(427)

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

X(40889) lies on these lines: {2, 3}, {112, 14712}, {232, 316}, {264, 7761}, {287, 18400}, {317, 2549}, {340, 538}, {385, 5523}, {648, 754}, {1235, 2896}, {1968, 7802}, {2211, 22151}, {3199, 7842}, {3849, 14581}, {4045, 36794}, {7737, 17907}, {7750, 27376}, {7785, 39575}, {7790, 10311}, {7797, 10312}, {7823, 8743}, {7830, 27371}, {12110, 39604}, {15048, 27377}, {32002, 33843}, {34163, 39359}

X(40889) = polar conjugate of trilinear pole of line X(523)X(7745) (the line through X(4) of the pedal triangles of PU(1))
X(40889) = anticomplement of X(15013)
X(40889) = Steiner-circumellipse-inverse of X(427)
X(40889) = {X(2479),X(2480)}-harmonic conjugate of X(427)

X(40890) = X(2)-HIRST INVERSE OF X(468)

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

X(40890) lies on these lines: {2, 3}, {6, 30227}, {69, 34360}, {112, 148}, {132, 10723}, {193, 525}, {287, 2777}, {317, 39062}, {340, 3849}, {393, 23582}, {543, 648}, {671, 6103}, {877, 32815}, {1560, 7665}, {2967, 13172}, {5095, 8593}, {5641, 23334}, {5984, 13200}, {7737, 16237}, {11161, 32250}, {14900, 38664}, {16077, 18823}, {32479, 37765}, {32827, 35088}

X(40890) = anticomplement of X(35923)
X(40890) = Steiner-circumellipse-inverse of X(468)
X(40890) = {X(2479),X(2480)}-harmonic conjugate of X(468)

X(40891) = X(2)-HIRST INVERSE OF X(551)

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

X(40891) lies on these lines: {1, 2}, {75, 4795}, {99, 2384}, {190, 28309}, {192, 37654}, {319, 17382}, {376, 29331}, {514, 4984}, {524, 903}, {536, 17487}, {545, 4969}, {740, 31310}, {742, 1992}, {752, 4716}, {1654, 4852}, {3656, 7384}, {3759, 17281}, {3765, 4479}, {3875, 17333}, {4034, 17396}, {4360, 17330}, {4361, 17378}, {4366, 4370}, {4371, 17379}, {4399, 28604}, {4402, 17375}, {4419, 17488}, {4440, 4715}, {4460, 4704}, {4464, 17260}, {4473, 4908}, {4687, 4910}, {4725, 31138}, {4856, 17116}, {4980, 17789}, {5839, 6646}, {6653, 32108}, {6999, 28204}, {9263, 17495}, {9812, 28909}, {17117, 20090}, {17151, 31300}, {17264, 28329}, {17271, 17302}, {17274, 17363}, {17297, 28337}, {17299, 17342}, {17313, 17377}, {17346, 24441}, {17772, 31151}, {26839, 36205}, {31178, 31314}, {32087, 37677}, {35076, 35121}

X(40891) = anticomplement of X(17310)
X(40891) = Steiner-circumellipse-inverse of X(551)

X(40892) = X(2)-HIRST INVERSE OF X(553)

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

X(40892) lies on these lines: {2, 7}, {77, 17389}, {241, 17264}, {269, 17294}, {279, 33935}, {320, 27526}, {519, 5018}, {536, 17078}, {664, 4971}, {1323, 28313}, {1418, 17359}, {1443, 6542}, {4670, 32007}, {4725, 6610}, {5263, 5434}, {6180, 17346}, {6648, 18821}, {17086, 17301}, {17092, 17280}, {17251, 33298}, {39353, 39358}

X(40892) = Steiner-circumellipse-inverse of X(553)

X(40893) = X(2)-HIRST INVERSE OF X(556)

Barycentrics    Csc[A/2]^2 - Csc[B/2]*Csc[C/2] : :

X(40893) lies on these lines: {2, 556}, {75, 236}, {173, 3729}, {174, 312}, {258, 30567}, {321, 8126}, {1089, 30408}, {3685, 8076}, {3705, 8379}, {3912, 21623}, {4358, 8125}, {4385, 8351}, {4692, 30411}, {4975, 30423}, {7028, 18743}, {7081, 7589}, {8083, 24349}

X(40893) = Steiner-circumellipse-inverse of X(556)

X(40894) = EULER LINE INTERCEPT OF X(74)X(32618)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*((a^2 - b^2 - c^2)*(5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4) + 2*Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)]*S) : :
Barycentrics    SB*SC*(SA*(2*S^2 - 3*SB*SC) + S*Sqrt[SA*SB*SC*SW]) : :
X(40894) = 3 X[2] - 4 X[31665], 3 X[186] - 2 X[5001], 4 X[31664] - 5 X[37952]

X(40894) lies on the curve Q163 and these lines: {2, 3}, {74, 32618}

X(40894) = reflection of X(i) in X(j) for these {i,j}: {4, 5000}, {5002, 3}, {40895, 10295}
X(40894) = circumcircle-inverse of X(40895)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35485, 40895}, {3, 378, 40895}, {3, 381, 31664}, {3, 5000, 186}, {4, 376, 40895}, {20, 18533, 40895}, {24, 21312, 40895}, {186, 7464, 40895}, {549, 35492, 40895}, {550, 7576, 40895}, {1113, 1114, 40895}, {3520, 35921, 40895}, {3524, 35483, 40895}, {3651, 4227, 40895}, {4221, 7414, 40895}, {4235, 7422, 40895}, {4238, 7429, 40895}, {4249, 7440, 40895}, {7421, 7436, 40895}, {7430, 7431, 40895}, {7454, 7463, 40895}, {7473, 36164, 40895}, {8703, 35484, 40895}, {11676, 35474, 40895}, {15160, 15161, 40895}, {36001, 37961, 40895}, {37960, 37979, 40895}

X(40895) = EULER LINE INTERCEPT OF X(74)X(32619)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*((a^2 - b^2 - c^2)*(5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4) - 2*Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)]*S) : :
Barycentrics    SB*SC*(SA*(2*S^2 - 3*SB*SC) - S*Sqrt[SA*SB*SC*SW])
X(40895) = 3 X[2] - 4 X[31664], 3 X[186] - 2 X[5000], 4 X[31665] - 5 X[37952]

X(40895) lies on the curve Q163 and these lines: {2, 3}, {74, 32619}

X(40895) = reflection of X(i) in X(j) for these {i,j}: {4, 5001}, {5003, 3}, {40895, 10295}
X(40895) = circumcircle-inverse of X(40894)
X(40895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35485, 40894}, {3, 378, 40894}, {3, 381, 31665}, {3, 5001, 186}, {4, 376, 40894}, {20, 18533, 40894}, {24, 21312, 40894}, {186, 7464, 40894}, {549, 35492, 40894}, {550, 7576, 40894}, {1113, 1114, 40894}, {3520, 35921, 40894}, {3524, 35483, 40894}, {3651, 4227, 40894}, {4221, 7414, 40894}, {4235, 7422, 40894}, {4238, 7429, 40894}, {4249, 7440, 40894}, {7421, 7436, 40894}, {7430, 7431, 40894}, {7454, 7463, 40894}, {7473, 36164, 40894}, {8703, 35484, 40894}, {11676, 35474, 40894}, {15160, 15161, 40894}, {36001, 37961, 40894}, {37960, 37979, 40894}

leftri

Perspectors associated with Gemini triangle 111: X(40896)-X(40908)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, January 3, 2021.

Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC, and let A'B'C' be the triangle given by

A' = t*(q + r) : - q : - r
B' = - p : t*(r + p) : - r
C' = - p : - q : t*(p + q),

where t is a function symmetric in a,b,c and of degree 0 of homogeneity. If t = 0, then A'B'C' = cevian triangle of P. Taking t = -2 gives A' = 2(q+r) : - q : - r, and in this case, A'B'C' is perspective to the Gemini 111 triangle A"B'C', where A" = -3 b c : a c : a b, and B" and C" are defined cyclically. The perspector is given by

M(P) = p q + p r - 3 q r : q r + q p - 3 r p : r p + r q - 3 p q.

Let A1B1 C1 = medial triangle of ABC and A2B2 C2 = anticomplementary triangle of ABC. Let M* be the collineation that maps (X(2), A1, B1, C1) onto (X(2), A2, B2, C2. Then M(P) = M*(tP), where tP = isotomic conjugate of P.

Equivalently, M(P) is the M(P) is the anticomplement of the anticomplement of the isotomic conjugate of P. Let M* denote the inverse of M. Then M*(P) is the isotomic conjugate of the complement of the complement of P, given by.

M*(P) = (p + 2*q + r)*(p + q + 2*r) : :

The appearance of (i,j) in the following list means that M(X(i)) = X(j):

(1,1278), (2,2), (4,20080), (6,20081), (7,3621), (8,20059), (9,20089), (10,20090), (20,20218), (27,20017), (69,3146), (75,145), (76,193), (83,69), (85,144), (86,8), (87,21219), (92,20078), (95,4), (98,7779), (99,523), (100,26824), (141,20088), (189,20214), (190,514), (253,5059), (257,31300), (261,20060), (264,20), (273,20013), (274,192), (276,3164), (287,40853), (290,511), (302,22113), (303,22114), (304,20061), (305,7500), (306,31292), (307,31294), (308,194), (309,20070), (310,20011), (312,9965), (313,20077), (315,20079), (318,20082), (319,20084), (320,20085), (321,20086), (325,5984), (327,6776), (329,20215), (330,4788), (331,20110), (333,5905), (335,20016), (491,12221), (492,12222), (513,31298), (523,20094), (524,8596), (561,20064), (598,11160), (648,525), (658,25259), (664,522), (666,918), (668,513), (670,512), (671,524), (673,6542), (693,20095), (799,31290), (801,6515), (850,14683), (870,31302), (871,19994), (886,888), (889,891), (892,690), (903,519), (1016,4440), (1032,20217), (1034,20216), (1088,20015), (1121,527), (1220,6646), (1221,21226), (1240,20040), (1241,8267), (1268,1), (1275,39351), (1441,15680), (1494,30), (1502,20065), (1577,31297), (1581,30662), (1698,31313), (1799,7391), (1930,20087), (1969,20074), (1978,26853), (2373,5189), (2481,518), (2966,2799), (2986,37779), (2998,20105), (3112,20068), (3224,32747), (3225,698), (3226,726), (3227,536), (3228,538), (3261,20096), (3263,20097), (3264,20098), (3266,20099), (3596,20076), (3634,31334), (4358,20092), (4373,20014), (4554,17494), (4555,900), (4562,812), (4569,3900), (4577,826), (4586,824), (4590,148), (4597,4777), {4598,20295), {4609,31299), {4632,14779), {4671,20093), {4997,3218), {4998,149), {5641,542), {5936,3623), {6063,20075), {6189,3414), {6190,3413), {6330,401), {6331,31296), {6335,17496), {6376,20091), {6384,20012), {6385,20109), {6386,31291), {6528,520), {6540,4977), (6548,20058), (6606,6362), (6635,6550), (6648,3910), (7018,20101), (7035,17154), (7249,20056), (8777,17950), (8781,385), (8797,3522), (9229,6658), (9293,31372), (9476,40867), (10159,6), (10302,1992), (11117,532), (11118,533), (11119,616), (11120,617), (14061,14588), (14534,2895), (14616,758), (14621,20055), (14728,33906), (14942,40868), (14970,732), (15164,2574), (15165,2575), (15455,4560), (15466,20213), (16077,9033), (16080,323), (18018,20062), (18019,20063), (18020,3448), (18023,14712), (18025,516), (18026,521), (18299,25304), (18816,517), (18821,528), (18822,537), (18823,543), (18824,696), (18825,712), (18826,714), (18827,740), (18828,782), (18829,804), (18830,4083), (18831,6368), (20336,31293), (20563,31304), (20564,31305), (20565,20066), (20566,20067), (20567,20071), (20568,20072), (20569,20073), (20948,20100), (23582,39352), (23895,23870), (23896,23871), (27191,32094), (27483,29588), (27805,7192), (28626,4678), (28650,3622), (30598,3617), (30610,693), (30663,30668), (30701,4452), (30705,30695), (30710,17147), (30712,20052), (30786,23), (31002,19998), (31008,21223), (31360,14035), (31617,17035), (31618,3177), (31619,17036), (31621,39358), (31623,6360), (31624,21225), (31625,9263), (31643,3869), (32008,7), (32009,75), (32010,17165), (32011,10453), (32012,320), (32014,1654), (32015,9), (32016,3952), (32017,3210), (32020,17759), (32021,3681), (32023,17784), (32036,23872), (32037,23873), (32038,23880), (32039,23886), (32041,4762), (32042,4802), (33665,8272), (34234,17484), (34393,515), (34403,17037), (34404,20211), (34537,25047), (35136,3566), (35137,7927), (35138,3906), (35139,526), (35140,1503), (35141,17768), (35142,3564), (35143,35101), (35144,35102), (35145,8680), (35146,5969), (35147,2787), (35148,2786), (35149,2792), (35150,2784), (35151,2783), (35152,2795), (35153,2796), (35154,2785), (35155,35103), (35156,8674), (35157,6366), (35158,5845), (35159,35104), (35160,5853), (35162,17770), (35164,2801), (35168,545), (35170,4715), (35171,3887), (35172,9055), (35174,3738), (35175,2802), (35179,1499), (35181,4160), (35511,35369), (36588,20049), (36804,21222), (36805,17495), (36807,239), (36889,15683), (36897,8782), (36948,3832), (37213,5195), (37870,28605), (38342,15412), (39287,3410), (39626,39624), (39700,20046), (39704,31145), (39707,20054), (39710,20050), (39717,24349), (39738,4821), (39968,76), (40301,40327), (40410,3), (40411,4329), (40412,2475), (40413,1370), (40414,3151), (40415,6327), (40416,315), (40417,962), (40418,17135), (40419,3434), (40420,329), (40421,5596), (40422,3868), (40423,146), (40424,12649), (40425,2896), (40427,18301), (40428,147), (40429,99), (40435,17483), (40702,20212), (40706,3181), (40707,3180), (40829,19569), (40834,17493), (40835,17485).

X(40896) = M(X(3))

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

X(40896) lies on these lines: {2, 216}, {20, 31388}, {23, 16313}, {69, 40853}, {253, 1972}, {317, 39352}, {401, 9308}, {458, 15851}, {511, 3146}, {2395, 38262}, {3091, 30258}, {3832, 39530}, {7391, 7779}, {7486, 10003}, {17484, 20017}, {17578, 20218}, {20477, 36748}

X(40896) = isogonal conjugate of X(36617)
X(40896) = isotomic conjugate of X(38256)
X(40896) = polar conjugate of X(38264)
X(40896) = anticomplement of X(3164)

X(40897) = M(X(5))

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

X(40897) lies on these lines: {2, 95}, {4, 31389}, {20, 31388}, {53, 40853}, {340, 22052}, {1204, 3098}, {1249, 39081}, {2992, 19779}, {2993, 19778}, {6636, 7779}, {14966, 20088}, {20017, 26934}

X(40897) = anticomplement of X(17035)

X(40898) = M(X(13))

Barycentrics    Sqrt[3]*(5*a^2 - 3*b^2 - 3*c^2) - 2*S : :

X(40898) lies on these lines: {2, 6}, {14, 634}, {383, 11898}, {531, 20094}, {532, 621}, {533, 616}, {628, 16242}, {633, 10653}, {1080, 34380}, {3091, 20425}, {5965, 6770}, {9115, 11129}, {17364, 40714}, {19776, 19778}, {19924, 36344}, {22495, 33560}, {33412, 33464}, {33561, 36368}

X(40898) = reflection of X(40899) in X(7779)
X(40898) = anticomplement of X(3180)
X(40898) = {X(2),X(20080)}-harmonic conjugate of X(40899)

X(40899) = M(X(14))

Barycentrics    Sqrt[3]*(5*a^2 - 3*b^2 - 3*c^2) + 2*S : :

X(40899) lies on these lines: {2, 6}, {13, 633}, {383, 34380}, {530, 20094}, {532, 617}, {533, 622}, {627, 16241}, {634, 10654}, {1080, 11898}, {3091, 20426}, {5965, 6773}, {9117, 11128}, {17364, 40713}, {19777, 19779}, {19924, 36319}, {22496, 33561}, {33413, 33465}, {33560, 36366}

X(40899) = reflection of X(40898) in X(7779)
X(40899) = anticomplement of X(3181)
X(40899) = {X(2),X(20080)}-harmonic conjugate of X(40898)

X(40900) = M(X(17))

Barycentrics    5*a^2 - 3*b^2 - 3*c^2 - 2*Sqrt[3]*S : :

X(40900) lies on these lines: {2, 6}, {4, 5487}, {15, 627}, {16, 633}, {20, 5872}, {532, 16808}, {533, 10645}, {616, 6777}, {617, 10646}, {618, 34754}, {622, 16809}, {634, 18581}, {636, 16961}, {2993, 19778}, {3098, 6773}, {3104, 5334}, {3146, 5864}, {3642, 34755}, {5059, 5868}, {5335, 7900}, {5471, 11128}, {5472, 40707}, {5488, 22237}, {5617, 37517}, {6646, 37794}, {7893, 11486}, {7906, 11485}, {7941, 11542}, {11543, 17129}, {11898, 37464}, {14144, 25236}, {17116, 40714}, {18582, 22113}, {19107, 22844}, {19779, 36300}, {34380, 37463}

X(40900) = anticomplement of anticomplement of X(302)
X(40900) = {X(2),X(20080)}-harmonic conjugate of X(40901)
X(40900) = {X(6),X(7779)}-harmonic conjugate of X(40901)

X(40901) = M(X(18))

Barycentrics    5*a^2 - 3*b^2 - 3*c^2 + 2*Sqrt[3]*S : :

X(40901) lies on these lines: {2, 6}, {4, 5488}, {15, 634}, {16, 628}, {20, 5873}, {532, 10646}, {533, 16809}, {616, 10645}, {617, 6778}, {619, 34755}, {621, 16808}, {633, 18582}, {635, 16960}, {2992, 19779}, {3098, 6770}, {3105, 5335}, {3146, 5865}, {3643, 34754}, {5059, 5869}, {5334, 7900}, {5471, 40706}, {5472, 11129}, {5487, 22235}, {5613, 37517}, {6646, 37795}, {7893, 11485}, {7906, 11486}, {7941, 11543}, {11542, 17129}, {11898, 37463}, {14145, 25235}, {17116, 40713}, {18581, 22114}, {19106, 22845}, {19778, 36301}, {34380, 37464}

X(40901) = anticomplement of anticomplement of X(303)
X(40901) = {X(2),X(20080)}-harmonic conjugate of X(40900)
X(40901) = {X(6),X(7779)}-harmonic conjugate of X(40900)

X(40902) = M(X(19))

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

X(40902) lies on these lines: {2, 304}, {144, 25270}, {145, 740}, {350, 20936}, {3177, 3797}, {3210, 7187}, {4704, 26759}, {5905, 6542}, {6360, 25257}, {10453, 33890}, {17230, 21071}, {18156, 26274}, {20080, 34381}, {20245, 20535}, {20929, 30699}, {21281, 35101}, {28594, 33936}

X(40902) = anticomplement of X(21216)

X(40903) = M(X(21))

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

X(40903) lies on these lines: {2, 92}, {145, 3332}, {346, 20444}, {518, 1278}, {1108, 17895}, {1826, 3007}, {3218, 17117}, {5905, 20017}, {8680, 21271}, {9312, 26651}, {9965, 31303}, {17221, 24315}, {17242, 31053}, {17479, 30690}, {18663, 28605}, {25243, 30807}

X(40903) = anticomplement of anticomplement of X(1441)
X(40903) = anticomplement of polar conjugate of isogonal conjugate of X(23171)

X(40904) = M(X(25))

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

X(40904) lies on these lines: {2, 39}, {1278, 33090}, {1369, 32747}, {1799, 7781}, {3620, 23642}, {5133, 7906}, {7391, 7779}, {7485, 17129}, {7754, 16951}, {7758, 8878}, {7766, 16949}, {7855, 16275}, {8681, 12058}, {11324, 22253}, {20062, 20094}, {21219, 33091}

X(40904) = anticomplement of anticomplement of X(305)
X(40904) = anticomplement of polar conjugate of isogonal conjugate of X(19597)

X(40905) = M(X(29))

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

X(40905) lies on these lines: {2, 7}, {320, 27396}, {347, 20110}, {991, 17364}, {1333, 4644}, {2287, 17347}, {2323, 17075}, {2895, 18663}, {2897, 6360}, {3418, 37106}, {3621, 20216}, {3868, 5751}, {3927, 25015}, {3998, 32859}, {4073, 6327}, {4257, 5703}, {4419, 5738}, {5736, 17365}, {5742, 7228}, {8680, 21270}, {9028, 17134}, {15936, 16777}, {17220, 24316}, {17276, 17863}, {17334, 18635}, {17343, 20089}, {20013, 20080}

X(40905) = anticomplement of anticomplement of X(307)

X(40906) = M(X(31))

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

X(40906) lies on these lines: {2, 561}, {145, 718}, {193, 720}, {712, 20055}, {714, 1278}, {722, 20064}, {724, 20065}, {8620, 20945}, {9017, 20080}, {31000, 36645}, {33796, 39345}

X(40906) = isotomic conjugate of X(38276)
X(40906) = anticomplement of X(17486)

X(40907) = M(X(32))

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

X(40907) lies on these lines: {2, 308}, {69, 698}, {145, 704}, {193, 706}, {194, 9230}, {264, 9865}, {700, 1278}, {708, 20064}, {710, 20065}, {2998, 3978}, {3852, 20079}, {7391, 7779}, {7774, 15437}, {20080, 34383}, {33797, 39346}

X(40907) = anticomplement of X(8264)

X(40908) = M(X(37))

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

X(40908) lies on these lines: {2, 39}, {8, 31298}, {69, 33823}, {75, 21226}, {145, 740}, {321, 7187}, {330, 4441}, {385, 17693}, {404, 17129}, {536, 25303}, {764, 26824}, {1475, 17029}, {1654, 33831}, {1909, 17759}, {1975, 16998}, {2227, 3617}, {2475, 7779}, {2476, 7906}, {2795, 15680}, {3552, 17002}, {3761, 26752}, {3933, 33841}, {4044, 26113}, {4189, 16996}, {4373, 27447}, {4440, 17152}, {4754, 33296}, {7200, 17762}, {7754, 16915}, {7766, 16919}, {7783, 37670}, {7839, 17686}, {7893, 17579}, {7941, 17577}, {8682, 17141}, {9263, 17143}, {9965, 20089}, {10459, 17116}, {11321, 22253}, {17001, 33062}, {17007, 33829}, {17128, 33854}, {17155, 33890}, {17684, 31859}, {20080, 37435}, {20109, 31300}, {20888, 26801}, {21024, 33947}, {24215, 31027}, {24275, 33955}, {24330, 34063}, {25244, 33889}, {30571, 39740}, {39028, 40598}

X(40908) = anticomplement of X(1655)

X(40909) = X(4)-CEVA CONJUGATE OF X(381)

Barycentrics    (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(3*a^6 - 5*a^4*b^2 + a^2*b^4 + b^6 - 5*a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 - b^2*c^4 + c^6) : :
X(40909) = 4 X[5] - 3 X[32620], 2 X[141] - 3 X[18420], 3 X[381] - 2 X[4550], 3 X[568] - X[10938], X[3426] - 3 X[3830], 5 X[3763] - 4 X[33533], 2 X[4549] - 3 X[32620], 3 X[9818] - 4 X[19130], 3 X[14852] - 2 X[34801], X[18440] - 3 X[18494]

Let P = p : q : r be a point in the plane of ABC, and let A'B'C' be the central triangle defined by A' = q + r + - 2 q : - 2 r, so that A'B'C' is the triangle discussed in the preamble just before X(40896) in the case t = -1/2. If P = X(376), the triangle A'B'C' is perspective to the orthic triangle, and the perspector is X(40909).

X(40909) lies on the Feuerbach circumhyperbola of the orthic triangle and these lines: {2, 35254}, {3, 3574}, {4, 3580}, {5, 4549}, {6, 30}, {20, 37506}, {25, 113}, {52, 12173}, {74, 31133}, {79, 7986}, {110, 10294}, {141, 18420}, {155, 3575}, {185, 382}, {265, 541}, {376, 14389}, {381, 1531}, {394, 38321}, {567, 1657}, {648, 16263}, {1154, 40341}, {1192, 13371}, {1351, 5095}, {1498, 11819}, {1499, 38359}, {1533, 18534}, {1598, 22550}, {1620, 23336}, {1829, 40263}, {1843, 13754}, {1853, 10264}, {1858, 5727}, {1986, 3060}, {1993, 2914}, {2777, 23049}, {2904, 6240}, {3089, 15751}, {3146, 12022}, {3517, 5448}, {3534, 14805}, {3543, 37644}, {3627, 5895}, {3763, 33533}, {5073, 11820}, {5076, 18430}, {5094, 32110}, {5654, 37458}, {5663, 25335}, {5889, 6152}, {5892, 18536}, {5925, 34798}, {6000, 34775}, {7487, 22660}, {7519, 32111}, {7576, 18451}, {9786, 18569}, {9818, 19130}, {10113, 18434}, {10296, 11002}, {10605, 31723}, {10982, 18563}, {10996, 33543}, {11817, 12111}, {12118, 31802}, {12160, 13431}, {12161, 17845}, {12225, 36752}, {12302, 18532}, {12362, 15805}, {13352, 34397}, {13568, 14790}, {14070, 18388}, {14791, 37475}, {15053, 31180}, {15068, 38322}, {15311, 18382}, {15361, 39484}, {16625, 34786}, {17814, 31830}, {18281, 37487}, {18400, 34777}, {18531, 37648}, {18533, 37645}, {19457, 20127}, {20125, 35264}, {20304, 26958}, {21659, 37493}, {22948, 35490}, {31152, 37470}, {31724, 37490}, {31815, 37498}, {32395, 40686}, {37483, 38323}

X(40909) = midpoint of X(5073) and X(11820)
X(40909) = reflection of X(i) in X(j) for these {i,j}: {3, 7706}, {1657, 8717}, {4549, 5}, {11472, 4}, {34802, 10113}, {35237, 4846}
X(40909) = complement of X(41465)
X(40909) = anticomplement of X(35254)
X(40909) = orthic-isogonal conjugate of X(381)
X(40909) = X(4)-Ceva conjugate of X(381)
X(40909) = crosspoint of X(4) and X(18533)
X(40909) = crosssum of X(3) and X(34801)
X(40909) = X(4549)-of-Johnson-triangle
X(40909) = barycentric product X(i)*X(j) for these {i,j}: {381, 37645}, {18533, 37638}
X(40909) = barycentric quotient X(5158)/X(34801)
X(40909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 37489, 14852}, {5, 4549, 32620}, {52, 12173, 12293}, {381, 3581, 37638}, {382, 568, 18396}, {1531, 34417, 381}

X(40910) = X(1)X(3)∩X(7)X(24309)

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

X(40910) is the homothetic center of the 3rd antipedal triangle of X(1) and the 3rd antipedal triangle of X(3). (Randy Hutson, January 22, 2021)

X(40910) lies on the curve Q165 and these lines: {1, 3}, {7, 24309}, {8, 37254}, {9, 1486}, {10, 4223}, {11, 19512}, {22, 3870}, {23, 3935}, {25, 200}, {31, 4253}, {42, 251}, {44, 16686}, {100, 2725}, {101, 2340}, {105, 3008}, {109, 28838}, {154, 22153}, {169, 28043}, {197, 3158}, {198, 6600}, {210, 17744}, {220, 35273}, {228, 3970}, {238, 39979}, {269, 1037}, {497, 7397}, {516, 28071}, {518, 3220}, {527, 1633}, {572, 2293}, {573, 1253}, {579, 21059}, {610, 3174}, {651, 29353}, {667, 3900}, {672, 1438}, {674, 692}, {902, 5030}, {910, 16550}, {927, 40864}, {936, 11365}, {993, 36479}, {1001, 29598}, {1011, 16783}, {1103, 11398}, {1110, 2356}, {1260, 20991}, {1279, 5096}, {1283, 1580}, {1376, 17284}, {1479, 36670}, {1490, 9911}, {1593, 12651}, {1610, 12437}, {1621, 5314}, {1631, 15624}, {1699, 28053}, {1736, 6211}, {1743, 7083}, {1757, 20678}, {1766, 4319}, {1802, 1973}, {1855, 36010}, {1930, 32932}, {2172, 2187}, {2175, 3779}, {2177, 4262}, {2182, 15733}, {2183, 3939}, {2245, 19624}, {2251, 4258}, {2329, 15621}, {2330, 21746}, {2809, 7291}, {3052, 5022}, {3085, 7390}, {3243, 22769}, {3474, 24797}, {3555, 20833}, {3556, 11523}, {3583, 36654}, {3684, 20875}, {3689, 5525}, {3751, 7295}, {3873, 7293}, {3938, 5322}, {3941, 4497}, {3955, 14520}, {3957, 6636}, {4220, 13405}, {4222, 21075}, {4224, 4847}, {4228, 25006}, {4298, 37328}, {4314, 37399}, {4384, 26241}, {4421, 29573}, {4433, 32847}, {4512, 7085}, {4557, 23854}, {4666, 7485}, {4876, 34067}, {5014, 24587}, {5020, 8580}, {5120, 21002}, {5223, 24320}, {5251, 29659}, {5259, 29633}, {5526, 20683}, {5531, 13222}, {5534, 7387}, {6745, 33849}, {6762, 22654}, {6765, 9798}, {6996, 14942}, {7193, 9052}, {7289, 8271}, {7290, 36741}, {7297, 21889}, {7484, 10582}, {7718, 17562}, {7951, 36526}, {8053, 16503}, {8540, 20958}, {9451, 20871}, {9778, 17170}, {11019, 19649}, {11491, 36012}, {11500, 37412}, {12527, 28029}, {13588, 33953}, {15246, 29817}, {16547, 21867}, {16688, 36743}, {16779, 20992}, {16781, 21000}, {16782, 17735}, {16818, 17687}, {16845, 19784}, {16876, 18206}, {17192, 33068}, {17266, 31073}, {17582, 19836}, {17780, 26262}, {17784, 24604}, {18108, 32927}, {18528, 18534}, {18529, 18535}, {19557, 23398}, {20344, 28757}, {20556, 24630}, {20831, 34790}, {20857, 35342}, {22464, 40576}, {24047, 35270}, {26015, 37449}

X(40910) = reflection of X(i) in X(j) for these (i,j): {2323, 692}, {3220, 20872}
X(40910) = isogonal conjugate of anticomplement of X(39048)
X(40910) = isogonal conjugate of the isotomic conjugate of X(32850)
X(40910) = anticomplement of complementary conjugate of X(39048)
X(40910) = X(i)-Ceva conjugate of X(j) for these (i,j): {1280, 6}, {26703, 9}, {39714, 5280}
X(40910) = cevapoint of X(55) and X(20468)
X(40910) = crosspoint of X(i) and X(j) for these (i,j): {59, 6078}, {1477, 3451}
X(40910) = crosssum of X(i) and X(j) for these (i,j): {11, 6084}, {2254, 3942}, {3452, 5853}
X(40910) = crossdifference of every pair of points on line {650, 3752}
X(40910) = barycentric product X(i)*X(j) for these {i,j}: {6, 32850}, {9, 4318}, {1280, 39048}
X(40910) = barycentric quotient X(i)/X(j) for these {i,j}: {4318, 85}, {32850, 76}
X(40910) = {X(i), X(j)}-harmonic conjugate for these (i, j, k): {55, 17798, 2223}, {55, 37538, 5269}, {55, 37580, 1}, {55, 37586, 35}, {1486, 12329, 9}, {2223, 17798, 36}, {6600, 36641, 198}

leftri

Points associated the special dilation triangle: X(40911)-X(40920)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, January 4, 2021.

Let T denote the pedal triangle of X(2). There is exactly one dilation of T from X(2) that is perspective to ABC. It is the (-1)-dilation, here named the special dilation triangle, with vertices given by

A' = 4 a^2 : a^2 - b^2 + c^2 : a^2 + b^2 - c^2
B' = b^2 + c^2 - a^2 : 4 b^2 : b^2 - c^2 + a^2
C' = c^2 - a^2 + b^2 : c^2 + a^2 - b^2 : 4 c^2.

See Clark Kimberling and Peter J. C. Moses, "Dilation-Induced Perspectivities among Triangles", Journal for Geometry and Graphics 14 (2010) 1-14.

X(2)-of-A'B'C' = X(5650), and X(3)-of-A'B'C' = X(141)

The appearance of (T,i) in the following list means that A'B'C' is perspective to T, and the perspector is X(i):

(ABC, 69)
(tangential. 31521)
(2nd Euler, 125)
(4th extouch, 69)
(Artzt, 2)
(infinite altitude, 2)
(anti-Artzt, 2)
(orthic-of medial, 69)
(anti-Atik, 69)
(triangle of reflection of ABC in X(3)
, 40911)
(reflection of X(3)
in A,B,C, 40912)
(Kosnita, 40913)
Trnh, 40914)
(1st Parry, 40915)
(1st Ehrmann, 40916)
(anti-Hutson-intouch, 40917)
(anti-incircle-circles triangle, 40918)
(antiAOA, 40919)
(Ehrmann side-triangle,40920)

X(40911) = X(2)X(1350)∩X(20)X(7998)

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

X(40911) is the perspector of the special dilation triangle and the triangle of reflection of ABC in X(3).

X(40911) lies on these lines: {2, 1350}, {20, 7998}, {110, 10304}, {125, 10519}, {193, 22829}, {376, 11820}, {1498, 3522}, {3098, 4232}, {3292, 33750}, {3523, 7691}, {3528, 6090}, {3564, 40912}, {3620, 16063}, {3819, 6995}, {3917, 6776}, {4190, 14110}, {4430, 20020}, {4576, 15589}, {5059, 14490}, {5447, 12118}, {6800, 21734}, {7378, 24206}, {7496, 31521}, {10691, 11898}, {10989, 15431}, {12121, 13416}, {14683, 32233}, {15080, 38942}, {15448, 31884}, {15705, 40112}, {15717, 37645}, {17818, 18931}, {21735, 26864}, {23061, 33748}, {26637, 37267}, {31099, 31124}

X(40911) = isotomic conjugate of polar conjugate of X(14482)
X(40911) = anticomplement of anticomplement of X(5646)

X(40912) = X(2)X(1351)∩X(69)X(10300)

Barycentrics    (a^2 - b^2 - c^2)*(3*a^4 - 15*a^2*b^2 + 2*b^4 - 15*a^2*c^2 - 4*b^2*c^2 + 2*c^4): :

X(40912) is the perspector of the special dilation triangle and the A'B'C', where A' = reflection of X(3) in A, and B' and C' are defined cyclically.

X(40912) lies on these lines: {2, 1351}, {69, 10300}, {195, 15720}, {382, 1216}, {546, 18489}, {550, 13093}, {3529, 11469}, {3531, 3851}, {3564, 40911}, {3629, 31521}, {3631, 15583}, {3917, 11898}, {4549, 15681}, {5447, 11850}, {10301, 33878}, {10620, 15688}, {11935, 15700}, {17847, 19140}, {18440, 33884}

X(40913) = X(2)X(161)∩X(3)X(14389)

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

X(40913) is the perspector of the special dilation triangle and the Kosnita triangle. (See X(1658.)

X(40913) lies on these lines: {2, 161}, {3, 14389}, {54, 7516}, {69, 7496}, {110, 7514}, {125, 12584}, {140, 8907}, {182, 1993}, {378, 8717}, {631, 9932}, {1147, 7509}, {5085, 17847}, {5422, 9967}, {6759, 6800}, {7393, 14516}, {7484, 18911}, {7526, 8718}, {11427, 15246}, {15106, 19381}, {17928, 23358}, {26864, 34864}, {37126, 37645}, {38396, 40112}

X(40914) = X(2)X(32125)∩X(23)X(69)

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

X(40914) is the perspector of the special dilation triangle and the Trinh triangle. (See X(7688.)

X(40914) lies on these lines: {2, 32125}, {3, 32111}, {4, 9938}, {6, 12824}, {22, 1495}, {23, 69}, {24, 7689}, {25, 3580}, {26, 7691}, {74, 6644}, {113, 378}, {125, 1995}, {157, 669}, {206, 6800}, {381, 12412}, {403, 18532}, {1843, 32127}, {2931, 12825}, {3357, 15030}, {3448, 32282}, {3569, 21397}, {3579, 11337}, {6090, 37928}, {7485, 32600}, {7503, 32401}, {7526, 16013}, {7699, 31861}, {8907, 10539}, {9308, 33294}, {9927, 10594}, {10117, 34778}, {10516, 23330}, {11413, 22978}, {12111, 22750}, {13289, 15078}, {13595, 37643}, {15577, 35265}, {15818, 26881}, {18438, 40114}, {19588, 40316}, {21284, 32124}, {26156, 34207}, {32120, 36176}, {35228, 35266}

X(40915) = X(2)X(98)∩X(6)X(10552)

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

X(40915) is the perspector of the special dilation triangle and the 1st Parry triangle. (See X(9122.)

X(40915) lies on these lines: {2, 98}, {6, 10552}, {66, 10836}, {67, 9129}, {69, 111}, {115, 5468}, {141, 2502}, {148, 9146}, {316, 352}, {524, 3124}, {543, 4576}, {597, 20976}, {599, 20998}, {620, 35356}, {671, 4563}, {1641, 39691}, {1992, 39024}, {2482, 10330}, {2780, 32312}, {2872, 9131}, {3589, 39689}, {3618, 10553}, {3620, 7665}, {5106, 7810}, {5108, 11646}, {5191, 8369}, {6090, 11318}, {6388, 9172}, {7799, 14660}, {7811, 9998}, {7812, 9463}, {7833, 7998}, {7841, 15066}, {8288, 11053}, {8359, 9155}, {9035, 9979}, {10555, 17948}, {10754, 38940}, {11336, 18440}, {14928, 31128}, {15080, 33274}, {15993, 26276}, {16055, 34507}, {21766, 35955}, {32984, 37645}, {33228, 40112}, {35265, 35295}

X(40916) = X(2)X(3)∩X(6)X(5888)

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

X(40916) is the perspector of the special dilation triangle and the 1st Ehrmann triangle. (See X(8537.)

X(40916 lies on these lines: {2, 3}, {6, 5888}, {55, 7292}, {56, 5297}, {69, 31521}, {74, 32620}, {110, 5085}, {125, 12584}, {126, 34010}, {141, 8546}, {182, 3292}, {183, 3266}, {184, 20190}, {197, 9342}, {251, 22331}, {323, 5050}, {373, 3098}, {394, 11422}, {511, 21766}, {574, 3291}, {575, 1993}, {576, 3917}, {612, 5563}, {614, 3746}, {620, 34013}, {1036, 27625}, {1078, 11059}, {1125, 37546}, {1180, 1611}, {1196, 31652}, {1350, 5640}, {1351, 15018}, {1383, 15655}, {1473, 27065}, {1495, 16187}, {1560, 34105}, {1853, 15581}, {1975, 26235}, {2353, 7930}, {2453, 9159}, {2482, 13233}, {2930, 6698}, {2979, 10601}, {3055, 8553}, {3060, 5643}, {3066, 15107}, {3303, 7191}, {3304, 3920}, {3448, 32254}, {3455, 22247}, {3589, 37827}, {3964, 33861}, {5012, 10541}, {5013, 9465}, {5024, 11580}, {5092, 5651}, {5116, 32526}, {5201, 11174}, {5284, 37577}, {5314, 5437}, {5329, 17124}, {5347, 37682}, {5354, 9605}, {5359, 7772}, {5363, 17122}, {5550, 8193}, {5642, 32305}, {5968, 15899}, {6090, 11003}, {7085, 27003}, {7293, 7308}, {7295, 17125}, {7301, 17123}, {7754, 31088}, {7815, 30749}, {7940, 30785}, {7999, 36752}, {8585, 8589}, {9142, 12093}, {9149, 15271}, {9695, 13785}, {9716, 11402}, {9723, 37688}, {9798, 19877}, {10170, 11456}, {10192, 15579}, {10717, 33900}, {11002, 33878}, {11063, 31489}, {11412, 15805}, {11441, 37515}, {11444, 37514}, {11451, 33586}, {11480, 37775}, {11481, 37776}, {11482, 34545}, {12279, 33537}, {13339, 15068}, {13363, 37494}, {14810, 34417}, {14912, 38396}, {14997, 37492}, {15024, 37486}, {15028, 17834}, {15054, 20791}, {15080, 35259}, {15355, 36751}, {15574, 34803}, {15582, 23332}, {15815, 39576}, {16261, 35237}, {17004, 19577}, {19862, 37557}, {21167, 32269}, {21356, 32621}, {21448, 40103}, {22352, 35264}, {23858, 31073}, {24320, 35595}, {30950, 37576}, {32142, 36753}, {32599, 38064}, {33854, 37503}, {34482, 39951}, {34573, 35707}, {36740, 37680}, {36741, 37633}, {36743, 37675}

X(40916) = isogonal conjugate of X(38005)

X(40917) = X(2)X(1619)∩X(3)X(15448)

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 + 16*a^6*b^2*c^2 - 10*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 10*a^4*b^2*c^4 + 46*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 8*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10): :

X(40917) is the perspector of the special dilation triangle and the anti-Hutson-intouch triangle. (See X(11363.)

X(40917) lies on these lines: {2, 1619}, {3, 15448}, {5, 12301}, {25, 1350}, {40, 37034}, {64, 15030}, {69, 1995}, {110, 32621}, {125, 10516}, {159, 35259}, {1352, 5020}, {1593, 22549}, {1624, 37344}, {3066, 29959}, {4232, 37485}, {5544, 11579}, {5642, 38396}, {5907, 6642}, {5972, 9818}, {7387, 33543}, {7395, 32345}, {7506, 12307}, {7517, 33542}, {9914, 17928}, {10546, 26283}, {11344, 15622}, {11472, 16187}, {12083, 33544}, {13394, 31521}, {15082, 33540}, {15259, 32000}, {32237, 35243}

X(40918) = X(2)X(39879)∩X(125)X(32254)

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 - 32*a^6*b^2*c^2 + 26*a^4*b^4*c^2 + 16*a^2*b^6*c^2 - 9*b^8*c^2 - 2*a^6*c^4 + 26*a^4*b^2*c^4 + 190*a^2*b^4*c^4 + 10*b^6*c^4 + 2*a^4*c^6 + 16*a^2*b^2*c^6 + 10*b^4*c^6 + a^2*c^8 - 9*b^2*c^8 - c^10): :

X(40918) is the perspector of the special dilation triangle and the anti-incircle-circles triangle. (See X(11363.)

X(40918) lies on these lines: {2, 39879}, {125, 32254}, {1351, 3917}, {3526, 12309}, {3527, 13340}, {12315, 15030}

X(40919) = X(2)X(15141)∩X(67)X(3917)

Barycentrics    a^20 - 4*a^18*b^2 + 3*a^16*b^4 + 8*a^14*b^6 - 14*a^12*b^8 + 14*a^8*b^12 - 8*a^6*b^14 - 3*a^4*b^16 + 4*a^2*b^18 - b^20 - 4*a^18*c^2 + 10*a^16*b^2*c^2 - 3*a^14*b^4*c^2 - 14*a^12*b^6*c^2 + 15*a^10*b^8*c^2 - 5*a^6*b^12*c^2 + 2*a^4*b^14*c^2 - 3*a^2*b^16*c^2 + 2*b^18*c^2 + 3*a^16*c^4 - 3*a^14*b^2*c^4 + 11*a^12*b^4*c^4 - 8*a^10*b^6*c^4 - 9*a^8*b^8*c^4 + 20*a^6*b^10*c^4 - 8*a^4*b^12*c^4 - 9*a^2*b^14*c^4 + 3*b^16*c^4 + 8*a^14*c^6 - 14*a^12*b^2*c^6 - 8*a^10*b^4*c^6 + 18*a^8*b^6*c^6 - 15*a^6*b^8*c^6 - 2*a^4*b^10*c^6 + 5*a^2*b^12*c^6 - 8*b^14*c^6 - 14*a^12*c^8 + 15*a^10*b^2*c^8 - 9*a^8*b^4*c^8 - 15*a^6*b^6*c^8 + 22*a^4*b^8*c^8 + 3*a^2*b^10*c^8 - 2*b^12*c^8 + 20*a^6*b^4*c^10 - 2*a^4*b^6*c^10 + 3*a^2*b^8*c^10 + 12*b^10*c^10 + 14*a^8*c^12 - 5*a^6*b^2*c^12 - 8*a^4*b^4*c^12 + 5*a^2*b^6*c^12 - 2*b^8*c^12 - 8*a^6*c^14 + 2*a^4*b^2*c^14 - 9*a^2*b^4*c^14 - 8*b^6*c^14 - 3*a^4*c^16 - 3*a^2*b^2*c^16 + 3*b^4*c^16 + 4*a^2*c^18 + 2*b^2*c^18 - c^20: :

X(40919) is the perspector of the special dilation triangle and the antiAOA triangle. (See X(15015.)

X(40919) lies on these lines: {2, 15141}, {67, 3917}, {125, 10510}, {524, 15137}, {2393, 32353}, {7574, 9019}, {7579, 19377}, {15462, 19381}

X(40920) = X(2)X(8780)∩X(69)X(5159)

Barycentrics    (a^2 - b^2 - c^2)*(5*a^4 - a^2*b^2 - 10*b^4 - a^2*c^2 + 20*b^2*c^2 - 10*c^4): :

X(40920) is the perspector of the special dilation triangle and the Ehrmann side-triangle. (See the preamble just before X(18300).)

X(40920) lies on these lines: {2, 8780}, {69, 5159}, {125, 32272}, {381, 6699}, {511, 31856}, {974, 22584}, {1112, 5094}, {1656, 15030}, {3090, 11469}, {3526, 6288}, {3763, 14913}, {3818, 6723}, {3917, 18438}, {4846, 5055}, {5642, 21968}, {6640, 12429}, {11550, 21974}, {15074, 33879}, {15113, 34778}, {15723, 18475}, {16881, 31283}, {18445, 19348}, {21849, 26958}, {21850, 30775}, {21970, 31670}, {32223, 34609}

leftri

Perspectors involving the obverse triangle of X(69): X(40921)-X(40926)

rightri

This preamble is based on notes from Peter Moses, January 6, 2021.

As in the preamble just before X(24307), the obverse triangle, A'B'C', of a point P = p : q : r is given by

A' = p : r : q,      B' = r : q : p,      C' = q : p : r.

In particular, for P = X(69),

A' = b^2 + c^2 - a^2 : a^2 + b^2 - c^2 : c^2 + a^2 - b^2.

X(40921) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND OUTER NAPOLEON

Barycentrics    (a^2 + b^2 + c^2)*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*Sqrt[3]*a^2*(a^2 - 3*b^2 - 3*c^2)*S : :

X(40921) lies on these lines: {2, 33410}, {3, 395}, {4, 6114}, {5, 16634}, {16, 6776}, {17, 37178}, {39, 22715}, {62, 1992}, {140, 3767}, {3107, 18906}, {5613, 10653}, {5869, 8721}, {6695, 14001}, {6775, 16635}, {7763, 11289}, {9763, 34511}, {9771, 11305}, {11154, 32985}, {22236, 38064}, {33387, 37177}

X(40921) = anticomplement of X(33410)
X(40921) = {X(140),X(5013)}-harmonic conjugate of X(40922)

X(40922) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND INNER NAPOLEON

Barycentrics    (a^2 + b^2 + c^2)*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4) + 2*Sqrt[3]*a^2*(a^2 - 3*b^2 - 3*c^2)*S : :

X(40922) lies on these lines: {2, 33411}, {3, 396}, {4, 6115}, {5, 16635}, {15, 6776}, {18, 37177}, {39, 22714}, {61, 1992}, {140, 3767}, {3106, 18906}, {5617, 10654}, {5868, 8721}, {6694, 14001}, {6772, 16634}, {7763, 11290}, {9761, 34511}, {9771, 11306}, {11153, 32985}, {22238, 38064}, {33386, 37178}

X(40922) = anticomplement of X(33411)
X(40922) = {X(140),X(5013)}-harmonic conjugate of X(40921)

X(40923) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND 1ST NEUBERG

Barycentrics    5*a^10*b^2 - 8*a^8*b^4 + 10*a^6*b^6 - 8*a^4*b^8 + a^2*b^10 + 5*a^10*c^2 - 13*a^8*b^2*c^2 + 10*a^6*b^4*c^2 - 2*a^4*b^6*c^2 - 7*a^2*b^8*c^2 - b^10*c^2 - 8*a^8*c^4 + 10*a^6*b^2*c^4 + 8*a^4*b^4*c^4 + 6*a^2*b^6*c^4 + 10*a^6*c^6 - 2*a^4*b^2*c^6 + 6*a^2*b^4*c^6 + 2*b^6*c^6 - 8*a^4*c^8 - 7*a^2*b^2*c^8 + a^2*c^10 - b^2*c^10 : :

X(40923) lies on these lines: {2, 6248}, {3, 18906}, {4, 2023}, {39, 6776}, {98, 194}, {511, 39647}, {1992, 3095}, {2782, 6337}, {3767, 40924}, {5013, 7709}, {5188, 8182}, {6287, 11171}, {6308, 30270}, {8667, 12251}, {9744, 37336}, {13357, 14853}, {14229, 21736}, {18860, 31981}, {20088, 36998}

X(40924) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND 2ND NEUBERG

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

X(40924) lies on these lines: {2, 9737}, {4, 38654}, {32, 6776}, {147, 12110}, {194, 9753}, {262, 7836}, {1992, 22143}, {3095, 3926}, {3767, 40923}, {5286, 7709}, {6309, 35437}, {6337, 37466}, {7797, 13334}, {7932, 9754}, {10336, 32467}

X(40925) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND McCAY

Barycentrics    5*a^8 - 32*a^6*b^2 + 42*a^4*b^4 - 16*a^2*b^6 + b^8 - 32*a^6*c^2 + 56*a^4*b^2*c^2 + 16*a^2*b^4*c^2 + 42*a^4*c^4 + 16*a^2*b^2*c^4 - 2*b^4*c^4 - 16*a^2*c^6 + c^8 : :
X(40925) = X[4] - 3 X[14494]

X(40925) lies on these lines: {2, 20398}, {3, 1992}, {4, 3815}, {20, 9774}, {140, 6337}, {194, 3523}, {574, 6776}, {575, 11151}, {631, 7610}, {1656, 39143}, {3522, 9737}, {3533, 7789}, {5024, 14853}, {7607, 10303}, {8550, 15815}, {11171, 18906}, {15717, 32467}, {18860, 33750}, {32990, 34507}

X(40926) = PERSPECTOR OF THESE TRIANGLES: OBVERSE OF X(69) AND ARTZT

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

X(40926) lies on these lines: {2, 10256}, {4, 22664}, {194, 9772}, {262, 1007}, {1513, 6776}, {1992, 6054}, {6390, 37071}, {7709, 9743}, {7774, 9742}, {9741, 9877}, {9749, 10654}, {9750, 10653}, {9751, 9754}, {9770, 14853}, {9774, 25406}

X(40927) = EULER LINE INTERCEPT OF X(230)X(22682)

Barycentrics    2*a^8+(b^2+c^2)*a^6+(9*(b^2+c^2)^2-4*b^2*c^2)*a^4-13*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-10*b^2*c^2+c^4)*(b^2-c^2)^2 : :
Barycentrics    3*S^2*(S^2+SB*SC)+4*SB*SC*SW^2 : :
X(40927) = X(4)+3*X(13860) = 5*X(1656)-3*X(37451)

Let AmBmCm be the medial triangle of ABC and A', A" the intersections of BmCm with the circumcircle of ABC. Denote as A* the intersection of the Simson lines of A' and A", and define B* and C* cyclically. Then K*, the symmedian point X(6)-of-A*B*C*, lies on the Euler line of ABC. (Kadir Altintas, January 6, 2021).

The point K* is X(40927) (César Lozada, January 8, 2021).

X(40927) lies on these lines: {2, 3}, {230, 22682}, {5305, 11623}, {7745, 10991}, {9756, 18907}, {9830, 38745}, {16334, 18575}, {31415, 36990}, {32834, 40268}

X(40927) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3091, 11318, 5), (36655, 36656, 3832), (36657, 36658, 3861)

X(40928) = X(4)X(16227) ∩ X(5)X(2883)

Barycentrics    (SB+SC)*((6*R^2-5*SA-SW)*S^2-(12*R^2*(16*R^2-2*SA-5*SW)+5*SA^2-5*SB*SC+4*SW^2)*SA) : :
X(40928) = 2*X(185)+X(5894) = X(2883)-4*X(40647) = 2*X(5893)-5*X(10574) = X(6241)+2*X(6696) = X(6247)+2*X(13491) = X(11381)-4*X(32184) = 2*X(17854)+X(23315)

Let O, H be the circumcenter and orthocenter of ABC, respectively, and A*B*C* be the circumcevian triangle of H. Denote (Ab, Ac) the orthogonal projections of A* in AC, AB, respectively. The triangle AAbAc is named here the A-circumorthiac triangle of ABC. Assume P(t) is a point on the Euler line of ABC such as OP/OH=t (number) and let Pa(t), Pb(t), Pc(t) be the same points P(t)-of-the three circumorthiac triangles. Then P(t), Pa(t), Pb(t), Pc(t) are concyclic. (Kadir Altintas, January 5, 2021)

This note and centers X(40928)-X(40932) were contributed by César Lozada, January 9, 2021:

The radius of the circle through the above mentioned points is rather complicated. Its center has barycentric coordinates:

O(t) = a^2*(((-a^2+b^2+c^2)^2-b^2*c^2)*((b^2+c^2)*a^2-(b^2-c^2)^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))*t^3-((b^2+c^2)*a^12-2*(2*b^4+3*b^2*c^2+2*c^4)*a^10+(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^8-(5*b^4+12*b^2*c^2+5*c^4)*b^2*c^2*a^6-(b^2+c^2)*(5*b^8+5*c^8-b^2*c^2*(11*b^4-5*b^2*c^2+11*c^4))*a^4+(b^2-c^2)^2*(4*b^8+4*c^8-b^2*c^2*(b^4-b^2*c^2+c^4))*a^2-(b^6+c^6)*(b^2-c^2)^4)*t^2-b^2*c^2*(2*a^10-3*(b^2+c^2)*a^8-2*(b^4-5*b^2*c^2+c^4)*a^6+(b^2+c^2)*(4*b^4-13*b^2*c^2+4*c^4)*a^4-(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3)*t-((b^2+c^2)*a^2-(b^2-c^2)^2)*a^2*b^4*c^4) : :

The appearance of (t, i, j) in the following list means that, for the given t, P(t)=X(i) and O(t)=X(j): (1/3, 2, 40928), (0, 3, 5), (1, 4, 5), (1/2, 5, 13630), (-1, 20, 40929), (1/4, 140, 40930), (2/3, 381, 40931), (2, 382, 40932)

X(40928) lies on these lines: {4, 16227}, {5, 2883}, {64, 7527}, {154, 22467}, {185, 1205}, {1147, 10226}, {1181, 3520}, {1503, 7729}, {2393, 22967}, {2935, 15032}, {3357, 32046}, {5663, 23328}, {5890, 12241}, {5893, 10574}, {6241, 6696}, {11381, 32184}, {11430, 11598}, {12061, 37196}, {12162, 38795}, {12897, 34798}, {14130, 35450}, {16534, 25563}, {17854, 23315}, {23329, 34331}, {32064, 34007}, {35497, 35602}

X(40928) = midpoint of X(7729) and X(15072)

X(40929) = X(3)X(11216) ∩ X(5)X(141)

Barycentrics    (SB+SC)*((18*R^2-5*SW)*S^2-(16*R^2+SA-4*SW)*SA*SW) : :
X(40929) = 9*X(141)-8*X(11793) = 4*X(389)-3*X(8584) = 3*X(576)-4*X(16881) = 2*X(5562)-3*X(22165) = X(5889)-3*X(37473) = 3*X(8550)-4*X(13630) = 2*X(9967)-3*X(21167) = 9*X(11188)-5*X(11439) = 3*X(25329)-4*X(25711) = 2*X(31834)-3*X(34507)

See X(40928).

X(40929) lies on these lines: {3, 11216}, {5, 141}, {6, 22467}, {20, 8705}, {24, 32217}, {30, 12061}, {69, 34007}, {185, 2854}, {389, 8584}, {524, 5889}, {550, 11649}, {575, 40931}, {576, 16881}, {1350, 3520}, {1657, 11663}, {2393, 22967}, {2781, 12162}, {3098, 10226}, {3629, 19161}, {5562, 22165}, {6143, 10519}, {6403, 18560}, {8550, 13630}, {9019, 35240}, {9967, 21167}, {9970, 18350}, {11188, 11439}, {11477, 37827}, {12038, 33851}, {14130, 33878}, {16042, 17811}, {19221, 37200}, {25329, 25711}, {31834, 34507}, {31884, 35497}, {34787, 35707}, {37126, 37283}

X(40929) = midpoint of X(1657) and X(11663)
X(40929) = reflection of X(3629) in X(19161)

X(40930) = X(5)X(10575) ∩ X(399)X(9705)

Barycentrics    (SB+SC)*((61*R^2-22*SA-10*SW)*S^2-(R^2*(770*R^2-115*SA-218*SW)+22*SA^2-22*SB*SC+12*SW^2)*SA) : :

See X(40928).

X(40930) lies on these lines: {5, 10575}, {399, 9705}, {1154, 13420}, {5012, 33541}, {13630, 38790}

X(40931) = X(5)X(5890) ∩ X(575)X(40929)

Barycentrics    (SB+SC)*((87*R^2-4*SA-20*SW)*S^2+(3*R^2*(110*R^2+5*SA-48*SW)-4*SA^2+4*SB*SC+16*SW^2)*SA) : :
X(40931) = X(11704)+5*X(37481)

See X(40928).

X(40931) lies on these lines: {5, 5890}, {575, 40929}, {3627, 16227}, {7527, 11999}, {16880, 37118}

X(40932) = X(4)X(12062) ∩ X(5)X(568)

Barycentrics    (SB+SC)*((9*R^2-20*SA-4*SW)*S^2-(R^2*(378*R^2-81*SA-160*SW)+4*(5*SA^2-5*SB*SC+4*SW^2))*SA) : :

X(40932) lies on these lines: {4, 12062}, {5, 568}, {3431, 10226}, {3520, 11935}, {3853, 12063}

See X(40928).

X(40932) = reflection of X(i) in X(j) for these (i, j): (12062, 4), (12063, 3853)

X(40933) = CROSSPOINT OF X(1) AND X(20)

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

X(40933) lies on these lines: {1, 64}, {20, 1097}, {25, 34}, {65, 2357}, {72, 1020}, {73, 8811}, {154, 1394}, {227, 11214}, {354, 2654}, {658, 1043}, {934, 1297}, {1010, 6359}, {1402, 10376}, {1446, 3424}, {1448, 7053}, {1461, 2360}, {2260, 2262}, {2646, 20277}, {3172, 3213}, {3198, 5930}, {3333, 12114}, {4313, 31526}, {4319, 17808}, {7011, 21147}, {32065, 32319}

X(40933) = crosspoint of X(1) and X(20)
X(40933) = crosssum of X(1) and X(64)
X(40933) = X(1)-Ceva conjugate of X(1042)

X(40934) = CROSSPOINT OF X(1) AND X(25)

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

X(40934) lies on these lines: {1, 69}, {8, 26041}, {25, 21148}, {31, 1486}, {37, 42}, {43, 344}, {55, 2277}, {65, 23492}, {71, 3747}, {73, 1284}, {238, 24575}, {354, 28022}, {497, 614}, {674, 16685}, {740, 20336}, {899, 17279}, {1001, 1193}, {1064, 4116}, {1086, 15320}, {1100, 4749}, {1184, 8020}, {1201, 1279}, {1334, 21035}, {1400, 1918}, {1472, 11365}, {1580, 16800}, {1616, 10387}, {1631, 5301}, {1826, 23663}, {1914, 23868}, {1961, 25420}, {2176, 3779}, {2183, 2209}, {2223, 17053}, {2239, 21330}, {2269, 3764}, {2275, 20992}, {2300, 21746}, {2333, 23664}, {2356, 5336}, {2611, 21322}, {2654, 3924}, {3056, 21769}, {3120, 16580}, {3214, 3932}, {3230, 3688}, {3248, 20978}, {3271, 20228}, {3293, 4078}, {3720, 4657}, {3744, 28366}, {4039, 21257}, {4368, 21080}, {4433, 21857}, {4443, 16690}, {7083, 16502}, {7191, 17302}, {8299, 27633}, {8610, 20990}, {11398, 22263}, {12530, 24445}, {12723, 20227}, {17384, 30950}, {20964, 22172}, {21278, 40886}, {21750, 21813}, {23381, 28266}, {24653, 30953}, {25124, 40718}, {25916, 25941}, {26237, 26971}, {27127, 33131}, {27180, 33134}, {28011, 38053}, {28420, 33137}, {38986, 40613}

X(40934) = isogonal conjugate of isotomic conjugate of X(3914)
X(40934) = isotomic conjugate of polar conjugate of X(8020)
X(40934) = crosspoint of X(1) and X(25)
X(40934) = crosssum of X(1) and X(69)
X(40934) = crossdifference of every pair of points on line X(1019)X(2484)
X(40934) = X(1)-Ceva conjugate of X(17441)

X(40935) = CROSSPOINT OF X(1) AND X(32)

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

X(40935) lies on these lines: {1, 76}, {8, 26043}, {10, 23629}, {32, 1917}, {37, 2231}, {38, 2232}, {42, 2229}, {65, 4128}, {72, 4093}, {73, 1284}, {172, 694}, {695, 1914}, {700, 38844}, {718, 27801}, {737, 825}, {766, 2085}, {1193, 1386}, {1201, 28359}, {1475, 3248}, {1613, 23863}, {1923, 8618}, {2233, 3721}, {2238, 23652}, {2241, 30495}, {2275, 3056}, {2887, 16889}, {3117, 8022}, {3720, 30955}, {3747, 22061}, {3780, 20464}, {4531, 16584}, {7184, 33947}, {8622, 23851}, {14963, 23626}, {18170, 26801}, {21751, 21815}, {21759, 40729}, {23619, 23664}

X(40935) = isogonal conjugate of isotomic conjugate of X(3778)
X(40935) = isotomic conjugate of isogonal conjugate of X(8022)
X(40935) = crosspoint of X(1) and X(32)
X(40935) = crosssum of X(1) and X(76)
X(40935) = X(1)-Ceva conjugate of X(3721)

X(40936) = CROSSPOINT OF X(1) AND X(39)

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

X(40936) lies on these lines: {1, 83}, {37, 2231}, {42, 2240}, {172, 1691}, {512, 1500}, {518, 872}, {660, 40432}, {695, 2276}, {730, 1089}, {732, 16720}, {756, 1107}, {1015, 14992}, {1215, 1237}, {1964, 33299}, {2275, 19586}, {2295, 4128}, {2667, 20706}, {3952, 21226}, {3954, 4093}, {4557, 21008}, {16600, 23629}, {17103, 18787}, {21021, 27880}, {21752, 21818}, {21805, 22292}, {27954, 27974}

X(40936) = crosspoint of X(1) and X(39)
X(40936) = crosssum of X(1) and X(83)
X(40936) = X(1)-Ceva conjugate of X(2295)

X(40937) = CROSSPOINT OF X(2) AND X(21)

Barycentrics    a*(a - b - c)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :
Trilinears    (b + c) cos A + b cos B + c cos C : :

X(40937) lies on these lines: {1, 6}, {2, 92}, {3, 19}, {5, 1826}, {7, 24554}, {8, 2335}, {10, 5721}, {21, 270}, {30, 1839}, {33, 13615}, {34, 37224}, {36, 1781}, {46, 37500}, {48, 1385}, {55, 39943}, {58, 15823}, {63, 7190}, {65, 579}, {71, 517}, {75, 25083}, {85, 25521}, {86, 1944}, {123, 1560}, {140, 8756}, {142, 241}, {169, 198}, {192, 26059}, {199, 2355}, {210, 3190}, {216, 828}, {233, 4187}, {242, 6998}, {243, 8748}, {268, 7129}, {282, 8583}, {284, 1731}, {307, 16608}, {346, 3702}, {355, 26063}, {379, 17134}, {380, 3601}, {387, 3931}, {440, 1848}, {442, 1838}, {516, 5829}, {572, 1630}, {573, 2262}, {608, 37228}, {610, 3576}, {648, 40412}, {672, 2171}, {905, 21202}, {910, 2919}, {942, 2260}, {946, 5798}, {965, 997}, {991, 5784}, {998, 2336}, {1011, 1824}, {1030, 7297}, {1038, 19520}, {1071, 15656}, {1125, 3002}, {1146, 1213}, {1196, 3290}, {1229, 28797}, {1249, 6857}, {1253, 28125}, {1334, 17452}, {1375, 40530}, {1400, 17451}, {1411, 2983}, {1418, 6173}, {1427, 25525}, {1442, 6510}, {1474, 36011}, {1575, 25075}, {1612, 5266}, {1617, 15288}, {1714, 4261}, {1737, 21933}, {1741, 5706}, {1742, 17668}, {1752, 36744}, {1754, 4640}, {1761, 3916}, {1765, 6001}, {1766, 3428}, {1804, 2002}, {1829, 37225}, {1855, 8728}, {1859, 8021}, {1867, 37056}, {1901, 12047}, {1959, 28287}, {2047, 14121}, {2074, 38852}, {2082, 4254}, {2092, 41015}, {2170, 2269}, {2173, 13624}, {2178, 7742}, {2183, 31786}, {2197, 24987}, {2201, 20834}, {2223, 21804}, {2245, 17443}, {2252, 34339}, {2272, 11227}, {2275, 20227}, {2276, 25074}, {2277, 16583}, {2285, 5120}, {2287, 4511}, {2289, 24929}, {2293, 15733}, {2302, 24299}, {2321, 3693}, {2328, 2361}, {2333, 13731}, {2340, 21039}, {2345, 5831}, {2354, 9840}, {2551, 5725}, {2947, 5927}, {2975, 5279}, {3007, 37111}, {3008, 25065}, {3163, 15670}, {3169, 4051}, {3177, 26125}, {3211, 37615}, {3271, 7062}, {3303, 15954}, {3332, 5698}, {3452, 5718}, {3485, 5746}, {3486, 5802}, {3496, 37570}, {3612, 37504}, {3616, 27382}, {3666, 3946}, {3682, 5044}, {3686, 3965}, {3691, 21033}, {3692, 3872}, {3730, 21871}, {3739, 4858}, {3743, 18249}, {3752, 31187}, {3869, 4047}, {3939, 15837}, {3949, 34790}, {3991, 17314}, {3998, 5271}, {4000, 37597}, {4007, 4515}, {4019, 30059}, {4167, 38408}, {4269, 18180}, {4300, 18251}, {4329, 14021}, {4341, 6180}, {4357, 15595}, {4512, 7070}, {4516, 11997}, {4552, 25001}, {4687, 30854}, {4698, 34852}, {5011, 37508}, {5084, 17916}, {5124, 5341}, {5236, 6356}, {5249, 16585}, {5257, 5930}, {5273, 24597}, {5307, 7522}, {5398, 31445}, {5723, 6666}, {5724, 5795}, {5744, 26635}, {5747, 11375}, {5749, 26690}, {5776, 6261}, {5830, 17355}, {5837, 37548}, {5903, 21866}, {6129, 20516}, {6600, 28043}, {6601, 40779}, {6675, 37565}, {6690, 8608}, {6913, 15831}, {7079, 11108}, {7146, 27626}, {7713, 37320}, {7987, 18594}, {8680, 34830}, {9816, 11347}, {9956, 21011}, {10246, 20818}, {10319, 21483}, {10391, 17194}, {10571, 30456}, {12723, 20992}, {13728, 34823}, {15178, 17438}, {15253, 23988}, {15263, 32664}, {15587, 35338}, {15624, 21867}, {16577, 25091}, {16587, 17055}, {16596, 18643}, {16696, 17189}, {16826, 27420}, {16831, 27384}, {17043, 26006}, {17077, 20905}, {17278, 24779}, {17303, 19854}, {17306, 25887}, {17321, 27509}, {17322, 18721}, {17353, 25099}, {17440, 21748}, {17442, 37613}, {17614, 18599}, {17625, 22163}, {17863, 25255}, {18230, 26669}, {18589, 30810}, {18675, 34586}, {18726, 24471}, {19684, 28950}, {20263, 25063}, {20305, 24317}, {20895, 27514}, {21014, 21798}, {21318, 37319}, {21495, 27059}, {21511, 26998}, {22070, 24541}, {22147, 37624}, {24315, 25523}, {24316, 26130}, {24915, 25461}, {24944, 25679}, {26068, 26107}, {26242, 40179}, {30478, 37592}, {33047, 37895}, {35066, 35508}, {35084, 39014}, {37275, 38860}, {37519, 37618}

X(40937) = isogonal conjugate of X(2982)
X(40937) = isotomic conjugate of polar conjugate of X(1859)
X(40937) = polar conjugate of isogonal conjugate of X(23207)
X(40937) = complement of X(1441)
X(40937) = complementary conjugate of complement of X(2194)
X(40937) = crosspoint of X(i) and X(j) for these {i,j}: {2, 21}, {8, 31623}, {9, 7110}
X(40937) = crosssum of X(i) and X(j) for these {i,j}: {6, 65}, {56, 1409}, {57, 2003}
X(40937) = crossdifference of every pair of points on line X(513)X(1946) (the polar of X(65) wrt the circumcircle)
X(40937) = {X(1),X(9)}-harmonic conjugate of X(219)
X(40937) = X(2)-Ceva conjugate of X(442)
X(40937) = X(21)-Ceva conjugate of X(8021)

X(40938) = CROSSPOINT OF X(2) AND X(22)

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

X(40938) lies on these lines: {2, 1235}, {3, 3162}, {4, 1180}, {6, 25}, {22, 8743}, {24, 5359}, {32, 21213}, {39, 427}, {112, 6636}, {132, 40588}, {141, 19595}, {186, 1627}, {216, 7499}, {233, 37439}, {343, 14994}, {418, 9475}, {428, 3199}, {468, 1196}, {648, 1799}, {1249, 7494}, {1289, 40357}, {1368, 1560}, {1370, 14961}, {2211, 20859}, {2489, 18311}, {2967, 7467}, {3291, 37453}, {3520, 38862}, {3917, 35325}, {5094, 8893}, {5133, 5523}, {5306, 11062}, {6353, 9465}, {6515, 37187}, {7392, 15355}, {7408, 33885}, {7667, 22401}, {8746, 9609}, {8792, 21284}, {8878, 40889}, {10312, 34482}, {14715, 39569}, {14768, 39530}, {15661, 33584}, {17068, 24789}, {17453, 21812}, {17907, 34254}, {23208, 27373}, {39172, 40144}

X(40938) = isogonal conjugate of X(40404)
X(40938) = isotomic conjugate of polar conjugate of X(27373)
X(40938) = polar conjugate of isogonal conjugate of X(23208)
X(40938) = polar conjugate of isotomic conjugate of X(3313)
X(40938) = complement of X(18018)
X(40938) = complementary conjugate of X(6697)
X(40938) = crosspoint of X(2) and X(22)
X(40938) = crosssum of X(6) and X(66)
X(40938) = crossdifference of every pair of points on the polar of X(66) wrt the circumcircle
X(40938) = X(2)-Ceva conjugate of X(427)
X(40938) = X(22)-Ceva conjugate of Danneels point of X(22)
X(40938) = trilinear product X(i)*X(j) for these {i,j}: {19, 3313}, {22, 17442}, {38, 8743}, {42, 16715}, {63, 27373}, {92, 23208}, {206, 20883}, {427, 2172}, {1235, 17453}, {1760, 1843}, {1930, 17409}, {1964, 17907}, {16747, 21034}, {19595, 19616}, {20641, 27369}, {23881, 32676}

X(40939) = CROSSPOINT OF X(2) AND X(24)

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

X(40939) lies on these lines: {2, 311}, {6, 1147}, {24, 571}, {39, 233}, {53, 136}, {216, 7542}, {231, 10018}, {232, 1609}, {577, 12095}, {800, 3163}, {1249, 3003}, {1594, 1879}, {1990, 14091}, {2079, 2965}, {7393, 15827}, {7509, 14806}, {8882, 14586}, {13351, 13881}

X(40939) = complement of X(20563)
X(40939) = crosspoint of X(2) and X(24)
X(40939) = crosssum of X(6) and X(68)
X(40939) = crossdifference of every pair of points on the polar of X(68) wrt the circumcircle
X(40939) = X(2)-Ceva conjugate of X(11585)
X(40939) = X(24)-Ceva conjugate of Danneels point of X(24)

X(40940) = CROSSPOINT OF X(2) AND X(27)

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

X(40940) lies on these lines: {1, 2}, {4, 204}, {6, 226}, {7, 23681}, {9, 4656}, {11, 4989}, {19, 57}, {27, 58}, {31, 516}, {34, 37388}, {35, 1612}, {36, 1817}, {38, 4353}, {44, 4415}, {55, 3755}, {56, 5930}, {63, 1723}, {69, 25527}, {72, 34937}, {75, 2064}, {76, 19803}, {81, 3664}, {86, 7058}, {92, 17861}, {116, 1560}, {142, 940}, {169, 40179}, {171, 1738}, {193, 26132}, {197, 1617}, {210, 17602}, {213, 22000}, {216, 1108}, {219, 3452}, {223, 4341}, {225, 1451}, {233, 37662}, {238, 2328}, {269, 18623}, {279, 18624}, {312, 17353}, {321, 5294}, {329, 1743}, {333, 4357}, {345, 3875}, {347, 5435}, {379, 1468}, {440, 950}, {442, 5717}, {443, 37554}, {445, 35201}, {464, 4304}, {497, 7070}, {515, 5721}, {518, 17061}, {527, 3782}, {553, 1086}, {573, 27659}, {579, 24310}, {593, 18653}, {595, 10624}, {648, 40414}, {673, 1416}, {726, 19791}, {740, 6679}, {896, 33145}, {908, 32911}, {942, 6678}, {946, 5706}, {967, 14377}, {968, 4356}, {988, 24609}, {993, 16368}, {1015, 35075}, {1021, 26146}, {1100, 17056}, {1150, 32774}, {1172, 1848}, {1191, 12053}, {1196, 16583}, {1203, 12047}, {1211, 3686}, {1266, 19796}, {1269, 19821}, {1284, 20967}, {1376, 16413}, {1386, 2886}, {1400, 1730}, {1427, 23653}, {1448, 37102}, {1449, 5712}, {1479, 37185}, {1659, 18991}, {1682, 28275}, {1699, 3332}, {1707, 24248}, {1708, 8557}, {1724, 12572}, {1731, 1762}, {1757, 33152}, {1763, 2082}, {1764, 28274}, {1780, 36250}, {1785, 37279}, {1824, 32118}, {1826, 17902}, {1829, 35650}, {1839, 18688}, {1864, 16870}, {2006, 2982}, {2194, 5137}, {2221, 12610}, {2256, 5316}, {2292, 18249}, {2308, 3120}, {2321, 32777}, {2325, 3175}, {2347, 21361}, {2550, 5269}, {2650, 12563}, {2887, 3791}, {3064, 21184}, {3162, 21621}, {3210, 25252}, {3218, 21376}, {3219, 33155}, {3220, 5324}, {3306, 24175}, {3338, 24171}, {3361, 24604}, {3428, 16435}, {3488, 16485}, {3589, 9022}, {3596, 19806}, {3610, 17279}, {3662, 37683}, {3666, 3946}, {3672, 5273}, {3673, 18750}, {3677, 24477}, {3683, 4854}, {3707, 19723}, {3710, 39589}, {3717, 19815}, {3729, 26065}, {3739, 6703}, {3742, 11221}, {3744, 5853}, {3745, 3925}, {3751, 33144}, {3759, 4417}, {3769, 4429}, {3821, 19834}, {3826, 4682}, {3838, 17070}, {3846, 4974}, {3879, 18134}, {3883, 32773}, {3891, 33114}, {3915, 12575}, {3929, 4419}, {3936, 4856}, {3944, 16468}, {3950, 17776}, {3977, 17147}, {3982, 17365}, {3998, 25078}, {4001, 16704}, {4008, 17860}, {4021, 28606}, {4035, 30811}, {4054, 26223}, {4138, 32946}, {4256, 7573}, {4257, 7560}, {4297, 37419}, {4314, 14021}, {4346, 28610}, {4359, 18698}, {4360, 33116}, {4385, 19814}, {4398, 19830}, {4416, 27184}, {4422, 35652}, {4438, 32921}, {4644, 4654}, {4649, 33130}, {4650, 33149}, {4657, 5737}, {4667, 24694}, {4684, 33124}, {4716, 33160}, {4719, 4999}, {4722, 32856}, {4859, 9776}, {4862, 9965}, {4886, 30832}, {4887, 33146}, {4967, 19808}, {5173, 15253}, {5196, 33774}, {5219, 26063}, {5224, 19812}, {5247, 12527}, {5248, 37323}, {5257, 19732}, {5266, 37326}, {5267, 27174}, {5295, 17698}, {5315, 30384}, {5743, 17348}, {5750, 31993}, {5837, 37614}, {5850, 32912}, {6381, 19801}, {6591, 23723}, {6684, 37528}, {6692, 16610}, {6848, 9121}, {7129, 8808}, {7262, 33154}, {7291, 10521}, {7308, 37650}, {7322, 38057}, {7681, 40658}, {7742, 11350}, {7952, 10396}, {7965, 21955}, {8609, 16577}, {8728, 37594}, {8755, 40149}, {8756, 18676}, {9816, 24162}, {9955, 9958}, {10859, 17604}, {12436, 37522}, {12577, 23675}, {13390, 18992}, {14206, 24208}, {14213, 24209}, {14331, 21174}, {14552, 17272}, {14829, 16706}, {14996, 27186}, {14997, 27131}, {16054, 24178}, {16056, 37609}, {16471, 21616}, {16475, 17064}, {16477, 33096}, {16478, 37445}, {16516, 37673}, {16609, 41015}, {16670, 28609}, {16685, 23112}, {16757, 23799}, {17067, 37520}, {17074, 30379}, {17111, 23300}, {17126, 33131}, {17127, 33134}, {17132, 32933}, {17182, 27644}, {17185, 28287}, {17197, 40153}, {17278, 37674}, {17282, 18141}, {17302, 38000}, {17340, 22034}, {17352, 18743}, {17469, 33136}, {17716, 32865}, {17721, 24386}, {17723, 31245}, {17781, 33151}, {17917, 18634}, {18163, 22097}, {18206, 24214}, {18228, 37681}, {18311, 21187}, {18589, 22119}, {19517, 22770}, {19564, 37891}, {19742, 26580}, {19787, 20888}, {19798, 33945}, {19799, 33937}, {19800, 21443}, {19925, 21935}, {20237, 26665}, {20335, 37676}, {20882, 26538}, {20985, 23682}, {21627, 37542}, {22383, 23725}, {23292, 26011}, {23512, 37570}, {24023, 24026}, {24167, 26991}, {24231, 32913}, {24391, 37549}, {25496, 38049}, {26128, 32853}, {26267, 40128}, {26685, 30568}, {26724, 37633}, {27064, 37759}, {30961, 37657}, {31019, 37685}, {31229, 33113}, {31237, 32852}, {32775, 32864}, {32919, 33123}, {32922, 33121}, {32924, 33119}, {32928, 33115}, {32929, 35263}, {33064, 34379}, {35092, 35112}, {37263, 37583}, {37266, 37539}, {37274, 37608}, {37280, 37552}

X(40940) = isogonal conjugate of X(2983)
X(40940) = isotomic conjugate of polar conjugate of X(1842)
X(40940) = polar conjugate of X(40445)
X(40940) = complement of X(306)
X(40940) = anticomplement of X(20106)
X(40940) = complementary conjugate of complement of X(1474)
X(40940) = crosspoint of X(i) and X(j) for these {i,j}: {2, 27}, {86, 279}
X(40940) = cevapoint of X(1104) and X(2264)
X(40940) = crosssum of X(i) and X(j) for these {i,j}: {6, 71}, {42, 220}
X(40940) = crossdifference of every pair of points on line X(649)X(8676) (the polar of X(71) wrt the circumcircle)
X(40940) = pole wrt polar circle of trilinear polar of X(40445) (line X(3239)X(4064))
X(40940) = X(2)-Ceva conjugate of X(440)
X(40940) = X(27)-Ceva conjugate of Danneels point of X(27)
X(40940) = barycentric product X(1)*X(17863)

X(40941) = CROSSPOINT OF X(2) AND X(28)

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

X(40941) lies on these lines: {1, 19285}, {2, 37}, {6, 169}, {9, 3670}, {11, 1560}, {19, 614}, {28, 1104}, {31, 8775}, {44, 5279}, {48, 3924}, {56, 1880}, {63, 19728}, {71, 24443}, {216, 7561}, {233, 21796}, {238, 1761}, {244, 2260}, {284, 30117}, {304, 25504}, {517, 16685}, {579, 24046}, {899, 3949}, {910, 2220}, {960, 4016}, {965, 37549}, {975, 4255}, {992, 3721}, {1015, 3163}, {1086, 1901}, {1100, 2303}, {1107, 16817}, {1108, 1148}, {1119, 1427}, {1193, 2294}, {1201, 1953}, {1212, 5069}, {1325, 38858}, {1386, 2214}, {1393, 1400}, {1409, 18838}, {1462, 28615}, {1716, 24445}, {1722, 5227}, {1778, 3218}, {1781, 16470}, {1826, 23536}, {2176, 21853}, {2178, 5336}, {2257, 5573}, {2300, 3125}, {2509, 21187}, {3008, 24219}, {3216, 22021}, {3686, 16611}, {3770, 25994}, {3823, 21858}, {3914, 23305}, {3959, 21769}, {4150, 37096}, {5299, 16547}, {5747, 24159}, {5750, 16600}, {8804, 24177}, {11997, 39688}, {15474, 36428}, {16502, 18596}, {16605, 17275}, {16700, 18607}, {16968, 36743}, {16970, 20367}, {17189, 18734}, {17355, 24176}, {17448, 19851}, {17452, 22072}, {21216, 26106}, {21530, 23537}, {30456, 37566}, {37151, 37528}

X(40941) = isogonal conjugate of X(40406)
X(40941) = polar conjugate of isotomic conjugate of X(18732)
X(40941) = complement of X(20336)
X(40941) = complementary conjugate of complement of X(2203)
X(40941) = crosspoint of X(i) and X(j) for these {i,j}: {2, 28}, {274, 39732}
X(40941) = crosssum of X(i) and X(j) for these {i,j}: {6, 72}, {37, 5687}, {213, 12329}, {3990, 11517}
X(40941) = crossdifference of every pair of points on line X(667)X(15313) (the polar of X(72) wrt the circumcircle)
X(40941) = X(2)-Ceva conjugate of X(21530)
X(40941) = X(28)-Ceva conjugate of Danneels point of X(28)
X(40941) = trilinear product X(i)*X(j) for these {i,j}: {6, 23537}, {19, 18732}, {25, 18651}, {28, 18674}, {1474, 21530}

X(40942) = CROSSPOINT OF X(2) AND X(29)

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

Trilinears    (b + c) sec A + b sec B + c sec C : :

X(40942) lies on these lines: {1, 281}, {2, 7}, {3, 8804}, {4, 610}, {6, 1210}, {10, 219}, {11, 2264}, {19, 946}, {29, 284}, {37, 216}, {44, 233}, {48, 515}, {71, 1715}, {77, 24553}, {78, 2321}, {86, 37774}, {124, 1560}, {198, 3149}, {220, 17303}, {222, 8808}, {268, 9376}, {346, 27383}, {355, 20818}, {380, 497}, {406, 1712}, {461, 10382}, {499, 1723}, {572, 5179}, {594, 6603}, {650, 14749}, {857, 18650}, {936, 2324}, {938, 1449}, {940, 20205}, {942, 9119}, {949, 24388}, {958, 16416}, {966, 5705}, {1012, 1436}, {1070, 17904}, {1071, 1903}, {1100, 1146}, {1125, 3002}, {1172, 17188}, {1196, 20227}, {1212, 17398}, {1419, 5932}, {1441, 26006}, {1699, 18594}, {1702, 14121}, {1703, 7090}, {1765, 6705}, {1766, 21068}, {1781, 12047}, {1838, 18599}, {1839, 2173}, {1848, 18598}, {1901, 4292}, {1953, 8756}, {2182, 6831}, {2256, 31397}, {2257, 3086}, {2262, 8074}, {2268, 27378}, {2287, 2323}, {2294, 34591}, {2303, 3194}, {2325, 27385}, {2327, 11103}, {3211, 5778}, {3247, 5703}, {3330, 14597}, {3589, 34852}, {3601, 27402}, {3618, 28827}, {3664, 18635}, {3668, 17073}, {3687, 27398}, {3692, 5552}, {3694, 6745}, {3946, 4858}, {4007, 20007}, {4150, 30882}, {4304, 37504}, {4311, 37519}, {4336, 23529}, {4667, 5738}, {5227, 21075}, {5704, 16670}, {5776, 6245}, {5790, 22147}, {5795, 27410}, {5802, 9581}, {5819, 38150}, {5883, 31896}, {6180, 20206}, {6700, 17355}, {7079, 21620}, {9028, 20305}, {10175, 26063}, {10469, 27802}, {11344, 15817}, {12616, 19350}, {13607, 17438}, {15844, 15849}, {16470, 40129}, {16596, 41003}, {17134, 35290}, {17381, 30854}, {18589, 24315}, {18655, 24580}, {18690, 26165}, {20264, 38015}, {21011, 22356}, {21039, 28060}, {21044, 21748}, {30456, 34050}, {30808, 41004}, {34830, 40530}, {37380, 38860}

X(40942) = isogonal conjugate of X(40407)
X(40942) = complement of X(307)
X(40942) = complementary conjugate of complement of X(2299)
X(40942) = cevapoint of X(1901) and X(18675)
X(40942) = crosspoint of X(i) and X(j) for these {i,j}: {2, 29}, {280, 333}
X(40942) = crosssum of X(i) and X(j) for these {i,j}: {6, 73}, {48, 1399}, {221, 1400}
X(40942) = crossdifference of every pair of points on line X(663)X(39199} (the polar of X(73) wrt the circumcircle)
X(40942) = X(2)-Ceva conjugate of X(18641)
X(40942) = X(29)-Ceva conjugate of Danneels point of X(29)
X(40942) = trilinear product X(i)*X(j) for these {i,j}: {6, 23661}, {9, 4292}, {21, 1901}, {29, 18675}, {33, 18652}, {650, 14544}, {1172, 18641}

X(40943) = CROSSPOINT OF X(2) AND X(40)

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

X(40943) lies on these lines: {2, 309}, {6, 57}, {9, 1720}, {37, 158}, {198, 208}, {216, 1212}, {800, 1108}, {966, 25939}, {1033, 36103}, {1104, 8573}, {1213, 18592}, {1604, 21147}, {1785, 15849}, {1901, 2092}, {2262, 22063}, {2271, 15836}, {3192, 3198}, {3553, 4646}, {15852, 36744}, {17102, 20262}, {20307, 21857}

X(40943) = complement of X(309)
X(40943) = complementary conjugate of complement of X(2187)
X(40943) = crosspoint of X(2) and X(40)
X(40943) = crosssum of X(6) and X(84)
X(40943) = crossdifference of every pair of points on line X(3900)X(23224) (the polar of X(84) wrt the circumcircle)
X(40943) = X(2)-Ceva conjugate of X(946)
X(40943) = X(40)-Ceva conjugate of Danneels point of X(40)

X(40944) = CROSSPOINT OF X(3) AND X(8)

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

X(40944) lies on these lines: {1, 389}, {3, 1364}, {51, 2654}, {55, 581}, {72, 519}, {73, 185}, {212, 22076}, {219, 4266}, {223, 7355}, {517, 1887}, {970, 1936}, {1425, 20277}, {1490, 6285}, {1682, 6056}, {1745, 6000}, {1837, 23638}, {1875, 5909}, {2269, 2288}, {2635, 11381}, {2807, 10571}, {3042, 17555}, {3326, 31847}, {3917, 22072}, {5562, 22350}, {5690, 39004}, {5907, 37694}, {6238, 37700}, {6254, 17831}, {7004, 23154}, {7358, 21031}, {9729, 37523}, {11214, 15498}, {17102, 30493}, {22071, 22073}, {24430, 29958}

X(40944) = isogonal conjugate of polar conjugate of complement of X(222)
X(40944) = crosspoint of X(3) and X(8)
X(40944) = crosssum of X(4) and X(56)

X(40945) = CROSSPOINT OF X(3) AND X(9)

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

X(40945) lies on these lines: {3, 1433}, {6, 31}, {9, 7003}, {11, 21912}, {19, 6618}, {33, 2183}, {40, 34231}, {48, 19354}, {73, 185}, {198, 2192}, {201, 3057}, {228, 3270}, {497, 26013}, {573, 7070}, {1040, 22097}, {1208, 34046}, {1364, 22053}, {1697, 3488}, {1856, 20262}, {1864, 2347}, {2082, 30223}, {2182, 2357}, {2328, 15629}, {4266, 10382}, {7004, 17441}, {16870, 21361}, {22072, 22076}, {22079, 22080}

X(40945) = isogonal conjugate of polar conjugate of X(20262)
X(40945) = isotomic conjugate of polar conjugate of X(40957)
X(40945) = crosspoint of X(3) and X(9)
X(40945) = crosssum of X(4) and X(57)

X(40946) = CROSSPOINT OF X(3) AND X(21)

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

X(40946) lies on these lines: {1, 3}, {11, 18641}, {19, 28348}, {21, 243}, {33, 13738}, {34, 37195}, {58, 11428}, {71, 216}, {73, 185}, {204, 1011}, {225, 13734}, {227, 15622}, {296, 40442}, {386, 11435}, {408, 2654}, {497, 37180}, {580, 11429}, {581, 11436}, {855, 1842}, {856, 1125}, {859, 1859}, {1075, 6875}, {1364, 4303}, {1410, 20277}, {1465, 1888}, {1848, 28381}, {1852, 13442}, {1854, 3185}, {1858, 20967}, {1869, 27622}, {1870, 7421}, {1951, 2332}, {2269, 18591}, {2360, 10535}, {2550, 25876}, {2968, 21677}, {3100, 4225}, {3164, 4189}, {3198, 23361}, {4267, 10391}, {5218, 27407}, {5432, 7515}, {6097, 35201}, {6198, 37115}, {6906, 37790}, {7004, 18673}, {7066, 22350}, {7355, 10571}, {7508, 15912}, {9816, 28383}, {10058, 14679}, {10393, 19763}, {11376, 17073}, {11399, 37310}, {22060, 22361}, {22344, 26934}, {25910, 32942}

X(40946) = isogonal conjugate of polar conjugate of X(6708)
X(40946) = crosspoint of X(3) and X(21)
X(40946) = crosssum of X(4) and X(65)
X(40946) = X(3)-Ceva conjugate of X(408)

X(40947) = CROSSPOINT OF X(3) AND X(25)

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

X(40947) lies on these lines: {3, 69}, {6, 157}, {22, 385}, {24, 1075}, {25, 393}, {26, 15912}, {32, 1843}, {98, 264}, {125, 33926}, {159, 237}, {160, 8553}, {184, 216}, {193, 37183}, {206, 3003}, {317, 36998}, {378, 3438}, {394, 20819}, {426, 1899}, {523, 1485}, {571, 2393}, {577, 6467}, {648, 9307}, {800, 1974}, {1181, 10608}, {1316, 34978}, {1384, 5938}, {1602, 20999}, {1995, 7806}, {2452, 36176}, {2965, 9973}, {3053, 9924}, {3068, 3155}, {3069, 3156}, {3167, 22087}, {3314, 7485}, {3442, 3443}, {3515, 15258}, {5063, 32366}, {5065, 40673}, {5158, 21637}, {5201, 33801}, {5800, 37237}, {6179, 33802}, {6644, 17423}, {6751, 39643}, {7083, 23402}, {7467, 15574}, {7503, 9863}, {7931, 40916}, {8550, 37813}, {9230, 38907}, {9308, 9512}, {9407, 19118}, {9969, 13345}, {10323, 12251}, {10602, 15905}, {10607, 22085}, {13335, 14913}, {13558, 23216}, {14003, 23300}, {18935, 37188}, {19165, 30549}, {19595, 27369}, {20477, 39646}, {20897, 20987}, {22151, 23163}, {22263, 40801}, {23852, 23868}, {32621, 37457}, {33237, 38909}

X(40947) = isogonal conjugate of X(34405)
X(40947) = complement of X(41757)
X(40947) = polar conjugate of isotomic conjugate of X(39643)
X(40947) = crosspoint of X(3) and X(25)
X(40947) = crosssum of X(4) and X(69)
X(40947) = crossdifference of every pair of points on line X(2489)X(2799)
X(40947) = X(3)-Ceva conjugate of X(426)
X(40947) = X(25)-Ceva conjugate of the barycentric product of the intersections of the nine-point circle and orthosymmedial circle

X(40948) = CROSSPOINT OF X(3) AND X(30)

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

X(40948) lies on these lines: {3, 74}, {20, 1075}, {30, 34334}, {112, 1033}, {113, 1650}, {122, 15063}, {146, 10745}, {216, 3269}, {250, 2693}, {376, 3164}, {417, 10575}, {418, 14855}, {548, 31388}, {550, 15912}, {648, 1294}, {852, 14915}, {974, 20975}, {1364, 4303}, {1553, 16177}, {1576, 2935}, {2071, 11587}, {2970, 36179}, {3184, 9033}, {3284, 9408}, {4240, 38605}, {6053, 34842}, {9475, 14961}, {9934, 14703}, {11455, 38283}, {13445, 15781}, {13611, 36518}, {16003, 35442}, {35311, 38621}, {39174, 39376}

X(40948) = crosspoint of X(3) and X(30)
X(40948) = crosssum of X(4) and X(74)
X(40948) = crossdifference of every pair of points on line X(1637)X(8749)

X(40949) = CROSSPOINT OF X(4) AND X(23)

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

X(40949) lies on these lines: {4, 67}, {6, 1112}, {23, 6593}, {25, 15141}, {26, 15462}, {51, 1205}, {52, 542}, {110, 20987}, {113, 511}, {125, 9969}, {155, 2930}, {159, 19504}, {182, 16222}, {185, 36201}, {193, 2854}, {599, 11591}, {895, 3060}, {1162, 32281}, {1163, 32280}, {1503, 1986}, {1614, 2904}, {1843, 13417}, {2393, 5095}, {2777, 19161}, {2836, 14054}, {2914, 12367}, {3313, 5972}, {3567, 5622}, {3763, 13416}, {3818, 7723}, {5085, 9826}, {5169, 6698}, {5621, 9786}, {5648, 32605}, {5663, 25335}, {6403, 10752}, {7716, 17847}, {8538, 34155}, {10516, 12358}, {11188, 32244}, {11560, 29012}, {11579, 12236}, {11806, 21852}, {13202, 34146}, {13431, 13433}, {14448, 32250}, {15107, 25489}, {15116, 35370}, {15321, 18125}, {15463, 15577}, {15761, 16789}, {18947, 36851}, {19041, 32252}, {19042, 32253}, {19136, 34470}, {25711, 32233}, {29959, 32257}, {32251, 32262}

X(40949) = reflection of X(6) in X(1112)
X(40949) = crosspoint of X(4) and X(23)
X(40949) = crosssum of X(3) and X(67)
X(40949) = X(4)-Ceva conjugate of X(37981)
X(40949) = orthic-isogonal conjugate of X(37981)
X(40949) = antigonal conjugate of X(6) wrt orthic triangle
X(40949) = antipode of X(6) in Hatzipolakis-Lozada hyperbola
X(40949) = X(3254)-of-orthic-triangle if ABC is acute

X(40950) = CROSSPOINT OF X(4) AND X(29)

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

X(40950) lies on these lines: {1, 4}, {6, 1826}, {8, 28950}, {10, 212}, {11, 429}, {12, 37368}, {19, 2269}, {25, 23361}, {28, 4276}, {29, 270}, {30, 37565}, {35, 37117}, {36, 7414}, {52, 1866}, {55, 1869}, {56, 37194}, {65, 185}, {113, 20620}, {155, 7524}, {186, 14794}, {193, 5942}, {204, 4207}, {222, 5787}, {227, 6253}, {235, 26481}, {281, 3686}, {318, 4673}, {354, 1426}, {355, 7078}, {377, 1040}, {378, 36152}, {406, 26363}, {407, 2646}, {427, 3011}, {461, 5231}, {475, 10198}, {517, 1825}, {519, 41013}, {580, 1737}, {603, 6245}, {607, 1855}, {860, 1125}, {942, 1835}, {1038, 6836}, {1060, 1076}, {1062, 1074}, {1100, 1865}, {1172, 2323}, {1210, 1451}, {1427, 7354}, {1452, 5709}, {1453, 5292}, {1465, 20420}, {1593, 15622}, {1724, 10395}, {1771, 3215}, {1824, 3057}, {1827, 1828}, {1829, 1831}, {1830, 1887}, {1836, 1854}, {1839, 1841}, {1844, 1845}, {1856, 5155}, {1858, 41011}, {1861, 5174}, {1864, 1868}, {1867, 10950}, {1873, 12433}, {1876, 1888}, {1878, 18839}, {1883, 28027}, {1885, 8758}, {1894, 11363}, {1904, 26475}, {1935, 3561}, {2006, 7559}, {2202, 2332}, {2342, 3072}, {2475, 3100}, {2478, 9817}, {2905, 15476}, {2906, 3194}, {3064, 18013}, {3087, 8557}, {3340, 14257}, {3576, 37414}, {3601, 14018}, {3691, 7079}, {4186, 10966}, {4194, 10527}, {4212, 29640}, {4213, 33140}, {4214, 7071}, {4219, 37583}, {4231, 24239}, {4292, 7004}, {4296, 6895}, {4653, 14016}, {5256, 37181}, {5705, 7498}, {5706, 19354}, {5707, 5722}, {5733, 37723}, {5745, 22361}, {6284, 15852}, {6738, 40149}, {6748, 8609}, {6835, 19372}, {6928, 37696}, {7069, 12572}, {7282, 22464}, {7378, 26228}, {7412, 11012}, {7497, 11398}, {9628, 13273}, {10902, 37305}, {11249, 11399}, {11401, 17516}, {11403, 33925}, {11471, 37550}, {11510, 37391}, {13202, 38357}, {13411, 37381}, {13473, 16272}, {13734, 22341}, {14004, 26015}, {17102, 37468}, {17555, 24541}, {18455, 37230}, {20832, 37564}, {23207, 37225}, {23537, 33178}, {24537, 34822}, {27378, 34823}

X(40950) = isogonal conjugate of X(40442)
X(40950) = polar conjugate of isogonal conjugate of X(21748)
X(40950) = polar conjugate of isotomic conjugate of X(5745)
X(40950) = polar conjugate of X(19)-isoconjugate of X(22361)
X(40950) = cevapoint of X(407) and X(2650)
X(40950) = crosspoint of X(4) and X(29)
X(40950) = crosssum of X(3) and X(73)
X(40950) = crossdifference of every pair of points on line X(652)X(17975)
X(40950) = X(4)-Ceva conjugate of X(407)
X(40950) = orthic-isogonal conjugate of X(407)
X(40950) = trilinear product X(i)*X(j) for these {i,j}: {4, 2646}, {19, 5745}, {21, 407}, {27, 21811}, {28, 21677}, {29, 2650}, {33, 3664}, {34, 6737}, {92, 21748}, {158, 22361}, {270, 21674}, {1172, 17056}, {2299, 18698}, {17136, 18344}

X(40951) = CROSSPOINT OF X(4) AND X(32)

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

X(40951) lies on these lines: {4, 290}, {5, 5167}, {6, 695}, {25, 32445}, {32, 2909}, {39, 2387}, {51, 460}, {184, 15270}, {185, 1503}, {187, 211}, {194, 34383}, {217, 9418}, {263, 9292}, {325, 3491}, {511, 7750}, {512, 7747}, {626, 14962}, {682, 3051}, {1154, 32151}, {2679, 31850}, {2882, 13330}, {2979, 7904}, {3060, 7823}, {3289, 11326}, {3767, 6784}, {5140, 10110}, {5889, 9863}, {6102, 39846}, {6746, 13166}, {6786, 7763}, {7807, 35060}, {7885, 33873}, {8265, 14820}, {8541, 34980}, {8870, 12110}, {10551, 21243}, {11674, 37334}, {13491, 16983}, {14581, 15897}, {15257, 19558}, {18322, 37243}, {22416, 23635}, {39646, 40254}

X(40951) = isogonal conjugate of isotomic conjugate of X(23635)
X(40951) = polar conjugate of isotomic conjugate of complement of X(18022)
X(40951) = crosspoint of X(4) and X(32)
X(40951) = crosssum of X(3) and X(76)
X(40951) = orthic-isotomic conjugate of X(6)
X(40951) = X(85)-of-orthic-triangle if ABC is acute

X(40952) = CROSSPOINT OF X(4) AND X(37)

Barycentrics    a^2*(b + c)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :
Trilinears    a (cos B + cos C) (b (cos A + cos B) + c (cos A + cos C)) : :

X(40952) lies on these lines: {1, 10974}, {2, 3786}, {6, 25}, {9, 4204}, {22, 5138}, {31, 21746}, {33, 11435}, {37, 209}, {39, 33718}, {41, 6186}, {42, 181}, {55, 2245}, {57, 851}, {58, 3145}, {63, 4199}, {65, 225}, {72, 4205}, {81, 511}, {89, 26910}, {108, 2982}, {125, 15904}, {185, 5706}, {198, 4272}, {199, 284}, {208, 19366}, {210, 1213}, {213, 21813}, {226, 3136}, {354, 17056}, {373, 4383}, {386, 13738}, {387, 37384}, {389, 7412}, {430, 1827}, {440, 5728}, {442, 942}, {518, 1211}, {579, 1011}, {612, 3779}, {661, 2488}, {674, 3745}, {756, 3954}, {758, 4425}, {872, 21936}, {912, 30444}, {926, 2610}, {940, 3917}, {970, 19767}, {1185, 1196}, {1210, 3142}, {1228, 17141}, {1230, 17165}, {1401, 36570}, {1475, 23636}, {1500, 2198}, {1621, 39543}, {1859, 1865}, {1899, 5800}, {1962, 4890}, {2171, 21807}, {2238, 40131}, {2260, 14547}, {2264, 2355}, {2308, 3271}, {2624, 6139}, {2650, 5360}, {2653, 20456}, {2979, 14996}, {3002, 36020}, {3060, 35988}, {3120, 39793}, {3122, 3725}, {3124, 20455}, {3291, 21779}, {3454, 3874}, {3555, 41014}, {3688, 5311}, {3705, 3873}, {3736, 20845}, {3781, 5287}, {3792, 4038}, {3819, 37633}, {3868, 5051}, {3920, 9052}, {3948, 32937}, {4026, 22275}, {4111, 8013}, {4207, 5746}, {4223, 5943}, {4239, 40571}, {4418, 6007}, {4430, 31037}, {4661, 27081}, {4854, 20718}, {5044, 17514}, {5135, 5347}, {5231, 11021}, {5262, 12109}, {5310, 19133}, {5364, 19587}, {5396, 7420}, {5446, 36750}, {5462, 37509}, {5562, 5707}, {5650, 37674}, {5738, 26052}, {5751, 7580}, {5752, 37320}, {5902, 17889}, {6057, 21865}, {6688, 37680}, {7102, 18391}, {7140, 21933}, {7235, 17874}, {10396, 37324}, {10399, 27553}, {10473, 11269}, {10527, 35620}, {11451, 14997}, {13367, 20837}, {14839, 32928}, {14973, 22279}, {16052, 24473}, {16574, 37329}, {17049, 32914}, {18165, 35466}, {19684, 37149}, {20129, 33329}, {20861, 23439}, {20963, 23639}, {21748, 23201}, {21969, 37516}, {22276, 37593}, {24475, 30449}, {25441, 35637}, {27003, 40649}, {29837, 35614}, {32783, 38485}, {32913, 36572}, {33171, 35892}, {33586, 37492}, {38389, 41011}

X(40952) = isogonal conjugate of X(40412)
X(40952) = polar conjugate of isotomic conjugate of X(18591)
X(40952) = polar conjugate of isogonal conjugate of X(6)-Ceva conjugate of X(40956)
X(40952) = crosspoint of X(i) and X(j) for these {i,j}: {4, 37}, {6, 65}, {3668, 17758}
X(40952) = crosssum of X(i) and X(j) for these {i,j}: {2, 21}, {3, 81}, {2328, 4251}
X(40952) = crossdifference of every pair of points on line X(448)X(525)
X(40952) = trilinear product X(i)*X(j) for these {i,j}: {6, 2294}, {19, 18591}, {31, 442}, {33, 39791}, {37, 2260}, {42, 942}, {48, 1865}, {65, 14547}, {71, 1841}, {73, 1859}, {213, 5249}, {225, 23207}, {228, 1838}, {560, 1234}, {692, 23752}, {1020, 33525}, {1254, 8021}, {1333, 21675}, {1402, 6734}, {1824, 4303}, {1826, 14597}, {2333, 18607}
X(40952) = X(4)-Ceva conjugate of X(1841)
X(40952) = orthic-isogonal conjugate of X(1841)

X(40953) = CROSSPOINT OF X(4) AND X(40)

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

X(40953) lies on these lines: {1, 6611}, {4, 1903}, {10, 118}, {19, 15811}, {25, 12335}, {33, 64}, {34, 7973}, {40, 198}, {73, 3057}, {84, 5908}, {185, 1864}, {225, 10374}, {226, 10373}, {516, 2901}, {517, 1490}, {1071, 2823}, {1436, 9120}, {1498, 1753}, {1593, 22778}, {1785, 15498}, {1824, 11381}, {1861, 2883}, {1871, 13474}, {1872, 6000}, {1887, 6285}, {1890, 16656}, {2340, 3198}, {2807, 14872}, {5101, 12920}, {5784, 5907}, {5928, 12324}, {6198, 12262}, {6223, 34371}, {10914, 35666}, {15836, 37413}, {17102, 33811}, {18439, 40263}, {37305, 40658}

X(40953) = crosspoint of X(4) and X(40)
X(40953) = crosssum of X(3) and X(84)

X(40954) = CROSSPOINT OF X(4) AND X(42)

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

X(40954) lies on these lines: {4, 1246}, {6, 25}, {39, 33722}, {65, 1874}, {86, 511}, {181, 1918}, {238, 10974}, {373, 17259}, {512, 21131}, {674, 1213}, {1001, 22076}, {1230, 17142}, {2092, 3764}, {2200, 20970}, {2238, 3779}, {2245, 20992}, {2293, 22369}, {2294, 5360}, {2309, 20961}, {2388, 4111}, {3060, 17379}, {3122, 39780}, {3136, 14053}, {3688, 16589}, {3728, 22196}, {3917, 15668}, {4204, 26893}, {4259, 27623}, {4269, 16372}, {5462, 37510}, {5640, 17349}, {5929, 29957}, {5943, 17277}, {8053, 22080}, {10381, 12109}, {11002, 37677}, {21035, 21838}, {23444, 23633}, {23639, 23660}, {31757, 33682}

X(40954) = isogonal conjugate of isotomic conjugate of X(3136)
X(40954) = polar conjugate of isogonal conjugate of X(6)-Ceva conjugate of X(40955)
X(40954) = polar conjugate of isotomic conjugate of crosspoint of X(2) and X(71)
X(40954) = crosspoint of X(4) and X(42)
X(40954) = crosssum of X(3) and X(86)
X(40954) = crossdifference of every pair of points on line X(525)X(23145)
X(40954) = X(4)-Ceva conjugate of X(1860)
X(40954) = orthic-isogonal conjugate of X(1860)

X(40955) = CROSSPOINT OF X(6) AND X(27)

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

X(40955) lies on these lines: {6, 2200}, {19, 1148}, {32, 1400}, {39, 2309}, {51, 23362}, {57, 3500}, {58, 1474}, {65, 2179}, {71, 5248}, {72, 24511}, {172, 23531}, {184, 1475}, {284, 35206}, {354, 4020}, {674, 16683}, {942, 1755}, {1015, 9419}, {1125, 14964}, {1451, 1973}, {1509, 17209}, {1572, 23535}, {1914, 23530}, {2183, 2302}, {2225, 21808}, {2241, 23415}, {2242, 20460}, {2304, 13738}, {3874, 20785}, {5021, 7083}, {6210, 33682}, {14963, 31757}, {20963, 20967}, {20968, 23442}, {23630, 23653}

X(40955) = isogonal conjugate of isotomic conjugate of X(34830)
X(40955) = isogonal conjugate of polar conjugate of X(1860)
X(40955) = crosspoint of X(6) and X(27)
X(40955) = crosssum of X(2) and X(71)
X(40955) = crossdifference of every pair of points on line X(4064)X(20294)
X(40955) = X(6)-Ceva conjugate of isogonal conjugate of polar conjugate of X(40954)

X(40956) = CROSSPOINT OF X(6) AND X(28)

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

X(40956) lies on these lines: {1, 1011}, {2, 37609}, {3, 5256}, {6, 228}, {11, 430}, {25, 34}, {28, 36419}, {31, 184}, {32, 21750}, {36, 199}, {39, 42}, {48, 5320}, {51, 1400}, {55, 1100}, {58, 4215}, {63, 37507}, {209, 583}, {210, 20990}, {237, 23197}, {239, 13588}, {306, 1009}, {386, 19343}, {518, 16687}, {579, 26893}, {612, 1107}, {667, 8034}, {672, 3690}, {942, 8021}, {956, 37059}, {958, 37060}, {968, 20992}, {993, 29644}, {999, 4666}, {1001, 37869}, {1108, 1824}, {1125, 4204}, {1284, 41011}, {1333, 2203}, {1386, 16678}, {1404, 13366}, {1405, 15004}, {1412, 26884}, {1451, 22341}, {1453, 13738}, {1468, 33718}, {1977, 9419}, {1999, 4203}, {2082, 15496}, {2194, 7113}, {2260, 14547}, {2269, 22080}, {2308, 3724}, {2999, 4191}, {3086, 4207}, {3187, 11322}, {3190, 4253}, {3198, 40133}, {3218, 3794}, {3286, 3666}, {3310, 23225}, {3745, 16679}, {3870, 37590}, {3917, 28274}, {4184, 17011}, {4210, 17012}, {4252, 22344}, {4257, 23205}, {5053, 26890}, {5088, 16750}, {5120, 7085}, {5222, 37262}, {5271, 11358}, {5287, 16058}, {5310, 17798}, {5563, 29820}, {5650, 28272}, {5717, 37225}, {5728, 23171}, {7290, 16878}, {7292, 7453}, {8053, 37593}, {8609, 21807}, {8624, 23533}, {8666, 29652}, {11679, 16405}, {16056, 26723}, {16287, 37594}, {16372, 37607}, {16373, 17022}, {16395, 16834}, {16475, 16778}, {16478, 33714}, {16688, 37553}, {17017, 37575}, {17023, 37329}, {17126, 37619}, {20777, 33863}, {20968, 21749}, {21318, 32118}, {23567, 23654}, {26626, 37175}, {29683, 40109}, {36077, 40395}

X(40956) = isogonal conjugate of X(40422)
X(40956) = polar conjugate of isotomic conjugate of X(14597)
X(40956) = crosspoint of X(i) and X(j) for these {i,j}: {6, 28}, {57, 1333}, {31, 6186}, {269, 39950}
X(40956) = crosssum of X(i) and X(j) for these {i,j}: {2, 72}, {8, 321}, {75, 319}, {200, 3294}
X(40956) = crossdifference of every pair of points on line X(4391)X(17494)
X(40956) = X(6)-Ceva conjugate of isogonal conjugate of isotomic conjugate of X(18591)
X(40956) = X(6)-Ceva conjugate of isogonal conjugate of polar conjugate of X(40952)
X(40956) = trilinear product X(i)*X(j) for these {i,j}: {6, 2260}, {19, 14597}, {25, 4303}, {31, 942}, {32, 5249}, {34, 23207}, {48, 1841}, {56, 14547}, {184, 1838}, {442, 2206}, {500, 6186}, {603, 1859}, {1042, 8021}, {1333, 2294}, {1397, 6734}, {1461, 33525}, {1474, 18591}, {1576, 23752}, {1973, 18607}, {2299, 39791}

X(40957) = CROSSPOINT OF X(6) AND X(33)

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

X(40957) lies on these lines: {1, 8074}, {6, 603}, {25, 41}, {73, 910}, {169, 22350}, {212, 32561}, {213, 14936}, {223, 24604}, {672, 37195}, {800, 1400}, {1042, 8776}, {1055, 4322}, {1193, 2082}, {1200, 10460}, {1201, 2170}, {1212, 22072}, {1334, 16588}, {1410, 1475}, {1450, 40133}, {1802, 5452}, {1951, 20838}, {2092, 2347}, {2238, 37324}, {2262, 22063}, {2270, 27621}, {2280, 7124}, {3192, 7156}, {3195, 7154}, {4258, 14547}, {5819, 37694}, {20229, 20970}, {40117, 40396}

X(40957) = isogonal conjugate of isotomic conjugate of X(20262)
X(40957) = isogonal conjugate of polar conjugate of X(1856)
X(40957) = polar conjugate of isotomic conjugate of X(40945)
X(40957) = crosspoint of X(6) and X(33)
X(40957) = crosssum of X(2) and X(77)
X(40957) = crossdifference of every pair of points on line X(4025)X(8058)
X(40957) = X(6)-Ceva conjugate of isogonal conjugate of isotomic conjugate of X(17102)
X(40957) = X(6)-Ceva conjugate of isogonal conjugate of polar conjugate of X(2262)

X(40958) = CROSSPOINT OF X(6) AND X(34)

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

X(40958) lies on these lines: {1, 329}, {6, 31}, {25, 604}, {33, 3087}, {34, 207}, {44, 2318}, {51, 1400}, {56, 17810}, {73, 1104}, {108, 40397}, {184, 1404}, {223, 614}, {228, 2347}, {238, 25941}, {244, 1427}, {375, 20990}, {386, 30282}, {511, 28274}, {580, 2077}, {581, 1193}, {748, 25893}, {756, 1212}, {909, 2206}, {991, 2999}, {1035, 1106}, {1108, 1864}, {1149, 16485}, {1402, 3271}, {1405, 9777}, {1451, 1470}, {1475, 23636}, {1612, 3074}, {1724, 3682}, {1743, 3190}, {1818, 4383}, {1824, 2170}, {1841, 2181}, {2183, 2352}, {2187, 7083}, {2194, 2317}, {2223, 23638}, {2340, 16572}, {2356, 3195}, {2635, 3772}, {3009, 16968}, {3119, 20311}, {3720, 5712}, {3752, 22053}, {3915, 7078}, {3917, 28272}, {4300, 30503}, {4642, 15852}, {5053, 5285}, {5320, 21748}, {5716, 10459}, {5943, 37609}, {6186, 34068}, {7069, 8609}, {14936, 20230}, {17056, 30950}, {17123, 25889}, {17809, 38296}, {18591, 20966}, {20974, 23624}, {22097, 37516}, {22350, 37817}, {23640, 33718}

X(40958) = isogonal conjugate of X(40424)
X(40958) = polar conjugate of isotomic conjugate of isogonal conjugate of X(40444)
X(40958) = crosspoint of X(i) and X(j) for these {i,j}: {1, 1436}, {6, 34}
X(40958) = crosssum of X(i) and X(j) for these {i,j}: {1, 329}, {2, 78}
X(40958) = crossdifference of every pair of points on line X(514)X(40863)
X(40958) = trilinear product X(i)*X(j) for these {i,j}: {4, 23204}, {6, 1108}, {25, 1071}, {28, 3611}, {31, 1210}, {32, 17862}, {55, 37566}, {56, 1864}, {560, 1226}, {1333, 21933}, {1532, 34858}, {2208, 6260}
X(40958) = X(6)-Ceva conjugate of isogonal conjugate of polar conjugate of X(1864)

X(40959) = CROSSPOINT OF X(7) AND X(28)

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

X(40959) lies on these lines: {1, 3}, {2, 4463}, {6, 17441}, {11, 132}, {19, 614}, {31, 3827}, {38, 71}, {58, 18732}, {63, 24476}, {72, 32777}, {159, 1473}, {209, 306}, {226, 12618}, {238, 1762}, {345, 3868}, {427, 16580}, {497, 4329}, {595, 1782}, {604, 20277}, {950, 35650}, {1100, 4280}, {1104, 1829}, {1122, 10374}, {1193, 18673}, {1357, 34228}, {1401, 39796}, {1423, 24430}, {1427, 1876}, {1428, 11428}, {1458, 37755}, {1469, 11435}, {1760, 25494}, {1824, 3772}, {1836, 12723}, {1869, 23536}, {1888, 1891}, {2260, 18674}, {2264, 16470}, {2294, 3720}, {2550, 4359}, {3011, 40635}, {3101, 7191}, {3198, 3752}, {3286, 18607}, {3693, 22021}, {3706, 18697}, {3739, 3925}, {3779, 33088}, {4030, 5836}, {4123, 37099}, {4259, 16465}, {4376, 34377}, {4523, 25453}, {4641, 34381}, {4854, 11997}, {5272, 9816}, {5324, 7291}, {7069, 28387}, {8609, 21318}, {9004, 32912}, {10391, 18650}, {10974, 14054}, {11031, 22097}, {11683, 32942}, {12835, 39045}, {16974, 21861}, {17194, 18726}, {17481, 26096}, {17495, 17784}, {19785, 20243}, {20271, 21775}, {21342, 21866}, {35466, 40656}

X(40959) = polar conjugate of isotomic conjugate of X(18734)
X(40959) = complement of X(4463)
X(40959) = crosspoint of X(7) and X(28)
X(40959) = crosssum of X(55) and X(72)

X(40960) = CROSSPOINT OF X(7) AND X(29)

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

X(40960) lies on these lines: {1, 4}, {2, 7070}, {7, 8809}, {8, 27413}, {10, 7074}, {11, 4989}, {55, 11347}, {56, 37046}, {57, 3332}, {65, 10373}, {77, 10431}, {142, 1040}, {219, 3686}, {221, 21628}, {347, 9812}, {354, 1439}, {387, 9581}, {516, 1214}, {527, 24430}, {553, 7004}, {938, 10365}, {942, 6000}, {1125, 18641}, {1172, 17188}, {1210, 5706}, {1394, 37434}, {1456, 7965}, {1498, 6245}, {1754, 3911}, {1827, 17441}, {1836, 3668}, {1837, 10367}, {1839, 18675}, {1854, 3671}, {1936, 2328}, {2192, 8808}, {2269, 2270}, {2310, 41011}, {2807, 5173}, {2822, 15902}, {3022, 39793}, {3075, 6705}, {3100, 5249}, {3182, 3333}, {3190, 5853}, {3452, 9817}, {3664, 10391}, {3691, 28070}, {3706, 6737}, {3817, 37695}, {3914, 4336}, {4648, 10383}, {5728, 8807}, {5929, 21334}, {5932, 10580}, {8727, 34050}, {9539, 31019}, {10395, 16471}, {10624, 37528}, {10903, 11029}, {11022, 15498}, {11365, 13737}, {14557, 17642}, {14746, 14749}, {14942, 33073}, {22079, 37319}, {33536, 40152}, {37310, 40292}, {39130, 39585}

X(40960) = crosspoint of X(7) and X(29)
X(40960) = crosssum of X(55) and X(73)

X(40961) = CROSSPOINT OF X(7) AND X(34)

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

X(40961) lies on these lines: {1, 28037}, {7, 4388}, {10, 12}, {11, 21621}, {31, 36570}, {34, 7337}, {56, 1448}, {57, 238}, {354, 1122}, {388, 5015}, {392, 26128}, {497, 3673}, {517, 33144}, {608, 3162}, {612, 28038}, {614, 1473}, {942, 26098}, {960, 25527}, {982, 6210}, {1042, 1426}, {1214, 1284}, {1279, 1626}, {1357, 1360}, {1365, 3320}, {1397, 1456}, {1402, 1427}, {1403, 1465}, {1460, 2263}, {1463, 17625}, {1464, 11214}, {1469, 5173}, {1824, 3120}, {1828, 3924}, {1836, 12723}, {2354, 20227}, {2886, 24476}, {3212, 21609}, {3555, 4865}, {3660, 7248}, {3706, 33936}, {3752, 20967}, {3772, 3827}, {3782, 12721}, {3869, 26132}, {3914, 17441}, {3966, 30617}, {4654, 31178}, {5266, 36508}, {5439, 25496}, {7175, 16800}, {7225, 17017}, {10473, 24471}, {10914, 32920}, {16583, 23620}, {17114, 37566}, {18838, 34029}, {21321, 37597}, {21867, 21949}, {24218, 29057}, {25760, 36503}, {29668, 36538}, {33137, 34381}

X(40961) = crosspoint of X(7) and X(34)
X(40961) = crosssum of X(55) and X(78)

X(40962) = CROSSPOINT OF X(8) AND X(19)

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

X(40962) lies on these lines: {1, 25514}, {6, 40635}, {19, 1460}, {31, 1824}, {33, 7083}, {37, 20967}, {41, 5311}, {42, 2170}, {55, 39943}, {56, 21370}, {63, 12721}, {72, 4362}, {181, 2262}, {200, 3061}, {210, 3686}, {392, 3741}, {517, 33137}, {607, 3162}, {612, 2082}, {756, 2347}, {960, 10480}, {1108, 1402}, {1397, 2182}, {1828, 21935}, {1829, 5230}, {1864, 3271}, {1871, 3072}, {1872, 3073}, {2171, 10460}, {2223, 3198}, {2264, 3745}, {3011, 17441}, {3057, 4847}, {3185, 8609}, {3753, 25453}, {3772, 3827}, {4512, 11997}, {5439, 29654}, {5711, 9895}, {6047, 7957}, {10914, 29673}, {14936, 16584}, {16583, 17442}, {17061, 24476}, {33144, 34381}

X(40962) = crosspoint of X(8) and X(19)
X(40962) = crosssum of X(56) and X(63)

X(40963) = CROSSPOINT OF X(8) AND X(27)

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

X(40963) lies on these lines: {1, 5802}, {2, 380}, {4, 2257}, {6, 946}, {9, 497}, {11, 2264}, {19, 1210}, {71, 10624}, {218, 21068}, {219, 12053}, {281, 9581}, {284, 1125}, {515, 1108}, {516, 579}, {608, 34050}, {610, 3086}, {960, 3686}, {966, 31435}, {1001, 4254}, {1172, 1848}, {1449, 3485}, {1479, 1723}, {1699, 5746}, {1741, 30223}, {1743, 9614}, {1838, 8748}, {1839, 2260}, {1901, 18483}, {2082, 20262}, {2287, 41012}, {2886, 5750}, {3008, 18589}, {3554, 6261}, {3663, 24316}, {3664, 17197}, {3694, 5853}, {3817, 5747}, {3946, 41003}, {4000, 41010}, {5274, 27382}, {5279, 26015}, {5572, 11028}, {5838, 27508}, {7289, 24213}, {8074, 24005}, {8557, 10445}, {10165, 37504}, {11373, 20818}, {21866, 28194}, {26063, 31397}, {31730, 37500}

X(40963) = crosspoint of X(8) and X(27)
X(40963) = crosssum of X(56) and X(71)

X(40964) = CROSSPOINT OF X(8) AND X(28)

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

X(40964) lies on these lines: {9, 3057}, {56, 3420}, {65, 1453}, {517, 1724}, {942, 18191}, {960, 41002}, {1104, 1829}, {1828, 3772}, {1858, 3271}, {1890, 7354}, {2170, 2654}, {3710, 3880}, {3827, 3924}, {5262, 5324}, {5294, 5836}, {8609, 13724}, {18732, 30117}

X(40964) = crosspoint of X(8) and X(28)
X(40964) = crosssum of X(56) and X(72)

X(40965) = CROSSPOINT OF X(8) AND X(33)

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

X(40965) lies on these lines: {6, 12723}, {8, 3718}, {9, 11997}, {11, 24388}, {19, 37580}, {33, 6059}, {37, 4068}, {41, 1827}, {42, 1824}, {55, 39943}, {65, 1439}, {69, 18252}, {72, 740}, {193, 12530}, {210, 2321}, {354, 3946}, {392, 32941}, {497, 4012}, {517, 3779}, {518, 3875}, {523, 10099}, {527, 17635}, {758, 4780}, {960, 3886}, {1071, 24257}, {1108, 2223}, {1334, 21039}, {1402, 3198}, {1441, 13576}, {1864, 23638}, {1871, 37529}, {1872, 37699}, {2082, 4319}, {2170, 2293}, {2175, 2264}, {2262, 21746}, {2268, 28125}, {2310, 2347}, {2340, 17452}, {2805, 17351}, {2809, 3663}, {3056, 15733}, {3057, 3059}, {3169, 4073}, {3271, 14100}, {3555, 32921}, {3678, 4133}, {3694, 4433}, {3696, 18697}, {3697, 3773}, {3740, 17286}, {3751, 15076}, {3753, 4085}, {3896, 4463}, {3914, 17441}, {4018, 4743}, {4343, 17451}, {4517, 40659}, {4527, 4533}, {4847, 21233}, {4878, 21801}, {7300, 16686}, {8609, 15624}, {9004, 17276}, {12689, 28849}, {12722, 16475}, {15185, 20358}, {17447, 37597}, {17610, 19029}, {17872, 20227}, {20683, 21871}, {21035, 22308}, {21853, 22277}, {24248, 34381}

X(40965) = crosspoint of X(8) and X(33)
X(40965) = crosssum of X(56) and X(77)
X(40965) = excentral-to-ABC barycentric image of X(69)
X(40965) = X(69)com(extouch triangle)

X(40966) = CROSSPOINT OF X(8) AND X(37)

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

X(40966) lies on these lines: {1, 8731}, {9, 23638}, {10, 37865}, {12, 22000}, {21, 10544}, {37, 181}, {42, 213}, {55, 219}, {71, 1402}, {72, 4028}, {200, 3208}, {209, 37593}, {210, 2321}, {312, 9564}, {333, 35104}, {345, 35628}, {375, 16814}, {511, 846}, {756, 21804}, {960, 1682}, {968, 21746}, {1460, 2256}, {1756, 13097}, {1962, 4890}, {2092, 3725}, {2187, 37586}, {2269, 20967}, {2292, 22076}, {2323, 20959}, {3030, 3740}, {3056, 4512}, {3057, 4847}, {3178, 10381}, {3240, 26911}, {3271, 3683}, {3705, 3877}, {3724, 22080}, {3743, 10974}, {3750, 9052}, {3757, 14839}, {3779, 37553}, {3781, 17594}, {3819, 17596}, {3878, 29671}, {3884, 29655}, {3890, 29843}, {3917, 4414}, {3931, 10822}, {3932, 22325}, {3943, 14973}, {4260, 17592}, {5091, 37261}, {5745, 21334}, {8013, 21044}, {9565, 31359}, {17056, 20718}, {17185, 18235}, {17593, 40649}, {18178, 18253}, {20958, 26890}, {22206, 36197}, {23630, 31442}, {24264, 37090}, {25941, 37575}, {29309, 33109}

X(40966) = isogonal conjugate of isotomic conjugate of X(3704)
X(40966) = crosspoint of X(8) and X(37)
X(40966) = crosssum of X(56) and X(81)
X(40966) = crossdifference of every pair of points on line X(7178)X(7192)
X(40966) = X(8)-Ceva conjugate of X(3965)

X(40967) = CROSSPOINT OF X(9) AND X(10)

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

X(40967) lies on these lines: {1, 12867}, {9, 33}, {10, 201}, {37, 42}, {38, 3772}, {55, 21811}, {71, 1824}, {181, 21804}, {199, 2173}, {209, 2171}, {220, 3715}, {240, 18679}, {429, 21671}, {612, 8557}, {960, 2654}, {984, 33135}, {1212, 1864}, {1393, 5705}, {1762, 4220}, {1834, 2292}, {1865, 21675}, {1897, 40435}, {1953, 26893}, {2265, 26890}, {2310, 3683}, {3191, 3678}, {3474, 24341}, {3690, 21801}, {3700, 6608}, {3740, 25091}, {3925, 6354}, {5044, 22350}, {5273, 24430}, {5745, 7004}, {5784, 22053}, {7102, 26063}, {7140, 21011}, {7465, 21367}, {11031, 35466}, {15823, 22361}, {16560, 37261}, {18673, 37225}, {20966, 21014}, {21694, 21699}, {21911, 26942}, {24316, 26052}, {24476, 28388}, {25024, 33118}, {27472, 37109}

X(40967) = crosspoint of X(9) and X(10)
X(40967) = crosssum of X(57) and X(58)
X(40967) = X(10)-Ceva conjugate of X(21675)

X(40968) = CROSSPOINT OF X(9) AND X(19)

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

X(40968) lies on these lines: {1, 21748}, {6, 1411}, {8, 9}, {19, 208}, {31, 1824}, {37, 41}, {44, 21871}, {48, 8609}, {55, 21811}, {63, 30699}, {219, 17452}, {220, 21809}, {281, 21044}, {294, 983}, {379, 4032}, {579, 16548}, {604, 1108}, {607, 21148}, {610, 1055}, {672, 1723}, {1200, 40131}, {1404, 2261}, {1405, 2262}, {1423, 7291}, {1475, 2257}, {1707, 7996}, {1743, 11531}, {1751, 6358}, {1763, 28387}, {1781, 21381}, {2087, 34543}, {2173, 2178}, {2175, 4336}, {2310, 7071}, {2330, 28125}, {2345, 36568}, {2348, 6602}, {2911, 21801}, {3101, 27659}, {3168, 26000}, {3219, 25269}, {3663, 16551}, {3690, 23638}, {3713, 33299}, {3731, 3961}, {3772, 26934}, {4000, 16560}, {4319, 8647}, {4362, 5282}, {5279, 20460}, {5320, 21807}, {5324, 24430}, {5783, 39244}, {6766, 16572}, {7117, 17053}, {7229, 36483}, {7368, 34524}, {7614, 37787}, {8608, 40590}, {8756, 24005}, {8897, 25894}, {9309, 9355}, {14936, 21796}, {17355, 29673}, {17439, 20818}, {18785, 38991}, {19785, 21367}, {21368, 24597}, {21371, 26998}, {23640, 27368}, {24591, 26538}

X(40968) = crosspoint of X(9) and X(19)
X(40968) = crosssum of X(57) and X(63)

X(40969) = CROSSPOINT OF X(9) AND X(25)

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

X(40969) lies on these lines: {9, 1265}, {32, 2333}, {41, 5452}, {71, 4426}, {213, 3271}, {220, 3056}, {604, 22654}, {958, 2269}, {1104, 1400}, {1212, 2347}, {1334, 3683}, {2082, 3486}, {2183, 16968}, {2354, 5336}, {5716, 40131}, {22072, 23637}

X(40969) = crosspoint of X(9) and X(25)
X(40969) = crosssum of X(57) and X(69)

X(40970) = CROSSPOINT OF X(9) AND X(28)

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

X(40970) lies on these lines: {25, 1108}, {31, 2262}, {44, 26893}, {55, 1212}, {56, 5338}, {154, 3554}, {910, 2352}, {1100, 2194}, {1104, 1829}, {1279, 17441}, {1395, 1455}, {1851, 3772}, {2264, 14547}, {2269, 3683}, {2280, 37593}, {2318, 2348}, {3666, 5324}, {3745, 17451}, {8557, 17810}, {37538, 40133}

X(40970) = crosspoint of X(9) and X(28)
X(40970) = crosssum of X(57) and X(72)

X(40971) = CROSSPOINT OF X(9) AND X(40)

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

X(40971) lies on these lines: {1, 947}, {4, 1697}, {9, 7003}, {19, 25}, {34, 1902}, {40, 196}, {63, 1897}, {71, 8802}, {84, 18283}, {108, 165}, {204, 212}, {207, 5584}, {220, 7007}, {225, 11471}, {281, 1856}, {318, 5250}, {475, 12053}, {497, 1861}, {610, 2188}, {1040, 19649}, {1075, 1712}, {1096, 1253}, {1334, 1857}, {1435, 23710}, {1763, 16870}, {1766, 7070}, {1785, 5119}, {1864, 2082}, {1870, 7962}, {1872, 3295}, {1876, 17642}, {1887, 3303}, {2182, 2192}, {2261, 19354}, {2269, 7102}, {2331, 3195}, {3197, 17832}, {3576, 15500}, {3601, 6198}, {3895, 5081}, {4200, 9785}, {5279, 6060}, {5930, 16389}, {12514, 22027}, {15845, 26020}, {31393, 34231}, {35445, 37441}

X(40971) = polar conjugate of isotomic conjugate of X(2324)
X(40971) = crosspoint of X(9) and X(40)
X(40971) = crosssum of X(57) and X(84)
X(40971) = X(9)-Ceva conjugate of X(7079)

X(40972) = CROSSPOINT OF X(9) AND X(41)

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

X(40972) lies on these lines: {1, 28592}, {6, 5364}, {9, 312}, {31, 39258}, {37, 24511}, {38, 17456}, {39, 1401}, {41, 212}, {42, 2225}, {55, 7077}, {190, 1965}, {228, 9454}, {354, 20459}, {672, 4641}, {740, 21369}, {872, 1185}, {1196, 21936}, {1197, 3774}, {1212, 23640}, {1334, 3683}, {1500, 2179}, {1755, 2276}, {1843, 21035}, {1964, 3051}, {2185, 2311}, {2235, 21877}, {2347, 2348}, {2646, 20460}, {3057, 23544}, {3208, 39250}, {3219, 3797}, {3271, 21795}, {3303, 23535}, {3508, 32926}, {3725, 40728}, {3730, 7262}, {3748, 23415}, {4697, 16549}, {5132, 20778}, {8026, 17336}, {8679, 23636}, {9315, 35445}, {16555, 34997}, {16584, 21760}, {16588, 23638}, {21788, 30646}, {23443, 37080}, {23531, 24929}, {24578, 32939}, {32912, 36808}

X(40972) = isogonal conjugate of isotomic conjugate of X(33299)
X(40972) = crosspoint of X(9) and X(41)
X(40972) = crosssum of X(57) and X(85)
X(40972) = X(9)-Ceva conjugate of X(33299)

X(40973) = CROSSPOINT OF X(10) AND X(19)

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

X(40973) lies on these lines: {10, 321}, {19, 31}, {38, 5271}, {42, 2294}, {197, 3724}, {210, 4016}, {240, 26000}, {500, 11259}, {612, 1962}, {647, 11124}, {740, 10327}, {896, 21376}, {899, 4137}, {968, 25081}, {1068, 1148}, {1193, 9895}, {1714, 24443}, {1824, 16583}, {2173, 2206}, {2643, 3725}, {2650, 11529}, {3924, 18673}, {3959, 26893}, {5336, 15496}, {8143, 32148}, {17446, 32914}, {17470, 28247}, {20966, 21014}, {21796, 21807}

X(40973) = crosspoint of X(10) and X(19)
X(40973) = crosssum of X(58) and X(63)
X(40973) = X(10)-Ceva conjugate of X(21678)
X(40973) = homothetic center of incentral triangle and medial triangle of Ayme triangle

X(40974) = CROSSPOINT OF X(19) AND X(21)

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

X(40974) lies on these lines: {1, 3167}, {3, 1711}, {9, 55}, {31, 1824}, {44, 22276}, {65, 1726}, {517, 3073}, {1100, 20959}, {1108, 1397}, {1195, 2355}, {1402, 2182}, {1460, 8557}, {1707, 12717}, {1776, 5324}, {1962, 21748}, {2083, 16583}, {2161, 40635}, {2170, 2308}, {2179, 2333}, {2328, 11997}, {3434, 26065}, {5842, 39591}, {8609, 20986}, {17127, 20243}

X(40974) = crosspoint of X(19) and X(21)
X(40974) = crosssum of X(63) and X(65)

X(40975) = CROSSPOINT OF X(19) AND X(27)

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

X(40975) lies on these lines: {4, 9}, {27, 310}, {28, 2201}, {278, 34497}, {407, 8735}, {501, 1474}, {607, 4214}, {862, 2355}, {1475, 33128}, {1824, 2356}, {1836, 23620}, {1840, 38462}, {1848, 15149}, {1860, 4196}, {1900, 5089}, {1973, 4185}, {2082, 41011}, {2260, 23537}, {2635, 23619}, {3691, 21020}, {9816, 37169}, {12723, 16583}, {15975, 17462}, {16589, 22369}, {20963, 39793}

X(40975) = polar conjugate of isotomic conjugate of X(3720)
X(40975) = crosspoint of X(19) and X(27)
X(40975) = crosssum of X(63) and X(71)

X(40976) = CROSSPOINT OF X(19) AND X(29)

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

X(40976) lies on these lines: {1, 6353}, {4, 43}, {6, 1395}, {19, 2258}, {25, 41}, {28, 4281}, {31, 1195}, {33, 200}, {34, 2999}, {55, 2212}, {73, 1452}, {78, 1039}, {199, 1951}, {284, 2189}, {386, 7713}, {427, 899}, {430, 8735}, {444, 1193}, {451, 32783}, {468, 3720}, {608, 10460}, {800, 1402}, {902, 14975}, {1172, 3684}, {1197, 2211}, {1201, 11396}, {1824, 2181}, {1843, 23638}, {1870, 29821}, {1905, 22350}, {2092, 2354}, {2202, 4206}, {2203, 21748}, {3089, 37529}, {3195, 11406}, {3214, 5090}, {3240, 6995}, {3517, 37698}, {3783, 4213}, {3961, 6198}, {4232, 17018}, {4270, 5338}, {5089, 21840}, {7487, 37699}, {8889, 16569}, {16583, 17442}, {26102, 38282}, {30950, 37453}

X(40976) = polar conjugate of isotomic conjugate of X(2269)
X(40976) = crosspoint of X(19) and X(29)
X(40976) = crosssum of X(63) and X(73)

X(40977) = CROSSPOINT OF X(19) AND X(37)

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

X(40977) lies on these lines: {1, 2287}, {6, 2294}, {9, 2292}, {19, 31}, {37, 42}, {41, 5336}, {43, 27396}, {44, 3958}, {58, 1781}, {71, 4642}, {198, 3724}, {213, 2171}, {244, 2260}, {281, 5230}, {346, 740}, {579, 24443}, {607, 21148}, {614, 2257}, {758, 1743}, {896, 1761}, {992, 39244}, {1042, 30456}, {1104, 2264}, {1108, 1201}, {1213, 20654}, {1253, 3747}, {1254, 1400}, {1333, 2173}, {1475, 20227}, {1731, 16470}, {1769, 2492}, {1826, 21935}, {1834, 21671}, {1901, 3120}, {1959, 27644}, {2092, 21798}, {2170, 2300}, {2176, 17452}, {2197, 21796}, {2268, 16968}, {2269, 41015}, {2322, 26000}, {2331, 8557}, {2345, 21020}, {2999, 25080}, {3008, 18698}, {3214, 3694}, {3216, 25078}, {3553, 31880}, {3731, 3743}, {3914, 8804}, {4000, 8680}, {4264, 16547}, {4647, 17355}, {5222, 25255}, {5247, 5279}, {5269, 11221}, {16600, 21061}, {17352, 35550}, {17353, 18697}, {21014, 21965}

X(40977) = crosspoint of X(19) and X(37)
X(40977) = crosssum of X(63) and X(81)

X(40978) = CROSSPOINT OF X(19) AND X(42)

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

X(40978) lies on these lines: {19, 2215}, {31, 1932}, {41, 3724}, {42, 2198}, {58, 1755}, {71, 3931}, {213, 1402}, {579, 986}, {583, 4016}, {672, 2292}, {740, 3501}, {758, 3061}, {942, 2260}, {1254, 1400}, {1334, 1962}, {1468, 4020}, {1475, 2650}, {1500, 2667}, {1724, 24511}, {2175, 40370}, {2304, 2352}, {3673, 8680}, {3730, 3743}, {3747, 14974}, {4515, 22317}, {4647, 16549}

X(40978) = crosspoint of X(19) and X(42)
X(40978) = crosssum of X(63) and X(86)

X(40979) = CROSSPOINT OF X(21) AND X(27)

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

X(40979) lies on these lines: {1, 15656}, {3, 1713}, {6, 1012}, {7, 17197}, {9, 21}, {20, 579}, {27, 57}, {28, 10396}, {40, 4269}, {58, 84}, {63, 3187}, {71, 4304}, {81, 7190}, {216, 37732}, {223, 18603}, {380, 4267}, {610, 859}, {1071, 1108}, {1333, 8557}, {1445, 14953}, {1723, 2193}, {1741, 2082}, {1766, 14964}, {1778, 16572}, {1839, 2260}, {1901, 37447}, {1953, 18389}, {2150, 2326}, {2194, 30223}, {2294, 10122}, {3286, 5732}, {3486, 15830}, {4254, 16552}, {4266, 17576}, {5327, 11372}, {5358, 7719}, {5746, 37434}, {5747, 6837}, {5776, 37252}, {8021, 10382}, {8822, 18206}, {10391, 17194}, {15149, 18634}, {16465, 22021}, {17171, 41010}, {37426, 37500}

X(40979) = crosspoint of X(21) and X(27)
X(40979) = crosssum of X(65) and X(71)

X(40980) = CROSSPOINT OF X(21) AND X(28)

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

X(40980) lies on these lines: {1, 1762}, {3, 1714}, {6, 13733}, {8, 21}, {11, 29}, {12, 25516}, {19, 1333}, {27, 7354}, {28, 56}, {40, 15952}, {58, 65}, {68, 3560}, {81, 11101}, {86, 409}, {209, 5247}, {272, 1441}, {283, 18178}, {284, 1731}, {385, 17522}, {388, 37113}, {405, 1211}, {517, 1780}, {859, 5358}, {1010, 3925}, {1104, 1829}, {1108, 1474}, {1319, 2360}, {1654, 16865}, {1724, 5752}, {1817, 5204}, {1834, 3145}, {2328, 3057}, {2550, 11115}, {2975, 11683}, {3086, 37168}, {3286, 16049}, {3430, 6045}, {3924, 26934}, {4184, 33139}, {4195, 27319}, {4215, 23361}, {4221, 5584}, {4227, 12114}, {4228, 26228}, {4234, 34612}, {4252, 37397}, {4293, 31900}, {4383, 13732}, {4516, 38336}, {4653, 37080}, {4658, 37816}, {5047, 30832}, {5230, 23843}, {5251, 37322}, {5292, 11334}, {7419, 28353}, {8021, 26357}, {11376, 17188}, {11398, 30733}, {12513, 30614}, {12635, 40571}, {12943, 31902}, {16609, 36017}, {16948, 37567}, {17061, 36560}, {17524, 37601}, {17539, 17784}, {20989, 27652}, {21120, 21789}, {22768, 37277}, {24883, 37311}, {25681, 27412}, {26543, 36740}, {37052, 37538}, {37259, 37646}

X(40980) = crosspoint of X(21) and X(28)
X(40980) = crosssum of X(65) and X(72)

X(40981) = CROSSPOINT OF X(25) AND X(32)

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

X(40981) lies on these lines: {2, 22062}, {3, 3618}, {4, 10790}, {5, 17500}, {6, 160}, {22, 16989}, {25, 393}, {32, 682}, {51, 216}, {69, 11328}, {98, 32085}, {141, 5201}, {143, 3133}, {159, 20897}, {193, 37465}, {206, 13345}, {217, 27374}, {317, 15143}, {384, 24729}, {511, 20819}, {571, 19136}, {604, 22096}, {669, 6041}, {800, 1843}, {852, 6389}, {1084, 36425}, {1176, 3398}, {1576, 2965}, {1609, 3148}, {1634, 3629}, {1992, 20794}, {1995, 17008}, {2271, 23212}, {2352, 23218}, {3003, 9969}, {3051, 8265}, {3053, 11326}, {3060, 22087}, {3135, 9777}, {3589, 8266}, {4230, 27377}, {5007, 23208}, {5020, 34229}, {5188, 22078}, {6375, 8623}, {6638, 40680}, {7467, 7792}, {7736, 20885}, {8553, 37457}, {8882, 32713}, {9301, 22138}, {9753, 17907}, {9917, 14001}, {11008, 22152}, {12110, 36794}, {14573, 14574}, {15004, 23195}, {15270, 19132}, {16285, 34811}, {18907, 21177}, {19125, 20960}, {20233, 34980}, {22369, 33718}, {30258, 40681}, {34098, 37689}, {37344, 37491}

X(40981) = isogonal conjugate of X(34384)
X(40981) = crosspoint of X(i) and X(j) for these {i,j}: {6, 2980}, {25, 32}
X(40981) = crosssum of X(i) and X(j) for these {i,j}: {2, 2979}, {69, 76}, {75, 21579}
X(40981) = crossdifference of every pair of points on line X(3267)X(7799)
X(40981) = X(25)-Ceva conjugate of X(3199)
X(40981) = trilinear product X(i)*X(j) for these {i,j}: {5, 560}, {6, 2179}, {19, 217}, {31, 51}, {32, 1953}, {48, 3199}, {53, 9247}, {82, 27374}, {184, 2181}, {216, 1973}, {311, 1917}, {418, 1096}, {669, 2617}, {798, 1625}, {1087, 14573}, {1393, 2175}, {1397, 7069}, {1501, 14213}, {1918, 18180}, {1923, 17500}, {1924, 14570}, {2205, 17167}, {2206, 21807}, {2212, 30493}, {2290, 11060}, {2618, 14574}, {15451, 32676}

X(40982) = CROSSPOINT OF X(25) AND X(33)

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

X(40982) lies on these lines: {4, 978}, {25, 31}, {33, 200}, {899, 5101}, {1193, 4186}, {1201, 1828}, {1450, 1877}, {1883, 27627}, {2299, 2316}, {2333, 3199}, {2356, 3192}, {3452, 22072}, {3846, 11105}, {3915, 26378}, {9368, 37588}, {28044, 28050}

X(40982) = polar conjugate of isotomic conjugate of X(2347)
X(40982) = crosspoint of X(25) and X(33)
X(40982) = crosssum of X(69) and X(77)

X(40983) = CROSSPOINT OF X(25) AND X(34)

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

X(40983) lies on these lines: {4, 1742}, {19, 2356}, {25, 31}, {28, 3736}, {33, 4326}, {34, 269}, {42, 2355}, {142, 17194}, {1064, 36009}, {1193, 17523}, {1438, 1474}, {1827, 2293}, {1843, 2333}, {1973, 9454}, {4270, 5338}, {4300, 37387}, {20857, 28266}, {34261, 37385}

X(40983) = isogonal conjugate of isotomic conjugate of polar conjugate of X(32008)
X(40983) = polar conjugate of isotomic conjugate of X(1475)
X(40983) = crosspoint of X(25) and X(34)
X(40983) = crosssum of X(69) and X(78)

X(40984) = CROSSPOINT OF X(25) AND X(42)

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

X(40984) lies on these lines: {1, 4199}, {6, 2328}, {25, 32}, {31, 21746}, {33, 5336}, {39, 1011}, {42, 213}, {55, 2092}, {58, 20834}, {115, 430}, {181, 1918}, {187, 199}, {200, 2238}, {228, 21796}, {386, 16058}, {440, 950}, {461, 7735}, {511, 38832}, {595, 10974}, {612, 4204}, {614, 851}, {800, 3192}, {902, 20966}, {980, 37175}, {1211, 3744}, {1230, 20045}, {1279, 17056}, {1495, 2206}, {2176, 3190}, {2205, 21813}, {2209, 23638}, {2245, 3052}, {2300, 14547}, {2352, 17053}, {3011, 3136}, {3198, 16583}, {3291, 7453}, {3454, 29656}, {3767, 4207}, {3915, 22076}, {4205, 5266}, {4257, 20841}, {4279, 5943}, {5019, 37538}, {5051, 29634}, {7172, 27040}, {15447, 16700}, {16502, 23653}, {32237, 34476}

X(40984) = crosspoint of X(25) and X(42)
X(40984) = crosssum of X(69) and X(86)

X(40985) = CROSSPOINT OF X(28) AND X(34)

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

X(40985) lies on these lines: {1, 1824}, {2, 5130}, {4, 1385}, {11, 431}, {19, 1100}, {24, 26286}, {25, 34}, {28, 60}, {33, 4214}, {36, 20832}, {57, 37236}, {65, 184}, {92, 16066}, {225, 1319}, {239, 14013}, {242, 4248}, {388, 11391}, {394, 39598}, {405, 1868}, {406, 5155}, {407, 2646}, {427, 1891}, {429, 1125}, {468, 4999}, {475, 5090}, {515, 37368}, {517, 37117}, {551, 39579}, {604, 1880}, {607, 20963}, {608, 2300}, {860, 17614}, {999, 26377}, {1062, 37241}, {1068, 24928}, {1107, 5089}, {1147, 24474}, {1214, 13733}, {1448, 1473}, {1465, 37259}, {1474, 1841}, {1475, 2333}, {1598, 16203}, {1825, 11011}, {1826, 17398}, {1835, 32636}, {1838, 1884}, {1842, 2969}, {1848, 11281}, {1861, 12135}, {1867, 5136}, {1869, 37080}, {1876, 3449}, {1878, 4222}, {1890, 25557}, {1900, 6198}, {1902, 31786}, {2187, 4332}, {2650, 21748}, {3089, 10785}, {3144, 5081}, {3304, 11401}, {3338, 7713}, {3516, 11471}, {3576, 37194}, {4186, 37618}, {4200, 5101}, {4212, 5174}, {4224, 4296}, {4231, 5253}, {4860, 5338}, {5307, 19701}, {5342, 37055}, {5436, 11323}, {6505, 19782}, {6917, 24301}, {7009, 11109}, {7292, 7438}, {7354, 16580}, {7414, 13624}, {7497, 37615}, {7501, 37623}, {8071, 11398}, {11012, 20837}, {11383, 37245}, {12138, 20418}, {13373, 36009}, {14014, 17011}, {14018, 24929}, {15844, 37432}, {17102, 37397}, {19372, 37366}, {20323, 23710}, {25639, 37982}, {26889, 37544}, {33597, 37381}, {34231, 37384}, {34489, 34492}, {37034, 37697}, {37227, 37565}, {37231, 37613}

X(40985) = polar conjugate of isotomic conjugate of complement of X(33066)
X(40985) = crosspoint of X(28) and X(34)
X(40985) = crosssum of X(72) and X(78)

X(40986) = CROSSPOINT OF X(31) AND X(37)

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

X(40986) lies on these lines: {32, 3724}, {37, 1089}, {39, 2292}, {213, 1402}, {758, 2275}, {762, 3728}, {893, 5277}, {1015, 2650}, {1213, 21685}, {1500, 1962}, {1574, 21020}, {1575, 4647}, {1959, 18167}, {1964, 2085}, {2092, 6155}, {2197, 21796}, {2276, 3743}, {2294, 17053}, {2667, 3774}, {3121, 20970}, {3670, 4016}, {3747, 37586}, {8574, 21837}, {14815, 23639}, {18697, 27633}, {18698, 28358}, {21816, 21838}, {21839, 23447}, {24254, 27634}, {27091, 35544}

X(40986) = crosspoint of X(31) and X(37)
X(40986) = crosssum of X(75) and X(81)

X(40987) = CROSSPOINT OF X(33) AND X(34)

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

X(40987) lies on these lines: {1, 242}, {4, 1716}, {19, 2195}, {25, 1096}, {29, 314}, {33, 200}, {34, 269}, {204, 1395}, {273, 34018}, {607, 1827}, {614, 1851}, {653, 1041}, {862, 2092}, {1027, 7649}, {1842, 36103}, {1863, 4319}, {2299, 39943}, {2331, 2356}, {2999, 28100}, {3779, 5185}, {7083, 16583}, {8756, 23050}

X(40987) = polar conjugate of isotomic conjugate of X(2082)
X(40987) = crosspoint of X(33) and X(34)
X(40987) = crosssum of X(77) and X(78)

X(40988) = CROSSPOINT OF X(37) AND X(44)

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

X(40988) lies on these lines: {6, 5429}, {9, 1030}, {10, 21943}, {37, 1018}, {44, 2251}, {45, 846}, {100, 2161}, {190, 24324}, {210, 228}, {214, 17455}, {594, 21012}, {758, 2245}, {900, 1635}, {1213, 21018}, {1962, 3122}, {2486, 3925}, {2610, 21828}, {2642, 39256}, {2783, 17281}, {3158, 8557}, {3880, 8609}, {3882, 7202}, {3943, 21942}, {4271, 25078}, {4459, 5432}, {4512, 16686}, {4881, 7113}, {4919, 16777}, {5257, 21090}, {6184, 23980}, {15015, 16554}, {16732, 22003}, {21814, 21835}

X(40988) = crosspoint of X(37) and X(44)
X(40988) = crosssum of X(81) and X(88)
X(40988) = X(37)-Ceva conjugate of X(4053)

X(40989) = X(4)X(3413) ∩ X(6)X(39163)

Barycentrics    -2*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)*(3*SA-2*SW+sqrt(-3*S^2+SW^2))+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2 : :

Centers X(40989)-X(40994) were contributed by César Lozada, January 12, 2021.

X(40989) lies on the Thomson cubic (K002) and these lines: {2, 40991}, {4, 3413}, {6, 39163}, {39023, 39162}

X(40989) = isogonal conjugate of X(40991)
X(40989) = complement of the isotomic conjugate of X(39158)
X(40989) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(40851)}} and {{A, B, C, X(4), X(39158)}}
X(40989) = crosspoint of X(2) and X(39158)
X(40989) = X(2)-Ceva conjugate of-X(39163)
X(40989) = X(31)-complementary conjugate of-X(39163)

X(40990) = X(4)X(3413) ∩ X(6)X(39162)

Barycentrics    2*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)*(3*SA-2*SW+sqrt(-3*S^2+SW^2))+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2 : :

X(40990) lies on the Thomson cubic (K002) and these lines: {2, 40992}, {4, 3413}, {6, 39162}, {1380, 32443}, {39023, 39163}

X(40990) = isogonal conjugate of X(40992)
X(40990) = complement of the isotomic conjugate of X(39159)
X(40990) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(40852)}} and {{A, B, C, X(4), X(32443)}}
X(40990) = crosspoint of X(2) and X(39159)
X(40990) = X(2)-Ceva conjugate of-X(39162)
X(40990) = X(31)-complementary conjugate of-X(39162)

X(40991) = ISOGONAL CONJUGATE OF X(40989)

Barycentrics    a^2/(-2*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)*(3*SA-2*SW+sqrt(-3*S^2+SW^2))+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2) : :

X(40991) lies on the Thomson cubic (K002) and these lines: {2, 40989}, {3, 39162}, {1249, 40992}

X(40991) = isogonal conjugate of X(40989)
X(40991) = X(6)-cross conjugate of-X(39162)

X(40992) = ISOGONAL CONJUGATE OF X(40990)

Barycentrics    a^2/(2*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)*(3*SA-2*SW+sqrt(-3*S^2+SW^2))+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2) : :

X(40992) lies on the Thomson cubic (K002) and these lines: {2, 40990}, {3, 39163}, {1249, 40991}

X(40992) = isogonal conjugate of X(40990)
X(40992) = X(6)-cross conjugate of-X(39163)

X(40993) = ISOGONAL CONJUGATE OF X(40851)

Barycentrics    a^2/(sqrt(-3*S^2+SW^2)*(S^2-3*SB*SC)-3*(SB+SC)*(SA^2-SB*SC)+2*S^2*(3*SA-SW)+2*SB*SC*sqrt(-3*S^2+18*sqrt(-3*S^2+SW^2)*R^2-4*sqrt(-3*S^2+SW^2)*SW-18*R^2*SW+5*SW^2)) : :

X(40993) lies on the Darboux cubic (K004) and these lines: {3, 40989}, {4, 40991}, {20, 39158}, {3183, 40994}

X(40993) = isogonal conjugate of X(40851)
X(40993) = X(64)-vertex conjugate of-X(40994)

X(40994) = ISOGONAL CONJUGATE OF X(40852)

Barycentrics    a^2/(sqrt(-3*S^2+SW^2)*(S^2-3*SB*SC)-3*(SB+SC)*(SA^2-SB*SC)+2*S^2*(3*SA-SW)-2*SB*SC*sqrt(-3*S^2+18*sqrt(-3*S^2+SW^2)*R^2-4*sqrt(-3*S^2+SW^2)*SW-18*R^2*SW+5*SW^2)) : :

X(40994) lies on the Darboux cubic (K004) and these lines: {3, 40990}, {4, 40992}, {20, 39159}, {3183, 40993}

X(40994) = isogonal conjugate of X(40852)
X(40994) = X(64)-vertex conjugate of-X(40993)

X(40995) = CROSSPOINT OF X(69) AND X(253)

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

X(40995) lies on these lines: {2, 15851}, {3, 69}, {4, 253}, {5, 32000}, {6, 15526}, {7, 2968}, {8, 6356}, {30, 6527}, {95, 15720}, {193, 441}, {216, 599}, {264, 339}, {317, 382}, {325, 30771}, {340, 1657}, {524, 6389}, {577, 40341}, {966, 18643}, {1073, 13567}, {1214, 3965}, {1368, 37668}, {1503, 34815}, {1589, 32814}, {1975, 22468}, {2794, 30549}, {2895, 21482}, {2897, 7580}, {2972, 26869}, {3284, 6144}, {3630, 34828}, {3686, 17073}, {3763, 5158}, {5076, 32002}, {5094, 35442}, {5702, 20200}, {6515, 6617}, {6676, 15589}, {7386, 10513}, {7536, 37655}, {7776, 14615}, {9307, 35140}, {9308, 39352}, {10605, 15394}, {11008, 33636}, {12241, 40675}, {14269, 36889}, {14376, 30435}, {14907, 38437}, {15108, 37068}, {15533, 36748}, {15594, 34774}, {16096, 23291}, {16196, 32818}, {17037, 33630}, {17102, 17272}, {17375, 21940}, {18437, 33878}, {20080, 37188}, {20204, 40138}, {30737, 34609}, {31829, 32830}

X(40995) = midpoint of X(6527) and X(32001)
X(40995) = reflection of X(15905) in X(6389)
X(40995) = isotomic conjugate of X(18848)
X(40995) = isotomic conjugate of the anticomplement of X(33553)
X(40995) = isotomic conjugate of the isogonal conjugate of X(1204)
X(40995) = isotomic conjugate of the polar conjugate of X(26958)
X(40995) = X(i)-cross conjugate of X(j) for these (i,j): {1204, 26958}, {33553, 2}
X(40995) = X(31)-isoconjugate of X(18848)
X(40995) = cevapoint of X(5895) and X(26958)
X(40995) = crosspoint of X(69) and X(253)
X(40995) = crosssum of X(i) and X(j) for these (i,j): {6, 8778}, {25, 154}
X(40995) = crossdifference of every pair of points on line {2489, 2881}
X(40995) = barycentric product X(i)*X(j) for these {i,j}: {63, 18691}, {69, 26958}, {76, 1204}, {3926, 37197}, {5895, 34403}
X(40995) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18848}, {1204, 6}, {5895, 1249}, {15126, 5523}, {18691, 92}, {26958, 4}, {34403, 34410}, {37197, 393}
X(40995) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 15526, 20208}, {193, 441, 38292}, {2972, 26869, 37072}

X(40996) = CROSSPOINT OF X(69) AND X(1494)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^6 - a^4*b^2 - 4*a^2*b^4 + 3*b^6 - a^4*c^2 + 8*a^2*b^2*c^2 - 3*b^4*c^2 - 4*a^2*c^4 - 3*b^2*c^4 + 3*c^6) : :
X(40996) = X[340] + 3 X[1494], 3 X[441] - 2 X[3284], X[3284] - 3 X[15526]

X(40996) lies on these lines: {2, 5702}, {3, 69}, {30, 340}, {95, 12108}, {141, 5158}, {193, 20208}, {216, 3631}, {253, 3146}, {264, 546}, {297, 39352}, {317, 3627}, {319, 6356}, {320, 2968}, {325, 5159}, {339, 3260}, {441, 524}, {577, 3630}, {1273, 10257}, {1368, 7788}, {1654, 18643}, {2897, 33557}, {3091, 32000}, {3265, 8057}, {3529, 6527}, {3580, 14919}, {3589, 15860}, {3619, 15851}, {3830, 36889}, {5921, 34815}, {6389, 40341}, {6676, 37671}, {7768, 12362}, {7796, 16196}, {7799, 16976}, {11008, 38292}, {11160, 37188}, {12102, 32002}, {12113, 36890}, {15704, 20477}, {15905, 20080}, {16051, 37668}

X(40996) = midpoint of X(297) and X(39352)
X(40996) = reflection of X(441) in X(15526)
X(40996) = isotomic conjugate of the isogonal conjugate of X(21663)
X(40996) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2155, 39358}, {2159, 17037}, {2184, 146}, {2349, 6225}, {36119, 14361}
X(40996) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34570}, {204, 5896}
X(40996) = crosspoint of X(69) and X(1494)
X(40996) = crosssum of X(25) and X(1495)
X(40996) = crossdifference of every pair of points on line {2489, 3172}
X(40996) = barycentric product X(i)*X(j) for these {i,j}: {63, 18699}, {76, 21663}, {305, 40135}, {3267, 5502}, {3926, 10151}
X(40996) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34570}, {1073, 5896}, {5502, 112}, {10151, 393}, {11598, 15262}, {13202, 1990}, {18699, 92}, {21663, 6}, {40135, 25}

X(40997) = CROSSPOINT OF X(8) AND X(76)

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

X(40997) lies on these lines: {1, 5305}, {2, 21049}, {6, 12649}, {8, 220}, {9, 1837}, {10, 3693}, {11, 3061}, {30, 1759}, {37, 24987}, {45, 346}, {56, 24247}, {69, 30694}, {72, 5179}, {75, 25002}, {80, 17744}, {141, 1229}, {169, 3419}, {200, 23058}, {257, 312}, {281, 3713}, {321, 13567}, {345, 19732}, {355, 17742}, {495, 3970}, {517, 21073}, {524, 4950}, {728, 3679}, {1018, 5690}, {1098, 27415}, {1212, 6734}, {1213, 27396}, {1329, 21044}, {1737, 25066}, {2170, 3813}, {2276, 21965}, {2321, 3965}, {2329, 10950}, {2345, 21933}, {2886, 17451}, {3207, 26258}, {3496, 6284}, {3501, 40663}, {3509, 7354}, {3665, 17046}, {3670, 15048}, {3692, 17275}, {3703, 4136}, {3704, 4165}, {3721, 3782}, {3816, 21928}, {3826, 21921}, {3869, 17747}, {3876, 38930}, {3878, 21090}, {3930, 12607}, {3959, 21956}, {3991, 10039}, {4051, 4534}, {4364, 25261}, {4515, 6735}, {4520, 5837}, {4671, 23970}, {4847, 4875}, {5175, 5819}, {5199, 6743}, {5244, 37445}, {5286, 37549}, {5525, 37710}, {5695, 21915}, {5794, 40131}, {5845, 21285}, {6737, 40869}, {7738, 17595}, {7748, 36283}, {9575, 17721}, {10527, 34522}, {10587, 16777}, {12433, 16783}, {13881, 28808}, {16588, 16699}, {16788, 37730}, {16815, 32851}, {16968, 35466}, {17044, 28734}, {17211, 33184}, {17280, 26671}, {17362, 17796}, {17736, 18990}, {20007, 27541}, {20171, 26543}, {20271, 40688}, {20880, 21258}, {21096, 31397}, {21808, 25466}, {25242, 33298}, {25244, 25355}, {26015, 40133}, {27068, 34772}

X(40997) = reflection of X(3665) in X(17046)
X(40997) = isotomic conjugate of the isogonal conjugate of X(16588)
X(40997) = X(20236)-Ceva conjugate of X(2886)
X(40997) = X(i)-isoconjugate of X(j) for these (i,j): {57, 3449}, {604, 40419}
X(40997) = cevapoint of X(6) and X(2919)
X(40997) = crosspoint of X(8) and X(76)
X(40997) = crosssum of X(32) and X(56)
X(40997) = barycentric product X(i)*X(j) for these {i,j}: {8, 2886}, {9, 20236}, {76, 16588}, {312, 17451}, {314, 21804}, {321, 16699}, {333, 21029}, {1502, 9449}, {3596, 21746}, {3699, 21118}, {3701, 18165}, {3703, 18088}, {6385, 21819}, {7017, 22070}, {18022, 22368}
X(40997) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 40419}, {55, 3449}, {2886, 7}, {9449, 32}, {16588, 6}, {16699, 81}, {17451, 57}, {18165, 1014}, {20236, 85}, {21029, 226}, {21118, 3676}, {21746, 56}, {21804, 65}, {21819, 213}, {22070, 222}, {22368, 184}
X(40997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 6554, 37658}, {2321, 20262, 3965}, {3721, 5254, 3782}, {17451, 21029, 2886}, {21044, 33299, 1329}

X(40998) = CROSSPOINT OF X(8) AND X(86)

Barycentrics    (a - b - c)*(2*a^2 + a*b + b^2 + a*c - 2*b*c + c^2) : :
X(40998) = 2 X[1] + X[12527], X[1] + 2 X[12572], X[8] + 2 X[12575], X[8] - 4 X[18250], 2 X[10] + X[10624], X[950] + 2 X[960], 2 X[950] + X[6737], 4 X[960] - X[6737], 4 X[1001] - X[12573], 4 X[1125] - X[4292], X[1770] - 7 X[3624], X[1770] - 4 X[12436], X[3057] + 2 X[5795], X[3059] + 2 X[15006], X[3555] - 4 X[40270], 5 X[3616] - 2 X[4298], 7 X[3622] - 4 X[12577], 7 X[3624] - 4 X[12436], X[3868] - 4 X[6744], X[3869] + 2 X[6738], 5 X[3876] - 2 X[6743], X[4018] - 4 X[17706], 2 X[5044] + X[15171], X[6284] + 5 X[25917], X[12527] - 4 X[12572], X[12575] + 2 X[18250], 2 X[15172] + X[34790], 2 X[31838] + X[37290]

X(40998) lies on these lines: {1, 329}, {2, 165}, {3, 25893}, {4, 31435}, {7, 10582}, {8, 4082}, {9, 497}, {10, 1479}, {11, 3683}, {20, 8583}, {21, 36}, {31, 39595}, {35, 6700}, {40, 5084}, {46, 9843}, {51, 29311}, {55, 3452}, {57, 5698}, {63, 11019}, {78, 4314}, {86, 552}, {100, 20103}, {142, 1836}, {144, 10580}, {149, 25006}, {200, 390}, {210, 3058}, {226, 1001}, {238, 2328}, {242, 28137}, {312, 3883}, {354, 527}, {392, 515}, {404, 12512}, {405, 946}, {428, 29024}, {442, 18483}, {443, 3646}, {461, 1890}, {474, 31730}, {496, 31445}, {517, 5943}, {519, 3681}, {522, 11193}, {524, 4891}, {528, 3740}, {529, 10179}, {551, 31156}, {553, 3742}, {614, 3663}, {631, 25522}, {748, 3008}, {846, 24239}, {908, 1621}, {936, 4294}, {938, 12526}, {950, 960}, {958, 12053}, {962, 5129}, {993, 22767}, {997, 4304}, {1005, 15931}, {1006, 1519}, {1155, 6692}, {1210, 12514}, {1279, 4415}, {1376, 5316}, {1512, 6965}, {1697, 2551}, {1698, 6919}, {1738, 17123}, {1770, 3624}, {1837, 5837}, {1848, 4183}, {1851, 28120}, {2321, 3966}, {2325, 3703}, {2475, 24564}, {2476, 12571}, {2550, 7308}, {2886, 14022}, {3057, 5795}, {3059, 15006}, {3062, 10430}, {3086, 31424}, {3158, 10385}, {3219, 26015}, {3246, 17061}, {3271, 21334}, {3295, 21075}, {3421, 31393}, {3474, 5437}, {3475, 28609}, {3485, 5436}, {3486, 15829}, {3555, 40270}, {3576, 11111}, {3579, 17527}, {3616, 4298}, {3622, 12577}, {3634, 4193}, {3635, 5330}, {3636, 3897}, {3664, 3720}, {3671, 11415}, {3677, 4419}, {3685, 3687}, {3686, 3706}, {3707, 4042}, {3715, 4863}, {3717, 4514}, {3741, 4368}, {3753, 28194}, {3755, 4383}, {3757, 17777}, {3794, 17770}, {3813, 5302}, {3816, 3911}, {3819, 15310}, {3828, 31159}, {3846, 4432}, {3848, 28534}, {3868, 6744}, {3869, 6738}, {3870, 21060}, {3872, 4342}, {3873, 5850}, {3876, 6743}, {3884, 17615}, {3886, 4061}, {3890, 11678}, {3912, 4388}, {3917, 29353}, {3925, 6666}, {3929, 24477}, {3944, 15485}, {3946, 4854}, {3950, 33088}, {3957, 26792}, {3982, 25557}, {4001, 29824}, {4009, 4030}, {4018, 17706}, {4021, 17017}, {4078, 4865}, {4096, 17765}, {4104, 32941}, {4138, 29642}, {4187, 6684}, {4297, 6872}, {4301, 19860}, {4307, 17022}, {4312, 9776}, {4349, 5287}, {4353, 7191}, {4356, 5256}, {4357, 4872}, {4416, 10453}, {4654, 38053}, {4666, 5542}, {4684, 33066}, {4689, 37663}, {4853, 9785}, {4881, 15677}, {4887, 33098}, {4914, 6057}, {5044, 15171}, {5046, 19925}, {5087, 6690}, {5121, 17596}, {5217, 24954}, {5218, 30827}, {5223, 36845}, {5231, 5273}, {5248, 8069}, {5251, 30384}, {5272, 24177}, {5281, 5328}, {5325, 11238}, {5435, 31249}, {5574, 28123}, {5705, 10591}, {5717, 6051}, {5791, 9669}, {5846, 35652}, {5880, 8167}, {5886, 16418}, {5918, 17612}, {5919, 34606}, {6172, 31146}, {6284, 25917}, {6361, 17559}, {6675, 9955}, {6688, 29309}, {6734, 18249}, {6836, 21628}, {6848, 10268}, {6857, 8227}, {6865, 12705}, {6910, 19862}, {6930, 37611}, {7172, 8055}, {7262, 24217}, {7292, 33100}, {7484, 24309}, {7667, 29050}, {7700, 31673}, {7965, 37363}, {7987, 17576}, {8273, 12679}, {8580, 17784}, {8728, 22793}, {9614, 19843}, {9623, 30305}, {9954, 9957}, {9965, 10980}, {10165, 16370}, {10172, 17533}, {10175, 17556}, {10248, 37161}, {10389, 25568}, {10473, 24705}, {10691, 29291}, {10863, 19541}, {10882, 28376}, {11108, 12699}, {11112, 28150}, {11114, 28164}, {11496, 16293}, {12511, 37282}, {12545, 13736}, {12565, 37423}, {12608, 37284}, {14552, 35613}, {15082, 29229}, {15172, 34790}, {16020, 23681}, {16058, 31394}, {16435, 20991}, {16484, 33096}, {16865, 24541}, {16866, 18493}, {17125, 33094}, {17132, 17155}, {17261, 29840}, {17280, 39597}, {17334, 21342}, {17337, 21949}, {17350, 29843}, {17353, 32773}, {17355, 32930}, {17484, 29817}, {17525, 34123}, {17561, 38021}, {17567, 35242}, {17579, 28158}, {19836, 27505}, {19864, 27506}, {19868, 24552}, {20106, 25760}, {20196, 35445}, {20214, 30350}, {20344, 32947}, {21246, 21321}, {21692, 24086}, {24165, 28526}, {24178, 24851}, {24231, 29820}, {24558, 30389}, {24709, 32918}, {24982, 37162}, {25072, 33104}, {25101, 29641}, {25531, 33068}, {25935, 31049}, {26006, 37076}, {26723, 33134}, {27537, 30176}, {28364, 37575}, {30988, 32023}, {31838, 37290}, {33110, 35595}, {36568, 39589}

X(40998) = midpoint of X(i) and X(j) for these {i,j}: {210, 3058}, {392, 11113}, {3873, 17781}, {5919, 34606}
X(40998) = reflection of X(553) in X(3742)
X(40998) = X(4633)-Ceva conjugate of X(4765)
X(40998) = X(i)-isoconjugate of X(j) for these (i,j): {37, 38811}, {57, 38825}
X(40998) = crosspoint of X(8) and X(86)
X(40998) = crosssum of X(i) and X(j) for these (i,j): {42, 56}, {2174, 21059}
X(40998) = barycentric product X(i)*X(j) for these {i,j}: {8, 3946}, {86, 38930}, {333, 4854}, {346, 10521}, {1509, 21673}, {3699, 23729}
X(40998) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 38825}, {58, 38811}, {3946, 7}, {4854, 226}, {10521, 279}, {21673, 594}, {23729, 3676}, {38930, 10}
X(40998) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12572, 12527}, {2, 35258, 10164}, {8, 30568, 4082}, {9, 497, 4847}, {11, 3683, 5745}, {40, 5084, 8582}, {55, 3452, 6745}, {55, 4679, 3452}, {149, 27065, 25006}, {390, 18228, 200}, {748, 3914, 3008}, {908, 1621, 13405}, {950, 960, 6737}, {1001, 24703, 226}, {1697, 2551, 6736}, {1770, 3624, 12436}, {1836, 4423, 142}, {2478, 5250, 10}, {3305, 3434, 10}, {3715, 4863, 24393}, {3816, 4640, 3911}, {3870, 31018, 21060}, {3886, 14555, 4061}, {3966, 4387, 2321}, {4666, 5905, 5542}, {5046, 24987, 19925}, {5057, 5284, 5249}, {5248, 21616, 13411}, {5249, 5284, 1125}, {5259, 12047, 1125}, {5272, 24248, 24177}, {5273, 5274, 5231}, {5698, 26105, 57}, {6872, 19861, 4297}, {7308, 9580, 2550}, {10389, 31142, 25568}, {12575, 18250, 8}, {17123, 33095, 1738}, {21060, 30331, 3870}, {28609, 38316, 3475}, {29820, 33099, 24231}

X(40999) = CROSSPOINT OF X(75) AND X(95)

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

X(40999) lies on these lines: {1, 5740}, {2, 219}, {3, 21270}, {5, 17220}, {7, 12}, {9, 25000}, {10, 307}, {30, 20289}, {48, 24581}, {69, 5552}, {71, 857}, {75, 311}, {77, 4551}, {100, 2893}, {141, 17077}, {181, 7179}, {241, 17239}, {261, 4998}, {319, 1273}, {347, 3617}, {355, 17134}, {379, 26063}, {445, 3219}, {498, 5736}, {594, 4552}, {651, 1654}, {664, 32025}, {952, 17221}, {966, 28739}, {1014, 30966}, {1332, 28755}, {1400, 16603}, {1439, 3697}, {1444, 21277}, {1445, 17308}, {1458, 3775}, {1737, 17863}, {1953, 24317}, {2003, 3578}, {2171, 27691}, {2197, 3661}, {2269, 26012}, {3085, 5738}, {3579, 20291}, {3584, 15936}, {3663, 23521}, {3672, 37715}, {3781, 37050}, {3876, 7066}, {3969, 16577}, {4329, 5657}, {4357, 24982}, {4466, 21012}, {4566, 6356}, {4643, 28968}, {4967, 22464}, {5080, 8822}, {5228, 17327}, {5257, 25004}, {5587, 18655}, {5690, 21271}, {5737, 28774}, {6180, 17251}, {6684, 18650}, {7269, 17322}, {8680, 21011}, {17074, 37653}, {17189, 17734}, {17228, 31225}, {17250, 39126}, {17252, 40862}, {17256, 29007}, {17259, 28741}, {17277, 28780}, {17289, 37787}, {17861, 18395}, {17950, 28604}, {18210, 22170}, {18357, 18661}, {19732, 28776}, {21403, 23518}, {21617, 24603}, {21933, 25255}, {22065, 30033}, {23151, 27507}, {27339, 32782}, {30832, 37797}

X(40999) = isotomic conjugate of X(3615)
X(40999) = isotomic conjugate of the isogonal conjugate of X(2594)
X(40999) = X(i)-Ceva conjugate of X(j) for these (i,j): {1275, 4552}, {17095, 16577}
X(40999) = X(i)-cross conjugate of X(j) for these (i,j): {3678, 3969}, {6741, 7265}, {17886, 4467}
X(40999) = X(i)-isoconjugate of X(j) for these (i,j): {21, 6186}, {25, 1789}, {28, 8606}, {31, 3615}, {58, 7073}, {79, 2194}, {284, 2160}, {476, 8648}, {654, 32678}, {663, 13486}, {1333, 7110}, {1989, 4282}, {2150, 8818}, {2299, 7100}, {3738, 14560}, {14936, 35049}, {21789, 26700}
X(40999) = cevapoint of X(i) and X(j) for these (i,j): {3678, 16577}, {6741, 7265}
X(40999) = crosspoint of X(75) and X(95)
X(40999) = crosssum of X(31) and X(51)
X(40999) = trilinear pole of line {7265, 32679}
X(40999) = X(75)-Ceva conjugate of isotomic conjugate of isogonal conjugate of X(2599)
X(40999) = barycentric product X(i)*X(j) for these {i,j}: {7, 3969}, {10, 17095}, {12, 34016}, {35, 349}, {65, 33939}, {75, 16577}, {76, 2594}, {85, 3678}, {226, 319}, {304, 1825}, {306, 7282}, {310, 21794}, {313, 2003}, {321, 1442}, {561, 21741}, {655, 3268}, {664, 7265}, {1231, 6198}, {1275, 6741}, {1399, 27801}, {1434, 7206}, {1441, 3219}, {1446, 4420}, {1969, 22342}, {4467, 4552}, {4551, 18160}, {4564, 17886}, {4620, 21054}, {4998, 8287}, {32679, 35174}, {34388, 40214}
X(40999) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3615}, {10, 7110}, {12, 8818}, {35, 284}, {37, 7073}, {63, 1789}, {65, 2160}, {71, 8606}, {226, 79}, {319, 333}, {349, 20565}, {526, 654}, {651, 13486}, {655, 476}, {1020, 26700}, {1214, 7100}, {1399, 1333}, {1400, 6186}, {1441, 30690}, {1442, 81}, {1825, 19}, {2003, 58}, {2174, 2194}, {2222, 32678}, {2594, 6}, {2599, 1953}, {2605, 7252}, {2611, 2170}, {2624, 8648}, {3219, 21}, {3268, 3904}, {3678, 9}, {3969, 8}, {4420, 2287}, {4467, 4560}, {4552, 6742}, {4566, 38340}, {6149, 4282}, {6198, 1172}, {6358, 6757}, {6741, 1146}, {7045, 35049}, {7202, 18191}, {7206, 2321}, {7265, 522}, {7279, 40214}, {7282, 27}, {8287, 11}, {9404, 21789}, {11107, 2326}, {14838, 3737}, {14975, 2204}, {16577, 1}, {17095, 86}, {17104, 2150}, {17886, 4858}, {18160, 18155}, {20982, 3271}, {21054, 21044}, {21141, 21132}, {21741, 31}, {21794, 42}, {21824, 4516}, {22094, 7117}, {22342, 48}, {27691, 14844}, {32675, 14560}, {32679, 3738}, {33939, 314}, {34016, 261}, {35057, 1021}, {35174, 32680}, {35193, 7054}, {40214, 60}
X(40999) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 307, 1441}, {71, 20305, 857}, {319, 17095, 1442}, {4466, 21012, 21231}, {5224, 33298, 7}

X(41000) = CROSSPOINT OF X(76) AND X(300)

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

X(41000) lies on these lines: {2, 39}, {94, 40707}, {99, 11146}, {298, 300}, {301, 302}, {303, 1232}, {308, 6151}, {396, 14922}, {471, 1235}, {532, 8014}, {850, 20579}, {1078, 11145}, {3770, 5362}, {5980, 34008}, {13483, 37779}, {19772, 30737}, {22687, 34394}

X(41000) = isotomic conjugate of X(2981)
X(41000) = isotomic conjugate of the isogonal conjugate of X(396)
X(41000) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2981}, {560, 40707}, {798, 10409}, {2151, 16459}, {2152, 2380}, {3439, 19300}, {6149, 11084}, {9247, 38428}
X(41000) = cevapoint of X(6) and X(2925)
X(41000) = crosspoint of X(76) and X(300)
X(41000) = crosssum of X(32) and X(34394)
X(41000) = crossdifference of every pair of points on line {669, 34395}
X(41000) = barycentric product X(i)*X(j) for these {i,j}: {76, 396}, {94, 14922}, {300, 618}, {301, 532}, {305, 463}, {850, 35314}, {6671, 34389}, {9115, 18023}, {19294, 20573}
X(41000) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2981}, {13, 16459}, {14, 2380}, {17, 34321}, {76, 40707}, {99, 10409}, {264, 38428}, {298, 38403}, {300, 11119}, {301, 11117}, {396, 6}, {463, 25}, {532, 16}, {618, 15}, {621, 39262}, {622, 1337}, {1989, 11084}, {3479, 3439}, {6671, 61}, {8014, 3457}, {9115, 187}, {14446, 6138}, {14922, 323}, {16256, 2378}, {19294, 50}, {23714, 8740}, {33526, 3201}, {34296, 3440}, {35314, 110}, {35329, 1576}, {35343, 35327}, {35345, 35330}, {35443, 6137}, {36304, 21461}, {38931, 3438}, {39134, 3489}
X(41000) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 76, 41001}, {298, 300, 3260}, {302, 34389, 311}

X(41001) = CROSSPOINT OF X(76) AND X(301)

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

X(41001) lies on these lines: {2, 39}, {94, 40706}, {99, 11145}, {299, 301}, {300, 303}, {302, 1232}, {308, 2981}, {395, 14921}, {470, 1235}, {533, 8015}, {850, 20578}, {1078, 11146}, {3770, 5367}, {5981, 34009}, {13484, 37779}, {19773, 30737}, {22689, 34395}

X(41001) = isotomic conjugate of X(6151)
X(41001) = isotomic conjugate of the isogonal conjugate of X(395)
X(41001) = X(i)-isoconjugate of X(j) for these (i,j): {31, 6151}, {560, 40706}, {798, 10410}, {2151, 2381}, {2152, 16460}, {3438, 19301}, {6149, 11089}, {9247, 38427}
X(41001) = cevapoint of X(6) and X(2926)
X(41001) = crosspoint of X(76) and X(301)
X(41001) = crosssum of X(32) and X(34395)
X(41001) = crossdifference of every pair of points on line {669, 34394}
X(41001) = barycentric product X(i)*X(j) for these {i,j}: {76, 395}, {94, 14921}, {300, 533}, {301, 619}, {305, 462}, {850, 35315}, {6672, 34390}, {9117, 18023}, {19295, 20573}
X(41001) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6151}, {13, 2381}, {14, 16460}, {18, 34322}, {76, 40706}, {99, 10410}, {264, 38427}, {299, 38404}, {300, 11118}, {301, 11120}, {395, 6}, {462, 25}, {533, 15}, {619, 16}, {621, 1338}, {622, 39261}, {1989, 11089}, {3480, 3438}, {6672, 62}, {8015, 3458}, {9117, 187}, {14447, 6137}, {14921, 323}, {16255, 2379}, {19295, 50}, {23715, 8739}, {33527, 3200}, {34295, 3441}, {35315, 110}, {35330, 1576}, {35344, 35327}, {35345, 35329}, {35444, 6138}, {36305, 21462}, {38932, 3439}, {39135, 3490}
X(41001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 76, 41000}, {299, 301, 3260}, {303, 34390, 311}

X(41002) = CROSSPOINT OF X(8) AND X(261)

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

X(41002) lies on these lines: {8, 3058}, {9, 3703}, {11, 333}, {31, 5743}, {38, 17332}, {45, 33088}, {55, 4023}, {69, 4423}, {141, 748}, {171, 5241}, {210, 3883}, {238, 1211}, {239, 4854}, {306, 15254}, {354, 4416}, {391, 497}, {524, 3720}, {528, 4651}, {594, 32930}, {597, 29647}, {614, 4643}, {756, 5846}, {1001, 5739}, {1086, 4683}, {1125, 37631}, {1150, 3816}, {1203, 4205}, {1213, 32772}, {1621, 37656}, {1654, 32942}, {1836, 4384}, {2308, 6703}, {2886, 5278}, {2895, 4966}, {3305, 3416}, {3578, 29824}, {3649, 16817}, {3683, 3687}, {3685, 4046}, {3686, 3706}, {3707, 4847}, {3717, 4914}, {3742, 4001}, {3744, 4104}, {3782, 4703}, {3826, 6327}, {3846, 35466}, {3914, 17348}, {3925, 4388}, {3932, 27065}, {4026, 32911}, {4113, 5853}, {4359, 17768}, {4364, 17017}, {4365, 4399}, {4395, 33145}, {4415, 32914}, {4422, 15523}, {4425, 4974}, {4432, 21085}, {4655, 40688}, {4679, 11679}, {4714, 28174}, {4883, 34379}, {5233, 5432}, {5235, 33107}, {5271, 24703}, {5741, 6690}, {5852, 17140}, {5943, 22275}, {6284, 9534}, {6533, 24470}, {6536, 17045}, {7263, 33098}, {10371, 31435}, {10453, 17346}, {11246, 19804}, {13567, 25885}, {14552, 26105}, {15485, 33084}, {16684, 21319}, {16823, 33066}, {16885, 33163}, {17056, 32843}, {17061, 26580}, {17123, 33082}, {17125, 33080}, {17155, 17334}, {17243, 32852}, {17245, 32949}, {17246, 32924}, {17257, 17599}, {17260, 33073}, {17330, 31330}, {17335, 29641}, {17337, 25957}, {17349, 32773}, {17362, 32915}, {17366, 32776}, {19723, 33137}, {19732, 26098}, {21363, 29207}, {24542, 31037}, {24697, 29821}, {25557, 32859}, {25960, 37646}, {26034, 37679}, {31143, 33173}, {32842, 33761}, {32916, 37663}, {32917, 37662}, {33078, 35595}, {33083, 37680}, {33086, 37687}

X(41002) = crosspoint of X(8) and X(261)
X(41002) = crosssum of X(56) and X(181)
X(41002) = barycentric product X(i)*X(j) for these {i,j}: {8, 17045}, {261, 6537}, {314, 6155}, {333, 6536}, {3699, 23731}
X(41002) = barycentric quotient X(i)/X(j) for these {i,j}: {6155, 65}, {6536, 226}, {6537, 12}, {17045, 7}, {23731, 3676}
X(41002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 3715, 4126}, {9, 3966, 3703}, {55, 14555, 4023}, {210, 3883, 4030}, {391, 497, 4042}, {2895, 5284, 4966}, {3683, 3687, 3712}, {3685, 4886, 4046}, {4388, 17277, 3925}, {4703, 16825, 3782}, {27065, 33075, 3932}

X(41003) = CROSSPOINT OF X(7) AND X(1441)

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

X(41003) lies on these lines: {1, 1503}, {2, 11683}, {5, 17861}, {6, 24316}, {7, 21}, {11, 17863}, {12, 313}, {37, 226}, {55, 4329}, {57, 1761}, {65, 307}, {69, 12635}, {75, 325}, {77, 1464}, {141, 7146}, {142, 25887}, {273, 15844}, {322, 12607}, {344, 5226}, {347, 388}, {442, 18698}, {495, 31395}, {497, 3672}, {594, 16603}, {857, 25255}, {908, 25099}, {946, 3663}, {960, 3674}, {986, 5799}, {1086, 11672}, {1211, 21033}, {1213, 16609}, {1361, 1367}, {1375, 1781}, {1400, 5244}, {1423, 4364}, {1429, 17045}, {1432, 15985}, {1439, 12709}, {1442, 26751}, {1446, 6046}, {1447, 17322}, {1565, 2792}, {1836, 18655}, {1848, 3666}, {1959, 26543}, {2286, 6180}, {2294, 4466}, {2646, 18650}, {3007, 15888}, {3212, 5224}, {3614, 17895}, {3704, 18697}, {3911, 17384}, {3947, 4078}, {4021, 29040}, {4205, 35650}, {4328, 39688}, {4360, 21276}, {4442, 4452}, {4554, 31643}, {4733, 7235}, {4890, 4934}, {5219, 17279}, {5886, 24179}, {5901, 24202}, {6063, 6385}, {7173, 23521}, {7190, 11553}, {7201, 24357}, {7354, 17134}, {7359, 25651}, {7951, 17885}, {9436, 39793}, {10436, 28628}, {10950, 21270}, {11238, 15956}, {12610, 37597}, {15338, 20291}, {16593, 21617}, {16597, 40615}, {17246, 24424}, {17274, 34647}, {17398, 25363}, {17451, 25964}, {18644, 24884}, {18714, 37796}, {18726, 26932}, {19758, 24248}, {20305, 21933}, {21471, 40688}, {24683, 37504}, {24993, 33864}, {25004, 27689}, {25078, 37326}, {25252, 37445}, {25254, 27052}, {27559, 38930}, {31394, 39542}, {39780, 39919}

X(41003) = complement of X(11683)
X(41003) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 24471}, {934, 17094}, {4554, 7178}, {6063, 20911}
X(41003) = X(2292)-cross conjugate of X(1211)
X(41003) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1169}, {33, 1798}, {41, 14534}, {55, 2363}, {284, 2298}, {961, 2328}, {1021, 8687}, {1172, 2359}, {1220, 2194}, {1791, 2299}, {2150, 14624}, {3737, 32736}, {7252, 36147}, {9447, 40827}, {21789, 36098}
X(41003) = cevapoint of X(1211) and X(4918)
X(41003) = crosspoint of X(i) and X(j) for these (i,j): {7, 1441}, {1446, 6063}
X(41003) = crosssum of X(55) and X(2194)
X(41003) = crossdifference of every pair of points on line {3709, 21789}
X(41003) = barycentric product X(i)*X(j) for these {i,j}: {7, 1211}, {10, 3674}, {12, 16705}, {56, 1228}, {57, 18697}, {65, 20911}, {85, 2292}, {226, 4357}, {279, 3704}, {307, 1848}, {321, 24471}, {331, 22076}, {348, 429}, {349, 1193}, {664, 21124}, {960, 1446}, {1088, 21033}, {1231, 1829}, {1434, 20653}, {1441, 3666}, {2092, 6063}, {2171, 16739}, {3004, 4552}, {3665, 27067}, {3668, 3687}, {3725, 20567}, {3882, 4077}, {3910, 4566}, {4509, 4551}, {4918, 27818}, {7249, 27697}, {34388, 40153}
X(41003) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 14534}, {12, 14624}, {56, 1169}, {57, 2363}, {65, 2298}, {73, 2359}, {222, 1798}, {226, 1220}, {349, 1240}, {429, 281}, {960, 2287}, {1020, 36098}, {1193, 284}, {1211, 8}, {1214, 1791}, {1228, 3596}, {1427, 961}, {1441, 30710}, {1446, 31643}, {1829, 1172}, {1848, 29}, {2092, 55}, {2269, 2328}, {2292, 9}, {2300, 2194}, {2354, 2299}, {3004, 4560}, {3666, 21}, {3674, 86}, {3687, 1043}, {3704, 346}, {3725, 41}, {3882, 643}, {3910, 7253}, {4267, 7054}, {4357, 333}, {4509, 18155}, {4551, 36147}, {4552, 8707}, {4559, 32736}, {4566, 6648}, {4918, 3161}, {6042, 21033}, {6063, 40827}, {6371, 7252}, {7178, 4581}, {16705, 261}, {17080, 40452}, {17094, 15420}, {17185, 1098}, {17420, 1021}, {18697, 312}, {20653, 2321}, {20911, 314}, {21033, 200}, {21124, 522}, {21810, 210}, {22076, 219}, {22097, 283}, {22345, 2193}, {24471, 81}, {27697, 7081}, {40153, 60}
X(41003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 17084, 86}, {1781, 24780, 1375}, {2294, 4466, 18635}, {3674, 4357, 24471}, {7179, 7249, 325}, {16609, 27691, 1213}

X(41004) = CROSSPOINT OF X(7) AND X(69)

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

X(41004) lies on these lines: {1, 1503}, {3, 307}, {4, 7}, {6, 16580}, {9, 141}, {11, 24179}, {19, 16608}, {30, 18655}, {37, 18734}, {48, 4466}, {57, 19542}, {63, 440}, {69, 72}, {75, 150}, {77, 1060}, {80, 17885}, {84, 6355}, {86, 1437}, {142, 5781}, {169, 25964}, {219, 9028}, {222, 226}, {269, 1490}, {320, 18147}, {329, 4358}, {347, 944}, {355, 1441}, {394, 18651}, {405, 4357}, {442, 10436}, {497, 14523}, {511, 24701}, {515, 3668}, {517, 4329}, {527, 8804}, {610, 1375}, {894, 37445}, {950, 3663}, {1122, 1864}, {1367, 34194}, {1370, 16465}, {1429, 20270}, {1750, 7271}, {1751, 24789}, {1760, 37796}, {1763, 13567}, {1836, 21746}, {1837, 17861}, {1848, 37543}, {1899, 17441}, {1901, 17365}, {2261, 36949}, {2263, 29207}, {2809, 30620}, {3007, 37727}, {3173, 18588}, {3487, 3945}, {3488, 3672}, {3586, 4862}, {3655, 17221}, {3662, 37086}, {3665, 10393}, {3772, 17189}, {4000, 5802}, {4341, 6261}, {4554, 34399}, {4644, 5746}, {4675, 14597}, {4851, 18733}, {4872, 10446}, {4888, 9612}, {5090, 17492}, {5173, 36844}, {5175, 31995}, {5228, 12610}, {5249, 7522}, {5307, 15946}, {5691, 20618}, {5736, 11374}, {5794, 18698}, {5812, 37536}, {5905, 37185}, {6675, 28627}, {6703, 25525}, {7179, 7413}, {7291, 26540}, {7580, 9436}, {8048, 13577}, {8226, 40719}, {9119, 34371}, {10167, 26929}, {10319, 26942}, {10373, 12324}, {10404, 39791}, {10444, 39598}, {10452, 10477}, {11018, 26118}, {11020, 37456}, {11113, 17274}, {11376, 24202}, {11433, 14557}, {11677, 15733}, {12625, 17151}, {12699, 17220}, {12701, 33551}, {12779, 15071}, {15624, 24713}, {16053, 17248}, {16888, 24268}, {17134, 18481}, {17257, 37169}, {17278, 24618}, {17671, 27420}, {18596, 18636}, {20227, 28078}, {20305, 24315}, {20818, 26006}, {20930, 21277}, {21011, 31163}, {21286, 30806}, {24316, 26130}, {24317, 25523}, {24682, 34830}, {25878, 34847}, {27382, 30809}, {31015, 40905}

X(41004) = reflection of X(i) in X(j) for these {i,j}: {19, 16608}, {219, 18589}, {5781, 142}
X(41004) = isotomic conjugate of X(34406)
X(41004) = isotomic conjugate of the polar conjugate of X(3772)
X(41004) = X(4554)-Ceva conjugate of X(905)
X(41004) = X(i)-isoconjugate of X(j) for these (i,j): {25, 40436}, {31, 34406}, {2212, 34399}
X(41004) = crosspoint of X(i) and X(j) for these (i,j): {7, 69}, {272, 286}
X(41004) = crosssum of X(i) and X(j) for these (i,j): {25, 55}, {209, 228}
X(41004) = barycentric product X(i)*X(j) for these {i,j}: {63, 17861}, {69, 3772}, {72, 16749}, {75, 26934}, {304, 3924}, {306, 17189}, {348, 1837}, {3718, 36570}, {17206, 21935}
X(41004) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34406}, {63, 40436}, {348, 34399}, {1837, 281}, {3772, 4}, {3924, 19}, {16749, 286}, {17189, 27}, {17861, 92}, {21935, 1826}, {26934, 1}, {36570, 34}
X(41004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 5738, 942}, {7, 5932, 1119}, {7, 7282, 10400}, {48, 4466, 17073}, {75, 2893, 3419}, {307, 18650, 3}, {610, 18634, 1375}, {1441, 21270, 355}, {5932, 21279, 1535}

X(41005) = CROSSPOINT OF X(69) AND X(264)

Barycentrics    (a^2 - b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
Barycentrics    (csc 2A) (sec^2 B + sec^2 C) : :
Barycentrics    (cot A) (cos^2 B + cos^2 C) : :
Barycentrics    (cot A) (1 - cos A cos(B - C)) : :

In the plane of a triangle ABC, let
DEF = anticevian triangle of X(3);
Ab = orthogonal projection of D on AB;
Ac = orthogonal projection of D on AC;
Db = BD∩AbAc;
Dc = CD∩AbAc;
A'=BDc∩CDb, and define B' and C' cyclically.
The lines AA', BB', CC' concur in X(41005). See figure (Angel Montesdeoca, January 2, 2022) and the general case in HG291221.

X(41005) lies on these lines: {2, 253}, {3, 69}, {4, 6527}, {5, 264}, {6, 441}, {20, 32001}, {30, 317}, {75, 2968}, {76, 6823}, {95, 549}, {99, 22468}, {127, 33184}, {141, 216}, {154, 15594}, {183, 6676}, {193, 15905}, {286, 8727}, {297, 3164}, {298, 465}, {299, 466}, {305, 40032}, {309, 17181}, {311, 339}, {315, 12362}, {325, 1368}, {338, 9722}, {340, 550}, {343, 26906}, {401, 27377}, {411, 2897}, {417, 16035}, {426, 11245}, {427, 13409}, {440, 4417}, {523, 23333}, {524, 577}, {599, 36751}, {800, 6509}, {1007, 30771}, {1214, 3687}, {1232, 13381}, {1235, 7399}, {1270, 1589}, {1271, 1590}, {1272, 18354}, {1513, 3186}, {1975, 31829}, {1992, 38292}, {2972, 30739}, {3172, 26204}, {3260, 11585}, {3267, 40494}, {3284, 3629}, {3589, 5158}, {3618, 15851}, {3627, 32002}, {3630, 22052}, {3631, 10979}, {3917, 26905}, {4357, 17102}, {4384, 17073}, {4869, 25932}, {5055, 8797}, {5157, 39129}, {5232, 25876}, {5305, 28405}, {5596, 33582}, {5739, 10432}, {5743, 18592}, {6394, 14601}, {6617, 11433}, {6643, 7776}, {7282, 10538}, {7386, 37668}, {7494, 15589}, {7515, 25650}, {7536, 14829}, {7762, 28723}, {7763, 16196}, {7788, 10691}, {7819, 14376}, {8266, 16789}, {8362, 19595}, {8721, 9924}, {9534, 18641}, {9605, 28406}, {10996, 32830}, {15013, 18907}, {15574, 15818}, {15717, 35510}, {15740, 34403}, {17300, 21940}, {18639, 29960}, {28407, 31406}, {28696, 30435}, {36748, 40341}

X(41005) = midpoint of X(317) and X(20477)
X(41005) = reflection of X(577) in X(34828)
X(41005) = isotomic conjugate of X(1105)
X(41005) = complement of X(9308)
X(41005) = complement of the isotomic conjugate of X(9289)
X(41005) = isotomic conjugate of the isogonal conjugate of X(185)
X(41005) = isotomic conjugate of the polar conjugate of X(13567)
X(41005) = pole wrt polar circle of line X(6587)X(39201)
X(41005) = polar conjugate of the isogonal conjugate of X(6509)
X(41005) = X(i)-complementary conjugate of X(j) for these (i,j): {9255, 141}, {9258, 5}, {9289, 2887}, {9292, 226}, {9307, 20305}
X(41005) = X(6331)-Ceva conjugate of X(525)
X(41005) = X(185)-cross conjugate of X(13567)
X(41005) = X(i)-isoconjugate of X(j) for these (i,j): {25, 775}, {31, 1105}, {184, 821}, {801, 1973}
X(41005) = cevapoint of X(i) and X(j) for these (i,j): {185, 6509}, {2883, 13567}
X(41005) = crosspoint of X(i) and X(j) for these (i,j): {2, 9289}, {69, 264}, {305, 34403}
X(41005) = crosssum of X(i) and X(j) for these (i,j): {6, 1968}, {25, 184}, {1974, 3172}
X(41005) = barycentric product X(i)*X(j) for these {i,j}: {63, 17858}, {69, 13567}, {75, 6508}, {76, 185}, {235, 3926}, {264, 6509}, {304, 774}, {305, 800}, {311, 19180}, {343, 19166}, {417, 18027}, {820, 1969}, {1624, 3267}, {2883, 34403}, {16035, 28706}, {18603, 20336}
X(41005) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1105}, {63, 775}, {69, 801}, {92, 821}, {185, 6}, {235, 393}, {305, 40830}, {417, 577}, {774, 19}, {800, 25}, {820, 48}, {1624, 112}, {2883, 1249}, {6508, 1}, {6509, 3}, {13567, 4}, {16035, 8882}, {17773, 16229}, {17858, 92}, {18603, 28}, {19166, 275}, {19180, 54}
X(41005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 253, 32000}, {6, 6389, 441}, {69, 40680, 3}, {69, 40697, 3964}, {193, 37188, 15905}, {216, 15526, 141}, {2968, 6356, 75}, {3964, 40697, 6390}, {10519, 26870, 3}

X(41006) = CROSSPOINT OF X(8) AND X(85)

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

X(41006) lies on these lines: {1, 6554}, {2, 3160}, {3, 8074}, {6, 6738}, {7, 30695}, {8, 9}, {10, 1146}, {34, 281}, {85, 142}, {101, 5882}, {144, 32003}, {169, 515}, {220, 519}, {226, 17451}, {405, 7367}, {514, 34847}, {527, 6604}, {672, 4848}, {910, 4297}, {946, 5179}, {958, 15288}, {960, 21071}, {993, 32561}, {1015, 9367}, {1107, 16588}, {1111, 24181}, {1121, 6606}, {1125, 5199}, {1222, 1438}, {1420, 40127}, {1449, 27382}, {1909, 36796}, {1944, 4667}, {1953, 21068}, {2170, 12053}, {2262, 8804}, {2267, 8756}, {2348, 10950}, {2391, 17170}, {2481, 20257}, {3039, 30618}, {3057, 4534}, {3061, 3452}, {3119, 5316}, {3158, 28057}, {3177, 9436}, {3244, 6603}, {3247, 27508}, {3496, 8558}, {3616, 27541}, {3668, 25964}, {3684, 12437}, {3693, 6736}, {3721, 17435}, {3730, 11362}, {3811, 34526}, {3825, 6506}, {3879, 27420}, {3946, 27509}, {4012, 4326}, {4051, 21627}, {4187, 5514}, {4301, 17747}, {4357, 27288}, {4384, 5745}, {4391, 40465}, {4847, 4875}, {4858, 20880}, {4967, 26059}, {5011, 31730}, {5013, 6181}, {5248, 31896}, {5257, 5930}, {5540, 10572}, {5574, 26105}, {5665, 5746}, {5691, 5819}, {5814, 18250}, {6173, 32086}, {6735, 25082}, {6737, 37658}, {8568, 24982}, {10106, 40131}, {10324, 18227}, {10481, 21258}, {11019, 21049}, {16572, 18391}, {16601, 31397}, {16831, 28827}, {17023, 37774}, {17030, 26068}, {17095, 31640}, {17102, 40616}, {17732, 28194}, {18228, 29616}, {20258, 30030}, {21246, 29966}, {21384, 24391}, {21950, 23649}, {24541, 27068}, {24635, 26001}, {25716, 26658}, {25719, 39351}, {29571, 34852}, {29627, 30827}, {30694, 40719}, {31169, 33298}

X(41006) = midpoint of X(6604) and X(30625)
X(41006) = reflection of X(10481) in X(21258)
X(41006) = isotomic conjugate of X(23618)
X(41006) = complement of X(9312)
X(41006) = complement of the isogonal conjugate of X(9439)
X(41006) = isotomic conjugate of the isogonal conjugate of X(1200)
X(41006) = X(i)-complementary conjugate of X(j) for these (i,j): {6169, 20335}, {9309, 2886}, {9311, 17046}, {9315, 142}, {9439, 10}, {20287, 20338}, {32023, 17047}
X(41006) = X(i)-Ceva conjugate of X(j) for these (i,j): {4554, 522}, {20905, 11019}
X(41006) = X(14100)-cross conjugate of X(11019)
X(41006) = X(i)-isoconjugate of X(j) for these (i,j): {31, 23618}, {222, 14493}
X(41006) = crosspoint of X(8) and X(85)
X(41006) = crosssum of X(i) and X(j) for these (i,j): {6, 9316}, {41, 56}
X(41006) = barycentric product X(i)*X(j) for these {i,j}: {8, 11019}, {9, 20905}, {75, 14100}, {76, 1200}, {312, 40133}, {318, 10167}, {333, 21049}, {2321, 26818}, {3596, 20978}, {7017, 22088}
X(41006) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 23618}, {33, 14493}, {1200, 6}, {10167, 77}, {11019, 7}, {14100, 1}, {20905, 85}, {20978, 56}, {21049, 226}, {22088, 222}, {26818, 1434}, {40133, 57}
X(41006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6554, 40869}, {2, 10405, 31994}, {1146, 1212, 10}, {3177, 26531, 9436}, {3912, 30854, 3452}, {21049, 40133, 11019}, {24982, 26690, 8568}, {25935, 30807, 226}

X(41007) = CROSSPOINT OF X(7) AND X(264)

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

X(41007) lies on these lines: {1, 1503}, {3, 4329}, {4, 347}, {5, 1441}, {7, 104}, {11, 17861}, {19, 1375}, {30, 17134}, {48, 17043}, {57, 20270}, {69, 5730}, {75, 17181}, {86, 37227}, {150, 4360}, {219, 24316}, {241, 12610}, {273, 6831}, {281, 16596}, {286, 37357}, {307, 517}, {322, 17757}, {392, 4357}, {427, 21318}, {448, 2905}, {469, 6360}, {496, 17863}, {550, 20291}, {946, 3668}, {952, 21270}, {962, 37046}, {1058, 3672}, {1119, 6847}, {1214, 1848}, {1385, 18650}, {1439, 12672}, {1442, 21740}, {1726, 23292}, {1736, 5480}, {1953, 4466}, {1985, 2969}, {2886, 18698}, {3101, 7536}, {3616, 37052}, {3663, 3665}, {3666, 21621}, {3732, 27547}, {4021, 29215}, {4858, 21239}, {4934, 21746}, {5690, 21271}, {5729, 14853}, {5736, 37737}, {5740, 5797}, {6349, 11347}, {6354, 10478}, {6676, 26260}, {6996, 17086}, {7719, 25915}, {7741, 17885}, {8609, 16580}, {9840, 18659}, {10593, 17895}, {11376, 24179}, {12116, 21279}, {12699, 18655}, {13728, 37613}, {13730, 17170}, {13745, 34643}, {15299, 38035}, {15829, 17272}, {17151, 24392}, {17220, 22791}, {17221, 34773}, {17366, 24618}, {17442, 18639}, {18041, 37796}, {18161, 26932}, {18589, 30810}, {20061, 24580}, {20254, 37360}, {20895, 33864}, {21231, 24317}, {21495, 26789}, {21511, 26837}, {31014, 40903}, {40687, 40688}

X(41007) = reflection of X(48) in X(17043)
X(41007) = crosspoint of X(7) and X(264)
X(41007) = crosssum of X(55) and X(184)
X(41007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 31600, 7053}, {19, 17073, 1375}, {1214, 1848, 19542}, {1953, 4466, 16608}

X(41008) = CROSSPOINT OF X(69) AND X(95)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^6 - 3*a^4*b^2 + b^6 - 3*a^4*c^2 - b^4*c^2 - b^2*c^4 + c^6) : :
X(41008) = X[3164] - 3 X[35937]

X(41008) lies on these lines: {2, 15905}, {3, 69}, {5, 317}, {20, 32000}, {30, 264}, {76, 12362}, {95, 140}, {97, 37636}, {141, 441}, {150, 40417}, {183, 1368}, {216, 524}, {253, 3522}, {286, 20420}, {298, 466}, {299, 465}, {311, 12605}, {315, 6823}, {319, 2968}, {320, 6356}, {325, 6676}, {339, 1225}, {343, 37867}, {376, 6527}, {391, 25932}, {394, 26906}, {401, 40897}, {440, 14829}, {464, 37655}, {546, 32002}, {550, 20477}, {599, 6389}, {1078, 16196}, {1270, 1590}, {1271, 1589}, {1494, 34200}, {1654, 21940}, {1992, 15851}, {2193, 24632}, {2897, 6909}, {3164, 35937}, {3284, 3589}, {3547, 7776}, {3575, 26166}, {3618, 38292}, {3620, 20208}, {3629, 5158}, {3630, 10979}, {3631, 15526}, {3851, 8797}, {3879, 17102}, {3945, 25876}, {4417, 7536}, {5065, 13567}, {5159, 37688}, {6748, 14767}, {7386, 15589}, {7494, 37668}, {7499, 23606}, {7542, 39113}, {7667, 30737}, {7750, 14615}, {7762, 37186}, {7771, 16976}, {7819, 10316}, {8359, 14961}, {8362, 23115}, {10691, 37671}, {11245, 16030}, {11331, 20204}, {11574, 34383}, {12812, 40410}, {14152, 16197}, {15394, 16096}, {15689, 36889}, {15694, 36948}, {17073, 17298}, {17300, 18643}, {18840, 28717}, {27420, 40616}, {30258, 34380}, {30771, 34229}, {36751, 40341}

X(41008) = reflection of X(6748) in X(14767)
X(41008) = isotomic conjugate of X(14860)
X(41008) = isotomic conjugate of the isogonal conjugate of X(13367)
X(41008) = isotomic conjugate of the polar conjugate of X(23292)
X(41008) = isogonal conjugate of the polar conjugate of X(26166)
X(41008) = complement of X(27377)
X(41008) = X(26166)-Ceva conjugate of X(23292)
X(41008) = X(13367)-cross conjugate of X(23292)
X(41008) = X(31)-isoconjugate of X(14860)
X(41008) = crosspoint of X(69) and X(95)
X(41008) = crosssum of X(25) and X(51)
X(41008) = barycentric product X(i)*X(j) for these {i,j}: {3, 26166}, {63, 17859}, {69, 23292}, {76, 13367}, {276, 31388}, {3574, 34386}, {3575, 3926}, {3933, 10548}
X(41008) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14860}, {3574, 53}, {3575, 393}, {10548, 32085}, {13367, 6}, {17859, 92}, {23292, 4}, {26166, 264}, {31388, 216}
X(41008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3964, 3933}, {141, 577, 441}, {599, 36748, 6389}, {3620, 37188, 20208}, {3631, 34828, 15526}, {15526, 22052, 34828}

X(41009) = CROSSPOINT OF X(264) AND X(305)

Barycentrics    b^2*c^2*(-a^2 + b^2 + c^2)*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :
Barycentrics    A'-power of Brocard circle : : , where A'B'C' = orthic triangle

X(41009) lies on these lines: {2, 1235}, {3, 76}, {4, 30737}, {5, 264}, {6, 28695}, {32, 15013}, {39, 28407}, {68, 69}, {95, 7516}, {127, 5025}, {194, 14961}, {216, 3934}, {230, 28697}, {274, 34120}, {287, 23128}, {305, 1368}, {311, 3547}, {315, 18531}, {316, 18404}, {317, 14790}, {325, 11585}, {328, 39170}, {338, 13881}, {340, 14791}, {350, 1062}, {385, 10316}, {441, 5305}, {464, 1230}, {525, 9289}, {538, 22401}, {577, 7751}, {626, 15526}, {631, 26166}, {1038, 3761}, {1040, 3760}, {1060, 1909}, {1184, 28719}, {1228, 37179}, {1236, 3548}, {1799, 15818}, {2072, 7752}, {3089, 6527}, {3260, 32816}, {3266, 16051}, {3284, 7805}, {3546, 3926}, {3549, 32832}, {3767, 6389}, {5158, 7808}, {5159, 11059}, {5276, 28718}, {5286, 28406}, {5304, 28717}, {5354, 28702}, {5359, 28701}, {5523, 26154}, {6179, 10317}, {6334, 14295}, {6381, 34823}, {6390, 16196}, {6640, 7769}, {6676, 40022}, {6804, 32000}, {7386, 8024}, {7387, 20477}, {7400, 32834}, {7494, 39998}, {7542, 37688}, {7735, 28696}, {7748, 35923}, {7750, 12605}, {7754, 23115}, {7760, 22120}, {7767, 12362}, {7776, 14615}, {7793, 35952}, {7796, 37452}, {7797, 28433}, {7802, 18563}, {7851, 37073}, {7866, 20208}, {7870, 34897}, {8538, 39099}, {8743, 26226}, {11057, 18564}, {11574, 14994}, {13409, 37988}, {14001, 26164}, {14603, 40073}, {14965, 32451}, {16043, 26214}, {16925, 26179}, {17128, 35928}, {17984, 18437}, {18835, 20254}, {18906, 37511}, {20888, 34822}, {22151, 28704}, {22365, 26244}, {26216, 37125}, {31276, 37186}

X(41009) = isotomic conjugate of the isogonal conjugate of X(1899)
X(41009) = isotomic conjugate of polar conjugate of X(41760)
X(41009) = polar conjugate of the isogonal conjugate of X(6389)
X(41009) = X(264)-Ceva conjugate of polar conjugate of isogonal conjugate of X(3767)
X(41009) = X(30450)-Ceva conjugate of X(850)
X(41009) = X(560)-isoconjugate of X(34405)
X(41009) = cevapoint of X(1899) and X(6389)
X(41009) = crosspoint of X(264) and X(305)
X(41009) = crosssum of X(184) and X(1974)
X(41009) = barycentric product X(i)*X(j) for these {i,j}: {76, 1899}, {264, 6389}, {304, 17871}, {305, 3767}, {426, 18027}, {561, 2083}, {1632, 3267}, {18018, 28405}, {18022, 39643}
X(41009) = barycentric quotient X(i)/X(j) for these {i,j}: {76, 34405}, {426, 577}, {1632, 112}, {1899, 6}, {2083, 31}, {2450, 232}, {3767, 25}, {6389, 3}, {6751, 217}, {17871, 19}, {18953, 7592}, {27362, 14576}, {28405, 22}, {39643, 184}
X(41009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 339, 76}, {385, 28723, 10316}, {3767, 6389, 28405}, {32828, 40680, 3547}

X(41010) = CROSSPOINT OF X(7) AND X(253)

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

X(41010) lies on these lines: {1, 1503}, {4, 3668}, {7, 84}, {9, 18589}, {19, 4466}, {40, 307}, {57, 1848}, {71, 31158}, {77, 6261}, {150, 3875}, {196, 8808}, {223, 5928}, {253, 39130}, {269, 1565}, {282, 16596}, {309, 34393}, {347, 515}, {497, 3663}, {610, 17073}, {693, 2995}, {960, 17272}, {1001, 3220}, {1119, 6245}, {1158, 7013}, {1375, 18594}, {1394, 3485}, {1439, 6001}, {1441, 5587}, {1490, 6356}, {1763, 30674}, {2883, 7992}, {2886, 25590}, {3007, 5881}, {3182, 4295}, {3339, 5799}, {3576, 18650}, {3953, 4862}, {4357, 17170}, {4872, 10444}, {5480, 10398}, {5738, 11529}, {5813, 25019}, {6173, 34830}, {6354, 10888}, {7053, 18237}, {7195, 24213}, {7719, 36907}, {8557, 16580}, {9581, 17861}, {9948, 14256}, {10436, 17181}, {10826, 17885}, {12779, 20618}, {17220, 31162}, {18713, 37796}, {18725, 26932}, {21279, 22464}, {25361, 25525}

X(41010) = reflection of X(610) in X(17073)
X(41010) = isotomic conjugate of X(34414)
X(41010) = X(31)-isoconjugate of X(34414)
X(41010) = crosspoint of X(i) and X(j) for these (i,j): {7, 253}, {273, 309}
X(41010) = crosssum of X(i) and X(j) for these (i,j): {55, 154}, {205, 1253}, {212, 2187}
X(41010) = barycentric product X(85)*X(1854)
X(41010) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34414}, {1854, 9}
X(41010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 4466, 18634}, {307, 4329, 40}, {18589, 24316, 9}

X(41011) = CROSSPOINT OF X(4) AND X(86)

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

X(41011) lies on these lines: {1, 5905}, {2, 1707}, {6, 1836}, {7, 614}, {10, 6327}, {25, 23359}, {31, 226}, {38, 527}, {42, 516}, {44, 3925}, {51, 65}, {57, 34029}, {58, 12047}, {63, 24695}, {79, 1203}, {81, 5057}, {142, 748}, {171, 908}, {190, 33073}, {193, 17156}, {213, 23682}, {223, 4331}, {238, 5249}, {244, 553}, {306, 3923}, {320, 32942}, {321, 5847}, {329, 612}, {354, 17365}, {386, 1770}, {497, 4644}, {513, 23811}, {519, 4365}, {524, 3706}, {595, 13407}, {750, 3452}, {752, 1215}, {851, 20967}, {894, 4388}, {896, 5745}, {902, 13405}, {940, 24703}, {946, 1468}, {950, 2650}, {968, 5698}, {984, 17781}, {1046, 6734}, {1070, 3157}, {1072, 37826}, {1100, 4854}, {1104, 3649}, {1125, 6536}, {1155, 37662}, {1191, 10404}, {1193, 4292}, {1201, 4298}, {1266, 32924}, {1386, 3782}, {1430, 1848}, {1456, 6354}, {1699, 11269}, {1724, 12609}, {1738, 20292}, {1757, 25006}, {1892, 3195}, {1957, 30687}, {2223, 21319}, {2285, 2354}, {2292, 5717}, {2308, 3120}, {2309, 18650}, {2321, 32852}, {2796, 4970}, {2886, 4641}, {2887, 4672}, {2999, 4312}, {3012, 20277}, {3052, 17718}, {3218, 24239}, {3219, 33112}, {3271, 39793}, {3338, 28018}, {3434, 3751}, {3626, 28599}, {3663, 17017}, {3664, 3720}, {3666, 17768}, {3671, 3924}, {3683, 17056}, {3685, 17778}, {3686, 21020}, {3687, 4418}, {3703, 17351}, {3717, 32938}, {3729, 33088}, {3741, 4001}, {3745, 4415}, {3752, 11246}, {3755, 33094}, {3758, 32773}, {3771, 35263}, {3816, 37520}, {3817, 29662}, {3838, 35466}, {3846, 4697}, {3879, 32915}, {3883, 32771}, {3896, 28580}, {3912, 32930}, {3915, 21620}, {3920, 17484}, {3946, 33145}, {3966, 4363}, {3977, 29671}, {4028, 31034}, {4030, 28566}, {4035, 33156}, {4054, 4362}, {4133, 20017}, {4252, 11375}, {4349, 4656}, {4353, 29819}, {4357, 4683}, {4383, 5880}, {4387, 4851}, {4416, 31330}, {4423, 4675}, {4425, 33682}, {4640, 5718}, {4645, 27064}, {4649, 33095}, {4650, 17717}, {4654, 7290}, {4655, 25496}, {4676, 18134}, {4679, 37674}, {4684, 32943}, {4722, 33136}, {4847, 32912}, {4849, 34612}, {4865, 32935}, {4888, 10582}, {4892, 6679}, {5087, 37634}, {5121, 27003}, {5180, 17015}, {5230, 9612}, {5256, 24248}, {5262, 14450}, {5263, 33066}, {5264, 21077}, {5268, 31018}, {5269, 28609}, {5297, 26792}, {5316, 17124}, {6682, 28558}, {6685, 28508}, {7076, 30686}, {7081, 20101}, {7191, 17483}, {7262, 33111}, {7292, 26842}, {7649, 9718}, {9352, 37651}, {10453, 17364}, {10459, 12527}, {10473, 26892}, {11551, 30117}, {12436, 27627}, {12608, 37530}, {12723, 17441}, {13408, 31937}, {15523, 17355}, {15569, 37631}, {16466, 23536}, {16468, 17889}, {16469, 23681}, {16475, 19785}, {16477, 33132}, {16669, 21949}, {17011, 33100}, {17012, 33102}, {17023, 32776}, {17064, 24597}, {17126, 31053}, {17127, 31019}, {17135, 34379}, {17147, 28526}, {17276, 17599}, {17350, 29641}, {17353, 25957}, {17469, 32856}, {17605, 37646}, {17716, 33101}, {18249, 21674}, {20064, 26227}, {21255, 29677}, {21321, 22060}, {21616, 37522}, {24169, 24692}, {24177, 30424}, {24552, 32859}, {25101, 29854}, {26015, 32913}, {26061, 31134}, {26065, 29857}, {26132, 29855}, {27385, 37603}, {29665, 30652}, {29681, 30653}, {29821, 32857}, {29840, 31300}, {31164, 33144}, {31266, 36277}, {32774, 38049}, {32844, 32940}, {32933, 33070}, {32939, 33071}, {32944, 33067}, {33134, 37685}

X(41011) = reflection of X(4001) in X(3741)
X(41011) = crosspoint of X(4) and X(86)
X(41011) = crosssum of X(3) and X(42)
X(41011) = barycentric product X(i)*X(j) for these {i,j}: {81, 17874}, {1509, 21698}
X(41011) = barycentric quotient X(i)/X(j) for these {i,j}: {17874, 321}, {21698, 594}
X(41011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1836, 3914}, {31, 226, 3011}, {31, 24725, 226}, {63, 26098, 29639}, {79, 1203, 23537}, {81, 5057, 24210}, {171, 33096, 908}, {238, 33097, 5249}, {329, 4307, 612}, {595, 13407, 28027}, {896, 33105, 5745}, {1191, 10404, 23675}, {1757, 33109, 25006}, {2887, 4672, 5294}, {2887, 5294, 30768}, {3218, 33107, 24239}, {3923, 32946, 306}, {4349, 4656, 5311}, {4418, 32843, 3687}, {4683, 32772, 4357}, {5698, 5712, 968}, {6327, 26223, 10}, {7191, 17483, 24231}, {16468, 17889, 26723}, {17017, 33098, 3663}, {20292, 32911, 1738}, {24695, 26098, 63}, {31034, 32929, 4028}, {32912, 33104, 4847}, {32913, 33106, 26015}, {32930, 32949, 3912}, {32938, 33072, 3717}

X(41012) = CROSSPOINT OF X(86) AND X(312)

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

X(41012) lies on these lines: {1, 908}, {2, 40}, {3, 1519}, {4, 19861}, {5, 392}, {8, 3452}, {9, 10527}, {10, 3877}, {11, 960}, {12, 5087}, {20, 35262}, {21, 36}, {29, 1842}, {30, 17614}, {46, 10200}, {51, 35631}, {55, 25681}, {56, 24703}, {57, 11415}, {63, 3086}, {65, 3816}, {72, 496}, {78, 497}, {100, 6700}, {142, 5550}, {145, 21075}, {165, 6921}, {191, 3582}, {210, 3813}, {226, 452}, {238, 283}, {329, 10396}, {355, 17556}, {377, 1699}, {380, 27395}, {382, 35272}, {390, 27383}, {404, 516}, {405, 5812}, {442, 9955}, {474, 12699}, {499, 12514}, {515, 5046}, {517, 4187}, {518, 37722}, {519, 5330}, {529, 20323}, {548, 35271}, {551, 3897}, {553, 14450}, {631, 35258}, {936, 3434}, {950, 4511}, {956, 11373}, {958, 4679}, {968, 30007}, {970, 24996}, {978, 3914}, {982, 28018}, {988, 30078}, {997, 1479}, {1001, 11344}, {1036, 27388}, {1058, 3870}, {1155, 6691}, {1191, 17720}, {1193, 24210}, {1201, 13161}, {1203, 3193}, {1210, 3869}, {1329, 3057}, {1376, 12701}, {1385, 11113}, {1484, 31835}, {1532, 31786}, {1537, 31788}, {1621, 13411}, {1697, 5552}, {1698, 6931}, {1706, 20196}, {1737, 3825}, {1738, 27627}, {1836, 25524}, {1837, 5289}, {1902, 26020}, {2057, 10388}, {2292, 24239}, {2475, 18483}, {2476, 3817}, {2551, 3872}, {2886, 25917}, {2975, 12572}, {3035, 37568}, {3085, 30852}, {3120, 24178}, {3218, 31888}, {3305, 19843}, {3306, 4295}, {3333, 5905}, {3339, 31249}, {3359, 6967}, {3419, 9669}, {3421, 36846}, {3428, 25875}, {3485, 26105}, {3487, 4666}, {3576, 6872}, {3579, 13747}, {3583, 17647}, {3622, 21620}, {3624, 4512}, {3626, 21630}, {3634, 31262}, {3647, 32557}, {3649, 3742}, {3662, 26093}, {3683, 4999}, {3685, 21246}, {3687, 3702}, {3705, 3710}, {3741, 37373}, {3753, 17527}, {3814, 3884}, {3820, 10914}, {3832, 10863}, {3847, 17606}, {3868, 11019}, {3871, 6745}, {3873, 10399}, {3876, 4847}, {3880, 21031}, {3885, 4342}, {3889, 21625}, {3890, 11681}, {3895, 5328}, {3916, 15325}, {3944, 21214}, {3951, 24477}, {3957, 40270}, {4101, 10453}, {4188, 31730}, {4189, 10165}, {4294, 4855}, {4297, 11114}, {4301, 8582}, {4310, 28016}, {4330, 15015}, {4357, 17183}, {4420, 5853}, {4423, 28628}, {4640, 5433}, {4646, 37663}, {4652, 5698}, {4673, 5233}, {4696, 30566}, {4719, 4854}, {4861, 5795}, {4870, 11281}, {4875, 38930}, {4881, 15680}, {5044, 7743}, {5080, 10106}, {5084, 5603}, {5119, 26364}, {5121, 24443}, {5128, 31190}, {5154, 10175}, {5177, 9779}, {5178, 10707}, {5180, 6692}, {5187, 5587}, {5251, 37735}, {5258, 16173}, {5316, 9780}, {5428, 33594}, {5438, 9580}, {5439, 39542}, {5440, 15171}, {5533, 18254}, {5534, 10806}, {5554, 7982}, {5690, 17619}, {5692, 10916}, {5717, 33107}, {5720, 12116}, {5722, 5730}, {5731, 6260}, {5749, 21068}, {5794, 10896}, {5919, 12607}, {6361, 17567}, {6762, 11240}, {6842, 31838}, {6890, 12705}, {6904, 9812}, {6922, 12672}, {6933, 7988}, {6962, 10268}, {7191, 34937}, {7330, 10785}, {7483, 11230}, {7498, 30687}, {7504, 10171}, {7681, 14110}, {8728, 38034}, {9711, 13463}, {9791, 30097}, {9856, 37374}, {9956, 17533}, {9957, 17757}, {10072, 17781}, {10164, 17566}, {10176, 24387}, {10179, 15888}, {10198, 37692}, {10478, 37314}, {10528, 31393}, {10529, 31018}, {10531, 37531}, {10572, 30144}, {11106, 31019}, {11108, 18493}, {11110, 17167}, {11112, 22793}, {11239, 37556}, {11260, 34606}, {11362, 25005}, {11682, 18391}, {12436, 20292}, {12512, 13587}, {12571, 17577}, {12611, 13624}, {13464, 37162}, {13724, 37620}, {13741, 25904}, {15254, 24953}, {15507, 22345}, {15670, 33592}, {15908, 22835}, {16202, 37713}, {16342, 24220}, {17174, 17588}, {17185, 19863}, {17452, 23637}, {17526, 21062}, {17594, 30006}, {17605, 25466}, {17719, 28027}, {17768, 32636}, {18589, 27407}, {19862, 37572}, {19864, 26092}, {19925, 37375}, {22076, 24997}, {22172, 24231}, {22753, 37248}, {23708, 26363}, {24171, 33146}, {24248, 30037}, {24318, 24460}, {24556, 37422}, {24705, 25369}, {25055, 31156}, {25527, 27505}, {25881, 33833}, {25893, 37282}, {26006, 37086}, {26029, 27130}, {26094, 27506}, {26116, 27184}, {26470, 37359}, {27625, 33131}, {28011, 33144}, {28150, 37256}, {31141, 32049}, {31246, 37828}, {33950, 40869}, {34716, 38314}, {37548, 37662}

X(41012) = reflection of X(24982) in X(4187)
X(41012) = crosspoint of X(i) and X(j) for these (i,j): {86, 312}, {4373, 30690}
X(41012) = crosssum of X(i) and X(j) for these (i,j): {42, 604}, {56, 18360}, {2174, 3052}
X(41012) = barycentric product X(i)*X(j) for these {i,j}: {1509, 21694}, {2321, 16714}
X(41012) = barycentric quotient X(i)/X(j) for these {i,j}: {16714, 1434}, {21694, 594}
X(41012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 21616, 908}, {2, 20070, 26062}, {5, 392, 24987}, {11, 960, 6734}, {40, 25522, 2}, {55, 25681, 27385}, {72, 496, 26015}, {145, 27131, 21075}, {405, 5886, 24541}, {936, 9614, 3434}, {1125, 4292, 5253}, {1125, 11813, 12047}, {1125, 12047, 5249}, {1329, 3057, 6735}, {1697, 30827, 5552}, {1699, 8583, 377}, {3120, 28352, 24178}, {3452, 12053, 8}, {3622, 31053, 21620}, {3624, 4512, 6910}, {3624, 18393, 12609}, {3705, 19582, 3710}, {3753, 17527, 25011}, {3814, 3884, 10039}, {3825, 3878, 1737}, {3877, 4193, 10}, {3890, 11681, 31397}, {3944, 21214, 23536}, {4342, 6736, 3885}, {4679, 11376, 958}, {5044, 7743, 24390}, {5044, 24390, 25006}, {5057, 5253, 4292}, {5084, 5603, 19860}, {5259, 5443, 1125}, {5328, 9785, 7080}, {5692, 37720, 10916}, {5698, 7288, 4652}, {5905, 10586, 3333}, {6700, 10624, 100}, {6745, 12575, 3871}, {7080, 9785, 3895}, {8227, 31435, 2}, {9581, 15829, 8}, {12701, 24954, 1376}, {17527, 22791, 3753}

X(41013) = CROSSPOINT OF X(92) AND X(264)

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

Let A'B'C' be the 2nd circumperp triangle. Let A" be the pole, wrt the polar circle, of line B'C'; define B" and C" cyclically. The lines AA", BB", CC" concur in X(41013). (Randy Hutson, January 22, 2021)

X(41013) lies on these lines: {1, 5136}, {2, 1068}, {3, 23661}, {4, 8}, {10, 201}, {19, 1759}, {25, 26227}, {28, 1791}, {29, 1807}, {33, 2901}, {34, 996}, {37, 158}, {40, 17860}, {57, 20320}, {65, 17869}, {75, 14018}, {85, 38461}, {100, 1300}, {108, 1940}, {210, 1882}, {213, 1783}, {226, 39130}, {228, 7412}, {242, 4222}, {254, 5552}, {264, 1969}, {273, 5936}, {278, 475}, {280, 6847}, {286, 31902}, {333, 14016}, {337, 16747}, {349, 20235}, {388, 14257}, {427, 3006}, {429, 3695}, {442, 1441}, {469, 33077}, {594, 1865}, {648, 2906}, {668, 35142}, {758, 1825}, {847, 11681}, {942, 17862}, {943, 1896}, {944, 20220}, {946, 24026}, {956, 4185}, {1060, 24537}, {1062, 27378}, {1074, 34823}, {1076, 34822}, {1089, 1826}, {1093, 6521}, {1105, 4219}, {1109, 21935}, {1125, 23710}, {1210, 4858}, {1214, 37154}, {1217, 3998}, {1231, 17864}, {1309, 2687}, {1389, 36921}, {1426, 3753}, {1698, 37805}, {1785, 10039}, {1792, 7283}, {1835, 3754}, {1838, 1861}, {1840, 2333}, {1848, 22020}, {1855, 3970}, {1869, 4647}, {1870, 11109}, {1873, 4015}, {1880, 14624}, {1881, 21011}, {1883, 2969}, {1884, 12135}, {1893, 3697}, {1895, 5703}, {2322, 37383}, {2968, 6831}, {2973, 34337}, {2975, 37117}, {3142, 21318}, {3144, 17927}, {3159, 10915}, {3175, 34619}, {3219, 3559}, {3294, 7079}, {3585, 13532}, {3616, 38295}, {3702, 5730}, {3822, 6757}, {3868, 37235}, {3935, 14004}, {3990, 40402}, {3995, 4194}, {4036, 18808}, {4066, 21060}, {4200, 31025}, {4231, 7081}, {4296, 37157}, {4404, 24006}, {5044, 26591}, {5089, 17911}, {5179, 22005}, {5230, 17871}, {5247, 24432}, {5439, 20905}, {5657, 37414}, {5687, 37194}, {5709, 20223}, {5903, 23580}, {5906, 37826}, {6335, 17983}, {6350, 6889}, {6360, 26027}, {6734, 14213}, {6906, 10538}, {7004, 14058}, {7270, 20920}, {7378, 31091}, {7518, 20013}, {7559, 14860}, {8889, 30741}, {9436, 21413}, {11363, 37168}, {16230, 18003}, {17134, 37063}, {17170, 21588}, {17924, 21108}, {18026, 35141}, {18283, 37028}, {21664, 21666}, {22000, 39574}, {23987, 37043}, {24390, 37368}, {26704, 37227}, {26877, 34234}

X(41013) = reflection of X(20222) in X(37565)
X(41013) = isogonal conjugate of X(1437)
X(41013) = isotomic conjugate of X(1444)
X(41013) = complement of X(20222)
X(41013) = anticomplement of X(37565)
X(41013) = polar conjugate of X(81)
X(41013) = isotomic conjugate of the isogonal conjugate of X(1824)
X(41013) = polar conjugate of the isotomic conjugate of X(321)
X(41013) = polar conjugate of the isogonal conjugate of X(37)
X(41013) = X(i)-Ceva conjugate of X(j) for these (i,j): {92, 1826}, {318, 10}, {40447, 281}
X(41013) = X(i)-cross conjugate of X(j) for these (i,j): {12, 10}, {37, 321}, {429, 4}, {1826, 40149}, {1865, 2052}, {4705, 1783}, {7140, 1826}, {17874, 75}, {21807, 37}
X(41013) = cevapoint of X(i) and X(j) for these (i,j): {1, 1710}, {4, 451}, {10, 21077}, {37, 1824}
X(41013) = crosspoint of X(92) and X(264)
X(41013) = crosssum of X(i) and X(j) for these (i,j): {3, 22458}, {48, 184}, {603, 7335}
X(41013) = trilinear pole of line {424, 2501}
X(41013) = crossdifference of every pair of points on line {22383, 23224}
X(41013) = X(264)-Ceva conjugate of isotomic conjugate of X(1790)
X(41013) = X(264)-Ceva conjugate of polar conjugate of X(58)
X(41013) = trilinear product of vertices of 2nd extouch triangle
X(41013) = perspector of ABC and orthoanticevian triangle of X(321)
X(41013) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1437}, {3, 58}, {6, 1790}, {19, 18604}, {21, 603}, {27, 577}, {28, 255}, {29, 7335}, {31, 1444}, {32, 17206}, {48, 81}, {56, 283}, {57, 2193}, {60, 73}, {63, 1333}, {69, 2206}, {71, 593}, {72, 849}, {77, 2194}, {78, 1408}, {86, 184}, {101, 7254}, {109, 23189}, {110, 1459}, {112, 4091}, {162, 23224}, {163, 905}, {212, 1014}, {219, 1412}, {222, 284}, {228, 757}, {248, 17209}, {270, 22341}, {274, 9247}, {285, 7114}, {295, 5009}, {310, 14575}, {326, 2203}, {332, 1397}, {345, 16947}, {394, 1474}, {513, 4575}, {514, 32661}, {604, 1812}, {608, 6514}, {647, 4556}, {649, 4558}, {652, 4565}, {662, 22383}, {667, 4592}, {741, 7193}, {859, 1795}, {906, 1019}, {967, 4288}, {1092, 8747}, {1098, 1410}, {1101, 18210}, {1106, 1792}, {1169, 22097}, {1171, 22054}, {1172, 7125}, {1175, 4303}, {1176, 17187}, {1178, 3955}, {1193, 1798}, {1214, 2150}, {1326, 17972}, {1331, 3733}, {1396, 2289}, {1399, 1789}, {1407, 2327}, {1409, 2185}, {1413, 1819}, {1414, 1946}, {1428, 1808}, {1433, 2360}, {1461, 23090}, {1509, 2200}, {1576, 4025}, {1797, 3285}, {1804, 2299}, {1805, 6502}, {1806, 2067}, {1813, 7252}, {1820, 18605}, {1919, 4563}, {2148, 16697}, {2169, 18180}, {2189, 40152}, {2204, 7183}, {2287, 7099}, {2318, 7341}, {2328, 7053}, {2359, 40153}, {2363, 22345}, {3049, 4610}, {3216, 15409}, {3286, 36057}, {3453, 11573}, {3710, 7342}, {3737, 36059}, {3784, 38813}, {3937, 4570}, {4131, 32676}, {4466, 23357}, {4560, 32660}, {4591, 22086}, {4600, 22096}, {5317, 6507}, {6629, 14908}, {7100, 17104}, {7192, 32656}, {10547, 16887}, {13486, 23226}, {14376, 17186}, {14533, 17167}, {14953, 32657}, {15373, 27644}, {15419, 32739}, {15958, 21102}, {16702, 36060}, {16704, 32659}, {18206, 32658}, {18268, 20769}, {18653, 18877}, {22128, 34079}, {22133, 40142}, {23086, 38832}, {23092, 34071}, {23201, 40438}, {30493, 35196}
X(41013) = barycentric product X(i)*X(j) for these {i,j}: {4, 321}, {8, 40149}, {10, 92}, {12, 31623}, {19, 313}, {25, 27801}, {27, 1089}, {28, 28654}, {29, 6358}, {33, 349}, {34, 30713}, {37, 264}, {42, 1969}, {65, 7017}, {72, 2052}, {75, 1826}, {76, 1824}, {81, 7141}, {100, 14618}, {158, 306}, {190, 24006}, {210, 331}, {213, 18022}, {225, 312}, {226, 318}, {228, 18027}, {273, 2321}, {274, 7140}, {276, 21807}, {278, 3701}, {281, 1441}, {286, 594}, {314, 8736}, {338, 5379}, {393, 20336}, {429, 30710}, {430, 32018}, {442, 40447}, {523, 6335}, {561, 2333}, {648, 4036}, {653, 4086}, {668, 2501}, {811, 4024}, {823, 4064}, {850, 1783}, {860, 18359}, {862, 18895}, {1093, 3998}, {1096, 40071}, {1172, 34388}, {1231, 1857}, {1235, 18098}, {1446, 7046}, {1577, 1897}, {1840, 7018}, {1847, 4082}, {1865, 40422}, {1867, 34258}, {1880, 3596}, {1896, 26942}, {2489, 6386}, {2970, 4567}, {3112, 21016}, {3668, 7101}, {3682, 6521}, {3700, 18026}, {3952, 17924}, {3992, 6336}, {4033, 7649}, {4080, 38462}, {4601, 8754}, {4705, 6331}, {6591, 27808}, {8750, 20948}, {11611, 17987}, {15065, 17923}, {15742, 16732}, {16082, 17757}, {18082, 20883}, {21011, 40440}, {21046, 23999}
X(41013) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1790}, {2, 1444}, {3, 18604}, {4, 81}, {5, 16697}, {6, 1437}, {8, 1812}, {9, 283}, {10, 63}, {12, 1214}, {19, 58}, {24, 18605}, {25, 1333}, {27, 757}, {28, 593}, {29, 2185}, {33, 284}, {34, 1412}, {37, 3}, {42, 48}, {53, 18180}, {55, 2193}, {65, 222}, {71, 255}, {72, 394}, {73, 7125}, {75, 17206}, {78, 6514}, {92, 86}, {100, 4558}, {101, 4575}, {108, 4565}, {115, 18210}, {158, 27}, {162, 4556}, {181, 1409}, {190, 4592}, {200, 2327}, {201, 40152}, {210, 219}, {213, 184}, {225, 57}, {226, 77}, {227, 7011}, {228, 577}, {235, 18603}, {240, 17209}, {264, 274}, {273, 1434}, {278, 1014}, {281, 21}, {286, 1509}, {306, 326}, {307, 7183}, {312, 332}, {313, 304}, {318, 333}, {321, 69}, {346, 1792}, {349, 7182}, {393, 28}, {403, 18609}, {406, 27174}, {427, 16696}, {429, 3666}, {430, 1100}, {442, 18607}, {451, 40592}, {468, 16702}, {512, 22383}, {513, 7254}, {523, 905}, {525, 4131}, {594, 72}, {607, 2194}, {608, 1408}, {647, 23224}, {650, 23189}, {653, 1414}, {656, 4091}, {661, 1459}, {668, 4563}, {692, 32661}, {693, 15419}, {740, 20769}, {756, 71}, {758, 22128}, {762, 3690}, {811, 4610}, {850, 15413}, {860, 3218}, {862, 1914}, {872, 2200}, {968, 4288}, {1018, 1331}, {1042, 7099}, {1089, 306}, {1096, 1474}, {1109, 4466}, {1118, 1396}, {1172, 60}, {1213, 3916}, {1214, 1804}, {1231, 7055}, {1235, 16703}, {1334, 212}, {1395, 16947}, {1396, 7341}, {1400, 603}, {1409, 7335}, {1426, 1407}, {1427, 7053}, {1441, 348}, {1446, 7056}, {1474, 849}, {1500, 228}, {1577, 4025}, {1594, 16698}, {1783, 110}, {1784, 18653}, {1824, 6}, {1825, 2003}, {1826, 1}, {1829, 40153}, {1832, 37772}, {1833, 37773}, {1840, 171}, {1855, 17194}, {1857, 1172}, {1861, 18206}, {1865, 942}, {1867, 940}, {1874, 1429}, {1880, 56}, {1882, 37543}, {1893, 5228}, {1897, 662}, {1903, 1433}, {1918, 9247}, {1962, 22054}, {1969, 310}, {1973, 2206}, {2052, 286}, {2092, 22345}, {2171, 73}, {2197, 22341}, {2201, 5009}, {2205, 14575}, {2207, 2203}, {2238, 7193}, {2250, 1795}, {2292, 22097}, {2294, 4303}, {2295, 3955}, {2298, 1798}, {2299, 2150}, {2318, 2289}, {2321, 78}, {2322, 1098}, {2324, 1819}, {2331, 2360}, {2333, 31}, {2358, 1413}, {2489, 667}, {2501, 513}, {2967, 16725}, {2968, 16731}, {2969, 16726}, {2970, 16732}, {2971, 3121}, {2972, 16730}, {2973, 16727}, {3064, 3737}, {3120, 3942}, {3121, 22096}, {3125, 3937}, {3136, 18606}, {3142, 18608}, {3198, 15905}, {3668, 7177}, {3682, 6507}, {3690, 3990}, {3694, 1259}, {3695, 3998}, {3696, 23151}, {3700, 521}, {3701, 345}, {3709, 1946}, {3710, 3719}, {3721, 3784}, {3728, 22065}, {3753, 22129}, {3900, 23090}, {3914, 7289}, {3930, 1818}, {3932, 25083}, {3943, 5440}, {3949, 3682}, {3950, 4855}, {3952, 1332}, {3954, 3917}, {3962, 23140}, {3971, 22370}, {3990, 1092}, {3992, 3977}, {3998, 3964}, {4016, 11573}, {4024, 656}, {4033, 4561}, {4036, 525}, {4041, 652}, {4055, 4100}, {4058, 3984}, {4064, 24018}, {4069, 4587}, {4079, 810}, {4082, 3692}, {4083, 23092}, {4086, 6332}, {4132, 22154}, {4145, 23141}, {4183, 7054}, {4213, 38814}, {4397, 15411}, {4463, 20806}, {4515, 1260}, {4516, 7117}, {4551, 1813}, {4552, 6516}, {4557, 906}, {4559, 36059}, {4647, 4001}, {4674, 1797}, {4705, 647}, {4730, 22086}, {4849, 20818}, {4858, 17219}, {4876, 1808}, {5089, 3286}, {5257, 4652}, {5360, 3289}, {5379, 249}, {6057, 3694}, {6059, 2204}, {6198, 40214}, {6331, 4623}, {6335, 99}, {6354, 1439}, {6358, 307}, {6378, 22381}, {6520, 8747}, {6524, 5317}, {6535, 3949}, {6591, 3733}, {6753, 34948}, {7003, 285}, {7017, 314}, {7046, 2287}, {7079, 2328}, {7101, 1043}, {7102, 2303}, {7110, 1789}, {7133, 1805}, {7140, 37}, {7141, 321}, {7237, 20727}, {7649, 1019}, {7952, 1817}, {8013, 3958}, {8735, 18191}, {8736, 65}, {8748, 270}, {8750, 163}, {8754, 3125}, {8818, 7100}, {9278, 17972}, {13149, 4616}, {13576, 1814}, {14208, 30805}, {14273, 14419}, {14571, 859}, {14618, 693}, {14624, 1791}, {15742, 4567}, {16583, 1473}, {16589, 22060}, {16600, 7293}, {16606, 23086}, {16732, 1565}, {17314, 14868}, {17442, 17187}, {17742, 1801}, {17869, 26871}, {17905, 4228}, {17916, 4184}, {17924, 7192}, {17927, 1931}, {17987, 19623}, {18022, 6385}, {18026, 4573}, {18082, 34055}, {18098, 1176}, {18344, 7252}, {18785, 36057}, {20336, 3926}, {20680, 20749}, {20681, 20750}, {20682, 20751}, {20683, 20752}, {20684, 20753}, {20685, 20754}, {20686, 20755}, {20687, 20756}, {20688, 20757}, {20689, 20758}, {20690, 20759}, {20691, 20760}, {20692, 20761}, {20693, 17976}, {20694, 20762}, {20695, 20763}, {20697, 20765}, {20698, 20766}, {20699, 20767}, {20700, 20768}, {20701, 20770}, {20702, 20728}, {20703, 20729}, {20704, 20730}, {20706, 20731}, {20707, 20732}, {20708, 20733}, {20709, 20734}, {20710, 20735}, {20711, 20736}, {20712, 20737}, {20713, 20739}, {20714, 20740}, {20715, 20741}, {20716, 20742}, {20717, 20743}, {20718, 20744}, {20719, 20745}, {20720, 20746}, {20723, 20747}, {20724, 20748}, {20883, 16887}, {20902, 17216}, {20970, 23201}, {21016, 38}, {21035, 4020}, {21043, 3708}, {21044, 7004}, {21046, 2632}, {21049, 10167}, {21051, 25098}, {21077, 6505}, {21794, 22342}, {21795, 22079}, {21796, 22344}, {21797, 22346}, {21798, 22347}, {21799, 22348}, {21800, 22349}, {21801, 22350}, {21802, 22352}, {21803, 22061}, {21804, 22070}, {21805, 22356}, {21806, 22357}, {21807, 216}, {21808, 22053}, {21809, 22072}, {21810, 22076}, {21811, 22361}, {21812, 22362}, {21813, 22363}, {21814, 20775}, {21815, 22364}, {21816, 22080}, {21817, 22078}, {21818, 22367}, {21819, 22368}, {21820, 22369}, {21821, 22371}, {21822, 22372}, {21823, 22373}, {21824, 22094}, {21825, 22375}, {21826, 22376}, {21827, 22378}, {21828, 22379}, {21829, 22098}, {21830, 20777}, {21831, 22382}, {21832, 22384}, {21833, 20975}, {21834, 22090}, {21835, 22386}, {21836, 22387}, {21837, 22388}, {21838, 22389}, {21839, 3292}, {21840, 22390}, {21853, 3157}, {21854, 20764}, {21855, 22457}, {21856, 20793}, {21857, 20805}, {21858, 22458}, {21859, 23067}, {21860, 20803}, {21861, 23068}, {21862, 23069}, {21863, 23070}, {21864, 23071}, {21865, 22164}, {21866, 23072}, {21867, 22131}, {21868, 22149}, {21870, 23073}, {21871, 7078}, {21872, 22117}, {21873, 22136}, {21874, 3167}, {21875, 23074}, {21876, 23075}, {21877, 20794}, {21878, 23076}, {21879, 22139}, {21880, 22138}, {21881, 23077}, {21882, 23078}, {21883, 23079}, {21884, 23080}, {21885, 23081}, {21886, 23082}, {21887, 23083}, {21888, 22148}, {21889, 22144}, {21890, 22156}, {21891, 23084}, {21892, 23085}, {21893, 22158}, {21894, 23087}, {21895, 23088}, {21896, 23089}, {21897, 20796}, {21899, 22143}, {21900, 23091}, {21901, 23093}, {21902, 23094}, {21904, 23095}, {21933, 1071}, {21935, 26934}, {21956, 34381}, {22167, 22066}, {22171, 22413}, {22195, 20824}, {22201, 23219}, {22205, 20781}, {22271, 22126}, {22275, 23124}, {22276, 22134}, {22299, 23131}, {22310, 23137}, {22321, 22145}, {22322, 22159}, {23493, 15373}, {24006, 514}, {27801, 305}, {28594, 5314}, {28654, 20336}, {30713, 3718}, {30730, 4571}, {31623, 261}, {31900, 30581}, {34336, 16733}, {34337, 16728}, {34388, 1231}, {34922, 35049}, {36118, 4637}, {36197, 3270}, {36797, 4612}, {36910, 1793}, {37168, 30576}, {38462, 16704}, {39579, 5256}, {39964, 15409}, {40149, 7}, {40447, 40412}, {40501, 23139}, {40521, 4574}, {40717, 30940}
X(41013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20222, 37565}, {4, 318, 38462}, {10, 225, 860}, {29, 1897, 6198}, {92, 318, 4}, {281, 7952, 406}, {1824, 1867, 4}, {1872, 39529, 4}, {17911, 17915, 5089}

X(41014) = CROSSPOINT OF X(69) AND X(306)

Barycentrics    (b + c)*(2*a + b + c)*(-a^2 + b^2 + c^2) : :
X(41014) = X[1046] - 3 X[33160], 2 X[1834] - 3 X[16052], 4 X[3454] - 3 X[16052], 3 X[4234] - X[20077]

X(41014) lies on these lines: {1, 1211}, {3, 69}, {4, 31782}, {5, 4417}, {6, 17698}, {8, 442}, {10, 4035}, {12, 10474}, {21, 2895}, {30, 1043}, {58, 524}, {72, 306}, {73, 26942}, {75, 6147}, {78, 1060}, {140, 14829}, {141, 386}, {145, 5051}, {191, 3712}, {193, 37176}, {226, 5295}, {317, 7546}, {319, 5719}, {320, 24470}, {325, 33297}, {333, 6675}, {345, 3927}, {391, 16845}, {404, 32863}, {405, 5739}, {429, 5730}, {430, 1230}, {496, 10453}, {517, 10381}, {518, 10974}, {519, 1834}, {525, 21134}, {599, 4255}, {631, 37655}, {758, 3704}, {851, 5687}, {857, 29616}, {936, 17296}, {942, 3687}, {956, 37225}, {964, 31034}, {966, 16844}, {975, 4851}, {976, 32852}, {978, 33087}, {1010, 17778}, {1046, 33160}, {1100, 1125}, {1150, 7483}, {1193, 33081}, {1482, 30444}, {1503, 3430}, {1654, 11110}, {1698, 4023}, {1714, 30811}, {1812, 22136}, {1901, 2321}, {2049, 5712}, {2092, 37592}, {2238, 16502}, {2292, 4062}, {2475, 4720}, {2650, 20653}, {2901, 4415}, {3085, 5774}, {3136, 17135}, {3142, 17751}, {3159, 3943}, {3295, 4199}, {3416, 3811}, {3578, 15670}, {3616, 17514}, {3622, 27081}, {3630, 4257}, {3631, 4256}, {3649, 4046}, {3650, 4427}, {3678, 3932}, {3682, 18643}, {3694, 18591}, {3696, 12609}, {3703, 5904}, {3706, 12047}, {3714, 21077}, {3846, 35633}, {3868, 33077}, {3876, 32858}, {3879, 37594}, {3882, 10461}, {3886, 12699}, {3912, 5044}, {3916, 4001}, {3931, 4028}, {3940, 21530}, {3949, 22073}, {4019, 20733}, {4042, 19854}, {4187, 5741}, {4202, 31017}, {4234, 20077}, {4252, 40341}, {4281, 15985}, {4340, 19276}, {4361, 24159}, {4388, 15171}, {4416, 31445}, {4420, 33078}, {4658, 6703}, {4673, 22791}, {4684, 5045}, {4869, 17582}, {4886, 16817}, {5047, 37656}, {5233, 17527}, {5235, 24936}, {5266, 5847}, {5293, 32846}, {5703, 32099}, {5718, 10479}, {5752, 10477}, {5763, 16284}, {5827, 18391}, {5844, 30449}, {6515, 37248}, {6542, 26601}, {6707, 28620}, {6790, 36155}, {6857, 14552}, {6882, 32128}, {7080, 37154}, {7283, 33066}, {7762, 37100}, {7776, 36659}, {8227, 35613}, {8728, 9534}, {10026, 21024}, {11108, 14555}, {11245, 37247}, {11374, 11679}, {11433, 37244}, {12635, 37346}, {13728, 19767}, {13745, 26064}, {14005, 37635}, {14007, 26109}, {14986, 27039}, {15108, 37293}, {16062, 20018}, {16299, 37502}, {16408, 18141}, {16466, 33171}, {17372, 21245}, {17529, 18139}, {17733, 34528}, {17742, 39690}, {17747, 21070}, {17770, 24850}, {19270, 37653}, {20007, 25015}, {24883, 30831}, {25253, 33329}, {25526, 37631}, {25645, 35466}, {26243, 37047}, {33082, 37573}

X(41014) = midpoint of X(1043) and X(1330)
X(41014) = reflection of X(i) in X(j) for these {i,j}: {1834, 3454}, {3704, 21081}
X(41014) = isotomic conjugate of the isogonal conjugate of X(22080)
X(41014) = isotomic conjugate of the polar conjugate of X(1213)
X(41014) = isogonal conjugate of the polar conjugate of X(1230)
X(41014) = X(i)-Ceva conjugate of X(j) for these (i,j): {69, 4001}, {1230, 1213}, {3702, 4647}, {4001, 3958}, {4561, 525}
X(41014) = X(22080)-cross conjugate of X(1213)
X(41014) = X(i)-isoconjugate of X(j) for these (i,j): {19, 1171}, {25, 40438}, {27, 28615}, {28, 1126}, {1255, 1474}, {1268, 2203}, {1396, 33635}, {1796, 5317}, {1973, 32014}, {4608, 32676}, {4629, 6591}
X(41014) = crosspoint of X(69) and X(306)
X(41014) = crosssum of X(i) and X(j) for these (i,j): {25, 1474}, {28, 4222}
X(41014) = barycentric product X(i)*X(j) for these {i,j}: {3, 1230}, {10, 4001}, {63, 4647}, {69, 1213}, {71, 1269}, {72, 4359}, {75, 3958}, {76, 22080}, {304, 1962}, {305, 20970}, {306, 1125}, {307, 3686}, {313, 22054}, {321, 3916}, {345, 3649}, {348, 4046}, {430, 3926}, {525, 4427}, {553, 3710}, {1100, 20336}, {1214, 3702}, {1231, 3683}, {1332, 30591}, {2308, 40071}, {3267, 35327}, {3695, 8025}, {3949, 16709}, {4025, 4115}, {4561, 4988}, {4563, 6367}, {8013, 17206}, {14208, 35342}, {17094, 30729}, {23201, 27801}
X(41014) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1171}, {63, 40438}, {69, 32014}, {71, 1126}, {72, 1255}, {228, 28615}, {306, 1268}, {430, 393}, {525, 4608}, {1100, 28}, {1125, 27}, {1213, 4}, {1230, 264}, {1331, 4629}, {1332, 4596}, {1839, 8747}, {1962, 19}, {2308, 1474}, {2318, 33635}, {2355, 5317}, {3649, 278}, {3682, 1796}, {3683, 1172}, {3686, 29}, {3694, 32635}, {3695, 6539}, {3702, 31623}, {3710, 4102}, {3775, 31909}, {3916, 81}, {3958, 1}, {4001, 86}, {4046, 281}, {4064, 31010}, {4115, 1897}, {4359, 286}, {4427, 648}, {4558, 6578}, {4561, 4632}, {4574, 8701}, {4647, 92}, {4856, 4248}, {4966, 15149}, {4969, 37168}, {4974, 31905}, {4977, 17925}, {4983, 6591}, {4988, 7649}, {4990, 17926}, {4991, 31912}, {4992, 17921}, {5625, 31904}, {6367, 2501}, {8013, 1826}, {8040, 1839}, {8663, 2489}, {20336, 32018}, {20970, 25}, {21816, 1824}, {22054, 58}, {22080, 6}, {23201, 1333}, {30591, 17924}, {30729, 36797}, {31900, 36419}, {32636, 1396}, {35327, 112}, {35342, 162}
X(41014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1211, 4205}, {8, 3936, 442}, {72, 306, 3695}, {145, 31037, 5051}, {306, 4101, 72}, {333, 25650, 6675}, {1834, 3454, 16052}, {3649, 4046, 4647}, {4417, 10449, 5}, {4658, 24931, 6703}, {9534, 18134, 8728}, {19767, 32782, 13728}

X(41015) = CROSSPOINT OF X(4) AND X(274)

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

X(41015) lies on these lines: {1, 2271}, {2, 16605}, {6, 19}, {8, 37}, {9, 37598}, {10, 4167}, {25, 18619}, {42, 17451}, {43, 3061}, {44, 33950}, {46, 5021}, {55, 16968}, {58, 5011}, {72, 3735}, {75, 21216}, {81, 16716}, {85, 4000}, {101, 15955}, {145, 26242}, {172, 910}, {213, 517}, {218, 9620}, {239, 257}, {241, 2275}, {244, 17474}, {294, 17097}, {304, 27299}, {350, 25994}, {354, 20271}, {518, 3721}, {519, 4168}, {536, 17489}, {614, 1611}, {672, 4642}, {872, 20593}, {899, 39244}, {942, 3125}, {960, 2238}, {986, 21384}, {1015, 35066}, {1086, 4059}, {1100, 2303}, {1104, 1914}, {1108, 2277}, {1125, 16611}, {1146, 1834}, {1155, 33863}, {1193, 2170}, {1212, 2276}, {1319, 21008}, {1449, 20227}, {1475, 24443}, {1500, 16601}, {1509, 16756}, {1572, 16466}, {1574, 25068}, {1697, 16970}, {1755, 23640}, {1858, 38345}, {2176, 3057}, {2178, 8192}, {2280, 3924}, {2292, 3691}, {2295, 5836}, {2363, 9278}, {2646, 18755}, {2999, 7146}, {3214, 33299}, {3230, 9957}, {3496, 5247}, {3554, 37523}, {3626, 28594}, {3672, 27288}, {3693, 20691}, {3702, 27040}, {3706, 21024}, {3720, 21921}, {3726, 34791}, {3739, 17497}, {3744, 16974}, {3753, 17750}, {3765, 19791}, {3812, 21951}, {3828, 25089}, {3869, 21874}, {3914, 5254}, {3931, 5283}, {3954, 34790}, {3987, 16549}, {4136, 29673}, {4165, 36568}, {4254, 5336}, {4255, 34522}, {4272, 17443}, {4383, 39248}, {4386, 37539}, {4421, 39255}, {4424, 16552}, {4681, 28598}, {4868, 25092}, {4955, 17365}, {5045, 16971}, {5119, 14974}, {5276, 17016}, {5280, 5540}, {5286, 17905}, {5530, 37661}, {5710, 16972}, {5919, 16969}, {6051, 16589}, {6392, 40028}, {6734, 21965}, {7968, 30385}, {7969, 30386}, {8682, 21240}, {9259, 20323}, {9593, 16572}, {10459, 21840}, {11997, 23668}, {16604, 16610}, {16706, 20955}, {16973, 37549}, {16975, 37592}, {17720, 28807}, {17735, 37568}, {17737, 27068}, {17752, 20363}, {17754, 24440}, {17756, 26690}, {18607, 23632}, {18904, 25978}, {19622, 38858}, {20310, 23058}, {21029, 33136}, {21334, 21779}, {21876, 32773}, {24214, 35102}, {25091, 30646}, {25917, 37673}, {26562, 30941}, {26978, 30806}, {28358, 40797}, {30107, 33936}, {37614, 37658}

X(41015) = midpoint of X(3721) and X(3780)
X(41015) = X(i)-Ceva conjugate of X(j) for these (i,j): {4573, 513}, {24210, 11997}
X(41015) = X(23668)-cross conjugate of X(24210)
X(41015) = crosspoint of X(i) and X(j) for these (i,j): {4, 274}, {57, 256}, {81, 9309}
X(41015) = crosssum of X(i) and X(j) for these (i,j): {3, 213}, {6, 4640}, {9, 171}, {37, 1376}
X(41015) = crossdifference of every pair of points on line {521, 7234}
X(41015) = barycentric product X(i)*X(j) for these {i,j}: {1, 24210}, {7, 11997}, {86, 23668}, {693, 16680}, {7192, 22280}
X(41015) = barycentric quotient X(i)/X(j) for these {i,j}: {11997, 8}, {16680, 100}, {22280, 3952}, {23668, 10}, {24210, 75}
X(41015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16583, 3290}, {6, 3959, 65}, {304, 27299, 30748}, {1212, 4646, 2276}, {2238, 3727, 960}, {3125, 20963, 942}, {3666, 4875, 1107}, {3752, 40133, 2275}, {3869, 37657, 21874}, {17497, 26965, 20911}, {20911, 26965, 3739}, {21921, 39247, 3720}, {21951, 24512, 3812}

leftri

Altintas-isodynamic triangles: X(41016)-X(41071)

rightri

This preamble and centers X(41016)-X(41071) were contributed by César Eliud Lozada, January 11, 2021.

Let ABC be a triangle. The parallel line to BC from P = X(15) (1st isodynamic point) intersects the circumcircle of ABC in A1, A2 and the Simson lines of A1, A2 cut at A'. Define B' and C' cyclically. Then A'B'C' is equilateral and its center lies on the Euler line of ABC. (Kadir Altintas, January 9, 2021)

A' has barycentric coordinates:
   A' = -((S^2+SB*SC)*sqrt(3)*S+(SA+3*SW)*S^2-2*(-SW*SA+S^2+SW^2)*SA)/(S^2-sqrt(3)*S*SA-2*SW*SA) : SC : SB

and squared-sidelength:
   L'2 = S^2*sqrt(3)/(3*S+sqrt(3)*SW)

When P=X(16) (2nd isodynamic point), the parallel line to BC from P intersect the circumcircle of ABC in two imaginary points, but the Simson lines of these points still intersect in a real point A". The triangle A"B"C" built in this way is also equilateral and:

   A" = -((-S^2+SB*SC)*sqrt(3)*S+(SA+3*SW)*S^2-2*(-SW*SA+S^2+SW^2)*SA)/(S^2-sqrt(3)*S*SA-2*SW*SA) : SC : SB

and its squared-sidelength is:
   L"2 = S^2*sqrt(3)/(-3*S+sqrt(3)*SW)

Triangles A'B'C' and A"B"C" are referred here as 1st- and 2nd- Altintas-isodynamic triangles, respectively. A list of relations among these triangles and others can be seen here.

X(41016) = CENTER OF 1st ALTINTAS-ISODYNAMIC EQUILATERAL TRIANGLE

Barycentrics    2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S+sqrt(3)*(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(41016) = 2*X(4)+X(41035)

X(41016) lies on these lines: {2, 3}, {13, 1503}, {14, 5480}, {15, 36763}, {115, 5318}, {230, 22513}, {396, 7684}, {397, 5309}, {398, 7753}, {511, 22796}, {524, 41042}, {531, 41060}, {532, 41056}, {533, 41054}, {538, 41062}, {542, 22573}, {543, 41046}, {754, 41064}, {1499, 9201}, {2794, 6109}, {3564, 20425}, {5321, 5475}, {5334, 15484}, {5463, 41019}, {5479, 22682}, {5868, 40693}, {5982, 10723}, {6770, 11542}, {6775, 9750}, {7694, 10654}, {11645, 41045}, {14538, 36765}, {16267, 41020}, {16635, 41039}, {20426, 21850}, {22511, 36969}, {22691, 31710}, {22891, 36962}, {25184, 29012}, {32419, 41048}, {32421, 41050}, {36761, 36764}, {36967, 36992}, {41058, 41068}

X(41016) = midpoint of X(i) and X(j) for these {i, j}: {4, 1080}, {15, 36961}, {3543, 35931}, {36967, 36992}, {41024, 41036}, {41058, 41068}, {41060, 41070}
X(41016) = reflection of X(i) in X(j) for these (i, j): (396, 7684), (5318, 5478), (6770, 11542), (31693, 381), (41035, 1080)
X(41016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 381, 41017), (3091, 11290, 5), (41026, 41028, 2), (41030, 41032, 2)

X(41017) = CENTER OF 2nd ALTINTAS-ISODYNAMIC EQUILATERAL TRIANGLE

Barycentrics    -2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S+sqrt(3)*(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(41017) = 2*X(4)+X(41034)

X(41017) lies on these lines: {2, 3}, {13, 5480}, {14, 1503}, {16, 36962}, {115, 5321}, {230, 22512}, {395, 7685}, {397, 7753}, {398, 5309}, {511, 22797}, {524, 41043}, {530, 41061}, {532, 41055}, {533, 41057}, {538, 41063}, {542, 22574}, {543, 41047}, {754, 41065}, {1499, 9200}, {2794, 6108}, {3564, 20426}, {5318, 5475}, {5335, 15484}, {5478, 22682}, {5869, 40694}, {5983, 10723}, {6772, 9749}, {6773, 11543}, {7694, 10653}, {11645, 41044}, {16268, 41021}, {16634, 41038}, {20425, 21850}, {22510, 36970}, {22692, 31709}, {22846, 36961}, {25188, 29012}, {32419, 41051}, {32421, 41049}, {36968, 36994}, {41059, 41069}

X(41017) = midpoint of X(i) and X(j) for these {i, j}: {4, 383}, {16, 36962}, {3543, 35932}, {36968, 36994}, {41025, 41037}, {41059, 41069}, {41061, 41071}
X(41017) = reflection of X(i) in X(j) for these (i, j): (395, 7685), (5321, 5479), (6773, 11543), (31694, 381), (41034, 383)
X(41017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 381, 41016), (3091, 11289, 5), (41027, 41029, 2), (41031, 41033, 2)

X(41018) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND BANKOFF

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

X(41018) lies on these lines: {2, 3}, {621, 9732}, {1151, 41038}, {1503, 35740}, {3390, 7684}, {5335, 12256}, {9738, 20428}, {35686, 41046}, {35730, 41020}, {35731, 41022}, {35733, 41024}, {35739, 41036}, {35741, 41042}, {35742, 41044}, {35743, 41048}, {35744, 41050}, {35745, 41052}, {35746, 41054}, {35747, 41056}, {35748, 41060}, {35755, 41062}, {35756, 41064}, {35759, 41070}, {35760, 41068}, {35761, 41058}, {36762, 41019}

X(41018) = {X(18586), X(36656)}-harmonic conjugate of X(4)

X(41019) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND LARGEST-CIRCUMSCRIBED-EQUILATERAL

Barycentrics    2*sqrt(3)*(3*a^6+2*(b^2+c^2)*a^4-(b^4-10*b^2*c^2+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2))*S+3*a^8-3*(b^2+c^2)*a^6+3*(5*b^4+6*b^2*c^2+5*c^4)*a^4-17*(b^4-c^4)*(b^2-c^2)*a^2+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2 : :

X(41019) lies on these lines: {4, 618}, {13, 11623}, {531, 1080}, {1503, 36771}, {5463, 41016}, {6115, 7710}, {6777, 41070}, {7684, 9112}, {9114, 41060}, {14538, 38412}, {21156, 37463}, {22490, 41061}, {23006, 41036}, {35751, 41026}, {36762, 41018}, {36763, 41020}, {36764, 41022}, {36766, 41024}, {36767, 41028}, {36768, 41030}, {36769, 41032}, {36770, 41034}, {36772, 41038}, {36773, 41068}, {36777, 41046}, {36778, 41048}, {36779, 41050}, {36780, 41052}, {36781, 41054}, {36782, 41056}, {36783, 41058}, {36784, 41062}, {36785, 41064}, {36788, 41066}, {36961, 41035}

X(41020) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 3rd FERMAT-DAO

Barycentrics    -2*sqrt(3)*(-a^2+b^2+c^2)*S*a^2+5*a^6-4*(b^2+c^2)*a^4+(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(41020) = 4*X(3)-3*X(5463) = 2*X(4)-3*X(13) = 5*X(4)-6*X(5478) = X(4)-3*X(6770) = 4*X(4)-3*X(36961) = 8*X(5)-9*X(22489) = 4*X(5)-3*X(41042) = 5*X(13)-4*X(5478) = 3*X(14)-4*X(11623) = 3*X(15)-2*X(41035) = 3*X(17)-2*X(41056) = 4*X(140)-3*X(5617) = 8*X(140)-9*X(21156) = 16*X(140)-15*X(36770) = 3*X(5464)-2*X(14981) = 2*X(5478)-5*X(6770) = 8*X(5478)-5*X(36961) = 4*X(6770)-X(36961) = 3*X(22489)-2*X(41042) = 4*X(30714)-3*X(37752) = X(35752)-4*X(36383)

X(41020) lies on these lines: {3, 67}, {4, 13}, {5, 22489}, {14, 11623}, {15, 1503}, {16, 6776}, {17, 5868}, {18, 98}, {20, 530}, {30, 22495}, {62, 8550}, {115, 5339}, {140, 5617}, {147, 32552}, {376, 35751}, {381, 20415}, {382, 16001}, {397, 9112}, {398, 5305}, {511, 23000}, {531, 38664}, {549, 36363}, {550, 5473}, {575, 37332}, {616, 3522}, {618, 3523}, {628, 5979}, {633, 5980}, {1080, 16962}, {1656, 6771}, {1657, 5865}, {2784, 5699}, {2794, 5869}, {3091, 5459}, {3105, 11257}, {3438, 10619}, {3448, 14170}, {3515, 9916}, {3524, 36344}, {3564, 14538}, {3627, 25154}, {3830, 32907}, {3851, 22796}, {3858, 20252}, {4857, 10078}, {5056, 6669}, {5073, 13103}, {5189, 11127}, {5237, 22998}, {5270, 10062}, {5318, 36992}, {5334, 41037}, {5340, 5472}, {5470, 41060}, {5474, 10992}, {5611, 23001}, {5882, 7975}, {6033, 25559}, {6108, 40694}, {6268, 14813}, {6270, 14814}, {6636, 40711}, {6773, 16242}, {6800, 40710}, {7684, 16960}, {9114, 23235}, {9115, 36843}, {9862, 22532}, {10304, 36769}, {10645, 39874}, {10657, 11456}, {11131, 14683}, {11177, 34508}, {11306, 38745}, {11485, 36990}, {11522, 11705}, {11542, 41036}, {11632, 16002}, {12042, 22507}, {12820, 21845}, {14880, 37824}, {15683, 35749}, {15692, 36768}, {16267, 41016}, {16529, 41070}, {16772, 36764}, {16808, 41038}, {19106, 22900}, {19107, 41039}, {22490, 38740}, {22509, 37825}, {22510, 41044}, {22571, 41046}, {22602, 41048}, {22631, 41050}, {22688, 41052}, {22846, 41054}, {22855, 36993}, {22997, 36967}, {25151, 41058}, {25157, 41062}, {25158, 41064}, {25217, 41068}, {32553, 34473}, {33388, 36766}, {33517, 36994}, {35730, 41018}, {36763, 41019}

X(41020) = midpoint of X(i) and X(j) for these {i, j}: {376, 36318}, {15683, 35749}
X(41020) = reflection of X(i) in X(j) for these (i, j): (13, 6770), (147, 32552), (382, 16001), (3830, 32907), (6033, 25559), (6777, 98), (22507, 12042), (35751, 376), (36363, 549), (36961, 13), (36992, 5318), (36994, 33517), (41021, 10991)
X(41020) = intersection, other than A,B,C, of conics {{A, B, C, X(13), X(34897)}} and {{A, B, C, X(67), X(8737)}}
X(41020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3448, 14170, 40709), (5617, 21156, 36770), (8550, 41034, 62), (16960, 41024, 7684)

X(41021) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 4th FERMAT-DAO

Barycentrics    2*sqrt(3)*(-a^2+b^2+c^2)*S*a^2+5*a^6-4*(b^2+c^2)*a^4+(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(41021) = 4*X(3)-3*X(5464) = 2*X(4)-3*X(14) = 5*X(4)-6*X(5479) = X(4)-3*X(6773) = 4*X(4)-3*X(36962) = 8*X(5)-9*X(22490) = 4*X(5)-3*X(41043) = 3*X(13)-4*X(11623) = 5*X(14)-4*X(5479) = 3*X(16)-2*X(41034) = 3*X(18)-2*X(41057) = 4*X(140)-3*X(5613) = 8*X(140)-9*X(21157) = 3*X(5463)-2*X(14981) = 2*X(5479)-5*X(6773) = 8*X(5479)-5*X(36962) = 4*X(6773)-X(36962) = 3*X(22490)-2*X(41043) = 4*X(30714)-3*X(37753) = X(36330)-4*X(36382)

X(41021) lies on these lines: {3, 67}, {4, 14}, {5, 22490}, {13, 11623}, {15, 6776}, {16, 1503}, {17, 98}, {18, 5869}, {20, 531}, {30, 22496}, {61, 8550}, {115, 5340}, {140, 5613}, {147, 32553}, {376, 36320}, {381, 20416}, {382, 16002}, {383, 16963}, {397, 5305}, {398, 9113}, {511, 23009}, {530, 38664}, {549, 36362}, {550, 5474}, {575, 37333}, {617, 3522}, {619, 3523}, {627, 5978}, {634, 5981}, {1656, 6774}, {1657, 5864}, {2784, 5700}, {2794, 5868}, {3091, 5460}, {3104, 11257}, {3439, 10619}, {3448, 14169}, {3515, 9915}, {3524, 36319}, {3564, 14539}, {3627, 25164}, {3830, 32909}, {3851, 22797}, {3858, 20253}, {4857, 10077}, {5056, 6670}, {5073, 13102}, {5189, 11126}, {5238, 22997}, {5270, 10061}, {5321, 36994}, {5335, 41036}, {5339, 5471}, {5469, 41061}, {5473, 10992}, {5615, 23010}, {5882, 7974}, {6033, 25560}, {6109, 40693}, {6269, 14814}, {6271, 14813}, {6636, 40712}, {6770, 16241}, {6800, 40709}, {7685, 16961}, {9116, 23235}, {9117, 36836}, {9862, 22531}, {10646, 39874}, {10658, 11456}, {11130, 14683}, {11177, 34509}, {11305, 38745}, {11486, 36990}, {11522, 11706}, {11543, 41037}, {11632, 16001}, {12042, 22509}, {12256, 35739}, {12821, 21846}, {14880, 37825}, {15683, 36327}, {16268, 41017}, {16530, 41071}, {16809, 41039}, {19106, 41038}, {19107, 22856}, {22489, 38740}, {22507, 37824}, {22511, 41045}, {22572, 41047}, {22604, 41051}, {22633, 41049}, {22690, 41053}, {22891, 41055}, {22901, 36995}, {22998, 36968}, {25161, 41059}, {25167, 41063}, {25168, 41065}, {25214, 41069}, {32552, 34473}, {33518, 36992}

X(41021) = midpoint of X(i) and X(j) for these {i, j}: {376, 36320}, {15683, 36327}
X(41021) = reflection of X(i) in X(j) for these (i, j): (14, 6773), (147, 32553), (382, 16002), (3830, 32909), (6033, 25560), (6778, 98), (22509, 12042), (36329, 376), (36362, 549), (36962, 14), (36992, 33518), (36994, 5321), (41020, 10991)
X(41021) = intersection, other than A,B,C, of conics {{A, B, C, X(14), X(34897)}} and {{A, B, C, X(67), X(8738)}}
X(41021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3448, 14169, 40710), (8550, 41035, 61), (16961, 41025, 7685)

X(41022) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 7th FERMAT-DAO

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

X(41022) lies on these lines: {2, 9749}, {3, 618}, {4, 13}, {5, 6669}, {6, 22513}, {14, 98}, {15, 1080}, {16, 6773}, {18, 33420}, {20, 616}, {23, 11092}, {30, 511}, {40, 2945}, {55, 12942}, {56, 12952}, {99, 5474}, {110, 36186}, {114, 619}, {115, 5321}, {125, 32461}, {147, 617}, {298, 14538}, {376, 5463}, {381, 5459}, {382, 5869}, {395, 10991}, {396, 7684}, {397, 7765}, {398, 5007}, {473, 1629}, {485, 13917}, {486, 13982}, {546, 20252}, {621, 5980}, {624, 5613}, {631, 36770}, {671, 41047}, {944, 7975}, {946, 11705}, {1352, 3643}, {1478, 10062}, {1479, 10078}, {1495, 32460}, {1513, 6109}, {1525, 32111}, {1546, 10657}, {1587, 19074}, {1588, 19073}, {2043, 6302}, {2044, 6306}, {2456, 12213}, {3023, 18975}, {3027, 13075}, {3104, 9873}, {3284, 40665}, {3398, 37332}, {3448, 11078}, {3534, 36363}, {3545, 22489}, {3575, 12142}, {3627, 16001}, {3830, 25154}, {5085, 11298}, {5238, 22532}, {5304, 5334}, {5318, 5472}, {5335, 9112}, {5460, 6055}, {5464, 6054}, {5469, 14651}, {5470, 14639}, {5691, 9901}, {5870, 6268}, {5871, 6270}, {5978, 5999}, {6036, 6670}, {6226, 6269}, {6227, 6271}, {6284, 13076}, {6298, 8721}, {6582, 34508}, {6672, 6774}, {6772, 36990}, {6776, 10653}, {6778, 10722}, {6779, 36995}, {6998, 37834}, {7354, 18974}, {7464, 13859}, {7710, 9762}, {7753, 31702}, {7768, 11129}, {7970, 7974}, {8174, 16806}, {8259, 10611}, {8703, 36768}, {8980, 13916}, {9116, 12117}, {9117, 41070}, {9760, 22664}, {9834, 12472}, {9835, 12473}, {9838, 12990}, {9839, 12991}, {9860, 9900}, {9861, 9915}, {9863, 9989}, {9864, 12780}, {10053, 10061}, {10069, 10077}, {10210, 11127}, {10516, 11297}, {10645, 36766}, {10676, 39849}, {11001, 35751}, {11177, 22574}, {11303, 12203}, {11488, 36771}, {11500, 12337}, {11600, 32627}, {11632, 25164}, {11646, 22512}, {11706, 11710}, {12022, 31898}, {12110, 12205}, {12113, 12793}, {12114, 12922}, {12115, 13105}, {12116, 13107}, {12131, 12141}, {12154, 41043}, {12176, 12204}, {12177, 12214}, {12178, 12336}, {12179, 12470}, {12180, 12471}, {12181, 12792}, {12182, 12921}, {12183, 12931}, {12184, 12941}, {12185, 12951}, {12186, 12988}, {12187, 12989}, {12188, 13102}, {12189, 13104}, {12190, 13106}, {12383, 37752}, {13349, 25560}, {13748, 18586}, {13749, 18587}, {13876, 36656}, {13929, 36655}, {13967, 13981}, {14145, 22843}, {14830, 33389}, {15640, 35749}, {15682, 35752}, {15687, 32907}, {16241, 37463}, {16267, 41036}, {16626, 33421}, {16644, 41040}, {16772, 22892}, {16962, 41024}, {19055, 19075}, {19056, 19076}, {19708, 36767}, {20377, 22795}, {20428, 30485}, {20429, 22509}, {21157, 34473}, {22504, 22774}, {22505, 22797}, {22507, 38741}, {22573, 41046}, {22691, 41052}, {22831, 22846}, {22847, 41054}, {22998, 36968}, {23006, 39874}, {23721, 34296}, {23722, 38932}, {25178, 41058}, {25183, 41062}, {25184, 41064}, {25220, 41068}, {32553, 38749}, {33529, 34008}, {35020, 35021}, {35731, 41018}, {35753, 35820}, {35754, 35821}, {35824, 35850}, {35825, 35851}, {36764, 41019}, {36963, 36969}, {36967, 36993}, {37464, 37835}, {39838, 41061}

X(41022) = X(523)-Hirst inverse of-X(41023)

X(41023) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 8th FERMAT-DAO

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

X(41023) lies on these lines: {2, 9750}, {3, 619}, {4, 14}, {5, 6670}, {6, 22512}, {13, 98}, {15, 6770}, {16, 383}, {17, 33421}, {20, 617}, {23, 11078}, {30, 511}, {40, 2946}, {55, 12941}, {56, 12951}, {99, 5473}, {110, 36185}, {114, 618}, {115, 5318}, {125, 32460}, {147, 616}, {299, 14539}, {376, 5464}, {381, 5460}, {382, 5868}, {395, 7685}, {396, 10991}, {397, 5007}, {398, 7765}, {472, 1629}, {485, 13916}, {486, 13981}, {546, 20253}, {622, 5981}, {623, 5617}, {671, 41046}, {944, 7974}, {946, 11706}, {1352, 3642}, {1478, 10061}, {1479, 10077}, {1495, 32461}, {1513, 6108}, {1524, 32111}, {1545, 10658}, {1587, 19076}, {1588, 19075}, {2043, 6307}, {2044, 6303}, {2456, 12214}, {3023, 18974}, {3027, 13076}, {3105, 9873}, {3284, 40666}, {3398, 37333}, {3448, 11092}, {3534, 36362}, {3545, 22490}, {3575, 12141}, {3627, 16002}, {3830, 25164}, {5085, 11297}, {5237, 22531}, {5304, 5335}, {5321, 5471}, {5334, 9113}, {5459, 6055}, {5463, 6054}, {5469, 14639}, {5470, 14651}, {5691, 9900}, {5870, 6269}, {5871, 6271}, {5979, 5999}, {6036, 6669}, {6226, 6268}, {6227, 6270}, {6284, 13075}, {6295, 34509}, {6299, 8721}, {6671, 6771}, {6775, 36990}, {6776, 10654}, {6777, 10722}, {6780, 36993}, {6998, 37831}, {7354, 18975}, {7464, 13858}, {7710, 9760}, {7753, 31701}, {7768, 11128}, {7970, 7975}, {8175, 16807}, {8260, 10612}, {8980, 13917}, {9114, 12117}, {9115, 41071}, {9762, 22664}, {9834, 12470}, {9835, 12471}, {9838, 12988}, {9839, 12989}, {9860, 9901}, {9861, 9916}, {9863, 9988}, {9864, 12781}, {10053, 10062}, {10069, 10078}, {10516, 11298}, {10617, 34602}, {10675, 39849}, {11001, 36319}, {11177, 22573}, {11304, 12203}, {11500, 12336}, {11601, 32628}, {11632, 25154}, {11646, 22513}, {11705, 11710}, {12022, 31899}, {12110, 12204}, {12113, 12792}, {12114, 12921}, {12115, 13104}, {12116, 13106}, {12131, 12142}, {12155, 41042}, {12176, 12205}, {12177, 12213}, {12178, 12337}, {12179, 12472}, {12180, 12473}, {12181, 12793}, {12182, 12922}, {12183, 12932}, {12184, 12942}, {12185, 12952}, {12186, 12990}, {12187, 12991}, {12188, 13103}, {12189, 13105}, {12190, 13107}, {12383, 37753}, {13350, 25559}, {13748, 18587}, {13749, 18586}, {13875, 36656}, {13928, 36655}, {13967, 13982}, {14144, 22890}, {14830, 33388}, {15640, 36327}, {15682, 36320}, {15687, 32909}, {16242, 37464}, {16268, 41037}, {16627, 33420}, {16645, 41041}, {16773, 22848}, {16963, 41025}, {19055, 19073}, {19056, 19074}, {20378, 22794}, {20428, 22507}, {20429, 30486}, {21156, 34473}, {22504, 22773}, {22505, 22796}, {22509, 38741}, {22574, 41047}, {22692, 41053}, {22832, 22891}, {22893, 41055}, {22997, 36967}, {23013, 39874}, {23721, 38931}, {23722, 34295}, {25173, 41059}, {25187, 41063}, {25188, 41065}, {25219, 41069}, {32552, 38749}, {33530, 34009}, {34552, 35748}, {35019, 35021}, {35753, 35824}, {35754, 35825}, {35820, 35850}, {35821, 35851}, {36964, 36970}, {36968, 36995}, {37463, 37832}, {39838, 41060}

X(41023) = X(523)-Hirst inverse of-X(41022)
X(41023) = X(3)-vertex conjugate of-X(23871)

X(41024) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 15th FERMAT-DAO

Barycentrics    2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+5*a^6+2*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(41024) = 4*X(4)-X(19106) = X(13)-4*X(41016) = 2*X(1080)+X(36961) = 4*X(1080)-X(36967) = X(6777)-4*X(41044) = 8*X(7684)-5*X(16960) = 4*X(7684)-X(41020) = 5*X(16960)-2*X(41020) = X(22493)-4*X(41042) = X(22577)-4*X(41046) = X(22607)-4*X(41048) = X(22636)-4*X(41050) = X(22695)-4*X(41052) = X(22849)-4*X(41054) = X(22894)-4*X(41056) = X(25166)-4*X(41060) = X(25182)-4*X(41058) = X(25199)-4*X(41062) = X(25200)-4*X(41064) = 2*X(36961)+X(36967)

X(41024) lies on these lines: {4, 16}, {13, 1503}, {30, 36765}, {262, 12817}, {381, 5085}, {382, 22795}, {511, 22493}, {1080, 9749}, {5469, 14639}, {6777, 10753}, {7684, 16960}, {16808, 36990}, {16962, 41022}, {16964, 41038}, {16966, 41040}, {22577, 41046}, {22607, 41048}, {22636, 41050}, {22681, 22714}, {22695, 41052}, {22796, 29317}, {22849, 41054}, {22894, 41056}, {25166, 41060}, {25182, 41058}, {25199, 41062}, {25200, 41064}, {25228, 41068}, {25236, 41070}, {33416, 41034}, {35733, 41018}, {36766, 41019}, {36992, 41035}

X(41024) = reflection of X(i) in X(j) for these (i, j): (13, 41036), (41036, 41016)
X(41024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1080, 36961, 36967), (7684, 41020, 16960)

X(41025) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 16th FERMAT-DAO

Barycentrics    -2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+5*a^6+2*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(41025) = 4*X(4)-X(19107) = X(14)-4*X(41017) = 2*X(383)+X(36962) = 4*X(383)-X(36968) = X(6778)-4*X(41045) = 8*X(7685)-5*X(16961) = 4*X(7685)-X(41021) = 5*X(16961)-2*X(41021) = X(22494)-4*X(41043) = X(22578)-4*X(41047) = X(22608)-4*X(41051) = X(22637)-4*X(41049) = X(22696)-4*X(41053) = X(22850)-4*X(41057) = X(22895)-4*X(41055) = X(25156)-4*X(41061) = X(25177)-4*X(41059) = X(25203)-4*X(41063) = X(25204)-4*X(41065) = 2*X(36962)+X(36968)

X(41025) lies on these lines: {4, 15}, {14, 1503}, {262, 12816}, {381, 5085}, {382, 22794}, {383, 9750}, {511, 22494}, {5470, 14639}, {6778, 10753}, {7685, 16961}, {16809, 36990}, {16963, 41023}, {16965, 41039}, {16967, 41041}, {22578, 41047}, {22608, 41051}, {22637, 41049}, {22681, 22715}, {22696, 41053}, {22797, 29317}, {22850, 41057}, {22895, 41055}, {25156, 41061}, {25177, 41059}, {25203, 41063}, {25204, 41065}, {25227, 41069}, {25235, 41071}, {33417, 41035}, {36994, 41034}

X(41025) = reflection of X(i) in X(j) for these (i, j): (14, 41037), (41037, 41017)
X(41025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (383, 36962, 36968), (7685, 41021, 16961)

X(41026) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 3rd INNER-FERMAT-DAO-NHI

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

X(41026) lies on these lines: {2, 3}, {5859, 41042}, {5868, 16267}, {33624, 41054}, {35692, 41046}, {35751, 41019}, {36329, 41060}, {36330, 41070}, {36362, 41044}, {36364, 41052}, {36366, 41056}, {36367, 41058}, {36370, 41050}, {36371, 41048}, {36373, 41062}, {36375, 41064}, {36387, 41068}

X(41026) = {X(41028), X(41040)}-harmonic conjugate of X(2)
X(41026) = {X(4),X(5066)}-harmonic conjugate of X(41027)

X(41027) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 3rd OUTER-FERMAT-DAO-NHI

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

X(41027) lies on these lines: {2, 3}, {5858, 41043}, {5869, 16268}, {33622, 41055}, {35696, 41047}, {35751, 41061}, {35752, 41071}, {36363, 41045}, {36365, 41053}, {36368, 41057}, {36369, 41059}, {36372, 41049}, {36374, 41051}, {36378, 41063}, {36379, 41065}, {36389, 41069}

X(41027) = {X(4),X(5066)}-harmonic conjugate of X(41026)
X(41027) = {X(41029), X(41041)}-harmonic conjugate of X(2)

X(41028) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 4th INNER-FERMAT-DAO-NHI

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

X(41028) lies on these lines: {2, 3}, {5469, 36962}, {5478, 36990}, {5858, 41042}, {12816, 41039}, {13102, 22505}, {19107, 36771}, {33627, 41054}, {35693, 41046}, {36329, 41070}, {36330, 41060}, {36367, 41068}, {36382, 41044}, {36384, 41052}, {36386, 41056}, {36387, 41058}, {36390, 41050}, {36391, 41048}, {36393, 41062}, {36395, 41064}, {36523, 41061}, {36767, 41019}

X(41028) = midpoint of X(3543) and X(37172)
X(41028) = reflection of X(11305) in X(381)
X(41028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3845, 41029), (4, 41030, 2), (3839, 11304, 381)

X(41029) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 4th OUTER-FERMAT-DAO-NHI

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

X(41029) lies on these lines: {2, 3}, {5470, 36961}, {5479, 36990}, {5859, 41043}, {12817, 41038}, {13103, 22505}, {33626, 41055}, {35697, 41047}, {35751, 41071}, {35752, 41061}, {36369, 41069}, {36383, 41045}, {36385, 41053}, {36388, 41057}, {36389, 41059}, {36392, 41049}, {36394, 41051}, {36398, 41063}, {36399, 41065}, {36523, 41060}

X(41029) = midpoint of X(3543) and X(37173)
X(41029) = reflection of X(11306) in X(381)
X(41029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3845, 41028), (4, 41031, 2), (3839, 11303, 381)

X(41030) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 1st OUTER-FERMAT-DAO-NHI

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

X(41030) lies on these lines: {2, 3}, {5862, 41042}, {33622, 41056}, {35694, 41046}, {36320, 41044}, {36321, 41068}, {36323, 41052}, {36324, 41054}, {36325, 41058}, {36327, 41060}, {36331, 41070}, {36334, 41050}, {36335, 41048}, {36338, 41062}, {36339, 41064}, {36768, 41019}

X(41030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 41028, 4), (4, 41032, 2), (41016, 41028, 2)

X(41031) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 1st INNER-FERMAT-DAO-NHI

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

X(41031) lies on these lines: {2, 3}, {5863, 41043}, {33624, 41057}, {35690, 41047}, {35749, 41061}, {35750, 41071}, {36318, 41045}, {36322, 41053}, {36326, 41055}, {36328, 41059}, {36332, 41049}, {36333, 41051}, {36336, 41063}, {36337, 41065}, {36354, 41069}

X(41031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 41029, 4), (4, 41033, 2), (41017, 41029, 2)

X(41032) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 2nd OUTER-FERMAT-DAO-NHI

Barycentrics    (5*a^4+8*(b^2+c^2)*a^2-13*(b^2-c^2)^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

X(41032) lies on these lines: {2, 3}, {5863, 41042}, {33626, 41056}, {35695, 41046}, {36319, 41044}, {36321, 41058}, {36325, 41068}, {36327, 41070}, {36331, 41060}, {36345, 41052}, {36346, 41054}, {36348, 41050}, {36349, 41048}, {36350, 41062}, {36351, 41064}, {36769, 41019}

X(41032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 41030, 4), (41016, 41026, 2)

X(41033) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 2nd INNER-FERMAT-DAO-NHI

Barycentrics    -(5*a^4+8*(b^2+c^2)*a^2-13*(b^2-c^2)^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

X(41033) lies on these lines: {2, 3}, {5862, 41043}, {33627, 41057}, {35691, 41047}, {35749, 41071}, {35750, 41061}, {36328, 41069}, {36344, 41045}, {36347, 41053}, {36352, 41055}, {36354, 41059}, {36356, 41049}, {36357, 41051}, {36358, 41063}, {36359, 41065}

X(41033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 41031, 4), (41017, 41027, 2)

X(41034) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 1st HALF-DIAMONDS-CENTRAL

Barycentrics    -2*sqrt(3)*(-a^2+b^2+c^2)*S*a^2+(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(41034) = 2*X(3)-3*X(35303) = X(4)-3*X(383) = 2*X(4)-3*X(41017) = 4*X(5)-3*X(31694) = X(20)-3*X(35932)

X(41034) lies on these lines: {2, 3}, {15, 5480}, {16, 1503}, {18, 41054}, {32, 398}, {39, 397}, {62, 8550}, {141, 14538}, {187, 5321}, {395, 10991}, {530, 14981}, {574, 5318}, {619, 41070}, {622, 6390}, {624, 18860}, {629, 41056}, {633, 7767}, {634, 3933}, {1384, 5334}, {2482, 41061}, {2794, 6114}, {3053, 5339}, {3564, 5615}, {3815, 9749}, {3818, 9736}, {3867, 11515}, {5013, 5340}, {5024, 5335}, {5206, 5349}, {5350, 37512}, {5460, 41060}, {5611, 21850}, {5869, 8721}, {6108, 11623}, {6774, 41044}, {6776, 11486}, {7684, 23302}, {9886, 41043}, {11481, 36990}, {11485, 14853}, {11645, 21402}, {13349, 29012}, {13350, 19130}, {14170, 14389}, {14539, 29181}, {15513, 22831}, {16809, 36992}, {16966, 41036}, {18581, 41038}, {18582, 40922}, {19107, 41037}, {22570, 36776}, {23005, 36771}, {24206, 36755}, {29317, 36756}, {32269, 40710}, {33416, 41024}, {33444, 41050}, {33445, 41048}, {33474, 41042}, {33476, 41046}, {33478, 41052}, {33481, 41058}, {33482, 41062}, {33484, 41064}, {33491, 41068}, {36770, 41019}, {36961, 37835}, {36994, 41025}, {41045, 41055}

X(41034) = reflection of X(i) in X(j) for these (i, j): (5321, 7685), (41017, 383)
X(41034) = orthocentroidal circle-inverse of-X(41040)
X(41034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 4, 41040), (3, 4, 41035), (3, 37332, 5), (4, 37464, 5), (20, 37173, 3), (427, 3132, 465), (37334, 37464, 140)

X(41035) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 2nd HALF-DIAMONDS-CENTRAL

Barycentrics    2*sqrt(3)*(-a^2+b^2+c^2)*S*a^2+(a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(41035) = 2*X(3)-3*X(35304) = X(4)-3*X(1080) = 2*X(4)-3*X(41016) = 4*X(5)-3*X(31693) = X(20)-3*X(35931)

X(41035) lies on these lines: {2, 3}, {15, 1503}, {16, 5480}, {17, 41055}, {32, 397}, {39, 398}, {61, 8550}, {141, 14539}, {187, 5318}, {396, 10991}, {531, 14981}, {574, 5321}, {618, 41071}, {621, 6390}, {623, 18860}, {630, 41057}, {633, 3933}, {634, 7767}, {1384, 5335}, {2482, 41060}, {2794, 6115}, {3053, 5340}, {3564, 5611}, {3815, 9750}, {3818, 9735}, {3867, 11516}, {5013, 5339}, {5024, 5334}, {5206, 5350}, {5349, 37512}, {5459, 41061}, {5615, 21850}, {5868, 8721}, {6109, 11623}, {6771, 41045}, {6776, 11485}, {7685, 23303}, {9885, 41042}, {11480, 36990}, {11486, 14853}, {11645, 21401}, {13349, 19130}, {13350, 29012}, {14169, 14389}, {14538, 29181}, {15513, 22832}, {16808, 36994}, {16967, 41037}, {18581, 40921}, {18582, 41039}, {19106, 41036}, {24206, 36756}, {29317, 36755}, {32269, 40709}, {33417, 41025}, {33446, 41049}, {33447, 41051}, {33475, 41043}, {33477, 41047}, {33479, 41053}, {33480, 41059}, {33483, 41063}, {33485, 41065}, {33490, 41069}, {36961, 41019}, {36962, 37832}, {36992, 41024}, {41044, 41054}

X(41035) = reflection of X(i) in X(j) for these (i, j): (5318, 7684), (41016, 1080)
X(41035) = orthocentroidal circle-inverse of-X(41041)
X(41035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 4, 41041), (3, 4, 41034), (3, 37333, 5), (4, 37463, 5), (20, 37172, 3), (427, 3131, 466), (37334, 37463, 140)

X(41036) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 1st ISODYNAMIC-DAO

Barycentrics    2*sqrt(3)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S+a^6+4*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(41036) = 2*X(4)+X(15) = X(4)+2*X(7684) = 4*X(4)-X(36992) = 5*X(4)+X(36993) = 4*X(5)-X(14538) = 8*X(5)-5*X(40334) = X(13)+2*X(41016) = X(15)-4*X(7684) = 2*X(15)+X(36992) = 5*X(15)-2*X(36993) = X(20)-4*X(6671) = 2*X(187)+X(36994) = X(382)+2*X(13350) = 2*X(396)+X(36961) = 4*X(546)-X(20428) = X(621)-7*X(3832) = 8*X(7684)+X(36992) = 10*X(7684)-X(36993) = 2*X(14538)-5*X(40334) = 5*X(36992)+4*X(36993)

X(41036) lies on these lines: {2, 33379}, {4, 15}, {5, 14538}, {13, 1503}, {14, 14853}, {16, 41040}, {20, 6671}, {30, 21156}, {61, 41038}, {187, 36994}, {381, 511}, {382, 13350}, {396, 36961}, {531, 3839}, {542, 22571}, {546, 16627}, {621, 3832}, {623, 3091}, {1080, 5478}, {1513, 23005}, {1656, 36755}, {2794, 22510}, {3146, 33413}, {3564, 22495}, {3843, 5611}, {5076, 21401}, {5102, 22496}, {5335, 41021}, {5464, 41046}, {5470, 14651}, {5480, 16809}, {5691, 11707}, {5965, 20425}, {6109, 36962}, {6778, 41044}, {7612, 12816}, {10646, 37463}, {11542, 41020}, {16267, 41022}, {16631, 36759}, {16966, 41034}, {18860, 40335}, {19106, 41035}, {21167, 37352}, {22511, 39663}, {22609, 41048}, {22638, 41050}, {22701, 41052}, {22855, 41054}, {22900, 41056}, {22997, 41060}, {22999, 41058}, {23000, 41062}, {23001, 41064}, {23004, 41070}, {23006, 41019}, {23007, 41068}, {25158, 29012}, {29181, 31693}, {35739, 41018}, {36761, 36763}

X(41036) = midpoint of X(13) and X(41024)
X(41036) = reflection of X(i) in X(j) for these (i, j): (22511, 39663), (39554, 38227), (41024, 41016)
X(41036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 15, 36992), (4, 7684, 15), (5, 14538, 40334), (1080, 5478, 36969)

X(41037) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 2nd ISODYNAMIC-DAO

Barycentrics    -2*sqrt(3)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S+a^6+4*(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2) : :
X(41037) = 2*X(4)+X(16) = X(4)+2*X(7685) = 4*X(4)-X(36994) = 5*X(4)+X(36995) = 4*X(5)-X(14539) = 8*X(5)-5*X(40335) = X(14)+2*X(41017) = X(16)-4*X(7685) = 2*X(16)+X(36994) = 5*X(16)-2*X(36995) = X(20)-4*X(6672) = 2*X(187)+X(36992) = X(382)+2*X(13349) = X(383)+2*X(5479) = 2*X(383)+X(36970) = 4*X(5479)-X(36970) = 8*X(7685)+X(36994) = 10*X(7685)-X(36995) = 2*X(14539)-5*X(40335) = 5*X(36994)+4*X(36995)

X(41037) lies on these lines: {2, 33378}, {4, 16}, {5, 14539}, {13, 14853}, {14, 1503}, {15, 41041}, {20, 6672}, {30, 21157}, {62, 41039}, {187, 36992}, {381, 511}, {382, 13349}, {383, 5479}, {395, 36962}, {530, 3839}, {542, 22572}, {546, 16626}, {622, 3832}, {624, 3091}, {1513, 23004}, {1656, 36756}, {2794, 22511}, {3146, 33412}, {3564, 22496}, {3843, 5615}, {5076, 21402}, {5102, 22495}, {5334, 41020}, {5463, 41047}, {5469, 14651}, {5480, 16808}, {5691, 11708}, {5965, 20426}, {6108, 36961}, {6777, 41045}, {7612, 12817}, {10645, 37464}, {11543, 41021}, {16268, 41023}, {16630, 36760}, {16967, 41035}, {18860, 40334}, {19107, 41034}, {21167, 37351}, {22510, 39663}, {22610, 41051}, {22639, 41049}, {22702, 41053}, {22856, 41057}, {22901, 41055}, {22998, 41061}, {23005, 41071}, {23008, 41059}, {23009, 41063}, {23010, 41065}, {23014, 41069}, {25168, 29012}, {29181, 31694}

X(41037) = midpoint of X(14) and X(41025)
X(41037) = reflection of X(i) in X(j) for these (i, j): (22510, 39663), (39555, 38227), (41025, 41017)
X(41037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 16, 36994), (4, 7685, 16), (5, 14539, 40335), (383, 5479, 36970)

X(41038) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND 1st LEMOINE-DAO

Barycentrics    2*sqrt(3)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S+2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2) : :
X(41038) = 2*X(4)+X(5868) = 4*X(4)-X(5869) = 2*X(5868)+X(5869) = X(36761)-3*X(36765)

X(41038) lies on these lines: {3, 623}, {4, 6}, {14, 2794}, {15, 41040}, {16, 36992}, {20, 302}, {30, 9761}, {61, 41036}, {115, 16942}, {154, 473}, {381, 5459}, {382, 5615}, {383, 9756}, {472, 1853}, {511, 25191}, {542, 22575}, {621, 1350}, {622, 15069}, {1151, 41018}, {3106, 22693}, {3543, 37785}, {3830, 25164}, {5085, 11303}, {5471, 39838}, {7684, 11485}, {7694, 10654}, {9749, 22796}, {9750, 22665}, {10516, 11304}, {11295, 41042}, {11301, 36761}, {11480, 36993}, {12257, 35740}, {12817, 41029}, {13102, 22728}, {16630, 36760}, {16634, 41017}, {16808, 41020}, {16809, 41041}, {16964, 41024}, {18440, 20429}, {18581, 41034}, {18586, 22634}, {18587, 22605}, {19106, 41021}, {19781, 38227}, {22512, 41044}, {22579, 41046}, {22611, 41048}, {22640, 41050}, {22707, 41052}, {22861, 41054}, {22906, 41056}, {23013, 41070}, {23017, 41058}, {23018, 41062}, {23019, 41064}, {23022, 41068}, {25192, 29012}, {36772, 41019}

X(41038) = midpoint of X(5868) and X(41039)
X(41038) = reflection of X(i) in X(j) for these (i, j): (5869, 41039), (9749, 22796), (41039, 4)
X(41038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 5334, 5480), (4, 5868, 5869), (4, 6776, 5318), (13748, 13749, 5868), (36993, 37463, 11480)

X(41039) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND 2nd LEMOINE-DAO

Barycentrics    -2*sqrt(3)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S+2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2) : :
X(41039) = 4*X(4)-X(5868) = 2*X(4)+X(5869) = X(5868)+2*X(5869)

X(41039) lies on these lines: {3, 624}, {4, 6}, {13, 2794}, {15, 36994}, {16, 41041}, {20, 303}, {30, 9763}, {62, 41037}, {115, 16943}, {154, 472}, {381, 5460}, {382, 5611}, {473, 1853}, {511, 25195}, {542, 22576}, {621, 15069}, {622, 1350}, {1080, 9756}, {3107, 22694}, {3543, 37786}, {3830, 25154}, {5085, 11304}, {5472, 39838}, {7685, 11486}, {7694, 10653}, {9749, 22666}, {9750, 22797}, {10516, 11303}, {11296, 41043}, {11481, 36995}, {12816, 41028}, {13103, 22728}, {16631, 36759}, {16635, 41016}, {16808, 41040}, {16809, 41021}, {16965, 41025}, {18440, 20428}, {18582, 41035}, {18586, 22606}, {18587, 22635}, {19107, 41020}, {19780, 38227}, {21736, 35740}, {22513, 41045}, {22580, 41047}, {22612, 41051}, {22641, 41049}, {22708, 41053}, {22862, 41057}, {22907, 41055}, {23006, 41071}, {23023, 41059}, {23024, 41063}, {23025, 41065}, {23028, 41069}, {25196, 29012}

X(41039) = midpoint of X(5869) and X(41038)
X(41039) = reflection of X(i) in X(j) for these (i, j): (5868, 41038), (9750, 22797), (41038, 4)
X(41039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 5335, 5480), (4, 5869, 5868), (4, 6776, 5321), (13748, 13749, 5869), (36995, 37464, 11481)

X(41040) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC AND OUTER-NAPOLEON

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

X(41040) lies on these lines: {2, 3}, {6, 7684}, {13, 11623}, {14, 41070}, {15, 41038}, {16, 41036}, {17, 5868}, {115, 9750}, {397, 3767}, {398, 2548}, {618, 33411}, {623, 1350}, {624, 10516}, {625, 14539}, {628, 41054}, {633, 7776}, {1503, 18582}, {3105, 22693}, {5321, 31415}, {5340, 13881}, {5460, 38072}, {5461, 41061}, {5464, 41060}, {5476, 20416}, {5480, 18581}, {5613, 41044}, {5865, 16626}, {6296, 41064}, {6300, 41048}, {6304, 41050}, {6581, 41062}, {6776, 11542}, {6800, 8838}, {8550, 40693}, {9762, 14981}, {9763, 38745}, {9886, 41046}, {10645, 36992}, {11477, 34508}, {11543, 14853}, {14182, 41058}, {14188, 41068}, {15069, 34509}, {16241, 36961}, {16644, 41022}, {16808, 41039}, {16943, 22832}, {16966, 41024}, {22513, 37637}, {22715, 41052}, {33461, 36776}, {33476, 41043}, {35230, 36969}, {36765, 41071}

X(41040) = orthocentroidal circle-inverse of-X(41034)
X(41040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 5, 41041), (4, 37463, 3), (37446, 37463, 1656), (41016, 41034, 4)

X(41041) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC AND INNER-NAPOLEON

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

X(41041) lies on these lines: {2, 3}, {6, 7685}, {13, 41071}, {14, 11623}, {15, 41037}, {16, 41039}, {18, 5869}, {115, 9749}, {397, 2548}, {398, 3767}, {619, 33410}, {623, 10516}, {624, 1350}, {625, 14538}, {627, 41055}, {634, 7776}, {1503, 18581}, {3104, 22694}, {5318, 31415}, {5339, 13881}, {5459, 38072}, {5461, 41060}, {5463, 41061}, {5476, 20415}, {5480, 18582}, {5617, 41045}, {5864, 16627}, {6294, 41063}, {6297, 41065}, {6301, 41051}, {6305, 41049}, {6776, 11543}, {6800, 8836}, {8550, 40694}, {9760, 14981}, {9761, 38745}, {9885, 41047}, {10646, 36994}, {11477, 34509}, {11542, 14853}, {14178, 41059}, {14186, 41069}, {15069, 34508}, {16242, 36962}, {16645, 41023}, {16809, 41038}, {16942, 22831}, {16967, 41025}, {22512, 37637}, {22714, 41053}, {33477, 41042}, {35229, 36970}

X(41041) = orthocentroidal circle-inverse of-X(41035)
X(41041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 5, 41040), (4, 37464, 3), (37446, 37464, 1656), (41017, 41035, 4)

X(41042) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO ANTI-ARTZT

Barycentrics    2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+5*a^6-2*(b^2+c^2)*a^4+(b^4+6*b^2*c^2+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : :
X(41042) = 4*X(2)-3*X(21156) = 2*X(2)-3*X(36765) = 4*X(5)-3*X(22489) = 4*X(5)-X(41020) = X(13)-4*X(22796) = 5*X(13)-4*X(32907) = 5*X(381)-2*X(32907) = 4*X(549)-5*X(36770) = 2*X(619)-3*X(23234) = 2*X(3534)-5*X(36767) = 3*X(3545)-2*X(5459) = 3*X(3545)-X(6770) = 3*X(5469)-2*X(11632) = X(5473)-4*X(5617) = X(5473)+2*X(36961) = 2*X(5617)+X(36961) = 2*X(6033)+X(6777) = 3*X(22489)-X(41020) = X(22493)+3*X(41024) = 5*X(22796)-X(32907)

The reciprocal orthologic center of these triangles is X(12155)

X(41042) lies on these lines: {2, 9749}, {4, 530}, {5, 22489}, {6, 13}, {15, 36764}, {18, 37332}, {30, 5463}, {98, 5460}, {114, 5464}, {376, 618}, {383, 10033}, {395, 22513}, {396, 36771}, {511, 22493}, {524, 41016}, {531, 1080}, {543, 41060}, {546, 22572}, {549, 36770}, {616, 3543}, {619, 23234}, {621, 5979}, {671, 5479}, {1503, 31693}, {2044, 36762}, {2482, 5474}, {2782, 22693}, {2794, 11296}, {3058, 12942}, {3534, 36767}, {3545, 5459}, {3564, 22495}, {3830, 35751}, {3839, 5478}, {3843, 16001}, {3845, 25154}, {3851, 20415}, {5055, 6771}, {5066, 36383}, {5071, 6669}, {5321, 6772}, {5434, 12952}, {5480, 22580}, {5613, 22566}, {5858, 41028}, {5859, 41026}, {5862, 41030}, {5863, 41032}, {5868, 11305}, {6055, 22490}, {6108, 18581}, {6115, 10654}, {6302, 36436}, {6306, 36454}, {6321, 22577}, {6774, 14830}, {6779, 19106}, {6782, 10653}, {7684, 37786}, {7759, 41056}, {7975, 28204}, {8724, 9114}, {9116, 23698}, {9760, 13860}, {9761, 41071}, {9763, 38745}, {9885, 41035}, {10613, 16962}, {10722, 32553}, {11001, 36768}, {11180, 22492}, {11295, 41038}, {11705, 38021}, {12155, 41023}, {12350, 13075}, {12351, 18975}, {12781, 28194}, {12922, 34697}, {12932, 34746}, {13083, 37463}, {13103, 14269}, {13859, 31861}, {15682, 36769}, {16964, 37333}, {20252, 38071}, {20423, 22496}, {22505, 22507}, {22998, 23006}, {25167, 36365}, {25560, 38744}, {33474, 41034}, {33477, 41041}, {35304, 36761}, {35741, 41018}, {36766, 36967}

X(41042) = midpoint of X(i) and X(j) for these {i, j}: {616, 3543}, {5463, 36961}, {25154, 36363}
X(41042) = reflection of X(i) in X(j) for these (i, j): (13, 381), (98, 5460), (376, 618), (381, 22796), (671, 5479), (5463, 5617), (5464, 114), (5473, 5463), (5474, 2482), (5613, 22566), (6770, 5459), (9114, 8724), (14830, 6774), (21156, 36765), (22577, 6321), (22580, 5480), (25154, 3845), (35752, 25154), (36775, 9762), (36776, 6054), (37786, 7684)
X(41042) = orthocentroidal circle-inverse of-X(41043)
X(41042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1080, 6054, 9762), (3545, 6770, 5459), (3845, 36363, 35752), (5617, 36961, 5473), (22998, 36969, 23006), (31862, 31863, 41043)

X(41043) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO ANTI-ARTZT

Barycentrics   5*a^6-2*a^4*(b^2+c^2)-4*(b^2-c^2)*(b^4-c^4)+a^2*(b^4+6*b^2*c^2+c^4)-2*sqrt(3)*(a^4-2*(b^2-c^2)^2+a^2*(b^2+c^2))*S : :
X(41043) = 4*X(2)-3*X(21157) = 4*X(5)-3*X(22490) = 4*X(5)-X(41021) = X(14)-4*X(22797) = 5*X(14)-4*X(32909) = 5*X(381)-2*X(32909) = 2*X(618)-3*X(23234) = 3*X(3545)-2*X(5460) = 3*X(3545)-X(6773) = 2*X(3830)+X(36329) = 3*X(5470)-2*X(11632) = X(5474)-4*X(5613) = X(5474)+2*X(36962) = 2*X(5613)+X(36962) = 2*X(6033)+X(6778) = 3*X(22490)-X(41021) = X(22494)+3*X(41025) = X(22496)-3*X(41037) = 5*X(22797)-X(32909) = X(36776)+2*X(41061)

The reciprocal orthologic center of these triangles is X(12154)

X(41043) lies on these lines: {2, 9750}, {4, 531}, {5, 22490}, {6, 13}, {17, 37333}, {30, 5464}, {98, 5459}, {114, 5463}, {376, 619}, {383, 530}, {396, 22512}, {511, 22494}, {524, 41017}, {543, 36776}, {546, 22571}, {617, 3543}, {618, 23234}, {622, 5978}, {671, 5478}, {1080, 10033}, {1503, 31694}, {2482, 5473}, {2782, 22694}, {2794, 11295}, {3058, 12941}, {3545, 5460}, {3564, 22496}, {3830, 36329}, {3839, 5479}, {3843, 16002}, {3845, 25164}, {3851, 20416}, {5055, 6774}, {5066, 36382}, {5071, 6670}, {5318, 6775}, {5434, 12951}, {5480, 22579}, {5617, 22566}, {5858, 41027}, {5859, 41029}, {5862, 41033}, {5863, 41031}, {5869, 11306}, {6055, 22489}, {6109, 18582}, {6114, 10653}, {6303, 36454}, {6307, 36436}, {6321, 22578}, {6771, 14830}, {6780, 19107}, {6783, 10654}, {7685, 37785}, {7759, 41057}, {7974, 28204}, {8724, 9116}, {9114, 23698}, {9761, 38745}, {9762, 13860}, {9763, 41070}, {9886, 41034}, {10614, 16963}, {10722, 32552}, {11180, 22491}, {11296, 41039}, {11706, 38021}, {12154, 41022}, {12350, 13076}, {12351, 18974}, {12780, 28194}, {12921, 34697}, {12931, 34746}, {13084, 37464}, {13102, 14269}, {13858, 31861}, {16965, 37332}, {20253, 38071}, {20423, 22495}, {22505, 22509}, {22997, 23013}, {25157, 36364}, {25559, 38744}, {33475, 41035}, {33476, 41040}

X(41043) = midpoint of X(i) and X(j) for these {i, j}: {617, 3543}, {5464, 36962}, {25164, 36362}
X(41043) = reflection of X(i) in X(j) for these (i, j): (14, 381), (98, 5459), (376, 619), (381, 22797), (671, 5478), (5463, 114), (5464, 5613), (5473, 2482), (5474, 5464), (5617, 22566), (6773, 5460), (9116, 8724), (14830, 6771), (22578, 6321), (22579, 5480), (25164, 3845), (36330, 25164), (37785, 7685)
X(41043) = orthocentroidal circle-inverse of-X(41042)
X(41043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (383, 6054, 9760), (3545, 6773, 5460), (3845, 36362, 36330), (5613, 36962, 5474), (22997, 36970, 23013), (31862, 31863, 41042)

X(41044) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 1st ANTI-BROCARD

Barycentrics    -2*(2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2))*sqrt(3)*S+2*a^8+(b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(41044) = X(6777)+3*X(41024) = X(6778)-3*X(41036) = 3*X(22510)-X(41020)

The reciprocal orthologic center of these triangles is X(5979).

X(41044) lies on these lines: {4, 14}, {30, 41060}, {115, 1503}, {381, 40672}, {383, 5460}, {530, 41046}, {531, 1080}, {542, 22573}, {619, 37463}, {1513, 6109}, {3850, 41057}, {5471, 5480}, {5613, 41040}, {6114, 13860}, {6303, 6813}, {6307, 6811}, {6670, 37464}, {6774, 41034}, {6777, 10753}, {6778, 41036}, {6783, 7684}, {7417, 30465}, {9113, 14853}, {9880, 22574}, {11645, 41017}, {14482, 31684}, {22510, 41020}, {22512, 41038}, {33518, 38383}, {35742, 41018}, {36319, 41032}, {36320, 41030}, {36362, 41026}, {36382, 41028}, {41035, 41054}, {41056, 41070}

X(41044) = reflection of X(i) in X(j) for these (i, j): (6783, 7684), (31709, 5479)
X(41044) = X(16)-pedal-to-X(15)-pedal similarity image of X(4)

X(41045) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 1st ANTI-BROCARD

Barycentrics    2*(2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2))*sqrt(3)*S+2*a^8+(b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(41045) = X(6777)-3*X(41037) = X(6778)+3*X(41025) = 3*X(22511)-X(41021)

The reciprocal orthologic center of these triangles is X(5978).

X(41045) lies on these lines: {4, 13}, {30, 41061}, {115, 1503}, {381, 40671}, {383, 530}, {531, 41047}, {542, 22574}, {618, 37464}, {1080, 5459}, {1513, 6108}, {3850, 41056}, {5472, 5480}, {5617, 41041}, {6115, 13860}, {6302, 6813}, {6306, 6811}, {6669, 37463}, {6771, 41035}, {6777, 41037}, {6778, 10753}, {6782, 7685}, {7417, 30468}, {9112, 14853}, {9880, 22573}, {11645, 41016}, {14482, 31683}, {22492, 36775}, {22511, 41021}, {22513, 41039}, {33517, 38383}, {36318, 41031}, {36344, 41033}, {36363, 41027}, {36383, 41029}, {41034, 41055}, {41057, 41071}

X(41045) = reflection of X(i) in X(j) for these (i, j): (6782, 7685), (31710, 5478)
X(41045) = X(15)-pedal-to-X(16)-pedal similarity image of X(4)

X(41046) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO ANTI-MCCAY

Barycentrics    -2*sqrt(3)*(a^2+b^2+c^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*S+10*a^8-9*(b^2+c^2)*a^6+(b^4+22*b^2*c^2+c^4)*a^4+(b^2+c^2)*(9*b^4-26*b^2*c^2+9*c^4)*a^2-(11*b^4-14*b^2*c^2+11*c^4)*(b^2-c^2)^2 : :
X(41046) = 3*X(3839)-X(5978) = X(5464)-3*X(41036) = 3*X(22571)-X(41020) = X(22577)+3*X(41024)

The reciprocal orthologic center of these triangles is X(8595)

X(41046) lies on these lines: {4, 531}, {30, 115}, {524, 41060}, {530, 41044}, {543, 41016}, {671, 41023}, {1080, 22576}, {1503, 31695}, {3839, 5978}, {5464, 41036}, {9886, 41040}, {10723, 35931}, {22238, 36252}, {22571, 41020}, {22573, 41022}, {22577, 41024}, {22579, 41038}, {25164, 33518}, {33476, 41034}, {35686, 41018}, {35692, 41026}, {35693, 41028}, {35694, 41030}, {35695, 41032}, {36777, 41019}

X(41046) = midpoint of X(10723) and X(35931)
X(41046) = reflection of X(i) in X(j) for these (i, j): (31709, 9880), (33518, 25164)

X(41047) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO ANTI-MCCAY

Barycentrics    2*sqrt(3)*(a^2+b^2+c^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*S+10*a^8-9*(b^2+c^2)*a^6+(b^4+22*b^2*c^2+c^4)*a^4+(b^2+c^2)*(9*b^4-26*b^2*c^2+9*c^4)*a^2-(11*b^4-14*b^2*c^2+11*c^4)*(b^2-c^2)^2 : :
X(41047) = 3*X(3839)-X(5979) = X(5463)-3*X(41037) = 3*X(22572)-X(41021) = X(22578)+3*X(41025)

The reciprocal orthologic center of these triangles is X(8594)

X(41047) lies on these lines: {4, 530}, {30, 115}, {383, 22575}, {524, 41061}, {531, 41045}, {543, 41017}, {671, 41022}, {1503, 31696}, {3839, 5979}, {5463, 41037}, {9885, 41041}, {10723, 35932}, {22236, 36251}, {22572, 41021}, {22574, 41023}, {22578, 41025}, {22580, 41039}, {25154, 33517}, {33477, 41035}, {35690, 41031}, {35691, 41033}, {35696, 41027}, {35697, 41029}

X(41047) = midpoint of X(10723) and X(35932)
X(41047) = reflection of X(i) in X(j) for these (i, j): (31710, 9880), (33517, 25154)

X(41048) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 3rd ANTI-TRI-SQUARES

Barycentrics    ((12*a^6+10*(b^2+c^2)*a^4-8*(b^4-4*b^2*c^2+c^4)*a^2-14*(b^4-c^4)*(b^2-c^2))*S-5*(b^2+c^2)*a^6-(3*b^2+2*b*c+3*c^2)*(3*b^2-2*b*c+3*c^2)*a^4+17*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4-14*b^2*c^2+3*c^4)*(b^2-c^2)^2)*sqrt(3)+(-24*a^6-14*(b^2+c^2)*a^4+4*(3*b^4-14*b^2*c^2+3*c^4)*a^2+26*(b^4-c^4)*(b^2-c^2))*S+2*a^8+7*(b^2+c^2)*a^6+(15*b^4+26*b^2*c^2+15*c^4)*a^4-27*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-22*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(41048) = 3*X(22602)-X(41020) = X(22607)+3*X(41024) = X(22609)-3*X(41036)

The reciprocal orthologic center of these triangles is X(22601)

X(41048) lies on these lines: {4, 22605}, {115, 3071}, {1503, 31697}, {6300, 41040}, {13929, 36655}, {22602, 41020}, {22607, 41024}, {22609, 41036}, {22611, 41038}, {32419, 41016}, {33445, 41034}, {35743, 41018}, {36335, 41030}, {36349, 41032}, {36371, 41026}, {36391, 41028}, {36778, 41019}

X(41049) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 4th ANTI-TRI-SQUARES

Barycentrics    (-(12*a^6+10*(b^2+c^2)*a^4-8*(b^4-4*b^2*c^2+c^4)*a^2-14*(b^4-c^4)*(b^2-c^2))*S-5*(b^2+c^2)*a^6-(3*b^2+2*b*c+3*c^2)*(3*b^2-2*b*c+3*c^2)*a^4+17*(b^4-c^4)*(b^2-c^2)*a^2-(3*b^4-14*b^2*c^2+3*c^4)*(b^2-c^2)^2)*sqrt(3)-(-24*a^6-14*(b^2+c^2)*a^4+4*(3*b^4-14*b^2*c^2+3*c^4)*a^2+26*(b^4-c^4)*(b^2-c^2))*S+2*a^8+7*(b^2+c^2)*a^6+(15*b^4+26*b^2*c^2+15*c^4)*a^4-27*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-22*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
X(41049) = 3*X(22633)-X(41021) = X(22637)+3*X(41025) = X(22639)-3*X(41037)

The reciprocal orthologic center of these triangles is X(22603).

X(41049) lies on these lines: {4, 22635}, {115, 3070}, {1503, 31700}, {6305, 41041}, {13875, 36656}, {22633, 41021}, {22637, 41025}, {22639, 41037}, {22641, 41039}, {32421, 41017}, {33446, 41035}, {36332, 41031}, {36356, 41033}, {36372, 41027}, {36392, 41029}

X(41050) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 4th ANTI-TRI-SQUARES

Barycentrics    (24*a^6+14*(b^2+c^2)*a^4-(12*b^4-8*(7+2*sqrt(3))*b^2*c^2+12*c^4)*a^2-26*(b^4-c^4)*(b^2-c^2))*S+2*a^8+7*(b^2+c^2)*a^6+(15*b^4+26*b^2*c^2+15*c^4)*a^4-27*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-22*b^2*c^2+3*c^4)*(b^2-c^2)^2+((12*a^6+10*(b^2+c^2)*a^4-8*(b^2-c^2)^2*a^2-14*(b^4-c^4)*(b^2-c^2))*S+5*(b^2+c^2)*a^6+(3*b^2+2*b*c+3*c^2)*(3*b^2-2*b*c+3*c^2)*a^4-17*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-14*b^2*c^2+3*c^4)*(b^2-c^2)^2)*sqrt(3) : :
X(41050) = 3*X(22631)-X(41020) = X(22636)+3*X(41024) = X(22638)-3*X(41036)

The reciprocal orthologic center of these triangles is X(22630).

X(41050) lies on these lines: {4, 22634}, {115, 3070}, {1503, 31699}, {6304, 41040}, {13876, 36656}, {22631, 41020}, {22636, 41024}, {22638, 41036}, {22640, 41038}, {32421, 41016}, {33444, 41034}, {35744, 41018}, {36334, 41030}, {36348, 41032}, {36370, 41026}, {36390, 41028}, {36779, 41019}

X(41051) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 3rd ANTI-TRI-SQUARES

Barycentrics    -(24*a^6+14*(b^2+c^2)*a^4-(12*b^4-8*(7+2*sqrt(3))*b^2*c^2+12*c^4)*a^2-26*(b^4-c^4)*(b^2-c^2))*S+2*a^8+7*(b^2+c^2)*a^6+(15*b^4+26*b^2*c^2+15*c^4)*a^4-27*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-22*b^2*c^2+3*c^4)*(b^2-c^2)^2+(-(12*a^6+10*(b^2+c^2)*a^4-8*(b^2-c^2)^2*a^2-14*(b^4-c^4)*(b^2-c^2))*S+5*(b^2+c^2)*a^6+(3*b^2+2*b*c+3*c^2)*(3*b^2-2*b*c+3*c^2)*a^4-17*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4-14*b^2*c^2+3*c^4)*(b^2-c^2)^2)*sqrt(3) : :
X(41051) = 3*X(22604)-X(41021) = X(22608)+3*X(41025) = X(22610)-3*X(41037)

The reciprocal orthologic center of these triangles is X(22632).

X(41051) lies on these lines: {4, 22606}, {115, 3071}, {1503, 31698}, {6300, 35742}, {6301, 41041}, {13928, 36655}, {22604, 41021}, {22608, 41025}, {22610, 41037}, {22612, 41039}, {32419, 41017}, {33447, 41035}, {36333, 41031}, {36357, 41033}, {36374, 41027}, {36394, 41029}

X(41052) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 1st BROCARD-REFLECTED

Barycentrics    sqrt(3)*(3*(b^2+c^2)*a^8-(b^2-c^2)^2*a^6+(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^4-3*(b^4-c^4)^2*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2)+2*(3*(b^2+c^2)*a^6+2*(b^4+4*b^2*c^2+c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2)*S : :
X(41052) = 3*X(22688)-X(41020) = X(22695)+3*X(41024) = X(22701)-3*X(41036) = 4*X(22796)-X(25183)

The reciprocal orthologic center of these triangles is X(22687).

X(41052) lies on these lines: {4, 3104}, {115, 5052}, {511, 22796}, {1503, 31701}, {2782, 41060}, {3106, 36961}, {13860, 33388}, {16809, 22694}, {22688, 41020}, {22691, 41022}, {22695, 41024}, {22701, 41036}, {22707, 41038}, {22715, 41040}, {25167, 36347}, {33478, 41034}, {33518, 38383}, {35745, 41018}, {36323, 41030}, {36345, 41032}, {36364, 41026}, {36384, 41028}, {36780, 41019}

X(41052) = midpoint of X(3106) and X(36961)

X(41053) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 1st BROCARD-REFLECTED

Barycentrics    sqrt(3)*(3*(b^2+c^2)*a^8-(b^2-c^2)^2*a^6+(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^4-3*(b^4-c^4)^2*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2)-2*(3*(b^2+c^2)*a^6+2*(b^4+4*b^2*c^2+c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2)*S : :
X(41053) = 3*X(22690)-X(41021) = X(22696)+3*X(41025) = X(22702)-3*X(41037) = 4*X(22797)-X(25187)

The reciprocal orthologic center of these triangles is X(22689).

X(41053) lies on these lines: {4, 3105}, {115, 5052}, {511, 22797}, {1503, 31702}, {2782, 41061}, {3107, 36962}, {13860, 33389}, {16808, 22693}, {22690, 41021}, {22692, 41023}, {22696, 41025}, {22702, 41037}, {22708, 41039}, {22714, 41041}, {25157, 36345}, {33479, 41035}, {33517, 38383}, {36322, 41031}, {36347, 41033}, {36365, 41027}, {36385, 41029}

X(41053) = midpoint of X(3107) and X(36962)

X(41054) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO INNER-FERMAT

Barycentrics    -2*(8*a^6-7*(b^2+c^2)*a^4+4*(b^4+b^2*c^2+c^4)*a^2-5*(b^4-c^4)*(b^2-c^2))*S+sqrt(3)*(4*a^8-3*(b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+7*(b^4-c^4)*(b^2-c^2)*a^2-(5*b^4-6*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :
X(41054) = 3*X(10612)-X(10991) = 3*X(22846)-X(41020) = X(22849)+3*X(41024) = X(22855)-3*X(41036)

The reciprocal orthologic center of these triangles is X(616)

X(41054) lies on these lines: {4, 3181}, {5, 33376}, {18, 41034}, {115, 398}, {533, 41016}, {628, 41040}, {1503, 31703}, {5480, 22856}, {10612, 10991}, {10753, 41055}, {22846, 41020}, {22847, 41022}, {22849, 41024}, {22855, 41036}, {22861, 41038}, {33624, 41026}, {33627, 41028}, {35746, 41018}, {36324, 41030}, {36346, 41032}, {36781, 41019}, {41035, 41044}

X(41055) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO OUTER-FERMAT

Barycentrics    2*(8*a^6-7*(b^2+c^2)*a^4+4*(b^4+b^2*c^2+c^4)*a^2-5*(b^4-c^4)*(b^2-c^2))*S+sqrt(3)*(4*a^8-3*(b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+7*(b^4-c^4)*(b^2-c^2)*a^2-(5*b^4-6*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :
X(41055) = 3*X(10611)-X(10991) = 3*X(22891)-X(41021) = X(22895)+3*X(41025) = X(22901)-3*X(41037)

The reciprocal orthologic center of these triangles is X(617).

X(41055) lies on these lines: {4, 3180}, {5, 33377}, {17, 41035}, {115, 397}, {532, 41017}, {627, 41041}, {1503, 31704}, {5480, 22900}, {10611, 10991}, {10753, 41054}, {22891, 41021}, {22893, 41023}, {22895, 41025}, {22901, 41037}, {22907, 41039}, {33622, 41027}, {33626, 41029}, {36326, 41031}, {36352, 41033}, {41034, 41045}

X(41056) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO OUTER-FERMAT

Barycentrics    2*(4*a^6+(b^2+c^2)*a^4-(b^4-8*b^2*c^2+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2)*sqrt(3) : :
X(41056) = 3*X(17)-X(41020) = 2*X(11623)-3*X(22891) = X(22894)+3*X(41024) = X(22900)-3*X(41036)

The reciprocal orthologic center of these triangles is X(13).

X(41056) lies on these lines: {4, 616}, {17, 5868}, {115, 397}, {532, 41016}, {546, 41061}, {629, 41034}, {1503, 31705}, {3850, 41045}, {7759, 41042}, {11623, 22891}, {14981, 41060}, {16772, 22892}, {20377, 33561}, {22894, 41024}, {22900, 41036}, {22906, 41038}, {33622, 41030}, {33626, 41032}, {35747, 41018}, {36366, 41026}, {36386, 41028}, {36782, 41019}, {41044, 41070}

X(41057) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO INNER-FERMAT

Barycentrics    -2*(4*a^6+(b^2+c^2)*a^4-(b^4-8*b^2*c^2+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2)*sqrt(3) : :
X(41057) = 3*X(18)-X(41021) = X(10991)-3*X(34602) = 2*X(11623)-3*X(22846) = X(22850)+3*X(41025) = X(22856)-3*X(41037)

The reciprocal orthologic center of these triangles is X(14).

X(41057) lies on these lines: {4, 617}, {18, 5869}, {115, 398}, {533, 41017}, {546, 41060}, {630, 41035}, {1503, 31706}, {3850, 41044}, {7759, 41043}, {10991, 34602}, {11623, 22846}, {14981, 41061}, {16773, 22848}, {20378, 33560}, {22850, 41025}, {22856, 41037}, {22862, 41039}, {33624, 41031}, {33627, 41033}, {36368, 41027}, {36388, 41029}, {41045, 41071}

X(41058) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 2nd FERMAT-DAO

Barycentrics    a^2*(-2*(2*(b^2+c^2)*a^6-2*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+5*(b^2-c^2)^2*b^2*c^2)*S+sqrt(3)*(6*a^6-3*(b^2+c^2)*a^4+4*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2)*((b^2+c^2)*a^2-b^4-c^4) : :
X(41058) = X(22999)-3*X(41036) = 3*X(25151)-X(41020) = X(25182)+3*X(41024).

The reciprocal orthologic center of these triangles is X(25207).

X(41058) lies on these lines: {4, 25180}, {114, 325}, {512, 41060}, {1503, 31707}, {7684, 25220}, {14182, 41040}, {22999, 41036}, {23017, 41038}, {25151, 41020}, {25178, 41022}, {25182, 41024}, {33481, 41034}, {35761, 41018}, {36321, 41032}, {36325, 41030}, {36367, 41026}, {36387, 41028}, {36783, 41019}, {41016, 41068}

X(41058) = reflection of X(i) in X(j) for these (i, j): (25220, 7684), (41068, 41016)

X(41059) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 1st FERMAT-DAO

Barycentrics    a^2*(2*(2*(b^2+c^2)*a^6-2*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+5*(b^2-c^2)^2*b^2*c^2)*S+sqrt(3)*(6*a^6-3*(b^2+c^2)*a^4+4*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2)*((b^2+c^2)*a^2-b^4-c^4) : :
X(41059) = X(23008)-3*X(41037) = 3*X(25161)-X(41021) = X(25177)+3*X(41025)

The reciprocal orthologic center of these triangles is X(25208).

X(41059) lies on these lines: {4, 25175}, {114, 325}, {512, 41061}, {1503, 31708}, {7685, 25219}, {14178, 41041}, {23008, 41037}, {23023, 41039}, {25161, 41021}, {25173, 41023}, {25177, 41025}, {33480, 41035}, {36328, 41031}, {36354, 41033}, {36369, 41027}, {36389, 41029}, {41017, 41069}

X(41059) = reflection of X(i) in X(j) for these (i, j): (25219, 7685), (41069, 41017)

X(41060) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 4th FERMAT-DAO

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))*S+sqrt(3)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(41060) = 2*X(619)-3*X(36519) = 2*X(620)-3*X(36765) = 3*X(5469)-2*X(11623) = 3*X(5470)-X(41020) = X(5474)-3*X(36765) = 4*X(6670)-3*X(38737) = 4*X(6722)-3*X(21156) = X(6770)-3*X(14639) = 2*X(6771)-3*X(23514) = 3*X(21157)-2*X(38747) = X(22997)-3*X(41036) = X(25166)+3*X(41024)

The reciprocal orthologic center of these triangles is X(5469).

X(41060) lies on these lines: {4, 542}, {14, 2794}, {30, 41044}, {114, 1080}, {115, 5321}, {381, 12154}, {383, 6055}, {511, 41068}, {512, 41058}, {524, 41046}, {531, 41016}, {543, 41042}, {546, 41057}, {616, 10723}, {618, 38738}, {619, 36519}, {620, 5474}, {1503, 31709}, {2482, 41035}, {2782, 41052}, {5318, 5477}, {5339, 6034}, {5460, 41034}, {5461, 41041}, {5464, 41040}, {5469, 11623}, {5470, 41020}, {5617, 23698}, {5969, 41062}, {6033, 13102}, {6670, 38737}, {6722, 21156}, {6770, 14639}, {6771, 23514}, {6773, 10722}, {6774, 38749}, {6777, 36962}, {7684, 9117}, {9114, 41019}, {12042, 20253}, {14981, 41056}, {16180, 41067}, {21157, 38747}, {22831, 34602}, {22997, 41036}, {22998, 36994}, {23871, 41066}, {25166, 41024}, {35748, 41018}, {36327, 41030}, {36329, 41026}, {36330, 41028}, {36331, 41032}, {36523, 41029}, {36776, 38745}, {39838, 41023}

X(41060) = midpoint of X(i) and X(j) for these {i, j}: {14, 36961}, {616, 10723}, {6033, 13102}, {6773, 10722}, {6777, 36962}, {22998, 36994}
X(41060) = reflection of X(i) in X(j) for these (i, j): (114, 22796), (115, 5479), (5474, 620), (9117, 7684), (12042, 20253), (31695, 9880), (31696, 25164), (34602, 22831), (36776, 38745), (38738, 618), (38749, 6774), (41061, 4), (41070, 41016)
X(41060) = {X(5474), X(36765)}-harmonic conjugate of X(620)

X(41061) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 3rd FERMAT-DAO

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))*S+sqrt(3)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(41061) = 2*X(618)-3*X(36519) = 3*X(5469)-X(41021) = 3*X(5470)-2*X(11623) = 4*X(6669)-3*X(38737) = 4*X(6722)-3*X(21157) = X(6773)-3*X(14639) = 2*X(6774)-3*X(23514) = 3*X(21156)-2*X(38747) = X(22998)-3*X(41037) = X(25156)+3*X(41025) = X(36776)-3*X(41043)

The reciprocal orthologic center of these triangles is X(5470).

X(41061) lies on these lines: {4, 542}, {13, 2794}, {30, 41045}, {114, 383}, {115, 5318}, {381, 12155}, {511, 41069}, {512, 41059}, {524, 41047}, {530, 41017}, {543, 36776}, {546, 41056}, {617, 10723}, {618, 36519}, {619, 38738}, {620, 5473}, {1080, 6055}, {1503, 31710}, {2482, 41034}, {2782, 41053}, {5321, 5477}, {5340, 6034}, {5459, 41035}, {5461, 41040}, {5463, 41041}, {5469, 41021}, {5470, 11623}, {5613, 23698}, {5969, 41063}, {6033, 13103}, {6669, 38737}, {6722, 21157}, {6770, 10722}, {6771, 38749}, {6773, 14639}, {6774, 23514}, {6778, 36961}, {7685, 9115}, {12042, 20252}, {14981, 41057}, {16179, 41066}, {21156, 38747}, {22490, 41019}, {22997, 36992}, {22998, 41037}, {23870, 41067}, {25156, 41025}, {35749, 41031}, {35750, 41033}, {35751, 41027}, {35752, 41029}, {36523, 41028}, {39838, 41022}

X(41061) = midpoint of X(i) and X(j) for these {i, j}: {13, 36962}, {617, 10723}, {6033, 13103}, {6770, 10722}, {6778, 36961}, {22997, 36992}
X(41061) = reflection of X(i) in X(j) for these (i, j): (114, 22797), (115, 5478), (5473, 620), (9115, 7685), (12042, 20252), (31695, 25154), (31696, 9880), (38738, 619), (38749, 6771), (41060, 4), (41071, 41017)

X(41062) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 1st NEUBERG

Barycentrics    2*((b^2+c^2)*a^6-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2)*sqrt(3)*S+(a^2+b^2+c^2)*(3*(b^2+c^2)*a^6-2*(b^2+c^2)^2*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2) : :
X(41062) = 4*X(5)-3*X(22691) = X(23000)-3*X(41036) = 3*X(25157)-X(41020) = X(25199)+3*X(41024)

The reciprocal orthologic center of these triangles is X(6582).

X(41062) lies on these lines: {4, 25191}, {5, 39}, {538, 41016}, {1503, 31711}, {5969, 41060}, {6581, 41040}, {11303, 36776}, {22687, 39646}, {23000, 41036}, {23018, 41038}, {25157, 41020}, {25183, 41022}, {25199, 41024}, {33482, 41034}, {35755, 41018}, {36338, 41030}, {36350, 41032}, {36373, 41026}, {36393, 41028}, {36784, 41019}

X(41063) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 1st NEUBERG

Barycentrics    -2*((b^2+c^2)*a^6-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2)*sqrt(3)*S+(a^2+b^2+c^2)*(3*(b^2+c^2)*a^6-2*(b^2+c^2)^2*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2) : :
X(41063) = 4*X(5)-3*X(22692) = X(23009)-3*X(41037) = 3*X(25167)-X(41021) = X(25203)+3*X(41025)

The reciprocal orthologic center of these triangles is X(6295).

X(41063) lies on these lines: {4, 25195}, {5, 39}, {538, 41017}, {1503, 31712}, {5969, 41061}, {6294, 41041}, {22689, 39646}, {23009, 41037}, {23024, 41039}, {25167, 41021}, {25187, 41023}, {25203, 41025}, {33483, 41035}, {36336, 41031}, {36358, 41033}, {36378, 41027}, {36398, 41029}

X(41064) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 2nd NEUBERG

Barycentrics    2*(4*(b^2+c^2)*a^6+(b^4+10*b^2*c^2+c^4)*a^4-2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4))*sqrt(3)*S+(a^2+b^2+c^2)*(4*a^8+2*(b^2+c^2)*a^6+(b^4+6*b^2*c^2+c^4)*a^4-4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-3*(b^4-c^4)^2) : :
X(41064) = X(23001)-3*X(41036) = 3*X(25158)-X(41020) = X(25200)+3*X(41024)

The reciprocal orthologic center of these triangles is X(6298).

X(41064) lies on these lines: {4, 25192}, {115, 546}, {754, 41016}, {1503, 31713}, {6296, 41040}, {23001, 41036}, {23019, 41038}, {25158, 41020}, {25184, 41022}, {25200, 41024}, {33484, 41034}, {35756, 41018}, {36339, 41030}, {36351, 41032}, {36375, 41026}, {36395, 41028}, {36785, 41019}

X(41065) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 2nd NEUBERG

Barycentrics    -2*(4*(b^2+c^2)*a^6+(b^4+10*b^2*c^2+c^4)*a^4-2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4))*sqrt(3)*S+(a^2+b^2+c^2)*(4*a^8+2*(b^2+c^2)*a^6+(b^4+6*b^2*c^2+c^4)*a^4-4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-3*(b^4-c^4)^2) : :
X(41065) = X(23010)-3*X(41037) = 3*X(25168)-X(41021) = X(25204)+3*X(41025)

The reciprocal orthologic center of these triangles is X(6299).

X(41065) lies on these lines: {4, 25196}, {115, 546}, {754, 41017}, {1503, 31714}, {6297, 41041}, {23010, 41037}, {23025, 41039}, {25168, 41021}, {25188, 41023}, {25204, 41025}, {33485, 41035}, {36337, 41031}, {36359, 41033}, {36379, 41027}, {36399, 41029}

X(41066) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO VU-DAO-X(16)-ISODYNAMIC

Barycentrics    (8*a^12-12*(b^2+c^2)*a^10-6*(-4*b^2*c^2+(b^2-c^2)^2)*a^8+4*(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4)*a^6-12*(b^2-c^2)^2*(2*b^4+3*b^2*c^2+2*c^4)*a^4+12*(b^4-c^4)*(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-2*(5*b^4+8*b^2*c^2+5*c^4)*(b^2-c^2)^4)*S+(4*a^14-6*(b^2+c^2)*a^12-3*(-4*b^2*c^2+(b^2-c^2)^2)*a^10+(b^2+c^2)*(13*b^4-28*b^2*c^2+13*c^4)*a^8-6*(b^2-c^2)^2*(3*b^4+4*b^2*c^2+3*c^4)*a^6+12*(b^8-c^8)*(b^2-c^2)*a^4+(b^2-c^2)^6*a^2+(b^4-c^4)*(b^2-c^2)^3*(-3*b^4-3*c^4))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(13).

X(41066) lies on these lines: {4, 19776}, {396, 10991}, {16076, 41067}, {16179, 41061}, {23871, 41060}, {36788, 41019}

X(41067) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO VU-DAO-X(15)-ISODYNAMIC

Barycentrics    -(8*a^12-12*(b^2+c^2)*a^10-6*(-4*b^2*c^2+(b^2-c^2)^2)*a^8+4*(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4)*a^6-12*(b^2-c^2)^2*(2*b^4+3*b^2*c^2+2*c^4)*a^4+12*(b^4-c^4)*(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-2*(5*b^4+8*b^2*c^2+5*c^4)*(b^2-c^2)^4)*S+(4*a^14-6*(b^2+c^2)*a^12-3*(-4*b^2*c^2+(b^2-c^2)^2)*a^10+(b^2+c^2)*(13*b^4-28*b^2*c^2+13*c^4)*a^8-6*(b^2-c^2)^2*(3*b^4+4*b^2*c^2+3*c^4)*a^6+12*(b^8-c^8)*(b^2-c^2)*a^4+(b^2-c^2)^6*a^2+(b^4-c^4)*(b^2-c^2)^3*(-3*b^4-3*c^4))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(14).

X(41067) lies on these lines: {4, 19777}, {395, 10991}, {16076, 41066}, {16180, 41060}, {23870, 41061}

X(41068) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 2nd FERMAT-DAO

Barycentrics    (sqrt(3)*(9*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^2-b^4-c^4)*b^2*c^2+(4*(b^2-c^2)^2*a^8-12*(b^2+c^2)*(b^4+c^4)*a^6+6*(2*b^4-3*b^2*c^2+2*c^4)*(b^2+c^2)^2*a^4-4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*(b^4+b^2*c^2+c^4)*a^2+2*(b^2-c^2)^2*(5*b^4-4*b^2*c^2+5*c^4)*b^2*c^2)*S)*a^2 : :
X(41068) = X(23007)-3*X(41036) = 3*X(25217)-X(41020) = X(25228)+3*X(41024)

The reciprocal parallelogic center of these triangles is X(25216).

X(41068) lies on these lines: {4, 25224}, {511, 41060}, {512, 41070}, {1503, 31719}, {7684, 25178}, {14188, 41040}, {23007, 41036}, {23022, 41038}, {25217, 41020}, {25220, 41022}, {25228, 41024}, {33491, 41034}, {35760, 41018}, {36321, 41030}, {36325, 41032}, {36367, 41028}, {36387, 41026}, {36773, 41019}, {41016, 41058}

X(41068) = reflection of X(i) in X(j) for these (i, j): (25178, 7684), (41058, 41016)

X(41069) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 1st FERMAT-DAO

Barycentrics    a^2*(-(4*(b^2-c^2)^2*a^8-12*(b^2+c^2)*(b^4+c^4)*a^6+6*(2*b^4-3*b^2*c^2+2*c^4)*(b^2+c^2)^2*a^4-4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*(b^4+b^2*c^2+c^4)*a^2+2*(b^2-c^2)^2*(5*b^4-4*b^2*c^2+5*c^4)*b^2*c^2)*S+sqrt(3)*(9*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^2-b^4-c^4)*b^2*c^2) : :
X(41069) = X(23014)-3*X(41037) = 3*X(25214)-X(41021) = X(25227)+3*X(41025)

The reciprocal parallelogic center of these triangles is X(25213).

X(41069) lies on these lines: {4, 25223}, {511, 41061}, {512, 41071}, {1503, 31720}, {7685, 25173}, {14186, 41041}, {23014, 41037}, {23028, 41039}, {25214, 41021}, {25219, 41023}, {25227, 41025}, {33490, 41035}, {36328, 41033}, {36354, 41031}, {36369, 41029}, {36389, 41027}, {41017, 41059}

X(41069) = reflection of X(i) in X(j) for these (i, j): (25173, 7685), (41059, 41017)

X(41070) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ALTINTAS-ISODYNAMIC TO 4th FERMAT-DAO

Barycentrics    (-2*sqrt(3)*S*a^2+3*a^4+(b^2-c^2)^2)*((b^2+c^2)*a^2-b^4-c^4) : :
X(41070) = 2*X(623)-3*X(36519) = 4*X(6671)-3*X(38737) = 2*X(11623)-3*X(22510) = 3*X(16529)-X(41020) = 3*X(21158)-2*X(38747) = X(23004)-3*X(41036) = X(25236)+3*X(41024) = 2*X(36755)-3*X(38748)

The reciprocal parallelogic center of these triangles is X(16530).

X(41070) lies on these lines: {4, 617}, {6, 9750}, {14, 41040}, {15, 2794}, {30, 41045}, {114, 325}, {115, 7684}, {383, 19924}, {396, 10991}, {512, 41068}, {531, 41016}, {542, 1080}, {619, 41034}, {620, 14538}, {623, 36519}, {1503, 6783}, {5480, 6114}, {5611, 6033}, {5617, 10753}, {6109, 14136}, {6671, 38737}, {6774, 25555}, {6777, 41019}, {6780, 36961}, {9117, 41022}, {9763, 41043}, {10336, 16941}, {10722, 36993}, {11623, 22510}, {13350, 38749}, {16529, 41020}, {21158, 38747}, {22715, 24206}, {23004, 41036}, {23013, 41038}, {25236, 41024}, {25559, 29012}, {35759, 41018}, {36327, 41032}, {36329, 41028}, {36330, 41026}, {36331, 41030}, {36755, 38748}, {36962, 36967}, {41044, 41056}

X(41070) = midpoint of X(i) and X(j) for these {i, j}: {5611, 6033}, {6780, 36961}, {10722, 36993}, {36962, 36967}
X(41070) = reflection of X(i) in X(j) for these (i, j): (115, 7684), (14538, 620), (38749, 13350), (41060, 41016), (41071, 1513)
X(41070) = orthojoin of X(395)
X(41070) = X(325)-Daleth conjugate of-X(41071)
X(41070) = {X(14501), X(14502)}-harmonic conjugate of X(41071)

X(41071) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ALTINTAS-ISODYNAMIC TO 3rd FERMAT-DAO

Barycentrics    (2*sqrt(3)*S*a^2+3*a^4+(b^2-c^2)^2)*((b^2+c^2)*a^2-b^4-c^4) : :
X(41071) = 2*X(624)-3*X(36519) = 4*X(6672)-3*X(38737) = 2*X(11623)-3*X(22511) = 3*X(16530)-X(41021) = 3*X(21159)-2*X(38747) = X(23005)-3*X(41037) = X(25235)+3*X(41025) = 2*X(36756)-3*X(38748)

The reciprocal parallelogic center of these triangles is X(16529).

X(41071) lies on these lines: {4, 616}, {6, 9749}, {13, 41041}, {16, 2794}, {30, 41044}, {114, 325}, {115, 7685}, {383, 542}, {395, 10991}, {512, 41069}, {530, 41017}, {618, 41035}, {620, 14539}, {624, 36519}, {1080, 19924}, {1503, 6782}, {5480, 6115}, {5613, 10753}, {5615, 6033}, {6108, 14137}, {6672, 38737}, {6771, 25555}, {6779, 36962}, {9115, 41023}, {9761, 41042}, {10336, 16940}, {10722, 36995}, {11623, 22511}, {13349, 38749}, {16530, 41021}, {21159, 38747}, {22714, 24206}, {23005, 41037}, {23006, 41039}, {25235, 41025}, {25560, 29012}, {35749, 41033}, {35750, 41031}, {35751, 41029}, {35752, 41027}, {36756, 38748}, {36765, 41040}, {36961, 36968}, {41045, 41057}

X(41071) = midpoint of X(i) and X(j) for these {i, j}: {5615, 6033}, {6779, 36962}, {10722, 36995}, {36961, 36968}
X(41071) = reflection of X(i) in X(j) for these (i, j): (115, 7685), (14539, 620), (38749, 13349), (41061, 41017), (41070, 1513)
X(41071) = orthojoin of X(396)
X(41071) = X(325)-Daleth conjugate of-X(41070)
X(41071) = {X(14501), X(14502)}-harmonic conjugate of X(41070)

X(41072) = X(99)X(4613)∩X(190)X(789)

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

X(41072) lies on the Steiner circumellipse and these lines: {99, 4613}, {190, 789}, {334, 4495}, {660, 668}, {670, 4589}, {870, 18822}, {874, 4562}, {1921, 9470}, {1966, 30663}, {3226, 32922}, {14621, 35172}, {18827, 30940}, {19564, 40835}, {35165, 40766}, {35166, 40718}, {35173, 40740}

X(41072) = isotomic conjugate of X(30665)
X(41072) = isotomic conjugate of the anticomplement of X(30665)
X(41072) = isotomic conjugate of the complement of X(30665)
X(41072) = isotomic conjugate of the isogonal conjugate of X(30664)
X(41072) = X(i)-cross conjugate of X(j) for these (i,j): {3805, 30663}, {30665, 2}, {30671, 40834}
X(41072) = X(i)-isoconjugate of X(j) for these (i,j): {31, 30665}, {32, 4486}, {238, 788}, {350, 8630}, {649, 16514}, {659, 869}, {667, 3783}, {810, 17569}, {812, 40728}, {824, 14599}, {875, 3802}, {893, 30654}, {1491, 2210}, {1914, 3250}, {1919, 3797}, {2276, 8632}, {3736, 4455}, {3766, 18900}
X(41072) = cevapoint of X(i) and X(j) for these (i,j): {2, 30665}, {812, 24325}, {824, 3836}, {1966, 3805}
X(41072) = trilinear pole of line {2, 292}
X(41072) = barycentric product X(i)*X(j) for these {i,j}: {75, 37207}, {76, 30664}, {291, 37133}, {334, 4586}, {335, 789}, {813, 871}, {870, 4562}, {876, 5388}, {1492, 18895}, {4583, 14621}, {4613, 40017}, {4639, 40718}
X(41072) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30665}, {75, 4486}, {100, 16514}, {171, 30654}, {190, 3783}, {291, 3250}, {292, 788}, {334, 824}, {335, 1491}, {648, 17569}, {660, 2276}, {668, 3797}, {716, 30640}, {752, 30655}, {789, 239}, {813, 869}, {825, 2210}, {870, 812}, {985, 8632}, {1492, 1914}, {1920, 30639}, {1922, 8630}, {3570, 3802}, {4444, 4475}, {4562, 984}, {4583, 3661}, {4584, 3736}, {4586, 238}, {4589, 40773}, {4613, 2238}, {4639, 30966}, {4817, 27846}, {5388, 874}, {14621, 659}, {18827, 4481}, {30663, 30671}, {30664, 6}, {30669, 3805}, {32922, 29955}, {34067, 40728}, {34069, 14599}, {35148, 40793}, {37133, 350}, {37207, 1}, {40718, 21832}, {40740, 9508}, {40745, 4164}, {40747, 4455}, {40766, 5029}

X(41073) = X(99)X(17938)∩X(290)X(34238)

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

X(41073) lies on the Steiner circumellipse and these lines: {99, 17938}, {290, 34238}, {670, 805}, {880, 18829}, {3114, 35146}, {9467, 14603}

X(41073) = X(i)-isoconjugate of X(j) for these (i,j): {1580, 17415}, {1924, 9865}, {1966, 9006}, {3116, 5027}, {8022, 30639}, {16584, 30654}
X(41073) = cevapoint of X(804) and X(24256)
X(41073) = trilinear pole of line {2, 3114}
X(41073) = barycentric product X(i)*X(j) for these {i,j}: {694, 9063}, {3114, 18829}, {18896, 33514}
X(41073) = barycentric quotient X(i)/X(j) for these {i,j}: {670, 9865}, {694, 17415}, {805, 3117}, {3114, 804}, {3407, 5027}, {9063, 3978}, {9468, 9006}, {17938, 18899}, {18829, 3094}, {33514, 1691}, {37134, 3116}, {40415, 30654}, {40834, 3805}, {40835, 30665}

X(41074) = X(99)X(685)∩X(648)X(17932)

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

X(41074) lies on the Steiner circumellipse and these lines: {99, 685}, {648, 17932}, {670, 22456}, {877, 2966}, {5641, 11180}

X(41074) = X(5999)-cross conjugate of X(23582)
X(41074) = X(i)-isoconjugate of X(j) for these (i,j): {810, 1513}, {4008, 39469}
X(41074) = cevapoint of X(1352) and X(2799)
X(41074) = trilinear pole of line {2, 6394}
X(41074) = barycentric product X(i)*X(j) for these {i,j}: {685, 40824}, {16081, 35575}, {22456, 40802}
X(41074) = barycentric quotient X(i)/X(j) for these {i,j}: {648, 1513}, {685, 7735}, {2966, 6776}, {16081, 30735}, {17932, 37188}, {20031, 6620}, {22456, 40814}, {32696, 40825}, {35575, 36212}, {40799, 39469}, {40801, 3569}, {40802, 684}, {40824, 6333}

X(41075) = X(190)X(927)∩X(664)X(36802)

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

X(410) lies on the Steiner circumellipse and these lines: {190, 927}, {664, 36802}, {666, 883}, {668, 34085}, {2481, 40704}, {18025, 32850}, {21446, 35167}, {35158, 39749}

X(41075) = X(926)-isoconjugate of X(7290)
X(41075) = cevapoint of X(918) and X(2550)
X(41075) = trilinear pole of line {2, 6559}
X(41075) = barycentric product X(i)*X(j) for these {i,j}: {927, 39749}, {34018, 37223}, {34085, 39959}
X(41075) = barycentric quotient X(i)/X(j) for these {i,j}: {666, 390}, {927, 5222}, {14942, 14330}, {21446, 2254}, {34018, 30804}, {36146, 7290}, {37223, 3693}

X(41076) = X(190)X(17930)∩X(17934)X(35148)

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

X(41076) lies on the Steiner circumellipse and these lines: {190, 17930}, {17934, 35148}, {32846, 35162}

X(41076) = trilinear pole of line {2, 6543}
X(41076) = barycentric quotient X(i)/X(j) for these {i,j}: {17930, 29586}, {35148, 25354}

X(41077) = X(20)X(2848)∩X(99)X(112)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)^2*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(41077) = X[3265] - 3 X[20580], 2 X[6334] - 3 X[14417], 4 X[8552] - 3 X[14417]

X(41077) lies on these lines: {20, 2848}, {99, 112}, {110, 39447}, {441, 525}, {684, 690}, {826, 22089}, {1294, 5897}, {1637, 5664}, {2416, 14919}, {2482, 23976}, {2797, 16230}, {3184, 9033}, {3269, 15526}, {3926, 35911}, {9003, 32233}, {10718, 35140}, {18557, 18558}

X(41077) = reflection of X(i) in X(j) for these {i,j}: {1637, 5664}, {6334, 8552}
X(41077) = isogonal conjugate of X(32695)
X(41077) = isotomic conjugate of X(15459)
X(41077) = isotomic conjugate of the isogonal conjugate of X(1636)
X(41077) = isotomic conjugate of the polar conjugate of X(9033)
X(41077) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 34549}, {1294, 21294}, {32646, 5906}, {36043, 317}
X(41077) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 11598}, {11744, 21253}, {22239, 20305}
X(41077) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 16163}, {2407, 11064}, {18878, 1092}
X(41077) = X(i)-cross conjugate of X(j) for these (i,j): {1636, 9033}, {34601, 1494}
X(41077) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32695}, {4, 36131}, {19, 1304}, {31, 15459}, {74, 24019}, {92, 32715}, {107, 2159}, {112, 36119}, {158, 32640}, {162, 8749}, {393, 36034}, {811, 40354}, {823, 40352}, {1973, 16077}, {2349, 32713}, {2433, 24000}, {6529, 35200}, {16080, 32676}, {18877, 36126}, {32712, 36063}, {35908, 36104}
X(41077) = cevapoint of X(9033) and X(14345)
X(41077) = crosspoint of X(i) and X(j) for these (i,j): {525, 15421}, {2407, 11064}
X(41077) = crosssum of X(i) and X(j) for these (i,j): {25, 14398}, {512, 14581}, {523, 32125}, {2433, 8749}
X(41077) = crossdifference of every pair of points on line {25, 8749}
X(41077) = barycentric product X(i)*X(j) for these {i,j}: {30, 3265}, {69, 9033}, {76, 1636}, {99, 1650}, {304, 2631}, {305, 9409}, {323, 18557}, {326, 36035}, {520, 3260}, {525, 11064}, {1231, 14395}, {1637, 3926}, {1990, 4143}, {2407, 15526}, {2420, 36793}, {3267, 3284}, {6333, 35912}, {7799, 18558}, {14206, 24018}, {14345, 34403}, {14391, 34386}, {16163, 34767}
X(41077) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 15459}, {3, 1304}, {6, 32695}, {30, 107}, {48, 36131}, {69, 16077}, {125, 18808}, {184, 32715}, {255, 36034}, {520, 74}, {525, 16080}, {577, 32640}, {647, 8749}, {656, 36119}, {684, 35908}, {822, 2159}, {1495, 32713}, {1568, 35360}, {1636, 6}, {1637, 393}, {1650, 523}, {1784, 36126}, {1990, 6529}, {2173, 24019}, {2407, 23582}, {2420, 23964}, {2631, 19}, {2972, 14380}, {3049, 40354}, {3260, 6528}, {3265, 1494}, {3269, 2433}, {3284, 112}, {4240, 32230}, {5489, 12079}, {5562, 36831}, {5664, 14165}, {6793, 23977}, {8057, 10152}, {9033, 4}, {9409, 25}, {11064, 648}, {11125, 8747}, {11589, 1301}, {12113, 31510}, {14206, 823}, {14345, 1249}, {14391, 53}, {14395, 1172}, {14396, 8743}, {14397, 8745}, {14398, 2207}, {14399, 5317}, {14400, 8748}, {14401, 1990}, {14919, 34568}, {15526, 2394}, {16163, 4240}, {18557, 94}, {18558, 1989}, {24018, 2349}, {32320, 18877}, {32663, 32712}, {35906, 20031}, {35912, 685}, {36035, 158}, {36062, 36117}, {38401, 10421}, {39008, 1637}, {39201, 40352}
X(41077) = {X(6334),X(8552)}-harmonic conjugate of X(14417)

X(41078) = X(2)X(24978)∩X(115)X(127)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    (A-power of B-Yiu circle) - (A-power of C-Yiu circle) : :
X(41078) = 3 X[3268] - 2 X[8552], 3 X[5664] - 4 X[8552], 2 X[9409] - 3 X[18556], 2 X[12077] - 3 X[18314], 4 X[12077] - 3 X[20577]

X(41078) lies on these lines: {2, 24978}, {115, 127}, {128, 18402}, {323, 401}, {399, 6069}, {523, 7574}, {690, 10992}, {1625, 14570}, {2394, 13582}, {3268, 5664}, {7265, 32679}, {9409, 18556}, {12077, 18314}

X(41078) = reflection of X(i) in X(j) for these {i,j}: {5664, 3268}, {20577, 18314}
X(41078) = anticomplement of X(24978)
X(41078) = isotomic conjugate of the isogonal conjugate of X(2081)
X(41078) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {33565, 21294}, {34448, 21221}
X(41078) = X(i)-Ceva conjugate of X(j) for these (i,j): {525, 5664}, {1494, 35442}, {3904, 32679}
X(41078) = X(i)-isoconjugate of X(j) for these (i,j): {54, 32678}, {162, 11077}, {163, 1141}, {476, 2148}, {1989, 36134}, {2166, 14586}, {2167, 14560}, {2190, 32662}, {8882, 36061}, {14533, 36129}
X(41078) = crosspoint of X(99) and X(20573)
X(41078) = crosssum of X(512) and X(19627)
X(41078) = crossdifference of every pair of points on line {51, 1576}
X(41078) = barycentric product X(i)*X(j) for these {i,j}: {5, 3268}, {50, 15415}, {76, 2081}, {311, 526}, {323, 18314}, {324, 8552}, {340, 6368}, {523, 1273}, {525, 14918}, {850, 1154}, {2290, 20948}, {3267, 11062}, {6369, 40999}, {7799, 12077}, {14213, 32679}, {23870, 33530}, {23871, 33529}
X(41078) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 476}, {50, 14586}, {51, 14560}, {186, 933}, {216, 32662}, {311, 35139}, {323, 18315}, {340, 18831}, {523, 1141}, {526, 54}, {647, 11077}, {1154, 110}, {1273, 99}, {1953, 32678}, {2081, 6}, {2088, 2623}, {2290, 163}, {2594, 36078}, {2599, 2222}, {2618, 2166}, {2624, 2148}, {3268, 95}, {4467, 39277}, {6116, 36306}, {6117, 36309}, {6149, 36134}, {6368, 265}, {6369, 3615}, {8552, 97}, {8562, 1157}, {10412, 14859}, {11062, 112}, {12077, 1989}, {14165, 16813}, {14213, 32680}, {14570, 39295}, {14918, 648}, {15415, 20573}, {16186, 23286}, {18314, 94}, {20577, 30529}, {22115, 15958}, {23290, 6344}, {32679, 2167}, {33529, 23896}, {33530, 23895}, {36831, 15395}

X(41079) = X(2)X(2411)∩X(3)X(2797)

Barycentrics    b^2*(b^2 - c^2)*c^2*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    (csc 2A) (tan B - tan C) (tan B + tan C - 2 tan A) : :
X(41079) = X[850] - 3 X[14618], 2 X[850] - 3 X[18314], 3 X[9979] - 2 X[24978], 3 X[19912] - 2 X[21731]

X(41079) lies on these lines: {2, 2411}, {3, 2797}, {4, 9517}, {5, 684}, {30, 9409}, {76, 14223}, {94, 2394}, {107, 39447}, {113, 133}, {115, 127}, {247, 2970}, {265, 526}, {290, 671}, {297, 525}, {520, 23290}, {523, 11799}, {648, 1625}, {690, 16003}, {804, 12188}, {1499, 30735}, {1636, 14920}, {1637, 5664}, {2395, 2510}, {2782, 31953}, {2881, 12918}, {3268, 14566}, {6086, 38577}, {8057, 39533}, {8673, 16229}, {17702, 32119}, {18487, 23878}, {23283, 23870}, {23284, 23871}

X(41079) = midpoint of X(2592) and X(2593)
X(41079) = reflection of X(i) in X(j) for these {i,j}: {3, 6130}, {684, 5}, {3268, 14566}, {5664, 1637}, {18314, 14618}, {35522, 18312}
X(41079) = isogonal conjugate of X(32640)
X(41079) = anticomplement of X(8552)
X(41079) = polar conjugate of X(1304)
X(41079) = isotomic conjugate of the isogonal conjugate of X(1637)
X(41079) = polar conjugate of the isogonal conjugate of X(9033)
X(41079) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 14731}, {162, 1272}, {476, 4329}, {2166, 13219}, {6344, 21294}, {14560, 6360}, {18384, 21221}, {24019, 12383}, {32676, 18301}, {32678, 20}, {32680, 1370}, {36061, 6527}, {36116, 35520}, {36129, 69}
X(41079) = X(i)-Ceva conjugate of X(j) for these (i,j): {94, 338}, {6528, 34334}, {14592, 18314}, {35139, 14254}
X(41079) = X(14391)-cross conjugate of X(9033)
X(41079) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32640}, {3, 36131}, {6, 36034}, {48, 1304}, {63, 32715}, {74, 163}, {110, 2159}, {112, 35200}, {162, 18877}, {255, 32695}, {662, 40352}, {1101, 2433}, {1576, 2349}, {2148, 36831}, {2394, 23995}, {2624, 15395}, {4575, 8749}, {4592, 40354}, {5063, 36083}, {9247, 16077}, {9717, 36142}, {14385, 32678}, {14574, 33805}, {14919, 32676}, {32661, 36119}, {35192, 36064}
X(41079) = cevapoint of X(i) and X(j) for these (i,j): {525, 14566}, {1637, 9033}
X(41079) = crosspoint of X(i) and X(j) for these (i,j): {76, 35139}, {648, 2986}
X(41079) = crosssum of X(i) and X(j) for these (i,j): {32, 14270}, {647, 3003}
X(41079) = trilinear pole of line {1650, 3258}
X(41079) = crossdifference of every pair of points on line {184, 1576}
X(41079) = pole wrt polar circle of trilinear polar of X(1304) (line X(6)X(74))
X(41079) = X(684)-of-Johnson-triangle
X(41079) = barycentric product X(i)*X(j) for these {i,j}: {30, 850}, {75, 36035}, {76, 1637}, {94, 5664}, {264, 9033}, {276, 14391}, {313, 11125}, {338, 2407}, {339, 4240}, {349, 14400}, {523, 3260}, {1502, 14398}, {1577, 14206}, {1636, 18027}, {1650, 6528}, {1784, 14208}, {1969, 2631}, {1990, 3267}, {2173, 20948}, {2394, 36789}, {2420, 23962}, {3258, 35139}, {3268, 14254}, {6148, 10412}, {9214, 35522}, {9409, 18022}, {11064, 14618}, {14165, 18557}, {14399, 27801}, {14592, 14920}, {20902, 24001}, {34334, 34767}
X(41079) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36034}, {4, 1304}, {5, 36831}, {6, 32640}, {19, 36131}, {25, 32715}, {30, 110}, {94, 39290}, {113, 15329}, {115, 2433}, {125, 14380}, {264, 16077}, {338, 2394}, {339, 34767}, {393, 32695}, {476, 15395}, {512, 40352}, {523, 74}, {525, 14919}, {526, 14385}, {647, 18877}, {656, 35200}, {661, 2159}, {690, 9717}, {850, 1494}, {868, 32112}, {1495, 1576}, {1539, 30510}, {1568, 23181}, {1577, 2349}, {1636, 577}, {1637, 6}, {1650, 520}, {1784, 162}, {1990, 112}, {2052, 15459}, {2173, 163}, {2394, 40384}, {2407, 249}, {2420, 23357}, {2433, 40353}, {2489, 40354}, {2501, 8749}, {2631, 48}, {2682, 351}, {2799, 35910}, {2970, 18808}, {3163, 2420}, {3258, 526}, {3260, 99}, {3284, 32661}, {3471, 1291}, {3700, 15627}, {4240, 250}, {5466, 9139}, {5642, 5467}, {5664, 323}, {6148, 10411}, {6357, 4565}, {6587, 15291}, {7359, 5546}, {9033, 3}, {9214, 691}, {9407, 14574}, {9409, 184}, {10412, 5627}, {11064, 4558}, {11125, 58}, {11251, 7480}, {13202, 5502}, {13857, 9145}, {14206, 662}, {14254, 476}, {14345, 15905}, {14391, 216}, {14395, 2193}, {14396, 10316}, {14397, 571}, {14398, 32}, {14399, 1333}, {14400, 284}, {14401, 3284}, {14582, 11079}, {14583, 14560}, {14618, 16080}, {14920, 14590}, {15328, 10419}, {15454, 10420}, {15475, 40355}, {16080, 34568}, {16230, 35908}, {16240, 23347}, {18039, 14989}, {18653, 4556}, {20948, 33805}, {23097, 3233}, {23105, 12079}, {24006, 36119}, {34288, 32681}, {34334, 4240}, {35522, 36890}, {35906, 2715}, {36035, 1}, {36130, 36117}, {36298, 5994}, {36299, 5995}, {36789, 2407}, {36891, 10425}, {39008, 1636}, {39176, 14591}

X(41080) = X(2)X(3351)∩X(8)X(14362)

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

X(41080) lies on the cubic K007 and these lines: {2, 3351}, {8, 14362}, {20, 3182}, {69, 34162}, {329, 8894}

X(41080) = isogonal conjugate of X(28784)
X(41080) = isotomic conjugate of X(34162)
X(41080) = cyclocevian conjugate of X(5932)
X(41080) = anticomplement of X(3351)
X(41080) = anticomplement of the isogonal conjugate of X(3352)
X(41080) = isotomic conjugate of the anticomplement of X(3342)
X(41080) = isotomic conjugate of the isogonal conjugate of X(34167)
X(41080) = X(3352)-anticomplementary conjugate of X(8)
X(41080) = cevapoint of X(3347) and X(3352)
X(41080) = X(i)-cross conjugate of X(j) for these (i,j): {4, 329}, {3342, 2}
X(41080) = perspector of ABC and pedal triangle of X(3347)
X(41080) = antigonal conjugate of isogonal conjugate of Darboux quintic point of X(3347)
X(41080) = X(i)-isoconjugate of X(j) for these (i,j): {1, 28784}, {31, 34162}, {1433, 8802}, {2192, 3182}
X(41080) = barycentric product X(i)*X(j) for these {i,j}: {76, 34167}, {3347, 40702}
X(41080) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34162}, {6, 28784}, {223, 3182}, {2331, 8802}, {3342, 3351}, {3347, 282}, {3352, 3341}, {7952, 8894}, {34167, 6}

X(41081) = X(1)X(280)∩X(2)X(77)

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

X(41081) lies on the cubics K317 and K972 and these lines: {1, 280}, {2, 77}, {9, 7125}, {21, 84}, {57, 36100}, {63, 268}, {65, 9376}, {78, 271}, {86, 309}, {326, 345}, {348, 34400}, {1436, 2339}, {1815, 3870}, {2062, 7364}, {2417, 6332}, {4303, 8583}, {5256, 34277}, {5587, 5923}, {6513, 22128}, {6612, 7131}, {7020, 11109}, {7099, 34591}, {7190, 7361}, {8059, 26703}, {18743, 34404}, {19860, 39130}, {35258, 39558}

X(41081) = isogonal conjugate of X(2331)
X(41081) = isotomic conjugate of the polar conjugate of X(84)
X(41081) = isogonal conjugate of the polar conjugate of X(309)
X(41081) = X(309)-Ceva conjugate of X(84)
X(41081) = X(i)-cross conjugate of X(j) for these (i,j): {1, 77}, {222, 63}, {268, 271}, {905, 37141}, {1071, 69}, {22063, 3}, {24031, 4025}
X(41081) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2331}, {2, 3195}, {4, 198}, {6, 7952}, {8, 3209}, {9, 208}, {19, 40}, {25, 329}, {28, 21871}, {33, 223}, {34, 2324}, {37, 3194}, {41, 342}, {55, 196}, {57, 40971}, {92, 2187}, {108, 14298}, {221, 281}, {227, 1172}, {278, 7074}, {318, 2199}, {322, 1973}, {347, 607}, {393, 7078}, {608, 7080}, {1103, 7129}, {1119, 7368}, {1252, 38362}, {1474, 21075}, {1783, 6129}, {1817, 1824}, {1826, 2360}, {1857, 7011}, {2175, 40701}, {2212, 40702}, {2333, 8822}, {6611, 7046}, {7008, 40212}, {7071, 14256}, {7115, 38357}, {7358, 23985}, {8058, 32674}, {8750, 14837}, {8894, 34167}, {10397, 36127}, {14571, 15501}, {40396, 40943}
X(41081) = cevapoint of X(i) and X(j) for these (i,j): {1, 282}, {3, 22124}, {268, 1433}, {1459, 34591}
X(41081) = trilinear pole of line {521, 4091}
X(41081) = barycentric product X(i)*X(j) for these {i,j}: {3, 309}, {7, 271}, {9, 34400}, {63, 189}, {69, 84}, {75, 1433}, {77, 280}, {78, 1440}, {85, 268}, {222, 34404}, {282, 348}, {285, 307}, {304, 1436}, {305, 2208}, {326, 40836}, {345, 1422}, {1098, 6355}, {1413, 3718}, {1444, 39130}, {1804, 7020}, {1812, 8808}, {1903, 17206}, {2188, 6063}, {2192, 7182}, {3926, 7129}, {4025, 13138}, {6332, 37141}, {7003, 7183}, {7008, 7055}, {8059, 35518}, {15413, 36049}, {30805, 40117}
X(41081) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7952}, {3, 40}, {6, 2331}, {7, 342}, {31, 3195}, {48, 198}, {55, 40971}, {56, 208}, {57, 196}, {58, 3194}, {63, 329}, {69, 322}, {71, 21871}, {72, 21075}, {73, 227}, {77, 347}, {78, 7080}, {84, 4}, {85, 40701}, {184, 2187}, {189, 92}, {212, 7074}, {219, 2324}, {222, 223}, {244, 38362}, {255, 7078}, {268, 9}, {271, 8}, {280, 318}, {282, 281}, {285, 29}, {309, 264}, {348, 40702}, {521, 8058}, {603, 221}, {604, 3209}, {652, 14298}, {905, 14837}, {1071, 6260}, {1256, 40836}, {1413, 34}, {1422, 278}, {1433, 1}, {1436, 19}, {1437, 2360}, {1440, 273}, {1444, 8822}, {1459, 6129}, {1790, 1817}, {1795, 15501}, {1802, 7368}, {1804, 7013}, {1812, 27398}, {1903, 1826}, {2188, 55}, {2192, 33}, {2208, 25}, {2357, 1824}, {3341, 3176}, {3576, 37410}, {4025, 17896}, {6001, 1528}, {6612, 1435}, {7004, 38357}, {7008, 1857}, {7011, 40212}, {7078, 1103}, {7099, 6611}, {7118, 607}, {7125, 7011}, {7129, 393}, {7151, 1096}, {7177, 14256}, {7335, 7114}, {7367, 7079}, {8059, 108}, {8808, 40149}, {8886, 1712}, {10884, 37421}, {13138, 1897}, {22063, 40943}, {24031, 7358}, {24560, 25022}, {28784, 8802}, {32652, 8750}, {34400, 85}, {34404, 7017}, {34591, 5514}, {36049, 1783}, {36054, 10397}, {37141, 653}, {39130, 41013}, {40836, 158}
X(41081) = {X(282),X(1422)}-harmonic conjugate of X(189)

X(41082) = X(1)X(19611)∩X(2)X(253)

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

X(41082) lies on the cubic K317 and these lines: {1, 19611}, {2, 253}, {21, 77}, {86, 5931}

X(41082) = X(3194)-cross conjugate of X(1817)
X(41082) = X(i)-isoconjugate of X(j) for these (i,j): {20, 2357}, {84, 3198}, {154, 39130}, {282, 30456}, {610, 1903}, {1436, 8804}, {2192, 5930}, {6587, 36049}, {7367, 36908}, {8059, 14308}, {17898, 32652}
X(41082) = cevapoint of X(1) and X(1073)
X(41082) = barycentric product X(i)*X(j) for these {i,j}: {223, 5931}, {253, 1817}, {2184, 8822}, {3194, 34403}, {8809, 27398}
X(41082) = barycentric quotient X(i)/X(j) for these {i,j}: {40, 8804}, {64, 1903}, {198, 3198}, {221, 30456}, {223, 5930}, {1301, 40117}, {1817, 20}, {2155, 2357}, {2184, 39130}, {2360, 610}, {3194, 1249}, {5931, 34404}, {6129, 6587}, {6611, 40933}, {8809, 8808}, {8822, 18750}, {14298, 14308}, {14837, 17898}

X(41083) = X(1)X(29)∩X(2)X(253)

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

X(41083) lies on the cubic K317 and these lines: {1, 29}, {2, 253}, {6, 37279}, {7, 27}, {21, 7149}, {28, 999}, {86, 309}, {112, 35935}, {162, 2328}, {196, 347}, {223, 342}, {226, 8748}, {243, 5327}, {273, 5256}, {297, 17778}, {321, 1897}, {322, 2331}, {329, 3194}, {333, 648}, {387, 5125}, {393, 5712}, {412, 5706}, {445, 37635}, {653, 1214}, {954, 4183}, {1010, 23661}, {1068, 37383}, {1108, 18603}, {1838, 36250}, {1886, 30686}, {1990, 17056}, {2299, 23710}, {2326, 36419}, {2906, 14016}, {5721, 7541}, {5739, 17555}, {6336, 40574}, {7513, 19767}, {8743, 37086}, {11109, 19684}, {11544, 31902}, {16054, 17903}, {17907, 18134}, {17917, 18644}, {17923, 40940}, {18703, 21459}, {25986, 26109}, {26003, 32911}, {30733, 38295}, {33144, 34856}

X(41083) = isogonal conjugate of X(41087)
X(41083) = pole wrt polar circle of trilinear polar of X(39130) (line X(656)X(3700))
X(41083) = polar conjugate of X(39130)
X(41083) = polar conjugate of the isotomic conjugate of X(8822)
X(41083) = polar conjugate of the isogonal conjugate of X(2360)
X(41083) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 29}, {31623, 27}
X(41083) = X(i)-cross conjugate of X(j) for these (i,j): {223, 1817}, {2331, 3194}, {2360, 8822}
X(41083) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1903}, {37, 1433}, {48, 39130}, {63, 2357}, {65, 268}, {71, 84}, {72, 1436}, {73, 282}, {189, 228}, {212, 8808}, {226, 2188}, {271, 1400}, {280, 1409}, {285, 2197}, {306, 2208}, {307, 7118}, {309, 2200}, {520, 40117}, {525, 32652}, {647, 13138}, {656, 36049}, {1214, 2192}, {1259, 2358}, {1413, 3694}, {1422, 2318}, {1439, 7367}, {3682, 7129}, {3990, 40836}, {3998, 7151}, {7003, 22341}, {7008, 40152}, {8059, 8611}
X(41083) = cevapoint of X(i) and X(j) for these (i,j): {1, 1249}, {196, 223}, {1172, 8885}, {2331, 7952}
X(41083) = trilinear product X(i)*X(j) for these {i,j}: {2, 3194}, {4, 1817}, {19, 8822}, {21, 196}, {27, 40}, {28, 329}, {29, 223}, {34, 27398}, {81, 7952}, {86, 2331}, {92, 2360}, {112, 17896}, {162, 14837}, {198, 286}, {208, 333}, {221, 31623}, {274, 3195}, {284, 342}, {314, 3209}, {322, 1474}, {347, 1172}, {648, 6129}, {1396, 7080}, {1434, 40971}, {1896, 7011}, {2194, 40701}, {2299, 40702}, {4183, 14256}, {4567, 38362}, {7013, 8748}
X(41083) = barycentric product X(i)*X(j) for these {i,j}: {4, 8822}, {21, 342}, {27, 329}, {28, 322}, {29, 347}, {40, 286}, {75, 3194}, {86, 7952}, {92, 1817}, {162, 17896}, {196, 333}, {208, 314}, {223, 31623}, {264, 2360}, {274, 2331}, {278, 27398}, {284, 40701}, {310, 3195}, {648, 14837}, {811, 6129}, {1172, 40702}, {1896, 7013}, {2322, 14256}, {3209, 28660}, {4600, 38362}
X(41083) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 39130}, {19, 1903}, {21, 271}, {25, 2357}, {27, 189}, {28, 84}, {29, 280}, {40, 72}, {58, 1433}, {112, 36049}, {162, 13138}, {196, 226}, {198, 71}, {208, 65}, {221, 73}, {223, 1214}, {227, 201}, {270, 285}, {278, 8808}, {284, 268}, {286, 309}, {322, 20336}, {329, 306}, {342, 1441}, {347, 307}, {1172, 282}, {1396, 1422}, {1434, 34400}, {1474, 1436}, {1817, 63}, {1819, 1259}, {1896, 7020}, {2187, 228}, {2194, 2188}, {2199, 1409}, {2203, 2208}, {2204, 7118}, {2299, 2192}, {2324, 3694}, {2331, 37}, {2332, 7367}, {2360, 3}, {3194, 1}, {3195, 42}, {3209, 1400}, {3668, 6355}, {5317, 7129}, {6129, 656}, {7011, 40152}, {7074, 2318}, {7078, 3682}, {7080, 3710}, {7114, 22341}, {7952, 10}, {8747, 40836}, {8748, 7003}, {8822, 69}, {8885, 3341}, {14298, 8611}, {14837, 525}, {17896, 14208}, {21075, 3695}, {21871, 3949}, {24019, 40117}, {27398, 345}, {31623, 34404}, {32676, 32652}, {38362, 3120}, {40701, 349}, {40702, 1231}, {40971, 210}
X(41083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 8747, 29}, {278, 1172, 27}, {1990, 17056, 18679}

X(41084) = X(1)X(280)∩X(2)X(271)

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

X(41084) lies on the cubic K317 and these lines: {1, 280}, {2, 271}, {3, 37141}, {7, 84}, {8, 13138}, {20, 7338}, {21, 8886}, {29, 81}, {55, 9376}, {86, 5931}, {1097, 37669}, {1528, 6223}, {1895, 6616}, {3176, 15524}, {5703, 24553}, {5731, 8059}, {6060, 35602}, {14548, 34400}, {27382, 36413}

X(41084) = X(i)-cross conjugate of X(j) for these (i,j): {1249, 18623}, {1394, 20}
X(41084) = X(i)-isoconjugate of X(j) for these (i,j): {40, 64}, {198, 2184}, {223, 30457}, {253, 2187}, {322, 33581}, {329, 2155}, {1073, 2331}, {3195, 19611}, {7074, 8809}, {7952, 19614}
X(41084) = cevapoint of X(i) and X(j) for these (i,j): {1, 3341}, {1433, 8886}
X(41084) = barycentric product X(i)*X(j) for these {i,j}: {20, 189}, {84, 18750}, {280, 18623}, {282, 33673}, {309, 610}, {1394, 34404}, {1433, 15466}, {1436, 14615}, {1440, 27382}, {37669, 40836}
X(41084) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 329}, {84, 2184}, {154, 198}, {189, 253}, {204, 2331}, {610, 40}, {1249, 7952}, {1394, 223}, {1422, 8809}, {1433, 1073}, {1436, 64}, {2192, 30457}, {2208, 2155}, {3172, 3195}, {3198, 21871}, {3213, 208}, {7070, 2324}, {7156, 40971}, {8804, 21075}, {8886, 3343}, {14331, 8058}, {15905, 7078}, {18623, 347}, {18750, 322}, {21172, 14837}, {27382, 7080}, {30456, 227}, {33673, 40702}, {40616, 7358}, {40836, 459}
X(41084) = {X(1433),X(40836)}-harmonic conjugate of X(189)

X(41085) = X(3)X(1301)∩X(4)X(1073)

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

X(41085) lies on the cubic K445 and these lines: {3, 1301}, {4, 1073}, {6, 31956}, {19, 30457}, {25, 64}, {34, 7008}, {235, 393}, {459, 3089}, {1498, 3343}, {1593, 14379}, {3515, 11589}, {5198, 8798}, {6527, 6616}, {13526, 13613}

X(41085) = reflection of X(64) in X(33583)
X(41085) = polar conjugate of the isotomic conjugate of X(3343)
X(41085) = orthic-isogonal conjugate of X(64)
X(41085) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 64}, {3343, 1033}
X(41085) = X(i)-isoconjugate of X(j) for these (i,j): {63, 3344}, {610, 1032}, {18750, 28783}
X(41085) = crosspoint of X(i) and X(j) for these (i,j): {4, 6523}, {1498, 3349}
X(41085) = crosssum of X(3346) and X(3350)
X(41085) = barycentric product X(i)*X(j) for these {i,j}: {4, 3343}, {64, 14361}, {253, 1033}, {459, 1498}, {1073, 6523}, {1712, 2184}, {3349, 40839}, {6526, 6617}
X(41085) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 3344}, {64, 1032}, {1033, 20}, {1498, 37669}, {1712, 18750}, {3343, 69}, {6523, 15466}, {14361, 14615}, {33581, 28783}
X(41085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 31942, 64}, {1301, 39268, 3}

X(41086) = X(1)X(1073)∩X(6)X(33)

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

X(41086) lies on the cubic K362 and these lines: {1, 1073}, {6, 33}, {55, 28783}, {65, 2357}, {84, 5706}, {154, 7156}, {387, 40836}, {1097, 37669}, {1436, 37538}, {2328, 36049}, {2331, 31956}, {7070, 15905}

X(41086) = X(i)-isoconjugate of X(j) for these (i,j): {64, 8822}, {221, 5931}, {253, 2360}, {1817, 2184}, {3194, 19611}
X(41086) = crosspoint of X(1) and X(1249)
X(41086) = crosssum of X(1) and X(1073)
X(41086) = barycentric product X(i)*X(j) for these {i,j}: {20, 1903}, {84, 8804}, {189, 3198}, {280, 30456}, {282, 5930}, {610, 39130}, {2357, 18750}, {6587, 13138}, {7070, 8808}, {8057, 40117}, {14308, 37141}, {17898, 36049}
X(41086) = barycentric quotient X(i)/X(j) for these {i,j}: {154, 1817}, {282, 5931}, {610, 8822}, {1903, 253}, {2357, 2184}, {3172, 3194}, {3198, 329}, {5930, 40702}, {6587, 17896}, {7070, 27398}, {8804, 322}, {30456, 347}, {40933, 14256}
X(41086) = {X(1),X(20225)}-harmonic conjugate of X(1073)

X(41087) = X(1)X(281)∩X(37)X(73)

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

X(41087) lies on the cubic K362 and these lines: {1, 281}, {6, 7367}, {37, 73}, {48, 55}, {71, 22341}, {84, 991}, {219, 255}, {326, 345}, {1108, 34591}, {1172, 26701}, {1333, 32652}, {1413, 2286}, {2169, 33629}, {2294, 2358}, {2318, 3990}, {3682, 3694}, {6355, 18643}, {7008, 14547}, {13853, 17056}, {15291, 15627}, {22124, 38288}

X(41087) = isogonal conjugate of X(41083)
X(41087) = isotomic conjugate of the polar conjugate of X(2357)
X(41087) = isogonal conjugate of the polar conjugate of X(39130)
X(41087) = X(i)-Ceva conjugate of X(j) for these (i,j): {282, 1903}, {39130, 2357}, {40117, 652}
X(41087) = X(i)-cross conjugate of X(j) for these (i,j): {42, 73}, {1409, 71}, {3611, 72}
X(41087) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3194}, {4, 1817}, {19, 8822}, {21, 196}, {27, 40}, {28, 329}, {29, 223}, {34, 27398}, {81, 7952}, {86, 2331}, {92, 2360}, {112, 17896}, {162, 14837}, {198, 286}, {208, 333}, {221, 31623}, {274, 3195}, {284, 342}, {314, 3209}, {322, 1474}, {347, 1172}, {648, 6129}, {1396, 7080}, {1434, 40971}, {1896, 7011}, {2194, 40701}, {2299, 40702}, {4183, 14256}, {4567, 38362}, {7013, 8748}
X(41087) = crosspoint of X(i) and X(j) for these (i,j): {1, 1073}, {268, 282}
X(41087) = crosssum of X(i) and X(j) for these (i,j): {1, 1249}, {196, 223}, {1172, 8885}, {2331, 7952}
X(41087) = trilinear product X(i)*X(j) for these {i,j}: {3, 1903}, {37, 1433}, {48, 39130}, {63, 2357}, {65, 268}, {71, 84}, {72, 1436}, {73, 282}, {189, 228}, {212, 8808}, {226, 2188}, {271, 1400}, {280, 1409}, {285, 2197}, {306, 2208}, {307, 7118}, {309, 2200}, {520, 40117}, {525, 32652}, {647, 13138}, {656, 36049}, {1214, 2192}, {1259, 2358}, {1413, 3694}, {1422, 2318}, {1439, 7367}, {3682, 7129}, {3990, 40836}, {3998, 7151}, {7003, 22341}, {7008, 40152}, {8059, 8611}
X(41087) = barycentric product X(i)*X(j) for these {i,j}: {3, 39130}, {10, 1433}, {63, 1903}, {65, 271}, {69, 2357}, {71, 189}, {72, 84}, {73, 280}, {201, 285}, {219, 8808}, {226, 268}, {228, 309}, {282, 1214}, {306, 1436}, {307, 2192}, {525, 36049}, {656, 13138}, {1231, 7118}, {1334, 34400}, {1409, 34404}, {1413, 3710}, {1422, 3694}, {1440, 2318}, {1441, 2188}, {1819, 7157}, {2208, 20336}, {2327, 13853}, {2328, 6355}, {2358, 3719}, {3682, 40836}, {3998, 7129}, {7003, 40152}, {7020, 22341}, {8611, 37141}, {14208, 32652}, {24018, 40117}
X(41087) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8822}, {31, 3194}, {42, 7952}, {48, 1817}, {65, 342}, {71, 329}, {72, 322}, {73, 347}, {84, 286}, {184, 2360}, {213, 2331}, {219, 27398}, {226, 40701}, {228, 40}, {268, 333}, {271, 314}, {282, 31623}, {647, 14837}, {656, 17896}, {810, 6129}, {1214, 40702}, {1400, 196}, {1402, 208}, {1409, 223}, {1433, 86}, {1436, 27}, {1903, 92}, {1918, 3195}, {2188, 21}, {2192, 29}, {2200, 198}, {2208, 28}, {2318, 7080}, {2357, 4}, {3122, 38362}, {3611, 6260}, {3690, 21075}, {4055, 7078}, {6056, 1819}, {7008, 1896}, {7118, 1172}, {7151, 8747}, {7154, 8748}, {7367, 2322}, {8808, 331}, {13138, 811}, {22341, 7013}, {32652, 162}, {36049, 648}, {39130, 264}, {40117, 823}
X(41087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 282, 7129}, {1, 20226, 1249}, {37, 836, 73}

X(41088) = X(1)X(1073)∩X(12)X(6526)

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

X(41088) lies on the cubic K362 and these lines: {1, 1073}, {12, 6526}, {37, 2331}, {55, 64}, {56, 14379}, {227, 40971}, {459, 3085}, {1301, 11399}, {3303, 8798}, {5217, 11589}, {7071, 31942}, {7160, 8809}, {10056, 13157}

X(41088) = X(1073)-Ceva conjugate of X(30457)
X(41088) = X(i)-cross conjugate of X(j) for these (i,j): {42, 2331}, {3195, 198}
X(41088) = X(i)-isoconjugate of X(j) for these (i,j): {20, 84}, {154, 309}, {189, 610}, {280, 1394}, {282, 18623}, {285, 5930}, {1422, 27382}, {1433, 1895}, {1436, 18750}, {1440, 7070}, {2192, 33673}, {2208, 14615}, {7129, 37669}, {7156, 34400}, {13138, 21172}, {14331, 37141}
X(41088) = crosspoint of X(1) and X(3342)
X(41088) = crosssum of X(i) and X(j) for these (i,j): {1, 3341}, {1433, 8886}
X(41088) = barycentric product X(i)*X(j) for these {i,j}: {40, 2184}, {64, 329}, {198, 253}, {322, 2155}, {347, 30457}, {459, 7078}, {1073, 7952}, {2324, 8809}, {2331, 19611}, {3195, 34403}
X(41088) = barycentric quotient X(i)/X(j) for these {i,j}: {40, 18750}, {64, 189}, {198, 20}, {221, 18623}, {223, 33673}, {329, 14615}, {2155, 84}, {2184, 309}, {2187, 610}, {2199, 1394}, {2331, 1895}, {3195, 1249}, {7074, 27382}, {7078, 37669}, {7952, 15466}, {14642, 1433}, {30457, 280}, {33581, 1436}

X(41089) = X(3)-CEVA CONJUGATE OF X(61)

a^2*(-a^2 + b^2 + c^2 + 2*Sqrt[3]*S)*((a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*Sqrt[3]*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S) : :

X(41089) lies on the cubic K349 and these lines: {3, 10640}, {16, 1614}, {61, 3518}, {62, 184}, {577, 41090}, {3165, 6759}, {3166, 10282}, {5238, 5669}, {8837, 11244}, {8839, 32078}

X(41089) = X(3)-Ceva conjugate of X(61)
X(41089) = barycentric product X(302)*X(30402)
X(41089) = barycentric quotient X(30402)/X(17)

X(41090) = X(3)-CEVA CONJUGATE OF X(62)

Barycentrics    a^2*(-a^2 + b^2 + c^2 - 2*Sqrt[3]*S)*((a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*Sqrt[3]*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S) : :

X(41090) lies on the cubic K349 and these lines: {3, 10639}, {15, 1614}, {61, 184}, {62, 3518}, {577, 41089}, {3165, 10282}, {3166, 6759}, {5237, 5668}, {8837, 32078}, {8839, 11243}

X(41090) = X(3)-Ceva conjugate of X(62)
X(41090) = barycentric product X(303)*X(30403)
X(41090) = barycentric quotient X(30403)/X(18)

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Miscellaneous centers: X(41091)-X(41132)

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This preamble and centers X(41091)-X(41132) were contributed by César Eliud Lozada, January 20, 2021.

For definitions of triangles mentioned in these centers, see the index of triangles.

X(41091) = X(2)X(8010) ∩ X(41093)X(41122)

Barycentrics    13*U-5*V-5*W : : ,
  where U : V : W = sin(A)*(4*sqrt(3)*sin((B-C)/3)^2*(cos((B-C)/3)-cos(A+Pi/3))-3*sin(A)) : :
X(41091) = 5*X(2)-6*X(8010) = 7*X(2)-6*X(8011) = 11*X(2)-12*X(33492) = 13*X(2)-12*X(33493) = 7*X(8010)-5*X(8011) = 11*X(8010)-10*X(33492) = 13*X(8010)-10*X(33493) = 3*X(8010)-5*X(36402) = 9*X(8010)-5*X(36403) = 12*X(8010)-5*X(41130) = 11*X(8011)-14*X(33492) = 13*X(8011)-14*X(33493) = 3*X(8011)-7*X(36402) = 9*X(8011)-7*X(36403) = 12*X(8011)-7*X(41130) = 13*X(33492)-11*X(33493) = 6*X(33492)-11*X(36402) = 18*X(33492)-11*X(36403) = 6*X(33493)-13*X(36402) = 18*X(33493)-13*X(36403)

X(41091) lies on these lines: {2, 8010}, {41093, 41122}, {41096, 41099}, {41106, 41123}, {41108, 41131}

X(41091) = reflection of X(i) in X(j) for these (i, j): (2, 36402), (41130, 2)
X(41091) = anticomplement of X(36403)
X(41091) = orthologic center of 1st inner-Fermat-Dao-Nhi to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler, 1st half-squares, 3rd tri-squares, 4th anti-tri-tri-squares, outer-Fermat, outer-Vecten
X(41091) = parallelogic center of 2nd inner-Fermat-Dao-Nhi to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler

X(41092) = X(15682)X(36320) ∩ X(23871)X(36329)

Barycentrics    9*(12*R^2-SA-2*SW)*S^2-9*(36*R^2-7*SW)*SB*SC+sqrt(3)*(12*(9*SA-5*SW)*R^2-26*S^2-27*SA^2+18*SB*SC+15*SW^2)*S : :
X(41092) = X(35751)-3*X(36788) = 5*X(36767)-3*X(36774)

X(41092) lies on these lines: {2, 41132}, {6111, 41112}, {15682, 36320}, {23871, 36329}, {35751, 36788}, {36767, 36774}, {41026, 41066}

X(41092) = complement of X(41132)
X(41092) = orthologic center 3rd inner-Fermat-Dao-Nhi to Vu-Dao-X(16)-isodynamic
X(41092) = parallelogic center 4th inner-Fermat-Dao-Nhi to Vu-Dao-X(16)-isodynamic

X(41093) = X(8010)X(16967) ∩ X(8011)X(16241)

Barycentrics    -sqrt(3)*(8*R^2*(v+w)+sqrt(3)*SW)*(3*S^2+SB*SC)+8*sqrt(3)*(5*S^2-SB*SC)*R^2*u+(8*(u+v+w)*R^2+12*S+sqrt(3)*SW)*S*a^2 : : ,
  where u:v:w = -2*sin(A)*sin(2*A/3+B/3)*sin((B-C)/3)^2*sin(2*A/3+C/3) : :
X(41093) = 3*X(14)-4*X(41124) = 5*X(16961)-6*X(41103) = 5*X(16961)-4*X(41126) = 3*X(16963)-4*X(41095) = 3*X(41103)-2*X(41126) = 8*X(41124)-3*X(41131)

X(41093) lies on these lines: {14, 41124}, {8010, 16967}, {8011, 16241}, {16808, 41123}, {16961, 41103}, {16963, 41095}, {16965, 41097}, {33417, 33493}, {36403, 41101}, {41091, 41122}, {41108, 41130}

X(41093) = reflection of X(41131) in X(14)
X(41093) = parallelogic center of 16th Fermat-Dao to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41093) = {X(41103), X(41126)}-harmonic conjugate of X(16961)

X(41094) = X(2)X(23005) ∩ X(5)X(3104)

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

X(41094) lies on these lines: {2, 23005}, {5, 3104}, {6, 22891}, {14, 37786}, {15, 381}, {16, 13103}, {39, 16631}, {62, 13881}, {115, 3107}, {511, 39601}, {624, 25167}, {625, 25157}, {1080, 16808}, {3105, 16630}, {3106, 7603}, {3181, 18581}, {3643, 16967}, {3734, 40335}, {5475, 22510}, {5613, 16809}, {5869, 41040}, {6114, 7777}, {6671, 11361}, {6770, 18582}, {6782, 41105}, {7617, 9761}, {7684, 22693}, {11300, 33417}, {13084, 25156}, {13860, 41036}, {15534, 41122}, {16943, 36969}, {16966, 31704}, {18424, 23004}, {22492, 22998}, {37637, 39554}

X(41094) = reflection of X(41098) in X(39601)
X(41094) = center of inverse similitude of these pairs of triangles: (2nd isodynamic-Dao, outer-Napoleon), (12th Fermat-Dao, 1st isodynamic-Dao)
X(41094) = {X(3105), X(39565)}-harmonic conjugate of X(16630)

X(41095) = X(6)X(8011) ∩ X(396)X(33493)

Barycentrics    -4*(sqrt(3)*a^2-4*S)*U+2*(3*a*sin(A)-2*SW/R)*S*sqrt(3)+3*a^2*SW/R-4*a*sin((B-C)/3)^2*(cos((B-C)/3)-cos(A+Pi/3))*(sqrt(3)*a^2+2*S) : : ,
  where U : V : W = (cos(A/3-C/3)-cos(B+Pi/3))*sin(A/3-C/3)^2*b+(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :
X(41095) = X(14)-3*X(41103) = 5*X(16961)-3*X(41127) = 5*X(16961)-X(41131) = 3*X(16963)+X(41093) = 3*X(41127)-X(41131)

X(41095) lies on these lines: {6, 8011}, {14, 41103}, {30, 41124}, {381, 41097}, {395, 41126}, {396, 33493}, {8010, 16645}, {10653, 41123}, {11486, 41116}, {16961, 41127}, {16963, 41093}, {18581, 41096}, {23303, 33492}

X(41095) = reflection of X(41126) in X(395)
X(41095) = parallelogic center of 8th Fermat-Dao to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41095) = {X(16961), X(41131)}-harmonic conjugate of X(41127)

X(41096) = X(3)X(33492) ∩ X(5)X(8011)

Barycentrics    (a*(5*S^2+3*SB*SC)*sin(A)-(S^2-3*SB*SC)*(b*sin(B)+c*sin(C)))*sqrt(3)-4*a*sin((B-C)/3)^2*(5*S^2+3*SB*SC)*(-cos(A+Pi/3)+cos((B-C)/3))+4*(S^2-3*SB*SC)*U : : ,
  where U:V:W = (-cos(B+Pi/3)+cos(A/3-C/3))*sin(A/3-C/3)^2*b+(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :
X(41096) = 2*X(3845)+X(36402) = 3*X(5055)-2*X(33493) = 4*X(5066)-X(36403) = X(41091)+5*X(41099) = 7*X(41106)-X(41130)

X(41096) lies on these lines: {3, 33492}, {5, 8011}, {30, 8010}, {62, 41127}, {381, 41123}, {395, 41097}, {3845, 36402}, {5055, 33493}, {5066, 36403}, {5318, 41116}, {10653, 41126}, {16808, 41131}, {18581, 41095}, {41091, 41099}, {41103, 41122}, {41106, 41130}

X(41096) = reflection of X(i) in X(j) for these (i, j): (3, 33492), (8011, 5), (41123, 381)
X(41096) = X(8011)-of-Johnson-triangle
X(41096) = orthologic center of 12th Fermat-Dao to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler

X(41097) = X(13)X(8011) ∩ X(16)X(8010)

Barycentrics    a*((3*(S^2+SB*SC)*sin(A)-3*a*b*SA*sin(B)-3*c*a*SA*sin(C))*sqrt(3)+6*sin(A)*SW*S-4*sin((B-C)/3)^2*(-cos(A+Pi/3)+cos((B-C)/3))*(3*S^2+3*SB*SC+sqrt(3)*S*a^2)-4*a*U*(-3*SA+sqrt(3)*S)) : : ,
  where U:V:W = (-cos(B+Pi/3)+cos(A/3-C/3))*sin(A/3-C/3)^2*b+(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :

X(41097) lies on these lines: {4, 41124}, {13, 8011}, {16, 8010}, {381, 41095}, {395, 41096}, {5318, 41123}, {10653, 41116}, {11486, 41126}, {16809, 41103}, {16965, 41093}, {18582, 33493}, {36402, 41100}, {36403, 41107}

X(41097) = reflection of X(41116) in X(10653)
X(41097) = parallelogic center of 2nd Lemoine-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler

X(41098) = X(2)X(23004) ∩ X(5)X(3105)

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

X(41098) lies on these lines: {2, 23004}, {5, 3105}, {6, 22846}, {13, 37785}, {15, 13102}, {16, 381}, {39, 16630}, {61, 13881}, {115, 3106}, {383, 16809}, {511, 39601}, {623, 25157}, {625, 25167}, {3104, 16631}, {3107, 7603}, {3180, 18582}, {3642, 16966}, {3734, 40334}, {5475, 22511}, {5617, 16808}, {5868, 41041}, {6115, 7777}, {6672, 11361}, {6773, 18581}, {6783, 41104}, {7617, 9763}, {7685, 22694}, {11299, 33416}, {13083, 25166}, {13860, 41037}, {15534, 41121}, {16942, 36970}, {16967, 31703}, {18424, 23005}, {22491, 22997}, {30472, 36766}, {37637, 39555}

X(41098) = reflection of X(41094) in X(39601)
X(41098) = center of inverse similitude of these pairs of triangles: (11th Fermat-Dao, 2nd isodynamic-Dao), (1st isodynamic-Dao, inner-Napoleon triangles)
X(41098) = {X(3104), X(39565)}-harmonic conjugate of X(16631)

X(41099) = EULER LINE INTERCEPT OF X(13)X(33603)

Barycentrics    7*a^4+4*(b^2+c^2)*a^2-11*(b^2-c^2)^2 : :
X(41099) = 11*X(2)-6*X(3) = 2*X(2)+3*X(4) = 7*X(2)-12*X(5) = 13*X(2)-3*X(20) = 8*X(2)-3*X(376) = X(2)-6*X(381) = 19*X(2)+6*X(382) = 17*X(2)-12*X(549) = 4*X(2)-3*X(631) = 13*X(2)-12*X(632) = 5*X(2)-6*X(1656) = X(2)-3*X(3091) = 17*X(2)+3*X(3146) = 7*X(2)-3*X(3522) = 14*X(2)-9*X(3524) = 7*X(2)-2*X(3534) = 7*X(2)+3*X(3543) = 4*X(2)-9*X(3545) = 3*X(2)+2*X(3830) = X(2)+9*X(3839) = X(2)+6*X(3843) = X(2)+4*X(3845) = X(2)-12*X(3858) = X(2)-16*X(3860) = 13*X(2)-18*X(5055) = 3*X(2)-8*X(5066) = 2*X(2)-3*X(5071) = 7*X(2)+6*X(5076) = 9*X(2)-4*X(8703) = 11*X(2)-16*X(10109) = 19*X(2)-9*X(10304) = 6*X(2)-X(11001) = 13*X(2)-8*X(12100) = 7*X(2)+8*X(12101) = 13*X(2)-6*X(14093) = 7*X(2)+18*X(14269) = 9*X(2)+X(15640) = 4*X(2)+X(15682) = 17*X(2)-2*X(15685) = 13*X(2)+12*X(15687) = 5*X(2)-3*X(15692) = 7*X(2)-6*X(15694) = 5*X(2)-2*X(15695) = 17*X(2)-6*X(15696) = 12*X(2)-7*X(15698) = 19*X(2)-14*X(15701) = 7*X(2)-4*X(15711) = 19*X(2)-12*X(15712) = 5*X(2)-4*X(15713) = 16*X(2)-11*X(15719) = 10*X(2)-3*X(17538) = 5*X(2)+3*X(17578) = 19*X(2)-4*X(19710) = 11*X(2)+4*X(33699) = 5*X(2)+6*X(35403) = 11*X(2)+6*X(35434) = 17*X(2)+18*X(38335) = 2*X(2)-7*X(41106)

As a point on the Euler line, X(41099) has Shinagawa coefficients (2, 9).

X(41099) lies on these lines: {2, 3}, {13, 33603}, {14, 33602}, {40, 38076}, {51, 16261}, {69, 14487}, {104, 38077}, {115, 14075}, {148, 22566}, {262, 32532}, {325, 32896}, {355, 34631}, {515, 30308}, {519, 18492}, {541, 15081}, {944, 12571}, {946, 34627}, {1056, 11238}, {1058, 11237}, {1131, 6498}, {1132, 6499}, {1327, 6436}, {1328, 6435}, {1699, 4677}, {1992, 3818}, {2548, 39563}, {3058, 10590}, {3241, 18480}, {3583, 8164}, {3616, 28208}, {3618, 11645}, {3654, 9812}, {3679, 18483}, {3767, 34681}, {4669, 12245}, {4745, 5587}, {5032, 18440}, {5225, 10056}, {5229, 10072}, {5261, 15170}, {5309, 34571}, {5346, 39590}, {5422, 12112}, {5434, 10591}, {5461, 10722}, {5475, 39593}, {5476, 14912}, {5478, 35752}, {5479, 36330}, {5480, 11180}, {5485, 14492}, {5550, 33697}, {5603, 28236}, {5613, 36331}, {5617, 35750}, {5640, 16194}, {5656, 23324}, {5657, 28232}, {5759, 38075}, {5818, 28194}, {5862, 22491}, {5863, 22492}, {5943, 11455}, {5965, 20423}, {6172, 18482}, {6241, 16226}, {6249, 12156}, {6250, 22485}, {6251, 22484}, {6361, 19875}, {6776, 38072}, {7581, 35823}, {7582, 35822}, {7620, 9766}, {7687, 10706}, {7735, 10033}, {7736, 11648}, {7737, 18362}, {7750, 32885}, {7767, 32893}, {7773, 32836}, {7776, 32869}, {7788, 32827}, {7799, 32822}, {7967, 9779}, {8584, 14853}, {8591, 22515}, {8667, 20112}, {9143, 10113}, {9166, 9862}, {9300, 14482}, {9760, 35695}, {9762, 35691}, {9781, 14831}, {9955, 38314}, {9956, 10248}, {10588, 18514}, {10589, 18513}, {10595, 28204}, {10598, 34697}, {10599, 34746}, {10653, 12816}, {10654, 12817}, {10783, 13794}, {10784, 13674}, {11002, 18435}, {11017, 37484}, {11057, 34229}, {11160, 21850}, {11177, 22505}, {11206, 18376}, {11451, 14915}, {11459, 21969}, {11488, 36970}, {11489, 36969}, {11491, 38078}, {11694, 15046}, {12117, 36519}, {12243, 14639}, {13172, 23234}, {13474, 15024}, {13570, 15030}, {13637, 22807}, {13757, 22806}, {13886, 23261}, {13939, 23251}, {14094, 15004}, {14458, 18842}, {14494, 17503}, {14711, 22682}, {14845, 15072}, {14848, 39884}, {14927, 38064}, {15031, 32006}, {15045, 32062}, {15052, 39522}, {15432, 18911}, {16616, 31165}, {16626, 33622}, {16627, 33624}, {16628, 36324}, {16629, 36326}, {16808, 37640}, {16809, 37641}, {17008, 19569}, {17487, 24827}, {18406, 34607}, {18581, 41100}, {18582, 41101}, {19130, 39874}, {19876, 31730}, {20049, 37705}, {20070, 38066}, {21356, 25561}, {22575, 35694}, {22576, 35690}, {22605, 36335}, {22606, 36333}, {22634, 36334}, {22635, 36332}, {22693, 36323}, {22694, 36322}, {22791, 31145}, {22793, 34632}, {22796, 36363}, {22797, 36362}, {22831, 36368}, {22832, 36366}, {23249, 32788}, {23259, 32787}, {23269, 35786}, {23275, 31412}, {23294, 32601}, {24817, 36522}, {25055, 31673}, {25154, 35749}, {25164, 36327}, {25175, 36328}, {25180, 36325}, {25191, 36338}, {25192, 36339}, {25193, 36340}, {25194, 36342}, {25195, 36336}, {25196, 36337}, {25197, 36341}, {25198, 36343}, {25223, 36354}, {25224, 36321}, {31423, 34638}, {32431, 37654}, {32819, 32837}, {32823, 32833}, {34718, 40273}, {36319, 41043}, {36344, 41042}, {36765, 36768}, {36996, 38073}, {38084, 38753}, {41091, 41096}, {41123, 41130}

X(41099) = midpoint of X(i) and X(j) for these {i, j}: {3, 35434}, {4, 5071}, {381, 3843}, {632, 15687}, {1656, 35403}, {3522, 3543}, {3627, 15714}, {3830, 15693}, {5076, 15694}, {15692, 17578}
X(41099) = reflection of X(i) in X(j) for these (i, j): (2, 19709), (20, 14093), (376, 631), (381, 3858), (549, 12812), (631, 5071), (3091, 381), (3522, 15694), (3534, 15711), (3543, 5076), (5071, 3091), (10303, 35382), (11001, 15697), (14093, 632), (15692, 1656), (15694, 5), (15695, 15713), (15696, 549), (15697, 15693), (15712, 547), (17538, 15692), (17578, 35403), (19708, 2)
X(41099) = anticomplement of X(15693)
X(41099) = complement of X(15697)
X(41099) = X(15694)-of-Johnson-triangle
X(41099) = homothetic center of these pairs of triangles: (11th Fermat-Dao, 1st outer-Fermat-Dao-Nhi), (12th Fermat-Dao, 1st inner-Fermat-Dao-Nhi)
X(41099) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 3845, 4), (3, 10109, 2), (4, 382, 3545), (4, 41106, 2), (5, 3534, 2), (381, 3839, 4), (381, 3845, 2), (546, 3832, 4), (547, 15701, 2), (1656, 15713, 2), (3090, 15719, 2), (3091, 3843, 4), (3146, 3861, 4), (3545, 15682, 2), (3830, 5066, 2), (5055, 12100, 2), (5071, 19708, 2), (10109, 33699, 3), (11540, 15703, 2), (12816, 41122, 10653), (12817, 41121, 10654), (14893, 38071, 3), (18586, 18587, 548), (25561, 31670, 21356), (31412, 35787, 23275), (34648, 38021, 944), (38321, 39484, 2), (41030, 41031, 4)

X(41100) = X(2)X(13) ∩ X(3)X(3412)

Barycentrics    6*a^2*sqrt(3)*S+7*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(41100) = 5*X(62)-2*X(398) = 4*X(62)-X(16964) = 3*X(62)-X(41108) = 8*X(398)-5*X(16964) = 6*X(398)-5*X(41108) = 3*X(16964)-4*X(41108)

X(41100) lies on these lines: {2, 13}, {3, 3412}, {4, 3411}, {6, 3534}, {14, 3830}, {15, 8703}, {17, 5054}, {18, 381}, {30, 62}, {61, 376}, {182, 5473}, {202, 3058}, {395, 3845}, {396, 10646}, {397, 549}, {472, 6111}, {532, 11300}, {533, 37008}, {541, 36209}, {547, 16773}, {574, 9112}, {754, 5858}, {1152, 35734}, {1351, 5474}, {3060, 30440}, {3105, 7757}, {3107, 36385}, {3365, 15764}, {3391, 36470}, {3392, 36452}, {3524, 5351}, {3525, 10188}, {3543, 5365}, {5055, 5340}, {5066, 5318}, {5238, 10304}, {5321, 33699}, {5334, 15640}, {5339, 15684}, {5350, 23046}, {5352, 34200}, {5434, 7006}, {5464, 5859}, {5469, 41115}, {5470, 6774}, {5969, 9114}, {6560, 36467}, {6561, 36450}, {6773, 36961}, {6775, 6777}, {7801, 22494}, {7860, 11303}, {8015, 15743}, {9736, 21156}, {9763, 36366}, {10109, 23303}, {10645, 19708}, {10654, 11001}, {11302, 21360}, {11481, 15693}, {11485, 15695}, {11488, 15719}, {11489, 41106}, {11542, 11812}, {11543, 12101}, {12150, 35931}, {13349, 20425}, {14093, 36836}, {15534, 25236}, {15682, 36970}, {15688, 22236}, {15701, 16644}, {15713, 23302}, {15765, 35730}, {16530, 41042}, {16645, 16808}, {16772, 17504}, {18581, 41099}, {20423, 36758}, {22495, 33458}, {22511, 25154}, {22513, 22998}, {22577, 35693}, {22580, 35696}, {22607, 36391}, {22612, 36374}, {22636, 36390}, {22641, 36372}, {22695, 36384}, {22708, 36365}, {22849, 33627}, {22862, 36368}, {22894, 36386}, {22907, 33622}, {23023, 36369}, {23024, 36378}, {23025, 36379}, {23026, 36380}, {23027, 36381}, {23028, 36389}, {25166, 36330}, {25182, 36387}, {25199, 36393}, {25200, 36395}, {25201, 36396}, {25202, 36397}, {25228, 36367}, {33417, 33607}, {34509, 41104}, {35731, 35822}, {35733, 35735}, {36402, 41097}, {36403, 41116}, {36962, 41029}, {41024, 41028}, {41027, 41039}

X(41100) = homothetic center of these pairs of triangles: (3rd outer-Fermat-Dao-Nhi, 2nd Lemoine-Dao), (15th Fermat-Dao, 4th inner-Fermat-Dao-Nhi)
X(41100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 5335, 41119), (2, 10653, 41107), (2, 41107, 13), (2, 41112, 41121), (2, 41119, 37832), (6, 3534, 41101), (6, 36968, 36967), (13, 16, 16242), (13, 16242, 16966), (13, 33416, 37832), (16, 5335, 33416), (16, 10653, 13), (16, 41107, 2), (381, 16963, 18), (3534, 41101, 36967), (5335, 37832, 13), (12817, 19106, 3830), (16268, 41120, 33606), (34755, 36969, 395), (36968, 41101, 3534), (41107, 41121, 41112), (41112, 41121, 13)

X(41101) = X(2)X(14) ∩ X(3)X(3411)

Barycentrics    -6*a^2*sqrt(3)*S+7*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(41101) = 5*X(61)-2*X(397) = 4*X(61)-X(16965) = 3*X(61)-X(41107) = 8*X(397)-5*X(16965) = 6*X(397)-5*X(41107) = 3*X(16965)-4*X(41107)

X(41101) lies on these lines: {2, 14}, {3, 3411}, {4, 3412}, {6, 3534}, {13, 3830}, {16, 8703}, {17, 381}, {18, 5054}, {30, 61}, {62, 376}, {182, 5474}, {203, 3058}, {395, 10645}, {396, 3845}, {398, 549}, {473, 6110}, {532, 37007}, {533, 11299}, {541, 36208}, {547, 16772}, {574, 9113}, {754, 5859}, {1151, 35734}, {1351, 5473}, {3060, 30439}, {3104, 7757}, {3106, 36384}, {3366, 36453}, {3367, 36469}, {3389, 15764}, {3524, 5352}, {3525, 10187}, {3543, 5366}, {5055, 5339}, {5066, 5321}, {5237, 10304}, {5318, 33699}, {5335, 15640}, {5340, 15684}, {5349, 23046}, {5351, 34200}, {5434, 7005}, {5463, 5858}, {5469, 6771}, {5470, 41114}, {5969, 9116}, {6560, 36449}, {6561, 36468}, {6770, 36962}, {6772, 6778}, {6782, 36768}, {7801, 22493}, {7860, 11304}, {8014, 11586}, {9735, 21157}, {9761, 36368}, {9766, 36775}, {10109, 23302}, {10646, 19708}, {10653, 11001}, {11301, 21359}, {11480, 15693}, {11486, 15695}, {11488, 41106}, {11489, 15719}, {11542, 12101}, {11543, 11812}, {12150, 35932}, {13350, 20426}, {14093, 36843}, {15534, 25235}, {15682, 36969}, {15688, 22238}, {15701, 16645}, {15713, 23303}, {16529, 41043}, {16644, 16809}, {16773, 17504}, {18582, 41099}, {20423, 36757}, {22496, 33459}, {22510, 25164}, {22512, 22997}, {22578, 35697}, {22579, 35692}, {22608, 36394}, {22611, 36371}, {22637, 36392}, {22640, 36370}, {22696, 36385}, {22707, 36364}, {22850, 36388}, {22861, 33624}, {22895, 33626}, {22906, 36366}, {22998, 36769}, {23017, 36367}, {23018, 36373}, {23019, 36375}, {23020, 36376}, {23021, 36377}, {23022, 36387}, {25156, 35752}, {25177, 36389}, {25203, 36398}, {25204, 36399}, {25205, 36400}, {25206, 36401}, {25227, 36369}, {33416, 33606}, {34508, 41105}, {36402, 41131}, {36403, 41093}, {36961, 41028}, {41025, 41029}, {41026, 41038}

X(41101) = homothetic center of these pairs of triangles: (3rd inner-Fermat-Dao-Nhi, 1st Lemoine-Dao), (16th Fermat-Dao, 4th outer-Fermat-Dao-Nhi)
X(41101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 5334, 41120), (2, 10654, 41108), (2, 41108, 14), (2, 41113, 41122), (2, 41120, 37835), (6, 3534, 41100), (6, 36967, 36968), (14, 15, 16241), (14, 16241, 16967), (14, 33417, 37835), (15, 5334, 33417), (15, 10654, 14), (15, 41108, 2), (381, 16962, 17), (3534, 41100, 36968), (5334, 37835, 14), (12816, 19107, 3830), (16267, 41119, 33607), (34754, 36970, 396), (36967, 41100, 3534), (41108, 41122, 41113), (41113, 41122, 14)

X(41102) = X(111)X(6054) ∩ X(113)X(14698)

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

X(41102) lies on these lines: {111, 6054}, {113, 14698}, {114, 9185}, {126, 9191}, {325, 3233}, {5996, 36170}

X(41102) = center of direct similitude of these pairs of triangles: (Euler, 1st Parry), (medial, 2nd Parry)

X(41103) = X(18)X(8010) ∩ X(61)X(33493)

Barycentrics    (a^3*SA*sin(A)-(3*S^2+SB*SC)*(b*sin(B)+c*sin(C)))*sqrt(3)+6*a*SW*sin(A)*S+4*U*(3*S^2+SB*SC-S*a^2*sqrt(3))-4*a^3*sin((B-C)/3)^2*(cos((B-C)/3)-cos(A+Pi/3))*(SA+sqrt(3)*S) : : ,
  where U:V:W = (cos(A/3-C/3)-cos(B+Pi/3))*sin(A/3-C/3)^2*b+(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :
X(41103) = X(14)+2*X(41095) = X(16)+2*X(41124) = 5*X(16961)+X(41093) = 5*X(16961)-2*X(41126) = X(41093)+2*X(41126)

X(41103) lies on these lines: {14, 41095}, {16, 41124}, {18, 8010}, {61, 33493}, {62, 41123}, {16268, 41127}, {16809, 41097}, {16961, 41093}, {41096, 41122}

X(41103) = reflection of X(41127) in X(16268)
X(41103) = parallelogic center of 4th Fermat-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41103) = {X(16961), X(41093)}-harmonic conjugate of X(41126)

X(41104) = X(61)X(381) ∩ X(3105)X(33458)

Barycentrics    2*(8*a^4-7*(b^2+c^2)*a^2+8*(b^2-c^2)^2)*S+sqrt(3)*(8*a^6-11*(b^2+c^2)*a^4+7*(b^4+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2)) : :
X(41104) = 3*X(16631)-2*X(41115)

X(41104) lies on these lines: {61, 381}, {531, 41114}, {3105, 33458}, {6783, 41098}, {9763, 41107}, {16634, 22997}, {34509, 41100}

X(41104) = center of inverse similitude of these pairs of triangles: (4th outer-Fermat-Dao-Nhi, 1st isodynamic-Dao), (2nd inner-Fermat-Dao-Nhi, 15th Fermat-Dao)

X(41105) = X(62)X(381) ∩ X(3104)X(33459)

Barycentrics    -2*(8*a^4-7*(b^2+c^2)*a^2+8*(b^2-c^2)^2)*S+sqrt(3)*(8*a^6-11*(b^2+c^2)*a^4+7*(b^4+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2)) : :
X(41105) = 3*X(16630)-2*X(41114)

X(41105) lies on these lines: {62, 381}, {530, 41115}, {3104, 33459}, {6782, 41094}, {9761, 41108}, {16635, 22998}, {34508, 41101}

X(41105) = center of inverse similitude of these pairs of triangles: (2nd outer-Fermat-Dao-Nhi, 16th Fermat-Dao), (4th inner-Fermat-Dao-Nhi, 2nd isodynamic-Dao)

X(41106) = EULER LINE INTERCEPT OF X(6)X(14226)

Barycentrics    5*a^4+8*(b^2+c^2)*a^2-13*(b^2-c^2)^2 : :
Barycentrics    4 S^2 + 9 SB SC : :
X(41106) = 13*X(2)-6*X(3) = 4*X(2)+3*X(4) = 5*X(2)-12*X(5) = 17*X(2)-3*X(20) = 10*X(2)-3*X(376) = X(2)+6*X(381) = 19*X(2)-12*X(549) = 2*X(2)-3*X(3090) = X(2)-15*X(3091) = 5*X(2)-3*X(3523) = 16*X(2)-9*X(3524) = 7*X(2)-6*X(3526) = 8*X(2)-3*X(3528) = 9*X(2)-2*X(3534) = 11*X(2)+3*X(3543) = 2*X(2)-9*X(3545) = 5*X(2)+2*X(3830) = X(2)+3*X(3832) = 5*X(2)+9*X(3839) = 3*X(2)+4*X(3845) = X(2)-6*X(3851) = X(2)+12*X(3857) = 5*X(2)+16*X(3860) = 11*X(2)-18*X(5055) = X(2)-8*X(5066) = 8*X(2)-15*X(5071) = 11*X(2)-4*X(8703) = 9*X(2)-16*X(10109) = 8*X(2)-X(11001) = 15*X(2)-8*X(12100) = 13*X(2)+8*X(12101) = 17*X(2)+18*X(14269) = 17*X(2)-12*X(14869) = 13*X(2)+X(15640) = 6*X(2)+X(15682) = 17*X(2)-10*X(15693) = 19*X(2)-5*X(15697) = 11*X(2)-6*X(15700) = 4*X(2)-3*X(15702) = 5*X(2)-6*X(15703) = 18*X(2)-11*X(15719) = 12*X(2)-5*X(19708) = 3*X(2)-10*X(19709) = 7*X(2)-4*X(19711) = 17*X(2)+4*X(33699) = 2*X(2)+5*X(41099)

As a point on the Euler line, X(41106) has Shinagawa coefficients (4, 9)

X(41106) lies on these lines: {2, 3}, {6, 14226}, {13, 33605}, {14, 33604}, {69, 25561}, {373, 11455}, {551, 18492}, {597, 39874}, {946, 4677}, {1285, 14537}, {1327, 3069}, {1328, 3068}, {1699, 38127}, {1992, 19130}, {2548, 39593}, {3058, 8164}, {3241, 9955}, {3316, 23261}, {3317, 23251}, {3582, 5229}, {3584, 5225}, {3619, 19924}, {3656, 9779}, {3817, 7967}, {3828, 6361}, {4669, 5587}, {4745, 5818}, {5334, 33603}, {5335, 33602}, {5461, 9862}, {5478, 35751}, {5479, 36329}, {5480, 15533}, {5485, 9766}, {5603, 30308}, {5613, 36327}, {5617, 35749}, {5862, 22492}, {5863, 22491}, {5921, 14848}, {5943, 16261}, {6054, 36523}, {6241, 27355}, {6248, 11055}, {6441, 23273}, {6442, 23267}, {6476, 6561}, {6477, 6560}, {6478, 35787}, {6479, 35786}, {6515, 18489}, {6564, 19053}, {6565, 19054}, {7687, 11427}, {7735, 18362}, {7736, 18424}, {7738, 39563}, {7776, 32874}, {7951, 10385}, {7989, 28194}, {8176, 9741}, {8227, 34648}, {8584, 11180}, {9306, 13482}, {9760, 35694}, {9762, 35690}, {9770, 18546}, {9780, 28198}, {9956, 34632}, {10155, 17503}, {10168, 14927}, {10248, 28202}, {10516, 22165}, {10590, 11238}, {10591, 11237}, {10595, 19925}, {10601, 12112}, {10706, 15081}, {10711, 38077}, {10722, 14971}, {11002, 15060}, {11160, 18358}, {11451, 16194}, {11459, 21849}, {11465, 13474}, {11488, 41101}, {11489, 41100}, {11648, 31415}, {12816, 37835}, {12817, 37832}, {13468, 23334}, {13678, 32807}, {13846, 23259}, {13847, 23249}, {14494, 32532}, {14644, 18950}, {14830, 15092}, {14831, 15058}, {14845, 15305}, {14853, 15534}, {15031, 32833}, {15300, 23234}, {16626, 33626}, {16627, 33627}, {16628, 36346}, {16629, 36352}, {16808, 37641}, {16809, 37640}, {18357, 31145}, {18376, 35260}, {18480, 38314}, {18483, 19875}, {18581, 41107}, {18582, 41108}, {22575, 35695}, {22576, 35691}, {22605, 36349}, {22606, 36357}, {22634, 36348}, {22635, 36356}, {22682, 33706}, {22693, 36345}, {22694, 36347}, {22796, 36383}, {22797, 36382}, {22831, 36388}, {22832, 36386}, {25154, 35750}, {25164, 36331}, {25175, 36354}, {25180, 36321}, {25191, 36350}, {25192, 36351}, {25193, 36353}, {25194, 36355}, {25195, 36358}, {25196, 36359}, {25197, 36360}, {25198, 36361}, {25223, 36328}, {25224, 36325}, {31412, 35823}, {32816, 32892}, {32818, 32896}, {32822, 32837}, {32823, 32836}, {32827, 37671}, {33630, 36412}, {34623, 38228}, {36318, 41042}, {36320, 41043}, {36519, 36521}, {36765, 36769}, {37749, 40340}, {38066, 40273}, {41091, 41123}, {41096, 41130}

X(41106) = midpoint of X(i) and X(j) for these {i, j}: {4, 15702}, {381, 3851}
X(41106) = reflection of X(i) in X(j) for these (i, j): (376, 3523), (381, 3857), (3523, 15703), (3528, 15702), (3832, 381), (14869, 547), (15698, 2), (15702, 3090), (15703, 5), (16434, 37299)
X(41106) = anticomplement of X(15701)
X(41106) = homothetic center of these pairs of triangles: (11th Fermat-Dao, 2nd outer-Fermat-Dao-Nhi), (12th Fermat-Dao, 2nd inner-Fermat-Dao-Nhi)
X(41106) = X(15703)-of-Johnson-triangle
X(41106) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 15640, 3), (2, 41099, 4), (5, 3830, 2), (376, 3839, 4), (381, 3545, 4), (381, 5066, 2), (546, 631, 4), (547, 15693, 2), (1656, 11812, 2), (3090, 3832, 4), (3090, 15698, 2), (3091, 3855, 4), (3529, 3843, 4), (3534, 10109, 2), (3545, 41099, 2), (3845, 19709, 2), (5055, 8703, 2), (5056, 15697, 2), (5071, 11001, 2), (11737, 23046, 3), (14226, 14241, 6), (34627, 38021, 10595), (41032, 41033, 4), (41953, 41954, 6)

X(41107) = X(2)X(13) ∩ X(3)X(16267)

Barycentrics    6*S*a^2*sqrt(3)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(41107) = X(61)-4*X(397) = X(61)+2*X(16965) = 3*X(61)-2*X(41101) = 2*X(397)+X(16965) = 6*X(397)-X(41101) = 3*X(16965)+X(41101)

X(41107) lies on these lines: {2, 13}, {3, 16267}, {4, 12817}, {5, 16963}, {6, 3830}, {14, 3845}, {15, 3534}, {17, 549}, {18, 3545}, {30, 61}, {51, 30440}, {62, 381}, {182, 13103}, {202, 11238}, {302, 33560}, {376, 5238}, {395, 5066}, {396, 8703}, {398, 15687}, {473, 6111}, {532, 7768}, {538, 3105}, {550, 3412}, {1152, 35735}, {1327, 36467}, {1328, 36450}, {2549, 9112}, {3091, 3411}, {3389, 35822}, {3390, 35823}, {3543, 16964}, {3839, 5344}, {3860, 11543}, {5054, 5237}, {5055, 22238}, {5097, 13102}, {5321, 12101}, {5339, 38335}, {5350, 14893}, {5352, 15688}, {5464, 22900}, {5469, 5478}, {5470, 22891}, {5476, 36758}, {5480, 36962}, {5862, 22493}, {6560, 36468}, {6561, 36449}, {6775, 6778}, {6776, 36961}, {6777, 36320}, {7006, 11237}, {7812, 22496}, {8584, 22579}, {8740, 18559}, {9114, 35695}, {9763, 41104}, {10109, 16967}, {10646, 15693}, {10654, 15682}, {10706, 36209}, {11001, 34754}, {11300, 34509}, {11480, 15695}, {11481, 15701}, {11485, 15685}, {11486, 19709}, {11488, 15698}, {11542, 12100}, {11812, 23302}, {12820, 33603}, {14539, 20425}, {15681, 22236}, {15689, 36836}, {15694, 36843}, {15699, 16773}, {15708, 22235}, {15713, 33417}, {15764, 35731}, {16645, 34755}, {16772, 34200}, {16809, 37641}, {16960, 19708}, {18581, 41106}, {19107, 33699}, {21359, 37170}, {21360, 37341}, {21969, 36981}, {22491, 31703}, {22513, 36383}, {22577, 35694}, {22580, 35697}, {22607, 36335}, {22609, 36371}, {22612, 36394}, {22636, 36334}, {22638, 36370}, {22641, 36392}, {22695, 36323}, {22701, 36364}, {22708, 36385}, {22849, 36324}, {22855, 33624}, {22862, 36388}, {22894, 33622}, {22907, 33626}, {22997, 36329}, {22999, 36367}, {23000, 36373}, {23001, 36375}, {23002, 36376}, {23003, 36377}, {23007, 36387}, {23023, 36389}, {23024, 36398}, {23025, 36399}, {23026, 36400}, {23027, 36401}, {23028, 36369}, {25166, 36327}, {25182, 36325}, {25199, 36338}, {25200, 36339}, {25201, 36340}, {25202, 36342}, {25228, 36321}, {25236, 36331}, {31693, 33459}, {35733, 35736}, {35734, 35739}, {36402, 41116}, {36403, 41097}, {37173, 41129}, {41016, 41021}, {41024, 41030}, {41026, 41036}, {41029, 41039}

X(41107) = reflection of X(i) in X(j) for these (i, j): (14904, 5459), (41108, 39593)
X(41107) = homothetic center of these pairs of triangles: (3rd inner-Fermat-Dao-Nhi, 1st isodynamic-Dao), (4th outer-Fermat-Dao-Nhi, 2nd Lemoine-Dao), (15th Fermat-Dao, 1st outer-Fermat-Dao-Nhi)
X(41107) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 13, 41121), (2, 5335, 41112), (2, 10653, 41100), (2, 41100, 16), (2, 41112, 13), (2, 41121, 37832), (4, 41113, 12817), (6, 3830, 41108), (6, 36969, 36970), (13, 16, 37832), (13, 6779, 6115), (13, 10653, 16), (13, 16242, 18582), (13, 41100, 2), (16, 41121, 2), (3830, 41108, 36970), (5335, 10653, 13), (10653, 41112, 2), (36969, 41108, 3830), (41099, 41120, 16809), (41100, 41112, 41121)

X(41108) = X(2)X(14) ∩ X(3)X(16268)

Barycentrics    -6*S*a^2*sqrt(3)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(41108) = X(62)-4*X(398) = X(62)+2*X(16964) = 3*X(62)-2*X(41100) = 2*X(398)+X(16964) = 6*X(398)-X(41100) = 3*X(16964)+X(41100)

X(41108) lies on these lines: {2, 14}, {3, 16268}, {4, 12816}, {5, 16962}, {6, 3830}, {13, 3845}, {16, 3534}, {17, 3545}, {18, 549}, {30, 62}, {51, 30439}, {61, 381}, {182, 13102}, {203, 11238}, {303, 33561}, {376, 5237}, {395, 8703}, {396, 5066}, {397, 15687}, {472, 6110}, {533, 7768}, {538, 3104}, {550, 3411}, {1151, 35735}, {1327, 36449}, {1328, 36468}, {2549, 9113}, {3091, 3412}, {3364, 35822}, {3365, 35823}, {3543, 16965}, {3839, 5343}, {3860, 11542}, {5054, 5238}, {5055, 22236}, {5097, 13103}, {5318, 12101}, {5340, 38335}, {5349, 14893}, {5351, 15688}, {5463, 22856}, {5469, 22846}, {5470, 5479}, {5476, 36757}, {5480, 36961}, {5863, 22494}, {6560, 36450}, {6561, 36467}, {6772, 6777}, {6776, 36962}, {6778, 36318}, {6782, 36769}, {7005, 11237}, {7812, 22495}, {8584, 22580}, {8739, 18559}, {9116, 35691}, {9761, 41105}, {10109, 16966}, {10645, 15693}, {10653, 15682}, {10706, 36208}, {11001, 34755}, {11299, 34508}, {11480, 15701}, {11481, 15695}, {11485, 19709}, {11486, 15685}, {11489, 15698}, {11543, 12100}, {11812, 23303}, {12821, 33602}, {14538, 20426}, {15681, 22238}, {15689, 36843}, {15694, 36836}, {15699, 16772}, {15708, 22237}, {15713, 33416}, {16644, 34754}, {16773, 34200}, {16808, 37640}, {16961, 19708}, {18582, 41106}, {19106, 33699}, {21359, 37340}, {21360, 37171}, {21969, 36979}, {22492, 31704}, {22512, 36382}, {22578, 35690}, {22579, 35693}, {22608, 36333}, {22610, 36374}, {22611, 36391}, {22637, 36332}, {22639, 36372}, {22640, 36390}, {22696, 36322}, {22702, 36365}, {22707, 36384}, {22850, 33624}, {22861, 33627}, {22895, 36326}, {22901, 33622}, {22906, 36386}, {22998, 35751}, {23008, 36369}, {23009, 36378}, {23010, 36379}, {23011, 36380}, {23012, 36381}, {23014, 36389}, {23017, 36387}, {23018, 36393}, {23019, 36395}, {23020, 36396}, {23021, 36397}, {23022, 36367}, {25156, 35749}, {25177, 36328}, {25203, 36336}, {25204, 36337}, {25205, 36341}, {25206, 36343}, {25227, 36354}, {25235, 35750}, {31694, 33458}, {35734, 36470}, {36767, 36772}, {37172, 41128}, {41017, 41020}, {41025, 41031}, {41027, 41037}, {41028, 41038}, {41091, 41131}, {41093, 41130}

X(41108) = reflection of X(i) in X(j) for these (i, j): (14905, 5460), (41107, 39593)
X(41108) = homothetic center of these pairs of triangles: (3rd outer-Fermat-Dao-Nhi, 2nd isodynamic-Dao), (4th inner-Fermat-Dao-Nhi, 1st Lemoine-Dao), (16th Fermat-Dao, 1st inner-Fermat-Dao-Nhi)
X(41108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14, 41122), (2, 5334, 41113), (2, 10654, 41101), (2, 41101, 15), (2, 41113, 14), (2, 41122, 37835), (4, 41112, 12816), (6, 3830, 41107), (6, 36970, 36969), (14, 15, 37835), (14, 6780, 6114), (14, 10654, 15), (14, 16241, 18581), (14, 41101, 2), (15, 41122, 2), (3830, 41107, 36969), (5334, 10654, 14), (10654, 41113, 2), (36970, 41107, 3830), (41099, 41119, 16808), (41101, 41113, 41122)

X(41109) = X(3)X(3276) ∩ X(358)X(5454)

Barycentrics    a*(((y^2-z^2)*u*x-((2*x*y-z)*v-(2*x*z-y)*w)*y*z)*SA*a+(2*y*z-x)*((u*x-w*z)*SB*b*y-(u*x-v*y)*SC*c*z)) : : ,
  where x : y : z = cos(A/3+Pi/3) : : and u : v : w = -cos(A/3+2*Pi/3) : :

X(41109) lies on these lines: {3, 3276}, {358, 5454}, {1135, 1136}

X(41109) = orthologic center of 2nd Morley-midpoint triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41109) = {X(31931), X(31934)}-harmonic conjugate of X(3)

X(41110) = X(3)X(3277) ∩ X(358)X(1134)

Barycentrics    a*(((y^2-z^2)*u*x-((2*x*y-z)*v-(2*x*z-y)*w)*y*z)*SA*a+(2*y*z-x)*((u*x-w*z)*SB*b*y-(u*x-v*y)*SC*c*z)) : : ,
  where x : y : z = cos(A/3-Pi/3) : : and u : v : w = -cos(A/3) : :

X(41111) is the center of the equilateral triangle described in Hyacinthos #21394 (Antreas Hatzipolakis, January 12, 2013). (Randy Hutson, April 13, 2021)

X(41110) lies on these lines: {3, 3277}, {358, 1134}, {3605, 7309}

X(41110) = orthologic center of 3rd Morley-midpoint triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41110) = {X(31932), X(31935)}-harmonic conjugate of X(3)

X(41111) = X(3)X(356) ∩ X(357)X(1137)

Barycentrics    a*(((y^2-z^2)*u*x-((2*x*y-z)*v-(2*x*z-y)*w)*y*z)*SA*a+(2*y*z-x)*((u*x-w*z)*SB*b*y-(u*x-v*y)*SC*c*z)) : :, where x : y : z = -cos(A/3) : : and u : v : w = -cos(A/3-2*Pi/3) : :

X(41111) lies on these lines: {3, 356}, {357, 1137}, {3606, 5456}

X(41111) = orthologic center of 1st Morley-midpoint triangle to these triangles: 1st-/2nd-/3rd- Morley, 2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41111) = {X(31930), X(31933)}-harmonic conjugate of X(3)

X(41112) = X(2)X(13) ∩ X(4)X(12816)

Barycentrics    6*S*a^2*sqrt(3)+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(41112) = 5*X(5340)+X(22236) = 2*X(5340)+X(40693) = 2*X(22236)-5*X(40693)

X(41112) lies on these lines: {2, 13}, {4, 12816}, {6, 1327}, {14, 33602}, {15, 11001}, {17, 3524}, {20, 16962}, {30, 5340}, {61, 3543}, {62, 3545}, {115, 36382}, {376, 5352}, {381, 397}, {395, 19709}, {396, 3534}, {398, 14269}, {532, 37170}, {547, 22238}, {1503, 41028}, {2043, 8960}, {3091, 16268}, {3411, 5068}, {3412, 3529}, {3830, 5318}, {5066, 18581}, {5071, 16963}, {5237, 15702}, {5334, 12817}, {5339, 14893}, {5350, 38335}, {5351, 15708}, {5366, 16964}, {5464, 35695}, {5470, 6773}, {5472, 11648}, {5478, 6776}, {5480, 41029}, {5858, 31693}, {5863, 22495}, {6111, 41092}, {6564, 36467}, {6565, 36450}, {6778, 36319}, {6783, 36329}, {8703, 11542}, {9763, 41129}, {10109, 16645}, {10646, 15719}, {11296, 33458}, {11480, 15690}, {11481, 11812}, {11488, 19708}, {11539, 36843}, {12100, 16644}, {15640, 19106}, {15682, 36969}, {15686, 36836}, {15688, 16772}, {15692, 22235}, {15697, 16960}, {15698, 16241}, {15701, 23302}, {15703, 16773}, {16808, 37641}, {22609, 36349}, {22638, 36348}, {22701, 36345}, {22855, 36346}, {22894, 36352}, {22900, 33626}, {22907, 36366}, {22997, 36331}, {22999, 36321}, {23000, 36350}, {23001, 36351}, {23002, 36353}, {23003, 36355}, {23004, 36327}, {23007, 36325}, {31695, 35693}, {31697, 36391}, {31699, 36390}, {31701, 36384}, {31703, 33627}, {31705, 36386}, {31707, 36387}, {31709, 36330}, {31711, 36393}, {31713, 36395}, {31715, 36396}, {31717, 36397}, {31719, 36367}, {32787, 36466}, {32788, 36448}, {35735, 35740}, {35737, 35739}, {41032, 41036}

X(41112) = homothetic center of these pairs of triangles: (4th inner-Fermat-Dao-Nhi, 3rd isodynamic-Dao), (2nd outer-Fermat-Dao-Nhi, 1st isodynamic-Dao)
X(41112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 13, 41119), (2, 5335, 41107), (2, 41107, 10653), (2, 41119, 18582), (6, 3845, 41113), (13, 5335, 10653), (13, 10653, 18582), (13, 41100, 41121), (13, 41107, 2), (10653, 41119, 2), (12816, 41108, 4), (15682, 37640, 41101), (16267, 16965, 376), (16808, 41122, 41106), (36969, 41101, 15682), (37641, 41106, 41122), (41100, 41121, 2), (41107, 41121, 41100)

X(41113) = X(2)X(14) ∩ X(4)X(12817)

Barycentrics    -6*S*a^2*sqrt(3)+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(41113) = 5*X(5339)+X(22238) = 2*X(5339)+X(40694) = 2*X(22238)-5*X(40694)

X(41113) lies on these lines: {2, 14}, {4, 12817}, {6, 1327}, {13, 33603}, {16, 11001}, {18, 3524}, {20, 16963}, {30, 5339}, {61, 3545}, {62, 3543}, {115, 36383}, {376, 5351}, {381, 398}, {395, 3534}, {396, 19709}, {397, 14269}, {533, 37171}, {547, 22236}, {1503, 41029}, {2044, 8960}, {3091, 16267}, {3411, 3529}, {3412, 5068}, {3830, 5321}, {5066, 18582}, {5071, 16962}, {5238, 15702}, {5335, 12816}, {5340, 14893}, {5349, 38335}, {5352, 15708}, {5365, 16965}, {5463, 35691}, {5469, 6770}, {5471, 11648}, {5479, 6776}, {5480, 41028}, {5859, 31694}, {5862, 22496}, {6564, 36449}, {6565, 36468}, {6777, 36344}, {6782, 35751}, {8703, 11543}, {9761, 41128}, {10109, 16644}, {10645, 15719}, {11295, 33459}, {11480, 11812}, {11481, 15690}, {11489, 19708}, {11539, 36836}, {12100, 16645}, {15640, 19107}, {15682, 36970}, {15686, 36843}, {15688, 16773}, {15692, 22237}, {15697, 16961}, {15698, 16242}, {15701, 23303}, {15703, 16772}, {16809, 37640}, {22610, 36357}, {22639, 36356}, {22702, 36347}, {22850, 36346}, {22856, 33627}, {22861, 36368}, {22901, 36352}, {22998, 35750}, {23005, 35749}, {23008, 36354}, {23009, 36358}, {23010, 36359}, {23011, 36360}, {23012, 36361}, {23014, 36328}, {31696, 35697}, {31698, 36394}, {31700, 36392}, {31702, 36385}, {31704, 33626}, {31706, 36388}, {31708, 36389}, {31710, 35752}, {31712, 36398}, {31714, 36399}, {31716, 36401}, {31718, 36400}, {31720, 36369}, {32787, 36448}, {32788, 36466}, {36403, 41124}, {41033, 41037}

X(41113) = homothetic center of these pairs of triangles: (2nd inner-Fermat-Dao-Nhi, 2nd isodynamic-Dao), (4th outer-Fermat-Dao-Nhi, 4th isodynamic-Dao)
X(41113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14, 41120), (2, 5334, 41108), (2, 41108, 10654), (2, 41120, 18581), (6, 3845, 41112), (14, 5334, 10654), (14, 10654, 18581), (14, 41101, 41122), (14, 41108, 2), (10654, 41120, 2), (12817, 41107, 4), (15682, 37641, 41100), (16268, 16964, 376), (16809, 41121, 41106), (36970, 41100, 15682), (37640, 41106, 41121), (41101, 41122, 2), (41108, 41122, 41101)

X(41114) = X(30)X(16631) ∩ X(62)X(381)

Barycentrics    -2*(4*a^4+(b^2+c^2)*a^2-14*(b^2-c^2)^2)*S+sqrt(3)*(4*a^6-(b^2+c^2)*a^4-(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)) : :
X(41114) = 3*X(16630)-X(41105)

X(41114) lies on these lines: {30, 16631}, {62, 381}, {531, 41104}, {3830, 19781}, {5470, 41101}, {9763, 18546}, {11645, 41115}, {18362, 36969}, {22491, 41117}

X(41114) = center of inverse similitude of these pairs of triangles: (1st inner-Fermat-Dao-Nhi, 15th Fermat-Dao), (3rd outer-Fermat-Dao-Nhi, 1st isodynamic-Dao)

X(41115) = X(30)X(16630) ∩ X(61)X(381)

Barycentrics    2*(4*a^4+(b^2+c^2)*a^2-14*(b^2-c^2)^2)*S+sqrt(3)*(4*a^6-(b^2+c^2)*a^4-(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)) : :
X(41115) = 3*X(16631)-X(41104)

X(41115) lies on these lines: {30, 16630}, {61, 381}, {530, 41105}, {3830, 19780}, {5469, 41100}, {9761, 18546}, {11645, 41114}, {18362, 36970}, {22492, 41118}

X(41115) = center of inverse similitude of these pairs of triangles: (3rd inner-Fermat-Dao-Nhi, 2nd isodynamic-Dao), (1st outer-Fermat-Dao-Nhi, 16th Fermat-Dao)

X(41116) = X(13)X(8010) ∩ X(16)X(8011)

Barycentrics    (-a*(5*S^2-3*SB*SC)*sin(A)+(S^2+3*SB*SC)*(b*sin(B)+c*sin(C)))*sqrt(3)-4*U*(S^2+3*SB*SC+S*a^2*sqrt(3))-4*a*sin((B-C)/3)^2*(cos((B-C)/3)-cos(A+Pi/3))*(S*a^2*sqrt(3)-5*S^2+3*SB*SC)+6*S*a*SW*sin(A) : : ,
  where U:V:W = (cos(A/3-C/3)-cos(B+Pi/3))*sin(A/3-C/3)^2*b+(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :

X(41116) lies on these lines: {13, 8010}, {16, 8011}, {381, 41126}, {395, 41123}, {5318, 41096}, {10653, 41097}, {11486, 41095}, {16809, 41127}, {16965, 41131}, {18582, 33492}, {36402, 41107}, {36403, 41100}, {37641, 41124}

X(41116) = reflection of X(41097) in X(10653)
X(41116) = orthologic center of 2nd Lemoine-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler

X(41117) = X(381)X(397) ∩ X(22491)X(41114)

Barycentrics    -6*S*(5*a^6-(b^2+c^2)*a^4-7*(b^2-c^2)^2*a^2-9*(b^4-c^4)*(b^2-c^2))*sqrt(3)+25*a^8+20*(b^2+c^2)*a^6-80*b^2*c^2*a^4-4*(b^2+c^2)*(13*b^4-29*b^2*c^2+13*c^4)*a^2+(b^2-c^2)^2*(7*b^4-74*b^2*c^2+7*c^4) : :

X(41117) lies on these lines: {381, 397}, {531, 41129}, {9763, 41119}, {22491, 41114}

X(41117) = center of inverse similitude of these pairs of triangles: (2nd inner-Fermat-Dao-Nhi, 1st isodynamic-Dao), (4th outer-Fermat-Dao-Nhi, 3rd isodynamic-Dao)

X(41118) = X(381)X(398) ∩ X(22492)X(41115)

Barycentrics    6*S*(5*a^6-(b^2+c^2)*a^4-7*(b^2-c^2)^2*a^2-9*(b^4-c^4)*(b^2-c^2))*sqrt(3)+25*a^8+20*(b^2+c^2)*a^6-80*b^2*c^2*a^4-4*(b^2+c^2)*(13*b^4-29*b^2*c^2+13*c^4)*a^2+(b^2-c^2)^2*(7*b^4-74*b^2*c^2+7*c^4) : :

X(41118) lies on these lines: {381, 398}, {530, 41128}, {9761, 41120}, {22492, 41115}

X(41118) = center of inverse similitude of these pairs of triangles: (4th inner-Fermat-Dao-Nhi, 4th isodynamic-Dao), (2nd outer-Fermat-Dao-Nhi, 2nd isodynamic-Dao)

X(41119) = X(2)X(13) ∩ X(4)X(3412)

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

X(41119) lies on these lines: {2, 13}, {4, 3412}, {6, 5066}, {14, 33604}, {15, 12816}, {17, 376}, {30, 36836}, {61, 3839}, {62, 5071}, {69, 33560}, {115, 36362}, {381, 398}, {396, 3830}, {397, 5055}, {549, 5340}, {1352, 20252}, {1503, 41026}, {3090, 16963}, {3411, 15022}, {3524, 16965}, {3534, 5318}, {3543, 16962}, {3545, 40694}, {3845, 10654}, {5237, 15709}, {5238, 5366}, {5339, 23046}, {5344, 10304}, {5350, 15684}, {5351, 15721}, {5464, 35694}, {5472, 18362}, {5480, 41027}, {5859, 31693}, {5862, 22495}, {5863, 31705}, {6564, 36468}, {6565, 36449}, {6778, 36320}, {6783, 36330}, {8703, 16644}, {9763, 41117}, {10124, 36843}, {10645, 15697}, {11001, 11488}, {11480, 19710}, {11481, 15713}, {11624, 18435}, {15640, 36967}, {15681, 16772}, {15687, 22236}, {15693, 23302}, {15698, 36968}, {15699, 22238}, {16241, 19708}, {16635, 36383}, {16808, 37640}, {18581, 19709}, {18587, 31454}, {22609, 36335}, {22638, 36334}, {22701, 36323}, {22855, 36324}, {22894, 36326}, {22900, 33622}, {22997, 36327}, {22999, 36325}, {23000, 36338}, {23001, 36339}, {23002, 36340}, {23003, 36342}, {23004, 36331}, {23007, 36321}, {31695, 35692}, {31697, 36371}, {31699, 36370}, {31701, 36364}, {31703, 33624}, {31707, 36367}, {31709, 36329}, {31711, 36373}, {31713, 36375}, {31715, 36376}, {31717, 36377}, {31719, 36387}, {34509, 37170}, {35734, 35740}, {35736, 35739}, {41030, 41036}

X(41119) = homothetic center of these pairs of triangles: (3rd inner-Fermat-Dao-Nhi, 3rd isodynamic-Dao), (1st outer-Fermat-Dao-Nhi, 1st isodynamic-Dao)
X(41119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 13, 41112), (2, 5335, 41100), (2, 41112, 10653), (2, 41121, 18582), (6, 5066, 41120), (13, 18582, 10653), (13, 37832, 5335), (13, 41121, 2), (15, 12816, 15682), (16808, 41108, 41099), (18582, 41112, 2), (33607, 41101, 16267), (37640, 41099, 41108), (37832, 41100, 2)

X(41120) = X(2)X(14) ∩ X(4)X(3411)

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

X(41120) lies on these lines: {2, 14}, {4, 3411}, {6, 5066}, {13, 33605}, {16, 12817}, {18, 376}, {30, 36843}, {61, 5071}, {62, 3839}, {69, 33561}, {115, 36363}, {381, 397}, {395, 3830}, {398, 5055}, {549, 5339}, {1352, 20253}, {1503, 41027}, {3090, 16962}, {3412, 15022}, {3524, 16964}, {3534, 5321}, {3543, 16963}, {3545, 40693}, {3845, 10653}, {5237, 5365}, {5238, 15709}, {5340, 23046}, {5343, 10304}, {5349, 15684}, {5352, 15721}, {5463, 35690}, {5471, 18362}, {5480, 41026}, {5858, 31694}, {5862, 31706}, {5863, 22496}, {6564, 36450}, {6565, 36467}, {6777, 36318}, {6782, 35752}, {8703, 16645}, {9761, 41118}, {10124, 36836}, {10646, 15697}, {11001, 11489}, {11480, 15713}, {11481, 19710}, {11626, 18435}, {15640, 36968}, {15681, 16773}, {15687, 22238}, {15693, 23303}, {15698, 36967}, {15699, 22236}, {16242, 19708}, {16634, 36382}, {16809, 37641}, {18582, 19709}, {18586, 31454}, {22610, 36333}, {22639, 36332}, {22702, 36322}, {22850, 36324}, {22856, 33624}, {22901, 36326}, {22998, 35749}, {23005, 35750}, {23008, 36328}, {23009, 36336}, {23010, 36337}, {23011, 36341}, {23012, 36343}, {23014, 36354}, {31696, 35696}, {31698, 36374}, {31700, 36372}, {31702, 36365}, {31704, 33622}, {31708, 36369}, {31710, 35751}, {31712, 36378}, {31714, 36379}, {31716, 36381}, {31718, 36380}, {31720, 36389}, {34508, 37171}, {36402, 41124}, {41031, 41037}

X(41120) = homothetic center of these pairs of triangles: (3rd outer-Fermat-Dao-Nhi, 4th isodynamic-Dao), (1st inner-Fermat-Dao-Nhi, 2nd isodynamic-Dao)
X(41120) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14, 41113), (2, 5334, 41101), (2, 41113, 10654), (2, 41122, 18581), (6, 5066, 41119), (14, 18581, 10654), (14, 37835, 5334), (14, 41122, 2), (16, 12817, 15682), (16809, 41107, 41099), (18581, 41113, 2), (33606, 41100, 16268), (37641, 41099, 41107), (37835, 41101, 2)

X(41121) = X(2)X(13) ∩ X(4)X(16962)

Barycentrics    6*sqrt(3)*S*a^2+a^4+7*(b^2+c^2)*a^2-8*(b^2-c^2)^2 : :
X(41121) = 4*X(17)-X(5238) = 7*X(17)+2*X(5350) = 3*X(17)+X(12816) = 5*X(17)-2*X(16772) = 7*X(5238)+8*X(5350) = 3*X(5238)+4*X(12816) = 5*X(5238)-8*X(16772) = 6*X(5350)-7*X(12816) = 5*X(5350)+7*X(16772) = 5*X(12816)+6*X(16772)

X(41121) lies on these lines: {2, 13}, {4, 16962}, {5, 16268}, {6, 18362}, {14, 5066}, {15, 3830}, {17, 30}, {18, 5071}, {61, 381}, {62, 5055}, {299, 33560}, {395, 10109}, {396, 3845}, {397, 547}, {398, 38071}, {524, 36366}, {531, 31705}, {542, 10611}, {546, 3412}, {549, 16965}, {635, 33413}, {1992, 36368}, {3105, 9466}, {3411, 5056}, {3534, 10645}, {3545, 40693}, {3839, 16964}, {3855, 33603}, {3860, 5321}, {5054, 5340}, {5237, 15694}, {5318, 8703}, {5344, 15692}, {5352, 15681}, {5464, 35693}, {5469, 14136}, {5470, 5613}, {5858, 7775}, {5863, 22493}, {5943, 30440}, {6564, 36466}, {6565, 36448}, {6777, 36319}, {6778, 36382}, {7849, 11305}, {9114, 35694}, {9760, 22571}, {9763, 18546}, {10188, 10303}, {10646, 15701}, {10654, 12817}, {11001, 19106}, {11178, 22846}, {11180, 41128}, {11480, 15685}, {11488, 15682}, {11624, 15060}, {11812, 33417}, {12100, 23302}, {12101, 19107}, {14269, 22236}, {14568, 22496}, {15534, 41098}, {15684, 36836}, {15698, 33602}, {15703, 22238}, {16529, 25164}, {16809, 37640}, {18435, 30439}, {18586, 35813}, {18587, 35812}, {21360, 37352}, {21849, 36981}, {22235, 33606}, {22510, 41043}, {22577, 35695}, {22607, 36349}, {22609, 36391}, {22636, 36348}, {22638, 36390}, {22695, 36345}, {22701, 36384}, {22849, 36346}, {22855, 33627}, {22894, 33626}, {22900, 36386}, {22997, 36330}, {22999, 36387}, {23000, 36393}, {23001, 36395}, {23002, 36396}, {23003, 36397}, {23004, 36329}, {23007, 36367}, {25151, 25224}, {25166, 36331}, {25180, 25217}, {25182, 36321}, {25199, 36350}, {25200, 36351}, {25201, 36353}, {25202, 36355}, {25228, 36325}, {25236, 36327}, {31693, 33458}, {31703, 36388}, {35733, 35737}, {35735, 35739}, {41024, 41032}, {41028, 41036}

X(41121) = homothetic center of these pairs of triangles: (4th inner-Fermat-Dao-Nhi, 1st isodynamic-Dao), (15th Fermat-Dao, 2nd outer-Fermat-Dao-Nhi)
X(41121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 13, 41107), (2, 41107, 16), (2, 41112, 41100), (2, 41119, 13), (6, 19709, 41122), (13, 16242, 5335), (13, 16966, 10653), (13, 18582, 37832), (13, 37832, 16), (13, 41100, 41112), (381, 16267, 61), (396, 3845, 41101), (396, 16808, 36970), (396, 36970, 34754), (397, 547, 16963), (3845, 41101, 36970), (16808, 41101, 3845), (18582, 41119, 2), (37832, 41107, 2), (41100, 41112, 41107), (41106, 41113, 16809)

X(41122) = X(2)X(14) ∩ X(4)X(16963)

Barycentrics    -6*sqrt(3)*S*a^2+a^4+7*(b^2+c^2)*a^2-8*(b^2-c^2)^2 : :
X(41122) = 4*X(18)-X(5237) = 7*X(18)+2*X(5349) = 3*X(18)+X(12817) = 5*X(18)-2*X(16773) = 7*X(5237)+8*X(5349) = 3*X(5237)+4*X(12817) = 5*X(5237)-8*X(16773) = 6*X(5349)-7*X(12817) = 5*X(5349)+7*X(16773) = 5*X(12817)+6*X(16773)

X(41122) lies on these lines: {2, 14}, {4, 16963}, {5, 16267}, {6, 18362}, {13, 5066}, {16, 3830}, {17, 5071}, {18, 30}, {61, 5055}, {62, 381}, {298, 33561}, {395, 3845}, {396, 10109}, {397, 38071}, {398, 547}, {524, 36368}, {530, 31706}, {542, 10612}, {546, 3411}, {549, 16964}, {636, 33412}, {1992, 36366}, {3104, 9466}, {3412, 5056}, {3534, 10646}, {3545, 40694}, {3839, 16965}, {3855, 33602}, {3860, 5318}, {5054, 5339}, {5238, 15694}, {5321, 8703}, {5343, 15692}, {5351, 15681}, {5463, 35697}, {5469, 5617}, {5470, 14137}, {5859, 7775}, {5862, 22494}, {5943, 30439}, {6564, 36448}, {6565, 36466}, {6777, 36383}, {6778, 36344}, {7849, 11306}, {9116, 35690}, {9761, 18546}, {9762, 22572}, {10187, 10303}, {10645, 15701}, {10653, 12816}, {11001, 19107}, {11178, 22891}, {11180, 41129}, {11481, 15685}, {11489, 15682}, {11626, 15060}, {11812, 33416}, {12100, 23303}, {12101, 19106}, {14269, 22238}, {14568, 22495}, {15534, 41094}, {15684, 36843}, {15698, 33603}, {15703, 22236}, {16530, 25154}, {16808, 37641}, {18435, 30440}, {18586, 35812}, {18587, 35813}, {21359, 37351}, {21849, 36979}, {22237, 33607}, {22511, 41042}, {22578, 35691}, {22608, 36357}, {22610, 36394}, {22637, 36356}, {22639, 36392}, {22696, 36347}, {22702, 36385}, {22850, 33627}, {22856, 36388}, {22895, 36352}, {22901, 33626}, {22998, 35752}, {23005, 35751}, {23008, 36389}, {23009, 36398}, {23010, 36399}, {23011, 36400}, {23012, 36401}, {23014, 36369}, {25156, 35750}, {25161, 25223}, {25175, 25214}, {25177, 36354}, {25203, 36358}, {25204, 36359}, {25205, 36360}, {25206, 36361}, {25227, 36328}, {25235, 35749}, {31694, 33459}, {31704, 36386}, {31710, 36769}, {35739, 36452}, {41025, 41033}, {41029, 41037}, {41091, 41093}, {41096, 41103}, {41123, 41127}, {41130, 41131}

X(41122) = homothetic center of these pairs of triangles: (16th Fermat-Dao, 2nd inner-Fermat-Dao-Nhi), (4th outer-Fermat-Dao-Nhi, 2nd isodynamic-Dao)
X(41122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14, 41108), (2, 41108, 15), (2, 41113, 41101), (2, 41120, 14), (6, 19709, 41121), (14, 16241, 5334), (14, 16967, 10654), (14, 18581, 37835), (14, 37835, 15), (14, 41101, 41113), (381, 16268, 62), (395, 3845, 41100), (395, 16809, 36969), (395, 36969, 34755), (398, 547, 16962), (3845, 41100, 36969), (16809, 41100, 3845), (18581, 41120, 2), (37835, 41108, 2), (41101, 41113, 41108), (41106, 41112, 16808)

X(41123) = X(3)X(33493) ∩ X(5)X(8010)

Barycentrics    (-a^3*SA*sin(A)+(S^2+SB*SC)*(b*sin(B)+c*sin(C)))*sqrt(3)+4*a^3*sin((B-C)/3)^2*SA*(cos((B-C)/3)-cos(A+Pi/3))-4*U*(S^2+SB*SC) : : ,
  where U:V:W = (-cos(B+Pi/3)+cos(A/3-C/3))*b*sin(A/3-C/3)^2+(cos(A/3-B/3)-cos(C+Pi/3))*c*sin(A/3-B/3)^2 : :
X(41123) = 2*X(3845)+X(36403) = 3*X(5055)-2*X(33492) = 4*X(5066)-X(36402) = X(41091)-7*X(41106) = 5*X(41099)+X(41130)

X(41123) lies on these lines: {3, 33493}, {5, 8010}, {6, 41124}, {30, 8011}, {62, 41103}, {381, 41096}, {395, 41116}, {3845, 36403}, {5055, 33492}, {5066, 36402}, {5318, 41097}, {10653, 41095}, {16808, 41093}, {18581, 41126}, {41091, 41106}, {41099, 41130}, {41122, 41127}

X(41123) = reflection of X(i) in X(j) for these (i, j): (3, 33493), (8010, 5), (41096, 381)
X(41123) = X(8010)-of-Johnson-triangle
X(41123) = parallelogic center of 12th Fermat-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-, 3rd- Morley-midpoint; circumnormal, circumtangential, Roussel, Stammler

X(41124) = X(15)X(33493) ∩ X(8010)X(18581)

Barycentrics    -2*(a*SW*sin(A)*S-2*a^3*sin((B-C)/3)^2*SA*(cos((B-C)/3)-cos(A+Pi/3))+(b^2+c^2)*U*a^2-(b^2-c^2)^2*U)*sqrt(3)-3*a^3*SA*sin(A)+3*(S^2+SB*SC)*(b*sin(B)+c*sin(C))+4*S*a^2*((cos((B-C)/3)-cos(A+Pi/3))*sin((B-C)/3)^2*a+(-cos(B+Pi/3)+cos(A/3-C/3))*b*sin(A/3-C/3)^2+(cos(A/3-B/3)-cos(C+Pi/3))*c*sin(A/3-B/3)^2) : : ,
  where U:V:W = (-cos(B+Pi/3)+cos(A/3-C/3))*b*sin(A/3-C/3)^2+(cos(A/3-B/3)-cos(C+Pi/3))*c*sin(A/3-B/3)^2 : :
X(41124) = 3*X(14)+X(41093) = 5*X(14)-X(41131) = X(16)-3*X(41103) = 5*X(41093)+3*X(41131)

X(41124) lies on these lines: {4, 41097}, {6, 41123}, {14, 41093}, {15, 33493}, {16, 41103}, {30, 41095}, {8010, 18581}, {8011, 10654}, {11543, 41126}, {33492, 37835}, {36402, 41120}, {36403, 41113}, {37641, 41116}

X(41124) = reflection of X(41126) in X(11543)
X(41124) = paralleogic center of 4th isodynamic-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-, 3rd- Morley-midpoint; circumnormal, circumtangential, Roussel, Stammler

X(41125) = X(110)X(671) ∩ X(115)X(9123)

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

X(41125) lies on these lines: {110, 671}, {115, 9123}, {3124, 8599}, {8352, 40915}, {15724, 31644}

X(41125) = center of direct similitude of these pairs of triangles: (medial, 1st Parry), (Euler, 2nd Parry)

X(41126) = X(6)X(8010) ∩ X(396)X(33492)

Barycentrics    a*(-2*(3*S*sin(A)+2*(cos((B-C)/3)-cos(A+Pi/3))*a^2*sin((B-C)/3)^2+2*U*a)*sqrt(3)+24*(cos((B-C)/3)-cos(A+Pi/3))*S*sin((B-C)/3)^2+6*SW*sin(A)) : : ,
  where U:V:W = (-cos(B+Pi/3)+cos(A/3-C/3))*b*sin(A/3-C/3)^2+(cos(A/3-B/3)-cos(C+Pi/3))*c*sin(A/3-B/3)^2 : :
X(41126) = X(14)-3*X(41127) = 5*X(16961)-X(41093) = 5*X(16961)-3*X(41103) = 3*X(16963)+X(41131) = X(41093)-3*X(41103)

X(41126) lies on these lines: {6, 8010}, {14, 41127}, {381, 41116}, {395, 41095}, {396, 33492}, {5637, 16271}, {8011, 16645}, {10653, 41096}, {11486, 41097}, {11543, 41124}, {16961, 41093}, {16963, 41131}, {18581, 41123}, {23303, 33493}

X(41126) = reflection of X(i) in X(j) for these (i, j): (41095, 395), (41124, 11543)
X(41126) = orthologic center of 8th Fermat-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41126) = {X(16961), X(41093)}-harmonic conjugate of X(41103)

X(41127) = X(18)X(8011) ∩ X(61)X(33492)

Barycentrics    b*c*(2*a^2*(3*sqrt(3)*S*a^2-13*S^2-3*SB*SC)*x+2*a*(b*y+c*z)*(3*sqrt(3)*S*a^2-S^2-3*SB*SC)+sin(A)*((S^2*(7*SW-6*SA)+3*SB*SW*SC)*sqrt(3)-9*S*SW*a^2)) : : ,
  where x : y : z = (cos((B-C)/3)-cos(A+Pi/3))*sin((B-C)/3)^2 : :
X(41127) = X(14)+2*X(41126) = 5*X(16961)-2*X(41095) = 5*X(16961)+X(41131) = 2*X(41095)+X(41131)

X(41127) lies on these lines: {14, 41126}, {18, 8011}, {61, 33492}, {62, 41096}, {16268, 41103}, {16809, 41116}, {16961, 41095}, {41122, 41123}

X(41127) = reflection of X(41103) in X(16268)
X(41127) = orthologic center of 4th Fermat-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41127) = {X(16961), X(41131)}-harmonic conjugate of X(41095)

X(41128) = X(381)X(397) ∩ X(530)X(41118)

Barycentrics    -2*(7*a^4-5*(b^2+c^2)*a^2+16*(b^2-c^2)^2)*S+sqrt(3)*(14*a^6-17*(b^2+c^2)*a^4+10*(b^4+c^4)*a^2-7*(b^4-c^4)*(b^2-c^2)) : :

X(41128) lies on these lines: {381, 397}, {530, 41118}, {9761, 41113}, {11180, 41121}, {37172, 41108}

X(41128) = center of inverse similitude of these pairs of triangles: (1st outer-Fermat-Dao-Nhi, 2nd isodynamic-Dao), (3rd inner-Fermat-Dao-Nhi, 4th isodynamic-Dao)

X(41129) = X(381)X(398) ∩ X(531)X(41117)

Barycentrics    2*(7*a^4-5*(b^2+c^2)*a^2+16*(b^2-c^2)^2)*S+sqrt(3)*(14*a^6-17*(b^2+c^2)*a^4+10*(b^4+c^4)*a^2-7*(b^4-c^4)*(b^2-c^2)) : :

X(41129) lies on these lines: {381, 398}, {531, 41117}, {9763, 41112}, {11180, 41122}, {37173, 41107}

X(41129) = center of inverse similitude of these pairs of triangles: (1st inner-Fermat-Dao-Nhi, 1st isodynamic-Dao), (3rd outer-Fermat-Dao-Nhi, 3rd isodynamic-Dao)

X(41130) = X(2)X(8010) ∩ X(41099)X(41123)

Barycentrics    (-11*a*sin(A)+7*b*sin(B)+7*c*sin(C))*sqrt(3)+44*(cos((B-C)/3)-cos(A+Pi/3))*sin((B-C)/3)^2*a-28*(cos(-C/3+A/3)-cos(B+Pi/3))*sin(-C/3+A/3)^2*b-28*(cos(A/3-B/3)-cos(C+Pi/3))*sin(A/3-B/3)^2*c : :
X(41130) = 7*X(2)-6*X(8010) = 5*X(2)-6*X(8011) = 13*X(2)-12*X(33492) = 11*X(2)-12*X(33493) = 5*X(8010)-7*X(8011) = 13*X(8010)-14*X(33492) = 11*X(8010)-14*X(33493) = 9*X(8010)-7*X(36402) = 3*X(8010)-7*X(36403) = 12*X(8010)-7*X(41091) = 13*X(8011)-10*X(33492) = 11*X(8011)-10*X(33493) = 9*X(8011)-5*X(36402) = 3*X(8011)-5*X(36403) = 12*X(8011)-5*X(41091) = 11*X(33492)-13*X(33493) = 18*X(33492)-13*X(36402) = 6*X(33492)-13*X(36403) = 18*X(33493)-11*X(36402) = 6*X(33493)-11*X(36403)

X(41130) lies on these lines: {2, 8010}, {41093, 41108}, {41096, 41106}, {41099, 41123}, {41122, 41131}

X(41130) = reflection of X(i) in X(j) for these (i, j): (2, 36403), (41091, 2)
X(41130) = anticomplement of X(36402)
X(41130) = orthologic center of 1st- and 2nd- inner-Fermat-Dao-Nhi triangles to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler

X(41131) = X(14)X(41093) ∩ X(8010)X(16241)

Barycentrics    (S^2+sqrt(3)*SA*S-3*(SB+SC)*SA)*x+(4*S^2-sqrt(3)*(sqrt(3)*SA+S)*(SB+SC))*(y+z) : : ,
  where x:y:z = X(41093)
X(41131) = 5*X(14)-4*X(41124) = 5*X(16961)-4*X(41095) = 5*X(16961)-6*X(41127) = 3*X(16963)-4*X(41126) = 5*X(41093)-8*X(41124) = 2*X(41095)-3*X(41127)

X(41131) lies on these lines: {14, 41093}, {8010, 16241}, {8011, 16967}, {16808, 41096}, {16961, 41095}, {16963, 41126}, {16965, 41116}, {33417, 33492}, {36402, 41101}, {41091, 41108}, {41122, 41130}

X(41131) = reflection of X(41093) in X(14)
X(41131) = orthologic center of 16th Fermat-Dao triangle to these triangles: 1st-/2nd-/3rd- Morley, 1st-/2nd-/3rd- Morley-midpoint, circumnormal, circumtangential, Roussel, Stammler
X(41131) = {X(41095), X(41127)}-harmonic conjugate of X(16961)

X(41132) = X(36768)X(36788) ∩ X(36769)X(36774)

Barycentrics    9*(12*R^2-SA-2*SW)*S^2-9*(36*R^2-7*SW)*SB*SC-S*sqrt(3)*(23*S^2-12*R^2*(9*SA-2*SW)+27*SA^2-18*SB*SC-6*SW^2) : :
X(41132) = 4*X(36768)-3*X(36788) = 2*X(36769)-3*X(36774)

X(41132) lies on these lines: {2, 41092}, {23871, 36327}, {36331, 41023}, {36768, 36788}, {36769, 36774}, {41030, 41066}

X(41132) = anticomplement of X(41092)
X(41132) = orthologic center of triangles 1st outer-Fermat-Dao-Nhi to Vu-Dao-X(16)-isodynamic
X(41132) = parallelogic center of triangles 2nd outer-Fermat-Dao-Nhi to Vu-Dao-X(16)-isodynamic

leftri

Inverses-in-permutation ellipses: X(41133)-X(41154)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, January 25, 2021.

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC. Suppose further that p,q,r are distinct homogeneous functions of a,b,c. The permutation ellipse of P, denoted by E(P), is introduced in the preamble just before X(34341) as the ellipse that passes through the six points p : q : r, q : r : p, r : p : q, p : r : q, q : p : r, r : q : p, given by

(q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

The centroid, X(2), is the center and the perspector of E(P).

In C. Kimberling and P. Moses, 'Permutation Ellipses', Journal for Geometry and Graphics 24 (2020) 233-247, the following three special permutation ellipses are discussed: (1) trisector ellipse, (2) self-dual permutation ellipse; and (3) Steiner midway ellipse. Following are definitions and equations:

(1) The trisector ellipse passes through the points 0:1:2, 0:2:1, 1:0:2, 2:0:1, 1:2:0, 2:1:0. These six points trisect the sides of ABC. The ellipse is given by 2(x^2 + y^2 + z^2) - 5(y z + z x + x y) = 0.

(2) The unique self-dual permutation ellipse is given by x^2 + y^2 + z^2 - 4 (y z + z x + x y) = 0.

(3) The Steiner midway ellipse (SME), is, loosely speaking, the ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE). That is, for each P on SCE, let P' be the intersection of the ray GP with SIE, and let P'' be the midpoint of PP'. Then SME is the set of all such midpoints. SME is given by 7(x^2 + y^2 + z^2) - 34(y z + z x + x y) = 0.

In general, the inverse of a point P = p : q : r in an ellipse j (x^2 + y^2 + z^2) - k (y z + z x + x y) = 0 is the point k*p^2 - j*p*q + j*q^2 - j*p*r - k*q*r + j*r^2 : : .

The appearance of (m,n) in the following list means that the trisector-ellipse-inverse of X(m) is X(n):

(6,8859), (69,41133), (99,41134), (115,41135), (141,41136), (183,41137), (190,41138), (193,41139), (148,14971), (230,5032), (239,25055), (297,3545), (325,21356), (401,5054), (441,10304), (448,15671), (671,9166), (1635,24508), (3524,40884), (5055,40885), (5215,40871), (7840,21358), (8591,9167), (11078,22489), (11092,22490), (15699,40853), (17310,19875), (19883,40891), (22510,25165), (22511,25155), (37907,40856), (39663,39908)

The appearance of (m,n) in the following list means that the self-dual-permutation-ellipse-inverse of X(m) is X(n):

(1,41140), (8,41141), (37,41142), (39,41143), (75,41144), (98,41145), (183,41146)

The appearance of (m,n) in the following list means that the midway-ellipse-inverse of X(m) is X(n):

(115,41147), (148,41148), (230,41149), (239,41150), (287,41151), (297,3860), (325,41152), (385,41153), (671,41154), (441,15690), (15759,40884)

X(41133) = TRISECTOR-ELLIPSE-INVERSE OF X(69)

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

X(41133) lies on these lines: {2, 6}, {4, 11164}, {5, 7870}, {20, 11147}, {30, 10242}, {76, 16509}, {83, 8365}, {99, 37350}, {114, 11645}, {140, 7883}, {187, 22247}, {316, 27088}, {441, 5641}, {538, 10150}, {543, 33228}, {598, 7752}, {620, 8598}, {625, 2482}, {626, 7619}, {632, 7922}, {671, 6390}, {754, 5215}, {1153, 7810}, {1506, 14762}, {1513, 19924}, {1975, 7620}, {2393, 12093}, {2548, 8366}, {3524, 10256}, {3628, 7608}, {3788, 8176}, {3849, 9167}, {3926, 5485}, {5025, 32480}, {5031, 9830}, {5055, 39663}, {5159, 16092}, {5461, 31275}, {5475, 35954}, {5503, 8781}, {5569, 7818}, {5968, 22087}, {6055, 40336}, {7617, 7801}, {7618, 7841}, {7622, 8356}, {7750, 8182}, {7763, 11165}, {7769, 8359}, {7773, 23334}, {7775, 7807}, {7789, 33013}, {7799, 9166}, {7809, 26613}, {7812, 7940}, {7821, 34506}, {7827, 8361}, {7832, 8367}, {7887, 34511}, {7888, 33249}, {7899, 8360}, {7912, 33274}, {7936, 14869}, {9189, 33906}, {9606, 14065}, {9741, 32837}, {9855, 32459}, {10989, 16320}, {11059, 40074}, {11148, 32831}, {12040, 33184}, {15699, 32515}, {15709, 21445}, {15850, 40107}, {17070, 37756}, {32819, 33006}, {32820, 32961}, {32821, 32969}, {32825, 32958}, {32829, 33190}, {32833, 40727}, {32841, 39143}, {33229, 34504}, {33245, 34604}, {35088, 40884}

X(41133) = midpoint of X(8859) and X(41136)
X(41133) = midpoint of X(21356) and X(41137)
X(41133) = reflection of X(8859) in X(41139)
X(41133) = complement of X(8859)
X(41133) = anticomplement of X(41139)
X(41133) = {X(2),X(8859)}-harmonic conjugate of X(41139)

X(41134) = TRISECTOR-ELLIPSE-INVERSE OF X(99)

Barycentrics    5*a^4 - 5*a^2*b^2 + 2*b^4 - 5*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(41134) = X(2) + X(99)

X(41134) lies on these lines: {2, 99}, {3, 6054}, {4, 38751}, {20, 20399}, {30, 10242}, {69, 18800}, {76, 7610}, {83, 5503}, {98, 549}, {114, 376}, {140, 11632}, {141, 11161}, {147, 15692}, {186, 20774}, {187, 7840}, {230, 11054}, {249, 524}, {302, 12154}, {303, 12155}, {315, 25486}, {316, 8598}, {325, 27088}, {381, 10723}, {385, 39785}, {491, 13642}, {492, 13761}, {523, 39061}, {530, 21360}, {531, 21359}, {538, 5215}, {542, 3524}, {547, 6321}, {551, 7983}, {597, 10754}, {598, 1003}, {599, 5026}, {618, 5464}, {619, 5463}, {625, 8597}, {631, 6055}, {1078, 5569}, {1153, 9466}, {1494, 40866}, {1641, 9170}, {1975, 40727}, {2782, 5054}, {2794, 10304}, {2795, 15671}, {2796, 19883}, {2936, 7485}, {3090, 10992}, {3096, 33215}, {3363, 37647}, {3455, 15246}, {3522, 38745}, {3523, 11177}, {3533, 20398}, {3534, 10722}, {3543, 38738}, {3545, 23698}, {3552, 7775}, {3582, 10086}, {3584, 10089}, {3619, 14928}, {3654, 7970}, {3679, 9884}, {3788, 7833}, {3815, 35954}, {3828, 13178}, {3839, 36519}, {3845, 38730}, {3849, 5149}, {3926, 9740}, {3972, 11163}, {4995, 12351}, {5013, 8366}, {5023, 7871}, {5025, 34504}, {5032, 14645}, {5055, 14639}, {5067, 38734}, {5071, 6721}, {5077, 7934}, {5152, 9830}, {5206, 7917}, {5210, 7850}, {5298, 12350}, {5459, 9116}, {5460, 9114}, {5477, 11160}, {5642, 11006}, {5969, 33220}, {5972, 9144}, {5976, 7757}, {5978, 35304}, {5979, 35303}, {5982, 37351}, {5983, 37352}, {6033, 8703}, {6036, 12243}, {6248, 32414}, {6337, 7857}, {6390, 22329}, {6778, 36769}, {7496, 34013}, {7752, 33007}, {7760, 16925}, {7763, 7812}, {7764, 34604}, {7769, 8370}, {7774, 37809}, {7778, 35955}, {7782, 7841}, {7783, 7817}, {7786, 33237}, {7796, 32964}, {7802, 33208}, {7803, 33197}, {7807, 7827}, {7810, 7836}, {7811, 8182}, {7814, 33235}, {7816, 33013}, {7821, 33276}, {7824, 10159}, {7831, 35705}, {7832, 8359}, {7847, 8360}, {7863, 33259}, {7869, 33022}, {7877, 32831}, {7880, 8290}, {7888, 33014}, {7897, 8588}, {7925, 9855}, {7930, 15815}, {7931, 8589}, {7940, 11318}, {7944, 37512}, {7945, 15515}, {7947, 15513}, {8176, 11361}, {8787, 15533}, {8997, 19054}, {9143, 15357}, {9164, 9182}, {9760, 11299}, {9762, 11300}, {9772, 21163}, {9862, 15698}, {9875, 19876}, {10303, 11623}, {10352, 12150}, {10991, 15717}, {11001, 38736}, {11049, 12347}, {11238, 15452}, {11539, 38224}, {11646, 20582}, {11812, 38739}, {12042, 15693}, {12100, 14830}, {12188, 15701}, {12258, 13174}, {12355, 15703}, {13188, 15694}, {13846, 19058}, {13847, 19057}, {13989, 19053}, {14041, 32479}, {14043, 31652}, {14093, 38744}, {14113, 31878}, {14269, 38635}, {14444, 38239}, {14651, 15709}, {15681, 22505}, {15683, 39838}, {15688, 38743}, {15708, 38737}, {16895, 31457}, {19108, 32787}, {19109, 32788}, {19708, 38749}, {19709, 22515}, {23055, 32817}, {23334, 35927}, {26619, 33341}, {26620, 33340}, {30471, 36775}, {32458, 32837}, {32552, 36768}, {32553, 36767}, {32833, 33216}, {33233, 34505}, {33474, 40707}, {33475, 40706}, {33799, 36953}, {34200, 38741}

X(41134) = reflection of X(2) in X(9167)
X(41134) = reflection of X(9166) in X(2)
X(41134) = reflection of X(41135) in X(14971)

X(41135) = TRISECTOR-ELLIPSE-INVERSE OF X(115)

Barycentrics    a^4 - a^2*b^2 - 5*b^4 - a^2*c^2 + 11*b^2*c^2 - 5*c^4 : :

X(41135) lies on these lines: {2, 99}, {4, 11177}, {20, 6055}, {30, 8859}, {98, 3543}, {147, 381}, {193, 11161}, {194, 9770}, {230, 9855}, {376, 6321}, {385, 8352}, {524, 5207}, {530, 5469}, {531, 5470}, {542, 3839}, {547, 13188}, {549, 12355}, {551, 9875}, {598, 5309}, {616, 5460}, {617, 5459}, {618, 22577}, {619, 22578}, {1327, 35825}, {1328, 35824}, {1916, 5485}, {1992, 11646}, {2023, 11152}, {2782, 3545}, {2796, 19875}, {2896, 7841}, {2996, 5503}, {3090, 38628}, {3091, 6054}, {3146, 11623}, {3153, 39120}, {3180, 22573}, {3181, 22574}, {3241, 12258}, {3329, 3363}, {3448, 9144}, {3523, 20398}, {3524, 26614}, {3620, 19662}, {3679, 11599}, {3828, 13174}, {3830, 9862}, {3832, 38664}, {3845, 12188}, {3849, 14568}, {3854, 38745}, {5025, 34505}, {5054, 38635}, {5055, 38229}, {5056, 23235}, {5068, 14981}, {5071, 8724}, {5254, 20112}, {5465, 9143}, {5569, 7748}, {5969, 21356}, {6034, 9830}, {6036, 12117}, {6669, 9114}, {6670, 9116}, {6770, 25164}, {6773, 25154}, {7610, 7833}, {7757, 8176}, {7765, 33024}, {7775, 13571}, {7779, 11054}, {7783, 12040}, {7793, 33192}, {7797, 8370}, {7806, 11159}, {7817, 10583}, {7827, 16044}, {7836, 11318}, {7840, 37350}, {7925, 8355}, {7983, 31145}, {8182, 33264}, {8356, 16509}, {8360, 17128}, {8587, 32532}, {8592, 32983}, {8597, 14712}, {8703, 38733}, {8782, 9466}, {9140, 16278}, {9168, 9183}, {9180, 18007}, {9740, 9939}, {9741, 32984}, {9878, 11361}, {9884, 11725}, {10303, 10992}, {10304, 23698}, {10385, 13183}, {10723, 15683}, {10754, 11160}, {10991, 17578}, {11001, 12042}, {11006, 15359}, {11050, 13179}, {11164, 33246}, {11608, 31164}, {13083, 22846}, {13084, 22891}, {13233, 14002}, {13881, 33274}, {13908, 19109}, {13968, 19108}, {14830, 15682}, {14893, 38744}, {15597, 33273}, {15640, 39809}, {15698, 38739}, {15702, 33813}, {15705, 38737}, {15708, 21166}, {15709, 34127}, {15710, 38731}, {15717, 38740}, {16092, 36174}, {16508, 33215}, {16984, 35954}, {17004, 35955}, {19708, 38730}, {21636, 30308}, {21969, 39836}, {23234, 23514}, {24711, 36224}, {26613, 32479}, {31276, 33190}, {32458, 32869}, {32552, 36331}, {32553, 35750}, {32966, 34511}, {33259, 34504}, {33260, 34506}, {33272, 39652}, {38071, 38743}, {38220, 38314}

X(41135) = reflection of X(2) in X(9166)
X(41135) = reflection of X(41134) in X(14971)
X(41135) = anticomplement of X(9166)

X(41136) = TRISECTOR-ELLIPSE-INVERSE OF X(141)

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

X(41136) lies on these lines: {2, 6}, {76, 8176}, {147, 11645}, {148, 31173}, {315, 7618}, {316, 8591}, {543, 7809}, {598, 7785}, {625, 11054}, {671, 7813}, {1078, 7619}, {1153, 7769}, {2482, 14712}, {2896, 15810}, {3545, 32515}, {3552, 32825}, {3849, 7799}, {3926, 23334}, {3933, 33013}, {4938, 37764}, {5207, 9830}, {5485, 20081}, {5503, 11606}, {5641, 40853}, {5978, 9116}, {5979, 9114}, {6054, 19924}, {6390, 9855}, {6655, 32480}, {6658, 11164}, {7617, 7752}, {7620, 32816}, {7622, 7811}, {7759, 7870}, {7762, 19661}, {7763, 8182}, {7764, 7883}, {7775, 7796}, {7776, 7833}, {7794, 10302}, {7810, 7917}, {7812, 7836}, {7817, 7905}, {7821, 7827}, {7839, 8360}, {7841, 7906}, {7860, 34504}, {7895, 14762}, {7900, 32818}, {7921, 33237}, {7929, 33215}, {7939, 8359}, {7941, 8370}, {7946, 33259}, {7947, 8369}, {8352, 8596}, {8592, 9866}, {8597, 20094}, {9166, 19570}, {9464, 40074}, {9741, 33017}, {11147, 33014}, {12040, 33273}, {15708, 21445}, {25383, 37756}, {25486, 32458}, {32831, 33208}, {32993, 34505}, {35077, 40858}

X(41136) = reflection of X(8859) in X(41133)
X(41136) = reflection of X(5032) in X(41137)
X(41136) = anticomplement of X(8859)

X(41137) = TRISECTOR-ELLIPSE-INVERSE OF X(183)

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

X(41137) lies on these lines: {2, 6}, {316, 18800}, {542, 22525}, {543, 12215}, {575, 7883}, {576, 7870}, {625, 11161}, {3849, 5182}, {5026, 9855}, {7909, 22330}, {7922, 22234}, {8593, 31173}, {9774, 13355}, {9830, 12151}, {10754, 39785}, {11645, 12177}, {11676, 19924}, {14848, 32515}, {21445, 38064}

X(41137) = midpoint of X(5032) and X(41136)
X(41137) = reflection of X(21356) in X(41133)

X(41138) = TRISECTOR-ELLIPSE-INVERSE OF X(190)

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

X(41138) lies on these lines: {2, 45}, {9, 17228}, {37, 24625}, {44, 17310}, {86, 25101}, {238, 519}, {239, 4908}, {335, 4755}, {344, 17378}, {376, 24828}, {514, 9460}, {528, 38057}, {537, 25055}, {547, 24833}, {549, 24813}, {551, 24841}, {597, 32029}, {599, 17488}, {894, 31333}, {1121, 40865}, {1213, 31057}, {1227, 18743}, {1268, 17260}, {1992, 4437}, {2161, 27065}, {2325, 17160}, {2796, 9166}, {3161, 17352}, {3305, 16561}, {3545, 29243}, {3582, 24845}, {3584, 24846}, {3679, 4432}, {3799, 16482}, {3828, 24715}, {3923, 24452}, {3943, 40891}, {3973, 17240}, {4360, 26685}, {4664, 17755}, {4715, 17297}, {4795, 17244}, {5071, 24817}, {5845, 21356}, {6172, 16593}, {6633, 35168}, {6651, 16590}, {9041, 38315}, {15492, 17268}, {15694, 24844}, {16814, 17307}, {16885, 17295}, {17233, 37654}, {17261, 17382}, {17266, 31138}, {17273, 17279}, {17274, 17283}, {17277, 17281}, {17280, 17330}, {17313, 17350}, {17320, 17353}, {17341, 25728}, {17793, 35030}, {19053, 24843}, {19054, 24842}, {19709, 24827}, {19876, 25351}, {24508, 33908}, {24807, 34362}, {24818, 32787}, {24819, 32788}, {27747, 31205}, {27776, 30832}, {28301, 37756}, {28604, 31311}, {32028, 36954}, {36807, 39704}

X(41139) = TRISECTOR-ELLIPSE-INVERSE OF X(193)

Barycentrics    14*a^4 - 11*a^2*b^2 + 11*b^4 - 11*a^2*c^2 - 14*b^2*c^2 + 11*c^4 : :

X(41139) lies on these lines: {2, 6}, {30, 5215}, {187, 8355}, {598, 7857}, {671, 32459}, {754, 10150}, {1003, 20112}, {1153, 7886}, {3053, 23334}, {3524, 39663}, {3767, 11165}, {5254, 7618}, {5309, 12040}, {5461, 27088}, {5569, 33184}, {6036, 11645}, {6055, 10011}, {6390, 22247}, {6680, 14762}, {6722, 37350}, {7615, 11288}, {7617, 8369}, {7619, 7817}, {7620, 13881}, {7745, 8176}, {7746, 16509}, {7749, 8360}, {7907, 32480}, {8182, 11318}, {8361, 34506}, {8598, 14061}, {9166, 35297}, {9607, 32977}, {10256, 11539}, {11147, 32989}, {11164, 16925}, {12815, 19697}, {12829, 25486}, {26613, 33228}, {32970, 34505}

X(41139) = complement of X(41133)
X(41139) = {X(2),X(8859)}-harmonic conjugate of X(41133)

X(41140) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(1)

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

X(41140) lies on these lines: {1, 2}, {6, 4795}, {44, 545}, {57, 17079}, {81, 17180}, {89, 24184}, {142, 3759}, {190, 28301}, {193, 4859}, {238, 28580}, {320, 4700}, {333, 16712}, {391, 17304}, {514, 1635}, {524, 31138}, {527, 666}, {536, 4370}, {549, 29331}, {553, 24588}, {597, 742}, {620, 35466}, {644, 31171}, {664, 3911}, {726, 31349}, {752, 1738}, {908, 17761}, {1015, 16610}, {1016, 36915}, {1086, 4715}, {1724, 11352}, {1992, 6173}, {2321, 17342}, {2325, 17160}, {2796, 20142}, {3218, 5540}, {3589, 4967}, {3619, 4034}, {3663, 17333}, {3664, 17121}, {3686, 16706}, {3707, 4389}, {3717, 28503}, {3729, 4402}, {3817, 28909}, {3834, 4969}, {3875, 25101}, {3879, 17278}, {3943, 6687}, {3946, 17277}, {3950, 17338}, {3975, 18145}, {3986, 17396}, {4000, 4416}, {4021, 17260}, {4044, 4479}, {4054, 14997}, {4058, 17358}, {4060, 17285}, {4357, 17330}, {4358, 4986}, {4359, 20893}, {4360, 6666}, {4361, 4431}, {4364, 16590}, {4371, 17286}, {4399, 17357}, {4422, 4908}, {4452, 25728}, {4454, 36588}, {4464, 17243}, {4667, 39704}, {4780, 15485}, {4850, 25057}, {4852, 17337}, {4856, 17300}, {4887, 20072}, {4899, 32922}, {4921, 24632}, {4989, 5263}, {5257, 17380}, {5258, 21516}, {5723, 9436}, {5839, 17282}, {5847, 31151}, {6547, 27751}, {6629, 24617}, {6996, 28194}, {7228, 16671}, {7263, 16669}, {7402, 38074}, {7757, 31169}, {14422, 33920}, {15828, 25269}, {16412, 40726}, {16666, 34824}, {16704, 17205}, {16711, 18206}, {16752, 17179}, {17117, 17355}, {17132, 17487}, {17133, 17264}, {17151, 26685}, {17301, 24441}, {17319, 25072}, {17356, 17362}, {17363, 21255}, {17374, 40480}, {17781, 24612}, {18821, 32040}, {18822, 32041}, {19796, 37086}, {19804, 20924}, {24175, 37683}, {24177, 37652}, {24187, 33129}, {24590, 24608}, {27478, 31178}, {27776, 33155}, {28562, 32096}, {32043, 35092}, {32094, 36591}, {36589, 37771}, {36662, 38021}

X(41140) = midpoint of X(2) and X(239)
X(41140) = midpoint of X(8859) and X(41133)
X(41140) = reflection of X(2) in X(3008)
X(41140) = reflection of X(3912) in X(2)
X(41140) = reflection of X(17310) in X(41141)
X(41140) = complement of X(17310)
X(41140) = anticomplement of X(41141)
X(41140) = {X(2),X(17310)}-harmonic conjugate of X(41141)

X(41141) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(8)

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

X(41141) lies on these lines: {1, 2}, {141, 25072}, {142, 17118}, {312, 20893}, {344, 17274}, {514, 1639}, {516, 31151}, {527, 4370}, {545, 2325}, {547, 29331}, {742, 4755}, {903, 4582}, {908, 31171}, {1086, 4908}, {1111, 4358}, {1146, 35121}, {2321, 17265}, {3589, 4909}, {3619, 3986}, {3664, 4795}, {3689, 11716}, {3836, 28580}, {3879, 17341}, {3943, 17067}, {3950, 17282}, {4021, 17243}, {4029, 17290}, {4098, 17304}, {4419, 36911}, {4422, 4715}, {4428, 21514}, {4664, 27487}, {4700, 6687}, {4856, 17352}, {4945, 6549}, {5316, 27739}, {6381, 30866}, {6666, 17231}, {6722, 15903}, {7238, 36522}, {7402, 38021}, {10022, 17359}, {14759, 27076}, {17232, 17333}, {17234, 17342}, {17241, 17353}, {17263, 17271}, {17268, 24199}, {17283, 17320}, {17296, 37654}, {17354, 39704}, {17678, 19815}, {18146, 18743}, {19512, 28204}, {21526, 40726}, {27475, 31178}, {28309, 40480}, {28313, 37756}, {30720, 32851}, {30849, 30857}, {31134, 40998}, {31647, 36591}, {36662, 38076}

X(41141) = midpoint of X(2) and X(3912)
X(41141) = midpoint of X(17310) and X(41140)
X(41141) = reflection of X(3008) in X(2)
X(41141) = complement of X(41140)
X(41141) = {X(2),X(17310)}-harmonic conjugate of X(41140)

X(41142) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(37)

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

X(41142) lies on these lines: {2, 37}, {39, 16829}, {43, 2230}, {190, 899}, {194, 25280}, {239, 20331}, {291, 519}, {513, 14404}, {537, 3783}, {538, 13466}, {671, 31160}, {726, 3807}, {730, 3097}, {1084, 2229}, {1266, 30997}, {1574, 25264}, {1992, 9025}, {3240, 3758}, {3634, 32026}, {3795, 31178}, {3943, 30967}, {4465, 32045}, {6172, 34343}, {6174, 22329}, {7786, 32104}, {9055, 31348}, {9596, 33031}, {9598, 33032}, {11163, 31140}, {16834, 17754}, {17028, 17119}, {17254, 36235}, {17310, 35123}, {17333, 24289}, {17793, 28554}, {18822, 32041}, {19998, 22323}, {20105, 40598}, {20691, 25303}, {21226, 21868}, {24512, 29584}, {24574, 35043}, {25102, 40908}, {25350, 31027}, {29577, 30945}, {30964, 32129}, {31462, 33033}

X(41142) = midpoint of X(2) and X(17759)
X(41142) = reflection of X(2) in X(1575)
X(41142) = reflection of X(350) in X(2)
X(41142) = anticomplement of X(41144)

X(41143) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(39)

Barycentrics    2*a^4*b^4 + 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 : :

X(41143) lies on these lines: {2, 39}, {99, 3231}, {385, 5106}, {512, 9147}, {524, 694}, {599, 706}, {671, 22735}, {702, 35073}, {1003, 1613}, {3051, 12150}, {3972, 9463}, {6179, 37465}, {6379, 39361}, {7760, 37338}, {7840, 35077}, {8623, 13586}, {8716, 21001}, {8859, 35078}, {9225, 17941}, {11196, 34364}, {11328, 14614}, {22329, 35146}

X(41143) = midpoint of X(2) and X(40858)
X(41143) = reflection of X(2) in X(3229)
X(41143) = reflection of X(3978) in X(2)

X(41144) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(75)

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

X(41144) lies on these lines: {2, 37}, {44, 4465}, {194, 24739}, {513, 4379}, {519, 13466}, {551, 730}, {599, 9025}, {716, 20363}, {1086, 4871}, {2230, 3720}, {3227, 18145}, {3760, 16604}, {3834, 30967}, {4428, 8556}, {4675, 30947}, {4721, 29750}, {4760, 26239}, {6173, 34363}, {6381, 33908}, {7801, 10199}, {16829, 18140}, {16833, 37673}, {16834, 21904}, {17144, 25107}, {17237, 25378}, {17310, 18822}, {17325, 29827}, {17374, 29824}, {17392, 30982}, {17448, 18135}, {24656, 31276}, {28554, 40533}, {28600, 31178}, {29584, 37678}, {31139, 35043}

X(41144) = reflection of X(2) in X(20530)
X(41144) = complement of X(41142)

X(41145) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(98)

Barycentrics    4*a^10 - 6*a^8*b^2 + 5*a^6*b^4 - 7*a^4*b^6 + 3*a^2*b^8 + b^10 - 6*a^8*c^2 + 2*a^6*b^2*c^2 + 5*a^4*b^4*c^2 - 4*a^2*b^6*c^2 + 3*b^8*c^2 + 5*a^6*c^4 + 5*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 4*b^6*c^4 - 7*a^4*c^6 - 4*a^2*b^2*c^6 - 4*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 + c^10 : :

X(41145) lies on these lines: {2, 98}, {297, 11645}, {401, 19924}, {441, 524}, {458, 5476}, {511, 40884}, {541, 40856}, {597, 3163}, {648, 5702}, {1494, 1992}, {1640, 2799}, {5032, 11348}, {5641, 9476}, {10112, 28433}, {21358, 40484}, {29012, 40885}

X(41145) = midpoint of X(2) and X(287)
X(41145) = reflection of X(15595) in X(2)

X(41146) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(183)

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

X(41146) lies on these lines: {2, 6}, {30, 12151}, {316, 8593}, {511, 2482}, {542, 15980}, {549, 22677}, {575, 7810}, {576, 7801}, {1503, 10488}, {2456, 14830}, {3564, 19905}, {3849, 18800}, {5017, 37809}, {5026, 8598}, {5038, 8359}, {5104, 15483}, {5107, 39785}, {6390, 8586}, {6784, 9027}, {7668, 39075}, {7794, 22330}, {7854, 22234}, {7873, 33749}, {8352, 9830}, {8369, 13330}, {8591, 12215}, {11150, 22503}, {11179, 22525}, {11646, 37350}, {14928, 32479}, {15303, 35088}, {18449, 34897}, {20423, 35930}

X(41146) = midpoint of X(2) and X(39099)
X(41146) = reflection of X(15993) in X(2)

X(41147) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(115)

Barycentrics    10*a^4 - 10*a^2*b^2 - 17*b^4 - 10*a^2*c^2 + 44*b^2*c^2 - 17*c^4 : :

X(41147) lies on these lines: {2, 99}, {542, 12101}, {2782, 3860}, {2794, 33699}, {3363, 9607}, {3534, 11623}, {3845, 38734}, {6036, 19711}, {6055, 15695}, {7837, 17503}, {9909, 13233}, {10109, 20399}, {10991, 15640}, {10992, 15701}, {11632, 15685}, {11812, 20398}, {12355, 38731}, {15690, 23698}, {15719, 38740}, {31695, 36330}, {31696, 35752}

X(41147) = reflection of X(2) in X(41154)
X(41147) = anticomplement of anticomplement of X(41148)
X(41147) = {X(2),X(36523)}-harmonic conjugate of X(41154)
X(41147) = {X(36521),X(36523)}-harmonic conjugate of X(115)

X(41148) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(148)

Barycentrics    14*a^4 - 14*a^2*b^2 + 41*b^4 - 14*a^2*c^2 - 68*b^2*c^2 + 41*c^4 : :

X(41148) lies on these lines: {2, 99}, {2794, 3860}, {5066, 20398}, {6036, 19710}, {11623, 19709}, {12355, 35381}, {15682, 38740}, {15711, 38229}, {15713, 38734}

X(41148) = midpoint of X(2) and X(41154)
X(41148) = complement of complement of X(41147)
X(41148) = {X(36521),X(36523)}-harmonic conjugate of X(148)

X(41149) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(230)

Barycentrics    22*a^2 - 5*b^2 - 5*c^2 : :

X(41149) lies on these lines: {2, 6}, {182, 19711}, {428, 5095}, {511, 15690}, {542, 12101}, {547, 22330}, {575, 11812}, {576, 3845}, {1353, 19710}, {1503, 33699}, {2854, 21849}, {3534, 8550}, {3564, 3860}, {4663, 4669}, {4856, 4912}, {7745, 11054}, {7760, 8352}, {8705, 21969}, {8787, 15300}, {11001, 11477}, {11055, 13330}, {11179, 15695}, {11539, 22234}, {11540, 40107}, {12007, 14810}, {12100, 20190}, {15685, 29181}, {15711, 17508}, {18800, 25608}, {19569, 33683}, {27088, 35007}, {33749, 34200}

X(41149) = reflection of X(41152) in X(41153)
X(41149) = anticomplement of X(41152)
X(41149) = {X(41152),X(41153)}-harmonic conjugate of X(2)

X(41150) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(239)

Barycentrics    34*a + 7*b + 7*c : :

X(41150) lies on these lines: {1, 2}, {515, 3860}, {516, 15690}, {553, 39782}, {1385, 19710}, {3653, 15716}, {3655, 12571}, {4301, 19708}, {5066, 15178}, {5882, 19709}, {5901, 12101}, {8703, 13464}, {10222, 15713}, {10283, 15711}, {13607, 38022}, {15685, 28158}, {15697, 30389}, {15719, 31425}, {15721, 16189}, {15759, 28194}, {28164, 33699}, {33812, 38026}

X(41151) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(287)

Barycentrics    34*a^8 - 48*a^6*b^2 + 55*a^4*b^4 - 48*a^2*b^6 + 7*b^8 - 48*a^6*c^2 - 20*a^4*b^2*c^2 + 34*a^2*b^4*c^2 - 48*b^6*c^2 + 55*a^4*c^4 + 34*a^2*b^2*c^4 + 82*b^4*c^4 - 48*a^2*c^6 - 48*b^2*c^6 + 7*c^8 : :

X(41151) lies on these lines: {2, 98}, {543, 15759}, {2482, 15722}, {2794, 3860}, {3845, 20398}, {8703, 11623}, {10991, 19709}, {11001, 38734}, {12042, 19710}, {15300, 15716}, {15685, 38732}, {15690, 23698}, {15695, 38730}, {19711, 36521}

X(41152) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(325)

Barycentrics    10*a^2 - 17*b^2 - 17*c^2 : :

X(41152) lies on these lines: {2, 6}, {511, 3860}, {542, 15759}, {1503, 15690}, {7238, 29615}, {7794, 27088}, {8550, 15701}, {8703, 34507}, {9872, 38862}, {10691, 32257}, {11054, 32027}, {11179, 15722}, {12100, 40107}, {15069, 19708}, {29181, 33699}

X(41152) = reflection of X(41149) in X(41153)
X(41152) = complement of X(41149)
X(41152) = anticomplement of X(41153)
X(41152) = {X(41149),X(2)}-harmonic conjugate of X(41153)

X(41153) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(385)

Barycentrics    34*a^2 + 7*b^2 + 7*c^2 : :

X(41153) lies on these lines: {2, 6}, {182, 19710}, {575, 5066}, {576, 15713}, {1503, 3860}, {5476, 33699}, {7829, 8355}, {8550, 19709}, {9731, 40344}, {10109, 25555}, {10124, 22330}, {10128, 32300}, {10541, 15697}, {11737, 33749}, {12007, 38079}, {12101, 18583}, {12834, 15826}, {14848, 15695}, {15690, 29181}, {15699, 22234}, {15716, 38064}, {27088, 31652}

X(41153) = midpoint of X(41149) and X(41152)
X(41153) = complement of X(41152)
X(41153) = {X(41149),X(2)}-harmonic conjugate of X(41152)

X(41154) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(671)

Barycentrics    2*a^4 - 2*a^2*b^2 - 25*b^4 - 2*a^2*c^2 + 52*b^2*c^2 - 25*c^4 : :

X(41154) lies on these lines: {2, 99}, {542, 3860}, {547, 38628}, {2794, 12101}, {3830, 11623}, {6036, 15711}, {6055, 15685}, {8703, 38734}, {12100, 20398}, {15695, 38732}, {15698, 38740}, {15716, 38224}, {15759, 23698}, {19711, 38735}

X(41154) = midpoint of X(2) and X(41147)
X(41154) = reflection of X(2) in X(41148)
X(41154) = {X(2),X(36523)}-harmonic conjugate of X(41147)

leftri

Inverses in circles: X(41155)-X(41161)

rightri

This preamble is based on notes from Predrag Terzic and Peter Moses, January 28-31, 2021.

Terzic observed that the following 8 points lie on a circle, here named the Terzic circle:

X(15), X(16), X(55), X(109), X(654), X(1155), X(2291), X(41155).

Moses found that the center of this circle is X(6139), and also

A-power = (b^2*c^2*(-a + b + c)*(-2*a^2 + a*b + b^2 + a*c - 2*b*c + c^2))/(2*(-a + b)*(a - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c - a*c^2 - b*c^2 + c^3))

squared radius = (a^2*b^2*c^2*(a^4 - a^3*b - a*b^3 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4)^2)/(4*(a - b)^2*(a - c)^2*(b - c)^2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c - a*c^2 - b*c^2 + c^3)^2)

See also the preamble just before X(41163).

The appearance of {n1, n2, ...} in the following list means that the points X(n1), X(n2), ... lie on a circle:

{1,15,16,36,3465,4040,5526,5529}
{2,15,16,23,110,111,352,353,5638,5639,6141,6142,7598,7599,7601,7602,7711,9138,9147,9153,9156,9157,9158,9162,9163,9212,9213,9978,9980,9998,9999,11199,11673,13114,13242,14660,14704,14705,32072,32073,32074,32526,33502,33503} (Parry circe)
{4,15,16,186,3484,11674,13509,15412}
{13,15,16,3165,5616,5669,6104,10658} {14,15,16,3166,5612,5668,6105,10657}
{15,16,55,109,654,1155,2291,41155,41162,41163,41164,41165,41166,41167} (Terzic circle)
{15,16,74,112,5667,9862,11587,40894,40895}
{15,16,101,106,214,9321,11716,38013,38014,41183,41184,41185,41186,41187,41188,41189,41190,41191,41192,41193} (Moses isodynamic circle)
{15,16,115,128,399,1263,1511,2079,10277,14367}
{15,16,501,3743,5127,14838,14873,39149}
{15,16,647,1495,14685,16319,35901}
{15,16,667,1083,3230,11650,11651,11652}
{15,16,1138,2132,6794,12112,14354}
{15,16,5000,5001,6112,6113,6114,6115,6116,6117} (Moses radical circle)

Every circle that passes through X(15) and X(16) has a center on the Lemoine Axis and is orthogonal to every circle ini the Schoute coaxal family, including these: circumcircle, Brocard circle, Lucas inner circle, Lucas circles radical circle, outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle.

X(41155) = CIRCUMCIRCLE-INVERSE OF X(654)

Barycentrics    (pending)

X(41155) lies on the Terzic circle and these lines: {3, 654}, {55, 184}

X(41156) = BROCARD-CIRCLE-INVERSE OF X(55)

Barycentrics    (pending)

X(41156) lies the Terzic circle and these lines: {55, 182}, {654, 8659}, {7077, 20999}

X(41157) = BROCARD-CIRCLE-INVERSE OF X(109)

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

X(41157) lies the Terzic circle and these lines: {2, 5075}, {109, 182}, {574, 2291}, {3094, 14936}, {7709, 17596}

X(41158) = BROCARD-CIRCLE-INVERSE OF X(654)

Barycentrics    (pending)

X(41158) lies the Terzic circle and this line: {182, 654}

X(41159) = BROCARD-CIRCLE-INVERSE OF X(1155)

Barycentrics    (pending)

0 X(41159) lies on this line: {182, 1155}

X(41160) = BROCARD-CIRCLE-INVERSE OF X(2291)

Barycentrics    a^2*(4*a^8 - 4*a^7*b - 7*a^6*b^2 + 6*a^5*b^3 + 5*a^4*b^4 - 3*a^3*b^5 - 4*a^2*b^6 + 5*a*b^7 - 2*b^8 - 4*a^7*c + 10*a^6*b*c + 2*a^5*b^2*c - 14*a^4*b^3*c + a^3*b^4*c + 11*a^2*b^5*c - 5*a*b^6*c - b^7*c - 7*a^6*c^2 + 2*a^5*b*c^2 + 5*a^4*b^2*c^2 + 7*a^3*b^3*c^2 - 7*a^2*b^4*c^2 - 4*a*b^5*c^2 + 2*b^6*c^2 + 6*a^5*c^3 - 14*a^4*b*c^3 + 7*a^3*b^2*c^3 - 5*a^2*b^3*c^3 + 4*a*b^4*c^3 + 6*b^5*c^3 + 5*a^4*c^4 + a^3*b*c^4 - 7*a^2*b^2*c^4 + 4*a*b^3*c^4 - 10*b^4*c^4 - 3*a^3*c^5 + 11*a^2*b*c^5 - 4*a*b^2*c^5 + 6*b^3*c^5 - 4*a^2*c^6 - 5*a*b*c^6 + 2*b^2*c^6 + 5*a*c^7 - b*c^7 - 2*c^8) : :

X(41160) lies the Terzic circle and these lines: {109, 574}, {182, 2291}, {353, 5075}, {10485, 14936}

X(41161) = LUCAS-CIRCLES-RADICAL-CIRCLE-INVERSE OF X(109)

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

X(41161) lies the Terzic circle and these lines: {109, 1151}, {2291, 6221}, {5075, 7601}, {6425, 14936}

X(41162) = REFLECTION OF X(1155) IN X(6139)

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

X(41162) lies the Terzic circle and these lines: {55, 513}, {109, 1308}, {517, 654}, {650, 2291}, {902, 1769}, {1155, 1638}, {1946, 23404}, {4449, 7004}

X(41162) = reflection of X(1155) in X(6139)
X(41162) = X(901)-Ceva conjugate of X(1155)






  Points on the Terzic circle: X(41163) - X(41166)  

This preamble is based on notes from Peter Moses, January 30, 2021.

Points X(41163)-X(41165) were found using the following theorem, which was introduced in the preamble just before X(3027):

Suppose that O1 and O2 are circles. Let P1 be a point on O1, and let P2 be the O1-antipode of P1. Let
Si = internal center of similitude of O1 and O2
Se = external center of similitude of O1 and O2
Q1 = P1Si∩P2Se
Q2 = P1Se∩P2Si

Then Q1 and Q2 are a pair of antipodes on O2, and the lines P1P2 and P2Q2 are parallel.

For example, for a point P = p : q : r on the Parry circle, the following point is on the Terzic circle:

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

X(41163) = X(98)X(109)∩X(99)X(333)

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

X(41163) lies on the Terzic circle and these lines: {55, 2784}, {63, 2319}, {98, 109}, {99, 333}, {385, 527}, {515, 11676}, {654, 2786}, {804, 6139}, {940, 15903}, {1155, 16609}, {1211, 6174}, {1751, 11608}, {3550, 9862}, {3666, 11998}, {3911, 5991}, {5075, 9147}, {5745, 8290}, {7709, 17596}

X(41164) = X(35)X(73)∩X(55)X(2772)

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

X(41164) lies on the Terzic circle and these lines: {35, 73}, {55, 2772}, {110, 284}, {526, 6139}, {654, 2774}, {942, 1511}, {1155, 18593}, {1495, 20129}, {3743, 37080}, {5075, 9138}

X(41165) = X(55)X(2813)∩X(109)X(111)

Barycentrics    a^2*(2*a^7 - 2*a^6*b - 7*a^5*b^2 + a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + 7*a*b^6 - b^7 - 2*a^6*c + 6*a^2*b^4*c + 4*b^6*c - 7*a^5*c^2 + 32*a^3*b^2*c^2 - 7*a^2*b^3*c^2 - 15*a*b^4*c^2 + 5*b^5*c^2 + a^4*c^3 - 7*a^2*b^2*c^3 - 8*b^4*c^3 - 2*a^3*c^4 + 6*a^2*b*c^4 - 15*a*b^2*c^4 - 8*b^3*c^4 + 2*a^2*c^5 + 5*b^2*c^5 + 7*a*c^6 + 4*b*c^6 - c^7) : :

X(41165) lies on the Terzic circle and these lines: {55, 2813}, {109, 111}, {654, 2824}, {1155, 16611}, {1296, 2291}, {5075, 9156}, {6088, 6139}

X(41166) = MIDPOINT OF X(80) AND X(4302)

Barycentrics    a*(2*a^5 - 3*a^4*b - 2*a^3*b^2 + 4*a^2*b^3 - b^5 - 3*a^4*c + 10*a^3*b*c - 5*a^2*b^2*c - 2*a*b^3*c - 2*a^3*c^2 - 5*a^2*b*c^2 + 4*a*b^2*c^2 + b^3*c^2 + 4*a^2*c^3 - 2*a*b*c^3 + b^2*c^3 - c^5)

X(41166) = {X[1] - 3 X[10058], X[100] - 3 X[35258], 2 X[5126] - 3 X[17010]}

X(41166) lies on the Terzic circle and these lines: {1, 104}, {9, 100}, {11, 516}, {35, 20117}, {44, 14936}, {55, 2801}, {80, 4302}, {149, 5744}, {214, 392}, {497, 11219}, {528, 4640}, {553, 38055}, {654, 3887}, {900, 6139}, {952, 4304}, {956, 2802}, {1000, 7972}, {1145, 3626}, {1387, 4031}, {1512, 6246}, {1697, 38669}, {1776, 5537}, {2078, 18450}, {2310, 23703}, {2932, 5217}, {3035, 5316}, {3579, 5840}, {3621, 12641}, {3683, 6174}, {4309, 12750}, {4652, 13279}, {4860, 18240}, {5075, 9980}, {5126, 17010}, {5218, 5660}, {5220, 13205}, {5225, 6903}, {6265, 37606}, {6702, 12764}, {7675, 13243}, {9580, 10707}, {9951, 22560}, {10389, 14151}, {10624, 37726}, {10711, 31434}, {11684, 12695}, {12053, 20418}, {12515, 12736}, {12831, 13405}, {16578, 35281}, {23845, 33848}, {31190, 31272}, {38133, 39692}

X(41166) = midpoint of X(80) and X(4302)
X(41166) = reflection of X(12831) in X(13405)

X(41167) = X(110)-CEVA CONJUGATE OF X(511)

Barycentrics    a^2*(b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :
X(41167) = 3 X[34291] + X[35364]

Let P and U be the two points on the Brocard axis whose trilinear polars are parallel to the Brocard axis. X(41167) is the crosspoint of P and U. (Randy Hutson, April 13, 2021)

X(41167) lies on the Jerabek circumhyperbola of the medial triangle and these lines: {2, 879}, {3, 512}, {5, 525}, {6, 520}, {110, 5649}, {113, 114}, {141, 523}, {206, 924}, {237, 38354}, {327, 850}, {511, 23350}, {526, 6593}, {647, 2422}, {684, 2491}, {826, 1209}, {868, 35088}, {1147, 8673}, {1510, 14318}, {1511, 9517}, {2421, 4230}, {2433, 11284}, {2883, 3566}, {3733, 40589}, {3800, 32204}, {3906, 7697}, {4550, 30209}, {5027, 6132}, {5113, 19576}, {5181, 9033}, {5968, 32112}, {6041, 9210}, {6072, 23098}, {6333, 16230}, {6368, 23285}, {7625, 8542}, {11178, 23878}, {11597, 14675}, {12073, 18556}, {18312, 24206}, {21187, 34830}, {21789, 40602}, {34290, 35922}, {34383, 39528}, {36471, 38971}

X(41167) = midpoint of X(i) and X(j) for these {i,j}: {110, 35909}, {684, 3569}, {6333, 16230}
X(41167) = reflection of X(i) in X(j) for these {i,j}: {879, 40550}, {5027, 6132}, {18312, 24206}
X(41167) = isogonal conjugate of X(41173)
X(41167) = complement of X(879)
X(41167) = anticomplement of X(40550)
X(41167) = complement of the isogonal conjugate of X(4230)
X(41167) = complement of the isotomic conjugate of X(877)
X(41167) = medial-isogonal conjugate of X(3150)
X(41167) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3150}, {112, 16609}, {162, 511}, {163, 441}, {232, 8287}, {237, 16573}, {240, 125}, {297, 21253}, {511, 34846}, {811, 21531}, {877, 2887}, {1755, 15526}, {1959, 127}, {2211, 16592}, {2421, 18589}, {3289, 16595}, {4230, 10}, {14966, 1214}, {17994, 24040}, {23997, 3}, {23999, 39469}, {24000, 6130}, {32676, 230}, {34859, 16583}
X(41167) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 511}, {850, 2799}, {15631, 36790}, {40173, 14966}
X(41167) = X(i)-isoconjugate of X(j) for these (i,j): {98, 36084}, {163, 34536}, {287, 36104}, {293, 685}, {336, 32696}, {1580, 18858}, {1821, 2715}, {1910, 2966}, {1976, 36036}
X(41167) = crosspoint of X(i) and X(j) for these (i,j): {2, 877}, {110, 511}, {850, 2799}, {4230, 39265}, {15631, 36790}
X(41167) = crosssum of X(i) and X(j) for these (i,j): {6, 878}, {98, 523}, {879, 34156}, {1576, 2715}
X(41167) = crossdifference of every pair of points on line {98, 230} (the tangent to the Kiepert hyperbola at X(98))
X(41167) = trilinear product X(i)*X(j) for these {i,j}: {163, 35088}, {240, 684}, {523, 23996}, {656, 2967}, {661, 36790}, {798, 32458}, {822, 36426}, {868, 23997}, {1355, 4086}, {1577, 11672}, {1755, 2799}, {1959, 3569}, {2643, 15631}, {4024, 16725}, {4077, 7062}, {9419, 20948}, {39469, 40703}
X(41167) = barycentric product X(i)*X(j) for these {i,j}: {110, 35088}, {115, 15631}, {232, 6333}, {297, 684}, {325, 3569}, {327, 33569}, {511, 2799}, {512, 32458}, {520, 36426}, {523, 36790}, {525, 2967}, {850, 11672}, {868, 2421}, {1577, 23996}, {4036, 16725}, {6393, 17994}, {16230, 36212}
X(41167) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 685}, {237, 2715}, {297, 22456}, {511, 2966}, {523, 34536}, {684, 287}, {694, 18858}, {1355, 4565}, {1755, 36084}, {1959, 36036}, {2211, 32696}, {2491, 1976}, {2799, 290}, {2967, 648}, {3569, 98}, {7062, 5546}, {8430, 9154}, {9419, 1576}, {11672, 110}, {15631, 4590}, {16230, 16081}, {17994, 6531}, {23098, 2421}, {23611, 14966}, {23996, 662}, {32458, 670}, {33569, 182}, {34854, 20031}, {35088, 850}, {36212, 17932}, {36425, 14574}, {36426, 6528}, {36790, 99}, {39469, 248}, {40810, 39291}
X(41167) = {X(2),X(879)}-harmonic conjugate of X(40550)

X(41168) = X(3)X(66)∩X(216)X(40588)

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

X(41168) lies on the cubic K1181 and these lines: {3, 66}, {216, 40588}, {264, 5133}, {287, 40404}, {1289, 7576}, {1843, 15526}, {3574, 8798}, {5064, 17407}, {5562, 27372}, {7539, 13854}, {13599, 14788}, {15318, 15559}, {35442, 40981}

X(41168) = isotomic conjugate of the isogonal conjugate of X(27372)
X(41168) = X(217)-cross conjugate of X(343)
X(41168) = X(i)-isoconjugate of X(j) for these (i,j): {22, 2190}, {206, 40440}, {275, 2172}, {276, 17453}, {1760, 8882}, {2148, 17907}, {2167, 8743}
X(41168) = cevapoint of X(15451) and X(35442)
X(41168) = crosspoint of X(14376) and X(18018)
X(41168) = crosssum of X(206) and X(8743)
X(41168) = barycentric product X(i)*X(j) for these {i,j}: {5, 14376}, {66, 343}, {76, 27372}, {216, 18018}, {217, 40421}, {2156, 18695}, {2353, 28706}
X(41168) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 17907}, {51, 8743}, {66, 275}, {216, 22}, {217, 206}, {343, 315}, {418, 10316}, {1289, 16813}, {2156, 2190}, {2353, 8882}, {5562, 20806}, {6368, 33294}, {13854, 8884}, {14376, 95}, {15451, 2485}, {17434, 8673}, {18018, 276}, {18695, 20641}, {23181, 4611}, {27372, 6}, {28706, 40073}, {35442, 127}, {40404, 39287}, {40981, 17409}

X(41169) = X(3)X(6)∩X(5)X(311)

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

X(41169) lies on the cubic K1181 and these lines: {3, 6}, {5, 311}, {51, 23181}, {140, 20819}, {143, 3133}, {324, 427}, {1353, 20975}, {1994, 14652}, {3060, 3135}, {3148, 19139}, {3564, 23635}, {3613, 5576}, {5946, 22087}, {6101, 22062}, {6321, 31656}, {7399, 26214}, {9407, 19155}, {12161, 40947}, {14853, 40697}, {15358, 38321}, {15912, 23335}, {19150, 34218}, {32191, 34990}

X(41169) = reflection of X(i) in X(j) for these {i,j}: {3, 570}, {311, 5}
X(41169) = X(262)-Ceva conjugate of X(5)
X(41169) = X(311)-of-Johnson-triangle
X(41169) = barycentric product X(343)*X(6403)
X(41169) = barycentric quotient X(6403)/X(275)
X(41169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3095, 30258, 1351}, {5611, 5615, 567}

X(41170) = X(4)X(6)∩X(52)X(15897)

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

X(41170) lies on the cubic K1181 and these lines: {4, 6}, {52, 15897}, {132, 40588}, {343, 35360}, {1209, 14363}, {6636, 12384}, {30258, 37347}, {32344, 32713}

X(41170) = barycentric product X(324)*X(15577)
X(41170) = barycentric quotient X(15577)/X(97)

X(41171) = X(4)X(69)∩X(5)X(49)

Barycentrics    a^10 - 2*a^8*b^2 + a^6*b^4 - a^4*b^6 + 2*a^2*b^8 - b^10 - 2*a^8*c^2 + 3*a^6*b^2*c^2 - 3*a^4*b^4*c^2 - a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - 3*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - 2*b^6*c^4 - 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 : :
X(41171) = 3*R^2*(J^2 - 5)*X[4] - 2*SW*X[69], 2*(J - 2)*(2 + J)*X[5] + (2 - J^2)*X[49]

X(41169) lies on the cubic K1181 and these lines: {2, 11464}, {3, 18432}, {4, 69}, {5, 49}, {24, 37638}, {30, 37636}, {52, 2888}, {68, 3567}, {70, 4846}, {74, 38323}, {113, 15052}, {146, 12162}, {182, 14789}, {185, 12317}, {186, 21243}, {216, 15340}, {323, 7565}, {343, 7576}, {376, 11550}, {381, 1993}, {539, 1994}, {542, 15032}, {569, 34799}, {575, 25320}, {631, 18381}, {1209, 7488}, {1495, 7552}, {1539, 7723}, {1594, 11064}, {1614, 12134}, {1899, 15045}, {1995, 14852}, {2979, 31723}, {3060, 11818}, {3090, 18925}, {3091, 5654}, {3146, 4549}, {3153, 5891}, {3410, 13754}, {3448, 9730}, {3518, 32223}, {3520, 16163}, {3545, 11427}, {3549, 26882}, {3581, 38322}, {3589, 6146}, {3763, 7509}, {5012, 37347}, {5133, 15033}, {5169, 13352}, {5576, 34148}, {5651, 23325}, {5890, 11442}, {5944, 13565}, {6145, 34782}, {6242, 18428}, {6642, 26917}, {6644, 23293}, {6699, 22467}, {6815, 11457}, {6823, 8718}, {7383, 36989}, {7399, 34224}, {7401, 15024}, {7503, 12289}, {7526, 8907}, {7527, 17702}, {7528, 9781}, {7540, 15107}, {7547, 17814}, {7550, 24206}, {7558, 9833}, {7566, 36747}, {7569, 19357}, {7574, 15067}, {7577, 9306}, {7592, 39899}, {7703, 18281}, {7998, 14791}, {7999, 37444}, {9825, 26879}, {10282, 32354}, {10516, 18396}, {10545, 23410}, {10574, 32140}, {11430, 12383}, {11444, 18569}, {11456, 18440}, {11572, 11793}, {11591, 22804}, {11750, 37126}, {12082, 36990}, {12086, 18488}, {12088, 13419}, {12225, 35254}, {12318, 18489}, {12325, 14531}, {12363, 14128}, {12429, 19139}, {13413, 40111}, {14094, 14982}, {14157, 15760}, {15028, 18952}, {15034, 15133}, {15035, 37118}, {15043, 25738}, {15056, 18404}, {15060, 18403}, {16042, 36253}, {16658, 39884}, {17811, 31180}, {17928, 23294}, {18356, 37481}, {18358, 34664}, {18383, 32346}, {18400, 35921}, {18405, 34787}, {20574, 40410}, {21659, 35500}, {22115, 39504}, {31133, 37483}, {33332, 37495}, {34331, 38794}

X(41171) = reflection of X(i) in X(j) for these {i,j}: {567, 5}, {5012, 37347}, {15033, 5133}
X(41171) = anticomplement of X(37513)
X(41171) = X(567)-of-Johnson-triangle
X(41171) = crossdifference of every pair of points on line {2081, 3049}
X(41171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18474, 25739}, {4, 1352, 11459}, {5, 14516, 54}, {68, 7544, 3567}, {621, 622, 311}, {3091, 5654, 7699}, {6403, 11459, 11412}, {6823, 16659, 8718}, {7401, 18912, 15024}, {11442, 18420, 5890}, {11591, 22804, 31724}, {12134, 13160, 1614}, {15806, 20584, 5}

X(41172) = MIDPOINT OF X(14966) AND X(14967)

Barycentrics    a^2*(b^2 - c^2)^2*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
Trilinears    (sin 2A) (sin 2B sin(A - C) csc(A - B) - sin 2C sin(A - B) csc(A - C)) : :
Trilinears    sin(C - A) csc(A - B) csc 2C - sin(A - B) csc(C - A) csc 2B : :
X(41172) = X[14967] + 2 X[34349]

X(41172) lies on the cubic K203 and these lines: {2, 877}, {3, 6}, {4, 38525}, {111, 14919}, {115, 127}, {122, 6388}, {232, 7418}, {248, 10766}, {446, 39073}, {647, 16186}, {852, 20977}, {1084, 35071}, {1648, 1650}, {1972, 16081}, {2501, 3134}, {2549, 35923}, {2679, 38974}, {2972, 3124}, {3269, 9409}, {5025, 40822}, {5489, 21134}, {5877, 7748}, {6394, 10754}, {8041, 23611}, {8430, 23350}, {10097, 14380}, {10311, 37930}, {13409, 20859}, {15449, 39019}, {15525, 39020}, {15915, 22240}, {23992, 39008}, {34147, 40350}, {35442, 39691}

X(41172) = midpoint of X(14966) and X(14967)
X(41172) = reflection of X(14966) in X(34349)
X(41172) = isotomic conjugate of X(41174)
X(41172) = complement of X(877)
X(41172) = complement of the isogonal conjugate of X(878)
X(41172) = complement of the isotomic conjugate of X(879)
X(41172) = isogonal conjugate of the polar conjugate of X(868)
X(41172) = X(i)-complementary conjugate of X(j) for these (i,j): {98, 21259}, {248, 4369}, {293, 512}, {336, 23301}, {798, 15595}, {810, 114}, {878, 10}, {879, 2887}, {1910, 30476}, {1976, 8062}, {2395, 20305}, {2422, 226}, {3049, 16591}, {3708, 36471}, {14600, 14838}, {14601, 16612}, {32696, 23998}
X(41172) = X(i)-Ceva conjugate of X(j) for these (i,j): {232, 3569}, {248, 647}, {511, 39469}, {1297, 512}, {1916, 525}, {1972, 523}, {2987, 520}, {3269, 38974}, {35098, 34983}, {36212, 684}
X(41172) = X(i)-isoconjugate of X(j) for these (i,j): {99, 36104}, {112, 36036}, {162, 2966}, {163, 22456}, {248, 23999}, {249, 36120}, {250, 1821}, {287, 24000}, {293, 23582}, {336, 23964}, {648, 36084}, {662, 685}, {799, 32696}, {811, 2715}, {1101, 16081}, {1910, 18020}, {4592, 20031}, {6531, 24041}, {17932, 24019}
X(41172) = crosspoint of X(i) and X(j) for these (i,j): {2, 879}, {6, 34212}, {232, 3569}, {248, 647}, {511, 2799}, {512, 39644}, {684, 36212}, {3267, 40708}
X(41172) = crosssum of X(i) and X(j) for these (i,j): {2, 34211}, {6, 4230}, {98, 2715}, {110, 10313}, {112, 19128}, {287, 2966}, {297, 648}, {685, 6531}
X(41172) = crossdifference of every pair of points on line {250, 523} (the line through X(648) perpendicular to the trilinear polar of X(648))
X(41172) = barycentric product X(i)*X(j) for these {i,j}: {3, 868}, {115, 36212}, {125, 511}, {232, 15526}, {237, 339}, {240, 2632}, {248, 35088}, {297, 3269}, {325, 20975}, {338, 3289}, {512, 6333}, {520, 16230}, {523, 684}, {525, 3569}, {647, 2799}, {850, 39469}, {1650, 35908}, {1755, 20902}, {1959, 3708}, {1972, 38974}, {2211, 36793}, {2491, 3267}, {2679, 40708}, {2972, 6530}, {3124, 6393}, {3265, 17994}, {4230, 5489}, {8430, 14417}, {9033, 32112}, {14356, 16186}, {17209, 21046}, {19189, 35442}, {23107, 34859}, {34138, 38356}
X(41172) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 16081}, {125, 290}, {232, 23582}, {237, 250}, {240, 23999}, {339, 18024}, {511, 18020}, {512, 685}, {520, 17932}, {523, 22456}, {647, 2966}, {656, 36036}, {669, 32696}, {684, 99}, {798, 36104}, {810, 36084}, {868, 264}, {2211, 23964}, {2489, 20031}, {2491, 112}, {2632, 336}, {2643, 36120}, {2679, 419}, {2799, 6331}, {2972, 6394}, {3049, 2715}, {3124, 6531}, {3269, 287}, {3289, 249}, {3569, 648}, {3708, 1821}, {5360, 5379}, {6333, 670}, {6393, 34537}, {16230, 6528}, {17994, 107}, {20975, 98}, {23216, 14601}, {32112, 16077}, {34854, 32230}, {34980, 17974}, {36212, 4590}, {38356, 31636}, {38974, 401}, {39469, 110}

X(41173) = X(523)-CROSS CONJUGATE OF X(98)

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

X(41173) lies on these lines: {2, 9476}, {98, 868}, {110, 18858}, {685, 2395}, {878, 4230}, {879, 2966}, {1316, 14265}, {1640, 35278}, {2396, 17932}, {2715, 6037}, {5304, 9154}, {5967, 14912}, {35912, 35922}

X(41173) = isogonal conjugate of X(41167)
X(41173) = isogonal conjugate of the anticomplement of X(40550)
X(41173) = isogonal conjugate of the complement of X(879)
X(41173) = X(i)-cross conjugate of X(j) for these (i,j): {523, 98}, {1576, 2715}, {1632, 22456}, {2966, 18858}, {35278, 6037}
X(41173) = X(i)-isoconjugate of X(j) for these (i,j): {163, 35088}, {240, 684}, {523, 23996}, {656, 2967}, {661, 36790}, {798, 32458}, {822, 36426}, {868, 23997}, {1355, 4086}, {1577, 11672}, {1755, 2799}, {1959, 3569}, {2643, 15631}, {4024, 16725}, {4077, 7062}, {9419, 20948}, {39469, 40703}
X(41173) = cevapoint of X(i) and X(j) for these (i,j): {6, 878}, {98, 523}, {879, 34156}, {1576, 2715}
X(41173) = trilinear pole of line {98, 230} (the tangent to the Kiepert hyperbola at X(98))
X(41173) = trilinear product X(i)*X(j) for these {i,j}: {98, 36084}, {163, 34536}, {287, 36104}, {293, 685}, {336, 32696}, {1580, 18858}, {1821, 2715}, {1910, 2966}, {1976, 36036}
X(41173) = barycentric product X(i)*X(j) for these {i,j}: {98, 2966}, {110, 34536}, {248, 22456}, {287, 685}, {290, 2715}, {336, 36104}, {385, 18858}, {1821, 36084}, {1910, 36036}, {6394, 20031}, {6531, 17932}, {39291, 40820}
X(41173) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 2799}, {99, 32458}, {107, 36426}, {110, 36790}, {112, 2967}, {163, 23996}, {248, 684}, {249, 15631}, {287, 6333}, {523, 35088}, {685, 297}, {1576, 11672}, {1976, 3569}, {2395, 868}, {2715, 511}, {2966, 325}, {6531, 16230}, {14574, 9419}, {14600, 39469}, {14601, 2491}, {14966, 23098}, {17932, 6393}, {18858, 1916}, {20031, 6530}, {32696, 232}, {34396, 33569}, {34536, 850}, {36084, 1959}, {36104, 240}
X(41173) = {X(32545),X(40820)}-harmonic conjugate of X(1316)

X(41174) = X(401)-CROSS CONJUGATE OF X(99)

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

X(41174) lies on these lines: {76, 249}, {250, 264}, {313, 4570}, {327, 5651}, {685, 32717}, {691, 22456}, {877, 879}, {1502, 4590}, {1968, 10684}, {4567, 27801}, {5649, 6331}, {6531, 18896}, {14960, 32697}, {16081, 18023}, {18879, 40832}, {20573, 39295}

X(41174) = isotomic conjugate of X(41172)
X(41174) = isotomic conjugate of the complement of X(877)
X(41174) = polar conjugate of isogonal conjugate of isotomic conjugate of X(868)
X(41174) = X(i)-cross conjugate of X(j) for these (i,j): {384, 39291}, {385, 648}, {401, 99}, {10313, 4577}, {16081, 22456}, {30737, 670}
X(41174) = X(i)-isoconjugate of X(j) for these (i,j): {125, 9417}, {237, 3708}, {656, 2491}, {661, 39469}, {684, 798}, {810, 3569}, {822, 17994}, {868, 9247}, {1755, 20975}, {1924, 6333}, {2211, 2632}, {2643, 3289}, {4117, 6393}, {9418, 20902}, {34854, 37754}
X(41174) = cevapoint of X(i) and X(j) for these (i,j): {2, 877}, {112, 419}, {290, 2966}, {9306, 14966}, {16081, 22456}
X(41174) = trilinear pole of line {110, 685}
X(41174) = barycentric product X(i)*X(j) for these {i,j}: {99, 22456}, {250, 18024}, {290, 18020}, {336, 23999}, {670, 685}, {811, 36036}, {2966, 6331}, {4590, 16081}, {4602, 36104}, {4609, 32696}, {6528, 17932}, {6531, 34537}, {24037, 36120}
X(41174) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 20975}, {99, 684}, {107, 17994}, {110, 39469}, {112, 2491}, {249, 3289}, {250, 237}, {264, 868}, {287, 3269}, {290, 125}, {336, 2632}, {401, 38974}, {419, 2679}, {648, 3569}, {670, 6333}, {685, 512}, {1821, 3708}, {2715, 3049}, {2966, 647}, {4590, 36212}, {5379, 5360}, {6331, 2799}, {6394, 2972}, {6528, 16230}, {6531, 3124}, {14601, 23216}, {16077, 32112}, {16081, 115}, {17932, 520}, {17974, 34980}, {18020, 511}, {18024, 339}, {20031, 2489}, {22456, 523}, {23582, 232}, {23964, 2211}, {23999, 240}, {31636, 38356}, {32230, 34854}, {32696, 669}, {34537, 6393}, {36036, 656}, {36084, 810}, {36104, 798}, {36120, 2643}

X(41175) = MIDPOINT OF X(98) AND X(2715)

Barycentrics    (a^2 - b^2 - c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(2*a^8 - 2*a^6*b^2 + a^4*b^4 - b^8 - 2*a^6*c^2 + 2*b^6*c^2 + a^4*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8) : :
X(41175) = X[2710] - 3 X[34473]

X(41175) lies on these lines: {3, 525}, {67, 11653}, {98, 230}, {468, 1976}, {2710, 34473}, {3564, 17974}, {5967, 26869}, {6036, 36471}, {6393, 15407}, {8550, 14355}, {13137, 36166}, {16925, 18337}

X(41175) = midpoint of X(98) and X(2715)
X(41175) = reflection of X(36471) in X(6036)
X(41175) = circumcircle-inverse of X(878)
X(41175) = X(240)-isoconjugate of X(2710)
X(41175) = barycentric product X(287)*X(2794)
X(41175) = barycentric quotient X(i)/X(j) for these {i,j}: {248, 2710}, {2794, 297}

X(41176) = X(2)X(99)∩X(1648)X(1649)

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

X(41176) lies on the cubics K203 and K219 and on these lines: {2, 99}, {1648, 1649}, {5099, 9193}, {14443, 14444}

X(41176) = complement of X(34760)
X(41176) = complement of the isotomic conjugate of X(34763)
X(41176) = tripolar centroid of X(690)
X(41176) = X(i)-complementary conjugate of X(j) for these (i,j): {9180, 21256}, {34763, 2887}
X(41176) = X(i)-Ceva conjugate of X(j) for these (i,j): {543, 33921}, {8371, 14423}, {18823, 690}
X(41176) = crosspoint of X(i) and X(j) for these (i,j): {2, 34763}, {543, 33921}, {690, 18823}, {1641, 8371}
X(41176) = crosssum of X(i) and X(j) for these (i,j): {6, 23348}, {691, 2502}
X(41176) = crossdifference of every pair of points on line {351, 691}
X(41176) = barycentric product X(i)*X(j) for these {i,j}: {543, 23992}, {690, 33921}, {1641, 1648}, {1649, 8371}, {5468, 14423}, {9182, 14443}, {14444, 17948}, {18007, 33915}
X(41176) = barycentric quotient X(i)/X(j) for these {i,j}: {1649, 9170}, {2502, 34539}, {9171, 34574}, {14423, 5466}, {14443, 9180}, {23992, 18823}, {33921, 892}
X(41176) = {X(1648),X(1649)}-harmonic conjugate of X(23992)

X(41177) = CROSSPOINT OF X(2) AND X(18007)

Barycentrics    (b^2 - c^2)^2*(-2*a^2 + b^2 + c^2)*(-2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 4*b^2*c^2 + c^4)*(5*a^4 - 5*a^2*b^2 + 2*b^4 - 5*a^2*c^2 + b^2*c^2 + 2*c^4) : :
X(41177) = 2 X[1648] - 3 X[23991]

X(41177) lies on the cubic K203) and these lines: {115, 524}, {690, 23992}, {1648, 23991}, {14444, 33906}

X(41177) = complement of the isogonal conjugate of X(17993)
X(41177) = complement of the isotomic conjugate of X(18007)
X(41177) = X(i)-complementary conjugate of X(j) for these (i,j): {9171, 16597}, {17955, 512}, {17964, 4369}, {17993, 10}, {18007, 2887}, {23348, 21254}
X(41177) = X(524)-Ceva conjugate of X(33921)
X(41177) = crosspoint of X(i) and X(j) for these (i,j): {2, 18007}, {690, 1641}
X(41177) = barycentric product X(9168)*X(33921)

X(41178) = CROSSPOINT OF X(2) AND X(882)

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

X(41178) lies on the cubic K203 and these lines: {2, 880}, {39, 141}, {115, 804}, {733, 11606}, {782, 15449}, {1645, 1648}, {2086, 2679}, {5661, 8265}, {6071, 21906}, {7790, 14700}, {9468, 11646}, {21725, 21823}, {23992, 39010}, {35971, 39691}

X(41178) = complement of X(880)
X(41178) = complement of the isogonal conjugate of X(881)
X(41178) = complement of the isotomic conjugate of X(882)
X(41178) = X(i)-complementary conjugate of X(j) for these (i,j): {669, 19563}, {798, 39080}, {881, 10}, {882, 2887}, {1581, 23301}, {1916, 21263}, {1924, 5976}, {1927, 523}, {1967, 512}, {4117, 35078}, {8789, 14838}, {9468, 4369}, {17938, 21254}, {17980, 21259}, {37134, 36950}, {40729, 27854}
X(41178) = X(i)-Ceva conjugate of X(j) for these (i,j): {115, 2679}, {11606, 512}, {20021, 688}, {39939, 523}
X(41178) = X(i)-isoconjugate of X(j) for these (i,j): {82, 39292}, {733, 24037}, {805, 4593}, {4577, 37134}, {4599, 18829}, {14970, 24041}, {17938, 37204}
X(41178) = crosspoint of X(i) and X(j) for these (i,j): {2, 882}, {732, 804}
X(41178) = crosssum of X(i) and X(j) for these (i,j): {6, 17941}, {733, 805}, {4576, 7779}
X(41178) = crossdifference of every pair of points on line {1634, 4577}
X(41178) = barycentric product X(i)*X(j) for these {i,j}: {115, 8623}, {141, 2086}, {688, 14295}, {732, 3124}, {804, 3005}, {826, 5027}, {1084, 35540}, {1691, 39691}, {2236, 2643}, {2679, 20021}, {11606, 39079}, {16587, 39786}, {21818, 27918}
X(41178) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 39292}, {688, 805}, {732, 34537}, {804, 689}, {1084, 733}, {2084, 37134}, {2086, 83}, {2236, 24037}, {2679, 20022}, {3005, 18829}, {3124, 14970}, {5027, 4577}, {8623, 4590}, {9494, 17938}, {21755, 39276}, {39079, 7779}, {39691, 18896}

X(41179) = CROSSPOINT OF X(2) AND X(18003)

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

X(41179) lies on the cubic K203 and these lines: {2, 17929}, {115, 513}, {891, 23992}, {1648, 14434}, {2679, 9297}, {3835, 8287}, {6371, 23991}

X(41179) = complement of X(17929)
X(41179) = complement of the isogonal conjugate of X(17989)
X(41179) = complement of the isotomic conjugate of X(18003)
X(41179) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 2787}, {872, 2511}, {2787, 3741}, {5006, 21196}, {5040, 1125}, {5291, 4369}, {17763, 512}, {17944, 21254}, {17987, 21259}, {17989, 10}, {18003, 2887}
X(41179) = X(1029)-Ceva conjugate of X(2787)
X(41179) = crosspoint of X(2) and X(18003)
X(41179) = crosssum of X(6) and X(17939)

X(41180) = CROSSPOINT OF X(2) AND X(18004)

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

X(41180) lies on the cubic K203 and these lines: {2, 17930}, {115, 514}, {650, 16592}, {661, 6627}, {900, 23992}, {1648, 6544}, {2679, 6373}, {3120, 4988}, {3936, 6651}, {4977, 23991}, {10026, 17770}

X(41180) = complement of X(17930)
X(41180) = complement of the isogonal conjugate of X(17990)
X(41180) = complement of the isotomic conjugate of X(18004)
X(41180) = X(i)-complementary conjugate of X(j) for these (i,j): {213, 2786}, {798, 239}, {1757, 512}, {2786, 21240}, {4079, 20337}, {4705, 20546}, {5029, 3739}, {6541, 21260}, {9508, 3741}, {17735, 4369}, {17927, 21259}, {17943, 21254}, {17990, 10}, {18004, 2887}, {18266, 523}, {20693, 3835}, {20947, 23301}
X(41180) = X(6625)-Ceva conjugate of X(2786)
X(41180) = crosspoint of X(i) and X(j) for these (i,j): {2, 18004}, {514, 17770}
X(41180) = crosssum of X(i) and X(j) for these (i,j): {6, 17940}, {101, 28482}

X(41181) = CROSSPOINT OF X(2) AND X(16230)

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

X(41181) lies on the cubic K203 and these lines: {2, 17932}, {114, 230}, {115, 525}, {125, 647}, {520, 23991}, {1648, 14401}, {2679, 39469}, {2715, 11005}, {3124, 17434}, {6388, 6587}, {8673, 14113}, {9033, 23992}, {11672, 39021}, {15449, 39009}, {16230, 38970}, {35441, 39691}

X(41181) = complement of X(17932)
X(41181) = complement of the isogonal conjugate of X(17994)
X(41181) = complement of the isotomic conjugate of X(16230)
X(41181) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 24284}, {158, 39469}, {232, 4369}, {240, 512}, {798, 441}, {1096, 6130}, {1973, 2799}, {2211, 14838}, {2489, 16609}, {2491, 1214}, {2643, 3150}, {3569, 18589}, {4230, 21254}, {5360, 20315}, {6530, 21259}, {16230, 2887}, {17994, 10}, {24006, 21531}, {34854, 8062}, {34859, 16599}, {40703, 23301}
X(41181) = X(i)-Ceva conjugate of X(j) for these (i,j): {68, 39469}, {2996, 2799}
X(41181) = X(i)-isoconjugate of X(j) for these (i,j): {2715, 36105}, {10425, 36104}, {32697, 36084}
X(41181) = crosspoint of X(i) and X(j) for these (i,j): {2, 16230}, {525, 3564}
X(41181) = crosssum of X(112) and X(3563)
X(41181) = crossdifference of every pair of points on line {4230, 32696}
X(41181) = barycentric product X(i)*X(j) for these {i,j}: {114, 125}, {868, 3564}, {17462, 20902}
X(41181) = barycentric quotient X(i)/X(j) for these {i,j}: {114, 18020}, {125, 40428}, {684, 10425}, {868, 35142}, {3569, 32697}, {20975, 2065}

X(41182) = CROSSPOINT OF X(2) AND X(18013)

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

X(411821) lies on the cubic K203 and these lines: {2, 17933}, {115, 124}, {1566, 35583}, {1648, 33573}, {3664, 5745}, {23992, 35091}

X(41182) = complement of X(17933)
X(41182) = complement of the isogonal conjugate of X(18000)
X(41182) = complement of the isotomic conjugate of X(18013)
X(41182) = X(i)-complementary conjugate of X(j) for these (i,j): {798, 39035}, {2648, 512}, {2652, 17072}, {17963, 4369}, {18000, 10}, {18013, 2887}
X(41182) = crosspoint of X(2) and X(18013)
X(41182) = crosssum of X(6) and X(17942)

X(41183) = X(101)X(842)∩X(106)X(691)

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

The circle passing through X(101) and the isodynamic points, X(15) and X(16), is here named the Moses isodynamic circle. The points X(41183)-X(41193) lies on this circle, as do the points kX(i) for i = 15,16,101,106,214,9321,11716,38013,38014.

X(41183) lies on the Moses isodynamic circle and these lines: {23, 5168}, {101, 842}, {106, 691}, {1960, 20403}, {5029, 9213}

X(41184) = X(42)X(101)∩X(106)X(1296)

Barycentrics    a^2*(2*a^5 + 2*a^4*b - 5*a^3*b^2 + a^2*b^3 - 7*a*b^4 - b^5 + 2*a^4*c - 2*a^2*b^2*c - 4*b^4*c - 5*a^3*c^2 - 2*a^2*b*c^2 + 22*a*b^2*c^2 + 4*b^3*c^2 + a^2*c^3 + 4*b^2*c^3 - 7*a*c^4 - 4*b*c^4 - c^5) : :

X(41184) lies on the Moses isodynamic circle and these lines: {42, 101}, {106, 1296}, {214, 2805}, {1960, 6088}, {5029, 9156}, {9522, 11716}

X(41185) = X(101)X(691)∩X(106)X(842)

Barycentrics    a^2*(b - c)*(2*a^11 - 2*a^10*b - 3*a^9*b^2 + a^8*b^3 + 2*a^7*b^4 + a^6*b^5 + a^4*b^7 - 4*a^3*b^8 + a^2*b^9 + 3*a*b^10 - 2*b^11 - 2*a^10*c + 2*a^9*b*c + 2*a^8*b^2*c + a^6*b^4*c - 4*a^5*b^5*c - a^4*b^6*c + a^2*b^8*c + 2*a*b^9*c - b^10*c - 3*a^9*c^2 + 2*a^8*b*c^2 + 8*a^7*b^2*c^2 - 2*a^6*b^3*c^2 - 10*a^5*b^4*c^2 + a^4*b^5*c^2 + 12*a^3*b^6*c^2 - 5*a^2*b^7*c^2 - 6*a*b^8*c^2 + 5*b^9*c^2 + a^8*c^3 - 2*a^6*b^2*c^3 - 4*a^5*b^3*c^3 + a^4*b^4*c^3 + 8*a^3*b^5*c^3 - a^2*b^6*c^3 - 6*a*b^7*c^3 + b^8*c^3 + 2*a^7*c^4 + a^6*b*c^4 - 10*a^5*b^2*c^4 + a^4*b^3*c^4 + 2*a^3*b^4*c^4 - 2*a^2*b^5*c^4 - 2*b^7*c^4 + a^6*c^5 - 4*a^5*b*c^5 + a^4*b^2*c^5 + 8*a^3*b^3*c^5 - 2*a^2*b^4*c^5 + 2*a*b^5*c^5 + 2*b^6*c^5 - a^4*b*c^6 + 12*a^3*b^2*c^6 - a^2*b^3*c^6 + 2*b^5*c^6 + a^4*c^7 - 5*a^2*b^2*c^7 - 6*a*b^3*c^7 - 2*b^4*c^7 - 4*a^3*c^8 + a^2*b*c^8 - 6*a*b^2*c^8 + b^3*c^8 + a^2*c^9 + 2*a*b*c^9 + 5*b^2*c^9 + 3*a*c^10 - b*c^10 - 2*c^11) : :

X(41185) lies on the Moses isodynamic circle and these lines: {23, 5029}, {101, 691}, {106, 842}, {523, 24350}, {1960, 20403}, {5168, 9213}

X(41186) = X(101)X(1296)∩X(106)X(111)

Barycentrics    a^2*(b - c)*(2*a^7 - 2*a^6*b - 5*a^5*b^2 - 4*a^3*b^4 + 3*a*b^6 - 2*b^7 - 2*a^6*c + 2*a^5*b*c + a^4*b^2*c + 4*a^3*b^3*c + 2*a^2*b^4*c + 2*a*b^5*c - b^6*c - 5*a^5*c^2 + a^4*b*c^2 + 34*a^3*b^2*c^2 - 10*a^2*b^3*c^2 - 12*a*b^4*c^2 + 10*b^5*c^2 + 4*a^3*b*c^3 - 10*a^2*b^2*c^3 - 14*a*b^3*c^3 + 2*b^4*c^3 - 4*a^3*c^4 + 2*a^2*b*c^4 - 12*a*b^2*c^4 + 2*b^3*c^4 + 2*a*b*c^5 + 10*b^2*c^5 + 3*a*c^6 - b*c^6 - 2*c^7) : :

X(41186) lies on the Moses isodynamic circle and these lines: {101, 1296}, {106, 111}, {214, 2830}, {665, 41165}, {1960, 6088}, {2780, 8659}, {2837, 11716}, {5168, 9156}

X(41187) = X(101)X(112)∩X(106)X(1297)

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

X(41187) lies on the Moses isodynamic circle and these lines: {101, 112}, {106, 1297}, {214, 2806}, {1960, 2881}, {2853, 19162}, {5029, 9157}, {5168, 13114}, {9523, 11716}

X(41188) = X(101)X(1297)∩X(106)X(112)

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

X(41188) lies on the Moses isodynamic circle and these lines: {101, 1297}, {106, 112}, {214, 2831}, {1960, 2881}, {2838, 11716}, {5029, 13114}, {5168, 9157}, {8659, 9517}, {9532, 19159}

X(41189) = X(74)X(106)∩X(101)X(110)

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

X(1189) lies on the Moses isodynamic circle and these lines: {74, 106}, {101, 110}, {214, 8674}, {526, 1960}, {665, 41164}, {2609, 9321}, {2773, 22586}, {2775, 11709}, {5168, 9138}, {8659, 9517}

X(41190) = X(98)X(106)∩X(99)X(101)

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

X(41190) lies on the Moses isodynamic circle and these lines: {2, 4107}, {98, 106}, {99, 101}, {214, 2787}, {385, 514}, {665, 41163}, {804, 1960}, {1575, 9321}, {2785, 8301}, {2788, 11710}, {3667, 11676}, {5168, 9147}

X(41191) = X(100)X(101)∩X(104)X(106)

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

X(41191) lies on the Moses isodynamic circle and these lines: {1, 13266}, {44, 4794}, {100, 101}, {104, 106}, {214, 900}, {659, 2802}, {665, 41166}, {1320, 21385}, {2800, 19916}, {2826, 11715}, {3738, 4491}, {3837, 32557}, {5029, 9978}, {5168, 9980}, {6161, 13277}

X(41192) = X(58)X(106)∩X(71)X(74)

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

X(41192) lies on the Moses isodynamic circle and these lines: {6, 15049}, {58, 106}, {71, 74}, {214, 960}, {323, 2392}, {399, 2779}, {526, 1960}, {542, 15985}, {758, 1495}, {846, 2948}, {1017, 2092}, {2305, 21781}, {2780, 8659}, {2836, 5049}, {3743, 37080}, {3960, 35053}, {5029, 9138}, {5092, 10176}, {5663, 35203}, {5692, 15080}, {5693, 11464}, {5902, 10546}, {11430, 31803}, {13605, 25354}, {20117, 37513}, {25058, 38482}, {31817, 37478}

X(41192) = 4th-Brocard-to-circumsymmedial similarity image of X(10)
X(41192) = X(10)-of-anti-orthocentroidal-triangle

X(41193) = X(10)X(98)∩X(86)X(99)

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

X(41193) lies on the Moses isodynamic circle and these lines: {1, 1281}, {2, 5168}, {10, 98}, {86, 99}, {214, 2783}, {385, 519}, {516, 11676}, {523, 24350}, {542, 15985}, {804, 1960}, {993, 2792}, {1015, 33682}, {1125, 5988}, {2795, 11711}, {2805, 24714}, {3616, 5992}, {3647, 25607}, {3741, 27950}, {3821, 10000}, {3840, 10069}, {3923, 7709}, {4027, 30038}, {4660, 9862}, {4871, 5990}, {5029, 9147}, {8618, 23398}, {9321, 24036}, {11364, 17766}, {11599, 17962}, {15953, 40763}, {16468, 27348}

X(41194) = POLAR CONJUGATE OF X(5000)

Barycentrics    1/(a^2*SA*(S*SB*SC + SA*Sqrt[SA*SB*SC*SW])) : :

X(41194) lies on the Kiepert circumhyperbola and these lines: {2, 41197}, {4, 32618}, {6, 264}, {98, 5000}, {262, 5001}, {401, 41196}

X(41194) = isogonal conjugate of X(41196)
X(41194) = isotomic conjugate of X(41198)
X(41194) = polar conjugate of X(5000)
X(41194) = polar conjugate of the isogonal conjugate of X(32618)
X(41194) = X(290)-Ceva conjugate of X(34239)
X(41194) = X(5000)-cross conjugate of X(264)
X(41194) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41196}, {48, 5000}, {1755, 32619}
X(41194) = cevapoint of X(6) and X(5000)
X(41194) = trilinear pole of line {523, 5001}
X(41194) = trilinear product X(i)*X(j) for these {i,j}: {92, 32618}, {1821, 5001}
X(41194) = barycentric product X(i)*X(j) for these {i,j}: {264, 32618}, {290, 5001}
X(41194) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5000}, {6, 41196}, {98, 32619}, {5001, 511}, {32618, 3}, {34239, 5003}, {41195, 3289}
X(41194) = {X(4),X(32618)}-harmonic conjugate of X(34239)

X(41195) = POLAR CONJUGATE OF X(5001)

Barycentrics    1/(a^2*SA*(S*SB*SC - SA*Sqrt[SA*SB*SC*SW])) : :

X(41195) lies on these lines: {2, 41196}, {4, 32619}, {6, 264}, {98, 5001}, {262, 5000}, {401, 41197}

X(41195) = isogonal conjugate of X(41197)
X(41195) = isotomic conjugate of X(41199)
X(41195) = polar conjugate of X(5001)
X(41195) = polar conjugate of the isogonal conjugate of X(32619)
X(41195) = X(290)-Ceva conjugate of X(34240)
X(41195) = X(5001)-cross conjugate of X(264)
X(41195) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41197}, {48, 5001}, {1755, 32618}
X(41195) = cevapoint of X(6) and X(5001)
X(41195) = trilinear pole of line {523, 5000}
X(41195) = trilinear product X(i)*X(j) for these {i,j}: {92, 32619}, {1821, 5000}
X(41195) = barycentric product X(i)*X(j) for these {i,j}: {264, 32619}, {290, 5000}
X(41195) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5001}, {6, 41197}, {98, 32618}, {5000, 511}, {32619, 3}, {34240, 5002}, {41196, 3289}
X(41195) = {X(4),X(32619)}-harmonic conjugate of X(34240)

X(41196) = ISOGONAL CONJUGATE OF X(41194)

Barycentrics    a^2*(Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)] + 2*(-a^2 + b^2 + c^2)*S) : :
Barycentrics    a^2*(S*SA + Sqrt[SA*SB*SC*SW]) : :
Barycentrics    a^2*(SA + S*Sqrt[Cot[A]*Cot[B]*Cot[C]*Cot[w]]) : :
X(41196) = S^3*X[3] + SW*Sqrt[SA*SB*SC*SW]*X[6]

X(41196) lies on these lines: {3, 6}, {112, 40894}, {232, 5000}, {248, 32618}, {5001, 10311}, {5002, 10313}, {5003, 22240}

X(41196) = isogonal conjugate of X(41194)
X(41196) = isogonal conjugate of the polar conjugate of X(5000)
X(41196) = crosssum of X(6) and X(5000)
X(41196) = complement of the isotomic conjugate of X(32618)
X(41196) = X(32618)-complementary conjugate of X(2887)
X(41196) = X(i)-Ceva conjugate of X(j) for these (i,j): {248, 41147}, {32618, 184}
X(41196) = X(11672)-cross conjugate of X(41147)
X(41196) = crosspoint of X(2) and X(32618)
X(41196) = crossdifference of every pair of points on line {523, 5001}
X(41196) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41194}, {92, 32618}, {1821, 5001}
X(41196) = trilinear product X(i)*X(j) for these {i,j}: {48, 5000}, {1755, 32619}
X(41196) = barycentric product X(i)*X(j) for these {i,j}: {3, 5000}, {511, 32619}
X(41196) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 32618}, {237, 5001}, {5000, 264}, {32619, 290}
X(41196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 41147}, {32, 577, 41147}, {39, 216, 41147}, {50, 10317, 41147}, {187, 3284, 41147}, {571, 10316, 41147}, {574, 5158, 41147}, {800, 22401, 41147}, {1609, 23115, 41147}, {3003, 14961, 41147}, {3053, 15905, 41147}, {5007, 22052, 41147}, {5023, 38292, 41147}, {7772, 10979, 41147}, {8553, 22120, 41147}, {8589, 15860, 41147}, {9605, 36751, 41147}, {11063, 22121, 41147}, {11574, 13357, 41147}, {13356, 19126, 41147}, {15166, 15167, 41147}, {15815, 15851, 41147}, {30435, 36748, 41147}, {34870, 37893, 41147}, {40135, 40349, 41147}

X(41197) = ISOGONAL CONJUGATE OF X(41195)

Barycentrics    a^2*(Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)] - 2*(-a^2 + b^2 + c^2)*S) : :
Barycentrics   
Barycentrics    a^2*(S*SA - Sqrt[SA*SB*SC*SW]) : :
a^2*(SA - S*Sqrt[Cot[A]*Cot[B]*Cot[C]*Cot[w]]) : :
X(41197) = S^3*X[3] + SW*Sqrt[SA*SB*SC*SW]*X[6]

X(41197) lies on these lines: {3, 6}, {112, 40895}, {232, 5001}, {248, 32619}, {5000, 10311}, {5002, 22240}, {5003, 10313}

X(41197) = isogonal conjugate of X(41145)
X(41197) = isogonal conjugate of the polar conjugate of X(5001)
X(41197) = crosssum of X(6) and X(5001)
X(41197) = complement of the isotomic conjugate of X(32619)
X(41197) = X(32619)-complementary conjugate of X(2887)
X(41197) = X(i)-Ceva conjugate of X(j) for these (i,j): {248, 41146}, {32619, 184}
X(41197) = X(11672)-cross conjugate of X(41146)
X(41197) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41195}, {92, 32619}, {1821, 5000}
X(41197) = crosspoint of X(2) and X(32619)
X(41197) = crossdifference of every pair of points on line {523, 5000}
X(41197) = trilinear product X(i)*X(j) for these {i,j}: {48, 5001}, {1755, 32618}
X(41197) = barycentric product X(i)*X(j) for these {i,j}: {3, 5001}, {511, 32618}
X(41197) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 32619}, {237, 5000}, {5001, 264}, {32618, 290}
X(41197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 41146}, {32, 577, 41146}, {39, 216, 41146}, {50, 10317, 41146}, {187, 3284, 41146}, {571, 10316, 41146}, {574, 5158, 41146}, {800, 22401, 41146}, {1609, 23115, 41146}, {3003, 14961, 41146}, {3053, 15905, 41146}, {5007, 22052, 41146}, {5023, 38292, 41146}, {7772, 10979, 41146}, {8553, 22120, 41146}, {8589, 15860, 41146}, {9605, 36751, 41146}, {11063, 22121, 41146}, {11574, 13357, 41146}, {13356, 19126, 41146}, {15166, 15167, 41146}, {15815, 15851, 41146}, {30435, 36748, 41146}, {34870, 37893, 41146}, {40135, 40349, 41146}

X(41197) = trilinear product X(i)*X(j) for these {i,j}: {48, 5001}, {1755, 32618}
&cap

X(41198) = X(2)X(6)∩X(511)X(5000)

Barycentrics    Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)] + 2*(-a^2 + b^2 + c^2)*S : :
X(41198) = 3*(S*SW + Sqrt[SA*SB*SC*SW])*X[2] - 2*S*SW*X[6]

X(41198) lies on these lines: {2, 6}, {3, 32619}, {76, 41195}, {511, 5000}, {1350, 5003}, {1352, 5001}, {1503, 5002}, {3564, 32618}, {36212, 41196}, {40802, 41201}

X(41198) = reflection of X(41199) in X(11064)
X(41198) = isogonal conjugate of X(41200)
X(41198) = isotomic conjugate of X(41194)
X(41198) = isotomic conjugate of the isogonal conjugate of X(41196)
X(41198) = isotomic conjugate of the polar conjugate of X(5000)
X(41198) = X(i)-Ceva conjugate of X(j) for these (i,j): {287, 41199}, {35140, 5003}
X(41198) = X(i)-cross conjugate of X(j) for these (i,j): {36790, 41199}, {41196, 5000}
X(41198) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41200}, {19, 32618}, {31, 41194}, {1910, 5001}, {36120, 41197}
X(41198) = trilinear product X(i)*X(j) for these {i,j}: {63, 5000}, {75, 41196}, {1959, 32619}
X(41198) = barycentric product X(i)*X(j) for these {i,j}: {69, 5000}, {76, 41196}, {325, 32619}, {6393, 41201}, {36212, 41195}
X(41198) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 41194}, {3, 32618}, {6, 41200}, {511, 5001}, {3289, 41197}, {5000, 4}, {5003, 34239}, {32619, 98}, {36212, 41199}, {41195, 16081}, {41196, 6}, {41201, 6531}
X(41198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 41199}, {6, 394, 41199}, {141, 343, 41199}, {193, 37669, 41199}, {323, 22151, 41199}, {599, 37638, 41199}, {1812, 15988, 41199}, {1993, 20806, 41199}, {2063, 40318, 41199}, {6515, 28419, 41199}, {8115, 8116, 41199}

X(41199) = X(2)X(6)∩X(511)X(5001)

Barycentrics    Sqrt[(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + c^2)] - 2*(-a^2 + b^2 + c^2)*S : :
X(41199) = 3*(S*SW - Sqrt[SA*SB*SC*SW])*X[2] - 2*S*SW*X[6]

X(41199) lies on these lines: {2, 6}, {3, 32618}, {76, 41194}, {511, 5001}, {1350, 5002}, {1352, 5000}, {1503, 5003}, {3564, 32619}, {36212, 41197}, {40802, 41200}

X(41199) = reflection of X(41198) in X(11064)
X(41199) = isogonal conjugate of X(41201)
X(41199) = isotomic conjugate of X(41195)
X(41199) = isotomic conjugate of the isogonal conjugate of X(41197)
X(41199) = isotomic conjugate of the polar conjugate of X(5001)
X(41199) = X(i)-Ceva conjugate of X(j) for these (i,j): {287, 41198}, {35140, 5002}
X(41199) = X(i)-cross conjugate of X(j) for these (i,j): {36790, 41198}, {41197, 5001}
X(41199) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41201}, {19, 32619}, {31, 41195}, {1910, 5000}, {36120, 41196}
X(41199) = trilinear product X(i)*X(j) for these {i,j}: {63, 5001}, {75, 41197}, {1959, 32618}
X(41199) = barycentric product X(i)*X(j) for these {i,j}: {69, 5001}, {76, 41197}, {325, 32618}, {6393, 41200}, {36212, 41194}
X(41199) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 41195}, {3, 32619}, {6, 41201}, {511, 5000}, {3289, 41196}, {5001, 4}, {5002, 34240}, {32618, 98}, {36212, 41198}, {41194, 16081}, {41197, 6}, {41200, 6531}
X(41199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 41198}, {6, 394, 41198}, {141, 343, 41198}, {193, 37669, 41198}, {323, 22151, 41198}, {599, 37638, 41198}, {1812, 15988, 41198}, {1993, 20806, 41198}, {2063, 40318, 41198}, {6515, 28419, 41198}, {8115, 8116, 41198}

X(41200) = X(4)X(32)∩X(6)X(5001)

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

X(41200) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 41194}, {4, 32}, {6, 5001}, {187, 40894}, {230, 5000}, {2549, 40895}, {5002, 10313}, {5304, 34239}, {40802, 41199}

X(41200) = isogonal conjugate of X(41198)
X(41200) = isogonal conjugate of the isotomic conjugate of X(41194)
X(41200) = polar conjugate of the isotomic conjugate of X(32618)
X(41200) = X(41194)-Ceva conjugate of X(32618)
X(41200) = X(232)-cross conjugate of X(41201)
X(41200) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41198}, {63, 5000}, {75, 41196}, {1959, 32619}
X(41200) = trilinear product X(i)*X(j) for these {i,j}: {19, 32618}, {31, 41194}, {1910, 5001}, {36120, 41197}
X(41200) = barycentric product X(i)*X(j) for these {i,j}: {4, 32618}, {6, 41194}, {98, 5001}, {6531, 41199}, {16081, 41197}, {34135, 34239}
X(41200) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 41198}, {25, 5000}, {32, 41196}, {1976, 32619}, {5001, 325}, {6531, 41195}, {32618, 69}, {41194, 76}, {41197, 36212}, {41199, 6393}
X(41200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7735, 41201}, {32, 10311, 41201}, {98, 6531, 41201}, {115, 6103, 41201}

X(41201) = X(4)X(32)∩X(6)X(5000)

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

X(41201) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 41195}, {4, 32}, {6, 5000}, {187, 40895}, {230, 5001}, {2549, 40894}, {5003, 10313}, {5304, 34240}, {40802, 41198}

X(41201) = isogonal conjugate of X(41199)
X(41201) = isogonal conjugate of the isotomic conjugate of X(41195)
X(41201) = polar conjugate of the isotomic conjugate of X(32619)
X(41201) = X(41195)-Ceva conjugate of X(32619)
X(41201) = X(232)-cross conjugate of X(41200)
X(41201) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41199}, {63, 5001}, {75, 41197}, {1959, 32618}
X(41201) = trilinear product X(i)*X(j) for these {i,j}: {19, 32619}, {31, 41195}, {1910, 5000}, {36120, 41196}
X(41201) = barycentric product X(i)*X(j) for these {i,j}: {4, 32619}, {6, 41195}, {98, 5000}, {6531, 41198}, {16081, 41196}, {34136, 34240}
X(41201) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 41199}, {25, 5001}, {32, 41197}, {1976, 32618}, {5000, 325}, {6531, 41194}, {32619, 69}, {41195, 76}, {41196, 36212}, {41198, 6393}
X(41201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7735, 41200}, {32, 10311, 41200}, {98, 6531, 41200}, {115, 6103, 41200}






  Christopher J. Bradley bicevian points and axes: X(41202) - X(41224)  

This preamble and centers X(41202)-X(41224) were contributed by César Eliud Lozada, Februry 05, 2021.

This section is based on the work On the Nine Intersections of Two Cevian Triangles (Article 20), by Christopher J. Bradley (with symmetrical notations by César Lozada).

Let U, X be two distinct points on the plane of ABC, neither on its sidelines nor on a line parallel to a sideline through the opposite vertex. Let AuBuCu and AxBxCx be the cevian triangles of U and X, respectively. We introduce the nine intersections of these two triangles:

  A1 = BuCu ∩ BxCx, and cyclically B1, C1

  A2 = AuBu ∩ AxCx, and cyclically B2, C2

  A3 = AxBx ∩ AuCu, and cyclically B3, C3

Then, assuming that U = u : v : w and X = x : y z (barycentrics):

  1. A, B1, C1 are collinear on a line a1; B, C1, A1 are collinear on a line b1; C, A1, B1 are collinear on a line c1.
  2. BC, B2C2, BuCx concrr in a point A'; CA, C2A2, CuAx concur in a point B'; AB, A2B2, AuBx concur in a point C'.
  3. BC, B3C3, BxCu concur in a point A"; CA, C3A3, CxAu concur in a point B"; AB, A3B3, AxBu concur in a point C".
  4. A', B', C' are collinear on a line L' and also A", B", C" are collinear on a line L". Moreover, these two lines are concurrent with UX at a point O(U,X), here named the Bradley bicevian point of (U, X).
    Barycentric coordinates of this point are O(U, X) = u^2*y*z-x^2*v*w : : . In other words, O(U, X) is the U-Hirst inverse-of-X = X-Hirst inverse-of-U (see the ETC glossary).
  5. A, A2, A3 are collinear, as are B, B2, B3 and C, C2, C3.
  6. The last result means that ABC, A2B2 and A3B3 are perspective by pairs with common perspector Q(U,X) = (v*z+w*y)*u*x ::.
  7. A1B1C1 is perspective to these triangles: ABC, AuBuCu, AxBxCx, A2B2C2 and A3B3C3. Let Q2(U,X) and Q3(U,X) be the perspectors of A1B1C1 with A2B2C2 and A3B3C3, respectively.
  8. Q(U,X), Q2(U,X) and Q3(U,X) lie on a line, here named the Bradley bicevian axis of (U, X), whose trilinear pole has barycentric coordinates:

     T(U, X) = (x*v^2*z-y^2*u*w)*(x^2*w^2-z^2*u^2)*(y*w^2*x-z^2*v*u)*(y^2*u^2-x^2*v^2) : :

Note that A1B1C1 is the side-triangle of the U- and X-cevian triangles. For shortening, the term side-of-cevians-of-(U,X) will be used here for this triangle.

The appearance of (i, j, k) in the following list means that the Bradley bicevian point of (X(i), X(j)) is X(k):
 (1, 2, 239), (1, 3, 1936), (1, 4, 243), (1, 5, 2596), (1, 6, 238), (2, 3, 401), (2, 4, 297), (2, 5, 40853), (2, 6, 385), (3, 4, 450), (3, 5, 41202), (3, 6, 511), (4, 5, 41203), (4, 6, 41204), (5, 6, 41205)

The appearance of (i, j, k) in the following list means that the trilinear pole of the Bradley bicevian axis of (X(i), X(j)) is X(k):
 (1, 2, 4589), (1, 3, 41206), (1, 4, 41207), (1, 6, 4584), (2, 3, 41208), (2, 4, 22456), (2, 6, 41209), (3, 6, 2966), (4, 6, 41210)

The appearance of (i, j, m, n) in the following list means that the perspectors of the side-cevians-of-(X(i), X(j)) with the cevian-of-X(i) and the cevian-of-X(j) are X(m) and X(n), respectively:

(1, 2, 244, 1015), (1, 3, 2638, 3270), (1, 4, 2310, 3270), (1, 5, 41211, 41218), (1, 6, 3248, 1015), (1, 7, 2310, 3022), (1, 8, 2310, 3271), (2, 3, 35071, 2972), (2, 4, 115, 125), (2, 5, 39019, 35442), (2, 6, 1084, 3124), (2, 7, 1086, 11), (2, 8, 1146, 11), (3, 4, 20975, 34980), (3, 5, 41212, 41219), (3, 6, 20975, 3269), (3, 7, 41214, 15616), (3, 8, 41215, 41220), (4, 5, 24862, 41221), (4, 6, 34980, 3269), (4, 7, 3270, 3022), (4, 8, 3270, 3271), (5, 6, 41213, 41222), (5, 7, 41216, 31889), (5, 8, 41217, 41223), (6, 7, 35505, 15615), (6, 8, 35506, 41224), (7, 8, 3022, 3271)

X(41202) = BRADLEY BICEVIAN POINT OF (X(3), X(5))

Barycentrics    a^12-3*(b^2+c^2)*a^10+(3*b^4+7*b^2*c^2+3*c^4)*a^8-(b^2+c^2)^3*a^6-2*(b^2-c^2)^2*b^2*c^2*a^4+(b^2-c^2)^4*b^2*c^2 : :
X(41202) = 2*X(23)-3*X(37926) = 4*X(468)-X(14460) = 4*X(858)-3*X(18870)

X(41202) lies on these lines: {2, 3}, {110, 32428}, {195, 15912}, {476, 1298}, {523, 32320}, {2052, 23606}, {2055, 13450}, {4993, 10003}, {6663, 14627}, {12012, 31626}

X(41202) = reflection of X(36162) in X(7464)
X(41202) = isogonal conjugate of the antigonal conjugate of X(8795)
X(41202) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(32439)}} and {{A, B, C, X(186), X(1298)}}
X(41202) = circumcircle-inverse of-X(418)
X(41202) = nine-point circle-inverse of-X(34965)
X(41202) = MacBeath-circumconic-inverse of X(5)
X(41202) = crossdifference of every pair of points on line {X(389), X(647)}
X(41202) = X(i)-Hirst inverse of-X(j) for these (i, j): {3, 5}, {5, 3}, {195, 15912}
X(41202) = X(418)-vertex conjugate of-X(523)
X(41202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (401, 450, 852), (436, 8613, 418), (1113, 1114, 418), (37974, 37975, 4)

X(41203) = BRADLEY BICEVIAN POINT OF (X(4), X(5))

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

X(41203) lies on these lines: {2, 3}, {52, 3462}, {107, 32223}, {110, 14918}, {340, 3292}, {511, 14165}, {1629, 21243}, {3580, 41204}, {5667, 32110}, {6530, 32269}, {10112, 38808}, {14920, 23061}, {17401, 37081}, {19189, 23181}, {32225, 37765}, {33971, 37638}, {35311, 37779}, {35360, 37766}

X(41203) = polar conjugate of the antigonal conjugate of X(275)
X(41203) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(18020)}} and {{A, B, C, X(94), X(40853)}}
X(41203) = crossdifference of every pair of points on line {X(647), X(13367)}
X(41203) = X(4)-Daleth conjugate of-X(23047)
X(41203) = X(i)-Hirst inverse of-X(j) for these (i, j): {4, 5}, {5, 4}, {52, 3462}
X(41203) = {X(297), X(468)}-harmonic conjugate of X(450)

X(41204) = BRADLEY BICEVIAN POINT OF (X(4), X(6))

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

Barycentrics    (tan A) (sin^2 2A - sin 2B sin 2C) : :

X(41204) = X(4)-4*X(1990) = 2*X(648)+X(35474) = 3*X(37765)-2*X(39569)

The trilinear polar of X(41204) passes through X(6130).

X(41204) lies on the cubics K223, K380, K785 and these lines: {3, 3164}, {4, 6}, {20, 1941}, {23, 35360}, {24, 1075}, {25, 3168}, {30, 15351}, {51, 1629}, {54, 8795}, {64, 38264}, {74, 15459}, {98, 232}, {107, 1495}, {110, 450}, {112, 11676}, {125, 14165}, {182, 264}, {184, 436}, {186, 523}, {275, 13366}, {297, 3564}, {323, 35311}, {324, 5012}, {340, 5965}, {378, 7709}, {389, 8884}, {401, 32428}, {403, 6761}, {419, 685}, {421, 2970}, {458, 5050}, {459, 7612}, {511, 648}, {542, 37765}, {575, 36794}, {631, 32000}, {933, 32439}, {1092, 14615}, {1093, 6759}, {1300, 2713}, {1351, 37200}, {1352, 17907}, {1353, 27377}, {1513, 16318}, {1594, 3462}, {1614, 13450}, {1692, 6531}, {1843, 39872}, {1899, 11547}, {1947, 3955}, {1948, 7193}, {1968, 11257}, {2322, 6998}, {2450, 14721}, {2456, 39931}, {2782, 15014}, {2967, 5999}, {3448, 37766}, {3580, 41203}, {4230, 34810}, {4240, 35265}, {5622, 37778}, {5667, 10295}, {6103, 38227}, {6344, 14560}, {6353, 14361}, {6524, 11206}, {7413, 41083}, {8613, 23606}, {8794, 26887}, {9306, 15466}, {12243, 37855}, {13353, 14978}, {13367, 38808}, {14157, 32230}, {14249, 26883}, {14634, 35475}, {22456, 39058}, {26864, 37070}, {30506, 34545}, {31623, 37527}, {32046, 37127}, {35719, 36153}, {37334, 39575}

X(41204) = reflection of X(i) in X(j) for these (i, j): (4, 6530), (6530, 1990)
X(41204) = isogonal conjugate of X(14941)
X(41204) = polar conjugate of X(1972)
X(41204) = X(6530)-of-anti-Euler triangle
X(41204) = barycentric product X(i)*X(j) for these {i, j}: {4, 401}, {6, 16089}, {92, 1955}, {264, 1971}, {275, 32428}, {297, 32545}
X(41204) = barycentric quotient X(i)/X(j) for these (i, j): (4, 1972), (19, 1956), (25, 1987), (232, 40804), (401, 69), (1971, 3)
X(41204) = trilinear product X(i)*X(j) for these {i, j}: {4, 1955}, {19, 401}, {31, 16089}, {92, 1971}, {162, 6130}, {240, 32545}, {275, 2313}
X(41204) = trilinear quotient X(i)/X(j) for these (i, j): (4, 1956), (19, 1987), (92, 1972), (240, 40804), (401, 63), (1971, 48)
X(41204) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(32445)}} and {{A, B, C, X(4), X(401)}}
X(41204) = pole of the trilinear polar of X(41210) with respect to Jerabek hyperbola
X(41204) = crossdifference of every pair of points on line {X(216), X(520)}
X(41204) = crosspoint of X(i) and X(j) for these (i, j): {98, 32545}, {685, 32230}
X(41204) = crosssum of X(i) and X(j) for these (i, j): {418, 3289}, {511, 40804}, {684, 2972}
X(41204) = X(i)-Ceva conjugate of-X(j) for these (i, j): (98, 4), (232, 419)
X(41204) = X(4)-Daleth conjugate of-X(6748)
X(41204) = X(i)-Hirst inverse of-X(j) for these (i, j): {4, 6}, {6, 4}, {25, 3168}
X(41204) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 14941}, {3, 1956}, {48, 1972}, {63, 1987}, {293, 40804}
X(41204) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 1972), (19, 1956), (25, 1987), (232, 40804)
X(41204) = X(4)-vertex conjugate of-X(39201)
X(41204) = pole wrt polar circle of trilinear polar of X(1972) (line X(5)X(525), or PU(38))
X(41204) = perspector of conic {{A,B,C,X(107),X(275),PU(157)}}
X(41204) = {P,U}-harmonic conjugate of X(4), where P and U are the circumcircle intercepts of the van Aubel line
X(41204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (182, 264, 37124), (184, 2052, 436), (393, 6776, 4), (393, 15258, 6776), (575, 39530, 36794), (1990, 35907, 8744), (2207, 39646, 4), (2970, 34397, 421), (5480, 16264, 4), (36302, 36303, 5523)

X(41205) = BRADLEY BICEVIAN POINT OF (X(5), X(6))

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

X(41205) lies on these lines: {5, 6}, {511, 9512}, {523, 2070}, {577, 39910}, {3164, 7488}, {3186, 3518}, {7709, 35921}, {14570, 32762}

X(41205) = crossdifference of every pair of points on line {X(570), X(924)}
X(41205) = X(i)-Hirst inverse of-X(j) for these (i, j): {5, 6}, {6, 5}
X(41205) = X(5)-vertex conjugate of-X(34952)

X(41206) = TRILINEAR POLE OF THE BRADLEY BICEVIAN AXIS OF (X(1), X(3))

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

X(41206) lies on these lines: {296, 416}, {448, 1952}, {643, 4564}, {645, 4998}, {648, 1020}, {662, 7045}, {927, 2249}, {1813, 4612}, {18206, 35145}, {37139, 37142}

X(41206) = barycentric product X(i)*X(j) for these {i, j}: {63, 41207}, {99, 1937}, {296, 811}, {648, 40843}, {651, 35145}, {662, 1952}
X(41206) = barycentric quotient X(i)/X(j) for these (i, j): (73, 9391), (109, 851), (110, 1936), (112, 2202), (162, 243), (163, 1951)
X(41206) = trilinear product X(i)*X(j) for these {i, j}: {3, 41207}, {99, 1945}, {109, 35145}, {110, 1952}, {162, 40843}, {296, 648}
X(41206) = trilinear quotient X(i)/X(j) for these (i, j): (99, 1944), (110, 1951), (162, 2202), (296, 647), (645, 7360), (648, 243)
X(41206) = trilinear pole of the line {21, 73}
X(41206) = cevapoint of X(652) and X(1936)
X(41206) = X(648)-Hirst inverse of-X(41207)
X(41206) = X(i)-isoconjugate-of-X(j) for these {i, j}: {243, 647}, {512, 1944}, {523, 1951}, {650, 851}
X(41206) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (73, 9391), (109, 851), (110, 1936), (112, 2202)

X(41207) = TRILINEAR POLE OF THE BRADLEY BICEVIAN AXIS OF (X(1), X(4))

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

X(41207) lies on these lines: {415, 1937}, {648, 1020}, {823, 17926}, {4551, 36797}

X(41207) = barycentric product X(i)*X(j) for these {i, j}: {92, 41206}, {296, 6528}, {648, 1952}, {653, 35145}, {811, 1937}, {823, 40843}
X(41207) = barycentric quotient X(i)/X(j) for these (i, j): (65, 9391), (107, 243), (108, 851), (112, 1951), (162, 1936), (296, 520)
X(41207) = trilinear product X(i)*X(j) for these {i, j}: {4, 41206}, {107, 40843}, {108, 35145}, {162, 1952}, {296, 823}, {648, 1937}
X(41207) = trilinear quotient X(i)/X(j) for these (i, j): (99, 6518), (107, 2202), (162, 1951), (226, 9391), (296, 822), (648, 1936)
X(41207) = trilinear pole of the line {29, 65}
X(41207) = cevapoint of X(243) and X(650)
X(41207) = X(648)-Hirst inverse of-X(41206)
X(41207) = X(i)-isoconjugate-of-X(j) for these {i, j}: {243, 822}, {284, 9391}, {512, 6518}, {520, 2202}
X(41207) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (65, 9391), (107, 243), (108, 851), (112, 1951)

X(41208) = TRILINEAR POLE OF THE BRADLEY BICEVIAN AXIS OF (X(2), X(3))

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

X(41208) lies on the Johnson circumconic and these lines: {1298, 22456}, {1625, 18831}, {18020, 23181}

X(41208) = isotomic conjugate of the polar conjugate of X(41210)
X(41208) = barycentric product X(i)*X(j) for these {i, j}: {69, 41210}, {1298, 6331}
X(41208) = barycentric quotient X(i)/X(j) for these (i, j): (162, 2313), (275, 6130), (648, 32428), (933, 1971), (1298, 647), (1972, 6368)
X(41208) = trilinear product X(i)*X(j) for these {i, j}: {63, 41210}, {811, 1298}, {1956, 18831}
X(41208) = trilinear quotient X(i)/X(j) for these (i, j): (648, 2313), (811, 32428), (1298, 810), (1956, 15451)
X(41208) = trilinear pole of the line {95, 216}
X(41208) = cevapoint of X(401) and X(520)
X(41208) = X(i)-isoconjugate-of-X(j) for these {i, j}: {647, 2313}, {810, 32428}
X(41208) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (162, 2313), (275, 6130), (648, 32428), (933, 1971)

X(41209) = TRILINEAR POLE OF THE BRADLEY BICEVIAN AXIS OF (X(2), X(6))

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

X(41209) lies on these lines: {2, 39079}, {67, 18896}, {648, 18828}, {689, 805}, {694, 40850}, {733, 9150}, {783, 3222}, {892, 14970}, {1634, 4577}, {4553, 4594}, {4600, 4603}, {5207, 20021}, {9170, 10130}, {18020, 35325}

X(41209) = anticomplement of X(39079)
X(41209) = isogonal conjugate of the centroid of the (degenerate) cross-triangle of the antipedal triangles of PU(1) X(41209) = isotomic conjugate of the anticomplement of X(5113)
X(41209) = barycentric product X(i)*X(j) for these {i, j}: {83, 18829}, {99, 14970}, {308, 805}, {384, 18828}, {670, 733}, {689, 694}
X(41209) = barycentric quotient X(i)/X(j) for these (i, j): (83, 804), (99, 732), (110, 8623), (251, 5027), (308, 14295), (384, 782)
X(41209) = trilinear product X(i)*X(j) for these {i, j}: {82, 18829}, {83, 37134}, {662, 14970}, {689, 1967}, {694, 4593}, {733, 799}
X(41209) = trilinear quotient X(i)/X(j) for these (i, j): (82, 5027), (99, 2236), (660, 40936), (661, 41178), (662, 8623), (689, 1966)
X(41209) = trilinear pole of the line {39, 83}
X(41209) = intersection, other than A,B,C, of conics {{A, B, C, X(67), X(110)}} and {{A, B, C, X(99), X(39685)}}
X(41209) = cevapoint of X(i) and X(j) for these (i, j): {141, 9479}, {385, 512}, {523, 5103}, {805, 18829}
X(41209) = crossdifference of every pair of points on line {X(39079), X(41178)}
X(41209) = X(i)-cross conjugate of-X(j) for these (i, j): (110, 39291), (660, 37134), (694, 39292)
X(41209) = X(i)-isoconjugate-of-X(j) for these {i, j}: {38, 5027}, {385, 2084}, {512, 2236}, {659, 40936}
X(41209) = X(2)-line conjugate of-X(39079)
X(41209) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (83, 804), (99, 732), (110, 8623), (251, 5027)

X(41210) = TRILINEAR POLE OF THE BRADLEY BICEVIAN AXIS OF (X(4), X(6))

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

X(41210) lies on these lines: {1625, 18831}, {14570, 15414}, {15412, 34538}, {16813, 23286}

X(41210) = polar conjugate of the isotomic conjugate of X(41208)
X(41210) = barycentric product X(i)*X(j) for these {i, j}: {4, 41208}, {1298, 6528}, {1972, 16813}
X(41210) = barycentric quotient X(i)/X(j) for these (i, j): (107, 32428), (1298, 520), (1987, 17434)
X(41210) = trilinear product X(i)*X(j) for these {i, j}: {19, 41208}, {823, 1298}
X(41210) = trilinear quotient X(i)/X(j) for these (i, j): (107, 2313), (823, 32428), (1298, 822), (1956, 17434)
X(41210) = trilinear pole of the line {51, 107}
X(41210) = cevapoint of X(647) and X(41204)
X(41210) = X(i)-isoconjugate-of-X(j) for these {i, j}: {520, 2313}, {822, 32428}
X(41210) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (107, 32428), (1298, 520)

X(41211) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(1), X(5)) AND CEVIAN-OF -X(1)

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

X(41211) lies on these lines: {5, 2603}, {1090, 2618}

X(41211) = barycentric product X(1087)*X(35128)
X(41211) = trilinear product X(5)*X(41218)
X(41211) = crosspoint of X(1) and X(2600)
X(41211) = X(1)-Ceva conjugate of-X(2600)

X(41212) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(3), X(5)) AND CEVIAN-OF -X(3)

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

X(41212) lies on these lines: {3, 18315}, {130, 38976}, {137, 20625}, {6368, 15526}, {24862, 39019}

X(41212) = isogonal conjugate of the polar conjugate of X(39019)
X(41212) = barycentric product X(i)*X(j) for these {i, j}: {3, 39019}, {216, 35442}, {324, 41219}, {394, 24862}, {525, 34983}, {1087, 37754}
X(41212) = barycentric quotient X(130)/X(21449)
X(41212) = trilinear product X(i)*X(j) for these {i, j}: {48, 39019}, {255, 24862}, {656, 34983}, {1087, 34980}
X(41212) = crossdifference of every pair of points on line {X(14586), X(16813)}
X(41212) = crosspoint of X(3) and X(17434)
X(41212) = crosssum of X(4) and X(16813)
X(41212) = X(3)-Ceva conjugate of-X(17434)
X(41212) = X(130)-reciprocal conjugate of-X(21449)

X(41213) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(5), X(6)) AND CEVIAN-OF -X(5)

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

X(41213) lies on these lines: {115, 2971}, {6754, 39013}

X(41213) = barycentric product X(i)*X(j) for these {i, j}: {5, 39013}, {54, 41222}, {115, 3133}, {216, 34338}, {343, 6754}
X(41213) = trilinear product X(i)*X(j) for these {i, j}: {1953, 39013}, {2148, 41222}
X(41213) = crosspoint of X(52) and X(6753)

X(41214) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(3), X(7)) AND CEVIAN-OF -X(3)

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

X(41214) lies on the line {2972, 7004}

X(41214) = barycentric product X(331)*X(15616)
X(41214) = trilinear product X(273)*X(15616)

X(41215) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(3), X(8)) AND CEVIAN-OF -X(3)

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

X(41215) lies on these lines: {656, 2972}, {3270, 38353}

X(41215) = barycentric product X(i)*X(j) for these {i, j}: {219, 3326}, {346, 35012}
X(41215) = trilinear product X(i)*X(j) for these {i, j}: {200, 35012}, {212, 3326}, {318, 41220}
X(41215) = crossdifference of every pair of points on line {X(2401), X(2405)}
X(41215) = crosssum of X(108) and X(34051)

X(41216) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(5), X(7)) AND CEVIAN-OF -X(5)

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

X(41216) lies on these lines: {}

X(41217) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(5), X(8)) AND CEVIAN-OF -X(5)

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

X(41217) lies on these lines: {}

X(41218) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(1), X(5)) AND CEVIAN-OF -X(5)

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

X(41218) lies on these lines: {1, 2597}, {56, 32352}, {6075, 18210}, {17433, 31947}

X(41218) = barycentric product X(i)*X(j) for these {i, j}: {5, 35128}, {654, 6369}, {2167, 41211}
X(41218) = barycentric quotient X(51)/X(23592)
X(41218) = trilinear product X(i)*X(j) for these {i, j}: {54, 41211}, {654, 2600}, {1953, 35128}
X(41218) = trilinear quotient X(1953)/X(23592)
X(41218) = crosspoint of X(i) and X(j) for these (i, j): {5, 2600}, {54, 654}
X(41218) = crosssum of X(5) and X(655)
X(41218) = X(i)-Ceva conjugate of-X(j) for these (i, j): (1, 2081), (5, 2600), (54, 654)
X(41218) = X(51)-reciprocal conjugate of-X(23592)

X(41219) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(3), X(5)) AND CEVIAN-OF -X(5)

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

X(41219) lies on these lines: {3, 32438}, {122, 125}, {130, 38976}, {648, 14941}, {34980, 35071}

X(41219) = barycentric product X(i)*X(j) for these {i, j}: {5, 35071}, {97, 41212}, {216, 2972}, {343, 34980}, {418, 15526}, {520, 17434}
X(41219) = barycentric quotient X(i)/X(j) for these (i, j): (51, 34538), (217, 32230), (418, 23582)
X(41219) = trilinear product X(i)*X(j) for these {i, j}: {216, 37754}, {418, 2632}, {822, 17434}, {1363, 7069}, {1393, 7065}, {1953, 35071}
X(41219) = trilinear quotient X(i)/X(j) for these (i, j): (51, 24021), (418, 24000), (822, 16813), (1953, 34538), (2179, 23590)
X(41219) = crossdifference of every pair of points on line {X(112), X(15352)}
X(41219) = crosspoint of X(i) and X(j) for these (i, j): {5, 17434}, {216, 520}
X(41219) = crosssum of X(i) and X(j) for these (i, j): {54, 16813}, {107, 275}, {110, 8613}
X(41219) = X(5)-Ceva conjugate of-X(17434)
X(41219) = X(i)-isoconjugate-of-X(j) for these {i, j}: {95, 24021}, {823, 16813}
X(41219) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (51, 34538), (217, 32230), (418, 23582)

X(41220) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(3), X(8)) AND CEVIAN-OF -X(8)

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

X(41220) lies on the Mandart inellipse and these lines: {11, 39004}, {1363, 38985}, {7065, 38353}

X(41220) = barycentric product X(i)*X(j) for these {i, j}: {219, 35012}, {222, 41215}, {577, 3326}, {1361, 35072}, {1364, 23980}
X(41220) = trilinear product X(i)*X(j) for these {i, j}: {212, 35012}, {603, 41215}, {1361, 2638}
X(41220) = trilinear quotient X(1361)/X(24032)

X(41221) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(4), X(5)) AND CEVIAN-OF -X(5)

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

X(41221) lies on these lines: {4, 9512}, {5, 14570}, {25, 34448}, {51, 23607}, {53, 40981}, {115, 2971}, {137, 20625}, {338, 868}, {460, 9407}, {523, 38394}, {648, 14639}, {1576, 1989}, {1879, 23635}, {2165, 2934}, {3133, 27361}, {3843, 11061}, {4558, 6321}, {11596, 17983}, {14060, 38224}, {16310, 23200}, {18105, 31644}, {22085, 23698}, {22143, 38732}

X(41221) = barycentric product X(i)*X(j) for these {i, j}: {5, 115}, {51, 338}, {53, 125}, {137, 2963}, {216, 2970}, {275, 24862}
X(41221) = barycentric quotient X(i)/X(j) for these (i, j): (5, 4590), (32, 14587), (51, 249), (53, 18020), (115, 95), (125, 34386)
X(41221) = trilinear product X(i)*X(j) for these {i, j}: {5, 2643}, {51, 1109}, {53, 3708}, {115, 1953}, {125, 2181}, {338, 2179}
X(41221) = trilinear quotient X(i)/X(j) for these (i, j): (5, 24041), (31, 14587), (51, 1101), (115, 2167), (311, 24037), (512, 36134)
X(41221) = intersection, other than A,B,C, of conics {{A, B, C, X(51), X(8029)}} and {{A, B, C, X(53), X(34294)}}
X(41221) = crossdifference of every pair of points on line {X(4558), X(14586)}
X(41221) = crosspoint of X(i) and X(j) for these (i, j): {5, 12077}, {51, 512}, {53, 23290}, {115, 2970}
X(41221) = crosssum of X(i) and X(j) for these (i, j): {54, 18315}, {95, 99}, {97, 15958}, {110, 1994}
X(41221) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 12077), (512, 8029), (1263, 2081)
X(41221) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 24041}, {75, 14587}, {95, 1101}, {99, 36134}
X(41221) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 4590), (32, 14587), (51, 249), (53, 18020)
X(41221) = X(115)-Waw conjugate of-X(39691)
X(41221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (115, 8754, 20975), (338, 34981, 868)

X(41222) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(5), X(6)) AND CEVIAN-OF -X(6)

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

X(41222) lies on the Brocard inellipse and the line {6, 2383}

X(41222) = barycentric product X(311)*X(41213)
X(41222) = trilinear product X(1087)*X(39013)

X(41223) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(5), X(8)) AND CEVIAN-OF -X(8)

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

X(41223) lies on the Mandart inellipse and these lines: {}

X(41224) = PERSPECTOR OF THESE TRIANGLES: SIDE-OF-CEVIANS-OF-(X(6), X(8)) AND CEVIAN-OF -X(8)

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

X(41224) lies on the Mandart inellipse and these lines: {11, 38992}, {1356, 3248}, {3271, 11998}, {7063, 38364}

X(41224) = barycentric product X(i)*X(j) for these {i, j}: {8, 39015}, {56, 35506}, {1015, 1682}
X(41224) = barycentric quotient X(1682)/X(31625)
X(41224) = trilinear product X(i)*X(j) for these {i, j}: {9, 39015}, {604, 35506}, {1682, 3248}
X(41224) = trilinear quotient X(1682)/X(7035)
X(41224) = crosssum of X(56) and X(6648)

X(41225) = X(1)X(16)∩X(9)X(46)

Barycentrics    Sin[A]*Tan[A/2 - Pi/6] : :
Barycentrics    a*(a^2 - b^2 - c^2 + b*c)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*b*c - 2*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*Sqrt[3]*a*(b + c)*S) : :

X(41225) lies on these lines: {1, 16}, {9, 46}, {14, 484}, {17, 1652}, {36, 5239}, {57, 554}, {498, 30328}, {2154, 4674}, {2163, 11073}, {4295, 30414}, {5011, 10651}, {5131, 11752}, {5240, 5903}, {5242, 12047}, {5353, 37773}, {7051, 39151}, {7127, 39153}

X(41225) = reflection of X(1) in X(10647)
X(41225) = isogonal conjugate of X(7150)
X(41225) = X(14)-Ceva conjugate of X(1)
X(41225) = X(203)-cross conjugate of X(1)
X(41225) = crosssum of X(16) and X(7005)
X(41225) = excentral-isogonal conjugate of ABC-to-excentral barycentric image of X(14)
X(41225) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7150}, {14, 7005}, {16, 14359}, {80, 5357}, {559, 7126}, {3219, 11072}
X(41225) = barycentric product X(i)*X(j) for these {i,j}: {320, 11073}, {554, 5239}, {5353, 30690}
X(41225) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7150}, {2152, 7005}, {2154, 14359}, {5239, 40714}, {5353, 3219}, {6186, 11072}, {7051, 559}, {7113, 5357}, {11073, 80}, {33653, 7043}
X(41225) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 46, 3179}, {14, 3376, 7150}, {79, 2160, 3179}, {1277, 1653, 1}, {33653, 33654, 39152}, {33653, 39152, 1}

X(41226) = X(2)X(2006)∩X(80)X(3617)

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

X(41226) lies on these lines: {2, 2006}, {80, 3617}, {655, 2349}, {1807, 9538}, {2161, 4473}, {4354, 4420}, {4671, 28829}, {6539, 24624}, {7080, 36626}, {17776, 20566}, {31079, 36815}, {32849, 36804}

X(41226) = X(i)-isoconjugate of X(j) for these (i,j): {36, 2160}, {79, 7113}, {654, 26700}, {3218, 6186}, {6742, 21758}, {8648, 38340}, {13486, 21828}
X(41226) = trilinear pole of line {3678, 35057}
X(41226) = barycentric product X(i)*X(j) for these {i,j}: {35, 20566}, {80, 319}, {2161, 33939}, {3219, 18359}, {3678, 14616}, {3969, 24624}, {4420, 18815}, {6740, 40999}, {14838, 36804}, {17095, 36910}, {35057, 35174}
X(41226) = barycentric quotient X(i)/X(j) for these {i,j}: {35, 36}, {80, 79}, {319, 320}, {655, 38340}, {1442, 1443}, {1793, 1789}, {1807, 7100}, {1825, 1835}, {2161, 2160}, {2174, 7113}, {2222, 26700}, {2594, 1464}, {3219, 3218}, {3647, 4973}, {3678, 758}, {3969, 3936}, {4420, 4511}, {4467, 4453}, {6187, 6186}, {6198, 1870}, {6740, 3615}, {7150, 3179}, {7265, 4707}, {9404, 654}, {11107, 17515}, {14838, 3960}, {15065, 6757}, {16577, 18593}, {17095, 17078}, {18359, 30690}, {20566, 20565}, {23226, 22379}, {33939, 20924}, {35057, 3738}, {35192, 4282}, {36804, 15455}, {36910, 7110}

X(41227) = X(1)X(19)∩X(4)X(12)

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

X(41227) lies on these lines: {1, 19}, {3, 278}, {4, 12}, {5, 37799}, {11, 7537}, {21, 92}, {24, 108}, {25, 7952}, {29, 1621}, {31, 3194}, {34, 40}, {35, 1838}, {54, 65}, {56, 196}, {71, 3074}, {100, 5125}, {104, 7049}, {105, 7154}, {109, 1715}, {112, 19849}, {158, 2218}, {186, 39751}, {207, 1420}, {225, 7412}, {255, 1396}, {273, 6986}, {281, 405}, {318, 26227}, {378, 37601}, {388, 37395}, {404, 17923}, {474, 17917}, {475, 2550}, {495, 7511}, {498, 5142}, {608, 7078}, {651, 5889}, {946, 37380}, {954, 37377}, {962, 37258}, {1001, 7498}, {1006, 40149}, {1010, 17913}, {1104, 14571}, {1125, 30686}, {1214, 37275}, {1435, 15803}, {1453, 2331}, {1467, 1767}, {1479, 37372}, {1612, 8747}, {1724, 1783}, {1735, 1782}, {1748, 3868}, {1753, 6769}, {1785, 1842}, {1848, 13411}, {1855, 5259}, {1859, 6198}, {1871, 24929}, {1880, 37399}, {1888, 37568}, {1891, 31397}, {1895, 4233}, {2975, 17515}, {3086, 7521}, {3101, 37231}, {3295, 7497}, {3468, 37755}, {3616, 37253}, {3871, 5174}, {4183, 5248}, {4185, 11406}, {4200, 17784}, {4223, 5089}, {4224, 17102}, {4292, 5236}, {4295, 5757}, {4306, 32714}, {5146, 32760}, {5253, 37304}, {5301, 5317}, {6906, 37790}, {7007, 17831}, {7071, 37387}, {7156, 7719}, {7503, 37800}, {7510, 37621}, {7549, 37695}, {7559, 7951}, {8069, 14017}, {8251, 37697}, {9786, 34032}, {10310, 37417}, {10319, 37431}, {11109, 26115}, {11363, 15500}, {11383, 37384}, {11396, 37238}, {11399, 17562}, {11507, 14018}, {14015, 18677}, {14127, 38564}, {15171, 15763}, {22467, 37798}

X(41227) = X(59)-Ceva conjugate of X(108)
X(41227) = X(15443)-cross conjugate of X(4)
X(41227) = crosspoint of X(107) and X(7012)
X(41227) = crosssum of X(520) and X(7004)
X(41227) = crossdifference of every pair of points on line {656, 40628}
X(41227) = barycentric product X(i)*X(j) for these {i,j}: {1, 37279}, {27, 3191}, {92, 580}
X(41227) = barycentric quotient X(i)/X(j) for these {i,j}: {580, 63}, {3191, 306}, {15443, 26942}, {37279, 75}
X(41227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 38300, 3085}, {12, 1852, 4}, {24, 1068, 108}, {35, 1838, 4219}, {1118, 37579, 108}, {1859, 37080, 6198}, {1870, 6197, 65}, {5248, 39585, 4183}

X(41228) = X(7)X(8)∩X(9)X(21)

Barycentrics    a*(a - b - c)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - a*c^3 - b*c^3 + c^4) : :
X(41228) = 2 X[7] - 3 X[10861], 4 X[9] - 5 X[3876], 2 X[390] - 3 X[3877], 4 X[1001] - 3 X[7671], 5 X[3616] - 4 X[5572], 3 X[3681] - 2 X[5223], X[3868] - 4 X[5784], X[3868] - 3 X[10861], X[3869] + 2 X[5696], 3 X[3873] - 4 X[5542], 5 X[3876] - 2 X[10394], 5 X[3889] - 6 X[11038], 5 X[3889] - 4 X[15185], 5 X[3890] - 4 X[30331], 3 X[5686] - X[40269], 3 X[5686] - 4 X[40659], 4 X[5732] - 3 X[11220], 4 X[5784] - 3 X[10861], 5 X[11025] - 6 X[38053], 3 X[11038] - 2 X[15185], 3 X[11220] - 2 X[12669], 5 X[15016] - 6 X[38123], 5 X[18398] - 6 X[38054], 3 X[21168] - 4 X[31837], 2 X[30329] - 3 X[38052], X[40269] - 4 X[40659]

X(41228) lies on these lines: {1, 5785}, {2, 955}, {7, 8}, {9, 21}, {10, 18412}, {20, 72}, {40, 9960}, {43, 11031}, {63, 100}, {142, 4197}, {210, 5218}, {219, 3100}, {224, 2975}, {329, 10431}, {379, 10477}, {390, 3877}, {404, 1445}, {443, 14054}, {480, 1259}, {516, 3869}, {527, 17579}, {528, 7673}, {673, 28916}, {758, 4312}, {908, 10883}, {912, 6916}, {936, 10398}, {942, 4208}, {954, 34772}, {956, 18444}, {960, 4313}, {984, 2340}, {991, 24635}, {997, 15299}, {1001, 4511}, {1004, 3218}, {1012, 3940}, {1071, 5657}, {1145, 9964}, {1150, 7360}, {1260, 3219}, {1350, 7291}, {1698, 10122}, {1736, 26669}, {1812, 4123}, {1864, 18228}, {2292, 4335}, {2318, 24430}, {2476, 21617}, {2478, 5809}, {2800, 36922}, {2951, 9961}, {2968, 33077}, {3057, 12536}, {3062, 31803}, {3174, 3871}, {3190, 28606}, {3243, 3872}, {3358, 6909}, {3419, 5805}, {3452, 10392}, {3476, 9846}, {3555, 11036}, {3616, 5572}, {3678, 31424}, {3679, 5833}, {3697, 27525}, {3713, 5279}, {3729, 9962}, {3751, 28043}, {3781, 20533}, {3811, 15298}, {3826, 37796}, {3873, 4847}, {3885, 5853}, {3889, 11038}, {3890, 30331}, {3897, 30284}, {3916, 37105}, {3927, 37426}, {3949, 15656}, {3962, 31391}, {4015, 31446}, {4073, 28287}, {4292, 5850}, {4304, 5692}, {4326, 5250}, {4649, 28125}, {4661, 9965}, {4853, 11520}, {4855, 21153}, {4861, 28965}, {5044, 17558}, {5128, 8544}, {5208, 11679}, {5231, 20116}, {5440, 31658}, {5552, 38057}, {5686, 7080}, {5698, 11015}, {5729, 37248}, {5735, 14923}, {5762, 37468}, {5777, 37434}, {5815, 9799}, {5817, 6837}, {5843, 31775}, {5851, 12532}, {6067, 25557}, {6666, 27385}, {6735, 24393}, {6884, 38108}, {8270, 34035}, {8583, 30330}, {10384, 15829}, {10427, 12755}, {10527, 11025}, {11678, 21060}, {15016, 38123}, {15066, 37782}, {15823, 15837}, {18230, 27383}, {18398, 38054}, {20059, 37435}, {20330, 24390}, {21151, 37112}, {21168, 31837}, {27131, 37358}, {27475, 28797}, {30329, 38052}, {31672, 37433}, {35614, 39553}, {37300, 37787}

X(41228) = anticomplement of X(5728)
X(41228) = midpoint of X(i) and X(j) for these {i,j}: {3869, 25722}, {3962, 31391}
X(41228) = reflection of X(i) in X(j) for these {i,j}: {7, 5784}, {8, 3059}, {65, 15587}, {144, 72}, {3062, 31803}, {3868, 7}, {7672, 2550}, {9961, 2951}, {10394, 9}, {12669, 5732}, {12755, 10427}, {14100, 960}, {18412, 10}, {25722, 5696}, {30628, 1}
X(41228) = crossdifference of every pair of points on line {3063, 4017}
X(41228) = barycentric product X(i)*X(j) for these {i,j}: {8, 24635}, {78, 37448}, {312, 991}
X(41228) = barycentric quotient X(i)/X(j) for these {i,j}: {991, 57}, {24635, 7}, {37448, 273}
X(41228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16465, 11020}, {7, 5784, 10861}, {9, 7675, 21}, {210, 10391, 5273}, {3868, 10861, 7}, {5223, 5732, 63}, {5732, 12669, 11220}, {11038, 15185, 3889}

X(41229) = X(1)X(6)∩X(8)X(90)

Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 3*b^2*c - a*c^2 - 3*b*c^2 - c^3) : :
X(41229) = X[1] - 4 X[5302], 3 X[165] - 2 X[37426], 4 X[1125] - 5 X[31259], 5 X[1698] - 4 X[8728], 5 X[1698] - 2 X[10404], 5 X[3617] - X[31295], X[4338] - 4 X[9710]

X(41229) lies on these lines: {1, 6}, {2, 3338}, {3, 210}, {5, 12704}, {7, 19855}, {8, 90}, {10, 46}, {12, 57}, {20, 5686}, {21, 3681}, {30, 40}, {35, 200}, {36, 936}, {43, 37329}, {55, 31445}, {56, 3715}, {58, 612}, {65, 3927}, {78, 993}, {84, 165}, {100, 9859}, {144, 4295}, {169, 3691}, {201, 21147}, {226, 19854}, {329, 12047}, {333, 4385}, {354, 11108}, {388, 1708}, {404, 7284}, {443, 38057}, {474, 3740}, {484, 1706}, {496, 4679}, {498, 5745}, {499, 3452}, {517, 37234}, {519, 5250}, {756, 975}, {758, 3951}, {894, 19853}, {908, 26363}, {920, 3421}, {944, 26878}, {962, 6172}, {964, 4981}, {967, 5268}, {971, 5584}, {976, 37817}, {988, 3216}, {990, 21039}, {997, 2975}, {999, 25917}, {1009, 3789}, {1089, 11679}, {1125, 3305}, {1150, 3701}, {1155, 3983}, {1158, 5657}, {1210, 18250}, {1214, 9370}, {1329, 17437}, {1330, 29641}, {1376, 3697}, {1423, 19879}, {1445, 4298}, {1479, 4847}, {1482, 31165}, {1490, 32159}, {1697, 3632}, {1699, 5812}, {1707, 5264}, {1711, 4424}, {1714, 13161}, {1722, 3670}, {1727, 6735}, {1737, 2551}, {1763, 32778}, {1764, 5788}, {1768, 9588}, {1770, 2550}, {1836, 31419}, {1867, 1889}, {1935, 8270}, {2136, 4677}, {2276, 31442}, {2478, 10916}, {2646, 3940}, {2801, 10884}, {2900, 37284}, {2901, 17156}, {2961, 12618}, {3057, 7082}, {3085, 5273}, {3086, 18228}, {3097, 12933}, {3158, 27787}, {3189, 11111}, {3214, 4414}, {3218, 9780}, {3220, 10830}, {3293, 17594}, {3295, 3683}, {3306, 3634}, {3333, 3624}, {3336, 3928}, {3337, 5437}, {3339, 4880}, {3358, 10864}, {3361, 30393}, {3428, 5777}, {3454, 29857}, {3467, 3680}, {3475, 16845}, {3560, 37569}, {3576, 32153}, {3577, 15910}, {3579, 18518}, {3582, 25522}, {3587, 35250}, {3600, 37787}, {3616, 27065}, {3617, 31295}, {3625, 3895}, {3633, 31393}, {3646, 5506}, {3647, 8715}, {3671, 8545}, {3690, 16980}, {3695, 10371}, {3698, 36279}, {3703, 5814}, {3711, 5217}, {3729, 4647}, {3742, 16842}, {3746, 4512}, {3848, 16854}, {3868, 5260}, {3869, 25415}, {3870, 5248}, {3872, 3878}, {3873, 5047}, {3881, 4666}, {3884, 36846}, {3889, 5284}, {3894, 11518}, {3899, 7982}, {3901, 11529}, {3953, 5272}, {3971, 17733}, {3976, 17123}, {3984, 4134}, {4015, 4652}, {4042, 5295}, {4067, 30147}, {4189, 4420}, {4302, 24393}, {4304, 6743}, {4305, 20007}, {4309, 5853}, {4311, 12447}, {4338, 9710}, {4357, 19784}, {4383, 37592}, {4413, 37582}, {4423, 5045}, {4430, 16859}, {4533, 5440}, {4640, 4662}, {4641, 5711}, {4661, 16865}, {4668, 11010}, {4673, 17336}, {4853, 5697}, {4855, 5267}, {4857, 24392}, {4863, 15171}, {4968, 5278}, {5022, 25068}, {5051, 33114}, {5082, 5698}, {5084, 24477}, {5130, 7713}, {5178, 11114}, {5231, 7741}, {5255, 7262}, {5262, 7226}, {5285, 8185}, {5293, 36504}, {5325, 10056}, {5438, 7280}, {5531, 35204}, {5534, 10902}, {5536, 7989}, {5550, 35595}, {5563, 8583}, {5587, 5709}, {5705, 7951}, {5720, 11012}, {5726, 15932}, {5749, 19866}, {5779, 12688}, {5795, 10573}, {5837, 12647}, {5903, 9623}, {5905, 12609}, {5918, 12684}, {6048, 17596}, {6675, 17718}, {6684, 10786}, {6734, 10522}, {6769, 15104}, {6796, 21165}, {6857, 25568}, {7085, 9798}, {7171, 35242}, {7289, 12587}, {7322, 37554}, {7719, 37396}, {7958, 38108}, {7987, 33597}, {7991, 12705}, {8192, 26867}, {8193, 24320}, {8227, 31142}, {8273, 31658}, {8580, 15803}, {8666, 10176}, {9548, 33167}, {9578, 18962}, {9581, 10953}, {9589, 11372}, {9841, 16192}, {9956, 11929}, {10058, 14740}, {10175, 10599}, {10436, 16828}, {10527, 21616}, {10582, 25542}, {10942, 21031}, {10954, 31434}, {11024, 28610}, {11037, 18230}, {11038, 17554}, {11194, 17614}, {11231, 37612}, {11263, 31164}, {11507, 15823}, {11520, 30143}, {11530, 13089}, {11551, 28629}, {12183, 24469}, {12329, 13730}, {12520, 12528}, {12607, 17699}, {12678, 37424}, {12687, 37837}, {12738, 15015}, {13100, 18259}, {13272, 37718}, {13411, 21060}, {13728, 38047}, {15071, 30503}, {15624, 17524}, {15843, 17700}, {15908, 37822}, {16062, 33118}, {16408, 32636}, {16418, 37080}, {16817, 24349}, {17206, 30758}, {17248, 19865}, {17270, 21277}, {17353, 19836}, {17527, 17728}, {17552, 38053}, {17605, 31493}, {17688, 27495}, {17691, 27484}, {17754, 19856}, {18206, 25526}, {18249, 31397}, {18446, 40661}, {18480, 37584}, {18492, 24468}, {18517, 18540}, {18761, 37585}, {19846, 25527}, {19851, 31302}, {19877, 27003}, {20653, 33161}, {21153, 35202}, {21364, 33165}, {22758, 31837}, {22793, 31140}, {24161, 33101}, {24390, 24703}, {24443, 36263}, {24541, 31458}, {24851, 32865}, {26064, 29667}, {26094, 26688}, {26487, 31423}, {27368, 32925}, {31446, 37719}, {33932, 34016}, {35239, 40263}, {36478, 36540}, {36483, 36486}, {36499, 36572}, {38127, 40256}

X(41229) = midpoint of X(i) and X(j) for these {i,j}: {8, 6872}, {3951, 19860}
X(41229) = reflection of X(i) in X(j) for these {i,j}: {1, 405}, {377, 10}, {405, 5302}, {10404, 8728}, {11520, 30143}
X(41229) = X(966)-Ceva conjugate of X(5268)
X(41229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1728, 15299}, {1, 1743, 1203}, {1, 3731, 27785}, {1, 5223, 5904}, {1, 5234, 5251}, {8, 3219, 12514}, {8, 12514, 5119}, {9, 5223, 15298}, {9, 6762, 31435}, {10, 63, 46}, {10, 3436, 10827}, {10, 12527, 1478}, {12, 5791, 1698}, {21, 3681, 3811}, {40, 3929, 191}, {40, 7330, 1709}, {56, 3715, 5044}, {72, 958, 1}, {78, 993, 3612}, {191, 3679, 40}, {200, 31424, 35}, {329, 19843, 12047}, {355, 11827, 5691}, {355, 21677, 3679}, {355, 26921, 40}, {392, 12513, 1}, {756, 1468, 975}, {908, 26363, 37692}, {956, 960, 1}, {958, 5220, 72}, {984, 5247, 1}, {993, 3678, 78}, {997, 2975, 37618}, {1001, 3555, 1}, {1155, 3983, 9709}, {1453, 7174, 1}, {1698, 6763, 57}, {1768, 9588, 37560}, {2646, 4005, 3940}, {2975, 3876, 997}, {3333, 7308, 3624}, {3647, 8715, 35258}, {3691, 5282, 169}, {3697, 3916, 1376}, {3820, 24914, 1698}, {3872, 3878, 30323}, {3927, 9708, 65}, {4134, 22836, 3984}, {4512, 6765, 3746}, {4640, 4662, 5687}, {4677, 37563, 2136}, {4847, 12572, 1479}, {5223, 5234, 1}, {5251, 5904, 1}, {5258, 5692, 1}, {5273, 5815, 3085}, {5506, 25055, 3646}, {5745, 21075, 498}, {6762, 31435, 1}, {8666, 10176, 19861}, {9623, 12526, 5903}, {10527, 21616, 23708}, {10527, 31018, 21616}, {11374, 24953, 3624}, {15325, 24954, 3624}, {16474, 27785, 1}, {21677, 34606, 355}, {31445, 34790, 55}

X(41230) = X(1)X(19)∩X(2)X(11)

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

X(41230) lies on these lines: {1, 19}, {2, 11}, {3, 1612}, {7, 36740}, {12, 7557}, {21, 75}, {22, 19785}, {23, 33155}, {31, 24310}, {35, 1738}, {37, 4223}, {38, 1762}, {40, 595}, {41, 20706}, {56, 347}, {63, 5324}, {65, 82}, {71, 238}, {192, 17522}, {209, 32911}, {226, 1890}, {229, 1963}, {321, 37325}, {344, 16048}, {345, 25494}, {387, 37547}, {388, 4198}, {404, 16706}, {405, 2345}, {496, 7561}, {518, 2264}, {612, 9816}, {613, 19350}, {614, 10319}, {651, 37516}, {943, 19763}, {950, 1861}, {954, 4254}, {982, 26934}, {1036, 3485}, {1058, 7521}, {1086, 4265}, {1104, 37399}, {1183, 12635}, {1279, 40941}, {1284, 23868}, {1429, 22390}, {1479, 5142}, {1633, 7295}, {1760, 3868}, {1782, 3670}, {1817, 2352}, {1842, 13161}, {2347, 23693}, {2975, 11683}, {3101, 7191}, {3198, 3744}, {3220, 3663}, {3246, 37568}, {3295, 7535}, {3478, 23707}, {3562, 18178}, {3666, 4224}, {3672, 37254}, {3685, 20336}, {3752, 19649}, {3755, 40910}, {3772, 4220}, {3871, 32850}, {3914, 5310}, {4185, 5716}, {4221, 37817}, {4228, 28606}, {4251, 20605}, {4267, 37113}, {4294, 11677}, {4310, 22769}, {4419, 24320}, {4644, 37492}, {4850, 37449}, {4854, 20988}, {5009, 33772}, {5047, 17289}, {5096, 17366}, {5222, 36741}, {5259, 19857}, {5285, 40940}, {5314, 26723}, {5327, 17220}, {5698, 7083}, {6636, 33150}, {7291, 24476}, {7354, 31293}, {7465, 33129}, {9536, 17024}, {11337, 37579}, {13723, 38871}, {15171, 21530}, {15569, 37080}, {16471, 40572}, {16548, 32118}, {17321, 19310}, {17370, 17531}, {17371, 17536}, {17549, 37756}, {17562, 27802}, {17602, 20989}, {17720, 33849}, {17863, 36018}, {19845, 37176}, {19849, 38858}, {23381, 23383}, {24789, 37261}, {25513, 28439}, {26253, 33134}, {26998, 34772}, {33133, 35996}, {37275, 37528}, {37581, 37642}

X(41230) = crossdifference of every pair of points on line {656, 665}
X(41230) = barycentric product X(1)*X(37086)
X(41230) = barycentric quotient X(37086)/X(75)
X(41230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3101, 7191, 40959}, {7295, 24248, 1633}

leftri

Gibert quadratic transformations and inverses: X(41231)-X(41278)

rightri

This preamble is based on notes from Bernard Gibert, and related points contributed by Peter Moses. (February 11, 2021)

Let X x : y : z (barycentrics) be a point in the plane of a triangle ABC. Define

F(X) = b^4 c^2 x^2-b^2 c^4 x^2-a^4 c^2 x y+b^2 c^4 x y+a^4 b^2 x z-b^4 c^2 x z-a^4 b^2 y z+a^4 c^2 y z : : ,

with inverse given by

invF(X) = a^2 (b^4 x^2-c^4 x^2-a^2 b^2 x y+c^4 x y-b^4 x z+a^2 c^2 x z+a^2 b^2 y z-a^2 c^2 y z) : :

In the sequel, these transformations are denoted by BGF and invBGF.

The singular points of BGF are X(2), X(6), and X(32). If X is on X(2)X(6), then F(X) = X(6). If X is on X(2)X(32), then F(X) = X(83). If X is on X(6)X(32), then F(X) = X(2).

The fixed points of F; i.e., points X such that BG(X) = invBG(X) = X, are the points on the circumconic having perspector X(669).

BGF maps every line through X(2) onto itself; in particular, BGF(Euler line) = Euler line.

BGF maps the circumconic having perspector X(512) onto the Kiepert hyperbola.

BGF maps the circumconic having perspector X(688) onto the circumconic having perspector X(512).

The transformations BGF and invBGF have many connections among pk cubics. In order to state a few of them, certain notations will be used: "/" denotes barycentric division, and "*" denotes barycentric product, and if X is a point, then gX = isogonal conjugate of X, and tX = isotomic conjugate of X.

Let pk1 = pK(Ω,P) be a pK cubic and let pk2 = pK(Ω',P') be another pK cubic such that the isopivot of Q' is Ω'/P'.

(1) The cubics pk1 and pk2 meet the line at infinity at the same points if and only if Q' lies on a psK cubic that is a pK cubic if and only if the points X(2), Ω, and P are collinear.

(2) The cubics pk1 and pk2 meet the circumcircle at the same points if and only if the points X(6), Ω/X(6), and P are collinear; or, equivalently, the points X(32), Ω, and P*X(6) are collinear. Note that Ω/X(6) = tgΩ and P*X(6) = gtΩ.

(3) Conditions (1) and (2) hold simultaneously if and only if
       P lies on the lines X(2)Ω and X(6)tgΩ meeting at BGF(Ω) and
       Ω lies on the lines X(2) P and X(32) gtP meeting at invBGF(P).

X(41231) = BGF(5)

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

X(41231) lies on these lines: {2, 3}, {6, 311}, {51, 10358}, {76, 1993}, {83, 5392}, {184, 6248}, {264, 8745}, {315, 37636}, {324, 2207}, {343, 7745}, {1176, 18092}, {1994, 7754}, {3734, 36212}, {5012, 39646}, {5254, 37649}, {11185, 14389}, {25051, 32621}, {26164, 40684}, {26206, 36794}

X(41232) = BGF(8)

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

X(41232) lies on these lines: {1, 2}, {6, 3596}, {81, 5209}, {150, 17778}, {171, 730}, {181, 14839}, {190, 4274}, {192, 573}, {213, 3975}, {333, 1573}, {538, 32939}, {668, 37676}, {981, 40747}, {985, 4112}, {1191, 11353}, {2092, 4360}, {2176, 30830}, {2985, 40153}, {3765, 17499}, {3997, 27064}, {4260, 32029}, {4279, 4366}, {4852, 21857}, {5105, 26042}, {7976, 21010}, {16685, 25660}, {18206, 21226}, {20162, 37502}

X(41233) = BGF(10)

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

X(41233) lies on these lines: {1, 2}, {6, 313}, {31, 11320}, {65, 19791}, {71, 192}, {81, 1909}, {209, 32926}, {213, 3948}, {321, 2295}, {385, 40744}, {672, 31036}, {730, 20985}, {940, 26634}, {1332, 27644}, {1334, 3995}, {1400, 4552}, {1655, 3219}, {1918, 4366}, {2260, 17148}, {3101, 18667}, {3208, 27659}, {3691, 19742}, {3780, 25298}, {3879, 17197}, {3896, 4433}, {3936, 26589}, {3975, 32911}, {3997, 4044}, {4463, 20715}, {4851, 29981}, {4852, 21858}, {5711, 19281}, {17137, 17184}, {17147, 20247}, {19785, 21281}, {19806, 24524}, {21080, 33888}, {22218, 40729}, {22277, 32029}, {24514, 31060}, {24656, 37595}, {26601, 37715}, {28660, 40394}

X(41234) = BGF(19)

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

X(41234) lies on these lines: {2, 19}, {6, 92}, {8, 28950}, {29, 960}, {33, 3685}, {83, 40149}, {243, 14006}, {281, 14555}, {314, 1172}, {894, 5307}, {1824, 5263}, {1826, 27064}, {3064, 28960}, {4676, 11323}, {5174, 5835}, {6703, 17923}

X(41235) = BGF(20)

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

X(41235) lies on these lines: {2, 3}, {6, 14615}, {76, 801}, {83, 17825}, {154, 12203}, {315, 13567}, {343, 7879}, {394, 7754}, {2207, 15466}, {3978, 40822}, {5286, 37669}, {6391, 19222}, {6688, 10358}, {7760, 37672}, {7762, 11433}, {7784, 26958}, {7803, 23292}, {9306, 39646}, {9308, 26206}, {26006, 26667}, {31859, 36212}

X(41236) = BGF(21)

Barycentrics    a^5 + a^4*b + a^3*b^2 + a^2*b^3 + a^4*c + 2*a^3*b*c + a^2*b^2*c + 2*a*b^3*c + a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 + 2*b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + 2*b^2*c^3 : :

X(41236) lies on these lines: {2, 3}, {6, 314}, {76, 940}, {81, 7754}, {83, 4383}, {183, 5337}, {218, 27064}, {226, 24549}, {238, 4384}, {239, 16466}, {315, 1211}, {894, 20171}, {980, 1975}, {1999, 4385}, {2207, 31623}, {4270, 4483}, {4357, 24702}, {4417, 33954}, {4911, 27184}, {5247, 11679}, {5254, 6703}, {5275, 30830}, {5276, 28809}, {5739, 7762}, {5743, 7745}, {5816, 25978}, {7146, 24291}, {7776, 31089}, {7879, 32782}, {9709, 26048}, {10446, 25898}, {15668, 25536}, {16877, 24630}, {16999, 30863}, {17023, 24210}, {17308, 26687}, {23151, 24514}, {24552, 24612}, {25510, 25524}, {29456, 37522}, {37527, 39646}

X(41237) = BGF(26)

Barycentrics    a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 - 2*a^4*b^2*c^2 - a^2*b^4*c^2 - 2*b^6*c^2 - a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - 2*b^2*c^6 + c^8 : :

X(41237) lies on these lines: {2, 3}, {76, 5392}, {83, 40393}, {95, 35142}, {141, 311}, {315, 1993}, {316, 14389}, {324, 26214}, {343, 5254}, {394, 7784}, {570, 39113}, {626, 36212}, {1176, 16890}, {1501, 7745}, {1972, 9229}, {1994, 7762}, {3580, 7790}, {5286, 6515}, {5422, 7803}, {8745, 17907}, {10349, 39516}, {11090, 39661}, {11091, 39660}, {11427, 32006}, {11442, 39646}, {26561, 26639}, {33529, 36252}, {33530, 36251}, {34854, 39604}, {37895, 39081}, {39466, 40643}

X(41238) = BGF(30)

Barycentrics    a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - a^6*c^2 + 2*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 - 4*b^4*c^4 + a^2*c^6 + 2*b^2*c^6 : :

X(41238) lies on these lines: {2, 3}, {6, 3260}, {76, 2986}, {83, 34289}, {110, 39646}, {264, 26206}, {287, 14265}, {315, 3580}, {317, 26156}, {323, 7754}, {373, 10358}, {1993, 7760}, {2207, 26212}, {5254, 11064}, {5286, 37645}, {5422, 7878}, {5651, 6248}, {6800, 12203}, {7745, 37648}, {7762, 37644}, {7781, 36212}, {7784, 37638}, {7803, 14389}, {9308, 15262}, {14615, 40318}, {14918, 26162}, {32006, 37643}

X(41239) = BGF(41)

Barycentrics    a*(a^3 - a^2*b - a^2*c - a*b*c - b^2*c - b*c^2) : :

X(41239) lies on these lines: {1, 6}, {2, 41}, {3, 17754}, {4, 1973}, {7, 17691}, {8, 2280}, {10, 3684}, {21, 672}, {32, 171}, {35, 16549}, {43, 2271}, {55, 3501}, {57, 11343}, {65, 3496}, {83, 226}, {85, 24249}, {86, 9454}, {101, 1125}, {142, 17682}, {145, 4390}, {150, 17062}, {172, 24512}, {284, 1010}, {304, 16822}, {329, 26626}, {333, 1174}, {384, 894}, {442, 19557}, {572, 4297}, {584, 17303}, {595, 3997}, {604, 3600}, {609, 37522}, {626, 30837}, {673, 17050}, {728, 10389}, {910, 3812}, {936, 16849}, {942, 3509}, {943, 4876}, {950, 1220}, {965, 19282}, {993, 4253}, {1018, 3746}, {1055, 5253}, {1193, 33854}, {1334, 1621}, {1376, 4258}, {1447, 17048}, {1475, 2975}, {1500, 3750}, {1571, 17601}, {1575, 18755}, {1580, 4199}, {1708, 7146}, {1742, 9305}, {1759, 5902}, {1861, 2332}, {1910, 34076}, {1914, 2295}, {1944, 25371}, {2082, 19860}, {2172, 37093}, {2174, 17398}, {2202, 11109}, {2241, 37588}, {2246, 21921}, {2251, 5277}, {2260, 38871}, {2268, 4195}, {2275, 37617}, {2276, 37573}, {2345, 3189}, {2356, 4183}, {2548, 17717}, {3053, 37603}, {3207, 25524}, {3208, 3295}, {3217, 5296}, {3219, 19237}, {3303, 4513}, {3339, 36643}, {3419, 29659}, {3486, 24247}, {3488, 36479}, {3494, 16606}, {3616, 9310}, {3636, 9327}, {3662, 33827}, {3673, 24333}, {3691, 5260}, {3693, 37080}, {3730, 5248}, {3754, 5011}, {3831, 26244}, {3868, 5282}, {4071, 5015}, {4168, 16086}, {4262, 25440}, {4511, 39244}, {4653, 25092}, {4657, 18162}, {4919, 9957}, {5030, 5267}, {5257, 37035}, {5262, 21840}, {5264, 7031}, {5294, 16050}, {5305, 37715}, {5437, 15509}, {5440, 25068}, {5711, 30435}, {5737, 17284}, {5746, 19766}, {5777, 38572}, {5792, 25525}, {5793, 12625}, {5819, 28629}, {6205, 37572}, {6645, 20147}, {6685, 7413}, {7787, 14621}, {8616, 14974}, {9593, 17594}, {9620, 37598}, {9729, 34457}, {10382, 28053}, {10436, 11321}, {11320, 26223}, {11517, 19584}, {13615, 37580}, {14949, 17120}, {16048, 40131}, {16394, 34701}, {16600, 30117}, {16604, 21008}, {16705, 18723}, {16827, 17000}, {16912, 17260}, {16914, 17350}, {16916, 24514}, {16991, 30174}, {16998, 17033}, {17015, 39247}, {17277, 29960}, {17289, 33954}, {17316, 37652}, {17353, 33821}, {17368, 17688}, {17451, 33950}, {17681, 20335}, {17736, 18398}, {18047, 25303}, {18659, 36023}, {19225, 26059}, {21044, 27068}, {21984, 25083}, {24036, 35016}, {24929, 25066}, {25361, 36019}, {25957, 26085}, {26242, 28082}, {27164, 33953}, {27662, 37633}, {27950, 29614}, {29631, 37330}, {30108, 33942}, {30175, 31090}, {31448, 37574}, {33299, 34772}, {37284, 37576}, {39035, 40861}

X(41240) = BGF(43)

Barycentrics    a^3*b + a^3*c + a^2*b*c + a*b^2*c + a*b*c^2 - b^2*c^2 : :

X(41240) lies on these lines: {1, 2}, {6, 6376}, {9, 27269}, {31, 16916}, {41, 16997}, {76, 894}, {83, 21759}, {171, 384}, {192, 3501}, {194, 17754}, {213, 18140}, {238, 16918}, {335, 942}, {350, 2295}, {668, 20963}, {672, 1655}, {742, 33944}, {750, 16915}, {940, 19806}, {1100, 25102}, {1107, 37686}, {1258, 32020}, {1400, 3503}, {1475, 21226}, {1500, 17319}, {1575, 16744}, {1834, 26582}, {1909, 24512}, {2176, 30963}, {2670, 16589}, {2887, 33834}, {3169, 20168}, {3224, 18832}, {3496, 21371}, {3508, 28287}, {3552, 37603}, {3589, 21025}, {3730, 17261}, {3758, 20943}, {3780, 25280}, {3836, 17673}, {3905, 19584}, {3913, 20162}, {3948, 27064}, {3995, 25248}, {4307, 32971}, {4360, 20691}, {4366, 5255}, {4385, 31317}, {4721, 18145}, {4766, 17669}, {4852, 21868}, {5711, 7770}, {6196, 23493}, {6381, 17120}, {6645, 37607}, {6656, 37715}, {7901, 30837}, {9263, 17474}, {10448, 17684}, {11024, 39721}, {16549, 25264}, {16604, 34063}, {16706, 20255}, {16906, 25957}, {16917, 17122}, {16920, 17126}, {16921, 17717}, {16924, 26098}, {16927, 32917}, {17048, 33891}, {17116, 20888}, {17137, 26100}, {17280, 21071}, {17287, 33297}, {17289, 21024}, {17291, 21240}, {17300, 29967}, {17312, 29981}, {17348, 25614}, {18135, 24514}, {18299, 40758}, {19810, 20913}, {20247, 31087}, {20332, 23525}, {20486, 33073}, {20891, 33941}, {20923, 33938}, {20970, 27076}, {21760, 27644}, {21904, 25107}, {21935, 33841}, {23534, 34252}, {24254, 33939}, {24586, 33827}, {24602, 33826}, {24699, 24726}, {26223, 31060}, {26686, 37634}, {33045, 33111}

X(41241) = BGF(44)

Barycentrics    2*a^3 + 2*a^2*b + 2*a^2*c - 2*a*b*c + b^2*c + b*c^2 : :

X(41241) lies on these lines: {2, 44}, {6, 4358}, {42, 4432}, {58, 30905}, {75, 14997}, {81, 645}, {83, 213}, {86, 35595}, {171, 9458}, {190, 17012}, {329, 32774}, {597, 29833}, {748, 24331}, {894, 24589}, {899, 4672}, {940, 26688}, {1100, 31035}, {1107, 29570}, {1150, 1743}, {1203, 3701}, {1386, 3952}, {1757, 32944}, {2999, 32933}, {3240, 4676}, {3305, 16552}, {3589, 26580}, {3618, 31018}, {3681, 36534}, {3759, 4671}, {3840, 4722}, {3896, 4693}, {3936, 17353}, {3967, 17150}, {4090, 17469}, {4096, 29816}, {4359, 4363}, {4434, 21747}, {4519, 17162}, {4663, 29824}, {4664, 17013}, {4679, 29829}, {4696, 16466}, {4698, 19740}, {4703, 29663}, {4753, 31136}, {4850, 17350}, {4980, 17119}, {4981, 25496}, {5219, 31229}, {5287, 19738}, {5294, 5741}, {5739, 29611}, {7191, 24841}, {14996, 30829}, {15254, 29822}, {16468, 32931}, {16477, 17763}, {16669, 16704}, {16706, 17484}, {17020, 32939}, {17120, 37633}, {17263, 37635}, {17279, 31034}, {17289, 37656}, {17292, 31143}, {17348, 31025}, {17351, 17495}, {17352, 31019}, {17354, 33077}, {17357, 31017}, {17367, 33151}, {17385, 27081}, {17779, 32845}, {18743, 37685}, {19741, 28639}, {19743, 37595}, {19786, 26792}, {24821, 29821}, {28996, 37543}, {29613, 32782}, {29850, 33096}, {29852, 33101}, {32772, 36531}, {32843, 33159}, {33071, 33166}, {33107, 33118}

X(41242) = BGF(45)

Barycentrics    a^3 + a^2*b + a^2*c - a*b*c + 2*b^2*c + 2*b*c^2 : :

X(41242) lies on these lines: {1, 3994}, {2, 45}, {6, 4671}, {9, 5235}, {10, 3245}, {11, 33170}, {38, 24821}, {42, 4693}, {75, 37680}, {81, 312}, {83, 213}, {86, 31035}, {100, 3923}, {141, 17484}, {226, 33157}, {329, 29611}, {330, 1255}, {536, 17012}, {594, 37656}, {726, 32944}, {756, 36531}, {846, 31264}, {894, 4358}, {908, 17355}, {984, 4756}, {996, 1320}, {1150, 17350}, {1203, 4066}, {1211, 26792}, {1215, 1621}, {1220, 25253}, {1376, 24344}, {1836, 29679}, {2345, 31018}, {2607, 4418}, {2886, 33166}, {3120, 33159}, {3175, 17011}, {3218, 17351}, {3240, 5695}, {3305, 16832}, {3315, 24349}, {3589, 33155}, {3661, 31143}, {3703, 33107}, {3729, 4850}, {3739, 35595}, {3741, 32938}, {3773, 32843}, {3790, 33070}, {3838, 29873}, {3840, 32940}, {3920, 3967}, {3932, 33112}, {3936, 17280}, {3944, 26061}, {3952, 5263}, {3971, 32772}, {3980, 9342}, {4009, 5297}, {4011, 5284}, {4054, 17353}, {4090, 32945}, {4135, 32928}, {4144, 17281}, {4359, 37687}, {4361, 14997}, {4383, 17119}, {4387, 17018}, {4519, 4663}, {4527, 14459}, {4670, 17021}, {4672, 17763}, {4676, 26227}, {4721, 17292}, {4754, 29569}, {4867, 25697}, {4921, 11679}, {4942, 17599}, {5241, 7227}, {5253, 25591}, {5294, 33133}, {5333, 16831}, {5718, 17340}, {5905, 33172}, {6057, 33093}, {6535, 32861}, {6685, 32936}, {7238, 17483}, {10129, 29857}, {10707, 33120}, {11680, 33163}, {15523, 33096}, {15988, 26612}, {17013, 17318}, {17019, 35652}, {17116, 24589}, {17165, 24841}, {17233, 31034}, {17243, 37635}, {17277, 31025}, {17279, 31019}, {17284, 31164}, {17285, 31017}, {17289, 26580}, {17605, 29872}, {17717, 33161}, {17740, 37651}, {17768, 33086}, {18228, 19822}, {19717, 34064}, {19739, 25417}, {19804, 26688}, {21093, 24295}, {24552, 32937}, {24703, 29667}, {24723, 26251}, {24725, 29674}, {24943, 33101}, {25385, 33115}, {25496, 32925}, {26098, 32862}, {26591, 37659}, {26627, 30829}, {27065, 31993}, {27184, 29613}, {27958, 30576}, {29637, 32856}, {29663, 33154}, {29677, 33103}, {29687, 33097}, {29827, 36263}, {30831, 31053}, {30942, 32935}, {32781, 33099}, {33078, 41011}, {33098, 33174}, {33104, 33165}, {33105, 33164}, {33106, 33162}, {33134, 38047}, {33168, 37662}

X(41243) = BGF(48)

Barycentrics    a*(a - b - c)*(a^4 + 2*a^3*b + a^2*b^2 + 2*a^3*c + 3*a^2*b*c + 2*a*b^2*c + b^3*c + a^2*c^2 + 2*a*b*c^2 + b*c^3) : :

X(41243) lies on these lines: {2, 48}, {6, 63}, {21, 14547}, {41, 14555}, {78, 210}, {83, 226}, {238, 3725}, {261, 284}, {312, 2329}, {345, 2268}, {386, 993}, {515, 1220}, {1211, 20769}, {1812, 21748}, {2174, 5743}, {2202, 31623}, {2267, 26065}, {2323, 17185}, {4199, 7193}, {4269, 27174}, {4383, 4426}, {5307, 7119}, {5905, 17302}, {6703, 7113}, {15988, 22097}, {18162, 27184}, {18228, 28922}, {21246, 31631}, {26889, 37329}

X(41244) = BGF(53)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 4*b^6*c^2 + 3*a^4*c^4 + 5*a^2*b^2*c^4 + 8*b^4*c^4 - a^2*c^6 - 4*b^2*c^6) : :

X(41244) lies on these lines: {2, 53}, {3, 37871}, {4, 343}, {6, 324}, {25, 39530}, {26, 35719}, {83, 458}, {141, 37192}, {155, 14978}, {264, 394}, {275, 9308}, {381, 34836}, {393, 37649}, {427, 14593}, {436, 35259}, {1593, 6503}, {3796, 33971}, {5422, 8746}, {6515, 6748}, {6641, 11197}, {6819, 37648}, {7395, 8887}, {7539, 39569}, {7745, 11433}, {12429, 35717}, {14561, 14569}, {17810, 30506}, {17811, 40684}, {19357, 37127}, {26898, 32428}

X(41245) = BGF(56)

Barycentrics    (a + b - c)*(a - b + c)*(a^3 + a*b^2 + a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(41245) lies on these lines: {1, 6996}, {2, 12}, {6, 7}, {11, 7384}, {27, 34}, {55, 37416}, {57, 85}, {65, 239}, {75, 2285}, {83, 226}, {86, 604}, {92, 20613}, {241, 1107}, {469, 11392}, {496, 36728}, {497, 7406}, {553, 24588}, {658, 37208}, {664, 7146}, {938, 5786}, {1014, 17077}, {1056, 7397}, {1317, 29588}, {1319, 16826}, {1388, 29570}, {1400, 17277}, {1420, 16831}, {1466, 37274}, {1470, 11329}, {1478, 7377}, {1909, 14829}, {1999, 34791}, {2099, 4393}, {2171, 4360}, {3008, 4298}, {3086, 36662}, {3188, 17080}, {3218, 24633}, {3339, 16833}, {3340, 16834}, {3361, 16832}, {3476, 17316}, {3485, 26626}, {3661, 5252}, {3662, 10401}, {3669, 24601}, {3687, 7270}, {3868, 28916}, {3869, 26621}, {3911, 24603}, {3912, 10106}, {4032, 17755}, {4267, 14953}, {4281, 29775}, {4293, 36698}, {4297, 23512}, {4307, 25406}, {4308, 5308}, {4313, 5716}, {4315, 29571}, {5088, 37597}, {5091, 40724}, {5176, 26575}, {5219, 29603}, {5221, 16816}, {5236, 7119}, {5290, 17681}, {5323, 26643}, {5435, 5737}, {5717, 37088}, {5718, 17966}, {5793, 29611}, {5839, 5933}, {6542, 10944}, {6762, 11679}, {6999, 7354}, {7153, 27424}, {7175, 30097}, {7195, 21454}, {7365, 24605}, {9578, 17308}, {9655, 36731}, {10404, 17367}, {10408, 29467}, {10481, 24803}, {11011, 29584}, {11349, 23361}, {11353, 25531}, {11375, 17397}, {12577, 39595}, {12607, 27526}, {14942, 37580}, {15509, 24604}, {15950, 29586}, {16367, 37579}, {16815, 32636}, {17086, 41003}, {17294, 37709}, {17389, 37738}, {17789, 32939}, {21008, 37662}, {21554, 37609}, {24623, 30725}, {24914, 29576}, {26001, 34831}, {27058, 38869}, {27064, 30618}, {28757, 33839}, {28827, 40127}, {30854, 40131}, {37677, 38296}

X(41245) = anticomplement of X(30847)

X(41246) = BGF(57)

Barycentrics    (a + b - c)*(a - b + c)*(a^3 - a^2*b - a^2*c - a*b*c - b^2*c - b*c^2) : :

X(41246) lies on these lines: {2, 7}, {6, 85}, {19, 27000}, {29, 1876}, {65, 5263}, {75, 3713}, {77, 17379}, {78, 4327}, {83, 1446}, {86, 241}, {192, 7190}, {239, 1441}, {273, 608}, {274, 1170}, {281, 26531}, {342, 458}, {347, 26626}, {572, 5088}, {604, 7176}, {610, 4209}, {651, 17120}, {664, 1100}, {673, 2264}, {938, 4307}, {948, 3618}, {980, 5736}, {988, 4310}, {1010, 37544}, {1418, 4670}, {1429, 4032}, {1449, 9312}, {1462, 2298}, {1471, 16823}, {1713, 27142}, {1766, 17753}, {1892, 5125}, {1999, 37543}, {2345, 6604}, {3668, 17023}, {3693, 21453}, {3729, 4328}, {3758, 6180}, {4318, 37558}, {4335, 24283}, {4363, 39126}, {4552, 7269}, {4645, 6734}, {5018, 33682}, {5173, 35617}, {5572, 14942}, {5728, 13727}, {6354, 19786}, {6999, 18650}, {7196, 24512}, {7377, 41004}, {9317, 21748}, {13411, 24231}, {14189, 16503}, {14564, 17345}, {15668, 31225}, {16560, 34830}, {17095, 17398}, {17302, 22464}, {17303, 33298}, {17367, 37800}, {17369, 32007}, {17825, 20921}, {18655, 37416}, {25964, 37774}, {26802, 40979}, {29610, 40999}, {29833, 37798}

X(41247) = BGF(63)

Barycentrics    a^5 + a^4*b + a^3*b^2 + a^2*b^3 + a^4*c + 3*a^3*b*c + a*b^3*c + b^4*c + a^3*c^2 + b^3*c^2 + a^2*c^3 + a*b*c^3 + b^2*c^3 + b*c^4 : :

X(41247) lies on these lines: {2, 7}, {6, 304}, {19, 27299}, {37, 33821}, {83, 2281}, {238, 16817}, {239, 16470}, {940, 3718}, {1509, 2303}, {1716, 3980}, {2256, 35274}, {2298, 3263}, {3685, 5262}, {5120, 25918}, {5227, 27248}, {7290, 19851}, {27396, 33819}, {30758, 34261}

X(41248) = BGF(71)

Barycentrics    (b + c)*(2*a^4 + a^3*b + a*b^3 + a^3*c + a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 + a*c^3 + b*c^3) : :

X(41248) lies on these lines: {2, 71}, {6, 306}, {10, 82}, {42, 3618}, {86, 2983}, {313, 17354}, {458, 1826}, {2200, 16061}, {2209, 26061}, {3219, 27270}, {3682, 37176}, {3759, 16478}, {3840, 29453}, {3997, 22008}, {4651, 37681}, {4657, 27638}, {8053, 35263}, {17142, 26230}, {17381, 21035}, {18230, 19874}, {19284, 20780}, {20159, 29653}, {20174, 26723}, {26223, 27252}, {27644, 32779}

X(41249) = BGF(72)

Barycentrics    (b + c)*(2*a^4 + 2*a^3*b + 2*a^3*c + a^2*b*c + b^3*c + b*c^3) : :

X(41249) lies on these lines: {1, 11342}, {2, 72}, {6, 20336}, {37, 3618}, {81, 40406}, {83, 213}, {218, 19281}, {228, 21495}, {458, 41013}, {894, 2287}, {1265, 17316}, {2901, 16834}, {3187, 3702}, {3954, 17397}, {3995, 17014}, {3998, 26065}, {4393, 17697}, {4641, 16720}, {5222, 19791}, {5294, 33299}, {9596, 14555}, {16050, 34772}, {16826, 26689}, {17353, 22021}, {21839, 29576}, {24631, 28254}, {29585, 35274}

X(41250) = BGF(75)

Barycentrics    b*c*(-a^2 + b*c)*(a^2*b + a^2*c + a*b*c + b^2*c + b*c^2) : :

X(41250) lies on these lines: {2, 37}, {6, 561}, {83, 18833}, {239, 35544}, {1100, 18059}, {1914, 1966}, {1920, 24512}, {1921, 2238}, {2517, 4010}, {3681, 30473}, {3684, 4495}, {4485, 17027}, {7244, 16503}, {7304, 7307}, {16517, 18078}, {16973, 18056}, {17034, 27801}, {20174, 21020}, {20889, 21840}, {33854, 35545}

X(41251) = BGF(78)

Barycentrics    a^5 + a^4*b + a^3*b^2 + a^2*b^3 + a^4*c + 3*a^3*b*c + 2*a^2*b^2*c + a*b^3*c - b^4*c + a^3*c^2 + 2*a^2*b*c^2 - b^3*c^2 + a^2*c^3 + a*b*c^3 - b^2*c^3 - b*c^4 : :

X(41251) lies on these lines: {1, 2}, {6, 3718}, {75, 34261}, {192, 1766}, {304, 940}, {314, 2303}, {344, 5336}, {1332, 2300}, {4552, 39714}, {5208, 24606}, {5280, 27064}, {11353, 18743}, {19791, 37233}, {20715, 21334}, {24632, 35623}, {30710, 33941}

X(41252) = BGF(101)

Barycentrics    2*a^5 - a^3*b^2 + a^2*b^3 - 2*a^3*b*c - a^3*c^2 - 2*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + b^2*c^3 : :

X(41252) lies on these lines: {2, 101}, {6, 190}, {81, 18047}, {83, 4080}, {145, 595}, {284, 26764}, {1016, 1252}, {1150, 17230}, {1429, 26982}, {2174, 27102}, {2210, 20352}, {4112, 17165}, {7113, 26975}, {9263, 37685}, {16738, 25536}, {18042, 26963}, {18268, 30669}, {20769, 27044}

X(41253) = BGF(112)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^10 - 2*a^8*b^2 + 2*a^4*b^6 - a^2*b^8 - 2*a^8*c^2 + a^6*b^2*c^2 + a^4*b^4*c^2 + a^2*b^6*c^2 - b^8*c^2 + a^4*b^2*c^4 - 4*a^2*b^4*c^4 + b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) : :

X(41253) lies on these lines: {2, 112}, {4, 110}, {6, 264}, {83, 16080}, {249, 297}, {275, 671}, {401, 14961}, {427, 6033}, {468, 2080}, {842, 7473}, {1316, 2967}, {2207, 36789}, {7784, 11331}, {9730, 37124}, {10564, 35474}, {10745, 34664}, {19165, 35278}, {22151, 37778}, {37200, 37497}, {37855, 40112}, {39176, 40879}, {40393, 40684}

X(41254) = BGF(115)

Barycentrics    a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 + 3*b^6*c^2 - a^4*c^4 - a^2*b^2*c^4 - 6*b^4*c^4 + a^2*c^6 + 3*b^2*c^6 : :

X(41254) lies on these lines: {2, 99}, {4, 74}, {6, 264}, {76, 2986}, {83, 94}, {98, 1316}, {110, 38664}, {187, 401}, {230, 40884}, {297, 39062}, {316, 3580}, {323, 1236}, {419, 1495}, {523, 9139}, {842, 34175}, {868, 14639}, {1494, 6128}, {1632, 7668}, {1640, 14223}, {2052, 6529}, {3098, 37190}, {3288, 34536}, {4226, 34473}, {5210, 35941}, {5286, 14920}, {5523, 14165}, {5642, 12243}, {6321, 11007}, {6723, 38734}, {7760, 11004}, {7770, 36789}, {7812, 37644}, {7827, 14389}, {7841, 37638}, {8370, 37648}, {9134, 32121}, {9155, 23235}, {10159, 13582}, {10723, 36163}, {11005, 16280}, {11054, 40112}, {11303, 40710}, {11304, 40709}, {11430, 37124}, {11623, 35282}, {11632, 34094}, {12203, 15080}, {14957, 15107}, {15118, 17983}, {21531, 35002}, {21639, 38294}, {23698, 35922}, {24284, 41079}, {26591, 26678}, {26864, 39646}, {37200, 37487}

X(41254) = polar conjugate of isogonal conjugate of X(5622)

X(41255) = BGF(125)

Barycentrics    a^12 - a^10*b^2 - a^4*b^8 + a^2*b^10 - a^10*c^2 + a^8*b^2*c^2 - a^2*b^8*c^2 + b^10*c^2 + 2*a^4*b^4*c^4 - 2*b^6*c^6 - a^4*c^8 - a^2*b^2*c^8 + a^2*c^10 + b^2*c^10 : :

X(41255) lies on these lines: {2, 98}, {6, 339}, {32, 14376}, {69, 4611}, {83, 648}, {141, 32661}, {511, 15013}, {1289, 1843}, {1494, 12150}, {2072, 2456}, {2781, 40856}, {3098, 35952}, {7787, 39352}, {7808, 23583}, {7815, 40484}, {13355, 18531}, {29012, 40889}

X(41256) = BGF(159)

Barycentrics    (a^2 - b^2 - c^2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + 3*a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 - b^2*c^4 + c^6) : :

X(41256) lies on these lines: {2, 159}, {4, 83}, {6, 1370}, {69, 305}, {141, 8547}, {155, 6643}, {184, 28708}, {193, 16063}, {239, 11677}, {511, 18916}, {1368, 19459}, {1495, 7392}, {1503, 6816}, {3313, 6515}, {3537, 16163}, {3589, 3796}, {3619, 21243}, {3867, 10601}, {5050, 14790}, {5085, 6815}, {5157, 23327}, {5182, 39842}, {5596, 26206}, {5622, 12319}, {6804, 18381}, {7528, 38110}, {7667, 37491}, {7716, 37648}, {10519, 12359}, {11513, 26945}, {11514, 26873}, {12017, 18420}, {12220, 18911}, {14853, 34938}, {16051, 28408}, {18531, 19125}, {19119, 22151}, {19137, 31383}, {24270, 32986}, {32114, 32255}, {34774, 34944}

X(41257) = BGF(161)

Barycentrics    (a^2 - b^2 - c^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 - 6*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 6*a^2*b^6*c^2 - 3*b^8*c^2 - 2*a^6*c^4 - 2*a^4*b^2*c^4 + 10*a^2*b^4*c^4 + 2*b^6*c^4 - 2*a^4*c^6 - 6*a^2*b^2*c^6 + 2*b^4*c^6 + a^2*c^8 - 3*b^2*c^8 + c^10) : :

X(41257) lies on these lines: {2, 161}, {4, 83}, {6, 37444}, {68, 69}, {125, 7386}, {511, 18912}, {1350, 3580}, {1351, 14791}, {1352, 12283}, {1370, 3060}, {1503, 26206}, {3567, 31670}, {3589, 7544}, {3796, 7394}, {5050, 18569}, {5085, 18382}, {5422, 5480}, {6146, 20806}, {6776, 18445}, {6804, 23325}, {6816, 36851}, {6997, 26881}, {7401, 37513}, {7503, 23300}, {9730, 34938}, {11511, 18935}, {11585, 28408}, {12319, 25320}, {14790, 14853}, {16072, 39879}, {18583, 31723}, {18925, 28708}, {26156, 34787}, {32273, 38727}

X(41258) = BGF(172)

Barycentrics    a^5 + a^4*b + a^3*b^2 + a^2*b^3 + a^4*c + 2*a^3*b*c + 2*a^2*b^2*c + a*b^3*c + a^3*c^2 + 2*a^2*b*c^2 + a*b^2*c^2 + b^3*c^2 + a^2*c^3 + a*b*c^3 + b^2*c^3 : :

X(41258) lies on these lines: {1, 4112}, {2, 172}, {6, 75}, {10, 58}, {27, 7119}, {42, 14012}, {81, 1909}, {83, 226}, {85, 222}, {86, 29967}, {291, 33732}, {306, 33954}, {334, 18268}, {345, 4195}, {384, 3666}, {940, 19806}, {958, 16830}, {1580, 40718}, {1910, 31636}, {1911, 36800}, {2064, 33941}, {2162, 27424}, {2298, 17787}, {3975, 5276}, {4267, 19842}, {4366, 20166}, {5337, 27020}, {5737, 19827}, {6645, 37596}, {11320, 28606}, {11321, 19804}, {16050, 33116}, {16062, 29633}, {16800, 25124}, {16913, 17490}, {17034, 19792}, {17103, 17209}, {17688, 32777}, {18134, 24549}, {19767, 19840}, {19799, 33938}, {19812, 29614}, {19822, 37652}, {20769, 37678}, {21384, 36483}, {24271, 25264}

X(41259) = BGF(194)

Barycentrics    -(b^2*c^2*(2*a^4 + a^2*b^2 + a^2*c^2 - b^2*c^2)) : :

X(41259) lies on these lines: {1, 7033}, {2, 39}, {4, 14135}, {6, 670}, {83, 3224}, {99, 11328}, {182, 8920}, {237, 7782}, {239, 6382}, {262, 6234}, {316, 35060}, {327, 37647}, {458, 6331}, {597, 30736}, {598, 14608}, {850, 11183}, {886, 30229}, {894, 6383}, {1207, 24734}, {1502, 3589}, {1613, 7760}, {1978, 4393}, {3051, 7894}, {3114, 3329}, {3148, 5152}, {3360, 7770}, {3403, 4384}, {3618, 9230}, {3763, 33769}, {5149, 33336}, {6179, 8623}, {6375, 8264}, {7752, 21531}, {7754, 21001}, {7771, 14096}, {7777, 18872}, {7860, 20022}, {7875, 35540}, {7878, 9490}, {7918, 33734}, {8265, 40478}, {9211, 18024}, {10010, 32449}, {13331, 14603}, {14897, 32746}, {16816, 40087}, {17350, 34086}, {17367, 18891}, {17941, 34396}, {18906, 34236}, {20284, 25264}, {20608, 33790}, {31639, 34811}

X(41260) = BGF(198)

Barycentrics    (a - b - c)*(a^5 + 2*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 + 2*a^4*c + 3*a^3*b*c + a^2*b^2*c + a*b^3*c + b^4*c + 2*a^3*c^2 + a^2*b*c^2 - b^3*c^2 + 2*a^2*c^3 + a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4) : :

X(41260) lies on these lines: {2, 198}, {4, 1220}, {6, 329}, {9, 312}, {41, 27398}, {55, 20487}, {72, 32926}, {75, 2339}, {83, 226}, {405, 32942}, {452, 497}, {950, 4514}, {1848, 37279}, {2478, 27540}, {4123, 10393}, {4251, 22020}, {4422, 5737}, {4429, 26052}, {5175, 5793}, {5247, 12572}, {5749, 19645}, {5750, 37092}, {5928, 25898}, {6996, 10888}, {10445, 23512}, {10478, 16788}, {11343, 27339}, {13615, 14942}, {17681, 25527}, {26590, 27319}

X(41261) = BGF(200)

Barycentrics    a^4 + 2*a^3*b + a^2*b^2 + 2*a^3*c + a^2*b*c + 2*a*b^2*c - b^3*c + a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - b*c^3 : :

X(41261) lies on these lines: {1, 2}, {6, 341}, {12, 33073}, {40, 192}, {65, 32926}, {81, 4696}, {193, 5815}, {228, 3871}, {312, 5710}, {894, 4385}, {958, 3769}, {979, 23524}, {989, 7155}, {1036, 26264}, {1071, 37088}, {1191, 18743}, {1203, 3992}, {1220, 3745}, {1329, 33071}, {1468, 9369}, {1706, 3875}, {1824, 37390}, {1834, 32850}, {1943, 9578}, {2650, 32927}, {3333, 17480}, {3685, 5255}, {3701, 27064}, {3714, 5263}, {3786, 34790}, {3812, 32922}, {3913, 34247}, {3931, 17319}, {4195, 5269}, {4360, 4646}, {4642, 32928}, {4645, 13161}, {4692, 37559}, {4852, 21896}, {4860, 34860}, {5015, 37715}, {5264, 7283}, {11681, 33070}, {12514, 17261}, {12527, 20077}, {17300, 21620}, {21935, 33072}, {24440, 32921}, {34064, 37548}

X(41262) = BGF(211)

Barycentrics    a^2*(a^4*b^4 - a^2*b^6 + a^4*b^2*c^2 - b^6*c^2 + a^4*c^4 - a^2*c^6 - b^2*c^6) : :

X(41262) lies on these lines: {2, 211}, {3, 6}, {22, 40643}, {51, 7808}, {83, 3060}, {98, 11412}, {315, 2387}, {458, 27370}, {626, 37988}, {1078, 2979}, {1154, 14880}, {1186, 20859}, {1993, 3203}, {3491, 7818}, {3917, 7815}, {4173, 7759}, {5167, 7825}, {5446, 10358}, {5889, 12203}, {6101, 10104}, {6337, 35704}, {7752, 33873}, {7761, 40951}, {7770, 27375}, {7834, 27374}, {7855, 34383}, {7857, 11673}, {9465, 38854}, {10263, 10796}, {10352, 39835}, {13207, 17129}, {15897, 17907}

X(41263) = BGF(212)

Barycentrics    a*(a^5 - a^4*b - a^3*b^2 + a^2*b^3 - a^4*c - a^3*b*c - a*b^3*c - b^4*c - a^3*c^2 - 4*a*b^2*c^2 - b^3*c^2 + a^2*c^3 - a*b*c^3 - b^2*c^3 - b*c^4) : :

X(41263) lies on these lines: {2, 212}, {6, 78}, {10, 82}, {21, 2183}, {31, 14555}, {86, 1167}, {171, 4417}, {750, 30828}, {894, 23693}, {908, 5327}, {1001, 3057}, {1010, 3074}, {1818, 15988}, {3764, 37573}, {3826, 25903}, {3923, 20927}, {4357, 13329}, {4649, 22836}, {4676, 30854}, {5135, 20769}, {9441, 24723}, {10573, 32941}, {11517, 37502}, {16484, 30147}, {16503, 24486}, {17614, 36942}, {26013, 32942}, {31837, 37510}

X(41264) = invBGF(7)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2*b + b^3 + a^2*c - a*b*c + c^3) : :

X(41264) lies on these lines: {2, 7}, {6, 1401}, {32, 56}, {169, 24178}, {171, 24309}, {573, 982}, {604, 5037}, {1460, 1486}, {1500, 38286}, {2269, 3677}, {2277, 20967}, {3290, 40961}, {4253, 39048}, {5277, 36508}, {7195, 18785}, {21240, 30617}, {28036, 31466}

X(41265) = invBGF(8)

Barycentrics    a^2*(a^2*b + b^3 + a^2*c + a*b*c + 2*b^2*c + 2*b*c^2 + c^3) : :

X(41265) lies on these lines: {1, 2}, {6, 3688}, {19, 2356}, {31, 3730}, {32, 55}, {39, 21010}, {45, 4749}, {69, 39712}, {76, 32926}, {171, 12782}, {172, 37586}, {197, 3207}, {213, 4517}, {573, 20964}, {579, 21035}, {872, 4270}, {968, 17742}, {980, 4447}, {981, 4876}, {991, 18788}, {1911, 3864}, {1930, 32771}, {1964, 5105}, {2092, 34247}, {2177, 4262}, {2223, 2276}, {2242, 17798}, {2258, 2318}, {2273, 19133}, {3247, 40934}, {3501, 5269}, {3735, 20715}, {3745, 17750}, {3891, 20913}, {4253, 20985}, {4261, 20990}, {4277, 4557}, {5069, 16679}, {16672, 39688}, {17192, 33069}, {17499, 32937}, {17602, 20486}, {17720, 20544}, {18082, 18147}, {33771, 37576}

X(41266) = invBGF(20)

Barycentrics    a^2*(a^6 + 5*a^4*b^2 - 5*a^2*b^4 - b^6 + 5*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - 5*a^2*c^4 + b^2*c^4 - c^6) : :

X(41266) lies on these lines: {2, 3}, {6, 33578}, {32, 154}, {39, 17810}, {51, 9605}, {159, 8573}, {184, 30435}, {187, 33979}, {216, 7716}, {574, 31860}, {800, 9924}, {1384, 1495}, {1609, 7669}, {1661, 3053}, {1974, 15905}, {2223, 20991}, {3511, 19588}, {5007, 17809}, {5024, 34417}, {5188, 17811}, {7767, 14826}, {8721, 13567}, {12167, 15851}, {17813, 40135}, {17825, 37479}, {19118, 38292}, {19459, 40981}, {20775, 33580}, {21309, 26864}, {33586, 36212}

X(41267) = invBGF(42)

Barycentrics    a^4*(b + c)*(b^2 + c^2) : :

X(41267) lies on these lines: {1, 2}, {6, 16683}, {32, 560}, {39, 1964}, {58, 1911}, {100, 719}, {101, 733}, {213, 40935}, {740, 21412}, {872, 20970}, {1575, 22328}, {1923, 3051}, {2085, 5369}, {2176, 4116}, {2230, 4721}, {2238, 23629}, {2309, 25092}, {3159, 20681}, {3230, 4161}, {3499, 21792}, {3510, 17499}, {3896, 21435}, {3954, 4093}, {3997, 23493}, {4253, 7032}, {6196, 8865}, {16604, 22279}, {17448, 22292}, {18082, 31622}, {18266, 18756}, {18755, 21788}, {20683, 23447}, {21080, 32453}, {21278, 27262}

X(41268) = invBGF(43)

Barycentrics    a^3*(a*b^3 - b^3*c - b^2*c^2 + a*c^3 - b*c^3) : :

X(41268) lies on these lines: {1, 2}, {3, 1911}, {6, 23534}, {9, 23652}, {31, 23525}, {32, 2209}, {39, 7032}, {192, 6196}, {194, 3510}, {941, 3495}, {1500, 4116}, {2276, 40935}, {3056, 23414}, {3224, 22061}, {3501, 23493}, {4253, 23524}, {5283, 23629}, {16969, 38986}, {18755, 20996}, {20456, 26893}, {20464, 21384}, {23547, 40736}

X(41269) = invBGF(75)

Barycentrics    a*(a^2*b + a*b^2 + b^3 + a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(41269) lies on these lines: {1, 32}, {2, 37}, {6, 38}, {9, 29821}, {21, 16974}, {39, 29646}, {41, 16519}, {44, 17025}, {55, 199}, {63, 16972}, {83, 30134}, {86, 4376}, {230, 17602}, {256, 4493}, {292, 980}, {335, 37632}, {386, 3954}, {594, 32860}, {626, 30122}, {756, 37673}, {869, 4093}, {893, 4362}, {976, 18755}, {982, 24512}, {984, 2238}, {986, 2295}, {1078, 30138}, {1100, 3873}, {1213, 3703}, {1403, 2171}, {1500, 29674}, {1961, 3247}, {2092, 32778}, {2176, 2292}, {2275, 37592}, {2321, 4970}, {3125, 30116}, {3240, 20693}, {3670, 17750}, {3681, 21904}, {3723, 3744}, {3730, 21802}, {3821, 4071}, {3896, 17299}, {3920, 4386}, {3959, 10459}, {4030, 17388}, {4085, 4119}, {4144, 4389}, {4277, 32842}, {4372, 16060}, {4392, 36409}, {4414, 17735}, {4426, 5262}, {4429, 20483}, {4760, 20538}, {4981, 17275}, {5275, 8301}, {5277, 30142}, {5283, 16600}, {5750, 24165}, {5904, 20970}, {6654, 40791}, {6680, 30118}, {6685, 21101}, {7226, 37657}, {9502, 16518}, {10026, 33065}, {13738, 21808}, {16062, 16886}, {16503, 17598}, {16555, 16946}, {16672, 31477}, {16705, 17489}, {17053, 21827}, {17246, 17747}, {17469, 21793}, {17591, 17754}, {18061, 30112}, {19329, 25090}, {20536, 40776}, {21764, 29819}, {24426, 25804}, {25263, 27162}, {25497, 30126}, {25499, 33945}, {26244, 32926}, {27274, 33935}, {27324, 33932}, {33295, 40773}

X(41270) = invBGF(95)

Barycentrics    a^4*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(41270) lies on these lines: {2, 95}, {6, 16030}, {32, 54}, {50, 647}, {160, 184}, {230, 8901}, {232, 19189}, {1298, 2715}, {1968, 8884}, {2207, 19173}, {3053, 16035}, {6423, 16029}, {6424, 16034}, {8721, 8883}, {9380, 14587}, {10316, 19210}, {26887, 33629}

X(41271) = invBGF(96)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^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^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(41271) lies on these lines: {2, 54}, {6, 2351}, {37, 2168}, {51, 14573}, {52, 8883}, {97, 2987}, {111, 32692}, {184, 2165}, {308, 34385}, {343, 15366}, {393, 14593}, {421, 8794}, {1989, 32734}, {5012, 5392}, {5647, 11402}, {16030, 40802}, {16391, 37476}

X(41272) = invBGF(111)

Barycentrics    a^4*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(b^2 + c^2) : :

X(41272) lies on these lines: {2, 99}, {3, 36821}, {32, 1084}, {39, 1634}, {351, 3455}, {691, 733}, {694, 5118}, {881, 887}, {895, 7772}, {923, 34067}, {2021, 21177}, {2936, 20998}, {5206, 6091}, {5651, 30495}, {8623, 36827}, {34294, 36157}

X(41273) = invBGF(115)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 - 2*b^6 - a^4*c^2 + 3*a^2*b^2*c^2 + 2*b^4*c^2 - a^2*c^4 + 2*b^2*c^4 - 2*c^6) : :

X(41273) lies on these lines: {2, 99}, {3, 8450}, {6, 3506}, {32, 3124}, {39, 20998}, {110, 7772}, {187, 6660}, {251, 9999}, {694, 3098}, {1084, 9142}, {2021, 40350}, {2054, 37586}, {3229, 35002}, {3291, 5162}, {8623, 15107}, {12367, 33875}, {14660, 15080}, {15066, 32452}

X(41274) = invBGF(125)

Barycentrics    a^2*(a^10 - a^8*b^2 + a^4*b^6 - a^2*b^8 - a^8*c^2 + a^6*b^2*c^2 - a^4*b^4*c^2 + a^2*b^6*c^2 - 2*b^8*c^2 - a^4*b^2*c^4 + 2*b^6*c^4 + a^4*c^6 + a^2*b^2*c^6 + 2*b^4*c^6 - a^2*c^8 - 2*b^2*c^8) : :

X(41274) lies on these lines: {2, 98}, {6, 39832}, {32, 3455}, {99, 19126}, {115, 1974}, {148, 19121}, {206, 11646}, {578, 10753}, {1177, 16278}, {1691, 2393}, {1692, 14917}, {1843, 39857}, {2387, 2458}, {2456, 13754}, {2782, 19131}, {2794, 19124}, {5026, 5157}, {5477, 39834}, {6034, 19136}, {12188, 19129}, {14061, 19137}, {14651, 19128}, {19125, 39849}, {37488, 39817}

X(41275) = invBGF(140)

Barycentrics    a^2*(a^6 - 5*a^4*b^2 + 5*a^2*b^4 - b^6 - 5*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + 5*a^2*c^4 + b^2*c^4 - c^6) : :

X(41275) lies on these lines: {2, 3}, {6, 30534}, {32, 13366}, {39, 15004}, {51, 574}, {154, 5191}, {157, 19596}, {160, 8553}, {184, 187}, {216, 8541}, {352, 15655}, {394, 9155}, {1384, 11402}, {1495, 8588}, {1609, 20775}, {1843, 10979}, {1974, 22052}, {1993, 2080}, {2351, 3455}, {2936, 19911}, {3053, 17809}, {3164, 38294}, {3917, 8722}, {5024, 9777}, {5171, 36212}, {5422, 11171}, {6800, 38225}, {8589, 34417}, {9723, 39099}, {9969, 14806}, {14075, 34566}, {15300, 33972}, {18472, 34397}, {19153, 23200}, {23635, 36751}

X(41275) = {X(3131),X(3132)}-harmonic conjugate of X(22)

X(41276) = invBGF(145)

Barycentrics    a^2*(a^2*b - 2*a*b^2 + b^3 + a^2*c - a*b*c + 2*b^2*c - 2*a*c^2 + 2*b*c^2 + c^3) : :

X(41276) lies on these lines: {1, 2}, {31, 5526}, {32, 220}, {37, 15733}, {38, 35293}, {55, 101}, {218, 38825}, {241, 17625}, {573, 3688}, {579, 20990}, {984, 991}, {1108, 40659}, {1376, 14839}, {1449, 4878}, {1573, 34522}, {1818, 7174}, {1864, 16601}, {2192, 10482}, {2223, 3730}, {2256, 6600}, {2293, 3731}, {2318, 3997}, {3973, 20978}, {4253, 20683}, {4266, 4557}, {4343, 16673}, {4419, 35338}, {4568, 32937}, {4712, 35269}, {7064, 20992}, {7322, 14547}, {9310, 40910}, {14523, 25067}, {15624, 18611}, {24250, 36544}, {24264, 36528}

X(41277) = invBGF(184)

Barycentrics    a^4*(a^6*b^2 - a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 - a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6) : :

X(41277) lies on these lines: {2, 98}, {32, 682}, {83, 19137}, {206, 1691}, {1078, 19126}, {1092, 13355}, {1147, 2456}, {1692, 3202}, {3203, 5034}, {5033, 40643}, {5116, 39832}, {7793, 19121}, {9292, 33632}, {9418, 40825}, {10104, 19131}, {12212, 19136}, {16285, 27374}

X(41278) = invBGF(187)

Barycentrics    a^4*(2*a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2 - c^4) : :

X(41278) lies on these lines: {2, 187}, {25, 34096}, {32, 184}, {39, 3060}, {51, 2021}, {263, 1692}, {571, 18371}, {574, 20965}, {1384, 1613}, {1915, 2080}, {2965, 9233}, {3053, 8601}, {3229, 35007}, {3552, 3978}, {3788, 20022}, {5017, 36213}, {5206, 14096}, {6179, 40858}, {6680, 33734}, {7747, 37988}, {7816, 20023}, {9489, 23610}, {10631, 11673}

leftri

2nd Lozada perspectors: X(41279)-X(41292)

rightri

This preamble is contributed by Vu Thanh Tung. (February 14, 2021)

As a sequel to the preamble just before X(6056), in which Lozada perspectors are introduced, this section is based on the following definition: Let A1B1C1 be the cevian triangle of a point P = p : q : r (barycentrics) in the plane of a triangle ABC. Let LA be the line, other than BC, that passes through A1 and is tangent to the A-excircle. Let A' be the touch point, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector, here named the 2nd Lozada perspector of P, denoted by V(P), is given by

V(P) = (a + b - c)(a - b + c) p^2 : (b + c - a)(b - c + a) q^2 : (c + a - b)(c - a + b) r^2.

See 2nd Lozada Perspector.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):

(1,56), (2,7), (3,7335), (5,41279), (6,1397), (7,479), (8,8), (9,55), (10,12), (31,41280), (32,41281), (36,41282), (75,6063), (76,41283), (83,41284), (141,41285), (560,41286), (561,41287), (1501,41288), (1502, 41289), (1928,41290), (2321,6057), (2887,41291), (3676,451292)

X(41279) = 2ND LOZADA PERSPECTOR OF X(5)

Barycentrics   (a+b-c)*(a-b+c)*(-(b^2-c^2)^2+a^2*(b^2+c^2))^2 : :

X(41279) lies on these lines: {56, 759}, {278, 7335}, {331, 1367}, {1361, 34434}, {2596, 10112}, {18180, 30493}

X(41279) = barycentric product X(i)*X(j) for these {i, j}: {7, 36412}, {57, 1087}, {324, 30493}, {1393, 14213}
X(41279) = barycentric quotient X(1087)/X(312)
X(41279) = trilinear product X(i)*X(j) for these {i, j}: {5, 1393}, {56, 1087}, {57, 36412}
X(41279) = trilinear quotient X(i)/X(j) for these (i, j): (1087, 8), (1393, 54)
X(41279) = X(1087)-Beth conjugate of-X(1087)
X(41279) = X(1087)-reciprocal conjugate of-X(312)

X(41280) = 2ND LOZADA PERSPECTOR OF X(31)

Barycentrics   a^6*(a+b-c)*(a-b+c) : :

X(41280) lies on these lines: {109, 697}, {560, 9448}, {1397, 2206}, {7122, 34396}, {17075, 41291}

X(41280) = isogonal conjugate of X(40363)
X(41280) = isotomic conjugate of the isogonal conjugate of X(41281)
X(41280) = barycentric product X(i)*X(j) for these {i, j}: {6, 1397}, {7, 1501}, {31, 604}, {32, 56}, {34, 9247}, {41, 1106}
X(41280) = barycentric quotient X(i)/X(j) for these (i, j): (7, 40362), (31, 28659), (32, 3596), (56, 1502), (57, 1928), (222, 40050)
X(41280) = trilinear product X(i)*X(j) for these {i, j}: {7, 1917}, {31, 1397}, {32, 604}, {34, 14575}, {56, 560}, {57, 1501}
X(41280) = trilinear quotient X(i)/X(j) for these (i, j): (6, 28659), (7, 1928), (31, 3596), (32, 312), (34, 18022), (56, 561)
X(41280) = crosssum of X(75) and X(21594)
X(41280) = X(560)-Beth conjugate of-X(560)
X(41280) = X(1917)-cross conjugate of-X(1501)
X(41280) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 40363}, {2, 28659}, {8, 561}, {9, 1502}, {10, 40072}
X(41280) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 40362), (31, 28659), (32, 3596), (56, 1502)
X(41280) = {X(560), X(14575)}-harmonic conjugate of X(9448)

X(41281) = 2ND LOZADA PERSPECTOR OF X(32)

Barycentrics   a^8*(a+b-c)*(a-b+c) : :

X(41281) lies on these lines: {1397, 7251}, {1917, 40373}

X(41281) = isogonal conjugate of the isotomic conjugate of X(41280)
X(41281) = isotomic conjugate of the isogonal conjugate of X(41286)
X(41281) = barycentric product X(i)*X(j) for these {i, j}: {6, 41280}, {7, 9233}, {32, 1397}, {56, 1501}, {57, 1917}, {76, 41286}
X(41281) = barycentric quotient X(i)/X(j) for these (i, j): (7, 40359), (32, 40363), (56, 40362), (222, 40360), (279, 41289), (560, 28659)
X(41281) = trilinear product X(i)*X(j) for these {i, j}: {31, 41280}, {34, 40373}, {56, 1917}, {57, 9233}, {75, 41286}, {560, 1397}
X(41281) = trilinear quotient X(i)/X(j) for these (i, j): (31, 40363), (32, 28659), (56, 1928), (57, 40362), (77, 40360), (85, 40359)
X(41281) = X(1917)-Beth conjugate of-X(1917)
X(41281) = X(i)-isoconjugate-of-X(j) for these {i, j}: {8, 1928}, {9, 40362}, {33, 40360}, {41, 40359}
X(41281) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 40359), (32, 40363), (56, 40362), (222, 40360)

X(41282) = 2ND LOZADA PERSPECTOR OF X(36)

Barycentrics   a^4*(a+b-c)*(a-b+c)*(a^2-b^2+b*c-c^2)^2 : :

X(41282) lies on these lines: {56, 3937}, {603, 2477}, {1357, 1408}, {1397, 1417}, {12903, 26933}

X(41282) = barycentric product X(i)*X(j) for these {i, j}: {215, 279}, {269, 34544}, {593, 3028}, {1262, 3025}, {1407, 4996}, {1443, 7113}
X(41282) = barycentric quotient X(i)/X(j) for these (i, j): (215, 346), (1106, 34535)
X(41282) = trilinear product X(i)*X(j) for these {i, j}: {215, 269}, {849, 3028}, {1106, 4996}, {1407, 34544}
X(41282) = trilinear quotient X(i)/X(j) for these (i, j): (215, 200), (1407, 34535), (1443, 20566), (1464, 15065)
X(41282) = X(849)-Beth conjugate of-X(1357)
X(41282) = X(346)-isoconjugate-of-X(34535)
X(41282) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (215, 346), (1106, 34535)

X(41283) = 2ND LOZADA PERSPECTOR OF X(76)

Barycentrics   ((a+b-c)*(a-b+c))/a^4 : :

X(41283) lies on these lines: {76, 40997}, {85, 21608}, {310, 15467}, {331, 6385}, {349, 1502}, {1928, 40050}, {3665, 6063}, {4572, 33298}, {7217, 41289}

X(41283) = isogonal conjugate of X(9448)
X(41283) = isotomic conjugate of X(2175)
X(41283) = barycentric product X(i)*X(j) for these {i, j}: {6, 41287}, {7, 1502}, {32, 41289}, {56, 40362}, {57, 1928}, {75, 20567}
X(41283) = barycentric quotient X(i)/X(j) for these (i, j): (1, 9447), (7, 32), (8, 14827), (12, 7109), (56, 1501), (57, 560)
X(41283) = trilinear product X(i)*X(j) for these {i, j}: {2, 20567}, {7, 561}, {31, 41287}, {34, 40050}, {56, 1928}, {57, 1502}
X(41283) = trilinear quotient X(i)/X(j) for these (i, j): (2, 9447), (7, 560), (56, 1917), (57, 1501), (76, 41), (77, 14575)
X(41283) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(3665)}} and {{A, B, C, X(8), X(40997)}}
X(41283) = cevapoint of X(i) and X(j) for these (i, j): {2, 21280}, {7, 17076}, {75, 20922}, {561, 20567}
X(41283) = X(1928)-Beth conjugate of-X(1928)
X(41283) = X(561)-cross conjugate of-X(1502)
X(41283) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 9448}, {6, 9447}, {8, 1917}, {9, 1501}, {32, 41}
X(41283) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 9447), (7, 32), (8, 14827), (12, 7109)
X(41283) = {X(1928), X(40050)}-harmonic conjugate of X(40363)

X(41284) = 2ND LOZADA PERSPECTOR OF X(83)

Barycentrics   (a^2+b^2)^2*(a+b-c)*(a-b+c)*(a^2+c^2)^2 : :

X(41284) lies on the line {7198, 17088}

X(41284) = barycentric quotient X(i)/X(j) for these (i, j): (7, 7794), (56, 8041), (82, 33299), (83, 3703), (251, 3688), (279, 41285)
X(41284) = trilinear quotient X(i)/X(j) for these (i, j): (57, 8041), (82, 3688), (83, 33299), (85, 7794), (251, 40972), (1088, 41285)
X(41284) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(7198)}} and {{A, B, C, X(56), X(7340)}}
X(41284) = X(i)-isoconjugate-of-X(j) for these {i, j}: {9, 8041}, {38, 3688}, {39, 33299}, {41, 7794}
X(41284) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 7794), (56, 8041), (82, 33299), (83, 3703)

X(41285) = 2ND LOZADA PERSPECTOR OF X(141)

Barycentrics   (a+b-c)*(a-b+c)*(b^2+c^2)^2 : :

X(41285) lies on these lines: {222, 1366}, {348, 1397}, {1365, 6063}, {1401, 3665}, {3323, 36482}, {4573, 17083}, {17625, 24798}, {21329, 35094}

X(41285) = barycentric product X(i)*X(j) for these {i, j}: {7, 7794}, {141, 3665}, {278, 4175}, {1401, 8024}, {2528, 4573}
X(41285) = barycentric quotient X(i)/X(j) for these (i, j): (279, 41284), (1401, 251), (2528, 3700)
X(41285) = trilinear product X(i)*X(j) for these {i, j}: {34, 4175}, {38, 3665}, {57, 7794}, {85, 8041}, {1401, 1930}, {1414, 2528}
X(41285) = trilinear quotient X(i)/X(j) for these (i, j): (1088, 41284), (2528, 4041)
X(41285) = X(1253)-isoconjugate-of-X(41284)
X(41285) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (279, 41284), (1401, 251)
X(41285) = {X(6063), X(7217)}-harmonic conjugate of X(1365)

X(41286) = 2ND LOZADA PERSPECTOR OF X(560)

Barycentrics   a^10*(a+b-c)*(a-b+c) : :

X(41286) lies on these lines: {}

X(41286) = isogonal conjugate of the isotomic conjugate of X(41281)
X(41286) = isotomic conjugate of the isogonal conjugate of X(41288)
X(41286) = barycentric product X(i)*X(j) for these {i, j}: {6, 41281}, {32, 41280}, {56, 9233}, {76, 41288}, {604, 1917}, {608, 40373}
X(41286) = barycentric quotient X(i)/X(j) for these (i, j): (56, 40359), (279, 41290), (1397, 40362), (1407, 41289), (1501, 40363), (1917, 28659)
X(41286) = trilinear product X(i)*X(j) for these {i, j}: {31, 41281}, {75, 41288}, {560, 41280}, {604, 9233}, {1395, 40373}, {1397, 1917}
X(41286) = trilinear quotient X(i)/X(j) for these (i, j): (57, 40359), (269, 41289), (560, 40363), (603, 40360), (604, 40362), (1088, 41290)
X(41286) = X(i)-isoconjugate-of-X(j) for these {i, j}: {9, 40359}, {200, 41289}, {312, 40362}, {318, 40360}
X(41286) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (56, 40359), (279, 41290), (1397, 40362), (1407, 41289)

X(41287) = 2ND LOZADA PERSPECTOR OF X(561)

Barycentrics   ((a+b-c)*(a-b+c))/a^6 : :

X(41287) lies on this line: {16888, 20567}

X(41287) = isogonal conjugate of the isotomic conjugate of X(41289)
X(41287) = isotomic conjugate of X(9448)
X(41287) = barycentric product X(i)*X(j) for these {i, j}: {6, 41289}, {7, 40362}, {32, 41290}, {56, 40359}, {76, 41283}, {85, 1928}
X(41287) = barycentric quotient X(i)/X(j) for these (i, j): (7, 1501), (56, 9233), (57, 1917), (75, 9447), (76, 2175), (85, 560)
X(41287) = trilinear product X(i)*X(j) for these {i, j}: {7, 1928}, {31, 41289}, {34, 40360}, {57, 40362}, {75, 41283}, {76, 20567}
X(41287) = trilinear quotient X(i)/X(j) for these (i, j): (7, 1917), (57, 9233), (76, 9447), (77, 40373), (85, 1501), (269, 41281)
X(41287) = X(1928)-cross conjugate of-X(40362)
X(41287) = X(i)-isoconjugate-of-X(j) for these {i, j}: {9, 9233}, {32, 9447}, {33, 40373}, {41, 1501}
X(41287) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 1501), (56, 9233), (57, 1917), (75, 9447)

X(41288) = 2ND LOZADA PERSPECTOR OF X(1501)

Barycentrics   a^12*(a+b-c)*(a-b+c) : :

X(41288) lies on these lines: {}

X(41288) = isogonal conjugate of the isotomic conjugate of X(41286)
X(41288) = barycentric product X(i)*X(j) for these {i, j}: {6, 41286}, {32, 41281}, {1397, 9233}, {1501, 41280}
X(41288) = barycentric quotient X(i)/X(j) for these (i, j): (1397, 40359), (1407, 41290)
X(41288) = trilinear product X(i)*X(j) for these {i, j}: {31, 41286}, {560, 41281}, {1917, 41280}
X(41288) = trilinear quotient X(i)/X(j) for these (i, j): (269, 41290), (604, 40359), (1106, 41289), (1917, 40363)
X(41288) = X(i)-isoconjugate-of-X(j) for these {i, j}: {200, 41290}, {312, 40359}, {341, 41289}, {1928, 40363}
X(41288) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1397, 40359), (1407, 41290)

X(41289) = 2ND LOZADA PERSPECTOR OF X(1502)

Barycentrics   ((a+b-c)*(a-b+c))/a^8 : :

X(41289) lies on this line: {7217, 41283}

X(41289) = isogonal conjugate of the isotomic conjugate of X(41290)
X(41289) = isotomic conjugate of the isogonal conjugate of X(41287)
X(41289) = barycentric product X(i)*X(j) for these {i, j}: {6, 41290}, {7, 40359}, {76, 41287}, {331, 40360}, {1502, 41283}, {1928, 20567}
X(41289) = barycentric quotient X(i)/X(j) for these (i, j): (7, 9233), (76, 9448), (85, 1917), (279, 41281), (348, 40373), (561, 9447)
X(41289) = trilinear product X(i)*X(j) for these {i, j}: {31, 41290}, {57, 40359}, {75, 41287}, {85, 40362}, {273, 40360}, {561, 41283}
X(41289) = trilinear quotient X(i)/X(j) for these (i, j): (85, 9233), (269, 41286), (561, 9448), (1088, 41281), (1106, 41288), (1502, 9447)
X(41289) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 9233}, {200, 41286}, {341, 41288}, {560, 9448}
X(41289) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 9233), (76, 9448), (85, 1917), (279, 41281)

X(41290) = 2ND LOZADA PERSPECTOR OF X(1928)

Barycentrics   ((a+b-c)*(a-b+c))/a^10 : :

X(41290) lies on these lines: {}

X(41290) = isotomic conjugate of the isogonal conjugate of X(41289)
X(41290) = barycentric product X(i)*X(j) for these {i, j}: {76, 41289}, {1502, 41287}
X(41290) = barycentric quotient X(i)/X(j) for these (i, j): (279, 41286), (1407, 41288), (1502, 9448), (1928, 9447)
X(41290) = trilinear product X(i)*X(j) for these {i, j}: {75, 41289}, {85, 40359}, {561, 41287}, {1928, 41283}
X(41290) = trilinear quotient X(i)/X(j) for these (i, j): (269, 41288), (1088, 41286), (1928, 9448)
X(41290) = X(i)-isoconjugate-of-X(j) for these {i, j}: {200, 41288}, {1253, 41286}
X(41290) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (279, 41286), (1407, 41288), (1502, 9448)

X(41291) = 2ND LOZADA PERSPECTOR OF X(2887)

Barycentrics   (a+b-c)*(a-b+c)*(b+c)^2*(b^2-b*c+c^2)^2 : :

X(41291) lies on these lines: {12835, 17086}, {17075, 41280}

X(41292) = 2ND LOZADA PERSPECTOR OF X(3676)

Barycentrics   (b-c)^2*(a+b-c)^3*(a-b+c)^3 : :

X(41292) lies on these lines: {479, 4626}, {6046, 7233}

X(41292) = barycentric product X(i)*X(j) for these {i, j}: {244, 23062}, {279, 1358}, {479, 1086}, {738, 1111}, {764, 36838}
X(41292) = barycentric quotient X(i)/X(j) for these (i, j): (244, 728), (279, 4076), (479, 1016), (738, 765), (764, 4130), (1015, 480)
X(41292) = trilinear product X(i)*X(j) for these {i, j}: {244, 479}, {269, 1358}, {738, 1086}, {764, 4626}, {1015, 23062}, {1088, 1357}
X(41292) = trilinear quotient X(i)/X(j) for these (i, j): (244, 480), (269, 6065), (479, 765), (738, 1252), (764, 4105), (1015, 6602)
X(41292) = X(i)-isoconjugate-of-X(j) for these {i, j}: {200, 6065}, {341, 6066}, {480, 765}, {728, 1252}
X(41292) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (244, 728), (279, 4076), (479, 1016), (738, 765)

X(41293) = X(2)X(37893)∩X(32)X(682)

Barycentrics    a^6*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 + a^2*c^2 - b^2*c^2) : :

X(41293) lies on the cubic K1182 and these lines: {2, 37893}, {32, 682}, {9917, 20806}, {11325, 38834}, {19118, 23163}

X(41293) = X(i)-isoconjugate of X(j) for these (i,j): {304, 40162}, {305, 18832}, {1928, 3504}, {2998, 40364}, {3223, 40050}, {34248, 40360}
X(41293) = barycentric product X(i)*X(j) for these {i,j}: {32, 11325}, {112, 9491}, {1501, 3186}, {1613, 1974}, {20794, 36417}, {23503, 32676}, {27369, 38834}
X(41293) = barycentric quotient X(i)/X(j) for these {i,j}: {194, 40360}, {1613, 40050}, {1974, 40162}, {3186, 40362}, {9233, 3504}, {9491, 3267}, {11325, 1502}

X(41294) = X(2)X(4590)∩X(32)X(1084)

Barycentrics    a^6*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :

X(41294) lies on the cubic K1182 and these lines: {2, 4590}, {32, 1084}, {1501, 9426}, {1645, 33875}

X(41294) = X(i)-isoconjugate of X(j) for these (i,j): {561, 14608}, {14210, 34087}
X(41294) = barycentric product X(i)*X(j) for these {i,j}: {32, 14609}, {111, 33875}, {538, 19626}, {691, 887}, {888, 32729}, {3231, 32740}
X(41294) = barycentric quotient X(i)/X(j) for these {i,j}: {887, 35522}, {1501, 14608}, {14609, 1502}, {19626, 3228}, {32729, 886}, {32740, 34087}, {33875, 3266}, {33918, 33919}

X(41295) = X(2)X(40425)∩X(6)X(22)

Barycentrics    a^2*(a^2 + b^2)*(a^2 + c^2)*(a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :

>X(41295) lies on the cubic K1182 and these lines: {2, 40425}, {6, 22}, {32, 733}, {83, 7761}, {1627, 35222}, {1799, 16986}, {3053, 9481}, {3314, 40850}, {4577, 7766}, {4628, 21793}, {5007, 14247}, {5309, 38946}, {7822, 40003}, {14885, 30435}

X(41295) = barycentric product X(i)*X(j) for these {i,j}: {83, 12212}, {251, 3329}, {4577, 14318}
X(41295) = barycentric quotient X(i)/X(j) for these {i,j}: {3329, 8024}, {12212, 141}, {14318, 826}
X(41295) = {X(1799),X(40000)}-harmonic conjugate of X(16986)

X(41296) = X(2)X(9233)∩X(83)X(316)

Barycentrics    (a^2 + b^2)*(a^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2) : :

X(41296) lies on the cubic K1183 and these lines: {2, 9233}, {83, 316}, {95, 325}, {251, 3329}, {290, 308}, {460, 32085}, {1502, 5157}, {5012, 33769}, {7931, 10130}, {14617, 40425}, {16987, 39668}, {18092, 33301}

X(41296) = X(3203)-cross conjugate of X(251)
X(41296) = X(i)-isoconjugate of X(j) for these (i,j): {38, 27375}, {1964, 3613}, {2084, 11794}
X(41296) = cevapoint of X(1078) and X(5012)
X(41296) = barycentric product X(i)*X(j) for these {i,j}: {82, 33764}, {83, 1078}, {251, 33769}, {308, 5012}, {689, 3050}, {1799, 36794}, {3112, 18042}, {4577, 31296}
X(41296) = barycentric quotient X(i)/X(j) for these {i,j}: {83, 3613}, {251, 27375}, {1078, 141}, {1629, 27376}, {1799, 36952}, {3050, 3005}, {4577, 11794}, {5012, 39}, {7668, 39691}, {10312, 1843}, {18042, 38}, {30506, 27371}, {31296, 826}, {33764, 1930}, {33769, 8024}, {36794, 427}
X(41296) = {X(308),X(1176)}-harmonic conjugate of X(4577)

X(41297) = X(6)X(35530)∩X(83)X(1207)

Barycentrics    b^2*c^2*(a^2 + b^2)*(a^2 + c^2)*(a^4 + a^2*b^2 + a^2*c^2 - b^2*c^2) : :

X(41297) lies on the cubic K1183 and these lines: {6, 35530}, {83, 1207}, {251, 689}, {308, 3108}, {8753, 32581}, {14603, 34482}

X(41297) = X(1923)-isoconjugate of X(6664)
X(41297) = barycentric product X(i)*X(j) for these {i,j}: {308, 7760}, {1627, 40016}, {3112, 18064}, {4593, 20953}, {18833, 33760}
X(41297) = barycentric quotient X(i)/X(j) for these {i,j}: {308, 6664}, {1627, 3051}, {7760, 39}, {8711, 2531}, {18064, 38}, {20953, 8061}, {21006, 688}, {33760, 1964}

X(41298) = ISOTOMIC CONJUGATE OF X(930)

Barycentrics    (b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4) : :
X(41298) = 4 X[647] - 3 X[9979], 2 X[669] - 3 X[9131], 3 X[850] - 4 X[3265], 2 X[850] - 3 X[3268], 5 X[850] - 6 X[30474], 8 X[3265] - 9 X[3268], 2 X[3265] - 3 X[6563], 10 X[3265] - 9 X[30474], 3 X[3268] - 4 X[6563], 5 X[3268] - 4 X[30474], 5 X[6563] - 3 X[30474], 8 X[8651] - 9 X[9123], 6 X[14417] - 5 X[31072], 2 X[33294] - 3 X[36900]

X(41298) lies on these lines: {2, 12077}, {99, 1291}, {325, 523}, {339, 24977}, {476, 18020}, {647, 9979}, {648, 36830}, {669, 9131}, {1993, 2623}, {2799, 31296}, {3804, 8704}, {6333, 7927}, {6368, 23061}, {7192, 21273}, {8651, 9123}, {10562, 14977}, {11450, 39469}, {14417, 31072}, {14920, 18558}, {31299, 32473}, {33294, 36900}, {36901, 38987}

X(41298) = anticomplement of X(12077)
X(41298) = reflection of X(850) in X(6563)
X(41298) = isogonal conjugate of X(32737)
X(41298) = isotomic conjugate of X(930)
X(41298) = intersection of trilinear polars of X(302) and X(303)
X(41298) = anticomplement of the isogonal conjugate of X(18315)
X(41298) = isotomic conjugate of the anticomplement of X(137)
X(41298) = isotomic conjugate of the complement of X(11671)
X(41298) = isotomic conjugate of the isogonal conjugate of X(1510)
X(41298) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {54, 21221}, {95, 21294}, {163, 17035}, {662, 2888}, {933, 5905}, {2148, 148}, {2167, 3448}, {2169, 39352}, {14586, 192}, {14587, 4560}, {15958, 6360}, {16813, 5906}, {18315, 8}, {18831, 21270}, {35196, 37781}, {36134, 2}
X(41298) = X(137)-cross conjugate of X(2)
X(41298) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32737}, {6, 36148}, {31, 930}, {163, 2963}, {1576, 2962}, {3519, 32676}, {9247, 38342}
X(41298) = cevapoint of X(2) and X(11671)
X(41298) = crosspoint of X(i) and X(j) for these (i,j): {99, 40410}, {264, 18831}
X(41298) = crosssum of X(i) and X(j) for these (i,j): {184, 15451}, {512, 13366}
X(41298) = crossdifference of every pair of points on line {32, 8565}
X(41298) = barycentric product X(i)*X(j) for these {i,j}: {76, 1510}, {95, 20577}, {302, 23873}, {303, 23872}, {523, 7769}, {525, 32002}, {850, 1994}, {1273, 2413}, {2964, 20948}, {3267, 3518}, {3268, 30529}, {15415, 25044}, {18027, 37084}
X(41298) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36148}, {2, 930}, {6, 32737}, {49, 32661}, {61, 16807}, {62, 16806}, {137, 12077}, {143, 1625}, {264, 38342}, {302, 32037}, {303, 32036}, {523, 2963}, {525, 3519}, {850, 11140}, {1154, 2439}, {1493, 35324}, {1510, 6}, {1577, 2962}, {1994, 110}, {2413, 1141}, {2964, 163}, {2965, 1576}, {3459, 39419}, {3518, 112}, {7769, 99}, {14129, 35360}, {14618, 93}, {15412, 252}, {18314, 25043}, {20577, 5}, {23872, 18}, {23873, 17}, {24978, 19552}, {25044, 14586}, {30529, 476}, {32002, 648}, {36978, 35331}, {36980, 35332}, {37084, 577}, {39183, 1487}
X(41298) = {X(850),X(6563)}-harmonic conjugate of X(3268)

X(41299) = X(514)X(661)∩X(850)X(17496)

Barycentrics    b*c*(b - c)*(-a + b + c)*(a^3 + a^2*b + a^2*c + 3*a*b*c + b^2*c + b*c^2) : :

X(41299) lies on these lines: {514, 661}, {850, 17496}, {905, 24622}, {3261, 3669}, {3910, 18155}, {18071, 29162}, {23684, 29037}, {23880, 35519}

X(41299) = isotomic conjugate of trilinear pole of line X(1)X(181)
X(41299) = X(40827)-Ceva conjugate of X(34387)
X(41299) = barycentric product X(35519)*X(37607)
X(41299) = barycentric quotient X(37607)/X(109)

X(41300) = MONTESDEOCA HOMOTHETIC CENTER

Barycentrics    (b^2-c^2)(3 a^4 - 3 a^2 (b^2 + c^2)- 2 b^2 c^2) : :

X(41300) is the homothetic center of the triangles AbBcCa and AcBaCb defined at PU(196) in Bicentric Pairs of Points.

X(41300) lies on these lines: {2,647}, {523,8651}, {550,30209}, {3005,32472}, {3629,8675}, {3631,9030}, {3804,9147}, {3906,32450}, {7950,14316}, {9404,19750}

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Points associated with Steiner parabolas: X(41301)-X(41308)

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This preamble and centers X(41301)-X(41308) were contributed by César Eliud Lozada, February 21, 2021.

Let ABC be a triangle and P a point on its plane. Let Ab, Ac be the points at which the parallel line to BC through P cuts AC and AB, respectively and denote Bc, Ba, Ca, Cb cyclically. It is well known that these six points lie on a conic and also that this conic is a parabola SP(P) if and only if P lies on the Steiner inellipse of ABC. This parabola is here named the Steiner parabola of P.

The following algebraic method for calculating the focus F(P) of the Steiner parabola SP(P) follows from the preambles just before X(31644) and X(38017):

  1. Let X = 3*PG/5, where G=X(2), and let X' = barycentric product of P and X.
  2. Let Y = X(2)-Ceva conjugate-of-P. Since P lies on the Steiner inellipse, Y lies on the line at infinity. Let A'B'C' be the orthic triangle of the cevian triangle of Y. Then A'B'C' and ABC are perspective. Denote their perspector by Y'.
  3. The focus F(P) is the midpoint of X' and Y'.
  4. The axis of SP(P) passes through Y.

X(41301) = FOCUS OF THE STEINER PARABOLA OF X(115)

Barycentrics    (a^18-4*(b^2+c^2)*a^16+4*(2*b^4+3*b^2*c^2+2*c^4)*a^14-2*(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^12+2*(4*b^8+4*c^8+9*(b^4+b^2*c^2+c^4)*b^2*c^2)*a^10-4*(b^2+c^2)*(b^8+c^8+b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*a^8+4*(3*b^8+3*c^8-b^2*c^2*(3*b^4-7*b^2*c^2+3*c^4))*b^2*c^2*a^6+(b^2+c^2)*(b^12+c^12-(8*b^8+8*c^8-b^2*c^2*(17*b^4-24*b^2*c^2+17*c^4))*b^2*c^2)*a^4+(b^16+c^16-(9*b^12+9*c^12-(35*b^8+35*c^8-b^2*c^2*(73*b^4-93*b^2*c^2+73*c^4))*b^2*c^2)*b^2*c^2)*a^2-(b^2+c^2)*(b^2-c^2)^8)*(b^2-c^2) : :

X(41301) lies on these lines: {523, 41302}, {12064, 31644}

X(41301) = midpoint of X(12064) and X(31644)

X(41302) = VERTEX OF THE STEINER PARABOLA OF X(115)

Barycentrics    (2*a^12-5*(b^2+c^2)*a^10+5*(b^2+c^2)^2*a^8-2*(2*b^4+b^2*c^2+7*c^4)*b^2*a^6+(b^8-3*c^8+2*b^2*c^2*(b^4+5*c^4))*a^4+(3*b^10-c^10-2*(7*b^6-5*c^6-13*(b^2-c^2)*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^2*(2*b^2-c^2)*(b^6+2*c^6-b^2*c^2*(b^2+c^2)))*(2*a^12-5*(b^2+c^2)*a^10+5*(b^2+c^2)^2*a^8-2*(7*b^4+b^2*c^2+2*c^4)*c^2*a^6-(3*b^8-c^8-2*b^2*c^2*(5*b^4+c^4))*a^4-(b^10-3*c^10-2*(5*b^6-7*c^6-13*(b^2-c^2)*b^2*c^2)*b^2*c^2)*a^2+(b^2-c^2)^2*(b^2-2*c^2)*(2*b^6+c^6-b^2*c^2*(b^2+c^2)))*(b^2-c^2) : :

X(41302) lies on these lines: {523, 41301}, {12065, 31644}

X(41302) = midpoint of X(12065) and X(31644)

X(41303) = FOCUS OF THE STEINER PARABOLA OF X(1015)

Barycentrics    a^2*((b^6+c^6-(7*b^4+7*c^4-b*c*(17*b^2-23*b*c+17*c^2))*b*c)*a^8-(b+c)*(b^6+c^6-(8*b^4+8*c^4-b*c*(17*b^2-24*b*c+17*c^2))*b*c)*a^7-(b^8+c^8-2*(4*b^6+4*c^6-(20*b^4+20*c^4-b*c*(34*b^2-49*b*c+34*c^2))*b*c)*b*c)*a^6+(b+c)*(b^8+c^8-2*(4*b^6+4*c^6-b*c*(4*b^2-5*b*c+4*c^2)*(4*b^2-b*c+4*c^2))*b*c)*a^5-2*(4*b^4+4*c^4+9*b*c*(b^2+b*c+c^2))*b^3*c^3*a^4+2*(b+c)*(5*b^2+4*b*c+5*c^2)*b^4*c^4*a^3-4*(2*b^2+3*b*c+2*c^2)*b^5*c^5*a^2+4*(b+c)*b^6*c^6*a-b^7*c^7)*(b-c) : :

X(41303) lies on the line {31645, 38018}

X(41303) = midpoint of X(31645) and X(38018)

X(41304) = VERTEX OF THE STEINER PARABOLA OF X(1015)

Barycentrics    a*(b-c)*((3*b^3-c^3-6*b*c*(b-c))*a^6-(3*b^4+8*b*c^3-c^4)*a^5-(3*b^5-c^5-(18*b^3-6*c^3-b*c*(21*b-31*c))*b*c)*a^4+(3*b^6-c^6-(8*b^4-8*c^4-b*c*(3*b^2-8*b*c-17*c^2))*b*c)*a^3-(b+c)*(2*b^4+2*c^4-b*c*(b+2*c)*(2*b+c))*b*c*a^2+2*(b^4-3*b^2*c^2+c^4)*b^2*c^2*a-(b^2-c^2)*(b-c)*b^3*c^3)*((b^3-3*c^3-6*b*c*(b-c))*a^6-(b^4-8*b^3*c-3*c^4)*a^5-(b^5-3*c^5-(6*b^3-18*c^3-b*c*(31*b-21*c))*b*c)*a^4+(b^6-3*c^6-(8*b^4-8*c^4-b*c*(17*b^2+8*b*c-3*c^2))*b*c)*a^3+(b+c)*(2*b^4+2*c^4-b*c*(b+2*c)*(2*b+c))*b*c*a^2-2*(b^4-3*b^2*c^2+c^4)*b^2*c^2*a+(b^2-c^2)*(b-c)*b^3*c^3) : :

X(41304) lies on the line {31645, 38242}

X(41304) = midpoint of X(31645) and X(38242)

X(41305) = FOCUS OF THE STEINER PARABOLA OF X(1084)

Barycentrics    a^2*((b^6-c^6-(b^4+c^4+b*c*(3*b^2-b*c-3*c^2))*b*c)*(b^6-c^6+(b^4+c^4-b*c*(3*b^2+b*c-3*c^2))*b*c)*a^16-(b^2+c^2)*(b^12+c^12-(8*b^8+8*c^8-b^2*c^2*(17*b^4-24*b^2*c^2+17*c^4))*b^2*c^2)*a^14+(b^16+c^16-2*(4*b^12+4*c^12-b^2*c^2*(2*b^2-c^2)*(b^2-2*c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2))*b^2*c^2)*a^12+4*(b^2+c^2)*(b^8+c^8+b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*b^4*c^4*a^10-2*(4*b^8+4*c^8+9*(b^4+b^2*c^2+c^4)*b^2*c^2)*b^6*c^6*a^8+2*(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*b^8*c^8*a^6-4*(2*b^4+3*b^2*c^2+2*c^4)*b^10*c^10*a^4+4*(b^2+c^2)*b^12*c^12*a^2-b^14*c^14)*(b^2-c^2) : :

X(41305) lies on these lines: {512, 41306}, {31646, 38017}

X(41305) = midpoint of X(31646) and X(38017)

X(41306) = VERTEX OF THE STEINER PARABOLA OF X(1084)

Barycentrics    a^2*((b^6-3*c^6-6*(b^2-c^2)*b^2*c^2)*a^10+(5*b^6+4*c^6+b^2*c^2*(4*b^2-3*c^2))*c^2*a^8+(b^10-3*c^10-(11*b^6-5*c^6-4*b^2*c^2*(2*b^2-5*c^2))*b^2*c^2)*a^6+2*(b^8+c^8+2*b^2*c^2*(b^2+c^2)^2)*b^2*c^2*a^4-(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*b^4*c^4*a^2+b^6*c^6*(b^4+c^4))*((3*b^6-c^6-6*(b^2-c^2)*b^2*c^2)*a^10-(4*b^6+5*c^6-b^2*c^2*(3*b^2-4*c^2))*b^2*a^8+(3*b^10-c^10-(5*b^6-11*c^6-4*b^2*c^2*(5*b^2-2*c^2))*b^2*c^2)*a^6-2*(b^8+c^8+2*b^2*c^2*(b^2+c^2)^2)*b^2*c^2*a^4+(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*b^4*c^4*a^2-b^6*c^6*(b^4+c^4))*(b^2-c^2) : :

X(41306) lies on these lines: {512, 41305}, {31646, 38241}

X(41306) = midpoint of X(31646) and X(38241)

X(41307) = FOCUS OF THE STEINER PARABOLA OF X(1086)

Barycentrics    (a^10-4*(b+c)*a^9+4*(2*b^2+3*b*c+2*c^2)*a^8-2*(b+c)*(5*b^2+4*b*c+5*c^2)*a^7+2*(4*b^4+4*c^4+9*b*c*(b^2+b*c+c^2))*a^6-4*(b+c)*(b^4+c^4+b*c*(2*b^2+b*c+2*c^2))*a^5+4*(3*b^4+3*c^4-b*c*(3*b^2-7*b*c+3*c^2))*b*c*a^4+(b+c)*(b^6+c^6-(8*b^4+8*c^4-b*c*(17*b^2-24*b*c+17*c^2))*b*c)*a^3-(b^6+c^6-(7*b^4+7*c^4-b*c*(17*b^2-23*b*c+17*c^2))*b*c)*b*c*a^2+(b^2-c^2)*(b-c)^7*a-(b^2+b*c+c^2)*(b-c)^8)*(b-c) : :

X(41307) lies on these lines: {514, 41308}, {31647, 38019}

X(41307) = midpoint of X(31647) and X(38019)

X(41308) = VERTEX OF THE STEINER PARABOLA OF X(1086)

Barycentrics    (b-c)*(2*a^7-5*(b+c)*a^6+2*(3*b^2+4*b*c+3*c^2)*a^5-(3*b^3+7*c^3+b*c*(11*b-c))*a^4+(b^4+5*c^4+4*b^2*c*(2*b-c))*a^3-(2*b^5+2*c^5-(b^3-3*c^3-4*b*c*(b-2*c))*b*c)*a^2-(b^2-c^2)*(b-c)*(b^3-3*c^3-b*c*(7*b-5*c))*a+(2*b^5-2*c^5-(5*b^3-3*c^3-2*b*c*(b-c))*b*c)*(b-c)^2)*(2*a^7-5*(b+c)*a^6+2*(3*b^2+4*b*c+3*c^2)*a^5-(7*b^3+3*c^3-b*c*(b-11*c))*a^4+(5*b^4+c^4-4*b*c^2*(b-2*c))*a^3-(2*b^5+2*c^5+(3*b^3-c^3-4*b*c*(2*b-c))*b*c)*a^2+(b^2-c^2)*(b-c)*(3*b^3-c^3-b*c*(5*b-7*c))*a-(2*b^5-2*c^5-(3*b^3-5*c^3-2*b*c*(b-c))*b*c)*(b-c)^2) : :

X(41308) lies on this line: {514, 41307}

X(41309) = X(2)X(5970)∩X(6)X(669)

Barycentrics    a^2*(2*a^2 - b^2 - c^2)*(2*a^2*b^2 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - 2*a^2*c^2 + b^2*c^2) : :

X(41309) lies on the cubic K1182 and these lines: {2, 5970}, {6, 669}, {32, 110}, {187, 5468}, {1383, 3228}, {4576, 33756}, {5467, 14567}, {9233, 23357}, {11003, 32717}

X(41309) = isogonal conjugate of the isotomic conjugate of X(14608)
X(41309) = X(i)-isoconjugate of X(j) for these (i,j): {75, 14609}, {538, 897}, {671, 2234}, {923, 30736}, {1928, 41294}, {9148, 36085}, {23342, 23894}
X(41309) = crossdifference of every pair of points on line {538, 9148}
X(41309) = barycentric product X(i)*X(j) for these {i,j}: {6, 14608}, {187, 3228}, {351, 9150}, {524, 729}, {690, 32717}, {896, 37132}, {2642, 36133}, {14567, 34087}
X(41309) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 14609}, {187, 538}, {351, 9148}, {524, 30736}, {729, 671}, {922, 2234}, {3228, 18023}, {5467, 23342}, {9233, 41294}, {14567, 3231}, {14608, 76}, {16702, 30938}, {32717, 892}

X(41310) = X(2)X(37)∩X(6)X(29573)

Barycentrics    2*a^2 - 3*a*b + 2*b^2 - 3*a*c + 2*c^2 : :
X(41310) = 2 X[2] + X[4908], 2 X[10] + X[4702], X[44] + 2 X[3912], X[44] - 4 X[4422], 2 X[44] + X[17374], X[190] + 2 X[3834], X[190] + 5 X[17266], 2 X[239] + X[4727], X[239] - 4 X[6687], X[320] + 5 X[4473], X[1086] + 2 X[2325], 2 X[1086] - 5 X[31243], 2 X[1125] + X[4439], X[1266] - 4 X[40480], X[1279] + 2 X[3932], 5 X[1698] + X[4693], 4 X[2325] + 5 X[31243], 2 X[3008] + X[3943], 2 X[3246] + X[32847], X[3685] + 2 X[3823], 2 X[3717] + X[4864], X[3797] + 5 X[4687], 2 X[3834] - 5 X[17266], X[3912] + 2 X[4422], 4 X[3912] - X[17374], 2 X[4370] + X[31138], X[4370] + 2 X[41141], 8 X[4422] + X[17374], X[4480] + 2 X[7238], X[4727] + 8 X[6687], 3 X[4908] + 2 X[37756], X[6541] + 2 X[31289], X[17160] - 7 X[29607], 3 X[17264] + X[37756], X[17297] + 3 X[41138], X[20358] + 2 X[40521], X[31138] - 4 X[41141]

X(41310) = lies on these lines: {2, 37}, {6, 29573}, {9, 599}, {10, 4702}, {44, 524}, {45, 17237}, {69, 15492}, {86, 29620}, {141, 16814}, {142, 17340}, {190, 3834}, {238, 28538}, {239, 4727}, {244, 4141}, {320, 4473}, {519, 1279}, {527, 4370}, {551, 4078}, {594, 6666}, {597, 1100}, {899, 4933}, {903, 28322}, {908, 16581}, {1001, 3679}, {1086, 2325}, {1125, 4439}, {1213, 25072}, {1266, 40480}, {1577, 4762}, {1698, 4693}, {1743, 15534}, {1992, 4851}, {2178, 21539}, {2321, 17337}, {2796, 3836}, {3008, 3943}, {3161, 17276}, {3246, 32847}, {3589, 3723}, {3683, 29687}, {3685, 3823}, {3717, 4864}, {3729, 17265}, {3731, 3763}, {3740, 33158}, {3742, 33164}, {3758, 29572}, {3828, 4026}, {3848, 33167}, {3879, 8584}, {3908, 8540}, {3950, 17366}, {3967, 29642}, {3973, 40341}, {4009, 29632}, {4029, 17395}, {4357, 20582}, {4364, 29596}, {4384, 17269}, {4387, 21949}, {4389, 29629}, {4416, 22165}, {4432, 28562}, {4480, 7238}, {4643, 21356}, {4670, 17244}, {4675, 29627}, {4690, 17230}, {4708, 17292}, {4715, 17297}, {4725, 17310}, {4852, 17242}, {4873, 17119}, {4883, 33166}, {4889, 17121}, {4891, 33118}, {4918, 25967}, {4971, 41140}, {5032, 29583}, {5087, 27759}, {6541, 31289}, {6745, 12035}, {7232, 25728}, {7611, 11231}, {7801, 25066}, {9025, 16482}, {13466, 35125}, {15254, 29674}, {15481, 33087}, {15533, 16885}, {15569, 33159}, {16590, 17251}, {16666, 17316}, {16668, 17390}, {16672, 29598}, {16675, 17306}, {16676, 17325}, {17118, 20195}, {17160, 29607}, {17229, 17268}, {17232, 17336}, {17233, 17338}, {17234, 17339}, {17235, 17261}, {17239, 17260}, {17240, 17349}, {17241, 17350}, {17245, 17355}, {17256, 29587}, {17259, 17286}, {17262, 17282}, {17275, 18230}, {17299, 37650}, {17330, 29594}, {17334, 21255}, {17346, 29577}, {17368, 28639}, {17369, 29571}, {17378, 29582}, {17392, 29600}, {18073, 27265}, {20358, 40521}, {21331, 35103}, {21689, 24086}, {24603, 31285}, {25536, 37792}, {26626, 39260}, {26774, 27036}, {27065, 31143}, {27751, 31171}, {28534, 31151}, {30566, 30823}, {31139, 38093}

X(41310) = complement of X(37756)
X(41310) = midpoint of X(2) and X(17264)
X(41310) = reflection of X(4908) in X(17264)
X(41310) = complement of the isotomic conjugate of X(34892)
X(41310) = X(i)-complementary conjugate of X(j) for these (i,j): {2748, 3835}, {34892, 2887}, {34893, 141}
X(41310) = crosspoint of X(2) and X(34892)
X(41310) = crosssum of X(6) and X(16784)
X(41310) = crossdifference of every pair of points on line {667, 3941}
X(41310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4664, 17382}, {2, 17281, 4688}, {2, 17342, 17359}, {9, 17231, 17344}, {9, 17267, 17231}, {37, 17279, 17357}, {37, 17357, 17384}, {44, 3912, 17374}, {45, 17284, 17237}, {141, 25101, 16814}, {190, 17266, 3834}, {192, 17341, 17356}, {344, 17279, 37}, {346, 17278, 4686}, {597, 17243, 29574}, {597, 29574, 1100}, {3912, 4422, 44}, {4029, 31191, 17395}, {4370, 41141, 31138}, {4687, 17358, 17385}, {4851, 26685, 16669}, {4873, 31183, 17119}, {17230, 17335, 4690}, {17232, 17336, 17345}, {17233, 17338, 17348}, {17234, 17339, 17351}, {17240, 17349, 17372}, {17241, 17350, 17376}, {17242, 17352, 4852}, {17243, 17353, 1100}, {17244, 17354, 4670}, {17260, 17285, 17239}, {17261, 17283, 17235}, {17263, 17280, 3739}, {17268, 17277, 17229}, {17285, 31333, 17260}, {17353, 29574, 597}, {17371, 27268, 25498}, {38312, 38313, 3290}

X(41311) = X(1)X(599)∩X(2)X(37)

Barycentrics    2*a^2 + 3*a*b + 2*b^2 + 3*a*c + 2*c^2 : :
X(41311) = X[594] + 2 X[4021], X[1100] + 2 X[4357], X[1100] - 4 X[17045], 2 X[1100] + X[17344], 5 X[3616] + X[24723], X[4357] + 2 X[17045], 4 X[4357] - X[17344], X[4360] + 2 X[17239], X[4360] + 5 X[17326], X[4464] + 2 X[4478], 2 X[5750] + X[17246], 8 X[17045] + X[17344], 2 X[17239] - 5 X[17326], 5 X[17326] - X[29615]

X(41311) lies on these lines: {1, 599}, {2, 37}, {10, 17395}, {36, 1001}, {44, 597}, {45, 29598}, {86, 16702}, {141, 3723}, {190, 29614}, {239, 4708}, {320, 29586}, {519, 3775}, {524, 1100}, {551, 752}, {553, 41003}, {594, 4021}, {744, 10180}, {894, 4912}, {903, 24358}, {1086, 1125}, {1104, 13745}, {1213, 3946}, {1266, 4472}, {1284, 4870}, {1449, 15534}, {1486, 31158}, {1698, 17119}, {1992, 4643}, {2178, 21509}, {3247, 3763}, {3589, 16814}, {3616, 4675}, {3618, 15492}, {3661, 4727}, {3662, 28639}, {3663, 17398}, {3679, 4716}, {3720, 39688}, {3828, 3932}, {3834, 16826}, {3875, 17327}, {3879, 22165}, {3912, 20582}, {3943, 29604}, {3986, 17337}, {4360, 17239}, {4363, 29603}, {4389, 4670}, {4393, 4690}, {4395, 24603}, {4410, 39995}, {4416, 8584}, {4464, 4478}, {4715, 17254}, {4725, 17271}, {4748, 17014}, {4758, 4887}, {4851, 21356}, {4852, 5224}, {4889, 17287}, {5249, 16581}, {5257, 17366}, {5266, 7810}, {5308, 26104}, {5750, 17132}, {6687, 29630}, {6707, 24199}, {7611, 11230}, {7801, 37592}, {8287, 17062}, {10436, 17323}, {15254, 29646}, {15533, 16884}, {15668, 17304}, {16669, 17257}, {16671, 17332}, {16672, 17284}, {16673, 17267}, {16705, 16726}, {16777, 17231}, {16831, 17290}, {16834, 17251}, {17011, 31143}, {17117, 28633}, {17160, 29610}, {17184, 37869}, {17227, 29570}, {17229, 17307}, {17234, 29622}, {17236, 17376}, {17238, 17372}, {17247, 17351}, {17248, 17348}, {17249, 17345}, {17276, 35578}, {17291, 29620}, {17297, 29580}, {17308, 17318}, {17313, 29597}, {17316, 39260}, {17329, 37677}, {18140, 25629}, {19883, 28542}, {22110, 24239}, {24338, 34363}, {25534, 27020}, {27191, 29578}, {27777, 29825}, {28558, 33682}, {29571, 31243}, {35089, 35119}

X(41311) = midpoint of X(i) and X(j) for these {i,j}: {2, 17320}, {4360, 29615}, {17271, 29584}
X(41311) = reflection of X(29615) in X(17239)
X(41311) = complement of the isotomic conjugate of X(34914)
X(41311) = X(i)-complementary conjugate of X(j) for these (i,j): {1919, 35135}, {8691, 3835}, {34914, 2887}, {34916, 141}, {35181, 21262}, {37210, 21260}
X(41311) = X(35181)-Ceva conjugate of X(513)
X(41311) = crosspoint of X(2) and X(34914)
X(41311) = crosssum of X(6) and X(16785)
X(41311) = crossdifference of every pair of points on line {667, 2515}
X(41311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17237, 17374}, {1, 17325, 17237}, {2, 4664, 17359}, {2, 17301, 4688}, {2, 17302, 37756}, {2, 17399, 17382}, {2, 37756, 3739}, {37, 4657, 17384}, {37, 17384, 17357}, {86, 17324, 17235}, {192, 17400, 17385}, {1100, 4357, 17344}, {3672, 17303, 4686}, {4357, 17045, 1100}, {4360, 17326, 17239}, {4364, 17023, 44}, {4389, 17397, 4670}, {4393, 17250, 4690}, {4395, 25358, 24603}, {4643, 26626, 16666}, {4657, 17321, 37}, {4664, 17359, 4908}, {4687, 17383, 17356}, {5224, 17396, 4852}, {16777, 17306, 17231}, {16777, 21358, 29573}, {16826, 17305, 3834}, {17236, 17394, 17376}, {17238, 17393, 17372}, {17247, 17381, 17351}, {17248, 17380, 17348}, {17249, 17379, 17345}, {17302, 17322, 3739}, {17306, 29573, 21358}, {17307, 17319, 17229}, {17318, 25503, 17308}, {17322, 37756, 2}, {21358, 29573, 17231}

X(41312) = X(1)X(524)∩X(2)X(37)

Barycentrics    a^2 + 3*a*b + b^2 + 3*a*c + c^2 : :
X(41312) = X[1] + 2 X[4364], 2 X[1] + X[4643], X[8] - 4 X[4708], 2 X[10] + X[17318], 2 X[37] + X[24357], X[145] + 2 X[4690], X[145] + 5 X[4748], 4 X[551] - X[4795], 2 X[551] + X[24441], 4 X[1125] - X[4363], 8 X[1125] - 5 X[4798], 5 X[1698] - 2 X[4665], 5 X[1698] - 8 X[25358], 5 X[3616] + X[4419], 5 X[3616] - 2 X[4670], 5 X[3616] - X[35578], 7 X[3622] - X[4644], 7 X[3624] - 4 X[4472], 7 X[3624] - X[4659], 4 X[3636] - X[4667], 2 X[4363] - 5 X[4798], 4 X[4364] - X[4643], X[4419] + 2 X[4670], 5 X[4470] - 11 X[5550], 4 X[4472] - X[4659], X[4665] - 4 X[25358], 5 X[4687] - 2 X[25384], 2 X[4690] - 5 X[4748], 4 X[4758] - 7 X[15808], X[4795] + 2 X[24441]

X(41312) lies on these lines: {1, 524}, {2, 37}, {7, 28639}, {8, 4708}, {9, 597}, {10, 17133}, {44, 26626}, {45, 17023}, {69, 3723}, {86, 17247}, {141, 3247}, {142, 17323}, {145, 4690}, {190, 17397}, {238, 38023}, {320, 29570}, {392, 34377}, {519, 9348}, {527, 551}, {534, 1486}, {545, 24358}, {599, 4357}, {966, 4852}, {980, 8610}, {1086, 16831}, {1100, 1992}, {1125, 4363}, {1213, 3875}, {1279, 4715}, {1449, 8584}, {1654, 17393}, {1698, 4665}, {2178, 16436}, {2321, 17327}, {3241, 4725}, {3589, 3731}, {3616, 4419}, {3618, 16814}, {3622, 4644}, {3624, 4472}, {3636, 4667}, {3653, 28464}, {3654, 31395}, {3656, 31394}, {3662, 29622}, {3663, 15668}, {3664, 17255}, {3679, 4026}, {3717, 38087}, {3720, 24691}, {3729, 17398}, {3758, 29586}, {3763, 16674}, {3828, 4078}, {3834, 5308}, {3879, 15533}, {3912, 16672}, {3932, 19875}, {3943, 17308}, {3945, 17345}, {3946, 3986}, {3950, 17293}, {3966, 6536}, {3973, 6329}, {4021, 4361}, {4029, 17269}, {4352, 25130}, {4360, 17248}, {4377, 18135}, {4384, 17395}, {4389, 4675}, {4393, 17256}, {4395, 16832}, {4416, 15534}, {4422, 16676}, {4440, 29592}, {4470, 5550}, {4648, 17235}, {4654, 41003}, {4758, 15808}, {4889, 32099}, {4910, 17362}, {5032, 16666}, {5224, 17299}, {5232, 17372}, {5296, 17348}, {5750, 17262}, {5794, 24319}, {5845, 38316}, {5886, 7611}, {5905, 37869}, {6542, 17250}, {6646, 17394}, {6707, 25590}, {7174, 9041}, {7618, 37599}, {8182, 37589}, {8680, 10180}, {9055, 38047}, {10022, 28297}, {10179, 34371}, {10186, 15726}, {10436, 17246}, {11160, 17344}, {11184, 24239}, {13161, 34505}, {13632, 20430}, {16581, 31019}, {16593, 38093}, {16667, 20583}, {16673, 17243}, {16675, 17353}, {16677, 25101}, {16705, 24652}, {16834, 17330}, {17119, 24603}, {17160, 29576}, {17227, 29569}, {17233, 17326}, {17234, 17324}, {17236, 17317}, {17237, 17316}, {17238, 17315}, {17239, 17314}, {17242, 17307}, {17244, 17305}, {17245, 17304}, {17249, 17300}, {17252, 17377}, {17254, 17378}, {17258, 17379}, {17260, 17380}, {17261, 17381}, {17270, 17388}, {17271, 17389}, {17272, 17390}, {17273, 17391}, {17274, 17392}, {17277, 17396}, {17290, 29571}, {17297, 31332}, {17329, 20090}, {17346, 29584}, {17354, 29614}, {17360, 29588}, {17369, 29603}, {17374, 29585}, {17781, 19722}, {19883, 28301}, {24424, 28628}, {24690, 29814}, {24699, 28562}, {24703, 29644}, {25349, 26102}, {25350, 25502}, {25354, 32921}, {26104, 29627}, {26580, 31179}, {27191, 29581}, {27759, 29657}, {27811, 33122}, {28633, 32087}, {31183, 31285}, {34511, 37592}, {34522, 40880}

X(41312) = midpoint of X(4419) and X(35578)
X(41312) = reflection of X(35578) in X(4670)
X(41312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4364, 4643}, {2, 4664, 17281}, {2, 17320, 17301}, {37, 4657, 17279}, {37, 17321, 4657}, {37, 17384, 344}, {86, 17247, 17276}, {145, 4748, 4690}, {192, 17322, 17303}, {551, 24441, 4795}, {599, 16777, 29574}, {599, 29574, 4851}, {1125, 4363, 4798}, {1213, 3875, 28634}, {3616, 4419, 4670}, {3624, 4659, 4472}, {3946, 3986, 17259}, {4021, 5257, 4361}, {4029, 29604, 17269}, {4357, 16777, 4851}, {4357, 29574, 599}, {4360, 17248, 17275}, {4360, 31144, 29617}, {4389, 16826, 4675}, {4665, 25358, 1698}, {4681, 25498, 2345}, {4687, 17302, 17278}, {4755, 17382, 2}, {5224, 17319, 17299}, {16672, 17325, 3912}, {16673, 17306, 17243}, {16676, 29598, 4422}, {17248, 29617, 31144}, {17254, 29580, 17378}, {17269, 25503, 29604}, {17274, 29597, 17392}, {17374, 39260, 29585}, {29617, 31144, 17275}

X(41313) = X(1)X(597)∩X(2)X(37)

Barycentrics    a^2 - 3*a*b + b^2 - 3*a*c + c^2 : :
X(41313) = 2 X[9] + X[4851], X[9] + 2 X[17243], 2 X[142] + X[17262], X[144] + 2 X[17376], X[1001] + 2 X[4078], 2 X[3950] + X[4361], X[3950] + 2 X[6666], X[4361] - 4 X[6666], X[4851] - 4 X[17243], X[6173] + 3 X[36911], 2 X[7263] - 5 X[20195], X[17314] + 2 X[17348], X[17314] + 5 X[18230], 2 X[17348] - 5 X[18230]

X(41313) = lies on these lines: {1, 597}, {2, 37}, {6, 25101}, {7, 4912}, {9, 524}, {10, 17269}, {43, 39688}, {44, 1992}, {45, 599}, {69, 16814}, {86, 17339}, {141, 3731}, {142, 17132}, {144, 17376}, {190, 4675}, {193, 15492}, {320, 29572}, {391, 17372}, {519, 1001}, {527, 17313}, {545, 6173}, {894, 29620}, {966, 17229}, {1100, 26685}, {1213, 17286}, {1279, 3241}, {1486, 4421}, {1654, 17240}, {1743, 8584}, {1836, 29854}, {2140, 22031}, {2178, 16431}, {2321, 17259}, {2325, 4363}, {2796, 5880}, {3008, 4029}, {3161, 4648}, {3247, 3589}, {3618, 3723}, {3629, 3973}, {3654, 31394}, {3656, 31395}, {3661, 31144}, {3663, 17265}, {3679, 3932}, {3686, 17309}, {3729, 17245}, {3748, 4952}, {3758, 4473}, {3763, 16677}, {3834, 4419}, {3875, 17337}, {3879, 15534}, {3943, 4384}, {3946, 4098}, {3950, 4361}, {3986, 17327}, {4026, 19875}, {4141, 30950}, {4357, 16675}, {4360, 17338}, {4364, 16676}, {4370, 4795}, {4377, 28809}, {4389, 17266}, {4395, 31183}, {4416, 15533}, {4437, 36404}, {4439, 24331}, {4644, 29621}, {4653, 30906}, {4659, 34824}, {4665, 4873}, {4667, 29606}, {4670, 5308}, {4679, 29643}, {4690, 29616}, {4708, 29611}, {4715, 6172}, {4725, 37654}, {4798, 16831}, {4852, 37650}, {4869, 17345}, {4884, 10582}, {4956, 33108}, {4969, 29605}, {4971, 16833}, {5032, 16669}, {5222, 6687}, {5224, 17268}, {5257, 17293}, {5296, 17239}, {5393, 13663}, {5405, 13783}, {5749, 28639}, {5750, 28640}, {6542, 17335}, {6646, 17241}, {6703, 25430}, {7263, 20195}, {7321, 25269}, {7611, 26446}, {9053, 38190}, {9055, 38186}, {10436, 17340}, {11160, 17374}, {13633, 20430}, {15254, 28538}, {15668, 17355}, {16503, 38088}, {16581, 31053}, {16666, 29585}, {16670, 20583}, {16672, 17023}, {16673, 17045}, {16777, 17353}, {16826, 17354}, {17056, 30568}, {17160, 29628}, {17230, 17256}, {17231, 17257}, {17232, 17258}, {17233, 17260}, {17234, 17261}, {17237, 29579}, {17242, 17277}, {17246, 17282}, {17247, 17283}, {17248, 17285}, {17250, 29587}, {17251, 29594}, {17255, 21255}, {17271, 29577}, {17294, 17330}, {17295, 17331}, {17296, 17332}, {17297, 17333}, {17298, 17334}, {17300, 17336}, {17305, 29629}, {17310, 17346}, {17312, 17347}, {17314, 17348}, {17315, 17349}, {17317, 17350}, {17319, 17352}, {17325, 29596}, {17365, 25728}, {17378, 29575}, {17387, 20072}, {24338, 36235}, {24441, 41141}, {24656, 27523}, {24703, 29653}, {25066, 34511}, {25349, 30822}, {26738, 30578}, {27777, 29640}, {28297, 38093}, {28530, 38052}, {28557, 38204}, {28581, 38057}, {28582, 38053}, {29016, 38108}, {31143, 32858}, {37589, 37809}

X(41313) = midpoint of X(9) and X(29573)
X(41313) = reflection of X(i) in X(j) for these {i,j}: {4851, 29573}, {17313, 29600}, {29573, 17243}
X(41313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 192, 37756}, {2, 4664, 17301}, {2, 17264, 17281}, {2, 37756, 17278}, {9, 17243, 4851}, {37, 344, 17279}, {37, 17279, 4657}, {37, 17357, 17321}, {45, 3912, 4643}, {190, 17244, 4675}, {192, 17263, 17278}, {2321, 17259, 28634}, {2321, 25072, 17259}, {2325, 29571, 4363}, {3008, 4029, 17318}, {3161, 4648, 17351}, {3950, 6666, 4361}, {4360, 31333, 17338}, {4419, 29627, 3834}, {4473, 29569, 3758}, {4665, 31285, 16832}, {4687, 17280, 17303}, {4755, 17359, 2}, {4873, 16832, 4665}, {16675, 17267, 4357}, {16676, 17284, 4364}, {16831, 17369, 4798}, {17233, 17260, 17275}, {17234, 17261, 17276}, {17242, 17277, 17299}, {17263, 37756, 2}, {17314, 18230, 17348}, {17333, 29582, 17297}, {20072, 29589, 17387}

X(41314) = X(2)X(37)∩X(100)X(9067)

Barycentrics    b*c*(-a + b)*(a - c)*(a*b + a*c - 2*b*c) : :
X(41314) = X[2] + 2 X[8031], X[1646] + 3 X[8031], X[1646] - 3 X[36847]

X(41314) lies on the cubic K015 and these lines: {2, 37}, {100, 9067}, {190, 24623}, {334, 4080}, {668, 891}, {874, 7035}, {4009, 35543}, {4583, 6548}, {4601, 5468}, {4937, 6381}, {7283, 37018}, {14433, 23891}, {17154, 18149}, {20345, 30578}

X(41314) = midpoint of X(8031) and X(36847)
X(41314) = reflection of X(2) in X(36847)
X(41314) = isogonal conjugate of X(23349)
X(41314) = anticomplement of X(1646)
X(41314) = anticomplement of the isogonal conjugate of X(5381)
X(41314) = isotomic conjugate of the anticomplement of X(14434)
X(41314) = isotomic conjugate of the isogonal conjugate of X(23343)
X(41314) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {765, 39360}, {889, 150}, {898, 4440}, {4607, 149}, {5381, 8}, {32718, 21224}, {34075, 9263}
X(41314) = X(i)-Ceva conjugate of X(j) for these (i,j): {889, 668}, {31625, 13466}
X(41314) = X(i)-cross conjugate of X(j) for these (i,j): {891, 536}, {13466, 31625}, {14426, 899}, {14434, 2}, {14441, 9295}
X(41314) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23349}, {6, 23892}, {244, 32718}, {649, 739}, {667, 37129}, {898, 3248}, {1015, 34075}, {1919, 3227}, {1977, 4607}, {1980, 31002}, {2206, 35353}, {3249, 5381}
X(41314) = cevapoint of X(i) and X(j) for these (i,j): {536, 891}, {3994, 4728}, {8031, 14434}
X(41314) = crosspoint of X(668) and X(889)
X(41314) = crosssum of X(667) and X(890)
X(41314) = trilinear pole of line {536, 6381}
X(41314) = crossdifference of every pair of points on line {667, 1977}
X(41314) = barycentric product X(i)*X(j) for these {i,j}: {75, 23891}, {76, 23343}, {100, 35543}, {190, 6381}, {536, 668}, {799, 3994}, {889, 13466}, {891, 31625}, {899, 1978}, {3230, 6386}, {4009, 4554}, {4465, 4583}, {4601, 14431}, {4728, 7035}
X(41314) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23892}, {6, 23349}, {100, 739}, {190, 37129}, {321, 35353}, {536, 513}, {646, 36798}, {668, 3227}, {765, 34075}, {890, 1977}, {891, 1015}, {899, 649}, {1016, 898}, {1252, 32718}, {1646, 8027}, {1978, 31002}, {3230, 667}, {3768, 3248}, {3994, 661}, {4009, 650}, {4465, 659}, {4526, 3271}, {4706, 4790}, {4728, 244}, {4937, 4893}, {6381, 514}, {7035, 4607}, {8031, 14434}, {13466, 891}, {14404, 3121}, {14426, 6377}, {14430, 2170}, {14431, 3125}, {14433, 27846}, {14434, 1646}, {19945, 21143}, {23343, 6}, {23891, 1}, {24004, 36872}, {30583, 2087}, {31625, 889}, {35543, 693}, {36816, 1027}, {36847, 14474}, {39011, 33917}
X(41314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {874, 7035, 17780}, {1978, 36863, 3952}, {4601, 17935, 5468}

X(41315) = X(2)X(37)∩X(646)X(21433)

Barycentrics    b*c*(a - b)*(a - c)*(a^2*b - 3*a*b^2 + a^2*c + 2*b^2*c - 3*a*c^2 + 2*b*c^2) : :

X(41315) lies on the cubic K090 and these lines: {2, 37}, {646, 21433}, {3888, 3952}

X(41315) = isotomic conjugate of X(23834)
X(41315) = isotomic conjugate of the isogonal conjugate of X(23830)
X(41315) = X(i)-isoconjugate of X(j) for these (i,j): {31, 23834}, {649, 28583}
X(41315) = barycentric product X(i)*X(j) for these {i,j}: {76, 23830}, {668, 28582}
X(41315) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 23834}, {100, 28583}, {23830, 6}, {28582, 513}

X(41316) = X(2)X(37)∩X(4)X(6335)

Barycentrics    b*c*(-3*a^3 - a*b^2 - 2*a*b*c + 2*b^2*c - a*c^2 + 2*b*c^2) : :

X(41316) lies on the cubi K295 and these lines: {2, 37}, {4, 6335}, {7, 18044}, {144, 18133}, {145, 4033}, {190, 18135}, {193, 30473}, {194, 26076}, {313, 5749}, {668, 1992}, {889, 34343}, {1269, 7229}, {3264, 5222}, {3596, 3618}, {3664, 18065}, {3882, 29400}, {3945, 18040}, {4266, 29511}, {4454, 39995}, {4470, 20913}, {4494, 17023}, {9965, 18136}, {17354, 28809}, {18739, 21454}, {29616, 30939}

X(41316) = X(649)-isoconjugate of X(28474)
X(41316) = barycentric product X(668)*X(28475)
X(41316) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 28474}, {28475, 513}
X(41316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18147, 29423, 346}

X(41317) = X(2)X(37)∩X(100)X(187)

Barycentrics    a*(a^3*b^2 + a*b^4 - a^3*b*c + a^2*b^2*c + 2*a*b^3*c - 2*b^4*c + a^3*c^2 + a^2*b*c^2 - a*b^2*c^2 - 2*b^3*c^2 + 2*a*b*c^3 - 2*b^2*c^3 + a*c^4 - 2*b*c^4) : :

X(41337) lies on the cubic K792 and these lines: {2, 37}, {100, 187}, {352, 3908}, {750, 16785}, {1054, 5525}, {16611, 33115}, {16975, 33089}, {26278, 28503}

X(41317) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(31087)

X(41318) = X(1)X(7033)∩X(2)X(37)

Barycentrics    b*c*(a^2 + b*c)*(a*b + a*c - b*c) : :

X(41318) lies on the cubic K992 and these lines: {1, 7033}, {2, 37}, {10, 7018}, {42, 1978}, {43, 6382}, {210, 25280}, {226, 334}, {325, 20487}, {354, 18149}, {561, 32931}, {668, 4090}, {726, 34020}, {799, 32938}, {899, 40087}, {1215, 1237}, {1934, 4518}, {1965, 27064}, {1966, 7081}, {2329, 27963}, {3009, 32926}, {3112, 32944}, {3212, 6376}, {3229, 25264}, {3971, 23824}, {4514, 20352}, {4903, 18135}, {6384, 24349}, {7196, 7211}, {10009, 16569}, {16606, 19565}, {17149, 19586}, {18064, 32927}, {20340, 24210}, {20449, 40493}, {20965, 21762}, {21327, 26974}, {21352, 32942}, {21412, 27324}, {21590, 33930}, {21780, 24343}, {25123, 35544}, {29679, 30632}, {30964, 32925}, {32918, 33764}, {35538, 37678}, {39080, 39929}

X(41318) = X(894)-Ceva conjugate of X(1909)
X(41318) = X(i)-isoconjugate of X(j) for these (i,j): {32, 27447}, {87, 904}, {256, 7121}, {330, 7104}, {893, 2162}, {1178, 23493}, {1431, 2053}, {1967, 34252}, {9468, 39914}, {21759, 40432}, {40736, 40738}
X(41318) = crosspoint of X(894) and X(17752)
X(41318) = barycentric product X(i)*X(j) for these {i,j}: {37, 27891}, {43, 1920}, {75, 17752}, {171, 6382}, {192, 1909}, {894, 6376}, {1215, 31008}, {1237, 27644}, {1966, 40848}, {1978, 24533}, {2533, 36860}, {3208, 7205}, {3212, 17787}, {3963, 33296}, {3971, 8033}, {4110, 7176}, {4369, 36863}, {4374, 4595}, {4554, 30584}, {7081, 30545}, {7122, 40367}, {7196, 27538}, {18047, 20906}
X(41318) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 893}, {75, 27447}, {171, 2162}, {172, 7121}, {192, 256}, {385, 34252}, {894, 87}, {1215, 16606}, {1423, 1431}, {1909, 330}, {1920, 6384}, {1966, 39914}, {2176, 904}, {2209, 7104}, {2295, 23493}, {2329, 2053}, {3212, 1432}, {3955, 15373}, {4110, 4451}, {4579, 34071}, {4595, 3903}, {6376, 257}, {6382, 7018}, {7081, 2319}, {7176, 7153}, {7205, 7209}, {7304, 7303}, {17741, 3494}, {17752, 1}, {17787, 7155}, {18047, 932}, {20760, 7116}, {20964, 21759}, {21021, 7148}, {21803, 6378}, {22061, 22381}, {22370, 7015}, {24533, 649}, {27644, 1178}, {27891, 274}, {30545, 7249}, {30584, 650}, {31008, 32010}, {33296, 40432}, {33890, 3865}, {36860, 4594}, {36863, 27805}, {40848, 1581}
X(41318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 192, 20284}, {1215, 1920, 1909}, {3978, 40790, 1909}, {31008, 36863, 3971}

X(41319) = X(1)X(187)∩X(4)X(9)

Barycentrics    a*(2*a^3 + a^2*b - a*b^2 - 2*b^3 + a^2*c - 3*a*b*c + b^2*c - a*c^2 + b*c^2 - 2*c^3) : :

X(41319) lies on the cubic K751 and these lines: {1, 187}, {4, 9}, {63, 8591}, {484, 6205}, {986, 1100}, {1572, 17596}, {1697, 36643}, {1759, 3208}, {1761, 3169}, {2329, 12702}, {3061, 3579}, {3125, 8616}, {3218, 4393}, {3245, 16788}, {3306, 17397}, {3509, 5119}, {3550, 3735}, {3727, 37603}, {3899, 35342}, {3916, 4051}, {4165, 4450}, {5255, 16777}, {9441, 34522}, {9778, 24247}, {16503, 36279}, {16558, 24578}, {17736, 37563}, {37567, 41239}

X(41319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 3496, 3501}, {1276, 1277, 6210}, {1759, 11010, 3208}, {6212, 6213, 7609}

X(41320) = X(4)X(9)∩X(24)X(101)

Barycentrics    a^2*(a - b - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c + a*b*c - c^3) : :

X(41320) lies on the cubic K620 and these lines: {4, 9}, {24, 101}, {25, 220}, {33, 1334}, {34, 672}, {35, 32756}, {55, 607}, {65, 5089}, {213, 3192}, {218, 11398}, {219, 1474}, {232, 2176}, {235, 17747}, {378, 24047}, {403, 24045}, {406, 3294}, {475, 16549}, {595, 8743}, {995, 39575}, {1039, 40779}, {1212, 1829}, {1398, 5022}, {1452, 40131}, {1724, 1783}, {1782, 8558}, {1802, 1973}, {1870, 4253}, {1902, 21872}, {1905, 16601}, {1968, 17735}, {2207, 8750}, {2266, 5280}, {3052, 3172}, {3101, 31015}, {3207, 3515}, {4207, 40586}, {5125, 5190}, {5338, 8012}, {6603, 11363}, {10319, 14021}, {11383, 32561}, {11396, 34522}, {16290, 40937}, {16970, 36103}, {20367, 37382}, {20613, 37550}, {28348, 35072}, {32674, 37579}

X(41320) = polar conjugate of X(15467)
X(41320) = polar conjugate of the isotomic conjugate of X(3190)
X(41320) = X(i)-Ceva conjugate of X(j) for these (i,j): {1252, 8750}, {8748, 33}
X(41320) = X(i)-isoconjugate of X(j) for these (i,j): {48, 15467}, {77, 1751}, {81, 28786}, {222, 2997}, {272, 1214}, {348, 2218}, {603, 40011}, {905, 1305}, {1014, 40161}
X(41320) = crosssum of X(1565) and X(4091)
X(41320) = crossdifference of every pair of points on line {1459, 17094}
X(41320) = barycentric product X(i)*X(j) for these {i,j}: {4, 3190}, {19, 27396}, {29, 209}, {33, 3868}, {55, 5125}, {281, 579}, {318, 2352}, {607, 18134}, {1172, 22021}, {1252, 5190}, {1897, 8676}, {2198, 31623}, {4306, 7046}, {8750, 20294}
X(41320) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 15467}, {33, 2997}, {42, 28786}, {209, 307}, {281, 40011}, {579, 348}, {607, 1751}, {1334, 40161}, {2198, 1214}, {2212, 2218}, {2299, 272}, {2352, 77}, {3190, 69}, {3868, 7182}, {4306, 7056}, {5125, 6063}, {5190, 23989}, {8676, 4025}, {8750, 1305}, {22021, 1231}, {27396, 304}
X(41320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 7719, 19}, {55, 607, 2332}, {2207, 14974, 8750}

X(41321) = X(4)X(9)∩X(101)X(26705)

Barycentrics    (a - b)*(a - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) : :

X(41321) lies on the cubic K406 and these lines: {4, 9}, {101, 26705}, {651, 653}, {877, 17934}, {927, 40116}, {1292, 40117}, {2398, 4241}, {4246, 17926}, {4250, 28132}, {4557, 6135}, {7115, 23987}, {26704, 28847}

X(41321) = polar conjugate of X(2400)
X(41321) = polar conjugate of the isotomic conjugate of X(2398)
X(41321) = polar conjugate of the isogonal conjugate of X(2426)
X(41321) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {32701, 149}, {36109, 150}
X(41321) = X(2426)-cross conjugate of X(2398)
X(41321) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2400}, {63, 2424}, {103, 905}, {513, 1815}, {514, 36056}, {677, 3942}, {693, 32657}, {906, 15634}, {911, 4025}, {1459, 36101}, {1565, 36039}, {2968, 32668}, {4091, 36122}, {18025, 22383}, {24016, 34591}
X(41321) = crosssum of X(20752) and X(22346)
X(41321) = trilinear pole of line {1886, 17747}
X(41321) = crossdifference of every pair of points on line {1459, 3270}
X(41321) = pole wrt polar circle of trilinear polar of X(2400) (line X(116)X(514), the Simson line of X(103))
X(41321) = barycentric product X(i)*X(j) for these {i,j}: {4, 2398}, {10, 4241}, {190, 1886}, {264, 2426}, {516, 1897}, {648, 17747}, {653, 40869}, {676, 15742}, {677, 21665}, {910, 6335}, {1783, 30807}, {7046, 23973}, {7079, 24015}, {8750, 35517}
X(41321) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2400}, {25, 2424}, {101, 1815}, {516, 4025}, {676, 1565}, {692, 36056}, {910, 905}, {1783, 36101}, {1886, 514}, {1897, 18025}, {2398, 69}, {2426, 3}, {3234, 26006}, {4241, 86}, {7649, 15634}, {8750, 103}, {14953, 15419}, {17747, 525}, {23972, 39470}, {23973, 7056}, {26006, 30805}, {30807, 15413}, {32699, 15380}, {32739, 32657}, {40869, 6332}

X(41322) = X(1)X(574)∩X(4)X(9)

Barycentrics    a*(a^3 + 2*a^2*b - 2*a*b^2 - b^3 + 2*a^2*c - 3*a*b*c + 2*b^2*c - 2*a*c^2 + 2*b*c^2 - c^3) : :

X(41322) lies on the cubic K751 and these lines: {1, 574}, {4, 9}, {36, 4919}, {46, 3208}, {57, 29574}, {63, 29615}, {165, 24264}, {214, 39041}, {484, 1018}, {728, 36643}, {986, 16777}, {1046, 20691}, {1054, 3230}, {1100, 5255}, {1757, 5184}, {2329, 3579}, {2802, 5030}, {2938, 17792}, {3061, 12702}, {3218, 6542}, {3306, 16826}, {3550, 9620}, {3693, 5183}, {5080, 21013}, {5119, 17754}, {5195, 24318}, {5524, 21839}, {5541, 24578}, {6603, 9356}, {9593, 16667}, {11010, 16549}, {13587, 17439}, {14974, 24440}, {17207, 33770}, {17735, 21888}, {31398, 33106}, {31443, 37617}, {37568, 41239}

X(41322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 3501, 3496}, {484, 1018, 3509}, {1276, 1277, 6211}

X(41323) = X(4)X(9)∩X(101)X(511)

Barycentrics    a^2*(a^3*b - b^4 + a^3*c - a^2*b*c + b^2*c^2 - c^4) : :

X(41323) lies on the cubic K289 and these lines: {4, 9}, {23, 32739}, {101, 511}, {386, 20861}, {514, 19565}, {910, 15310}, {1308, 35106}, {1400, 5018}, {1756, 18785}, {1916, 5991}, {2702, 17735}, {3294, 5195}, {3509, 17770}, {3563, 35182}, {4251, 21746}, {4253, 5144}, {5074, 17248}, {5088, 16827}, {5280, 23659}, {7291, 20373}, {10446, 27547}, {14963, 20733}, {17364, 17736}, {17368, 27300}, {17798, 20670}, {20470, 24484}, {20672, 37510}, {21813, 38832}, {22197, 30362}

X(41323) =reflection of X(2702) in X(20666)
X(41323) = symgonal image of X(20666)
X(41323) =X(35162)-Ceva conjugate of X(42)
X(41323) =crossdifference of every pair of points on line {1459, 2309}

X(41324) = X(2)X(17735)∩X(11)X(37686)

Barycentrics    a^3*b - b^4 + a^3*c - a^2*b*c + b^2*c^2 - c^4 : :

X(41324) lies on these lines: {2, 17735}, {11, 37686}, {76, 24045}, {92, 264}, {99, 5134}, {101, 316}, {115, 40859}, {190, 325}, {220, 7773}, {305, 24054}, {350, 4766}, {595, 7828}, {995, 7790}, {1191, 7851}, {2176, 5025}, {2295, 17669}, {2886, 17277}, {3261, 3835}, {3266, 24055}, {3314, 4713}, {3570, 20553}, {3730, 7752}, {3759, 33141}, {3846, 17289}, {4071, 20947}, {4109, 17762}, {4153, 33939}, {4568, 24079}, {4595, 17757}, {4645, 20531}, {4687, 33111}, {5074, 20924}, {5080, 18047}, {5224, 20541}, {5254, 34063}, {6542, 6543}, {6655, 21008}, {7112, 20454}, {7763, 17732}, {7769, 24047}, {7806, 21793}, {7887, 14974}, {9166, 37854}, {10025, 17347}, {11813, 18061}, {14061, 17734}, {17234, 20335}, {17381, 25496}, {17737, 33295}, {17976, 20558}, {20159, 26098}, {20179, 33106}, {20255, 33837}, {20337, 27272}, {21241, 41248}, {26590, 37678}, {30941, 31058}, {33940, 40690}, {35144, 35171}

X(41324) = cevapoint of X(306) and X(20500)
X(41324) = {X(325),X(17747)}-harmonic conjugate of X(190)

X(41325) = X(4)X(9)∩X(7)X(37)

Barycentrics    a^4 + 4*a^3*b - 4*a^2*b^2 - b^4 + 4*a^3*c - 4*a^2*b*c - 4*a^2*c^2 + 2*b^2*c^2 - c^4 : :

X(41325) lies on the cubic K6166 and these lines: {2, 17747}, {4, 9}, {6, 390}, {7, 37}, {8, 21872}, {20, 220}, {44, 5838}, {69, 144}, {101, 376}, {142, 21068}, {165, 40869}, {198, 1633}, {218, 4294}, {329, 3693}, {347, 39063}, {348, 27129}, {388, 1334}, {443, 3294}, {497, 672}, {518, 17314}, {527, 29573}, {528, 37654}, {594, 5686}, {631, 24047}, {673, 37650}, {728, 12527}, {910, 9778}, {954, 5746}, {962, 1212}, {971, 3781}, {1018, 3421}, {1058, 4253}, {1100, 8236}, {1155, 40127}, {1213, 40333}, {1249, 8750}, {1268, 17354}, {1449, 30331}, {1818, 2324}, {1901, 8232}, {2176, 7738}, {2267, 16503}, {2276, 21856}, {2280, 10385}, {2321, 5223}, {3052, 5304}, {3090, 24045}, {3161, 4645}, {3177, 26790}, {3207, 3522}, {3247, 5542}, {3474, 40131}, {3553, 7675}, {3598, 24352}, {3684, 34607}, {3731, 4312}, {3779, 14100}, {3950, 5850}, {4000, 16970}, {4295, 16601}, {4302, 5526}, {4309, 17745}, {4515, 5815}, {4676, 5749}, {5022, 14986}, {5082, 16552}, {5084, 16549}, {5257, 38052}, {5286, 14974}, {5296, 24723}, {5731, 6603}, {5839, 5853}, {6172, 17281}, {6184, 36698}, {6817, 40586}, {7672, 21853}, {7676, 36744}, {7677, 36743}, {7735, 17735}, {7991, 41006}, {8732, 37500}, {9785, 40133}, {10509, 34820}, {10624, 16572}, {11038, 16777}, {11111, 16788}, {11415, 25082}, {12329, 28071}, {15254, 26039}, {15984, 29181}, {16369, 24364}, {16672, 30340}, {16676, 30424}, {17300, 20059}, {17303, 18230}, {17754, 26105}, {17784, 37658}, {20672, 37416}, {21153, 40942}, {21801, 24484}, {27508, 37499}

X(41325) = reflection of X(5819) in X(9)
X(41325) = barycentric product X(10)*X(4229)
X(41325) = barycentric quotient X(4229)/X(86)
X(41325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 2550, 966}, {144, 20533, 69}, {3730, 17732, 4}

X(41326) = X(4)X(9)∩X(101)X(550)

Barycentrics    a^4 + 3*a^3*b - 3*a^2*b^2 - b^4 + 3*a^3*c - 3*a^2*b*c - 3*a^2*c^2 + 2*b^2*c^2 - c^4 : :

X(41326) lies on the cubic K618 and these lines: {4, 9}, {101, 550}, {140, 17747}, {190, 7768}, {220, 1657}, {514, 26790}, {672, 4857}, {1334, 5270}, {1656, 24045}, {7755, 17735}, {16549, 37162}, {17310, 31015}, {17483, 22002}, {27186, 29578}

X(41326) = {X(3730),X(17732)}-harmonic conjugate of X(5134)

X(41327) = X(4)X(9)∩X(5)X(38857)

Barycentrics    a^6 - a^5*b + a^3*b^3 - b^6 - a^5*c + a^4*b*c + a^3*b^2*c - a^2*b^3*c + a^3*b*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + a^3*c^3 - a^2*b*c^3 + b^2*c^4 - c^6 : :

X(41327) lies on the cubic K288 and these lines: {4, 9}, {5, 38857}, {98, 35182}, {101, 1503}, {514, 16086}, {542, 17976}, {858, 1331}, {1060, 5820}, {1213, 38868}, {1260, 1853}, {1899, 3190}, {1944, 29307}, {2328, 21015}, {3685, 20502}, {5018, 5293}, {5285, 21912}, {12588, 41276}, {16415, 24931}, {17734, 39026}

X(41327) = reflection of X(242) in X(31897)
X(41327) = polar-circle-inverse of X(1842)

X(41328) = X(2)X(160)∩X(3)X(6)

Barycentrics    a^2*(b^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2) : :

X(41328) lies on the cubic K975 and these lines: {2, 160}, {3, 6}, {22, 11174}, {35, 18170}, {83, 39968}, {95, 290}, {99, 308}, {141, 1634}, {183, 1232}, {237, 3589}, {384, 18092}, {385, 15246}, {394, 15648}, {524, 22062}, {597, 40981}, {599, 20794}, {1078, 33769}, {1176, 1576}, {1624, 13394}, {2071, 32224}, {2531, 39495}, {2916, 6660}, {2963, 38224}, {3051, 31613}, {3148, 33801}, {3329, 6636}, {3613, 14957}, {3618, 37184}, {3815, 7467}, {3867, 27369}, {3917, 16030}, {4045, 21177}, {4074, 23210}, {4184, 37686}, {4210, 37678}, {4218, 4996}, {4366, 8053}, {5010, 18194}, {7484, 8891}, {7495, 23181}, {7496, 9149}, {7499, 23195}, {7503, 20792}, {7509, 39646}, {7834, 11360}, {8362, 23208}, {8667, 19568}, {9418, 22352}, {9722, 37451}, {14533, 17974}, {15270, 16043}, {17710, 23635}, {17798, 23374}, {18311, 37895}, {19189, 37124}, {19308, 20148}, {23333, 41237}, {31355, 32990}, {31521, 37344}, {34888, 40000}, {36422, 38737}

X(41328) = isogonal conjugate of X(30505)
X(41328) = isotomic conjugate of the isogonal conjugate of X(3203)
X(41328) = isogonal conjugate of the polar conjugate of X(37125)
X(41328) = X(i)-Ceva conjugate of X(j) for these (i,j): {83, 3051}, {95, 141}, {99, 31296}, {5012, 3203}, {39968, 6}
X(41328) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30505}, {82, 3613}, {3112, 27375}
X(41328) = cevapoint of X(39) and X(34452)
X(41328) = crosspoint of X(1078) and X(5012)
X(41328) = crosssum of X(i) and X(j) for these (i,j): {512, 34294}, {3613, 27375}
X(41328) = Brocard-circle-inverse of X(8266)
X(41328) = barycentric product X(i)*X(j) for these {i,j}: {3, 37125}, {38, 18042}, {39, 1078}, {76, 3203}, {141, 5012}, {1634, 31296}, {1923, 33778}, {1964, 33764}, {3050, 4576}, {3051, 33769}, {3917, 36794}, {3933, 10312}, {8041, 41296}, {27370, 34386}
X(41328) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 30505}, {39, 3613}, {1078, 308}, {1634, 11794}, {3051, 27375}, {3203, 6}, {3917, 36952}, {5012, 83}, {10312, 32085}, {18042, 3112}, {27370, 53}, {33764, 18833}, {33769, 40016}, {37125, 264}
X(41328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 8266}, {6, 8266, 5201}, {141, 20775, 1634}, {237, 3589, 35222}, {570, 11574, 3001}, {577, 37479, 5157}, {11574, 13334, 570}, {14096, 20775, 141}

X(41329) = X(3)X(6)∩X(8)X(79)

Barycentrics    a^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^3*b*c + 2*a^2*b^2*c - b^4*c + a^3*c^2 + 2*a^2*b*c^2 + a*b^2*c^2 + a^2*c^3 - a*c^4 - b*c^4 - c^5) : :
X(41329) = 3 X[3017] - 2 X[18178], 3 X[3794] - 4 X[6693]

X(41329) lies on the cubic K680 and these lines: {3, 6}, {8, 79}, {171, 31737}, {199, 501}, {209, 1782}, {238, 31757}, {484, 1046}, {540, 17579}, {994, 5794}, {1125, 3792}, {1654, 3678}, {1724, 3060}, {1754, 5889}, {2099, 11553}, {2476, 3454}, {2915, 17104}, {2979, 37522}, {3017, 18178}, {3145, 5127}, {3240, 37572}, {3754, 26051}, {3794, 6693}, {3811, 3882}, {3868, 3909}, {3874, 17778}, {3878, 26117}, {3936, 35637}, {3954, 20708}, {4653, 22076}, {4658, 40952}, {5051, 18417}, {5259, 20961}, {5266, 9047}, {5429, 37616}, {5496, 30362}, {5692, 26064}, {5902, 26131}, {6224, 20040}, {6830, 7683}, {10381, 17532}, {10544, 34471}, {11412, 37530}, {15556, 37098}, {16471, 33586}, {18180, 24880}, {18389, 26054}, {18398, 37635}, {20077, 37256}, {21853, 40263}, {23156, 32913}, {31732, 37570}

X(41329) = reflection of X(58) in X(10974)
X(41329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {500, 2245, 4278}, {4259, 5752, 386}

X(41330) = X(3)X(6)∩X(5)X(6787)

Barycentrics    a^2*(a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10 - a^8*b^2*c^2 + 3*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 4*a^2*b^8*c^2 - b^10*c^2 + a^8*c^4 + 3*a^6*b^2*c^4 + a^4*b^4*c^4 - 2*a^2*b^6*c^4 + 2*b^8*c^4 - 3*a^6*c^6 - 5*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 2*b^6*c^6 + 3*a^4*c^8 + 4*a^2*b^2*c^8 + 2*b^4*c^8 - a^2*c^10 - b^2*c^10) : :
X(41330) = X[3] + 2 X[31850], 4 X[5] - X[18321], 2 X[187] + X[18322], 5 X[1656] - 2 X[31848], 2 X[6786] - 3 X[15561], X[9301] + 2 X[14962], 8 X[11554] - 5 X[37481], X[11674] - 4 X[14693], 2 X[14881] + X[38527], X[35002] - 4 X[35060]

Let P13 and P14 be the orthogonal projections of X(13) and X(14) on the Brocard axis, resp. Then X(41330) = {P13,P14}-harmonic conjugate of X(6). (Randy Hutson, April 13, 2021)

X(41330) lies on the cubic K508 and these lines: {3, 6}, {5, 6787}, {13, 25151}, {14, 25161}, {30, 6785}, {83, 13363}, {98, 5663}, {381, 512}, {476, 1316}, {1078, 15067}, {1656, 31848}, {2854, 12177}, {4226, 10788}, {5182, 14984}, {5475, 23017}, {5476, 14186}, {5946, 12150}, {6784, 11632}, {6786, 15561}, {6800, 21525}, {7706, 38953}, {8724, 34383}, {9151, 33330}, {9218, 15033}, {10104, 11459}, {10358, 18307}, {11674, 14693}, {11676, 13207}, {13137, 35930}, {14675, 16223}, {14880, 15072}, {14881, 38527}, {16279, 20403}, {18583, 36157}

X(41330) = midpoint of X(i) and X(j) for these {i,j}: {3111, 31850}, {11676, 13207}
X(41330) = reflection of X(i) in X(j) for these {i,j}: {3, 3111}, {568, 15544}, {6787, 5}, {11632, 6784}, {15536, 13363}, {18321, 6787}
X(41330) = X(6787)-of-Johnson-triangle
X(41330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1687, 1688, 32761}

X(41331) = X(3)X(6)∩X(99)X(6579)

Barycentrics    a^6*(b^2 + c^2) : :
Trilinears    sin^4 A sin(A + ω) : :
Trilinears    sin A sin(A + ω) sin(A - ω) : :
Trilinears    (sin A) (1 - sec^2 ω cos^2 A) : :

Let circles (O)A, (O)B, (O)C be as defined at X(8160). Let TA be the touch point of the outer Montesdeoca-Lemoine circle and (O)A, and define TB and TC cyclically. Let T'A be the touch point of the inner Montesdeoca-Lemoine circle and (O)A, and define T'B and T'C cyclically. The lines TAT'A, TBT'B, TCT'C concur in X(39). Let P1 be the trilinear product TA*TB*TC. Let P2 be the trilinear product T'A*T'B*T'C. X(41331) is the {P1,P2}-harmonic conjugate of X(32). (Randy Hutson, April 13, 2021)

Let E be the inellipse that is the barycentric square of the Brocard axis. Then X(41331) is the intersection of the tangents to E at the barycentric squares of X(32) and X(39). (Randy Hutson, April 13, 2021)

X(41331) lies on the cubic K1034 and these lines: {3, 6}, {99, 6579}, {110, 703}, {141, 8623}, {160, 3117}, {183, 8891}, {237, 8265}, {251, 3329}, {308, 384}, {385, 1627}, {560, 14598}, {670, 19585}, {682, 31355}, {1084, 36425}, {1501, 9233}, {1576, 9468}, {1914, 18170}, {2237, 20234}, {3051, 20775}, {3552, 8264}, {4577, 9497}, {6375, 35222}, {7031, 18194}, {7804, 18092}, {8023, 23200}, {8041, 22078}, {11174, 15822}, {14614, 19568}

X(41331) = isogonal conjugate of X(40016)
X(41331) = isogonal conjugate of the isotomic conjugate of X(3051)
X(41331) = isogonal conjugate of the polar conjugate of X(27369)
X(41331) = X(i)-Ceva conjugate of X(j) for these (i,j): {32, 3051}, {251, 3203}, {1576, 9426}, {3108, 3202}, {9468, 9418}
X(41331) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40016}, {2, 18833}, {75, 308}, {76, 3112}, {82, 1502}, {83, 561}, {251, 1928}, {523, 37204}, {670, 18070}, {689, 1577}, {850, 4593}, {1799, 1969}, {1926, 14970}, {3115, 20627}, {3405, 18024}, {4577, 20948}, {6385, 18082}, {6386, 10566}, {18022, 34055}, {18090, 38812}, {18097, 40072}, {20889, 31622}, {30505, 33778}, {32085, 40364}, {37221, 40074}
X(41331) = crosspoint of X(i) and X(j) for these (i,j): {32, 1501}, {3051, 27369}
X(41331) = crosssum of X(i) and X(j) for these (i,j): {75, 18050}, {76, 1502}
X(41331) = crossdifference of every pair of points on line {523, 14603}
X(41331) = barycentric product X(i)*X(j) for these {i,j}: {1, 1923}, {3, 27369}, {6, 3051}, {25, 20775}, {31, 1964}, {32, 39}, {38, 560}, {54, 27374}, {58, 41267}, {99, 9494}, {110, 688}, {141, 1501}, {163, 2084}, {184, 1843}, {187, 41272}, {427, 14575}, {604, 40972}, {669, 1634}, {732, 8789}, {826, 14574}, {827, 2531}, {1147, 27367}, {1235, 40373}, {1333, 21814}, {1397, 3688}, {1401, 2175}, {1576, 3005}, {1613, 19606}, {1917, 1930}, {1918, 17187}, {1927, 2236}, {1973, 4020}, {1974, 3917}, {1980, 4553}, {2205, 16696}, {2206, 21035}, {2353, 23208}, {3049, 35325}, {3118, 38826}, {3202, 27366}, {3203, 27375}, {3313, 40146}, {3404, 9417}, {3665, 9448}, {3703, 41280}, {4576, 9426}, {7813, 19626}, {8024, 9233}, {8623, 9468}, {9247, 17442}, {9418, 20021}, {14406, 32717}, {14585, 27376}, {14604, 35540}, {14617, 18899}, {16030, 40981}, {21123, 32739}, {23963, 39691}
X(41331) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40016}, {31, 18833}, {32, 308}, {38, 1928}, {39, 1502}, {141, 40362}, {163, 37204}, {560, 3112}, {688, 850}, {732, 18901}, {1401, 41283}, {1501, 83}, {1576, 689}, {1634, 4609}, {1843, 18022}, {1917, 82}, {1923, 75}, {1924, 18070}, {1964, 561}, {2084, 20948}, {2531, 23285}, {3051, 76}, {3203, 33769}, {3665, 41287}, {3688, 40363}, {3917, 40050}, {3933, 40360}, {4020, 40364}, {8022, 16889}, {8024, 40359}, {8623, 14603}, {8789, 14970}, {9233, 251}, {9418, 20022}, {9427, 34294}, {9494, 523}, {14573, 39287}, {14574, 4577}, {14575, 1799}, {14604, 733}, {19606, 40162}, {20775, 305}, {21814, 27801}, {23208, 40073}, {27369, 264}, {27374, 311}, {40373, 1176}, {40377, 5025}, {40972, 28659}, {41267, 313}, {41272, 18023}
X(41331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {160, 16285, 3117}, {1342, 1343, 1207}, {1501, 14575, 9233}, {1501, 18899, 9418}

X(41332) = X(3)X(6)∩X(19)X(2204)

Barycentrics    a^3*(a + b)*(a + c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :

X(41331) lies on the cubic K431 and these lines: {3, 6}, {19, 2204}, {28, 40570}, {112, 915}, {212, 2206}, {228, 2174}, {230, 37362}, {759, 29041}, {1172, 8609}, {1474, 2178}, {1630, 16685}, {1778, 5546}, {1780, 2911}, {1914, 40959}, {2164, 2299}, {2203, 2352}, {2249, 15440}, {2303, 7054}

X(41332) = isogonal conjugate of the isotomic conjugate of X(40571)
X(41332) = isogonal conjugate of the polar conjugate of X(30733)
X(41332) = polar conjugate of isotomic conjugate of X(41608)
X(41332) = X(i)-Ceva conjugate of X(j) for these (i,j): {28, 2194}, {40570, 6}
X(41332) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23604}, {10, 15474}, {92, 28787}, {306, 39267}, {1441, 39943}, {1577, 13397}
X(41332) = crosspoint of X(30733) and X(40571)
X(41332) = crosssum of X(525) and X(16732)
X(41332) = barycentric product X(i)*X(j) for these {i,j}: {1, 1780}, {3, 30733}, {6, 40571}, {21, 37579}, {28, 11517}, {29, 3215}, {58, 3811}, {81, 2911}, {110, 15313}, {284, 1708}, {1172, 3173}, {1175, 14054}, {1333, 17776}, {2328, 4341}
X(41332) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 23604}, {184, 28787}, {1333, 15474}, {1576, 13397}, {1708, 349}, {1780, 75}, {2203, 39267}, {2911, 321}, {3173, 1231}, {3215, 307}, {3811, 313}, {11517, 20336}, {14054, 1234}, {15313, 850}, {17776, 27801}, {30733, 264}, {37579, 1441}, {40571, 76}
X(41332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {284, 5060, 4269}, {1333, 19622, 2193}, {4282, 33628, 1333}

X(41333) = X(3)X(6)∩X(31)X(21753)

Barycentrics    a^3*(b + c)*(a^2 - b*c) : :

X(41333) lies on the cubics K987 and K993 and these lines: {3, 6}, {31, 21753}, {37, 4523}, {41, 904}, {42, 2205}, {55, 21838}, {83, 20148}, {86, 5277}, {100, 2229}, {101, 9264}, {172, 4649}, {213, 872}, {228, 16584}, {238, 1914}, {251, 39971}, {385, 30940}, {518, 8624}, {667, 788}, {746, 35551}, {851, 16592}, {1001, 2241}, {1015, 20470}, {1284, 39786}, {1580, 8845}, {1911, 18263}, {1933, 18038}, {2210, 14599}, {2240, 3936}, {2352, 23543}, {3121, 3724}, {3882, 15994}, {3948, 4366}, {4557, 21830}, {6600, 14974}, {7031, 16468}, {8033, 16956}, {8852, 39690}, {14827, 40984}, {16678, 22199}, {16687, 23533}, {17277, 33821}, {17349, 17696}, {17379, 27162}, {18792, 35342}, {19308, 37128}, {20179, 27042}, {20553, 27273}, {20760, 21775}, {21024, 32941}, {21779, 38832}, {26085, 27252}, {28243, 29964}

X(41333) = isogonal conjugate of X(40017)
X(41333) = isogonal conjugate of the isotomic conjugate of X(2238)
X(41333) = isogonal conjugate of the polar conjugate of X(862)
X(41333) = X(i)-Ceva conjugate of X(j) for these (i,j): {699, 2176}, {919, 512}, {1914, 3747}, {1976, 2175}, {18268, 31}, {34067, 669}
X(41333) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40017}, {2, 18827}, {7, 36800}, {27, 337}, {58, 18895}, {75, 37128}, {76, 741}, {81, 334}, {86, 335}, {99, 4444}, {274, 291}, {292, 310}, {331, 1808}, {333, 7233}, {513, 4639}, {514, 4589}, {561, 18268}, {660, 7199}, {670, 3572}, {693, 4584}, {799, 876}, {875, 4602}, {982, 40834}, {1019, 4583}, {1434, 4518}, {1577, 36066}, {1581, 8033}, {1911, 6385}, {1916, 17103}, {1930, 39276}, {2311, 6063}, {4017, 36806}, {4369, 18829}, {4374, 37134}, {4481, 41072}, {4562, 7192}, {4610, 35352}, {5378, 16727}, {17096, 36801}, {30663, 30940}, {30669, 32010}, {33295, 40098}, {39747, 40093}, {39950, 40094}
X(41333) = crosspoint of X(i) and X(j) for these (i,j): {31, 18268}, {862, 2238}, {1252, 34067}, {1914, 2210}
X(41333) = crosssum of X(i) and X(j) for these (i,j): {2, 30941}, {6, 16876}, {75, 3948}, {334, 335}, {514, 23822}, {1086, 3766}, {18827, 36800}
X(41333) = crossdifference of every pair of points on line {75, 523}
X(41333) = barycentric product X(i)*X(j) for these {i,j}: {1, 3747}, {3, 862}, {6, 2238}, {10, 2210}, {31, 740}, {32, 3948}, {37, 1914}, {41, 16609}, {42, 238}, {55, 1284}, {56, 4433}, {71, 2201}, {82, 4093}, {100, 4455}, {101, 21832}, {110, 4155}, {210, 1428}, {212, 1874}, {213, 239}, {228, 242}, {313, 18892}, {321, 14599}, {350, 1918}, {385, 40729}, {512, 3573}, {560, 35544}, {604, 3985}, {659, 4557}, {669, 874}, {692, 4010}, {727, 20681}, {741, 4094}, {756, 5009}, {798, 3570}, {872, 33295}, {904, 4039}, {1018, 8632}, {1252, 39786}, {1333, 4037}, {1334, 1429}, {1400, 3684}, {1402, 3685}, {1824, 7193}, {1911, 4368}, {1921, 2205}, {1924, 27853}, {1967, 4154}, {2054, 8298}, {2194, 7235}, {2333, 20769}, {3724, 36815}, {3903, 5027}, {4435, 4559}, {4839, 34074}, {6654, 39258}, {7109, 30940}, {16369, 25426}, {16514, 40747}, {18264, 20716}, {18268, 35068}, {18756, 39926}, {18786, 20964}, {18793, 20663}, {18894, 27801}
X(41333) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40017}, {31, 18827}, {32, 37128}, {37, 18895}, {41, 36800}, {42, 334}, {101, 4639}, {213, 335}, {228, 337}, {238, 310}, {239, 6385}, {560, 741}, {669, 876}, {692, 4589}, {740, 561}, {798, 4444}, {862, 264}, {874, 4609}, {1284, 6063}, {1402, 7233}, {1501, 18268}, {1576, 36066}, {1691, 8033}, {1914, 274}, {1918, 291}, {1924, 3572}, {1933, 17103}, {2205, 292}, {2210, 86}, {2238, 76}, {3570, 4602}, {3573, 670}, {3684, 28660}, {3685, 40072}, {3747, 75}, {3948, 1502}, {3985, 28659}, {4010, 40495}, {4037, 27801}, {4093, 1930}, {4094, 35544}, {4154, 1926}, {4155, 850}, {4368, 18891}, {4433, 3596}, {4455, 693}, {4557, 4583}, {5009, 873}, {5027, 4374}, {5546, 36806}, {8632, 7199}, {9426, 875}, {9447, 2311}, {14599, 81}, {16609, 20567}, {18892, 58}, {18894, 1333}, {20681, 35538}, {21832, 3261}, {32739, 4584}, {35544, 1928}, {39258, 40217}, {39786, 23989}, {40729, 1916}
X(41333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3053, 37507}, {6, 5132, 39}, {6, 18755, 3736}, {41, 2209, 40728}, {228, 21750, 16584}, {2240, 21341, 3936}, {2251, 21760, 9454}, {4251, 4279, 6}

X(41334) = X(3)X(6)∩X(5)X(217)

Barycentrics    a^2*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(41334) lies on the cubic K976 and these lines: {3, 6}, {5, 217}, {49, 35324}, {53, 40449}, {54, 32661}, {83, 290}, {112, 13434}, {140, 3289}, {232, 5462}, {276, 648}, {324, 458}, {381, 32445}, {401, 34545}, {546, 3331}, {1498, 8799}, {1976, 10547}, {1993, 37067}, {2211, 18583}, {2421, 7769}, {3269, 13630}, {3526, 40805}, {3567, 22240}, {3589, 34850}, {3843, 38297}, {5012, 10312}, {5890, 26216}, {6102, 22416}, {7592, 9755}, {9419, 11272}, {10314, 10539}, {10575, 33843}, {14389, 35325}, {14585, 32046}, {15024, 15355}, {15043, 39575}, {19156, 20968}, {20965, 21531}, {21177, 40951}, {22159, 39500}, {35318, 37124}

X(41334) = isogonal conjugate of the polar conjugate of X(30506)
X(41334) = X(i)-Ceva conjugate of X(j) for these (i,j): {83, 5}, {95, 418}, {648, 31296}, {36794, 30506}
X(41334) = X(i)-isoconjugate of X(j) for these (i,j): {2167, 3613}, {2190, 36952}, {2616, 11794}
X(41334) = crosspoint of X(5012) and X(36794)
X(41334) = crossdifference of every pair of points on line {523, 21646}
X(41334) = barycentric product X(i)*X(j) for these {i,j}: {3, 30506}, {5, 5012}, {51, 1078}, {216, 36794}, {343, 10312}, {1625, 31296}, {1629, 5562}, {1799, 27370}, {1953, 18042}, {2179, 33764}, {3050, 14570}, {33769, 40981}
X(41334) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 3613}, {216, 36952}, {1078, 34384}, {1625, 11794}, {1629, 8795}, {3050, 15412}, {3203, 16030}, {5012, 95}, {10312, 275}, {27370, 427}, {30506, 264}, {36794, 276}, {40981, 27375}
X(41334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 217, 1625}, {6, 182, 14965}, {6, 37514, 23115}, {14630, 14631, 5661}

X(41335) = X(2)X(18375)∩X(3)X(6)

Barycentrics    a^2*(a^6 - 3*a^2*b^4 + 2*b^6 - a^2*b^2*c^2 - 2*b^4*c^2 - 3*a^2*c^4 - 2*b^2*c^4 + 2*c^6) : :

X(41335) lies on the cubic K487 and these lines: {2, 18375}, {3, 6}, {5, 1989}, {53, 35895}, {54, 11077}, {858, 9300}, {1506, 3018}, {1594, 1990}, {2493, 16042}, {3091, 9220}, {3163, 9698}, {3470, 11079}, {3518, 11062}, {3520, 39176}, {3628, 16310}, {3815, 30789}, {5094, 11058}, {5306, 7495}, {6128, 7765}, {6240, 6749}, {6593, 37335}, {7574, 7753}, {8749, 9606}, {9607, 38323}, {9971, 20897}, {10312, 16328}, {11134, 36296}, {11137, 36297}, {14791, 34288}, {14881, 32224}, {15018, 34834}, {34319, 37345}, {37119, 40138}

X(41335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 216, 2965}, {6, 566, 50}, {6, 3003, 13338}, {6, 18573, 32}, {61, 62, 567}, {371, 372, 14805}

X(41336) = X(3)X(6)∩X(112)X(230)

Barycentrics    a^4*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 5*a^2*b^2*c^2 - 2*b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :

X(41336) lies on the cubic K478 and these lines: {3, 6}, {112, 230}, {115, 31726}, {172, 9627}, {232, 2079}, {248, 11744}, {468, 40347}, {682, 15257}, {1611, 17409}, {1968, 13881}, {2493, 8744}, {3291, 38463}, {4558, 32459}, {4611, 35297}, {5475, 18373}, {5523, 16306}, {5866, 37784}, {8770, 36417}, {10312, 18560}, {10313, 16386}, {11580, 36415}, {13509, 17854}, {14908, 18374}, {14910, 16303}, {31255, 39602}, {32640, 32654}

X(41336) = isogonal conjugate of the isotomic conjugate of X(37784)
X(41336) = isogonal conjugate of the polar conjugate of X(37777)
X(41336) = X(468)-Ceva conjugate of X(18374)
X(41336) = X(75)-isoconjugate of X(40347)
X(41336) = polar conjugate of isotomic conjugate of X(41615)
X(41336) = crosspoint of X(i) and X(j) for these (i,j): {249, 10423}, {2373, 40405}, {37777, 37784}
X(41336) = crosssum of X(i) and X(j) for these (i,j): {6, 37928}, {1196, 2393}
X(41336) = crossdifference of every pair of points on line {523, 1368}
X(41336) = barycentric product X(i)*X(j) for these {i,j}: {3, 37777}, {6, 37784}, {25, 5866}, {32, 37803}, {74, 20772}, {468, 39169}
X(41336) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 40347}, {5866, 305}, {20772, 3260}, {37777, 264}, {37784, 76}, {37803, 1502}, {39169, 30786}
X(41336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 187, 10317}, {32, 1384, 2965}, {32, 40320, 3053}, {187, 10317, 50}, {187, 40135, 40349}, {1384, 1609, 3053}

X(41337) = X(3)X(6)∩X(99)X(670)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2*b^4 - b^4*c^2 + a^2*c^4 - b^2*c^4) : :

Let E be the ellipse described at X(7757). Let P and U be the Brocard axis intercepts of E. Then X(41337) is the {P,U}-harmonic conjugate of X(6). (Randy Hutson, April 13, 2021)

X(41337) lies on the cubic K723 and these lines: {3, 6}, {99, 670}, {110, 25424}, {112, 30254}, {446, 29181}, {669, 5468}, {698, 32540}, {805, 881}, {865, 11053}, {3229, 36821}, {5108, 11332}, {5969, 21444}, {7766, 32531}, {9149, 14931}, {17938, 17941}, {26714, 39639}, {31128, 38998}

X(41337) = X(i)-Ceva conjugate of X(j) for these (i,j): {17938, 1634}, {17941, 2421}, {39292, 6}
X(41337) = X(i)-isoconjugate of X(j) for these (i,j): {661, 3225}, {699, 1577}
X(41337) = crosspoint of X(99) and X(805)
X(41337) = crosssum of X(i) and X(j) for these (i,j): {512, 804}, {4155, 21051}
X(41337) = trilinear pole of line {3229, 32748}
X(41337) = crossdifference of every pair of points on line {523, 1084}
X(41337) = barycentric product X(i)*X(j) for these {i,j}: {99, 3229}, {110, 698}, {662, 2227}, {670, 32748}, {805, 39080}, {1576, 35524}, {2396, 32540}, {5468, 36821}, {9429, 34537}
X(41337) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 3225}, {698, 850}, {1576, 699}, {2227, 1577}, {3229, 523}, {4558, 8858}, {9429, 3124}, {32540, 2395}, {32748, 512}, {36821, 5466}, {39080, 14295}

X(41338) = X(1)X(3)∩X(63)X(516)

Barycentrics    a*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c - 2*a^3*b*c + 2*a^2*b^2*c - 2*a*b^3*c + 3*b^4*c - 2*a^3*c^2 + 2*a^2*b*c^2 + 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 + a*c^4 + 3*b*c^4 - c^5) : :
X(41338) = 3 X[40] - X[12703], 2 X[55] - 3 X[165], 5 X[1698] - 4 X[7680], 3 X[1699] - 4 X[2886], 3 X[3576] - 2 X[37533], 3 X[5119] - 2 X[12703], X[5119] - 4 X[37584], 5 X[7987] - 4 X[24929], 9 X[7988] - 10 X[31245], 3 X[9778] - X[20075], X[12703] - 6 X[37584], 4 X[32613] - 5 X[35242]

X(41338) lies on the cubic K609 and these lines: {1, 3}, {4, 41229}, {9, 1699}, {10, 6835}, {20, 10085}, {38, 990}, {63, 516}, {84, 5842}, {103, 13397}, {105, 39947}, {169, 8012}, {170, 2942}, {191, 9589}, {200, 15104}, {210, 19541}, {212, 34036}, {219, 910}, {226, 15298}, {238, 10900}, {329, 20588}, {347, 9533}, {411, 3811}, {497, 1708}, {518, 2900}, {528, 1768}, {573, 40131}, {583, 2257}, {614, 13329}, {672, 1723}, {674, 7289}, {946, 6832}, {962, 5273}, {972, 36082}, {1002, 37400}, {1158, 6361}, {1282, 1763}, {1445, 11019}, {1448, 1496}, {1456, 22117}, {1465, 7074}, {1479, 1728}, {1490, 5904}, {1698, 7680}, {1707, 12652}, {1742, 32913}, {1750, 5223}, {1752, 14268}, {1753, 1838}, {1780, 5324}, {1836, 5762}, {1837, 31799}, {2136, 5855}, {2262, 15288}, {2328, 4228}, {2717, 6099}, {2951, 15733}, {2954, 37319}, {3190, 24611}, {3218, 9778}, {3219, 9812}, {3305, 3817}, {3306, 10164}, {3419, 5691}, {3474, 35514}, {3681, 36002}, {3715, 10157}, {3838, 15296}, {3868, 12520}, {3873, 7411}, {3874, 10884}, {3925, 5805}, {3927, 12688}, {3929, 11372}, {3951, 31803}, {4301, 5250}, {4333, 11826}, {4423, 31658}, {4430, 35986}, {4679, 7956}, {5220, 5927}, {5274, 37787}, {5288, 12650}, {5338, 36009}, {5437, 6690}, {5573, 19624}, {5603, 6878}, {5657, 6854}, {5732, 16465}, {5758, 12047}, {5763, 11375}, {5812, 15908}, {6211, 33138}, {6601, 24477}, {6825, 7162}, {6836, 10916}, {6838, 21077}, {6908, 13407}, {6948, 7284}, {6974, 28228}, {6985, 17857}, {7308, 7988}, {7330, 37820}, {7713, 37387}, {7965, 12699}, {8257, 26105}, {9580, 30223}, {9588, 17529}, {9779, 27065}, {10167, 11495}, {10382, 18412}, {10404, 37424}, {10582, 21153}, {11365, 26935}, {11522, 31435}, {12516, 12777}, {12565, 15071}, {12618, 33163}, {12651, 31424}, {12675, 37426}, {12717, 21375}, {13174, 24469}, {14872, 37411}, {15430, 24430}, {15493, 24779}, {16139, 16617}, {16438, 39592}, {17156, 29016}, {17728, 37364}, {17860, 20223}, {18407, 18540}, {18499, 28146}, {28850, 32853}, {31730, 37000}, {33118, 36652}, {34625, 34632}, {34628, 34740}

X(41338) = reflection of X(i) in X(j) for these {i,j}: {1, 3428}, {40, 37584}, {1709, 63}, {5119, 40}, {5691, 3419}, {10679, 3579}, {11531, 2099}, {37000, 31730}, {37569, 3}
X(41338) = Bevan-circle-inverse of X(5537)
X(41338) = excentral-isogonal conjugate of X(3174)
X(41338) = X(i)-Ceva conjugate of X(j) for these (i,j): {6601, 1}, {24477, 5272}
X(41338) = crosspoint of X(7045) and X(37206)
X(41338) = X(22)-of-excentral-triangle
X(41338) = X(55)-of-tangential-of-excentral-triangle
X(41338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 15931}, {3, 7964, 165}, {3, 12704, 3338}, {40, 57, 165}, {40, 3333, 37551}, {40, 3576, 3587}, {40, 5535, 3359}, {40, 5709, 46}, {40, 6766, 1697}, {40, 12704, 3}, {40, 24468, 5709}, {55, 5173, 1}, {57, 354, 3338}, {165, 5536, 57}, {165, 7991, 7994}, {165, 7994, 5537}, {165, 10980, 10857}, {165, 30350, 7987}, {191, 9589, 12705}, {354, 7964, 3}, {497, 1708, 15299}, {1155, 6244, 165}, {1155, 17642, 1617}, {1617, 17642, 1}, {1697, 6766, 11531}, {2448, 2449, 5537}, {3333, 37551, 7987}, {3337, 16192, 37526}, {3359, 5535, 46}, {3359, 5709, 5535}, {3513, 3514, 34489}, {3579, 13373, 3}, {3874, 12511, 10884}, {3928, 10860, 1768}, {6244, 22770, 1617}, {8186, 8187, 10902}, {11012, 37531, 3612}, {11227, 37532, 57}, {11252, 11253, 3}, {12702, 22770, 14110}, {14110, 22770, 1}, {32622, 32623, 46}

X(41339) = X(1)X(3)∩X(33)X(210)

Barycentrics    a*(a - b - c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) : :
X(41339) = X[7291] - 3 X[35280]

X(41339) = lies on the cubic K984 and these lines: {1, 3}, {6, 4319}, {7, 30621}, {8, 37774}, {11, 3008}, {12, 37160}, {31, 40133}, {33, 210}, {37, 1253}, {44, 2310}, {77, 11495}, {169, 35273}, {200, 4513}, {212, 1212}, {218, 1864}, {219, 3059}, {222, 5918}, {239, 14942}, {279, 3474}, {294, 2348}, {390, 1386}, {480, 2324}, {497, 5222}, {516, 1456}, {518, 677}, {650, 663}, {651, 15726}, {674, 3270}, {692, 2182}, {910, 9502}, {948, 1836}, {990, 8581}, {1100, 2293}, {1108, 21059}, {1191, 10866}, {1279, 39077}, {1376, 25930}, {1419, 2951}, {1442, 7676}, {1443, 30295}, {1449, 4326}, {1486, 2262}, {1633, 34371}, {1709, 22117}, {1721, 6180}, {1743, 4907}, {1827, 21867}, {1854, 3962}, {1859, 36010}, {1886, 17747}, {2115, 3693}, {2124, 34033}, {2170, 8647}, {2175, 2264}, {2187, 3198}, {2323, 15733}, {2328, 16699}, {2330, 11997}, {2346, 7269}, {2398, 30807}, {3000, 6610}, {3022, 20683}, {3160, 7056}, {3242, 7221}, {3554, 21002}, {3562, 9943}, {3681, 9539}, {3694, 4149}, {3827, 20468}, {3925, 40960}, {4012, 27382}, {4046, 6743}, {4383, 17604}, {4640, 24635}, {4663, 10394}, {4682, 5281}, {4919, 19589}, {5218, 5308}, {5432, 29571}, {5526, 9629}, {5543, 10578}, {5779, 15430}, {5904, 9576}, {6169, 7077}, {6510, 35338}, {6554, 28124}, {7071, 12329}, {7073, 10482}, {7078, 12688}, {7291, 35280}, {8053, 22079}, {8609, 19624}, {9577, 15104}, {9848, 16466}, {10178, 17074}, {10384, 16469}, {10481, 11246}, {11376, 16020}, {12907, 37482}, {15251, 30384}, {15587, 37659}, {16572, 30223}, {20613, 40971}, {22464, 38454}, {27509, 30620}, {27541, 28131}

X(41339) = reflection of X(2182) in X(692)
X(41339) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 9502}, {516, 910}, {518, 2348}, {1814, 5452}, {2736, 513}, {28071, 55}, {36086, 650}
X(41339) = X(i)-isoconjugate of X(j) for these (i,j): {7, 103}, {56, 18025}, {57, 36101}, {59, 15634}, {77, 36122}, {85, 911}, {109, 2400}, {241, 9503}, {273, 36056}, {278, 1815}, {279, 2338}, {331, 32657}, {522, 24016}, {664, 2424}, {677, 3676}, {4391, 32668}, {24002, 36039}
X(41339) = crosspoint of X(i) and X(j) for these (i,j): {1, 294}, {516, 40869}, {644, 5377}
X(41339) = crosssum of X(i) and X(j) for these (i,j): {1, 241}, {57, 1456}, {665, 3022}, {3669, 3675}
X(41339) = crossdifference of every pair of points on line {57, 650}
X(41339) = X(1)-line conjugate of X(57)
X(41339) = barycentric product X(i)*X(j) for these {i,j}: {1, 40869}, {8, 910}, {9, 516}, {21, 17747}, {33, 26006}, {41, 35517}, {55, 30807}, {78, 1886}, {210, 14953}, {346, 1456}, {644, 676}, {650, 2398}, {1566, 5377}, {2338, 24014}, {2426, 4391}, {3254, 28345}, {4105, 24015}, {4130, 23973}, {4241, 8611}, {9502, 14942}, {28071, 39063}
X(41339) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 18025}, {41, 103}, {55, 36101}, {212, 1815}, {516, 85}, {607, 36122}, {650, 2400}, {676, 24002}, {910, 7}, {1253, 2338}, {1415, 24016}, {1456, 279}, {1886, 273}, {2170, 15634}, {2175, 911}, {2195, 9503}, {2398, 4554}, {2426, 651}, {3063, 2424}, {9502, 9436}, {17747, 1441}, {23973, 36838}, {26006, 7182}, {30807, 6063}, {35517, 20567}, {40869, 75}
X(41339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5228, 354}, {1, 9441, 241}, {6, 4319, 14100}, {33, 7074, 210}, {37, 1253, 15837}, {55, 38285, 21010}, {220, 28043, 210}, {241, 9441, 1155}, {1253, 4336, 37}, {1721, 6180, 31391}, {1936, 9371, 1155}, {2175, 40965, 2264}

X(41340) = X(1)X(3)∩X(2)X(9895)

Barycentrics    a*(b + c)*(a^2 - b^2 - c^2)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(41340) lies on the cubic K737 and these lines: {1, 3}, {2, 9895}, {4, 1441}, {5, 1848}, {8, 37179}, {10, 4523}, {12, 18588}, {19, 7535}, {28, 3101}, {30, 1891}, {37, 4456}, {63, 18732}, {71, 18674}, {72, 306}, {377, 20243}, {392, 7515}, {405, 1829}, {442, 1824}, {464, 3868}, {572, 8555}, {916, 1071}, {1125, 7561}, {1231, 17170}, {1368, 31419}, {1409, 3157}, {1465, 19543}, {1500, 18591}, {1782, 2328}, {1828, 11113}, {1870, 37399}, {1872, 6907}, {1876, 37062}, {1880, 37415}, {1882, 6928}, {1900, 17532}, {1902, 7580}, {2193, 18178}, {2359, 7100}, {2915, 11363}, {3682, 18673}, {3753, 18641}, {3784, 13369}, {3812, 17073}, {3827, 12514}, {3869, 6350}, {3877, 27407}, {3927, 4047}, {3991, 21871}, {4220, 6198}, {4221, 4296}, {4642, 22057}, {4901, 34790}, {5046, 5146}, {5082, 7386}, {5174, 37098}, {5295, 18697}, {5439, 7536}, {5717, 32118}, {5810, 5928}, {5837, 34823}, {5909, 8807}, {6197, 37275}, {6643, 36844}, {6827, 40149}, {7719, 11108}, {9052, 11574}, {10198, 40635}, {10391, 37482}, {11396, 37246}, {11471, 15951}, {14054, 26893}, {16290, 40937}, {17094, 30209}, {18517, 18531}, {18719, 33116}, {19544, 37696}, {19860, 30675}, {21318, 37225}, {24611, 37034}, {30444, 39579}

X(41340) = reflection of X(1871) in X(5)
X(41340) = anticomplement of X(9895)
X(41340) = X(i)-complementary conjugate of X(j) for these (i,j): {835, 20316}, {1459, 5515}, {2214, 226}
X(41340) = X(1871)-of-Johnson-triangle
X(41340) = barycentric product X(i)*X(j) for these {i,j}: {72, 19785}, {1214, 2478}
X(41340) = barycentric quotient X(i)/X(j) for these {i,j}: {2478, 31623}, {19785, 286}
X(41340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40, 37547}, {1, 46, 37538}, {1, 10319, 3}, {3, 12702, 15941}, {3, 18453, 3579}, {3, 20254, 37565}, {10, 18589, 21530}, {11471, 30265, 15951}

X(41341) = X(1)X(3)∩X(23)X(105)

Barycentrics    a^2*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 - 2*a^3*c + a^2*b*c - a*b^2*c + 2*b^3*c - a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + 2*b*c^3 - c^4) : :
X(41341) = 6 X[36] + X[3303]

X(41341) lies on the cubic K903 and these lines: {1, 3}, {11, 7677}, {23, 105}, {59, 6066}, {103, 34921}, {104, 5427}, {110, 840}, {215, 1362}, {404, 9710}, {405, 535}, {497, 35986}, {513, 8642}, {528, 36003}, {551, 37286}, {672, 3446}, {1001, 5057}, {1004, 31140}, {1006, 5434}, {1458, 2361}, {1621, 11246}, {1626, 20988}, {1776, 18450}, {2246, 19297}, {2717, 26700}, {3058, 7411}, {3321, 32624}, {3651, 37722}, {3715, 17615}, {3813, 35976}, {3814, 16842}, {3925, 35985}, {4325, 13743}, {4423, 5087}, {4857, 16117}, {4999, 17531}, {5047, 5080}, {5096, 9049}, {5298, 6905}, {5433, 6946}, {6584, 28471}, {6681, 16862}, {6883, 11237}, {6912, 7354}, {6986, 15888}, {7580, 11238}, {7965, 21669}, {9671, 37411}, {11108, 31160}, {11194, 37300}, {12513, 37301}, {14733, 38451}, {16064, 18613}, {16355, 30981}, {16373, 40109}, {16855, 31263}, {16865, 20067}, {17572, 26040}, {20470, 20989}, {20835, 28534}, {23854, 38863}, {23855, 38530}, {34610, 37313}, {34612, 35977}, {38665, 40663}

X(41341) = isogonal conjugate of antigonal conjugate of X(2346)
X(41341) = circumcircle-inverse of X(354)
X(41341) = X(34578)-Ceva conjugate of X(6)
X(41341) = crosspoint of X(59) and X(1308)
X(41341) = crosssum of X(11) and X(3887)
X(41341) = crossdifference of every pair of points on line {650, 16601}
X(41341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 34879, 55}, {3, 1617, 33925}, {3, 33925, 55}, {36, 2078, 1155}, {36, 5126, 56}, {36, 32760, 5122}, {55, 56, 4860}, {354, 1155, 5536}, {354, 15931, 55}, {1155, 1319, 18839}, {1155, 2078, 55}, {1381, 1382, 354}, {1617, 37578, 55}, {3513, 3514, 5172}, {5126, 7742, 5172}, {5597, 5598, 5902}, {11492, 11493, 999}, {14798, 37582, 14882}, {20470, 20999, 20989}, {32622, 32623, 35000}, {33925, 37578, 3}

X(41342) = X(1)X(3)∩X(2)X(15830)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a^5 - 2*a^3*b^2 + a*b^4 - a^3*b*c - a^2*b^2*c + a*b^3*c + b^4*c - 2*a^3*c^2 - a^2*b*c^2 - b^3*c^2 + a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4) : :

X(41342) lies on the cubics K109 and K457 and and these lines: {1, 3}, {2, 15830}, {19, 1708}, {63, 1441}, {71, 226}, {109, 1396}, {209, 4551}, {223, 1409}, {278, 579}, {284, 2982}, {580, 41227}, {851, 3611}, {1427, 3990}, {1630, 37263}, {1728, 1882}, {1736, 1859}, {1744, 1762}, {1765, 5307}, {1782, 7098}, {1836, 33536}, {1943, 18206}, {2245, 6354}, {2270, 8807}, {2294, 16577}, {2357, 8808}, {3191, 15443}, {3197, 11347}, {3668, 40152}, {6358, 21061}, {15556, 18673}, {18593, 37755}

X(41342) = X(27)-Ceva conjugate of X(226)
X(41342) = barycentric product X(i)*X(j) for these {i,j}: {7, 3191}, {86, 15443}, {307, 41227}, {580, 1441}, {1214, 37279}
X(41342) = barycentric quotient X(i)/X(j) for these {i,j}: {580, 21}, {3191, 8}, {15443, 10}, {37279, 31623}, {41227, 29}
X(41342) = {X(10319),X(24310)}-harmonic conjugate of X(1764)

X(41343) = X(1)X(3)∩X(2)X(8686)

Barycentrics    a*(2*a^6 - 3*a^5*b - 4*a^4*b^2 + 6*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5 - 3*a^5*c + 14*a^4*b*c - 7*a^3*b^2*c - 13*a^2*b^3*c + 10*a*b^4*c - b^5*c - 4*a^4*c^2 - 7*a^3*b*c^2 + 22*a^2*b^2*c^2 - 7*a*b^3*c^2 + 6*a^3*c^3 - 13*a^2*b*c^3 - 7*a*b^2*c^3 + 2*b^3*c^3 + 2*a^2*c^4 + 10*a*b*c^4 - 3*a*c^5 - b*c^5) : :
X(41342) = 5 X[1] + 4 X[14000]

X(41342) lies on the cubic K921 and these lines: {1, 3}, {102, 8686}, {104, 106}, {105, 28233}, {244, 11715}, {515, 1647}, {516, 14028}, {759, 28219}, {944, 6788}, {953, 2718}, {1054, 6264}, {1071, 15854}, {1458, 34913}, {3086, 18340}, {4297, 23869}, {4551, 10074}, {4674, 12737}, {5400, 12773}, {5657, 24864}, {5886, 24871}, {6713, 24222}, {14127, 37815}

X(41342) = barycentric product X(34234)*X(39756)
X(41342) = barycentric quotient X(39756)/X(908)
X(41342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 23703}, {104, 106, 32486}, {999, 5126, 23890}

X(41344) = X(1)X(3)∩X(2)X(3562)

Barycentrics    a*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - 2*a^2*b^3*c - a*b^4*c + 2*b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 4*b^3*c^3 + a^2*c^4 - a*b*c^4 + a*c^5 + 2*b*c^5) : :

X(41344) lies on the cubic K520 and these lines: {1, 3}, {2, 3562}, {4, 222}, {5, 3157}, {6, 1210}, {7, 412}, {10, 17811}, {20, 17074}, {29, 81}, {33, 1071}, {37, 1741}, {73, 3149}, {109, 11496}, {221, 946}, {255, 405}, {381, 8757}, {394, 6734}, {406, 26932}, {474, 22350}, {499, 35466}, {515, 34046}, {578, 36059}, {603, 1012}, {613, 36574}, {651, 3091}, {912, 37696}, {936, 25934}, {950, 36746}, {1124, 14121}, {1249, 17905}, {1335, 7090}, {1393, 20277}, {1406, 1836}, {1407, 4292}, {1498, 6245}, {1656, 23071}, {1698, 25878}, {1699, 34043}, {1712, 5728}, {1728, 4641}, {1745, 19541}, {1818, 5687}, {1838, 5805}, {1854, 5884}, {1905, 18732}, {1935, 6913}, {2003, 9581}, {2049, 23131}, {2360, 15509}, {2594, 11502}, {2956, 11372}, {3074, 11108}, {3085, 4648}, {3086, 16466}, {3173, 17814}, {3487, 7567}, {3577, 34039}, {3911, 36745}, {3955, 37415}, {4000, 11023}, {4185, 26884}, {4186, 26892}, {4194, 26871}, {4295, 7365}, {4303, 7580}, {4304, 37501}, {5262, 37253}, {5399, 11499}, {5587, 9370}, {5603, 34040}, {5703, 7572}, {5704, 32911}, {5713, 15844}, {5715, 34032}, {5722, 7524}, {5777, 9817}, {5783, 10479}, {5906, 24983}, {5930, 34042}, {6147, 15252}, {6180, 9612}, {6684, 7074}, {6831, 19349}, {6918, 37694}, {7538, 14996}, {7680, 34030}, {7681, 34029}, {7686, 21147}, {7959, 9948}, {8614, 10896}, {9843, 17825}, {10571, 22753}, {10629, 37715}, {11019, 14058}, {11398, 18180}, {12680, 36985}, {13374, 34036}, {13411, 37674}, {13740, 20745}, {14986, 15501}, {15079, 35197}, {15524, 37393}, {18240, 30148}, {21370, 40660}, {22053, 37426}, {22097, 37320}, {24475, 37729}

X(41344) = barycentric product X(189)*X(37413)
X(41344) = barycentric quotient X(37413)/X(329)
X(41344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 17102}, {1, 1771, 55}, {1, 3075, 3}, {2, 3562, 7078}, {5, 3157, 34048}, {381, 23070, 8757}, {603, 2654, 1012}, {940, 5710, 37554}, {942, 5707, 37543}, {1936, 37523, 3}, {6913, 23072, 1935}, {11108, 22117, 3074}

X(41345) = X(1)X(3)∩X(21)X(18990)

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*b^3*c + b^4*c - 2*a^3*c^2 - 2*a*b^2*c^2 + 2*a^2*c^3 + a*b*c^3 + a*c^4 + b*c^4 - c^5) : :
X(41345) = 4 X[36] + X[3295], 4 X[2078] - X[3295]

X(41345) lies on the cubic K904 and these lines: {1, 3}, {21, 18990}, {25, 5146}, {47, 23070}, {404, 31419}, {405, 5080}, {411, 496}, {495, 1006}, {499, 18517}, {513, 22160}, {535, 16418}, {580, 5399}, {859, 20999}, {934, 32624}, {956, 37300}, {1001, 11813}, {1056, 37106}, {1068, 20837}, {1324, 20470}, {1451, 37698}, {1478, 7489}, {1621, 5180}, {1731, 19297}, {1737, 18524}, {1788, 32141}, {2070, 14667}, {2361, 23071}, {2594, 37509}, {2932, 13587}, {2964, 8614}, {2975, 37308}, {3002, 22144}, {3286, 3446}, {3421, 37313}, {3560, 9655}, {3600, 6875}, {3651, 15171}, {3814, 11108}, {4188, 5082}, {4191, 33142}, {4293, 6914}, {5047, 10592}, {5057, 37284}, {5096, 9052}, {5176, 9708}, {5248, 37292}, {5253, 19524}, {5265, 6942}, {5298, 10090}, {5427, 12773}, {5434, 28443}, {5687, 37301}, {5938, 16693}, {6284, 16117}, {6681, 16408}, {6876, 14986}, {6883, 31479}, {6905, 7677}, {6924, 7288}, {6985, 9669}, {7098, 24475}, {7354, 13743}, {7484, 29664}, {7580, 9668}, {9037, 37492}, {9709, 37282}, {10016, 14703}, {10058, 15326}, {12331, 40663}, {16058, 29661}, {16059, 24892}, {16287, 24936}, {16345, 30981}, {16414, 24880}, {16453, 24883}, {16853, 31263}, {16857, 31160}, {19537, 38901}, {20831, 23383}, {20832, 35220}, {20834, 29689}, {20918, 38863}, {24390, 35979}, {25466, 33961}, {28377, 28383}, {31493, 37229}

X(41345) = midpoint of X(36) and X(2078)
X(41345) = isogonal conjugate of X(24298)
X(41345) = X(21907)-Ceva conjugate of X(6)
X(41345) = X(1)-isoconjugate of X(24298)
X(41345) = crosspoint of X(59) and X(1290)
X(41345) = crosssum of X(11) and X(8674)
X(41345) = circumcircle-inverse of X(942)
X(41345) = barycentric quotient X(6)/X(24298)
X(41345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 14799, 55}, {3, 1617, 999}, {3, 6767, 40292}, {3, 7373, 26357}, {35, 36, 5131}, {36, 1319, 22765}, {36, 2077, 5122}, {36, 5172, 3}, {36, 32760, 1155}, {55, 56, 5902}, {56, 10246, 999}, {56, 11507, 5708}, {56, 36152, 3}, {65, 14798, 37621}, {484, 1319, 15934}, {1155, 1319, 5570}, {1155, 32760, 35000}, {1319, 22765, 999}, {1381, 1382, 942}, {3336, 14795, 14882}, {6905, 7677, 15325}, {7742, 8069, 37578}, {7742, 37579, 3}, {8069, 37578, 3}, {11492, 11493, 4860}, {23383, 23850, 20831}, {33925, 40292, 6767}, {37578, 37579, 8069}

X(41346) = X(1)X(3)∩X(6)X(23438)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 - a*b + b^2 - a*c + b*c + c^2) : :

X(41346) lies on the cubic K1001 and these lines: {1, 3}, {6, 23438}, {12, 13740}, {21, 28386}, {31, 1469}, {42, 1428}, {82, 23868}, {100, 29840}, {109, 1401}, {181, 4279}, {226, 29656}, {345, 3476}, {388, 4195}, {404, 28385}, {595, 23850}, {604, 2276}, {893, 21008}, {902, 16064}, {978, 38903}, {995, 1324}, {1001, 29634}, {1191, 23843}, {1201, 37259}, {1284, 1621}, {1376, 3705}, {1397, 5145}, {1400, 1914}, {1405, 5332}, {1423, 8616}, {1432, 6660}, {1486, 20473}, {1626, 3052}, {1964, 34250}, {2260, 10315}, {2274, 23380}, {2305, 23379}, {2308, 19369}, {2975, 18235}, {3145, 3915}, {3485, 36508}, {3703, 10944}, {3869, 36560}, {3911, 29655}, {4030, 40663}, {4203, 8299}, {4224, 28353}, {4234, 5434}, {4428, 17320}, {5014, 36497}, {5096, 15621}, {5252, 32777}, {5989, 30545}, {7235, 32914}, {7248, 9316}, {7677, 29837}, {9259, 20284}, {10459, 37247}, {11237, 11354}, {11334, 16483}, {12588, 33171}, {20988, 23404}, {23369, 40153}, {28364, 33849}, {36529, 38000}

X(41346) = X(1432)-Ceva conjugate of X(6)
X(41346) = X(i)-isoconjugate of X(j) for these (i,j): {8, 7194}, {9, 39724}, {55, 40038}, {3502, 7155}
X(41346) = crosspoint of X(59) and X(29055)
X(41346) = crosssum of X(11) and X(3907)
X(41346) = crossdifference of every pair of points on line {650, 3810}
X(41346) = barycentric product X(i)*X(j) for these {i,j}: {56, 17280}, {57, 3961}, {604, 33938}, {1400, 33954}, {1423, 3494}, {1431, 17741}, {3212, 34249}
X(41346) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 39724}, {57, 40038}, {604, 7194}, {3494, 27424}, {3961, 312}, {17280, 3596}, {33938, 28659}, {33954, 28660}, {34249, 7155}
X(41346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 56, 1403}, {1460, 1617, 56}, {5329, 23853, 17798}, {10475, 37583, 56}

X(41347) = X(1)X(3)∩X(5)X(12615)

Barycentrics    a*(2*a^6 - a^5*b - 5*a^4*b^2 + 2*a^3*b^3 + 4*a^2*b^4 - a*b^5 - b^6 - a^5*c + 2*a^4*b*c + a^3*b^2*c - 4*a^2*b^3*c + 2*b^5*c - 5*a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 + b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 + a*b^2*c^3 - 4*b^3*c^3 + 4*a^2*c^4 + b^2*c^4 - a*c^5 + 2*b*c^5 - c^6) : :
X(41347) = X[3] - 3 X[5131], 3 X[3] - X[5538], 4 X[1155] - X[3579], 5 X[1155] - X[13528], 2 X[3245] + X[11278], 3 X[3576] - X[35457], 5 X[3579] - 4 X[13528], 8 X[5122] - 3 X[17502], 3 X[5131] + X[5535], 9 X[5131] - X[5538], 2 X[5176] - 3 X[38176], X[5180] - 3 X[5886], 3 X[5535] + X[5538], X[5536] + 2 X[31663], 5 X[10222] - 4 X[23960], 5 X[10225] - 2 X[13528], 3 X[11230] - 2 X[11813], 3 X[17502] - 4 X[23961], 2 X[31160] - 3 X[38083]

X(41347) lies on the cubic K798 and these lines: {1, 3}, {5, 12615}, {30, 10265}, {140, 11263}, {355, 20067}, {546, 26202}, {758, 22935}, {952, 4973}, {2771, 6905}, {3149, 31828}, {3628, 3647}, {3652, 6915}, {3814, 31445}, {3822, 11231}, {3916, 5080}, {4640, 11230}, {5057, 6852}, {5176, 38176}, {5180, 5886}, {5694, 6924}, {5841, 12619}, {5844, 33337}, {5883, 7508}, {6839, 38140}, {6888, 9955}, {6940, 16139}, {6952, 16159}, {9352, 26446}, {9803, 28204}, {10738, 15228}, {13513, 34464}, {13587, 39778}, {17613, 28198}, {24470, 31659}, {26201, 26877}, {31160, 38083}, {33697, 34862}

X(41347) = midpoint of X(i) and X(j) for these {i,j}: {3, 5535}, {355, 20067}, {484, 22765}, {5536, 35000}, {10738, 15228}
X(41347) = reflection of X(i) in X(j) for these {i,j}: {1385, 36}, {3579, 10225}, {5080, 9956}, {10225, 1155}, {23961, 5122}, {35000, 31663}
X(41347) = reflection of X(10225) in the anti-Orthic axis
X(41347) = X(6595)-Ceva conjugate of X(1)
X(41347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3336, 5885}, {3, 5885, 1385}, {46, 26286, 35004}, {1381, 1382, 37621}, {1385, 11278, 11567}, {3336, 5131, 36}, {5131, 5535, 3}, {11567, 37605, 1385}

X(41348) = X(1)X(3)∩X(63)X(4678)

Barycentrics    a*(5*a^3 + 5*a^2*b - 5*a*b^2 - 5*b^3 + 5*a^2*c - 6*a*b*c + 5*b^2*c - 5*a*c^2 + 5*b*c^2 - 5*c^3) : :

X(41348) lies on the cubic K156 and these lines: {1, 3}, {63, 4678}, {728, 36643}, {962, 31231}, {1706, 3929}, {1788, 5493}, {1836, 9588}, {2136, 3218}, {2270, 16885}, {3474, 9578}, {3632, 15326}, {3654, 9613}, {3828, 12514}, {3911, 20070}, {3928, 31145}, {4848, 9778}, {5657, 9579}, {5727, 31730}, {6361, 9581}, {6762, 20053}, {7288, 28228}, {7308, 19877}, {7330, 38138}, {7966, 26877}, {9582, 38235}, {9589, 24914}, {10591, 28232}, {10895, 38121}, {12053, 34632}, {19878, 31435}, {20196, 26062}

X(41348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 46, 1697}, {40, 484, 5128}, {40, 3333, 11010}, {40, 3359, 37551}, {40, 5128, 57}, {46, 1697, 57}, {46, 5119, 3337}, {46, 11010, 3333}, {165, 37567, 3340}, {1155, 7991, 1420}, {1697, 5128, 46}, {1788, 5493, 9580}, {2093, 3579, 3601}, {2448, 2449, 5048}, {3333, 11010, 1697}, {3339, 37568, 10389}, {5903, 35242, 13384}, {12702, 15803, 7962}

X(41349) = X(1)X(3)∩X(108)X(511)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^6 - a^5*b - a^4*b^2 + a^3*b^3 - a^5*c + a^4*b*c + a^3*b^2*c - b^5*c - a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + a^3*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - b*c^5) : :

X(41349) lies on the cubic K908 and these lines: {1, 3}, {108, 511}, {110, 37798}, {278, 9306}, {651, 3292}, {851, 1813}, {1020, 17975}, {1068, 1092}, {1275, 5379}, {1425, 3193}, {3167, 34032}, {4551, 17976}, {4552, 17977}, {4554, 12215}, {5651, 37800}, {5972, 37799}, {7193, 34050}, {7952, 13346}, {20980, 36054}, {22464, 26884}

X(41349) = X(522)-isoconjugate of X(2714)
X(41349) = crosspoint of X(425) and X(23695)
X(41349) = crossdifference of every pair of points on line {650, 1858}
X(41349) = barycentric product X(i)*X(j) for these {i,j}: {226, 23695}, {425, 1214}, {651, 2798}
X(41349) = barycentric quotient X(i)/X(j) for these {i,j}: {425, 31623}, {1415, 2714}, {2798, 4391}, {23695, 333}

X(41350) = X(1)X(7)∩X(56)X(20676)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - a*c^3 + b*c^3) : :

X(41350) lies on the cubic K775 and these lines: {1, 7}, {56, 20676}, {57, 7032}, {109, 7122}, {292, 694}, {604, 1403}, {741, 1014}, {1409, 8779}, {1423, 3009}, {1427, 20359}, {1428, 23578}, {1740, 3212}, {1911, 7175}, {1964, 24471}, {2114, 20978}, {2309, 7146}, {17792, 28391}, {21352, 30097}

X(41350) = isogonal conjugate of X(39924)
X(41350) = X(i)-Ceva conjugate of X(j) for these (i,j): {1911, 1458}, {7175, 1400}
X(41350) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39924}, {55, 18299}
X(41350) = crosssum of X(1) and X(24728)
X(41350) = crossdifference of every pair of points on line {657, 4147}
X(41350) = barycentric product X(i)*X(j) for these {i,j}: {1, 28391}, {56, 17760}, {57, 17792}, {85, 18758}, {292, 39919}, {1403, 27436}, {4334, 40785}
X(41350) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39924}, {57, 18299}, {8844, 3975}, {17760, 3596}, {17792, 312}, {18269, 2053}, {18758, 9}, {28391, 75}, {39919, 1921}

X(41351) = X(1)X(7)∩X(44)X(658)

Barycentrics    (a + b - c)^2*(a - b + c)^2*(a^4 - 2*a^3*b + a^2*b^2 - 2*a^3*c + 5*a^2*b*c - 2*a*b^2*c - b^3*c + a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 - b*c^3) : :

X(41351) lies on the cubic K137 and these lines: {1, 7}, {44, 658}, {239, 4569}, {241, 4626}, {657, 7658}, {1088, 24352}, {1275, 10025}, {1931, 4616}, {3662, 30705}, {4644, 7056}, {5222, 36888}, {6180, 23062}, {17113, 37681}, {23618, 25067}, {27420, 40702}

X(41351) = barycentric product X(i)*X(j) for these {i,j}: {279, 40872}, {4569, 9511}
X(41351) = barycentric quotient X(i)/X(j) for these {i,j}: {9511, 3900}, {40872, 346}

X(41352) = X(1)X(7)∩X(37)X(17084)

Barycentrics    (a + b - c)*(a - b + c)*(a^4 - a^3*b + 2*a^2*b^2 - a*b^3 - b^4 - a^3*c + a^2*b*c - a*b^2*c + b^3*c + 2*a^2*c^2 - a*b*c^2 - a*c^3 + b*c^3 - c^4) : :

X(41352) lies on the cubic K769 and these lines: {1, 7}, {37, 17084}, {335, 39919}, {348, 24349}, {350, 40704}, {664, 4645}, {3212, 3959}, {3662, 9312}, {7174, 7179}, {7265, 22042}, {31038, 31527}, {35312, 37798}, {36620, 39749}

X(41352) = X(335)-Ceva conjugate of X(7)
X(41352) = barycentric product X(85)*X(18788)
X(41352) = barycentric quotient X(i)/X(j) for these {i,j}: {8932, 3684}, {18788, 9}
X(41352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {279, 4310, 7}

X(41353) = X(1)X(7)∩X(57)X(9319)

Barycentrics    a*(a - b)*(a - c)*(a + b - c)^2*(a - b + c)^2*(a*b - b^2 + a*c - c^2) : :

X(41353) lies on the cubic K407 and these lines: {1, 7}, {57, 9319}, {100, 4617}, {101, 651}, {109, 6183}, {241, 17435}, {658, 4551}, {664, 4569}, {883, 1026}, {1025, 2284}, {2736, 24016}, {3939, 6516}, {4564, 23704}, {4998, 23705}, {7045, 23703}, {9436, 36819}, {34230, 34855}

,p> X(41353) = X(i)-cross conjugate of X(j) for these (i,j): {2254, 241}, {2283, 1025}, {34253, 1275}
X(41353) = cevapoint of X(241) and X(2254)
X(41353) = trilinear pole of line {241, 672}
X(41353) = crossdifference of every pair of points on line {657, 2310}
X(41353) = X(i)-isoconjugate of X(j) for these (i,j): {6, 28132}, {8, 884}, {9, 1024}, {33, 23696}, {55, 885}, {105, 3900}, {200, 1027}, {294, 650}, {513, 28071}, {522, 2195}, {649, 6559}, {657, 673}, {663, 14942}, {666, 14936}, {919, 1146}, {927, 3022}, {1021, 18785}, {1416, 4163}, {1438, 3239}, {1462, 4130}, {2310, 36086}, {2481, 8641}, {3063, 36796}, {3119, 36146}, {3271, 36802}, {4081, 32735}, {4183, 10099}, {13576, 21789}, {24026, 32666}
X(41353) = barycentric product X(i)*X(j) for these {i,j}: {7, 1025}, {57, 883}, {85, 2283}, {109, 40704}, {190, 34855}, {241, 664}, {279, 1026}, {518, 658}, {651, 9436}, {672, 4569}, {918, 7045}, {934, 3912}, {1020, 30941}, {1088, 2284}, {1275, 2254}, {1362, 34085}, {1458, 4554}, {1461, 3263}, {1818, 13149}, {2340, 36838}, {2414, 4350}, {3693, 4626}, {3717, 4617}, {3930, 4616}, {3932, 4637}, {4566, 18206}, {4635, 20683}, {5236, 6516}, {25083, 36118}
X(41353) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 28132}, {56, 1024}, {57, 885}, {100, 6559}, {101, 28071}, {109, 294}, {222, 23696}, {241, 522}, {518, 3239}, {604, 884}, {651, 14942}, {658, 2481}, {664, 36796}, {665, 2310}, {672, 3900}, {883, 312}, {918, 24026}, {926, 3119}, {934, 673}, {1020, 13576}, {1025, 8}, {1026, 346}, {1262, 36086}, {1407, 1027}, {1415, 2195}, {1458, 650}, {1461, 105}, {1876, 3064}, {2223, 657}, {2254, 1146}, {2283, 9}, {2284, 200}, {2340, 4130}, {3286, 1021}, {3693, 4163}, {3912, 4397}, {4238, 2322}, {4350, 2402}, {4447, 4529}, {4564, 36802}, {4569, 18031}, {4619, 5377}, {4626, 34018}, {6614, 1462}, {7045, 666}, {7339, 36146}, {8299, 4148}, {9436, 4391}, {9454, 8641}, {14439, 4528}, {17435, 23615}, {18206, 7253}, {20683, 4171}, {23979, 32666}, {24016, 9503}, {24027, 919}, {32714, 36124}, {34253, 3716}, {34855, 514}, {39258, 4524}, {40704, 35519}

X(41354) = X(1)X(7)∩X(2)X(2114)

Barycentrics    (a + b - c)*(a - b + c)*(2*a^4 - a^3*b - a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a*b*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(41354) lies on the cubic K1018 and these lines: {1, 7}, {2, 2114}, {6, 3212}, {85, 1456}, {348, 4645}, {664, 24349}, {894, 1419}, {985, 1471}, {1447, 7290}, {1462, 2162}, {4648, 17084}, {5749, 31994}, {34059, 41246}, {36620, 39716}


X(41354) = X(14621)-Ceva conjugate of X(7)
X(41354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {77, 14189, 3160}, {279, 4307, 7}

X(41355) = X(1)X(7)∩X(42)X(7056)

Barycentrics    a*(a + b - c)^2*(a - b + c)^2*(a^2 - a*b - a*c + 2*b*c)*(a*b - b^2 + a*c - c^2) : :

X(41355) lies on the cubic K981 and these lines: {1, 7}, {42, 7056}, {43, 9533}, {238, 934}, {657, 905}, {658, 899}, {1743, 17106}, {2669, 4635}, {3751, 7177}, {4617, 9364}, {6168, 39066}, {6180, 9310}, {24011, 33677}

X(41355) = crossdifference of every pair of points on line {657, 4319}
X(41355) = X(i)-isoconjugate of X(j) for these (i,j): {9, 6169}, {6559, 9315}, {8641, 14727}, {9309, 28071}, {9439, 14942}
X(41355) = barycentric product X(i)*X(j) for these {i,j}: {7, 6168}, {241, 9312}, {269, 40883}, {3729, 34855}, {6180, 9436}, {9316, 40704}
X(41355) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 6169}, {658, 14727}, {1376, 6559}, {4449, 28132}, {6168, 8}, {6180, 14942}, {9310, 28071}, {9312, 36796}, {9316, 294}, {34855, 9311}, {40883, 341}

X(41356) = X(1)X(7)∩X(2)X(30705)

Barycentrics    (a + b - c)^2*(a - b + c)^2*(a^4 - 2*a^3*b + 2*a^2*b^2 - 2*a*b^3 + b^4 - 2*a^3*c + 6*a^2*b*c - 2*a*b^2*c - 2*b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :

X(41356) lies on the cubic K689 and these lines: {1, 7}, {2, 30705}, {6, 7056}, {241, 30682}, {479, 40131}, {658, 37650}, {948, 23062}, {1418, 17093}, {7347, 16663}, {7348, 16662}, {9533, 37681}

X(41356) = {X(7),X(3160)}-harmonic conjugate of X(4319)

X(41357) = MIDPOINT OF X(468) AND X(2501)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(3*a^6 - 4*a^4*b^2 + b^6 - 4*a^4*c^2 + 7*a^2*b^2*c^2 - 2*b^4*c^2 - 2*b^2*c^4 + c^6) : :
X(41357) = X[16230] - 9 X[39606]

X(41357) lies on the cubic K218 and these lines: {4, 8371}, {25, 10278}, {30, 39533}, {98, 40119}, {107, 691}, {125, 3566}, {132, 31655}, {230, 231}, {297, 31953}, {427, 10189}, {669, 37777}, {685, 31510}, {1499, 37984}, {3265, 37803}, {4232, 5466}, {5159, 14341}, {6353, 8029}, {6756, 10280}, {9007, 32119}, {9180, 16080}, {10190, 37453}, {10279, 21841}, {11123, 38282}, {18020, 31998}, {19577, 33294}, {23301, 37981}

X(41357) = midpoint of X(468) and X(2501)
X(41357) = reflection of X(5159) in X(14341)
X(41357) = Dao-Moses-Telv circle inverse of X(14273)
X(41357) = barycentric product X(i)*X(j) for these {i,j}: {523, 40890}, {648, 16278}, {14273, 16093}
X(41357) = barycentric quotient X(i)/X(j) for these {i,j}: {14273, 16103}, {16278, 525}, {40890, 99}
X(41357) = {X(24007),X(24008)}-harmonic conjugate of X(14273)

X(41358) = X(4)X(5007)∩X(230)X(231)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^6 - a^4*b^2 - a^2*b^4 - 2*b^6 - a^4*c^2 + 2*b^4*c^2 - a^2*c^4 + 2*b^2*c^4 - 2*c^6) : :

X(41358) lies on the cubic K487 and these lines: {4, 5007}, {23, 18487}, {230, 231}, {393, 1383}, {648, 7840}, {858, 3163}, {3284, 5189}, {5094, 11058}, {5169, 15860}, {5306, 10301}, {6032, 37665}, {7735, 40350}, {10311, 13338}, {13596, 39593}

X(41358) = polar conjugate of the isotomic conjugate of X(11645)
X(41358) = X(63)-isoconjugate of X(14388)
X(41358) = barycentric product X(4)*X(11645)
X(41358) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 14388}, {11645, 69}
X(41358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1990, 6103, 232}, {1990, 16318, 6103}, {3018, 16306, 3291}

X(41359) = X(3)X(67)∩X(4)X(1632)

Barycentrics    2*a^8 - 2*a^6*b^2 + 3*a^4*b^4 - 4*a^2*b^6 + b^8 - 2*a^6*c^2 - 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 3*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - 4*a^2*c^6 + c^8 : :

X(41359) lies on the cubic K818 and these lines: {3, 67}, {4, 1632}, {6, 35282}, {113, 14687}, {115, 2453}, {125, 9142}, {187, 32113}, {230, 231}, {248, 5063}, {338, 11623}, {1316, 6128}, {1576, 5095}, {9145, 32114}, {15000, 15118}, {15258, 35486}, {15268, 34158}, {15589, 35520}, {23976, 35133}

X(41359) = crossdifference of every pair of points on line {3, 2492}
X(41359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {67, 7669, 10991}, {10991, 15526, 67}, {15000, 20975, 15118}

X(41360) = X(4)X(99)∩X(110)X(193)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6 - 4*a^4*b^2 + 5*a^2*b^4 - b^6 - 4*a^4*c^2 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 - c^6) : :

X(41360) lies on the cubic K482 and these lines: {2, 8754}, {4, 99}, {25, 1634}, {110, 193}, {126, 136}, {230, 231}, {620, 2971}, {648, 6353}, {3515, 14900}, {11005, 32250}, {17907, 17983}

X(41360) = polar conjugate of the isotomic conjugate of X(14645)
X(41360) = X(63)-isoconjugate of X(14659)
X(41360) = crossdifference of every pair of points on line {3, 2510}
X(41360) = barycentric product X(4)*X(14645)
X(41360) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 14659}, {14645, 69}
X(41360) = {X(2),X(36898)}-harmonic conjugate of X(8754)

X(41361) = X(4)X(6)∩X(20)X(112)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6) : :

X(41361) lies on the cubics K141 and K617 and these lines: {2, 1235}, {3, 16318}, {4, 6}, {19, 23537}, {20, 112}, {22, 8879}, {24, 7735}, {25, 5305}, {30, 3172}, {32, 18533}, {39, 3541}, {69, 19595}, {76, 17907}, {83, 21459}, {92, 3673}, {132, 8721}, {161, 6525}, {186, 16306}, {216, 7383}, {230, 3147}, {232, 3542}, {254, 6531}, {264, 7803}, {297, 6515}, {315, 648}, {317, 7760}, {343, 14361}, {378, 7738}, {427, 9605}, {459, 18840}, {550, 8778}, {1033, 11414}, {1072, 8755}, {1075, 12251}, {1217, 37476}, {1289, 26269}, {1352, 39604}, {1370, 3162}, {1562, 5878}, {1593, 15048}, {1594, 7736}, {1783, 3436}, {1895, 4385}, {1968, 2549}, {2138, 37201}, {2139, 15466}, {2322, 16062}, {2393, 27373}, {2493, 7505}, {3199, 5309}, {3547, 22240}, {3575, 30435}, {3926, 34129}, {5094, 31406}, {5179, 36103}, {5280, 11392}, {5299, 11393}, {5304, 7487}, {5319, 10311}, {6103, 35486}, {6331, 40050}, {6527, 28425}, {6620, 40366}, {6656, 9308}, {7404, 26216}, {7500, 17409}, {7577, 31404}, {7748, 14581}, {7772, 27371}, {8750, 17732}, {8753, 15591}, {8779, 9833}, {8793, 17407}, {10313, 31305}, {10897, 21737}, {11206, 22135}, {11547, 40814}, {12173, 18907}, {14790, 22120}, {15454, 32708}, {15484, 23047}, {17911, 18683}, {20065, 40889}, {28405, 38652}, {28406, 30737}, {31400, 37119}, {32000, 32956}, {37186, 40807}, {37445, 41083}

X(41361) = isotomic conjugate of the isogonal conjugate of X(3162)
X(41361) = isotomic conjugate of polar conjugate of X(41766)
X(41361) = anticomplement of X(14376)
X(41361) = polar conjugate of X(13575)
X(41361) = anticomplement of the isogonal conjugate of X(8743)
X(41361) = anticomplement of the isotomic conjugate of X(17907)
X(41361) = polar conjugate of the isotomic conjugate of X(1370)
X(41361) = polar conjugate of the isogonal conjugate of X(159)
X(41361) = pole wrt polar circle of trilinear polar of X(13575) (line X(525)X(2485))
X(41361) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {4, 17492}, {19, 7391}, {22, 4329}, {25, 17481}, {28, 18656}, {82, 18018}, {92, 66}, {162, 3267}, {206, 6360}, {1760, 1370}, {1973, 20065}, {2172, 20}, {8743, 8}, {11605, 21274}, {17186, 20222}, {17409, 192}, {17453, 3164}, {17907, 6327}, {21034, 18666}, {24019, 8673}, {40938, 21289}
X(41361) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 4}, {17907, 2}
X(41361) = X(159)-cross conjugate of X(1370)
X(41361) = X(i)-isoconjugate of X(j) for these (i,j): {48, 13575}, {63, 34207}, {184, 39733}, {326, 40144}, {9247, 40009}, {24018, 39417}
X(41361) = cevapoint of X(i) and X(j) for these (i,j): {159, 3162}, {19595, 40938}
X(41361) = crosspoint of X(6331) and X(32230)
X(41361) = crosssum of X(2972) and X(3049)
X(41361) = barycentric product X(i)*X(j) for these {i,j}: {4, 1370}, {19, 21582}, {76, 3162}, {92, 18596}, {159, 264}, {281, 18629}, {315, 17407}, {393, 28419}, {455, 40009}, {1235, 8793}, {2052, 23115}, {14615, 33584}
X(41361) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 13575}, {25, 34207}, {92, 39733}, {159, 3}, {206, 39172}, {264, 40009}, {427, 39129}, {455, 159}, {1370, 69}, {2207, 40144}, {3162, 6}, {5286, 40185}, {8743, 40358}, {8793, 1176}, {17407, 66}, {17810, 33579}, {18596, 63}, {18629, 348}, {21582, 304}, {23115, 394}, {28419, 3926}, {32713, 39417}, {33584, 64}, {40357, 40404}
X(41361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1249, 8743}, {6, 27376, 4}, {232, 3767, 3542}, {393, 5286, 4}, {1249, 15262, 40138}, {1249, 18685, 34266}, {1990, 5254, 2207}, {2207, 5254, 4}, {5304, 7487, 10312}, {5523, 8743, 4}, {6530, 39646, 4}, {13854, 40938, 2}

X(41362) = X(2)X(17845)∩X(4)X(6)

Barycentrics    4*a^10 - 7*a^8*b^2 + 2*a^4*b^6 + 4*a^2*b^8 - 3*b^10 - 7*a^8*c^2 + 8*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 9*b^8*c^2 - 2*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 6*b^6*c^4 + 2*a^4*c^6 - 8*a^2*b^2*c^6 - 6*b^4*c^6 + 4*a^2*c^8 + 9*b^2*c^8 - 3*c^10 : :
X(41362) = 2 X[3] - 3 X[23332], 3 X[4] - X[1498], 7 X[4] - 3 X[5656], 3 X[4] - 2 X[5893], X[4] - 3 X[18405], 5 X[4] - X[34781], 4 X[5] - 3 X[10192], 3 X[5] - 2 X[10282], 2 X[5] - 3 X[23324], X[20] - 3 X[1853], 3 X[20] - 5 X[8567], X[64] - 3 X[32064], 2 X[140] - 3 X[23325], 3 X[154] - 5 X[3091], X[195] + 3 X[32402], 3 X[376] - 5 X[40686], 3 X[381] - X[9833], 9 X[381] - 5 X[14530], 3 X[381] - 2 X[16252], 3 X[382] + X[13093], 2 X[546] - 3 X[18376], 2 X[548] - 3 X[23329], 3 X[549] - 4 X[32767], 2 X[550] - 3 X[23328], 2 X[1498] - 3 X[2883], 7 X[1498] - 9 X[5656], X[1498] - 9 X[18405], 5 X[1498] - 3 X[34781], 3 X[1853] - 2 X[6696], 9 X[1853] - 5 X[8567], 7 X[2883] - 6 X[5656], 3 X[2883] - 4 X[5893], X[2883] - 6 X[18405], 5 X[2883] - 2 X[34781], 3 X[3060] - X[6293], 7 X[3090] - 5 X[17821]

X(41362) lies on the cubic K127 and these lines: {2, 17845}, {3, 16254}, {4, 6}, {5, 5944}, {20, 1853}, {30, 3357}, {51, 11743}, {64, 3146}, {66, 29181}, {140, 23325}, {141, 12362}, {154, 3091}, {159, 11479}, {161, 7503}, {184, 23047}, {185, 6746}, {195, 31724}, {221, 5229}, {235, 13851}, {343, 6145}, {376, 40686}, {381, 9833}, {382, 13093}, {403, 18394}, {427, 11572}, {459, 34286}, {524, 12429}, {546, 6759}, {548, 23329}, {549, 32767}, {550, 20299}, {567, 32379}, {578, 15872}, {599, 11821}, {858, 12278}, {1192, 23291}, {1594, 12289}, {1595, 13403}, {1596, 13419}, {1619, 5198}, {1885, 11550}, {1899, 12173}, {1994, 17824}, {2192, 5225}, {2393, 5907}, {2777, 14864}, {2917, 35921}, {3060, 6293}, {3090, 17821}, {3153, 14516}, {3529, 10606}, {3543, 5895}, {3564, 34788}, {3575, 13567}, {3589, 36989}, {3627, 5446}, {3628, 11202}, {3629, 10112}, {3830, 5878}, {3832, 11206}, {3853, 22802}, {5068, 35260}, {5073, 20427}, {5076, 12315}, {5889, 34751}, {5891, 12363}, {5925, 33703}, {6001, 31673}, {6225, 17578}, {6240, 15138}, {6288, 9935}, {6697, 21167}, {6756, 15873}, {6823, 23300}, {7395, 15577}, {7399, 20300}, {7487, 18918}, {7488, 32345}, {7507, 19467}, {7509, 35228}, {7530, 32321}, {7576, 38848}, {7687, 15647}, {7729, 12279}, {7973, 9812}, {8703, 25563}, {8883, 11077}, {9920, 34864}, {9924, 33537}, {10024, 18430}, {10151, 26883}, {10193, 32903}, {10295, 23294}, {10297, 10539}, {10323, 15578}, {10516, 15585}, {10575, 40928}, {11204, 12103}, {11381, 21652}, {11457, 35480}, {11536, 32365}, {11745, 18494}, {11750, 15760}, {11801, 13289}, {12082, 15579}, {12084, 32048}, {12134, 18404}, {12161, 40285}, {12164, 34777}, {12250, 15682}, {12262, 28164}, {12293, 14790}, {12605, 18474}, {13148, 13202}, {13371, 30522}, {13394, 18434}, {13598, 34146}, {14389, 32354}, {15074, 39884}, {15139, 34148}, {15153, 18533}, {15761, 18379}, {15805, 18420}, {16266, 18569}, {17702, 23315}, {17819, 31412}, {17823, 34007}, {18323, 18439}, {18377, 22660}, {18388, 31804}, {18483, 40658}, {18559, 26879}, {18568, 32139}, {18583, 34776}, {18951, 40909}, {19506, 32423}, {19925, 40660}, {20376, 32330}, {25328, 36201}, {26882, 35487}, {26937, 37196}, {31304, 32269}, {31383, 37197}, {32062, 36982}, {32743, 34153}, {35491, 40242}

X(41362) = midpoint of X(i) and X(j) for these {i,j}: {64, 3146}, {382, 14216}, {5073, 20427}, {5878, 34780}, {5895, 12324}, {5925, 33703}, {12293, 14790}, {18381, 34786}, {36851, 36990}
X(41362) = reflection of X(i) in X(j) for these {i,j}: {5, 18383}, {20, 6696}, {550, 20299}, {1498, 5893}, {2883, 4}, {5480, 18382}, {5894, 6247}, {6247, 18381}, {6759, 546}, {9833, 16252}, {10192, 23324}, {13289, 11801}, {15647, 7687}, {15761, 18379}, {16254, 38443}, {22660, 18377}, {22802, 3853}, {32330, 20376}, {32391, 32393}, {32392, 16625}, {34153, 32743}, {34774, 5480}, {34776, 18583}, {34782, 5}, {34785, 140}, {36989, 3589}, {40658, 18483}, {40660, 19925}
X(41362) = complement of X(17845)
X(41362) = X(34782)-of-Johnson-triangle
X(41362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1498, 5893}, {4, 6146, 12233}, {4, 12241, 5480}, {4, 18396, 12241}, {4, 18945, 6}, {4, 36990, 16656}, {5, 18383, 23324}, {5, 34782, 10192}, {20, 1853, 6696}, {381, 9833, 16252}, {550, 20299, 23328}, {1498, 5893, 2883}, {1498, 10982, 34117}, {1899, 12173, 13568}, {3146, 32064, 64}, {3543, 12324, 5895}, {3830, 34780, 5878}, {6146, 12233, 8550}, {6756, 18390, 15873}, {6759, 18376, 546}, {7507, 19467, 23292}, {10112, 31802, 3629}, {11572, 21659, 427}, {18396, 18405, 18382}, {18494, 39571, 11745}, {23324, 34782, 5}, {23325, 34785, 140}

X(41363) = X(4)X(6)∩X(112)X(511)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + a^2*c^4 - c^6) : :
X(41363) = 4 X[6] - X[13509]

X(41363) lies on the cubic K289 and these lines: {4, 6}, {24, 40825}, {32, 6403}, {69, 34138}, {110, 14580}, {112, 511}, {182, 39575}, {186, 1691}, {232, 1692}, {323, 35325}, {427, 34945}, {525, 2451}, {729, 935}, {1351, 3172}, {1570, 14581}, {1597, 39524}, {1843, 10312}, {1968, 5028}, {1993, 3162}, {2138, 40318}, {2393, 15388}, {2456, 2967}, {3060, 17409}, {3094, 3520}, {3124, 37777}, {3564, 16318}, {5012, 40938}, {5039, 8541}, {5052, 8537}, {5207, 22151}, {5305, 39871}, {5921, 23128}, {5938, 12167}, {5999, 14965}, {6515, 8879}, {8548, 18337}, {8778, 33878}, {10316, 12220}, {10766, 34146}, {11442, 13854}, {14064, 26206}, {19131, 22240}, {39081, 40601}

X(41363) = reflection of X(i) in X(j) for these {i,j}: {69, 34138}, {2715, 1692}, {13509, 34137}, {34137, 6}
X(41363) = 2nd-Lemoine-circle-inverse of X(4)
X(41363) = polar circle inverse of X(5254)
X(41363) = symgonal image of X(1692)
X(41363) = polar conjugate of the isotomic conjugate of X(37183)
X(41363) = X(35142)-Ceva conjugate of X(25)
X(41363) = X(i)-isoconjugate of X(j) for these (i,j): {304, 39644}, {326, 39645}
X(41363) = crosspoint of X(23964) and X(32697)
X(41363) = crosssum of X(684) and X(38356)
X(41363) = crossdifference of every pair of points on line {520, 6467}
X(41363) = barycentric product X(i)*X(j) for these {i,j}: {4, 37183}, {35142, 39072}
X(41363) = barycentric quotient X(i)/X(j) for these {i,j}: {1974, 39644}, {2207, 39645}, {37183, 69}, {39072, 3564}
X(41363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {232, 1692, 19128}

X(41364) = X(4)X(6)∩X(19)X(58)

Barycentrics    a*(a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(41364) lies on the cubic K379 and these lines: {4, 6}, {19, 58}, {21, 40457}, {27, 17903}, {28, 961}, {81, 92}, {108, 40590}, {112, 1295}, {196, 1396}, {204, 2328}, {281, 2303}, {284, 2331}, {314, 648}, {478, 14257}, {859, 3209}, {998, 1474}, {1010, 2322}, {1333, 14571}, {1783, 2287}, {2092, 7414}, {2298, 41013}, {2360, 3213}, {3436, 22132}, {5019, 37117}, {5236, 17189}, {5336, 30733}, {8885, 40937}, {9308, 41236}, {14534, 31623}, {16318, 37360}, {17408, 34263}, {18687, 35466}

X(41364) = polar conjugate of the isotomic conjugate of X(16049)
X(41364) = X(i)-Ceva conjugate of X(j) for these (i,j): {14534, 4}, {31623, 28}
X(41364) = X(i)-isoconjugate of X(j) for these (i,j): {71, 8048}, {73, 34277}, {226, 39167}, {306, 3435}, {24018, 40097}
X(41364) = barycentric product X(i)*X(j) for these {i,j}: {4, 16049}, {21, 14257}, {27, 1766}, {28, 3436}, {29, 21147}, {162, 21186}, {197, 286}, {314, 17408}, {478, 31623}, {648, 6588}, {1325, 39990}, {1474, 20928}
X(41364) = barycentric quotient X(i)/X(j) for these {i,j}: {28, 8048}, {197, 72}, {205, 71}, {478, 1214}, {1172, 34277}, {1766, 306}, {2194, 39167}, {2203, 3435}, {3436, 20336}, {6588, 525}, {14257, 1441}, {16049, 69}, {17408, 65}, {20928, 40071}, {21147, 307}, {21186, 14208}, {22132, 3998}, {32713, 40097}
X(41364) = {X(4),X(1249)}-harmonic conjugate of X(34266)

X(41365) = X(3)X(2052)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 4*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 3*a^4*c^4 + a^2*b^2*c^4 + 4*b^4*c^4 - a^2*c^6 - 2*b^2*c^6) : :

X(41365) lies on the cubic K1057 and these lines: {3, 2052}, {4, 6}, {5, 11547}, {24, 13450}, {25, 1093}, {107, 3517}, {158, 11500}, {183, 18027}, {264, 7395}, {275, 11426}, {324, 7503}, {436, 19357}, {578, 8887}, {648, 12160}, {1075, 9786}, {1105, 21312}, {1192, 40664}, {1217, 7386}, {1598, 1629}, {1656, 14165}, {1941, 37498}, {3527, 19169}, {3575, 14569}, {5562, 9308}, {6524, 7487}, {6750, 18390}, {7399, 17907}, {10282, 37070}, {11479, 21447}, {37127, 37506}

X(41365) = X(i)-isoconjugate of X(j) for these (i,j): {255, 13599}, {6570, 24018}
X(41365) = crosssum of X(680) and X(1364)
X(41365) = barycentric product X(i)*X(j) for these {i,j}: {275, 8887}, {578, 2052}
X(41365) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 13599}, {578, 394}, {8884, 37872}, {8887, 343}, {32713, 6570}
X(41365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 41204, 1181}, {53, 12241, 4}, {1093, 8884, 25}, {1629, 14249, 1598}

X(41366) = X(4)X(6)∩X(112)X(550)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6 + a^4*b^2 - 2*a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 - c^6) : :

X(41366) lies on the cubic K and these lines: {4, 6}, {112, 550}, {140, 16318}, {232, 7755}, {427, 3108}, {648, 7768}, {1657, 3172}, {3162, 16063}, {3518, 5306}, {5007, 7576}, {5158, 14788}, {5319, 10594}, {5368, 10985}, {6103, 10018}, {6793, 10282}, {7495, 40938}, {7739, 35502}, {7760, 37765}, {7772, 15559}, {8750, 41326}, {9607, 14865}, {11331, 37636}, {14581, 18560}, {17409, 37900}, {18487, 34613}, {22240, 34002}

X(41366) = X(10159)-Ceva conjugate of X(4)
X(41366) = polar conjugate of isotomic conjugate of Lucas-isogonal conjugate of X(54)
X(41366) = polar conjugate of isotomic conjugate of anticomplement of X(428)

X(41367) = X(4)X(6)∩X(20)X(1625)

Barycentrics    a^2*(2*a^6*b^2 - 4*a^4*b^4 + 2*a^2*b^6 + 2*a^6*c^2 + 3*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 3*b^6*c^2 - 4*a^4*c^4 - 2*a^2*b^2*c^4 + 6*b^4*c^4 + 2*a^2*c^6 - 3*b^2*c^6) : :

X(41367) lies on the cubic K664 and these lines: {4, 6}, {20, 1625}, {32, 14157}, {39, 12290}, {112, 6759}, {187, 26882}, {216, 15058}, {232, 6241}, {389, 33885}, {577, 8718}, {1614, 1968}, {1987, 11270}, {3172, 32063}, {3199, 5890}, {3289, 3529}, {3528, 40805}, {3832, 41334}, {5206, 37114}, {6000, 39575}, {8778, 14530}, {9781, 33842}, {9968, 10766}, {10312, 26883}, {12162, 22240}, {12279, 14961}, {15355, 40647}

X(41367) = crossdifference of every pair of points on line {520, 31277}
X(41367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {217, 3331, 38297}, {217, 38297, 4}, {1498, 8743, 13509}, {3331, 32445, 4}, {32445, 38297, 217}

X(41368) = X(4)X(6)∩X(19)X(7107)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^4 - a^2*b^2 + a^2*b*c - b^3*c - a^2*c^2 + 2*b^2*c^2 - b*c^3)*(a^4 - a^2*b^2 - a^2*b*c + b^3*c - a^2*c^2 + 2*b^2*c^2 + b*c^3) : :

X(41368) lies on the cubic K222 and these lines: {4, 6}, {19, 7107}, {20, 36434}, {32, 14249}, {99, 40887}, {107, 187}, {115, 34170}, {232, 6529}, {248, 20031}, {316, 36426}, {385, 6528}, {401, 23582}, {450, 39034}, {1093, 1968}, {1629, 33842}, {6587, 39201}, {7735, 36876}, {10317, 34334}, {17966, 36127}

X(41368) = polar conjugate of the isotomic conjugate of X(450)
X(41368) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1942}, {2713, 24018}, {7016, 40843}
X(41368) = crossdifference of every pair of points on line {520, 6509}
X(41368) = barycentric product X(i)*X(j) for these {i,j}: {4, 450}, {107, 2797}, {243, 1940}, {393, 40888}, {1947, 2202}, {1948, 7120}
X(41368) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1942}, {450, 69}, {2797, 3265}, {7120, 40843}, {32713, 2713}, {35236, 2972}, {40888, 3926}
X(41368) = {P,U}-harmonic conjugate of X(393), where P and U are the circumcircle intercepts of the van Aubel line

X(41369) = X(4)X(6)∩X(39)X(23976)

Barycentrics    4*a^10 - 3*a^8*b^2 - 6*a^4*b^6 + 4*a^2*b^8 + b^10 - 3*a^8*c^2 + 6*a^4*b^4*c^2 - 3*b^8*c^2 + 6*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 - 6*a^4*c^6 + 2*b^4*c^6 + 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(41369) lies on the cubic K858 and these lines: {4, 6}, {39, 23976}, {112, 5894}, {169, 23982}, {185, 12145}, {1073, 9605}, {1970, 9607}, {3172, 15311}, {3346, 11425}, {5304, 9786}, {5305, 13567}, {6247, 16318}, {6793, 23328}, {8779, 34782}, {10192, 39575}, {11348, 11427}, {11437, 14917}, {13526, 15048}, {13568, 30435}

X(41369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5286, 12241}, {8743, 15341, 2883}

X(41370) = X(2)X(112)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6) : :

X(41370) lies on the cubic K2183 and these lines: {2, 112}, {4, 6}, {5, 3172}, {20, 14961}, {25, 18907}, {32, 3542}, {132, 7694}, {140, 8778}, {187, 35486}, {232, 7737}, {235, 30435}, {297, 37645}, {315, 28419}, {316, 17907}, {317, 7812}, {378, 7736}, {381, 16318}, {403, 7735}, {427, 15484}, {459, 18842}, {468, 1384}, {574, 35485}, {648, 11185}, {1235, 32971}, {1285, 6353}, {1383, 4232}, {1783, 3434}, {1885, 9605}, {1968, 2548}, {1992, 37855}, {3053, 3147}, {3089, 10312}, {3162, 6997}, {3199, 37122}, {3516, 31406}, {3520, 31400}, {5133, 8879}, {5304, 6623}, {5305, 37197}, {5475, 14581}, {6525, 20410}, {6995, 33885}, {7500, 40938}, {7738, 18560}, {7774, 15014}, {8370, 9308}, {8753, 14593}, {13575, 26209}, {13596, 39662}, {14361, 41244}, {14376, 26218}, {15472, 15920}, {23115, 37201}, {28708, 32006}, {31404, 37119}, {32661, 34518}, {32674, 34029}, {37174, 37766}

X(41370) = polar conjugate of the isotomic conjugate of X(7493)
X(41370) = polar conjugate of the isogonal conjugate of X(19153)
X(41370) = X(598)-Ceva conjugate of X(4)
X(41370) = X(19153)-cross conjugate of X(7493)
X(41370) = X(24018)-isoconjugate of X(39382)
X(41370) = barycentric product X(i)*X(j) for these {i,j}: {4, 7493}, {264, 19153}, {4232, 34165}, {16789, 32085}
X(41370) = barycentric quotient X(i)/X(j) for these {i,j}: {7493, 69}, {16789, 3933}, {19153, 3}, {32713, 39382}
X(41370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1249, 5523}, {4, 8744, 393}, {232, 7737, 18533}, {1968, 2548, 3541}, {2207, 7745, 4}, {2207, 8746, 8744}, {5523, 8743, 1249}, {35913, 35914, 35902}

X(41371) = X(2)X(1972)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 4*a^4*b^4 - b^8 - 4*a^6*c^2 + 2*a^4*b^2*c^2 + 2*b^6*c^2 + 4*a^4*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8) : :
Barycentrics    (tan A) (tan A - tan B - tan C - cot ω) : :

X(41371) lies on the cubic K791 and these lines: {2, 1972}, {4, 6}, {5, 9308}, {24, 40981}, {51, 11547}, {132, 9744}, {186, 16324}, {232, 9753}, {263, 6403}, {264, 14561}, {297, 1351}, {315, 39604}, {317, 576}, {403, 2452}, {436, 6524}, {450, 37645}, {458, 18583}, {459, 14494}, {511, 17907}, {648, 1352}, {1075, 3541}, {1941, 6815}, {2322, 7380}, {3090, 32000}, {3186, 3462}, {3618, 37124}, {5093, 27377}, {5476, 39530}, {5667, 35485}, {6619, 18950}, {6747, 15004}, {7792, 40801}, {8889, 14361}, {10788, 18533}, {11002, 37766}, {13860, 16318}, {20423, 37765}, {21850, 37200}

X(41371) =X(262)-Ceva conjugate of X(4)
X(41371) = polar conjugate of isotomic conjugate of anticomplement of X(458)
X(41371) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1249, 41204}, {6, 6530, 4}, {6, 15274, 6530}, {393, 14853, 4}, {576, 39569, 317}, {1990, 5480, 33971}, {5480, 33971, 4}, {6524, 11427, 436}, {6776, 10002, 4}, {10002, 40138, 6776}

X(41372) = X(3)X(107)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 12*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - 2*b^6*c^2 + 3*a^4*c^4 - 7*a^2*b^2*c^4 + 4*b^4*c^4 - a^2*c^6 - 2*b^2*c^6) : :

X(41372) lies on the cubic K297 and these lines: {2, 36876}, {3, 107}, {4, 6}, {64, 1075}, {183, 6528}, {376, 6525}, {381, 34170}, {450, 37497}, {631, 6523}, {1093, 1593}, {1596, 11547}, {1597, 2052}, {1629, 18535}, {1853, 6761}, {1941, 17814}, {3090, 6526}, {3168, 10605}, {3357, 14363}, {5198, 8884}, {7738, 36434}, {7841, 36426}, {9308, 15030}, {10606, 40664}, {11430, 37070}, {11441, 35311}, {13450, 35502}, {14530, 38808}, {15466, 21312}, {17538, 34286}, {28783, 40448}, {31859, 40887}

X(41372) = barycentric product X(2052)*X(37480)
X(41372) = barycentric quotient X(37480)/X(394)
X(41372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1249, 5656}, {4, 5656, 1515}

X(41373) = X(3)X(8612)∩X(4)X(6)

Barycentrics    a^2*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 + a^10*c^2 - a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 2*a^4*b^6*c^2 + a^2*b^8*c^2 - b^10*c^2 - 4*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 + 4*b^8*c^4 + 6*a^6*c^6 + 2*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 6*b^6*c^6 - 4*a^4*c^8 + a^2*b^2*c^8 + 4*b^4*c^8 + a^2*c^10 - b^2*c^10) : :
X(41373) = 3 X[154] - 2 X[160], 3 X[154] - X[31382], 3 X[1853] - 4 X[34845]

Let P1 and P2 be the two points with tripolar coordinates |csc 2A| : : (or, equivalently, csc A' : :, where A'B'C' is the orthic triangle). Then X(41373) is the {P1,P2}-harmonic conjugate of X(3). (Randy Hutson, April 13, 2021)

X(41373) lies on the cubic K075 and these lines: {3, 8612}, {4, 6}, {25, 6752}, {31, 10535}, {48, 1950}, {64, 26897}, {154, 160}, {184, 35709}, {216, 6000}, {233, 20299}, {571, 1613}, {577, 6759}, {1609, 9244}, {1619, 10329}, {1625, 15905}, {1853, 34845}, {1988, 14533}, {3003, 36982}, {3289, 11206}, {3357, 10979}, {5895, 31353}, {8439, 39849}, {9833, 13322}, {10132, 10533}, {10133, 10534}, {10182, 36422}, {10282, 22052}, {15087, 18416}, {17821, 26876}

X(41373) = reflection of X(31382) in X(160)
X(41373) = isogonal conjugate of X(34287)
X(41373) = isogonal conjugate of the polar conjugate of X(1075)
X(41373) = tangential-isogonal conjugate of X(35225)
X(41373) = tangential-isotomic conjugate of X(3)
X(41373) = X(75)-of-tangential-triangle if ABC is acute
X(41373) = X(i)-Ceva conjugate of X(j) for these (i,j): {577, 6}, {6759, 154}
X(41373) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34287}, {92, 13855}
X(41373) = crosspoint of X(110) and X(34538)
X(41373) = crosssum of X(i) and X(j) for these (i,j): {216, 13322}, {523, 35071}, {6368, 38976}
X(41373) = barycentric product X(3)*X(1075)
X(41373) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34287}, {184, 13855}, {1075, 264}
X(41373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1498, 17849}, {6, 38297, 53}, {154, 31382, 160}, {217, 3087, 6}, {8745, 39643, 6}, {8761, 21767, 48}, {10675, 10676, 11456}, {12964, 12970, 1181}, {20233, 40138, 6}

X(41374) = X(3)X(253)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(11*a^8 - 20*a^6*b^2 + 10*a^4*b^4 - 4*a^2*b^6 + 3*b^8 - 20*a^6*c^2 + 12*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 4*b^6*c^2 + 10*a^4*c^4 + 4*a^2*b^2*c^4 - 14*b^4*c^4 - 4*a^2*c^6 + 4*b^2*c^6 + 3*c^8) : :
X(41374) = 3 X[4] - 4 X[10002], X[4] - 4 X[15258], 5 X[4] - 8 X[15274], 3 X[4] - 8 X[15576], 3 X[154] - X[5922], 5 X[631] - 4 X[20208], 3 X[1249] - 2 X[10002], 5 X[1249] - 4 X[15274], 3 X[1249] - 4 X[15576], 7 X[3090] - 8 X[20204], 5 X[3522] - X[20218], 11 X[3525] - 10 X[20200], X[10002] - 3 X[15258], 5 X[10002] - 6 X[15274], X[14927] + 3 X[31887], 5 X[15258] - 2 X[15274], 3 X[15258] - 2 X[15576], 3 X[15274] - 5 X[15576]

X(41374) lies on the cubic K047 and these lines: {3, 253}, {4, 6}, {20, 15312}, {98, 38253}, {107, 3079}, {112, 15428}, {154, 459}, {542, 6330}, {631, 20208}, {648, 14927}, {1370, 35311}, {2777, 33702}, {3090, 20204}, {3183, 34782}, {3522, 20218}, {3524, 12096}, {3525, 20200}, {5759, 18283}, {5895, 22049}, {5897, 30247}, {5984, 37187}, {6621, 6759}, {6624, 32063}, {9530, 11001}, {15005, 17845}, {25406, 32000}

X(41374) = midpoint of X(20) and X(17037)
X(41374) = reflection of X(i) in X(j) for these {i,j}: {4, 1249}, {253, 3}, {1249, 15258}, {10002, 15576}
X(41374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 41204, 33630}, {10002, 15258, 15576}, {10002, 15576, 1249}, {10783, 10784, 6146}, {11206, 14361, 3079}, {39874, 41204, 4}

X(41375) = X(2)X(34427)∩X(4)X(6)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(b^2 + c^2)*(a^4 - b^4 - c^4) : :

X(41375) lies on the cubic K975 and these lines: {2, 34427}, {4, 6}, {22, 17907}, {25, 26269}, {107, 7495}, {132, 216}, {264, 5133}, {297, 40052}, {308, 6528}, {1176, 32713}, {1235, 39129}, {3313, 27373}, {6525, 7494}, {7403, 14978}, {11574, 39604}, {12253, 35921}, {13160, 14249}, {15760, 34334}, {34603, 37765}, {36426, 41237}

X(41375) = polar conjugate of X(40404)
X(41375) = isotomic conjugate of the isogonal conjugate of X(27373)
X(41375) = polar conjugate of the isogonal conjugate of X(40938)
X(41375) = X(6528)-Ceva conjugate of X(33294)
X(41375) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40404}, {255, 16277}, {2156, 28724}
X(41375) = cevapoint of X(27373) and X(40938)
X(41375) = crosspoint of X(264) and X(17907)
X(41375) = barycentric product X(i)*X(j) for these {i,j}: {76, 27373}, {107, 23881}, {264, 40938}, {315, 27376}, {427, 17907}, {1235, 8743}, {2052, 3313}, {16715, 41013}, {18027, 23208}
X(41375) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40404}, {22, 28724}, {393, 16277}, {427, 14376}, {3313, 394}, {8743, 1176}, {16715, 1444}, {17409, 10547}, {17907, 1799}, {23208, 577}, {23881, 3265}, {27371, 41168}, {27373, 6}, {27376, 66}, {40938, 3}
X(41375) = {X(4),X(1249)}-harmonic conjugate of X(5596)

X(41376) = X(4)X(6)∩X(112)X(154)

Barycentrics    a^2*(a^8 + 4*a^6*b^2 - 10*a^4*b^4 + 4*a^2*b^6 + b^8 + 4*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 8*b^6*c^2 - 10*a^4*c^4 - 4*a^2*b^2*c^4 + 14*b^4*c^4 + 4*a^2*c^6 - 8*b^2*c^6 + c^8) : :

X(41376) lies on the cubic K810 and these lines: {4, 6}, {64, 39575}, {112, 154}, {232, 10605}, {394, 1625}, {1294, 26714}, {1350, 38689}, {1384, 1495}, {1968, 19357}, {3172, 6759}, {8778, 10282}, {8779, 32063}, {9605, 11381}, {11550, 15484}, {15355, 37475}, {17810, 33885}, {18907, 31383}, {26883, 30435}, {34108, 35259}

X(41376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1498, 13509}, {1249, 5656, 15341}, {2207, 32445, 1181}, {8743, 13509, 6}, {8744, 11456, 6}

X(41377) = X(4)X(6)∩X(98)X(186)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^10 - 2*a^6*b^4 + a^2*b^8 - 5*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - a^2*b^6*c^2 + 2*b^8*c^2 - 2*a^6*c^4 + 4*a^4*b^2*c^4 - 2*b^6*c^4 - a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 2*b^2*c^8) : :
X(41377) = 3 X[186] - 2 X[935], 2 X[35907] - 3 X[41204]

X(41377) lies on the cubic K302 and these lines: {4, 6}, {24, 5938}, {30, 12384}, {98, 186}, {297, 3448}, {458, 11003}, {525, 35474}, {1235, 12203}, {2697, 7464}, {2741, 32704}, {3520, 11257}, {5984, 40889}, {7577, 9744}, {10295, 14654}, {14957, 35311}, {34797, 36998}, {35503, 39647}, {37946, 38672}

X(41377) = reflection of X(i) in X(j) for these {i,j}: {4, 5523}, {7464, 2697}, {13509, 18338}
X(41377) = polar-circle-inverse of X(5480)
X(41377) = pole wrt polar circle of the line perpendicular to X(4)X(6) at the midpoint of X(4) and X(6) (line X(525)X(5480))
X(41377) = {X(4),X(41204)}-harmonic conjugate of X(8744)

X(41378) = X(2)X(1343)∩X(32)X(184)

Barycentrics    a^4*(a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 - c^4)*(a^4 + 2*a^2*b^2 + 2*a^2*c^2 + 2*b^2*c^2 - (2*a^2 + b^2 + c^2)*Sqrt[a^2*b^2 + a^2*c^2 + b^2*c^2]) : :

X(41378) lies on the cubic K177 and these lines: {2, 1343}, {6, 15247}, {32, 184}, {154, 15248}, {1180, 1670}, {1671, 2979}, {15080, 15244}


X(41378) = isogonal conjugate of the isotomic conjugate of X(1671)
X(41378) = X(1343)-Ceva conjugate of X(6)
X(41378) = X(75)-isoconjugate of X(1676)
X(41378) = barycentric product X(i)*X(j) for these {i,j}: {6, 1671}, {160, 1677}, {184, 16246}
X(41378) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1676}, {1671, 76}, {3202, 1670}, {16246, 18022}
X(41378) = {X(32),X(3051)}-harmonic conjugate of X(41379)
X(41378) = {X(184),X(237)}-harmonic conjugate of X(41379)

X(41379) = X(2)X(1342)∩X(32)X(184)

Barycentrics    a^4*(a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 - c^4)*(a^4 + 2*a^2*b^2 + 2*a^2*c^2 + 2*b^2*c^2 + (2*a^2 + b^2 + c^2)*Sqrt[a^2*b^2 + a^2*c^2 + b^2*c^2]) : :

X(41379) lies on the cubic K and these lines: {2, 1342}, {6, 15248}, {32, 184}, {154, 15247}, {1180, 1671}, {1670, 2979}, {15080, 15245}

X(41379) = isogonal conjugate of the isotomic conjugate of X(1670)
X(41379) = X(1342)-Ceva conjugate of X(6)
X(41379) = X(75)-isoconjugate of X(1677)
X(41379) = barycentric product X(i)*X(j) for these {i,j}: {6, 1670}, {160, 1676}
X(41379) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1677}, {1670, 76}, {3202, 1671}
X(41379) = {X(32),X(3051)}-harmonic conjugate of X(41378)
X(41379) = {X(184),X(237)}-harmonic conjugate of X(41378)

leftri

Centers of antiparallels conics: X(41380)-X(41388)

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This preamble and centers X(41380)-X(41388) were contributed by César Eliud Lozada, February 22, 2021.

Draw antiparallels through the symmedian point K. The points where these lines intersect the sides then lie on a circle, known as the cosine circle (or sometimes the second Lemoine circle). (Reference: Weisstein, Eric W. "Cosine Circle." From MathWorld--A Wolfram Web Resource)

More generally, if P is a point on the Jerabek circumhyperbola of a triangle ABC then the antiparallel lines through P cut the sidelines of the triangle in six points lying on a conic, here named the antiparallels conic of P and denoted as 𝕁(P).

The appearance of (i, j) in the following list means that the center of 𝕁(X(i)) is X(j):
 (3, 37864), (4, 6), (6, 6), (54, 41380), (64, 33584), (65, 41381), (66, 41382), (67, 41383), (68, 41384), (69, 41385), (70, 41386), (71, 41387), (72, 41388), (74, 40355).

𝕁(P) is a circle for P=X(6) and a degenerated conic for P in {X(1987), X(8678), X(8679)}. 𝕁(P) is never a parabola.

X(41380) = CENTER OF THE ANTIPARALLELS CONIC OF X(54)

Barycentrics    (SB+SC)*(S^2+SA*SC)*(S^2+SA*SB)*(3*S^2-SB^2)*(3*S^2-SC^2)*(3*S^2-4*R^2*(SA-SW)-SW^2) : :

X(41380) lies on the line {570, 8603}

X(41381) = CENTER OF THE ANTIPARALLELS CONIC OF X(65)

Barycentrics    a^2*((b^2+c^2)*a^4-(b^2-c^2)*(b-c)*a^3-(b^2+c^2)*(b^2+b*c+c^2)*a^2+(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c)*(b+c)*(a-b+c)*(a+b-c) : :

X(41381) lies on these lines: {42, 181}, {2171, 2310}, {3271, 21741}

X(41382) = CENTER OF THE ANTIPARALLELS CONIC OF X(66)

Barycentrics    a^2*(a^8+2*(b^2+c^2)*a^6-2*(b^4+c^4)*a^4-2*(b^6+c^6)*a^2+(b^4+c^4)*(b^2-c^2)^2)*(a^4-b^4+c^4)*(a^4+b^4-c^4) : :

X(41382) lies on these lines: {32, 1843}, {66, 7737}, {5017, 34138}

X(41383) = CENTER OF THE ANTIPARALLELS CONIC OF X(67)

Barycentrics    a^2*(a^10+2*(b^2+c^2)*a^8-5*(b^4+b^2*c^2+c^4)*a^6-(b^2+c^2)*(b^4-9*b^2*c^2+c^4)*a^4+(4*b^8+4*c^8-(7*b^4-2*b^2*c^2+7*c^4)*b^2*c^2)*a^2-(b^6+c^6)*(b^2-c^2)^2)*(a^2+c^2-2*b^2)*(a^4-c^2*a^2+c^4-b^4)*(a^2+b^2-2*c^2)*(a^4-b^2*a^2+b^4-c^4) : :

X(41383) lies on these lines: {187, 2393}, {842, 10422}, {10630, 20410}

X(41384) = CENTER OF THE ANTIPARALLELS CONIC OF X(68)

Barycentrics    (2*R^2-SC)*(2*R^2-SB)*(S^2-SB*SC)*(S^2-SB^2)*(S^2-SC^2)*((2*R^2+SW)*S^2+(2*R^2-SW)*(2*(5*SA-2*SW)*R^2-2*SA^2+2*SB*SC+SW^2)) : :

X(41384) lies on the line {577, 2351}

X(41385) = CENTER OF THE ANTIPARALLELS CONIC OF X(69)

Barycentrics    (2*SB-SW)*(S^2-SB*SC)*(2*SC-SW)*(8*R^2*S^2-((6*SA-4*SW)*R^2-SA^2+SB*SC+SW^2)*SW) : :

X(41385) lies on these lines: {3, 6391}, {3565, 12272}

X(41386) = CENTER OF THE ANTIPARALLELS CONIC OF X(70)

Barycentrics    a^2*(a^14-(b^2+c^2)*a^12-(5*b^4+6*b^2*c^2+5*c^4)*a^10+(b^2+c^2)*(9*b^4+4*b^2*c^2+9*c^4)*a^8-(b^8+c^8+2*(4*b^4+5*b^2*c^2+4*c^4)*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)*(7*b^4-2*b^2*c^2+7*c^4)*a^4+(5*b^8+2*b^4*c^4+5*c^8)*(b^2-c^2)^2*a^2-(b^8-c^8)*(b^2-c^2)^3)*(a^6-(3*b^2+c^2)*a^4+(3*b^4-c^4)*a^2-(b^2-c^2)^3)*(a^8-2*(b^2+c^2)*a^6+2*c^4*a^4+2*(b^6-c^6)*a^2-(b^4-c^4)*(b^2-c^2)^2)*(a^6-(b^2+3*c^2)*a^4-(b^4-3*c^4)*a^2+(b^2-c^2)^3)*(a^8-2*(b^2+c^2)*a^6+2*b^4*a^4-2*(b^6-c^6)*a^2+(b^4-c^4)*(b^2-c^2)^2) : :

X(41386) lies on these lines: {}

X(41387) = CENTER OF THE ANTIPARALLELS CONIC OF X(71)

Barycentrics    a^2*(a^3-(b+c)*c*a+(b^2-c^2)*b)*(a+b-c)*(a^3-(b+c)*b*a-(b^2-c^2)*c)*(a-b+c)*((b+c)^2*a^11-(b+c)^3*a^10-(3*b^4+3*c^4+4*(b^2+b*c+c^2)*b*c)*a^9+2*(b+c)*(2*b^4+2*c^4+(2*b^2+b*c+2*c^2)*b*c)*a^8+(2*b^6+2*c^6+(b^4+c^4+5*(b+c)^2*b*c)*b*c)*a^7-(b^3+c^3)*(6*b^4+6*c^4+(7*b^2+4*b*c+7*c^2)*b*c)*a^6+(2*b^6+2*c^6-(3*b^4+3*c^4+(b^2+c^2)*b*c)*b*c)*(b+c)^2*a^5+(4*b^6+4*c^6-(9*b^4+9*c^4-(17*b^2-20*b*c+17*c^2)*b*c)*b*c)*(b+c)^3*a^4-(b^2-c^2)^2*(3*b^6+3*c^6-(b^2-c^2)^2*b*c)*a^3-(b^2-c^2)^2*(b+c)*(b^6+c^6+(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*b*c)*a^2+(b^4+c^4-(b^2-b*c+c^2)*b*c)*(b^2-c^2)^4*a+(b^3+c^3)*(b^2-c^2)^4*b*c)*(b+c)*(-a^2+b^2+c^2) : :

X(41387) lies on the line {2197, 2200}

X(41388) = CENTER OF THE ANTIPARALLELS CONIC OF X(72)

Barycentrics    a^2*(a^3+(b-c)*a^2+(b^2-2*b*c-c^2)*a+(b-c)*(b^2-c^2))*(a^3-(b-c)*a^2-(b^2+2*b*c-c^2)*a+(b-c)*(b^2-c^2))*((b^2+c^2)*a^10-(b+c)^3*a^9-(2*b^2+3*b*c+2*c^2)*(b^2+c^2)*a^8+2*(b^2+c^2)*(b+c)^3*a^7+2*(2*b^2-b*c+2*c^2)*(b+c)^2*b*c*a^6-2*(b+c)*(2*b^2+3*b*c+2*c^2)*(b^2+c^2)*b*c*a^5+2*(b^6+c^6-(b^3+c^3)*(b+c)*b*c)*(b+c)^2*a^4-2*(b^6+c^6-(4*b^4+4*c^4-(5*b^2-6*b*c+5*c^2)*b*c)*b*c)*(b+c)^3*a^3-(b^2-c^2)^2*(b^2+4*b*c+c^2)*(b^4+c^4)*a^2+(b^2-c^2)^3*(b-c)*(b^4+c^4)*a+(b^4-c^4)*(b^2-c^2)^3*b*c)*(b+c)*(-a^2+b^2+c^2) : :

X(41388) lies on the line {228, 18591}

X(41389) = X(1)X(6)∩X(8)X(6893)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(41389) = 3 X[210] - X[36920], 2 X[3660] - 3 X[34123], 3 X[4511] + X[12532], X[4867] + 3 X[5692], 3 X[6735] - X[39776]

X(41389) lies on the curve Q101 and these lines: {1, 6}, {8, 6893}, {63, 10269}, {65, 26364}, {78, 5887}, {80, 3880}, {104, 912}, {119, 517}, {191, 14803}, {200, 12703}, {210, 36920}, {329, 12115}, {474, 12709}, {519, 15558}, {758, 3911}, {952, 17615}, {971, 10609}, {997, 1470}, {1000, 3421}, {1858, 22836}, {2057, 5687}, {2077, 2932}, {2800, 6745}, {2975, 24927}, {3419, 26333}, {3660, 34123}, {3689, 17638}, {3753, 5219}, {3811, 26358}, {3812, 37701}, {3869, 5552}, {3876, 5554}, {3878, 10915}, {3916, 37561}, {3927, 16203}, {3935, 13278}, {3940, 10679}, {3984, 37622}, {4358, 38955}, {5044, 24982}, {5087, 39692}, {5123, 8068}, {5316, 10176}, {5587, 10893}, {5744, 10202}, {5836, 7951}, {5882, 12059}, {5902, 31190}, {6256, 14110}, {6261, 11517}, {6282, 12686}, {6737, 20117}, {10200, 25917}, {11415, 37585}, {12608, 31806}, {14077, 30700}, {14740, 28234}, {21077, 26482}, {21616, 26476}, {25253, 41013}, {27385, 34339}, {35251, 40266}

X(41389) = midpoint of X(3689) and X(17638)
X(41389) = X(1000)-Ceva conjugate of X(1145)
X(41389) = X(i)-isoconjugate of X(j) for these (i,j): {104, 998}, {513, 36090}, {514, 32685}
X(41389) = crossdifference of every pair of points on line {513, 2423}
X(41389) = intersection of Simson line of X(100) and trilinear polar of X(100)
X(41389) = barycentric product X(i)*X(j) for these {i,j}: {517, 17740}, {908, 997}, {2397, 9001}, {17757, 26637}
X(41389) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 36090}, {692, 32685}, {997, 34234}, {1470, 34051}, {2183, 998}, {2427, 9058}, {9001, 2401}, {17740, 18816}
X(41389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 392, 956}, {908, 6735, 119}, {2077, 2950, 17613}, {3869, 5552, 37562}, {5440, 17613, 2932}, {12648, 31018, 30513}

X(41390) = X(6)X(13)∩X(476)X(541)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(4*a^6 - 7*a^4*b^2 + 2*a^2*b^4 + b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4 + c^6)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :

X(41390) lies on the curve Q101 and these lines: {6, 13}, {476, 541}, {477, 15396}, {1522, 1523}, {2777, 34192}, {14254, 15063}, {16003, 39170}, {32423, 33505}

X(41390) = crossdifference of every pair of points on line {526, 2436}
X(41390) = X(i)-isoconjugate of X(j) for these (i,j): {526, 36097}, {32679, 32690}
X(41390) = intersection of Simson line of X(476) and trilinear polar of X(476)
X(41390) = barycentric product X(i)*X(j) for these {i,j}: {2410, 9003}, {34209, 40112}
X(41390) = barycentric quotient X(i)/X(j) for these {i,j}: {2437, 9060}, {9003, 2411}, {14560, 32690}, {32678, 36097}

X(41391) = X(1)X(6)∩X(39)X(9434)

Barycentrics    a*(2*a^3 - 3*a^2*b + 2*a*b^2 - b^3 - 3*a^2*c + 4*a*b*c - b^2*c + 2*a*c^2 - b*c^2 - c^3) : :

X(41391) lies on the curve Q101 and these lines: {1, 6}, {39, 9434}, {40, 4936}, {41, 3991}, {101, 2751}, {104, 6078}, {169, 4513}, {190, 5088}, {200, 35273}, {517, 644}, {519, 2348}, {649, 3309}, {728, 5687}, {758, 25095}, {910, 1018}, {1055, 14439}, {1145, 3234}, {1319, 24036}, {1385, 25082}, {1759, 21872}, {2182, 2325}, {2264, 3950}, {2291, 2748}, {2371, 2736}, {2752, 29127}, {2932, 32625}, {3039, 38455}, {3161, 5731}, {3730, 3916}, {3732, 40872}, {3753, 40131}, {3880, 5540}, {3911, 36954}, {4358, 5773}, {4752, 5011}, {5316, 31191}, {5514, 17757}, {5657, 7390}, {5744, 14154}, {7580, 24771}, {9310, 17614}, {9957, 33950}, {10167, 38876}, {17350, 35274}, {18343, 40127}, {24928, 26690}, {26258, 26446}, {28740, 30616}

X(41391) = midpoint of X(5525) and X(5526)
X(41391) = incircle-inverse of X(14523)
X(41391) = crossdifference of every pair of points on line {513, 614}
X(41391) = barycentric product X(72)*X(14954)
X(41391) = barycentric quotient X(14954)/X(286)
X(41391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {101, 3693, 5440}, {169, 4513, 10914}, {220, 17742, 72}, {9310, 25066, 17614}

X(41392) = X(6)X(13)∩X(112)X(476)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(41392) lies on the curve Q053 and these lines: {6, 13}, {99, 5649}, {112, 476}, {648, 34568}, {1637, 2420}, {1640, 23588}, {2407, 41079}, {2493, 15454}, {5618, 5994}, {5619, 5995}, {6103, 39170}, {14254, 35906}, {14401, 32650}, {14560, 32738}, {14920, 18883}, {14993, 23967}, {16760, 31378}, {34334, 39176}

X(41392) = X(i)-Ceva conjugate of X(j) for these (i,j): {23588, 1989}, {39290, 476}, {39295, 30}
X(41392) = X(i)-cross conjugate of X(j) for these (i,j): {512, 15454}, {1637, 1989}
X(41392) = X(i)-isoconjugate of X(j) for these (i,j): {74, 32679}, {526, 2349}, {1494, 2624}, {1577, 14385}, {2159, 3268}, {2394, 6149}, {8552, 36119}, {14270, 33805}
X(41392) = cevapoint of X(i) and X(j) for these (i,j): {1637, 3163}, {3284, 14401}
X(41392) = crosspoint of X(476) and X(39290)
X(41392) = trilinear pole of line {1495, 3081}
X(41392) = crossdifference of every pair of points on line {526, 16186}
X(41392) = barycentric product X(i)*X(j) for these {i,j}: {30, 476}, {94, 2420}, {99, 14583}, {110, 14254}, {265, 4240}, {328, 23347}, {1495, 35139}, {1637, 39295}, {1784, 36061}, {1989, 2407}, {2173, 32680}, {3163, 39290}, {3233, 5627}, {3260, 14560}, {5664, 23588}, {9214, 14559}, {14206, 32678}, {15329, 39375}, {18557, 23964}, {18558, 23582}, {23895, 36298}, {23896, 36299}
X(41392) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 3268}, {265, 34767}, {476, 1494}, {1495, 526}, {1576, 14385}, {1989, 2394}, {2173, 32679}, {2407, 7799}, {2420, 323}, {3163, 5664}, {3233, 6148}, {3284, 8552}, {4240, 340}, {5664, 23965}, {9406, 2624}, {9407, 14270}, {9409, 16186}, {11060, 2433}, {14254, 850}, {14398, 2088}, {14559, 36890}, {14560, 74}, {14583, 523}, {15475, 12079}, {18384, 18808}, {18557, 36793}, {18558, 15526}, {23347, 186}, {23588, 39290}, {32662, 14919}, {32678, 2349}, {32680, 33805}, {36298, 23870}, {36299, 23871}, {39290, 31621}

X(41393) = X(1)X(3)∩X(12)X(6046)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)^2*(a^2 - b^2 - c^2)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(41393) lies on the curve Q121 and these lines: {1, 3}, {12, 6046}, {201, 1425}, {216, 24443}, {227, 21867}, {331, 1441}, {440, 3649}, {1782, 10536}, {1835, 37225}, {1869, 21318}, {1882, 39531}, {2292, 18592}, {2294, 18591}, {3101, 37405}, {4296, 11101}, {6350, 28629}, {6829, 40149}, {8555, 11428}, {15526, 20653}, {17073, 25447}, {18210, 18673}, {18641, 21677}

X(41393) = X(1214)-Ceva conjugate of X(18591)
X(41393) = crosspoint of X(i) and X(j) for these (i,j): {1214, 6356}, {1441, 26942}
X(41393) = crosssum of X(2189) and X(2194)
X(41393) = X(i)-isoconjugate of X(j) for these (i,j): {29, 1175}, {270, 943}, {284, 40395}, {333, 40570}, {2150, 40447}, {2189, 40435}, {2299, 40412}, {2326, 2982}, {4636, 14775}, {7054, 40573}
X(41393) = barycentric product X(i)*X(j) for these {i,j}: {12, 18607}, {77, 21675}, {201, 5249}, {307, 2294}, {321, 39791}, {442, 1214}, {942, 26942}, {1231, 40952}, {1234, 1409}, {1441, 18591}, {4303, 6358}, {6356, 40937}, {6734, 37755}, {14597, 34388}
X(41393) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 40447}, {65, 40395}, {201, 40435}, {442, 31623}, {1214, 40412}, {1254, 40573}, {1402, 40570}, {1409, 1175}, {1425, 2982}, {1859, 36421}, {1865, 1896}, {2197, 943}, {2260, 270}, {2294, 29}, {4303, 2185}, {14547, 2326}, {14597, 60}, {18591, 21}, {18607, 261}, {21675, 318}, {23207, 7054}, {26942, 40422}, {39791, 81}, {40952, 1172}, {40956, 2189}, {40967, 2322}, {40978, 2299}
X(41393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {201, 37755, 7066}

leftri

Perspectors involving g-triangles: X(41394)-X(41430)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, February 25, 2021.

Mappings g and h are defined in the preamble just before X(33628) as follows. Suppose that P = p : q : r is a point. The points

q - r : r - p : p - q    and     2p - q - r : 2q - r - p : 2r - p - q

clearly lie on the line at infinity, so that their isogonal conjugates,

g(P) = a^2/(q-r) : b^2/(r-p) : c^2/(p-q)     and     h(P) = a^2/(2p-q-r) : b^2/(2q-r-p) : c^2/(2r-p-q),

lie on the circumcircle. Thus, if T is a central triangle, then g(T) and h(T) are central triangles inscribed in the circumcircle.

For many choices of triangle T, there are triangles T' such that T is perspective to g(T'). This section lists associated perspectors.

The appearance of (T,T',i) in the following list means that T is perspective to g(T') and the perspector is X(i):

(ABC, 3rd Brocard, 32)
(ABC, orthocentroidal, 6)
(ABC, 2nd Parry, 187
(ABC, centers of the Apollonius circles, 32)
(ABC, Gemini 19, 649
(medial, MacBeath, 4)
(anticomplementary, Steiner, 99)
(tangential, orthic, 1611)
(tangential, tangential, 1627)
(tangential, incentral, 1613)
(tangential, excentral, 595)
(tangential, intouch, 1616)
(tangential, extouch, 1615)
(tangential, half-altitude,1498)
(tangential, 1st Brocard, 33704)
(tangential, 2nd Brocard, 33704)
(tangential, 3rd Brocard, 33786)
(tangential, MacBeath, 24)
(tangential, orthocentroidal, 6)
(tangential, anti-1st-Brocard,33704)
(tangential, Steiner, 110)
(tangential, symmedial, 33786)
(tangential, anticevian of X(523), 110)
(tangential, 2nd Conway, 595
(tangential, submedial, 1611)
(tangential, orthic of anticomplementary, 1611)
(tangential, tangential of anticomplementary, 1627)
(tangential, orthic of medial, 1611)
(tangential, Yff contact, 101)
(tangential, inner Conway, 1616)
(tangential, 1st Zaniah, 1615)
(tangential, 2nd Zaniah, 1615)
(tangential, Wasat, 595)
(tangential, centyers of the Apollonius circles, 33786)
(tangential, Gemini 3 (and 5,16,18,26), 1613)
(tangential, Gemini 19, 41395)
(tangential, Gemini 29, 1615)
(tangential, Gemini 30, 1616)
(tangential, Gemini 62, 41396)
(tangential, Gemini 112, 1498)
(tangential, Gemini 113, 1611)
(tangential, Gemini 114, 41397)
(tangential, Lemoine, 41394)
(tangential triangle of 2nd circumperp, excentral, 41401)
(tangential triangle of 2nd circumperp, Soddy, 41402)
(incentral, Soddy, 57)
(excentral, Yff contact, 100)
(excentral, Gemini 7 (and 15, 17, 25), 165)
(hexyl, Soddy, 41403)
(intouch, Soddy, 56)
(ABC reflected in X(3), outer Napoleon, 15)
(ABC reflected in X(3), inner Napoleon, 16)
(ABC reflected in X(3), outer Fermat, 61)
(ABC reflected in X(3), inner Fermat, 62)
(ABC reflected in X(3), half altitude, 1498)
(ABC reflected in X(3), inner Vecten, 372)
(ABC reflected in X(3), outer Vecten, 371)
(ABC reflected in X(3), 1st Neuberg, 32)
(ABC reflected in X(3), 2nd Neuberg, 39)
(ABC reflected in X(3), MacBeath, 4)
(ABC reflected in X(3), McCay (and anti-McCay, and 1st Parry), 187)
(ABC reflected in X(3), Artzt (and anti-Artzt, and infinite altitude, 6)
(ABC reflected in X(3), reflected 1st Brocard, 182)
(ABC reflected in X(3), Gemini 7 (and 15, 17, 25), 40)
(ABC reflected in X(3), Soddy, 1)
(ABC reflected in X(3), Moses-Steiner osculating reflection, 41398)
(intangents, Soddy, 103)
(circum-medial, tangential, 251)
(circum-medial, MacBeath, 25)
(circum-orthic, MacBeath, 24)
(circum-orthic, tangential of tangential, 41399)
(circum-orthic, 9th Brocard, 41400)
(2nd circumperp, excentral, 58)
(2nd circumperp, 2nd Sharygin, 8624)
(2nd circumperp, 2nd Conway (and Wasat), 58)
(2nd circumperp, inverse of ABC in excircles, 1191)
(2nd circumperp, Soddy, 56)
(circumnormal, circumtangential, 32)
(tangential of 1st circumperp, Yff contact, 3939)
(2nd Brocard, orthocentroidal, 6)
(3rd Brocard, 3rd Brocard, 32)
(3rd Brocard, orthocentroidal, 6195)
(centers of the Apollonius circles, 32)
(MacBeath, MacBeath, 25)
(Lucas central, orthocentroidal, 6199)
(Lucas tangents, orthocentroidal, 6200)
(Lucas inner, orthocentroidal, 6221)
(1st Sharygin, orthocentroidal, 8296)
(2nd Sharygin, orthocentroidal, 8297)
(anti-1st-Brocard, orthocentroidal, 8289)
(symmedial, 1st Brocard (and 2nd Brocard, anti-1st-Brocard, Steiner, and anticevian of X(523)), 9218)
(symmedial, 2nd Parry, 41404)
(symmedial, Yff contct (and Gemioni 19), 41405)
(Aries, half altitude, 1498)
(Aries, Gemini 112, 1498)
(inner Grebe, orthocentroidal, 6)
(outer Grebe, orthocentroidal, 6)
(anticevian of X(523), X-parabola-tangential (at X(12064)), 99)
(circum-symmedial, Euler, 38920)
(circum-symmedial, ABC reflected in X(3), 38920)
(circum-symmedial, 1st Brocard (and 2nd Brocard), 187
(circum-symmedial, McCay (and anti-McCay), 2030)
(circum-symmedial, anti-1st-Brocard, 187)
(circum-symmedial, Artzt (and anti-Artzt, and infinite altitude), 1384)
(circum-symmedial, outer Napoleon (and 1st half-diamonds-central equilateral), 41406)
(circum-symmedial, 1st inner Fermat-Dao-Nhi (and 3rd, and 4th), 41406)
(circum-symmedial, 1st outer Fermat-Dao-Nhi (and 2nd), 41406)
(circum-symmedial, inner Napoleon (and 2nd half-diamonds-central equilateral), 41407)
(circum-symmedial, 1st inner Fermat-Dao-Nhi (and 2nd), 41407)
(circum-symmedial, 3rd outer Fermat-Dao-Nhi (and 4th), 41407)
(circum-symmedial, outer Fermat (and 2nd half-diamonds), 41408)
(circum-symmedial, inner Fermat (and 1st half-diamonds), 41409)
(circum-symmedial, inner Vecten (and 2nd half-squares), 41410)
(circum-symmedial, outer Vecten (and 1st half-squares), 41411)
(circum-symmedial, 1st Neuberg, 41412)
(circum-symmedial, 2nd Neuberg, 41413)
(circum-symmedial, reflection of ABC in X(5), 41414)
(circum-symmedial, Aquilla (and T(1,2 in TCCT acrticle 6.40), 41415)
(circum-symmedial, outer Garcia (and anti-Aquila), 41416)
(circum-symmedial, tangential of excentral, 41417)
(circum-symmedial, inverse of ABC in excircles, 41418)
(circum-symmedial, 3rd tri-squares central (and anti-outer-Grebe), 41419)
(circum-symmedial, Fermat-Dao, 41420)
(circum-symmedial, 8th Vijay-Paasche-Hutson, 41421)
(3rd mixtilinear, intouch (and inner Conway, and 2nd Zaniah), 1616)
(3rd mixtilinear, Gemini 30, 1616)
(3rd mixtilinear, Gemini 71, 3445)
(3rd mixtilinear, Soddy, 1420)
(4th mixtilinear, extouch, 1615)
(4th mixtilinear, 1st Zaniah, 1615)
(4th mixtilinear, Soddy, 57)
(4th mixtilinear, 2nd extouch (and 1st Conway and Ascella), 41422)
(4th mixtilinear, Germini 63, 41423)
(Thomson, Thomson (and orthic of Thomson), 6)
(Thomson, medial of Thomson, 1384)
(tangential of Thomson, Thomson (and orthic of Thomson), 41424) (5th Brocard, 3rd Brocard, 32)
(Lucas-Brocard, orthocentroidal, 8375)
(Lucas_-1)-Brocard, orthocentroidal, 8376)
(orthic of intouch, Soddy, 1)
(1st Kenmotu diagonal, orthocentroidal, 6)
(2nd Kenmotu diagonal, orthocentroidal, 6)
(anti-orthocentroidal, anti-orthocentroidal, 110)
(infinite altitude, 1st Brocard, 98)
(infinite altitude, 2nd Brocard, 98)
(inner tri-equilateral, orthocentroidal, 6)
(outer tri-equilateral, orthocentroidal, 6)
(anti-Artzt, orthocentroidal, 99)
(anti-Conway, orthocentroidal, 6)
(medial of orthic, orthocentroidal, 6)
(anti-1st Euler, MacBeath, 41425)
(anti-3rd Euler, Steiner, 110)
(anti-3rd Euler, anticevian of X(523), 110)
(anti-5th Brocard, 3rd Brocard, 32)
(anti-5th Brocard, reflected 1st Brocard, 182)
(anti-5th Brocard, centers of the Apollonius circles, 32)
(anti-5th Brocard, 1str Brocard (and 2nd and anti-1st Brocard), 41429
(anti-6th Brocard, Gemini 112, 98)
(Lucas inner, orthocentroidal, 6221)
(Lucas (-1)-inner, orthocentroidal, 6398)
(Lucas inner tangential, orthocentroidal, 6433)
(Lucas (-1)-inner tangential, orthocentroidal, 6434)
(Lucas (-1) central, orthocentroidal, 6495)
(Lucas reflection, orthocentroidal, 22785)
(Lucas (-1) reflection, orthocentroidal, 6496)
(anti-AOA, orthocentroidal, 19379)
(Garcia reflection, Soddy, 104)
(anti-inner Grebe, inner Vecten (and 1st half-squares), 372)
(anti-inner Grebe, orthocentroidal, 6)
(anti-Honsberger, 1st (and 2nd and 3rd and anti-1st Brocard, 1691)
(anti-Honsberger, orthocentroidal, 6)
(anti-Honsberger, centers of the Apollonius circles, 1691)
(Wasat, Wasat (and excentral and 2nd Conway), 98)
(centers of the Apollonius circles, 1st (and 2nd and anti-1st) Brocard, 110)
(centers of the Apollonius circles, 2nd Parry, 111)
(centers of the Apollonius circles, Gemini 19, 101)
(Gemini 3, Soddy, 226)
(Gemini 4, 1st circumperp, 41430)
(Gemini 5, Soddy, 3911)
(Gemini 29, Soddy, 100)
(Gemini 44, tangential (and tangential of anticomplementary), 251)
(Gemini 44, MacBeath, 25)
(Gemini 107, orthocentroidal, 99)
(Gemini 112, MacBeath, 110)
(Gemini 114, incentral, 789)
(9th Fermat-Dao, orthocentroidal, 6)
(10th Fermat-Dao, orthocentroidal, 6)
(13th Fermat-Dao, orthocentroidal, 6)
(14th Fermat-Dao, orthocentroidal, 6)
(Walsmith, orthocentroidal, 6)
(Moses-Steiner reflection, orthocentroidal, 99)
(Moses-Steiner osculating reflection, orthocentroidal, 99)
(Bevan-antipodal, Soddy, 1420)
(8th Brocard, orthocentroidal (and Artzt and infinite altitude and anti-Artzt), 1384)
(inner mixtilinear tangents, Soddy, 41426)
(anti-Hutson intouch, half-altitude (and Gemini 112), 41427)
(anti-Hutson intouch, Moses-Steiner osculating reflection, 41428)

Following are two examples of pairs, (T,T') of named triangles T and T' such that T = g(T'):

orthocentroidal triangle = g(cirum-symmedial triangle)
1st cirumperp triangle = g(Gemini 7) = g(Gemini 15) = g(Gemini 17) = g(Gemini 25)

The appearance of T in the following list means that T is perspective to g(T) = 1st cirumperp triangle, and the perspector is X(3): medial, tangential, 2nd circumperp, inner Napoleon, outer Napoleon, inner Fermat, outer Fermat, inner Vecten, outer Vecten, 1st Neuberg, 2nd Neuberg, Fuhrmann, 1st Brocard, Kosnita, McCay, Trihn, reflection of ABC in X(5), Ara, 2nd Euler

The appearance of (T,i) in the following list means that T is perspectiive to g(T) = 1st cirumperp triangle, and the perspector is X(i):

(3rd Euler, 2), (4th Euler, 4), (intouch, 55), (hexyl, 40), (Yff central, 7589), (inner tangential mid-arc, 8075), (tangential of 1st circumperp, 11495), (tangential of 2nd circumperp, 12513), (1st Sharygin, 4220), (2nd Sharygin, 105), (Honsberger, 7676), (2nd Pamfilos-Zhou, 8224), (2nd extouch, 7580), (3rd mixtilinear, 1), (4th mixtilinear, 165), (6th mixtilinear, 165), (outer tangential mid-arc, 8076), (1st Conway, 7411), (incircle-inverse of ABC, 57), (inner Hutson, 8107), (outer Hutson, 8108), (T(-2,1) in TCCT, Art. 6.41, 9), (T(-1,3) in TCCT, Art. 6.41, 7991), (Hutson intouch, 56), (1st EhrmannT, 165), (Atik, 10860),

The appearance of (T,i) in the following list means that T is perspective to g(T), and the persepctor is X(i):

(tangential,1627), (3rd Brocard, 32), (MacBeath, 25), (Thomson, 6), (anti-orthocentroidal, 110), (Wasat, 1691)

The locus of a point X such that for T = anticevian triangle of X, we have g(t) = T is the cubic K141; see Bernard Gibert, K141: pK(X(2),X(76)).

The locus of a point X such that for T = cevian triangle of X, we have g(t) = T is the cubic pK(X(308), X(380)), which passes through X(i) for these i: 2, 76, 83, 264, 308, 1799, 17907.

See César Lozada, Perspectivities involving the g and h mappings.

X(41394) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND g(LEMOINE)

Barycentrics    a^2*(7*a^4 + 8*a^2*b^2 + b^4 + 8*a^2*c^2 - 16*b^2*c^2 + c^4) : :

X(41394) lies on these lines: {2, 3793}, {6, 5888}, {22, 187}, {32, 20481}, {111, 3053}, {352, 40825}, {858, 37689}, {1184, 38862}, {1383, 1384}, {2502, 38880}, {5024, 5354}, {5210, 9465}, {6636, 15603}, {6800, 38010}, {7484, 22246}, {7754, 31128}, {8860, 23297}, {10546, 33979}, {15655, 40126}, {22331, 39576}

X(41395) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND g(GEMINI 19)

Barycentrics    a^2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3 - b^4 - 2*a^3*c + 8*a^2*b*c - 4*a*b^2*c + 2*b^3*c - a^2*c^2 - 4*a*b*c^2 - b^2*c^2 + 2*a*c^3 + 2*b*c^3 - c^4) : :

X(41395) lies on these lines: {6, 1252}, {649, 9259}, {902, 17798}, {1279, 9356}, {1914, 2183}, {5078, 38863}, {16686, 21005}, {20998, 23404}

X(41396) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND g(GEMINI 62)

Barycentrics    a^2*(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(41396) lies on these lines: {6, 40720}, {31, 172}, {87, 3501}, {213, 40753}, {330, 17475}, {727, 3224}, {932, 16969}, {2241, 34071}, {3230, 17105}, {16468, 21759}

X(41397) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND g(GEMINI 114)

Barycentrics    a^2*(a^2*b^2 - 6*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2) : :

X(41397) lies on these lines: {6, 31}, {238, 21780}, {1613, 36647}, {1979, 21001}, {2162, 8616}, {4512, 16515}

X(41398) = PERSPECTOR OF THESE TRIANGLES: ABC-REFLECTED-IN-X(3) AND g(MOSES-STEINER OSCULATING REFLECTION TRIANGLE)

Barycentrics    a^2*(4*a^8 - 6*a^6*b^2 - 6*a^4*b^4 + 14*a^2*b^6 - 6*b^8 - 6*a^6*c^2 + 9*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 - 6*a^4*c^4 - 4*a^2*b^2*c^4 + 10*b^4*c^4 + 14*a^2*c^6 + b^2*c^6 - 6*c^8) : :

X(41398) lies on these lines: {2, 4549}, {3, 323}, {4, 18442}, {20, 11454}, {22, 11820}, {23, 11472}, {69, 32233}, {74, 7492}, {110, 10298}, {146, 7493}, {376, 3448}, {548, 11270}, {1071, 31663}, {1350, 2071}, {1498, 7488}, {2888, 32330}, {2889, 21735}, {2979, 35493}, {3098, 11579}, {3520, 15110}, {3522, 12118}, {3523, 11821}, {3528, 12325}, {3543, 15431}, {3581, 11002}, {3906, 6194}, {4550, 14002}, {5562, 38942}, {5645, 5946}, {5663, 7712}, {5921, 36989}, {6031, 32228}, {6101, 23040}, {6676, 34796}, {6776, 10304}, {7502, 10620}, {7691, 37480}, {7722, 23039}, {8547, 31884}, {10296, 37638}, {10625, 35494}, {11004, 39242}, {11440, 38435}, {11564, 17702}, {11801, 18564}, {12063, 12084}, {12219, 19374}, {12319, 16063}, {13620, 18451}, {14118, 37489}, {14809, 22089}, {15053, 22112}, {15054, 35268}, {15066, 37952}, {15305, 32237}, {15692, 40911}, {15915, 21733}, {16042, 32620}, {17538, 32210}, {32269, 37077}, {35259, 37957}, {35473, 37477}, {37126, 37475}

X(41399) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-ORTHIC AND g(TANGENTIAL OF TANGENTIAL)

Barycentrics    a^2*(a^14 - 2*a^12*b^2 - a^10*b^4 + 4*a^8*b^6 - a^6*b^8 - 2*a^4*b^10 + a^2*b^12 - 2*a^12*c^2 - 3*a^10*b^2*c^2 + 5*a^8*b^4*c^2 + 4*a^6*b^6*c^2 - 2*a^4*b^8*c^2 - a^2*b^10*c^2 - b^12*c^2 - a^10*c^4 + 5*a^8*b^2*c^4 + 8*a^6*b^4*c^4 + a^2*b^8*c^4 + 3*b^10*c^4 + 4*a^8*c^6 + 4*a^6*b^2*c^6 - 2*a^2*b^6*c^6 - 2*b^8*c^6 - a^6*c^8 - 2*a^4*b^2*c^8 + 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(41399) lies on these lines: {5, 251}, {6, 5889}, {20, 19220}, {24, 1627}, {32, 7544}, {112, 3575}

X(41400) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-ORTHIC AND g(9TH BROCARD)

Barycentrics    15*a^8 - 30*a^6*b^2 + 20*a^4*b^4 - 2*a^2*b^6 - 3*b^8 - 30*a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 8*b^6*c^2 + 20*a^4*c^4 + 2*a^2*b^2*c^4 - 10*b^4*c^4 - 2*a^2*c^6 + 8*b^2*c^6 - 3*c^8 : :

X(41400) lies on these lines: {4, 230}, {20, 9755}, {24, 40321}, {187, 7694}, {376, 511}, {439, 9742}, {631, 7752}, {682, 22655}, {1285, 22521}, {3090, 3972}, {3398, 33226}, {3524, 11184}, {3533, 7831}, {3564, 35927}, {6353, 35278}, {6462, 12510}, {6463, 12509}, {6776, 8719}, {7697, 14033}, {7710, 36998}, {9880, 14651}, {10155, 31404}, {10256, 32816}, {11676, 15428}, {18950, 35941}, {33215, 40108}, {33216, 38225}

X(41401) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF 2ND CIRCUMPERP AND g(EXCENTRAL)

Barycentrics    a^2*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4 + 2*a^4*c - a^3*b*c + a^2*b^2*c + a*b^3*c - 3*b^4*c + a^2*b*c^2 + 3*b^3*c^2 - 2*a^2*c^3 + a*b*c^3 + 3*b^2*c^3 - a*c^4 - 3*b*c^4) : :

X(41401) is also the perspector of these pairs of triangles: tangential of 2nd circumperp and g(2nd Conway); tangential of 2nd circumperp and g(Wasat).

X(41401) lies on these lines: {1, 19}, {24, 102}, {25, 10571}, {56, 7143}, {58, 14529}, {101, 12635}, {109, 28348}, {221, 37260}, {595, 23383}, {1042, 3220}, {1191, 17053}, {1616, 3053}, {2187, 3340}, {4224, 37558}, {4306, 22654}, {10570, 16066}, {12563, 18162}, {20991, 34040}, {37226, 38945}

X(41402) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF 2ND CIRCUMPERP AND g(SODDY)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c - 2*a^4*b^2*c + 2*a^2*b^4*c + 2*b^6*c - a^5*c^2 - 2*a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 4*b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 - 4*b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 + a*c^6 + 2*b*c^6) : :

X(41402) lies on these lines: {1, 84}, {3, 5930}, {4, 1035}, {21, 18623}, {28, 56}, {34, 1712}, {57, 40933}, {73, 34032}, {77, 37228}, {104, 7149}, {223, 405}, {347, 2975}, {387, 1466}, {603, 5706}, {956, 34039}, {958, 1214}, {1001, 10571}, {1249, 1436}, {1295, 3346}, {1376, 34030}, {1410, 4185}, {1440, 24565}, {1612, 23404}, {1617, 1661}, {1804, 16049}, {1838, 22753}, {2217, 2385}, {3183, 41227}, {3428, 37404}, {3560, 34052}, {3682, 9370}, {4293, 37379}, {7070, 37022}, {25524, 37695}, {34042, 37523}

X(41403) = PERSPECTOR OF THESE TRIANGLES: HEXYL AND g(SODDY)

Barycentrics    a*(3*a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 - 6*a^5*b^4 + 6*a^4*b^5 + 12*a^3*b^6 - 4*a^2*b^7 - 5*a*b^8 + b^9 + a^8*c + 4*a^7*b*c + 4*a^6*b^2*c - 4*a^5*b^3*c - 10*a^4*b^4*c - 4*a^3*b^5*c + 4*a^2*b^6*c + 4*a*b^7*c + b^8*c - 4*a^7*c^2 + 4*a^6*b*c^2 + 20*a^5*b^2*c^2 + 4*a^4*b^3*c^2 - 12*a^3*b^4*c^2 - 4*a^2*b^5*c^2 - 4*a*b^6*c^2 - 4*b^7*c^2 - 4*a^6*c^3 - 4*a^5*b*c^3 + 4*a^4*b^2*c^3 + 8*a^3*b^3*c^3 + 4*a^2*b^4*c^3 - 4*a*b^5*c^3 - 4*b^6*c^3 - 6*a^5*c^4 - 10*a^4*b*c^4 - 12*a^3*b^2*c^4 + 4*a^2*b^3*c^4 + 18*a*b^4*c^4 + 6*b^5*c^4 + 6*a^4*c^5 - 4*a^3*b*c^5 - 4*a^2*b^2*c^5 - 4*a*b^3*c^5 + 6*b^4*c^5 + 12*a^3*c^6 + 4*a^2*b*c^6 - 4*a*b^2*c^6 - 4*b^3*c^6 - 4*a^2*c^7 + 4*a*b*c^7 - 4*b^2*c^7 - 5*a*c^8 + b*c^8 + c^9) : :

X(41403) lies on these lines: {1, 64}, {56, 84}, {57, 207}, {154, 2956}, {196, 40657}, {278, 3183}, {942, 17832}, {1033, 2257}, {1214, 40675}, {1394, 3576}, {1490, 7011}, {1838, 6525}, {4292, 41010}

X(41404) = PERSPECTOR OF THESE TRIANGLES: SYMMEDIAL AND g(2ND PARRY)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(5*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 7*b^2*c^2 - c^4) : :

X(41404) lies on these lines: {23, 111}, {512, 10562}, {574, 14246}, {671, 14712}, {843, 39024}, {895, 8586}, {923, 16784}, {1384, 14263}, {2502, 9218}, {8591, 39061}, {10766, 21639}, {13492, 32648}

X(41404) = isogonal conjugate of isotomic conjugate of crosssum of PU(107)
X(41404) = perspector of circumcevian triangle of X(187) and cross-triangle of ABC and circumcevian triangle of X(187)

X(41405) = PERSPECTOR OF THESE TRIANGLES: SYMMEDIAL AND g(YFF CONTACT)

Barycentrics    a^2*(a - b)*(a - c)*(a^2 - a*b - b^2 - a*c + 3*b*c - c^2) : :

X(41405) is also the perspector of these triangles: symmedial and g(Gemini 19).

X(41405) lies on these lines: {55, 24484}, {100, 661}, {101, 649}, {109, 813}, {165, 9357}, {513, 14589}, {672, 5537}, {902, 5030}, {919, 1293}, {1580, 9323}, {1635, 5375}, {1914, 5053}, {2245, 17735}, {3509, 21801}, {3835, 4998}, {4369, 40865}, {4813, 14513}, {6014, 28875}, {6078, 8699}, {8640, 34067}

X(41406) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(OUTER NAPOLEON)

Barycentrics    a^2*(5*a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :

X(41406) is also the perspector of these triangles:
circum-symmedial and g(1st half-diamonds-central-equilateral)
circum-symmedial and g(3rd inner-Fermat-Dao-Nhi)
circum-symmedial and g(4th inner-Fermat-Dao-Nhi)
circum-symmedial and g(1st outer-Fermat-Dao-Nhi)
circum-symmedial and g(2nd outer-Fermat-Dao-Nhi)

X(41406) lies on these lines: {3, 6}, {13, 230}, {14, 7737}, {18, 7745}, {111, 3457}, {112, 2378}, {115, 36969}, {172, 7006}, {202, 1914}, {299, 7835}, {302, 7812}, {385, 25183}, {395, 18907}, {396, 5463}, {531, 1285}, {609, 7127}, {623, 11489}, {843, 5994}, {1250, 16785}, {1383, 37775}, {1992, 11154}, {2379, 2715}, {2549, 36968}, {3054, 16966}, {3055, 33416}, {3170, 39689}, {3767, 16965}, {3815, 16242}, {5306, 41100}, {5318, 41036}, {5321, 36992}, {5335, 7684}, {5472, 16267}, {5475, 37835}, {5477, 22997}, {6109, 7735}, {6671, 11488}, {8744, 10633}, {9136, 9203}, {10617, 23302}, {10641, 10986}, {11543, 20428}, {11580, 37776}, {12155, 22329}, {14537, 41122}, {15484, 16645}, {15993, 22998}, {16241, 21843}, {16784, 19373}, {16967, 31415}, {20194, 35303}, {22580, 35304}, {23303, 40334}, {26613, 37786}, {27088, 36775}, {32732, 39422}, {33518, 40694}, {37637, 37832}, {38431, 39410}

X(41406) = isogonal conjugate of X(42035)
X(41406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 187, 15), (6, 1384, 41407), (187, 2030, 41407), (574, 41412, 41407), (5008, 41413, 41407)

X(41407) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(INNER NAPOLEON)

Barycentrics    a^2*(5*a^2 - b^2 - c^2 - 2*Sqrt[3]*S) : :

X(41407) is also the perspector of these triangles:
circum-symmedial and g(2nd half-diamonds-central-equilateral)
circum-symmedial and g(1st inner-Fermat-Dao-Nhi)
circum-symmedial and g(2nd inner-Fermat-Dao-Nhi)
circum-symmedial and g(3rd outer-Fermat-Dao-Nhi)
circum-symmedial and g(4th outer-Fermat-Dao-Nhi)

X(41407) lies on these lines: {3, 6}, {13, 7737}, {14, 230}, {17, 7745}, {111, 3458}, {112, 2379}, {115, 36970}, {172, 7005}, {203, 1914}, {298, 7835}, {303, 7812}, {385, 25187}, {395, 5464}, {396, 18907}, {530, 1285}, {624, 11488}, {843, 5995}, {1383, 37776}, {1992, 11153}, {2307, 7031}, {2378, 2715}, {2549, 36967}, {3054, 16967}, {3055, 33417}, {3171, 39689}, {3767, 16964}, {3815, 16241}, {5306, 41101}, {5318, 36994}, {5321, 41037}, {5334, 7685}, {5471, 16268}, {5475, 37832}, {5477, 22998}, {6108, 7735}, {6672, 11489}, {6782, 36776}, {7051, 16784}, {8744, 10632}, {9136, 9202}, {10616, 23303}, {10638, 16785}, {10642, 10986}, {11542, 20429}, {11580, 37775}, {12154, 22329}, {14537, 41121}, {15484, 16644}, {15993, 22997}, {16242, 21843}, {16966, 31415}, {20194, 35304}, {22579, 35303}, {23302, 40335}, {26613, 37785}, {32732, 39423}, {33517, 40693}, {37637, 37835}, {38432, 39411}

X(41407) = isogonal conjugate of X(42036)
X(41407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 187, 18), (6, 1384, 41406), (187, 2030, 41406), (574, 41412, 41406), (5008, 41413, 41406)

X(41408) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(OUTER FERMAT)

Barycentrics    a^2*(15*a^2 - 3*b^2 - 3*c^2 + 2*Sqrt[3]*S) : :

X(41408) is also the perspector of these triangles: circum-symmedial and g(2nd half-diamonds).

X(41408) lies on these lines: {3, 6}, {112, 2380}, {230, 16808}, {396, 36366}, {609, 1250}, {616, 9112}, {1285, 11489}, {2548, 33416}, {3643, 11488}, {3767, 19106}, {5306, 36968}, {5478, 37689}, {7031, 19373}, {7737, 16809}, {7745, 16967}, {10312, 11476}, {18907, 23303}

X(41408) = {X(6),X(1384)}-harmonic conjugate of X(41409)

X(41409) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(INNER FERMAT)

Barycentrics    a^2*(15*a^2 - 3*b^2 - 3*c^2 - 2*Sqrt[3]*S) : :

X(41409) is also the perspector of these triangles: circum-symmedial and g(1st half-diamonds).

X(41409) lies on these lines: {3, 6}, {112, 2381}, {230, 16809}, {395, 36368}, {609, 10638}, {617, 9113}, {1285, 11488}, {2548, 33417}, {3642, 11489}, {3767, 19107}, {5306, 36967}, {5479, 37689}, {7031, 7051}, {7737, 16808}, {7745, 16966}, {10312, 11475}, {18907, 23302}

X(41409) = {X(6),X(1384)}-harmonic conjugate of X(41408)

X(41410) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(INNER VECTEN)

Barycentrics    a^2*(5*a^2 - b^2 - c^2 - 2*S) : :

X(41410) is also the perspector of these triangles: circum-symmedial and g(2nd half-squares).

X(41410) lies on these lines: {3, 6}, {4, 13711}, {112, 5412}, {172, 35808}, {230, 6565}, {590, 18907}, {609, 2066}, {637, 13941}, {639, 32786}, {754, 13763}, {1285, 3068}, {1572, 35763}, {1914, 35768}, {1968, 35764}, {2067, 7031}, {3069, 13770}, {3093, 8778}, {3767, 35821}, {5304, 9541}, {5861, 13650}, {6460, 19105}, {6561, 7735}, {6564, 7737}, {7000, 13834}, {7375, 32785}, {7745, 10576}, {7747, 35786}, {7761, 13644}, {8253, 15484}, {10312, 11473}, {12222, 13651}, {12829, 35878}, {13881, 35787}, {23273, 26441}, {31411, 35815}

X(41410) = isogonal conjugate of X(42024)
X(41410) = {X(6),X(1384)}-harmonic conjugate of X(41411)

X(41411) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(OUTER VECTEN)

Barycentrics    a^2*(5*a^2 - b^2 - c^2 + 2*S) : :

X(41411) is also the perspector of these triangles: circum-symmedial and g(1st half-squares).

X(41411) lies on these lines: {3, 6}, {4, 13834}, {112, 5413}, {172, 35809}, {230, 6564}, {609, 5414}, {615, 18907}, {638, 8972}, {640, 32785}, {754, 13644}, {1285, 3069}, {1572, 35762}, {1914, 35769}, {1968, 35765}, {3068, 13651}, {3092, 8778}, {3767, 35820}, {5860, 13771}, {6459, 19102}, {6502, 7031}, {6560, 7735}, {6565, 7737}, {7374, 13711}, {7376, 32786}, {7745, 10577}, {7747, 35787}, {7761, 13763}, {8252, 15484}, {8982, 23267}, {10312, 11474}, {12221, 13770}, {12829, 35879}, {13881, 35786}, {21843, 31463}, {31411, 35812}

X(41411) = isogonal conjugate of X(42023)
X(41411) = {X(6),X(1384)}-harmonic conjugate of X(41410)

X(41412) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(1ST NEUBERG)

Barycentrics    a^2*(3*a^4 + a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :

X(41412) lies on these lines: {3, 6}, {69, 7835}, {83, 18841}, {98, 3424}, {99, 32474}, {112, 729}, {141, 8368}, {184, 1627}, {193, 5182}, {230, 3818}, {597, 8358}, {609, 2330}, {699, 26714}, {713, 32722}, {727, 26716}, {1078, 3619}, {1204, 34137}, {1285, 3618}, {1383, 10545}, {1428, 7031}, {1501, 3231}, {2715, 5970}, {3331, 38926}, {3506, 5106}, {3589, 7761}, {3620, 7793}, {3734, 8177}, {3767, 29012}, {4048, 7751}, {5207, 7857}, {5354, 15080}, {5359, 22352}, {5651, 8617}, {6144, 12151}, {6179, 12215}, {6800, 40130}, {7737, 19130}, {7745, 38317}, {7787, 33202}, {7808, 8364}, {7815, 33185}, {7853, 15484}, {9136, 32694}, {9292, 33632}, {9465, 35268}, {10104, 18358}, {10312, 19124}, {10546, 11580}, {11364, 16496}, {12194, 16491}, {14561, 32152}, {14614, 33685}, {17222, 30554}, {20080, 33205}, {28343, 34217}, {31670, 38749}

X(41412) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(1350)
X(41412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1384, 41413), (1687, 1688, 1350), (41406, 41407, 574)

X(41413) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(2ND NEUBERG)

Barycentrics    a^2*(2*a^4 + 3*a^2*b^2 - b^4 + 3*a^2*c^2 - c^4) : :

X(41413) lies on these lines: {3, 6}, {23, 40130}, {51, 1627}, {69, 1285}, {112, 755}, {141, 754}, {211, 3852}, {230, 19130}, {251, 3917}, {315, 3619}, {384, 14994}, {609, 3056}, {626, 34573}, {698, 7805}, {732, 7816}, {733, 17970}, {1383, 10546}, {1469, 7031}, {1495, 9463}, {1501, 34986}, {1915, 3787}, {3051, 8627}, {3552, 32451}, {3618, 7771}, {3620, 20065}, {3629, 5026}, {3631, 19697}, {3763, 7818}, {3818, 7737}, {5031, 7843}, {5103, 7886}, {5254, 29317}, {5305, 29181}, {5306, 19924}, {5354, 15107}, {6179, 18906}, {7735, 31670}, {7745, 24206}, {7780, 24256}, {8289, 36849}, {8623, 34236}, {9753, 37689}, {10312, 12294}, {10545, 11580}, {11676, 39872}, {12039, 30489}, {12042, 21850}, {14537, 25561}, {18373, 29959}, {20080, 33201}, {21167, 31406}, {21444, 40981}, {26716, 28485}

X(41413) = isogonal conjugate of isotomic conjugate of X(41624) X(41413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1384, 41412), (41406, 41407, 5008)

X(41414) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(JOHSON TRIANGLE)

Barycentrics    a^2*(4*a^8 - 4*a^6*b^2 - 6*a^4*b^4 + 8*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 7*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + 5*b^6*c^2 - 6*a^4*c^4 - 8*a^2*b^2*c^4 - 6*b^4*c^4 + 8*a^2*c^6 + 5*b^2*c^6 - 2*c^8) : :

X(41414) is also the perspector of these triangles: circum-symmedial and g(anti-1st-Euler).

X(41414) lies on these lines: {6, 186}, {25, 38920}, {74, 10311}, {112, 34417}, {187, 6785}, {1383, 2433}, {1384, 3148}, {1495, 10986}, {2420, 11002}, {9781, 35007}, {10312, 11438}, {10313, 37470}, {11004, 32661}

X(41415) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(AQUILLA)

Barycentrics    a^2*(5*a^2 + 13*a*b + 2*b^2 + 13*a*c + b*c + 2*c^2) : :

X(41415) lies on these lines: {6, 6767}, {44, 31514}, {58, 8649}, {101, 21747}, {1475, 5315}, {4752, 16477}

X(41416) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(ANTI-AQUILLA)

Barycentrics    a^2*(4*a^2 - 4*a*b - 2*b^2 - 4*a*c - b*c - 2*c^2) : :

X(41416) is also the perspector of these triangles: circum-symmedial and g(outer Garcia).

X(41416) lies on these lines: {1, 8297}, {6, 36}, {41, 4256}, {101, 2177}, {106, 2280}, {869, 902}, {1017, 2276}, {2271, 16474}, {3684, 16499}, {4258, 16483}, {5315, 7031}, {9336, 21008}, {15015, 36404}, {16785, 20989}, {30644, 35342}

X(41416) = trilinear pole, wrt circumsymmedial triangle, of antiorthic axis

X(41417) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(TANGENTIAL OF EXCENTRAL)

Barycentrics    a^2*(3*a^3 + 6*a^2*b + 3*a*b^2 + 6*a^2*c + a*b*c - 3*b^2*c + 3*a*c^2 - 3*b*c^2) : :

X(41417) lies on these lines: {6, 595}, {109, 8700}, {3247, 3915}, {3723, 40091}, {4290, 16677}, {5315, 27787}

X(41418) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(INVERSE OF ABC IN EXCIRCLES)

Barycentrics    a^2*(2*a^2*b + 2*a*b^2 + 2*a^2*c - a*b*c - b^2*c + 2*a*c^2 - b*c^2) : :

X(41418) lies on these lines: {1, 6}, {48, 7031}, {106, 32693}, {572, 3915}, {573, 1201}, {595, 604}, {603, 38855}, {614, 10439}, {739, 28162}, {902, 10434}, {992, 3679}, {995, 2269}, {1572, 1781}, {1999, 14997}, {2260, 9336}, {2268, 40091}, {3169, 3216}, {3928, 18186}, {4856, 37657}, {4859, 28350}, {5037, 22356}, {5783, 37542}, {5903, 20227}, {9315, 33628}, {10447, 17117}, {10452, 20080}, {10882, 37508}, {11679, 37680}, {16833, 27623}, {16834, 27644}, {23681, 28368}, {25590, 28365}

X(41419) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(ANTI-OUTER-GREBE)

Barycentrics    a^2*(13*a^4 - 14*a^2*b^2 + b^4 - 14*a^2*c^2 + 10*b^2*c^2 + c^4 + 8*a^2*S - 4*b^2*S - 4*c^2*S) : :

X(41419) is also the perspector of these triangles: circum-symmedial and g(3rd tri-squares central).

X(41419) lies on these lines: {353, 9600}, {493, 6199}, {3167, 6398}, {5408, 6412}, {6200, 10132}, {9690, 32568}

X(41420) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(13TH FERMAT-DAO)

Barycentrics    a^2*(4*a^6*b^2 - 8*a^4*b^4 + 4*a^2*b^6 + 4*a^6*c^2 - 3*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 2*b^6*c^2 - 8*a^4*c^4 - 3*a^2*b^2*c^4 + 4*b^4*c^4 + 4*a^2*c^6 - 2*b^2*c^6 - 6*Sqrt[3]*a^2*b^2*c^2*S) : :

X(41420) lies on these lines: {6, 13}, {729, 16807}, {2378, 26714}, {2421, 36775}, {2493, 30439}, {5994, 32730}

X(41421) = PERSPECTOR OF THESE TRIANGLES: CIRCUM-SYMMEDIAL AND g(8TH VIJAY-PAASCHE-HUTSON)

Barycentrics    a^2*(3*a*b*c - a*S + 2*b*S + 2*c*S) : :

X(41421) lies on these lines: {6, 31}, {106, 6135}, {3069, 16484}, {3750, 19053}, {4256, 18991}, {9350, 13846}

X(41422) = PERSPECTOR OF THESE TRIANGLES: 4TH MIXTILINEAR AND g(1ST CONWAY)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 14*a^2*b*c - 4*a*b^2*c + 2*b^3*c - 2*a^2*c^2 - 4*a*b*c^2 - 6*b^2*c^2 + 2*b*c^3 + c^4) : :

X(41422) is also the perspector of these triangles:
4th mixtilinear and g(2nd extouch)
4th mixtilinear and g(Ascella)

X(41422) lies on these lines: {3, 10460}, {6, 165}, {55, 58}, {57, 221}, {171, 218}, {354, 16483}, {386, 10824}, {601, 6244}, {1191, 10980}, {1616, 30350}, {3215, 37541}, {4252, 15931}, {5273, 5711}, {5292, 7965}

X(41423) = PERSPECTOR OF THESE TRIANGLES: 4TH MIXTILINEAR AND g(GEMINI 63)

Barycentrics    a^2*(a^2 + a*b - 2*b^2 + a*c - 2*b*c - 2*c^2) : :

X(41423) lies on these lines: {1, 5030}, {3, 1055}, {6, 31}, {9, 100}, {19, 35993}, {21, 3501}, {35, 41}, {37, 750}, {39, 3915}, {40, 17451}, {46, 21808}, {48, 34879}, {57, 1255}, {63, 3930}, {101, 5010}, {103, 8693}, {105, 2108}, {106, 6016}, {165, 846}, {213, 31451}, {220, 5217}, {238, 17756}, {344, 24602}, {354, 3723}, {498, 17732}, {573, 5537}, {574, 3230}, {594, 33161}, {604, 2078}, {614, 9574}, {644, 17549}, {748, 1575}, {799, 17786}, {813, 840}, {894, 35292}, {910, 16814}, {993, 1018}, {1150, 2321}, {1193, 14974}, {1201, 5013}, {1212, 37568}, {1400, 37541}, {1468, 1500}, {1475, 3295}, {1571, 24443}, {1621, 17754}, {1696, 34820}, {1743, 39251}, {2112, 6184}, {2170, 5119}, {2289, 6602}, {2295, 10448}, {2329, 4189}, {2345, 32917}, {2348, 9353}, {2646, 21872}, {2975, 3208}, {3217, 36744}, {3290, 31443}, {3294, 25440}, {3303, 5022}, {3496, 25082}, {3550, 5276}, {3570, 17336}, {3576, 17439}, {3579, 16601}, {3691, 5687}, {3693, 4640}, {3722, 16973}, {3729, 37670}, {3746, 4253}, {3871, 21384}, {3905, 25248}, {3916, 3991}, {3973, 31508}, {4071, 33113}, {4257, 16785}, {4271, 34140}, {4396, 17262}, {4421, 37658}, {4517, 20672}, {4642, 16968}, {4860, 9345}, {4877, 5235}, {5024, 16483}, {5124, 41341}, {5134, 7951}, {5218, 41325}, {5248, 16549}, {5250, 39244}, {5264, 25092}, {5432, 17747}, {5749, 35261}, {5750, 35263}, {5883, 6205}, {6014, 28317}, {6603, 37600}, {8589, 8649}, {8616, 33854}, {8715, 16552}, {9350, 37673}, {9351, 15515}, {9441, 40779}, {9598, 21935}, {12514, 33299}, {16466, 31461}, {16603, 31015}, {16781, 23649}, {16969, 32577}, {16997, 17261}, {17084, 26790}, {17314, 32919}, {17594, 21840}, {17596, 26242}, {17603, 21871}, {17692, 17743}, {20677, 28471}, {21029, 26066}, {21044, 26446}, {21101, 32933}, {21956, 24892}, {23407, 24578}, {25269, 33889}, {33104, 37661}, {33844, 39258}, {33925, 36743}, {38874, 40155}

X(41424) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF THOMSON AND g(THOMSON)

Barycentrics    a^2*(7*a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 - 5*c^4) : :

X(41424) is also the perspector of these triangles: tangential of Thomson and g(orthic of Thomson).

X(41424) lies on these lines: {3, 5646}, {4, 15448}, {6, 25}, {22, 10546}, {23, 1350}, {24, 64}, {26, 14128}, {110, 11477}, {187, 33979}, {237, 5210}, {323, 33586}, {373, 10541}, {381, 32267}, {382, 5972}, {394, 15107}, {399, 37489}, {468, 36990}, {542, 21970}, {599, 37904}, {842, 33988}, {902, 20991}, {1192, 26883}, {1352, 37897}, {1498, 3517}, {1503, 4232}, {1511, 7530}, {1514, 18533}, {1598, 11430}, {1620, 11381}, {1853, 6353}, {1995, 5085}, {2207, 9408}, {3053, 5191}, {3066, 6800}, {3089, 17845}, {3098, 9909}, {3129, 11480}, {3130, 11481}, {3155, 6411}, {3156, 6412}, {3167, 37517}, {3426, 10606}, {3515, 8567}, {3518, 9786}, {3523, 16656}, {3531, 10282}, {3581, 18451}, {3619, 10565}, {3631, 14826}, {3796, 7712}, {4257, 37260}, {4550, 14070}, {5020, 5092}, {5102, 11004}, {5480, 35260}, {5544, 20190}, {5651, 31884}, {6437, 10132}, {6438, 10133}, {6525, 33630}, {6644, 35237}, {6995, 10192}, {7426, 37638}, {7488, 33537}, {7493, 10516}, {7494, 34573}, {7517, 37483}, {7529, 37513}, {7545, 37506}, {7575, 11472}, {7687, 18405}, {7714, 23292}, {8549, 15647}, {8780, 37672}, {9306, 20850}, {9714, 17814}, {10154, 18358}, {10244, 11793}, {10249, 37962}, {10545, 10601}, {10564, 18534}, {10594, 11425}, {10605, 12112}, {10979, 26909}, {10982, 26882}, {10986, 41376}, {11002, 12061}, {11202, 18535}, {11284, 35268}, {11414, 40917}, {12017, 17825}, {12106, 37475}, {13202, 15131}, {13567, 39874}, {13861, 37476}, {14389, 38072}, {14924, 22112}, {15051, 35904}, {15068, 17834}, {15069, 32269}, {15578, 37977}, {16654, 35486}, {18440, 32223}, {19219, 23259}, {19357, 34484}, {26255, 37648}, {26958, 31383}, {28348, 33846}, {34147, 34815}, {35266, 37645}, {37254, 37633}, {40350, 40825}

X(41425) = PERSPECTOR OF THESE TRIANGLES: ANTI-1ST-EULER AND g(MACBEATH)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^12 - 6*a^10*b^2 - 7*a^8*b^4 + 28*a^6*b^6 - 27*a^4*b^8 + 10*a^2*b^10 - b^12 - 6*a^10*c^2 + 22*a^8*b^2*c^2 - 28*a^6*b^4*c^2 + 12*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - 2*b^10*c^2 - 7*a^8*c^4 - 28*a^6*b^2*c^4 + 30*a^4*b^4*c^4 - 12*a^2*b^6*c^4 + 17*b^8*c^4 + 28*a^6*c^6 + 12*a^4*b^2*c^6 - 12*a^2*b^4*c^6 - 28*b^6*c^6 - 27*a^4*c^8 + 2*a^2*b^2*c^8 + 17*b^4*c^8 + 10*a^2*c^10 - 2*b^2*c^10 - c^12) : :

X(41425) lies on these lines: {3, 14361}, {4, 64}, {24, 33582}, {107, 12324}, {186, 15512}, {196, 3487}, {216, 631}, {1093, 18931}, {3079, 34781}, {3176, 5657}, {3357, 36876}, {3462, 3533}, {3525, 33924}, {3529, 6761}, {5067, 8888}, {5656, 6621}, {5667, 23241}, {6000, 6616}, {6225, 6624}, {6524, 26937}, {6618, 18909}, {6889, 18687}, {8966, 13886}, {10002, 40686}, {13939, 13960}, {15258, 17821}, {20207, 40675}

X(41426) = PERSPECTOR OF THESE TRIANGLES: INNER-MIXTILINEAR TANGENTS AND g(SODDY)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 8*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 + c^3) : :

X(41426) lies on these lines: {1, 3}, {2, 1476}, {6, 38855}, {11, 33898}, {100, 6049}, {104, 10309}, {106, 1413}, {108, 1398}, {109, 1616}, {144, 7677}, {222, 1201}, {388, 4187}, {474, 3476}, {603, 16483}, {651, 28370}, {956, 7288}, {958, 5316}, {962, 24465}, {995, 34046}, {1035, 16485}, {1056, 6967}, {1106, 1149}, {1145, 1788}, {1404, 23073}, {1407, 3445}, {1457, 2122}, {1458, 15287}, {1532, 3086}, {1604, 3554}, {1837, 30283}, {2478, 3600}, {2975, 18228}, {3911, 6736}, {4308, 5253}, {4311, 22753}, {4315, 9843}, {4413, 37709}, {5083, 12635}, {5252, 16408}, {5265, 6921}, {5298, 34689}, {5330, 18419}, {5433, 9708}, {5573, 34039}, {5704, 38669}, {6614, 38866}, {6700, 8666}, {6925, 14986}, {6929, 18990}, {6959, 15325}, {8582, 10106}, {9363, 21214}, {9669, 18961}, {9709, 10944}, {10074, 37725}, {10957, 17528}, {11108, 22759}, {11501, 16417}, {14594, 17480}, {14923, 37789}, {15347, 36846}, {17625, 19861}, {18238, 37252}, {21578, 37411}, {28077, 28080}, {28348, 34051}

X(41427) = PERSPECTOR OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND g(HALF-ALTITUDE)

Barycentrics    a^2*(7*a^8 - 16*a^6*b^2 + 6*a^4*b^4 + 8*a^2*b^6 - 5*b^8 - 16*a^6*c^2 + 44*a^4*b^2*c^2 - 24*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 - 24*a^2*b^2*c^4 + 18*b^4*c^4 + 8*a^2*c^6 - 4*b^2*c^6 - 5*c^8) : :

X(41) lies on these lines: {3, 6}, {20, 5893}, {64, 2063}, {154, 11413}, {378, 22549}, {382, 18418}, {394, 8567}, {1092, 10606}, {1204, 37672}, {3292, 34469}, {3516, 17811}, {5059, 15751}, {5562, 22967}, {5894, 37669}, {5895, 11064}, {6696, 15069}, {7396, 27082}, {12084, 32137}, {12086, 35259}, {12163, 34152}, {12293, 15122}, {17810, 22467}, {17821, 21312}, {22951, 22968}, {22971, 22973}, {32171, 35237}

X(41428) = PERSPECTOR OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND g(MOSES-STEINER OSCULATING REFLECTION TRIANGLE)

Barycentrics    a^2*(7*a^8 - 6*a^6*b^2 - 24*a^4*b^4 + 38*a^2*b^6 - 15*b^8 - 6*a^6*c^2 + 54*a^4*b^2*c^2 - 34*a^2*b^4*c^2 - 14*b^6*c^2 - 24*a^4*c^4 - 34*a^2*b^2*c^4 + 58*b^4*c^4 + 38*a^2*c^6 - 14*b^2*c^6 - 15*c^8) : :

X(41428) lies on these lines: {22, 11820}, {64, 11454}, {74, 1995}, {110, 10606}, {378, 7722}, {1350, 13445}, {2070, 33887}, {3357, 5562}, {5921, 16386}, {6000, 38446}, {7503, 13339}, {11440, 33524}, {11442, 20725}, {12041, 15078}, {13093, 38438}, {14927, 16789}, {15021, 35259}, {15062, 37475}, {32063, 38441}

X(41429) = PERSPECTOR OF THESE TRIANGLES: ANTI-5TH BROCARD AND g(1ST BROCARD)

Barycentrics    a^2*(2*a^8*b^8 - 2*a^6*b^10 + a^12*b^2*c^2 - 2*a^10*b^4*c^2 + 3*a^8*b^6*c^2 + a^6*b^8*c^2 - 2*a^4*b^10*c^2 + a^2*b^12*c^2 - 2*a^10*b^2*c^4 - 4*a^8*b^4*c^4 - a^6*b^6*c^4 + a^4*b^8*c^4 - a^2*b^10*c^4 + b^12*c^4 + 3*a^8*b^2*c^6 - a^6*b^4*c^6 + 3*a^4*b^6*c^6 - b^10*c^6 + 2*a^8*c^8 + a^6*b^2*c^8 + a^4*b^4*c^8 - 2*a^6*c^10 - 2*a^4*b^2*c^10 - a^2*b^4*c^10 - b^6*c^10 + a^2*b^2*c^12 + b^4*c^12) : :

X(41429) is also the perspector of these triangles:
anti-5th Brocard and g(2nd Brocard)
anti-5th Brocard and g(anti-1st Brocard)

X(41429) lies on these lines: {32, 34214}, {83, 9467}, {98, 2679}, {99, 511}, {512, 34238}, {805, 10352}, {1576, 1691}, {8870, 12110}, {10350, 38527}

X(41430) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND g(GEMINI 7)

Barycentrics    a*(2*a^3*b - 2*a^2*b^2 + 2*a^3*c - a*b^2*c - b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - b*c^3) : :

X(41430) is also the perspector of these triangles:
Gemini 4 and g(Gemini 15)
Gemini 4 and g(Gemini 17)
Gemini 4 and g(Gemini 25)

X(41430) lies on these lines: {3, 142}, {10, 37425}, {20, 19853}, {40, 991}, {43, 165}, {55, 3664}, {71, 35338}, {75, 29036}, {100, 4416}, {511, 3579}, {517, 13476}, {524, 4097}, {527, 15624}, {572, 9441}, {759, 1292}, {1155, 21746}, {1293, 6015}, {1385, 29309}, {2223, 3663}, {2293, 20367}, {2321, 4436}, {2807, 14690}, {2938, 13610}, {3008, 20992}, {3286, 3755}, {3522, 10882}, {3941, 3946}, {3950, 4447}, {4000, 16688}, {4021, 21010}, {4191, 40998}, {4192, 10164}, {4353, 37590}, {4356, 37609}, {4512, 37262}, {4689, 15447}, {4847, 22060}, {6210, 15601}, {7676, 40910}, {8299, 21255}, {8715, 34379}, {9778, 10434}, {12723, 25065}, {14636, 34638}, {15310, 31663}, {18252, 25078}, {20257, 23370}, {23407, 24199}, {24705, 40600}, {29347, 30273}, {35258, 35980}, {35263, 35984}, {37582, 39543}

leftri

Perspectors involving g-triangles: X(41431)-X(41479)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, February 28, 2021.

Mappings g and h are defined in the preamble just before X(33628) as follows. Suppose that P = p : q : r is a point. The points

q - r : r - p : p - q    and     2p - q - r : 2q - r - p : 2r - p - q

clearly lie on the line at infinity, so that their isogonal conjugates,

g(P) = a^2/(q-r) : b^2/(r-p) : c^2/(p-q)     and     h(P) = a^2/(2p-q-r) : b^2/(2q-r-p) : c^2/(2r-p-q),

lie on the circumcircle. Thus, if T is a central triangle, then g(T) and h(T) are central triangles inscribed in the circumcircle.

For many choices of triangle T, there are triangles T' such that T is perspective to h(T'). This section lists associated perspectors.

The appearance of (T,T',i) in the following list means that T is perspective to h(T') and the perspector is X(i):

See César Lozada, Perspectivities involving the g and h mappings.

X(41431) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(TANGENTIAL OF 1ST CIRCUMPERP)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 + 2*b^2 - 2*a*c + c^2)*(a^2 - 2*a*b + b^2 + 2*c^2) : :

X(41431) is also the perspector of these triangles: ABC and h(anti-Mandart-incircle)

X(41431) lies on these lines: {35, 1458}, {404, 518}, {1442, 3748}, {1876, 5045}, {2772, 7343}, {3286, 35193}, {5434, 6740}

X(41431) = isogonal conjugate of X(3058)

X(41432) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(TANGENTIAL OF 2ND CIRCUMPERP)

Barycentrics    a^2*(a - b - c)*(a^2 + 2*b^2 + 2*a*c + c^2)*(a^2 + 2*a*b + b^2 + 2*c^2) : :

X(41432) lies on these lines: {1, 15107}, {35, 595}, {106, 40148}, {960, 3689}, {1319, 1442}, {1829, 4222}, {3058, 6740}, {3961, 40172}, {4267, 35193}, {7343, 41192}, {10545, 30116}

X(41432) = isogonal conjugate of X(5434)

X(41433) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(GOSSARD)

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^8 + 2*a^6*b^2 - 6*a^4*b^4 + 2*a^2*b^6 + b^8 - 4*a^6*c^2 + 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 4*b^6*c^2 + 3*a^4*c^4 - 8*a^2*b^2*c^4 + 3*b^4*c^4 + 2*a^2*c^6 + 2*b^2*c^6 - 2*c^8)*(a^8 - 4*a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6 - 2*b^8 + 2*a^6*c^2 + 4*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 + 4*a^2*b^2*c^4 + 3*b^4*c^4 + 2*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(41433) lies on these lines: {4, 15459}, {6, 1304}, {67, 10762}, {74, 34568}, {265, 16075}, {3426, 35908}, {9139, 10097}, {14380, 40384}

X(41433) = isogonal conjugate of X(1651)

X(41434) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(AQUILLA)

Barycentrics    a^2*(a + 4*b + c)*(a + b + 4*c) : :

X(41434) lies on these lines: {1, 4015}, {6, 6767}, {42, 106}, {56, 4256}, {58, 902}, {86, 519}, {101, 21754}, {145, 19741}, {269, 18421}, {292, 16971}, {386, 3445}, {512, 23345}, {518, 34916}, {996, 3241}, {1027, 4160}, {1126, 5315}, {1220, 3244}, {1222, 3635}, {1386, 34893}, {1438, 16785}, {2163, 2177}, {2242, 21782}, {2334, 16483}, {2802, 7312}, {3230, 25426}, {4482, 29584}, {4649, 37129}, {4658, 39949}, {4752, 16666}, {8649, 17962}, {9432, 10761}, {10013, 30116}, {15485, 39972}, {16484, 40433}, {16499, 17018}

X(41434) = isogonal conjugate of X(551)

X(41435) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(ARA)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 + 2*b^2 + c^2)*(a^2 + b^2 + 2*c^2) : :

X(41435) lies on the Jerabek hyperbola and these lines: {4, 3096}, {6, 1627}, {54, 5092}, {64, 15577}, {65, 33844}, {66, 3410}, {67, 3631}, {68, 10519}, {69, 4175}, {71, 7293}, {74, 7953}, {110, 34437}, {141, 15321}, {182, 13472}, {248, 22052}, {265, 28725}, {290, 1232}, {511, 1173}, {542, 5900}, {895, 11574}, {1176, 3917}, {1245, 5315}, {3313, 12039}, {3426, 33533}, {3527, 7393}, {3564, 34483}, {3630, 13622}, {4558, 22078}, {4846, 28419}, {5157, 33884}, {5447, 40441}, {5486, 20080}, {5888, 22336}, {5965, 18368}, {7691, 21167}, {11008, 17040}, {11464, 34439}, {11515, 36296}, {11516, 36297}, {12017, 16266}, {14060, 22062}, {14810, 16835}, {15066, 34207}, {15080, 34436}, {15107, 34573}, {15328, 31065}, {15812, 18124}, {17711, 34507}, {20188, 35364}, {21766, 37485}, {22334, 31884}, {38005, 40911}

X(41435) = isogonal conjugate of X(428)
X(41435) = isotomic conjugate of polar conjugate of X(3108)

X(41436) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(CAELUM)

Barycentrics    a^2*(a + b - 5*c)*(a - 5*b + c) : :

X(41436) lies on these lines: {1, 3689}, {2, 1120}, {6, 1149}, {34, 1388}, {55, 106}, {56, 902}, {58, 3304}, {86, 16711}, {87, 16484}, {269, 1319}, {551, 996}, {663, 23345}, {870, 40029}, {937, 16485}, {999, 2163}, {1001, 37129}, {1027, 14421}, {1201, 2334}, {1220, 3622}, {1222, 3616}, {1411, 40172}, {1438, 8649}, {2177, 8162}, {2279, 3230}, {2297, 3247}, {2665, 16497}, {3242, 34893}, {3303, 3445}, {3636, 19746}, {3669, 37627}, {4423, 16499}, {5315, 7373}, {6065, 9268}, {9259, 21782}, {13541, 17063}, {15485, 36598}, {34916, 38315}

X(41436) = isogonal conjugate of X(3241)

X(41437) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(INNER GREBE)

Barycentrics    a^2*(3*b^2 - S)*(-3*c^2 + S) : :

X(41437) lies on these lines: {25, 6396}, {39, 41438}, {393, 32786}, {589, 15066}, {615, 34288}, {1989, 8252}, {2165, 32790}, {6221, 8576}, {6395, 8577}, {8253, 30537}, {31473, 39974}

X(41437) = isogonal conjugate of X(19053)

X(41438) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(OUTER GREBE)

Barycentrics    a^2*(3*b^2 + S)*(3*c^2 + S) : :

X(41438) lies on these lines: {25, 6200}, {39, 41437}, {111, 9600}, {393, 32785}, {588, 15066}, {590, 34288}, {1989, 8253}, {2165, 32789}, {6199, 8576}, {6398, 8577}, {8252, 30537}, {31473, 39981}

X(41438) = isogonal conjugate of X(19054)

X(41439) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(MANDART-INCIRCLE)

Barycentrics    a*(3*a*b - 3*b^2 - 2*a*c + 3*b*c)*(2*a*b - 3*a*c - 3*b*c + 3*c^2) : :

X(41439) lies on these lines: {518, 1278}, {672, 3973}, {1458, 33633}, {3286, 7419}, {4014, 9309}, {14261, 16189}

X(41439) = isogonal conjugate of X(4421)

X(41440) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(5TH BROCARD)

Barycentrics    a^2*(a^2*b^2 + 3*b^4 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + a^2*c^2 + b^2*c^2 + 3*c^4) : :

X(41440) lies on these lines: {39, 3231}, {141, 538}, {511, 30489}, {1843, 33843}, {3094, 14609}, {3111, 5970}, {24256, 25322}

X(41440) = isogonal conjugate of X(12150)
X(41440) = trilinear pole of line X(888)X(3005)

X(41441) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(ORTHIC OF INTOUCH)

Barycentrics    a*(3*a^2 - 2*a*b + 3*b^2 - 3*c^2)*(3*a^2 - 3*b^2 - 2*a*c + 3*c^2) : :

X(41441) lies on these lines: {6, 3340}, {9, 3617}, {55, 3731}, {57, 40968}, {284, 3247}, {333, 3729}, {1436, 37583}, {2160, 8557}, {2161, 2270}, {2339, 7308}, {3928, 4373}, {3973, 7991}, {6169, 18785}, {16548, 39943}, {16814, 34820}

X(41441) = isogonal conjugate of X(3928)
X(41441) = isotomic conjugate of X(21605)

X(41442) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(OUTER YFF)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*b*c + 2*a*b^2*c - 2*a^2*c^2 - 4*a*b*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*b*c - 4*a*b^2*c - 2*a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 + c^4) : :

X(41442) lies on these lines: {1, 22128}, {222, 1411}, {998, 5902}, {1167, 4256}, {3445, 36746}, {14793, 36052}

X(41442) = isogonal conjugate of complement of X(34625)

X(41443) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(ANTI-5TH BROCARD)

Barycentrics    a^2*(a^4 + 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 + a^2*c^2 + b^2*c^2 + c^4) : :

X(41443) lies on these lines: {30, 141}, {39, 1495}, {74, 14458}, {76, 15107}, {1843, 14581}, {5039, 30489}, {5309, 34417}, {7884, 10545}

X(41443) = isogonal conjugate of X(7811)

X(41444) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(3RD TRI-SQUARES CENTRAL)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2 + S)*(a^2 - 2*b^2 + c^2 + S) : :

X(41444) is also the perspector of these triangles: ABC and h(anti-outer Grebe).

X(41444) lies on these lines: {2, 13832}, {6, 41419}, {25, 41411}, {493, 5407}, {6396, 21448}, {8854, 39951}

X(41445) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(4TH TRI-SQUARES CENTRAL)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2 - S)*(a^2 - 2*b^2 + c^2 - S) : :

X(41445) is also the perspector of these triangles: ABC and h(anti-inner Grebe).

X(41445) lies on these lines: {2, 9600}, {25, 41410}, {494, 5406}, {2502, 41419}, {6200, 21448}, {8855, 39951}

X(41446) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(2ND ANTI-CIRCUMPERP TANGENTIAL)

Barycentrics    a*(3*a^2*b - 3*b^3 + 2*a^2*c - 4*a*b*c + 2*a*c^2 + 3*b*c^2)*(2*a^2*b + 2*a*b^2 + 3*a^2*c - 4*a*b*c + 3*b^2*c - 3*c^3) : :

X(41446) lies on these lines: {960, 3617}, {1193, 3340}, {2269, 3731}, {3303, 4267}, {6048, 10563}, {23839, 24471}

X(41446) = isogonal conjugate of X(11194)

X(41447) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(EULER)

Barycentrics    a^2*(23*a^8 - 44*a^6*b^2 - 6*a^4*b^4 + 52*a^2*b^6 - 25*b^8 - 44*a^6*c^2 + 68*a^4*b^2*c^2 - 52*a^2*b^4*c^2 + 28*b^6*c^2 - 6*a^4*c^4 - 52*a^2*b^2*c^4 - 6*b^4*c^4 + 52*a^2*c^6 + 28*b^2*c^6 - 25*c^8) : :

X(41447) is also the perspector of these triangles: tangential and h(ABC reflected in X(3)).
X(41447) lies on these lines: {3, 14924}, {6, 186}, {1192, 1614}, {1204, 1495}, {1511, 37672}, {2070, 33534}, {3426, 10606}, {7575, 35237}, {10545, 38446}, {11424, 15750}, {11438, 17821}, {15051, 33586}, {17825, 18324}, {18405, 37460}, {18579, 40909}, {32534, 38848}

X(41448) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(X(3) REFLECTED IN ABC)

Barycentrics    a^2*(7*a^8 - 13*a^6*b^2 - 3*a^4*b^4 + 17*a^2*b^6 - 8*b^8 - 13*a^6*c^2 + 7*a^4*b^2*c^2 - 17*a^2*b^4*c^2 + 23*b^6*c^2 - 3*a^4*c^4 - 17*a^2*b^2*c^4 - 30*b^4*c^4 + 17*a^2*c^6 + 23*b^2*c^6 - 8*c^8) : :

X(41448) lies on these lines: {24, 31860}, {25, 74}, {54, 22233}, {389, 1495}, {1620, 10594}, {2070, 10545}, {3431, 11202}, {3517, 11464}, {4550, 14002}, {5092, 37939}, {5890, 41424}, {10752, 38851}, {10986, 38867}, {12106, 13340}

X(41448) = Vu tangential transform of X(381)

X(41449) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(1ST BROCARD)

Barycentrics    a^2*(7*a^8 - 14*a^6*b^2 + 33*a^4*b^4 - 26*a^2*b^6 + b^8 - 14*a^6*c^2 - 24*a^4*b^2*c^2 + 12*a^2*b^4*c^2 + 22*b^6*c^2 + 33*a^4*c^4 + 12*a^2*b^2*c^4 - 39*b^4*c^4 - 26*a^2*c^6 + 22*b^2*c^6 + c^8) : :

X(41449) is also the perspector of these triangles: tangential and h(2nd Brocard).
X(41449) is also the perspector of these triangles: tangential and h(anti-1st Euler).

X(41449) lies on these lines: {6, 10630}, {187, 9225}, {316, 10488}, {2393, 8586}, {2502, 33704}, {39689, 41404}

X(41450) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(JOHNSON TRIANGLE)

Barycentrics    a^2*(7*a^8 - 22*a^6*b^2 + 24*a^4*b^4 - 10*a^2*b^6 + b^8 - 22*a^6*c^2 - 2*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 14*b^6*c^2 + 24*a^4*c^4 + 10*a^2*b^2*c^4 - 30*b^4*c^4 - 10*a^2*c^6 + 14*b^2*c^6 + c^8) : :

X(41450) lies on these lines: {3, 41428}, {6, 14157}, {22, 399}, {24, 185}, {74, 154}, {110, 35237}, {323, 12082}, {378, 3426}, {403, 39874}, {1181, 31860}, {1498, 11464}, {1614, 11425}, {1620, 6241}, {4550, 6800}, {5890, 41424}, {7509, 15052}, {10294, 10721}, {10323, 15068}, {10594, 11432}, {10752, 19149}, {11004, 18534}, {11206, 35480}, {12174, 35479}, {14530, 32534}, {15080, 18451}, {15577, 35218}, {15919, 33803}, {20125, 31152}, {26882, 37487}, {26883, 37505}, {35264, 37470}

X(41451) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(AQUILLA)

Barycentrics    a^2*(7*a^2 - a*b - 8*b^2 - a*c - 7*b*c - 8*c^2) : :

X(41451) lies on these lines: {35, 595}, {55, 106}, {101, 21782}, {574, 38865}, {2163, 2177}, {2320, 4792}, {3295, 33804}, {4257, 5217}, {26285, 38857}, {41416, 41423}

X(41452) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(CAELUM)

Barycentrics    a^2*(23*a^2 - 2*a*b - 25*b^2 - 2*a*c + 22*b*c - 25*c^2) : :

X(41452) lies on these lines: {6, 36}, {106, 21000}, {595, 8572}, {902, 1616}, {5210, 38865}, {7280, 16486}

X(41453) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND h(OUTER GARCIA)

Barycentrics    a^2*(7*a^2 + 8*a*b + b^2 + 8*a*c - 16*b*c + c^2) : :

X(41453) is also the perspector of these triangles: tangential and h(anti-Aquilla).

X(41453) lies on these lines: {3, 902}, {6, 6767}, {106, 3052}, {517, 16487}, {595, 7373}, {999, 2163}, {1159, 1279}, {1191, 33771}, {1384, 8649}, {1482, 16485}, {1616, 4257}, {3230, 21309}, {3246, 40587}, {3295, 5312}, {5255, 16863}, {8158, 38857}, {8692, 9708}, {15485, 16857}, {15934, 35227}

X(41454) = PERSPECTOR OF THESE TRIANGLES: 1ST CIRCUMPERP AND h(1ST CIRCUMPERP)

Barycentrics    a^2*(a^3 - a^2*b + 2*a*b^2 - 2*b^3 - a^2*c + 2*a*b*c - b^2*c + 2*a*c^2 - b*c^2 - 2*c^3) : :

X(41454) is also the perspector of these triangles: 1st circumperp and h(3rd Euler).

X(41454) lies on these lines: {3, 9049}, {6, 31}, {46, 16496}, {74, 30257}, {210, 34417}, {517, 4550}, {518, 3098}, {942, 30145}, {1350, 9026}, {1631, 2340}, {3242, 5221}, {3620, 12586}, {3630, 5848}, {3681, 15107}, {3826, 34573}, {4471, 4878}, {5092, 9052}, {8679, 33878}, {9047, 37517}, {15624, 37508}, {17454, 37586}

X(41455) = PERSPECTOR OF THESE TRIANGLES: 1ST CIRCUMPERP AND h(2ND CIRCUMPERP)

Barycentrics    a^2*(a^5 + a^4*b + a^3*b^2 + a^2*b^3 - 2*a*b^4 - 2*b^5 + a^4*c - 2*a^2*b^2*c + b^4*c + a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 - 5*b^3*c^2 + a^2*c^3 - 5*b^2*c^3 - 2*a*c^4 + b*c^4 - 2*c^5) : :

X(41455) is also the perspector of these triangles: 1st circumperp and h(4th Euler).

X(41455) lies on these lines: {3, 902}, {30, 4660}, {71, 3426}, {74, 9070}, {517, 3098}, {960, 3579}, {2177, 30269}, {3654, 30615}, {3753, 34417}, {7986, 12702}, {11438, 31788}, {14915, 35203}, {16474, 37482}, {23361, 35237}, {37478, 37562}

X(41455) = center of similitude of anti-orthocentroidal and outer Garcia triangles

X(41456) = PERSPECTOR OF THESE TRIANGLES: 1ST CIRCUMPERP AND h(2ND EXTOUCH)

Barycentrics    a*(7*a^4 + 14*a^3*b - 4*a^2*b^2 - 14*a*b^3 - 3*b^4 + 14*a^3*c - 4*a^2*b*c - 10*a*b^2*c - 4*a^2*c^2 - 10*a*b*c^2 + 6*b^2*c^2 - 14*a*c^3 - 3*c^4) : :

X(41456) is also the perspector of these triangles: 1st circumperp and h(1st Conway).
X(41456) is also the perspector of these triangles: 1st circumperp and h(Ascella).

X(41456) lies on these lines: {6, 165}, {9, 3579}, {40, 2294}, {74, 15322}, {376, 4034}, {1449, 35242}, {3723, 7991}, {3731, 34820}

X(41457) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF 1ST CIRCUMPERP AND h(EXCENTRAL)

Barycentrics    a^2*(a - b - c)*(a^2 - 3*a*b + 2*b^2 - 3*a*c - b*c + 2*c^2) : :

X(41457) is also the perspector of these triangles: tangential of 1st circumperp and h(2nd Conway).
X(41457) is also the perspector of these triangles: tangential of 1st circumperp and h(Wasat).

X(41457) lies on these lines: {6, 3939}, {55, 7064}, {101, 12033}, {612, 1962}, {1995, 8694}, {3174, 3731}, {3689, 40937}, {4097, 19589}, {4256, 37592}, {5853, 16484}, {15624, 37508}, {15733, 16814}

X(41458) = PERSPECTOR OF THESE TRIANGLES: ANTIPEDAL OF X(14) AND h(ARTZT)

Barycentrics    Sqrt[3]*(9*a^10 - 24*a^8*b^2 + a^6*b^4 + 15*a^4*b^6 + 2*a^2*b^8 - 3*b^10 - 24*a^8*c^2 - 32*a^6*b^2*c^2 + 33*a^4*b^4*c^2 + 10*a^2*b^6*c^2 + 13*b^8*c^2 + a^6*c^4 + 33*a^4*b^2*c^4 - 24*a^2*b^4*c^4 - 10*b^6*c^4 + 15*a^4*c^6 + 10*a^2*b^2*c^6 - 10*b^4*c^6 + 2*a^2*c^8 + 13*b^2*c^8 - 3*c^10) + 6*(a^8 + 13*a^6*b^2 - 10*a^4*b^4 - 3*a^2*b^6 - b^8 + 13*a^6*c^2 - 8*a^4*b^2*c^2 - a^2*b^4*c^2 - 4*b^6*c^2 - 10*a^4*c^4 - a^2*b^2*c^4 + 10*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 - c^8)*S : :

X(41458) is also the perspector of these triangles: antipedal of X(14) and h(infinite altitude).
X(41458) is also the perspector of these triangles: tangential of 1st circumperp and h(anti-Artzt).

X(41458) lies on these lines: {376, 5464}, {383, 9750}, {1503, 5474}, {2794, 5473}, {3104, 11257}, {9116, 12117}, {11302, 41039}, {21157, 37351}, {35303, 41043}, {36761, 38738}

X(41459) = PERSPECTOR OF THESE TRIANGLES: TRINH AND h(13TH FERMAT-DAO)

Barycentrics    a^2*(Sqrt[3]*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 + a^2*b^6*c^2 - b^8*c^2 - 3*a^6*c^4 + 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 + a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) + 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8)*S) : :

X(41459) lies on these lines: {3, 13}, {30, 36248}, {95, 99}, {542, 3439}, {1605, 18400}, {5995, 13350}, {14369, 40707}, {16459, 36755}

X(41460) = PERSPECTOR OF THESE TRIANGLES: TRINH AND h(14TH FERMAT-DAO)

Barycentrics    a^2*(Sqrt[3]*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 + a^2*b^6*c^2 - b^8*c^2 - 3*a^6*c^4 + 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 + a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8)*S) : :

X(41460) lies on these lines: {3, 14}, {30, 36249}, {95, 99}, {542, 3438}, {1606, 18400}, {5994, 13349}, {14368, 40706}, {16460, 36756}

X(41461) = PERSPECTOR OF THESE TRIANGLES: SYMMEDIAL AND h(GEMINI 19)

Barycentrics    a^2*(a + b - 2*c)*(a - 2*b + c)*(5*a^2 - 5*a*b - b^2 - 5*a*c + 7*b*c - c^2) : :

X(41461) lies on these lines: {36, 106}, {2177, 40215}, {2245, 17969}, {3218, 4792}, {8649, 41405}, {9324, 9326}, {14193, 17960}

X(41462) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF THOMSON AND h(TANGENTIAL OF THOMSON)

Barycentrics    a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 5*b^2*c^2 - 2*c^4) : :

X(41462) lies on these lines: {2, 3098}, {3, 74}, {6, 1627}, {22, 10546}, {23, 5650}, {111, 12149}, {141, 12367}, {182, 11004}, {323, 3917}, {352, 8589}, {376, 4550}, {426, 39243}, {511, 7496}, {549, 3581}, {574, 8623}, {631, 37478}, {1078, 4576}, {1216, 15032}, {1350, 5640}, {1370, 3619}, {1473, 26911}, {1495, 3819}, {1583, 35246}, {1584, 35247}, {1613, 38862}, {1993, 12017}, {1995, 31884}, {2981, 3105}, {3051, 12055}, {3060, 7484}, {3104, 6151}, {3448, 40107}, {3523, 7691}, {3524, 37470}, {3528, 15062}, {3620, 11442}, {3763, 31133}, {4210, 37508}, {5017, 15302}, {5085, 11422}, {5104, 33873}, {5133, 34573}, {5189, 24206}, {5324, 37680}, {5447, 34148}, {5643, 11002}, {5651, 7492}, {5891, 12112}, {6030, 7712}, {6101, 15037}, {7085, 26910}, {7386, 23293}, {7465, 33844}, {7495, 21167}, {7509, 37483}, {7516, 13434}, {7533, 29317}, {7552, 12900}, {7667, 18358}, {7703, 31152}, {7771, 9146}, {7810, 13210}, {9781, 13154}, {9998, 41273}, {10304, 13445}, {10510, 37283}, {10519, 18911}, {10564, 35921}, {10691, 37636}, {11003, 17508}, {11145, 36755}, {11146, 36756}, {11412, 40913}, {11430, 37126}, {11439, 37198}, {11451, 16419}, {11673, 14096}, {11793, 15052}, {12100, 40112}, {12294, 37977}, {13336, 15801}, {13587, 26637}, {14002, 16187}, {14490, 33537}, {15028, 37486}, {15033, 37496}, {15082, 16042}, {15305, 35237}, {15360, 37648}, {16261, 33532}, {17811, 26881}, {18392, 18536}, {35283, 37900}

X(41463) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL OF THOMSON AND h(ANTICOMPLEMENTARY OF THOMSON)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 - 34*a^4*b^2*c^2 + 30*a^2*b^4*c^2 + 6*b^6*c^2 + 30*a^2*b^2*c^4 - 10*b^4*c^4 + 2*a^2*c^6 + 6*b^2*c^6 - c^8) : :

X(41463) lies on these lines: {2, 3}, {74, 2930}, {576, 36987}, {1296, 33900}, {5889, 33543}, {7998, 35237}, {8717, 15066}, {10541, 15033}, {10606, 15581}, {11422, 37483}, {12290, 16936}, {13233, 38738}, {14915, 21766}, {16163, 32305}, {16261, 33534}, {30256, 33998}

X(41464) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(TANGENTIAL)

Barycentrics    a^2*(a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6 + 2*a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 - 2*c^6) : :

X(41464) is also the perspector of these triangles: orthic of anticomplementary and h(tangential of anticomplementary).
X(41464) is also the perspector of these triangles: Gemini 113 and h(tangential).
X(41464) is also the perspector of these triangles: Gemini 113 and h(tangential of anticomplementary).

X(41464) lies on these lines: {3, 11387}, {6, 22}, {20, 1352}, {23, 11574}, {110, 3313}, {141, 15321}, {159, 2979}, {182, 38435}, {511, 12226}, {755, 3565}, {858, 5888}, {1350, 12111}, {1351, 11423}, {1370, 3619}, {1503, 7691}, {1843, 6636}, {1974, 37913}, {2071, 32600}, {2854, 34437}, {2937, 19128}, {3057, 3100}, {3630, 32244}, {3631, 12367}, {3867, 7495}, {5092, 7488}, {5104, 32464}, {6200, 8408}, {6396, 8420}, {7485, 7716}, {7492, 19126}, {7493, 10545}, {7555, 19129}, {7667, 26156}, {8718, 11412}, {9715, 12017}, {9822, 15246}, {9909, 26206}, {9918, 22424}, {9967, 12088}, {10546, 26283}, {10565, 34417}, {10979, 37184}, {11411, 37478}, {11440, 14927}, {11515, 34008}, {11516, 34009}, {12087, 12294}, {12272, 37485}, {14810, 24860}, {14958, 32085}, {15531, 37491}, {16049, 33844}, {16932, 31360}, {20806, 26881}, {21512, 23635}, {22052, 37183}, {23061, 35707}, {27866, 40949}, {38110, 38848}

X(41465) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(CIRCUM-ORTHIC)

Barycentrics    7*a^10 - 7*a^8*b^2 - 18*a^6*b^4 + 26*a^4*b^6 - 5*a^2*b^8 - 3*b^10 - 7*a^8*c^2 + 4*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 4*a^2*b^6*c^2 + 9*b^8*c^2 - 18*a^6*c^4 - 2*a^4*b^2*c^4 + 18*a^2*b^4*c^4 - 6*b^6*c^4 + 26*a^4*c^6 - 4*a^2*b^2*c^6 - 6*b^4*c^6 - 5*a^2*c^8 + 9*b^2*c^8 - 3*c^10 : :

X(41465) is also the perspector of these triangles: orthic of anticomplementary and h(2nd Euler).
X(41465) is also the perspector of these triangles: Gemini 113 and h(circum-orthic).
X(41465) is also the perspector of these triangles: Gemini 113 and h(2nd Euler).

X(41465) lies on these lines: {2, 35254}, {4, 1209}, {20, 155}, {30, 69}, {74, 1370}, {376, 3431}, {399, 11206}, {541, 11001}, {631, 7706}, {1154, 10938}, {1480, 4294}, {1511, 37669}, {1657, 34781}, {2777, 38885}, {3091, 32620}, {3146, 11472}, {3529, 10625}, {3546, 37487}, {3575, 11487}, {3581, 18531}, {3619, 18420}, {4293, 6580}, {5059, 16659}, {5663, 14927}, {6643, 11438}, {7386, 37470}, {8717, 17538}, {11411, 12225}, {11427, 14805}, {11820, 15704}, {11821, 31833}, {12220, 13754}, {12900, 38282}, {14641, 32601}, {14791, 18931}, {15052, 31304}, {15066, 18533}, {15311, 33534}, {16051, 32110}, {17702, 32244}, {18358, 18494}, {18537, 34417}

X(41465) = anticomplement of X(40909)

X(41466) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(KOSNITA)

Barycentrics    a^2*(a^14 - 2*a^12*b^2 - 3*a^10*b^4 + 10*a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10 + 7*a^2*b^12 - 2*b^14 - 2*a^12*c^2 - a^10*b^2*c^2 + 10*a^8*b^4*c^2 - 8*a^6*b^6*c^2 + 4*a^4*b^8*c^2 - 7*a^2*b^10*c^2 + 4*b^12*c^2 - 3*a^10*c^4 + 10*a^8*b^2*c^4 - a^6*b^4*c^4 - 4*a^4*b^6*c^4 - 2*a^2*b^8*c^4 + 10*a^8*c^6 - 8*a^6*b^2*c^6 - 4*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - 2*b^8*c^6 - 5*a^6*c^8 + 4*a^4*b^2*c^8 - 2*a^2*b^4*c^8 - 2*b^6*c^8 - 6*a^4*c^10 - 7*a^2*b^2*c^10 + 7*a^2*c^12 + 4*b^2*c^12 - 2*c^14) : :

X(41466) is also the perspector of these triangles: Gemini 113 and h(Kosnita).

X(41466) lies on these lines: {3, 6242}, {22, 399}, {156, 38435}, {323, 1147}, {1154, 18882}, {2071, 3098}, {3153, 18427}, {3431, 3581}, {4550, 10296}, {7691, 34785}, {11413, 33542}, {11438, 38448}, {12225, 18432}, {12380, 18474}, {23330, 37444}

X(41467) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(5TH MIXTILINEAR OF ORTHIC)

Barycentrics    91*a^10 - 145*a^8*b^2 - 90*a^6*b^4 + 230*a^4*b^6 - 65*a^2*b^8 - 21*b^10 - 145*a^8*c^2 + 484*a^6*b^2*c^2 - 278*a^4*b^4*c^2 - 124*a^2*b^6*c^2 + 63*b^8*c^2 - 90*a^6*c^4 - 278*a^4*b^2*c^4 + 378*a^2*b^4*c^4 - 42*b^6*c^4 + 230*a^4*c^6 - 124*a^2*b^2*c^6 - 42*b^4*c^6 - 65*a^2*c^8 + 63*b^2*c^8 - 21*c^10 : :

X(41467) is also the perspector of these triangles: Gemini 113 and h(5th mixtilinear of orthic).

X(41467) lies on these lines: {2, 13202}, {1620, 3146}, {2883, 3522}, {12111, 13348}, {15717, 15751}, {16386, 20080}

X(41468) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(ANTI-HUTSON INTOUCH)

Barycentrics    a^2*(a^14 - 2*a^12*b^2 - 3*a^10*b^4 + 10*a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10 + 7*a^2*b^12 - 2*b^14 - 2*a^12*c^2 + 35*a^10*b^2*c^2 - 53*a^8*b^4*c^2 - 26*a^6*b^6*c^2 + 76*a^4*b^8*c^2 - 25*a^2*b^10*c^2 - 5*b^12*c^2 - 3*a^10*c^4 - 53*a^8*b^2*c^4 + 278*a^6*b^4*c^4 - 166*a^4*b^6*c^4 - 83*a^2*b^8*c^4 + 27*b^10*c^4 + 10*a^8*c^6 - 26*a^6*b^2*c^6 - 166*a^4*b^4*c^6 + 202*a^2*b^6*c^6 - 20*b^8*c^6 - 5*a^6*c^8 + 76*a^4*b^2*c^8 - 83*a^2*b^4*c^8 - 20*b^6*c^8 - 6*a^4*c^10 - 25*a^2*b^2*c^10 + 27*b^4*c^10 + 7*a^2*c^12 - 5*b^2*c^12 - 2*c^14) : :

X(41468) is also the perspector of these triangles: Gemini 113 and h(anti-Hutson intouch).

X(41468) lies on these lines: {20, 11438}, {69, 13445}, {2071, 5888}, {3426, 15066}, {11413, 33537}, {12058, 12219}

X(41469) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(ANTI-INCIRCLE-CIRCLES)

Barycentrics    a^2*(a^14 - 2*a^12*b^2 - 3*a^10*b^4 + 10*a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10 + 7*a^2*b^12 - 2*b^14 - 2*a^12*c^2 - 37*a^10*b^2*c^2 + 73*a^8*b^4*c^2 + 10*a^6*b^6*c^2 - 68*a^4*b^8*c^2 + 11*a^2*b^10*c^2 + 13*b^12*c^2 - 3*a^10*c^4 + 73*a^8*b^2*c^4 + 206*a^6*b^4*c^4 - 166*a^4*b^6*c^4 - 83*a^2*b^8*c^4 - 27*b^10*c^4 + 10*a^8*c^6 + 10*a^6*b^2*c^6 - 166*a^4*b^4*c^6 + 130*a^2*b^6*c^6 + 16*b^8*c^6 - 5*a^6*c^8 - 68*a^4*b^2*c^8 - 83*a^2*b^4*c^8 + 16*b^6*c^8 - 6*a^4*c^10 + 11*a^2*b^2*c^10 - 27*b^4*c^10 + 7*a^2*c^12 + 13*b^2*c^12 - 2*c^14) : :

X(414) lies on these lines: {20, 4550}, {1173, 33524}, {11412, 35237}, {11820, 12111}, {12220, 13391}, {14483, 15018}

X(41469) is also the perspector of these triangles: Gemini 113 and h(anti-incircle-circles).

X(41470) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(EHRMANN SIDE)

Barycentrics    16*a^10 - 25*a^8*b^2 - 15*a^6*b^4 + 35*a^4*b^6 - 5*a^2*b^8 - 6*b^10 - 25*a^8*c^2 + 49*a^6*b^2*c^2 - 23*a^4*b^4*c^2 - 19*a^2*b^6*c^2 + 18*b^8*c^2 - 15*a^6*c^4 - 23*a^4*b^2*c^4 + 48*a^2*b^4*c^4 - 12*b^6*c^4 + 35*a^4*c^6 - 19*a^2*b^2*c^6 - 12*b^4*c^6 - 5*a^2*c^8 + 18*b^2*c^8 - 6*c^10 : :

X(41470) is also the perspector of these triangles: Gemini 113 and h(Ehrmann side).

X(41470) lies on these lines: {20, 5654}, {30, 7703}, {69, 11001}, {399, 15681}, {1657, 7691}, {3146, 15432}, {3534, 18550}, {6241, 12226}, {12111, 13340}, {15683, 18474}, {19710, 40112}, {32244, 33878}, {32348, 34797}

X(41471) = PERSPECTOR OF THESE TRIANGLES: ORTHIC OF ANTICOMPLEMENTARY AND h(EHRMANN VERTEX)

Barycentrics    a^2*(4*a^14 - 8*a^12*b^2 - 12*a^10*b^4 + 40*a^8*b^6 - 20*a^6*b^8 - 24*a^4*b^10 + 28*a^2*b^12 - 8*b^14 - 8*a^12*c^2 + 20*a^10*b^2*c^2 - 2*a^8*b^4*c^2 - 44*a^6*b^6*c^2 + 64*a^4*b^8*c^2 - 40*a^2*b^10*c^2 + 10*b^12*c^2 - 12*a^10*c^4 - 2*a^8*b^2*c^4 + 47*a^6*b^4*c^4 - 34*a^4*b^6*c^4 - 17*a^2*b^8*c^4 + 18*b^10*c^4 + 40*a^8*c^6 - 44*a^6*b^2*c^6 - 34*a^4*b^4*c^6 + 58*a^2*b^6*c^6 - 20*b^8*c^6 - 20*a^6*c^8 + 64*a^4*b^2*c^8 - 17*a^2*b^4*c^8 - 20*b^6*c^8 - 24*a^4*c^10 - 40*a^2*b^2*c^10 + 18*b^4*c^10 + 28*a^2*c^12 + 10*b^2*c^12 - 8*c^14) : :

X(41471) is also the perspector of these triangles: Gemini 113 and h(Ehrmann vertex).

X(41471) lies on these lines: {1511, 10298}, {3098, 32244}, {10564, 11412}, {12111, 38435}, {12226, 13340}, {20421, 35493}

X(41472) = PERSPECTOR OF THESE TRIANGLES: OUTER LE VIET AN AND h(OUTER NAPOLEON)

Barycentrics    a^2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 - b^2*c^2 - c^4 + 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(41472) is also the perspector of the outer Le Viet An triangle and each of the following triangles:
1st half-diamonds-central equilateral triangle (see X(33338)),
3rd inner-Fermat-Dao-Nhi triangle (see X(33602)),
4th inner-Fermat-Dao-Nhi triangle (see X(33602)),
1st outer-Fermat-Dao-Nhi triangle (see X(33602)),
2nd outer-Fermat-Dao-Nhi triangle (see X(33602)).

X(41472) lies on these lines: {2, 22796}, {3, 74}, {15, 1337}, {16, 5012}, {22, 11480}, {23, 13350}, {30, 8838}, {98, 14185}, {99, 14177}, {111, 36515}, {323, 36755}, {616, 14922}, {1495, 11146}, {1627, 19781}, {2379, 14183}, {3060, 11485}, {3130, 10545}, {3170, 10646}, {5092, 11145}, {6636, 36756}, {7492, 9735}, {9140, 40709}, {9142, 17402}, {9203, 14175}, {9736, 11003}, {10645, 34009}, {11078, 15768}, {12042, 40854}, {14181, 35314}, {14538, 23061}, {16642, 38403}, {18582, 37847}, {21158, 38431}

X(41472) = {X(3),X(15080)}-harmonic conjugate of X(41473)

X(41473) = PERSPECTOR OF THESE TRIANGLES: INNER LE VIET AN AND h(INNER NAPOLEON)

Barycentrics    a^2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 - b^2*c^2 - c^4 - 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(41473) is also the perspector of the inner Le Viet An triangle and each of the following triangles:
2nd half-diamonds-central equilateral triangle (see X(33338)),
1st inner-Fermat-Dao-Nhi triangle (see X(33602)),
2nd inner-Fermat-Dao-Nhi triangle (see X(33602)),
3rd outer-Fermat-Dao-Nhi triangle (see X(33602)),
4th outer-Fermat-Dao-Nhi triangle (see X(33602)).

X(41473) lies on these lines: {2, 22797}, {3, 74}, {15, 5012}, {16, 1338}, {22, 11481}, {23, 13349}, {30, 8836}, {98, 14187}, {99, 14181}, {111, 36514}, {323, 36756}, {617, 14921}, {1495, 11145}, {1627, 19780}, {2378, 14184}, {3060, 11486}, {3129, 10545}, {3171, 10645}, {5092, 11146}, {6636, 36755}, {7492, 9736}, {9140, 40710}, {9142, 17403}, {9202, 14176}, {9735, 11003}, {10646, 34008}, {11092, 15769}, {12042, 40855}, {14177, 35315}, {14539, 23061}, {16643, 38404}, {18581, 37849}, {21159, 38432}

X(41473) = {X(3),X(15080)}-harmonic conjugate of X(41472)

X(41474) = PERSPECTOR OF THESE TRIANGLES: 5th FERMAT-DAO AND h(3RD INNER FERMAT-DAO-NHI)

Barycentrics    a^2*(Sqrt[3]*(a^12 - 4*a^10*b^2 + 6*a^8*b^4 - 4*a^6*b^6 + a^4*b^8 - 4*a^10*c^2 + 12*a^8*b^2*c^2 - 7*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 10*a^2*b^8*c^2 - 7*b^10*c^2 + 6*a^8*c^4 - 7*a^6*b^2*c^4 + 5*a^4*b^4*c^4 - 7*a^2*b^6*c^4 + 22*b^8*c^4 - 4*a^6*c^6 - 4*a^4*b^2*c^6 - 7*a^2*b^4*c^6 - 30*b^6*c^6 + a^4*c^8 + 10*a^2*b^2*c^8 + 22*b^4*c^8 - 7*b^2*c^10) + 2*(a^10 - 3*a^8*b^2 + 5*a^6*b^4 - 7*a^4*b^6 + 6*a^2*b^8 - 2*b^10 - 3*a^8*c^2 + 14*a^6*b^2*c^2 - 10*a^4*b^4*c^2 + 5*a^2*b^6*c^2 + 3*b^8*c^2 + 5*a^6*c^4 - 10*a^4*b^2*c^4 - 7*a^2*b^4*c^4 - b^6*c^4 - 7*a^4*c^6 + 5*a^2*b^2*c^6 - b^4*c^6 + 6*a^2*c^8 + 3*b^2*c^8 - 2*c^10)*S) : :

X(41474) is also the perspector of the 5th Fermat-Dao triangle and each of the following triangles:
4th inner Fermat-Dao-Nhi (see X(33602)),
1st outer Fermat-Dao-Nhi triangle (see X(33602)),
2nd outer Fermat-Dao-Nhi triangle (see X(33602)).

X(414) lies on these lines: {3, 16461}, {13, 15}, {16, 16459}, {98, 25230}, {99, 25162}, {2379, 25218}, {3457, 11146}, {9203, 25171}

X(41475) = PERSPECTOR OF THESE TRIANGLES: 6th FERMAT-DAO AND h(2ND INNER FERMAT-DAO-NHI)

Barycentrics    a^2*(Sqrt[3]*(a^12 - 4*a^10*b^2 + 6*a^8*b^4 - 4*a^6*b^6 + a^4*b^8 - 4*a^10*c^2 + 12*a^8*b^2*c^2 - 7*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 10*a^2*b^8*c^2 - 7*b^10*c^2 + 6*a^8*c^4 - 7*a^6*b^2*c^4 + 5*a^4*b^4*c^4 - 7*a^2*b^6*c^4 + 22*b^8*c^4 - 4*a^6*c^6 - 4*a^4*b^2*c^6 - 7*a^2*b^4*c^6 - 30*b^6*c^6 + a^4*c^8 + 10*a^2*b^2*c^8 + 22*b^4*c^8 - 7*b^2*c^10) - 2*(a^10 - 3*a^8*b^2 + 5*a^6*b^4 - 7*a^4*b^6 + 6*a^2*b^8 - 2*b^10 - 3*a^8*c^2 + 14*a^6*b^2*c^2 - 10*a^4*b^4*c^2 + 5*a^2*b^6*c^2 + 3*b^8*c^2 + 5*a^6*c^4 - 10*a^4*b^2*c^4 - 7*a^2*b^4*c^4 - b^6*c^4 - 7*a^4*c^6 + 5*a^2*b^2*c^6 - b^4*c^6 + 6*a^2*c^8 + 3*b^2*c^8 - 2*c^10)*S) : :

X(41475) lies on these lines: {3, 16462}, {14, 16}, {15, 16460}, {98, 25229}, {99, 25152}, {2378, 25215}, {3458, 11145}, {9202, 25172}

X(41476) = PERSPECTOR OF THESE TRIANGLES: 9th FERMAT-DAO AND h(4TH INNER FERMAT-DAO-NHI)

Barycentrics    a^2*(Sqrt[3]*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + 8*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) - 2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 8*b^2*c^2 + c^4)*S) : :

X(41476) is also perspective to these triangles: 9th Fermat-Dao and 1st outer-Fermat-Dao-Nhi.

X(41476) lies on these lines: {3, 11624}, {6, 2981}, {13, 15}, {98, 25232}, {99, 25163}, {399, 11485}, {2379, 25222}, {9145, 11626}, {9203, 25179}, {9722, 23302}, {11130, 16644}, {25153, 40879}

X(41477) = PERSPECTOR OF THESE TRIANGLES: 13th FERMAT-DAO AND h(OUTER NAPOLEON)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 8*b^2*c^2 + c^4 + 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(41477) is also the perspector of the 13th Fermat-Dao triangle and each of the following triangles:
3rd inner-Fermat-Dao-Nhi triangle (see X(33602)),
4th inner-Fermat-Dao-Nhi triangle (see X(33602)),
1st outer-Fermat-Dao-Nhi triangle (see X(33602)),
2nd outer-Fermat-Dao-Nhi triangle (see X(33602)).

X(41477) lies on these lines: {2, 14}, {3, 5640}, {6, 2981}, {16, 11146}, {51, 36755}, {61, 323}, {98, 25234}, {99, 25165}, {182, 14169}, {373, 13350}, {470, 10632}, {1995, 14170}, {2379, 25226}, {3129, 15080}, {3130, 10545}, {3132, 11451}, {3580, 37340}, {5092, 34009}, {5422, 11486}, {5611, 7998}, {5615, 15019}, {5980, 25155}, {6151, 20859}, {7496, 14539}, {8838, 11081}, {9203, 25181}, {9735, 22112}, {10601, 11481}, {10645, 11145}, {11002, 14538}, {11078, 11299}, {11127, 11485}, {11420, 19773}, {34008, 34417}, {35469, 37470}

X(41478) = PERSPECTOR OF THESE TRIANGLES: 14th FERMAT-DAO AND h(INNER NAPOLEON)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 8*b^2*c^2 + c^4 - 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(41478) is also the perspector of the 14th Fermat-Dao triangle and each of the following triangles:
2nd half-diamonds central equilateral,
1st inner-Fermat-Dao-Nhi triangle (see X(33602)),
2nd inner-Fermat-Dao-Nhi triangle (see X(33602)),
3rd outer-Fermat-Dao-Nhi triangle (see X(33602)),
4th outer-Fermat-Dao-Nhi triangle (see X(33602)).

X(41478) lies on these lines: {2, 13}, {3, 5640}, {6, 6151}, {15, 11145}, {51, 36756}, {62, 323}, {98, 25233}, {99, 25155}, {182, 14170}, {373, 13349}, {471, 10633}, {1995, 14169}, {2378, 25225}, {2981, 20859}, {3129, 10545}, {3130, 15080}, {3131, 11451}, {3580, 37341}, {5092, 34008}, {5422, 11485}, {5611, 15019}, {5615, 7998}, {5981, 25165}, {7496, 14538}, {8836, 11086}, {9202, 25176}, {9736, 22112}, {10601, 11480}, {10646, 11146}, {11002, 14539}, {11092, 11300}, {11126, 11486}, {11421, 19772}, {34009, 34417}, {35470, 37470}

X(41479) = PERSPECTOR OF THESE TRIANGLES: 3rd VIJAY-PAASCHE-HUTSON AND h(ARTZT)

Barycentrics    a^2*(b^2*c^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(a + b - c)*(a - b + c)*S^3) : :

X(41479) is also the perspector of the 3rd Vijay-Paasche-Hutson triangle and each of the following triangles:
infinite altitude,
anti-Artzt.

X(41479) lies on these lines: {3, 3083}, {4, 11}, {20, 38016}, {388, 6808}, {1181, 6502}, {2067, 10982}, {6807, 7288}, {7592, 18995}

X(41480) = X(76)-CEVA CONJUGATE OF X(5)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 - c^4)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(41480) lies on these lines: {3, 6}, {22, 23128}, {30, 22416}, {53, 14978}, {112, 7691}, {217, 1154}, {232, 1216}, {297, 324}, {550, 3269}, {1209, 27371}, {1625, 5562}, {1971, 2937}, {1987, 3519}, {2871, 23208}, {2888, 15340}, {2979, 39575}, {3199, 5891}, {3289, 6101}, {3331, 5876}, {3933, 36790}, {5305, 20859}, {7488, 32661}, {7502, 14585}, {7999, 15355}, {8041, 31406}, {10282, 35324}, {11412, 22240}, {13564, 22146}, {15056, 33885}, {18435, 38297}, {18436, 32445}, {35318, 41366}

X(41480) = reflection of X(1235) in X(36952)
X(41480) = isotomic conjugate of the isogonal conjugate of X(40588)
X(41480) = X(76)-Ceva conjugate of X(5)
X(41480) = X(2167)-isoconjugate of X(2980)
X(41480) = crosspoint of X(76) and X(7796)
X(41480) = barycentric product X(i)*X(j) for these {i,j}: {5, 2979}, {51, 7796}, {76, 40588}, {160, 311}, {305, 15897}, {343, 39575}
X(41480) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 2980}, {160, 54}, {2979, 95}, {7796, 34384}, {15897, 25}, {39575, 275}, {40588, 6}
X(41480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {52, 216, 41334}

X(41481) = X(5562)-CEVA CONJUGATE OF X(5)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 + a^10*c^2 - a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 2*a^4*b^6*c^2 + a^2*b^8*c^2 - b^10*c^2 - 4*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 + 4*b^8*c^4 + 6*a^6*c^6 + 2*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 6*b^6*c^6 - 4*a^4*c^8 + a^2*b^2*c^8 + 4*b^4*c^8 + a^2*c^10 - b^2*c^10) : :
X(41481) = 3 X[5] - 4 X[6663], 3 X[549] - 2 X[6662], 2 X[6663] - 3 X[15912]

X(41481) lies on these lines: {3, 3164}, {5, 324}, {30, 5889}, {52, 6751}, {343, 15780}, {523, 34782}, {549, 6662}, {648, 2055}, {2782, 21651}, {3168, 38281}, {14059, 15466}

X(41481) = reflection of X(i) in X(j) for these {i,j}: {5, 15912}, {13322, 52}
X(41481) = X(5562)-Ceva conjugate of X(5)
X(41481) = X(i)-isoconjugate of X(j) for these (i,j): {2148, 34287}, {2190, 13855}
X(41481) = barycentric product X(i)*X(j) for these {i,j}: {311, 41373}, {343, 1075}
X(41481) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 34287}, {216, 13855}, {1075, 275}, {41373, 54}

X(41482) = ANTICOMPLEMENT OF X(11572)

Barycentrics    3*a^10 - 6*a^8*b^2 + a^6*b^4 + 3*a^4*b^6 - b^10 - 6*a^8*c^2 + 5*a^6*b^2*c^2 - a^4*b^4*c^2 - a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 2*b^6*c^4 + 3*a^4*c^6 - a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :

X(41482) lies on these lines: {2, 11572}, {3, 18432}, {4, 18475}, {20, 2979}, {22, 12278}, {23, 21659}, {26, 12289}, {30, 49}, {52, 12254}, {110, 12225}, {186, 11750}, {265, 12107}, {550, 13445}, {1503, 11440}, {1657, 32139}, {1658, 25739}, {1853, 38438}, {1994, 10619}, {2937, 30522}, {3060, 19467}, {3153, 10282}, {3515, 26913}, {3575, 5012}, {5640, 7487}, {5944, 31724}, {6030, 6823}, {6800, 12173}, {6816, 10546}, {7488, 18400}, {7527, 13419}, {7556, 9927}, {7574, 32171}, {7576, 13434}, {7691, 14516}, {10201, 18394}, {10298, 18381}, {10574, 18533}, {10575, 13619}, {10625, 12383}, {11422, 18925}, {11439, 31383}, {11449, 37444}, {11454, 14216}, {11464, 18569}, {11745, 15019}, {11819, 15033}, {12086, 29012}, {12088, 17702}, {12121, 14094}, {12220, 36989}, {13353, 38322}, {14157, 18563}, {14927, 30552}, {15062, 16659}, {15072, 35471}, {16655, 34005}, {18324, 23294}, {18404, 26882}, {20299, 38448}, {22528, 38885}, {23293, 38444}, {32338, 32354}, {38446, 40686}

X(41482) = reflection of X(31724) in X(5944)
X(41482) = anticomplement of X(11572)
X(41482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 9833, 12111}, {22, 17845, 12278}, {11454, 40241, 14216}, {12225, 34782, 110}, {19467, 31304, 3060}

X(41483) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(1st KENMOTU-FREE-VERTICES)

Barycentrics    a^2*(-2*(a^2-2*b^2+c^2)*S+a^4-(5*b^2+2*c^2)*a^2+(b^2-c^2)*(4*b^2-c^2))*(-2*(a^2+b^2-2*c^2)*S+a^4-(2*b^2+5*c^2)*a^2+(b^2-c^2)*(b^2-4*c^2)) : :

X(41483) lies on these lines: {371, 5640}, {5413, 33885}

X(41483) = isogonal conjugate of X(41490)
X(41483) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(32420)}} and {{A, B, C, X(6), X(371)}}

X(41484) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(2nd KENMOTU-FREE-VERTICES)

Barycentrics    a^2*(2*(a^2-2*b^2+c^2)*S+a^4-(5*b^2+2*c^2)*a^2+(b^2-c^2)*(4*b^2-c^2))*(2*(a^2+b^2-2*c^2)*S+a^4-(2*b^2+5*c^2)*a^2+(b^2-c^2)*(b^2-4*c^2)) : :

X(41484) lies on these lines: {372, 5640}, {5412, 33885}

X(41484) = isogonal conjugate of X(41491)
X(41484) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(32422)}} and {{A, B, C, X(6), X(372)}}

X(41485) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(LUCAS HOMOTHETIC)

Barycentrics    (SB+SC)*(S^2+(SA+2*SB-SC)*S+2*SA^2)*(S^2+(SA-SB+2*SC)*S+2*SA^2)*(S+SB+SC)^2 : :

X(41485) lies on this line: {6, 8820}

X(41486) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(LUCAS(-1) HOMOTHETIC)

Barycentrics    (SB+SC)*(S^2-(SA+2*SB-SC)*S+2*SA^2)*(S^2-(SA-SB+2*SC)*S+2*SA^2)*(-S+SB+SC)^2 : :

X(41486) lies on this line: {6, 8821}

X(41487) = PERSPECTOR OF THESE TRIANGLES: ABC AND h(INNER-YFF)

Barycentrics    a^2*(a^4-2*(b^2+b*c+c^2)*a^2+2*(2*b-c)*b*c*a+(b^2-c^2)^2)*(a^4-2*(b^2+b*c+c^2)*a^2-2*(b-2*c)*b*c*a+(b^2-c^2)^2) : :

X(41487) lies on these lines: {902, 41442}, {998, 5315}, {1411, 16483}, {3445, 36745}

X(41487) = complement of the anticomplementary conjugate of X(34619)
X(41487) = isogonal conjugate of the complement of X(34619)
X(41487) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(6)}} and {{A, B, C, X(4), X(840)}}
X(41487) = X(6)-vertex conjugate of-X(41442)

X(41488) = ISOGONAL CONJUGATE OF X(27374)

Barycentrics    b^2*(a^2 + b^2)*c^2*(a^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + 2*a^2*c^2 + b^2*c^2 - c^4) : :

X(41488) lies on these lines: {54, 34384}, {83, 3289}, {96, 40016}, {276, 308}, {689, 1141}, {1799, 8884}

X(41488) = isogonal conjugate of X(27374)
X(41488) = isotomic conjugate of the isogonal conjugate of X(39287)
X(41488) = X(95)-cross conjugate of X(39287)
X(41488) = X(i)-isoconjugate of X(j) for these (i,j): {1, 27374}, {5, 1923}, {38, 40981}, {39, 2179}, {51, 1964}, {217, 17442}, {688, 2617}, {798, 35319}, {1625, 2084}, {1953, 3051}, {2181, 20775}, {3199, 4020}, {9247, 27371}, {14213, 41331}, {18180, 41267}
X(41488) = cevapoint of X(i) and X(j) for these (i,j): {76, 1078}, {95, 34384}, {308, 1799}
X(41488) = trilinear pole of line {2623, 39182}
X(41488) = barycentric product X(i)*X(j) for these {i,j}: {54, 40016}, {76, 39287}, {83, 34384}, {95, 308}, {276, 1799}, {670, 39182}, {689, 15412}, {2167, 18833}, {2616, 37204}, {20023, 39283}
X(41488) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 27374}, {54, 3051}, {82, 2179}, {83, 51}, {95, 39}, {97, 20775}, {99, 35319}, {251, 40981}, {264, 27371}, {275, 1843}, {276, 427}, {308, 5}, {689, 14570}, {1176, 217}, {1799, 216}, {2148, 1923}, {2167, 1964}, {2616, 2084}, {2623, 688}, {3112, 1953}, {4577, 1625}, {4580, 15451}, {4593, 2617}, {8795, 27376}, {8882, 27369}, {15412, 3005}, {18831, 35325}, {18833, 14213}, {28724, 418}, {32085, 3199}, {34384, 141}, {34386, 3917}, {36794, 27370}, {39182, 512}, {39283, 263}, {39287, 6}, {40016, 311}, {40404, 27372}, {40440, 17442}, {41296, 41334}

X(41489) = X(64)-CEVA CONJUGATE OF X(25)

Barycentrics    a^2*(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 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :
Barycentrics    (tan A)/(cot A - csc A cos B cos C) : :
Barycentrics    a^2(tan A)/(tan B + tan C - tan A) : :
Barycentrics    a^3(sec A)/(S^2 - 2 SB SC) : :
Trilinears    (sin A)/(tan B tan C - 2) : :
Trilinears    (tan A)/(cos A - cos B cos C) : :

The trilinear polar of X(41489) meets the line at infinity at X(512).

X(41489) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 253}, {6, 64}, {19, 1427}, {25, 800}, {37, 2331}, {42, 3195}, {53, 34288}, {111, 1301}, {112, 5896}, {115, 36424}, {216, 14379}, {230, 40323}, {232, 8770}, {235, 393}, {263, 12167}, {493, 15210}, {494, 15213}, {577, 11589}, {588, 15198}, {589, 15201}, {607, 1400}, {608, 7154}, {1609, 14910}, {1976, 19118}, {1990, 2165}, {2207, 31942}, {2257, 8809}, {2433, 6753}, {2489, 34212}, {2987, 40318}, {3087, 15433}, {3162, 39951}, {3457, 11409}, {3458, 11408}, {3516, 5065}, {5158, 8798}, {5410, 8576}, {5411, 8577}, {6103, 40347}, {6356, 17903}, {6391, 15143}, {6622, 33630}, {6823, 41361}, {8573, 8882}, {8743, 11479}, {8745, 8749}, {8778, 11413}, {8879, 15661}, {13342, 17409}, {14580, 21448}, {15394, 40802}, {15466, 40032}, {16081, 21447}, {26958, 40221}, {32713, 33582}, {35974, 39975}

X(41489) = isogonal conjugate of X(37669)
X(41489) = polar conjugate of X(14615)
X(41489) = isogonal conjugate of the anticomplement of X(26958)
X(41489) = isogonal conjugate of the isotomic conjugate of X(459)
X(41489) = isogonal conjugate of the polar conjugate of X(6526)
X(41489) = polar conjugate of the isotomic conjugate of X(64)
X(41489) = polar conjugate of the isogonal conjugate of X(33581)
X(41489) = X(i)-Ceva conjugate of X(j) for these (i,j): {64, 25}, {253, 33584}, {459, 64}, {1073, 41085}
X(41489) = X(i)-cross conjugate of X(j) for these (i,j): {32, 33585}, {2207, 25}, {3269, 2501}, {33581, 64}
X(41489) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37669}, {3, 18750}, {20, 63}, {48, 14615}, {69, 610}, {75, 15905}, {77, 27382}, {78, 18623}, {92, 35602}, {154, 304}, {162, 20580}, {204, 3926}, {219, 33673}, {255, 15466}, {326, 1249}, {332, 30456}, {345, 1394}, {348, 7070}, {394, 1895}, {656, 36841}, {662, 8057}, {1073, 1097}, {1102, 6525}, {1264, 3213}, {1332, 21172}, {1444, 8804}, {1562, 24041}, {1792, 36908}, {1812, 5930}, {3198, 17206}, {4558, 17898}, {4592, 6587}, {6507, 14249}, {6516, 14331}, {7045, 40616}, {7055, 7156}, {18695, 33629}, {19611, 36413}
X(41489) = cevapoint of X(2489) and X(20975)
X(41489) = crosspoint of X(i) and X(j) for these (i,j): {6, 31956}, {459, 6526}
X(41489) = crosssum of X(i) and X(j) for these (i,j): {20, 36413}, {394, 6617}, {1249, 6616}, {1562, 8057}, {15905, 35602}
X(41489) = crossdifference of every pair of points on line {8057, 15427}
X(41489) = pole wrt polar circle of trilinear polar of X(14615) (line X(6587)X(20580))
X(41489) = barycentric product X(i)*X(j) for these {i,j}: {3, 6526}, {4, 64}, {6, 459}, {19, 2184}, {20, 31942}, {25, 253}, {33, 8809}, {92, 2155}, {125, 15384}, {158, 19614}, {264, 33581}, {278, 30457}, {393, 1073}, {523, 1301}, {1039, 10375}, {1093, 14379}, {1096, 19611}, {2052, 14642}, {2207, 34403}, {3346, 41085}, {5896, 10151}, {6524, 15394}, {8798, 8884}, {8882, 13157}, {13575, 33584}, {31956, 40839}, {32001, 33585}, {40836, 41088}
X(41489) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14615}, {6, 37669}, {19, 18750}, {25, 20}, {32, 15905}, {34, 33673}, {64, 69}, {112, 36841}, {184, 35602}, {204, 1097}, {253, 305}, {393, 15466}, {459, 76}, {512, 8057}, {607, 27382}, {608, 18623}, {647, 20580}, {1073, 3926}, {1096, 1895}, {1301, 99}, {1395, 1394}, {1973, 610}, {1974, 154}, {2155, 63}, {2184, 304}, {2207, 1249}, {2212, 7070}, {2333, 8804}, {2489, 6587}, {3124, 1562}, {3172, 36413}, {6524, 14249}, {6526, 264}, {7151, 41084}, {8809, 7182}, {13157, 28706}, {14379, 3964}, {14390, 2063}, {14398, 14345}, {14642, 394}, {14936, 40616}, {15384, 18020}, {15394, 4176}, {17510, 11413}, {19614, 326}, {20975, 122}, {30457, 345}, {31942, 253}, {33581, 3}, {33584, 1370}, {33585, 15077}, {36417, 3172}, {40354, 15291}, {41085, 6527}
X(41489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 64, 14642}, {6, 1033, 3172}, {1593, 2138, 3172}, {2138, 14091, 1033}

X(41490) = X(2)X(372)∩X(3)X(591)

Barycentrics    4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4 + 2*S*(2*a^2 - b^2 - c^2) : :
X(41490) = 3 X[2] + X[26288], 5 X[631] - X[5861], X[1991] - 3 X[5054], 3 X[3524] + X[5860], 5 X[15692] - X[26289]

X(41490) lies on these lines: {2, 372}, {3, 591}, {30, 9739}, {39, 13701}, {115, 615}, {141, 35256}, {182, 524}, {376, 13786}, {381, 13687}, {486, 13712}, {490, 10577}, {492, 6396}, {530, 36439}, {531, 36457}, {631, 5861}, {639, 1152}, {1991, 5054}, {2459, 35953}, {3069, 26619}, {3524, 5860}, {3594, 11315}, {3629, 35255}, {5062, 32787}, {5254, 13880}, {6420, 39387}, {6426, 11313}, {6454, 7389}, {6560, 32805}, {6561, 26617}, {6564, 32807}, {6566, 31173}, {7388, 35813}, {7618, 13700}, {7690, 8703}, {7749, 13921}, {7763, 32809}, {9300, 18993}, {9767, 12256}, {9892, 22726}, {11287, 15884}, {11288, 19146}, {11648, 13988}, {13710, 37343}, {13757, 35823}, {13821, 32808}, {13941, 38426}, {15692, 26289}, {22872, 31718}, {22917, 31715}, {33458, 35741}, {34508, 34552}, {34509, 34551}

X(41490) = midpoint of X(i) and X(j) for these {i,j}: {3, 591}, {9767, 12256}
X(41490) = reflection of X(41491) in X(549)
X(41490) = isogonal conjugate of X(41483)
X(41490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {488, 5420, 640}, {13757, 35949, 35823}

X(41491) = X(2)X(371)∩X(3)X(1991)

Barycentrics    4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*S*(2*a^2 - b^2 - c^2) : :
X(41491) = 3 X[2] + X[26289], X[591] - 3 X[5054], 5 X[631] - X[5860], 3 X[3524] + X[5861], 5 X[15692] - X[26288]

X(41491) lies on these lines: {2, 371}, {3, 1991}, {30, 9738}, {39, 13821}, {115, 590}, {141, 35255}, {182, 524}, {376, 13666}, {381, 13807}, {485, 13835}, {489, 10576}, {491, 6200}, {530, 36457}, {531, 36439}, {591, 5054}, {631, 5860}, {640, 1151}, {3068, 26620}, {3524, 5861}, {3592, 11316}, {3595, 9541}, {3629, 35256}, {5058, 32788}, {5254, 13921}, {6419, 39388}, {6425, 11314}, {6453, 7388}, {6560, 26618}, {6561, 32806}, {6567, 31173}, {7389, 35812}, {7618, 13820}, {7692, 8703}, {7749, 13880}, {7763, 32808}, {8960, 11293}, {8972, 38425}, {9300, 18994}, {9680, 11292}, {9768, 12257}, {9894, 22727}, {11287, 15883}, {11288, 19145}, {11648, 13848}, {13637, 35822}, {13701, 32809}, {13830, 37342}, {15692, 26288}, {22718, 31463}, {22874, 31716}, {22919, 31717}, {34508, 34551}, {34509, 34552}

X(41491) = midpoint of X(i) and X(j) for these {i,j}: {3, 1991}, {9768, 12257}
X(41491) = reflection of X(41490) in X(549)
X(41491) = isogonal conjugate of X(41484)
X(41491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {487, 5418, 639}, {13637, 35948, 35822}

X(41492) = X(1)X(30)∩X(28)X(1851)

Barycentrics    (a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3)*(2*a^4 + a^3*b - a^2*b^2 - a*b^3 - b^4 + a^3*c + a*b^2*c - a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 - c^4) : :

X(41492) lies on the cubic K1184 and these lines: {1, 30}, {28, 1851}, {442, 16585}, {1873, 15763}, {2911, 8757}, {3651, 18625}, {5179, 14873}, {6841, 18593}, {17768, 22136}, {26723, 37582}

X(41492) = orthic-isogonal conjugate of X(942)
X(41492) = X(4)-Ceva conjugate of X(942)
X(41492) = barycentric product X(1770)*X(5249)
X(41492) = barycentric quotient X(1770)/X(40435)

X(41493) = X(1)X(8818)∩X(4)X(1029)

Barycentrics    (b + c)*(-a^3 - a^2*b + a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c + a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(-(a^2*b) + b^3 - a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(41493) lies on the cubic K1184 and these lines: {1, 8818}, {4, 1029}, {12, 502}, {46, 267}, {442, 16585}, {1844, 1865}

X(41493) = X(942)-cross conjugate of X(442)
X(41493) = X(i)-isoconjugate of X(j) for these (i,j): {191, 1175}, {501, 943}, {1794, 2906}, {2259, 40592}
X(41493) = barycentric product X(i)*X(j) for these {i,j}: {442, 1029}, {502, 5249}, {1234, 3444}
X(41493) = barycentric quotient X(i)/X(j) for these {i,j}: {442, 2895}, {502, 40435}, {942, 40592}, {1029, 40412}, {1841, 2906}, {1865, 451}, {2260, 501}, {2294, 191}, {3444, 1175}, {18591, 22136}, {21353, 943}, {21675, 21081}, {23752, 21192}, {40952, 1030}

X(41494) = X(1)X(2074)∩X(442)X(1844)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3)*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c - a*b^2*c - a*b*c^2 - c^4)*(a^4 - b^4 - a^2*b*c - a*b^2*c - 2*a^2*c^2 - a*b*c^2 + c^4) : :

X(41494) lies on the cubic K1184 and these lines: {1, 2074}, {442, 1844}

X(41494) = barycentric quotient X(39772)/X(28754)

X(41495) = X(28)X(36)∩X(35)X(6011)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 - a*b*c - b^2*c - b*c^2 + c^3)*(a^3 + b^3 - a^2*c - a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^4 - b^4 + a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(41495) lies on the cubic K1184 and these lines: {28, 36}, {35, 6011}, {46, 267}, {79, 6598}, {5902, 34243}

X(41495) = X(34435)-isoconjugate of X(34772)
X(41495) = barycentric product X(i)*X(j) for these {i,j}: {2475, 37887}, {6598, 18625}
X(41495) = barycentric quotient X(i)/X(j) for these {i,j}: {1781, 34772}, {2475, 33116}

X(41496) = X(4)-CEVA CONJUGATE OF X(79)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*b*c - a^3*b^2*c + a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 + a^2*b*c^3 - a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 + a*b*c^4 - b^2*c^4 + 2*b*c^5 + c^6) : :

X(41496) lies on the cubic K1184 and these lines: {1, 8818}, {79, 942}, {1770, 38340}, {2478, 6757}, {5259, 7110}, {6742, 21077}, {39267, 41494}

X(41496) = orthic-isogonal conjugate of X(79)
X(41496) = X(4)-Ceva conjugate of X(79)
X(41496) = barycentric product X(1717)*X(30690)
X(41496) = barycentric quotient X(1717)/X(3219)

X(41497) = X(1)X(29)∩X(4)X(7016)

Barycentrics    b*c*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(-a^4 + a^2*b^2 - a^2*b*c + b^3*c + a^2*c^2 - 2*b^2*c^2 + b*c^3)*(a^4 - a^2*b^2 - a^2*b*c + b^3*c - a^2*c^2 + 2*b^2*c^2 + b*c^3) : :

X(41497) lies on the cubics K221 and K1185 and these lines" {1, 29}, {4, 7016}, {240, 36126}, {293, 36043}, {822, 17898}, {823, 896}, {1957, 6521}, {23999, 24021}

X(41497) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1942}, {296, 7016}, {520, 2713}, {7107, 40843}
X(41497) = crossdifference of every pair of points on line {820, 822}
X(41497) = barycentric product X(i)*X(j) for these {i,j}: {75, 41368}, {92, 450}, {158, 40888}, {243, 1947}, {823, 2797}, {1940, 1948}
X(41497) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 1942}, {450, 63}, {1940, 40843}, {2202, 7016}, {2797, 24018}, {7120, 296}, {24019, 2713}, {35236, 37754}, {40888, 326}, {41368, 1}

X(41498) = X(6)-CROSS CONJUGATE OF X(524)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^6 - 3*a^4*b^2 - 3*a^2*b^4 + b^6 + a^4*c^2 + 5*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 - 3*a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 + b^2*c^4 + c^6) : :

X(41498) lies on the the cubics K042, K273, K531, K565, and these lines: {23, 524}, {67, 111}, {542, 23720}, {671, 25328}, {690, 2492}, {895, 34898}, {2482, 6593}, {5523, 13492}, {7664, 36792}, {14444, 32740}, {20099, 32255}

X(41498) = isogonal conjugate of X(15899)
X(41498) = isogonal conjugate of the complement of X(13574)
X(41498) = X(i)-cross conjugate of X(j) for these (i,j): {6, 524}, {14357, 468}
X(41498) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15899}, {111, 16563}, {897, 2930}, {923, 14360}, {18310, 36142}
X(41498) = cevapoint of X(i) and X(j) for these (i,j): {6, 22259}, {351, 14444}, {524, 38304}
X(41498) = barycentric product X(i)*X(j) for these {i,j}: {524, 13574}, {3266, 22259}
X(41498) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 15899}, {187, 2930}, {524, 14360}, {690, 18310}, {896, 16563}, {13574, 671}, {22259, 111}

X(41499) = X(4)X(1046)∩X(29)X(65)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - 2*a^2*b + b^3 - 2*a^2*c + a*b*c + c^3)*(a^4 - a^2*b^2 + a^2*b*c - b^3*c - a^2*c^2 + 2*b^2*c^2 - b*c^3) : :

X(41499) lies on the cubic K1185 and these lines: {4, 1046}, {29, 65}, {46, 3144}, {242, 1845}, {243, 1430}, {1758, 17985}, {1846, 17768}, {1901, 39033}

X(41499) = polar-circle inverse of X(11608)
X(41499) = polar conjugate of the isotomic conjugate of X(39035)
X(41499) = orthic isogonal conjugate of X(243)
X(41499) = X(4)-Ceva conjugate of X(243)
X(41499) = X(i)-isoconjugate of X(j) for these (i,j): {296, 2648}, {1937, 17973}, {1949, 17947}, {17963, 40843}
X(41499) = crosspoint of X(4) and X(17985)
X(41499) = crosssum of X(3) and X(17973)
X(41499) = barycentric product X(i)*X(j) for these {i,j}: {4, 39035}, {243, 17950}, {415, 8680}, {1758, 1948}, {1944, 17985}, {1981, 2785}
X(41499) = barycentric quotient X(i)/X(j) for these {i,j}: {243, 17947}, {415, 35145}, {1758, 40843}, {1951, 17973}, {1981, 35154}, {2202, 2648}, {17966, 296}, {17985, 1952}, {39035, 69}

X(41500) = X(1)X(8764)∩X(4)X(296)

Barycentrics    b*c*(-a + b + c)^2*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(-a^4 + a^2*b^2 - a^2*b*c + b^3*c + a^2*c^2 - 2*b^2*c^2 + b*c^3)^2 : :

X(41500) lies on the cubic K1185 and these lines: {1, 8764}, {4, 296}, {29, 2638}, {46, 158}, {243, 851}, {821, 1254}, {823, 2310}, {1896, 2648}

X(41500) = X(1949)-isoconjugate of X(40843)
X(41500) = barycentric product X(243)*X(1948)
X(41500) = barycentric quotient X(i)/X(j) for these {i,j}: {243, 40843}, {2202, 296}

X(41501) = X(1)X(442)∩X(2)X(40430)

Barycentrics    (b + c)*(a^3 - a^2*b - a*b^2 + b^3 - a*b*c - b^2*c - b*c^2 + c^3)*(a^3 + b^3 - a^2*c - a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(41501) lies on the cubic K259 and K383 and these lines: {1, 442}, {2, 40430}, {3, 759}, {8, 8286}, {9, 32580}, {19, 407}, {37, 21029}, {46, 267}, {65, 21949}, {75, 1234}, {82, 18088}, {115, 220}, {158, 860}, {377, 2363}, {499, 6739}, {851, 2217}, {994, 10974}, {1247, 26066}, {1837, 27687}, {2166, 18395}, {2218, 37225}, {2911, 8818}, {3017, 17528}, {3553, 5949}, {3945, 40438}, {4190, 24898}, {5051, 31359}, {5127, 37230}, {5880, 13610}, {7483, 24902}, {7952, 36119}, {8287, 27686}, {10826, 30447}, {17606, 27685}, {24899, 37233}, {24914, 36195}

X(41501) = X(i)-cross conjugate of X(j) for these (i,j): {55, 8818}, {1365, 523}, {21811, 226}, {40967, 37}
X(41501) = X(i)-isoconjugate of X(j) for these (i,j): {3, 13739}, {21, 37583}, {58, 34772}, {60, 15556}, {110, 6003}, {759, 27086}, {1101, 8286}, {1175, 39772}, {1333, 33116}, {1437, 5174}
X(41501) = cevapoint of X(115) and X(4041)
X(41501) = barycentric product X(i)*X(j) for these {i,j}: {10, 37887}, {226, 6598}, {1577, 6011}
X(41501) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 33116}, {19, 13739}, {37, 34772}, {115, 8286}, {661, 6003}, {1365, 40622}, {1400, 37583}, {1826, 5174}, {2171, 15556}, {2245, 27086}, {2294, 39772}, {6011, 662}, {6598, 333}, {7178, 31603}, {21043, 21961}, {21131, 23775}, {37887, 86}

X(41502) = X(1)X(19)∩X(6)X(74)

Barycentrics    a^2*(a + b)*(a - b - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + c^2) : :

X(41502) lies on the cubic K1186 and these lines: {1, 19}, {4, 584}, {6, 74}, {35, 35192}, {46, 2906}, {162, 3240}, {186, 17454}, {281, 2326}, {583, 3520}, {607, 2189}, {1333, 10623}, {1442, 40214}, {2174, 6198}, {2193, 4184}, {2194, 11189}, {2900, 4183}, {4420, 11107}

X(41502) = polar conjugate of the isotomic conjugate of X(35193)
X(41502) = X(933)-Ceva conjugate of X(21789)
X(41502) = X(2174)-cross conjugate of X(35192)
X(41502) = X(i)-isoconjugate of X(j) for these (i,j): {73, 30690}, {77, 8818}, {79, 1214}, {125, 35049}, {222, 6757}, {226, 7100}, {265, 18593}, {307, 2160}, {525, 26700}, {656, 38340}, {1231, 6186}, {1409, 20565}, {1439, 7110}, {1446, 8606}, {1789, 6354}, {3615, 37755}
X(41502) = crossdifference of every pair of points on line {656, 9033}
X(41502) = barycentric product X(i)*X(j) for these {i,j}: {1, 11107}, {4, 35193}, {21, 6198}, {28, 4420}, {29, 35}, {92, 35192}, {162, 35057}, {186, 6740}, {250, 6741}, {270, 3678}, {281, 40214}, {314, 14975}, {318, 17104}, {319, 2299}, {607, 34016}, {648, 9404}, {1098, 1825}, {1172, 3219}, {1442, 4183}, {2003, 2322}, {2174, 31623}, {2189, 3969}, {2204, 33939}, {2326, 16577}, {2328, 7282}, {2332, 17095}, {2605, 36797}
X(41502) = barycentric quotient X(i)/X(j) for these {i,j}: {29, 20565}, {33, 6757}, {35, 307}, {112, 38340}, {607, 8818}, {1172, 30690}, {1399, 1439}, {2174, 1214}, {2194, 7100}, {2204, 2160}, {2299, 79}, {2332, 7110}, {2594, 6356}, {2605, 17094}, {3219, 1231}, {4420, 20336}, {6198, 1441}, {6740, 328}, {6741, 339}, {9404, 525}, {11107, 75}, {14975, 65}, {17104, 77}, {21741, 37755}, {22094, 1367}, {32676, 26700}, {32715, 36064}, {34397, 1464}, {35057, 14208}, {35192, 63}, {35193, 69}, {40214, 348}
X(41502) = {X(284),X(2332)}-harmonic conjugate of X(1172)

X(41503) = X(1)X(2074)∩X(9)X(33)

Barycentrics    a^2*(a + b)*(a - b - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3) : :

X(41503) lies on the cubic K1186 and these lines: {1, 2074}, {6, 38852}, {9, 33}, {24, 38850}, {28, 46}, {110, 3173}, {112, 3052}, {162, 278}, {219, 2189}, {270, 35193}, {1754, 37372}, {2073, 40214}, {2194, 11435}, {2911, 40570}, {2947, 23692}, {14017, 16471}, {15451, 21789}

X(41503) = X(40431)-Ceva conjugate of X(284)
X(41503) = X(i)-isoconjugate of X(j) for these (i,j): {1214, 37887}, {1439, 6598}, {6011, 17094}
X(41503) = barycentric product X(i)*X(j) for these {i,j}: {9, 13739}, {284, 5174}, {1172, 34772}, {2299, 33116}, {2322, 37583}, {2326, 15556}
X(41503) = barycentric quotient X(i)/X(j) for these {i,j}: {2299, 37887}, {2332, 6598}, {5174, 349}, {13739, 85}, {34772, 1231}
X(41503) = {X(2299),X(2328)}-harmonic conjugate of X(1172)

X(41504) = X(1)X(30)∩X(55)X(8818)

Barycentrics    (a - b - c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2)*(2*a^4 + a^3*b - a^2*b^2 - a*b^3 - b^4 + a^3*c + a*b^2*c - a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 - c^4) : :

X(41504) lies on the cubic K1186 and these lines: {1, 30}, {55, 8818}, {265, 18499}, {1989, 3052}, {3474, 38340}, {4512, 7110}, {5905, 6742}

X(41504) = orthic-isogonal conjugate of X(8818)
X(41504) = X(4)-Ceva conjugate of X(8818)
X(41504) = barycentric product X(1770)*X(7110)
X(41504) = barycentric quotient X(1770)/X(17095)

X(41505) = X(1)X(39267)∩X(28)X(46)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^3*c - 2*b^3*c + 2*a*b*c^2 + 2*a*c^3 + 2*b*c^3 - c^4)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 + 2*a*b^2*c + 2*b^3*c - 2*a^2*c^2 - 2*b*c^3 + c^4) : :

X(41505) lies on the cubics K619 and K1187 and these lines: {1, 39267}, {4, 2900}, {19, 2911}, {28, 46}, {34, 34430}, {40, 915}, {286, 20930}, {1118, 28076}, {1119, 4341}, {11248, 11471}, {17220, 39695}, {18451, 24474}

X(41505) = isogonal conjugate of X(224)
X(41505) = isogonal conjugate of the anticomplement of X(10395)
X(41505) = polar conjugate of the isotomic conjugate of X(39947)
X(41505) = X(55)-cross conjugate of X(19)
X(41505) = X(i)-isoconjugate of X(j) for these (i,j): {1, 224}, {2, 3211}, {3, 12649}, {63, 1723}, {77, 2900}, {78, 34489}
X(41505) = barycentric product X(i)*X(j) for these {i,j}: {4, 39947}, {19, 39695}, {92, 34430}
X(41505) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 224}, {19, 12649}, {25, 1723}, {31, 3211}, {607, 2900}, {608, 34489}, {34430, 63}, {39695, 304}, {39947, 69}

X(41506) = X(10)X(55)∩X(12)X(42)

Barycentrics    (b + c)*(a^3 + b^3 - a*b*c - a*c^2 - b*c^2)*(-a^3 + a*b^2 + a*b*c + b^2*c - c^3) : :

X(41506) lies on the cubic K1187 and these lines: {4, 209}, {8, 313}, {10, 55}, {11, 3682}, {12, 42}, {65, 28786}, {71, 6284}, {210, 1089}, {272, 1268}, {334, 7270}, {355, 5706}, {377, 21276}, {388, 1002}, {594, 1334}, {607, 1826}, {814, 35352}, {1220, 5086}, {1305, 28471}, {1824, 1882}, {1829, 22272}, {2334, 5252}, {2357, 40663}, {2551, 4651}, {3293, 5587}, {4013, 21075}, {4055, 7299}, {4685, 31141}, {15467, 39741}, {19642, 19842}

X(41506) = polar conjugate of the isotomic conjugate of X(40161)
X(41506) = X(i)-cross conjugate of X(j) for these (i,j): {3690, 37}, {21671, 12}
X(41506) = X(i)-isoconjugate of X(j) for these (i,j): {21, 4306}, {58, 3868}, {81, 579}, {86, 2352}, {110, 23800}, {209, 757}, {593, 22021}, {1014, 3190}, {1333, 18134}, {1412, 27396}, {1414, 8676}, {1437, 5125}, {1509, 2198}
X(41506) = crosspoint of X(1751) and X(2997)
X(41506) = crosssum of X(579) and X(2352)
X(41506) = trilinear pole of line {3709, 4024}
X(41506) = barycentric product X(i)*X(j) for these {i,j}: {4, 40161}, {10, 1751}, {37, 2997}, {42, 40011}, {272, 594}, {281, 28786}, {321, 2218}, {1305, 3700}, {1334, 15467}, {3695, 40574}, {4552, 23289}
X(41506) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 18134}, {37, 3868}, {42, 579}, {210, 27396}, {213, 2352}, {272, 1509}, {661, 23800}, {756, 22021}, {872, 2198}, {1305, 4573}, {1334, 3190}, {1400, 4306}, {1500, 209}, {1751, 86}, {1826, 5125}, {2218, 81}, {2997, 274}, {3700, 20294}, {3709, 8676}, {23289, 4560}, {28786, 348}, {40011, 310}, {40161, 69}, {40590, 19367}

X(41507) = X(4)X(17776)∩X(10)X(55)

Barycentrics    (b + c)*(-a^2 + b^2 + c^2)*(a^4 + 2*a^3*b + b^4 + 2*a^3*c - 2*a*b^2*c - 2*a*b*c^2 - 2*b^2*c^2 + c^4) : :

X(41507) lies on the buic K1187 and these lines: {4, 17776}, {10, 55}, {37, 442}, {71, 2083}, {72, 306}, {100, 30733}, {209, 14054}, {226, 3159}, {345, 37179}, {406, 1863}, {2325, 8804}, {3178, 21080}, {3488, 17526}, {3998, 21530}, {5142, 27396}, {5439, 37326}, {5440, 7515}, {5791, 37323}, {6734, 16290}, {7283, 37098}, {11400, 37318}, {25516, 33116}

X(41507) = orthic-isogonal conjugate of X(72)
X(41507) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 72}, {17776, 37}
X(41507) = X(28)-isoconjugate of X(39945)
X(41507) = barycentric product X(306)*X(1714)
X(41507) = barycentric quotient X(i)/X(j) for these {i,j}: {71, 39945}, {1714, 27}
X(41507) = {X(440),X(3695)}-harmonic conjugate of X(72)

X(41508) = X(4)X(2911)∩X(37)X(442)

Barycentrics    (b + c)^2*(a^3 + a^2*b + a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(41508) lies on the cubic K1187 and these lines: {4, 2911}, {37, 442}, {209, 1824}, {321, 1234}, {594, 32580}, {756, 21675}, {1255, 15474}, {1766, 2161}, {21353, 41493}

X(41508) = X(i)-cross conjugate of X(j) for these (i,j): {3690, 12}, {8754, 4036}
X(41508) = X(i)-isoconjugate of X(j) for these (i,j): {58, 40571}, {60, 1708}, {81, 1780}, {86, 41332}, {270, 3173}, {593, 3811}, {757, 2911}, {849, 17776}, {1790, 30733}, {2185, 37579}, {4341, 7054}, {4556, 15313}, {17877, 23357}
X(41508) = crosssum of X(1780) and X(41332)
X(41508) = barycentric product X(i)*X(j) for these {i,j}: {10, 23604}, {594, 15474}, {3695, 39267}, {4036, 13397}, {6358, 39943}, {28787, 41013}
X(41508) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 40571}, {42, 1780}, {181, 37579}, {213, 41332}, {594, 17776}, {756, 3811}, {1109, 17877}, {1254, 4341}, {1500, 2911}, {1824, 30733}, {2171, 1708}, {2197, 3173}, {3690, 11517}, {4705, 15313}, {8754, 5521}, {15474, 1509}, {23604, 86}, {28787, 1444}, {39943, 2185}

X(41509) = X(4)X(15443)∩X(5)X(226)

Barycentrics    a*(a - b - c)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c + a^3*b*c - a^2*b^2*c - a*b^3*c - a^3*c^2 - a*b^2*c^2 - 2*b^3*c^2 - a^2*c^3 + a*b*c^3 + a*c^4 + b*c^4)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + a^3*b*c + a*b^3*c + b^4*c - a^2*b*c^2 - a*b^2*c^2 - 2*a^2*c^3 - a*b*c^3 - 2*b^2*c^3 + c^5) : :

X(41509) lies on these lines: {4, 15443}, {5, 226}, {9, 8021}, {37, 1864}, {209, 1826}, {672, 1903}

X(41509) = X(3690)-cross conjugate of X(55)
X(41509) = X(i)-isoconjugate of X(j) for these (i,j): {7, 580}, {77, 41227}, {81, 41342}, {222, 37279}, {757, 15443}, {1014, 3191}
X(41509) = trilinear pole of line {4041, 33525}
X(41509) = barycentric quotient X(i)/X(j) for these {i,j}: {33, 37279}, {41, 580}, {42, 41342}, {607, 41227}, {1334, 3191}, {1500, 15443}

X(41510) = X(4)X(3190)∩X(12)X(42)

Barycentrics    a^2*(b + c)*(a^2 - b^2 - c^2)*(a^6*b - a^5*b^2 - 2*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 - a*b^6 + a^6*c - 3*a^4*b^2*c + 3*a^2*b^4*c - b^6*c - a^5*c^2 - 3*a^4*b*c^2 - 4*a^3*b^2*c^2 + a*b^4*c^2 - b^5*c^2 - 2*a^4*c^3 + 2*b^4*c^3 + 2*a^3*c^4 + 3*a^2*b*c^4 + a*b^2*c^4 + 2*b^3*c^4 + a^2*c^5 - b^2*c^5 - a*c^6 - b*c^6) : :

X(41510) lies on the cubic K1187 and these lines: {4, 3190}, {12, 42}, {37, 1864}, {71, 228}, {72, 6508}, {220, 1011}, {440, 3682}, {2340, 3198}, {3588, 26893}, {4064, 38360}

X(41510) = orthic-isogonal conjugate of X(71)
X(41510) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 71}, {3190, 42}, {3191, 37}, {22000, 40586}
X(41510) = X(27)-isoconjugate of X(39944)
X(41510) = crossdifference of every pair of points on line {17925, 23723}
X(41510) = barycentric product X(72)*X(1713)
X(41510) = barycentric quotient X(i)/X(j) for these {i,j}: {228, 39944}, {1713, 286}

X(41511) = X(6)-CROSS CONJUGATE OF X(895)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - b^2 - c^2)*(a^2 - 2*b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 2*a^2*b^2*c^2 - a^2*c^4 - b^2*c^4)*(a^6 - a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 + c^6) : :

X(41511) lies on the the cubics K113, K273, K335, the curve Q120, and these lines: {23, 895}, {524, 2373}, {671, 5523}, {2506, 36696}, {5622, 30209}, {5866, 10766}, {32740, 37801}

X(41511) = midpoint of X(895) and X(14909)
X(41511) = isogonal conjugate of X(1560)
X(41511) = isogonal conjugate of the complement of X(2373)
X(41511) = isotomic conjugate of the polar conjugate of X(10422)
X(41511) = X(i)-cross conjugate of X(j) for these (i,j): {3, 2373}, {6, 895}, {647, 10423}, {38356, 10097}
X(41511) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1560}, {19, 5181}, {468, 18669}, {896, 5523}, {14210, 14580}
X(41511) = cevapoint of X(6) and X(1177)
X(41511) = trilinear pole of line {9517, 14908}
X(41511) = barycentric product X(i)*X(j) for these {i,j}: {69, 10422}, {671, 18876}, {895, 2373}, {1177, 30786}, {36060, 37220}
X(41511) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 5181}, {6, 1560}, {111, 5523}, {895, 858}, {1177, 468}, {10415, 39269}, {10422, 4}, {14908, 2393}, {18876, 524}, {30786, 1236}, {32740, 14580}, {36060, 18669}

X(41512) = ISOGONAL CONJUGATE OF X(15470)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
X(41512) = 4 X[5972] - 3 X[33927], 3 X[14643] - 2 X[14670]

X(41512) lies on the cubics K186, K210, K613, and these lines: {4, 94}, {107, 35189}, {110, 476}, {113, 14264}, {125, 14356}, {542, 14583}, {648, 15395}, {925, 14884}, {1302, 39290}, {1640, 23588}, {1989, 3124}, {5627, 10706}, {5655, 14993}, {5972, 33927}, {6740, 7978}, {12121, 16168}, {13754, 39985}, {14643, 14670}, {15454, 15468}, {18883, 34310}, {34209, 36169}

X(41512) = reflection of X(i) in X(j) for these {i,j}: {265, 14254}, {14264, 113}
X(41512) = isogonal conjugate of X(15470)
X(41512) = antigonal image of X(14264)
X(41512) = X(41392)-anticomplementary conjugate of X(21221)
X(41512) = X(i)-Ceva conjugate of X(j) for these (i,j): {476, 15329}, {648, 41392}, {15395, 265}
X(41512) = X(21731)-cross conjugate of X(1989)
X(41512) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15470}, {255, 14222}, {526, 36053}, {656, 38936}, {2624, 2986}, {6149, 15328}, {14910, 32679}, {16186, 36114}
X(41512) = crosspoint of X(35139) and X(39290)
X(41512) = trilinear pole of line {3003, 34104}
X(41512) = barycentric product X(i)*X(j) for these {i,j}: {94, 15329}, {113, 39290}, {265, 16237}, {476, 3580}, {648, 39170}, {1725, 32680}, {2410, 39985}, {3003, 35139}
X(41512) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 15470}, {112, 38936}, {113, 5664}, {265, 15421}, {393, 14222}, {476, 2986}, {686, 16186}, {1725, 32679}, {1989, 15328}, {2410, 39988}, {2420, 39371}, {2437, 39986}, {3003, 526}, {3580, 3268}, {13754, 8552}, {14560, 14910}, {15329, 323}, {16237, 340}, {21731, 2088}, {32662, 5504}, {32678, 36053}, {35139, 40832}, {39170, 525}, {39290, 40423}, {39295, 18878}, {39985, 2411}, {41392, 15454}
X(41512) = {X(36839),X(36840)}-harmonic conjugate of X(41392)

X(41513) = CYCLOCEVIAN CONJUGATE OF X(76)

Barycentrics    (a^6 + a^4*b^2 + a^2*b^4 + b^6 + a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 + c^6) : :

X(41513) lies on the curves K539, K655, Q066 and these lines: {2, 14247}, {141, 2916}, {427, 8792}, {1369, 5189}, {3448, 17949}, {8267, 39346}, {8877, 8878}, {15523, 16555}, {21289, 33091}, {39726, 39728}, {40002, 40043}

X(41513) = isogonal conjugate of X(2916)
X(41513) = isotomic conjugate of X(1369)
X(41513) = cyclocevian conjugate of X(76)
X(41513) = isotomic conjugate of the anticomplement of X(251)
X(41513) = isogonal conjugate of the isotomic conjugate of X(40036)
X(41513) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {39727, 76}, {40036, 21278}
X(41513) = X(i)-cross conjugate of X(j) for these (i,j): {251, 2}, {15321, 4}
X(41513) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2916}, {6, 21378}, {19, 23133}, {31, 1369}, {32, 21598}, {63, 8792}
X(41513) = cevapoint of X(512) and X(15449)
X(41513) = barycentric product X(i)*X(j) for these {i,j}: {1, 39727}, {6, 40036}
X(41513) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21378}, {2, 1369}, {3, 23133}, {6, 2916}, {25, 8792}, {75, 21598}, {39727, 75}, {40036, 76}

X(41514) = ISOGONAL CONJUGATE OF X(3197)

Barycentrics    (a^6 + 2*a^5*b - a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5 + b^6 - 2*a^5*c + 2*a^4*b*c + 2*a*b^4*c - 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 + 4*b^3*c^3 - a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6)*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 + 2*a^5*c + 2*a^4*b*c - 2*a*b^4*c - 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 4*a^3*c^3 + 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 + 2*a*c^5 - 2*b*c^5 + c^6) : :

X(41514) lies on the cubics K154, K332, K334, and these lines: {2, 342}, {7, 41081}, {20, 78}, {21, 7149}, {63, 347}, {92, 280}, {189, 253}, {196, 268}, {322, 345}, {459, 40616}, {1812, 7152}, {6360, 30680}, {8811, 34259}, {10431, 34188}

X(41514) = isogonal conjugate of X(3197)
X(41514) = anticomplement of X(40837)
X(41514) = polar conjugate of X(3176)
X(41514) = isotomic conjugate of the anticomplement of X(278)
X(41514) = isotomic conjugate of the isogonal conjugate of X(7152)
X(41514) = isotomic conjugate of the polar conjugate of X(7149)
X(41514) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {268, 34162}, {1034, 21270}, {7007, 6515}, {7037, 5905}, {7152, 12649}, {40838, 5906}
X(41514) = X(i)-cross conjugate of X(j) for these (i,j): {84, 7}, {278, 2}, {2184, 189}, {7152, 7149}
X(41514) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3197}, {6, 1490}, {9, 1035}, {32, 33672}, {41, 5932}, {48, 3176}, {71, 8885}, {198, 3341}, {207, 219}, {212, 40837}, {1400, 13614}, {1415, 14302}, {8063, 32652}
X(41514) = cevapoint of X(i) and X(j) for these (i,j): {9, 6769}, {513, 35072}, {514, 16596}, {522, 40616}, {650, 3318}
X(41514) = trilinear pole of line {521, 14837}
X(41514) = barycentric product X(i)*X(j) for these {i,j}: {7, 1034}, {69, 7149}, {75, 3345}, {76, 7152}, {86, 8806}, {309, 3342}, {314, 8811}, {348, 40838}, {6063, 7037}, {7007, 7182}
X(41514) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1490}, {4, 3176}, {6, 3197}, {7, 5932}, {21, 13614}, {28, 8885}, {34, 207}, {56, 1035}, {75, 33672}, {84, 3341}, {278, 40837}, {522, 14302}, {1034, 8}, {3318, 13612}, {3342, 40}, {3345, 1}, {7007, 33}, {7037, 55}, {7149, 4}, {7152, 6}, {8064, 36049}, {8806, 10}, {8811, 65}, {14837, 8063}, {40838, 281}
X(41514) = {X(3345),X(8806)}-harmonic conjugate of X(1034)

X(41515) = ISOGONAL CONJUGATE OF X(5408)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + b^2 - c^2 + 2*S)*(a^2 - b^2 + c^2 + 2*S) : :

Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically (as at X(1327)). Let A' be the barycentric product of BA and CA, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(41515). (Randy Hutson, April 13, 2021)

X(41515) lies on the cubics K070a, K233, K678, and these lines: {2, 24244}, {4, 371}, {6, 6219}, {25, 53}, {30, 26953}, {68, 1322}, {193, 13439}, {225, 2362}, {254, 372}, {264, 492}, {393, 5200}, {486, 847}, {491, 35142}, {590, 8563}, {925, 24245}, {1007, 13441}, {1217, 11474}, {1249, 8037}, {1300, 6560}, {1826, 7133}, {1990, 5411}, {3069, 8944}, {3070, 12231}, {3087, 8035}, {3093, 10665}, {3128, 8801}, {3536, 26361}, {5254, 19006}, {5410, 6748}, {6423, 12148}, {7090, 41013}, {7586, 21463}, {8735, 34125}, {8736, 34121}, {13051, 32497}, {13834, 32422}, {15211, 37873}, {17819, 23251}, {18855, 26468}, {26462, 40065}

X(41515) = reflection of X(26945) in X(8969)
X(41515) = isogonal conjugate of X(5408)
X(41515) = polar conjugate of X(492)
X(41515) = isogonal conjugate of the complement of X(13439)
X(41515) = polar conjugate of the isotomic conjugate of X(485)
X(41515) = polar conjugate of the isogonal conjugate of X(8577)
X(41515) = X(8577)-cross conjugate of X(485)
X(41515) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5408}, {47, 11091}, {48, 492}, {63, 371}, {75, 8911}, {255, 1585}, {326, 5413}, {563, 34392}, {1748, 26922}, {3378, 5409}
X(41515) = cevapoint of X(485) and X(8944)
X(41515) = barycentric product X(i)*X(j) for these {i,j}: {4, 485}, {25, 34391}, {53, 16032}, {264, 8577}, {273, 13455}, {372, 847}, {393, 11090}, {486, 13440}, {491, 14593}, {1131, 1321}, {1586, 2165}, {2052, 6413}, {5392, 5412}, {8944, 34208}, {14618, 39383}, {18819, 32588}
X(41515) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 492}, {6, 5408}, {25, 371}, {32, 8911}, {372, 9723}, {393, 1585}, {485, 69}, {486, 13430}, {847, 34392}, {1321, 1270}, {1586, 7763}, {2165, 11091}, {2207, 5413}, {2351, 26922}, {5200, 39387}, {5412, 1993}, {5413, 1599}, {6413, 394}, {8576, 10666}, {8577, 3}, {8944, 6337}, {11090, 3926}, {13440, 491}, {13455, 78}, {14593, 486}, {16032, 34386}, {24246, 8222}, {34391, 305}, {39383, 4558}
X(41515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 485, 1321}, {4, 13440, 485}, {25, 53, 41516}, {393, 5200, 5413}

X(41516) = ISOGONAL CONJUGATE OF X(5409)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + b^2 - c^2 - 2*S)*(a^2 - b^2 + c^2 - 2*S) : :

Let BBACAC be the internal square on side BC, and define CCBABA and AACBCB cyclically (as at X(1328)). Let A' be the barycentric product of BA and CA, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(41516). (Randy Hutson, April 13, 2021)

X(41516) lies on the cubics K070b, K233, K678, and these lines: {2, 24243}, {4, 372}, {6, 6220}, {25, 53}, {68, 1321}, {193, 13428}, {225, 16232}, {254, 371}, {264, 491}, {393, 5412}, {485, 847}, {492, 35142}, {615, 8564}, {925, 24246}, {1007, 13430}, {1217, 11473}, {1249, 8038}, {1300, 6561}, {1990, 5410}, {3068, 8940}, {3071, 12232}, {3087, 8036}, {3092, 10666}, {3127, 8801}, {3535, 26362}, {5254, 19005}, {5411, 6748}, {6424, 12147}, {7585, 21464}, {8735, 34121}, {8736, 34125}, {13052, 32494}, {13711, 32420}, {14121, 41013}, {15212, 37873}, {17820, 23261}, {18855, 26469}, {26457, 40065}

X(41516) = isogonal conjugate of X(5409)
X(41516) = polar conjugate of X(491)
X(41516) = isogonal conjugate of the complement of X(13428)
X(41516) = polar conjugate of the isotomic conjugate of X(486)
X(41516) = polar conjugate of the isogonal conjugate of X(8576)
X(41516) = X(8576)-cross conjugate of X(486)
X(41516) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5409}, {47, 11090}, {48, 491}, {63, 372}, {75, 26920}, {255, 1586}, {326, 5412}, {563, 34391}, {603, 13461}, {3377, 5408}
X(41516) = cevapoint of X(486) and X(8940)
X(41516) = barycentric product X(i)*X(j) for these {i,j}: {4, 486}, {25, 34392}, {53, 16037}, {264, 8576}, {371, 847}, {393, 11091}, {485, 13429}, {492, 14593}, {1093, 26922}, {1132, 1322}, {1585, 2165}, {2052, 6414}, {5392, 5413}, {8940, 34208}, {14618, 39384}, {18820, 32587}
X(41516) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 491}, {6, 5409}, {25, 372}, {32, 26920}, {281, 13461}, {371, 9723}, {393, 1586}, {485, 13441}, {486, 69}, {847, 34391}, {1322, 1271}, {1585, 7763}, {2165, 11090}, {2207, 5412}, {5412, 1600}, {5413, 1993}, {6414, 394}, {8576, 3}, {8577, 10665}, {8940, 6337}, {11091, 3926}, {13429, 492}, {14593, 485}, {16037, 34386}, {24245, 8223}, {26922, 3964}, {34392, 305}, {39384, 4558}
X(41516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 486, 1322}, {4, 13429, 486}, {25, 53, 41515}

X(41517) = ISOGONAL CONJUGATE OF X(4027)

Barycentrics    a^2*(b^2 - a*c)^2*(b^2 + a*c)^2*(a*b - c^2)^2*(a*b + c^2)^2 : :

X(41517) lies on the cubics K787, K828, K1023, and these lines: {6, 14251}, {39, 3493}, {232, 16068}, {292, 30658}, {325, 698}, {334, 30643}, {511, 694}, {732, 18829}, {733, 805}, {893, 30648}, {1581, 18904}, {1691, 9467}, {2023, 36897}, {3094, 40810}, {3224, 38527}, {5031, 14603}, {5207, 19566}, {5968, 36821}, {7018, 30633}, {13330, 18873}, {13331, 18872}, {15391, 34130}, {18905, 30663}

X(41517) = midpoint of X(5207) and X(19566)
X(41517) = reflection of X(14603) in X(5031)
X(41517) = isogonal conjugate of X(4027)
X(41517) = X(i)-cross conjugate of X(j) for these (i,j): {3005, 18829}, {3124, 882}, {20859, 14970}, {40810, 694}
X(41517) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4027}, {238, 27982}, {385, 1580}, {1691, 1966}, {1926, 14602}, {1933, 3978}, {6645, 8300}, {24041, 35078}
X(41517) = cevapoint of X(882) and X(3124)
X(41517) = trilinear pole of line {882, 3569}
X(41517) = barycentric product X(i)*X(j) for these {i,j}: {694, 1916}, {882, 18829}, {1581, 1581}, {1934, 1967}, {9468, 18896}, {17980, 40708}, {36897, 40810}
X(41517) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 4027}, {292, 27982}, {694, 385}, {805, 17941}, {881, 5027}, {882, 804}, {1581, 1966}, {1916, 3978}, {1927, 1933}, {1934, 1926}, {1967, 1580}, {3124, 35078}, {3493, 19571}, {8789, 14602}, {8871, 19585}, {9468, 1691}, {14251, 36213}, {14604, 18902}, {17980, 419}, {18829, 880}, {18872, 5026}, {18896, 14603}, {30657, 7369}, {30658, 6652}, {34238, 40820}, {36214, 12215}, {36897, 14382}, {40810, 5976}
X(41517) = {X(9467),X(9468)}-harmonic conjugate of X(1691)

X(41518) = ISOGONAL CONJUGATE OF X(1344)

Barycentrics    a^2/(2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + a^2*(-a^2 + b^2 + c^2)*(1 + J)) : :

X(41518) lies on the Jerabek circumhyperbola, the cubics K280, K292, K393, and these lines: {2, 2574}, {3, 8115}, {6, 1113}, {20, 14375}, {64, 14710}, {74, 13415}, {182, 15460}, {184, 15461}, {376, 511}, {895, 13414}, {1345, 3426}, {2575, 6776}, {10751, 18550}, {11002, 15158}, {31955, 35485}

X(41518) = reflection of X(41519) in X(11179)
X(41518) = isogonal conjugate of X(1344)
X(41518) = trilinear pole of line {647, 2575}
X(41518) = barycentric quotient X(6)/X(1344)

X(41519) = ISOGONAL CONJUGATE OF X(1345)

Barycentrics    a^2/(2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + a^2*(-a^2 + b^2 + c^2)*(1 - J)) : :

X(41519) lies on the Jerabek circumhyperbola, the cubics K280, K295, K394, and these lines: {2, 2575}, {3, 8116}, {6, 1114}, {20, 14374}, {64, 14709}, {74, 13414}, {182, 15461}, {184, 15460}, {376, 511}, {895, 13415}, {1344, 3426}, {2574, 6776}, {10750, 18550}, {11002, 15159}, {31954, 35485}

X(41519) = reflection of X(41518) in X(11179)
X(41519) = isogonal conjugate of X(1345)
X(41519) = trilinear pole of line {647, 2574}
X(41519) = barycentric quotient X(6)/X(1345)

X(41520) = ISOGONAL CONJUGATE OF X(3511)

Barycentrics    (a^6*b^4 - a^4*b^6 + a^6*b^2*c^2 + a^4*b^4*c^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 + a^2*b^2*c^6 + b^4*c^6)*(a^6*b^4 + a^4*b^6 - a^6*b^2*c^2 + a^4*b^4*c^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 + a^2*b^2*c^6 + b^4*c^6) : :

X(41520) lies on the curves K128, K355, Q066 and these lines: {2, 14251}, {32, 8870}, {237, 385}, {419, 2211}, {511, 3978}, {3508, 4039}, {8782, 39684}, {20021, 34214}, {34238, 34536}, {35078, 36897}

X(41520) = reflection of X(36897) in X(35078)
X(41520) = isogonal conjugate of X(3511)
X(41520) = isotomic conjugate of X(25332)
X(41520) = anticomplement of X(39092)
X(41520) = antitomic conjugate of X(36897)
X(41520) = cyclocevian conjugate of X(11606)
X(41520) = isotomic conjugate of the anticomplement of X(694)
X(41520) = X(i)-cross conjugate of X(j) for these (i,j): {290, 4}, {694, 2}
X(41520) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3511}, {31, 25332}, {1580, 39092}, {1755, 39941}
X(41520) = cevapoint of X(i) and X(j) for these (i,j): {512, 35078}, {523, 2679}
X(41520) = trilinear pole of line {804, 2023}
X(41520) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 25332}, {6, 3511}, {98, 39941}, {694, 39092}

X(41521) = ISOGONAL CONJUGATE OF X(5866)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6)*(a^6 - 2*a^4*b^2 - 2*a^2*b^4 + b^6 - a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(41521) = 3 X[16072] - 2 X[31842]

X(41521) lies on the cubics K025, K475, K480, and these lines: {4, 14984}, {25, 1560}, {30, 3563}, {468, 6091}, {670, 683}, {1368, 3565}, {5523, 8753}, {8754, 10602}, {8946, 35821}, {8948, 35820}, {10152, 34174}, {11605, 34171}, {16072, 31842}, {23698, 40801}, {34170, 34175}, {34173, 38949}

X(41521) = reflection of X(i) in X(j) for these {i,j}: {25, 5139}, {3565, 1368}
X(41521) = isogonal conjugate of X(5866)
X(41521) = polar conjugate of X(37803)
X(41521) = antigonal image of X(25)
X(41521) = symgonal image of X(1368)
X(41521) = polar conjugate of the isotomic conjugate of X(40347)
X(41521) = X(i)-cross conjugate of X(j) for these (i,j): {2393, 25}, {32740, 8791}, {40350, 393}
X(41521) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5866}, {48, 37803}, {63, 37784}, {304, 41336}, {326, 37777}, {14210, 39169}
X(41521) = cevapoint of X(6388) and X(14273)
X(41521) = trilinear pole of line {1196, 2489}
X(41521) = barycentric product X(4)*X(40347)
X(41521) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37803}, {6, 5866}, {25, 37784}, {1974, 41336}, {2207, 37777}, {14581, 20772}, {32740, 39169}, {40347, 69}

X(41522) = ISOGONAL CONJUGATE OF X(14094)

Barycentrics    (2*a^8 + 3*a^6*b^2 - 10*a^4*b^4 + 3*a^2*b^6 + 2*b^8 - 7*a^6*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 7*b^6*c^2 + 9*a^4*c^4 - a^2*b^2*c^4 + 9*b^4*c^4 - 5*a^2*c^6 - 5*b^2*c^6 + c^8)*(2*a^8 - 7*a^6*b^2 + 9*a^4*b^4 - 5*a^2*b^6 + b^8 + 3*a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 - 5*b^6*c^2 - 10*a^4*c^4 + 3*a^2*b^2*c^4 + 9*b^4*c^4 + 3*a^2*c^6 - 7*b^2*c^6 + 2*c^8) : :
X(41522) = 3 X[74] + 5 X[14536], 3 X[1138] + X[36164], 5 X[3154] - 3 X[5627]

X(41522) lies on the cubics K808, K809, K811, and these lines: {3, 9214}, {4, 9717}, {30, 3292}, {74, 14536}, {140, 14254}, {187, 1990}, {523, 20417}, {550, 15454}, {1138, 36164}, {3154, 5627}, {3260, 6390}, {3471, 33923}, {15048, 35906}, {16168, 31945}

X(41522) = isogonal conjugate of X(14094)
X(41522) = isogonal conjugate of the anticomplement of X(16003)
X(41522) = X(10990)-cross conjugate of X(4)
X(41522) = barycentric quotient X(6)/X(14094)

X(41523) = X(5)X(6)∩X(52)X(53)

Barycentrics    (a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(3*a^8 - 6*a^6*b^2 + 4*a^4*b^4 - 2*a^2*b^6 + b^8 - 6*a^6*c^2 + 2*a^2*b^4*c^2 - 4*b^6*c^2 + 4*a^4*c^4 + 2*a^2*b^2*c^4 + 6*b^4*c^4 - 2*a^2*c^6 - 4*b^2*c^6 + c^8)::

X(41523) lies on the cubic K627 and these lines: {5, 6}, {52, 53}, {216, 343}, {393, 467}, {418, 40947}, {570, 12359}, {577, 6146}, {1609, 3133}, {1879, 22660}, {6748, 18474}, {6755, 18130}, {8799, 36412}, {8882, 14516}

X(41523) = polar conjugate of the isotomic conjugate of X(8905)
X(41523) = X(i)-Ceva conjugate of X(j) for these (i,j): {393, 216}, {467, 5}, {6515, 52}
X(41523) = X(i)-isoconjugate of X(j) for these (i,j): {63, 14518}, {2167, 34428}
X(41523) = barycentric product X(i)*X(j) for these {i,j}: {4, 8905}, {5, 6193}, {52, 40698}, {155, 39117}, {39111, 39113}
X(41523) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 14518}, {51, 34428}, {6193, 95}, {8905, 69}, {39111, 96}, {40698, 34385}

X(41524) = X(6)X(14593)∩X(216)X(2165)

Barycentrics    (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^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 - 2*a^8*b^2*c^2 + 8*a^6*b^4*c^2 - 8*a^4*b^6*c^2 + 10*a^2*b^8*c^2 - 6*b^10*c^2 - a^8*c^4 + 8*a^6*b^2*c^4 + 2*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 15*b^8*c^4 + 4*a^6*c^6 - 8*a^4*b^2*c^6 - 8*a^2*b^4*c^6 - 20*b^6*c^6 - a^4*c^8 + 10*a^2*b^2*c^8 + 15*b^4*c^8 - 2*a^2*c^10 - 6*b^2*c^10 + c^12)::

X(41524) lies on the cubic K627 and these lines: {6, 14593}, {216, 2165}, {32132, 40939}

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X(41524) = polar conjugate of the isotomic conjugate of X(8906)
X(41524) = X(393)-Ceva conjugate of X(2165)
X(41524) = X(63)-isoconjugate of X(14517)
X(41524) = barycentric product X(i)*X(j) for these {i,j}: {4, 8906}, {847, 9937}
X(41524) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 14517}, {8906, 69}, {9937, 9723}

X(41525) = X(6)X(39110)∩X(393)X(467)

Barycentrics    a^2*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^8 - 2*a^6*b^2 + 4*a^4*b^4 - 6*a^2*b^6 + 3*b^8 - 4*a^6*c^2 + 2*a^4*b^2*c^2 - 6*b^6*c^2 + 6*a^4*c^4 + 2*a^2*b^2*c^4 + 4*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 2*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 2*b^6*c^2 + 4*a^4*c^4 + 4*b^4*c^4 - 6*a^2*c^6 - 6*b^2*c^6 + 3*c^8)::

X(41525) lies on the cubic K627 and these lines: {6, 39110}, {393, 467}, {1609, 2207}, {8745, 14518}

X(41525) = polar conjugate of the isotomic conjugate of X(34428)
X(41525) = X(216)-cross conjugate of X(393)
X(41525) = X(i)-isoconjugate of X(j) for these (i,j): {63, 6193}, {2167, 8905}
X(41525) = barycentric product X(i)*X(j) for these {i,j}: {4, 34428}, {5, 14518}, {847, 39110}, {39109, 39115}
X(41525) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 6193}, {51, 8905}, {14518, 95}, {14593, 40698}, {34428, 69}, {39110, 9723}

X(41526) = X(1)X(20594)∩X(6)X(41)

Barycentrics    a^3*(a + b - c)*(a - b + c)*(a*b + a*c - b*c)::

X(41526) lies on the cubic K285, I789, K1021 and these lines: {1, 20594}, {2, 1429}, {6, 41}, {31, 237}, {57, 1258}, {100, 32468}, {101, 978}, {651, 1424}, {699, 29055}, {869, 1402}, {959, 985}, {1106, 1922}, {1319, 17448}, {1397, 2210}, {1403, 2176}, {1415, 34077}, {1423, 27644}, {1447, 16827}, {1958, 27982}, {2238, 28386}, {3684, 20036}, {6602, 28272}, {10571, 41264}, {14829, 25940}, {15389, 34248}, {18755, 41346}, {20769, 37683}, {24512, 28385}, {26244, 31339}

X(41526) = isogonal conjugate of X(27424)
X(41526) = isogonal conjugate of the complement of X(36858)
X(41526) = isogonal conjugate of the isotomic conjugate of X(1423)
X(41526) = X(i)-Ceva conjugate of X(j) for these (i,j): {1397, 604}, {1403, 2209}
X(41526) = X(1403)-cross conjugate of X(604)
X(41526) = X(i)-isoconjugate of X(j) for these (i,j): {1, 27424}, {2, 7155}, {8, 330}, {9, 6384}, {11, 5383}, {55, 6383}, {75, 2319}, {76, 2053}, {87, 312}, {200, 7209}, {314, 16606}, {341, 7153}, {522, 4598}, {650, 18830}, {932, 4391}, {2162, 3596}, {3680, 27496}, {4147, 32039}, {4518, 39914}, {6378, 18021}, {7017, 23086}, {7081, 27447}, {7121, 28659}, {21759, 40072}, {23493, 28660}, {27436, 39924}, {34071, 35519}
X(41526) = crosspoint of X(1415) and X(4564)
X(41526) = crosssum of X(i) and X(j) for these (i,j): {1, 20606}, {2, 20348}, {8, 27523}, {2170, 4391}
X(41526) = barycentric product X(i)*X(j) for these {i,j}: {1, 1403}, {6, 1423}, {7, 2209}, {31, 3212}, {32, 30545}, {34, 20760}, {43, 56}, {57, 2176}, {59, 3123}, {65, 38832}, {108, 22090}, {109, 4083}, {192, 604}, {608, 22370}, {651, 20979}, {664, 8640}, {1106, 27538}, {1397, 6376}, {1400, 27644}, {1402, 33296}, {1407, 3208}, {1408, 3971}, {1412, 20691}, {1415, 3835}, {2149, 21138}, {3502, 41346}, {4551, 16695}, {4559, 18197}, {4564, 6377}, {4565, 21834}, {4620, 21835}, {4998, 38986}, {7132, 20284}, {9316, 20287}, {24533, 29055}, {25098, 32674}
X(41526) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 27424}, {31, 7155}, {32, 2319}, {43, 3596}, {56, 6384}, {57, 6383}, {109, 18830}, {192, 28659}, {560, 2053}, {604, 330}, {1397, 87}, {1403, 75}, {1407, 7209}, {1415, 4598}, {1423, 76}, {2149, 5383}, {2176, 312}, {2209, 8}, {3123, 34387}, {3212, 561}, {4083, 35519}, {6376, 40363}, {6377, 4858}, {8640, 522}, {16695, 18155}, {20691, 30713}, {20760, 3718}, {20979, 4391}, {21762, 2170}, {21835, 21044}, {22090, 35518}, {22386, 7004}, {27644, 28660}, {30545, 1502}, {33296, 40072}, {38832, 314}, {38986, 11}, {41280, 7121}
X(41526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 20606, 20594}, {6, 20471, 20460}, {1055, 20460, 20471}

X(41527) = X(1)X(1447)∩X(2)X(4876)

Barycentrics    (a^2*b - a*b^2 - a^2*c - a*b*c - b^2*c - a*c^2 + b*c^2)*(a^2*b + a*b^2 - a^2*c + a*b*c - b^2*c + a*c^2 + b*c^2)::

X(41527) lies on the Feuerbach circumhyperbola, the cubics K1037 and K1038 and these lines: {1, 1447}, {2, 4876}, {4, 30546}, {7, 3056}, {8, 350}, {9, 192}, {21, 8844}, {75, 7155}, {79, 7272}, {194, 3495}, {256, 3672}, {257, 4051}, {294, 6654}, {330, 33890}, {497, 7261}, {870, 18906}, {982, 7233}, {983, 1253}, {1107, 40785}, {1172, 31905}, {1320, 27922}, {1476, 36638}, {2276, 5222}, {2280, 2344}, {3112, 32937}, {3551, 3663}, {3661, 30998}, {3677, 17082}, {3680, 27813}, {4087, 7034}, {4392, 7192}, {4441, 20023}, {5088, 7284}, {7033, 27538}, {7091, 7176}, {9311, 17448}, {16816, 33889}, {17321, 24752}, {17752, 32095}, {17787, 32087}, {21299, 30479}, {24203, 36477}, {25834, 36258}, {37416, 37555}
X(41527) = isogonal conjugate of X(21010)
X(41527) = isotomic conjugate of X(24349)
X(41527) = isotomic conjugate of the anticomplement of X(984)
X(41527) = isotomic conjugate of the complement of X(31302)
X(41527) = anticomplement of X(19584)
X(41527) = X(25425)-anticomplementary conjugate of X(1330)
X(41527) = X(984)-cross conjugate of X(2)
X(41527) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21010}, {6, 17754}, {19, 22163}, {31, 24349}, {32, 20917}, {55, 4334}, {692, 24720}, {985, 19586}, {1333, 21101}, {8926, 17798}, {14621, 19587}, {19584, 40746}, {25429, 40747}
X(41527) = cevapoint of X(i) and X(j) for these (i,j): {1, 37555}, {2, 31302}, {11, 824}, {788, 6377}, {3022, 14330}, {30665, 35119}
X(41527) = trilinear pole of line {650, 812}
X(41527) = barycentric product X(i)*X(j) for these {i,j}: {85, 7220}, {18299, 40785}, {23605, 40845}, {25425, 30966}
X(41527) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17754}, {2, 24349}, {3, 22163}, {6, 21010}, {10, 21101}, {57, 4334}, {75, 20917}, {514, 24720}, {869, 19587}, {984, 19584}, {2276, 19586}, {3512, 8926}, {3736, 25429}, {7220, 9}, {23605, 3509}, {25425, 40718}, {40785, 17792}, {40787, 28600}

X(41528) = X(6)X(20466)∩X(239)X(8301)

Barycentrics    a^3*(a^3*b - b^4 + a*b^2*c - 2*a^2*c^2 + b*c^3)*(-2*a^2*b^2 + a^3*c + b^3*c + a*b*c^2 - c^4)::

X(41528) lies on the cubics K1003 and K1021 and these lines: {6, 20466}, {239, 8301}, {1931, 2110}, {2109, 17962}, {2223, 16514}, {9454, 18264}

X(41528) = isogonal conjugate of X(20345)
X(41528) = isogonal conjugate of the anticomplement of X(292)
X(41528) = isogonal conjugate of the isotomic conjugate of X(2113)
X(41528) = X(9472)-Ceva conjugate of X(6)
X(41528) = X(1922)-cross conjugate of X(6)
X(41528) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20345}, {2, 17738}, {6, 20446}, {75, 8301}, {76, 2112}, {81, 20496}, {86, 20716}, {92, 20742}, {100, 20518}, {238, 18034}, {335, 27916}, {350, 9470}
X(41528) = crosssum of X(i) and X(j) for these (i,j): {8301, 20742}, {20496, 20716}
X(41528) = barycentric product X(i)*X(j) for these {i,j}: {1, 18783}, {6, 2113}, {292, 9472}, {335, 18264}
X(41528) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 20446}, {6, 20345}, {31, 17738}, {32, 8301}, {42, 20496}, {184, 20742}, {213, 20716}, {292, 18034}, {560, 2112}, {649, 20518}, {1922, 9470}, {2113, 76}, {2210, 27916}, {9472, 1921}, {18264, 239}, {18783, 75}

X(41529) = ISOGONAL CONJUGATE OF X(39148)

Barycentrics    (2*a - b - c)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3 + a^2*c + 5*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - 3*a^2*c + 5*a*b*c - b^2*c - 3*a*c^2 + b*c^2 + c^3) : :
X(41529) = 8 X[1387] - 9 X[14028]

X(41529) lies on the cubics K058, K510, K575, and these lines: on lines {10, 36936}, {80, 88}, {214, 4370}, {484, 519}, {900, 1387}, {903, 21630}, {952, 14000}, {1145, 36924}, {1320, 24858}, {9802, 20092}

X(41529) = reflection of X(36945) in X(214)
X(41529) = isogonal conjugate of X(39148)
X(41529) = X(1)-cross conjugate of X(519)
X(41529) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39148}, {6, 40594}, {88, 3196}, {106, 5541}, {9456, 30578}, {16944, 36909}, {21198, 32665}, {22141, 36125}
X(41529) = trilinear product X(44)*X(8046)
X(41529) = trilinear quotient X(i)/X(j) for these (i,j): (44, 3196), (8046, 88)
X(41529) = barycentric product X(519)*X(8046)
X(41529) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40594}, {6, 39148}, {44, 5541}, {519, 30578}, {900, 21198}, {902, 3196}, {3943, 21087}, {4358, 20937}, {8046, 903}, {22356, 22141}

X(41530) = ISOTOMIC CONJUGATE OF X(154)

Barycentrics    b^2*c^2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(-a^4 - 2*a^2*b^2 + 3*b^4 + 2*a^2*c^2 - 2*b^2*c^2 - c^4) : :

X(41530) lies on these lines: {76, 459}, {253, 305}, {1241, 41489}, {1503, 34412}, {1853, 35140}, {2052, 36793}, {14572, 40022}, {15394, 34384}

X(41530) = isotomic conjugate of X(154)
X(41530) = polar conjugate of X(3172)
X(41530) = isotomic conjugate of the anticomplement of X(23332)
X(41530) = isotomic conjugate of the complement of X(32064)
X(41530) = isotomic conjugate of the isogonal conjugate of X(253)
X(41530) = polar conjugate of the isogonal conjugate of X(34403)
X(41530) = X(i)-cross conjugate of X(j) for these (i,j): {264, 76}, {23332, 2}
X(41530) = X(i)-isoconjugate of X(j) for these (i,j): {20, 560}, {31, 154}, {32, 610}, {48, 3172}, {184, 204}, {1249, 9247}, {1394, 2175}, {1397, 7070}, {1501, 18750}, {1895, 14575}, {1917, 14615}, {1924, 36841}, {1973, 15905}, {2179, 33629}, {2206, 3198}, {9406, 15291}, {9447, 18623}, {9448, 33673}, {14574, 17898}
X(41530) = cevapoint of X(i) and X(j) for these (i,j): {2, 32064}, {69, 32831}, {253, 34403}, {850, 36793}
X(41530) = trilinear pole of line {3267, 14638}
X(41530) = barycentric product X(i)*X(j) for these {i,j}: {64, 1502}, {76, 253}, {264, 34403}, {305, 459}, {349, 5931}, {561, 2184}, {1073, 18022}, {1928, 2155}, {1969, 19611}, {6528, 14638}, {8809, 28659}, {13157, 34384}, {15394, 18027}, {30457, 41283}, {33581, 40362}, {40050, 41489}
X(41530) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 154}, {4, 3172}, {64, 32}, {69, 15905}, {75, 610}, {76, 20}, {85, 1394}, {92, 204}, {95, 33629}, {253, 6}, {264, 1249}, {273, 3213}, {276, 38808}, {305, 37669}, {312, 7070}, {313, 8804}, {318, 7156}, {321, 3198}, {339, 1562}, {349, 5930}, {459, 25}, {561, 18750}, {670, 36841}, {850, 6587}, {1073, 184}, {1441, 30456}, {1446, 40933}, {1494, 15291}, {1502, 14615}, {1853, 20232}, {1969, 1895}, {2052, 6525}, {2155, 560}, {2184, 31}, {3261, 21172}, {3267, 8057}, {3596, 27382}, {3926, 35602}, {5931, 284}, {6063, 18623}, {6526, 2207}, {8798, 217}, {8809, 604}, {13157, 51}, {14379, 14585}, {14615, 36413}, {14638, 520}, {14642, 14575}, {15394, 577}, {15466, 3079}, {16096, 8779}, {18022, 15466}, {18027, 14249}, {19611, 48}, {19614, 9247}, {20567, 33673}, {20948, 17898}, {30457, 2175}, {33581, 1501}, {34403, 3}, {35519, 14331}, {36793, 122}, {37874, 40174}, {38956, 9408}, {41489, 1974}

X(41531) = ISOGONAL CONJUGATE OF X(34252)

Barycentrics    a*(-b^2 + a*c)*(a*b + a*c - b*c)*(a*b - c^2) : :

X(41531) lies on the cubics K136, K982, K990, and these lines: {1, 7077}, {2, 38}, {9, 87}, {37, 256}, {43, 6377}, {171, 8932}, {192, 3123}, {238, 660}, {240, 420}, {241, 1463}, {334, 1221}, {665, 24462}, {668, 19581}, {726, 36801}, {813, 9082}, {876, 6085}, {983, 1582}, {1054, 14200}, {1613, 1922}, {1967, 7081}, {2227, 33889}, {2303, 2311}, {2664, 3229}, {3252, 9309}, {3717, 20340}, {3799, 20681}, {3971, 23824}, {4858, 19953}, {6211, 19522}, {17257, 26135}, {20331, 24413}, {23354, 27809}, {24173, 40093}, {33680, 40844}

X(41531) = isogonal conjugate of X(34252)
X(41531) = X(i)-Ceva conjugate of X(j) for these (i,j): {292, 291}, {33680, 192}
X(41531) = X(40848)-cross conjugate of X(291)
X(41531) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34252}, {6, 39914}, {87, 238}, {239, 2162}, {242, 23086}, {330, 1914}, {350, 7121}, {659, 932}, {812, 34071}, {1428, 7155}, {1429, 2319}, {1447, 2053}, {1691, 27447}, {1929, 8843}, {2210, 6384}, {3684, 7153}, {4598, 8632}, {6383, 14599}, {21759, 30940}, {23493, 33295}
X(41531) = crosspoint of X(5378) and X(8684)
X(41531) = crosssum of X(i) and X(j) for these (i,j): {239, 39914}, {3808, 27846}
X(41531) = trilinear pole of line {4083, 20691}
X(41531) = crossdifference of every pair of points on line {4164, 8632}
X(41531) = barycentric product X(i)*X(j) for these {i,j}: {1, 40848}, {43, 335}, {192, 291}, {292, 6376}, {334, 2176}, {660, 3835}, {694, 41318}, {813, 20906}, {876, 4595}, {1423, 4518}, {1575, 33680}, {1581, 17752}, {1911, 6382}, {2209, 18895}, {3208, 7233}, {3212, 4876}, {3572, 36863}, {3971, 37128}, {4083, 4562}, {4583, 20979}, {4584, 21051}, {4589, 21834}, {5378, 21138}, {7077, 30545}, {14598, 40367}, {18827, 20691}, {40155, 40844}
X(41531) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 39914}, {6, 34252}, {43, 239}, {192, 350}, {291, 330}, {292, 87}, {334, 6383}, {335, 6384}, {660, 4598}, {813, 932}, {1403, 1429}, {1423, 1447}, {1581, 27447}, {1911, 2162}, {1922, 7121}, {2176, 238}, {2196, 23086}, {2209, 1914}, {3123, 27918}, {3208, 3685}, {3212, 10030}, {3501, 14199}, {3835, 3766}, {3971, 3948}, {4083, 812}, {4110, 4087}, {4518, 27424}, {4562, 18830}, {4595, 874}, {4876, 7155}, {5378, 5383}, {6376, 1921}, {6377, 27846}, {6382, 18891}, {7077, 2319}, {7233, 7209}, {8640, 8632}, {14408, 4448}, {14426, 14433}, {16362, 40741}, {17735, 8843}, {17752, 1966}, {20691, 740}, {20760, 20769}, {20979, 659}, {21834, 4010}, {24533, 4107}, {27538, 3975}, {27644, 33295}, {30545, 18033}, {33296, 30940}, {33680, 32020}, {34067, 34071}, {36863, 27853}, {40155, 40881}, {40848, 75}, {41318, 3978}, {41527, 1428}
X(41531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 3862, 4876}

X(41532) = ISOGONAL CONJUGATE OF X(7061)

Barycentrics    a^2*(b^2 + a*c)*(a*b + c^2)*(a^3 - b^3 + a*b*c - c^3) : :

Let A1B1C1 and A2B2C2 be the 1st and 2nd bicentrics of the anticomplementary triangle. The points A1, B1, C1, A2, B2, C2 lie on a common hyperbola, H, centered at X(9468). Let A' be the intersection of the tangents to H at A1 and A2; define B' and C' cyclically. Let A" be the intersection of the tangents to H at B1 and C2; define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(41532). (Randy Hutson, April 13, 2021)

X(41532) lies on the cubics K040, K155, K864 and these lines: {1, 256}, {31, 893}, {105, 805}, {238, 1581}, {257, 5263}, {292, 30658}, {512, 3287}, {516, 19637}, {518, 3903}, {694, 2054}, {904, 1964}, {1155, 37137}, {1178, 1326}, {1386, 40432}, {1965, 4388}, {2108, 8936}, {2110, 2115}, {2113, 17493}, {2651, 4603}, {2700, 29055}, {5031, 25760}, {5205, 27805}, {5360, 7077}, {7260, 14195}, {17451, 25844}, {17792, 27954}, {18904, 34214}, {20666, 40729}

X(41532) = reflection of X(30648) in X(238)
X(41532) = isogonal conjugate of X(7061)
X(41532) = X(1581)-Ceva conjugate of X(893)
X(41532) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7061}, {6, 40846}, {171, 7261}, {172, 40845}, {385, 24479}, {894, 3512}, {1909, 8852}, {1966, 30648}, {7122, 18036}, {7176, 7281}, {8875, 39933}
X(41532) = crosssum of X(i) and X(j) for these (i,j): {4039, 27697}, {4107, 4459}
X(41532) = crossdifference of every pair of points on line {3287, 28369}
X(41532) = barycentric product X(i)*X(j) for these {i,j}: {1, 40873}, {256, 3509}, {257, 17798}, {694, 1281}, {893, 4645}, {904, 17789}, {1178, 4071}, {1581, 19557}, {1916, 19561}, {1934, 18038}, {1967, 18037}, {7018, 19554}, {20715, 40432}
X(41532) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40846}, {6, 7061}, {256, 40845}, {257, 18036}, {893, 7261}, {904, 3512}, {1281, 3978}, {1967, 24479}, {3509, 1909}, {4071, 1237}, {4645, 1920}, {5018, 7196}, {7104, 8852}, {9468, 30648}, {17798, 894}, {18037, 1926}, {18038, 1580}, {18262, 172}, {19554, 171}, {19557, 1966}, {19561, 385}, {20715, 3963}, {40873, 75}
X(41532) = {X(1581),X(9467)}-harmonic conjugate of X(30648)

X(41533) = ISOGONAL CONJUGATE OF X(5207)

Barycentrics    a^2*(a^6 + b^6 - a^2*b^2*c^2 - c^6)*(a^6 - b^6 - a^2*b^2*c^2 + c^6) : :

X(41533) lies on the cubic K699, K1001, K1033, and theser lines: {3, 147}, {6, 19575}, {22, 3504}, {32, 39857}, {115, 3456}, {184, 3094}, {228, 21880}, {1281, 8857}, {1502, 5989}, {1799, 5976}, {1916, 6660}, {2076, 3852}, {2351, 20885}, {2353, 10828}, {2491, 17997}, {3425, 9861}, {3778, 18038}, {4027, 9229}, {8884, 12131}, {10547, 14885}

X(41533) = isogonal conjugate of X(5207)
X(41533) = isogonal conjugate of the anticomplement of X(1691)
X(41533) = X(i)-cross conjugate of X(j) for these (i,j): {14602, 6}, {14946, 39932}
X(41533) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5207}, {2, 19555}, {75, 6660}, {76, 19559}, {561, 19558}, {662, 14316}, {694, 19574}, {1502, 19560}, {1581, 19571}, {1916, 19572}, {1928, 19556}, {1934, 19576}, {1965, 3505}, {1966, 3493}, {1967, 8783}, {18896, 19578}
X(41533) = cevapoint of X(i) and X(j) for these (i,j): {5027, 20975}, {8623, 23642}
X(41533) = crosspoint of X(14370) and X(34130)
X(41533) = crosssum of X(147) and X(2896)
X(41533) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5207}, {31, 19555}, {32, 6660}, {385, 8783}, {512, 14316}, {560, 19559}, {1501, 19558}, {1580, 19574}, {1691, 19571}, {1917, 19560}, {1933, 19572}, {9233, 19556}, {9468, 3493}, {14602, 19576}, {18902, 19575}

X(41534) = ISOGONAL CONJUGATE OF X(40873)

Barycentrics    a*(a^2 + b*c)*(a^3 + b^3 - a*b*c - c^3)*(a^3 - b^3 - a*b*c + c^3) : :

X(41534) lies on the cubics K155, K225, K252, and these lines: {1, 40777}, {2, 2112}, {3, 2053}, {6, 2114}, {171, 19576}, {238, 1581}, {662, 17289}, {672, 1931}, {1193, 29055}, {1429, 1910}, {1920, 19574}, {2329, 7061}, {2330, 36213}, {16822, 40845}, {27958, 40846}, {27982, 30669}

X(41534) = isogonal conjugate of X(40873)
X(41534) = isogonal conjugate of the isotomic conjugate of X(40846)
X(41534) = X(1691)-cross conjugate of X(171)
X(41534) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40873}, {256, 3509}, {257, 17798}, {694, 1281}, {893, 4645}, {904, 17789}, {1178, 4071}, {1581, 19557}, {1916, 19561}, {1934, 18038}, {1967, 18037}, {7018, 19554}, {20715, 40432}
X(41534) = crosssum of X(256) and X(8936)
X(41534) = barycentric product X(i)*X(j) for these {i,j}: {1, 7061}, {6, 40846}, {171, 7261}, {172, 40845}, {385, 24479}, {894, 3512}, {1909, 8852}, {1966, 30648}, {7122, 18036}, {7176, 7281}, {8875, 39933}
X(41534) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40873}, {171, 4645}, {172, 3509}, {385, 18037}, {894, 17789}, {1580, 1281}, {1691, 19557}, {1933, 19561}, {2295, 4071}, {3512, 257}, {4107, 27951}, {4367, 4458}, {7061, 75}, {7122, 17798}, {7261, 7018}, {7281, 4451}, {8852, 256}, {8875, 40849}, {14602, 18038}, {20964, 20715}, {24479, 1916}, {30648, 1581}, {40846, 76}

X(41535) = ISOTOMIC CONJUGATE OF X(2107)

Barycentrics    b^2*(a + b)*c^2*(a + c)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 - b^2*c^2) : :

X(41535) lies on the cubic K1020 and these lines: {2, 34021}, {38, 75}, {141, 18152}, {274, 1573}, {350, 670}, {661, 7199}, {672, 799}, {1575, 30938}, {1921, 40017}, {1978, 4037}, {2276, 34022}, {2669, 39028}, {3978, 39367}, {4602, 6381}, {4639, 36906}, {17759, 40874}, {20345, 30941}, {27164, 27188}, {31004, 40849}

X(41535) = isotomic conjugate of X(2107)
X(41535) = isotomic conjugate of the isogonal conjugate of X(2669)
X(41535) = X(40017)-Ceva conjugate of X(310)
X(41535) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2107}, {1918, 2665}, {2205, 39925}
X(41535) = barycentric product X(i)*X(j) for these {i,j}: {75, 40874}, {76, 2669}, {310, 17759}, {561, 2106}, {2664, 6385}, {15148, 40364}, {39028, 40017}
X(41535) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2107}, {274, 2665}, {310, 39925}, {2106, 31}, {2664, 213}, {2669, 6}, {15148, 1973}, {17759, 42}, {20796, 2200}, {21788, 1918}, {21897, 872}, {27854, 4455}, {30940, 40769}, {39028, 2238}, {39916, 3747}, {40874, 1}
X(41535) = {X(6385),X(30966)}-harmonic conjugate of X(310)

X(41536) = X(6)X(14593)∩X(24)X(254)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(-a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 + 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)*(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)::

X(41536) lies on the cubic K627 and these lines: {6, 14593}, {24, 254}, {52, 53}, {193, 6504}, {216, 40678}, {324, 39113}, {847, 9722}, {2165, 34428}, {2383, 39416}, {14569, 14576}

X(41536) = polar conjugate of the isotomic conjugate of X(8800)
X(41536) = X(39114)-Ceva conjugate of X(5)
X(41536) = X(216)-cross conjugate of X(53)
X(41536) = X(i)-isoconjugate of X(j) for these (i,j): {63, 8883}, {97, 920}, {155, 2167}, {2148, 40697}, {2169, 6515}, {2190, 6503}, {14533, 33808}
X(41536) = barycentric product X(i)*X(j) for these {i,j}: {4, 8800}, {5, 254}, {53, 6504}, {311, 39109}, {847, 40678}, {2165, 39114}, {13398, 23290}, {13450, 15316}, {34428, 39117}
X(41536) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 40697}, {25, 8883}, {51, 155}, {53, 6515}, {216, 6503}, {254, 95}, {2181, 920}, {3199, 1609}, {6504, 34386}, {8800, 69}, {14569, 3542}, {39109, 54}, {39114, 7763}, {40678, 9723}

leftri

Pairs of orthologic pedal triangles: X(41537)-X(41679)

rightri

This preamble and centers X(41537)-X(41679) were contributed by César Eliud Lozada, March 07, 2021.

Let P', P" be any two distinct, finite points not both on the circumcircle of a triangle ABC and such that they are collinear with its circumcenter O=X(3). Then the pedal triangles A'B'C' and A"B"C" of P' and P" are orthologic.

Let Q' and Q" be the orthologic centers (A'B'C' to A"B"C") and (A"B"C" to A'B'C'), respectively. If P' is fixed and P" moves along the line OP', then Q' describes a rectangular hyperbola ℍ(P') circumscribed to A'B'C', whilst Q" moves on a line 𝕃(P'). ℍ(P) is named here the bipedal circumcentral conic of P' (shortened to the BPC-conic of P') and 𝕃(P') is named here the bipedal circumcentral line of P', abbreviated as the BPC-line of P'.

If P' = x:y:z (barycentrics), then

The appearance of (i, j, m, n) in the following lists means that the orthological centers of the pedal triangles of X(i) and X(j) are X(m) and X(n):

For selected points P', P" on the line X(1)X(3):
(1, 3, 1, 10), (1, 35, 3649, 12), (1, 36, 1317, 40663), (1, 40, 1071, 72), (1, 46, 41537, 41538), (1, 55, 7, 226), (1, 56, 145, 4848), (1, 57, 15185, 41539), (1, 65, 39772, 15556), (3, 35, 3647, 35), (3, 36, 214, 36), (3, 40, 6260, 40), (3, 46, 41540, 46), (3, 55, 9, 55), (3, 56, 1, 56), (3, 57, 142, 57), (3, 65, 442, 65), (35, 36, 41541, 41542), (35, 40, 41543, 3650), (35, 46, 41544, 41545), (35, 55, 29007, 41546), (35, 56, 34772, 41547), (35, 57, 41548, 41549), (35, 65, 41550, 41551), (36, 40, 1532, 1145), (36, 46, 41552, 12832), (36, 55, 37787, 41553), (36, 56, 38460, 41554), (36, 57, 41555, 41556), (36, 65, 41557, 41558), (40, 46, 41559, 41560), (40, 55, 144, 41561), (40, 56, 8, 1210), (40, 57, 3059, 1864), (40, 65, 31938, 41562), (46, 55, 41563, 41564), (46, 56, 12649, 41565), (46, 57, 41566, 41567), (46, 65, 41568, 41569), (55, 56, 3870, 1445), (55, 57, 41570, 12848), (55, 65, 41571, 41572), (56, 57, 41573, 36845), (56, 65, 41574, 41575), (57, 65, 41576, 41577)

For selected points P', P" on the line X(2)X(3):

(2, 3, 2, 141), (2, 4, 51, 1843), (2, 5, 41578, 41579), (2, 20, 41580, 3313), (2, 21, 41581, 41582), (2, 22, 19153, 16789), (2, 23, 15303, 41583), (2, 24, 3167, 41584), (2, 25, 1992, 41585), (3, 4, 5, 4), (3, 5, 1209, 5), (3, 20, 2883, 20), (3, 21, 960, 21), (3, 22, 206, 22), (3, 23, 6593, 23), (3, 24, 1147, 24), (3, 25, 6, 25), (4, 5, 6152, 143), (4, 20, 185, 5562), (4, 21, 1829, 18180), (4, 22, 6, 343), (4, 23, 5095, 41586), (4, 24, 155, 41587), (4, 25, 193, 41588), (5, 20, 41589, 41590), (5, 21, 41591, 41592), (5, 22, 41593, 41594), (5, 23, 41595, 41596), (5, 24, 41597, 41598), (5, 25, 3629, 41599), (20, 21, 41600, 41601), (20, 22, 159, 41602), (20, 23, 5181, 41603), (20, 24, 3, 235), (20, 25, 69, 13567), (21, 22, 41604, 41605), (21, 23, 41606, 41607), (21, 24, 41608, 41609), (21, 25, 41610, 41611), (22, 23, 41612, 41613), (22, 24, 184, 1974), (22, 25, 41614, 19136), (23, 24, 41615, 41616), (23, 25, 41617, 41618), (24, 25, 40318, 41619)

For selected points P', P" on the line X(3)X(6):

(3, 6, 2, 6), (3, 15, 618, 15), (3, 16, 619, 16), (3, 32, 39, 32), (3, 39, 6292, 39), (3, 50, 34834, 50), (3, 52, 34835, 52), (3, 58, 1125, 58), (6, 15, 41620, 37786), (6, 16, 41621, 37785), (6, 32, 41622, 14614), (6, 39, 41623, 41624), (6, 50, 41625, 41626), (6, 52, 41627, 41628), (6, 58, 4856, 41629), (15, 16, 9117, 9115), (15, 32, 41630, 41631), (15, 39, 41632, 41633), (15, 50, 41634, 41635), (15, 52, 41636, 41637), (15, 58, 41638, 41639), (16, 32, 41640, 41641), (16, 39, 41642, 41643), (16, 50, 41644, 41645), (16, 52, 41646, 41647), (16, 58, 41648, 41649), (32, 39, 41650, 41651), (32, 50, 41652, 41653), (32, 52, 41654, 41655), (32, 58, 41656, 41657), (39, 50, 41658, 41659), (39, 52, 41660, 41661), (39, 58, 41662, 41663), (50, 52, 41664, 41665), (50, 58, 41666, 41667), (52, 58, 41668, 41669)

The appearance of (i, j) in the following list means that the center of the BPC-conic of X(i) is X(j):

(1, 5083), (2, 41670), (4, 1112), (5, 41671), (6, 41672), (20, 41673), (23, 32269), (26, 41674), (32, 41675), (36, 3911), (40, 14740), (186, 468), (1319, 3660), (1691, 2030), (2070, 32223), (2071, 11064), (2077, 6745)

The appearance of (i, j) in the following list means that the tripole of the BPC-line of X(i) is X(j):

(1, 4552), (2, 41676), (4, 14570), (5, 41677), (6, 99), (15, 35314), (16, 35315), (20, 41678), (22, 112), (23, 4235), (24, 4558), (25, 99), (26, 41679), (32, 4576), (39, 10330), (55, 664), (56, 190), (58, 4427), (64, 648), (66, 41676), (159, 1289), (186, 2407), (187, 5468), (237, 877), (284, 17136), (399, 39290), (574, 35356), (878, 6333), (911, 883), (1177, 4235), (1319, 2397), (1350, 35278), (1384, 9146), (1403, 33946), (1420, 25268), (1436, 664), (1617, 644), (1676, 11794), (1677, 11794), (1691, 2396), (1987, 877), (2070, 14590), (2076, 17941), (2217, 4552), (2221, 33952), (2223, 883), (2283, 918), (2353, 110), (2420, 3268), (3053, 4563), (3286, 3573), (3433, 644), (3435, 651), (3437, 4610), (3445, 4582), (3446, 5376), (3447, 39295), (3455, 5468), (3456, 10330), (3515, 36841), (3532, 36841), (4230, 2799), (5104, 34245), (5118, 9147), (5172, 4585), (5201, 5118), (5467, 690), (5611, 14185), (5615, 14187), (5989, 39291), (6145, 41677), (9181, 9168), (9217, 39292), (10355, 39296), (11063, 10411), (11634, 14273), (11744, 16237), (14380, 41077), (15470, 2394), (16678, 4559), (18755, 4610), (20832, 4612), (21213, 4611), (22259, 39296), (23703, 21222), (23832, 30725), (23981, 3904), (34130, 39291), (34159, 32029), (34178, 39290), (34179, 39293), (34183, 39272), (34189, 39294), (34190, 39297), (34207, 112), (34250, 18047), (34427, 1289), (34435, 4612), (34436, 4611), (34438, 41679), (34442, 4585), (34448, 10411), (35364, 2799), (35365, 23887), (36176, 40866), (37579, 1332), (39644, 4226), (41337, 5027), (41346, 18047)

X(41537) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(1) TO PEDAL OF X(46)

Barycentrics    a*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b+c)^2*a+(b+c)*(b^2+c^2))*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41538)

X(41537) lies on these lines: {1, 90}, {7, 7702}, {12, 5728}, {56, 224}, {354, 10957}, {518, 11510}, {942, 10044}, {1071, 6284}, {1317, 3555}, {1537, 12688}, {1708, 3811}, {1770, 5840}, {1836, 10052}, {3174, 37550}, {3649, 10949}, {3683, 18232}, {3873, 10941}, {3962, 39781}, {10427, 37566}, {10916, 41540}, {10940, 12649}, {10944, 39779}, {10966, 12675}, {12116, 18839}, {12831, 41560}, {15015, 37583}, {39776, 41575}

X(41537) = reflection of X(41559) in X(41540)
X(41537) = barycentric product X(1708)*X(10916)
X(41537) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(41540)}} and {{A, B, C, X(90), X(3811)}}
X(41537) = crosspoint of X(7) and X(1708)
X(41537) = crosssum of X(55) and X(39943)
X(41537) = intouch-isogonal conjugate of X(3)
X(41537) = X(70)-of-intouch-triangle
X(41537) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (14054, 37579, 41538), (41540, 41564, 41544), (41540, 41565, 41552), (41544, 41552, 41540), (41564, 41565, 41540), (41564, 41569, 41559), (41565, 41569, 41567)

X(41538) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(1)

Barycentrics    a*(a+b-c)*(a-b+c)*(b+c)*(a^3-(b+c)*a^2-(b+c)^2*a+(b+c)*(b^2+c^2)) : :
X(41538) = 3*X(210)-2*X(21075)

The reciprocal orthologic center of these triangles is X(41537)

X(41538) lies on these lines: {1, 6883}, {7, 26060}, {8, 20928}, {10, 12}, {40, 1858}, {42, 201}, {43, 37591}, {46, 912}, {55, 12710}, {56, 78}, {57, 5904}, {63, 11509}, {73, 4878}, {100, 7098}, {191, 3256}, {200, 11501}, {227, 4849}, {354, 5433}, {388, 3681}, {392, 11011}, {484, 16143}, {498, 942}, {516, 1898}, {517, 1479}, {527, 12059}, {603, 32912}, {692, 11363}, {920, 11248}, {960, 2099}, {976, 1451}, {997, 26437}, {1038, 3751}, {1071, 1155}, {1214, 2594}, {1254, 21805}, {1319, 3555}, {1376, 1454}, {1388, 34791}, {1399, 4641}, {1400, 3949}, {1437, 31811}, {1697, 15104}, {1708, 3811}, {1728, 37569}, {1737, 24474}, {1757, 1935}, {1770, 40263}, {1788, 3868}, {1824, 1882}, {1825, 1826}, {1836, 5777}, {1864, 6284}, {2093, 5693}, {2551, 3869}, {2911, 41609}, {3057, 28234}, {3086, 18839}, {3340, 5692}, {3474, 12528}, {3485, 3876}, {3522, 40269}, {3600, 4661}, {3601, 18412}, {3779, 19366}, {3870, 11510}, {3873, 7288}, {3874, 3911}, {3927, 37541}, {4430, 5265}, {4559, 21874}, {4640, 14882}, {4847, 10957}, {5044, 5173}, {5128, 15071}, {5172, 16465}, {5204, 12675}, {5217, 10391}, {5252, 34790}, {5570, 10320}, {5587, 5887}, {5709, 11502}, {5728, 15837}, {5794, 18962}, {5905, 7702}, {6001, 37567}, {6253, 12664}, {6666, 15950}, {6684, 18389}, {6737, 10944}, {6745, 37566}, {6769, 30223}, {7082, 11496}, {7235, 21867}, {7354, 14872}, {10202, 31659}, {10370, 37613}, {10393, 37601}, {10404, 37544}, {10572, 37585}, {10831, 12329}, {10914, 36920}, {10950, 14110}, {10956, 14740}, {11507, 26921}, {12672, 18483}, {12680, 15326}, {12711, 37568}, {12832, 41565}, {13750, 26446}, {15867, 34339}, {17609, 20116}, {17615, 18961}, {17625, 32636}, {17642, 37722}, {17659, 17668}, {18398, 31231}, {18967, 19861}, {20989, 40660}, {22760, 37531}, {24167, 26741}, {30274, 31423}, {31730, 41562}, {31786, 37740}, {31835, 39542}, {31938, 41572}

X(41538) = reflection of X(i) in X(j) for these (i, j): (65, 4848), (11682, 960)
X(41538) = barycentric product X(i)*X(j) for these {i, j}: {10, 1708}, {12, 40571}, {65, 17776}, {226, 3811}, {321, 37579}, {1441, 2911}
X(41538) = barycentric quotient X(i)/X(j) for these (i, j): (42, 39943), (65, 15474), (1708, 86), (1780, 2185), (1880, 39267), (2171, 23604)
X(41538) = trilinear product X(i)*X(j) for these {i, j}: {10, 37579}, {12, 1780}, {37, 1708}, {65, 3811}, {201, 30733}, {210, 4341}
X(41538) = trilinear quotient X(i)/X(j) for these (i, j): (12, 23604), (37, 39943), (201, 28787), (225, 39267), (226, 15474), (1708, 81)
X(41538) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(12609)}} and {{A, B, C, X(10), X(3811)}}
X(41538) = X(i)-isoconjugate-of-X(j) for these {i, j}: {60, 23604}, {81, 39943}, {270, 28787}, {283, 39267}
X(41538) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (42, 39943), (65, 15474), (1708, 86), (1780, 2185)
X(41538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 15556, 65), (40, 18397, 1858), (65, 210, 12), (72, 41539, 65), (200, 37550, 11501), (226, 12432, 65), (1708, 3811, 37579), (1788, 3868, 18838), (1864, 7957, 6284), (3678, 12432, 226), (3876, 7672, 3485), (5044, 5173, 11375), (14054, 37579, 41537)

X(41539) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(57) TO PEDAL OF X(1)

Barycentrics    a*(a+b-c)*(a-b+c)*(b+c)*(a^2-2*(b+c)*a+b^2+c^2) : :
X(41539) = 3*X(210)-2*X(21060) = 4*X(18227)-3*X(31142)

The reciprocal orthologic center of these triangles is X(15185)

X(41539) lies on these lines: {1, 5920}, {2, 5173}, {6, 8270}, {7, 3681}, {10, 12}, {40, 9786}, {42, 1214}, {43, 1465}, {46, 1071}, {55, 1708}, {56, 3555}, {57, 200}, {63, 37541}, {100, 16465}, {109, 4641}, {158, 1872}, {165, 10391}, {201, 3931}, {218, 7719}, {222, 3751}, {227, 3293}, {354, 3911}, {388, 34790}, {392, 2099}, {497, 517}, {516, 1864}, {553, 8581}, {612, 37543}, {631, 16193}, {908, 18236}, {912, 36279}, {942, 1788}, {950, 7957}, {960, 3340}, {971, 3474}, {1155, 10167}, {1254, 3214}, {1400, 3694}, {1402, 23067}, {1420, 34791}, {1427, 4551}, {1439, 22277}, {1441, 4651}, {1445, 1617}, {1448, 9370}, {1451, 5266}, {1471, 3938}, {1478, 18908}, {1621, 10177}, {1699, 5903}, {1728, 11496}, {1737, 7680}, {1750, 2093}, {1824, 1893}, {1836, 5927}, {1858, 37567}, {1887, 39585}, {1999, 14594}, {2003, 4663}, {2263, 34048}, {3057, 6738}, {3198, 41342}, {3240, 17080}, {3256, 4640}, {3339, 5904}, {3485, 5044}, {3486, 31793}, {3579, 7098}, {3660, 3873}, {3696, 6358}, {3740, 5219}, {3742, 31231}, {3744, 14523}, {3779, 14524}, {3827, 14557}, {3868, 7080}, {3869, 13601}, {3874, 37566}, {3889, 5265}, {3896, 4552}, {3916, 11509}, {3957, 7677}, {4003, 26740}, {4292, 14872}, {4295, 5777}, {4318, 32911}, {4383, 34036}, {4463, 22294}, {4524, 17094}, {4661, 21454}, {4662, 9578}, {4666, 11526}, {4860, 10569}, {4863, 10914}, {5045, 7288}, {5083, 6174}, {5128, 9943}, {5218, 11018}, {5229, 9947}, {5281, 11020}, {5439, 10198}, {5563, 17624}, {5572, 10389}, {5584, 10393}, {5687, 14054}, {5692, 18421}, {5729, 30223}, {5887, 6867}, {5902, 31434}, {5905, 17615}, {6284, 9844}, {6604, 21609}, {6743, 10106}, {6769, 10396}, {7276, 14973}, {7982, 17622}, {7994, 10398}, {8543, 27065}, {9316, 32912}, {9364, 32913}, {9778, 10394}, {10164, 17603}, {10369, 37613}, {10399, 12710}, {11011, 30143}, {11019, 17642}, {11220, 40269}, {12647, 39779}, {12675, 15803}, {12848, 15733}, {15253, 26723}, {16577, 37593}, {17614, 26437}, {17626, 17728}, {17634, 31803}, {17668, 41572}, {17747, 21853}, {18227, 31142}, {18593, 21870}, {18838, 24473}, {21867, 22278}, {27186, 30312}, {28774, 33114}, {37585, 37730}

X(41539) = midpoint of X(2093) and X(18397)
X(41539) = reflection of X(i) in X(j) for these (i, j): (17625, 57), (17642, 11019)
X(41539) = barycentric product X(i)*X(j) for these {i, j}: {7, 3991}, {10, 1445}, {12, 41610}, {37, 6604}, {42, 21609}, {65, 344}
X(41539) = barycentric quotient X(i)/X(j) for these (i, j): (37, 6601), (65, 277), (218, 21), (344, 314), (1042, 17107), (1427, 40154)
X(41539) = trilinear product X(i)*X(j) for these {i, j}: {7, 4878}, {10, 1617}, {37, 1445}, {42, 6604}, {57, 3991}, {59, 21945}
X(41539) = trilinear quotient X(i)/X(j) for these (i, j): (10, 6601), (65, 2191), (218, 284), (226, 277), (344, 333), (1427, 17107)
X(41539) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(3870)}} and {{A, B, C, X(37), X(3925)}}
X(41539) = crosspoint of X(1445) and X(6604)
X(41539) = X(100)-Beth conjugate of-X(1427)
X(41539) = X(1446)-Ceva conjugate of-X(37)
X(41539) = X(i)-isoconjugate-of-X(j) for these {i, j}: {21, 2191}, {58, 6601}, {277, 284}, {1292, 3737}
X(41539) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (37, 6601), (65, 277), (218, 21), (344, 314)
X(41539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 7672, 5173), (10, 12432, 65), (65, 72, 12709), (65, 210, 226), (65, 40663, 3753), (65, 41538, 72), (165, 18412, 10391), (1427, 4849, 4551), (1445, 3870, 1617), (3873, 5435, 3660), (4848, 15556, 65), (13405, 30329, 354), (17728, 18839, 17626), (34790, 37544, 388)

X(41540) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(3) TO PEDAL OF X(46)

Barycentrics    (a^3+(b+c)*a^2-(b^2+c^2)*a-(b^2-c^2)*(b-c))*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(46)

X(41540) lies on these lines: {1, 224}, {2, 90}, {3, 12608}, {9, 2252}, {10, 912}, {46, 5552}, {65, 1145}, {79, 27385}, {119, 12616}, {142, 25639}, {214, 946}, {226, 7702}, {442, 1864}, {997, 6897}, {1125, 22768}, {1329, 40296}, {1519, 6934}, {1770, 35976}, {2886, 13373}, {3338, 10530}, {3452, 3647}, {3753, 10955}, {3814, 6260}, {4190, 12047}, {4870, 11112}, {5259, 5444}, {5794, 12645}, {5836, 32213}, {5880, 6600}, {5904, 6735}, {6594, 30424}, {6831, 10427}, {6931, 36599}, {10528, 13407}, {10916, 41537}, {12514, 37112}, {12640, 21620}, {14054, 18838}, {17057, 24982}, {22754, 28628}, {27186, 31418}, {34822, 34825}

X(41540) = midpoint of X(i) and X(j) for these {i, j}: {7702, 11517}, {41537, 41559}
X(41540) = complement of X(90)
X(41540) = complementary conjugate of X(21616)
X(41540) = trilinear product X(46)*X(10916)
X(41540) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(41537)}} and {{A, B, C, X(9), X(41559)}}
X(41540) = crosspoint of X(2) and X(20930)
X(41540) = X(i)-complementary conjugate of-X(j) for these (i, j): (1, 21616), (6, 63), (46, 10), (56, 499)
X(41540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5552, 10940, 46), (5553, 26364, 18232), (41537, 41544, 41564), (41537, 41552, 41565), (41544, 41552, 41537), (41552, 41559, 10916), (41564, 41565, 41537)

X(41541) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(35) TO PEDAL OF X(36)

Barycentrics    a*(2*a-b-c)*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)*(b^2+c^2)) : :
X(41541) = X(80)-3*X(3584)

The reciprocal orthologic center of these triangles is X(41542)

X(41541) lies on these lines: {1, 6797}, {3, 17660}, {11, 13411}, {35, 2771}, {55, 6326}, {56, 15015}, {65, 100}, {80, 3584}, {104, 37600}, {119, 12743}, {149, 11375}, {214, 519}, {354, 10090}, {404, 13751}, {518, 4996}, {528, 4870}, {952, 2646}, {1385, 7972}, {1387, 3748}, {1470, 6600}, {1768, 5217}, {1836, 13199}, {2099, 5541}, {2800, 33597}, {2801, 15837}, {2802, 11011}, {3035, 6734}, {3057, 6265}, {3485, 20095}, {3579, 11571}, {3601, 5531}, {3612, 12773}, {3683, 18254}, {3870, 22560}, {3918, 17614}, {4302, 16128}, {4640, 12532}, {4855, 34880}, {4861, 20586}, {5048, 19907}, {5083, 12432}, {5172, 33667}, {5218, 9803}, {5252, 6224}, {5432, 10265}, {5660, 12764}, {5719, 33593}, {5919, 12740}, {6264, 34471}, {6284, 21635}, {7993, 13384}, {9897, 37571}, {10058, 12738}, {10073, 17606}, {10074, 37605}, {10572, 11698}, {10609, 10956}, {10698, 37837}, {10738, 17605}, {10827, 12747}, {10896, 15017}, {10944, 33337}, {11570, 33814}, {12119, 12763}, {12332, 18446}, {12619, 24299}, {12675, 18861}, {12688, 12775}, {12735, 20323}, {12767, 35445}, {12831, 24466}, {14803, 35451}, {14882, 34600}, {15338, 41543}, {15950, 21630}, {17652, 25439}, {26287, 37707}

X(41541) = midpoint of X(100) and X(34772)
X(41541) = reflection of X(i) in X(j) for these (i, j): (11, 13411), (6734, 3035)
X(41541) = barycentric product X(1319)*X(32849)
X(41541) = barycentric quotient X(i)/X(j) for these (i, j): (44, 11604), (1319, 21907)
X(41541) = trilinear product X(i)*X(j) for these {i, j}: {519, 5172}, {1404, 32849}
X(41541) = trilinear quotient X(519)/X(11604)
X(41541) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(33812)}} and {{A, B, C, X(214), X(5424)}}
X(41541) = X(519)-Beth conjugate of-X(41558)
X(41541) = X(i)-isoconjugate-of-X(j) for these {i, j}: {106, 11604}, {1290, 23838}
X(41541) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (44, 11604), (1319, 21907)
X(41541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 12331, 17636), (55, 6326, 17638), (100, 12739, 65), (214, 1317, 1319), (214, 41553, 1317), (1317, 6174, 12832), (1317, 40663, 41558), (1319, 3689, 36920), (6265, 10087, 3057), (10609, 10956, 18976), (11570, 33814, 1155), (15015, 37736, 56), (33667, 35204, 41542)

X(41542) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(36) TO PEDAL OF X(35)

Barycentrics    a*(a+b-c)*(a-b+c)*(2*a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)*(b^2+c^2)) : :
X(41542) = X(3065)+3*X(5131) = X(3649)-3*X(5298)

The reciprocal orthologic center of these triangles is X(41541)

X(41542) lies on these lines: {1, 22937}, {3, 17637}, {21, 65}, {30, 1155}, {36, 1749}, {46, 13743}, {56, 191}, {79, 17605}, {140, 14526}, {381, 16118}, {442, 18977}, {484, 6797}, {499, 16159}, {529, 18253}, {553, 1125}, {758, 1319}, {997, 37308}, {1388, 16126}, {1708, 37286}, {1770, 16160}, {1788, 15680}, {1864, 3651}, {2099, 5426}, {2475, 9352}, {2594, 4722}, {2646, 5428}, {3057, 16139}, {3065, 5131}, {3336, 7489}, {3485, 15676}, {3579, 5441}, {3648, 24703}, {5127, 16164}, {5172, 33667}, {5204, 16132}, {5221, 16418}, {5265, 31888}, {5433, 11263}, {6940, 14883}, {7082, 7701}, {7288, 14450}, {8226, 10123}, {10021, 12047}, {10122, 37080}, {10543, 37568}, {11011, 35016}, {11375, 15674}, {11509, 37292}, {11684, 31165}, {15254, 15671}, {15325, 33593}, {16113, 37623}, {17606, 37230}, {17660, 41345}, {17768, 30379}, {20118, 41557}, {21161, 33857}, {28453, 36279}, {33856, 41347}, {33858, 37605}

X(41542) = midpoint of X(i) and X(j) for these {i, j}: {36, 1749}, {3647, 4973}, {33856, 41347}
X(41542) = reflection of X(i) in X(j) for these (i, j): (1319, 5427), (33593, 15325)
X(41542) = barycentric quotient X(1100)/X(11604)
X(41542) = trilinear product X(i)*X(j) for these {i, j}: {553, 17796}, {1125, 5172}
X(41542) = trilinear quotient X(i)/X(j) for these (i, j): (553, 21907), (1125, 11604)
X(41542) = X(1126)-isoconjugate-of-X(11604)
X(41542) = X(1100)-reciprocal conjugate of-X(11604)
X(41542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1125, 41551, 3649), (3647, 15670, 3683), (3647, 41547, 3649), (3649, 41547, 32636), (3683, 32636, 4870), (4973, 5298, 32636), (15670, 41549, 3649), (21161, 33857, 37600), (22936, 37582, 79), (33667, 35204, 41541)

X(41543) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(35) TO PEDAL OF X(40)

Barycentrics    (2*a^3+(b+c)*a^2-2*(b^2+b*c+c^2)*a-(b^2-c^2)*(b-c))*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3650)

X(41543) lies on these lines: {1, 18243}, {3, 13257}, {4, 6147}, {5, 27186}, {30, 11015}, {55, 16127}, {78, 37429}, {226, 37447}, {329, 37426}, {442, 40263}, {515, 11011}, {944, 1537}, {952, 37437}, {971, 6831}, {1012, 5703}, {1071, 1210}, {1490, 9579}, {2801, 15908}, {3090, 31657}, {3149, 5658}, {3529, 5763}, {3543, 3623}, {3579, 3650}, {3649, 31673}, {4187, 13369}, {5719, 21669}, {5779, 6889}, {5852, 24468}, {5905, 37411}, {6001, 10039}, {6175, 18357}, {6256, 13273}, {6259, 18446}, {6261, 12678}, {6284, 12831}, {6690, 7701}, {6833, 12684}, {6848, 36996}, {6907, 12528}, {6922, 11220}, {6925, 20013}, {6949, 13226}, {6960, 13243}, {9669, 38055}, {9848, 21620}, {10427, 26364}, {10884, 37822}, {11114, 34773}, {11551, 16616}, {12608, 12680}, {12699, 31164}, {15071, 18242}, {15338, 41541}, {17484, 33557}, {30147, 34697}

X(41543) = barycentric product X(1210)*X(17781)
X(41543) = barycentric quotient X(1108)/X(10308)
X(41543) = trilinear product X(i)*X(j) for these {i, j}: {1108, 17781}, {1210, 3579}
X(41543) = trilinear quotient X(1210)/X(10308)
X(41543) = X(1167)-isoconjugate-of-X(10308)
X(41543) = X(1108)-reciprocal conjugate of-X(10308)
X(41543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1071, 6260, 1532), (6260, 41561, 1071)

X(41544) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(35) TO PEDAL OF X(46)

Barycentrics    ((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*(2*a^4-(b+c)*a^3-(3*b^2+4*b*c+3*c^2)*a^2+(b+c)^3*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(41545)

X(41544) lies on these lines: {912, 10039}, {4654, 7702}, {10916, 41537}, {11248, 12831}, {15338, 33597}

X(41544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41537, 41540, 41552), (41540, 41564, 41537)

X(41545) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(35)

Barycentrics    (2*a+b+c)*(2*a^4-(b+c)*a^3-(3*b^2+4*b*c+3*c^2)*a^2+(b+c)^3*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(41544)

X(41545) lies on these lines: {21, 26437}, {30, 10573}, {191, 5290}, {553, 1125}, {1445, 15297}, {1749, 6841}, {7957, 17637}, {11684, 34605}, {12704, 16155}, {12860, 16141}, {16137, 28463}, {18977, 21677}

X(41545) = reflection of X(3649) in X(41547)
X(41545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3647, 41551, 3649), (3650, 41549, 3649)

X(41546) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(35)

Barycentrics    a*(a+b-c)*(a-b+c)*(2*a+b+c)*(a^2-2*(b+c)*a+b^2+3*b*c+c^2) : :
X(41546) = X(2099)+5*X(16140) = 3*X(10056)-X(16154)

The reciprocal orthologic center of these triangles is X(29007)

X(41546) lies on these lines: {1, 28461}, {21, 1412}, {30, 31397}, {57, 16133}, {79, 498}, {109, 1961}, {191, 1708}, {226, 4640}, {553, 1125}, {758, 956}, {846, 18593}, {942, 19919}, {1709, 18446}, {2771, 10391}, {3256, 29007}, {3295, 16138}, {3652, 10122}, {3748, 27778}, {3929, 11684}, {4654, 10032}, {5083, 29817}, {5126, 28463}, {5219, 9352}, {5221, 17542}, {5441, 10385}, {8545, 35989}, {8582, 18253}, {10056, 16154}, {11246, 13159}, {11544, 22937}, {12650, 21669}, {12831, 13405}, {14646, 16116}, {15174, 26202}, {15702, 37524}, {16137, 22936}, {17625, 35016}

X(41546) = barycentric product X(1125)*X(29007)
X(41546) = barycentric quotient X(1100)/X(3255)
X(41546) = trilinear product X(i)*X(j) for these {i, j}: {1100, 29007}, {1125, 3256}
X(41546) = trilinear quotient X(1125)/X(3255)
X(41546) = X(1126)-isoconjugate-of-X(3255)
X(41546) = X(1100)-reciprocal conjugate of-X(3255)
X(41546) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3647, 3649, 41547), (3649, 3650, 41551), (3649, 41549, 553)

X(41547) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(56) TO PEDAL OF X(35)

Barycentrics    a*(a+b-c)*(a-b+c)*(2*a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+b^3+c^3) : :
X(41547) = 3*X(10072)-X(16155)

The reciprocal orthologic center of these triangles is X(34772)

X(41547) lies on these lines: {1, 21161}, {3, 10122}, {7, 15674}, {21, 57}, {30, 1210}, {36, 22347}, {56, 758}, {58, 18593}, {65, 5427}, {79, 499}, {191, 3361}, {226, 6675}, {376, 5441}, {404, 31938}, {442, 3911}, {553, 1125}, {936, 1708}, {942, 5428}, {999, 16139}, {1420, 34195}, {1445, 35979}, {1466, 37286}, {2078, 31660}, {2475, 5435}, {3336, 6906}, {3338, 21165}, {3339, 5426}, {3579, 15174}, {3651, 10382}, {3928, 11684}, {4292, 6841}, {4654, 15671}, {5083, 35204}, {5172, 12432}, {5204, 33857}, {5221, 16370}, {5708, 28443}, {5902, 6875}, {6147, 31650}, {6738, 10543}, {6876, 10399}, {7288, 26725}, {9841, 33557}, {10021, 24470}, {10072, 16155}, {10106, 21677}, {10123, 37447}, {10167, 17637}, {11544, 22936}, {13411, 28465}, {13462, 16126}, {13743, 37545}, {15325, 33592}, {15556, 27086}, {15676, 21454}, {16137, 22937}, {26877, 40249}, {31231, 31254}

X(41547) = midpoint of X(3649) and X(41545)
X(41547) = barycentric product X(553)*X(34772)
X(41547) = barycentric quotient X(1100)/X(6598)
X(41547) = trilinear product X(1125)*X(37583)
X(41547) = trilinear quotient X(i)/X(j) for these (i, j): (553, 37887), (1125, 6598)
X(41547) = X(1126)-isoconjugate-of-X(6598)
X(41547) = X(1100)-reciprocal conjugate of-X(6598)
X(41547) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (65, 5427, 35016), (3647, 3649, 41546), (3649, 41542, 3647), (3649, 41549, 41551), (3916, 15670, 3647), (3916, 32636, 553), (32636, 41542, 3649)

X(41548) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(35) TO PEDAL OF X(57)

Barycentrics    ((b+c)*a-(b-c)^2)*(2*a^3-3*(b+c)*a^2+2*b*c*a+(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(41549)

X(41548) lies on these lines: {7, 100}, {9, 6690}, {35, 17768}, {142, 354}, {226, 17668}, {516, 33597}, {518, 10039}, {527, 4995}, {1145, 5902}, {3243, 3632}, {3754, 5542}, {3811, 5832}, {4675, 8271}, {5728, 17619}, {5853, 11011}, {6601, 30275}, {7373, 38053}, {10247, 20330}, {13257, 16112}, {15733, 21617}, {18412, 38211}, {21635, 36868}, {41549, 41571}

X(41548) = reflection of X(6067) in X(142)
X(41548) = barycentric product X(354)*X(28974)
X(41548) = trilinear product X(i)*X(j) for these {i, j}: {1212, 41572}, {1475, 28974}
X(41548) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(41555)}} and {{A, B, C, X(142), X(41572)}}
X(41548) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (142, 15185, 41555), (142, 41570, 15185)

X(41549) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(57) TO PEDAL OF X(35)

Barycentrics    (2*a+b+c)*(2*a^3-3*(b+c)*a^2+2*b*c*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(41548)

X(41549) lies on these lines: {30, 3474}, {46, 16154}, {57, 3816}, {191, 4355}, {442, 1454}, {553, 1125}, {758, 4315}, {1532, 3336}, {1709, 4338}, {1776, 6841}, {3929, 18253}, {5221, 11113}, {5918, 17637}, {6049, 34195}, {8148, 10385}, {8226, 11246}, {11684, 28610}, {12848, 35990}, {14646, 21669}, {16133, 21454}, {16137, 28443}, {17605, 30424}, {19919, 24470}, {35989, 37541}, {41548, 41571}

X(41549) = barycentric product X(1125)*X(41572)
X(41549) = trilinear product X(1100)*X(41572)
X(41549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (553, 41546, 3649), (3649, 41542, 15670), (3649, 41545, 3650), (41547, 41551, 3649)

X(41550) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(35) TO PEDAL OF X(65)

Barycentrics    ((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(a^4-(2*b^2+3*b*c+2*c^2)*a^2+(b+c)*b*c*a+(b^2-c^2)^2) : :
X(41550) = X(191)-3*X(3584)

The reciprocal orthologic center of these triangles is X(41551)

X(41550) lies on these lines: {10, 12909}, {12, 20612}, {21, 908}, {30, 33595}, {40, 14450}, {191, 3584}, {226, 2475}, {442, 942}, {519, 5178}, {758, 10039}, {2646, 33961}, {3452, 15674}, {3601, 15680}, {3649, 10956}, {4861, 21620}, {5270, 34600}, {5426, 21616}, {6260, 37433}, {8261, 24982}, {10106, 39778}, {11114, 37571}, {11281, 20323}, {11520, 31019}, {12639, 12913}, {15676, 27131}, {15837, 17768}, {20084, 35445}, {27385, 37308}, {33179, 33592}, {33668, 37562}, {35016, 41012}

X(41550) = midpoint of X(2475) and X(34772)
X(41550) = reflection of X(i) in X(j) for these (i, j): (21, 13411), (6734, 442)
X(41550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (442, 39772, 41557), (442, 41571, 39772)

X(41551) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(35)

Barycentrics    (2*a+b+c)*(a^4-(2*b^2+3*b*c+2*c^2)*a^2+(b+c)*b*c*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :
X(41551) = 3*X(553)-2*X(3649)

The reciprocal orthologic center of these triangles is X(41550)

X(41551) lies on these lines: {7, 31888}, {57, 14450}, {65, 17643}, {79, 1737}, {191, 226}, {484, 18244}, {516, 17637}, {527, 11684}, {553, 1125}, {758, 10106}, {1210, 16159}, {1749, 12047}, {2475, 4848}, {2771, 4292}, {3336, 6949}, {3340, 15680}, {3474, 16143}, {3911, 11263}, {3982, 16140}, {4295, 7701}, {5172, 12909}, {5221, 17556}, {5441, 28194}, {6260, 16116}, {6684, 14526}, {8261, 17768}, {11246, 31803}, {12267, 14882}, {12639, 12913}, {13411, 22937}, {13995, 37568}, {16118, 31673}, {22936, 39542}, {33668, 37582}, {37005, 37702}

X(41551) = barycentric quotient X(1100)/X(6597)
X(41551) = trilinear quotient X(1125)/X(6597)
X(41551) = X(1126)-isoconjugate-of-X(6597)
X(41551) = X(1100)-reciprocal conjugate of-X(6597)
X(41551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3649, 3650, 41546), (3649, 41542, 1125), (3649, 41545, 3647), (3649, 41549, 41547)

X(41552) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(36) TO PEDAL OF X(46)

Barycentrics    ((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*((b+c)*a^3-(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(12832)

X(41552) lies on these lines: {5, 10052}, {36, 5533}, {56, 10943}, {57, 79}, {119, 912}, {1319, 10609}, {1470, 3086}, {1532, 20118}, {4187, 18232}, {5553, 26476}, {5927, 10395}, {10916, 41537}, {35976, 37579}

X(41552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10916, 41540, 41559), (41537, 41540, 41544), (41540, 41565, 41537)

X(41553) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(36)

Barycentrics    a*(2*a-b-c)*(a+b-c)*(a-b+c)*(a^2-2*(b+c)*a+b^2+b*c+c^2) : :
X(41553) = X(80)-3*X(10056) = X(2099)-3*X(12739) = X(5119)-3*X(10087) = X(6224)+3*X(11239) = 2*X(6702)-3*X(10197)

The reciprocal orthologic center of these triangles is X(37787)

X(41553) lies on these lines: {1, 6946}, {11, 3748}, {55, 2801}, {57, 100}, {80, 3488}, {104, 30282}, {149, 5226}, {214, 519}, {226, 528}, {484, 11570}, {497, 5660}, {498, 12750}, {516, 12831}, {527, 18801}, {664, 35962}, {678, 14191}, {950, 37725}, {952, 24929}, {1260, 14740}, {1768, 31508}, {2078, 3935}, {2099, 2802}, {2800, 5119}, {3035, 4847}, {3254, 30275}, {3295, 12738}, {3586, 10711}, {3601, 38669}, {3722, 4551}, {3887, 15730}, {3961, 16577}, {3968, 4413}, {3982, 6154}, {4114, 24465}, {4292, 10993}, {5122, 33814}, {5218, 7967}, {5219, 10707}, {5531, 10382}, {5541, 18421}, {5658, 34789}, {5856, 41570}, {6224, 11239}, {6326, 15558}, {6702, 10197}, {10106, 10609}, {10167, 17660}, {10596, 16174}, {11491, 40249}, {12331, 12736}, {12735, 22935}, {13411, 37726}, {13462, 15015}, {14628, 40172}, {25416, 33598}, {39776, 39778}

X(41553) = midpoint of X(i) and X(j) for these {i, j}: {100, 3870}, {7972, 12647}
X(41553) = reflection of X(i) in X(j) for these (i, j): (11, 13405), (4847, 3035), (41166, 55)
X(41553) = barycentric product X(i)*X(j) for these {i, j}: {519, 37787}, {1319, 17264}
X(41553) = barycentric quotient X(i)/X(j) for these (i, j): (44, 3254), (1319, 34578)
X(41553) = trilinear product X(i)*X(j) for these {i, j}: {44, 37787}, {519, 2078}, {1319, 3935}, {1404, 17264}
X(41553) = trilinear quotient X(519)/X(3254)
X(41553) = X(i)-isoconjugate-of-X(j) for these {i, j}: {106, 3254}, {1308, 23838}
X(41553) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (44, 3254), (1319, 34578)
X(41553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (57, 14151, 5083), (57, 37736, 14151), (100, 14151, 57), (100, 37736, 5083), (214, 1317, 41554), (1145, 1317, 41558), (1317, 6174, 41556), (1317, 41541, 214), (6174, 41556, 3911)

X(41554) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(56) TO PEDAL OF X(36)

Barycentrics    a*(2*a-b-c)*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2-5*b*c+c^2)*a+(b+c)*(b^2-3*b*c+c^2)) : :
X(41554) = X(80)-3*X(10072) = X(6224)+3*X(11240) = 2*X(6700)-3*X(34123) = 2*X(6702)-3*X(10199)

The reciprocal orthologic center of these triangles is X(38460)

X(41554) lies on these lines: {1, 104}, {9, 14151}, {11, 10106}, {12, 32557}, {56, 2802}, {57, 1320}, {65, 15999}, {78, 37736}, {80, 3476}, {100, 1420}, {145, 12641}, {149, 4308}, {214, 519}, {226, 1387}, {388, 16173}, {499, 12749}, {912, 19907}, {952, 1210}, {956, 1388}, {999, 12736}, {1000, 37525}, {1125, 10956}, {1376, 11256}, {1385, 12735}, {1478, 16174}, {1617, 22560}, {1697, 38693}, {2078, 4996}, {2801, 12740}, {2829, 12053}, {3035, 6736}, {3086, 12751}, {3304, 18240}, {3361, 12653}, {4293, 14217}, {4311, 5840}, {4315, 21630}, {4564, 6551}, {4848, 5854}, {5048, 17613}, {5126, 33814}, {5193, 37789}, {5252, 6702}, {5261, 32558}, {5533, 6246}, {5541, 13462}, {5657, 21842}, {6049, 12649}, {6224, 11240}, {6594, 7677}, {6700, 34123}, {6713, 31397}, {7743, 22799}, {7972, 10573}, {9578, 31272}, {9614, 10728}, {9957, 38602}, {10039, 38133}, {10087, 37618}, {10475, 38484}, {10624, 38761}, {10742, 11373}, {10944, 15863}, {11376, 12763}, {12513, 14740}, {12531, 31190}, {13411, 38032}, {15906, 24201}, {17460, 23703}, {17625, 17638}, {24028, 41343}, {30725, 41191}

X(41554) = midpoint of X(i) and X(j) for these {i, j}: {1, 10074}, {56, 20586}, {100, 36846}, {1317, 12832}, {7972, 10573}
X(41554) = reflection of X(6736) in X(3035)
X(41554) = barycentric product X(i)*X(j) for these {i, j}: {519, 37789}, {1319, 37758}
X(41554) = barycentric quotient X(44)/X(12641)
X(41554) = trilinear product X(i)*X(j) for these {i, j}: {44, 37789}, {519, 5193}, {1319, 38460}, {1404, 37758}
X(41554) = trilinear quotient X(519)/X(12641)
X(41554) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1145)}} and {{A, B, C, X(104), X(519)}}
X(41554) = X(765)-Ceva conjugate of-X(23703)
X(41554) = X(106)-isoconjugate-of-X(12641)
X(41554) = X(44)-reciprocal conjugate of-X(12641)
X(41554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 104, 15558), (1, 11570, 25485), (104, 10698, 2950), (104, 15558, 41166), (214, 1317, 41553), (999, 12737, 12736), (1317, 1319, 214), (1317, 41556, 41558), (10944, 20118, 15863)

X(41555) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(36) TO PEDAL OF X(57)

Barycentrics    ((b+c)*a-(b-c)^2)*((b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c)) : :
X(41555) = X(10427)+2*X(26015)

The reciprocal orthologic center of these triangles is X(41556)

X(41555) lies on these lines: {7, 11680}, {9, 3816}, {11, 527}, {36, 528}, {142, 354}, {145, 30312}, {149, 30295}, {390, 17549}, {442, 18398}, {496, 5698}, {516, 4973}, {518, 1737}, {999, 2550}, {1001, 10072}, {1319, 5853}, {1532, 2801}, {1617, 6601}, {2095, 14647}, {2476, 30340}, {2886, 6173}, {3243, 3679}, {3254, 5536}, {3338, 5880}, {3361, 3813}, {3660, 10427}, {3822, 5542}, {4187, 5220}, {4860, 37363}, {4915, 38200}, {5784, 10916}, {5856, 37787}, {6362, 23599}, {7677, 27086}, {7678, 20059}, {8257, 17728}, {8271, 17278}, {8580, 20195}, {10177, 11019}, {15934, 38053}, {17668, 24389}, {18236, 21060}, {24387, 30424}

X(41555) = midpoint of X(i) and X(j) for these {i, j}: {149, 30295}, {26015, 30379}
X(41555) = reflection of X(i) in X(j) for these (i, j): (10427, 30379), (38211, 1737)
X(41555) = barycentric product X(i)*X(j) for these {i, j}: {142, 26015}, {354, 37788}, {1212, 38468}, {1229, 3660}
X(41555) = barycentric quotient X(i)/X(j) for these (i, j): (1212, 34894), (1418, 15728)
X(41555) = trilinear product X(i)*X(j) for these {i, j}: {354, 26015}, {1212, 30379}, {1475, 37788}
X(41555) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(41548)}} and {{A, B, C, X(142), X(23599)}}
X(41555) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1212, 34894), (1418, 15728)
X(41555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (142, 15185, 41548), (142, 41573, 15185)

X(41556) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(57) TO PEDAL OF X(36)

Barycentrics    (2*a-b-c)*((b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :
X(41556) = 2*X(997)-3*X(34123) = 4*X(6667)-5*X(31249) = 4*X(20103)-5*X(31235)

The reciprocal orthologic center of these triangles is X(41555)

X(41556) lies on these lines: {1, 11219}, {2, 14151}, {7, 10707}, {11, 118}, {56, 10609}, {57, 528}, {100, 1617}, {104, 3488}, {149, 21454}, {200, 3035}, {214, 519}, {388, 12019}, {664, 34342}, {938, 38669}, {942, 37726}, {952, 999}, {997, 12739}, {1210, 37725}, {1260, 24477}, {1387, 15934}, {1421, 33970}, {1465, 24216}, {1537, 5570}, {2829, 3586}, {3315, 15253}, {3338, 12750}, {3660, 10427}, {3756, 4551}, {4848, 13996}, {5218, 10246}, {5219, 17051}, {5533, 18393}, {5554, 22754}, {5856, 12848}, {6667, 31249}, {9581, 38757}, {10051, 10074}, {10738, 18541}, {10956, 20118}, {10957, 13751}, {10993, 37582}, {12649, 41426}, {15950, 38026}, {18839, 37374}, {20103, 31235}, {20586, 25416}, {30304, 34789}, {33814, 41345}, {35312, 40615}

X(41556) = midpoint of X(i) and X(j) for these {i, j}: {100, 36845}, {1864, 17660}, {30304, 34789}
X(41556) = reflection of X(i) in X(j) for these (i, j): (11, 11019), (200, 3035), (17625, 5083)
X(41556) = barycentric product X(i)*X(j) for these {i, j}: {44, 38468}, {519, 30379}, {1319, 37788}
X(41556) = barycentric quotient X(44)/X(34894)
X(41556) = trilinear product X(i)*X(j) for these {i, j}: {44, 30379}, {519, 3660}, {902, 38468}, {1319, 26015}, {1404, 37788}
X(41556) = trilinear quotient X(519)/X(34894)
X(41556) = X(106)-isoconjugate-of-X(34894)
X(41556) = X(44)-reciprocal conjugate of-X(34894)
X(41556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (11, 354, 38055), (11, 17660, 13257), (1317, 6174, 41553), (1317, 12832, 1145), (3911, 41553, 6174), (10956, 20118, 34122), (18801, 30379, 10427), (41554, 41558, 1317)

X(41557) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(36) TO PEDAL OF X(65)

Barycentrics    ((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(a^4-2*(b+c)*a^3+b*c*a^2+(b+c)*(2*b^2-3*b*c+2*c^2)*a-(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41558)

X(41557) lies on these lines: {10, 5425}, {21, 1210}, {30, 13226}, {57, 2475}, {191, 2478}, {442, 942}, {758, 908}, {1319, 26015}, {1532, 2771}, {1835, 37371}, {2476, 5902}, {3091, 14450}, {3740, 21677}, {3911, 27086}, {3984, 25005}, {4711, 25006}, {5499, 10202}, {5535, 6840}, {5563, 10916}, {5722, 37286}, {5855, 11281}, {6598, 35979}, {6684, 31660}, {6838, 16132}, {11277, 13151}, {11520, 25525}, {11529, 31266}, {14988, 33594}, {16126, 18395}, {20118, 41542}, {37797, 39778}

X(41557) = reflection of X(27086) in X(3911)
X(41557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (442, 39772, 41550), (442, 41574, 39772)

X(41558) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(36)

Barycentrics    (a+b-c)*(a-b+c)*(2*a-b-c)*(a^4-2*(b+c)*a^3+b*c*a^2+(b+c)*(2*b^2-3*b*c+2*c^2)*a-(b^2-c^2)^2) : :
X(41558) = 4*X(12447)-5*X(31235)

The reciprocal orthologic center of these triangles is X(41557)

X(41558) lies on these lines: {1, 6952}, {7, 20085}, {8, 37736}, {10, 12739}, {11, 6738}, {35, 17009}, {56, 33337}, {57, 6224}, {65, 33667}, {80, 226}, {100, 4848}, {142, 12531}, {149, 3340}, {153, 5727}, {214, 519}, {388, 9897}, {497, 13253}, {515, 11570}, {516, 12743}, {942, 952}, {946, 10073}, {950, 2800}, {1125, 20118}, {1210, 6265}, {1768, 3486}, {1788, 15015}, {1837, 21635}, {2093, 13199}, {2099, 21630}, {2475, 12657}, {2771, 37730}, {2802, 12432}, {3035, 6737}, {3244, 20586}, {3485, 37718}, {3671, 13273}, {4298, 18976}, {4301, 13274}, {4304, 12515}, {5533, 13464}, {5563, 7972}, {5853, 39776}, {5882, 10074}, {5883, 10944}, {6326, 18391}, {6713, 12735}, {10052, 40264}, {10057, 21620}, {10087, 11362}, {10572, 11571}, {10698, 12053}, {10950, 17660}, {10956, 15863}, {11019, 12740}, {12447, 31235}, {12619, 13411}, {12773, 37739}, {15950, 33709}, {17636, 30329}, {17663, 39781}, {19914, 31397}, {33898, 41561}, {37797, 39778}

X(41558) = midpoint of X(i) and X(j) for these {i, j}: {100, 41575}, {10572, 11571}, {10950, 17660}
X(41558) = reflection of X(i) in X(j) for these (i, j): (11, 6738), (6737, 3035), (10106, 5083), (18976, 4298)
X(41558) = barycentric product X(519)*X(37797)
X(41558) = barycentric quotient X(44)/X(6596)
X(41558) = trilinear product X(44)*X(37797)
X(41558) = trilinear quotient X(519)/X(6596)
X(41558) = X(519)-Beth conjugate of-X(41541)
X(41558) = X(106)-isoconjugate-of-X(6596)
X(41558) = X(44)-reciprocal conjugate of-X(6596)
X(41558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (80, 5425, 33593), (214, 12832, 3911), (1145, 1317, 41553), (1317, 1319, 33812), (1317, 12832, 214), (1317, 40663, 41541), (1317, 41556, 41554)

X(41559) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(40) TO PEDAL OF X(46)

Barycentrics    a*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*(a^5-(b+c)*a^4-2*(b^2-b*c+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c)) : :
X(41559) = 3*X(210)-2*X(18232)

The reciprocal orthologic center of these triangles is X(41560)

X(41559) lies on these lines: {8, 912}, {9, 35}, {72, 11826}, {210, 18232}, {224, 1385}, {518, 10052}, {1145, 14872}, {1898, 18254}, {5784, 24390}, {5840, 5887}, {10395, 17619}, {10916, 41537}, {11248, 41560}, {17615, 35448}

X(41559) = reflection of X(41537) in X(41540)
X(41559) = X(8)-Ceva conjugate of-X(10916)
X(41559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10916, 41540, 41552), (41564, 41569, 41537)

X(41560) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(40)

Barycentrics    a*(a^5-(b+c)*a^4-2*(b^2-b*c+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c))*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(41560) = 2*X(6736)-3*X(18908) = 3*X(10072)-2*X(12675)

The reciprocal orthologic center of these triangles is X(41559)

X(41560) lies on these lines: {4, 912}, {9, 37287}, {55, 32159}, {78, 1012}, {515, 1898}, {519, 12672}, {946, 10949}, {971, 1445}, {1071, 1210}, {1490, 37302}, {1537, 3555}, {1858, 6256}, {2800, 12688}, {5570, 10893}, {5784, 6700}, {5927, 6831}, {6001, 10573}, {6736, 18908}, {6826, 7700}, {6834, 13369}, {6918, 17616}, {6927, 11220}, {6938, 31837}, {6941, 10202}, {9856, 11278}, {10072, 12675}, {10306, 17615}, {11248, 41559}, {12664, 37468}, {12666, 18391}, {12831, 41537}, {31156, 31838}, {37437, 37562}

X(41560) = midpoint of X(12528) and X(12649)
X(41560) = reflection of X(i) in X(j) for these (i, j): (78, 5777), (1071, 1210)
X(41560) = barycentric quotient X(1108)/X(5553)
X(41560) = trilinear product X(1210)*X(11248)
X(41560) = trilinear quotient X(1210)/X(5553)
X(41560) = X(1167)-isoconjugate-of-X(5553)
X(41560) = X(1108)-reciprocal conjugate of-X(5553)
X(41560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1864, 18239, 1071), (6260, 41562, 1071)

X(41561) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(40)

Barycentrics    (3*a^2-2*(b+c)*a-(b-c)^2)*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(41561) = 3*X(226)-2*X(8727) = 3*X(4304)-2*X(6938) = X(6938)-3*X(18446) = X(10431)-3*X(31164)

The reciprocal orthologic center of these triangles is X(144)

X(41561) lies on these lines: {1, 6223}, {2, 11407}, {4, 4654}, {7, 1750}, {10, 12528}, {12, 9948}, {57, 5658}, {84, 6935}, {142, 5927}, {144, 165}, {222, 16870}, {226, 971}, {329, 5732}, {515, 1836}, {516, 3870}, {519, 6925}, {527, 7580}, {551, 6912}, {553, 19541}, {908, 11220}, {944, 9580}, {946, 12680}, {950, 6259}, {991, 4656}, {1071, 1210}, {1125, 10085}, {1490, 4292}, {1699, 5542}, {1709, 8545}, {1768, 3219}, {2801, 4847}, {2823, 17441}, {2947, 3668}, {3058, 5882}, {3085, 7992}, {3146, 3241}, {3452, 10167}, {3475, 11372}, {3485, 10864}, {3601, 12246}, {3671, 5691}, {3715, 6684}, {3742, 10863}, {3817, 31019}, {3982, 5805}, {4297, 4511}, {4304, 6938}, {4311, 6261}, {4640, 5851}, {5249, 12669}, {5316, 11227}, {5400, 24175}, {5439, 9842}, {5493, 33557}, {5811, 8726}, {5850, 41338}, {6001, 31397}, {6245, 6830}, {6881, 40263}, {6992, 10884}, {7308, 21151}, {7411, 17781}, {9612, 9799}, {9778, 20214}, {9851, 11522}, {9943, 21075}, {10157, 31657}, {10265, 33519}, {10394, 11019}, {10427, 18236}, {10431, 31164}, {10857, 18228}, {10860, 25568}, {11106, 30389}, {11374, 12684}, {12520, 12527}, {12675, 18243}, {12688, 21620}, {13407, 21628}, {14151, 34789}, {14646, 35445}, {15726, 41570}, {21629, 33064}, {33898, 41558}, {37161, 37714}

X(41561) = reflection of X(i) in X(j) for these (i, j): (1709, 13405), (4304, 18446)
X(41561) = barycentric product X(i)*X(j) for these {i, j}: {144, 1210}, {165, 17862}, {1108, 16284}, {1226, 3207}
X(41561) = barycentric quotient X(i)/X(j) for these (i, j): (144, 40424), (165, 40399), (1108, 3062), (1210, 10405), (1864, 19605)
X(41561) = trilinear product X(i)*X(j) for these {i, j}: {144, 1108}, {165, 1210}
X(41561) = trilinear quotient X(i)/X(j) for these (i, j): (144, 40399), (165, 1167), (1108, 11051), (1210, 3062)
X(41561) = X(1167)-isoconjugate-of-X(3062)
X(41561) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (144, 40424), (165, 40399), (1108, 3062), (1210, 10405)
X(41561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1071, 6260, 1210), (1071, 18239, 41562), (1071, 41543, 6260), (5658, 36996, 57), (10167, 13257, 3452)

X(41562) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(40)

Barycentrics    a*(a^2-b^2-b*c-c^2)*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2) : :
X(41562) = 3*X(4292)-4*X(37544) = 4*X(12436)-3*X(17616)

The reciprocal orthologic center of these triangles is X(31938)

X(41562) lies on these lines: {1, 651}, {4, 18389}, {5, 9946}, {7, 10399}, {11, 12005}, {12, 13995}, {20, 18397}, {30, 15556}, {35, 3219}, {58, 3465}, {65, 16006}, {72, 4304}, {80, 2475}, {90, 5248}, {144, 4294}, {185, 2823}, {226, 6841}, {484, 33557}, {496, 5083}, {498, 15064}, {500, 16577}, {515, 1858}, {518, 10624}, {527, 14054}, {546, 942}, {581, 24430}, {758, 10572}, {912, 950}, {946, 1898}, {962, 40269}, {971, 4292}, {1071, 1210}, {1479, 3874}, {1483, 15558}, {1727, 35989}, {1728, 10884}, {1736, 4303}, {1750, 9960}, {1770, 12432}, {1776, 10902}, {1837, 5884}, {1844, 7282}, {2003, 6198}, {2093, 9961}, {2646, 20117}, {2771, 37730}, {2800, 10950}, {2802, 37706}, {3091, 30274}, {3146, 5903}, {3151, 41329}, {3486, 5693}, {3586, 3868}, {3612, 10176}, {3614, 33519}, {3811, 12059}, {3873, 9614}, {3876, 17574}, {3878, 6872}, {3881, 30384}, {3911, 13369}, {3951, 5119}, {4127, 5441}, {4295, 18412}, {4305, 5692}, {4311, 12680}, {5080, 20612}, {5172, 16141}, {5697, 20050}, {5727, 10728}, {5777, 10391}, {5883, 10826}, {5885, 12019}, {6001, 13601}, {6223, 15071}, {6906, 12691}, {6925, 10573}, {7004, 37732}, {7330, 10393}, {7741, 31019}, {8715, 14740}, {9623, 12529}, {9797, 30305}, {9964, 13729}, {10265, 10958}, {10398, 12669}, {10591, 18398}, {11107, 17104}, {11220, 15803}, {11545, 13145}, {11570, 37702}, {11849, 41166}, {12047, 31871}, {12436, 17616}, {12564, 13407}, {12609, 16120}, {12711, 14872}, {13750, 19925}, {15325, 26201}, {15528, 26476}, {16865, 37525}, {17010, 33597}, {17638, 37734}, {17660, 37722}, {17768, 41577}, {18240, 37720}, {24929, 31649}, {31445, 40661}, {31730, 41538}, {31828, 39542}, {35986, 37572}, {37739, 40266}

X(41562) = reflection of X(1770) in X(12432)
X(41562) = barycentric product X(i)*X(j) for these {i, j}: {35, 17862}, {319, 1108}, {1210, 3219}, {1226, 2174}
X(41562) = barycentric quotient X(i)/X(j) for these (i, j): (35, 40399), (1108, 79), (1210, 30690), (1864, 7110)
X(41562) = trilinear product X(i)*X(j) for these {i, j}: {35, 1210}, {319, 40958}, {1071, 6198}, {1108, 3219}, {1442, 1864}
X(41562) = trilinear quotient X(i)/X(j) for these (i, j): (35, 1167), (319, 40424), (1071, 7100), (1108, 2160), (1210, 79), (1226, 20565)
X(41562) = X(79)-isoconjugate-of-X(1167)
X(41562) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (35, 40399), (1108, 79), (1210, 30690)
X(41562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (500, 35194, 16577), (1071, 1532, 40249), (1071, 1864, 1210), (1071, 18239, 41561), (1071, 41560, 6260), (1837, 5884, 12736), (3219, 31938, 3678), (5777, 10391, 13411), (10394, 12528, 1), (12711, 14872, 31397)

X(41563) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(55)

Barycentrics    (3*a^3-5*(b+c)*a^2+(b^2+4*b*c+c^2)*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(41564)

X(41563) lies on these lines: {2, 7}, {20, 18397}, {56, 5852}, {80, 3543}, {193, 4552}, {279, 5526}, {390, 5697}, {516, 10573}, {518, 36977}, {651, 2911}, {954, 11048}, {971, 6934}, {1001, 18967}, {1442, 3553}, {1743, 22464}, {1744, 14543}, {1757, 4331}, {1788, 7702}, {2099, 31156}, {3436, 5857}, {3600, 5692}, {4312, 18395}, {4454, 20236}, {5220, 18962}, {5228, 17334}, {5552, 7098}, {5698, 7672}, {5727, 20070}, {5728, 6936}, {5729, 5762}, {5759, 6868}, {5779, 6917}, {5805, 11662}, {5817, 6867}, {5843, 6924}, {5850, 30144}, {6872, 15556}, {6875, 21168}, {6942, 36996}, {6992, 18389}, {14100, 36976}, {14564, 16814}, {37681, 37771}

X(41563) = reflection of X(7) in X(1445)
X(41563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 6172, 29007), (9, 41572, 7), (144, 12848, 7), (8732, 20059, 7), (17350, 17950, 28739)

X(41564) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(46)

Barycentrics    (a+b-c)*(a-b+c)*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*(3*a^3-5*(b+c)*a^2+(b^2+4*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(41563)

X(41564) lies on these lines: {224, 10941}, {912, 31397}, {2099, 3244}, {4302, 18446}, {4338, 10052}, {5083, 16465}, {10175, 10395}, {10916, 41537}

X(41564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41537, 41540, 41565), (41537, 41544, 41540), (41537, 41559, 41569)

X(41565) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(56) TO PEDAL OF X(46)

Barycentrics    (a^4-2*(b+c)*a^3-2*b*c*a^2+2*(b^3+c^3)*a-(b^2-c^2)^2)*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(12649)

X(41565) lies on these lines: {5, 226}, {224, 2900}, {553, 7702}, {1479, 17437}, {3911, 11517}, {10052, 10598}, {10916, 41537}, {12832, 41538}

X(41565) = midpoint of X(224) and X(12649)
X(41565) = reflection of X(10395) in X(1210)
X(41565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41537, 41540, 41564), (41537, 41552, 41540), (41537, 41567, 41569)

X(41566) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(57)

Barycentrics    a*((b+c)*a-(b-c)^2)*(a^5-3*(b+c)*a^4+2*(b^2+3*b*c+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2-(3*b^4+3*c^4+2*(b^2-3*b*c+c^2)*b*c)*a+(b^2-c^2)^2*(b+c)) : :

The reciprocal orthologic center of these triangles is X(41567)

X(41566) lies on these lines: {142, 354}, {518, 10573}, {1445, 15733}, {2078, 3174}, {5784, 30329}

X(41566) = reflection of X(15185) in X(41573)
X(41566) = {X(142), X(41577)}-harmonic conjugate of X(15185)

X(41567) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(57) TO PEDAL OF X(46)

Barycentrics
a*(a+b-c)*(a-b+c)*(a^5-3*(b+c)*a^4+2*(b^2+3*b*c+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2-(3*b^4+3*c^4+2*(b^2-3*b*c+c^2)*b*c)*a+(b^2-c^2)^2*(b+c))*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41566)

X(41567) lies on these lines: {912, 18391}, {10916, 41537}

X(41567) = {X(41565), X(41569)}-harmonic conjugate of X(41537)

X(41568) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(46) TO PEDAL OF X(65)

Barycentrics
a*((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(a^6-2*(b+c)*a^5-(b^2-b*c+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3-(b^4+c^4+2*(b^2+c^2)*b*c)*a^2-2*(b+c)*(b^4-b^2*c^2+c^4)*a+(b+c)*(b^2-c^2)*(b^3-c^3)) : :

The reciprocal orthologic center of these triangles is X(41569)

X(41568) lies on these lines: {21, 1728}, {442, 942}, {758, 3436}, {1445, 35979}, {3984, 40661}

X(41568) = reflection of X(39772) in X(41574)
X(41568) = {X(31938), X(41576)}-harmonic conjugate of X(39772)

X(41569) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(46)

Barycentrics
a*(a+b-c)*(a-b+c)*(a^6-2*(b+c)*a^5-(b^2-b*c+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3-(b^4+c^4+2*(b^2+c^2)*b*c)*a^2-2*(b+c)*(b^4-b^2*c^2+c^4)*a+(b+c)*(b^2-c^2)*(b^3-c^3))*((b+c)*a^3-(b-c)^2*a^2-(b+c)*(b^2+c^2)*a+(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41568)

X(41569) lies on these lines: {224, 1420}, {5691, 11571}, {10916, 41537}

X(41569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41537, 41559, 41564), (41537, 41567, 41565)

X(41570) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(57)

Barycentrics    ((b+c)*a-(b-c)^2)*(3*a^3-5*(b+c)*a^2+(b+c)^2*a+(b^2-c^2)*(b-c)) : :
X(41570) = X(3872)-3*X(11038) = X(5223)-3*X(10056)

The reciprocal orthologic center of these triangles is X(12848)

X(41570) lies on these lines: {7, 3174}, {9, 13405}, {55, 527}, {65, 10427}, {142, 354}, {226, 15733}, {516, 18446}, {518, 8255}, {519, 1056}, {908, 7671}, {1389, 3244}, {2099, 5853}, {3189, 3671}, {3452, 10177}, {3664, 8271}, {3816, 5572}, {3872, 11038}, {3873, 30379}, {5223, 10056}, {5528, 30424}, {5696, 13407}, {5728, 17757}, {5784, 21620}, {5856, 41553}, {6745, 8257}, {10481, 35338}, {11680, 21617}, {15726, 41561}, {25557, 34791}, {30275, 36845}, {34919, 36973}

X(41570) = midpoint of X(7) and X(3870)
X(41570) = reflection of X(i) in X(j) for these (i, j): (9, 13405), (4847, 142)
X(41570) = trilinear product X(1212)*X(12848)
X(41570) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(41573)}} and {{A, B, C, X(142), X(12848)}}
X(41570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (142, 15185, 41573), (3059, 15185, 41577), (15185, 41548, 142), (21617, 30628, 24389)

X(41571) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(55) TO PEDAL OF X(65)

Barycentrics    ((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(2*a^3-3*(b+c)*a^2+2*b*c*a+(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(41572)

X(41571) lies on these lines: {7, 8730}, {21, 329}, {30, 18446}, {55, 5905}, {144, 8255}, {145, 388}, {191, 17699}, {226, 16465}, {442, 942}, {758, 31397}, {908, 11018}, {1466, 35979}, {2886, 3873}, {2900, 4654}, {3219, 6690}, {3304, 11281}, {3305, 6675}, {3419, 6147}, {3681, 12607}, {5057, 10543}, {5927, 6841}, {7680, 12528}, {10394, 31053}, {14450, 33557}, {20122, 21452}, {25557, 31245}, {35016, 40998}, {41548, 41549}

X(41571) = midpoint of X(5905) and X(35989)
X(41571) = barycentric product X(942)*X(28974)
X(41571) = trilinear product X(2260)*X(28974)
X(41571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (442, 39772, 41574), (39772, 41550, 442)

X(41572) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(55)

Barycentrics    (2*a^3-3*(b+c)*a^2+2*b*c*a+(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :
X(41572) = 5*X(7)-6*X(553) = 5*X(5728)-4*X(12433) = 5*X(11662)+4*X(12433)

The reciprocal orthologic center of these triangles is X(41571)

X(41572) lies on these lines: {2, 7}, {6, 22464}, {12, 15481}, {37, 14564}, {65, 17768}, {77, 3553}, {80, 34648}, {85, 17347}, {241, 17365}, {388, 3951}, {390, 7982}, {515, 40269}, {516, 5903}, {518, 10944}, {651, 3668}, {653, 1826}, {971, 37468}, {1001, 26437}, {1014, 18645}, {1156, 15909}, {1441, 4416}, {1442, 4667}, {1471, 24231}, {1512, 36279}, {1743, 37800}, {1770, 12432}, {1813, 7175}, {1836, 16112}, {1866, 1890}, {2093, 6925}, {2263, 24695}, {2475, 4848}, {2911, 6180}, {3091, 5825}, {3146, 5727}, {3255, 3256}, {3340, 6872}, {3585, 4312}, {3671, 5251}, {3674, 16788}, {3751, 4331}, {3879, 4552}, {4032, 27492}, {4067, 5850}, {4292, 12528}, {4298, 5692}, {4315, 4867}, {4323, 11106}, {4326, 36976}, {4419, 7190}, {4641, 6354}, {5228, 17276}, {5526, 10481}, {5537, 7676}, {5542, 5563}, {5572, 18839}, {5723, 16669}, {5728, 5762}, {5729, 5805}, {5735, 10398}, {5759, 7675}, {5832, 6734}, {5843, 37281}, {5851, 31391}, {5852, 8581}, {5856, 15185}, {8255, 15837}, {8544, 36996}, {11433, 20223}, {12730, 26726}, {14100, 38454}, {14151, 33812}, {15932, 21077}, {17086, 17120}, {17364, 25250}, {17668, 41539}, {20122, 41381}, {20927, 39126}, {31673, 36991}, {31938, 41538}, {36595, 37654}, {37545, 37713}, {41548, 41549}

X(41572) = midpoint of X(5728) and X(11662)
X(41572) = barycentric product X(i)*X(j) for these {i, j}: {57, 28974}, {664, 28473}, {1268, 41549}
X(41572) = trilinear product X(i)*X(j) for these {i, j}: {56, 28974}, {651, 28473}, {1170, 41548}, {1255, 41549}
X(41572) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(28974)}} and {{A, B, C, X(142), X(41548)}}
X(41572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 9, 21617), (7, 144, 8545), (7, 1445, 30379), (7, 6172, 8232), (7, 12848, 1445), (7, 18230, 30275), (7, 29007, 226), (7, 37787, 142), (7, 41563, 9), (894, 17950, 307)

X(41573) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(56) TO PEDAL OF X(57)

Barycentrics    ((b+c)*a-(b-c)^2)*(a^3-3*(b+c)*a^2+(3*b^2-2*b*c+3*c^2)*a-(b^2-c^2)*(b-c)) : :
X(41573) = X(480)-3*X(17728)

The reciprocal orthologic center of these triangles is X(36845)

X(41573) lies on these lines: {7, 24389}, {9, 11019}, {10, 3243}, {56, 5853}, {57, 6601}, {142, 354}, {390, 4652}, {480, 17728}, {516, 12116}, {518, 1210}, {527, 11238}, {1617, 8730}, {2191, 3008}, {2550, 3333}, {3174, 8732}, {3826, 34791}, {3873, 21617}, {3911, 6600}, {3983, 8582}, {4326, 31146}, {5435, 7674}, {5542, 10916}, {6173, 15841}, {6734, 11038}, {7682, 18529}, {9843, 38057}, {10390, 25525}, {21625, 30478}, {30379, 30628}

X(41573) = midpoint of X(15185) and X(41566)
X(41573) = barycentric product X(i)*X(j) for these {i, j}: {142, 36845}, {354, 20946}, {1233, 21002}
X(41573) = trilinear product X(i)*X(j) for these {i, j}: {142, 16572}, {354, 36845}, {1212, 8732}, {1475, 20946}
X(41573) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(41570)}} and {{A, B, C, X(142), X(8732)}}
X(41573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 26015, 24389), (142, 15185, 41570), (354, 6067, 142), (8732, 36845, 3174), (15185, 41555, 142)

X(41574) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(56) TO PEDAL OF X(65)

Barycentrics    ((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(2*a^4-3*(b+c)*a^3-(b^2+c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41575)

X(41574) lies on these lines: {2, 11281}, {21, 938}, {30, 37532}, {56, 12649}, {57, 6598}, {442, 942}, {758, 1210}, {1998, 34489}, {2475, 21454}, {3649, 10129}, {3873, 25466}, {5303, 10543}, {5705, 26725}, {6986, 31660}, {8256, 20013}, {10893, 14450}, {24391, 40661}, {31821, 33592}, {34772, 40663}, {39783, 41575}

X(41574) = midpoint of X(i) and X(j) for these {i, j}: {12649, 35979}, {39772, 41568}
X(41574) = trilinear product X(942)*X(41575)
X(41574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (442, 39772, 41571), (39772, 41557, 442)

X(41575) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(56)

Barycentrics    2*a^4-3*(b+c)*a^3-(b^2+c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^2-c^2)^2 : :
X(41575) = 3*X(2)-4*X(6738) = 9*X(2)-8*X(12447) = 5*X(3617)-4*X(6743) = 7*X(3622)-8*X(6744) = 3*X(6737)-4*X(12447) = 3*X(6738)-2*X(12447)

The reciprocal orthologic center of these triangles is X(41574)

X(41575) lies on these lines: {1, 2}, {30, 4018}, {63, 3486}, {65, 16465}, {72, 37730}, {80, 21077}, {92, 40950}, {100, 4848}, {149, 4301}, {185, 517}, {225, 1897}, {226, 5086}, {285, 3193}, {322, 17377}, {333, 40430}, {377, 11529}, {388, 11520}, {392, 12433}, {497, 11682}, {515, 3868}, {518, 10950}, {644, 21096}, {758, 10572}, {908, 1837}, {944, 5709}, {950, 3869}, {952, 3555}, {956, 37739}, {958, 37724}, {1331, 5247}, {1441, 3879}, {1478, 12559}, {1482, 18544}, {1512, 37700}, {1621, 5837}, {1770, 4084}, {1788, 4855}, {2078, 12640}, {2475, 3671}, {2551, 3984}, {2975, 24391}, {3057, 5855}, {3189, 37550}, {3218, 4297}, {3243, 37709}, {3339, 4190}, {3340, 3434}, {3436, 5727}, {3488, 5250}, {3577, 5881}, {3585, 16126}, {3586, 11415}, {3674, 21285}, {3681, 5795}, {3689, 8256}, {3692, 8557}, {3813, 11011}, {3871, 10902}, {3873, 10106}, {3875, 22464}, {3885, 28234}, {3895, 12245}, {3913, 37579}, {3962, 17781}, {4305, 4652}, {4311, 6224}, {4312, 31295}, {4313, 35258}, {4416, 25255}, {4430, 28236}, {4640, 10543}, {4867, 21616}, {4930, 9669}, {5048, 26475}, {5082, 11041}, {5128, 34701}, {5229, 31164}, {5249, 5794}, {5425, 12609}, {5493, 20066}, {5690, 24299}, {5691, 5905}, {5722, 5730}, {5853, 7672}, {5882, 11012}, {5902, 17647}, {6245, 9803}, {6601, 11526}, {6872, 12526}, {7674, 12630}, {7991, 20075}, {8609, 17388}, {8715, 36152}, {9776, 18221}, {10107, 34612}, {10222, 26470}, {10609, 37582}, {10680, 18524}, {10896, 34647}, {10941, 16236}, {10944, 34791}, {10957, 33895}, {11015, 31730}, {11112, 31794}, {11249, 37727}, {11260, 37734}, {12513, 26357}, {12536, 17784}, {12563, 31019}, {12690, 22793}, {12704, 36977}, {12750, 26726}, {13601, 15733}, {15829, 37723}, {16865, 18249}, {17364, 20089}, {18839, 33956}, {18990, 24473}, {19925, 31053}, {24216, 32577}, {25416, 37726}, {26437, 37738}, {31435, 36922}, {31503, 37631}, {39776, 41537}, {39783, 41574}

X(41575) = reflection of X(i) in X(j) for these (i, j): (72, 37730), (100, 41558), (1770, 4084), (3869, 950), (6737, 6738), (10944, 34791)
X(41575) = anticomplement of X(6737)
X(41575) = trilinear product X(943)*X(41574)
X(41575) = X(643)-Beth conjugate of-X(37583)
X(41575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 8, 24987), (1, 3679, 10198), (1, 5705, 3616), (1, 6734, 24541), (1, 12649, 26015), (8, 145, 3870), (8, 19860, 25006), (78, 18391, 24982), (145, 12649, 1), (145, 36845, 36846), (145, 38460, 3244), (1737, 22836, 27385), (1837, 12635, 908), (1998, 36845, 26015), (3241, 10527, 1), (3244, 10916, 1), (3632, 9623, 8), (3811, 10573, 6735), (4848, 12437, 100), (5086, 34195, 226), (5554, 20013, 200), (5727, 11523, 3436), (6737, 6738, 2), (9797, 20050, 145)

X(41576) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(57) TO PEDAL OF X(65)

Barycentrics    a*((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(a^5-3*(b+c)*a^4+(b+2*c)*(2*b+c)*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2-(3*b^2-5*b*c+3*c^2)*(b+c)^2*a+(b^2-c^2)*(b^3-c^3)) : :

The reciprocal orthologic center of these triangles is X(41577)

X(41576) lies on these lines: {21, 10396}, {57, 35990}, {329, 758}, {442, 942}, {1864, 17768}, {3434, 6598}, {3681, 11523}, {3873, 25525}, {5177, 5902}, {10382, 35989}, {19843, 26725}

X(41576) = {X(39772), X(41568)}-harmonic conjugate of X(31938)

X(41577) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(65) TO PEDAL OF X(57)

Barycentrics    a*((b+c)*a-(b-c)^2)*(a^5-3*(b+c)*a^4+(b+2*c)*(2*b+c)*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2-(3*b^2-5*b*c+3*c^2)*(b+c)^2*a+(b^2-c^2)*(b^3-c^3)) : :

The reciprocal orthologic center of these triangles is X(41576)

X(41577) lies on these lines: {9, 2266}, {65, 17668}, {100, 1445}, {142, 354}, {518, 14454}, {2550, 3754}, {5572, 6690}, {17768, 41562}

X(41577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3059, 15185, 41570), (15185, 41566, 142)

X(41578) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(2) TO PEDAL OF X(5)

Barycentrics    a^2*(a^4-b^4+3*b^2*c^2-c^4)*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41578) = X(54)-4*X(9827) = 5*X(54)-11*X(15024) = 2*X(973)+X(2888) = 5*X(1209)-2*X(1216) = 2*X(1209)+X(6152) = X(1209)+2*X(6153) = 4*X(1209)-X(41590) = 4*X(1216)+5*X(6152) = X(1216)+5*X(6153) = 8*X(1216)-5*X(41590) = X(1493)-4*X(13365) = 5*X(3091)-2*X(15739) = X(6152)-4*X(6153) = 2*X(6152)+X(41590) = 8*X(6153)+X(41590) = 5*X(6288)+X(34783) = X(7691)+2*X(11576) = 20*X(9827)-11*X(15024) = X(12606)+2*X(13368) = X(12606)-4*X(13565)

The reciprocal orthologic center of these triangles is X(41579)

X(41578) lies on these lines: {2, 34751}, {51, 5965}, {54, 5422}, {184, 9977}, {195, 9777}, {381, 1154}, {973, 2888}, {1209, 1216}, {1493, 13365}, {1593, 7691}, {1992, 16776}, {1995, 9972}, {3091, 15739}, {3167, 5640}, {5642, 5943}, {6288, 11442}, {9703, 15026}, {10255, 12606}, {11188, 19153}, {11245, 32423}, {12363, 13423}, {13595, 41579}, {14389, 27365}, {14769, 34836}, {15004, 19150}, {18400, 38323}, {21660, 32396}, {22804, 32137}

X(41578) = barycentric product X(1225)*X(40633)
X(41578) = barycentric quotient X(570)/X(13622)
X(41578) = X(570)-reciprocal conjugate of-X(13622)
X(41578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1209, 6152, 41590), (1209, 6153, 6152), (1209, 41594, 37636), (1209, 41599, 41594), (13368, 13565, 12606), (37636, 41596, 41594), (41594, 41599, 41596)

X(41579) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(2)

Barycentrics    a^2*(b^2+c^2)*(a^4-b^4+3*b^2*c^2-c^4) : :
X(41579) = 3*X(2)+X(9973) = 5*X(6)-9*X(5640) = X(6)+3*X(11188) = 3*X(6)+X(12272) = 7*X(6)-3*X(15531) = X(6)-3*X(16776) = 3*X(51)-X(3629) = X(69)+3*X(9971) = 3*X(141)-X(3313) = 5*X(141)-3*X(3917) = X(141)-3*X(29959) = 3*X(1843)+X(3313) = 5*X(1843)+3*X(3917) = X(1843)+3*X(29959) = 3*X(5640)+5*X(11188) = 3*X(5640)-5*X(16776) = 9*X(11188)-X(12272) = 7*X(11188)+X(15531) = 7*X(12272)+9*X(15531) = X(12272)+9*X(16776) = X(15531)-7*X(16776)

The reciprocal orthologic center of these triangles is X(41578)

X(41579) lies on these lines: {2, 9973}, {6, 110}, {51, 3629}, {69, 7394}, {141, 427}, {143, 5965}, {182, 5944}, {206, 9813}, {511, 546}, {524, 9969}, {542, 11561}, {597, 6467}, {1176, 19596}, {1350, 35502}, {1352, 18436}, {1503, 31833}, {1539, 2781}, {2393, 3589}, {3060, 40341}, {3564, 16881}, {3763, 12220}, {4611, 11327}, {5020, 34777}, {5097, 10095}, {5102, 9781}, {5157, 32154}, {5159, 8705}, {5462, 12007}, {5943, 6329}, {6403, 7547}, {6697, 6698}, {7392, 34751}, {7761, 16983}, {8681, 32455}, {9967, 12061}, {9972, 18369}, {9977, 34149}, {10224, 24206}, {10272, 13364}, {11002, 11008}, {11702, 34155}, {12294, 40929}, {13488, 29181}, {13595, 41578}, {14984, 19130}, {15026, 39561}, {15074, 38317}, {15806, 18583}, {16105, 16261}, {19127, 20987}, {20975, 35222}, {23635, 34990}, {25329, 32260}

X(41579) = midpoint of X(i) and X(j) for these {i, j}: {110, 32299}, {141, 1843}, {7761, 16983}, {9967, 12061}, {9969, 14913}, {9973, 17710}, {11188, 16776}, {12294, 40929}, {25329, 32260}
X(41579) = reflection of X(i) in X(j) for these (i, j): (3589, 9822), (5097, 10095), (11574, 34573), (12007, 5462), (32366, 6329)
X(41579) = complement of X(17710)
X(41579) = barycentric product X(141)*X(13595)
X(41579) = barycentric quotient X(39)/X(13622)
X(41579) = trilinear product X(38)*X(13595)
X(41579) = trilinear quotient X(38)/X(13622)
X(41579) = intersection, other than A,B,C, of conics {{A, B, C, X(111), X(427)}} and {{A, B, C, X(141), X(895)}}
X(41579) = X(82)-isoconjugate-of-X(13622)
X(41579) = X(39)-reciprocal conjugate of-X(13622)
X(41579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 9973, 17710), (1843, 29959, 141), (3589, 9822, 40670), (5943, 32366, 6329)

X(41580) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(2) TO PEDAL OF X(20)

Barycentrics    a^2*(a^4-b^4-c^4)*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41580) = X(52)+2*X(6759) = X(64)-4*X(9729) = X(185)+2*X(2883) = 2*X(185)+X(36982) = X(185)-4*X(41589) = 4*X(206)-X(3313) = 3*X(373)-2*X(23332) = 2*X(389)+X(1498) = X(1843)+2*X(34774) = 4*X(2883)-X(36982) = X(2883)+2*X(41589) = X(2979)-3*X(35260) = 5*X(3567)+X(34781) = 2*X(5446)+X(9833) = 4*X(5462)-X(14216) = X(5562)-4*X(16252) = 4*X(10282)-X(10625) = 3*X(14845)-2*X(23325) = 2*X(19149)+X(19161) = X(36982)+8*X(41589)

The reciprocal orthologic center of these triangles is X(3313)

X(41580) lies on these lines: {2, 34146}, {4, 14542}, {6, 1619}, {22, 206}, {25, 19149}, {51, 428}, {52, 161}, {64, 9729}, {66, 6997}, {154, 511}, {159, 33586}, {184, 11470}, {185, 235}, {373, 23332}, {381, 1853}, {389, 1498}, {568, 32063}, {569, 32321}, {1624, 6509}, {1660, 1993}, {1843, 34774}, {1992, 2393}, {1995, 9968}, {2777, 14855}, {2781, 3917}, {2937, 10282}, {2979, 35260}, {3567, 34781}, {3796, 19153}, {3981, 32445}, {5012, 41593}, {5157, 34207}, {5446, 9833}, {5462, 14216}, {5562, 16252}, {5596, 6995}, {5640, 32064}, {5655, 10628}, {5656, 5890}, {5878, 34944}, {5893, 11381}, {5907, 6293}, {6225, 10574}, {6241, 23291}, {6243, 14530}, {6697, 37990}, {7378, 15011}, {7484, 34778}, {7500, 36989}, {7540, 18400}, {8549, 9777}, {8567, 17704}, {9306, 34779}, {9934, 11557}, {9967, 10117}, {10024, 12162}, {10243, 17834}, {10575, 18404}, {10606, 16836}, {11064, 12058}, {11574, 19132}, {11695, 40686}, {11746, 15010}, {11808, 32359}, {12111, 32392}, {12239, 12970}, {12240, 12964}, {12294, 23292}, {12315, 37481}, {12324, 15043}, {12824, 36201}, {13402, 14810}, {13417, 15647}, {13598, 17845}, {14787, 23329}, {15126, 26913}, {15311, 34664}, {15644, 17821}, {21849, 34751}, {21851, 41424}, {21969, 34750}, {29012, 34659}, {29181, 34658}, {35450, 40280}, {38356, 40938}

X(41580) = midpoint of X(i) and X(j) for these {i, j}: {568, 32063}, {3060, 11206}, {5656, 5890}, {21969, 34750}
X(41580) = reflection of X(i) in X(j) for these (i, j): (1853, 5943), (3917, 10192), (10606, 16836), (34751, 21849)
X(41580) = barycentric product X(i)*X(j) for these {i, j}: {22, 13567}, {185, 17907}, {235, 20806}, {315, 800}, {774, 1760}, {1624, 33294}
X(41580) = barycentric quotient X(i)/X(j) for these (i, j): (22, 801), (185, 14376), (315, 40830), (800, 66)
X(41580) = trilinear product X(i)*X(j) for these {i, j}: {22, 774}, {206, 17858}, {800, 1760}
X(41580) = trilinear quotient X(i)/X(j) for these (i, j): (22, 775), (774, 66), (800, 2156), (1760, 801)
X(41580) = intersection, other than A,B,C, of conics {{A, B, C, X(22), X(235)}} and {{A, B, C, X(185), X(14542)}}
X(41580) = X(i)-isoconjugate-of-X(j) for these {i, j}: {66, 775}, {801, 2156}
X(41580) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (22, 801), (185, 14376), (315, 40830), (800, 66)
X(41580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (185, 2883, 36982), (2883, 13567, 41602), (2883, 41589, 185), (6225, 10574, 31978), (13567, 41602, 41603)

X(41581) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(2) TO PEDAL OF X(21)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a^3-(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : :
X(41581) = 2*X(960)+X(1829) = X(960)+2*X(41591) = 4*X(960)-X(41600) = 4*X(1125)-X(18732) = X(1829)-4*X(41591) = 2*X(1829)+X(41600) = 5*X(25917)-2*X(37613) = 8*X(41591)+X(41600)

The reciprocal orthologic center of these triangles is X(41582)

X(41581) lies on these lines: {1, 40964}, {2, 3827}, {51, 518}, {65, 1722}, {72, 3966}, {169, 5452}, {210, 381}, {238, 1762}, {354, 3167}, {429, 960}, {614, 24476}, {908, 40635}, {942, 1203}, {1001, 17441}, {1125, 18732}, {1386, 5320}, {1824, 24703}, {1828, 5794}, {1992, 3873}, {2262, 37658}, {2836, 3742}, {3434, 20927}, {3666, 20967}, {3869, 14555}, {3878, 4104}, {4228, 41582}, {4914, 34790}, {5813, 36844}, {9895, 12047}, {10914, 30615}, {17447, 28364}, {17720, 40962}, {18607, 20470}, {24789, 40961}, {25917, 37613}

X(41581) = barycentric product X(i)*X(j) for these {i, j}: {169, 4357}, {960, 37800}, {1193, 20927}, {1211, 4228}, {1486, 20911}, {1829, 28420}
X(41581) = barycentric quotient X(i)/X(j) for these (i, j): (169, 1220), (1486, 2298)
X(41581) = trilinear product X(i)*X(j) for these {i, j}: {169, 3666}, {960, 34036}, {1193, 3434}, {1486, 4357}
X(41581) = trilinear quotient X(i)/X(j) for these (i, j): (169, 2298), (1193, 3433)
X(41581) = X(1220)-isoconjugate-of-X(3433)
X(41581) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (169, 1220), (1486, 2298)
X(41581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (960, 1829, 41600), (960, 41591, 1829), (960, 41611, 41605), (41605, 41611, 41607)

X(41582) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(2)

Barycentrics    a*(a+b)*(a+c)*(b^2+c^2)*(a^3-(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : :

The reciprocal orthologic center of these triangles is X(41581)

X(41582) lies on these lines: {6, 18165}, {21, 3827}, {58, 24476}, {141, 427}, {511, 6841}, {518, 18180}, {1634, 16696}, {2393, 26543}, {2836, 16164}, {3242, 18178}, {4228, 41581}, {7289, 17194}, {9004, 41610}, {34381, 41608}

X(41582) = barycentric product X(i)*X(j) for these {i, j}: {141, 4228}, {169, 16887}, {1486, 16703}, {1634, 26546}
X(41582) = barycentric quotient X(i)/X(j) for these (i, j): (169, 18082), (1486, 18098)
X(41582) = trilinear product X(i)*X(j) for these {i, j}: {38, 4228}, {169, 16696}, {1486, 16887}, {1634, 21185}
X(41582) = trilinear quotient X(169)/X(18098)
X(41582) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (169, 18082), (1486, 18098)

X(41583) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(2)

Barycentrics    (b^2+c^2)*(4*a^6-(b^2+c^2)*a^4-2*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41583) = 5*X(3618)-3*X(11416) = 3*X(15360)-X(41617)

The reciprocal orthologic center of these triangles is X(15303)

X(41583) lies on these lines: {6, 21970}, {67, 11559}, {69, 15107}, {110, 27085}, {113, 511}, {125, 8262}, {141, 427}, {193, 206}, {524, 1495}, {542, 3581}, {599, 3818}, {1533, 2781}, {1632, 14712}, {1634, 7813}, {2080, 41359}, {2393, 3580}, {2854, 41586}, {2930, 5898}, {3618, 11416}, {5095, 32217}, {5449, 11663}, {5640, 16511}, {5648, 19140}, {5972, 10510}, {6593, 32218}, {7426, 15303}, {7495, 12039}, {7579, 24206}, {9971, 10254}, {13169, 37901}, {13340, 40107}, {15360, 41617}, {20304, 32246}, {20850, 20987}, {21637, 32455}, {29317, 32257}, {32267, 34319}, {34507, 37494}

X(41583) = midpoint of X(i) and X(j) for these {i, j}: {69, 15107}, {13169, 37901}
X(41583) = reflection of X(i) in X(j) for these (i, j): (6, 32223), (125, 8262), (5095, 32217), (5181, 32113), (6593, 32218), (10510, 5972), (15303, 7426), (34319, 32267)
X(41583) = barycentric product X(141)*X(7426)
X(41583) = barycentric quotient X(39)/X(5505)
X(41583) = trilinear product X(38)*X(7426)
X(41583) = trilinear quotient X(38)/X(5505)
X(41583) = X(82)-isoconjugate-of-X(5505)
X(41583) = X(39)-reciprocal conjugate of-X(5505)
X(41583) = {X(16789), X(41585)}-harmonic conjugate of X(29959)

X(41584) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(24) TO PEDAL OF X(2)

Barycentrics    (3*a^2-b^2-c^2)*(b^2+c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

The reciprocal orthologic center of these triangles is X(3167)

X(41584) lies on these lines: {2, 12167}, {3, 39871}, {4, 3620}, {5, 6403}, {6, 468}, {24, 3564}, {25, 69}, {49, 1353}, {125, 15583}, {140, 39588}, {141, 427}, {159, 21213}, {184, 15585}, {193, 3167}, {235, 511}, {237, 41005}, {297, 3186}, {317, 460}, {343, 14913}, {403, 21850}, {419, 27377}, {428, 599}, {524, 1974}, {550, 41464}, {1112, 5181}, {1204, 1503}, {1350, 1885}, {1351, 3542}, {1352, 3575}, {1368, 12220}, {1495, 34774}, {1593, 10519}, {1595, 13340}, {1596, 23039}, {1899, 9924}, {1906, 12294}, {1907, 40107}, {3147, 5050}, {3148, 41008}, {3416, 12135}, {3515, 6776}, {3517, 11898}, {3580, 12272}, {3581, 37458}, {3589, 8541}, {3618, 37453}, {3619, 5094}, {3631, 10301}, {3933, 27369}, {4232, 20080}, {4437, 24814}, {5064, 21356}, {5847, 11363}, {6240, 39884}, {6393, 11325}, {6467, 13567}, {6515, 19588}, {6620, 32001}, {6677, 26206}, {6756, 37494}, {7505, 18583}, {7507, 40330}, {7667, 15812}, {8901, 19197}, {9822, 37439}, {9973, 23300}, {10018, 38110}, {10151, 31670}, {10154, 19121}, {10192, 21637}, {10477, 37376}, {10516, 23047}, {11245, 19459}, {11405, 38282}, {11574, 30739}, {12017, 35486}, {12061, 20300}, {12143, 14994}, {12283, 26879}, {12828, 32114}, {13171, 32241}, {15647, 32285}, {15750, 25406}, {15809, 37636}, {16285, 36879}, {16868, 38136}, {18440, 18533}, {18907, 37912}, {18935, 26869}, {18947, 32240}, {19119, 26864}, {19129, 34351}, {20897, 40996}, {20960, 23172}, {21841, 34380}, {32263, 40337}, {34381, 41609}, {34382, 41587}, {37460, 39874}, {40317, 40318}, {40995, 41266}

X(41584) = midpoint of X(40317) and X(40318)
X(41584) = polar conjugate of the isogonal conjugate of X(3787)
X(41584) = barycentric product X(i)*X(j) for these {i, j}: {141, 6353}, {193, 427}, {264, 3787}, {1235, 3053}, {1707, 20883}
X(41584) = barycentric quotient X(i)/X(j) for these (i, j): (39, 6391), (141, 6340), (193, 1799), (427, 2996), (1707, 34055), (1843, 8770)
X(41584) = trilinear product X(i)*X(j) for these {i, j}: {38, 6353}, {92, 3787}, {193, 17442}, {427, 1707}
X(41584) = trilinear quotient X(i)/X(j) for these (i, j): (38, 6391), (193, 34055), (427, 8769), (1707, 1176), (1843, 38252)
X(41584) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(29959)}} and {{A, B, C, X(141), X(193)}}
X(41584) = X(i)-isoconjugate-of-X(j) for these {i, j}: {82, 6391}, {1176, 8769}
X(41584) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (39, 6391), (141, 6340), (193, 1799), (427, 2996)
X(41584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (141, 1843, 427), (141, 41585, 1843), (193, 6353, 19118), (12220, 26156, 1368)

X(41585) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(2)

Barycentrics    (5*a^2-b^2-c^2)*(b^2+c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

The reciprocal orthologic center of these triangles is X(1992)

X(41585) lies on these lines: {4, 599}, {6, 6353}, {24, 8550}, {25, 524}, {53, 3186}, {67, 6240}, {69, 6995}, {141, 427}, {182, 37935}, {193, 35264}, {235, 5181}, {343, 11188}, {403, 5480}, {419, 6749}, {428, 22165}, {468, 597}, {511, 1596}, {542, 37458}, {576, 21841}, {1352, 18494}, {1503, 10605}, {1595, 40107}, {1634, 27369}, {1974, 3629}, {1992, 4232}, {2393, 13567}, {2854, 12828}, {2883, 37473}, {2930, 3518}, {3089, 11477}, {3589, 12167}, {3620, 7409}, {3627, 32257}, {5094, 20582}, {5476, 37942}, {6676, 9813}, {6677, 11511}, {6756, 34507}, {7378, 21356}, {7487, 15069}, {7714, 15533}, {8681, 41588}, {8889, 21358}, {9004, 41611}, {9973, 15583}, {12061, 20303}, {12241, 34787}, {15152, 19149}, {15448, 19153}, {19118, 32455}, {19124, 21167}, {20192, 37962}, {20987, 34774}, {21213, 35707}, {26156, 31101}, {32269, 41614}, {37931, 39871}

X(41585) = midpoint of X(69) and X(33586)
X(41585) = reflection of X(11511) in X(6677)
X(41585) = barycentric product X(i)*X(j) for these {i, j}: {141, 4232}, {427, 1992}, {1235, 1384}, {1843, 11059}
X(41585) = barycentric quotient X(i)/X(j) for these (i, j): (427, 5485), (1384, 1176), (1499, 4580), (1843, 21448)
X(41585) = trilinear product X(i)*X(j) for these {i, j}: {38, 4232}, {427, 36277}, {1384, 20883}
X(41585) = trilinear quotient X(1992)/X(34055)
X(41585) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (427, 5485), (1384, 1176), (1499, 4580)
X(41585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (141, 1843, 3867), (468, 8541, 597), (1843, 41584, 141), (16789, 29959, 141), (29959, 41583, 16789)

X(41586) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(4)

Barycentrics    (2*a^2-b^2-c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(41586) = 4 X[5] - 3 X[1568], X[23] - 3 X[15360], 3 X[110] - 5 X[37760], 3 X[125] - 2 X[858], X[382] + 3 X[32608], 4 X[468] - 3 X[5642], 2 X[468] - 3 X[32225], 5 X[631] - 3 X[43574], X[858] - 3 X[3580], 3 X[1495] - 4 X[37897], 3 X[1511] - 4 X[22249], 3 X[2070] - X[23236], 2 X[3292] - 3 X[5642], X[3292] - 3 X[32225], 3 X[3448] + X[20063], 7 X[3526] - 6 X[14156], 4 X[5159] - 3 X[13857], X[5189] - 3 X[9140], 2 X[10564] - 3 X[38727], 3 X[11064] - 4 X[37911], 3 X[13399] - 4 X[16003], 3 X[15061] - X[37496], 3 X[15107] - X[20063], 4 X[15122] - 5 X[38729], 3 X[15360] + X[41724], 3 X[15361] - 2 X[18571], 3 X[20126] - X[35001], 4 X[20396] - 3 X[37938], 3 X[22151] - 4 X[32300], X[24981] - 4 X[32269], 3 X[24981] - 8 X[37897], 6 X[32223] - 5 X[37760], 2 X[32223] + X[37779], 3 X[32269] - 2 X[37897], 5 X[37760] + 3 X[37779]

The reciprocal orthologic center of these triangles is X(5095)

See Antreas Hatzipolakis and Peter Moses, euclid 1735.

X(41586) lies on these lines: {2, 576}, {3, 43573}, {5, 51}, {6, 16511}, {20, 1204}, {23, 542}, {25, 15069}, {30, 6070}, {69, 5651}, {110, 5965}, {115, 20977}, {125, 511}, {141, 373}, {182, 37644}, {184, 6515}, {237, 14981}, {323, 5972}, {340, 450}, {382, 9927}, {427, 21969}, {468, 524}, {530, 37975}, {531, 37974}, {532, 32460}, {533, 32461}, {538, 5112}, {539, 2070}, {541, 18325}, {548, 12370}, {575, 7495}, {578, 631}, {599, 11284}, {648, 41203}, {754, 1316}, {826, 32312}, {852, 15526}, {895, 10416}, {1205, 9019}, {1350, 26869}, {1351, 37638}, {1352, 34417}, {1353, 13394}, {1493, 34577}, {1495, 3564}, {1503, 37899}, {1506, 13410}, {1511, 22249}, {1533, 5663}, {1994, 41594}, {1995, 34507}, {2081, 2600}, {2777, 41738}, {2781, 41603}, {2836, 41607}, {2854, 41583}, {2937, 10116}, {3060, 5169}, {3098, 18911}, {3231, 6388}, {3291, 6791}, {3448, 15107}, {3518, 41598}, {3519, 13621}, {3526, 10601}, {3528, 18950}, {3567, 14789}, {3581, 17702}, {3917, 13567}, {4121, 7796}, {5012, 11225}, {5094, 11477}, {5097, 14389}, {5133, 21849}, {5159, 13857}, {5189, 9140}, {5449, 6243}, {5477, 14567}, {5609, 25338}, {5611, 40709}, {5615, 40710}, {5640, 24206}, {5648, 16510}, {5650, 37648}, {5943, 37636}, {6090, 40341}, {6676, 13366}, {6689, 14627}, {6698, 12099}, {6699, 37477}, {6776, 35268}, {7464, 20417}, {7488, 10112}, {7519, 11442}, {7574, 36253}, {7575, 30714}, {7765, 20859}, {7794, 37338}, {7813, 9155}, {9027, 32113}, {9143, 32267}, {9225, 10418}, {9698, 20965}, {9705, 10274}, {9714, 9908}, {10510, 15118}, {10564, 38727}, {10754, 19577}, {10984, 18951}, {10991, 37916}, {11002, 19130}, {11061, 27085}, {11064, 34380}, {11245, 22352}, {11411, 26883}, {11645, 37900}, {11799, 13754}, {11898, 21970}, {12325, 43598}, {12827, 32271}, {13160, 16625}, {13352, 18580}, {13391, 20379}, {13403, 35240}, {13431, 18282}, {13434, 32348}, {13564, 18128}, {14449, 34826}, {14831, 15760}, {14918, 35360}, {15061, 37496}, {15122, 38729}, {15139, 32235}, {15361, 18571}, {15605, 38848}, {15644, 26879}, {16163, 32110}, {16311, 32515}, {16789, 40673}, {16982, 33332}, {18947, 19128}, {20126, 35001}, {20191, 37495}, {20396, 37938}, {20427, 32064}, {21284, 32233}, {22151, 32300}, {23293, 31857}, {25335, 32262}, {32306, 37972}, {34002, 34564}, {34004, 36153}, {34565, 37649}, {34986, 41628}, {35500, 40240}, {37453, 37672}, {37804, 39099}, {41617, 41721}

X(41586) = midpoint of X(i) and X(j) for these {i,j}: {23, 41724}, {110, 37779}, {3448, 15107}, {41617, 41721}
X(41586) = reflection of X(i) in X(j) for these {i,j}: {110, 32223}, {125, 3580}, {323, 5972}, {1495, 32269}, {3292, 468}, {5181, 8262}, {5609, 25338}, {5642, 32225}, {7464, 20417}, {7574, 36253}, {9143, 32267}, {10510, 15118}, {15063, 11799}, {15137, 6689}, {16163, 32110}, {24981, 1495}, {30714, 7575}, {32114, 32113}, {37477, 6699}
X(41586) = complement of X(23061)
X(41586) = X(5467)-Ceva conjugate of X(690)
X(41586) = crosssum of X(i) and X(j) for these (i, j): {6, 39231}, {111, 14908}
X(41586) = crossdifference of every pair of points on line {X(54), X(2623)}
X(41586) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 671), (51, 111), (53, 17983), (187, 54)
X(41586) = X(524)-Waw conjugate of-X(7813)
X(41586) = Johnson-circumconic-inverse of X(51)
X(41586) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(468)}} and {{A, B, C, X(52), X(187)}}
X(41586) = X(i)-isoconjugate of X(j) for these (i,j): {54, 897}, {95, 923}, {97, 36128}, {111, 2167}, {275, 36060}, {671, 2148}, {691, 2616}, {895, 2190}, {2169, 17983}, {2623, 36085}, {5466, 36134}, {14908, 40440}, {15412, 36142}, {18315, 23894}
X(41586) = trilinear product X(i)*X(j) for these {i, j}: {5, 896}, {51, 14210}, {187, 14213}, {311, 922}, {524, 1953}, {690, 2617}
X(41586) = trilinear quotient X(i)/X(j) for these (i, j): (5, 897), (51, 923), (53, 36128), (187, 2148), (216, 36060), (468, 2190)
X(41586) = barycentric product X(i)*X(j) for these {i,j}: {5, 524}, {51, 3266}, {53, 6390}, {187, 311}, {324, 3292}, {343, 468}, {690, 14570}, {896, 14213}, {1154, 43084}, {1625, 35522}, {1953, 14210}, {2618, 23889}, {4062, 17167}, {4235, 6368}, {5467, 18314}, {5468, 12077}, {5562, 37778}, {6629, 21011}, {7813, 17500}, {14417, 35360}, {14559, 41078}, {16741, 21807}, {18180, 42713}, {27364, 32459}
X(41586) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 671}, {51, 111}, {53, 17983}, {187, 54}, {216, 895}, {217, 14908}, {311, 18023}, {343, 30786}, {351, 2623}, {468, 275}, {524, 95}, {690, 15412}, {896, 2167}, {922, 2148}, {1625, 691}, {1648, 8901}, {1953, 897}, {2081, 9213}, {2179, 923}, {2181, 36128}, {2617, 36085}, {2642, 2616}, {3199, 8753}, {3266, 34384}, {3292, 97}, {4235, 18831}, {5467, 18315}, {6368, 14977}, {6390, 34386}, {12077, 5466}, {14570, 892}, {15451, 10097}, {22105, 39182}, {23200, 14533}, {27374, 41272}, {32225, 4993}, {35319, 36827}, {37778, 8795}, {40981, 32740}
X(41586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15019, 25555}, {343, 41588, 51}, {468, 3292, 5642}, {3060, 38397, 5169}, {3292, 32225, 468}, {5169, 38397, 21243}, {7488, 10112, 10619}, {11898, 21970, 35259}, {14918, 35360, 39569}, {15360, 41724, 23}

X(41587) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(24) TO PEDAL OF X(4)

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-3*(b^2+c^2)*a^4+(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(155)

X(41587) lies on these lines: {2, 36747}, {3, 12241}, {4, 3580}, {5, 51}, {6, 3549}, {20, 26879}, {22, 18912}, {23, 34224}, {25, 68}, {26, 6146}, {30, 1204}, {49, 10192}, {113, 16879}, {125, 23335}, {140, 13142}, {141, 1351}, {155, 3542}, {156, 32358}, {182, 34002}, {184, 13292}, {185, 23331}, {195, 3629}, {235, 13754}, {265, 41362}, {381, 15873}, {382, 6247}, {389, 15760}, {403, 5889}, {427, 5446}, {454, 1609}, {467, 8800}, {468, 1147}, {511, 11585}, {550, 18555}, {568, 10024}, {569, 6676}, {578, 7542}, {858, 26917}, {1092, 16238}, {1181, 18951}, {1199, 7552}, {1211, 6861}, {1352, 7529}, {1353, 21637}, {1368, 10625}, {1498, 18917}, {1503, 7517}, {1594, 3060}, {1596, 12162}, {1658, 12370}, {1829, 12259}, {1885, 7689}, {1899, 7387}, {1974, 3564}, {1993, 7505}, {2070, 9920}, {2072, 6243}, {2883, 11799}, {2888, 13595}, {3089, 11411}, {3090, 37636}, {3133, 8905}, {3448, 16659}, {3515, 12118}, {3517, 12429}, {3518, 14516}, {3546, 37483}, {3547, 11433}, {3548, 26958}, {3567, 13160}, {3575, 9927}, {3581, 18563}, {3627, 11572}, {5097, 12242}, {5133, 9781}, {5410, 19061}, {5411, 19062}, {5422, 7558}, {5447, 30739}, {5462, 7399}, {5480, 5576}, {5640, 14788}, {5654, 12160}, {5943, 7405}, {6102, 15761}, {6193, 6353}, {6639, 23292}, {6643, 37486}, {6756, 18474}, {6759, 37971}, {6823, 9730}, {6825, 26540}, {6887, 32782}, {6958, 26005}, {7383, 15805}, {7403, 10110}, {7426, 26882}, {7488, 12022}, {7526, 16657}, {7528, 17810}, {7530, 16655}, {7545, 32599}, {7553, 18381}, {7592, 37644}, {7691, 35254}, {9140, 34613}, {9714, 9833}, {9914, 14216}, {10018, 34148}, {10055, 11399}, {10071, 11398}, {10111, 15647}, {10112, 10282}, {10154, 31804}, {10201, 12161}, {10224, 14449}, {10257, 13346}, {10263, 13371}, {10264, 10990}, {10323, 18911}, {10594, 11442}, {10620, 15105}, {10982, 37638}, {10984, 16618}, {11064, 16266}, {11414, 26869}, {12085, 26937}, {12293, 18533}, {12362, 37478}, {12605, 18390}, {12828, 25711}, {13336, 16197}, {13367, 32225}, {13394, 32046}, {13421, 20304}, {13568, 37490}, {13598, 20299}, {14070, 19467}, {14363, 39569}, {14530, 39899}, {14576, 41523}, {14790, 33586}, {15311, 31725}, {15559, 23293}, {16003, 30443}, {16252, 18445}, {16625, 18388}, {17834, 18531}, {20192, 23410}, {21213, 32048}, {21451, 37779}, {23291, 34938}, {23324, 31724}, {26932, 37532}, {26944, 39568}, {34382, 41584}, {34777, 38260}, {37452, 37484}, {37826, 39585}, {37943, 41628}, {38848, 41171}

X(41587) = midpoint of X(7517) and X(25738)
X(41587) = reflection of X(i) in X(j) for these (i, j): (1092, 16238), (10539, 21841)
X(41587) = barycentric product X(i)*X(j) for these {i, j}: {5, 6515}, {52, 39116}, {53, 40697}, {155, 324}, {311, 1609}, {343, 3542}
X(41587) = barycentric quotient X(i)/X(j) for these (i, j): (5, 6504), (53, 254), (155, 97), (216, 15316), (920, 2167), (1609, 54)
X(41587) = trilinear product X(i)*X(j) for these {i, j}: {5, 920}, {51, 33808}, {1609, 14213}
X(41587) = trilinear quotient X(i)/X(j) for these (i, j): (5, 921), (155, 2169), (920, 54), (1087, 8800), (1609, 2148)
X(41587) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(3542)}} and {{A, B, C, X(52), X(53)}}
X(41587) = X(467)-Ceva conjugate of-X(53)
X(41587) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 921}, {254, 2169}
X(41587) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 6504), (53, 254), (155, 97), (216, 15316)
X(41587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3580, 12359), (5, 41588, 52), (25, 68, 12134), (140, 13142, 13352), (403, 5889, 22660), (568, 10024, 12233), (1993, 7505, 9820), (3089, 11411, 18451), (3542, 6515, 155), (3547, 11433, 36752), (5446, 5449, 427), (6146, 32269, 26), (6639, 36749, 23292), (7530, 32140, 16655), (10110, 21243, 7403), (10112, 32223, 10282), (11799, 34783, 2883), (12429, 21970, 3517), (13292, 13383, 184), (26958, 37498, 3548)

X(41588) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(4)

Barycentrics    (3*a^2-b^2-c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(41588) = X(1370)-3*X(26869)

The reciprocal orthologic center of these triangles is X(193)

X(41588) lies on these lines: {2, 1351}, {3, 11433}, {4, 11469}, {5, 51}, {6, 6676}, {20, 18950}, {22, 11245}, {25, 3564}, {26, 13292}, {30, 1899}, {53, 27364}, {68, 6756}, {69, 5020}, {110, 41628}, {125, 21969}, {140, 10601}, {141, 5943}, {154, 37897}, {155, 21841}, {184, 1353}, {193, 3167}, {235, 5889}, {237, 23158}, {297, 3168}, {324, 6755}, {372, 8964}, {382, 12250}, {389, 6823}, {394, 6677}, {419, 7762}, {427, 3060}, {428, 11442}, {436, 27377}, {465, 5611}, {466, 5615}, {467, 14569}, {468, 1993}, {511, 1368}, {524, 8263}, {549, 13352}, {568, 15760}, {576, 23292}, {1112, 12827}, {1350, 10691}, {1352, 17810}, {1370, 26869}, {1589, 12313}, {1590, 12314}, {1595, 5446}, {1596, 13754}, {1598, 11411}, {1853, 31670}, {1906, 12111}, {2895, 4223}, {2979, 30739}, {3053, 8940}, {3066, 10128}, {3089, 12164}, {3448, 34603}, {3517, 6193}, {3518, 22550}, {3526, 5644}, {3527, 7404}, {3542, 12160}, {3547, 11432}, {3549, 37493}, {3567, 7399}, {3627, 11550}, {3629, 10192}, {3787, 6388}, {3917, 37648}, {3933, 11328}, {4232, 8780}, {5012, 15360}, {5050, 7494}, {5092, 32068}, {5093, 11427}, {5133, 11002}, {5159, 11477}, {5422, 7499}, {5480, 21243}, {5640, 37439}, {5654, 37942}, {5739, 25514}, {5907, 15873}, {5921, 7714}, {6243, 11585}, {6247, 13598}, {6638, 41005}, {6688, 40107}, {6776, 9909}, {6995, 18440}, {6997, 18358}, {7386, 33878}, {7387, 18910}, {7426, 9544}, {7487, 12429}, {7493, 11402}, {7495, 34545}, {7542, 36749}, {7553, 25738}, {7561, 36750}, {7667, 18911}, {7715, 12134}, {8550, 11225}, {8681, 41585}, {8703, 32110}, {9786, 31829}, {9861, 39804}, {9919, 18932}, {10112, 34782}, {10263, 23335}, {10519, 16419}, {10565, 14912}, {11206, 20850}, {11414, 18916}, {11484, 11487}, {11548, 14561}, {11898, 14826}, {12161, 13383}, {12163, 13488}, {12233, 16625}, {12295, 33699}, {12310, 18947}, {12362, 17834}, {12828, 14984}, {13175, 39833}, {13321, 37347}, {13366, 13394}, {13371, 14449}, {13470, 15704}, {13561, 16982}, {13595, 37779}, {13861, 31831}, {15004, 37649}, {15048, 20859}, {15108, 16042}, {15534, 32113}, {15818, 37488}, {16195, 18925}, {16196, 37498}, {16197, 36752}, {16238, 16266}, {16976, 37497}, {16981, 31074}, {18445, 37971}, {18534, 18917}, {18909, 39568}, {18918, 34725}, {20965, 31406}, {23291, 34609}, {26005, 37521}, {26906, 30258}, {26944, 34938}, {28383, 41014}, {30771, 37643}, {32358, 37440}, {34002, 36753}, {34381, 41611}, {37453, 37645}

X(41588) = midpoint of X(i) and X(j) for these {i, j}: {25, 6515}, {1899, 33586}, {18534, 18917}
X(41588) = reflection of X(i) in X(j) for these (i, j): (394, 6677), (1368, 13567)
X(41588) = barycentric product X(i)*X(j) for these {i, j}: {5, 193}, {53, 6337}, {311, 3053}, {324, 3167}, {343, 6353}, {439, 27364}
X(41588) = barycentric quotient X(i)/X(j) for these (i, j): (5, 2996), (51, 8770), (53, 34208), (193, 95), (216, 6391), (217, 40319)
X(41588) = trilinear product X(i)*X(j) for these {i, j}: {5, 1707}, {51, 18156}, {193, 1953}
X(41588) = trilinear quotient X(i)/X(j) for these (i, j): (5, 8769), (51, 38252), (193, 2167), (1087, 27364), (1707, 54)
X(41588) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(31802)}} and {{A, B, C, X(5), X(6353)}}
X(41588) = X(53)-Ceva conjugate of-X(5)
X(41588) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 8769}, {95, 38252}
X(41588) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 2996), (51, 8770), (53, 34208), (193, 95)
X(41588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 9777, 18583), (5, 52, 31802), (22, 37644, 11245), (26, 13292, 31804), (51, 343, 5), (51, 41586, 343), (52, 41587, 5), (184, 32269, 10154), (193, 6353, 3167), (427, 3060, 21850), (428, 11442, 39884), (467, 35360, 14569), (1353, 10154, 184), (3060, 3580, 427), (3167, 21970, 6353), (3629, 10192, 34986), (3787, 6388, 40326), (5446, 12359, 1595), (8940, 8944, 3053), (17834, 39571, 12362), (20850, 39899, 11206), (21243, 21849, 5480), (32223, 34986, 10192)

X(41589) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(20)

Barycentrics    a^2*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^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)) : :
X(41589) = 3*X(2)+X(6293) = 3*X(51)-X(41362) = X(64)-5*X(10574) = 3*X(154)+X(5889) = 3*X(185)+X(36982) = X(185)+3*X(41580) = 3*X(568)+X(9833) = X(1498)+3*X(5890) = 3*X(1853)-7*X(15043) = 3*X(2883)-X(36982) = X(2883)-3*X(41580) = 3*X(3060)+X(17845) = X(5562)-3*X(10192) = X(5895)+3*X(15072) = 3*X(5946)-X(18381) = X(6101)-3*X(11202) = X(6225)+3*X(7729) = X(6247)-3*X(9730) = 3*X(9730)-2*X(32184) = X(36982)-9*X(41580)

The reciprocal orthologic center of these triangles is X(41590)

X(41589) lies on these lines: {2, 6293}, {3, 1177}, {5, 32364}, {51, 11743}, {52, 3629}, {64, 10574}, {66, 9815}, {110, 17824}, {143, 11262}, {154, 5889}, {156, 20773}, {185, 235}, {389, 1503}, {546, 5462}, {568, 9833}, {973, 7576}, {974, 6241}, {1112, 21659}, {1154, 10282}, {1498, 5890}, {1539, 13491}, {1614, 1986}, {1853, 7566}, {2070, 32379}, {2393, 16625}, {2777, 11561}, {3060, 17845}, {3589, 6696}, {5449, 5663}, {5562, 10192}, {5895, 15072}, {5907, 32392}, {5944, 10274}, {5946, 18381}, {6101, 11202}, {6102, 6759}, {6225, 7729}, {6247, 7403}, {7401, 34118}, {7488, 32391}, {8567, 20791}, {9781, 18405}, {9786, 19149}, {10095, 18383}, {10182, 32142}, {10263, 34785}, {10272, 10628}, {11412, 17821}, {12006, 20299}, {12111, 37638}, {13363, 32767}, {13383, 13754}, {13434, 32345}, {13861, 40285}, {14216, 37481}, {14862, 25338}, {14865, 15138}, {15026, 23325}, {15045, 40686}, {15311, 40647}, {15577, 17834}, {15578, 37515}, {15582, 31810}, {15738, 16868}, {16223, 23315}, {19161, 34774}, {31728, 40658}, {31732, 40660}, {32401, 34149}

X(41589) = midpoint of X(i) and X(j) for these {i, j}: {52, 34782}, {185, 2883}, {1986, 15647}, {5907, 32392}, {6102, 6759}, {10263, 34785}, {13289, 38898}, {13491, 22802}, {19161, 34774}, {31728, 40658}, {31732, 40660}
X(41589) = reflection of X(i) in X(j) for these (i, j): (6247, 32184), (6696, 9729), (18383, 10095), (20299, 12006)
X(41589) = complement of the complement of X(6293)
X(41589) = barycentric quotient X(i)/X(j) for these (i, j): (800, 6145), (1624, 16039)
X(41589) = trilinear product X(774)*X(7488)
X(41589) = trilinear quotient X(774)/X(6145)
X(41589) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(41603)}} and {{A, B, C, X(235), X(1177)}}
X(41589) = X(775)-isoconjugate-of-X(6145)
X(41589) = X(800)-reciprocal conjugate of-X(6145)
X(41589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (185, 41580, 2883), (389, 11745, 32191), (6247, 9730, 32184), (10274, 13289, 5944)

X(41590) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(20) TO PEDAL OF X(5)

Barycentrics    a^2*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(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)) : :
X(41590) = 3*X(51)-4*X(32396) = 3*X(1209)-2*X(6153) = 4*X(1209)-3*X(41578) = 4*X(1216)-X(6152) = 3*X(1216)-X(6153) = 8*X(1216)-3*X(41578) = 3*X(2979)-X(7691) = 3*X(2979)+X(32338) = 5*X(3091)-4*X(11743) = 3*X(3917)-2*X(32348) = 3*X(3917)-X(32352) = 2*X(6101)+X(12606) = 3*X(6152)-4*X(6153) = 2*X(6152)-3*X(41578) = 8*X(6153)-9*X(41578) = X(6288)-3*X(23039) = 2*X(10625)+X(12300) = X(11412)+2*X(12363) = 5*X(11444)-2*X(11576) = 3*X(32391)-2*X(41589)

The reciprocal orthologic center of these triangles is X(41589)

X(41590) lies on these lines: {2, 973}, {3, 54}, {20, 32330}, {22, 32379}, {51, 32396}, {52, 6689}, {69, 1225}, {70, 3519}, {159, 11441}, {343, 32351}, {394, 2917}, {511, 3574}, {858, 15606}, {1092, 23358}, {1209, 1216}, {1350, 17824}, {1370, 32337}, {3091, 9971}, {3153, 32369}, {3917, 32348}, {5447, 11802}, {5562, 12225}, {5907, 32340}, {5965, 6467}, {6193, 12254}, {6240, 10625}, {6242, 37119}, {6288, 18569}, {6403, 7507}, {7386, 32334}, {7488, 32391}, {7495, 12242}, {7568, 8254}, {7730, 7999}, {9630, 13079}, {9967, 12585}, {10628, 15644}, {11591, 22804}, {11793, 11808}, {12318, 12325}, {12824, 13383}, {13371, 21230}, {13565, 15067}, {15133, 33565}, {15800, 37484}, {16063, 18910}, {20806, 32344}, {32368, 41614}, {36752, 40913}

X(41590) = midpoint of X(i) and X(j) for these {i, j}: {54, 11412}, {2888, 12226}, {7691, 32338}, {12307, 22815}, {15800, 37484}
X(41590) = reflection of X(i) in X(j) for these (i, j): (52, 6689), (54, 12363), (1209, 1216), (6152, 1209), (11802, 5447), (11808, 11793), (22804, 11591), (32196, 13565), (32340, 5907), (32352, 32348)
X(41590) = anticomplement of X(973)
X(41590) = barycentric quotient X(570)/X(6145)
X(41590) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(1209)}} and {{A, B, C, X(54), X(1594)}}
X(41590) = X(69)-Ceva conjugate of-X(37636)
X(41590) = X(570)-reciprocal conjugate of-X(6145)
X(41590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (69, 32346, 2888), (1209, 6152, 41578), (2888, 37444, 6145), (2979, 32338, 7691), (3917, 32352, 32348), (15067, 32196, 13565)

X(41591) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(21)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a^4-b*c*a^2+(b+c)*b*c*a-(b^2-c^2)^2) : :
X(41591) = X(960)-3*X(41581) = 3*X(960)-X(41600) = X(1829)+3*X(41581) = 3*X(1829)+X(41600) = 3*X(3742)-X(18732) = 9*X(41581)-X(41600)

The reciprocal orthologic center of these triangles is X(41592)

X(41591) lies on these lines: {65, 32911}, {429, 960}, {517, 546}, {518, 9969}, {942, 2836}, {1539, 2778}, {2771, 11561}, {3589, 3812}, {3629, 3874}, {3742, 18732}, {3838, 9895}, {3869, 37656}, {6001, 40647}, {6583, 41597}, {8286, 12047}, {15254, 41340}, {16547, 22122}

X(41591) = midpoint of X(960) and X(1829)
X(41591) = barycentric product X(1193)*X(20919)
X(41591) = trilinear product X(1848)*X(22122)
X(41591) = trilinear quotient X(1193)/X(34441)
X(41591) = X(1220)-isoconjugate-of-X(34441)
X(41591) = {X(1829), X(41581)}-harmonic conjugate of X(960)

X(41592) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(5)

Barycentrics    a*(a+b)*(a+c)*(a^4-b*c*a^2+(b+c)*b*c*a-(b^2-c^2)^2)*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41591)

X(41592) lies on these lines: {1154, 6841}, {1209, 1216}, {4219, 7691}

X(41593) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(22)

Barycentrics    a^2*(2*a^6-(b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41593) = 5*X(6)+3*X(154) = 3*X(6)+X(159) = 7*X(6)+X(9924) = 7*X(6)-3*X(11216) = 11*X(6)-3*X(17813) = 3*X(6)+5*X(19132) = X(6)+3*X(19153) = 5*X(6)-X(34777) = 9*X(154)-5*X(159) = 3*X(154)-5*X(206) = 7*X(154)+5*X(11216) = 11*X(154)+5*X(17813) = X(154)-5*X(19153) = 3*X(154)+X(34777) = 6*X(154)+5*X(39125) = X(159)-3*X(206) = 7*X(159)-3*X(9924) = 7*X(159)+9*X(11216) = 11*X(159)+9*X(17813) = X(159)-5*X(19132) = X(159)-9*X(19153) = 5*X(159)+3*X(34777) = 2*X(159)+3*X(39125)

The reciprocal orthologic center of these triangles is X(41594)

X(41593) lies on these lines: {6, 25}, {39, 15257}, {49, 22550}, {52, 32344}, {66, 3618}, {69, 19122}, {141, 5972}, {143, 5097}, {157, 5158}, {160, 3284}, {182, 3357}, {193, 9716}, {216, 1576}, {373, 15139}, {511, 1658}, {546, 575}, {576, 15577}, {597, 23300}, {692, 22122}, {1176, 1177}, {1351, 23041}, {1539, 36201}, {2781, 5092}, {3003, 14575}, {3292, 40341}, {3313, 19121}, {3518, 32367}, {3564, 19155}, {3589, 6697}, {3629, 10192}, {3827, 31794}, {5007, 15270}, {5012, 41580}, {5050, 12315}, {5093, 34787}, {5102, 17821}, {5157, 26206}, {5476, 18382}, {5480, 13403}, {5596, 23327}, {5650, 17847}, {5965, 41597}, {6153, 10274}, {6759, 39561}, {6776, 19123}, {7503, 32392}, {9407, 23635}, {9822, 12039}, {10169, 15583}, {11202, 37517}, {11422, 35260}, {11557, 12228}, {11574, 19127}, {11645, 18566}, {12007, 16252}, {12017, 34778}, {13417, 15140}, {13622, 32226}, {14853, 36989}, {15462, 37511}, {15578, 20190}, {15581, 22234}, {15582, 22330}, {15585, 32218}, {18472, 37808}, {19128, 19161}, {19510, 28408}, {20300, 25555}, {23333, 23583}, {31166, 36851}, {34107, 34137}, {34118, 38317}, {34782, 37505}

X(41593) = midpoint of X(i) and X(j) for these {i, j}: {6, 206}, {182, 34117}, {576, 15577}, {5097, 10282}, {12007, 16252}, {15585, 32455}, {18382, 34776}, {23300, 34774}
X(41593) = reflection of X(i) in X(j) for these (i, j): (6697, 3589), (15578, 20190), (20300, 25555), (39125, 6)
X(41593) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(34777)}} and {{A, B, C, X(54), X(32366)}}
X(41593) = crosssum of X(2) and X(31074)
X(41593) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 154, 34777), (6, 184, 32366), (6, 1974, 9969), (6, 9924, 11216), (6, 18374, 1843), (6, 19118, 19136), (6, 19132, 159), (6, 19153, 206), (6, 20987, 8541), (159, 19132, 206), (159, 19153, 19132), (576, 23042, 15577), (597, 34774, 23300), (5476, 34776, 18382), (19121, 22151, 3313)

X(41594) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(22) TO PEDAL OF X(5)

Barycentrics    ((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(2*a^6-(b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41593)

X(41594) lies on these lines: {20, 2888}, {54, 6515}, {184, 5965}, {343, 27365}, {1154, 15760}, {1209, 1216}, {1994, 41586}, {2979, 7703}, {3060, 3574}, {3519, 34439}, {5422, 6689}, {10610, 11245}, {21230, 23335}

X(41594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1209, 41599, 41578), (37636, 41578, 1209), (37636, 41596, 41578), (41578, 41596, 41599)

X(41595) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(23)

Barycentrics    (2*a^2-b^2-c^2)*(3*a^6-(b^2+c^2)*a^4-(3*b^4-5*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41595) = 3*X(2)+X(16176) = X(5)-3*X(34155) = 5*X(6)-X(3448) = 3*X(6)+X(11061) = 7*X(6)-3*X(25320) = X(6)+3*X(25321) = 3*X(6)-X(25328) = 11*X(6)-3*X(25330) = 5*X(6)+3*X(25331) = 9*X(6)-X(25335) = 7*X(6)+X(25336) = 3*X(3448)+5*X(11061) = 7*X(3448)-15*X(25320) = X(3448)+15*X(25321) = 3*X(3448)-5*X(25328) = X(3448)+5*X(25329) = 11*X(3448)-15*X(25330) = X(3448)+3*X(25331) = 9*X(3448)-5*X(25335) = 7*X(3448)+5*X(25336)

The reciprocal orthologic center of these triangles is X(41596)

X(41595) lies on these lines: {2, 16176}, {5, 9977}, {6, 3448}, {51, 32299}, {67, 597}, {110, 3629}, {125, 6329}, {141, 32244}, {468, 524}, {511, 11561}, {542, 546}, {576, 38322}, {858, 15140}, {895, 8584}, {1352, 15046}, {1353, 19140}, {1503, 1539}, {1992, 2930}, {2781, 40647}, {2854, 9969}, {3564, 25556}, {3589, 6698}, {3631, 5972}, {5032, 32255}, {5097, 32423}, {5102, 12383}, {5663, 12007}, {5965, 10272}, {6240, 32233}, {8550, 9970}, {8705, 40949}, {9769, 15491}, {9971, 32248}, {10125, 40107}, {10264, 39561}, {12241, 15063}, {13193, 13196}, {13248, 34774}, {13417, 17710}, {14763, 15118}, {15069, 16868}, {16776, 32260}, {20380, 41498}, {20582, 32257}, {34153, 37517}, {37760, 41596}

X(41595) = midpoint of X(i) and X(j) for these {i, j}: {6, 25329}, {110, 3629}, {1353, 19140}, {5095, 6593}, {8550, 9970}, {8584, 34319}, {11061, 25328}, {13248, 34774}, {13417, 17710}, {34153, 37517}
X(41595) = reflection of X(i) in X(j) for these (i, j): (125, 6329), (3631, 5972), (6698, 32300)
X(41595) = complement of the complement of X(16176)
X(41595) = barycentric product X(524)*X(37760)
X(41595) = trilinear product X(896)*X(37760)
X(41595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 11061, 25328), (6, 25321, 25329), (6, 25331, 3448), (6, 25336, 25320), (5095, 15303, 6593), (6698, 32300, 3589), (25328, 25329, 11061)

X(41596) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(5)

Barycentrics    (3*a^6-(b^2+c^2)*a^4-(3*b^4-5*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41595)

X(41596) lies on these lines: {54, 34351}, {524, 27085}, {1154, 11799}, {1209, 1216}, {1495, 5965}, {1533, 10628}, {2888, 17834}, {2917, 6515}, {3060, 13406}, {3580, 32316}, {3581, 12380}, {3627, 6288}, {7495, 9977}, {8254, 15019}, {15360, 41615}, {21230, 37484}, {32223, 32226}, {37760, 41595}

X(41596) = reflection of X(32226) in X(32223)
X(41596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41578, 41594, 37636), (41594, 41599, 41578)

X(41597) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(5) TO PEDAL OF X(24)

Barycentrics    a^2*(-a^2+b^2+c^2)*(2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)) : :
X(41597) = 3*X(2)+X(9936) = X(3)+3*X(155) = X(3)-3*X(1147) = X(3)-9*X(3167) = 5*X(3)-3*X(7689) = 2*X(3)-3*X(12038) = 7*X(3)-3*X(12163) = 5*X(3)+3*X(12164) = X(155)+3*X(3167) = 5*X(155)+X(7689) = 2*X(155)+X(12038) = 7*X(155)+X(12163) = 5*X(155)-X(12164) = 3*X(155)-X(15083) = X(1147)-3*X(3167) = 5*X(1147)-X(7689) = 7*X(1147)-X(12163) = 5*X(1147)+X(12164) = 3*X(1147)+X(15083) = 15*X(3167)-X(7689) = 6*X(3167)-X(12038) = 15*X(3167)+X(12164) = 9*X(3167)+X(15083)

The reciprocal orthologic center of these triangles is X(41598)

X(41597) lies on these lines: {2, 9936}, {3, 49}, {5, 539}, {51, 195}, {52, 110}, {54, 5891}, {61, 10662}, {62, 10661}, {68, 3090}, {97, 20574}, {156, 511}, {182, 13154}, {235, 16534}, {323, 1614}, {399, 11381}, {524, 13383}, {542, 13371}, {546, 5448}, {567, 14926}, {569, 10170}, {575, 3564}, {576, 9925}, {578, 15068}, {632, 12359}, {912, 15178}, {1069, 3303}, {1154, 10282}, {1173, 1994}, {1368, 18128}, {1495, 6243}, {1539, 3627}, {1598, 12002}, {1656, 13366}, {1993, 5446}, {1995, 12235}, {2070, 14531}, {2931, 15039}, {3091, 5654}, {3146, 12118}, {3157, 3304}, {3525, 11411}, {3529, 41482}, {3592, 8909}, {4550, 11425}, {5072, 12429}, {5079, 14852}, {5092, 32142}, {5097, 10095}, {5198, 36747}, {5462, 9306}, {5504, 12086}, {5650, 37471}, {5651, 36753}, {5876, 11430}, {5892, 7592}, {5965, 41593}, {6090, 36752}, {6102, 40111}, {6240, 30714}, {6241, 10564}, {6419, 10666}, {6420, 10665}, {6583, 41591}, {6759, 16266}, {7387, 37672}, {7393, 17809}, {7488, 9705}, {7525, 15606}, {7530, 34966}, {7772, 23128}, {7982, 9928}, {7999, 11003}, {8548, 22234}, {8681, 22330}, {9544, 11412}, {9545, 11459}, {9706, 35921}, {9707, 37478}, {9730, 12364}, {9781, 11004}, {9972, 11536}, {10116, 11585}, {10627, 23060}, {11232, 18912}, {11271, 37943}, {11284, 19458}, {11403, 18451}, {11441, 13352}, {11444, 37513}, {11449, 32110}, {11456, 14641}, {11477, 20987}, {11482, 19588}, {11793, 32046}, {11898, 21637}, {12088, 23061}, {12106, 16625}, {12111, 35475}, {12162, 14865}, {12379, 37497}, {12893, 15034}, {12901, 15054}, {13336, 15066}, {13346, 14915}, {13431, 17713}, {13450, 35311}, {14643, 32226}, {14869, 20191}, {14984, 16982}, {15030, 37472}, {15038, 27355}, {15091, 21660}, {15585, 34380}, {18356, 32767}, {18378, 21969}, {18383, 32423}, {19123, 20080}, {20303, 33547}, {22333, 33556}, {22952, 22955}, {23236, 31724}, {23292, 31831}, {26864, 37486}, {32661, 35007}, {34153, 34798}, {35259, 37493}

X(41597) = midpoint of X(i) and X(j) for these {i, j}: {3, 15083}, {155, 1147}, {576, 9925}, {6193, 9927}, {6759, 16266}, {7689, 12164}, {13346, 32139}
X(41597) = reflection of X(i) in X(j) for these (i, j): (5449, 9820), (12038, 1147), (18356, 32767)
X(41597) = complement of the complement of X(9936)
X(41597) = isogonal conjugate of the polar conjugate of X(41628)
X(41597) = barycentric product X(3)*X(41628)
X(41597) = trilinear product X(i)*X(j) for these {i, j}: {48, 41628}, {1749, 12011}
X(41597) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(2383)}} and {{A, B, C, X(4), X(15083)}}
X(41597) = pole of the trilinear polar of X(1173) with respect to MacBeath circumconic
X(41597) = pole of the trilinear polar of X(1994) with respect to Johnson circumconic
X(41597) = X(1173)-Ceva conjugate of-X(3)
X(41597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 155, 15083), (5, 1493, 37505), (49, 5562, 18475), (110, 15801, 3518), (155, 3167, 1147), (195, 18350, 51), (323, 1614, 10625), (399, 37495, 11381), (1092, 18445, 40647), (1147, 15083, 3), (1993, 10539, 5446), (3518, 15801, 52), (3628, 32136, 575), (5654, 6193, 9927), (9306, 12161, 5462), (9703, 18436, 13367), (9925, 19139, 576), (34986, 37505, 1493)

X(41598) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(24) TO PEDAL OF X(5)

Barycentrics    (2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

The reciprocal orthologic center of these triangles is X(41597)

X(41598) lies on these lines: {24, 539}, {25, 3519}, {54, 3147}, {186, 10619}, {235, 1154}, {468, 1493}, {1204, 11457}, {1209, 1216}, {1595, 11576}, {1899, 9935}, {1974, 5965}, {2888, 7487}, {3518, 41586}, {3542, 15801}, {3574, 6242}, {6288, 12173}, {6353, 11271}, {7405, 9827}, {7505, 12242}, {8537, 14940}, {11577, 13567}, {11803, 37942}, {13431, 17713}

X(41598) = barycentric product X(1594)*X(41628)
X(41598) = {X(1209), X(41599)}-harmonic conjugate of X(6152)

X(41599) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(5)

Barycentrics    (4*a^2-b^2-c^2)*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

The reciprocal orthologic center of these triangles is X(3629)

X(41599) lies on these lines: {25, 5965}, {468, 34565}, {539, 6515}, {1154, 1596}, {1209, 1216}, {1907, 11576}, {3060, 5448}, {3574, 15873}, {9777, 12175}, {10605, 18400}, {10619, 11245}, {15585, 21637}, {32068, 37920}

X(41599) = barycentric product X(1594)*X(3629)
X(41599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6152, 41598, 1209), (41578, 41594, 1209), (41578, 41596, 41594)

X(41600) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(20) TO PEDAL OF X(21)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a^4+2*b*c*a^2-2*(b+c)*b*c*a-(b^2-c^2)^2) : :
X(41600) = 4*X(960)-3*X(41581) = 3*X(960)-2*X(41591) = 2*X(1829)-3*X(41581) = 3*X(1829)-4*X(41591) = 9*X(41581)-8*X(41591)

The reciprocal orthologic center of these triangles is X(41601)

X(41600) lies on these lines: {1, 3}, {8, 36844}, {69, 3827}, {72, 5928}, {238, 40964}, {306, 22282}, {429, 960}, {478, 1766}, {511, 1858}, {518, 6467}, {758, 18732}, {1231, 18659}, {1350, 1854}, {1610, 3101}, {1824, 5794}, {1828, 24703}, {1891, 1902}, {2292, 22097}, {2778, 16163}, {3436, 20928}, {3962, 34381}, {4513, 5227}, {4872, 17762}, {5562, 6001}, {5836, 30778}, {8897, 11682}, {10401, 12709}, {10459, 17470}, {12635, 17441}, {15556, 29311}, {16049, 41601}, {18650, 20718}, {21333, 28386}, {24987, 40635}

X(41600) = reflection of X(i) in X(j) for these (i, j): (65, 37613), (1829, 960)
X(41600) = isogonal conjugate of X(40454)
X(41600) = barycentric product X(i)*X(j) for these {i, j}: {197, 20911}, {1193, 20928}, {1211, 16049}, {1766, 4357}
X(41600) = barycentric quotient X(i)/X(j) for these (i, j): (197, 2298), (478, 961), (960, 34277), (1766, 1220)
X(41600) = trilinear product X(i)*X(j) for these {i, j}: {197, 4357}, {205, 20911}, {478, 3687}, {960, 21147}, {1193, 3436}, {1766, 3666}
X(41600) = trilinear quotient X(i)/X(j) for these (i, j): (1193, 3435), (1766, 2298)
X(41600) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1766)}} and {{A, B, C, X(3), X(960)}}
X(41600) = X(69)-Ceva conjugate of-X(1211)
X(41600) = X(1220)-isoconjugate-of-X(3435)
X(41600) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (197, 2298), (478, 961), (960, 34277)
X(41600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (65, 3057, 37614), (960, 1829, 41581)

X(41601) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(20)

Barycentrics    a*(a+b)*(a+c)*(a^4+2*b*c*a^2-2*(b+c)*b*c*a-(b^2-c^2)^2)*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41600)

X(41601) lies on these lines: {185, 235}, {221, 18178}, {517, 41608}, {1498, 7497}, {1854, 18165}, {2778, 16164}, {3827, 41610}, {6000, 6841}, {6001, 18180}, {16049, 41600}, {26543, 34146}

X(41601) = trilinear product X(i)*X(j) for these {i, j}: {774, 16049}, {1624, 21186}, {1766, 18603}

X(41602) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(22) TO PEDAL OF X(20)

Barycentrics    (a^6+(b^2+c^2)*a^4-(b^2+c^2)^2*a^2-(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(159)

X(41602) lies on these lines: {2, 1619}, {3, 34944}, {5, 1498}, {30, 161}, {64, 6823}, {154, 1368}, {159, 1370}, {184, 427}, {185, 235}, {343, 34146}, {403, 5656}, {858, 11206}, {1594, 34781}, {1596, 17810}, {1660, 11064}, {1899, 19149}, {1906, 5893}, {1907, 41362}, {2072, 32063}, {3618, 5133}, {5576, 34780}, {5642, 23315}, {5878, 12163}, {6000, 15760}, {6247, 7399}, {6759, 11585}, {7378, 32621}, {7403, 18381}, {7405, 20299}, {7485, 35219}, {7542, 32321}, {7667, 15577}, {9722, 41373}, {9833, 23335}, {10024, 12315}, {10117, 10154}, {10192, 22352}, {10984, 16252}, {11245, 34117}, {11442, 41614}, {12324, 13160}, {14530, 37452}, {15583, 40673}, {15809, 36990}, {16165, 36201}, {21850, 34751}, {23332, 37439}, {34609, 39879}

X(41602) = barycentric product X(i)*X(j) for these {i, j}: {235, 28419}, {774, 21582}, {1370, 13567}
X(41602) = barycentric quotient X(i)/X(j) for these (i, j): (800, 34207), (1370, 801)
X(41602) = trilinear product X(i)*X(j) for these {i, j}: {159, 17858}, {774, 1370}, {800, 21582}
X(41602) = trilinear quotient X(i)/X(j) for these (i, j): (774, 34207), (1370, 775)
X(41602) = X(775)-isoconjugate-of-X(34207)
X(41602) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (800, 34207), (1370, 801)
X(41602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (154, 32125, 1368), (2883, 13567, 41580), (5133, 32064, 23300), (41580, 41603, 13567)

X(41603) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(20)

Barycentrics    ((b^2+c^2)*a^4-2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2))*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :
X(41603) = 3*X(403)-2*X(15125) = 3*X(14157)-X(34781) = 6*X(15113)-5*X(30745)

The reciprocal orthologic center of these triangles is X(5181)

X(41603) lies on these lines: {2, 1660}, {3, 161}, {23, 36201}, {25, 15121}, {125, 468}, {185, 235}, {206, 18911}, {343, 12058}, {403, 15125}, {511, 32125}, {542, 15139}, {858, 2393}, {1177, 9140}, {1498, 26944}, {1619, 26958}, {1885, 11572}, {2777, 3581}, {2781, 41586}, {3542, 11457}, {3580, 34146}, {3619, 5888}, {5094, 8549}, {5449, 14216}, {5965, 17847}, {6000, 11799}, {7493, 23293}, {7689, 20427}, {10117, 32223}, {11206, 26913}, {13203, 15107}, {13445, 37201}, {13561, 34351}, {15113, 30745}, {15138, 20417}, {16618, 34826}, {17823, 32396}, {18128, 32379}, {18383, 18488}, {19149, 26869}, {22802, 37490}, {23294, 35486}, {23332, 30739}, {29012, 37928}, {31152, 34787}

X(41603) = midpoint of X(13203) and X(15107)
X(41603) = reflection of X(i) in X(j) for these (i, j): (858, 15126), (10117, 32223), (15138, 20417)
X(41603) = barycentric product X(i)*X(j) for these {i, j}: {774, 20884}, {800, 1236}, {858, 13567}
X(41603) = barycentric quotient X(i)/X(j) for these (i, j): (185, 18876), (800, 1177), (858, 801), (1236, 40830)
X(41603) = trilinear product X(i)*X(j) for these {i, j}: {774, 858}, {800, 20884}
X(41603) = trilinear quotient X(i)/X(j) for these (i, j): (774, 1177), (858, 775)
X(41603) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(41589)}} and {{A, B, C, X(235), X(858)}}
X(41603) = X(775)-isoconjugate-of-X(1177)
X(41603) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (185, 18876), (800, 1177), (858, 801), (1236, 40830)
X(41603) = {X(13567), X(41602)}-harmonic conjugate of X(41580)

X(41604) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(22)

Barycentrics    a^3*(a+b)*(a+c)*(a^6-2*(b+c)*a^5+(b+c)^2*a^4-2*(b+c)*b*c*a^3-(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41605)

X(41604) lies on these lines: {6, 25}, {518, 41608}, {1437, 22769}, {1503, 6841}, {1576, 2193}, {3827, 18180}, {19128, 31385}, {32713, 41364}

X(41604) = barycentric product X(1169)*X(41605)

X(41605) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(22) TO PEDAL OF X(21)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a^6-2*(b+c)*a^5+(b+c)^2*a^4-2*(b+c)*b*c*a^3-(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41604)

X(41605) lies on these lines: {63, 10829}, {72, 9798}, {78, 10830}, {184, 518}, {210, 355}, {343, 3827}, {429, 960}, {517, 15760}, {5794, 11391}, {11390, 24703}

X(41605) = barycentric product X(1228)*X(41604)
X(41605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (960, 41611, 41581), (41581, 41607, 41611)

X(41606) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(23)

Barycentrics    a*(a+b)*(a+c)*(2*a^2-b^2-c^2)*(a^6-2*(b+c)*a^5+(b^2+b*c+c^2)*a^4-(b+c)*b*c*a^3-(b^4+c^4-(b+c)^2*b*c)*a^2+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41607)

X(41606) lies on these lines: {110, 9061}, {468, 524}, {518, 16164}, {542, 6841}, {2836, 18180}, {3827, 12826}, {7469, 41607}

X(41606) = midpoint of X(110) and X(41610)
X(41606) = barycentric product X(524)*X(7469)
X(41606) = barycentric quotient X(187)/X(10101)
X(41606) = trilinear product X(896)*X(7469)
X(41606) = trilinear quotient X(896)/X(10101)
X(41606) = crossdifference of every pair of points on line {X(10097), X(10101)}
X(41606) = X(897)-isoconjugate-of-X(10101)
X(41606) = X(187)-reciprocal conjugate of-X(10101)

X(41607) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(21)

Barycentrics    a*((b+c)*a+b^2+c^2)*(a^6-2*(b+c)*a^5+(b^2+b*c+c^2)*a^4-(b+c)*b*c*a^3-(b^4+c^4-(b+c)^2*b*c)*a^2+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41606)

X(41607) lies on these lines: {210, 18480}, {429, 960}, {511, 32126}, {517, 11799}, {518, 1495}, {1533, 2778}, {2771, 3581}, {2836, 41586}, {3580, 3827}, {5904, 8185}, {7469, 41606}, {15904, 32223}

X(41607) = reflection of X(15904) in X(32223)
X(41607) = barycentric product X(1211)*X(7469)
X(41607) = {X(41605), X(41611)}-harmonic conjugate of X(41581)

X(41608) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(24)

Barycentrics    a^3*(a+b)*(a+c)*(-a^2+b^2+c^2)*(a^3-(b+c)*a^2-(b+c)^2*a+(b+c)*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(41609)

X(41608) lies on these lines: {1, 1762}, {3, 49}, {21, 912}, {28, 110}, {54, 7549}, {58, 3157}, {68, 6861}, {81, 1175}, {284, 1069}, {517, 41601}, {518, 41604}, {916, 10902}, {1408, 36059}, {1614, 36029}, {1780, 3173}, {1993, 14017}, {2193, 3990}, {2328, 10267}, {2360, 11249}, {3561, 22458}, {3564, 6675}, {4219, 34148}, {5012, 7523}, {5707, 7535}, {6193, 6824}, {6237, 32613}, {6881, 9820}, {7193, 18606}, {7497, 10539}, {7520, 9544}, {7562, 18350}, {9895, 11428}, {9928, 37227}, {10202, 37277}, {12649, 37168}, {13346, 15951}, {14016, 37826}, {14054, 30733}, {16049, 37585}, {16415, 37530}, {17188, 26470}, {34381, 41582}

X(41608) = isogonal conjugate of the polar conjugate of X(40571)
X(41608) = isotomic conjugate of the polar conjugate of X(41332)
X(41608) = barycentric product X(i)*X(j) for these {i, j}: {3, 40571}, {21, 3173}, {63, 1780}, {69, 41332}, {81, 11517}, {283, 1708}
X(41608) = barycentric quotient X(i)/X(j) for these (i, j): (48, 23604), (228, 41508), (577, 28787), (1333, 39267), (1437, 15474)
X(41608) = trilinear product X(i)*X(j) for these {i, j}: {3, 1780}, {21, 3215}, {48, 40571}, {58, 11517}, {63, 41332}, {255, 30733}
X(41608) = trilinear quotient X(i)/X(j) for these (i, j): (3, 23604), (58, 39267), (71, 41508), (255, 28787), (1708, 40149)
X(41608) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(915)}} and {{A, B, C, X(283), X(1780)}}
X(41608) = pole of the trilinear polar of X(1175) with respect to MacBeath circumconic
X(41608) = X(i)-Ceva conjugate of-X(j) for these (i, j): (81, 2193), (1175, 3)
X(41608) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 23604}, {10, 39267}, {27, 41508}, {158, 28787}
X(41608) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (48, 23604), (228, 41508), (577, 28787), (1333, 39267)
X(41608) = {X(28), X(3193)}-harmonic conjugate of X(24474)

X(41609) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(24) TO PEDAL OF X(21)

Barycentrics    a*(a^2+b^2-c^2)*(a^2-b^2+c^2)*((b+c)*a+b^2+c^2)*(a^3-(b+c)*a^2-(b+c)^2*a+(b+c)*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(41608)

X(41609) lies on these lines: {4, 3876}, {24, 912}, {25, 72}, {186, 13369}, {210, 5090}, {213, 5089}, {235, 517}, {392, 11396}, {427, 5044}, {429, 960}, {468, 942}, {518, 1974}, {976, 2212}, {997, 22479}, {1071, 3515}, {1204, 6001}, {1824, 39585}, {1825, 8756}, {1858, 32126}, {1885, 31793}, {1902, 3690}, {2292, 40976}, {2354, 21033}, {2911, 41538}, {3136, 9895}, {3147, 10202}, {3542, 24474}, {3575, 5777}, {3681, 7718}, {3868, 6353}, {5266, 14975}, {5439, 37453}, {5692, 7713}, {5927, 12173}, {6756, 31835}, {10157, 23047}, {10167, 15750}, {11383, 12514}, {12135, 34790}, {14054, 30733}, {18533, 40263}, {34381, 41584}

X(41609) = barycentric product X(i)*X(j) for these {i, j}: {429, 40571}, {1211, 30733}
X(41609) = barycentric quotient X(1829)/X(15474)
X(41609) = trilinear product X(429)*X(1780)
X(41609) = trilinear quotient X(429)/X(23604)
X(41609) = {X(960), X(41611)}-harmonic conjugate of X(1829)

X(41610) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(21) TO PEDAL OF X(25)

Barycentrics    a*(a+b)*(a+c)*(a^2-2*(b+c)*a+b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(41611)

X(41610) lies on these lines: {2, 6}, {21, 518}, {58, 1792}, {85, 3759}, {100, 22277}, {110, 9061}, {218, 344}, {284, 1444}, {286, 648}, {511, 3651}, {583, 21495}, {584, 21511}, {644, 17315}, {662, 1014}, {1332, 3879}, {1333, 2991}, {1350, 37105}, {1351, 6985}, {1352, 6990}, {1353, 37356}, {1431, 4603}, {1503, 37433}, {1582, 32913}, {1760, 3868}, {1943, 2982}, {2260, 20769}, {2402, 2440}, {2651, 3056}, {2836, 12826}, {2911, 17316}, {2966, 35152}, {3120, 26731}, {3193, 17584}, {3564, 6841}, {3729, 4483}, {3827, 41601}, {3870, 21059}, {3873, 4228}, {3874, 5358}, {4184, 12329}, {4225, 22769}, {4233, 41611}, {4251, 16574}, {4259, 35976}, {4273, 16696}, {4420, 4663}, {4653, 16496}, {4658, 16475}, {5208, 5324}, {6604, 23144}, {6776, 6851}, {6849, 14853}, {6899, 14912}, {8539, 17792}, {9004, 41582}, {11683, 39765}, {14005, 38047}, {14570, 32029}, {16973, 40773}, {17201, 18230}, {17321, 23151}, {17353, 17745}, {17390, 17796}, {17454, 35276}, {18180, 34381}, {18714, 34195}, {28619, 38049}, {34378, 35637}, {36740, 37285}, {37284, 37492}, {38871, 40744}

X(41610) = reflection of X(i) in X(j) for these (i, j): (69, 26543), (110, 41606), (15988, 6)
X(41610) = isotomic conjugate of the polar conjugate of X(4233)
X(41610) = barycentric product X(i)*X(j) for these {i, j}: {21, 6604}, {69, 4233}, {81, 344}, {86, 3870}, {99, 3309}, {218, 274}
X(41610) = barycentric quotient X(i)/X(j) for these (i, j): (21, 6601), (58, 2191), (81, 277), (110, 1292), (218, 37), (344, 321)
X(41610) = trilinear product X(i)*X(j) for these {i, j}: {21, 1445}, {29, 23144}, {58, 344}, {63, 4233}, {81, 3870}, {86, 218}
X(41610) = trilinear quotient X(i)/X(j) for these (i, j): (81, 2191), (86, 277), (99, 37206), (218, 42), (333, 6601), (344, 10)
X(41610) = trilinear pole of the line {3309, 8642}
X(41610) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(4648)}} and {{A, B, C, X(2), X(344)}}
X(41610) = X(i)-isoconjugate-of-X(j) for these {i, j}: {42, 277}, {210, 17107}, {512, 37206}, {661, 1292}
X(41610) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (21, 6601), (58, 2191), (81, 277), (110, 1292)
X(41610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (81, 2287, 86), (81, 37783, 26637), (81, 40571, 1812), (284, 18206, 1444), (26637, 37783, 1812), (26637, 40571, 37783)

X(41611) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(21)

Barycentrics    a*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^2-2*(b+c)*a+b^2+c^2)*((b+c)*a+b^2+c^2) : :

The reciprocal orthologic center of these triangles is X(41610)

X(41611) lies on these lines: {4, 210}, {19, 37658}, {24, 12675}, {25, 518}, {34, 4383}, {72, 7713}, {218, 7719}, {242, 1859}, {354, 6353}, {427, 3740}, {429, 960}, {468, 3742}, {517, 1596}, {1395, 4641}, {2212, 3744}, {2333, 22275}, {2354, 3965}, {2836, 12828}, {3059, 7717}, {3542, 13374}, {3666, 40976}, {3681, 6995}, {3789, 4207}, {3827, 13567}, {3848, 37453}, {3873, 4232}, {4233, 41610}, {4640, 11383}, {4662, 5090}, {6001, 10605}, {7487, 14872}, {9004, 41585}, {11363, 34791}, {34381, 41588}

X(41611) = barycentric product X(i)*X(j) for these {i, j}: {344, 1829}, {429, 41610}, {1211, 4233}
X(41611) = barycentric quotient X(218)/X(1791)
X(41611) = trilinear product X(i)*X(j) for these {i, j}: {218, 1848}, {344, 2354}
X(41611) = trilinear quotient X(218)/X(2359)
X(41611) = X(277)-isoconjugate-of-X(2359)
X(41611) = X(218)-reciprocal conjugate of-X(1791)
X(41611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1829, 41609, 960), (41581, 41605, 960), (41581, 41607, 41605)

X(41612) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(22) TO PEDAL OF X(23)

Barycentrics
a^2*(2*a^2-b^2-c^2)*(a^10-(b^2+c^2)*a^8-(2*b^4-3*b^2*c^2+2*c^4)*a^6+(b^2-2*c^2)*(2*b^2-c^2)*(b^2+c^2)*a^4+(b^8+c^8-(3*b^4-8*b^2*c^2+3*c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)) : :

The reciprocal orthologic center of these triangles is X(41613)

X(41612) lies on these lines: {6, 1511}, {110, 19153}, {184, 2854}, {468, 524}, {542, 15760}, {895, 32621}, {2393, 16165}, {3047, 40228}, {5609, 15069}, {9813, 32246}, {10201, 34319}, {11179, 32233}, {18374, 20772}, {21637, 32114}, {37980, 41613}

X(41612) = midpoint of X(110) and X(41614)
X(41612) = barycentric product X(524)*X(37980)
X(41612) = trilinear product X(896)*X(37980)
X(41612) = {X(6593), X(41618)}-harmonic conjugate of X(15303)

X(41613) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(22)

Barycentrics
a^4*(a^10-(b^2+c^2)*a^8-(2*b^4-3*b^2*c^2+2*c^4)*a^6+(b^2-2*c^2)*(2*b^2-c^2)*(b^2+c^2)*a^4+(b^8+c^8-(3*b^4-8*b^2*c^2+3*c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)) : :
X(41613) = 2*X(206)-3*X(18374) = X(9924)-3*X(19596)

The reciprocal orthologic center of these triangles is X(41612)

X(41613) lies on these lines: {6, 25}, {30, 19138}, {182, 23049}, {265, 1177}, {511, 2931}, {568, 34117}, {597, 38321}, {1533, 36201}, {1576, 10317}, {2781, 3581}, {2854, 41615}, {5012, 10169}, {5523, 32713}, {6593, 11597}, {8705, 13248}, {10249, 35237}, {10274, 22330}, {11645, 34775}, {15139, 32235}, {17835, 32262}, {18382, 19127}, {18583, 32344}, {19121, 23300}, {32113, 38851}, {32734, 32741}, {37980, 41612}

X(41613) = reflection of X(i) in X(j) for these (i, j): (159, 1495), (1177, 32217)
X(41613) = isogonal conjugate of the isotomic conjugate of X(37980)
X(41613) = barycentric product X(i)*X(j) for these {i, j}: {6, 37980}, {111, 41612}
X(41613) = trilinear product X(i)*X(j) for these {i, j}: {31, 37980}, {923, 41612}
X(41613) = intersection, other than A,B,C, of conics {{A, B, C, X(25), X(37980)}} and {{A, B, C, X(265), X(2393)}}
X(41613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 18374, 34397), (1495, 1974, 18374)

X(41614) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(22) TO PEDAL OF X(25)

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^4-b^4+4*b^2*c^2-c^4) : :
X(41614) = 5*X(3618)-4*X(23292) = 3*X(6800)-4*X(19127)

The reciprocal orthologic center of these triangles is X(19136)

X(41614) lies on these lines: {2, 6}, {3, 895}, {20, 8549}, {22, 2393}, {25, 11188}, {49, 9925}, {51, 9813}, {76, 648}, {110, 19153}, {159, 12272}, {182, 32127}, {184, 8681}, {216, 9723}, {311, 8745}, {338, 19221}, {378, 511}, {403, 1352}, {458, 3260}, {468, 8263}, {542, 11456}, {569, 32284}, {575, 1092}, {576, 5562}, {691, 23701}, {858, 23327}, {1078, 36841}, {1154, 1351}, {1176, 6391}, {1216, 8538}, {1332, 22131}, {1350, 2071}, {1568, 5476}, {1974, 14913}, {1975, 14570}, {1995, 8542}, {2854, 6800}, {2888, 11061}, {2904, 5095}, {2979, 11216}, {3092, 12323}, {3093, 12322}, {3153, 23049}, {3547, 19458}, {3564, 15760}, {3785, 23115}, {3917, 11511}, {5012, 15531}, {5093, 15038}, {5157, 32366}, {5158, 36212}, {5486, 7495}, {5907, 11470}, {5921, 19149}, {6101, 11255}, {6152, 9972}, {6403, 37488}, {6467, 19126}, {7386, 18919}, {7488, 34787}, {7527, 11477}, {7767, 22120}, {8537, 11412}, {8749, 35910}, {9976, 16163}, {10250, 37480}, {10313, 15574}, {10607, 36751}, {10627, 32155}, {11180, 18451}, {11185, 37855}, {11442, 41602}, {11898, 19139}, {12167, 37491}, {12215, 35928}, {12220, 34777}, {13169, 15106}, {14615, 36794}, {14853, 18537}, {15014, 18906}, {15141, 32244}, {16387, 16789}, {18374, 35264}, {18449, 23039}, {18475, 19131}, {19125, 19588}, {20811, 23127}, {22123, 23151}, {25051, 39646}, {32255, 34799}, {32257, 34470}, {32269, 41585}, {32368, 41590}, {35259, 37962}, {35901, 41511}, {37893, 40947}, {38263, 41435}

X(41614) = reflection of X(i) in X(j) for these (i, j): (69, 343), (110, 41612), (1993, 6)
X(41614) = isogonal conjugate of the polar conjugate of X(11185)
X(41614) = isotomic conjugate of the polar conjugate of X(1995)
X(41614) = barycentric product X(i)*X(j) for these {i, j}: {3, 11185}, {69, 1995}, {99, 30209}, {305, 19136}, {662, 14209}
X(41614) = barycentric quotient X(i)/X(j) for these (i, j): (3, 5486), (110, 30247), (662, 37217)
X(41614) = trilinear product X(i)*X(j) for these {i, j}: {48, 11185}, {63, 1995}, {110, 14209}, {304, 19136}, {662, 30209}
X(41614) = trilinear quotient X(i)/X(j) for these (i, j): (63, 5486), (99, 37217), (662, 30247), (691, 36115)
X(41614) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(895)}} and {{A, B, C, X(3), X(524)}}
X(41614) = crossdifference of every pair of points on line {X(512), X(14273)}
X(41614) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 5486}, {512, 37217}, {661, 30247}, {690, 36115}
X(41614) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 5486), (110, 30247), (662, 37217)
X(41614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 37784, 6), (6, 69, 20806), (6, 141, 26206), (6, 394, 22151), (69, 3618, 28419), (69, 22151, 394), (394, 22151, 20806), (1350, 10249, 2071), (2979, 11443, 11416), (3917, 21639, 11511), (5012, 15531, 32621), (5032, 34545, 6), (8542, 19136, 29959), (11416, 11443, 11216), (12272, 19121, 159), (15069, 34117, 11441), (19136, 29959, 1995), (34777, 37485, 12220)

X(41615) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(24)

Barycentrics    a^4*(-a^2+b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-5*b^2*c^2+c^4)*a^2+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)) : :
X(41615) = 4*X(1147)-3*X(22115) = 5*X(1656)-4*X(15123) = 3*X(3167)-2*X(3292) = X(3581)+2*X(12364) = 2*X(15136)-3*X(22115)

The reciprocal orthologic center of these triangles is X(41616)

X(41615) lies on these lines: {2, 15121}, {3, 49}, {22, 34966}, {23, 14984}, {25, 27365}, {54, 34664}, {68, 18350}, {110, 468}, {156, 6193}, {206, 9027}, {235, 14516}, {265, 32123}, {323, 3047}, {511, 10117}, {520, 8651}, {524, 1177}, {525, 13290}, {542, 15139}, {567, 5654}, {1495, 12310}, {1533, 17702}, {1656, 15123}, {1660, 9909}, {1974, 6391}, {1995, 8548}, {2393, 37972}, {2854, 41613}, {2930, 8681}, {3581, 12364}, {5012, 30739}, {5063, 9604}, {5159, 5622}, {5504, 10293}, {5925, 13346}, {6146, 22808}, {7493, 9544}, {7506, 19458}, {7517, 15316}, {9306, 15069}, {9820, 13353}, {9938, 18439}, {10539, 12429}, {10540, 11799}, {11064, 13198}, {12235, 13621}, {12271, 26882}, {12420, 25738}, {14567, 32661}, {15360, 41596}, {19154, 35260}, {20772, 37777}, {22660, 37472}, {22661, 31383}, {32223, 32263}

X(41615) = reflection of X(i) in X(j) for these (i, j): (265, 32123), (6391, 32127), (12310, 1495), (15136, 1147), (32263, 32223), (37477, 5504)
X(41615) = isogonal conjugate of the isotomic conjugate of X(5866)
X(41615) = isotomic conjugate of the polar conjugate of X(41336)
X(41615) = barycentric product X(i)*X(j) for these {i, j}: {3, 37784}, {6, 5866}, {69, 41336}, {184, 37803}, {394, 37777}, {524, 39169}
X(41615) = barycentric quotient X(i)/X(j) for these (i, j): (32, 41521), (184, 40347)
X(41615) = trilinear product X(i)*X(j) for these {i, j}: {31, 5866}, {48, 37784}, {63, 41336}, {255, 37777}, {896, 39169}
X(41615) = trilinear quotient X(i)/X(j) for these (i, j): (31, 41521), (48, 40347)
X(41615) = pole of the trilinear polar of X(1177) with respect to MacBeath circumconic
X(41615) = crossdifference of every pair of points on line {X(2501), X(5254)}
X(41615) = X(i)-Ceva conjugate of-X(j) for these (i, j): (524, 10317), (1177, 3)
X(41615) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 41521}, {92, 40347}
X(41615) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 41521), (184, 40347)
X(41615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (155, 19908, 18436), (184, 41619, 3167), (1147, 15136, 22115)

X(41616) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(24) TO PEDAL OF X(23)

Barycentrics    a^2*(2*a^2-b^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-5*b^2*c^2+c^4)*a^2+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(41615)

X(41616) lies on these lines: {3, 19388}, {6, 1112}, {24, 14984}, {25, 895}, {110, 19118}, {143, 8537}, {235, 542}, {427, 15118}, {468, 524}, {1204, 2781}, {1511, 19128}, {1593, 5622}, {1598, 39562}, {1885, 34563}, {1899, 32264}, {1974, 2854}, {2892, 15128}, {5050, 15472}, {5609, 32234}, {6467, 15647}, {8541, 37827}, {9970, 13148}, {10602, 38885}, {11416, 37928}, {11579, 12133}, {12099, 32251}, {12164, 35603}, {13416, 26206}, {13567, 32285}, {15462, 35602}, {18947, 25321}, {19136, 32246}, {19153, 32245}, {19504, 20806}, {20771, 34382}, {20772, 37777}, {25328, 32239}, {30739, 32300}

X(41616) = midpoint of X(110) and X(40318)
X(41616) = barycentric product X(i)*X(j) for these {i, j}: {468, 37784}, {524, 37777}
X(41616) = trilinear product X(896)*X(37777)
X(41616) = crosssum of X(3) and X(34470)
X(41616) = {X(6593), X(41618)}-harmonic conjugate of X(5095)

X(41617) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(23) TO PEDAL OF X(25)

Barycentrics    a^2*(a^6-(b^2+c^2)*a^4-(b^4-9*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-5*b^2*c^2+c^4)) : :
X(41617) = 4*X(6)-3*X(22151) = 2*X(6)-3*X(37784) = 3*X(23)-2*X(12367) = 3*X(249)-4*X(2030) = 2*X(323)-3*X(22151) = X(323)-3*X(37784) = 2*X(858)-3*X(25320) = 2*X(2930)-3*X(35265) = 5*X(3618)-4*X(11064) = 3*X(5622)-2*X(10564) = 2*X(5648)-3*X(37907) = 5*X(15059)-4*X(19510) = 3*X(15360)-2*X(41583) = 4*X(32217)-3*X(35265) = 4*X(33851)-5*X(37952) = X(37496)-3*X(39562)

The reciprocal orthologic center of these triangles is X(41618)

X(41617) lies on these lines: {2, 6}, {23, 2854}, {74, 511}, {110, 9027}, {111, 6096}, {187, 4558}, {249, 2030}, {340, 15262}, {399, 3564}, {542, 1533}, {574, 34883}, {576, 11459}, {648, 8744}, {858, 25320}, {1351, 31861}, {1495, 8681}, {1503, 17812}, {1691, 14972}, {2393, 15107}, {2452, 32515}, {2781, 12379}, {2914, 5095}, {2930, 32217}, {3260, 41254}, {3291, 36696}, {3581, 14984}, {5092, 40673}, {5505, 37980}, {5622, 10564}, {5640, 8542}, {5648, 37907}, {7492, 8547}, {7570, 20113}, {8541, 13596}, {8675, 9137}, {9136, 35575}, {9142, 37183}, {9145, 35298}, {9301, 22143}, {9872, 15256}, {10545, 29959}, {10546, 19136}, {10602, 33878}, {10752, 13754}, {11188, 34417}, {12039, 15019}, {12358, 18449}, {14927, 33534}, {15052, 15069}, {15059, 19510}, {15073, 37478}, {15080, 15531}, {15122, 19348}, {15360, 41583}, {17813, 34778}, {18931, 37483}, {20975, 35002}, {21309, 22241}, {32114, 32223}, {32284, 37513}, {33851, 37952}, {37496, 39562}, {37962, 41618}

X(41617) = midpoint of X(193) and X(37779)
X(41617) = reflection of X(i) in X(j) for these (i, j): (69, 3580), (323, 6), (895, 32127), (2930, 32217), (7464, 11579), (11061, 32220), (22151, 37784), (32114, 32223)
X(41617) = isotomic conjugate of the polar conjugate of X(37962)
X(41617) = barycentric product X(i)*X(j) for these {i, j}: {69, 37962}, {99, 2780}
X(41617) = barycentric quotient X(110)/X(2696)
X(41617) = trilinear product X(i)*X(j) for these {i, j}: {63, 37962}, {662, 2780}
X(41617) = trilinear quotient X(662)/X(2696)
X(41617) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(9139)}} and {{A, B, C, X(74), X(524)}}
X(41617) = crossdifference of every pair of points on line {X(512), X(40673)}
X(41617) = X(661)-isoconjugate-of-X(2696)
X(41617) = X(110)-reciprocal conjugate of-X(2696)
X(41617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 323, 22151), (6, 3231, 5166), (323, 37784, 6), (385, 2407, 41626), (2930, 32217, 35265), (3580, 40318, 37784), (8115, 8116, 1992)

X(41618) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(23)

Barycentrics    a^2*(2*a^2-b^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-9*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-5*b^2*c^2+c^4)) : :
X(41618) = X(6515)+3*X(25321)

The reciprocal orthologic center of these triangles is X(41617)

X(41618) lies on these lines: {6, 74}, {25, 2854}, {186, 33851}, {468, 524}, {542, 1596}, {1112, 8541}, {1177, 32621}, {1539, 3867}, {1597, 11579}, {5467, 40078}, {5648, 6353}, {6515, 25321}, {6623, 14982}, {7378, 25320}, {8681, 20772}, {11470, 13148}, {12824, 37784}, {19126, 25487}, {37962, 41617}

X(41618) = midpoint of X(i) and X(j) for these {i, j}: {5095, 12828}, {10605, 10752}
X(41618) = barycentric product X(i)*X(j) for these {i, j}: {468, 41617}, {524, 37962}
X(41618) = trilinear product X(896)*X(37962)
X(41618) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(5642)}} and {{A, B, C, X(74), X(524)}}
X(41618) = crossdifference of every pair of points on line {X(9033), X(10097)}
X(41618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5095, 41616, 6593), (15303, 41612, 6593)

X(41619) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(25) TO PEDAL OF X(24)

Barycentrics    a^4*(-a^2+b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-8*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4)) : :
X(41619) = X(394)-3*X(3167)

The reciprocal orthologic center of these triangles is X(40318)

X(41619) lies on these lines: {3, 49}, {25, 34382}, {68, 14826}, {110, 6353}, {206, 524}, {511, 1660}, {541, 5504}, {569, 6804}, {1974, 8780}, {1993, 40914}, {3089, 6193}, {3518, 12271}, {3564, 6677}, {5446, 15316}, {5462, 19458}, {5654, 11427}, {5943, 8548}, {6642, 17836}, {7506, 21651}, {8681, 19136}, {9544, 10565}, {10116, 12420}, {11206, 12118}, {13346, 15311}, {13488, 22660}, {17702, 31383}, {19139, 34986}, {40318, 40337}

X(41619) = midpoint of X(10605) and X(12164)
X(41619) = isogonal conjugate of the polar conjugate of X(40318)
X(41619) = isotomic conjugate of the polar conjugate of X(40320)
X(41619) = barycentric product X(i)*X(j) for these {i, j}: {3, 40318}, {69, 40320}, {193, 15261}
X(41619) = barycentric quotient X(i)/X(j) for these (i, j): (32, 15591), (184, 40323)
X(41619) = trilinear product X(i)*X(j) for these {i, j}: {48, 40318}, {63, 40320}, {1707, 15261}
X(41619) = trilinear quotient X(i)/X(j) for these (i, j): (31, 15591), (48, 40323)
X(41619) = X(193)-Ceva conjugate of-X(32)
X(41619) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 15591}, {92, 40323}
X(41619) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 15591), (184, 40323)
X(41619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (184, 3167, 1147), (3167, 41615, 184)

X(41620) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(15)

Barycentrics    (2*sqrt(3)*S+5*a^2-b^2-c^2)*(sqrt(3)*a^2+2*S) : :
X(41620) = 3*X(6)-X(6772)

The reciprocal orthologic center of these triangles is X(37786)

X(41620) lies on these lines: {2, 9112}, {6, 530}, {13, 3545}, {14, 5478}, {61, 6779}, {62, 6108}, {99, 22488}, {395, 5459}, {396, 532}, {531, 5477}, {543, 8584}, {620, 33458}, {671, 9113}, {1384, 9885}, {1992, 12155}, {5008, 36769}, {5306, 32553}, {5335, 41042}, {5463, 37640}, {6115, 22998}, {6669, 16645}, {6776, 10653}, {6777, 36320}, {7735, 9762}, {8595, 41407}, {10654, 23006}, {11489, 22489}, {16530, 37832}, {19073, 33441}, {19074, 33440}, {26613, 37786}, {31683, 41099}, {37785, 40671}

X(41620) = midpoint of X(i) and X(j) for these {i, j}: {1992, 12155}, {10654, 23006}
X(41620) = barycentric product X(i)*X(j) for these {i, j}: {396, 37786}, {618, 21466}
X(41620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (395, 5472, 5459), (396, 9115, 618), (8584, 18907, 41621), (13646, 13765, 2), (41631, 41633, 618)

X(41621) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(16)

Barycentrics    (-2*sqrt(3)*S+5*a^2-b^2-c^2)*(sqrt(3)*a^2-2*S) : :
X(41621) = 3*X(6)-X(6775)

The reciprocal orthologic center of these triangles is X(37785)

X(41621) lies on these lines: {2, 9113}, {6, 531}, {13, 5479}, {14, 3545}, {61, 6109}, {62, 6780}, {99, 22487}, {395, 533}, {396, 5460}, {530, 5477}, {543, 8584}, {620, 33459}, {671, 9112}, {1384, 9886}, {1992, 12154}, {5306, 32552}, {5334, 41043}, {5464, 37641}, {6114, 22997}, {6670, 16644}, {6776, 10654}, {6778, 36318}, {7735, 9760}, {8594, 41406}, {10653, 23013}, {11488, 22490}, {16529, 37835}, {19075, 33443}, {19076, 33442}, {26613, 37785}, {31684, 41099}, {37786, 40672}

X(41621) = midpoint of X(i) and X(j) for these {i, j}: {1992, 12154}, {10653, 23013}
X(41621) = barycentric product X(i)*X(j) for these {i, j}: {395, 37785}, {619, 21467}
X(41621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (395, 9117, 619), (396, 5471, 5460), (8584, 18907, 41620), (13645, 13764, 2), (41641, 41643, 619)

X(41622) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(32)

Barycentrics    (b^2+c^2)*(3*a^4+(b^2+c^2)*a^2-2*b^2*c^2) : :
X(41622) = 3*X(39)-2*X(141) = 5*X(39)-4*X(10007) = X(69)-3*X(7757) = 3*X(76)-5*X(3618) = 5*X(141)-6*X(10007) = 4*X(141)-3*X(14994) = X(141)-3*X(32449) = X(193)+3*X(194) = X(193)-3*X(32451) = 3*X(3094)-X(40341) = 3*X(3095)-X(18440) = 4*X(3589)-3*X(9466) = 2*X(3934)-3*X(13331) = 3*X(5052)-4*X(32455) = 3*X(6248)-4*X(19130) = 8*X(10007)-5*X(14994) = 2*X(10007)-5*X(32449) = X(14994)-4*X(32449) = 2*X(21850)-3*X(35439) = 6*X(32450)-X(40341)

The reciprocal orthologic center of these triangles is X(14614)

X(41622) lies on these lines: {2, 13647}, {6, 538}, {20, 185}, {39, 141}, {51, 8267}, {69, 7757}, {76, 3618}, {99, 41413}, {182, 7754}, {385, 5092}, {524, 8354}, {597, 14711}, {698, 5052}, {1285, 32474}, {1691, 7805}, {1975, 5039}, {1992, 11055}, {2024, 8149}, {2782, 21850}, {3051, 19568}, {3094, 32450}, {3095, 18440}, {3098, 31859}, {3266, 40130}, {3589, 5355}, {3629, 5477}, {3763, 7908}, {3793, 5188}, {3818, 7774}, {3934, 13331}, {4048, 5007}, {5017, 7781}, {5116, 7780}, {5207, 7905}, {5650, 31088}, {6248, 19130}, {6309, 13357}, {6329, 24256}, {6392, 14561}, {7735, 9764}, {7760, 12215}, {7762, 29012}, {7783, 14810}, {7816, 12212}, {7823, 29323}, {7837, 11645}, {8891, 11205}, {9870, 10545}, {13354, 32515}, {13571, 18553}, {14614, 33685}, {15819, 17008}, {18907, 41623}, {19089, 33453}, {19090, 33452}, {34236, 40858}, {35700, 40825}

X(41622) = midpoint of X(i) and X(j) for these {i, j}: {194, 32451}, {1992, 11055}
X(41622) = reflection of X(i) in X(j) for these (i, j): (39, 32449), (3094, 32450), (14711, 597), (14994, 39)
X(41622) = barycentric product X(141)*X(14614)
X(41622) = barycentric quotient X(1634)/X(39639)
X(41622) = trilinear product X(38)*X(14614)
X(41622) = intersection, other than A,B,C, of conics {{A, B, C, X(39), X(9292)}} and {{A, B, C, X(141), X(9307)}}
X(41622) = crossdifference of every pair of points on line {X(2451), X(9009)}
X(41622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (13647, 13766, 2), (41630, 41640, 39)

X(41623) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(39)

Barycentrics    (2*a^2+b^2+c^2)*(2*a^4+3*(b^2+c^2)*a^2-b^4-c^4) : :
X(41623) = X(69)-3*X(83) = 4*X(3589)-3*X(6292) = 5*X(3618)-3*X(31168) = 5*X(3763)-6*X(6704) = 2*X(3818)-3*X(6249) = X(6144)+3*X(24273) = 3*X(13111)+X(39899)

The reciprocal orthologic center of these triangles is X(41624)

X(41623) lies on these lines: {2, 13648}, {6, 754}, {69, 83}, {576, 3146}, {732, 3629}, {1992, 12156}, {3589, 5007}, {3618, 31168}, {3763, 6704}, {3818, 6249}, {5103, 5368}, {5477, 32455}, {6144, 24273}, {7735, 9765}, {7766, 19130}, {7838, 12212}, {7839, 29317}, {7921, 24206}, {12206, 12216}, {13111, 39899}, {18907, 41622}, {19091, 33455}, {19092, 33454}, {41413, 41624}

X(41623) = midpoint of X(1992) and X(12156)
X(41623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (13648, 13767, 2), (41632, 41642, 6292)

X(41624) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(6)

Barycentrics    2*a^4+3*(b^2+c^2)*a^2-b^4-c^4 : :
X(41624) = 4*X(39)-X(7750) = 2*X(39)+X(7762) = 5*X(39)-2*X(7830) = X(39)+2*X(7838) = X(194)+2*X(7745) = X(194)+5*X(7921) = 2*X(194)+X(32819) = 2*X(7745)-5*X(7921) = 4*X(7745)-X(32819) = X(7750)+2*X(7762) = 5*X(7750)-8*X(7830) = X(7750)+8*X(7838) = 5*X(7762)+4*X(7830) = X(7762)-4*X(7838) = X(7830)+5*X(7838) = 4*X(7830)-5*X(8356) = X(7837)+2*X(9300) = 2*X(7837)+X(37671) = 4*X(7838)+X(8356) = 4*X(9300)-X(37671)

The reciprocal orthologic center of these triangles is X(41623)

X(41624) lies on these lines: {2, 6}, {5, 7760}, {30, 3095}, {32, 35297}, {39, 754}, {51, 34383}, {83, 3933}, {98, 1353}, {99, 12156}, {114, 5097}, {140, 6179}, {147, 5480}, {190, 4884}, {194, 7745}, {232, 27377}, {251, 4558}, {262, 3564}, {315, 9605}, {316, 15048}, {384, 13571}, {427, 648}, {538, 7753}, {542, 38383}, {543, 14537}, {549, 3398}, {575, 37450}, {576, 1513}, {598, 11055}, {620, 5008}, {625, 5355}, {626, 5041}, {671, 3845}, {702, 19568}, {881, 25423}, {1003, 34511}, {1078, 31406}, {1351, 9744}, {1506, 7805}, {1916, 12830}, {1975, 14033}, {2548, 7754}, {3096, 7949}, {3102, 32421}, {3103, 32419}, {3363, 11054}, {3407, 13196}, {3705, 3758}, {3759, 7179}, {3793, 7771}, {3849, 8353}, {3926, 14039}, {3934, 7890}, {3972, 6390}, {4045, 7845}, {4479, 17224}, {5007, 7764}, {5013, 20065}, {5024, 14907}, {5039, 6393}, {5052, 5976}, {5093, 9753}, {5254, 7785}, {5286, 7773}, {5305, 7752}, {5309, 7775}, {5319, 7887}, {5332, 26629}, {5346, 7862}, {5368, 7886}, {5475, 7798}, {5477, 5939}, {5987, 25329}, {5999, 8550}, {6292, 7882}, {6419, 35685}, {6420, 35684}, {6655, 9607}, {6656, 7759}, {6661, 7801}, {6683, 7826}, {7296, 26686}, {7737, 31859}, {7738, 33272}, {7739, 7841}, {7747, 32450}, {7751, 32992}, {7755, 33249}, {7758, 7770}, {7763, 11288}, {7765, 7843}, {7767, 7786}, {7768, 8362}, {7776, 7803}, {7780, 9698}, {7781, 19687}, {7783, 20088}, {7787, 7789}, {7790, 7926}, {7796, 7819}, {7797, 7941}, {7799, 8369}, {7804, 7813}, {7808, 7855}, {7809, 7827}, {7811, 8359}, {7814, 7856}, {7821, 7829}, {7822, 7916}, {7823, 33264}, {7824, 9606}, {7831, 14929}, {7833, 14976}, {7834, 7903}, {7846, 7871}, {7851, 32816}, {7859, 7917}, {7860, 8357}, {7864, 7900}, {7870, 8368}, {7874, 34571}, {7876, 7946}, {7884, 8360}, {7889, 7895}, {7909, 33185}, {7912, 7920}, {7922, 8364}, {7947, 10583}, {7976, 34715}, {8024, 35549}, {8176, 18362}, {8352, 11648}, {8354, 11057}, {8703, 35002}, {8716, 33007}, {9478, 9765}, {9741, 11164}, {9764, 22486}, {10352, 12212}, {10754, 35705}, {10983, 36998}, {11185, 15484}, {11285, 14023}, {11286, 32833}, {11477, 37182}, {11482, 37071}, {12040, 26613}, {12100, 26316}, {12782, 34645}, {13586, 34604}, {14001, 32821}, {14069, 32825}, {14581, 35920}, {14970, 36214}, {17365, 33891}, {17369, 30179}, {17388, 33889}, {19569, 32480}, {19570, 33013}, {19661, 41134}, {21485, 37503}, {22331, 32964}, {22332, 32965}, {22712, 34380}, {31173, 39593}, {31407, 32975}, {31492, 33012}, {32006, 33210}, {32837, 33191}, {33016, 34505}, {34612, 41142}, {35954, 39785}, {41413, 41623}, {41630, 41643}, {41633, 41640}

X(41624) = midpoint of X(i) and X(j) for these {i, j}: {2, 7837}, {194, 11361}, {7757, 7812}, {7762, 8356}, {7823, 33264}, {7976, 34715}, {11257, 34733}, {12782, 34645}
X(41624) = reflection of X(i) in X(j) for these (i, j): (2, 9300), (7750, 8356), (7811, 8359), (8356, 39), (8370, 7753), (11057, 8354), (11361, 7745), (32819, 11361), (37671, 2)
X(41624) = isotomic conjugate of the isogonal conjugate of X(41413)
X(41624) = barycentric product X(i)*X(j) for these {i, j}: {76, 41413}, {99, 32473}
X(41624) = trilinear product X(i)*X(j) for these {i, j}: {75, 41413}, {662, 32473}
X(41624) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(41413)}} and {{A, B, C, X(98), X(13468)}}
X(41624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 385, 13468), (2, 1992, 14614), (2, 7774, 9766), (2, 9766, 325), (2, 14614, 22329), (6, 325, 7792), (6, 7774, 325), (6, 7778, 16989), (6, 9766, 2), (69, 37665, 11174), (193, 7736, 183), (230, 32455, 7766), (3329, 7779, 141), (3618, 37668, 7868), (3629, 3815, 385), (3815, 13468, 2), (7752, 7894, 5305), (7759, 7772, 6656), (7760, 7858, 5), (7766, 7777, 230), (7785, 7839, 5254), (7786, 7877, 7767), (11163, 14614, 2), (14930, 37668, 3618), (15484, 22253, 11185), (37785, 37786, 597)

X(41625) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(50)

Barycentrics    ((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41626)

X(41625) lies on these lines: {1986, 16237}, {3003, 3580}, {3018, 14389}, {5663, 6776}

X(41625) = trilinear product X(1725)*X(41626)
X(41625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41634, 41644, 34834), (41652, 41658, 34834)

X(41626) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(6)

Barycentrics    3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :

The reciprocal orthologic center of these triangles is X(41625)

X(41626) lies on these lines: {2, 6}, {30, 9512}, {50, 35933}, {186, 648}, {538, 4558}, {671, 18316}, {1513, 11061}, {7550, 7760}, {7757, 35921}, {13586, 14570}

X(41626) = reflection of X(35933) in X(50)
X(41626) = {X(385), X(2407)}-harmonic conjugate of X(41617)

X(41627) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(6) TO PEDAL OF X(52)

Barycentrics    (2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

The reciprocal orthologic center of these triangles is X(41628)

X(41627) lies on these lines: {6, 539}, {34835, 41636}

X(41627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41636, 41646, 34835), (41654, 41660, 34835)

X(41628) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(6)

Barycentrics    2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(41628) = 4*X(52)-X(14516) = 4*X(193)-X(40316) = 2*X(428)-3*X(3060) = 3*X(2979)-4*X(10691) = 5*X(3567)-4*X(10127) = 9*X(5640)-8*X(10128) = 3*X(5650)-4*X(32068) = X(6243)+2*X(32358) = 2*X(6243)+X(34224) = 7*X(9781)-4*X(31831) = 4*X(10112)-X(12225) = 2*X(10112)+X(14531) = 4*X(10263)-X(16659) = 2*X(10691)-3*X(11245) = X(11412)-4*X(13292) = X(12111)-4*X(13142) = X(12225)+2*X(14531) = 4*X(32358)-X(34224)

The reciprocal orthologic center of these triangles is X(41627).

Let A'B'C' be the reflection triangle. Let AB, AC be the orthogonal projections of A' on CA, AB, resp. Define BC, BA, CA, CB cyclically. Let A" = CAAC∩ABBA, and define B" and C" cyclically. Triangle A"B"C" is homothetic to ABC at X(6), and X(41628) is the centroid of A"B"C". (Randy Hutson, April 13, 2021)

X(41628) lies on these lines: {2, 6}, {5, 1173}, {30, 5889}, {51, 5965}, {52, 539}, {61, 40712}, {62, 40711}, {70, 38260}, {95, 288}, {110, 41588}, {297, 41366}, {324, 27377}, {376, 37486}, {381, 12160}, {397, 19778}, {398, 19779}, {427, 8537}, {428, 3060}, {465, 11127}, {466, 11126}, {467, 648}, {542, 13417}, {549, 13353}, {576, 5133}, {1154, 12022}, {1238, 13345}, {1351, 5064}, {1353, 5012}, {1368, 23061}, {2979, 10691}, {3410, 5480}, {3518, 11271}, {3567, 10127}, {3860, 36852}, {3917, 11225}, {4995, 9637}, {5640, 10128}, {5650, 32068}, {6636, 8550}, {6676, 11422}, {7391, 11477}, {7394, 15069}, {7488, 19468}, {7495, 13366}, {7539, 11482}, {7540, 31810}, {7760, 41237}, {7877, 40814}, {7890, 36212}, {9544, 32269}, {9777, 11898}, {9781, 31831}, {9936, 10594}, {10112, 12225}, {10154, 15360}, {10263, 16659}, {10733, 33699}, {11412, 13292}, {11423, 34002}, {11550, 37517}, {12111, 13142}, {12370, 18564}, {13431, 17713}, {13432, 18369}, {14645, 19568}, {14769, 27362}, {14831, 38323}, {15004, 34507}, {15019, 37439}, {16266, 26879}, {17846, 34782}, {24206, 34565}, {25738, 31181}, {32002, 39284}, {34986, 41586}, {35937, 41480}, {37943, 41587}

X(41628) = reflection of X(i) in X(j) for these (i, j): (2979, 11245), (3917, 11225), (7576, 52), (14516, 7576), (18564, 12370), (34603, 21969), (38323, 14831)
X(41628) = anticomplement of the isogonal conjugate of X(33631)
X(41628) = polar conjugate of the isogonal conjugate of X(41597)
X(41628) = anticomplement of isotomic conjugate of X(39284)
X(41628) = anticomplementary conjugate of the anticomplement of X(33631)
X(41628) = barycentric product X(i)*X(j) for these {i, j}: {99, 20184}, {264, 41597}
X(41628) = barycentric quotient X(110)/X(20185)
X(41628) = trilinear product X(i)*X(j) for these {i, j}: {92, 41597}, {662, 20184}
X(41628) = trilinear quotient X(662)/X(20185)
X(41628) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(1487)}} and {{A, B, C, X(288), X(5422)}}
X(41628) = pole of the trilinear polar of X(39284) with respect to Steiner circumellipse
X(41628) = X(i)-anticomplementary conjugate of-X(j) for these (i, j): (19, 2889), (1173, 4329)
X(41628) = X(661)-isoconjugate-of-X(20185)
X(41628) = X(110)-reciprocal conjugate of-X(20185)
X(41628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 37672, 40112), (193, 6515, 1993), (343, 1994, 14389), (343, 3629, 1994), (1993, 6515, 3580), (1994, 37779, 343), (3629, 37779, 14389), (6243, 32358, 34224), (10112, 14531, 12225), (11143, 11144, 5), (15004, 34507, 37990)

X(41629) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(6)

Barycentrics    (3*a-b-c)*(a+c)*(a+b) : :
X(41629) = 4*X(58)-X(1043) = 2*X(1834)+X(20077)

The reciprocal orthologic center of these triangles is X(4856)

X(41629) lies on these lines: {2, 6}, {21, 3241}, {27, 648}, {57, 3759}, {58, 519}, {63, 4360}, {99, 17222}, {110, 9083}, {145, 3052}, {171, 4685}, {190, 1999}, {274, 16833}, {320, 40940}, {345, 17377}, {518, 3794}, {540, 3017}, {551, 4658}, {553, 37756}, {593, 30606}, {645, 17195}, {662, 1412}, {757, 7058}, {1010, 3679}, {1014, 8051}, {1100, 38000}, {1121, 19607}, {1171, 4102}, {1330, 16052}, {1434, 16711}, {1743, 18743}, {1757, 4096}, {1834, 20077}, {2308, 32919}, {2319, 2669}, {2329, 5325}, {2403, 2441}, {2651, 3058}, {2966, 35153}, {3187, 17160}, {3193, 11240}, {3219, 34064}, {3286, 4421}, {3736, 18192}, {3751, 3769}, {3752, 17121}, {3758, 11679}, {3772, 17364}, {3780, 21792}, {3791, 32913}, {3828, 14007}, {3840, 16477}, {3879, 33116}, {3928, 9311}, {3929, 4664}, {3996, 17126}, {4001, 17273}, {4052, 17936}, {4184, 15621}, {4252, 20018}, {4267, 11194}, {4398, 9965}, {4415, 20072}, {4483, 17133}, {4643, 29841}, {4663, 7081}, {4676, 39594}, {4722, 17763}, {4831, 4854}, {4849, 18211}, {4856, 33628}, {4862, 19830}, {5208, 18191}, {5222, 33947}, {5263, 32853}, {5294, 17285}, {5327, 11235}, {5437, 17207}, {5847, 33121}, {6996, 7760}, {7419, 18613}, {8822, 28610}, {9534, 19290}, {10449, 11354}, {11115, 31145}, {14534, 33766}, {16050, 17310}, {16054, 41140}, {16670, 30567}, {16696, 25059}, {16701, 18603}, {16712, 17206}, {17020, 24593}, {17022, 17335}, {17233, 26065}, {17270, 19827}, {17272, 19812}, {17294, 33954}, {17295, 32777}, {17354, 34255}, {17361, 25527}, {17448, 29584}, {17539, 20049}, {17553, 38314}, {17770, 33135}, {17771, 33152}, {17772, 33167}, {18180, 24473}, {18792, 36634}, {18827, 32040}, {19808, 32025}, {19870, 37559}, {19875, 25526}, {20012, 37540}, {20942, 30939}, {21747, 32943}, {26723, 27191}, {28194, 37422}, {29594, 33297}, {32912, 32926}, {33126, 34379}, {37442, 40726}, {37870, 39948}

X(41629) = reflection of X(i) in X(j) for these (i, j): (1043, 4234), (1330, 16052), (4234, 58), (17677, 3017)
X(41629) = isotomic conjugate of X(4052)
X(41629) = barycentric product X(i)*X(j) for these {i, j}: {21, 39126}, {69, 4248}, {75, 16948}, {76, 33628}, {81, 18743}, {86, 145}
X(41629) = barycentric quotient X(i)/X(j) for these (i, j): (21, 3680), (58, 3445), (86, 4373), (110, 1293), (145, 10), (163, 34080)
X(41629) = trilinear product X(i)*X(j) for these {i, j}: {2, 16948}, {21, 5435}, {27, 4855}, {58, 18743}, {63, 4248}, {75, 33628}
X(41629) = trilinear quotient X(i)/X(j) for these (i, j): (58, 38266), (81, 3445), (99, 27834), (110, 34080), (145, 37), (274, 4373)
X(41629) = trilinear pole of the line {3667, 4881}
X(41629) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(37674)}} and {{A, B, C, X(2), X(145)}}
X(41629) = cevapoint of X(145) and X(1743)
X(41629) = crosssum of X(37) and X(21892)
X(41629) = X(333)-Beth conjugate of-X(4921)
X(41629) = X(1434)-Ceva conjugate of-X(86)
X(41629) = X(2)-Hirst inverse of-X(37792)
X(41629) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 38266}, {37, 3445}, {210, 40151}, {213, 4373}
X(41629) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (21, 3680), (58, 3445), (86, 4373), (110, 1293)
X(41629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 4921, 333), (2, 16704, 4921), (2, 19723, 17277), (2, 37652, 19723), (2, 37685, 19738), (6, 37683, 14829), (81, 333, 86), (81, 4921, 2), (81, 5235, 8025), (81, 5333, 26860), (81, 16704, 333), (193, 37642, 4417), (333, 25507, 5235), (940, 19723, 2), (940, 37652, 17277), (1150, 19738, 2), (1999, 4641, 190), (5235, 8025, 25507), (6189, 6190, 37792), (8025, 25507, 86), (37785, 37786, 31144)

X(41630) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(15) TO PEDAL OF X(32)

Barycentrics    (2*sqrt(3)*a^2*S+3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*(b^2+c^2) : :
X(41630) = 2*X(623)-3*X(22691) = 5*X(16960)-3*X(25157) = 5*X(16960)-X(25199) = 3*X(25157)-X(25199)

The reciprocal orthologic center of these triangles is X(41631)

X(41630) lies on these lines: {6, 6581}, {15, 385}, {39, 141}, {193, 3104}, {194, 628}, {299, 7757}, {396, 538}, {623, 22691}, {1916, 23001}, {2782, 41070}, {3106, 32451}, {3390, 35755}, {5306, 25187}, {5969, 9117}, {6783, 32552}, {7684, 41062}, {9112, 36784}, {11485, 23018}, {11542, 31711}, {12203, 14539}, {16960, 25157}, {18582, 25191}, {18806, 33466}, {23302, 33482}, {41624, 41643}

X(41630) = midpoint of X(15) and X(23000)
X(41630) = reflection of X(i) in X(j) for these (i, j): (25183, 396), (31711, 11542), (41062, 7684)
X(41630) = X(141)-Hirst inverse of-X(41640)
X(41630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 41622, 41640), (16960, 25199, 25157)

X(41631) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(15)

Barycentrics    (2*sqrt(3)*a^2*S+3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*(sqrt(3)*a^2+2*S) : :

The reciprocal orthologic center of these triangles is X(41630)

X(41631) lies on these lines: {6, 6582}, {62, 37334}, {99, 41408}, {299, 7870}, {385, 25183}, {396, 532}, {530, 5306}, {5477, 32552}, {5969, 41413}

X(41631) = {X(618), X(41620)}-harmonic conjugate of X(41633)

X(41632) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(15) TO PEDAL OF X(39)

Barycentrics    (2*a^4+3*(b^2+c^2)*a^2-b^4-c^4+2*(b^2+c^2)*S*sqrt(3))*(2*a^2+b^2+c^2) : :
X(41632) = 5*X(16960)-3*X(25158) = 5*X(16960)-X(25200) = 3*X(25158)-X(25200)

The reciprocal orthologic center of these triangles is X(41633)

X(41632) lies on these lines: {6, 6296}, {15, 23001}, {83, 303}, {396, 754}, {3390, 35756}, {3589, 5007}, {7684, 41064}, {9112, 36785}, {9300, 25188}, {11485, 23019}, {11542, 31713}, {16960, 25158}, {18582, 25192}, {23302, 33484}

X(41632) = midpoint of X(15) and X(23001)
X(41632) = reflection of X(i) in X(j) for these (i, j): (25184, 396), (31713, 11542), (41064, 7684)
X(41632) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6292, 41623, 41642), (16960, 25200, 25158)

X(41633) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(15)

Barycentrics    (2*a^4+3*(b^2+c^2)*a^2-b^4-c^4+2*(b^2+c^2)*S*sqrt(3))*(sqrt(3)*a^2+2*S) : :

The reciprocal orthologic center of these triangles is X(41632)

X(41633) lies on these lines: {6, 6298}, {13, 6054}, {303, 9112}, {396, 532}, {530, 9300}, {3329, 14904}, {38383, 41023}, {41624, 41640}

X(41633) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (618, 41620, 41631), (3329, 14904, 25184)

X(41634) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(15) TO PEDAL OF X(50)

Barycentrics
(2*((-a^2+b^2+c^2)^2-b^2*c^2)*a^2*S*sqrt(3)+3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41635)

X(41634) lies on these lines: {15, 5916}, {3003, 3580}

X(41634) = {X(34834), X(41625)}-harmonic conjugate of X(41644)

X(41635) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(15)

Barycentrics    (2*((-a^2+b^2+c^2)^2-b^2*c^2)*a^2*S*sqrt(3)+3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2)*(sqrt(3)*a^2+2*S) : :

The reciprocal orthologic center of these triangles is X(41634)

X(41635) lies on the line {396, 532}

X(41636) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(15) TO PEDAL OF X(52)

Barycentrics
(2*(2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*sqrt(3))*(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

The reciprocal orthologic center of these triangles is X(41637)

X(41636) lies on these lines: {396, 539}, {34835, 41627}

X(41636) = {X(34835), X(41627)}-harmonic conjugate of X(41646)

X(41637) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(15)

Barycentrics    (2*(2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*sqrt(3))*(sqrt(3)*a^2+2*S) : :

The reciprocal orthologic center of these triangles is X(41636)

X(41637) lies on the line {396, 532}

X(41638) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(15) TO PEDAL OF X(58)

Barycentrics    (2*sqrt(3)*S+(3*a-b-c)*(a+b+c))*(2*a+b+c) : :

The reciprocal orthologic center of these triangles is X(41639)

X(41638) lies on these lines: {396, 519}, {1100, 1125}, {2784, 41070}, {2796, 9117}, {3244, 5246}

X(41638) = barycentric product X(1125)*X(37795)
X(41638) = trilinear product X(1100)*X(37795)
X(41638) = X(1213)-Hirst inverse of-X(41648)
X(41638) = {X(1125), X(4856)}-harmonic conjugate of X(41648)

X(41639) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(15)

Barycentrics    (2*sqrt(3)*S+(3*a-b-c)*(a+b+c))*(sqrt(3)*a^2+2*S)*(a+b)*(a+c) : :

The reciprocal orthologic center of these triangles is X(41638)

X(41639) lies on these lines: {396, 532}, {2796, 41649}

X(41640) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(16) TO PEDAL OF X(32)

Barycentrics    (-2*sqrt(3)*a^2*S+3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*(b^2+c^2) : :
X(41640) = 2*X(624)-3*X(22692) = 5*X(16961)-3*X(25167) = 5*X(16961)-X(25203) = 3*X(25167)-X(25203)

The reciprocal orthologic center of these triangles is X(41641)

X(41640) lies on these lines: {6, 6294}, {16, 385}, {39, 141}, {193, 3105}, {194, 627}, {298, 7757}, {395, 538}, {624, 22692}, {1916, 23010}, {2782, 41071}, {3107, 32451}, {5306, 25183}, {5969, 9115}, {6782, 32553}, {7685, 41063}, {11486, 23024}, {11543, 31712}, {12203, 14538}, {16961, 25167}, {18581, 25195}, {18806, 33467}, {23303, 33483}, {41624, 41633}

X(41640) = midpoint of X(16) and X(23009)
X(41640) = reflection of X(i) in X(j) for these (i, j): (25187, 395), (31712, 11543), (41063, 7685)
X(41640) = X(141)-Hirst inverse of-X(41630)
X(41640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 41622, 41630), (16961, 25203, 25167)

X(41641) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(16)

Barycentrics    (-2*sqrt(3)*a^2*S+3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*(sqrt(3)*a^2-2*S) : :

The reciprocal orthologic center of these triangles is X(41640)

X(41641) lies on these lines: {6, 6295}, {61, 37334}, {99, 41409}, {298, 7870}, {385, 25187}, {395, 533}, {531, 5306}, {5477, 32553}, {5969, 41413}

X(41641) = {X(619), X(41621)}-harmonic conjugate of X(41643)

X(41642) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(16) TO PEDAL OF X(39)

Barycentrics    (2*a^4+3*(b^2+c^2)*a^2-b^4-c^4-2*(b^2+c^2)*S*sqrt(3))*(2*a^2+b^2+c^2) : :
X(41642) = 5*X(16961)-3*X(25168) = 5*X(16961)-X(25204) = 3*X(25168)-X(25204)

The reciprocal orthologic center of these triangles is X(41643)

X(41642) lies on these lines: {6, 6297}, {16, 23010}, {83, 302}, {395, 754}, {3589, 5007}, {7685, 41065}, {9300, 25184}, {11486, 23025}, {11543, 31714}, {16961, 25168}, {18581, 25196}, {23303, 33485}

X(41642) = midpoint of X(16) and X(23010)
X(41642) = reflection of X(i) in X(j) for these (i, j): (25188, 395), (31714, 11543), (41065, 7685)
X(41642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6292, 41623, 41632), (16961, 25204, 25168)

X(41643) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(16)

Barycentrics    (2*a^4+3*(b^2+c^2)*a^2-b^4-c^4-2*(b^2+c^2)*S*sqrt(3))*(sqrt(3)*a^2-2*S) : :

The reciprocal orthologic center of these triangles is X(41642)

X(41643) lies on these lines: {6, 6299}, {14, 6054}, {302, 9113}, {395, 533}, {531, 9300}, {3329, 14905}, {38383, 41022}, {41624, 41630}

X(41643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (619, 41621, 41641), (3329, 14905, 25188)

X(41644) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(16) TO PEDAL OF X(50)

Barycentrics
(-2*((-a^2+b^2+c^2)^2-b^2*c^2)*a^2*S*sqrt(3)+3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41645)

X(41644) lies on these lines: {16, 5917}, {3003, 3580}

X(41644) = {X(34834), X(41625)}-harmonic conjugate of X(41634)

X(41645) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(16)

Barycentrics    (-2*((-a^2+b^2+c^2)^2-b^2*c^2)*a^2*S*sqrt(3)+3*a^8-5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-2*(b^2-c^2)^2*b^2*c^2)*(sqrt(3)*a^2-2*S) : :

The reciprocal orthologic center of these triangles is X(41644)

X(41645) lies on the line {395, 533}

X(41646) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(16) TO PEDAL OF X(52)

Barycentrics
(-2*(2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*sqrt(3))*(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

The reciprocal orthologic center of these triangles is X(41647)

X(41646) lies on these lines: {395, 539}, {34835, 41627}

X(41646) = {X(34835), X(41627)}-harmonic conjugate of X(41636)

X(41647) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(16)

Barycentrics    (-2*(2*a^6-5*(b^2+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*S+((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*sqrt(3))*(sqrt(3)*a^2-2*S) : :

The reciprocal orthologic center of these triangles is X(41646)

X(41647) lies on the line {395, 533}

X(41648) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(16) TO PEDAL OF X(58)

Barycentrics    (-2*sqrt(3)*S+(3*a-b-c)*(a+b+c))*(2*a+b+c) : :

The reciprocal orthologic center of these triangles is X(41649)

X(41648) lies on these lines: {395, 519}, {1100, 1125}, {2784, 41071}, {2796, 9115}, {3244, 5245}

X(41648) = barycentric product X(1125)*X(37794)
X(41648) = trilinear product X(1100)*X(37794)
X(41648) = X(1213)-Hirst inverse of-X(41638)
X(41648) = {X(1125), X(4856)}-harmonic conjugate of X(41638)

X(41649) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(16)

Barycentrics    (-2*sqrt(3)*S+(3*a-b-c)*(a+b+c))*(sqrt(3)*a^2-2*S)*(a+b)*(a+c) : :

The reciprocal orthologic center of these triangles is X(41648)

X(41649) lies on these lines: {395, 533}, {2796, 41639}

X(41650) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(39)

Barycentrics    (2*a^2+b^2+c^2)*(a^6+3*(b^2+c^2)*a^4+b^2*c^2*a^2-(b^2+c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41651)

X(41650) lies on these lines: {6, 6308}, {32, 732}, {83, 183}, {385, 12206}, {754, 5306}, {2896, 16989}, {3091, 6287}, {3398, 37455}, {3589, 5007}, {6704, 7780}, {7766, 9821}, {7878, 31268}, {9755, 14880}

X(41651) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(32)

Barycentrics    (b^2+c^2)*(a^6+3*(b^2+c^2)*a^4+b^2*c^2*a^2-(b^2+c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41650)

X(41651) lies on these lines: {3, 32451}, {4, 147}, {6, 6309}, {32, 5026}, {39, 141}, {76, 9605}, {193, 9821}, {385, 12054}, {538, 9300}, {736, 8357}, {1353, 35430}, {2023, 7764}, {3094, 7758}, {5007, 13196}, {5188, 8550}, {5976, 7760}, {6308, 34873}, {6390, 13357}, {6392, 7697}, {7757, 7788}, {7772, 24256}, {7779, 32476}, {7786, 10159}, {7795, 13331}, {7798, 8149}, {7838, 12830}, {7839, 9865}, {8725, 14712}, {9983, 33202}, {11257, 14532}, {11646, 31982}, {14981, 35437}

X(41651) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(732)}} and {{A, B, C, X(39), X(17980)}}
X(41651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 7794, 10007), (39, 14994, 8362), (194, 13571, 1916)

X(41652) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(50)

Barycentrics
((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(2*a^10-3*(b^2+c^2)*a^8+(b^4+c^4)*a^6-(b^4-c^4)*(b^2-c^2)*a^4+(b^4+c^4)*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41653)

X(41652) lies on the line {3003, 3580}

X(41652) = {X(34834), X(41625)}-harmonic conjugate of X(41658)

X(41653) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(32)

Barycentrics    (b^2+c^2)*(2*a^10-3*(b^2+c^2)*a^8+(b^4+c^4)*a^6-(b^4-c^4)*(b^2-c^2)*a^4+(b^4+c^4)*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41652)

X(41653) lies on the line {39, 141}

X(41654) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(52)

Barycentrics
(a^8-2*(b^2+c^2)*a^6+(b^4-5*b^2*c^2+c^4)*a^4+2*(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^2*b^2*c^2)*(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

The reciprocal orthologic center of these triangles is X(41655)

X(41654) lies on these lines: {539, 5306}, {34835, 41627}

X(41654) = {X(34835), X(41627)}-harmonic conjugate of X(41660)

X(41655) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(32)

Barycentrics    (b^2+c^2)*(a^8-2*(b^2+c^2)*a^6+(b^4-5*b^2*c^2+c^4)*a^4+2*(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^2*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41654)

X(41655) lies on these lines: {39, 141}, {76, 5422}, {2782, 10263}

X(41656) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(32) TO PEDAL OF X(58)

Barycentrics    (2*a+b+c)*(2*a^3-(b+c)*a^2+(b^2+b*c+c^2)*a-(b+c)*b*c) : :
X(41656) = X(3905)+3*X(14614) = X(4136)-3*X(5306)

The reciprocal orthologic center of these triangles is X(41657)

X(41656) lies on these lines: {6, 8669}, {32, 726}, {519, 4136}, {1100, 1125}, {3053, 8720}, {3840, 16787}, {3905, 14614}, {4258, 32921}, {5008, 22036}, {5266, 17355}, {7296, 21101}, {7766, 17760}, {16780, 29649}

X(41656) = {X(1125), X(4856)}-harmonic conjugate of X(41662)

X(41657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(32)

Barycentrics    (b^2+c^2)*(2*a^3-(b+c)*a^2+(b^2+b*c+c^2)*a-(b+c)*b*c)*(a+c)*(a+b) : :

The reciprocal orthologic center of these triangles is X(41656)

X(41657) lies on these lines: {39, 141}, {58, 726}, {81, 17760}, {194, 1764}, {5021, 22253}, {16704, 33890}, {17351, 22036}

X(41657) = barycentric quotient X(1634)/X(28469)

X(41658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(50)

Barycentrics
((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^10+(b^2+c^2)*a^8-(5*b^4+7*b^2*c^2+5*c^4)*a^6+3*(b^2+c^2)*(b^4+c^4)*a^4+2*b^4*c^4*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41659)

X(41658) lies on the line {3003, 3580}

X(41658) = {X(34834), X(41625)}-harmonic conjugate of X(41652)

X(41659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(39)

Barycentrics    (2*a^2+b^2+c^2)*(a^10+(b^2+c^2)*a^8-(5*b^4+7*b^2*c^2+5*c^4)*a^6+3*(b^2+c^2)*(b^4+c^4)*a^4+2*b^4*c^4*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41658)

X(41659) lies on the line {3589, 5007}

X(41660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(52)

Barycentrics
(a^8-(b^2+c^2)*a^6-(2*b^4+7*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^6-c^6)*(b^2-c^2))*(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

The reciprocal orthologic center of these triangles is X(41661)

X(41660) lies on these lines: {539, 9300}, {34835, 41627}

X(41660) = {X(34835), X(41627)}-harmonic conjugate of X(41654)

X(41661) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(39)

Barycentrics    (2*a^2+b^2+c^2)*(a^8-(b^2+c^2)*a^6-(2*b^4+7*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^6-c^6)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41660)

X(41661) lies on these lines: {83, 37636}, {2896, 34545}, {3589, 5007}, {20088, 37779}

X(41662) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(39) TO PEDAL OF X(58)

Barycentrics    (2*a+b+c)*(a^3+(2*b^2-b*c+2*c^2)*a-b^3-c^3) : :
X(41662) = X(4095)-3*X(9300)

The reciprocal orthologic center of these triangles is X(41663)

X(41662) lies on these lines: {39, 17766}, {519, 4095}, {1100, 1125}, {2784, 38383}, {3178, 16784}, {16502, 29671}, {16825, 31405}

X(41662) = barycentric product X(1125)*X(29840)
X(41662) = trilinear product X(1100)*X(29840)
X(41662) = {X(1125), X(4856)}-harmonic conjugate of X(41656)

X(41663) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(39)

Barycentrics    (2*a^2+b^2+c^2)*(a^3+(2*b^2-b*c+2*c^2)*a-b^3-c^3)*(a+c)*(a+b) : :

The reciprocal orthologic center of these triangles is X(41662)

X(41663) lies on these lines: {58, 17766}, {83, 14829}, {3589, 5007}, {16704, 17741}

X(41664) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(52)

Barycentrics
(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4)*(a^12-4*(b^2+c^2)*a^10+(6*b^4+7*b^2*c^2+6*c^4)*a^8-(b^2+c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^6+(b^8+c^8)*a^4+(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41665)

X(41664) lies on the line {34835, 41627}

X(41665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(50)

Barycentrics
((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^12-4*(b^2+c^2)*a^10+(6*b^4+7*b^2*c^2+6*c^4)*a^8-(b^2+c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^6+(b^8+c^8)*a^4+(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2) : :

The reciprocal orthologic center of these triangles is X(41664)

X(41665) lies on these lines: {94, 275}, {1885, 5663}, {3003, 3580}

X(41666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(50) TO PEDAL OF X(58)

Barycentrics
(2*a+b+c)*(2*a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(2*b^2-b*c+2*c^2)*a^4-(2*b^2-b*c+2*c^2)*b*c*a^3-(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^2*(b+c)*b*c) : :

The reciprocal orthologic center of these triangles is X(41667)

X(41666) lies on the line {1100, 1125}

X(41667) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(50)

Barycentrics
(a+c)*(a+b)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(2*a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(2*b^2-b*c+2*c^2)*a^4-(2*b^2-b*c+2*c^2)*b*c*a^3-(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^2*(b+c)*b*c) : :

The reciprocal orthologic center of these triangles is X(41666)

X(41667) lies on the line {3003, 3580}

X(41668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(52) TO PEDAL OF X(58)

Barycentrics    (2*a+b+c)*(a^6+(b+c)*a^5-(3*b^2+b*c+3*c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+(3*b^4+3*c^4+2*(b^2+c^2)*b*c)*a^2+(b^2-c^2)^2*(b+c)*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :

The reciprocal orthologic center of these triangles is X(41669)

X(41668) lies on the line {1100, 1125}

X(41669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: PEDAL OF X(58) TO PEDAL OF X(52)

Barycentrics
(2*a^8-5*(b^2+c^2)*a^6+(5*b^4+2*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4)*(a^6+(b+c)*a^5-(3*b^2+b*c+3*c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+(3*b^4+3*c^4+2*(b^2+c^2)*b*c)*a^2+(b^2-c^2)^2*(b+c)*a-(b+c)*(b^2-c^2)*(b^3-c^3))*(a+c)*(a+b) : :

The reciprocal orthologic center of these triangles is X(41668)

X(41669) lies on the line {34835, 41627}

X(41670) = CENTER OF THE BPC-CONIC OF X(2)

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^2-c^2)^2*a^6-3*(b^2+c^2)*b^2*c^2*a^4+2*(b^8+c^8-3*(b^2-c^2)^2*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)) : :
X(41670) = 2*X(3)+X(16105) = 4*X(5)-X(15738) = 2*X(5)+X(25711) = X(52)+5*X(38795) = X(110)+3*X(5640) = X(110)+2*X(11746) = 2*X(113)+X(974) = X(113)+2*X(9826) = X(125)-3*X(373) = X(974)-4*X(9826) = 3*X(5640)-2*X(11746) = 2*X(6593)+X(32246) = 5*X(6593)+X(32299) = X(15030)+3*X(16223) = X(15030)-3*X(36518) = X(15063)+2*X(16270) = X(15738)+2*X(25711) = 5*X(16776)-X(32299) = X(20379)-4*X(32205) = 5*X(32246)-2*X(32299)

See the preamble just before X(41537).

Let A'B'C' be the orthic triangle. Let LA, LB, LC be the orthic axes of AB'C', BC'A', CA'B', resp. Let A" = LB∩LC, B" = LC∩LA, C" = LA∩LB. Triangle A"B"C" is inversely similar to ABC, with similitude center X(6). A"B"C" is perspective to ABC at X(11744), and to the orthic triangle at X(113). X(41670) is the centroid of A"B"C". (Randy Hutson, April 13, 2021)

X(41670) lies on the Hutson centroidal ellipse and these lines: {2, 2781}, {3, 16105}, {4, 36789}, {5, 113}, {6, 110}, {24, 15035}, {25, 15462}, {51, 5642}, {52, 38795}, {74, 37475}, {107, 41253}, {146, 15151}, {184, 20772}, {343, 12828}, {468, 511}, {526, 14697}, {541, 5892}, {542, 5943}, {568, 5654}, {1154, 32225}, {1511, 12106}, {1514, 10297}, {1637, 9517}, {1986, 11459}, {2393, 35266}, {2777, 16836}, {2842, 37999}, {2935, 7503}, {3233, 12052}, {3448, 14982}, {3527, 5504}, {4846, 7728}, {5050, 13198}, {5085, 10117}, {5462, 16534}, {5544, 10620}, {5609, 15026}, {5621, 17825}, {5622, 10601}, {5650, 13416}, {5907, 13148}, {6090, 19504}, {6388, 15544}, {6723, 37454}, {6800, 15647}, {7426, 9019}, {7493, 40949}, {7528, 15133}, {7575, 10564}, {8705, 15448}, {9027, 35370}, {9140, 11451}, {9155, 23181}, {9306, 34155}, {9729, 38791}, {9781, 15034}, {9970, 11284}, {9971, 26255}, {10170, 11557}, {10272, 12236}, {10706, 15045}, {11002, 37645}, {11694, 13451}, {11695, 20417}, {11744, 13203}, {12041, 22112}, {12111, 15029}, {12228, 20771}, {12292, 16261}, {12825, 37643}, {12827, 13567}, {13201, 33879}, {13340, 38794}, {13754, 16227}, {14094, 15024}, {14787, 15061}, {15028, 15054}, {15113, 34146}, {15303, 29959}, {15329, 34990}, {16163, 21971}, {16657, 17702}, {18374, 37980}, {19136, 41612}, {22151, 37962}, {23410, 32423}, {30739, 32271}, {34351, 38793}, {41614, 41618}

X(41670) = midpoint of X(i) and X(j) for these {i, j}: {2, 12824}, {51, 5642}, {113, 9730}, {1986, 11459}, {6593, 16776}, {10170, 11557}, {11694, 13451}, {14643, 16222}, {15303, 29959}, {16223, 36518}
X(41670) = reflection of X(i) in X(j) for these (i, j): (974, 9730), (9730, 9826), (10170, 12900), (12099, 5943), (12358, 10170), (32246, 16776)
X(41670) = crossdifference of every pair of points on line {X(690), X(15106)}
X(41670) = X(2)-of-pedal-triangle-of-X(113)
X(41670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 25711, 15738), (25, 15462, 16165), (110, 35904, 35259), (113, 9826, 974), (373, 9730, 37648), (1112, 5972, 41673), (5972, 41671, 1112), (11557, 12900, 12358), (15035, 15472, 37497)

X(41671) = CENTER OF THE BPC-CONIC OF X(5)

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^4-b^2*c^2+c^4)*a^6-(b^2+c^2)*b^2*c^2*a^4+(2*b^8+2*c^8-(5*b^4-8*b^2*c^2+5*c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(41671) = 9*X(2)-X(13201) = 3*X(2)+X(13417) = X(4)+3*X(16223) = 3*X(5)+X(38898) = 3*X(51)+X(110) = 3*X(51)-X(11800) = X(52)+3*X(14643) = X(113)+3*X(16222) = X(125)-3*X(5943) = X(125)+3*X(12824) = X(146)+7*X(15043) = 9*X(373)-5*X(15059) = 3*X(381)+X(11562) = X(389)-3*X(16222) = X(1112)+3*X(41670) = 3*X(1112)+X(41673) = X(5972)-3*X(41670) = 3*X(5972)-X(41673) = 3*X(11557)-X(38898) = X(13201)+3*X(13417)

X(41671) lies on these lines: {2, 13201}, {3, 11807}, {4, 16223}, {5, 10628}, {23, 27866}, {51, 110}, {52, 14643}, {113, 389}, {125, 5133}, {143, 10272}, {146, 15043}, {182, 10117}, {373, 15059}, {381, 11562}, {468, 511}, {512, 12064}, {542, 11746}, {546, 11561}, {575, 13198}, {974, 15012}, {1205, 3618}, {1352, 18947}, {1511, 5446}, {1539, 40647}, {1624, 18114}, {1986, 5907}, {2777, 9729}, {2781, 6688}, {3047, 13366}, {3090, 7731}, {3091, 21650}, {3448, 3818}, {3521, 22816}, {3545, 12281}, {3567, 21649}, {3832, 12270}, {3850, 5462}, {5020, 17847}, {5095, 14913}, {5642, 21849}, {5890, 18418}, {5892, 12041}, {5946, 11806}, {6000, 14708}, {6153, 11702}, {6593, 9969}, {6699, 11695}, {7687, 25711}, {7722, 15030}, {7728, 9730}, {9306, 19504}, {9781, 12383}, {9919, 37514}, {10095, 13163}, {10110, 17702}, {10114, 12134}, {10264, 15026}, {10282, 12228}, {10601, 13171}, {10625, 38794}, {10706, 16226}, {11432, 17838}, {11479, 17835}, {11574, 40949}, {11597, 11808}, {11649, 15448}, {11793, 12900}, {11801, 13364}, {12099, 13402}, {12236, 16534}, {12244, 15045}, {12310, 17810}, {13365, 23409}, {13376, 18400}, {13598, 16163}, {15036, 36987}, {15038, 15089}, {15046, 18436}, {15141, 19137}, {15644, 38793}, {16105, 17704}, {16111, 16836}, {16776, 25329}, {21451, 32352}, {25321, 32260}, {30551, 32196}, {32205, 40685}, {32438, 40641}, {37481, 38789}, {41579, 41595}

X(41671) = midpoint of X(i) and X(j) for these {i, j}: {3, 11807}, {5, 11557}, {110, 11800}, {113, 389}, {143, 10272}, {546, 11561}, {974, 38791}, {1112, 5972}, {1511, 5446}, {1539, 40647}, {1986, 5907}, {5095, 14913}, {5642, 21849}, {5943, 12824}, {6153, 11702}, {6593, 9969}, {7687, 25711}, {7728, 17855}, {10114, 12134}, {11574, 40949}, {11597, 11808}, {11802, 11805}, {12236, 16534}, {13598, 16163}, {16105, 37853}, {41579, 41595}
X(41671) = reflection of X(i) in X(j) for these (i, j): (974, 15012), (6699, 11695), (9729, 9826), (11793, 12900), (37853, 17704), (40685, 32205)
X(41671) = complement of the complement of X(13417)
X(41671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (51, 110, 11800), (113, 16222, 389), (1112, 41670, 5972), (1986, 36518, 5907), (7728, 9730, 17855)

X(41672) = CENTER OF THE BPC-CONIC OF X(6)

Barycentrics    6*a^6-6*(b^2+c^2)*a^4+(5*b^4-4*b^2*c^2+5*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(41672) = 3*X(6)-X(115) = 7*X(6)-3*X(6034) = 5*X(6)-X(11646) = X(99)+3*X(1992) = X(99)-3*X(18800) = X(115)+3*X(5477) = 7*X(115)-9*X(6034) = 5*X(115)-3*X(11646) = X(148)-9*X(5032) = X(148)+3*X(8593) = X(193)+3*X(5182) = 3*X(597)-2*X(6722) = 3*X(599)-5*X(31274) = 3*X(1351)+X(38730) = 3*X(5032)+X(8593) = 7*X(5477)+3*X(6034) = 5*X(5477)+X(11646) = 15*X(6034)-7*X(11646) = 4*X(6722)-3*X(19662) = 3*X(6776)+X(10722)

X(41672) lies on these lines: {2, 10153}, {6, 13}, {32, 32135}, {39, 33749}, {98, 37665}, {99, 1285}, {110, 6791}, {112, 5095}, {114, 7735}, {148, 5032}, {193, 5182}, {511, 38736}, {524, 620}, {543, 8584}, {575, 3815}, {576, 7737}, {597, 6722}, {599, 31274}, {1249, 20774}, {1351, 38730}, {1384, 2482}, {1562, 6776}, {1569, 13330}, {1570, 29012}, {1648, 5972}, {1692, 5965}, {2079, 12584}, {2393, 39835}, {2548, 20398}, {2794, 8550}, {3068, 13760}, {3069, 13640}, {3292, 5913}, {3629, 5026}, {5039, 12177}, {5050, 38739}, {5093, 38733}, {5107, 19924}, {5111, 29317}, {5304, 6054}, {5305, 38745}, {5334, 41061}, {5335, 41060}, {5642, 6792}, {5969, 32455}, {6055, 7736}, {6321, 11482}, {6721, 34507}, {6781, 8586}, {6793, 11005}, {7603, 25555}, {7745, 22330}, {7813, 12151}, {7845, 41146}, {7890, 38905}, {8724, 21309}, {9115, 41407}, {9117, 41406}, {9167, 15533}, {9605, 10991}, {9830, 20583}, {9862, 10753}, {10418, 39689}, {10542, 38741}, {10754, 14928}, {11177, 14930}, {11179, 38749}, {11477, 38738}, {13196, 16385}, {13639, 13761}, {13642, 13759}, {13645, 13765}, {13646, 13764}, {13653, 19054}, {13773, 19053}, {14853, 31683}, {14981, 30435}, {15069, 36519}, {15092, 18583}, {15471, 16183}, {19108, 26289}, {19109, 26288}, {20423, 39809}, {22165, 22247}, {23234, 37689}, {24206, 39764}, {24981, 39024}, {32305, 34866}, {32621, 39857}, {36521, 41149}, {39846, 40673}

X(41672) = midpoint of X(i) and X(j) for these {i, j}: {6, 5477}, {1569, 13330}, {1992, 18800}, {2482, 15534}, {3629, 5026}, {6781, 8586}, {8584, 8787}, {10754, 14928}, {11477, 38738}, {41620, 41621}
X(41672) = reflection of X(i) in X(j) for these (i, j): (6036, 575), (19662, 597), (22165, 22247), (34507, 6721)
X(41672) = {X(9112), X(9113)}-harmonic conjugate of X(115)

X(41673) = CENTER OF THE BPC-CONIC OF X(20)

Barycentrics    a^2*(-a^2+b^2+c^2)*((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+(b^6-c^6)*(b^2-c^2)) : :
X(41673) = X(52)-3*X(38793) = X(110)+3*X(2979) = 3*X(110)+X(13201) = X(125)-3*X(3917) = 3*X(376)-X(17854) = 2*X(1112)-3*X(41670) = 3*X(1112)-4*X(41671) = 4*X(1216)-X(15738) = 3*X(1511)-X(38898) = 9*X(2979)-X(13201) = X(3448)-9*X(33884) = 3*X(3917)-2*X(13416) = 4*X(5972)-3*X(41670) = 3*X(5972)-2*X(41671) = 2*X(6101)+X(25711) = 3*X(6101)+X(38898) = 2*X(9826)-3*X(38793) = 2*X(10625)+X(16105) = 3*X(25711)-2*X(38898) = 9*X(41670)-8*X(41671)

X(41673) lies on these lines: {2, 11746}, {3, 974}, {20, 12825}, {22, 110}, {26, 20771}, {52, 9826}, {67, 69}, {113, 10625}, {125, 343}, {140, 12236}, {141, 32246}, {323, 37978}, {376, 17854}, {468, 511}, {542, 7667}, {550, 5562}, {895, 34817}, {1092, 1511}, {1154, 14708}, {1177, 37485}, {1205, 32114}, {1216, 12358}, {1568, 11563}, {1986, 11412}, {2777, 15644}, {2972, 23181}, {3047, 6636}, {3313, 5181}, {3580, 32282}, {3819, 6723}, {4549, 7723}, {5169, 41579}, {5446, 12900}, {5447, 6699}, {5642, 10154}, {5889, 15051}, {5890, 15036}, {5891, 12295}, {5907, 12133}, {6000, 20725}, {6243, 16222}, {6333, 9517}, {6593, 20806}, {7687, 11793}, {7728, 13340}, {7731, 15034}, {7998, 15059}, {7999, 14644}, {9019, 32113}, {9934, 11414}, {9973, 31099}, {10111, 11577}, {10113, 15067}, {10574, 14528}, {10628, 15606}, {10733, 11444}, {11413, 11598}, {11449, 41589}, {11459, 12292}, {11720, 31737}, {11744, 37201}, {12228, 16266}, {12273, 15055}, {12824, 37669}, {12827, 23315}, {12902, 40912}, {13293, 37480}, {13348, 37853}, {13394, 37511}, {13562, 32239}, {13754, 38726}, {14643, 37484}, {15462, 19504}, {15472, 37498}, {15739, 31833}, {16186, 23217}, {16270, 21649}, {18436, 38723}, {20304, 32142}, {21766, 41614}, {23061, 27866}, {23332, 27365}, {28419, 40949}, {35360, 36789}, {37473, 37645}

X(41673) = midpoint of X(i) and X(j) for these {i, j}: {20, 12825}, {113, 10625}, {1205, 32114}, {1511, 6101}, {1986, 11412}, {3313, 5181}, {5562, 16163}, {7723, 12121}, {11720, 31737}
X(41673) = reflection of X(i) in X(j) for these (i, j): (52, 9826), (125, 13416), (974, 3), (1112, 5972), (5446, 12900), (6699, 5447), (7687, 11793), (11800, 6723), (12099, 3819), (12133, 5907), (12236, 140), (12358, 1216), (15738, 12358), (16105, 113), (20304, 32142), (21649, 16270), (25711, 1511), (32239, 13562), (32246, 141), (37853, 13348)
X(41673) = anticomplement of X(11746)
X(41673) = crosspoint of X(69) and X(249)
X(41673) = crosssum of X(25) and X(115)
X(41673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (22, 110, 15647), (52, 38793, 9826), (69, 4576, 36793), (125, 3917, 13416), (1092, 22109, 1511), (1112, 5972, 41670), (3819, 11800, 6723), (5972, 41674, 468), (6243, 38794, 16222), (6723, 11800, 12099), (11412, 15035, 1986), (12121, 23039, 7723), (21649, 38727, 16270), (25407, 25408, 394)

X(41674) = CENTER OF THE BPC-CONIC OF X(26)

Barycentrics    S^4+(R^2*(96*R^2-5*SA-48*SW)+SA^2-SB*SC+6*SW^2)*S^2+(13*R^2-3*SW)*SB*SC*SW : :
Barycentrics
2*a^12-2*(b^2+c^2)*a^10-(5*b^4-8*b^2*c^2+5*c^4)*a^8+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^6+2*(b^8+c^8-2*(3*b^4-4*b^2*c^2+3*c^4)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41674) = 3*X(10154)-X(15647) = X(13346)-3*X(38793) = 3*X(14643)+X(17834) = X(37498)-5*X(38794)

X(41674) lies on these lines: {22, 125}, {468, 511}, {542, 10154}, {550, 6699}, {1368, 6723}, {1658, 17702}, {6676, 11746}, {7493, 13198}, {7687, 12605}, {13346, 38793}, {14643, 17834}, {15059, 16063}, {15113, 29317}, {15118, 17710}, {15646, 38726}, {16163, 32534}, {16534, 38898}, {19121, 32285}, {37498, 38794}

X(41674) = {X(468), X(41673)}-harmonic conjugate of X(5972)

X(41675) = CENTER OF THE BPC-CONIC OF X(32)

Barycentrics    4*a^8-3*(b^2+c^2)*a^6+3*(b^2-c^2)^2*a^4+(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(41675) = X(99)+3*X(14614) = X(115)-3*X(5306)

X(41675) lies on these lines: {6, 12042}, {30, 2031}, {32, 2782}, {98, 30435}, {99, 1384}, {112, 5186}, {115, 3845}, {148, 1285}, {187, 33694}, {524, 620}, {609, 3023}, {732, 16385}, {1569, 35007}, {1625, 9427}, {2023, 5007}, {2032, 3564}, {2548, 34127}, {2794, 5305}, {2871, 39835}, {3027, 7031}, {3053, 33813}, {3767, 22505}, {5026, 41412}, {5286, 38741}, {5304, 9862}, {5475, 15092}, {5969, 41413}, {5976, 6179}, {6033, 7735}, {7736, 38739}, {7737, 22515}, {7738, 38742}, {7787, 36864}, {9605, 34473}, {10312, 12131}, {12188, 21309}, {14061, 15484}, {15048, 38749}, {31406, 38737}

X(41675) = midpoint of X(i) and X(j) for these {i, j}: {32, 12829}, {41631, 41641}

X(41676) = TRILINEAR POLE OF THE BPC-LINE OF X(2)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(b^2+c^2)*(a^2-b^2)*(a^2-c^2) : :
X(41676) = 3*X(376)-2*X(38553)

See the preamble just before X(41537).

X(41676) lies on these lines: {2, 339}, {4, 147}, {20, 12253}, {24, 7754}, {25, 8267}, {39, 1235}, {76, 39575}, {99, 112}, {107, 907}, {110, 1289}, {132, 14981}, {186, 385}, {193, 1986}, {232, 538}, {250, 827}, {264, 7757}, {325, 5523}, {376, 3164}, {378, 9308}, {420, 40858}, {427, 31125}, {451, 1655}, {523, 7482}, {525, 1625}, {620, 6103}, {698, 2211}, {811, 4583}, {826, 35362}, {1112, 25047}, {1249, 32817}, {1843, 41622}, {1897, 4238}, {1968, 7781}, {1975, 8743}, {2073, 20045}, {2396, 4609}, {2966, 14586}, {3266, 14580}, {3520, 7783}, {3542, 6392}, {3565, 30247}, {3615, 25240}, {3732, 4244}, {3793, 37931}, {3926, 34129}, {4226, 35311}, {4230, 11794}, {4576, 35325}, {5094, 31088}, {5286, 40691}, {5938, 21458}, {5999, 41377}, {6198, 25264}, {6240, 7762}, {6241, 9289}, {6353, 7665}, {6390, 16318}, {6403, 32451}, {6995, 20099}, {7473, 14611}, {7476, 17494}, {7576, 27377}, {7577, 7777}, {7760, 10312}, {7779, 40889}, {7793, 21844}, {7798, 10311}, {7823, 34797}, {7837, 18559}, {8024, 40938}, {8744, 15014}, {9512, 39832}, {9870, 37962}, {11055, 33885}, {11107, 25237}, {12215, 41363}, {13219, 35923}, {13619, 14712}, {14015, 17148}, {14900, 38738}, {14961, 30737}, {17907, 32833}, {19570, 37943}, {20065, 35471}, {20094, 40890}, {25238, 35193}, {28728, 32818}, {32661, 34211}, {32815, 41370}, {32820, 41366}, {33294, 37937}, {35486, 37667}, {35925, 40807}

X(41676) = reflection of X(i) in X(j) for these (i, j): (4, 2967), (30737, 14961)
X(41676) = anticomplement of X(339)
X(41676) = isotomic conjugate of X(4580)
X(41676) = polar conjugate of the anticomplement of X(3005)
X(41676) = barycentric product X(i)*X(j) for these {i, j}: {4, 4576}, {27, 4568}, {38, 811}, {39, 6331}, {76, 35325}, {99, 427}
X(41676) = barycentric quotient X(i)/X(j) for these (i, j): (25, 18105), (27, 10566), (28, 18108), (38, 656), (39, 647), (92, 18070)
X(41676) = trilinear product X(i)*X(j) for these {i, j}: {19, 4576}, {27, 4553}, {28, 4568}, {38, 648}, {39, 811}, {75, 35325}
X(41676) = trilinear quotient X(i)/X(j) for these (i, j): (19, 18105), (27, 18108), (38, 647), (39, 810), (99, 34055), (141, 656)
X(41676) = trilinear pole of the line {141, 427}
X(41676) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(827)}} and {{A, B, C, X(39), X(2396)}}
X(41676) = pole of the trilinear polar of X(250) with respect to Steiner circumellipse
X(41676) = pole wrt polar circle of line X(115)X(804)
X(41676) = cevapoint of X(i) and X(j) for these (i, j): {39, 826}, {141, 2525}, {428, 2489}, {523, 1194}
X(41676) = crossdifference of every pair of points on line {X(20975), X(23216)}
X(41676) = crosspoint of X(648) and X(6331)
X(41676) = crosssum of X(647) and X(3049)
X(41676) = X(i)-anticomplementary conjugate of-X(j) for these (i, j): (112, 21294), (163, 13219), (250, 6327), (1101, 1370)
X(41676) = X(i)-Ceva conjugate of-X(j) for these (i, j): (250, 2), (648, 35325), (935, 4235)
X(41676) = X(i)-cross conjugate of-X(j) for these (i, j): (826, 1235), (1634, 4576)
X(41676) = X(648)-Daleth conjugate of-X(877)
X(41676) = X(i)-isoconjugate-of-X(j) for these {i, j}: {63, 18105}, {71, 18108}, {82, 647}, {83, 810}
X(41676) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (25, 18105), (27, 10566), (28, 18108), (38, 656)
X(41676) = X(648)-Waw conjugate of-X(35360)
X(41676) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 1235, 37125), (99, 112, 4235), (99, 648, 112), (648, 14570, 41678), (648, 41677, 14570), (2971, 5186, 4), (9308, 31859, 378)

X(41677) = TRILINEAR POLE OF THE BPC-LINE OF X(5)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*((b^2+c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

X(41677) lies on these lines: {2, 9381}, {99, 112}, {110, 1288}, {930, 933}, {3164, 40897}, {6331, 30450}, {16039, 18315}, {35360, 36829}

X(41677) = barycentric product X(i)*X(j) for these {i, j}: {99, 1594}, {107, 1238}, {570, 6331}, {648, 37636}, {933, 1225}, {1209, 18831}
X(41677) = barycentric quotient X(i)/X(j) for these (i, j): (107, 1179), (110, 40441), (162, 2216), (570, 647), (648, 40393), (933, 1166)
X(41677) = trilinear product X(i)*X(j) for these {i, j}: {162, 37636}, {570, 811}, {662, 1594}, {823, 1216}, {1238, 24019}
X(41677) = trilinear quotient X(i)/X(j) for these (i, j): (570, 810), (648, 2216), (662, 40441), (811, 40393), (823, 1179), (1216, 822)
X(41677) = trilinear pole of the line {1209, 1216}
X(41677) = cevapoint of X(647) and X(32377)
X(41677) = crosspoint of X(648) and X(38342)
X(41677) = X(933)-anticomplementary conjugate of-X(21294)
X(41677) = X(i)-isoconjugate-of-X(j) for these {i, j}: {647, 2216}, {661, 40441}, {810, 40393}, {822, 1179}
X(41677) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (107, 1179), (110, 40441), (162, 2216), (570, 647)
X(41677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (99, 648, 41679), (648, 14570, 16237), (14570, 41676, 648)

X(41678) = TRILINEAR POLE OF THE BPC-LINE OF X(20)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :

X(41678) lies on these lines: {4, 1942}, {99, 112}, {107, 1301}, {250, 39190}, {401, 15262}, {523, 37937}, {823, 17926}, {1249, 3164}, {1632, 2409}, {2404, 15352}, {3003, 37778}, {15329, 35360}, {15412, 15459}, {18667, 18687}, {26645, 41083}

X(41678) = anticomplement of the polar conjugate of X(15384)
X(41678) = polar conjugate of the isogonal conjugate of X(1624)
X(41678) = barycentric product X(i)*X(j) for these {i, j}: {99, 235}, {107, 41005}, {162, 17858}, {185, 6528}, {264, 1624}, {648, 13567}
X(41678) = barycentric quotient X(i)/X(j) for these (i, j): (107, 1105), (162, 775), (185, 520), (235, 523), (648, 801), (774, 656)
X(41678) = trilinear product X(i)*X(j) for these {i, j}: {92, 1624}, {107, 6508}, {112, 17858}, {162, 13567}, {185, 823}, {235, 662}
X(41678) = trilinear quotient X(i)/X(j) for these (i, j): (185, 822), (235, 661), (648, 775), (774, 647), (800, 810), (811, 801)
X(41678) = trilinear pole of the line {185, 235}
X(41678) = intersection, other than A,B,C, of conics {{A, B, C, X(107), X(36841)}} and {{A, B, C, X(235), X(4235)}}
X(41678) = cevapoint of X(647) and X(1885)
X(41678) = crosspoint of X(648) and X(15352)
X(41678) = crosssum of X(i) and X(j) for these (i, j): {520, 22089}, {647, 32320}
X(41678) = X(1301)-anticomplementary conjugate of-X(21294)
X(41678) = X(i)-isoconjugate-of-X(j) for these {i, j}: {647, 775}, {801, 810}, {821, 32320}, {822, 1105}
X(41678) = X(4)-line conjugate of-X(20975)
X(41678) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (107, 1105), (162, 775), (185, 520), (235, 523)
X(41678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (648, 14570, 41676), (648, 16237, 14570), (648, 41679, 2407), (1632, 32713, 2409)

X(41679) = TRILINEAR POLE OF THE BPC-LINE OF X(26)

Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*(a^2-c^2)*(a^2-b^2+c^2)*(a^2-b^2)*(a^2+b^2-c^2) : :

In the plane of a triangle ABC, let
MaMbMc = medial triangle
DEF = orthic triangle
H = X(4) = orthocenter
Hb = line through H perpendicular to BH
Hab = Hb∩BC
Hc = line through H perpendicular to CH
Hac = Hc∩BC
Pb = line through Hac parallel to DMb
Pc = line through Hab parallel to DMc
Ab = Pb∩AC; define Bc and Ca cyclically
Ac = Pc∩AB; define Ba and Cb cyclically
A'B'C' = triangle with edges AbAc, BcBa, CaCb
Then X(41679) is the unique finite fixed point of the affine transformation that maps ABC onto A'B'C'. The triangles ABC and A'B'C' are perspective, and their perspector is X(51776). The perspector of the perspeconic of ABC and A'B'C' is X(51777), and the center of that conic is X(51778). (Angel Montesdeoca, September 12, 2022)

X(41679) lies on these lines: {2, 36423}, {50, 297}, {99, 112}, {107, 13398}, {110, 933}, {186, 22463}, {250, 7468}, {317, 571}, {338, 38872}, {467, 18883}, {577, 17907}, {687, 15352}, {691, 39382}, {1576, 4230}, {1968, 33007}, {2965, 27377}, {2967, 22085}, {6528, 16813}, {6748, 34989}, {8882, 32002}, {15143, 23200}, {34990, 39176}

X(41679) = barycentric product X(i)*X(j) for these {i, j}: {24, 99}, {47, 811}, {52, 18831}, {107, 9723}, {110, 317}, {112, 7763}
X(41679) = barycentric quotient X(i)/X(j) for these (i, j): (24, 523), (47, 656), (52, 6368), (99, 20563), (107, 847), (110, 68)
X(41679) = trilinear product X(i)*X(j) for these {i, j}: {24, 662}, {47, 648}, {110, 1748}, {162, 1993}, {163, 317}, {467, 36134}
X(41679) = trilinear quotient X(i)/X(j) for these (i, j): (24, 661), (47, 647), (110, 1820), (162, 2165), (163, 2351), (250, 36145)
X(41679) = trilinear pole of the line {24, 52}
X(41679) = intersection, other than A,B,C, of conics {{A, B, C, X(24), X(4235)}} and {{A, B, C, X(99), X(18315)}}
X(41679) = cevapoint of X(i) and X(j) for these (i, j): {24, 6753}, {571, 924}
X(41679) = X(1288)-anticomplementary conjugate of-X(21294)
X(41679) = X(924)-cross conjugate of-X(317)
X(41679) = X(i)-isoconjugate-of-X(j) for these {i, j}: {68, 661}, {91, 647}, {125, 36145}, {523, 1820}
X(41679) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (24, 523), (47, 656), (52, 6368), (99, 20563)
X(41679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (99, 648, 41677), (112, 4558, 648), (648, 14590, 4558), (2407, 41678, 648), (39298, 39299, 16237)

X(41680) = X(1)X(1362)∩X(8)X(144)

Barycentrics    a*(a^5*b - 6*a^3*b^3 + 8*a^2*b^4 - 3*a*b^5 + a^5*c - 5*a^4*b*c + 8*a^3*b^2*c - 4*a^2*b^3*c - a*b^4*c + b^5*c + 8*a^3*b*c^2 - 8*a^2*b^2*c^2 + 4*a*b^3*c^2 - 4*b^4*c^2 - 6*a^3*c^3 - 4*a^2*b*c^3 + 4*a*b^2*c^3 + 6*b^3*c^3 + 8*a^2*c^4 - a*b*c^4 - 4*b^2*c^4 - 3*a*c^5 + b*c^5) : :
X(41680) = 3 X[165] - 2 X[170], 3 X[165] - 4 X[3730], 3 X[1699] - 2 X[17753], 3 X[1699] - 4 X[34848], 8 X[2140] - 9 X[7988]

X(41680) lies on these lines: {1,1362}, {8,144}, {165,170}, {169,39156}, {1334,1742}, {1699,17753}, {1709,17742}, {2082,9355}, {2140,7988}, {7987,8835}, {8245,37225}, {15726,21872}

X(41680) = reflection of X(i) in X(j) for these {i,j}: {170, 3730}, {17753, 34848}}. X(41680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{170, 3730, 165}, {17753, 34848, 1699}

X(41681) = X(2)X(6)∩X(9)X(18046)

Barycentrics    a^4*b - a^2*b^3 + a^4*c + 2*a^3*b*c - a^2*b^2*c - a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 : :

X(41681) lies on these lines: {2,6}, {9,18046}, {75,16549}, {190,34017}, {239,21858}, {292,39693}, {350,40586}, {583,20913}, {672,1269}, {1500,4360}, {1964,3216}, {4283,32922}, {10449,16301}, {16552,18133}, {16574,29484}, {17031,21035}, {17755,18137}, {18143,18206}, {18147,29455}, {20367,29756}, {21038,29673}, {22277,26237}

X(41681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 29453, 29559}, {9, 29561, 18046}

X(41682) = X(1)X(2390)∩X(6)X(41)

Barycentrics    a^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - 2*a^3*b*c + 2*a*b^3*c + a^3*c^2 - 6*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - c^5) : :
X(41682) = 3 X[354] - X[1828]

X(41682) lies on these lines: {1,2390}, {6,41}, {35,36058}, {78,9026}, {354,1828}, {375,25524}, {513,12053}, {517,550}, {1106,2933}, {1319,23154}, {1496,1626}, {2098,38496}, {3057,3937}, {3304,26892}, {3436,18141}, {3784,12513}, {4315,34434}, {4322,23846}, {8666,11573}, {10544,21342}, {10915,35059}, {13540,30618}, {14923,26910}, {16980,32636}, {18191,23675}

X(41683) = X(1)X(190)∩X(10)X(3122)

Barycentrics    (b + c)*(-(a*b) + 2*a*c - b*c)*(-2*a*b + a*c + b*c) : :
X(41683) = 5 X[4687] - 4 X[24003]

X(41683) lies on these lines: {1,190}, {10,3122}, {19,1897}, {37,3121}, {65,4552}, {75,244}, {145,25048}, {192,13476}, {291,24004}, {313,22214}, {335,876}, {513,31061}, {536,3999}, {596,3976}, {668,24517}, {714,3994}, {726,39697}, {739,835}, {740,4674}, {759,898}, {889,4639}, {897,4607}, {994,2802}, {1015,24487}, {1910,34075}, {3123,18150}, {3644,39742}, {4010,4080}, {4043,22045}, {4145,27809}, {4389,39712}, {4613,40747}, {4632,40438}, {4687,17038}, {9263,24482}, {18832,20943}, {21900,22227}, {24575,29396}, {28611,39708}

X(41683) = midpoint of X(192) and X(17154)
X(41683) = reflection of X(i) in X(j) for these {i,j}: {75, 244}, {3952, 37}
X(41683) = X(i)-cross conjugate of X(j) for these (i,j): {714, 75}, {3994, 10}
X(41683) = cevapoint of X(10) and X(3994)
X(41683) = crosspoint of X(3227) and X(31002)
X(41683) = trilinear pole of line {10, 661}
X(41683) = X(i)-isoconjugate of X(j) for these (i,j): {58, 899}, {81, 3230}, {99, 890}, {110, 891}, {163, 4728}, {536, 1333}, {662, 3768}, {849, 3994}, {1408, 4009}, {1646, 4567}, {2206, 6381}, {3733, 23343}, {4465, 18268}, {4526, 4565}, {4570, 19945}, {4591, 14437}
X(41683) = barycentric product X(i)*X(j) for these {i,j}: {10, 3227}, {37, 31002}, {190, 35353}, {226, 36798}, {313, 739}, {321, 37129}, {523, 4607}, {661, 889}, {850, 34075}, {898, 1577}, {3120, 5381}, {4080, 36872}, {20948, 32718}, {23892, 27808}
X(41683) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 536}, {37, 899}, {42, 3230}, {313, 35543}, {321, 6381}, {512, 3768}, {523, 4728}, {594, 3994}, {661, 891}, {739, 58}, {740, 4465}, {798, 890}, {889, 799}, {898, 662}, {1018, 23343}, {2321, 4009}, {3122, 1646}, {3125, 19945}, {3227, 86}, {3700, 14430}, {3952, 23891}, {3994, 13466}, {4010, 14433}, {4024, 14431}, {4033, 41314}, {4041, 4526}, {4079, 14404}, {4120, 30583}, {4607, 99}, {4730, 14437}, {4931, 28603}, {4988, 30592}, {5257, 4706}, {5381, 4600}, {13576, 36816}, {21834, 14426}, {23892, 3733}, {31002, 274}, {32718, 163}, {34075, 110}, {35353, 514}, {36798, 333}, {36872, 16704}, {37129, 81}
X(41683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3122, 21100, 4033}, {3227, 36798, 37129}, {36798, 37129, 36872}

leftri

Pairs of orthologic reflection triangles: X(41684)-X(41756)

rightri

This preamble and centers X(41684)-X(41756) were contributed by César Eliud Lozada, March 08, 2021.

If P is point on the plane of ABC, the reflections of P in the sidelines of ABC are the vertices of a new triangle named the reflection triangle of P.

Let P', P" be any two distinct points on the plane of a triangle ABC and such that they are collinear with its circumcenter O=X(3). Then the reflection triangles A'B'C' and A"B"C" of P' and P" are orthologic.

Let Q' and Q" be the orthologic centers (A'B'C' to A"B"C") and (A"B"C" to A'B'C'), respectively. If P' is fixed and P" moves along the line OP', then Q' describes a rectangular hyperbola ℍ(P') circumscribed to A'B'C', whilst Q" moves on a line 𝕃(P'). ℍ(P') is named here the bireflection circumcentral conic of P' (shortened to the BRC-conic of P') and 𝕃(P') is named here the bireflection circumcentral line of P', abbreviated as the BRC-line of P'.

If P' = x : y : z (barycentrics), then:

Note: The orthologic center (reflection-of-P' to reflection-of-P") is the reflection in P' of the orthologic center (pedal-of-P' to pedal-of-P"). (See X(41537))

The appearance of (i, j, m, n) in the following lists means that the orthological centers of the reflection triangles of X(i) and X(j) are X(m) and X(n):

For selected points P', P" on the line X(1)X(3):
(1, 3, 1, 355), (1, 35, 79, 3585), (1, 36, 7972, 41684), (1, 40, 15071, 5693), (1, 46, 41685, 41686), (1, 55, 4312, 1836), (1, 56, 3633, 41687), (3, 35, 3652, 35), (3, 36, 6265, 36), (3, 40, 6259, 40), (3, 46, 41688, 46), (3, 55, 5779, 55), (3, 56, 1482, 56), (35, 36, 41689, 1749), (35, 40, 41690, 41691), (35, 46, 41692, 41693), (35, 55, 41694, 41695), (35, 56, 41696, 41697), (36, 40, 41698, 12751), (36, 46, 41699, 10073), (36, 55, 41700, 41701), (36, 56, 41702, 20586), (40, 46, 41703, 41704), (40, 55, 41705, 41706), (40, 56, 5881, 1837), (46, 55, 41707, 41708), (46, 56, 41709, 41710), (55, 56, 41711, 41712)
For selected points P', P" on the line X(2)X(3):
(2, 3, 2, 1352), (2, 4, 3060, 6403), (2, 5, 41713, 41714), (2, 20, 41715, 41716), (2, 21, 41717, 41718), (2, 22, 41719, 69), (2, 23, 41720, 41721), (3, 4, 4, 4), (3, 5, 6288, 5), (3, 20, 5878, 20), (3, 21, 5887, 21), (3, 22, 19149, 22), (3, 23, 9970, 23), (4, 5, 6242, 52), (4, 20, 6241, 12111), (4, 21, 41722, 41723), (4, 22, 6776, 11442), (4, 23, 32234, 41724), (5, 20, 41725, 41726), (5, 21, 41727, 41728), (5, 22, 41729, 41730), (5, 23, 41731, 41732), (20, 21, 41733, 41734), (20, 22, 41735, 41736), (20, 23, 41737, 41738), (21, 22, 41739, 41740), (21, 23, 41741, 41742), (22, 23, 41743, 41744)

For selected points P', P" on the line X(3)X(6):

(3, 6, 381, 6), (3, 15, 5617, 15), (3, 16, 5613, 16), (3, 32, 3095, 32), (3, 39, 6287, 39), (6, 15, 41745, 22495), (6, 16, 41746, 22496), (6, 32, 41747, 41748), (6, 39, 41749, 41750), (15, 16, 22997, 22998), (15, 32, 23000, 41751), (15, 39, 23001, 41752), (16, 32, 23009, 41753), (16, 39, 23010, 41754), (32, 39, 41755, 41756)

The appearance of (i, j) in the following list means that the center of the BRC-conic of X(i) is X(j):

{{1, 11570}, {2, 12824}, {4, 1986}, {5, 11557}, {6, 5477}, {15, 6783}, {16, 6782}, {20, 12825}, {21, 12826}, {22, 12827}, {23, 3580}, {25, 12828}, {30, 13754}, {32, 12829}, {36, 1737}, {39, 12830}, {40, 12665}, {54, 27423}, {55, 12831}, {56, 12832}, {64, 38956}, {74, 34150}, {98, 13137}, {99, 12833}, {100, 34151}, {101, 34805}, {110, 7471}, {111, 34806}, {186, 403}, {187, 230}, {511, 3564}, {512, 525}, {513, 521}, {516, 916}, {517, 912}, {518, 34381}, {520, 8057}, {523, 520}, {524, 8681}, {525, 8673}, {526, 9033}, {542, 14984}, {674, 9028}, {690, 9517}, {804, 39469}, {834, 23874}, {900, 8677}, {924, 523}, {926, 39470}, {1154, 539}, {1157, 40631}, {1319, 18838}, {1498, 3184}, {1499, 30209}, {1503, 511}, {1510, 6368}, {1691, 1692}, {2390, 519}, {2393, 524}, {2574, 2574}, {2575, 2575}, {2777, 5663}, {2778, 2771}, {2781, 542}, {2818, 952}, {2881, 39473}, {3564, 34382}, {3566, 512}, {3667, 32475}, {3827, 518}, {3852, 732}, {5663, 17702}, {6000, 30}, {6001, 517}, {6003, 30212}, {6085, 39472}, {6150, 10615}, {6759, 38605}, {8057, 30211}, {8674, 2850}, {8675, 9007}, {8676, 514}, {8677, 39471}, {9002, 9031}, {10628, 32423}, {12095, 27087}, {13152, 30210}, {13557, 12095}, {15311, 6000}, {15313, 513}, {17511, 10689}, {18400, 1154}, {20184, 1510}, {20186, 1499}, {20403, 39474}, {34146, 1503}, {36201, 2781}}

The appearance of (i, j) in the following list means that the tripole of the BRC-line of X(i) is X(j):

(2, 6331), (4, 30450), (5, 38342), (20, 15352), (21, 6335), (22, 107), (36, 655), (74, 648), (98, 6331), (99, 16081), (100, 16082), (102, 653), (103, 1897), (104, 6335), (110, 16080), (112, 6330), (187, 17708), (249, 2966), (511, 476), (523, 40427), (972, 13149), (1141, 38342), (1293, 6336), (1294, 15352), (1296, 17983), (1297, 107), (1300, 30450), (1691, 30530), (1817, 13149), (2071, 15459), (2693, 15459), (2710, 685), (3431, 35178), (3447, 476), (3455, 110), (3658, 16082), (4184, 1897), (4225, 653), (4226, 16081), (4230, 6330), (5152, 39291), (5663, 39290), (5961, 4558), (6091, 4563), (7488, 16813), (11634, 17983), (11643, 35178), (15329, 16080), (15468, 39295), (15469, 2986), (18401, 16813), (32710, 687), (34008, 36306), (34009, 36309), (34418, 18315), (34442, 651), (37183, 685), (40080, 287), (40083, 2987), (40084, 2991)

X(41684) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(36) TO REFLECTION OF X(1)

Barycentrics    a^4-2*(b+c)*a^3+3*b*c*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(41684) = 2*X(1)-3*X(3582) = 4*X(1737)-3*X(3582) = 4*X(3626)-X(3935) = X(3632)+2*X(26015) = 3*X(3679)-2*X(6735)

The reciprocal orthologic center of these triangles is X(7972)

X(41684) lies on these lines: {1, 2}, {3, 37706}, {5, 11009}, {11, 5844}, {12, 38178}, {30, 3245}, {35, 5690}, {36, 952}, {40, 1727}, {46, 4325}, {56, 12645}, {57, 37708}, {65, 2962}, {80, 517}, {140, 24926}, {355, 1836}, {484, 515}, {495, 5425}, {518, 10057}, {522, 21102}, {528, 41700}, {529, 4880}, {758, 1109}, {912, 11571}, {944, 7280}, {946, 11280}, {950, 26878}, {956, 14793}, {962, 18514}, {1145, 32760}, {1146, 5526}, {1155, 28204}, {1159, 11237}, {1317, 15325}, {1319, 7972}, {1385, 5445}, {1457, 6127}, {1478, 11552}, {1479, 12245}, {1482, 7741}, {1483, 5433}, {1519, 13253}, {1697, 37721}, {1706, 17700}, {1725, 24028}, {1728, 12625}, {1733, 32850}, {1739, 26727}, {1772, 4695}, {1776, 4330}, {1837, 4857}, {2067, 35843}, {2093, 37712}, {2098, 37720}, {2099, 5790}, {2136, 41709}, {2342, 40437}, {2800, 41698}, {2964, 5247}, {2975, 14792}, {3036, 4867}, {3057, 18527}, {3336, 4848}, {3337, 10106}, {3338, 37709}, {3340, 10827}, {3419, 18397}, {3474, 34627}, {3476, 37587}, {3555, 32537}, {3746, 37730}, {3825, 5330}, {3880, 10073}, {3901, 17890}, {4084, 20060}, {4744, 17483}, {4792, 36590}, {4881, 33337}, {5010, 5657}, {5048, 16173}, {5119, 5727}, {5127, 6740}, {5131, 9803}, {5172, 12331}, {5183, 15228}, {5193, 12832}, {5204, 18526}, {5252, 5902}, {5432, 37728}, {5441, 22937}, {5442, 37605}, {5443, 9956}, {5444, 11231}, {5533, 5854}, {5559, 9957}, {5563, 10944}, {5587, 18393}, {5710, 16472}, {5836, 16152}, {6149, 6741}, {6502, 35842}, {6684, 37616}, {6762, 17437}, {6971, 21398}, {7354, 37705}, {7982, 10826}, {8070, 24390}, {8148, 10896}, {8275, 10051}, {9219, 18406}, {9581, 30323}, {10090, 12531}, {10222, 17606}, {10483, 18525}, {10543, 28463}, {10954, 31419}, {11376, 15079}, {12607, 41696}, {12749, 18838}, {15326, 28224}, {15446, 26285}, {15950, 38042}, {16200, 23708}, {18481, 37572}, {21075, 39599}, {21842, 24914}, {26446, 37525}, {28234, 30384}, {31263, 34122}, {34690, 34717}, {37571, 37739}, {37606, 38066}, {37701, 38176}, {38063, 38213}

X(41684) = midpoint of X(i) and X(j) for these {i, j}: {484, 9897}, {3245, 37006}
X(41684) = reflection of X(i) in X(j) for these (i, j): (1, 1737), (11, 11545), (36, 40663), (1317, 15325), (3583, 80), (4316, 484), (4511, 10), (4867, 17757), (5176, 15863), (7972, 1319), (13253, 1519), (15228, 5183), (17757, 3036), (36975, 1155)
X(41684) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(34535)}} and {{A, B, C, X(8), X(2962)}}
X(41684) = crosspoint of X(80) and X(5559)
X(41684) = crosssum of X(36) and X(5563)
X(41684) = X(8)-Beth conjugate of-X(4511)
X(41684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1737, 3582), (8, 10573, 1), (8, 18391, 12647), (10, 24541, 1698), (56, 12645, 37707), (65, 37710, 5270), (140, 37734, 24926), (145, 499, 1), (355, 5903, 3585), (355, 41687, 5903), (1837, 5697, 4857), (2099, 5790, 7951), (3626, 25006, 3679), (3632, 3679, 200), (4677, 30286, 1), (5690, 10950, 35), (10572, 11010, 4330), (10572, 11362, 11010), (10573, 12647, 18391), (12647, 18391, 1)

X(41685) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(1) TO REFLECTION OF X(46)

Barycentrics
a*((b+c)*a^8-(2*b^2+b*c+2*c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6+2*(b+c)*(b^2+c^2)*a^4*b*c+(6*b^4+6*c^4+(b+c)^2*b*c)*a^5-(6*b^6+6*c^6-(b^2+4*b*c+c^2)*(b-c)^2*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4)*a^2+(2*b^4+2*c^4-(b-c)^2*b*c)*(b^2-c^2)^2*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :

The reciprocal orthologic center of these triangles is X(41686)

X(41685) lies on these lines: {1, 90}, {36, 224}, {46, 2900}, {79, 10052}, {5840, 11571}, {5902, 41569}, {5903, 41575}, {5904, 14798}, {6734, 41540}, {7951, 10395}, {7972, 36977}, {10680, 17660}, {11517, 36152}, {11570, 12649}, {12750, 34789}, {15016, 41544}, {18412, 37710}, {37579, 41686}, {41688, 41692}

X(41685) = reflection of X(i) in X(j) for these (i, j): (1, 41537), (41703, 41688)
X(41685) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41688, 41708, 41692), (41688, 41710, 41699), (41692, 41699, 41688), (41708, 41710, 41688)

X(41686) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(46) TO REFLECTION OF X(1)

Barycentrics    a*((b+c)*a^5-(b^2+b*c+c^2)*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3+(2*b^4+2*c^4+(b^2+c^2)*b*c)*a^2+(b^3-c^3)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
X(41686) = 4*X(5572)-5*X(15299) = 3*X(5692)-2*X(5730) = 6*X(17728)-5*X(18398)

The reciprocal orthologic center of these triangles is X(41685)

X(41686) lies on these lines: {1, 6}, {12, 5902}, {35, 26921}, {36, 37700}, {46, 912}, {65, 5790}, {80, 10526}, {90, 37569}, {210, 13750}, {355, 1836}, {484, 11500}, {498, 3678}, {499, 3874}, {517, 1898}, {758, 3436}, {920, 3811}, {971, 4333}, {1478, 15556}, {1727, 11248}, {1737, 3868}, {1770, 12528}, {1788, 11570}, {1858, 5119}, {2099, 5694}, {2771, 18518}, {2801, 4299}, {3583, 5812}, {3612, 31837}, {3681, 10039}, {3901, 17615}, {3927, 11507}, {3940, 22766}, {4134, 13411}, {4302, 41562}, {4305, 40269}, {5445, 15016}, {5690, 10955}, {5697, 5844}, {5791, 30274}, {5883, 10585}, {5884, 10786}, {5887, 25415}, {6253, 11661}, {6763, 14793}, {7280, 33597}, {8148, 17638}, {10073, 12649}, {10599, 31870}, {10826, 24474}, {10942, 40663}, {10953, 37702}, {10954, 21677}, {11010, 15104}, {11374, 17728}, {11375, 31835}, {11552, 30290}, {11571, 37725}, {12647, 15862}, {12738, 37524}, {16152, 31938}, {18962, 37710}, {21842, 37733}, {24475, 24914}, {25413, 36920}, {37579, 41685}

X(41686) = reflection of X(i) in X(j) for these (i, j): (46, 41538), (5903, 41687)
X(41686) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (72, 958, 5692), (920, 3811, 32760), (3678, 18389, 498), (5904, 18397, 1)

X(41687) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(56) TO REFLECTION OF X(1)

Barycentrics    (a^2-3*(b+c)*a+(b+c)^2)*(a-b+c)*(a+b-c) : :
X(41687) = 2*X(1)-3*X(17728) = 2*X(1479)-3*X(1837) = X(1479)-3*X(10573) = 4*X(1479)-3*X(12701) = 4*X(4848)-X(37738) = 4*X(10573)-X(12701)

The reciprocal orthologic center of these triangles is X(3633)

X(41687) lies on these lines: {1, 140}, {2, 11011}, {3, 37740}, {5, 25415}, {7, 8}, {10, 2099}, {11, 7982}, {12, 3340}, {20, 5183}, {30, 37711}, {34, 34893}, {35, 3654}, {36, 32141}, {40, 10950}, {46, 952}, {55, 11362}, {56, 519}, {57, 3632}, {78, 5855}, {80, 5560}, {145, 1319}, {226, 3626}, {355, 1836}, {443, 3922}, {474, 18967}, {484, 18481}, {496, 30323}, {499, 10222}, {515, 37567}, {517, 1479}, {553, 34641}, {613, 5710}, {938, 5919}, {942, 11045}, {950, 5729}, {956, 11509}, {960, 5554}, {978, 26727}, {993, 14882}, {1058, 3057}, {1145, 3811}, {1159, 13407}, {1210, 2098}, {1317, 1420}, {1320, 20118}, {1329, 11682}, {1376, 26437}, {1388, 3244}, {1393, 4695}, {1400, 17299}, {1405, 17281}, {1452, 12135}, {1457, 3214}, {1466, 24391}, {1470, 12513}, {1482, 1737}, {1483, 37618}, {1697, 15299}, {1698, 15950}, {1728, 12703}, {1770, 18525}, {1825, 5130}, {1866, 5101}, {2093, 5881}, {2136, 13996}, {2171, 17275}, {2285, 17362}, {2551, 31165}, {2594, 24806}, {2646, 5657}, {2800, 12679}, {3085, 11041}, {3086, 5048}, {3241, 7288}, {3256, 5258}, {3295, 34718}, {3303, 6738}, {3336, 37707}, {3339, 4677}, {3419, 15556}, {3421, 3962}, {3476, 3621}, {3485, 3617}, {3486, 37568}, {3555, 18838}, {3576, 37734}, {3600, 31145}, {3612, 37728}, {3625, 5221}, {3649, 4668}, {3655, 7280}, {3656, 7741}, {3671, 4669}, {3826, 11526}, {3869, 26792}, {3877, 26127}, {3878, 4679}, {3880, 12649}, {3889, 13751}, {3913, 37579}, {4018, 18961}, {4084, 7702}, {4298, 4701}, {4299, 28204}, {4301, 10896}, {4308, 20053}, {4323, 4870}, {4333, 28186}, {4511, 37828}, {4534, 16572}, {4856, 38296}, {4863, 10914}, {5119, 10386}, {5128, 15326}, {5172, 8715}, {5204, 5882}, {5250, 15297}, {5289, 24954}, {5435, 20050}, {5603, 17606}, {5697, 5722}, {5727, 6284}, {5734, 10589}, {5763, 9581}, {5790, 12047}, {5818, 17605}, {5843, 9579}, {5854, 12832}, {5886, 11009}, {6684, 34471}, {6735, 12635}, {6744, 8162}, {6863, 15867}, {6907, 10955}, {6922, 10959}, {7080, 37829}, {7173, 11522}, {7181, 25716}, {7223, 25719}, {7294, 25055}, {7962, 37722}, {7967, 37605}, {8148, 30384}, {8197, 18956}, {8204, 18955}, {9282, 14584}, {9588, 13384}, {9613, 11246}, {9623, 21677}, {9819, 37723}, {9897, 10483}, {10039, 17718}, {10306, 22760}, {10529, 33895}, {10572, 12702}, {10826, 11545}, {10827, 39542}, {10912, 26015}, {11502, 22770}, {11523, 26482}, {11529, 15888}, {11681, 34647}, {11871, 12459}, {11872, 12458}, {12247, 17636}, {12526, 34606}, {12531, 18976}, {12645, 36279}, {12648, 34791}, {12667, 41706}, {12953, 28194}, {13273, 15863}, {13411, 38127}, {13601, 34790}, {15171, 37721}, {18526, 21578}, {18990, 37708}, {22759, 37541}, {25005, 25681}, {29617, 41245}, {33956, 36977}, {37692, 38042}, {37737, 38112}

X(41687) = midpoint of X(5903) and X(41686)
X(41687) = reflection of X(i) in X(j) for these (i, j): (56, 4848), (78, 8256), (1837, 10573), (2098, 1210), (5730, 10), (11682, 1329), (12701, 1837), (20586, 12832), (30323, 496), (37738, 56)
X(41687) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(7321)}} and {{A, B, C, X(69), X(34893)}}
X(41687) = X(8)-Beth conjugate of-X(5730)
X(41687) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 40663, 24914), (8, 65, 5252), (8, 3868, 32049), (10, 2099, 11375), (57, 3632, 10944), (65, 5252, 10404), (65, 36920, 8), (145, 1788, 1319), (484, 37706, 18481), (1420, 3633, 1317), (1482, 1737, 11376), (2093, 5881, 7354), (3244, 3911, 1388), (3339, 4677, 37709), (3339, 37709, 5434), (3340, 3679, 12), (3654, 37739, 35), (5903, 41684, 355), (7741, 11280, 3656), (9578, 18421, 3649), (11009, 18395, 5886), (11545, 22791, 10826), (12245, 18391, 3057), (36928, 36929, 17361)

X(41688) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(3) TO REFLECTION OF X(46)

Barycentrics
a^10-(b+c)*a^9-(3*b^2-4*b*c+3*c^2)*a^8+2*(b^2-c^2)*(b-c)*a^7+2*(2*b^2-3*b*c+2*c^2)*(b^2+c^2)*a^6+4*(b^3+c^3)*b*c*a^5-2*(b^2+c^2)*(2*b^4+2*c^4-(b^2+c^2)*b*c)*a^4-2*(b^2-c^2)*(b-c)*(b^4+c^4)*a^3+(3*b^4+3*c^4-2*(b^2-b*c+c^2)*b*c)*(b^2-c^2)^2*a^2+(b^2-c^2)^3*(b-c)^3*a-(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(46)

X(41688) lies on these lines: {3, 12608}, {5, 90}, {46, 119}, {65, 68}, {79, 5720}, {224, 1537}, {377, 5887}, {442, 3652}, {3338, 26470}, {5553, 30513}, {5570, 41537}, {5779, 5880}, {5794, 25413}, {5812, 11517}, {5886, 22766}, {11499, 21077}, {12609, 22758}, {12676, 16128}, {12679, 37356}, {16465, 37820}, {18232, 26066}, {28628, 37535}, {41685, 41692}

X(41688) = midpoint of X(41685) and X(41703)
X(41688) = reflection of X(i) in X(j) for these (i, j): (3, 41540), (90, 5)
X(41688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41685, 41692, 41708), (41685, 41699, 41710), (41692, 41699, 41685), (41708, 41710, 41685)

X(41689) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(35) TO REFLECTION OF X(36)

Barycentrics    a*(a^6-2*(b+c)*a^5-(b-c)^2*a^4+4*(b^3+c^3)*a^3-(b^2+3*b*c+c^2)*(b^2-b*c+c^2)*a^2-2*(b^4-c^4)*(b-c)*a+(b^4-c^4)*(b^2-c^2)) : :
X(41689) = 2*X(11)-3*X(37701) = 2*X(4996)-3*X(15015) = X(6763)-3*X(15015) = 4*X(8068)-3*X(37718) = 3*X(16173)-4*X(37737) = 5*X(31272)-6*X(38062)

The reciprocal orthologic center of these triangles is X(1749)

X(41689) lies on these lines: {1, 5}, {10, 39778}, {35, 2771}, {36, 17660}, {72, 35204}, {73, 33565}, {78, 4996}, {100, 484}, {104, 7161}, {149, 18393}, {191, 12532}, {214, 2975}, {498, 9803}, {518, 33598}, {943, 3065}, {1749, 5172}, {1768, 5010}, {2800, 11010}, {2802, 11280}, {2948, 10260}, {3336, 11570}, {3337, 10090}, {3583, 21635}, {3585, 34772}, {3746, 17638}, {3811, 5541}, {3870, 12653}, {4302, 9809}, {4324, 41690}, {4511, 33337}, {5425, 6797}, {5497, 13604}, {5538, 5841}, {5694, 14795}, {5842, 34789}, {5903, 12331}, {6126, 33649}, {6224, 20060}, {6261, 13253}, {6596, 13272}, {10087, 37563}, {10573, 18467}, {12332, 15071}, {12515, 16132}, {12758, 41553}, {12773, 37525}, {13143, 17097}, {13146, 16118}, {14151, 20116}, {14794, 33597}, {14800, 35451}, {18406, 33594}, {18513, 37533}, {19860, 38215}, {27778, 38602}, {30143, 38219}, {31272, 38062}, {37615, 38135}

X(41689) = midpoint of X(6224) and X(20060)
X(41689) = reflection of X(i) in X(j) for these (i, j): (35, 41541), (80, 12), (2975, 214), (6763, 4996)
X(41689) = intersection, other than A,B,C, of conics {{A, B, C, X(36), X(38458)}} and {{A, B, C, X(80), X(33565)}}
X(41689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 5531, 9897), (1, 6127, 38458), (12, 5719, 37701), (80, 12739, 1), (100, 11571, 484), (5660, 7972, 10073), (6265, 7972, 1), (6265, 12738, 11698), (6265, 41701, 7972), (6763, 15015, 4996), (12738, 12739, 80), (17660, 22935, 36), (37710, 37733, 1)

X(41690) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(35) TO REFLECTION OF X(40)

Barycentrics    a^7+(b+c)*a^6-(3*b^2-7*b*c+3*c^2)*a^5-3*(b^3+c^3)*a^4+3*(b^3-c^3)*(b-c)*a^3+(b^2-c^2)*(b-c)*(3*b^2+b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(41691)

X(41690) lies on these lines: {1, 16127}, {12, 41694}, {30, 41696}, {35, 41543}, {36, 18243}, {84, 37692}, {515, 11280}, {1490, 4333}, {1768, 6260}, {1837, 6259}, {3579, 41691}, {4295, 5691}, {4297, 9809}, {4314, 41561}, {4324, 41689}, {5128, 41707}, {6001, 37710}, {11037, 11522}, {11415, 34628}, {11552, 16116}, {12680, 34789}

X(41690) = reflection of X(35) in X(41543)
X(41690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6259, 15071, 41698), (6259, 41706, 15071), (16116, 31673, 11552)

X(41691) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(40) TO REFLECTION OF X(35)

Barycentrics
3*a^7-(8*b^2-3*b*c+8*c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+(7*b^4+7*c^4-(b^2+c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^2-2*(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :
X(41691) = 4*X(5)-3*X(79) = 2*X(5)-3*X(3652) = X(20)-3*X(3648) = 2*X(20)-3*X(16113) = 9*X(191)-7*X(9588) = 3*X(191)-2*X(37401) = 5*X(631)-6*X(3647) = 5*X(631)-3*X(16116) = 3*X(3065)-2*X(37726) = 6*X(3649)-7*X(9624) = 7*X(3832)-6*X(16125) = 7*X(3832)-3*X(20084) = 5*X(3843)-3*X(16150) = 5*X(3843)-6*X(22798) = 2*X(4301)-3*X(21669) = 13*X(5067)-12*X(6701) = 3*X(5441)-2*X(37727) = 2*X(5882)-3*X(15678) = 3*X(7701)-2*X(37447) = 7*X(9588)-6*X(37401)

The reciprocal orthologic center of these triangles is X(41690)

X(41691) lies on these lines: {5, 79}, {20, 3648}, {21, 17168}, {30, 4677}, {40, 3650}, {191, 3359}, {516, 10308}, {548, 6326}, {631, 3647}, {912, 4330}, {2771, 12680}, {2951, 17857}, {3065, 16155}, {3579, 41690}, {3649, 9624}, {3832, 16125}, {3843, 16150}, {4301, 21669}, {4325, 5887}, {4338, 7330}, {5067, 6701}, {5441, 37727}, {5735, 7701}, {5882, 15678}, {6175, 31399}, {6278, 16131}, {6281, 16130}, {8227, 11544}, {9657, 18977}, {9670, 16142}, {9715, 16119}, {10032, 33557}, {11362, 11684}, {12665, 12682}, {15774, 16129}, {15888, 16140}, {16005, 17781}, {16006, 35242}, {16118, 37714}, {16139, 37725}, {16141, 37722}, {16152, 37719}, {16153, 24467}, {22936, 26725}, {31938, 41703}

X(41691) = reflection of X(i) in X(j) for these (i, j): (40, 3650), (79, 3652), (16113, 3648), (16116, 3647), (16150, 22798), (20084, 16125)

X(41692) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(35) TO REFLECTION OF X(46)

Barycentrics
a^10-(5*b^2-3*b*c+5*c^2)*a^8-4*(b+c)*b*c*a^7+5*(2*b^4+2*c^4-(b-c)^2*b*c)*a^6+2*(b+c)*(3*b^2-2*b*c+3*c^2)*b*c*a^5-(10*b^6+10*c^6-(3*b^2+4*b*c+3*c^2)*(b-c)^2*b*c)*a^4+(5*b^4+5*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*(b^2-c^2)^2*a^2-2*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(41693)

X(41692) lies on these lines: {35, 41544}, {90, 5219}, {912, 37710}, {1770, 3811}, {3585, 20612}, {9613, 10052}, {41685, 41688}

X(41692) = reflection of X(35) in X(41544)
X(41692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41685, 41688, 41699), (41688, 41708, 41685)

X(41693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(46) TO REFLECTION OF X(35)

Barycentrics    3*a^7-(8*b^2-b*c+8*c^2)*a^5-(b^3+c^3)*a^4+(7*b^4+7*c^4-(b+c)^2*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a^2-2*(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(41692)

X(41693) lies on these lines: {5, 79}, {30, 37711}, {46, 41545}, {191, 16152}, {758, 36977}, {3647, 6910}, {3648, 6872}, {5441, 37740}, {6880, 16116}, {10955, 16139}, {10959, 16141}, {11681, 18232}, {13465, 18977}, {15299, 16153}

X(41693) = reflection of X(i) in X(j) for these (i, j): (46, 41545), (79, 41697), (16155, 16141)

X(41694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(35) TO REFLECTION OF X(55)

Barycentrics    a*(a^5-(4*b^2-5*b*c+4*c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+3*(b^3-c^3)*(b-c)*a+(b^2-c^2)*(b-c)*(-2*b^2-6*b*c-2*c^2)) : :

The reciprocal orthologic center of these triangles is X(41695)

X(41694) lies on these lines: {1, 16112}, {7, 7741}, {12, 41690}, {35, 29007}, {79, 41572}, {484, 3921}, {516, 37710}, {518, 11280}, {527, 31159}, {971, 2646}, {1156, 5542}, {3062, 7160}, {3256, 41695}, {3585, 17768}, {4312, 5587}, {5425, 16133}, {5843, 26470}, {5851, 8068}, {7962, 11372}, {8581, 12773}, {12560, 18412}

X(41694) = reflection of X(35) in X(29007)
X(41694) = {X(4312), X(5779)}-harmonic conjugate of X(41700)

X(41695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(55) TO REFLECTION OF X(35)

Barycentrics    a*(a^5-(b+c)*a^4-(2*b-c)*(b-2*c)*a^3+2*(b^3+c^3)*a^2+(b^2+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)) : :
X(41695) = 2*X(5119)-5*X(16140)

The reciprocal orthologic center of these triangles is X(41694)

X(41695) lies on these lines: {1, 16138}, {5, 79}, {21, 37605}, {30, 5119}, {46, 19919}, {55, 41546}, {63, 1836}, {191, 5128}, {354, 13243}, {377, 3648}, {474, 3647}, {495, 16154}, {1012, 33857}, {1319, 28461}, {1387, 3065}, {1709, 17718}, {1837, 18540}, {3219, 10032}, {3256, 41694}, {3333, 3649}, {5123, 6175}, {5440, 17653}, {5836, 11684}, {6833, 16116}, {6871, 20084}, {7082, 41549}, {9342, 27065}, {9656, 40256}, {10543, 10864}, {10827, 16118}, {10895, 18232}, {10957, 14883}, {16113, 31775}, {16142, 37433}, {16153, 16160}, {17605, 27003}, {24926, 33858}, {28190, 37563}, {41571, 41708}

X(41695) = reflection of X(i) in X(j) for these (i, j): (55, 41546), (16154, 495)
X(41695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (79, 3652, 41697), (3649, 7701, 16141)

X(41696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(35) TO REFLECTION OF X(56)

Barycentrics    a*(a^3+2*b^3+2*c^3-2*(b+c)*a^2-(b^2+b*c+c^2)*a) : :
X(41696) = 4*X(6734)-5*X(31262) = 2*X(6734)-3*X(37701) = 4*X(12047)-3*X(31159) = 5*X(31262)-6*X(37701)

The reciprocal orthologic center of these triangles is X(41697)

X(41696) lies on these lines: {1, 6}, {3, 3901}, {8, 3822}, {10, 5425}, {21, 4067}, {30, 41690}, {35, 758}, {36, 3868}, {56, 3894}, {63, 37571}, {65, 16126}, {78, 5902}, {80, 21077}, {100, 4084}, {145, 1479}, {191, 3962}, {484, 4018}, {519, 5086}, {908, 37702}, {997, 11520}, {1071, 5538}, {1320, 17501}, {1482, 1699}, {1698, 3940}, {1770, 12437}, {1793, 3193}, {1837, 31160}, {2098, 4930}, {2099, 3632}, {2646, 6763}, {2650, 30115}, {3219, 4127}, {3245, 8715}, {3295, 3899}, {3336, 5440}, {3337, 24473}, {3621, 31418}, {3622, 31458}, {3624, 15934}, {3635, 5330}, {3681, 30147}, {3746, 3869}, {3754, 4420}, {3811, 5903}, {3870, 5697}, {3873, 30144}, {3874, 4511}, {3876, 30143}, {3880, 11280}, {3881, 37602}, {3884, 3957}, {3925, 16137}, {3947, 17097}, {4134, 5260}, {4324, 17768}, {4325, 10609}, {4333, 34701}, {4855, 37524}, {5313, 37549}, {5426, 31445}, {5443, 10916}, {5534, 7982}, {5536, 37837}, {5693, 37533}, {5719, 21677}, {5844, 15908}, {5905, 10483}, {6326, 24474}, {6734, 31262}, {6737, 13407}, {6765, 25415}, {7688, 33858}, {7741, 12649}, {8666, 24926}, {10176, 25542}, {10573, 25568}, {10591, 20008}, {10698, 26726}, {11012, 37733}, {11278, 12653}, {12607, 41684}, {12739, 35204}, {15015, 37582}, {15071, 37531}, {15079, 30852}, {15931, 31806}, {16132, 37585}, {18444, 35202}, {24475, 37561}, {26015, 37735}, {37625, 37700}, {37783, 37816}

X(41696) = reflection of X(i) in X(j) for these (i, j): (35, 34772), (5288, 1), (6763, 2646), (11012, 37733)
X(41696) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(15173)}} and {{A, B, C, X(44), X(17501)}}
X(41696) = X(643)-Beth conjugate of-X(4084)
X(41696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 72, 5251), (1, 5692, 5259), (1, 5904, 5258), (1, 11523, 5904), (1, 12635, 4867), (3, 3901, 4880), (10, 34195, 5425), (78, 12559, 5902), (997, 11520, 18398), (1482, 3633, 41702), (1482, 41711, 3633), (1699, 3633, 41709), (2650, 30115, 37559), (3868, 22836, 36), (3874, 4511, 5563), (3962, 24929, 191), (4127, 35016, 3219), (6734, 37701, 31262), (21077, 41575, 80)

X(41697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(56) TO REFLECTION OF X(35)

Barycentrics    a*(a+b-c)*(a-b+c)*(a^4-(2*b^2+3*b*c+2*c^2)*a^2-(b+c)*b*c*a+(b^3+c^3)*(b+c)) : :
X(41697) = 2*X(46)+X(16141) = X(5730)-3*X(37308) = X(16153)-4*X(34753)

The reciprocal orthologic center of these triangles is X(41696)

X(41697) lies on these lines: {1, 5424}, {3, 33857}, {5, 79}, {12, 15932}, {21, 65}, {30, 46}, {36, 33858}, {40, 10543}, {55, 10122}, {56, 758}, {57, 191}, {63, 10404}, {100, 33667}, {405, 3647}, {442, 1454}, {484, 5441}, {496, 16155}, {499, 33592}, {846, 11553}, {920, 1836}, {942, 22937}, {960, 3218}, {1046, 1464}, {1158, 37447}, {1319, 34195}, {1376, 31938}, {1402, 35637}, {1406, 16471}, {1420, 16126}, {1445, 15297}, {1725, 38336}, {1727, 16160}, {1728, 7701}, {1737, 37230}, {1758, 2594}, {1768, 3374}, {1788, 2475}, {1882, 31902}, {2099, 35016}, {2478, 3648}, {2646, 21161}, {2771, 10081}, {3058, 24468}, {3065, 12019}, {3193, 34977}, {3337, 5298}, {3338, 16137}, {3340, 5426}, {3474, 37433}, {3485, 15674}, {3650, 4679}, {3651, 17637}, {3911, 11263}, {4870, 15671}, {5119, 15174}, {5172, 15556}, {5187, 20084}, {5252, 21677}, {5434, 6763}, {5435, 14450}, {5442, 16763}, {5499, 14883}, {5535, 16113}, {5536, 37722}, {5918, 33557}, {6834, 16116}, {7677, 13751}, {7686, 21669}, {8614, 18593}, {10021, 39542}, {10123, 10395}, {10826, 16118}, {11509, 37286}, {11544, 19919}, {11604, 20118}, {12005, 41341}, {12432, 14882}, {12514, 15670}, {12739, 35204}, {13465, 37545}, {13743, 36279}, {15254, 16133}, {15803, 16132}, {16153, 34753}, {16478, 37591}, {17718, 26921}, {17728, 37532}, {18259, 31254}, {26877, 37837}, {37292, 37541}, {37579, 39772}, {41574, 41710}

X(41697) = midpoint of X(79) and X(41693)
X(41697) = reflection of X(i) in X(j) for these (i, j): (56, 41547), (16155, 496)
X(41697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (57, 191, 3649), (65, 41542, 21), (79, 1749, 3652), (79, 3652, 41695), (191, 3649, 16140), (1155, 17637, 3651), (1454, 1708, 24914), (1749, 3336, 79), (3649, 5433, 26725), (3649, 6675, 11375), (3911, 41551, 11263)

X(41698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(36) TO REFLECTION OF X(40)

Barycentrics
a^7-(b+c)*a^6-(b^2-5*b*c+c^2)*a^5+(b+c)*(b^2-3*b*c+c^2)*a^4-(b-c)^2*(b^2+3*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(41698) = 3*X(4)-2*X(24042) = 2*X(104)-3*X(3582) = 3*X(1699)-2*X(30384) = 3*X(3583)-4*X(24042) = 2*X(4297)-3*X(4881) = X(4316)+2*X(10728) = 2*X(5440)-3*X(5660) = X(5537)-4*X(38757) = 4*X(6681)-3*X(38693) = 3*X(16173)-4*X(22835) = X(35000)-3*X(38755)

The reciprocal orthologic center of these triangles is X(12751)

X(41698) lies on these lines: {1, 4}, {3, 31246}, {5, 37561}, {10, 37437}, {11, 5193}, {20, 26364}, {30, 119}, {35, 18242}, {36, 1532}, {40, 37821}, {56, 40267}, {65, 22792}, {79, 7686}, {80, 6001}, {84, 10826}, {104, 3582}, {149, 28236}, {153, 519}, {165, 6925}, {355, 12679}, {377, 7989}, {381, 10269}, {382, 11248}, {484, 1512}, {516, 5080}, {517, 10742}, {908, 5538}, {952, 41702}, {971, 13273}, {993, 6932}, {1012, 7951}, {1071, 37702}, {1125, 13729}, {1158, 18395}, {1319, 1538}, {1470, 12943}, {1593, 26309}, {1698, 6850}, {1709, 3359}, {1737, 1768}, {1749, 27247}, {1837, 6259}, {2475, 19925}, {2478, 7987}, {2551, 9588}, {2800, 41684}, {3062, 30513}, {3070, 26465}, {3071, 26459}, {3091, 10200}, {3146, 5552}, {3149, 10483}, {3434, 37712}, {3436, 7991}, {3576, 6929}, {3624, 6893}, {3627, 6253}, {3634, 37163}, {3667, 21186}, {3814, 6909}, {3822, 6912}, {3830, 10679}, {3843, 16203}, {4297, 4881}, {4299, 6848}, {4301, 20060}, {4316, 6905}, {4324, 6796}, {4330, 11491}, {4511, 21635}, {5010, 6938}, {5048, 12763}, {5073, 35251}, {5076, 18545}, {5086, 31803}, {5123, 17613}, {5176, 39776}, {5251, 6907}, {5258, 15908}, {5267, 6960}, {5434, 7956}, {5440, 5660}, {5441, 33597}, {5450, 6941}, {5537, 17757}, {5553, 5560}, {5563, 7681}, {5692, 37822}, {5722, 12678}, {5836, 34918}, {5881, 10525}, {6244, 31141}, {6284, 26482}, {6506, 32625}, {6681, 38693}, {6826, 21164}, {6831, 14803}, {6834, 7280}, {6840, 15017}, {6894, 35010}, {6895, 27385}, {6898, 34595}, {6917, 18492}, {6951, 10175}, {6957, 7988}, {6965, 10165}, {7354, 26476}, {7741, 12114}, {7971, 37711}, {7995, 37714}, {9581, 10085}, {9590, 10731}, {9812, 12648}, {9955, 24927}, {9956, 26202}, {10528, 17578}, {10724, 25438}, {10738, 28204}, {10827, 12705}, {10834, 11403}, {10893, 37720}, {10902, 37290}, {10948, 40290}, {10958, 20420}, {11012, 37406}, {11113, 15931}, {11238, 30283}, {11496, 37719}, {11729, 28186}, {12173, 26378}, {12672, 37710}, {12688, 18480}, {12953, 26358}, {13271, 33956}, {13532, 38462}, {14988, 16128}, {15687, 32213}, {16173, 22835}, {17606, 34862}, {19047, 23261}, {19048, 23251}, {21031, 31777}, {22765, 38756}, {22793, 23340}, {22938, 28224}, {23961, 38753}, {31159, 34697}, {33110, 38155}, {34339, 37230}, {35000, 38755}, {40663, 41700}

X(41698) = midpoint of X(i) and X(j) for these {i, j}: {382, 18524}, {6905, 10728}, {22765, 38756}
X(41698) = reflection of X(i) in X(j) for these (i, j): (1, 1519), (36, 1532), (484, 1512), (1319, 1538), (1768, 1737), (2077, 119), (3583, 4), (4316, 6905), (4511, 21635), (5537, 17757), (5538, 908), (6909, 3814), (17613, 5123), (17757, 38757), (38753, 23961)
X(41698) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(34901)}} and {{A, B, C, X(34), X(17101)}}
X(41698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 1478, 1699), (4, 6256, 1), (4, 12115, 26333), (4, 12667, 1479), (382, 18542, 11248), (1837, 6259, 15071), (6256, 26333, 12115), (6259, 15071, 41690), (6923, 18516, 5587), (12115, 26333, 1)

X(41699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(36) TO REFLECTION OF X(46)

Barycentrics
a^10-2*(b+c)*a^9-(b^2-5*b*c+c^2)*a^8+4*(b^3+c^3)*a^7-(2*b^4+2*c^4+(7*b^2-6*b*c+7*c^2)*b*c)*a^6+2*(b^2-c^2)*(b-c)*b*c*a^5+(2*b^6+2*c^6+(b^2-4*b*c+c^2)*(b-c)^2*b*c)*a^4-4*(b^2-c^2)*(b-c)*(b^4+c^4)*a^3+(b^2-c^2)^2*(b-c)*(b^3-c^3)*a^2+2*(b^2-c^2)^2*(b-c)^2*(b^3+c^3)*a-(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(10073)

X(41699) lies on these lines: {1, 224}, {36, 5533}, {80, 912}, {90, 9581}, {1479, 31515}, {1749, 41557}, {1768, 3583}, {3585, 10052}, {5902, 7702}, {10073, 18838}, {10993, 32760}, {13274, 18857}, {14793, 37720}, {41685, 41688}

X(41699) = reflection of X(36) in X(41552)
X(41699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41685, 41688, 41692), (41688, 41710, 41685)

X(41700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(36) TO REFLECTION OF X(55)

Barycentrics    a*(a^5-(4*b^2+b*c+4*c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(3*b^2+5*b*c+3*c^2)*(b-c)^2*a+(b+c)*(b^3-c^3)*(-2*b+2*c)) : :
X(41700) = 4*X(9)-X(4867) = 3*X(36)-2*X(18450) = 2*X(1156)+X(3245) = 3*X(3582)-2*X(38055) = X(18450)-3*X(37787)

The reciprocal orthologic center of these triangles is X(41701)

X(41700) lies on these lines: {1, 6}, {7, 7951}, {35, 10394}, {36, 2801}, {46, 30353}, {80, 516}, {119, 5843}, {381, 36971}, {484, 15726}, {527, 1737}, {528, 41684}, {908, 5850}, {920, 5696}, {971, 1155}, {1478, 12848}, {1776, 5537}, {2078, 41701}, {3577, 24644}, {3582, 38055}, {3583, 38454}, {3584, 8255}, {3911, 5660}, {3935, 14740}, {4312, 5587}, {4860, 5219}, {5126, 6326}, {5506, 16193}, {5542, 37701}, {5698, 10573}, {5727, 36920}, {5735, 10826}, {5762, 12019}, {5852, 8068}, {5880, 18395}, {5902, 8545}, {6172, 18391}, {6594, 35204}, {7677, 10074}, {8544, 37524}, {15104, 30223}, {15346, 17057}, {15909, 18483}, {17530, 33558}, {29007, 30329}, {30424, 41572}, {31658, 37600}, {40663, 41698}

X(41700) = reflection of X(36) in X(37787)
X(41700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (9, 5223, 5692), (4312, 5779, 41694), (5220, 5729, 1)

X(41701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(55) TO REFLECTION OF X(36)

Barycentrics    a*(a^5-3*(b+c)*a^4+(b+2*c)*(2*b+c)*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)*(b^3-c^3)) : :
X(41701) = 2*X(1)-3*X(12739) = 2*X(11)-3*X(17718) = 3*X(55)-2*X(41166) = 4*X(5719)-3*X(16173) = X(41166)-3*X(41553)

The reciprocal orthologic center of these triangles is X(41700)

X(41701) lies on these lines: {1, 5}, {45, 38358}, {55, 2801}, {65, 38665}, {100, 518}, {104, 37600}, {153, 12743}, {200, 6174}, {214, 956}, {354, 14151}, {480, 6594}, {528, 1836}, {1156, 2346}, {1768, 27778}, {2078, 41700}, {2246, 38375}, {2646, 38669}, {2771, 10065}, {3174, 5528}, {3245, 11571}, {3811, 10609}, {3957, 10707}, {4420, 37605}, {4511, 10031}, {4860, 5083}, {5126, 10074}, {5316, 33812}, {5432, 11219}, {6765, 13996}, {9809, 30332}, {10389, 33519}, {11570, 12331}, {12531, 39778}, {12773, 37606}, {12776, 37837}, {13274, 21635}, {13279, 34791}, {17728, 41556}, {31397, 33857}

X(41701) = reflection of X(i) in X(j) for these (i, j): (55, 41553), (80, 495), (956, 214), (1836, 12831), (37740, 1317)
X(41701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (495, 37703, 17718), (1317, 6326, 12740), (5531, 37736, 11), (6265, 7972, 20586), (7972, 41689, 6265)

X(41702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(36) TO REFLECTION OF X(56)

Barycentrics    a*(a^3-2*(b+c)*a^2-(b^2-9*b*c+c^2)*a+2*(b+c)*(b^2-3*b*c+c^2)) : :
X(41702) = 3*X(1)-2*X(5440) = 2*X(1145)-3*X(3582) = 4*X(1320)-X(4867) = X(4880)+4*X(12653) = 2*X(6735)-3*X(16173) = 4*X(6735)-5*X(31263) = 3*X(15015)-4*X(25405) = 6*X(16173)-5*X(31263) = 4*X(30384)-3*X(31160)

The reciprocal orthologic center of these triangles is X(20586)

X(41702) lies on these lines: {1, 474}, {8, 3825}, {35, 3885}, {36, 2802}, {63, 5288}, {80, 519}, {84, 11531}, {145, 1478}, {149, 37006}, {517, 1768}, {912, 7982}, {952, 41698}, {1145, 3582}, {1149, 10700}, {1319, 5541}, {1482, 1699}, {2077, 12737}, {2098, 3632}, {2099, 4654}, {3057, 5258}, {3241, 5425}, {3555, 11280}, {3583, 38455}, {3585, 13463}, {3625, 5330}, {3635, 24200}, {3679, 20196}, {3746, 4861}, {3871, 24926}, {3872, 5251}, {3884, 27065}, {3895, 37525}, {3901, 8148}, {4051, 21373}, {4677, 5289}, {4919, 5526}, {5193, 20586}, {5252, 31159}, {5259, 9957}, {5533, 5854}, {5559, 6734}, {5563, 14923}, {5692, 7962}, {5720, 16200}, {5844, 37726}, {5903, 36846}, {5904, 12629}, {6001, 7993}, {6735, 16173}, {7951, 12648}, {9669, 36972}, {9897, 33956}, {10483, 36977}, {10915, 37735}, {11010, 11260}, {11278, 40263}, {13541, 16489}, {13996, 15325}, {15015, 25405}, {15955, 17469}, {16465, 25415}, {17460, 30117}, {17757, 32426}, {18393, 34640}, {21343, 33905}, {25055, 40587}, {34699, 37728}

X(41702) = reflection of X(i) in X(j) for these (i, j): (36, 38460), (2077, 12737), (5176, 21630), (5541, 1319), (13996, 15325), (37006, 149)
X(41702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1482, 3633, 41696), (3885, 22837, 35), (6735, 16173, 31263), (12629, 30323, 5904)

X(41703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(40) TO REFLECTION OF X(46)

Barycentrics
a*((b+c)*a^11-(3*b^2+b*c+3*c^2)*a^10-(b^3+c^3)*a^9+(11*b^4+11*c^4-2*(2*b^2-7*b*c+2*c^2)*b*c)*a^8-2*(b+c)*(3*b^4+3*c^4-2*(b-c)^2*b*c)*a^7-2*(b^2-b*c+c^2)*(7*b^4+7*c^4-2*(b^2+c^2)*b*c)*a^6+2*(b^3+c^3)*(7*b^4+7*c^4-2*(b^2+c^2)*b*c)*a^5+2*(3*b^8+3*c^8-2*(5*b^2-3*b*c+5*c^2)*(b^4-b^2*c^2+c^4)*b*c)*a^4-(b^2-c^2)*(b-c)*(11*b^6+11*c^6+(2*b^4+2*c^4+(b^2+8*b*c+c^2)*b*c)*b*c)*a^3+(b^2-c^2)^2*(b^6+c^6+(7*b^4+7*c^4-(9*b^2-10*b*c+9*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^3*(b-c)*(3*b^4+3*c^4-(b^2+c^2)*b*c)*a-(b^2-c^2)^6) : :

The reciprocal orthologic center of these triangles is X(41704)

X(41703) lies on these lines: {40, 41559}, {90, 5720}, {912, 4338}, {1709, 11248}, {5696, 41705}, {31938, 41691}, {41685, 41688}

X(41703) = reflection of X(i) in X(j) for these (i, j): (40, 41559), (41685, 41688)

X(41704) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(46) TO REFLECTION OF X(40)

Barycentrics
a*((b+c)*a^8-(2*b^2-b*c+2*c^2)*a^7-2*(b^3+c^3)*a^6-2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^4*b*c+(3*b^2-4*b*c+3*c^2)*(2*b^2+b*c+2*c^2)*a^5-(6*b^4+6*c^4+(5*b^2+4*b*c+5*c^2)*b*c)*(b-c)^2*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4+(3*b^2+b*c+3*c^2)*b*c)*a^2+(2*b^4+2*c^4-(3*b^2+2*b*c+3*c^2)*b*c)*(b^2-c^2)^2*a-(b^2-c^2)^4*(b+c)) : :

The reciprocal orthologic center of these triangles is X(41703)

X(41704) lies on these lines: {4, 10052}, {46, 41560}, {56, 971}, {912, 41709}, {1071, 10893}, {1482, 12688}, {1709, 11248}, {1750, 3336}, {1837, 6259}, {2771, 37001}, {3913, 17661}, {5691, 5903}, {10320, 34293}, {10572, 12666}, {10598, 15528}, {11508, 18239}, {14450, 36991}, {41538, 41707}

X(41704) = reflection of X(i) in X(j) for these (i, j): (46, 41560), (10085, 1898), (15071, 1837), (17857, 40263)

X(41705) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(40) TO REFLECTION OF X(55)

Barycentrics    5*a^6-3*(b+c)*a^5-12*(b^2-b*c+c^2)*a^4+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^3+(9*b^2+10*b*c+9*c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b+3*c)*(3*b+c)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(41705) = 4*X(7)-5*X(8227) = 8*X(9)-7*X(31423) = 3*X(3576)-2*X(36996) = 2*X(4312)-3*X(5587) = 3*X(5587)-4*X(5779) = 3*X(5817)-2*X(30424) = 3*X(10861)-4*X(20117) = 4*X(11372)-3*X(31162) = 6*X(21168)-5*X(35242) = 13*X(34595)-12*X(38111)

The reciprocal orthologic center of these triangles is X(41706)

X(41705) lies on these lines: {1, 5843}, {7, 8227}, {9, 2252}, {40, 144}, {63, 9809}, {165, 41706}, {516, 3625}, {527, 11372}, {946, 20059}, {971, 5693}, {1768, 31142}, {2094, 10863}, {2829, 36922}, {2951, 17857}, {3062, 5762}, {3576, 36996}, {4301, 28647}, {4312, 5587}, {5223, 12751}, {5696, 41703}, {5732, 5851}, {5735, 16112}, {5777, 31391}, {5817, 30424}, {5850, 7982}, {10860, 17781}, {10861, 20117}, {11246, 30326}, {14646, 21060}, {15017, 31231}, {21168, 35242}, {34595, 38111}

X(41705) = reflection of X(i) in X(j) for these (i, j): (40, 144), (4312, 5779), (5735, 16112), (20059, 946), (31391, 5777)
X(41705) = {X(4312), X(5779)}-harmonic conjugate of X(5587)

X(41706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(55) TO REFLECTION OF X(40)

Barycentrics    a^6+(b+c)*a^5-(5*b^2-6*b*c+5*c^2)*a^4+(5*b^2+6*b*c+5*c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2 : :
X(41706) = 2*X(1709)-3*X(17718) = 3*X(5252)-4*X(12115) = 2*X(12115)-3*X(12678)

The reciprocal orthologic center of these triangles is X(41705)

X(41706) lies on these lines: {11, 30304}, {12, 7992}, {55, 41561}, {63, 5851}, {65, 6223}, {84, 11375}, {144, 7964}, {165, 41705}, {329, 5918}, {354, 36996}, {516, 41711}, {971, 1836}, {1071, 12679}, {1158, 41543}, {1709, 17718}, {1750, 11246}, {1837, 6259}, {2646, 12246}, {2801, 4863}, {3062, 4654}, {3654, 12767}, {4679, 10167}, {5249, 16112}, {5252, 6001}, {5843, 41338}, {5905, 15726}, {6260, 24914}, {7971, 37738}, {7995, 15888}, {9355, 24789}, {9809, 11220}, {9948, 10895}, {10085, 11376}, {10178, 31018}, {10404, 12688}, {10596, 12675}, {11495, 17781}, {12047, 12684}, {12667, 41687}, {12680, 12701}, {20070, 28647}

X(41706) = reflection of X(i) in X(j) for these (i, j): (55, 41561), (5252, 12678)
X(41706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1071, 16127, 12679), (3062, 4654, 7965), (6259, 15071, 1837), (9809, 11220, 24703), (15071, 41690, 6259)

X(41707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(46) TO REFLECTION OF X(55)

Barycentrics    5*a^6-3*(b+c)*a^5-6*(2*b^2-b*c+2*c^2)*a^4+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^3+3*(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^2-3*(b^2-c^2)^2*(b+c)*a-2*(b^2-c^2)^2*(b-c)^2 : :
X(41707) = 6*X(10072)-7*X(15299)

The reciprocal orthologic center of these triangles is X(41708)

X(41707) lies on these lines: {1, 5852}, {7, 37692}, {9, 17700}, {46, 41563}, {144, 12514}, {516, 37711}, {527, 10072}, {4312, 5587}, {5128, 41690}, {5850, 11682}, {10398, 28646}, {11662, 16112}, {41538, 41704}

X(41707) = reflection of X(46) in X(41563)

X(41708) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(55) TO REFLECTION OF X(46)

Barycentrics
a^9-(7*b^2-2*b*c+7*c^2)*a^7+(b+c)*(5*b^2-6*b*c+5*c^2)*a^6+(11*b^4+11*c^4-2*(4*b^2-7*b*c+4*c^2)*b*c)*a^5-(b+c)*(11*b^4+11*c^4-2*(8*b^2-9*b*c+8*c^2)*b*c)*a^4-(5*b^4-2*b^2*c^2+5*c^4)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(7*b^4-2*b^2*c^2+7*c^4)*a^2-4*(b^2-c^2)^2*(b-c)^2*b*c*a-(b^2-c^2)^3*(b-c)^3 : :

The reciprocal orthologic center of these triangles is X(41707)

X(41708) lies on these lines: {55, 41564}, {90, 11374}, {912, 5252}, {1836, 16465}, {41571, 41695}, {41685, 41688}

X(41708) = reflection of X(55) in X(41564)
X(41708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41685, 41688, 41710), (41685, 41692, 41688)

X(41709) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(46) TO REFLECTION OF X(56)

Barycentrics    3*a^4-3*(b+c)*a^3-(b^2+4*b*c+c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-2*(b^2-c^2)^2 : :
X(41709) = 3*X(78)-4*X(3825)

The reciprocal orthologic center of these triangles is X(41710)

X(41709) lies on these lines: {1, 442}, {8, 7162}, {10, 31259}, {46, 12649}, {78, 3825}, {80, 6765}, {145, 12047}, {200, 37702}, {499, 12437}, {519, 1479}, {912, 41704}, {950, 41229}, {1482, 1699}, {1698, 37080}, {1737, 3189}, {2136, 41684}, {2771, 3901}, {3057, 3632}, {3086, 12536}, {3158, 18395}, {3187, 26783}, {3241, 31418}, {3243, 5270}, {3295, 3679}, {3583, 11523}, {3586, 5904}, {3612, 10916}, {3625, 40998}, {3649, 36867}, {3811, 10826}, {3870, 10827}, {3875, 21276}, {3894, 9579}, {3928, 4324}, {3962, 9668}, {4295, 20008}, {4302, 24391}, {4863, 37730}, {4867, 9614}, {5175, 13407}, {5231, 37571}, {5853, 10573}, {7280, 34701}, {9897, 26726}, {11011, 11237}, {12629, 37706}, {12630, 38037}, {15908, 37727}, {18514, 28609}, {20013, 21616}, {22836, 23708}, {25055, 31493}, {25415, 41575}, {26015, 37618}, {34489, 41710}, {34640, 34747}, {34772, 37692}

X(41709) = reflection of X(i) in X(j) for these (i, j): (46, 12649), (20013, 21616)
X(41709) = {X(1699), X(3633)}-harmonic conjugate of X(41696)

X(41710) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(56) TO REFLECTION OF X(46)

Barycentrics
(a^8-3*(b+c)*a^7+2*(b+c)^2*a^6+(b+c)*(3*b^2+2*b*c+3*c^2)*a^5-2*(3*b^4+3*c^4+4*(b^2+c^2)*b*c)*a^4+(b+c)*(3*b^4+3*c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^3+2*(b+c)^2*(b^4+c^4)*a^2-(b^2-c^2)^2*(b+c)*(3*b^2-2*b*c+3*c^2)*a+(b^2-c^2)^4)*(a-b+c)*(a+b-c) : :

The reciprocal orthologic center of these triangles is X(41709)

X(41710) lies on these lines: {56, 41565}, {90, 3652}, {224, 41552}, {912, 1837}, {942, 10044}, {3419, 41540}, {4863, 22837}, {5252, 14563}, {5840, 37532}, {11517, 24914}, {34489, 41709}, {41574, 41697}, {41685, 41688}

X(41710) = reflection of X(56) in X(41565)
X(41710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (41685, 41688, 41708), (41685, 41699, 41688)

X(41711) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(55) TO REFLECTION OF X(56)

Barycentrics    a*(a^2+2*b^2+2*c^2-3*(b+c)*a) : :
X(41711) = 3*X(55)-2*X(63) = 5*X(55)-4*X(4640) = 7*X(55)-6*X(35258) = X(63)-3*X(3870) = 5*X(63)-6*X(4640) = 7*X(63)-9*X(35258) = 4*X(226)-3*X(31140) = 2*X(3419)-3*X(11237) = 5*X(3870)-2*X(4640) = 7*X(3870)-3*X(35258) = 14*X(4640)-15*X(35258) = 2*X(4847)-3*X(17718) = 4*X(4847)-5*X(31245) = 2*X(4863)-3*X(31140) = 6*X(17718)-5*X(31245)

The reciprocal orthologic center of these triangles is X(41712)

X(41711) lies on these lines: {1, 210}, {2, 3711}, {3, 15104}, {6, 3930}, {7, 34612}, {8, 3475}, {9, 3748}, {11, 5748}, {42, 3242}, {43, 17597}, {55, 63}, {56, 3555}, {57, 3689}, {65, 6765}, {69, 4030}, {72, 3303}, {78, 3304}, {100, 4430}, {144, 10385}, {145, 497}, {200, 354}, {226, 519}, {329, 3058}, {344, 4126}, {392, 8162}, {474, 3881}, {480, 8257}, {516, 41706}, {528, 5905}, {599, 33074}, {614, 4849}, {896, 21000}, {908, 11238}, {936, 17609}, {938, 21031}, {940, 3961}, {984, 3979}, {1001, 3681}, {1155, 3158}, {1211, 36479}, {1260, 33925}, {1376, 3873}, {1388, 22836}, {1482, 1699}, {1617, 37736}, {1621, 4661}, {1697, 3962}, {1750, 7982}, {1757, 17715}, {1836, 5853}, {1998, 18839}, {2550, 20015}, {2646, 6762}, {2999, 21870}, {3052, 3722}, {3057, 10382}, {3189, 7354}, {3210, 24841}, {3214, 17054}, {3218, 4421}, {3219, 4428}, {3241, 5289}, {3244, 4679}, {3295, 5904}, {3340, 3893}, {3474, 6154}, {3476, 34749}, {3485, 6764}, {3488, 34606}, {3621, 34195}, {3649, 5082}, {3666, 16496}, {3679, 15934}, {3683, 5223}, {3698, 4882}, {3740, 4666}, {3744, 3751}, {3746, 3927}, {3749, 4641}, {3757, 4042}, {3775, 29669}, {3868, 3913}, {3874, 5221}, {3889, 4420}, {3891, 13576}, {3894, 36279}, {3901, 12702}, {3912, 30615}, {3970, 21373}, {3996, 24349}, {4005, 31435}, {4015, 16842}, {4113, 4384}, {4360, 30946}, {4361, 30949}, {4363, 32945}, {4387, 32937}, {4677, 5425}, {4753, 19750}, {4847, 17718}, {4851, 4952}, {4867, 31142}, {4883, 5268}, {4930, 34747}, {4966, 10327}, {4995, 5744}, {5048, 17604}, {5432, 24477}, {5435, 6174}, {5524, 17063}, {5531, 19541}, {5538, 30283}, {5692, 6767}, {5694, 12000}, {5695, 17165}, {5722, 31141}, {5780, 9624}, {5836, 11520}, {5855, 12648}, {5918, 7994}, {6510, 8271}, {6600, 37578}, {6603, 28070}, {6745, 17728}, {6769, 12680}, {7174, 37593}, {7232, 32948}, {8167, 29817}, {9026, 26892}, {9049, 26893}, {9053, 33088}, {9709, 18398}, {9812, 12630}, {9965, 34607}, {10157, 10222}, {10388, 14100}, {10896, 21077}, {10912, 20050}, {10914, 12559}, {11011, 12629}, {11038, 26040}, {11235, 31053}, {11246, 17784}, {11510, 14054}, {12513, 34471}, {12607, 12649}, {14872, 18540}, {15228, 34707}, {15950, 34625}, {16884, 29816}, {17377, 21276}, {17388, 17747}, {17450, 37682}, {17484, 34611}, {17605, 24392}, {17716, 36483}, {17724, 33137}, {17765, 32946}, {17768, 20075}, {17783, 33140}, {19701, 36480}, {19732, 29651}, {20012, 32922}, {21805, 37679}, {23404, 29531}, {29670, 37660}, {29673, 30811}, {30305, 34699}, {30827, 31146}, {31165, 31393}, {31493, 37731}, {32049, 41575}, {32913, 37540}, {34595, 36946}, {34619, 40663}, {34748, 35457}, {35272, 37602}

X(41711) = reflection of X(i) in X(j) for these (i, j): (55, 3870), (4863, 226)
X(41711) = X(643)-Beth conjugate of-X(23958)
X(41711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 210, 4423), (8, 3475, 3925), (42, 3242, 17599), (78, 34791, 3304), (145, 12635, 2098), (200, 354, 4413), (200, 3243, 354), (226, 4863, 31140), (1001, 3681, 3715), (1376, 3873, 4860), (1621, 4661, 5220), (3555, 3811, 56), (3633, 41696, 1482), (3681, 3957, 1001), (3722, 32912, 3052), (3873, 3935, 1376), (4849, 4864, 614), (4882, 11518, 3698), (5223, 10389, 3683), (25568, 36845, 11)

X(41712) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(56) TO REFLECTION OF X(55)

Barycentrics    a*(a+b-c)*(a-b+c)*(a^3-3*(b+c)^2*a+2*(b+c)*(b^2+c^2)) : :
X(41712) = 5*X(56)-2*X(30318) = 5*X(1445)-X(30318) = 2*X(5729)+X(37567)

The reciprocal orthologic center of these triangles is X(41711)

X(41712) lies on these lines: {1, 15837}, {3, 18412}, {7, 12}, {9, 65}, {40, 10398}, {46, 971}, {55, 1708}, {56, 78}, {57, 210}, {77, 4663}, {142, 24914}, {144, 2551}, {226, 3715}, {241, 3751}, {307, 38047}, {388, 5686}, {405, 12432}, {499, 20330}, {516, 1837}, {517, 15299}, {527, 31141}, {942, 15298}, {954, 15556}, {984, 5228}, {1001, 2099}, {1155, 5732}, {1319, 3243}, {1376, 41228}, {1388, 7677}, {1407, 32912}, {1454, 5784}, {1456, 1743}, {1458, 4878}, {1471, 3242}, {1617, 37736}, {1697, 30330}, {1737, 5805}, {1757, 6180}, {1770, 31672}, {2093, 11372}, {2283, 37507}, {2550, 12848}, {2646, 21153}, {2951, 5128}, {3035, 5435}, {3059, 37550}, {3254, 20118}, {3303, 5572}, {3339, 3927}, {3474, 36991}, {3485, 18230}, {3649, 8232}, {3911, 4860}, {3913, 30628}, {4295, 5817}, {4312, 5587}, {4321, 32636}, {4326, 37568}, {4423, 5173}, {4429, 17950}, {5217, 7675}, {5252, 12573}, {5433, 38053}, {5708, 11231}, {5759, 18391}, {5809, 6284}, {5850, 21075}, {5853, 41687}, {5856, 12832}, {5880, 41572}, {6594, 12739}, {6600, 37579}, {6604, 27549}, {6666, 11375}, {6766, 10866}, {7098, 10394}, {7288, 11038}, {7957, 10396}, {7964, 10382}, {7991, 10384}, {8545, 15481}, {9709, 15932}, {10039, 38126}, {10826, 18482}, {11011, 38316}, {12047, 38108}, {12635, 18467}, {12702, 31795}, {13411, 38130}, {17606, 38150}, {17768, 41563}, {17857, 37582}, {27484, 41245}, {31230, 38186}, {37544, 41229}, {37692, 38318}, {37737, 38113}

X(41712) = midpoint of X(4312) and X(41707)
X(41712) = reflection of X(56) in X(1445)
X(41712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 38057, 12), (40, 10398, 14100), (57, 5223, 8581), (1001, 7672, 2099), (1708, 41539, 55), (4312, 41700, 5779), (5779, 36279, 4312), (7672, 37787, 1001), (12573, 24393, 5252)

X(41713) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(2) TO REFLECTION OF X(5)

Barycentrics    a^2*((b^2+c^2)*a^8-(2*b^4+3*b^2*c^2+2*c^4)*a^6+2*(b^2+c^2)*b^2*c^2*a^4+(b^4+b^2*c^2+c^4)*(2*b^2-c^2)*(b^2-2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(41713) = 5*X(54)-8*X(5462) = X(54)-4*X(6153) = 10*X(1209)-7*X(7999) = 4*X(1209)-X(12226) = 2*X(1209)+X(13423) = X(2888)+2*X(6152) = 5*X(3519)+4*X(13421) = X(3519)+2*X(32196) = 2*X(5462)-5*X(6153) = 4*X(5876)+5*X(6242) = 2*X(5876)-5*X(6288) = X(5876)+5*X(13368) = 8*X(5876)-5*X(41726) = X(6242)+2*X(6288) = X(6242)-4*X(13368) = 2*X(6242)+X(41726) = X(6288)+2*X(13368) = 4*X(6288)-X(41726) = 8*X(13368)+X(41726) = 2*X(13421)-5*X(32196)

The reciprocal orthologic center of these triangles is X(41714)

X(41713) lies on these lines: {2, 34751}, {4, 93}, {23, 9972}, {51, 110}, {54, 5462}, {193, 9971}, {195, 13861}, {343, 41596}, {539, 7730}, {1176, 2393}, {1209, 7999}, {2979, 7703}, {3060, 5965}, {3574, 12280}, {3819, 30745}, {3832, 15739}, {5012, 9977}, {5890, 10116}, {6403, 41599}, {6689, 12291}, {7691, 12086}, {9973, 37913}, {11451, 15073}, {11808, 15801}, {13365, 15532}, {13371, 21357}, {13622, 41579}, {15072, 18400}, {22467, 23358}, {32423, 38321}

X(41713) = reflection of X(2) in X(41578)
X(41713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1209, 13423, 12226), (3410, 41732, 41730), (6242, 6288, 41726), (6288, 13368, 6242), (6288, 41730, 3410)

X(41714) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(5) TO REFLECTION OF X(2)

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^2+c^2)^2*a^6+2*(b^2+c^2)*b^2*c^2*a^4+2*(b^8+c^8-(b^2+c^2)^2*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(41714) = 3*X(51)-2*X(5097) = 5*X(141)-4*X(32142) = 5*X(182)-6*X(5892) = X(1351)-3*X(9971) = X(1352)-3*X(11188) = 5*X(1352)-3*X(11459) = 3*X(1352)-X(41716) = 4*X(5462)-3*X(39561) = 3*X(5946)-2*X(12007) = X(6403)+3*X(11188) = 5*X(6403)+3*X(11459) = 3*X(6403)+X(41716) = X(9967)-3*X(29959) = 5*X(11188)-X(11459) = 9*X(11188)-X(41716) = 9*X(11459)-5*X(41716) = 5*X(12061)+4*X(32142) = 2*X(24206)-3*X(29959) = 2*X(32366)-3*X(39561) = 3*X(34155)-4*X(41671)

The reciprocal orthologic center of these triangles is X(41713)

X(41714) lies on these lines: {3, 9973}, {4, 69}, {5, 41579}, {6, 49}, {51, 110}, {52, 5965}, {140, 17710}, {141, 12061}, {143, 3629}, {182, 2393}, {427, 41673}, {524, 13490}, {542, 11562}, {575, 6467}, {576, 9925}, {1351, 9971}, {1353, 2854}, {1503, 13491}, {1634, 41169}, {2071, 14810}, {2072, 9967}, {2781, 39884}, {3098, 12084}, {3313, 40107}, {3518, 9972}, {3548, 11574}, {3564, 31830}, {3589, 15074}, {3631, 6101}, {3917, 31074}, {5020, 34751}, {5092, 22467}, {5446, 15068}, {5462, 32366}, {5480, 14984}, {5504, 12584}, {5650, 30745}, {5654, 10110}, {5891, 15432}, {5946, 12007}, {6153, 10274}, {6243, 40341}, {6329, 15026}, {6642, 34777}, {7723, 16194}, {7730, 11271}, {8705, 15122}, {9822, 38317}, {9977, 10282}, {10516, 18438}, {11179, 12283}, {11557, 41731}, {11663, 12220}, {14561, 15073}, {15516, 40673}, {15520, 32284}, {16625, 21651}, {16776, 18583}, {18440, 37473}, {19137, 34788}, {21850, 22800}, {22234, 22829}, {22463, 37813}, {29012, 37511}, {32299, 32423}

X(41714) = midpoint of X(i) and X(j) for these {i, j}: {3, 9973}, {141, 12061}, {1352, 6403}, {6243, 40341}, {11663, 12220}, {18440, 37473}
X(41714) = reflection of X(i) in X(j) for these (i, j): (5, 41579), (576, 9969), (1353, 32191), (3313, 40107), (3629, 143), (6101, 3631), (6467, 575), (9967, 24206), (15074, 3589), (17710, 140), (32366, 5462), (34507, 14913), (37517, 5446), (41731, 11557)
X(41714) = 1st Droz-Farny circle-inverse of-X(316)
X(41714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5097, 34986, 19150), (5462, 32366, 39561), (6403, 11188, 1352), (9967, 29959, 24206)

X(41715) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(2) TO REFLECTION OF X(20)

Barycentrics    a^2*((b^2+c^2)*a^8-(2*b^4-3*b^2*c^2+2*c^4)*a^6-(b^2+c^2)*b^2*c^2*a^4+(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4)) : :
X(41715) = 2*X(52)+X(34781) = 2*X(64)-5*X(10574) = X(64)-4*X(41589) = 4*X(143)-X(34780) = 2*X(185)+X(6225) = 4*X(389)-X(12324) = 2*X(2883)+X(6293) = 4*X(2883)-X(12111) = 5*X(3567)-2*X(14216) = 2*X(5878)+X(6241) = 4*X(5878)-X(36983) = X(5878)+2*X(41725) = 2*X(6241)+X(36983) = X(6241)-4*X(41725) = 7*X(9781)-4*X(18381) = 5*X(10574)-8*X(41589) = X(12290)-4*X(22802) = X(13201)-4*X(15647) = 4*X(19149)-X(41716) = X(36983)+8*X(41725)

The reciprocal orthologic center of these triangles is X(41716)

X(41715) lies on these lines: {2, 34146}, {4, 51}, {22, 110}, {23, 9968}, {52, 34781}, {54, 32321}, {64, 10574}, {66, 7394}, {143, 34780}, {161, 15107}, {184, 34779}, {193, 2393}, {206, 6636}, {323, 1660}, {343, 2883}, {511, 11206}, {1154, 7387}, {1176, 19153}, {1498, 5889}, {1503, 3060}, {1619, 1993}, {1853, 5640}, {3357, 35500}, {3547, 5891}, {3580, 41602}, {3819, 7494}, {3917, 35260}, {5012, 34117}, {5133, 7703}, {5562, 33522}, {5644, 33541}, {5656, 13754}, {5892, 7404}, {5893, 11439}, {5895, 12279}, {6101, 14530}, {6102, 12315}, {6247, 15043}, {6403, 31383}, {6515, 41735}, {6759, 11412}, {6995, 19161}, {7408, 9969}, {7485, 34778}, {7503, 10606}, {7512, 11202}, {7553, 13142}, {7731, 9934}, {7998, 10192}, {9818, 35450}, {11204, 35921}, {11427, 12294}, {11444, 16252}, {11744, 12270}, {12058, 37669}, {12118, 31305}, {12220, 34774}, {12250, 40647}, {12283, 13417}, {13093, 13630}, {14826, 37511}, {15024, 20299}, {15028, 40686}, {15072, 15311}, {15257, 40368}, {15760, 18435}, {17409, 34137}, {18911, 34944}, {19119, 40673}, {20062, 36989}, {20859, 32445}, {32392, 36982}, {34783, 41588}

X(41715) = reflection of X(i) in X(j) for these (i, j): (2, 41580), (2979, 154), (32064, 51)
X(41715) = crosssum of X(3) and X(34609)
X(41715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (64, 41589, 10574), (1899, 5878, 41736), (1899, 41736, 41738), (2883, 6293, 12111), (5878, 6241, 36983), (5878, 41725, 6241)

X(41716) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(20) TO REFLECTION OF X(2)

Barycentrics    -a^2*(-a^2+b^2+c^2)*((b^2+c^2)*a^6-(b^4-b^2*c^2+c^4)*a^4-(b^2+c^2)*(b^4+c^4)*a^2+(b^6-c^6)*(b^2-c^2)) : :
X(41716) = 2*X(52)-3*X(14853) = 4*X(141)-5*X(11444) = 4*X(182)-3*X(5890) = 2*X(185)-3*X(25406) = 4*X(389)-5*X(3618) = 3*X(568)-4*X(18583) = 2*X(1350)-3*X(2979) = 4*X(1352)-3*X(11188) = 2*X(1352)-3*X(11459) = 3*X(1352)-2*X(41714) = 3*X(3060)-4*X(5480) = 4*X(3818)-5*X(15058) = 2*X(6403)-3*X(11188) = X(6403)-3*X(11459) = 3*X(6403)-4*X(41714) = 9*X(11188)-8*X(41714) = 5*X(11444)-2*X(37473) = 9*X(11459)-4*X(41714) = 4*X(11574)-3*X(25406) = 4*X(19149)-3*X(41715)

The reciprocal orthologic center of these triangles is X(41715)

X(41716) lies on these lines: {2, 19161}, {3, 1176}, {4, 69}, {6, 5889}, {20, 3313}, {22, 110}, {52, 7404}, {66, 37444}, {68, 18124}, {141, 11444}, {159, 11441}, {182, 5890}, {185, 11574}, {206, 7488}, {343, 3060}, {389, 3618}, {542, 12281}, {568, 18583}, {576, 35500}, {631, 28708}, {895, 34801}, {1092, 3098}, {1154, 1351}, {1216, 3547}, {1369, 12384}, {1503, 12111}, {1568, 7699}, {1899, 41257}, {2393, 5921}, {2854, 12273}, {3091, 9969}, {3094, 3289}, {3564, 12605}, {3567, 14561}, {3575, 13562}, {3589, 15043}, {3619, 11793}, {3631, 40929}, {3852, 9863}, {3917, 7494}, {4549, 6776}, {5050, 6102}, {5085, 10574}, {5157, 37126}, {5392, 35098}, {5640, 32191}, {5876, 18440}, {5891, 40330}, {5969, 39807}, {6000, 14927}, {6101, 7387}, {6243, 7403}, {6293, 34774}, {6457, 11514}, {6458, 11513}, {7495, 7998}, {7553, 37484}, {7558, 7999}, {7566, 11743}, {7691, 19121}, {7723, 11898}, {7731, 9970}, {9019, 15305}, {9781, 19130}, {10316, 34137}, {10516, 15056}, {10565, 41580}, {10625, 31305}, {10628, 11061}, {11413, 34778}, {11426, 31810}, {11449, 35228}, {11454, 15578}, {11579, 12284}, {12017, 13630}, {12164, 19459}, {12177, 39837}, {12244, 15102}, {12270, 32233}, {12272, 15069}, {12290, 15103}, {12362, 26926}, {12825, 41737}, {15024, 38317}, {15072, 16775}, {15074, 39899}, {15107, 26284}, {15644, 25712}, {15760, 23039}, {16475, 31732}, {18435, 39884}, {18909, 41256}, {19124, 30100}, {19132, 41589}, {19140, 22109}, {19154, 38898}, {19649, 28711}, {20300, 23293}, {21554, 28712}, {21649, 25320}, {22658, 34207}, {23041, 38444}, {28754, 37521}, {31884, 35602}, {35934, 41169}, {37481, 38110}

X(41716) = midpoint of X(i) and X(j) for these {i, j}: {12111, 12220}, {18436, 18438}
X(41716) = reflection of X(i) in X(j) for these (i, j): (20, 3313), (69, 5562), (185, 11574), (1843, 5907), (3575, 13562), (5889, 6), (6243, 21850), (6293, 34774), (6403, 1352), (6776, 9967), (7731, 9970), (11188, 11459), (12270, 32233), (12272, 15069), (12284, 11579), (15073, 18438), (18440, 5876), (19459, 31807), (21851, 11793), (26926, 12362), (33878, 6101), (37473, 141), (37511, 1216), (39837, 12177), (39899, 15074), (40929, 3631), (41737, 12825)
X(41716) = anticomplement of X(19161)
X(41716) = isotomic conjugate of the polar conjugate of X(22240)
X(41716) = barycentric product X(69)*X(22240)
X(41716) = trilinear product X(63)*X(22240)
X(41716) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(1235)}} and {{A, B, C, X(4), X(10547)}}
X(41716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (185, 11574, 25406), (1216, 37511, 10519), (1350, 19149, 22), (1352, 6403, 11188), (3060, 33523, 343), (6403, 11459, 1352)

X(41717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(2) TO REFLECTION OF X(21)

Barycentrics    a*(2*b*c*a^3-2*a*b^2*c^2-(b+c)*a^4+(b^4-c^4)*(b-c)) : :
X(41717) = X(65)-4*X(41591) = 2*X(1829)+X(3869) = 5*X(3616)-2*X(18732) = 2*X(5887)+X(41722) = X(5887)+2*X(41727) = 4*X(5887)-X(41733) = X(41722)-4*X(41727) = 2*X(41722)+X(41733) = 8*X(41727)+X(41733)

The reciprocal orthologic center of these triangles is X(41718)

X(41717) lies on these lines: {2, 3827}, {4, 8}, {65, 32911}, {81, 105}, {165, 5314}, {193, 4430}, {238, 21367}, {518, 3060}, {608, 22122}, {912, 30438}, {959, 15474}, {960, 32782}, {1203, 5902}, {1621, 17441}, {3616, 18732}, {3740, 7703}, {3873, 34381}, {4523, 32947}, {5698, 20243}, {6001, 15072}, {15507, 21318}, {17127, 40959}, {21867, 33110}, {23155, 34371}, {31053, 40635}, {31143, 31165}, {33129, 40961}, {33133, 40962}

X(41717) = reflection of X(2) in X(41581)
X(41717) = intersection, other than A,B,C, of conics {{A, B, C, X(105), X(41013)}} and {{A, B, C, X(321), X(1814)}}
X(41717) = crossdifference of every pair of points on line {X(22383), X(24290)}
X(41717) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5887, 41722, 41733), (5887, 41727, 41722)

X(41718) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(2)

Barycentrics    a*(a+b)*(c+a)*((b^2+b*c+c^2)*a^4-(b+c)*b*c*a^3-(b^2+c^2)*b*c*a^2+(b+c)*(b^2+c^2)*b*c*a-(b^4-c^4)*(b^2-c^2)) : :

The reciprocal orthologic center of these triangles is X(41717)

X(41718) lies on these lines: {4, 69}, {21, 3827}, {81, 105}, {518, 41723}, {3868, 32922}, {5820, 12272}, {5847, 35637}, {12826, 41741}, {18180, 34381}, {41592, 41604}

X(41718) = reflection of X(i) in X(j) for these (i, j): (21, 41582), (41610, 18180), (41741, 12826)
X(41718) = crossdifference of every pair of points on line {X(3049), X(24290)}

X(41719) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(2) TO REFLECTION OF X(22)

Barycentrics    5*a^8-2*(b^2+c^2)*a^6-4*(b^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 : :
X(41719) = X(4)-4*X(34117) = 2*X(6)+X(5596) = 5*X(6)-2*X(15583) = 3*X(6)-2*X(23326) = X(6)+2*X(34774) = 4*X(6)-X(36851) = X(1498)+2*X(8550) = 5*X(5596)+4*X(15583) = 3*X(5596)+4*X(23326) = X(5596)-4*X(34774) = 2*X(5596)+X(36851) = X(5656)+4*X(41729) = X(6776)+2*X(19149) = X(6776)-4*X(41729) = 2*X(6776)+X(41735) = 2*X(8549)+X(34781) = 3*X(14853)-2*X(23049) = 3*X(15583)-5*X(23326) = X(15583)+5*X(34774) = 8*X(15583)-5*X(36851)

The reciprocal orthologic center of these triangles is X(69)

X(41719) lies on these lines: {2, 19153}, {4, 6}, {23, 159}, {26, 34380}, {66, 3618}, {69, 110}, {141, 19132}, {154, 524}, {376, 2781}, {575, 14216}, {576, 9833}, {597, 1853}, {599, 10192}, {1353, 7530}, {1370, 22151}, {1576, 37188}, {1619, 32621}, {1992, 2393}, {2777, 9970}, {2892, 6593}, {3090, 34118}, {3284, 8721}, {3313, 27082}, {3529, 10510}, {3549, 13562}, {3566, 18311}, {3619, 31267}, {3629, 9924}, {5032, 11216}, {5085, 23328}, {5093, 7540}, {5095, 38885}, {5169, 15431}, {5201, 15270}, {5965, 32379}, {6000, 11179}, {6225, 9968}, {6241, 35371}, {6353, 18374}, {6389, 35282}, {6696, 10541}, {7488, 37485}, {7492, 32262}, {7494, 19127}, {7519, 34777}, {7556, 15577}, {7714, 9971}, {8263, 8780}, {8541, 31383}, {8584, 17813}, {9019, 34608}, {10117, 25329}, {10182, 19126}, {10519, 23041}, {10565, 16789}, {11433, 19136}, {11470, 19467}, {11477, 34782}, {11574, 41725}, {11799, 39899}, {12324, 13434}, {14001, 15257}, {14561, 23325}, {15069, 16252}, {15072, 25406}, {15138, 35483}, {15139, 40132}, {15141, 16063}, {15303, 36201}, {15585, 40341}, {18400, 20423}, {19118, 26869}, {19122, 28708}, {19459, 35219}, {20968, 28696}, {23324, 38072}, {23327, 32064}, {23329, 38064}, {31670, 34776}, {32241, 38851}, {37473, 41589}, {37644, 41613}

X(41719) = midpoint of X(i) and X(j) for these {i, j}: {1992, 11206}, {5656, 6776}, {15131, 32264}
X(41719) = reflection of X(i) in X(j) for these (i, j): (2, 19153), (599, 10192), (1853, 597), (2892, 15131), (5656, 19149), (10519, 23041), (11206, 31166), (15131, 6593), (17813, 8584), (18405, 5480), (32064, 23327), (41735, 5656)
X(41719) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(5523)}} and {{A, B, C, X(393), X(2373)}}
X(41719) = crossdifference of every pair of points on line {X(520), X(7652)}
X(41719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5596, 36851), (6, 34774, 5596), (66, 41593, 3618), (6593, 32264, 2892), (6776, 14853, 12022), (6776, 19149, 41735), (19149, 41729, 6776)

X(41720) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(2) TO REFLECTION OF X(23)

Barycentrics    7*a^8-6*(b^2+c^2)*a^6-(5*b^4-13*b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-2*(b^4-c^4)^2 : :
X(41720) = X(2)-3*X(25321) = X(67)-4*X(41595) = 3*X(110)-2*X(5648) = X(110)-4*X(25329) = X(895)-4*X(5095) = X(895)+2*X(11061) = 5*X(895)-2*X(32255) = 5*X(1992)-X(32255) = 2*X(5095)+X(11061) = 10*X(5095)-X(32255) = 2*X(9970)+X(32234) = X(9970)+2*X(41731) = 4*X(9970)-X(41737) = X(10706)+4*X(41731) = 5*X(11061)+X(32255) = X(13169)-4*X(15303) = X(13169)-6*X(25321) = 2*X(15303)-3*X(25321) = X(32234)-4*X(41731) = 2*X(32234)+X(41737) = 8*X(41731)+X(41737)

The reciprocal orthologic center of these triangles is X(41721)

X(41720) lies on these lines: {2, 9769}, {4, 542}, {6, 6032}, {23, 16510}, {30, 10752}, {67, 597}, {69, 5642}, {74, 11179}, {110, 524}, {113, 11180}, {193, 9143}, {541, 6776}, {567, 5622}, {599, 6593}, {690, 8593}, {2407, 6054}, {2781, 15072}, {2854, 3060}, {2930, 40342}, {3448, 5032}, {3564, 5655}, {4558, 8724}, {5181, 11160}, {5465, 11161}, {5476, 14644}, {5641, 35138}, {5972, 21356}, {8550, 15054}, {9041, 32298}, {9759, 22329}, {10721, 11645}, {11006, 18800}, {11008, 32114}, {11178, 25556}, {11188, 12824}, {11579, 15033}, {11694, 34351}, {12117, 14570}, {12584, 37953}, {13857, 22151}, {14848, 32306}, {15068, 41614}, {19140, 41617}, {20583, 25328}, {28538, 32278}, {31166, 38885}, {32238, 38023}, {32248, 40949}, {32274, 38072}, {37909, 41583}

X(41720) = midpoint of X(i) and X(j) for these {i, j}: {193, 9143}, {599, 16176}, {1992, 11061}, {10706, 32234}
X(41720) = reflection of X(i) in X(j) for these (i, j): (2, 15303), (67, 597), (69, 5642), (74, 11179), (110, 34319), (597, 41595), (599, 6593), (895, 1992), (1992, 5095), (9140, 6), (10706, 9970), (11006, 18800), (11160, 5181), (11161, 5465), (11178, 25556), (11180, 113), (11188, 12824), (13169, 2), (14833, 9144), (25328, 20583), (32244, 599), (34319, 25329), (38885, 31166), (41721, 7426), (41737, 10706)
X(41720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 25321, 15303), (5095, 11061, 895), (6593, 16176, 32244), (9970, 32234, 41737), (9970, 41731, 32234)

X(41721) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(2)

Barycentrics    a^8-3*(b^2+c^2)*a^6+3*b^2*c^2*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4-c^4)^2 : :
X(41721) = 5*X(3620)-4*X(19510) = 4*X(5972)-3*X(22151) = 3*X(10519)-2*X(10564) = 3*X(11416)-5*X(15059) = 2*X(13857)-3*X(21356) = 2*X(15303)-3*X(37907) = 3*X(18374)-2*X(25329) = 3*X(18374)-4*X(32218) = 3*X(18449)-4*X(18583) = 3*X(19596)-X(25336) = 3*X(25321)-5*X(37760)

The reciprocal orthologic center of these triangles is X(41720)

X(41721) lies on these lines: {2, 12039}, {4, 69}, {6, 8262}, {23, 41583}, {30, 13169}, {67, 8705}, {110, 524}, {141, 10510}, {193, 9716}, {323, 5181}, {895, 3580}, {1353, 34351}, {1495, 11061}, {1503, 15054}, {1992, 32225}, {2393, 3448}, {2854, 41724}, {3564, 3581}, {3620, 19510}, {5095, 32223}, {5166, 6388}, {5486, 37644}, {5965, 25714}, {5972, 22151}, {6776, 32110}, {7699, 20423}, {9027, 37779}, {10519, 10564}, {10752, 11799}, {11416, 15059}, {11645, 12244}, {11649, 25739}, {11676, 35520}, {13622, 32455}, {13857, 21356}, {14915, 32247}, {15018, 16511}, {15107, 32244}, {15303, 37907}, {15533, 33586}, {18449, 18583}, {19596, 25336}, {25321, 37760}, {41586, 41617}, {41596, 41613}

X(41721) = midpoint of X(15107) and X(32244)
X(41721) = reflection of X(i) in X(j) for these (i, j): (6, 8262), (23, 41583), (110, 32113), (323, 5181), (895, 3580), (1992, 32225), (5095, 32223), (6776, 32110), (10510, 141), (10752, 11799), (11061, 1495), (25329, 32218), (32220, 32269), (41617, 41586), (41720, 7426)
X(41721) = crossdifference of every pair of points on line {X(3049), X(39499)}
X(41721) = {X(25329), X(32218)}-harmonic conjugate of X(18374)

X(41722) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(4) TO REFLECTION OF X(21)

Barycentrics    a*(a^2+b^2-c^2)*(a^2-b^2+c^2)*((b+c)*a^4-2*b*c*a^3-2*(b^3+c^3)*a^2+2*(b^2-b*c+c^2)*b*c*a+(b^4-c^4)*(b-c)) : :
X(41722) = 3*X(4)-2*X(1902) = 5*X(631)-4*X(37613) = 3*X(1829)-X(1902) = 2*X(5887)-3*X(41717) = 3*X(41717)-4*X(41727) = 3*X(41717)-X(41733) = 4*X(41727)-X(41733)

The reciprocal orthologic center of these triangles is X(41723).

X(41722) lies on these lines: {1, 24}, {3, 11396}, {4, 8}, {10, 1594}, {19, 2323}, {25, 1482}, {28, 110}, {33, 5697}, {34, 5903}, {40, 378}, {54, 65}, {74, 12262}, {145, 7487}, {165, 35477}, {186, 1385}, {225, 1845}, {235, 22791}, {392, 451}, {403, 946}, {406, 3877}, {427, 5690}, {468, 5901}, {515, 6240}, {516, 18560}, {518, 6403}, {519, 7576}, {631, 37613}, {912, 5889}, {942, 7501}, {944, 18533}, {952, 3575}, {953, 2766}, {1006, 41340}, {1125, 10018}, {1214, 37115}, {1386, 19128}, {1398, 36279}, {1483, 37458}, {1572, 8743}, {1593, 12702}, {1598, 8148}, {1614, 40660}, {1699, 35488}, {1785, 1866}, {1848, 31806}, {1885, 28174}, {1905, 3057}, {1986, 31732}, {1993, 9928}, {2095, 37245}, {2098, 11399}, {2099, 11398}, {2771, 7722}, {2778, 10721}, {2836, 32234}, {3147, 3616}, {3515, 10246}, {3517, 10247}, {3518, 10222}, {3520, 3579}, {3541, 5657}, {3542, 5603}, {3576, 32534}, {3580, 12259}, {3817, 35487}, {3827, 6776}, {3868, 6193}, {4219, 37585}, {4222, 23340}, {4297, 10295}, {4663, 8537}, {5064, 34718}, {5142, 31837}, {5330, 35973}, {5412, 35641}, {5413, 35642}, {5587, 7547}, {5691, 35480}, {5790, 7507}, {5844, 6756}, {5886, 7505}, {6001, 6241}, {6143, 11231}, {6353, 10595}, {6684, 37118}, {6829, 9895}, {6868, 20243}, {7414, 14110}, {7488, 24301}, {7577, 9956}, {7686, 7699}, {7713, 7982}, {7714, 34631}, {7718, 37122}, {7968, 10881}, {7969, 10880}, {7987, 35472}, {7991, 35502}, {9626, 37932}, {9955, 16868}, {10151, 40273}, {10573, 11393}, {11230, 14940}, {11248, 22479}, {11249, 11383}, {11252, 11385}, {11253, 11384}, {11278, 31948}, {11362, 15559}, {11392, 12647}, {11473, 35610}, {11474, 35611}, {12173, 18525}, {12645, 18494}, {12778, 15463}, {13488, 28212}, {13624, 21844}, {14017, 37533}, {14157, 40658}, {17502, 17506}, {18357, 23047}, {18481, 35471}, {18504, 37372}, {18559, 28204}, {21740, 31384}, {22750, 36009}, {26446, 37119}, {28160, 34797}, {31663, 35473}, {31730, 35491}, {31786, 37441}, {37305, 37562}

X(41722) = reflection of X(i) in X(j) for these (i, j): (4, 1829), (5887, 41727), (12135, 6756), (41733, 5887)
X(41722) = Zosma transform of X(18395)
X(41722) = Kosnita-to-orthic similarity image of X(1)
X(41722) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(54)}} and {{A, B, C, X(56), X(37821)}}
X(41722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1828, 1872, 4), (1870, 6197, 37117), (1905, 3057, 6198), (5887, 41727, 41717), (41717, 41733, 5887)

X(41723) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(4)

Barycentrics    a*(a+b)*(c+a)*((b^2+b*c+c^2)*a^2-(b+c)*b*c*a-(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(41722)

X(41723) lies on these lines: {1, 994}, {2, 970}, {4, 52}, {10, 17167}, {21, 517}, {22, 5706}, {27, 3187}, {28, 110}, {40, 4184}, {51, 5046}, {58, 5903}, {65, 81}, {79, 2392}, {100, 22300}, {145, 5208}, {185, 6895}, {278, 19367}, {283, 11101}, {286, 17220}, {314, 1240}, {333, 3869}, {343, 5799}, {377, 2979}, {389, 6840}, {392, 17557}, {404, 18465}, {429, 3580}, {443, 7998}, {484, 4278}, {511, 2475}, {518, 41718}, {519, 35637}, {674, 5178}, {758, 27368}, {859, 1482}, {912, 31902}, {942, 1817}, {946, 14008}, {960, 5235}, {962, 14956}, {1010, 31778}, {1043, 14923}, {1125, 38474}, {1154, 37230}, {1216, 6901}, {1243, 7549}, {1325, 1437}, {1351, 4214}, {1425, 37798}, {1434, 23839}, {1722, 27643}, {1778, 21853}, {1790, 4658}, {1829, 40571}, {1896, 35097}, {1904, 41588}, {1953, 4269}, {1993, 4185}, {2099, 4267}, {2262, 2287}, {2360, 33325}, {2476, 5752}, {2478, 5640}, {2836, 41741}, {2895, 10381}, {3057, 18165}, {3212, 16749}, {3286, 37567}, {3340, 18163}, {3436, 17139}, {3562, 41227}, {3567, 6928}, {3583, 31757}, {3615, 18330}, {3617, 3786}, {3622, 35645}, {3658, 31849}, {3736, 4642}, {3753, 14005}, {3754, 25526}, {3794, 11115}, {3812, 5333}, {3827, 41601}, {3833, 28618}, {3872, 10461}, {3877, 11110}, {3897, 37303}, {3924, 38832}, {4188, 37521}, {4219, 11440}, {4221, 37562}, {4227, 25713}, {4393, 14953}, {4653, 5697}, {4720, 10914}, {4848, 17197}, {5012, 37231}, {5084, 11451}, {5142, 23293}, {5196, 5884}, {5260, 22299}, {5422, 37415}, {5462, 6902}, {5482, 13587}, {5562, 6839}, {5707, 11337}, {5800, 12220}, {5883, 28619}, {5885, 37294}, {5891, 6900}, {5907, 6894}, {5908, 13614}, {5943, 37162}, {6000, 37433}, {6001, 41734}, {6826, 11444}, {6827, 15043}, {6835, 15056}, {6836, 10574}, {6851, 15072}, {6899, 20791}, {6903, 9730}, {6905, 39271}, {6917, 11412}, {6929, 9781}, {6947, 15028}, {6951, 10625}, {7419, 7982}, {7497, 11441}, {7501, 11449}, {7504, 34466}, {7511, 14516}, {7991, 17194}, {10110, 13729}, {10404, 23155}, {10431, 12279}, {10439, 19861}, {10446, 37191}, {10458, 37598}, {10459, 35623}, {10473, 25059}, {10480, 25058}, {10822, 33139}, {10974, 24883}, {11344, 19771}, {11684, 18722}, {12435, 17185}, {12702, 17524}, {14011, 17174}, {14616, 15232}, {15489, 37291}, {15644, 37163}, {15952, 25413}, {16054, 26621}, {16948, 18191}, {16980, 20060}, {17168, 33815}, {17182, 24982}, {17483, 23154}, {17484, 29958}, {17518, 26637}, {17523, 35264}, {17579, 37482}, {19782, 37301}, {22791, 37357}, {24987, 29311}, {27653, 34772}, {31788, 37402}, {31870, 37158}, {34148, 37117}, {35193, 36011}, {35631, 37442}, {38480, 41575}

X(41723) = reflection of X(i) in X(j) for these (i, j): (21, 18180), (110, 12826)
X(41723) = anticomplement of X(22076)
X(41723) = barycentric product X(81)*X(11681)
X(41723) = trilinear product X(58)*X(11681)
X(41723) = X(i)-anticomplementary conjugate of-X(j) for these (i, j): (28, 5484), (961, 3152), (1169, 6360)
X(41723) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (28, 3193, 110), (51, 15488, 5046), (65, 18178, 81), (404, 37536, 33852), (17174, 25005, 14011)

X(41724) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(4)

Barycentrics    a^6-2*(b^2+c^2)*a^4+(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(41724) = 2*X(23)-3*X(15360) = 3*X(110)-4*X(468) = 6*X(125)-5*X(30745) = 4*X(140)-3*X(22115) = 3*X(186)-2*X(30714) = 3*X(265)-2*X(18572) = 3*X(323)-5*X(30745) = 2*X(468)-3*X(3580) = 2*X(858)-3*X(9140) = 3*X(2071)-4*X(20417) = 3*X(3448)-X(5189) = 5*X(3522)-6*X(21663) = 2*X(5095)-3*X(37784) = 4*X(5159)-3*X(40112) = X(5189)+3*X(37779) = 2*X(7574)-3*X(25739) = 3*X(9140)-X(23061) = 3*X(9143)-5*X(37760) = 3*X(15107)-2*X(37900) = 3*X(15360)-4*X(41586)

The reciprocal orthologic center of these triangles is X(32234)

X(41724) lies on these lines: {2, 575}, {3, 11264}, {4, 52}, {5, 15019}, {6, 20113}, {23, 542}, {30, 15054}, {51, 3410}, {54, 140}, {67, 524}, {69, 3266}, {110, 468}, {125, 323}, {186, 539}, {193, 7703}, {195, 34826}, {265, 11564}, {340, 520}, {343, 5012}, {389, 2888}, {394, 26913}, {427, 8537}, {511, 3448}, {532, 36186}, {533, 36185}, {538, 36163}, {568, 41171}, {576, 5169}, {599, 40916}, {633, 41000}, {634, 41001}, {1154, 7574}, {1199, 1209}, {1216, 12325}, {1352, 5640}, {1353, 14389}, {1495, 14683}, {1503, 15107}, {1594, 15801}, {1648, 9225}, {1656, 5422}, {1657, 12289}, {1899, 2979}, {1993, 5094}, {1994, 21243}, {1995, 15069}, {2071, 20417}, {2393, 32255}, {2781, 41738}, {2836, 41742}, {2854, 41721}, {2930, 8262}, {3291, 6792}, {3522, 18913}, {3581, 12380}, {3631, 5888}, {3818, 11002}, {3819, 15108}, {3851, 9777}, {3978, 7768}, {5095, 12827}, {5111, 39691}, {5159, 40112}, {5468, 37803}, {5663, 18325}, {6070, 36188}, {6143, 11271}, {6146, 7691}, {6193, 11449}, {6243, 18356}, {6776, 15080}, {6800, 39899}, {7464, 16003}, {7496, 40107}, {7505, 9936}, {7512, 10116}, {7542, 9706}, {7570, 11225}, {7575, 23236}, {7860, 33796}, {7999, 18952}, {8681, 32248}, {9143, 32225}, {9705, 10020}, {9932, 32534}, {9979, 39474}, {10112, 14118}, {10264, 37477}, {10301, 41588}, {10540, 25338}, {10619, 41594}, {10991, 37183}, {11064, 15059}, {11078, 37974}, {11092, 37975}, {11232, 37513}, {11412, 14791}, {11433, 11451}, {11444, 18912}, {11454, 35485}, {11477, 31133}, {11645, 20063}, {11646, 20977}, {11799, 14094}, {11898, 15066}, {12278, 12429}, {12307, 13470}, {12317, 14915}, {12359, 15136}, {12383, 32110}, {12828, 37777}, {12834, 37990}, {13137, 20021}, {13292, 13434}, {13399, 37944}, {14157, 16619}, {14165, 35311}, {14460, 32428}, {14516, 37458}, {14918, 41204}, {14940, 41597}, {14981, 35298}, {15004, 37353}, {15018, 24206}, {15043, 18951}, {15056, 39571}, {15072, 18917}, {15083, 16868}, {15118, 22151}, {15361, 37958}, {16266, 23294}, {16770, 20428}, {16771, 20429}, {17986, 36831}, {20126, 37950}, {21028, 24149}, {21849, 37349}, {22533, 32338}, {23292, 41730}, {24981, 32223}, {30522, 32608}, {30744, 37672}, {31305, 40241}, {34008, 41020}, {34009, 41021}, {34799, 41482}, {41603, 41617}

X(41724) = midpoint of X(3448) and X(37779)
X(41724) = reflection of X(i) in X(j) for these (i, j): (23, 41586), (110, 3580), (193, 32127), (323, 125), (2930, 8262), (7464, 16003), (9143, 32225), (10510, 25328), (12383, 32110), (14094, 11799), (14683, 1495), (15136, 12359), (23061, 858), (23236, 7575), (24981, 32223), (36188, 6070), (37477, 10264), (37944, 13399)
X(41724) = anticomplement of X(3292)
X(41724) = anticomplementary conjugate of the anticomplement of X(17983)
X(41724) = intersection, other than A,B,C, of conics {{A, B, C, X(252), X(847)}} and {{A, B, C, X(316), X(23061)}}
X(41724) = crossdifference of every pair of points on line {X(217), X(30451)}
X(41724) = crosspoint of X(95) and X(671)
X(41724) = crosssum of X(51) and X(187)
X(41724) = X(i)-anticomplementary conjugate of-X(j) for these (i, j): (19, 8591), (92, 14360), (111, 6360), (671, 4329)
X(41724) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (23, 41586, 15360), (51, 18553, 7533), (69, 18911, 7998), (343, 8550, 7495), (1352, 37644, 5640), (3410, 7533, 18553), (6515, 11442, 3060), (7495, 8550, 5012), (7570, 34545, 25555), (9140, 23061, 858), (11422, 38397, 2), (11898, 26869, 15066), (21230, 32165, 13353), (24981, 32223, 35265)

X(41725) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(5) TO REFLECTION OF X(20)

Barycentrics
a^2*((b^2+c^2)*a^12-4*(b^4+c^4)*a^10+(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^8-2*(b^4-4*b^2*c^2+c^4)*b^2*c^2*a^6-(b^4-c^4)*(b^2-c^2)*(5*b^4-4*b^2*c^2+5*c^4)*a^4+2*(b^2-c^2)^2*(2*b^8+2*c^8+(b^4+c^4)*b^2*c^2)*a^2-(b^4-c^4)^3*(b^2-c^2)) : :
X(41725) = 3*X(51)-2*X(18383) = 3*X(154)-X(18436) = 2*X(1216)-3*X(11202) = 3*X(1853)-5*X(37481) = 4*X(5462)-3*X(23325) = 3*X(5878)-X(36983) = X(5878)-3*X(41715) = 3*X(5890)-X(14216) = 3*X(6241)+X(36983) = X(6241)+3*X(41715) = X(6759)+2*X(32392) = 3*X(7729)-X(13093) = 4*X(9729)-3*X(23329) = 3*X(9730)-2*X(20299) = 4*X(10095)-3*X(23324) = 4*X(10110)-3*X(18376) = 3*X(10192)-2*X(11591) = X(12162)-3*X(41580) = 3*X(32379)-X(41726) = X(36983)-9*X(41715)

The reciprocal orthologic center of these triangles is X(41726)

X(41725) lies on these lines: {3, 6293}, {4, 51}, {5, 32364}, {25, 40285}, {26, 6759}, {49, 17824}, {52, 10112}, {64, 36752}, {110, 5562}, {143, 41362}, {154, 18436}, {182, 3357}, {511, 12118}, {550, 2781}, {567, 32345}, {1147, 38450}, {1154, 34782}, {1181, 32321}, {1216, 11202}, {1495, 40276}, {1498, 7517}, {1503, 6102}, {1660, 15083}, {1853, 37481}, {1986, 34224}, {2777, 10575}, {2883, 5663}, {3549, 5907}, {5446, 34786}, {5462, 7564}, {5576, 9730}, {5876, 16252}, {5889, 9833}, {5891, 32348}, {6243, 17845}, {6247, 13630}, {7565, 16226}, {7576, 32332}, {7722, 9934}, {7729, 13093}, {9729, 23329}, {10024, 12162}, {10095, 23324}, {10192, 11591}, {10274, 18475}, {11271, 14531}, {11557, 19506}, {11561, 23315}, {11574, 41719}, {11585, 25711}, {11799, 36982}, {12006, 23332}, {12163, 19149}, {13491, 15311}, {15072, 20427}, {15105, 40928}, {16223, 32743}, {17821, 23039}, {23300, 38136}, {26913, 32767}, {31861, 31978}, {32401, 40441}, {32903, 36987}

X(41725) = midpoint of X(i) and X(j) for these {i, j}: {3, 6293}, {1498, 34783}, {5878, 6241}, {5889, 9833}, {6243, 17845}, {7722, 9934}
X(41725) = reflection of X(i) in X(j) for these (i, j): (5, 41589), (3357, 40647), (5562, 10282), (5876, 16252), (6247, 13630), (18381, 389), (19506, 11557), (23315, 11561), (34786, 5446), (41362, 143)
X(41725) = crosssum of X(3) and X(18569)
X(41725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5878, 14216, 41736), (6241, 41715, 5878), (7488, 32379, 10282)

X(41726) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(20) TO REFLECTION OF X(5)

Barycentrics
a^2*((b^2+c^2)*a^12-(4*b^4+3*b^2*c^2+4*c^4)*a^10+(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*b^2*c^2*a^6-(b^2+c^2)*(5*b^8+5*c^8-4*(2*b^4-3*b^2*c^2+2*c^4)*b^2*c^2)*a^4+(4*b^8+4*c^8+(5*b^4+3*b^2*c^2+5*c^4)*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)^3*(b^2-c^2)) : :
X(41726) = 4*X(973)-5*X(3091) = 2*X(1209)-3*X(11459) = 9*X(3839)-8*X(11743) = 4*X(5876)-X(6242) = 3*X(5876)-X(13368) = 8*X(5876)-3*X(41713) = 3*X(5890)-4*X(6689) = 3*X(5891)-2*X(11802) = 3*X(6242)-4*X(13368) = 2*X(6242)-3*X(41713) = 3*X(6288)-2*X(13368) = 4*X(6288)-3*X(41713) = 3*X(7730)-5*X(15058) = 5*X(11444)-4*X(32348) = 3*X(11459)-X(32339) = 2*X(11808)-3*X(15030) = 2*X(12111)+X(12226) = 8*X(13368)-9*X(41713) = 3*X(18435)-2*X(22804) = 3*X(32379)-2*X(41725)

The reciprocal orthologic center of these triangles is X(41725)

X(41726) lies on these lines: {3, 8154}, {4, 93}, {20, 32330}, {26, 12307}, {54, 13754}, {110, 5562}, {195, 7526}, {973, 3091}, {1209, 5448}, {1568, 14076}, {2917, 11441}, {3153, 6145}, {3574, 5889}, {3839, 11743}, {5576, 20424}, {5663, 18442}, {5890, 6689}, {5891, 11802}, {5907, 32352}, {6293, 38435}, {6816, 32334}, {7503, 32341}, {7527, 15801}, {7722, 11597}, {7723, 33565}, {7730, 15058}, {10024, 21230}, {10298, 32391}, {10610, 11003}, {11440, 32401}, {11444, 32348}, {11808, 15030}, {12111, 12226}, {12164, 32333}, {12254, 12606}, {15043, 32396}, {15305, 32340}, {18563, 22584}, {23039, 32171}, {32337, 37444}

X(41726) = midpoint of X(12111) and X(32338)
X(41726) = reflection of X(i) in X(j) for these (i, j): (20, 41590), (5889, 3574), (6242, 6288), (6288, 5876), (7691, 5562), (7722, 11597), (12226, 32338), (12254, 12606), (21230, 31834), (32339, 1209), (32352, 5907), (33565, 7723), (34783, 10610)
X(41726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (110, 7691, 23358), (6242, 6288, 41713), (6288, 41730, 2888), (7691, 32379, 7488), (11459, 32339, 1209)

X(41727) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(5) TO REFLECTION OF X(21)

Barycentrics
a*(-2*a^7*b*c+(b+c)*a^8+3*(b^2+c^2)*a^5*b*c-2*(b^2+c^2)*a^3*b^2*c^2-(b+c)*(2*b^2-b*c+2*c^2)*a^6-(b^2-c^2)^2*(b-c)^2*b*c*a+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(b+c)^2*b*c)*a^2-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
X(41727) = X(5887)-3*X(41717) = 3*X(5887)-X(41733) = 3*X(41717)+X(41722) = 9*X(41717)-X(41733) = 3*X(41722)+X(41733)

The reciprocal orthologic center of these triangles is X(41728)

X(41727) lies on these lines: {4, 8}, {5, 41591}, {65, 1718}, {110, 6583}, {182, 3827}, {912, 31732}, {942, 34116}, {2771, 11562}, {2836, 24475}, {3073, 21381}, {6001, 13491}, {9956, 32126}, {13373, 18732}

X(41727) = midpoint of X(5887) and X(41722)
X(41727) = reflection of X(i) in X(j) for these (i, j): (5, 41591), (18732, 13373)
X(41727) = {X(41717), X(41722)}-harmonic conjugate of X(5887)

X(41728) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(5)

Barycentrics
a*(a+b)*(c+a)*((b^2+b*c+c^2)*a^8-(b+c)*b*c*a^7-2*(b^4+c^4+(b^2+b*c+c^2)*b*c)*a^6+2*(b+c)*(b^2+c^2)*b*c*a^5+(b^4+b^2*c^2+c^4)*b*c*a^4-(b^3+c^3)*(b^2+b*c+c^2)*b*c*a^3+2*(b^6-c^6)*(b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3) : :

The reciprocal orthologic center of these triangles is X(41727)

X(41728) lies on these lines: {4, 93}, {21, 41592}, {110, 6583}

X(41728) = reflection of X(21) in X(41592)

X(41729) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(5) TO REFLECTION OF X(22)

Barycentrics
4*a^12-9*(b^2+c^2)*a^10+3*(b^2+c^2)^2*a^8+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^6-2*(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41729) = X(66)-3*X(5050) = X(69)-3*X(23041) = 5*X(182)-3*X(23329) = 3*X(597)-2*X(20300) = X(1352)-3*X(19153) = 3*X(5102)+X(17845) = X(5596)+3*X(14912) = 3*X(5656)+5*X(6776) = 3*X(5656)-5*X(19149) = X(5656)-5*X(41719) = 9*X(5656)-5*X(41735) = 2*X(5893)-5*X(34117) = 2*X(6697)-3*X(38110) = X(6776)+3*X(41719) = 3*X(6776)+X(41735) = X(8549)-3*X(14912) = 3*X(14853)-X(34775) = X(19149)-3*X(41719) = 3*X(19149)-X(41735) = 9*X(41719)-X(41735)

The reciprocal orthologic center of these triangles is X(41730)

X(41729) lies on these lines: {4, 6}, {5, 41593}, {52, 3629}, {66, 5050}, {68, 16252}, {69, 23041}, {110, 343}, {141, 7542}, {154, 6515}, {156, 206}, {159, 9714}, {161, 15580}, {182, 23329}, {193, 34787}, {524, 14070}, {569, 6247}, {575, 23300}, {576, 34776}, {597, 20300}, {1112, 41580}, {1351, 36989}, {1352, 19153}, {1353, 2393}, {2781, 9967}, {2917, 11271}, {3589, 34118}, {5097, 18400}, {5102, 17845}, {5894, 6293}, {5965, 10282}, {6696, 37476}, {6697, 38110}, {6759, 13292}, {9833, 34777}, {10169, 15516}, {10628, 35254}, {12107, 32367}, {13289, 41731}, {13491, 34146}, {15069, 19132}, {15139, 37648}, {15432, 23324}, {17821, 40341}, {17835, 25331}, {18381, 39561}, {18911, 23332}, {19506, 34155}, {20079, 33748}, {23042, 34507}, {23328, 37513}, {25406, 34778}, {31166, 39879}, {31804, 32366}, {32217, 37971}, {32379, 41587}, {34785, 37517}, {35228, 38444}

X(41729) = midpoint of X(i) and X(j) for these {i, j}: {193, 34787}, {576, 34776}, {1351, 36989}, {3629, 34782}, {5596, 8549}, {6776, 19149}, {8550, 34774}, {9833, 34777}, {13289, 41731}, {34785, 37517}
X(41729) = reflection of X(i) in X(j) for these (i, j): (5, 41593), (23300, 575), (34118, 3589)
X(41729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5596, 14912, 8549), (6776, 41719, 19149)

X(41730) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(22) TO REFLECTION OF X(5)

Barycentrics
a^12-3*(b^2+c^2)*a^10-2*(b^2+c^2)*b^2*c^2*a^6+(3*b^4+5*b^2*c^2+3*c^4)*a^8-(b^4+b^2*c^2+c^4)*(3*b^4-4*b^2*c^2+3*c^4)*a^4+3*(b^8-c^8)*(b^2-c^2)*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(41729)

X(41730) lies on these lines: {4, 93}, {22, 41594}, {54, 12359}, {110, 343}, {1352, 41578}, {1993, 5965}, {6145, 32338}, {7542, 21230}, {7691, 15138}, {8907, 23358}, {12358, 33565}, {14531, 18428}, {18442, 30522}, {23292, 41724}

X(41730) = reflection of X(22) in X(41594)
X(41730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2888, 41726, 6288), (3410, 41713, 6288), (3410, 41732, 41713)

X(41731) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(5) TO REFLECTION OF X(23)

Barycentrics
3*a^12-8*(b^2+c^2)*a^10+(5*b^4+11*b^2*c^2+5*c^4)*a^8+(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^6-(7*b^8+7*c^8-4*(2*b^4-b^2*c^2+2*c^4)*b^2*c^2)*a^4+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^2-c^2)^2 : :
X(41731) = 2*X(5)-3*X(34155) = 3*X(6)-2*X(20301) = 3*X(6)-X(32306) = 5*X(6)-3*X(38724) = 2*X(125)-3*X(39561) = 5*X(182)-4*X(6699) = X(399)-3*X(25331) = 4*X(5095)-X(32273) = 5*X(9970)-3*X(10706) = X(9970)-3*X(41720) = 3*X(9970)-X(41737) = 3*X(10706)+5*X(32234) = X(10706)-5*X(41720) = 9*X(10706)-5*X(41737) = 10*X(20301)-9*X(38724) = X(32234)+3*X(41720) = 3*X(32234)+X(41737) = 5*X(32306)-9*X(38724) = 3*X(34155)-4*X(41595) = 9*X(41720)-X(41737)

The reciprocal orthologic center of these triangles is X(41732)

X(41731) lies on these lines: {3, 16176}, {4, 542}, {5, 9977}, {6, 7579}, {20, 14448}, {52, 23236}, {67, 575}, {110, 5965}, {125, 39561}, {143, 32299}, {182, 6699}, {265, 5097}, {399, 25331}, {511, 11562}, {524, 7575}, {578, 16003}, {1352, 25321}, {1353, 9976}, {2781, 13491}, {3564, 19140}, {3629, 32423}, {5102, 12902}, {5181, 34116}, {5476, 32274}, {5609, 40342}, {5622, 33749}, {6593, 34507}, {8550, 32305}, {10168, 13169}, {10264, 12007}, {10752, 29012}, {11178, 15303}, {11179, 32247}, {11180, 25566}, {11536, 25328}, {11557, 41714}, {13289, 41729}, {15118, 22234}, {15132, 32317}, {15462, 32244}, {15545, 15548}, {17702, 37517}, {18553, 32272}, {25335, 39562}, {32235, 37644}, {32609, 40341}

X(41731) = midpoint of X(i) and X(j) for these {i, j}: {3, 16176}, {9970, 32234}
X(41731) = reflection of X(i) in X(j) for these (i, j): (5, 41595), (67, 575), (265, 5097), (576, 5095), (1352, 25556), (5609, 40342), (9976, 1353), (10264, 12007), (11178, 15303), (11180, 25566), (13169, 10168), (13289, 41729), (19140, 25329), (32244, 40107), (32272, 18553), (32273, 576), (32299, 143), (32305, 8550), (32306, 20301), (34507, 6593), (41714, 11557)
X(41731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 41595, 34155), (6, 32306, 20301), (1352, 25321, 25556), (15462, 32244, 40107), (32234, 41720, 9970)

X(41732) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(5)

Barycentrics
a^12-4*(b^2+c^2)*a^10+(5*b^4+8*b^2*c^2+5*c^4)*a^8-4*(b^2+c^2)*b^2*c^2*a^6-(b^4+b^2*c^2+c^4)*(5*b^4-9*b^2*c^2+5*c^4)*a^4+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^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(41731)

X(41732) lies on these lines: {2, 9977}, {4, 93}, {23, 41596}, {110, 5965}, {1493, 13418}, {6515, 9716}, {7575, 25714}, {8254, 34545}, {9972, 21243}, {15054, 18400}, {15137, 21230}, {15331, 36966}

X(41732) = reflection of X(i) in X(j) for these (i, j): (23, 41596), (15137, 21230)
X(41732) = {X(41713), X(41730)}-harmonic conjugate of X(3410)

X(41733) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(20) TO REFLECTION OF X(21)

Barycentrics
a*(2*a^7*b*c+6*b^2*c^2*a^5-(b+c)*a^8+2*(b^2-c^2)*(b-c)*a^6+6*(b^2-c^2)*(b-c)*a^4*b*c-2*(3*b^4+3*c^4+2*(b^2-3*b*c+c^2)*b*c)*a^3*b*c+2*(b^2-c^2)^2*(2*b^2-b*c+2*c^2)*a*b*c-2*(b^2-c^2)*(b-c)*(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^2+(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
X(41733) = 4*X(5887)-3*X(41717) = 3*X(5887)-2*X(41727) = 3*X(41717)-2*X(41722) = 9*X(41717)-8*X(41727) = 3*X(41722)-4*X(41727)

The reciprocal orthologic center of these triangles is X(41734)

X(41733) lies on these lines: {1, 1774}, {4, 8}, {20, 41600}, {40, 11337}, {110, 1295}, {946, 37983}, {1858, 5889}, {2000, 37625}, {3827, 5921}, {6001, 12111}, {13121, 23340}

X(41733) = reflection of X(i) in X(j) for these (i, j): (20, 41600), (5889, 1858), (41722, 5887)
X(41733) = {X(5887), X(41722)}-harmonic conjugate of X(41717)

X(41734) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(20)

Barycentrics
a*(a+b)*(c+a)*((b^2+b*c+c^2)*a^8-(b+c)*b*c*a^7-(2*b^4+2*c^4-(b^2+4*b*c+c^2)*b*c)*a^6-(b+c)*(b^2+c^2)*b*c*a^5-5*(b^2-c^2)^2*b*c*a^4+5*(b^2-c^2)^2*(b+c)*b*c*a^3+(2*b^4+2*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c*a-(b^4-c^4)*(b^2-c^2)^3) : :

The reciprocal orthologic center of these triangles is X(41733)

X(41734) lies on these lines: {4, 51}, {21, 41601}, {110, 1295}, {6001, 41723}, {32125, 37983}

X(41734) = reflection of X(21) in X(41601)

X(41735) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(20) TO REFLECTION OF X(22)

Barycentrics    a^12-3*(-4*b^2*c^2+(b^2-c^2)^2)*a^8-8*(b^2+c^2)*b^2*c^2*a^6+(3*b^4+2*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^4-8*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41735) = 5*X(4)-4*X(18382) = 4*X(206)-3*X(25406) = 3*X(376)-4*X(15577) = 5*X(3091)-4*X(23300) = 9*X(3524)-8*X(15578) = 11*X(3525)-8*X(15579) = 7*X(3528)-8*X(35228) = 3*X(5656)-X(6776) = 3*X(5656)-2*X(19149) = 9*X(5656)-4*X(41729) = 2*X(6776)-3*X(41719) = 3*X(6776)-4*X(41729) = 2*X(8549)-3*X(14853) = 3*X(11206)-X(14927) = 3*X(11206)-2*X(36989) = 3*X(14912)-4*X(34117) = 8*X(18382)-5*X(36851) = 4*X(19149)-3*X(41719) = 3*X(19149)-2*X(41729) = 9*X(41719)-8*X(41729)

The reciprocal orthologic center of these triangles is X(41736)

X(41735) lies on these lines: {2, 1619}, {3, 35219}, {4, 6}, {20, 159}, {30, 39879}, {64, 141}, {66, 6815}, {69, 6225}, {110, 1370}, {154, 7386}, {161, 34608}, {182, 18537}, {206, 6816}, {376, 15577}, {511, 5878}, {518, 12779}, {962, 3827}, {1350, 15311}, {1352, 6000}, {1469, 12940}, {1853, 7392}, {1885, 19459}, {1974, 19142}, {2777, 38885}, {2854, 11744}, {2892, 14982}, {3056, 12950}, {3091, 23300}, {3098, 20427}, {3524, 15578}, {3525, 15579}, {3528, 35228}, {3529, 15581}, {3537, 10606}, {3538, 17821}, {3545, 20300}, {3556, 27509}, {3763, 6696}, {3818, 7401}, {5085, 6804}, {5621, 38282}, {5846, 7973}, {5894, 15585}, {5895, 9924}, {6001, 24476}, {6247, 6803}, {6285, 12588}, {6515, 41715}, {6643, 6759}, {6997, 11451}, {7355, 12589}, {7528, 34780}, {7716, 13568}, {8567, 21167}, {8721, 22401}, {9833, 13346}, {9914, 37485}, {10519, 12250}, {11001, 15580}, {11413, 28419}, {11433, 41580}, {11441, 15438}, {12174, 26926}, {12315, 18440}, {14862, 23042}, {15080, 35260}, {15582, 17538}, {16051, 32125}, {18420, 39884}, {18531, 32063}, {22802, 31670}, {25712, 34782}, {31166, 41257}, {31725, 39899}, {32605, 37444}, {37643, 41603}

X(41735) = midpoint of X(i) and X(j) for these {i, j}: {69, 6225}, {5895, 9924}, {12315, 18440}
X(41735) = reflection of X(i) in X(j) for these (i, j): (6, 2883), (20, 159), (64, 141), (2892, 14982), (5596, 1498), (5894, 15585), (6776, 19149), (12250, 34778), (12324, 66), (14216, 3818), (14927, 36989), (15583, 5893), (20427, 3098), (31670, 22802), (36851, 4), (41719, 5656)
X(41735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (154, 34944, 7386), (1619, 41602, 2), (5656, 6776, 19149), (5656, 34781, 11456), (5894, 15585, 31884), (6776, 19149, 41719), (10519, 12250, 34778), (11206, 14927, 36989), (11206, 37669, 1660), (11206, 41736, 1370), (12324, 15740, 31978)

X(41736) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(22) TO REFLECTION OF X(20)

Barycentrics    a^12-(3*b^4-8*b^2*c^2+3*c^4)*a^8-2*(b^2+c^2)*b^2*c^2*a^6+(3*b^4+4*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^4-6*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41736) = 4*X(11550)-3*X(32064)

The reciprocal orthologic center of these triangles is X(41735)

X(41736) lies on these lines: {2, 32125}, {4, 51}, {20, 8907}, {22, 41602}, {110, 1370}, {154, 16063}, {161, 20062}, {343, 15311}, {858, 1619}, {1498, 37444}, {1503, 1993}, {1853, 3066}, {1992, 36851}, {5596, 22151}, {5656, 18531}, {5895, 33586}, {6247, 7544}, {6997, 26913}, {7386, 35260}, {7691, 33522}, {9932, 31305}, {11442, 34146}, {12118, 34938}, {12315, 18569}, {14790, 32139}, {14791, 32063}, {18911, 41580}, {19149, 41257}, {28408, 35219}, {31725, 41588}, {32321, 37119}, {34778, 37636}

X(41736) = reflection of X(i) in X(j) for these (i, j): (22, 41602), (20062, 161)
X(41736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1370, 41735, 11206), (1899, 5878, 41715), (1899, 41738, 32064), (5878, 14216, 41725), (5878, 36983, 6225), (11206, 13203, 1370), (41715, 41738, 1899)

X(41737) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(20) TO REFLECTION OF X(23)

Barycentrics
3*a^12-4*(b^2+c^2)*a^10-(2*b^4-19*b^2*c^2+2*c^4)*a^8+(b^2+c^2)*(2*b^4-13*b^2*c^2+2*c^4)*a^6+(b^8+c^8+(b^4+4*b^2*c^2+c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-2*(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41737) = 4*X(5)-3*X(5622) = 3*X(110)-2*X(32233) = 4*X(141)-3*X(15055) = 3*X(2071)-4*X(19510) = 5*X(3091)-4*X(15118) = 3*X(9140)-2*X(16010) = 3*X(9140)-4*X(32274) = 2*X(9970)-3*X(10706) = 4*X(9970)-3*X(41720) = 3*X(9970)-2*X(41731) = 3*X(10706)-X(32234) = 9*X(10706)-4*X(41731) = 3*X(11180)-X(32247) = 3*X(11180)-2*X(32275) = 3*X(13169)-2*X(32247) = 3*X(13169)-4*X(32275) = 3*X(14982)-X(32233) = 2*X(32234)-3*X(41720) = 3*X(32234)-4*X(41731) = 9*X(41720)-8*X(41731)

The reciprocal orthologic center of these triangles is X(41738)

X(41737) lies on these lines: {4, 542}, {5, 5622}, {20, 1632}, {67, 15054}, {69, 2777}, {74, 1352}, {110, 858}, {113, 6776}, {125, 35904}, {141, 15055}, {146, 5921}, {265, 39884}, {382, 14984}, {399, 19139}, {511, 10721}, {541, 11180}, {1205, 5907}, {1351, 1539}, {1514, 32220}, {1614, 15462}, {1995, 9140}, {2071, 19510}, {2393, 10296}, {2781, 5895}, {2836, 12688}, {2854, 10733}, {2883, 32264}, {3024, 39891}, {3028, 39892}, {3091, 15118}, {3564, 7728}, {3618, 36518}, {3619, 38727}, {3818, 11579}, {3843, 39562}, {5621, 6698}, {5642, 16051}, {5655, 10297}, {5663, 11188}, {5889, 40949}, {5972, 25406}, {6642, 20379}, {6699, 40330}, {7464, 11645}, {7493, 12827}, {7687, 25320}, {9143, 31099}, {9833, 25712}, {10516, 15059}, {10519, 16111}, {10990, 32257}, {11413, 15581}, {11441, 15141}, {11898, 38790}, {12168, 39879}, {12373, 39873}, {12374, 39897}, {12825, 41716}, {14927, 16163}, {15061, 18358}, {16003, 18913}, {18553, 32305}, {23236, 37495}, {25335, 32246}, {33878, 34584}, {37197, 41616}, {38789, 39899}

X(41737) = midpoint of X(i) and X(j) for these {i, j}: {146, 5921}, {11898, 38790}
X(41737) = reflection of X(i) in X(j) for these (i, j): (20, 5181), (74, 1352), (110, 14982), (265, 39884), (895, 4), (1205, 5907), (1351, 1539), (5095, 38791), (5889, 40949), (6776, 113), (10733, 36990), (10752, 7728), (10990, 32257), (11061, 15063), (11579, 3818), (13169, 11180), (14927, 16163), (15054, 67), (16010, 32274), (32220, 1514), (32234, 9970), (32244, 15069), (32247, 32275), (32264, 2883), (32305, 18553), (41716, 12825), (41720, 10706)
X(41737) = anticomplement of the circumperp conjugate of X(2936)
X(41737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3818, 11579, 14644), (5621, 6698, 15057), (9970, 32234, 41720), (10706, 32234, 9970), (11180, 32247, 32275), (16010, 32274, 9140), (32247, 32275, 13169)

X(41738) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(20)

Barycentrics
a^12-(b^2+c^2)*a^10-(b^4-5*b^2*c^2+c^4)*a^8-(b^2+c^2)*b^2*c^2*a^6+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2*a^4+(b^4-c^4)*(b^2-c^2)*(b^4-5*b^2*c^2+c^4)*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(41738) = 3*X(2)-4*X(15126) = 5*X(3091)-4*X(15125) = 4*X(18381)-3*X(25739)

The reciprocal orthologic center of these triangles is X(41737)

X(41738) lies on these lines: {2, 15126}, {4, 51}, {23, 36201}, {66, 11188}, {110, 858}, {125, 37777}, {343, 5894}, {511, 13203}, {924, 30735}, {1660, 31101}, {1853, 1995}, {2393, 2892}, {2777, 41586}, {2781, 41724}, {3091, 15125}, {3448, 34146}, {3818, 40132}, {5622, 37981}, {6247, 15138}, {7464, 18400}, {8549, 31133}, {9716, 31099}, {9927, 12250}, {10540, 11585}, {11206, 16051}, {11413, 17845}, {11442, 34944}, {12085, 30522}, {12827, 37929}, {14580, 35902}, {15054, 15311}, {17928, 40686}, {20079, 39125}, {23294, 32321}, {31978, 34007}, {32139, 34780}, {34609, 34966}

X(41738) = reflection of X(i) in X(j) for these (i, j): (23, 41603), (110, 32125), (15138, 6247)
X(41738) = anticomplement of the anticomplement of X(15126)
X(41738) = crosssum of X(3) and X(37928)
X(41738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1899, 41736, 41715), (32064, 41736, 1899)

X(41739) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(22)

Barycentrics
a*(a+b)*(c+a)*(a^9-(b+c)*a^8-2*(b+c)^2*a^7+2*(b+c)*(b^2+c^2)*a^6-4*b^2*c^2*a^5+4*(b+c)*b^2*c^2*a^4+2*(b^2-c^2)^2*(b+c)^2*a^3-2*(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)) : :

The reciprocal orthologic center of these triangles is X(41740)

X(41739) lies on these lines: {4, 6}, {21, 41604}, {110, 1812}, {3827, 41601}, {12675, 25713}

X(41739) = reflection of X(21) in X(41604)

X(41740) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(22) TO REFLECTION OF X(21)

Barycentrics
a*((b+c)*a^7-(b+c)^2*a^6-(b^2-c^2)*(b-c)*a^5+(b^2-c^2)^2*a^4-(b^2-c^2)^2*(b+c)*a^3+(b^6+c^6+(2*b^4+2*c^4-(b^2+c^2)*b*c)*b*c)*a^2+(b^4-c^4)*(b^2+c^2)*(b-c)*a-(b^4-c^4)^2) : :

The reciprocal orthologic center of these triangles is X(41739)

X(41740) lies on these lines: {4, 8}, {22, 41605}, {78, 15177}, {110, 1812}, {518, 1993}, {3827, 11442}, {3984, 37546}, {4511, 24301}, {32126, 40914}

X(41740) = reflection of X(22) in X(41605)

X(41741) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(21) TO REFLECTION OF X(23)

Barycentrics
a*(a+b)*(c+a)*(a^9-(b+c)*a^8-3*(b+c)^2*a^7+3*(b+c)*(b^2+c^2)*a^6+(b^4+c^4+(3*b^2+b*c+3*c^2)*b*c)*a^5-(b^3+c^3)*(b^2+b*c+c^2)*a^4+(3*b^6+3*c^6+2*(3*b^4+3*c^4-(b^2+6*b*c+c^2)*b*c)*b*c)*a^3-(b+c)*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(2*b^2+3*b*c+2*c^2)*a+(b^4-c^4)^2*(2*b+2*c)) : :

The reciprocal orthologic center of these triangles is X(41742)

X(41741) lies on these lines: {4, 542}, {21, 41606}, {110, 518}, {2771, 41610}, {2836, 41723}, {4663, 32286}, {12826, 41718}

X(41741) = reflection of X(i) in X(j) for these (i, j): (21, 41606), (41718, 12826)

X(41742) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(21)

Barycentrics    a*((b+c)*a^7-(b+c)^2*a^6-(b^3+c^3)*a^5+(b^4+c^4)*a^4-(b^2-c^2)^2*(b+c)*a^3+(b^6+c^6+(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c)*a^2+(b^3-c^3)*(b^4-c^4)*a-(b^4-c^4)^2) : :

The reciprocal orthologic center of these triangles is X(41741)

X(41742) lies on these lines: {4, 8}, {23, 41607}, {110, 518}, {2836, 41724}, {3448, 3827}, {4122, 8702}, {6001, 15054}

X(41742) = reflection of X(i) in X(j) for these (i, j): (23, 41607), (110, 32126)

X(41743) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(22) TO REFLECTION OF X(23)

Barycentrics
a^2*(a^12-3*(b^2+c^2)*a^10+7*b^2*c^2*a^8+2*(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^6-(3*b^8+3*c^8+4*(b^4-5*b^2*c^2+c^4)*b^2*c^2)*a^4-(b^2+c^2)*(3*b^8+3*c^8-2*(7*b^4-12*b^2*c^2+7*c^4)*b^2*c^2)*a^2+(2*b^4-3*b^2*c^2+2*c^4)*(b^4-c^4)^2) : :

The reciprocal orthologic center of these triangles is X(41744)

X(41743) lies on these lines: {4, 542}, {6, 12824}, {22, 41612}, {69, 12827}, {110, 2393}, {125, 9813}, {399, 12596}, {1993, 2854}, {2070, 19138}, {2781, 41614}, {5609, 11255}, {5622, 9730}, {5642, 11511}, {5648, 10510}, {5655, 18449}, {5890, 11579}, {7728, 32220}, {8538, 16534}, {8907, 34787}, {10752, 13754}, {11433, 25320}, {11464, 15462}, {34155, 40673}

X(41743) = reflection of X(i) in X(j) for these (i, j): (22, 41612), (69, 12827), (895, 8541)

X(41744) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(23) TO REFLECTION OF X(22)

Barycentrics
a^2*(a^12-2*(b^2+c^2)*a^10-(b^4-5*b^2*c^2+c^4)*a^8+2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^6-(b^8+c^8+2*(2*b^4-7*b^2*c^2+2*c^4)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)*a^2+(b^4-b^2*c^2+c^4)*(b^4-c^4)^2) : :
X(41744) = 2*X(159)-3*X(35265) = 2*X(15647)-3*X(18374)

The reciprocal orthologic center of these triangles is X(41743)

X(41744) lies on these lines: {4, 6}, {23, 41613}, {110, 2393}, {154, 8547}, {159, 35265}, {323, 15141}, {524, 17847}, {1495, 34470}, {1994, 11216}, {2071, 41612}, {2854, 15139}, {2892, 32125}, {3448, 32220}, {6000, 11579}, {9716, 34777}, {10117, 32217}, {10169, 34545}, {10510, 25714}, {10628, 32599}, {11003, 19153}, {11574, 23358}, {15054, 34146}, {15647, 18374}, {18911, 23327}, {19118, 35219}, {19128, 35371}, {41603, 41617}

X(41744) = reflection of X(i) in X(j) for these (i, j): (23, 41613), (323, 15141), (2892, 32125), (10117, 32217), (38885, 1495), (41735, 32111)

X(41745) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(6) TO REFLECTION OF X(15)

Barycentrics    9*a^4+2*sqrt(3)*(5*a^2-b^2-c^2)*S-3*(b^2-c^2)^2 : :
X(41745) = X(6772)-4*X(41620)

The reciprocal orthologic center of these triangles is X(22495).

X(41745) lies on these lines: {2, 5472}, {5, 13}, {6, 530}, {14, 25154}, {15, 6779}, {30, 23006}, {32, 616}, {99, 3180}, {110, 21466}, {115, 37641}, {187, 8595}, {230, 9762}, {298, 7845}, {299, 7801}, {381, 6782}, {396, 5463}, {524, 12155}, {532, 22687}, {542, 6775}, {543, 1992}, {576, 10654}, {590, 13646}, {615, 13765}, {618, 16644}, {671, 5471}, {3105, 36968}, {3181, 7812}, {5318, 41042}, {5459, 16645}, {5475, 37785}, {6108, 11486}, {6109, 20425}, {6298, 41631}, {6582, 41633}, {6777, 36969}, {6783, 8724}, {7785, 40707}, {8787, 12154}, {9761, 40671}, {10617, 16960}, {11129, 40899}, {13103, 31710}, {14537, 35697}, {16001, 40694}, {16267, 36766}, {20423, 22512}, {22113, 22892}, {22489, 23303}, {22601, 22630}, {31415, 33477}, {36363, 41107}, {36383, 41100}

X(41745) = reflection of X(i) in X(j) for these (i, j): (6, 41620), (6772, 6), (12154, 8787), (22513, 10653), (41746, 5477)
X(41745) = X(13)-of-reflection-triangle-of-X(6)
X(41745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (13, 16530, 37835), (13, 22998, 5617), (1992, 7737, 41746), (5463, 9112, 396), (5472, 9115, 2), (8595, 37786, 9885), (22495, 41406, 396), (22601, 22630, 22796), (41751, 41752, 5617)

X(41746) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(6) TO REFLECTION OF X(16)

Barycentrics    9*a^4-2*sqrt(3)*(5*a^2-b^2-c^2)*S-3*(b^2-c^2)^2 : :
X(41746) = X(6775)-4*X(41621)

The reciprocal orthologic center of these triangles is X(22496)

X(41746) lies on these lines: {2, 5471}, {5, 14}, {6, 531}, {13, 25164}, {16, 6780}, {30, 23013}, {32, 617}, {99, 3181}, {110, 21467}, {115, 37640}, {187, 8594}, {230, 9760}, {298, 7801}, {299, 7845}, {381, 6783}, {395, 5464}, {524, 12154}, {533, 22689}, {542, 6772}, {543, 1992}, {576, 10653}, {590, 13645}, {615, 13764}, {619, 16645}, {671, 5472}, {3104, 36967}, {3180, 7812}, {5321, 41043}, {5460, 16644}, {5475, 37786}, {6108, 20426}, {6109, 11485}, {6295, 41643}, {6299, 41641}, {6778, 36970}, {6782, 8724}, {7785, 40706}, {8787, 12155}, {9763, 40672}, {10616, 16961}, {11128, 40898}, {13102, 31709}, {14537, 35693}, {16002, 40693}, {20423, 22513}, {22114, 22848}, {22490, 23302}, {22603, 22632}, {31415, 33476}, {36362, 41108}, {36382, 41101}

X(41746) = reflection of X(i) in X(j) for these (i, j): (6, 41621), (6775, 6), (12155, 8787), (22512, 10654), (41745, 5477)
X(41746) = X(14)-of-reflection-triangle-of-X(6)
X(41746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (14, 16529, 37832), (14, 22997, 5613), (1992, 7737, 41745), (5464, 9113, 395), (5471, 9117, 2), (8594, 37785, 9886), (22496, 41407, 395), (22603, 22632, 22797), (41753, 41754, 5613)

X(41747) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(6) TO REFLECTION OF X(32)

Barycentrics    2*(b^2+c^2)*a^4+(b^4+b^2*c^2+c^4)*a^2-2*(b^2+c^2)*b^2*c^2 : :
X(41747) = 6*X(39)-5*X(3763) = X(69)-3*X(194) = 2*X(69)-3*X(3094) = 3*X(76)-4*X(3589) = 2*X(76)-3*X(13331) = 4*X(76)-5*X(40332) = 2*X(141)-3*X(7757) = 3*X(3095)-2*X(3818) = 8*X(3589)-9*X(13331) = 2*X(3589)-3*X(32449) = 16*X(3589)-15*X(40332) = 4*X(3629)-3*X(13330) = 2*X(3629)-3*X(32451) = 3*X(13331)-4*X(32449) = 6*X(13331)-5*X(40332) = 3*X(22486)-4*X(32455) = 8*X(32449)-5*X(40332) = 3*X(32520)+X(39899)

The reciprocal orthologic center of these triangles is X(41748)

X(41747) lies on these lines: {6, 538}, {39, 3763}, {69, 194}, {76, 3589}, {141, 7757}, {183, 12055}, {193, 5969}, {230, 9764}, {511, 1657}, {524, 8353}, {590, 13647}, {615, 13766}, {698, 3629}, {1613, 19568}, {1691, 7754}, {1975, 12212}, {2076, 7781}, {2782, 31670}, {3095, 3818}, {3981, 8267}, {4048, 7760}, {4074, 40904}, {5031, 7906}, {5116, 7751}, {6294, 41630}, {6309, 6390}, {6581, 41640}, {7737, 41749}, {7772, 24273}, {7806, 9865}, {7890, 29012}, {9821, 32429}, {14031, 18906}, {14881, 22613}, {14994, 32450}, {15480, 33706}, {20081, 24256}, {22486, 32455}, {31981, 41651}, {41412, 41748}

X(41747) = midpoint of X(18906) and X(20105)
X(41747) = reflection of X(i) in X(j) for these (i, j): (6, 41622), (76, 32449), (3094, 194), (9821, 32429), (13330, 32451), (14994, 32450), (20081, 24256)
X(41747) = crossdifference of every pair of points on line {X(9009), X(39520)}
X(41747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (76, 13331, 40332), (76, 32449, 13331), (22613, 22642, 14881), (23000, 23009, 3095)

X(41748) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(32) TO REFLECTION OF X(6)

Barycentrics    3*a^4+2*(b^2+c^2)*a^2-4*b^2*c^2 : :
X(41748) = 3*X(32)-2*X(1003) = 5*X(32)-2*X(1975) = X(32)+2*X(7754) = X(32)-4*X(7805) = 7*X(32)-4*X(7816) = 5*X(1003)-3*X(1975) = X(1003)+3*X(7754) = X(1003)-6*X(7805) = 7*X(1003)-6*X(7816) = X(1003)-3*X(14614) = X(1975)+5*X(7754) = X(1975)-10*X(7805) = 7*X(1975)-10*X(7816) = X(1975)-5*X(14614) = 3*X(5309)-2*X(33184) = X(7754)+2*X(7805) = 7*X(7754)+2*X(7816) = 7*X(7805)-X(7816) = 2*X(7816)-7*X(14614) = 3*X(7818)-4*X(33184)

The reciprocal orthologic center of these triangles is X(41747)

X(41748) lies on these lines: {2, 3108}, {6, 9466}, {32, 538}, {39, 8667}, {69, 5355}, {115, 193}, {187, 8716}, {194, 5206}, {381, 576}, {385, 574}, {524, 5028}, {542, 35389}, {543, 33193}, {754, 11648}, {1992, 7753}, {3363, 41149}, {3524, 32467}, {3543, 38664}, {3629, 5475}, {3734, 7766}, {3767, 7890}, {3933, 5346}, {5007, 11286}, {5033, 41622}, {5102, 22682}, {5111, 31173}, {5286, 7826}, {5304, 7820}, {5305, 7855}, {5306, 7801}, {5368, 7795}, {6144, 7845}, {6179, 7781}, {6392, 7747}, {7617, 32994}, {7697, 15520}, {7735, 7813}, {7739, 7810}, {7759, 33228}, {7765, 14023}, {7768, 7902}, {7775, 7837}, {7779, 7844}, {7788, 7817}, {7797, 7896}, {7804, 14075}, {7806, 7908}, {7808, 7894}, {7812, 18546}, {7815, 7839}, {7821, 33240}, {7825, 7877}, {7827, 7865}, {7828, 7916}, {7851, 7882}, {7853, 40341}, {7862, 7905}, {7863, 33191}, {7872, 7893}, {7914, 7920}, {8358, 9607}, {8556, 9605}, {8588, 31859}, {9167, 32837}, {9764, 41756}, {11054, 11361}, {11152, 36859}, {11287, 39593}, {11288, 39785}, {12188, 37517}, {12243, 34733}, {14537, 34505}, {15048, 15480}, {15300, 35927}, {15515, 32450}, {31274, 37689}, {32451, 39764}, {33216, 34511}, {33266, 36521}, {41412, 41747}

X(41748) = midpoint of X(7754) and X(14614)
X(41748) = reflection of X(i) in X(j) for these (i, j): (32, 14614), (7788, 7817), (7801, 5306), (7818, 5309), (14614, 7805)
X(41748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (69, 5355, 7913), (381, 15534, 41750), (385, 7798, 574), (3767, 7890, 7903), (5007, 14711, 11286), (5286, 7826, 7935), (5305, 7855, 7867), (6179, 11055, 13586), (7751, 7760, 7772), (7754, 7805, 32), (7755, 7758, 7888), (7775, 14568, 18362), (7812, 19570, 18546), (7837, 14568, 7775), (7894, 17129, 7808), (11055, 13586, 7781), (11286, 14711, 17130)

X(41749) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(6) TO REFLECTION OF X(39)

Barycentrics    3*a^6+5*(b^2+c^2)*a^4+3*b^2*c^2*a^2-(b^2+c^2)*(b^4+c^4) : :
X(41749) = 3*X(83)-2*X(141) = X(193)+3*X(20088) = 3*X(2896)-5*X(3618) = 4*X(3589)-3*X(31168) = 3*X(6287)-4*X(19130) = 3*X(13111)-X(18440) = 3*X(24273)-X(40341)

The reciprocal orthologic center of these triangles is X(41750)

X(41749) lies on these lines: {6, 754}, {83, 141}, {193, 732}, {230, 9765}, {524, 12156}, {590, 13648}, {615, 13767}, {1351, 5073}, {1992, 19569}, {2076, 7838}, {2896, 3618}, {3589, 31168}, {3629, 10754}, {3793, 6308}, {5017, 35701}, {5111, 19695}, {6287, 19130}, {6292, 30435}, {6296, 41642}, {6297, 41632}, {6704, 7776}, {7737, 41747}, {13111, 18440}, {13331, 20065}, {14023, 40332}, {14712, 32476}, {21850, 35427}, {22614, 22643}, {24273, 40341}, {31982, 41650}, {41413, 41750}

X(41749) = reflection of X(6) in X(41623)
X(41749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (22614, 22643, 22803), (23001, 23010, 6287)

X(41750) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(39) TO REFLECTION OF X(6)

Barycentrics    4*a^4+3*(b^2+c^2)*a^2-2*b^4-2*c^4 : :
X(41750) = 5*X(39)-2*X(7750) = X(39)+2*X(7762) = 7*X(39)-4*X(7830) = X(39)-4*X(7838) = 3*X(39)-2*X(8356) = 3*X(262)-X(34623) = X(7750)+5*X(7762) = 7*X(7750)-10*X(7830) = X(7750)-10*X(7838) = 3*X(7750)-5*X(8356) = X(7750)-5*X(41624) = 7*X(7762)+2*X(7830) = X(7762)+2*X(7838) = 3*X(7762)+X(8356) = 3*X(7812)-X(11361) = X(7830)-7*X(7838) = 6*X(7830)-7*X(8356) = 2*X(7830)-7*X(41624) = 6*X(7838)-X(8356) = X(8356)-3*X(41624)

The reciprocal orthologic center of these triangles is X(41749)

X(41750) lies on these lines: {2, 5007}, {6, 7818}, {32, 9766}, {39, 754}, {83, 7882}, {115, 3629}, {141, 41623}, {187, 7774}, {193, 5475}, {262, 34623}, {315, 5041}, {325, 5008}, {381, 576}, {384, 12156}, {385, 7603}, {524, 5052}, {538, 7812}, {542, 35439}, {591, 5062}, {599, 5039}, {625, 7766}, {1003, 39785}, {1506, 13468}, {1570, 1992}, {1991, 5058}, {3329, 7848}, {3564, 22682}, {3849, 7757}, {3934, 7877}, {5215, 9770}, {5319, 33285}, {5346, 32816}, {5355, 32455}, {6144, 15484}, {6683, 7893}, {7735, 31275}, {7739, 33210}, {7745, 7890}, {7754, 18546}, {7758, 14033}, {7760, 7843}, {7764, 35007}, {7772, 7873}, {7775, 14614}, {7776, 7852}, {7779, 7804}, {7785, 7805}, {7787, 7895}, {7796, 14036}, {7799, 34604}, {7801, 14039}, {7809, 7817}, {7810, 9300}, {7811, 15810}, {7813, 18907}, {7816, 7905}, {7823, 32450}, {7826, 31239}, {7834, 34571}, {7839, 7842}, {7840, 7880}, {7841, 39593}, {7861, 7894}, {7874, 7903}, {7879, 39784}, {7886, 7941}, {7915, 7917}, {8370, 14711}, {8584, 33184}, {9765, 41755}, {9939, 40344}, {11171, 34734}, {11614, 17005}, {13571, 33265}, {14839, 34645}, {20065, 37512}, {27377, 33842}, {31652, 33008}, {32837, 37809}, {34511, 35927}, {34682, 35002}, {37350, 41149}, {41413, 41749}

X(41750) = midpoint of X(i) and X(j) for these {i, j}: {7762, 41624}, {7812, 7837}
X(41750) = reflection of X(i) in X(j) for these (i, j): (39, 41624), (7810, 9300), (9466, 7753), (14537, 7812), (14711, 8370), (41624, 7838)
X(41750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 7845, 7853), (381, 15534, 41748), (5007, 7759, 7821), (6144, 15484, 17131), (7762, 7838, 39), (7766, 7926, 625), (7785, 7805, 39565), (7787, 7949, 7895), (7840, 12150, 7880), (7877, 7921, 3934), (7878, 7946, 7849), (7894, 7900, 7861), (7903, 30435, 7874), (7905, 20088, 7816)

X(41751) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(32) TO REFLECTION OF X(15)

Barycentrics    2*(3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*S+a^2*(a^4+2*(b^2+c^2)*a^2-b^4-c^4)*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(23000)

X(41751) lies on these lines: {5, 13}, {6, 5980}, {32, 41631}, {39, 6582}, {99, 19780}, {530, 5309}, {3643, 9115}, {3818, 6782}, {5017, 5969}, {5149, 25187}, {7751, 25183}, {11486, 12188}, {22892, 22914}, {32465, 39554}

X(41751) = reflection of X(i) in X(j) for these (i, j): (32, 41631), (41753, 12829)
X(41751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (395, 41752, 5617), (5017, 14614, 41753), (5617, 41745, 41752)

X(41752) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(39) TO REFLECTION OF X(15)

Barycentrics    2*(2*a^4+3*(b^2+c^2)*a^2-b^4-c^4)*S+(2*a^6+(b^2+c^2)*a^4+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(23001)

X(41752) lies on these lines: {5, 13}, {32, 6298}, {39, 41633}, {83, 14904}, {396, 1691}, {530, 7753}, {624, 5472}, {1916, 12830}, {3933, 22915}, {5318, 6033}, {6782, 19130}, {10617, 36766}, {12214, 34509}

X(41752) = reflection of X(i) in X(j) for these (i, j): (39, 41633), (41754, 12830)
X(41752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5617, 41745, 41751), (5617, 41751, 395)

X(41753) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(32) TO REFLECTION OF X(16)

Barycentrics    -2*(3*a^4+(b^2+c^2)*a^2-2*b^2*c^2)*S+a^2*(a^4+2*(b^2+c^2)*a^2-b^4-c^4)*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(23009)

X(41753) lies on these lines: {5, 14}, {6, 5981}, {32, 41641}, {39, 6295}, {99, 19781}, {531, 5309}, {3642, 9117}, {3818, 6783}, {5017, 5969}, {5149, 25183}, {7751, 25187}, {11485, 12188}, {22848, 22869}, {32466, 39555}

X(41753) = reflection of X(i) in X(j) for these (i, j): (32, 41641), (41751, 12829)
X(41753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (396, 41754, 5613), (5017, 14614, 41751), (5613, 41746, 41754)

X(41754) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(39) TO REFLECTION OF X(16)

Barycentrics    -2*(2*a^4+3*(b^2+c^2)*a^2-b^4-c^4)*S+(2*a^6+(b^2+c^2)*a^4+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(23010)

X(41754) lies on these lines: {5, 14}, {32, 6299}, {39, 41643}, {83, 14905}, {395, 1691}, {531, 7753}, {623, 5471}, {1916, 12830}, {3933, 22870}, {5321, 6033}, {6783, 19130}, {12213, 34508}

X(41754) = reflection of X(i) in X(j) for these (i, j): (39, 41643), (41752, 12830)
X(41754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5613, 41746, 41753), (5613, 41753, 396)

X(41755) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(32) TO REFLECTION OF X(39)

Barycentrics    a^8+4*(b^2+c^2)*a^6+(b^2+2*c^2)*(2*b^2+c^2)*a^4-(b^2+c^2)*b^2*c^2*a^2-(b^2+c^2)^2*b^2*c^2 : :
X(41755) = 2*X(7882)-5*X(31268)

The reciprocal orthologic center of these triangles is X(41756)

X(41755) lies on these lines: {6, 6292}, {32, 732}, {39, 6308}, {76, 12206}, {83, 385}, {183, 6704}, {754, 5309}, {1992, 13086}, {2896, 7766}, {3329, 7882}, {6287, 19130}, {7760, 32476}, {7762, 9478}, {7805, 36849}, {7838, 32190}, {8725, 9301}, {9765, 41750}, {20088, 33018}, {24273, 30435}

X(41755) = reflection of X(32) in X(41650)

X(41756) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION OF X(39) TO REFLECTION OF X(32)

Barycentrics    (b^2+c^2)*(a^6+2*(b^2+c^2)*a^4-(b^2+c^2)*b^2*c^2) : :
X(41756) = 3*X(7757)-X(9983)

The reciprocal orthologic center of these triangles is X(41755)

X(41756) lies on these lines: {32, 6309}, {39, 141}, {76, 3329}, {194, 315}, {511, 7890}, {538, 7753}, {1916, 7905}, {2782, 3627}, {3094, 7855}, {3095, 3818}, {3934, 5355}, {5041, 24256}, {5976, 7805}, {7754, 8149}, {7757, 7865}, {7760, 9865}, {7768, 32476}, {7786, 7908}, {7822, 13331}, {7863, 13357}, {8597, 11055}, {9764, 41748}, {12177, 35386}, {14981, 35436}

X(41756) = reflection of X(39) in X(41651)
X(41756) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 14994, 6292), (194, 7758, 32452), (3933, 32449, 39), (6309, 32451, 32), (7760, 9865, 18806)

X(41757) = X(2)X(40947)∩X(4)X(193)

Barycentrics    a^8 - b^8 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 2*b^6*c^2 + 2*a^2*b^2*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8 : :

X(41757) lies on these lines: {2, 40947}, {4, 193}, {66, 3260}, {69, 2871}, {110, 17907}, {147, 3164}, {264, 11442}, {315, 12220}, {317, 12272}, {340, 40317}, {1370, 6527}, {5304, 6997}, {6530, 11441}, {7391, 7779}, {7394, 7766}, {7897, 16063}, {14516, 33971}, {14957, 36851}, {18531, 36875}, {19459, 41237}, {28419, 33314}, {30506, 41523}

X(41757) = anticomplement of X(40947)
X(41757) = anticomplement of the isogonal conjugate of X(34405)
X(41757) = X(34405)-anticomplementary conjugate of X(8)

X(41758) = X(6)X(186)∩X(24)X(254)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 - b^6*c^2 + 6*a^4*c^4 - 4*a^2*c^6 - b^2*c^6 + c^8) : :

X(41758) lies on these lines: {3, 3087}, {4, 8553}, {6, 186}, {24, 254}, {53, 1601}, {112, 2383}, {216, 7488}, {232, 1627}, {317, 35296}, {570, 10312}, {577, 22467}, {590, 15219}, {615, 15220}, {2965, 11062}, {3053, 8745}, {3068, 15207}, {3069, 15208}, {3515, 8573}, {3520, 6748}, {6240, 9722}, {6636, 10311}, {6749, 15109}, {7512, 36751}, {7585, 15195}, {7586, 15196}, {7735, 21213}, {13345, 39575}, {13595, 36412}, {15905, 37814}, {21844, 40065}

X(41758) = barycentric product X(76)*X(41759)
X(41758) = barycentric quotient X(41759)/X(6)
X(41758) = trilinear product X(75)*X(41759)
X(41758) = trilinear quotient X(41759)/X(31)
X(41758) = {X(24),X(1609)}-harmonic conjugate of X(393)

X(41759) = X(32)X(34397)∩X(186)X(9603)

Barycentrics    a^4*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 - b^6*c^2 + 6*a^4*c^4 - 4*a^2*c^6 - b^2*c^6 + c^8) : :

X(41759) lies on these lines: {32, 34397}, {186, 9603}, {2207, 39109}, {9604, 10312}

X(41759) = barycentric product X(6)*X(41758)
X(41759) = barycentric quotient X(41758)/X(76)
X(41759) = trilinear product X(31)*X(41758)
X(41759) = trilinear quotient X(41758)/X(75)

leftri

Points on the cubic K1188: X(41760)-X(41780)

rightri

Points X(41759)-X(41780) are contributed by Peter Moses, March 9, 2021.

X(41760) = X(2)X(311)∩X(4)X(66)

Barycentrics    b^2*c^2*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

X(41760) lies on the cubic K1188 and these lines: {2, 311}, {4, 66}, {6, 264}, {39, 14767}, {53, 2052}, {76, 141}, {94, 34289}, {98, 157}, {159, 39646}, {160, 11257}, {193, 3260}, {206, 419}, {230, 34828}, {262, 3613}, {305, 7778}, {316, 33797}, {324, 11433}, {339, 20208}, {389, 39530}, {401, 571}, {460, 26926}, {523, 9307}, {524, 14615}, {847, 18912}, {966, 26592}, {1232, 3620}, {1235, 5286}, {1249, 37778}, {1316, 14575}, {1609, 20477}, {1632, 40947}, {1899, 6751}, {1990, 21447}, {2345, 34388}, {2393, 3186}, {2970, 26869}, {3003, 3164}, {3266, 37690}, {3313, 37190}, {3767, 6389}, {3972, 40416}, {3978, 40073}, {4000, 34387}, {4648, 26541}, {5117, 6697}, {5596, 6620}, {5640, 17500}, {6179, 33768}, {6248, 9822}, {7735, 30737}, {7738, 26166}, {9465, 36901}, {11427, 40684}, {11432, 14978}, {12272, 25051}, {14064, 28706}, {15271, 40022}, {15466, 26958}, {20775, 39906}, {20806, 41238}

X(41760) = isotomic conjugate of the isogonal conjugate of X(3767)
X(41760) = polar conjugate of the isotomic conjugate of X(41009)
X(41760) = polar conjugate of the isogonal conjugate of X(1899)
X(41760) = X(1899)-cross conjugate of X(41009)
X(41760) = X(9247)-isoconjugate of X(34405)
X(41760) = cevapoint of X(1899) and X(3767)
X(41760) = crosspoint of X(76) and X(2052)
X(41760) = crosssum of X(i) and X(j) for these (i,j): {32, 577}, {23208, 41331}
X(41760) = crossdifference of every pair of points on line {9426, 34952}
X(41760) = barycentric product X(i)*X(j) for these {i,j}: {4, 41009}, {75, 17871}, {76, 3767}, {264, 1899}, {290, 2450}, {850, 1632}, {1969, 2083}, {2052, 6389}, {18022, 40947}, {18027, 39643}, {27362, 34385}
X(41760) = barycentric quotient X(i)/X(j) for these {i,j}: {264, 34405}, {426, 1092}, {1632, 110}, {1899, 3}, {2083, 48}, {2450, 511}, {3767, 6}, {6389, 394}, {6751, 418}, {17871, 1}, {27362, 52}, {28405, 20806}, {39643, 577}, {40947, 184}, {41009, 69}
X(41760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 338, 264}, {264, 40814, 6}

X(41761) = X(4)X(6)∩X(66)X(264)

Barycentrics    (a^4 - b^4 - c^4)*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :
X(41761) = 3 X[2] - 4 X[23333], 3 X[4] - 4 X[18380], 5 X[631] - 4 X[37813], 5 X[3618] - 4 X[19156]

X(41761) lies on the cubic K1188 and these lines: {2, 157}, {4, 6}, {66, 264}, {69, 2871}, {98, 2165}, {159, 297}, {206, 17907}, {311, 11442}, {315, 3313}, {317, 2393}, {325, 1370}, {458, 23300}, {570, 9744}, {571, 36998}, {577, 2794}, {631, 37813}, {1513, 1609}, {1632, 6389}, {1899, 6751}, {2450, 40947}, {2790, 6033}, {2980, 7394}, {2998, 32528}, {3001, 37444}, {3398, 7528}, {3618, 19156}, {5065, 36997}, {6759, 39569}, {6815, 20792}, {6997, 7792}, {7386, 7778}, {7391, 7774}, {7791, 41328}, {8266, 37182}, {9753, 13345}, {14790, 32428}, {16063, 31076}, {16924, 18092}, {18381, 39530}, {19174, 19212}, {23327, 36794}, {27377, 34777}, {28408, 33314}, {31166, 37765}, {32006, 35549}, {33797, 40074}

X(41761) = anticomplement of X(157)
X(41761) = reflection of X(157) in X(23333)
X(41761) = circumcircle-of-anticomplementary triangle-inverse of X(13509)
X(41761) = polar conjugate of the isotomic conjugate of X(28405)
X(41761) = X(1485)-anticomplementary conjugate of X(192)
X(41761) = X(99)-Ceva conjugate of X(33294)
X(41761) = cevapoint of X(14713) and X(39643)
X(41761) = barycentric product X(i)*X(j) for these {i,j}: {4, 28405}, {315, 3767}, {1632, 33294}, {1760, 17871}, {1899, 17907}, {2450, 31636}, {8743, 41009}
X(41761) = barycentric quotient X(i)/X(j) for these {i,j}: {1899, 14376}, {2450, 34138}, {3767, 66}, {17907, 34405}, {28405, 69}
X(41761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {157, 23333, 2}, {1899, 6751, 18953}

X(41762) = X(4)X(69)∩X(66)X(338)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

X(41762) lies on the cubic K1188 and these lines: {4, 69}, {6, 460}, {25, 53}, {66, 338}, {114, 40697}, {133, 5139}, {235, 30549}, {393, 1974}, {419, 17907}, {427, 7778}, {428, 8667}, {1513, 20477}, {1899, 6751}, {1990, 19118}, {2450, 6389}, {2549, 20775}, {3087, 8541}, {3089, 6750}, {3542, 39569}, {3767, 40947}, {5254, 19459}, {6748, 12167}, {6995, 37667}, {7487, 39647}, {9229, 33300}, {10151, 16312}, {10551, 17500}, {15809, 41244}, {18533, 38749}

X(41762) = polar conjugate of the isotomic conjugate of X(3767)
X(41762) = orthic isogonal conjugate of X(1899)
X(41762) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1899}, {1289, 2501}, {15352, 2489}
X(41762) = X(255)-isoconjugate of X(34405)
X(41762) = crosspoint of X(4) and X(6524)
X(41762) = crosssum of X(3) and X(3964)
X(41762) = barycentric product X(i)*X(j) for these {i,j}: {4, 3767}, {19, 17871}, {158, 2083}, {393, 1899}, {1093, 39643}, {1632, 2501}, {2052, 40947}, {2207, 41009}, {2450, 6531}, {6389, 6524}, {6751, 8794}
X(41762) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 34405}, {1632, 4563}, {1899, 3926}, {2083, 326}, {2450, 6393}, {3767, 69}, {6389, 4176}, {17871, 304}, {39643, 3964}, {40947, 394}
X(41762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3186, 317}, {393, 6620, 1974}, {1974, 8754, 393}

X(41763) = X(6)X(66)∩X(264)X(7828)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 + b^4 - 2*b^2*c^2 + c^4)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 - b^2*c^4 - c^6) : :

X(41763) lies on the cubic K1188 and these lines: {6, 66}, {264, 7828}, {2165, 39645}, {3767, 40947}, {8361, 41005}, {40697, 41361}

X(41763) = polar conjugate of the isogonal conjugate of X(14713)
X(41763) = barycentric product X(264)*X(14713)
X(41763) = barycentric quotient X(14713)/X(3)

X(41764) = X(66)X(69)∩X(338)X(30549)

Barycentrics    b^2*c^2*(a^4 + b^4 - 2*b^2*c^2 + c^4)*(-3*a^6 + 3*a^4*b^2 - a^2*b^4 + b^6 + 3*a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :

X(41764) lies on the cubic K1188 and these lines: {66, 69}, {338, 30549}, {393, 847}, {1632, 40947}, {7387, 32428}, {11585, 14059}

X(41765) = X(157)X(264)∩X(7796)X(40073)

Barycentrics    b^2*c^2*(a^4 + b^4 - 2*b^2*c^2 + c^4)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(-a^6 + a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :

X(41765) lies on the cubic K1188 and these lines: {157, 264}, {7796, 40073}

X(41765) = X(523)-cross conjugate of X(1632)
X(41765) = barycentric quotient X(i)/X(j) for these {i,j}: {1899, 23128}, {3767, 157}, {17871, 21374}, {40947, 22391}

X(41766) = X(2)X(34427)∩X(4)X(66)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6) : :

X(41766) lies on the cubic K1188 and these lines: {2, 34427}, {4, 66}, {5, 6523}, {25, 393}, {107, 7493}, {132, 6389}, {159, 41361}, {206, 1249}, {264, 6997}, {1235, 15435}, {1502, 6528}, {2052, 40178}, {3542, 6530}, {6526, 7507}, {6620, 8745}, {6816, 14249}, {7528, 14978}, {7716, 27376}, {12918, 18531}, {19125, 40138}, {33971, 37122}

X(41766) = polar conjugate of the isotomic conjugate of X(41361)
X(41766) = polar conjugate of the isogonal conjugate of X(3162)
X(41766) = X(264)-Ceva conjugate of X(393)
X(41766) = X(i)-cross conjugate of X(j) for these (i,j): {3162, 41361}, {17407, 4}
X(41766) = X(i)-isoconjugate of X(j) for these (i,j): {255, 13575}, {326, 34207}, {577, 39733}, {1102, 40144}
X(41766) = crosspoint of X(6528) and X(23590)
X(41766) = barycentric product X(i)*X(j) for these {i,j}: {4, 41361}, {158, 18596}, {159, 2052}, {264, 3162}, {393, 1370}, {1093, 23115}, {1096, 21582}, {1857, 18629}, {6524, 28419}, {15466, 33584}, {17407, 17907}, {40357, 41375}
X(41766) = barycentric quotient X(i)/X(j) for these {i,j}: {158, 39733}, {159, 394}, {393, 13575}, {455, 23115}, {1370, 3926}, {2052, 40009}, {2207, 34207}, {3162, 3}, {8793, 28724}, {17407, 14376}, {17409, 39172}, {18596, 326}, {18629, 7055}, {23115, 3964}, {27376, 39129}, {28419, 4176}, {33584, 1073}, {41361, 69}

X(41767) = X(5)X(6)∩X(66)X(2871)

Barycentrics    (a^2 - b^2 - c^2)*(a^4 + b^4 - 2*b^2*c^2 + c^4)*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 2*a^6*c^2 + 4*a^4*b^2*c^2 - 2*b^6*c^2 + 2*a^4*c^4 + 2*b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) : :

X(41767) lies on the cubic K1188 and these lines: {5, 6}, {66, 2871}, {264, 11442}, {426, 1899}

X(41767) = X(264)-Ceva conjugate of X(6389)

X(41768) = CROSSSUM OF X(155) AND X(159)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - 3*c^6)*(a^6 - a^4*b^2 + 3*a^2*b^4 - 3*b^6 - a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :

X(41768) lies on the cubic K1188 and these lines: {155, 159}, {157, 40801}, {232, 1609}, {325, 1370}, {1576, 34233}, {3542, 6530}, {6389, 11585}, {8883, 19189}, {15594, 37971}

X(41768) = isogonal conjugate of the anticomplement of X(1899)
X(41768) = X(40947)-cross conjugate of X(6)
X(41768) = crosspoint of X(254) and X(13575)
X(41768) = crosssum of X(155) and X(159)
X(41768) = trilinear pole of line {3569, 30442}

X(41769) = X(2)X(157)∩X(69)X(23128)

Barycentrics    (a^2 - b^2 - c^2)*(a^6 + a^4*b^2 + a^2*b^4 + b^6 - a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 - b^2*c^4 + c^6) : :

X(41769) lies on the cubic K1188 and these lines: {2, 157}, {69, 23128}, {95, 3096}, {264, 7828}, {305, 28412}, {6340, 28427}, {7735, 13575}, {20563, 28442}, {28408, 40708}

X(41769) = X(40947)-cross conjugate of X(69)
X(41769) = cevapoint of X(127) and X(647)
X(41769) = barycentric quotient X(40947)/X(14713)

X(41770) = X(2)X(6)∩X(66)X(98)

Barycentrics    (a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

X(41770) lies on the cubic K1188 and these lines: {2, 6}, {4, 157}, {66, 98}, {159, 1513}, {264, 2165}, {297, 1609}, {317, 571}, {393, 41678}, {458, 9722}, {2450, 40947}, {3003, 17907}, {3542, 6530}, {3549, 30258}, {3767, 6389}, {5254, 34828}, {7505, 41371}, {7746, 14767}, {8968, 10962}, {9308, 16310}, {9753, 9969}, {11547, 14576}, {16321, 37943}, {18953, 39643}, {20065, 34827}

X(41770) = X(275)-Ceva conjugate of X(39643)
X(41770) = crosspoint of X(7763) and X(11547)
X(41770) = barycentric product X(i)*X(j) for these {i,j}: {24, 41009}, {95, 27362}, {317, 1899}, {1632, 6563}, {2450, 31635}, {3767, 7763}, {6389, 11547}
X(41770) = barycentric quotient X(i)/X(j) for these {i,j}: {317, 34405}, {426, 16391}, {1632, 925}, {1899, 68}, {2083, 1820}, {3767, 2165}, {17871, 91}, {27362, 5}, {40947, 2351}, {41009, 20563}

X(41771) = X(1)X(87) ∩ X(2)X(17090)

Barycentrics    (2*a^2 - a*b - a*c + b*c)*(b^2 - b*c + c^2) : :

X(41771) lies on these lines: {1, 87}, {2, 17090}, {257, 25918}, {514, 27091}, {664, 17743}, {960, 17333}, {1212, 17338}, {2275, 33890}, {3061, 3662}, {3160, 26685}, {3752, 17367}, {3948, 17056}, {4019, 17363}, {6505, 27064}, {7200, 18055}, {8256, 9311}, {14951, 40598}, {17242, 18156}, {17248, 25895}, {21435, 33933}, {21608, 30054}, {24514, 39046}, {26653, 39047}, {29982, 30022}

X(41772) = X(44)X(63) ∩ X(144)X(145)

Barycentrics    a*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 8*a^2*b*c + 5*a*b^2*c + 2*b^3*c + a^2*c^2 + 5*a*b*c^2 - 6*b^2*c^2 - a*c^3 + 2*b*c^3 - c^4) : :

X(41772) lies on these lines: {44, 63}, {144, 145}, {190, 34371}, {320, 329}, {513, 4468}, {517, 4480}, {752, 12527}, {894, 3812}, {908, 3834}, {960, 17333}, {1122, 26685}, {1463, 17257}, {2975, 3246}, {3257, 36100}, {3305, 25891}, {3729, 10914}, {4416, 15310}, {4499, 15726}, {4715, 17781}, {5744, 31233}, {6180, 7131}, {16561, 18735}, {16596, 40880}, {17350, 24471}, {20245, 29982}, {20348, 30090}, {21362, 25083}, {27834, 37131}, {29698, 40937}, {30852, 31243}

X(41773) = X(192)X(519) ∩ X(514)X(4374)

Barycentrics    3*a^2*b^2 - 3*a*b^3 - 6*a^2*b*c + 4*a*b^2*c + b^3*c + 3*a^2*c^2 + 4*a*b*c^2 - 4*b^2*c^2 - 3*a*c^3 + b*c^3 : :

X(41773) lies on these lines: {192, 519}, {514, 4374}, {517, 4480}, {3061, 3452}, {3752, 41140}, {4737, 17760}, {20924, 30090}, {30038, 33934}

X(41774) = X(1)X(87) ∩ X(1654)X(25895)

Barycentrics    a^3*b + 3*a^2*b^2 - a*b^3 + a^3*c - 7*a^2*b*c + a*b^2*c + b^3*c + 3*a^2*c^2 + a*b*c^2 - b^2*c^2 - a*c^3 + b*c^3 : :

X(41774) lies on these lines: {1, 87}, {1654, 25895}, {3765, 18743}, {6692, 17367}, {14759, 40533}

X(41775) = X(8)X(192) ∩ X(194)X(3987)

Barycentrics    a^2*b^2 - 2*a*b^3 - 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 - 3*b^2*c^2 - 2*a*c^3 : :

X(41775) lies on these lines: {8, 192}, {194, 3987}, {330, 24174}, {3061, 40598}, {3662, 14951}, {3752, 16816}, {3959, 21219}, {4051, 30998}, {4699, 17090}, {16284, 17232}, {17230, 18743}, {17379, 25975}, {20955, 30090}, {24440, 32005}

X(41776) = X(2)X(17090) ∩ X(10)X(192)

Barycentrics    a^3*b + 2*a*b^3 + a^3*c - a^2*b*c - 3*a*b^2*c - 3*a*b*c^2 + 3*b^2*c^2 + 2*a*c^3 : :

X(41776) lies on these lines: {2, 17090}, {10, 192}, {257, 25917}, {894, 3812}, {3752, 16815}, {16605, 17117}, {17292, 18743}, {17333, 18250}, {30090, 33944}

X(41777) = X(1)X(5575) ∩ X(6)X(57)

Barycentrics    a*(a + b - c)*(a - b + c)*(b^2 - b*c + c^2) : :

X(41777) lies on these lines: {1, 5575}, {6, 57}, {7, 192}, {56, 5018}, {63, 26657}, {65, 4334}, {77, 1429}, {85, 1921}, {86, 36538}, {87, 17063}, {141, 36482}, {226, 17234}, {241, 1122}, {279, 1432}, {553, 17378}, {604, 1443}, {982, 3056}, {984, 1463}, {986, 17114}, {1014, 16947}, {1086, 18161}, {1400, 17092}, {1442, 7225}, {2099, 13541}, {2186, 3668}, {3061, 3662}, {3340, 4321}, {3676, 21143}, {3911, 17352}, {3942, 4000}, {4346, 17452}, {4357, 25895}, {4654, 17313}, {4699, 17090}, {5219, 17265}, {6173, 18726}, {6180, 20601}, {7195, 24484}, {10436, 25975}, {16609, 27343}, {17074, 36570}, {17447, 24341}, {17861, 24237}, {18168, 24445}, {19604, 21446}, {20090, 21454}, {20248, 26997}, {24635, 28351}, {27502, 33891}

X(41777) = X(7185)-Ceva conjugate of X(982)
X(41777) = X(2275)-cross conjugate of X(982)
X(41777) = cevapoint of X(2275) and X(7248)
X(41777) = X(i)-isoconjugate of X(j) for these (i,j): {9, 983}, {41, 7033}, {55, 17743}, {200, 7132}, {663, 4621}, {1334, 40415}, {2321, 38813}, {3239, 8685}, {3407, 4517}, {3790, 18898}, {4435, 8684}, {7034, 9447}
X(41777) = barycentric product X(i)*X(j) for these {i,j}: {1, 7185}, {7, 982}, {56, 33930}, {57, 3662}, {65, 33947}, {75, 7248}, {81, 16888}, {85, 2275}, {269, 3705}, {273, 3784}, {279, 3061}, {479, 4073}, {552, 7237}, {651, 3776}, {664, 3777}, {934, 3810}, {1014, 2887}, {1088, 3056}, {1214, 31917}, {1412, 20234}, {1414, 3801}, {1432, 7187}, {1434, 3721}, {3668, 3794}, {3669, 33946}, {3676, 3888}, {3863, 7196}, {3865, 7176}, {6063, 7032}, {7153, 33890}, {7184, 7249}, {7209, 20284}, {7239, 17096}, {7305, 41291}
X(41777) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 7033}, {56, 983}, {57, 17743}, {651, 4621}, {982, 8}, {1014, 40415}, {1407, 7132}, {1408, 38813}, {1434, 38810}, {2275, 9}, {2887, 3701}, {3056, 200}, {3061, 346}, {3116, 4517}, {3662, 312}, {3705, 341}, {3721, 2321}, {3776, 4391}, {3777, 522}, {3778, 210}, {3784, 78}, {3794, 1043}, {3801, 4086}, {3808, 3716}, {3810, 4397}, {3865, 4451}, {3888, 3699}, {4073, 5423}, {4787, 3711}, {6063, 7034}, {7032, 55}, {7184, 7081}, {7185, 75}, {7186, 4420}, {7187, 17787}, {7203, 7255}, {7237, 6057}, {7239, 30730}, {7248, 1}, {7341, 7305}, {12836, 4073}, {16584, 1334}, {16888, 321}, {18904, 3985}, {18905, 4095}, {20234, 30713}, {20284, 3208}, {20665, 220}, {20684, 4515}, {20727, 3694}, {20753, 1260}, {31917, 31623}, {33890, 4110}, {33891, 3975}, {33930, 3596}, {33946, 646}, {33947, 314}, {40499, 4578}
X(41777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 7146, 7201}, {57, 269, 7175}, {77, 28017, 1429}, {85, 7209, 7205}, {241, 1122, 1423}, {1418, 24471, 57}, {3662, 7185, 16888}

X(41778) = X(192)X(20356) ∩ X(3061)X(3662)

Barycentrics    (b^2 - b*c + c^2)*(2*a^4*b^2 - a^3*b^3 - a^4*b*c - 2*a^3*b^2*c + a^2*b^3*c + 2*a^4*c^2 - 2*a^3*b*c^2 + 5*a^2*b^2*c^2 - 2*a*b^3*c^2 - a^3*c^3 + a^2*b*c^3 - 2*a*b^2*c^3 + b^3*c^3) : :

X(41778) lies on these lines: {192, 20356}, {3061, 3662}, {7777, 29641}

X(41779) = X(8)X(192) ∩ X(333)X(3752)

Barycentrics    a^2*b^3 + a*b^4 + 3*a^3*b*c - a^2*b^2*c - a*b^3*c - a^2*b*c^2 - a*b^2*c^2 + b^3*c^2 + a^2*c^3 - a*b*c^3 + b^2*c^3 + a*c^4 : :

X(41779) lies on these lines: {8, 192}, {333, 3752}, {385, 614}, {894, 3732}, {1211, 17228}, {1722, 17739}, {7202, 27111}, {9311, 25895}, {15985, 30090}, {16609, 27343}, {25121, 31090}, {25898, 25975}, {31233, 35466}

X(41780) = X(1)X(87) ∩ X(3752)X(26746)

Barycentrics    2*a^3*b^2 + a^2*b^3 - a*b^4 - 2*a^3*b*c - 2*a^2*b^2*c + b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + a*b^2*c^2 + a^2*c^3 - a*c^4 + b*c^4 : :

X(41780) lies on these lines: {1, 87}, {3752, 26746}, {5718, 18040}, {7777, 29641}, {9311, 25895}

X(41781) = X(2)X(17090) ∩ X(8)X(192)

Barycentrics    a^3*b - 2*a^2*b^2 + 3*a*b^3 + a^3*c + a^2*b*c - 3*a*b^2*c - b^3*c - 2*a^2*c^2 - 3*a*b*c^2 + 4*b^2*c^2 + 3*a*c^3 - b*c^3 : :

X(41781) lies on these lines: {2, 17090}, {8, 192}, {3212, 26685}, {3662, 41006}, {5795, 17333}, {16816, 38000}, {18743, 20955}, {30090, 33930}

X(41782) = X(192)X(1100) ∩ X(6384)X(18743)

Barycentrics    2*a^4*b^3 - 2*a^2*b^5 - 2*a^4*b^2*c + 7*a^3*b^3*c + a^2*b^4*c - 2*a^4*b*c^2 - 2*a^3*b^2*c^2 + a^2*b^3*c^2 - 2*a*b^4*c^2 + 2*b^5*c^2 + 2*a^4*c^3 + 7*a^3*b*c^3 + a^2*b^2*c^3 - 5*a*b^3*c^3 + b^4*c^3 + a^2*b*c^4 - 2*a*b^2*c^4 + b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 : :

X(41782) lies on these lines: {192, 1100}, {6384, 18743}, {18144, 29982}, {30056, 30090}

X(41783) = X(190)X(3752) ∩ X(192)X(8055)

Barycentrics    2*a^4*b^2 + 3*a^3*b^3 - a*b^5 + a^4*b*c - 8*a^3*b^2*c - 5*a^2*b^3*c + 4*a*b^4*c + 2*a^4*c^2 - 8*a^3*b*c^2 + 27*a^2*b^2*c^2 - 9*a*b^3*c^2 + b^4*c^2 + 3*a^3*c^3 - 5*a^2*b*c^3 - 9*a*b^2*c^3 + 2*b^3*c^3 + 4*a*b*c^4 + b^2*c^4 - a*c^5 : :

X(41783) lies on these lines: {190, 3752}, {192, 8055}, {17243, 17786}

X(41784) = X(2)X(17090) ∩ X(42)X(192)

Barycentrics    2*a^3*b^3 - a^2*b^4 - a^3*b^2*c - a^3*b*c^2 + 3*a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 + 2*a^3*c^3 - a*b^2*c^3 - b^3*c^3 - a^2*c^4 + b^2*c^4 : :

X(41784) lies on these lines: {2, 17090}, {42, 192}, {3752, 16728}, {18743, 20945}, {22230, 31004}

leftri

Points on the cubic K1189: X(41785)-X(41791)

rightri

Points X(41785)-X(41791) are contributed by Peter Moses, March 9, 2021.

X(41785) = X(1)X(2)∩X(4)X(673)

Barycentrics    (a^2 - 2*a*b + b^2 - 2*a*c + c^2)*(a^2 + b^2 - 2*b*c + c^2) : :

X(41785) lies on the cubic K1189 and these lines: {1, 2}, {4, 673}, {6, 21258}, {7, 169}, {56, 26007}, {85, 277}, {218, 4904}, {344, 3991}, {346, 33937}, {348, 20269}, {377, 24596}, {388, 17682}, {496, 17675}, {497, 17671}, {948, 24781}, {1111, 30694}, {1445, 7719}, {1446, 37800}, {1724, 26961}, {2082, 17170}, {2280, 26101}, {2348, 30617}, {2550, 33838}, {2551, 17681}, {3194, 15149}, {3304, 31203}, {3434, 33839}, {3672, 16600}, {3673, 4000}, {3732, 7195}, {3759, 28753}, {3772, 27539}, {3812, 38186}, {3813, 30825}, {3875, 21096}, {3913, 16593}, {4209, 4293}, {4295, 27000}, {4429, 32956}, {4515, 17279}, {4911, 5819}, {5358, 14953}, {5526, 20111}, {6706, 16605}, {6921, 24582}, {7713, 24590}, {8074, 8732}, {9436, 16572}, {9798, 11349}, {9965, 20602}, {10900, 26065}, {16502, 17060}, {17062, 26036}, {17366, 21049}, {17753, 18785}, {24181, 40719}, {27129, 30305}, {28739, 37681}

X(41785) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 17170}, {658, 31605}, {21453, 4319}
X(41785) = X(i)-isoconjugate of X(j) for these (i,j): {277, 7084}, {2191, 7123}
X(41785) = crosspoint of X(344) and X(17093)
X(41785) = crosssum of X(657) and X(14935)
X(41785) = barycentric product X(i)*X(j) for these {i,j}: {344, 4000}, {497, 6604}, {2082, 21609}, {3673, 3870}, {3732, 4468}, {3991, 16750}, {4233, 20235}, {6554, 17093}
X(41785) = barycentric quotient X(i)/X(j) for these {i,j}: {218, 7123}, {344, 30701}, {497, 6601}, {614, 2191}, {1445, 7131}, {1617, 1037}, {1633, 1292}, {3732, 37206}, {4000, 277}, {6604, 8817}, {7195, 40154}, {17093, 30705}, {21059, 7084}, {28017, 17107}, {41610, 40403}
X(41785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 145, 28740}, {2, 10528, 28742}, {2, 10529, 28734}, {2, 27304, 19843}, {218, 4904, 6604}, {4000, 6554, 3673}, {28813, 31189, 2}

X(41786) = X(1)X(4)∩X(85)X(16706)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 + b^2 - 2*b*c + c^2)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c + a*c^2 - b*c^2 - c^3) : :

X(41786) lies on the cubic K1189 and these lines: {1, 4}, {85, 16706}, {221, 39063}, {2082, 4000}, {18743, 28736}, {24584, 41245}, {26528, 26533}, {28962, 28967}

X(41786) = X(85)-Ceva conjugate of X(7195)
X(41786) = X(15487)-cross conjugate of X(11677)
X(41786) = crosssum of X(3063) and X(14935)
X(41786) = barycentric product X(i)*X(j) for these {i,j}: {7, 11677}, {85, 15487}, {278, 28409}, {3673, 8270}, {4000, 28739}, {7195, 10327}
X(41786) = barycentric quotient X(i)/X(j) for these {i,j}: {7195, 39732}, {11677, 8}, {15487, 9}, {28017, 40188}, {28409, 345}, {28739, 30701}

X(41787) = X(4)X(8)∩X(277)X(279)

Barycentrics    b*c*(a^2 + b^2 - 2*b*c + c^2)*(-3*a^3 + 3*a^2*b - a*b^2 + b^3 + 3*a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(41787) lies on the cubic K1189 and these lines: {4, 8}, {277, 279}, {2082, 3673}, {2262, 17753}, {8074, 40702}

X(41787) = X(85)-Ceva conjugate of X(3673)
X(41787) = barycentric product X(i)*X(j) for these {i,j}: {3673, 17784}, {3732, 25009}

X(41788) = X(4)X(39732)∩X(8)X(13577)

Barycentrics    b*c*(a^2 + b^2 - 2*b*c + c^2)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c + a*c^2 + b*c^2 - c^3)*(-a^3 + a^2*b - a*b^2 + b^3 + a^2*c - b^2*c + a*c^2 + b*c^2 - c^3) : :

X(41788) lies on the cubic K1189 and these lines: {4, 39732}, {8, 13577}, {85, 169}, {304, 17233}, {3433, 5088}, {11677, 17170}

X(41788) =X(i)-cross conjugate of X(j) for these (i,j): {514, 3732}, {2082, 3673}
X(41788) =X(i)-isoconjugate of X(j) for these (i,j): {169, 7084}, {1037, 5452}, {1486, 7123}
X(41788) =barycentric product X(3673)*X(13577)
X(41788) =barycentric quotient X(i)/X(j) for these {i,j}: {614, 1486}, {2082, 5452}, {3433, 7084}, {3673, 3434}, {3914, 21867}, {4000, 169}, {7195, 34036}, {7289, 22131}

X(41789) = X(1)X(142)∩X(4)X(2809)

Barycentrics    (a - b - c)*(a^2 + b^2 - 2*b*c + c^2)*(a^4 - 2*a^3*b + 2*a^2*b^2 - 2*a*b^3 + b^4 - 2*a^3*c + 4*a^2*b*c - 2*b^3*c + 2*a^2*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :

X(41789) lies on the cubic K1189 and these lines: {1, 142}, {4, 2809}, {85, 3434}, {120, 3913}, {149, 30694}, {497, 2082}, {528, 9305}, {962, 3827}, {11677, 17170}, {14256, 36845}

X(41789) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 6554}, {3434, 11677}
X(41789) = barycentric product X(497)*X(28740)
X(41789) = barycentric quotient X(i)/X(j) for these {i,j}: {8271, 7131}, {28740, 8817}

X(41790) = X(4)X(18725)∩X(40)X(518)

Barycentrics    a*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c + 3*a*c^2 + 3*b*c^2 - 3*c^3)*(a^3 - a^2*b + 3*a*b^2 - 3*b^3 - a^2*c - 2*a*b*c + 3*b^2*c - a*c^2 - b*c^2 + c^3) : :

X(41790) lies on the cubic K1189 and these lines: {4, 18725}, {40, 518}, {169, 21446}, {196, 5236}, {223, 241}, {269, 7719}, {329, 3912}, {1422, 40397}, {1817, 18206}, {3942, 7079}, {8074, 8732}, {17107, 38375}

X(41790) = X(i)-cross conjugate of X(j) for these (i,j): {1864, 7}, {2082, 1}
X(41790) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17784}, {692, 25009}
X(41790) = cevapoint of X(i) and X(j) for these (i,j): {513, 38375}, {650, 3942}
X(41790) = trilinear pole of line {2254, 6129}
X(41790) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17784}, {514, 25009}

X(41791) = X(2)X(169)∩X(7)X(20613)

Barycentrics    (a - b - c)*(a^3 + a^2*b + a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(a^3 - a^2*b + a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c + a*c^2 - b*c^2 + c^3) : :

X(41785) lies on the cubic K1189 and these lines: {2, 169}, {7, 20613}, {8, 3100}, {85, 16706}, {92, 3673}, {189, 5222}, {239, 2994}, {312, 27540}, {377, 1220}, {607, 26932}, {1311, 3086}, {1760, 28739}, {1952, 3212}, {2988, 5773}, {4518, 5552}, {4872, 28736}, {4997, 28793}, {5224, 32008}, {6557, 27539}, {17489, 18359}, {17492, 17743}, {24597, 34234}, {28740, 31637}

X(41791) = isotomic conjugate of X(28739)
X(41791) = X(i)-cross conjugate of X(j) for these (i,j): {33, 7}, {2082, 8}, {17880, 4560}
X(41791) = X(i)-isoconjugate of X(j) for these (i,j): {3, 20613}, {6, 8270}, {31, 28739}, {56, 17742}, {57, 12329}, {109, 2509}, {222, 23050}, {604, 10327}, {1037, 15487}, {1801, 1880}
X(41791) = cevapoint of X(i) and X(j) for these (i,j): {6, 10829}, {650, 26932}
X(41791) = trilinear pole of line {522, 11934}
X(41791) = barycentric product X(i)*X(j) for these {i,j}: {8, 39732}, {312, 40188}, {333, 36907}
X(41791) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8270}, {2, 28739}, {8, 10327}, {9, 17742}, {19, 20613}, {33, 23050}, {55, 12329}, {283, 1801}, {497, 11677}, {650, 2509}, {2082, 15487}, {4560, 17498}, {27509, 28409}, {36907, 226}, {39732, 7}, {40184, 2285}, {40188, 57}
X(41791) = {X(36907),X(40188)}-harmonic conjugate of X(39732)

X(41792) = X(2)X(9311) ∩ X(57)X(239)

Barycentrics    a^3*b + 2*a^2*b^2 - 3*a*b^3 + a^3*c - 9*a^2*b*c + 5*a*b^2*c + 3*b^3*c + 2*a^2*c^2 + 5*a*b*c^2 - 6*b^2*c^2 - 3*a*c^3 + 3*b*c^3 : :

X(41792) lies on these lines: {2, 9311}, {57, 239}, {312, 10405}, {514, 25242}, {518, 1278}, {2098, 14942}, {3061, 3119}, {3177, 21272}, {4051, 17090}, {14951, 27538}, {16610, 40133}

X(41793) = X(2)X(9311) ∩ X(312)X(1909)

Barycentrics    a^3*b + 2*a^2*b^2 - a*b^3 + a^3*c - 6*a^2*b*c + a*b^2*c + 2*b^3*c + 2*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + 2*b*c^3 : :

X(41793) lies on these lines: {2, 9311}, {312, 1909}, {1278, 3623}, {3057, 7187}, {3666, 27499}, {4009, 21219}, {14951, 24003}, {16604, 16610}, {17014, 17490}, {18157, 20892}, {29584, 33296}

X(41794) = X(2)X(37) ∩ X(239)X(2348)

Barycentrics    a^3*b + a*b^3 + a^3*c - 4*a^2*b*c + a*b^2*c - 2*b^3*c + a*b*c^2 + 2*b^2*c^2 + a*c^3 - 2*b*c^3 : :

X(41794) lies on these lines: {2, 37}, {239, 2348}, {3057, 33890}, {3663, 26590}, {3684, 3875}, {3689, 32926}, {4119, 4357}, {4440, 4872}, {4514, 6646}, {10025, 32029}, {17116, 32942}, {17158, 21216}, {17247, 32773}

X(41795) = X(1)X(9367) ∩ X(2)X(2898)

Barycentrics    a*(a - b - c)*(a^4 - 4*a^2*b^2 + 4*a*b^3 - b^4 + 6*a^2*b*c - 4*a*b^2*c - 2*b^3*c - 4*a^2*c^2 - 4*a*b*c^2 + 6*b^2*c^2 + 4*a*c^3 - 2*b*c^3 - c^4) : :

X(41795) lies on these lines: {1, 9367}, {2, 2898}, {9, 165}, {63, 3119}, {100, 28070}, {200, 14943}, {281, 6353}, {282, 7015}, {650, 1040}, {1375, 16832}, {2884, 16593}, {3305, 32578}, {3848, 34522}, {5268, 16588}, {6181, 17594}, {6692, 41006}, {13609, 24703}, {17435, 18193}

X(41796) = X(2)X(3119) ∩ X(9)X(165)

Barycentrics    a*(a - b - c)*(a^2*b^2 - 2*a*b^3 + b^4 - 3*a^2*b*c + 2*a*b^2*c + b^3*c + a^2*c^2 + 2*a*b*c^2 - 4*b^2*c^2 - 2*a*c^3 + b*c^3 + c^4) : :

X(41796) lies on these lines: {2, 3119}, {9, 165}, {43, 20310}, {200, 28053}, {281, 4213}, {650, 24430}, {846, 6181}, {1146, 3816}, {2886, 13609}, {3061, 17284}, {3239, 17860}, {4051, 24386}, {4521, 25128}, {10589, 38375}, {11680, 33573}, {16588, 25061}, {17063, 17435}, {17784, 28123}, {24341, 35508}, {27065, 32578}, {27541, 33299}

X(41797) = X(2)X(4551) ∩ X(5)X(10)

Barycentrics    (a - b - c)*(a^3*b^2 - a*b^4 + 2*a^3*b*c - a*b^3*c - b^4*c + a^3*c^2 + 4*a*b^2*c^2 + b^3*c^2 - a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :

X(41797) lies on these lines: {2, 4551}, {5, 10}, {9, 4418}, {124, 3925}, {522, 7069}, {756, 4858}, {958, 16414}, {1125, 5399}, {1698, 1745}, {3634, 34831}, {3842, 16579}, {3902, 6735}, {4023, 4113}, {4521, 25627}, {5745, 16056}, {8679, 40687}, {17197, 23638}, {18236, 31993}, {20262, 40606}, {30827, 31330}

X(41798) = X(2)X(664) ∩ X(9)X(100)

Barycentrics    a*(a - b - c)*(a^2 - 2*a*b + b^2 + a*c + b*c - 2*c^2)*(a^2 + a*b - 2*b^2 - 2*a*c + b*c + c^2) : :

X(41798) lies on these lines: {2, 664}, {9, 100}, {88, 17435}, {149, 13609}, {200, 644}, {281, 1897}, {282, 1998}, {346, 3699}, {527, 15727}, {643, 2287}, {1280, 35348}, {1320, 23893}, {1639, 14942}, {3218, 25954}, {3219, 15855}, {3306, 21446}, {3689, 28071}, {3870, 19605}, {3935, 6603}, {4358, 36796}, {4614, 27174}, {5011, 36002}, {5199, 6745}, {6559, 17780}, {6742, 7110}, {8580, 24036}, {10025, 35157}, {10707, 33573}, {14733, 37763}, {15726, 15731}, {20905, 31618}, {26540, 30705}, {34068, 36147}, {36910, 40869}

X(41798) = isogonal conjugate of X(6610)
X(41798) = isotomic conjugate of X(37780)
X(41798) = polar conjugate of X(38461)
X(41798) = trilinear pole of line X(9)X(3900) (the line through X(9) parallel to the trilinear polar of X(9))

X(41799) = X(1)X(164) ∩ X(2)X(174)

Barycentrics    Sin[A]*(Sin[A/2] - Sin[B/2])*(Sin[A/2] - Sin[C/2])*(Sin[B] - Sin[C]) : :

X(41799) lies on these lines: {1, 164}, {2, 174}, {57, 289}, {173, 8056}, {259, 20183}, {277, 8729}, {361, 12445}, {2091, 14596}, {5935, 8080}, {6553, 12646}, {6724, 18258}, {7371, 19296}, {7707, 16015}, {10490, 16011}, {16018, 39694}, {32017, 40893}

X(41800) = X(2)X(525) ∩ X(241)X(514)

Barycentrics    (b - c)*(-2*a^3 - a^2*b + 2*a*b^2 + b^3 - a^2*c + 2*a*b*c - b^2*c + 2*a*c^2 - b*c^2 + c^3) : :

X(41800) lies on these lines: {2, 525}, {241, 514}, {523, 24920}, {656, 11125}, {676, 1734}, {900, 21189}, {1019, 2487}, {1577, 17069}, {1639, 23875}, {2401, 39962}, {3700, 21192}, {3835, 28493}, {4041, 34958}, {4129, 4897}, {4142, 25380}, {4773, 29270}, {4777, 21186}, {4789, 26146}, {4823, 4976}, {4904, 24791}, {4927, 29302}, {6004, 26275}, {6008, 28590}, {16287, 22089}, {16451, 39228}, {16453, 39201}, {17899, 19804}, {17924, 26023}, {21124, 24924}, {21173, 28209}, {21195, 28867}, {24178, 29264}, {28473, 30574}, {28588, 28898}

X(41801) = X(2)X(222) ∩ X(7)X(528)

Barycentrics    (2*a - b - c)*(a + b - c)*(a - b + c)*(a^2 - b^2 + b*c - c^2) : :

X(41801) lies on these lines: {2, 222}, {7, 528}, {77, 17274}, {85, 14584}, {241, 4715}, {320, 1443}, {524, 34253}, {545, 4552}, {752, 1458}, {1099, 1111}, {1275, 18821}, {1366, 1447}, {1442, 17320}, {2398, 10427}, {3000, 28854}, {3738, 4453}, {3870, 35338}, {3945, 24203}, {4872, 38385}, {6180, 17313}, {6610, 31138}, {7232, 17075}, {9312, 36595}, {14020, 37523}, {17077, 17330}, {17092, 17364}, {17271, 40999}, {17281, 28968}, {17310, 40862}, {25718, 36588}, {30379, 41140}, {39771, 39775}

X(41802) = X(2)X(45) ∩ X(333)X(16723)

Barycentrics    7*a^3 - 6*a^2*b - 9*a*b^2 + 4*b^3 - 6*a^2*c + 21*a*b*c - 3*b^2*c - 9*a*c^2 - 3*b*c^2 + 4*c^3 : :

X(41802) lies on these lines: {2, 45}, {333, 16723}, {664, 3911}, {1644, 3699}, {3241, 5854}, {3828, 10713}, {4453, 14425}, {4595, 17310}, {4604, 37222}, {17191, 41629}, {17277, 25057}, {24281, 35124}, {30720, 32851}

X(41803) = X(1)X(36595) ∩ X(2)X(2006)

Barycentrics    (a + b - c)*(a - b + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 5*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 - c^3) : :

X(41803) lies on these lines: {1, 36595}, {2, 2006}, {7, 528}, {75, 1272}, {279, 24858}, {347, 34625}, {519, 22464}, {545, 651}, {1111, 3672}, {1266, 1443}, {1278, 17075}, {1441, 17320}, {1465, 36914}, {2804, 4453}, {3160, 36588}, {3582, 24209}, {4318, 28580}, {4370, 5723}, {5736, 15956}, {5932, 36640}, {17147, 18625}, {17160, 39751}, {17281, 28780}, {17950, 40891}, {27754, 37695}, {37787, 41140}

X(41804) = X(1)X(15936) ∩ X(2)X(7359)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 + b*c - c^2) : :

X(41804) lies on these lines: {1, 15936}, {2, 7359}, {6, 17075}, {7, 21}, {9, 25584}, {10, 307}, {30, 18661}, {57, 21376}, {75, 3260}, {85, 17250}, {145, 347}, {190, 28757}, {219, 40905}, {226, 30588}, {241, 3834}, {320, 1443}, {333, 18625}, {379, 24316}, {517, 3007}, {522, 693}, {527, 26006}, {651, 17950}, {857, 4466}, {1210, 23521}, {1367, 3319}, {1375, 14543}, {1400, 16888}, {1439, 4018}, {1737, 17895}, {3012, 28849}, {3218, 14920}, {3580, 18668}, {3662, 17092}, {3663, 3953}, {3868, 39791}, {3911, 24183}, {3936, 18593}, {3943, 4552}, {4329, 6361}, {4346, 14986}, {4357, 24564}, {4440, 40892}, {4854, 10580}, {4890, 11025}, {5740, 17861}, {5905, 6349}, {6357, 16704}, {10436, 27187}, {11064, 39767}, {11553, 17320}, {16099, 37311}, {17077, 34824}, {17134, 18481}, {17160, 39751}, {17220, 22791}, {17274, 19861}, {18525, 21270}, {18635, 25255}, {18655, 41010}, {24315, 24581}, {24541, 24999}, {25529, 37797}, {29614, 41246}, {37787, 40622}

X(41804) = isotomic conjugate of X(6740)
X(41804) = anticomplement of X(7359)

X(41805) = X(1)X(75) ∩ X(76)X(25591)

Barycentrics    a^3*b + a^2*b^2 + a^3*c - a^2*b*c + b^3*c + a^2*c^2 - b^2*c^2 + b*c^3 : :

X(41805) lies on these lines: {1, 75}, {76, 25591}, {83, 9317}, {213, 7187}, {758, 33947}, {978, 33934}, {995, 33930}, {1193, 20924}, {1201, 33940}, {2292, 16712}, {3216, 20955}, {7200, 17499}, {16600, 33946}, {16609, 31225}, {16711, 17164}, {16720, 40859}, {17048, 27195}, {17103, 17104}, {17946, 39724}, {18061, 26978}, {18145, 25079}, {24882, 38941}

X(41806) = X(2)X(6) ∩ X(190)X(33133)

Barycentrics    3*a^3 - a*b^2 + 2*b^3 + a*b*c - b^2*c - a*c^2 - b*c^2 + 2*c^3 : :

X(41806) lies on these lines: {2, 6}, {190, 33133}, {226, 25529}, {257, 29630}, {404, 40980}, {1010, 6693}, {1043, 24883}, {1762, 3306}, {1834, 25459}, {3011, 33121}, {3616, 21677}, {3699, 33118}, {3758, 31266}, {3769, 29857}, {3772, 32939}, {3996, 33137}, {4257, 17678}, {4360, 33113}, {4438, 29658}, {5263, 24892}, {5745, 19786}, {6679, 32942}, {6682, 29859}, {8258, 24161}, {11110, 25441}, {11683, 16706}, {16609, 31225}, {17160, 33168}, {17356, 27002}, {18698, 19804}, {19270, 20083}, {24899, 33954}, {24902, 25526}, {26228, 30614}, {27003, 27191}, {29665, 33114}, {29683, 33115}, {29856, 32916}, {29863, 32917}, {29865, 32919}, {29867, 32918}, {31189, 31203}, {31215, 31224}, {32851, 40940}, {32922, 33119}, {33116, 34064}

X(41807) = X(7)X(21) ∩ X(10)X(664)

Barycentrics    (a + b - c)*(a - b + c)*(3*a^2 - a*b - 2*b^2 - a*c - b*c - 2*c^2) : :

X(41807) lies on these lines: {7, 21}, {10, 664}, {77, 17250}, {78, 17360}, {85, 3624}, {140, 18329}, {145, 25592}, {190, 28969}, {241, 29614}, {551, 32007}, {1125, 17078}, {1323, 19878}, {3160, 19877}, {3635, 25723}, {3636, 9436}, {4668, 25716}, {4892, 40723}, {5088, 9955}, {5550, 17079}, {16609, 31225}, {17044, 32008}, {17086, 34824}, {17181, 18481}

X(41808) = X(2)X(7110) ∩ X(7)X(21)

Barycentrics    (a + b - c)*(a - b + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :

X(41808) lies on these lines: {2, 7110}, {7, 21}, {75, 1272}, {77, 997}, {150, 17221}, {241, 17384}, {307, 1442}, {347, 18698}, {651, 17332}, {664, 32025}, {934, 6356}, {1214, 18632}, {1441, 17095}, {1443, 4357}, {2071, 4329}, {2893, 17136}, {3160, 5932}, {3582, 24208}, {3664, 36589}, {4452, 10529}, {4460, 36845}, {4554, 34388}, {4657, 17092}, {5235, 6357}, {5273, 6349}, {5936, 25583}, {6224, 21276}, {6516, 7279}, {7269, 9436}, {12383, 41004}, {16609, 24909}, {17073, 24635}, {17077, 17086}, {17078, 17322}, {17134, 17181}, {17279, 28780}, {20930, 27529}

X(41809) = X(2)X(6) ∩ X(8)X(4205)

Barycentrics    (b + c)*(a^2 + 2*a*b + b^2 + 2*a*c + b*c + c^2) : :

X(41809) lies on these lines: {2, 6}, {8, 4205}, {9, 21376}, {10, 321}, {37, 3969}, {75, 1230}, {92, 860}, {100, 4199}, {169, 857}, {226, 40999}, {306, 5257}, {319, 17019}, {329, 442}, {594, 3995}, {661, 10196}, {740, 6536}, {984, 20966}, {1010, 26064}, {1046, 1698}, {1125, 4938}, {1255, 6542}, {1268, 17484}, {1330, 14005}, {1577, 21198}, {1621, 4204}, {1834, 3617}, {1962, 21085}, {2092, 17248}, {2245, 3219}, {2610, 30565}, {3120, 27798}, {3136, 33108}, {3187, 17275}, {3266, 16739}, {3616, 17514}, {3661, 16589}, {3666, 4708}, {3681, 40952}, {3720, 3775}, {3739, 17184}, {3740, 26251}, {3765, 31339}, {3770, 30599}, {3828, 31177}, {3842, 15523}, {3846, 30970}, {3876, 10974}, {3963, 40603}, {4026, 4651}, {4042, 29829}, {4046, 27804}, {4062, 8040}, {4129, 23825}, {4272, 4886}, {4357, 4359}, {4364, 17147}, {4384, 32774}, {4415, 31025}, {4418, 24697}, {4419, 19825}, {4467, 21209}, {4683, 24342}, {4687, 32858}, {4690, 37595}, {4699, 33146}, {4733, 4854}, {4751, 27186}, {4967, 4980}, {5249, 17746}, {5260, 37225}, {5287, 17270}, {5296, 17776}, {5657, 30444}, {5810, 37151}, {5813, 21530}, {5816, 19645}, {5949, 26792}, {6537, 27064}, {6539, 30582}, {6554, 26605}, {6626, 19308}, {6627, 6651}, {7308, 17052}, {8300, 24542}, {10436, 32859}, {14007, 26131}, {16568, 17289}, {16603, 28387}, {16777, 20017}, {16815, 26724}, {16830, 33075}, {17250, 19804}, {17257, 19822}, {17258, 19797}, {17260, 33157}, {17273, 26842}, {17303, 26223}, {17397, 20970}, {17557, 25650}, {18098, 27067}, {18202, 21810}, {19281, 26085}, {19767, 37039}, {19856, 32772}, {21024, 29593}, {24175, 24589}, {24988, 32781}, {25001, 25013}, {25015, 30807}, {25383, 30566}, {26035, 37096}, {26037, 32784}, {26580, 31993}, {26601, 27040}, {26747, 28244}, {27025, 27038}, {27184, 29576}, {27476, 31322}, {27541, 31043}, {27942, 40468}, {28653, 33066}, {31330, 33141}, {32025, 34064}, {33069, 40328}, {33072, 36531}

X(41809) = complement of X(8025)

X(41810) = X(3)X(86) ∩ X(20)X(2895)

Barycentrics    a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6 + 4*a^6*c + 11*a^5*b*c + 3*a^4*b^2*c - 8*a^3*b^3*c - 6*a^2*b^4*c - 3*a*b^5*c - b^6*c + 5*a^5*c^2 + 3*a^4*b*c^2 - 6*a^3*b^2*c^2 - 2*a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 - 8*a^3*b*c^3 - 2*a^2*b^2*c^3 + 6*a*b^3*c^3 + 2*b^4*c^3 - 5*a^3*c^4 - 6*a^2*b*c^4 + a*b^2*c^4 + 2*b^3*c^4 - 4*a^2*c^5 - 3*a*b*c^5 - b^2*c^5 - a*c^6 - b*c^6 : :

X(41810) lies on these lines: {3, 86}, {20, 2895}, {40, 4647}, {500, 4229}, {946, 37508}, {2792, 3430}, {3579, 7413}, {3651, 5758}, {4220, 6361}, {6998, 12699}, {8822, 41014}, {22080, 25516}, {36754, 37416}

X(41811) = X(3)X(4647) ∩ X(55)X(86)

Barycentrics    a*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4 + 2*a^4*c + 3*a^3*b*c - 3*a^2*b^2*c - 2*a*b^3*c - 3*a^2*b*c^2 + 2*b^3*c^2 - 2*a^2*c^3 - 2*a*b*c^3 + 2*b^2*c^3 - a*c^4) : :

X(41811) lies on these lines: {3, 4647}, {55, 86}, {100, 2895}, {199, 4436}, {1030, 4418}, {1444, 4046}, {1817, 3712}, {3474, 14450}, {4387, 11350}, {5124, 32860}, {5695, 11340}, {19297, 32936}

X(41812) = X(1)X(75) ∩ X(2)X(191)

Barycentrics    a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + 3*a^3*c + 7*a^2*b*c + 7*a*b^2*c + 2*b^3*c + 3*a^2*c^2 + 7*a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3 : :

X(41812) lies on these lines: {1, 75}, {2, 191}, {3, 16124}, {10, 2895}, {79, 4205}, {142, 19881}, {758, 14005}, {846, 3624}, {894, 16828}, {946, 2941}, {1046, 1698}, {1125, 4418}, {1203, 3739}, {1224, 4472}, {1781, 5750}, {1962, 28620}, {2049, 5902}, {2938, 9589}, {2939, 38052}, {2948, 4197}, {3175, 25431}, {3337, 19863}, {3647, 17557}, {3743, 5333}, {4427, 5550}, {4658, 21020}, {4798, 24335}, {5131, 19270}, {5426, 11115}, {5692, 16458}, {6763, 19858}, {10180, 28618}, {11684, 17551}, {15015, 19284}, {15668, 27785}, {16118, 26117}, {16126, 17589}, {17514, 17768}, {19273, 37524}, {19276, 37571}, {19684, 28612}, {19871, 41229}, {21081, 37635}, {31993, 37559}, {37150, 37702}

X(41813) = X(1)X(75) ∩ X(8)X(4205)

Barycentrics    a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + 3*a^3*c + 7*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(41813) lies on these lines: {1, 75}, {8, 4205}, {10, 34064}, {21, 27804}, {30, 944}, {58, 4065}, {145, 2895}, {191, 41629}, {239, 6051}, {333, 3743}, {474, 4734}, {594, 19865}, {664, 3671}, {1126, 3159}, {1220, 2901}, {1255, 19874}, {1330, 4854}, {1621, 37322}, {1962, 11110}, {1999, 3931}, {3178, 33135}, {3244, 38456}, {3295, 37327}, {3702, 17011}, {3993, 5247}, {3996, 30142}, {4393, 16466}, {4442, 26131}, {4664, 41229}, {4970, 37607}, {6155, 26244}, {11321, 27480}, {13740, 32915}, {14005, 17163}, {14007, 21020}, {15569, 16817}, {16777, 19853}, {16834, 31435}, {16912, 31308}, {17233, 19784}, {17277, 27785}, {17380, 19836}, {17551, 27812}, {17557, 27811}, {17592, 17733}, {19870, 25431}, {21081, 30832}, {32932, 37594}

X(41814) = X(8)X(79) ∩ X(10)X(86)

Barycentrics    a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 2*a^3*c + a^2*b*c - 5*a*b^2*c - 3*b^3*c - 5*a*b*c^2 - 4*b^2*c^2 - 2*a*c^3 - 3*b*c^3 - c^4 : :

X(41814) lies on these lines: {8, 79}, {10, 86}, {58, 21085}, {69, 28612}, {191, 3578}, {316, 4385}, {333, 21081}, {1125, 4886}, {1654, 3743}, {2891, 3881}, {3632, 11533}, {3931, 4690}, {4042, 30172}, {4065, 24697}, {5814, 35104}, {8013, 25526}, {11024, 32099}, {11281, 41014}, {28611, 32863}

X(41815) = X(1)X(75) ∩ X(519)X(27790)

Barycentrics    2*a^4 + 6*a^3*b + 6*a^2*b^2 + 2*a*b^3 + 6*a^3*c + 14*a^2*b*c + 8*a*b^2*c + b^3*c + 6*a^2*c^2 + 8*a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 + b*c^3 : :

X(41815) lies on these lines: {1, 75}, {519, 27790}, {1089, 17019}, {1100, 25512}, {1125, 4938}, {1224, 6542}, {2895, 3616}, {3622, 20077}, {3636, 12579}, {3647, 27811}, {3743, 8025}, {6533, 17011}, {14450, 15677}, {16828, 28639}, {17379, 27785}, {19308, 27787}, {19717, 27784}, {19863, 37869}, {27368, 28620}, {29597, 41229}

X(41816) = X(2)X(6) ∩ X(8)X(4854)

Barycentrics    a^3 - 3*a*b^2 - 2*b^3 - 5*a*b*c - 3*b^2*c - 3*a*c^2 - 3*b*c^2 - 2*c^3 : :

X(41816) lies on these lines: {2, 6}, {8, 4854}, {10, 33066}, {63, 17328}, {257, 3175}, {306, 17256}, {312, 17270}, {319, 34064}, {321, 32025}, {527, 19797}, {540, 1010}, {1043, 26064}, {1762, 31153}, {1999, 4690}, {3210, 17253}, {3305, 17228}, {3666, 17252}, {3679, 4385}, {3686, 19786}, {3699, 4104}, {3775, 32942}, {4046, 9791}, {4357, 4886}, {4359, 17273}, {4407, 32866}, {4416, 19808}, {4643, 32939}, {4654, 16609}, {4683, 8013}, {5256, 17250}, {5287, 17360}, {6175, 11236}, {9534, 11359}, {11683, 17781}, {15526, 25908}, {16858, 40980}, {17239, 27064}, {17272, 19804}, {17275, 27184}, {17285, 27065}, {17331, 32777}, {17347, 19822}, {17484, 32101}, {21085, 24697}, {25280, 30713}

X(41817) = X(2)X(6) ∩ X(10)X(34064)

Barycentrics    a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3 + 4*a^2*c + 11*a*b*c + 5*b^2*c + 5*a*c^2 + 5*b*c^2 + 2*c^3 : :

X(41817) lies on these lines: {2, 6}, {10, 34064}, {312, 1698}, {321, 1268}, {1125, 4886}, {1656, 9535}, {1761, 3305}, {2160, 35595}, {3704, 9780}, {4914, 16830}, {5257, 19808}, {8055, 19877}, {14450, 18228}, {17019, 32025}, {17248, 32939}, {19786, 24603}, {19856, 32942}, {33124, 39580}

X(41818) = X(2)X(6) ∩ X(846)X(3624)

Barycentrics    (2*a + b + c)*(a^2 + 2*a*b + b^2 + 2*a*c + 3*b*c + c^2) : :

X(41818) lies on these lines: {2, 6}, {846, 3624}, {896, 19862}, {1125, 1962}, {1230, 16709}, {1255, 28604}, {3650, 6675}, {3969, 16826}, {3995, 4472}, {4697, 8040}, {4798, 26223}, {5564, 27790}, {5625, 8013}, {6693, 34595}, {8298, 24988}, {17019, 28653}, {17397, 36812}, {19804, 30598}, {19883, 24177}, {24583, 30007}, {24931, 28618}, {25660, 30599}, {28606, 29612}, {28651, 29617}, {29578, 33157}

X(41819) = X(1)X(5180) ∩ X(2)X(6)

Barycentrics    3*a^3 + 5*a^2*b + a*b^2 - b^3 + 5*a^2*c + 5*a*b*c + b^2*c + a*c^2 + b*c^2 - c^3 : :

X(41819) lies on these lines: {1, 5180}, {2, 6}, {20, 5453}, {58, 15676}, {144, 16585}, {145, 4647}, {1100, 33150}, {1449, 27186}, {3146, 13408}, {3219, 4667}, {3475, 14683}, {3622, 20077}, {3664, 17011}, {3672, 17482}, {3743, 31888}, {3957, 4349}, {4038, 33107}, {4340, 37256}, {4656, 17019}, {4658, 26131}, {4683, 5625}, {5287, 26792}, {7277, 33761}, {7960, 20214}, {11036, 31293}, {16666, 26724}, {16884, 33146}, {17012, 24175}, {17013, 40612}, {17018, 20095}, {17391, 26223}, {17394, 32859}, {18593, 21454}, {19785, 21907}, {20043, 30712}, {20078, 25080}, {25256, 29585}, {26064, 28619}, {29588, 40908}, {33112, 33141}, {33173, 33682}

X(41820) = X(2)X(594) ∩ X(3)X(962)

Barycentrics    (2*a + b + c)*(a^2 + 2*a*b + b^2 + 2*a*c + b*c + c^2) : :

X(41820) lies on these lines: {2, 594}, {3, 962}, {39, 17397}, {81, 6626}, {86, 26842}, {551, 17469}, {1100, 3578}, {1125, 1962}, {2482, 35076}, {3666, 26747}, {3782, 19740}, {3970, 17023}, {4021, 4980}, {4026, 28599}, {4272, 4886}, {4364, 19717}, {4657, 18139}, {4974, 8040}, {4988, 6546}, {5278, 26626}, {5333, 17302}, {5905, 17321}, {6292, 16826}, {6509, 26635}, {8025, 30581}, {10180, 24542}, {15808, 24177}, {15819, 29634}, {17147, 17398}, {17184, 37869}, {17257, 19738}, {17327, 20017}, {17332, 19743}, {17392, 30562}, {17399, 27186}, {17400, 32858}, {25507, 33150}, {26223, 41312}, {29570, 33172}, {29614, 33157}

X(41821) = X(2)X(594) ∩ X(8)X(2891)

Barycentrics    a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 5*a*b*c - 5*b^2*c - a*c^2 - 5*b*c^2 - c^3 : :

X(41821) lies on these lines: {2, 594}, {8, 2891}, {75, 2895}, {81, 4399}, {149, 17163}, {319, 26842}, {321, 18151}, {329, 14213}, {1655, 17147}, {3219, 5540}, {3419, 20243}, {3696, 20344}, {4359, 5564}, {4371, 19822}, {4431, 27065}, {4647, 5180}, {4651, 17794}, {4665, 32911}, {4671, 8055}, {4699, 20017}, {4886, 4980}, {4967, 17011}, {5271, 33168}, {5361, 7560}, {8025, 20016}, {9263, 39747}, {17117, 33150}, {17119, 32782}, {17362, 20086}, {19825, 37685}, {20879, 37781}, {21020, 32842}, {21085, 33148}, {25525, 33077}, {26840, 39348}, {28606, 28634}

X(41822) = X(2)X(31321) ∩ X(8)X(2891)

Barycentrics    a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 2*a^3*c + a^2*b*c - 3*a*b^2*c - 3*b^3*c - 3*a*b*c^2 - 4*b^2*c^2 - 2*a*c^3 - 3*b*c^3 - c^4 : :

X(41822) lies on these lines: {2, 31321}, {8, 2891}, {10, 86}, {79, 20290}, {191, 3969}, {1051, 1698}, {1089, 2895}, {1224, 8025}, {2901, 33082}, {3416, 9052}, {3634, 4886}, {3679, 27790}, {3743, 6542}, {3931, 17372}, {4066, 33066}, {4385, 7768}, {4445, 5711}, {4553, 34790}, {4647, 27786}, {5086, 36974}, {5815, 21270}, {10449, 21278}, {10479, 32852}, {12514, 17294}, {21081, 40605}

X(41823) = X(2)X(594) ∩ X(333)X(4852)

Barycentrics    3*a^3 + 6*a^2*b + 3*a*b^2 + 6*a^2*c + a*b*c - 3*b^2*c + 3*a*c^2 - 3*b*c^2 : :

X(41823) lies on these lines: {2, 594}, {333, 4852}, {553, 664}, {1051, 28516}, {1126, 6534}, {3189, 3241}, {3699, 32928}, {3929, 16834}, {4021, 4886}, {4035, 19829}, {4393, 32005}, {4740, 19722}, {4980, 17011}, {17117, 25507}, {17305, 20017}, {29584, 33296}, {29588, 40688}

X(41824) = X(7)X(3436) ∩ X(8)X(57)

Barycentrics    (a + b - c)*(a - b + c)*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 14*a^2*b^2*c + b^4*c + 2*a^3*c^2 - 14*a^2*b*c^2 + 6*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 - 3*a*c^4 + b*c^4 + c^5) : :

X(41824) lies on these lines: {7, 3436}, {8, 57}, {65, 497}, {100, 4308}, {226, 8165}, {388, 3698}, {1420, 5281}, {1466, 5731}, {1767, 4198}, {1788, 4413}, {1837, 31391}, {3339, 4292}, {3340, 10580}, {3893, 6764}, {3911, 18231}, {4295, 12736}, {4313, 37541}, {4321, 6736}, {5221, 9803}, {5253, 5435}, {5554, 21454}, {5768, 37544}, {5807, 14524}, {5812, 31794}, {8732, 24987}, {9578, 40333}, {9785, 13601}, {10578, 34489}, {11037, 37566}, {11362, 12842}, {12541, 14923}, {12709, 18228}, {31777, 36279}

X(41825) = X(1)X(3146) ∩ X(2)X(1743)

Barycentrics    3*a^3 + 7*a^2*b + a*b^2 - 3*b^3 + 7*a^2*c + 6*a*b*c + 3*b^2*c + a*c^2 + 3*b*c^2 - 3*c^3 : :

X(41825) lies on these lines: {1, 3146}, {2, 1743}, {7, 941}, {8, 2650}, {226, 1419}, {306, 7229}, {329, 5308}, {345, 35578}, {651, 940}, {748, 5550}, {3160, 6354}, {3241, 30699}, {3475, 4344}, {3487, 13442}, {3616, 13736}, {3672, 4654}, {3677, 30340}, {3749, 4307}, {3982, 4346}, {4340, 5703}, {4644, 5273}, {4648, 18228}, {4656, 29624}, {4667, 25525}, {4675, 16602}, {4813, 6545}, {4888, 21454}, {5222, 5249}, {5296, 33066}, {5328, 37674}, {5435, 5718}, {5713, 9799}, {5717, 11036}, {5748, 37633}, {5749, 18134}, {5905, 25080}, {5984, 5988}, {10446, 32067}, {10580, 26098}, {12221, 31583}, {12222, 31582}, {14949, 29570}, {17014, 23681}, {17257, 26109}, {17316, 17760}, {17365, 28610}, {17379, 26132}, {17483, 25094}, {18625, 37631}, {18663, 25255}, {20088, 26626}, {22117, 33993}, {27064, 29627}, {29621, 30568}, {31993, 32099}, {32946, 39581}, {33112, 36845}

X(41826) = X(2)X(169) ∩ X(4)X(26563)

Barycentrics    a^4 - b^4 + 2*a*b^2*c + 2*b^3*c + 2*a*b*c^2 - 2*b^2*c^2 + 2*b*c^3 - c^4 : :

X(41826) lies on these lines: {2, 169}, {4, 26563}, {7, 8}, {29, 17183}, {78, 3674}, {81, 277}, {105, 1036}, {142, 2082}, {150, 5554}, {226, 28795}, {279, 6904}, {294, 17683}, {315, 33934}, {329, 29611}, {348, 404}, {443, 18732}, {474, 1565}, {631, 27187}, {962, 13727}, {1111, 20552}, {1376, 3665}, {1447, 10527}, {1930, 10327}, {2478, 4872}, {3086, 26229}, {3434, 3673}, {3436, 4911}, {3661, 5905}, {3662, 27000}, {3732, 33838}, {3933, 17740}, {3945, 5262}, {4190, 5088}, {4295, 20347}, {4329, 4357}, {4675, 41015}, {4847, 10521}, {5218, 25581}, {5226, 28813}, {5552, 7179}, {6554, 33839}, {6921, 17095}, {7198, 12513}, {9310, 28967}, {9800, 37201}, {10481, 28043}, {11415, 30946}, {12526, 17272}, {17238, 17481}, {17292, 31018}, {17300, 21216}, {17316, 17489}, {18228, 37774}, {18391, 21285}, {24635, 37280}, {25940, 28922}, {27025, 31080}, {28734, 40127}, {28740, 40131}

X(41827) = X(2)X(1766) ∩ X(4)X(3753)

Barycentrics    a^6 + 2*a^5*b + a^4*b^2 - a^2*b^4 - 2*a*b^5 - b^6 + 2*a^5*c + 6*a^4*b*c + 6*a^3*b^2*c + 2*a^2*b^3*c + a^4*c^2 + 6*a^3*b*c^2 + 6*a^2*b^2*c^2 + 10*a*b^3*c^2 + b^4*c^2 + 2*a^2*b*c^3 + 10*a*b^2*c^3 - a^2*c^4 + b^2*c^4 - 2*a*c^5 - c^6 : :

X(41827) lies on these lines: {2, 1766}, {4, 3753}, {7, 321}, {329, 1211}, {894, 36850}, {962, 964}, {1086, 9776}, {3434, 5807}, {3661, 5905}, {5051, 11024}, {5813, 26223}, {25003, 30444}

X(41828) = X(2)X(7) ∩ X(8)X(3781)

Barycentrics    a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + a^3*b*c - 2*a^2*b^2*c + a*b^3*c - b^4*c + a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4 : :

X(41828) lies on these lines: {2, 7}, {8, 3781}, {69, 3975}, {75, 21871}, {192, 21809}, {962, 29309}, {978, 24248}, {1469, 2551}, {1958, 37416}, {3487, 19518}, {3616, 19283}, {3739, 21866}, {3869, 20905}, {3965, 27489}, {4000, 27623}, {4334, 12527}, {4335, 40998}, {4384, 10446}, {4388, 7386}, {5175, 37191}, {5222, 17183}, {5815, 29983}, {9785, 20036}, {10327, 17153}, {17170, 29960}, {17298, 29968}, {17300, 26101}, {17753, 24199}, {21281, 30090}, {21296, 29982}, {24612, 37659}

X(41829) = X(2)X(1475) ∩ X(7)X(30966)

Barycentrics    a^4*b^2 - a^2*b^4 + 2*a^4*b*c - 6*a^2*b^3*c - 2*a*b^4*c + a^4*c^2 - 3*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 - 6*a^2*b*c^3 - 4*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - 2*a*b*c^4 - b^2*c^4 : :

X(41829) lies on these lines: {2, 1475}, {7, 30966}, {8, 17149}, {69, 26069}, {1699, 3741}, {1909, 26038}, {3786, 17082}, {3789, 7196}, {6646, 36861}, {9780, 31341}, {10453, 31008}, {20537, 33082}, {30973, 36850}

X(41830) = X(2)X(3882) ∩ X(7)X(528)

Barycentrics    3*a^4*b + 3*a^3*b^2 - 3*a^2*b^3 - 3*a*b^4 + 3*a^4*c - 10*a^3*b*c - 2*a^2*b^2*c + 8*a*b^3*c - 3*b^4*c + 3*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 + 3*b^3*c^2 - 3*a^2*c^3 + 8*a*b*c^3 + 3*b^2*c^3 - 3*a*c^4 - 3*b*c^4 : :

X(41830) lies on these lines: {2, 3882}, {7, 528}, {69, 39996}, {86, 36006}, {668, 32099}, {1266, 20039}, {3543, 10446}, {3738, 21297}, {3945, 17205}, {9785, 37038}, {17018, 35338}, {17183, 17271}, {18228, 31171}

X(41831) = X(40)X(194) ∩ X(55)X(192)

Barycentrics    a^5*b^2 - a^4*b^3 - a^3*b^4 + a^2*b^5 - 2*a^5*b*c - 3*a^4*b^2*c + a^2*b^4*c + a^5*c^2 - 3*a^4*b*c^2 + a^3*b^2*c^2 - a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 - a^4*c^3 - a^2*b^2*c^3 + 2*a*b^3*c^3 + b^4*c^3 - a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + b^3*c^4 + a^2*c^5 - b^2*c^5 : :

X(41831) lies on these lines: {40, 194}, {55, 192}, {65, 330}, {193, 3779}, {2550, 7774}, {3101, 3164}, {3212, 19591}, {3550, 25252}, {3925, 7777}, {5584, 7783}, {6253, 7823}, {7754, 10306}, {7766, 10315}, {7793, 10902}, {7837, 34612}, {9308, 11406}, {32005, 37567}, {32095, 37080}, {32107, 37568}, {36858, 39355}

X(41832) = X(2)X(9547) ∩ X(10)X(7777)

Barycentrics    a^5*b^2 + 2*a^4*b^3 - a^2*b^5 + 2*a^5*b*c + 2*a^4*b^2*c - a^3*b^3*c + a^5*c^2 + 2*a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + 2*a^4*c^3 - a^3*b*c^3 - a*b^3*c^3 - b^4*c^3 - a*b^2*c^4 - b^3*c^4 - a^2*c^5 : :

X(41832) lies on these lines: {2, 9547}, {10, 7777}, {76, 9561}, {181, 330}, {192, 1682}, {193, 4260}, {194, 970}, {385, 386}, {573, 7783}, {1916, 34454}, {6625, 34258}, {7754, 9567}, {7757, 9560}, {7774, 9534}, {9566, 31859}, {17033, 23637}, {24598, 37683}

X(41833) = X(55)X(330) ∩ X(57)X(192)

Barycentrics    a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + 6*a^4*b*c - 6*a^3*b^2*c - 4*a^2*b^3*c + a^4*c^2 - 6*a^3*b*c^2 + 5*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - 4*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 - b^2*c^4 : :

X(41833) lies on these lines: {55, 330}, {57, 192}, {165, 194}, {354, 32095}, {1155, 32107}, {6244, 31859}, {7777, 7965}, {10389, 31999}, {17594, 27340}, {17596, 25242}, {32005, 35445}

X(41834) = X(1)X(87) ∩ X(2)X(20167)

Barycentrics    a^3*b^2 + a^2*b^3 - 2*a^3*b*c + a^2*b^2*c + a^3*c^2 + a^2*b*c^2 - 3*a*b^2*c^2 - b^3*c^2 + a^2*c^3 - b^2*c^3 : :

X(41834) lies on these lines: {1, 87}, {2, 20167}, {6, 3210}, {9, 24621}, {63, 17349}, {190, 28365}, {193, 3779}, {291, 21299}, {1654, 2227}, {1740, 32937}, {3177, 11683}, {3770, 26042}, {3971, 25528}, {4440, 20348}, {4671, 17178}, {6376, 26077}, {16710, 27268}, {17000, 26274}, {17147, 20168}, {17155, 22343}, {17248, 27318}, {17277, 17595}, {17350, 27644}, {17787, 36858}, {20146, 28606}, {24310, 37652}, {25535, 39798}, {26082, 34284}, {26963, 30998}, {32922, 36635}

X(41835) = X(2)X(20167) ∩ X(63)X(192)

Barycentrics    a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - 2*a^4*b*c + 4*a^3*b^2*c + 6*a^2*b^3*c + a^4*c^2 + 4*a^3*b*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 + 2*a^3*c^3 + 6*a^2*b*c^3 - 2*b^3*c^3 + a^2*c^4 - b^2*c^4 : :

X(4135) lies on these lines: {2, 20167}, {63, 192}, {190, 940}, {194, 4384}, {330, 37870}, {333, 3210}, {894, 17022}, {17147, 24616}, {19732, 24620}, {20170, 41629}, {39703, 39980}

X(41836) = X(2)X(37) ∩ X(8)X(31999)

Barycentrics    a^2*b^2 - 2*a^2*b*c - 4*a*b^2*c + a^2*c^2 - 4*a*b*c^2 - b^2*c^2 : :

X(41836) lies on these lines: {2, 37}, {8, 31999}, {10, 330}, {42, 24661}, {43, 17379}, {145, 28600}, {194, 1698}, {385, 4413}, {894, 16569}, {1125, 32095}, {1376, 17000}, {1500, 39738}, {1654, 26038}, {1655, 19877}, {1909, 40598}, {3616, 3795}, {3634, 27269}, {3925, 7777}, {4426, 33062}, {5224, 25350}, {7774, 26040}, {9342, 16997}, {9780, 32005}, {16710, 30966}, {16816, 37686}, {16912, 31448}, {16991, 24988}, {17005, 31245}, {17319, 25502}, {17343, 24691}, {17349, 17754}, {17350, 37673}, {17373, 30962}, {17398, 20168}, {20146, 21857}, {20532, 29593}, {20943, 25109}, {21904, 37677}, {24621, 29610}, {25121, 26037}, {31276, 32092}

X(41837) = X(2)X(20935) ∩ X(55)X(3177)

Barycentrics    a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 2*a^4*b*c + 4*a^3*b^2*c - 2*a^2*b^3*c + a^4*c^2 + 4*a^3*b*c^2 - a^2*b^2*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 + 2*b^3*c^3 + a^2*c^4 - b^2*c^4 : :

X(41837) lies on these lines: {2, 20935}, {55, 3177}, {57, 239}, {192, 3870}, {193, 3779}, {385, 26241}, {3210, 32029}, {4430, 9443}, {9311, 20359}, {20533, 20537}, {27340, 28043}

X(41838) = X(1)X(20464) ∩ X(2)X(330)

Barycentrics    a^2*b^2 + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 - b^2*c^2 : :

X(41838) lies on these lines: {1, 20464}, {2, 330}, {8, 192}, {9, 17752}, {10, 194}, {37, 24524}, {56, 16999}, {75, 20081}, {76, 1573}, {145, 32095}, {172, 16995}, {183, 31490}, {321, 21608}, {385, 958}, {668, 5283}, {993, 7793}, {1078, 31456}, {1329, 7777}, {1376, 7783}, {1574, 7757}, {1698, 27318}, {1914, 16914}, {1921, 4699}, {2276, 25280}, {2295, 17350}, {2551, 7774}, {2975, 16997}, {3061, 17316}, {3177, 3436}, {3208, 17261}, {3248, 24662}, {3314, 26558}, {3329, 26687}, {3552, 4386}, {3616, 9263}, {3617, 17759}, {3662, 30030}, {3679, 25264}, {3691, 17033}, {3760, 16829}, {3761, 16819}, {3780, 4393}, {3995, 20055}, {4400, 16996}, {4426, 7766}, {4474, 21225}, {4489, 4734}, {4517, 20535}, {4518, 27340}, {4671, 18050}, {4687, 24656}, {4741, 17137}, {4999, 17004}, {5260, 16998}, {5275, 6645}, {5739, 6542}, {5839, 20168}, {6381, 17030}, {6646, 21281}, {6653, 9598}, {7187, 30758}, {7226, 20352}, {7754, 9708}, {7786, 27076}, {7824, 31449}, {7864, 26582}, {7933, 20541}, {9597, 17565}, {9709, 31859}, {9780, 32005}, {10459, 24514}, {11185, 31416}, {11285, 31468}, {16502, 16918}, {16552, 30114}, {16705, 16722}, {16706, 24735}, {16738, 28660}, {16921, 31466}, {16924, 31405}, {16975, 18140}, {17008, 30478}, {17128, 20172}, {17148, 31339}, {17232, 29966}, {17280, 27523}, {17300, 26101}, {17362, 20170}, {17379, 23579}, {17383, 26965}, {17448, 30963}, {17480, 39581}, {17499, 30116}, {18055, 21332}, {18135, 26801}, {20553, 33824}, {20943, 21264}, {21384, 41240}, {22343, 40720}, {24215, 30063}, {24621, 29576}, {25298, 28606}, {26752, 40598}, {27184, 30059}, {27538, 40790}, {27891, 34021}, {29593, 31036}, {29595, 31061}, {30036, 31004}, {31298, 39738}, {34063, 37673}, {37652, 41233}, {39713, 39729}

X(41838) = anticomplement of X(31997)

X(41839) = X(1)X(979) ∩ X(2)X(37)

Barycentrics    a^2*b + a*b^2 + a^2*c + 3*a*b*c - b^2*c + a*c^2 - b*c^2 : :

X(41839) lies on these lines: {1, 979}, {2, 37}, {6, 34064}, {8, 756}, {9, 1999}, {42, 27538}, {43, 3993}, {45, 333}, {55, 26264}, {63, 17261}, {81, 17350}, {145, 960}, {190, 940}, {194, 16826}, {210, 20012}, {239, 3305}, {244, 26103}, {306, 17242}, {329, 1655}, {330, 1255}, {341, 37548}, {392, 20037}, {452, 20009}, {561, 18135}, {612, 3685}, {726, 26102}, {748, 32928}, {750, 32936}, {846, 29649}, {894, 5287}, {899, 4734}, {968, 7081}, {975, 7283}, {980, 32026}, {982, 30947}, {984, 10453}, {1001, 32926}, {1089, 27785}, {1125, 4135}, {1150, 33761}, {1211, 17233}, {1442, 28997}, {1500, 30830}, {1961, 3923}, {1962, 4903}, {2176, 4393}, {2667, 25294}, {2895, 17373}, {2901, 9534}, {3007, 6360}, {3120, 29854}, {3161, 26065}, {3187, 17349}, {3212, 28387}, {3219, 37683}, {3240, 3725}, {3247, 21101}, {3452, 4029}, {3485, 7211}, {3616, 3994}, {3617, 3714}, {3622, 17480}, {3661, 21071}, {3683, 3769}, {3687, 3950}, {3700, 25258}, {3720, 24349}, {3729, 17022}, {3731, 11679}, {3735, 22171}, {3745, 4676}, {3758, 37595}, {3782, 17234}, {3836, 33154}, {3846, 33092}, {3873, 31302}, {3875, 7308}, {3876, 20018}, {3886, 7322}, {3891, 5284}, {3912, 4656}, {3930, 18228}, {3932, 32773}, {3943, 5743}, {3944, 29653}, {3952, 17018}, {3967, 15569}, {3989, 30942}, {4001, 17333}, {4009, 37593}, {4038, 32935}, {4044, 27255}, {4059, 25237}, {4078, 24210}, {4360, 4383}, {4365, 26037}, {4385, 6051}, {4387, 5263}, {4398, 40688}, {4415, 17243}, {4417, 31056}, {4419, 18141}, {4423, 32922}, {4425, 29674}, {4429, 4854}, {4432, 17716}, {4439, 33169}, {4552, 5226}, {4560, 4931}, {4641, 17336}, {4651, 9330}, {4679, 33071}, {4703, 32846}, {4712, 10580}, {4741, 32863}, {4851, 33066}, {4871, 17591}, {4885, 25271}, {4886, 17299}, {4970, 16569}, {5047, 19851}, {5205, 17594}, {5211, 26105}, {5222, 28598}, {5249, 17244}, {5250, 41261}, {5256, 17319}, {5268, 32932}, {5271, 17260}, {5294, 17339}, {5296, 26044}, {5297, 32929}, {5308, 25242}, {5311, 32930}, {5333, 29595}, {5422, 23135}, {5698, 20101}, {5737, 16675}, {5739, 6542}, {5748, 18662}, {5905, 17300}, {6374, 34088}, {6382, 18140}, {6541, 32778}, {6703, 17340}, {7028, 16018}, {7226, 29824}, {8610, 26746}, {9345, 32940}, {9780, 21020}, {9791, 26034}, {9965, 20073}, {10436, 25430}, {14555, 17314}, {14973, 20011}, {16484, 32920}, {16831, 24621}, {17012, 26688}, {17019, 17379}, {17032, 19565}, {17122, 32934}, {17123, 32921}, {17124, 32845}, {17125, 32924}, {17155, 30950}, {17165, 29814}, {17184, 17232}, {17230, 32782}, {17236, 33172}, {17257, 34255}, {17262, 32939}, {17277, 20170}, {17318, 37679}, {17338, 26723}, {17364, 17781}, {17375, 32859}, {17486, 31052}, {17489, 29611}, {17742, 41251}, {17777, 26098}, {18139, 29572}, {18163, 29531}, {18359, 27290}, {19587, 27481}, {19743, 25417}, {20017, 37656}, {20075, 40635}, {21024, 29593}, {21226, 29585}, {21416, 33942}, {21438, 27193}, {21611, 26854}, {24165, 25502}, {24627, 30567}, {24703, 33073}, {25098, 27139}, {25101, 40940}, {25249, 27475}, {25268, 26070}, {25269, 32933}, {25960, 32848}, {25961, 33145}, {26038, 32860}, {26083, 29647}, {26093, 37592}, {26150, 29677}, {26243, 30661}, {26580, 32858}, {26792, 31034}, {27158, 27351}, {27288, 29616}, {27318, 29612}, {27340, 29624}, {27659, 41240}, {27789, 35058}, {27809, 39967}, {29575, 31164}, {29635, 33164}, {29642, 33152}, {29687, 32776}, {29829, 33166}, {29830, 33153}, {29837, 33163}, {29845, 33161}, {29851, 33143}, {31161, 38314}, {32943, 36534}

X(41839) = anticomplement of X(19804)

X(41840) = X(2)X(330) ∩ X(42)X(192)

Barycentrics    a^3*b^3 - a^3*b^2*c + a^2*b^3*c - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 + a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 - b^3*c^3 : :

X(41840) lies on these lines: {2, 330}, {42, 192}, {43, 194}, {144, 17759}, {310, 16722}, {329, 17493}, {350, 24528}, {899, 32005}, {1654, 2227}, {2229, 40598}, {3210, 33888}, {3240, 32107}, {3720, 31999}, {3952, 17486}, {4090, 32453}, {4440, 39741}, {4699, 34086}, {8026, 21884}, {10327, 17485}, {17018, 32095}, {17232, 29976}, {17350, 30661}, {17379, 27663}, {17490, 27484}, {18152, 30998}, {19565, 27538}, {20350, 26032}, {20533, 20537}, {21877, 25287}, {25121, 26037}, {26974, 31000}, {39703, 39925}

X(41840) = polar conjugate of isogonal conjugate of X(23177)
X(41840) = anticomplement of X(6384)

X(41841) = X(2)X(21093) ∩ X(37)X(4473)

Barycentrics    (3*a^2 + a*b - b^2 + a*c - 3*b*c - c^2)*(a^2 + a*b - b^2 + a*c - b*c - c^2) : :

X(41841) lies on these lines: {2, 21093}, {37, 4473}, {190, 1213}, {335, 31310}, {1757, 6541}, {3161, 4393}, {4366, 4370}, {4440, 28653}, {5852, 16593}, {6329, 32029}, {15481, 20533}, {17755, 28516}

X(41842) = X(2)X(846) ∩ X(7)X(192)

Barycentrics    a^4 + a^3*b - 2*a^2*b^2 - a*b^3 - b^4 + a^3*c - a^2*b*c + a*b^2*c + b^3*c - 2*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 + b*c^3 - c^4 : :

X(41842) lies on these lines: {2, 846}, {7, 192}, {8, 6653}, {190, 2345}, {194, 33867}, {239, 516}, {329, 36861}, {528, 32029}, {543, 24281}, {545, 599}, {673, 5698}, {726, 6542}, {1086, 15668}, {1284, 21495}, {2475, 25248}, {2549, 24282}, {3011, 35292}, {3661, 3729}, {3663, 16826}, {3797, 4645}, {3912, 28526}, {3977, 30763}, {4080, 31020}, {4353, 29580}, {4366, 26626}, {4473, 5296}, {4655, 27474}, {4766, 32845}, {5211, 31348}, {6999, 29057}, {7783, 17084}, {8424, 21511}, {17132, 17310}, {17304, 29612}, {17355, 29610}, {17493, 40873}, {17755, 24715}, {17766, 20016}, {18794, 25805}, {19645, 29243}, {20679, 29593}, {24631, 33095}, {25264, 33865}, {25270, 33823}, {26590, 32939}, {26601, 27707}, {26840, 31027}, {28550, 29590}, {28562, 40891}, {30225, 35103}, {31023, 33168}, {31041, 32849}, {31310, 31329}, {31347, 39581}

X(41842) = anticomplement of X(17738)

X(41843) = X(2)X(846) ∩ X(37)X(4473)

Barycentrics    3*a^4 + 2*a^3*b - a^2*b^2 + b^4 + 2*a^3*c - 6*a^2*b*c - 4*a*b^2*c + 2*b^3*c - a^2*c^2 - 4*a*b*c^2 + 3*b^2*c^2 + 2*b*c^3 + c^4 : :

X(41843) lies on these lines: {2, 846}, {37, 4473}, {190, 17398}, {238, 6542}, {335, 29592}, {594, 4366}, {597, 32029}, {3797, 28484}, {4393, 26685}, {4432, 6653}, {4440, 17322}, {6541, 20016}, {6646, 24358}, {15254, 20533}, {16020, 27494}, {17264, 28329}, {20132, 29569}, {31310, 31319}

X(41844) = X(2)X(21093) ∩ X(75)X(1654)

Barycentrics    a^4 + 2*a^3*b - 3*a^2*b^2 - 4*a*b^3 - b^4 + 2*a^3*c + 2*a^2*b*c + 2*b^3*c - 3*a^2*c^2 + 5*b^2*c^2 - 4*a*c^3 + 2*b*c^3 - c^4 : :

X(41844) lies on these lines: {2, 21093}, {75, 1654}, {190, 17398}, {335, 29569}, {527, 40891}, {537, 6653}, {545, 4366}, {726, 6542}, {894, 29586}, {3629, 32029}, {4393, 31300}, {6651, 29592}, {17147, 21220}, {17487, 41312}, {17770, 20016}

X(41845) = X(2)X(11) ∩ X(7)X(4393)

Barycentrics    3*a^4 - a^3*b - a*b^3 - b^4 - a^3*c - 5*a^2*b*c + 3*a*b^2*c - b^3*c + 3*a*b*c^2 + 4*b^2*c^2 - a*c^3 - b*c^3 - c^4 : :

X(41845) lies on these lines: {2, 11}, {7, 4393}, {8, 17738}, {142, 29586}, {144, 1278}, {145, 335}, {190, 391}, {192, 5819}, {193, 4440}, {239, 516}, {518, 20016}, {527, 40891}, {1086, 3945}, {3056, 25722}, {4294, 27304}, {4307, 17014}, {4312, 16834}, {4441, 18037}, {5011, 20373}, {5082, 17691}, {5223, 29617}, {5542, 29584}, {5698, 27484}, {5838, 17350}, {5853, 6542}, {6009, 20092}, {7677, 19308}, {8236, 29570}, {10386, 17687}, {16816, 30332}, {16826, 30331}, {17300, 20162}, {17349, 41325}, {17397, 38052}, {21454, 35312}, {24599, 32096}, {26839, 34772}, {29592, 38316}, {29609, 38204}, {31329, 31352}, {35514, 37416}, {36101, 39351}

X(41845) = anticomplement of X(20533)

X(41846) = X(2)X(2245) ∩ X(7)X(1319)

Barycentrics    3*a^4*b + 3*a^3*b^2 - 3*a^2*b^3 - 3*a*b^4 + 3*a^4*c + a^3*b*c + 2*a^2*b^2*c + a*b^3*c - 3*b^4*c + 3*a^3*c^2 + 2*a^2*b*c^2 + 2*a*b^2*c^2 + 3*b^3*c^2 - 3*a^2*c^3 + a*b*c^3 + 3*b^2*c^3 - 3*a*c^4 - 3*b*c^4 : :

X(41846) lies on these lines: {2, 2245}, {7, 1319}, {30, 10446}, {86, 16370}, {320, 1836}, {903, 14260}, {3838, 17250}, {4389, 39542}, {4715, 5905}, {10468, 17274}, {16590, 31018}, {17271, 17532}, {17484, 17488}

X(41847) = X(1)X(75) ∩ X(2)X(44)

Barycentrics    2*a^2 + 2*a*b + 2*a*c + 3*b*c : :

X(41847) lies on these lines: {1, 75}, {2, 44}, {6, 4751}, {7, 5550}, {10, 17360}, {37, 24063}, {45, 894}, {63, 25507}, {69, 3844}, {85, 1443}, {88, 26627}, {141, 29608}, {142, 17370}, {190, 16676}, {192, 28639}, {312, 17021}, {319, 3617}, {524, 29576}, {536, 29570}, {551, 1266}, {594, 17386}, {597, 29628}, {599, 29610}, {679, 27922}, {899, 37632}, {903, 4432}, {1086, 17397}, {1100, 4699}, {1125, 4389}, {1213, 17328}, {1268, 17270}, {1278, 3723}, {1698, 17271}, {2345, 17240}, {3210, 37869}, {3579, 10446}, {3589, 27147}, {3616, 17320}, {3621, 5564}, {3624, 4672}, {3625, 4967}, {3626, 3879}, {3634, 3664}, {3644, 16777}, {3661, 4472}, {3662, 17398}, {3663, 15808}, {3720, 4479}, {3739, 3759}, {3943, 10022}, {4346, 7321}, {4357, 4896}, {4359, 17013}, {4363, 4664}, {4364, 29612}, {4393, 4688}, {4406, 4794}, {4410, 31060}, {4422, 29581}, {4440, 29592}, {4470, 17316}, {4648, 17241}, {4657, 26806}, {4659, 29597}, {4665, 17389}, {4667, 17346}, {4697, 17196}, {4698, 17350}, {4702, 38314}, {4753, 19875}, {4758, 17023}, {4759, 19883}, {4764, 17118}, {4772, 4852}, {4850, 19740}, {4851, 28604}, {4888, 17273}, {5232, 28650}, {5249, 19812}, {5257, 17347}, {5297, 30758}, {5308, 17264}, {5749, 17263}, {5750, 17234}, {6173, 17305}, {6707, 17248}, {7222, 28641}, {7227, 17242}, {7228, 17247}, {7232, 17326}, {7263, 17396}, {7277, 17331}, {8822, 28627}, {16670, 17277}, {16694, 23407}, {16700, 28606}, {16884, 17117}, {17012, 19684}, {17022, 20942}, {17028, 24512}, {17119, 29584}, {17120, 17259}, {17215, 20954}, {17228, 17300}, {17232, 17385}, {17233, 29601}, {17236, 25498}, {17238, 17376}, {17239, 17375}, {17244, 17342}, {17245, 17341}, {17269, 29575}, {17272, 19872}, {17275, 20090}, {17276, 28640}, {17281, 29569}, {17288, 17327}, {17290, 29614}, {17292, 17313}, {17293, 17312}, {17297, 17308}, {17298, 17307}, {17301, 29586}, {17318, 29580}, {17325, 29609}, {17348, 37677}, {17349, 31238}, {17351, 27268}, {17354, 29571}, {17359, 29572}, {17367, 34824}, {17374, 29593}, {17380, 24199}, {17595, 19701}, {17786, 29388}, {18073, 20917}, {18134, 19827}, {18137, 20150}, {18735, 24315}, {22279, 25279}, {24487, 30950}, {26626, 37756}, {27191, 29598}, {28626, 39707}, {29599, 41310}, {33682, 40328}, {37129, 39044}, {40434, 41242}

X(41848) = X(2)X(44) ∩ X(75)X(28301)

Barycentrics    4*a^2 - 10*a*b - 2*b^2 - 10*a*c - 7*b*c - 2*c^2 : :

X(41848) lies on these lines: {2, 44}, {75, 28301}, {903, 16832}, {966, 17240}, {3828, 17354}, {4370, 29576}, {4664, 28309}, {4687, 17330}, {4751, 7228}, {5296, 17320}, {7229, 17336}, {16815, 24441}, {16834, 31332}, {17241, 17271}, {17249, 17259}, {17260, 17293}, {17313, 17328}, {17394, 37654}, {19875, 41138}

X(41849) = X(2)X(21024) ∩ X(3)X(86)

Barycentrics    (2*a^2 + a*b + a*c + b*c)*(a*b + b^2 + a*c + b*c + c^2) : :

X(41849) lies on these lines: {2, 21024}, {3, 86}, {6, 6626}, {39, 17381}, {75, 1125}, {114, 7380}, {386, 5224}, {978, 17322}, {986, 15903}, {1403, 17084}, {1909, 30964}, {2276, 27642}, {3061, 5745}, {3616, 8299}, {3622, 21281}, {3666, 16604}, {6292, 17234}, {6376, 6685}, {7379, 7710}, {16060, 20159}, {16342, 33295}, {16705, 37632}, {16887, 17378}, {17030, 17380}, {17103, 19684}, {17233, 27274}, {17346, 20970}, {17354, 25092}, {17379, 17689}, {17398, 27318}, {17400, 29991}, {17592, 17762}, {19701, 40605}, {19765, 33954}, {19767, 30966}, {21010, 40418}, {23905, 33046}, {25506, 27020}, {29612, 31993}, {30022, 30963}

X(41850) = X(2)X(319) ∩ X(72)X(354)

Barycentrics    4*a^3 + 7*a^2*b + 5*a*b^2 + 2*b^3 + 7*a^2*c + 10*a*b*c + 4*b^2*c + 5*a*c^2 + 4*b*c^2 + 2*c^3 : :

X(41850) lies on these lines: {2, 319}, {72, 354}, {81, 25498}, {333, 29609}, {662, 5333}, {940, 29603}, {1255, 17359}, {1746, 11230}, {3175, 5750}, {3616, 32777}, {3624, 4038}, {3666, 16604}, {4641, 17322}, {4670, 17483}, {4708, 37685}, {4798, 19785}, {6707, 26723}, {7522, 19701}, {8025, 17237}, {17019, 17385}, {17023, 36812}, {17398, 31993}, {17598, 25055}, {19808, 29586}, {19832, 26109}, {24614, 29614}, {27195, 37870}

X(41850) = complement of complement of X(25417)

X(41851) = X(2)X(319) ∩ X(6)X(31028)

Barycentrics    a^3*b - a*b^3 + a^3*c + 3*a^2*b*c - b^3*c - b^2*c^2 - a*c^3 - b*c^3 : :

X(41851) lies on these lines: {2, 319}, {6, 31028}, {7, 4479}, {8, 28600}, {43, 17377}, {69, 26069}, {75, 354}, {86, 3741}, {192, 24691}, {238, 17731}, {239, 24663}, {320, 350}, {325, 4966}, {518, 20947}, {672, 17264}, {942, 17762}, {1125, 33297}, {1246, 6384}, {1575, 6542}, {2238, 30967}, {2276, 17315}, {3555, 33938}, {3720, 17322}, {3840, 3879}, {3873, 33931}, {3874, 33939}, {3881, 33941}, {3912, 37686}, {4360, 17598}, {4441, 7321}, {4713, 17364}, {5224, 26102}, {5259, 34016}, {5564, 17135}, {10449, 31997}, {16706, 17027}, {17026, 17234}, {17149, 18147}, {17233, 17754}, {17258, 24690}, {17270, 25502}, {17289, 24512}, {17297, 20335}, {17300, 21264}, {17319, 25349}, {17352, 30822}, {17360, 30947}, {17361, 30946}, {17363, 37673}, {17374, 20530}, {17375, 30998}, {17378, 31137}, {17388, 25350}, {17751, 25303}, {18398, 33935}, {19786, 30965}, {20943, 36854}, {26103, 32099}, {28653, 31330}, {30942, 37632}, {32919, 33295}, {33296, 35633}, {33954, 37607}

X(41852) = X(8)X(3543) ∩ X(9)X(1125)

Barycentrics    (a + 2*b + 2*c)*(5*a^3 + a^2*b - 5*a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - 5*a*c^2 + b*c^2 - c^3) : :

X(41852) lies on these lines: {8, 3543}, {9, 1125}, {10, 28647}, {79, 2093}, {144, 31730}, {200, 3650}, {1145, 12526}, {1698, 3715}, {3057, 3633}, {3059, 40263}, {3951, 5587}, {5220, 28645}, {5223, 12699}, {5234, 16137}, {5815, 16127}, {5905, 11024}, {6068, 12738}, {12527, 36922}, {16008, 41691}, {16855, 18217}

X(41853) = X(1)X(37426) ∩ X(3)X(1699)

Barycentrics    a*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c - 3*a^3*b*c + 2*a^2*b^2*c + a*b^3*c + 2*b^4*c + 2*a^2*b*c^2 - 2*b^3*c^2 + 2*a^2*c^3 + a*b*c^3 - 2*b^2*c^3 - a*c^4 + 2*b*c^4) : :

X(41853) lies on these lines: {1, 37426}, {3, 1699}, {9, 165}, {10, 33557}, {20, 993}, {30, 3925}, {35, 79}, {36, 376}, {40, 912}, {46, 10382}, {55, 4312}, {63, 5696}, {72, 16143}, {100, 10032}, {170, 2947}, {191, 210}, {411, 6700}, {484, 41539}, {495, 34618}, {516, 1621}, {548, 37561}, {550, 10943}, {920, 37572}, {946, 35202}, {956, 34628}, {971, 7964}, {1004, 4512}, {1006, 28150}, {1478, 37427}, {1541, 36012}, {1698, 37411}, {1742, 1754}, {1764, 38485}, {1768, 5918}, {1839, 4219}, {2077, 37713}, {3149, 16192}, {3475, 3746}, {3534, 18499}, {3583, 37428}, {3587, 5692}, {3841, 37433}, {3901, 12702}, {3970, 18788}, {4018, 7991}, {4220, 41430}, {4297, 11015}, {4316, 37429}, {4863, 5288}, {4880, 37584}, {5248, 37105}, {5258, 18519}, {5450, 17538}, {5536, 10167}, {5537, 5905}, {5584, 5691}, {5658, 6796}, {5732, 16465}, {6899, 7741}, {6912, 28158}, {6985, 7308}, {7489, 28154}, {7676, 13405}, {7992, 11500}, {8167, 30308}, {8273, 11522}, {9342, 10164}, {9441, 17745}, {9579, 37601}, {9580, 37578}, {9943, 14054}, {10157, 31663}, {11551, 28194}, {13369, 24468}, {13615, 38052}, {15338, 37583}, {15690, 38602}, {15696, 26286}, {16132, 37585}, {18228, 25440}, {18483, 25542}, {19541, 38318}, {24309, 37400}, {25439, 34632}, {28174, 34486}, {31142, 35238}, {31262, 37356}, {35977, 40998}

X(41854) = X(1)X(30) ∩ X(3)X(9)

Barycentrics    a*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c - 6*a^4*b*c + 4*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 + 4*a^2*b*c^3 - 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6) : :

Let AB, AC, BC, BA, CA, CB be the points on the Conway conic, as at X(40589). Let A' = CBAB∩ACBC, B' = ACBC∩BACA, C' = BACA∩CBAB. Triangle A'B'C' is homothetic to the hexyl triangle at X(41854). (Randy Hutson, April 13, 2021)

X(41854) lies on these lines: {1, 30}, {3, 9}, {4, 5249}, {5, 1750}, {8, 37427}, {10, 18528}, {20, 5758}, {33, 4303}, {40, 912}, {57, 6985}, {63, 3651}, {72, 3587}, {77, 6198}, {78, 376}, {140, 10857}, {142, 6849}, {165, 17857}, {200, 3579}, {223, 1062}, {226, 6851}, {355, 30503}, {381, 3824}, {382, 37615}, {405, 18540}, {411, 11220}, {515, 6850}, {517, 12565}, {527, 3811}, {550, 6282}, {581, 990}, {582, 1743}, {908, 6899}, {942, 10382}, {943, 8545}, {944, 36846}, {1038, 3465}, {1040, 1745}, {1066, 4319}, {1071, 1998}, {1467, 5722}, {1657, 37533}, {1706, 18518}, {1708, 41562}, {1709, 10902}, {1721, 37529}, {1898, 37578}, {2771, 12526}, {2801, 12511}, {2900, 16117}, {2951, 5762}, {3146, 18444}, {3149, 10167}, {3157, 7070}, {3158, 35448}, {3219, 37105}, {3359, 9943}, {3428, 12680}, {3487, 7675}, {3560, 3576}, {3586, 34489}, {3654, 4882}, {3868, 33557}, {3870, 6361}, {3929, 35242}, {4292, 6869}, {4297, 6261}, {4321, 5045}, {4512, 37292}, {4652, 6876}, {4853, 28204}, {4855, 37403}, {5219, 37356}, {5436, 13151}, {5531, 12515}, {5553, 37000}, {5584, 14872}, {5587, 37438}, {5658, 6865}, {5665, 15934}, {5691, 6917}, {5768, 37421}, {5787, 6907}, {5811, 37423}, {5918, 10310}, {6068, 12738}, {6223, 6987}, {6245, 6825}, {6256, 41540}, {6259, 31789}, {6260, 6827}, {6326, 38761}, {6705, 6954}, {6765, 12702}, {6841, 25525}, {6845, 31266}, {6846, 36991}, {6847, 10430}, {6864, 21151}, {6908, 9799}, {6911, 37526}, {6918, 11227}, {6924, 21164}, {7411, 12528}, {7688, 41229}, {7742, 30223}, {7966, 23340}, {7992, 10268}, {8583, 13624}, {9581, 37406}, {9623, 18525}, {9643, 20277}, {9940, 19541}, {9955, 10582}, {10085, 11012}, {10267, 12705}, {10383, 11374}, {10476, 38485}, {10860, 11248}, {10864, 22758}, {11111, 19861}, {11344, 17616}, {11496, 15726}, {11499, 37560}, {11523, 37585}, {11827, 12678}, {12127, 37727}, {12629, 18526}, {12650, 18499}, {12651, 28146}, {14986, 18450}, {15446, 36599}, {15763, 25365}, {16004, 17784}, {17528, 18480}, {17745, 36754}, {18243, 24703}, {18491, 40296}, {24467, 30304}, {31019, 37433}, {31435, 31937}, {31837, 37551}, {33178, 40909}, {33597, 37022}, {34059, 38461}, {36985, 37558}, {37428, 41543}

X(41855) = X(9)X(173) ∩ X(79)X(1127)

Barycentrics    (a + b - c)*(a - b + c)*(a + 2*b + 2*c) - 2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 4*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*Sin[A/2] : :

X(41855) lies on these lines: {9, 173}, {79, 1127}, {174, 4654}, {259, 30370}, {5249, 8125}, {5259, 7587}, {5904, 12445}, {5905, 11891}, {8126, 17781}, {8389, 21623}, {12491, 40263}, {35627, 38485}

X(41856) = X(9)X(43) ∩ X(21)X(36)

Barycentrics    a*(a^4*b - a^2*b^3 + a^4*c + a^3*b*c - 3*a^2*b^2*c - 3*a*b^3*c - b^4*c - 3*a^2*b*c^2 - 5*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - 3*a*b*c^3 - b^2*c^3 - b*c^4) : :

X(41856) lies on these lines: {9, 43}, {21, 36}, {55, 24697}, {191, 40952}, {405, 17889}, {1001, 33097}, {1284, 4654}, {1580, 16783}, {1621, 4683}, {1756, 4512}, {2292, 5904}, {3914, 5251}, {4184, 6536}, {4204, 24342}, {4220, 41430}, {5248, 27184}, {5905, 9791}, {8040, 35983}, {9959, 40263}, {11688, 17781}, {12567, 19853}, {13588, 25354}, {20472, 37327}, {20834, 36554}, {35623, 38485}

X(41857) = X(2)X(7) ∩ X(79)X(516)

Barycentrics    (a + b - c)*(a - b + c)*(a^2*b - 2*a*b^2 + b^3 + a^2*c - 6*a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3) : :

X(41857) lies on these lines: {2, 7}, {79, 516}, {85, 17233}, {390, 21620}, {480, 5880}, {518, 3649}, {946, 11038}, {948, 7190}, {954, 37426}, {971, 37447}, {1001, 10404}, {1441, 4431}, {1519, 20330}, {1839, 5236}, {2475, 5853}, {2801, 33593}, {3146, 8236}, {3487, 7675}, {3671, 5904}, {3674, 3970}, {3838, 6067}, {3873, 24389}, {3947, 7679}, {3951, 38057}, {4298, 5259}, {4328, 37800}, {5223, 12609}, {5290, 6765}, {5542, 10394}, {5728, 6147}, {5809, 11036}, {5850, 11263}, {6764, 11526}, {6872, 38316}, {7671, 41561}, {7676, 13405}, {7678, 11019}, {8255, 31391}, {8388, 21624}, {8389, 21623}, {8543, 12573}, {10399, 12528}, {11372, 16127}, {11495, 17718}, {11551, 30329}, {11680, 41573}, {12608, 38036}, {12611, 38055}, {13159, 14526}, {14151, 21630}, {15733, 41571}, {17092, 29571}, {17246, 22464}, {17277, 32007}, {20116, 41562}, {21075, 40333}, {21077, 38052}, {24181, 37681}, {25722, 41570}, {30424, 37572}, {31671, 37411}, {35617, 38485}, {38053, 41012}

X(41858) = X(4)X(5259) ∩ X(5)X(165)

Barycentrics    a^4*b^2 - 2*a^3*b^3 + 2*a*b^5 - b^6 - 3*a^4*b*c - a^2*b^3*c + 2*a*b^4*c + 2*b^5*c + a^4*c^2 + 2*a^2*b^2*c^2 - 4*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 + 2*a*b*c^4 + b^2*c^4 + 2*a*c^5 + 2*b*c^5 - c^6 : :

X(41858) lies on these lines: {4, 5259}, {5, 165}, {9, 1699}, {11, 4654}, {35, 6849}, {55, 381}, {57, 79}, {354, 9955}, {403, 5338}, {496, 30350}, {946, 5904}, {1617, 3585}, {1709, 38150}, {1839, 37372}, {2280, 32431}, {3091, 3814}, {3545, 26040}, {3624, 37447}, {3683, 18482}, {3748, 18480}, {3817, 5249}, {3820, 7989}, {3822, 3839}, {3851, 6244}, {5173, 18393}, {5273, 25639}, {5905, 9779}, {6828, 12571}, {6851, 25542}, {6990, 18483}, {7678, 11019}, {7958, 7987}, {7964, 22793}, {7988, 8727}, {8012, 24045}, {10383, 37692}, {10388, 10827}, {10389, 18492}, {10399, 12047}, {10434, 36654}, {10478, 38485}, {10483, 37234}, {11018, 17605}, {11680, 17781}, {34879, 37230}

X(41859) = X(1)X(3925) ∩ X(2)X(35)

Barycentrics    a^2*b^2 - b^4 + 3*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4 : :

X(41859) lies on these lines: {1, 3925}, {2, 35}, {3, 18406}, {5, 165}, {9, 46}, {10, 3681}, {11, 34595}, {12, 3339}, {36, 443}, {40, 6881}, {78, 26725}, {142, 18398}, {210, 3824}, {377, 5251}, {474, 36152}, {498, 26040}, {499, 17582}, {516, 6991}, {594, 39708}, {750, 24880}, {936, 37701}, {1010, 19846}, {1125, 33108}, {1210, 38204}, {1478, 4208}, {1656, 10310}, {1714, 37559}, {1737, 10399}, {1838, 1855}, {1839, 5142}, {2077, 6861}, {2476, 3634}, {2550, 3746}, {2886, 3624}, {2901, 29854}, {3085, 40333}, {3216, 33111}, {3336, 5791}, {3454, 26037}, {3526, 18499}, {3582, 31493}, {3583, 11108}, {3584, 9709}, {3585, 17528}, {3632, 9710}, {3678, 31019}, {3679, 25466}, {3705, 6533}, {3743, 33131}, {3812, 14054}, {3814, 19877}, {3820, 34501}, {3822, 5905}, {3828, 11681}, {3836, 10479}, {3874, 27186}, {3876, 11263}, {3914, 27785}, {3947, 7679}, {3970, 29674}, {4187, 19872}, {4202, 19858}, {4302, 16845}, {4324, 16418}, {4359, 30172}, {4413, 11507}, {4423, 4857}, {4668, 15888}, {4680, 16817}, {4894, 16823}, {5010, 6675}, {5258, 20076}, {5270, 9708}, {5312, 17056}, {5425, 28629}, {5506, 24703}, {5550, 24387}, {5563, 19843}, {5587, 37438}, {5659, 12704}, {5692, 12609}, {5745, 37524}, {5837, 5903}, {6051, 21949}, {6684, 6829}, {6701, 31053}, {6828, 10164}, {6841, 35242}, {6842, 37560}, {6894, 12511}, {6907, 7989}, {6933, 31263}, {6937, 10175}, {6941, 10172}, {6989, 15931}, {6990, 31730}, {7280, 24953}, {7535, 37557}, {7680, 9588}, {7742, 37271}, {7988, 15908}, {8069, 16408}, {8583, 37735}, {8727, 16192}, {9778, 12558}, {9956, 40263}, {10826, 25973}, {10883, 12512}, {10896, 16853}, {11680, 19862}, {12679, 38108}, {12953, 16857}, {13160, 21160}, {13750, 18395}, {14647, 16127}, {15932, 24914}, {16056, 39578}, {16062, 16828}, {16569, 37693}, {16783, 29633}, {16784, 31416}, {17057, 24982}, {17564, 31260}, {17745, 37657}, {17749, 33105}, {18393, 25917}, {18492, 37401}, {19784, 37153}, {19804, 30171}, {19860, 37706}, {19863, 33833}, {24210, 31318}, {24390, 25055}, {24931, 31237}, {25453, 25526}, {25512, 32773}, {25525, 37731}, {25992, 37150}, {26363, 37462}, {26481, 31231}, {27784, 33134}, {29661, 33771}, {30103, 33035}, {30142, 33129}, {32092, 37664}, {33138, 37522}, {35202, 37407}, {36845, 36946}, {37075, 37576}

X(41860) = X(1)X(30) ∩ X(9)X(165)

Barycentrics    a*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 6*a^3*b*c + 2*a^2*b^2*c + 2*a*b^3*c + 5*b^4*c + 2*a^3*c^2 + 2*a^2*b*c^2 + 2*a*b^2*c^2 - 6*b^3*c^2 + 2*a^2*c^3 + 2*a*b*c^3 - 6*b^2*c^3 - 3*a*c^4 + 5*b*c^4 + c^5) : :

X(41860) lies on these lines: {1, 30}, {9, 165}, {20, 997}, {40, 18518}, {46, 1864}, {90, 6985}, {200, 17781}, {516, 3870}, {971, 41338}, {990, 17017}, {1012, 5259}, {1214, 15430}, {1490, 16127}, {1697, 30290}, {1699, 5249}, {1742, 1961}, {2900, 17768}, {3146, 12520}, {3149, 16209}, {3339, 10399}, {3529, 6261}, {3579, 3715}, {3838, 30308}, {3925, 31672}, {4312, 10382}, {4333, 6869}, {4511, 15683}, {4512, 35989}, {5250, 16120}, {5531, 6154}, {5536, 30304}, {5691, 6925}, {5762, 41706}, {5779, 7964}, {5904, 6001}, {5918, 19541}, {6836, 41540}, {6907, 7989}, {6923, 18406}, {7688, 18540}, {7988, 8727}, {8544, 11019}, {9352, 36002}, {9812, 29817}, {10391, 10980}, {11372, 15931}, {11500, 12686}, {12514, 33557}, {12515, 33519}, {12526, 31938}, {12705, 16208}, {12953, 34489}, {14054, 15071}, {15685, 35459}, {18446, 28150}, {18499, 28160}, {18529, 19875}, {28146, 37569}, {28154, 37533}, {35621, 38485}

X(41861) = X(1)X(6) ∩ X(7)X(79)

Barycentrics    a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c + a^3*b*c - 4*a^2*b^2*c + a*b^3*c + b^4*c - 2*a^3*c^2 - 4*a^2*b*c^2 - 6*a*b^2*c^2 + a*b*c^3 + 2*a*c^4 + b*c^4 - c^5) : :

X(41861) lies on these lines: {1, 6}, {7, 79}, {10, 30628}, {35, 1445}, {36, 7675}, {46, 4326}, {142, 5696}, {144, 3874}, {354, 971}, {390, 5903}, {516, 5902}, {758, 15933}, {938, 5883}, {942, 4312}, {991, 21346}, {1071, 3062}, {1125, 41228}, {1210, 38204}, {1478, 5809}, {1698, 3059}, {2771, 15934}, {2801, 11038}, {3295, 41712}, {3336, 11495}, {3338, 5732}, {3339, 12710}, {3475, 5817}, {3616, 31938}, {3670, 4335}, {3873, 5850}, {3947, 10865}, {4008, 24799}, {4328, 15430}, {5045, 8581}, {5249, 10861}, {5439, 15587}, {5533, 38055}, {5542, 10394}, {5697, 7672}, {5722, 18406}, {5785, 10582}, {5905, 10580}, {6001, 24644}, {6173, 31159}, {7676, 37572}, {7677, 37525}, {7741, 21617}, {10056, 38057}, {10072, 26725}, {10384, 11529}, {10389, 15104}, {10390, 36599}, {10391, 10980}, {10392, 21620}, {10404, 31672}, {10572, 12573}, {10826, 17620}, {11018, 31245}, {11036, 30290}, {11372, 11518}, {12005, 16127}, {15009, 20059}, {15733, 38052}, {15908, 31657}, {17625, 30350}, {17718, 38108}, {17728, 38122}, {18530, 36971}, {20330, 37722}, {21842, 30284}, {31397, 38210}, {31658, 37080}

X(41862) = X(1)X(21949) ∩ X(2)X(79)

Barycentrics    (a + 2*b + 2*c)*(a^2*b - b^3 + a^2*c + 3*a*b*c + b^2*c + b*c^2 - c^3) : :

X(41862) lies on these lines: {1, 21949}, {2, 79}, {3, 1699}, {9, 3336}, {10, 3681}, {119, 15016}, {142, 5696}, {149, 214}, {377, 5441}, {405, 16118}, {442, 10399}, {443, 26725}, {474, 35204}, {498, 6594}, {1145, 25466}, {1656, 1768}, {1698, 3715}, {3090, 16127}, {3120, 31318}, {3632, 40587}, {3634, 17781}, {3649, 5692}, {3812, 17057}, {3833, 15079}, {3834, 10472}, {3838, 34595}, {3841, 18398}, {3970, 19584}, {4668, 11530}, {4725, 25543}, {5087, 5131}, {5437, 31262}, {5883, 31938}, {5905, 9782}, {6260, 6829}, {6600, 38052}, {6881, 15071}, {7951, 25011}, {10129, 19878}, {11517, 25525}, {12609, 24564}, {13089, 16763}, {16783, 19557}, {17889, 27785}, {18406, 18481}, {27558, 28612}, {31320, 33145}, {37462, 37701}

X(41863) = X(1)X(6) ∩ X(8)X(4208)

Barycentrics    a*(a^3 - 3*a^2*b - a*b^2 + 3*b^3 - 3*a^2*c - 6*a*b*c + b^2*c - a*c^2 + b*c^2 + 3*c^3) : :

X(41863) lies on these lines: {1, 6}, {8, 4208}, {10, 3475}, {35, 3928}, {40, 3868}, {46, 3158}, {57, 3811}, {65, 6765}, {78, 3333}, {79, 3633}, {84, 16465}, {145, 515}, {200, 942}, {226, 31418}, {354, 936}, {386, 3677}, {388, 519}, {443, 5542}, {474, 10980}, {480, 16410}, {496, 31146}, {517, 12565}, {527, 4294}, {529, 30323}, {758, 1697}, {912, 12705}, {938, 8165}, {946, 36845}, {971, 12651}, {976, 37554}, {997, 3881}, {1046, 3749}, {1056, 6737}, {1066, 3190}, {1071, 6769}, {1210, 25568}, {1419, 4347}, {1420, 22836}, {1467, 41539}, {1478, 12625}, {1479, 28609}, {1482, 9856}, {1621, 3951}, {1706, 5902}, {1998, 17857}, {2093, 3913}, {2098, 9848}, {2099, 12629}, {2136, 5903}, {3085, 24391}, {3189, 4292}, {3216, 5573}, {3218, 35242}, {3241, 11682}, {3244, 3486}, {3295, 12526}, {3303, 3962}, {3306, 4420}, {3338, 5438}, {3339, 5687}, {3359, 24475}, {3361, 5440}, {3419, 5290}, {3421, 6738}, {3487, 4847}, {3488, 12527}, {3576, 4430}, {3577, 5881}, {3585, 41709}, {3600, 30318}, {3601, 5267}, {3616, 3984}, {3621, 11525}, {3632, 5794}, {3646, 3876}, {3649, 4863}, {3678, 7308}, {3679, 25466}, {3689, 5221}, {3753, 4882}, {3869, 31393}, {3872, 8000}, {3875, 17753}, {3878, 37556}, {3879, 17170}, {3889, 19861}, {3901, 5119}, {3927, 4512}, {3929, 5248}, {3940, 5045}, {3957, 5250}, {4005, 4423}, {4018, 7991}, {4255, 21342}, {4293, 12437}, {4295, 5853}, {4299, 34701}, {4314, 5850}, {4321, 37544}, {4355, 11112}, {4533, 16842}, {4668, 11530}, {4848, 34619}, {4849, 17054}, {5044, 10582}, {5084, 6744}, {5128, 8715}, {5219, 10916}, {5231, 11374}, {5437, 18398}, {5439, 8580}, {5534, 18518}, {5587, 12649}, {5705, 17718}, {5732, 7957}, {5735, 6253}, {6282, 12675}, {6326, 13279}, {6734, 10585}, {6764, 11526}, {6835, 38036}, {7994, 9943}, {8168, 10107}, {8227, 26015}, {8666, 13384}, {9581, 21077}, {9624, 10529}, {9965, 31730}, {10222, 31821}, {10388, 12711}, {10389, 12514}, {10390, 17529}, {10914, 18421}, {11019, 25522}, {11037, 20007}, {11224, 12127}, {11280, 34710}, {11372, 12528}, {12005, 37526}, {12047, 24392}, {13253, 25416}, {13411, 24477}, {15934, 34790}, {16200, 36846}, {16856, 36835}, {18163, 35637}, {19855, 24393}, {26321, 37533}, {31424, 37080}, {32913, 37552}, {34749, 37738}, {36867, 37730}

X(41864) = X(1)X(30) ∩ X(4)X(10389)

Barycentrics    (a - b - c)*(5*a^3 + 4*a^2*b + a*b^2 + 2*b^3 + 4*a^2*c - 2*a*b*c - 2*b^2*c + a*c^2 - 2*b*c^2 + 2*c^3) : :

X(41864) lies on these lines: {1, 30}, {4, 10389}, {8, 9}, {10, 10385}, {11, 34595}, {21, 24392}, {35, 31231}, {40, 4309}, {55, 1698}, {57, 4294}, {145, 17781}, {320, 10889}, {354, 31805}, {377, 38316}, {380, 7359}, {388, 30331}, {443, 497}, {496, 30282}, {515, 37556}, {516, 11518}, {517, 10399}, {938, 5128}, {1058, 1420}, {1210, 35445}, {1479, 5219}, {1657, 5049}, {1699, 9670}, {1706, 20075}, {2093, 12433}, {2478, 3158}, {3057, 3633}, {3146, 8236}, {3189, 40998}, {3244, 3486}, {3295, 3586}, {3296, 4292}, {3303, 5691}, {3306, 20066}, {3333, 4302}, {3338, 4330}, {3340, 3488}, {3361, 15338}, {3434, 5436}, {3474, 6744}, {3612, 37704}, {3622, 4313}, {3623, 5905}, {3624, 11238}, {3746, 5587}, {3748, 5290}, {3811, 31142}, {3877, 31938}, {3894, 17637}, {4293, 40270}, {4319, 33178}, {4326, 5880}, {4512, 18253}, {4857, 8227}, {4863, 5234}, {5047, 38200}, {5057, 10393}, {5225, 13405}, {5560, 10827}, {5722, 10386}, {5790, 31436}, {5882, 16127}, {5919, 9856}, {6762, 6872}, {6765, 11113}, {6767, 9613}, {7987, 37722}, {8582, 34607}, {9612, 9668}, {9614, 18493}, {9624, 37571}, {9819, 10950}, {9957, 18526}, {10056, 18492}, {10480, 38485}, {10572, 31393}, {10857, 31777}, {10944, 30337}, {11519, 34699}, {11531, 37724}, {13606, 36599}, {16192, 17728}, {18530, 37582}, {19860, 34611}, {19861, 34701}, {23536, 35227}, {24914, 31508}, {30143, 34649}, {31480, 38140}, {37563, 37721}

X(41865) = X(9)X(3634) ∩ X(20)X(946)

Barycentrics    3*a^4 + 5*a^3*b + 3*a^2*b^2 - 5*a*b^3 - 6*b^4 + 5*a^3*c + 14*a^2*b*c + 13*a*b^2*c + 3*a^2*c^2 + 13*a*b*c^2 + 12*b^2*c^2 - 5*a*c^3 - 6*c^4 : :

X(41865) lies on these lines: {9, 3634}, {20, 946}, {65, 3679}, {79, 4679}, {1056, 3244}, {1699, 31805}, {3306, 7701}, {3621, 11525}, {3671, 36922}, {3820, 34501}, {3824, 4312}, {5204, 5259}, {5219, 7702}, {5587, 5884}, {5905, 11024}, {9581, 17637}, {9940, 38107}, {12632, 21620}, {12667, 18406}, {19883, 28617}

X(41866) = X(8)X(79) ∩ X(9)X(165)

Barycentrics    a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c + a^3*b*c + a*b^3*c - 3*b^4*c - 2*a^3*c^2 + 2*a*b^2*c^2 + 4*b^3*c^2 + a*b*c^3 + 4*b^2*c^3 + 2*a*c^4 - 3*b*c^4 - c^5) : :

X(41866) lies on these lines: {8, 79}, {9, 165}, {10, 9961}, {226, 5696}, {958, 16143}, {1699, 5784}, {1864, 38052}, {2550, 13227}, {3612, 5259}, {4654, 5173}, {5231, 17616}, {5249, 10861}, {5691, 18251}, {5790, 9947}, {9856, 14110}, {10399, 13750}, {10855, 17603}, {10864, 22758}, {11678, 17781}, {12447, 28158}, {13405, 25722}, {18398, 31418}, {25440, 27065}, {25639, 27186}, {35613, 38485}

X(41867) = X(1)X(3925) ∩ X(2)X(7)

Barycentrics    a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c - 6*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + 2*c^3 : :

X(41867) lies on these lines: {1, 3925}, {2, 7}, {3, 1699}, {5, 1750}, {10, 3475}, {11, 10383}, {12, 1467}, {37, 23681}, {40, 6989}, {55, 37271}, {79, 4679}, {84, 6832}, {165, 5805}, {200, 3826}, {210, 942}, {277, 25430}, {278, 1855}, {284, 5333}, {312, 32015}, {333, 17298}, {345, 24199}, {377, 5436}, {405, 9579}, {442, 9581}, {443, 497}, {610, 17398}, {936, 17529}, {946, 37407}, {950, 4208}, {1001, 9580}, {1211, 16832}, {1449, 26723}, {1621, 35985}, {1656, 9940}, {1738, 37553}, {1839, 7490}, {1848, 34847}, {2095, 11231}, {2136, 10587}, {2550, 10389}, {2886, 10582}, {2951, 7965}, {2999, 4272}, {3008, 5712}, {3090, 5658}, {3243, 25006}, {3247, 19785}, {3333, 19854}, {3340, 28629}, {3361, 24953}, {3434, 38316}, {3526, 37623}, {3576, 6826}, {3586, 17528}, {3587, 31162}, {3616, 37436}, {3622, 12437}, {3634, 21060}, {3646, 12047}, {3666, 4859}, {3679, 15934}, {3683, 4312}, {3731, 3782}, {3742, 5231}, {3772, 17022}, {3812, 41539}, {3816, 37363}, {3824, 9612}, {3834, 5737}, {3838, 8167}, {3848, 31249}, {3870, 38200}, {3970, 17284}, {4007, 32858}, {4197, 37723}, {4292, 16845}, {4383, 17745}, {4384, 4886}, {4512, 5880}, {4641, 4888}, {4648, 40940}, {4657, 37266}, {4659, 17776}, {4666, 24392}, {4677, 36867}, {4847, 38053}, {4873, 28605}, {5234, 10404}, {5256, 26724}, {5268, 33130}, {5271, 17296}, {5272, 33111}, {5284, 35977}, {5287, 33129}, {5425, 36922}, {5438, 37462}, {5439, 5705}, {5444, 23708}, {5550, 6904}, {5573, 29639}, {5587, 6881}, {5708, 19872}, {5709, 31423}, {5718, 23511}, {5732, 8226}, {5768, 10175}, {5787, 7989}, {5791, 30393}, {5817, 41561}, {5881, 37615}, {5886, 6282}, {6678, 31312}, {6705, 16127}, {6824, 37526}, {6837, 9841}, {6856, 9843}, {6857, 12436}, {6861, 37534}, {6991, 10884}, {7322, 33144}, {7580, 38150}, {7987, 20420}, {7988, 8727}, {8580, 17718}, {8583, 28628}, {9578, 25466}, {9624, 37531}, {9780, 24391}, {9816, 16545}, {10200, 41540}, {10578, 40333}, {10585, 25011}, {10855, 17603}, {11018, 31245}, {11347, 17384}, {11522, 31793}, {11679, 17234}, {12572, 17552}, {12609, 31435}, {13226, 15017}, {13405, 26040}, {13411, 17582}, {15104, 24474}, {15601, 41011}, {15829, 24564}, {16054, 25500}, {16577, 24554}, {16667, 37631}, {16783, 19701}, {16831, 19786}, {17064, 26102}, {17237, 19744}, {17263, 30568}, {17272, 19732}, {17367, 26109}, {17400, 37265}, {17917, 37276}, {18588, 25915}, {19520, 25524}, {19812, 37092}, {19855, 21620}, {19860, 37709}, {21258, 26942}, {21921, 23058}, {24929, 25055}, {25590, 32777}, {25878, 37543}, {25917, 37544}, {25961, 29828}, {26128, 39586}, {26738, 37687}, {29609, 37274}, {30304, 31657}, {30326, 38108}, {31242, 40649}, {31254, 31938}, {35612, 38485}, {38036, 41338}, {40677, 41010}

X(41867) = {X(2),X(226)}-harmonic conjugate of X(7308)

X(41868) = X(1)X(30) ∩ X(9)X(1764)

Barycentrics    a^6 + 3*a^5*b + 4*a^4*b^2 - 5*a^2*b^4 - 3*a*b^5 + 3*a^5*c + 4*a^4*b*c - 2*a^3*b^2*c - a*b^4*c - 4*b^5*c + 4*a^4*c^2 - 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + 4*a*b^3*c^2 + 4*a*b^2*c^3 + 8*b^3*c^3 - 5*a^2*c^4 - a*b*c^4 - 3*a*c^5 - 4*b*c^5 : :

X(41868) lies on these lines: {1, 30}, {9, 1764}, {516, 29651}, {1999, 5905}, {5249, 10444}, {5259, 10882}, {5904, 12435}, {10439, 38485}, {10441, 40263}, {11679, 17781}, {16127, 35635}

X(41869) = X(1)X(30) ∩ X(2)X(18483)

Barycentrics    3*a^4 + a^3*b - a^2*b^2 - a*b^3 - 2*b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + 4*b^2*c^2 - a*c^3 - 2*c^4 : :

Let (OA), (OB), (OC) be the extraversions of the Conway circle, and AB, AC, BC, BA, CA, CB be the points on the Conway conic, as at X(40589). Let A' = CBAB∩ACBC, B' = ACBC∩BACA, C' = BACA∩CBAB. Let A" be the intersection of the tangents to (OB) at AB and to (OC) at AC. Define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(41869). (Randy Hutson, April 13, 2021)

X(41869) lies on these lines: {1, 30}, {2, 18483}, {3, 1699}, {4, 9}, {5, 165}, {8, 3543}, {11, 15803}, {20, 946}, {35, 5219}, {36, 4333}, {46, 3583}, {55, 9612}, {56, 9614}, {57, 1479}, {63, 2894}, {65, 3586}, {78, 5057}, {80, 36599}, {84, 10431}, {140, 7988}, {145, 515}, {156, 9586}, {191, 18407}, {226, 4294}, {265, 9904}, {354, 9670}, {355, 3627}, {376, 1125}, {377, 24564}, {380, 1901}, {381, 1698}, {382, 517}, {388, 10624}, {390, 21620}, {404, 25522}, {405, 7688}, {411, 10129}, {442, 4512}, {443, 3646}, {484, 10826}, {485, 9616}, {496, 3361}, {497, 3333}, {498, 35445}, {519, 15682}, {528, 6765}, {529, 12629}, {546, 7989}, {548, 38034}, {549, 30308}, {550, 5886}, {551, 11001}, {590, 9582}, {631, 3817}, {758, 12625}, {936, 24703}, {942, 4312}, {944, 3635}, {950, 4295}, {952, 11531}, {968, 27577}, {988, 33106}, {990, 29050}, {993, 21669}, {1001, 37426}, {1006, 12511}, {1012, 11012}, {1046, 29097}, {1054, 15522}, {1056, 12575}, {1058, 4298}, {1071, 5735}, {1155, 10896}, {1158, 5535}, {1210, 3474}, {1282, 10741}, {1330, 3886}, {1351, 39878}, {1385, 1657}, {1420, 4299}, {1453, 3914}, {1478, 1697}, {1482, 5073}, {1483, 16189}, {1490, 5842}, {1506, 31422}, {1519, 6934}, {1537, 12119}, {1539, 12778}, {1571, 5475}, {1572, 7748}, {1593, 9911}, {1597, 8193}, {1614, 9622}, {1621, 16125}, {1656, 31663}, {1702, 3070}, {1703, 3071}, {1709, 5709}, {1721, 29291}, {1722, 24715}, {1737, 5128}, {1750, 5812}, {1754, 3073}, {1768, 10738}, {1777, 1936}, {1837, 2093}, {1838, 7070}, {1858, 5727}, {1898, 18397}, {1902, 12173}, {2077, 3149}, {2100, 10751}, {2101, 10750}, {2328, 31902}, {2475, 5250}, {2476, 35258}, {2478, 25011}, {2548, 9574}, {2549, 9575}, {2771, 3901}, {2777, 33535}, {2800, 10724}, {2802, 10728}, {2809, 10727}, {2816, 33650}, {2817, 10732}, {2829, 6264}, {2886, 16113}, {2948, 7728}, {2951, 5805}, {2979, 31751}, {3057, 9613}, {3060, 31728}, {3062, 5762}, {3090, 10164}, {3091, 6684}, {3097, 14881}, {3158, 21077}, {3241, 15640}, {3295, 5290}, {3338, 4857}, {3339, 5722}, {3340, 10572}, {3359, 6928}, {3419, 12526}, {3485, 4304}, {3487, 4314}, {3488, 3671}, {3522, 10165}, {3523, 9779}, {3524, 19862}, {3525, 10171}, {3529, 3636}, {3534, 13624}, {3545, 3634}, {3585, 5119}, {3601, 4302}, {3612, 4324}, {3617, 34632}, {3622, 15683}, {3625, 34627}, {3626, 34648}, {3633, 8148}, {3651, 5248}, {3653, 15686}, {3654, 15687}, {3679, 3830}, {3729, 5015}, {3751, 31670}, {3753, 16616}, {3811, 28609}, {3815, 31421}, {3825, 31190}, {3828, 41099}, {3832, 10175}, {3839, 9780}, {3841, 6990}, {3843, 9588}, {3845, 19875}, {3851, 11231}, {3853, 5690}, {3858, 30315}, {3861, 38042}, {3869, 31938}, {3875, 29040}, {3895, 20060}, {3911, 10591}, {3916, 5231}, {3922, 31788}, {3928, 10916}, {3944, 37552}, {3951, 5178}, {3958, 4034}, {3988, 31871}, {4004, 7686}, {4005, 5777}, {4018, 5895}, {4127, 31803}, {4190, 41012}, {4192, 29825}, {4217, 25904}, {4293, 12053}, {4309, 10389}, {4316, 37618}, {4336, 8555}, {4338, 5902}, {4339, 34937}, {4347, 6198}, {4355, 5045}, {4389, 10444}, {4533, 5927}, {4547, 15064}, {4640, 5705}, {4652, 11680}, {4677, 33699}, {4678, 11362}, {4679, 34630}, {4691, 38074}, {4701, 12245}, {4880, 11665}, {4882, 34746}, {4915, 34697}, {5010, 37692}, {5055, 19872}, {5059, 5731}, {5066, 19876}, {5068, 10172}, {5076, 5790}, {5082, 12527}, {5180, 11682}, {5195, 9312}, {5217, 17605}, {5223, 31672}, {5229, 31397}, {5234, 31419}, {5251, 35239}, {5258, 18761}, {5288, 18519}, {5293, 36512}, {5436, 12609}, {5437, 6899}, {5438, 21616}, {5442, 7741}, {5531, 16128}, {5536, 24467}, {5537, 11499}, {5540, 15521}, {5541, 10742}, {5550, 10304}, {5556, 10578}, {5584, 6913}, {5660, 10993}, {5692, 31937}, {5697, 37709}, {5708, 18527}, {5714, 13405}, {5715, 7580}, {5732, 25557}, {5734, 13607}, {5745, 31418}, {5767, 29223}, {5787, 7992}, {5840, 6326}, {5880, 37428}, {5882, 20057}, {5884, 9961}, {5890, 31757}, {5901, 15704}, {5918, 9940}, {5919, 9657}, {5925, 12262}, {6033, 13174}, {6221, 13888}, {6241, 31732}, {6256, 12703}, {6261, 11015}, {6282, 11826}, {6321, 9860}, {6398, 13942}, {6560, 18992}, {6561, 18991}, {6564, 13893}, {6565, 13947}, {6744, 30424}, {6796, 36002}, {6826, 37551}, {6827, 37560}, {6829, 12558}, {6836, 10860}, {6845, 25639}, {6849, 7308}, {6907, 10268}, {6922, 10270}, {6925, 26332}, {6999, 17244}, {7280, 23708}, {7290, 23537}, {7330, 37820}, {7373, 31776}, {7380, 9746}, {7381, 39592}, {7387, 9626}, {7390, 39605}, {7517, 9590}, {7681, 37374}, {7745, 9593}, {7747, 9620}, {7756, 9619}, {7951, 37406}, {7956, 21164}, {7967, 11541}, {7994, 37822}, {8185, 18534}, {8583, 11112}, {8727, 15908}, {8983, 9541}, {9355, 29105}, {9573, 18453}, {9576, 18455}, {9596, 31426}, {9621, 34148}, {9650, 31433}, {9655, 9957}, {9656, 31436}, {9669, 37582}, {9818, 37557}, {9841, 10531}, {9856, 14110}, {9862, 11599}, {9864, 39838}, {9896, 12293}, {9899, 14216}, {9905, 15800}, {9943, 15016}, {10106, 30305}, {10167, 13374}, {10222, 28168}, {10246, 17800}, {10310, 19541}, {10439, 37482}, {10441, 15310}, {10526, 41698}, {10733, 12407}, {10739, 39156}, {10749, 12408}, {10789, 14880}, {10827, 11010}, {10893, 17613}, {10895, 31434}, {10980, 24470}, {11037, 40270}, {11108, 18482}, {11114, 19860}, {11220, 12005}, {11224, 28186}, {11238, 32636}, {11278, 18526}, {11363, 37196}, {11365, 21312}, {11373, 13462}, {11375, 15338}, {11376, 15326}, {11459, 31737}, {11495, 38150}, {11520, 14450}, {12103, 38028}, {12244, 13605}, {12248, 21630}, {12261, 20127}, {12295, 13211}, {12368, 13202}, {12436, 26105}, {12515, 22938}, {12611, 15015}, {12751, 13996}, {12785, 32340}, {12918, 13221}, {13172, 21636}, {13178, 39809}, {13199, 21635}, {13369, 18398}, {13665, 31439}, {13883, 23249}, {13912, 31412}, {13936, 23259}, {14927, 39870}, {15017, 33814}, {15058, 31752}, {15071, 24474}, {15488, 29349}, {15691, 38022}, {15696, 17502}, {15829, 17647}, {15934, 31795}, {16173, 38761}, {16174, 38693}, {16783, 28897}, {16832, 36728}, {17284, 36731}, {17579, 19861}, {17768, 28646}, {17784, 21075}, {17845, 40658}, {18421, 37730}, {18491, 35448}, {18517, 18540}, {18529, 34612}, {18655, 41010}, {18788, 36685}, {19708, 19883}, {20053, 28234}, {20066, 31053}, {20196, 35238}, {22753, 37022}, {22836, 34701}, {22837, 34716}, {24210, 37554}, {24309, 37431}, {24644, 37424}, {24883, 36277}, {25440, 30827}, {26129, 37267}, {27558, 32929}, {28212, 37712}, {28534, 34706}, {29032, 37823}, {29614, 37416}, {30503, 31789}, {31394, 37425}, {31428, 31443}, {34464, 40100}, {35262, 37256}, {35403, 38066}, {35490, 41722}, {35774, 35821}, {35775, 35820}, {36478, 36551}, {36490, 36531}, {36499, 36583}, {36990, 39885}, {37407, 38037}, {38038, 38759}, {38220, 38749}

X(41869) = anticomplement of X(31730)

X(41870) = X(1)X(30) ∩ X(5)X(30291)

Barycentrics    a^4 - 5*a^3*b - 3*a^2*b^2 + 5*a*b^3 + 2*b^4 - 5*a^3*c - 14*a^2*b*c - 5*a*b^2*c - 3*a^2*c^2 - 5*a*b*c^2 - 4*b^2*c^2 + 5*a*c^3 + 2*c^4 : :

X(41870) lies on these lines: {1, 30}, {5, 30291}, {7, 31730}, {8, 4208}, {9, 1125}, {40, 3475}, {210, 942}, {226, 10591}, {354, 5777}, {388, 18406}, {496, 30350}, {553, 35242}, {936, 25557}, {946, 11038}, {999, 5259}, {1001, 28645}, {1056, 3244}, {1490, 38036}, {1697, 11551}, {3174, 5880}, {3243, 12609}, {3336, 31425}, {3361, 5719}, {3576, 5758}, {3579, 5586}, {3616, 17781}, {3622, 5905}, {3650, 4512}, {3671, 31393}, {3678, 20195}, {3811, 6173}, {3873, 31938}, {3874, 25525}, {3982, 4294}, {4355, 24929}, {4888, 5266}, {5045, 8581}, {5049, 11522}, {5219, 18398}, {5231, 14054}, {5261, 17706}, {5290, 15934}, {5425, 37709}, {5587, 11518}, {5603, 10864}, {5703, 30340}, {5714, 6744}, {5763, 38030}, {5771, 17718}, {10980, 11374}, {11263, 24392}, {11373, 30343}, {12559, 36922}, {13464, 16127}, {13867, 31822}, {15933, 31673}, {17582, 38054}, {22760, 37602}, {31231, 37731}, {35620, 38485}

X(41871) = X(5)X(41540) ∩ X(9)X(165)

Barycentrics    a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c + 2*a^3*b*c - 3*b^4*c - 2*a^3*c^2 + 4*b^3*c^2 + 4*b^2*c^3 + 2*a*c^4 - 3*b*c^4 - c^5) : :

X(41871) lies on these lines: {5, 41540}, {9, 165}, {11, 3742}, {30, 960}, {79, 6598}, {226, 41564}, {355, 5836}, {377, 1898}, {474, 36599}, {518, 1836}, {950, 16120}, {971, 2886}, {1004, 7082}, {1776, 35990}, {1858, 2475}, {1864, 5880}, {3035, 10157}, {3660, 3829}, {3683, 35989}, {3812, 10826}, {3848, 10584}, {3880, 37708}, {4654, 11235}, {4677, 5904}, {5087, 8727}, {5229, 12529}, {5259, 17614}, {5696, 28609}, {5784, 10431}, {5794, 6925}, {5795, 12446}, {5832, 41706}, {6907, 9943}, {8256, 9947}, {10861, 26105}, {10943, 12675}, {10944, 34699}, {11928, 13374}, {12625, 30290}, {13369, 25639}, {15254, 20835}, {15297, 37270}, {15310, 40635}, {17615, 17781}, {25568, 25722}, {34791, 39542}, {35626, 38485}, {38389, 41581}

X(41872) = X(1)X(6) ∩ X(2)X(79)

Barycentrics    a*(2*a^3 + a^2*b - 2*a*b^2 - b^3 + a^2*c - 5*a*b*c - 5*b^2*c - 2*a*c^2 - 5*b*c^2 - c^3) : :

X(41872) lies on these lines: {1, 6}, {2, 79}, {3, 5506}, {35, 3305}, {58, 31318}, {63, 25542}, {191, 5221}, {381, 1698}, {631, 16127}, {758, 16859}, {1125, 17781}, {1155, 19872}, {1621, 32635}, {1770, 6666}, {2476, 3634}, {3219, 18398}, {3336, 16842}, {3337, 8167}, {3624, 4654}, {3916, 34595}, {4309, 38057}, {4420, 5248}, {4423, 6763}, {4676, 16828}, {5046, 9780}, {5047, 5902}, {5054, 22936}, {5055, 22937}, {5131, 16862}, {5249, 19862}, {5260, 5697}, {5426, 16866}, {5438, 36599}, {5441, 31156}, {5550, 5905}, {5659, 10531}, {5883, 17570}, {6883, 15071}, {6985, 7308}, {6990, 18483}, {9352, 31253}, {10176, 16865}, {11095, 37834}, {11096, 37831}, {11344, 35204}, {13624, 18515}, {15015, 17571}, {15079, 37162}, {15175, 18233}, {16173, 31458}, {16845, 26725}, {16858, 22836}, {16861, 30143}, {17057, 18406}, {17590, 17768}, {18357, 34746}, {19875, 34706}, {25440, 35595}, {31018, 37731}

X(41873) = X(75)X(150) ∩ X(86)X(99)

Barycentrics    (a^2 - b^2 + b*c - c^2)*(a^3 + b^3 + a*b*c - 2*b^2*c - 2*b*c^2 + c^3) : :

X(41873) lies on these lines: {75, 150}, {86, 99}, {190, 226}, {320, 758}, {497, 4459}, {545, 17056}, {846, 4389}, {1086, 6703}, {4440, 26109}, {17487, 27754}, {25361, 32939}

X(41874) = X(2)X(1762) ∩ X(7)X(21)

Barycentrics    a^5 - a^3*b^2 + a^2*b^3 - b^5 - a^3*b*c - 4*a^2*b^2*c - a*b^3*c + 2*b^4*c - a^3*c^2 - 4*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 + a^2*c^3 - a*b*c^3 - b^2*c^3 + 2*b*c^4 - c^5 : :

X(41874) lies on these lines: {2, 1762}, {7, 21}, {8, 35550}, {45, 329}, {69, 9399}, {320, 960}, {325, 30741}, {497, 4459}, {894, 18589}, {2647, 3668}, {2893, 18698}, {3152, 4329}, {4346, 33100}, {4429, 11024}, {4440, 24424}, {5905, 26109}, {6703, 9776}, {10436, 17181}, {11683, 18635}, {12635, 17378}, {15680, 18661}, {15936, 34195}, {17274, 31435}, {24199, 40963}, {24683, 37274}, {26806, 34830}, {27524, 27689}, {34647, 39704}

X(41875) = X(1)X(75) ∩ X(2)X(20955)

Barycentrics    b*c*(-3*a^2 - a*b + b^2 - a*c - b*c + c^2) : :

X(41875) lies on these lines: {1, 75}, {2, 20955}, {37, 7187}, {85, 11375}, {192, 24654}, {312, 26109}, {320, 960}, {325, 9711}, {344, 27340}, {551, 33940}, {668, 7278}, {1125, 20924}, {1212, 17263}, {1655, 7200}, {1909, 3701}, {2292, 33947}, {3061, 17234}, {3263, 25303}, {3616, 33930}, {3624, 33934}, {3636, 20893}, {3666, 24663}, {3673, 15903}, {3743, 16712}, {3766, 24099}, {3948, 17056}, {4037, 40908}, {4066, 33939}, {4485, 30092}, {4657, 39724}, {4687, 27296}, {4760, 17693}, {5333, 20929}, {6625, 20529}, {6645, 24358}, {6703, 17367}, {6757, 20951}, {7185, 41003}, {7249, 40027}, {15668, 17788}, {16826, 17789}, {17090, 26103}, {17758, 18061}, {18140, 18159}, {20090, 27705}, {20432, 29580}, {21609, 40723}, {21808, 33946}, {24524, 30758}, {24786, 27195}, {25647, 38941}, {27487, 30038}, {29672, 40038}, {30827, 30854}, {33938, 33942}

X(41876) = X(2)X(6) ∩ X(37)X(17789)

Barycentrics    a^4*b + a*b^4 + a^4*c - a^3*b*c - a^2*b^2*c + a*b^3*c + b^4*c - a^2*b*c^2 - a*b^2*c^2 + a*b*c^3 + a*c^4 + b*c^4 : :

X(41876) lies on these lines: {2, 6}, {37, 17789}, {87, 29858}, {192, 35550}, {238, 1330}, {257, 25132}, {319, 27321}, {344, 1655}, {894, 41248}, {1001, 26117}, {1740, 3771}, {1918, 4645}, {2092, 25470}, {2209, 25957}, {2309, 29632}, {2893, 26081}, {3736, 25650}, {3770, 17279}, {3882, 17282}, {4388, 16690}, {4675, 27349}, {4687, 27296}, {5263, 25466}, {17084, 27097}, {17142, 33148}, {17353, 17499}, {18040, 27295}, {18792, 25645}, {19785, 20170}, {19834, 20161}, {22008, 40859}, {22343, 29869}, {23751, 27293}, {24505, 27289}, {25361, 27184}, {26125, 41003}, {33833, 37502}

X(41877) = X(2)X(6) ∩ X(37)X(29967)

Barycentrics    2*a^3*b^2 + a^2*b^3 - a*b^4 + 2*a^3*b*c + 3*a^2*b^2*c + a*b^3*c - b^4*c + 2*a^3*c^2 + 3*a^2*b*c^2 + 2*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4 : :

X(41877) lies on these lines: {2, 6}, {37, 29967}, {142, 27633}, {238, 6675}, {442, 3736}, {1045, 21926}, {1193, 11281}, {1740, 33111}, {1865, 31909}, {1918, 6690}, {2209, 29678}, {2274, 25466}, {2309, 33105}, {3002, 28278}, {3136, 10458}, {3286, 37225}, {3666, 34830}, {3831, 25113}, {5110, 37233}, {5132, 27622}, {5156, 7483}, {5336, 16831}, {6693, 33682}, {8731, 20992}, {17084, 29993}, {17317, 41240}, {17390, 41233}, {17717, 37370}, {18165, 40954}, {20891, 35550}, {21246, 40937}, {29991, 36812}, {30017, 34586}, {40784, 41003}

X(41878) = X(2)X(6) ∩ X(8)X(11281)

Barycentrics    a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c - a*b*c - b^2*c - a*c^2 - b*c^2 + 2*c^3 : :

X(41878) lies on these lines: {2, 6}, {8, 11281}, {12, 36926}, {75, 25525}, {190, 226}, {312, 31266}, {319, 4035}, {320, 5745}, {344, 5226}, {442, 1043}, {846, 4892}, {908, 25361}, {1010, 25645}, {1125, 36974}, {1215, 29862}, {1222, 15888}, {1330, 6675}, {1962, 31280}, {1997, 17084}, {2886, 29839}, {2887, 29640}, {3011, 33073}, {3178, 24161}, {3452, 17263}, {3454, 11110}, {3475, 30741}, {3685, 3838}, {3750, 21241}, {3752, 27191}, {3757, 4914}, {3771, 5263}, {3772, 4360}, {3996, 33108}, {4138, 24723}, {4359, 27757}, {4389, 26132}, {4645, 6690}, {4653, 17677}, {4865, 29675}, {4997, 15455}, {5051, 24936}, {5219, 18044}, {5249, 32851}, {5273, 17347}, {5719, 16086}, {6686, 31252}, {7249, 27805}, {10453, 31245}, {11679, 17295}, {11680, 29830}, {13405, 32850}, {13741, 37693}, {14007, 24931}, {14061, 15903}, {14206, 20921}, {17122, 41263}, {17228, 18229}, {17241, 30567}, {17273, 38000}, {17289, 20106}, {17305, 25527}, {17317, 39595}, {17336, 28609}, {17717, 29642}, {17718, 29641}, {17719, 29653}, {17722, 29672}, {21363, 29474}, {24542, 33107}, {24552, 29866}, {25385, 33158}, {25446, 41014}, {25496, 29858}, {25526, 25669}, {25529, 33133}, {25531, 29851}, {25663, 26051}, {25665, 33954}, {25760, 29661}, {25957, 29678}, {26128, 29657}, {26223, 26738}, {27690, 34195}, {28742, 28771}, {29571, 30851}, {29632, 32942}, {29639, 33124}, {29643, 32926}, {29664, 33122}, {29671, 32922}, {29681, 33070}, {29682, 32775}, {29688, 33123}, {29689, 32844}, {29865, 32772}, {29869, 32944}, {30827, 30854}, {31019, 32939}

X(41879) = X(2)X(20955) ∩ X(58)X(86)

Barycentrics    a^4 + a^3*b + a*b^3 + b^4 + a^3*c + 4*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 + b^2*c^2 + a*c^3 + c^4 : :

X(41879) lies on these lines: {2, 20955}, {58, 86}, {65, 17084}, {319, 5293}, {325, 16830}, {3743, 7799}, {4920, 7321}, {6703, 29614}, {14949, 23905}, {16604, 16706}, {16700, 16744}, {17056, 29578}, {17320, 37592}, {29633, 40533}

X(41880) = ISOGONAL CONJUJGATE OF X(30508)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4 - (b^2 + c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(41880) lies on the cubics K162, K229, K1067 and these lines: {6, 5638}, {110, 351}, {381, 3413}, {512, 1380}, {523, 6189}, {30508, 35345}, {38583, 38597}

X(41880) = isogonal conjugate of X(30508)
X(41880) = isogonal conjugate of the anticomplement of X(13636)
X(41880) = isogonal conjugate of the isotomic conjugate of X(30509)
X(41880) = X(110)-Ceva conjugate of X(1380)
X(41880) = X(512)-cross conjugate of X(5638)
X(41880) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30508}, {662, 39022}, {799, 2029}, {14501, 36084}
X(41880) = cevapoint of X(512) and X(5638)
X(41880) = crosspoint of X(110) and X(1380)
X(41880) = crosssum of X(523) and X(3414)
X(41880) = trilinear pole of line {2028, 3124}
X(41880) = crossdifference of every pair of points on line {115, 2029}
X(41880) = barycentric product X(i)*X(j) for these {i,j}: {6, 30509}, {99, 2028}, {110, 39023}, {1380, 3413}, {2715, 14502}, {5638, 6189}
X(41880) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 30508}, {512, 39022}, {669, 2029}, {1380, 6190}, {2028, 523}, {3569, 14501}, {5638, 3414}, {30509, 76}, {39023, 850}
X(41880) = {X(110),X(5467)}-harmonic conjugate of X(41881)

X(41881) = ISOGONAL CONJUJGATE OF X(30509)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4 + (b^2 + c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(41881) lies on the cubics K162, K229, K1067 and these lines: {6, 5639}, {110, 351}, {381, 3414}, {512, 1379}, {523, 6190}, {30509, 35345}, {38583, 38596}

X(41881) = isogonal conjugate of X(30509)
X(41881) = isogonal conjugate of the anticomplement of X(13722)
X(41881) = isogonal conjugate of the isotomic conjugate of X(30508)
X(41881) = X(110)-Ceva conjugate of X(1379)
X(41881) = X(512)-cross conjugate of X(5639)
X(41881) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30509}, {662, 39023}, {799, 2028}, {14502, 36084}
X(41881) = cevapoint of X(512) and X(5639)
X(41881) = crosspoint of X(110) and X(1379)
X(41881) = crosssum of X(523) and X(3413)
X(41881) = trilinear pole of line {2029, 3124}
X(41881) = crossdifference of every pair of points on line {115, 2028}
X(41881) = barycentric product X(i)*X(j) for these {i,j}: {6, 30508}, {99, 2029}, {110, 39022}, {1379, 3414}, {2715, 14501}, {5639, 6190}
X(41881) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 30509}, {512, 39023}, {669, 2028}, {1379, 6189}, {2029, 523}, {3569, 14502}, {5639, 3413}, {30508, 76}, {39022, 850}

X(41882) = ISOGONAL CONJUJGATE OF X(40846)

Barycentrics    a^3*(b^2 + a*c)*(a*b + c^2)*(a^3 - b^3 + a*b*c - c^3) : :

X(41882) lies on the cubics K225, K532, K771, and these lines: {1, 18784}, {6, 893}, {31, 40729}, {32, 904}, {256, 5276}, {257, 384}, {385, 40849}, {694, 1914}, {1055, 29055}, {1178, 21764}, {1967, 21760}, {2670, 21746}, {3051, 7104}, {4603, 17731}, {18783, 18786}, {19557, 40873}, {20179, 32010} X(41882) = isogonal conjugate of X(40846)
X(41882) = isogonal conjugate of the isotomic conjugate of X(40873)
X(41882) = X(694)-Ceva conjugate of X(904)
X(41882) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40846}, {2, 7061}, {75, 41534}, {171, 40845}, {172, 18036}, {894, 7261}, {1909, 3512}, {1920, 8852}, {1966, 24479}, {3978, 30648}, {7196, 7281}
X(41882) = crosssum of X(3512) and X(39920)
X(41882) = barycentric product X(i)*X(j) for these {i,j}: {1, 41532}, {6, 40873}, {256, 17798}, {257, 19554}, {694, 19557}, {893, 3509}, {904, 4645}, {1178, 20715}, {1281, 1967}, {1581, 19561}, {1916, 18038}, {7018, 18262}, {7104, 17789}, {9468, 18037}
X(41882) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40846}, {31, 7061}, {32, 41534}, {256, 18036}, {893, 40845}, {904, 7261}, {1281, 1926}, {1927, 30648}, {3509, 1920}, {5018, 7205}, {7104, 3512}, {9468, 24479}, {17798, 1909}, {18037, 14603}, {18038, 385}, {18262, 171}, {19554, 894}, {19557, 3978}, {19561, 1966}, {20715, 1237}, {40873, 76}, {41532, 75}

X(41883) = COMPLEMENT OF X(222)

Barycentrics    (a - b - c)*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - a^2*b^2*c + b^4*c + a^3*c^2 - a^2*b*c^2 + 2*a*b^2*c^2 + a^2*c^3 - a*c^4 + b*c^4 - c^5) : :

X(41883) lies on these lines: {2, 222}, {5, 34831}, {9, 440}, {10, 5777}, {57, 26005}, {69, 27539}, {72, 1905}, {81, 28836}, {118, 124}, {141, 3452}, {197, 29207}, {212, 33305}, {219, 5739}, {223, 17073}, {226, 6708}, {281, 329}, {343, 908}, {406, 7078}, {946, 5909}, {958, 5810}, {971, 34822}, {1146, 3735}, {1150, 28826}, {1329, 3454}, {1745, 18641}, {1848, 14557}, {1861, 5927}, {1944, 33066}, {2183, 19542}, {2635, 21912}, {2968, 24430}, {3074, 7515}, {3580, 31053}, {3687, 40880}, {3782, 4858}, {3816, 34589}, {3844, 18227}, {4364, 16579}, {5044, 34823}, {5226, 26540}, {5249, 37648}, {5278, 28796}, {5328, 33172}, {5723, 20268}, {5737, 5778}, {5743, 5745}, {5812, 39585}, {6180, 20266}, {6922, 14058}, {7046, 7358}, {7359, 28950}, {8679, 23304}, {8757, 34120}, {10570, 31832}, {11433, 37543}, {14555, 27509}, {17860, 38357}, {18228, 32782}, {23058, 28609}, {25525, 25964}, {26223, 26609}, {26579, 27064}, {27131, 37636}, {28122, 30706}, {28951, 32859}, {31445, 34851}, {33117, 38211}

X(41883) = midpoint of X(i) and X(j) for these {i,j}: {72, 1905}, {1763, 5928}
X(41883) = complement of X(222)
X(41883) = complement of the isogonal conjugate of X(281)
X(41883) = complement of the isotomic conjugate of X(7017)
X(41883) = polar conjugate of the isogonal conjugate of X(40944)
X(41883) = medial-isogonal conjugate of X(17073)
X(41883) = crosspoint of X(2) and X(7017)
X(41883) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 17073}, {2, 34822}, {4, 142}, {6, 17102}, {8, 18589}, {9, 3}, {10, 18642}, {19, 1}, {25, 3752}, {27, 3742}, {28, 3946}, {29, 3739}, {33, 2}, {34, 4000}, {37, 18641}, {41, 216}, {42, 18592}, {55, 1214}, {65, 18643}, {75, 18639}, {78, 6389}, {91, 18638}, {92, 2886}, {108, 7658}, {158, 16608}, {162, 17069}, {210, 440}, {212, 6509}, {225, 18635}, {264, 17046}, {270, 17045}, {273, 21258}, {278, 11019}, {281, 10}, {284, 37565}, {286, 17050}, {312, 1368}, {318, 141}, {346, 34823}, {393, 1210}, {607, 37}, {644, 20315}, {650, 2968}, {653, 3900}, {657, 35072}, {811, 17066}, {1021, 34588}, {1039, 4657}, {1096, 3772}, {1172, 1125}, {1334, 18591}, {1435, 5573}, {1783, 522}, {1824, 17056}, {1826, 442}, {1857, 226}, {1861, 17060}, {1880, 1834}, {1896, 34830}, {1897, 4885}, {1969, 17047}, {1973, 17053}, {2190, 17043}, {2207, 20227}, {2212, 39}, {2250, 856}, {2287, 34851}, {2299, 3666}, {2319, 20254}, {2321, 21530}, {2322, 960}, {2326, 4999}, {2331, 7952}, {2332, 40937}, {2333, 2092}, {2489, 16613}, {2501, 8286}, {3064, 11}, {3239, 123}, {3700, 34846}, {3709, 16573}, {3900, 16596}, {4041, 15526}, {4086, 127}, {4130, 40616}, {4183, 5745}, {6059, 16583}, {6335, 17072}, {6591, 3756}, {7003, 946}, {7007, 278}, {7008, 57}, {7012, 17044}, {7017, 2887}, {7020, 21239}, {7046, 3452}, {7071, 1212}, {7079, 9}, {7101, 1329}, {7115, 24025}, {7129, 3086}, {7133, 31534}, {7154, 1108}, {7156, 1249}, {7649, 4904}, {7719, 6600}, {7952, 20206}, {8611, 122}, {8748, 942}, {8750, 905}, {8756, 1145}, {13426, 31591}, {13427, 13389}, {13454, 31590}, {13456, 13388}, {15742, 21232}, {17442, 17055}, {17924, 17059}, {17926, 34589}, {18344, 1086}, {20613, 23050}, {31623, 3741}, {32085, 17048}, {32674, 6129}, {36119, 18644}, {36125, 17067}, {36128, 17070}, {36797, 4369}, {39943, 1062}, {40117, 8058}, {40169, 20266}, {40573, 11018}, {40838, 6245}, {40971, 223}, {41013, 17052}, {41505, 24779}
X(41883) = barycentric product X(i)*X(j) for these {i,j}: {8, 41007}, {264, 40944}
X(41883) = barycentric quotient X(i)/X(j) for these {i,j}: {40944, 3}, {41007, 7}
X(41883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 34048, 36949}, {2, 37781, 17074}, {226, 13567, 16608}, {226, 20262, 6708}, {5739, 27540, 219}

X(41884) = ISOGONAL CONJUGATE OF X(21355)

Barycentrics    (a^2 + b^2)*(a^2 + c^2)*(a^4 - a^2*b^2 - b^4 - a^2*c^2 - b^2*c^2 - c^4) : :
X(41884) = 3 X[2] + X[8272]

X(41884) lies on the cubic K252 and these lines: {2, 1031}, {3, 8928}, {6, 40850}, {82, 171}, {83, 316}, {95, 39182}, {141, 4577}, {206, 3096}, {308, 338}, {385, 1194}, {419, 32085}, {689, 9230}, {733, 6375}, {827, 7831}, {1078, 8265}, {1176, 36213}, {2329, 34055}, {2896, 14885}, {7664, 10130}, {7760, 40003}, {7761, 16095}, {7875, 41295}, {8290, 8856}, {9233, 10347}, {9481, 33021}, {10329, 40035}, {20998, 39999}

X(41884) = complement of X(1031)
X(41884) = midpoint of X(i) and X(j) for these {i,j}: {1031, 8272}, {4577, 9483}
X(41884) = isogonal conjugate of X(21355)
X(41884) = isotomic conjugate of X(33665)
X(41884) = complement of the isogonal conjugate of X(10329)
X(41884) = complement of the isotomic conjugate of X(2896)
X(41884) = isotomic conjugate of the complement of X(13511)
X(41884) = isotomic conjugate of the isogonal conjugate of X(14885)
X(41884) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 83}, {2896, 2887}, {10329, 10}, {14885, 1215}, {16556, 141}, {17083, 17046}, {20934, 626}, {21083, 21245}, {21194, 21252}, {21880, 3454}, {22138, 18589}, {40035, 21235}
X(41884) = X(2)-Ceva conjugate of X(83)
X(41884) = X(10329)-cross conjugate of X(14885)
X(41884) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21355}, {31, 33665}, {38, 14370}, {39, 39725}, {1031, 1964}, {3051, 18834}
X(41884) = cevapoint of X(i) and X(j) for these (i,j): {2, 13511}, {2896, 10329}
X(41884) = crosspoint of X(2) and X(2896)
X(41884) = crosssum of X(6) and X(14370)
X(41884) = barycentric product X(i)*X(j) for these {i,j}: {76, 14885}, {82, 20934}, {83, 2896}, {251, 40035}, {308, 10329}, {3112, 16556}, {39938, 40850}
X(41884) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 33665}, {6, 21355}, {82, 39725}, {83, 1031}, {251, 14370}, {2896, 141}, {3112, 18834}, {10329, 39}, {14885, 6}, {16556, 38}, {17083, 3665}, {20934, 1930}, {21083, 15523}, {21194, 16892}, {21880, 3954}, {22138, 3917}, {39938, 17949}, {40035, 8024}
X(41884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8272, 1031}, {16890, 41296, 83}

X(41885) = COMPLEMENT OF X(1143)

Barycentrics    Tan[B/4] + Tan[C/4] : :
Barycentrics    1 + cos(A/2) - sin(A/2) : :

X(41885) lies on the cubic K363 and these lines: {1, 188}, {2, 1143}, {174, 557}, {236, 21465}, {258, 3082}, {2090, 14121}, {13388, 15495}

X(41885) = complement of X(1143)
X(41885) = complement of the isotomic conjugate of X(1274)
X(41885) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1489}, {1274, 2887}, {7001, 141}
X(41885) = X(2)-Ceva conjugate of X(1489)
X(41885) = X(258)-cross conjugate of X(1489)
X(41885) = crosspoint of X(2) and X(1274)
X(41885) = barycentric product X(1274)*X(1489)
X(41885) = barycentric quotient X(i)/X(j) for these {i,j}: {258, 1489}, {1489, 1143}
X(41885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16015, 1489}, {188, 7028, 1489}

X(41886) = COMPLEMENT OF X(7155)

Barycentrics    a*(a*b + a*c - b*c)*(b^2 - b*c + c^2) : :

X(41886) lies on the Feuerbach circumhyperbola of the medial triangle, the cubics K1012 and K1035, and on these lines: {1, 17792}, {2, 256}, {3, 238}, {6, 7166}, {7, 291}, {9, 1575}, {10, 75}, {31, 27678}, {37, 19584}, {38, 17236}, {42, 28402}, {43, 1403}, {69, 24575}, {100, 983}, {119, 5518}, {141, 4443}, {142, 17063}, {171, 19591}, {192, 3123}, {214, 995}, {330, 26135}, {442, 20338}, {579, 1755}, {599, 24437}, {714, 18144}, {982, 2887}, {1045, 3795}, {1086, 4446}, {1581, 7146}, {1739, 17057}, {1756, 3216}, {1909, 7275}, {2053, 30649}, {2176, 3502}, {2234, 3647}, {2236, 7262}, {2275, 7184}, {2276, 24458}, {2345, 24463}, {3009, 25279}, {3061, 3094}, {3122, 17234}, {3126, 21189}, {3728, 17238}, {3752, 25135}, {3755, 12640}, {3764, 16706}, {3782, 20487}, {3888, 7032}, {3912, 22214}, {3914, 24997}, {3936, 23444}, {3944, 20545}, {4000, 24478}, {4022, 17227}, {4283, 4655}, {4484, 7232}, {4492, 17293}, {4735, 17235}, {6184, 21796}, {6374, 18277}, {6377, 23643}, {6383, 19567}, {6686, 17353}, {7174, 11530}, {7239, 21815}, {9025, 18194}, {10472, 17064}, {14815, 33833}, {15287, 22754}, {16604, 25528}, {16609, 19222}, {17122, 34261}, {17127, 27658}, {17230, 22167}, {17232, 21330}, {17244, 22172}, {17257, 26038}, {17262, 24338}, {17327, 24450}, {17364, 20456}, {17367, 23659}, {17786, 25140}, {18168, 35552}, {20340, 30090}, {20361, 20686}, {20498, 33101}, {20544, 24230}, {20547, 24341}, {20923, 21257}, {20966, 33125}, {21935, 25010}, {23524, 24625}, {25277, 26756}, {25521, 33130}, {25928, 32932}, {26042, 33159}, {28248, 28287}

X(41886) = reflection of X(17786) in X(25140)
X(41886) = complement of X(7155)
X(41886) = complement of the isogonal conjugate of X(1403)
X(41886) = complement of the isotomic conjugate of X(3212)
X(41886) = polar conjugate of the isogonal conjugate of X(20783)
X(41886) = medial isogonal conjugate of X(20545)
X(41886) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20545}, {6, 20258}, {31, 3061}, {34, 20256}, {43, 1329}, {56, 3840}, {57, 20255}, {109, 4083}, {192, 21244}, {603, 20254}, {604, 75}, {651, 21191}, {1106, 17063}, {1395, 20271}, {1397, 16604}, {1402, 21024}, {1403, 10}, {1407, 20257}, {1415, 31286}, {1423, 141}, {1436, 20259}, {2176, 3452}, {2209, 9}, {3212, 2887}, {4083, 124}, {4564, 40562}, {6611, 20260}, {8640, 1146}, {16695, 34589}, {20760, 34823}, {20979, 26932}, {22090, 123}, {27644, 21246}, {30545, 626}, {38832, 960}, {41526, 2}
X(41886) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3061}, {100, 4083}, {2275, 982}
X(41886) = X(33890)-cross conjugate of X(982)
X(41886) = X(i)-isoconjugate of X(j) for these (i,j): {87, 983}, {2162, 17743}, {2319, 7132}, {3114, 40736}, {7033, 7121}, {7148, 7305}, {21759, 38810}, {23493, 40415}
X(41886) = crosspoint of X(i) and X(j) for these (i,j): {2, 3212}, {87, 39742}, {2275, 20284}, {20332, 33680}
X(41886) = crosssum of X(i) and X(j) for these (i,j): {6, 2053}, {43, 8616}, {87, 17105}
X(41886) = crossdifference of every pair of points on line {1919, 21348}
X(41886) = barycentric product X(i)*X(j) for these {i,j}: {1, 33890}, {43, 3662}, {75, 20284}, {192, 982}, {264, 20783}, {1423, 3705}, {2176, 33930}, {2275, 6376}, {2887, 27644}, {3056, 30545}, {3061, 3212}, {3208, 7185}, {3721, 33296}, {3777, 4595}, {3778, 31008}, {3835, 3888}, {3863, 41318}, {3865, 17752}, {4083, 33946}, {4110, 7248}, {6382, 7032}, {7237, 7304}, {7239, 17217}, {20234, 38832}, {20691, 33947}, {33891, 41531}
X(41886) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 17743}, {192, 7033}, {982, 330}, {1403, 7132}, {2176, 983}, {2275, 87}, {3056, 2319}, {3061, 7155}, {3662, 6384}, {3705, 27424}, {3778, 16606}, {3865, 27447}, {3888, 4598}, {6382, 7034}, {7032, 2162}, {7185, 7209}, {7248, 7153}, {16584, 23493}, {18197, 7255}, {20284, 1}, {20665, 2053}, {20783, 3}, {27644, 40415}, {33296, 38810}, {33890, 75}, {33930, 6383}, {33946, 18830}, {40935, 21759}
X(41886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {142, 17065, 17063}, {192, 3123, 4941}, {978, 6210, 238}, {1716, 27626, 238}, {3094, 18904, 3061}, {3116, 33891, 982}, {3662, 3778, 982}, {17792, 28358, 1}, {25279, 28395, 3009}

X(41887) = COMPLEMENT OF X(11078)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S) : :
X(41887) = X[1546] + 2 X[10564]

X(41887) lies on the cubics K472 and K867a and these lines: {2, 13}, {14, 2986}, {30, 113}, {110, 36186}, {298, 340}, {299, 1494}, {323, 533}, {395, 6128}, {396, 3003}, {511, 32460}, {531, 40112}, {532, 3580}, {617, 5669}, {619, 34834}, {858, 41023}, {3260, 36298}, {3268, 6137}, {3642, 15066}, {5972, 32461}, {6106, 6670}, {6110, 14920}, {6671, 37851}, {6782, 30465}, {8836, 33561}, {10654, 37645}, {11094, 14165}, {15769, 36756}, {19774, 40711}
X(41887) = midpoint of X(i) and X(j) for these {i,j}: {110, 36186}, {299, 23896}, {323, 11092}
X(41887) = reflection of X(i) in X(j) for these {i,j}: {1524, 113}, {32461, 5972}
X(41887) = complement of X(11078)
X(41887) = isotomic conjugate of X(36308)
X(41887) = complement of the isogonal conjugate of X(11086)
X(41887) = complement of the isotomic conjugate of X(11092)
X(41887) = isotomic conjugate of the polar conjugate of X(6110)
X(41887) = X(i)-complementary conjugate of X(j) for these (i,j): {2151, 619}, {2154, 623}, {11086, 10}, {11092, 2887}, {23284, 21253}
X(41887) = X(i)-Ceva conjugate of X(j) for these (i,j): {299, 533}, {23896, 23870}, {40710, 33529}
X(41887) = X(i)-isoconjugate of X(j) for these (i,j): {13, 2159}, {19, 39377}, {31, 36308}, {74, 2153}, {2152, 5627}, {2349, 3457}, {8737, 35200}, {20578, 36034}, {36119, 36296}
X(41887) = crosspoint of X(2) and X(11092)
X(41887) = crosssum of X(6) and X(11081)
X(41887) = crossdifference of every pair of points on line {2433, 3457}
X(41887) = barycentric product X(i)*X(j) for these {i,j}: {14, 6148}, {15, 3260}, {30, 298}, {69, 6110}, {301, 1511}, {470, 11064}, {2407, 23870}, {5664, 23896}, {6782, 36891}, {7799, 36298}, {11129, 36299}, {14920, 40710}, {17402, 41079}
X(41887) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36308}, {3, 39377}, {14, 5627}, {15, 74}, {30, 13}, {298, 1494}, {470, 16080}, {1495, 3457}, {1511, 16}, {1637, 20578}, {1990, 8737}, {2151, 2159}, {2173, 2153}, {2407, 23895}, {2420, 5995}, {3163, 36299}, {3258, 30468}, {3260, 300}, {3284, 36296}, {3458, 40355}, {4240, 36306}, {5616, 3470}, {5664, 23871}, {6110, 4}, {6137, 2433}, {6148, 299}, {6782, 36875}, {8739, 8749}, {9214, 36307}, {11064, 40709}, {11092, 36311}, {14920, 471}, {23870, 2394}, {23896, 39290}, {30465, 12079}, {34394, 40352}, {36296, 39380}, {36297, 11079}, {36298, 1989}, {36299, 11080}, {39176, 8740}
X(41887) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 616, 40709}, {2, 8838, 6669}, {2, 11131, 618}

X(41888) = COMPLEMENT OF X(11092)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S) : :
X(41888) = X[1545] + 2 X[10564]

X(41888) lies on the cubics kK472 and K867b and these lines: {2, 14}, {13, 2986}, {30, 113}, {110, 36185}, {298, 1494}, {299, 340}, {323, 532}, {395, 3003}, {396, 6128}, {511, 32461}, {530, 40112}, {533, 3580}, {616, 5668}, {618, 34834}, {858, 41022}, {3260, 36299}, {3268, 6138}, {3643, 15066}, {5972, 32460}, {6107, 6669}, {6111, 14920}, {6672, 37852}, {6783, 30468}, {8838, 33560}, {10653, 37645}, {11093, 14165}, {15768, 36755}, {19775, 40712}

X(41888) = midpoint of X(i) and X(j) for these {i,j}: {110, 36185}, {298, 23895}, {323, 11078}
X(41888) = reflection of X(i) in X(j) for these {i,j}: {1525, 113}, {32460, 5972}
X(41888) = isotomic conjugate of X(36311)
X(41888) = complement of X(11092)
X(41888) = complement of the isogonal conjugate of X(11081)
X(41888) = complement of the isotomic conjugate of X(11078)
X(41888) = isotomic conjugate of the polar conjugate of X(6111)
X(41888) = X(i)-complementary conjugate of X(j) for these (i,j): {2152, 618}, {2153, 624}, {11078, 2887}, {11081, 10}, {23283, 21253}
X(41888) = X(i)-Ceva conjugate of X(j) for these (i,j): {298, 532}, {23895, 23871}, {40709, 33530}
X(41888) = X(i)-isoconjugate of X(j) for these (i,j): {14, 2159}, {19, 39378}, {31, 36311}, {74, 2154}, {2151, 5627}, {2349, 3458}, {8738, 35200}, {20579, 36034}, {36119, 36297}
X(41888) = crosspoint of X(2) and X(11078)
X(41888) = crosssum of X(6) and X(11086)
X(41888) = crossdifference of every pair of points on line {2433, 3458}
X(41888) = barycentric product X(i)*X(j) for these {i,j}: {13, 6148}, {16, 3260}, {30, 299}, {69, 6111}, {300, 1511}, {471, 11064}, {2407, 23871}, {5664, 23895}, {6783, 36891}, {7799, 36299}, {11128, 36298}, {14920, 40709}, {17403, 41079}
X(41888) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36311}, {3, 39378}, {13, 5627}, {16, 74}, {30, 14}, {299, 1494}, {471, 16080}, {1495, 3458}, {1511, 15}, {1637, 20579}, {1990, 8738}, {2152, 2159}, {2173, 2154}, {2407, 23896}, {2420, 5994}, {3163, 36298}, {3258, 30465}, {3260, 301}, {3284, 36297}, {3457, 40355}, {4240, 36309}, {5612, 3470}, {5664, 23870}, {6111, 4}, {6138, 2433}, {6148, 298}, {6783, 36875}, {8740, 8749}, {9214, 36310}, {11064, 40710}, {11078, 36308}, {14920, 470}, {23871, 2394}, {23895, 39290}, {30468, 12079}, {34395, 40352}, {36296, 11079}, {36297, 39381}, {36298, 11085}, {36299, 1989}, {39176, 8739}
X(41888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 617, 40710}, {2, 8836, 6670}, {2, 11130, 619}

X(41889) = X(6)X(13) ∩ X(472)X(648)

Barycentrics    (2*sqrt(3)*a^2*S+a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*(2*sqrt(3)*(-a^2+b^2+c^2)*S+5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(41889) = X(38943)+2*X(50465)

X(41889) lies on the cubic K419a and these lines: {2,2993}, {6,13}, {30,5668}, {298,1494}, {395,11081}, {472,648}, {531,5995}, {533,33499}, {549,50468}, {1081,2153}, {2986,40158}, {3181,11078}, {3457,37641}, {5663,46466}, {5872,8919}, {6104,16963}, {6107,16267}, {9033,23283}, {11083,43229}, {11142,49948}, {17702,46465}

X(41889) = inverse of X(36208) in MacBeath circumconic
X(41889) = intersection, other than A, B, C, of circumconics {A, B, C, X(2), X(51277)} and {A, B, C, X(6), X(40157)}
X(41889) = barycentric product X(i)*X(j) for these {i, j}: {13, 617}, {619, 39133}
X(41889) = barycentric quotient X(i)/X(j) for these (i, j): (13, 19777), (617, 298), (1989, 40159)
X(41889) = trilinear product X(i)*X(j) for these {i, j}: {13, 19299}, {617, 2153}
X(41889) = trilinear quotient X(2153)/X(3441)
X(41889) = {X(14), X(36208)}-harmonic conjugate of X(13)




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Centers from some problems in Russian Sharygin Olympiads: X(41890)-X(41911)

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This preamble and centers X(41890)-X(41911) were contributed by César Eliud Lozada, March 12, 2021.

  1. A line ℓ meets the sidelines BC, CA, AB of a triangle ABC in A', B', C', respectively. The perpendicular to BC through A' cuts AC, AB at Ab and Ac, respectively and Oa is the circumcenter of AAbAc. Build Ob and Oc cyclically. Then Oa, Ob and Oc are collinear.. (Source: XII Geometrical Olympiad in honour of I. F. Sharygin (2016) - Correspondence round, problem 13) (*).

    If ℓ is the trilinear polar of P = x : y : z (barycentrics) then the given circumcenters are collinear on a line whose trilinear pole has coordinates:

     Q(P) = a^2*(SA*x + SB*y)*(SA*x + SC*z) : :

    Q(P) is named here the Muscovite point of P. The appearance of (i, j) in the following partial list means that Q(X(i)) = X(j):

    (1, 284), (2, 2), (3, 41890), (4, 6), (5, 41891), (6, 54), (9, 1167), (10, 2983), (13, 41892), (14, 41893), (19, 58), (20, 41894), (24, 8882), (25, 251), (27, 1171), (28, 1169), (30, 34570), (33, 1174), (34, 3451), (37, 943), (53, 1173), (75, 40403), (76, 40405), (92, 81), (95, 2984), (107, 23964), (108, 7115), (111, 10422), (112, 250), (158, 1172), (186, 14910), (196, 222), (216, 40448), (226, 40407), (230, 3563), (231, 2383), (232, 98), (240, 2311), (242, 1438), (264, 83), (273, 1170), (275, 288), (278, 57), (281, 1), (286, 40408), (297, 2987), (318, 23617), (321, 40406), (324, 40393), (393, 4), (403, 8749), (406, 941), (419, 1976), (420, 694), (427, 3108), (428, 34572), (451, 37), (458, 30535), (459, 1073), (468, 111), (470, 2981), (471, 6151), (475, 39956)

    Note: Q(P) = IsogonalConjugate(Complement(IsotomicConjugate(PolarConjugate(P))))

  2. Given a triangle ABC and an arbitrary point P, the lines passing through P and perpendicular to segments AP, BP, CP meet lines BC, CA, AB at points A', B', C', respectively. Then the respective midpoints A", B", C" of segments AA', BB', CC' are collinear. (Source: X Geometrical Olympiad in honour of I. F. Sharygin (2014) - Correspondence round, problem 16) (*).

    If P = x : y : z (barycentrics) then the trilinear pole of the line through A", B", C" is:

     Q(P) = (SA*x^2+SB*y^2-(2*x+2*y+z)*SC*z)*(SA*x^2+SC*z^2-(2*x+y+2*z)*SB*y) : :

    Q(P) is named here the Siberian point of P. Q(P)=X(2) for P in the line at infinity. For other P, the appearance of (i, j) in the following list means that Q(X(i)) = X(j):

    (1, 189), (2, 41895), (3, 13579), (4, 2), (5, 13585), (6, 41896), (13, 19776), (14, 19777), (15, 41897), (16, 41898), (20, 41899), (107, 15351), (112, 13485), (115, 13485), (125, 15351), (186, 13585), (403, 13579), (468, 41895), (1113, 13580), (1114, 13581), (1312, 13581), (1313, 13580), (1785, 189), (5523, 41896), (6110, 19776), (6111, 19777), (6116, 41897), (6117, 41898), (10151, 41899), (39162, 39159), (39163, 39158)
  3. Given are a triangle ABC and a line ℓ intersecting BC, CA, AB at A', B', C', respectively. Ab and Ac are the orthogonal projections of A' in AC and AB (resp.). A" is the midpoint of Ab and Ac and B", C" are built cyclically. Then A", B", C" are collinear on a line 𝓂. (Source: VII Geometrical Olympiad in honour of I. F. Sharygin (2011) - Correspondence round, problem 23) (*).

    Assume the line ℓ is the tripolar of P = x : y : z (barycentrics) and Q(P) is the tripole of the line 𝓂. Point Q(P), here named the Caucasian point of P, has coordinates:

     Q(P) = ((a^2*y*z-b^2*x*z+c^2*x*y)*S^2-2*(SC*c^2*x+SA*a^2*z)*SB*y) * ((a^2*y*z+b^2*x*z-c^2*x*y)*S^2-2*(SB*b^2*x+SA*a^2*y)*SC*z) : :

    The appearance of (i, j) in the following partial list means that Q(X(i)) = X(j):

    (1, 280), (2, 2), (3, 20), (4, 3346), (6, 253), (9, 347), (13, 41900), (14, 41901), (15, 41902), (16, 41903), (54, 4), (57, 41514), (58, 7219), (74, 1294), (81, 7361), (98, 1297), (99, 3565), (100, 13397), (101, 1305), (102, 41904), (103, 41905), (104, 1295), (105, 26703), (106, 2370), (107, 110), (108, 100), (109, 41906), (110, 925), (111, 2373), (112, 99), (249, 15351), (250, 13573), (251, 13575), (275, 34287), (284, 7), (371, 1131), (372, 1132), (476, 10420), (477, 2693), (759, 39435), (801, 35061), (842, 2697), (847, 254), (915, 104), (917, 103), (925, 13398), (930, 20185), (933, 930), (935, 691), (943, 1), (953, 2734), (961, 40457), (1113, 1113), (1114, 1114), (1141, 18401), (1167, 8), (1173, 15318), (1249, 6527), (1287, 11635), (1289, 112), (1294, 5897), (1296, 20187), (1297, 34168), (1299, 1300), (1300, 74), (1301, 107), (1304, 476), (1309, 901), (1461, 190), (1487, 3459), (2374, 111), (2383, 1141)

    Note: If P lies on the circumcircle of ABC:
     Q(P) = IsotomicConjugate(PolarConjugate(IsogonalConjugate(Complement(PolarConjugate(P)))))
    otherwise:
     Q(P) = IsotomicConjugate(Anticomplement(PolarConjugate(IsotomicConjugate(Anticomplement(IsogonalConjugate(P)))))).

  4. In a triangle ABC, let A'B'C' be the cevian triangle of a point P. Let Ab, Ac be the points where the circle {{P, B', C'}} cuts again the sidelines AC and AB, respectively, and build (Bc, Ba), (Ca, Cb) cyclically. Let a1 be the line joining the midpoints of AbAc and BcCb and define lines b1 and c1 similarly. Then these three lines concur. (Source: XIII Geometrical Olympiad in honour of I. F. Sharygin (2017) - Correspondence round, problem 19) (*).

    If P = x : y : z (barycentrics) then the given point of concurrence Q'(P) has coordinates:

      Q'(P) = a^2*(x+y)*(x+z)*y*z : :

    Added by César Lozada: If a2 is the line joining the midpoints of BaCa and BcCb and b2 and c2 are defined cyclically, then these three lines also concur in a point Q"(P).

    Q'(P) and Q"(P) are named here the 1st Crimean point of P and 2nd Crimean point of P, respectively.

    The appearance of (i, j) in the following partial list means that Q'(X(i)) = X(j):

    (1, 81), (2, 6), (3, 275), (4, 2), (5, 288), (6, 83), (7, 1), (8, 57), (9, 1170), (10, 1171), (11, 38809), (13, 2981), (14, 6151), (15, 41907), (16, 41908), (19, 40403), (20, 1073), (21, 2982), (22, 40404), (25, 40405), (27, 2983), (28, 40406), (29, 40407), (30, 40384), (31, 38810), (32, 38830), (37, 40408), (42, 40409), (53, 2984), (54, 40393), (56, 2985), (57, 23617), (58, 40394), (63, 1172), (64, 801), (65, 14534), (66, 76), (67, 671), (68, 2052), (69, 4), (70, 5392), (72, 40395), (74, 2986), (75, 58), (76, 251), (77, 40396), (78, 40397), (79, 1255), (80, 88), (81, 2298), (82, 40398), (83, 3108), (84, 40399), (85, 1174), (86, 1126), (88, 40400), (89, 40401), (92, 284), (94, 14910), (95, 1173), (97, 40402), (98, 2987), (99, 110), (100, 651), (102, 2988), (103, 2989), (104, 2990), (110, 648), (111, 41909), (145, 40151), (189, 9), (190, 101), (192, 7121), (193, 14248), (253, 3), (254, 1993), (255, 829), (256, 1258), (261, 18772), (262, 30535), (263, 3114), (264, 54), (265, 16080), (273, 1167), (275, 41891), (279, 7123), (280, 222), (286, 943), (290, 98), (291, 20332), (298, 16459), (299, 16460), (309, 947), (311, 1166), (312, 3451), (313, 3453), (314, 961), (315, 18018), (316, 10415), (317, 847), (320, 1168), (321, 1169), (325, 2065), (328, 38534), (329, 282), (330, 31), (335, 1438), (340, 5627), (347, 1433), (385, 34238), (388, 34260), (459, 41894), (470, 41892), (471, 41893), (485, 588), (486, 589)

    Note: Q'(P) = Complement(IsotomicConjugate(CyclocevianConjugate(P)))

(*): Ennunciates of cited problem have been slightly modified. Sources in Russian and English can be downloaded from geometry.ru.

X(41890) = MUSCOVITE POINT OF X(3)

Barycentrics    a^2*(a^6-(b^2+2*c^2)*a^4-(b^4-4*b^2*c^2-c^4)*a^2+(b^2-c^2)^2*b^2)*(a^6-(2*b^2+c^2)*a^4+(b^4+4*b^2*c^2-c^4)*a^2+(b^2-c^2)^2*c^2) : :

X(41890) lies on these lines: {2, 801}, {3, 41489}, {6, 2929}, {20, 393}, {25, 10313}, {32, 36413}, {37, 775}, {39, 41891}, {97, 8794}, {216, 8749}, {308, 26166}, {571, 34288}, {800, 22467}, {1880, 1951}, {1976, 6467}, {2165, 5063}, {2433, 17434}, {3108, 13341}, {3284, 8882}, {5523, 12362}, {7386, 13854}, {8791, 30739}, {9306, 14642}, {10312, 38292}, {15394, 17811}, {22240, 40144}, {26206, 40802}, {40065, 40132}

X(41890) = complement of the anticomplementary conjugate of X(394)
X(41890) = isogonal conjugate of X(13567)
X(41890) = barycentric product X(i)*X(j) for these {i, j}: {1, 775}, {3, 1105}, {6, 801}, {32, 40830}, {255, 821}
X(41890) = barycentric quotient X(i)/X(j) for these (i, j): (1, 17858), (3, 41005), (25, 235), (31, 774), (32, 800), (48, 6508)
X(41890) = trilinear product X(i)*X(j) for these {i, j}: {6, 775}, {31, 801}, {48, 1105}, {560, 40830}, {577, 821}
X(41890) = trilinear quotient X(i)/X(j) for these (i, j): (2, 17858), (3, 6508), (6, 774), (19, 235), (31, 800), (48, 185)
X(41890) = 1st Saragossa point of X(393)
X(41890) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(3), X(20)}}
X(41890) = cevapoint of X(i) and X(j) for these (i, j): {3, 9306}, {6, 577}, {32, 154}
X(41890) = crosspoint of X(801) and X(1105)
X(41890) = crosssum of X(i) and X(j) for these (i, j): {6, 2929}, {185, 800}
X(41890) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 13567}, {2, 774}, {4, 6508}, {6, 17858}, {10, 18603}
X(41890) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 17858), (3, 41005), (25, 235), (31, 774)

X(41891) = MUSCOVITE POINT OF X(5)

Barycentrics    a^2*(a^6-b^2*a^4-(b^4+3*c^4)*a^2+(b^2+2*c^2)*(b^2-c^2)^2)*(a^6-c^2*a^4-(3*b^4+c^4)*a^2+(2*b^2+c^2)*(b^2-c^2)^2) : :

X(41891) lies on these lines: {6, 5889}, {25, 22240}, {39, 41890}, {216, 7488}, {251, 800}, {324, 4993}, {393, 3091}, {570, 14910}, {1249, 13854}, {1297, 19157}, {1976, 21637}, {2963, 41335}, {5065, 41894}, {7772, 36413}, {8743, 11479}, {8791, 37454}, {10986, 16195}, {13342, 36414}, {14577, 34818}, {33579, 40144}

X(41891) = isogonal conjugate of X(23292)
X(41891) = isotomic conjugate of X(26166)
X(41891) = complement of the anticomplementary conjugate of X(343)
X(41891) = barycentric product X(3)*X(14860)
X(41891) = barycentric quotient X(i)/X(j) for these (i, j): (1, 17859), (2, 26166), (3, 41008), (25, 3575), (51, 3574), (184, 13367), (251, 10548)
X(41891) = cevapoint of X(i) and X(j) for these (i, j): {3, 34986}, {6, 216}
X(41891) = trilinear product X(48)*X(14860)
X(41891) = trilinear quotient X(i)/X(j) for these (i, j): (2, 17859), (19, 3575), (48, 13367), (63, 41008), (82, 10548), (1953, 3574)
X(41891) = 1st Saragossa point of X(8882)
X(41891) = trilinear pole of the line {512, 34983}
X(41891) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(3), X(253)}}
X(41891) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 23292}, {6, 17859}, {19, 41008}, {31, 26166}, {38, 10548}, {63, 3575}
X(41891) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 17859), (3, 41008), (25, 3575), (51, 3574)

X(41892) = MUSCOVITE POINT OF X(13)

Barycentrics    (SB+SC)*(S+sqrt(3)*SB)*(S+sqrt(3)*SC)*((SA+SB)*S+2*sqrt(3)*SA*SB)*((SA+SC)*S+2*sqrt(3)*SA*SC) : :

X(41892) lies on these lines: {61, 39377}, {3284, 8603}, {8739, 36296}, {8741, 36299}

X(41892) = isogonal conjugate of the complement of X(40709)
X(41892) = 1st Saragossa point of X(8739)
X(41892) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(14)}} and {{A, B, C, X(13), X(36296)}}
X(41892) = cevapoint of X(6) and X(36296)
X(41892) = X(647)-cross conjugate of-X(5995)

X(41893) = MUSCOVITE POINT OF X(14)

Barycentrics    (SB+SC)*(-S+sqrt(3)*SB)*(-S+sqrt(3)*SC)*(-(SA+SB)*S+2*sqrt(3)*SA*SB)*(-(SA+SC)*S+2*sqrt(3)*SA*SC) : :

X(41893) lies on these lines: {62, 39378}, {3284, 8604}, {8740, 36297}, {8742, 36298}

X(41893) = isogonal conjugate of the complement of X(40710)
X(41893) = trilinear quotient X(2154)/X(5321)
X(41893) = 1st Saragossa point of X(8740)
X(41893) = cevapoint of X(6) and X(36297)
X(41893) = X(647)-cross conjugate of-X(5994)

X(41894) = MUSCOVITE POINT OF X(20)

Barycentrics    a^2*(2*a^6-(2*b^2+3*c^2)*a^4-2*(b^2-3*c^2)*b^2*a^2+(2*b^2+c^2)*(b^2-c^2)^2)*(2*a^6-(3*b^2+2*c^2)*a^4+2*(3*b^2-c^2)*c^2*a^2+(b^2+2*c^2)*(b^2-c^2)^2) : :

X(41894) lies on these lines: {2, 22468}, {6, 22467}, {20, 18213}, {25, 38292}, {393, 3146}, {577, 8749}, {2433, 32320}, {5065, 41891}, {7396, 13854}, {8778, 11413}, {8791, 16051}

X(41894) = isogonal conjugate of X(26958)
X(41894) = barycentric product X(i)*X(j) for these {i, j}: {3, 18848}, {154, 34410}
X(41894) = barycentric quotient X(i)/X(j) for these (i, j): (1, 18691), (3, 40995), (25, 37197), (154, 5895), (184, 1204)
X(41894) = trilinear product X(48)*X(18848)
X(41894) = trilinear quotient X(i)/X(j) for these (i, j): (2, 18691), (19, 37197), (48, 1204), (63, 40995), (610, 5895)
X(41894) = 1st Saragossa point of X(41489)
X(41894) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(3), X(3146)}}
X(41894) = cevapoint of X(i) and X(j) for these (i, j): {6, 15905}, {32, 1660}
X(41894) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 18691}, {19, 40995}, {63, 37197}, {92, 1204}
X(41894) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 18691), (3, 40995), (25, 37197), (154, 5895)

X(41895) = SIBERIAN POINT OF X(2)

Barycentrics    (5*a^2+5*c^2-7*b^2)*(5*a^2+5*b^2-7*c^2) : :
X(41895) = 2*X(1327)+X(33457) = 2*X(1328)+X(33456)

X(41895) lies on the Kiepert hyperbola and these lines: {2, 11147}, {4, 5032}, {20, 7607}, {30, 7612}, {76, 7620}, {98, 3543}, {148, 5503}, {193, 671}, {262, 3839}, {381, 14494}, {439, 41139}, {524, 2996}, {543, 8781}, {1327, 33457}, {1328, 33456}, {3091, 7608}, {3146, 8859}, {3523, 10185}, {3545, 10155}, {3566, 5466}, {5254, 18845}, {5485, 8352}, {7519, 10511}, {7615, 32885}, {7617, 32867}, {7618, 32839}, {7825, 32890}, {7833, 32870}, {7841, 18840}, {8370, 18841}, {8587, 37689}, {8597, 11172}, {9166, 35927}, {10159, 32974}, {10302, 11185}, {11317, 18842}, {11668, 15692}, {14041, 40824}, {16080, 37174}, {21356, 32982}, {26617, 35873}, {26618, 35874}, {32479, 35287}, {32835, 33006}, {32893, 33017}, {32980, 41133}, {32996, 41136}

X(41895) = anticomplement of X(11147)
X(41895) = isogonal conjugate of X(5210)
X(41895) = isotomic conjugate of X(11160)
X(41895) = polar conjugate of the complement of X(30769)
X(41895) = antigonal conjugate of the isogonal conjugate of X(5107)
X(41895) = barycentric quotient X(1992)/X(11147)
X(41895) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(3), X(11482)}}
X(41895) = cevapoint of X(115) and X(1499)

X(41896) = SIBERIAN POINT OF X(6)

Barycentrics    (a^6-(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))*(a^6+(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(41896) lies on these lines: {2, 8744}, {4, 18019}, {23, 69}, {287, 37644}, {305, 316}, {6340, 31074}, {7394, 18018}, {7752, 14246}, {7806, 41769}, {10561, 14977}, {13575, 13595}

X(41896) = isotomic conjugate of X(16063)
X(41896) = trilinear pole of the line {525, 2492}
X(41896) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(4), X(23)}}
X(41896) = cevapoint of X(2) and X(7519)
X(41896) = X(1995)-cross conjugate of-X(2)
X(41896) = X(2)-vertex conjugate of-X(1485)

X(41897) = SIBERIAN POINT OF X(15)

Barycentrics    (SA*SC-S*sqrt(3)*SB)*(SA*SB-sqrt(3)*SC*S) : :

X(41897) lies on the Jerabek hyperbola and these lines: {2, 10639}, {3, 302}, {6, 473}, {64, 41038}, {68, 622}, {264, 2992}, {317, 2993}, {634, 3519}, {1899, 41898}, {8838, 36296}, {11143, 11421}, {16771, 36297}, {19712, 40693}

X(41897) = anticomplement of X(10639)
X(41897) = isogonal conjugate of X(3132)
X(41897) = isotomic conjugate of X(634)
X(41897) = cyclocevian conjugate of X(19713)
X(41897) = antigonal conjugate of the isogonal conjugate of X(37975)
X(41897) = barycentric product X(76)*X(3443)
X(41897) = barycentric quotient X(62)/X(10639)
X(41897) = trilinear product X(75)*X(3443)
X(41897) = trilinear pole of the line {647, 23872}
X(41897) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(302)}} and Jerabek hyperbola
X(41897) = cevapoint of X(125) and X(23873)
X(41897) = X(62)-cross conjugate of-X(2)
X(41897) = X(62)-reciprocal conjugate of-X(10639)

X(41898) = SIBERIAN POINT OF X(16)

Barycentrics    (SA*SC+S*sqrt(3)*SB)*(SA*SB+sqrt(3)*SC*S) : :

X(41898) lies on the Jerabek hyperbola and these lines: {2, 10640}, {3, 303}, {6, 472}, {64, 41039}, {68, 621}, {264, 2993}, {317, 2992}, {633, 3519}, {1899, 41897}, {8836, 36297}, {11144, 11420}, {16770, 36296}, {19713, 40694}

X(41898) = anticomplement of X(10640)
X(41898) = isogonal conjugate of X(3131)
X(41898) = isotomic conjugate of X(633)
X(41898) = cyclocevian conjugate of X(19712)
X(41898) = antigonal conjugate of the isogonal conjugate of X(37974)
X(41898) = barycentric product X(76)*X(3442)
X(41898) = barycentric quotient X(61)/X(10640)
X(41898) = trilinear product X(75)*X(3442)
X(41898) = trilinear pole of the line {647, 23873}
X(41898) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(303)}} and Jerabek hyperbola
X(41898) = cevapoint of X(125) and X(23872)
X(41898) = X(61)-cross conjugate of-X(2)
X(41898) = X(61)-reciprocal conjugate of-X(10640)

X(41899) = SIBERIAN POINT OF X(20)

Barycentrics    (5*a^6-(5*b^2+9*c^2)*a^4-(5*b^4-10*b^2*c^2-3*c^4)*a^2+(5*b^2+c^2)*(b^2-c^2)^2)*(5*a^6-(9*b^2+5*c^2)*a^4+(3*b^4+10*b^2*c^2-5*c^4)*a^2+(b^2+5*c^2)*(b^2-c^2)^2) : :

X(41899) lies on the Kiepert hyperbola and these lines: {4, 8780}, {193, 459}, {1916, 18287}, {2996, 37669}, {5395, 23292}, {6677, 14494}, {7612, 30771}, {8796, 37645}

X(41899) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(193), X(287)}}

X(41900) = CAUCASIAN POINT OF X(13)

Barycentrics    (sqrt(3)*(S^2-2*SB*(4*R^2-SB))+(4*R^2-SW)*S)*(sqrt(3)*(S^2-2*SC*(4*R^2-SC))+(4*R^2-SW)*S) : :

X(41900) lies on these lines: {393, 41901}, {394, 10676}, {473, 1073}

X(41900) = isogonal conjugate of X(10675)

X(41901) = CAUCASIAN POINT OF X(14)

Barycentrics    (sqrt(3)*(S^2-2*SB*(4*R^2-SB))-(4*R^2-SW)*S)*(sqrt(3)*(S^2-2*SC*(4*R^2-SC))-(4*R^2-SW)*S) : :

X(41901) lies on these lines: {393, 41900}, {394, 10675}, {472, 1073}

X(41901) = isogonal conjugate of X(10676)

X(41902) = CAUCASIAN POINT OF X(15)

Barycentrics    (S^2-sqrt(3)*(4*R^2-SW)*S-2*(12*R^2-2*SW-SB)*SB)*(S^2-sqrt(3)*(4*R^2-SW)*S-2*(12*R^2-2*SW-SC)*SC) : :

X(41902) lies on these lines: {20, 8919}, {30, 36302}, {11064, 19772}, {40138, 41903}

X(41903) = CAUCASIAN POINT OF X(16)

Barycentrics    (S^2+sqrt(3)*(4*R^2-SW)*S-2*(12*R^2-2*SW-SB)*SB)*(S^2+sqrt(3)*(4*R^2-SW)*S-2*(12*R^2-2*SW-SC)*SC) : :

X(41903) lies on these lines: {20, 8918}, {30, 36303}, {11064, 19773}, {40138, 41902}

X(41904) = CAUCASIAN POINT OF X(102)

Barycentrics
(a^8-b*a^7+(2*b+3*c)*(b-c)*a^6+(b-c)*(b-2*c)*b*a^5-(b^2-c^2)*(6*b^2-2*b*c+3*c^2)*a^4+(b-c)*(b^3+c^3+(3*b-c)*b*c)*b*a^3+(b^2-c^2)*(2*b^4+c^4-(3*b^2-5*b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*(b-c)*b^2*a+(b^2-c^2)^3*b^2)*(a^8-c*a^7-(3*b+2*c)*(b-c)*a^6+(2*b-c)*(b-c)*c*a^5+(b^2-c^2)*(3*b^2-2*b*c+6*c^2)*a^4-(b-c)*(b^3+c^3-(b-3*c)*b*c)*c*a^3-(b^2-c^2)*(b^4+2*c^4-(b^2-5*b*c+3*c^2)*b*c)*a^2+(b^2-c^2)^2*(b-c)*c^2*a-(b^2-c^2)^3*c^2) : :

X(41904) lies on the circumcircle and these lines: {3, 26704}, {4, 38977}, {7, 36079}, {20, 109}, {22, 9056}, {29, 1301}, {101, 27382}, {107, 4225}, {108, 411}, {112, 7415}, {515, 35183}, {915, 7456}, {917, 7455}, {934, 17134}, {1289, 7436}, {1300, 7454}, {1304, 7424}, {2071, 2689}, {2075, 22239}, {2374, 7439}, {2765, 10538}, {3563, 7441}, {5731, 8059}, {7412, 40097}, {7420, 26705}, {7443, 15344}, {7457, 40101}, {7488, 26709}, {9107, 35996}, {11413, 41906}, {29067, 30265}, {36067, 37420}

X(41904) = reflection of X(i) in X(j) for these (i, j): (4, 38977), (26704, 3)
X(41904) = isotomic conjugate of the anticomplement of X(8755)
X(41904) = circumperp conjugate of X(26704)
X(41904) = circumnormal-isogonal conjugate of the isogonal conjugate of X(26704)
X(41904) = circumtangential-isogonal conjugate of the isogonal conjugate of X(41904)
X(41904) = Collings transform of X(38977)
X(41904) = V-transform of X(26704)
X(41904) = trilinear pole of the line {6, 14331}
X(41904) = cevapoint of X(3) and X(515)
X(41904) = circumcircle-antipode of X(26704)
X(41904) = Λ(tangent to hyperbola {{A,B,C,X(4),X(58)}} at X(4) or X(102))

X(41905) = CAUCASIAN POINT OF X(103)

Barycentrics
(a^7-c*a^6+2*(b^2-c^2)*a^5-(b-c)*(3*b^2+2*b*c+2*c^2)*a^4-(b^2-c^2)*(3*b^2+c^2)*a^3+(b-c)*(2*b^4+c^4+(3*b^2+5*b*c+c^2)*b*c)*a^2+(b^2-c^2)^2*(b-c)*b^2)*(a^7-b*a^6-2*(b^2-c^2)*a^5+(b-c)*(2*b^2+2*b*c+3*c^2)*a^4+(b^2-c^2)*(b^2+3*c^2)*a^3-(b-c)*(b^4+2*c^4+(b^2+5*b*c+3*c^2)*b*c)*a^2-(b^2-c^2)^2*(b-c)*c^2) : :

X(41905) lies on the circumcircle and these lines: {2, 20622}, {3, 21665}, {20, 101}, {22, 9057}, {27, 1301}, {100, 18750}, {107, 4184}, {108, 7411}, {109, 9778}, {112, 4229}, {279, 36079}, {390, 8059}, {516, 35184}, {915, 7442}, {1289, 7431}, {1300, 7440}, {1304, 5196}, {1305, 11413}, {2071, 2690}, {2073, 22239}, {2374, 7432}, {2750, 20294}, {3563, 7433}, {4219, 40097}, {6183, 17134}, {7416, 26704}, {7445, 15344}, {7446, 40101}, {7455, 32706}, {7465, 9107}, {7474, 9064}, {7488, 26710}, {32710, 37166}, {37048, 40117}

X(41905) = reflection of X(26705) in X(3)
X(41905) = anticomplement of X(20622)
X(41905) = isotomic conjugate of the anticomplement of X(1886)
X(41905) = circumperp conjugate of X(26705)
X(41905) = circumnormal-isogonal conjugate of the isogonal conjugate of X(26705)
X(41905) = circumtangential-isogonal conjugate of the isogonal conjugate of X(41905)
X(41905) = barycentric quotient X(1886)/X(20622)
X(41905) = Collings transform of X(40616)
X(41905) = V-transform of X(26705)
X(41905) = trilinear pole of the line {6, 21172}
X(41905) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(4229)}} and {{A, B, C, X(3), X(4184)}}
X(41905) = circumcircle-antipode of-X(26705)
X(41905) = cevapoint of X(3) and X(516)
X(41905) = X(1886)-cross conjugate of-X(2)
X(41905) = X(1886)-reciprocal conjugate of-X(20622)

X(41906) = CAUCASIAN POINT OF X(109)

Barycentrics    (a^5-(b^2-b*c+2*c^2)*a^3-(b-c)^2*b*a^2+(b^2-c^2)*(b-c)*c*a+(b^2-c^2)^2*b)*(a-b)*(a^5-(2*b^2-b*c+c^2)*a^3-(b-c)^2*c*a^2+(b^2-c^2)*(b-c)*b*a+(b^2-c^2)^2*c)*(a-c) : :

X(41906) lies on the circumcircle and these lines: {2, 20620}, {3, 21666}, {20, 102}, {22, 1311}, {74, 28788}, {104, 37404}, {106, 24159}, {107, 7450}, {108, 7451}, {109, 14544}, {111, 26254}, {112, 7462}, {522, 35187}, {915, 6906}, {917, 7416}, {990, 29015}, {1289, 7463}, {1300, 7421}, {1301, 7452}, {2071, 2695}, {2374, 7449}, {2716, 10538}, {3101, 29056}, {3563, 7413}, {4224, 15344}, {4225, 39440}, {6099, 20294}, {7428, 40101}, {7460, 26705}, {7461, 40097}, {11413, 41904}, {37227, 39439}

X(41906) = reflection of X(32706) in X(3)
X(41906) = anticomplement of X(20620)
X(41906) = isotomic conjugate of the anticomplement of X(3064)
X(41906) = circumcircle-antipode of X(32706)
X(41906) = de-Longchamps-circle-inverse of X(33650)
X(41906) = circumperp conjugate of X(32706)
X(41906) = circumnormal-isogonal conjugate of the isogonal conjugate of X(32706)
X(41906) = circumtangential-isogonal conjugate of the isogonal conjugate of X(41906)
X(41906) = barycentric product X(648)*X(28788)
X(41906) = trilinear product X(162)*X(28788)
X(41906) = Collings transform of X(17073)
X(41906) = V-transform of X(32706)
X(41906) = trilinear pole of the line {6, 1210}
X(41906) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7462)}} and {{A, B, C, X(3), X(7450)}}
X(41906) = circumcircle-antipode of-X(32706)
X(41906) = cevapoint of X(i) and X(j) for these (i, j): {3, 522}, {513, 17102}, {514, 17073}, {523, 18641}

X(41907) = 1st CRIMEAN POINT OF X(15)

Barycentrics    (S+sqrt(3)*SB)*(S+sqrt(3)*SC)*(sqrt(3)*(S^2+SA*SC)+(SB+SW)*S)*(sqrt(3)*(S^2+SA*SB)+(SC+SW)*S) : :

X(41907) lies on these lines: {13, 14181}, {50, 396}, {323, 41000}, {463, 34397}, {5472, 23357}, {8014, 34395}

X(41907) = complement of the anticomplementary conjugate of X(300)
X(41907) = isogonal conjugate of X(40695)
X(41907) = barycentric quotient X(i)/X(j) for these (i, j): (13, 623), (476, 35316)
X(41907) = trilinear quotient X(2151)/X(8016)
X(41907) = 1st Saragossa point of X(34394)
X(41907) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(396)}} and {{A, B, C, X(6), X(50)}}
X(41907) = cevapoint of X(6) and X(13)
X(41907) = X(623)-isoconjugate-of-X(2151)
X(41907) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (13, 623), (476, 35316)

X(41908) = 1st CRIMEAN POINT OF X(16)

Barycentrics    (sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(S^2+SA*SC)-(SB+SW)*S)*(sqrt(3)*(S^2+SA*SB)-(SC+SW)*S) : :

X(41908) lies on these lines: {14, 14177}, {50, 395}, {323, 41001}, {462, 34397}, {5471, 23357}, {8015, 34394}

X(41908) = complement of the anticomplementary conjugate of X(301)
X(41908) = isogonal conjugate of X(40696)
X(41908) = barycentric quotient X(i)/X(j) for these (i, j): (14, 624), (476, 35317)
X(41908) = 1st Saragossa point of X(34395)
X(41908) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(395)}} and {{A, B, C, X(6), X(50)}}
X(41908) = cevapoint of X(6) and X(14)
X(41908) = X(624)-isoconjugate-of-X(2152)
X(41908) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (14, 624), (476, 35317)

X(41909) = 1st CRIMEAN POINT OF X(111)

Barycentrics    (a^4+(b-2*c)*(b+2*c)*a^2+(b^2+c^2)*c^2)*(a^4-(2*b-c)*(2*b+c)*a^2+(b^2+c^2)*b^2) : :
X(41909) = 4*X(32740)-3*X(35279)

X(41909) lies on the MacBeath circumconic, cubic K482 and these lines: {2, 39238}, {6, 4563}, {32, 1992}, {69, 6387}, {110, 193}, {213, 1332}, {287, 2422}, {524, 9225}, {648, 2207}, {690, 895}, {1331, 1918}, {2987, 39905}, {2996, 8754}, {7316, 17497}, {7783, 40673}, {8681, 36898}, {9289, 10602}, {9468, 39099}, {10765, 25052}, {11060, 41617}, {15534, 20380}, {17708, 37784}, {30454, 38414}, {30455, 38413}

X(41909) = reflection of X(i) in X(j) for these (i, j): (69, 6388), (4563, 6)
X(41909) = isogonal conjugate of X(3291)
X(41909) = isotomic conjugate of the anticomplement of X(6390)
X(41909) = isotomic conjugate of polar conjugate of X(2374)
X(41909) = complement of the anticomplementary conjugate of X(3266)
X(41909) = barycentric product X(i)*X(j) for these {i, j}: {69, 2374}, {98, 36892}, {671, 34161}
X(41909) = barycentric quotient X(i)/X(j) for these (i, j): (3, 8681), (25, 5140), (81, 16756), (98, 36874), (110, 11634), (111, 14263)
X(41909) = trilinear product X(i)*X(j) for these {i, j}: {63, 2374}, {897, 34161}, {1910, 36892}
X(41909) = trilinear quotient X(i)/X(j) for these (i, j): (19, 5140), (63, 8681), (86, 16756), (524, 17466), (662, 11634), (897, 14263)
X(41909) = 1st Saragossa point of X(32740)
X(41909) = orthocorrespondent of X(126)
X(41909) = trilinear pole of the line {3, 669}
X(41909) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(1992)}} and {{A, B, C, X(4), X(32985)}}
X(41909) = MacBeath circumconic-antipode of-X(4563)
X(41909) = cevapoint of X(i) and X(j) for these (i, j): {6, 524}, {525, 1648}, {690, 6388}
X(41909) = X(i)-cross conjugate of-X(j) for these (i, j): (6, 15387), (351, 99)
X(41909) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 8681}, {42, 16756}, {63, 5140}, {111, 17466}
X(41909) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 8681), (25, 5140), (81, 16756), (98, 36874)

X(41910) = 2nd CRIMEAN POINT OF X(1)

Barycentrics    a*(a^3+(b+c)*a^2+(b^2+b*c-c^2)*a+(b+c)*(b^2-c^2))*(a^3+(b+c)*a^2-(b^2-b*c-c^2)*a-(b+c)*(b^2-c^2))*(a^2-b^2+b*c-c^2)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(41910) lies on the line {267, 1046}

X(41910) = barycentric product X(1029)*X(16586)
X(41910) = barycentric quotient X(i)/X(j) for these (i, j): (267, 40437), (1845, 451)
X(41910) = trilinear product X(i)*X(j) for these {i, j}: {267, 16586}, {1029, 34586}
X(41910) = trilinear quotient X(1029)/X(40437)
X(41910) = X(1030)-isoconjugate-of-X(40437)
X(41910) = X(267)-reciprocal conjugate of-X(40437)

X(41911) = 2nd CRIMEAN POINT OF X(2)

Barycentrics    a^2*((b^2+c^2)*a^2-2*b^4+2*b^2*c^2-2*c^4)*(2*a^2-b^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(41911) lies on these lines: {1112, 1843}, {1560, 2679}, {3060, 10552}

X(41911) = barycentric product X(468)*X(20977)
X(41911) = crosssum of X(3) and X(30786)

X(41912) = X(1)X(41840)∩X(2)X(17144)

Barycentrics    a^3*b^3 + 7*a^3*b^2*c + a^2*b^3*c + 7*a^3*b*c^2 + 10*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 : :

See the preamble just before X(41680).

X(41912) lies on these lines: {1, 41840}, {2, 17144}, {192, 3720}, {330, 29814}, {1654, 10453}, {3210, 31308}, {17018, 38986}, {17027, 27484}, {33888, 41839}

X(41913) = X(2)X(1743)∩X(145)X(226)

Barycentrics    3*a^3 + 13*a^2*b + a*b^2 - 9*b^3 + 13*a^2*c + 2*a*b*c + 5*b^2*c + a*c^2 + 5*b*c^2 - 9*c^3 : :

X(41913) lies on these lines: {2, 1743}, {145, 226}, {3177, 29621}, {3210, 4373}, {3617, 31993}, {3622, 5484}, {3995, 6360}, {4090, 39570}, {14552, 26738}, {18134, 30833}, {20211, 31053}, {21454, 30577}

X(41914) = X(2)X(15905)∩X(20)X(343)

Barycentrics    3*a^12 - 22*a^10*b^2 + 49*a^8*b^4 - 36*a^6*b^6 - 11*a^4*b^8 + 26*a^2*b^10 - 9*b^12 - 22*a^10*c^2 + 46*a^8*b^2*c^2 - 28*a^6*b^4*c^2 + 28*a^4*b^6*c^2 - 46*a^2*b^8*c^2 + 22*b^10*c^2 + 49*a^8*c^4 - 28*a^6*b^2*c^4 - 34*a^4*b^4*c^4 + 20*a^2*b^6*c^4 - 7*b^8*c^4 - 36*a^6*c^6 + 28*a^4*b^2*c^6 + 20*a^2*b^4*c^6 - 12*b^6*c^6 - 11*a^4*c^8 - 46*a^2*b^2*c^8 - 7*b^4*c^8 + 26*a^2*c^10 + 22*b^2*c^10 - 9*c^12 : :

X(41914) lies on these lines: {2, 15905}, {20, 343}, {147, 10565}, {3091, 8799}, {6194, 7396}, {6503, 10298}, {7486, 34836}, {7493, 23608}, {7616, 30769}

X(41915) = X(2)X(2321)∩X(7)X(2895)

Barycentrics    a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 10*a*b*c - 7*b^2*c - a*c^2 - 7*b*c^2 - c^3 : :

X(41915) lies on these lines: {2, 2321}, {7, 2895}, {8, 443}, {63, 28638}, {75, 329}, {321, 8055}, {553, 4034}, {940, 4371}, {1655, 3210}, {2094, 14552}, {2550, 4914}, {2968, 6349}, {3241, 19332}, {3305, 4461}, {3616, 41813}, {3618, 19797}, {3679, 24177}, {3686, 9965}, {3696, 36845}, {3969, 29627}, {4046, 38053}, {4361, 6703}, {4460, 17019}, {4688, 5712}, {4699, 26109}, {4720, 15933}, {4858, 18228}, {4904, 33172}, {5222, 19822}, {5271, 5744}, {5294, 24599}, {5296, 17147}, {5564, 18141}, {5739, 31995}, {5749, 19825}, {7229, 32911}, {9780, 32862}, {10436, 20043}, {16816, 26065}, {17321, 41817}, {19785, 31247}, {19804, 34255}, {19826, 26580}, {21454, 31994}, {32860, 39581}, {37653, 39360}

X(41916) = X(2)X(3933)∩X(4)X(1369)

Barycentrics    a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 10*a^2*b^2*c^2 - 7*b^4*c^2 - a^2*c^4 - 7*b^2*c^4 - c^6 : :

X(41916) lies on these lines: {2, 3933}, {4, 1369}, {22, 6527}, {69, 3060}, {75, 10327}, {76, 1370}, {183, 40697}, {339, 7386}, {1285, 16952}, {5133, 32834}, {6515, 14994}, {6995, 30698}, {7392, 40002}, {7485, 32830}, {7500, 7767}, {8024, 19583}, {8267, 16043}, {13575, 40032}, {15246, 32817}, {16045, 34482}, {18750, 39732}, {26255, 33651}, {31133, 32874}, {34255, 41826}, {37668, 37990}, {39978, 40022}

X(41917) = X(2)X(5007)∩X(83)X(8267)

Barycentrics    4*a^6 + 9*a^4*b^2 + 5*a^2*b^4 + 9*a^4*c^2 + 12*a^2*b^2*c^2 + 3*b^4*c^2 + 5*a^2*c^4 + 3*b^2*c^4 : :

X(41917) lies on these lines: {2, 5007}, {83, 8267}, {3329, 8266}, {5596, 23327}, {7668, 37353}, {31128, 39951}

X(41918) = X(2)X(10481)∩X(7)X(3681)

Barycentrics    (a + b - c)*(a - b + c)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 4*a^3*c - 4*a^2*b*c + 16*a*b^2*c - 8*b^3*c + 6*a^2*c^2 + 16*a*b*c^2 + 14*b^2*c^2 - 4*a*c^3 - 8*b*c^3 + c^4) : :

X(41918) lies on these lines: {2, 10481}, {7, 3681}, {85, 36845}, {347, 40719}, {1111, 10580}, {3870, 4328}, {5226, 40615}, {9776, 30623}

X(41919) = X(2)X(12167)∩X(6)X(18287)

Barycentrics    7*a^8 - 28*a^6*b^2 - 14*a^4*b^4 + 20*a^2*b^6 - b^8 - 28*a^6*c^2 + 4*a^4*b^2*c^2 - 36*a^2*b^4*c^2 - 4*b^6*c^2 - 14*a^4*c^4 - 36*a^2*b^2*c^4 - 6*b^4*c^4 + 20*a^2*c^6 - 4*b^2*c^6 - c^8 : :

X(41919) lies on these lines: {2, 12167}, {6, 18287}, {69, 8892}, {193, 7745}, {3164, 32981}, {7665, 39024}, {8878, 20080}, {14035, 17037}, {17035, 32991}

X(41920) = X(2)X(3340)∩X(145)X(3664)

Barycentrics    7*a^4 - 28*a^3*b - 14*a^2*b^2 + 20*a*b^3 - b^4 - 28*a^3*c + 4*a^2*b*c - 36*a*b^2*c - 4*b^3*c - 14*a^2*c^2 - 36*a*b*c^2 - 6*b^2*c^2 + 20*a*c^3 - 4*b*c^3 - c^4 : :

X(41920) lies on these lines: {2, 3340}, {145, 3664}, {3210, 3623}, {3315, 30577}, {3621, 17778}

X(41921) = X(2)X(3702)∩X(8)X(3879)

Barycentrics    a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 + 4*a^3*c + 12*a^2*b*c + 24*a*b^2*c + 8*b^3*c + 6*a^2*c^2 + 24*a*b*c^2 + 14*b^2*c^2 + 4*a*c^3 + 8*b*c^3 + c^4 : :

X(41921) lies on these lines: {2, 3702}, {8, 3879}, {10, 329}, {145, 41821}, {346, 19874}, {2895, 3617}, {7080, 31993}, {7384, 20070}, {7396, 29667}, {8055, 9780}, {18229, 26062}, {19822, 24635}, {19866, 28612}

X(41922) = X(69)X(7762)∩X(141)X(1370)

Barycentrics    a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 4*a^6*c^2 + 12*a^4*b^2*c^2 + 24*a^2*b^4*c^2 + 8*b^6*c^2 + 6*a^4*c^4 + 24*a^2*b^2*c^4 + 14*b^4*c^4 + 4*a^2*c^6 + 8*b^2*c^6 + c^8 : :

X(41922) lies on these lines: {69, 7762}, {141, 1370}, {193, 40002}, {1369, 3620}, {3619, 19583}, {7795, 40697}, {8891, 28419}

X(41923) = X(1)X(5564)∩X(10)X(30562)

Barycentrics    4*a^4 + 15*a^3*b + 19*a^2*b^2 + 10*a*b^3 + 2*b^4 + 15*a^3*c + 40*a^2*b*c + 35*a*b^2*c + 10*b^3*c + 19*a^2*c^2 + 35*a*b*c^2 + 16*b^2*c^2 + 10*a*c^3 + 10*b*c^3 + 2*c^4 : :

X(41923) lies on these lines: {1, 5564}, {10, 30562}, {63, 3337}, {140, 1764}, {1125, 17147}, {3216, 41818}, {5550, 30579}, {13174, 41812}, {16552, 17398}, {19862, 30564}

X(41924) = PERSPECTOR OF THESE TRIANGLES: X(2)X(21868)∩X(192)X(24165)

Barycentrics    a^3*b^3 + 11*a^3*b^2*c + a^2*b^3*c + 11*a^3*b*c^2 + 26*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(41924) lies on these lines: {2, 21868}, {192, 24165}, {3720, 31999}, {17490, 31308}, {25502, 32095}

X(41925) = X(2)X(5023)∩X(193)X(8889)

Barycentrics    5*a^6 - 29*a^4*b^2 - 9*a^2*b^4 + 25*b^6 - 29*a^4*c^2 + 14*a^2*b^2*c^2 - 21*b^4*c^2 - 9*a^2*c^4 - 21*b^2*c^4 + 25*c^6 : :

X(41925) lies on these lines: {2, 5023}, {193, 8889}, {427, 18287}, {5094, 8892}, {7378, 7665}

X(41926) = X(2)X(3950)∩X(8)X(354)

Barycentrics    a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 26*a*b*c - 11*b^2*c - a*c^2 - 11*b*c^2 - c^3 : :

X(41926) lies on these lines: {2, 3950}, {8, 354}, {75, 8055}, {329, 4359}, {1655, 17490}, {2895, 21296}, {3039, 5273}, {3617, 24175}, {4384, 30695}, {17794, 26038}, {20881, 30578}, {34255, 41821}

X(41927) = X(2)X(9606)∩X(51)X(69)

Barycentrics    a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 26*a^2*b^2*c^2 - 11*b^4*c^2 - a^2*c^4 - 11*b^2*c^4 - c^6 : :

X(41927) lies on these lines: {2, 9606}, {51, 69}, {75, 3974}, {76, 19583}, {183, 6527}, {1272, 7494}, {1369, 7394}, {1370, 39998}, {7195, 20934}, {7398, 37671}, {7499, 34229}, {11548, 32885}, {15589, 30698}, {16419, 32836}, {16990, 32747}, {40002, 40123}

X(41928) = X(2)X(7762)∩X(22)X(83)

Barycentrics    3*a^6 + 8*a^4*b^2 + 5*a^2*b^4 + 8*a^4*c^2 + 12*a^2*b^2*c^2 + 4*b^4*c^2 + 5*a^2*c^4 + 4*b^2*c^4 : :

X(41978) lies on these lines: {2, 7762}, {6, 39668}, {22, 83}, {3329, 8264}, {3618, 5133}, {5024, 16953}, {5359, 7808}, {7485, 8266}, {7571, 7792}, {7770, 8267}

X(41929) = X(7)X(210)∩X(85)X(31527)

Barycentrics    (a + b - c)*(a - b + c)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 4*a^3*c - 20*a^2*b*c + 36*a*b^2*c - 12*b^3*c + 6*a^2*c^2 + 36*a*b*c^2 + 22*b^2*c^2 - 4*a*c^3 - 12*b*c^3 + c^4) : :

X(41929) lies on these lines: {7, 210}, {85, 31527}, {4452, 40719}, {10578, 41803}, {17093, 41808}, {25585, 32086}, {31994, 36845}

X(41930) = X(1)X(4720)∩X(2)X(1449)

Barycentrics    3*a^3 + 8*a^2*b + 5*a*b^2 + 8*a^2*c + 12*a*b*c + 4*b^2*c + 5*a*c^2 + 4*b*c^2 : :

X(41930) lies on these lines: {1, 4720}, {2, 1449}, {9, 30561}, {20, 946}, {63, 86}, {147, 29634}, {194, 16826}, {321, 29597}, {329, 28644}, {551, 19785}, {968, 13174}, {1125, 32946}, {1211, 28640}, {1764, 3306}, {2896, 17397}, {3187, 30562}, {3305, 16552}, {3624, 32782}, {4654, 41808}, {4698, 19722}, {5256, 15668}, {5271, 17394}, {5287, 19701}, {5625, 17156}, {5712, 28641}, {8025, 30564}, {10436, 17147}, {17304, 41820}, {17778, 29612}, {18139, 29603}, {25055, 37817}, {27064, 29595}, {27184, 29592}, {31266, 32431}

X(41931) = X(2)X(6748)∩X(3)X(11197)

Barycentrics    3*a^12 - 17*a^10*b^2 + 38*a^8*b^4 - 42*a^6*b^6 + 23*a^4*b^8 - 5*a^2*b^10 - 17*a^10*c^2 + 46*a^8*b^2*c^2 - 28*a^6*b^4*c^2 - 18*a^4*b^6*c^2 + 21*a^2*b^8*c^2 - 4*b^10*c^2 + 38*a^8*c^4 - 28*a^6*b^2*c^4 - 10*a^4*b^4*c^4 - 16*a^2*b^6*c^4 + 16*b^8*c^4 - 42*a^6*c^6 - 18*a^4*b^2*c^6 - 16*a^2*b^4*c^6 - 24*b^6*c^6 + 23*a^4*c^8 + 21*a^2*b^2*c^8 + 16*b^4*c^8 - 5*a^2*c^10 - 4*b^2*c^10 : :

X(41931) lies on these lines: {2, 6748}, {3, 11197}, {95, 1993}, {631, 6193}, {5422, 37067}

leftri

Points on the square of the circumcircle: X(41932)-X(41937)

rightri

Points X(41932)-X(41937) are contributed by Clark Kimberling and Peter Moses, March 13, 2021.

The square of the circumcircle is given by the barycentric equation

a^8 y^2 z^2 - 2 b^4 c^4 x^2 y*z + (cyclic) = 0.

The appearance of (i,j) in the following list means that X(i) is on the circumcircle, and X(i)^2 = X(j): (74,40353), (98, 41932), (99,4590), (100,1252), (101,23990), (104,41933), (105, 41934), (106, 31935), (107,23590), (108,23985), (109,23979), (110,23357), (111, 41936), (112, 41937)(476,23588), (934,23971)

The cube of the circumcircle is given by

a^18 y^3 z^3 + 3 a^12 x y^2 z^2 (c^6 y + b^6 z) - 7 a^6 b^6 c^6 x^2 y^2 z^2 + (cyclic) = 0.

X(41932) = BARYCENTRIC SQUARE OF X(98)

Barycentrics    (a^4 + b^4 - a^2*c^2 - b^2*c^2)^2*(a^4 - a^2*b^2 - b^2*c^2 + c^4)^2 : :

X(41932) lies on these lines: {2, 40428}, {6, 36897}, {98, 230}, {232, 685}, {248, 290}, {287, 325}, {460, 6531}, {1976, 9418}, {5254, 8861}, {5304, 9154}, {5967, 7736}, {5970, 18858}, {8870, 34870}, {9473, 9476}, {9755, 32545}, {11177, 23967}, {11672, 15391}

X(41932) = isogonal conjugate of X(36790)
X(41932) = isotomic conjugate of X(32458)
X(41932) = cevapoint of X(i) and X(j) for these {i,j}: {32, 1976}, {115, 2395}
X(41932) = trilinear pole of line X(1976)X(2395)
X(41932) = crossdifference of every pair of points on line X(6072)X(23098)
X(41932) = trilinear product X(i)*X(j) for these {i,j}: {31, 34536}, {98, 1910}, {248, 36120}, {293, 6531}, {879, 36104}, {1821, 1976}, {2395, 36084}, {2422, 36036}

X(41933) = BARYCENTRIC SQUARE OF X(104)

Barycentrics    a^2*(a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)^2*(a^3 - a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + c^3)^2 : :

X(41933) lies on these lines: {44, 14578}, {104, 8609}, {902, 2342}, {909, 1404}, {1795, 2323}, {2720, 7113}, {32723, 34858}

X(41933) = isogonal conjugate of X(26611)

X(41934) = BARYCENTRIC SQUARE OF X(105)

Barycentrics    a^2*(a^2 + b^2 - a*c - b*c)^2*(a^2 - a*b - b*c + c^2)^2 : :

X(41934) lies on these lines: {105, 910}, {672, 2195}, {1438, 2210}, {6185, 14953}, {7191, 40754}, {8751, 32703}, {9455, 32724}, {9500, 35505}, {32644, 32658}, {33854, 36086}

X(41934) = isogonal conjugate of X(4437)

X(41935) = BARYCENTRIC SQUARE OF X(106)

Barycentrics    a^4*(a + b - 2*c)^2*(a - 2*b + c)^2 : :

X(41935) lies on these lines: {106, 5053}, {2226, 3285}, {2251, 7113}, {8752, 32705}, {9259, 35129}, {9459, 32719}, {32645, 32659}

X(41935) = isogonal conjugate of X(36791)
X(41935) = trilinear product X(i)*X(j) for these {i,j}: {31, 2226}, {32, 679}, {106, 9456}, {604, 1318}, {667, 4638}, {1022, 32719}, {1417, 2316}, {1919, 4618}, {2206, 30575}, {2441, 36042}, {8752, 36058}, {23345, 32665}, {32659, 36125}

X(41936) = BARYCENTRIC SQUARE OF X(111)

Barycentrics    a^4*(a^2 + b^2 - 2*c^2)^2*(a^2 - 2*b^2 + c^2)^2 : :

X(41936) lies on these lines: {2, 34161}, {23, 111}, {115, 2770}, {858, 34169}, {923, 32672}, {1196, 8877}, {1995, 14263}, {8753, 14580}, {9225, 32583}, {9465, 14246}, {10422, 10561}, {13574, 23992}, {14567, 18374}, {14908, 32648}, {34574, 41309}, {36877, 40132}, {39602, 40343}

X(41936) = isogonal conjugate of X(36792)
X(41936) = cevapoint of X(32) and X(32740)
X(41936) = crossdifference of every pair of points on line X(1649)X(6077)
X(41936) = trilinear product X(i)*X(j) for these {i,j}: {31, 10630}, {111, 923}, {798, 34574}, {897, 32740}, {1973, 15398}, {2444, 36045}, {8753, 36060}, {9178, 36142}, {14908, 36128}, {23894, 32729}

X(41937) = BARYCENTRIC SQUARE OF X(112)

Barycentrics    a^4*(a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2 : :

X(41937) lies on these lines: {23, 232}, {112, 2485}, {393, 32230}, {1576, 32649}, {2211, 18374}, {2489, 32696}, {3049, 32715}, {14581, 23975}, {15388, 34137}, {23582, 40889}, {32673, 32676}

X(41937) = isogonal conjugate of X(36793)
X(41937) = cevapoint of X(206) and X(1576)
X(41937) = trilinear pole of line X(14574)X(34859)
X(41937) = crossdifference of every pair of points on line X(5489)X(23107)
X(41937) = trilinear product X(i)*X(j) for these {i,j}: {31, 23964}, {32, 24000}, {112, 32676}, {158, 23963}, {163, 32713}, {250, 1973}, {255, 23975}, {393, 23995}, {560, 23582}, {577, 24022}, {823, 14574}, {1096, 23357}, {1101, 2207}, {1110, 36420}, {1501, 23999}, {1576, 24019}, {2445, 36046}, {9247, 32230}, {14585, 24021}, {23347, 36131}, {24041, 36417}, {34859, 36084}

X(41938) = X(1)X(3346)∩X(46)X(20226)

Barycentrics    a*(a*b*(SA^2*SB^2 + SA^2*SC^2 - SB^2*SC^2)*(SA^2*SB^2 - SA^2*SC^2 + SB^2*SC^2) + a*c*(SA^2*SB^2 + SA^2*SC^2 - SB^2*SC^2)*(-(SA^2*SB^2) + SA^2*SC^2 + SB^2*SC^2) - b*c*(SA^2*SB^2 - SA^2*SC^2 + SB^2*SC^2)*(-(SA^2*SB^2) + SA^2*SC^2 + SB^2*SC^2)) : :

X(41938) lies on the cubic K1190 and these lines: {1, 3346}, {46, 20226}, {1044, 1711}, {1745, 20225}, {1779, 15803}

X(41938) = X(1498)-Ceva conjugate of X(1)

X(41939) = X(2)X(6)∩X(39)X(647)

Barycentrics    2*a^6 - 2*a^4*b^2 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6 : :

X(41939) lies on these lines: {2, 6}, {39, 647}, {110, 11646}, {111, 6034}, {115, 2502}, {125, 5477}, {187, 13857}, {468, 20977}, {542, 8288}, {858, 14567}, {868, 5475}, {1316, 2549}, {2030, 39602}, {2420, 11007}, {3124, 5972}, {4226, 13394}, {5107, 32225}, {5182, 30786}, {5471, 30465}, {5472, 30468}, {5476, 14694}, {5480, 7417}, {5650, 15544}, {5652, 10097}, {5969, 7664}, {6781, 8627}, {7737, 36194}, {7812, 40877}, {7831, 17941}, {7833, 40871}, {8356, 34245}, {8586, 15360}, {9167, 12036}, {12830, 31127}, {13192, 37907}, {14061, 35279}, {15048, 31945}, {31125, 35356}

X(41939) = midpoint of X(i) and X(j) for these {i,j}: {8288, 39689}, {31125, 35356}
X(41939) = X(6) of {X(2),X(13),X(14)}
X(41939) = X(39450)-complementary conjugate of X(21256)
X(41939) = crossdifference of every pair of points on line {23, 512}
X(41939) = barycentric product X(542)*X(36825)
X(41939) = barycentric quotient X(36825)/X(5641)
X(41939) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 1648}, {2, 5468, 141}, {2, 14916, 21358}, {115, 5642, 2502}, {3589, 11053, 2}, {15048, 34094, 35606}

X(41940) = X(3)X(6)∩X(4)X(39593)

Barycentrics    a^2*(4*a^2 + 5*b^2 + 5*c^2) : :

X(41940) lies on these lines: {3, 6}, {4, 39593}, {69, 39784}, {115, 3857}, {538, 7878}, {546, 7753}, {597, 7794}, {625, 7920}, {632, 5306}, {940, 21543}, {1194, 14002}, {1196, 16042}, {1569, 38628}, {1992, 7854}, {2548, 3544}, {2549, 11541}, {3090, 5319}, {3091, 5309}, {3108, 7496}, {3146, 7739}, {3163, 7403}, {3199, 26863}, {3329, 7805}, {3589, 7890}, {3618, 7855}, {3627, 7765}, {3628, 7755}, {3629, 6292}, {3767, 14930}, {3815, 5368}, {3934, 7894}, {4383, 21535}, {5032, 14023}, {5076, 11648}, {5283, 17544}, {5286, 39590}, {5304, 31455}, {5305, 7603}, {5346, 7736}, {5355, 12811}, {6329, 7889}, {6656, 41750}, {6683, 7766}, {7745, 12102}, {7746, 37665}, {7760, 9466}, {7770, 14711}, {7774, 7852}, {7787, 32450}, {7803, 7845}, {7804, 7839}, {7810, 20583}, {7817, 7858}, {7821, 7829}, {7826, 32455}, {7827, 7843}, {7837, 7849}, {7838, 7853}, {7856, 10150}, {7859, 7882}, {7861, 7921}, {7874, 16989}, {7875, 7895}, {7880, 13571}, {7902, 31173}, {7905, 7915}, {8362, 8584}, {9606, 14869}, {9607, 15704}, {12156, 33256}, {13196, 15870}, {14001, 39785}, {14581, 35502}, {15004, 20897}, {17546, 33854}, {21513, 34986}, {34396, 34566}

X(41940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 32, 34571}, {6, 5041, 39}, {6, 7772, 5007}, {6, 22246, 14075}, {32, 39, 8589}, {32, 15815, 187}, {39, 5007, 35007}, {39, 5008, 15513}, {61, 62, 5092}, {187, 9605, 39}, {575, 576, 13354}, {3329, 7805, 31239}, {5007, 5041, 7772}, {5007, 7772, 39}, {5007, 31652, 32}, {5007, 35007, 5008}, {5210, 30435, 32}, {6419, 6420, 182}, {6419, 11824, 3592}, {6420, 11825, 3594}, {7829, 41624, 7821}, {31652, 34571, 5007}

X(41941) = X(23)X(232)∩X(112)X(1113)

Barycentrics    (a^2*(1 - J)*SA - 2*SB*SC)^2 : :
Barycentrics    a^2*(a^2 - b^2)^2*(a^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*(-a^2 + b^2 + c^2)*(-1 + J) + (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(1 + J)) : :

X(41941) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 39298}, {6, 15461}, {23, 232}, {112, 1113}, {1312, 8791}, {2553, 36412}, {3163, 15158}, {8749, 15167}

X(41941) = isogonal conjugate of the complement of X(8115)
X(41941) = polar conjugate of the isotomic conjugate of X(15461)
X(41941) = X(i)-cross conjugate of X(j) for these (i,j): {25, 1113}, {8106, 112}
X(41941) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1313}, {75, 15166}, {2349, 14499}, {2574, 2582}, {2578, 22339}, {2580, 23109}, {2584, 2592}, {15460, 20902}
X(41941) = crosspoint of X(250) and X(39298)
X(41941) = crossdifference of every pair of points on line {5489, 14499}
X(41941) = barycentric square of X(1113)
X(41941) = barycentric product X(i)*X(j) for these {i,j}: {4, 15461}, {250, 1312}, {1113, 1113}, {1822, 2586}, {2576, 2580}, {8106, 39298}, {15167, 23582}
X(41941) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1313}, {32, 15166}, {1113, 22339}, {1312, 339}, {1495, 14499}, {2576, 2582}, {15167, 15526}, {15461, 69}

X(41942) = X(23)X(232)∩X(112)X(1114)

Barycentrics    (a^2*(1 + J)*SA - 2*SB*SC)^2 : :
Barycentrics    a^2*(a^2 - b^2)^2*(a^2 - c^2)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(-((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(1 - J)) + a^2*(-a^2 + b^2 + c^2)*(1 + J)) : :

X(41942) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 39299}, {6, 15460}, {23, 232}, {112, 1114}, {1313, 8791}, {2552, 36412}, {3163, 15159}, {8749, 15166}

X(41942) = isogonal conjugate of the complement of X(8116)
X(41942) = polar conjugate of the isotomic conjugate of X(15460)
X(41942) = X(i)-cross conjugate of X(j) for these (i,j): {25, 1114}, {8105, 112}
X(41942) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1312}, {75, 15167}, {2349, 14500}, {2575, 2583}, {2579, 22340}, {2581, 23110}, {2585, 2593}, {15461, 20902}
X(41942) = crosspoint of X(250) and X(39299)
X(41942) = crossdifference of every pair of points on line {5489, 14500}
X(41942) = barycentric square of X(1114)
X(41942) = barycentric product X(i)*X(j) for these {i,j}: {4, 15460}, {250, 1313}, {1114, 1114}, {1823, 2587}, {2577, 2581}, {8105, 39299}, {15166, 23582}
X(41942) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1312}, {32, 15167}, {1114, 22340}, {1313, 339}, {1495, 14500}, {2577, 2583}, {15166, 15526}, {15460, 69}

leftri

Points on the Evans conic: X(41943)-X(41980)

rightri

Points X(41943)-X(41980) are contributed by Peter Moses, March 14-16, 2021.

For a discussion of the Evans conic, see Evans Conic.

If k1 and k2 are functions symmetric in a,b,c and homogeneous of degree 0, then the points

E1(k1,k2) = 3*a^2*(a^2 - b^2 - c^2)*k2 - 4*k1*S^2 - 2*a^2*Sqrt[k1^2 + 2*k1*k2 + 3*k2^2]*S : : and E2(k1,k2) = 3*a^2*(a^2 - b^2 - c^2)*k2 - 4*k1*S^2 + 2*a^2*Sqrt[k1^2 + 2*k1*k2 + 3*k2^2]*S : :

lie on the Evans conic. The lines tangent to the Evans conic at the two points meet on the Euler line in the point

T(k1,k2) = (k1 + 3*k2)*X[2] - (k1 + k2)*X[3].

The appearance of (k1,k2) → (i,j); h in the following list means that E1(k1,k2) = X(i), E2(k1,k2) = X(j), and T(k1,k2) = X(h).

(-2,1) → (14,13); 549
(0, 1) → (15,16); 5
(-6,1) → (18,17); 550
(-3,2) → (3071,3070); 140
(4, 1) → (41943,41944); 15687
(-1,2) → (41945,41946); 547
(1, 0) → (590,615); 30
(-39,8) → (41947,41948); 41981
(-23,6) → (41949,41950); 41982
(-11,4) → (41951, 41952); 14891
(-13,6) → (419543 41954); 41983
(-13,12) → (41955, 41956); 41984
(-11,12) → (41957, 41958); 41985
(-1,24) → (41959, 41960); 41986
(1,6) → (41961, 41962); 14892
(3,4) → (41963, 41964); 3850
(11,6) → (41965, 41966); 41987
(23,8) → (41967, 41968); 41988
(3,-20) → (41969, 41970); 41989
(8,-13) → (41971, 41972); 41990
(12,-11) → (41973, 41974); 41991
(3,1) → (41975, 41976); 4
(24,23) → (41977, 41978); 41992

The points E! and E2 defined above are also Gibert points, as defined in the preamble just before X(42085); specifically, in the notation of that preamble,

E1(k1,k2) = (Sqrt[3*(k1^2 + 2*k1*k2 + 3*k2^2)] , k1, 2*k1 + 3*k2) Gibert point;
E2(k1,k2) = (-Sqrt[3*(k1^2 + 2*k1*k2 + 3*k2^2)], k1, 2*k1 + 3*k2) Gibert point.}

X(41943) = REFLECTION OF X(41121) IN X(17)

Barycentrics    7*a^4 - 11*a^2*b^2 + 4*b^4 - 11*a^2*c^2 - 8*b^2*c^2 + 4*c^4 - 6*Sqrt[3]*a^2*S : :
X(41943) = X[16] - 4 X[10616], 2 X[17] + X[5238], 11 X[17] - 2 X[5350], 5 X[17] - X[12816], X[17] + 2 X[16772], 11 X[5238] + 4 X[5350], 5 X[5238] + 2 X[12816], X[5238] - 4 X[16772], 10 X[5350] - 11 X[12816], X[5350] + 11 X[16772], 4 X[5350] - 11 X[41121], X[12816] + 10 X[16772], 2 X[12816] - 5 X[41121], 4 X[16772] + X[41121]

X(41943) lies on Evans conic and these lines: {2, 18}, {3, 16267}, {5, 41101}, {6, 15694}, {13, 376}, {14, 547}, {15, 381}, {16, 396}, {17, 30}, {20, 41119}, {62, 5054}, {69, 36770}, {140, 3412}, {203, 3584}, {299, 6671}, {303, 7811}, {371, 36456}, {372, 36438}, {395, 10124}, {397, 12100}, {398, 15699}, {530, 30559}, {531, 14138}, {542, 22892}, {548, 33607}, {599, 36757}, {617, 5469}, {1656, 41122}, {3055, 9113}, {3070, 36457}, {3071, 35731}, {3090, 41113}, {3091, 12817}, {3106, 11302}, {3364, 13847}, {3365, 13846}, {3411, 3525}, {3524, 5351}, {3534, 5352}, {3543, 18582}, {3545, 16964}, {3582, 7005}, {3830, 36836}, {5055, 22236}, {5067, 10188}, {5071, 10654}, {5237, 15693}, {5318, 15686}, {5321, 11737}, {5340, 15688}, {5344, 15697}, {5418, 36468}, {5420, 36449}, {5464, 22510}, {5859, 7780}, {5892, 30440}, {6771, 14539}, {7753, 41407}, {8703, 16965}, {9112, 21843}, {9117, 32909}, {9763, 11301}, {10168, 36758}, {10304, 41112}, {10646, 15700}, {10653, 15692}, {11295, 35229}, {11299, 39555}, {11300, 13083}, {11480, 15681}, {11481, 15718}, {11485, 15703}, {11542, 34200}, {14893, 19107}, {15683, 19106}, {15687, 16808}, {15701, 22238}, {15702, 16242}, {15707, 36843}, {15713, 16773}, {15723, 16645}, {18585, 41963}, {21360, 37340}, {22489, 37170}, {23039, 30439}, {33416, 37641}, {33475, 37351}

X(41943) = midpoint of X(5238) and X(41121)
X(41943) = reflection of X(41121) in X(17)
X(41943) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 61, 16268}, {2, 16962, 61}, {3, 16267, 41107}, {6, 15694, 41944}, {15, 16644, 37832}, {15, 37832, 36970}, {17, 16772, 5238}, {396, 16241, 16}, {3524, 40693, 41100}, {3524, 41100, 5351}, {5055, 22236, 41108}, {36453, 36469, 15}

X(41944) = REFLECTION OF X(41122) IN X(18)

Barycentrics    7*a^4 - 11*a^2*b^2 + 4*b^4 - 11*a^2*c^2 - 8*b^2*c^2 + 4*c^4 + 6*Sqrt[3]*a^2*S : :
X(41944) = X[15] - 4 X[10617], 2 X[18] + X[5237], 11 X[18] - 2 X[5349], 5 X[18] - X[12817], X[18] + 2 X[16773], 11 X[5237] + 4 X[5349], 5 X[5237] + 2 X[12817], X[5237] - 4 X[16773], 10 X[5349] - 11 X[12817], X[5349] + 11 X[16773], 4 X[5349] - 11 X[41122], X[12817] + 10 X[16773], 2 X[12817] - 5 X[41122], 4 X[16773] + X[41122]

X(41944) lies on Evans conic and these lines: {2, 17}, {3, 16268}, {5, 41100}, {6, 15694}, {13, 547}, {14, 376}, {15, 395}, {16, 381}, {18, 30}, {20, 41120}, {61, 5054}, {140, 3411}, {202, 3584}, {298, 6672}, {302, 7811}, {371, 36438}, {372, 36456}, {396, 10124}, {397, 15699}, {398, 12100}, {530, 14139}, {531, 30560}, {542, 22848}, {548, 33606}, {599, 36758}, {616, 5470}, {1656, 41121}, {3055, 9112}, {3070, 36439}, {3071, 36457}, {3090, 41112}, {3091, 12816}, {3107, 11301}, {3389, 13847}, {3390, 13846}, {3412, 3525}, {3524, 5352}, {3534, 5351}, {3543, 18581}, {3545, 16965}, {3582, 7006}, {3589, 36770}, {3830, 36843}, {5055, 22238}, {5067, 10187}, {5071, 10653}, {5238, 15693}, {5318, 11737}, {5321, 15686}, {5339, 15688}, {5343, 15697}, {5418, 36450}, {5420, 36467}, {5463, 22511}, {5858, 7780}, {5892, 30439}, {6774, 14538}, {7753, 41406}, {8703, 16964}, {9113, 21843}, {9115, 32907}, {9761, 11302}, {10168, 36757}, {10304, 41113}, {10645, 15700}, {10654, 15692}, {11296, 35230}, {11299, 13084}, {11300, 39554}, {11480, 15718}, {11481, 15681}, {11486, 15703}, {11543, 34200}, {14893, 19106}, {15683, 19107}, {15687, 16809}, {15701, 22236}, {15702, 16241}, {15707, 36836}, {15713, 16772}, {15723, 16644}, {15765, 41963}, {21359, 37341}, {22490, 37171}, {23039, 30440}, {33417, 37640}, {33474, 37352}, {35739, 36455}, {36251, 36768}

X(41944) = midpoint of X(5237) and X(41122)
X(41944) = reflection of X(41122) in X(18)
X(41944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 62, 16267}, {2, 16963, 62}, {3, 16268, 41108}, {6, 15694, 41943}, {16, 16645, 37835}, {16, 37835, 36969}, {18, 16773, 5237}, {395, 16242, 15}, {3524, 40694, 41101}, {3524, 41101, 5352}, {5055, 22238, 41107}, {36452, 36470, 16}

X(41945) = REFLECTION OF X(32787) IN X(371)

Barycentrics    3*a^2*(a^2 - b^2 - c^2) - 3*a^2*S + 2*S^2 : :
X(41945) = 4 X[371] - X[3070], 5 X[371] - 2 X[7583], 7 X[371] - X[35820], 3 X[371] - X[35822], 5 X[3070] - 8 X[7583], 7 X[3070] - 4 X[35820], 3 X[3070] - 4 X[35822], 4 X[7583] - 5 X[32787], 14 X[7583] - 5 X[35820], 6 X[7583] - 5 X[35822], 7 X[32787] - 2 X[35820], 3 X[32787] - 2 X[35822], 3 X[35820] - 7 X[35822]

X(41945) lies on Evans conic and these lines: {2, 489}, {3, 9681}, {4, 6425}, {5, 6453}, {6, 376}, {15, 36439}, {16, 36457}, {17, 18585}, {18, 15765}, {20, 3592}, {30, 371}, {372, 8703}, {381, 590}, {382, 1327}, {395, 36437}, {396, 36455}, {428, 11473}, {485, 3830}, {486, 5054}, {524, 35949}, {546, 35812}, {547, 6480}, {548, 6420}, {549, 615}, {550, 6419}, {597, 35948}, {599, 26289}, {1152, 10304}, {1328, 5055}, {1587, 11001}, {1588, 3524}, {1656, 6519}, {1992, 26617}, {2066, 5434}, {2067, 3058}, {2549, 8375}, {3068, 3543}, {3069, 6411}, {3090, 9693}, {3298, 10385}, {3311, 3534}, {3312, 15688}, {3364, 41100}, {3389, 15764}, {3522, 3594}, {3525, 10147}, {3528, 6426}, {3545, 6429}, {3627, 8960}, {3655, 35775}, {3679, 9616}, {3845, 8981}, {4995, 9662}, {5056, 9692}, {5066, 10576}, {5067, 10141}, {5071, 6468}, {5072, 10195}, {5073, 31487}, {5298, 9649}, {5306, 12963}, {5309, 9675}, {5318, 36436}, {5321, 36454}, {5415, 34618}, {5420, 6455}, {5655, 10819}, {6199, 6560}, {6396, 34200}, {6398, 14093}, {6410, 7582}, {6412, 7586}, {6417, 15689}, {6418, 15695}, {6424, 7739}, {6427, 15696}, {6431, 6460}, {6433, 8252}, {6444, 37665}, {6445, 13785}, {6451, 15700}, {6454, 33923}, {6470, 7581}, {6476, 11737}, {6478, 23046}, {6484, 10577}, {6486, 11812}, {6488, 10303}, {6496, 15706}, {6564, 15687}, {6567, 31173}, {7584, 12100}, {7585, 15683}, {7801, 32419}, {7969, 28194}, {8976, 14269}, {8980, 9880}, {9542, 32785}, {9582, 13973}, {9583, 31162}, {9584, 19876}, {9585, 30308}, {9615, 25055}, {9685, 18362}, {9690, 15703}, {9691, 19709}, {10124, 18762}, {11241, 15311}, {11294, 32809}, {11477, 35945}, {11916, 12124}, {12101, 13925}, {12117, 19056}, {12123, 26348}, {12239, 21969}, {13665, 15684}, {13903, 38335}, {13935, 15698}, {13939, 15719}, {13951, 15701}, {13961, 15716}, {13966, 17504}, {13968, 38737}, {14241, 23253}, {14891, 35256}, {14893, 18538}, {15170, 35768}, {15534, 26288}, {15682, 23251}, {15690, 35771}, {15712, 35813}, {15721, 32786}, {19057, 34473}, {19117, 19710}, {19145, 20423}, {22236, 36468}, {22238, 36450}, {23263, 41106}, {28204, 31439}, {33699, 35815}, {35787, 38071}

X(41945) = reflection of X(i) in X(j) for these {i,j}: {3070, 32787}, {32787, 371}
X(41945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6425, 31454}, {6, 376, 41946}, {549, 35823, 615}, {1151, 6459, 3071}, {1328, 5418, 5055}, {1588, 3524, 13847}, {1656, 6519, 9680}, {6200, 35823, 549}, {6221, 6561, 590}, {6409, 13847, 3524}, {6429, 23261, 9540}, {6480, 6565, 35255}, {6565, 35255, 32789}, {10304, 19053, 1152}, {26289, 26619, 599}

X(41946) = REFLECTION OF X(32788) IN X(372)

Barycentrics    3*a^2*(a^2 - b^2 - c^2) + 3*a^2*S + 2*S^2 : :
X(41946) = 4 X[372] - X[3071], 5 X[372] - 2 X[7584], 7 X[372] - X[35821], 3 X[372] - X[35823], 5 X[3071] - 8 X[7584], 7 X[3071] - 4 X[35821], 3 X[3071] - 4 X[35823], 4 X[7584] - 5 X[32788], 14 X[7584] - 5 X[35821], 6 X[7584] - 5 X[35823], 7 X[32788] - 2 X[35821], 3 X[32788] - 2 X[35823], 3 X[35821] - 7 X[35823]

X(41946) lies on Evans conic and these lines: {2, 490}, {3, 9680}, {4, 6426}, {5, 6454}, {6, 376}, {15, 36457}, {16, 36439}, {17, 15765}, {18, 18585}, {20, 3594}, {30, 372}, {371, 8703}, {381, 615}, {382, 1328}, {395, 36455}, {396, 36437}, {428, 11474}, {485, 5054}, {486, 3830}, {524, 35948}, {546, 35813}, {547, 6481}, {548, 6419}, {549, 590}, {550, 6420}, {597, 35949}, {599, 26288}, {1151, 10304}, {1327, 5055}, {1587, 3524}, {1588, 11001}, {1656, 6522}, {1992, 26618}, {2549, 8376}, {3058, 6502}, {3068, 6412}, {3069, 3543}, {3090, 17852}, {3297, 10385}, {3311, 15688}, {3312, 3534}, {3365, 15764}, {3390, 41101}, {3522, 3592}, {3525, 10148}, {3528, 6425}, {3530, 8960}, {3545, 6430}, {3655, 35774}, {3845, 13966}, {5066, 10577}, {5067, 10142}, {5071, 6469}, {5072, 10194}, {5306, 12968}, {5318, 36454}, {5321, 36436}, {5414, 5434}, {5416, 34618}, {5418, 6456}, {5655, 10820}, {6200, 34200}, {6221, 14093}, {6395, 6561}, {6409, 7581}, {6411, 7585}, {6417, 15695}, {6418, 15689}, {6423, 7739}, {6427, 9681}, {6428, 15696}, {6432, 6459}, {6434, 8253}, {6443, 37665}, {6446, 13665}, {6452, 15700}, {6453, 33923}, {6471, 7582}, {6477, 11737}, {6479, 23046}, {6485, 10576}, {6487, 11812}, {6489, 10303}, {6497, 15706}, {6565, 15687}, {6566, 31173}, {7583, 12100}, {7586, 15683}, {7801, 32421}, {7968, 28194}, {8976, 15701}, {8981, 17504}, {9540, 15698}, {9880, 13967}, {10124, 18538}, {11242, 15311}, {11293, 32808}, {11477, 35944}, {11917, 12123}, {12101, 13993}, {12117, 19055}, {12124, 26341}, {12240, 21969}, {13785, 15684}, {13886, 15719}, {13903, 15716}, {13908, 38737}, {13951, 14269}, {13961, 38335}, {14226, 23263}, {14891, 35255}, {14893, 18762}, {15170, 35769}, {15534, 26289}, {15682, 23261}, {15690, 35770}, {15712, 35812}, {15721, 32785}, {19058, 34473}, {19116, 19710}, {19146, 20423}, {21736, 39664}, {22236, 36449}, {22238, 36467}, {23253, 41106}, {31156, 31473}, {33699, 35814}, {35786, 38071}

X(41946) = reflection of X(i) in X(j) for these {i,j}: {3071, 32788}, {32788, 372}
X(41946) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 376, 41945}, {549, 35822, 590}, {1152, 6460, 3070}, {1327, 5420, 5055}, {1587, 3524, 13846}, {6396, 35822, 549}, {6398, 6560, 615}, {6410, 13846, 3524}, {6430, 23251, 13935}, {6481, 6564, 35256}, {6564, 35256, 32790}, {10304, 19054, 1151}, {26288, 26620, 599}

X(41947) = X(6)X(3590)∩X(20)X(615)

Barycentrics    4*a^2*(a^2 - b^2 - c^2) - 11*a^2*S + 26*S^2 : :

X(41947) lies on the Evans conic and these lines: {2, 41967}, {3, 41953}, {4, 41970}, {5, 41954}, {6, 3590}, {20, 615}, {30, 41968}, {140, 41969}, {372, 3859}, {486, 5054}, {590, 3628}, {1588, 3533}, {3070, 3851}, {3071, 15712}, {3317, 9541}, {3845, 13966}, {5071, 7581}, {5076, 6522}, {5418, 41965}, {6565, 41966}, {8252, 41959}, {8976, 42603}, {13941, 42574}, {13993, 14892}, {15684, 42261}, {15698, 42258}, {15765, 41971}, {18585, 41972}, {18762, 41964}, {23253, 41958}, {32789, 42522}, {34200, 41951}, {35813, 41960}, {35823, 42607}, {41949, 42266}, {41950, 42583}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {486, 6449, 42573}, {5076, 41962, 42259}

X(41948) = X(6)X(3590)∩X(20)X(590)

Barycentrics    4*a^2*(a^2 - b^2 - c^2) + 11*a^2*S + 26*S^2 : :

X(41948) lies on the Evans conic and these lines: {2, 41968}, {3, 41954}, {4, 41969}, {5, 41953}, {6, 3590}, {20, 590}, {30, 41967}, {140, 41970}, {371, 3859}, {485, 5054}, {615, 3628}, {1587, 3533}, {3070, 15712}, {3071, 3851}, {3316, 41956}, {3845, 8981}, {5071, 7582}, {5076, 6519}, {5420, 41966}, {6564, 41965}, {8253, 17852}, {8972, 42575}, {9543, 42271}, {10195, 17851}, {13925, 14892}, {13951, 42602}, {15684, 42260}, {15698, 42259}, {15765, 41972}, {18538, 41963}, {18585, 41971}, {23263, 41957}, {32790, 42523}, {34200, 41952}, {35812, 41959}, {35822, 42606}, {41949, 42582}, {41950, 42267}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {485, 6450, 42572}, {5076, 41961, 42258}

X(41949) = X(590)X(15699)∩X(615)X(1328)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) - 19*a^2*S + 46*S^2 : :

X(41949) lies on the Evans conic and these lines: {2, 41965}, {6, 41950}, {30, 41966}, {486, 41963}, {590, 15699}, {615, 1328}, {3069, 3070}, {3071, 3530}, {3316, 42583}, {6396, 41968}, {6433, 8252}, {6440, 41962}, {6485, 6565}, {13785, 41955}, {13951, 41970}, {14893, 18762}, {15705, 41953}, {15714, 41951}, {23263, 42569}, {32786, 42607}, {32788, 41954}, {32790, 41961}, {41947, 42266}, {41948, 42582}, {41952, 42277}, {41957, 42525}, {41958, 42284}, {41959, 42601}, {41967, 42603}, {41969, 42215}

X(41950) = X(590)X(1327)∩X(615)X(15699)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) + 19*a^2*S + 46*S^2 : :

X(41950) lies on the Evans conic and these lines: {{2, 41966}, {6, 41949}, {30, 41965}, {485, 41964}, {590, 1327}, {615, 15699}, {3068, 3071}, {3070, 3530}, {3317, 42582}, {3590, 9543}, {6200, 41967}, {6434, 8253}, {6439, 9541}, {6484, 6564}, {8976, 9681}, {13665, 41956}, {14893, 18538}, {15705, 41954}, {15714, 41952}, {23253, 42568}, {32785, 42606}, {32787, 41953}, {32789, 41962}, {41947, 42583}, {41948, 42267}, {41951, 42274}, {41957, 42283}, {41958, 42524}, {41960, 42600}, {41968, 42602}, {41970, 42216}

X(41951) = X(2)X(6425)∩X(17)X(36439)

Barycentrics    6*a^2*(a^2 - b^2 - c^2) - 9*a^2*S + 22*S^2 : :

X(41951) lies on the Evans conic and these lines: {2, 6425}, {3, 42527}, {6, 41952}, {17, 36439}, {18, 36457}, {20, 42577}, {30, 35813}, {140, 42417}, {372, 14893}, {376, 615}, {381, 486}, {547, 590}, {549, 3071}, {1328, 13951}, {3069, 41958}, {3317, 15715}, {3534, 42609}, {3543, 6430}, {3594, 41099}, {5055, 31487}, {5071, 7582}, {5420, 14093}, {6200, 41957}, {6396, 41966}, {6398, 35434}, {6409, 42571}, {6419, 10109}, {6420, 38071}, {6445, 13785}, {6454, 33699}, {6460, 42579}, {6476, 41961}, {6493, 42272}, {6561, 15700}, {6565, 15687}, {7584, 11737}, {8252, 15721}, {10124, 10577}, {10194, 15701}, {12100, 42607}, {13966, 35404}, {14226, 15702}, {14269, 42418}, {14869, 42525}, {15683, 23261}, {15684, 41962}, {15691, 35821}, {15692, 42258}, {15699, 31454}, {15703, 42573}, {15714, 41949}, {15765, 41973}, {18510, 42602}, {18585, 41974}, {19053, 42273}, {23273, 41959}, {32785, 42605}, {32789, 41967}, {34200, 41947}, {35256, 41968}, {35400, 42261}, {35403, 42268}, {35822, 41954}, {36436, 42153}, {36437, 42251}, {36454, 42156}, {36455, 42250}, {41950, 42274}, {41956, 42284}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13785, 42603, 41945}, {41945, 42603, 32790}

X(41952) = X(2)X(6426)∩X(17)X(36457)

Barycentrics    6*a^2*(a^2 - b^2 - c^2) + 9*a^2*S + 22*S^2 : :

X(41952) lies on the Evans conic and these lines: {2, 6426}, {3, 42526}, {6, 41951}, {17, 36457}, {18, 36439}, {20, 42576}, {30, 35812}, {140, 42418}, {371, 14893}, {376, 590}, {381, 485}, {547, 615}, {549, 3070}, {1327, 8976}, {3068, 41957}, {3316, 15715}, {3534, 42608}, {3543, 6429}, {3592, 41099}, {3830, 31454}, {3845, 8960}, {5055, 42527}, {5071, 7581}, {5418, 14093}, {6200, 41965}, {6221, 35434}, {6396, 41958}, {6410, 42570}, {6419, 38071}, {6420, 10109}, {6446, 13665}, {6453, 33699}, {6459, 42578}, {6477, 41962}, {6492, 42271}, {6560, 15700}, {6564, 15687}, {7583, 11737}, {8253, 15721}, {8981, 35404}, {9543, 42537}, {9680, 15685}, {9690, 15684}, {10124, 10576}, {10195, 15701}, {12100, 42606}, {14241, 15702}, {14269, 42417}, {14869, 42524}, {15683, 23251}, {15691, 35820}, {15692, 42259}, {15703, 42572}, {15714, 41950}, {15765, 41974}, {18512, 42603}, {18585, 41973}, {19054, 42270}, {23267, 41960}, {32786, 42604}, {32790, 41968}, {34200, 41948}, {35255, 41967}, {35400, 42260}, {35403, 42269}, {35823, 41953}, {36436, 42156}, {36437, 42252}, {36454, 42153}, {36455, 42253}, {41949, 42277}, {41955, 42283}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13665, 42602, 41946}, {41946, 42602, 32789}

X(41953) = X(6)X(14226)∩X(486)X(1657)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) - 11*a^2*S + 26*S^2 : :

X(41953) lies on the Evans conic and these lines: {2, 41961}, {3, 41947}, {5, 41948}, {6, 14226}, {30, 41962}, {486, 1657}, {546, 3070}, {590, 5055}, {615, 8703}, {631, 3071}, {1132, 42263}, {1328, 17851}, {3069, 3543}, {3529, 42579}, {5068, 7585}, {6200, 41955}, {6435, 11737}, {6436, 23046}, {6448, 12819}, {6519, 41969}, {6561, 15700}, {10124, 18762}, {10138, 22615}, {11541, 17852}, {14813, 41978}, {14814, 41977}, {15705, 41949}, {18510, 42270}, {23259, 41966}, {32787, 41950}, {32788, 41956}, {32789, 41965}, {32790, 41957}, {35382, 42602}, {35823, 41952}, {41963, 42215}, {41970, 42267}, {42262, 42571}, {42522, 42582}

{X(6),X(41106)}-harmonic conjugate of X(41954)

X(41954) = X(6)X(14226)∩X(485)X(1657)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) + 11*a^2*S + 26*S^2 : :

X(41954) lies on the Evans conic and these lines: {2, 41962}, {3, 41948}, {5, 41947}, {6, 14226}, {30, 41961}, {485, 1657}, {546, 3071}, {590, 8703}, {615, 5055}, {631, 3070}, {1131, 42264}, {1327, 41959}, {3068, 3543}, {3529, 42578}, {5068, 7586}, {6396, 41956}, {6435, 23046}, {6436, 11737}, {6447, 12818}, {6522, 41970}, {6560, 15700}, {6561, 31487}, {9541, 35409}, {10124, 18538}, {10137, 22644}, {11541, 23251}, {14813, 41977}, {14814, 41978}, {15705, 41950}, {18512, 42273}, {23249, 41965}, {31414, 42523}, {31454, 41969}, {32787, 41955}, {32788, 41949}, {32789, 41958}, {32790, 41966}, {35382, 42603}, {35822, 41951}, {41964, 42216}, {42265, 42570}

{X(6),X(41106)}-harmonic conjugate of X(41953)

X(41955) = X(6)X(15640)∩X(371)X(3859)

Barycentrics    18*a^2*(a^2 - b^2 - c^2) - 17*a^2*S + 26*S^2 : :

X(41955) lies on the Evans conic and these lines: {2, 41957}, {6, 15640}, {30, 41958}, {371, 3859}, {590, 3545}, {615, 12100}, {3069, 3522}, {3070, 3627}, {3071, 3526}, {3311, 12818}, {3317, 9541}, {6200, 41953}, {6395, 6561}, {6459, 41969}, {6522, 18510}, {6565, 35018}, {7582, 42263}, {8252, 15721}, {13785, 41949}, {23273, 41968}, {31412, 42522}, {32787, 41954}, {32788, 41960}, {32789, 41961}, {35255, 41959}, {41952, 42283}

{X(6),X(15640)}-harmonic conjugate of X(41956)

X(41956) = X(6)X(15640)∩X(372)X(3859)

Barycentrics    18*a^2*(a^2 - b^2 - c^2) + 17*a^2*S + 26*S^2 : :

X(41956) lies on the Evans conic and these lines: {2, 41958}, {6, 15640}, {30, 41957}, {372, 3859}, {590, 12100}, {615, 3545}, {3068, 3522}, {3070, 3526}, {3071, 3627}, {3312, 12819}, {3316, 41948}, {6199, 6560}, {6396, 41954}, {6460, 17852}, {6519, 18512}, {6564, 35018}, {7581, 42264}, {8253, 15721}, {13665, 41950}, {17851, 32790}, {23267, 41967}, {32787, 41959}, {32788, 41953}, {35256, 41960}, {41951, 42284}, {42523, 42561}

{X(6),X(15640)}-harmonic conjugate of X(41955)

X(41957) = X(382)X(3070)∩X(590)X(5066)

Barycentrics    18*a^2*(a^2 - b^2 - c^2) - 17*a^2*S + 22*S^2 : :

X(41957) lies on the Evans conic and these lines: {2, 41955}, {6, 41958}, {30, 41956}, {382, 3070}, {590, 5066}, {615, 3524}, {632, 3071}, {3068, 41952}, {5056, 6429}, {6200, 41951}, {6221, 41965}, {6426, 7586}, {6446, 15695}, {6477, 41960}, {6565, 41961}, {7584, 33923}, {9690, 15703}, {10194, 13785}, {15686, 41946}, {23259, 41967}, {23263, 41948}, {32789, 41959}, {32790, 41953}, {35255, 41969}, {35823, 41968}, {41949, 42525}, {41950, 42283}

X(41958) = X(382)X(3071)∩X(590)X(3524)

Barycentrics    18*a^2*(a^2 - b^2 - c^2) + 17*a^2*S + 22*S^2 : :

X(41958) lies on the Evans conic and these lines: {2, 41956}, {6, 41957}, {30, 41955}, {382, 3071}, {590, 3524}, {615, 5066}, {632, 3070}, {3069, 41951}, {5056, 6430}, {6396, 41952}, {6398, 41966}, {6425, 7585}, {6445, 15695}, {6476, 41959}, {6564, 41962}, {7583, 33923}, {10195, 13665}, {15686, 41945}, {15703, 41946}, {23249, 41968}, {23253, 41947}, {32789, 41954}, {32790, 41960}, {35256, 41970}, {35822, 41967}, {41949, 42284}, {41950, 42524}

X(41959) = X(590)X(3839)∩X(615)X(6480)

Barycentrics    36*a^2*(a^2 - b^2 - c^2) - 41*a^2*S + 2*S^2 : :

X(41959) lies on the Evans conic and these lines: {6, 41960}, {371, 41970}, {590, 3839}, {615, 6480}, {1151, 10299}, {1327, 41954}, {3068, 42576}, {3069, 6439}, {3070, 6453}, {3071, 5070}, {6200, 41962}, {6221, 14093}, {6429, 7585}, {6476, 41958}, {8252, 41947}, {9690, 41966}, {9693, 23263}, {23273, 41951}, {32787, 41956}, {32789, 41957}, {34089, 41969}, {35255, 41955}, {35812, 41948}, {41949, 42601}, {41965, 41990}, {41967, 42283}

X(41960) = X(590)X(6481)∩X(615)X(3839)

Barycentrics    36*a^2*(a^2 - b^2 - c^2) + 41*a^2*S + 2*S^2 : :

X(41960) lies on the Evans conic and these lines: {6, 41959}, {372, 41969}, {590, 6481}, {615, 3839}, {1152, 10299}, {1328, 17851}, {3068, 6440}, {3069, 42577}, {3070, 5070}, {3071, 6454}, {6396, 41961}, {6398, 14093}, {6430, 7586}, {6477, 41957}, {8253, 17852}, {23267, 41952}, {32788, 41955}, {32790, 41958}, {34091, 41970}, {35256, 41956}, {35813, 41947}, {41950, 42600}, {41966, 41990}, {41968, 42284}

X(41961) = X(6)X(15698)∩X(20)X(1151)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) - 11*a^2*S - 2*S^2 : :

X(41961) lies on the Evans conic and these lines: {2, 41953}, {3, 41970}, {5, 41969}, {6, 15698}, {20, 1151}, {30, 41954}, {371, 15712}, {381, 41967}, {549, 41968}, {590, 3845}, {615, 5054}, {1587, 10147}, {3071, 3628}, {3533, 6429}, {3851, 6407}, {5071, 6468}, {5076, 6519}, {6200, 34200}, {6396, 41960}, {6425, 7582}, {6435, 17504}, {6436, 14891}, {6437, 7586}, {6439, 9541}, {6445, 15695}, {6460, 6488}, {6476, 41951}, {6478, 7584}, {6484, 41981}, {6565, 41957}, {8976, 10137}, {9680, 9691}, {9690, 15684}, {9693, 23261}, {10145, 42273}, {11541, 42578}, {32788, 41966}, {32789, 41955}, {32790, 41949}, {36439, 41971}, {36457, 41972}, {41965, 42283}, {42226, 42525}, {42276, 42572}, {42568, 42583}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 15698, 41962}, {41948, 42258, 5076}

X(41962) = X(6)X(15698)∩X(20)X(1152)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) + 11*a^2*S - 2*S^2 : :

X(41962) lies on the Evans conic and these lines: {2, 41954}, {3, 41969}, {5, 41970}, {6, 15698}, {20, 1152}, {30, 41953}, {372, 15712}, {381, 41968}, {549, 41967}, {590, 5054}, {615, 3845}, {1588, 10148}, {3070, 3628}, {3533, 6430}, {3851, 6408}, {5071, 6469}, {5076, 6522}, {6200, 41959}, {6396, 34200}, {6426, 7581}, {6435, 14891}, {6436, 17504}, {6438, 7585}, {6440, 41949}, {6446, 15695}, {6459, 6489}, {6477, 41952}, {6479, 7583}, {6485, 41981}, {6564, 41958}, {10138, 13951}, {10146, 42270}, {11541, 42579}, {13935, 17852}, {15684, 41951}, {17851, 32790}, {32787, 41965}, {32789, 41950}, {36439, 41972}, {36457, 41971}, {41966, 42284}, {42225, 42524}, {42275, 42573}, {42569, 42582}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 15698, 41961}, {41947, 42259, 5076}

X(41963) = X(2)X(6425)∩X(3)X(9680)

Barycentrics    2*a^2*(a^2 - b^2 - c^2) - 3*a^2*S - 2*S^2 : :

X(41963) lies on the Evans conic and these lines: {2, 6425}, {3, 9680}, {4, 590}, {5, 6453}, {6, 3523}, {13, 14814}, {14, 14813}, {17, 42222}, {18, 42221}, {20, 13846}, {30, 35812}, {140, 371}, {372, 15712}, {381, 6519}, {485, 1657}, {486, 41949}, {546, 41967}, {548, 35822}, {549, 6419}, {550, 3070}, {631, 3592}, {632, 35823}, {1152, 10299}, {1327, 17800}, {1328, 5079}, {1350, 26516}, {1587, 6411}, {1588, 3533}, {1656, 3071}, {2045, 16773}, {2046, 16772}, {2066, 9648}, {2067, 9663}, {3068, 3522}, {3091, 9692}, {3146, 3590}, {3311, 15720}, {3524, 3594}, {3526, 6447}, {3529, 6488}, {3530, 6420}, {3545, 9693}, {3832, 10141}, {3843, 42602}, {3845, 6482}, {3850, 6480}, {3851, 6407}, {3854, 9543}, {3858, 35821}, {4857, 9661}, {5055, 42417}, {5056, 6429}, {5059, 6433}, {5068, 6468}, {5073, 6445}, {5270, 9646}, {5420, 6199}, {5493, 8983}, {5882, 13912}, {6396, 41970}, {6410, 7585}, {6412, 7581}, {6426, 15717}, {6428, 15693}, {6431, 13935}, {6448, 15700}, {6451, 42261}, {6454, 12100}, {6455, 6560}, {6460, 42542}, {6470, 7586}, {6484, 6564}, {6486, 13925}, {6490, 23275}, {6496, 18512}, {6565, 35018}, {6696, 11241}, {7583, 33923}, {8375, 31401}, {8980, 10992}, {8991, 10533}, {8994, 30714}, {8997, 10991}, {8998, 10990}, {9585, 30315}, {9615, 13911}, {9616, 11522}, {9690, 42277}, {10303, 13847}, {10993, 13913}, {11292, 39657}, {12101, 42606}, {13886, 42264}, {13918, 14900}, {14869, 35813}, {15765, 41944}, {18538, 41948}, {18585, 41943}, {18762, 42566}, {18965, 31500}, {30389, 31440}, {33364, 40341}, {34200, 42418}, {35256, 35771}, {35403, 42526}, {35738, 36470}, {35815, 42216}, {41953, 42215}, {42126, 42194}, {42127, 42193}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 31454, 32787}, {5, 6453, 41945}, {6, 3523, 41964}, {6, 42569, 42523}, {381, 6519, 9681}, {550, 8960, 3070}, {550, 8981, 8960}, {590, 1151, 42258}, {590, 42258, 42273}, {590, 42271, 42265}, {631, 3592, 32788}, {1151, 9540, 590}, {1151, 42265, 9541}, {3068, 6409, 42259}, {3071, 5418, 32789}, {3851, 10195, 42582}, {5418, 6221, 3071}, {6200, 8960, 550}, {6200, 8981, 3070}, {6429, 8253, 6459}, {6445, 8976, 42260}, {6455, 13903, 6560}, {6459, 8253, 42270}, {6459, 9542, 6429}, {6561, 10195, 3851}, {8976, 42260, 42284}, {9541, 42265, 42271}, {9541, 42271, 42258}, {15717, 19054, 6426}

X(41964) = X(2)X(6426)∩X(3)X(9681)

Barycentrics    2*a^2*(a^2 - b^2 - c^2) + 3*a^2*S - 2*S^2 : :

X(41964) lies on the Evans conic and these lines: {2, 6426}, {3, 9681}, {4, 615}, {5, 6454}, {6, 3523}, {13, 14813}, {14, 14814}, {17, 42224}, {18, 42223}, {20, 13847}, {30, 35813}, {140, 372}, {371, 15712}, {381, 6522}, {485, 41950}, {486, 1657}, {546, 41968}, {548, 35823}, {549, 6420}, {550, 3071}, {631, 3594}, {632, 35822}, {1151, 10299}, {1327, 5079}, {1328, 17800}, {1350, 26521}, {1587, 3533}, {1588, 6412}, {1656, 3070}, {2045, 16772}, {2046, 16773}, {3069, 3522}, {3091, 17852}, {3146, 3591}, {3312, 15720}, {3524, 3592}, {3526, 6448}, {3529, 6489}, {3530, 6419}, {3832, 10142}, {3843, 42603}, {3845, 6483}, {3850, 6481}, {3851, 6408}, {3858, 35820}, {5055, 42418}, {5056, 6430}, {5059, 6434}, {5068, 6469}, {5073, 6446}, {5418, 6395}, {5493, 13971}, {5882, 13975}, {6200, 41969}, {6409, 7586}, {6411, 7582}, {6425, 15717}, {6427, 9680}, {6432, 9540}, {6447, 15700}, {6452, 42260}, {6453, 12100}, {6456, 6561}, {6459, 42541}, {6471, 7585}, {6485, 6565}, {6487, 13993}, {6491, 23269}, {6497, 18510}, {6564, 35018}, {6696, 11242}, {7584, 33923}, {8376, 31401}, {10303, 13846}, {10534, 13980}, {10990, 13990}, {10991, 13989}, {10992, 13967}, {10993, 13977}, {11291, 39664}, {12101, 42607}, {13939, 42263}, {13969, 30714}, {13985, 14900}, {14869, 35812}, {15701, 31487}, {15765, 41943}, {18538, 42567}, {18585, 41944}, {18762, 41947}, {33365, 40341}, {34200, 42417}, {35255, 35770}, {35403, 42527}, {35738, 36469}, {35814, 42215}, {41954, 42216}, {42126, 42192}, {42127, 42191}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 6454, 41946}, {6, 3523, 41963}, {6, 42568, 42522}, {549, 6420, 31454}, {615, 1152, 42259}, {615, 42259, 42270}, {615, 42272, 42262}, {631, 3594, 32787}, {1152, 13935, 615}, {3069, 6410, 42258}, {3070, 5420, 32790}, {3851, 10194, 42583}, {5420, 6398, 3070}, {6396, 13966, 3071}, {6427, 15693, 9680}, {6430, 8252, 6460}, {6446, 13951, 42261}, {6456, 13961, 6561}, {6460, 8252, 42273}, {6560, 10194, 3851}, {13951, 42261, 42283}, {15717, 19053, 6425}

X(41965) = X(548)X(3070)∩X(590)X(3830)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) - 19*a^2*S - 22*S^2 : :

X(41965) lies on the Evans conic and these lines: {2, 41949}, {6, 41966}, {30, 41950}, {548, 3070}, {590, 3830}, {615, 11539}, {3068, 6412}, {3071, 3090}, {3312, 15720}, {3523, 6493}, {3590, 5059}, {3858, 35821}, {5418, 41947}, {6200, 41952}, {6221, 41957}, {6430, 7581}, {6476, 11737}, {6561, 41969}, {6564, 41948}, {7586, 31454}, {8981, 41970}, {9542, 42582}, {23249, 41954}, {32787, 41962}, {32789, 41953}, {41959, 41990}, {41961, 42283}, {42264, 42570}

X(41966) = X(548)X(3071)∩X(590)X(11539)

Barycentrics    9*a^2*(a^2 - b^2 - c^2) + 19*a^2*S - 22*S^2 : :

X(41966) lies on the Evans conic and these lines: {2, 41950}, {6, 41965}, {30, 41949}, {548, 3071}, {590, 11539}, {615, 3830}, {3069, 6411}, {3070, 3090}, {3311, 15720}, {3523, 6492}, {3591, 5059}, {3858, 35820}, {5420, 41948}, {6396, 41951}, {6398, 41958}, {6429, 7582}, {6477, 11737}, {6560, 41970}, {6565, 41947}, {9690, 41959}, {13966, 41969}, {23259, 41953}, {32788, 41961}, {32790, 41954}, {41960, 41990}, {41962, 42284}, {42263, 42571}

X(41967) = X(590)X(3543)∩X(615)X(10124)

Barycentrics    12*a^2*(a^2 - b^2 - c^2) - 33*a^2*S - 46*S^2 : :

X(41967) lies on the Evans conic and these lines: {2, 41947}, {6, 41968}, {30, 41948}, {381, 41961}, {546, 41963}, {549, 41962}, {590, 3543}, {615, 10124}, {631, 3594}, {3068, 6440}, {3070, 8703}, {3071, 5055}, {6200, 41950}, {6452, 15700}, {6485, 41970}, {9540, 41106}, {10194, 31454}, {13846, 15705}, {15765, 41977}, {18585, 41978}, {23259, 41957}, {23267, 41956}, {32789, 41951}, {35255, 41952}, {35822, 41958}, {41949, 42603}, {41959, 42283}, {42266, 42606}

X(41968) = X(590)X(10124)∩X(615)X(3543)

Barycentrics    12*a^2*(a^2 - b^2 - c^2) + 33*a^2*S - 46*S^2 : :

X(41968) lies on the Evans conic and these lines: {2, 41948}, {6, 41967}, {30, 41947}, {381, 41962}, {546, 41964}, {549, 41961}, {590, 10124}, {615, 3543}, {631, 3592}, {3069, 6439}, {3070, 5055}, {3071, 8703}, {6396, 41949}, {6451, 15700}, {6484, 41969}, {9691, 15722}, {13847, 15705}, {13935, 41106}, {15765, 41978}, {18585, 41977}, {23249, 41958}, {23273, 41955}, {32790, 41952}, {35256, 41951}, {35823, 41957}, {41950, 42602}, {41960, 42284}, {42267, 42607}

X(41969) = X(3)X(41962)∩X(4)X(41948)

Barycentrics    10*a^2*(a^2 - b^2 - c^2) - 11*a^2*S + 2*S^2 : :

X(41969) lies on the Evans conic and these lines: {3, 41962}, {4, 41948}, {5, 41961}, {6, 41970}, {140, 41947}, {371, 8703}, {372, 41960}, {546, 590}, {615, 631}, {1328, 5055}, {1657, 3070}, {3069, 10147}, {3071, 6480}, {3543, 6429}, {5068, 6468}, {6200, 41964}, {6409, 15705}, {6425, 7585}, {6437, 7581}, {6449, 15700}, {6459, 41955}, {6476, 8981}, {6478, 7583}, {6484, 41968}, {6488, 13935}, {6490, 23253}, {6519, 41953}, {6561, 41965}, {8253, 9692}, {8976, 9681}, {9540, 41106}, {9541, 11541}, {9690, 32790}, {10124, 10577}, {10137, 13785}, {10145, 42583}, {10303, 42579}, {13886, 35409}, {13966, 41966}, {23275, 32789}, {31454, 41954}, {34089, 41959}, {35255, 41957}, {41949, 42215}, {42571, 42605}

X(41970) = X(3)X(41961)∩X(4)X(41947)

Barycentrics    10*a^2*(a^2 - b^2 - c^2) + 11*a^2*S + 2*S^2 : :

X(41970) lies on the Evans conic and these lines: {3, 41961}, {4, 41947}, {5, 41962}, {6, 41969}, {140, 41948}, {371, 41959}, {372, 8703}, {546, 615}, {590, 631}, {1327, 5055}, {1657, 3071}, {3068, 10148}, {3070, 6481}, {3543, 6430}, {5068, 6469}, {6396, 41963}, {6410, 15705}, {6426, 7586}, {6438, 7582}, {6450, 15700}, {6460, 17852}, {6477, 13966}, {6479, 7584}, {6485, 41967}, {6489, 9540}, {6491, 23263}, {6522, 41954}, {6560, 41966}, {8981, 41965}, {10124, 10576}, {10138, 13665}, {10146, 42582}, {10303, 42578}, {13935, 41106}, {13939, 35409}, {13951, 41949}, {17851, 42284}, {23269, 32790}, {34091, 41960}, {35256, 41958}, {41950, 42216}, {41953, 42267}, {42570, 42604}

X(41971) = X(13)X(3543)∩X(14)X(10124)

Barycentrics    39*a^2*(a^2 - b^2 - c^2) - 22*Sqrt[3]*a^2*S + 32*S^2 : :

X(41971) lies on the Evans conic and these lines: {3, 41977}, {5, 41978}, {6, 41972}, {13, 3543}, {14, 10124}, {15, 5055}, {16, 8703}, {17, 546}, {18, 631}, {61, 1657}, {3643, 33624}, {5068, 16964}, {10645, 15700}, {11480, 15722}, {11541, 37640}, {12817, 37832}, {15765, 41947}, {16241, 41973}, {16268, 41983}, {16962, 42142}, {16967, 42594}, {18585, 41948}, {19106, 35409}, {19107, 33607}, {19708, 42479}, {36439, 41961}, {36457, 41962}, {36967, 42120}, {37835, 42490}, {41107, 42119}, {41108, 42129}, {41121, 42154}, {41943, 42117}, {42108, 42506}, {42112, 42516}, {42150, 42433}

X(41972) = X(13)X(10124)∩X(14)X(3543)

Barycentrics    39*a^2*(a^2 - b^2 - c^2) + 22*Sqrt[3]*a^2*S + 32*S^2 : :

X(41972) lies on the Evans conic and these lines: {3, 41978}, {5, 41977}, {6, 41971}, {13, 10124}, {14, 3543}, {15, 8703}, {16, 5055}, {17, 631}, {18, 546}, {62, 1657}, {3642, 33622}, {5068, 16965}, {10646, 15700}, {11481, 15722}, {11541, 37641}, {12816, 37835}, {15765, 41948}, {16242, 41974}, {16267, 41983}, {16963, 42139}, {16966, 42595}, {18585, 41947}, {19106, 33606}, {19107, 35409}, {19708, 42478}, {36439, 41962}, {36457, 41961}, {36968, 42119}, {37832, 42491}, {41107, 42132}, {41108, 42120}, {41122, 42155}, {41944, 42118}, {42109, 42507}, {42113, 42517}, {42151, 42434}

X(41973) = X(4)X(13)∩X(14)X(140)

Barycentrics    11*a^2*(a^2 - b^2 - c^2) - 6*Sqrt[3]*a^2*S + 16*S^2 : :

X(41973) lies on the Evans conic and these lines: {3, 16268}, {4, 13}, {5, 41101}, {6, 5073}, {14, 140}, {15, 1656}, {16, 398}, {17, 3850}, {18, 3523}, {20, 42510}, {30, 42420}, {62, 1657}, {203, 4857}, {376, 3411}, {382, 41107}, {395, 33923}, {396, 3858}, {397, 19107}, {531, 11290}, {546, 3412}, {548, 16963}, {549, 42503}, {631, 41113}, {3090, 10188}, {3091, 16962}, {3389, 42231}, {3390, 42232}, {3522, 5351}, {3525, 41120}, {3526, 41122}, {3529, 41100}, {3533, 18581}, {3832, 12817}, {3839, 42532}, {3843, 16267}, {3851, 22236}, {5054, 42509}, {5056, 42159}, {5059, 42085}, {5068, 5343}, {5270, 7005}, {5340, 42126}, {5344, 42104}, {5349, 16808}, {5350, 42136}, {5352, 15720}, {5365, 18582}, {6656, 12154}, {7685, 22532}, {7755, 41407}, {10299, 16242}, {10304, 42507}, {10646, 41977}, {12102, 12816}, {12155, 33280}, {14813, 41945}, {14814, 41946}, {14893, 42506}, {15681, 42613}, {15686, 42533}, {15765, 41951}, {16002, 38740}, {16241, 41971}, {16772, 35018}, {16773, 42529}, {16960, 42133}, {16961, 42122}, {16965, 42164}, {16966, 42473}, {18585, 41952}, {21735, 42119}, {22496, 36769}, {30440, 40647}, {33703, 42589}, {34755, 42130}, {36330, 37340}, {36836, 37835}, {37641, 42433}, {41978, 42124}, {41987, 42502}, {42092, 42495}, {42099, 42151}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5073, 41974}, {6, 42432, 42431}, {13, 16964, 42160}, {14, 5238, 42489}, {14, 42147, 5238}, {18, 42150, 10645}, {61, 16964, 36970}, {140, 42599, 10187}, {398, 42117, 42157}, {398, 42157, 16}, {546, 3412, 41121}, {5073, 41974, 42431}, {5334, 42150, 18}, {5343, 42152, 16809}, {10654, 16964, 61}, {22237, 42089, 18}, {36967, 40694, 5351}, {41974, 42432, 5073}

X(41974) = X(4)X(14)∩X(13)X(140)

Barycentrics    11*a^2*(a^2 - b^2 - c^2) + 6*Sqrt[3]*a^2*S + 16*S^2 : :

X(41974) lies on the Evans conic and these lines: {3, 16267}, {4, 14}, {5, 41100}, {6, 5073}, {13, 140}, {15, 397}, {16, 1656}, {17, 3523}, {18, 3850}, {20, 42511}, {30, 42419}, {61, 1657}, {202, 4857}, {376, 3412}, {382, 41108}, {395, 3858}, {396, 33923}, {398, 19106}, {530, 11289}, {546, 3411}, {548, 16962}, {549, 42502}, {631, 41112}, {3090, 10187}, {3091, 16963}, {3364, 42233}, {3365, 42234}, {3522, 5352}, {3525, 41119}, {3526, 41121}, {3529, 41101}, {3533, 18582}, {3832, 12816}, {3839, 42533}, {3843, 16268}, {3851, 22238}, {5054, 42508}, {5056, 42162}, {5059, 42086}, {5068, 5344}, {5270, 7006}, {5339, 42127}, {5343, 42105}, {5349, 42137}, {5350, 16809}, {5351, 15720}, {5366, 18581}, {6656, 12155}, {7684, 22531}, {7755, 41406}, {10299, 16241}, {10304, 42506}, {10645, 41978}, {12102, 12817}, {12154, 33280}, {14813, 41946}, {14814, 41945}, {14893, 42507}, {15681, 42612}, {15686, 42532}, {15765, 41952}, {16001, 38740}, {16242, 41972}, {16772, 42528}, {16773, 35018}, {16960, 42123}, {16961, 42134}, {16964, 42165}, {16967, 42472}, {18585, 41951}, {21735, 42120}, {22511, 41019}, {30439, 40647}, {33703, 42588}, {34754, 42131}, {35739, 42229}, {35752, 37341}, {36843, 37832}, {37640, 42434}, {41977, 42121}, {41987, 42503}, {42089, 42494}, {42100, 42150}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5073, 41973}, {6, 42431, 42432}, {13, 5237, 42488}, {13, 42148, 5237}, {14, 16965, 42161}, {17, 42151, 10646}, {62, 16965, 36969}, {140, 42598, 10188}, {397, 42118, 42158}, {397, 42158, 15}, {546, 3411, 41122}, {5073, 41973, 42432}, {5335, 42151, 17}, {5344, 42149, 16808}, {10653, 16965, 62}, {22235, 42092, 17}, {36968, 40693, 5352}, {41973, 42431, 5073}

X(41975) = X(6)X(140)∩X(590)X(3386)

Barycentrics    a^2*(a^2 - b^2 - c^2) + 2*Sqrt[2]*a^2*S - 4*S^2 : :

X(41975) lies on the Evans conic and these lines: {3, 41979}, {5, 41980}, {6, 140}, {590, 3386}, {615, 3372}, {3374, 35821}, {3388, 35820}, {9540, 14782}, {13935, 14783}, {14784, 31412}, {14785, 42561}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 140, 41976}, {5418, 8981, 41976}, {5420, 13966, 41976}, {31401, 38110, 41976}, {42121, 42149, 41976}, {42124, 42152, 41976}

X(41976) = X(6)X(140)∩X(590)X(3385)

Barycentrics    a^2*(a^2 - b^2 - c^2) - 2*Sqrt[2]*a^2*S - 4*S^2 : :

X(41976) lies on the Evans conic and these lines: {3, 41980}, {5, 41979}, {6, 140}, {590, 3385}, {615, 3371}, {3373, 35821}, {3387, 35820}, {9540, 14783}, {13935, 14782}, {14784, 42561}, {14785, 31412}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 140, 41975}, {5418, 8981, 41975}, {5420, 13966, 41975}, {31401, 38110, 41975}, {42121, 42149, 41975}, {42124, 42152, 41975}

X(41977) = X(14)X(20)∩X(62)X(5054)

Barycentrics    23*a^2*(a^2 - b^2 - c^2) + 22*Sqrt[3]*a^2*S - 32*S^2 : :

X(41977) lies on the Evans conic and these lines: {3, 41971}, {5, 41972}, {6, 41978}, {13, 3628}, {14, 20}, {15, 15712}, {16, 3851}, {17, 3533}, {18, 42101}, {62, 5054}, {398, 34200}, {1656, 42477}, {3845, 41944}, {3859, 36969}, {5071, 42510}, {5351, 15695}, {10646, 41973}, {14813, 41954}, {14814, 41953}, {14892, 42148}, {15684, 36843}, {15698, 16963}, {15765, 41967}, {16267, 42420}, {18585, 41968}, {41974, 42121}, {41981, 42087}

X(41978) = X(13)X(20)∩X(61)X(5054)

Barycentrics    23*a^2*(a^2 - b^2 - c^2) - 22*Sqrt[3]*a^2*S - 32*S^2 : :

X(41978) lies on the Evans conic and these lines: {3, 41972}, {5, 41971}, {6, 41977}, {13, 20}, {14, 3628}, {15, 3851}, {16, 15712}, {17, 42102}, {18, 3533}, {61, 5054}, {397, 34200}, {1656, 42476}, {3845, 41943}, {3859, 36970}, {5071, 42511}, {5352, 15695}, {10645, 41974}, {14813, 41953}, {14814, 41954}, {14892, 42147}, {15684, 36836}, {15698, 16962}, {15765, 41968}, {16268, 42419}, {18585, 41967}, {41973, 42124}, {41981, 42088}

X(41979) = X(6)X(30)∩X(3070)X(3371)

Barycentrics    3*a^2*(a^2 - b^2 - c^2) - 2 Sqrt[2]*a^2*S + 4*S^2 : :

The lines tangent to the Evans conic at X(41979) and X(41980) intersect in X(2)

X(41979) lies on the Evans conic and these lines: {3, 41975}, {5, 41976}, {6, 30}, {3070, 3371}, {3071, 3385}, {3372, 42258}, {3386, 42259}, {6459, 14784}, {6460, 14785}

X(41979) = reflection of X(41980) in X(6)
X(41979) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6560, 42216, 41980}, {6561, 42215, 41980}, {7737, 21850, 41980}, {10653, 42118, 41980}, {10654, 42117, 41980}, {18907, 31670, 41980}

X(41980) = X(6)X(30)∩X(3070)X(3372)

Barycentrics    3*a^2*(a^2 - b^2 - c^2) + 2 Sqrt[2]*a^2*S + 4*S^2 : :

The lines tangent to the Evans conic at X(41979) and X(41980) intersect in X(2)

X(41980) lies on the Evans conic and these lines: {3, 41976}, {5, 41975}, {6, 30}, {3070, 3372}, {3071, 3386}, {3371, 42258}, {3385, 42259}, {6459, 14785}, {6460, 14784}}

X(41980) = reflection of X(41979) in X(6)
X(41980) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6560, 42216, 41979}, {6561, 42215, 41979}, {7737, 21850, 41979}, {10653, 42118, 41979}, {10654, 42117, 41979}, {18907, 31670, 41979}

X(41981) = X(2)X(3)∩X(5493)X(11278)

Barycentrics    26*a^4 - 21*a^2*b^2 - 5*b^4 - 21*a^2*c^2 + 10*b^2*c^2 - 5*c^4 : :
X(41981) = 15 X[2] - 31 X[3]

X(41981) lies on these lines: {2, 3}, {17, 42584}, {18, 42585}, {5351, 42497}, {5352, 42496}, {5493, 11278}, {5844, 12512}, {6429, 42216}, {6430, 42215}, {6437, 42261}, {6438, 42260}, {6484, 41961}, {6485, 41962}, {6486, 7583}, {6487, 7584}, {7745, 15602}, {12002, 17704}, {12041, 13393}, {13464, 28216}, {15172, 37587}, {28212, 33179}, {31666, 34638}, {34754, 42123}, {34755, 42122}, {41977, 42087}, {41978, 42088}

X(41982) = X(2)X(3)∩X(3653)X(28216)

Barycentrics    46*a^4 - 41*a^2*b^2 - 5*b^4 - 41*a^2*c^2 + 10*b^2*c^2 - 5*c^4 : :
X(41982) = 5 X[2] - 17 X[3]

X(41982) lies on these lines: {2, 3}, {3653, 28216}, {11694, 37853}, {16962, 42123}, {16963, 42122}, {19883, 28182}, {28190, 38068}, {28194, 31662}, {28212, 30392}

X(41983) = X(2)X(3)∩X(165)X(38022)

Barycentrics    26*a^4 - 31*a^2*b^2 + 5*b^4 - 31*a^2*c^2 - 10*b^2*c^2 + 5*c^4 : :
X(41983) = 5 X[2] + 7 X[3]

X(41983) lies on these lines: {2, 3}, {165, 38022}, {230, 15602}, {519, 31662}, {3653, 16200}, {3655, 4668}, {4669, 31666}, {4691, 13624}, {5102, 38064}, {5731, 38081}, {5844, 30392}, {6429, 13966}, {6430, 8981}, {6484, 41968}, {6485, 41967}, {6486, 7584}, {6487, 7583}, {10137, 13961}, {10138, 13903}, {11480, 42497}, {11481, 42496}, {11693, 38727}, {13393, 15020}, {13925, 41946}, {13993, 41945}, {16267, 41972}, {16268, 41971}, {16772, 42480}, {16773, 42481}, {16962, 34755}, {16963, 34754}, {17502, 38068}, {17704, 31834}, {21167, 39561}, {25055, 28212}, {31884, 38079}, {42143, 42529}, {42146, 42528}, {42433, 42590}, {42434, 42591}, {42490, 42510}, {42491, 42511}, {42586, 42610}, {42587, 42611}

X(41984) = X(2)X(3)∩X(5844)X(19883)

Barycentrics    26*a^4 - 49*a^2*b^2 + 23*b^4 - 49*a^2*c^2 - 46*b^2*c^2 + 23*c^4 : :
X(41984) = 23 X[2] + X[3]

X(41984) lies on these lines: {2, 3}, {15, 42594}, {16, 42595}, {5844, 19883}, {12045, 13391}, {16962, 42497}, {16963, 42496}, {28212, 38068}, {28224, 38083}, {42143, 42498}, {42146, 42499}, {42510, 42610}, {42511, 42611}

X(41985) = X(2)X(3)∩X(1131)X(10138)

Barycentrics    22*a^4 - 47*a^2*b^2 + 25*b^4 - 47*a^2*c^2 - 50*b^2*c^2 + 25*c^4 : :
X(41985) = 25 X[2] - X[3]

X(41985) lies on these lines: {2, 3}, {1131, 10138}, {1132, 10137}, {1154, 12045}, {3828, 33179}, {5097, 20582}, {5102, 38079}, {6482, 42417}, {6483, 42418}, {10139, 42262}, {10140, 42265}, {10172, 31662}, {16200, 38022}, {16644, 42477}, {16645, 42476}, {36967, 42594}, {36968, 42595}, {37640, 42492}, {37641, 42493}, {38083, 38155}

X(41986) = (pending)

Barycentrics    2*a^4 - 73*a^2*b^2 + 71*b^4 - 73*a^2*c^2 - 142*b^2*c^2 + 71*c^4 : :
X(41986) = 71 X[2] - 23 X[3]

X(41986) lies on this lines: {2, 3}

X(41987) = X(2)X(3)∩X(9812)X(38081)

Barycentrics    -22*a^4 - 7*a^2*b^2 + 29*b^4 - 7*a^2*c^2 - 58*b^2*c^2 + 29*c^4 : :
X(41987) = 29 X[2] - 17 X[3]

X(41987) lies on these lines: {2, 3}, {9812, 38081}, {16962, 42101}, {16963, 42102}, {41973, 42502}, {41974, 42503}, {42093, 42496}, {42094, 42497}

X(41988) = (pending)

Barycentrics    -46*a^4 - a^2*b^2 + 47*b^4 - a^2*c^2 - 94*b^2*c^2 + 47*c^4 : :
X(41988) = 47 X[2] - 31 X[3]

X(41988) lies on this line: {2, 3}

X(41989) = X(2)X(3)∩X(5965)X(30531)

Barycentrics    2*a^4 + 19*a^2*b^2 - 21*b^4 + 19*a^2*c^2 + 42*b^2*c^2 - 21*c^4 : :

X(41989) lies on these lines: {2, 3}, {5339, 42512}, {5340, 42513}, {5965, 30531}, {6468, 42268}, {6469, 42269}, {6470, 13925}, {6471, 13993}, {7173, 37602}, {8162, 10592}, {10095, 40247}, {14845, 31834}, {15520, 18358}, {16625, 18874}, {16960, 42163}, {16961, 42166}, {42164, 42592}, {42165, 42593}

X(41990) = X(2)X(3)∩X(3630)X(25561)

Barycentrics    16*a^4 + 31*a^2*b^2 - 47*b^4 + 31*a^2*c^2 + 94*b^2*c^2 - 47*c^4 : :

X(41990) lies on these lines: {2, 3}, {3630, 25561}, {30308, 38138}, {41959, 41965}, {41960, 41966}, {42159, 42419}, {42162, 42420}

X(41991) = X(2)X(3)∩X(1483)X(18492)

Barycentrics    8*a^4 + 7*a^2*b^2 - 15*b^4 + 7*a^2*c^2 + 30*b^2*c^2 - 15*c^4 : :

X(41991) lies on these lines: {2, 3}, {1483, 18492}, {1539, 38725}, {3592, 42578}, {3594, 42579}, {5097, 38136}, {5343, 42496}, {5344, 42497}, {6429, 42277}, {6430, 42274}, {6431, 42268}, {6432, 42269}, {6433, 22615}, {6434, 22644}, {6447, 23263}, {6448, 23253}, {6482, 35821}, {6483, 35820}, {6484, 42225}, {6485, 42226}, {6486, 42582}, {6487, 42583}, {7843, 20112}, {7982, 38138}, {9880, 38628}, {10113, 38792}, {10147, 10195}, {10148, 10194}, {10222, 12571}, {10263, 13570}, {10593, 37587}, {11278, 19925}, {11531, 18357}, {13451, 15058}, {14845, 32137}, {16194, 18874}, {16200, 37705}, {18483, 38112}, {22505, 38735}, {22515, 38746}, {22791, 38155}, {22938, 38758}, {31662, 31673}, {32062, 32205}, {33179, 38034}, {34754, 42110}, {34755, 42107}, {38141, 38757}, {39561, 39884}, {42101, 42581}, {42102, 42580}, {42135, 42166}, {42138, 42163}, {42143, 42161}, {42146, 42160}, {42492, 42585}, {42493, 42584}, {42530, 42598}, {42531, 42599}

X(41992) = X(2)X(3)∩X(373)X(16982)

Barycentrics    16*a^4 - 31*a^2*b^2 + 15*b^4 - 31*a^2*c^2 - 30*b^2*c^2 + 15*c^4 : :

X(41992) lies on these lines: {2, 3}, {373, 16982}, {395, 42592}, {396, 42593}, {3054, 5041}, {5351, 42499}, {5352, 42498}, {5609, 38725}, {6429, 18762}, {6430, 18538}, {6484, 42583}, {6485, 42582}, {7294, 37587}, {10172, 31666}, {10219, 10263}, {10222, 19878}, {10283, 34595}, {15026, 15082}, {19862, 33179}, {20582, 22234}, {22251, 36253}, {30389, 38138}, {30392, 37705}, {31253, 38028}, {32789, 35770}, {32790, 35771}, {34573, 39561}

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Points on Simmons inconics: X(41993)-X(42004)

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These points are contributed by Peter Moses, March 17-18, 2021.

See Bernard Gibert, Simmons Conics. In this section, the Simmons inconic with foci X(13) and X(15) is the 1st Simmons inconic and the Simmons inconic with foci X(14) and (16) is the 2nd Simmons inconic.

Suppose that IC is an inconic, and let P be its perspector. Then the barycentric product P*(Steiner inellipse) is on IC. In particular, if U is on the line at infinity, then the barycentric product X(13)*U^2 is on the 1st Simmons inconic, and X(14)*U^2 is on the 2nd Simmons inconic..

X(41993) = X(13)X(34087)∩X(694)X(3457)

Barycentrics    a^4*(b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :

X(41993) lies on the 1st Simmons inconic and these lines: {13, 34087}, {694, 3457}, {25322, 30454}, {30452, 34294}, {30460, 38994}

X(41993) = X(i)-isoconjugate of X(j) for these (i,j): {298, 24037}, {4602, 17402}
X(41993) = barycentric product X(i)*X(j) for these {i,j}: {13, 1084}, {32, 30452}, {300, 9427}, {669, 20578}, {2971, 36296}, {3124, 3457}, {5995, 22260}, {23099, 23895}
X(41993) = barycentric quotient X(i)/X(j) for these {i,j}: {1084, 298}, {3457, 34537}, {9426, 17402}, {9427, 15}, {20578, 4609}, {23099, 23870}, {23610, 6137}, {30452, 1502}

X(41994) = X(14)X(34087)∩X(694)X(3458)

Barycentrics    a^4*(b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :

X(41994) lies on the 2nd Simmons inconic and these lines: {14, 34087}, {694, 3458}, {25322, 30455}, {30453, 34294}, {30463, 38993}

X(41994) = X(i)-isoconjugate of X(j) for these (i,j): {299, 24037}, {4602, 17403}
X(41994) = barycentric product X(i)*X(j) for these {i,j}: {14, 1084}, {32, 30453}, {301, 9427}, {669, 20579}, {2971, 36297}, {3124, 3458}, {5994, 22260}, {23099, 23896}
X(41994) = barycentric quotient X(i)/X(j) for these {i,j}: {1084, 299}, {3458, 34537}, {9426, 17403}, {9427, 16}, {20579, 4609}, {23099, 23871}, {23610, 6138}, {30453, 1502}

X(41995) = X(13)X(470)∩X(1989)X(3457)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :

X(41995) lies on the 1st Simmons inconic and these lines: {13, 470}, {1989, 3457}, {1990, 36299}, {5318, 8014}, {9408, 36435}

X(41995) = X(13)-Ceva conjugate of X(36299)
X(41995) = X(2151)-isoconjugate of X(31621)
X(41995) = crosspoint of X(13) and X(36299)
X(41995) = barycentric product X(i)*X(j) for these {i,j}: {13, 3163}, {30, 36299}, {300, 9408}, {1099, 2153}, {3081, 36308}, {3233, 20578}, {3457, 36789}, {8737, 16163}, {14401, 36306}, {14583, 41888}, {16240, 40709}, {34334, 36296}
X(41995) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 31621}, {3081, 41887}, {3163, 298}, {3457, 40384}, {9408, 15}, {14583, 36311}, {16240, 470}, {36299, 1494}

X(41996) = X(14)X(471)∩X(1989)X(3458)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :

X(41996) lies on the 2nd Simmons inconic and these lines: {14, 471}, {1989, 3458}, {1990, 36298}, {5321, 8015}, {9408, 36435}

X(41996) = X(14)-Ceva conjugate of X(36298)
X(41996) = X(2152)-isoconjugate of X(31621)
X(41996) = crosspoint of X(14) and X(36298)
X(41996) = barycentric product X(i)*X(j) for these {i,j}: {14, 3163}, {30, 36298}, {301, 9408}, {1099, 2154}, {3081, 36311}, {3233, 20579}, {3458, 36789}, {8738, 16163}, {14401, 36309}, {14583, 41887}, {16240, 40710}, {34334, 36297}
X(41996) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 31621}, {3081, 41888}, {3163, 299}, {3458, 40384}, {9408, 16}, {14583, 36308}, {16240, 471}, {36298, 1494}

X(41997) = X(13)X(470)∩X(67)X(30454)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 + c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :

X(41997) lies on the 1st Simmons inconic and these lines: {13, 470}, {67, 30454}, {287, 38414}, {338, 30452}, {401, 11078}, {3269, 17434}, {30466, 37975}

X(41997) = X(i)-isoconjugate of X(j) for these (i,j): {15, 24000}, {2151, 23582}, {17402, 24019}, {23999, 34394}
X(41997) = barycentric product X(i)*X(j) for these {i,j}: {13, 15526}, {125, 40709}, {300, 3269}, {339, 36296}, {1650, 36308}, {2153, 17879}, {3265, 20578}, {3457, 36793}, {3926, 30452}, {5489, 23895}, {23616, 36306}
X(41997) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 23582}, {125, 470}, {520, 17402}, {1650, 41887}, {2153, 24000}, {3269, 15}, {3457, 23964}, {5489, 23870}, {8737, 32230}, {14582, 36309}, {15526, 298}, {20578, 107}, {20975, 8739}, {30452, 393}, {30468, 14165}, {35442, 33529}, {36296, 250}, {40709, 18020}

X(41998) = X(14)X(471)∩X(67)X(30455)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 + c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :

X(41998) lies on the 2nd Simmons inconic and these lines: {14, 471}, {67, 30455}, {287, 38413}, {338, 30453}, {401, 11092}, {3269, 17434}, {30469, 37974}

X(41998) = X(i)-isoconjugate of X(j) for these (i,j): {16, 24000}, {2152, 23582}, {17403, 24019}, {23999, 34395}
X(41998) = barycentric product X(i)*X(j) for these {i,j}: {14, 15526}, {125, 40710}, {301, 3269}, {339, 36297}, {1650, 36311}, {2154, 17879}, {3265, 20579}, {3458, 36793}, {3926, 30453}, {5489, 23896}, {23616, 36309}
X(41998) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 23582}, {125, 471}, {520, 17403}, {1650, 41888}, {2154, 24000}, {3269, 16}, {3458, 23964}, {5489, 23871}, {8738, 32230}, {14582, 36306}, {15526, 299}, {20579, 107}, {20975, 8740}, {30453, 393}, {30465, 14165}, {35442, 33530}, {36297, 250}, {40710, 18020}

X(41999) = X(13)X(11600)∩X(23)X(6104)

Barycentrics    (b - c)^2*(b + c)^2*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S)^2*(-a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S) : :

X(41999) lies on the 1st Simmons inconic and these lines: {13, 11600}, {23, 6104}, {115, 12077}, {137, 15609}, {546, 11555}, {16021, 30459}

X(41999) = barycentric product X(13)*X(13)*X(23872)

X(42000) = X(14)X(11601)∩X(23)X(6105)

Barycentrics    (b - c)^2*(b + c)^2*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S)^2*(-a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S) : :

X(42000) lies on the 2nd Simmons inconic and these lines: {14, 11601}, {23, 6105}, {115, 12077}, {137, 15610}, {546, 11556}, {16022, 30462}

X(42000) = barycentric product X(14)*X(14)*X(23872)

This is the end of PART 21: Centers X(40001) - X(42000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)