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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)


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}



underbar

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.

underbar

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).

underbar



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) = ZANA POINT

Barycentrics    sin A * (sec(B-C) / (sec(C-A) + sec(A-B) - sec(B-C)) : :
Trilinears    sec(B-C) / (sec(C-A) + sec(A-B) - sec(B-C)) : :

X(40140) lies on these lines: (pending)

X(40140) = hodpiece of X(5)


X(40141) = DEDALUS 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)

underbar



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.

underbar



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 CÚsar 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)

The Z-coordinate of D, Z(D) = D 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, 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.

underbar

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)

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).

underbar



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.

underbar



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)

underbar

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.

underbar



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.

underbar



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)

underbar



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(40385) = 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 x + q z)(p y + q x) + w (p y + 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}}

underbar



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}

underbar



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(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}






leftri   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)

underbar



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    (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) : :
X(40579) = X[3181] + 3 X[19773]

X(40579) lies on these lines: {2, 19777}, {3, 5624}, {13, 39170}, {14, 16}, {299, 1494}, {471, 8738}, {476, 36299}, {2154, 37773}, {2993, 3181}, {3642, 40580}, {8015, 37641}, {16241, 40695}, {16644, 18776}

X(40579) = midpoint of X(23896) and X(36311)
X(40579) = reflection of X(14) in X(10218)
X(40579) = complement of X(19777)
X(40579) = complementary conjugate of complement of X(16)-Ceva conjugate of X(6)
X(40579) = X(2)-Ceva conjugate of X(14)
X(40579) = perspector of circumconic centered at X(14)
X(40579) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 14}, {617, 2887}, {19299, 141}
X(40579) = X(15769)-cross conjugate of X(617)
X(40579) = X(i)-isoconjugate of X(j) for these (i,j): {1095, 40159}, {2152, 19777}
X(40579) = crosspoint of X(2) and X(617)
X(40579) = crosssum of X(6) and X(3441)
X(40579) = barycentric product X(i)*X(j) for these {i,j}: {14, 617}, {533, 39133}, {11092, 40579}, {11120, 15769}
X(40579) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 19777}, {617, 299}, {3458, 3441}, {8015, 34295}, {11085, 40159}, {15769, 619}, {39133, 11118}, {40579, 11078}
X(40579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16, 36210, 14}, {395, 11085, 14}, {395, 11549, 11085}, {11582, 16268, 14}


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) lies on these lines: {1, 3689}, {2, 1000}, {3, 5836}, {8, 442}, {9, 374}, {10, 1482}, {45, 4752}, {55, 5426}, {80, 31140}, {100, 2320}, {119, 2886}, {142, 519}, {145, 37462}, {210, 25415}, {214, 1376}, {355, 6260}, {405, 14923}, {474, 4861}, {514, 996}, {518, 1159}, {529, 18541}, {936, 10222}, {942, 4853}, {952, 2550}, {956, 3218}, {958, 3647}, {960, 8148}, {997, 10247}, {999, 3306}, {1001, 2802}, {1100, 34261}, {1125, 10912}, {1319, 16417}, {1329, 18493}, {1385, 1706}, {1538, 5587}, {1698, 2098}, {2092, 16777}, {2099, 3679}, {2551, 22791}, {2800, 5779}, {3057, 11108}, {3059, 4915}, {3126, 4825}, {3245, 16558}, {3295, 3895}, {3338, 3922}, {3340, 34790}, {3421, 39542}, {3434, 12690}, {3452, 3656}, {3526, 37828}, {3617, 5730}, {3626, 12635}, {3654, 5745}, {3680, 31792}, {3697, 11682}, {3715, 3899}, {3754, 5708}, {3812, 7373}, {3820, 5328}, {3826, 5854}, {3877, 35595}, {3880, 6600}, {3890, 16842}, {3898, 8167}, {3913, 30147}, {3918, 22837}, {3925, 12647}, {3927, 5903}, {4002, 19861}, {4304, 34707}, {4555, 20569}, {4674, 16499}, {4677, 5425}, {4731, 5048}, {5044, 7982}, {5045, 12629}, {5055, 5123}, {5082, 37730}, {5119, 16418}, {5176, 17532}, {5221, 5288}, {5252, 17528}, {5258, 37567}, {5438, 15178}, {5439, 36846}, {5554, 24390}, {5690, 6862}, {5698, 28212}, {5719, 34619}, {5774, 16821}, {5791, 11362}, {5794, 12645}, {5795, 12699}, {6184, 34522}, {6547, 24864}, {6735, 31479}, {6736, 11374}, {6762, 31794}, {6832, 12245}, {6923, 33898}, {7080, 37737}, {7171, 31788}, {7686, 8158}, {7971, 9947}, {7991, 31445}, {8256, 26363}, {8580, 16200}, {8582, 11373}, {8666, 37545}, {9819, 15837}, {10915, 28628}, {11278, 15829}, {12515, 22758}, {12609, 32049}, {12650, 31787}, {14151, 38092}, {15347, 16863}, {17060, 36479}, {17571, 37568}, {17573, 37618}, {17647, 18526}, {18357, 31418}, {18393, 31141}, {20085, 33110}, {21888, 31449}, {24870, 31139}, {25917, 30323}, {26727, 29676}, {28444, 35460}, {31246, 37735}, {31485, 35775}, {35457, 38066}

X(40587) = complement of X(1000)
X(40587) = complementary conjugate of X(3820)
X(40587) = X(2)-Ceva conjugate of X(45)
X(40587) = perspector of circumconic centered at X(45)


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: {8, 4553}, {75, 4552}, {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}

X(40624) = polar conjugate of X(4)-cross conjugate of X(108)
X(40624) = complement of isotomic conjugate of X(17496)
X(40624) = complement of trilinear pole of line X(5)X(10)
X(40624) = X(2)-Ceva conjugate of X(4391)
X(40624) = perspector of circumconic centered at X(4391)


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)

underbar



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(6213)
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)

underbar



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.

underbar



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 image 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 image 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 image 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}






leftri   Points associated with CCC cubics: X(40718) - X(40725)  rightri

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. /p>

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

underbar



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}, {2