X(40) = BEVAN POINT

Trilinears

\(cos B + cos C - cos A - 1 : :\)

\(b/(c + a - b) + c/(a + b - c) - a/(b + c -a) : :\)

\(sin2(B/2) + sin2(C/2) - sin2(A/2) : :\)

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

\(b(tan B/2) + c(tan C/2) - a(tan A/2) : :\)

\((r/R) - 2 cos A : :\)

Notes

If you have GeoGebra, you can view X(40).

This point is mentioned in a problem proposal by Benjamin Bevan, published in Leybourn’s Mathematical Repository, 1804, p. 18.

Constructions received from Randy Hutson, January 29, 2015: (1) Let A’B’C’ be the extangents triangle. Let A″ be the reflection of A’ in BC, and define B″, C″ cyclically. A’A″, B’B″, C’C″ concur in X(40). (2) Let A’B’C’ be the extangents triangle. Let A″ be the cevapoint of B’ and C’, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40). (3) Let A’B’C’ be the hexyl triangle and A″B″C″ be the side-triangle of ABC and hexyl triangle. Let A* be the {B,C}-harmonic conjugate of A″, and define B*, C* cyclically. The lines A’A*, B’B*, C’C* concur in X(40). (4) Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib, Ic cyclically.

X(40) = X(104)-of-IaIbIc. (5) Let A’B’C’ be the cevian triangle of X(189). Let A″ be the orthocenter of AB’C’, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40). (6) Let A’B’C’ be the mixtilinear incentral triangle. Let A″ be the trilinear product B’*C’, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40).

Let A’B’C’ be the intouch triangle of the extangents triangle, if ABC is acute. Then A’B’C’ is homothetic to the Aquila triangle at X(40). (Randy Hutson, December 2, 2017)

Let A’B’C’ be the excentral triangle. Let A″ be the isogonal conjugate, wrt A’BC, of A. Define B″ and C″ cyclically. (A″ is also the reflection of A’ in BC, and cyclically for B″ and C″). The lines A’A″, B’B″, C’C″ concur in X(40). (Randy Hutson, December 2, 2017)

Let OA be the circle centered at the A-vertex of the anti-Mandart-incircle triangle and passing through A; define OB and OC cyclically. X(40) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the Gemini triangle 4 and passing through A; define OB and OC cyclically. X(40) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let I = X(1) and P = X(40), and let DEF = pedal triangle of P, Ab = AI∩PE, and define Bc and Ca cyclically, Ac = AI∩PF, and define Ba and Cb cyclically. The centers of the circles {Ab, Bc, Ca}} and {Ac, Ba, Cb}} are the bicentric pair PU(44). (Angel Montesdeoca, April 12, 2021)

X(40) lies on the following curves: Q010, Q122, K004, K033, K100, K133, K179, K199, K269, K308, K333, K338, K343, K384, K414, K619, K667, K679, K692, K710, K736, K750, K806, K807, K815, K826, the Jerabek hyhperbola of the excentral triangle, the Mandart hyperbola, and these lines: {1,3}, {2,946}, {4,9}, {5,1698}, {6,380}, {7,7160}, {8,20}, {11,6922}, {12,1836}, {15,10636}, {16,10637}, {21,3577}, {22,9626}, {24,9625}, {25,1902}, {26,9590}, {28,2328}, {30,191}, {31,580}, {32,9620}, {33,201}, {34,212}, {39,1571}, {41,2301}, {42,581}, {43,970}, {47,1774}, {49,9586}, {58,601}, {64,72}, {74,6011}, {75,10444}, {77,947}, {78,100}, {79,4338}, {80,90}, {92,412}, {98,6010}, {101,972}, {103,1292}, {104,1293}, {108,207}, {109,255}, {140,3624}, {144,5815}, {145,3218}, {149,10265}, {151,2816}, {153,3648}, {154,7973}, {164,188}, {168,9837}, {170,1282}, {173,8351}, {182,1700}, {184,9622}, {185,3611}, {187,10988}, {190,341}, {196,208}, {197,3556}, {198,2324}, {209,3293}, {210,1750}, {214,10698}, {219,610}, {220,910}, {221,223}, {226,3085}, {228,3191}, {238,1722}, {256,989}, {258,8092}, {269,8829}, {307,4329}, {322,8822}, {329,6260}, {347,7013}, {371,5415}, {372,5416}, {376,519}, {381,7989}, {382,5790}, {386,1064}, {387,579}, {388,3474}, {389,11435}, {390,938}, {392,474}, {402,12696}, {404,3877}, {405,1730}, {442,5715}, {483,3645}, {485,13893}, {486,13947}, {495,4312}, {497,1210}, {498,5219}, {499,6891}, {500,8274}, {511,1045}, {518,1071}, {542,9881}, {548,3633}, {549,3656}, {550,952}, {551,3524}, {572,1449}, {574,9619}, {576,8539}, {578,11428}, {595,602}, {612,2292}, {614,3915}, {631,1125}, {653,1895}, {659,2821}, {664,7183}, {672,2082}, {726,12251}, {728,1018}, {730,11257}, {738,1323}, {758,3158}, {774,4319}, {813,2724}, {846,9840}, {901,2716}, {902,3924}, {908,5552}, {912,5534}, {920,4302}, {936,960}, {943,5665}, {950,1708}, {953,2743}, {954,12560}, {956,3916}, {958,1012}, {971,2951}, {978,1050}, {979,9359}, {984,1721}, {990,7174}, {993,6906}, {997,3878}, {1000,4315}, {1001,3812}, {1006,3754}, {1043,7415}, {1054,11512}, {1056,4298}, {1058,11019}, {1066,4306}, {1104,3052}, {1118,1785}, {1130,6585}, {1145,2829}, {1147,9621}, {1151,7969}, {1152,7968}, {1154,6255}, {1160,5588}, {1161,5589}, {1181,2323}, {1191,3752}, {1253,1254}, {1256,9376}, {1320,11715}, {1329,1532}, {1330,2792}, {1334,7390}, {1386,5085}, {1419,3157}, {1421,1772}, {1423,13161}, {1448,7273}, {1473,8192}, {1475,11200}, {1478,1770}, {1479,1737}, {1480,5315}, {1483,3655}, {1496,4320}, {1503,1761}, {1519,6834}, {1537,3035}, {1587,13883}, {1588,13936}, {1593,1829}, {1621,6986}, {1630,2289}, {1633,2823}, {1656,7988}, {1657,4668}, {1707,5247}, {1712,3176}, {1723,2955}, {1724,3073}, {1725,1775}, {1726,2949}, {1727,4324}, {1728,1837}, {1736,4907}, {1739,12659}, {1743,2264}, {1745,2818}, {1746,4714}, {1748,5174}, {1765,3696}, {1769,9525}, {1777,1935}, {1783,7156}, {1790,3193}, {1817,1819}, {1834,2245}, {1859,1872}, {1870,4347}, {1888,3074}, {2066,2362}, {2123,3421}, {2130,3354}, {2131,3472}, {2177,2650}, {2218,4674}, {2222,2745}, {2254,2814}, {2266,4251}, {2269,2285}, {2294,3247}, {2331,3194}, {2717,2742}, {2771,5531}, {2777,10119}, {2778,2915}, {2782,9860}, {2784,4050}, {2794,4769}, {2796,12243}, {2801,5528}, {2807,5562}, {2835,3939}, {2886,5705}, {2900,10605}, {2939,2947}, {2940,2948}, {2945,2953}, {2946,2952}, {2956,9370}, {2957,14026}, {2975,3872}, {3008,7397}, {3062,4866}, {3065,12747}, {3068,13912}, {3069,13975}, {3070,13911}, {3071,13973}, {3086,3911}, {3090,3634}, {3091,3305}, {3095,3097}, {3098,9941}, {3099,9821}, {3100,9610}, {3146,3219}, {3160,7177}, {3169,5847}, {3182,3346}, {3185,7420}, {3207,6603}, {3208,3509}, {3220,9798}, {3241,10304}, {3243,3874}, {3244,3528}, {3306,3523}, {3309,4063}, {3348,3353}, {3355,3473}, {3358,5787}, {3398,10789}, {3434,6734}, {3436,6256}, {3452,6848}, {3467,5560}, {3476,4311}, {3485,5218}, {3486,4304}, {3487,3671}, {3488,4314}, {3515,11363}, {3516,11396}, {3526,11230}, {3529,3626}, {3534,4677}, {3545,3828}, {3555,10167}, {3560,5251}, {3575,5090}, {3583,6928}, {3585,6923}, {3622,5734}, {3636,10299}, {3640,11825}, {3641,11824}, {3647,11530}, {3653,12100}, {3661,6999}, {3663,10521}, {3681,3951}, {3683,3698}, {3689,3962}, {3690,11381}, {3692,5279}, {3697,5927}, {3710,10327}, {3711,4005}, {3714,5695}, {3715,3983}, {3719,7270}, {3729,4385}, {3781,5907}, {3784,13348}, {3814,6941}, {3822,6937}, {3825,6963}, {3827,12329}, {3839,10248}, {3841,6829}, {3844,10516}, {3868,3870}, {3873,12005}, {3880,12513}, {3884,6940}, {3886,10449}, {3890,5253}, {3914,5230}, {3918,6920}, {3947,5714}, {3955,13346}, {3980,12545}, {3984,4420}, {4026,5799}, {4047,5776}, {4084,12559}, {4187,7681}, {4293,10106}, {4299,6948}, {4326,5728}, {4333,5841}, {4384,6996}, {4413,6918}, {4414,10459}, {4421,12635}, {4450,5016}, {4511,4855}, {4654,10056}, {4662,5220}, {4663,11477}, {4669,11001}, {4678,5059}, {4816,12103}, {4847,5082}, {4880,13369}, {5013,9592}, {5044,8580}, {5056,9779}, {5057,6932}, {5067,10171}, {5084,7682}, {5088,9312}, {5171,11364}, {5180,6960}, {5234,9708}, {5252,7354}, {5259,6883}, {5260,6912}, {5261,8545}, {5267,6950}, {5274,5704}, {5281,5703}, {5295,5774}, {5312,5396}, {5314,7503}, {5316,6964}, {5330,13587}, {5426,5428}, {5432,11375}, {5433,11376}, {5435,9785}, {5439,10582}, {5440,5730}, {5442,6713}, {5445,6882}, {5550,10303}, {5554,6872}, {5559,7284}, {5561,7161}, {5688,5870}, {5689,5871}, {5692,5720}, {5722,10384}, {5726,9654}, {5729,9844}, {5735,5880}, {5744,6705}, {5745,6847}, {5763,11374}, {5768,5853}, {5791,8727}, {5804,8257}, {5805,8728}, {5806,11108}, {5905,10528}, {5909,10374}, {5918,12680}, {6043,11991}, {6048,9566}, {6068,6259}, {6198,9611}, {6200,9615}, {6221,9618}, {6237,9928}, {6241,11460}, {6246,10724}, {6265,13253}, {6407,9584}, {6700,6927}, {6759,10536}, {6764,12516}, {6842,7951}, {6889,10198}, {6890,10527}, {6897,10532}, {6899,10916}, {6943,11680}, {6947,10531}, {6949,11813}, {6967,10200}, {6990,12558}, {6998,9746}, {7082,12953}, {7387,8185}, {7413,12544}, {7589,7590}, {7596,8231}, {7672,7675}, {7673,7677}, {7970,11711}, {7978,11720}, {7983,11710}, {7984,11709}, {7993,12773}, {8075,8081}, {8076,8082}, {8078,8091}, {8089,8099}, {8090,8100}, {8107,8111}, {8108,8112}, {8140,12488}, {8144,9576}, {8188,10669}, {8189,10673}, {8197,9834}, {8204,9835}, {8214,9838}, {8215,9839}, {8224,8234}, {8244,12490}, {8245,9959}, {8423,12491}, {8616,13732}, {8981,13888}, {8983,9540}, {9521,10015}, {9751,12264}, {9786,10382}, {9857,9873}, {9896,12417}, {9906,12662}, {9907,12663}, {10087,11570}, {10090,12758}, {10197,11263}, {10373,11347}, {10386,12433}, {10404,11246}, {10436,10446}, {10437,10447}, {10578,11036}, {10606,12262}, {10695,11714}, {10696,11700}, {10697,11712}, {10703,11713}, {10705,12265}, {10738,12619}, {10791,12110}, {10899,10900}, {10912,11194}, {10915,12115}, {11251,11852}, {11445,12111}, {11722,13099}, {11754,11756}, {11763,11765}, {11772,11774}, {11781,11783}, {11828,12440}, {11829,12441}, {11900,12113}, {11919,12648}, {11920,12649}, {12059,12665}, {12247,13199}, {12387,12398}, {12407,12661}, {12408,13221}, {12517,12843}, {12518,12844}, {12519,12845}, {12530,12683}, {12556,12660}, {12653,12737}, {12670,12671}, {12756,12757}, {12786,13465}, {13935,13971}, {13942,13966}

X(40) = midpoint of X(i) and X(j) for these {i,j}: {1, 7991}, {3, 12702}, {4, 6361}, {8, 20}, {10, 5493}, {65, 7957}, {944, 12245}, {1768, 5541}, {2093, 7994}, {2100, 2101}, {2136, 6762}, {2448, 2449}, {2948, 9904}, {2951, 5223}, {3245, 5537}, {5531, 12767}, {6764, 12632}, {9860, 13174}, {9961, 12528}, {11826, 11827}, {12247, 13199}, {12408, 13221}, {12488, 12489}, {12526, 12565}, {12697, 12698}

X(40) = reflection of X(i) in X(j) for these {i,j}: {1, 3}, {3, 3579}, {4, 10}, {8, 11362}, {57, 3359}, {84, 1158}, {145, 5882}, {149, 10265}, {355, 5690}, {944, 4297}, {946, 6684}, {962, 946}, {1012, 4640}, {1071, 9943}, {1320, 11715}, {1482, 1385}, {1490, 11500}, {1537, 3035}, {1768, 12515}, {1836, 6907}, {2077, 13528}, {2948, 12778}, {3062, 5779}, {3555, 12675}, {3576, 165}, {3655, 8703}, {3656, 549}, {3679, 3654}, {3811, 8715}, {3868, 5884}, {4297, 12512}, {4301, 1125}, {5531, 12331}, {5535, 484}, {5587, 5657}, {5603, 10164}, {5691, 355}, {5693, 72}, {5732, 11495}, {5735, 5880}, {5881, 8}, {6210, 573}, {6261, 6796}, {6264, 104}, {6282, 6244}, {6326, 100}, {6361, 5493}, {6765, 3913}, {6769, 10306}, {7688, 7964}, {7701, 191}, {7970, 11711}, {7971, 6261}, {7978, 11720}, {7982, 1}, {7983, 11710}, {7984, 11709}, {7991, 12702}, {7993, 12773}, {8148, 10222}, {9579, 6850}, {9580, 6827}, {9589, 12699}, {9799, 9948}, {9812, 10175}, {9845, 9841}, {9856, 5044}, {10222, 13624}, {10695, 11714}, {10696, 11700}, {10697, 11712}, {10698, 214}, {10703, 11713}, {10705, 12265}, {10724, 6246}, {10738, 12619}, {10864, 84}, {10912, 11260}, {11014, 11012}, {11224, 10246}, {11372, 9}, {11477, 4663}, {11523, 3811}, {11531, 1482}, {12398, 12387}, {12407, 13211}, {12520, 12511}, {12629, 12513}, {12650, 12114}, {12651, 11496}, {12653, 12737}, {12665,14740}, {12672, 960}, {12688, 5777}, {12696, 402}, {12699, 5}, {12701, 6922}, {12703, 5119}, {12704, 46}, {12705, 12514}, {12717, 1766}, {12751, 1145}, {12842, 12516}, {12843, 12517}, {12844, 12518}, {12845, 12519}, {13099, 11722}, {13253, 6265}

X(40) = isogonal conjugate of X(84)

X(40) = isotomic conjugate of X(309)

X(40) = inverse-in-circumcircle of X(2077)

X(40) = complement of X(962)

X(40) = anticomplement of X(946)

X(40) = X(963)-complementary conjugate of X(10)

X(40) = X(947)-anticomplementary conjugate of X(8)

X(40) = X(i)-Ceva conjugate of X(j) for these (i,j): (4, 2910), (8, 1), (20, 1490), (63, 9), (329, 2324), (347, 223), (515, 6326), (1817, 198), (7080, 1103), (7128, 101), (8822, 329), (9369, 1050), (9778, 2951)

X(40) = X(i)-cross conjugate of X(j) for these (i,j): (64, 3354), (198, 223), (208, 3342), (221, 1), (227, 7952), (2187, 2331), (7074, 2324)

X(40) = crosspoint of X(i) and X(j) for these (i,j): (329,347)

X(40) = crosssum of X(i) and X(j) for these (i,j): {19, 7008}, {56, 1413}, {513, 7004}, {649, 2310}, {1436, 2192}, {1903, 2357}

X(40) = crossdifference of every pair of points on line X(650)X(1459)

X(40) = cevapoint of X(i) and X(j) for these (i,j): {1, 2956}, {19, 8802}, {55, 3197}, {65, 8803}, {71, 3198}, {198, 7074}

X(40) = crosspoint of X(i) and X(j) for these (i,j): {8, 7080}, {63, 7013}, {100, 7012}, {190, 7045}, {329, 347}, {1817, 8822}

X(40) = trilinear pole of line {6129, 10397}

X(40) = point of concurrence of the perpendiculars from the excenters to the respective sides

X(40) = circumcenter of the excentral triangle

X(40) = incenter of the extangents triangle if triangle ABC is acute

X(40) = perspector of the excentral and extangents triangles

X(40) = perspector of the excentral and extouch triangles

X(40) = X(4)-of-hexyl-triangle

X(40) = X(4)-of-1st-circumperp-triangle

X(40) = X(20)-of-2nd-circumperp-triangle

X(40) = Miquel associate of X(8)

X(40) = perspector of hexyl triangle and cevian triangle of X(63)

X(40) = perspector of hexyl triangle and anticevian triangle of X(9)

X(40) = perspector of hexyl triangle and antipedal triangle of X(84)

X(40) = perspector of ABC and the reflection in X(57) of the antipedal triangle of X(57)

X(40) = excentral isogonal conjugate of X(1)

X(40) = excentral isotomic conjugate of X(1742)

X(40) = hexyl isogonal conjugate of X(1)

X(40) = perspector of ABC and extraversion triangle of X(84)

X(40) = trilinear product of extraversions of X(84)

X(40) = homothetic center of extangents triangle and reflection of intangents triangle in X(3)

X(40) = trilinear product of centers of mixtilinear incircles

X(40) = intangents-to-extangents similarity image of X(1)

X(40) = X(26)-of-reflection-triangle of X(1)

X(40) = {X(56),:ref:X(3057) <X(3057)>}-harmonic conjugate of X(1)

X(40) = perspector of extangents triangle and cross-triangle of ABC and extangents triangle

X(40) = perspector of ABC and cross-triangle of ABC and hexyl triangle

X(40) = circumcircle-inverse of X(2077)

X(40) = inverse-in-incircle-of-anticomplementary-triangle of X(10538)

X(40) = X(1)-Hirst inverse of X(9371)

X(40) = outer-Garcia-to-ABC similarity image of X(4)

X(40) = Cundy-Parry Phi transform of X(57)

X(40) = Cundy-Parry Psi transform of X(9)

X(40) = anticevian isogonal conjugate of X(1)

X(40) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 3345}, {513, 2077}, {2077, 513}, {3345, 3}

X(40) = X(5)-of-6th-mixtilinear-triangle

X(40) = orthic-to-ABC functional image of X(3), if ABC is acute

X(40) = intouch-to-ABC barycentric image of X(4)

X(40) = intouch-to-excentral similarity image of X(1)

X(40) = ABC-to-excentral barycentric image of X(1)

X(40) = antipode of X(12665) in the Mandart hyperbola

X(40) = extouch-isogonal conjugate of X(14872)

X(40) = X(10306)-of-Mandart-incircle-triangle

X(40) = X(65)-of-anti-Mandart-incircle-triangle

X(40) = endo-homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle; the homothetic center is X(382).

X(40) = X(i)-isoconjugate of X(j) for these (i,j): {1, 84}, {2, 1436}, {4, 1433}, {6, 189}, {7, 2192}, {8, 1413}, {9, 1422}, {31, 309}, {34, 271}, {40, 1256}, {55, 1440}, {56, 280}, {57, 282}, {63, 7129}, {65, 285}, {69, 7151}, {75, 2208}, {77, 7008}, {81, 1903}, {85, 7118}, {86, 2357}, {222, 7003}, {268, 278}, {273, 2188}, {279, 7367}, {284, 8808}, {346, 6612}, {348, 7154}, {513, 13138}, {522, 8059}, {603, 7020}, {1174, 13156}, {1812, 2358}, {3341, 3345}, {3346, 8886}, {7054, 13853}, {9375, 9376}

X(40) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 978}, {2, 57}, {8, 40}, {9, 1742}, {100, 4551}, {188, 1}, {259, 1740}, {366, 1743}, {522, 2957}, {556, 63}, {4146, 1445}, {4182, 165}, {6728, 1052}, {6731, 2951}, {7025, 361}, {7043, 7150}

X(40) = X(i)-beth conjugate of X(j) for these (i,j): {8, 4}, {21, 1420}, {40, 221}, {100, 40}, {643, 78}, {644, 728}, {13138, 3341}

X(40) = X(i)-gimel conjugate of X(j) for these (i,j): {8, 84}, {78, 40}, {521, 10085}, {522, 40}, {3717, 40}, {4041, 40}, {4086, 40}, {4147, 40}, {4163, 40}, {4391, 40}, {4397, 40}, {4723, 40}, {4768, 40}, {4811, 40}, {4985, 40}, {6615, 40}, {6735, 40}, {7628, 40}, {7629, 40}, {7646, 40}, {7647, 40}

X(40) = X(i)-he conjugate of X(j) for these (i,j): {2, 516}, {190, 40}, {312, 6211}, {645, 40}, {646, 40}, {3699, 40}, {4518, 1766}, {4554, 40}, {4582, 40}, {4621, 40}, {4633, 40}, {4876, 165}, {4997, 40}, {6335, 40}, {6559, 10860}, {8707, 40}, {9365, 1}, {11609, 3}, {13136, 40}

X(40) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 84}, {8, 40}, {10, 1158}, {11, 1768}, {21, 7701}, {55, 1709}, {65, 7992}, {72, 1490}, {145, 10864}, {200, 10860}, {210, 165}, {390, 11372}, {497, 57}, {517, 6001}, {518, 971}, {519, 515}, {521, 513}, {522, 3667}, {950, 4}, {952, 2829}, {958, 7330}, {960, 3}, {1001, 3358}, {1145, 2950}, {1697, 12705}, {1837, 46}, {1864, 1750}, {2098, 10085}, {2321, 1766}, {2802, 2800}, {2804, 900}, {3036, 12515}, {3057, 1}, {3058, 1699}, {3059, 2951}, {3239, 649}, {3271, 9355}, {3678, 6796}, {3680, 6762}, {3686, 573}, {3688, 1742}, {3706, 1764}, {3717, 6211}, {3738, 2827}, {3877, 3576}, {3878, 6261}, {3880, 517}, {3883, 6210}, {3884, 5450}, {3885, 7982}, {3886, 12717}, {3893, 7991}, {3900, 3309}, {3907, 6002}, {4046, 2941}, {4111, 2938}, {4520, 3294}, {4534, 5540}, {4662, 3579}, {4673, 10476}, {4847, 63}, {5119, 12686}, {5245, 1277}, {5246, 1276}, {5289, 7171}, {5697, 7971}, {5795, 10}, {5837, 12514}, {5853, 516}, {5854, 952}, {5856, 5851}, {6366, 2826}, {6737, 20}, {6738, 9948}, {8058, 522}, {8275, 7966}, {9119, 5776}, {9785, 3333}, {9898, 7160}, {9957, 12114}, {10106, 12246}, {10866, 3361}, {10950, 5691}, {12448, 8158}, {12527, 6223}, {12541, 6766}, {12572, 6260}, {12575, 946}

X(40) = barycentric product X(i)*X(j) for these {i,j}: {1, 329}, {6, 322}, {7, 2324}, {8, 223}, {9, 347}, {10, 1817}, {37, 8822}, {57, 7080}, {63, 7952}, {69, 2331}, {75, 198}, {76, 2187}, {78, 196}, {85, 7074}, {92, 7078}, {189, 1103}, {190, 6129}, {208, 345}, {219, 342}, {221, 312}, {227, 333}, {281, 7013}, {304, 3195}, {306, 3194}, {318, 7011}, {321, 2360}, {341, 6611}, {651, 8058}, {1088, 7368}, {2199, 3596}, {3209, 3718}, {5514, 7045}, {7017, 7114}, {7128, 7358}

X(40) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 189}, {2, 309}, {6, 84}, {9, 280}, {25, 7129}, {31, 1436}, {32, 2208}, {33, 7003}, {41, 2192}, {42, 1903}, {48, 1433}, {55, 282}, {56, 1422}, {57, 1440}, {65, 8808}, {101, 13138}, {196, 273}, {198, 1}, {208, 278}, {212, 268}, {213, 2357}, {219, 271}, {221, 57}, {223, 7}, {227, 226}, {281, 7020}, {284, 285}, {322, 76}, {329, 75}, {342, 331}, {347, 85}, {354, 13156}, {604, 1413}, {607, 7008}, {1103, 329}, {1106, 6612}, {1253, 7367}, {1254, 13853}, {1415, 8059}, {1436, 1256}, {1817, 86}, {1819, 1812}, {1973, 7151}, {2175, 7118}, {2187, 6}, {2199, 56}, {2212, 7154}, {2324, 8}, {2331, 4}, {2360, 81}, {3194, 27}, {3195, 19}, {3197, 3341}, {3209, 34}, {6129, 514}, {6611, 269}, {7011, 77}, {7013, 348}, {7074, 9}, {7078, 63}, {7080, 312}, {7114, 222}, {7368, 200}, {7952, 92}, {8058, 4391}, {8822, 274}, {10397, 521}

X(40) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 3576), (1, 35, 3601), (1, 36, 1420), (1, 46, 57), (1, 57, 3333), (1, 65, 11529), (1, 165, 3), (1, 484, 46), (1, 1764, 10476), (1, 2093, 65), (1, 3336, 3338), (1, 3339, 942), (1, 3361, 999), (1, 3612, 13384), (1, 3670, 3677), (1, 3746, 10389), (1, 5010, 3612), (1, 5119, 1697), (1, 5264, 5269), (1, 5697, 7962), (1, 5709, 12704), (1, 5902, 11518), (1, 5903, 3340), (1, 7987, 1385), (1, 7994, 6769), (1, 9819, 9957), (1, 10268, 10902), (1, 10980, 5045), (1, 11010, 5119), (1, 11224, 10222), (1, 11531, 1482), (2, 946, 8227), (2, 962, 946), (3, 55, 10902), (3, 942, 8726), (3, 1385, 7987), (3, 1482, 1385), (3, 2095, 9940), (3, 3428, 11012), (3, 3579, 165), (3, 5584, 7688), (3, 5708, 11227), (3, 5709, 57), (3, 6244, 10310), (3, 7991, 7982), (3, 8148, 10246), (3, 8158, 999), (3, 8251, 10319), (3, 9940, 10857), (3, 10246, 13624), (3, 10306, 55), (3, 10310, 2077), (3, 10679, 10267), (3, 10680, 10269), (3, 11248, 35), (3, 11249, 36), (4, 10, 5587), (4, 5657, 10), (4, 6197, 19), (4, 12705, 11372), (5, 12699, 1699), (8, 9778, 20), (8, 10860, 10864), (9, 1706, 10), (10, 573, 9548), (10, 12514, 9), (10, 12572, 2551), (11, 12701, 9614), (12, 1836, 9612), (19, 71, 9), (19, 11471, 4), (20, 63, 84), (20, 9537, 3101), (20, 11362, 5881), (35, 3245, 5903), (35, 5537, 11248), (35, 5903, 1), (36, 5697, 1), (39, 1571, 9574), (39, 1572, 9575), (42, 4300, 581), (43, 1695, 970), (46, 484, 5128), (46, 1697, 3333), (46, 3338, 3336), (46, 5119, 1), (46, 5709, 5535), (46, 11010, 1697), (55, 65, 1), (55, 2093, 11529), (55, 5183, 2093), (55, 5584, 3), (55, 7957, 6769), (55, 7964, 165), (56, 3057, 1), (57, 1697, 1), (57, 5128, 46), (57, 7991, 6766), (63, 9778, 10860), (64, 3198, 1490), (65, 3579, 7688), (65, 6769, 7982), (65, 7964, 5584), (65, 10268, 3576), (72, 5687, 200), (72, 7580, 1490), (78, 6261, 6326), (100, 411, 6796), (100, 3869, 78), (140, 5886, 3624), (145, 3522, 5731), (145, 5731, 5882), (164, 505, 188), (165, 484, 3359), (165, 5538, 5010), (165, 6282, 2077), (165, 6769, 10902), (165, 7991, 1), (165, 7994, 55), (165, 11531, 7987), (165, 12702, 7982), (169, 3730, 9), (191, 2960, 1710), (191, 5691, 7330), (200, 12526, 72), (200, 12565, 1490), (210, 12688, 5777), (221, 227, 223), (221, 7074, 7078), (227, 7074, 1103), (354, 3303, 1), (355, 3654, 5690), (355, 5690, 3679), (376, 944, 4297), (376, 12245, 944), (381, 9956, 7989), (388, 3474, 4292), (392, 474, 8583), (411, 3869, 6261), (484, 5119, 57), (484, 7991, 5709), (484, 11010, 1), (497, 1788, 1210), (498, 12047, 5219), (573, 1766, 9), (595, 13329, 602), (631, 4301, 9624), (631, 5603, 1125), (942, 3295, 1), (946, 6684, 2), (950, 1708, 10396), (956, 10914, 4853), (958, 5836, 9623), (960, 1376, 936), (962, 6684, 8227), (986, 5255, 1), (999, 9957, 1), (1125, 4301, 5603), (1125, 5603, 9624), (1125, 10164, 631), (1151, 7969, 9583), (1155, 3057, 56), (1210, 10624, 497), (1276, 1277, 9), (1276, 6192, 6191), (1277, 6191, 6192), (1319, 2098, 1), (1381, 1382, 2077), (1385, 1482, 1), (1385, 7987, 3576), (1388, 5048, 1), (1402, 10480, 1), (1420, 7962, 1), (1467, 10388, 1), (1478, 1770, 9579), (1478, 10039, 9578), (1479, 1737, 9581), (1482, 11531, 7982), (1490, 12526, 5693), (1571, 1572, 39), (1656, 9955, 7988), (1697, 5128, 57), (1698, 1699, 5), (1698, 9589, 1699), (1699, 9589, 12699), (1700, 1701, 182), (1702, 1703, 6), (1704, 1705, 182), (1706, 12705, 5587), (1709, 7330, 7701), (1750, 7995, 12688), (1754, 5264, 3072), (1770, 10039, 1478), (1837, 6284, 3586), (2017, 2018, 39), (2077, 11012, 3), (2093, 10306, 7982), (2098, 5204, 1319), (2099, 2646, 1), (2099, 5217, 2646), (2136, 3928, 6762), (2136, 9841, 944), (2292, 4220, 8235), (2551, 5698, 12572), (2572, 2573, 3), (2975, 6909, 5450), (3057, 10270, 3576), (3085, 4295, 226), (3091, 9780, 10175), (3158, 11523, 3811), (3303, 5221, 354), (3304, 5919, 1), (3336, 3338, 57), (3340, 3601, 1), (3359, 3587, 165), (3359, 5119, 3576), (3359, 5709, 46), (3359, 7991, 12704), (3361, 9819, 1), (3428, 6244, 2077), (3428, 6282, 3576), (3428, 10310, 3), (3428, 13528, 165), (3436, 6925, 6256), (3485, 5218, 13411), (3496, 3501, 9), (3523, 3616, 10165), (3555, 10167, 12675), (3576, 5535, 57), (3576, 7982, 1), (3576, 12704, 3333), (3579, 7957, 10902), (3579, 7991, 3576), (3579, 10306, 10268), (3579, 12702, 1), (3587, 5709, 3), (3587, 12702, 1697), (3624, 11522, 5886), (3634, 3817, 3090), (3666, 5710, 1), (3671, 13405, 3487), (3679, 5691, 355), (3681, 9961, 12528), (3681, 11684, 3951), (3730, 5011, 169), (3746, 5902, 1), (3811, 8715, 3158), (3868, 3871, 3870), (3868, 7411, 10884), (3869, 6796, 6326), (3872, 4652, 2975), (3890, 9352, 5253), (3895, 5731, 7966), (3911, 12053, 3086), (3916, 10914, 956), (3931, 5711, 1), (4297, 6762, 9845), (4297, 12512, 376), (4301, 10164, 1125), (4302, 10573, 10572), (4314, 6738, 3488), (4424, 5264, 1), (4512, 12651, 11496), (4640, 5836, 958), (4853, 10914, 11525), (4855, 11682, 4511), (5044, 9709, 8580), (5045, 5708, 10980), (5045, 6767, 1), (5119, 5128, 3333), (5119, 5709, 7982), (5221, 8273, 9940), (5252, 7354, 9613), (5535, 7982, 12704), (5536, 7987, 3338), (5541, 6763, 3632), (5552, 11415, 908), (5584, 6769, 3576), (5584, 7957, 1), (5584, 7991, 11529), (5584, 10306, 10902), (5657, 6361, 4), (5687, 7580, 11500), (5708, 6767, 5045), (5709, 11010, 12703), (5714, 8164, 3947), (5758, 6908, 226), (5812, 6907, 9612), (5887, 11499, 5720), (6191, 6192, 9), (6210, 6211, 9), (6210, 12717, 11372), (6212, 6213, 9), (6252, 6404, 3779), (6736, 12527, 3421), (6769, 10268, 55), (6838, 11415, 12608), (6922, 12700, 9614), (6923, 10526, 3585), (6928, 10525, 3583), (7589, 12445, 7590), (7672, 7676, 7675), (7688, 10902, 3), (7742, 11508, 2078), (7957, 7964, 3), (7987, 7991, 11531), (7987, 11531, 1), (7991, 7994, 7957), (8075, 8093, 8081), (8076, 8094, 8082), (8107, 9805, 8111), (8108, 9806, 8112), (8148, 10222, 11224), (8148, 10246, 10222), (8148, 11224, 7982), (8148, 13624, 1), (8158, 9819, 7982), (8224, 9808, 8234), (9572, 9573, 8141), (9574, 9575, 39), (9576, 9577, 8144), (9578, 9579, 1478), (9580, 9581, 1479), (9582, 9583, 1151), (9584, 9585, 6407), (9586, 9587, 49), (9588, 9589, 5), (9590, 9591, 26), (9780, 9812, 3091), (9955, 11231, 1656), (10222, 10246, 1), (10222, 13624, 10246), (10267, 10679, 3746), (10269, 10680, 5563), (10306, 12702, 7957), (10389, 11518, 1), (10434, 12435, 1), (10470, 11521, 1), (10572, 10573, 5727), (10912, 11194, 11260), (11019, 12575, 1058), (11822, 11823, 55), (12000, 13373, 1), (12703, 12704, 7982)