X₄₀

BEVAN POINT

X₄₀ = point of concurrence of the perpendiculars from the excenters to the respective sides

5, 6, 7 3, 4, 5 Kepler Kimberling

X₄₀ (BEVAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Excentral Triangle △J_aJ_bJ_c — vertices J_a, J_b, J_c (Anticevian triangle of Incenter X₁ (the three excenters))

• Circle: Bevan Circle (bevan_circle) — Circumcircle of the excentral triangle △J_a J_b J_c; centered at X₄₀ BEVAN POINT

• Concurrence Lines: J_a–X₄₀, J_b–X₄₀, J_c–X₄₀ concurring at X₄₀

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

Search Value (6, 9, 13): \(11.874417278936\)

▸ Alternate Coordinates5 forms

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

• Trilinears: \(\sin^{2}(B/2) + \sin^{2}(C/2) - \sin^{2}(A/2) : :\)

• Trilinears: \(a^{3} + a^{2}(b + c) - a(b + c)^{2} - (b + c)(b - c)^{2} : :\)

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

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

▸ Linear Combinations2 combinations

• X₄₀ = X₁ - \(2\) X₃ = \(2R\) X₄ - \((r + 4R)\) X₉

• X₄₀ = X₁ - \(3\) X₁₆₅ = \(2\) X₃ - \(3\) X₁₆₅ = \(3\) X₃₇₆ - X₉₄₄ = \(5\) X₆₃₁ - \(4\) X₁₁₂₅ = \(3\) X₁ - \(4\) X₁₃₈₅ = \(9\) X₁₆₅ - \(4\) X₁₃₈₅ = \(3\) X₃ - \(2\) X₁₃₈₅ = \(3\) X₁ - \(2\) X₁₄₈₂ = \(9\) X₁₆₅ - \(2\) X₁₄₈₂ = \(3\) X₃ - X₁₄₈₂ = \(4\) X₅ - \(5\) X₁₆₉₈ = \(4\) X₅ - \(3\) X₁₆₉₉ = \(5\) X₁₆₉₈ - \(3\) X₁₆₉₉ = \(2\) X₃₀₉₅ - \(3\) X₃₀₉₇ = X₁₄₅ - \(5\) X₃₅₂₂ = \(2\) X₅₅₁ - \(3\) X₃₅₂₄ = \(2\) X₃₂₄₄ - \(7\) X₃₅₂₈ = \(8\) X₁₃₈₅ - \(9\) X₃₅₇₆ = \(4\) X₁₄₈₂ - \(9\) X₃₅₇₆ = \(2\) X₁ - \(3\) X₃₅₇₆ = \(4\) X₃ - \(3\) X₃₅₇₆ = \(3\) X₃₅₇₆ - \(8\) X₃₅₇₉

X₄₀ 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.

▸ Incident Central Lines532 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}

▸ Functional Relations69 relations

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = isogonal conjugate of X₈₄

• X₄₀ = isotomic conjugate of X₃₀₉

• X₄₀ = inverse-in-circumcircle of X₂₀₇₇

• X₄₀ = complement of X₉₆₂

• X₄₀ = anticomplement of X₉₄₆

• X₄₀ = X₉₆₃-complementary conjugate of X₁₀

• X₄₀ = X₉₄₇-anticomplementary conjugate of X₈

• X₄₀ = 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)

• X₄₀ = 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)

• X₄₀ = crosspoint of X_i and X_j for these (i,j): (329, 347)

• X₄₀ = crosssum of X_i and X_j for these (i,j): {19, 7008}, {56, 1413}, {513, 7004}, {649, 2310}, {1436, 2192}, {1903, 2357}

• X₄₀ = crossdifference of every pair of points on line X₆₅₀X₁₄₅₉

• X₄₀ = cevapoint of X_i and X_j for these (i,j): {1, 2956}, {19, 8802}, {55, 3197}, {65, 8803}, {71, 3198}, {198, 7074}

• X₄₀ = crosspoint of X_i and X_j for these (i,j): {8, 7080}, {63, 7013}, {100, 7012}, {190, 7045}, {329, 347}, {1817, 8822}

• X₄₀ = trilinear pole of line {6129, 10397}

• X₄₀ = point of concurrence of the perpendiculars from the excenters to the respective sides

• X₄₀ = circumcenter of the excentral triangle

• X₄₀ = incenter of the extangents triangle if triangle ABC is acute

• X₄₀ = perspector of the excentral and extangents triangles

• X₄₀ = perspector of the excentral and extouch triangles

• X₄₀ = X₄-of-hexyl-triangle

• X₄₀ = X₄-of-1st-circumperp-triangle

• X₄₀ = X₂₀-of-2nd-circumperp-triangle

• X₄₀ = Miquel associate of X₈

• X₄₀ = perspector of hexyl triangle and cevian triangle of X₆₃

• X₄₀ = perspector of hexyl triangle and anticevian triangle of X₉

• X₄₀ = perspector of hexyl triangle and antipedal triangle of X₈₄

• X₄₀ = perspector of ABC and the reflection in X₅₇ of the antipedal triangle of X₅₇

• X₄₀ = excentral isogonal conjugate of X₁

• X₄₀ = excentral isotomic conjugate of X₁₇₄₂

• X₄₀ = hexyl isogonal conjugate of X₁

• X₄₀ = perspector of ABC and extraversion triangle of X₈₄

• X₄₀ = trilinear product of extraversions of X₈₄

• X₄₀ = homothetic center of extangents triangle and reflection of intangents triangle in X₃

• X₄₀ = trilinear product of centers of mixtilinear incircles

• X₄₀ = intangents-to-extangents similarity image of X₁

• X₄₀ = X₂₆-of-reflection-triangle of X₁

• X₄₀ = {X₅₆,X₃₀₅₇}-harmonic conjugate of X₁

• X₄₀ = perspector of extangents triangle and cross-triangle of ABC and extangents triangle

• X₄₀ = perspector of ABC and cross-triangle of ABC and hexyl triangle

• X₄₀ = circumcircle-inverse of X₂₀₇₇

• X₄₀ = inverse-in-incircle-of-anticomplementary-triangle of X₁₀₅₃₈

• X₄₀ = X₁-Hirst inverse of X₉₃₇₁

• X₄₀ = outer-Garcia-to-ABC similarity image of X₄

• X₄₀ = Cundy-Parry Phi transform of X₅₇

• X₄₀ = Cundy-Parry Psi transform of X₉

• X₄₀ = anticevian isogonal conjugate of X₁

• X₄₀ = X_i-vertex conjugate of X_j for these (i,j): {3, 3345}, {513, 2077}, {2077, 513}, {3345, 3}

• X₄₀ = X₅-of-6th-mixtilinear-triangle

• X₄₀ = orthic-to-ABC functional image of X₃, if ABC is acute

• X₄₀ = intouch-to-ABC barycentric image of X₄

• X₄₀ = intouch-to-excentral similarity image of X₁

• X₄₀ = ABC-to-excentral barycentric image of X₁

• X₄₀ = antipode of X₁₂₆₆₅ in the Mandart hyperbola

• X₄₀ = extouch-isogonal conjugate of X₁₄₈₇₂

• X₄₀ = X₁₀₃₀₆-of-Mandart-incircle-triangle

• X₄₀ = X₆₅-of-anti-Mandart-incircle-triangle

• X₄₀ = endo-homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle; the homothetic center is X₃₈₂.

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = 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}

• X₄₀ = {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)

Notes

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 X₄₀.

• 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 X₄₀.

• 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 X₄₀.

• 4. Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b, I_c cyclically. X₄₀ = X₁₀₄-of-I_aI_bI_c.

• 5. Let A'B'C' be the cevian triangle of X₁₈₉. Let A″ be the orthocenter of AB'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₀. (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 X₄₀.

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 X₄₀. (Randy Hutson, December 2, 2017)

Let J_aJ_bJ_c be the excentral triangle. Let A″ be the isogonal conjugate, wrt J_aBC, of A. Define B″ and C″ cyclically. (A″ is also the reflection of J_a in BC, and cyclically for B″ and C″). The lines J_aA″, J_bB″, J_cC″ concur in X₄₀. (Randy Hutson, December 2, 2017)

Let O_A be the circle centered at the A-vertex of the anti-Mandart-incircle triangle and passing through A; define O_B and O_C cyclically. X₄₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the Gemini triangle 4 and passing through A; define O_B and O_C cyclically. X₄₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let I = X₁ and P = X₄₀, and let

• DEF = pedal triangle of P,

• A_b = AI∩PE, and define B_c and C_a cyclically,

• A_c = AI∩PF, and define B_a and C_b cyclically.

The centers of the circles {A_b, B_c, C_a}} and {A_c, B_a, C_b}} are the bicentric pair PU(44). (Angel Montesdeoca, April 12, 2021)