X₆₅₀

CROSSDIFFERENCE OF X₁ AND X₃

X₆₅₀ = crossdifference of every pair of points on line {1, 3}

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

X₆₅₀ (CROSSDIFFERENCE OF X₁ AND X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {230, 231}, {44, 513}

• Related Centers: X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈), X₄₄ (-LINE CONJUGATE OF X₁), X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀)

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

Barycentrics: \(\sin A (\cos B - \cos C) : \sin B (\cos C - \cos A) : \sin C (\cos A - \cos B)\)

Search Value (6, 9, 13): \(-18.029957434208\)

▸ Alternate Coordinates5 forms

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

• Trilinears: \(\cos(A/2) \sin(B/2 - C/2) : :\)

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

• Trilinears: \(\cos^{2}(B/2) - \cos^{2}(C/2) : :\)

• Barycentrics: \(a(b - c)(b + c - a) : b(c - a)(c + a - b) : c(a - b)(a + b - c)\)

▸ Linear Combinations1 combination

• X₆₅₀ = \(3\) X₂ + X₁₇₄₉₄, \(3\) X₂ + \(5\) X₂₆₇₇₇, \(9\) X₂ - X₂₆₈₂₄, \(9\) X₂ - \(5\) X₂₆₉₈₅, \(3\) X₂ - \(7\) X₂₇₁₁₅, \(3\) X₂ - \(5\) X₃₁₂₀₉, \(6\) X₂ - \(5\) X₃₁₂₅₀, \(3\) X₂ - \(4\) X₃₁₂₈₇, X₆₄₉ - \(3\) X₁₆₃₅, X₆₄₉ - \(4\) X₂₅₁₆, \(3\) X₆₄₉ + X₄₈₁₃, X₆₄₉ + \(3\) X₄₈₉₃, \(3\) X₆₄₉ - X₄₉₇₉, \(2\) X₆₅₉ + X₂₅₂₆, X₆₆₁ + \(3\) X₁₆₃₅, X₆₆₁ + \(4\) X₂₅₁₆, X₆₆₁ + \(2\) X₄₃₉₄, \(2\) X₆₆₁ + X₄₇₉₀, \(3\) X₆₆₁ - X₄₈₁₃, X₆₆₁ - \(3\) X₄₈₉₃, \(3\) X₆₆₁ + X₄₉₇₉, \(3\) X₆₆₃ + X₄₈₁₄, \(3\) X₆₆₃ - X₄₈₉₅, \(5\) X₆₆₃ - X₄₉₅₉, X₆₉₃ + \(5\) X₂₆₇₇₇, \(3\) X₆₉₃ - X₂₆₈₂₄, \(3\) X₆₉₃ - \(5\) X₂₆₉₈₅, X₆₉₃ - \(7\) X₂₇₁₁₅, X₆₉₃ + \(3\) X₃₁₁₅₀, X₆₉₃ - \(5\) X₃₁₂₀₉, \(2\) X₆₉₃ - \(5\) X₃₁₂₅₀, X₆₉₃ - \(4\) X₃₁₂₈₇, X₆₉₃ - \(6\) X₄₄₅₆₇, \(2\) X₆₉₃ - \(3\) X₄₅₃₂₀, \(3\) X₉₀₅ - \(2\) X₃₉₆₀, \(3\) X₁₆₃₅ - \(4\) X₂₅₁₆, \(3\) X₁₆₃₅ - \(2\) X₄₃₉₄, \(6\) X₁₆₃₅ - X₄₇₉₀, \(9\) X₁₆₃₅ + X₄₈₁₃, \(9\) X₁₆₃₅ - X₄₉₇₉, \(3\) X₁₆₃₈ - \(2\) X₃₆₇₆, \(3\) X₁₆₃₈ - \(4\) X₇₆₅₈, \(3\) X₁₆₃₈ - X₂₁₁₀₄, \(3\) X₁₆₃₉ - \(2\) X₃₂₃₉, \(3\) X₁₆₃₉ - X₃₇₀₀, \(3\) X₁₆₃₉ - \(4\) X₄₅₂₁, \(3\) X₁₆₃₉ + \(2\) X₄₇₆₅, \(6\) X₁₆₃₉ - X₄₈₂₀, \(3\) X₁₆₃₉ + X₄₉₇₆, \(4\) X₂₄₉₀ - X₆₅₉₀, \(2\) X₂₄₉₀ - \(3\) X₁₄₄₂₅, \(4\) X₂₄₉₀ + X₄₅₇₄₅, \(8\) X₂₅₁₆ - X₄₇₉₀, \(12\) X₂₅₁₆ + X₄₈₁₃, \(4\) X₂₅₁₆ + \(3\) X₄₈₉₃, \(12\) X₂₅₁₆ - X₄₉₇₉, X₃₀₀₄ + \(2\) X₁₁₀₆₈, \(4\) X₃₂₃₉ - X₄₈₂₀, \(4\) X₃₂₃₉ - \(3\) X₄₉₄₄, \(2\) X₃₂₃₉ + X₄₉₇₆, \(3\) X₃₆₆₉ - \(4\) X₃₉₆₀, X₃₆₆₉ - \(4\) X₁₄₈₃₈, X₃₇₀₀ - \(4\) X₄₅₂₁, X₃₇₀₀ + \(2\) X₄₇₆₅, \(2\) X₃₇₀₀ - \(3\) X₄₉₄₄, \(3\) X₃₇₀₉ - X₄₅₂₆, X₃₇₆₂ + \(3\) X₄₅₆₇₁, X₃₉₆₀ - \(3\) X₁₄₈₃₈, X₄₀₂₄ - \(9\) X₆₅₄₄, \(2\) X₄₀₄₁ + X₄₁₆₂, \(3\) X₄₀₄₁ - X₄₈₁₄, \(3\) X₄₀₄₁ + X₄₈₉₅, \(5\) X₄₀₄₁ + X₄₉₅₉, X₄₁₀₅ - \(3\) X₁₄₃₉₂, X₄₁₀₆ - \(4\) X₂₅₆₆₆, \(3\) X₄₁₆₂ + \(2\) X₄₈₁₄, \(3\) X₄₁₆₂ - \(2\) X₄₈₉₅, \(5\) X₄₁₆₂ - \(2\) X₄₉₅₉, \(2\) X₄₁₆₃ - \(3\) X₄₄₇₂₉, X₄₃₆₉ - \(3\) X₄₇₆₃, X₄₃₇₈ - \(3\) X₁₄₄₁₉, \(3\) X₄₃₇₉ - \(5\) X₂₄₉₂₄, \(3\) X₄₃₇₉ - \(7\) X₃₁₂₀₇, X₄₃₈₀ + \(3\) X₄₇₇₆, X₄₃₈₀ + \(2\) X₄₉₄₀, X₄₃₈₂ - \(3\) X₄₇₂₈, X₄₃₈₂ - \(5\) X₃₀₈₃₅, \(4\) X₄₃₉₄ - X₄₇₉₀, \(6\) X₄₃₉₄ + X₄₈₁₃, \(2\) X₄₃₉₄ + \(3\) X₄₈₉₃, \(6\) X₄₃₉₄ - X₄₉₇₉, X₄₄₆₇ - \(3\) X₂₇₄₈₆, X₄₄₆₇ + \(3\) X₃₀₅₆₅, X₄₄₆₈ + \(2\) X₁₇₀₆₉, X₄₄₇₄ - \(3\) X₁₄₄₃₀, \(2\) X₄₅₂₁ + X₄₇₆₅, \(8\) X₄₅₂₁ - X₄₈₂₀, \(8\) X₄₅₂₁ - \(3\) X₄₉₄₄, \(4\) X₄₅₂₁ + X₄₉₇₆, X₄₅₆₀ + \(2\) X₂₀₃₁₇, X₄₇₀₅ + \(2\) X₆₀₅₀, \(3\) X₄₇₂₈ - \(2\) X₂₃₈₁₃, \(3\) X₄₇₂₈ - \(5\) X₃₀₈₃₅, \(7\) X₄₇₅₁ - \(3\) X₄₈₂₈, \(3\) X₄₇₆₃ - \(2\) X₃₁₂₈₆, \(6\) X₄₇₆₃ - X₄₃₀₆₇, \(4\) X₄₇₆₅ + X₄₈₂₀, \(4\) X₄₇₆₅ + \(3\) X₄₉₄₄, \(3\) X₄₇₇₆ - \(2\) X₄₉₄₀, \(3\) X₄₇₇₆ - X₂₀₂₉₅, \(3\) X₄₇₉₀ + \(2\) X₄₈₁₃, X₄₇₉₀ + \(6\) X₄₈₉₃, \(3\) X₄₇₉₀ - \(2\) X₄₉₇₉, \(2\) X₄₇₉₁ - \(3\) X₄₅₆₆₄, X₄₈₁₃ - \(9\) X₄₈₉₃, \(5\) X₄₈₁₄ + \(3\) X₄₉₅₉, X₄₈₂₀ - \(3\) X₄₉₄₄, X₄₈₂₀ + \(2\) X₄₉₇₆, X₄₈₄₁ + \(8\) X₃₁₁₈₂, X₄₈₄₁ + \(4\) X₄₃₀₆₁, \(2\) X₄₈₈₅ + X₁₇₄₉₄, \(2\) X₄₈₈₅ + \(5\) X₂₆₇₇₇, \(6\) X₄₈₈₅ - X₂₆₈₂₄, \(6\) X₄₈₈₅ - \(5\) X₂₆₉₈₅, \(2\) X₄₈₈₅ - \(7\) X₂₇₁₁₅, \(2\) X₄₈₈₅ + \(3\) X₃₁₁₅₀, \(2\) X₄₈₈₅ - \(5\) X₃₁₂₀₉, \(4\) X₄₈₈₅ - \(5\) X₃₁₂₅₀, X₄₈₈₅ - \(3\) X₄₄₅₆₇, \(4\) X₄₈₈₅ - \(3\) X₄₅₃₂₀, \(9\) X₄₈₉₃ + X₄₉₇₉, \(5\) X₄₈₉₅ - \(3\) X₄₉₅₉, X₄₉₃₂ - \(3\) X₄₅₃₁₃, \(3\) X₄₉₄₄ + \(2\) X₄₉₇₆, \(2\) X₄₉₄₉ + \(3\) X₄₉₈₄, \(3\) X₆₅₄₆ + X₁₆₈₉₂, X₆₅₉₀ - \(6\) X₁₄₄₂₅, X₆₆₀₈ + \(3\) X₁₄₃₉₂, X₇₁₉₂ - \(5\) X₂₇₀₁₃, \(4\) X₇₆₅₃ - \(3\) X₃₁₁₄₈, \(4\) X₇₆₅₈ - X₂₁₁₀₄, X₇₆₅₉ - \(4\) X₉₅₀₈, \(3\) X₁₀₁₉₆ + X₂₁₁₉₆, \(3\) X₁₁₁₉₃ - \(2\) X₁₁₉₃₄, \(3\) X₁₁₁₉₃ - \(4\) X₁₇₁₁₅, X₁₃₄₀₁ + \(2\) X₁₄₂₉₈, X₁₃₄₀₁ + \(4\) X₄₀₁₃₇, X₁₄₃₀₀ + \(2\) X₄₀₁₃₇, \(3\) X₁₄₃₉₅ - X₃₆₀₅₄, \(3\) X₁₄₄₀₄ - X₂₀₉₈₃, \(6\) X₁₄₄₂₅ + X₄₅₇₄₅, X₁₇₄₉₄ - \(5\) X₂₆₇₇₇, \(3\) X₁₇₄₉₄ + X₂₆₈₂₄, \(3\) X₁₇₄₉₄ + \(5\) X₂₆₉₈₅, X₁₇₄₉₄ + \(7\) X₂₇₁₁₅, X₁₇₄₉₄ - \(3\) X₃₁₁₅₀, X₁₇₄₉₄ + \(5\) X₃₁₂₀₉, \(2\) X₁₇₄₉₄ + \(5\) X₃₁₂₅₀, X₁₇₄₉₄ + \(4\) X₃₁₂₈₇, X₁₇₄₉₄ + \(6\) X₄₄₅₆₇, \(2\) X₁₇₄₉₄ + \(3\) X₄₅₃₂₀, \(2\) X₂₁₁₈₈ - \(3\) X₄₁₈₀₀, X₂₁₂₂₂ - \(3\) X₄₄₅₅₀, \(3\) X₂₁₂₉₇ - \(7\) X₂₇₁₃₈, \(2\) X₂₃₈₁₃ - \(5\) X₃₀₈₃₅, \(5\) X₂₄₉₂₄ - \(7\) X₃₁₂₀₇, X₂₅₂₅₉ + \(3\) X₂₇₄₈₆, X₂₅₂₅₉ - \(3\) X₃₀₅₆₅, \(15\) X₂₆₇₇₇ + X₂₆₈₂₄, \(3\) X₂₆₇₇₇ + X₂₆₉₈₅, \(5\) X₂₆₇₇₇ + \(7\) X₂₇₁₁₅, \(5\) X₂₆₇₇₇ - \(3\) X₃₁₁₅₀, \(2\) X₂₆₇₇₇ + X₃₁₂₅₀, \(5\) X₂₆₇₇₇ + \(4\) X₃₁₂₈₇, \(5\) X₂₆₇₇₇ + \(6\) X₄₄₅₆₇, \(10\) X₂₆₇₇₇ + \(3\) X₄₅₃₂₀, X₂₆₈₂₄ - \(5\) X₂₆₉₈₅, X₂₆₈₂₄ - \(21\) X₂₇₁₁₅, X₂₆₈₂₄ + \(9\) X₃₁₁₅₀, X₂₆₈₂₄ - \(15\) X₃₁₂₀₉, \(2\) X₂₆₈₂₄ - \(15\) X₃₁₂₅₀, X₂₆₈₂₄ - \(12\) X₃₁₂₈₇, X₂₆₈₂₄ - \(18\) X₄₄₅₆₇, \(2\) X₂₆₈₂₄ - \(9\) X₄₅₃₂₀, \(5\) X₂₆₉₈₅ - \(21\) X₂₇₁₁₅, \(5\) X₂₆₉₈₅ + \(9\) X₃₁₁₅₀, X₂₆₉₈₅ - \(3\) X₃₁₂₀₉, \(2\) X₂₆₉₈₅ - \(3\) X₃₁₂₅₀, \(5\) X₂₆₉₈₅ - \(12\) X₃₁₂₈₇, \(5\) X₂₆₉₈₅ - \(18\) X₄₄₅₆₇, \(10\) X₂₆₉₈₅ - \(9\) X₄₅₃₂₀, \(7\) X₂₇₁₁₅ + \(3\) X₃₁₁₅₀, \(7\) X₂₇₁₁₅ - \(5\) X₃₁₂₀₉, \(14\) X₂₇₁₁₅ - \(5\) X₃₁₂₅₀, \(7\) X₂₇₁₁₅ - \(4\) X₃₁₂₈₇, \(7\) X₂₇₁₁₅ - \(6\) X₄₄₅₆₇, \(14\) X₂₇₁₁₅ - \(3\) X₄₅₃₂₀, \(3\) X₃₁₁₅₀ + \(5\) X₃₁₂₀₉, \(6\) X₃₁₁₅₀ + \(5\) X₃₁₂₅₀, \(3\) X₃₁₁₅₀ + \(4\) X₃₁₂₈₇, X₃₁₁₅₀ + \(2\) X₄₄₅₆₇, \(2\) X₃₁₁₅₀ + X₄₅₃₂₀, \(5\) X₃₁₂₀₉ - \(4\) X₃₁₂₈₇, \(5\) X₃₁₂₀₉ - \(6\) X₄₄₅₆₇, \(10\) X₃₁₂₀₉ - \(3\) X₄₅₃₂₀, \(5\) X₃₁₂₅₀ - \(8\) X₃₁₂₈₇, \(5\) X₃₁₂₅₀ - \(12\) X₄₄₅₆₇, \(5\) X₃₁₂₅₀ - \(3\) X₄₅₃₂₀, \(4\) X₃₁₂₈₆ - X₄₃₀₆₇, \(2\) X₃₁₂₈₇ - \(3\) X₄₄₅₆₇, \(8\) X₃₁₂₈₇ - \(3\) X₄₅₃₂₀, \(3\) X₃₈₂₃₈ - \(2\) X₄₃₉₃₁, \(4\) X₄₄₅₆₇ - X₄₅₃₂₀, X₄₅₆₆₉ + \(2\) X₄₅₆₇₀

X₆₅₀ lies on the GEOS circle, the cubic K925.

▸ Incident Central Lines251 lines

{1, 1643}, {2, 693}, {3, 8760}, {6, 2423}, {9, 15737}, {10, 21721}, {11, 1566}, {19, 2432}, {20, 8142}, {35, 11247}, {37, 4024}, {39, 7626}, {42, 21727}, {44, 513}, {45, 24457}, {55, 884}, {56, 9373}, {65, 1938}, {75, 21438}, {88, 37131}, {100, 919}, {101, 2222}, {105, 43079}, {108, 40116}, {110, 9090}, {111, 2752}, {112, 2766}, {115, 5520}, {123, 38972}, {141, 9015}, {226, 23806}, {230, 231}, {241, 514}, {244, 38375}, {354, 9443}, {512, 2499}, {518, 5098}, {521, 1021}, {522, 1639}, {644, 5548}, {651, 7045}, {663, 861}, {666, 4998}, {667, 4705}, {812, 3835}, {814, 21051}, {824, 3666}, {830, 3803}, {876, 23768}, {885, 5218}, {900, 14321}, {918, 4025}, {926, 2488}, {949, 23696}, {1015, 4534}, {1022, 39963}, {1027, 3126}, {1040, 41795}, {1054, 39344}, {1084, 35079}, {1086, 24198}, {1146, 7117}, {1212, 21132}, {1376, 5452}, {1415, 1783}, {1418, 23730}, {1427, 23727}, {1459, 24532}, {1577, 23882}, {1734, 3309}, {1758, 21189}, {1814, 25954}, {1919, 38469}, {1946, 2202}, {1960, 4770}, {1980, 5040}, {2170, 33646}, {2174, 21860}, {2276, 4448}, {2291, 2717}, {2457, 2529}, {2517, 26080}, {2520, 6139}, {2523, 2527}, {2532, 6371}, {2605, 17412}, {2646, 2649}, {2820, 15599}, {3035, 15914}, {3057, 9366}, {3119, 7004}, {3122, 45743}, {3210, 25271}, {3246, 39308}, {3250, 4083}, {3251, 4825}, {3261, 24622}, {3569, 32126}, {3570, 42717}, {3572, 16606}, {3667, 4773}, {3699, 36801}, {3730, 7634}, {3739, 4411}, {3746, 32195}, {3752, 6546}, {3762, 45671}, {3798, 4897}, {3837, 25126}, {3887, 4794}, {3904, 28938}, {3907, 4147}, {3910, 6332}, {3975, 4391}, {4063, 14349}, {4120, 4926}, {4129, 29013}, {4132, 42664}, {4140, 4397}, {4142, 23877}, {4160, 30234}, {4163, 44729}, {4171, 42312}, {4367, 4490}, {4370, 35129}, {4378, 14419}, {4379, 24924}, {4380, 4776}, {4382, 4728}, {4449, 17427}, {4455, 8640}, {4462, 17496}, {4467, 25259}, {4474, 14430}, {4481, 18197}, {4498, 8712}, {4500, 44307}, {4546, 8710}, {4551, 35326}, {4554, 30610}, {4562, 9362}, {4730, 4775}, {4751, 4828}, {4785, 45315}, {4791, 45664}, {4802, 4988}, {4806, 29328}, {4834, 4983}, {4838, 28165}, {4843, 4990}, {4850, 31992}, {4927, 44432}, {4931, 28205}, {4932, 28840}, {4949, 4984}, {4962, 14350}, {4995, 21795}, {5190, 13999}, {5432, 15584}, {5514, 10017}, {5540, 34460}, {6004, 8659}, {6164, 21893}, {6174, 6184}, {6367, 42653}, {6506, 8735}, {6545, 16602}, {6615, 14418}, {6666, 23810}, {7192, 16751}, {7212, 26146}, {7642, 10567}, {7650, 21960}, {7653, 31148}, {8056, 37626}, {8632, 21901}, {8642, 23687}, {9010, 14404}, {9013, 15990}, {9320, 20683}, {9355, 9357}, {9364, 20980}, {9590, 39227}, {10164, 40606}, {11502, 30706}, {11672, 35083}, {13259, 45695}, {13405, 14746}, {13609, 38357}, {14296, 20906}, {14475, 31197}, {14735, 17355}, {14749, 40942}, {14812, 16670}, {14935, 40062}, {15313, 17796}, {16592, 41180}, {16601, 21201}, {16706, 25603}, {16757, 26248}, {16975, 30583}, {17072, 29051}, {17161, 28606}, {17780, 42723}, {17924, 37799}, {17926, 44426}, {18004, 29078}, {18108, 26249}, {20316, 24718}, {20954, 27293}, {21052, 25637}, {21053, 29274}, {21105, 40133}, {21183, 44902}, {21192, 23875}, {21222, 44550}, {21260, 29070}, {21297, 27138}, {21758, 39521}, {22091, 44408}, {23100, 24774}, {23758, 28151}, {23972, 35116}, {24002, 37757}, {24098, 25067}, {24115, 25091}, {24141, 25099}, {24430, 41796}, {24484, 34583}, {24720, 25380}, {24789, 24793}, {25900, 25955}, {27009, 40619}, {27345, 28758}, {28161, 40500}, {28292, 38324}, {28798, 36796}, {28851, 45674}, {28867, 45679}, {29226, 40464}, {30023, 42327}, {30764, 30865}, {31615, 31628}, {32688, 40097}, {33573, 35015}, {34591, 38345}, {35091, 35508}, {36067, 40117}, {37998, 42322}, {40131, 42758}, {45669, 45670}

▸ Functional Relations46 relations

• X₆₅₀ = midpoint of X_i and X_j for these {i,j}: {2, 31150}, {649, 661}, {657, 21127}, {659, 1491}, {663, 4041}, {667, 4705}, {693, 17494}, {1635, 4893}, {1734, 4040}, {1960, 4770}, {2254, 4724}, {2488, 4524}, {3239, 4765}, {3250, 21832}, {3251, 4825}, {3700, 4976}, {3716, 4913}, {4025, 4468}, {4063, 14349}, {4105, 6608}, {4367, 4490}, {4380, 20295}, {4391, 4560}, {4462, 17496}, {4467, 25259}, {4730, 4775}, {4813, 4979}, {4814, 4895}, {4834, 4983}, {6590, 45745}, {14298, 14300}, {17418, 17420}, {20906, 21225}, {26777, 31209}, {27486, 30565}

• X₆₅₀ = reflection of X_i in X_j for these {i,j}: {2, 44567}, {11, 10006}, {20, 8142}, {649, 4394}, {667, 6050}, {693, 4885}, {905, 14838}, {2526, 1491}, {3239, 4521}, {3669, 905}, {3676, 7658}, {3700, 3239}, {3776, 21212}, {3803, 4401}, {3835, 25666}, {4025, 17069}, {4106, 3835}, {4162, 663}, {4369, 31286}, {4382, 23813}, {4391, 20317}, {4394, 2516}, {4411, 3739}, {4790, 649}, {4820, 3700}, {4885, 31287}, {4897, 3798}, {4927, 44432}, {4944, 1639}, {4976, 4765}, {7178, 14837}, {7655, 656}, {7662, 4874}, {11934, 17115}, {13401, 14300}, {14298, 40137}, {20295, 4940}, {21104, 3676}, {21183, 44902}, {21348, 6586}, {24720, 25380}, {31250, 31209}, {43051, 24782}, {43052, 10015}, {43061, 31182}, {43067, 4369}, {45320, 2}

• X₆₅₀ = isogonal conjugate of X₆₅₁

• X₆₅₀ = isotomic conjugate of X₄₅₅₄

• X₆₅₀ = complement of X₆₉₃

• X₆₅₀ = complementary conjugate of X₂₁₂₅₂

• X₆₅₀ = X_i-anticomplementary conjugate of X_j for these (i,j): {692, 41792}, {9309, 150}, {9311, 21293}, {9315, 149}, {9439, 37781}, {30610, 6327}

• X₆₅₀ = X_i-complementary conjugate of X_j for these (i,j): {1, 21252}, {6, 116}, {31, 11}, {32, 1086}, {37, 21253}, {41, 26932}, {42, 125}, {55, 124}, {56, 17059}, {59, 17072}, {71, 127}, {100, 2887}, {101, 141}, {109, 2886}, {110, 3741}, {112, 34830}, {163, 3739}, {184, 2968}, {190, 626}, {212, 123}, {213, 8287}, {228, 34846}, {251, 44312}, {560, 1015}, {604, 4904}, {644, 21244}, {651, 17046}, {662, 21240}, {663, 46100}, {664, 17047}, {668, 21235}, {692, 10}, {765, 21260}, {813, 20541}, {825, 21264}, {901, 21241}, {902, 3259}, {906, 18589}, {919, 20335}, {1016, 21262}, {1018, 21245}, {1110, 513}, {1252, 3835}, {1253, 5514}, {1331, 1368}, {1333, 17761}, {1397, 3756}, {1402, 8286}, {1415, 142}, {1461, 21258}, {1501, 6377}, {1576, 1125}, {1783, 20305}, {1813, 18639}, {1897, 21243}, {1918, 115}, {1919, 6547}, {1978, 40379}, {2149, 4885}, {2175, 1146}, {2177, 15614}, {2187, 7358}, {2194, 34589}, {2200, 15526}, {2205, 16592}, {2206, 244}, {2209, 5518}, {2210, 38989}, {2212, 6506}, {2284, 20540}, {2426, 118}, {3052, 5510}, {3573, 20542}, {3939, 1329}, {4055, 122}, {4557, 3454}, {4559, 17052}, {4565, 17050}, {4567, 42327}, {4570, 512}, {4588, 21242}, {4600, 23301}, {4601, 21263}, {4628, 3934}, {4630, 29654}, {5379, 21259}, {5380, 21256}, {5384, 788}, {5546, 21246}, {6066, 4521}, {7109, 6627}, {7122, 40608}, {8685, 17792}, {8750, 5}, {9454, 35094}, {9459, 35092}, {14598, 39786}, {14599, 35119}, {14827, 13609}, {17943, 20339}, {21059, 5511}, {23344, 121}, {23357, 21196}, {23979, 7658}, {23990, 514}, {23995, 31947}, {24027, 3900}, {32642, 516}, {32652, 946}, {32653, 24220}, {32656, 3}, {32660, 17073}, {32665, 3834}, {32666, 518}, {32674, 16608}, {32676, 942}, {32682, 674}, {32699, 916}, {32718, 4871}, {32719, 519}, {32736, 3831}, {32739, 2}, {34067, 3836}, {34069, 24325}, {34071, 20255}, {34073, 34824}, {34080, 21255}, {36049, 21239}, {36059, 34822}, {36086, 20544}, {41267, 15449}, {41280, 16614}, {46148, 21248}, {46288, 21208}

• X₆₅₀ = X_i-Ceva conjugate of X_j for these (i,j): {1, 2310}, {2, 11}, {4, 3270}, {6, 11998}, {7, 3022}, {8, 3271}, {9, 2170}, {11, 11193}, {19, 38345}, {21, 4516}, {33, 38358}, {55, 38347}, {57, 7004}, {99, 11997}, {100, 55}, {101, 37}, {107, 1859}, {108, 33}, {174, 10501}, {189, 1364}, {190, 3057}, {200, 38375}, {264, 11988}, {274, 4459}, {277, 1086}, {278, 38357}, {280, 4081}, {281, 1146}, {282, 34591}, {294, 17435}, {330, 24840}, {333, 18191}, {346, 4534}, {513, 4162}, {514, 513}, {522, 3900}, {598, 11936}, {643, 3683}, {644, 9}, {645, 960}, {646, 8}, {648, 1858}, {651, 1}, {653, 65}, {655, 517}, {658, 354}, {662, 2646}, {664, 14100}, {666, 518}, {668, 3056}, {670, 42397}, {693, 11934}, {799, 21334}, {885, 926}, {905, 6129}, {929, 44670}, {979, 3248}, {1018, 2269}, {1019, 42312}, {1020, 2654}, {1021, 652}, {1024, 4435}, {1029, 3024}, {1043, 45743}, {1119, 11918}, {1222, 40528}, {1247, 2643}, {1290, 9629}, {1292, 4319}, {1783, 6}, {1897, 1864}, {2006, 35015}, {2401, 900}, {3239, 14298}, {3257, 5048}, {3362, 2638}, {3699, 210}, {3737, 663}, {3882, 8240}, {3911, 33646}, {3939, 1212}, {4069, 3691}, {4373, 36639}, {4391, 521}, {4551, 14547}, {4552, 950}, {4554, 497}, {4559, 14749}, {4560, 522}, {4573, 10391}, {4581, 512}, {4582, 3880}, {4598, 20359}, {4603, 17611}, {4606, 1697}, {4612, 21}, {4626, 10939}, {4627, 6051}, {5380, 8540}, {5546, 40937}, {6135, 7133}, {6136, 42013}, {6335, 1837}, {7040, 42069}, {7110, 21044}, {7658, 17427}, {8046, 3025}, {8051, 1357}, {8056, 244}, {8817, 14935}, {10492, 6728}, {13138, 30223}, {13149, 1836}, {13486, 7073}, {14838, 31947}, {15352, 42385}, {17924, 15313}, {17926, 3064}, {18830, 23497}, {19605, 3119}, {23838, 4895}, {24002, 3309}, {26704, 1824}, {26705, 1827}, {26706, 7071}, {27527, 25128}, {27818, 4014}, {27834, 2098}, {30610, 2}, {30730, 3686}, {31343, 200}, {31628, 100}, {32038, 3486}, {32041, 390}, {32641, 8609}, {32714, 1854}, {34085, 20358}, {34277, 2968}, {34529, 35604}, {35326, 14746}, {36049, 1108}, {36050, 31}, {36086, 41339}, {36098, 3745}, {36100, 35014}, {36796, 4124}, {36838, 7}, {36910, 4530}, {37136, 1319}, {37137, 3666}, {37139, 1155}, {37141, 56}, {37143, 18839}, {37206, 17642}, {37222, 14115}, {37887, 3120}, {38340, 942}, {39272, 28071}, {39956, 1015}, {40097, 25}, {40116, 5089}, {40117, 19}, {40400, 2087}, {41207, 243}, {41514, 3318}, {41790, 3942}, {41791, 26932}, {42343, 5274}, {42380, 42378}, {42381, 42379}, {42388, 42386}, {42389, 42387}, {42408, 14923}, {43069, 37548}, {43760, 3675}, {44178, 17463}, {44426, 18344}, {45877, 6729}

• X₆₅₀ = X_i-cross conjugate of X_j for these (i,j): {55, 42552}, {647, 652}, {657, 3900}, {661, 3064}, {663, 513}, {926, 885}, {1566, 5089}, {1946, 521}, {2170, 9}, {2310, 1}, {3022, 7}, {3063, 18344}, {3119, 33}, {3270, 4}, {3271, 8}, {3310, 2432}, {3709, 663}, {4041, 522}, {4516, 21}, {4895, 23838}, {7117, 6}, {10581, 657}, {11124, 100}, {14936, 55}, {17435, 294}, {21044, 37}, {21127, 514}, {33525, 21789}, {36197, 7073}, {42454, 38347}

• X₆₅₀ = cevapoint of X_i and X_j for these (i,j): {1, 9355}, {9, 4919}, {11, 42454}, {514, 21195}, {522, 4147}, {647, 661}, {657, 663}, {1946, 3063}, {3126, 42341}, {3709, 4041}, {10581, 21127}, {21044, 42462}

• X₆₅₀ = crosspoint of X_i and X_j for these (i,j): {1, 651}, {2, 100}, {7, 36838}, {8, 646}, {9, 644}, {21, 4612}, {29, 653}, {57, 108}, {81, 13486}, {101, 284}, {107, 40395}, {112, 1169}, {190, 1222}, {274, 4594}, {280, 37141}, {281, 1783}, {282, 40117}, {333, 3699}, {514, 522}, {655, 40437}, {658, 21453}, {662, 40430}, {664, 23618}, {1018, 14624}, {1021, 17926}, {1897, 40444}, {3737, 4560}, {3939, 10482}, {4391, 44426}, {4554, 8817}, {6135, 7133}, {6136, 42013}, {6335, 34406}, {8056, 31343}, {9503, 36086}, {10492, 10495}, {13149, 34398}, {19607, 26704}, {34277, 40097}, {39272, 43760}

• X₆₅₀ = crosssum of X_i and X_j for these (i,j): {1, 650}, {2, 17496}, {3, 36054}, {6, 513}, {9, 521}, {57, 3669}, {73, 652}, {101, 109}, {213, 7234}, {218, 43049}, {221, 14298}, {222, 905}, {226, 514}, {408, 32320}, {500, 9404}, {520, 18591}, {522, 20262}, {525, 1211}, {649, 1201}, {654, 34586}, {656, 18675}, {657, 2293}, {661, 2650}, {663, 1200}, {665, 1362}, {1019, 40153}, {1400, 43924}, {1415, 36059}, {2254, 9502}, {2850, 40584}, {3063, 7083}, {3287, 28369}, {3676, 10481}, {4551, 4559}, {6180, 20980}, {6364, 13389}, {6365, 13388}, {8677, 23980}, {15730, 43050}, {30719, 45204}, {39063, 39470}

• X₆₅₀ = trilinear pole of line {926, 2170}

• X₆₅₀ = crossdifference of every pair of points on line {1, 3}

• X₆₅₀ = polar conjugate of X₁₈₀₂₆

• X₆₅₀ = orthojoin of X₁₅₂₁

• X₆₅₀ = trilinear pole of line X₉₂₆X₂₁₇₀ (the tangent to the incircle at X₃₀₂₂)

• X₆₅₀ = anticomplement of X₄₈₈₅

• X₆₅₀ = perspector of the Feuerbach hyperbola

• X₆₅₀ = center of circumconic that is locus of trilinear poles of lines passing through X₁₁

• X₆₅₀ = bicentric difference of PU(i) for i in (15, 57, 59, 60, 80, 94, 125)

• X₆₅₀ = PU(15)-harmonic conjugate of X₆₅

• X₆₅₀ = PU(57)-harmonic conjugate of X₁₁₅₅

• X₆₅₀ = PU(59)-harmonic conjugate of X₃₀₅₇

• X₆₅₀ = PU(60)-harmonic conjugate of X₅₆

• X₆₅₀ = PU(80)-harmonic conjugate of X₂₆₄₆

• X₆₅₀ = PU(94)-harmonic conjugate of X₃₅₄

• X₆₅₀ = PU(125)-harmonic conjugate of X₃

• X₆₅₀ = medial-isogonal conjugate of X₂₁₂₅₂

• X₆₅₀ = X₆-isoconjugate of X₆₆₄

• X₆₅₀ = crosspoint of X₁ and X₆₅₁

• X₆₅₀ = radical center of {circumcircle, nine-point circle, Bevan circle}

• X₆₅₀ = radical center of {circumcircle, nine-point circle, Apollonius circle}

• X₆₅₀ = bicentric difference of PU(112)

• X₆₅₀ = PU(112)-harmonic conjugate of X₅₅

• X₆₅₀ = X₂₅₀₁-of-excentral-triangle

• X₆₅₀ = PU(4)-harmonic conjugate of X₅₀₈₉

• X₆₅₀ = trilinear square root of X₂₃₁₀

• X₆₅₀ = trilinear product of Feuerbach hyperbola intercepts of antiorthic axis

• X₆₅₀ = crosssum of Feuerbach hyperbola intercepts of antiorthic axis

• X₆₅₀ = excentral-to-ABC barycentric image of X₆₄₉

• X₆₅₀ = X_i-isoconjugate of X_j for these (i,j): {1, 651}, {2, 109}, {3, 653}, {4, 1813}, {6, 664}, {7, 101}, {8, 1461}, {9, 934}, {10, 4565}, {11, 4619}, {12, 4556}, {19, 6516}, {21, 1020}, {27, 23067}, {31, 4554}, {32, 4572}, {34, 1332}, {35, 38340}, {36, 655}, {37, 1414}, {40, 37141}, {41, 4569}, {42, 4573}, {48, 18026}, {55, 658}, {56, 190}, {57, 100}, {58, 4552}, {59, 514}, {60, 4605}, {63, 108}, {65, 662}, {69, 32674}, {73, 648}, {75, 1415}, {77, 1783}, {78, 32714}, {81, 4551}, {83, 46153}, {85, 692}, {86, 4559}, {88, 23703}, {92, 36059}, {99, 1400}, {102, 2406}, {104, 24029}, {105, 1025}, {107, 40152}, {110, 226}, {112, 307}, {145, 38828}, {162, 1214}, {163, 1441}, {171, 37137}, {174, 6733}, {181, 4610}, {200, 4617}, {210, 4637}, {212, 13149}, {213, 4625}, {219, 36118}, {220, 4626}, {221, 44327}, {222, 1897}, {223, 13138}, {225, 4558}, {241, 36086}, {244, 31615}, {264, 32660}, {269, 644}, {273, 906}, {278, 1331}, {279, 3939}, {284, 4566}, {294, 41353}, {296, 1981}, {320, 32675}, {329, 8059}, {331, 32656}, {346, 6614}, {347, 36049}, {348, 8750}, {349, 1576}, {393, 6517}, {394, 36127}, {512, 4620}, {513, 4564}, {517, 37136}, {518, 36146}, {521, 7128}, {522, 1262}, {527, 14733}, {553, 8701}, {579, 1305}, {603, 6335}, {604, 668}, {608, 4561}, {643, 1427}, {645, 1042}, {646, 1106}, {649, 4998}, {650, 7045}, {660, 1429}, {663, 1275}, {665, 39293}, {666, 1458}, {672, 927}, {673, 2283}, {677, 43035}, {693, 2149}, {738, 4578}, {757, 21859}, {765, 3669}, {799, 1402}, {811, 1409}, {813, 1447}, {823, 22341}, {825, 7179}, {851, 41206}, {883, 1438}, {893, 6649}, {894, 29055}, {901, 3911}, {905, 7012}, {908, 2720}, {919, 9436}, {932, 1423}, {951, 14543}, {961, 3882}, {1014, 1018}, {1016, 43924}, {1026, 1462}, {1037, 3732}, {1038, 36099}, {1055, 35157}, {1110, 24002}, {1119, 4587}, {1121, 23346}, {1155, 37139}, {1156, 23890}, {1170, 35338}, {1174, 35312}, {1193, 6648}, {1231, 32676}, {1252, 3676}, {1253, 36838}, {1254, 4612}, {1268, 36075}, {1284, 4584}, {1292, 1445}, {1293, 5435}, {1308, 37787}, {1310, 2285}, {1317, 4638}, {1319, 3257}, {1334, 4616}, {1357, 6632}, {1397, 1978}, {1399, 15455}, {1403, 4598}, {1404, 4555}, {1405, 4597}, {1407, 3699}, {1408, 4033}, {1411, 4585}, {1412, 3952}, {1416, 42720}, {1417, 24004}, {1420, 27834}, {1428, 4562}, {1431, 18047}, {1432, 4579}, {1434, 4557}, {1435, 4571}, {1457, 13136}, {1459, 46102}, {1460, 37215}, {1465, 36037}, {1468, 32038}, {1469, 4586}, {1471, 32041}, {1475, 6606}, {1476, 21362}, {1492, 7146}, {1617, 37206}, {1633, 7131}, {1634, 18097}, {1708, 13397}, {1880, 4592}, {1983, 18815}, {2002, 44059}, {2003, 6742}, {2078, 37143}, {2082, 8269}, {2099, 4604}, {2221, 14594}, {2222, 3218}, {2223, 34085}, {2338, 23973}, {2347, 6613}, {2425, 34393}, {2701, 17950}, {2742, 30379}, {2743, 37789}, {3212, 34071}, {3219, 26700}, {3239, 7339}, {3262, 32669}, {3361, 4606}, {3451, 21272}, {3649, 4629}, {3662, 8685}, {3665, 4628}, {3666, 36098}, {3668, 5546}, {3671, 4627}, {3674, 32736}, {3888, 7132}, {3903, 7175}, {3912, 32735}, {3982, 28176}, {4017, 4567}, {4025, 7115}, {4031, 28210}, {4079, 7340}, {4103, 7341}, {4105, 23586}, {4114, 28214}, {4130, 24013}, {4163, 23971}, {4357, 8687}, {4391, 24027}, {4570, 7178}, {4575, 40149}, {4588, 5219}, {4591, 40663}, {4600, 7180}, {4621, 7248}, {4636, 6354}, {4654, 8652}, {5221, 37211}, {5226, 28162}, {5228, 37138}, {5249, 15439}, {5257, 5545}, {5364, 34083}, {5750, 29279}, {5905, 36082}, {6063, 32739}, {6183, 40131}, {6558, 7023}, {6577, 17077}, {6734, 32651}, {7013, 40117}, {7113, 35174}, {7649, 44717}, {8545, 14074}, {8677, 39294}, {8690, 28387}, {8693, 40719}, {8694, 21454}, {8697, 31231}, {9268, 30725}, {9316, 30610}, {9357, 9358}, {9454, 46135}, {10030, 34067}, {10436, 32693}, {10571, 44765}, {11608, 17942}, {12848, 28291}, {13486, 16577}, {14544, 40407}, {16947, 27808}, {17080, 36050}, {17484, 34921}, {17966, 35154}, {18830, 41526}, {19369, 35180}, {21105, 38809}, {21446, 35280}, {21453, 35326}, {22464, 32641}, {23113, 40446}, {23353, 40843}, {23845, 40420}, {23979, 35519}, {23981, 34234}, {24016, 40869}, {24032, 36054}, {24035, 36055}, {24471, 36147}, {27339, 28624}, {27382, 36079}, {29052, 41246}, {29363, 36538}, {30598, 36074}, {30806, 36141}, {32636, 37212}, {32643, 35516}, {32652, 40702}, {32666, 40704}, {32689, 33864}, {34055, 46152}, {34075, 43037}, {34080, 39126}, {36048, 40937}, {37607, 43069}, {40151, 43290}, {40576, 44178}

• X₆₅₀ = barycentric product X_i*X_j for these {i,j}: {1, 522}, {3, 44426}, {4, 521}, {6, 4391}, {7, 3900}, {8, 513}, {9, 514}, {10, 3737}, {11, 100}, {19, 6332}, {21, 523}, {25, 35518}, {27, 8611}, {29, 656}, {31, 35519}, {33, 4025}, {37, 4560}, {41, 3261}, {42, 18155}, {48, 46110}, {55, 693}, {56, 4397}, {57, 3239}, {58, 4086}, {59, 42455}, {60, 4036}, {63, 3064}, {65, 7253}, {69, 18344}, {75, 663}, {76, 3063}, {78, 7649}, {79, 35057}, {80, 3738}, {81, 3700}, {84, 8058}, {85, 657}, {86, 4041}, {87, 4147}, {88, 1639}, {89, 4944}, {92, 652}, {99, 4516}, {101, 4858}, {102, 14304}, {104, 2804}, {106, 4768}, {108, 2968}, {109, 24026}, {115, 4612}, {123, 40097}, {124, 36050}, {149, 42552}, {174, 6730}, {178, 10495}, {188, 6728}, {189, 14298}, {190, 2170}, {200, 3676}, {210, 7192}, {212, 46107}, {219, 17924}, {220, 24002}, {226, 1021}, {241, 28132}, {244, 3699}, {256, 3907}, {257, 3287}, {261, 4705}, {264, 1946}, {269, 4163}, {270, 4064}, {274, 3709}, {279, 4130}, {280, 6129}, {281, 905}, {282, 14837}, {283, 24006}, {284, 1577}, {291, 3716}, {294, 918}, {312, 649}, {314, 512}, {318, 1459}, {321, 7252}, {333, 661}, {335, 4435}, {341, 43924}, {345, 6591}, {346, 3669}, {517, 43728}, {518, 885}, {519, 23838}, {520, 1896}, {525, 1172}, {527, 23893}, {556, 6729}, {607, 15413}, {608, 15416}, {643, 3120}, {644, 1086}, {645, 3125}, {646, 1015}, {647, 31623}, {651, 1146}, {653, 34591}, {654, 18359}, {658, 3119}, {659, 4518}, {660, 4124}, {662, 21044}, {664, 2310}, {665, 36796}, {666, 17435}, {667, 3596}, {668, 3271}, {669, 40072}, {679, 4543}, {692, 34387}, {751, 4474}, {764, 4076}, {765, 21132}, {798, 28660}, {810, 44130}, {812, 4876}, {824, 2344}, {850, 2194}, {875, 4087}, {876, 3685}, {884, 3263}, {891, 36798}, {897, 14432}, {900, 1320}, {903, 4895}, {926, 2481}, {934, 4081}, {941, 23880}, {952, 46041}, {960, 4581}, {983, 3810}, {985, 4522}, {1018, 17197}, {1019, 2321}, {1022, 2325}, {1024, 3912}, {1027, 3717}, {1036, 2517}, {1039, 23874}, {1043, 4017}, {1088, 4105}, {1109, 4636}, {1111, 3939}, {1126, 4985}, {1156, 6366}, {1214, 17926}, {1220, 17420}, {1222, 6615}, {1252, 40166}, {1255, 4976}, {1265, 43923}, {1293, 4939}, {1295, 14312}, {1309, 35014}, {1332, 8735}, {1334, 7199}, {1358, 4578}, {1392, 4926}, {1415, 23978}, {1432, 4529}, {1434, 4171}, {1441, 21789}, {1476, 42337}, {1635, 4997}, {1638, 41798}, {1783, 26932}, {1785, 37628}, {1807, 44428}, {1809, 39534}, {1812, 2501}, {1857, 4131}, {1861, 23696}, {1880, 15411}, {1897, 7004}, {1919, 28659}, {1980, 40363}, {2052, 36054}, {2053, 20906}, {2087, 4582}, {2090, 45877}, {2161, 3904}, {2175, 40495}, {2182, 2399}, {2184, 14331}, {2185, 4024}, {2191, 44448}, {2192, 17896}, {2193, 14618}, {2204, 3267}, {2218, 20294}, {2254, 14942}, {2287, 7178}, {2298, 3910}, {2299, 14208}, {2316, 3762}, {2319, 3835}, {2320, 4777}, {2328, 4077}, {2334, 4811}, {2335, 23882}, {2339, 6590}, {2341, 4707}, {2342, 36038}, {2346, 6362}, {2349, 14400}, {2364, 4791}, {2400, 41339}, {2509, 41791}, {2516, 38255}, {2520, 34409}, {2618, 35196}, {2648, 2785}, {2651, 18013}, {2771, 14224}, {2787, 11609}, {2798, 43746}, {2806, 43735}, {2826, 34894}, {2827, 12641}, {2969, 4571}, {2997, 8676}, {3022, 4569}, {3122, 7257}, {3124, 4631}, {3223, 25128}, {3227, 4526}, {3254, 3887}, {3257, 4530}, {3270, 18026}, {3307, 3308}, {3309, 6601}, {3310, 36795}, {3345, 14302}, {3572, 3975}, {3667, 3680}, {3675, 36802}, {3683, 4608}, {3684, 4444}, {3689, 6548}, {3694, 17925}, {3701, 3733}, {3703, 18108}, {3756, 31343}, {3766, 7077}, {3837, 8851}, {3868, 23289}, {3871, 40086}, {3876, 43927}, {3880, 23836}, {3903, 4459}, {3952, 18191}, {3960, 36910}, {4009, 43928}, {4051, 25576}, {4069, 17205}, {4082, 7203}, {4083, 7155}, {4102, 4979}, {4140, 40432}, {4162, 4373}, {4183, 17094}, {4367, 4451}, {4394, 6557}, {4458, 7281}, {4467, 7073}, {4477, 7249}, {4515, 17096}, {4521, 8056}, {4534, 27834}, {4542, 4618}, {4546, 19604}, {4553, 18101}, {4554, 14936}, {4564, 42462}, {4573, 36197}, {4594, 40608}, {4617, 23970}, {4626, 24010}, {4723, 23345}, {4762, 40779}, {4765, 25430}, {4778, 4866}, {4801, 34820}, {4813, 42030}, {4814, 39704}, {4820, 25417}, {4893, 30608}, {4900, 6006}, {4913, 30571}, {4919, 42555}, {4957, 5549}, {4959, 39707}, {4977, 32635}, {4978, 33635}, {5423, 43932}, {5514, 37141}, {5546, 16732}, {5853, 35355}, {6001, 43737}, {6003, 6598}, {6063, 8641}, {6335, 7117}, {6336, 14418}, {6364, 13454}, {6365, 13426}, {6516, 42069}, {6588, 34277}, {6605, 21104}, {6607, 42311}, {6608, 21453}, {6741, 13486}, {6745, 35348}, {7017, 22383}, {7045, 23615}, {7105, 8062}, {7106, 17899}, {7110, 14838}, {7220, 24720}, {7658, 19605}, {8047, 11193}, {8648, 20566}, {8674, 11604}, {8678, 30479}, {8702, 10266}, {8713, 42015}, {8748, 24018}, {8750, 17880}, {8817, 17115}, {9001, 30513}, {9404, 30690}, {9439, 20907}, {10309, 30201}, {10492, 16016}, {10566, 33299}, {10570, 21189}, {10581, 31618}, {11125, 44693}, {11934, 13577}, {13138, 38357}, {14077, 34919}, {14330, 21446}, {14395, 16080}, {14430, 37129}, {14589, 31611}, {15313, 43740}, {16596, 40117}, {16606, 27527}, {16726, 30730}, {17418, 31359}, {17424, 34523}, {18210, 36797}, {20317, 39956}, {20979, 27424}, {21011, 39177}, {21102, 44687}, {21120, 23617}, {21127, 32008}, {21172, 44692}, {21272, 40528}, {21302, 40505}, {21666, 36059}, {21832, 36800}, {21859, 26856}, {23090, 40149}, {23104, 24027}, {23189, 41013}, {23351, 30806}, {24031, 36127}, {26546, 40141}, {26704, 34588}, {27475, 45755}, {27538, 43931}, {27846, 36801}, {28071, 43042}, {31628, 46101}, {33573, 37139}, {34860, 42312}, {34896, 40577}, {35015, 36037}, {35508, 36838}, {36121, 39471}, {37206, 38375}, {38345, 44765}, {38358, 43190}, {40444, 40628}, {40716, 42657}

• X₆₅₀ = barycentric quotient X_i/X_j for these {i,j}: {1, 664}, {2, 4554}, {3, 6516}, {4, 18026}, {6, 651}, {7, 4569}, {8, 668}, {9, 190}, {11, 693}, {19, 653}, {21, 99}, {25, 108}, {29, 811}, {31, 109}, {32, 1415}, {33, 1897}, {34, 36118}, {37, 4552}, {41, 101}, {42, 4551}, {48, 1813}, {55, 100}, {56, 934}, {57, 658}, {58, 1414}, {65, 4566}, {75, 4572}, {78, 4561}, {80, 35174}, {81, 4573}, {86, 4625}, {100, 4998}, {101, 4564}, {105, 927}, {109, 7045}, {171, 6649}, {184, 36059}, {200, 3699}, {210, 3952}, {212, 1331}, {213, 4559}, {219, 1332}, {220, 644}, {228, 23067}, {244, 3676}, {255, 6517}, {261, 4623}, {269, 4626}, {278, 13149}, {279, 36838}, {281, 6335}, {282, 44327}, {283, 4592}, {284, 662}, {294, 666}, {312, 1978}, {314, 670}, {333, 799}, {346, 646}, {354, 35312}, {480, 4578}, {512, 65}, {513, 7}, {514, 85}, {518, 883}, {521, 69}, {522, 75}, {523, 1441}, {525, 1231}, {604, 1461}, {607, 1783}, {608, 32714}, {612, 14594}, {643, 4600}, {644, 1016}, {645, 4601}, {646, 31625}, {647, 1214}, {649, 57}, {651, 1275}, {652, 63}, {654, 3218}, {656, 307}, {657, 9}, {659, 1447}, {661, 226}, {662, 4620}, {663, 1}, {665, 241}, {667, 56}, {669, 1402}, {672, 1025}, {673, 34085}, {692, 59}, {693, 6063}, {728, 6558}, {764, 1358}, {788, 1469}, {798, 1400}, {810, 73}, {812, 10030}, {822, 40152}, {830, 7247}, {876, 7233}, {884, 105}, {885, 2481}, {891, 43037}, {893, 37137}, {902, 23703}, {904, 29055}, {905, 348}, {906, 44717}, {909, 37136}, {918, 40704}, {926, 518}, {941, 32038}, {1014, 4616}, {1015, 3669}, {1019, 1434}, {1021, 333}, {1024, 673}, {1036, 1310}, {1037, 8269}, {1043, 7257}, {1055, 23890}, {1086, 24002}, {1096, 36127}, {1106, 6614}, {1146, 4391}, {1156, 35157}, {1172, 648}, {1252, 31615}, {1253, 3939}, {1260, 4571}, {1261, 8706}, {1318, 4618}, {1320, 4555}, {1333, 4565}, {1334, 1018}, {1357, 43932}, {1364, 4131}, {1400, 1020}, {1407, 4617}, {1412, 4637}, {1415, 1262}, {1434, 4635}, {1436, 37141}, {1438, 36146}, {1456, 23973}, {1458, 41353}, {1459, 77}, {1476, 6613}, {1486, 40576}, {1491, 7179}, {1500, 21859}, {1577, 349}, {1635, 3911}, {1638, 37780}, {1639, 4358}, {1643, 5723}, {1734, 33298}, {1769, 22464}, {1783, 46102}, {1802, 4587}, {1812, 4563}, {1843, 46152}, {1896, 6528}, {1919, 604}, {1944, 15418}, {1946, 3}, {1960, 1319}, {1964, 46153}, {1973, 32674}, {1980, 1397}, {2053, 932}, {2082, 3732}, {2087, 30725}, {2149, 4619}, {2150, 4556}, {2160, 38340}, {2161, 655}, {2170, 514}, {2171, 4605}, {2175, 692}, {2182, 2406}, {2183, 24029}, {2185, 4610}, {2192, 13138}, {2193, 4558}, {2194, 110}, {2195, 36086}, {2202, 1981}, {2204, 112}, {2208, 8059}, {2212, 8750}, {2218, 1305}, {2223, 2283}, {2249, 41206}, {2254, 9436}, {2264, 14543}, {2269, 3882}, {2287, 645}, {2291, 37139}, {2293, 35338}, {2298, 6648}, {2299, 162}, {2310, 522}, {2311, 4584}, {2316, 3257}, {2319, 4598}, {2320, 4597}, {2321, 4033}, {2323, 4585}, {2325, 24004}, {2328, 643}, {2329, 18047}, {2330, 4579}, {2339, 37215}, {2340, 1026}, {2342, 36037}, {2344, 4586}, {2346, 6606}, {2347, 21362}, {2364, 4604}, {2423, 34051}, {2424, 43736}, {2432, 36100}, {2481, 46135}, {2484, 2285}, {2488, 354}, {2489, 1880}, {2501, 40149}, {2509, 28739}, {2520, 1836}, {2530, 3665}, {2605, 1442}, {2646, 17136}, {2648, 35154}, {2651, 17933}, {2804, 3262}, {2826, 38468}, {2968, 35518}, {2978, 10473}, {3022, 3900}, {3049, 1409}, {3056, 3888}, {3057, 21272}, {3061, 33946}, {3063, 6}, {3064, 92}, {3119, 3239}, {3120, 4077}, {3121, 7180}, {3122, 4017}, {3125, 7178}, {3158, 43290}, {3208, 4595}, {3239, 312}, {3248, 43924}, {3250, 7146}, {3251, 1317}, {3254, 35171}, {3261, 20567}, {3270, 521}, {3271, 513}, {3287, 894}, {3309, 6604}, {3310, 1465}, {3423, 6183}, {3452, 21580}, {3596, 6386}, {3667, 39126}, {3669, 279}, {3675, 43042}, {3676, 1088}, {3683, 4427}, {3684, 3570}, {3685, 874}, {3688, 4553}, {3689, 17780}, {3693, 42720}, {3699, 7035}, {3700, 321}, {3701, 27808}, {3709, 37}, {3711, 4767}, {3712, 42721}, {3715, 4756}, {3716, 350}, {3733, 1014}, {3737, 86}, {3738, 320}, {3766, 18033}, {3777, 7185}, {3790, 4505}, {3810, 33930}, {3835, 30545}, {3876, 33948}, {3900, 8}, {3904, 20924}, {3907, 1909}, {3910, 20911}, {3939, 765}, {3960, 17078}, {3975, 27853}, {4009, 41314}, {4017, 3668}, {4024, 6358}, {4025, 7182}, {4036, 34388}, {4041, 10}, {4079, 2171}, {4081, 4397}, {4083, 3212}, {4086, 313}, {4091, 7183}, {4092, 4036}, {4105, 200}, {4124, 3766}, {4130, 346}, {4131, 7055}, {4140, 3963}, {4147, 6376}, {4148, 3975}, {4155, 7235}, {4162, 145}, {4163, 341}, {4171, 2321}, {4183, 36797}, {4367, 7176}, {4369, 7196}, {4374, 7205}, {4378, 7223}, {4382, 7243}, {4390, 4482}, {4391, 76}, {4394, 5435}, {4397, 3596}, {4435, 239}, {4449, 9312}, {4455, 1284}, {4459, 4374}, {4468, 21609}, {4474, 3761}, {4477, 7081}, {4501, 4361}, {4502, 7201}, {4515, 30730}, {4516, 523}, {4517, 3799}, {4518, 4583}, {4521, 18743}, {4522, 33931}, {4524, 210}, {4526, 536}, {4528, 4723}, {4529, 17787}, {4530, 3762}, {4534, 4462}, {4543, 4738}, {4546, 44720}, {4560, 274}, {4578, 4076}, {4581, 31643}, {4612, 4590}, {4617, 23586}, {4626, 24011}, {4631, 34537}, {4636, 24041}, {4705, 12}, {4724, 40719}, {4729, 4848}, {4730, 40663}, {4765, 19804}, {4768, 3264}, {4775, 2099}, {4790, 21454}, {4813, 4654}, {4814, 3679}, {4820, 28605}, {4822, 3671}, {4827, 391}, {4834, 5221}, {4858, 3261}, {4876, 4562}, {4893, 5219}, {4895, 519}, {4919, 6631}, {4936, 30720}, {4944, 4671}, {4959, 3632}, {4976, 4359}, {4979, 553}, {4983, 3649}, {4985, 1269}, {4990, 3702}, {5040, 5061}, {5075, 1758}, {5276, 14612}, {5375, 31633}, {5532, 42455}, {5546, 4567}, {5547, 5380}, {5548, 5376}, {5549, 5385}, {6004, 30617}, {6050, 7288}, {6129, 347}, {6139, 1155}, {6182, 2550}, {6186, 26700}, {6187, 2222}, {6332, 304}, {6362, 20880}, {6363, 1122}, {6364, 13453}, {6365, 13436}, {6366, 30806}, {6371, 24471}, {6372, 4059}, {6373, 1463}, {6377, 43051}, {6589, 17080}, {6591, 278}, {6607, 3059}, {6608, 4847}, {6614, 24013}, {6615, 3663}, {6728, 4146}, {6729, 174}, {6730, 556}, {7004, 4025}, {7054, 4612}, {7058, 4631}, {7064, 40521}, {7073, 6742}, {7077, 660}, {7083, 1633}, {7110, 15455}, {7117, 905}, {7118, 36049}, {7154, 40117}, {7155, 18830}, {7178, 1446}, {7180, 1427}, {7252, 81}, {7253, 314}, {7359, 42716}, {7649, 273}, {7658, 31627}, {8012, 35341}, {8027, 1357}, {8058, 322}, {8611, 306}, {8632, 1429}, {8638, 2223}, {8640, 1403}, {8641, 55}, {8642, 1617}, {8643, 1420}, {8645, 2078}, {8646, 1460}, {8648, 36}, {8653, 37593}, {8655, 16878}, {8656, 13462}, {8676, 3868}, {8678, 388}, {8735, 17924}, {8748, 823}, {8750, 7012}, {8755, 24035}, {8851, 8709}, {9247, 32660}, {9355, 10001}, {9404, 3219}, {9447, 32739}, {10581, 1212}, {11124, 3035}, {11193, 149}, {11604, 35156}, {11609, 35147}, {11934, 3434}, {11998, 17496}, {14224, 46141}, {14298, 329}, {14300, 9776}, {14302, 33672}, {14304, 35516}, {14330, 30854}, {14331, 18750}, {14349, 33949}, {14392, 6745}, {14395, 11064}, {14399, 6357}, {14400, 14206}, {14411, 1642}, {14413, 1323}, {14418, 3977}, {14419, 7181}, {14421, 43038}, {14427, 2325}, {14430, 6381}, {14432, 14210}, {14837, 40702}, {14838, 17095}, {15997, 45876}, {16686, 40577}, {16726, 17096}, {16757, 17076}, {17115, 497}, {17197, 7199}, {17418, 10436}, {17420, 4357}, {17425, 34522}, {17427, 31527}, {17435, 918}, {17924, 331}, {17926, 31623}, {18000, 2652}, {18155, 310}, {18191, 7192}, {18210, 17094}, {18265, 34067}, {18344, 4}, {20229, 35326}, {20293, 44139}, {20317, 18135}, {20684, 7239}, {20979, 1423}, {20980, 6180}, {20981, 7175}, {21044, 1577}, {21120, 26563}, {21124, 45196}, {21127, 142}, {21132, 1111}, {21172, 33673}, {21195, 40593}, {21761, 1935}, {21789, 21}, {21795, 35310}, {21811, 22003}, {21828, 18593}, {21832, 16609}, {22108, 37787}, {22382, 7364}, {22383, 222}, {23090, 1812}, {23189, 1444}, {23224, 1804}, {23289, 2997}, {23351, 1156}, {23572, 1424}, {23615, 24026}, {23696, 31637}, {23838, 903}, {23864, 13588}, {23880, 34284}, {23893, 1121}, {24010, 4163}, {24012, 4105}, {24026, 35519}, {25128, 17149}, {25430, 4624}, {26932, 15413}, {27527, 31008}, {27538, 36863}, {27837, 27829}, {27846, 43041}, {28071, 36802}, {28132, 36796}, {28660, 4602}, {29226, 17090}, {29278, 4968}, {30584, 41318}, {31623, 6331}, {31628, 31619}, {31947, 41808}, {32635, 6540}, {32674, 7128}, {32739, 2149}, {33299, 4568}, {33525, 40937}, {33635, 37212}, {33950, 33951}, {34068, 14733}, {34387, 40495}, {34591, 6332}, {34820, 4606}, {34858, 2720}, {35015, 36038}, {35057, 319}, {35355, 35160}, {35508, 4130}, {35518, 305}, {35519, 561}, {36054, 394}, {36086, 39293}, {36127, 24032}, {36197, 3700}, {36796, 36803}, {36798, 889}, {36800, 4639}, {36910, 36804}, {37908, 4238}, {38266, 38828}, {38347, 17494}, {38357, 17896}, {38358, 25259}, {38365, 4040}, {38375, 4468}, {38991, 21390}, {39201, 22341}, {39687, 36054}, {39786, 7212}, {40072, 4609}, {40137, 18228}, {40166, 23989}, {40495, 41283}, {40608, 2533}, {40627, 45208}, {40779, 32041}, {40869, 42719}, {40972, 46148}, {41339, 2398}, {42067, 43923}, {42069, 44426}, {42084, 39771}, {42312, 3875}, {42317, 32040}, {42325, 32007}, {42337, 20895}, {42454, 40619}, {42455, 34387}, {42462, 4858}, {42552, 8047}, {42649, 3336}, {42657, 484}, {42670, 5172}, {42771, 42758}, {43035, 24015}, {43049, 17093}, {43050, 37757}, {43728, 18816}, {43921, 43930}, {43923, 1119}, {43924, 269}, {43925, 1396}, {43929, 1462}, {43932, 479}, {44426, 264}, {45755, 4384}, {45902, 28391}, {46041, 46136}, {46110, 1969}

• X₆₅₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 693, 4885}, {2, 4885, 31250}, {2, 17494, 693}, {2, 24562, 25925}, {2, 26114, 25511}, {2, 26641, 24562}, {2, 26777, 17494}, {2, 26824, 26985}, {2, 26854, 27193}, {2, 27115, 31209}, {2, 27648, 27674}, {2, 28132, 28143}, {2, 31209, 31287}, {11, 14936, 38347}, {75, 21611, 21438}, {100, 1252, 14589}, {100, 5375, 1252}, {647, 3310, 6589}, {647, 6590, 21348}, {647, 6591, 6129}, {649, 652, 654}, {649, 657, 652}, {649, 1635, 4394}, {649, 4813, 4979}, {649, 4893, 661}, {649, 21127, 14300}, {651, 9358, 7045}, {652, 657, 9404}, {652, 14300, 4790}, {652, 21127, 13401}, {654, 9404, 652}, {654, 13401, 4790}, {657, 661, 14298}, {657, 17418, 3287}, {661, 1635, 649}, {661, 4394, 4790}, {661, 4979, 4813}, {663, 4814, 4895}, {663, 45755, 4435}, {665, 7180, 43060}, {693, 4885, 45320}, {693, 24948, 27674}, {693, 26641, 28984}, {693, 26695, 29005}, {693, 27115, 31287}, {693, 30024, 18154}, {693, 31150, 17494}, {693, 31209, 2}, {693, 31287, 31250}, {1021, 3737, 7252}, {1146, 7117, 11998}, {1638, 21104, 3676}, {1639, 3700, 3239}, {1639, 4765, 4820}, {1639, 4976, 3700}, {2516, 4394, 1635}, {2516, 4893, 4790}, {2590, 2591, 654}, {3119, 7004, 38358}, {3239, 3700, 4944}, {3239, 4521, 1639}, {3239, 4976, 4820}, {3310, 40134, 6129}, {3676, 7658, 1638}, {3676, 43049, 3669}, {3676, 43050, 43049}, {4041, 4895, 4814}, {4369, 4763, 31286}, {4379, 31207, 24924}, {4380, 4776, 20295}, {4382, 4728, 23813}, {4382, 30835, 4728}, {4394, 14298, 654}, {4394, 40137, 652}, {4467, 30565, 25259}, {4521, 4765, 3700}, {4521, 4976, 4944}, {4560, 27527, 18155}, {4776, 20295, 4940}, {4820, 4944, 3700}, {4885, 31287, 2}, {4885, 44567, 31287}, {6586, 6589, 647}, {6588, 6589, 6129}, {6591, 40134, 6588}, {6608, 14392, 4105}, {7045, 37139, 9358}, {11934, 17115, 11193}, {15280, 15283, 11}, {17494, 26114, 29771}, {17494, 26777, 31150}, {17494, 26985, 26824}, {17494, 27014, 25667}, {17494, 27115, 2}, {17494, 27139, 29739}, {17494, 27346, 18071}, {17494, 27648, 28374}, {17494, 28834, 40166}, {17494, 31209, 4885}, {17494, 31287, 45320}, {17494, 44567, 31250}, {18154, 25511, 4885}, {18154, 29771, 693}, {24900, 24948, 2}, {24900, 27648, 4885}, {24948, 31150, 28374}, {25259, 27486, 4467}, {26049, 27014, 2}, {26114, 30024, 4885}, {26777, 27115, 693}, {26824, 26985, 693}, {27014, 30061, 4885}, {27115, 31150, 4885}, {27115, 31209, 44567}, {28423, 28714, 24562}, {28984, 31287, 25925}, {29427, 29739, 693}, {31150, 31209, 693}, {31150, 44567, 45320}, {31250, 45320, 4885}, {31287, 44567, 31209}

Notes

X₆₅₀ is the perspector of triangle ABC and the tangential triangle of the Feuerbach hyperbola. (Randy Hutson, 9/23/2011)

Let T₁, T₂, T₃ denote the intouch, extouch,and incentral triangles. Let T₄ be the side triangle of T₂ and T₃; T₅ that of T₃ and T₁, and T₆ that of T₁ and T₂. Then X₆₅₀ is the perspector of triangle ABC and T₄, of ABC and T₅, and of ABC and T₆. (Randy Hutson, 9/23/2011)

X₆₅₀ is the point of intersection of orthic axis, antiorthic axis and Gergonne line, these being the trilinear polars of X₄, X₁ and X₇, respectively. More generally, X₆₅₀ is the intersection of the trilinear polars of every pair of points on the Feuerbach hyperbola. (Randy Hutson, December 26, 2015)