X₆₄₉

CROSSDIFFERENCE OF X₁ and X₂

X₆₄₉ = crossdifference of every pair of points on line {1, 2}

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

X₆₄₉ (CROSSDIFFERENCE OF X₁ and X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {187, 237}, {44, 513}

• Related Centers: X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂), X₄₄ (-LINE CONJUGATE OF X₁), X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀)

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

Barycentrics: \(a^{2}(b - c) : b^{2}(c - a) : c^{2}(a - b)\)

Search Value (6, 9, 13): \(13.522468075656\)

▸ Alternate Coordinates2 forms

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

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

▸ Linear Combinations1 combination

• X₆₄₉ = \(9\) X₂ - \(5\) X₂₆₇₉₈, \(3\) X₂ + X₂₆₈₅₃, \(3\) X₂ - \(5\) X₂₇₀₁₃, \(9\) X₂ - \(7\) X₂₇₁₃₈, \(6\) X₂ - \(5\) X₃₀₈₃₅, \(6\) X₂ - \(7\) X₃₁₂₀₇, \(3\) X₂ - \(4\) X₃₁₂₈₆, \(5\) X₂ - \(4\) X₄₅₃₃₉, \(3\) X₁₆₅ - \(2\) X₁₅₅₉₉, \(3\) X₃₅₁ - X₈₆₆₃, \(2\) X₆₅₀ - \(3\) X₁₆₃₅, \(3\) X₆₅₀ - \(4\) X₂₅₁₆, \(4\) X₆₅₀ - X₄₈₁₃, \(4\) X₆₅₀ - \(3\) X₄₈₉₃, \(2\) X₆₅₀ + X₄₉₇₉, \(3\) X₆₅₇ - \(4\) X₂₂₁₀₈, X₆₆₁ - \(3\) X₁₆₃₅, \(3\) X₆₆₁ - \(8\) X₂₅₁₆, X₆₆₁ - \(4\) X₄₃₉₄, X₆₆₁ + \(2\) X₄₇₉₀, \(2\) X₆₆₁ - \(3\) X₄₈₉₃, \(3\) X₆₆₃ - \(4\) X₁₉₆₀, \(3\) X₆₆₃ - \(2\) X₄₇₇₅, X₆₆₃ + \(2\) X₄₈₃₄, \(2\) X₆₆₃ - \(3\) X₈₆₄₃, \(3\) X₆₆₃ - \(5\) X₈₆₅₆, \(3\) X₆₆₇ - \(2\) X₁₉₆₀, \(3\) X₆₆₇ - X₄₇₇₅, \(4\) X₆₆₇ - \(3\) X₈₆₄₃, \(6\) X₆₆₇ - \(5\) X₈₆₅₆, \(2\) X₆₉₃ - \(3\) X₄₃₇₉, \(3\) X₇₉₈ - X₃₇₆₈, \(2\) X₁₀₁₉ + X₄₄₉₈, \(3\) X₁₀₁₉ + X₂₁₃₈₅, \(9\) X₁₆₃₅ - \(8\) X₂₅₁₆, \(3\) X₁₆₃₅ - \(4\) X₄₃₉₄, \(3\) X₁₆₃₅ + \(2\) X₄₇₉₀, \(6\) X₁₆₃₅ - X₄₈₁₃, \(3\) X₁₆₃₅ + X₄₉₇₉, \(3\) X₁₆₃₈ - \(4\) X₂₄₈₇, \(3\) X₁₆₃₈ - X₂₃₇₂₉, \(3\) X₁₆₃₉ - \(4\) X₂₄₉₀, \(3\) X₁₆₃₉ - \(2\) X₁₄₃₂₁, \(2\) X₁₉₆₀ + \(3\) X₄₈₃₄, \(8\) X₁₉₆₀ - \(9\) X₈₆₄₃, \(4\) X₁₉₆₀ - \(5\) X₈₆₅₆, \(3\) X₂₄₈₃ - \(2\) X₂₅₁₅, \(3\) X₂₄₈₄ - \(4\) X₂₅₁₅, \(4\) X₂₄₈₇ - X₂₃₇₂₉, \(2\) X₂₅₁₆ - \(3\) X₄₃₉₄, \(4\) X₂₅₁₆ + \(3\) X₄₇₉₀, \(16\) X₂₅₁₆ - \(3\) X₄₈₁₃, \(16\) X₂₅₁₆ - \(9\) X₄₈₉₃, \(8\) X₂₅₁₆ + \(3\) X₄₉₇₉, \(4\) X₂₅₂₇ - X₃₇₀₀, \(8\) X₂₅₂₉ - X₄₈₃₈, \(4\) X₃₂₃₉ - \(3\) X₄₁₂₀, \(4\) X₃₆₇₆ - \(3\) X₆₅₄₅, \(4\) X₃₇₃₉ - \(3\) X₂₇₄₈₅, \(2\) X₃₇₆₈ - \(3\) X₂₀₉₇₉, \(2\) X₃₇₇₆ - \(3\) X₄₄₅₃, \(4\) X₃₇₉₈ - \(3\) X₄₇₅₀, \(2\) X₃₇₉₈ - \(3\) X₄₇₈₆, \(4\) X₃₇₉₈ - X₁₆₈₉₂, \(6\) X₃₈₃₅ - \(5\) X₂₆₇₉₈, \(2\) X₃₈₃₅ + X₂₆₈₅₃, \(2\) X₃₈₃₅ - \(5\) X₂₇₀₁₃, \(6\) X₃₈₃₅ - \(7\) X₂₇₁₃₈, \(4\) X₃₈₃₅ - \(5\) X₃₀₈₃₅, \(4\) X₃₈₃₅ - \(3\) X₃₁₁₄₇, \(4\) X₃₈₃₅ - \(7\) X₃₁₂₀₇, X₃₈₃₅ - \(3\) X₄₅₃₁₃, \(5\) X₃₈₃₅ - \(6\) X₄₅₃₃₉, X₄₀₂₄ + \(3\) X₄₉₈₄, \(2\) X₄₀₂₅ - \(3\) X₄₇₅₀, X₄₀₂₅ - \(3\) X₄₇₈₆, \(3\) X₄₀₆₃ - X₂₁₃₈₅, \(2\) X₄₁₀₆ - \(3\) X₄₇₂₈, \(2\) X₄₁₀₆ - \(5\) X₂₄₉₂₄, \(3\) X₄₁₂₀ - \(8\) X₄₃₀₆₁, \(3\) X₄₃₆₇ - X₂₁₃₄₃, \(4\) X₄₃₆₉ - \(3\) X₄₃₇₉, \(2\) X₄₃₆₉ + X₄₃₈₀, \(4\) X₄₃₆₉ - X₄₃₈₂, \(3\) X₄₃₇₉ + \(2\) X₄₃₈₀, \(3\) X₄₃₇₉ - X₄₃₈₂, \(2\) X₄₃₈₀ + X₄₃₈₂, \(2\) X₄₃₉₄ + X₄₇₉₀, \(8\) X₄₃₉₄ - X₄₈₁₃, \(8\) X₄₃₉₄ - \(3\) X₄₈₉₃, \(4\) X₄₃₉₄ + X₄₉₇₉, \(3\) X₄₄₄₉ - \(2\) X₂₁₃₄₃, \(2\) X₄₄₆₈ - \(3\) X₆₅₄₆, \(3\) X₄₄₉₈ - \(2\) X₂₁₃₈₅, \(2\) X₄₅₀₀ - \(3\) X₄₇₈₉, \(2\) X₄₅₀₇ + \(3\) X₈₀₂₇, \(8\) X₄₅₂₁ - \(9\) X₆₅₄₄, X₄₇₂₄ - \(4\) X₄₇₈₂, X₄₇₂₄ + \(2\) X₄₇₈₄, \(3\) X₄₇₂₈ - \(4\) X₄₈₈₅, \(3\) X₄₇₂₈ - \(5\) X₂₄₉₂₄, \(3\) X₄₇₅₀ - X₁₆₈₉₂, \(3\) X₄₇₆₃ - \(2\) X₂₅₆₆₆, \(6\) X₄₇₆₃ - \(5\) X₃₁₂₀₉, \(4\) X₄₇₆₅ - X₄₉₈₈, X₄₇₆₅ - \(4\) X₁₄₃₅₁, \(4\) X₄₇₆₅ - \(9\) X₁₄₄₃₅, \(3\) X₄₇₇₃ - X₄₉₇₆, X₄₇₇₅ + \(3\) X₄₈₃₄, \(4\) X₄₇₇₅ - \(9\) X₈₆₄₃, \(2\) X₄₇₇₅ - \(5\) X₈₆₅₆, \(3\) X₄₇₇₆ - \(4\) X₂₅₆₆₆, \(3\) X₄₇₇₆ - \(5\) X₃₁₂₀₉, \(2\) X₄₇₈₂ + X₄₇₈₄, \(6\) X₄₇₈₆ - X₁₆₈₉₂, \(4\) X₄₇₉₀ + X₄₈₁₃, \(4\) X₄₇₉₀ + \(3\) X₄₈₉₃, X₄₈₁₃ - \(3\) X₄₈₉₃, X₄₈₁₃ + \(2\) X₄₉₇₉, \(2\) X₄₈₂₀ - \(3\) X₄₉₃₁, X₄₈₂₂ - \(4\) X₆₀₅₀, \(4\) X₄₈₃₄ + \(3\) X₈₆₄₃, \(6\) X₄₈₃₄ + \(5\) X₈₆₅₆, \(4\) X₄₈₈₅ - \(5\) X₂₄₉₂₄, \(3\) X₄₈₉₃ + \(2\) X₄₉₇₉, \(2\) X₄₉₃₂ + X₁₇₄₉₄, \(3\) X₄₉₈₄ + \(2\) X₆₅₉₀, X₄₉₈₈ - \(16\) X₁₄₃₅₁, X₄₉₈₈ - \(9\) X₁₄₄₃₅, \(3\) X₆₅₄₆ - \(4\) X₁₁₀₆₈, \(9\) X₈₆₄₃ - \(10\) X₈₆₅₆, \(3\) X₈₆₄₄ - \(2\) X₈₆₅₃, \(16\) X₁₄₃₅₁ - \(9\) X₁₄₄₃₅, \(8\) X₁₄₃₅₁ - X₄₅₇₄₅, \(9\) X₁₄₄₃₅ - \(2\) X₄₅₇₄₅, \(3\) X₂₀₂₉₅ - \(5\) X₂₆₇₉₈, X₂₀₂₉₅ - \(5\) X₂₇₀₁₃, \(3\) X₂₀₂₉₅ - \(7\) X₂₇₁₃₈, \(2\) X₂₀₂₉₅ - \(5\) X₃₀₈₃₅, \(2\) X₂₀₂₉₅ - \(3\) X₃₁₁₄₇, \(2\) X₂₀₂₉₅ - \(7\) X₃₁₂₀₇, X₂₀₂₉₅ - \(4\) X₃₁₂₈₆, X₂₀₂₉₅ - \(6\) X₄₅₃₁₃, \(5\) X₂₀₂₉₅ - \(12\) X₄₅₃₃₉, \(2\) X₂₁₁₉₆ - \(3\) X₂₇₄₈₆, X₂₁₁₉₆ - \(3\) X₄₅₆₇₉, \(4\) X₂₁₂₁₂ - \(3\) X₄₄₄₃₅, \(2\) X₂₁₂₁₂ - \(3\) X₄₅₆₇₄, \(3\) X₂₁₂₉₇ - \(5\) X₂₆₉₈₅, \(2\) X₂₃₈₁₃ - \(3\) X₄₅₃₂₀, \(4\) X₂₅₃₈₀ - \(3\) X₄₄₄₂₉, \(4\) X₂₅₆₆₆ - \(5\) X₃₁₂₀₉, \(5\) X₂₆₇₇₇ - X₃₁₂₉₀, \(5\) X₂₆₇₉₈ + \(3\) X₂₆₈₅₃, X₂₆₇₉₈ - \(3\) X₂₇₀₁₃, \(5\) X₂₆₇₉₈ - \(7\) X₂₇₁₃₈, \(2\) X₂₆₇₉₈ - \(3\) X₃₀₈₃₅, \(10\) X₂₆₇₉₈ - \(9\) X₃₁₁₄₇, \(10\) X₂₆₇₉₈ - \(21\) X₃₁₂₀₇, \(5\) X₂₆₇₉₈ - \(12\) X₃₁₂₈₆, \(5\) X₂₆₇₉₈ - \(18\) X₄₅₃₁₃, \(25\) X₂₆₇₉₈ - \(36\) X₄₅₃₃₉, X₂₆₈₅₃ + \(5\) X₂₇₀₁₃, \(3\) X₂₆₈₅₃ + \(7\) X₂₇₁₃₈, \(2\) X₂₆₈₅₃ + \(5\) X₃₀₈₃₅, \(2\) X₂₆₈₅₃ + \(3\) X₃₁₁₄₇, \(2\) X₂₆₈₅₃ + \(7\) X₃₁₂₀₇, X₂₆₈₅₃ + \(4\) X₃₁₂₈₆, X₂₆₈₅₃ + \(6\) X₄₅₃₁₃, \(5\) X₂₆₈₅₃ + \(12\) X₄₅₃₃₉, \(15\) X₂₇₀₁₃ - \(7\) X₂₇₁₃₈, \(10\) X₂₇₀₁₃ - \(3\) X₃₁₁₄₇, \(10\) X₂₇₀₁₃ - \(7\) X₃₁₂₀₇, \(5\) X₂₇₀₁₃ - \(4\) X₃₁₂₈₆, \(5\) X₂₇₀₁₃ - \(6\) X₄₅₃₁₃, \(25\) X₂₇₀₁₃ - \(12\) X₄₅₃₃₉, \(14\) X₂₇₁₃₈ - \(15\) X₃₀₈₃₅, \(14\) X₂₇₁₃₈ - \(9\) X₃₁₁₄₇, \(2\) X₂₇₁₃₈ - \(3\) X₃₁₂₀₇, \(7\) X₂₇₁₃₈ - \(12\) X₃₁₂₈₆, \(7\) X₂₇₁₃₈ - \(18\) X₄₅₃₁₃, \(35\) X₂₇₁₃₈ - \(36\) X₄₅₃₃₉, \(3\) X₂₇₄₈₆ - X₄₅₇₄₆, \(3\) X₂₈₃₉₈ - \(2\) X₂₉₇₃₈, \(3\) X₃₀₅₆₅ - X₄₄₄₄₉, \(5\) X₃₀₈₃₅ - \(3\) X₃₁₁₄₇, \(5\) X₃₀₈₃₅ - \(7\) X₃₁₂₀₇, \(5\) X₃₀₈₃₅ - \(8\) X₃₁₂₈₆, \(5\) X₃₀₈₃₅ - \(12\) X₄₅₃₁₃, \(25\) X₃₀₈₃₅ - \(24\) X₄₅₃₃₉, \(3\) X₃₁₁₄₇ - \(7\) X₃₁₂₀₇, \(3\) X₃₁₁₄₇ - \(8\) X₃₁₂₈₆, X₃₁₁₄₇ - \(4\) X₄₅₃₁₃, \(5\) X₃₁₁₄₇ - \(8\) X₄₅₃₃₉, \(3\) X₃₁₁₄₈ - \(2\) X₄₃₀₆₇, \(7\) X₃₁₂₀₇ - \(8\) X₃₁₂₈₆, \(7\) X₃₁₂₀₇ - \(12\) X₄₅₃₁₃, \(35\) X₃₁₂₀₇ - \(24\) X₄₅₃₃₉, \(2\) X₃₁₂₈₆ - \(3\) X₄₅₃₁₃, \(5\) X₃₁₂₈₆ - \(3\) X₄₅₃₃₉, \(5\) X₄₅₃₁₃ - \(2\) X₄₅₃₃₉, \(6\) X₄₅₆₇₉ - X₄₅₇₄₆

X₆₄₉ lies on the Yff parabola.

▸ Incident Central Lines262 lines

{1, 3249}, {2, 3835}, {3, 14825}, {6, 2441}, {8, 25301}, {9, 4521}, {10, 21720}, {19, 3064}, {27, 46107}, {31, 884}, {32, 8578}, {40, 28292}, {42, 788}, {44, 513}, {45, 14437}, {55, 23865}, {57, 1024}, {58, 43926}, {63, 4468}, {75, 20909}, {81, 18200}, {89, 1022}, {100, 660}, {101, 901}, {103, 43079}, {106, 2384}, {109, 919}, {110, 2702}, {111, 2712}, {165, 15599}, {171, 18098}, {187, 237}, {190, 889}, {239, 514}, {244, 38346}, {522, 3509}, {523, 4773}, {604, 2423}, {614, 9315}, {662, 17929}, {693, 812}, {727, 9111}, {739, 2382}, {764, 1475}, {802, 4374}, {814, 2533}, {824, 4467}, {830, 1734}, {834, 1459}, {840, 2291}, {854, 1404}, {876, 40747}, {891, 4378}, {893, 17187}, {894, 21211}, {900, 2527}, {905, 28372}, {918, 4897}, {926, 4105}, {995, 30650}, {1015, 43922}, {1018, 4781}, {1027, 2279}, {1125, 19949}, {1146, 39006}, {1201, 9433}, {1293, 6078}, {1334, 6161}, {1415, 32675}, {1436, 2432}, {1461, 7339}, {1577, 29013}, {1638, 2487}, {1639, 2490}, {1768, 21382}, {1769, 6588}, {1797, 10756}, {1977, 6377}, {1978, 4598}, {2082, 23764}, {2160, 21131}, {2161, 14442}, {2170, 15635}, {2269, 24118}, {2284, 23832}, {2308, 8034}, {2319, 17155}, {2340, 9320}, {2473, 2521}, {2504, 7178}, {2529, 4777}, {2664, 4040}, {2666, 3720}, {2786, 21391}, {2787, 4474}, {3004, 17069}, {3051, 23575}, {3120, 43920}, {3219, 10196}, {3239, 3667}, {3246, 39310}, {3259, 46101}, {3261, 10566}, {3306, 14475}, {3309, 3803}, {3666, 14751}, {3669, 7216}, {3709, 4502}, {3726, 4132}, {3739, 27485}, {3762, 29148}, {3766, 24601}, {3776, 4453}, {3801, 29025}, {3837, 24719}, {3900, 4729}, {3904, 28468}, {3929, 45670}, {3937, 14936}, {3960, 14438}, {4010, 4874}, {4014, 38991}, {4017, 6591}, {4041, 8678}, {4057, 4079}, {4083, 4367}, {4106, 4728}, {4122, 29078}, {4129, 27045}, {4139, 4501}, {4142, 29118}, {4148, 4462}, {4191, 20674}, {4266, 14812}, {4384, 14433}, {4391, 6002}, {4414, 25800}, {4432, 40614}, {4481, 16751}, {4486, 26248}, {4491, 14407}, {4500, 4789}, {4584, 4600}, {4586, 35009}, {4588, 28875}, {4603, 4610}, {4640, 40586}, {4672, 35353}, {4730, 4814}, {4761, 29066}, {4762, 31148}, {4763, 4776}, {4774, 29236}, {4791, 29178}, {4804, 7662}, {4820, 4926}, {4822, 6050}, {4823, 29270}, {4841, 4977}, {4879, 23506}, {4940, 31287}, {4944, 4958}, {4978, 29302}, {5701, 34583}, {6003, 23691}, {6004, 24290}, {6014, 6017}, {6084, 21104}, {6085, 23650}, {6182, 6608}, {6332, 28478}, {6363, 20980}, {6372, 21763}, {6373, 20983}, {6615, 40134}, {7004, 21339}, {7045, 36146}, {7199, 29771}, {7265, 29216}, {7289, 23730}, {7649, 21190}, {8056, 23834}, {8611, 15313}, {8677, 36054}, {8697, 28891}, {9085, 29242}, {9097, 17222}, {9259, 9262}, {9294, 17759}, {9359, 9361}, {10015, 29126}, {10459, 30203}, {14349, 14838}, {14377, 23100}, {14425, 39386}, {14474, 30950}, {14991, 23657}, {15487, 28589}, {16231, 42403}, {16557, 32925}, {16560, 24129}, {16612, 23800}, {16695, 40627}, {17029, 26824}, {17072, 21053}, {17159, 21225}, {17204, 17215}, {17412, 23224}, {17439, 38242}, {18154, 20954}, {18278, 21791}, {20517, 29158}, {21003, 21005}, {21051, 25636}, {21122, 43925}, {21146, 29362}, {21204, 27003}, {21212, 31095}, {21260, 24960}, {21297, 26985}, {21302, 28470}, {21348, 21834}, {21349, 21350}, {22092, 45902}, {22094, 23647}, {22224, 26048}, {23301, 30968}, {23726, 26934}, {23794, 30024}, {23803, 26983}, {23813, 45320}, {23845, 35326}, {24462, 26249}, {24484, 46125}, {24560, 25902}, {24562, 25900}, {24592, 27855}, {24756, 30957}, {24804, 35355}, {25008, 25981}, {25009, 26017}, {25128, 31330}, {25380, 44429}, {25511, 42327}, {25604, 40474}, {25627, 26037}, {26049, 27527}, {26777, 31290}, {27009, 44312}, {27014, 28758}, {27451, 30095}, {28255, 28286}, {28840, 31150}, {28867, 30565}, {29179, 43361}, {32669, 32674}, {33570, 35270}, {35595, 45684}, {38325, 42322}, {38344, 38345}, {38348, 40789}, {38367, 40733}

▸ Functional Relations41 relations

• X₆₄₉ = midpoint of X_i and X_j for these {i,j}: {650, 4790}, {659, 4784}, {661, 4979}, {667, 4834}, {693, 4380}, {1019, 4063}, {7192, 17494}, {17159, 21225}, {20295, 26853}, {21302, 31291}, {29545, 29807}

• X₆₄₉ = reflection of X_i in X_j for these {i,j}: {2, 45313}, {650, 4394}, {659, 4782}, {661, 650}, {663, 667}, {693, 4369}, {1459, 3733}, {1491, 9508}, {2484, 2483}, {3004, 17069}, {3239, 43061}, {3250, 665}, {3835, 31286}, {4010, 4874}, {4024, 6590}, {4025, 3798}, {4040, 4401}, {4079, 6586}, {4106, 4885}, {4382, 693}, {4449, 4367}, {4468, 11068}, {4498, 4063}, {4502, 3709}, {4724, 659}, {4750, 4786}, {4775, 1960}, {4776, 4763}, {4804, 7662}, {4813, 661}, {4814, 4730}, {4893, 1635}, {4940, 31287}, {4958, 4944}, {4979, 4790}, {4988, 45745}, {7192, 4932}, {14321, 2490}, {14349, 14838}, {16892, 4025}, {17217, 21191}, {20295, 3835}, {20979, 798}, {21301, 17072}, {21834, 21348}, {23655, 7234}, {24719, 3837}, {27486, 45679}, {30835, 27013}, {31147, 2}, {38325, 42322}, {42664, 647}, {44435, 45674}, {45745, 4765}, {45746, 21196}

• X₆₄₉ = reflection of X_i in X_j for these (i,j): (661, 650), (663, 667)

• X₆₄₉ = isogonal conjugate of X₁₉₀

• X₆₄₉ = isotomic conjugate of X₁₉₇₈

• X₆₄₉ = complement of X₂₀₂₉₅

• X₆₄₉ = anticomplement of X₃₈₃₅

• X₆₄₉ = excentral-isogonal conjugate of X₉₃₅₅

• X₆₄₉ = tangential-isogonal conjugate of X₁₆₈₇₃

• X₆₄₉ = trilinear pole of line {1015, 1960}

• X₆₄₉ = Parry-circle-inverse of X₅₀₂₉

• X₆₄₉ = Parry-isodynamic-circle-inverse of X₅₁₆₈

• X₆₄₉ = crossdifference of every pair of points on line {1, 2}

• X₆₄₉ = excentral isogonal conjugate of X₉₃₅₅

• X₆₄₉ = tangential isogonal conjugate of X₁₆₈₇₃

• X₆₄₉ = bicentric difference of PU(i) for i in (8, 48, 58, 84, 92, 98)

• X₆₄₉ = PU(8)-harmonic conjugate of X₄₂

• X₆₄₉ = trilinear pole of PU(25) (line X₁₀₁₅X₁₉₆₀)

• X₆₄₉ = barycentric product of PU(34)

• X₆₄₉ = PU(48)-harmonic conjugate of X₃₁

• X₆₄₉ = PU(58)-harmonic conjugate of X₈₉₉

• X₆₄₉ = PU(84)-harmonic conjugate of X₃₇₂₀

• X₆₄₉ = PU(92)-harmonic conjugate of X₁₂₀₁

• X₆₄₉ = PU(98)-harmonic conjugate of X₁₁₄₉

• X₆₄₉ = intersection of antiorthic and Lemoine axes (trilinear polars of X₁ and X₆)

• X₆₄₉ = pole, with respect to Bevan circle, of line X₂X₇

• X₆₄₉ = X₆-isoconjugate of X₆₆₈

• X₆₄₉ = X₆₄₇-of-excentral-triangle

• X₆₄₉ = perspector of the excentral Jerabek hyperbola

• X₆₄₉ = complement of polar conjugate of isogonal conjugate of X₂₂₁₅₄

• X₆₄₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {87, 150}, {109, 20537}, {110, 17149}, {330, 21293}, {651, 20350}, {664, 20559}, {692, 21219}, {932, 69}, {1576, 36857}, {2053, 37781}, {2162, 149}, {2319, 33650}, {4598, 6327}, {5383, 21301}, {7121, 4440}, {16606, 3448}, {18830, 315}, {21759, 148}, {22381, 39352}, {23493, 21221}, {32739, 41840}, {34071, 8}, {42027, 21294}

• X₆₄₉ = X_i-complementary conjugate of X_j for these (i,j): {31, 8054}, {596, 21252}, {692, 4075}, {8050, 2887}, {20615, 17059}, {34594, 3741}, {37205, 21240}, {39798, 116}, {40085, 21253}, {40148, 11}, {40519, 10}

• X₆₄₉ = X_i-Ceva conjugate of X_j for these (i,j): {1, 3248}, {2, 8054}, {6, 1015}, {19, 2170}, {25, 20974}, {27, 3120}, {31, 38346}, {56, 3271}, {57, 244}, {58, 3122}, {64, 3270}, {84, 2310}, {87, 38986}, {99, 2309}, {100, 42}, {101, 6}, {108, 40958}, {109, 31}, {110, 2308}, {163, 2260}, {190, 1}, {267, 2643}, {513, 663}, {514, 1459}, {651, 1201}, {653, 3924}, {658, 614}, {660, 3009}, {662, 1193}, {664, 7032}, {667, 20979}, {673, 27846}, {692, 1475}, {799, 3720}, {813, 672}, {825, 21764}, {893, 3121}, {901, 902}, {932, 22343}, {934, 20978}, {1018, 1100}, {1019, 513}, {1020, 1104}, {1086, 8578}, {1292, 2293}, {1293, 55}, {1407, 3937}, {1415, 1400}, {1436, 7117}, {1461, 56}, {1576, 40955}, {1919, 23572}, {2156, 21339}, {2160, 3125}, {2161, 2087}, {2162, 1977}, {2176, 23560}, {2226, 43922}, {2248, 3124}, {2423, 21758}, {2702, 1914}, {2705, 8540}, {3224, 21762}, {3257, 1149}, {3345, 2638}, {3437, 20975}, {3444, 20982}, {3500, 3123}, {3572, 5029}, {3669, 43924}, {3733, 667}, {3903, 23532}, {4556, 58}, {4557, 20963}, {4559, 2300}, {4584, 238}, {4588, 21747}, {4591, 36}, {4598, 2}, {4603, 81}, {4604, 995}, {4607, 899}, {4610, 17187}, {4628, 5299}, {4629, 1203}, {4638, 106}, {4817, 4724}, {6010, 2269}, {6011, 14547}, {6014, 2177}, {6577, 1918}, {7087, 23646}, {7096, 17463}, {7097, 34591}, {7252, 22383}, {8050, 40148}, {8052, 5262}, {8056, 17477}, {8699, 3052}, {8709, 20464}, {8750, 16502}, {8917, 24012}, {9500, 35505}, {10566, 514}, {11051, 14936}, {13478, 11}, {13610, 4128}, {14377, 1086}, {16695, 23464}, {18830, 23493}, {21362, 20323}, {23345, 1960}, {23355, 6373}, {23892, 3768}, {28469, 3056}, {28477, 2268}, {28486, 2330}, {28847, 2280}, {29014, 71}, {29217, 5301}, {30554, 5332}, {32039, 87}, {32665, 2183}, {32674, 604}, {32675, 1404}, {32714, 1042}, {34067, 20459}, {34071, 2275}, {34080, 2347}, {34183, 38363}, {34234, 1647}, {34440, 3269}, {34445, 23456}, {35342, 4272}, {36146, 1458}, {36614, 23470}, {37133, 21352}, {37138, 869}, {37205, 1125}, {37209, 30950}, {37211, 386}, {37215, 17017}, {38828, 32577}, {39950, 16726}, {40151, 1357}, {40188, 3942}, {40519, 39}, {40737, 4117}, {40770, 21755}, {41405, 41395}, {42467, 7004}, {43077, 2276}, {43739, 11998}, {43924, 8643}, {43929, 8632}

• X₆₄₉ = X_i-cross conjugate of X_j for these (i,j): {6, 9262}, {244, 2350}, {512, 513}, {665, 3572}, {667, 43924}, {798, 667}, {1015, 6}, {1960, 23345}, {1977, 40148}, {3063, 663}, {3121, 42}, {3122, 58}, {3125, 1400}, {3248, 1}, {3249, 3248}, {3271, 56}, {3768, 23892}, {4083, 20979}, {6373, 23355}, {6377, 2}, {7180, 6591}, {7234, 18108}, {8659, 43929}, {8661, 43922}, {14936, 25}, {16614, 39956}, {21123, 514}, {21143, 1015}, {21191, 23572}, {23456, 34445}, {23470, 30651}, {38986, 87}

• X₆₄₉ = X_i-isoconjugate of X_j for these (i,j): {1, 190}, {2, 100}, {3, 6335}, {4, 1332}, {6, 668}, {7, 644}, {8, 651}, {9, 664}, {10, 662}, {11, 31615}, {12, 4612}, {19, 4561}, {21, 4552}, {31, 1978}, {32, 6386}, {35, 15455}, {36, 36804}, {37, 99}, {40, 44327}, {41, 4572}, {42, 799}, {43, 4598}, {44, 4555}, {45, 4597}, {55, 4554}, {56, 646}, {57, 3699}, {58, 4033}, {59, 4391}, {63, 1897}, {65, 645}, {69, 1783}, {71, 811}, {72, 648}, {74, 42716}, {75, 101}, {76, 692}, {78, 653}, {80, 4585}, {81, 3952}, {82, 4568}, {83, 4553}, {85, 3939}, {86, 1018}, {87, 4595}, {88, 17780}, {89, 4767}, {92, 1331}, {98, 42717}, {102, 42718}, {103, 42719}, {104, 2397}, {105, 42720}, {106, 24004}, {107, 3998}, {108, 345}, {109, 312}, {110, 321}, {111, 42721}, {112, 20336}, {145, 27834}, {162, 306}, {163, 313}, {171, 27805}, {181, 4631}, {192, 932}, {200, 658}, {210, 4573}, {213, 670}, {219, 18026}, {220, 4569}, {226, 643}, {228, 6331}, {238, 4562}, {239, 660}, {241, 36802}, {244, 6632}, {249, 4036}, {256, 18047}, {257, 4579}, {261, 21859}, {264, 906}, {269, 6558}, {273, 4587}, {274, 4557}, {278, 4571}, {279, 4578}, {281, 6516}, {286, 4574}, {291, 3570}, {292, 874}, {294, 883}, {304, 8750}, {314, 4559}, {318, 1813}, {322, 36049}, {329, 13138}, {333, 4551}, {335, 3573}, {341, 1461}, {344, 1292}, {346, 934}, {350, 813}, {386, 37218}, {476, 42701}, {480, 36838}, {512, 4601}, {513, 1016}, {514, 765}, {517, 13136}, {518, 666}, {519, 3257}, {521, 46102}, {522, 4564}, {523, 4567}, {524, 5380}, {525, 5379}, {536, 898}, {556, 6733}, {561, 32739}, {598, 3908}, {612, 37215}, {649, 7035}, {650, 4998}, {655, 4511}, {661, 4600}, {667, 31625}, {673, 1026}, {675, 42723}, {677, 30807}, {689, 21814}, {691, 42713}, {693, 1252}, {728, 4626}, {739, 41314}, {740, 4584}, {751, 4482}, {752, 5386}, {754, 5389}, {756, 4610}, {757, 4103}, {788, 5388}, {789, 2276}, {812, 5378}, {823, 3682}, {824, 5384}, {825, 33931}, {831, 17289}, {833, 32777}, {835, 28606}, {839, 4261}, {840, 42722}, {869, 37133}, {889, 3230}, {891, 5381}, {892, 21839}, {894, 3903}, {899, 4607}, {900, 5376}, {901, 4358}, {903, 1023}, {905, 15742}, {908, 36037}, {914, 36106}, {918, 5377}, {919, 3263}, {925, 42700}, {927, 3693}, {931, 31993}, {933, 42698}, {958, 32038}, {960, 6648}, {982, 4621}, {983, 33946}, {984, 4586}, {985, 3807}, {1001, 32041}, {1014, 30730}, {1020, 1043}, {1025, 14942}, {1042, 7258}, {1089, 4556}, {1098, 4605}, {1100, 6540}, {1110, 3261}, {1125, 37212}, {1212, 6606}, {1213, 4596}, {1214, 36797}, {1215, 4603}, {1220, 3882}, {1222, 21362}, {1255, 4427}, {1257, 14543}, {1260, 13149}, {1262, 4397}, {1265, 32714}, {1267, 6135}, {1268, 35342}, {1275, 3900}, {1278, 29227}, {1290, 32849}, {1293, 18743}, {1296, 42724}, {1301, 42699}, {1302, 42704}, {1305, 27396}, {1308, 17264}, {1310, 2345}, {1319, 4582}, {1333, 27808}, {1334, 4625}, {1376, 30610}, {1400, 7257}, {1414, 2321}, {1415, 3596}, {1427, 7256}, {1429, 36801}, {1434, 4069}, {1441, 5546}, {1473, 42384}, {1476, 25268}, {1492, 3661}, {1500, 4623}, {1509, 40521}, {1575, 8709}, {1576, 27801}, {1577, 4570}, {1633, 30701}, {1698, 37211}, {1757, 35148}, {1824, 4563}, {1826, 4592}, {1911, 27853}, {1914, 4583}, {1918, 4602}, {1921, 34067}, {1930, 4628}, {1962, 4632}, {1969, 32656}, {1983, 20566}, {1997, 30236}, {2087, 6635}, {2149, 35519}, {2162, 36863}, {2176, 18830}, {2205, 4609}, {2214, 33948}, {2222, 32851}, {2223, 36803}, {2238, 4589}, {2283, 36796}, {2284, 2481}, {2287, 4566}, {2295, 4594}, {2323, 35174}, {2339, 14594}, {2340, 34085}, {2398, 36101}, {2427, 18816}, {2702, 20947}, {2703, 17790}, {2715, 42703}, {2742, 37788}, {2743, 37758}, {2748, 37756}, {2753, 37857}, {2832, 5387}, {3006, 36087}, {3035, 31628}, {3112, 46148}, {3175, 8690}, {3198, 44326}, {3219, 6742}, {3222, 21877}, {3227, 23343}, {3239, 7045}, {3240, 37209}, {3262, 32641}, {3264, 32665}, {3290, 35574}, {3293, 37205}, {3616, 4606}, {3666, 8707}, {3667, 5382}, {3668, 7259}, {3669, 4076}, {3672, 6574}, {3679, 4604}, {3681, 43190}, {3687, 36098}, {3692, 36118}, {3701, 4565}, {3717, 36146}, {3718, 32674}, {3719, 36127}, {3739, 8708}, {3747, 4639}, {3752, 8706}, {3762, 9268}, {3783, 37207}, {3797, 30664}, {3799, 14621}, {3869, 44765}, {3870, 37206}, {3888, 17743}, {3909, 40394}, {3912, 36086}, {3935, 37143}, {3943, 4622}, {3954, 4577}, {3969, 13486}, {3990, 6528}, {3992, 4591}, {3995, 34594}, {4024, 24041}, {4037, 36066}, {4039, 37134}, {4041, 4620}, {4043, 43076}, {4062, 36085}, {4079, 24037}, {4082, 4637}, {4083, 5383}, {4115, 40438}, {4357, 36147}, {4359, 8701}, {4370, 4618}, {4384, 37138}, {4417, 36050}, {4420, 38340}, {4421, 42343}, {4436, 32009}, {4441, 8693}, {4463, 44766}, {4505, 40746}, {4512, 4624}, {4515, 4616}, {4558, 41013}, {4576, 18098}, {4588, 4671}, {4590, 4705}, {4593, 21035}, {4599, 15523}, {4613, 40773}, {4614, 5257}, {4615, 21805}, {4617, 5423}, {4619, 24026}, {4629, 4647}, {4633, 37593}, {4636, 6358}, {4638, 4738}, {4663, 35177}, {4664, 29351}, {4687, 6013}, {4699, 29199}, {4752, 39704}, {4756, 25417}, {4777, 5385}, {4781, 40434}, {4812, 29026}, {4850, 9059}, {4980, 28176}, {4997, 23703}, {5222, 37223}, {5223, 32040}, {5291, 35147}, {5293, 8052}, {5297, 37210}, {5360, 43187}, {5375, 8047}, {5391, 6136}, {5435, 31343}, {5526, 35171}, {5545, 42712}, {6011, 33116}, {6012, 17279}, {6014, 30829}, {6065, 24002}, {6079, 16610}, {6163, 6630}, {6164, 6634}, {6332, 7012}, {6376, 34071}, {6381, 34075}, {6542, 37135}, {6554, 8269}, {6559, 41353}, {6577, 18137}, {6603, 35157}, {6605, 35312}, {6614, 30693}, {6631, 9282}, {6735, 37136}, {6745, 37139}, {6790, 46119}, {7017, 36059}, {7080, 37141}, {7081, 37137}, {7115, 35518}, {7239, 40415}, {7260, 20964}, {8050, 32911}, {8056, 43290}, {8652, 28605}, {8684, 33891}, {8694, 19804}, {8699, 20942}, {9058, 17740}, {9067, 17756}, {9070, 32779}, {9265, 9296}, {9266, 9295}, {9271, 17487}, {9278, 17934}, {9361, 9362}, {11124, 31619}, {11495, 42303}, {11611, 17944}, {13396, 17281}, {13397, 17776}, {14947, 40865}, {15322, 28653}, {15624, 31624}, {16514, 41072}, {16593, 39272}, {16777, 32042}, {16785, 35181}, {17459, 35572}, {17787, 29055}, {17796, 35156}, {17863, 29163}, {17930, 20693}, {18140, 40519}, {18147, 29014}, {19604, 30720}, {19799, 32691}, {20332, 23354}, {20440, 20640}, {20453, 20696}, {20568, 23344}, {20901, 31616}, {20911, 32736}, {20940, 40150}, {21272, 23617}, {21453, 35341}, {21802, 35137}, {21833, 31614}, {21874, 35136}, {21899, 37880}, {22003, 40430}, {22456, 42702}, {23067, 31623}, {23493, 36860}, {23704, 35160}, {23832, 36805}, {23845, 32017}, {23891, 37129}, {23981, 36795}, {23990, 40495}, {24589, 28210}, {25001, 43344}, {25660, 29151}, {26700, 42033}, {26706, 28420}, {26711, 33168}, {26714, 42711}, {28148, 42029}, {28162, 42034}, {28474, 41316}, {28583, 41315}, {28847, 30758}, {30555, 31130}, {30625, 42301}, {30963, 43077}, {31633, 42552}, {32008, 35338}, {32018, 35327}, {32676, 40071}, {32718, 35543}, {32931, 35009}, {33113, 33637}, {33157, 43348}, {35280, 39749}, {35517, 36039}, {36077, 42706}, {36080, 44140}, {37204, 41267}, {38828, 44720}, {40728, 46132}, {41839, 43350}, {44426, 44717}

• X₆₄₉ = cevapoint of X_i and X_j for these (i,j): {1, 9359}, {2, 21224}, {6, 9259}, {512, 798}, {513, 4083}, {514, 21191}, {667, 3063}, {1015, 21143}, {3121, 8027}, {3248, 3249}

• X₆₄₉ = crosspoint of X_i and X_j for these (i,j): {1, 190}, {2, 8050}, {6, 101}, {19, 32674}, {28, 32714}, {56, 1461}, {57, 109}, {58, 4556}, {81, 100}, {87, 32039}, {99, 40409}, {106, 4638}, {108, 40397}, {110, 1171}, {274, 18830}, {513, 3669}, {514, 7649}, {651, 1476}, {662, 2363}, {799, 40439}, {893, 4603}, {901, 2226}, {1019, 3733}, {1293, 40151}, {1333, 1415}, {2162, 4598}, {2702, 9506}, {15397, 32682}

• X₆₄₉ = crosssum of X_i and X_j for these (i,j): {1, 649}, {2, 514}, {6, 4057}, {7, 31605}, {8, 3239}, {9, 522}, {10, 4024}, {37, 513}, {57, 30719}, {63, 6332}, {71, 4064}, {75, 20954}, {100, 644}, {101, 1331}, {115, 12078}, {145, 31182}, {192, 20979}, {213, 8640}, {239, 4375}, {321, 4391}, {440, 525}, {512, 21838}, {519, 6544}, {523, 1213}, {650, 3057}, {656, 18674}, {657, 4319}, {659, 17475}, {661, 2292}, {663, 20665}, {667, 21757}, {668, 36863}, {693, 20880}, {798, 2667}, {812, 17755}, {900, 4370}, {918, 16593}, {1016, 6634}, {1018, 3952}, {1019, 8025}, {1252, 14887}, {1635, 17460}, {1960, 23552}, {2183, 23757}, {2254, 17464}, {2398, 3234}, {2786, 6651}, {3059, 4130}, {3161, 3667}, {3219, 7265}, {3294, 17494}, {3699, 30720}, {3700, 21677}, {3730, 25259}, {3768, 42083}, {3835, 33890}, {3995, 4063}, {4025, 17170}, {4079, 21035}, {4115, 4427}, {4368, 21832}, {4473, 24129}, {4498, 30568}, {4560, 17185}, {4568, 46148}, {4775, 23553}, {4777, 16590}, {4785, 27481}, {4893, 17461}, {5513, 23887}, {6006, 36911}, {6364, 40651}, {6545, 31647}, {6546, 32094}, {7192, 17175}, {8058, 38015}, {8714, 40586}, {16892, 17192}, {21362, 25268}, {24979, 24980}, {28840, 31336}

• X₆₄₉ = barycentric product X_i*X_j for these {i,j}: {1, 513}, {3, 7649}, {4, 1459}, {6, 514}, {7, 663}, {8, 43924}, {9, 3669}, {10, 3733}, {11, 109}, {19, 905}, {21, 4017}, {25, 4025}, {27, 647}, {28, 656}, {31, 693}, {32, 3261}, {34, 521}, {37, 1019}, {38, 18108}, {39, 10566}, {41, 24002}, {42, 7192}, {43, 43931}, {44, 1022}, {48, 17924}, {54, 21102}, {55, 3676}, {56, 522}, {57, 650}, {58, 523}, {59, 21132}, {63, 6591}, {64, 21172}, {65, 3737}, {71, 17925}, {74, 11125}, {75, 667}, {76, 1919}, {77, 18344}, {78, 43923}, {79, 2605}, {81, 661}, {82, 2530}, {83, 21123}, {84, 6129}, {85, 3063}, {86, 512}, {87, 4083}, {88, 1635}, {89, 4893}, {91, 34948}, {92, 22383}, {99, 3122}, {100, 244}, {101, 1086}, {103, 676}, {104, 1769}, {105, 2254}, {106, 900}, {108, 7004}, {110, 3120}, {111, 4750}, {112, 4466}, {115, 4556}, {158, 23224}, {162, 18210}, {163, 16732}, {174, 6729}, {184, 46107}, {190, 1015}, {200, 43932}, {210, 7203}, {213, 7199}, {222, 3064}, {225, 23189}, {226, 7252}, {238, 876}, {239, 3572}, {241, 1024}, {249, 21131}, {250, 21134}, {251, 16892}, {256, 4367}, {257, 20981}, {266, 6728}, {267, 31947}, {269, 3900}, {273, 1946}, {274, 798}, {278, 652}, {279, 657}, {284, 7178}, {286, 810}, {291, 659}, {292, 812}, {306, 43925}, {310, 669}, {330, 20979}, {333, 7180}, {335, 8632}, {350, 875}, {386, 43927}, {393, 4091}, {479, 4105}, {516, 2424}, {518, 1027}, {519, 23345}, {520, 8747}, {525, 1474}, {536, 23892}, {560, 40495}, {561, 1980}, {593, 4024}, {595, 40086}, {596, 4057}, {603, 44426}, {604, 4391}, {608, 6332}, {651, 2170}, {653, 7117}, {654, 2006}, {658, 14936}, {660, 27846}, {662, 3125}, {664, 3271}, {665, 673}, {668, 3248}, {679, 3251}, {692, 1111}, {694, 4107}, {726, 23355}, {727, 3837}, {738, 4130}, {739, 4728}, {741, 4010}, {751, 4378}, {753, 4809}, {757, 4705}, {764, 765}, {788, 870}, {799, 3121}, {813, 27918}, {824, 40746}, {832, 977}, {834, 43531}, {849, 4036}, {850, 2206}, {871, 8630}, {884, 9436}, {885, 1458}, {890, 31002}, {891, 37129}, {893, 4369}, {897, 14419}, {898, 19945}, {899, 43928}, {901, 1647}, {902, 6548}, {903, 1960}, {904, 4374}, {908, 2423}, {909, 10015}, {918, 1438}, {932, 3123}, {934, 2310}, {959, 17418}, {961, 17420}, {963, 7661}, {967, 45745}, {983, 3777}, {985, 1491}, {996, 9002}, {998, 9001}, {1002, 4724}, {1014, 4041}, {1016, 21143}, {1018, 16726}, {1021, 1427}, {1026, 43921}, {1042, 7253}, {1043, 7250}, {1054, 6164}, {1088, 8641}, {1096, 4131}, {1106, 4397}, {1120, 6085}, {1126, 4977}, {1146, 1461}, {1149, 23836}, {1155, 35348}, {1156, 14413}, {1169, 21124}, {1170, 21127}, {1171, 4988}, {1173, 21103}, {1174, 21104}, {1175, 23752}, {1176, 21108}, {1177, 21109}, {1178, 2533}, {1193, 4581}, {1220, 6371}, {1222, 6363}, {1252, 6545}, {1255, 4979}, {1262, 42462}, {1279, 35355}, {1293, 3756}, {1318, 39771}, {1319, 23838}, {1323, 23351}, {1326, 18014}, {1331, 2969}, {1333, 1577}, {1334, 17096}, {1357, 3699}, {1358, 3939}, {1364, 36127}, {1365, 4636}, {1395, 35518}, {1396, 8611}, {1397, 35519}, {1400, 4560}, {1402, 18155}, {1407, 3239}, {1408, 4086}, {1411, 3738}, {1412, 3700}, {1413, 8058}, {1414, 4516}, {1415, 4858}, {1417, 4768}, {1421, 42552}, {1422, 14298}, {1431, 3907}, {1432, 3287}, {1434, 3709}, {1436, 14837}, {1437, 24006}, {1457, 43728}, {1472, 2517}, {1476, 6615}, {1486, 26721}, {1492, 4475}, {1509, 4079}, {1565, 8750}, {1576, 21207}, {1581, 4164}, {1638, 2291}, {1643, 37131}, {1646, 4607}, {1751, 43060}, {1783, 3942}, {1790, 2501}, {1795, 39534}, {1813, 8735}, {1826, 7254}, {1875, 37628}, {1876, 23696}, {1897, 3937}, {1911, 3766}, {1914, 4444}, {1924, 6385}, {1929, 9508}, {1967, 14296}, {1973, 15413}, {1977, 1978}, {2087, 3257}, {2109, 25381}, {2149, 40166}, {2156, 16757}, {2160, 14838}, {2161, 3960}, {2162, 3835}, {2163, 4777}, {2164, 21188}, {2183, 2401}, {2191, 3309}, {2194, 4077}, {2195, 43042}, {2203, 14208}, {2207, 30805}, {2208, 17896}, {2214, 14349}, {2215, 23882}, {2217, 21189}, {2218, 23800}, {2221, 6590}, {2226, 6544}, {2248, 21196}, {2258, 43067}, {2276, 4817}, {2279, 4762}, {2287, 7216}, {2296, 2978}, {2297, 8712}, {2299, 17094}, {2308, 4608}, {2311, 7212}, {2316, 30725}, {2319, 43051}, {2333, 15419}, {2334, 4778}, {2340, 43930}, {2348, 37626}, {2349, 14399}, {2350, 17494}, {2353, 21178}, {2354, 15420}, {2364, 43052}, {2382, 36848}, {2384, 14475}, {2395, 17209}, {2426, 15634}, {2432, 34050}, {2433, 18653}, {2486, 43076}, {2488, 21453}, {2489, 17206}, {2504, 9085}, {2509, 40188}, {2516, 36603}, {2526, 39958}, {2611, 13486}, {2616, 18180}, {2623, 17167}, {2718, 24457}, {2720, 35015}, {2786, 17962}, {2787, 17954}, {2832, 34893}, {2973, 32656}, {2983, 29162}, {2985, 23751}, {2998, 23572}, {3011, 35365}, {3022, 4626}, {3049, 44129}, {3119, 4617}, {3124, 4610}, {3224, 21191}, {3226, 6373}, {3227, 3768}, {3249, 31625}, {3250, 14621}, {3270, 36118}, {3285, 4049}, {3310, 34234}, {3433, 21185}, {3435, 21186}, {3437, 21187}, {3444, 21192}, {3445, 3667}, {3446, 21201}, {3447, 21203}, {3449, 21118}, {3450, 21119}, {3451, 21120}, {3453, 21121}, {3455, 21205}, {3500, 21348}, {3668, 21789}, {3675, 36086}, {3762, 9456}, {3778, 7255}, {3798, 8770}, {3801, 38813}, {3803, 23051}, {3805, 40763}, {3862, 23597}, {3880, 37627}, {3912, 43929}, {3954, 39179}, {4040, 13476}, {4062, 43926}, {4063, 39798}, {4081, 6614}, {4128, 4594}, {4132, 39949}, {4142, 34250}, {4160, 34916}, {4162, 19604}, {4163, 7023}, {4303, 14775}, {4306, 23289}, {4373, 8643}, {4379, 30650}, {4382, 30651}, {4394, 8056}, {4401, 7241}, {4440, 9262}, {4449, 9309}, {4453, 6187}, {4455, 18827}, {4458, 8852}, {4459, 29055}, {4462, 38266}, {4467, 6186}, {4481, 40747}, {4491, 39697}, {4498, 39956}, {4521, 40151}, {4534, 38828}, {4551, 18191}, {4557, 17205}, {4559, 17197}, {4561, 42067}, {4565, 21044}, {4584, 39786}, {4598, 6377}, {4600, 8034}, {4603, 16592}, {4618, 42084}, {4637, 36197}, {4638, 35092}, {4707, 34079}, {4775, 39704}, {4784, 30571}, {4786, 21448}, {4790, 25430}, {4791, 28607}, {4813, 25417}, {4823, 34819}, {4834, 30598}, {4885, 9315}, {4932, 39967}, {4943, 16079}, {4957, 34073}, {4960, 28625}, {4978, 28615}, {4983, 40438}, {5009, 35352}, {5029, 6650}, {5209, 18002}, {5317, 24018}, {5331, 8672}, {5620, 42741}, {6005, 10013}, {6006, 41436}, {6149, 43082}, {6336, 22086}, {6372, 40433}, {6381, 23349}, {6384, 8640}, {6549, 23344}, {6550, 9268}, {6551, 24188}, {6586, 14377}, {6588, 42467}, {6589, 13478}, {6610, 23893}, {6629, 9178}, {6730, 7370}, {7035, 8027}, {7077, 43041}, {7087, 20517}, {7121, 20906}, {7169, 21174}, {7234, 32010}, {7260, 21755}, {7316, 14432}, {7339, 23615}, {7658, 11051}, {8050, 8054}, {8059, 38357}, {8578, 44184}, {8648, 18815}, {8656, 36588}, {8659, 36807}, {8677, 36123}, {8690, 21963}, {8713, 10579}, {8714, 34444}, {8917, 17427}, {9217, 21200}, {9259, 42555}, {9265, 21211}, {9267, 9359}, {9269, 9325}, {9292, 17215}, {9299, 18149}, {9311, 20980}, {9361, 38238}, {9505, 38348}, {9506, 27929}, {10428, 23757}, {10490, 10495}, {10492, 18888}, {10509, 10581}, {14370, 21194}, {14554, 21786}, {15378, 21133}, {15382, 20504}, {16099, 42662}, {16606, 18197}, {16695, 42027}, {16702, 23894}, {16737, 40729}, {16887, 18105}, {17216, 32713}, {17217, 23493}, {17222, 45677}, {17435, 36146}, {17731, 18001}, {17758, 21007}, {17780, 43922}, {18018, 21122}, {18101, 46153}, {18359, 21758}, {18771, 21105}, {18772, 21106}, {18830, 38986}, {20295, 40148}, {20516, 34183}, {20518, 41528}, {20908, 34077}, {20974, 43190}, {21003, 39714}, {21110, 38826}, {21113, 42346}, {21138, 34071}, {21173, 34434}, {21175, 34436}, {21176, 34437}, {21179, 34441}, {21180, 34442}, {21183, 34446}, {21190, 34427}, {21202, 34179}, {21206, 36615}, {21208, 40519}, {21385, 39981}, {21763, 42328}, {21828, 24624}, {21832, 37128}, {22084, 26705}, {22108, 34578}, {22350, 43933}, {23100, 23990}, {23707, 30691}, {23723, 34429}, {23729, 38825}, {23807, 34248}, {23845, 40451}, {23989, 32739}, {24027, 42455}, {25426, 28840}, {26932, 32674}, {26933, 32691}, {28209, 41434}, {29198, 39972}, {29226, 36598}, {30723, 34820}, {30724, 33635}, {32039, 40610}, {32641, 42754}, {32714, 34591}, {34858, 36038}, {35014, 36110}, {36037, 42753}, {40397, 40628}, {40409, 40627}, {40738, 45882}, {41799, 45877}, {42290, 45755}

• X₆₄₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 668}, {2, 1978}, {3, 4561}, {6, 190}, {7, 4572}, {9, 646}, {10, 27808}, {11, 35519}, {19, 6335}, {21, 7257}, {25, 1897}, {27, 6331}, {28, 811}, {31, 100}, {32, 101}, {34, 18026}, {37, 4033}, {39, 4568}, {41, 644}, {42, 3952}, {43, 36863}, {44, 24004}, {48, 1332}, {55, 3699}, {56, 664}, {57, 4554}, {58, 99}, {75, 6386}, {81, 799}, {86, 670}, {87, 18830}, {100, 7035}, {101, 1016}, {106, 4555}, {109, 4998}, {110, 4600}, {163, 4567}, {172, 18047}, {184, 1331}, {190, 31625}, {212, 4571}, {213, 1018}, {220, 6558}, {238, 874}, {239, 27853}, {244, 693}, {269, 4569}, {274, 4602}, {284, 645}, {291, 4583}, {292, 4562}, {310, 4609}, {351, 4062}, {386, 33948}, {512, 10}, {513, 75}, {514, 76}, {521, 3718}, {522, 3596}, {523, 313}, {525, 40071}, {560, 692}, {593, 4610}, {603, 6516}, {604, 651}, {608, 653}, {647, 306}, {650, 312}, {652, 345}, {654, 32851}, {656, 20336}, {657, 346}, {659, 350}, {661, 321}, {662, 4601}, {663, 8}, {665, 3912}, {667, 1}, {669, 42}, {672, 42720}, {673, 36803}, {676, 35517}, {688, 21035}, {692, 765}, {693, 561}, {727, 8709}, {738, 36838}, {739, 4607}, {741, 4589}, {757, 4623}, {764, 1111}, {788, 984}, {798, 37}, {810, 72}, {812, 1921}, {822, 3998}, {830, 33941}, {834, 5224}, {869, 3799}, {870, 46132}, {872, 40521}, {875, 291}, {876, 334}, {884, 14942}, {890, 899}, {891, 6381}, {893, 27805}, {896, 42721}, {899, 41314}, {900, 3264}, {902, 17780}, {904, 3903}, {905, 304}, {909, 13136}, {910, 42719}, {923, 5380}, {926, 3717}, {984, 4505}, {985, 789}, {1014, 4625}, {1015, 514}, {1019, 274}, {1022, 20568}, {1024, 36796}, {1027, 2481}, {1042, 4566}, {1084, 4079}, {1086, 3261}, {1106, 934}, {1111, 40495}, {1126, 6540}, {1171, 4632}, {1178, 4594}, {1201, 21272}, {1252, 6632}, {1253, 4578}, {1326, 17934}, {1333, 662}, {1334, 30730}, {1357, 3676}, {1395, 108}, {1397, 109}, {1398, 36118}, {1400, 4552}, {1402, 4551}, {1407, 658}, {1408, 1414}, {1411, 35174}, {1412, 4573}, {1415, 4564}, {1416, 927}, {1435, 13149}, {1436, 44327}, {1437, 4592}, {1438, 666}, {1458, 883}, {1459, 69}, {1460, 14594}, {1461, 1275}, {1462, 34085}, {1472, 1310}, {1474, 648}, {1491, 33931}, {1500, 4103}, {1501, 32739}, {1576, 4570}, {1577, 27801}, {1635, 4358}, {1646, 4728}, {1734, 33932}, {1755, 42717}, {1769, 3262}, {1790, 4563}, {1911, 660}, {1912, 12782}, {1914, 3570}, {1918, 4557}, {1919, 6}, {1922, 813}, {1924, 213}, {1946, 78}, {1960, 519}, {1964, 4553}, {1973, 1783}, {1974, 8750}, {1979, 9362}, {1980, 31}, {2084, 3954}, {2087, 3762}, {2149, 31615}, {2150, 4612}, {2160, 15455}, {2161, 36804}, {2162, 4598}, {2163, 4597}, {2170, 4391}, {2173, 42716}, {2175, 3939}, {2176, 4595}, {2177, 4767}, {2182, 42718}, {2183, 2397}, {2185, 4631}, {2194, 643}, {2195, 36802}, {2200, 4574}, {2203, 162}, {2206, 110}, {2208, 13138}, {2210, 3573}, {2214, 37218}, {2221, 37215}, {2223, 1026}, {2225, 42723}, {2242, 4482}, {2246, 42722}, {2251, 1023}, {2254, 3263}, {2275, 33946}, {2276, 3807}, {2279, 32041}, {2287, 7258}, {2299, 36797}, {2300, 3882}, {2308, 4427}, {2310, 4397}, {2316, 4582}, {2328, 7256}, {2347, 25268}, {2423, 34234}, {2424, 18025}, {2440, 31638}, {2441, 31227}, {2483, 17289}, {2484, 2345}, {2485, 4150}, {2488, 4847}, {2489, 1826}, {2492, 21094}, {2516, 20942}, {2520, 17860}, {2522, 19799}, {2530, 1930}, {2533, 1237}, {2605, 319}, {2624, 42701}, {2642, 42713}, {2643, 4036}, {2969, 46107}, {2978, 31330}, {3005, 15523}, {3009, 23354}, {3022, 4163}, {3049, 71}, {3051, 46148}, {3052, 43290}, {3063, 9}, {3064, 7017}, {3120, 850}, {3121, 661}, {3122, 523}, {3123, 20906}, {3124, 4024}, {3125, 1577}, {3221, 21080}, {3230, 23891}, {3248, 513}, {3249, 1015}, {3250, 3661}, {3251, 4738}, {3261, 1502}, {3271, 522}, {3287, 17787}, {3310, 908}, {3572, 335}, {3669, 85}, {3676, 6063}, {3700, 30713}, {3709, 2321}, {3716, 4087}, {3733, 86}, {3737, 314}, {3752, 21580}, {3766, 18891}, {3768, 536}, {3777, 33930}, {3803, 39731}, {3835, 6382}, {3837, 35538}, {3900, 341}, {3937, 4025}, {3939, 4076}, {3942, 15413}, {3960, 20924}, {4010, 35544}, {4014, 20907}, {4017, 1441}, {4024, 28654}, {4025, 305}, {4040, 17143}, {4041, 3701}, {4057, 4360}, {4063, 18140}, {4079, 594}, {4083, 6376}, {4091, 3926}, {4105, 5423}, {4107, 3978}, {4128, 2533}, {4130, 30693}, {4142, 18835}, {4162, 44720}, {4164, 1966}, {4258, 30728}, {4367, 1909}, {4369, 1920}, {4378, 3761}, {4391, 28659}, {4394, 18743}, {4435, 3975}, {4444, 18895}, {4453, 40075}, {4455, 740}, {4466, 3267}, {4491, 17160}, {4498, 18135}, {4516, 4086}, {4521, 44723}, {4524, 4082}, {4556, 4590}, {4560, 28660}, {4565, 4620}, {4581, 1240}, {4586, 5388}, {4610, 34537}, {4636, 6064}, {4705, 1089}, {4724, 4441}, {4728, 35543}, {4730, 3992}, {4750, 3266}, {4762, 21615}, {4770, 4125}, {4775, 3679}, {4782, 30963}, {4785, 10009}, {4786, 11059}, {4790, 19804}, {4802, 30596}, {4809, 35548}, {4813, 28605}, {4832, 5257}, {4834, 1698}, {4874, 4485}, {4879, 25280}, {4893, 4671}, {4895, 4723}, {4905, 33933}, {4977, 1269}, {4979, 4359}, {4983, 4647}, {4988, 1230}, {5027, 4039}, {5029, 6542}, {5040, 17763}, {5299, 33951}, {5317, 823}, {5532, 23104}, {6004, 33937}, {6085, 1266}, {6129, 322}, {6139, 6745}, {6161, 4986}, {6186, 6742}, {6363, 3663}, {6371, 4357}, {6372, 20888}, {6373, 726}, {6377, 3835}, {6544, 36791}, {6545, 23989}, {6551, 42372}, {6586, 17233}, {6588, 20928}, {6589, 4417}, {6591, 92}, {6615, 20895}, {6729, 556}, {7004, 35518}, {7023, 4626}, {7032, 3888}, {7077, 36801}, {7113, 4585}, {7117, 6332}, {7121, 932}, {7122, 4579}, {7178, 349}, {7180, 226}, {7192, 310}, {7199, 6385}, {7202, 18160}, {7216, 1446}, {7234, 1215}, {7250, 3668}, {7252, 333}, {7254, 17206}, {7335, 6517}, {7337, 36127}, {7366, 4617}, {7649, 264}, {8027, 244}, {8034, 3120}, {8042, 16727}, {8054, 20295}, {8574, 21092}, {8578, 150}, {8630, 869}, {8631, 17033}, {8632, 239}, {8633, 4362}, {8634, 30167}, {8635, 3920}, {8636, 976}, {8637, 386}, {8638, 2340}, {8640, 43}, {8641, 200}, {8642, 3870}, {8643, 145}, {8645, 3935}, {8646, 612}, {8648, 4511}, {8649, 6633}, {8650, 7292}, {8651, 4028}, {8653, 4061}, {8654, 3938}, {8655, 17018}, {8656, 3241}, {8657, 16834}, {8658, 41140}, {8659, 3008}, {8660, 1149}, {8661, 1647}, {8662, 2999}, {8663, 8013}, {8678, 4385}, {8735, 46110}, {8747, 6528}, {8750, 15742}, {9002, 4389}, {9008, 14620}, {9247, 906}, {9259, 6631}, {9262, 6630}, {9268, 6635}, {9297, 2228}, {9299, 9361}, {9313, 4429}, {9315, 30610}, {9359, 9296}, {9404, 42033}, {9426, 1918}, {9454, 2284}, {9456, 3257}, {9459, 23344}, {9494, 41267}, {9508, 20947}, {10566, 308}, {11125, 3260}, {14296, 1926}, {14349, 33935}, {14377, 31624}, {14399, 14206}, {14404, 3994}, {14407, 3943}, {14413, 30806}, {14419, 14210}, {14428, 4156}, {14575, 32656}, {14598, 34067}, {14621, 37133}, {14623, 9065}, {14838, 33939}, {14936, 3239}, {14991, 22011}, {15413, 40364}, {16502, 3732}, {16584, 7239}, {16612, 2064}, {16695, 33296}, {16702, 24039}, {16726, 7199}, {16732, 20948}, {16757, 20641}, {16892, 8024}, {16947, 4565}, {17186, 4611}, {17187, 4576}, {17209, 2396}, {17411, 34528}, {17424, 30827}, {17458, 30473}, {17477, 4106}, {17494, 18152}, {17924, 1969}, {17925, 44129}, {17954, 35147}, {17962, 35148}, {17990, 6541}, {18001, 11599}, {18105, 18082}, {18108, 3112}, {18155, 40072}, {18191, 18155}, {18196, 34022}, {18197, 31008}, {18200, 8033}, {18210, 14208}, {18268, 4584}, {18344, 318}, {20228, 21362}, {20229, 35341}, {20295, 40087}, {20517, 40365}, {20954, 40088}, {20970, 4115}, {20974, 25259}, {20975, 4064}, {20979, 192}, {20980, 3729}, {20981, 894}, {20982, 7265}, {20983, 32925}, {21003, 32922}, {21005, 32926}, {21007, 17277}, {21102, 311}, {21103, 1232}, {21104, 1233}, {21108, 1235}, {21109, 1236}, {21110, 44166}, {21122, 22}, {21123, 141}, {21124, 1228}, {21126, 42554}, {21127, 1229}, {21131, 338}, {21132, 34387}, {21134, 339}, {21143, 1086}, {21172, 14615}, {21178, 40073}, {21191, 6374}, {21194, 40035}, {21205, 40074}, {21207, 44173}, {21297, 40089}, {21348, 17786}, {21385, 18145}, {21747, 4781}, {21758, 3218}, {21761, 7283}, {21762, 20979}, {21763, 25264}, {21789, 1043}, {21814, 35309}, {21828, 3936}, {21832, 3948}, {21835, 21834}, {21837, 4006}, {22086, 3977}, {22096, 1459}, {22108, 17264}, {22260, 21043}, {22383, 63}, {22386, 22090}, {23189, 332}, {23220, 22350}, {23224, 326}, {23225, 1818}, {23345, 903}, {23349, 37129}, {23355, 3226}, {23472, 17350}, {23503, 21877}, {23506, 25287}, {23524, 4499}, {23572, 194}, {23649, 25272}, {23655, 32937}, {23751, 3782}, {23752, 1234}, {23807, 18837}, {23892, 3227}, {23979, 4619}, {24002, 20567}, {24533, 41318}, {25142, 8026}, {27644, 36860}, {27846, 3766}, {27929, 18035}, {28607, 4604}, {28615, 37212}, {29226, 20943}, {31947, 20932}, {32660, 44717}, {32665, 5376}, {32666, 5377}, {32674, 46102}, {32676, 5379}, {32702, 39294}, {32719, 9268}, {32735, 39293}, {32739, 1252}, {33917, 19945}, {34067, 5378}, {34069, 5384}, {34071, 5383}, {34073, 5385}, {34075, 5381}, {34080, 5382}, {34591, 15416}, {34819, 37211}, {34858, 36037}, {34916, 35181}, {34948, 44179}, {35519, 40363}, {36054, 3719}, {37128, 4639}, {37129, 889}, {38238, 18149}, {38266, 27834}, {38346, 17494}, {38367, 17475}, {38389, 17894}, {38986, 4083}, {39201, 3682}, {40148, 8050}, {40432, 7260}, {40495, 1928}, {40610, 23886}, {40627, 21024}, {40746, 4586}, {41405, 6634}, {41935, 4638}, {42067, 7649}, {42078, 15632}, {42336, 1122}, {42462, 23978}, {42649, 27529}, {42653, 21081}, {42661, 20653}, {42662, 16086}, {42753, 36038}, {43041, 18033}, {43049, 21609}, {43051, 30545}, {43060, 18134}, {43922, 6548}, {43923, 273}, {43924, 7}, {43925, 27}, {43928, 31002}, {43929, 673}, {43931, 6384}, {43932, 1088}, {45755, 28809}, {45902, 17760}, {46107, 18022}, {46288, 4628}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3835, 30835}, {2, 20295, 3835}, {2, 26798, 27138}, {2, 26853, 20295}, {2, 27013, 31286}, {2, 31286, 31207}, {75, 20952, 20909}, {101, 41405, 1252}, {190, 9362, 7035}, {650, 652, 657}, {650, 654, 652}, {650, 661, 4893}, {650, 4394, 1635}, {650, 4979, 4813}, {650, 14300, 21127}, {661, 1635, 650}, {663, 667, 8643}, {663, 8656, 1960}, {665, 8632, 5029}, {665, 8658, 1960}, {665, 8659, 8632}, {667, 669, 8655}, {667, 1960, 8656}, {667, 3250, 5029}, {667, 4775, 1960}, {667, 8640, 669}, {667, 8641, 8642}, {667, 8646, 8635}, {669, 890, 8640}, {669, 2978, 663}, {693, 4369, 4379}, {798, 2484, 657}, {901, 1252, 41405}, {1019, 18197, 7192}, {1021, 4765, 45755}, {1635, 4790, 4813}, {1635, 4979, 661}, {1919, 21123, 1459}, {1960, 4775, 663}, {1960, 8656, 8643}, {1960, 8658, 8632}, {1960, 8659, 8658}, {1977, 6377, 8054}, {2488, 6139, 8641}, {2488, 8641, 663}, {2490, 14321, 1639}, {2590, 2591, 1635}, {3063, 20981, 23472}, {3250, 8632, 663}, {3733, 21007, 1919}, {3798, 4025, 4750}, {3835, 20295, 31147}, {3835, 27013, 31207}, {3835, 31286, 2}, {3835, 45313, 31286}, {3937, 14936, 20974}, {4025, 4786, 3798}, {4106, 4885, 4728}, {4369, 4380, 4382}, {4379, 4382, 693}, {4394, 4790, 661}, {4394, 4979, 4893}, {4468, 11068, 6546}, {4498, 45755, 21832}, {4607, 7035, 9362}, {4728, 24924, 4885}, {4750, 16892, 4025}, {4763, 25666, 31209}, {4776, 31209, 25666}, {4782, 4784, 4724}, {4813, 4893, 661}, {4817, 24623, 4379}, {5638, 5639, 5029}, {6589, 43060, 21828}, {7252, 43060, 1459}, {8641, 8642, 8645}, {14351, 45745, 14435}, {20295, 27013, 2}, {20295, 27138, 26798}, {20295, 27673, 28398}, {20295, 31286, 30835}, {20295, 45313, 31207}, {22044, 24089, 4024}, {25924, 25955, 2}, {26798, 27138, 3835}, {26853, 27013, 3835}, {26853, 31286, 31147}, {26853, 45313, 30835}, {27293, 30023, 30835}, {27486, 45746, 21196}, {27673, 29487, 30835}, {27673, 29807, 31147}, {29487, 29545, 20295}, {29545, 31286, 28398}, {30835, 31147, 3835}, {30835, 31207, 2}, {31147, 31207, 30835}, {31286, 45313, 27013}, {39665, 39666, 42662}

Notes

X₆₄₉ is the perspector of triangle ABC and the tangential triangle of the conic {A, B, C, X₁, X₆}}. (Randy Hutson, 9/23/2011)

A'B'C' be the excentral triangle and H the hyperbola {A',B',C',X₁,X(9}}, so that H is the Jerabek hyperbola of A'B'C'. Let TT be the tangential triangle, wrt A'B'C', of H. Then T and TT are perspective, and their perspector is X₆₄₉. (Randy Hutson, December 26, 2015)

The circle that has center X₆₄₉ and passes through the isodynamic points, X₁₅ and X₁₆, also passes through the isodynamic points of the excentral triangle (X1276) and X₁₂₇₇, as well as X₃₂₆₂₂ and X₃₂₆₂₃. This circle is here named the Gheorghe circle, in recognition of the work of Liliana Gheorghe. The Gheorghe circle is coaxial with the three Apollonius circles and also the Moses radical circle, which has center X₆₄₇ and passes through X₆₁₁₆ and X₆₁₁₇, these being the isodynamic points of the orthic triangle. The centers of the Gheorghe circle, Moses radical circle, and the three Apollonius circles all lie on the Lemoine axis, X₁₈₇X₂₃₇; for related circles, see X₁₅. (Dan Reznik, April 16, 2021)

See Gheorghe Circle and Gheorghe Circle (zoomed in). See also Gheorghe Circle in motion.

The Moses radical circle is the Gheorghe circle of the orthic triangle. (Dan Reznik, April 20, 2021)