X₆₆₀

TRILINEAR POLE OF LINE X₁X_39

X₆₆₀ = trilinear pole of line {1, 39}

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

X₆₆₀ (TRILINEAR POLE OF LINE X₁X_39) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {44, 292}, {88, 291}

• Related Centers: X₆₅₉ (CROSSDIFFERENCE OF X₁ AND X₃₉), X₄₄ (-LINE CONJUGATE OF X₁), X₂₉₂ (-HIRST INVERSE OF X₂₉₁), X₈₈ (ISOGONAL CONJUGATE OF X₄₄), X₂₉₁ (2nd SHARYGIN POINT)

Trilinears: \(1/[(a^{2} - bc)(b - c)] : :\)

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

Search Value (6, 9, 13): \(0.356797529576\)

▸ Linear Combinations1 combination

• X₆₆₀ = \(3\) X₂ - \(4\) X₄₀₅₃₈

X₆₆₀ lies on the curve CC₉.

▸ Incident Central Lines89 lines

{2, 38989}, {9, 3252}, {42, 40796}, {43, 40155}, {44, 292}, {45, 24482}, {63, 39344}, {67, 37221}, {88, 291}, {100, 649}, {101, 1492}, {109, 8684}, {110, 4599}, {162, 35325}, {190, 513}, {238, 1911}, {239, 335}, {295, 4518}, {320, 334}, {337, 37214}, {512, 1016}, {651, 6163}, {653, 43923}, {658, 43932}, {662, 765}, {663, 9266}, {668, 37133}, {670, 37204}, {672, 36906}, {694, 43763}, {799, 3952}, {805, 8707}, {823, 46151}, {874, 23354}, {875, 4607}, {876, 3257}, {883, 34085}, {891, 4555}, {897, 46154}, {898, 1960}, {926, 36802}, {932, 40499}, {984, 25800}, {1026, 3572}, {1155, 14200}, {1156, 4876}, {1757, 2664}, {1821, 4645}, {2144, 27920}, {2235, 3862}, {2254, 36238}, {2284, 3573}, {2349, 46147}, {2580, 46166}, {2581, 46167}, {3570, 37207}, {3681, 24586}, {3699, 4598}, {3766, 36803}, {3903, 4613}, {4083, 6631}, {4436, 37212}, {4444, 37143}, {4551, 37137}, {4715, 7245}, {4767, 37209}, {5220, 22116}, {6005, 32094}, {6372, 32028}, {6633, 29350}, {7233, 43762}, {8701, 36066}, {9458, 37222}, {14621, 19586}, {17763, 24624}, {17944, 37140}, {18278, 43761}, {20072, 30669}, {20683, 40098}, {23691, 36100}, {27495, 40740}, {29936, 36256}, {32680, 46155}, {36085, 36827}, {37132, 46156}, {37134, 46161}, {37218, 43927}, {37220, 46165}, {40432, 40936}

▸ Functional Relations16 relations

• X₆₆₀ = reflection of X₃₈₉₈₉ in X₄₀₅₃₈

• X₆₆₀ = isogonal conjugate of X₆₅₉

• X₆₆₀ = isotomic conjugate of X₃₇₆₆

• X₆₆₀ = anticomplement of X₃₈₉₈₉

• X₆₆₀ = X_i-anticomplementary conjugate of X_j for these (i,j): {813, 39353}, {919, 39362}, {5377, 17794}, {5378, 20344}

• X₆₆₀ = X_i-Ceva conjugate of X_j for these (i,j): {4584, 813}, {4589, 4562}, {5378, 291}, {36081, 34067}

• X₆₆₀ = X_i-cross conjugate of X_j for these (i,j): {291, 5378}, {511, 59}, {659, 1}, {665, 2}, {875, 292}, {876, 291}, {926, 3252}, {1026, 100}, {1027, 1280}, {1757, 765}, {2238, 1016}, {2664, 7035}, {3509, 4564}, {3570, 3903}, {3572, 37128}, {4447, 4998}, {6165, 513}, {6211, 7012}, {6373, 6}, {17990, 37}, {18788, 7045}, {20683, 1252}, {21391, 82}, {21830, 31625}, {24462, 75}

• X₆₆₀ = cevapoint of X_i and X_j for these (i,j): {1, 659}, {6, 21003}, {9, 926}, {141, 918}, {291, 876}, {292, 875}, {512, 2238}, {513, 518}, {514, 20335}, {649, 3009}, {665, 40730}, {2284, 4557}, {3570, 18047}, {3952, 23354}, {4705, 20708}

• X₆₆₀ = crosspoint of X_i and X_j for these (i,j): {666, 8709}, {4584, 4589}, {37134, 41209}

• X₆₆₀ = crosssum of X_i and X_j for these (i,j): {665, 6373}, {4455, 21832}

• X₆₆₀ = trilinear pole of line {1, 39}

• X₆₆₀ = crossdifference of every pair of points on line {27846, 38989}

• X₆₆₀ = X_i-isoconjugate of X_j for these (i,j): {1, 659}, {2, 8632}, {4, 22384}, {6, 812}, {31, 3766}, {55, 43041}, {56, 3716}, {57, 4435}, {58, 4010}, {81, 21832}, {86, 4455}, {100, 27846}, {101, 27918}, {106, 4448}, {109, 4124}, {238, 513}, {239, 649}, {242, 1459}, {244, 3573}, {256, 4164}, {284, 7212}, {292, 4375}, {350, 667}, {512, 33295}, {514, 1914}, {520, 34856}, {522, 1428}, {523, 5009}, {647, 31905}, {650, 1429}, {662, 39786}, {663, 1447}, {665, 6654}, {693, 2210}, {739, 14433}, {740, 3733}, {757, 4155}, {798, 30940}, {804, 1178}, {813, 35119}, {874, 3248}, {875, 39044}, {876, 8300}, {884, 39775}, {893, 4107}, {904, 14296}, {905, 2201}, {983, 3808}, {985, 30665}, {1015, 3570}, {1019, 2238}, {1024, 34253}, {1027, 8299}, {1284, 3737}, {1407, 4148}, {1474, 24459}, {1874, 23189}, {1911, 27855}, {1919, 1921}, {1929, 38348}, {1960, 27922}, {1977, 27853}, {1980, 18891}, {2163, 4800}, {2276, 23597}, {2334, 4830}, {3063, 10030}, {3253, 6373}, {3261, 14599}, {3572, 4366}, {3669, 3684}, {3685, 43924}, {3747, 7192}, {4083, 34252}, {4367, 18786}, {4432, 23345}, {4433, 7203}, {4465, 23892}, {4486, 40746}, {4508, 30650}, {4817, 16514}, {5027, 32010}, {5029, 40725}, {6591, 20769}, {7193, 7649}, {7199, 41333}, {7252, 16609}, {9262, 27912}, {9508, 40767}, {17493, 20981}, {17755, 43929}, {17793, 23355}, {17962, 27929}, {18264, 20518}, {18892, 40495}, {20979, 39914}, {30654, 40738}, {32020, 38367}, {36086, 38989}

• X₆₆₀ = barycentric product X_i*X_j for these {i,j}: {1, 4562}, {6, 4583}, {10, 4584}, {37, 4589}, {42, 4639}, {57, 36801}, {75, 813}, {76, 34067}, {100, 335}, {101, 334}, {141, 36081}, {181, 36806}, {190, 291}, {292, 668}, {295, 6335}, {337, 1783}, {514, 5378}, {594, 36066}, {644, 7233}, {651, 4518}, {662, 43534}, {664, 4876}, {666, 22116}, {692, 18895}, {741, 4033}, {765, 4444}, {789, 3862}, {805, 3963}, {875, 31625}, {876, 1016}, {932, 40848}, {984, 37207}, {1018, 18827}, {1025, 33676}, {1215, 37134}, {1581, 18047}, {1911, 1978}, {1916, 4579}, {1922, 6386}, {2276, 41072}, {2295, 18829}, {3570, 30663}, {3572, 7035}, {3573, 40098}, {3661, 30664}, {3662, 8684}, {3864, 4586}, {3903, 30669}, {3952, 37128}, {4551, 36800}, {4554, 7077}, {4557, 40017}, {4567, 35352}, {4598, 41531}, {5384, 23596}, {16587, 41209}, {18268, 27808}, {18787, 27805}, {32739, 44172}, {35148, 40794}, {36086, 40217}, {36803, 40730}

• X₆₆₀ = barycentric quotient X_i/X_j for these {i,j}: {1, 812}, {2, 3766}, {6, 659}, {9, 3716}, {31, 8632}, {37, 4010}, {42, 21832}, {44, 4448}, {45, 4800}, {48, 22384}, {55, 4435}, {57, 43041}, {65, 7212}, {72, 24459}, {99, 30940}, {100, 239}, {101, 238}, {109, 1429}, {162, 31905}, {163, 5009}, {171, 4107}, {172, 4164}, {190, 350}, {200, 4148}, {213, 4455}, {238, 4375}, {239, 27855}, {291, 514}, {292, 513}, {295, 905}, {334, 3261}, {335, 693}, {337, 15413}, {512, 39786}, {513, 27918}, {644, 3685}, {646, 4087}, {649, 27846}, {650, 4124}, {651, 1447}, {659, 35119}, {662, 33295}, {664, 10030}, {665, 38989}, {668, 1921}, {692, 1914}, {741, 1019}, {750, 4508}, {765, 3570}, {805, 40432}, {813, 1}, {875, 1015}, {876, 1086}, {894, 14296}, {899, 14433}, {906, 7193}, {932, 39914}, {984, 4486}, {985, 23597}, {1016, 874}, {1018, 740}, {1023, 4432}, {1025, 39775}, {1026, 17755}, {1252, 3573}, {1331, 20769}, {1415, 1428}, {1449, 4830}, {1500, 4155}, {1757, 27929}, {1783, 242}, {1911, 649}, {1922, 667}, {1978, 18891}, {2196, 1459}, {2275, 3808}, {2276, 30665}, {2283, 34253}, {2284, 8299}, {2295, 804}, {2311, 3737}, {2427, 15507}, {2664, 27854}, {2702, 40767}, {3252, 2254}, {3257, 27922}, {3570, 39044}, {3572, 244}, {3573, 4366}, {3699, 3975}, {3799, 3797}, {3862, 1491}, {3864, 824}, {3888, 33891}, {3903, 17493}, {3939, 3684}, {3952, 3948}, {3963, 14295}, {4033, 35544}, {4069, 3985}, {4169, 4783}, {4444, 1111}, {4482, 4495}, {4518, 4391}, {4551, 16609}, {4554, 18033}, {4557, 2238}, {4559, 1284}, {4562, 75}, {4579, 385}, {4583, 76}, {4584, 86}, {4589, 274}, {4639, 310}, {4645, 27951}, {4752, 4693}, {4876, 522}, {5378, 190}, {6163, 27912}, {6335, 40717}, {6386, 44169}, {7035, 27853}, {7077, 650}, {7233, 24002}, {7245, 4411}, {8684, 17743}, {8750, 2201}, {14598, 1919}, {16777, 4810}, {17735, 38348}, {18047, 1966}, {18265, 3063}, {18268, 3733}, {18787, 4369}, {18827, 7199}, {18895, 40495}, {18897, 1980}, {21003, 40623}, {21143, 24193}, {21801, 42767}, {21859, 7235}, {22116, 918}, {23343, 4465}, {24019, 34856}, {30663, 4444}, {30664, 14621}, {30669, 4374}, {30671, 4475}, {32739, 2210}, {34067, 6}, {34071, 34252}, {35342, 4974}, {35352, 16732}, {36066, 1509}, {36081, 83}, {36086, 6654}, {36800, 18155}, {36801, 312}, {36806, 18021}, {37128, 7192}, {37134, 32010}, {37135, 40725}, {37207, 870}, {37593, 4839}, {39026, 27943}, {40093, 20949}, {40095, 21606}, {40521, 4037}, {40730, 665}, {40794, 2786}, {40848, 20906}, {41531, 3835}, {43262, 4406}, {43534, 1577}

• X₆₆₀ = {X₃₈₉₈₉,X₄₀₅₃₈}-harmonic conjugate of X₂