X₆₅₉

CROSSDIFFERENCE OF X₁ AND X₃₉

X₆₅₉ = X₉₈-Ceva conjugate of X₁₁

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

X₆₅₉ (CROSSDIFFERENCE OF X₁ AND X₃₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {100, 190}, {23, 385}

• Related Centers: X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₉₀ (YFF PARABOLIC POINT), X₂₃ (FAR-OUT POINT), X₃₈₅ (-HIRST INVERSE OF X₆)

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

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

Search Value (6, 9, 13): \(-6.519761393620\)

▸ Linear Combinations1 combination

• X₆₅₉ = \(2\) X₁ - \(3\) X₂₅₅₆₉, \(6\) X₂ - \(5\) X₃₀₇₉₅, \(5\) X₂ - \(4\) X₄₅₃₄₀, \(3\) X₆₅₀ - X₂₅₂₆, \(3\) X₆₆₇ - X₄₃₇₈, \(2\) X₆₇₆ - \(3\) X₂₆₂₇₅, X₇₆₄ - \(3\) X₁₄₄₁₉, \(3\) X₁₄₉₁ - \(2\) X₂₅₂₆, \(3\) X₁₆₃₅ - X₂₂₅₄, \(3\) X₁₆₃₅ - \(2\) X₉₅₀₈, \(2\) X₁₇₆₉ - \(3\) X₂₈₃₉₆, \(4\) X₁₉₆₀ - X₂₁₃₄₃, \(2\) X₁₉₆₀ + X₂₁₃₈₅, \(4\) X₁₉₆₀ - \(3\) X₂₅₅₆₉, \(2\) X₃₇₁₆ - \(3\) X₄₄₄₈, \(4\) X₃₇₁₆ - \(3\) X₄₈₀₀, \(4\) X₃₇₁₆ - X₄₈₁₀, X₃₇₇₇ - \(4\) X₆₀₅₀, \(2\) X₃₈₀₃ + X₄₄₉₀, \(4\) X₃₈₃₇ - \(5\) X₃₀₇₉₅, X₃₈₃₇ - \(3\) X₄₅₃₁₄, \(5\) X₃₈₃₇ - \(6\) X₄₅₃₄₀, \(2\) X₃₉₆₀ - \(3\) X₁₄₄₁₉, X₄₀₁₀ - \(3\) X₄₄₄₈, \(2\) X₄₀₁₀ - \(3\) X₄₈₀₀, X₄₀₁₀ + \(2\) X₄₈₃₀, X₄₀₈₈ - \(3\) X₆₅₄₆, \(3\) X₄₃₆₇ - \(2\) X₄₃₇₈, X₄₃₆₇ - \(4\) X₄₄₀₁, X₄₃₇₈ - \(6\) X₄₄₀₁, \(6\) X₄₄₄₈ - X₄₈₁₀, \(3\) X₄₄₄₈ + \(2\) X₄₈₃₀, X₄₄₄₉ - \(3\) X₈₆₄₃, \(2\) X₄₄₅₈ - \(3\) X₄₈₀₉, \(2\) X₄₄₉₈ + X₄₈₇₉, X₄₇₂₄ + \(2\) X₄₇₈₂, \(2\) X₄₇₂₄ + X₄₇₈₄, \(3\) X₄₇₆₃ - \(2\) X₂₅₃₈₀, \(4\) X₄₇₈₂ - X₄₇₈₄, \(3\) X₄₈₀₀ - X₄₈₁₀, \(3\) X₄₈₀₀ + \(4\) X₄₈₃₀, \(3\) X₄₈₀₉ - \(4\) X₁₃₂₄₆, X₄₈₁₀ + \(4\) X₄₈₃₀, \(2\) X₁₈₀₀₄ - \(3\) X₃₀₅₆₅, X₂₁₃₄₃ + \(2\) X₂₁₃₈₅, X₂₁₃₄₃ - \(3\) X₂₅₅₆₉, \(2\) X₂₁₃₈₅ + \(3\) X₂₅₅₆₉, X₂₃₇₇₀ - \(3\) X₂₆₂₇₅, \(4\) X₂₅₃₈₀ - \(3\) X₃₆₈₄₈, \(5\) X₃₀₇₉₅ - \(12\) X₄₅₃₁₄, \(25\) X₃₀₇₉₅ - \(24\) X₄₅₃₄₀, \(5\) X₃₁₂₀₉ - \(3\) X₄₄₄₂₉, \(5\) X₄₅₃₁₄ - \(2\) X₄₅₃₄₀

▸ Incident Central Lines172 lines

{1, 891}, {2, 3837}, {3, 2826}, {10, 21722}, {11, 33311}, {23, 385}, {36, 6550}, {40, 2821}, {44, 513}, {56, 24097}, {75, 21439}, {88, 16505}, {100, 190}, {105, 676}, {145, 25574}, {244, 3248}, {291, 875}, {292, 665}, {512, 4040}, {514, 667}, {521, 13256}, {522, 4122}, {663, 4083}, {666, 34067}, {678, 38349}, {693, 4874}, {764, 2832}, {784, 8637}, {804, 1281}, {812, 3716}, {814, 4391}, {830, 4705}, {834, 2978}, {885, 8638}, {898, 4555}, {905, 3777}, {918, 8301}, {926, 1282}, {952, 19916}, {1001, 6009}, {1019, 6372}, {1022, 14422}, {1054, 6085}, {1280, 23834}, {1283, 6089}, {1376, 4925}, {1577, 29070}, {1734, 6004}, {1757, 6165}, {1769, 5075}, {1929, 18001}, {2517, 6133}, {2530, 14838}, {2533, 29051}, {2782, 19926}, {2785, 5592}, {2787, 3762}, {2802, 41191}, {2804, 13222}, {2808, 19921}, {2814, 38324}, {2827, 14664}, {2975, 24093}, {3004, 3733}, {3126, 19593}, {3218, 8661}, {3226, 23355}, {3667, 13252}, {3669, 23765}, {3709, 21389}, {3722, 4145}, {3737, 6371}, {3738, 13277}, {3751, 9032}, {3766, 39044}, {3803, 4490}, {3808, 4164}, {3835, 24719}, {3887, 4730}, {3900, 13165}, {4088, 6546}, {4093, 4132}, {4147, 28470}, {4155, 8298}, {4369, 21146}, {4380, 29328}, {4435, 8632}, {4449, 8643}, {4462, 29324}, {4474, 29236}, {4508, 14433}, {4560, 16695}, {4583, 8709}, {4707, 29102}, {4728, 44304}, {4750, 8650}, {4761, 29188}, {4762, 7662}, {4763, 25380}, {4765, 8662}, {4774, 29066}, {4775, 4794}, {4778, 4932}, {4785, 45673}, {4791, 29033}, {4802, 8655}, {4806, 20295}, {4824, 7234}, {4834, 6005}, {4885, 24747}, {4948, 31150}, {5027, 19580}, {5253, 24099}, {5276, 25808}, {6164, 14122}, {6363, 21173}, {6631, 9266}, {7265, 29106}, {8297, 24286}, {8424, 24141}, {8636, 29025}, {8648, 21132}, {8659, 24578}, {8660, 21222}, {8674, 38325}, {8712, 35683}, {9318, 24447}, {9451, 42341}, {10015, 29240}, {14296, 27855}, {14667, 15914}, {17930, 36066}, {17954, 18015}, {17990, 21391}, {19903, 28850}, {19915, 28915}, {20142, 28209}, {20316, 25636}, {21051, 21301}, {21053, 25638}, {21115, 28195}, {21201, 42670}, {21789, 29162}, {22379, 24126}, {23301, 26049}, {23569, 23650}, {23768, 43060}, {23815, 31288}, {23888, 30580}, {24193, 27846}, {24350, 24353}, {24488, 24499}, {24512, 25815}, {24666, 24768}, {24720, 31286}, {24721, 25381}, {25259, 29078}, {25299, 26033}, {25634, 33115}, {25686, 27293}, {26073, 26076}, {26114, 44451}, {27045, 31946}, {27167, 40086}, {27628, 28283}, {27648, 28241}, {27915, 27952}, {28373, 29274}, {28602, 31073}, {31209, 44429}, {33660, 33663}, {43051, 43924}

▸ Functional Relations24 relations

• X₆₅₉ = midpoint of X_i and X_j for these {i,j}: {1, 21385}, {100, 13266}, {649, 4724}, {663, 4498}, {3716, 4830}, {4040, 4063}, {4730, 6161}

• X₆₅₉ = reflection of X_i in X_j for these {i,j}: {1, 1960}, {2, 45314}, {3, 44805}, {649, 4782}, {667, 4401}, {693, 4874}, {764, 3960}, {876, 665}, {905, 6050}, {1022, 14422}, {1491, 650}, {2254, 9508}, {2517, 6133}, {2530, 14838}, {3777, 905}, {3801, 4142}, {4010, 3716}, {4367, 667}, {4458, 13246}, {4486, 27929}, {4728, 45666}, {4775, 4794}, {4784, 649}, {4800, 4448}, {4810, 4010}, {4879, 663}, {4948, 31150}, {20295, 4806}, {21146, 4369}, {21301, 21051}, {21343, 1}, {23765, 3669}, {23770, 676}, {23815, 31288}, {24097, 30725}, {24462, 17990}, {24719, 3835}, {24720, 31286}, {24721, 25381}, {31131, 28602}, {36848, 4763}, {38348, 8632}

• X₆₅₉ = isogonal conjugate of X₆₆₀

• X₆₅₉ = isotomic conjugate of X₄₅₈₃

• X₆₅₉ = complement of isotomic conjugate of isogonal conjugate of X₂₁₀₀₃

• X₆₅₉ = X₉₈-Ceva conjugate of X₁₁

• X₆₅₉ = crosspoint of X₁₀₀ and X₁₀₅

• X₆₅₉ = crosssum of X_i and X_j for these (i,j): (1, 659), (9, 926), (141, 918), (291, 876), (292, 875), (513, 518)

• X₆₅₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {101, 20355}, {692, 39354}, {727, 149}, {3226, 21293}, {8709, 6327}, {8851, 33650}, {18793, 3448}, {20332, 150}, {27809, 21294}, {34077, 4440}

• X₆₅₉ = X_i-complementary conjugate of X_j for these (i,j): {31, 40623}, {39714, 21252}, {39979, 116}

• X₆₅₉ = X_i-Ceva conjugate of X_j for these (i,j): {2, 40623}, {98, 11}, {99, 4366}, {100, 8299}, {190, 17475}, {238, 27846}, {242, 4124}, {513, 38348}, {660, 1}, {666, 6}, {812, 4435}, {874, 239}, {927, 6654}, {1026, 1279}, {1027, 513}, {1447, 27918}, {1929, 244}, {2665, 3248}, {3225, 21762}, {3512, 2170}, {3570, 2238}, {3572, 4367}, {3573, 238}, {7166, 3123}, {7168, 38986}, {7351, 7004}, {8709, 2}, {17930, 81}, {18793, 8054}, {37128, 1015}, {37207, 24512}, {40737, 38978}, {43747, 2310}

• X₆₅₉ = X_i-cross conjugate of X_j for these (i,j): {4455, 8632}, {21832, 812}, {27846, 238}

• X₆₅₉ = cevapoint of X_i and X_j for these (i,j): {665, 6373}, {4455, 21832}

• X₆₅₉ = crosspoint of X_i and X_j for these (i,j): {1, 660}, {57, 927}, {99, 37128}, {100, 105}, {190, 3226}, {238, 3573}, {239, 874}, {251, 919}, {812, 43041}, {3570, 33295}, {3733, 23355}, {17930, 40725}

• X₆₅₉ = crosssum 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}

• X₆₅₉ = trilinear pole of line {27846, 38989}

• X₆₅₉ = crossdifference of every pair of points on line {1, 39}

• X₆₅₉ = bicentric difference of PU(134)

• X₆₅₉ = PU(134)-harmonic conjugate of X₈₂₉₉

• X₆₅₉ = complement of polar conjugate of isogonal conjugate of X₂₂₁₅₅

• X₆₅₉ = X_i-isoconjugate of X_j for these (i,j): {1, 660}, {2, 813}, {6, 4562}, {31, 4583}, {37, 4584}, {38, 36081}, {42, 4589}, {56, 36801}, {75, 34067}, {100, 291}, {101, 335}, {109, 4518}, {110, 43534}, {190, 292}, {213, 4639}, {295, 1897}, {334, 692}, {337, 8750}, {513, 5378}, {651, 4876}, {664, 7077}, {666, 3252}, {668, 1911}, {694, 18047}, {741, 3952}, {756, 36066}, {765, 876}, {805, 1215}, {869, 41072}, {875, 7035}, {919, 40217}, {932, 41531}, {982, 8684}, {984, 30664}, {1016, 3572}, {1018, 37128}, {1237, 17938}, {1252, 4444}, {1492, 3864}, {1581, 4579}, {1922, 1978}, {2196, 6335}, {2276, 37207}, {2283, 33676}, {2295, 37134}, {2311, 4552}, {3573, 30663}, {3862, 4586}, {3903, 18787}, {3939, 7233}, {4033, 18268}, {4557, 18827}, {4559, 36800}, {4570, 35352}, {4572, 18265}, {6386, 14598}, {8709, 40155}, {18099, 46161}, {18829, 20964}, {18895, 32739}, {22116, 36086}, {33888, 39420}, {34071, 40848}, {35309, 39276}, {37135, 40794}, {40093, 40519}, {40936, 41209}

• X₆₅₉ = barycentric product X_i*X_j for these {i,j}: {1, 812}, {6, 3766}, {7, 4435}, {9, 43041}, {21, 7212}, {28, 24459}, {57, 3716}, {75, 8632}, {81, 4010}, {86, 21832}, {88, 4448}, {89, 4800}, {92, 22384}, {99, 39786}, {100, 27918}, {190, 27846}, {238, 514}, {239, 513}, {242, 905}, {244, 3570}, {256, 4107}, {257, 4164}, {269, 4148}, {274, 4455}, {291, 4375}, {292, 27855}, {350, 649}, {512, 30940}, {522, 1429}, {650, 1447}, {651, 4124}, {656, 31905}, {660, 35119}, {661, 33295}, {663, 10030}, {666, 38989}, {667, 1921}, {693, 1914}, {740, 1019}, {751, 4508}, {804, 40432}, {862, 15419}, {874, 1015}, {876, 4366}, {885, 34253}, {893, 14296}, {984, 23597}, {985, 4486}, {1022, 4432}, {1024, 39775}, {1027, 17755}, {1086, 3573}, {1284, 4560}, {1428, 4391}, {1509, 4155}, {1577, 5009}, {1635, 27922}, {1919, 18891}, {1929, 27929}, {1980, 44169}, {2201, 4025}, {2210, 3261}, {2238, 7192}, {2254, 6654}, {2401, 15507}, {2665, 27854}, {2786, 40767}, {3063, 18033}, {3248, 27853}, {3572, 39044}, {3669, 3685}, {3676, 3684}, {3733, 3948}, {3737, 16609}, {3747, 7199}, {3783, 4817}, {3808, 17743}, {3835, 34252}, {3960, 36815}, {3975, 43924}, {3985, 7203}, {4083, 39914}, {4367, 17493}, {4369, 18786}, {4433, 17096}, {4444, 8300}, {4465, 43928}, {4810, 25417}, {4830, 25430}, {6164, 27912}, {6632, 24193}, {6650, 38348}, {7193, 17924}, {7255, 18904}, {7649, 20769}, {8852, 27951}, {9508, 40725}, {14433, 37129}, {14599, 40495}, {14621, 30665}, {22383, 40717}, {24018, 34856}, {24685, 35348}

• X₆₅₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 4562}, {2, 4583}, {6, 660}, {9, 36801}, {31, 813}, {32, 34067}, {58, 4584}, {81, 4589}, {86, 4639}, {101, 5378}, {238, 190}, {239, 668}, {242, 6335}, {244, 4444}, {251, 36081}, {261, 36806}, {350, 1978}, {513, 335}, {514, 334}, {593, 36066}, {649, 291}, {650, 4518}, {661, 43534}, {663, 4876}, {665, 22116}, {667, 292}, {693, 18895}, {740, 4033}, {788, 3862}, {804, 3963}, {812, 75}, {874, 31625}, {876, 40098}, {905, 337}, {985, 37207}, {1015, 876}, {1019, 18827}, {1024, 33676}, {1178, 37134}, {1284, 4552}, {1428, 651}, {1429, 664}, {1447, 4554}, {1580, 18047}, {1691, 4579}, {1914, 100}, {1919, 1911}, {1921, 6386}, {1977, 875}, {1980, 1922}, {2201, 1897}, {2210, 101}, {2238, 3952}, {2254, 40217}, {3063, 7077}, {3125, 35352}, {3248, 3572}, {3250, 3864}, {3261, 44172}, {3570, 7035}, {3572, 30663}, {3573, 1016}, {3669, 7233}, {3684, 3699}, {3685, 646}, {3716, 312}, {3733, 37128}, {3737, 36800}, {3747, 1018}, {3766, 76}, {3783, 3807}, {3797, 4505}, {3808, 3662}, {3948, 27808}, {4010, 321}, {4063, 40093}, {4083, 40848}, {4093, 35309}, {4107, 1909}, {4124, 4391}, {4148, 341}, {4155, 594}, {4164, 894}, {4366, 874}, {4367, 30669}, {4375, 350}, {4378, 7245}, {4432, 24004}, {4433, 30730}, {4435, 8}, {4448, 4358}, {4455, 37}, {4465, 41314}, {4475, 23596}, {4486, 33931}, {4508, 3761}, {4760, 42721}, {4800, 4671}, {4810, 28605}, {4830, 19804}, {5009, 662}, {5027, 2295}, {5029, 40794}, {7192, 40017}, {7193, 1332}, {7212, 1441}, {7255, 40834}, {8299, 42720}, {8300, 3570}, {8632, 1}, {10030, 4572}, {14296, 1920}, {14433, 6381}, {14599, 692}, {14621, 41072}, {15507, 2397}, {16514, 3799}, {17475, 23354}, {17494, 40094}, {18786, 27805}, {18892, 32739}, {20769, 4561}, {20979, 41531}, {20981, 18787}, {21385, 40095}, {21832, 10}, {22383, 295}, {22384, 63}, {23597, 870}, {24193, 6545}, {24459, 20336}, {27846, 514}, {27855, 1921}, {27918, 693}, {27929, 20947}, {30654, 40790}, {30665, 3661}, {30940, 670}, {31905, 811}, {33295, 799}, {34252, 4598}, {34253, 883}, {34856, 823}, {35119, 3766}, {36815, 36804}, {38348, 6542}, {38367, 3009}, {38989, 918}, {39044, 27853}, {39786, 523}, {39914, 18830}, {40432, 18829}, {40495, 44170}, {40746, 30664}, {40767, 35148}, {41333, 4557}, {43041, 85}

• X₆₅₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 1960, 25569}, {2, 3837, 30795}, {2, 26855, 27194}, {75, 21612, 21439}, {764, 14419, 3960}, {1635, 2246, 22108}, {1635, 2254, 9508}, {1635, 3768, 20331}, {1960, 21385, 21343}, {3716, 4010, 4800}, {4010, 4448, 3716}, {4448, 4830, 4810}, {4458, 13246, 4809}, {4724, 4782, 4784}, {4800, 4810, 4010}, {8632, 21832, 4435}, {13245, 13246, 45695}, {21343, 25569, 1}, {23770, 26275, 676}, {27015, 27075, 2}, {27294, 30025, 30865}, {31596, 31597, 45695}