X₆₅₈

TRILINEAR POLE OF LINE X₁X_7

X₆₅₈ = trilinear pole of line {1, 7}

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

X₆₅₈ (TRILINEAR POLE OF LINE X₁X_7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 11}, {88, 279}

• Related Centers: X₆₅₇ (CROSSDIFFERENCE OF X₁ AND X₇), X₇ (GERGONNE POINT), X₁₁ (FEUERBACH POINT), X₈₈ (ISOGONAL CONJUGATE OF X₄₄), X₂₇₉ (ISOGONAL CONJUGATE OF X₂₂₀)

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

Barycentrics: \(1/((b - c) (a - b - c)^{2}) : :\)

Search Value (6, 9, 13): \(0.092942878743\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1/(\sec^{2}(B/2) - \sec^{2}(C/2)) : :\)

• Trilinears: \(1/(\tan^{2}(B/2) - \tan^{2}(C/2)) : :\)

▸ Linear Combinations1 combination

• X₆₅₈ = \(3\) X₂ - \(4\) X₄₀₅₃₇

X₆₅₈ lies on the curve CC₉.

▸ Incident Central Lines94 lines

{1, 24012}, {2, 7056}, {7, 11}, {44, 41351}, {55, 31526}, {57, 673}, {63, 31627}, {64, 45239}, {77, 23707}, {85, 3306}, {86, 1439}, {88, 279}, {100, 664}, {108, 6183}, {109, 927}, {144, 17113}, {162, 1414}, {190, 1020}, {269, 37129}, {273, 43764}, {333, 37202}, {348, 5744}, {354, 9446}, {479, 5435}, {514, 37139}, {651, 1638}, {653, 13149}, {655, 24002}, {657, 24011}, {660, 43932}, {662, 1461}, {799, 4635}, {883, 3699}, {897, 3668}, {899, 41355}, {1042, 37132}, {1043, 40933}, {1097, 3182}, {1155, 14189}, {1262, 36087}, {1275, 3257}, {1407, 20332}, {1427, 37128}, {1434, 1446}, {1445, 23062}, {1447, 34855}, {1565, 13226}, {1821, 7196}, {1847, 37203}, {1897, 4025}, {2349, 17095}, {2898, 3474}, {2968, 18025}, {3218, 37780}, {3599, 10580}, {3732, 24015}, {3817, 10136}, {3911, 37131}, {4417, 7055}, {4551, 37138}, {4552, 4606}, {4605, 37212}, {4998, 43290}, {6172, 36888}, {6614, 36098}, {7013, 33673}, {7070, 45742}, {7182, 14829}, {7183, 40702}, {7339, 36094}, {7658, 23586}, {9057, 24016}, {9312, 17106}, {9358, 21104}, {9778, 31527}, {9812, 36620}, {13257, 28344}, {14837, 24013}, {15252, 45276}, {17277, 37214}, {17352, 30705}, {17605, 42386}, {17728, 30623}, {18240, 24203}, {23587, 23972}, {24009, 39156}, {27834, 27836}, {30988, 40593}, {32040, 44551}, {32714, 36099}, {37208, 41245}, {37650, 41356}, {38285, 43750}, {41798, 44005}

▸ Functional Relations17 relations

• X₆₅₈ = reflection of X₁₃₆₀₉ in X₄₀₅₃₇

• X₆₅₈ = isogonal conjugate of X₆₅₇

• X₆₅₈ = isotomic conjugate of X₃₂₃₉

• X₆₅₈ = anticomplement of X₁₃₆₀₉

• X₆₅₈ = X₇₃₃₉-anticomplementary conjugate of X₃₁₅₂₇

• X₆₅₈ = X_i-Ceva conjugate of X_j for these (i,j): {1275, 279}, {4569, 664}, {4616, 934}, {4635, 4569}, {4998, 17093}, {7045, 33765}, {23586, 7056}, {24011, 1}, {34085, 24015}, {36838, 4626}

• X₆₅₈ = X_i-cross conjugate of X_j for these (i,j): {1, 24011}, {57, 7045}, {169, 7012}, {223, 24032}, {279, 1275}, {514, 7}, {650, 21453}, {651, 664}, {657, 1}, {905, 86}, {934, 4626}, {1020, 934}, {1445, 4564}, {1461, 36118}, {3008, 39293}, {3676, 1088}, {3732, 190}, {4025, 7056}, {4091, 77}, {4253, 59}, {4566, 4569}, {5435, 4998}, {7658, 2}, {9533, 23586}, {10015, 903}, {14282, 5558}, {14331, 34402}, {14543, 99}, {14837, 75}, {16572, 765}, {20521, 6384}, {21188, 273}, {21212, 7249}, {21390, 1476}, {23511, 7035}, {23730, 514}, {23799, 310}, {24002, 1434}, {24782, 40418}, {37681, 1016}, {39470, 18025}, {41800, 1268}

• X₆₅₈ = cevapoint of X_i and X_j for these (i,j): {1, 657}, {2, 4025}, {6, 44408}, {7, 514}, {57, 3676}, {77, 4091}, {354, 650}, {513, 40133}, {522, 23058}, {614, 649}, {651, 934}, {652, 20277}, {905, 1439}, {1020, 4566}, {1122, 3669}, {1427, 43932}, {1446, 24002}, {1638, 3321}, {7658, 9533}, {31605, 41785}, {39063, 43042}

• X₆₅₈ = crosspoint of X_i and X_j for these (i,j): {799, 44326}, {4569, 36838}, {4616, 4635}

• X₆₅₈ = crossdifference of every pair of points on line {3022, 14936}

• X₆₅₈ = trilinear pole of line {1, 7}

• X₆₅₈ = trilinear pole wrt tangential triangle of line X₁X₇

• X₆₅₈ = BSS(a^2→a) of X₁₀₇

• X₆₅₈ = X_i-isoconjugate of X_j for these (i,j): {1, 657}, {2, 8641}, {6, 3900}, {8, 3063}, {9, 663}, {21, 3709}, {31, 3239}, {32, 4397}, {33, 652}, {37, 21789}, {41, 522}, {42, 1021}, {55, 650}, {56, 4130}, {57, 4105}, {58, 4171}, {81, 4524}, {100, 14936}, {101, 2310}, {106, 14427}, {109, 3119}, {110, 36197}, {200, 649}, {210, 7252}, {212, 3064}, {213, 7253}, {219, 18344}, {220, 513}, {228, 17926}, {281, 1946}, {284, 4041}, {294, 926}, {341, 1919}, {346, 667}, {480, 3669}, {512, 2287}, {514, 1253}, {521, 607}, {604, 4163}, {644, 3271}, {647, 4183}, {651, 3022}, {656, 2332}, {658, 24012}, {661, 2328}, {665, 28071}, {692, 1146}, {693, 14827}, {728, 43924}, {798, 1043}, {810, 2322}, {884, 3693}, {893, 4477}, {904, 4529}, {905, 7071}, {906, 42069}, {934, 35508}, {943, 33525}, {949, 6182}, {1015, 4578}, {1024, 2340}, {1098, 4079}, {1110, 42462}, {1170, 6607}, {1174, 6608}, {1260, 6591}, {1334, 3737}, {1415, 4081}, {1459, 7079}, {1461, 24010}, {1783, 3270}, {1792, 2489}, {1802, 7649}, {1824, 23090}, {1857, 36054}, {1911, 4148}, {1974, 15416}, {2149, 23615}, {2170, 3939}, {2175, 4391}, {2192, 14298}, {2194, 3700}, {2212, 6332}, {2223, 28132}, {2291, 14392}, {2299, 8611}, {2316, 4895}, {2334, 4827}, {2346, 10581}, {2364, 4814}, {2488, 6605}, {3121, 7256}, {3122, 7259}, {3248, 6558}, {3676, 6602}, {3733, 4515}, {4435, 7077}, {4516, 5546}, {4528, 9456}, {4546, 38266}, {4631, 7063}, {4705, 7054}, {4953, 34080}, {4990, 28615}, {5514, 32652}, {6066, 40166}, {6129, 7367}, {6139, 41798}, {6366, 18889}, {6603, 23351}, {6726, 6729}, {7008, 10397}, {7046, 22383}, {7050, 40137}, {7073, 9404}, {7118, 8058}, {7123, 17115}, {7162, 17412}, {8638, 36796}, {8648, 36910}, {8750, 34591}, {9447, 35519}, {10482, 21127}, {11934, 40141}, {22108, 42064}, {23990, 42455}, {24026, 32739}, {30692, 32726}, {34526, 46006}

• X₆₅₈ = barycentric product X_i*X_j for these {i,j}: {1, 4569}, {7, 664}, {8, 4626}, {9, 36838}, {10, 4616}, {37, 4635}, {56, 4572}, {57, 4554}, {63, 13149}, {65, 4625}, {69, 36118}, {75, 934}, {76, 1461}, {77, 18026}, {85, 651}, {86, 4566}, {99, 3668}, {100, 1088}, {108, 7182}, {109, 6063}, {190, 279}, {226, 4573}, {241, 34085}, {269, 668}, {273, 6516}, {274, 1020}, {304, 32714}, {312, 4617}, {321, 4637}, {331, 1813}, {348, 653}, {349, 4565}, {479, 3699}, {514, 1275}, {644, 23062}, {646, 738}, {655, 17078}, {662, 1446}, {670, 1042}, {693, 7045}, {789, 7204}, {799, 1427}, {811, 1439}, {927, 9436}, {1025, 34018}, {1106, 6386}, {1119, 4561}, {1254, 4623}, {1262, 3261}, {1323, 35157}, {1332, 1847}, {1407, 1978}, {1414, 1441}, {1415, 20567}, {1434, 4552}, {1443, 35174}, {1458, 46135}, {1509, 4605}, {1897, 7056}, {2481, 41353}, {3239, 23586}, {3596, 6614}, {3663, 6613}, {3673, 8269}, {3674, 6648}, {3676, 4998}, {3732, 30705}, {3900, 24011}, {4131, 24032}, {4397, 24013}, {4564, 24002}, {4601, 7216}, {4610, 6354}, {4619, 23989}, {4620, 7178}, {4624, 21454}, {4631, 7147}, {6335, 7177}, {6606, 10481}, {6649, 7249}, {7035, 43932}, {7055, 36127}, {7128, 15413}, {7196, 37137}, {7205, 29055}, {7339, 35519}, {7365, 37215}, {10004, 32040}, {14256, 44327}, {14615, 36079}, {14727, 41355}, {17093, 37206}, {17095, 38340}, {18025, 23973}, {21453, 35312}, {23984, 30805}, {24015, 36101}, {24016, 35517}, {24027, 40495}, {32041, 42309}, {35171, 38459}, {35338, 42311}, {36146, 40704}, {36908, 44326}, {37139, 37780}, {37141, 40702}, {37143, 37757}, {39293, 43042}

• X₆₅₈ = barycentric quotient X_i/X_j for these {i,j}: {1, 3900}, {2, 3239}, {6, 657}, {7, 522}, {8, 4163}, {9, 4130}, {11, 23615}, {27, 17926}, {31, 8641}, {34, 18344}, {37, 4171}, {42, 4524}, {44, 14427}, {55, 4105}, {56, 663}, {57, 650}, {58, 21789}, {59, 3939}, {65, 4041}, {75, 4397}, {77, 521}, {81, 1021}, {85, 4391}, {86, 7253}, {99, 1043}, {100, 200}, {101, 220}, {108, 33}, {109, 55}, {110, 2328}, {112, 2332}, {145, 4546}, {162, 4183}, {171, 4477}, {174, 6730}, {190, 346}, {222, 652}, {223, 14298}, {226, 3700}, {239, 4148}, {269, 513}, {273, 44426}, {278, 3064}, {279, 514}, {304, 15416}, {331, 46110}, {347, 8058}, {348, 6332}, {354, 6608}, {479, 3676}, {513, 2310}, {514, 1146}, {519, 4528}, {522, 4081}, {553, 4976}, {603, 1946}, {604, 3063}, {614, 17115}, {644, 728}, {646, 30693}, {648, 2322}, {649, 14936}, {650, 3119}, {651, 9}, {653, 281}, {655, 36910}, {657, 35508}, {661, 36197}, {662, 2287}, {663, 3022}, {664, 8}, {666, 6559}, {668, 341}, {673, 28132}, {692, 1253}, {693, 24026}, {738, 3669}, {765, 4578}, {883, 3717}, {894, 4529}, {905, 34591}, {906, 1802}, {927, 14942}, {934, 1}, {1014, 3737}, {1016, 6558}, {1018, 4515}, {1020, 37}, {1025, 3693}, {1042, 512}, {1086, 42462}, {1088, 693}, {1106, 667}, {1111, 42455}, {1119, 7649}, {1122, 6615}, {1125, 4990}, {1155, 14392}, {1214, 8611}, {1254, 4705}, {1262, 101}, {1275, 190}, {1308, 42064}, {1317, 4543}, {1319, 4895}, {1323, 6366}, {1331, 1260}, {1332, 3692}, {1358, 21132}, {1388, 4959}, {1400, 3709}, {1407, 649}, {1410, 810}, {1412, 7252}, {1414, 21}, {1415, 41}, {1416, 884}, {1418, 21127}, {1420, 4162}, {1421, 11193}, {1427, 661}, {1429, 4435}, {1434, 4560}, {1435, 6591}, {1439, 656}, {1441, 4086}, {1442, 35057}, {1443, 3738}, {1446, 1577}, {1447, 3716}, {1449, 4827}, {1458, 926}, {1459, 3270}, {1461, 6}, {1462, 1024}, {1475, 10581}, {1633, 4319}, {1638, 33573}, {1783, 7079}, {1790, 23090}, {1813, 219}, {1847, 17924}, {1897, 7046}, {2003, 9404}, {2099, 4814}, {2260, 33525}, {2263, 6182}, {2283, 2340}, {2293, 6607}, {2635, 30692}, {2720, 2342}, {2999, 40137}, {3212, 4147}, {3239, 23970}, {3261, 23978}, {3663, 42337}, {3667, 4953}, {3668, 523}, {3669, 2170}, {3671, 4843}, {3674, 3910}, {3676, 11}, {3699, 5423}, {3732, 6554}, {3882, 3965}, {3888, 4073}, {3900, 24010}, {3911, 1639}, {3939, 480}, {3952, 4082}, {4017, 4516}, {4025, 2968}, {4032, 4140}, {4091, 35072}, {4131, 24031}, {4298, 29278}, {4306, 8676}, {4308, 8710}, {4320, 8678}, {4341, 15313}, {4350, 3309}, {4551, 210}, {4552, 2321}, {4554, 312}, {4556, 7054}, {4558, 2327}, {4559, 1334}, {4561, 1265}, {4564, 644}, {4565, 284}, {4566, 10}, {4567, 7259}, {4569, 75}, {4572, 3596}, {4573, 333}, {4592, 1792}, {4600, 7256}, {4601, 7258}, {4605, 594}, {4610, 7058}, {4616, 86}, {4617, 57}, {4619, 1252}, {4620, 645}, {4625, 314}, {4626, 7}, {4635, 274}, {4636, 6061}, {4637, 81}, {4654, 4820}, {4848, 44729}, {4998, 3699}, {5219, 4944}, {5222, 14330}, {5228, 45755}, {5435, 4521}, {5930, 14308}, {5932, 14302}, {6049, 4943}, {6063, 35519}, {6335, 7101}, {6354, 4024}, {6356, 4064}, {6357, 14400}, {6359, 8062}, {6516, 78}, {6517, 1259}, {6604, 44448}, {6613, 1222}, {6614, 56}, {6649, 7081}, {6733, 6726}, {7011, 10397}, {7023, 43924}, {7045, 100}, {7053, 1459}, {7056, 4025}, {7099, 22383}, {7125, 36054}, {7128, 1783}, {7175, 3287}, {7176, 3907}, {7177, 905}, {7178, 21044}, {7179, 4522}, {7181, 14432}, {7182, 35518}, {7185, 3810}, {7203, 18191}, {7204, 1491}, {7216, 3125}, {7223, 4474}, {7225, 4501}, {7250, 3122}, {7339, 109}, {7365, 6590}, {7370, 6729}, {7371, 6728}, {7649, 42069}, {7658, 13609}, {8059, 2192}, {8641, 24012}, {8750, 7071}, {9533, 7658}, {10481, 6362}, {13149, 92}, {13444, 16012}, {14256, 14837}, {14594, 3974}, {14733, 4845}, {14837, 5514}, {16727, 40213}, {17078, 3904}, {17082, 25128}, {17093, 4468}, {17096, 17197}, {17136, 6737}, {17206, 15411}, {18026, 318}, {18623, 14331}, {21132, 5532}, {21188, 6506}, {21272, 6736}, {21454, 4765}, {22464, 2804}, {23062, 24002}, {23067, 2318}, {23224, 2638}, {23703, 3689}, {23723, 34969}, {23788, 14010}, {23890, 6603}, {23971, 1461}, {23973, 516}, {23979, 32739}, {24002, 4858}, {24011, 4569}, {24013, 934}, {24015, 30807}, {24016, 103}, {24027, 692}, {24471, 17420}, {26700, 7073}, {30719, 4534}, {30725, 4530}, {30805, 23983}, {32651, 2259}, {32668, 911}, {32674, 607}, {32714, 19}, {32735, 2195}, {32739, 14827}, {33765, 17494}, {34036, 11934}, {34056, 23893}, {34085, 36796}, {34387, 23104}, {34496, 45743}, {34855, 2254}, {35312, 4847}, {35326, 8012}, {35338, 3059}, {35341, 45791}, {36048, 943}, {36049, 7367}, {36059, 212}, {36079, 64}, {36082, 7072}, {36086, 28071}, {36118, 4}, {36127, 1857}, {36141, 18889}, {36146, 294}, {36838, 85}, {36908, 6587}, {37139, 41798}, {37141, 282}, {37757, 30565}, {38340, 7110}, {38459, 3887}, {38859, 4040}, {39293, 36802}, {39771, 4542}, {41353, 518}, {41355, 42341}, {42309, 4762}, {43037, 14430}, {43041, 4124}, {43049, 38375}, {43290, 6555}, {43924, 3271}, {43932, 244}, {44408, 14714}, {44717, 4587}, {46107, 21666}, {46153, 3688}

• X₆₅₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 9533, 7056}, {57, 1088, 33765}, {100, 35312, 664}, {354, 9446, 21453}, {479, 5435, 17093}, {934, 4566, 664}, {1439, 6359, 86}, {4573, 4616, 4637}, {13609, 40537, 2}