X₆₆₇

CROSSDIFFERENCE OF X₂ AND X₃₇

X₆₆₇ = reflection of X₆₆₇ in the Lemoine axis

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

X₆₆₇ (CROSSDIFFERENCE OF X₂ AND X₃₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {36, 238}, {187, 237}

• Related Centers: X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₂₃₈ (-LINE CONJUGATE OF X₃₇), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂)

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

Barycentrics: \(a^{3}(b - c) : :\)

Search Value (6, 9, 13): \(2.897671730498\)

▸ Linear Combinations1 combination

• X₆₆₇ = X₁ + \(2\) X₄₇₈₂, \(6\) X₂ - \(5\) X₃₁₂₅₁, \(3\) X₂ - \(4\) X₃₁₂₈₈, \(3\) X₂ + X₃₁₂₉₁, \(3\) X₃₅₁ - \(2\) X₄₂₆₅₃, \(3\) X₃₅₁ - X₄₂₆₆₁, X₆₄₉ + \(2\) X₁₉₆₀, \(2\) X₆₄₉ + X₄₇₇₅, X₆₄₉ + \(3\) X₈₆₄₃, X₆₄₉ + \(5\) X₈₆₅₆, \(2\) X₆₅₉ + X₄₃₇₈, \(2\) X₆₆₃ + X₄₈₃₄, X₆₆₃ - \(3\) X₈₆₄₃, X₆₆₃ - \(5\) X₈₆₅₆, \(2\) X₉₀₅ - \(3\) X₁₄₄₁₉, X₉₀₅ - \(3\) X₃₀₂₃₄, \(4\) X₁₁₂₅ - X₂₄₇₁₉, \(3\) X₁₆₃₅ - X₄₀₄₁, \(4\) X₁₆₃₅ - X₄₈₂₅, \(4\) X₁₉₆₀ - X₄₇₇₅, \(4\) X₁₉₆₀ + X₄₈₃₄, \(2\) X₁₉₆₀ - \(3\) X₈₆₄₃, \(2\) X₁₉₆₀ - \(5\) X₈₆₅₆, X₂₅₃₀ + \(2\) X₃₈₀₃, X₂₅₃₀ - \(3\) X₁₄₄₁₉, X₂₅₃₀ - \(6\) X₃₀₂₃₄, \(3\) X₃₂₅₁ - \(2\) X₄₁₆₂, \(3\) X₃₇₃₃ + X₄₄₉₁, X₃₈₀₁ - \(3\) X₄₈₀₉, \(2\) X₃₈₀₃ + \(3\) X₁₄₄₁₉, X₃₈₀₃ + \(3\) X₃₀₂₃₄, \(4\) X₄₀₄₁ - \(3\) X₄₈₂₅, \(3\) X₄₀₅₇ - X₄₄₉₁, X₄₃₆₇ + \(2\) X₄₄₀₁, X₄₃₇₈ + \(4\) X₄₄₀₁, \(4\) X₄₃₉₄ - X₄₇₃₀, X₄₇₀₅ - \(4\) X₆₀₅₀, X₄₇₇₅ - \(6\) X₈₆₄₃, X₄₇₇₅ - \(10\) X₈₆₅₆, X₄₇₈₄ + \(2\) X₄₇₉₄, \(3\) X₄₈₀₉ - \(2\) X₂₀₅₁₇, X₄₈₃₄ + \(6\) X₈₆₄₃, X₄₈₃₄ + \(10\) X₈₆₅₆, X₄₈₇₉ - \(3\) X₂₅₅₆₉, \(3\) X₈₆₄₃ - \(5\) X₈₆₅₆, \(4\) X₂₁₂₆₀ - \(3\) X₃₁₁₄₉, \(4\) X₂₁₂₆₀ - \(5\) X₃₁₂₅₁, \(2\) X₂₁₂₆₀ + X₃₁₂₉₁, \(2\) X₂₁₃₀₁ - \(3\) X₃₁₁₄₉, \(2\) X₂₁₃₀₁ - \(5\) X₃₁₂₅₁, X₂₁₃₀₁ - \(4\) X₃₁₂₈₈, X₂₁₃₀₂ - \(5\) X₂₇₀₁₃, \(3\) X₃₁₁₄₉ - \(5\) X₃₁₂₅₁, \(3\) X₃₁₁₄₉ - \(8\) X₃₁₂₈₈, \(3\) X₃₁₁₄₉ + \(2\) X₃₁₂₉₁, \(5\) X₃₁₂₅₁ - \(8\) X₃₁₂₈₈, \(5\) X₃₁₂₅₁ + \(2\) X₃₁₂₉₁, \(4\) X₃₁₂₈₈ + X₃₁₂₉₁

▸ Incident Central Lines164 lines

{1, 4063}, {2, 21260}, {3, 1083}, {6, 9010}, {10, 21724}, {21, 29150}, {23, 9980}, {25, 18344}, {28, 17924}, {31, 8027}, {36, 238}, {55, 3251}, {56, 764}, {75, 21440}, {100, 898}, {101, 813}, {104, 2726}, {105, 14665}, {106, 2382}, {110, 2703}, {111, 35107}, {171, 38238}, {187, 237}, {197, 11124}, {213, 875}, {274, 23807}, {514, 659}, {522, 8045}, {644, 8671}, {650, 4705}, {656, 832}, {668, 932}, {676, 29162}, {692, 1110}, {693, 18108}, {750, 14474}, {784, 4560}, {788, 798}, {814, 1577}, {830, 1491}, {834, 2605}, {838, 1980}, {876, 37609}, {885, 37254}, {891, 4449}, {900, 4990}, {956, 30583}, {958, 20317}, {993, 4448}, {999, 14421}, {1001, 6008}, {1022, 37587}, {1125, 24719}, {1262, 32735}, {1379, 11651}, {1380, 11652}, {1415, 32666}, {1459, 6371}, {1635, 4041}, {1734, 9508}, {1979, 9265}, {2163, 23345}, {2217, 42455}, {2254, 6004}, {2300, 18002}, {2483, 6586}, {2484, 3709}, {2533, 29066}, {2787, 4391}, {2826, 44811}, {2832, 23765}, {2975, 4462}, {3271, 22096}, {3398, 39519}, {3508, 21390}, {3667, 39210}, {3700, 29232}, {3716, 6002}, {3762, 29324}, {3766, 26277}, {3777, 3960}, {3835, 25537}, {3900, 4394}, {4010, 29013}, {4086, 6133}, {4122, 29062}, {4155, 4501}, {4160, 4490}, {4170, 29328}, {4223, 40551}, {4369, 29051}, {4474, 29268}, {4477, 43061}, {4557, 40522}, {4707, 29082}, {4724, 6372}, {4729, 4895}, {4761, 29366}, {4784, 4794}, {4785, 45316}, {4790, 16874}, {4791, 29344}, {4800, 29178}, {4810, 29270}, {4822, 4979}, {4823, 29033}, {4879, 25569}, {4978, 29362}, {5251, 45666}, {5253, 19947}, {6003, 39225}, {6085, 22379}, {6363, 7250}, {6373, 20459}, {7117, 23646}, {7178, 29240}, {7255, 14296}, {7265, 29078}, {7497, 39536}, {7662, 23882}, {8034, 40956}, {8652, 29341}, {9029, 22108}, {9048, 23087}, {9218, 17940}, {9269, 37602}, {10006, 33849}, {11171, 39502}, {11193, 20988}, {11247, 20831}, {14431, 28475}, {14434, 17125}, {16158, 29298}, {17072, 19522}, {17126, 43928}, {17166, 17494}, {17197, 24234}, {17989, 29058}, {18102, 18107}, {18173, 18197}, {20456, 20962}, {20982, 22373}, {21007, 21788}, {21146, 29186}, {21302, 27013}, {21348, 21389}, {21385, 29226}, {22224, 22381}, {22386, 39025}, {23394, 29198}, {23656, 23657}, {23770, 34958}, {24286, 37575}, {24698, 25356}, {25259, 29090}, {26148, 27345}, {26275, 29126}, {27673, 27677}, {28401, 30094}, {28468, 30580}, {28473, 44805}, {34467, 38389}, {39476, 42325}, {40519, 40521}

▸ Functional Relations36 relations

• X₆₆₇ = radical center of the circumcircle, Brocard circle, and the circle with (diameter = segment X₁X₃) (Wilson Stothers, 3/31/2003)

• X₆₆₇ = midpoint of X_i and X_j for these {i,j}: {1, 4063}, {649, 663}, {659, 4367}, {669, 8639}, {905, 3803}, {1019, 4040}, {3733, 4057}, {4449, 4498}, {4729, 4895}, {4775, 4834}, {4822, 4979}, {17166, 17494}, {21301, 31291}

• X₆₆₇ = reflection of X_i in X_j for these {i,j}: {3, 39227}, {650, 6050}, {659, 4401}, {663, 1960}, {764, 3669}, {1491, 14838}, {1577, 4874}, {1734, 9508}, {2530, 905}, {3230, 42655}, {3777, 3960}, {3801, 20517}, {4063, 4782}, {4086, 6133}, {4142, 13246}, {4378, 4367}, {4705, 650}, {4775, 663}, {4834, 649}, {8640, 8637}, {14419, 30234}, {17072, 31286}, {21260, 31288}, {21301, 21260}, {23224, 34948}, {23770, 34958}, {24698, 25356}, {31149, 2}, {42661, 42653}

• X₆₆₇ = reflection of X₆₆₇ in the Lemoine axis

• X₆₆₇ = isogonal conjugate of X₆₆₈

• X₆₆₇ = isotomic conjugate of X₆₃₈₆

• X₆₆₇ = complement of X₂₁₃₀₁

• X₆₆₇ = anticomplement of X₂₁₂₆₀

• X₆₆₇ = circumcircle-inverse of X₁₀₈₃

• X₆₆₇ = Parry-circle-inverse of X₅₀₄₀

• X₆₆₇ = Parry-isodynamic-circle-inverse of X₅₁₆₃

• X₆₆₇ = polar conjugate of the isotomic conjugate of X₂₂₃₈₃

• X₆₆₇ = tangential-isogonal conjugate of X₂₃₄₀₂

• X₆₆₇ = X_i-Ceva conjugate of X_j for these (i,j): {6, 1977}, {25, 3271}, {28, 3125}, {31, 3248}, {56, 1015}, {99, 20963}, {100, 6}, {101, 213}, {104, 2087}, {109, 20228}, {110, 2300}, {190, 21757}, {513, 22383}, {649, 3063}, {662, 1197}, {692, 31}, {789, 23660}, {813, 21760}, {898, 3230}, {932, 1}, {934, 16502}, {1333, 3121}, {1415, 32}, {1436, 14936}, {1576, 40956}, {1919, 8640}, {2162, 21762}, {2176, 23573}, {2217, 2170}, {2248, 21755}, {2353, 23646}, {3435, 7117}, {3437, 20982}, {3444, 3124}, {3733, 649}, {3903, 23525}, {4557, 2308}, {4598, 21759}, {4623, 81}, {6186, 3122}, {7087, 20974}, {7152, 39687}, {7169, 3270}, {7255, 514}, {8693, 40728}, {8709, 20669}, {16695, 23572}, {18108, 513}, {23345, 21758}, {23349, 890}, {23355, 8632}, {26703, 17435}, {29227, 2176}, {29351, 16971}, {30664, 16782}, {32665, 2251}, {34067, 2223}, {34080, 41}, {34183, 35505}, {34443, 8054}, {34444, 38346}, {34445, 23470}, {34594, 1100}, {36614, 23560}, {38470, 16470}, {40519, 42}, {40770, 9427}, {43350, 1449}

• X₆₆₇ = X_i-cross conjugate of X_j for these (i,j): {6, 9299}, {669, 1919}, {798, 649}, {890, 23349}, {1977, 6}, {3121, 1333}, {3122, 1402}, {3248, 31}, {8027, 3248}, {20979, 8640}, {21762, 2162}, {21835, 893}, {22096, 1397}, {23560, 36614}, {38367, 8632}, {38986, 1}

• X₆₆₇ = cevapoint of X_i and X_j for these (i,j): {6, 1979}, {649, 20979}, {669, 798}, {3248, 8027}

• X₆₆₇ = crosspoint of X_i and X_j for these (i,j): {6, 100}, {31, 692}, {56, 1415}, {58, 101}, {81, 4623}, {86, 4598}, {99, 40408}, {109, 3451}, {110, 1169}, {513, 6591}, {649, 43924}, {919, 15382}, {932, 7121}, {16945, 34080}, {32735, 41934}

• X₆₆₇ = crosssum of X_i and X_j for these (i,j): {1, 4063}, {2, 513}, {6, 21005}, {8, 4391}, {10, 514}, {42, 20979}, {69, 35518}, {75, 693}, {100, 1332}, {120, 918}, {190, 3699}, {192, 25142}, {210, 4140}, {312, 4397}, {313, 35519}, {321, 4036}, {341, 15416}, {350, 27855}, {512, 16589}, {522, 3452}, {523, 1211}, {525, 21530}, {536, 14434}, {650, 3056}, {661, 3728}, {812, 17793}, {850, 1234}, {891, 13466}, {900, 16594}, {1016, 11607}, {1233, 3261}, {1577, 4647}, {1654, 24381}, {1909, 2533}, {2397, 15632}, {3762, 4738}, {3789, 4762}, {3900, 6554}, {3952, 4033}, {3954, 4705}, {4024, 21728}, {4083, 6376}, {4462, 44720}, {4651, 20954}, {4791, 4793}, {4824, 27495}, {6084, 40609}, {7192, 16709}, {7199, 16748}, {8027, 31645}, {20983, 30473}, {24533, 28369}, {28195, 28651}, {29226, 40598}

• X₆₆₇ = trilinear pole of line {890, 1977}

• X₆₆₇ = crossdifference of every pair of points on line {2, 37}

• X₆₆₇ = intersection of tangents to circumcircle at intersections with line X₁X₆

• X₆₆₇ = bicentric difference of PU(i) for i in (9, 26)

• X₆₆₇ = PU(9)-harmonic conjugate of X₂₁₃

• X₆₆₇ = trilinear product of PU(25)

• X₆₆₇ = PU(26)-harmonic conjugate of X₃₂₃₀

• X₆₆₇ = vertex conjugate of PU(26)

• X₆₆₇ = trilinear pole of PU(42)

• X₆₆₇ = pole, wrt circumcircle, of line X₁X₆

• X₆₆₇ = inverse-in-Parry-circle of X₅₀₄₀

• X₆₆₇ = homothetic center of 4th Euler triangle and extraversion triangle of X₈

• X₆₆₇ = inverse-in-Parry-isodynamic-circle of X₅₁₆₃; see X₂

• X₆₆₇ = polar conjugate of isotomic conjugate of X₂₂₃₈₃

• X₆₆₇ = X_i-isoconjugate of X_j for these (i,j): {1, 668}, {2, 190}, {4, 4561}, {6, 1978}, {7, 3699}, {8, 664}, {9, 4554}, {10, 99}, {31, 6386}, {37, 799}, {42, 670}, {43, 18830}, {55, 4572}, {57, 646}, {58, 27808}, {59, 35519}, {63, 6335}, {65, 7257}, {69, 1897}, {71, 6331}, {72, 811}, {75, 100}, {76, 101}, {78, 18026}, {81, 4033}, {83, 4568}, {85, 644}, {86, 3952}, {87, 36863}, {88, 24004}, {92, 1332}, {108, 3718}, {109, 3596}, {110, 313}, {112, 40071}, {162, 20336}, {163, 27801}, {192, 4598}, {200, 4569}, {210, 4625}, {213, 4602}, {226, 645}, {238, 4583}, {239, 4562}, {257, 18047}, {264, 1331}, {273, 4571}, {274, 1018}, {279, 6558}, {291, 874}, {292, 27853}, {304, 1783}, {305, 8750}, {306, 648}, {307, 36797}, {308, 46148}, {310, 4557}, {312, 651}, {314, 4551}, {318, 6516}, {319, 6742}, {321, 662}, {322, 13138}, {329, 44327}, {330, 4595}, {331, 4587}, {333, 4552}, {334, 3573}, {335, 3570}, {341, 934}, {344, 37206}, {345, 653}, {346, 658}, {349, 5546}, {350, 660}, {391, 4624}, {513, 7035}, {514, 1016}, {519, 4555}, {522, 4998}, {523, 4600}, {536, 4607}, {561, 692}, {594, 4610}, {643, 1441}, {649, 31625}, {655, 32851}, {661, 4601}, {666, 3912}, {672, 36803}, {673, 42720}, {677, 35517}, {689, 21035}, {693, 765}, {726, 8709}, {728, 36838}, {740, 4589}, {756, 4623}, {789, 984}, {813, 1921}, {823, 3998}, {831, 33941}, {835, 5224}, {850, 4570}, {869, 46132}, {870, 3799}, {873, 40521}, {883, 14942}, {889, 899}, {892, 4062}, {894, 27805}, {897, 42721}, {898, 6381}, {901, 3264}, {903, 17780}, {906, 1969}, {908, 13136}, {927, 3717}, {932, 6376}, {985, 4505}, {1023, 20568}, {1025, 36796}, {1026, 2481}, {1043, 4566}, {1086, 6632}, {1088, 4578}, {1110, 40495}, {1125, 6540}, {1213, 4632}, {1215, 4594}, {1222, 21272}, {1230, 4629}, {1252, 3261}, {1264, 36127}, {1265, 36118}, {1266, 6079}, {1268, 4427}, {1269, 8701}, {1275, 3239}, {1310, 4385}, {1414, 3701}, {1415, 28659}, {1427, 7258}, {1434, 30730}, {1446, 7259}, {1447, 36801}, {1492, 33931}, {1502, 32739}, {1509, 4103}, {1577, 4567}, {1647, 6635}, {1698, 32042}, {1738, 35574}, {1813, 7017}, {1821, 42717}, {1826, 4563}, {1909, 3903}, {1918, 4609}, {2171, 4631}, {2238, 4639}, {2276, 37133}, {2284, 18031}, {2295, 7260}, {2321, 4573}, {2340, 46135}, {2345, 37215}, {2349, 42716}, {2397, 34234}, {2398, 18025}, {2414, 31638}, {2415, 31227}, {3112, 4553}, {3218, 36804}, {3219, 15455}, {3222, 21080}, {3226, 23354}, {3227, 23891}, {3250, 5388}, {3257, 4358}, {3262, 36037}, {3263, 36086}, {3661, 4586}, {3662, 4621}, {3663, 8706}, {3668, 7256}, {3676, 4076}, {3679, 4597}, {3682, 6528}, {3687, 6648}, {3692, 13149}, {3693, 34085}, {3700, 4620}, {3729, 30610}, {3730, 31624}, {3732, 30701}, {3762, 5376}, {3766, 5378}, {3783, 41072}, {3797, 37207}, {3807, 14621}, {3835, 5383}, {3882, 30710}, {3888, 7033}, {3911, 4582}, {3935, 35171}, {3939, 6063}, {3943, 4615}, {3948, 4584}, {3954, 4593}, {3963, 4603}, {3992, 4622}, {3995, 37205}, {4024, 4590}, {4025, 15742}, {4028, 35136}, {4036, 24041}, {4039, 18829}, {4064, 18020}, {4079, 34537}, {4082, 4616}, {4115, 32014}, {4150, 44766}, {4357, 8707}, {4359, 37212}, {4360, 8050}, {4373, 43290}, {4384, 32041}, {4389, 9059}, {4391, 4564}, {4397, 7045}, {4417, 44765}, {4441, 37138}, {4451, 6649}, {4462, 5382}, {4511, 35174}, {4515, 4635}, {4556, 28654}, {4559, 28660}, {4565, 30713}, {4574, 44129}, {4576, 18082}, {4577, 15523}, {4579, 7018}, {4585, 18359}, {4592, 41013}, {4596, 4647}, {4604, 4671}, {4605, 7058}, {4606, 19804}, {4612, 6358}, {4613, 30966}, {4617, 30693}, {4618, 4738}, {4619, 23978}, {4626, 5423}, {4628, 8024}, {4633, 5257}, {4634, 21805}, {4636, 34388}, {4638, 36791}, {4664, 37209}, {4705, 24037}, {4728, 5381}, {4752, 20569}, {4756, 30598}, {4767, 39704}, {4791, 5385}, {4847, 6606}, {4858, 31615}, {5297, 35181}, {5379, 14208}, {5380, 14210}, {6012, 33937}, {6332, 46102}, {6382, 34071}, {6541, 17930}, {6542, 35148}, {6550, 42372}, {6613, 6736}, {6630, 6631}, {6633, 35168}, {6634, 42555}, {6745, 35157}, {7012, 35518}, {7128, 15416}, {7239, 38810}, {7289, 42384}, {8652, 30596}, {8693, 21615}, {8708, 20888}, {8804, 44326}, {9065, 14620}, {9295, 9362}, {9296, 9361}, {9436, 36802}, {10009, 43077}, {11599, 17934}, {11679, 32038}, {14594, 30479}, {16086, 35169}, {16606, 36860}, {17233, 43190}, {17264, 37143}, {17295, 43191}, {17708, 21094}, {17743, 33946}, {17763, 35147}, {17787, 37137}, {18022, 32656}, {18743, 27834}, {18891, 34067}, {19799, 36099}, {20881, 31628}, {20911, 36147}, {20943, 29227}, {20947, 37135}, {21043, 31614}, {21089, 37880}, {21362, 32017}, {21580, 23617}, {21814, 37204}, {23067, 44130}, {23343, 31002}, {24029, 36795}, {25268, 40420}, {25728, 42343}, {27818, 30720}, {28605, 37211}, {28606, 37218}, {29574, 35177}, {29615, 35180}, {29616, 32040}, {30625, 42303}, {31343, 39126}, {31618, 35341}, {32018, 35342}, {32028, 36954}, {32680, 42701}, {32931, 35008}, {33948, 43531}, {34075, 35543}, {34832, 35572}, {36084, 42703}, {36085, 42713}, {36100, 42718}, {36101, 42719}, {36570, 42380}, {37129, 41314}, {37130, 42723}, {37131, 42722}, {37216, 42724}, {37764, 46143}, {38340, 42033}, {38828, 44723}, {40087, 40519}, {41267, 42371}, {44717, 46110}

• X₆₆₇ = barycentric product X_i*X_j for these {i,j}: {1, 649}, {3, 6591}, {4, 22383}, {6, 513}, {7, 3063}, {9, 43924}, {11, 1415}, {19, 1459}, {21, 7180}, {25, 905}, {27, 810}, {28, 647}, {31, 514}, {32, 693}, {34, 652}, {37, 3733}, {39, 18108}, {41, 3676}, {42, 1019}, {44, 23345}, {48, 7649}, {50, 43082}, {55, 3669}, {56, 650}, {57, 663}, {58, 661}, {65, 7252}, {72, 43925}, {74, 14399}, {75, 1919}, {76, 1980}, {80, 21758}, {81, 512}, {82, 21123}, {86, 798}, {87, 20979}, {88, 1960}, {89, 4775}, {99, 3121}, {100, 1015}, {101, 244}, {104, 3310}, {105, 665}, {106, 1635}, {108, 7117}, {109, 2170}, {110, 3125}, {111, 14419}, {112, 18210}, {163, 3120}, {184, 17924}, {190, 3248}, {213, 7192}, {219, 43923}, {220, 43932}, {222, 18344}, {228, 17925}, {238, 3572}, {239, 875}, {241, 884}, {251, 2530}, {256, 20981}, {266, 6729}, {269, 657}, {274, 669}, {277, 8642}, {278, 1946}, {279, 8641}, {284, 4017}, {286, 3049}, {291, 8632}, {292, 659}, {310, 1924}, {330, 8640}, {393, 23224}, {517, 2423}, {518, 43929}, {520, 5317}, {521, 608}, {522, 604}, {523, 1333}, {525, 2203}, {536, 23349}, {560, 3261}, {593, 4705}, {603, 3064}, {644, 1357}, {651, 3271}, {654, 1411}, {656, 1474}, {662, 3122}, {668, 1977}, {672, 1027}, {676, 911}, {692, 1086}, {694, 4164}, {738, 4105}, {739, 891}, {741, 21832}, {753, 14438}, {757, 4079}, {759, 21828}, {764, 1252}, {765, 21143}, {788, 14621}, {812, 1911}, {813, 27846}, {822, 8747}, {825, 4475}, {834, 2214}, {840, 1643}, {849, 4024}, {869, 4817}, {876, 1914}, {890, 3227}, {893, 4367}, {898, 1646}, {899, 23892}, {900, 9456}, {901, 2087}, {902, 1022}, {904, 4369}, {906, 2969}, {909, 1769}, {910, 2424}, {919, 3675}, {923, 4750}, {926, 1462}, {932, 6377}, {934, 14936}, {985, 3250}, {1014, 3709}, {1016, 8027}, {1021, 1042}, {1023, 43922}, {1024, 1458}, {1054, 9262}, {1055, 35348}, {1084, 4623}, {1096, 4091}, {1101, 21131}, {1106, 3239}, {1110, 6545}, {1111, 32739}, {1118, 36054}, {1126, 4979}, {1170, 2488}, {1171, 4983}, {1219, 8662}, {1280, 8659}, {1332, 42067}, {1334, 7203}, {1356, 4631}, {1395, 6332}, {1397, 4391}, {1400, 3737}, {1402, 4560}, {1404, 23838}, {1407, 3900}, {1408, 3700}, {1410, 17926}, {1412, 4041}, {1413, 14298}, {1417, 1639}, {1426, 23090}, {1427, 21789}, {1431, 3287}, {1436, 6129}, {1437, 2501}, {1438, 2254}, {1444, 2489}, {1455, 2432}, {1461, 2310}, {1472, 6590}, {1491, 40746}, {1501, 40495}, {1575, 23355}, {1576, 16732}, {1577, 2206}, {1638, 34068}, {1647, 32665}, {1783, 3937}, {1824, 7254}, {1880, 23189}, {1918, 7199}, {1922, 3766}, {1929, 5029}, {1931, 18001}, {1964, 10566}, {1967, 4107}, {1973, 4025}, {1974, 15413}, {1979, 9267}, {2006, 8648}, {2053, 43051}, {2148, 21102}, {2149, 21132}, {2155, 21172}, {2159, 11125}, {2160, 2605}, {2162, 4083}, {2163, 4893}, {2165, 34948}, {2175, 24002}, {2176, 43931}, {2194, 7178}, {2204, 17094}, {2207, 4131}, {2208, 14837}, {2210, 4444}, {2217, 6589}, {2218, 43060}, {2220, 40086}, {2221, 8678}, {2226, 3251}, {2251, 6548}, {2279, 4724}, {2284, 43921}, {2287, 7250}, {2291, 14413}, {2298, 6371}, {2300, 4581}, {2328, 7216}, {2334, 4790}, {2350, 4040}, {2353, 16757}, {2384, 14421}, {2427, 15635}, {2623, 18180}, {2643, 4556}, {2787, 17961}, {3022, 4617}, {3119, 6614}, {3123, 34071}, {3126, 41934}, {3223, 23572}, {3230, 43928}, {3249, 7035}, {3270, 32714}, {3420, 40134}, {3435, 6588}, {3444, 31947}, {3445, 4394}, {3451, 6615}, {3500, 23655}, {3551, 23472}, {3667, 38266}, {3756, 34080}, {3768, 37129}, {3798, 38252}, {3803, 39951}, {3835, 7121}, {3837, 34077}, {3942, 8750}, {3960, 6187}, {4010, 18268}, {4057, 39798}, {4063, 40148}, {4086, 16947}, {4128, 4603}, {4130, 7023}, {4162, 40151}, {4163, 7366}, {4374, 7104}, {4378, 30650}, {4449, 9315}, {4455, 37128}, {4466, 32676}, {4491, 39981}, {4516, 4565}, {4521, 16945}, {4557, 16726}, {4559, 18191}, {4567, 8034}, {4594, 21755}, {4598, 38986}, {4638, 42084}, {4777, 28607}, {4784, 25426}, {4785, 40735}, {4802, 34819}, {4833, 28658}, {4834, 25417}, {4840, 28625}, {4977, 28615}, {5006, 18015}, {5040, 17946}, {5376, 8661}, {5381, 33917}, {6085, 40400}, {6139, 34056}, {6164, 9259}, {6186, 14838}, {6335, 22096}, {6363, 23617}, {6373, 20332}, {6385, 9426}, {6610, 23351}, {7004, 32674}, {7050, 8712}, {7139, 21184}, {7234, 40432}, {7255, 16584}, {8056, 8643}, {8615, 16612}, {8638, 34018}, {8639, 37870}, {8645, 34578}, {8647, 37626}, {8650, 34892}, {8656, 39963}, {8660, 36805}, {8735, 36059}, {9002, 40401}, {9178, 16702}, {9247, 46107}, {9263, 9299}, {9265, 38238}, {9297, 18825}, {9309, 20980}, {9468, 14296}, {9506, 38348}, {9508, 17962}, {10015, 34858}, {13476, 21007}, {13486, 20982}, {14422, 28317}, {14578, 39534}, {14597, 14775}, {16082, 23220}, {16606, 16695}, {16696, 18105}, {16892, 46289}, {17212, 40729}, {17217, 21759}, {17435, 32735}, {17921, 22381}, {18002, 19623}, {18197, 23493}, {18757, 21196}, {18830, 21762}, {19945, 34075}, {20517, 40145}, {21003, 39979}, {21035, 39179}, {21191, 34248}, {21448, 30234}, {21839, 43926}, {21907, 42670}, {22086, 36125}, {23524, 25576}, {23979, 42455}, {24027, 42462}, {27918, 34067}, {29226, 36614}, {30691, 32726}, {32641, 42753}, {32669, 35015}, {32702, 35014}, {40143, 42653}, {40763, 45882}, {41799, 45878}, {41933, 42757}

• X₆₆₇ = barycentric quotient X_i/X_j for these {i,j}: {1, 1978}, {2, 6386}, {6, 668}, {25, 6335}, {28, 6331}, {31, 190}, {32, 100}, {37, 27808}, {41, 3699}, {42, 4033}, {48, 4561}, {55, 646}, {56, 4554}, {57, 4572}, {58, 799}, {60, 4631}, {81, 670}, {86, 4602}, {100, 31625}, {101, 7035}, {105, 36803}, {110, 4601}, {163, 4600}, {184, 1332}, {187, 42721}, {213, 3952}, {237, 42717}, {238, 27853}, {244, 3261}, {274, 4609}, {284, 7257}, {292, 4583}, {351, 42713}, {512, 321}, {513, 76}, {514, 561}, {522, 28659}, {523, 27801}, {560, 101}, {593, 4623}, {604, 664}, {608, 18026}, {647, 20336}, {649, 75}, {650, 3596}, {652, 3718}, {656, 40071}, {657, 341}, {659, 1921}, {661, 313}, {663, 312}, {665, 3263}, {669, 37}, {688, 3954}, {692, 1016}, {693, 1502}, {739, 889}, {741, 4639}, {764, 23989}, {788, 3661}, {798, 10}, {810, 306}, {812, 18891}, {834, 33935}, {838, 32782}, {849, 4610}, {869, 3807}, {872, 4103}, {875, 335}, {876, 18895}, {884, 36796}, {890, 536}, {891, 35543}, {902, 24004}, {904, 27805}, {905, 305}, {985, 37133}, {1015, 693}, {1019, 310}, {1027, 18031}, {1084, 4705}, {1086, 40495}, {1106, 658}, {1110, 6632}, {1178, 7260}, {1201, 21580}, {1253, 6558}, {1333, 99}, {1357, 24002}, {1395, 653}, {1397, 651}, {1398, 13149}, {1402, 4552}, {1407, 4569}, {1408, 4573}, {1412, 4625}, {1415, 4998}, {1416, 34085}, {1437, 4563}, {1459, 304}, {1462, 46135}, {1472, 37215}, {1474, 811}, {1492, 5388}, {1495, 42716}, {1501, 692}, {1576, 4567}, {1635, 3264}, {1911, 4562}, {1914, 874}, {1917, 32739}, {1918, 1018}, {1919, 1}, {1922, 660}, {1923, 46148}, {1924, 42}, {1946, 345}, {1960, 4358}, {1964, 4568}, {1973, 1897}, {1974, 1783}, {1977, 513}, {1979, 9296}, {1980, 6}, {2084, 15523}, {2162, 18830}, {2170, 35519}, {2175, 644}, {2176, 36863}, {2194, 645}, {2203, 648}, {2204, 36797}, {2205, 4557}, {2206, 662}, {2208, 44327}, {2209, 4595}, {2210, 3570}, {2223, 42720}, {2251, 17780}, {2276, 4505}, {2328, 7258}, {2423, 18816}, {2483, 33941}, {2484, 4385}, {2488, 1229}, {2489, 41013}, {2530, 8024}, {2605, 33939}, {3049, 72}, {3051, 4553}, {3063, 8}, {3120, 20948}, {3121, 523}, {3122, 1577}, {3124, 4036}, {3125, 850}, {3221, 22028}, {3230, 41314}, {3248, 514}, {3249, 244}, {3250, 33931}, {3251, 36791}, {3261, 1928}, {3270, 15416}, {3271, 4391}, {3288, 42711}, {3310, 3262}, {3569, 42703}, {3572, 334}, {3669, 6063}, {3676, 20567}, {3709, 3701}, {3733, 274}, {3737, 28660}, {3766, 44169}, {3768, 6381}, {3803, 40022}, {3937, 15413}, {3960, 40075}, {4017, 349}, {4025, 40364}, {4040, 18152}, {4041, 30713}, {4057, 18140}, {4063, 40087}, {4079, 1089}, {4083, 6382}, {4105, 30693}, {4107, 1926}, {4117, 4079}, {4162, 44723}, {4164, 3978}, {4367, 1920}, {4391, 40363}, {4435, 4087}, {4444, 44172}, {4455, 3948}, {4491, 18145}, {4556, 24037}, {4560, 40072}, {4623, 44168}, {4705, 28654}, {4724, 21615}, {4775, 4671}, {4782, 10009}, {4809, 30874}, {4813, 30596}, {4817, 871}, {4826, 4066}, {4834, 28605}, {4979, 1269}, {4983, 1230}, {5006, 17935}, {5029, 20947}, {5040, 17790}, {5317, 6528}, {6186, 15455}, {6187, 36804}, {6363, 26563}, {6371, 20911}, {6377, 20906}, {6586, 33932}, {6591, 264}, {7023, 36838}, {7032, 33946}, {7104, 3903}, {7109, 40521}, {7117, 35518}, {7121, 4598}, {7122, 18047}, {7180, 1441}, {7192, 6385}, {7234, 3963}, {7250, 1446}, {7252, 314}, {7366, 4626}, {7649, 1969}, {8027, 1086}, {8034, 16732}, {8054, 20949}, {8578, 20940}, {8618, 42723}, {8630, 2276}, {8632, 350}, {8635, 17289}, {8636, 32777}, {8637, 28606}, {8638, 3693}, {8639, 31993}, {8640, 192}, {8641, 346}, {8642, 344}, {8643, 18743}, {8644, 42724}, {8645, 17264}, {8646, 2345}, {8648, 32851}, {8650, 37756}, {8653, 42712}, {8654, 17279}, {8655, 4687}, {8656, 30829}, {8660, 16610}, {8662, 3672}, {9002, 33934}, {9247, 1331}, {9297, 712}, {9299, 9295}, {9426, 213}, {9447, 3939}, {9454, 1026}, {9455, 2284}, {9456, 4555}, {9459, 1023}, {9491, 21877}, {9494, 21814}, {10566, 18833}, {11125, 46234}, {14270, 42701}, {14296, 14603}, {14399, 3260}, {14407, 3992}, {14419, 3266}, {14436, 4439}, {14438, 35548}, {14575, 906}, {14598, 813}, {14599, 3573}, {14621, 46132}, {14827, 4578}, {14936, 4397}, {15413, 40050}, {15451, 42698}, {16692, 32026}, {16695, 31008}, {16732, 44173}, {16757, 40073}, {16947, 1414}, {17494, 40088}, {17924, 18022}, {17961, 35147}, {18002, 11611}, {18108, 308}, {18210, 3267}, {18268, 4589}, {18344, 7017}, {18897, 34067}, {20228, 21272}, {20906, 40367}, {20979, 6376}, {20981, 1909}, {20983, 30473}, {21007, 17143}, {21122, 1760}, {21123, 1930}, {21131, 23994}, {21143, 1111}, {21191, 18837}, {21385, 40089}, {21755, 2533}, {21758, 320}, {21760, 23354}, {21762, 4083}, {21828, 35550}, {21832, 35544}, {21835, 21051}, {22096, 905}, {22383, 69}, {22386, 25098}, {23099, 21833}, {23224, 3926}, {23225, 25083}, {23227, 22095}, {23345, 20568}, {23349, 3227}, {23355, 32020}, {23470, 23794}, {23472, 24524}, {23503, 21080}, {23560, 29226}, {23572, 17149}, {23655, 17786}, {23892, 31002}, {24002, 41283}, {26884, 15418}, {28607, 4597}, {28615, 6540}, {30234, 11059}, {32718, 5381}, {32719, 5376}, {32739, 765}, {32740, 5380}, {34077, 8709}, {34819, 32042}, {34858, 13136}, {34948, 7763}, {34952, 42700}, {36054, 1264}, {38346, 20954}, {38348, 18035}, {38367, 17793}, {38832, 36860}, {38986, 3835}, {38996, 22322}, {39201, 3998}, {40495, 40362}, {40728, 3799}, {40746, 789}, {40935, 7239}, {41267, 35309}, {41280, 1415}, {41935, 4618}, {42067, 17924}, {42653, 42710}, {42658, 42699}, {42660, 42704}, {42662, 42709}, {42664, 42714}, {42670, 32849}, {43082, 20573}, {43923, 331}, {43924, 85}, {43925, 286}, {43929, 2481}, {43931, 6383}

• X₆₆₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 21260, 31251}, {2, 21301, 21260}, {2, 26823, 27168}, {2, 31291, 21301}, {75, 21613, 21440}, {100, 9266, 1016}, {351, 42661, 42653}, {649, 669, 8640}, {649, 1960, 4775}, {649, 5029, 3250}, {649, 8635, 8646}, {649, 8642, 8641}, {649, 8643, 663}, {649, 8655, 669}, {649, 8656, 1960}, {663, 1946, 8641}, {663, 8643, 1960}, {663, 8648, 1946}, {663, 8656, 8643}, {669, 8646, 8641}, {669, 8654, 8642}, {798, 1919, 3063}, {898, 1016, 9266}, {1960, 8648, 42670}, {2530, 14419, 905}, {3733, 16695, 1019}, {3801, 4809, 20517}, {3803, 30234, 905}, {4775, 42670, 8641}, {5638, 5639, 5040}, {8630, 8632, 8640}, {8635, 8655, 8642}, {8636, 8637, 42670}, {8636, 8639, 8646}, {8637, 8639, 4834}, {8641, 8662, 8640}, {8645, 8650, 890}, {8648, 8656, 8642}, {8656, 8660, 8640}, {20979, 20981, 20980}, {20979, 23572, 21763}, {21260, 21301, 31149}, {21260, 31288, 2}, {21301, 31288, 31251}, {26984, 27046, 2}, {28255, 28373, 21260}, {29458, 29770, 24601}, {31149, 31251, 21260}, {31288, 31291, 31149}, {42667, 42668, 42670}