X₆₆₉

CROSSDIFFERENCE OF X₂ AND X₃₉

X₆₆₉ = crossdifference of every pair of points on line {2, 39}

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

X₆₆₉ (CROSSDIFFERENCE OF X₂ AND X₃₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {23, 385}, {187, 237}

• Related Centers: X₂₃ (FAR-OUT POINT), X₃₈₅ (-HIRST INVERSE OF X₆), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂)

Trilinears: \(a^{3}(b^{2} - c^{2}) : :\)

Barycentrics: \(a^{4}(b^{2} - c^{2}) : :\)

Search Value (6, 9, 13): \(1.708101230609\)

▸ Linear Combinations1 combination

• X₆₆₉ = \(6\) X₂ - \(5\) X₃₁₂₇₉, \(3\) X₂ + X₃₁₂₉₉, \(3\) X₂ - \(4\) X₄₄₄₅₁, \(3\) X₃₅₁ - \(2\) X₆₄₇, \(3\) X₃₅₁ - X₃₀₀₅, \(3\) X₃₅₁ + \(2\) X₃₈₀₄, \(3\) X₃₅₁ - \(4\) X₈₆₅₁, \(3\) X₃₅₁ + X₈₆₆₄, \(6\) X₃₅₁ - X₈₆₆₅, X₆₄₇ - \(3\) X₈₆₄₄, \(2\) X₆₄₇ + X₈₆₆₄, \(4\) X₆₄₇ - X₈₆₆₅, \(4\) X₆₄₇ - \(3\) X₁₇₄₁₄, X₈₅₀ - \(3\) X₄₁₀₈, \(2\) X₈₈₇ - \(3\) X₉₄₉₁, X₈₈₇ - \(3\) X₉₄₉₄, \(3\) X₁₆₃₇ - \(2\) X₁₂₀₇₅, \(5\) X₁₆₅₆ - \(4\) X₃₉₅₁₁, \(4\) X₂₅₀₁ - \(3\) X₈₀₂₉, X₃₀₀₅ + \(2\) X₃₈₀₄, X₃₀₀₅ - \(6\) X₈₆₄₄, X₃₀₀₅ - \(4\) X₈₆₅₁, \(2\) X₃₀₀₅ - \(3\) X₁₇₄₁₄, \(4\) X₃₂₆₅ - \(3\) X₁₄₄₂₄, X₃₂₈₈ + \(2\) X₁₄₃₁₈, X₃₈₀₄ + \(3\) X₈₆₄₄, X₃₈₀₄ + \(2\) X₈₆₅₁, \(4\) X₃₈₀₄ + X₈₆₆₅, \(4\) X₃₈₀₄ + \(3\) X₁₇₄₁₄, \(4\) X₅₁₁₃ - \(3\) X₉₂₁₀, X₅₉₉₆ - \(3\) X₁₅₇₂₄, \(2\) X₆₅₆₃ - \(3\) X₁₁₁₂₃, \(3\) X₈₆₄₄ - \(2\) X₈₆₅₁, \(6\) X₈₆₄₄ + X₈₆₆₄, \(12\) X₈₆₄₄ - X₈₆₆₅, \(4\) X₈₆₄₄ - X₁₇₄₁₄, \(4\) X₈₆₅₁ + X₈₆₆₄, \(8\) X₈₆₅₁ - X₈₆₆₅, \(8\) X₈₆₅₁ - \(3\) X₁₇₄₁₄, \(4\) X₈₆₅₃ - X₈₆₆₃, \(2\) X₈₆₆₄ + X₈₆₆₅, \(2\) X₈₆₆₄ + \(3\) X₁₇₄₁₄, X₈₆₆₅ - \(3\) X₁₇₄₁₄, \(3\) X₉₁₃₁ - X₄₁₂₉₈, \(3\) X₉₁₄₇ - X₃₁₂₉₆, \(3\) X₉₁₄₈ - \(4\) X₃₀₄₇₆, \(4\) X₉₄₈₉ - \(3\) X₉₄₉₁, \(2\) X₉₄₈₉ - \(3\) X₉₄₉₄, \(2\) X₁₁₁₇₆ - \(3\) X₁₅₇₂₄, \(2\) X₁₄₂₇₀ - \(3\) X₃₄₉₅₂, \(4\) X₁₄₂₇₀ - \(3\) X₃₉₂₀₁, \(3\) X₁₄₄₂₀ - \(2\) X₃₃₂₉₄, \(4\) X₂₃₃₀₁ - \(3\) X₃₁₁₇₆, \(4\) X₂₃₃₀₁ - \(5\) X₃₁₂₇₉, \(2\) X₂₃₃₀₁ + X₃₁₂₉₉, X₂₃₃₀₁ - \(3\) X₄₅₃₁₇, \(3\) X₃₁₁₇₆ - \(5\) X₃₁₂₇₉, \(3\) X₃₁₁₇₆ + \(2\) X₃₁₂₉₉, \(3\) X₃₁₁₇₆ - \(2\) X₄₄₄₄₅, \(3\) X₃₁₁₇₆ - \(8\) X₄₄₄₅₁, X₃₁₁₇₆ - \(4\) X₄₅₃₁₇, \(5\) X₃₁₂₇₉ + \(2\) X₃₁₂₉₉, \(5\) X₃₁₂₇₉ - \(2\) X₄₄₄₄₅, \(5\) X₃₁₂₇₉ - \(8\) X₄₄₄₅₁, \(5\) X₃₁₂₇₉ - \(12\) X₄₅₃₁₇, X₃₁₂₉₉ + \(4\) X₄₄₄₅₁, X₃₁₂₉₉ + \(6\) X₄₅₃₁₇, \(3\) X₃₄₉₅₂ - X₄₂₆₆₀, \(3\) X₃₉₂₀₁ - \(2\) X₄₂₆₆₀, X₄₄₄₄₅ - \(4\) X₄₄₄₅₁, X₄₄₄₄₅ - \(6\) X₄₅₃₁₇, \(2\) X₄₄₄₅₁ - \(3\) X₄₅₃₁₇

X₆₆₉ lies on the Kiepert parabola.

▸ Incident Central Lines129 lines

{1, 18197}, {2, 23301}, {3, 1499}, {6, 9009}, {10, 21726}, {22, 6563}, {23, 385}, {25, 878}, {31, 875}, {32, 23099}, {55, 25819}, {75, 21441}, {99, 886}, {100, 9362}, {110, 805}, {111, 5970}, {112, 9091}, {157, 40914}, {160, 7712}, {184, 30451}, {187, 237}, {353, 9212}, {511, 30219}, {513, 27469}, {514, 23786}, {525, 2528}, {661, 4455}, {684, 924}, {688, 864}, {690, 2525}, {804, 850}, {865, 1648}, {881, 3051}, {1252, 34067}, {1383, 9178}, {1402, 7180}, {1491, 27648}, {1510, 6132}, {1576, 23357}, {1598, 39533}, {1634, 14607}, {1637, 12075}, {1656, 39511}, {1977, 4117}, {1995, 8371}, {2106, 3733}, {2308, 23570}, {2353, 5489}, {2373, 2868}, {2422, 46319}, {2451, 3221}, {2485, 2514}, {2487, 20470}, {2489, 44099}, {2513, 3050}, {2930, 30220}, {2937, 8151}, {3124, 15630}, {3222, 4609}, {3233, 7468}, {3265, 3566}, {3398, 39518}, {3455, 14443}, {3700, 17989}, {3800, 20854}, {3806, 7927}, {4010, 27731}, {4083, 23506}, {4128, 22386}, {4132, 4782}, {4151, 4401}, {4367, 5937}, {4467, 23401}, {4724, 8672}, {4897, 16678}, {5020, 14341}, {5026, 38998}, {5466, 14002}, {5468, 41337}, {5940, 35296}, {5996, 11176}, {6041, 22260}, {6130, 30735}, {6636, 10190}, {6753, 17994}, {7203, 16878}, {7253, 23400}, {7492, 9168}, {8617, 44007}, {8704, 11616}, {8723, 35268}, {9125, 39477}, {9131, 41298}, {9137, 20403}, {9148, 30476}, {9213, 9999}, {9313, 43060}, {10278, 13595}, {10279, 13621}, {10280, 18369}, {11171, 39501}, {11182, 37338}, {11183, 14096}, {11328, 34290}, {11450, 26881}, {11479, 44931}, {11634, 34245}, {11641, 24974}, {12106, 16220}, {13564, 32204}, {14417, 32478}, {15080, 39495}, {20965, 38237}, {20976, 38241}, {20981, 23467}, {21051, 27045}, {21301, 27293}, {21448, 34204}, {21763, 23574}, {23964, 32696}, {25862, 45882}, {28255, 30023}, {28373, 28470}, {32320, 39469}, {32473, 45687}, {34347, 34396}, {36182, 40871}, {37777, 41357}, {41939, 45900}

▸ Functional Relations38 relations

• X₆₆₉ = midpoint of X_i and X_j for these {i,j}: {647, 3804}, {3005, 8664}, {5027, 14318}, {18105, 21006}, {31299, 44445}

• X₆₆₉ = reflection of X_i in X_j for these {i,j}: {2, 45317}, {3, 5926}, {351, 8644}, {647, 8651}, {887, 9489}, {2514, 2485}, {3005, 647}, {3049, 9426}, {3288, 5027}, {5996, 11176}, {8639, 667}, {8664, 3804}, {8665, 3005}, {9178, 46001}, {9491, 9494}, {17414, 351}, {17415, 2491}, {18117, 6140}, {23301, 44451}, {30735, 6130}, {31176, 2}, {39201, 34952}, {42660, 14270}, {44445, 23301}

• X₆₆₉ = isogonal conjugate of X₆₇₀

• X₆₆₉ = isotomic conjugate of X₄₆₀₉

• X₆₆₉ = complement of X₄₄₄₄₅

• X₆₆₉ = anticomplement of X₂₃₃₀₁

• X₆₆₉ = circumcircle-inverse of X₅₁₀₈

• X₆₆₉ = Parry-circle-inverse of X₅₀₂₇

• X₆₆₉ = Stammler-circle-inverse of X₁₈₃₄₆

• X₆₆₉ = Parry-isodynamic-circle-inverse of X₅₁₀₆

• X₆₆₉ = tangential-isogonal conjugate of X₇₆₆₉

• X₆₆₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {163, 32747}, {662, 32548}, {2998, 21294}, {3222, 6327}, {3223, 3448}, {3224, 21221}, {4599, 19562}, {34248, 148}

• X₆₆₉ = X_i-complementary conjugate of X_j for these (i,j): {31, 38996}, {163, 6665}, {6573, 21238}, {6664, 21253}

• X₆₆₉ = X_i-Ceva conjugate of X_j for these (i,j): {1, 21762}, {2, 38996}, {6, 9427}, {25, 3124}, {31, 1977}, {32, 1084}, {56, 21755}, {98, 2086}, {99, 6}, {100, 21753}, {107, 42295}, {110, 3051}, {512, 3049}, {667, 798}, {670, 9490}, {689, 20965}, {805, 32748}, {931, 1185}, {932, 23551}, {1402, 3121}, {1576, 32}, {1634, 5007}, {1919, 1924}, {2353, 20975}, {2373, 1648}, {2980, 115}, {3222, 2}, {3455, 21906}, {3903, 23546}, {4557, 213}, {4559, 21751}, {6573, 3589}, {7087, 20982}, {7954, 16285}, {8640, 23503}, {8709, 2238}, {9150, 3231}, {9426, 9491}, {10425, 3289}, {14574, 40981}, {16277, 39691}, {17938, 237}, {18105, 512}, {26714, 18899}, {32696, 2211}, {32729, 14567}, {32734, 217}, {34067, 41333}, {34207, 3269}, {34444, 1015}, {34445, 23554}, {39644, 44114}, {40519, 20970}

• X₆₆₉ = X_i-cross conjugate of X_j for these (i,j): {688, 512}, {1084, 32}, {2491, 881}, {3221, 9491}, {4117, 7109}, {9427, 6}, {9494, 9426}, {23099, 1084}, {23216, 1501}

• X₆₆₉ = cevapoint of X_i and X_j for these (i,j): {6, 9431}, {512, 3221}, {688, 9494}, {1084, 23099}

• X₆₆₉ = crosspoint of X_i and X_j for these (i,j): {6, 99}, {32, 1576}, {110, 251}, {213, 4557}, {512, 2489}, {667, 1919}, {691, 15387}, {14573, 14574}, {32729, 41936}

• X₆₆₉ = crosssum of X_i and X_j for these (i,j): {1, 18197}, {2, 512}, {6, 21006}, {69, 3267}, {75, 7199}, {76, 850}, {99, 4563}, {110, 4611}, {126, 690}, {141, 523}, {274, 7192}, {513, 3739}, {514, 3741}, {520, 6389}, {522, 21246}, {525, 1368}, {650, 42397}, {668, 1978}, {669, 9490}, {693, 20911}, {740, 27854}, {799, 7257}, {804, 39080}, {826, 21248}, {888, 35073}, {1269, 3261}, {2396, 15631}, {2786, 20339}, {3051, 9491}, {3221, 6374}, {3794, 4560}, {6333, 32458}, {8672, 10472}, {9402, 34021}, {23285, 42554}, {23610, 31646}, {35522, 36792}

• X₆₆₉ = trilinear pole of line {887, 1084}

• X₆₆₉ = crossdifference of every pair of points on line {2, 39}

• X₆₆₉ = circumcircle-pole of line X₂X₆

• X₆₆₉ = trilinear pole of line X₈₈₇X₁₀₈₄

• X₆₆₉ = complement of isotomic conjugate of isogonal conjugate of X₂₁₀₀₆

• X₆₆₉ = perspector of hyperbola {A, B, C,X₆,X₃₂}}, which is the locus of barycentric product of circumcircle-X₅₁₂-antipodes

• X₆₆₉ = perspector of the tangential triangle and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola

• X₆₆₉ = X₆₅₀-of-the-tangential-triangle

• X₆₆₉ = radical center of {circumcircle, 1st Brocard circle, 1st anti-Brocard circle}

• X₆₆₉ = radical center of {circumcircle, anti-Brocard circle, anti-McCay circumcircle}

• X₆₆₉ = trilinear pole of PU(91)

• X₆₆₉ = barycentric product of PU(105)

• X₆₆₉ = bicentric difference of PU(148)

• X₆₆₉ = PU(148)-harmonic conjugate of X₂

• X₆₆₉ = pole wrt circumcircle of line X₂X₆

• X₆₆₉ = complement of polar conjugate of isogonal conjugate of X₂₂₁₅₉

• X₆₆₉ = X_i-isoconjugate of X_j for these (i,j): {1, 670}, {2, 799}, {6, 4602}, {7, 7257}, {8, 4625}, {10, 4623}, {21, 4572}, {31, 4609}, {38, 689}, {39, 37204}, {58, 6386}, {63, 6331}, {69, 811}, {75, 99}, {76, 662}, {81, 1978}, {85, 645}, {86, 668}, {92, 4563}, {100, 310}, {101, 6385}, {109, 40072}, {110, 561}, {112, 40364}, {141, 4593}, {162, 305}, {163, 1502}, {190, 274}, {226, 4631}, {239, 4639}, {249, 20948}, {264, 4592}, {279, 7258}, {286, 4561}, {304, 648}, {312, 4573}, {314, 664}, {321, 4610}, {325, 36036}, {326, 6528}, {330, 36860}, {332, 18026}, {333, 4554}, {336, 877}, {341, 4616}, {346, 4635}, {349, 4612}, {350, 4589}, {514, 4601}, {519, 4634}, {523, 24037}, {525, 46254}, {643, 6063}, {646, 1434}, {651, 28660}, {661, 34537}, {666, 18157}, {671, 24039}, {693, 4600}, {757, 27808}, {789, 30966}, {798, 44168}, {805, 1926}, {823, 3926}, {850, 24041}, {873, 3952}, {874, 18827}, {880, 1581}, {886, 2234}, {892, 14210}, {894, 7260}, {1016, 7199}, {1019, 31625}, {1043, 4569}, {1088, 7256}, {1101, 44173}, {1102, 15352}, {1109, 31614}, {1269, 4596}, {1310, 44154}, {1332, 44129}, {1414, 3596}, {1509, 4033}, {1576, 1928}, {1577, 4590}, {1634, 18833}, {1821, 2396}, {1909, 4594}, {1920, 4603}, {1921, 4584}, {1930, 4577}, {1934, 17941}, {1959, 43187}, {1964, 42371}, {1966, 18829}, {1969, 4558}, {2407, 33805}, {2421, 46273}, {2617, 34384}, {2966, 46238}, {3112, 4576}, {3116, 9063}, {3222, 17149}, {3261, 4567}, {3264, 4622}, {3265, 23999}, {3266, 36085}, {3570, 40017}, {3736, 46132}, {3882, 40827}, {3978, 37134}, {4077, 6064}, {4086, 7340}, {4176, 36126}, {4358, 4615}, {4359, 4632}, {4391, 4620}, {4481, 5388}, {4551, 18021}, {4555, 30939}, {4556, 27801}, {4562, 30940}, {4565, 28659}, {4570, 40495}, {4575, 18022}, {4583, 33295}, {4598, 31008}, {4599, 8024}, {4611, 46244}, {4633, 19804}, {4998, 18155}, {5209, 35147}, {5468, 46277}, {5546, 20567}, {6335, 17206}, {6516, 44130}, {6540, 16709}, {6632, 16727}, {7035, 7192}, {7182, 36797}, {7239, 7307}, {7799, 32680}, {8033, 27805}, {8707, 16739}, {10159, 18062}, {11059, 37216}, {11794, 33764}, {14208, 18020}, {15418, 35145}, {15455, 34016}, {16609, 36806}, {17446, 35567}, {17708, 20944}, {17871, 42297}, {17930, 20947}, {17932, 40703}, {17934, 18032}, {18023, 23889}, {18024, 23997}, {18140, 37205}, {18156, 35136}, {18206, 36803}, {18695, 18831}, {18750, 44326}, {18830, 33296}, {20641, 44766}, {20939, 37880}, {27853, 37128}, {30736, 36133}, {32676, 40050}, {33809, 36953}, {33946, 38810}, {34594, 40087}, {35544, 36066}, {37133, 40773}, {40088, 43076}, {44179, 46134}, {44769, 46234}

• X₆₆₉ = barycentric product X_i*X_j for these {i,j}: {1, 798}, {3, 2489}, {4, 3049}, {6, 512}, {10, 1919}, {19, 810}, {25, 647}, {31, 661}, {32, 523}, {37, 667}, {39, 18105}, {41, 4017}, {42, 649}, {50, 15475}, {51, 2623}, {55, 7180}, {56, 3709}, {58, 4079}, {65, 3063}, {74, 14398}, {75, 1924}, {76, 9426}, {82, 2084}, {83, 688}, {98, 2491}, {99, 1084}, {100, 3121}, {101, 3122}, {106, 14407}, {110, 3124}, {111, 351}, {112, 20975}, {115, 1576}, {163, 2643}, {181, 7252}, {184, 2501}, {187, 9178}, {213, 513}, {220, 7250}, {228, 6591}, {232, 878}, {237, 2395}, {248, 17994}, {249, 22260}, {251, 3005}, {263, 3288}, {292, 4455}, {308, 9494}, {321, 1980}, {338, 14574}, {385, 881}, {393, 39201}, {418, 15422}, {511, 2422}, {514, 1918}, {520, 2207}, {525, 1974}, {526, 11060}, {560, 1577}, {574, 46001}, {604, 4041}, {645, 1356}, {650, 1402}, {656, 1973}, {657, 1042}, {663, 1400}, {670, 9427}, {690, 32740}, {691, 21906}, {692, 3125}, {693, 2205}, {694, 5027}, {729, 888}, {739, 14404}, {755, 14428}, {788, 40747}, {799, 4117}, {804, 9468}, {805, 2086}, {822, 1096}, {826, 46288}, {842, 6041}, {843, 9171}, {850, 1501}, {865, 9091}, {872, 1019}, {875, 2238}, {876, 41333}, {879, 2211}, {882, 1691}, {887, 3228}, {893, 7234}, {922, 23894}, {923, 2642}, {932, 21835}, {941, 8639}, {1015, 4557}, {1018, 3248}, {1027, 39258}, {1106, 4171}, {1169, 42661}, {1171, 8663}, {1245, 2484}, {1252, 8034}, {1253, 7216}, {1333, 4705}, {1334, 43924}, {1383, 17414}, {1395, 8611}, {1397, 3700}, {1407, 4524}, {1409, 18344}, {1415, 4516}, {1427, 8641}, {1459, 2333}, {1495, 2433}, {1499, 39238}, {1500, 3733}, {1637, 40352}, {1645, 9150}, {1648, 32729}, {1649, 41936}, {1692, 35364}, {1824, 22383}, {1880, 1946}, {1911, 21832}, {1917, 20948}, {1922, 4010}, {1923, 18070}, {1976, 3569}, {1977, 3952}, {1989, 14270}, {2028, 41881}, {2029, 41880}, {2054, 5029}, {2088, 14560}, {2165, 34952}, {2175, 7178}, {2179, 2616}, {2200, 7649}, {2206, 4024}, {2281, 8678}, {2334, 4832}, {2351, 6753}, {2353, 2485}, {2394, 9407}, {2421, 15630}, {2451, 9292}, {2492, 3455}, {2519, 15369}, {2533, 7104}, {2715, 44114}, {2799, 14601}, {2971, 4558}, {2987, 42663}, {2998, 9491}, {3050, 27375}, {3108, 8664}, {3114, 9006}, {3120, 32739}, {3199, 23286}, {3221, 3224}, {3223, 23503}, {3225, 9429}, {3265, 36417}, {3267, 44162}, {3269, 32713}, {3271, 4559}, {3407, 17415}, {3444, 42653}, {3447, 8574}, {3457, 6137}, {3458, 6138}, {3572, 3747}, {3690, 43925}, {3708, 32676}, {3804, 39951}, {3903, 21755}, {4049, 9459}, {4077, 9447}, {4083, 21759}, {4155, 18268}, {4367, 40729}, {4563, 42068}, {4573, 7063}, {4574, 42067}, {4580, 27369}, {4590, 23099}, {4630, 39691}, {4729, 38266}, {4730, 9456}, {4770, 28607}, {4775, 28658}, {4834, 28625}, {4983, 28615}, {5106, 14606}, {5113, 46286}, {5191, 14998}, {5291, 18002}, {5466, 14567}, {5489, 41937}, {5638, 5639}, {6140, 14579}, {6187, 21828}, {6331, 23216}, {6378, 16695}, {6524, 32320}, {6529, 34980}, {6531, 39469}, {6562, 10318}, {6587, 33581}, {6784, 26714}, {7109, 7192}, {7121, 21834}, {7212, 18265}, {7255, 21815}, {8029, 23357}, {8061, 46289}, {8105, 42667}, {8106, 42668}, {8541, 30491}, {8640, 16606}, {8644, 21448}, {8651, 8770}, {8665, 39955}, {8749, 9409}, {8754, 32661}, {8789, 14295}, {8791, 42659}, {8882, 15451}, {8884, 42293}, {9033, 40354}, {9218, 19610}, {9233, 44173}, {9247, 24006}, {9297, 27810}, {9299, 21893}, {9402, 40770}, {9418, 43665}, {9462, 9489}, {10097, 44102}, {10412, 19627}, {10566, 41267}, {11166, 11186}, {14273, 14908}, {14380, 14581}, {14569, 46088}, {14573, 18314}, {14575, 14618}, {14582, 34397}, {14586, 41221}, {14593, 30451}, {14599, 35352}, {14600, 16230}, {14642, 44705}, {14910, 21731}, {15321, 37085}, {15387, 21905}, {15412, 40981}, {17735, 18001}, {17961, 17989}, {17962, 17990}, {17963, 17992}, {17965, 17995}, {17966, 18000}, {18108, 21814}, {19626, 35522}, {20578, 34394}, {20579, 34395}, {20683, 43929}, {20979, 23493}, {21758, 34857}, {21830, 23355}, {22105, 41272}, {22227, 29227}, {23105, 23963}, {23610, 34537}, {23878, 46319}, {27374, 39182}, {30442, 32319}, {30735, 40823}, {32641, 42752}, {32696, 41172}, {33294, 40146}, {34067, 39786}, {34079, 42666}, {34212, 42671}, {34288, 42660}, {41489, 42658}

• X₆₆₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 4602}, {2, 4609}, {6, 670}, {25, 6331}, {31, 799}, {32, 99}, {37, 6386}, {41, 7257}, {42, 1978}, {82, 37204}, {83, 42371}, {99, 44168}, {110, 34537}, {115, 44173}, {163, 24037}, {184, 4563}, {213, 668}, {237, 2396}, {251, 689}, {351, 3266}, {512, 76}, {513, 6385}, {523, 1502}, {525, 40050}, {560, 662}, {604, 4625}, {647, 305}, {649, 310}, {650, 40072}, {656, 40364}, {661, 561}, {663, 28660}, {667, 274}, {688, 141}, {692, 4601}, {729, 886}, {798, 75}, {804, 14603}, {810, 304}, {850, 40362}, {872, 4033}, {875, 40017}, {881, 1916}, {882, 18896}, {887, 538}, {888, 30736}, {904, 7260}, {922, 24039}, {1084, 523}, {1106, 4635}, {1253, 7258}, {1333, 4623}, {1356, 7178}, {1397, 4573}, {1400, 4572}, {1402, 4554}, {1500, 27808}, {1501, 110}, {1576, 4590}, {1577, 1928}, {1645, 9148}, {1691, 880}, {1911, 4639}, {1917, 163}, {1918, 190}, {1919, 86}, {1922, 4589}, {1924, 1}, {1927, 37134}, {1973, 811}, {1974, 648}, {1976, 43187}, {1977, 7192}, {1980, 81}, {2084, 1930}, {2086, 14295}, {2175, 645}, {2194, 4631}, {2200, 4561}, {2205, 100}, {2206, 4610}, {2207, 6528}, {2209, 36860}, {2211, 877}, {2395, 18024}, {2422, 290}, {2484, 44154}, {2485, 40073}, {2489, 264}, {2491, 325}, {2492, 40074}, {2501, 18022}, {2531, 7794}, {2623, 34384}, {2643, 20948}, {2971, 14618}, {3005, 8024}, {3049, 69}, {3050, 33769}, {3051, 4576}, {3063, 314}, {3121, 693}, {3122, 3261}, {3124, 850}, {3125, 40495}, {3221, 6374}, {3248, 7199}, {3249, 17205}, {3267, 40360}, {3288, 20023}, {3407, 9063}, {3700, 40363}, {3709, 3596}, {3747, 27853}, {3774, 4505}, {3804, 40022}, {4010, 44169}, {4017, 20567}, {4041, 28659}, {4079, 313}, {4117, 661}, {4455, 1921}, {4557, 31625}, {4705, 27801}, {4826, 30596}, {5027, 3978}, {7063, 3700}, {7104, 4594}, {7109, 3952}, {7178, 41283}, {7180, 6063}, {7234, 1920}, {7252, 18021}, {8027, 16727}, {8029, 23962}, {8034, 23989}, {8630, 40773}, {8639, 34284}, {8640, 31008}, {8644, 11059}, {8660, 16711}, {8663, 1230}, {8664, 39998}, {8789, 805}, {9006, 3094}, {9171, 45809}, {9178, 18023}, {9233, 1576}, {9247, 4592}, {9407, 2407}, {9418, 2421}, {9419, 15631}, {9426, 6}, {9427, 512}, {9429, 698}, {9431, 9428}, {9447, 643}, {9448, 5546}, {9456, 4634}, {9468, 18829}, {9489, 7757}, {9491, 194}, {9494, 39}, {11060, 35139}, {14270, 7799}, {14295, 18901}, {14398, 3260}, {14404, 35543}, {14407, 3264}, {14428, 35549}, {14567, 5468}, {14573, 18315}, {14574, 249}, {14575, 4558}, {14598, 4584}, {14600, 17932}, {14601, 2966}, {14602, 17941}, {14604, 17938}, {14618, 44161}, {14827, 7256}, {15451, 28706}, {15475, 20573}, {15630, 43665}, {17414, 9464}, {17415, 3314}, {17938, 39292}, {17994, 44132}, {18105, 308}, {19626, 691}, {19627, 10411}, {20968, 4611}, {20975, 3267}, {21051, 40367}, {21751, 3888}, {21755, 4374}, {21759, 18830}, {21762, 17217}, {21828, 40075}, {21832, 18891}, {21835, 20906}, {21837, 33932}, {21906, 35522}, {22096, 15419}, {22260, 338}, {23099, 115}, {23216, 647}, {23357, 31614}, {23503, 17149}, {23610, 3124}, {25423, 10010}, {27369, 41676}, {32320, 4176}, {32676, 46254}, {32696, 41174}, {32739, 4600}, {32740, 892}, {33581, 44326}, {33875, 23342}, {34952, 7763}, {34980, 4143}, {35352, 44170}, {36417, 107}, {37085, 7768}, {38996, 44445}, {39201, 3926}, {39238, 35179}, {39469, 6393}, {40146, 44766}, {40351, 1304}, {40354, 16077}, {40373, 32661}, {40747, 46132}, {40823, 35575}, {40935, 33946}, {40981, 14570}, {41221, 15415}, {41267, 4568}, {41280, 4565}, {41331, 1634}, {41333, 874}, {41993, 20578}, {41994, 20579}, {42068, 2501}, {42659, 37804}, {42660, 32833}, {42661, 1228}, {44112, 15418}, {44162, 112}, {44173, 40359}, {46001, 40826}, {46104, 42395}, {46288, 4577}, {46289, 4593}

• X₆₆₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 23301, 31279}, {2, 26148, 31003}, {2, 31299, 44445}, {2, 44445, 23301}, {75, 21614, 21441}, {351, 3005, 647}, {351, 3804, 8665}, {351, 8664, 3005}, {351, 42652, 3231}, {647, 3005, 17414}, {647, 8644, 8651}, {647, 8651, 351}, {647, 8664, 8665}, {649, 663, 2978}, {649, 8640, 890}, {649, 8655, 667}, {667, 8640, 649}, {667, 8641, 8646}, {667, 8642, 8654}, {887, 9489, 9491}, {887, 9494, 9489}, {1995, 44823, 8371}, {3804, 8644, 647}, {3804, 8651, 3005}, {4057, 23865, 21005}, {4455, 7234, 661}, {5638, 5639, 5027}, {5996, 15724, 11176}, {6137, 6138, 9210}, {7492, 9168, 44822}, {8644, 42659, 34952}, {8651, 8664, 17414}, {8665, 17414, 3005}, {14270, 42660, 39201}, {18773, 18774, 887}, {20979, 23655, 20983}, {23301, 44445, 31176}, {23301, 44451, 2}, {23301, 45317, 44451}, {24533, 25537, 2}, {24533, 28401, 31003}, {25473, 25686, 2}, {27016, 27077, 2}, {31176, 31279, 23301}, {31299, 44451, 31176}, {31299, 45317, 31279}, {34952, 42660, 14270}, {42667, 42668, 351}, {44445, 44451, 31279}