X₆₄₈

TRILINEAR POLE OF EULER LINE

X₆₄₈ is constructed as the pole of the Euler line L as follows: let A″, B″, C″ be the points where L meets the sidelines BC, CA, AB of ABC. Let A', B', C' be the harmonic conjugates of A″, B″, C″ with respect to {B,C}, {C,A}, {A,B}, respectively, The lines AA', BB', CC' concur in X₆₄₈.

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

X₆₄₈ (TRILINEAR POLE OF EULER LINE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {99, 112}, {107, 110}

• Related Centers: X₉₉ (STEINER POINT), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

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

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

Search Value (6, 9, 13): \(-0.963168126295\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\csc 2A \csc(B - C) : :\)

• Trilinears: \((\csc A)/(\tan B - \tan C) : :\)

▸ Linear Combinations1 combination

• X₆₄₈ = \(3\) X₂ - \(4\) X₂₃₅₈₃, \(7\) X₂ - \(8\) X₄₀₄₇₇, \(9\) X₂ - \(8\) X₄₀₄₈₄, \(3\) X₂₅₀ - \(2\) X₇₄₇₃, \(2\) X₂₉₇ - \(3\) X₃₇₇₆₅, X₃₄₀ - \(4\) X₁₉₉₀, X₃₄₀ - \(3\) X₃₇₇₆₅, X₁₄₉₄ - \(4\) X₃₁₆₃, \(3\) X₁₄₉₄ - \(4\) X₁₅₅₂₆, \(3\) X₁₄₉₄ - \(8\) X₂₃₅₈₃, \(3\) X₁₄₉₄ - \(2\) X₃₉₃₅₂, X₁₄₉₄ + \(2\) X₃₉₃₅₈, \(7\) X₁₄₉₄ - \(16\) X₄₀₄₇₇, \(9\) X₁₄₉₄ - \(16\) X₄₀₄₈₄, \(4\) X₁₅₇₆ - \(3\) X₃₅₂₇₈, \(2\) X₁₆₃₂ - \(3\) X₃₅₂₇₈, \(4\) X₁₉₉₀ - \(3\) X₃₇₇₆₅, \(3\) X₃₁₆₃ - X₁₅₅₂₆, \(3\) X₃₁₆₃ - \(2\) X₂₃₅₈₃, \(6\) X₃₁₆₃ - X₃₉₃₅₂, \(2\) X₃₁₆₃ + X₃₉₃₅₈, \(7\) X₃₁₆₃ - \(4\) X₄₀₄₇₇, \(9\) X₃₁₆₃ - \(4\) X₄₀₄₈₄, \(3\) X₆₅₃₀ - \(2\) X₄₄₂₂₈, \(4\) X₇₄₇₃ - \(3\) X₃₀₇₁₆, X₉₄₁₀ - \(4\) X₃₆₄₃₅, \(2\) X₁₅₅₂₆ + \(3\) X₃₉₃₅₈, \(7\) X₁₅₅₂₆ - \(12\) X₄₀₄₇₇, \(3\) X₁₅₅₂₆ - \(4\) X₄₀₄₈₄, X₁₆₀₇₇ - \(3\) X₂₃₅₈₂, \(2\) X₁₆₀₇₇ - \(3\) X₃₉₀₆₂, \(4\) X₂₃₅₈₃ - X₃₉₃₅₂, \(4\) X₂₃₅₈₃ + \(3\) X₃₉₃₅₈, \(7\) X₂₃₅₈₃ - \(6\) X₄₀₄₇₇, \(3\) X₂₃₅₈₃ - \(2\) X₄₀₄₈₄, X₃₅₄₇₄ - \(3\) X₄₁₂₀₄, X₃₉₃₅₂ + \(3\) X₃₉₃₅₈, \(7\) X₃₉₃₅₂ - \(24\) X₄₀₄₇₇, \(3\) X₃₉₃₅₂ - \(8\) X₄₀₄₈₄, \(7\) X₃₉₃₅₈ + \(8\) X₄₀₄₇₇, \(9\) X₃₉₃₅₈ + \(8\) X₄₀₄₈₄, \(9\) X₄₀₄₇₇ - \(7\) X₄₀₄₈₄, \(3\) X₄₄₅₇₉ - \(2\) X₄₅₃₁₂

X₆₄₈ lies on the Steiner circumellipse, the MacBeath circumconic, the cubics K256 and K676.

▸ Incident Central Lines352 lines

{1, 1982}, {2, 1494}, {4, 542}, {5, 42873}, {6, 264}, {19, 18827}, {20, 9530}, {24, 6179}, {25, 3228}, {27, 903}, {28, 3227}, {29, 1121}, {30, 16075}, {33, 35144}, {34, 35176}, {52, 8884}, {53, 3629}, {69, 1249}, {76, 8743}, {81, 16082}, {83, 1235}, {86, 1815}, {92, 14616}, {94, 275}, {95, 216}, {98, 2452}, {99, 112}, {100, 36077}, {107, 110}, {108, 931}, {132, 147}, {133, 6053}, {146, 10152}, {155, 1093}, {162, 190}, {183, 45141}, {185, 1105}, {186, 41626}, {193, 317}, {194, 1968}, {217, 9291}, {225, 2907}, {232, 385}, {233, 40410}, {242, 35173}, {243, 2651}, {249, 687}, {250, 523}, {253, 45245}, {273, 3759}, {276, 41334}, {281, 35141}, {283, 8764}, {286, 1172}, {297, 340}, {305, 3162}, {314, 41364}, {315, 34163}, {316, 5523}, {318, 3758}, {323, 46106}, {325, 16318}, {333, 34393}, {378, 7757}, {394, 15466}, {399, 34334}, {401, 3284}, {403, 14568}, {415, 23710}, {419, 35146}, {422, 35155}, {423, 8756}, {427, 41624}, {436, 34986}, {445, 42045}, {447, 519}, {450, 3292}, {459, 44877}, {467, 41628}, {468, 18823}, {470, 23713}, {471, 23712}, {472, 8737}, {473, 8738}, {511, 35474}, {514, 35169}, {525, 15459}, {527, 44331}, {530, 6110}, {531, 6111}, {532, 11094}, {533, 11093}, {536, 44330}, {538, 14581}, {543, 40890}, {575, 37124}, {577, 3164}, {599, 11331}, {608, 35159}, {621, 36302}, {622, 36303}, {645, 668}, {651, 823}, {653, 662}, {666, 5379}, {670, 2421}, {677, 7253}, {691, 30247}, {754, 40889}, {799, 36099}, {827, 1289}, {847, 2904}, {850, 17708}, {889, 43925}, {892, 2501}, {925, 933}, {935, 11636}, {1020, 1021}, {1033, 3964}, {1043, 44698}, {1075, 1092}, {1078, 39575}, {1179, 6152}, {1196, 40413}, {1294, 40948}, {1300, 1986}, {1302, 1304}, {1350, 15576}, {1351, 33971}, {1352, 41371}, {1441, 40582}, {1474, 3226}, {1503, 44704}, {1560, 30786}, {1568, 6761}, {1585, 45420}, {1586, 45421}, {1594, 7858}, {1625, 6528}, {1629, 3060}, {1634, 4230}, {1792, 8885}, {1799, 40938}, {1826, 2905}, {1843, 14970}, {1855, 32004}, {1861, 35163}, {1886, 17731}, {1896, 3193}, {1936, 2659}, {1947, 2003}, {1948, 2323}, {1973, 18826}, {1974, 3186}, {1975, 3172}, {1993, 2052}, {2055, 41481}, {2201, 35166}, {2203, 18825}, {2207, 7754}, {2211, 17984}, {2287, 46137}, {2326, 4360}, {2331, 27958}, {2332, 33296}, {2404, 23977}, {2451, 43187}, {2489, 18829}, {2592, 8115}, {2593, 8116}, {2633, 21187}, {2777, 30227}, {2782, 22143}, {2906, 41013}, {2914, 6344}, {2970, 19504}, {2990, 40571}, {2991, 5317}, {3003, 44375}, {3064, 35154}, {3087, 43981}, {3168, 9306}, {3187, 36419}, {3199, 7805}, {3260, 15262}, {3270, 37142}, {3289, 16089}, {3564, 6530}, {3574, 14860}, {3580, 14165}, {3618, 32000}, {3732, 24019}, {4238, 32041}, {4241, 32040}, {4552, 5546}, {4555, 17906}, {4560, 4565}, {4562, 15742}, {4567, 32698}, {4569, 4573}, {4570, 32699}, {4577, 42396}, {4586, 32676}, {4590, 32697}, {4591, 32705}, {4616, 15419}, {4664, 11107}, {5089, 35152}, {5094, 11163}, {5097, 39530}, {5467, 35178}, {5468, 35179}, {5649, 18311}, {5655, 11251}, {5664, 30528}, {5667, 16163}, {5702, 41145}, {5895, 18848}, {5921, 10002}, {5965, 39569}, {5994, 23873}, {5995, 23872}, {6353, 41360}, {6394, 9475}, {6515, 11547}, {6526, 32605}, {6527, 17037}, {6591, 35147}, {6748, 32455}, {6749, 8584}, {7258, 42384}, {7480, 14480}, {7488, 38808}, {7649, 35148}, {7669, 36176}, {7728, 18507}, {7762, 27376}, {7766, 10311}, {7768, 41366}, {7772, 37337}, {7838, 27371}, {7840, 41358}, {8057, 32230}, {8681, 34854}, {8744, 41617}, {8745, 21447}, {8755, 35149}, {8879, 34254}, {9182, 46144}, {9229, 37895}, {9307, 40947}, {9381, 40393}, {9410, 36435}, {9487, 15471}, {9514, 12077}, {9755, 40801}, {10602, 39646}, {10762, 18338}, {10979, 43980}, {11008, 32001}, {11054, 37855}, {11062, 44376}, {11185, 41370}, {11416, 14957}, {11441, 14249}, {11477, 37200}, {11511, 37190}, {12160, 41365}, {12829, 14772}, {13582, 39284}, {14024, 35167}, {14060, 34473}, {14361, 37669}, {14379, 45255}, {14401, 42308}, {14571, 19623}, {14615, 20806}, {14618, 18557}, {14627, 14978}, {14639, 41221}, {14918, 37766}, {14927, 31887}, {14941, 41219}, {15069, 15274}, {15149, 18821}, {15384, 39191}, {15395, 41512}, {15796, 42456}, {15905, 20477}, {15912, 19210}, {16175, 20410}, {16263, 40909}, {16264, 21850}, {16276, 17409}, {16704, 46136}, {16813, 18315}, {17035, 36412}, {17346, 17555}, {17378, 37448}, {17402, 32036}, {17403, 32037}, {17778, 18679}, {17911, 33770}, {17924, 35156}, {17926, 35157}, {17932, 20031}, {18019, 40583}, {18024, 40601}, {18487, 40885}, {18685, 21287}, {18687, 28754}, {18822, 31905}, {18828, 41209}, {19041, 24243}, {19042, 24244}, {20022, 21459}, {21166, 22085}, {22052, 45845}, {22456, 26714}, {23895, 36306}, {23896, 36309}, {23964, 33294}, {25054, 36849}, {25986, 37631}, {27369, 43094}, {30450, 46134}, {31150, 37966}, {31510, 44552}, {31909, 43097}, {32711, 39295}, {34545, 40684}, {34568, 39290}, {34810, 35908}, {34990, 40879}, {35168, 37168}, {35174, 46102}, {36212, 40888}, {36830, 41298}, {37644, 43462}, {37778, 37784}, {37783, 46141}, {38253, 46206}, {38342, 46139}, {40412, 40937}, {40414, 40715}, {40448, 42441}, {40996, 44334}, {41203, 41586}, {41363, 44132}, {42401, 42405}, {43093, 44129}, {44579, 45312}

▸ Functional Relations36 relations

• X₆₄₈ = midpoint of X_i and X_j for these {i,j}: {2, 39358}, {193, 40867}

• X₆₄₈ = reflection of X_i in X_j for these {i,j}: {2, 3163}, {69, 15595}, {95, 39081}, {99, 40866}, {287, 6}, {297, 1990}, {317, 36426}, {340, 297}, {401, 3284}, {1494, 2}, {1632, 1576}, {1972, 216}, {6330, 1249}, {15014, 14581}, {15526, 23583}, {30716, 250}, {39062, 23582}, {39352, 15526}, {40715, 40940}, {40885, 18487}, {40996, 44334}, {46239, 40938}

• X₆₄₈ = isogonal conjugate of X₆₄₇

• X₆₄₈ = isotomic conjugate of X₅₂₅

• X₆₄₈ = complement of X₃₉₃₅₂

• X₆₄₈ = anticomplement of X₁₅₅₂₆

• X₆₄₈ = cevapoint of X₁₁₀ and X₁₁₂

• X₆₄₈ = X_i-cross conjugate of X_j for these (i,j): (6, 250), (110, 99), (112, 107), (520, 95), (523, 264)

• X₆₄₈ = trilinear pole of PU(30)

• X₆₄₈ = polar-circle-inverse of X₁₆₂₇₈

• X₆₄₈ = polar conjugate of X₅₂₃

• X₆₄₈ = cyclocevian conjugate of X₁₃₅₇₃

• X₆₄₈ = crossdifference of PU(109)

• X₆₄₈ = MacBeath circumconic antipode of X₂₈₇

• X₆₄₈ = pole wrt polar circle of trilinear polar of X₅₂₃ (line X₁₁₅X₁₂₅)

• X₆₄₈ = X_i-anticomplementary conjugate of X_j for these (i,j): {107, 21294}, {162, 13219}, {163, 34186}, {250, 4329}, {1101, 6527}, {23582, 6327}, {23590, 5906}, {23964, 8}, {23999, 315}, {24000, 69}, {24019, 3448}, {24021, 317}, {24022, 6515}, {32230, 21270}, {32676, 39352}, {32713, 21221}, {36131, 45289}, {41937, 192}

• X₆₄₈ = X_i-complementary conjugate of X_j for these (i,j): {31, 39062}, {9390, 141}, {9392, 127}, {9406, 32750}, {15351, 2887}

• X₆₄₈ = X_i-Ceva conjugate of X_j for these (i,j): {2, 39062}, {250, 36794}, {264, 30716}, {687, 14590}, {811, 36797}, {4590, 317}, {6331, 99}, {6528, 107}, {15459, 16237}, {16077, 4240}, {16813, 41679}, {18020, 4}, {18831, 110}, {22456, 4230}, {23582, 2}, {32230, 17907}, {33513, 35360}, {34538, 15466}, {42308, 30}, {42396, 112}, {42405, 6528}, {44181, 20}

• X₆₄₈ = X_i-cross conjugate of X_j for these (i,j): {2, 23582}, {4, 18020}, {6, 250}, {19, 15742}, {20, 44181}, {30, 42308}, {69, 44183}, {92, 46102}, {110, 99}, {112, 107}, {193, 4590}, {385, 41174}, {512, 40413}, {514, 40414}, {520, 95}, {521, 40412}, {523, 264}, {525, 2}, {647, 1105}, {651, 662}, {653, 823}, {850, 83}, {1172, 5379}, {1249, 32230}, {1499, 10603}, {1503, 39297}, {1625, 110}, {1636, 1294}, {1783, 162}, {1897, 811}, {1993, 249}, {2407, 687}, {2420, 476}, {2421, 32697}, {2451, 6}, {2489, 32085}, {2501, 4}, {2799, 6330}, {3187, 1016}, {3288, 40801}, {3562, 7045}, {3569, 98}, {3580, 39295}, {3732, 799}, {3868, 4998}, {4230, 22456}, {4240, 16077}, {4391, 14534}, {4560, 31623}, {5664, 46106}, {6368, 40410}, {6587, 18848}, {6753, 8884}, {7192, 40411}, {7253, 86}, {7754, 34537}, {8057, 69}, {8115, 39299}, {8116, 39298}, {8673, 1799}, {8743, 23964}, {9033, 1494}, {9209, 18850}, {9517, 2373}, {9979, 671}, {10015, 24624}, {12077, 14860}, {14273, 17983}, {14316, 1916}, {14391, 265}, {14401, 30}, {14543, 190}, {14544, 664}, {14570, 30450}, {14618, 275}, {15639, 2409}, {16230, 35142}, {17434, 40448}, {17498, 274}, {17924, 40395}, {17925, 27}, {17926, 29}, {18311, 37765}, {18314, 40393}, {20577, 324}, {20580, 15466}, {21300, 40415}, {23090, 21}, {23616, 46115}, {23878, 42330}, {24978, 94}, {25259, 32014}, {31296, 276}, {32320, 3}, {32661, 925}, {33294, 76}, {34211, 2966}, {35311, 18831}, {35318, 35360}, {35324, 930}, {35325, 112}, {35360, 6528}, {35441, 14938}, {36828, 11636}, {39470, 37202}, {40571, 4567}, {41079, 2986}, {41676, 6331}, {41677, 38342}, {41678, 15352}, {43925, 28}, {44409, 7}, {44427, 16080}, {45292, 31621}

• X₆₄₈ = cevapoint of X_i and X_j for these (i,j): {1, 1021}, {2, 525}, {3, 32320}, {4, 2501}, {6, 523}, {21, 23090}, {27, 17925}, {28, 43925}, {29, 17926}, {30, 14401}, {52, 6753}, {81, 4560}, {110, 112}, {185, 647}, {216, 520}, {230, 38359}, {233, 6368}, {323, 5664}, {324, 14618}, {394, 20580}, {512, 1196}, {513, 40941}, {514, 40940}, {521, 40937}, {522, 40942}, {524, 18311}, {650, 1858}, {651, 653}, {850, 1235}, {924, 40939}, {1249, 8057}, {1625, 35360}, {1636, 40948}, {1637, 13202}, {1783, 1897}, {1829, 6591}, {1839, 7649}, {1843, 2489}, {1994, 20577}, {2322, 7253}, {2409, 15639}, {2451, 9308}, {2492, 40949}, {2574, 8106}, {2575, 8105}, {2592, 2593}, {2799, 15595}, {2967, 3569}, {3049, 40947}, {3064, 40950}, {3163, 9033}, {3288, 9755}, {3574, 12077}, {3737, 40979}, {3800, 40179}, {5095, 14273}, {5895, 6587}, {7252, 40980}, {8673, 40938}, {8743, 33294}, {14091, 30211}, {14920, 44427}, {15412, 19170}, {17434, 42441}, {31296, 41334}, {35311, 35318}, {35325, 41676}, {36054, 40946}, {41392, 41512}

• X₆₄₈ = crosspoint of X_i and X_j for these (i,j): {2, 15351}, {99, 44326}, {110, 44828}, {6331, 6528}, {18831, 42405}

• X₆₄₈ = crosssum of X_i and X_j for these (i,j): {3, 22143}, {3049, 39201}, {15451, 42293}

• X₆₄₈ = trilinear pole of line {2, 3}

• X₆₄₈ = crossdifference of every pair of points on line {3269, 9409}

• X₆₄₈ = Feuerbach image of X₄ if ABC is acute

• X₆₄₈ = Brianchon point (perspector) of inscribed parabola with focus X₁₁₂

• X₆₄₈ = perspector of hyperbola {A, B, C,X₁₆₀₇₇,X₁₈₀₂₀,X₂₂₄₅₆}} (the circumconic through the polar conjugates of PU(40))

• X₆₄₈ = X₉₂-isoconjugate of X₃₉₂₀₁

• X₆₄₈ = Steiner-circumellipse-X₁-antipode of X₃₅₁₄₅

• X₆₄₈ = Steiner-circumellipse-X₄-antipode of X₆₇₁

• X₆₄₈ = Steiner-circumellipse-X₆-antipode of X₂₉₀

• X₆₄₈ = trilinear product of circumcircle intercepts of line X₁₉X₂₇

• X₆₄₈ = X_i-isoconjugate of X_j for these (i,j): {1, 647}, {2, 810}, {3, 661}, {4, 822}, {6, 656}, {10, 22383}, {19, 520}, {25, 24018}, {31, 525}, {32, 14208}, {37, 1459}, {41, 17094}, {42, 905}, {48, 523}, {56, 8611}, {63, 512}, {65, 652}, {69, 798}, {71, 513}, {72, 649}, {73, 650}, {74, 2631}, {75, 3049}, {77, 3709}, {78, 7180}, {91, 30451}, {92, 39201}, {101, 18210}, {107, 37754}, {110, 3708}, {112, 2632}, {115, 4575}, {125, 163}, {158, 32320}, {162, 3269}, {184, 1577}, {201, 7252}, {212, 7178}, {213, 4025}, {216, 2616}, {219, 4017}, {222, 4041}, {225, 36054}, {226, 1946}, {228, 514}, {244, 4574}, {255, 2501}, {265, 2624}, {293, 3569}, {295, 21832}, {304, 669}, {305, 1924}, {306, 667}, {307, 3063}, {326, 2489}, {336, 2491}, {521, 1400}, {522, 1409}, {560, 3267}, {577, 24006}, {603, 3700}, {657, 1439}, {662, 20975}, {663, 1214}, {672, 10099}, {684, 1910}, {686, 36053}, {690, 36060}, {692, 4466}, {693, 2200}, {756, 7254}, {823, 34980}, {850, 9247}, {872, 15419}, {878, 1959}, {879, 1755}, {895, 2642}, {896, 10097}, {906, 3120}, {922, 14977}, {923, 14417}, {924, 1820}, {1018, 3937}, {1019, 3690}, {1020, 3270}, {1021, 1425}, {1109, 32661}, {1176, 8061}, {1245, 2522}, {1254, 23090}, {1260, 7216}, {1331, 3125}, {1332, 3122}, {1333, 4064}, {1402, 6332}, {1410, 3239}, {1437, 4024}, {1444, 4079}, {1576, 20902}, {1636, 36119}, {1637, 35200}, {1650, 36131}, {1790, 4705}, {1796, 4983}, {1797, 4730}, {1799, 2084}, {1807, 21828}, {1813, 4516}, {1821, 39469}, {1824, 4091}, {1826, 23224}, {1911, 24459}, {1918, 15413}, {1919, 20336}, {1953, 23286}, {1964, 4580}, {1967, 24284}, {1973, 3265}, {1980, 40071}, {2088, 36061}, {2148, 6368}, {2155, 8057}, {2156, 8673}, {2157, 9517}, {2159, 9033}, {2167, 15451}, {2169, 12077}, {2170, 23067}, {2171, 23189}, {2173, 14380}, {2184, 42658}, {2190, 17434}, {2196, 4010}, {2197, 3737}, {2247, 35909}, {2249, 9391}, {2250, 8677}, {2252, 3657}, {2281, 23874}, {2312, 2435}, {2314, 43709}, {2315, 15328}, {2318, 3669}, {2333, 4131}, {2349, 9409}, {2451, 9255}, {2524, 3223}, {2525, 46289}, {2533, 7116}, {2574, 2579}, {2575, 2578}, {2582, 42667}, {2583, 42668}, {2584, 8106}, {2585, 8105}, {2618, 14533}, {2623, 44706}, {2643, 4558}, {2972, 24019}, {3005, 34055}, {3064, 22341}, {3121, 4561}, {3124, 4592}, {3292, 23894}, {3682, 6591}, {3692, 7250}, {3694, 43924}, {3725, 15420}, {3733, 3949}, {3835, 22381}, {3942, 4557}, {3990, 7649}, {4033, 22096}, {4049, 23202}, {4055, 17924}, {4088, 32658}, {4120, 36058}, {4171, 7053}, {4524, 7177}, {4551, 7117}, {4559, 7004}, {4602, 23216}, {4674, 22086}, {6129, 41087}, {6149, 14582}, {6587, 19614}, {6729, 7591}, {7084, 21107}, {7138, 17926}, {7148, 23092}, {8766, 34212}, {8818, 23226}, {9258, 22089}, {9396, 22143}, {9406, 34767}, {9426, 40364}, {9456, 14429}, {14575, 20948}, {14642, 17898}, {15352, 42080}, {15373, 21051}, {15389, 20910}, {15409, 21720}, {15526, 32676}, {16186, 32678}, {16606, 22090}, {16732, 32656}, {17438, 39180}, {18070, 20775}, {18344, 40152}, {18793, 22092}, {18877, 36035}, {21044, 36059}, {21789, 37755}, {21834, 23086}, {22373, 27805}, {23201, 31010}, {23493, 25098}, {23503, 43714}, {24290, 36057}, {30491, 36263}, {35071, 36126}, {36084, 41172}, {36516, 43963}, {40440, 42293}

• X₆₄₈ = barycentric product X_i*X_j for these {i,j}: {1, 811}, {3, 6528}, {4, 99}, {5, 18831}, {6, 6331}, {7, 36797}, {19, 799}, {21, 18026}, {24, 46134}, {25, 670}, {27, 190}, {28, 668}, {29, 664}, {30, 16077}, {33, 4625}, {34, 7257}, {63, 823}, {69, 107}, {75, 162}, {76, 112}, {81, 6335}, {83, 41676}, {86, 1897}, {92, 662}, {94, 14590}, {95, 35360}, {98, 877}, {100, 286}, {101, 44129}, {108, 314}, {109, 44130}, {110, 264}, {140, 33513}, {141, 42396}, {158, 4592}, {163, 1969}, {186, 35139}, {216, 42405}, {232, 43187}, {240, 36036}, {242, 4589}, {249, 14618}, {250, 850}, {273, 643}, {274, 1783}, {275, 14570}, {276, 1625}, {278, 645}, {281, 4573}, {290, 4230}, {297, 2966}, {298, 36306}, {299, 36309}, {304, 24019}, {305, 32713}, {308, 35325}, {310, 8750}, {311, 933}, {315, 1289}, {316, 935}, {317, 925}, {318, 1414}, {324, 18315}, {325, 685}, {326, 36126}, {331, 5546}, {332, 36127}, {333, 653}, {340, 476}, {343, 16813}, {393, 4563}, {394, 15352}, {403, 18878}, {415, 35154}, {419, 18829}, {422, 35147}, {423, 35148}, {427, 4577}, {428, 35137}, {447, 35169}, {459, 36841}, {468, 892}, {470, 23895}, {471, 23896}, {472, 32037}, {473, 32036}, {511, 22456}, {523, 18020}, {525, 23582}, {561, 32676}, {646, 1396}, {651, 31623}, {656, 23999}, {658, 2322}, {661, 46254}, {666, 15149}, {671, 4235}, {687, 3580}, {689, 1843}, {691, 44146}, {693, 5379}, {801, 41678}, {805, 17984}, {827, 1235}, {880, 17980}, {930, 32002}, {1016, 17925}, {1043, 36118}, {1113, 15165}, {1114, 15164}, {1119, 7256}, {1172, 4554}, {1236, 10423}, {1249, 44326}, {1275, 17926}, {1288, 44128}, {1301, 14615}, {1302, 44134}, {1303, 9291}, {1304, 3260}, {1309, 17139}, {1435, 7258}, {1474, 1978}, {1494, 4240}, {1513, 41074}, {1576, 18022}, {1632, 34405}, {1634, 46104}, {1733, 36105}, {1799, 46151}, {1824, 4623}, {1826, 4610}, {1839, 4632}, {1847, 7259}, {1880, 4631}, {1896, 6516}, {1944, 41207}, {1948, 41206}, {1973, 4602}, {1974, 4609}, {1981, 35145}, {1993, 30450}, {1994, 38342}, {2052, 4558}, {2074, 35156}, {2201, 4639}, {2203, 6386}, {2287, 13149}, {2299, 4572}, {2349, 24001}, {2396, 6531}, {2407, 16080}, {2409, 35140}, {2421, 16081}, {2479, 2480}, {2481, 4238}, {2489, 34537}, {2501, 4590}, {2580, 2581}, {2592, 39298}, {2593, 39299}, {2617, 40440}, {2715, 44132}, {2986, 16237}, {3064, 4620}, {3186, 3222}, {3265, 32230}, {3267, 23964}, {3518, 46139}, {3569, 41174}, {3658, 46133}, {3732, 40411}, {3926, 6529}, {4143, 23590}, {4183, 4569}, {4226, 35142}, {4232, 35179}, {4241, 18025}, {4242, 14616}, {4246, 18816}, {4249, 43093}, {4552, 46103}, {4555, 37168}, {4560, 46102}, {4561, 8747}, {4562, 31905}, {4565, 7017}, {4567, 17924}, {4570, 46107}, {4576, 32085}, {4586, 31909}, {4591, 46109}, {4593, 17442}, {4594, 7009}, {4599, 20883}, {4600, 7649}, {4601, 6591}, {4611, 43678}, {4612, 40149}, {4614, 5342}, {4615, 8756}, {4616, 7046}, {4621, 31917}, {4622, 38462}, {4635, 7079}, {4637, 7101}, {5094, 35138}, {5117, 33514}, {5383, 17921}, {5392, 41679}, {5467, 46111}, {5468, 17983}, {5641, 7473}, {6035, 6103}, {6330, 34211}, {6344, 10411}, {6353, 35136}, {6393, 20031}, {6530, 17932}, {6540, 31900}, {6578, 44143}, {7012, 18155}, {7119, 7260}, {7192, 15742}, {7452, 34393}, {7953, 44142}, {8057, 44181}, {8754, 31614}, {8795, 23181}, {9033, 42308}, {9064, 44133}, {9308, 43188}, {10301, 42367}, {10420, 44138}, {10425, 44145}, {11064, 15459}, {11185, 30247}, {11412, 39418}, {11794, 36794}, {13485, 30716}, {14052, 33799}, {14208, 24000}, {14221, 40118}, {14543, 40414}, {14574, 44161}, {14591, 20573}, {14920, 39290}, {15351, 39062}, {15411, 23984}, {17515, 35174}, {17569, 41072}, {17708, 37765}, {17907, 44766}, {17927, 17930}, {17929, 17987}, {17931, 17985}, {17934, 17982}, {17935, 17981}, {18027, 32661}, {20884, 36095}, {24006, 24041}, {24039, 36128}, {26705, 33297}, {26714, 44144}, {27369, 42371}, {28660, 32674}, {30737, 44770}, {31617, 35318}, {31625, 43925}, {31902, 32042}, {31926, 32041}, {32038, 44734}, {32428, 41208}, {32691, 44154}, {33294, 44183}, {33640, 44136}, {34336, 34574}, {34568, 36789}, {35311, 40410}, {35314, 38428}, {35315, 38427}, {35575, 43976}, {36077, 44140}, {36084, 40703}, {36104, 46238}, {36131, 46234}, {36831, 43752}, {37908, 46135}, {37966, 46141}, {39291, 39931}, {39295, 44427}, {40393, 41677}, {41083, 44327}, {41331, 42395}, {41762, 42297}, {44769, 46106}

• X₆₄₈ = barycentric quotient X_i/X_j for these {i,j}: {1, 656}, {2, 525}, {3, 520}, {4, 523}, {5, 6368}, {6, 647}, {7, 17094}, {9, 8611}, {10, 4064}, {19, 661}, {20, 8057}, {21, 521}, {22, 8673}, {23, 9517}, {24, 924}, {25, 512}, {27, 514}, {28, 513}, {29, 522}, {30, 9033}, {31, 810}, {32, 3049}, {33, 4041}, {34, 4017}, {48, 822}, {49, 37084}, {51, 15451}, {53, 12077}, {54, 23286}, {58, 1459}, {59, 23067}, {60, 23189}, {63, 24018}, {69, 3265}, {74, 14380}, {75, 14208}, {76, 3267}, {81, 905}, {83, 4580}, {86, 4025}, {92, 1577}, {94, 14592}, {98, 879}, {99, 69}, {100, 72}, {101, 71}, {105, 10099}, {107, 4}, {108, 65}, {109, 73}, {110, 3}, {111, 10097}, {112, 6}, {125, 5489}, {141, 2525}, {154, 42658}, {158, 24006}, {162, 1}, {163, 48}, {184, 39201}, {186, 526}, {190, 306}, {216, 17434}, {217, 42293}, {232, 3569}, {233, 35441}, {237, 39469}, {239, 24459}, {242, 4010}, {249, 4558}, {250, 110}, {264, 850}, {265, 43083}, {270, 3737}, {273, 4077}, {274, 15413}, {275, 15412}, {278, 7178}, {281, 3700}, {284, 652}, {286, 693}, {288, 39181}, {297, 2799}, {314, 35518}, {317, 6563}, {318, 4086}, {323, 8552}, {324, 18314}, {325, 6333}, {333, 6332}, {340, 3268}, {376, 9007}, {378, 8675}, {385, 24284}, {393, 2501}, {415, 2785}, {419, 804}, {420, 9479}, {422, 2787}, {423, 2786}, {425, 2798}, {427, 826}, {428, 7927}, {430, 6367}, {441, 39473}, {450, 2797}, {458, 23878}, {461, 4843}, {468, 690}, {469, 23879}, {470, 23870}, {471, 23871}, {472, 23873}, {473, 23872}, {476, 265}, {477, 14220}, {511, 684}, {512, 20975}, {513, 18210}, {514, 4466}, {519, 14429}, {520, 2972}, {523, 125}, {524, 14417}, {525, 15526}, {526, 16186}, {571, 30451}, {577, 32320}, {593, 7254}, {607, 3709}, {608, 7180}, {643, 78}, {644, 3694}, {645, 345}, {647, 3269}, {651, 1214}, {653, 226}, {656, 2632}, {661, 3708}, {662, 63}, {664, 307}, {668, 20336}, {670, 305}, {671, 14977}, {685, 98}, {687, 2986}, {691, 895}, {692, 228}, {799, 304}, {805, 36214}, {811, 75}, {822, 37754}, {823, 92}, {827, 1176}, {842, 35909}, {850, 339}, {851, 9391}, {859, 8677}, {860, 6370}, {862, 4155}, {877, 325}, {892, 30786}, {906, 3990}, {907, 34817}, {915, 3657}, {925, 68}, {930, 3519}, {931, 34259}, {933, 54}, {934, 1439}, {935, 67}, {1010, 23874}, {1018, 3949}, {1019, 3942}, {1020, 37755}, {1021, 34591}, {1101, 4575}, {1113, 2575}, {1114, 2574}, {1172, 650}, {1173, 39180}, {1235, 23285}, {1249, 6587}, {1252, 4574}, {1286, 18124}, {1287, 18125}, {1288, 70}, {1289, 66}, {1291, 43704}, {1294, 43701}, {1297, 2435}, {1299, 43709}, {1300, 15328}, {1301, 64}, {1302, 4846}, {1304, 74}, {1305, 28786}, {1309, 38955}, {1325, 2850}, {1331, 3682}, {1332, 3998}, {1333, 22383}, {1383, 30491}, {1396, 3669}, {1398, 7250}, {1414, 77}, {1415, 1409}, {1435, 7216}, {1437, 23224}, {1444, 4131}, {1474, 649}, {1494, 34767}, {1495, 9409}, {1509, 15419}, {1576, 184}, {1577, 20902}, {1611, 2519}, {1613, 2524}, {1624, 185}, {1625, 216}, {1627, 22159}, {1632, 1899}, {1633, 17441}, {1634, 3917}, {1783, 37}, {1784, 36035}, {1790, 4091}, {1813, 40152}, {1822, 2585}, {1823, 2584}, {1824, 4705}, {1826, 4024}, {1838, 23752}, {1839, 4988}, {1843, 3005}, {1845, 42768}, {1848, 21124}, {1861, 4088}, {1877, 30572}, {1884, 6089}, {1895, 17898}, {1896, 44426}, {1897, 10}, {1957, 17478}, {1968, 2451}, {1969, 20948}, {1973, 798}, {1974, 669}, {1976, 878}, {1978, 40071}, {1981, 8680}, {1989, 14582}, {1990, 1637}, {1995, 30209}, {2052, 14618}, {2073, 2774}, {2074, 8674}, {2075, 2773}, {2173, 2631}, {2189, 7252}, {2190, 2616}, {2193, 36054}, {2194, 1946}, {2201, 21832}, {2203, 667}, {2204, 3063}, {2207, 2489}, {2211, 2491}, {2299, 663}, {2303, 2522}, {2322, 3239}, {2326, 1021}, {2332, 657}, {2333, 4079}, {2355, 4983}, {2393, 42665}, {2396, 6393}, {2407, 11064}, {2409, 1503}, {2420, 3284}, {2421, 36212}, {2445, 42671}, {2452, 22264}, {2485, 38356}, {2489, 3124}, {2501, 115}, {2576, 2579}, {2577, 2578}, {2580, 2583}, {2581, 2582}, {2586, 2589}, {2587, 2588}, {2605, 22094}, {2617, 44706}, {2633, 2629}, {2690, 38535}, {2713, 1942}, {2715, 248}, {2722, 43723}, {2766, 10693}, {2905, 21196}, {2906, 31947}, {2914, 8562}, {2966, 287}, {2967, 41167}, {2970, 23105}, {2971, 22260}, {2986, 15421}, {3003, 686}, {3050, 38352}, {3064, 21044}, {3068, 17431}, {3069, 17432}, {3120, 21134}, {3163, 14401}, {3186, 23301}, {3194, 6129}, {3222, 43714}, {3233, 16163}, {3266, 45807}, {3267, 36793}, {3284, 1636}, {3285, 22086}, {3518, 1510}, {3563, 35364}, {3565, 6391}, {3569, 41172}, {3580, 6334}, {3581, 14314}, {3658, 912}, {3699, 3710}, {3732, 18589}, {3733, 3937}, {3737, 7004}, {3867, 3806}, {3926, 4143}, {3939, 2318}, {3952, 3695}, {4000, 21107}, {4024, 21046}, {4025, 17216}, {4143, 23974}, {4183, 3900}, {4185, 8672}, {4186, 4139}, {4206, 8678}, {4221, 9051}, {4222, 4132}, {4226, 3564}, {4227, 9001}, {4230, 511}, {4232, 1499}, {4233, 3309}, {4234, 9031}, {4235, 524}, {4236, 34381}, {4237, 9028}, {4238, 518}, {4240, 30}, {4241, 516}, {4242, 758}, {4243, 916}, {4244, 3827}, {4246, 517}, {4247, 9002}, {4248, 3667}, {4249, 674}, {4250, 20718}, {4427, 41014}, {4551, 201}, {4552, 26942}, {4554, 1231}, {4556, 1790}, {4557, 3690}, {4558, 394}, {4559, 2197}, {4560, 26932}, {4563, 3926}, {4565, 222}, {4566, 6356}, {4567, 1332}, {4570, 1331}, {4573, 348}, {4575, 255}, {4576, 3933}, {4577, 1799}, {4589, 337}, {4590, 4563}, {4591, 1797}, {4592, 326}, {4594, 7019}, {4599, 34055}, {4600, 4561}, {4602, 40364}, {4609, 40050}, {4610, 17206}, {4611, 20806}, {4612, 1812}, {4616, 7056}, {4625, 7182}, {4629, 1796}, {4630, 10547}, {4636, 283}, {4637, 7177}, {5009, 22384}, {5064, 7950}, {5089, 24290}, {5094, 3906}, {5095, 1649}, {5101, 4808}, {5317, 6591}, {5338, 4822}, {5342, 4815}, {5379, 100}, {5467, 3292}, {5468, 6390}, {5502, 21663}, {5546, 219}, {5994, 36297}, {5995, 36296}, {6011, 43708}, {6103, 1640}, {6330, 43673}, {6331, 76}, {6335, 321}, {6336, 4049}, {6344, 10412}, {6353, 3566}, {6368, 35442}, {6525, 44705}, {6528, 264}, {6529, 393}, {6530, 16230}, {6531, 2395}, {6587, 1562}, {6591, 3125}, {6663, 34979}, {6733, 7591}, {6799, 42059}, {6995, 3800}, {7009, 2533}, {7012, 4551}, {7054, 23090}, {7058, 15411}, {7071, 4524}, {7076, 21831}, {7079, 4171}, {7115, 4559}, {7128, 1020}, {7192, 1565}, {7252, 7117}, {7253, 2968}, {7256, 1265}, {7257, 3718}, {7259, 3692}, {7419, 32475}, {7431, 9000}, {7435, 6001}, {7436, 8999}, {7452, 515}, {7463, 8679}, {7468, 14984}, {7471, 17702}, {7473, 542}, {7476, 2836}, {7480, 5663}, {7482, 2854}, {7649, 3120}, {7799, 45792}, {7953, 41435}, {8057, 122}, {8541, 17414}, {8737, 20578}, {8738, 20579}, {8739, 6137}, {8740, 6138}, {8743, 2485}, {8744, 2492}, {8745, 6753}, {8747, 7649}, {8748, 3064}, {8749, 2433}, {8750, 42}, {8753, 9178}, {8754, 8029}, {8756, 4120}, {8882, 2623}, {9033, 1650}, {9060, 34802}, {9064, 3426}, {9218, 22143}, {9291, 42331}, {9306, 22089}, {9308, 30476}, {9390, 9392}, {9426, 23216}, {9544, 39228}, {10015, 42761}, {10098, 5505}, {10101, 10100}, {10295, 9003}, {10301, 12073}, {10311, 3288}, {10312, 3050}, {10330, 7767}, {10420, 5504}, {10423, 1177}, {10425, 43705}, {11062, 2081}, {11064, 41077}, {11101, 30212}, {11107, 35057}, {11325, 3221}, {11413, 30211}, {11634, 8681}, {11636, 43697}, {11794, 36952}, {12092, 16867}, {12111, 40494}, {12383, 38401}, {13149, 1446}, {13397, 28787}, {13398, 15316}, {13450, 23290}, {13486, 7100}, {13621, 30210}, {13739, 6003}, {14004, 4151}, {14006, 3907}, {14013, 28846}, {14015, 9013}, {14024, 3716}, {14052, 36955}, {14129, 20577}, {14165, 44427}, {14208, 17879}, {14273, 1648}, {14401, 39008}, {14480, 15061}, {14533, 46088}, {14534, 15420}, {14543, 440}, {14544, 18641}, {14570, 343}, {14574, 14575}, {14581, 14398}, {14586, 14533}, {14587, 15958}, {14590, 323}, {14591, 50}, {14611, 6699}, {14618, 338}, {14918, 41078}, {14920, 5664}, {14953, 39470}, {14966, 3289}, {15014, 9035}, {15149, 918}, {15164, 22340}, {15165, 22339}, {15329, 13754}, {15352, 2052}, {15384, 1301}, {15411, 23983}, {15459, 16080}, {15471, 9125}, {15526, 23616}, {15639, 23976}, {15742, 3952}, {15958, 19210}, {16077, 1494}, {16080, 2394}, {16081, 43665}, {16166, 11559}, {16230, 868}, {16237, 3580}, {16757, 18187}, {16806, 32585}, {16807, 32586}, {16813, 275}, {17094, 1367}, {17104, 23226}, {17171, 16892}, {17206, 30805}, {17402, 44718}, {17403, 44719}, {17442, 8061}, {17515, 3738}, {17520, 832}, {17569, 30665}, {17708, 34897}, {17906, 4415}, {17907, 33294}, {17921, 21138}, {17923, 4707}, {17924, 16732}, {17925, 1086}, {17926, 1146}, {17927, 18004}, {17932, 6394}, {17938, 17970}, {17939, 17971}, {17940, 17972}, {17941, 12215}, {17942, 17975}, {17943, 17976}, {17944, 17977}, {17945, 17978}, {17980, 882}, {17981, 18015}, {17982, 18014}, {17983, 5466}, {17984, 14295}, {17985, 18006}, {17987, 18003}, {17994, 44114}, {18020, 99}, {18022, 44173}, {18026, 1441}, {18047, 4019}, {18155, 17880}, {18315, 97}, {18344, 4516}, {18374, 42659}, {18384, 15475}, {18808, 12079}, {18829, 40708}, {18831, 95}, {18879, 43755}, {19118, 8651}, {20031, 6531}, {20189, 34483}, {20626, 6145}, {21789, 3270}, {22239, 11744}, {22456, 290}, {23067, 7066}, {23090, 35072}, {23181, 5562}, {23189, 1364}, {23347, 1495}, {23353, 851}, {23357, 32661}, {23590, 6529}, {23710, 30574}, {23712, 9200}, {23713, 9201}, {23714, 14446}, {23715, 14447}, {23895, 40709}, {23896, 40710}, {23964, 112}, {23977, 16318}, {23999, 811}, {24000, 162}, {24001, 14206}, {24006, 1109}, {24019, 19}, {24021, 36126}, {24041, 4592}, {26283, 30213}, {26704, 15232}, {26705, 15320}, {26714, 43718}, {27369, 688}, {27644, 25098}, {30221, 38729}, {30247, 5486}, {30248, 13418}, {30249, 43695}, {30450, 5392}, {30512, 44665}, {30716, 3448}, {30733, 15313}, {31510, 2777}, {31623, 4391}, {31900, 4977}, {31901, 28195}, {31902, 4802}, {31903, 4778}, {31904, 28840}, {31905, 812}, {31907, 28859}, {31908, 28882}, {31909, 824}, {31912, 4785}, {31914, 28855}, {31916, 30519}, {31917, 3776}, {31921, 28890}, {31925, 30520}, {31926, 4762}, {32002, 41298}, {32036, 40712}, {32037, 40711}, {32230, 107}, {32320, 35071}, {32640, 18877}, {32656, 4055}, {32661, 577}, {32674, 1400}, {32676, 31}, {32687, 43717}, {32691, 1245}, {32695, 8749}, {32696, 1976}, {32697, 2987}, {32708, 14910}, {32710, 15453}, {32713, 25}, {32714, 1427}, {32715, 40352}, {32729, 14908}, {32734, 2351}, {32739, 2200}, {33294, 127}, {33513, 40410}, {33640, 11270}, {33803, 14060}, {34211, 441}, {34386, 15414}, {34397, 14270}, {34403, 14638}, {34484, 20188}, {34538, 15352}, {34568, 40384}, {34574, 15398}, {34854, 17994}, {34859, 2211}, {34983, 41212}, {35136, 6340}, {35139, 328}, {35140, 2419}, {35169, 40715}, {35265, 44810}, {35278, 6776}, {35311, 140}, {35318, 233}, {35324, 22052}, {35325, 39}, {35327, 22080}, {35342, 3958}, {35360, 5}, {35907, 6103}, {35908, 32112}, {36034, 35200}, {36036, 336}, {36049, 41087}, {36059, 22341}, {36084, 293}, {36092, 8767}, {36104, 1910}, {36105, 8773}, {36114, 36053}, {36118, 3668}, {36126, 158}, {36127, 225}, {36128, 23894}, {36129, 2166}, {36131, 2159}, {36134, 2169}, {36142, 36060}, {36145, 1820}, {36306, 13}, {36309, 14}, {36419, 17925}, {36420, 43925}, {36421, 17926}, {36793, 23107}, {36794, 31296}, {36797, 8}, {36829, 23039}, {36830, 22146}, {36831, 44715}, {36839, 10217}, {36840, 10218}, {36841, 37669}, {36898, 34290}, {37168, 900}, {37669, 20580}, {37765, 9979}, {37766, 24978}, {37908, 926}, {37937, 2781}, {37943, 45147}, {37962, 2780}, {37963, 2775}, {37966, 2771}, {38294, 45689}, {38342, 11140}, {38832, 22090}, {38861, 13198}, {38936, 15470}, {39062, 39352}, {39201, 34980}, {39284, 39183}, {39297, 2867}, {39298, 8115}, {39299, 8116}, {39352, 38240}, {39383, 6413}, {39384, 6414}, {39417, 34207}, {39534, 42759}, {40097, 43703}, {40117, 1903}, {40138, 9209}, {40596, 10117}, {40979, 40628}, {41013, 4036}, {41083, 14837}, {41174, 43187}, {41204, 6130}, {41206, 40843}, {41207, 1952}, {41321, 17747}, {41357, 16278}, {41364, 6588}, {41502, 9404}, {41512, 39170}, {41610, 24562}, {41676, 141}, {41677, 37636}, {41678, 13567}, {41679, 1993}, {41906, 28788}, {42067, 8034}, {42068, 23099}, {42308, 16077}, {42396, 83}, {42405, 276}, {43188, 9289}, {43351, 42021}, {43717, 34212}, {43754, 17974}, {43925, 1015}, {43976, 30735}, {44060, 3532}, {44077, 34952}, {44080, 42660}, {44084, 21731}, {44089, 5027}, {44090, 5113}, {44091, 8664}, {44092, 42661}, {44097, 42653}, {44099, 42663}, {44100, 8653}, {44102, 351}, {44103, 42664}, {44113, 42666}, {44123, 42667}, {44124, 42668}, {44129, 3261}, {44130, 35519}, {44134, 30474}, {44146, 35522}, {44162, 9426}, {44183, 44766}, {44326, 34403}, {44695, 14308}, {44698, 21172}, {44734, 23880}, {44766, 14376}, {44769, 14919}, {44770, 1297}, {44828, 40800}, {45215, 22416}, {45662, 39474}, {46102, 4552}, {46103, 4560}, {46106, 41079}, {46107, 21207}, {46134, 20563}, {46151, 427}, {46254, 799}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 39352, 15526}, {6, 264, 36794}, {6, 9308, 264}, {53, 3629, 27377}, {53, 27377, 32002}, {69, 1249, 17907}, {98, 2452, 13479}, {98, 11596, 20975}, {99, 36841, 4558}, {110, 35360, 107}, {112, 4558, 41679}, {112, 41676, 99}, {162, 1897, 36797}, {185, 1941, 1105}, {193, 393, 317}, {297, 1990, 37765}, {324, 1994, 275}, {340, 37765, 297}, {1576, 1632, 35278}, {2407, 4558, 36841}, {2407, 14570, 4558}, {2407, 16237, 14590}, {2407, 41678, 41679}, {2452, 9512, 20975}, {2452, 20975, 11596}, {3163, 15526, 23583}, {3163, 39358, 1494}, {4558, 14570, 99}, {4558, 41679, 14590}, {9512, 20975, 98}, {11596, 20975, 13479}, {14570, 41676, 41677}, {14570, 41678, 16237}, {15526, 23583, 2}, {15526, 39352, 1494}, {16237, 41677, 14570}, {16813, 18831, 18315}, {17037, 36413, 6527}, {32713, 46151, 107}, {35311, 35360, 110}, {37766, 37779, 14918}, {39298, 39299, 14590}, {41194, 41195, 41254}, {41676, 41678, 14570}, {41677, 41679, 99}

Notes

Let T be the anticevian triangle of X₄, and let U be the bianticevian conic of X₁ and X₄. Let TT be the tangential triangle of U with respect to T. Then ABC and TT are perspective, and X₆₄₈ is their perspector. (Randy Hutson, December 26, 2015)