X₄₆₈

X₂-LINE CONJUGATE OF X₃

X₄₆₈ = circumcircle-inverse of X₂₅

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

X₄₆₈ (X₂-LINE CONJUGATE OF X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {230, 231}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈)

Trilinears: \((\sec A)(\cot B + \cot C - 2 \cot A) : :\)

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

Search Value (6, 9, 13): \(1.208634964898\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec A - 3 \csc A \tan \omega : :\)

• Trilinears: \(3 \csc A - \sec A \cot \omega : :\)

▸ Linear Combinations1 combination

• X₄₆₈ = \(3\) X₂ + X₂₃, \(9\) X₂ - X₅₁₈₉, \(5\) X₂ - X₁₀₉₈₉, \(15\) X₂ + X₂₀₀₆₃, \(9\) X₂ - \(5\) X₃₀₇₄₅, \(3\) X₂ + \(5\) X₃₇₇₆₀, \(3\) X₂ + \(2\) X₃₇₈₉₇, \(6\) X₂ + X₃₇₈₉₉, \(9\) X₂ + X₃₇₉₀₀, \(7\) X₂ + X₃₇₉₀₁, \(2\) X₂ + X₃₇₉₀₄, X₂ + \(3\) X₃₇₉₀₇, \(5\) X₂ + \(3\) X₃₇₉₀₉, \(9\) X₂ + \(2\) X₃₇₉₁₀, \(3\) X₂ - \(4\) X₃₇₉₁₁, \(6\) X₂ - X₄₆₅₁₇, \(3\) X₃ + X₁₈₃₂₅, X₃ - \(3\) X₄₄₂₁₄, X₄ + \(3\) X₁₈₆, X₄ - \(3\) X₄₀₃, \(2\) X₄ - \(3\) X₁₀₁₅₁, \(4\) X₄ - \(3\) X₁₃₄₇₃, \(5\) X₄ + \(3\) X₁₃₆₁₉, \(7\) X₄ + \(9\) X₃₅₄₈₉, \(2\) X₄ + \(3\) X₃₇₉₃₁, X₄ + \(2\) X₃₇₉₃₄, X₄ + \(6\) X₃₇₉₃₅, X₄ - \(6\) X₃₇₉₄₂, X₄ - \(9\) X₃₇₉₄₃, \(3\) X₅ - X₁₈₅₇₂, X₅ - \(3\) X₄₄₂₈₂, X₂₀ - \(5\) X₃₇₉₅₂, X₂₀ - \(3\) X₄₄₂₈₀, X₂₃ + \(2\) X₅₁₅₉, \(3\) X₂₃ + X₅₁₈₉, X₂₃ - \(3\) X₇₄₂₆, \(5\) X₂₃ + \(3\) X₁₀₉₈₉, \(5\) X₂₃ - X₂₀₀₆₃, \(3\) X₂₃ + \(5\) X₃₀₇₄₅, X₂₃ - \(5\) X₃₇₇₆₀, \(3\) X₂₃ - X₃₇₉₀₀, \(7\) X₂₃ - \(3\) X₃₇₉₀₁, \(2\) X₂₃ - \(3\) X₃₇₉₀₄, X₂₃ - \(9\) X₃₇₉₀₇, \(5\) X₂₃ - \(9\) X₃₇₉₀₉, \(3\) X₂₃ - \(2\) X₃₇₉₁₀, X₂₃ + \(4\) X₃₇₉₁₁, \(2\) X₂₃ + X₄₆₅₁₇, X₆₇ + \(3\) X₁₈₃₇₄, \(3\) X₁₁₀ + X₄₁₇₂₄, X₁₂₅ + \(2\) X₁₅₄₄₈, X₁₄₀ + \(3\) X₁₀₀₉₆, \(4\) X₁₄₀ - \(3\) X₁₀₂₅₇, \(2\) X₁₄₀ + X₁₆₆₁₉, \(4\) X₁₄₀ + \(3\) X₃₇₉₇₁, X₁₄₀ - \(3\) X₄₄₂₃₄, \(2\) X₁₄₀ - \(3\) X₄₄₄₅₂, X₁₄₀ - \(6\) X₄₄₉₀₀, \(2\) X₁₈₆ + X₁₀₁₅₁, \(3\) X₁₈₆ - X₁₀₂₉₅, \(4\) X₁₈₆ + X₁₃₄₇₃, \(5\) X₁₈₆ - X₁₃₆₁₉, \(7\) X₁₈₆ - \(3\) X₃₅₄₈₉, \(3\) X₁₈₆ - \(2\) X₃₇₉₃₄, X₁₈₆ + \(2\) X₃₇₉₄₂, X₁₈₆ + \(3\) X₃₇₉₄₃, \(3\) X₁₈₆ + \(2\) X₃₇₉₈₄, \(2\) X₂₃₀ + X₁₆₃₁₆, \(3\) X₂₅₀ + X₃₄₀, \(3\) X₃₈₁ - X₁₈₃₂₃, \(3\) X₃₈₁ + \(5\) X₃₇₉₅₈, \(2\) X₃₈₉ - \(3\) X₁₆₂₂₇, \(3\) X₄₀₃ + X₁₀₂₉₅, \(4\) X₄₀₃ - X₁₃₄₇₃, \(5\) X₄₀₃ + X₁₃₆₁₉, \(7\) X₄₀₃ + \(3\) X₃₅₄₈₉, \(2\) X₄₀₃ + X₃₇₉₃₁, \(3\) X₄₀₃ + \(2\) X₃₇₉₃₄, X₄₀₃ + \(2\) X₃₇₉₃₅, X₄₀₃ - \(3\) X₃₇₉₄₃, \(3\) X₄₀₃ - \(2\) X₃₇₉₈₄, \(3\) X₅₄₉ - X₃₇₉₅₀, X₅₅₀ + \(3\) X₁₁₅₆₃, X₅₅₀ - \(3\) X₁₅₆₄₆, X₅₅₀ - \(6\) X₁₆₅₃₁, X₅₅₀ - \(9\) X₁₆₅₃₂, \(3\) X₅₉₇ - X₁₅₈₂₆, \(5\) X₆₃₁ - X₇₄₆₄, \(5\) X₆₃₂ + X₃₇₉₆₇, X₈₄₂ + \(3\) X₃₈₂₂₇, \(3\) X₈₅₈ - X₅₁₈₉, X₈₅₈ + \(3\) X₇₄₂₆, \(5\) X₈₅₈ - \(3\) X₁₀₉₈₉, \(5\) X₈₅₈ + X₂₀₀₆₃, \(3\) X₈₅₈ - \(5\) X₃₀₇₄₅, X₈₅₈ + \(5\) X₃₇₇₆₀, X₈₅₈ + \(2\) X₃₇₈₉₇, \(2\) X₈₅₈ + X₃₇₈₉₉, \(3\) X₈₅₈ + X₃₇₉₀₀, \(7\) X₈₅₈ + \(3\) X₃₇₉₀₁, \(2\) X₈₅₈ + \(3\) X₃₇₉₀₄, X₈₅₈ + \(9\) X₃₇₉₀₇, \(5\) X₈₅₈ + \(9\) X₃₇₉₀₉, \(3\) X₈₅₈ + \(2\) X₃₇₉₁₀, X₈₅₈ - \(4\) X₃₇₉₁₁

X₄₆₈ lies on the Darboux quintic (Q071), the cubics K043, K164 ,K209, K217, K418, K452, K478, K533, K535, K600, K608, K698, K824, K869, K954, the GEOS circle.

▸ Incident Central Lines428 lines

{2, 3}, {6, 5486}, {10, 11363}, {32, 40102}, {33, 5160}, {34, 5433}, {49, 13292}, {51, 23292}, {52, 9820}, {53, 3054}, {64, 43903}, {67, 1177}, {69, 6090}, {74, 10293}, {81, 44097}, {98, 685}, {99, 37803}, {105, 2766}, {107, 842}, {108, 2752}, {110, 3564}, {111, 935}, {112, 2770}, {113, 32110}, {114, 3233}, {115, 40350}, {125, 1495}, {126, 5203}, {132, 3258}, {133, 46436}, {136, 16188}, {141, 1974}, {154, 1899}, {171, 14975}, {182, 13394}, {183, 44134}, {184, 8550}, {185, 16252}, {187, 1560}, {206, 26926}, {230, 231}, {242, 2969}, {243, 42069}, {250, 325}, {262, 43530}, {264, 37688}, {275, 7608}, {305, 32820}, {323, 19504}, {343, 9306}, {373, 1843}, {385, 35511}, {389, 16227}, {393, 21448}, {394, 44492}, {395, 8740}, {396, 8739}, {459, 43537}, {476, 3563}, {477, 9064}, {498, 11399}, {499, 11398}, {511, 1112}, {512, 22264}, {524, 3292}, {525, 14697}, {542, 35266}, {590, 5413}, {597, 8541}, {612, 10149}, {615, 5412}, {620, 5186}, {648, 18823}, {653, 17985}, {691, 2374}, {748, 1395}, {750, 2212}, {879, 2433}, {899, 2356}, {908, 27421}, {930, 23096}, {940, 44086}, {942, 41609}, {973, 12242}, {1001, 11383}, {1078, 11380}, {1125, 1829}, {1147, 41587}, {1155, 40560}, {1164, 1165}, {1172, 37675}, {1196, 7755}, {1204, 2883}, {1211, 2203}, {1213, 1474}, {1235, 26235}, {1249, 37689}, {1289, 37801}, {1290, 15344}, {1297, 22239}, {1299, 16167}, {1300, 9060}, {1301, 2697}, {1302, 32710}, {1351, 21970}, {1352, 35259}, {1353, 37644}, {1384, 41370}, {1398, 7288}, {1452, 11375}, {1473, 20266}, {1493, 41598}, {1498, 26937}, {1514, 2777}, {1531, 36518}, {1533, 38727}, {1611, 3162}, {1614, 18914}, {1620, 5895}, {1648, 5967}, {1692, 6388}, {1698, 5090}, {1799, 40413}, {1824, 6690}, {1828, 6691}, {1848, 2355}, {1851, 17917}, {1853, 31383}, {1861, 1862}, {1870, 7292}, {1876, 3911}, {1878, 6681}, {1892, 5219}, {1897, 17927}, {1902, 6684}, {1976, 41175}, {1986, 10272}, {1993, 41588}, {2052, 7607}, {2080, 41253}, {2204, 5277}, {2211, 3231}, {2299, 17056}, {2373, 10423}, {2393, 12099}, {2452, 7735}, {2453, 37637}, {2687, 9107}, {2690, 9085}, {2758, 9088}, {2931, 32123}, {2967, 14920}, {2970, 38552}, {3043, 13392}, {3055, 6748}, {3066, 14561}, {3068, 5411}, {3069, 5410}, {3092, 5418}, {3093, 5420}, {3167, 6515}, {3186, 7806}, {3192, 37646}, {3199, 7749}, {3220, 26933}, {3266, 6390}, {3448, 35265}, {3532, 43695}, {3569, 32120}, {3574, 11745}, {3581, 14643}, {3616, 11396}, {3618, 12167}, {3624, 7713}, {3712, 42713}, {3720, 40976}, {3742, 41611}, {3815, 5063}, {3818, 45303}, {3917, 44079}, {3934, 12143}, {4383, 44105}, {4422, 24814}, {4648, 44100}, {4999, 40985}, {5012, 45298}, {5130, 24953}, {5139, 31655}, {5140, 6719}, {5185, 6710}, {5205, 26231}, {5218, 7071}, {5275, 45786}, {5285, 21015}, {5297, 6198}, {5304, 5702}, {5305, 9465}, {5306, 18361}, {5370, 15326}, {5446, 43823}, {5449, 12134}, {5461, 12132}, {5462, 6746}, {5476, 20192}, {5480, 34417}, {5520, 20621}, {5622, 40114}, {5640, 6403}, {5650, 12294}, {5654, 37489}, {5656, 18931}, {5891, 44201}, {5901, 41722}, {5943, 11649}, {6000, 15738}, {6036, 12131}, {6091, 41521}, {6108, 6111}, {6109, 6110}, {6118, 12148}, {6119, 12147}, {6146, 10282}, {6152, 8254}, {6225, 34469}, {6242, 22051}, {6247, 26883}, {6331, 17984}, {6335, 17987}, {6336, 17982}, {6467, 15585}, {6669, 12142}, {6670, 12141}, {6673, 22482}, {6674, 22481}, {6689, 11576}, {6696, 11381}, {6698, 32239}, {6699, 12133}, {6701, 16114}, {6702, 12137}, {6703, 44092}, {6704, 12144}, {6705, 12136}, {6707, 17171}, {6713, 12138}, {6720, 13166}, {6723, 29012}, {6776, 26864}, {6800, 18911}, {7009, 7140}, {7302, 15338}, {7583, 10881}, {7584, 10880}, {7665, 19577}, {7703, 32124}, {7718, 9780}, {7746, 27376}, {7747, 15820}, {7767, 26233}, {7768, 33651}, {7777, 27377}, {7789, 30749}, {7846, 11386}, {7857, 38526}, {7891, 30793}, {8263, 41614}, {8428, 14729}, {8537, 15019}, {8737, 11537}, {8738, 11549}, {8744, 11580}, {8753, 15899}, {8754, 9172}, {8770, 13854}, {8827, 46111}, {8854, 8960}, {8859, 38294}, {8901, 19189}, {9061, 10101}, {9084, 10098}, {9128, 34512}, {9164, 22110}, {9173, 39241}, {9174, 39240}, {9176, 43084}, {9185, 44427}, {9225, 15993}, {9308, 17008}, {9544, 45968}, {9707, 18912}, {9745, 18907}, {9777, 11427}, {10006, 18344}, {10102, 30247}, {10117, 32125}, {10163, 11594}, {10182, 11430}, {10185, 39284}, {10198, 26377}, {10200, 26378}, {10214, 11701}, {10416, 15398}, {10420, 40120}, {10422, 10424}, {10511, 46105}, {10535, 26956}, {10536, 26957}, {10539, 12359}, {10546, 18358}, {10564, 38793}, {10601, 44503}, {10619, 32391}, {10632, 11543}, {10633, 11542}, {10641, 23303}, {10642, 23302}, {10990, 15311}, {10991, 42671}, {11202, 18390}, {11206, 23291}, {11402, 11433}, {11408, 11489}, {11409, 11488}, {11424, 15873}, {11442, 35264}, {11459, 44683}, {11464, 12022}, {11547, 14569}, {11550, 23332}, {11645, 32267}, {11746, 44668}, {12007, 44109}, {12140, 20304}, {12145, 34841}, {12162, 44158}, {12165, 20125}, {12174, 18913}, {12241, 13367}, {12290, 43607}, {12370, 32171}, {12421, 33563}, {12827, 20772}, {12900, 15473}, {13142, 34148}, {13148, 13754}, {13399, 15152}, {13414, 44123}, {13415, 44124}, {13419, 32767}, {13474, 25563}, {13568, 43831}, {13869, 26228}, {14052, 34827}, {14457, 14528}, {14530, 26944}, {14560, 34310}, {15010, 21969}, {15106, 32247}, {15126, 36201}, {15128, 38885}, {15270, 15652}, {15271, 32224}, {15360, 23061}, {15433, 39662}, {15472, 37477}, {15526, 34147}, {15806, 16881}, {15904, 32126}, {16081, 46316}, {16103, 23992}, {16178, 31842}, {16655, 20299}, {16659, 23294}, {16823, 26259}, {17398, 44103}, {17409, 40320}, {17718, 44095}, {17811, 43653}, {17821, 19467}, {18350, 31831}, {18553, 21243}, {18916, 19347}, {18947, 37779}, {19121, 26156}, {19124, 22112}, {19357, 39571}, {20376, 32340}, {20725, 37853}, {20771, 46085}, {20774, 23234}, {20977, 41939}, {20987, 23300}, {22479, 25524}, {24206, 35283}, {26880, 26905}, {26881, 26913}, {26882, 26917}, {26884, 26932}, {26885, 26942}, {26886, 26950}, {26887, 26954}, {26888, 26955}, {28419, 37491}, {30714, 44665}, {30716, 40429}, {31489, 44658}, {32112, 32119}, {32114, 32127}, {32133, 38532}, {32821, 34254}, {34565, 41599}, {34573, 44091}, {34656, 37546}, {35268, 44882}, {35466, 44113}, {36990, 41424}, {37761, 38461}, {37762, 38462}, {37892, 43528}, {38110, 39588}, {40144, 40323}, {40347, 41336}, {40511, 44381}, {43461, 43462}, {43896, 44544}, {44125, 46167}, {44126, 46166}, {44899, 46184}

▸ Functional Relations71 relations

• X₄₆₈ = midpoint of X_i and X_j for these {i,j}: midpoint of X_i and X_j for these {i,j}: {2, 7426}, {3, 11799}, {4, 10295}, {5, 7575}, {6, 32113}, {23, 858}, {69, 32220}, {74, 32111}, {110, 3580}, {113, 32110}, {125, 1495}, {140, 25338}, {141, 32217}, {186, 403}, {187, 5099}, {230, 16320}, {237, 36189}, {297, 7473}, {381, 44265}, {427, 37969}, {549, 44266}, {550, 44267}, {935, 5523}, {1316, 5112}, {1513, 36166}, {2070, 2072}, {2074, 37989}, {2770, 5913}, {2931, 32123}, {3292, 41586}, {3569, 32120}, {3589, 32218}, {3628, 44264}, {5000, 5001}, {5159, 37897}, {5189, 37900}, {5642, 32225}, {5972, 32223}, {6593, 8262}, {7703, 32124}, {8598, 36196}, {9128, 34512}, {10096, 44234}, {10117, 32125}, {10151, 37931}, {10257, 37971}, {11007, 37906}, {11064, 32269}, {11563, 15646}, {11657, 16319}, {14120, 36180}, {15122, 16619}, {15360, 40112}, {15904, 32126}, {16272, 16309}, {16303, 16334}, {16305, 16332}, {16315, 16316}, {16321, 16324}, {18571, 44961}, {21284, 37981}, {31726, 44246}, {32112, 32119}, {32114, 32127}, {32267, 45311}, {32460, 32461}, {34152, 43893}, {35266, 44569}, {36165, 37927}, {37899, 46517}, {37934, 37984}, {37935, 37942}, {37936, 37938}, {37974, 37975}

• X₄₆₈ = reflection of X_i in X_j for these {i,j}: {4, 37984}, {23, 37897}, {186, 37935}, {403, 37942}, {858, 5159}, {1112, 44084}, {1495, 15448}, {2071, 16976}, {2072, 44911}, {2501, 41357}, {5095, 15471}, {5159, 37911}, {10151, 403}, {10257, 44452}, {10295, 37934}, {10297, 5}, {11064, 5972}, {12105, 44264}, {13473, 10151}, {15122, 140}, {15646, 16531}, {16103, 23992}, {16272, 16332}, {16303, 16324}, {16304, 16309}, {16309, 16305}, {16312, 16334}, {16315, 230}, {16316, 16320}, {16334, 16321}, {16387, 6676}, {16619, 25338}, {18571, 22249}, {20725, 37853}, {23323, 46031}, {32246, 35370}, {32269, 32223}, {36170, 10011}, {37899, 23}, {37900, 37910}, {37904, 7426}, {37931, 186}, {37987, 44334}, {44234, 44900}, {44452, 44234}, {46517, 858}

• X₄₆₈ = isogonal conjugate of X₈₉₅

• X₄₆₈ = isotomic conjugate of X₃₀₇₈₆

• X₄₆₈ = complement of X₈₅₈

• X₄₆₈ = anticomplement of X₅₁₅₉

• X₄₆₈ = circumcircle-inverse of X₂₅

• X₄₆₈ = nine-point-circle-inverse of X₄₂₇

• X₄₆₈ = nine-point-circle-of-medial triangle-inverse of X₂

• X₄₆₈ = orthocentroidal-circle-inverse of X₅₀₉₄

• X₄₆₈ = Stevanovic-circle-inverse of X₅₀₈₉

• X₄₆₈ = polar-circle-inverse of X₂

• X₄₆₈ = Dao-Moses-Telv-circle-inverse of X₆₁₀₃

• X₄₆₈ = orthoptic-circle-of-Steiner-inellipse-inverse of X₄

• X₄₆₈ = orthoptic-circle-of-Steiner-circumellipe-inverse of X₃₁₄₆

• X₄₆₈ = Moses-radical-circle-inverse of X₂₃₂

• X₄₆₈ = Moses-Parry-circle-inverse of X₄₄₄₆₇

• X₄₆₈ = DeLongchamps-circle-inverse of X₇₃₉₆

• X₄₆₈ = Stammler-circle-inverse of X₄₄₄₅₄

• X₄₆₈ = {circumcircle, nine-point circle}-inverter-inverse of X₄

• X₄₆₈ = 1st-Droz-Farney-circle-inverse of X₄₄₄₄₁

• X₄₆₈ = 2nd-Droz-Farney circle inverse of X₄₄₄₃₈

• X₄₆₈ = circumcircle-of-anticomplementary-triangle-inverse of X₄₄₄₄₂

• X₄₆₈ = circumcircle-of-inner-Napoleon-triangle-inverse of X₄₄₄₅₉

• X₄₆₈ = circumcircle-of-outer-Napoleon-triangle-inverse of X₄₄₄₆₃

• X₄₆₈ = X₂-line conjugate of X₃

• X₄₆₈ = polar conjugate of X₆₇₁

• X₄₆₈ = polar conjugate of the isotomic conjugate of X₅₂₄

• X₄₆₈ = polar conjugate of the isogonal conjugate of X₁₈₇

• X₄₆₈ = cevapoint of X_i and X_j for these (i,j): {2, 7665}, {187, 44102}, {351, 1648}, {524, 32459}, {18374, 41336}

• X₄₆₈ = crosspoint of X_i and X_j for these (i,j): {2, 2373}, {4, 17983}, {250, 10423}, {264, 46105}, {37778, 44146}

• X₄₆₈ = crosssum of X_i and X_j for these (i,j): {3, 3292}, {6, 2393}, {184, 10317}, {8542, 10510}

• X₄₆₈ = crossdifference of every pair of points on line X₃X₆₄₇

• X₄₆₈ = X_i-complementary conjugate of X_j for these (i,j): 1, 15116}, {31, 1560}, {1177, 10}, {2373, 2887}, {10422, 4892}, {10423, 8062}, {18876, 18589}, {36095, 30476}, {37220, 626}, {46140, 21235}

• X₄₆₈ = X_i-Ceva conjugate of X_j for these (i,j): {2, 1560}, {4, 5095}, {935, 523}, {2374, 25}, {4235, 14273}, {8791, 427}, {17983, 4}, {37765, 5523}, {41521, 46444}, {44146, 524

• X₄₆₈ = X_i-cross conjugate of X_j for these (i,j): {187, 524}, {524, 5203}, {5095, 4}, {5099, 523}, {14273, 4235}, {14357, 41498}, {35282, 5967}, {44915, 671}}

• X₄₆₈ = X_i-Hirst inverse of X_j for these (i,j): {523, 6103}, {1990, 16230}

• X₄₆₈ = X_i-vertex conjugate of X_j for these (i,j): {23, 13574}, {25, 523}

• X₄₆₈ = trilinear pole of line X₆₉₀X₅₀₉₅

• X₄₆₈ = medial-isogonal conjugate of X₁₅₁₁₆

• X₄₆₈ = orthic-isogonal conjugate of X₅₀₉₅

• X₄₆₈ = psi-transform of X₆₇₇₆

• X₄₆₈ = X₁₁₅₅-of-orthic-triangle if ABC is acute

• X₄₆₈ = X₅₁₂₆-of-circumorthic-triangle

• X₄₆₈ = X₁₀₂₉₅-of-Euler-triangle

• X₄₆₈ = X₁₀₂₉₇-of-Carnot-triangle

• X₄₆₈ = X₁₀₁₅₁-of-Thomson-triangle

• X₄₆₈ = X₅₉₇₂-of-anti-Brocard-triangle

• X₄₆₈ = X₆₇₁₉-of-anti-McCay-triangle

• X₄₆₈ = X₁₃₁₉-of-orthic-triangle-of-Thomson-triangle)

• X₄₆₈ = X₂₃-of-intouch-triangle-of-orthic-triangle

• X₄₆₈ = centroid of ABCX₂₃

• X₄₆₈ = Kosnita(X₂₃,X₂) point

• X₄₆₈ = intersection of Euler line and orthic axis

• X₄₆₈ = bicentric sum of PU(4)

• X₄₆₈ = midpoint of PU(4)

• X₄₆₈ = perspector of circumconic centered at X₁₅₆₀

• X₄₆₈ = center of circumconic that is locus of trilinear poles of lines passing through X₁₅₆₀

• X₄₆₈ = intersection of tangents to hyperbola {A, B, C,X₂,X₆₉} at X₂ and X₂₃₇₃

• X₄₆₈ = trilinear pole of line X₆₉₀X₅₀₉₅ (the perspectrix of ABC and 4th Brocard triangle)

• X₄₆₈ = pole wrt polar circle of trilinear polar of X₆₇₁ (line X₂X₅₂₃)

• X₄₆₈ = X₄₈-isoconjugate (polar conjugate) of X₆₇₁

• X₄₆₈ = radical trace of circumcircle and tangential circle

• X₄₆₈ = radical trace of polar circle and {circumcircle, nine-point circle}-inverter

• X₄₆₈ = radical trace of Moses-Parry circle and Moses radical circle

• X₄₆₈ = excentral-to-ABC functional image of X₁₁₅₅

• X₄₆₈ = center of Walsmith rectangular hyperbola

• X₄₆₈ = X_i-isoconjugate of X_j for these {i,j}: {1, 895}, {2, 36060}, {3, 897}, {31, 30786}, {48, 671}, {63, 111}, {69, 923}, {75, 14908}, {77, 5547}, {78, 7316}, {163, 14977}, {184, 46277}, {255, 17983}, {293, 5968}, {304, 32740}, {326, 8753}, {394, 36128}, {525, 36142}, {647, 36085}, {656, 691}, {662, 10097}, {810, 892}, {896, 15398}, {1459, 5380}, {4558, 23894}, {4575, 5466}, {4592, 9178}, {6091, 8769}, {9213, 36061}, {9214, 35200}, {9247, 18023}, {14208, 32729}, {14209, 35188}, {18669, 41511}, {19626, 40364}, {34055, 46154}, {34158, 37220}

• X₄₆₈ = barycentric product X_i*X_j for these {i,j}: {3, 37778}, {4, 524}, {6, 44146}, {19, 14210}, {25, 3266}, {27, 4062}, {28, 42713}, {76, 44102}, {92, 896}, {99, 14273}, {107, 14417}, {111, 34336}, {112, 35522}, {126, 2374}, {186, 43084}, {187, 264}, {193, 5203}, {275, 41586}, {278, 3712}, {281, 7181}, {286, 21839}, {297, 5967}, {351, 6331}, {393, 6390}, {428, 31068}, {523, 4235}, {648, 690}, {653, 14432}, {671, 5095}, {811, 2642}, {922, 1969}, {935, 18311}, {1309, 42760}, {1560, 2373}, {1648, 18020}, {1824, 16741}, {1826, 6629}, {1839, 31013}, {1897, 4750}, {1990, 36890}, {2052, 3292}, {2482, 17983}, {2501, 5468}, {2986, 12828}, {3793, 8801}, {5467, 14618}, {5477, 35142}, {5485, 15471}, {5486, 37855}, {5642, 16080}, {6330, 35282}, {6335, 14419}, {6591, 42721}, {6593, 46105}, {7664, 8791}, {7813, 32085}, {8753, 36792}, {9115, 38428}, {9117, 38427}, {9155, 16081}, {9204, 36306}, {9205, 36309}, {9717, 46106}, {10603, 24855}, {14052, 45291}, {14357, 37765}, {14424, 42396}, {14559, 44427}, {14567, 18022}, {16103, 40890}, {16702, 41013}, {17984, 18872}, {18027, 23200}, {22105, 41676}, {23889, 24006}, {24038, 36128}, {32225, 43530}, {32459, 34208}, {32713, 45807}, {39689, 46111}, {44814, 46456}

• X₄₆₈ = barycentric quotient X_i/X_j for these {i,j}: {2, 30786}, {4, 671}, {6, 895}, {19, 897}, {25, 111}, {31, 36060}, {32, 14908}, {92, 46277}, {111, 15398}, {112, 691}, {162, 36085}, {187, 3}, {232, 5968}, {264, 18023}, {351, 647}, {393, 17983}, {427, 31125}, {512, 10097}, {523, 14977}, {524, 69}, {607, 5547}, {608, 7316}, {648, 892}, {690, 525}, {896, 63}, {922, 48}, {1096, 36128}, {1177, 41511}, {1503, 36894}, {1560, 858}, {1648, 125}, {1649, 14417}, {1783, 5380}, {1843, 46154}, {1973, 923}, {1974, 32740}, {1990, 9214}, {2052, 46111}, {2207, 8753}, {2374, 44182}, {2482, 6390}, {2489, 9178}, {2501, 5466}, {2642, 656}, {3053, 6091}, {3266, 305}, {3292, 394}, {3712, 345}, {3793, 3785}, {4062, 306}, {4235, 99}, {4750, 4025}, {5026, 12215}, {5094, 42008}, {5095, 524}, {5140, 14263}, {5203, 2996}, {5467, 4558}, {5468, 4563}, {5477, 3564}, {5642, 11064}, {5967, 287}, {6103, 16092}, {6390, 3926}, {6531, 9154}, {6593, 22151}, {6629, 17206}, {7181, 348}, {7664, 37804}, {7813, 3933}, {8428, 19330}, {8541, 42007}, {8737, 36307}, {8738, 36310}, {8744, 14246}, {8749, 9139}, {8753, 10630}, {8791, 10415}, {9155, 36212}, {9717, 14919}, {11183, 24284}, {12828, 3580}, {14210, 304}, {14273, 523}, {14357, 34897}, {14417, 3265}, {14419, 905}, {14424, 2525}, {14432, 6332}, {14567, 184}, {15471, 1992}, {16702, 1444}, {17994, 8430}, {18872, 36214}, {21839, 72}, {21906, 20975}, {22105, 4580}, {23200, 577}, {23889, 4592}, {24855, 16051}, {27369, 41272}, {32225, 37638}, {32459, 6337}, {32676, 36142}, {34336, 3266}, {35282, 441}, {35325, 36827}, {35522, 3267}, {37778, 264}, {37855, 11185}, {39689, 3292}, {40890, 16093}, {41336, 39169}, {41586, 343}, {41616, 37784}, {41618, 41617}, {41911, 20977}, {42713, 20336}, {43084, 328}, {43925, 43926}, {44102, 6}, {44146, 76}, {44162, 19626}, {44467, 46783}, {44814, 8552}, {44915, 32257}, {45808, 45792}, {46522, 14609}

• X₄₆₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3, 30739}, {2, 4, 5094}, {2, 5, 37454}, {2, 20, 16051}, {2, 22, 1368}, {2, 23, 858}, {2, 25, 427}, {2, 376, 32216}, {2, 858, 5159}, {2, 1113, 1312}, {2, 1114, 1313}, {2, 1370, 30771}, {2, 1995, 5}, {2, 2475, 30770}, {2, 3146, 30769}, {2, 3151, 30772}, {2, 3152, 30774}, {2, 3543, 30775}, {2, 4223, 37315}, {2, 4228, 37360}, {2, 4232, 4}, {2, 4233, 37362}, {2, 4239, 442}, {2, 5020, 37439}, {2, 5189, 30745}, {2, 6353, 25}, {2, 6655, 30777}, {2, 6676, 7499}, {2, 6872, 30776}, {2, 6995, 8889}, {2, 7386, 31255}, {2, 7391, 30744}, {2, 7392, 7539}, {2, 7394, 31236}, {2, 7417, 868}, {2, 7418, 3134}, {2, 7423, 3139}, {2, 7432, 3138}, {2, 7438, 429}, {2, 7439, 3137}, {2, 7448, 3141}, {2, 7449, 3142}, {2, 7453, 3136}, {2, 7458, 3140}, {2, 7471, 36170}, {2, 7493, 3}, {2, 7494, 7484}, {2, 7495, 140}, {2, 9832, 11007}, {2, 10154, 7667}, {2, 10565, 7386}, {2, 13595, 5133}, {2, 14002, 5169}, {2, 14790, 30803}, {2, 16387, 10257}, {2, 17522, 16067}, {2, 26252, 440}, {2, 26253, 21530}, {2, 26254, 18641}, {2, 26255, 381}, {2, 26256, 405}, {2, 26257, 6656}, {2, 26283, 11585}, {2, 26284, 13371}, {2, 32985, 11336}, {2, 34098, 37988}, {2, 35973, 25985}, {2, 36007, 37330}, {2, 36166, 3154}, {2, 36168, 14120}, {2, 37444, 30802}, {2, 37760, 23}, {2, 37777, 37981}, {2, 37897, 46517}, {2, 37907, 7426}, {2, 37909, 10989}, {2, 37913, 31101}, {2, 37962, 403}, {2, 37980, 2072}, {2, 37990, 11548}, {2, 38282, 37453}, {2, 40132, 11284}, {2, 44210, 43957}, {2, 46586, 867}, {2, 46589, 3143}, {3, 5, 34664}, {3, 235, 1885}, {3, 2070, 45171}, {3, 3542, 235}, {3, 7493, 44210}, {3, 30739, 43957}, {3, 37928, 37929}, {3, 37933, 186}, {3, 37973, 37928}, {4, 24, 37458}, {4, 25, 10301}, {4, 186, 10295}, {4, 403, 37984}, {4, 3147, 35486}, {4, 3523, 3516}, {4, 3524, 35483}, {4, 4232, 25}, {4, 5094, 427}, {4, 6353, 4232}, {4, 10018, 140}, {4, 10301, 428}, {4, 17506, 35491}, {4, 32534, 550}, {4, 35486, 3}, {4, 35487, 3850}, {4, 37458, 3575}, {4, 37460, 37196}, {4, 37984, 10151}, {5, 24, 3575}, {5, 1658, 12605}, {5, 3575, 23047}, {5, 10020, 7542}, {5, 34351, 3}, {5, 37458, 4}, {5, 38322, 546}, {5, 44212, 1995}, {10, 11363, 12135}, {20, 6622, 37197}, {20, 16051, 31152}, {20, 37952, 44280}, {22, 1368, 7667}, {22, 16063, 550}, {23, 186, 21284}, {23, 5159, 46517}, {23, 5189, 37900}, {23, 7426, 37897}, {23, 10989, 20063}, {23, 21284, 37969}, {23, 30745, 5189}, {23, 37760, 7426}, {23, 37777, 25}, {23, 37897, 37904}, {23, 37900, 37910}, {23, 37907, 37760}, {23, 37929, 37928}, {23, 37962, 37777}, {23, 37977, 186}, {23, 37978, 22}, {23, 37980, 37972}, {23, 40916, 37950}, {24, 186, 7575}, {24, 1995, 25}, {24, 3147, 34351}, {24, 7505, 5}, {24, 16868, 38322}, {25, 186, 37969}, {25, 427, 428}, {25, 5064, 6995}, {25, 5094, 4}, {25, 21284, 23}, {25, 30739, 1885}, {25, 32216, 44438}, {25, 37453, 2}, {25, 37454, 23047}, {25, 37920, 21284}, {27, 4213, 430}, {28, 451, 429}, {28, 7438, 25}, {29, 3144, 407}, {69, 19118, 46444}, {107, 14165, 6530}, {110, 19128, 34397}, {140, 6676, 7495}, {140, 7495, 7499}, {140, 10096, 25338}, {140, 13383, 16618}, {140, 15122, 10257}, {140, 16618, 3}, {140, 21841, 4}, {154, 26958, 1899}, {184, 13567, 11245}, {186, 419, 7473}, {186, 3542, 11799}, {186, 6353, 7426}, {186, 10295, 37934}, {186, 18533, 44265}, {186, 36191, 5112}, {186, 37777, 23}, {186, 37937, 237}, {186, 37942, 10151}, {186, 37943, 403}, {186, 37951, 2070}, {186, 37952, 15750}, {186, 37962, 25}, {186, 37970, 15646}, {186, 37977, 37920}, {186, 44893, 36189}, {186, 46619, 36180}, {230, 232, 16318}, {230, 1990, 6103}, {230, 2493, 16310}, {230, 10418, 16317}, {230, 16308, 16306}, {230, 16321, 16325}, {230, 16335, 16313}, {232, 3291, 14580}, {232, 6103, 1990}, {235, 30739, 427}, {235, 34664, 23047}, {237, 852, 44886}, {237, 44887, 441}, {242, 17923, 2969}, {297, 419, 460}, {376, 6623, 44438}, {403, 858, 37981}, {403, 1594, 45181}, {403, 7426, 25}, {403, 7473, 460}, {403, 7575, 3575}, {403, 10295, 4}, {403, 11799, 235}, {403, 37931, 13473}, {403, 37935, 37931}, {403, 37943, 37942}, {403, 37969, 428}, {403, 37989, 37982}, {403, 44214, 44281}, {406, 4185, 1904}, {406, 7521, 4185}, {415, 423, 422}, {419, 420, 297}, {419, 450, 421}, {419, 470, 462}, {419, 471, 463}, {427, 10301, 4}, {450, 41203, 297}, {451, 6353, 7438}, {462, 463, 460}, {470, 471, 297}, {548, 44226, 18560}, {549, 1596, 378}, {550, 1368, 16063}, {550, 11563, 44267}, {550, 16063, 7667}, {550, 44960, 4}, {597, 41585, 8541}, {631, 3089, 1593}, {632, 1595, 37119}, {851, 46554, 1375}, {851, 46555, 33305}, {852, 44888, 441}, {852, 44891, 44888}, {858, 7426, 23}, {858, 37760, 37897}, {858, 37897, 37899}, {858, 37900, 5189}, {858, 37981, 427}, {860, 37168, 1884}, {1113, 1114, 25}, {1113, 1312, 20405}, {1114, 1313, 20406}, {1304, 16080, 17986}, {1312, 1313, 427}, {1368, 10154, 22}, {1375, 33305, 851}, {1593, 3089, 1906}, {1594, 3518, 6756}, {1594, 14940, 3628}, {1598, 3526, 3541}, {1598, 3541, 1907}, {1614, 26879, 18914}, {1656, 2070, 7574}, {1656, 3517, 4}, {1656, 7574, 2072}, {1990, 6103, 16318}, {1995, 6639, 37439}, {1995, 10020, 7499}, {2045, 2046, 7395}, {2070, 5020, 37980}, {2070, 37972, 23}, {2072, 44907, 10257}, {2074, 37965, 37908}, {2211, 3231, 35325}, {2409, 7417, 25}, {2454, 2455, 40856}, {2479, 2480, 40890}, {2493, 11062, 232}, {3090, 7487, 7507}, {3147, 3542, 3}, {3147, 6353, 7493}, {3233, 16221, 16933}, {3291, 14580, 44467}, {3292, 32225, 41586}, {3515, 37196, 37460}, {3518, 14940, 1594}, {3530, 13488, 3520}, {3535, 5200, 32588}, {3542, 7493, 25}, {3542, 34351, 3575}, {3542, 35486, 4}, {3549, 6642, 7399}, {3575, 10297, 13473}, {3575, 34664, 1885}, {3575, 37454, 427}, {3628, 6756, 1594}, {3628, 23410, 5}, {3628, 34577, 7568}, {4207, 7490, 1889}, {4231, 4233, 25}, {4232, 5094, 10301}, {4232, 37934, 37904}, {4239, 30733, 25}, {4244, 7458, 25}, {5002, 5003, 16386}, {5004, 5005, 2071}, {5020, 7506, 1995}, {5064, 8889, 427}, {5095, 44102, 15471}, {5133, 7533, 3850}, {5142, 17562, 37398}, {5159, 6676, 16977}, {5159, 7426, 37899}, {5159, 16977, 10257}, {5159, 37760, 37904}, {5159, 37777, 10151}, {5159, 37911, 2}, {5189, 30745, 858}, {5640, 14389, 18583}, {5642, 41586, 3292}, {5972, 41674, 41673}, {6143, 34484, 15559}, {6240, 16868, 546}, {6240, 38322, 3575}, {6353, 7505, 1995}, {6353, 37453, 427}, {6353, 37943, 37962}, {6353, 38282, 2}, {6622, 37197, 45004}, {6623, 32216, 427}, {6639, 7506, 5}, {6640, 7517, 23335}, {6644, 10201, 15760}, {6676, 6677, 2}, {6676, 10096, 7426}, {6676, 13383, 7493}, {6676, 44232, 1995}, {6756, 45181, 10151}, {6776, 35260, 26864}, {6776, 37643, 26869}, {6995, 8889, 5064}, {7412, 7537, 37368}, {7418, 46587, 25}, {7423, 7435, 25}, {7426, 30745, 37910}, {7426, 37911, 46517}, {7426, 44214, 44210}, {7426, 44234, 7499}, {7493, 16238, 7499}, {7493, 37201, 10565}, {7519, 31133, 3627}, {7542, 10297, 10257}, {7570, 37990, 35018}, {7575, 15646, 1658}, {7575, 44234, 7542}, {7575, 44272, 24}, {7575, 44282, 5}, {7714, 35489, 37901}, {8105, 8106, 44467}, {8889, 13619, 10989}, {9707, 18912, 31804}, {9909, 21974, 2}, {9909, 30771, 1370}, {10020, 10096, 7575}, {10020, 13383, 34351}, {10020, 23410, 7568}, {10020, 44232, 5}, {10024, 45735, 31833}, {10096, 16238, 11799}, {10096, 34577, 44264}, {10096, 44452, 37971}, {10096, 44900, 44452}, {10128, 11548, 37990}, {10151, 10297, 23047}, {10192, 13567, 184}, {10295, 37934, 37931}, {10416, 15398, 16092}, {10594, 37119, 1595}, {11548, 35018, 7570}, {11563, 16532, 15646}, {11799, 44214, 3}, {11799, 44281, 1885}, {12362, 44277, 7488}, {13160, 44802, 9825}, {13371, 37440, 7553}, {13383, 16238, 3}, {13394, 37648, 182}, {13884, 13937, 6}, {14070, 18531, 44239}, {14807, 14808, 44442}, {14813, 14814, 11585}, {15078, 44440, 44241}, {15122, 44452, 140}, {15154, 15155, 44454}, {15646, 16532, 16531}, {15646, 44267, 550}, {15750, 37197, 20}, {15760, 44211, 6644}, {15765, 18585, 44218}, {16051, 37197, 427}, {16066, 17555, 37226}, {16080, 17986, 12079}, {16238, 16618, 140}, {16238, 34351, 7542}, {16272, 16304, 16322}, {16272, 16305, 16304}, {16272, 16323, 16309}, {16303, 16312, 16326}, {16303, 16321, 16312}, {16303, 16331, 16327}, {16306, 16308, 16303}, {16306, 16316, 16327}, {16306, 16321, 16313}, {16306, 16335, 16331}, {16308, 16313, 16327}, {16308, 16321, 16331}, {16308, 16335, 16316}, {16309, 16323, 16305}, {16313, 16314, 16303}, {16313, 16316, 16331}, {16313, 16331, 16312}, {16314, 16316, 16308}, {16314, 16321, 16327}, {16315, 16329, 16303}, {16316, 16325, 16312}, {16317, 16318, 44467}, {16323, 16332, 16304}, {16325, 16331, 16313}, {16387, 44452, 7499}, {16619, 25338, 37971}, {16619, 44452, 15122}, {16868, 44879, 6240}, {17506, 35491, 33923}, {18325, 37928, 37900}, {18533, 26255, 25}, {18560, 21844, 548}, {18560, 44958, 44226}, {18570, 44270, 46030}, {18571, 22249, 18579}, {18580, 31861, 44218}, {18580, 44275, 31861}, {20063, 37909, 23}, {20405, 20406, 428}, {20850, 34609, 7500}, {21284, 37920, 186}, {21841, 37934, 16619}, {21844, 44958, 18560}, {23410, 31830, 13163}, {23712, 23713, 1990}, {23712, 23715, 23713}, {23713, 23714, 23712}, {23714, 23715, 1990}, {24007, 24008, 6103}, {25338, 44234, 140}, {26864, 26869, 6776}, {26882, 26917, 34224}, {30739, 44210, 3}, {30745, 37910, 46517}, {31510, 36166, 25}, {31681, 31682, 5066}, {31726, 37955, 44246}, {32225, 44102, 12828}, {32460, 37974, 37975}, {32461, 37975, 37974}, {34477, 46030, 18570}, {34577, 44232, 23410}, {35259, 37638, 1352}, {35260, 37643, 6776}, {35471, 35488, 3627}, {35472, 35481, 8703}, {35479, 35488, 35471}, {35490, 35503, 15704}, {35491, 44959, 4}, {36168, 46619, 25}, {37165, 46549, 46552}, {37453, 37907, 37969}, {37453, 37981, 37911}, {37760, 37911, 37899}, {37777, 37920, 37969}, {37777, 37977, 21284}, {37897, 37911, 858}, {37899, 37904, 23}, {37899, 37981, 13473}, {37900, 37910, 37899}, {37904, 37931, 37969}, {37904, 46517, 37899}, {37907, 37962, 6353}, {37910, 37911, 30745}, {37917, 37954, 186}, {37928, 37973, 23}, {37933, 37943, 235}, {37934, 37942, 37984}, {37935, 37943, 10151}, {37935, 37962, 37904}, {37935, 37984, 37934}, {37937, 44893, 403}, {37942, 37977, 46517}, {37943, 37977, 37981}, {37951, 37980, 25}, {37951, 44911, 10151}, {37962, 37977, 23}, {42789, 42790, 7464}, {42807, 42808, 7526}, {44145, 46106, 2970}, {44234, 44264, 7568}, {44272, 44282, 403}, {44886, 44889, 852}, {44886, 44890, 237}, {44887, 44892, 44889}, {44888, 44892, 852}, {44889, 44890, 44886}, {44891, 44892, 441}, {45994, 45995, 3575}, {46484, 46548, 46553}, {46554, 46555, 851}, {46586, 46588, 25}, {46589, 46592, 25}

Notes

As a point on the Euler line, X₄₆₈ has Shinagawa coefficients \((3F,-E - F)\).

Let (P_a) be the A-power circle of the A-altimedial triangle, and define (P_b) and (P_c) cyclically. The radical center of (P_a), (P_b), (P_c) is X₄, and the harmonic traces of (P_a), (P_b), (P_c) are X₂ and X₄₆₈. (Randy Hutson, November 2, 2017)

Let P and P' be circumcircle antipodes. Let Q and Q' be the complements of P and P', resp. The rectangular hyperbola passing through P, P', Q, Q' has center X₄₆₈ for all P. (Randy Hutson, March 29, 2020)