X₈₅₈

COMPLEMENT OF X₂₃

X₈₅₈ is the complement of X₂₃.

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

X₈₅₈ (COMPLEMENT OF X₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 23}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₃ (FAR-OUT POINT), X₃ (CIRCUMCENTER)

Trilinears: \(\sin^{3} C \sin 2B \sin(C - A) + \sin^{3} B \sin 2C \sin(B - A) : :\)

Barycentrics: \(a^{4} b^{2} - b^{6} + a^{4} c^{2} - 2 a^{2} b^{2} c^{2} + b^{4} c^{2} + b^{2} c^{4} - c^{6} : :\)

Search Value (6, 9, 13): \(5.470836447671\)

▸ Alternate Coordinates2 forms

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

• Barycentrics: \(Sin[2 B] + Sin[2 C] - 3 Tan[w] : :\)

▸ Linear Combinations1 combination

• X₈₅₈ = \(3\) X₂ - \(4\) X₅₁₅₉, \(3\) X₂ + X₅₁₈₉, \(9\) X₂ - X₂₀₀₆₃, \(3\) X₂ - \(5\) X₃₀₇₄₅, \(9\) X₂ - \(5\) X₃₇₇₆₀, \(9\) X₂ - \(4\) X₃₇₈₉₇, \(9\) X₂ - \(2\) X₃₇₈₉₉, \(6\) X₂ - X₃₇₉₀₀, \(5\) X₂ - X₃₇₉₀₁, \(5\) X₂ - \(2\) X₃₇₉₀₄, \(5\) X₂ - \(3\) X₃₇₉₀₇, \(7\) X₂ - \(3\) X₃₇₉₀₉, \(15\) X₂ - \(4\) X₃₇₉₁₀, \(9\) X₂ - \(8\) X₃₇₉₁₁, \(3\) X₂ + \(2\) X₄₆₅₁₇, \(4\) X₃ - \(3\) X₄₄₂₈₀, \(4\) X₅ - \(3\) X₄₀₃, \(2\) X₅ - \(3\) X₂₀₇₂, \(5\) X₅ - \(3\) X₁₁₅₆₃, X₅ - \(3\) X₃₇₉₃₈, \(7\) X₅ - \(3\) X₄₃₈₉₃, \(3\) X₅ - \(2\) X₄₄₉₆₁, \(7\) X₅ - \(6\) X₄₆₀₃₁, X₂₀ - \(3\) X₂₀₇₁, X₂₀ + \(3\) X₃₁₅₃, \(2\) X₂₀ - \(3\) X₁₆₃₈₆, X₂₀ - \(9\) X₄₄₄₅₀, X₂₃ - \(4\) X₅₁₅₉, \(2\) X₂₃ - \(3\) X₇₄₂₆, X₂₃ + \(3\) X₁₀₉₈₉, \(3\) X₂₃ - X₂₀₀₆₃, X₂₃ - \(5\) X₃₀₇₄₅, \(3\) X₂₃ - \(5\) X₃₇₇₆₀, \(3\) X₂₃ - \(4\) X₃₇₈₉₇, \(3\) X₂₃ - \(2\) X₃₇₈₉₉, \(5\) X₂₃ - \(3\) X₃₇₉₀₁, \(5\) X₂₃ - \(6\) X₃₇₉₀₄, \(5\) X₂₃ - \(9\) X₃₇₉₀₇, \(7\) X₂₃ - \(9\) X₃₇₉₀₉, \(5\) X₂₃ - \(4\) X₃₇₉₁₀, \(3\) X₂₃ - \(8\) X₃₇₉₁₁, X₂₃ + \(2\) X₄₆₅₁₇, \(3\) X₁₂₅ - X₄₁₅₈₆, \(9\) X₁₄₀ - \(5\) X₂₂₂₄₈, \(3\) X₁₄₀ - \(2\) X₂₂₂₄₉, \(4\) X₁₄₀ - \(3\) X₄₄₂₁₄, \(3\) X₁₈₆ - \(5\) X₆₃₁, \(3\) X₃₈₁ - X₁₈₃₂₅, \(3\) X₃₈₁ + X₃₅₀₀₁, X₃₈₂ - \(3\) X₁₈₄₀₃, X₃₈₂ + \(3\) X₁₈₈₅₉, \(5\) X₄₀₃ - \(4\) X₁₁₅₆₃, \(3\) X₄₀₃ - \(2\) X₁₁₇₉₉, X₄₀₃ - \(4\) X₃₇₉₃₈, \(7\) X₄₀₃ - \(4\) X₄₃₈₉₃, \(9\) X₄₀₃ - \(8\) X₄₄₉₆₁, \(7\) X₄₀₃ - \(8\) X₄₆₀₃₁, \(2\) X₄₆₈ + X₅₁₈₉, \(4\) X₄₆₈ - \(3\) X₇₄₂₆, \(2\) X₄₆₈ + \(3\) X₁₀₉₈₉, \(6\) X₄₆₈ - X₂₀₀₆₃, \(2\) X₄₆₈ - \(5\) X₃₀₇₄₅, \(6\) X₄₆₈ - \(5\) X₃₇₇₆₀, \(3\) X₄₆₈ - \(2\) X₃₇₈₉₇, \(3\) X₄₆₈ - X₃₇₈₉₉, \(4\) X₄₆₈ - X₃₇₉₀₀, \(10\) X₄₆₈ - \(3\) X₃₇₉₀₁, \(5\) X₄₆₈ - \(3\) X₃₇₉₀₄, \(10\) X₄₆₈ - \(9\) X₃₇₉₀₇, \(14\) X₄₆₈ - \(9\) X₃₇₉₀₉, \(5\) X₄₆₈ - \(2\) X₃₇₉₁₀, \(3\) X₄₆₈ - \(4\) X₃₇₉₁₁, \(2\) X₅₄₈ - \(3\) X₃₄₁₅₂, \(4\) X₅₄₈ - \(3\) X₄₄₂₄₆, \(3\) X₅₄₉ - \(2\) X₁₈₅₇₁, \(5\) X₆₃₁ - \(6\) X₁₀₂₅₇, \(5\) X₆₃₁ + \(3\) X₄₆₄₅₀, \(5\) X₆₃₂ - \(2\) X₁₂₁₀₅

X₈₅₈ lies on the cubics K008, K288, K474, K475, K479, K794.

▸ Incident Central Lines367 lines

{2, 3}, {6, 16306}, {11, 3100}, {12, 4296}, {39, 15820}, {50, 230}, {51, 12058}, {52, 26879}, {53, 15355}, {66, 20806}, {67, 524}, {69, 23300}, {70, 15316}, {74, 16167}, {76, 15667}, {97, 23295}, {98, 2986}, {99, 37804}, {110, 1503}, {111, 24855}, {113, 14915}, {114, 3258}, {115, 3291}, {120, 5520}, {122, 42426}, {125, 511}, {126, 625}, {132, 16177}, {141, 7703}, {147, 14611}, {155, 11457}, {159, 28408}, {182, 14389}, {187, 40544}, {193, 18919}, {195, 43588}, {250, 37801}, {262, 16311}, {265, 37477}, {305, 7796}, {316, 691}, {323, 3448}, {325, 523}, {343, 2979}, {373, 19130}, {385, 16315}, {394, 1853}, {476, 1297}, {496, 9538}, {541, 44791}, {542, 3292}, {566, 3815}, {590, 11417}, {612, 37719}, {614, 9643}, {615, 11418}, {626, 30749}, {671, 15398}, {842, 925}, {930, 23319}, {1078, 11056}, {1086, 24164}, {1092, 14516}, {1147, 34224}, {1154, 20379}, {1180, 9607}, {1194, 7765}, {1209, 5447}, {1290, 26703}, {1294, 9060}, {1302, 2693}, {1304, 34168}, {1331, 41327}, {1350, 20300}, {1351, 26869}, {1352, 15066}, {1353, 11004}, {1495, 5972}, {1531, 2777}, {1533, 36518}, {1560, 5523}, {1568, 6000}, {1614, 9820}, {1619, 41736}, {1648, 20977}, {1843, 26156}, {1899, 1993}, {1994, 11245}, {2393, 5181}, {2452, 7774}, {2453, 7778}, {2549, 9745}, {2690, 30788}, {2694, 9058}, {2752, 13397}, {2770, 3565}, {2883, 12279}, {2886, 20243}, {2892, 11061}, {2972, 38552}, {3060, 13567}, {3101, 3925}, {3163, 41358}, {3164, 7777}, {3218, 26933}, {3219, 21015}, {3231, 39691}, {3284, 6103}, {3313, 6697}, {3314, 16313}, {3566, 14698}, {3574, 9729}, {3581, 15061}, {3589, 19121}, {3613, 14761}, {3788, 30747}, {3794, 26542}, {3818, 5651}, {3819, 43130}, {3917, 11649}, {3920, 15888}, {3933, 9464}, {4316, 5370}, {4324, 7302}, {4325, 5322}, {4329, 23305}, {4330, 5310}, {4576, 6393}, {5012, 23292}, {5103, 13518}, {5211, 31126}, {5254, 9465}, {5305, 5354}, {5318, 37776}, {5319, 5359}, {5321, 37775}, {5422, 44480}, {5446, 43817}, {5448, 10575}, {5449, 10625}, {5480, 5640}, {5562, 20299}, {5596, 28708}, {5642, 11645}, {5650, 24206}, {5654, 11456}, {5888, 32218}, {5971, 13219}, {6036, 22463}, {6090, 18440}, {6101, 13561}, {6106, 6109}, {6107, 6108}, {6146, 34148}, {6241, 22660}, {6243, 11660}, {6247, 12111}, {6340, 13575}, {6390, 14360}, {6515, 23291}, {6530, 46106}, {6688, 44300}, {6696, 11440}, {6698, 8262}, {6699, 32110}, {6723, 29317}, {6776, 37645}, {6795, 9744}, {6800, 46264}, {7191, 9630}, {7668, 22329}, {7689, 43607}, {7691, 32351}, {7706, 37470}, {7736, 16303}, {7750, 26233}, {7752, 11059}, {7779, 31372}, {7830, 15822}, {7832, 30785}, {7840, 39352}, {7849, 21248}, {7868, 16321}, {7912, 30793}, {7925, 16316}, {8024, 45201}, {8280, 35812}, {8281, 35813}, {8288, 15993}, {8550, 11422}, {8585, 18424}, {8680, 25344}, {8759, 43735}, {8791, 10317}, {8901, 43768}, {9076, 11635}, {9128, 10160}, {9189, 39509}, {9225, 11646}, {9300, 41335}, {9306, 11550}, {9479, 19375}, {9545, 31804}, {9681, 18289}, {9710, 29667}, {9711, 29679}, {9822, 46026}, {10102, 20187}, {10192, 26881}, {10264, 12219}, {10418, 40350}, {10423, 39436}, {10539, 16659}, {10564, 17702}, {10574, 12233}, {10627, 34826}, {10717, 22110}, {11002, 21850}, {11163, 16333}, {11174, 16324}, {11206, 41602}, {11411, 23307}, {11412, 12359}, {11420, 23302}, {11421, 23303}, {11441, 14216}, {11443, 23326}, {11449, 34782}, {11454, 23328}, {11580, 43291}, {11592, 13565}, {11605, 18876}, {11680, 23304}, {11750, 12038}, {11809, 29639}, {12022, 13352}, {12079, 31127}, {12121, 32227}, {12160, 26944}, {12163, 20302}, {12221, 23309}, {12222, 23310}, {12223, 23311}, {12224, 23312}, {12226, 21230}, {12272, 15583}, {12278, 41362}, {12370, 37495}, {12824, 32271}, {13009, 23313}, {13010, 23314}, {13292, 15134}, {13366, 33749}, {13367, 44829}, {13372, 14769}, {13391, 20396}, {13394, 15080}, {13398, 39119}, {13445, 15138}, {13470, 43394}, {13568, 43601}, {13573, 15262}, {13574, 34164}, {13754, 16003}, {13869, 29832}, {14156, 44407}, {14262, 32133}, {14538, 40709}, {14539, 40710}, {14567, 41939}, {14644, 43576}, {14852, 37483}, {14919, 17986}, {14927, 35260}, {14965, 35325}, {14981, 36212}, {15018, 18583}, {15043, 16227}, {15059, 15107}, {15132, 22115}, {15133, 15136}, {15140, 41595}, {15360, 44569}, {15448, 35904}, {15523, 21680}, {15606, 41590}, {15644, 32767}, {15880, 31401}, {15958, 46064}, {16103, 39356}, {16111, 33547}, {16163, 32743}, {16221, 31842}, {16266, 25738}, {16275, 33651}, {16319, 43460}, {16320, 30717}, {16325, 16990}, {16658, 46261}, {17134, 44412}, {17171, 26167}, {17712, 44516}, {17747, 24055}, {17834, 20303}, {18235, 27703}, {18279, 23097}, {18392, 23324}, {18396, 37497}, {18637, 21234}, {18669, 21017}, {18912, 36747}, {18914, 43590}, {18952, 36749}, {19122, 34774}, {19151, 45835}, {19379, 25329}, {19406, 23298}, {19407, 23299}, {19924, 32225}, {20403, 45689}, {20477, 23333}, {20481, 43620}, {21639, 32285}, {22112, 38317}, {22528, 23308}, {23325, 37480}, {23327, 41614}, {24284, 32120}, {24322, 24687}, {25320, 41617}, {25365, 31019}, {26875, 26951}, {26917, 41587}, {26958, 33586}, {28419, 36851}, {29323, 32237}, {29664, 39751}, {30741, 38514}, {31125, 46783}, {31383, 35264}, {31815, 37490}, {31843, 46439}, {32064, 37669}, {32068, 34565}, {32251, 37784}, {32300, 44102}, {34169, 41936}, {34237, 44766}, {34380, 37779}, {34545, 45298}, {34845, 37688}, {34989, 44401}, {35002, 38953}, {35259, 36990}, {35360, 44704}, {36983, 45014}, {37496, 38724}, {37649, 43815}, {37689, 41394}, {40111, 46114}, {41022, 41888}, {41023, 41887}, {41673, 44668}, {42329, 43461}, {43608, 44158}, {43650, 44491}, {44953, 46437}

▸ Functional Relations46 relations

• X₈₅₈ = midpoint of (I) and X_j for these (i,j): {2, 10989}, {3, 7574}, {4, 7464}, {20, 10296}, {23, 5189}, {67, 10510}, {186, 46450}, {265, 37477}, {316, 691}, {323, 3448}, {468, 46517}, {1370, 37980}, {2071, 3153}, {5002, 5003}, {5999, 36173}, {7468, 14957}, {14790, 45171}, {15133, 15136}, {17511, 36188}, {18325, 35001}, {18403, 18859}, {18572, 37950}, {23061, 41724}, {24322, 24687}, {25739, 43574}, {31726, 35452}, {35002, 38953}, {36185, 36186}

• X₈₅₈ = reflection of X_i in X_j for these {i,j}: {3, 15122}, {4, 10297}, {22, 16387}, {23, 468}, {110, 11064}, {186, 10257}, {187, 40544}, {297, 37987}, {385, 16315}, {403, 2072}, {468, 5159}, {1495, 5972}, {1513, 36170}, {2070, 44452}, {2072, 37938}, {3580, 125}, {5099, 625}, {5112, 11007}, {5181, 19510}, {5189, 46517}, {5523, 38971}, {5913, 31655}, {7426, 2}, {7473, 441}, {7575, 140}, {8262, 6698}, {9128, 10160}, {10295, 3}, {11799, 5}, {15107, 32269}, {15360, 44569}, {16320, 44377}, {16386, 2071}, {18323, 18572}, {20063, 37899}, {21284, 16977}, {25338, 3628}, {31726, 23323}, {32110, 6699}, {32111, 113}, {32113, 141}, {32120, 24284}, {32123, 23306}, {32125, 23315}, {32217, 3589}, {32218, 34573}, {32220, 6}, {32223, 6723}, {32224, 16324}, {32225, 45311}, {36189, 21531}, {36196, 37350}, {37897, 37911}, {37899, 37897}, {37900, 23}, {37901, 37904}, {37924, 16619}, {37925, 37971}, {37927, 36157}, {37931, 16976}, {37932, 44907}, {37936, 44234}, {37947, 10096}, {37967, 25338}, {37969, 6676}, {37971, 44911}, {39356, 16103}, {40111, 46114}, {40112, 13857}, {41603, 15126}, {43893, 46031}, {44246, 34152}, {44264, 16239}, {44265, 549}, {44266, 547}, {44267, 546}

• X₈₅₈ = isogonal conjugate of X₁₁₇₇

• X₈₅₈ = isotomic conjugate of X₂₃₇₃

• X₈₅₈ = complement of X₂₃

• X₈₅₈ = anticomplement of X₄₆₈

• X₈₅₈ = circumcircle-inverse of X₂₂

• X₈₅₈ = nine-point-circle-inverse of X₂

• X₈₅₈ = orthocentroidal-circle-inverse of X₁₉₉₅

• X₈₅₈ = polar-circle-inverse of X₂₅

• X₈₅₈ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃

• X₈₅₈ = orthoptic-circle-of-Steiner-circumellipse-inverse of X₂₀

• X₈₅₈ = 1st-Droz-Farney-circle-inverse of X₆₆₄₄

• X₈₅₈ = 2nd-Droz-Farney-circle-inverse of X₄₄₄₄₀

• X₈₅₈ = circumcircle-of-anticomplementary-triangle-inverse of X₁₃₇₀

• X₈₅₈ = Steiner-circle-inverse of X₂₀

• X₈₅₈ = de-Longchamps circle inverse of X₂

• X₈₅₈ = nine-point-circle-of-medial-triangle-inverse of X₇₄₉₉

• X₈₅₈ = Stammler-circle-inverse of X₄₄₄₅₇

• X₈₅₈ = Stammler-circles-radical-circle-inverse of X₄₄₄₅₈

• X₈₅₈ = circumcircle-of-inner-Napoleon-triangle-inverse of X₄₄₄₆₁

• X₈₅₈ = circumcircle-of-outer-Napoleon-triangle-inverse of X₄₄₄₆₅

• X₈₅₈ = {circumcircle, nine-point circle}-inverter-inverse of X₃

• X₈₅₈ = X_i-anticomplementary conjugate of X_j for these (i,j): {31, 7665}, {48, 8591}, {63, 14360}, {75, 34518}, {111, 5905}, {671, 21270}, {691, 7253}, {810, 39356}, {892, 21300}, {895, 8}, {897, 4}, {923, 193}, {4575, 44010}, {5380, 20293}, {5547, 5942}, {7316, 12649}, {10097, 21221}, {14908, 192}, {14977, 21294}, {15398, 17491}, {17983, 5906}, {30786, 6327}, {32729, 17498}, {32740, 21216}, {34055, 25052}, {36060, 2}, {36085, 850}, {36128, 6515}, {36142, 525}, {46277, 11442}

• X₈₅₈ = X_i-complementary conjugate of X_j for these (i,j): {1, 6593}, {67, 10}, {656, 38971}, {935, 8062}, {2157, 2}, {3455, 37}, {8791, 226}, {9076, 1215}, {10415, 4892}, {14357, 16597}, {17708, 4369}, {18019, 2887}, {34897, 18589}, {36820, 19563}, {37221, 3934}, {46105, 20305}

• X₈₅₈ = X_i-Ceva conjugate of X_j for these (i,j): {316, 524}, {691, 523}, {11605, 46442}, {17172, 18669}, {30786, 2}, {39436, 22}

• X₈₅₈ = X_i-cross conjugate of X_j for these (i,j): {1560, 2}, {2393, 5523}, {15116, 39269}, {21017, 20884}

• X₈₅₈ = cevapoint of X_i and X_j for these (i,j): {3, 15141}, {2393, 14961}

• X₈₅₈ = crosspoint of X_i and X_j for these (i,j): {2, 18019}, {264, 671}, {892, 23582}

• X₈₅₈ = crosssum of X_i and X_j for these (i,j): {3, 41615}, {6, 18374}, {184, 187}, {351, 3269}, {9426, 21906}

• X₈₅₈ = trilinear pole of line {5181, 21109}

• X₈₅₈ = crossdifference of every pair of points on line {32, 647}

• X₈₅₈ = reflection of X₂₃ in the orthic axis

• X₈₅₈ = homothetic center of complement of tangential triangle and anticomplement of orthic triangle

• X₈₅₈ = X₁₂₅-of-1st-anti-Brocard triangle

• X₈₅₈ = radical trace of circumcircle and de Longchamps circle

• X₈₅₈ = one of two harmonic traces of the power circles (X₂ is the other)

• X₈₅₈ = homothetic center of X₂- and X₄-Ehrmann triangles; see X₂₅

• X₈₅₈ = Euler line intercept, other than X₈₅₈, of circle {X₄₀₃,X₈₅₈,PU(4)}}

• X₈₅₈ = perspector of X₂-Ehrmann triangle and cross-triangle of ABC and X₂-Ehrmann triangle

• X₈₅₈ = intersection of orthic axes of 1st and 2nd Ehrmann circumscribing triangles

• X₈₅₈ = intersection of orthic axes of anticevian triangles of PU(4)

• X₈₅₈ = X_i-isoconjugate of X_j for these (i,j): {1, 1177}, {19, 18876}, {31, 2373}, {32, 37220}, {560, 46140}, {647, 36095}, {656, 10423}, {896, 10422}, {1910, 36823}, {46165, 46289}

• X₈₅₈ = barycentric product X_i*X_j for these {i,j}: {1, 20884}, {6, 1236}, {10, 17172}, {69, 5523}, {75, 18669}, {76, 2393}, {86, 21017}, {190, 21109}, {264, 14961}, {305, 14580}, {598, 19510}, {671, 5181}, {801, 41603}, {1560, 30786}, {2986, 12827}, {3267, 46592}, {3933, 21459}, {6331, 42665}, {22151, 39269}

• X₈₅₈ = barycentric quotient X_i/X_j for these {i,j}: {2, 2373}, {3, 18876}, {6, 1177}, {75, 37220}, {76, 46140}, {111, 10422}, {112, 10423}, {141, 46165}, {162, 36095}, {511, 36823}, {895, 41511}, {1236, 76}, {1560, 468}, {2393, 6}, {5181, 524}, {5523, 4}, {12827, 3580}, {14580, 25}, {14961, 3}, {15126, 26958}, {17172, 86}, {18669, 1}, {19510, 599}, {20410, 8744}, {20884, 75}, {21017, 10}, {21109, 514}, {21459, 32085}, {34158, 14908}, {39269, 46105}, {41603, 13567}, {42665, 647}, {46592, 112}

• X₈₅₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3, 7495}, {2, 4, 1995}, {2, 20, 7493}, {2, 23, 468}, {2, 427, 5133}, {2, 1370, 22}, {2, 2071, 16387}, {2, 2475, 4239}, {2, 3146, 4232}, {2, 3151, 26252}, {2, 3152, 26254}, {2, 3153, 37980}, {2, 3543, 26255}, {2, 5133, 37990}, {2, 5169, 5}, {2, 5189, 23}, {2, 6636, 6676}, {2, 6655, 26257}, {2, 6872, 26256}, {2, 7378, 6997}, {2, 7379, 7474}, {2, 7386, 7485}, {2, 7391, 25}, {2, 7394, 5020}, {2, 7396, 1370}, {2, 7409, 7398}, {2, 7496, 140}, {2, 7500, 6353}, {2, 7533, 16042}, {2, 7570, 3628}, {2, 8889, 31236}, {2, 13595, 6677}, {2, 14790, 26284}, {2, 14807, 1114}, {2, 14808, 1113}, {2, 15246, 7499}, {2, 16063, 3}, {2, 17511, 36166}, {2, 20063, 37760}, {2, 24585, 1375}, {2, 26097, 25495}, {2, 26761, 27029}, {2, 26809, 26968}, {2, 30745, 5159}, {2, 30889, 46531}, {2, 30952, 37370}, {2, 31022, 857}, {2, 31074, 427}, {2, 31099, 4}, {2, 31100, 442}, {2, 31101, 1368}, {2, 31102, 440}, {2, 31103, 21530}, {2, 31104, 18641}, {2, 31105, 381}, {2, 31106, 405}, {2, 31107, 6656}, {2, 31114, 13371}, {2, 31857, 5169}, {2, 34609, 34603}, {2, 36163, 9832}, {2, 36173, 7471}, {2, 36174, 36168}, {2, 37353, 37439}, {2, 37444, 26283}, {2, 37456, 4228}, {2, 37901, 37907}, {2, 46336, 40916}, {2, 46517, 37900}, {3, 4, 38323}, {3, 1594, 13160}, {3, 5094, 2}, {3, 10295, 44280}, {3, 13371, 1594}, {3, 18281, 37118}, {3, 18569, 6240}, {3, 21284, 37978}, {3, 26283, 22}, {3, 31152, 16063}, {3, 34725, 37196}, {3, 37444, 12225}, {3, 37972, 21284}, {4, 3546, 17928}, {4, 16051, 2}, {4, 31099, 31133}, {5, 427, 5169}, {5, 1368, 30739}, {5, 1907, 3832}, {5, 5169, 5133}, {5, 11799, 403}, {5, 30739, 2}, {20, 631, 38444}, {20, 3153, 10296}, {20, 7493, 22}, {20, 30769, 2}, {20, 35497, 548}, {21, 30770, 2}, {22, 30744, 2}, {23, 468, 7426}, {23, 2071, 37929}, {23, 10989, 5189}, {23, 20063, 37899}, {23, 30745, 2}, {23, 37760, 37897}, {23, 37901, 37910}, {23, 37929, 22}, {23, 37978, 21284}, {24, 30802, 2}, {25, 382, 7519}, {25, 7391, 34603}, {25, 30771, 2}, {25, 34609, 7391}, {25, 37928, 23}, {26, 6640, 10018}, {26, 30803, 2}, {27, 30772, 2}, {28, 30773, 2}, {29, 30774, 2}, {50, 44529, 230}, {66, 20806, 46442}, {115, 39602, 5913}, {140, 5576, 14788}, {140, 7575, 44214}, {140, 10300, 43957}, {140, 11819, 45735}, {140, 37454, 2}, {140, 43957, 7496}, {343, 23332, 23293}, {376, 7577, 15760}, {376, 30775, 2}, {377, 30776, 2}, {378, 31180, 18531}, {381, 21312, 44440}, {381, 32216, 2}, {381, 35001, 18325}, {382, 7519, 34603}, {384, 30777, 2}, {394, 1853, 11442}, {427, 468, 37981}, {427, 1312, 46698}, {427, 1313, 46699}, {427, 1368, 2}, {427, 5094, 1594}, {427, 15809, 7378}, {427, 23335, 31099}, {427, 30739, 5}, {427, 37454, 5576}, {428, 6677, 13595}, {468, 5159, 2}, {468, 5189, 37900}, {468, 37897, 37760}, {468, 37899, 37897}, {468, 37981, 403}, {549, 39504, 37347}, {549, 44239, 10298}, {550, 10224, 10024}, {550, 44210, 7492}, {851, 37165, 46488}, {851, 46484, 46487}, {851, 46485, 46484}, {851, 46486, 37165}, {1092, 18381, 14516}, {1113, 1114, 22}, {1312, 1313, 2}, {1346, 10720, 1312}, {1347, 10719, 1313}, {1351, 26869, 37644}, {1368, 11585, 16051}, {1368, 13371, 5094}, {1368, 31074, 5133}, {1370, 7493, 20}, {1370, 16387, 16386}, {1370, 26283, 12225}, {1375, 46553, 33325}, {1594, 7495, 37990}, {1594, 10295, 403}, {1657, 10255, 15761}, {1899, 1993, 45968}, {1995, 7485, 17928}, {1995, 11413, 22}, {1995, 31133, 4}, {2041, 2042, 24}, {2071, 5999, 46620}, {2071, 7464, 11413}, {2071, 7574, 12225}, {2071, 10296, 20}, {2071, 10989, 1370}, {2071, 37980, 22}, {2072, 7426, 37990}, {2072, 7575, 14788}, {2072, 10295, 13160}, {2072, 11799, 5}, {2072, 25338, 34939}, {2072, 44265, 37347}, {2450, 36189, 403}, {2479, 2480, 35923}, {2979, 23293, 343}, {3060, 26913, 13567}, {3153, 7396, 10989}, {3153, 37444, 7574}, {3153, 44450, 2071}, {3154, 36170, 2}, {3520, 12605, 34005}, {3541, 6643, 7503}, {3548, 14790, 24}, {3575, 16196, 22467}, {3627, 44212, 10301}, {3628, 25338, 44282}, {3917, 21243, 37636}, {4232, 30552, 22}, {5000, 5001, 403}, {5004, 5005, 186}, {5020, 5064, 7394}, {5020, 31255, 2}, {5064, 31255, 5020}, {5094, 7396, 12225}, {5094, 16063, 7495}, {5094, 31152, 3}, {5117, 37190, 41237}, {5133, 7426, 403}, {5133, 7495, 13160}, {5159, 5189, 7426}, {5159, 10989, 37900}, {5159, 37897, 37911}, {5159, 46517, 23}, {5169, 31074, 31857}, {5169, 31857, 427}, {5189, 10989, 46517}, {5189, 30745, 468}, {5189, 37760, 20063}, {5480, 37648, 5640}, {5576, 7575, 403}, {6143, 7512, 7542}, {6353, 44442, 7500}, {6644, 31181, 31723}, {6644, 31723, 7576}, {6676, 7667, 6636}, {7378, 37962, 10151}, {7386, 8889, 2}, {7386, 16051, 3546}, {7391, 7492, 18565}, {7391, 7519, 382}, {7396, 16051, 11413}, {7396, 30769, 20}, {7426, 16386, 22}, {7426, 37900, 23}, {7471, 46620, 22}, {7485, 31236, 2}, {7495, 12225, 22}, {7499, 10691, 15246}, {7539, 16419, 2}, {7542, 32144, 6143}, {7553, 16238, 3518}, {7570, 33332, 5133}, {7574, 15122, 10295}, {7574, 18403, 18569}, {7574, 37938, 1594}, {7667, 37969, 44246}, {8613, 10486, 7493}, {8889, 37119, 5094}, {9140, 23061, 41724}, {10255, 15761, 35487}, {10257, 37984, 6803}, {10297, 11585, 2072}, {10300, 37454, 7496}, {10301, 44212, 14002}, {10415, 15899, 16092}, {10415, 34320, 15899}, {10989, 30745, 23}, {10989, 37938, 5133}, {11250, 18563, 35491}, {11284, 35001, 23}, {11318, 11336, 2}, {11412, 23294, 12359}, {11585, 23335, 4}, {11585, 31099, 5133}, {12084, 18404, 18560}, {12085, 18859, 7464}, {12088, 14940, 13383}, {13371, 15122, 2072}, {13371, 16063, 5133}, {13394, 44882, 15080}, {14791, 18281, 3}, {14807, 14808, 1370}, {14957, 33314, 297}, {15122, 37444, 16386}, {15154, 15155, 44457}, {15329, 46594, 22}, {15899, 40343, 34320}, {15899, 42008, 10415}, {16049, 26253, 22}, {16051, 31099, 1995}, {16063, 37119, 7485}, {16063, 37444, 1370}, {16239, 44264, 44234}, {16387, 37980, 7426}, {16976, 37931, 37941}, {18531, 44441, 378}, {18586, 18587, 44275}, {19772, 19773, 401}, {20063, 30745, 37911}, {20063, 37760, 23}, {20063, 37899, 37900}, {20063, 37911, 7426}, {20405, 20406, 10691}, {20408, 20409, 20}, {21284, 37972, 23}, {21531, 21536, 2450}, {23047, 31829, 34007}, {23335, 37938, 10297}, {24887, 24940, 2}, {27099, 27151, 2}, {27387, 27511, 33305}, {28021, 28085, 33302}, {28049, 28127, 33306}, {28241, 28355, 859}, {28413, 28702, 441}, {29386, 29443, 46497}, {29499, 29702, 46575}, {29554, 29753, 46574}, {29971, 30041, 46514}, {30739, 31857, 5133}, {30745, 46517, 7426}, {30771, 34609, 25}, {31074, 31101, 2}, {31101, 31857, 30739}, {33305, 46552, 13589}, {34152, 37969, 6636}, {34320, 42008, 16092}, {34725, 37196, 3146}, {37050, 37330, 2}, {37165, 46484, 851}, {37165, 46485, 46487}, {37165, 46486, 46492}, {37353, 37901, 11563}, {37454, 43957, 140}, {37760, 37897, 7426}, {37897, 37899, 23}, {37897, 37911, 468}, {37899, 37911, 37760}, {37901, 37907, 37904}, {37904, 37907, 7426}, {37904, 37910, 23}, {37911, 46517, 20063}, {37925, 37943, 37971}, {37929, 37980, 23}, {37967, 44282, 25338}, {37971, 44911, 37943}, {39504, 44265, 403}, {40343, 42008, 15899}, {44309, 44310, 16386}, {46484, 46485, 46491}, {46484, 46486, 46488}, {46485, 46486, 851}, {46485, 46490, 46489}, {46486, 46489, 46490}, {46487, 46488, 851}, {46487, 46491, 46484}, {46487, 46492, 46488}, {46488, 46491, 46487}, {46488, 46492, 37165}, {46489, 46490, 851}, {46491, 46492, 851}, {46550, 46555, 46549}, {46551, 46554, 46548}, {46698, 46699, 5133}

Notes

As a point on the Euler line, X₈₅₈ has Shinagawa coefficients \((E - 2F,-2E - 2F)\).

Let P and P' be circumcircle antipodes. Let Q and Q' be the anticomplements of P and P', resp. The rectangular hyperbola passing through P, P', Q, Q' has center X₈₅₈ for all P. (Randy Hutson, March 29, 2020)