X₄₂₇

COMPLEMENT OF X₂₂

X₄₂₇ is the complement of X₂₂.

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

X₄₂₇ (COMPLEMENT OF X₂₂) 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, 22}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₂ (EXETER POINT), X₃ (CIRCUMCENTER)

Trilinears: \(\sec A + \cos(B - C)\)

Barycentrics: \(\tan A + \tan \omega : :\)

Search Value (6, 9, 13): \(16.125778007218\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₄₂₇ = \(3\) X₂ + X₇₃₉₁, \(9\) X₂ - X₂₀₀₆₂, X₂ + \(3\) X₃₁₁₀₅, \(3\) X₂ - \(5\) X₃₁₂₃₆, \(5\) X₂ - \(3\) X₄₇₅₉₆, \(3\) X₄ - X₃₅₄₈₀, \(3\) X₄ + X₃₅₄₈₁, \(3\) X₅ - \(4\) X₁₃₄₁₃, and many others

X₄₂₇ lies on the cubics K517, K533, K539, K823, K836.

▸ Incident Central Lines627 lines

{1, 5090}, {2, 3}, {6, 66}, {8, 11396}, {9, 21015}, {10, 1829}, {11, 33}, {12, 34}, {19, 3925}, {32, 47298}, {39, 12143}, {51, 125}, {52, 6746}, {53, 232}, {54, 31804}, {55, 11393}, {56, 11392}, {57, 1892}, {64, 14542}, {68, 36747}, {69, 12167}, {76, 8890}, {83, 11380}, {92, 2969}, {98, 275}, {107, 29011}, {110, 39884}, {112, 251}, {113, 12133}, {114, 136}, {115, 1196}, {116, 5185}, {119, 12138}, {120, 5521}, {122, 46831}, {126, 5139}, {127, 13166}, {135, 31842}, {137, 23319}, {141, 1843}, {143, 13561}, {154, 31383}, {155, 23307}, {182, 37649}, {183, 317}, {184, 1503}, {185, 3574}, {193, 8892}, {211, 27370}, {214, 12137}, {216, 26905}, {225, 26481}, {226, 1876}, {230, 571}, {238, 14975}, {242, 52412}, {262, 2052}, {264, 305}, {265, 15472}, {273, 7179}, {278, 1390}, {281, 1851}, {286, 37664}, {316, 33651}, {318, 3705}, {323, 3410}, {324, 2967}, {340, 37671}, {343, 511}, {371, 8280}, {372, 8281}, {373, 44079}, {385, 27377}, {388, 1398}, {389, 20299}, {393, 7736}, {394, 1352}, {395, 8739}, {396, 8740}, {459, 14484}, {485, 3093}, {486, 3092}, {487, 23309}, {488, 23310}, {495, 1870}, {496, 6198}, {497, 7071}, {498, 11398}, {499, 11399}, {515, 51707}, {524, 8541}, {542, 34986}, {578, 6146}, {590, 5412}, {597, 37875}, {608, 34261}, {615, 5413}, {618, 12142}, {619, 12141}, {625, 5140}, {629, 22482}, {630, 22481}, {641, 12148}, {642, 12147}, {648, 41624}, {748, 2212}, {750, 1395}, {804, 47230}, {899, 40976}, {908, 27409}, {930, 3563}, {940, 44105}, {946, 1902}, {958, 11391}, {973, 49108}, {1007, 19583}, {1031, 3329}, {1039, 1722}, {1086, 24163}, {1105, 14860}, {1125, 11363}, {1147, 12134}, {1172, 33854}, {1180, 5523}, {1181, 14216}, {1184, 3767}, {1194, 5254}, {1204, 6696}, {1209, 10625}, {1213, 44103}, {1217, 18855}, {1235, 3933}, {1241, 9230}, {1249, 37665}, {1291, 40118}, {1297, 20626}, {1329, 1828}, {1350, 43653}, {1351, 6515}, {1353, 1994}, {1369, 14929}, {1376, 11383}, {1401, 46152}, {1426, 15844}, {1447, 7282}, {1452, 24914}, {1474, 17398}, {1495, 10192}, {1506, 3199}, {1511, 12140}, {1568, 15030}, {1576, 39086}, {1611, 13881}, {1614, 16659}, {1627, 10312}, {1629, 14165}, {1698, 7713}, {1737, 1905}, {1753, 15908}, {1785, 24239}, {1799, 7750}, {1824, 1848}, {1826, 1841}, {1827, 21239}, {1838, 1867}, {1842, 30768}, {1865, 37661}, {1878, 3814}, {1890, 2355}, {1891, 25466}, {1897, 29840}, {1900, 25639}, {1916, 37892}, {1953, 21911}, {1968, 7745}, {1973, 29647}, {1974, 3589}, {1975, 34254}, {1986, 10264}, {1990, 9300}, {1992, 11405}, {1993, 3564}, {2023, 14715}, {2056, 11646}, {2183, 21912}, {2207, 2548}, {2211, 20965}, {2356, 3720}, {2453, 47177}, {2482, 12132}, {2486, 47232}, {2493, 34981}, {2501, 10278}, {2549, 15880}, {2550, 11406}, {2881, 47205}, {2883, 11381}, {2892, 32251}, {2904, 43588}, {2905, 17000}, {2971, 34336}, {2974, 34338}, {2979, 6403}, {3006, 41013}, {3011, 40950}, {3051, 20021}, {3054, 10985}, {3055, 14806}, {3060, 3580}, {3068, 5410}, {3069, 5411}, {3070, 11474}, {3071, 11473}, {3087, 7735}, {3096, 11386}, {3108, 41366}, {3167, 18440}, {3168, 51358}, {3172, 8879}, {3186, 3314}, {3192, 37662}, {3203, 27366}, {3258, 42426}, {3266, 44142}, {3291, 40326}, {3527, 45010}, {3567, 23294}, {3616, 7718}, {3618, 19118}, {3647, 16114}, {3703, 20883}, {3740, 41611}, {3757, 5174}, {3763, 7716}, {3796, 46264}, {3818, 9306}, {3819, 24206}, {3820, 29679}, {4383, 44086}, {4994, 8883}, {5012, 14389}, {5044, 41609}, {5081, 7081}, {5095, 8584}, {5097, 11225}, {5146, 29873}, {5155, 5268}, {5190, 5513}, {5203, 8770}, {5207, 37894}, {5249, 25365}, {5272, 7741}, {5275, 50036}, {5297, 10592}, {5304, 19041}, {5305, 5359}, {5306, 6103}, {5310, 6284}, {5318, 11476}, {5321, 11475}, {5322, 7354}, {5345, 10483}, {5422, 18583}, {5432, 52427}, {5446, 5449}, {5475, 9515}, {5490, 26373}, {5491, 26374}, {5552, 11400}, {5590, 11389}, {5591, 11388}, {5596, 19125}, {5599, 11384}, {5600, 11385}, {5640, 26913}, {5654, 18451}, {5690, 41722}, {5718, 44113}, {5743, 44092}, {5800, 44094}, {5889, 31802}, {5891, 51392}, {5913, 50718}, {5921, 15431}, {5943, 19130}, {5946, 52000}, {5966, 30248}, {5972, 16165}, {5996, 14618}, {6000, 18388}, {6033, 41253}, {6090, 14826}, {6108, 31687}, {6109, 31688}, {6114, 6117}, {6115, 6116}, {6152, 21230}, {6212, 16033}, {6213, 16027}, {6260, 12136}, {6288, 37495}, {6291, 23311}, {6292, 12144}, {6331, 8842}, {6340, 14248}, {6406, 23312}, {6467, 15583}, {6524, 10002}, {6525, 17830}, {6560, 18290}, {6561, 18289}, {6564, 8854}, {6565, 8855}, {6593, 32239}, {6697, 9969}, {6699, 15473}, {6759, 16655}, {6776, 11402}, {7009, 17923}, {7085, 50861}, {7292, 10593}, {7592, 11457}, {7603, 33842}, {7608, 39284}, {7649, 44316}, {7687, 32743}, {7699, 11455}, {7717, 18230}, {7722, 11804}, {7752, 47846}, {7753, 14581}, {7762, 8878}, {7774, 9308}, {7776, 40123}, {7778, 41762}, {7788, 44134}, {7789, 30747}, {7792, 36794}, {7840, 38294}, {8222, 11394}, {8223, 11395}, {8263, 11188}, {8428, 34866}, {8537, 41628}, {8550, 13366}, {8680, 25343}, {8753, 42008}, {8792, 41513}, {8796, 14494}, {8877, 10415}, {8968, 48467}, {8981, 10880}, {9134, 47236}, {9148, 17994}, {9157, 10735}, {9483, 42396}, {9544, 46818}, {9605, 41361}, {9732, 11091}, {9733, 11090}, {9744, 11547}, {9752, 35710}, {9766, 45921}, {9777, 11433}, {9781, 26917}, {9786, 26937}, {9792, 26954}, {9820, 10539}, {9833, 19357}, {9993, 43462}, {10110, 32767}, {10189, 41357}, {10190, 47627}, {10263, 34826}, {10282, 13419}, {10327, 17757}, {10516, 17811}, {10523, 15666}, {10527, 11401}, {10547, 16277}, {10550, 51862}, {10559, 51405}, {10576, 35764}, {10577, 35765}, {10601, 14561}, {10632, 42124}, {10633, 42121}, {10641, 23302}, {10642, 23303}, {10733, 32227}, {10881, 13966}, {10982, 39571}, {11174, 17907}, {11197, 44131}, {11206, 26864}, {11216, 47277}, {11408, 11488}, {11409, 11489}, {11411, 12160}, {11412, 33523}, {11424, 12241}, {11425, 19467}, {11430, 18400}, {11432, 18916}, {11435, 26957}, {11436, 26956}, {11438, 23329}, {11470, 15004}, {11472, 32123}, {11566, 40685}, {11572, 21659}, {11808, 14076}, {12022, 15033}, {12038, 45286}, {12058, 37511}, {12079, 35908}, {12139, 12864}, {12146, 13089}, {12161, 32140}, {12162, 18488}, {12165, 12317}, {12166, 12318}, {12168, 12319}, {12169, 12320}, {12170, 12321}, {12171, 12322}, {12172, 12323}, {12174, 12324}, {12175, 12325}, {12242, 14864}, {12295, 15432}, {12298, 45861}, {12299, 45860}, {12300, 20424}, {12618, 21062}, {12827, 14984}, {13007, 13025}, {13008, 13026}, {13051, 23313}, {13052, 23314}, {13148, 16003}, {13171, 13203}, {13292, 25738}, {13352, 18474}, {13367, 34782}, {13394, 29012}, {13403, 18383}, {13409, 23635}, {13428, 49029}, {13439, 49028}, {13562, 20806}, {13668, 13701}, {13788, 13821}, {13851, 23324}, {13857, 47354}, {14111, 35887}, {14156, 43586}, {14157, 16658}, {14376, 17407}, {14458, 43530}, {14492, 16080}, {14516, 34148}, {14583, 43090}, {14593, 40801}, {14627, 46443}, {14644, 43391}, {14767, 51412}, {14852, 44413}, {14918, 38429}, {14983, 45020}, {15038, 45967}, {15060, 51391}, {15066, 18358}, {15069, 37672}, {15116, 32246}, {15118, 41616}, {15152, 16252}, {15448, 44082}, {15449, 16102}, {15463, 32423}, {15484, 41370}, {15487, 46835}, {15523, 17442}, {15543, 15552}, {15544, 35605}, {15589, 32001}, {15819, 22480}, {16030, 19174}, {16188, 16221}, {16228, 47802}, {16229, 34964}, {16231, 39508}, {16276, 32819}, {16580, 40959}, {16607, 18636}, {16657, 18390}, {17451, 21717}, {17747, 24054}, {17810, 26958}, {17949, 32449}, {18287, 41925}, {18344, 21260}, {18356, 32358}, {18424, 24855}, {18475, 44407}, {18809, 46436}, {18853, 18854}, {18951, 37493}, {19119, 20079}, {19128, 38110}, {19347, 34780}, {19366, 26955}, {19404, 19420}, {19405, 19421}, {19446, 23298}, {19447, 23299}, {19459, 36851}, {19568, 47730}, {19724, 19755}, {19725, 19754}, {19784, 51686}, {19799, 19839}, {20185, 40120}, {20192, 45311}, {20306, 42448}, {20376, 52008}, {21072, 29016}, {21318, 41007}, {21637, 34774}, {21639, 23326}, {21663, 23328}, {21969, 41586}, {22240, 42459}, {22352, 44882}, {22483, 22966}, {22497, 22555}, {22750, 43614}, {22970, 23308}, {24321, 24686}, {26359, 26371}, {26360, 26372}, {26361, 26375}, {26362, 26376}, {26363, 26377}, {26364, 26378}, {26866, 26929}, {26867, 26939}, {26868, 26945}, {26892, 26932}, {26893, 26942}, {26894, 26950}, {26919, 26951}, {29667, 31419}, {30749, 40325}, {30786, 40413}, {31125, 41676}, {31127, 35360}, {31131, 44428}, {31406, 39575}, {31655, 48317}, {31670, 33586}, {31867, 35717}, {32000, 37668}, {32002, 37688}, {32046, 52432}, {32062, 51403}, {32085, 40425}, {32136, 45732}, {32184, 52003}, {32223, 48895}, {32240, 32255}, {32269, 48901}, {32332, 32391}, {32333, 32346}, {32337, 32341}, {32911, 44097}, {33547, 46686}, {34380, 45794}, {34417, 47296}, {34751, 44439}, {34781, 43841}, {34945, 41363}, {35540, 45211}, {35603, 36753}, {35718, 35729}, {35719, 35720}, {35723, 39565}, {35895, 35896}, {35971, 40377}, {36898, 38427}, {37472, 43595}, {37478, 44201}, {37650, 44100}, {37775, 42135}, {37776, 42138}, {38253, 43951}, {38396, 39879}, {38872, 44529}, {38970, 38975}, {39071, 47200}, {39511, 39533}, {39512, 51513}, {39532, 44432}, {39534, 48182}, {39662, 43448}, {39803, 39813}, {39832, 39842}, {39998, 44146}, {40149, 45964}, {40684, 44145}, {41171, 43574}, {41482, 51033}, {41603, 50649}, {41673, 41714}, {42353, 46832}, {42873, 51939}, {43817, 43823}, {44091, 51126}, {44426, 44429}, {45400, 45472}, {45401, 45473}, {45689, 47206}, {46147, 46151}, {46242, 46288}, {46261, 51425}, {47105, 51343}, {47392, 51385}

▸ Functional Relations59 relations

• X₄₂₇ = midpoint of X_i and X_j for these {i,j}: {2, 31133}, {3, 31723}, {4, 378}, {22, 7391}, {112, 11605}, {184, 11550}, {1993, 11442}, {13352, 18474}, {15463, 44795}, {18570, 44288}, {24321, 24686}, {35480, 35481}

• X₄₂₇ = reflection of X_i in X_j for these {i,j}: {1, 51718}, {3, 52262}, {5, 39504}, {6, 51744}, {22, 6676}, {184, 23292}, {235, 45179}, {343, 21243}, {1843, 51994}, {7502, 140}, {12083, 16618}, {15760, 5}, {16102, 15449}, {16165, 5972}, {16387, 5159}, {16789, 141}, {18570, 44236}, {19127, 3589}, {25337, 3628}, {34177, 6697}, {37478, 44201}, {37969, 468}, {44210, 2}, {44218, 44287}, {44239, 3}, {44249, 18570}, {44259, 16238}, {44260, 6677}, {44261, 549}, {44262, 547}, {44263, 546}, {44285, 44218}, {46029, 13413}, {51692, 1125}

• X₄₂₇ = isogonal conjugate of X₁₁₇₆

• X₄₂₇ = isotomic conjugate of X₁₇₉₉

• X₄₂₇ = complement of X₂₂

• X₄₂₇ = anticomplement of X₆₆₇₆

• X₄₂₇ = circumcircle-inverse of X₂₁₂₈₄

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

• X₄₂₇ = orthocentroidal-circle-inverse of X₂₅

• X₄₂₇ = polar-circle-inverse of X₂₃

• X₄₂₇ = orthoptic-circle-of-the-Steiner-inellipse inverse of X₁₈₆

• X₄₂₇ = 1st-Droz-Farney-circle-inverse of X₄₅₁₇₁

• X₄₂₇ = orthosymmedial-circle-inverse of X₁₁₁₂

• X₄₂₇ = polar conjugate of X₈₃

• X₄₂₇ = antitomic image of X₁₆₁₀₂

• X₄₂₇ = complement of the isogonal conjugate of X₆₆

• X₄₂₇ = complement of the isotomic conjugate of X₁₈₀₁₈

• X₄₂₇ = isotomic conjugate of the complement of X₈₈₇₈

• X₄₂₇ = isotomic conjugate of the isogonal conjugate of X₁₈₄₃

• X₄₂₇ = isogonal conjugate of the isotomic conjugate of X₁₂₃₅

• X₄₂₇ = isotomic conjugate of the polar conjugate of X₂₇₃₇₆

• X₄₂₇ = polar conjugate of the isotomic conjugate of X₁₄₁

• X₄₂₇ = polar conjugate of the isogonal conjugate of X₃₉

• X₄₂₇ = medial-isogonal conjugate of X₂₀₆

• X₄₂₇ = orthic-isogonal conjugate of X₁₈₄₃

• X₄₂₇ = X_i-complementary conjugate of X_j for these (i,j): {1, 206}, {2, 21247}, {6, 16582}, {31, 40938}, {57, 17068}, {66, 10}, {1289, 8062}, {2156, 2}, {2353, 37}, {13854, 226}, {14376, 18589}, {15388, 16599}, {16277, 1215}, {18018, 2887}, {40146, 16584}, {40421, 21235}, {43678, 20305}, {44766, 4369}, {46244, 626}

• X₄₂₇ = X_i-Ceva conjugate of X_j for these (i,j): {2, 40938}, {4, 1843}, {112, 523}, {264, 1235}, {1235, 141}, {6331, 2501}, {8791, 468}, {8889, 8893}, {16747, 20883}, {17171, 17442}, {27376, 41584}, {34129, 16318}, {46151, 826}

• X₄₂₇ = X_i-cross conjugate of X_j for these (i,j): {39, 141}, {141, 47730}, {826, 46151}, {1843, 27376}, {2530, 46152}, {3005, 35325}, {21016, 20883}, {23642, 76}, {23666, 75}, {42442, 6}, {46026, 4}, {46654, 31067}

• X₄₂₇ = X_i-isoconjugate of X_j for these (i,j): {1, 1176}, {3, 82}, {6, 34055}, {19, 28724}, {31, 1799}, {48, 83}, {63, 251}, {69, 46289}, {71, 52376}, {75, 10547}, {163, 4580}, {184, 3112}, {228, 52394}, {248, 3405}, {255, 32085}, {293, 51862}, {304, 46288}, {308, 9247}, {525, 34072}, {647, 4599}, {656, 827}, {810, 4577}, {822, 42396}, {905, 4628}, {906, 10566}, {1331, 18108}, {1437, 18082}, {1760, 46765}, {1790, 18098}, {2169, 17500}, {2172, 40404}, {2193, 18097}, {3049, 4593}, {4020, 52395}, {4574, 39179}, {4592, 18105}, {4630, 14208}, {10317, 37221}, {14575, 18833}, {18070, 32661}, {19611, 51508}, {22384, 36081}, {46104, 52430}

• X₄₂₇ = X_i-Dao conjugate of X_j for these (i,j): {2, 1799}, {2, 40938}, {3, 141}, {3, 1176}, {6, 28724}, {9, 34055}, {22, 427}, {25, 8793}, {39, 69}, {63, 40585}, {82, 36103}, {83, 1249}, {115, 4580}, {132, 51862}, {184, 34452}, {206, 10547}, {251, 3162}, {339, 3267}, {525, 15449}, {647, 3124}, {827, 40596}, {1194, 45201}, {1843, 15270}, {3405, 39039}, {3589, 7767}, {3933, 6665}, {3934, 22062}, {4577, 39062}, {4599, 39052}, {5139, 18105}, {5190, 10566}, {5521, 18108}, {6523, 32085}, {7664, 37804}, {8673, 47413}, {14363, 17500}, {18097, 47345}, {20775, 52042}, {21458, 50938}, {22105, 48317}, {23093, 44312}

• X₄₂₇ = cevapoint of X_i and X_j for these (i,j): {2, 8878}, {25, 8792}, {39, 1843}, {115, 2514}, {3005, 39691}, {28666, 46026}

• X₄₂₇ = crosspoint of X_i and X_j for these (i,j): {2, 18018}, {4, 264}, {16747, 17171}

• X₄₂₇ = crosssum of X_i and X_j for these (i,j): {3, 184}, {6, 206}, {25, 51509}, {251, 51508}

• X₄₂₇ = trilinear pole of line {826, 21108}

• X₄₂₇ = crossdifference of every pair of points on line {647, 8673}

• X₄₂₇ = X₂-line conjugate of X₄₄₈₉₄

• X₄₂₇ = X₄-Hirst inverse of X₄₂₀

• X₄₂₇ = X₅₅ of orthic triangle if ABC is acute

• X₄₂₇ = exsimilicenter of nine-point circle and incircle of orthic triangle if ABC is acute; the insimilicenter is X₂₃₅

• X₄₂₇ = intersection of tangents to nine-point circle at PU(4)

• X₄₂₇ = pole of orthic axis wrt the nine-point circle

• X₄₂₇ = pole wrt polar circle of trilinear polar of X₈₃ (line X₂₃X₃₈₅)

• X₄₂₇ = X₄₈-isoconjugate (polar conjugate) of X₈₃

• X₄₂₇ = inverse-in-polar-circle of X₂₃

• X₄₂₇ = inverse-in-orthosymmedial-circle of X₁₁₁₂

• X₄₂₇ = perspector of orthic and 5th Euler triangles

• X₄₂₇ = radical trace of anticomplementary circle and tangential circle

• X₄₂₇ = homothetic center of the medial triangle and the 2nd pedal triangle of X₄

• X₄₂₇ = perspector of 4th Brocard triangle and cross-triangle of ABC and 4th Brocard triangle

• X₄₂₇ = perspector of ABC and cross-triangle of ABC and 5th Euler triangle

• X₄₂₇ = harmonic center of nine-point circle and circle O(PU(4))

• X₄₂₇ = orthic-isogonal conjugate of X₁₈₄₃

• X₄₂₇ = intersection of tangents to Hatzipolakis-Lozada hyperbola at X₄ and X₆

• X₄₂₇ = crosspoint, wrt orthic triangle, of X₄ and X₆

• X₄₂₇ = excentral-to-ABC functional image of X₅₆

• X₄₂₇ = intersection of tangents to Walsmith rectangular hyperbola at X₁₁₃ and X₁₂₅

• X₄₂₇ = barycentric product X_i*X_j for these {i,j}: {1, 20883}, {4, 141}, {6, 1235}, {10, 17171}, {19, 1930}, {25, 8024}, {27, 15523}, {37, 16747}, {38, 92}, {39, 264}, {69, 27376}, {75, 17442}, {76, 1843}, {86, 21016}, {95, 27371}, {107, 2525}, {112, 23285}, {190, 21108}, {193, 47730}, {273, 33299}, {278, 3703}, {281, 3665}, {286, 3954}, {297, 20021}, {324, 16030}, {331, 3688}, {343, 19174}, {393, 3933}, {420, 17949}, {468, 31125}, {523, 41676}, {525, 46151}, {648, 826}, {653, 48278}, {811, 8061}, {850, 35325}, {1289, 23881}, {1401, 7017}, {1502, 27369}, {1634, 14618}, {1783, 48084}, {1824, 16703}, {1826, 16887}, {1897, 16892}, {1964, 1969}, {2052, 3917}, {2501, 4576}, {2528, 42396}, {2530, 6335}, {2592, 46166}, {2593, 46167}, {2996, 41584}, {3005, 6331}, {3051, 18022}, {3186, 42551}, {3313, 43678}, {3404, 40703}, {3613, 37125}, {3867, 18840}, {4074, 37892}, {4391, 46152}, {4553, 17924}, {4568, 7649}, {5094, 23297}, {5117, 14617}, {5485, 41585}, {5523, 46165}, {6531, 51371}, {7794, 32085}, {7813, 17983}, {8041, 46104}, {8362, 8801}, {9019, 46105}, {9229, 12143}, {10159, 46026}, {14376, 41375}, {16080, 51360}, {16102, 40889}, {16696, 41013}, {17980, 35540}, {18018, 40938}, {18020, 39691}, {18027, 20775}, {19568, 47847}, {21035, 44129}, {28666, 40425}, {31613, 42394}, {35140, 51434}, {39129, 41361}, {40416, 46508}, {41331, 44161}, {44132, 51869}, {44146, 46154}, {44427, 46155}, {46106, 46147}, {46107, 46148}, {46108, 46149}, {46109, 46150}, {46110, 46153}

• X₄₂₇ = barycentric quotient X_i/X_j for these {i,j}: {1, 34055}, {2, 1799}, {3, 28724}, {4, 83}, {6, 1176}, {19, 82}, {25, 251}, {27, 52394}, {28, 52376}, {32, 10547}, {38, 63}, {39, 3}, {53, 17500}, {66, 40404}, {92, 3112}, {107, 42396}, {112, 827}, {141, 69}, {162, 4599}, {225, 18097}, {232, 51862}, {240, 3405}, {264, 308}, {275, 39287}, {276, 41488}, {297, 20022}, {393, 32085}, {420, 40850}, {429, 27067}, {523, 4580}, {648, 4577}, {688, 3049}, {732, 12215}, {811, 4593}, {826, 525}, {1235, 76}, {1401, 222}, {1593, 26224}, {1634, 4558}, {1824, 18098}, {1826, 18082}, {1840, 18099}, {1843, 6}, {1883, 27005}, {1923, 9247}, {1930, 304}, {1964, 48}, {1969, 18833}, {1973, 46289}, {1974, 46288}, {2052, 46104}, {2084, 810}, {2353, 46765}, {2489, 18105}, {2525, 3265}, {2528, 2525}, {2530, 905}, {2971, 51906}, {3005, 647}, {3051, 184}, {3118, 4173}, {3162, 8793}, {3172, 51508}, {3313, 20806}, {3404, 293}, {3575, 10548}, {3665, 348}, {3688, 219}, {3703, 345}, {3787, 3167}, {3867, 3618}, {3914, 18084}, {3917, 394}, {3933, 3926}, {3954, 72}, {4020, 255}, {4074, 37894}, {4553, 1332}, {4568, 4561}, {4576, 4563}, {5064, 39668}, {5094, 10130}, {6292, 7767}, {6331, 689}, {6591, 18108}, {6995, 42037}, {7649, 10566}, {7794, 3933}, {7813, 6390}, {8024, 305}, {8041, 3917}, {8061, 656}, {8362, 3785}, {8711, 22159}, {8735, 18101}, {8750, 4628}, {8754, 34294}, {8791, 9076}, {9019, 22151}, {11105, 29534}, {11205, 22352}, {11325, 38834}, {12143, 384}, {13854, 16277}, {14273, 22105}, {14424, 14417}, {15523, 306}, {16030, 97}, {16318, 21458}, {16696, 1444}, {16747, 274}, {16887, 17206}, {16892, 4025}, {16893, 4121}, {17171, 86}, {17187, 1790}, {17407, 40357}, {17442, 1}, {17980, 733}, {18022, 40016}, {19118, 33632}, {19174, 275}, {19577, 51459}, {19595, 5596}, {19606, 15389}, {20021, 287}, {20775, 577}, {20883, 75}, {21016, 10}, {21035, 71}, {21108, 514}, {21123, 1459}, {21248, 45201}, {21336, 45220}, {21814, 228}, {23208, 10316}, {23285, 3267}, {23642, 11574}, {24006, 18070}, {27369, 32}, {27370, 41334}, {27371, 5}, {27373, 8743}, {27374, 217}, {27376, 4}, {28666, 6292}, {28667, 8788}, {29959, 41614}, {30489, 43697}, {31125, 30786}, {31390, 21637}, {32085, 52395}, {32676, 34072}, {33299, 78}, {35319, 23181}, {35325, 110}, {36794, 41296}, {37125, 1078}, {39284, 39289}, {39691, 125}, {40889, 16095}, {40936, 22061}, {40938, 22}, {40972, 212}, {41267, 2200}, {41272, 14908}, {41331, 14575}, {41375, 17907}, {41584, 193}, {41585, 1992}, {41676, 99}, {42551, 43714}, {44090, 46228}, {45211, 45210}, {46026, 3589}, {46147, 14919}, {46148, 1331}, {46149, 1814}, {46150, 1797}, {46151, 648}, {46152, 651}, {46153, 1813}, {46154, 895}, {46166, 8115}, {46167, 8116}, {46387, 22384}, {46508, 626}, {46509, 8023}, {47730, 2996}, {48084, 15413}, {48278, 6332}, {50521, 22383}, {51360, 11064}, {51371, 6393}, {51434, 1503}, {51869, 248}

• X₄₂₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 5090, 12135}, {2, 3, 7499}, {2, 4, 25}, {2, 5, 37439}, {2, 20, 7494}, {2, 22, 6676}, {2, 25, 468}, {2, 468, 52297}, {2, 858, 1368}, {2, 1368, 30739}, {2, 1370, 3}, {2, 1995, 6677}, {2, 3091, 7392}, {2, 3146, 10565}, {2, 3830, 37904}, {2, 3832, 7398}, {2, 4232, 38282}, {2, 5064, 428}, {2, 5133, 5}, {2, 5169, 5133}, {2, 5189, 6636}, {2, 6353, 37453}, {2, 6636, 7495}, {2, 6995, 6353}, {2, 6997, 5020}, {2, 7378, 4}, {2, 7386, 7484}, {2, 7391, 22}, {2, 7392, 11284}, {2, 7394, 1995}, {2, 7396, 7386}, {2, 7398, 40132}, {2, 7408, 4232}, {2, 7409, 6995}, {2, 7485, 140}, {2, 7500, 7493}, {2, 7571, 3628}, {2, 8889, 5094}, {2, 14957, 7467}, {2, 16051, 31255}, {2, 16063, 7485}, {2, 16951, 7807}, {2, 24584, 1375}, {2, 26032, 37099}, {2, 26096, 25494}, {2, 26760, 27028}, {2, 26808, 26967}, {2, 28684, 28664}, {2, 30744, 5159}, {2, 30888, 46531}, {2, 30951, 37370}, {2, 31021, 857}, {2, 31074, 858}, {2, 31099, 1370}, {2, 31105, 31133}, {2, 31152, 43957}, {2, 31857, 31074}, {2, 34603, 10154}, {2, 34609, 7667}, {2, 37174, 37187}, {2, 37349, 13595}, {2, 37353, 37990}, {2, 37362, 37432}, {2, 37456, 4224}, {2, 37913, 52300}, {2, 38282, 52292}, {2, 46336, 16419}, {2, 47314, 8703}, {2, 52284, 8889}, {2, 52285, 10301}, {2, 52299, 52298}, {3, 4, 3575}, {3, 5, 7399}, {3, 25, 21213}, {3, 381, 18420}, {3, 1370, 7667}, {3, 5576, 5}, {3, 18494, 18533}, {3, 18533, 37931}, {3, 31099, 46517}, {3, 34609, 1370}, {3, 44441, 47090}, {4, 5, 235}, {4, 20, 12173}, {4, 24, 6756}, {4, 25, 428}, {4, 28, 37398}, {4, 186, 7576}, {4, 235, 1906}, {4, 381, 10151}, {4, 403, 1596}, {4, 406, 4186}, {4, 429, 1904}, {4, 442, 37376}, {4, 451, 4222}, {4, 458, 460}, {4, 468, 10301}, {4, 469, 430}, {4, 470, 462}, {4, 471, 463}, {4, 475, 4185}, {4, 631, 7487}, {4, 860, 1894}, {4, 1346, 1312}, {4, 1347, 1313}, {4, 1585, 52286}, {4, 1586, 52287}, {4, 1593, 1885}, {4, 1594, 5}, {4, 1595, 1907}, {4, 2476, 431}, {4, 3088, 1593}, {4, 3089, 5198}, {4, 3090, 3089}, {4, 3091, 37197}, {4, 3127, 32588}, {4, 3128, 32587}, {4, 3144, 37390}, {4, 3147, 37122}, {4, 3520, 6240}, {4, 3535, 52291}, {4, 3536, 5200}, {4, 3541, 3}, {4, 3542, 1598}, {4, 3545, 6623}, {4, 4194, 17516}, {4, 4196, 1889}, {4, 4200, 4214}, {4, 4212, 27}, {4, 4213, 14004}, {4, 5064, 52285}, {4, 5094, 468}, {4, 5117, 297}, {4, 5125, 407}, {4, 5136, 1884}, {4, 5142, 429}, {4, 5169, 37981}, {4, 6143, 3518}, {4, 6353, 6995}, {4, 7378, 5064}, {4, 7498, 28076}, {4, 7505, 10594}, {4, 7507, 23047}, {4, 7547, 546}, {4, 7577, 403}, {4, 7714, 7408}, {4, 8889, 2}, {4, 10018, 7715}, {4, 11109, 37226}, {4, 14865, 18560}, {4, 14940, 34484}, {4, 15559, 1595}, {4, 18533, 18494}, {4, 18560, 3627}, {4, 26020, 37432}, {4, 30100, 12225}, {4, 35473, 18559}, {4, 35475, 34797}, {4, 35481, 35480}, {4, 35482, 3520}, {4, 35488, 44226}, {4, 35490, 3853}, {4, 35502, 13488}, {4, 37074, 1529}, {4, 37117, 7511}, {4, 37118, 37458}, {4, 37119, 24}, {4, 37125, 27369}, {4, 37337, 11325}, {4, 37388, 37385}, {4, 37943, 52294}, {4, 38282, 7714}, {4, 44287, 44281}, {4, 44438, 13473}, {4, 44958, 44803}, {4, 52249, 52280}, {4, 52252, 28}, {4, 52280, 6755}, {4, 52284, 5094}, {4, 52288, 6620}, {4, 52290, 52301}, {4, 52295, 1594}, {4, 52296, 21841}, {4, 52299, 6353}, {5, 140, 7405}, {5, 858, 30739}, {5, 1368, 2}, {5, 1595, 4}, {5, 1596, 403}, {5, 1906, 45004}, {5, 1907, 1906}, {5, 3627, 15761}, {5, 3845, 46030}, {5, 6823, 13160}, {5, 13371, 11585}, {5, 15559, 1907}, {5, 15687, 11563}, {5, 15809, 25}, {5, 23335, 3}, {5, 44960, 16868}, {6, 66, 26926}, {6, 1853, 1899}, {6, 1899, 11245}, {20, 13160, 6823}, {22, 6676, 44210}, {22, 31133, 7391}, {22, 31236, 2}, {22, 37353, 46029}, {24, 37119, 140}, {25, 428, 10301}, {25, 5064, 4}, {25, 5094, 2}, {25, 7378, 52285}, {25, 37453, 6353}, {25, 52298, 37453}, {33, 5101, 1862}, {39, 27371, 27376}, {51, 125, 13567}, {53, 3815, 232}, {54, 34224, 31804}, {115, 1560, 44467}, {125, 132, 47202}, {140, 6756, 24}, {140, 10691, 7485}, {140, 11548, 2}, {140, 16198, 6756}, {140, 47315, 16063}, {140, 50138, 5}, {141, 1843, 41584}, {141, 3867, 1843}, {154, 36990, 31383}, {185, 3574, 12233}, {186, 7576, 37458}, {186, 37118, 549}, {230, 6748, 10311}, {235, 1907, 4}, {235, 30739, 468}, {235, 44960, 44995}, {235, 45003, 44996}, {264, 18022, 42394}, {264, 51843, 18022}, {275, 23295, 8901}, {343, 45303, 21243}, {378, 35480, 35481}, {378, 37970, 3520}, {378, 44269, 18570}, {378, 44274, 44281}, {378, 45179, 15760}, {381, 1597, 4}, {381, 2072, 5}, {381, 5020, 6997}, {381, 30771, 5020}, {382, 7493, 37899}, {382, 9909, 7500}, {393, 7736, 45141}, {403, 1594, 7577}, {403, 1596, 235}, {403, 7577, 5}, {428, 468, 25}, {428, 5094, 52297}, {428, 8889, 52293}, {428, 52285, 4}, {429, 1883, 4}, {442, 8226, 7522}, {465, 41034, 3132}, {466, 41035, 3131}, {468, 5094, 52293}, {468, 52285, 428}, {470, 471, 52289}, {470, 472, 16250}, {471, 473, 16249}, {472, 473, 52281}, {546, 5159, 1995}, {546, 10224, 5}, {546, 13488, 4}, {549, 37458, 186}, {578, 18381, 6146}, {590, 5412, 13884}, {615, 5413, 13937}, {631, 7487, 3515}, {631, 44831, 44837}, {858, 1594, 5094}, {858, 2072, 47097}, {858, 5133, 2}, {858, 5169, 5}, {858, 5576, 37454}, {858, 7378, 15809}, {858, 31099, 23335}, {858, 37981, 468}, {858, 46698, 1312}, {858, 46699, 1313}, {867, 37354, 8727}, {1113, 1114, 21284}, {1125, 49542, 11363}, {1204, 6696, 43903}, {1312, 1313, 468}, {1346, 1347, 5094}, {1346, 46699, 5}, {1347, 46698, 5}, {1368, 1595, 15809}, {1368, 5133, 37439}, {1370, 31099, 34609}, {1370, 34609, 46517}, {1513, 52280, 25}, {1585, 1586, 458}, {1593, 7507, 4}, {1593, 18386, 44438}, {1594, 1595, 235}, {1594, 15559, 4}, {1594, 15809, 37439}, {1595, 52284, 30739}, {1596, 16868, 45000}, {1597, 52284, 47097}, {1598, 1656, 3542}, {1656, 16419, 2}, {1656, 50137, 5}, {1799, 16275, 7750}, {1843, 46026, 3867}, {1848, 1861, 1824}, {1883, 5142, 1904}, {1885, 13473, 44438}, {1885, 23047, 4}, {1907, 37439, 428}, {1994, 3448, 45968}, {1994, 45968, 1353}, {1995, 30744, 2}, {2041, 2042, 9715}, {2052, 6530, 14569}, {2071, 38323, 44241}, {2450, 37988, 5}, {2476, 37983, 5}, {2479, 2480, 40889}, {2969, 7140, 92}, {2979, 37636, 48876}, {3060, 3580, 41588}, {3060, 7703, 23293}, {3060, 23293, 3580}, {3088, 7507, 1885}, {3091, 16051, 11284}, {3091, 37197, 10019}, {3127, 3128, 4}, {3146, 10565, 34608}, {3147, 37122, 3517}, {3148, 14003, 441}, {3153, 7527, 52069}, {3162, 13854, 16318}, {3258, 42426, 47223}, {3516, 12173, 20}, {3517, 3526, 3147}, {3518, 6143, 10018}, {3518, 7496, 37920}, {3520, 6240, 550}, {3547, 34938, 11414}, {3548, 7528, 6642}, {3567, 23294, 26879}, {3575, 7499, 21213}, {3575, 37454, 468}, {3575, 37931, 18533}, {3627, 45177, 235}, {3628, 10300, 40916}, {3628, 21841, 7505}, {3628, 50136, 5}, {3830, 7579, 10254}, {3830, 10254, 11799}, {3832, 30769, 40132}, {3839, 30775, 47597}, {3850, 44226, 35488}, {3850, 49673, 5}, {3851, 50143, 5}, {3853, 37897, 7519}, {4074, 40379, 141}, {4232, 7408, 7714}, {4232, 7714, 25}, {4232, 52292, 468}, {5000, 5001, 5}, {5002, 5003, 12225}, {5004, 5005, 7488}, {5020, 30771, 2}, {5055, 50135, 5}, {5064, 5094, 25}, {5064, 8889, 468}, {5064, 37439, 1906}, {5064, 52292, 7408}, {5064, 52298, 6995}, {5066, 50140, 5}, {5072, 50139, 5}, {5094, 7378, 428}, {5094, 37197, 11284}, {5094, 37453, 52298}, {5094, 37920, 6143}, {5094, 37981, 5}, {5094, 52298, 52299}, {5133, 15559, 5064}, {5133, 15809, 235}, {5133, 31074, 1368}, {5133, 37990, 37353}, {5159, 6677, 2}, {5169, 31074, 2}, {5169, 31101, 37353}, {5169, 31857, 858}, {5169, 40916, 50136}, {5169, 46336, 50137}, {5169, 52284, 1594}, {5189, 6636, 52397}, {5189, 7495, 550}, {5422, 18911, 45298}, {5446, 5449, 41587}, {5480, 13567, 51}, {5480, 23332, 13567}, {5576, 23335, 7399}, {6143, 10018, 632}, {6240, 7495, 21284}, {6247, 12233, 185}, {6353, 6995, 25}, {6353, 7577, 37990}, {6353, 8889, 52299}, {6353, 35481, 22}, {6353, 37453, 468}, {6353, 52299, 2}, {6623, 32216, 468}, {6636, 52397, 550}, {6639, 7517, 13383}, {6640, 7506, 16238}, {6640, 44259, 44907}, {6643, 7404, 7395}, {6644, 18281, 10257}, {6676, 13413, 37990}, {6696, 13568, 1204}, {6747, 42400, 53}, {6756, 16198, 4}, {6776, 11427, 11402}, {6995, 7378, 7409}, {6995, 7391, 35480}, {6995, 7409, 4}, {6995, 8889, 52298}, {6995, 37353, 403}, {6995, 52298, 468}, {6995, 52299, 37453}, {7378, 8889, 25}, {7378, 15809, 1907}, {7378, 52284, 2}, {7386, 7396, 31152}, {7386, 7484, 43957}, {7386, 28412, 426}, {7391, 31236, 6676}, {7392, 11403, 428}, {7392, 16051, 2}, {7394, 30744, 6677}, {7396, 7539, 43957}, {7398, 30769, 2}, {7400, 52398, 37198}, {7403, 11585, 5}, {7408, 38282, 25}, {7409, 8889, 37453}, {7409, 37453, 428}, {7409, 37990, 1596}, {7409, 52284, 52299}, {7409, 52299, 25}, {7418, 46591, 21284}, {7484, 7539, 2}, {7484, 31152, 7386}, {7485, 16063, 10691}, {7493, 7500, 9909}, {7494, 44442, 20}, {7495, 52397, 6636}, {7499, 7667, 3}, {7499, 37454, 2}, {7499, 46517, 7667}, {7500, 9909, 37899}, {7503, 37444, 12362}, {7505, 10594, 21841}, {7505, 52296, 3628}, {7507, 44438, 18386}, {7514, 31181, 14791}, {7526, 18569, 12605}, {7530, 10201, 37971}, {7536, 49132, 199}, {7539, 31152, 7484}, {7542, 7553, 26}, {7544, 17928, 9825}, {7547, 35502, 4}, {7558, 10323, 16197}, {7566, 17928, 7544}, {7569, 10323, 7558}, {7570, 37977, 14940}, {7571, 40916, 2}, {7576, 37118, 186}, {7577, 18533, 37347}, {7577, 31101, 52298}, {7577, 35480, 46029}, {7579, 11799, 5}, {7592, 11457, 18914}, {7667, 37454, 7499}, {7667, 46517, 1370}, {7714, 38282, 4232}, {8370, 35920, 15014}, {8889, 15559, 15809}, {8889, 15809, 30739}, {8889, 52285, 52297}, {8891, 21248, 141}, {9777, 26869, 11433}, {9786, 40686, 26937}, {9818, 18531, 34664}, {9825, 16196, 17928}, {9909, 18364, 6636}, {10109, 50134, 5}, {10151, 47097, 468}, {10295, 35473, 8703}, {10301, 52293, 468}, {10594, 52296, 7505}, {10691, 11548, 140}, {11056, 16275, 1799}, {11284, 31255, 2}, {11381, 43831, 2883}, {11403, 37197, 4}, {11410, 37196, 376}, {11427, 32064, 6776}, {11432, 26944, 18916}, {11433, 14853, 9777}, {11433, 23291, 26869}, {11548, 47315, 10691}, {11572, 21659, 41362}, {11737, 50142, 5}, {11818, 18281, 6644}, {12086, 34007, 52071}, {12100, 37934, 35472}, {12106, 44452, 44211}, {13371, 33332, 7403}, {13413, 46029, 5}, {13490, 44452, 12106}, {13567, 23332, 125}, {14130, 31724, 18563}, {14782, 14783, 7401}, {14784, 14785, 3547}, {14813, 14814, 34002}, {14826, 37669, 6090}, {14853, 23291, 11433}, {15033, 25739, 12022}, {15559, 52295, 5}, {16051, 37197, 468}, {16238, 32144, 6640}, {16252, 16621, 26883}, {16387, 44260, 44210}, {17111, 23304, 11}, {18386, 44438, 4}, {18420, 44441, 3}, {18494, 18533, 3575}, {18559, 35473, 10295}, {18560, 45177, 15761}, {18570, 44236, 44218}, {18570, 44249, 44285}, {18570, 44269, 44281}, {18570, 44274, 44269}, {18570, 44287, 44236}, {18583, 45298, 5422}, {19041, 19042, 40065}, {21213, 37454, 52297}, {21850, 41588, 3060}, {23336, 31830, 37814}, {24886, 24939, 2}, {25738, 36749, 13292}, {25985, 26020, 2}, {25985, 37362, 25}, {27098, 27150, 2}, {27386, 27510, 33305}, {28020, 28084, 33302}, {28048, 28126, 33306}, {28240, 28354, 859}, {28412, 28701, 441}, {29385, 29442, 46497}, {29498, 29701, 46575}, {29553, 29752, 46574}, {29970, 30040, 46514}, {30739, 37439, 2}, {31074, 37353, 31101}, {31099, 52284, 3541}, {31101, 37353, 2}, {31105, 52284, 378}, {31133, 31236, 22}, {31236, 35480, 37453}, {31670, 37638, 47582}, {31723, 52262, 44239}, {32587, 32588, 4}, {34221, 34222, 19161}, {34797, 35475, 35491}, {34797, 35491, 15704}, {35471, 35477, 548}, {35480, 52299, 6676}, {35487, 44803, 44958}, {35502, 52295, 10224}, {37353, 37990, 5}, {37452, 50137, 1656}, {37453, 52298, 2}, {37454, 46517, 3}, {37472, 44076, 43595}, {39504, 46029, 13413}, {42789, 42790, 35921}, {43957, 47311, 31152}, {44218, 44249, 18570}, {44236, 44288, 44249}, {44287, 44288, 18570}, {44637, 44638, 6748}, {44996, 45003, 45006}, {46517, 47097, 47090}, {46698, 46699, 37981}, {47090, 47310, 47340}, {47612, 47613, 11585}, {52286, 52287, 460}, {52293, 52297, 2}

Notes

As a point on the Euler line, X₄₂₇ has Shinagawa coefficients \((F,E + F)\).

Let L_A be the line tangent to the nine-point circle at the midpoint of segment BC, and define L_B and L_C cyclically. The triangle formed by the lines L_A, L_B, L_C is homothetic to the orthic triangle, and the center of homothety is X₄₂₇. (Randy Hutson, 9/23/2011)

Let (O) be the circumcircle, (N) the nine-point circle, and (I_A, (I_B, (I_C the excircles of ABC. Let A' and A″ be the points of intersection of (O) and (I_A. Let F_A be the touchpoint of (N) and (I_A, and let (K_A) be the circle through A' and A″ that is internally tangent to (N); let L_A be the touchpoint. Define F_B, F_C and L_B, L_C cyclically. The lines F_AL_A, F_BL_B, F_CL_C concur ion X₄₂₇. (Tran Quang Hung ADGEOM #1458, August 5, 2014; see also #1459)

Let Cm_aCm_bCm_c be the circummedial triangle. Let A″ be the pole, wrt the polar circle, of line Cm_bCm_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₂₇. Moreover, X₄₂₇ is the Euler line intercept of radical axis of nine-point circle and every circle with center on orthic axis that is orthogonal to nine-point circle, and X₄₂₇ is the point in which the extended trapezoid legs (P(4),P(4)-Ceva conjugate of U(4)) and (U(4),U(4)-Ceva conjugate of P(4)) meet. Also, X₄₂₇ is the QA-P38 center (Montesdeoca-Hutson Point) of quadrangle ABCX₂. (Randy Hutson, October 13, 2015)

X₄₂₇ is the homothetic center of any pair of the following: polar triangle of nine-point circle, orthic axes triangle (see X₂₅₀₁), Yiu tangents triangle (see X₇₄₉₅). (Randy Hutson, August 19, 2019)