X₆₃₁

3/5*OG

X₆₃₁ = reflection of X₆₃₂ in X₁₄₀

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

X₆₃₁ (3/5*OG) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {632, 140}, {2, 3} (Euler line)

• Related Centers: X₆₃₂ (9/10*OG), X₁₄₀ (MIDPOINT OF X₃ AND X₅), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(2 \cos A + \cos B \cos C : 2 \cos B + \cos C \cos A : 2 \cos C + \cos A \cos B\)

Barycentrics: \((\sin A)(2 \cos A + \cos B \cos C) : (\sin B)(2 \cos B + \cos C \cos A) : (\sin C)(2 \cos C + \cos A \cos B)\)

Search Value (6, 9, 13): \(4.290566433172\)

▸ Alternate Coordinates7 forms

• Trilinears: \(\cos A + \sin B \sin C : \cos B + \sin C \sin A : \cos C + \sin A \sin B : :\)

• Trilinears: \(\sec A + 2 \sec B \sec C : \sec B + 2 \sec C \sec A : \sec C + 2 \sec A \sec B\)

• Trilinears: \(a/(b \sec B + c \sec C) + 2 \cos A : :\)

• Barycentrics: \(3a^{4} - 4a^{2}(b^{2} + c^{2}) + (b^{2} - c^{2})^{2} : :\)

• Barycentrics: \(2 S_{A} a^{2} + S_{B} S_{C} : :\)

• Barycentrics: \(2 S^{2} - S_{B} S_{C} : :\)

• Barycentrics: \(\cot B \cot C - 2 : :\)

▸ Incident Central Lines435 lines

{1, 1000}, {2, 3}, {6, 9540}, {7, 11374}, {8, 1385}, {9, 6700}, {10, 944}, {11, 4294}, {12, 4293}, {13, 5351}, {14, 5352}, {15, 11489}, {16, 11488}, {17, 5237}, {18, 5238}, {32, 7736}, {35, 497}, {36, 388}, {39, 5319}, {40, 1125}, {46, 3485}, {49, 11003}, {51, 11695}, {52, 2979}, {54, 69}, {55, 1058}, {56, 1056}, {57, 3487}, {61, 3411}, {62, 3412}, {64, 5646}, {68, 3431}, {72, 5744}, {74, 5972}, {76, 6337}, {83, 5171}, {84, 5658}, {98, 620}, {99, 6036}, {100, 5082}, {101, 6712}, {102, 6718}, {103, 6710}, {104, 958}, {109, 6711}, {110, 6699}, {113, 12244}, {114, 3096}, {115, 13172}, {119, 12248}, {122, 5667}, {125, 12383}, {127, 13200}, {132, 12253}, {141, 5085}, {142, 5735}, {143, 11592}, {145, 5690}, {146, 12041}, {147, 7945}, {154, 6247}, {165, 946}, {171, 602}, {174, 8127}, {183, 3926}, {184, 9705}, {185, 5650}, {187, 2548}, {193, 5050}, {194, 7616}, {196, 8762}, {210, 12675}, {212, 3075}, {214, 12247}, {216, 1075}, {230, 5013}, {238, 601}, {262, 5188}, {315, 1007}, {323, 12161}, {325, 3785}, {329, 3916}, {355, 5731}, {371, 3069}, {372, 3068}, {373, 10110}, {386, 9568}, {387, 4255}, {389, 3917}, {390, 496}, {394, 7592}, {485, 3316}, {486, 3317}, {487, 492}, {488, 491}, {495, 3600}, {511, 3567}, {515, 1698}, {516, 8227}, {517, 3616}, {551, 7982}, {568, 6101}, {569, 6515}, {572, 966}, {573, 9569}, {574, 3767}, {575, 1992}, {576, 10168}, {577, 10312}, {578, 11433}, {580, 5712}, {581, 3216}, {590, 1152}, {597, 11477}, {599, 8550}, {603, 3074}, {615, 1151}, {616, 6771}, {617, 6774}, {618, 6770}, {619, 6773}, {621, 13350}, {622, 13349}, {629, 3642}, {630, 3643}, {640, 8982}, {641, 5590}, {642, 5591}, {671, 10992}, {748, 3073}, {750, 3072}, {908, 4652}, {912, 3876}, {936, 5745}, {942, 5435}, {952, 3617}, {954, 8732}, {956, 7080}, {962, 3579}, {978, 1064}, {993, 2551}, {999, 5265}, {1001, 6691}, {1038, 1870}, {1040, 6198}, {1071, 5044}, {1131, 6452}, {1132, 6451}, {1141, 13372}, {1147, 5012}, {1153, 5485}, {1155, 4295}, {1192, 11821}, {1199, 1993}, {1209, 12254}, {1210, 3488}, {1216, 5889}, {1285, 3053}, {1292, 6714}, {1293, 6715}, {1294, 6716}, {1295, 6717}, {1296, 6719}, {1297, 6720}, {1350, 3589}, {1352, 3619}, {1376, 4999}, {1478, 4325}, {1479, 4330}, {1482, 3622}, {1483, 3621}, {1490, 6705}, {1498, 6696}, {1503, 3763}, {1506, 5206}, {1511, 3448}, {1512, 8582}, {1519, 4512}, {1614, 9306}, {1621, 10596}, {1703, 8983}, {1737, 3486}, {1837, 4305}, {1899, 13367}, {1986, 13416}, {2077, 5248}, {2080, 7787}, {2096, 3452}, {2482, 11623}, {2549, 7746}, {2550, 10902}, {2794, 7867}, {2883, 10606}, {2888, 10610}, {2975, 3421}, {3055, 5210}, {3060, 5462}, {3070, 6410}, {3071, 6409}, {3095, 6194}, {3098, 7846}, {3100, 9644}, {3183, 3462}, {3189, 10916}, {3241, 3653}, {3279, 3608}, {3295, 5281}, {3296, 3338}, {3303, 4995}, {3304, 5298}, {3305, 7330}, {3306, 5709}, {3311, 7586}, {3312, 7585}, {3333, 13405}, {3335, 3609}, {3357, 6225}, {3359, 5250}, {3398, 7774}, {3428, 6690}, {3430, 6693}, {3474, 4338}, {3476, 10039}, {3564, 3620}, {3582, 10385}, {3614, 12943}, {3623, 5844}, {3632, 13607}, {3634, 4297}, {3646, 12705}, {3654, 10222}, {3679, 5882}, {3746, 10072}, {3788, 7800}, {3796, 9707}, {3813, 4421}, {3817, 12512}, {3819, 5562}, {3868, 10202}, {3871, 10529}, {3873, 13373}, {3897, 5554}, {3933, 9755}, {3934, 11257}, {4045, 9754}, {4256, 5292}, {4292, 5219}, {4299, 5229}, {4302, 5225}, {4304, 9581}, {4308, 5126}, {4311, 9578}, {4313, 5704}, {4340, 5718}, {4413, 8273}, {4423, 11496}, {4648, 5733}, {4678, 12645}, {4846, 11270}, {4855, 6734}, {4996, 10522}, {5023, 7745}, {5024, 5305}, {5045, 10578}, {5096, 5800}, {5102, 6329}, {5122, 5226}, {5158, 5702}, {5180, 10225}, {5181, 5622}, {5251, 5450}, {5253, 10597}, {5259, 8166}, {5303, 11681}, {5304, 9605}, {5316, 6260}, {5326, 7354}, {5334, 11480}, {5335, 11481}, {5395, 10155}, {5436, 9843}, {5442, 5902}, {5444, 5903}, {5445, 10573}, {5446, 5640}, {5449, 12118}, {5461, 12117}, {5473, 6669}, {5474, 6670}, {5482, 5752}, {5563, 10056}, {5569, 7759}, {5599, 11843}, {5600, 11844}, {5651, 6759}, {5654, 5888}, {5691, 10175}, {5692, 5884}, {5693, 10176}, {5708, 5719}, {5720, 10884}, {5732, 5817}, {5758, 9776}, {5768, 5791}, {5777, 10167}, {5891, 12111}, {5893, 5925}, {5901, 12702}, {5904, 12005}, {5907, 6241}, {5943, 9781}, {5946, 6243}, {6054, 9167}, {6118, 12124}, {6119, 12123}, {6193, 12359}, {6197, 10319}, {6221, 7584}, {6224, 12619}, {6239, 12360}, {6242, 12363}, {6282, 6692}, {6284, 7294}, {6292, 7710}, {6398, 7583}, {6400, 12361}, {6403, 11574}, {6445, 9543}, {6450, 8976}, {6454, 8960}, {6560, 10576}, {6561, 10577}, {6680, 9753}, {6685, 9548}, {6688, 13598}, {6689, 7691}, {6702, 12119}, {6704, 12122}, {6722, 7872}, {6761, 12096}, {6769, 10582}, {7173, 12953}, {7317, 12735}, {7610, 9741}, {7619, 7775}, {7622, 7751}, {7694, 7853}, {7722, 12358}, {7739, 7755}, {7747, 8588}, {7748, 8589}, {7758, 7780}, {7782, 11185}, {7790, 9734}, {7803, 7857}, {7808, 8722}, {7810, 7888}, {7822, 8721}, {7830, 7862}, {7831, 7940}, {7835, 12203}, {7836, 10104}, {7870, 11179}, {7904, 7925}, {7905, 11008}, {7910, 13449}, {7914, 9873}, {7931, 9863}, {7957, 13374}, {7989, 10172}, {7991, 13464}, {8071, 10321}, {8125, 8130}, {8126, 8129}, {8148, 10283}, {8151, 9168}, {8222, 11846}, {8223, 11847}, {8254, 12307}, {8416, 8975}, {8537, 11511}, {8553, 9608}, {9655, 10592}, {9668, 10593}, {9740, 12040}, {9778, 11230}, {9785, 11373}, {9812, 9955}, {9820, 12163}, {9821, 10583}, {9829, 12506}, {9833, 11202}, {9932, 12318}, {10163, 12505}, {10170, 12162}, {10198, 10532}, {10263, 11002}, {10272, 10620}, {10282, 11206}, {10320, 10629}, {10470, 10479}, {10525, 10584}, {10526, 10585}, {10574, 11444}, {10586, 10679}, {10587, 10680}, {10598, 11826}, {10599, 11827}, {10632, 11515}, {10633, 11516}, {10706, 10990}, {10707, 10993}, {10880, 11513}, {10881, 11514}, {11194, 12607}, {11425, 13567}, {11442, 11449}, {11456, 13394}, {11557, 13201}, {11806, 12273}, {12026, 13512}, {12249, 12864}, {12255, 13089}, {12319, 12893}, {12320, 12972}, {12321, 12973}, {12322, 12974}, {12323, 12975}, {12528, 13369}, {13025, 13049}, {13026, 13050}, {13027, 13035}, {13028, 13036}, {13203, 13289}, {13352, 13434}

▸ Functional Relations18 relations

• X₆₃₁ = reflection of X₆₃₂ in X₁₄₀

• X₆₃₁ = isogonal conjugate of X₃₅₂₇

• X₆₃₁ = isotomic conjugate of X₈₇₉₇

• X₆₃₁ = complement of X₃₀₉₁

• X₆₃₁ = anticomplement of X₁₆₅₆

• X₆₃₁ = circumcircle-inverse of X₃₇₉₂₅

• X₆₃₁ = orthocentroidal-circle-inverse of X₃₀₉₀

• X₆₃₁ = insimilicenter of circumcircle and 1st Steiner circle; the exsimilicenter is X₂₀

• X₆₃₁ = pole wrt polar circle of trilinear polar of X₈₇₉₆

• X₆₃₁ = X₄₈-isoconjugate (polar conjugate)-of-X₈₇₉₆

• X₆₃₁ = {X₃,X₄}-harmonic conjugate of X₃₇₆

• X₆₃₁ = insimilicenter of circumcircles of Euler and anti-Euler triangles; the exsimilicenter is X₄

• X₆₃₁ = homothetic center of circumorthic triangle and mid-triangle of orthic and dual of orthic triangles

• X₆₃₁ = perspector of ABC and cross-triangle of ABC and anti-Euler triangle

• X₆₃₁ = orthocenter of cross-triangle of Euler and anti-Euler triangles

• X₆₃₁ = homothetic center of X₄-altimedial and X₂-anti-altimedial triangles

• X₆₃₁ = Euler line intercept, other than X₂₀, of conic {X₁₃,X₁₄,X₁₇,X₁₈,X₂₀}}

• X₆₃₁ = pole of Brocard axis wrt conic {X₂,X₁₅,X₁₆,X₁₇,X₁₈}}

Notes

As a point on the Euler line, X₆₃₁ has Shinagawa coefficients \((2,-1)\).

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). X₆₃₁ is the radical center of the circumcircles of triangles AAeAi, BBeBi, CCeCi. (Randy Hutson, June 7, 2019)

X₆₃₁ is the {X₂,X₃}-harmonic conjugate of X₄.