X₃₈₆

INVERSE-IN-BROCARD-CIRCLE OF X₅₈

X₃₈₆ = Brocard-circle-inverse of X₅₈

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

X₃₈₆ (INVERSE-IN-BROCARD-CIRCLE OF X₅₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {3, 6}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT)

Trilinears: \(a(b^{2} + c^{2} + bc + ca + ab) : :\)

Barycentrics: \(a^{2}(b^{2} + c^{2} + bc + ca + ab) : :\)

Search Value (6, 9, 13): \(1.215365916742\)

▸ Alternate Coordinates2 forms

• Trilinears: \(r \cos A + s \sin A : :\), where s = semiperimeter, r = inradius

• Trilinears: \(\sin(A + U) : :\), where cot \(U = \cot(A/2) \cot(B/2) \cot(C/2)\)

▸ Incident Central Lines277 lines

{1, 2}, {3, 6}, {4, 2051}, {5, 1834}, {9, 10467}, {11, 9555}, {12, 9552}, {20, 9535}, {21, 1724}, {22, 9571}, {24, 9570}, {31, 35}, {34, 14018}, {36, 1468}, {37, 5044}, {38, 5904}, {40, 1064}, {41, 2172}, {55, 595}, {56, 181}, {57, 73}, {60, 3453}, {65, 994}, {69, 16887}, {72, 3666}, {81, 404}, {99, 3029}, {100, 3032}, {101, 3033}, {105, 3034}, {106, 2334}, {109, 11509}, {110, 3031}, {165, 1695}, {184, 3437}, {191, 4414}, {192, 3159}, {194, 17499}, {213, 2276}, {222, 1466}, {223, 1448}, {226, 23537}, {238, 5248}, {244, 18398}, {312, 2901}, {333, 19270}, {388, 4551}, {405, 4383}, {411, 1754}, {442, 5718}, {443, 5712}, {474, 940}, {514, 22090}, {517, 4646}, {518, 4719}, {549, 5453}, {590, 8959}, {601, 2077}, {602, 10902}, {603, 2003}, {631, 9568}, {727, 11490}, {741, 5213}, {748, 5259}, {758, 986}, {759, 6044}, {810, 14838}, {872, 984}, {939, 13404}, {942, 3752}, {946, 3755}, {956, 16302}, {958, 9564}, {960, 3931}, {982, 3874}, {987, 5150}, {988, 3751}, {990, 1490}, {993, 5247}, {999, 5399}, {1001, 16288}, {1010, 5331}, {1036, 8193}, {1040, 10393}, {1042, 3339}, {1043, 13740}, {1045, 3923}, {1046, 17596}, {1060, 8555}, {1066, 3333}, {1086, 6147}, {1100, 5956}, {1104, 16290}, {1107, 21904}, {1147, 9562}, {1191, 3295}, {1211, 13728}, {1245, 2258}, {1330, 4201}, {1376, 5711}, {1386, 5266}, {1420, 1450}, {1437, 20842}, {1449, 1818}, {1453, 3601}, {1457, 3340}, {1458, 3361}, {1464, 5221}, {1500, 2176}, {1575, 17750}, {1616, 6767}, {1738, 12609}, {1742, 12512}, {1745, 4292}, {1780, 20846}, {1871, 14571}, {2050, 5786}, {2140, 3487}, {2177, 3746}, {2194, 11334}, {2221, 7085}, {2238, 5283}, {2242, 21008}, {2251, 7296}, {2273, 7193}, {2274, 4649}, {2275, 20963}, {2280, 5299}, {2292, 5692}, {2293, 16469}, {2299, 14017}, {2300, 3781}, {2308, 5010}, {2309, 16468}, {2328, 11344}, {2332, 8743}, {2345, 10469}, {2536, 2538}, {2537, 2539}, {2635, 9579}, {2646, 16455}, {2650, 5902}, {2653, 3981}, {2654, 9581}, {2915, 5347}, {2933, 20986}, {3072, 6796}, {3091, 5400}, {3100, 9550}, {3120, 6127}, {3145, 5320}, {3149, 5706}, {3191, 22020}, {3194, 4219}, {3303, 16296}, {3454, 4417}, {3501, 3997}, {3553, 5720}, {3576, 9548}, {3589, 17698}, {3596, 4360}, {3597, 13478}, {3647, 7262}, {3664, 12436}, {3670, 3868}, {3702, 3896}, {3725, 3743}, {3731, 8951}, {3772, 11374}, {3780, 16975}, {3795, 4116}, {3869, 4424}, {3873, 3953}, {3878, 4868}, {3881, 3976}, {3889, 4694}, {3914, 12047}, {3916, 4641}, {3936, 4202}, {3945, 17580}, {3993, 14823}, {4065, 19582}, {4188, 17187}, {4205, 5743}, {4216, 10457}, {4224, 5358}, {4225, 20966}, {4303, 15803}, {4322, 13462}, {4340, 6904}, {4642, 5903}, {4648, 17582}, {4749, 16300}, {4878, 7174}, {4991, 18194}, {5051, 5741}, {5091, 16382}, {5134, 9598}, {5251, 10448}, {5255, 8715}, {5275, 16852}, {5278, 10458}, {5439, 16610}, {5563, 16474}, {5584, 10823}, {5640, 7419}, {5687, 5710}, {5707, 6911}, {5713, 6826}, {5719, 17366}, {5721, 6831}, {5725, 5794}, {5732, 10443}, {5736, 17189}, {5737, 19273}, {6198, 9551}, {6284, 9554}, {6748, 7546}, {7031, 21764}, {7354, 9553}, {7513, 8747}, {7951, 21935}, {8069, 16294}, {8071, 16295}, {8300, 20862}, {8728, 17056}, {9310, 16785}, {9371, 12711}, {9441, 12511}, {9575, 20606}, {10407, 10895}, {10441, 19513}, {10470, 21363}, {11110, 17277}, {11263, 17889}, {11343, 19758}, {11993, 17467}, {13161, 21077}, {13407, 23536}, {13725, 14555}, {14793, 16473}, {14996, 17572}, {14997, 16865}, {15488, 19540}, {15654, 23638}, {15934, 17054}, {16343, 19757}, {16347, 19742}, {16351, 19723}, {16352, 19725}, {16353, 19724}, {16434, 19782}, {16454, 19684}, {16457, 19744}, {16458, 19701}, {16476, 23629}, {16498, 17477}, {16549, 17756}, {16589, 16846}, {16844, 17259}, {16917, 20132}, {16948, 17549}, {17379, 18792}, {17674, 18139}, {18169, 19278}, {18178, 19550}, {18185, 19261}, {19284, 19717}, {19289, 19750}, {19290, 19722}, {19331, 19739}, {19336, 19738}, {19337, 19741}, {19523, 19728}, {19645, 19752}, {19761, 21477}, {23414, 23659}

▸ Functional Relations6 relations

• X₃₈₆ = complement of X₁₀₄₄₉

• X₃₈₆ = crosssum of X₆ in X₁₀₁₁

• X₃₈₆ = crossdifference of every pair of points on line X₅₂₃X₆₄₉

• X₃₈₆ = Brocard-circle-inverse of X₅₈

• X₃₈₆ = intersection of tangents at X₂ and X₆ to Thomson cubic K002

• X₃₈₆ = intersection of Nagel line and Brocard axis

Notes

X₃₈₆ is the external center of similitude of the circumcircle and Apollonius circle. The internal center is X₅₇₃. (Peter J. C. Moses, 8/22/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the 2nd circumperp triangle at X₃₈₆. (Randy Hutson, November 18, 2015)

X₃₈₆ is the {X₃,X₆}-harmonic conjugate of X₅₈.