X991

INTERSECTION OF LINES X1X_7 AND X3X_6

The function p = (r + 4R)/s is the Tucker parameter for X991; see the preamble to X(13323.)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 573, 3 }: Central Line L(573, 3) { 1, 7 }: Soddy Line { 3, 6 }: Brocard Axis { 55, 103 }: Central Line L(55, 103) { 42, 165 }: Central Line L(42, 165) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_573 = ORTHOHARMONIC OF X(58) (0.4583, 0.2009) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_991 = INTERSECTION OF LINES X(1)X(7) AND X(3)X(6) (0.5514, 0.1870) A B C X573 X3 X1 X7 X6 X55 X103 X42 X165 X991 (a, b, c) = (5, 6, 7) | AB = 1

X991 (INTERSECTION OF LINES X1X_7 AND X3X_6) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a[a^{3}(b + c) - a^{2}(b^{2} - bc + c^{2}) - a(b + c)(b^{2} + c^{2}) + (b - c)(b^{3} - c^{3})] : :\)
Barycentrics: \((\sin A)[\cos(A - arccot(p))] : :\), where \(p = (r + 4R)/s\)
Search Value (6, 9, 13): \(3.457495508350\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sin(A + U) : :\), where tan \(U = \tan(A/2) \tan(B/2) \tan(C/2)\) (Joe Goggins,11/26/08)

▸ Linear Combinations1 combination

X991 lies on the cubic K382.

▸ Incident Central Lines79 lines

{1, 7}, {2, 5400}, {3, 6}, {4, 4648}, {9, 1818}, {22, 1790}, {35, 255}, {36, 1471}, {37, 971}, {42, 165}, {43, 10164}, {45, 5779}, {48, 3220}, {51, 4191}, {55, 103}, {57, 955}, {63, 3190}, {73, 1394}, {78, 4416}, {81, 1754}, {84, 2335}, {212, 2003}, {223, 10383}, {241, 5728}, {394, 2328}, {550, 5453}, {601, 10902}, {741, 6011}, {940, 7580}, {942, 1418}, {954, 6180}, {968, 1709}, {975, 1490}, {984, 2801}, {993, 6518}, {995, 1064}, {999, 8147}, {1011, 3917}, {1012, 4653}, {1038, 10393}, {1062, 8555}, {1193, 7987}, {1214, 10391}, {1279, 1385}, {1283, 5197}, {1308, 12032}, {1396, 4219}, {1427, 11018}, {1456, 2646}, {1469, 2223}, {1699, 3720}, {1724, 6986}, {1736, 10394}, {1768, 4414}, {1779, 2979}, {2177, 5537}, {2195, 7295}, {2318, 3929}, {2340, 5223}, {2594, 5217}, {2635, 5219}, {2700, 2701}, {2999, 10857}, {3060, 4210}, {3214, 9588}, {3216, 3523}, {3666, 10167}, {3730, 3781}, {3752, 11227}, {3912, 12618}, {3931, 9943}, {4551, 5218}, {4644, 5759}, {4649, 9441}, {4658, 5706}, {4675, 5805}, {5292, 6908}, {5713, 6851}, {6051, 12688}

▸ Functional Relations9 relations