X1139

OUTER PENTAGON POINT

Let A' be the outermost vertex of the regular pentagon erected outward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X1139. (Steve Sigur and Antreas Hatzipolakis, #5246, 12/31/02; generalizations to n-gons, for odd n, by Milorad R. Stevanovic, #5253, 5256)
< 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) { 6, 1140 }: Central Line L(6, 1140) { 1, 3369 }: Central Line L(1, 3369) { 4, 3368 }: Central Line L(4, 3368) { 3, 3370 }: Central Line L(3, 3370) { 5, 3393 }: Central Line L(5, 3393) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3394 = SIN(A + π/5) POINT (0.5720, 0.2210) X_3397 = SEC(A + 2π/5) POINT (0.5013, 0.1676) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_1140 = INNER PENTAGON POINT (0.5248, 1.3312) X_1 = INCENTER (0.5714, 0.2333) X_3369 = SIN(A - 2π/5) POINT (0.3449, -0.1541) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3368 = COS(A - 2π/5) POINT (0.5852, 0.2428) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_3370 = SEC(A - 2π/5) POINT (0.5569, 0.2237) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3393 = COS(A + π/5) POINT (0.8782, 0.7267) X_1139 = OUTER PENTAGON POINT (0.5970, 0.3096) A B C X3394 X3397 X6 X1140 X1 X3369 X4 X3368 X3 X3370 X5 X3393 X1139 (a, b, c) = (5, 6, 7) | AB = 1

X1139 (OUTER PENTAGON POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\csc A)/(\cot A + \cot 2\pi /5) : (\csc B)/(\cot B + \cot 2\pi /5) : (\csc C)/(\cot C + \cot 2\pi /5)\)
Barycentrics: \(1/(\cot A + \cot 2\pi /5) : 1/(\cot B + \cot 2\pi /5) : 1/(\cot C + \cot 2\pi /5)\)
Search Value (6, 9, 13): \(-0.882375056596\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\csc(A + 2\pi /5) : \csc(B + 2\pi /5) : \csc(C + 2\pi /5)\) (Joe Goggins, Oct. 19, 2005)

▸ Incident Central Lines5 lines

{1, 3369}, {3, 3370}, {4, 3368}, {5, 3393}, {6, 1140}

▸ Functional Relations2 relations