X3381

SEC(A - π/5) POINT

X3381 and related points occupy Bernard Gibert’s Table 38. See the note at X3364.
< 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) { 1140, 3368 }: Central Line L(1140, 3368) { 1139, 3379 }: Central Line L(1139, 3379) { 3369, 3370 }: Central Line L(3369, 3370) { 5, 3382 }: Central Line L(5, 3382) { 2, 3393 }: Central Line L(2, 3393) { 4, 3394 }: Central Line L(4, 3394) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3396 = SIN(A + 2π/5) POINT (0.5378, 0.1645) X_3382 = CSC(A - π/5) POINT (0.3759, 0.1017) X_1140 = INNER PENTAGON POINT (0.5248, 1.3312) X_3368 = COS(A - 2π/5) POINT (0.5852, 0.2428) X_1139 = OUTER PENTAGON POINT (0.5970, 0.3096) X_3379 = COS(A - π/5) POINT (0.5576, 0.1972) X_3369 = SIN(A - 2π/5) POINT (0.3449, -0.1541) X_3370 = SEC(A - 2π/5) POINT (0.5569, 0.2237) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_3393 = COS(A + π/5) POINT (0.8782, 0.7267) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3394 = SIN(A + π/5) POINT (0.5720, 0.2210) X_3381 = SEC(A - π/5) POINT (0.5836, 0.2714) A B C X3396 X3382 X1140 X3368 X1139 X3379 X3369 X3370 X5 X2 X3393 X4 X3394 X3381 (a, b, c) = (5, 6, 7) | AB = 1

X3381 (SEC(A - π/5) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sec(A - \pi /5) : :\)
Barycentrics: \(\sin A \sec(A - \pi /5) : :\)
Search Value (6, 9, 13): \(0.408157576208\)

X3381 lies on the Kiepert hyperbola.

▸ Incident Central Lines9 lines

{2, 3393}, {3, 5402}, {4, 3394}, {5, 3382}, {6, 5401}, {1139, 3379}, {1140, 3368}, {3369, 3370}, {3380, 3397}

▸ Functional Relations5 relations

Notes

On the sides of ABC construct internal regular pentagons ABC1C2C3, BCA1A2A3, CAB1B2B3. The Aubert lines of [ABB3A1], [BCC3B1], [CAA3C1] concur in X3381. These lines pass through A, B, C. (Aubert lines are defined at X45010; see also A construction of X3381 and X5401 using Aubert lines). For the case of external pentagons, see X5401. (Ivan Pavlov, April 15, 2022)