X3370

SEC(A - 2π/5) POINT

X3370 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) { 3, 1139 }: Central Line L(3, 1139) { 5, 1140 }: Central Line L(5, 1140) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3380 = SIN(A - π/5) POINT (0.6635, 0.3721) X_1140 = INNER PENTAGON POINT (0.5248, 1.3312) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1139 = OUTER PENTAGON POINT (0.5970, 0.3096) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3370 = SEC(A - 2π/5) POINT (0.5569, 0.2237) A B C X3380 X1140 X3 X1139 X5 X3370 (a, b, c) = (5, 6, 7) | AB = 1

X3370 (SEC(A - 2π/5) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sec(A - 2\pi /5)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.822509573072\)

X3370 lies on the Kiepert hyperbola.

▸ Incident Central Lines2 lines

{3, 1139}, {5, 1140}

▸ Functional Relations2 relations

Notes

Let A' be the center of the regular pentagon \(AA_{2}\)A3A4A5, with opposite orientation than ABC and such that B ∈ A2A3 and C ∈ A4A5. Build B' and C' cyclically. Then the lines AA', BB', CC' concur at X3370. (César E. Lozada, May 14, 2019), Hyacinthos #29013).