X₃₃₉₃

COS(A + π/5) POINT

X₃₃₉₃ and related points occupy Bernard Gibert’s Table 38. See the note at X₃₃₆₄.

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

X₃₃₉₃ (COS(A + π/5) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {5, 1139}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅ (NINE-POINT CENTER), X₁₁₃₉ (OUTER PENTAGON POINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos(A + \pi /5)\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-3.851033519701\)

▸ Incident Central Lines2 lines

{3, 6}, {5, 1139}

▸ Functional Relations3 relations

• X₃₃₉₃ = isogonal conjugate of X₅₄₀₁

• X₃₃₉₃ = inverse-in-Brocard-circle of X₃₃₇₉

• X₃₃₉₃ = X₁₁₄₀-Ceva conjugate of X₃₃₉₄