X₃₃₆₄

COS(A - 5π/12) POINT

X₃₃₆₄ and related points occupy Bernard Gibert’s Table 38 concerning these three loci: Brocard axis, Kiepert hyperbola, and the cubic K457. Trilinear equations for these curves, of degrees 1,2,3, respectively, in the variables a,b,c, are given as follows:

Curve Trilinears Brocard axis \(\cos(A + t) : \cos(B + t) : \cos(C + t)\) Kiepert hyperbola \(\sec(A + t) : \sec(B + t) : \sec(C + t)\) K457 \(\tan(A + t) : \tan(B + t) : \tan(C + t)\)

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

X₃₃₆₄ (COS(A - 5π/12) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {14, 485}

• Related Centers: X₃₃₆₆ (SEC(A - 5π/12) POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₁₄ (2nd ISOGONIC CENTER), X₄₈₅ (VECTEN POINT)

Trilinears: \(\cos(A - 5\pi /12) : :\)

Search Value (6, 9, 13): \(1.464511557309\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cos(A + \pi /3) - \sin(A + \pi /3) : :\)

▸ Incident Central Lines6 lines

{3, 6}, {14, 485}, {17, 486}, {18, 590}, {202, 2067}, {203, 1124}

▸ Functional Relations6 relations

• X₃₃₆₄ = isogonal conjugate of X₃₃₆₆

• X₃₃₆₄ = inverse-in-Brocard-circle of X₃₃₉₀

• X₃₃₆₄ = X₁₈-Ceva conjugate of X₃₃₆₅

• X₃₃₆₄ = X₃₂₀₆-cross conjugate of X₃₃₆₅

• X₃₃₆₄ = {X₁₅,X₆₂}-harmonic conjugate of X₃₃₆₅

• X₃₃₆₄ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₃₈₉