X₆₄₅

BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA

X₆₄₅ satisfies the equation X*(incircle) = Kiepert parabola, where * denotes barycentric multiplication, defined by

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

X₆₄₅ (BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {99, 101}, {9, 261}

• Related Centers: X₉₉ (STEINER POINT), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₉ (MITTENPUNKT), X₂₆₁ (ISOTOMIC CONJUGATE OF X₁₂)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b + c - a)/[a(b^{2} - c^{2})]\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b + c - a)/(b^{2} - c^{2})\)

Search Value (6, 9, 13): \(9.493327301254\)

▸ Incident Central Lines8 lines

{9, 261}, {99, 101}, {100, 931}, {294, 314}, {644, 646}, {648, 668}, {651, 799}, {666, 670}

▸ Functional Relations3 relations

• X₆₄₅ = anticomplement of X₁₇₀₅₈

• X₆₄₅ = trilinear pole of line X₈X₂₁

• X₆₄₅ = perspector of anticevian triangle of X₉₉ and unary cofactor triangle of intangents triangle

Notes

(\(u : v : w\)) * (\(x : y : z\)) =\(ux : vy : wz\) (barycentric coordinates; see note at X₂).