X₆₄₆

BARYCENTRIC MULTIPLIER FOR YFF PARABOLA

X₆₄₆ satisfies the equation X*(incircle) = Yff parabola, where * denotes barycentric multiplication, defined by

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

X₆₄₆ (BARYCENTRIC MULTIPLIER FOR YFF PARABOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {644, 645}, {190, 668}

• Related Centers: X₃₆₆₉ (com(INTOUCH TRIANGLE)), X₆₄₄ (TRILINEAR MULTIPLIER FOR YFF PARABOLA), X₆₄₅ (BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA), X₁₉₀ (YFF PARABOLIC POINT), X₆₆₈ (TRILINEAR POLE OF LINE X₂X₃₇)

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b + c - a)/[a(b - c)]\)

Search Value (6, 9, 13): \(9.403725070080\)

▸ Incident Central Lines2 lines

{190, 668}, {644, 645}

▸ Functional Relations5 relations

• X₆₄₆ = isotomic conjugate of X₃₆₆₉

• X₆₄₆ = trilinear pole of line X₈X₂₁₀ (the tangent to Feuerbach hyperbola at X₈)

• X₆₄₆ = X₆₅₀-cross conjugate of X₈

• X₆₄₆ = X₅₂₂-cross conjugate of X₃₁₄

• X₆₄₆ = PU(4)-harmonic conjugate of polar conjugate of X₃₅₁₅₈

Notes

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