X₆₄₃

TRILINEAR MULTIPLIER FOR KIEPERT PARABOLA

X₆₄₃ satisfies the equation X*(incircle) = Kiepert parabola, where * denotes trilinear multiplication, defined by

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

X₆₄₃ (TRILINEAR 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, 109}, {100, 110}

• Related Centers: X₉₉ (STEINER POINT), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(8.674646291873\)

▸ Incident Central Lines8 lines

{8, 1098}, {99, 109}, {100, 110}, {101, 931}, {162, 190}, {163, 1018}, {212, 312}, {283, 1043}

▸ Functional Relations3 relations

• X₆₄₃ = isogonal conjugate of X₄₀₁₇

• X₆₄₃ = isotomic conjugate of X₄₀₇₇

• X₆₄₃ = trilinear pole of line X₉X₂₁

Notes

(\(u : v : w\)) * (\(x : y : z\)) =\(ux : vy : wz\).