X₆₄₄

TRILINEAR MULTIPLIER FOR YFF PARABOLA

X₆₄₄ satisfies the equation X*(incircle) = Yff parabola, where * denotes trilinear multiplication, defined by

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

X₆₄₄ (TRILINEAR MULTIPLIER FOR YFF PARABOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {100, 101}, {8, 220}

• Related Centers: X₃₆₆₉ (com(INTOUCH TRIANGLE)), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₈ (NAGEL POINT), X₂₂₀ (-CEVA CONJUGATE OF X₅₅)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b + c - a)/(b - c)\)

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): \(9.556744265007\)

▸ Incident Central Lines11 lines

{8, 220}, {78, 728}, {100, 101}, {105, 1083}, {145, 218}, {190, 651}, {219, 346}, {645, 646}, {666, 668}, {813, 932}, {934, 1025}

▸ Functional Relations14 relations

• X₆₄₄ = reflection of X_i in X_j for these (i,j): (105, 1083), (1280, 1)

• X₆₄₄ = isogonal conjugate of X₃₆₆₉

• X₆₄₄ = isotomic conjugate of X₂₄₀₀₂

• X₆₄₄ = anticomplement of X₄₉₀₄

• X₆₄₄ = X₁₉₀-Ceva conjugate of X₁₀₀

• X₆₄₄ = crosssum of X₇₆₄ and X₁₀₁₅

• X₆₄₄ = crossdifference of every pair of points on line X₂₄₄X₁₃₅₇

• X₆₄₄ = trilinear pole of line X₉X₅₅ (tangent to Feuerbach hyperbola at X₉)

• X₆₄₄ = X₆₅₀-cross conjugate of X₉

• X₆₄₄ = eigencenter of Caelum triangle

• X₆₄₄ = eigencenter of 5th mixtilinear triangle

• X₆₄₄ = perspector of unary cofactor triangles of 3rd and 5th mixtilinear triangles

• X₆₄₄ = perspector of 5th mixtilinear triangle and unary cofactor triangle of 3rd mixtilinear triangle

• X₆₄₄ = intersection, other than vertices of Gemini triangle 29, of {ABC, Gemini 29}-circumconic and {Gemini 29, Gemini 30}-circumconic

Notes

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

X₆₄₄ is the perspector of the anticevian triangle of X₁₀₀ and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 15, 2019)