X₃₂₈₉

INTERSECTION OF LINES X₂X_6 AND X₃X_217

Let L denote the line through X₄ perpendicular to the Brocard axis, X₃X₆. Coefficients for a trilinear equation for L are the trilinears for X₃₂₈₉.

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

X₃₂₈₉ (INTERSECTION OF LINES X₂X_6 AND X₃X_217) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3331, 1625}

• Related Centers: X₃₃₃₁ (INTERSECTION OF LINES X), X₁₆₂₅ (TCC-PERSPECTOR OF X₁₆₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{3}(b^{2} + c^{2} - a^{2})(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2})\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.599722636790\)

▸ Incident Central Lines10 lines

{2, 6}, {3, 217}, {30, 1625}, {110, 1971}, {112, 2706}, {115, 1568}, {184, 418}, {232, 511}, {520, 647}, {571, 1501}

▸ Functional Relations6 relations

• X₃₂₈₉ = reflection of X₃₃₃₁ in X₁₆₂₅

• X₃₂₈₉ = isogonal conjugate of isotomic conjugate of X₃₆₂₁₂

• X₃₂₈₉ = X_i-Ceva conjugate of X_j for these (i,j): (287, 3), (2421, 684)

• X₃₂₈₉ = crosspoint of X₃ and X₂₈₇

• X₃₂₈₉ = crosssum of X₄ and X₂₃₂

• X₃₂₈₉ = crossdifference of every pair of points on the line X₄X₅₁₂