X₃₂₈₈

INTERSECTION OF LINES X₂₃₇X_351 AND X₅₂₃X_879

Let L denote the line through X₂ parallel to the Brocard axis, X₃X₆. Coefficients for a trilinear equation for L are the trilinears for X₃₂₈₈.

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

X₃₂₈₈ (INTERSECTION OF LINES X₂₃₇X_351 AND X₅₂₃X_879) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {187, 237}, {6, 523}

• Related Centers: X₉₉ (STEINER POINT), X₃₁₁₄ (TgX(32)), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂), X₆ (SYMMEDIAN POINT), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀)

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

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

Search Value (6, 9, 13): \(-5.820086186114\)

▸ Incident Central Lines6 lines

{6, 523}, {111, 2698}, {187, 237}, {323, 401}, {352, 1499}, {419, 2501}

▸ Functional Relations7 relations

• X₃₂₈₈ = reflection of X_i in X_j for these (i,j): (2451, 3049), (3049, 3050)

• X₃₂₈₈ = crosspoint of X₉₉ and X₃₁₁₄

• X₃₂₈₈ = crosssum of X₅₁₂ and X₃₁₁₇

• X₃₂₈₈ = crossdifference of every pair of points on the line X₂X₅₁

• X₃₂₈₈ = X₂-Ceva conjugate of X₃₈₉₉₇

• X₃₂₈₈ = perspector of hyperbola {{A, B, C,X₆,X₉₅,X₉₈}}

• X₃₂₈₈ = radical center of {circumcircle, Brocard circle, orthosymmedial circle}