X₃₂₉₂

INTERSECTION OF LINES X₃X_49 AND X₂₃X_110

Let L denote the line through X₄ perpendicular to the line X₂X₆. Coefficients for a trilinear equation for L are the trilinears for X₃₂₉₂.

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

X₃₂₉₂ (INTERSECTION OF LINES X₃X_49 AND X₂₃X_110) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {110, 323}

• Related Centers: X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₃₂₃ (ISOTOMIC CONJUGATE OF X₉₄)

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

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

Search Value (6, 9, 13): \(2.888127720090\)

▸ Incident Central Lines12 lines

{2, 575}, {3, 49}, {6, 237}, {23, 110}, {51, 576}, {112, 2763}, {352, 2030}, {450, 648}, {520, 647}, {539, 2072}, {542, 858}, {1209, 1493}

▸ Functional Relations9 relations

• X₃₂₉₂ = midpoint of X₁₁₀ and X₃₂₃

• X₃₂₉₂ = reflection of X₁₄₉₅ in X₁

• X₃₂₉₂ = X_i-Ceva conjugate of X_j for these (i,j): (524, 187), (895, 3)

• X₃₂₉₂ = isotomic conjugate of isogonal conjugate of X₂₃₂₀₀

• X₃₂₉₂ = isotomic conjugate of polar conjugate of X₁₈₇

• X₃₂₉₂ = crosspoint of X₃ and X₈₉₅

• X₃₂₉₂ = crosssum of X₄ and X₄₆₈

• X₃₂₉₂ = X₁₉-isoconjugate of X₆₇₁

• X₃₂₉₂ = crossdifference of every pair of points on the line X₄X₁₄₉₉

Notes

Let O* be the circle with segment X₁₅X₁₆ as diameter (and center X₁₈₇. Let P be the perspector of O*. Then X₃₂₉₂ is the trilinear pole of the polar of P with respect to O*. See X₅₆₄₂ for a similar property involving the segment X₁₃X₁₄. (Randy Hutson, July 18, 2014)