X₃₄₂₆

X₃-VERTEX CONJUGATE OF X₆

X₃₄₂₆ = isogonal conjugate of X₃₇₆

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

X₃₄₂₆ (X₃-VERTEX CONJUGATE OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {25, 74}, {30, 69}

• Related Centers: X₃₇₆ (CENTROID OF THE ANTIPEDAL TRIANGLE OF X), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₃₀ (EULER INFINITY POINT), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a/[a^{2}vwyz - ux(bw + cv)(bz + cy)]\), where \(u : v : w\) = X₃ and \(x : y : z\) = X₆

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

Search Value (6, 9, 13): \(-39.142190307047\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1/(3 \cos A - \sin B \sin C) : 1/(3 \cos B - \sin C \sin A) : 1/(3 \cos C - \sin A \sin B)\)

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}/[5a^{4} - 4a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] (E\). Danneels)

▸ Incident Central Lines14 lines

{3, 1495}, {6, 1597}, {25, 74}, {30, 69}, {54, 1593}, {64, 1598}, {67, 2777}, {68, 382}, {73, 1480}, {248, 1384}, {265, 541}, {381, 1514}, {879, 1499}, {895, 1351}

▸ Functional Relations1 relation

• X₃₄₂₆ = isogonal conjugate of X₃₇₆

Notes

Let P_A be the parabola with focus A and directrix BC. Let L_A be the polar of X₂₀ wrt P_A. Define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₄₂₆. (Randy Hutson, July 11, 2019)