X₃₄₅₈

X₁₄-VERTEX CONJUGATE OF X₁₆

Let A' be the free vertex of the equilateral triangle constructed inwardly on BC, and define B' and C' cyclically. X₃₄₅₈ is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)

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

X₃₄₅₈ (X₁₄-VERTEX CONJUGATE OF X₁₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 14}, {61, 110}

• Related Centers: X₁₄₉₅ (CROSSSUM OF X₂ AND X₃₀), X₃₄₅₇ (-VERTEX CONJUGATE OF X₁₅), X₂ (CENTROID), X₁₄ (2nd ISOGONIC CENTER), X₆₁ (ISOGONAL CONJUGATE OF X₁₇), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

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 \(u : v : w\) = X₁₄ and \(x : y : z\) = X₁₆

Search Value (6, 9, 13): \(-9.344732204999\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = (\sin^{2} A)(\sin A/2) \sec(A + \pi /6)\)

▸ Incident Central Lines7 lines

{2, 14}, {6, 3129}, {16, 1338}, {32, 3124}, {37, 1254}, {61, 110}, {301, 308}

▸ Functional Relations8 relations

• X₃₄₅₈ = isogonal conjugate of X₂₉₉

• X₃₄₅₈ = X₁₄₉₅-cross conjugate of X₃₄₅₇

• X₃₄₅₈ = crosspoint of X₆ and X₃₄₄₁

• X₃₄₅₈ = crosssum of X₂ and X₆₁₇

• X₃₄₅₈ = polar conjugate of isotomic conjugate of X₃₆₂₉₇

• X₃₄₅₈ = X₁₆-isoconjugate of X₇₅

• X₃₄₅₈ = X₆₃-iaoconjugate of X₄₇₁

• X₃₄₅₈ = barycentric product of circumcircle intercepts of outer Napoleon circle