X₃₄₅₇

X₁₃-VERTEX CONJUGATE OF X₁₅

Let A' be the free vertex of the equilateral triangle constructed outwardly on BC, and define B' and C' cyclically. X₃₄₅₇ is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)

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

X₃₄₅₇ (X₁₃-VERTEX CONJUGATE OF X₁₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 13}, {62, 110}

• Related Centers: X₁₄₉₅ (CROSSSUM OF X₂ AND X₃₀), X₃₄₅₈ (-VERTEX CONJUGATE OF X₁₆), X₂ (CENTROID), X₁₃ (1st ISOGONIC CENTER), X₆₂ (ISOGONAL CONJUGATE OF X₁₈), 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 \(u : v : w\) = X₁₃ and \(x : y : z\) = X₁₅

Search Value (6, 9, 13): \(0.016425956796\)

▸ 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, 13}, {6, 3130}, {15, 1337}, {32, 3124}, {37, 1250}, {62, 110}, {300, 308}

▸ Functional Relations8 relations

• X₃₄₅₇ = isogonal conjugate of X₂₉₈

• X₃₄₅₇ = X₁₄₉₅-cross conjugate of X₃₄₅₈

• X₃₄₅₇ = crosspoint of X₆ and X₃₄₄₀

• X₃₄₅₇ = crosssum of X₂ and X₆₁₆

• X₃₄₅₇ = polar conjugate of isotomic conjugate of X₃₆₂₉₆

• X₃₄₅₇ = X₁₅-isoconjugate of X₇₅

• X₃₄₅₇ = X₆₃-isoconjugate of X₄₇₀

• X₃₄₅₇ = barycentric product of circumcircle intercepts of inner Napoleon circle