X₃₄₅₂

ISOGONAL CONJUGATE OF X₃₄₅₁

X₃₄₅₂ is the isogonal conjugate of X₃₄₅₁.

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

X₃₄₅₂ (ISOGONAL CONJUGATE OF X₃₄₅₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {999, 1125}

• Related Centers: X₉₉₉ (MIDPOINT OF X₁ AND X₅₇), X₁₁₂₅ (COMPLEMENT OF X₁₀)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/(x(3451)\)

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

Search Value (6, 9, 13): \(3.753825542974\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines25 lines

{1, 2551}, {2, 7}, {4, 936}, {5, 10}, {11, 210}, {69, 1997}, {72, 1210}, {78, 950}, {118, 123}, {120, 124}, {200, 497}, {281, 1848}, {312, 2321}, {333, 645}, {345, 2325}, {515, 997}, {516, 1376}, {631, 2096}, {958, 999}, {962, 1706}, {1656, 2095}, {1698, 2093}, {1699, 2550}, {1764, 2183}, {2324, 2999}

▸ Functional Relations12 relations

• X₃₄₅₂ = midpoint of X_i and X_j for these (i,j): (1, 3421), (57, 329), (200, 497)

• X₃₄₅₂ = reflection of X₉₉₉ in X₁₁₂₅

• X₃₄₅₂ = isogonal conjugate of X₃₄₅₁

• X₃₄₅₂ = complement of X₅₇

• X₃₄₅₂ = complementary conjugate of X₃₄₅₀

• X₃₄₅₂ = X₆₆₈-Ceva conugate of X₅₂₂

• X₃₄₅₂ = crosspoint of X₂ and X₃₁₂

• X₃₄₅₂ = crosssum of X_i and X_j for these (i,j): (6, 604), (41, 3052)

• X₃₄₅₂ = complement of X₅₇

• X₃₄₅₂ = isotomic conjugate of isogonal conjugate of X₂₃₄₇

• X₃₄₅₂ = polar conjugate of isogonal conjugate of X₂₂₀₇₂

• X₃₄₅₂ = perspector of medial triangle and inverse-in-excircles triangle