X₃₄₁₆

ISOGONAL CONJUGATE OF X₃₄₁₅

X₃₄₁₆ is the isogonal conjugate of X₃₄₁₅.

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

X₃₄₁₆ (ISOGONAL CONJUGATE OF X₃₄₁₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {8, 69}

• Related Centers: X₈ (NAGEL POINT), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/x(3415)\)

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

Search Value (6, 9, 13): \(5.204889372943\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{3} - (b + c)(b^{2} + c^{2}) (E\). Danneels)

▸ Incident Central Lines13 lines

{1, 141}, {2, 1386}, {6, 10}, {7, 8}, {40, 1503}, {66, 72}, {200, 223}, {321, 1836}, {355, 511}, {608, 1861}, {612, 1211}, {613, 1737}, {1376, 1460}

▸ Functional Relations5 relations

• X₃₄₁₆ = midpoint of X₈ and X₆₉

• X₃₄₁₆ = reflection of X_i in X_j for these (i,j): (1, 141), (6, 10)

• X₃₄₁₆ = isogonal conjugate of X₃₄₁₅

• X₃₄₁₆ = anticomplement of X₁₃₈₆

• X₃₄₁₆ = crossdifference ofevery pair of points on the line X₈₃₄X₂₄₈₃