X₈₄₀

ISOGONAL CONJUGATE OF X₅₂₈

X₈₄₀ is the isogonal conjugate of X₅₂₈.

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

X₈₄₀ (ISOGONAL CONJUGATE OF X₅₂₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2742, 3}

• Related Centers: X₂₇₄₂ (SR(X₁₀₄, X₁₀₅)), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a/(2ax - by - cz)\), \(x = x(A,B,C) = 1 - \cos(B - C)\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(7.997672077475\)

X₈₄₀ lies on the circumcircle.

▸ Incident Central Lines10 lines

{6, 919}, {7, 927}, {36, 101}, {55, 901}, {100, 518}, {105, 513}, {106, 663}, {109, 902}, {759, 1019}, {898, 1083}

▸ Functional Relations6 relations

• X₈₄₀ = reflection of X₂₇₄₂ in X₃

• X₈₄₀ = isogonal conjugate of X₅₂₈

• X₈₄₀ = trilinear pole of line X₆X₆₆₅

• X₈₄₀ = Ψ(X₆, X₆₆₅)

• X₈₄₀ = reflection of X₁₀₅ in line X₁X₃

• X₈₄₀ = inverse-in-circle-O(3513,3514) of X₁₀₈

Notes

X₈₄₀ is the perspector of ABC and the (degenerate) cross-triangle of the circumcevian triangles of X₃₅₁₃ and X₃₅₁₄. (Randy Hutson, June 7, 2019)