X₆₇₅

TRILINEAR POLE OF LINE X₆X_514

X₆₇₅ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₁₁₆

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

X₆₇₅ (TRILINEAR POLE OF LINE X₆X_514) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 101}, {7, 109}

• Related Centers: X₆₇₄ (CROSSDIFFERENCE OF X₆ AND X₅₁₄), X₂ (CENTROID), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₇ (GERGONNE POINT), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER))

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc/[b^{3} + c^{3} - a(b^{2} + c^{2})]\)

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): \(7.622324049262\)

X₆₇₅ lies on the circumcircle.

▸ Incident Central Lines12 lines

{2, 101}, {7, 109}, {27, 112}, {75, 100}, {86, 110}, {99, 310}, {108, 273}, {335, 813}, {673, 919}, {789, 871}, {901, 903}, {934, 1088}

▸ Functional Relations3 relations

• X₆₇₅ = isogonal conjugate of X₆₇₄

• X₆₇₅ = isotomic conjugate of X₃₀₀₆

• X₆₇₅ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₁₁₆