X₆₈₅

TRILINEAR POLE OF LINE X₄X_98

X₆₈₅ = trilinear pole of line X₄X₃₂

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

X₆₈₅ (TRILINEAR POLE OF LINE X₄X_98) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {98, 468}, {287, 297}

• Related Centers: X₆₈₄ (CROSSDIFFERENCE OF X₄ AND X₃₂), X₉₈ (TARRY POINT), X₄₆₈ (-LINE CONJUGATE OF X₃), X₂₈₇ (-HIRST INVERSE OF X₉₈), X₂₉₇ (-HIRST INVERSE OF X₄)

Trilinears: \(f(A,B,C): f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 1/(\sec B \sin^{3} C - \sec C \sin^{3} B)\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sin A f(A,B,C)\)

Search Value (6, 9, 13): \(0.481231651426\)

▸ Incident Central Lines4 lines

{98, 468}, {110, 850}, {250, 523}, {287, 297}

▸ Functional Relations6 relations

• X₆₈₅ = isogonal conjugate of X₆₈₄

• X₆₈₅ = isotomic conjugate of X₆₃₃₃

• X₆₈₅ = polar conjugate of X₂₇₉₉

• X₆₈₅ = trilinear pole of line X₄X₃₂

• X₆₈₅ = trilinear product of circumcircle intercepts of line X₃₁X₉₂

• X₆₈₅ = vertex conjugate of MacBeath circumconic intercepts of Euler line