X₁₉₆₉

TRILINEAR PRODUCT OF PU(38)

X₁₉₆₉ = cevapoint of X₇₅ and X₉₂

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

X₁₉₆₉ (TRILINEAR PRODUCT OF PU(38)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {75, 158}, {1, 336}

• Related Centers: X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT), X₁₅₈ (-CROSS CONJUGATE OF X₉₂), X₁ (INCENTER), X₃₃₆ (ISOTOMIC CONJUGATE OF X₂₄₀)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{2}c^{2} \sin 2B \sin 2C\)

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

Search Value (6, 9, 13): \(7.127319360623\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = \sec A \csc^{3} A\)

▸ Incident Central Lines5 lines

{1, 336}, {75, 158}, {76, 331}, {92, 304}, {273, 1240}

▸ Functional Relations7 relations

• X₁₉₆₉ = isotomic conjugate of X₄₈

• X₁₉₆₉ = cevapoint of X₇₅ and X₉₂

• X₁₉₆₉ = trilinear pole of polar of X₃₁ wrt polar circle (line X₁₄₂₀₈X_20948)

• X₁₉₆₉ = pole wrt polar circle of trilinear polar of X₃₁ (line X₆₆₇X₇₈₈)

• X₁₉₆₉ = polar conjugate of X₃₁

• X₁₉₆₉ = trilinear product of vertices of Gemini triangle 37

• X₁₉₆₉ = trilinear product of vertices of Gemini triangle 38