X₁₁₀₆

TRILINEAR SQUARE OF X₅₆

X₁₁₀₆ = trilinear product of vertices of tangential mid-arc triangle

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

X₁₁₀₆ (TRILINEAR SQUARE OF X₅₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {31, 56}, {34, 244}

• Related Centers: X₃₄₁ (ISOTOMIC CONJUGATE OF X₂₆₉), X₃₁ (2nd POWER POINT), X₅₆ (EXSIMILICENTER), X₃₄ (-BETH CONJUGATE OF X₄), X₂₄₄ (-LINE CONJUGATE OF X₁₀₀)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (1 - \cos A)^{2}\)

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): \(0.011940767480\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sin^{4} A/2 : \sin^{4} B/2 : \sin^{4} C/2\)

▸ Incident Central Lines15 lines

{3, 1037}, {7, 987}, {31, 56}, {32, 604}, {34, 244}, {36, 255}, {38, 1038}, {57, 961}, {58, 269}, {77, 988}, {279, 985}, {388, 750}, {601, 999}, {651, 978}, {727, 934}

▸ Functional Relations4 relations

• X₁₁₀₆ = isogonal conjugate of X₃₄₁

• X₁₁₀₆ = complement of anticomplementary conjugate of X₁₇₄₈₀

• X₁₁₀₆ = X₉₂-isoconjugate of X₃₆₉₂

• X₁₁₀₆ = trilinear product of vertices of tangential mid-arc triangle