X₁₅₇

X₆-OF-TANGENTIAL-TRIANGLE

X₁₅₇ = X₆-of-tangential-triangle

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

X₁₅₇ (X₆-OF-TANGENTIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 66}, {25, 53}

• Related Centers: X₂₉₀₉ (ORTHIC-ISOGONAL CONJUGATE OF X₃₂), X₃ (CIRCUMCENTER), X₆₆ (ISOGONAL CONJUGATE OF X₂₂), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[b^{3}\cos B + c^{3}\cos C - a^{3}\cos A]\)

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

Search Value (6, 9, 13): \(-1.134813455477\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a[a^{6} - b^{6} - c^{6} - a^{4}(b^{2} + c^{2}) + b^{4}(c^{2} + a^{2}) + c^{4}(a^{2} + b^{2})]\)

▸ Incident Central Lines6 lines

{3, 66}, {6, 248}, {22, 183}, {25, 53}, {161, 418}, {206, 216}

▸ Functional Relations8 relations

• X₁₅₇ = isogonal conjugate of anticomplement of X₂₂₃₉₁

• X₁₅₇ = isotomic conjugate of isogonal conjugate of X₂₉₀₉

• X₁₅₇ = X₂₆₄-Ceva conjugate of X₆

• X₁₅₇ = crosssum of X₁₂₇ and X₅₂₀

• X₁₅₇ = X₆-of-tangential-triangle

• X₁₅₇ = perspector of circumcircle wrt Schroeter triangle

• X₁₅₇ = perspector of polar circle wrt tangential triangle

• X₁₅₇ = intouch-to-ABC functional image of X₆