X₁₃₉₇

X₃₁-BETH CONJUGATE OF X₃₁

X₁₃₉₇ = X₅₆₀-cross conjugate of X₃₂

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

X₁₃₉₇ (X₃₁-BETH CONJUGATE OF X₃₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {56, 58}, {6, 181}

• Related Centers: X₃₁ (2nd POWER POINT), X₅₆ (EXSIMILICENTER), X₅₈ (ISOGONAL CONJUGATE OF X₁₀), X₆ (SYMMEDIAN POINT), X₁₈₁ (APOLLONIUS POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (1 - \cos A)u(a,b,c)\), where \(u : v : w\) = X₃₁

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

Search Value (6, 9, 13): \(0.042545853970\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = [\sin A \sin A/2]^{2}\)

▸ Incident Central Lines15 lines

{1, 987}, {6, 181}, {31, 184}, {42, 1404}, {55, 572}, {56, 58}, {57, 985}, {60, 959}, {109, 727}, {171, 182}, {278, 1365}, {392, 993}, {602, 1092}, {603, 1472}, {1257, 1407}

▸ Functional Relations8 relations

• X₁₃₉₇ = X_i-Ceva conjugate of X_j for these (i,j): (59, 1415), (604, 32), (1408, 604)

• X₁₃₉₇ = anticomplement of complementary conjugate of X₁₇₀₅₃

• X₁₃₉₇ = X₅₆₀-cross conjugate of X₃₂

• X₁₃₉₇ = crosspoint of X_i and X_j for these (i,j): (56, 608), (59, 1415), (604, 1106)

• X₁₃₉₇ = crosssum of X_i and X_j for these (i,j): (8, 345), (75, 322), (312, 341)

• X₁₃₉₇ = isogonal conjugate of X₃₅₉₆

• X₁₃₉₇ = X₈-isoconjugate of X₇₅

• X₁₃₉₇ = trilinear product of extraversions of X₅₅

Notes

X₁₃₉₇ is the vertex conjugate of the foci of the inellipse that is the isogonal conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)