X₂₁₆₅

X₂-ISOCONJUGATE OF X₄₇

X₂₁₆₅ = cevapoint of X₁₁₅ and X₆₄₇

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

X₂₁₆₅ (X₂-ISOCONJUGATE OF X₄₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 6}, {25, 53}

• Related Centers: X₁₉₉₃ (ORTHOCORRESPONDENT OF X₃), X₅ (NINE-POINT CENTER), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE)

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X₄₇

Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)

Search Value (6, 9, 13): \(9.245704983081\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = \sin A \sec 2A\)

▸ Incident Central Lines14 lines

{2, 311}, {4, 96}, {5, 6}, {25, 53}, {37, 91}, {111, 925}, {115, 577}, {206, 1976}, {393, 847}, {493, 590}, {494, 615}, {1263, 2079}, {1321, 1322}, {1400, 1454}

▸ Functional Relations14 relations

• X₂₁₆₅ = isogonal conjugate of X₁₉₉₃

• X₂₁₆₅ = isotomic conjugate of X₇₇₆₃

• X₂₁₆₅ = X₉₆-Ceva conjugate of X₂₃₅₁

• X₂₁₆₅ = cevapoint of X₁₁₅ and X₆₄₇

• X₂₁₆₅ = X_i-cross conjugate of X_j for these (i,j): (184, 4), (216, 6), (2351, 68)

• X₂₁₆₅ = crosspoint of X₂ and X₂₅₄

• X₂₁₆₅ = crosssum of X_i and X_j for these (i,j): (6, 155), (571, 1147)

• X₂₁₆₅ = barycentric product of X₄₈₅ and X₄₈₆

• X₂₁₆₅ = complement of isogonal conjugate of X₃₉₁₀₉

• X₂₁₆₅ = orthic-isogonal conjugate of X₃₉₁₁₁

• X₂₁₆₅ = X₄-Ceva conjugate of X₃₉₁₁₁

• X₂₁₆₅ = polar conjugate of X₃₁₇

• X₂₁₆₅ = X₉₂-isoconjugate of X₁₁₄₇

• X₂₁₆₅ = X₆₁₀-of-orthic-triangle if ABC is acute