X₂₄₅₀

INVERSE-IN-NINE-POINT-CIRCLE OF X₁₃₁₆

X₂₄₅₀ = inverse-in-nine-point-circle of X₁₃₁₆

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

X₂₄₅₀ (INVERSE-IN-NINE-POINT-CIRCLE OF X₁₃₁₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {230, 248}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₄₈ (-LINE CONJUGATE OF X₁₃₂)

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

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

Search Value (6, 9, 13): \(9.649344720497\)

▸ Incident Central Lines5 lines

{2, 3}, {66, 2165}, {230, 248}, {1503, 1976}, {3566, 3569}

▸ Functional Relations5 relations

• X₂₄₅₀ = complement of X₃₇₁₈₃

• X₂₄₅₀ = inverse-in-nine-point-circle of X₁₃₁₆

• X₂₄₅₀ = inverse-in-orthocentroidal circle of X₃₁₄₈

• X₂₄₅₀ = crosspoint of X₉₈ and X₂₆₄

• X₂₄₅₀ = crosssum of X₁₈₄ and X₅₁₁

Notes

As a point on the Euler line, X₂₄₅₀ has Shinagawa coefficients \(((E + F)F - S^{2},(E + F)^{2} - S^{2})\).