X₂₄₈₉

RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE}

X₂₄₈₉ = X₁₁₂-Ceva conjugate of X₂₅

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

X₂₄₈₉ (RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE}) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {112, 250}, {230, 231}

• Related Centers: X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} - c^{2})/(b^{2} + c^{2} - a^{2})\)

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

Search Value (6, 9, 13): \(-0.534859981302\)

▸ Incident Central Lines10 lines

{6, 924}, {19, 876}, {112, 250}, {230, 231}, {393, 2395}, {512, 1692}, {520, 2451}, {882, 1843}, {1974, 2422}, {2507, 2524}

▸ Functional Relations12 relations

• X₂₄₈₉ = reflection of X_i in X_j for these (i,j): (647, 2485), (2485, 2492), (2524, 2507)

• X₂₄₈₉ = X₁₁₂-Ceva conjugate of X₂₅

• X₂₄₈₉ = cevapoint of X₆₄₇ and X₂₅₁₉

• X₂₄₈₉ = X_i-cross conjugate of X_j for these (i,j): (669, 512), (1084, 1974)

• X₂₄₈₉ = crosspoint of X₂₅ and X₁₁₂

• X₂₄₈₉ = perspector of hyperbola {{A, B, C,X₄,X₂₅}}

• X₂₄₈₉ = intersection of trilinear polars of X₄ and X₂₅

• X₂₄₈₉ = isogonal conjugate of X₄₅₆₃

• X₂₄₈₉ = pole wrt polar circle of trilinear polar of X₆₇₀ (line X₂X₃₉)

• X₂₄₈₉ = polar conjugate of X₆₇₀

• X₂₄₈₉ = barycentric product X₈₁₀₅*X₈₁₀₆

• X₂₄₈₉ = PU(4)-harmonic conjugate of X₃₂₉₁