X₁₂₉₇

COLLINGS TRANSFORM OF X₁₂₇

X₁₂₉₇ = X₂₃₂-cross conjugate of X₂

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

X₁₂₉₇ (COLLINGS TRANSFORM OF X₁₂₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 107}, {3, 112}

• Related Centers: X₂₃₂ (-CEVA CONJUGATE OF X₁₃₂), X₂ (CENTROID), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₃ (CIRCUMCENTER), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT))

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a/(b^{6} + c^{6} - 2a^{6} + a^{4}b^{2} + a^{4}c^{2} - b^{4}c^{2} - b^{2}c^{4})\) (M. Iliev, 5/13/07)

Search Value (6, 9, 13): \(13.765188509881\)

X₁₂₉₇ lies on the circumcircle, the hyperbolas {{A, B, C,X₄,X₂₂}} and {{A, B, C,X₂,X₃}}.

▸ Incident Central Lines11 lines

{2, 107}, {3, 112}, {4, 127}, {20, 99}, {22, 110}, {23, 1304}, {25, 1073}, {30, 935}, {97, 933}, {108, 1214}, {476, 858}

▸ Functional Relations21 relations

• X₁₂₉₇ = reflection of X_i in X_j for these (i,j): (4, 127), (112, 3)

• X₁₂₉₇ = X₂₃₂-cross conjugate of X₂

• X₁₂₉₇ = cevapoint of X₃ and X₅₁₁

• X₁₂₉₇ = crosssum of X₂₀ and X₁₄₇

• X₁₂₉₇ = isogonal conjugate of X₁₅₀₃

• X₁₂₉₇ = isotomic conjugate of X₃₀₇₃₇

• X₁₂₉₇ = complement of X₁₂₃₈₄

• X₁₂₉₇ = anticomplement of X₁₃₂

• X₁₂₉₇ = trilinear pole of line X₆X₅₂₀

• X₁₂₉₇ = Ψ(X₆, X₅₂₀)

• X₁₂₉₇ = Λ(X₄, X₆)

• X₁₂₉₇ = Λ(X₉₈, X₂₃₀)

• X₁₂₉₇ = X₁₃₉-of-hexyl triangle

• X₁₂₉₇ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₂₂

• X₁₂₉₇ = Cundy-Parry Phi transform of X₈₇₄₃

• X₁₂₉₇ = Cundy-Parry Psi transform of X₁₄₃₇₆

• X₁₂₉₇ = Thomson-isogonal conjugate of X₅₂₅

• X₁₂₉₇ = Lucas-isogonal conjugate of X₅₂₅

• X₁₂₉₇ = X₄-of-1st-anti-orthosymmedial-triangle

• X₁₂₉₇ = SR(P,U), where P and U are the circumcircle intercepts of the van Aubel line

• X₁₂₉₇ = orthoptic-circle-of-Steiner-circumellipse-inverse of X₃₄₁₈₆