X₁₃₀₀

COLLINGS TRANSFORM OF X₁₃₆

X₁₃₀₀ = cevapoint of X₄ and X₁₈₆

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

X₁₃₀₀ (COLLINGS TRANSFORM OF X₁₃₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 110}, {24, 107}

• Related Centers: X₄ (ORTHOCENTER), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER))

Trilinears: \((\sec A)/(1 + \cos 2B + \cos 2C) : :\)

Barycentrics: \(1/((a^{2} - b^{2} - c^{2}) (a^{4} (b^{2} + c^{2}) - 2 a^{2} (b^{4} - b^{2} c^{2} + c^{4}) + (b^{2} - c^{2})^{2} (b^{2} + c^{2})) : :\)

Search Value (6, 9, 13): \(14.115830694140\)

▸ Alternate Coordinates1 form

• Trilinears: \((\sec A)/(-1 + 2cos^{2} B + 2 \cos^{2} C) : :\)

X₁₃₀₀ lies on the circumcircle.

▸ Incident Central Lines13 lines

{2, 131}, {3, 847}, {4, 110}, {20, 254}, {24, 107}, {25, 1302}, {93, 930}, {99, 264}, {109, 225}, {112, 393}, {186, 476}, {403, 1304}, {687, 691}

▸ Functional Relations11 relations

• X₁₃₀₀ = reflection of X_i in X_j for these (i,j): (4, 136), (925, 3)

• X₁₃₀₀ = isogonal conjugate of X₁₃₇₅₄

• X₁₃₀₀ = cevapoint of X₄ and X₁₈₆

• X₁₃₀₀ = X_i-cross conjugate of X_j for these (i,j): (30, 4), (50, 275)

• X₁₃₀₀ = Λ(X₅, X₃₈₉)

• X₁₃₀₀ = anticomplement of X₁₃₁

• X₁₃₀₀ = inverse-in-polar-circle of X₁₁₃

• X₁₃₀₀ = pole wrt polar circle of trilinear polar of X₃₅₈₀

• X₁₃₀₀ = X₄₈-isoconjugate (polar conjugate) of X₃₅₈₀

• X₁₃₀₀ = eigencenter of circumanticevian triangle of X₃

• X₁₃₀₀ = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A, B, C,X₄,X₉₃}} (circumconic centered at X₁₃₆)