X₁₂₉₉

COLLINGS TRANSFORM OF X₁₃₅

X₁₂₉₉ = reflection of X₄ in X₁₃₅

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

X₁₂₉₉ (COLLINGS TRANSFORM OF X₁₃₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 135}

• Related Centers: X₄ (ORTHOCENTER), X₁₃₅ (INTERSECTION OF LINE X)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X₁₂₈₆)

Search Value (6, 9, 13): \(6.455078269818\)

X₁₂₉₉ lies on the circumcircle.

▸ Incident Central Lines5 lines

{4, 131}, {24, 110}, {99, 317}, {403, 476}, {459, 1302}

▸ Functional Relations5 relations

• X₁₂₉₉ = reflection of X₄ in X₁₃₅

• X₁₂₉₉ = Λ(X₄, X₁₅₅)

• X₁₂₉₉ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A, B, C,X₄,X₂₄}} (circumconic centered at X₁₃₅)

• X₁₂₉₉ = isogonal conjugate of crosspoint of X₁₅₅ and X₂₉₃₁ wrt both the excentral and tangential triangles

• X₁₂₉₉ = inverse-in-polar-circle of X₁₃₁