On Some Triads of Homothetic Triangles

To a given reference triangle Δ and three directions α, β, γ we construct four triads of homothetic triangles and investigate relations between their homothetic centers, centroids, midway triangles, medial triangles and areas. 1. Two triads of homothetic triangles Given an arbitrary triangle Δ = ABC with sidelines a, b, c and an ordered set {α, β, γ} of three directions in the plane of Δ. Let p1 = αβγ, p2 = γαβ, p3 = βγα, and p1 = γβα, p2 = βαγ, p3 = αγβ be the even and odd permutations of these directions. Each such permutation p = π1π2π3 defines three lines with directions π1 at A, π2 at B and π3 at C, which are the sidelines of two triads TΔ and TΔ of homothetic triangles Δi = UiViWi and Δi = U iV iW i (i = 1, 2, 3) with angles U, V,W . The assignment of the indexed symbols U, V,W to the vertices of these triangles is chosen so that homologous vertices have the same symbol (see Figure 1).