On two triads of triangles associated with the perpendicular bisectors of the sides of a triangle (Q2929579)

The author of the paper under review defines two triads of triangles associated with a given triangle \( \mathbb{T} = ABC\) and investigates their properties. Denoting the points where the perpendicular bisector of \(BC\) meets \(AB\) and \(AC\) by \(C_b\) and \(B_c\), respectively, and defining \(C_a\), \(A_c\), \(A_b\), and \(B_a\) similarly, the two triads are given by (i) \( \mathbb{T}_a = AA_bA_c\), \( \mathbb{T}_b = B_aBB_c\), \( \mathbb{T}_c = C_aC_bC\), and (ii) \( \mathbb{T}_a' = AB_aC_a\), \( \mathbb{T}_b' = A_bBC_b\), \( \mathbb{T}_c' = A_cB_cC\). For any \(P\) with barycentric coordinates \((x : y : z)\) with respect to \( \mathbb{T}\), the points \(A_P\), \(B_P\), and \(C_P\) are the points having the same barycentric coordinates \((x : y : z)\) with respect to \( \mathbb{T}_a\), \( \mathbb{T}_b\), and \( \mathbb{T}_c\). The triangle \(A_PB_BC_P\) is denoted by \( \mathbb{T} (P)\). The circumcenters of \( \mathbb{T}_a\), \( \mathbb{T}_b\), and \( \mathbb{T}_c\) are denoted by \(O_a\), \(O_b\), and \(O_c\), the centroids by \(G_a\), \(G_b\), and \(G_c\), and the orthocenters by \(H_a\), \(H_b\), and \(H_c\).NEWLINENEWLINEAmong other things, the author studies properties of triangle \( \mathbb{T} (P)\), showing, for example, that it is orthologic to \( \mathbb{T}\), that its orthocenter is the circumcenter of \( \mathbb{T}\), and that the points \(A_P\), \(B_P\), \(C_P\), and \(P\) are concyclic. He also studies properties of triangles \(O_aO_bO_c\), \(G_aG_bG_c\), and \(H_aH_bH_c\), proving, for example, that they are all orthologic to \( \mathbb{T}\), and finding conditions under which each is orthologic to the pedal triangle of \(P\). A parallel study on the second triad in (ii) is made. The main tool used is the calculation of barycentric coordinates of the points involved.