Some open problems in concurrences

摘要

An open problem In the list of 12 open problems posed on pages 132-133 of a recent book [1] Problem 4 states, or rather conjectures: If G_A, G_B and G_C are the orthogonal projections of the centroid G of triangle ABC on the sidelines BC, CA and AB, respectively, and if AG_A, BG_B and CG_C are concurrent, then ABC is isosceles. (*) The problem is attributed to Temistocle B?rsan. Needless to say, the converse is trivial and follows from symmetry, i.e., if ABC is isosceles, then AG_A, BG_B and CG_C are concurrent. In this note, we provide a proof of the problem in (*), and we place it in a more general context.