The isoperimetric point and the point(s) of equal detour in a triangle

Abstract

A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly 92 (1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits the relations of these points to the centers of the Soddy circles.