Isoperimetric point

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.

Abstract: In this paper, we consider Archimedes’ arbelos and the two identical Archimedean circles it contains, and we explore possible generalizations to three-dimensional Euclidean space. Thinking of the line segment joining the centers of the two smaller semicircles in the ordinary arbelos as its base, we devise a three-dimensional configuration having a triangle as base and containing three spheres that seem to play the roles of the two Archimedean circles. In Theorem 3.1, we find formulas for the radii of these spheres (in terms of the base triangle) and conditions under which two of them are equal, and we describe (in the last paragraph of Section 1) how these conditions can be viewed as an honest generalization of the ordinary arbelos theorem. These conditions also imply that the three spheres are equal if and only if the base triangle is equilateral. As a by-product, Theorem 3.4 shows that there is associated with the ordinary arbelos a pair of spheres that amazingly turn out to have the same radii as the Archimedean circles.