Concentric Circles Each Passing Through One Vertex of Each of Two Regular Polygons

Given a regular $n$-gon on the plane, it is evident that from any point on the plane, taken as a center, one can draw $n$ concentric circles such that each circle passes through one of the vertices of the polygon. Naturally, this raises the problem of whether such a construction is possible for any two given regular $n$-gons on the plane. In this paper, we establish the necessary and sufficient conditions for the existence of $n$ concentric circles such that each circle passes through one vertex of each of the two regular $n$-gons. Keywords and phrases: Polygonal distances, cyclic averages, concentric circles, two regular polygons, two equilateral triangles, two squares