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

TL;DR

This paper determines the precise conditions under which two regular polygons can be simultaneously intersected by a set of concentric circles, each passing through one vertex of each polygon.

Contribution

It establishes the necessary and sufficient conditions for two regular polygons to be intersected by concentric circles passing through their vertices.

Findings

Derived conditions for two regular polygons to share concentric circles passing through vertices

Extended the problem from a single polygon to pairs of polygons

Provided geometric criteria applicable to polygons like triangles and squares

Abstract

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