Two Regular Polygons with a Shared Vertex

TL;DR

This paper establishes a method to find points equidistant from vertices of two regular polygons sharing a vertex, proves the existence of such points, and generalizes Bottema's theorem for any two regular polygons.

Contribution

It introduces a new method for locating equidistant points, proves their existence for polygons sharing a vertex, and generalizes Bottema's theorem to all regular polygons.

Findings

Existence of two equidistant points for polygons sharing a vertex

A method to find points equidistant from polygon vertices

Generalization of Bottema's theorem for regular polygons

Abstract

For two non-congruent regular polygons of the same type, the method of finding the points in the plane at the equal distances to the vertices, is established. The existence of two points with this property is proved for two polygons with a shared vertex. For one of them, it is proved that it satisfies the Bottema theorem conditions and based on this, the generalized Bottema theorem for any two regular polygons is given.