Two Non-Congruent Regular Polygons Having Vertices at the Same Distances from the Point

TL;DR

This paper proves the existence of a second non-congruent regular polygon sharing the same set of distances from a point to its vertices, with a unique size determined by these distances, and provides a geometric construction.

Contribution

It introduces a novel geometric property linking two non-congruent regular polygons via shared distance sets from a point, including a construction method.

Findings

Existence of a second non-congruent regular polygon with identical vertex distances from a point.

Unique determination of polygon sizes based on these distances.

Identification of two points with identical distance sets to two such polygons.

Abstract

For the given regular plane polygon and an arbitrary point in the plane of the polygon, the distances from the point to the vertices of the polygon are defined. We proved that there is one more non-congruent regular polygon having the vertices at the same distances from the point. The sizes of both regular polygons are uniquely determined by these distances. In general case, geometrical construction of the second regular polygon is given. It is proved that there are two points in the plane, which separately have the same set of the distances to the vertices of two non-congruent regular polygons with a shared vertex.