New Sense of a Circle

TL;DR

This paper establishes a new condition under which the locus of points, based on the sum of even powers of distances to a regular polygon's vertices, forms a circle, providing explicit formulas for the radius.

Contribution

It introduces a novel geometric condition for circle loci related to sums of even powers of distances to polygon vertices, expanding understanding of geometric loci.

Findings

Locus is a circle when the sum exceeds a specific threshold.

Derived explicit formula for the circle's radius.

Provides conditions for the sum of powers to be constant.

Abstract

New condition is found for the set of points in the plane, for which the locus is a circle. It is proved: the locus of points, such that the sum of the $(2m)$-th powers $S_n^{(2m)}$}of the distances to the vertexes of fixed regular $n$-sided polygon is constant, is a circle if $$ S_n^{(2m)}>nr^{2m},\ {\rm where}\ m=1,2,\dots,n-1 $$ and $r$ is the distance from the center of the regular polygon to the vertex. The radius $\ell$ satisfies: $$ S_n^{(2m)}=n\Bigg[(r^2+\ell^2)^m+\sum_{k=1}^{[\frac{m}{2}]} {m\choose 2k} (r^2+\ell^2)^{m-2k}(r\ell)^{2k} {2k\choose m}\Bigg]. $$