New Sense of a Circle

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: $$ S_n^{(2m)}=n\Bigg[(r^2+\ell^2)^m+\sum_{k=1}^{[\frac{m}{2}]} {m\choose 2k} (r^2+\ell^2)^{m-2k}(r\ell)^{2k} {2k\choose m}\Bigg]. $$