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 $$S_n^{(2m)}>n{r}^{2m},\mathrm{where}m=1,2,\dots ,n-1$$ and $r$ is the distance from the center of the regular polygon to the vertex. The radius $\ell$ satisfies: $$S_n^{(2m)}=n[(r^2+{\ell }^2{)}^m+\sum _{k=1}^{[\frac{m}{2}]}\left(\genfrac{}{}{0ex}{}{m}{2k}\right)(r^2+{\ell }^2{)}^{m-2k}(r\ell {)}^{2k}\left(\genfrac{}{}{0ex}{}{2k}{m}\right)].$$

38 pages