{"paper":{"title":"New Sense of a Circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Mamuka Meskhishvili","submitted_at":"2019-05-24T18:54:58Z","abstract_excerpt":"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: $$ S_n^{(2m)}=n\\Bigg[(r^2+\\ell^2)^m+\\sum_{k=1}^{[\\frac{m}{2}]} {m\\choose 2k} (r^2+\\ell^2)^{m-2k}(r\\ell)^{2k} {2k\\choose m}\\Bigg]. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.10406","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}