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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: $$ S_n^{(2m)}=n\\Bigg[(r^2+\\ell^2)^m+\\sum_{k=1}^{[\\frac{m}{2}]} {m\\choose 2k} (r^2+\\ell^2)^{m-2k}(r\\ell)^{2k} {2k\\choose m}\\Bigg]. $$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.10406","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-05-24T18:54:58Z","cross_cats_sorted":[],"title_canon_sha256":"5f520bb8ee51cdc624a890058ecb852b8ca6f7f64e0bb52e68bb7645c5d98210","abstract_canon_sha256":"6f656c7055cccb9304d7f6fd279f6fd8e344961ecb3e3d1107b23b55f3c98471"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:58.139917Z","signature_b64":"ErcOq+qNcBU6ZKRsNKr+//Cus5O58sqMvSB7xsjVbg7XRT7wdXauOjpuSILMcX15FGQecipYHLx2SntzBSHaBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0845e72d4f671788c29b5db3bccb06981cbfe58877cabf8bec0a29baa16155e","last_reissued_at":"2026-05-17T23:42:58.139485Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:58.139485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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