{"paper":{"title":"Triangles and groups via cevians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"\\'Arp\\'ad B\\'enyi, Branko \\'Curgus","submitted_at":"2011-09-02T21:04:03Z","abstract_excerpt":"For a given triangle $T$ and a real number $\\rho$ we define Ceva's triangle $\\CT_\\rho(T)$ to be the triangle formed by three cevians each joining a vertex of $T$ to the point which divides the opposite side in the ratio $\\rho:(1-\\rho)$. We identify the smallest interval $\\nM_T \\subset \\nR$ such that the family $\\CT_\\rho(T), \\rho\\in \\nM_T$, contains all Ceva's triangles up to similarity. We prove that the composition of operators $\\CT_\\rho, \\rho \\in \\nR$, acting on triangles is governed by a certain group structure on $\\nR$. We use this structure to prove that two triangles have the same Brocar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0557","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"}