{"paper":{"title":"The Two Incenters of the Arbitrary Convex Quadrilateral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.HO","authors_text":"Dimitris M. Christodoulou, Nikolaos Dergiades","submitted_at":"2017-11-29T04:12:02Z","abstract_excerpt":"For an arbitrary convex quadrilateral $ABCD$ with area ${\\cal A}$ and perimeter $p$, we define two points $I_1, I_2$ on its Newton line that serve as incenters. These points are the centers of two circles with radii $r_1, r_2$ that are tangent to opposite sides of $ABCD$. We then prove that ${\\cal A}=pr/2$, where $r$ is the harmonic mean of $r_1$ and $r_2$. We also investigate the special cases with $I_1\\equiv I_2$ and/or $r_1=r_2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02207","kind":"arxiv","version":1},"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"}