{"paper":{"title":"On Parametric Spaces of Bicentric Quadrilaterals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Allan J. MacLeod, Arman Shamsi Zargar, Farzali Izadi, Foad Khoshnam","submitted_at":"2016-04-07T18:36:51Z","abstract_excerpt":"In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by $E_a:y^2=x^3+(a^4-4a^3-2a^2-4a+1)x^2+16a^4x$ which have, in general, \\(\\mathbb Z/8\\mathbb Z\\), and in rare cases \\(\\mathbb Z/2\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02104","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"}