{"paper":{"title":"An optimization problem for triangles","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Optimizing the product of distances from an interior point to a triangle's vertices falls into one of two cases, fully specified for isosceles triangles.","cross_cats":[],"primary_cat":"math.MG","authors_text":"Kevin Tran, Tommy Murphy","submitted_at":"2026-05-13T04:32:48Z","abstract_excerpt":"We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For isosceles triangles, we explicitly show exactly when both cases occur.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The optimization problem admits exactly two possible cases in general.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Optimizing the product of distances from a point to a triangle's vertices has two cases, resolved explicitly for isosceles triangles.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Optimizing the product of distances from an interior point to a triangle's vertices falls into one of two cases, fully specified for isosceles triangles.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2a0d5aa311bbe9c96c6c875ed7c1419e2298f04ae4f92b96dea7bd572ee1d52f"},"source":{"id":"2605.12985","kind":"arxiv","version":1},"verdict":{"id":"524860be-cf94-481e-b0bf-3077200edab2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:15:19.561863Z","strongest_claim":"For isosceles triangles, we explicitly show exactly when both cases occur.","one_line_summary":"Optimizing the product of distances from a point to a triangle's vertices has two cases, resolved explicitly for isosceles triangles.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The optimization problem admits exactly two possible cases in general.","pith_extraction_headline":"Optimizing the product of distances from an interior point to a triangle's vertices falls into one of two cases, fully specified for isosceles triangles."},"references":{"count":6,"sample":[{"doi":"","year":1994,"title":"Kimberling, C.Central Points and Central Lines in the Plane of a TriangleMath. Mag. 67 (1994), no. 3, 163–187","work_id":"68b8225f-823e-486a-850c-bc0f6f56ad8d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1941,"title":"and Robbins, H.What Is Mathematics?, 2nd ed","work_id":"cf813547-88c7-42c0-960f-19677c7f6b6e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"Hajja, M.An Advanced Calculus Approach to Finding the Fermat Point, Math. Mag., Vol. 67 (1994), no. 1, 29–34","work_id":"abd08101-025a-4233-b681-ae9dd4b1dca8","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Problems, Hints and Solutions, unpublished","work_id":"c6f039f8-e791-4b33-9cf3-68ffc623f98d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"A.,Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp","work_id":"97e840f1-541c-47f0-a4aa-d3e93b9b0487","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":6,"snapshot_sha256":"fd02a5d7fe5e93f040b4b8481cc8007248686df35c6b503bfbc2c2b59ea3a882","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"4b0dde7139e3fcb32b57048a1afc43eb2e52b59ba8aea81f9a0a66d3f487bf78"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}