{
  "abstract": "We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $θ$, with equal perimeter and equal area. Writing $ρ=\\cos(θ)\\in\\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $ρ$, a family of curves $C_ρ$ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for $ρ=\\frac{1}{2}$, while $ρ=\\frac{47}{49}$ admits infinitely many genuine pairs. We completely parametrize two natural one-parameter families in which the $θ$-triangle is isosceles, and prove several density results for values of $ρ$ admitting one or more genuine pairs. For thirty-five explicit values of $ρ$, Chabauty--Coleman computations determine $C_ρ(\\mathbb{Q})$ completely and show that the corresponding pair is unique up to homothety. We also study multiplicity when one triangle is fixed; the condition for a scalene $θ$-triangle to have two isosceles partners leads to an elliptic $K3$ surface and yields a dense set of such values of $ρ$. Finally, we construct infinite families with at least four genuine pairs and exhibit examples carrying at least five, and in two cases at least six, genuine pairs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28020",
  "author_total": 7,
  "authors": [
    {
      "name": "Jesse Allen"
    },
    {
      "name": "Olivier Beaudry-Ogden"
    },
    {
      "name": "Théophane Bouchard"
    },
    {
      "name": "Matéo Frély"
    },
    {
      "name": "Matilde Lalín"
    },
    {
      "name": "Abel-Jimmy Oyono-Montoki"
    },
    {
      "name": "Berend Ringeling"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a2mzj63dra5j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28020",
  "fulltext_url": null,
  "id": "2a2mzj63dra5j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28020",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214222091",
  "pdf_url": "https://arxiv.org/pdf/2609.28020",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a2mzj63dra5j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28020",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28020"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area",
  "title_text": "Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2a2mzj63dra5j",
  "venue": "arXiv",
  "versions": []
}