{
  "abstract": "In this paper we study Euclidean triangles associated with Diophantine $D(4)$-triples. For a $D(4)$-triple $\\{a,b,c\\}$, $a<b<c$, we prove that the lengths $r=\\sqrt{ab+4}$, $s=\\sqrt{ac+4}$ and $t=\\sqrt{bc+4}$ associated with this $D(4)$-triple form a non-degenerate triangle if and only if the triple is regular and $r>b-a$. For a fixed shortest side $r$, we describe all triangles associated with $D(4)$-triples in terms of the factorisations of $r^2-4$, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with $D(4)$-triples are necessarily congruent. Finally, for the family of $D(4)$-triples $\\{r-2,r+2,4r\\}$, we obtain a Pell-equation characterisation of the Heronian triangles associated with these $D(4)$-triples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38544",
  "author_total": 2,
  "authors": [
    {
      "name": "Marija Bliznac Trebješanin"
    },
    {
      "name": "Jelena Pleština"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59rb06zzvsmtz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38544",
  "fulltext_url": null,
  "id": "59rb06zzvsmtz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38544",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7215048169",
  "pdf_url": "https://arxiv.org/pdf/2609.38544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59rb06zzvsmtz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Euclidean Triangles Associated with Diophantine $D(4)$-triples",
  "title_text": "Euclidean Triangles Associated with Diophantine D(4)-triples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/59rb06zzvsmtz",
  "venue": "arXiv",
  "versions": []
}