{
  "abstract": "We study a variant of Napoleon's construction on the unit sphere, in which the triangles erected on the sides of a spherical triangle $ABC$ are required to have similar isosceles chord triangles with apex angle $β\\in(0,π/2)$. For a non-equilateral $ABC$, we prove that the triangle $XYZ$ of spherical circumcentres is equilateral if and only if $\\cos(θ_a+θ_b+θ_c\\pmΩ/2)=0$, where $θ_a,θ_b,θ_c$ are the base angles of the isosceles spherical triangles $XBC$, $YCA$, $ZAB$, $Ω$ is the area of $ABC$, and the sign $+$ (resp.~$-$) corresponds to the outward (resp.~inward) construction. We derive from this an explicit classification in terms of two quadrics and a cubic inequality. It recovers the theorem of Dipierro, Noakes and Valdinoci for $β=π/3$, and exhibits three families of solutions for $β>π/3$, together with a fourth one when $\\cosβ<1/4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21408",
  "author_total": 1,
  "authors": [
    {
      "name": "Quang Hung Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6p0b06g8ff4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21408",
  "fulltext_url": null,
  "id": "d6p0b06g8ff4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21408",
  "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.MG"
    ],
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7222495467",
  "pdf_url": "https://arxiv.org/pdf/2610.21408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6p0b06g8ff4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Napoleonic triangles on the sphere from similar chord triangles",
  "title_text": "Napoleonic triangles on the sphere from similar chord triangles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/d6p0b06g8ff4p",
  "venue": "arXiv",
  "versions": []
}