{
  "abstract": "Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \\[ M_T(I)=\\frac{2r}π \\log\\left(\\cot\\frac A4\\cot\\frac B4\\cot\\frac C4\\right). \\] Our main result is the sharp inequality \\[ M_T(P)\\le \\frac{p}{π\\sqrt3}\\log(2+\\sqrt3), \\] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11370",
  "author_total": 1,
  "authors": [
    {
      "name": "Stanley Rabinowitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/949h6zt28yqrp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11370",
  "fulltext_url": null,
  "id": "949h6zt28yqrp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11370",
  "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.GM"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7212255482",
  "pdf_url": "https://arxiv.org/pdf/2609.11370",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/949h6zt28yqrp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11370",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11370"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Average Chord Lengths in a Triangle",
  "title_text": "Average Chord Lengths in a Triangle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/949h6zt28yqrp",
  "venue": "arXiv",
  "versions": []
}