{
  "abstract": "Let $P$ and $Q$ be independent points chosen uniformly from the interior of a nondegenerate triangle $ABC$, and let $I$ be its incenter. We study the probability that the closed disk with diameter $PQ$ contains $I$. In the language of multivariate statistics, this is the spherical depth of $I$ with respect to the uniform distribution on the triangle. We first give an elementary planar form of the normalized cone-measure construction. If $O$ is an interior point of a convex polygon, then the direction from $O$ to a uniformly distributed interior point has the same law as the direction from $O$ to a boundary point whose density on each side is proportional to the distance from $O$ to that side. Consequently, this boundary point is uniform in arclength if and only if the polygon is tangential with incircle center $O$; for a triangle, this characterizes the incenter. Using this transfer principle, we obtain the closed formula \\[\\mathrm{SphD}(I) =\\left(\\frac r s\\right)^2 \\left[ \\frac{8R}{r}-1 -Γ(\\cos A)-Γ(\\cos B)-Γ(\\cos C) \\right], \\] where $r,R,s$ are the inradius, circumradius, and semiperimeter, and \\[ Γ(t)=\\frac1t-\\frac{1-t^2}{t^2}\\mathrm{arctanh}\\ t \\] with continuous values $Γ(0)=0$ and $Γ(\\pm1)=\\pm1$. Finally we prove the sharp inequality \\[\\mathrm{SphD}(I)\\le \\frac13+\\frac{\\log 3}{6}, \\] with equality if and only if $ABC$ is equilateral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27491",
  "author_total": 1,
  "authors": [
    {
      "name": "Stanley Rabinowitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8a86jp803gws.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27491",
  "fulltext_url": null,
  "id": "e8a86jp803gws",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27491",
  "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": "W7204814895",
  "pdf_url": "https://arxiv.org/pdf/2608.27491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8a86jp803gws.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk",
  "title_text": "The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/e8a86jp803gws",
  "venue": "arXiv",
  "versions": []
}