The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk

[2608.27491] The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk Skip to main content Search Submit Donate Log in Mathematics > General Mathematics arXiv:2608.27491 (math) [Submitted on 26 Aug 2026] Title: The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk Authors: Stanley Rabinowitz View PDF HTML (experimental) 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 \[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -\Gamma(\cos A)-\Gamma(\cos B)-\Gamma(\cos C) \right], \] where $r,R,s$ are the inradius, circumradius, and semiperimeter, and \[ \Gamma(t)=\frac1t-\frac{1-t^2}{t^2}\mathrm{arctanh}\ t \] with continuous values $\Gamma(0)=0$ and $\Gamma(\pm1)=\pm1$. Finally we prove the sharp inequality \[\mathrm{SphD}(I)\le \frac13+\frac{\log 3}{6}, \] with equality if and only if $ABC$ is equilateral. Subjects: General Mathematics (math.GM) MSC classes: 60D05 (primary) 51M16 secondary Cite as: arXiv:2608.27491 [math.GM] (or arXiv:2608.27491v1 [math.GM] for this version) https://doi.org/10.48550/arXiv.2608.27491 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Stanley Rabinowitz [ view email ] [v1] Wed, 26 Aug 2026 15:26:22 UTC (12 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GM < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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