The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk
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 SphD(I) =(r/s)2 [ 8R/r−1 −Γ(cos A)−Γ(cos B)−Γ(cos C) ], where r,R,s are the inradius, circumradius, and semiperimeter, and Γ(t)=1/t−(1−t2)/t2arctanh t with continuous values Γ(0)=0 and Γ(±1)=±1. Finally we prove the sharp inequality SphD(I)≤ 1/3+(log 3)/6, with equality if and only if ABC is equilateral.
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BibTeX
@misc{rabinowitz2026probability,
title = {{The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk}},
author = {Stanley Rabinowitz},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27491},
eprint = {2608.27491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27491},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk
AU - Stanley Rabinowitz
PY - 2026
DA - 2026/08/26
PB - arXiv
DO - 10.48550/arxiv.2608.27491
UR - https://arxiv.org/abs/2608.27491
AB - 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 SphD(I) =(r/s)^2 [ 8R/r−1 −Γ(cos A)−Γ(cos B)−Γ(cos C) ], where r,R,s are the inradius, circumradius, and semiperimeter, and Γ(t)=1/t−(1−t^2)/t^2arctanh t with continuous values Γ(0)=0 and Γ(±1)=±1. Finally we prove the sharp inequality SphD(I)≤ 1/3+(log 3)/6, with equality if and only if ABC is equilateral.
ER -