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 \[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -\Gamma(\cos A)-\Gamma(\cos B)-\Gamma(\cos C) \right], \] 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.