Average Chord Lengths in a Triangle

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 $$M_T(I)=\frac{2r}{\pi }\log (\cot \frac{A}{4}\cot \frac{B}{4}\cot \frac{C}{4})\mathrm{.}$$ Our main result is the sharp inequality $$M_T(P)\le \frac{p}{\pi \sqrt{3}}\log (2+\sqrt{3}),$$ 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.