Three points uniformly selected on the unit circle form a triangle containing a point $X$ at distance $r \in [0; 1]$ from its center with probability $P(r) = \frac{ 1}{ 4} - \frac{ 3}{ 2 \pi^2}\textrm{ Li}_2(r^2)$, where $\textrm{ Li}_2$ is the dilogarithm function (Jeremy Tan Jie Rui, 2018). In this paper we present an alternative proof of this fact. We also discuss a couple of other geometric probability problems where the dilogarithm function arises.
Categories: math.PR math.MG