With what probability does an inscribed triangle contain a given point?

Abstract

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 $P(r)=\frac{1}{4}-\frac{3}{2\ddot{I}{€}^2}{\text{Li}}_2(r^2)$, where ${\text{Li}}_2$ 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.