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

TL;DR

This paper provides an alternative proof for the probability that a triangle inscribed in a circle contains a specific point at a given distance from the center, involving the dilogarithm function, and explores related geometric probability problems.

Contribution

It offers a new proof of a known probability result and discusses additional geometric probability problems involving the dilogarithm function.

Findings

Probability formula involving dilogarithm function

Alternative proof of the inscribed triangle probability

Connection to other geometric probability problems

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 $P(r) = \frac{1}{4} - \frac{3}{2 \pi^2}\textrm{Li}_2(r^2)$, 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.