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arXiv · 2601.02929

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

Abdulamin Ismailov
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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.

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Abdulamin Ismailov. 2026-01-06. With what probability does an inscribed triangle contain a given point?. https://arxiv.org/abs/2601.02929

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