With what probability does an inscribed triangle contain a given point?
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914236100247552 |
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| author | Ismailov, Abdulamin |
| author_facet | Ismailov, Abdulamin |
| contents | 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 $P(r) = \frac{1}{4} - \frac{3}{2 π^2}\textrm{Li}_2(r^2)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02929 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | With what probability does an inscribed triangle contain a given point? Ismailov, Abdulamin Probability Metric Geometry 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 $P(r) = \frac{1}{4} - \frac{3}{2 π^2}\textrm{Li}_2(r^2)$, 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. |
| title | With what probability does an inscribed triangle contain a given point? |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2601.02929 |