Intersections of random chords of a circle

Abstract

Where are the intersection points of diagonals of a regular $n$-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

Unrefereed draft, final version to appear in The American Mathematical Monthly