Things to do with a broken stick

TL;DR

This paper explores the probabilities associated with various triangle elements derived from a broken stick model, analyzing their likelihood of forming specific triangle types and comparing geometric properties.

Contribution

It extends the broken stick problem to various triangle elements, calculating probabilities and analyzing the likelihood of resulting in acute or obtuse triangles.

Findings

Probabilities for medians, altitudes, and other elements are computed.

Comparison of probabilities for triangles being acute versus obtuse.

Provides a unified probabilistic framework for different triangle elements.

Abstract

We use the idea of the broken stick problem (which goes back to Poincare) and calculate the corresponding probabilities for the cases in which the three broken part are: the medians in a triangle, the altitudes, radii of excircles, angle bisectors, distances from I or O to the vertices, respectively sides, and some other three elements in a triangle which determine (more or less uniquely) the triangle. For each case we also look at the probability that the triangle that is (more or less uniquely) defined by the elements, being acute and compare to that of being obtuse.