Things to do with a broken stick

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.

💡 Research Summary

The paper revisits the classic broken‑stick problem—originally posed by Poincaré, which asks for the probability that three random pieces obtained by breaking a unit stick twice can form a triangle—and extends the idea to a variety of geometric quantities associated with a triangle. Instead of the three side lengths, the authors consider nine different triples of triangle‑defining elements: the three medians, the three altitudes, the three excircle radii (or equivalently the three circumradii), the three internal angle bisectors, the distances from the incenter (I) to the vertices, the distances from the circumcenter (O) to the vertices, the three sides themselves, and three other sets of quantities that uniquely (or almost uniquely) determine a triangle. For each triple they ask two questions: (1) what is the probability that three independent, uniformly distributed numbers on (