How doth the random triangle

TL;DR

This paper investigates the probability of randomly chosen triangles being obtuse in Euclidean spaces of various dimensions, establishing universal lower bounds and constructing specific distributions to demonstrate these bounds.

Contribution

It provides the first universal lower bounds on the probability of obtuse triangles in any Euclidean dimension and constructs distributions achieving these bounds.

Findings

Planar obtuse triangles have at least a 1/3 probability.

A distribution exists with a 4/9 probability of obtuse triangles.

Higher dimensions allow for even lower probabilities of obtuse triangles.

Abstract

Charles L. Dodgson, also known as Lewis Carroll, in his book "Pillow problems" from 1893 asked for the likelihood of a random triangle to be obtuse. Clearly, the answer to Dodgson's question depends strongly on the assumed random distribution. In this article, we show nevertheless that there are certain fundamental limitations imposed by the geometry of Euclidean space. Specifically, we give universal lower bounds for how improbable obtuse triangles can be, if drawn from a distribution in $\mathbb{R}^d$. We prove that planar obtuse triangles cannot be less likely than 1/3, and construct a distribution for which the probability is 4/9. Analogous results are provided in three and higher dimensions, where obtuse triangles can be increasingly less likely. Sharpness of the lower bounds are left as open problems.