How doth the random triangle

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.