A strong triangle inequality in hyperbolic geometry

Abstract

For a triangle in the hyperbolic plane, let $\alpha ,\beta ,\gamma$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $\alpha ,\beta ,\gamma$ can be chosen uniformly in the interval $(0,\pi )$ and it is given that $\alpha +\beta +\gamma <\pi$, we show that the strong triangle inequality $a+b>c+h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded.