A strong triangle inequality in hyperbolic geometry

For a triangle in the hyperbolic plane, let $α,β,γ$ 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 $α,β, γ$ can be chosen uniformly in the interval $(0,π)$ and it is given that $α+β+γ<π$, 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.

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