Csaba Biró and Robert C. Powers, A strong triangle inequality in hyperbolic geometry,
Forum Geometricorum, 16 (2016) 99--114.

Abstract. 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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