A strong triangle inequality in hyperbolic geometry
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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