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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