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arXiv · 1507.04033

A strong triangle inequality in hyperbolic geometry

Csaba Biró·Robert C. Powers
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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.

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Csaba Biró, Robert C. Powers. 2015-07-14. A strong triangle inequality in hyperbolic geometry. https://arxiv.org/abs/1507.04033

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