some existence theorems for hyperbolic triangles

 Suppose that we are given three lengths a,b,c with each less than the sum of the others. Then we can construct a triangle with sides of length a,b,c. We may as well suppose that a ≥b,c. We take a segment XY of length a, and draw circles round X of radius b, and Y of radius c. By the condition on the lengths, these must meet in two points. Either of these is a suitable Z. This, of course works in either euclidean or hyperbolic geometry. From earlier work in neutral geometry, we can deduce the following first existence theorem For a,b,c > 0, there is a hyperbolic triangle with sides of length a,b,c if and only if Δ2(a,b,c) > 0. Suppose instead, that we take three angles α,β,γ ≥ 0 with α + β + γ < π. It is then reasonable to ask whether there is a hyperbolic triangle with the given angles. The (AAA) condition shows that any two such triangles will be hyperbolic congruent, but does not guarantee the existence. If there is such a triangle, then the Second Cosine Rule can be used to calculate the lengths of the sides. Our approach is to use the formulae from the second Cosine Rule to define suitable lengths, and to verify that these give a triangle with the appropriate angles. second existence theorem If α,β,γ > 0, there is a hyperbolic triangle with angles α,β,γ if and only if α+β+γ<π. The first theorem relates to the (SSS) condition, the second to the (AAA) condition. We shall also require a result relating to the (ASA) condition. third existence theorem If α,β,c > 0 and α+β<π, there is a hyperbolic triangle ABC with