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 Δ^{2}(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.
proof
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 <BAC=α, <ABC=β and c = d(A,B)
if and only if (*) sin(α)sin(β)cosh(c)cos(α)cos(β) ≤ 1.
The triangle is asymptotic at C if and only if
we have equality in (*).
proof
