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
If there is such a triangle, then <ACB = γ, where, by the Second Cosine Rule,
(**) cos(γ) = sin(α)sin(β)cosh(c)-cos(α)cos(β).
The inequality follows as cos(γ) ≤ 1. This proves necessity.
Now suppose that (*) holds. Now, sin(α)sin(β)cosh(c)-cos(α)cos(β) ≥ -cos(α)cos(β) > -1.
Thus, we can define γε[0,π) by (**). Then
sin(α)sin(β)(cosh(c)-1) = cos(γ)+cos(β+γ) =
2cos(½(α+β+γ))cos(½(α+β-γ)).
As cosh(c) > 1, the left hand side is positive. Also ½(α+β-γ) > -½γ > = -½π and
½(α+β-γ) < ½(α+β) < = ½π, so cos(½(α+β-γ)) > 0.
It follows that cos(½(α+βγ)) > 0,
so ½(α+β+γ) < ½π. Thus we do have a triangle with angles
(α,β,γ). By the Second Cosine
Rule, the side opposite γ has length c' where cos(γ) = sin(α)sin(β)cosh(c')-cos(α)cos(β).
Comparing this with (**), we get c = c', and so this triangle is of the required type.
The triangle is asymptotic at C if and only if γ=0, i.e. we have equality in (*).
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