A trebly asymptotic triangle is on with vertices X, Y, Z on the boundary. The CabriJava window on the right allows you to experiment. It appears that such triangles come in a variety of shapes and sizes. You can drag X and Y round the boundary in any way.
In any position,
In fact, all such triangles are hyperbolic congruent. This may be regarded
As remarked earlier, each hyperbolic triangle has a hyperbolic incircle.

Since sinh(x) = ½(exp(x)exp(x)), the length a in (3) above satisfies the
equation exp(½a) + exp(½a) = 1, so exp(½a) = φ, the golden section.
Thus a = 2ln(φ).
As cos (α) = 3/5, α is the larger acute angle of a euclidean (3,4,5) triangle.