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, the angles at X, Y, Z are all zero, and the sides are all of infinite hyperbolic length. In fact, all such triangles are hyperbolic congruent. This may be regarded as the analogue of the various conditions for congruence of finite triangles such as (SSS) or (AAA). A proof is given below. As remarked earlier, each hyperbolic triangle has a hyperbolic incircle. Since all the trebly asymptotic triangles are congruent, this incircle must have a fixed radius. The triangle whose vertices are the points of contact of this circle must be equilateral (as we shall show). The angles and sides of this finite triangle are rather interesting.

properties of trebly asymptotic triangles
(1) All trebly asymptotic triangles are hyperbolic congruent.
(2) A trebly asymptotic triangle has a hyperbolic incircle.
(3) If XYZ is a trebly asymptotic triangle and the incircle touches YZ at A,
ZX at B and XY at C, then ABC is equilateral. The excircles of ΔABC are
horocycles. Each side of ABC has length a, where sinh(½a) = ½, and each
angle is α, where cos(α)=3/5. The hyperbolic radius of the incircle is r,
where tanh(r) = ½.

proof

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.

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