incircles and excircles in hyperbolic geometry

 We already know that a hyperbolic triangle ABC has an incircle, and may have up to three excircles. Each centre is determined by angle bisectors (either all internal, or two external). If it exists, the excircle arising from the external bisectors at B and C is the excircle opposite A. Its radius is denoted by RA. The radius of the incircle is R. The cabrijava applet allows you to experiment in the hyperbolic plane. We begin with some results in neutral geometry. lemma (1) Suppose that the incircle of ΔABC meets the sides as shown, and that AC'=x, BC'=y, CA'=z. Then x=s-a, y=s-b, z=s-c. (2) Suppose that the excircle opposite A of ΔABC exists, and meets the sides as shown, and that AC"=u, BC"=v, CA"=w. Then u=s, v=s-c, w=s-b. proof Both parts rely on the fact that, if P lies outside a circle C, then the tangents from P to C are of equal length since the figure is symmetric in the line through P and the centre of C. For (1) we then have x+y=c, y+z=a, z+x =b. Adding these x+y+z=s, and the results follow. (2) is similar. We now specialize to hyperbolic geometry, though there are euclidean analogues. inradius theorem The inradius R of the hyperbolic ΔABC satisfies (1) tanh(R) = sinh(s-a)tan(½α) = sinh(s-b)tan(½β) = sinh(s-c)tan(½γ), and (2) tanh(R) = Δ/2sinh(s), (3) tanh(R) = Φ/4cos(½α)cos(½β)cos(½γ). exradius theorem If the hyperbolic ΔABC has an excircle beyond A, then its radius RA is given by tanh(RA) = sinh(s)tan(½α)= Φ/4cos(½α)sin(½β)sin(½γ). This is easy - the first follows from the fact that d(A,B") = s, by the lemma, and an application of the Tangent Formula to ΔAB"(XA). The second follows from the first by the s-theorem. Notes (1)A simple consequence is that a hyperbolic triangle has two excircles of equal radius if and only if it is isosceles, and hence three if and only if it is equilateral. (2)It also gives a necessary condition for the existence of the excircle, namely that sinh(s)tan(½α) < 1. We wish to establish a necessary and sufficient condition for the existence of excircles. This requires an existence theorem for hyperbolic triangles. the existence theorem for excircles Suppose that the hyperbolic triangle ABC has