A quick way to define the hyperbolic metric in the Poincare disc is via the cross ratio: Given points a,b in the disc, let p,q be the endpoints of the hyperbolic line (halfcircle/line perpendicular to the circle) through a and b. Then the hyperbolic metric is given by
d(a,b) = log [a:b:p:q].
(Depending on the precise definition of cross ratio, the order of the four entries might vary.)
This definitions has many advantages, e.g. it is easy to show that the metric is invariant under Möbius transformations preserving the circle and that the corresponding geodesics are precisely the hyperbolic lines. However, the disadvantage seems to be that the triangle inequality is not quite obvious.
Does anybody know a quick and elegant proof of the triangle inequality for the above definition of the hyperbolic metric?