The triangle inequality is another basic property of distances, but a difficult one to prove directly.

In addition to the points , and , there are six edgepoints involved, two for each line joining the points in pairs. In Euclidean geometry, we can often simplify a difficult proof by applying an appropriate motion to the given configuration of points, lines etc.. Here the approach is the same, but first we must determine just what a motion in the Poincaré universe should be.