Dieter Ruoff - Proportionality in the non-Euclidean plane

The Euclidean proportionality theorems involving an angle that is intersected by a pair of parallel lines do not extend to the hyperbolic plane; on the the one hand the uniqueness of the parallel line and on the other a meaningful concept of similarity are missing. In fact, the proportionality theorems also fail when parallelism is interpreted in the narrower sense of being boundary parallel.

What is well-known is the theorem that a line which bisects two sides of a triangle is hyperparallel to the third side. Taking this as a rudimentary proportionality theorem one can ask whether the pair of lines that trisects two sides of a triangle would still be hyperparallel to the third, etc. As will be shown the proof that this is so is not straightforward but requires some interesting lemmas concerning the hyperbolic plane.