Before I introduce the problem let me just say that I am not sure which forum this should go in. I thought about putting it in several forums, but none had a clear advantage so here we are.

Let $\displaystyle l$ and $\displaystyle m$ be distinct hyperbolic lines in the hyperbolic plane. I am supposed to use the projective module to show that there cannot be more than one line orthogonal to both $\displaystyle l$ and $\displaystyle m$.

Unfortunately the theorem that any point-line pair in the hyperbolic plane gives a unique orthogonal line from the point to the line doesn't work. I've used some intuition about triangles and how the sum of the angles of a triangle is less than 180 degrees in hyperbolic geometry to get some intuition as to why this is true, but I am no closer to a proof.