(This uses a bit of physics)
So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their velocities after the collision will be perpendicular.
The proof goes like this: Let $v$ be the starting velocity, and let $v_1$ and $v_2$ be the velocities of the two pool balls after the collision. By conservation of momentum, $v_1+v_2=v$ and by conservation of energy, $|v_1|^2+|v_2|^2=|v|^2$. The first says that $v_1,v_2$ and $v$ form a triangle, and the second tells us that this triangle is right.
Is this still true for hyperbolic pool, pool played on the hyperbolic plane? (See http://www.geometrygames.org/HyperbolicGames/ for a hyperbolic pool app.)