This actually came up as a lemma I need to prove something else, but I'm having a hard time with it. Intuitively I know that if $r$ doesn't intersect, say, the line $AB$, then it must intersect $AC$ and $BC$, otherwise, I wouldn't have a triangle, but I haven't been able to prove it.
EDIT: Sorry for not making it clearer, but I can't use coordinates or anything like that here (I haven't defined what it means for two lines to be parallel yet either).
You will not be able to prove this, because it's false in hyperbolic geometry. (Of course, it's true in Euclidean geometry, so it's independent from neutral geometry.)
Consider the diagram below (in the Poincaré disk model). If $\triangle ABC$ is formed by the three red lines, then the blue line does not intersect any of them.