In Thomassen's great exposition of the Jordan-Schoenflies Theorem he proves Lemma 2.3, which is the Jordan Curve Theorem for polygonal curves:
In the proof a disc refers to a closed open ball in the plane. I got stuck on this proof at first, because I asked myself why does one know that one can always go from $q_i$ to the disk $D$? In particular it is not assumed that every one of the (supposedly more than two) regions has $C$ as boundary (one region could have a proper subset of $C$ as boundary for instance).
With a bit more careful look one realizes this has to be a consequence of the curve being simple. It's a minor detail but I thought since I got stuck the claim required at least some more words.
If for some reason one of the $q_i$ can't go to $D$, at least one has a polygonal curve from $q_i$ into some point of the curve $C$, say $p$. Now, we have a tubular neighborhood of $C$, since it is a Jordan curve and and a properly embedded submanifold. We can then (and this is when we must have the simple hypothesis) choose a direction and unambiguously walk along this tubular neighborhood (with a simple polygonal arc) to the part of $C$ where $D$ intersects it.
That is, simpleness of the curve is necessary because then $C$ is a loop graph, and we cover all of $C$ by walking along it. If it were not simple, then we would have trouble unambiguously reaching $D$, or might even never reach it.