A fixed point for the disk

Let $D$ be the closed unit disk in the plane and let $f:D\to D$ be continuous. Show that there is a point $x\in D$ with $f(x)=x$.

Proofs go through the fact that a disk cannot be continuously squeezed onto its own boundary circle, or through a combinatorial lemma about colouring triangulations. Which one do you find more believable?