State Brouwer's fixed point theorem on the plane, and also an equivalent version of it concerning continuous retractions. Prove the equivalence of the two statements.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous map with the property that $|f(x)| \leqslant 1$ whenever $|x|=1$. Show that $f$ has a fixed point. [Hint. Compose $f$ with the map that sends $x$ to the nearest point to $x$ inside the closed unit disc.]