Brouwer Fixed Point Theorem via the Jordan Curve Theorem

There is a proof of the Brouwer Fixed Point Theorem via the Jordan Curve Theorem ?

The Brouwer Fixed Point Theorem. Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . Any continuous function $f:B\rightarrow B$ has a fixed point.

The Jordan curve theorem. The image of a continuous injective mapping $J:S^1\rightarrow \mathbb R^2$ divides the plane into exactly two components, one of which is unbounded and the other bounded. Moreover, both of these components have the image of the mapping $J$ as their boundary.

The Jordan Curve Theorem via the Brouwer Fixed Point Theorem

$\partial B=S^{1}$

Any hints would be appreciated.