How do I show that brouwer's theorem holds for this domain?

Question

Define $\alpha:[0,1]\rightarrow \mathbb{R}^2:t\mapsto (\cos 2\pi t, \sin 2\pi t)$.

Let $\gamma:[0,1]\rightarrow\mathbb{R}^2$ be a loop at $(1,0)$ homotopic with $\alpha$.

Let $D$ be the inside region made by $\gamma$.

Then, how do I show that any continuous function $g:D\rightarrow D$ has a fixed point?