On the set of points "inside" a closed curve

Question

consider the following: one has a simple closed rectifiable curve $\gamma$ in the plane, and there is a point $a$ such that for all $p\in\gamma$ the segment $\overline{ap}$ intersects $\gamma$ only in $p$. Let $G$ be the set of points which are inside $\gamma$, i.e. those points $q$ such that the segment $\overline{aq}$ doesn't intersect $\gamma$. Then one has that $G$ is an open and connected set with boundary equal to $\gamma$.
Up to now I managed to prove openness, connectedness and that $\gamma$ is contained in the boundary. However I am quite clueless about how to prove the reverse inclusion, does somebody have any suggestion?
Thanks!