Consider a square $ABCD$. Let $M$ be the midpoint of segment $AB$, $\Gamma_1$ be the circle tangent to $\overline{AD}$, $\overline{AM}$ and $\overline{MC}$ with radius $r > 0$ and let $\Gamma_2$ be the circle tangent to $\overline{AD}$, $\overline{DC}$ and $\overline{MC}$ with radius $R > 0$. Prove that $R =\frac{2r}{r+1}$.