Fundamental Domain for Congruence (mod 2) Group

Question

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the Fuchsian group $\Gamma \leq PSL(2, \mathbb{R})$ consisting of integer matrices that are congruent to the identity mod 2. I'm pretty sure I can show that this set contains a fundamental domain. Now I need to show that no element of this set is translated into the set by an element of $\Gamma$. From the examples I've seen, it seems the proof of such things is usually annoyingly ad hoc. Also, if you have a better tag for this, please feel free to add it or to subtract an inappropriate tag.