geometry - Flex a square into a circle, and prove...

Let points $A$, $B$, $C$, and $D$ be the vertices on a square. Let $\overline{CD}$'s midpoint be $E$. Flex the square into a circle (so they'll have equal perimeter/circumference), and translate the circle so it touches point $E$ (i.e. if $\overline{CD}$ is the bottom line, then translate the circle up). Prove or disprove whether the points $A$ and $B$ will lie inside the circle or not. Using graphing tools I've determined that points $A$ and $B$ do lie inside the circle, but not by much. Need a nice, simple proof though. Thanks.