HMMT 2008 E1

Let $A B C D$ be a unit square (that is, the labels $A, B, C, D$ appear in that order around the square). Let $X$ be a point outside of the square such that the distance from $X$ to $A C$ is equal to the distance from $X$ to $B D$, and also that $A X=\frac{\sqrt{2}}{2}$. Determine the value of $C X^{2}$.