1. [2] 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 $C X^{2}$.
Answer:
$\frac{5}{2}$
Since $X$ is equidistant from $A C$ and $B D$, it must lie on either the perpendicular bisector of $A B$ or the perpendicular bisector of $A D$. It turns that the two cases yield the same answer, so we will just assume the first case. Let $M$ be the midpoint of $A B$ and $N$ the midpoint of $C D$. Then, $X M$ is perpendicular to $A B$, so $X M=\frac{1}{2}$ and thus $X N=\frac{3}{2}, N C=\frac{1}{2}$. By the Pythagorean Theorem we find $X C=\frac{\sqrt{10}}{2}$ and the answer follows.