A point is chosen at random from the interior of square ABCD. What is the probability that the point is closer to either A or C than it is to the midpoint of the diagonal connecting A and C.

My attempt at solving the problem follows.

Let $\displaystyle x$ be the length of a side of the square. Then, there are $\displaystyle x^2$ points inside of the square. Furthermore, the midpoint of the diagonal connecting A and C is at the point $\displaystyle (\dfrac{x}{2},\dfrac{x}{2})$.

For my square, point A is at point $\displaystyle (0,x)$ and point C is at point $\displaystyle (x,0)$. I found the distances between points A and C, point A and the midpoint, and point C and the midpoint. However, I don't think I am approaching the solution.