What is the probability that a point chosen at random within a square is closer to the centre than to the boundary?
You're probably referring to a square of side length $\displaystyle \frac{a}{2}$ located within the original square of side length $\displaystyle a$.
However, the corner of the inner square is $\displaystyle \frac{a}{2\sqrt{2}}$ units away from the centre whereas it is only $\displaystyle \frac{a}{4}$ units away from the boundary.
So I don't think the above approach is correct.
I will expand slightly on what Plato has said. Think about this:
$\displaystyle \sqrt{x^2 + y^2} = 1 - y \Rightarrow \, ....$
$\displaystyle \sqrt{x^2 + y^2} = 1 - x \Rightarrow \, ....$
Since these parabolas are inverses of each other, they intersect on the line $\displaystyle y = x \, ....$
By symmetry, the required area is four times the required area in one quadrant of the square.