Question 4
In the Cartesian plane let $A = (1, 0)$ and $B = (2, 2\sqrt{3}).$ Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P = (x, y)$ be the center of $\triangle ABC.$ Then $x \cdot y$ can be written as $\frac{p\sqrt{q}}{r},$ where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p + q + r.$