In the Cartesian plane let A=(1,0) and B=(2,2 \sqrt{3}). Equilateral triangle A B C is constructed so that C lies in the first quadrant. Let P=(x, y) be the center of \triangle A B C. 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.