2013 AIME II Problem 4

In the Cartesian plane let A = (1,0) and B = \left( 2, 2\sqrt{3} \right). 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 \tfrac{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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.