2006 AIME II Problem 9

Circles \mathcal{C}_{1}, \mathcal{C}_{2}, and \mathcal{C}_{3} have their centers at (0,0),(12,0), and (24,0), and
have radii 1,2 , and 4 , respectively. Line t_{1} is a common internal tangent to \mathcal{C}_{1} and \mathcal{C}_{2} and has a positive slope,
and line t_{2} is a common internal tangent to \mathcal{C}_{2} and \mathcal{C}_{3} and has a negative slope.
Given that lines t_{1} and t_{2} intersect at (x, y), and that x=p-q \sqrt{r}, where p, q, and r are
positive integers and r is not divisible by the square of any prime, find p+q+r.