2001 AIME II Problem 7

Let \triangle P Q R be a right triangle with P Q=90, P R=120, and Q R=150. Let C_{1} be the inscribed circle. Construct \overline{S T}, with S on \overline{P R} and T on \overline{Q R}, such that \overline{S T} is perpendicular to \overline{P R} and tangent to C_{1}. Construct \overline{U V} with U on \overline{P Q} and V on \overline{Q R} such that \overline{U V} is perpendicular to \overline{P Q} and tangent to C_{1}. Let C_{2} be the inscribed circle of \triangle R S T and C_{3} the inscribed circle of \triangle Q U V. The distance between the centers of C_{2} and C_{3} can be written as \sqrt{10 n}. What is n ?