2001 AIME II Problem 7

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

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.