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 \(n\)?

(第十九届AIME2 2001 第7题)