DE-MO 2008 Problem 2

The triangle $\triangle SFA$ has a right angle at $F$ . The points $P$ and $Q$ lie on the line $SF$ such that the point $P$ lies between $S$ and $F$ and the point $F$ is the midpoint of the segment $[PQ]$ . The circle $k_1$ is th incircle of the triangle $\triangle SPA$ . The circle $k_2$ lies outside the triangle $\triangle SQA$ and touches the segment $[QA]$ and the lines $SQ$ and $SA$ . Prove that the sum of the radii of the circles $k_1$ and $k_2$ equals the length of $[FA]$ .