APMO 1997 Problem 4

Triangle $A_1 A_2 A_3$ has a right angle at $A_3$ . A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ( $n \geq 3$ ), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$ . (a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$ , $n \geq 3$ . (b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$ .