NMC 1998 Problem 2

Let $C_{1}$ and $C_{2}$ be two circles intersecting at $A$ and $B$. Let $S$ and $T$ be the centres of $C_{1}$ and $C_{2}$, respectively. Let $P$ be a point on the segment $A B$ such that $|A P| \neq|B P|$ and $P \neq A, P \neq B$. We draw a line perpendicular to $S P$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_{1}$. We likewise draw a line perpendicular to TP through $P$ and denote by $E$ and $F$ the points at which this line intersects $C_{2}$. Show that $C, D, E$, and $F$ are the vertices of a rectangle.