RIOPLATENSE 2007 Problem 2

Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$ , with $E$ and $F$ between $A$ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$ , respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$ , respectively. If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$ , respectively, calculate the measurements of $B P$ and $B S$ .