Question 35
Let $ABC$ be a triangle and $I$ its incentre. The circle with centre $I$ passing through $B$ meets $AC$ at the points $E$ and $F$, with $E$ and $F$ between $A$ and $C$ and distinct from them. The circumcircle of triangle $IEF$ meets the segments $EB$ and $FB$ at $Q$ and $R$, respectively. The line $QR$ meets the sides $AB$ and $BC$ at $P$ and $S$, respectively.
If $a$, $b$ and $c$ are the lengths of the sides $BC$, $CA$ and $AB$, respectively, compute the lengths of $BP$ and $BS$.
If $a$, $b$ and $c$ are the lengths of the sides $BC$, $CA$ and $AB$, respectively, compute the lengths of $BP$ and $BS$.