Question 22
The incircle (with centre $O$) of an isosceles triangle $ABC$ with $AB = AC$ meets $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. Let $N$ be the intersection of the lines $OL$ and $KM$, and let $Q$ be the intersection of the lines $BN$ and $CA$. Let $P$ be the foot of the perpendicular from $A$ to $BQ$. Suppose $BP = AP + 2PQ$. Determine the possible value(s) of $\frac{AB}{BC}$.