Question 16
G4 (AUS-ARM), IMO2. $N$ is an arbitrary point on the bisector of $\angle BAC$. $P$ and $O$ are points on the lines $AB$ and $AN$, respectively, such that $\angle ANP=90^\circ=\angle APO$. $Q$ is an arbitrary point on $NP$, and an arbitrary line through $Q$ meets the lines $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $\angle OQE=90^\circ$ if and only if $QE=QF$.