CGMO 2016 Problem 7

In acute triangle $ABC, AB<AC$ , $I$ is its incenter, $D$ is the foot of perpendicular from $I$ to $BC$ , altitude $AH$ meets $BI,CI$ at $P,Q$ respectively. Let $O$ be the circumcenter of $\triangle IPQ$ , extend $AO$ to meet $BC$ at $L$ . Circumcircle of $\triangle AIL$ meets $BC$ again at $N$ . Prove that $\frac{BD}{CD}=\frac{BN}{CN}$ .

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