OMCC 2016 Problem 6

Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$ . Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$ , the line parallel to $MN$ through $I$ cuts $AB$ , $AC$ in $P$ and $Q$ . Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.