CGMO 2018 Problem 8

Let $I$ be the incenter of triangle $ABC$ . The tangent point of $\odot I$ on $AB,AC$ is $D,E$ , respectively. Let $BI \cap AC = F$ , $CI \cap AB = G$ , $DE \cap BI = M$ , $DE \cap CI = N$ , $DE \cap FG = P$ , $BC \cap IP = Q$ . Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$ .