Problem

Source: First Romanian JBMO TST 2023 P2

Tags: geometry, ROMANIA JBMO TST



Given is a triangle $ABC$. Let the points $P$ and $Q$ be on the sides $AB, AC$, respectively, so that $AP=AQ$, and $PQ$ passes through the incenter $I$. Let $(BPI)$ meet $(CQI)$ at $M$, $PM$ meets $BI$ at $D$ and $QM$ meets $CI$ at $E$. Prove that the line $MI$ passes through the midpoint of $DE$.