EGMO 2025 Problem 4

Let $A B C$ be an acute triangle with incentre $I$ and $A B \neq A C$ . Let lines $B I$ and $C I$ intersect the circumcircle of $A B C$ at $P \neq B$ and $Q \neq C$ , respectively. Consider points $R$ and $S$ such that $A Q R B$ and $A C S P$ are parallelograms (with $A Q \parallel R B$ , $A B \parallel Q R$ , $A C \parallel S P$ , and $A P \parallel C S$ ). Let $T$ be the point of intersection of lines $R B$ and $S C$ . Prove that points $R$ , $S$ , $T$ , and $I$ are concyclic.