ELMO 2025 Problem 4

In quadrilateral $ABCD$ with $\angle BAD=\angle BCD$ , diagonal $AC$ intersects the circumcircle of $\triangle BCD$ at a point $P \neq C$ . Let $Q$ , $R$ , $S$ and $T$ be the reflections of $P$ across $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ , respectively. Prove that the circumcircle of $\triangle AQT$ is tangent to line $RS$ .

Benny Wang