DE-BUNDESWETTBEWERB 2025 Problem 7

Two circles $k_1$ with center $M_1$ and $k_2$ with center $M_2$ intersect such that their two intersection points and their two centers are four vertices of a square. A point $P$ lies on $k_1$ , and two other distinct points $Q$ and $R$ lie on $k_2$ such that the segments $PQ$ and $PR$ have the same length as the segment $M_1M_2$ and $P$ and $R$ lie on the same side of the line $M_1M_2$ . Let the midpoints of the segments $PQ$ and $PR$ be denoted by $S$ and $T$ , respectively. Prove: The triangles $\triangle PSM_1$ and $\triangle PTM_2$ are similar.