Baltic Way 2022 · Problem 13

Let $ABCD$ be a cyclic quadrilateral with $AB<BC$ and $AD<DC$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $AB=BE$ and $AD=DF$. Let further $M$ denote the midpoint of the segment $EF$. Prove that $\angle BMD=90^\circ$.