Question 1
Given a scalene acute triangle $ABC$, let $M$ be the midpoint of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D, E$ its intersections with the bisector of angle $BAC$. Points $D', E'$ lie on $\omega$ such that $DED'E'$ is a rectangle. Prove that $D', E'$, $M, N$ lie on a single circle.