KöMaL 2023 C1792

In triangle $ABC$ let the midpoints of sides $AB$ and $AC$ be $F$ and $E$, respectively. Let $P$ and $Q$ be two arbitrary points in the plane of triangle $ABC$. Let $P'$ and $Q'$ be the reflection of $P$ and $Q$ across $E$ and $F$, respectively. Let $M$ and $N$ denote the midpoints of line segments $PB$ and $QC$, respectively. Prove that $MN\parallel P'Q'$ and $P'Q'=2MN$.