I would appreciate either an explanation or a citation to the following statement in Euclidean geometry.
$\overline{\mathit{AB}}$ and $\overline{\mathit{CD}}$ are chords of a circle such that $\overline{\mathit{AD}}$ and $\overline{\mathit{BC}}$ are parallel. If $M$ is the midpoint of $\overline{\mathit{AB}}$ and $N$ is the midpoint of $\overline{\mathit{CD}}$, and if $M$ and $N$ are distinct points, $\overline{\mathit{MN}}$ is also parallel to $\overline{\mathit{AD}}$ and $\overline{\mathit{BC}}$.
Is the center of the circle collinear with $M$ and $N$?