Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram? (M. Volchkevich)
Problem
Source: 2008 Oral Moscow Geometry Olympiad grades 10-11 p3
Tags: geometry, parallelogram, midpoints, equal segments