Let $ABCD$ and $AB'C'D'$ be two squares with common vertex $A$. Let $E$ and $G$ be the midpoints of $B'D$ and $D'B$ respectively, and let $F$ and $H$ be the centers of the two squares. Then the quadrilateral $EFGH$ is a square as well

Let $ABCD$ and $AB'C'D'$ be two squares with common vertex $A$. Let $E$ and $G$ be the midpoints of $B'D$ and $D'B$ respectively, and let $F$ and $H$ be the centers of the two squares. Then the quadrilateral $EFGH$ is a square as well