A rectangle question

If you take any rectangle, and take the midpoint of each side, and connect them in order, the result is a rhombus (quadrilateral with four equal sides). Cool, and pretty easy to show. (Lots of options – maybe the most accessible is to use the Pythagorean Theorem, since we have right angles, four times, and get four equal hypotenuses)

But what if someone gave you a rhombus, and told you that they formed the rhombus by connecting the four midpoints of some quadrilateral, BUT ONE THAT IS NOT A RECTANGLE. Could they be correct?

(Inspired by Patrick Honner’s cool post on proving the Varignon Theorem, with details that were new to me, including the name of the theorem!)

Below, precinct house around the corner from me. The boxes below the windows show the midpoints of rectangles forming rhombuses.