I came across this problem in Alfred Posamentier’s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.

In *any* convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.

See the Magic Parallelogram.

**(Update 5/15/2020)** Matthew Scroggs has a different proof of the Magic Parallelogram for The Aperiodical’s Big Math Lockdown contest.