Yesterday Patrick Honner posted a nice illustration of Varignon’s Theorem by one of his students:
It is particularly fun to move the points around to form a non-convex quadrilateral and see that the midpoints still form a parallelogram.
As with many advanced concepts in geometry, my introduction to Varignon’s theorem came from Geometry Revisited by Coxeter and Greitzer. I remember the theorem partially because of the lovely introductory statement in the book:
“The following theorem is so simple that one is surprised to find its date of publication to be as late as 1731. It is due to Pierre Varignon (1654 – 1722).
Theorem 3.11. The figure formed when the midpoints of the sides of a quadrangle are joined in order is a parallelogram, and its area is half that of the quadrangle.”
The chapter also presents three other wonderful theorems and some super problems including this one:
“1. [Show that] the perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrangle.”
So, this special parallelogram has some really interesting properties!
As a fun follow up this morning, my older son was working on some problems from the 2006 AMC 8. Looking over the test I noticed that problem #5 was a simple example of Varignon’s theorem:
Problem #5 from the 2006 AMC 8
I chose a different problem to go through with my older son, but thought my younger son would like this one. It was a challenge, but he eventually was able to work through it. It is kind of fun to think of this basic problem as one that opens the door to this beautiful theorem.
Mike's Math Page 
It was cool to see his conceptual breakthrough at the end. With these questions I’m always curious what the instinct is before trying to dive in and apply area formulas. Also AMC8 is definitely improved by removing the multiple choices.