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:
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.