This theorem can be stated:
Start with any planar quadrilateral. Draw squares outwards from each edge, and draw lines between the centres of opposite squares. The two lines thus drawn will be equal in length, and perpendicular.
Here are some illustrations (from answers.com):
As you see, one of the pleasant things about this theorem is its generality: the quadrilateral does not have to be convex; its edges may cross, and one or two of them can even have zero length. I like this theorem: it has a nice unexpectedness about it (to me, anyway); also it’s not hard to prove. I have a rather uninspiring proof using vectors. I think this would make a nice student project: verifying the theorem for different quadrilaterals, and (for the better students) proving it.
There’s a lovely interactive diagram of this theorem at http://www.mste.uiuc.edu/dildine/geometry/vanaubel.html.