This is one of the cutest pieces of geometry I know.  It has all the right ingredients, simple to understand, but astonishing!  It goes like this:

Take any old quadrilateral, and construct squares on each edge.  Now take the line segments joining the centres of the squares on opposite sides. These two line segments are at right angles and of equal length!  The wikipedia article shows a couple of pictures: http://en.wikipedia.org/wiki/Van_Aubel's_theorem. There is also a wondersful JavaScetch of it at http://mste.illinois.edu/dildine/geometry/vanaubel.html - you can move the vertices of the square around, and watch the lines dancing about but remaining the same length as each other and at right angles to each other.

This is straightforward Euclidean geometry, and it could have been discovered by Euclid himself, but we had to wait over 2000 years before someone discovered it (van Aubel of course).   It is such a surprising result that nobody thought to look anywhere near there for theorems. Nowadays it is very easy to prove using vectors.

There is some small print.  There are of course two squares on each side of the quadrilateral, and you have to choose the right ones.  If it is an ordinary convex quadrilateral, you must chose all the ones on the outside, or all the ones on the inside.  More generally, you go round the quadrilateral in one direction, constructing all the squares on your left, or all the ones on the right.  The wikipedia article illustrates this for a quadrilateral that crosses over itself.