My favorite theorem is the Theorem of Menelaus. It says this:
Let ABC be any triangle. Let A' be a point of BC other than B and C, let B' be a point of AC other than A and C, and let C' be a point of AB other than A and B. If A', B', and C' are collinear then $ \frac{A'B}{A'C} \frac{B'C}{B'A} \frac{C'A}{C'B} = 1. $
For a picture go to http://en.wikipedia.org/wiki/Menelaus%27_theorem
