This question grew out of my recent job interview. Since the interviewers were math professors, I had a hard time searching for interesting elementary theorems in case I got asked for one.
I thought topics such as the Banach-Tarski paradox, Godel's theorems, the Mandelbrot set, the Brouwer Fixed Point Theorem, etc were well-known and wouldn't do the job. However, after a cursory search, I found some to my taste:


1.<s>[Marden's theorem] [1] (or [ here][2])(It is not Marsden.)</s> [Gauss–Lucas theorem][3]

2.The identity $\int_{0}^1 \frac{x^4(1-x)^4}{1+ x^4} dx = \frac{22}{7}- \pi$


So, my  question here is an invitation to expand the list (of theorems that would get an interviewee accepted).

To recap, my criteria for selection are 

1. Not widely known,
2. Elementary- understandable to a first year grad student, and
3. Interesting-i.e. MOtizens, assuming they are the audience, will be delighted to have come across it.   


Thank you.


  [1]: http://en.wikipedia.org/wiki/Marden%2527s_theorem
  [2]: http://www.maa.org/joma/Volume8/Kalman/index.html
  [3]: http://en.wikipedia.org/wiki/Gauss-Lucas_theorem