I suggest the ham sandwich theorem as an example of using the intermediate value theorem.
I tried finding a nice easy to explain proof online, the best I could find is the following: http://everything2.com/title/proof+of+the+ham+sandwich+theorem
You might also be interested in reading a bit of the history of the ham sandwich theorem in this article published in the monthly: http://russell.lums.edu.pk/~cs611s09/slides/historyofhamdsandwich.pdf
