A ladder is placed against a (vertical) wall and the bottom of the ladder is moved away along the (horizontal) ground. What is the shape of the curve traced by the midpoint of the ladder?
It is very easy to work out the answer to this problem, and I won't bother to do that here. If you haven't seen the problem before, test your intuition. Try to sketch what you think curve looks like before solving the problem. (In particular, is the curve concave up or concave down?) The first time I saw this, my intuition was wrong.