Friday Math Problem

I am going to start posting clever math problems every Friday. These are in the style of the high school Math Olympiad problems, and are intended for recreational purposes. I think that this is a great idea because a) it will save me the trouble of having to come up with original content to get you through the weekend, and b) you might get smarter! I know, I know, this isn't fair to those of you who went to lower and middle school in the United States, but look at this as a productive challenge. And also, I'd like to mention that googling the answer is completely unethical, and goes against the very spirit of clever math problems.

The three frogs problem: Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the leapee is at the midpoint of the line segment whose endpoints are the starting and ending position of the leaper. Will a frog ever occupy the vertex of the square that was originally unoccupied? Show a proof.