This dialogue is from the West Wing episode, "Evidence of Things Not Seen."
C.J. There's a spot on the earth where the temperature is exactly the same as it would be if you drilled through the earth to the other side.
WILL No, there isn't. How about six dollars if you do it with a face card?
C.J. Yes, there is, and it's called the antipode. And if that's true, then why can't it be that you could stand an egg on end at the equinox?
Now, the problem. Prove that at any given moment, what C.J. is saying is true: there are two points on the earth exactly opposite each other which have the exact same temperature. You can assume that temperature is continuous (i.e., there's no spot on the earth where the temperature just "jumps" from 0 degrees at one spot to 100 degrees at a spot infinitesimally close to it).