geometry - Is it always possible to balance a 4-legged table?

A perfectly symmetrical small 4-legged table is standing in a large room with a continuous but uneven floor. Is it always possible to position the table in such a way that it doesn't wobble, i.e. all four legs are touching the floor?

No tricks. No lateral-thinking. Serious question (with real-life applications too!) with a serious answer.

_{This might look like it'd fit better on Lifehacks.SE, but the answer has a nice mathematical proof/formula [depending on} ^{whether it's yes/no; I won't give the game away!] which is surprisingly simple and elegant.}

Answer

The answer is

yes!

Here's why:

Imagine that the table can pass through the floor. We're going to call one leg the "floating leg" - the other three are going to always be on the floor. Now, after we rotate the table a quarter turn, the floating leg is going to be above the floor if it was originally below, or vice versa. By the intermediate value theorem, it will be exactly on the floor at one point in that rotation.

More detailed proof:

Three legs of the table define a plane. Define the "offset" of a leg to be the distance above the floor if the other three legs are placed on the floor directly - negative if below, positive if above. In any arbitrary placement, the offset of any two adjacent legs will be positive and negative. (WLOG assume the one on the left is positive.) Offset is a continuous function because it's the distance from the floor. If you start with a positive offset on leg A and rotate the table 90° to the right, the offset on leg A (now in the original position of another leg) will be negative. This means that at some point, offset was 0, therefore the table was completely touching the floor.