# How can we prove that a three legged chair will never be wobbly?

I am taking the geometry approach. We know from intuition that more than three legs on a chair will make it unstable if any of the legs have a different length than the others. So by "wobble" I mean the possibility that at least one of the legs will be in the air when one or more legs are made shorter/longer than others. Also, the "surface" must be perfectly flat.

A three legged chair is unaffected by any amount of change we make to its legs. So to prove this I started out connecting lines between each legs (diagonals). So far I haven't made any progress.

For a triangle there are no diagonals. Is it enough to show that all the legs must be in the same plane for the chair to be stable?

• A badly designed three-legged chair can fall over. Apr 22, 2015 at 21:06
• @Henry Nevertheless, it wouldn't wobble! Apr 22, 2015 at 21:07
• There is always a plane that passes through 3 points. Not so for more than 3 points. Apr 22, 2015 at 21:08
• You can have a chair with more than 3 unequal legs that isn't wobbly, if the surfaces of the feet touch a common plane. Apr 22, 2015 at 21:10
• In fact, even a four-legged chair can always be rotated to be stable on any smooth floor. The proof involves the intermediate value theorem. Apr 22, 2015 at 21:36