circle table and smooth ground

Prove that given a circular table with four legs, and given a smooth ground, there is always a spot to place the table such that it doesn't wobble.

Consider a smooth, circular curve C:θ→h.

Consider the curve traced out by the fourth point on this curve, (C_4). Need to show that (C_4)intersects C.

Can we show that if (C_4) is below C at some point, it must be above C at some other point? And vice versa?

(C_4)(θ)=C*(θ-90)+C*(θ+90)-C(θ)