ZA-SAMO 2001 Problem 5

Starting from a given cyclic quadrilateral $\mathcal{Q}_0$ , a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$ . The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$ , $B$ , $C$ and $D$ .) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.