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5

Part of 2001 South africa National Olympiad

Problems(1)

Sequence of quadrilaterals!

Source: South Africa 2001

10/1/2005
Starting from a given cyclic quadrilateral Q0\mathcal{Q}_0, a sequence of quadrilaterals is constructed so that Qk+1\mathcal{Q}_{k + 1} is the circumscribed quadrilateral of Qk\mathcal{Q}_k for k=0,1,…k = 0,1,\dots. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral ABCDABCD has sides that are tangent to the circumcircle of ABCDABCD at AA, BB, CC and DD.) Prove that the sequence always terminates, except when Q0\mathcal{Q}_0 is a square.
geometrycircumcircleparallelogramtrapezoidincentercyclic quadrilateralperpendicular bisector