Let A, B, and C be three circles in the plane, and let X be any circle touching B and C. Then build up a chain of circles such that Y:C A X, Z:A B Y, X' :B C Z, Y' :C A X', Z' :A B Y', X'' :B C Z', where C:C_1 C_2 C_3 denotes a circle C tangent to circles C_1, C_2, and C_3. Although there are a number of choices for each successive tangent circle in the chain, if the choice at each stage is made appropriately, then the ninth and final circle X'' coincides with the first circle X .
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