Steiner's Porism
Suppose that a circle C lies inside circle D, and that circle E touches both of them. We can draw a sequence of circles, each touching C and D and the previous member of the sequence.
There are two possibilities,
either
Steiner's Porism
The proof is quite simple.
Note that C and D are nonintersecting ilines.  
Since the figure is symmetric about the common centre P, it is clear that, if starting with E gives a Steiner Chain, then starting with F will give the Steiner Chain obtained by rotation. Clearly, it will contain the same number of circles.
The CabriJava pane on the right shows a typical Steiner Chain

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