Steiner chain of circles

Given two circles C1, and C2 inside C1.
In the area between these two circles, we draw a circle K1 touching both C1 and C2, then a circle K2 touching both C1, C2 and K1,
then K3 touching C1, C2 and K2 ... Kn touching C1, C2 and Kn-1.
Locus of centers of circles K    Solution

Sometimes, this chain closes and Kn also touches K1 !
Prove that in this case, we can choose the first circle anywhere, and the chain may then "rotate" freely
(Steiner porism).     Hint

Deduce a criterion for such a chain to exist.

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