On touching circles

The circles C1, … , Cn form a chain of length n, or an n-chain, if Ci touches Ci + 1, for i = 1, … , n − 1, and the chain is closed if also Cn touches C1. If Ci touches Ci + 1 at Qi, for i = 1, … , n (subscripts being interpreted modulo n), then Q1, … , Qn (assumed distinct) are the contact points of the chain. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain, and if Ci touches S at Pi, then P1, … , Pn are the base points of the chain.