Touching circles

When you start drawing two circle, touching and one inside the other, you can draw a whole chain of circles touching both, and each other. The contact points of that chain lie again on a circle.

Please enable Java for an interactive construction (with Cinderella).

This is most easily shown by an inversion relative to the big circle, and a classical example of this technique. Play around with the construction by dragging the red points!

Try using this construction to show that circle n in the chain, with radius r, is at a height of 2nr over the red base line.

Created with Cinderella by Leo Dorst 20010719.