Steiner chains of circles

These are chains of circles (sequence a0, a1, ...) tangent each to the previous and all to two circles c1, c2.
The problem here is to find the whole sequence given the two limiting circles and the first circle a0 of the chain. In SteinerChain.html we examined a construction method of this sequence using inversions with respect of circles orthogonal to c1, c2. Here we discuss another method based on a single inversion which transforms the two given circles to two intersecting lines e1, e2 correspondingly.

To construct the chain define first the inversion F with respect to the circle c(B,|BA|). Then apply the inversion to the two circles c1, c2 to obtain lines e1, e2 correspondingly. Circle a0 transforms via F to a circle b0 tangent to b0. Define subsequently the homothety H with center A and modulus (1-s)/(1+s), where s=sin(w/2) and w is the angle of lines e1, e2. This homothety transforms b0 to a circle b1 tangent to the two lines and b0 itself. By repeated application of the homothety H we obtain a chain of circles b0, b1, b2, ... each tangent to e1, e2 and its predecessor. Applying then F to this chain we obtain the desired chain of circles inscribed in c1, c2.
A similar argument is used in Steiner_Chain.html to prove a theorem of Steiner on finite chains lying between two circles, one containing the other.