SteinerChain2

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.

Steiner chains of circles

See Also