I would like to determine radius' of pappus chain circles which are tangent to Arbelos. Below I will put some photos from one book that showed how to determine it. The only thing that i don't understand, is the last line of determination of $r_n$. I don't get why $r_n$ would be equal to $\frac{k}{k_n} \cdot r'$.
Figure:
Determination:
