ILL 1972 Problem 30

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$, respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation $$r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)$$

where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that $$r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1$$

(b) Prove that the sequence of circles $K_1,K_2, \ldots$ is periodic.