Question 11
(NET 6) 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
$$r_1\cot\frac A2+2\sqrt{r_1r_2}+r_2\cot\frac B2=r\left(\cot\frac A2+\cot\frac B2\right),$$
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
$$r_1=r\cot\frac B2\cot\frac C2\sin^2t_1.$$
(b) Prove that the sequence of circles $K_1,K_2,\ldots$ is periodic.
(a) Prove the relation
$$r_1\cot\frac A2+2\sqrt{r_1r_2}+r_2\cot\frac B2=r\left(\cot\frac A2+\cot\frac B2\right),$$
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
$$r_1=r\cot\frac B2\cot\frac C2\sin^2t_1.$$
(b) Prove that the sequence of circles $K_1,K_2,\ldots$ is periodic.