ZA-SAMO 1997 Problem 1

From an initial triangle $\Delta A_0B_0C_0$ , a sequence of triangles $\Delta A_1B_1C_1$ , $A_2B_2C_2$ , ... is formed such that, at each stage, $A_{k + 1}$ , $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$ , $C_kA_k$ and $A_kB_k$ respectively.

(a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$ .

(b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$ .