Question 76
Let $A_0B_0C_0$ be a triangle. For a positive integer $n \ge 1$, we define $A_n$ on the segment $B_{n-1}C_{n-1}$ such that $B_{n-1}A_n : C_{n-1}A_n = 2 : 1$ and $B_n, C_n$ are defined cyclically in a similar manner. Show that there exists a unique point $P$ that lies in the interior of all triangles $A_nB_nC_n$.