Question 86
Let $ABC$ be a triangle. Let $C_1$, $A_1$ and $B_1$ be the points of $AB$, $BC$ and $AC$, respectively. Let $K$ be the projection of $B_1$ on the line $A_1C_1$. Let the points $M$ and $N$ lie on the rays $B_1A$ and $B_1C$ respectively, so that $\angle B_1A_1C_1 = 2\angle KNB_1$ and $\angle B_1C_1A_1 = 2\angle KMB_1$. Prove that the length of the segment $MN$ is not greater than the perimeter of the triangle $\Delta A_1B_1C_1$.