Question 5
Triangle $A_1A_2A_3$ is inscribed in a circle $(O)$. A circle $(K_1)$ is tangent to the sides $A_1A_2,A_1A_3$ and internally tangent to $(O)$ at $M_1,N_1,P_1$, respectively. The points $M_1,N_2,P_2$ and $M_3,N_3,P_3$ are defined similarly. Prove that $M_1N_1,M_2N_2,M_3N_3$ intersect at the midpoint of each segment.