Suppose the inscribed circle of $\triangle A_1A_2A_3$ touches the sides $A_2A_3, A_3A_1, A_1A_2$ at $T_1,T_2,T_3$. From the midpoints $M_1,M_2,M_3$ of $A_2A_3,A_3A_1,A_1A_2$, draw lines perpendicular to $T_2T_3,T_3T_1,T_1T_2$. Prove that these perpendicular lines are concurrent.
I have a solution but is rather long and is not elegant. Since it is a contest type problem, I guess there is an elegant way of solving it, may someone please help, thanks.
A picture of the construct is added below: