VN-TST 2009 Problem 1

Let an acute triangle $ABC$ with curcumcircle $(O)$ . Call $A_1,B_1,C_1$ are foots of perpendicular line from $A,B,C$ to opposite side. $A_2,B_2,C_2$ are reflect points of $A_1,B_1,C_1$ over midpoints of $BC,CA,AB$ respectively. Circle $(AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $(O)$ at $A_3,B_3,C_3$ respectively. Prove that: $A_1A_3,B_1B_3,C_1C_3$ are concurent.