RIOPLATENSE 1998 Problem 5

We say that $M$ is the midpoint of the open polygonal $XYZ$ , formed by the segments $XY, YZ$ , if $M$ belongs to the polygonal and divides its length by half. Let $ABC$ be a acute triangle with orthocenter $H$ . Let $A_1, B_1,C_1,A_2, B_2,C_2$ be the midpoints of the open polygonal $CAB, ABC, BCA, BHC, CHA, AHB$ , respectively. Show that the lines $A_1 A_2, B_1B_2$ and $C_1C_2$ are concurrent.