SHARYGIN 2006 Problem 20

Four points are given $A, B, C, D$ . Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD$ , $CDA$ , $DAB$ , $ABC$ and $A_2,$ $B_2$ , $C_2$ , $D_2$ are orthocenters of the triangles $B_1C_1D_1$ , $C_1D_1A_1$ , $D_1A_1B_1$ , $A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.