KöMaL 2017 B4896

Let $A_1$, $B_1$, $C_1$, $D_1$ denote the midpoints of the sides of a convex quadrilateral $ABCD$. Let $A_2$, $B_2$, $C_2$, $D_2$ denote the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. The procedure is continued. Prove that if quadrilateral $A_1B_1C_1D_1$ is cyclic then quadrilateral $A_{2017}B_{2017}C_{2017}D_{2017}$ is also cyclic.