KöMaL 2011 B4404

The angles of a polygon $A_1 A_2$...$A_{2n}$ are all equal, and its sides satisfy $A_1 A_2$=$A_3 A_4$=...=$A_{2n-1} A_{2n}$ and $A_2 A_3$=$A_4 A_5$=...=$A_{2n} A_1$. Prove that the polygon is cyclic.