Let P = A_0 A_1 ... A_n be a convex polygon on the plane. Define for all 1 <= k <= n-1 the operation f_k which replaces P with a new polygon f_k(P) = A_0 ... A_{k-1} A_k' A_{k+1} ... A_n where A_k' is the reflection of A_k across the perpendicular bisector of A_{k-1}A_{k+1}. Prove that (f_1 f_2 ... f_{n-1})^n P = P.