ISL 1970 Problem 33

The vertices of a given square are clockwise lettered $A,B,C,D$ . On the side $AB$ is situated a point $E$ such that $AE = AB/3$ . Starting from an arbitrarily chosen point $P_0$ on segment $AE$ and going clockwise around the perimeter of the square, a series of points $P_0, P_1, P_2, \ldots$ is marked on the perimeter such that $P_iP_{i+1} = AB/3$ for each $i$ . It will be clear that when $P_0$ is chosen in $A$ or in $E$ , then some $P_i$ will coincide with $P_0$ . Does this possibly also happen if $P_0$ is chosen otherwise?