ÖMO 2008 Problem 6

We are given a square $ABCD$ . Let $P$ be a point not equal to a corner of the square or to its center $M$ . For any such $P$ , we let $E$ denote the common point of the lines $PD$ and $AC$ , if such a point exists. Furthermore, we let $F$ denote the common point of the lines $PC$ and $BD$ , if such a point exists. All such points $P$ , for which $E$ and $F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $EF$ is parallel to $AD$ .