IT-CESENATICO 2008 Problem 1

Let $ABCDEFGHILMN$ be a regular dodecagon, let $P$ be the intersection point of the diagonals $AF$ and $DH$ . Let $S$ be the circle which passes through $A$ and $H$ , and which has the same radius of the circumcircle of the dodecagon, but is different from the circumcircle of the dodecagon. Prove that:

1. $P$ lies on $S$
2. the center of $S$ lies on the diagonal $HN$
3. the length of $PE$ equals the length of the side of the dodecagon