Sorry, the problem is messed up.  The correct version should be:

Let $P$ be the intersection of the diagonals of cyclic quadrilateral $ABCD$.  Let $K,L,M,N$ be the projection of $P$ at the four sides.  Show that quadrilateral $KLMN$ has an inscribed circle.  In addition, if $\overline{AC}\perp\overline{BD}$, let $R$ be the circumradius of $ABCD$ and $d$ be the distance between circumcenter $O$ and $P$, find the inradius of $KLMN$.

The packet has been updated.