VN-TST 2008 Problem 1

On the plane, given an angle $xOy$ . $M$ be a mobile point on ray $Ox$ and $N$ a mobile point on ray $Oy$ . Let $d$ be the external angle bisector of angle $xOy$ and $I$ be the intersection of $d$ with the perpendicular bisector of $MN$ . Let $P$ , $Q$ be two points lie on $d$ such that $IP = IQ = IM = IN$ , and let $K$ the intersection of $MQ$ and $NP$ .

$1.$ Prove that $K$ always lie on a fixed line.

$2.$ Let $d_1$ line perpendicular to $IM$ at $M$ and $d_2$ line perpendicular to $IN$ at $N$ . Assume that there exist the intersections $E$ , $F$ of $d_1$ , $d_2$ from $d$ . Prove that $EN$ , $FM$ and $OK$ are concurrent.