Question 1
In the plane, an angle $xOy$ is given. Let $M,N$ lie on the rays $Ox,Oy$, respectively. Let $d$ be the external angle-bisector line of $xOy$, and let $I$ be the intersection of $d$ with the perpendicular bisector of $MN$. Let $P,Q$ be distinct points on $d$ such that $IM=IN=IP=IQ$, and suppose $K=MQ\cap NP$.
1. Prove that $K$ lies on a fixed line.
2. Let $d_1$ be the line perpendicular to $IM$ at $M$, and $d_2$ the line perpendicular to $IN$ at $N$. Suppose $d_1,d_2$ intersect $d$ at $E,F$, respectively. Prove that $EN,FM,OK$ are concurrent.
1. Prove that $K$ lies on a fixed line.
2. Let $d_1$ be the line perpendicular to $IM$ at $M$, and $d_2$ the line perpendicular to $IN$ at $N$. Suppose $d_1,d_2$ intersect $d$ at $E,F$, respectively. Prove that $EN,FM,OK$ are concurrent.