Question 152
Given an angle $xOy$ on a plane. Let $M$ and $N$ are variable points on the rays $Ox$ and $Oy$, respectively. Denote by $d$ the outer angle-bisector of the angle $xOy$ and let $I$ denote the intersection of $d$ and perpendicular bisector of the segment $MN$. Choose two points $P$ and $Q$ on $d$ such that $IP = IQ = IM = IN$. Denote by $K$ the intersection of $MQ$ and $NP$.
1/ Prove that $K$ lies on a fixed line.
2/ Suppose $M$ and $N$ are such that both the line $d_1$ through $M$ and perpendicular with $IM$, and the line $d_2$ through $N$ and perpendicular with $IN$ intersect with $d$. Let $E$ and $F$ be denote the respective intersection. Prove that the three lines $EN, FM$ and $OK$ meet at a point.
1/ Prove that $K$ lies on a fixed line.
2/ Suppose $M$ and $N$ are such that both the line $d_1$ through $M$ and perpendicular with $IM$, and the line $d_2$ through $N$ and perpendicular with $IN$ intersect with $d$. Let $E$ and $F$ be denote the respective intersection. Prove that the three lines $EN, FM$ and $OK$ meet at a point.