Question 151
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E$, $AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is a convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN$, $Q$ be the intersection of $MN$ and $GH$.
1. Prove that $IK$ and $OE$ are parallel.
2. Prove that $PK$ is perpendicular to $IQ$.
1. Prove that $IK$ and $OE$ are parallel.
2. Prove that $PK$ is perpendicular to $IQ$.