Question 6
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. Connect $AC$ and $BD$ intersecting at $K$. $O_1$ is the circumcenter of triangle $ABK$ and $O_2$ is the circumcenter of triangle $CDK$. A line $l$ through $K$ intersect the two circumcircles at $E$ and $F$ respectively, and the circumcircle of $ABCD$ at $G$ and point $H$. Prove that $EG=FH$.
