KURSCHAK 1989 Problem 1

In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$ , define $A=\ell\cap a$ , and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$ . Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.