Question 91
Let $K$ and $L$ be points on a semicircle with diameter $\overline{AB}$. Bisector of the side $\overline{AB}$ intersects the segment $\overline{KL}$ at a point $U$ so that $A$ and $K$ lie on one side of the bisector and $B$ and $L$ on the other. Let $N$ be the foot of perpendicular from the intersection of lines $AK$ and $BL$ to the line $AB$, and let $V$ be a point on the line $KL$ such that $\angle VAU = \angle VBU$. Prove that the lines $NV$ and $KL$ are perpendicular. (Iran TST 2009, modified)