Question 129
$BB_1$ is an altitude of the acute-angled scalene triangle $ABC$. A point $D$ lies on the side $BC$ so that $\angle BAD = \angle CBB_1$. The segments $AD$ and $BB_1$ intersect at a point $F$. A line $l$ is drawn through the point $B$ perpendicular to the side $AB$. This line intersects the line $CF$ at a point $K$. Prove that the line $DK$ intersects the segment $BF$ at its midpoint.