Question 1
In an acute-angled triangle $ABC$, $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$, $E$ and $F$ are the feet of the perpendiculars from $D$ onto $AB$ and $AC$ respectively. The segments $BF$ and $CE$ intersect at $K$. Prove that $AK$ is perpendicular to $BC$.