Question 11
Let $ABC$ be a non-right triangle such that $AB \lt AC$. Denote by $H$ the projection of $A$ onto $(BC)$, and by $E, F$ the projections of $H$ onto $(AB)$ and $(AC)$ respectively. The line $(EF)$ meets $(BC)$ at the point $D$. Consider the semicircle with diameter $[CD]$ lying in the same half-plane bounded by $(CD)$ as $A$. Let $K$ be the point of this semicircle that projects onto $B$. Show that $(DK)$ is tangent to the circle $KEF$.