Question 116
Let $ABC$ be a triangle and let $E$ denote the midpoint of $BC$. The circle passing through the points $A$ and $B$ and tangent to the perpendicular bisector of $AC$ at point $D$ meets the side $AC$ again at point $K$ different from $A$. Let $F$ denote the midpoint of the segment $KC$. Prove that $DE$ is perpendicular to $EF$.