Question 94
Let $ABC$ be a triangle, and let $E$ be the midpoint of $[BC]$. Let $m$ be the perpendicular bisector of $[AC]$, and let $D$ be a point of $m$ such that the circumcircle of $ABD$ is tangent to $m$. Finally, let $K$ be the intersection point, other than $A$, of the circumcircle of $ABD$ and the line $(AC)$, and let $F$ be the midpoint of $[CK]$. Prove that the lines $(DE)$ and $(EF)$ are perpendicular.