Question 5
Let $ABC$ be a triangle with $AC \lt AB \lt BC$. Choose a point $D$ on the side $AB$. Suppose the circumcircle of $ABC$ meets the internal angle bisector of $\angle A$ and $CD$ at $E$ ($\neq A$) and $F$ ($\neq C$), respectively. Let $K$ be the intersection point of $BC$ and $DE$. Prove that $CK = AC$ if and only if $DK \cdot EF = AC \cdot DF$.
