Question 12
Let $ABC$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $BC$, $CA$ and $AB$ are $D$, $E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $AI$, and let $L$ be the second point of intersection of the circumcircles of the triangles $BFK$ and $CEK$. If $\frac{1}{3}BC = AC - AB$, prove that $DE = 2KL$.