Middle European Mathematical Olympiad 2011 Problem I-3

In a plane the circles $\mathcal{K}_1$ and $\mathcal{K}_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal{K}_1$ in $A$ intersects $\mathcal{K}_2$ again in $C$ and the tangent to $\mathcal{K}_2$ in $A$ intersects $\mathcal{K}_1$ again in $D$. Let $\mathcal{K}_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal{K}_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal{K}_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.