MEMO 2011 I3

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 $KL$ .