Question 50
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 $\angle I_{1} A I_{2}$ 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 $B C D$. Let $E$ be the midpoint of that arc $C D$ of $\mathcal{K}_{3}$ that contains $B$. The lines $A C$ and $A D$ intersect $\mathcal{K}_{3}$ again in $K$ and $L$, respectively. Prove that the line $A E$ is perpendicular to $K L$.