Construct for the triangle $ABC$ a circle $S$ passing through the point $B$ and touching the line $CA$ at the point $A$, a circle $T$ passing through the point $C$ and touches the line $BA$ at the point $A$. The second intersection point of the circles $S$ and $T$ is denoted by $D$. The intersection point of the line $AD$ and the circumscribed circle $\Delta ABC$ is denoted by $E$. Prove that $D$ is the midpoint of the segment $AE$.