DE-BUNDESWETTBEWERB 2018 Problem 7

Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$ . Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$ . Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$ .