IN-TST 2002 Problem 1

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$ . Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$ . Let $\Gamma$ be the circle which is tangent to the line $l$ , tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$ . The diameter of $\Gamma$ through $D$ meets $l$ in $E$ . Show that $AB=DE$ .