OMCPLP 2011 Problem 5

Consider two circles, tangent at $T$ , both inscribed in a rectangle of height $2$ and width $4$ . A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$ , and $D$ starts at $A$ , where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$ .