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 $D$.
1. **Define the Geometry and Initial Conditions:**
   - Consider two circles, each with radius \(1\), inscribed in a rectangle of height \(2\) and width \(4\).
   - The circles are tangent at point \(T\) and touch the top side of the rectangle at points \(A\) and \(B\).
   - Let \(O_1\) and \(O_2\) be the centers of the left and right circles, respectively.
   - Point \(E\) moves counterclockwise around the left circle starting from \(T\), and point \(D\) moves clockwise around the right circle starting from \(A\). Both points move with the same speed.

2. **Symmetry and Motion Analysis:**
   - Since \(E\) and \(D\) move with equal angular velocities, the angles they sweep out are equal in magnitude but opposite in direction.
   - Let \(\angle AO_1D = -\angle TO_2E\). This implies that the motion of \(D\) and \(E\) are symmetric with respect to the vertical line passing through \(T\).

3. **Transformation and Locus:**
   - The transformation \(\mathcal{L}\) that maps \(E\) to \(D\) can be decomposed into a series of reflections:
     - Reflect \(E\) about the vertical line through \(T\) (denoted as \(PT\)).
     - Reflect the result about the vertical line through \(O_1\) (denoted as \(PO_1\)).
     - Reflect the result about the horizontal line through \(A\) (denoted as \(AO_1\)).
   - The composition of these reflections results in a transformation that is equivalent to a reflection about a certain axis \(\ell\) and a translation parallel to \(\ell\).

4. **Determine the Axis \(\ell\):**
   - Since \(\mathcal{L}\) maps \(T\) to \(A\) and the antipode of \(B\) with respect to \((O_2)\) to \(T\), the axis \(\ell\) must be the line \(AT\).

5. **Locus of Midpoints:**
   - The midpoint \(M\) of the segment \(\overline{DE}\) lies on the line \(AT\).
   - As \(E\) and \(D\) move around their respective circles, the midpoint \(M\) traces out a segment on the line \(AT\).
   - Specifically, \(M\) will trace out the diameter of the circle centered at \(T\) with radius \(1\), intersected by the line \(AT\).

The final answer is the segment on the line \( \boxed{ AT } \) that corresponds to the diameter of the circle centered at \(T\) with radius \(1\).