Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.
1. **Define the centers and midpoints:**
   Let \( O_1 \) and \( O_2 \) be the centers of circles \( C_1 \) and \( C_2 \), respectively. Let \( A_1 \) be a fixed point on \( C_1 \) and \( A_2 \) be a fixed point on \( C_2 \). Let \( P_1 \) be a point on \( C_1 \) such that the chord \( A_1P_1 \) is parallel to the chord \( A_2P_2 \) on \( C_2 \). Let \( M \) be the midpoint of \( \overline{P_1P_2} \) and \( N \) be the midpoint of \( \overline{A_1A_2} \).

2. **Projection of centers:**
   Let \( X_1 \) and \( X_2 \) be the midpoints of \( \overline{A_1P_1} \) and \( \overline{A_2P_2} \), respectively. These points \( X_1 \) and \( X_2 \) are the projections of \( O_1 \) and \( O_2 \) onto the lines \( A_1P_1 \) and \( A_2P_2 \), respectively.

3. **Midpoint of midpoints:**
   Let \( K \) be the midpoint of \( \overline{MN} \). The perpendicular bisector \( \ell \) of \( \overline{MN} \) intersects \( A_1P_1 \) and \( A_2P_2 \) at points \( Y_1 \) and \( Y_2 \), respectively.

4. **Parallel chords and midpoints:**
   Since \( A_1P_1 \parallel A_2P_2 \), the segment \( MN \) is a midparallel of these parallel chords. Therefore, \( K \) is the midpoint of \( \overline{X_1X_2} \) and \( \overline{Y_1Y_2} \).

5. **Equidistant centers:**
   The equality \( X_1Y_1 = X_2Y_2 \) implies that \( O_1 \) and \( O_2 \) are equidistant from the line \( \ell \). This means that \( \ell \) passes through the midpoint \( O \) of \( \overline{O_1O_2} \).

6. **Locus of midpoint \( M \):**
   Since \( \ell \) passes through \( O \), the locus of \( M \) is a circle centered at \( O \) with radius equal to the distance \( ON \), where \( N \) is the midpoint of \( \overline{A_1A_2} \).

The final answer is the circle with center \( \boxed{ O } \) and radius \( ON \).