Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.
1. **Consider the given setup:**
   - Two parallel lines \( r \) and \( s \).
   - Points \( P \in r \) and \( Q \in s \).
   - Circles \( C_P \) and \( C_Q \) such that \( C_P \) touches \( r \) at \( P \) and \( C_Q \) touches \( s \) at \( Q \).
   - The circles \( C_P \) and \( C_Q \) touch each other externally at some point \( T \).

2. **Apply negative homothety:**
   - Consider a negative homothety \( h \) with center at \( T \) that maps \( C_P \) to \( C_Q \).
   - Since \( C_P \) and \( C_Q \) touch each other externally at \( T \), such a homothety exists.

3. **Analyze the effect of homothety on lines \( r \) and \( s \):**
   - Because \( r \) and \( s \) are parallel and each touches one of the circles, the homothety \( h \) maps \( r \) to \( s \).
   - This implies that the point \( P \) on \( r \) is mapped to the point \( Q \) on \( s \).

4. **Determine the collinearity of points \( P \), \( Q \), and \( T \):**
   - Since \( P \) is mapped to \( Q \) under the homothety centered at \( T \), the points \( P \), \( Q \), and \( T \) must be collinear.

5. **Identify the locus of point \( T \):**
   - The locus of \( T \) is the line segment joining \( P \) and \( Q \), excluding the endpoints \( P \) and \( Q \).
   - This is because for any point \( T \) on the open segment \( PQ \), we can construct circles \( C_P \) and \( C_Q \) that touch each other externally at \( T \) and satisfy the given conditions.

\[
\boxed{\text{The locus of } T \text{ is the open segment } PQ \text{ (excluding } P \text{ and } Q\text{).}}
\]