Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).
1. **Define the lines and the stripe:**
   Let the three parallel lines be denoted as \( r, s, \) and \( s' \). Assume \( r \) is the middle line, and \( s \) and \( s' \) are the lines on either side of \( r \). The stripe determined by \( s \) and \( s' \) is the region between these two lines.

2. **Incenter properties:**
   The incenter \( I \) of a triangle is the point where the angle bisectors of the triangle intersect. It is also the center of the incircle, which is tangent to all three sides of the triangle.

3. **Position of the incenter:**
   If the incenter \( I \) is within the stripe defined by \( r \) and \( s \), it must be closer to \( r \) than to \( s \). This is because the incircle must touch \( r \) and not cross \( s \). Similarly, if \( I \) is within the stripe defined by \( r \) and \( s' \), it must be closer to \( r \) than to \( s' \).

4. **Constructing the triangle:**
   Consider points \( A \in r \), \( B \in s \), and \( C \in s' \). We can move \( C \) along \( s' \) away from \( A \) and \( B \). As \( C \) moves far away, the triangle \( ABC \) tends towards an open rectangle with one side \( AB \) and the other sides being rays on \( r \) and \( s \). The incenter \( I \) of this triangle will tend towards a point on the line halfway between \( r \) and \( s \).

5. **Displacing points:**
   - Displace \( A \) along \( s \) away from \( B \) and \( C \). The incenter \( I \) will tend towards a point on the line halfway between \( r \) and \( s' \).
   - Since \( C \) and \( A \) can be moved continuously, and the incenter \( I \) moves continuously, for each line \( t \) parallel to \( r \) and strictly within the stripe defined by the lines halfway between \( r \) and \( s \), and halfway between \( r \) and \( s' \), there exists a position of \( A, B, \) and \( C \) such that the incenter is on \( t \).

6. **Conclusion:**
   By displacing \( A, B, \) and \( C \) along \( s, r, \) and \( s' \) respectively, the incenter \( I \) can reach any desired point on \( t \). Therefore, the locus of the incenters of triangles with vertices on these three lines is the region strictly between the lines halfway between \( r \) and \( s \), and halfway between \( r \) and \( s' \).

The final answer is the region strictly between the lines halfway between \( \boxed{ r } \) and \( s \), and halfway between \( r \) and \( s' \).