Given two parallel lines $r$ and $r'$ and a point $P$ on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point $P$ , and the other two, one on each of the two lines.
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1. **Identify the given elements:**
   - Two parallel lines \( r \) and \( r' \).
   - A point \( P \) not on either line.

2. **Choose a point \( Q \) on line \( r \):**
   - Let \( Q \) be a point on line \( r \).

3. **Rotate point \( Q \) around point \( P \) by \( \pm 60^\circ \):**
   - When \( Q \) is rotated around \( P \) by \( 60^\circ \) or \( -60^\circ \), the image of \( Q \) will lie on a circle centered at \( P \) with radius \( PQ \).
   - Since \( r \) and \( r' \) are parallel, the image of \( Q \) after rotation will lie on line \( r' \).

4. **Find the intersection point \( R \) on line \( r' \):**
   - Rotate line \( r \) around point \( P \) by \( 60^\circ \) or \( -60^\circ \). The intersection of the rotated line with \( r' \) will give the point \( R \).

5. **Determine the coordinates of \( R \):**
   - Let \( R \) be the intersection point of the rotated line and \( r' \).

6. **Rotate point \( R \) back by the opposite angle to find \( Q \):**
   - Rotate \( R \) around \( P \) by \( -60^\circ \) or \( 60^\circ \) (opposite to the initial rotation) to find the original point \( Q \) on line \( r \).

7. **Verify the equilateral triangle:**
   - Check that \( \triangle PQR \) is equilateral by ensuring \( PQ = PR = QR \).

The final answer is the coordinates of points \( \boxed{ Q } \) and \( R \) such that \( \triangle PQR \) is an equilateral triangle with \( P \) as one vertex and \( Q \) and \( R \) on lines \( r \) and \( r' \) respectively.