We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).
1. **Identify the center and radius of the circle:**
   Let \( O \) be the center of the circle and \( r \) be its radius.

2. **Draw the given line:**
   Let the given line be \( l \). Since the line does not intersect the circle, it is outside the circle.

3. **Project the center \( O \) onto the line \( l \):**
   Draw a perpendicular from \( O \) to the line \( l \). Let the foot of this perpendicular be \( T \). Thus, \( T \) is the projection of \( O \) onto \( l \).

4. **Construct a square \( ABCD \) with \( T \) as the midpoint of \( AB \):**
   - Draw a line segment \( AB \) on \( l \) such that \( T \) is the midpoint of \( AB \). This can be done by choosing points \( A \) and \( B \) on \( l \) such that \( AT = TB \).
   - The length of \( AB \) should be \( 2r \) because the diagonal of the square will be equal to the diameter of the circle.

5. **Draw lines \( TA \) and \( TB \):**
   - Draw lines \( TA \) and \( TB \) from \( T \) to \( A \) and \( B \) respectively.

6. **Find the intersection points of \( TA \) and \( TB \) with the circle:**
   - Extend \( TA \) and \( TB \) to intersect the circle at points \( U \) and \( V \) respectively. Ensure that \( U \) and \( V \) are on the same side of the line \( l \) as the circle.

7. **Construct the square \( PQVU \):**
   - Draw perpendiculars from \( U \) and \( V \) to the line \( l \). Let the feet of these perpendiculars be \( P \) and \( Q \) respectively.
   - Since \( T \) is the midpoint of \( AB \) and \( AB \) is parallel to \( PQ \), \( PQ \) will be equal in length to \( AB \).
   - Connect \( P \) to \( Q \), \( Q \) to \( V \), \( V \) to \( U \), and \( U \) to \( P \) to form the square \( PQVU \).

8. **Verify the construction:**
   - Ensure that \( PQVU \) is a square by checking that all sides are equal and all angles are right angles.
   - Verify that \( U \) and \( V \) lie on the circle and \( P \) and \( Q \) lie on the line \( l \).

\[
\boxed{PQVU}
\]