Prove that a cyclic parallelogram is a rectangle.

1. In any parallelogram, the opposite angles are equal. Thus, for parallelogram PQRS:

   ∠P = ∠R and ∠Q = ∠S  ... (1)

2. For a cyclic quadrilateral, the sum of either pair of opposite angles is always 180°. Therefore:

   ∠P + ∠R = 180°

3. From equation (1), we know that ∠P = ∠R. Substituting this in the above equation:

   ∠P + ∠P = 180°

4. This simplifies to:

   2∠P = 180°

   ∠P = 90°

5. In a parallelogram, if one interior angle is 90°, all the other angles must also be 90° due to the properties of parallelograms.

Hence, since all angles of parallelogram PQRS are 90°, we have successfully demonstrated that PQRS is a rectangle.

Conclusion

To summarize, the properties of cyclic quadrilaterals and parallelograms together establish that a cyclic parallelogram must always have right angles at all its vertices. Therefore, we can conclusively prove that a cyclic parallelogram is a rectangle.

The steps above provide a clear mathematical basis to prove that a cyclic parallelogram is a rectangle, reinforcing the unique relationship between cyclic quadrilaterals and rectangles.