prove that a cyclic parallelogram must be a rectangle .

To prove that a cyclic parallelogram must be a rectangle, we need to use the properties of cyclic quadrilaterals and parallelograms.

### Definitions and Properties:
1. **Cyclic Parallelogram**: A parallelogram is cyclic if all its vertices lie on a common circle.
2. **Parallelogram**: A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
3. **Cyclic Quadrilateral**: A quadrilateral is cyclic if its vertices lie on a single circle. In a cyclic quadrilateral, opposite angles sum up to 180 degrees.

### Proof:

1. **Properties of Cyclic Quadrilaterals**:
  - For a cyclic quadrilateral, the sum of the opposite angles is \(180^\circ\). That is, if \(ABCD\) is a cyclic quadrilateral, then:
    \[
    \angle A + \angle C = 180^\circ
    \]
    \[
    \angle B + \angle D = 180^\circ
    \]

2. **Properties of a Parallelogram**:
  - In a parallelogram, opposite angles are equal. So, if \(ABCD\) is a parallelogram, then:
    \[
    \angle A = \angle C
    \]
    \[
    \angle B = \angle D
    \]

3. **Combining the Properties**:
  - Since \(ABCD\) is both a parallelogram and a cyclic quadrilateral, we use both sets of properties.

  - From the property of the cyclic quadrilateral, we have:
    \[
    \angle A + \angle C = 180^\circ
    \]

  - Since opposite angles in a parallelogram are equal, we also have:
    \[
    \angle A = \angle C
    \]

  - Substitute \(\angle C\) from the parallelogram property into the cyclic quadrilateral property:
    \[
    \angle A + \angle A = 180^\circ
    \]

    \[
    2 \angle A = 180^\circ
    \]

    \[
    \angle A = 90^\circ
    \]

  - Thus, each angle in the parallelogram \(ABCD\) is \(90^\circ\), which means all the angles in the parallelogram are right angles.

4. **Conclusion**:
  - Since a parallelogram with all angles equal to \(90^\circ\) is a rectangle, we conclude that a cyclic parallelogram must be a rectangle.

### Summary:
In a cyclic parallelogram, the property of the cyclic quadrilateral (opposite angles sum to \(180^\circ\)) combined with the property of the parallelogram (opposite angles are equal) shows that each angle in the parallelogram is \(90^\circ\). Therefore, the parallelogram must be a rectangle.