Any parallelogram cannot be cyclic. For any quadrilateral to be cyclic the sum of the opposite angles should be 180 deg. So the quadrilaterals that can be cyclic are definitely a square and a rectangle.

Proof that a Cyclic Quadrilateral is a Rectangle

To prove that a cyclic quadrilateral is a rectangle, we need to show that all four angles of the quadrilateral are right angles.

Let's consider a cyclic quadrilateral ABCD, where the vertices A, B, C, and D lie on a circle.

1. Properties of a Cyclic Quadrilateral

Before we begin the proof, let's understand some properties of a cyclic quadrilateral:

- Opposite angles of a cyclic quadrilateral are supplementary. This means that the sum of the opposite angles is 180 degrees.
- The opposite angles of a cyclic quadrilateral are inscribed angles that intercept the same arc on the circle.
- The angle formed by a tangent and a chord is equal to the angle subtended by the chord in the opposite segment.
- The opposite sides of a cyclic quadrilateral are equal in length.

2. Proof

Now, let's proceed with the proof:

Step 1: Let ∠BAD and ∠BCD be two opposite angles of the quadrilateral ABCD.

Step 2: Since ABCD is a cyclic quadrilateral, ∠BAD and ∠BCD are inscribed angles that intercept the same arc AD on the circle.

Step 3: By the property of inscribed angles, ∠BAD = ∠BCD.

Step 4: Since opposite angles of a cyclic quadrilateral are equal, ∠BAD = ∠BCD = x (say).

Step 5: Sum of the angles of a quadrilateral is 360 degrees. In ABCD, we have:
∠BAD + ∠ABC + ∠BCD + ∠CDA = 360 degrees.

Step 6: Substituting the value of ∠BAD and ∠BCD as x, we get:
x + ∠ABC + x + ∠CDA = 360 degrees.

Step 7: Simplifying the equation, we have:
2x + ∠ABC + ∠CDA = 360 degrees.

Step 8: Since opposite angles of a cyclic quadrilateral are supplementary, ∠ABC + ∠CDA = 180 degrees.

Step 9: Substituting the value of ∠ABC + ∠CDA as 180 degrees in the equation from Step 7, we get:
2x + 180 degrees = 360 degrees.

Step 10: Simplifying the equation, we have:
2x = 180 degrees.

Step 11: Dividing both sides of the equation by 2, we get:
x = 90 degrees.

Step 12: Therefore, both ∠BAD and ∠BCD are right angles.

Step 13: Since all four angles of the quadrilateral ABCD are right angles, it can be concluded that ABCD is a rectangle.

Hence, we have proved that a cyclic quadrilateral is a rectangle.