Show that rectangle is the only parallelogram that can be inscribed in a circle.

Let ABCD be a parallelogram inscribed in a circle.

Since, ABCD is a cyclic quadrilateral.

$$\angle A + \angle C = 180^\circ \quad \text{...(i)}$$

[Opposite angles of a cyclic quadrilateral]

Since, ABCD is a parallelogram.

$$\angle A = \angle C \quad \text{...(ii)}$$

[Opposite angles of a parallelogram are equal]

From (i) and (ii),

$$2\angle A = 180^\circ$$

$$\angle A = 90^\circ$$

Similarly, $\angle B = \angle D = 90^\circ$.

A parallelogram with all angles $90^\circ$ is a rectangle.

Hence, only a rectangle among parallelograms can be inscribed in a circle.