Prove that a cyclic parallelogram is a rectangle.

Let ABCD be the cyclic parallelogram.

We know that opposite angles of a parallelogram are equal.

⇒ ∠A = ∠C …….(1)

⇒ ∠B = ∠D …….(2)

We know that the sum of opposite angles of a cyclic quadrilateral is 180°.

⇒ ∠A + ∠C = 180°

⇒ ∠A + ∠A = 180° (From equation (1))

⇒ 2∠A = 180°

⇒ ∠A = $\dfrac{180°}{2}$

⇒ ∠A = 90°.

We know that,

If one angle of a parallelogram is 90°, then it is a rectangle.

Thus, quadrilateral ABCD is a rectangle.

Hence, proved that a cyclic parallelogram is a rectangle.