Given – ABCD is a rectangle circumscribed in circle with centre O.

To prove – ABCD is a square.

Property – Lengths of the two tangents drawn from an external point to a circle are equal.

Answer –

We know that, opposite sides of a rectangle are equal.

∴ AB = CD & AD = BC ………(1)

As lengths of the two tangents drawn from an external point to a circle are equal.

∴ AP = AS ………(2)

∴ BP = BQ ………(3)

∴ CR = CQ ………(4)

∴ DR = DS ………(5)

Adding (1), (2), (3) & (4),

∴ AP + BP + CR + DR = AS + BQ + CQ + DS

∴ (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

∴ AB + CD = AD + BC ………from figure

∴ AB + AB = BC + BC ………from (1)

∴ 2AB = 2BC

∴ AB = BC

Therefore, adjacent sides of ABCD are equal.

Rectangle with equal adjacent sides is a square.

Hence, ABCD is a square.