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Prove that a cyclic rhombus is a square.

Circles

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Answer

Let ABCD be a cyclic rhombus.

Prove that a cyclic rhombus is a square. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

In rhombus opposite angles are equal.

∴ ∠A = ∠C and ∠B = ∠D.

Sum of opposite angles of a cyclic quadrilateral is 180°

⇒ ∠A + ∠C = 180°
⇒ ∠A + ∠A = 180°
⇒ 2∠A = 180°
⇒ ∠A = 90°.

∴ ∠C = 90°.

Similarly,

⇒ ∠B + ∠D = 180°
⇒ ∠B + ∠B = 180°
⇒ 2∠B = 180°
⇒ ∠B = 90°.

∴ ∠D = 90°.

Hence, ∠A = ∠B = ∠C = ∠D = 90°.

In rhombus all sides are equal i.e. AD = BC = AB = CD.

Hence, ABCD is a square as all sides are equal and all the angles are equal to 90°.

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