Question

Prove that a cyclic rhombus is a squre

Answer 1

Given---> ABCD is a cyclic rhombus .

To prove ---> ABCD is a square.

Proof---> To prove ABCD a square it is sufficient to prove that all sides of ABCD are equal and all angles of ABCD are right angles.

Now,

AB = BC = CD = DA , because all sides of rhombus are equal.

Now, opposite angles of rhombus are equal , so,

∠ A = ∠ C ( opposite angles of rhombus )

∠ A + ∠ C = 180° ( Opposite angles of cyclic ||gm )

Putting ∠A = ∠C , in it we get,

∠ A + ∠A = 180°

=> 2 ∠A = 180°

=> ∠ A = 180° / 2

=> ∠ A = 90°

So , ∠ A = ∠B = 90°

Similarly we can prove that,

∠ C = ∠ D = 90°

So , ABCD is a square.

So a cyclic rhombus is a square .

Hence proved

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