Solution

Given: ABCD is a cyclic quadrilateral. Its angle bisectors from a quadrilateral PQRS.

To Prove: PQRS is a cyclic quadrilateral.

Proof : ∠1 + ∠2 + ∠3 = 180o ...(i) [Sum of the angles of a Δ is 180o]

∠4 + ∠5 + ∠6 = 180o ...(ii) [Sum of the angles of a Δis  180o]

∴ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6  =360.. ....(iii) [Adding (i) and (ii)]

But ∠2 + ∠3 + ∠6 + ∠5 = [∠A + ∠B + ∠C + ∠D]

=. 360o = 180o [Sum of the angles of quadrilateral is 360o]

∴ ∠1 + ∠4 = 360o– (∠2 + ∠3 + ∠6 + ∠5)

∴ PQRS is a cyclic quadrilateral.

[If the sum of any pair of opposite angles of a quadrilateral is 180o, then the quadrilateral is a cyclic] Hence Proved.