The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

Let ABCD be a cyclic quadrilateral and PRQS be the quadrilateral formed by the angle bisectors of angle ∠A, ∠C, ∠B and ∠D.

In ∆APD,

⇒ ∠PAD + ∠ADP + ∠APD = 180° [By angle sum property of a triangle] …..(1)

In ∆BQC,

⇒ ∠QBC + ∠BCQ + ∠BQC = 180° [By angle sum property of a triangle] …..(2)

Adding (1) and (2), we get

⇒ ∠PAD + ∠ADP + ∠APD + ∠QBC + ∠BCQ + ∠BQC = 180° + 180° = 360° …..(3)

From figure,

⇒ ∠PAD + ∠QBC + ∠BCQ + ∠ADP = $\dfrac{1}{2}$[∠A + ∠B + ∠C + ∠D]

⇒ ∠PAD + ∠QBC + ∠BCQ + ∠ADP = $\dfrac{1}{2} \times 360°$ = 180°

Substituting above value in equation 3 we get :

⇒ ∠APD + ∠BQC + 180° = 360°

⇒ ∠APD + ∠BQC = 360° - 180°

⇒ ∠APD + ∠BQC = 180°

But, these are the sum of opposite angles of quadrilateral PRQS.

∴ PRQS is also a cyclic quadrilateral [As sum of opposite angles in a cyclic quadrilateral = 180°.]

Hence, proved that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.