#### Question

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

#### Solution

Given – ABCD is a cyclic quadrilateral and PQRS is a

Quadrilateral formed by the angle Bisectors of angle ∠A, ∠B , ∠C and ∠D

To prove – PQRS is a cyclic quadrilateral. Proof – In APD ∠PAD + ∠ADP + ∠APD = 180° .…. (1)

Similarly, IN ∆BQC,

∠QBC + ∠BCQ + ∠BQC = 180° …………(2)

Adding (1) and (2) .we get

∠PAD + ∠ADP + ∠APD + ∠QBC + ∠BCQ + ∠BQC = 180° +180°

∠PAD + ∠ADP + ∠QBC + ∠BCQ + ∠APD + ∠BQC = 360°

But ∠PAD + ∠ADP + ∠QBC + ∠BCQ =`1/2` [∠A + **∠**B + ∠C + ∠D]

= `1/2 xx 360° = 180°`

∴ ∠APD + ∠BQC = 360° -180° = 180° [from (3)]

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

∴ Quad. PRQS is a cyclic quadrilateral.