Question

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

Answer 1

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 ΔAPD,

∠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^circ`

= 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.