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