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

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

KnowledgeBoat · Accepted Answer

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 = [∠A + ∠B + ∠C + ∠D] ⇒ ∠PAD + ∠QBC + ∠BCQ + ∠ADP = = 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.

Mathematics

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

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