Question

Prove that angle bisectors of cyclic quadrilaterals formed a cyclic quadrilateral.

Answer 1

Given :

A cyclic quadrilateral ABCD.

AP, BP, CR and DR are respectively bisector of
∠A, ∠B, ∠C, and ∠D.

To Proved : PQRS is a cyclic quadrilateral.

Proof : To Prove PQRS a cyclic quadrilateral, we should prove

∠APB + ∠CRD = 180°

We know that sum of opposite angles of a cyclic quadrilateral is 180°

Now in ∆APB and ∆CRD

⇒ One pair of opposite angles of quadrilateral PQRS arc supplementary.

⇒ PQRS is a cyclic quadrilateral.