Question

Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral ABCD is also cyclic.

Answer 1

Given: In cyclic ABCD the bisectors formed a quadrilateral ABCD.

To prove: PQRS is a cyclic quadrilateral.

Proof: In cyclic quadrilateral ABCD, AR and BS be the bisectors of ∠A and ∠B.
So, ∠ 1 = ∠ A/2 and ∠ 2 = ∠ B/2

In Δ ASB, ∠ RSP is the exterior angle
So ∠ RSP = ∠1 + ∠2
∠ RSP = `"∠A"/2 + "∠B"/2`                        ....(i)
Similarly, ∠ PQR = `"∠C"/2 + "∠D"/2`       ....(ii)

Adding (i) and (ii),
∠ PQR + ∠ RSP = `1/2`(∠A + ∠B + ∠C + ∠D)
 = `1/2` x 360° = 180°

∠ PQR + ∠ RSP = 180°

But these are the opposite angles of quadrilateral PQRS 
Hence PQRS is a cyclic quadrilateral.
Hence proved.