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

Explanation

Given: `ABCD` is a cyclic quadrilateral.
To prove: The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.
Proof: `AH`, `BF`, `CF` and `DH` are the angle bisectors of `/_A`, `/_B`, `/_C` and `/_D`.
`=>/_FEH=/_AEB` .......(1) [Vertically opposite angles]
`=>/_FGH=/_DGC` .......(2) [Vertically opposite angles]
Adding (1) and (2),
`=>/_FEH+/_FGH=/_AEB+/_DGC` .......(3)
Now, By angle sum property of a triangle,
`=>/_AEB=180^@−(1/2/_A+1/2/_B)` ........ (4)
`=>/_DGC=180^@−(1/2/_C+1/2/_D)` .......(5)
Substituting equation (4) and equation (5) in equation (3)
`=>/_FEH+/_FGH=180^@−(1/2/_A+1/2/_B)+180^@−(1/2/_C+1/2/_D)`
`=>/_FEH+/_FGH=360^@−1/2(/_A+/_B+/_C+/_D)`
`=>/_FEH+/_FGH=360^@−1/2xx360^@`
`=>/_FEH+/_FGH=180^@`
Since The sum of opposite angles of quadrilateral `EFGH=180^@`
`∴ EFGH` is a cyclic quadrilateral.
`therefore` The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.