Question

prove that the quadilaterals formed by the angel bisectors of a cyclic quadrilateral is also cyclic

Answer 1

let ABCD be a cyclic quadrilateral and EFGH the quadrilateral formed by the angle bisectors of  ABCD 

<FEH = <AEB  = 180 - <EAB - <EBA  (ANGLE SUM PROPERTY)
                         = 180 - 1/2 (<A+<B)

<FGH = <CGD = 180 - <GCD - <GDC (ASP)\
                         =180 - (<C+<D)

<FEH + <FGH = 180 - 1/2(<A+<B) + 180 - 1/2(<C+<D)
                        = 360 - 1/2(<A + <B + <C + <D)
                        =360 - 180
                        =180°
∴ EFGH is a cyclic quadrilateral