Prove that the angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral.
 

A cyclic quadrilateral ABCD in which AP, BP, CR and DR are angle bisectors of angles A, B, C, D respectively and form a new quadrilateral PQRS (Fig.)
To Prove: PQRS is cyclic
i.e. ∠Q + ∠S = 180° and ∠R + ∠P = 180°
Proof: In ΔDRC,  ∠DRC + ∠RCD + ∠CDR = 180° ....................(i) 
In ΔAPB,  ∠APB + ∠PBA + ∠BAP = 180°            ....................(ii)
Adding equations (i) and (ii) we get,
∠DRC + ∠RCD + ∠CDR + ∠APB + ∠PBA + ∠BAP = 360°   ....................(iii)
⇒ ∠DRC + ∠C + ∠D + ∠APB + ∠B + ∠A = 360°
⇒ (∠DRC + ∠APB)+ (∠A + ∠B + ∠C + ∠D) = 360°
⇒ (∠DRC + ∠APB)+ × 360° = 360°
⇒ ∠DRC + ∠APB = 180°
but ∠DRC = ∠QRS (Vertically opposite angles)
and ∠APB = ∠QPS 
∴ ∠QRS + ∠QPS = 180°
Thus PQRS is cyclic, as opposite pairs of angles are supplementary.