Prove that quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. -Maths 9th

Prove that quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. -Maths 9th
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Solution :- Given: A cyclic quadrilateral ABCD in which the angle bisectors AR, BR, CP O and DP of internal angles A, B, C and D respectively form a quadrilateral PQRS. To prove: PQRS is a cyclic quadrilateral.  Proof: In △ARB, we have 1/2∠A + 1/2∠B + ∠R = 180°   ....(i)   (Since, AR, BR are bisectors of ∠A and ∠B) In △DPC, we have  1/2∠D + 1/2∠C +  ∠P = 180°  ....(ii)   (Since, DP,CP are bisectors of ∠D and ∠C respectively) Adding (i) and (ii),we get 1/2∠A + 1/2∠B + ∠R + 1/2∠D + 1/2∠C + ∠P = 180° + 180° ∠P + ∠R = 360° - 1/2(∠A  + ∠B + ∠C  + ∠D) ∠P + ∠R = 360° - 1/2 x 360° = 360° - 180° ⇒ ∠P + ∠R = 180° As the sum of a pair of opposite angles of quadrilateral PQRS is 180°. Therefore, quadrilateral PQRS is cyclic.
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