Question

Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic

Answer 1

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.