Question

prove that a quadrilateral made from bisector of angles of cyclic quadrilateral is also a cyclic quadrilateral.​

Answer 1

Step-by-step explanation:

→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°.

→Answers :

As the sum of a pair of opposite angles of quadrilateral PQRS is 180°. Therefore, quadrilateral PQRS is cyclic.