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Question

Prove that the quadrilateral formed by the angle bisectors of a cyclic quadrilateral is also cyclic.

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Solution


Given: ABCD is a cyclic quadrilateral.
To Prove: PQRS is a cyclic quadrilateral.
Proof:
C+A=180° (ABCD is a cyclic quadrilateral) Dividing both sides by 2, we get: 12C+12A=90° ...(i) Similarly, B+D=180° (ABCD is a cyclic quadrilateral) Dividing both sides by 2, we get: 12B+12D=90° ...(ii) Adding equations (i) and (ii), we get: 12C+12A+ 12B+12D=180° 12C+12B+12A+12D=180°QCB+QBC+SDA+SAD=180° As AR, BS, CP and DQ are the respective angle bisectors of A, B, C andDCQR+DSR=180° Exterior angle property180°-PQR+180°-PSR=180°PQR+PSR=180°

∴ PQRS is a cyclic quadrilateral.

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