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Question

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

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Solution

Given: ABCD  is a cyclic quadrilateral and PQRS  is a quadrilateral

d6In \triangle APD  :

PAD + ADP + APD = 180^o  … … … … … (i)

In \triangle BQC

QBC + BCQ + BQC = 180^o  … … … … … (ii)

Adding (i) and (ii)

PAD + ADP + APD + QBC + BCQ + BQC = 360^o

\frac{1}{2} (BAD + ADC + DCB + CBA) + APD + BQC = 360^o

180 + APD + BQC = 360^o \Rightarrow APD + BQC = 180^o

Therefore PQRS  is a cyclic quadrilateral as the opposite angles are supplementary.

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