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  • Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.

Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.

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Solution
According to question
 

To Prove : QORP is a cyclic quadrilateral.
OQ \perp PQ, OR \perp PR   (\because  PQ, PR are tangents)
Hence, \angle OQP+\angle ORP= 180^{\circ}\cdots (i)        
We know that sum of interior angles of quadrilateral is 360^{\circ}
\angle OPQ+\angle OPR+\angle ORP+\angle ROQ= 360^{\circ} [Given (i)]
180^{\circ}+\angle QPR+\angle ROQ= 360^{\circ}

\angle QPR+\angle ROQ= 180^{\circ}
Here we found that sum of opposite angles of quadrilateral is 180^{\circ}
Hence QORP is a cyclic quadrilateral.
Hence proved

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