Question

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

Answer 1

Since tangent at a point to a circle is perpendicular to the radius through the point
$\therefore OQ\perp QP\&OR\perp RP$
$\Rightarrow \angle OQP={90}^{\circ }\&\angle ORP={90}^{\circ }$
$\Rightarrow \angle OQP+\angle ORP={90}^{\circ }={180}^{\circ }----\left(i\right)$
In quadrilateral, OQPR,
$\angle OQP+\angle QPR+\angle QOR+\angle ORP={360}^{\circ }$
$\Rightarrow (\angle QPR+\angle QPR)+(\angle OQP+\angle ORP)={360}^{\circ }\left(From\right(i)$
$\Rightarrow \angle QPR+\angle QOR={180}^{\circ }-------\left(ii\right)$
From$\left(i\right)\&\left(ii\right)$, we can say that quadrilateral QORP is cyclic