Question

Prove that the length of tangent drawn from an external point to a circle are equal and also prove the following if a circle touches all the four sides of parallelogram show that the parallelogram is a rhombus?

Answer 1

Proof : We know a tangent to circle is perpendicular to the radius through the point of contact.
$\therefore \angle OPT=\angle OQT={90}^{\circ }$
In triangle OPT and OQT
OT=OT (Common)
OP=OQ (Radius of the circle)
$\angle OPT=\angle OQT\left({90}^{\circ }\right)$
$\therefore \Delta OPT$
$\approx$
$\Delta OQT$ (RHS congruence criterion)
$\Rightarrow PT=TQand\angle OTP=\angle OTQ\left(CPCT\right)PT=TQ$