How many tangents can a circle have?

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

$A$ quadrilateral $ABCD$ is drawn to circumscribe a circle (see figure). Prove that $AB + CD = AD + BC$.

From a point $Q$,the length of the tangent to a circle is $24 \, cm$ and the distance of $Q$ from the centre is $25 \, cm$. The radius of the circle is (in $cm$)

Prove that in two concentric circles,the chord of the larger circle,which touches the smaller circle,is bisected at the point of contact.