Prove that, in a cyclic quadrilateral, if one pair of opposite sides is equal, the other pair is parallel.

A circle is touching the side $\mathrm{BC}$ of $∆ABC$ at point $\mathrm{P}$ and touching $\mathrm{AB}$ and $\mathrm{AC}$ produced at $\mathrm{Q}$ and $R$, respectively. Prove that $\mathrm{AQ}=\frac{1}{2}$(perimeter $∆ABC$).

In the given figure, $\mathrm{O}$ is the centre of a circle of radius $5\mathrm{cm}$, $\mathrm{T}$ is a point such that $\mathrm{OT}=13 \mathrm{cm}$ and $\mathrm{OT}$ intersects the circle at $\mathrm{E}$. If  $\mathrm{AB}$ is the tangent to the circle at $\mathrm{E}$, then find the length of $\mathrm{AB}$.

In the given figure, a circle is inscribed in a quadrilateral $\mathrm{ABCD}$ in which $\angle \mathrm{B}=90°$. If  $\mathrm{AD}=23 \mathrm{cm}, \mathrm{AB}=29 \mathrm{cm}$ and $\mathrm{DS}=5 \mathrm{cm}$, then find the radius $\left(\mathrm{cm}\right)$ of the circle.

In the given figure, a circle touches all the four sides of a quadrilateral $\mathrm{ABCD}$, whose three sides are $\mathrm{AB}=6 \mathrm{cm}, \mathrm{BC}=7 \mathrm{cm}$ and $\mathrm{CD}=4 \mathrm{cm}$. Find $\mathrm{AD} \left(\mathrm{cm}\right)$.