In the given figure, two circles intersect at $\mathrm{S}$ and $T$. $\mathrm{STP}, \mathrm{BSC}$ and$\mathrm{BAP}$ are straight lines. Prove that $\mathrm{PATD}$ is a cyclic quadrilateral.

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 $\mathrm{AB}$.