A point $\mathrm{P}$ is at a distance of $13 \mathrm{cm}$ from the centre $\mathrm{O}$ of a circle. If the radius of the circle is $5 \mathrm{cm}$. Find the length of the tangent from $\mathrm{P}$ to the circle in $\mathrm{cm}$.

Prove that opposite sides of the quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

If the circle touches all the four sides of the quadrilateral $\mathrm{ABCD}$, prove that $\mathrm{AB}+\mathrm{CD}=\mathrm{BC}+\mathrm{AD}$.

$∆\mathrm{ABC}$ is isosceles with $\mathrm{AB}=\mathrm{AC}$. The incircle of the $∆\mathrm{ABC}$ touches $\mathrm{BC}$ at $\mathrm{P}$. Prove that $\mathrm{BP}=\mathrm{CP}$.