Two circles touch each other externally at $\mathrm{P}$. Prove that the common tangent bisects the other two common tangents. 

$PA$ and $PB$ are tangents from an external point $P$ to a circle with centre $O$. $LN$ touches the circle at $M$. Prove that $PL+LM=PN+MN.$

In the figure, a $∆ABC$ is drawn to circumscribe a circle of radius 4cm such that the segments $BD$ and $DC$ are of lengths $8 \mathrm{cm}$ and $6 \mathrm{cm}$ respectively. Find the lengths of sides $AB$ and $AC$, when area of $∆ABC$ is $84 {\mathrm{cm}}^2$.