Find the length of PC Here PE is the tangent of the two circle. PA = 12 ; CD/AB = 2
 Find the length of PC [Source: BDMO]
]1

It holds that $$ PE^2= 
\boxed{PA \cdot PD = PB \cdot PC}.$$ 
Taking advantage of the equality in the box, we have:
$\begin{array}[t]{l}
12\cdot (PC + CD) = (12+AB) \cdot PC\\
12 PC +12 CD  = 12 PC + AB \cdot PC\\
PC = \dfrac{12CD}{AB}=24
\end{array}$
