Theory:

Result \(6\):
The two direct common tangents drawn to the circles are equal in length.
Explanation:
 
r6.png
 
The two direct common tangents \(AC\) and \(BD\) from \(P\) drawn to the circles are equal in length.
 
\(\Rightarrow\) \(AC\) \(=\) \(BD\)
Proof for the result:
By the result \(3\), we have:
The lengths of the two tangents drawn from an exterior point to a circle are equal.
\(PA\) \(=\) \(PB\) and \(PC\) \(=\) \(PD\).
 
Subtract the above two equations.
 
\(PA\) \(-\) \(PC\) \(=\) \(PB\) \(-\) \(PD\)
 
\(AC\) \(=\) \(BD\)
Example:
In the above given figure if \(PB\) \(=\) \(9\) \(cm\) and \(AC\) \(=\) \(6\) \(cm\), find the length of the tangent \(PD\).
 
Solution:
 
By the result, we know that \(AC\) \(=\) \(BD\).
 
So, \(BD\) \(=\) \(6\) \(cm\).
 
From the figure it is observed that, \(PD\) \(=\) \(PB\) \(-\) \(BD\).
 
\(PD\) \(=\) \(9\) \(cm\) \(-\) \(6\) \(cm\)
 
\(PD\) \(=\) \(3\) \(cm\)
 
Therefore, the length of the tangent \(PD\) \(=\) \(3\) \(cm\).