Similitude Center

Example 1

The point of intersection of the two crossing tangents is called the internal similitude center. The point of intersection of the extensions of the other two tangents is called the external similitude center.

[PDF] [TEX]

\documentclass{standalone}
\usepackage{tkz-euclide} 
\begin{document}  
\begin{tikzpicture}
   \tkzDefPoints{0/0/O,4/-5/A}
   \tkzDrawCircle[R,fill=red!50,opacity=.3](O,3)
   \tkzDrawCircle[R,fill=blue!50,opacity=.3](A,1) 
   \tkzDefIntSimilitudeCenter[R](O,3)(A,1) \tkzGetPoint{I}
   \tkzDefExtSimilitudeCenter[R](O,3)(A,1) \tkzGetPoint{J}
   \tkzDefTangent[from with R= I](O,3)  \tkzGetPoints{D}{E}
   \tkzDefTangent[from with R= I](A,1)  \tkzGetPoints{D'}{E'}
   \tkzDefTangent[from  with R= J](O,3) \tkzGetPoints{F}{G}
   \tkzDefTangent[from with R= J](A,1)  \tkzGetPoints{F'}{G'}
   \tkzDrawSegments[add = .5 and .5,color=red](D,D' E,E')
   \tkzDrawSegments[add= 0 and 0.25,color=blue](J,F J,G)
   \tkzDrawPoints(O,A,I,J,D,E,F,G,D',E',F',G')
   \tkzLabelPoints(O,A,I,J,D,E,F,G,D',E',F',G')
  \end{tikzpicture} 
\end{document}