hmmm I'd say the center of the circle will end up at the center of the triangle, namely the Centroid of the triangle circumscribed in the circle.
[tex]\bf \textit{Centroid of a Triangle}
\begin{array}{llll}
\stackrel{E(2,2)~~F(2,-2)~~G(6,-2)}{\left(\cfrac{2+2+6}{3}\quad ,\cfrac{2-2-2}{3}\quad \right)}
\end{array}[/tex]
