\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{through}

\begin{document}
	\begin{tikzpicture}
		\coordinate[label=45:$A$]  (A) at (3,2);
		\coordinate[label=200:$B$] (B) at (-1, -0.5);
		\coordinate[label=0:$C$]   (C) at (4,-2);
		
		% Segments middle points
		\coordinate (AB) at ($(A)!.5!(B)$);
		\coordinate (AC) at ($(A)!.5!(C)$);
		% Points in perpendicular to above middle points
		% The numbers 1 and 2.8 are choosen for the drawing
		% of the help lines, but the center of the circle
		% is found also if we use any other number
		\coordinate (AAB) at ($(AB)!2!-90:(A)$);
		\coordinate (AAC) at ($(AC)!3!90:(A)$);
		
		% Center as intersection of mediatrix
		\coordinate (Center) at (intersection of AB--AAB and AC--AAC);
		
		% Draw the points
		\foreach \p in {A,B,C}  \draw[fill=red] (\p) circle (3pt);
		
		% Draw the center
		\draw[fill=blue] (Center) circle (4pt);
		
		% Draw final circle 
		\node at (Center) [draw=blue, circle through=(A)] {};
		
		% Optionally, draw help lines to see how it is built
		 \draw[help lines] (A)--(B) (A)--(C) (AB) node {AB} 
		   --(AAB) node {AAB} (AC) node{AC}--(AAC) node{AAC};
		\end{tikzpicture}
\end{document}

