%$ (x,y) =
%\Bigg(
%\frac{\left|\begin{array}{cc}b_1& c_1 \\ b_2 & c_2\end{array}\right|}
%     {\left|\begin{array}{cc}a_1& b_1 \\ a_2 & b_2\end{array}\right|},
%\frac{-\left|\begin{array}{cc}a_1& c_1 \\ a_2 & c_2\end{array}\right|}
%     {\phantom{-}\left|\begin{array}{cc}a_1& b_1 \\ a_2 & b_2\end{array}\right|}
%\Bigg)
%$
%$ (x,y,w) =
%\Bigg(
%{\left|\begin{array}{cc}b_1& c_1 \\ b_2 & c_2\end{array}\right|},
%{-\left|\begin{array}{cc}a_1& c_1 \\ a_2 & c_2\end{array}\right|},
%{\left|\begin{array}{cc}a_1& b_1 \\ a_2 & b_2\end{array}\right|}\Bigg)
%$
%${\mbox{\small sign}}\left|
%\begin{array}{ccc}
%{x_1} & {y_1} & w_1 \\
%{x_2} & {y_2} & w_2 \\
%{x_3} & {y_3} & w_3 \\
%\end{array}
%\right|
%$
$
{\mbox{\small sign}}\left|
\begin{array}{ccc}
{x_1} & {y_1} & 1 \\
{x_2} & {y_2} & 1 \\
{x_3} & {y_3} & 1 \\
\end{array}
\right| =
{\mbox{\small sign}}\left|
\begin{array}{ccc}
{x_1 - x_3} & {y_1 - y_3} & 0 \\
{x_2 - x_3} & {y_2 - y_3} & 0 \\
{x_3} & {y_3} & 1 \\
\end{array}
\right|
$