Find the equation of the circle circumscribing the triangle formed by the lines x + $y=6,2 x+y=4$ and $x+2 y=5$
Solving the equations we get the coordinates of the triangle:
The required circle equation
$\left|\begin{array}{cccc}x^{2}+y^{2} & x & y & 1 \\ (-2)^{2}+8^{2} & -2 & 8 & 1 \\ 1^{2}+2^{2} & 1 & 2 & 1 \\ 7^{2}+(-1)^{2} & 7 & -1 & 1\end{array}\right|=0$
Using Laplace Expansion
$\left(x^{2}+y^{2}\right)\left|\begin{array}{ccc}-2 & 8 & 1 \\ 1 & 2 & 1 \\ 7 & -1 & 1\end{array}\right|-x\left|\begin{array}{ccc}68 & 8 & 1 \\ 5 & 2 & 1 \\ 50 & -1 & 1\end{array}\right|+y\left|\begin{array}{ccc}68 & -2 & 1 \\ 5 & 1 & 1 \\ 50 & 7 & 1\end{array}\right|-$
$\left|\begin{array}{ccc}68 & -2 & 8 \\ 5 & 1 & 2 \\ 50 & 7 & -1\end{array}\right|=0$
$\Rightarrow 27\left(x^{2}+y^{2}\right)-459 x-513 y+1350=0$
$\Rightarrow x^{2}+y^{2}-17 x-19+50=0$