What is the easier way to find the circle given three points?

Given three points $({x}_{1},{y}_{1}),({x}_{2},{y}_{2})$, and $({x}_{3},{y}_{3})$, if

$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\ne \frac{{y}_{3}-{y}_{2}}{{x}_{3}-{x}_{2}}\ne \frac{{y}_{1}-{y}_{3}}{{x}_{1}-{x}_{3}},$

then there will be a circle passing through them. The general form of the circle is

${x}^{2}+{y}^{2}+dx+ey+f=0.$

By substituting $x={x}_{i}\text{}\text{and}\text{}y={y}_{i}$, there will be a system of equation in three variables, that is:

$\begin{array}{rl}\left(\begin{array}{ccc}{x}_{1}& {y}_{1}& 1\\ {x}_{2}& {y}_{2}& 1\\ {x}_{3}& {y}_{3}& 1\end{array}\right)\left(\begin{array}{c}d\\ e\\ f\end{array}\right)& =\left(\begin{array}{c}-({x}_{1}^{2}+{y}_{1}^{2})\\ -({x}_{2}^{2}+{y}_{2}^{2})\\ -({x}_{3}^{2}+{y}_{3}^{2})\end{array}\right).\end{array}$

As there are a lot of things going around, the solution is prone to errors. Maybe this solution also has an error.

Is there a better way to solve for the equation of the circle?