The homogenous representation of a circle is given by $x^2 + y^2 + 2gxz + 2fyz + cz^2 = 0$ (or, equivalently, if we set $z=1$, $x^2 + y^2 + 2gx + 2fy + c = 0$). Now, given 3 points (in a homogenous form), we can solve a system of linear equations and retrieve the unknowns $f$, $g$ and $c$.
This is all very nice (because of linear algebra), but what do these unknowns actually represent with respect to the circle? Which of these numbers represent the x and y coordinates of a circle and which one represents the radius?
Apparently, $-f$ and $-g$ would be the $x$ and $y$ coordinate of the center of the circle? Why is that the case? I would like to see a proof/derivation of it. Also, what is the radius then?
$$ \begin{aligned} 0&=x^2+y^2+2gx+2fy+c\\ &= x^2+2gx+g^2+y^2+2fy+f^2+(c-g^2-f^2)\\ &=(x+g)^2+(y+f)^2-(f^2+g^2-c) \end{aligned} $$ This equation says that the squared distance of the point $(x,y)$ from the point $(-g,-f)$ is $f^2+g^2-c$, which describes a circle centered at $(-g,-f)$ with radius $\sqrt{f^2+g^2-c}$.