The homogenous representation of a circle is given by ${x}^{2}+{y}^{2}+2gxz+2fyz+c{z}^{2}=0$ (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?