Example. Find the equation of the quadrilateral with vertices $A(-3,-1), B(0,-4)$, $C(4,0), D(-1,1)$.
Solving, by the two-point form, the equations of $A C$ and $B D$ are
$$
x-7 y-4=0, \quad 5 x+y+4=0 \text { . }
$$

Substituting (*) we get
$$
\begin{array}{l}
|x-7 y-4|+r|5 x+y+4|+a x+b y+ \\
c=0 .
\end{array}
$$

Substituting the coordinates of the vertices, we solve to get $r=1, a=-2, b=2$, $c=-16$. Thus, the equation of $A B C D$ is
$$
\begin{array}{l}
|x-7 y-4|+|5 x+y+4|-2 x-2 y- \\
16=0 .
\end{array}
$$