perspective transformation of a grid
We can apply a linear transformation to get into the situation
$$
\begin{align*}
x' &amp;= \frac{x+C}{Ax+By+1} \\
y' &amp;= \frac{y+D}{Ax+By+1}
\end{align*}
$$
The Jacobian of the mapping
$$
J = \begin{vmatrix} \frac{\partial x'}{\partial x} &amp; \frac{\partial x'}{\partial y} \\ \frac{\partial y'}{\partial x} &amp; \frac{\partial y'}{\partial y} \end{vmatrix}
$$
is the infinitesimal area element of the mapping. That is, it is the area of an epsilonic rectangle around the image of a point $(x,y)$.
In our case, the Jacobian turns out to be
$$
J(x,y) = \frac{1-AC-BD}{(Ax+By+1)^3}.
$$
This function is either increasing or decreasing in each of the inputs $x,y$, and so after the initial affine transformation, one of the corner rectangles has the maximal area (and same for the minimal area). The linear transformation might change the sense of the axes, but otherwise only affects the area by a constant factor.