This isn't so bad, so that's good. Your suggested set of points, however, do not submit to the same sort of analysis as what I'm about to give because it's a square - so the sides are parallel (so the f1 and f2 are... inconvenient, even in the projective plane; if you know projective geometry, mention it, and I'll update).
The general process is to find the equations of lines, find where they intersect, and then make more lines, and find where they intersect. Ok? Great.
First, the point $O$ is nothing more than the midpoint of DB. Perhaps the figure in general is not symmetric, in which case it's the intersection of DB and AC. Say the coordinates of A are $\left[ \begin{array}{cc} a_x \\ a_y \end{array} \right]$, the coordinates of B are $\left[ \begin{array}{cc} b_x \\ b_y \end{array} \right]$ and so on.
Then the lines DB and AC can be parameterized by the equations
$\overline{DB} = \left[ \begin{array}{cc} b_x \\ b_y \end{array} \right] + \left( \left[ \begin{array}{cc} d_x \\ b_y \end{array} \right] - \left[ \begin{array}{cc} b_x \\ b_y \end{array} \right] \right)t$
$\overline{AC} = \left[ \begin{array}{cc} a_x \\ a_y \end{array} \right] + \left( \left[ \begin{array}{cc} c_x \\ c_y \end{array} \right] - \left[ \begin{array}{cc} a_x \\ a_y \end{array} \right] \right)s$
Now you set equal and solve for s and t. How? Considering x and y components separately, you have two equations in 2 variables - use a matrix.
$\left[ \begin{array}{cc} d_x - b_x & c_x - a_x \\ d_y - b_y & c_y - a_y \end{array} \right] \left[ \begin{array}{cc} t \\ s \end{array} \right] = \left[ \begin{array}{cc} b_x - a_x \\ b_y - a_y \end{array} \right]$
And this procedure will give you the intersection of those two lines. In fact, this will give you the intersection between any two non-parallel lines. Good.
So with this, we found the point $O$. Do this with the lines AD and BC to find their focal point, f2. Repeat with the lines AD and BC to find their focal point, f1. Then we can repeat with the lines $\overline{f_1O}$ and $\overline{AD}$ to find z. $\overline{f_1O}$ and $\overline{BC}$ to find x. And because I've given almost everything away, rinse, wash, and repeat for w and y.
Does that make sense?