Problems
Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero.
Prove that every equation of the form ax+by=c, where a, b, and c are all rational numbers and a and b are not both zero, is in fact a PZ line.
Prove that if two distinct PZ lines m and n are not parallel in the usual Euclidean sense, then they have a PZ point in common.
In terms of the possible parallel axioms for geometry, which geometry does PZ geometry most resemble: projective, Euclidean, or hyperbolic geometry?
Progress
PZ means Pixel + Zoom geometry. The PZ points are $(x,y)$ of the coordinate plane such that $x$ and $y$ are rational numbers and the $PZ$ lines are the lines of Euclidean geometry which pass through (at least) two $PZ$ points.
I have tried using geometer's sketchpad to come up with PZ lines, but I'm not sure how to start the proof. Should I just use random variables to represent ratios for a and b? How do I show that a and b cannot be zero?
Hint for the first question: if $(x_1, y_1)$ and $(x_2,y_2)$ are on the same line, then either:
a) $x_1=x_2$ (what is the equation for the line in this case?)
or
b)this line has the form $y-y_1=\frac{y_1-y_2}{x_1-x_2}(x-x_1)$ And all the numbers in sight are rational. I'll leave it to you to manipulate this equation into the appropriate form.
For the second question: What does it mean, in terms of $a$, $b$, and $c$ for the lines to be parallel? When the lines are not parallel, can you find a common solution to the two equations? Keep track of your steps to show that the common solution has rational coordinates.
The last question is pretty much answered by the second.