back of the napkin: take the first 4 points to be a unit box with the lower left point at the origin. You can draw any of 4 lines to the 5th point (x,y). These 4 lines will have slopes x:y, x-1:y, x:y-1, x-1,y-1. You can always arrive at a slope ratio that is evenly divisible through some combination of -1 and reducing the ratio to simplest form. An evenly divisible slope ratio will pass through another point. exampl…
A real proof would work for an arbitrary first four points (with any segment between two not containing a lattice point), but I think nikln already provided this :)
doh, forgot to do the last and most important step, generalizing.
