What regular polygons can be constructed on the points of a regular orthogonal grid?

We answer Part $4$ for a square grid. We may assume that this is the grid of points with integer coordinates.

Then any line between grid points has rational coefficients. Thus, if two such lines meet, they meet at a point that has rational coordinates.

Suppose that we draw a finite number of distinct lines. Let $N$ be the least common multiple of the denominators of coordinates of intersection points of these lines

Now imagine constructing a figure that uses some of these intersection points (including possibly original grid points). If we scale this figure by the factor $N$, we have scaled it to have integer coordinates. Thus, up to similarity, no more figures can be drawn using intersection points than can be drawn using original grid points. In particular, intersection points do not help in drawing a regular octagon.