Polygon with n sides

Question

I'm wondering if this is true: If a polygon with n sides whose vertices are points of integer coordinates and the sides are equal, then n is even. can you prove or disprove it?

This is not a duplicate, as the polygon is not required to be regular. In particular the hexagon $(0,0),(5,0),(8,4),(5,8),(0,8),(-3,4)$ meets the requirement (and has an even number of sides) — Ross Millikan, Oct 17 '13 at 18:17

score 4 · Accepted Answer · answered Oct 18 '13 at 12:11

This is a very nice question. The answer to it is given by the paper

D.G. Ball, "Constructability of regular and equilateral polygons on square pinboards", Mathematical Gazette 57 (1973) p. 119-122.

Theorem. There exists an equilateral lattice $n$-gon in the plane if and only if $n$ is even. Furthermore, if $n$ is even, this polygon can be taken to be convex.

You can find a proof and much more here.

Note: A planar polygon is called a lattice polygon if its vertices belong to the usual square lattice on the plane.

Polygon with n sides

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