Question regarding polygons

Question

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?

Answer 1

The set of possible side lengths is exactly the set $S$ of all positive numbers of the form $\sqrt{a^2+b^2}$ for integers $a$ and $b$. In increasing order, its elements are $$\{1, \sqrt2, 2, \sqrt 5, \sqrt8, 3, \sqrt{10}, \ldots\}.$$ This set $S$ is well-ordered, which means that every nonempty subset of $S$ contains a minimal element. (It is well-ordered because it is order-isomorphic to $\Bbb N$, the positive integers, which is also well-ordered.)

Since any constructible lattice $n$-gon must have a side length from $S$, the set of constructible side lengths $C$ is a subset of $S$. Since $S$ is well-ordered, $C$ is either empty or has a minimum element, say $m$. So either there is no constructible lattice $n$-gon at all, or else there is at least one with the minimum side length $m$.

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