The Torus of Triangles

We prove the 2-torus $\mathbb T$, an abelian linear algebraic group, is a fine moduli space of labeled, oriented, possibly-degenerate inscribable similarity classes of triangles, where a triangle is {\it inscribable} if it can be inscribed in a circle. A natural action by the dihedral group $D_6$ defines a quotient stack $[\mathbb T/D_6]$, which is the stack of absolute (unlabeled, unoriented) possibly-degenerate inscribable classes. We show the main triangle types form distinguished algebraic substructures: subgroups, cosets, and elements of small order, and we apply the natural metric on $\mathbb T$ to compare them.