Affine regularity (Q1261191)

The author shows that an \(n\)-gon \(A_ 0A_ 1\ldots A_{n-1}\) in the plane is affinely regular (that is, the image of a metrically regular polygon under an affinity) if all the lines \(A_ \mu A_ \nu\) with \(\mu+\nu \equiv c \pmod n\) are parallel for each value of \(c\). In fact, a weaker condition is established: all that is needed is that, for each \(\mu\), the lines \(A_ \mu A_{\mu+3}\) and \(A_{\mu+1}A_{\mu+2}\) are parallel, as are the lines \(A \mu A_{\mu+4}\) and \(A_{\mu+1}A_{\mu+3}\) \((\mu\) and the suffixes are taken modulo \(n\); for \(n \leq 5\), some of the conditions are redundant).