Ruler and regular polygon constructions (Q2441419)

In this article the authors explore the constructions that can be made using a ruler and a given \(n\)-gon, inspired by constructions with a ruler and a given circle. A point is said to be a ruler and regular \(n\)-gon point (\(lr_n\)-point) if it is the last in a finite sequence \(Q_1,Q_2,\dots, Q_n\) of points such that each point is one of the vertices of the regular \(n\)-gon or is the intersection of two lines through points appearing earlier in the sequence. A real number \(x\) is said to be a ruler and regular \(n\)-gon number if there exist a \(rl_n\) point \(X\) on a line passing through the center \(O\) of the \(n\)-gon and a vertex of the polygon such that \(OX=|x|\). In the first sections, the authors present the basic constructions that can be carried out with this method (such as a parallel line to two given parallel lines). They show that starting with a trapezoid which is not a parallelogram, the sum, difference, product and quotient of two constructible numbers are \(lr_n\) numbers. They also study the algebraic structure of the set of \(lr_n\) numbers and determine which regular polygons can be constructed with ruler and a given regular \(n\)-gon.