Rectangles conformally inscribed in lines (Q2070308)

The aim of this paper is to study the set of rectangles that can be inscribed in a system \(\mathcal{C}\) of four lines \(A, B, C, D\) in the Euclidean plane \({\mathbb R}^2\), such that the four lines do not meet in a single point, \(B\) goes through the point \((0, 1)\), and \(C\) and \(D\) are distinct lines meeting in the origin (a situation which can be achieved whenever not all the lines are parallel to each other and they do not all have a point in common). To that end, it is found convenient to study parallelograms that are inscribed in scaled copies of \(\mathcal{C}\), given that the computations are less involved. The set of all such parallelograms inscribed in \(\mathcal{C}\) forms a three-dimensional Euclidean space \(\Pi\) with inner product determined by the diagonals of the parallelograms. A notion of \textit{norm} is introduced for parallelograms, and one of a \textit{conformally inscribed} parallelogram, which means that the parallelogram is inscribed in a scaled copy of \(\mathcal{C}\). Among conformally inscribed rectangles unit rectangles are singled out, those with norm \(1\). Since any rectangle inscribed in \(\mathcal{C}\) can be represented by a unit rectangle inscribed in a scaled copy of \(\mathcal{C}\), it suffices to study the latter. In the space \(\Pi\) of conformally inscribed parallelograms, the unit rectangles lie on an intersection of two cylinders and thus form an algebraic space curve, the union of two simple closed curves.