On polygons inscribed in other polygons (Q2912902)

Assume we are given a polygon \(p=P_0\dots P_{n-1}\), where \(P_i\) denotes the vertices of \(p\). We are also given a set of unit plane vectors \(\{v_1,\dots, v_{n-1}\}\). Denote by \(\alpha_i\) the straight line defined by the vertices \(P_i\) and \(P_{i+1}\).NEWLINENEWLINEStarting with a point \(X_0\in\alpha_0\) we construct a polygon \(X_0\dots X_{n-1}\) satisfying the following conditions:NEWLINENEWLINE(i) \(X_i\in\alpha_i\) andNEWLINENEWLINE(ii) the line \(X_iX_{i+1}\) is parallel to \(v_i\).NEWLINENEWLINEThe paper is concerned about the existence of closed polygons (i.e., such that \(X_0=X_{n-1}\)) constructed under the previous conditions. In the triangular case a close relation to Ceva's theorem is found.