91Ó°ÊÓ

A point \(M_{1}\) on the side \(A_{1} A_{2}\) of a regular \(n\)-gon \(A_{1} A_{2} \cdots A_{n}\) is projected from \(A_{n}\) to a point \(M_{2}\) on the side \(A_{2} A_{8} ; M_{2}\) is then projected from \(A_{1}\) to a point \(M_{3}\) on the side \(A_{3} A_{4}\); next \(M_{3}\) is projected from \(A_{2}\) to a point \(M_{4}\) on the side \(A_{4} A_{5}\), and so on. Prove that: (a) If \(n=4\) then the point \(M_{13}\), obtained after three circuits of the \(n\)-gon, coincides with the initial point \(M_{1}\) (and, thereiore, \(M_{16}\) coincides with \(M_{2}, M_{15}\) with \(M_{3}\), etc.). (b) If \(n=6\) then the point \(M_{13}\), obtained after two circuits of the \(n\)-gon, coincides with the initial point \(M_{1}\) (and, therefore, \(M_{14}\) coincides with \(M_{2}, M_{16}\) with \(M_{2}\), etc.). (c) If \(n=10\) then the point \(M_{11}\), obtained after one circuit of the \(n\)-gon, coincides with the initial point \(M_{1}\) (and, therefore, \(M_{12}\) coincides with \(M_{2}, M_{18}\) with \(M_{2}\), etc.).

(a) Let \(l\) be a line, \(P\) a point not on \(l\). Find a segment \(X Y\) on \(l\) which subtends a given angle \(\alpha\) at \(P\). (b) Let \(l_{1}\) and \(l_{2}\) be two lines, \(P\) and \(Q\) two points not on these lines. Find a point \(X\) on \(l_{1}\) and a point \(Y\) on \(l_{2}\) such that the segment \(X Y\) subtends a given angle \(\alpha\) at \(P\) and a given angle \(\beta\) at \(Q\).

Let \(n\) lines \(l_{1}, l_{2}, \cdots, l_{n}\) be given in a plane together with \(n\) points \(M_{1}, M_{2}, \cdots, M_{n}\) (Fig. 101). Construct an \(n\)-gon whose vertices lie on the given lines and whose sides pass through the given points (or some of the sides pass through given points and others have preassigned directions). [This problem can also be stated as follows: Inscribe in a given \(n\)-gon an \(n\)-gon whose sides pass through \(n\) given points (or some of the sides pass through given points and others have preassigned directions); or, in yet another form, as follows: Construct an \(n\)-gon inscribed in a given \(n\)-gon \(N_{1} N_{2} \cdots N_{n}\) and circumscribed about another given \(n\)-gon \(M_{1} M_{2} \cdots M_{n \cdot}\) ]

Let two circles \(S_{1}\) and \(S_{2}\) be given in a plane. Find a straightedge construction for the determination of their centers if the two circles (a) intersect in two points; (b) are tangent; (c) are concentric.