Problem
GEO-B3-M02-P012 A Circle as an Intermediate Line
B. New Original Problem. A circle \(\omega\), a line \(l\), and points \(M,N\in\omega\), not lying on \(l\), are given. For \(X\in l\), draw \(MX\), meeting \(\omega\) again at \(Y\); then \(NY\) meets \(l\) at \(X'\). Prove that the map \(X\mapsto X'\) preserves the cross-ratio of four points on \(l\).
C. Hint 1. Show that the map is a composition of two projections.
D. Hint 2. To speak about cross-ratio on a circle, project the circle to any auxiliary line.
E. Full Solution.
The first step \(X\mapsto Y\) is central projection from the line \(l\) to the circle with center \(M\). The second step \(Y\mapsto X'\) is central projection from the circle to the line \(l\) with center \(N\).
Choose an auxiliary line \(s\) and identify the circle \(\omega\) with \(s\) by a central projection from a point not on \(\omega\). Then both steps become projective maps between lines. Therefore their composition is a projective transformation of \(l\).
Every projective transformation of a line preserves cross-ratio. Hence for any four points \(X_1,X_2,X_3,X_4\) on \(l\), we have \((X_1X_2X_3X_4)=(X'_1X'_2X'_3X'_4)\).
It is important to stress that the circle is used as a carrier of a projective parametrisation, not as a metric object.