Problem
GEO-B3-M02-P001 Cross-Ratio on Two Transversals
B. New Original Problem. Four distinct lines pass through a point \(O\). They meet a line \(l\), not passing through \(O\), at \(A,B,C,D\), and a line \(m\), also not passing through \(O\), at \(A_1,B_1,C_1,D_1\). Prove that \((A B C D)=(A_1 B_1 C_1 D_1)\).
C. Hint 1. Do not compare the actual lengths on the two different lines.
D. Hint 2. Express the cross-ratio through the pencil of four lines with center \(O\).
E. Full Solution.
Denote the four lines by \(a,b,c,d\). For any transversal not passing through \(O\), the cross-ratio of the intersection points equals the cross-ratio of the pencil:
\[ (A B C D)=\frac{\sin(a,c)}{\sin(b,c)}:\frac{\sin(a,d)}{\sin(b,d)}. \]
The same formula is obtained for \(A_1,B_1,C_1,D_1\), since the angles between \(a,b,c,d\) do not depend on the transversal. Hence the two cross-ratios are equal to the same number.
A basic problem for understanding that the cross-ratio belongs to the pencil, not to a particular transversal.