Problem
GEO-B3-M02-P005 A Harmonic Quadruple Is Preserved
B. New Original Problem. On a line \(l\), points \(A,B,C,D\) form a harmonic quadruple: \((A B C D)=-1\). A central projection sends them to a line \(m\) as \(A_1,B_1,C_1,D_1\). Prove that \((A_1 B_1 C_1 D_1)=-1\).
C. Hint 1. Harmonicity is not about individual lengths, but about cross-ratio.
D. Hint 2. Use preservation of cross-ratio under projection.
E. Full Solution.
A central projection is a projective map from the line \(l\) to the line \(m\). Therefore it preserves cross-ratio:
\[ (A_1 B_1 C_1 D_1)=(A B C D). \]
By assumption \((A B C D)=-1\). Hence \((A_1 B_1 C_1 D_1)=-1\), so the image of a harmonic quadruple is harmonic.
After this task it is useful to ask the student to construct the harmonic conjugate with a straightedge via a complete quadrilateral.