Problem
GEO-B3-M06-P007 Trigonometric Menelaus
A line \(l\) meets sides \(BC,CA,AB\) or their extensions at \(A_1,B_1,C_1\). Prove that the product of the corresponding sine ratios is \(1\) in absolute value.
C. Hint 1. Start with ordinary Menelaus.
D. Hint 2. Express each segment ratio through sines of angles between \(l\) and the sides.
By Menelaus with directed segments, \[ \frac{BA_1}{CA_1}\cdot\frac{CB_1}{AB_1}\cdot\frac{AC_1}{BC_1}=-1. \]
In each small triangle formed by line \(l\) and two sides of the original triangle, the sine rule expresses a side-segment ratio as a ratio of sines of the angles that \(l\) makes with those sides. Substitution gives the trigonometric condition. Without directed angles, one obtains equality of absolute values to \(1\).
At this level, signs should be discussed even if later problems use absolute values.