Problem
GEO-B3-M06-P016 Collinearity from Sines
#16
★★★☆☆ Level 3 of 5
Points \(A_1,B_1,C_1\) lie on lines \(BC,CA,AB\), respectively. Suppose the directed trigonometric Menelaus product equals \(-1\). Prove that \(A_1,B_1,C_1\) are collinear.
Inspired by Prasolov trigonometric geometry method
C. Hint 1. Convert the sine ratios into segment ratios.
D. Hint 2. Then apply ordinary Menelaus.
For each point on a side, the sine rule in the corresponding small triangle gives equality between the sine ratio and the directed segment ratio on the side.
By assumption, the product of sine ratios is \(-1\), so the product of the corresponding directed segment ratios is also \(-1\). By the converse of Menelaus, \(A_1,B_1,C_1\) are collinear.
This reinforces the converse direction of trig Menelaus.