Problem
GEO-B2-M06-P023 Menelaus with Signs
#23
★★★★★ Level 5 of 5
Points \(D,E,F\) lie on lines \(BC,CA,AB\) of triangle \(ABC\). Given \(\frac{BD}{DC}=2\), \(\frac{CE}{EA}=-\frac{3}{5}\), \(\frac{AF}{FB}=\frac{5}{6}\). Prove that \(D,E,F\) are collinear.
For directed Menelaus, the product must be \(-1\).
Multiply: \(2\cdot\left(-\frac{3}{5}\right)\cdot\frac{5}{6}=-1\). Therefore, by Menelaus' theorem with directed ratios, points \(D,E,F\) lie on one line.
A final step toward conscious work with directed ratios.