Problem
GEO-B2-M06-P016 Why Menelaus Works
#16
★★★★☆ Level 4 of 5
In triangle \(ABC\), a line \(l\) meets lines \(BC,CA,AB\) at \(D,E,F\), respectively. Prove the directed form of Menelaus: \(\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}=-1\).
Compare the directed distances from vertices \(A,B,C\) to line \(l\).
Let the directed distances from \(A,B,C\) to line \(l\) be \(h_A,h_B,h_C\). Since \(D\) lies on both \(BC\) and \(l\), point \(D\) divides \(BC\) in the ratio of directed distances to \(l\): \(\frac{BD}{DC}=-\frac{h_B}{h_C}\). Similarly, \(\frac{CE}{EA}=-\frac{h_C}{h_A}\) and \(\frac{AF}{FB}=-\frac{h_A}{h_B}\). Multiplying gives \(\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}=-1\).
This proof is useful for stronger students: it explains the sign in the formula.