Problem
GEO-B2-M06-P005 Checking Collinearity
#5
★★☆☆☆ Level 2 of 5
In triangle \(ABC\), points \(D\in BC\), \(E\in CA\), and \(F\) lies on the extension of \(AB\) beyond \(B\). Given \(BD:DC=2:3\), \(CE:EA=3:4\), \(AF:FB=2:1\). Prove that \(D,E,F\) are collinear.
Check the product of ratios.
We have \(\frac{2}{3}\cdot\frac{3}{4}\cdot 2=1\). Exactly one point lies on an extension of a side, so by the positive form of Menelaus, points \(D,E,F\) are collinear.
A contrast with Ceva problems.