Problem
GEO-B2-M06-P017 A Directed Ratio
#17
★★★★☆ Level 4 of 5
In triangle \(ABC\), point \(D\) lies on \(BC\), \(BD:DC=2:3\). Point \(E\) lies on the extension of \(CA\) beyond \(A\), with directed ratio \(\frac{CE}{EA}=-\frac{3}{5}\). Find the directed ratio \(\frac{AF}{FB}\) for which \(AD\), \(BE\), \(CF\) are concurrent.
Use Ceva with directed ratios.
By Ceva, \(\frac{2}{3}\cdot\left(-\frac{3}{5}\right)\cdot\frac{AF}{FB}=1\). The first two factors give \(-\frac{2}{5}\). Hence \(\frac{AF}{FB}=-\frac{5}{2}\). The negative sign means that \(F\) lies on an extension of \(AB\).
A first careful exercise on directed ratios.