Problem
GEO-B2-M06-P022 Ceva with Two External Points
#22
★★★★★ Level 5 of 5
In triangle \(ABC\), points \(D,E,F\) lie on lines \(BC,CA,AB\). Directed ratios are given: \(\frac{BD}{DC}=-\frac{2}{3}\), \(\frac{CE}{EA}=-\frac{3}{4}\). Find \(\frac{AF}{FB}\) if \(AD\), \(BE\), \(CF\) are concurrent.
In directed Ceva, the product equals \(1\).
By Ceva, \(\left(-\frac{2}{3}\right)\left(-\frac{3}{4}\right)\frac{AF}{FB}=1\). The first two factors give \(\frac{1}{2}\). Hence \(\frac{AF}{FB}=2\). The positive sign shows that \(F\) lies on segment \(AB\).
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