Problem
GEO-B2-M06-P014 Side Ratios
#14
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D,E,F\) on \(BC,CA,AB\) are chosen so that \(\frac{BD}{DC}=\frac{AB}{AC}\), \(\frac{CE}{EA}=\frac{BC}{BA}\), \(\frac{AF}{FB}=\frac{CA}{CB}\). Prove that \(AD\), \(BE\), \(CF\) are concurrent.
Multiply the three given ratios.
The product is \(\frac{AB}{AC}\cdot\frac{BC}{BA}\cdot\frac{CA}{CB}=1\). By Ceva's theorem, lines \(AD\), \(BE\), \(CF\) are concurrent.
Algebraic cancellation in a geometry problem.