Points $D,E$ and $F$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$ respectively so that the segments $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB}$ .
Source: 2001 Estonia National Olympiad Final Round grade 11 p3
Tags: geometry, ratio
Points $D,E$ and $F$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$ respectively so that the segments $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB}$ .