DISCLAIMER: I am not a native speaker of English and I have never learned geometry in English. So my appologies, if I use a non-standard terminology in some place. (Although I did my best to avoid it.)
I will denote the area of triangle $XYZ$ by $|XYZ|$.
You have:
$\frac{|CGF|}{|AGF|} = \frac{|CBF|}{|ABF|} = \frac{|CGF|+|CBG|}{|AGF|+|ABG|}$
The first equality follows from the fact that ratio of areas of the triangles with the same height is the ratio of the lengths of their sides. The second one is additivity of area.
If you compare the first and the last fraction, you get that it is also equal to $\frac{|CBG|}{|ABG|}$. (Using $\frac ab=\frac{a+c}{b+d}$ $\Rightarrow$ $\frac ab=\frac cd$.)