Problem. Prove that if the cevians $(AA^\prime)$, $(BB^\prime)$, and $(CC^\prime)$ of $\triangle ABC$ are concurrent in a point $G$, then $$\frac {GA^\prime}{GA}=\frac {C^\prime B}{C^\prime A}+\frac {B^\prime C}{B^\prime A}$$
I have solved a similar problem where I had to prove that the medians of a triangle are concurrents in one point, the barycenter. To prove it I used a barycentric coordinate system to find the equations of the medians and then I just had to solve a simple system of linear equations in order to find the point which belongs to the three medians.
However, in this problem using a barycentric coordinate system does not seem to work. I have also tried to solve it using the results of Thales and Ceva theorems unsuccessfully.