Given any Euclidean triangle of vertexes $v_1,v_2,v_3$ and any point $P$ in the interior of the triangle, consider the following system of 6 real equations in $c_{12},c_{13},c_{23},d_1,d_2,d_3$:

$$c_{12}(v_1-v_2)+c_{13}(v_1-v_3)=d_1(v_1-P)$$ $$c_{12}(v_2-v_1)+c_{23}(v_2-v_3)=d_2(v_2-P)$$ $$c_{23}(v_3-v_2)+c_{13}(v_3-v_1)=d_3(v_3-P)$$

I am trying to prove the existence of a positive solution (i.e. such that $c_{12}>0,c_{13}>0,c_{23}>0,d_1>0,d_2>0,d_3>0$) to such system for every triangle, but until now I have not succeeded.

Notice that it is not sufficient to infer that couples of these vectors are linearly independent. Indeed, solving the first equation, for each value $c_{12}>0$ one finds $c_{13}(c_{12})>0$ (i.e. the value of $c_{13}$ depending on $c_{12}$). Solving the third equation given $c_{13}(c_{12})$ one finds $c_{23}(c_{12})$ and lastly one has to prove that such $c_{23}(c_{12})$ solves the second equation.

Do you know how to find such $c_{12}>0,c_{13}>0,c_{23}>0,d_1>0,d_2>0,d_3>0$ (or at least prove their existence)? I would be grateful if you could explain it to me.