Multiplying the equation by $a,b,c$ individually,

$$\begin{array}{l}\frac{a^{2}}{b+c}+\frac{a b}{a+c}+\frac{a c}{a+b}=a \\ \frac{b a}{b+c}+\frac{b^{2}}{a+c}+\frac{b c}{a+b}=b \\ \frac{c a}{b+c}+\frac{c b}{a+c}+\frac{c^{2}}{a+b}=c\end{array}$$