Not Really Big Power

$\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b}=\dfrac{a+b+c}{a+b+c}$ .

Multiplying both sides by $a+b+c$ , we get

$\dfrac{a^{2}}{b+c}+a+\dfrac{b^{2}}{a+c}+b+\dfrac{c^{2}}{a+b}+c=a+b+c$ .

Subtracting $a+b+c$ from both sides,

$\dfrac{a^{2}}{b+c}+\dfrac{b^{2}}{a+c}+\dfrac{c^{2}}{a+b}=\boxed{0}$

$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1$

Multiply the given equation separately by a, b & c. This yields 3 equations :

$\frac{a^2}{b+c}+\frac{ab}{a+c}+\frac{ac}{a+b}=a$

$\frac{ab}{b+c}+\frac{b^2}{a+c}+\frac{bc}{a+b}=b$

$\frac{ac}{b+c}+\frac{bc}{a+c}+\frac{c^2}{a+b}=c$

Now add the above three equations :

$L.H.S=\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}+\frac{a(b+c)}{b+c}+\frac{b(a+c)}{a+c}+\frac{c(a+b)}{a+b}$

$\Rightarrow L.H.S.=\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}+a+b+c$

$R.H.S.=a+b+c$

Equating L.H.S. and R.H.S., we get :

$\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}+a+b+c=a+b+c$

$\Rightarrow \frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}=0$