If a/(b+c)+b/(c+a)+c/(a+b)=1, prove a²/(b+c)+b²/(c+a)+c²/(a+b)=0

[anemone]

Prove that if $\displaystyle \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$, then $\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$

 

[Albert1]

$\dfrac {c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{c+a}=1----(1)$
let :
$\dfrac {c^2}{a+b}+\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}=k$
$(1)\times a+(1)\times b+(1)\times c$
we get
k+a+b+c=a+b+c
$\therefore k=0$