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Prove the sum equals 0 provided another given sum equals 1

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,681
Prove that if $\displaystyle \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$, then $\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$
 

Albert

Well-known member
Jan 25, 2013
1,225
$\dfrac {c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{c+a}=1----(1)$
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
$\therefore k=0$
 
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