Question

cho \(\frac{a}{b+c}\)+\(\frac{b}{c+a}\)+\(\frac{c}{a+b}\)=1. Chứng minh \(\frac{a^2}{b+c}\)+\(\frac{b^2}{c+a}\)+\(\frac{c^2}{a+b}\)=0

Answer 1

\(\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)\left(a+b+c\right)=a+b+c\)

\(\Leftrightarrow\left(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right)+\frac{ab}{c+a}+\frac{ac}{a+b}+\frac{ab}{b+c}+\frac{bc}{a+b}+\frac{ac}{b+c}+\frac{bc}{c+a}=a+b+c\)

\(\Leftrightarrow\left(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right)+\frac{ab+bc}{c+a}+\frac{ac+bc}{a+b}+\frac{ab+ac}{b+c}=a+b+c\)

\(\Leftrightarrow\left(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right)+a+b+c=a+b+c\)

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