Ta có : \(a+b+c=1\Leftrightarrow a+b=1-c\)
Lại có :
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1=0\)
\(\Leftrightarrow\frac{a+b}{ab}+\frac{1-c}{c}=0\)
\(\Leftrightarrow\frac{1-c}{ab}+\frac{1-c}{c}=0\)
\(\Leftrightarrow\left(1-c\right)\left(\frac{1}{ab}+\frac{1}{c}\right)=0\)
\(\Leftrightarrow\left(1-c\right).\frac{c+ab}{abc}=0\)
\(\Leftrightarrow\left(1-c\right).\frac{1-a-b+ab}{abc}=0\)
\(\Leftrightarrow\frac{\left(1-c\right)\left(1-a\right)\left(1-b\right)}{abc}=0\)
\(\Leftrightarrow\left(1-a\right)\left(1-b\right)\left(1-c\right)=0\left(đpcm\right)\)
