TH1: \(a+b+c=0\Rightarrow\left\{{}\begin{matrix}a+b=-c\\b+c=-a\\a+c=-b\end{matrix}\right.\)
\(P=\dfrac{\left(b+c\right)}{b}.\dfrac{\left(a+b\right)}{a}.\dfrac{\left(a+c\right)}{c}=\dfrac{-a}{b}.\dfrac{-c}{a}.\dfrac{-b}{c}=-1\)
TH2: \(a+b+c\ne0\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\dfrac{a-b+c}{2b}=\dfrac{c-a+b}{2a}=\dfrac{a-c+b}{2c}=\dfrac{a-b+c+c-a+b+a-c+b}{2b+2a+2c}=\dfrac{a+b+c}{2\left(a+b+c\right)}=\dfrac{1}{2}\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{a-b+c}{2b}=\dfrac{1}{2}\\\dfrac{c-a+b}{2a}=\dfrac{1}{2}\\\dfrac{a-c+b}{2c}=\dfrac{1}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a+c=2b\\c+b=2a\\a+b=2c\end{matrix}\right.\) \(\Rightarrow a=b=c\)
\(\Rightarrow P=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)
