\(\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}=\frac{a+b+c}{b+c+a+c+a+b}=\frac{1}{2}\)
\(\Rightarrow\left\{{}\begin{matrix}b+c=2a\\a+c=2b\\a+b=2c\end{matrix}\right.\)
\(\Rightarrow a=b=c\)
\(M=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}\)
\(\Rightarrow M=8\)
Ta có:
\(\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}\) và \(a+b+c=1\) (vì \(a+b+c\ne0\)).
Áp dụng tính chất dãy tỉ số bằng nhau ta được:
\(\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}=\frac{a+b+c}{b+c+a+c+a+b}=\frac{1}{\left(b+c+a\right)+\left(c+a+b\right)}=\frac{1}{1+1}=\frac{1}{2}.\)
\(\Rightarrow\left\{{}\begin{matrix}\frac{a}{b+c}=\frac{1}{2}\Rightarrow b+c=a:\frac{1}{2}\Rightarrow b+c=2a\\\frac{b}{a+c}=\frac{1}{2}\Rightarrow a+c=b:\frac{1}{2}\Rightarrow a+c=2b\\\frac{c}{a+b}=\frac{1}{2}\Rightarrow a+b=c:\frac{1}{2}\Rightarrow a+b=2c\end{matrix}\right.\)
\(\Rightarrow2a=2b=2c.\)
\(\Rightarrow a=b=c.\)
Có: \(M=\left(\frac{a}{b}+1\right).\left(\frac{b}{c}+1\right).\left(\frac{c}{a}+1\right)\)
\(\Rightarrow M=\left(\frac{a}{a}+1\right).\left(\frac{a}{a}+1\right).\left(\frac{a}{a}+1\right)\)
\(\Rightarrow M=\left(1+1\right).\left(1+1\right).\left(1+1\right)\)
\(\Rightarrow M=2.2.2\)
\(\Rightarrow M=8.\)
Vậy \(M=8.\)
Chúc bạn học tốt!
