\(\left(a+b\right):\left(b+c\right):\left(c+a\right)=6:7:8\\ \Rightarrow\dfrac{a+b}{6}=\dfrac{b+c}{7}=\dfrac{c+a}{8}\)
Áp dụng t/c dtsbn:
\(\dfrac{a+b}{6}=\dfrac{b+c}{7}=\dfrac{c+a}{8}=\dfrac{a+b+b+c+c+a}{6+7+8}=\dfrac{2\left(a+b+c\right)}{21}=\dfrac{28}{21}=\dfrac{4}{3}\\ \Rightarrow\left\{{}\begin{matrix}a+b=8\\b+c=\dfrac{28}{3}\\c+a=\dfrac{32}{3}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}c=14-8=6\\a=14-\dfrac{28}{3}=\dfrac{14}{3}\\b=14-\dfrac{32}{3}=\dfrac{10}{3}\end{matrix}\right.\)
\(\Rightarrow\dfrac{a+b}{6}=\dfrac{b+c}{7}=\dfrac{a+c}{8}=\dfrac{a+b+b+c+c+a}{6+7+8}=\dfrac{2\left(a+b+c\right)}{6+7+8}=\dfrac{28}{21}=\dfrac{4}{3}\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=\dfrac{4.6}{3}=8\\b+c=\dfrac{4.7}{3}=\dfrac{28}{3}\\a+c=\dfrac{4.8}{3}=\dfrac{32}{3}\end{matrix}\right.\)
Mà a+b+c=14
\(\Rightarrow\left\{{}\begin{matrix}c=14-8=6\\a=14-\dfrac{28}{3}=\dfrac{14}{3}\\b=14-\dfrac{32}{3}=\dfrac{10}{3}\end{matrix}\right.\)