Đặt \(\left\{{}\begin{matrix}b+c-a=x\\c+a-b=y\\a+b-c=z\end{matrix}\right.\)\(\left(x,y,z>0\right)\)\(\Rightarrow\left\{{}\begin{matrix}x+y=2c\\y+z=2a\\x+z=2b\end{matrix}\right.\)
Thì ta có: \(\dfrac{2\left(y+z\right)}{x}+\dfrac{9\left(x+z\right)}{2y}+\dfrac{8\left(x+y\right)}{z}\ge26\)
Áp dụng BĐT AM-GM ta có:
\(VT=\dfrac{2\left(y+z\right)}{x}+\dfrac{9\left(x+z\right)}{2y}+\dfrac{8\left(x+y\right)}{z}\)
\(=\dfrac{2y}{x}+\dfrac{2z}{x}+\dfrac{9x}{2y}+\dfrac{9z}{2y}+\dfrac{8x}{z}+\dfrac{8y}{z}\)
\(=\left(\dfrac{2y}{x}+\dfrac{9x}{2y}\right)+\left(\dfrac{2z}{x}+\dfrac{8x}{z}\right)+\left(\dfrac{9z}{2y}+\dfrac{8y}{z}\right)\)
\(\ge2\sqrt{\dfrac{2y}{x}\cdot\dfrac{9x}{2y}}+2\sqrt{\dfrac{2z}{x}\cdot\dfrac{8x}{z}}+2\sqrt{\dfrac{9z}{2y}\cdot\dfrac{8y}{z}}\)
\(\ge6+8+12=26=VP\)
