Question

Chứng minh rằng:

Nếu x²+y²+z²=xy+yz+zx thì x=y=z

Giúp mình nha. Thank you so much

Answer 1

x²+y²+z²=xy+yz+zx

<=>2(x²+y²+z²)=2(xy+yz+zx)

<=>2x²+2y²+2z²=2xy+2yz+2zx

<=>2x²+2y²+2z²-2xy-2yz-2zx=0

<=>(x²-2xy+y²)+(y²-2yz+z²)+(z²-2zx+x²)=0

<=>(x-y)²+(y-z)²+(z-x)²=0

Vì \(\hept{\begin{cases}\left(x-y\right)^2\ge0\\\left(y-z\right)^2\ge0\\\left(z-x\right)^2\ge0\end{cases}\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0}\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}x-y=0\\y-z=0\\z-x=0\end{cases}\Leftrightarrow x=y=z}\)(đpcm)