Question

2.CMR: Nếu x²+y²+z² = xy + yz + zx thì x=y=z

Answer 1

Có: x² + y² + z² = xy + yz + zx

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

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

<=> (x-y)² + (y-z)² + (z-x)² = 0

<=> x-y = y-z = z-x = 0

<=> x = y = z

Answer 2

\(x^2+y^2+z^2=xy+yz+xz\\ 2x^2+2y^2+2z^2=2xy+2yz+2xz\\ 2x^2+2y^2+2z^2-2xy-2yz-2xz=0\\ x^2-2xy+y^2+y^2-2yz+z^2+z^2+-2zx+x^2=0\\ \left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\\ \Rightarrow\left\{{}\begin{matrix}x-y=0\\y-z=0\\z-x=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=y\\y=z\\z=x\end{matrix}\right.\Rightarrow x=y=z\)