Cho x² + y² +z² = xy + yz + xz. Chứng minh z = x = y.

x² + y² +z² = xy + yz + xz

⇔ x² + y² +z² – xy – yz – xz = 0

⇔ 2x² + 2y² + 2z² – 2xy – 2yz – 2xz = 0

⇔ (x² – 2xy + y²) + (y² – 2yz + z²) + (z² – 2xz + x²) = 0

⇔ (x – y)² + (y – z)² + (z – x)² = 0

⇔  $\left[\begin{array}{l}x-y=0\\ y-z=0\\ z-x=0\end{array}\right.\iff \left[\begin{array}{l}x=y\\ y=z\\ z=x\end{array}\right.\iff x=y=z$(đpcm)