Cho x^2 + y^2 +z^2 = xy + yz + xz. Chứng minh z = x = y.
Giải thích
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
⇔ x−y=0y−z=0z−x=0⇔x=yy=zz=x⇔x=y=z(đpcm)