Mathematics
If x2 + y2 + z2 = xy + yz + zx, prove that x = y = z
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Answer
Given,
x2 + y2 + z2 = xy + yz + zx
⇒ x2 + y2 + z2 - (xy + yz + zx) = 0
⇒ x2 + y2 + z2 - xy - yz - zx = 0
Multiplying by 2 on both sides,
⇒ 2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx = 0
⇒ (x2 + y2 - 2xy) + (y2 + z2 - 2yz) + (z2 + x2 - 2xz) = 0
⇒ (x - y)2 + (y - z)2 + (z - x)2 = 0
⇒ x - y = 0, y - z = 0 and z - x = 0
⇒ x = y, y = z and z = x
⇒ x = y = z
Hence, proved that if x2 + y2 + z2 = xy + yz + zx, then x = y = z
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