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X² + y² + z² -xy - xz - yz = 0

multiplying 2 on both sides

2x² + 2y² + 2z² - 2xy - 2xz - 2yz = 0

x² + y² - 2xy+ x² + z² - 2xz + z² + y² - 2yz = 0

(x - y)² + (x - z)² + (y - z)² = 0

so the value are in square so they can't be negative so they must be indivudually zero to satisfy the equation

so x - y = 0 ⇒ x = y

x - z = 0 ⇒ x = z

y - z = 0 ⇒ y = z

so hence x = y = z

multiplying 2 on both sides

2x² + 2y² + 2z² - 2xy - 2xz - 2yz = 0

x² + y² - 2xy+ x² + z² - 2xz + z² + y² - 2yz = 0

(x - y)² + (x - z)² + (y - z)² = 0

so the value are in square so they can't be negative so they must be indivudually zero to satisfy the equation

so x - y = 0 ⇒ x = y

x - z = 0 ⇒ x = z

y - z = 0 ⇒ y = z

so hence x = y = z

__proved__