Well, I'll probably get jumped on here but here goes .......

The equation can be re-arranged in two ways:

x(x - y) + y(y - z) + z(z - x) = 0 .... (1)

x(x - z) + y(y - x) + z(z - y) = 0 ....(2)

Compare and equate coefficients of x, y and z:

x - y = x - z => y = z.

y - z = y - x => x = z.

Therefore x = y = z is the only solution.