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$$(b - c)(x - a)(y - a) + (c - a)(x - b)(y - b) + (a - b)(x - c)(y - c)$$ is
Expanding $$(x - a)(y - a) = xy - a(x + y) + a^2$$, the coefficient of $$xy$$ is $$\sum (b - c) = 0$$ and the coefficient of $$(x + y)$$ is $$-\sum a(b - c) = 0$$. Only the constant $$\sum a^2(b - c) = -(a - b)(b - c)(c - a)$$ remains. So the expression is independent of both $$x$$ and $$y$$.
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