$$a ( b - c ) + b ( c - a ) + c ( a - b )$$
$0$
$$ab-ac+b\left(c-a\right)+c\left(a-b\right)$$
$$ab-ac+bc-ba+c\left(a-b\right)$$
$$-ac+bc+c\left(a-b\right)$$
$$-ac+bc+ca-cb$$
$$bc-cb$$
$$0$$
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