If $x \ge 0$, $y \ge 0$, and $z \ge 0$, then $ma - nb = c(m - n)$ and analogously, we have
$$m(a - c) = n(b - c), \quad m(b - a) = n(c - a), \quad m(c - b) = n(a - b). \quad (*)$$
If all parentheses in $(*)$ are nonzero, then we obtain $m^3(a-c)(b-a)(c-b) = -n^3(b-c)(c-a)(a-b)$, and therefore $m = -n$, false. Thus at least one difference is zero, say $a - c = 0$. It follows that $b = c$, hence $a = b = c$.