Solution:
Let
$$f(t) = (t - x)(t - y)(t - z) - (t - a)(t - b)(t - c)$$
Then $f(t) = k t$ for some constant $k$. Note that $k a = f(a) = (a - x)(a - y)(a - z) \leq 0$ and hence $k \leq 0$. Similarly, $k c = f(c) = (c - x)(c - y)(c - z) \geq 0$ and hence $k \geq 0$. Combining the two, it follows that $k = 0$ and that $f(a) = f(c) = 0$. These equalities imply that $a = x$ and $c = z$, and then it also follows that $b = y$.