Assume on the contrary that $\frac{rs}{t}$, $\frac{rt}{s}$, $\frac{st}{r}$ are positive integers.
Without loss of generality we may assume that $a < b < c$. Since $\frac{rs}{t}$ is a positive integer, $\frac{r^2 s^2}{t^2}$ is also a positive integer, and hence
$$\frac{r^2 s^2}{t^2} = \frac{a^2 bc + ab + ac + 1}{bc + 1} = a^2 + \frac{ab + ac + 1 - a^2}{bc + 1}$$
is also a positive integer. Since $a < b < c$, we have
$$ab + ac + 1 - a^2 > ac + 1 = s^2 > 0.$$
Therefore, both the numerator and the denominator of the fraction $\frac{ab + ac + 1 - a^2}{bc + 1}$ are positive, so
$$ab + ac + 1 - a^2 \geq bc + 1 \implies (b-a)(c-a) \leq 0,$$
which is a contradiction.