Set $u := \frac{ab}{c}$, $v := \frac{ac}{b}$ and $w := \frac{bc}{a}$. By assumption, $u + v + w$ is an integer. It is easily seen that $uv + uw + vw = a^2 + b^2 + c^2$ and $uvw = abc$ are integers, too.

According to Vieta's formulae, the rational numbers $u, v, w$ are the roots of a cubic polynomial $x^3 + px^2 + qx + r$ with integer coefficients. As the leading coefficient is 1, these roots are integers.