Use a separation theorem to show: $Ax=b$ has a solution $\Leftrightarrow (c^TA=0\Rightarrow c^Tb=0)$

Question

Let $A\in \mathbb{R}^{n \times m}, b \in \mathbb{R}^{n}$ and $c \in \mathbb{R}^n.$ Show that $Ax=b$ has a solution if and only for all $c$ with $c^TA=0$ it holds that $c^Tb=0$ as well.

The exercise seems quiet easy using linear algebra, but I don't understand how to solve it with separation theorems.

EDIT: I need to use separation theorems for both directions