By Ceva's theorem, $AD$, $BE$, $CF$ are concurrent if and only if
$$\frac{\sin \angle CAD}{\sin \angle EAD} \times \frac{\sin \angle AEB}{\sin \angle CEB} \times \frac{\sin \angle ECF}{\sin \angle ACF} = 1.$$

By the extended sine law, we have
$$\frac{\sin \angle CAD}{\sin \angle EAD} = \frac{CD}{DE}, \quad \frac{\sin \angle AEB}{\sin \angle CEB} = \frac{AB}{BC}, \quad \frac{\sin \angle ECF}{\sin \angle ACF} = \frac{EF}{FA}.$$
Thus, $AD$, $BE$, $CF$ are concurrent if and only if
$$\frac{CD}{DE} \times \frac{AB}{BC} \times \frac{EF}{FA} = 1.$$
This is exactly $AB \cdot CD \cdot EF = BC \cdot DE \cdot FA$.