Let $CM$ be the median through vertex $C$, and let $G$ be the point of intersection of the triangle's medians.
Let $CH$ be the altitude of the triangle through vertex $C$ and $GP$ be the distance from $G$ to $AB$, with the point $P$ laying on $AB$.
Using Thales' intercept theorem, we derive the proportion:

$\frac{GP}{CH} = \frac{GM}{CM}$

The fraction $\frac{GM}{CM}$ in any triangle is equal to $\frac{1}{3}$ . Therefore $GP = \frac{CH}{3}$ .

Since the problem states that the vertex $C$ is moving on a straight line, the length of $CH$ is a constant value. That means that the length of $GP$ is also a constant. Therefore the point $G$ is moving on a straight line.
Answer: D