It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the centroid are not area bisectors. With other polygons there are area bisectors which pass through the centroid but not through a vertex (for instance any line which passes through the centroid of a square), but based on the examples that I am able to compute it seems that any line which passes through a vertex and the centroid is an area bisector. This leads me to pose the following general question:

Let $C$ be a convex body in the plane and let $L$ be a line which passes through the centroid of $C$ and an extreme point of $C$. Is $L$ necessarily an area bisector?

I couldn't find any relevant tools in my usual convex geometry references, but I apologize if I missed something obvious.