Prove ( or disprove) that for all kinds of simple polygon, the centroid lies inside the polygon

Question

Is it possible to prove that for all kinds of simple polygon, regardless of whether it is convex or concave and with no opening, the centroid of the polygon must ( or may not) lie inside the polygon?

The wiki link above gives example of polygon which has the centroid lying outside the polygon:

A non-convex object might have a centroid that is outside the figure itself. The centroid of a ring or a bowl, for example, lies in the object's central void.

But let's say my polygon has no opening, can it be proved that the centroid must lie inside the polygon?

Answer 1

Comment as answer as requested:

• Think of a boomerang.
• Alternatively, since you already understand the example of the polyon with an opening, imagine an arbitrarily narrow passage drilled to connect the opening to the outside and "simplify" the polygon. Since you can make the passage as narrow as you want, you can move the centroid as little as you want, so you can see it doesn't have to move inside the polygon.