# Can Centroid Lies on the Edge of a Polygon?

This question is similar to the one here.

Now the question is, given a simple polygon with $Area>0$, regardless of whether it is convex or concave and with no opening, can we prove that the centroid of the polygon can never lie on the exact edge of the polygon?

If so, how?

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You should be able to do it with a crescent-shaped polygon. You shouldn't be able to do it for a convex polygon (basically because if the centroid of a convex polygon lies on an edge then all of the vertices of the polygon must lie on the corresponding line). –  Qiaochu Yuan May 4 '11 at 7:28
@Qiaochu, are you saying that a concave polygon-- if carefully constructed-- can have its centroid lies on the edge of the polygon? –  Graviton May 4 '11 at 8:37