Centroid location

[Strike]

Don't ask why I started thinking of this, but I was looking for a formulaic way of determining the location of the centroid of any arbitrary polygon, and I found that one way of doing it was to use a common vertex to form n-2 triangles (from an n-gon), and to find the centroids of each of those and to then weight each coordinate with the percentage of the area formed by that triangle and summing the centroids.

But, for some weird reason, I thought this might be able to be done recursively by creating the n-2 triangles and connecting their centroids to get a new (n-2)-gon, and then getting an (n-4)-gon the same way ... until you get a triangle or a line at which point you fall back either on the midpoint formula or the centroid formula for a triangle.

Any geometers out there who know if this is valid or not? It seems like it would be, but I don't have the gusto to prove it.

[f'lar]

Maybe, but somehow it just doesn't seem right. I understand what you're asking, though. You could check to find out if it works with regular polygons, and then vary a regular polygon only slightly so you can still be fairly sure of the centeroid and see if it still holds. Just drawing it freehand a few times it seems as if it might. If nothing else, I looks like it would be good for an approximation.

[Strike]

Well, I don't think this algorithm is any better than the original one, as this one must run up to n/2 times for an n-gon. But, I thought it'd be a neat recursive approach. I'm working on a proof right now, actually (in spite of what I said above).