The positions of the geometric centroid of a planar non-self-intersecting polygon with vertices (x_1, y_1), ..., (x_n, y_n) are x^_ | = | 1/(6A) sum_(i = 1)^n(x_i + x_(i + 1))(x_i y_(i + 1) - x_(i + 1) y_i) y^_ | = | 1/(6A) sum_(i = 1)^n(y_i + y_(i + 1))(x_i y_(i + 1) - x_(i + 1) y_i), where A is the polygon area and x_(n + 1) = x_1 and y_(n + 1) = y_1.
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