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Consider a set of points, [itex]\{A_0, A_1, \ldots A_n\}[/itex]. The midpoint of ##A_k## and ##A_{k+1}## is denoted by ##a_k##, with ##a_n## as the midpoint of ##A_n## and ##A_0##.

Now form a set of ##n## vectors defined by [itex]\stackrel{\rightarrow}{A_ra_s}[/itex] such that each vertex and each midpoint is used once by a vector in the set.

Now that set of vectors will sum to zero.

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Sketch of messy proof

First step - the midpoints are a distraction. ##\stackrel{\rightarrow}{A_ra_s} = \stackrel{\rightarrow}{A_rA_s} + \stackrel{\rightarrow}{A_sa_s} = \stackrel{\rightarrow}{A_rA_s }+ \frac{1}2\stackrel{\rightarrow}{A_sA_{s+1}}##, so the vector sum contains a scaled loop from the midpoint steps which sums to zero.

Next step - The remaining vectors form a set of closed loops which also sum to zero.

Improvements on this welcome...