Van Aubel's Theorem
Construct squares , , , and externally on the sides of quadrilateral , and let the centroids of the four squares be and , respectively. Then and .
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Proof 1: Complex Numbers
Putting the diagram on the complex plane, let any point be represented by the complex number . Note that and that , and similarly for the other sides of the quadrilateral. Then we have
From this, we find that Similarly,
Finally, we have , which implies and , as desired.