Van Aubel's Theorem
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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Proofs
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.