Van Aubel's Theorem
Revision as of 20:06, 17 December 2008 by Archimedes1 (talk | contribs) (New page: = Theorem = Construct squares <math>ABA'B'</math>, <math>BCB'C'</math>, <math>CDC'D'</math>, and <math>DAD'A'</math> externally on the sides of quadrilateral <math>ABCD</math>, and let the...)
Theorem
Construct squares ,
,
, and
externally on the sides of quadrilateral
, and let the centroids of the four squares be
and
, respectively. Then
and
.
<geogebra>21cd94f930257bcbd188d1ed7139a9336b3eb9bc</geogebra>
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.