Difference between revisions of "Van Aubel's Theorem"
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Finally, we have <math>(p-r) = i(q-s) = e^{i \pi/2}(q-r)</math>, which implies <math>PR = QS</math> and <math>PR \perp QS</math>, as desired. | Finally, we have <math>(p-r) = i(q-s) = e^{i \pi/2}(q-r)</math>, which implies <math>PR = QS</math> and <math>PR \perp QS</math>, as desired. | ||
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Revision as of 12:17, 30 May 2019
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.