Difference between revisions of "Van Aubel's Theorem"
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...) |
Thomasihkim (talk | contribs) (→Theorem) |
||
Line 1: | Line 1: | ||
= Theorem = | = 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 centroids of the four squares be <math>P, Q, R,</math> and <math>S</math>, respectively. Then <math>PR = QS</math> and <math>PR \perp QS</math>. | 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 centroids of the four squares be <math>P, Q, R,</math> and <math>S</math>, respectively. Then <math>PR = QS</math> and <math>PR \perp QS</math>. | ||
− | <geogebra>21cd94f930257bcbd188d1ed7139a9336b3eb9bc</geogebra> | + | <math><geogebra>21cd94f930257bcbd188d1ed7139a9336b3eb9bc</geogebra></math> |
= Proofs = | = Proofs = |
Revision as of 10:58, 17 June 2015
Theorem
Construct squares ,
,
, and
externally on the sides of quadrilateral
, and let the centroids of the four squares be
and
, respectively. Then
and
.
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.