# Difference between revisions of "Van Aubel's Theorem"

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= 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> 21cd94f930257bcbd188d1ed7139a9336b3eb9bc <geogebra> |

= Proofs = | = Proofs = |

## Latest revision as of 23:17, 19 July 2020

# 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.