<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The focus of this note is the nontrivial determinant identities which typically underlie the complex analytic proofs of all the results in the plane geometry of lines and circles. After setting up a basic dictionary relating lines and circles to complex determinants we derive such identities in connection with four geometry problems: the Steiner line, a variant of Euler’s nine-point circle, the Johnson-Tzitzeica circles, and an extension of a certain geometry problem, proposed at the 52nd International Mathematical Olympiad, Amsterdam 2011.</p></abstract>

The focus of this note is the nontrivial determinant identities which typically underlie the complex analytic proofs of all the results in the plane geometry of lines and circles. After setting up a basic dictionary relating lines and circles to complex determinants we derive such identities in connection with four geometry problems: the Steiner line, a variant of Euler’s nine-point circle, the Johnson-Tzitzeica circles, and an extension of a certain geometry problem, proposed at the 52nd International Mathematical Olympiad, Amsterdam 2011.

Determinant Identities and the Geometry of Lines and Circles

Department of Mathematics, University of North Texas, Denton TX, 76203

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Determinant Identities and the Geometry of Lines and Circles"}

The focus of this note is the nontrivial determinant identities which typically underlie the complex analytic proofs of all the results in the plane geometry...

Determinant Identities and the Geometry of Lines and Circles

Published Online: Oct 20, 2015

Page range: 37 - 50

Received: Mar 01, 2013

Accepted: Jun 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0029

Keywords
Determinant, Line, Circle, Complex Numbers, Möbius Transformation, Steiner Line, Euler Nine-Point Circle, Johnson-Tzitzeica Circles, International Mathematical Olympiad

© 2014 Nicolae Anghel, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Determinant Identities and the Geometry of Lines and Circles

Published Online: Oct 20, 2015

Page range: 37 - 50

Received: Mar 01, 2013

Accepted: Jun 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0029

KeywordsDeterminant, Line, Circle, Complex Numbers, Möbius Transformation, Steiner Line, Euler Nine-Point Circle, Johnson-Tzitzeica Circles, International Mathematical Olympiad

© 2014 Nicolae Anghel, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Determinant, Line, Circle, Complex Numbers, Möbius Transformation, Steiner Line, Euler Nine-Point Circle, Johnson-Tzitzeica Circles, International Mathematical Olympiad