Applications of square-related theorems

Abstract

The square centre of a given square is the point of intersection of its two diagonals. When two squares of different side lengths share the same square centre, there are in general four diagonals that go through the same square centre. The Two Squares Theorem developed in this paper summarizes some nice theoretical conclusions that can be obtained when two squares of different side lengths share the same square centre. These results provide the theoretical basis for two of the constructions given in the book of H.S. Hall and F.H. Stevens , ‘A Shorter School Geometry, Part 1, Metric Edition’. In page 134 of this book, the authors present, in exercise 4, a practical construction which leads to a verification of the Pythagorean theorem. Subsequently in Theorems 29 and 30, the authors present the standard proofs of the Pythagorean theorem and its converse. In page 140, the authors present, in exercise 15, what amounts to a geometric construction, whose verification involves a simple algebraic identity. Both the constructions are of great importance and can be replicated by using the standard equipment provided in a ‘geometry toolbox’ carried by students in high schools. The author hopes that the results proved in this paper, in conjunction with the two constructions from the above-mentioned book, would provide high school students an appreciation of the celebrated theorem of Pythagoras. The diagrams that accompany this document are based on the free software GeoGebra. The author formally acknowledges his indebtedness to the creators of this free software at the end of this document.

Keywords:

• central square chords
• square centre
• Pythagorean theorem

Acknowledgements

The author is grateful to the authors and co-authors of the free software GeoGebra. The GeoGebra can be obtained from http://www.GeoGebra.org. However, my acknowledgment should not be construed as their endorsement of my work in any way. My thanks are for all the people included in the following list.

Main authors: Judith  Hohenwarter,  [email protected]; Markus Hohenwarter, [email protected].

Co-authors: Christina Biermayr, [email protected]; Corinna Kröhn, [email protected];  Melanie Tomaschko, melanie.tomaschko@geogebra, together with help from many other GeoGebra team members.

The author would like to thank the referees for their suggestions. In order to keep the length of the main document as short as possible, the author made substantial changes from the previous version and decided to omit many diagrams except the two crucial ones that illustrate the two verifications of the theorem of Pythagoras by H.S. Hall and F.H. Stevens.