A simple proof that the area of a Pythagorean triangle is square-free

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 5, 1999, Number 3, Pages 115–118
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Authors and affiliations

J. V. Leyendekkers
The University of Sydney, 2006, Australia

A. G. Shannon
University of Technology, Sydney, 2007, Australia

Abstract

By using the geometry of a pythagorean triangle with a circle inscribed, it can be proved by a simple geometric proof that the area of such a triangle can never be a square. The class structure of the modular ring Z4 can be used to illustrate the result for various Pythagorean triples.

AMS Classification

  • 51M04
  • 11A07

References

  1. C.B. Boyer. A History’ of Mathematics. Princeton: Princeton University Press, 1985.
  2. A.F. Horadam. A Guide to Undergraduate Projective Geometry. Sydney v Per gam on Press, 1970.
  3. J.V. Leyendekkers, J.M. Rybak and A.G. Shannon. Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. 3.2 (1997): 61-74.
  4. I. Todhunter. Plane Trigonometry. London: Macmillan, 1884.

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Cite this paper

Leyendekkers, J. V. & Shannon, A. G. (1999). A simple proof that the area of a Pythagorean triangle is square-free. Notes on Number Theory and Discrete Mathematics, 5(3), 115-118.

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