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Heron triangles and touching circles

Published online by Cambridge University Press:  01 August 2016

Christopher J. Bradley
Christopher J. Bradley*
Affiliation:
Clifton College, Bristol BS8 3JH
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Extract

Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.

Type
Articles
Information
The Mathematical Gazette , Volume 87 , Issue 508 , March 2003 , pp. 36 - 41
Copyright
Copyright © The Mathematical Association 2003

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References

1. Wain, Geoffrey and Willson, William Wynne, 13, 14, 15: an investigation, Math. Gaz. 71 (March 1987) pp. 32–37.Google Scholar
2. McLean, K. Robin , Heronian triangles are almost everywhere, Math. Gaz. 72 (March 1988), pp. 49–51.Google Scholar
3. Sastry, K. R. S., Heron problems, Math, and Comput. Education, 29 (Spring 1995), pp. 192–202.Google Scholar
4. Sastry, K. R. S., Heron triangles: a new perspective, Aust. Math. Soc. Gazette, 26 (Oct 1999), pp. 160–168.Google Scholar
5. Sastry, K. R. S., A Heron difference, Crux Mathematicorum, 21, (February 2001) pp. 22–26.Google Scholar
6. The correspondence of Descartes with the Princess Elizabeth in Adam and Tannery, Oeuvres de Descartes, Vol. 4 (Paris 1901), p. 63.Google Scholar
7. Pedoe, Dan, Geometry, a comprehensive course, Dover (1988) pp. 153–159.Google Scholar