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On the Number of Perfect Triangles with a Fixed Angle

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Abstract

Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle \(0<\theta < \pi \), the number of perfect triangles with an angle \(\theta \) is finite.

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Notes

  1. In general the conic \(ax^2+bxy+cy^2+dx+ey+f=0\) is an ellipse if \(b^2-4ac <0\). In our situation \(b=2\lambda =2\cos \theta \), where \(0<\theta <\pi \) and \(\theta \ne {\pi }/{2}\) and \(d=e=0\).

  2. https://math.stackexchange.com/questions/1479139/fiber-products-of-curves.

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Acknowledgements

The author was supported by the Austrian Science Fund (FWF): Project P 30405-N32. I am grateful to Matteo Gallet, Niels Lubbes, Oliver Roche-Newton, Josef Schicho, and Audie Warren for several helpful conversations and comments.

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    Mehdi Makhul

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Correspondence to Mehdi Makhul.

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Makhul, M. On the Number of Perfect Triangles with a Fixed Angle. Discrete Comput Geom 66, 1143–1149 (2021). https://doi.org/10.1007/s00454-020-00227-7

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