The Rational Distance Problem for Equilateral Triangles

Roy Barbara; Antoine Karam

doi:10.26713/cma.v9i2.659

Authors

Roy Barbara Department of Mathematics, Lebanese University, Faculty of Science II, Fanar
Antoine Karam Department of Mathematics, Lebanese University, Faculty of Science II, Fanar

DOI:

https://doi.org/10.26713/cma.v9i2.659

Keywords:

Equilateral triangle, Rational distance problem, Bi-quadric number, Legendre's symbol, Non-degenerated triangle, Primitive integral triangle

Abstract

We provide a complete characterization of all equilateral triangles \(T\) for which there exists a point in the plane of \(T\), that is at rational distance from each vertex of \(T\).

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References

T.G. Berry, Points at rational distance from the vertices of a triangle, Acta Arithmetica LXII (4) (1992), 391 – 398.

R. Barbara, The rational distance problem for polygons, Mathematical Gazette 97 (538) (2013), note 97.11.

R. Barbara and A. Karam, The rational distance problem for isosceles triangles with one rational side, Communications in Mathematics and Applications 4 (2) (2013), 169 – 179.

Wikipedia, Equilateral Triangle, https://en.wikipedia.org/wiki/Equilateral_triangle.

Wolfram Mathworld, Equilateral Triangle, http://mathworld.wolfram.com/EquilateralTraiangle.html.