Heronian triangles conjecture

Let my first blog post be about my six-year-old conjecture:

Every integer Heronian triangle is congruent to a lattice triangle.

A triangle is said to be an integer Heronian triangle, if it has integer side lengths and integer area. A triangle is said to be a lattice triangle, if all its vertices have integer coordinates.

Not much is known about this conjecture. Clearly:

  • An area of any lattice triangle is an integer multiple of ½. If in addition it has integer side lengths, then its area is integer. This follows from Heron’s formula.
  • Any counterexample must have a side with length greater than 1000. This was checked by enumerating all small Heronian triangles and verifying the conjecture directly.
  • Noteworthy, there are integer Heronian triangles which are not congruent to a lattice triangle with one side parallel to a coordinate axis, like in the following example (side lengths 5, 29, 30):An integer Heronian triangle

The problem was first posed on a Polish math Usenet group pl.sci.matematyka in 2004.

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3 Responses to Heronian triangles conjecture

  1. Johannes boot says:

    I think this conjecture was proven by Paul Yiu. See math.fau.edu/Yiu/AMM2001pages261–263.pdf

  2. Alexander Perlis says:

    For a different proof, as well as the corresponding result for tetrahedra, see http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.140

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