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):

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

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

Thanks for the link! The conjecture was proved well before I stated it. Well, I failed googling exam…

A stable link to the article is here: http://www.jstor.org/stable/2695390

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