Heronian triangles conjecture

Definitions

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.

Statement

Every integer Heronian triangle is congruent to a lattice triangle.

State of the art

• An area of a 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.
• I only know that the result is true for small enough triangles (side length less than 1000).
• Noteworthy, there are integer Heronian triangles which are not congruent to a lattice triangle with one side parallel to a coordinate axis. Example: 5, 29, 30.Explicit formula for side lengths of Heronian triangles does not seem to help much in such cases.

Origins

I first stated the conjecture on a Polish math usenet group pl.sci.matematyka in 2004.

Contact

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