Perfect rational triangles

Definitions and mathematical statement

For a nondegenerate Euclidean triangle let $a,b,c>0$ be its side lengths, $\Delta>0$ its area, and $m_a,m_b,m_c$ its medians, each joining a vertex to the midpoint of the opposite side. Its altitudes are $h_a=2\Delta/a$, $h_b=2\Delta/b$, and $h_c=2\Delta/c$. The question asks for a triangle for which all ten of these quantities are rational.

Equivalently, find positive rational $a,b,c$ satisfying the strict triangle inequalities, together with positive rational $\Delta,m_a,m_b,m_c$, such that \[ \begin{aligned} 16\Delta^2&=(a+b+c)(-a+b+c)(a-b+c)(a+b-c),\\ 4m_a^2&=2b^2+2c^2-a^2, \end{aligned} \] and the two cyclic versions of the median equation also hold. Rationality of the altitudes then follows from the displayed formulas. Conversely, rational sides and altitudes force rational area.

Uniform rational scaling preserves every rationality requirement and can clear all denominators simultaneously. Thus an equivalent existence question asks for integral sides, medians, altitudes, and area after a suitable scaling. Strict nondegeneracy is indispensable; collinear triples with zero area are excluded. Examples with rational medians but irrational area do not settle the question.

Short English statement

Does a genuine triangle exist whose sides, medians, altitudes, and area are all rational numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3900012 (AMR-038-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eppstein - The Geometry Junkyard: Open Problems, Perfect rational triangles. Original source indexed by AMR-038-0012. https://ics.uci.edu/~eppstein/junkyard/open.html.
• S3 David Eppstein, Geometry Junkyard, Rational triangles, including the original question. https://ics.uci.edu/~eppstein/junkyard/q-triangle.html.