3 problems

Rathbun's conjecture. Infinitely many square-sided Heron triangles exist.

Nonexistence conjecture for perfect Heron triangles. No perfect triangle exists.

Let $p$ be a prime satisfying $p\not\equiv 1\pmod{8}$ and suppose that $p^{2}+1=2q$ for a prime $q$. Consider triangles with rational side lengths, area $p$, and an angle $\theta$…