Heron triangles with two perfect-square sides are completely parametrized via an elliptic curve, producing infinitely many primitive examples, while all-three-squares cases reduce to rational points on genus-3 curves suggesting finiteness.
Sierpi´ nski, Elementary theory of numbers, Wroclawska Drukarnia Naukowa (Warzawa, 1964)
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A Complete Characterization of Heron Triangles with Two Perfect Square Sides and the All-Square Equivalence Condition
Heron triangles with two perfect-square sides are completely parametrized via an elliptic curve, producing infinitely many primitive examples, while all-three-squares cases reduce to rational points on genus-3 curves suggesting finiteness.