Spherical Heron triangles and elliptic curves

We define spherical Heron triangles (spherical triangles with "ra- tional" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number prob- lem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a ra- tional area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.