  We define spherical Heron triangles (spherical triangles with "rational"
side-lengths and angles) and parametrize them via rational points of certain
families of elliptic curves. We show that the congruent number problem 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 rational area
bisector (median splitting the triangle in half), and discuss various problems
involving isosceles spherical triangles.
