  We define hyperbolic Heron triangles (hyperbolic triangles with "rational"
side-lengths and area) and parametrize them in two ways as rational points of
certain elliptic curves. We show that there are infinitely many hyperbolic
Heron triangles with one angle $\alpha$ and area $A$ for any (admissible)
choice of $\alpha$ and $A$; in particular, the congruent number problem has
always infinitely many solutions in the hyperbolic setting. We also explore the
question of hyperbolic triangles with a rational median and a rational area
bisector (median splitting the triangle in half).
