In the 1770s Euler discovered an infinite family of triangles with three rational sides and three rational medians---however, this family did not contain all such triangles. Over 200 years later, Buchholz and Hayes found a parameterisation of all rational sided triangles with two rational medians. When one tries to force all of these triangles to have a third rational median one is forced to find rational points on an elliptic surface.

In fact, any rational point on the above surface would correspond to a triangle with all sides and medians rational.

For more details see Triangles with 3 rational medians.