A characterization of all integer-sided triangles with a rational median is given, similar to the categorization of Pythagorean triangles. An infinite family of integer-sided triangles with two rational medians is given, along with several examples of three rational medians. All examples come from solutions to systems of quadratic Diophantine equations.
Goddard, Bart and Mesner, Dale, "Integer Triangles with Rational Medians" (1991). Mathematical Sciences Technical Reports (MSTR). 76.