Elliptic curves and Triangles with three rational medians

In his paper Triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a one-parameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at least 4 and we prove the existence of infinitely many curves of rank 5 over Q parametrized by an elliptic curve of positive rank. Finally, we show particular examples of curves within those families having rank 9 and 10 over Q.