Geometric median of two disjoint sets of points lies on line between their respective medians
If $A$ and $B$ are two set of points, the geometric median of $A \cup B$ does not necessarily lie on the line passing between the geometric mean of A and the geometric mean of $B$.

In the following example the points of $A$ are red and the one in $B$ is blue. The geometric median of $A$ is the red square. The green line connects the geometric means of $A$ and $B$. The black square is the geometric mean of $A \cup B$, i.e., the Fermat point of the triangle whose vertices are in $A \cup B$.