The line $y = mx$ goes through $(0,0)$, which is one of the vertices of the triangle.
If a line goes through a vertex, then it splits the triangle in two triangles that have the same height from that vertex. Thus, the triangles have equal area if and only if the line is a median, in particular, the line goes through the centroid of the triangle.
Note we would have reached the same answer if we had used the midpoint of side $BC$ instead of the centroid of the triangle.