Triangle Median Problem: Finding AD=1/2AB and BE=1/2EC Relationship

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Here, we need to find such a segment in triangle $\triangle ABC$.

Let's analyze the given conditions:

• $AD = \frac{1}{2}AB$: Point $D$ is the midpoint of $AB$.
• $BE = \frac{1}{2}EC$: Point $E$ is the midpoint of $EC$.

Given that $D$ is the midpoint of $AB$, if we consider the line segment $DC$, it starts from vertex $D$ and ends at $C$, passing through the midpoint of $AB$ (which is $D$), fulfilling the condition for a median.

Therefore, the line segment $DC$ is the median from vertex $A$ to side $BC$.

In summary, the correct answer is the segment $DC$.