D and E are points on sides AB and AC respectively of ∆ ABC such that ar (DBC) = ar (EBC). Prove that DE || BC.

If two triangles are on a common base and have equal areas, then they will lie between the same parallel lines.

Let's draw the given triangle ABC.

Given that ar (DBC) = ar (EBC)

Since ΔEBC and ΔDBC lie on a common base BC and they also have equal areas, therefore, ΔEBC and ΔDBC will lie between the same parallel lines.

Hence, DE || BC proved.