The median goes from the middle of a side to the opposite vertex. Since that point is in the middle of that side, it divides it into two equal segments. (See the diagram below).
Compare the two triangles that are formed. They will be the same height because the distance from the base to the vertex has to be the same (a line segment is congruent to itself because of the reflexive property). And we already know they have congruent bases equal to ½ of the original triangle's base.
So if we have two triangles, each with the same base and the same height measurements, their areas will have to be equal. Area = ½bh
So if b₁ = b₂ and h = h, then the areas must be the same.