In a triangle ABC, E is the midpoint of median AD. Show that area(BED) = 1/4 area(ABC).

In a triangle ABC, we are asked to show that the area of triangle BED is equal to one-fourth of the area of triangle ABC, given that E is the midpoint of the median AD.

Step 1: Define the triangle and important points
Let triangle ABC be a triangle.
D is the midpoint of side BC, so AD is the median of the triangle.
E is the midpoint of median AD, as given in the problem.
Step 2: Properties of the median and midpoints
Since D is the midpoint of BC, the median AD divides triangle ABC into two smaller triangles, ABD and ACD, that have equal areas.
Since E is the midpoint of AD, the line segment AE is equal to ED. This divides the median AD into two equal parts.
Step 3: Analyzing the area
The area of triangle ABC can be written as the sum of the areas of triangles ABD and ACD. Since D is the midpoint of BC, both triangles ABD and ACD have equal areas, so the area of triangle ABD is half of the area of triangle ABC. Therefore, the area of triangle ABD = area of triangle ABC / 2.

Now, since E is the midpoint of AD, triangle BED is formed by connecting points B, E, and D. Triangle BED is a smaller triangle formed within triangle ABD. Specifically, triangle BED is one-fourth of triangle ABD because:

The area of a triangle is proportional to the base and the height. In this case, the base of triangle BED is BD, which is half of the base of triangle ABD (since E is the midpoint of AD).
The height from point E to base BD is half of the height from point A to base BC (since E divides the median AD into two equal parts).
Therefore, the area of triangle BED is one-fourth of the area of triangle ABD.

Step 4: Conclusion
Since the area of triangle ABD is half of the area of triangle ABC, the area of triangle BED is one-fourth of the area of triangle ABC.

Thus, we have shown that:

Area(BED) = (1/4) * Area(ABC)