In a triangle ABC,D and E are the mid points of BC and AD respectively.prove that the area of triangle ABC is four times to the area of triangle ABE

To prove that the area of triangle ABC is four times the area of triangle ABE, we can use the concept of similar triangles.

First, let's draw triangle ABC and mark point D as the midpoint of BC and point E as the midpoint of AD.

Since D is the midpoint of BC, we can say that BD = CD. Similarly, since E is the midpoint of AD, AE = ED.

Now, let's draw a line segment from point A to point E, and let's call the point where this line segment intersects BC as F.

By connecting points B and F, we create two triangles: ABE and BCF.

We know that DE is parallel to BC since D and E are the midpoints of the respective sides. Therefore, we can use the concept of similar triangles.

By the Midpoint Theorem, we can say that:

AE/AD = DE/BC

Since AE = ED, we can substitute these values into the equation:

ED/AD = DE/BC

Since triangle BED is similar to triangle BAC, we can say that the ratios of the corresponding sides are equal:

BE/BA = ED/DC

Since BD = CD, we can substitute these values into the equation:

BE/BA = ED/BD

Cross-multiplying the equation gives us:

BE * BD = ED * BA

Now, we can see that triangle ABC and triangle ABE share the same height, which is the perpendicular distance from point A to BC.

Since both triangles have the same height, the ratio of their areas is equal to the ratio of their bases.

Therefore, the area of triangle ABC is equal to (BE * BC) / 2, and the area of triangle ABE is equal to (BE * BA) / 2.

Now, we can substitute the values we obtained earlier:

Area of triangle ABC = (BE * BC) / 2
Area of triangle ABE = (BE * BA) / 2

Since we have BE * BD = ED * BA, we can rewrite the equation for the area of triangle ABE as:

Area of triangle ABE = (BE * BD) / 2

Substituting this value back into the equation for the area of triangle ABC:

Area of triangle ABC = (BE * BC) / 2 = (BE * 2 * BD) / 2 = BE * BD

Therefore, the area of triangle ABC is equal to four times the area of triangle ABE, as required to be proved.