I began this exploration by constructing a triangle and it's midpoints using the program GSP. Then I constructed another triangle with the same length size as the midpoints. I will now explore the sketch and the relationship between the two triangles.

I have reconstructed the small triangle with circles 3 different circles having radius of the line segments DE, EF, and DF.

First, I will determine if these triangles (ABC and DEF) are congruent. One way to tell if the two triangles will be congruent if the sides are of equal length. As you can see from the picture this will not be true but the angles are congruent.

Second, I will determine if these triangles are similar. One way to tell if the two triangles will be similar if the ratios of each side is equal and the angles are congruent.

Since,

Now that we have proved that they are similar then we will determine if their area is similar as well.