Exploring Triangles and its Medians
by
Rita Meyers

This write-up will explore a triangle and a triangle whose sides are equal to the lengths of the original triangle's medians.

We will first start by constructing a triangle from the medians of the original triangle.  I will call this triangle the medial triangle.
So, given this original triangle ABC and its medians.

We construct the medial triangle DEF.
Once we have both triangles let's look at their relationship to one another.
First we will measure the medians of the original triangle and verify that they are in fact equal.

As we see the lengths are in fact equal

Now, let's look at  the area of both the original triangle and the medial triangle.

Right off the bat we see no relation between the two...they are not equal.
Let's look at the ratio of the area of triangle DEF to triangle ABC

It appears that the area of triangle DEF is 3/4 the area of triangle ABC.

Is this always the case? Let's look!
Exploration #1

Moving pt B of our original triangle along a segment, we notice that it changes the size of not only the original triangle but also the medial triangle.

We can also see the changes in the calculation of the areas of both triangle.

More importantly, we notice that the ratio between the two areas of the triangles does not change.

Let's look at it another way.  This time we will move pt A of our original triangle along the circumference of a circle and look for changes in the ratio of the areas.

Exploration #2

No matter if the original triangle is acute or obtuse, the ratio of the area of the original triangle to the area of the triangle made up of its medians does not change.  The area of the medial triangle is always 3/4 of the original triangle.