Write Up 4: The "Centroid"

Centroid Exploration: "The medians of a triangle divide the triangle into 6 smaller triangles. Show that all of these triangles have the same area."

What is the Centroid?

The centroid of a triangle is the intersection point of the three medians of a triangles. A median of a triangle is a line segment from a vertex to the midpoint of the opposite side from that vertex. The centroid of a triangle is denoted by the capital letter "G." Let's observe a triangle and its constructed centroid.

Note: Click this link to use the GSP script tool for centroid:

As we can see above, the centroid (G) creates six smaller triangles within the larger triangle. One interesting characteristic of the centroid is that the areas of all of the 6 smaller triangles will have the same area. A proof of this characteristic can be seen below.

Area of Smaller Triangles

First, let's divide our outer triangle ABC into six smaller triangles each sharing a vertex at the centroid of the big triangle labeled "G."

Let's look first at the triangle FGD. This triangle is divided into two smaller triangles FGE and DGE. Let's prove that these two triangle areas are equal.

First, let's observe that the bases of triangles FGE and DGE are equal because of the midpoint E. Since point E is the midpoint of FD, the lenghts FE=ED. Therefore, the bases of the two triangles are the same. Now suppose that the line passing through point G is parallel to the base of the triangle FD. Then the perpendicular distance from G to the base FD is shared by both triangles FGE and DGE and serves as their altitude. Therefore, since both triangles FGE and DGE share the same base and altitude, their areas are the same. A similar argument can be used to prove that the areas are equal for triangles FGZ and BGZ and for the pair of triangles BGC and DGC. So the following is what we have proved so far:

Area of FGE=DGE, Area of FGZ=BGZ, Area of BGC=DGC

Now let's use what we have proved to show that all of these triangle areas are equal. Let's start by dividing the outer triangle ABC into two triangles BFE and BDE. Let's look more closely at triangle BFE. The area of this triangle can be found be adding the areas of triangle BGF and triangle FGE. Since we already know that the area of triangle FGE equals the area of triangle DGE, we can find the area of triangle BGF by subtracting triangle FGE from it. The argument to find the area of triangle BGD is similar by subtracting the area of DGE from triangle BGD. Again, we previously proved that the areas of triangles FGE and DGE are equal. Therefore, since the areas of triangles FBE and DBE are equal, the areas of BGF and BGD must also be equal. Let's look at this mathematically.

Area of FBE=Area of DBE (they have the same base length and share the same altitude)

Area of FBE= Area of FBG-Area of FGE

Area of DBE= Area of DBG-Area of DGE

Area of FGE=Area of DGE

Therefore, Area of FBG=Area of DBG