Properties of the Centroid

By: Sydney Roberts

A median is a line segment that extends from the vertex of a triangle to the midpoint of the opposite side of the triangle. Therefore, each triangle with have three medians that intersect at a point that is exactly two-thirds of the way along each median from each vertex. We call this point the Centroid.

How do we know this is true? Consider the following triangle XYZ.

We can think of each vertex of this triangle as a vector. Hence, we can label each side as such, where x  is the vector extending from O to X, y is the vector extending from O to Y, and similarly z is the vector extending from O to Z.

Now, hiding the original vectors in order to avoid clutter, we can also find the midpoints of each side of the triangle and label them as such.

Now that we have the midpoints, we can construct the medians.

Now we want to show that each of these medians intersect at a common point. Specifically, we want to show that these medians intersect at the point  which we refer to as the centroid of a triangle. As mentioned before, we want to show that this centroid lies two-thirds of the way down each median from the vertex. Well, now that we have the vectors for each side length, we can label the medians as follows.

Now we just need to show that

And

Well,

Similarly, this algebra will hold on the remaining two equations and we see that the centroid does indeed lie at a point two-thirds of the way along each median from the vertex.

The centroid is commonly referred to as the “center of mass” or “center of gravity” for a triangle. This idea comes from the fact that if you had a 2D triangle with the centroid labeled, that this centroid would be the balancing point and the triangle could rest on this point alone without toppling. This should only work if the area around this point is evenly dispersed.

Therefore, consider the three triangles that are formed by the intersection of the medians.

Now we want to show that the area of triangle ABG equals the area of triangles CGA and BGC. Conveniently, we can use Geometer’s Sketchpad to calculate this, and we can see that this is true.

If you want to change the triangle and see that the areas remain equal, use the following GSP file:

Equal Areas