Geometry

Properties of the Medians Proved Using the Center of Mass

Taken from Wikipedia

In this article, we will discuss a famous property of the medians and prove it using the properties of the center of mass.

For those who don’t know (or know very little) about the center of mass, I recommend to check out a handout I wrote on this topic.

What are we going to prove?

We will prove two quite well-known facts, namely, that medians concur at one point and that the centroid (their point of intersection) divides each median into parts in the ratio 2:1, counting from the vertex.

Prep

First, we need to remember two important properties:

1. If you replace a set of points with one point, located at their center of mass, and with a mass equal to the sum of their masses, then the center of mass of the whole system will stay the same.
2. The center of mass of points A and B with masses a and b, respectively, lies on segment AB and divides it in the ratio b:a.

Proof

Let the medians of the triangle ABC intersect sides AB, BC, and CA at points C₁, A₁, and B₁, respectively, and G be the centroid. Place unit masses into each vertex of the triangle ABC. Then A₁ is the center of mass of points B and C. Thus we can replace the points B and C with a point A₁ with mass being equal to 2, and the center of mass G of the triangle will lie on segment AA₁ and divide it in the ratio AG:GA₁ = 2:1. Similar statements are proven for the remaining medians and the conclusion follows.

I hope you liked this example of many applications of the properties of the center of mass. If you did, feel free to read my handout. It contains proofs of the main facts, some example problems (one of which I published in this article), and some problems for self-study.