Hello. In this mini-lesson, I’ll talk about a special point in a triangle – called the centroid.

Take any triangle, say ΔABC. Draw the three medians, AD, BE, and CF. Then:

AD , BE , and CF always intersect at a point. In other words, they are concurrent.

, , and always intersect at a point. In other words, they are concurrent. The point of concurrency of the three medians is known as the triangle’s centroid.

Let’s observe the same in the applet below. Press the play button to start.

Drag the vertices to see how the centroid (G) changes with their positions. Once you’re done, think about the following:

does the centroid always lie inside the triangle?

can the centroid lie on the (sides or vertices of the) triangle?

and why do the medians have to be concurrent anyways?

Go, play around with the vertices a bit more to see if you can find the answers.

A couple more questions for you. In terms of the side lengths (a, b, c) and angles (A, B, C),

what is the length of each median?

how far does the centroid lie from each vertex?

To find these answers, you’ll need to use Pythagoras’ theorem, along with a special property which I’m going to talk about next.

The centroid has a nice little property – it divides each median in the ratio 2 : 1, the larger part being that joining the vertex and the centroid.

Let’s observe the same in the applet below. Press the Play button to start.

Once again, you can drag the vertices to vary the triangle.

AG : GD = BG : GE = CG : GF = 2 : 1. Always. Awesome, right?

Try proving this property – it doesn’t involve much beyond high-school geometry.

Drop me a message here in case you need some direction in proving this, or discussing the answers of any of the previous questions.

Goodbye.