Centroid Proof

The centroid is the point of concurrency of the three medians (a line segment through a vertex and through the midpoint of the opposite side).  One property of a centroid is that it cuts each median in a 2:1 ratio.

                                                                       

Centroid P of ∆ABC

This property can be derived by considering the area of each triangle formed in the centroid diagram.  Since D is the midpoint of BC, the area of ∆BPD is equal to the area of ∆CPD (which we will call x), since they share the same height and have equal bases.  Similarly, since E is the midpoint of AC, the area of ∆BPE is equal to the area of ∆CPE (which we will call y); and since F is the midpoint of AB, the area of ∆BPF is equal to the area of ∆APF (which we will call z).

Now, since D is the midpoint of BC, it also means that the area of ∆BAD is equal to the area of ∆CAD, since they also share the same height and also have equal bases.  So according to our diagram, x + z + z = x + y + y, which simplifies to y = z.  We can use the same reasoning to show that x = y, and x = z, which means that x = y = z. 

So when we consider a triangle with any median as one of its sides, say ∆BEA, we see that it is composed of three equal areas (in this case z, z, and y), in which ∆BPA is composed of two of those equal areas (z and z) and ∆EPA is composed of the other area (y).  This means BP is twice as long as PE, proving that the centroid cuts the median in a 2:1 ratio.