Area Ratios from Medians and Centroid

AD,BE,CF are medians, G is the point of intersection, D,E,F are midpoints.

We showed in the "medians" post that the three medians concurred. Let's try to find some properties about this point of concurrence.

We make use of a very important idea: If two triangles have the same height, the ratio of their areas equals the ratio of their bases. If two triangles have the same base, the ratio of their areas equals the ratio of their heights.

Now, the ratio of AGF to BGF is $1:1$, the ratio of $AEG$ to $CEG$ is $1:1$, and the ratio CGD to GBD is $1:1$

Now we use the mass points technique: Assign masses of 1 to A,B,C , masses of 2 to D,E,F, and mass of 3 to G.

Then we find that $CG:GF = 2:1, BG:GE = 2:1, AG:GD= 2:1$

Now this means that the ratio of AGB to ACB is $1:3$, because GF is one third of CF.

Now.. if AGF is one third of AFC, and $CEG=AEG$, then we conclude all three triangles have equal area! By that logic, all six triangles have equal area! This is an important result.

Well... that's pretty much it for centroids.