By: Melissa Wilson
The three medians in a triangle divide the original triangle into six smaller triangles with a common vertex at the centroid of the original triangle.
We are going to prove that the six triangles all have the same area. First let us look at the triangle AOD and triangle BOD, shaded below. Suppose that there is a line through O that is parallel to AB.
For the areas to be equal they need to share the same base and altitude lengths. By construction of the median we know that D is at the midpoint of AB, hence the length AD = length DB. This shows that the bases are the same. Now we can see that the altitudes are the same since the perpendicular distance from O to AB will be the same for both triangles. Thus, the area of triangle AOD is the same as the area of triangle BOD.
We can do do the same analysis for triangle AOF and triangle COF showing that they share the same area. Also, it follows that the area of triangle COE is equal to the area triangle BOE.
Now, we need to show that each pair of the smaller triangles all have the same area, thus showing the six smaller ones have the same area.
We know that CD cuts triangle ABC into two equal triangles (using the similar argument as above). Let's look at triangle ACD. We can say the area of triangle AOC (shown below shaded) is equal to the area of ACD minus the area of AOD.
Looking at triangle CDB, we can also see that the area of triangle BOC (shaded below) is equal to the area of CDB minus the area of BOD.
Remember that we proved earlier that the area of triangle AOD is equal to triangle BOD. Hence, the area of triangle AOC is equal to triangle BOC.
In a similar process we can show that triangle COB has the same area as triangle AOB and that triangle AOC has the same area as triangle AOB.
Therefore, the three large triangles all have the same area. Previously we showed that each large triangle was divided equally into the two smaller triangle. Hence, all six smaller triangles have the same area.
