Objective: We will first look define the centroid. Then we will see that the medians divide the triangle into six smaller triangles and we will prove that these six smaller triangles all have the same area.
Vocabulary
Centroid: the point where the three medians of a triangle intersect; often referred to as the center of gravity of the triangle and it will always be contained inside the triangle; G is the centroid in both the acute and obtuse triangles below
Median: a line segment connecting a vertex to the midpoint of the opposite side; triangle contains three medians; line segments AE, DC, and BF are the medians in both the acute and obtuse triangles below
Proof
Claim: The six smaller triangles formed by the medians all have the same area.
We see that the claim states that the smaller triangles all have the same area. We are not proving that the six smaller triangles are congruent as evidence by both visuals above. For the proof we will use acute triangle representations; however, the same argument holds for obtuse and right triangles. Note that F, E, and D are the midpoints of the sides of the triangle.
First we will examine three triangles:∆AGC, ∆BGC, and ∆AGB.
We see that line segment AF=FC since F is the midpoint of the line segment AC. Thus, ∆AFG and ∆FGC have the same area since they have the same base and they share the same height of length FG. Similar arguments can be made for the two smaller triangles contained in ∆BGC and ∆AGB. This gives us the following visual representation with x, y, and z representing area values.
Next we will look at another set of triangles:∆ACE and ∆ABE.
The area of ∆ACE is given by 2x+y and the area of ∆ABE is given by 2z+y. We can show that ∆ACE and ∆ABE both have the same area since they share the same height and have the same base (BE=EC since E is the midpoint of the line segment BC). Therefore 2x+y=2z+y which tells us that x=z. Now we have the four of the smaller triangles have the same area.
Last we will look at the pair of triangles ∆BCF and ∆BAF.
The area of ∆BCF is given by 2y+x and the area of ∆BAF is given by 2z+x. In a similar manner, we can show that ∆BCF and ∆BAF both have the same area since they share the same height and have the same base (AF=FC since F is the midpoint of the line segment AC). Therefore 2y+x=2z+x which tells us y=z.
In summary, we found that x=z and y=z which tells us by the transitive property that x=z=y. In conclusion, the six smaller triangles formed by the division of the medians all have the same area. This is why the centroid (point G) is known as the center of gravity for a triangle since the mass is evenly distributed among the six smaller triangles.
