Elizabeth Nelli


The centroid of a triangle is the intersection of the three medians.
A median of a triangle is a straight line drawn from the vertex of an angle to the middle of the opposite side.


Let’s say triangle uwv is made up of the vectors u,v,w,d,e,f. We know by construction that e is the midpoint of v and w, d is the midpoint of v and u, and f is the midpoint of u and w. Therefore, e=1/2(v+w), d=1/2(u+v), and f=1/2(u+w).
By the definition of a median, we know that ue is a median, wd is a median, and vf is a median.

Assume that the centroid lies 2/3 of the way between each angle.
This means that

u + (2/3)((1/2)(v+ w) - u) = u + (1/3)(v + w) - (2/3)u = (1/3)( u + v + w)=
w+ (2/3)((1/2)(u+ v) - w) = w + (1/3)(u+ v) - (2/3)w = (1/3)( u + v + w)=
v + (2/3)((1/2)(w + u) - v) = v + (1/3)(w + u) - (2/3)v = (1/3)( u + v + w)

This means the centroid lies 2/3 of the way between ue, vf, and wd.



There is also a claim that states the 6 sub triangles formed by the medians with the common point at the centroid have the same area. Using GSP, we can prove this claim with measurements. Note the 6 different triangles formed: ABC, ABG, DBC, DBE, FBE, and FBG.

Based on the above proof, we know that A, D, and F are all midpoints of their respective sides, then we know that CD=DE=AC=AG=FG=FE.
Also, we know that the centroid is 2/3 of the way between CF, EA, and GB, therefore CB=EB=GB.
Lastly, we know that from D, A, and F, to be is equal to ½, therefore AB=DB=FB.
This means that all sides of each triangle are equal; meaning the area of each triangle is equal as well.

Here is a GSP construction and measurements showing this: