Centroid and Medians of a Triangle
By: Lacy Gainey
The CENTROID (G) of a triangle is the common intersection of the three medians.
A median of a triangle is the segment from a vertex to the midpoint of the opposite side. The medians divide the triangle into six small triangles.
We want to prove that the three medians of a triangle are concurrent.
We also want to show that the centroid is located ⅔ the distance from a vertex to the midpoint of the opposite side.
Observe the triangle below,
Draw a segment between point D and E.
We can see that 
.
Using the Side-Angle-Side theorem, we can see that ΔABC ≈ ΔDBE. (1)
We know 
since ΔABC and ΔDBE are similar.
Additionally, DE and AC are parallel.
since DE and AC are parallel.
Also, 
Hence, ΔDGE ≈ ΔAGC. (2)
Observe,
We can use similar reasoning to show BG is ⅔ the length of BF.
Additionally, the G mentioned in all of these proofs is the same, showing that the medians of the triangle are concurrent.
