Lesson # 4 Ė by Jan White
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.
I will show that the centroid is 2/3 the distance from any vertex to the midpoint of the other side. (To see a more animated version click here. Does the relationship always hold?)
Letís start by erasing the median BF and creating a segment DE.
AB/AD = CB/CE, therefore the new segment DE is parallel to segment AC.
Angle AGC=Angle DGE
Angle DEG=Angle GAC
Therefore triangle DGE is similar to triangle AGC.
By drawing a segment EF we can see that DE=AF and FE=AD. Since F is the midpoint of segment AC, DE/AC=1/2.
Looking back at the previous sketch DE/AC=DG/DC=EG/EA=1/2. Therefore G is 2/3 the distance from vertex A to side BC and vertex C to Side AB. Similar reasoning can be used to show G is 2/3 the distance from vertex B to side AC.