Suppose that ∆ABC is any triangle. To determine its centroid, draw the median passing through one of its vertices, A. Let D be the midpoint of the line segment BC. (So the median passes through A and D.) Similarly, draw the median passing through C and let the median meet AB at E, the midpoint of the line segment AB. AD intersects CE at T. Then T is the centroid of ∆ABC.
If the length of line segment AT is AT and the length of line segment TD is TD, what is AT:TD? Also, if the length of line segment CT is CT and the length of line segment TE is TE, what is CT:TE?
To answer this, let:
Note that:
Accordingly, we have the following:
……………………………. (*)
As 
, the equation (*) is equivalent to:
…………………………………….. (**)
The equation (**) implies 
and 
.
Hence AT : TD = CT : TE = 2 : 1.
The next question:
If the median passing through B is drawn, does it pass through T? In other words, are all the medians concurrent? Yes! Let’s prove it. Let the midpoint of the line segment AC is P. (See the figure below.)
Note that:
The fact that 
shows that: 1) the points B, T, and P are collinear, and 2) BT : BP = 2 : 1. To sum up, all the medians of ∆ABC are concurrent. All the medians meet at one and only one common point T.
The next question:
Given the coordinates of A, B, and C of the triangle ∆ABC on a cartesian space, how to determine the coordinate of the triangle’s centroid? The previous result shows that 
. The equation implies:
where 
, 
, and 
are the position vectors of T, E, and C, respectively. Since E is the midpoint of the line segment AB, it follows that:
Accordingly, the coordinate of the centroid can be determined easily from the position vector .
Example 1
Given ΔABC with A(1,5), B(-3,8), C(5,11). Find the coordinates of its centroid.
Answer
Let the centroid be T with the position vector .
Consequently, the coordinates of the centroid is (1,8).
Example 2
Given ΔABC with A(1,5,0), B(-3,8,4), C(5,11,2). Find the coordinates of its centroid.
Answer
Let the centroid be T with the position vector .
Consequently, the coordinates of the centroid is (1,8,2).
The next question:
What is the distance between the centroid of a triangle and the base? Consider the ΔABC below.
In the ΔABC, AD and CE are medians and T is the centroid. Our previous result states that CT : TE = 2 : 1. Let K be the point on the line segment AB such that CK ⊥ AB. Let L be the point on the line segment AB such that TL ⊥ AB. Note that ΔCKE is similar to ΔTLE, and consequently:
Since CT : TE = 2 : 1, CE = 3 TE and furthermore 
. This leads to the conclusion that the distance between the centroid of a triangle and the base of the triangle is one third of the length of the altitude drawn from the vertex opposite to the base.
