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.**