The centroid of a triangle is a point that represents the intersection of the three medians of the triangle. On the other hand, the medians are the segments that connect the vertices with the midpoints of the opposite side.

In this article, we will learn about the centroid of a triangle in detail. We will learn how to determine its position and we will solve some practical examples.

## Definition of the centroid of a triangle

The centroid of a triangle is the point where the three medians of the triangle intersect. In turn, let us remember that the medians of the triangle are the segments that connect a vertex with the midpoint of the opposite side. Each of the medians divides the triangle into two equal smaller triangles.

The following is a diagram of the centroid of a triangle:

The centroid is always located inside the triangle no matter what type of triangle we have. However, for equilateral triangles, the centroid, orthocenter, incenter, and circumcenter are located in the same position.

## Properties of the centroid of a triangle

The following are some of the important properties and characteristics of the centroid of a triangle:

- The centroid of a triangle is the point of intersection of the medians of the triangle.
- The centroid represents the geometric center of the triangle.
- The centroid of a triangle is always located inside the triangle.
- The centroid of an equilateral triangle is located in the same position as its incenter, orthocenter, and circumcenter.
- The centroid divides the medians in a 2:1 ratio.

## Formula for the centroid of a triangle

The centroid formula allows us to find the coordinates of the centroid of a triangle. To illustrate this formula we will use the following triangle.

We can only get the coordinates of the centroid if we know the coordinates of the three vertices of the triangle. The formula for the centroid is:

$latex C(x, y)=\left( \frac{x_{1}+x_{2}+x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3}\right)$

where, ($latex x_{1},~y_{1}$), ($latex x_{2},~y_{2}$), ($latex x_{3},~y_{3}$) are the vertices and C(*x, y*) is the centroid.

Basically, we have to add the three *x*-coordinates of the vertices and divide them by 3 to get the *x*-coordinate of the centroid. We do the same with the *y*-coordinates.

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## Finding the centroid of a triangle graphically

If we don’t know the coordinates of the vertices of the triangle, we can find the centroid graphically. The centroid can be easily found by plotting two of the medians of the triangle.

Therefore, using a ruler, we follow these steps:

**Step 1:** Locate the midpoint on side AB. Measure that side and mark the middle to form point D.

** Step 2:** Draw a line segment from vertex C to point D.

** Step 3:** Locate the midpoint on side AC to get point E

** Step 4:** Draw a line segment from vertex B to point E.

** Step 5:** Find the point of intersection of segments CD and BE.

The segments CD and BE connect the midpoints with their opposite vertices. This means that they are two medians of the triangle. Therefore, the point of intersection is the centroid of the triangle.

## Centroid of a triangle – Examples with answers

The following examples apply what you have learned about the centroid of a triangle.

### EXAMPLE 1

Determine the coordinates of the centroid of a triangle that has vertices A(3, 6), B(1, 1), and C(5, 2).

**Solution:** We have the following coordinates:

- $latex (x_{1},~y_{1})=(3, ~6)$
- $latex (x_{2},~y_{2})=(1,~1)$
- $latex (x_{3},~y_{3})=(5,~2)$

Therefore, using the centroid formula, we have:

$latex C(x, y)=\left(\frac{x_{1}+x_{2}+x_{3}}{3},~\frac{y_{1}+y_{2}+y_{3}}{3}\right)$

$latex C(x, y)=\left(\frac{3+1+5}{3},~\frac{6+1+2}{3}\right)$

$latex C(x, y)=\left(\frac{9}{3},~\frac{9}{3}\right)$

$latex C(x, y)=(3,~3)$

### EXAMPLE 2

Find the centroid of a triangle that has vertices A(5, 7), B(2, 3), and C(6, 4).

**Solution:** We use the formula for the centroid with the following coordinates:

- $latex (x_{1},~y_{1})=(5, ~7)$
- $latex (x_{2},~y_{2})=(2,~3)$
- $latex (x_{3},~y_{3})=(6,~4)$

Therefore, we have:

$latex C(x, y)=\left(\frac{x_{1}+x_{2}+x_{3}}{3},~\frac{y_{1}+y_{2}+y_{3}}{3}\right)$

$latex C(x, y)=\left(\frac{5+2+6}{3},~\frac{7+3+4}{3}\right)$

$latex C(x, y)=\left(\frac{13}{3},~\frac{14}{3}\right)$

## See also

Interested in learning more about the circumcenter, orthocenter, incenter, and centroid of a triangle? Take a look at these pages:

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