Centroid of a Triangle

The centroid of a triangle is the point where the three medians of the triangle intersect. A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians, and they are all concurrent at the centroid. The centroid is also known as the geometric center or barycenter of the triangle.

Properties of the Centroid

The centroid has several important properties:

1. It is the center of mass of a uniform-density triangle.
2. It divides each median into segments with a ratio of 2:1, with the longer segment being closer to the vertex.
3. It is always located inside the triangle, regardless of the type of triangle.

Formula for Finding the Centroid

The coordinates of the centroid (G) can be calculated as the average of the vertices' coordinates. If the vertices of the triangle are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), then the centroid's coordinates (G) are given by:

$$ G\left(\frac{x₁ + x₂ + x₃}{3}, \frac{y₁ + y₂ + y₃}{3}\right) $$

Table of Differences and Important Points

Property	Centroid	Circumcenter	Incenter	Orthocenter
Definition	Intersection of medians	Intersection of perpendicular bisectors	Intersection of angle bisectors	Intersection of altitudes
Location	Always inside the triangle	Inside for acute, on hypotenuse for right, outside for obtuse	Always inside the triangle	Inside for acute, on vertex for right, outside for obtuse
Ratio of Division	Divides medians in a 2:1 ratio	Equidistant from vertices	Equidistant from sides	No fixed ratio
Formula	Average of vertices' coordinates	Intersection point of perpendicular bisectors	Intersection point of angle bisectors	Intersection point of altitudes

Examples

Example 1: Finding the Centroid of a Triangle with Given Vertices

Given a triangle with vertices A(1, 2), B(3, -4), and C(-5, 6), find the centroid.

Using the formula for the centroid:

$$ G\left(\frac{1 + 3 - 5}{3}, \frac{2 - 4 + 6}{3}\right) = G\left(\frac{-1}{3}, \frac{4}{3}\right) = G\left(-\frac{1}{3}, \frac{4}{3}\right) $$

So, the centroid of the triangle is at (-1/3, 4/3).

Example 2: Verifying the Centroid Divides the Medians in a 2:1 Ratio

Let's use the same triangle from Example 1. We'll find the midpoint of side BC and verify that the centroid divides the median from vertex A in a 2:1 ratio.

The midpoint of BC, M, is:

$$ M\left(\frac{3 - 5}{2}, \frac{-4 + 6}{2}\right) = M\left(-1, 1\right) $$

Now, let's find the distances from A to G and G to M:

Distance AG:

$$ AG = \sqrt{\left(-\frac{1}{3} - 1\right)^2 + \left(\frac{4}{3} - 2\right)^2} $$

Distance GM:

$$ GM = \sqrt{\left(-\frac{1}{3} + 1\right)^2 + \left(\frac{4}{3} - 1\right)^2} $$

After calculating, we should find that AG is twice as long as GM, which confirms that the centroid divides the median in a 2:1 ratio.

Example 3: Using the Centroid to Find the Center of Mass

If a triangle is made of a uniform material, the centroid represents the center of mass. For instance, if you have a triangular cardboard of uniform density and you want to balance it on a pencil tip, you would place the pencil tip at the centroid of the triangle.

Conclusion

The centroid of a triangle is a fundamental concept in geometry that has applications in various fields, including physics, engineering, and computer graphics. Understanding its properties and how to calculate it is essential for solving problems related to the center of mass, balance, and symmetry in triangles.