Centroid of a Triangle

The centroid of a triangle is one of the most basic concepts in geometry. A triangle is a three-sided enclosed figure with three internal angles. The centroid of any triangle represents the point of intersection of its three medians. The centroid is most commonly referred to as the "centre of mass", also called the "centre of gravity" of a triangle since it balances the triangle when supported at that point. In order to solve geometrical and practical difficulties, professionals and students of different backgrounds can benefit from knowing the centroid of a triangle.

1.0What is the Centroid of a Triangle?

The centroid of any triangle is the point that divides each median in a 2:1 ratio, with the larger area nearer the vertex. It is a mathematical representation of the mean of all the triangle's vertex coordinates. The centroid will always be inside the triangle, regardless of whether it is scalene, isosceles, or equilateral.

According to the provided figure, a triangle's centroid "G" is where its three medians meet. A centroid is also known as the centre of gravity.

If the coordinates (values) of the vertices of a triangle are (x_1, y_1), (x_2, y_2), & (x_3, y_3), then the centroid of a triangle formula is given as:

$\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

Where x_1, x_2, & x_3 are the X co-ordinates of the vertices of a triangle and y_1, y_2, y_3 are the Y co-ordinates.

2.0Proof of the Centroid of a Triangle

The centroid of a triangle is where its three medians intersect. To prove this, consider a triangle ABC with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃). The medians are line segments connecting each vertex to the midpoint of the opposite side. 

Let D, E, and F be the midpoints of AB, BC, and AC, respectively.

Using the midpoint formula for BC, the coordinates of E are:

E ((x₂ + x₃) /2, (y₂ + y₃) /2).

Similarly, the coordinates of the medians AC at F and AB at D can be expressed parametrically. 

F ((x₁ + x₃) /2, (y₁ + y₃) /2)

D ((x₁ + x₂) /2, (y₁ + y₂) /2) 

By taking the mean of the above three coordinates 

G ((x₁ + x₂ + x₂ + x₃ + x₁ + x₃)/6 , (y₁ + y₂ + y₂ + y₃ + y₁ + y₃)/6) 

and solving these equations, it can be shown that all three medians intersect at a single point G:

= G ((2x₁ + 2x₂ + 2x₃) /6, (2y₁ + 2y₂ + 2y₃) /6)

= G (2(x₁ + x₂ + x₃) /6, 2(y₁ + y₂ + y₃) /6 

= G ((x₁ + x₂ + x₃) /3, (y₁ + y₂ + y₃) /3).

This point G, known as the centroid of a triangle, divides each median in a 2:1 ratio. 

This proof works for all types of triangles, including equilateral triangles, making the centroid a universal property.

3.0Properties of the Centroid of a Triangle

Here are some key properties of the centroid of a triangle which help distinguish it from other points of concurrency:

• The centroid is sometimes called the geometric centre of the triangle.
• It is the point of intersection of the three medians of a triangle.
• The centroid separates each median into segments with a ratio of 2:1, with the larger segment on the side closer to the vertex.
• The centroid of every triangle is always located inside the triangle, regardless of its type.

4.0How to Find the Centroid of a Triangle

Step 1: Identify the Vertices

Determine the coordinates of the triangle's three vertices. For instance, let the vertices be A (x_1,y_1), B (x_2,y_2), & C (x_3,y_3).

Step 2: Apply the Formula

Substitute the vertex coordinates into the centroid formula:

G (x, y) = (x_1+x_2+x_3)/3, (y_1+y_2+y_3) /3

Step 3: Simplify

Calculate the average of the x-coordinates and the y-coordinates to find G (x,y).

5.0Relation Between Orthocentre, Centroid, and Circumcentre

The centroid, orthocentre, and circumcentre of any triangle are all collinear. The Euler Line is the path along which this alignment takes place. There is a nice geometric link between these points since the distance between them follows certain ratios.

Let G, H, and O represent the centroid, orthocentre, and circumcentre, respectively. The relationship can be summarized as:

• G divides the line segment HO in the ratio 2:1. This means that the centroid is closer to the circumcentre than to the orthocentre.

In an equilateral triangle, all three points coincide, making it a unique case where the centroid of a triangle equilateral also acts as its circumcentre and orthocentre.

It is crucial to comprehend these relationships to solve complex geometry problems and recognise triangular symmetry in theoretical and real-world contexts. 

6.0Solved Problems

Problem 1: Find the Centroid of a Triangle.

Vertices: A(2, 3), B(4, 5), C(6, 7)

Solution:

Using the centroid formula:

G (x, y) =

$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

G (x, y) =

$(\frac{2+4+6}{3},\frac{3+5+7}{3})$

G (x, y) = (12/3, 15/3) = (4,5)

The centroid is at G (4,5).

Problem 2: The vertices of a triangle are (1, 2), (h, −3), & (−4, k), and the coordinates of the centroid of this triangle are (5, –1). Find the value of h & k.

Solution: We have

(x_1, y_1) = (1, 2)

(x_2, y_2) = (h, –3)

(x_3, y_3) = (–4, k)

Using the centroid of the triangle formula, we have

C(x, y) =

$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

(representing both points)

⇒ C(5, -1) = C((1 + h – 4)/3, (2 - 3 + k)/3)

after equating X and Y

⇒ (h - 3)/3 = 5 and (k - 1)/3 = -1

⇒ h - 3 = 15 & k - 1 = -3

which gives h = 18 & k = -2

7.0Why learn about the Centroid?

1. Real-world Applications: When it comes to physics, architecture, and engineering applications, knowing how to locate a triangle's centroid is helpful. In structural designs, for example, it determines the centre of gravity.
2. Logical Understanding: The centroid deepens our comprehension of symmetry, balance, and geometric connections.

Simplified Calculations: The formula for centroid makes complex calculations manageable, whether for scalene or equilateral triangles.