Types of Center in a Triangle : Understanding the types of center in a triangle is an important part of geometry that helps students grasp key concepts about triangles and their properties. A triangle has several notable centers, but the four common centers are the centroid, circumcenter, incenter, and orthocenter. By learning about these centers, students can better understand the relationships within triangles and apply this knowledge to solve various geometric problems.

In this article, we will understand the meaning of the center of the triangle and the types of centers of triangles: Centroid, Circumcenter, Incenter, Orthocenter, and Excenter.

What is Center of a Triangle?

The center of a triangle refers to a specific point that holds special geometric properties within the triangle. There are several notable centers in a triangle, each defined by different characteristics and constructions:

1. Centroid: The point where the three medians (the lines drawn from each vertex to the midpoint of the opposite side) intersect. It is the triangle’s center of mass or balance point.
2. Circumcenter: The point where the three perpendicular bisectors of the sides intersect. It is the center of the triangle’s circumscribed circle (circumcircle), which passes through all three vertices.
3. Incenter: The point where the three angle bisectors intersect. It is the center of the triangle’s inscribed circle (incircle), which touches all three sides.
4. Orthocenter: The point where the three altitudes (the perpendicular lines drawn from each vertex to the opposite side) intersect.
5. Excentre : The points where the internal angle bisector of one angle and the external angle bisectors of the other two angles intersect.

Types of center in a Triangle

The various types of center in a triangle are:

• Centroid of Triangle
• Circumcenter of Triangle
• Incenter of Triangle
• Orthocenter of Triangle
• Excenter of Triangle

Centroid of a Triangle (G)

Definition : The point where the three medians of a triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side.

Centroid Of a Triangle

Properties of Centroid of a Triangle

• It is the center of mass or balancing point of the triangle.
• It divides each median into a ratio of 2:1, with the longer segment being closer to the vertex.
• Always located inside the triangle.

Formula of Centroid of a Triangle With Example

The centroid G of a triangle with vertices at (x₁,y₁), (x₂,y₂), and (x₃,y₃) is given by:

G[Tex]\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)[/Tex]

For Example, for a triangle with vertices at (1, 2), (3, 4), and (5, 6)

G = [Tex]\left( \frac{1 + 3 + 5}{3},~\frac{2 + 4 + 6}{3} \right)[/Tex]

⇒ G = (9/3, 12/3)

⇒ G = (3, 4)

Circumcenter of Triangle (O)

Definition : The point where the three perpendicular bisectors of the sides intersect. A perpendicular bisector is a line that is perpendicular to a side of the triangle and bisects it.

Properties of Circumcenter of a Triangle

• It is the center of the circumcircle (the circle that passes through all three vertices of the triangle).
• It can be inside, outside, or on the triangle, depending on the type of triangle (acute, obtuse, or right, respectively).

Formula of Circumcenter of a Triangle With Example

For a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃), the coordinates of the circumcenter O are given by:

[Tex]O \left( \frac{ (x_1^2 + y_1^2)(y_2 – y_3) + (x_2^2 + y_2^2)(y_3 – y_1) + (x_3^2 + y_3^2)(y_1 – y_2) }{ D }, \frac{ (x_1^2 + y_1^2)(x_3 – x_2) + (x_2^2 + y_2^2)(x_1 – x_3) + (x_3^2 + y_3^2)(x_2 – x_1) }{ D } \right)[/Tex]

where,

D = 2{x_1​(y₂ ​- y₃​) + x₂​(y₃ ​- y₁​) + x₃​(y₁ ​- y₂​)}

For example, for a triangle with vertices at (0, 0), (4, 0), and (0, 3)

D = 2{0(0−3)+4(3−0)+0(0−0)}

= 2(0 + 12 + 0)

= 24

Coordinates of Circumcenter are:

[Tex]O \left( \frac{ (0^2 + 0^2)(0 – 3) + (4^2 + 0^2)(3 – 0) + (0^2 + 3^2)(0 – 0) }{ 24 }, \frac{ (0^2 + 0^2)(0 – 4) + (4^2 + 0^2)(0 – 0) + (0^2 + 3^2)(4 – 0) }{ 24 } \right)[/Tex]

[Tex]O \left( \frac{ 0 + 48 + 0 }{ 24 }, \frac{ 0 + 0 + 36 }{ 24 } \right)[/Tex]

= O(2, 1.5)

Incenter of a Triangle (I)

Definition : The point where the three angle bisectors intersect. An angle bisector is a line that divides an angle into two equal angles.

Properties of Incenter of a Triangle

• It is the center of the incircle (the circle that is tangent to all three sides of the triangle).
• Always located inside the triangle.

Formula of Incenter of a Triangle With Example

Definition : The incenter ‘I’ of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) and side lengths a, b, and c opposite these vertices is given by:

[Tex]I \left( \frac{a x_1 + b x_2 + c x_3}{a + b + c},~\frac{a y_1 + b y_2 + c y_3}{a + b + c} \right)[/Tex]

For example, for a triangle with vertices at (0, 0), (4, 0), and (0, 3), with sides:

• a = 4
• b = 5
• c = 3

Coordinates of Incenter are:

[Tex]I \left( \frac{4 \cdot 0 + 5 \cdot 4 + 3 \cdot 0}{4 + 5 + 3}, \frac{4 \cdot 0 + 5 \cdot 0 + 3 \cdot 3}{4 + 5 + 3} \right)[/Tex]

[Tex]I \left( \frac{0 + 20 + 0}{12}, \frac{0 + 0 + 9}{12} \right) = I \left( \frac{20}{12}, \frac{9}{12} \right)[/Tex]

= I(20/12, 9/12)

= I(5/3, 3/4)

Orthocenter of a Triangle (H)

Definition : The point where the three altitudes of a triangle intersect. An altitude is a perpendicular segment from a vertex to the opposite side (or the line containing the opposite side).

Properties of Orthocenter of a Triangle

• Orthocenter can be inside, on, or outside the triangle, depending on whether the triangle is acute, right, or obtuse.
• In an acute triangle, the orthocenter lies inside the triangle.
• In a right triangle, the orthocenter is at the vertex of the right angle.
• In an obtuse triangle, the orthocenter lies outside the triangle.

Formula of Orthocenter of a Triangle With Example

The orthocenter H of a triangle can be found by solving the system of equations derived from the slopes of the altitudes.

For example, for a triangle with vertices at (0, 0), (4, 0), and (0, 3):

• Slope of the side (4, 0) to (0, 3) is -3/4​, and the altitude from (0, 0) to this side has a slope of 4/3. The equation is y = 4/3x
• Slope of the side (0, 0) to (0, 3) is undefined, and the altitude from (4, 0) to this side is vertical with the equation x = 4

Solving the equations y = 4/3x and x = 4:

y = [Tex]\frac{4}{3}(4)[/Tex]

= 16/3

Thus, orthocenter is H(4, 16/3)

Excenters of a Triangle (E)

Definition : The points where the internal angle bisector of one angle and the external angle bisectors of the other two angles intersect.

Properties of Excenters of a Triangle

• There are three excenters for a triangle, each associated with one of the triangle’s vertices.
• Each excenter is the center of an excircle (a circle that is tangent to one side of the triangle and the extensions of the other two sides).

Formula of Excenters of a Triangle With Example

The excenter I_A opposite the vertex A for a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) and side lengths a, b, and c is given by:

[Tex]I_A \left( \frac{-a x_1 + b x_2 + c x_3}{-a + b + c}, \frac{-a y_1 + b y_2 + c y_3}{-a + b + c} \right)[/Tex]

For a triangle with vertices at (0, 0), (4, 0), and (0, 3), with sides:

• a = 4
• b = 5
• c = 3

The coordinates of the excenter opposite vertex (0, 0) are:

[Tex] I_A \left( \frac{-4 \cdot 0 + 5 \cdot 4 + 3 \cdot 0}{-4 + 5 + 3}, \frac{-4 \cdot 0 + 5 \cdot 0 + 3 \cdot 3}{-4 + 5 + 3} \right)[/Tex]

[Tex]I_A \left( \frac{0 + 20 + 0}{4}, \frac{0 + 0 + 9}{4} \right)[/Tex]

= I_A(20/4, 9/4)

= I_A(5, 9/4)

Types of center in a Triangle