Circumcenter of Triangle: Formula, Properties, Examples

The circumcenter of Triangle is a specific point where the perpendicular bisectors of the sides of the triangle intersect. This point is significant because it is equidistant from all three vertices of the triangle. It makes it the center of circle that can be circumscribed around the triangle which is known as circumcircle.

It is a point belonging to a triangle where the perpendicular bisector of the triangle meets. It is a point inside the triangle and is represented using P(x, y).

Let’s learn about the Circumcenter of triangle in detail, including its Definition, Properties and formula.

Circumcenter is the center of a circumcircle, whereas a circumcircle is a circle that passes through all three vertices of the triangle. It is a specific point where the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from all three vertices of the triangle.

Every triangle is cyclic, which means every triangle can be circumscribed by a circle. Hence, any type of triangle will have a circumcenter.

Circumcenter of a triangle, P(x, y) with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is given using the following formula:

P(x, y) = {(x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c), (x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c)}

While considering the triangle ABC with circumcenter O below, let’s discuss the various properties of circumcenter of a triangle.

• The circumcenter of a triangle is equidistant from all the vertices, i.e., OA = OB = OC.

• All the new triangles that are formed by joining the circumcenter of a triangle to its vertices are isosceles. ∆ AOB, ∆ BOC, and ∆ COA are isosceles triangles.

• In a triangle, if ∠A is acute or when O and A are on the same side of BC, then ∠BOC = 2 ∠A.

• In a triangle, if ∠A is obtuse or when O and A are on different sides of BC, then ∠BOC = 2(180° – ∠A).

• For an Acute-Angled Triangle, the circumcenter lies inside the triangle.

• In the case of an Obtuse-Angled Triangle, the circumcenter lies outside the triangle.

• For a Right-Angled Triangle, the circumcenter lies on the hypotenuse of the triangle.

The circumcenter of a Acute Angle Triangle, Right Angle Triangle, and Obtuse Angle Triangle can be understood with the help of diagram below:

Read More

• Acute-Angled Triangle
• Obtuse-Angled Triangle
• Right-Angled Triangle

To construct the circumcenter of any triangle, we need to draw the perpendicular bisectors of any two sides of the triangle. The following are the steps to construct the circumcenter of a triangle:

Step 1: For the given triangle, draw the perpendicular bisectors of any two sides using a compass.

Step 2: With the help of a ruler, extend the perpendicular bisectors till they intersect at a point.

Step 3: Now, mark the point of intersection, which will be the circumcenter of the given triangle.

We can get the circumcenter accurately by drawing the perpendicular bisector of the third side of the triangle.

Circumcenter of a Triangle is calculated using the following methods:

1. Using the Midpoint Formula
2. Using the Distance Formula
3. Using Extended Sin Law

Let’s discuss them in detail.

Using the Midpoint Formula: Method 1

Let us consider a triangle ABC whose vertices are A (x₁, y₁), B(x₂, y₂), and C (x₃, y₃), and O(x, y) is its circumcenter.

Step 1: Calculate the coordinates of the mid-points of the sides AB, BC, and AC using the mid-point formula.

M(x_m ,y_m) = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Step 2: Calculate the slopes of the sides AB, BC, and AC. Let the slope of a side be m, then the slope of its perpendicular bisector is “-1/m”.

Step 3: Now, using the coordinates of the midpoint (x_m, y_m) and the slope of the perpendicular bisector (-1/m), find out the equation of the perpendicular bisector using the point-slope form.

(y – y_m) = -1/m(x – x_m)

Step-4: Similarly, find out the equations of the other bisector lines too.

Step-5: Solve any of the two equations and find their intersection point.

The obtained intersection point is the circumcenter of the given triangle.

Using the Distance Formula: Method 2

Let us consider a triangle ABC whose vertices are A (x₁, y₁), B(x₂, y₂), and C (x₃, y₃), and O (x, y) is its circumcenter.

We know that the circumcenter of a triangle is equidistant from all the vertices, i.e., OA = OB = OC = circumradius.

Let OA = D₁, OB = D₂, and OC = D₃.

Step 1: Using the distance formula between two coordinates, find the values of D₁, i.e.,

(D₁)² = (x – x₁)² + (y – y₁)²

Similarly,

(D₂)² = (x – x₂)² + (y – y₂)²

(D₃)² = (x – x₃)² + (y – y₃)²

Step 2: Now, by equating D₁ = D₂ = D₃ we will get linear equations. By solving these linear equations, we can get the coordinates of the circumcenter O (x, y).

Method 3: Using Extended Sin Law

Let us consider a triangle ABC whose vertices are A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃), and ∠A, ∠B, and ∠C are their respective angles. O (x, y) is the circumcenter of the triangle.

Circumcenter Formula Using Extended Sin Law :

O(x, y) = {(x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c), (x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c)}

Read More

• Triangle
• Incenter of a Triangle
• Area of a Triangle
• Area of a Rectangle
• Polygon

Let’s solve some example problems on the Circumcenter of Triangle.

Examples 1: Determine the circumcenter of a triangle with vertices A (1,3), B (0,4), and C (-2,5).

Solution:

Given,

Vertices of a Triangle are A (x₁, y₁) = (1,3), B (x₂ , y₂) = (0,4), and C (x₃, y₃) = (-2,5)

Let “O” be the circumcenter of the triangle ABC and (x, y) be its coordinates.

Let D₁ be the distance from the circumcenter to vertex A, i.e., OA = D₁

Let D₂ be the distance from the circumcenter to vertex B, i.e., OB = D₂

Let D₃ be the distance from the circumcenter to vertex C, i.e., OC = D₃

By using the Distance Formula,

(D₁)² = (x – x₁)² + (y – y₁)² = (x – 1)² + ( y – 3)²

(D₂)² = (x – x₂)² + (y – y₂)² = ( x- 0)² + (y – 4)²

(D₃)² = (x – x₃)² + (y – y₃)² = (x + 2)² + (y – 5)²

We know that the distances from all the vertices to the circumcenter (O) are equal, i.e.,

OA = OB = OC = Circumradius

⇒ D₁ = D₂ = D₃

Now, take D₁ = D₂

 (x -1)² + (y – 3)² =  (x – 0)² + (y -4)²

⇒ x² – 2x + 1 + y² – 6y + 9 = x² + y² – 8y + 16

⇒ 2x – 2y = -6    ……(1)

Now, take D₂ = D₃

 ( x- 0)² + (y – 4)² = (x + 2)² + (y – 5)²

⇒ x² + y² – 8y + 16 = x² + 4x + 4 + y² – 10y + 25

⇒ 4x – 2y = -13  …..(2)

Now, solve both equations (1) and (2)

2x – 2y = – 6

4x – 2y = -13

Adding eq (1) and (2)

 – 2x = 7

⇒ x = -7/2 = – 3.5

Now, substitute the value of x in equation (1)

2 (-3.5) – 2y = -6

⇒ -7 – 2y = -6 

⇒ 2y = -1 ⇒ y = -1/2 = -0.5

Hence, the circumcenter of the triangle ABC is (-3.5, -0.5)

Example 2:  Determine the circumcenter of the triangle with vertices A (3, -6), B (1, 4), and C (5, 2).

Solution:

Given,

Vertices of a Triangle are A (x₁, y₁) = (3, -6), B (x₂ , y₂) = (1, 4), and C (x₃, y₃) = (5, 2).

Let “O” be the circumcenter of the triangle ABC and (x, y) be its coordinates.

For finding the circumcenter of a triangle, we can calculate the intersection point of any two perpendicular bisectors.

Now, the mid point of the side AB = [(3 + 1)/2, (-6 + 4)/2] = (2, -1)

Slope of AB = (y₂ – y₁)/(x₂ – x₁) = (4 + 6)/(1 – 3) = -5

We know that the product of the slopes of two perpendicular lines = -1.

So, the slope of the perpendicular bisector of the side AB = 1/5

Now, the equation of the perpendicular bisector of AB with slope = 1/5 and the coordinates (2,-1) is

(y – (-1)) = (1/5) (x – 2) [From point-slope form]

⇒ 5(y +1) = x – 2

⇒ x – 5y = 7    ……(1)

Mid-Point of the side BC is [(1 + 5)/2, (4 + 2)/2] = (3,3)

Slope of BC = (y₃ – y₂)/(x₃ – x₂) = (2 – 4)/(5 – 1) = -1/2

Now, the slope of the perpendicular bisector of the side BC = 2

Equation of the perpendicular bisector of BC with slope = 2 and the coordinates (3,3) is

(y – 3) = 2(x – 3)  [From point-slope form]

⇒ y – 3 = 2x – 6

⇒ 2x – y = 3…….(2)

Now, multiply equation (1) with “2” on both sides and subtract equation (2) from the obtained equation.

Subtracting eq. (1) and (2)

-9y = 11

⇒ y = -11/9

Now Substitute the Value of y in equation (2)

2x + 11/9 = 3

⇒ x = 8/9

Hence, the circumcenter of the triangle ABC is (8/9, -11/9).

Problem 3: Find the circumcenter of the ∆ ABC with vertices A (1, 3), B (3, 7), and C (5, 9), and the measures of the respective angles are 45°, 45°, and 90°.

Solution:

Given,

Vertices of a Triangle are A (x₁, y₁) = (0, 3), B (x_2, y₂) = (3, 7), and C (x₃, y₃) = (5, 9)

Measures of Angles are ∠A = 45°, ∠B = 45°, and ∠C = 90°

We know the formula for the circumcenter (O) of a triangle when its vertices and their respective angles are given, i.e.,

Circumcenter (O) = {(x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c), (x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c)}

O = {(1)sin2(45°) + (1)sin2(45°) + (1)sin2(45°)}/{sin2(45°) + sin2(45°) + sin2(45°), (3)sin2(45°) + (7)sin2(45°) + (9)sin2(45°)}/{sin2(45°) + sin2(45°) + sin2(45°)}

O = [{(1)(1) + (3)(0) + (5)(0)}/(1 + 0 + 0)}, {(3)(1) + (7)(0) + (9)(0)}/(1 + 0 + 0)]

Circumcenter (O) = (1, 3)

Hence, the circumcenter of the triangle ABC is (1,3).

What is Circumcenter of Triangle in Geometry?

Circumcenter of a triangle is a point belonging to the triangle which is the center of the circumcircle of the triangle.

What is the Formula of Circumcenter?

The formula for the circumcenter of the triangle is,

P(x, y) = {(x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c), (x₁ sin 2A + x₂ sin 2B + x₃ sin 2C)/(sin 2A + sin 2B + sin 2c)}

How to Find the Circumcenter of Triangle?

Circumcenter of a triangle is find using the intersection of the perpendicular bisector of any two lines of the triangle.

What is Difference Between Circumcenter and Incenter of Triangle?

Circumcenter of a triangle is the center of the circumcircle of the triangle where as incenter is the center of the encircle of the triangle.

Are Circumcenter and Centroid of Triangle the Same?

No, the circumcenter and centroid of a triangle are not the same. The circumcenter is the point where the perpendicular bisectors of the triangle’s sides intersect, while the centroid is the point where the triangle’s three medians intersect, typically located within the triangle.