How to find circumcenter of a triangle whose vertices are given

The point of concurrency of the perpendicular bisectors of the sides of a triangle is called the circumcenter of the triangle.

The following steps will be useful to find circumcenter of a triangle. 

Step 1 : 

Find the equations of the perpendicular bisectors of any two sides of the triangle. 

Step 2 : 

Solve the two equations found in step 2 for x and y. 

The solution (x, y) is the circumcenter of the triangle given.

Example : 

Find the co ordinates of the circumcenter of a triangle whose vertices are (2, -3), (8, -2) and (8, 6).

Solution : 

Let A(2, -3), B(8, -2) and C(8, 6) be the vertices of the triangle. 

D is the midpoint of AB and E is the midpoint of BC. 

All the polygons that possess circumcircles are identified as cyclic polygons. However, only regular polygons namely triangles, rectangles, and right-kites hold the circumcircle and hence the circumcenter also.

Circumcenter of a Triangle: Formula

The circumcentre of a triangle is specified as the point where the perpendicular bisectors of the sides of a given triangle intersect or meet. In other words, we can say that the point of concurrency of the bisector of the sides of a triangle is termed the circumcenter.

The circumcenter however is also said to be the centre of the circumcircle of a triangle which can be either inside/outside of the provided triangle. We can instantly find the circumcenter formula by using the below-discussed formula.

If \(A (x_1, y_1)\), \(B (x_2, y_2)\) and \(C (x_3, y_3)\) are the vertices of the given ∆ ABC with A, B, C as their respective angles. Then the circumcentre of a triangle formula is as follow:

\(O(x,y)=\frac{\left(x_1\sin2A+x_2\sin2B+x_3\sin2C\right)}{\left(\sin2A+\sin2B+\sin2C\right)},\frac{\left(y_1\sin2A+y_2\sin2B+y_3\sin2C\right)}{\left(\sin2A+\sin2B+\sin2C\right)}\)

Learn more about Area of a Triangle.

How do we Construct Circumcenter of a Triangle?

The circumcenter of a triangle can be located as the intersection of the perpendicular bisectors (these are the lines that stand at right angles to the midpoint of every side of the given triangle) of all sides of the triangle. This also indicates that the perpendicular bisectors of the triangle are concurrent (i.e. meeting at a single location).

The circumcenter of a triangle can be constructed by tracing the perpendicular bisector of any of the two sides of the given triangle. The basic steps to construct the circumcenter are discussed below:

Step 1: Outline the perpendicular bisectors of all the sides of the triangle applying a compass.

Step 2: Applying a ruler, extend the perpendicular bisectors until they meet each other at a point.

Step 3: Mark the intersection point as ‘O’, this denotes the circumcenter.

Step 4: With the help of the compass and keep ‘O’ as the center and any vertex of the triangle as a spot on the circumference, trace a circle, the circle formed is our circumcircle whose center is at ‘O’.

How to find the Circumcenter of a Triangle?

So far we saw the definition, formula and steps to calculate the circumcentre coordinates. Let us now learn how to find the circumcentre of a triangle. To locate or obtain the coordinates of the circumcentre of triangles, various formulas can be applied, one such is the direct circumcenter of a triangle formula as discussed in the previous heading.

Apart from that, the various methods through which we can locate the circumcenter of a triangle whose vertices are given by the coordinates; \(A (x_1, y_1)\), \(B (x_2, y_2)\) and \(C (x_3, y_3)\) are as follows.

Method 1: Applying the Midpoint Formula

• In this method, we first determine the midpoints of the line segments AB, AC, and BC by the midpoint formula.

• \(M\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)
• Next, obtain the slope of any of the line segments AB, AC, and BC.

• Then by adopting the midpoint and the slope of the perpendicular line, get the equation of the perpendicular bisector line.
• \(\left(y−y_1\right)=\left(\frac{−1}{m}\right)\left(x−x_1\right)\)

• Similarly, we can obtain the equation of the other perpendicular bisector line and by solving two perpendicular bisector equations we find the intersection point.

• This particular intersection point implies the circumcenter of the given triangle.

Also, read about Centroid of a Triangle.

Method 2: Applying the Distance Formula

By using the distance formula obtain \(d_1\), \(d_2\) and \(d_3\) as shown below:

\(d_1=\sqrt{(x−x_1)^2+(y−y_1)^2}\)

\(d_1\) is the distance between circumcenter and vertex A.

\(d_2=\sqrt{(x−x_2)^2+(y−y_2)^2}\)

\(d_2\) is the distance between circumcenter and vertex B.

\(d_3=\sqrt{(x−x_3)^2+(y−y_3)^2}\)

\(d_3\) is the distance between circumcenter and vertex C.

Now by calculating \(d_1=d_2 = d_3\) we can obtain the coordinates of the circumcenter for the circumcentre formula.

Know more about Mean Median Mode here.

Method 3: Practising Extended Sin Law

By applying the extended form of sin law, we can obtain the radius of the circumcircle, and furthermore by the distance formula can find the accurate location of the circumcenter.

\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R\)

For the above formula a, b, and c are lengths of the corresponding surfaces of the triangle and R is the radius of the circumcircle.

Here’s a brief summary regarding some of the other properties of triangle and terms.

TermsDefinitionIncenter:

The perpendicular bisector of a triangle is the lines drawn perpendicularly from the midpoint of the triangle.

The point at which the three medians of the triangle intersect or touch each other is recognised as the centroid of a triangle.

A line segment developed from one vertex to the opposite side, which is perpendicular to the opposite side, is known as the altitude of a triangle.

Check more topics of Mathematics here.

Circumcenter of a Triangle: Example Questions

Let us check out some of the examples regarding the topic for more practice and clarity:

Question1: If O is the circumcenter of a triangle △ABC and ∠BOC is 40° then what is the value of ∠BAC?

Solution:

Given:

O is the circumcenter of the triangle △ABC and ∠BOC = 40°

Concept Used:

If O is the circumcenter of the △ABC then the angle made at the circumcenter by joining any two vertices of the triangle is twice the angle at the third vertex of the triangle i.e. the angle made at the circumference at the circle.

Calculation:

O is the circumcenter of △ABC and ∠BOC = 40°

∠BAC = (1/2) × ∠BOC

∠BAC = (1/2) × 40°

∠BAC = 20°

∴ The value of ∠BAC is 20°

Question 2: Three sides of a triangle are 40 cm, 58 cm and 42 cm. Find the distance between the orthocentre and the circumcenter of a triangle.

Solution:

Sides: 40 cm, 58 cm and 42 cm

We can observe that:

\(58^2 = 42^2 + 40^2\)

∴ The triangle must be a right angle triangle.

In a right angle triangle: Orthocentre lies at the vertex at which the right angle is formed.

Hence, B is the orthocentre and O is the circumcentre as per the diagram.

BO = Circumradius = 58/2 = 29 cm.

Question 3: If the circumcenter of a triangle lies on one of the sides then the orthocenter of the triangle lies on?

1) One of the vertices

2) On the same side of the triangle

3) Outside the triangle

4) Strictly inside the triangle

Solution:

Given that the circum-center lies on one of the sides of a triangle. Hence the triangle is a right angled triangle. In a right angled triangle the orthocenter is the vertex where the angle is 90°. Therefore option 1 would be the answer.

Properties of Circumcentre

In the previous headings, we saw how to find the circumcenter of the triangle and the formula of circumcentre now let us learn some of the important properties of the circumcenter of a triangle.

• The circumcenter is also known as the centre of the circumcircle.
• All the vertices of the given triangle are equidistant from the circumcenter. Look at the image below to learn this better.

• Also, all the unique triangles developed by joining O to the vertices are isosceles triangles.
• In terms of angle ∠BOC = 2 ∠A when ∠A is acute/ when O and A are on the identical side of BC.
• ∠BOC = 2( 180° – ∠A) if ∠A is obtuse / O and A are on different sides of BC.

Check out this article on Number Systems.

The location for the circumcenter of a triangle is different for distinct types of triangles as follows.

• In an acute-angled triangle, the circumcenter rests within the triangle.

• In an obtuse-angled triangle, it rests outside of the triangle.

• The circumcenter rests at the midpoint of the hypotenuse side for the right-angled triangle.

• All the four points i.e. circumcenter, orthocenter, incenter, and centroid match with each other in an equilateral triangle.
• Moreover, except for the equilateral triangle, the orthocenter, circumcenter, and centroid rest in the same straight line are identified as the Euler Line for the other varieties of triangles.

We hope that the above article on Circumcenter of a Triangle is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Circumcenter of a Triangle article, also check the related maths articles in the table below:Order and degree of differential equationsTransitive relationsTypes of vectorsTypes of functionsLagrangs mean value theoremDomain of a function

Circumcenter of a Triangle FAQs

Q.1 What are the properties of a triangle?

Ans.1 Some of the important properties of a triangle are:
A triangle has three sides, angles, and vertices respectively.
The total of all internal angles of a triangle is always equal to 180 degrees.
The total of all exterior angles of any triangle is equivalent to 360°.

Q.2 Is a centroid equidistant from vertices?

Ans.2 The centroid intersects the vertices at the middle of the triangle, therefore it is the point that is equal in distance from all three vertices.

Q.3 What is the circumcentre of a triangle?

Ans.3 The circumcentre of a triangle is specified as the point where the perpendicular bisectors of the sides of a given triangle intersect or meet.

Q.4  How do we define the circumcircle of a triangle?

Ans.4 The circumcircle of a triangle is defined as the circle that moves through all of its vertices and the center of that particular circle is termed the circumcentre.

Q.5 What is the circumcentre of a triangle formula?

Ans.5 The circumcentre of a triangle is given by the formula:

\(O(x,y)=\frac{\left(x_1\sin2A+x_2\sin2B+x_3\sin2C\right)}{\left(\sin2A+\sin2B+\sin2C\right)},\frac{\left(y_1\sin2A+y_2\sin2B+y_3\sin2C\right)}{\left(\sin2A+\sin2B+\sin2C\right)}\)

Q.6 How to find a circumcenter?

Ans.6 The circumcenter of a triangle can be located as the intersection of the perpendicular bisectors of all sides of the triangle.

Q.7 What is the circumference of the triangle formula?

Ans.7 The circumference/ perimeter of a triangle=P(perimeter)=(a+b+c) units; where a,b and c are the sides of the triangle.

What is the formula of circumcircle of a triangle?

How do you find the coordinates of a Circumcentre?