Circumcenter of a Triangle: Definition, Types and Examples

In geometry, we deal with different geometrical figures, and with the help of these geometrical figures, we can visualize the concept of the circumcenter. The circumcenter is nothing it is just the center of a special type of circle which is called the circumcircle.

In this article, we will discuss the definition, geometrical views, and examples of the circumcenter of a triangle.

What is the Circumcenter?

The circumcenter is the center of a circumscribed circle. The circumscribed circle is a circle which passes through the vertex of the polygon. All polygons do not need a circumcenter. It is necessary the polygon must be a cyclic polygon.

Before discussing the position of the circumcenter we have an idea about the angles and their types.

What is an angle?

The angle is formed when two lines meet at a common end-point. The point of intersection is called vertex and two lines are called arms of an angle. 

Image showing angle between the two lines of 67 degrees
In this example the angle between the two lines is 67 degress

Types of Angles

There are three types of angles depending on their angle values in degrees.

  1. Acute angle
  2. Right angle
  3. Obtuse angle

Acute angle

Image 2

The angle which is lies between 00 to 900. Simply an angle less than 900 is called an acute angle.

Right angle 

Image 3

An angle of 900 is called a right angle. When we say a line is a perpendicular its means it makes an angle of 900.

Obtuse angle

Image 4

An angle greater than 900 but less than 1800 is called an obtuse angle.

The circumcenter of a Triangle

Now I will discuss here the circumcenter of a Triangle, the three-sided polygon. The circumcenter does not always lie in the closed polygon. It depends upon the angles of the polygon.

Let’s consider the three scenarios.

  • The circumcenter lies inside the polygon
  • The circumcenter lies on the side of the polygon
  • The circumcenter lies outside of the polygon.

Circumcenter lies inside the polygon

Lets we have a three-sided polygon called a triangle ABC. If the triangle is an acute angle triangle then the circumcenter lies inside the triangle.

Circumcenter lies inside the polygon

In the figure, if we select all the angles less than 900. then circumcenter lies inside the triangle.

Circumcenter lies on the side of the polygon

If the triangle is a right-angle triangle then the circumcenter lies on the hypotenuse of the triangle.

Circumcenter lies on the side of the polygon

In the figure, we see that a right angle at point c and circumcenter lies on the opposite side of the right angle which is known as hypotenuse.

Circumcenter lies outside of the polygon

If the triangle is an obtuse angle triangle then the circumcenter lies outside of the triangle.

In the figure, we have an obtuse angle of 1050. that’s why the circumcenter lies outside of the triangle.

Circumcenter lies outside of the polygon

How to Draw the Circumcenter in a Polygon?

  • Draw a closed polygon
  • Make the right  bisector of each side
  • Draw the lines which intersect each other

the point of intersection of these lines is known as the circumcenter of the polygon.

At the point of intersection, we draw a circle which is known as the circumcircle.

Also read: Triangle Inequality

Example Problem to Find the Circumcenter of a Triangle

We can find the circumcenter of a triangle when the vertex is given to us. The method is very easy to solve the questions.

Problem

Find the circumcenter of a triangle with three vertexes given as $A= (2,4)$, $B=(6,2)$, and $C=(3, -2)$.

Solution

Step 1: Write (and remember) the formula of mid-point and slope along with the values given.

  • Mid-point = $(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2})$
  • Slope = $\dfrac{y_1 + y_2}{x_1+x_2}$

Step 2: Find the mid-points by using the mid-point formula

For the mid-point of AB, the values given are $x_1 =2, \ x_2=6, \ y_1=4, \ y_2=2$

Thus the mid-point of AB = ((2+4)/2, (6+2)/2)

                 = (6/2, 8/2)

                              = (3, 4)

Similarly, the mid-point of BC = (6+2)/2, (3-2)/2

                              = (8/2), (1/2)

                              = (4, 0.5)

 Similarly, mid-point of CA = (3-2)/2, (2+4)/2

                              = (1/2), (6/2)

                              = (0.5, 3)

Step 3: Find the slope of each side of the triangle

Slope of AB = (2-4)/ (6-2)

                    = (-2/4)

                    = -0.5

Now, slope of BC = (-2-2)/ (3-6)

                    = (-4/-3)

                    = 1.33

Similarly, slope of CA= (4-(-2))/ (2-3)

                    = (6/-1)

                    = -6

Step 4: Find the slopes of perpendicular lines on AB, BC, and CA.

Slope of perpendicular line on AB; m = -1/Slope of AB

                                                             = -1/ (-0.5)

                                                             = 2 

Now, the slope of a perpendicular line on BC, m = -1/ Slope of BC

                                                                       = -1/1.33

                                                                       = -0.75

Similarly 

The slope of a perpendicular line on CA, m = -1/ Slope of CA

                                                                       = -1/-6

                                                                       = 0.17

Step 5: Now make point-slope equations for any two points. The values of x1 and y1 are from the midpoint values.

For AB

  • m=2 and x1=3 and y1=4
  • y- y1= m (x-x1)
  • y-4 = 2(x-3)
  • y -4 = 2x -6 
  • y = 2x -10

For BC

  • m= -0.75 x1 = 4 and y1 = 0.5
  • y- y1= m (x-x1)
  • y -0.5 = -0.75x – (-3)
  • y= -0.75+2.5

For CA

  • m = 0.17   x1= 0.5 and y1 = 3
  • y- y1= m (x-x1)
  • y-0.5 = 0.17(x-3)
  • y-0.5 = 0.17x -0.51
  • y = 0.17x -1.01

Step 6: Find the value of x and y by using any two equations

y = -0.75x +2.5 … (1)

y = 2x-10 … (2) 

2x – 10 = -0.75x +2.5

x = 3.05

Now put the value of x in equation (2)

y = 2x -10

y = 2(3.05) -10

y = 1.09

The circumcenter is = (3.05, 1.09)

The process of finding the circumcenter is a very long and time-consuming procedure. For quick and fast calculation, you can use a circumcenter calculator     

Image 8

Real-life use

In the summer holidays, a school arranges a tour for the students. The tour place is far three hours far from the school. If a bus arranges for the students it is very time-consuming to pick the students from their homes and drop them at the destination and also a lot of fuel is consumed in this process.

To overcome this difficulty, they decided to make three points in such a way the students are divided into three sectors. The distance of every student is equidistance from the fixed point. In this way, the bus just picks up the students from a given point. 

Image 9

This point is called circumcenter. And the homes of the students are called the vertex of the polygon. In this way, we save our time and fuel. Let’s A, B, and C are three villages. The circumcenter is the point of the tour place. 

Summary

This article discussed the definition of the circumcenter of the triangle. Additionally, I went through the types of angles and the circumcenter of the triangle. I have also discussed the examples of the circumcenter of the triangle and its real-life usage. I hope you can solve any questions about circumcenter with the help of this article.

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