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.

## Types of Angles

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

- Acute angle
- Right angle
- Obtuse angle

### Acute angle

The angle which is lies between 0^{0} to 90^{0}. Simply an angle less than 90^{0} is called an acute angle.

### Right angle

An angle of 90^{0} is called a right angle. When we say a line is a perpendicular its means it makes an angle of 90^{0}.

### Obtuse angle

An angle greater than 90^{0} but less than 180^{0} 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.

In the figure, if we select all the angles less than 90^{0}. 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.

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 105^{0}. that’s why the circumcenter lies outside of the triangle.

## 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 x_{1} and y_{1} are from the midpoint values.

**For AB**

- m=2 and x
_{1}=3 and y_{1}=4 - y- y
_{1}= m (x-x_{1}) - y-4 = 2(x-3)
- y -4 = 2x -6
- y = 2x -10

**For BC**

- m= -0.75 x
_{1}= 4 and y_{1}= 0.5 - y- y
_{1}= m (x-x_{1}) - y -0.5 = -0.75x – (-3)
- y= -0.75+2.5

**For CA**

- m = 0.17 x
_{1}= 0.5 and y_{1}= 3 - y- y
_{1}= m (x-x_{1}) - 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

## 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.

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.