Circumcenter Of Triangle - Definition, Properties, And Examples

Circumcenter of Triangle

Circumcenter of triangle is the point where three perpendicular bisectors from the sides of a triangle intersect or meet. The circumcenter of a triangle is also known as the point of concurrency of a triangle. The point of origin of a circumcircle i.e. a circle inscribed inside a triangle is also called the circumcenter. Let us learn more about the circumcenter of triangle, its properties, ways to locate and construct a triangle, and solve a few examples.

1.	What is the Circumcenter of Triangle?
2.	Properties of Circumcenter of Triangle
3.	Constructing Circumcenter of Triangle
4.	Formulas to Locate the Circumcenter of Triangle
5.	FAQs on Circumcenter of Triangle

What is the Circumcenter of Triangle?

The circumcenter of triangle can be found out as the intersection of the perpendicular bisectors (i.e., the lines that are at right angles to the midpoint of each side) of all sides of the triangle. This means that the perpendicular bisectors of the triangle are concurrent (i.e. meeting at one point). All triangles are cyclic and hence, can circumscribe a circle, therefore, every triangle has a circumcenter. To construct the circumcenter of any triangle, perpendicular bisectors of any two sides of a triangle are drawn.

Definition of Circumcenter

The circumcenter is the center point of the circumcircle drawn around a polygon. The circumcircle of a polygon is the circle that passes through all of its vertices and the center of that circle is called the circumcenter. All polygons that have circumcircles are known as cyclic polygons. However, all polygons need not have a circumcircle. Only regular polygons, triangles, rectangles, and right-kites can have the circumcircle and thus the circumcenter.

Properties of Circumcenter of Triangle

A circumcenter of triangle has many properties, let us take a look:

Consider any ΔABC with circumcenter O.

Property 1: All the vertices of the triangle are equidistant from the circumcenter. Let us look at the image below to understand this better. Join O to the vertices of the triangle.

AO = BO = CO. Hence, the vertices of the triangle are equidistant from the circumcenter.

Property 2. All the new triangles formed by joining O to the vertices are Isosceles triangles.

Property 3. ∠BOC = 2 ∠A when ∠A is acute or when O and A are on the same side of BC.

Property 4. ∠BOC = 2( 180° - ∠A) when ∠A is obtuse or O and A are on different sides of BC.

Property 5. Location for the circumcenter is different for different types of triangles.

Acute Angle Triangle: The location of the circumcenter of an acute angle triangle is inside the triangle. Here is an image for better understanding. Point O is the circumcenter.

Obtuse Angle Triangle: The circumcenter in an obtuse angle triangle is located outside the triangle. Point O is the circumcenter in the below-seen image.

Right Angled Triangle: The circumcenter in a right-angled triangle is located on the hypotenuse of a triangle. In the image below, O is the circumcenter.

Equilateral Triangle: All the four points i.e. circumcenter, incenter, orthocenter, and centroid coincide with each other in an equilateral triangle. The circumcenter divides the equilateral triangle into three equal triangles if joined with vertices of the triangle. Also, except for the equilateral triangle, the orthocenter, circumcenter, and centroid lie in the same straight line known as the Euler Line for the other types of triangles.

Constructing Circumcenter of Triangle

To construct the circumcenter of triangle, we use a geometric tool called the compass. The compass consists of two ends, where one end is placed on the hypotenuse of the triangle and the second end is on the vertex of the triangle. The steps to construct a circumcenter of triangle are:

• Step 1: Draw the perpendicular bisectors of all the sides of the triangle using a compass.
• Step 2: Extend all the perpendicular bisectors to meet at a point. Mark the intersection point as O, this is the circumcenter.
• Step 3: Using a compass and keeping O as the center and any vertex of the triangle as a point on the circumference, draw a circle, this circle is our circumcircle whose center is O.

Formulas to Locate the Circumcenter of Triangle

To locate or calculate the circumcenter of triangles, there are various formulas that can be applied. The various methods through which we can locate the circumcenter O(x,y) of a triangle whose vertices are given as \( \text A(x_1,y_1), \text B(x_2,y_2)\space \text and \space \text C(x_3,y_3)\) are as follows along with the steps.

Method 1: Using the Midpoint Formula

Step 1: Calculate the midpoints of the line segments AB, AC, and BC using the midpoint formula.

\( \begin{equation} M(x,y) = \left(\dfrac{ x_1 + x_2} { 2} , \dfrac{y_1 + y_2}{2}\right) \end{equation}\)

Step 2: Calculate the slope of any of the line segments AB, AC, and BC.

Step 3: By using the midpoint and the slope of the perpendicular line, find out the equation of the perpendicular bisector line.

\( (y-y_1) = \left(- \dfrac1m \right)(x-x_1)\)

Step 4: Similarly, find out the equation of the other perpendicular bisector line.

Step 5: Solve two perpendicular bisector equations to find out the intersection point.

This intersection point will be the circumcenter of the given triangle.

Method 2: Using the Distance Formula

\(\begin{equation} d = \sqrt{( x - x_1) {^2} + ( y - y_1) {^2}} \end{equation}\)

Step 1 : Find \(d_1, d_2\space and \space d_3\)

\[ \begin{equation} d_1= \sqrt{( x - x_1) {^2} + ( y - y_1) {^2}} \end{equation}\] \(d_1\) is the distance between circumcenter and vertex \(A\).

\[ \begin{equation} d_2= \sqrt{( x - x_2) {^2} + ( y - y_2) {^2}} \end{equation}\] \(d_2\) is the distance between circumcenter and vertex \(B\).

\[ \begin{equation} d_3= \sqrt{( x - x_3) {^2} + ( y - y_3) {^2}} \end{equation}\] \(d_3\) is the distance between circumcenter and vertex \(C\).

Step 2 : Now by computing, \(d_1 = d_2\space = \space d_3\) we can find out the coordinates of the circumcenter.

This is the widely used distance formula to determine the distance between any two points in the coordinate plane.

Method 3: Using Extended Sin Law

\(\begin{equation} \dfrac{ a}{ \sin A}=\dfrac{b}{ \sin B} =\dfrac{c} { \sin C} = 2R \end{equation}\)

Given that a, b, and c are lengths of the corresponding sides of the triangle and R is the radius of the circumcircle.

By using the extended form of sin law, we can find out the radius of the circumcircle, and using the distance formula can find the exact location of the circumcenter.

Method 4: Using the Circumcenter Formula

We can quickly find the circumcenter by using the circumcenter of a triangle formula:

\[\begin{equation} O(x, y)=\left(\frac{x_{1} \sin 2 A+x_{2} \sin 2 B+x_{3} \sin 2 C}{\sin 2 A+\sin 2B+\sin 2 C},\\ \frac{y_{1} \sin 2 A+y_{2} \sin 2 B+y_{3} \sin {2} C}{\sin 2 A+\sin 2 B+\sin 2 C}\right) \end{equation}\]

Where ∠A, ∠B, and ∠C are respective angles of ΔABC.

Related Topics

Listed below are a few topics related to the circumcenter of triangle, take a look.

• Incenter
• Orthocenter
• Parts of Circle