The Circumcenter Calculator is a free tool that displays the center of a triangle circumcircle. STUDYQUERIES’s circumcenter calculator tool makes circumcenter calculations faster and displays the coordinates of the circumcenter instantly.

Circumcenter Calculator Step By Step Of A Triangle

The procedure to use the circumcenter calculator is as follows:

  • Step 1: Enter the three coordinates of a triangle in the respective input fields
  • Step 2: Now click the button “Submit” to get the output
  • Step 3: Finally, the coordinates of the circumcenter will be displayed in the new window

Circumcenter Calculator

Circumcenter Of A Triangle

Definition

The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect. In other words, the point of concurrency of the bisector of the sides of a triangle is called the circumcenter. It is denoted by P(X, Y). The circumcenter is also the center of the circumcircle of that triangle and it can be either inside or outside the triangle.

Circumcenter Formula

$$\mathbf{P(X, Y) = \frac{(x_1 \sin 2A + x_2 \sin 2B + x_3 \sin 2C)}{(\sin 2A + \sin 2B + \sin 2C)}, \frac{(y_1 \sin 2A + y_2 \sin 2B + y_3 \sin 2C)}{(\sin 2A + \sin 2B + \sin 2C)}}$$

Here,

\(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) are the vertices of the triangle, and \(A,\ B,\ C\) are their respective angles.

Method to Calculate the Circumcenter of a Triangle

Steps to find the circumcenter of a triangle is:

  • Calculate the midpoint of given coordinates, i.e. midpoints of AB, AC, and BC
  • Calculate the slope of the particular line
  • By using the midpoint and the slope, find out the equation of the line \((y-y_1) = m (x-x_1)\)
  • Find out the equation of the other line in a similar manner
  • Solve two bisector equations by finding out the intersection point
  • The calculated intersection point will be the circumcenter of the given triangle

Finding Circumcenter Using Linear Equations Or Distance Formula

The circumcenter can also be calculated by forming linear equations using the distance formula. Let us take \((X, Y)\) be the coordinates of the circumcenter. According to the circumcenter properties, the distance of \((X, Y)\) from each vertex of a triangle would be the same.

Assume that \(D_1\) be the distance between the vertex \((x_1, y_1)\) and the circumcenter \((X, Y)\), then the formula is given by,

$$D_1= \sqrt{[(X−x_1)^2+(Y−y_1)^2]}$$

$$D2= \sqrt{[(X−x_2)^2+(Y−y_2)^2]}$$

$$D3= \sqrt{[(X−x_3)^2+(Y−y_3)^2]}$$

Now, since \(D_1=D_2\) and \(D_2=D_3\), we get

$$(X−x_1)^2 + (Y−y_1)^2 = (X−x_2)^2 + (Y−y_2)^2$$

From this, two linear equations are obtained. By solving the linear equations using the substitution or elimination method, the coordinates of the circumcenter can be obtained.

Finding Circumcenter Using The Midpoint Formula

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 \(A(x_1,y_1)\), \(B(x_2,y_2)\) and \(C(x_3,y_3)\) are as follows along with the steps.

Using the Midpoint Formula
Using the Midpoint Formula
  • Step 1: Calculate the midpoints of the line segments \(AB,\ AC,\ and\ BC\) using the midpoint formula.
    $$M(x,y)=\mathbf{(\frac{x_1+x_2)}{2},\frac{y_1+y_2}{2})}$$
  • 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.
    $$(y−y_1)=(\frac{−1}{m})(x−x_1)$$
  • 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.

Using Extended Sin Law

$$\mathbf{\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R}$$

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

Using Extended Sin Law
Using Extended Sin Law

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.

Properties of Circumcenter

Some of the properties of a triangle’s circumcenter are as follows:

  • The circumcenter is the center of the circumcircle.
  • All the vertices of a triangle are equidistant from the circumcenter.
  • In an acute-angled triangle, the circumcenter lies inside the triangle.
  • In an obtuse-angled triangle, it lies outside of the triangle.
  • Circumcenter lies at the midpoint of the hypotenuse side of a right-angled triangle.

How To Construct Circumcenter Of A Triangle?

The circumcenter of any triangle can be constructed by drawing the perpendicular bisector of any of the two sides of that triangle. The steps to construct the circumcenter are:

Constructing Circumcenter of Triangle
Constructing Circumcenter of Triangle
  • Step 1: Draw the perpendicular bisector of any two sides of the given triangle.
  • Step 2: Using a ruler, extend the perpendicular bisectors until they intersect each other.
  • Step 3: Mark the intersecting point as P which will be the circumcenter of the triangle. It should be noted that even the bisector of the third side will also intersect at P.

Step By Step Calculation Of Circumcenter Of A Triangle With Coordinates

\(\mathbf{\color{red}{Let\ the\ points\ of\ the\ sides\ be\ A(5,7),\ B(6,6),\ and\ C(2,-2).\ Consider\ the\ points\ of\\ the\ sides\ to\ be\ x_1,y_1,\ and\ x_2,y_2\ respectively.\ We\ need\ to\ find\ the\ equation\ of\ the\\ perpendicular\ bisectors\ to\ find\ the\ points\ of\ the\ Circumcenter.}}\)

Step 1: Lets calculate the midpoint of the sides \(AB,\ BC\ and\ CA\) which is the average of the \(x\ and\ y\) co-ordinates.

Midpoint of a line in the triangle = $$\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}$$

Midpoint of \(AB = \frac{5+6}{2}, \frac{7+6}{2} = (\frac{11}{2}, \frac{13}{2})\)

Midpoint of \(BC = \frac{6+2}{2}, \frac{6-2}{2} = (4, 2)\)

Midpoint of \(CA = \frac{2+5}{2}, \frac{-2+7}{2} = (7/2, 5/2)\)

Step 2: Next, we need to find the slope of the sides \(AB,\ BC,\ and\ CA\) using the formula \(\frac{y_2-y_1}{x_2-x_1}\). Kindly note that the slope is represented by the letter \(‘m’\).

Slope of \(AB\ (m) = \frac{6-7}{6-5} = -1.\)

Slope of \(BC\ (m) = \frac{-2-6}{2-6} = 2.\)

Slope of \(CA\ (m) = \frac{7+2}{5-2} = 3.\)

Step 3: Now, let’s calculate the slope of the perpendicular bisector of the lines \(AB,\ BC,\ and\ CA.\)

The slope of the perpendicular bisector = \(\frac{-1}{slope\ of\ the\ line}\)

Slope of the perpendicular bisector of \(AB = \frac{-1}{-1} = 1\)

The slope of the perpendicular bisector of \(BC = \frac{-1}{2}\)

The slope of the perpendicular bisector of \(CA = \frac{-1}{3}\)

Step 4: Once we find the slope of the perpendicular lines, we have to find the equation of the perpendicular bisectors with the slope and the midpoints.

Let’s find the equation of the perpendicular bisector of \(AB)\) with midpoints \((\frac{11}{2},\frac{13}{2})\) and \(slope\ 1\).

Formula to find the circumcenter equation \(y-y_1 = m(x-x_1)\)

\(y-\frac{13}{2} = 1(x-\frac{11}{2})\)

By solving the above, we get the equation \(-x + y = 1 \longrightarrow(1)\)

Similarly, we have to find the equation of the perpendicular bisectors of the lines BE and CF.

For \(BC\) with midpoints \((4,2)\) and slope \(\frac{-1}{2}\)

\(y-2 = \frac{-1}{2}(x-4)\)

By solving the above, we get the equation \(x + 2y = 8 \longrightarrow(2)\)

For \(CA\) with midpoints \((\frac{7}{2},\frac{5}{2}\) and slope \(\frac{-1}{3}\)

\(y-\frac{5}{2} = \frac{-1}{3}(x-\frac{7}{2})\)

By solving the above, we get the equation \(x + 3y = 11 \longrightarrow(3)\)

Step 5: Find the value of \(x\) and \(y\) by solving any \(2\) of the above \(3\) equations.
In this example, the values of \(x\ and\ y\ are\ (2,3)\) which are the coordinates of the Circumcenter \((o)\).

FAQs

What are circumcenter and Orthocenter?

The circumcenter is also the center of the circle passing through the three vertices, which circumscribes the triangle. The orthocenter is the point of intersection of the altitudes of the triangle, that is, the perpendicular lines between each vertex and the opposite side.

How do you find the circumcenter of a triangle?

To find the circumcenter of any triangle, draw the perpendicular bisectors of the sides and extend them. The point at which the perpendicular intersects each other will be the circumcenter of that triangle.

What are Circumcentre and Incentre?

circumcenter O, the point of which is equidistant from all the vertices of the triangle; incenter I, the point of which is equidistant from the sides of the triangle; centroid G, the point of intersection of the medians of the triangle.

What are circumcenter and example?

The circumcenter of a triangle is the point where three perpendicular bisectors from the sides of a triangle intersect or meet. The point of origin of a circumcircle i.e. a circle inscribed inside a triangle is also called the circumcenter.

Is the incenter the same as the circumcenter?

The incenter of a triangle is the point where the angle bisectors of a triangle run together (point of concurrency). The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of a triangle.