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.
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.
Examples on Circumcenter of Triangle
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Example 1: Shemron has a cake that is shaped like an equilateral triangle of sides \(\sqrt3 \text { inch}\) each. He wants to find out the dimension of the circular base of the cylindrical box which will contain this cake.
Solution
Since it is an equilateral triangle, \( \text {AD}\) (perpendicular bisector) will go through the circumcenter \(\text O \). Now using circumcenter facts that the Circumcenter will divide the equilateral triangle into three equal triangles if joined with the vertices.
i.e.
\[\begin{align*} \text {area of } \triangle AOC = \text {area of } \triangle AOB \\= \text {area of } \triangle BOC \end{align*}\]
Therefore,
\[\begin{align*} \text {area of } \triangle {ABC} {} &= 3 \times \text {area of } \triangle BOC \end{align*} \]
Using the formula for the area of an equilateral triangle \[\begin{align*} &= \dfrac{\sqrt3}{4} \times a^2 \end{align*} \]
Also, area of triangle \[\begin{align*} &= \dfrac{1}{2} \times \text { base } \times \text { height } \end{align*} \]
On substituting we get,
\[\begin{align*} {\dfrac{\sqrt3}{4}} \times a^2 &= 3\times \dfrac{1}{2} \times a\times OD\\OD &= \dfrac{1}{2{\sqrt3}} \times a \end{align*}\]
Now for \(\triangle \text{ ABC}\)
Again using formula for area of \(\triangle \text{ ABC}\) = \( \dfrac{1}{2} \times \text { base } \times \text { height } \) = \( \dfrac{\sqrt3}{4} \times a^2 \)
\[\begin{align*}\dfrac {1}2\times a\times (R+OD) &= \dfrac {\sqrt 3}4\times a^2 \\\dfrac12 a\times \left( R+\dfrac a{2\sqrt3}\right) &= \dfrac{\sqrt3}4\times a^2\\R &= \dfrac a{\sqrt3} \end{align*}\]
Substituting,
\[ \begin{align*}a & = \sqrt3 \end{align*}\]
R = 1 inch.
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Example 2: Charlie came to know that the circumcenter of a Right-angled triangle lies in the exact center of its hypotenuse. He wants to check this with a Right-angled triangle of sides L(0,5), M(0,0), and N(5,0)\). Can you help him in confirming this fact?
Solution:
Using the circumcenter formula or circumcenter of a triangle formula from circumcenter geometry:
\[ \begin{equation} O(x, y)=\left(\dfrac{x_{1} \sin 2 A+x_{2} \sin 2 B+x_{3} \sin 2 C}{\sin 2 A+\sin 2B+\sin 2 C},\\ \dfrac{y_{1} \sin 2 A+y_{2} \sin 2 B+y_{3} \sin {2} C}{\sin 2 A+\sin 2 B+\sin 2 C}\right) \end{equation}\]
Putting the corresponding values,
\[O(x,y) = \dfrac { (0 + 0 + 5 \times 1)}{ (0 + 1 + 1) }, \dfrac { (5 \times 1 + 0 + 0)}{(0 + 1 + 1)}\]
\[ O(x,y) = \dfrac {5}{2} , \dfrac {5}{2}\].
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Example 3: Thomas has triangular cardboard whose one side is 19 inches and the opposite angle to that side is 30°. He wants to know the base area of the cylindrical box so that he can fit this card in it completely.
Solution:
Using Extended sin law,
\[\begin{equation} \dfrac{ a}{ \sin A}=\dfrac{b}{ \sin B} =\dfrac{c} { \sin C} = 2R \end{equation}\]
\[\dfrac{19} { \sin30} = 2R\]
\[R = 19 \text { in}\]
Now, the area of the circumcircle:
\[\pi r^2 = \pi \times{19^2}\]
Therefore, Area = 1133.54 in2.
FAQs on Circumcenter of Triangle
What is the Circumcenter of Triangle?
Circumcenter of triangle is the point of intersection of three perpendicular lines from the sides of a triangle. The point of intersection can also be called as the point of concurrency.
How to Find the Circumcenter of a Triangle?
We can find circumcenter by using the circumcenter of a triangle formula, where the location of the circumcenter is O(x,y) and the coordinates of a triangle are given as \( \text A(x_1,y_1), \text B(x_2,y_2)\space \text and \space \text C(x_3,y_3)\).
\[\begin{equation} O(x, y)=\left(\frac{x_{1} \sin 2 A+x_{2} \sin 2 B+x_{3} \sin 2 C}{\sin 2 A+\sin 2B+\sin 2 C},\\ \frac{y_{1} \sin 2 A+y_{2} \sin 2 B+y_{3} \sin {2} C}{\sin 2 A+\sin 2 B+\sin 2 C}\right) \end{equation}\]
Where A, B, and C are the angles of the triangles.
How Do You Find the Circumcenter of Triangle with Vertices?
Using the Distance formula, where the vertices of the triangle are given as \( \text A(x_1,y_1), \text B(x_2,y_2)\space \text and \space \text C(x_3,y_3)\) and the coordinate of the circumcenter is O(x,y).
Find \(d_1, d_2\space and \space d_3\) by using following formula
\[ \begin{equation} d_1= \sqrt{( x - x_1) {^2} + ( y - y_1) {^2}} \end{equation}\] \(d_1\) 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 \(C\).
Does Every Triangle Have a Circumcenter?
Yes, as all the triangles are cyclic in nature which means that they can circumscribe a circle, and hence, every triangle has a circumcenter.
What is the Difference Between a Circumcenter and an Incenter of a Triangle?
The incenter is the center of the circle inscribed inside a triangle (incircle) and the circumcenter is the center of a circle drawn outside a triangle (circumcircle). The incenter can never lie outside the triangle, whereas, the circumcenter can lie outside of the triangle.
Are Circumcenter and Centroid of Triangle the Same?
Except for Equilateral triangles, the circumcenter and centroid are two distinct points as they do not coincide with each other.