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Circumcenter Formula

In this section, circumcenter refers to the circumcenter of a triangle. This can be defined as the point at which the perpendicular bisector of all sides of a triangle meets. If the coordinates of three points are given, one can find out the circumcenter by using the mid point and slope of two sides. Given figure illustrates the circumcenter of a triangle.

Circumcenter Image
The steps to find the circumcenter of a triangle:
  • Find and Calculate the midpoint of given coordinates or midpoints (AB, AC, BC)
  • Calculate the slope of the particular line
  • By using the midpoint and the slope, find out the equation of line (y-y1) = m (x-x1)
  • Find out the other line of equation in the similar manner
  • Solve the two bisector equation by finding out the intersection point
  • Calculated intersection point will be the circumcenter of the given triangle

Related Calculators
Circumcenter Calculator Acceleration Formula Calculator
Area of a Circle Formula Calculator Area of a Cylinder Formula Calculator
 

Circumcenter Examples

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Given problems will help you for better understanding the concept.

Solved Examples

Question 1: If three coordinates of a triangle are (3,2), (1,4), (5,4). Calculate the circumcenter of this triangle ?

Solution:
 
Given points are,
A = (3, 2), B = (1, 4), C = (5, 4)
To find out the circumcenter we have to solve any two bisector equations and find out the intersection points.

So, mid point of AB = ($\frac{3+1}{2}$,$\frac{2+4}{2}$) = (2,3)

Slope of AB = ($\frac{4-2}{1-3}$) = -1
Slope of the bisector is the negative reciprocal of the given slope.
So, the slope of the perpendicular bisector = 1
Equation of AB with slope 1 and the coordinates (2,3) is,
(y - 3) = 1(x - 2)
x - y = -1..................(1)
Similarly, for AC
Mid point of AC = ($\frac{3+5}{2}$,$\frac{2+4}{2}$) = (4,3)

Slope of AC = ($\frac{4-2}{5-3}$) = 1
Slope of the bisector is the negative reciprocal of the given slope.
So, the slope of the perpendicular bisector = -1
Equation of AC with slope -1 and the coordinates (4,3) is,
(y - 3) = -1(x - 4)
y - 3 = -x + 4
x + y = 7..................(2)
By solving equation (1) and (2),
(1) + (2) ⇒ 2x = 6; x = 3
Substitute the value of x in to (1)
3 - y = -1
y = 3 + 1 = 4
So the circumcenter is (3, 4)


 

Question 2: If three coordinates of a triangle are (2,1), (4, 5), (6, 3). Calculate the circumcenter of this triangle ?

Solution:
 
Given points are,
A = (2,1), B = (4, 5), C = (6, 3)
To find out the circumcenter we have to solve any two bisector equations and find out the intersection points.

So, mid point of AB = ($\frac{2+4}{2}$,$\frac{1+5}{2}$) = (3,3)

Slope of AB = ($\frac{5-1}{4-2}$) = $\frac{4}{2}$ = 2
Slope of the bisector is the negative reciprocal of the given slope.

So, the slope of the perpendicular bisector = $\frac{-1}{2}$

Equation of AB with slope $\frac{-1}{2}$ and the coordinates (3,3) is,

(y - 3) = $\frac{-1}{2}$(x - 3)
2y - 6 = -x + 3
x + 2y = 9..................(1)
Similarly, for AC
Mid point of AC = ($\frac{6+2}{2}$,$\frac{1+3}{2}$) = (4,2)

Slope of AC = ($\frac{3-1}{4-2}$) = $\frac{2}{4}$ = $\frac{1}{2}$
Slope of the bisector is the negative reciprocal of the given slope.
So, the slope of the perpendicular bisector = -2
Equation of AC with slope -2 and the coordinates (4,2) is,
(y - 2) = -2(x - 4)
y - 2 = -2x + 8
2x + y = 10..................(2)
By solving equation (1) and (2),
(2) × 2 ⇒ 4x + 2y = 20...............(3)
(3) - (2) ⇒ 3x = 11; x = $\frac{11}{3}$

Substitute the value of x in to (1)

$\frac{11}{3}$ + 2y = 9

y = $\frac{8}{3}$

So the circumcenter is ($\frac{11}{3}$,$\frac{8}{3}$)

 

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