# Circumcenter Formula

**Method to Calculate the Circumcenter of aTriangle**

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

### Solved Examples

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

** Solution: **

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)