**Definition of Circumcenter**

- Circumcenter of a triangle is the point of intersection of all the three perpendicular bisectors of the triangle.

**More about Circumcenter**

- The circle drawn around the triangle by taking circumcenter as the center is called a circumscribed circle.

**Example of Circumcenter**

- In the above diagram, the three perpendicular bisectors PO, QO, and RO of sides BC, AB, and AC of the triangle ABC intersect at the point O. So, the point O is called the circumcenter of the triangle ABC.

**Solved Example on Circumcenter**

Find the circumcenter of the triangle in the figure shown.

Choices:

A. (- , - )

B. (, )

C. (, - )

D. (- , )

Correct Answer: C

Solution:

Step 1:The point where all the perpendicular bisectors intersect is called circumcenter.

Step 2:To find the perpendicular bisector of , find the midpoint of and then find its slope.

Step 3:Midpoint of is (, ) = (, 0)

Step 4:Slope of is - 6.

Step 5:The slope of perpendicular bisector of is the negative reciprocal of - 6, .

Step 6:The perpendicular bisector of passes through the midpoint of .

Step 7:So, the equation of perpendicular bisector of is = implies 2x- 12y= 5.

Step 8:Similarly, the equation of perpendicular bisector of is 8x- 2y= 13.

Step 9:Solving 2x- 12y= 5 and 8x- 2y= 13 givesx= andy= - .

Step 10:So the circumcenter of the given triangle is (, - ).

**Related Terms for Circumcenter**

- Center
- Circle
- Circumscribed
- Perpendicular bisector
- Point of Intersection
- Polygon
- Triangle