On this page we are going to see how to find the circumcentre of a triangle. First, let us see the definition of circumcentre

__Definition__:

The point of concurrency of the perpendicular bisector of the sides of a triangle is called the circumcentre of the triangle. The circumcentre is denoted by S.

Let ABC be the triangle and D,E,F are the midpoint of BC,CA and AB.We need to find the slopes of the perpendicular bisectors of BC,CA and AB.Then we need to find the equation of the perpendicular bisectors.By solving any two equations we can get the circumcentre.

**Example 1:**

Find the coordinates of the circumcentre of a triangle whose vertices are (3,1) (2,2) and (2,0)

Let A,B and C are the vertices of the triangle

Now we need to find the midpoint of the side AB

Midpoint of AB = [(x₁ + x₂)/2 , (y₁ + y₂)/2]

A (3,1) and B (2,2)

Here x₁ = 3, x₂ = 2 and y₁ = 1,y₂ = 2

= [(3+2)/2,(1+2)/2]

= [5/2,3/2]

= [5/2,3/2]

So the vertices of D is (5/2,3/2)

Slope of AB = [(y₂ - y₁)/(x₂ - x₁)]

= [(2-1)/(2-3)]

= 1/(-1)

= -1

Slope of the perpendicular line through D = -1/slope of AB

= -1/(-1)

= 1

__Equation of the perpendicular line through D:__

(y-y₁) = m (x-x₁)

Here point D is (5/2,3/2)

x₁ = 5/2 ,y₁ = 3/2

(y-3/2) = 1(x-5/2)

(y-3/2) = 1(x-5/2)

(2y -3)/2 = (2x-5)/2

2y-3 = 2x-5

2x-2y= -3+5

2x-2y = 2

Equation of the perpendicular line through D is 2x -5 =0

Now we need to find the midpoint of the side BC

Midpoint of BC = [(x₁ + x₂)/2 , (y₁ + y₂)/2]

B (2,2) and C (2,0)

Here x₁ = 2, x₂ = 2 and y₁ = 2,y₂ = 0

= [(2+2)/2,(2+0)/2]

= [4/2,2/2]

= [2,1]

So the vertices of E is (2,1)

Slope of BC = [(y₂ - y₁)/(x₂ - x₁)]

= [(0-2)/(2-2)]

= -2/0

Slope of the perpendicular line through E = -1/slope of BC

= 1/(-2/0)

= 1/∞

= 0

__Equation of the perpendicular line through E:__

(y-y₁) = m (x-x₁)

Here point E is (2,1)

x₁ = 2 ,y₁ = 1

(y-1) = 0(x-2)

(y-1) = 0

y -1 = 0

Equation of the perpendicular line through E is y -1 = 0

Now we need to solve the equations of perpendicular bisectors D and E

2x-2y = 2 ---------(1)

y -1 = 0 ---------(2)

y = 1

Substitute y = 1 in the first equation we get 2x-2(1) = 2

2x-2 = 2

2x = 2+2

2x = 4

x = 4/2

x =2

So the circumcentre of a triangle ABC is **(2,1)**

**Related Topics**

- Section formula
- Area of triangle
- Area of triangle worksheets
- Area of quadrilateral
- Centroid
- Centroid of the triangle worksheets
- Finding missing vertex using centroid worksheet
- Midpoint
- Distance between two points
- Distance between two points worksheet
- Slope of the line
- Equation of the line
- Equation of line using two points worksheet
- Equation of the line using point and slope worksheets
- Point of intersection of two lines
- Point of intersection Worksheets
- concurrency of straight line
- Concurrency of straight lines worksheet
- Circumcentre of Triangle Worksheet
- Orthocentre of a triangle
- Orthocentre of Triangle Worksheet
- Incentre of a triangle
- Locus
- Perpendicular distance
- Angle between two straight lines
- Equation of a circle
- With center and radius
- With endpoints of a diameter
- Equation of a circle passing though three points
- Length of the tangent to a circle
- Equation of the tangent to a circle
- Family of circles
- Orthogonal circles
- Parabola
- Ellipse

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