3.2.8
Equation of a Circumcircle
Equation of a Circumcircle of a Triangle from Three Points
Equation of a Circumcircle of a Triangle from Three Points
We can find the general equation of a circumcircle of a triangle by using the properties of chords.
What is a circumcircle?
What is a circumcircle?
- The circumcircle of a triangle is the circumference of a circle upon which the three vertices of the triangle lie.
- A group of three points can only have one circumcircle.
What is a circumcircle?
What is a circumcircle?
- Each side of the triangle is a chord of the circle.
- The perpendicular bisectors of the triangle sides intersect at the centre of the circle.
How do you find the equation of a circumcircle?
How do you find the equation of a circumcircle?
- To write the general equation of the circumcircle, we need to find the centre and the radius of the circle.
How do you find the equation of a circumcircle?
How do you find the equation of a circumcircle?
- The centre is found by working out the point of intersection between the perpendicular bisectors of two of the sides of the triangle.
- The radius is found by working out the distance between the centre of the circle and one of the points of the triangle.
Example
Example
- The points A(-6,3), B(-3,2) and C(0,3) lie on the circumference of a circle.
- What is the equation of the circle?
Find the perpendicular bisector of AB
Find the perpendicular bisector of AB
- The gradient of the chord AB is equal to:
- The gradient of the perpendicular bisector of AB must be equal to 3.
- The midpoint of AB has coordinates:
- Midpoint =
Find the perpendicular bisector of AB
Find the perpendicular bisector of AB
- The intercept of the perpendicular bisector, is then found from:
Find the perpendicular bisector of AB
Find the perpendicular bisector of AB
- Substituting in the midpoint coordinates, we get:
- So the equation of the perpendicular bisector of AB is
Find the perpendicular bisector of AC
Find the perpendicular bisector of AC
- The gradient of the chord AC is equal to:
- This means that AC is horizontal.
- The equation for the perpendicular bisector must be:
Find the centre of the circumcircle
Find the centre of the circumcircle
- The centre of the circumcircle is the point of intersection between the perpendicular bisectors of AB and AC.
Find the centre of the circumcircle
Find the centre of the circumcircle
- Substituting in into
, we get:
- The centre of the circumcircle has coordinates (3,7).
Find the radius of the circumcircle
Find the radius of the circumcircle
- The radius of the circumcircle is equal to the distance between the centre and any one of the points of the triangle.
Find the radius of the circumcircle
Find the radius of the circumcircle
- The distance between the centre and point is found by using Pythagoras' theorem:
- Distance
- The radius of the circumcircle is equal to 5.
Write the equation of the circumcircle
Write the equation of the circumcircle
- We can now write the equation of the circumcircle in its general form:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Unlock your full potential with GoStudent tutoring
Affordable 1:1 tutoring from the comfort of your home
Tutors are matched to your specific learning needs
30+ school subjects covered