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Equation of a Circumcircle

Maths: Edexcel A Level Pure Maths, topic 3.2.8

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:
  • 2−3(−3)−(−6)=−13 \frac{2-3}{(-3) - (-6)}=-\frac{1}{3}
  • The gradient of the perpendicular bisector of AB must be equal to 3.
  • The midpoint of AB has coordinates:
  • Midpoint = ((−6)+(−3)2,3+22)=(−4.5,2.5)(\frac{(-6)+(-3)}{2},\frac{3+2}{2}) = (-4.5,2.5)
Find the perpendicular bisector of AB

Find the perpendicular bisector of AB

  • The intercept of the perpendicular bisector, cc is then found from:
  • y=3x+cy = 3x + c
Find the perpendicular bisector of AB

Find the perpendicular bisector of AB

  • Substituting in the midpoint coordinates, we get:
  • 2.5=(3×(−4.5))+c2.5 = (3 \times (-4.5)) + c
  • 2.5=−13.5+c→c=162.5 = -13.5 +c \rightarrow c = 16
  • So the equation of the perpendicular bisector of AB is y=3x+16y = 3x +16
Find the perpendicular bisector of AC

Find the perpendicular bisector of AC

  • The gradient of the chord AC is equal to:
  • 3−30−(−6)=0 \frac{3-3}{0 - (-6)}=0
  • This means that AC is horizontal.
  • The equation for the perpendicular bisector must be:
  • x=(−6)+02=−3x = \frac{(-6)+0}{2} = -3
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 x=3x=3 into y=3x+16y = 3x +16, we get:
  • y=(3×3)+16→y=7y = (3 \times 3) +16 \rightarrow y = 7
  • 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 AA is found by using Pythagoras' theorem:
  • Distance =(3−(−6))2+(7−3)2=5= \sqrt{(3-(-6))^2 + (7-3)^2} = 5
  • 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:
  • (x+3)2+(y−7)2=25(x+3)^2 + (y-7)^2 = 25

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