I’m Up and Down, and Round and Round · Lesson 2 of 8
How Many Circles?
“Give three non-collinear points a chance and exactly one circle shows up.”
• Determine how many circles pass through two fixed points. • Understand why their centres lie on a perpendicular bisector. • Understand why three non-collinear points determine one unique circle. • Define circumcircle and circumcentre. • Locate the circumcentre for acute, obtuse and right triangles.
A circle is determined by its centre and radius. If we require a circle to pass through one or more fixed points, those conditions restrict where its centre can be. The key idea is always equal distance: the centre must be equally far from every point lying on the circle.
Circles Through Two Points
Suppose a circle must pass through two points A and B. If its centre is O, then OA and OB are radii of the same circle, so OA=OB. Therefore O must be a point equidistant from A and B.
The locus of points equidistant from A and B is the perpendicular bisector of AB. Hence every possible centre lies somewhere on that perpendicular bisector.
Infinitely many circles pass through two distinct points. Their centres lie on the perpendicular bisector of the segment joining the points.
The Smallest Circle Through Two Points
The midpoint M of AB lies on the perpendicular bisector and is the closest possible centre to A and B. With M as centre, MA=MB=AB/2. In this circle, AB is a diameter.
As the centre moves farther away from AB along the perpendicular bisector, the radius increases. There is no largest radius because the centre can move arbitrarily far away.
What About Three Points?
Now suppose a circle must pass through three distinct points A, B and C. If the points are collinear, no circle can pass through all three because a line can meet a circle in at most two distinct points.
If A, B and C are non-collinear, the situation changes completely. There is exactly one circle through all three.
There is a unique circle passing through three non-collinear points.
Why the Circle Is Unique
The centre O must satisfy OA=OB, so O lies on the perpendicular bisector of AB. It must also satisfy OA=OC, so O lies on the perpendicular bisector of AC. Because the three points are not collinear, these two perpendicular bisectors intersect at exactly one point. That unique intersection is O.
Circumcircle and Circumcentre
The unique circle passing through all three vertices of a triangle is called the circumcircle of the triangle.
The centre of the circumcircle is called the circumcentre. It is the common intersection point of the perpendicular bisectors of the sides of the triangle.
Where Is the Circumcentre?
| Triangle type | Position of circumcentre |
|---|---|
| Acute-angled triangle | Inside the triangle |
| Right-angled triangle | At the midpoint of the hypotenuse |
| Obtuse-angled triangle | Outside the triangle |
Problem
A right triangle has hypotenuse 10 cm. What is the radius of its circumcircle?
- 1.For a right triangle, the circumcentre is the midpoint of the hypotenuse.
- 2.The radius is the distance from that midpoint to either end of the hypotenuse.
- 3.Therefore radius = 10/2 = 5 cm.
Practice Problems
- Explain why infinitely many circles can pass through two distinct points.
- If AB=12 cm, find the least possible radius of a circle passing through A and B.
- Explain why no circle can pass through three distinct collinear points.
- Construct an acute triangle and describe how to locate its circumcentre using perpendicular bisectors.
- Construct a right triangle and verify that the circumcentre lies at the midpoint of its hypotenuse.
- For an obtuse triangle, predict whether the circumcentre lies inside or outside before constructing it.
Key Takeaways
• Centres of circles through A and B lie on the perpendicular bisector of AB. • Infinitely many circles pass through two points. • The smallest such circle has AB as diameter. • Three non-collinear points determine exactly one circle. • The circumcentre is the intersection of perpendicular bisectors. • Its position depends on whether the triangle is acute, right or obtuse.
Next, we study chords and the angles they subtend at the centre.