Three points, one circle
Unique circle through three points
Three points
Mark three points A, B and C on a sheet of paper. Can you always draw a circle that passes through all three?
Is there always a circle through three points?
What this lesson covers
The idea
No circle passes through three collinear points, but exactly one circle passes through three non-collinear points; its centre is where the perpendicular bisectors of AB and AC meet.
Three points
Mark three points A, B and C on a sheet of paper. Can you always draw a circle that passes through all three?
Is there always a circle through three points?
- Yes, always, and there are many of them
- Yes, always, and there is only one
- No, sometimes there is none
Move C
A and B are fixed. The two dashed lines are the perpendicular bisectors of AB and AC. Drag C around the sheet.
One circle
No circle passes through three collinear points, but exactly one circle passes through three non-collinear points. Its centre is where the perpendicular bisectors of AB and AC meet.
The centre O has OA = OB and OA = OC, so it lies on both bisectors. Two lines that cross meet at one point, so there is only one O and one circle. If A, B and C are on a line, the bisectors are parallel and never meet.
Notes
No circle passes through three collinear points; exactly one passes through three non-collinear points. Its centre is where the perpendicular bisectors of AB and AC meet.
Check yourself
A, B and C are three points on one straight line. How many circles pass through all three?
A, B and C are not on one straight line. How many circles pass through all three?
Where is the centre of the circle through A, B and C?
The perpendicular bisectors of AB and AC meet at O. OB = 8 cm. What is OC, in cm?
Answer: 8 cm
OA = OB because O is on the bisector of AB, and OA = OC because O is on the bisector of AC. So OC = OB = 8 cm.
- Exactly one. One circle is for three points that are not on a line.
- Infinitely many. That is for two points. For three points on a line there are none.
- None — correct. Yes. The perpendicular bisectors of AB and AC are parallel, so no point can be the centre.
- Exactly one — correct. Yes. The two bisectors cross at one point, which is the centre of the only circle.
- None. For points not on a line, the bisectors do meet, and the meeting point is a centre.
- Infinitely many. The centre must be on two different bisectors at once. They cross at only one point.
- At the midpoint of AB. The midpoint of AB is equally far from A and B only. C is usually at a different distance.
- Where the perpendicular bisectors of AB and AC meet — correct. Yes. It is equally far from A, B and C.
- At the point C. C is on the circle. The centre is inside, equally far from all three points.