Three points give one circle, and the triangle shows where its centre is
Three dots, not in a line. Exactly one circle passes through all three. The perpendicular bisector of AB is the 90° line through its middle.
- A triangle, corners A, B, C.
- The perpendicular bisector of AB.
- The perpendicular bisector of BC.
- Where the two bisectors cross: the centre.
- The circle, through all three corners.
Where the centre has to be
Join the centre to the middle of chord AB. That line meets AB at 90°. So the centre sits on the perpendicular bisector of AB. It sits on the bisector of BC too. Two lines that are not parallel cross at one point. That point is the circumcentre. It is the centre of the circumcircle. That is the circle through A, B and C. Three points in a line give parallel bisectors. They never cross, so no circle fits.
Where the centre lies
| Corners | Widest angle | Centre | Where it sits |
|---|---|---|---|
| (0,0) (8,0) (4,8) | 63.4° | (4, 3) | Inside the triangle |
| (0,0) (8,0) (4,4) | 90° | (4, 0) | Middle of the long side |
| (0,0) (8,0) (4,2) | 126.9° | (4, −3) | Outside the triangle |
The widest angle tells where the centre sits. Under 90°, the centre is inside the triangle. At exactly 90°, it is on the longest side. Over 90°, it is outside.
A right angle at a corner doubles to 180° at the centre. So the opposite side is a diameter. A diameter's midpoint is the centre. So a right-angled triangle's centre is the midpoint of its longest side.
Lift corner C. Watch where the two lines cross.
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