Many circles fit through two points
Put two dots on a page. Draw a circle through both. Can you draw a different one?
- Two dots, A and B.
- The line that cuts AB in half at 90°.
- Pick a dot O on that line. OA and OB match.
- The circle with centre O passes through A and B.
- Another dot P on the line gives another circle through A and B.
- The midpoint of AB gives the smallest circle.
The rule
Yes: endless different circles pass through any two dots.
A circle's centre is the same distance from every point on it. Take a circle through A and B. Its centre O has OA = OB.
The dots with OA = OB make a line. It is the perpendicular bisector of AB. It cuts AB in half at 90°. A set of points that follow one rule is a locus.
Rule: A circle passes through A and B. Its centre lies on the perpendicular bisector of AB.
Every dot on that line can be a centre. A line has endless dots. So endless circles pass through two points.
The smallest circle has its centre at the midpoint of AB. Its radius is half of AB.
Two points, worked
Let A = (−3, 0) and B = (3, 0). The perpendicular bisector is the line x = 0. Let the centre be (0, k), any dot on that line. From the centre to A is 3 across and k up. So radius² = 3² + k².
| Centre | Radius² | Radius |
|---|---|---|
| (0, 0) | 9 + 0 = 9 | 3 |
| (0, 2) | 9 + 4 = 13 | 3.61 |
| (0, 4) | 9 + 16 = 25 | 5 |
| (0, 8) | 9 + 64 = 73 | 8.54 |
The radius never drops below 3, which is half of AB. The dot (2, 1) is off the line. Its distances are 5.1 to A and 1.41 to B. They differ. No circle with centre (2, 1) passes through both.
Test some dots. Could each one be a centre?
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