Circles through two points

"How many circles pass through two points A and B?" The centre O must be the same distance from both: OA = OB.

The midpoint of AB works. A circle centred there has AB as a diameter and radius AB/2.

But so does every point on the perpendicular bisector of AB. The book shows that this line is exactly the locus of points equidistant from A and B:

• every point equidistant from A and B is on it;
• every point on it is equidistant from A and B.

So there are infinitely many circles through A and B, one for each centre on the bisector.

The smallest has radius AB/2. As the centre slides away from AB along the bisector, the radius grows without limit, and near AB the circle looks less and less curved. There is no largest circle.

Compare squares: with A and B as corners there are exactly three (two with AB as a side, one with AB as a diagonal).