Three points and the circumcircle

"Now three points A, B, C. How many circles?" If they lie on one straight line (collinear), the answer is none at all.

Theorem 1: there is exactly one circle through three non-collinear points. Its centre O must be:

• on the perpendicular bisector of AB (so OA = OB), and
• on the perpendicular bisector of AC (so OA = OC).

Two lines that cross meet in exactly one point, so O is fixed, and so is the circle.

Why collinear points fail. Their two perpendicular bisectors are both at right angles to the same line, so they are parallel and never meet: no centre, no circle. This also shows that a straight line can cut a circle in at most two points.

Three non-collinear points make a triangle. The circle through its corners is the circumcircle, and its centre the circumcentre. The circle circumscribes the triangle; the triangle is inscribed in the circle.

Where is the circumcentre?

• inside an acute-angled triangle;
• outside an obtuse-angled one;
• exactly at the midpoint of the hypotenuse of a right-angled one.