One circle, three points
Two points do not define a circle: infinitely many circles pass through them. A third point, as long as it is not on the same line, fixes exactly one. This circle is the circumscribed circle of the triangle formed by the three points, and its center is the circumcenter.
The geometry
The center must be the same distance from all three points. The set of points equidistant from two points is the perpendicular bisector of the segment joining them. So the center is where the perpendicular bisectors of the triangle’s sides meet. All three bisectors cross at the same point, which is why the construction always works when the points are not aligned.
The algebra
With points (x₁, y₁), (x₂, y₂) and (x₃, y₃), let
D = 2 × [x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]
Then the center (h, k) is:
- h = [(x₁² + y₁²)(y₂ − y₃) + (x₂² + y₂²)(y₃ − y₁) + (x₃² + y₃²)(y₁ − y₂)] ÷ D
- k = [(x₁² + y₁²)(x₃ − x₂) + (x₂² + y₂²)(x₁ − x₃) + (x₃² + y₃²)(x₂ − x₁)] ÷ D
and the radius is the distance from the center to any of the points. If D = 0, the points are aligned and no circle exists.
Worked example
Take the points (0, 0), (4, 0) and (0, 3), the corners of a 3-4-5 right triangle.
- D = 2 × [0 + 4 × (3 − 0) + 0] = 24.
- h = [0 + 16 × 3 + 0] ÷ 24 = 2.
- k = [0 + 0 + 9 × 4] ÷ 24 = 1.5.
- r = √(2² + 1.5²) = 2.5.
The equation is (x − 2)² + (y − 1.5)² = 6.25, or x² + y² − 4x − 3y = 0 in general form. The circumference is 15.708 and the area 19.635.
The center is the midpoint of the hypotenuse, as Thales’ theorem predicts for any right triangle: the hypotenuse is a diameter of the circumscribed circle.
Uses
- Surveying and road design: fitting a curve through three surveyed points.
- Machining and inspection: finding the center and radius of a hole or a worn shaft from three probe points.
- Computer graphics: drawing an arc through three clicked points.
- Triangles: the circumradius links the sides and area: R = abc ÷ (4 × area).