Main problem

We now know that given three distinct points in the plane, there is (usually) a unique circle passing through them.

So if we are given three points, we would like to find the equation of the circle that passes through them. How many ways can you find to do this? There are suggestions available in the “Suggestion” section.

Here are some examples for you to try your ideas out on. For each set of three points, find the equation of the circle passing through them.

It is certainly worth sketching a graph to help you understand what is going on in each case!

Which of your approaches is the most effective?

  1. \(A(3,2)\), \(B(3,6)\), \(C(5,8)\)

  2. \(A(0,4)\), \(B(4,0)\), \(C(-6,0)\)

  3. \(A(-1,-5)\), \(B(-2,2)\), \(C(2,-1)\)

  4. \(A(3,3)\), \(B(1,-2)\), \(C(4,1)\)

If you want more practice, there is a Mathmo exercise for this.

If we join the three points up to form a triangle, then the circle which passes through the three points – the three vertices of the triangle – is known as the circumcircle of the triangle. The centre of this circle is known as the circumcentre of the triangle.