Finding the equation of a circle from chords

The  circumcentre can be used to find the equation of a circle.

1. Find three points on the circle.
2. Draw chords connecting these points.
3. Find the midpoints of each chord.
4. Find the equation of the perpendicular bisectors.
5. Find the point of intersection of these three lines.
6. The point of intersection is the centre of the circle.
   Use this in the relevant circle formula.

 

Example

Find the equation of the circle which passes through the points
A(-6,-4) , B(-1,-5),  C(-1,1)

Sketch the circle, draw in chords.

 

Using the midpoint formula :

 

Now find the gradients :

and use them to find the perpendicular bisectors:

Now solve the equations to find the centre:

 

Next,find the length of the radius,
using the distance formula with the centre of the circle .

     

Finally, put all together using the equation of a circle with known centre