### Circles

Many ways problem

## Suggestion

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?

#### An algebraic approach

Might the general equation of a circle be of use here?

#### A geometric approach

If we have two (distinct) points, then there are many circles that pass through both points. Can you say anything about the centres of these circles? Might that help us when we know a third point on the circle?

This GeoGebra applet might be helpful to explore this. The two points $A$ and $B$ (in red) are fixed, while the third point $C$ (in blue) can be moved. The centre of the circle through these three points, labelled $O$, can be seen, and leaves a trace as you move $C$.

Can you explain the behaviour you have observed? And how can this help us to find the equation of the circle when we know a third point on the circumference?