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# Finding Circles

#### An algebraic approach

We want to find an equation of the form $(x-a)^2 + (y-b)^2 = r^2$, where $(a,b)$ is the (unknown) centre of the circle and $r$ is the (unknown) radius. We have $3$ points that lie on the circle, so we

can use these to get some simultaneous equations...

A slightly more geometric way to think about this approach is that we know that the centre, say $(a,b)$, is at an equal distance from all points. So we could write down the distances from $(a,b)$ to our known points, and then equate those...

#### 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?

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Age 16 to 18

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We want to find an equation of the form $(x-a)^2 + (y-b)^2 = r^2$, where $(a,b)$ is the (unknown) centre of the circle and $r$ is the (unknown) radius. We have $3$ points that lie on the circle, so we

can use these to get some simultaneous equations...

A slightly more geometric way to think about this approach is that we know that the centre, say $(a,b)$, is at an equal distance from all points. So we could write down the distances from $(a,b)$ to our known points, and then equate those...

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?