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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?
Finding Circles
Age 16 to 18
Challenge Level 
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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