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Given three complex values (for example, $2i, 4, i+3$), how would you calculate the equation of the circle that contains those three points? I know it has something to do with the cross ratio of the three points and $z$ and the fact that the cross ratio is a real number, but I don't know what to do with that information.

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    Do you know what a [Möbius transformation](http://en.wikipedia.org/wiki/Mobius_transformation) is? If so, the inverse of the Möbius transformation given by the cross-ratio gives you a *parametrization* of the circle (up to one among $2i$, $4$, $i+3$ depending on your exact formula of the cross ratio) by the real numbers.2011-05-15
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    If you need to verify the result you get from Möbius, you can use the usual determinant for the Cartesian equation of a circle through three points (treating the complex plane as a Cartesian plane) and check that the parametrization you obtain (by replacing $x$ and $y$ with appropriate expressions) satisfies that Cartesian expression.2011-05-15
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    "the inverse of the Möbius transformation given by the cross-ratio" what do you mean by given by?2011-05-15
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    The cross-ratio is a special Möbius transformation (the unique one that maps $z_1, z_2, z_3$ to $1,0,\infty$, say - again depending on your conventions - it might be $0,1,\infty$ as well).2011-05-15
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    Googling for `circle three points` turned up [this link](http://paulbourke.net/geometry/circlefrom3/) as a first hit, which may help, too.2011-05-15
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    just take perpendicular bisectors of any two sides of the triangle..the point of intersection of two bisectors is the center..now find the distance between center and one of the vertices..it is the radius.. done.2011-05-15

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