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Is there any elegant way of solving this system of equations?

\begin{gather*} a_1 x^2+a_2 x+ a_3 y^2+a_4 y+a_5 z^2+a_6 z+a_7=0 \\
b_1 x^2+b_2 x+b_3 y^2+b_4 y+b_5 z^2+b_6 z+b_7=0\\
c_1 x^2+c_2 x+c_3 y^2+c_4 y+c_5 z^2+c_6 z+c_7=0 \end{gather*}

Still, any solution will be appreciable.

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    Hmm... intersecting ellipsoids? Where did your nonlinear system come from?2011-04-20
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    @J.M. first i would like to know how to put `a1`, secondly, it comes from circle in 3-d ... generalized to polynomials.. lets say a first step of the problem towards solution.2011-04-20
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    Could you elaborate on the application where it came up? We can either bash at the problem with Yuval's suggestion, or study what led you to your equation set and probably tease out a way to solve this without heavy machinery...2011-04-20
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    @J.M. well, currently comes from http://stackoverflow.com/questions/5712963/find-circum-center-of-three-point-of-triangle-not-using-compass/5713169#5713169 but still,it's an old problem of mine. .... still i would be more happy to solve it for myself than help him.2011-04-20
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    So you want the circumcenter of a tetrahedron? That is a much easier problem than what you've presented. It certainly won't need Gröbner.2011-04-20
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    @J.M. well it guess it is, myself have formulated few steps, but ... still this one is out of curiosity.2011-04-20

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Plug it in Wolfram Alpha.

More seriously, Wolfram Alpha will use Groebner bases to find all solutions to any polynomial system of equations.

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    can you help me with inputing on wolfam alpha. just generaize [this](stackoverflow.com/questions/5739777/sphere-that-surely-encompass-given-list-of-points-points-are-with-x-y-and-z-co/5739859#5739859)2011-04-21
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    sorry url can be found [here](http://www.wolframalpha.com/input/?i=%20%28x-5%29^2%2b%28y-8%29^2%2b%28z-1%29^2%20=%20r^2,%20%28x-2%29^2%2b%28y-0%29^2%2b%28z-4%29^2%20=%20r^2,%20%28x-3%29^2%2b%28y%2b3%29^2%2b%28z%2b1%29^2%20=%20r^2)2011-04-21