Baltic Way 2011 · Shortlist problem

No, it is not: for suppose you have an algorithm that constructs the centre of the given circle $\gamma$ such that each step of the algorithm consists in choosing three points and constructing their circumcircle. Consider some arbitrary circle $\Omega$ different from $\gamma$. Invert $\gamma$ in $\Omega$ to get a new circle $\gamma'$. If we apply our algorithm to $\gamma'$, then on one the hand we construct the centre of $\gamma'$, and on the other hand we construct the image of the centre of $\gamma$ under the inversion, since every step of the algorithm commutes with the inversion. Thus the centre of $\gamma$ is mapped to the centre of $\gamma'$ by the inversion, yielding a contradiction since $\Omega$ was arbitrary and inversion does not in general map the centre of a circle to the centre of its image.