I have a finite set of points A and another point C. I would like to compute a vector N so that the plane defined by C (point on plane) and N (normal of plane) divides all points in A with the sum of distances of points to the plane on either side of plane is equal.
What is this problem called? What are good algorithm to solve it?
I suggest you to have a look at this similar question.
The existence of such a plane is granted by the Borsuk-Ulam theorem, since the map $\varphi:S^2\to\mathbb{R}$ associating the direction of the normal vector with the difference of the "total distances" in the two half-spaces cut by the plane through $C$ orthogonal to $N$ is a continuous function (almost everywhere $C^{\infty}$) and it is antipodal on $S^2$.
Hence you can find $N$ through any algorithm for finding the zeroes of an almost-everywhere-$C^{\infty}$ function over a compact 2d-set.