Hi, I have a circle in 3D (common part for x^2+y^2+z^2<=R^2 and A*x+B*y+C*z=0). How to find random and uniformly distributed point in this circle? Thanks.

For the random part note that on a 2D surface (which is what you have) you can only vary 2 of the parameters independently. Vary those two with random perturbations and then compute the third such that your point is on the sphere. To generate a uniform distribution you may find it easier to write your expression in spherical coordinates since they provide a more natural description of the surface.