Equation for Circle in 3D Space Given Center, Radius, and Point

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I'm looking for how to derive the equation of a circle, in 3D space, given the following information:

  • The Center Point
  • The Radius
  • One point on the circle

I understand that this is functionally the intersection of a plane (x+y+z=d) and a sphere (x^2+y^2+z^2=r), but I'm at a loss how to combine those into a single equation for graphing purposes.

Thanks!