Elementary algebraic geometry?

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I'm not an algebraic geometer, so apologies in advance if this is a trivial question. I'm currently exploring some (Euclidean) geometry relating to various lines and circles, and their intersections. Here is one of the sorts of questions I'm looking at: given a fixed point (u,v) and a fixed circle C of radius R with centre (0,0), then given a general point (x,y) on C, find another point (X,Y) on C so that the line between (x,y) and (X,Y) is at distance d from (u,v). In other words, I'd like to express (X,Y) as a function of (x,y) in the nicest possible way. I can set up a system of equations and attempt to solve them, but this is slow and produces hugely messy results.

I was wondering if there's a more natural, or better way, in Sage.