Can you place a Work Point on the line at the point required. If so, then use the Work-Point to constain to. Or extrude the line as a surface and costrain to it. If you use the constraint distance to match the radius it will match.
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I couldn't figure out how to extrude the line as a surface, but I did a workaround and created the curved line as a part instead. The surface of this part was able to create the constrain, as you described. Worked perfectly!
Now I just have to make the part invisible, and I am good to go.