I am trying to prove the statement:
Reflection in a spherical line (i.e. the image of a great circle under stereographic projection) in the extended complex plane corresponds (via stereographic projection) to Euclidean reflection in the plane that cuts out the associated great circle on the unit sphere.
I have tried drawing it out but my diagrams start to get way too complicated. Is there a neat way to prove this? I am really not sure where to start.