If F is the stereographic projection of a sphere $ S\in S^2 $ to a plane $P$ (here take the plane as passing through the centre of the sphere S) then how can we prove that if $\ell$ is a great circle on S passing through the North Pole of the sphere (so this point is on the sphere such that the distance from the plane to the sphere is maximised) then $F(\ell)$ is a straight line in plane P passing through centre of the sphere.
This is an idea for your use:
For $x\in S^2$, you get $F(x)$ by drawing the line through the north pole and $x$ and finding its intersection with $P$. A great circle $\ell$ is given by intersecting a plane $\Pi$ with $S^2$. In particular, when $\ell$ passes through the north pole, so does $\Pi$ and $F(\ell)$ is precisely $\Pi\cap P$, which is a line in $P$. In fact, this line contains the center of the sphere, because $\ell$ passes through the south pole.