I'm looking for an argument that the $n$-dimensional stereographic projection maps circles (intersections of affine two-dimensional subspaces with $\mathbb S^n$) to circles in $\mathbb R^n$. I've looked around and the only argument I saw for the n-dimensional case is a generalization of the geometric proof for $n = 2$ (with the tangent cone) which I don't really feel comfortable with, even when $n = 2$. Is it possible to reduce it to the $n = 2$ case somehow or give a "direct", algebraic, proof?