In the figure below, the circle e is the image of the equator by a stereographic projection of the sphere, and the plane Euclidean circles c1 and c2 are images of orthogonal great circles on the sphere.
The circles c1 and c2 have Euclidean centers E1 and E2. These are not the images of the spherical centers, but they are useful in constructions. The point S is the projection of the south pole of the sphere; S is the Euclidean center of e.
Reflection in a great circle on the sphere appears in the map as inversion in the plane circle. Recall that if a circle c1 is orthogonal to a circle c2, then the lines through E2, the center of c2, intersect c1 in pairs of points that are inversion images in c2 of each other in. Thus if we have these orthogonal circles, inversion pairs of points can be constructed by drawing lines with an unmarked straightedge.
In the stereographic map, any line through S intersects a great circle in antipodal points, since a line through E is the image of a great circle through S, and great circles intersect each other in antipodal points.
The two poles of a great circle c1 are the intersection points of any two great circles orthogonal to c1. In this figure, c2 is one such circle. A second such circle is the Euclidean line SE1, since this is the image of a great circle, and any line through the Euclidean center of a circle is orthogonal to the circle.
Ruler Lines Construct Special Points on Orthogonal Great Circles