as point . (If the circles coincide, then can be taken as any point on either; otherwise is one of the two points of intersection.) Now is on the initial circle (the blue circle), so its image will be on the transported circle (red). He labels that image as point . Since is also on the transported circle (red), it is the image of another point that was on the initial circle (blue) and he labels that preimage as (see Figure 2). Then he considers the two arcs joining and to . These arcs have the same length because arc is mapped onto arc . Also, since is a fixed point, triangle is mapped onto triangle , so these triangles are isosceles, and arc bisects angle . Construction of the best candidate point Let us construct a point that could be invariant using the previous considerations. We start with the blue great circle and its image under the transformation, which is the red great circle as in the Figure 1. Let point be a point of intersection of those circles. If ’s image under the transformation is the same point then is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing is the axis of rotation and the theorem is proved. Otherwise we label ’s image as and its preimage as , and connect these two points to with arcs and . These arcs have the same length. Construct the great circle that bisects and locate point on that great circle so that arcs and have the same length, and call the region of the sphere containing and bounded by the blue and red great circles the interior of . (That is, the yellow region in Figure 3.) Then since and is on the bisector of , we also have . Proof of its invariance under the transformation Now let us suppose that is the image of . Then we know and orientation is preserved, so must be interior to . Now is transformed to , so . Since is also the same length as , . But , so and therefore is the same point as . In other words, is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing is the axis of rotation.