Editing Euler's rotation theorem (section)

====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 {{math|'''A'''}} be a point of intersection of those circles. If {{math|'''A'''}}’s image under the transformation is the same point then {{math|'''A'''}} is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing {{math|'''A'''}} is the axis of rotation and the theorem is proved.

Otherwise we label {{math|'''A'''}}’s image as {{math|'''a'''}} and its preimage as {{math|'''α'''}}, and connect these two points to {{math|'''A'''}} with arcs {{math|'''αA'''}} and {{math|'''Aa'''}}. These arcs have the same length. Construct the great circle that bisects {{math|∠'''αAa'''}} and locate point {{math|'''O'''}} on that great circle so that arcs {{math|'''AO'''}} and {{math|'''aO'''}} have the same length, and call the region of the sphere containing {{math|'''O'''}} and bounded by the blue and red great circles the interior of {{math|∠'''αAa'''}}. (That is, the yellow region in '''Figure 3'''.) Then since {{math|'''αA''' {{=}} '''Aa'''}} and {{math|'''O'''}} is on the bisector of {{math|∠'''αAa'''}}, we also have {{math|'''αO''' {{=}} '''aO'''}}.