Editing Euler's rotation theorem (section)

====Proof of its invariance under the transformation====
Now let us suppose that {{math|'''O′'''}} is the image of {{math|'''O'''}}. Then we know {{math|∠'''αAO''' {{=}} ∠'''AaO′'''}} and orientation is preserved,{{efn|Orientation is preserved in the sense that if {{math|'''αA'''}} is rotated about {{math|'''A'''}} counterclockwise to align with {{math|'''OA'''}}, then {{math|'''Aa'''}} must be rotated about {{math|'''a'''}} counterclockwise to align with {{math|'''O′a'''}}. Likewise if the rotations are clockwise.}} so {{math|'''O′'''}} must be interior to {{math|∠'''αAa'''}}. Now {{math|'''AO'''}} is transformed to {{math|'''aO′'''}}, so {{math|'''AO''' {{=}} '''aO′'''}}. Since {{math|'''AO'''}} is also the same length as {{math|'''aO'''}}, then {{math|'''aO''' {{=}} '''aO′'''}} and {{math|∠'''AaO''' {{=}} ∠'''aAO'''}}. But {{math|∠'''αAO''' {{=}} ∠'''aAO'''}}, so {{math|∠'''αAO''' {{=}} ∠'''AaO'''}} and {{math|∠'''AaO''' {{=}} ∠'''AaO′'''}}. Therefore {{math|'''O′'''}} is the same point as {{math|'''O'''}}. In other words, {{math|'''O'''}} is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing {{math|'''O'''}} is the axis of rotation.

{{clear}}