Editing Euler's rotation theorem (section)

====Final notes about the construction====
[[Image:Eulerrotation.svg|200px|left|thumb|Euler's original drawing where ABC is the blue circle and ACc is the red circle]]

Euler also points out that {{math|'''O'''}} can be found by intersecting the perpendicular bisector of {{math|'''Aa'''}} with the angle bisector of {{math|∠'''αAa'''}}, a construction that might be easier in practice. He also proposed the intersection of two planes:

*the symmetry plane of the angle {{math|∠'''αAa'''}} (which passes through the center {{math|'''C'''}} of the sphere), and
*the symmetry plane of the arc {{math|'''Aa'''}} (which also passes through {{math|'''C'''}}).

:'''Proposition'''. These two planes intersect in a diameter. This diameter is the one we are looking for.

:'''Proof'''. Let us call {{math|'''O'''}} either of the endpoints (there are two) of this diameter over the sphere surface. Since {{math|'''αA'''}} is mapped on {{math|'''Aa'''}} and the triangles have the same angles, it follows that the triangle {{math|'''OαA'''}} is transported onto the triangle {{math|'''OAa'''}}. Therefore the point {{math|'''O'''}} has to remain fixed under the movement.

:'''Corollaries'''. This also shows that the rotation of the sphere can be seen as two consecutive reflections about the two planes described above. Points in a mirror plane are invariant under reflection, and hence the points on their intersection (a line: the axis of rotation) are invariant under both the reflections, and hence under the rotation.

Another simple way to find the rotation axis is by considering the plane on which the points {{math|'''α'''}}, {{math|'''A'''}}, {{math|'''a'''}} lie. The rotation axis is obviously orthogonal to this plane, and passes through the center {{math|'''C'''}} of the sphere.

Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis.