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I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated.

In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the orthogonal matrix must have an Eigen Value of +1 for a proper rotation.

1. My question is how does this proves the theorem ?

I have understood the logic behind the +1 eigen value, but could not able to find any equivalence between 'Any Displacement' and 'The rotation'.

2. Another one is that, why should we need three Euler angles for the orientation of a body because from Euler's theorem it can be obtained by only one rotation about some axis ?

Thanks