I'd like to show the following statement:
Let $V$ be a euclidean vector space with finite dimension, $B(V) := \{\phi: V \rightarrow V | \phi$ is an isometry$\}$, $\phi \in B(V)$. Assume there exists a $\psi \in B(V)$ s.t. $\psi^{-1} \circ \phi \circ \psi \in O(V)$, where $O(V)$ is the orthogonal group. Show that $\phi$ has a fixed point.
Any hints?