Prove that if $P$ is a fixed point for an isometry $F$ then $P$ is also fixed for its inverse.

Question

I think this proof should be easy but I am having some troubles with it. I am approaching the problem by considering various cases: $F$ as a rotation $G$; $F$ as a reflection $R$; $F$ as a composite of rotation and reflection $G º R$ and $F$ as the identity $I$. From here I am a bit stuck on how to prove for each case that if $F(P)=P$ then $P$ is also fixed for the inverse. I was thinking about analyzing each case for example for $F$ as a rotation $G$, for $P$ to be fixed it has to be either rotation by 0 or 360 and its multiples or rotation from $P$. In all these cases its inverse would also leave $P$ fixed, then I do something similar for the other cases. I am new to proofs and this seems like an overly complicated and maybe even wrong way to approach this, what do you guys think?