1. ## Euclidean Isometries

"
Let P, Q, P' and Q' be 4 points in the euclidean plane where distance(P,Q) = distance(P', Q') and the distance is not 0. Show there are precisely 2 isometries $\displaystyle \phi$ so $\displaystyle \phi(P) = P'$ and $\displaystyle \phi(Q) = Q'$"

I think that there's at least one such isometry because the euclidean plane is isotropic, which means that a translation/rotation exists to map these points. But I don't know how to define 2 such phis, or what the differences are between them.

Thank you so much for any help!

2. with an extra reflection on the first one you get the second one.
So one is preserving orientation but the other is not.