# Homework Help: Geometry/Topology Isometry question

1. Sep 11, 2010

### kimberu

1. The problem statement, all variables and given/known data
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 $$\varphi$$ so $$\varphi(Q) = Q^{l}$$ and $$\varphi(P) = P^{l}$$

2. Relevant equations
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3. The attempt at a solution
So I know (I think) that there's at least one such isometry because the plane is isotropic, which means that a translation/rotation exists to map these points. But I don't know how to define 2 such $$\varphi$$s, or what the differences are between them.

Thank you so much for any help!