'Let $P_0,P_1,P_2$ and $P_0′,P_1′,P_2′$ in $\mathbb{E}^2$ be two ordered triples of distinct non-collinear points such that $d(Pi,Pj) = d(Pi′,Pj′)$ for all $i,j$. Prove that there exists a unique motion $T :\mathbb{E}^2\rightarrow\mathbb{E}^2$ taking $Pi$ to $Pi′$ for all $i$.' (We're assuming here that $d$ is the Euclidean metric).
I understand this idea intuitively, but am not sure how to formally prove it. Any hints are ideas on how to start are appreciated.