Exercise 3

The proof that a map that is distance-preserving and sends the zero vector to itself incidentally shows that such a map is one-to-one and onto (the point in the domain determined by $d_{0}$, $d_{1}$, and $d_{2}$ corresponds to the point in the codomain determined by those three). Therefore any distance-preserving map has an inverse. Show that the inverse is also distance-preserving.

Prove that congruence is an equivalence relation between plane figures.