Chapter 4

Prove that the composition of two isometries of the complex plane is an isometry.

An isometry of the complex plane has two fixed points \(A\) and \(B\). Prove that any point \(M\) of line \(A B\) is a fixed point of the transformation.

Prove that the mapping \(g: \mathbb{C} \rightarrow \mathbb{C}, g(z)=-i z+1+2 i\) is an isometry. Analyze \(g\) as in problem \(4 .\)