Similarity Mapping on Plane with Scale Factor Minus 1

Theorem
Let $s_{-1}: \R^2 \to \R^2$ be a similarity mapping on $\R^2$ whose scale factor is $-1$.

Then $s_{-1}$ is the same as the rotation $r_\pi$ of the plane about the origin one half turn.

Proof
Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.

Then: