Similarity Mapping on Plane Commutes with Half Turn about Origin

Theorem
Let $\beta \in \R_{>0}$ be a (strictly) positive real number.

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

Then $s_{-\beta}$ is the same as:
 * a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn

and:
 * a rotation one half turn followed by a stretching or contraction of scale factor $\beta$.

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

From Similarity Mapping on Plane with Negative Parameter, $s_{-\beta}$ is a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn.

Thus:

That is:
 * $s_\beta$ is a rotation one half turn followed by a stretching or contraction of scale factor $\beta$.