Rotation — preserves distances and angles, and corresponding points are equidistant from the centre

Dependencies: Ruler axiom, Protractor axiom, SAS congruence.

Statement

Let $O$ be a fixed point in the plane, and let $\theta\in[0,\,2\pi)$ be a fixed angle. The rotation about centre $O$ by angle $\theta$, $R_{O,\theta}$, sends each point $P$ to the point $P'$ such that

A rotation has three properties:

1. Distance-preserving: any two points $A$, $B$ and their images $A'=R_{O,\theta}(A)$, $B'=R_{O,\theta}(B)$ satisfy $|A'B'| = |AB|$;
2. Angle-preserving: any three points $A$, $B$, $C$ and their images $A'$, $B'$, $C'$ satisfy $\angle ABC = \angle A'B'C'$;
3. Equidistant from the centre: every point keeps its distance to the centre $O$, i.e. $|OP'|=|OP|$ (this is given directly by the definition).