Composition of the three transformations = isometry

Dependencies: Perpendicular bisector ⇔ equidistant from endpoints (perpendicular bisector ⇔ equidistance), Translation properties, Rotation properties, Reflection properties, Point symmetry (the four "properties" theorems K.01–K.04).

Statement

Let $T:\mathbb{R}^2 \to \mathbb{R}^2$ be a map of the plane. If for every pair of points $P$, $Q$,

then $T$ is called an isometry (a distance-preserving map, a rigid motion). This theorem ties the abstract notion of "isometry" to the four concrete, constructible basic transformations in a single sentence:

Translation, rotation, reflection (and any finite composition of these) = the planar isometries; every planar isometry can be written as the composition of at most $3$ reflections.

In particular, the isometry group $\mathrm{Isom}(\mathbb{R}^2)$ is generated by all reflections, and every element is a product of at most three reflections.