Translation: distance-preserving, angle-preserving; corresponding segments are parallel and equal

Dependencies: Ruler axiom, Protractor axiom, Parallelogram tests.

Statement

Let $\mathbf v$ be a fixed vector in the plane. The translation $T_{\mathbf v}$ sends each point $P$ to $P'$ with

Then for any two points $A$, $B$ in the plane, with images $A' = T_{\mathbf v}(A)$, $B' = T_{\mathbf v}(B)$:

• Distance-preserving: $|A'B'| = |AB|$;
• Parallel + equal: segments $AB$ and $A'B'$ are parallel and of equal length (in the degenerate collinear case, they are interpreted as two collinear segments pointing in the same direction);
• Angle-preserving: for any three points $A$, $B$, $C$ with images $A'$, $B'$, $C'$, $$\angle A'B'C' = \angle ABC.$$

Globally, $T_{\mathbf v}$ is a rigid motion — it "slides" each figure to a new location without rotating, flipping, or stretching it.