Point sym — Point symmetry · Principia

Dependencies: Parallelogram properties (opposite sides equal, diagonals bisect each other) (Parallelogram: diagonals bisect each other, F.01), Parallelogram tests (F.04, especially (d) Parallelogram: diagonals bisect each other ⇒ parallelogram).

Statement

Let $O$ be a fixed point. Central symmetry about $O$ (also called point symmetry) is the map of the plane to itself

\sigma_O:\;P\;\longmapsto\;P',\qquad O = \mathrm{midpoint}(PP').

Equivalently, $\sigma_O$ is rotation by $180^{\circ}$ about $O$ . Its geometric property is: for any two points $A$ and $B$ and their images $A' = \sigma_O(A)$ , $B' = \sigma_O(B)$ ,

|AB| = |A'B'|,\qquad AB \parallel A'B'.

$Central symmetry: O is simultaneously the midpoint of the three corresponding segments AA', BB', CC'; \triangle ABC and \triangle A'B'C' are central-symmetric images of each other about O.$

Point symmetry

Statement

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