## Friday, July 17, 2015

### Unizor - Geometry3D - Central Symmetry

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Recall that on a plane we had two types of symmetry - symmetry relative to a point (central symmetry) and symmetry relative to a straight line (symmetry about an axis or reflection).

Here we will discuss the central symmetry.

On a plane point A', centrally symmetrical to point A relative to a center of symmetry P, can be constructed by connecting A and P by a segment and extending this segment beyond point P (center of symmetry) by the same length, thus obtaining point A'.

Central symmetry in three-dimensional space requires analogous construction.
If we are given point A and a center of symmetry P, then point A', symmetrical to point A relatively to center P, is located on a continuation of a segment AP beyond point P by the same distance as the length of segment AP.

The construction of a symmetrical point is, obviously, reversible. If we start from point A', connect it to the center of symmetry P and extend by the same length, we will be at point A. So, if point A'
is centrally symmetrical to point A relatively to center P, point A is symmetrical to point A' relatively to the same center of symmetry P.

Theorem 1
A centrally symmetrical counterpart of a straight line is a straight line parallel to the original one. Original line and its symmetrical counterpart are equidistant from the center of symmetry.

Theorem 2
A centrally symmetrical counterpart of a plane is a plane parallel to the original one. Original plane and its symmetrical counterpart are equidistant from the center of symmetry.