MathDB

Given a figure

B

in the plane, consider the figures

A

, containing

\mathcal B

, with the property

(P)

: a composition of an odd number of central symmetries with centers in

A

is also a central symmetry with center in

A

. The task of this problem is to determine the smallest such figure, denoted by

\mathcal A

, that is contained in every figure

A

.
(a) Determine the figure

\mathcal A

B

consists of:

(1)

two distinct points

I,J

;

(2)

three non-collinear points

I,J,K

. (b) Determine

\mathcal A

B

is a circle (with nonzero radius). (c) Give some examples of figures

B

whose associated figures

\mathcal A

are mutually distinct and distinct from the above ones.

Problem Statement