MathDB

Let

A

be an infinite set of points in the plane such that inside each circle there are only a finite number of points of

A

, with the following properties:

\bullet

(0, 0)

belongs to

A

\bullet

(a, b)

and

(c, d)

belong to

A

, then

(a-c, b-d)

belongs to

A

\bullet

There is a value of

\alpha

such that by rotating the set

A

with center at

(0, 0)

and angle

\alpha

, the set

A

is obtained again. Prove that

\alpha

must be equal to

\pm 60^{\circ}

\pm 90^{\circ}

\pm 120^{\circ}

\pm 180^{\circ}

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