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Geometrically constructible points

Let O {\displaystyle O} and A {\displaystyle A} be two given distinct points in the Euclidean plane, and define S {\displaystyle S} to be the set of points that can be constructed with compass and straightedge starting with O {\displaystyle O} and A {\displaystyle A}. Then the points of S {\displaystyle S} are called constructible points. O {\displaystyle O} and A {\displaystyle A} are, by definition, elements of S {\displaystyle S}. To more precisely describe the remaining elements of S {\displaystyle S}, make the following two definitions:

Then, the points of S {\displaystyle S}, besides O {\displaystyle O} and A {\displaystyle A} are:

As an example, the midpoint of constructed segment O A {\displaystyle OA} is a constructible point. One construction for it is to construct two circles with O A {\displaystyle OA} as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment O A {\displaystyle OA} is the point where this segment is crossed by the constructed line.