Editing Constructible number (section)

===Geometrically constructible points===
Let <math>O</math> and <math>A</math> be two given distinct points in the [[plane (geometry)|Euclidean plane]], and define <math>S</math> to be the set of points that can be constructed with compass and straightedge starting with <math>O</math> and <math>A</math>. Then the points of <math>S</math> are called '''constructible points'''. <math>O</math> and <math>A</math> are, by definition, elements of <math>S</math>. To more precisely describe the remaining elements of <math>S</math>, make the following two definitions:{{sfnp|Kazarinoff|2003|p=10}}
* a line segment whose endpoints are in <math>S</math> is called a '''constructed segment''', and
* a circle whose center is in <math>S</math> and which passes through a point of <math>S</math> (alternatively, whose radius is the distance between some pair of distinct points of <math>S</math>) is called a '''constructed circle'''.
Then, the points of <math>S</math>, besides <math>O</math> and <math>A</math> are:{{sfnmp|Kazarinoff|2003|1p=10|Martin|1998|2pp=30–31|2loc=Definition 2.1}}
* the [[Line-line intersection|intersection]] of two non-parallel constructed segments, or lines through constructed segments,
* the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or
* the intersection points of two distinct constructed circles.

As an example, the midpoint of constructed segment <math>OA</math> is a constructible point. One construction for it is to construct two circles with <math>OA</math> as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment <math>OA</math> is the point where this segment is crossed by the constructed line.<ref>This construction for the midpoint is given in Book I, Proposition 10 of [[Euclid's Elements|Euclid's ''Elements'']].</ref>