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A constructible circle has a constructible point as its centre and radius equal to the distance between two constructible points. When those coordinates lie in a real field $F$, its equation $(x-a{)}^2+(y-b{)}^2=r^2$ has coefficients in $F$ because the squared radius is a sum of squared coordinate differences. Intersecting it with a constructible line reduces to a quadratic equation. Intersecting two distinct circles reduces to the same situation by subtracting their normalized equations to obtain the radical axis. Thus each isolated real intersection requires at most a real quadratic field extension.

A constructible line is a line through two distinct constructible points. If the available coordinates lie in a real field $F$, a line through $(a,b)$ and $(c,d)$ has equation $(d-b)(x-a)-(c-a)(y-b)=0$, with coefficients in $F$ and a nonzero normal vector. Intersecting two distinct such lines introduces no new field elements, because solving a nonsingular pair of linear equations uses only field operations.

Start with $O=(0,0)$ and $E=(1,0)$. A constructible point is a point obtained from these by finitely many ruler-and-compass steps. At each step one may draw the line through two previously obtained distinct points, or the circle with a previously obtained centre and radius equal to the distance between two previously obtained points, and take isolated intersection points of the drawn lines or circles. A constructible line is a line through two distinct constructible points; a constructible circle has constructible centre and a radius obtainable as such a distance. These conventions also permit copying a previously obtained segment with the compass.

Finite construction from a fixed unit segment is essential: a drawing through an arbitrary unspecified real point does not make that point constructible.