Definition 1.1.1.Let $\sum$ be a set of points in the plane $\mathbf{R}^2$. One says that a point $P$ is constructible with ruler and compass from $\sum$ if there is an integer $n$ and a sequence of points $(P_1,\cdots,P_n)$ with $P_n=P$ and such that for any $i\in\{1,\cdots,n\}$, denoting $\sum_i=\sum\bigcup\{P_1,\cdots,P_{i-1}\}$, one of the following holds:

1.there are four points $A, B, A'$ and $B'\in\sum_i$ such that $P_i$ is the intersection point of the two nonparallel lines $(AB)$ and $(A'B')$;

2.there are four points $A, B, C$, and $D\in\sum_i$ such that $P_i$ is one of the (at most) two intersection points of the line $(AB)$ and the circle with center $C$ and radius $CD$;

3.there are four points $O, M , O'$ and $M'\in\sum_i$ such that $P_i$ is one of the (at most) two intersection points of the distinct circles with, respectively, center $O$ and radius $OM$ ,and center $O$ radius $O'M'$ .

Remark 1. When $i<j$,$\sum_i\subseteq \sum_j$.Once $P$ is constructed,$P$ itself can be added into the "existing point set".And you start from this existing set,to creat even newer existing set.......

Remark 2.If $P$ is constructible from $\sum$,then you can get $P$ by using compass and ruler finitely many times,because $n$ is a natural number,a natural number is finite.

 

Remark 3.However,there is a  minor flaw in this definition:1.The definition fail the case of $i=1$.