User:AlexxMitch/sandbox

Deﬁnition. $$(P,L)$$ is an origami pair if $$P$$ is a set of points in $$R^2$$ and L is a collection of lines in $$R^2$$ satisfying:

L_1$$ and $$L_2$$ are lines in $$L$$, then the line which is equidistant from $$L_1$$ and $$L_2$$ is in $$L$$.
 * 1) The point of intersection of any two non-parallel lines in $$L$$ is a point in $$P$$.
 * 2) Given any two distinct points in $$P$$, there is a line in $$L$$ going through them.
 * 3) Given any two distinct points in $$P$$, the perpendicular bisector of the line segment with given endpoints is a line in $$L$$.
 * 4) If $$
 * 1) If $$L_1$$ and $$L_2$$ are lines in $$L$$, then there exists a line $$L_3$$ in $$L$$ such that $$L_3$$ is the mirror reﬂection of $$L_2$$ about $$L_1$$.

A subset, $$P \subset R^2$$, is said to be closed under origami constructions, if there exists a collection of lines, $$L$$, such that $$(P,L)$$ is an origami pair.

$$P_0 = \cap \{ P | (0,0),(0,1) \in P$$and$$P$$is closed  under origami constructions$$\}$$ is the set of origami constructible points.

Proof: Given a line $$L$$ and a point $$p$$, pick two points $$p_1$$ and $$p_2$$ on $$L$$. By property 2. in the deﬁnition of an origami pair, we may construct lines $$L_1$$ and $$L_2$$ running through $$p_1$$, $$p$$ and $$p_2$$, $$p$$, respectively. By property 5. we may reﬂect $$L_1$$ and $$L_2$$ through $$L$$ to obtain $$L_3$$ and $$L_4$$. Now the intersection of $$L_3$$ and $$L_4$$ is a constructible point, so there is a line, $$L_5$$ through this point and the given point, $$p$$, by properties 1. and 2. Call the point where $$L_5$$ and $$L$$ intersect $$p_3$$. To ﬁnish the construction, use property 3. to construct a perpendicular bisector to $$p$$, $$p_3$$, and reﬂect $$L$$ through this bisector with property 5. to obtain the desired line, $$L_6$$. It is a straightforward exercise to show that $$L_6$$ has the desired properties.