Ruler with two parallel edges
The constructions with a straightedge
alone do not allow much.
Except if we add something else : For instance some device to draw parallel lines etc.
To draw pairs of parallel lines, we may use the two parallel edges of a ruler, with width d.
The "axioms" are then :
- [C1] We may draw a line through two known points.
- [C2] We may draw a parallel line at distance d from a known line,
d being the width of the ruler.
- [C3] We may draw, through two known points A and B,
a line through A and a parallel line through B at distance d, if AB > d.
- Get new points by intersections of such together, or with known lines.
However we also allow to choose arbitrary points, if the final result doesn't depend on the exact choice of these points.
(for constructions including "draw any line" etc)
By using just these constructions [C1] [C2] [C3] above, to construct :
- A parallel through any given point to any given line
- A perpendicular through any given point to any given line
Can we discard axiom [C3] for these constructions ?
More complicated : to construct an equilateral triangle ? an hexagon ? octogon ? dodecagon ?
A regular pentagon ? decagon ? etc ?
Same question : can we discard axiom [C3] ?