Draw a parallel line with only a straightedge

Question

I'm trying to draw a line through point P, parallel to given line l, with only a blank ruler (a straightedge of a certain length). I know this is easy with a compass, but I don't know how to do it without anything else. Is this even possible?

Thanks!

Answer 1

It is not possible it appears.

However if allowed to use a rectangle vertices of a ruler (most trulers are rectangular) :) it may be possible, ... by keeping one vertex on P and the diagonally opposite vertex on line L, reversing the ruler inclination, drawing bottom edge lines,finding intersection point Q of lower parallel edges of ruler so that red vertical line PQ is perpendicular to L. Extend if needed.

Now align short edge of rectangular ruler along perpendicular PQ placing a vertex of ruler at P and draw the required red horizontal line along its top edge.

Answer 2

Whether or not its possible depends on what you have available to you. In general, with only a straightedge, it is not possible to make a line through an arbitrary point that is parallel to an arbitrary line. There are special conditions that do make it possible though...

Under the Poncelet-Steiner theorem,

1. Any figures you can construct with a straightedge and compass together you can also construct with a straightedge alone, provided that a circle with its center identified exists in the plane. You just need one circle with its center somewhere on the plane - no compass required. Its like having your compass break on you after you draw your first circle and cant be used again thereafter. All constructions are still possible with straightedge alone.

Variants on this theme also exist that restrict (or generalize?) the above even further. The center of the circle may be substituted for some other sufficient information. For example, instead of the circle center given to you, you can have:

• two concentric circles.
• two distinct circles intersecting in one or two points.
• any other case of two non-intersecting circles, with a centerline point (colinear to their centers) is known.
• any other case of two non-intersecting circles, with a known point on the radical axis.
• other variations exist involving one or two circles and some additional information. You can actually invent a few atypical but creative scenarios.
• any three non-intersecting circles, as it turns out, is sufficient.

From any of these scenarios the center of any or all of the circles can be constructed and the problem reduces to the aforementioned Poncelet-Steiner straightedge-only construction.

Whats more, any of the above can be modified further by eliminating a portion of the circle itself. As it turns out, any full circle is equivalent to any portion of the circle.

• any full circle may be substituted for any arc of that circle, no matter how small the arc, in any of the above theorems or its variants... with the caveat that the intersection points of two intersecting circles are provided if their arcs dont intersect.

Lets eliminate the circle entirely now.

2. If the line you wish to make a parallel of has three points on it, A,M,B, where M is the midpoint between A and B, then you can create a parallel of it.

3. If you have two parallel lines already, you can create a third parallel to them through any arbitrary point.

4. If you have an arbitrary parallelogram anywhere on the plane, you can also create a parallel to any arbitrary line through any arbitrary point.

There may indeed be other tricks and conditions, but these are the ones I am aware of. They are all pretty fun constructions.

The above are all restricted Euclidean constructions, obviously. I emphasize that fact because you mentioned "rulers" rather than just sticking to the traditional straightedges.

If you are broadening the scope to physical objects and tools... Rulers tend to provide two parallels and two perpendiculars right off the bat, plus the capacity to measure length. All of this is immensely powerful and I wont even bother getting into the various options you have.

I am embedding animated GIF files below to demonstrate the constructions of parallels...

If youre given three points on a line, one of which is the midpoint of the other two:

If, however, your line just happens to pass through the center of a circle, the translation into three points is a trivial property of the circle. The parallel is finished by the previous construction:

If your line does not go through the center of a circle then you must construct your three points. This is done by choosing an arbitrary line through the circle center and a parallel is constructed from that. Ultimately the previous two constructions are both used.

But if instead of a circle you are given two parallels and wish to construct a third:

Or if instead of a circle you are given a parallelogram (square in this case). Use the square to construct a second parallel then use your two parallels in the previous construction to get your desired third.