# For compass and straightedge problems, are you allowed to use the compass as a ruler?

For compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE.

If you placed the two points of the compass between A and B, and made a circle around C with the same radius, that would achieve this result.

But is this something you are allowed to do?

• Typically, constructions are described as being done with a "collapsible compass" or a "rusty (fixed) compass". With the former, it's not possible to transfer distances in the way you describe; with the latter, it is possible. Luckily, the "Compass Equivalence Theorem" says that whatever is constructible with one type of compass is constructible with the other. – Blue Apr 11 '18 at 8:09
• – Ethan Bolker Apr 11 '18 at 15:56
• @Blue: Actually, I don't think you even need a compass if a segment of a circle is drawn somewhere on the plane, and you can if and where that segment of the circle intersects a line. From what I recall, given such a segment and a line, one can determine if and where the line would intersect the whole circle. From there, one can determine whether and where a line defined by two arbitrary points would intersect a circle defined by two arbitrary points. – supercat Apr 11 '18 at 17:24
• Obligatory XKCD: xkcd.com/866 – Sam Weaver Apr 11 '18 at 19:54

## 2 Answers

Yes. Not by the rules about how to use compass and straightedge but because it can be proved that it's as if we could do it (that's proposition 2 of book I of Euclid's Elements).

Putting aside the specific construction of this query: The general rule in strictly classical Euclidean constructions is: The instruments cannot carry distance information from one step to another. So, in particular: (a) The straightedge cannot be marked, not even at a single point; (b) The compass does retain its spacing while drawing any one circle, but forgets the spacing once the compass is 'lifted off of the plane'. If you want to replicate a distance at a specific locale, you generally need auxiliary constructions (e.g parallel lines through a segment's endpoints).