Collapsing Compass to Rigid Compass
- Brad Ballinger
Using only a collapsing compass (circle with center through point) and not a rigid compass (circle with center and radius), can you construct a circle about C congruent to the circle about A, through B? Please spend as long as you like working on this problem before you scroll down for spoilers.
To read more about Euclid's solution, see David Joyce's free online presentation of Euclid's Elements. This is Book I Proposition 2: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI2.html. In the diagram above, I've emphasized different parts of the construction than Joyce/Euclid did, but I've planned all of my shown labels to mimic some of Joyce's in order to minimize distractions.
Brad's Solution: straightedge not needed.
I drew radii in the figure above for emphasis only, not because they were necessary for constructing the two circles. The idea of doing constructions with a compass alone (no straightedge) has some history to it as well. For example,
- Read about Mohr-Mascheroni constructions at cut-the-knot.org.
- Read about the Compass Equivalence Theorem at wikipedia. The construction in that article is the same as mine, except for labels.