# Collapsing Compass to Rigid Compass

- Author:
- Brad Ballinger

- Topic:
- Congruence, Constructions

## Task

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.

Spoilers below.

## Euclid's Solution

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.