Compass equivalence theorem

The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances.

The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument.

This theorem is Proposition II of Book I of Euclid's Elements.

Given points A, B, and C, construct a circle centered at A with the radius BC, using only a collapsing compass and no straightedge.

There are several proofs of the correctness of this construction and it is often left as an exercise for the reader.