Compass equivalence theorem

The compass equivalence theorem is an important statement in compass and straightedge constructions. In these constructions it is assumed that whenever a compass is lifted from a page, it collapses, so that it may not be directly used to transfer distances. While this might seem a difficult obstacle to surmount, the compass equivalence theorem states that any construction via a "fixed" compass may be attained with a collapsing compass. In other words, it is possible to construct a circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements.

We are given points A, B, and C, and wish to construct a circle centered at A with the same radius as BC (the first green circle).

It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.

We are given points A, B, and C, and wish to construct a circle centered at A with the same radius as BC, using only a collapsing compass and no straightedge.

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