Compass equivalence theorem

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The compass equivalence theorem is a statement of compass and straightedge constructions that whenever the compass is lifted from the page, it is assumed to collapse, so may not be directly used to transfer distances.

This might seem a difficult obstacle to surmount, but it is not. Any construction via a "fixed" compass may be attained with a collapsing compass. It is possible to construct a circle of equal radius to any other circle, at any point; see references.