page_title: Compass equivalence theorem
section_title: Alternative construction without straightedge
hierarchical_section_title: Compass equivalence theorem / Alternative construction without straightedge
caption_attribution_description: English: A proof of the equivalence of fixed and collapsible compasses, without using a straightedge.
context_page_description: The compass equivalence theorem is an important statement in compass and straightedge constructions. 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. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.
context_section_description: It is possible to prove compass equivalence without the use of the straightedge.
  This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.
  Given points A, B, and C, construct a circle centered at A with the radius BC, using only a collapsing compass and no straightedge.
  Draw a circle centered at A and passing through B and vice versa (the blue circles). They will intersect at points D and D'.
  Draw circles through C with centers at D and D' (the red circles). Label their other intersection E.
  Draw a circle (the green circle) with center A passing through E. This is the required circle.
  There are several proofs of the correctness of this construction and it is often left as an exercise for the reader. Here is a modern one using transformations.
  The line DD' is the perpendicular bisector of AB.  Thus A is the reflection of B through line DD'.
  By construction, E is the reflection of C through line DD'.
  Since reflection is an isometry, it follows that AE = BC as desired.
caption: Construction without using straightedge
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