Compass equivalence theorem

In geometry, 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.

Construction

File:Euclid2.svg

Diagram for proof of Euclid I.2

The following construction and proof of correctness are given by Euclid in his Elements. Although there appear to be several cases in Euclid's treatment, depending upon choices made when interpreting ambiguous instructions, they all lead to the same conclusion, and so, specific choices are given below.

Given points Template:Mvar, Template:Mvar, and Template:Mvar, construct a circle centered at Template:Mvar with radius the length of Template:Mvar (that is, equivalent to the solid green circle, but centered at Template:Mvar).

• Draw a circle centered at Template:Mvar and passing through Template:Mvar and vice versa (the red circles). They will intersect at point Template:Mvar and form the equilateral triangle △ABD.
• Extend Template:Mvar past Template:Mvar and find the intersection of Template:Mvar and the circle ◯BC, labeled Template:Mvar.
• Create a circle centered at Template:Mvar and passing through Template:Mvar (the blue circle).
• Extend Template:Mvar past Template:Mvar and find the intersection of Template:Mvar and the circle ◯DE, labeled Template:Mvar.
• Construct a circle centered at Template:Mvar and passing through Template:Mvar (the dotted green circle)
• Because △ADB is an equilateral triangle, DA = DB.
• Because Template:Mvar and Template:Mvar are on a circle around Template:Mvar, DE = DF.
• Therefore, AF = BE.
• Because Template:Mvar is on the circle ◯BC, BE = BC.
• Therefore, AF = BC.

Alternative construction without straightedge

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.

File:Compass-equivalence-no-straightedge.png

Construction without using straightedge

Given points Template:Mvar, Template:Mvar, and Template:Mvar, construct a circle centered at Template:Mvar with the radius Template:Mvar, using only a collapsing compass and no straightedge.

• Draw a circle centered at Template:Mvar and passing through Template:Mvar and vice versa (the blue circles). They will intersect at points Template:Mvar and Template:Mvar.
• Draw circles through Template:Mvar with centers at Template:Mvar and Template:Mvar (the red circles). Label their other intersection Template:Mvar.
• Draw a circle (the green circle) with center Template:Mvar passing through Template:Mvar. 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 Template:Mvar is the perpendicular bisector of Template:Mvar. Thus Template:Mvar is the reflection of Template:Mvar through line Template:Mvar.
• By construction, Template:Mvar is the reflection of Template:Mvar through line Template:Mvar.
• Since reflection is an isometry, it follows that AE = BC as desired.