The compass equivalence theorem is an important statement in compass and straightedge constructions. In these constructions it is assumed that whenever a compass

Compass (drafting)

A compass or pair of compasses is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as a tool to measure distances, in particular on maps...

 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

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 of equal radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

, centered at any given point on the plane. This theorem is known as Proposition II of Book I of Euclid's Elements

Euclid's Elements

Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

.

Proof

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).

• Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form equilateral triangle ABD.
• Extend DB past B and find the intersection of DB and the circle BC, labeled E.
• Create a Circle centered at D and passing through E (the blue circle).
• Extend DA past A and find the Intersection of the DA and the circle DE, labeled F.
• Construct a circle centered at A and passing through F (the second green circle)

• Because E is on the circle BC, BE=BC.
• Because ADB is an equilateral triangle, DA=DB.
• Because E and F are on a circle around D, DE=DF.
• Therefore, AF=BE and AF=BC.

Alternate Proof without Straightedge

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.

• Draw a circle centered at A and passing through B and vice versa (the blue circles). They will intersect at points D and D'.
• Now draw circles through C with centers at D and D' (the red circles). Find their other intersection and label it E.
• Draw a circle (the green circle) with center A passing through E.

• 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.

External links

• s:Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/32
• Minnesota State University Mathematics Department