geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the compass equivalence theorem is an important statement in

compass and straightedge constructions In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

. The tool advocated by

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

in these constructions is a ''divider'' or ''collapsing compass'', that is, a

compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...

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 A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

in the plane, it is possible to construct another circle of equal

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

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

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

of this

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

has had a chequered history.

Construction

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 , , and , construct a circle centered at with radius the length of (that is, equivalent to the solid green circle, but centered at ). * Draw a circle centered at and passing through and vice versa (the red circles). They will intersect at point and form the

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

. * Extend past and find the intersection of and the circle , labeled . * Create a circle centered at and passing through (the blue circle). * Extend past and find the intersection of and the circle , labeled . * Construct a circle centered at and passing through (the dotted green circle) * Because is an equilateral triangle, . * Because and are on a circle around , . * Therefore, . * Because is on the circle , . * Therefore, .

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. Compass-equivalence-no-straightedge

Given points , , and , construct a circle centered at with the radius , using only a collapsing compass and no straightedge. * Draw a circle centered at and passing through and vice versa (the blue circles). They will intersect at points and . * Draw circles through with centers at and (the red circles). Label their other intersection . * Draw a circle (the green circle) with center passing through . 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 is the

perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...

of . Thus is the reflection of through line . * By construction, is the reflection of through line . * Since reflection is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

, it follows that as desired.

References

{{reflist Straightedge and compass constructions Theorems in geometry