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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 .
*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, .
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 of . Thus is the reflection of through line .
*By construction, is the reflection of through line .
*Since reflection is an
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
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 ...
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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
in the plane, it is possible to construct another circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
of equal radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
. 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 con ...
of this theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
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 .
*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.
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 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'' mea ...
, it follows that as desired.
References
{{reflist Compass and straightedge constructions