Английская Википедия: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.[1]
Construction
The following construction and proof of correctness are given by Euclid in his Elements.[2] 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,[1] and so, specific choices are given below.
Given points Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar, construct a circle centered at Шаблон:Mvar with radius the length of Шаблон:Mvar (that is, equivalent to the solid green circle, but centered at Шаблон:Mvar).
- Draw a circle centered at Шаблон:Mvar and passing through Шаблон:Mvar and vice versa (the red circles). They will intersect at point Шаблон:Mvar and form the equilateral triangle Шаблон:Math.
- Extend Шаблон:Mvar past Шаблон:Mvar and find the intersection of Шаблон:Mvar and the circle Шаблон:Math, labeled Шаблон:Mvar.
- Create a circle centered at Шаблон:Mvar and passing through Шаблон:Mvar (the blue circle).
- Extend Шаблон:Mvar past Шаблон:Mvar and find the intersection of Шаблон:Mvar and the circle Шаблон:Math, labeled Шаблон:Mvar.
- Construct a circle centered at Шаблон:Mvar and passing through Шаблон:Mvar (the dotted green circle)
- Because Шаблон:Math is an equilateral triangle, Шаблон:Math.
- Because Шаблон:Mvar and Шаблон:Mvar are on a circle around Шаблон:Mvar, Шаблон:Math.
- Therefore, Шаблон:Math.
- Because Шаблон:Mvar is on the circle Шаблон:Math, Шаблон:Math.
- Therefore, Шаблон:Math.
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 Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar, construct a circle centered at Шаблон:Mvar with the radius Шаблон:Mvar, using only a collapsing compass and no straightedge.
- Draw a circle centered at Шаблон:Mvar and passing through Шаблон:Mvar and vice versa (the blue circles). They will intersect at points Шаблон:Mvar and Шаблон:Mvar.
- Draw circles through Шаблон:Mvar with centers at Шаблон:Mvar and Шаблон:Mvar (the red circles). Label their other intersection Шаблон:Mvar.
- Draw a circle (the green circle) with center Шаблон:Mvar passing through Шаблон:Mvar. This is the required circle.[3][4]
There are several proofs of the correctness of this construction and it is often left as an exercise for the reader.[3][4] Here is a modern one using transformations.
- The line Шаблон:Mvar is the perpendicular bisector of Шаблон:Mvar. Thus Шаблон:Mvar is the reflection of Шаблон:Mvar through line Шаблон:Mvar.
- By construction, Шаблон:Mvar is the reflection of Шаблон:Mvar through line Шаблон:Mvar.
- Since reflection is an isometry, it follows that Шаблон:Math as desired.
References
