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
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Euclid2.svg/300px-Euclid2.svg.png)
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 A, B, and C, construct a circle centered at A with radius the length of BC (that is, equivalent to the solid green circle, but centered at A).
- Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form the 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 DA and the circle ◯DE, labeled F.
- Construct a circle centered at A and passing through F (the dotted green circle)
- Because △ADB is an equilateral triangle, DA = DB.
- Because E and F are on a circle around D, DE = DF.
- Therefore, AF = BE.
- Because E is on the circle ◯BC, BE = BC.
- Therefore, AF = BC.
Alternative construction without straightedge
[edit]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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Compass-equivalence-no-straightedge.png/300px-Compass-equivalence-no-straightedge.png)
Given points A, B, and C, construct a circle centered at A with the radius 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'.
- Draw circles through C with centers at D and D' (the red circles). Label their other intersection E.
- Draw a circle (the green circle) with center A passing through E. 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 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.
References
[edit]- ^ a b Toussaint, Godfried T. (January 1993). "A New Look at Euclid's Second Proposition" (PDF). The Mathematical Intelligencer. 15 (3). Springer US: 12–24. doi:10.1007/bf03024252. eISSN 1866-7414. ISSN 0343-6993. S2CID 26811463.
- ^ Heath, Thomas L. (1956) [1925]. The Thirteen Books of Euclid's Elements (2nd ed.). New York: Dover Publications. p. 244. ISBN 0-486-60088-2.
- ^ a b Eves, Howard (1963), A survey of Geometry (Vol. I), Allyn Bacon, p. 185
- ^ a b Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 212, ISBN 0-534-35188-3