In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. The result was originally published by Georg Mohr in 1672, but his proof languished in obscurity until 1928. The theorem was independently discovered by Lorenzo Mascheroni in 1797.
To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be doable by compass alone. These are:
- Creating the line through two existing points
- Creating the circle through one point with centre another point
- Creating the point which is the intersection of two existing, non-parallel lines
- Creating the one or two points in the intersection of a line and a circle (if they intersect)
- Creating the one or two points in the intersection of two circles (if they intersect).
Since lines cannot be drawn without a straightedge (#1), a line is considered to be given by two points. #2 and #5 are directly doable with a compass. Thus there need to be constructions only for #3 and #4.
An alternative algebraic approach uses the isomorphism between the Euclidean plane and . This approach can be used to provide significantly stronger versions of the theorem. It also clearly shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
- Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).
- Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B , pages 1–7.
- Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A , pages 34–36.
- Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.
- Norbert Hungerbühler, "A Short Elementary Proof of the Mohr–Mascheroni Theorem," The American Mathematical Monthly, vol. 101, no. 8, p. 784, Oct. 1994.
- Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.
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