Mohr–Mascheroni theorem

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,^[1] but his proof languished in obscurity until 1928.^[2][3] The theorem was independently discovered by Lorenzo Mascheroni in 1797.^[4]

Proof approach

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:

1. Creating the line through two existing points
2. Creating the circle through one point with centre another point
3. Creating the point which is the intersection of two existing, non-parallel lines
4. Creating the one or two points in the intersection of a line and a circle (if they intersect)
5. 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.^[5]

See also

• Poncelet–Steiner theorem
• Napoleon's problem

Notes

1. Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).
2. 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.
3. Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A , pages 34–36.
4. Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.
5. Norbert Hungerbühler, "A Short Elementary Proof of the Mohr–Mascheroni Theorem," The American Mathematical Monthly, vol. 101, no. 8, p. 784, Oct. 1994.

External links

• Construction with the Compass Only
• A short elementary proof of the Mohr–Mascheroni Theorem