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, 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:

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 ${\displaystyle \mathbb {R} ^{2}}$. 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).

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