# 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]

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.[6] It also clearly shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).