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. It must be understood that by "any geometric construction", we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as: Any Euclidean construction, insofar as the given and required elements are points (or circles), may be completed with the compass alone if it can be completed with both the compass and the straightedge together. Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary. History 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 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered. Several proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool. Mohr's solution was different. In 1890, August Adler published a proof using the inversion transformation. An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space . In this way, a stronger version of the theorem was proven in 1990. It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language). Constructive proof Outline To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a compass alone, as these are the foundations of, or elementary steps for, all other constructions. 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