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.

It must be understood that "any geometric construction" refers 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.