Mohr–Mascheroni theorem

Remarks

A line counts as known in a compass-only construction when two of its points are known; the line itself is never drawn. The proof shows how to find with compass alone the intersection points of a circle and a line given by two points. It also shows how to find the intersection point of two lines given by two points each. A standard tool is inversion in a circle, which can be done with compass alone and turns lines that miss the centre into circles through the centre. Georg Mohr published the result in 1672, and Lorenzo Mascheroni found it again independently in 1797. For example, the midpoint of a segment $AB$ can be found with compass alone, although no line through $A$ and $B$ is ever drawn.

Going deeper

For a student

Algebraically, ruler and compass constructions from $0$ and $1$ give exactly the constructible numbers, the smallest subfield of $\C$ closed under square roots and complex conjugation. The theorem says that compass constructions already give all of them. The Poncelet–Steiner theorem is the companion result that keeps the ruler and replaces the compass by a single fixed circle.