---
tag: G9VT2J3ZVK
kind: statement
role: theorem
name: Mohr–Mascheroni theorem
msc: [51M15]
uses: [0DAECMZVRT, 060W5CBZP6]
about: [0DAECMZVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Mohr%E2%80%93Mascheroni_theorem
same_as:
  wikidata: Q1152521
  wikipedia: Mohr–Mascheroni theorem
---

## Statement

Let $S$ be a set of points in the Euclidean plane.
Every point that can be constructed from $S$ with [[straightedge and compass|0DAECMZVRT]] can also be constructed from $S$ with compass alone.

## 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|060W5CBZP6]] 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.

## Explanations

### L1 human

The ruler is not needed in classical constructions.
Anything one can construct with ruler and compass, one can also construct with a compass alone, as long as a line is described by two of its points.
So the field of constructible points is the same for both kinds of tools.

### L2 human

Algebraically, ruler and compass constructions from $0$ and $1$ give exactly the [[constructible numbers|0QNFRBJF4Q]], 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|2P0DP2TK6P]] is the companion result that keeps the ruler and replaces the compass by a single fixed circle.