---
tag: 2P0DP2TK6P
kind: statement
role: theorem
name: Poncelet–Steiner theorem
msc: [51M15]
uses: [0DAECMZVRT, 060W5CBZP6]
about: [0DAECMZVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Poncelet%E2%80%93Steiner_theorem
same_as:
  wikidata: Q1186134
  wikipedia: Poncelet–Steiner theorem
---

## Statement

Let $S$ be a set of points in the Euclidean plane, and suppose that one [[circle|060W5CBZP6]] and its centre are drawn in the plane.
Every point that can be constructed from $S$ with [[straightedge and compass|0DAECMZVRT]] can also be constructed from $S$ and the given circle with straightedge alone.

## Remarks

As for compass-only constructions, a circle counts as known when its centre and one of its points are known.
The proof shows how to find, with straightedge alone and the help of the fixed circle, the intersection points of a line and a circle, and of two circles.
Jean-Victor Poncelet stated the result in 1822, and Jakob Steiner proved it in 1833.
The centre is needed: with straightedge alone one cannot find the centre of a single given circle.
With straightedge alone and no circle, one obtains only points with coordinates in the field generated by the coordinates of $S$.
For example, $\sqrt 2$ cannot be constructed from $0$ and $1$ in this way.

## Explanations

### L1 human

A compass is needed only once.
If one circle and its centre are already on the page, a ruler alone can do every ruler-and-compass construction.
The single circle supplies all the square roots that the compass would otherwise give.

### L2 human

The theorem and the [[Mohr–Mascheroni theorem|G9VT2J3ZVK]] show that the constructible points depend on the algebraic operations available, not on the tools themselves.
Rational operations come from the straightedge, and square roots need some circle.